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+{"text":"\\section{Motivation}\\label{sec:motivation}\nModern cosmology has confronted the physics community with a number of unexplained observations, such as the accelerating expansion of the universe and the isotropy of the cosmological microwave background. The standard model of cosmology aims to explain these observations by the presence of dark energy in the $\\Lambda$CDM model and an inflationary phase in the early universe. However, the mechanisms behind these explanations, i.e., the constituents of dark energy and the driving force of the inflation, are not yet understood. This opens the possibility to create various models both from particle physics and gravity theory. The model we discuss here belongs to the gravitational category. The basic idea of this model is to replace the metric geometry of spacetime from general relativity by a Finsler length measure. In this article we derive a theory of fluid dynamics based on this Finsler geometric background.\n\nThe work we present in this article has two main ingredients. The first ingredient, which serves as the background geometry, is the Finsler spacetime framework.\\cite{Pfeifer:2011tk,Pfeifer:2011xi,Pfeifer:2013gha,Pfeifer:2014yua} This framework introduces a notion of Finsler geometry which provides a unified description of a Lorentzian causality, observers and gravity, and which can be used as a background for field theories such as electrodynamics. The second ingredient is the kinetic theory of fluids.\\cite{Ehlers:1971,Sarbach:2013fya,Sarbach:2013uba} This theory is based on the idea that fluids can be modeled as being constituted by particles, whose worldlines are geodesics on the background spacetime. In the continuum limit this theory yields equations of motion for a density function on a subspace of the tangent bundle of spacetime. In this article we generalize this kinetic theory of fluids to Finsler spacetimes. In this model the density function becomes a function on observer space,\\cite{Gielen:2012fz,Hohmann:2013fca,Hohmann:2014gpa,Hohmann:2015pva} which is the space of physically allowed four-velocities. We apply this formalism to two physically motivated examples.\n\nThe outline of this article is as follows. In section~\\ref{sec:finsler} we provide a brief introduction to the geometry of Finsler spacetimes. From this starting point we construct the space of physical observer velocities in section~\\ref{sec:observer} and discuss its geometric structure. We then use this structure to construct a model of fluid dynamics based on kinetic theory in section~\\ref{sec:fluids}. In section~\\ref{sec:special} we apply this model to two particular examples: a collisionless dust fluid and a fluid with cosmological symmetry. We end with a conclusion in section~\\ref{sec:conclusion}.\n\n\\section{Finsler spacetime geometry}\\label{sec:finsler}\nWe start our discussion with a brief review of the background geometry. In this section we introduce the basic geometric objects on a Finsler spacetime, which will later be used for the construction of fluid dynamics. Our starting point is the following definition:\\cite{Pfeifer:2011tk,Pfeifer:2011xi,Pfeifer:2013gha}\n\nA \\emph{Finsler spacetime} \\((M,L,F)\\) is a four-dimensional, connected, Hausdorff, paracompact, smooth manifold \\(M\\) equipped with continuous real functions \\(L, F\\) on the tangent bundle \\(TM\\) which has the following properties:\n\\begin{enumerate}\n\\item\\label{finsler:lsmooth}\n\\(L\\) is smooth on the tangent bundle without the zero section \\(TM \\setminus \\{0\\}\\).\n\\item\\label{finsler:lhomogeneous}\n\\(L\\) is positively homogeneous of real degree \\(n \\geq 2\\) with respect to the fiber coordinates of~\\(TM\\),\n\\begin{equation}\nL(x,\\lambda y) = \\lambda^nL(x,y) \\quad \\forall \\lambda > 0\\,,\n\\end{equation}\nand defines the Finsler function \\(F\\) via \\(F(x,y) = |L(x,y)|^{\\frac{1}{n}}\\).\n\\item\\label{finsler:lreversible}\n\\(L\\) is reversible: \\(|L(x,-y)| = |L(x,y)|\\).\n\\item\\label{finsler:lhessian}\nThe Hessian\n\\begin{equation}\ng^L_{ab}(x,y) = \\frac{1}{2}\\bar{\\partial}_a\\bar{\\partial}_bL(x,y)\n\\end{equation}\nof \\(L\\) with respect to the fiber coordinates is non-degenerate on \\(TM \\setminus X\\), where \\(X \\subset TM\\) has measure zero and does not contain the null set \\(\\{(x,y) \\in TM | L(x,y) = 0\\}\\).\n\\item\\label{finsler:timelike}\nThe unit timelike condition holds, i.e., for all \\(x \\in M\\) the set\n\\begin{multline}\n\\Omega_x = \\Bigg\\{y \\in T_xM \\Bigg| |L(x,y)| = 1,\\\\\ng^L_{ab}(x,y) \\text{ has signature } (\\epsilon,-\\epsilon,-\\epsilon,-\\epsilon), \\epsilon = \\frac{L(x,y)}{|L(x,y)|}\\Bigg\\}\n\\end{multline}\ncontains a non-empty closed connected component \\(S_x \\subseteq \\Omega_x \\subset T_xM\\).\n\\end{enumerate}\n\nHere we have used coordinates \\((x^a)\\) on \\(M\\) and the \\emph{induced coordinates} \\((x^a,y^a)\\) on \\(TM\\) such that \\((x,y) = y^a\\partial_a|_x \\in T_xM\\). The Finsler function \\(F\\) introduced above defines a length functional\n\\begin{equation}\\label{eqn:finslerlength}\ns[\\gamma] = \\int d\\tau\\,F(\\gamma(\\tau),\\dot{\\gamma}(\\tau))\\,,\n\\end{equation}\nwhich measures the length of a curve \\(\\tau \\mapsto \\gamma(\\tau)\\) on \\(M\\). An important class of curves called \\emph{Finsler geodesics} are those for which the length functional becomes extremal. They satisfy the geodesic equation\n\\begin{equation}\\label{eqn:geodesic}\n\\ddot{\\gamma}^a + N^a{}_b(\\gamma,\\dot{\\gamma})\\dot{\\gamma}^b = 0\\,.\n\\end{equation}\nHere we have introduced the \\emph{Cartan non-linear connection}\n\\begin{equation}\\label{eqn:nonlinconn}\nN^a{}_b = \\frac{1}{4}\\bar{\\partial}_b\\left[g^{F\\,ac}(y^d\\partial_d\\bar{\\partial}_cF^2 - \\partial_cF^2)\\right]\\,,\n\\end{equation}\nwhere \\(\\partial_a = \\partial\/\\partial x^a\\), \\(\\bar{\\partial}_a = \\partial\/\\partial y^a\\) and \\(g^{F\\,ab}\\) is the inverse of the \\emph{Finsler metric}\n\\begin{equation}\ng^F_{ab}(x,y) = \\frac{1}{2}\\bar{\\partial}_a\\bar{\\partial}_bF^2(x,y)\\,.\n\\end{equation}\nThe Cartan non-linear connection induces a unique split of the tangent bundle \\(TTM\\) into horizontal and vertical parts, \\(TTM = HTM \\oplus VTM\\). The horizontal tangent bundle \\(HTM\\) is spanned by the vector fields\n\\begin{equation}\n\\{\\delta_a = \\partial_a - N^b{}_a\\bar{\\partial}_b\\}\\,,\n\\end{equation}\nwhile the vertical tangent bundle \\(VTM\\) is spanned by \\(\\{\\bar{\\partial}_a\\}\\). The corresponding basis \\(\\{\\delta_a,\\bar{\\partial}_a\\}\\) that respects this split is called the \\emph{Berwald basis}. Its dual basis of \\(T^*TM\\) is given by\n\\begin{equation}\n\\{dx^a, \\delta y^a = dy^a + N^a{}_bdx^b\\}\\,.\n\\end{equation}\nUsing the Berwald basis we can find another description for Finsler geodesics. For this purpose we consider the canonical lift \\(\\Gamma: \\mathbb{R} \\to TM\\) of a geodesic \\(\\gamma\\) to the tangent bundle \\(TM\\). Writing \\(\\Gamma\\) in coordinates \\((\\Gamma^a,\\bar{\\Gamma}^a) = (\\gamma^a,\\dot{\\gamma}^a)\\) we find the geodesic equation\n\\begin{equation}\n\\dot{\\Gamma}^a = \\bar{\\Gamma}^a\\,, \\quad \\dot{\\bar{\\Gamma}}^a = -N^a{}_b(\\Gamma^a,\\bar{\\Gamma}^a)\\bar{\\Gamma}^b\\,.\n\\end{equation}\nSince this is a first order differential equation, the canonical lifts of Finsler geodesics take the form of integral curves of a vector field \\(\\mathbf{S} = y^a\\delta_a\\) on \\(TM\\), which we call the \\emph{geodesic spray}. Finally, the Berwald basis and the Finsler metric allow the construction of a metric\n\\begin{equation}\\label{eqn:sasakimetric}\nG = -g^F_{ab}\\,dx^a \\otimes dx^b - \\frac{g^F_{ab}}{F^2}\\,\\delta y^a \\otimes \\delta y^b\\,.\n\\end{equation}\non the tangent bundle, which is called the \\emph{Sasaki metric}.\n\n\\section{Observer space geometry}\\label{sec:observer}\nAfter discussing the most important Finsler geometric structures on the tangent bundle \\(TM\\) in the previous section we now restrict ourselves to a particular subspace of \\(TM\\). Recall from the definition of a Finsler spacetime that for each \\(x \\in M\\) there exists a shell \\(S_x \\subset T_xM\\) of future unit timelike vectors, which corresponds to the four-velocities of test masses and physical observers. Their union\n\\begin{equation}\\label{eqn:observerspace}\nO = \\bigcup_{x \\in M}S_x\n\\end{equation}\nis called \\emph{observer space}. This is the space on which we will define fluid dynamics in the remainder of this work.\n\nThe observer space is a seven-dimensional submanifold of the tangent bundle \\(TM\\) and thus inherits a number of geometric structures from the Finsler geometry. In particular, the geodesic spray \\(\\mathbf{S}\\) introduced in the previous section is tangent to the observer space, and thus restricts to a vector field \\(\\mathbf{r}\\) on \\(O\\), called the \\emph{Reeb vector field}. This relation between the geodesic spray and the observer space is in fact necessary for the interpretation of \\(O\\) as the space of physical four-velocities: it means that a test mass possessing a physical initial four-velocity and following a geodesic, and thus an integral curve of \\(\\mathbf{S}\\), will have a physical four-velocity at all times. The canonical lifts of all physical geodesics, corresponding to freely falling test masses, are thus given by integral curves of the Reeb vector field \\(\\mathbf{r}\\) on \\(O\\).\n\nThe second important structure on the observer space is the restriction \\(\\tilde{G}\\) of the Sasaki metric \\(G\\) to \\(O\\). This also equips \\(O\\) with a volume form \\(\\Sigma = \\mathrm{Vol}_{\\tilde{G}}\\). This volume form has a number of properties which are relevant for the construction of a theory of fluid dynamics. The most relevant property for the remainder of this article is the fact that it is preserved by the flow of the Reeb vector field, \\(\\mathcal{L}_{\\mathbf{r}}\\Sigma = 0\\).\n\nFrom the Reeb vector field \\(\\mathbf{r}\\) and the volume form \\(\\Sigma\\) we finally define the \\emph{particle measure} \\(\\omega = \\iota_{\\mathbf{r}}\\Sigma\\). It is the unique six-form (up to a constant factor) which has the following properties necessary for the construction of fluid dynamics from the kinetic theory. Most importantly, its restriction to a hypersurface \\(\\sigma \\subset O\\) which is not tangent to the Reeb vector field \\(\\mathbf{r}\\) is a non-vanishing volume form on \\(\\sigma\\). Further, it is closed, \\(d\\omega = 0\\), and preserved by the flow of the Reeb vector field, \\(\\mathcal{L}_{\\mathbf{r}}\\omega = 0\\). The relevance of these properties will become clear in the following section, when we use it as an ingredient to develop a theory of fluid dynamics.\n\n\\section{Kinetic theory and fluid dynamics}\\label{sec:fluids}\nWe now turn our attention to the kinetic theory of fluids\\cite{Ehlers:1971,Sarbach:2013fya,Sarbach:2013uba} and apply it to fluids on the Finsler spacetime background geometry detailed in the preceding section. For this purpose we first briefly review how fluids can be modeled by the geodesic motion of particles. From the geodesic motion on Finsler spacetimes we then derive the equations of motion for a Finsler fluid.\n\nThe basic idea of the kinetic theory of fluids is the assumption that fluids are constituted by particles. In the simplest possible case of a single component fluid, which we will discuss here, all particles have identical properties, such as mass and electric charge, and follow piecewise geodesic curves. This geodesic motion corresponds to the motion of freely falling test masses without any other interaction. The interaction between particles is modeled by collisions, which mark the endpoints of the geodesic pieces of the particle trajectories and correspond to instantaneous transfers of momentum between the particles. The physical background of this model of interactions is the assumption that the interaction distances are small compared to the distances between the particles and that the interaction times are short compared to the time between interactions, which is the case for a sufficiently low density.\n\nIn order to construct a continuum theory of fluids from the particle model one introduces the one-particle distribution function \\(\\phi: O \\to \\mathbb{R}^+\\) such that for each oriented hypersurface \\(\\sigma \\subset O\\) the integral\n\\begin{equation}\nN[\\sigma] = \\int_{\\sigma}\\phi\\omega\n\\end{equation}\nis the number \\(N[\\sigma]\\) of particle trajectories whose canonical lifts to \\(O\\) pass through \\(\\sigma\\). Here particle trajectories \\(\\gamma\\) are counted positively (negatively) if the tangent vector of their canonical lift and a positively oriented basis of the tangent space to \\(\\sigma\\) at the intersection point between \\(\\gamma\\) and \\(\\sigma\\) form a positively (negatively) oriented basis of the tangent space of \\(O\\) at that point. Further, \\(\\omega\\) denotes the particle measure introduced in the previous section.\n\nWe now take a closer look at the canonical lifts of the particle trajectories to the observer space. It follows from our assumption of instantaneous momentum transfer that these lifts are discontinuous at collisions, i.e., they have endpoints corresponding to the particles' velocities before and after the collision. To incorporate collisions into the continuum theory we define the collision transfer density \\(\\dot{\\phi}: O \\to \\mathbb{R}\\) such that for each hypervolume \\(V \\subset O\\) the integral\n\\begin{equation}\n\\dot{N}[V] = \\int_V\\dot{\\phi}\\Sigma\n\\end{equation}\nis the number of initial points minus the number of final points of canonical lifts of particle trajectories in \\(V\\). Here \\(\\Sigma\\) denotes the volume form of the Sasaki metric \\(\\tilde{G}\\) on \\(O\\).\n\nNote that we have defined the counting prescription for curves passing a hypersurface such that any curve which has an initial point, but no final point in the hypervolume \\(V\\) passes its boundary \\(\\partial V\\) in the positive direction, and in the negative direction in the opposite case. It thus follows that \\(\\dot{N}[V] = N[\\partial V]\\). From this we further find that\n\\begin{equation}\n\\int_V\\dot{\\phi}\\Sigma = \\dot{N}[V] = N[\\partial V] = \\int_{\\partial V}\\phi\\omega = \\int_Vd(\\phi\\omega) = \\int_V(\\mathcal{L}_{\\mathbf{r}}\\phi)\\Sigma\\,,\n\\end{equation}\nwhere we used Stokes' theorem and the properties of the particle measure \\(\\omega\\). Since this holds for any hypervolume \\(V\\) it follows that\n\\begin{equation}\n\\dot{\\phi} = \\mathcal{L}_{\\mathbf{r}}\\phi\\,.\n\\end{equation}\nThe right hand side of this equation describes the evolution of \\(\\phi\\). We thus obtain the equation of motion of the fluid by specifying a functional \\(\\dot{\\phi}[\\phi]\\) describing collisions between particles. For the simplest possible case of a collisionless fluid we have \\(\\dot{\\phi} = 0\\), and the fluid equation of motion reduces to the Liouville equation \\(\\mathcal{L}_{\\mathbf{r}}\\phi = 0\\).\n\nThis concludes our discussion of the kinetic theory of fluids in general. In the following section we will discuss particular examples of fluids and derive their equations of motion.\n\n\\section{Special cases}\\label{sec:special}\nAfter discussing general fluids on Finsler spacetimes in the previous section, we now consider a few special cases. The first case we display in section~\\ref{subsec:dust} is dust, which is conventionally described as a collisionless perfect fluid with vanishing pressure. Starting from the Liouville equation we derive the generalization of the Euler equations to dust on Finsler spacetimes. The second case shown in section~\\ref{subsec:cosmology} is the most general cosmological fluid. We start from a general Finsler spacetime with cosmological symmetry and derive the equations of motion for the most general fluid obeying the same symmetry.\n\n\\subsection{Dust fluid}\\label{subsec:dust}\nThe first example we discuss is a collisionless dust fluid characterized by a matter density \\(\\rho(x)\\) and four-velocity distribution \\(u^a(x)\\), which is a future timelike vector field normalized by the Finsler function, and thus a function \\(u: M \\to O\\). The aim of this section is to derive the equations of motion for \\(\\rho\\) and \\(u\\) from the kinetic theory introduced in the preceding section. For this purpose we will first construct the one-particle distribution function \\(\\phi\\) and then impose the Liouville equation \\(\\mathcal{L}_{\\mathbf{r}}\\phi = 0\\) for a collisionless fluid.\n\nWe start by introducing coordinates \\((\\hat{x}^a,\\theta^{\\alpha}) = (x^a,y^{\\alpha}\/y^0)\\) on \\(O\\) and writing the three-velocity as \\(v^{\\alpha} = u^{\\alpha}\/u^0\\). Together with the proper Dirac delta distribution on the unit timelike shell \\(S_{\\hat{x}}\\) we find the one-particle distribution function\n\\begin{equation}\\label{eqn:dustphi}\n\\phi(\\hat{x},\\theta) = \\frac{1}{m}\\rho(\\hat{x})\\frac{\\delta(\\theta - v(\\hat{x}))}{\\sqrt{h^F(\\hat{x},\\theta)}}\\,,\n\\end{equation}\nwhere \\(h^F\\) denotes the determinant of the restriction\n\\begin{equation}\nh^F_{\\alpha\\beta} = \\frac{\\partial y^a}{\\partial\\theta^{\\alpha}}\\frac{\\partial y^b}{\\partial\\theta^{\\beta}}g^F_{ab}\n\\end{equation}\nof the Sasaki metric to \\(S_{\\hat{x}}\\). A direct calculation of the Lie derivative \\(\\mathcal{L}_{\\mathbf{r}}\\phi\\) is obstructed by the presence of the delta distribution. We therefore introduce an arbitrary smooth test function \\(f: O \\to \\mathbb{R}\\). Using the properties of the particle measure \\(\\omega\\) and integrating by parts we then find\n\\begin{equation}\n0 = \\int_{O}f\\left(\\mathcal{L}_{\\mathbf{r}}\\phi\\right)\\Sigma = \\int_{O}d(\\phi\\omega)f = -\\int_{O}df \\wedge \\omega\\phi = -\\int_{O}d(f\\omega)\\phi =\n-\\int_{O}\\phi\\left(\\mathcal{L}_{\\mathbf{r}}f\\right)\\Sigma\\,.\n\\end{equation}\nWriting the Finsler function in the form\n\\begin{equation}\nF = y^0\\tilde{F}(x,y\/y^0) = y^0\\tilde{F}(\\hat{x},\\theta)\n\\end{equation}\nwe obtain the Reeb vector field\n\\begin{equation}\n\\mathbf{r} = y^a\\left[\\hat{\\partial}_a + \\tilde{F}\\left(N^0{}_a\\theta^{\\alpha} - N^{\\alpha}{}_a\\right)\\tilde{\\partial}_{\\alpha}\\right],\n\\end{equation}\nusing the partial derivatives \\(\\hat{\\partial}_a\\) and \\(\\tilde{\\partial}_{\\alpha}\\) with respect to \\(\\hat{x}^a\\) and \\(\\theta^{\\alpha}\\), and the volume form\n\\begin{equation}\n\\Sigma = \\sqrt{g^Fh^F}d^4\\hat{x}d^3\\theta\\,.\n\\end{equation}\nFinally inserting the explicit form for \\(\\phi\\) we arrive at\n\\begin{equation}\n0 = \\int_Od^4\\hat{x}d^3\\theta\\sqrt{g^F}\\delta(\\theta - v)\\rho y^a\\left[\\hat{\\partial}_a + \\tilde{F}\\left(N^0{}_a\\theta^{\\alpha} - N^{\\alpha}{}_a\\right)\\tilde{\\partial}_{\\alpha}\\right]f\\,.\n\\end{equation}\nIntegrating over \\(d^3\\theta\\) cancels the Dirac distribution and yields\n\\begin{equation}\n0 = \\left.\\int_Md^4\\hat{x}\\sqrt{g^F}\\rho u^a\\left[\\hat{\\partial}_af + \\tilde{F}\\left(N^0{}_av^{\\alpha} - N^{\\alpha}{}_a\\right)\\tilde{\\partial}_{\\alpha}f\\right]\\right|_{y^a = u^a(\\hat{x})}\\,,\n\\end{equation}\nwhere it is indicated that all objects on observer space are taken at the point \\((\\hat{x},v(\\hat{x}))\\). Observe that the partial derivative \\(\\hat{\\partial}_af(\\hat{x},v(\\hat{x}))\\) is fixed by \\(f(\\hat{x},v(\\hat{x}))\\) and thus not an independent quantity, while \\(\\tilde{\\partial}_{\\alpha}f(\\hat{x},v(\\hat{x}))\\) depends on the choice of \\(f\\) in a neighborhood of \\(v(\\hat{x})\\), which can be chosen independently. We therefore need to eliminate \\(\\hat{\\partial}_af\\) using integration by parts. Since \\(\\hat{\\partial}_a\\) is only a partial derivative, but \\(f(\\hat{x},v(\\hat{x}))\\) contains also an implicit dependence on \\(\\hat{x}\\) via \\(v(\\hat{x})\\), we first need to rewrite the partial derivative into a total derivative using\n\\begin{equation}\n\\hat{\\partial}_af(\\hat{x},v(\\hat{x})) = \\frac{d}{d\\hat{x}^a}f(\\hat{x},v(\\hat{x})) - \\hat{\\partial}_av^{\\alpha}\\tilde{\\partial}_{\\alpha}f(\\hat{x},v(\\hat{x}))\\,.\n\\end{equation}\nIntegration by parts and using the normalization \\(u^0\\tilde{F}(\\hat{x},v(\\hat{x})) = F(\\hat{x},u(\\hat{x})) = 1\\) then finally yields\n\\begin{multline}\n0 = \\int_Md^4\\hat{x}\\sqrt{g^F}\\bigg\\{\\left[\\frac{u^a}{u^0}\\left(N^0{}_a\\frac{u^{\\alpha}}{u^0} - N^{\\alpha}{}_a\\right) - u^a\\hat{\\partial}_a\\left(\\frac{u^{\\alpha}}{u^0}\\right)\\right]\\rho\\tilde{\\partial}_{\\alpha}f\\\\\n- \\left[\\hat{\\partial}_a(\\rho u^a) + \\frac{1}{2}\\rho u^ag^{F\\,bc}\\left(\\hat{\\partial}_ag^F_{bc} + \\hat{\\partial}_au^d\\bar{\\partial}_dg^F_{bc}\\right)\\right]f\\bigg\\}\\bigg|_{y^a = u^a(\\hat{x})}\\,.\n\\end{multline}\nSince now all spacetime derivatives act on objects which depend only on spacetime coordinates, we can rename the coordinates \\(\\hat{x}^a\\) back to \\(x^a\\) and read off the equations of motion\n\\begin{equation}\\label{eqn:dusteomraw1}\nu^a\\left(N^0{}_a\\frac{u^{\\alpha}}{u^0} - N^{\\alpha}{}_a\\right) - u^a\\partial_au^{\\alpha} + \\frac{u^au^{\\alpha}}{u^0}\\partial_au^0 = 0\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:dusteomraw2}\n\\partial_a(\\rho u^a) + \\frac{1}{2}\\rho u^ag^{F\\,bc}\\left(\\partial_ag^F_{bc} + \\partial_au^d\\bar{\\partial}_dg^F_{bc}\\right) = 0\\,.\n\\end{equation}\nThese equations can further be simplified. Using the properties of the Cartan non-linear connection we can rewrite the first equation~\\eqref{eqn:dusteomraw1} in the form\n\\begin{equation}\\label{eqn:dusteom1}\n0 = u^a(\\partial_au^b + N^b{}_a) = \\nabla u^b\\,,\n\\end{equation}\nwhere we have introduced the dynamical covariant derivative \\(\\nabla\\). Similarly, the second equation~\\eqref{eqn:dusteomraw2} can be written as\n\\begin{equation}\\label{eqn:dusteom2}\n0 = \\partial_a(\\rho u^a) + \\frac{1}{2}\\rho u^ag^{F\\,bc}\\delta_ag^F_{bc} = \\nabla_{\\delta_a}(\\rho u^a)\\,,\n\\end{equation}\nwhere we have introduced the covariant derivative \\(\\nabla_{\\delta_a}\\) of the Cartan linear connection. In the case of a metric Finsler function \\(F^2(x,y) = |g_{ab}(x)y^ay^b|\\) these equations reduce to the well-known Euler equations\n\\begin{equation}\nu^b\\nabla_bu^a = 0 \\quad \\text{and} \\quad \\nabla_a(\\rho u^a) = 0\n\\end{equation}\nfor a pressureless fluid.\n\n\\subsection{Cosmological fluid}\\label{subsec:cosmology}\nAs the final example we will derive the Liouville equation for a fluid with cosmological symmetry. For this purpose we introduce coordinates \\((t,r,\\vartheta,\\varphi)\\) on the spacetime manifold \\(M\\) and the corresponding induced coordinates \\((t,r,\\vartheta,\\varphi,y^t,y^r,y^{\\vartheta},y^{\\varphi})\\) on \\(TM\\). In these coordinates the cosmological symmetry generators take the form\n\\begin{gather}\n\\xi_1 = \\sqrt{1 - kr^2}\\left(\\sin\\vartheta\\cos\\varphi\\partial_r + \\frac{\\cos\\vartheta\\cos\\varphi}{r}\\partial_{\\vartheta} - \\frac{\\sin\\varphi}{r\\sin\\vartheta}\\partial_{\\varphi}\\right)\\,,\\nonumber\\\\\n\\xi_2 = \\sqrt{1 - kr^2}\\left(\\sin\\vartheta\\sin\\varphi\\partial_r + \\frac{\\cos\\vartheta\\sin\\varphi}{r}\\partial_{\\vartheta} + \\frac{\\cos\\varphi}{r\\sin\\vartheta}\\partial_{\\varphi}\\right)\\,,\\nonumber\\\\\n\\xi_3 = \\sqrt{1 - kr^2}\\left(\\cos\\vartheta\\partial_r - \\frac{\\sin\\vartheta}{r}\\partial_{\\vartheta}\\right)\\,, \\quad\n\\xi_6 = \\partial_{\\varphi}\\,,\\nonumber\\\\\n\\xi_4 = \\sin\\varphi\\partial_{\\vartheta} + \\frac{\\cos\\varphi}{\\tan\\vartheta}\\partial_{\\varphi}\\,, \\quad\n\\xi_5 = -\\cos\\varphi\\partial_{\\vartheta} + \\frac{\\sin\\varphi}{\\tan\\vartheta}\\partial_{\\varphi}\\,,\\label{eqn:cosmovect}\n\\end{gather}\nwhere \\(k \\in \\{-1,0,1\\}\\) determines the spatial curvature of the corresponding spacetime. For cosmologically symmetric fluid dynamics we require that both the background geometry and the one-particle distribution function of the fluid obey this symmetry.\n\nWe start by deriving the most general cosmologically symmetric background geometry. A Finsler spacetime is symmetric under the action of a vector field \\(\\xi\\) if and only if the geometry function \\(L\\), and thus also the Finsler function \\(F\\), is invariant under the complete lift of \\(\\xi\\) to the tangent bundle,\\cite{Pfeifer:2011xi,Pfeifer:2013gha,Hohmann:2015pva}\n\\begin{equation}\n\\left(\\xi^a\\partial_a + y^b\\partial_b\\xi^a\\bar{\\partial}_a\\right)F = 0\\,.\n\\end{equation}\nIn the present case of a cosmological symmetry it is most convenient to introduce new coordinates on \\(TM\\) via the definition\n\\begin{gather}\nt = \\hat{t}\\,, \\quad r = \\hat{r}\\,, \\quad \\vartheta = \\hat{\\vartheta}\\,, \\quad \\varphi = \\hat{\\varphi}\\,, \\quad y^t = \\hat{y}\\,,\\nonumber\\\\\ny^r = \\hat{w}\\cos\\hat{u}\\sqrt{1 - k\\hat{r}^2}\\,, \\quad y^{\\vartheta} = \\frac{\\hat{w}}{\\hat{r}}\\sin\\hat{u}\\cos\\hat{v}\\,, \\quad y^{\\varphi} = \\frac{\\hat{w}}{\\hat{r}\\sin\\hat{\\vartheta}}\\sin\\hat{u}\\sin\\hat{v}\\,.\n\\end{gather}\nAfter calculating the complete lifts of the vector fields~\\eqref{eqn:cosmovect} in these coordinates it turns out that the most general Finsler function with cosmological symmetry is given by \\(F = F(\\hat{t},\\hat{y},\\hat{w})\\). Further, from the fact that \\(F\\) is 1-homogeneous in the coordinates \\(\\hat{y}\\) and \\(\\hat{w}\\) follows that it must be of the form\n\\begin{equation}\nF(\\hat{t},\\hat{y},\\hat{w}) = \\hat{y}\\tilde{F}\\left(\\hat{t},\\hat{w}\/\\hat{y}\\right)\\,.\n\\end{equation}\nFor the discussion of geodesic motion it is convenient to introduce yet another set of coordinates on \\(TM\\) as\n\\begin{equation}\n\\tilde{t} = \\hat{t}\\,, \\quad \\tilde{r} = \\hat{r}\\,, \\quad \\tilde{\\vartheta} = \\hat{\\vartheta}\\,, \\quad \\tilde{\\varphi} = \\hat{\\varphi}\\,, \\quad \\tilde{u} = \\hat{u}\\,, \\quad \\tilde{v} = \\hat{v}\\,, \\quad \\tilde{y} = \\hat{y}\\tilde{F}\\left(\\hat{t},\\frac{\\hat{w}}{\\hat{y}}\\right)\\,, \\quad \\tilde{w} = \\frac{\\hat{w}}{\\hat{y}}\\,.\n\\end{equation}\nIn these coordinates the observer space \\(O\\) is given as a connected component of the submanifold \\(\\tilde{y} = 1\\), so that one can use the remaining seven coordinates to parametrize \\(O\\). The Reeb vector field then takes the form\n\\begin{multline}\n\\mathbf{r} = \\frac{1}{\\tilde{F}}\\Bigg(\\tilde{\\partial}_t + \\tilde{w}\\cos\\tilde{u}\\sqrt{1 - k\\tilde{r}^2}\\tilde{\\partial}_r + \\frac{\\tilde{w}\\sin\\tilde{u}\\cos\\tilde{v}}{\\tilde{r}}\\tilde{\\partial}_{\\vartheta} + \\frac{\\tilde{w}\\sin\\tilde{u}\\sin\\tilde{v}}{\\tilde{r}\\sin\\tilde{\\vartheta}}\\tilde{\\partial}_{\\varphi}\\\\\n- \\frac{\\tilde{w}\\sin\\tilde{u}\\sqrt{1 - k\\tilde{r}^2}}{\\tilde{r}}\\tilde{\\partial}_u - \\frac{\\tilde{w}\\sin\\tilde{u}\\sin\\tilde{v}}{\\tilde{r}\\tan\\tilde{\\vartheta}}\\tilde{\\partial}_v - \\frac{\\tilde{F}_{tw}}{\\tilde{F}_{ww}}\\tilde{\\partial}_w\\Bigg)\\,,\n\\end{multline}\nwhere the subscripts \\(t\\) and \\(w\\) indicate derivatives with respect to \\(\\tilde{t}\\) and \\(\\tilde{w}\\), respectively.\n\nWe finally come to the discussion of fluids on the Finsler spacetime background derived above. Since the background geometry obeys the cosmological symmetry defined by the vector fields~\\eqref{eqn:cosmovect}, their canonical lifts are tangent to the observer space \\(O\\). A fluid obeys the same symmetry if and only if its one-particle distribution function \\(\\phi\\) is invariant under the restriction of these canonical lifts to \\(O\\). In the present case the most general one-particle distribution function satisfying this condition takes the form \\(\\phi = \\phi(\\tilde{t},\\tilde{w})\\). Its Lie derivative with respect to the Reeb vector field is thus given by\n\\begin{equation}\n\\mathcal{L}_{\\mathbf{r}}\\phi = \\frac{1}{\\tilde{F}}\\left(\\phi_t - \\frac{\\tilde{F}_{tw}}{\\tilde{F}_{ww}}\\phi_w\\right)\\,.\n\\end{equation}\nFor the simplest possible case of a collisionless fluid the equation of motion hence takes the form\n\\begin{equation}\n\\tilde{F}_{ww}\\phi_t = \\tilde{F}_{tw}\\phi_w\\,.\n\\end{equation}\nThis is the Liouville equation for a fluid with cosmological symmetry.\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe have derived a model for fluid dynamics on Finsler geometric backgrounds based on the kinetic theory of fluids and applied our model to two important special cases. Our results show that Finsler spacetimes provide a suitable background geometry for fluid dynamics and that the obtained fluid equations of motion reduce to their well-known limits if the background geometry is metric.\n\nOf course any model of fluid dynamics must be complemented by a suitable model of gravitational dynamics in order to derive consistent solutions. For the Finsler spacetimes we used here an extension of general relativity exists.\\cite{Pfeifer:2011xi,Pfeifer:2013gha} The source of the gravitational field in this model is a scalar function on the tangent bundle of spacetime. Deriving this energy-momentum scalar for a kinetic fluid and the corresponding gravitational field equations will be a topic of future research.\n\n\\section*{Acknowledgments}\nThe author is happy to thank Christian Pfeifer for extensive discussions and fruitful collaboration. He gratefully acknowledges the full financial support of the Estonian Research Council through the Postdoctoral Research Grant ERMOS115 and the Startup Research Grant PUT790.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSuppose a point target is arbitrarily placed on the unit-circumference circle. The target then proceeds to move at some constant velocity $v$ (either known or unknown). An agent is interested to determine the target's position and velocity to within some resolution $\\delta$, with an error probability at most $\\varepsilon$, as quickly as possible. To that end, the agent can probe any region of his choosing (contiguous or non-contiguous) on the circle for the presence of the target, say once per second. He then receives a binary measurement pertaining to the presence of the target in the probed region, which is corrupted by additive binary noise. While the noise sequence is assumed to be independent over time, its magnitude will generally depend on the size of the probed region. This postulate is practically motivated if one imagines that the circle is densely covered by many small sensors; probing a region then corresponds to activating the relevant sensors and obtaining a measurement that is a (Boolean) function of the sum of the noisy signals from these sensors. We therefore further operate under the assumption that the larger the probed region, the higher the noise level. Our goal is to characterize the relation between $\\varepsilon$, $\\delta$, and the expected time $\\mathbb{E}(\\tau)$ until the agent's goal is met, for both adaptive and non-adaptive search strategies. \n\nThe case of stationary target search with measurement independent noise $p$ is well known (see e.g. \\cite{burnashev1974interval}) to be equivalent to the problem of channel coding with noiseless feedback over a Binary Symmetric Channel (BSC) with crossover probability $p$, where the message corresponds to the target, the number of messages pertains to inverse of the resolution, the channel noise plays the role of measurement noise, and the existence of noiseless feedback pertains to the fact that the agent may use past measurements to adapt his probing strategy. Based on the results of \\cite{burnashev_exp} it can be readily shown that using adaptive strategies one can achieve \n\\begin{equation*}\n\\mathbb{E}(\\tau) = \\frac{\\log{(1\\slash\\delta)}}{C(p)} + \\frac{\\log{(1\\slash\\varepsilon)}}{C_1(p)} + \\mathrm{O}(\\log\\log{\\frac{1}{\\delta\\varepsilon}})\n\\end{equation*}\nwhere $C(p)$ is the Shannon capacity of the BSC with crossover probability $p$, and $C_1(p) = D(p\\|1-p)$. This result is also the best possible up to sub-logarithmic terms. For non-adaptive strategies, standard channel coding results \\cite{GallagerBook} indicate for any fixed $0<R<C(p)$ there exists a strategy such that \n\\begin{equation*}\n\\tau = \\frac{\\log{(1\\slash\\delta)}}{R},\\qquad \\log{(1\\slash\\varepsilon)} = \\frac{E(R,p)}{R}\\cdot \\log{(1\\slash\\delta)}\n\\end{equation*}\nwhere $E(R,p)$ is the reliability function of the BSC, for which bounds are known \\cite{GallagerBook}. Hence, the minimal expected search time (with a vanishing error guarantee) is roughly the same for adaptive and non-adaptive strategies in the limit of high resolution $\\delta\\to 0$, and is given by $\\mathbb{E}(\\tau) \\approx \\frac{\\log{(1\\slash\\delta)}}{C(p)}$. This directly corresponds to the fact that feedback does not increase the capacity of a memoryless channel \\cite{shannon_zero_error}. Adaptive search strategies do however exhibit superior performance over non-adaptive strategies for a fixed resolution, attaining the same error probability with a lower expected search time. They are also asymptotically better if a certain exponential decay of the error probability is desired, which directly corresponds to the fact that the Burnashev exponent \\cite{burnashev_exp} exceeds the sphere packing bound \\cite{GallagerBook} at all rates below capacity. \n\nThe contribution of this work is threefold: \n\\begin{itemize}[leftmargin=*]\n\\item In contrast to the case of measurement independent noise, it is shown that for known velocity and measurement dependent noise there exists a multiplicative gap between the minimal expected search time for adaptive vs. non-adaptive strategies, in the limit of high resolution. This \\textit{targeting rate} gap generally depends on the variability of the measurement noise with the size of the probed region, and can be arbitrarily large. The source of the difference lies mainly in the fact that from a channel coding perspective, the channel associated with measurement dependent noise is time-varying in quite an unusual way; it depends on the choice of the \\textit{entire codebook}. The maximal targeting rates achievable using adaptive and non-adaptive strategies under known velocity are given.  \n\\item A rate-reliability tradeoff analysis is provided for the proposed adaptive and non-adaptive schemes, under known velocity. It is shown that the former attains the best possible tradeoff. \n\\item For unknown velocity, the maximal targeting rate achievable using non-adaptive schemes is shown to be reduced by a factor of two relative to the case of known velocity.  \n\\end{itemize}\n\n\n\\section{Preliminaries}\n\\subsection{Notations}\nThe Shannon entropy of a random variable (r.v.) $X$ is denoted by $H(X)$. The mutual information between two jointly distributed r.v.s $X$ and $Y$ is denoted $I(X;Y)$. A BSC($p$) is a BSC with crossover probability $p$.  When $X\\sim\\textrm{Bern}(q)$ and $Y$ is the output of a BSC($p$) with input $X$, we write $I(q,p)$ for $I(X;Y)$. We write $C(p)$ for the Shannon capacity of a BSC($p$), and $C_1(p)$ for the relative entropy $D(p\\|1-p)$. The cardinality of a finite set $S$ is denoted by $|S|$. The Lebesgue measure of a set $S\\subset\\mathbb{R}$ is similarly denoted by $|S|$. We write $\\mathds{1}(\\cdot)$ for the indicator function. The \\textit{cyclic distance} between $a,b\\in[0,1)$ is the associated angular distance on the unit-circumference circle, i.e., $|a-b|_c  \\stackrel{\\textnormal{def}}{=}  \\min\\{|a-b|,1-|a-b|\\}$.  \n\\subsection{Setup}\nLet $w_0\\in[0,1)$ be the initial position of the target, arbitrarily placed on the unit interval\\footnote{For simplicity of notation, we will think of the target as moving on the unit interval modulo 1, instead of on the circle.}. The target moves at a fixed but unknown velocity $v\\in[0,1)$, i.e., at time $n$ the position of the target is given by\n\\begin{equation*}\nw_n = w_0 + v\\cdot n \\quad (\\textrm{mod }1)\n\\end{equation*}     \n\nAt time $n$, the agent may seek the target by choosing (possibly at random) any measurable \\textit{query set} $S_n\\subset [0,1)$ to probe. Without loss of generality, we will assume throughout that $|S_n| \\leq \\frac{1}{2}$ almost surely. Let $X_n = \\mathds{1}(w_n\\in S_n)$ denote the clean binary signal indicating whether the target is in the probed region. The agent obtains a corrupted version $Y_n$ of $X_n$, with noise level that corresponds to the size of the region $S_n$. Specifically, \n\\begin{align*}\nY_n = X_n + Z_n \\;(\\textrm{mod}\\;2),  \n\\end{align*}\nwhere $Z_n \\sim \\textrm{Bern}\\left(p[|S_n|]\\right)$, and where $p:(0,1\/2]\\mapsto [0,1\/2)$ is a continuous and monotonically non-decreasing function.  \n\nA \\textit{search strategy} is a causal protocol for determining the sets $S_n = S_n(Y^{n-1})$, associated with a stopping time $\\tau$ and estimators $\\widehat{W}_\\tau = \\widehat{W}_\\tau(Y^\\tau),\\widehat{V}=\\widehat{V}(Y^\\tau)$ for the last position and the velocity. A strategy is said to be \\textit{non-adaptive} if the choice of the region $S_n$ is independent of $Y^{n-1}$, i.e., the sets we probe do not depend on the observations. In such a case, the stopping time is also fixed in advance. Otherwise, the strategy is said to be \\textit{adaptive}, and may have a variable stopping time. A strategy is said to have \\textit{search resolution} $\\delta$ and error probability $\\varepsilon$ if for any $w_0,v$,\n\\begin{equation*}\n\\Pr(\\max\\{|\\widehat{W}_\\tau-w_\\tau|_c,|\\widehat{V}-v|_c\\} \\leq \\delta) \\geq 1-\\varepsilon\n\\end{equation*}\nWe are interested in the expected search time $\\mathbb{E}(\\tau)$ for such strategies, and specifically in the \\textit{maximal targeting rate}, which is the maximal ratio $\\frac{\\log{1\\slash\\delta}}{\\mathbb{E}(\\tau)}$ such that $\\varepsilon\\to 0$ is possible as $\\delta\\to 0$. We say that a sequence of strategies indexed by $k$ achieves a \\textit{targeting rate} $R$ and an associated \\textit{targeting reliability} $E = E(R)$, if $\\delta_k\\to 0$ as $k\\to \\infty$ and\n\\begin{equation*}\n\\mathbb{E}(\\tau_k) \\leq \\frac{\\log{(1\\slash\\delta_k)}}{R},\\qquad \\log{(1\\slash\\varepsilon_k)} \\geq \\frac{E}{R}\\cdot \\log{(1\\slash\\delta_k)}\n\\end{equation*}\nfor all $k$ large enough. \n\n\n\\section{Non-adaptive strategies}\nWe state our main result for the non-adaptive case. Both known and unknown velocities are treated. In our proofs we will assume the latter; the former simpler case follows easily. \n\\begin{theorem}\\label{thrm:non-adapt}\nLet $p[\\cdot]$ be a measurement noise function. For non-adaptive search strategies, the maximal targeting rate is given by \n  \\begin{equation}\\label{eq:rate_non_adaptive}\n    \\max_{q\\in(0,\\frac{1}{2})} \\kappa I(q,p[q])\n  \\end{equation}\nwhere $\\kappa=\\frac{1}{2}$ for an unknown velocity, and $\\kappa=1$ for a known velocity. Moreover, for any $R$ below the maximal targeting rate, there exist non-adaptive search strategy such that \n\\begin{equation*}\n\\tau = \\frac{\\log{((1\\slash\\delta))}}{R},\\qquad \\log{(1\\slash\\varepsilon)} = \\frac{E_r(R,q^*)}{R}\\cdot \\log{(1\/\\delta)}\n\\end{equation*}\nwhere $q^*$ is the maximizer in \\eqref{eq:rate_non_adaptive}, and \n\\begin{equation*}\n  E_r(R,q^*) = \\max_{\\rho\\in (0,1)} E_0(\\rho,q^*) - \\rho R\/\\kappa\n\\end{equation*}\nis the random coding exponent \\cite{GallagerBook} for a  BSC$(p[q^*])$ with input distribution $q^*$, at rate $R\/\\kappa$. \n\\end{theorem}\n\\subsection{Proof of Converse}\nDenote the fixed stopping time by $\\tau = N$. Let  $\\{S_n\\}_{n=1}^N$ be any non-adaptive strategy achieving an error probability $\\varepsilon$ with search resolution $\\delta$. We prove the converse holds even under the less stringent requirement where the initial position and velocity are uniformly distributed $(W_0,V)\\sim\\textrm{Unif}([0,1)^2)$. Partition the unit interval into $\\lceil\\beta\/\\delta\\rceil$ equi-sized intervals for some constant $\\beta\\in(0,\\tfrac{1}{2})$, and let $W_N'$ be the index of the interval containing $W_N$. Similarly, let $V'$ be the index of the interval containing $V$. It is easy to see that the scheme $\\{S_n\\}$ can be made to return $W_N'$ with error probability at most $\\varepsilon' \\stackrel{\\textnormal{def}}{=}  \\varepsilon + 4\\beta(1-\\beta)$, where the latter addend stems from the probability that $(\\widehat{W}_N,\\widehat{V})$ is too close to a boundary point. \n\nNote that $X_n\\sim\\mathrm{Bern}(q_n)$ where $q_n \\stackrel{\\textnormal{def}}{=}  |S_n|$ and that $Y_n$ is obtained from $X_n$ through a memoryless binary symmetric channel with a time-varying crossover probability $p\\left[q_n\\right]$. Following the steps of the converse to the channel coding theorem, we have\n\\allowdisplaybreaks{\\begin{IEEEeqnarray}{rCl}\\label{eq:converse}\n2\\log\\left(\\frac{\\beta}{\\delta}\\right) &=& H(W_N',V') \\notag\\\\ \n&=& I(W_N',V';Y^N) + H(W_N',V'|Y^N) \\notag\\\\ \n&\\stackrel{(\\textrm{a})}{\\leq}& I(W_N',V';Y^N) + N\\varepsilon' \\notag\\\\ \n&=& \\sum_{n=1}^N I(W_N',V';Y_n|Y^{n-1}) + N\\varepsilon' \\notag\\\\ \n&\\leq& \\sum_{n=1}^N I(W_N',V',Y^{n-1};Y_n) + N\\varepsilon' \\notag\\\\ \n&\\leq& \\sum_{n=1}^N I(W_N',V',W_n,Y^{n-1};Y_n) + N\\varepsilon' \\notag\\\\ \n&\\stackrel{(\\textrm{b})}{=}& \\sum_{n=1}^N I(X_n;Y_n) + N\\varepsilon' \\notag\\\\ \n&\\stackrel{(\\textrm{c})}{=}& \\sum_{n=1}^N I(q_n,p[q_n]) + N\\varepsilon', \n\\end{IEEEeqnarray}}\nwhere (a) is by virtue of Fano's inequality, (b) follows since $X_n$ is a function of $W_n$ and the measurement noise is independent across time, and (c) stems from the fact that the  crossover probability sequence $p[q_n]$ is a fixed (time-varying) function of the codebook. Note that here, in contrast to the standard memoryless channel coding setup where the channel noise is strategy independent, (b) above does not generally hold when an adaptive strategy (i.e., feedback) is employed; this stems from the fact that in this case, the intensity of the observation noise would generally depend on $Y^{n-1}$, and therefore $Y_n-X_n-Y^{n-1}$ would not form a Markov chain. Dividing by $N$ we obtain \n\\begin{IEEEeqnarray*}{rCl}\nR &=& \\frac{\\log(1\/\\delta)}{N} \\leq  \\frac{1}{2N}\\left(\\sum_{n=1}^N I(q_n,p[q_n]) - 2\\log\\beta\\right) + \\frac{\\varepsilon'}{2} \\\\\n&\\leq&  \\frac{1}{2}\\left(\\sup_{q\\in(0,\\frac{1}{2})} I(q,p[q]) -\\frac{2\\log\\beta}{N}+ \\varepsilon + 4\\beta(1-\\beta)\\right).\n\\end{IEEEeqnarray*}\nNoting that the inequality above holds for any $\\beta\\in(0,\\tfrac 1 2)$, the converse now follows by taking the limit $N\\to\\infty$, and then requiring $\\varepsilon\\to 0$. \n\n\\subsection{Proof of Achievability}\nAchievability is obtained via random coding using an input distribution $q^*$ that achieves the supremum in \\eqref{eq:rate_non_adaptive}. A sketch of the proof is now given. We partition the unit interval into $M = \\frac{N}{\\delta}$ equi-sized subintervals $\\{b_m\\}$. Each pair of initial position and velocity $(w_0,v)$ naturally induces a \\textit{trajectory} $m(w_0,v)$ w.r.t. this partition. We say that two trajectories $m(w_0,v)$ and $m(w_0',v')$ are \\textit{$(\\delta,N)$-close} if $|w_0-w_0'|_c\\leq \n\\delta$ and $|v-v'|_c\\leq \\frac{\\delta}{N}$. Otherwise, we say the trajectories are $(\\delta,N)$-far.   \n\\begin{lemma}\\label{lem:trajec}\n  The number of different trajectories is upper bounded by $K = M^2\\cdot \\mathrm{O}(\\textrm{poly}(N))$. Moreover, if two trajectories intersect more than once, then their corresponding initial positions and velocities are $(\\delta,N)$-close.  \n\\end{lemma}\n\nWe now draw a codebook with $M$ rows, where each row $\\bx_m$ has $N$ bits. The codebook is drawn i.i.d. $\\textrm{Bern}(q^*)$. We define our random query set $S_n$ according to the codebook's columns:  \n\\begin{align*}\n  S_n  \\stackrel{\\textnormal{def}}{=}  A + Bn + \\bigcup_{m:x_{m,n}=1}b_m \n\\end{align*}\nwhere $(A,B)\\sim\\textrm{Unif}([0,1)^2)$ serve as a random ``dither'', mutually independent of the measurement noise. This dithering procedure renders our setting equivalent to the setup where the initial position and velocity are uniform and independent, and where the query sets are given by $A=B=0$. We shall proceed under this latter setup. \n\nThe next Lemma stems directly from Chernoff's bound.  \n\\begin{lemma}\nLet $\\mathcal{A}$ be the event where $||S_n|-q^*|\\leq \\epsilon$ for all $n$. Then for any $\\epsilon > 0$, $\\Pr(\\mathcal{A}^c) = 2^{-2^{O(N)}}$. \n\\end{lemma}\n\\begin{remark}\nUnder the event $\\mathcal{A}$, we can safely assume that the measurements are observed through a BSC$(p[q^*+\\epsilon])$, since we can always artificially add noise to the observations at any time $n$ for which $|S_n| < q^*+\\epsilon$.   \n\\end{remark}\n\nOur codebook induces a set of \\textit{trajectory codewords} $\\{x_{m(w_0,v),n}\\}_{n,w_0,v}$. Note that each trajectory codeword corresponds to a set of possible initial positions and velocities. With a slight abuse of notations, we denote the trajectory codewords by $\\{\\bx_k\\}_{k=1}^K$. After $N$ queries, we find the trajectory codeword that has the highest likelihood under the assumption that the measurements are observed through a BSC$(p[q^*+\\epsilon])$. We now show that the likelihood of the correct trajectory codeword is with high probability higher than that of all trajectory codewords whose associated initial position or velocity are at least $(\\delta,N)$-far. Hence, the initial position and velocity of the decoded trajectory will be $(\\delta,N)$-close to the correct one, with high probability.  Note that if the target had been stationary, we would have searched for the highest likelihood row just as in channel coding. \n\n\n\nWe write the average probability of error as \n\\begin{equation*}\n\\overline{P}_e = \\Pr(\\mathcal{A})\\Pr(e|\\mathcal{A}) + \\Pr(\\mathcal{A}^c)\\Pr(e|\\mathcal{A}^c).  \n\\end{equation*}\nThe second term vanishes double exponentially fast. For the other term we have\n\\begin{align*}\n \t\\Pr(e|\\mathcal{A})=\\sum_{\\bx_k}\\Pr(\\bx_k|\\mathcal{A})P_{\\mathcal{A}}(\\by|\\bx_k)\\Pr(e|\\bx_k,\\by,\\mathcal{A}),\n\\end{align*}\nwhere $\\by$ are the noisy observations and $P_{\\mathcal{A}}(\\by|\\bx_m)$ is the $BSC(q^*+\\epsilon)$ induced by the event $\\mathcal{A}$ (and possible randomization). Let $\\mathcal{E}_{k'}$ denote the event that the trajectory codeword $\\bx_{k'}$ is chosen instead of $\\bx_k$. Let $T_k$ be the set of all $k'$ for which either the velocity or the initial position of each of the trajectories associated with $\\bx_k'$, are more than  $\\delta$-far from those of $\\bx_k$. \n\\begin{align}\\label{eq:Tk}\n \t\\Pr(e|\\bx_k,\\by,\\mathcal{A}) \\leq \\sum_{k'\\in T_k}\\Pr(\\mathcal{E}_{k'}|\\mathcal{A})\n\\end{align}\nand \n\\begin{align}\n \t\\Pr(\\mathcal{E}_{k'}|\\mathcal{A}) = \\sum_{\\bx_{k'}: P_{\\mathcal{A}}(\\by|\\bx_k)\\leq  P_{\\mathcal{A}}(\\by|\\bx_{k'})} \\Pr(\\bx_{k'}|\\bx_k,\\mathcal{A})\\label{eq:CondErr}\n\\end{align}\nNote that unlike \\cite[eq. 5.6.8]{GallagerBook}, we cannot assume the trajectory codewords are independent under event $\\mathcal{A}$. Furthermore, for $k'\\in T_k$ the trajectories may intersect once. We therefore have that $\\Pr(\\bx_k, \\bx_{k'}|\\mathcal{A}))\\leq \\frac {\\Pr(\\bx_k,\\bx_{k'})}{1-\\Pr(\\mathcal{A}^c)} \\leq \\frac {Q(\\bx_k)Q(\\bx_{k'})}{(1-\\Pr(\\mathcal{A}^c))q_{min}}$ and $\\Pr(\\bx_k|\\mathcal{A})\\geq Q(\\bx_k) - \\Pr(\\mathcal{A}^c)$, where $Q(\\cdot)$ denotes the random coding prior, and $q_{min}$ denotes the probability of the least probable binary symbol under Q. Using this and Bayes rule, for $N$ large enough we have:\n\\begin{align}\n&P(\\bx_{k'}|\\bx_k,\\mathcal{A}) \\leq \\frac{Q(\\bx_k)Q(\\bx_{k'})}{(1-\\Pr(\\mathcal{A}^c))(Q(\\bx_k)-\\Pr(\\mathcal{A}^c))q_{min}} \\notag\n\\\\ &\\leq \\frac{Q(\\bx_{m'})}{(1-\\Pr(\\mathcal{A}^c)\/q_{min}^N)^2q_{min}} = \\frac{Q(\\bx_{m'})}{q_{min}}(1+2^{-2^{O(N)}})\\label{eq:CondProb}\n\\end{align}\nAfter substituting \\eqref{eq:CondProb} in \\eqref{eq:CondErr} and \\eqref{eq:Tk} and plugging in $\\delta=2^{-NR}$, we can follow Gallager's derivation of the random error exponent \\cite{GallagerBook} almost verbatim, with the following two distinctions: 1) By Lemma \\ref{lem:trajec} the effective number of messages is now $|T_k| = K = M^2\\cdot\\mathrm{O}(\\textrm{poly}(N))$; and 2) for any finite $N$, the exponent is multiplied by a constant pertaining to the double exponential penalty and to $q_{min}$, but this constant converges to unity as $N$ grows. The exponent is positive as long as $R\\leq I(q^*,p[q^*+\\epsilon])\/2$. As $\\epsilon$ is arbitrary, this concludes the proof of achievability.   \n\n\n\\section{Adaptive strategies}\nIn this section, we consider the gain to be reaped by allowing the search decisions to be made adaptively. For simplicity, we assume here that the velocity is known in advance, and hence without loss of generality can be assumed to be zero. We will again use dithering to make the initial position appear uniformly random. Here, the duration of search $\\tau$ will generally be a random stopping time dependent on the measurements sample path. Moreover, the choice of probing regions $S_n$, for $n$ up to the horizon $\\tau$, \ncan now depend on past measurements. We characterize this gain in terms of the maximal targeting rate, and the targeting rate-reliability tradeoff. As we shall see, adaptivity allows us to achieve the maximal possible rate and reliability, i.e., those associated with the minimal observation noise $p[0]$. \n\n\\subsection{Non Adaptive Search with Validation}\nAs a first attempt at an adaptive strategy, we continue with the non-adaptive search from the previous section, but allow the agent to validate the outcome of the search phase. We will consider two validation schemes, due to Forney  \\cite{Forney1968} and Yamamoto-Itoh \\cite{YamaItoh1980}. \n\nIn \\cite{Forney1968}, Forney considered a communication system in which a decoder, at the end of the transmission can signal the encoder to either repeat the message or continue to the next one. Namely, it is assumed that a one bit ``decision feedback'' can be sent back to the transmitter at the end of each message block. This is achieved by adding an erasure option to the decision regions, that allows the decoder\/agent to request a ``retransmission'' if uncertainty is too high, i.e., to restart the exact same coding\/search process from scratch. More concretely, given $Y^N$, a codeword $k$ will be declared as the output if $\\frac {P(y^N|\\bx_k)} {\\sum_{k'\\neq k}P(y^N|\\bx_k')}\\geq 2^{NT}$, where $T>0$ governs the tradeoff between the probability of error and the probability erasure. Let $\\mathcal{E}$ denote the event of erasure. The expected search duration will be $\\frac N {1-\\Pr(\\mathcal{E})}$. While having negligible effect on the rate (as long as $\\Pr(\\mathcal{E})$ vanishes as N grows), the results of  \\cite{Forney1968} immediately imply that such a scheme drastically improves the error exponent compared to non-adaptive schemes (see Fig.\\ref{Fig:Exponents}). \n\nThe second validation scheme we consider was proposed by Yamamoto and Itoh in \\cite{YamaItoh1980} in the context of channel coding with clean feedback. Unlike Forney's scheme which requires only one bit of feedback, this scheme requires the decoder to feed back its decision. While perfect feedback is impractical in a communication system, in our model it is inherent and can be readily harnessed. After completing the search phase with resolution $\\delta$, the agent continues to probe the estimated target location, namely an interval of size $\\delta$. If the probed region contains the target, the output of the validation phase should look like a sequence of '1's passing through a BSC$(p[\\delta])$. Thus, if the validation output is typical w.r.t. to a binary source with $\\Pr('1')=1-p[\\delta]$, the agent outputs that region as the final decision. Otherwise, the whole search is repeated from scratch. Specifically, After the $N$ queries of the non-adaptive search, we probe the aforementioned region $\\lambda N$ more times, where $0\\leq \\lambda\\leq \\infty$ determines the tradeoff between rate and reliability.  Let $\\mathcal{E}$ denote the event that the search is repeated. This happens if the wrong region has been chosen, or otherwise if the observations in the validation step were not typical. Both these events will have vanishing probabilities and therefore the rate will be negligibly affected; the average search length is now $\\mathbb{E}(\\tau) = \\frac {N(1+\\lambda)}{1-\\Pr(\\mathcal{E})}$. Following the derivations of \\cite{YamaItoh1980} with $\\lambda=\\frac {I(q^*;p[q^*])} {R} -1$, and noting that $\\delta$ can be made arbitrarily small, we obtain: \n\\begin{lemma}\nThe targeting rate-reliability tradeoff for non-adaptive scheme with a Yamamoto-Itoh validation is given by\n\\begin{equation*}\nE = C_1(p[0])\\cdot \\left(1-\\frac{R}{I(q^*;p[q^*])}\\right)  \n\\end{equation*}  \n\\end{lemma}\nNote that with this search strategy, we get better reliability than the optimal one for the $BSC(q^*)$ with feedback (given by Burnashev \\cite{burnashev1974interval}) since the validation is done over the least noisy channel (see Fig.\\ref{Fig:Exponents}).\n\n\n\\subsection{Two-Phase Search with Validation}\nIn this section, we show that a simple two-phase scheme with validation achieves the best possible performance, improving upon non-adaptive strategies (with and without validation) both in maximal targeting rate and in targeting rate-reliability tradeoff. \n      \n\\begin{theorem}\nLet $p[\\cdot]$ be a measurement noise function. For any $\\alpha\\in (0,\\tfrac{1}{2})$, there exists a search scheme with error probability $\\varepsilon$ and resolution $\\delta$, satisfying   \n\\begin{equation*}\n\\label{binC}\n\\mathbb{E} [\\tau]\\le  \\left(\\frac{\\log(1\\slash\\alpha)}{C(p[q^*])} + \\frac{\\log (1\\slash\\delta)}{C(p[\\alpha])} + \\frac{\\log(1\\slash\\epsilon)}{C_1(p[\\delta])}\\right)\\left(1+\\mathrm{o}(1)\\right).\n\\end{equation*}\n\\end{theorem}\n\\begin{corollary}\nBy letting $\\alpha$ vanish much slower than $\\delta$, we conclude that the maximal targeting rate for adaptive schemes is given by \n  \\begin{equation*}\n    C(p[0])  \\stackrel{\\textnormal{def}}{=}  \\max_{q\\in(0,\\frac{1}{2})} I(q,p[0])= I(\\tfrac{1}{2}, p[0]),\n  \\end{equation*}\nwhich is the capacity of the least noisy BSC associated with the measurements, which is the best possible. The associated targeting rate-reliability tradeoff is \n\\begin{equation*}\n  E(R) =  C_1(p[0])\\left(1 -\\frac{R}{C(p[0])}\\right).\n\\end{equation*}\nwhich is also the best possible.   \n\\end{corollary}\n\\begin{remark}\n  Juxtaposing Theorem \\ref{thrm:non-adapt} and the Corollary above, we conclude that (unlike the case of constant interval-independent noise) adaptive search strategies outperform the optimal non-adaptive strategy in both targeting rate and reliability.  \n\\end{remark}\n\\begin{proof}\nWe prove the theorem for a fixed $\\alpha$ and $\\delta,\\varepsilon\\to 0$. In the first search phase, the agent employs the optimal non-adaptive search strategy with $\\tau=\\log{N}$ and resolution $\\alpha$, i.e. with a vanishing rate $R = \\frac{\\log{1\\slash\\alpha}}{\\log{N}}$. At the end of this phase, the agent knows an interval of size $\\alpha$ containing the target with probability $1- \\textrm{o}(N)$.\n\nIn the second phase, the agent ``zooms-in'' and performs the search only within the interval obtained in the first phase. To that end, the agent employs the optimal non-adaptive search strategy with $\\tau=\\lambda N-\\log{N}$ and resolution $\\delta = 2^{-(\\lambda N-\\log{N})R}$, i.e. with  rate $R = \\frac{\\log{1\\slash\\delta}}{\\lambda N-\\log{N}}$, with the query sets properly shrunk by a factor of $\\alpha$. We note that in this phase, all queried sets are of size smaller than $\\alpha\/2$, hence the associated noise is less that $p[\\alpha]$. Therefore, if the rate $R< C[p[\\alpha]]$, then at the end of this phase the agent knows an interval of size $\\delta$ containing the target with probability $1- \\textrm{o}(N)$. \n\nAt this point, the agent perform the Yamamoto-Itoh validation step of length $(1-\\lambda)N$, which queries a fixed interval of size $\\delta$. If not successful, the agent repeats the whole two-phase search from scratch. The expected stopping time of this procedure is $\\frac{N}{1-o(N)}$, and the error probability  decays exponentially with an exponent controlled by trading off the search and validation as before, yielding the associated Burnashev behavior for the channel $p[\\delta]$. \n\\end{proof}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Exponents} \n\\caption{Error exponents (known velocity) for noise growing linearly with size: $p[0]=0.1, p[\\frac 1 2]=0.45$ (a) Random coding (b) Decision feedback (c) Burnashev's upper bound for BSC$(p[q*])$ (d) Yamamoto-Itoh validation for the non-adaptive scheme (e) Yamamoto-Itoh validation for BSC$(p[0])$}.\\label{Fig:Exponents}\n\\end{figure}\n\n\\section{Conclusions and Further Research}\nIn this paper, we considered the problem of acquiring a target moving with known\/unknown velocity on a circle starting from an unknown position, under the physically motivated observation model where the noise intensity increases with the size of the queried region. For a known velocity, we showed that unlike the constant noise model, there can be a large gap in performance (both in targeting rate and reliability) between adaptive and non-adaptive search strategies. The various rate-reliability tradeoffs discussed herein are depicted in Fig. \\ref{Fig:Exponents}.  Furthermore, we demonstrated that the cost of accommodating an unknown velocity in the non-adaptive setting, is a factor of two in the targeting rate, as intuition may suggest.  \n\nOne may also consider other search performance criteria, e.g., where the agent is cumulatively penalized by the size of either the queried region or its complement, according to the one containing the target. The rate-optimal scheme presented herein, which is based on a two-phase random search, may be far from optimal in this setup. In such cases we expect that sequential search strategies, e..g, ones based on posterior matching \\cite{Shayevitz11,naghshvar2013extrinsic}, would exhibit superior performance as they naturally shrink the queried region with time.\n\nOther research directions include more complex stochastic motion models, as well as searching for multiple targets (a ``multi-user'' setting). For the latter, preliminary results indicate that the gain reaped by using adaptive strategies vs. non-adaptive ones diminishes as the number of targets increases.   \n \n\\section{Acknowledgement}\nThe authors would like to thank an anonymous reviewer for throughly reading the paper and for many useful comments. \n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Model Hamiltonian}\n\n\\subsection*{Main Terms}\n\nOur model consists of two Ni-Ligand clusters, roughly equivalent to two NiO$_6$ octahedra, each containing a Ni $2p$ shell, a Ni $3d$ shell, and a ligand shell. The ligand shell orbitals are defined as linear combinations of the actual oxygen $2p$ like Wannier-orbitals having the same rotation properties as the Ni $d$ orbitals within the $O_h$ point group used here \\cite{Ballhausen1968S,Haverkort_Wannier_PRB_2010S}. The general form of our Hamiltonian is:\n\\begin{equation}\nH = H_{LF_A} + H_{LF_B} + H_{mix},\n\\end{equation}\nwhere $A$ and $B$ refer to the short and long bond sites, respectively (or identical sites when no breathing distortion is present) and $H_{mix}$ is the part of the Hamiltonian that couples the two clusters together. Taken separately, $H_{LF_A}$ and $H_{LF_B}$ are independent multiplet ligand field theory Hamiltonians. For $H_{LF_A}$, we have:\n\\begin{equation}\nH_{LF_A} = H_U^{dd} + H_U^{pd} + H_{\\bm{l} \\cdot \\bm{s}}^{d} + H_{\\bm{l} \\cdot \\bm{s}}^{p} + H_{o}^{p} + H_{o}^{d} + H_{o}^{L} + H_{hyb}^{dL},\n\\end{equation}\nwith, \\\\\n\\begin{tabular}{ll}\n$H_U^{dd}$ & the Coulomb repulsion between two Ni $3d$ electrons including all multiplet effects,\\\\\n$H_U^{pd}$ & the Coulomb repulsion between a Ni $2p$ core and $3d$ valence electron including all multiplet effects,\\\\\n$H_{\\bm{l} \\cdot \\bm{s}}^{d}$ & the Ni $3d$ spin orbit interaction,\\\\\n$H_{\\bm{l} \\cdot \\bm{s}}^{p}$ & the Ni $2p$ core level spin orbit interaction,\\\\\n$H_{o}^{p}$ & the onsite energy of the Ni $2p$ core orbitals,\\\\\n$H_{o}^{d}$ & the orbital dependent onsite energy of the Ni $3d$ valence orbitals,\\\\\n$H_{o}^{L}$ & the orbital dependent onsite energy of the Ligand orbitals, and\\\\\n$H_{hyb}^{dL}$ & the hybridization strength between the Ni $3d$ and Ligand orbitals.\n\\end{tabular}\n\nThe Hamiltonian for site $B$ is analogous to $H_{LF_A}$. Below we list each term in the Hamiltonian in more detail.\n\n\\subsection*{\\texorpdfstring{On-site Energy of the Ni $2p$ Orbitals - $H_{o}^{p}$}{}}\n\nThe onsite energy of the Ni $2p$ core electrons is given as:\n\\begin{equation}\nH_{o}^{p} = \\epsilon_p \\sum_{\\tau} \\bm{p}^{\\dag}_{\\tau} \\bm{p}^{\\phantom{\\dag}}_{\\tau},\n\\end{equation}\nwith $\\tau$ labeling the 6 different Ni $2p$ spin-orbitals with $m=-1,0,1$ and $\\sigma=\\pm1\/2$, $\\bm{p}^{\\dag}_{\\tau}$ ($\\bm{p}^{\\phantom{\\dag}}_{\\tau}$) the operator creating (annihilating) an electron in orbital $\\tau$ and $\\epsilon_p$ defined in terms of $U_{dd}$, $U_{pd}$ and $\\Delta$ as \\cite{Zaanen_ZSA_PRL_1985S, Zaanen_NiPES_PRB_1986S, degroot2008S}:\n\\begin{equation}\n\\epsilon_{p} = \\frac{ 10\\Delta + \\left(1+n_d\\right) \\left(n_d\\frac{U_{dd}}{2}-\\left(10+n_d\\right)U_{pd}\\right)}{16+n_d}\n\\end{equation}\nwhere $n_d$ is the formal number of $3d$ electrons per Ni ($n_d = 7$ for the nickelates studied here).\n\n\\subsection*{\\texorpdfstring{On-site Energy of the Ni $3d$ Orbitals - $H_{o}^{d}$}{}}\n\nThe onsite energy of the Ni $3d$ valence electrons is given as:\n\\begin{equation}\nH_{o}^{d} = \\epsilon_d \\sum_{\\tau} \\bm{d}^{\\dag}_{\\tau} \\bm{d}^{\\phantom{\\dag}}_{\\tau} + Dq_i \\left( 6 \\sum_{\\tau\\in e_g} \\bm{d}^{\\dag}_{\\tau} \\bm{d}^{\\phantom{\\dag}}_{\\tau} - 4 \\sum_{\\tau\\in t_{2g}} \\bm{d}^{\\dag}_{\\tau} \\bm{d}^{\\phantom{\\dag}}_{\\tau} \\right),\n\\end{equation}\nwith $\\tau$ labeling the 10 different Ni $3d$ spin-orbitals belonging either to the $t_{2g}$ irreducible representation ($yz$, $xz$, and $xy$) or to the $e_g$ irreducible representation ($3z^2-r^2$, and $x^2-y^2$) with either spin up or spin down, $\\bm{d}^{\\dag}_{\\tau}$ ($\\bm{d}^{\\phantom{\\dag}}_{\\tau}$) the operator creating (annihilating) an electron in orbital $\\tau$, $\\epsilon_d$ the shell average energy defined in terms of $U_{dd}$, $U_{pd}$ and $\\Delta$ as \\cite{Zaanen_ZSA_PRL_1985S, Zaanen_NiPES_PRB_1986S, degroot2008S}:\n\\begin{align}\n\\epsilon_d = \\frac{10\\Delta - n_d\\left(31+n_d\\right)\\frac{U_{dd}}{2}-90U_{pd}}{16+n_d},\n\\end{align}\nand $Dq_i$ the onsite part of the cubic crystal-field splitting. The value of $Dq_i$ depends on the the breathing distortion, being larger for the smaller site. In terms of the non-breathing value $Dq_0$, the breathing distortion is approximated by Harrison's rules for hybridization \\cite{Harrison1983S} as\n\\begin{align}\nDq_i = Dq_0 \\left(1 + \\frac{\\delta d_i}{d_0}\\right)^{-4}\n\\end{align}\nwhere, $d_0$ is the average bond length and $\\delta d$ is the positive (negative) displacement from average for the long (short) bond octahedron $B$ ($A$).\n\n\\subsection*{\\texorpdfstring{On-site Energy of the Ligand Orbitals - $H_{o}^{L}$}{}}\n\nThe ligand orbitals are linear combinations of the valence states of the infinite solid such that the local Ni $d$ orbitals directly interact only with these orbitals \\cite{Haverkort_Wannier_PRB_2010S}. For each Ni spin-orbital there is exactly one Ligand orbital, independent of crystal symmetry and breathing distortion. The ligand orbitals can have a different onsite energy depending if they belong to the local $t_{2g}$ or $e_g$ irreducible representation. The Hamiltonian thus takes a very similar form as $H_{o}^{d}$ namely:\n\\begin{equation}\nH_{o}^{L} = \\epsilon_L \\sum_{\\tau} \\bm{L}^{\\dag}_{\\tau} \\bm{L}^{\\phantom{\\dag}}_{\\tau} +  Tpp_i \\left( \\sum_{\\tau\\in e_g} \\bm{L}^{\\dag}_{\\tau} \\bm{L}^{\\phantom{\\dag}}_{\\tau} - \\sum_{\\tau\\in t_{2g}} \\bm{L}^{\\dag}_{\\tau} \\bm{L}^{\\phantom{\\dag}}_{\\tau} \\right),\n\\end{equation}\nwith $\\tau$ labeling the 10 different Ligand spin-orbitals belonging either to the $t_{2g}$ irreducible representation ($yz$, $xz$, and $xy$) or to the $e_g$ irreducible representation ($3z^2-r^2$, and $x^2-y^2$) with either spin up or spin down, $\\bm{L}^{\\dag}_{\\tau}$ ($\\bm{L}^{\\phantom{\\dag}}_{\\tau}$) the operator creating (annihilating) an electron in orbital $\\tau$, $\\epsilon_L$ the shell average energy defined in terms of $U_{dd}$, $U_{pd}$ and $\\Delta$ as \\cite{Zaanen_ZSA_PRL_1985S, Zaanen_NiPES_PRB_1986S, degroot2008S}:\n\\begin{align}\n\\epsilon_{L} = \\frac{\\left(1+n_d\\right)\\left(n_d\\frac{U_{dd}}{2}+6U_{pd}\\right) - \\left(6+n_d\\right)\\Delta}{16+n_d},\n\\end{align}\nand $T_{pp,i}$ roughly the hopping strength between two ligand O $2p$ orbitals \\cite{Haverkort_Wannier_PRB_2010S}. The value of $T_{pp,i}$ depends on the breathing distortion and can be expressed in terms of $\\delta d_i$, the positive (negative) displacement from average for the long (short) bond octahedron $B$ ($A$) and the non-breathing value $T_{pp}$, using rules defined by Harrison \\cite{Harrison1983S} as\n\\begin{align}\nT_{pp,i} = T_{pp} \\left(1 + \\frac{\\delta d_i}{d_0}\\right)^{-3}.\n\\end{align}\n\n\n\\subsection*{\\texorpdfstring{Hybridization Between Ni $3d$ and Ligand Orbitals - $H_{hyb}^{dL}$}{}}\n\nThe interaction between the Ni $3d$ orbitals and the Ligand orbitals is given as:\n\\begin{equation}\n\\label{Eqn:Hhyb}\nH_{hyb}^{dL} =  \\sqrt{1-x} \\left( V_{e_{g}} \\sum_{\\tau\\in e_g} \\left( \\bm{d}^{\\dag}_{\\tau} \\bm{L}^{\\phantom{\\dag}}_{\\tau} + \\bm{L}^{\\dag}_{\\tau} \\bm{d}^{\\phantom{\\dag}}_{\\tau} \\right) + V_{t_{2g}} \\sum_{\\tau\\in t_{2g}} \\left( \\bm{d}^{\\dag}_{\\tau} \\bm{L}^{\\phantom{\\dag}}_{\\tau} + \\bm{L}^{\\dag}_{\\tau} \\bm{d}^{\\phantom{\\dag}}_{\\tau} \\right) \\right),\n\\end{equation}\nwith $\\tau$ labeling the 10 different Ni $3d$ or Ligand spin-orbitals belonging either to the $t_{2g}$ irreducible representation ($yz$, $xz$, and $xy$) or to the $e_g$ irreducible representation ($3z^2-r^2$, and $x^2-y^2$) with either spin up or spin down, $\\bm{d}^{\\dag}_{\\tau}$, $\\bm{L}^{\\dag}_{\\tau}$ ($\\bm{d}^{\\phantom{\\dag}}_{\\tau}$, $\\bm{L}^{\\phantom{\\dag}}_{\\tau}$) the operator creating (annihilating) an electron in orbital $\\tau$ and either the $d$ or Ligand shell. $V_{e_{g}}$ and $V_{t_{2g}}$ are the individual hopping strengths between the $d$ and Ligand orbitals. The parameter $x$ determines the ratio between the hopping within a single ligand field cluster and between two ligand-field clusters.\n\n\\subsection*{\\texorpdfstring{Coupling Between Cluster $A$ and $B$ - $H_{mix}$}{}}\n\nThe interaction between cluster $A$ and cluster $B$ is given as:\n\\begin{align}\n\\label{Eqn:Hmix}\nH_{mix} =  \\sqrt{x} \\bigg( &V_{e_{g}} \\sum_{\\tau\\in e_g} \\left( \\bm{d}^{\\dag}_{A,\\tau} \\bm{L}^{\\phantom{\\dag}}_{B,\\tau} + \\bm{L}^{\\dag}_{B,\\tau} \\bm{d}^{\\phantom{\\dag}}_{A,\\tau} + \\bm{d}^{\\dag}_{B,\\tau} \\bm{L}^{\\phantom{\\dag}}_{A,\\tau} + \\bm{L}^{\\dag}_{A,\\tau} \\bm{d}^{\\phantom{\\dag}}_{B,\\tau} \\right) \\\\ \n\\nonumber &+  V_{t_{2g}} \\sum_{\\tau\\in t_{2g}} \\left( \\bm{d}^{\\dag}_{A,\\tau} \\bm{L}^{\\phantom{\\dag}}_{B,\\tau} + \\bm{L}^{\\dag}_{B,\\tau} \\bm{d}^{\\phantom{\\dag}}_{A,\\tau} + \\bm{d}^{\\dag}_{B,\\tau} \\bm{L}^{\\phantom{\\dag}}_{A,\\tau} + \\bm{L}^{\\dag}_{A,\\tau} \\bm{d}^{\\phantom{\\dag}}_{B,\\tau}\\right) \\bigg),\n\\end{align}\nwith the individual terms defined as in the previous section. The definition of the hybridization interaction using the parameter $x$ is such that the overall hopping strength is independent of the coupling between the two clusters. For perfect periodic boundary conditions $x=\\frac{1}{2}$. In this case one can create  bonding and anti-bonding linear combinations of the ligand orbitals of cluster $A$ and cluster $B$ and the anti-bonding linear combination will be non-bonding with respect to the Ni $3d$ orbitals. Although this senario on first sight looks like a good cluster model, it highly overestimates the Ni-Ni exchange interactions. For nearest neighbor clusters only one out of six oxygens is shared and one might expect $x\\approx 1\/6$ to yield reasonable results, assuming 180 degree Ni-O-Ni bonds. We used $x$ as an empirical parameter, as one can not claim convergence with respect to cluster size for a two site calculation. Due to the presence of non-local excitations, the XAS spectrum is quite sensitive to $x$ and we find best agreement with experiment using $x = 0.35^2 = 0.1225$, which is quite close to the expected value of $1\/6$. The fact that we find a value slightly smaller than $1\/6$ is likely due in part to the octahedral tilts in the real materials, which decrease the Ni-O-Ni bond angle from 180 degrees and reduce the effective Ni-Ni hopping. \n\n\\subsection*{\\texorpdfstring{Coulomb Repulsion Between Two Ni $3d$ Electrons - $H_{U}^{dd}$}{}}\n\nThe onsite Coulomb repulsion between two $d$ electrons is defined as:\n\\begin{align}\nH_{U}^{dd} &= \\sum_{i,j} \\frac{1}{2}\\frac{e^2}{|r_i-r_j|}\\\\\n\\nonumber  &= \\sum_{\\tau_1,\\tau_2,\\tau_3,\\tau_4} U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4} \\bm{d}^{\\dag}_{\\tau_1} \\bm{d}^{\\dag}_{\\tau_2} \\bm{d}^{\\phantom{\\dag}}_{\\tau_3} \\bm{d}^{\\phantom{\\dag}}_{\\tau_4},\n\\end{align}\nwith,\n\\begin{align}\n U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4} = -\\frac{1}{2} \\delta_{\\sigma_1,\\sigma_3} \\delta_{\\sigma_2,\\sigma_4}\n \\sum_{k=0,2,4} \n c^{\\left(k\\right)}\\left[l_1=2,m_1;l_3=2,m_3\\right]\n c^{\\left(k\\right)}\\left[l_4=2,m_4;l_2=2,m_2\\right]\n \\times F^{\\left(k\\right)},\n\\end{align}\nwhere $\\tau$ are combined spin and orbital indices, $\\sigma$ are spin indices, $l$ and $m$ are angular momentum indices ($l=2$ for $d$ electrons), $F^{(k)}$ are the radial (Slater) integrals, and \n\\begin{align}\nc^{\\left(k\\right)} \\left[l_1,m_1;l_2,m_2\\right] = \n\\Bra{Y_{m_1}^{\\left(l_1\\right) } }\nC^{\\left(k\\right)}_{m_1-m_2} \n\\Ket{Y_{m_2}^{\\left(l_2\\right)}}\n\\end{align}\nare angular integrals of spherical harmonics $Y_{m}^{\\left(l\\right) }$ and renormalized spherical harmonics $C^{\\left(k\\right)}_{m} = \\sqrt{\\frac{4 \\pi}{2 k +1}} Y_{m}^{\\left(k\\right)}$.\nThe Slater integrals $F^{(2)}$ and $F^{(4)}$ are related to the multipole interaction between two $d$ electrons and responsible for the multiplet splitting between the different levels. They can be approximated by $J_H$ albeit at the loss of the experimentally observed multiplet structure. $F^{(0)}$ is the spherical averaged Coulomb interaction, i.e. the monopole part of the interaction. $F^{(0)}$ is related to $U$ by:\n\\begin{equation}\nF^{(0)} = U + \\frac{2}{63}(F^{(2)} + F^{(4)}).\n\\end{equation}\n\n\n\\subsection*{\\texorpdfstring{Coulomb repulsion between a Ni $2p$ and Ni $3d$ electron - $H_{U}^{pd}$}{}}\n\nThe onsite interaction between the Ni $2p$ and Ni $3d$ electrons is given as:\n\\begin{align}\nH_{U}^{pd} &= \\sum_{\\tau_1,\\tau_2,\\tau_3,\\tau_4} 2 U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4}^G \\bm{d}^{\\dag}_{\\tau_1} \\bm{p}^{\\dag}_{\\tau_2} \\bm{p}^{\\phantom{\\dag}}_{\\tau_3} \\bm{d}^{\\phantom{\\dag}}_{\\tau_4} + 2 U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4}^F \\bm{d}^{\\dag}_{\\tau_1} \\bm{p}^{\\dag}_{\\tau_2} \\bm{d}^{\\phantom{\\dag}}_{\\tau_3} \\bm{p}^{\\phantom{\\dag}}_{\\tau_4},\n\\end{align}\nwith \n\\begin{align}\n U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4}^F = -\\frac{1}{2} \\delta_{\\sigma_1,\\sigma_3} \\delta_{\\sigma_2,\\sigma_4}\n \\sum_{k=0,2} \n c^{\\left(k\\right)}\\left[l_1=2,m_1;l_3=2,m_3\\right]\n c^{\\left(k\\right)}\\left[l_4=1,m_4;l_2=1,m_2\\right]\n \\times F^{\\left(k\\right)}_{pd},\n\\end{align}\nand \n\\begin{align}\n U_{\\tau_1,\\tau_2,\\tau_3,\\tau_4}^G = -\\frac{1}{2} \\delta_{\\sigma_1,\\sigma_3} \\delta_{\\sigma_2,\\sigma_4}\n \\sum_{k=1,3} \n c^{\\left(k\\right)}\\left[l_1=2,m_1;l_3=1,m_3\\right]\n c^{\\left(k\\right)}\\left[l_4=2,m_4;l_2=1,m_2\\right]\n \\times G^{\\left(k\\right)}_{pd}.\n\\end{align}\nThe monopole part of the $p$-$d$ interaction ($F^{(0)}_{pd}$) is related to $Q = U_{pd}$ by:\n\\begin{equation}\nF^{(0)}_{pd} = U_{pd} + \\frac{1}{15}G^{1}_{pd} + \\frac{3}{70}G^{3}_{pd}\n\\end{equation}\n\n\\vspace*{10pt}\n\n\\subsection*{\\texorpdfstring{Ni $3d$ and $2p$ spin-orbit coupling - $H_{\\bm{l} \\cdot \\bm{s}}^{d}$ and $H_{\\bm{l} \\cdot \\bm{s}}^{p}$}{}}\n\nThe spin-orbit coupling interaction within either the $2p$ or $3d$ shell of Ni is given as:\n\\begin{align}\nH_{\\bm{l \\cdot s}} = \\xi\\sum_{i} \\bm{l}_i \\cdot \\bm{s}_i =\n\\xi\\sum_{m=-l}^{m=l} \\sum_{\\sigma} m\\sigma \\bm{a}^{\\dagger}_{m\\sigma} \\bm{a}_{m\\sigma} +\n\\frac{\\xi}{2}\\sum_{m=-l}^{m=l-1} \\sqrt{l+m+1}\\sqrt{l-m}\n\\left( \n\\bm{a}^{\\dagger}_{m+1,\\downarrow} \\bm{a}_{m,\\uparrow} +  \n\\bm{a}^{\\dagger}_{m,\\uparrow} \\bm{a}_{m+1,\\downarrow}\n\\right)\n\\end{align}\nwhere here $m$ is the orbital index, $l$ the angular momentum, i.e. either $p$ or $d$, $\\sigma$ is the spin index, and $\\xi$ the coupling constant. The operator $\\bm{a}^{\\dagger}$ corresponds to $\\bm{d}^{\\dagger}$ or $\\bm{p}^{\\dagger}$ for the valence $3d$ or core $p$ shells respectively.\n\n\\vspace*{10pt}\n\n\\subsection*{Hopping Parameter Definitions}\n\nBoth the inter- and intra-cluster hybridization interactions are affected by the breathing distortion. This dependence is contained in the hopping integrals $V_{e_{g}}$ and $V_{t_{2g}}$ of equations \\ref{Eqn:Hhyb} and \\ref{Eqn:Hmix}. In terms of the non-breathing values $V_{e_{g},0}$ and $V_{t_{2g},0}$, the hybridization parameters are defined using Harrison's rules \\cite{Harrison1983S}, as\n\\begin{align}\nV_{e_{g}} &= V_{e_{g},0} \\left( 1 + \\frac{\\delta d}{d_0} \\right)^{-4} \\\\\nV_{t_{2g}} &= V_{t_{2g},0} \\left( 1 + \\frac{\\delta d}{d_0} \\right)^{-4}\n\\end{align}\nwhere $\\delta d$ is the bond length displacement as defined above. The values used for $V_{e_{g},0}$ and $V_{t_{2g},0}$ are provided in the section below. \n\n\\clearpage\n\n\\section*{Parameter Values}\n\nParameter values enter into our model in the form of Coulomb interactions, on-site energies, spin orbit interactions, and hopping integrals. The values for these parameters have been quite well established over several decades of core level spectroscopy and other techniques, and here we do not deviate in any way from standard values (discussions of typical values can be found in references \\onlinecite{degroot2008S, Zaanen_NiPES_PRB_1986S, Zaanen_ZSA_PRL_1985S, Haverkort_Wannier_PRB_2010S, Ballhausen1968S}, among many others). For the monopole Coulomb interaction parameters we use $U_{dd} = 6$ eV and $U_{pd} = 7$ eV. For the charge transfer energy we use $\\Delta = -0.5$ eV (see the following paragraph for a discussion of the exact definition of $\\Delta$). For the non-breathing-distorted values of on-site energies we use $10Dq = 0.95$ eV and $T_{pp} = 0.75$ eV. For non-breathing-distorted intra-cluster hopping integrals we use $V_{e_{g},0} = 3.0$ eV and $V_{t_{2g},0} = 1.74$ eV. For inter-cluster hopping, we find best agreement with experiment using $V_{I} = \\sqrt{x} = 0.35$. Spin orbit interaction parameters are taken as the atomic values for Ni $3d^7$, $\\xi_{2p} = 11.506$ eV  and $\\xi_{3d} = 91$ meV. Finally, the multipole Coulomb interaction parameters are taken as 80\\% of their atomic Hartree-Fock values for Ni $3d^7$ \\cite{Cowan1981S}: $F^2_{dd} = 10.622$, $F^4_{dd} = 6.636$, $F^2_{pd} = 6.680$, $G^1_{pd} = 5.066$, and $G^3_{pd} = 2.882$, all expressed in units of electron volts.\n\nThere exist several different ways to define the charge transfer energy $\\Delta$. As detailed in the preceding sections, in our model $\\Delta$ contributes to the shell average energies $\\epsilon_d$, $\\epsilon_L$, and $\\epsilon_p$. Thus, our value of $\\Delta = -0.5$ eV sets the energy of the $d^8\\underline{L}$ configuration 0.5 eV lower than the $d^7$ configuration \\emph{before} the inclusion of multiplet, spin-orbit, crystal field, and ligand-ligand hopping terms. These terms all modify the energy separation between the lowest $d^7$ and $d^8\\underline{L}$ states, thus affecting the effective charge transfer energy.  For example, the ligand-ligand hopping shifts the ligand $e_g$ orbitals up by $T_{pp}$ and shifts the ligand $t_{2g}$ orbitals down by $T_{pp}$, which ends up shifting the $d^8\\underline{L}$ configuration lower by $T_{pp}$ with respect to the $d^7$ configuration, thus decreasing the effective delta further. Once all of the on-site energy effects are considered, we find that the lowest $d^8\\underline{L}$ eigenstate is 0.6 eV lower than the lowest $d^7$ eigenstate (before the inclusion of hybridization), thus giving an effective charge transfer energy of $-0.6$ eV.\n\n\n\\section*{Spectroscopy Simulations}\n\nAfter finding the ground state wavefunction of our model, we simulate the core XAS and MCD spectra using a Lanczos-based Green's function method \\cite{Dagotto_RMP1994S, Haverkort_Wannier_PRB_2010S, Haverkort_DMFTXAS_EPL2014S}. For XAS we calculate the isotropic signal via the sum of spectra using $z$, left circular, and right circular polarized dipole transition operators. The MCD response (i.e. the fundamental MCD spectrum, or $f^{\\left(1\\right)}$ component of the atomic scattering tensor) is the difference between left and right circular polarized spectral functions.\n\nThe resonant magnetic diffraction signal of the nickelates has been intensively studied of late, with experiments probing the collinearity of moments via the azimuthal dependence of the diffraction signal \\cite{Frano_OrbControl_PRL_2013S}. For this work we are interested in the energy dependence of the RMD signal, so for simplicity we assume a collinear arrangement of spins with a structure factor of \n\\begin{align}\nS\\left[\\bm{q}=\\left(\\sfrac{1}{4},\\sfrac{1}{4},\\sfrac{1}{4}\\right)\\right] = \n1\\cdot f^{\\left(1\\right)}_{A1} + i\\cdot f^{\\left(1\\right)}_{B1} - 1\\cdot f^{\\left(1\\right)}_{A2} - i\\cdot f^{\\left(1\\right)}_{B2}\n\\end{align} \nwhere $f^{\\left(1\\right)}$ denote the complex valued, energy dependent, magnetic circular dichroic form factors and $A1$ and $A2$ refer to short bond sites in different antiferromagnetic planes (similar for long bond sites $B$). Given an antiferromagnetic alignment of $A1$ and $A2$ and thus a negation of the form factor ($f^{\\left(1\\right)}_{A1} = -f^{\\left(1\\right)}_{A2} \\equiv f^{\\left(1\\right)}_{A}$, and similar for $B$), our structure factor simplifies to\n\\begin{align}\nS\\left[\\bm{q}=\\left(\\sfrac{1}{4},\\sfrac{1}{4},\\sfrac{1}{4}\\right)\\right] = \n2 \\left( f^{\\left(1\\right)}_{A} + i\\cdot f^{\\left(1\\right)}_{B} \\right)\n\\end{align} \nand, neglecting polarization effects for simplicity, the spectral intensity is therefore\n\\begin{align}\n\\label{Eqn:RMD}\nI\\left(\\omega\\right) \\propto \\left| f^{\\left(1\\right)}_{A}\\left(\\omega\\right) + i f^{\\left(1\\right)}_{B}\\left(\\omega\\right) \\right|^2\n\\end{align}\nwhere we have restored the dependence on energy, $\\omega$. Note that this expression does assume a particular domain structure, as $S_A \\leq S_B$ in our model, i.e. the ordering in Eqn. \\ref{Eqn:RMD} is (~$\\SpinUp[6pt]~\\SpinUp[10pt]~\\SpinDn[6pt]~\\SpinDn[10pt]~$).  One can assume the alternate ordering (~$\\SpinUp[10pt]~\\SpinUp[6pt]~\\SpinDn[10pt]~\\SpinDn[6pt]~$) with $f^{\\left(1\\right)}_{A}$ and $f^{\\left(1\\right)}_{B}$ switched in Eqn. \\ref{Eqn:RMD} and obtain a slightly different energy dependence of the magnetic scattering. The spectra of both arrangements are shown along with the XAS for $\\delta d = 0.03$\\AA~in Fig. \\ref{Fig:SRMD} below. While both spectra are qualitatively similar, having the magnetic scattering intensity strongest at the energy of the first XAS peak and a small shoulder at higher energy, there are differences in overall intensity and in the fine details of the peaks. In the manuscript, we plot the average of these two responses.\n\n\n\\begin{figure}[t]\n\\includegraphics[width=4in]{Fig_Supp_XAS_RMD}\n\\caption{Comparison of resonant magnetic diffraction at a breathing distortion of $\\delta d = 0.03$\\AA ~for the two different spin arrangements shown. }\n\\label{Fig:SRMD}\n\\end{figure}\n\n\\vspace*{40pt}\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nArabish is the romanization of Arabic Dialects (ADs) used for informal messaging, especially in social networks.\\footnote{Also known as Arabizi (from the combination between the Arabic words \\textipa{\"}Arab\\textipa{\"}, \\textipa{[\"Qarab]} and \\textipa{\"}English\\textipa{\"} \\textipa{[i:n\"Zli:zi:]}, or the English one: \\textipa{\"}easy\\textipa{\"}, \\textipa{[\"i:zi]}), Franco-Arabic, Arabic Chat Alphabet, ACII-ized Arabic, and many others. \\textipa{\"}Arabish\\textipa{\"} is probably the result of the union between  \\textipa{[\"Qarab]} and \\textipa{\"}English\\textipa{\"}.} This writing system provides an interesting ground for linguistic research, computational as well as sociolinguistic, mainly due to the fact that it is a spontaneous representation of the ADs, and because it is a linguistic phenomenon in constant expansion on the web. \nDespite such potential, little research has been dedicated to Tunisian Arabish (TA). In this paper we describe the work we carried to develop a flexible and multi-purpose TA resource. This will include a TA corpus, together with some tools that could be useful for analyzing the corpus and for its extension with new data.\n\nFirst of all, the resource will be useful to give an overview of the TA. At the same time, it will be a reliable representation of the Tunisian dialect (TUN) evolution over the last ten years: the collected texts date from 2009 to present. This selection was done with the purpose to observe to what extent the TA orthographic system has evolved toward a writing convention.\nTherefore, the TArC will be suitable for phonological, morphological, syntactic and semantic studies, both in the linguistic and the Natural Language Processing (NLP) domains.\nFor these reasons, we decided to build a corpus which could highlight the structural characteristics of TA through different annotation levels, including Part of Speech (POS) tags and lemmatization. In particular, to facilitate the match with the already existing tools and studies for the Arabic language processing, we provide a transcription in Arabic characters at token level, following the Conventional Orthography for Dialectal Arabic guidelines \n\\emph{CODA*} (CODA \\emph{star}) \\cite{habash-etal-2018-unified} and taking into account the specific guidelines for TUN (CODA TUN) \\cite{DBLP:conf\/lrec\/ZribiBMEBH14}.\nFurthermore, even if the translation is not the main goal of this research, we have decided to provide an Italian translation of the TArC's texts.\\footnote{We considered the Italian translation as an integrated part of the annotation phase that would have cost us less effort in addition to us if carried out in our mother-tongue. The possibility of a TArC English translation is left open for a later time.}\n\nEven though in the last few years ADs have received an increasing attention by the NLP community, many aspects have not been studied yet and one of these is the Arabish code-system. The first reason for this lack of research is the relatively recent widespread of its use: before the advent of the social media, Arabish usage was basically confined to text messaging.\nHowever, the landscape has changed considerably, and particularly thanks to the massive registration of users on Facebook since 2008. At that time, in Tunisia there were still no Arabic keyboards, neither for Personal Computers, nor for phones, so Arabic-speaking users designed TA for writing in social media (Table \\ref{tab1}).  \nA second issue that has held back the study of Arabish is its lack of a standard orthography, and the informal context of use. It is important to note that also the ADs lack a standard code-system, mainly because of their oral nature.\nIn recent years the scientific community has been active in producing various sets of guidelines for dialectal Arabic writing in Arabic characters: CODA (Conventional Orthography for Dialectal Arabic)  \\cite{habash-etal-2012-conventional}.\n\\newline\n\nThe remainder of the paper is organized as follows:  section~\\ref{sec:StART} is an overview of NLP studies on TUN and TA; section~\\ref{sec:TD_TA} describes TUN and TA; section~\\ref{sec:TArC} presents the TArC corpus building process; section~\\ref{sec:pro} explains preliminary experiments with a semi-automatic transcription and annotation procedure, adopted for a faster and simpler construction of the TArC corpus; conclusions are drawn in section~\\ref{sec:concl}\n\n\\section{Related Work}\n\\label{sec:StART}\n\nIn this section, we provide an overview of work done on automatic processing of TUN and TA. As briefly outlined above, many studies on TUN and TA aim at solving the lack of standard orthography. The first Conventional Orthography for Dialectal Arabic (CODA) was for Egyptian Arabic  \\cite{habash-etal-2012-conventional} and it was used by \n \\newcite{bies2014transliteration} for Egyptian Arabish transliteration into Arabic script.\nThe CODA version for TUN (CODA TUN) was developed by  \\newcite{DBLP:conf\/lrec\/ZribiBMEBH14}, and was used in many studies, like \\newcite{boujelbane2015traitements}. Such work presents a research on automatic word recognition in TUN. Narrowing down to the specific field of TA, CODA TUN was used in \\newcite{masmoudi2015arabic} to realize a TA-Arabic script conversion tool, implemented with a rule-based approach.\nThe most extensive CODA is CODA*, a unified set of guidelines for 28 Arab city dialects  \\cite{habash-etal-2018-unified}. For the present research,  CODA* is considered the most convenient guideline to follow due to its extensive applicability, which will support comparative studies of corpora in different ADs. \nAs we already mentioned, there are few NLP tools available for Arabish processing in comparison to the amount of NLP tools realized for Arabic.\nConsidering the lack of spelling conventions for Arabish, previous effort has focused on automatic transliteration from Arabish to Arabic script, e.g.  \\newcite{chalabi2012romanized}, \\newcite{darwish2013arabizi}, and \\newcite{al2014automatic}.\nThese three work are based on a character-to-character mapping model that aims at generating a range of alternative words that must then be selected through a linguistic model. A different method is presented in  \\newcite{younes2018sequence}, in which the authors present a sequence-to-sequence-based approach for TA-Arabic characters transliteration in both directions \\cite{Sutskever-2014-SSL-2969033.2969173,younes2018sequence}.\n\nRegardless of the great number of work done on TUN automatic processing, there are not a lot of TUN corpora available for free  \\cite{younes2018survey}.\nTo the best of our knowledge there are only five TUN corpora freely downloadable: one of these is the PADIC  \\citelanguageresource{PADIC}, composed of 6,400 sentences in six Arabic dialects, translated in \\emph{Modern Standard Arabic} (MSA), and annotated at sentence level.\\footnote{The Arabic dialects of the PADIC are: TUN (Sfax), two dialects of Algeria, Syrian, Palestinian and Moroccan \\cite{meftouh2018padic}.} Two other corpora are the Tunisian Dialect Corpus Interlocutor (TuDiCoI) \\citelanguageresource{Tudicoi} and the Spoken Tunisian Arabic Corpus (STAC) \\citelanguageresource{stac}, which are both morpho-syntactically annotated. The first one is a spoken task-oriented dialogue corpus, which gathers a set of conversations between staff and clients recorded in a railway station. TuDiCoI consists of 21,682 words in client turns \\cite{graja2013discriminative}.\\footnote{The annotation was carried out only for 7,814 word.}\nThe STAC is composed of 42,388 words collected from audio files downloaded from the web (as TV channels and radio stations files)  \\cite{zribi2015spoken}. A different corpus is the TARIC \\citelanguageresource{Taric}, which contains 20 hours of TUN speech, transcribed in Arabic characters \\cite{masmoudi2014corpus}.\\footnote{The 20 hours recorded are equivalent to 71,684 words.}\n The last one is the TSAC \\citelanguageresource{Tsac}, containing 17k comments from Facebook, manually annotated to positive and negative polarities \\cite{medhaffar2017sentiment}. This corpus is the only one that contains TA texts as well as texts in Arabic characters. As far as we know there are no available corpora of TA transcribed in Arabic characters which are also morpho-syntactically annotated.\nIn order to provide an answer to the lack of resources for TA, we decided to create TArC, a corpus entirely dedicated to the TA writing system, transcribed in CODA TUN and provided with a lemmatization level and POS tag annotation. \n\n\\section{Characteristics of Tunisian Arabic and Tunisian Arabish}\n\\label{sec:TD_TA}\n\nThe Tunisian dialect (TUN) is the spoken language of Tunisian everyday life, commonly referred to as \\RL{\u0627\u0644\u062f\u064e\u0651\u0627\u0631\u0650\u062c\u064e\u0629}, \\textit{ad-d\u0101rija}, \\RL{\u0627\u0644\u0639\u064e\u0627\u0645\u0650\u0651\u064a\u064e\u0651\u0629}, \\textit{al-'\u0101mmiyya}, or \\RL{\u0627\u0644\u062a\u064f\u0651\u0648\u0646\u0652\u0633\u0650\u064a}, \\textit{\\textipa{@t-t\u016bns\u012b}}.\nAccording to the traditional diatopic classification, TUN belongs to the area of Maghrebi Arabic, of which the other main varieties are Libyan, Algerian, Moroccan and the\n\\textsubdot{H}ass\u0101n\u012bya variety of Mauritania\\footnote{The main geographical macro-areas, also called geolects, are the area of the Levant or Syro-Palestinian, Egypt and Sudan, Mesopotamia, Maghreb (North Africa) and the Arabian Peninsula. \n}  \\cite{durand2009dialettologia}.\nArabish is the transposition of ADs, which are mainly spoken systems, into written form, thus turning into a quasi-oral system (this topic will be discussed in section  \\ref{subsec:TA_system}). In addition, Arabish is not realized through Arabic script and consequently it is not subject to the Standard Arabic orthographic rules. As a result, it is possible to consider TA as a faithful written representation of the spoken TUN \\cite{akbar2019arabizi}. \n\n\\subsection{Tunisian Arabic}\n\nThe following list provides an excerpt of the principal features of TUN, which, through the TArC, would be researched in depth among many others. \\footnote{For a detailed description please refer to \\cite{durand2009dialettologia}, \\cite{marccais1977esquisse}.}\n\n\nAt the phonetic level, some of the main characteristics of TUN, and Maghrebi Arabic in general, are the following: \\\\\n\\begin{adjustwidth}{1em}{0pt}\n\\textbf{*} Strong influence of the Berber substratum, to which it is possible to attribute the conservative phonology of TUN consonants.\n\\end{adjustwidth}\n\n\\begin{adjustwidth}{1em}{0pt}\n\\textbf{*} Presence of new emphatic phonemes, above all [\\textsubdot{r}], [\\textsubdot{l}], [\\textsubdot{b}].\\\\ \n\\textbf{*} Realization of the voiced post-alveolar affricate [\\textdyoghlig]  as fricative \\textipa{[Z]}. \\\\\n\\textbf{*} Overlapping of the pharyngealized voiced alveolar stop \\textipa{[d\\super Q]}, <\\RL{\u0636}>, with the fricative \\textipa{[D\\super Q]}, <\\RL{\u0638}>. \\\\\n\\textbf{*} Preservation of a full glottal stop \\textipa{[P]} mainly in cases of loans from Classical Arabic (CA) or exclamations and interjections of frequent use.\n\\textbf{*} Loss of short vowels in open syllables.\n\\newpage\n\\textbf{*} Monophthongization.\\footnote{Reduction of the diphthongs \\textipa{[aw]} and \\textipa{[aj]} to \\textipa{[u:]} and \\textipa{[i:]} in pre-Hilalian dialects, and to \\textipa{[o:]} and \\textipa{[e:]} in the Hilalian ones.} In TUN <\\RL{\u0628\u064e\u064a\u062a}>, \\textipa{[\"baijt]}, \\textipa{\"}house\\textipa{\"}, becomes \\textipa{[\"bi:t]} meaning \\textipa{\"}room\\textipa{\"}. \\\\\n\\textbf{*} Palatalization of \u0101: Im\u0101la, <\\RL{\u0625\u0645\u0627\u0644\u0629}>, literally \\textipa{\"}inclination\\textipa{\"}.\n(In TUN the phenomenon is of medium intensity.)\nThereby the word <\\RL{\u0628\u0627\u0628}>, \\textipa{[\"ba:b]}, \\textipa{\"}door\\textipa{\"},\nbecomes \\textipa{[\"bE:b]}. \\\\ \n\\textbf{*} Metathesis.\n(Transposition of the first vowel of the word.\nIt occurs when non-conjugated verbs or names without suffix begin with the\nsequence CCvC, where C stands for ungeminated consonant, and\n\\textipa{\"}v\\textipa{\"} for short vowel. When a suffix is added to this type of\nname, or a verb of this type is conjugated, the first vowel changes position\ngiving rise to the CvCC sequence.)\nIn TUN it results in: \\textipa{\"}(he) has\nunderstood\\textipa{\"}: \\\\ <\\RL{\u0641\u0652\u0647\u0650\u0645}>, \\textipa{[\"fh@m]}, \\textipa{\"}(she) has\nunderstood\\textipa{\"}: <\\RL{\u0641\u0650\u0647\u0652\u0645\u0650\u062a}>, \\textipa{[\"f@hm@t]} or\n\\textipa{\"}leg\\textipa{\"}: <\\RL{\u0631\u0652\u062c\u0650\u0644}>, \\textipa{[\"rZ@l]}, \\textipa{\"}my\nleg\\textipa{\"}: <\\RL{\u0631\u0650\u062c\u0652\u0644\u0650\u064a}>, \\textipa{[\"r@Zli]}.\\\\\n\n\\end{adjustwidth} \n\nRegarding the morpho-syntactic level, TUN presents:\\\\\n\\begin{adjustwidth}{1em}{0pt}\n\\textbf{*} Addition of the prefix \/-n\/ to first person verbal morphology in \\textit{mu\\textsubdot{d}\u0101ri'} (imperfective).\\\\\n\\textbf{*} Realization of passive-reflexive verbs through the morpheme \/-t\/ \\footnote{The morpheme \/-t\/ can be traced back to the same morpheme present in the V and VI verbal patterns of CA \\cite{mion2004osservazioni}.} prefixed to the verb as in the example:\\\\ <\\RL{\u0633\u0648\u0631\u064a\u0651\u0629 \u0645\u0627\u0644\u062d\u064e\u0641\u0652\u0635\u064a\u0651\u0629 \u062a\u0652\u062a\u0650\u0644\u0652\u0628\u0650\u0633}>, \n\\textipa{[su:\"ri:j:a m@l-\\textcrh af\"s\\super Qij:a t-\"t@lb@s]}, \\textipa{\"}the shirts of \\textsubdot{H}af\\textsubdot{s}iya\\footnote{ \\textsubdot{H}af\\textsubdot{s}iya is a neighborhood in the Med\u012bna of Tunis, known for its great daily fr\u012bp (second-hand market).} are not bad\\textipa{\"}, (lit: \\textipa{\"}they dress\\textipa{\"}). \\\\\n\\textbf{*} Loss of gender distinction at the 2\\super{nd} and 3\\super{rd} persons, at verbal and pronominal level. \\\\\n\\textbf{*} Disappearance of the dual form from verbal and pronominal inflexion.\nThere is a residual of pseudo-dual in some words fixed in time in their dual form. \\\\\n\\textbf{*} Loss of relative pronouns flexion and replacement with the invariable form <\\RL{\u0627\u0650\u0644\u0651\u064a}>, \\textipa{[@l:i]}.\\\\\n\\textbf{*} Use of presentatives  \/\\textsubdot{r}\u0101-\/ and \/h\u0101-\/ with the meaning of \\textipa{\"}here\\textipa{\"}, \\textipa{\"}look\\textipa{\"}, as in the example in TUN: <\\RL{\u0631\u0627\u0646\u064a \u0645\u064e\u062e\u0652\u0646\u0648\u0642}>,\n\\textipa{[\"}\\textsubdot{r}\\textipa{a:ni: m@x\"nu:q]}, \\textipa{\"}here I am asphyxiated (by problems)\\textipa{\"}, or in <\\RL{\u0647\u0627\u0643 \u062f\u064e\u0628\u064e\u0651\u0631\u0652\u062a\u0652\u0647\u0627}>, \\textipa{[\"ha:-k d@\"b:@rt-ha:]}, \\textipa{\"}here you are, finding it (the solution)\\textipa{\"} hence: \\textipa{\"}you were lucky\\textipa{\"}. \\\\\n\\textbf{*} Presence of circumfix negation marks, such as <~<\\RL{\u0645\u0627}>, \\textipa{[ma]} + verb + <\\RL{\u0634}>, \\textipa{[S]}>. The last element of this structure must be omitted if there is another negation, such as the Tunisian adverb <\\RL{\u0639\u064f\u0645\u0652\u0631}>, \\textipa{[\"Qomr]}, \\textipa{\"}never\\textipa{\"}, as in the structure: <\\textipa{[\"Qomr]} + personal pronoun suffix + \\textipa{[m@]} + perfect verb>. This construction is used to express the concept of \\textipa{\"}never having done\\textipa{\"} the action in question, as in the example:\n<\\RL{\u0639\u064f\u0645\u0631\u064a \u0645\u0627 \u0643\u064f\u0646\u0652\u062a  \u0646\u0650\u062a\u0652\u0635\u064e\u0648\u064f\u0651\u0631...}>, \\textipa{[\"Qomr-i ma \"k@nt n@ts\\super Qaw:@r]}, \\textipa{\"}I never imagined that...\\textipa{\"}. \n\\newline Instead, to deny an action pointing out that it will never repeat itself again, a structure widely used is <[ma] + \\textipa{[\"Qa:d]} + \\textipa{[S]} + imperfective verb>, where the element within the circumfix marks is a grammaticalized  element of verbal origin from CA: <\\RL{\u0639\u0627\u062f}>, \\textipa {[\"Qa:d]}, meaning \\textipa{\"}to go back, to reoccur\\textipa{\"}, which gives the structure a sense of denied repetitiveness, as in the sentence:\n\\newline <\\RL{\u0647\u0648 \u0645\u0627 \u0639\u0627\u062f\u0650\u0634 \u064a\u064e\u0631\u0652\u062c\u064e\u0639}>,   \\textipa{[\"hu:wa ma \"Qa:d-S \"j@rZaQ]}, \n\n\\textipa{\"}he will not come back\\textipa{\"}.\n\\newline Finally, to deny the nominal phrase, in TUN both the\n<\\RL{\u0645\u0648\u0634}>, \\textipa{[\"mu:S]}, and the circumfix marks are frequently used. \nFor the negative form of the verb \\textipa{\"}to be\\textipa{\"} in the present, circumfix marks can be combined with the personal suffix pronoun, placed between the marks, as in <\\RL{\u0645\u064e\u0627\u0646\u0650\u064a\u0634}>, \\textipa{[ma\"ni:S]}, \\textipa{\"}I am not\\textipa{\"}.\\\\ Within the negation marks we can also find other types of nominal structures, such as: <\\textipa{[fi:]} + \\textipa{[\"bE:l]}(\\textipa{\"}mind\\textipa{\"})\n+ personal pronoun suffix>, which has a value equivalent to the verb \\textipa{\"}be aware of\\textipa{\"}, as in the example: \\\\ <\\RL{\u0645\u0627 \u0641\u064a \u0628\u0627\u0644\u064a\u0634}>, \\textipa{[ma fi: bE:l-\"i:-S]}, \\textipa{\"}I did not know\\textipa{\"}.\n\\end{adjustwidth}\n\n\\subsection{Tunisian Arabish}\n\\label{subsec:TA_system}\n\nAs previously mentioned, we consider Arabish a quasi-oral system.\nWith \\textit{quasi-orality} it is intended the form of communication typical of Computer-Mediated Communication (CMC), characterized by informal tones, dependence on context, lack of attention to spelling and especially the ability to create a sense of collectivity \\cite{hert1999quasi}\\footnote{Even though the CMC is generally a type of asynchronous communication.}. \n\nTA and TUN have not a standard orthography, with the exception of the CODA TUN. Nevertheless, TA is a spontaneous code-system used since more than ten years, and is being conventionalized by its daily usage. \n\nFrom the table~\\ref{tab1}, where the coding scheme of TA is illustrated, it is possible to observe that there is no one-to-one correspondence between TA and TUN characters and that often Arabish presents overlaps in the encoding possibilities. \nThe main issue is represented by the not proper representation by TA of the emphatic phones: \\textipa{{[D\\super Q]}}, \\textipa{[t\\super Q]} and \\textipa{[s\\super Q]}.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{c|c|c||c|c|c}\n\n      \\hline\n      \\textit{IPA}& \\textit{TUN} &  \\textit{TA} & \\textit{IPA}& \\textit{TUN}& \\textit{TA} \\\\\n      \\hline\n      \\hline\n      &&&&& \\\\\n      \\textipa{[a:]} & \\RL{\u0629} & a, e, h & \\textipa{[a][a:]} & \\RL{\u0649}, \\RL{\u0627} & a, e, \u00e9, \u00e8\\\\\n     \n      \\textipa{[P]} & \\RL{\u0621} & 2 & \\textipa{[D\\super Q]} & \\RL{\u0636} & dh,  th, d\\\\ \n     \n      \\textipa{[b]} & \\RL{\u0628} & b, p & \\textipa{[t\\super Q]} & \\RL{\u0637} & 6, t\\\\\n      \n      \\textipa{[t]} & \\RL{\u062a} & t & \\textipa{[D\\super Q]} & \\RL{\u0638} & th, dh\\\\\n      \n      \\textipa{[T]} & \\RL{\u062b} & th & \\textipa{[Q]} & \\RL{\u0639} & 3, a\\\\\n      \n      \\textipa{[Z]} & \\RL{\u062c} & j & \\textipa{[G]} & \\RL{\u063a} & 4, gh\\\\\n      \n      \\textipa{[\\textcrh]} & \\RL{\u062d} & 7, h & \\textipa{[f]} & \\RL{\u0641} & f\\\\\n      \n      \\textipa{[x]} & \\RL{\u062e} & 5, kh & \\textipa{[q]} & \\RL{\u0642} & 9, q\\\\\n      \n      \\textipa{[d]} & \\RL{\u062f} & d & \\textipa{[k]} & \\RL{\u0643} & k\\\\\n      \n      \\textipa{[D]} & \\RL{\u0630} & dh & \\textipa{[l]} & \\RL{\u0644} & l\\\\\n      \n      \\textipa{[r]} & \\RL{\u0631} & r & \\textipa{[m]} & \\RL{\u0645} & m\\\\\n      \n      \\textipa{[z]} & \\RL{\u0632} & z & \\textipa{[n]} & \\RL{\u0646} & n\\\\\n      \n      \\textipa{[s]} & \\RL{\u0633} & s & \\textipa{[h]} & \\RL{\u0647} & 8, h\\\\\n      \n      \\textipa{[S]} & \\RL{\u0634} & ch, (sh) & \\textipa{[w][u:]} & \\RL{\u0648} & ou, w\\\\\n      \n      \\textipa{[s\\super Q]} & \\RL{\u0635} &  s &  \\textipa{[j][i:]} & \\RL{\u064a} & i, y\\\\\n      &&&&& \\\\\n      \\hline\n\n\\end{tabularx}\n\\caption{Arabish code-system for TUN}\n\\label{tab1}\n \\end{center}\n\\end{table}\n\nOn the other hand, being TA not codified through the Arabic alphabet, it can well represent the phonetic realization of TUN, as shown by the\nfollowing examples: \n\n\\textbf{ *} The Arabic alphabet is generally used for formal\nconversations in Modern Standard Arabic (MSA), the Arabic of formal situations, or in that of \nClassical Arabic (CA), the Arabic of the Holy Qur'\u0101n, also known as\n'The Beautiful Language'. Like MSA and CA, also Arabic Dialects\n(ADs) can be written in the Arabic alphabet, but in this case it is\npossible to observe a kind of hypercorrection operated by the speakers in order to respect the writing rules of MSA. For example, in TUN texts written in Arabic script, it is possible to find a 'silent vowel' (namely an epenthetic \\textipa{\"}alif\n<\\RL{\u0627}>) written at the beginning of those words starting with\nthe sequence '\\#CCv', which is not allowed in MSA.\n\n\\textbf{ *} Writing TUN in Arabic script, the Code-Mixing or Switching in foreign language will be unnaturally reduced. \n\n\\textbf{ *} As described in table~\\ref{tab1}, the Arabic alphabet\nis provided with three short vowels, which correspond to the three long ones: \\textipa{[a:]}, \\textipa{[u:]}, \\textipa{[i:]}, but TUN presents\na wider range of vowels. Indeed, regarding the early presented\ncharacteristics of TUN, the TA range of vowels offers better possibility to represent most  of the TUN characteristics outlined in the previous subsection, in particular:\n\n\\begin{itemize}[nosep]\n    \\item Palatalization.\n    \\item Vowel metathesis.\n    \\item Monophthongization.\\footnote{Regarding the last two phenomena, they can be visible in Arabic script only in case of texts provided with short vowels, which are quite rare.}\n\\end{itemize}\n\n\n\n\\section{Tunisian Arabish Corpus}\n\\label{sec:TArC}\n\nIn order to analyze the TA system, we have built a TA Corpus based on social media data, considering this as the best choice to observe the quasi-oral nature of the TA system.\n\n\\subsection{Text collection}\n\\label{subsec:txt-coll}\nThe corpus collection procedure is composed of the following steps: \n\\begin{enumerate}[nosep]\n\\item Thematic categories detection.\n\\label{step1} \n\\item Match of categories with sets of semantically related TA keywords.\n\\label{step2}\n\\item Texts and metadata extraction. \\\\\n\\label{step3}\n\\end{enumerate}\n\n\\textbf{Step~\\ref{step1}.} In order to build a Corpus that was as representative as possible of the linguistic system, it was considered useful to identify wide thematic categories that could represent the most common topics of daily conversations on CMC.\n\nIn this regard, two instruments with a similar thematic organization have been employed: \n\\begin{itemize}[nosep]\n\\item\\textbf{'A Frequency Dictionary of Arabic'} \\\\ \\cite{buckwalter2014frequency} In particular its 'Thematic Vocabulary List' (TVL). \n\\item\\textbf{'Loanword Typology Meaning List'} \\\\ A list of 1460 meanings\\footnote{The 'Loanword Typology Meaning List' is a result of a joint project by Uri Tadmor and Martin Haspelmath: the 'Loanword Typology Project' (LWT), launched in 2004 and ended in 2008.} (LTML)  \\cite{haspelmath2009loanwords}. \n\\end{itemize}\nThe TVL consists of 30 groups of frequent words, each one represented by a thematic word.\nThe second consists of 23 groups of basic meanings sorted by representative word heading.\nConsidering that the boundaries between some categories are very blurred, some categories have been merged, such as \\textipa{\"}Body\\textipa{\"} and \\textipa{\"}Health\\textipa{\"}, (see table~\\ref{tab2}). Some others have been eliminated, being not relevant for the purposes of our research, e.g. \\textipa{\"}Colors\\textipa{\"}, \\textipa{\"}Opposites\\textipa{\"}, \\textipa{\"}Male names\\textipa{\"}. In the end, we obtained 15 macro-categories listed in table~\\ref{tab2}. \\\\\n\n\\textbf{Step~\\ref{step2}.} Aiming at easily detect texts and the respective \\textit{seed URLs}, without introducing relevant query biases, we decided to avoid using the category names as query keywords \\cite{schafer2013web}. Therefore, we associated to each category a set of TA keywords belonging to the basic Tunisian vocabulary. We found that \na semantic category with three meanings was enough to obtain a sufficient number of keywords and URLs for each category. For example, to the category \\textipa{\"}Family\\textipa{\"} the meanings: \\textipa{\"}son\\textipa{\"}, \\textipa{\"}wedding\\textipa{\"}, \\textipa{\"}divorce\\textipa{\"} have been associated in all their TA variants, obtaining a set of 11 keywords (table~\\ref{tab2}).\\\\\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|X|X|} \n\n      \\hline\n      \\textbf{Macro-Categories}& \\textbf{Words Associated} \\\\\n      \\hline\n      1. Family \\newline\\textit{son, wedding, divorce}& weld, wild, 3ars, 3ers, \\newline tla9, 6la9, tlaq, 6laq, tle9, tleq, 6leq\\\\\n      \\hline\n      2. Clothing \\newline\\textit{dress, shoes, t-shirt} &robe, lebsa, rouba, \\newline sabat, spedri, spadri, \\newline marioul, maryoul, \\newline meryoul, merioul\\\\\n      \\hline\n      3. Automobiles \\newline\\textit{gasoil, engine, \\newline occasion} & mazout, motor, moteur, motour, forsa\\\\\n      \\hline\n      4. Animals \\newline\\textit{cock, dog, cat} & sardouk, kelb, kalb, \\newline9attous, gattous\\\\\n      \\hline\n      5. Body and Health \\newline\\textit{sick, doctor, health} & maridh, marith, mridh, ettbib, tbib, sa77a, sa7a, sahha, saha  \\\\\n      \\hline\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n\n\\end{tabularx}\n\\caption{Example of the fifteen thematic categories}\n\\label{tab2}\n \\end{center}\n\\end{table}\n\n\\textbf{Step~\\ref{step3}.} \nWe collected about 25,000 words and the related metadata as first part of our corpus, which are being semi-automatically transcribed into Arabic characters (see next sections).\nWe planned to increase the size of the corpus at a later time.\nRegarding the metadata, we have extracted the information published by users, focusing on the three types of information generally used in ethnographic studies: \n\\begin{enumerate}[nosep]\n\\item Gender: Male (M) and Female (F). \n\\item Age range: [10-25], [25-35], [35-50], [50-90]. \n\\item City of origin.\n\\end{enumerate}\n\n\\subsection{Corpus Creation}\n\\label{subsec:Corpus_cr}\n\nIn order to create our corpus, we applied a word-level annotation. This phase was preceded by some data pre-processing steps, in particular tokenization. \nEach token has been associated with its annotations and metadata (table~\\ref{tab3}).\nIn order to obtain the correspondence between Arabish and Arabic morpheme transcriptions, tokens were segmented into morphemes.\nThis segmentation was carried out completely manually for a first group of tokens.\\footnote{Arabic, in general, is a language with a high level of synthesis, that means that it can concentrate within a token more syntactic and grammatical information through the addition of different morphemes.}\nIn its final version, each token is associated with a total of 11 different annotations, corresponding to the number of the annotation levels we chose.\nAn excerpt of the corpus after tokens annotation is depicted in table~\\ref{tab3}.\n\nFor the sake of clarity, in table~\\ref{tab3} we show:\\\\\n\\textbf{ *} The A column, \\textit{Cor}, indicates the token\\textipa{\"}s source code. For example, the code \\textit{3fE}, which stands for \\textit{3rab fi Europe}, is the forum from which the text was extracted.\\\\\n\\textbf{ *} The B column, \\textit{Textco}, is the publication date of the text. \\\\%e.g. '150902' stand for '2015\/09\/02'\n\\textbf{ *} The C column, \\textit{Par}, is the row index of the token in the paragraph.\\\\\n\\textbf{ *} The D column, \\textit{W}, is the index of the token in the sentence. When \\textipa{\"}W\\textipa{\"} corresponds to a range of numbers, it means that the token has been segmented in to its components, specified in the rows below.\\\\\n\\textbf{ *} The E column, \\textit{Arabi\\textipa{S}},  corresponds to the token transcription in Arabish.\\\\\n\\textbf{ *} The F column, \\textit{Tra}, is the transcription into Arabic characters.\\\\\n\\textbf{ *} The G column, \\textit{Ita},  is the translation to Italian. \\\\\n\\textbf{ *} The H column, \\textit{Lem},  corresponds to the lemma.\\\\\n\\textbf{ *} The I column, \\textit{POS}, is the Part-Of-Speech tag of the token. The tags that have been used for the POS tagging are conform to the annotation system of Universal Dependencies.\\\\\n\\textbf{ *} The last three columns (J, K, L) contain the metadata: \\textit{Var}, \\textit{Age}, \\textit{Gen}.\n\n\n\\begin{table}[htbp]\\small\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|XlllllX|}\n     \\hline \n     \\textbf{A} & \\textbf{B} & \\textbf{C} & \\textbf{D} & \\textbf{E} & \\textbf{F} & \\textbf{G}\\\\\n     \\hline\n     \\hline\n     Cor\\label{Cor} & Textco & Par & W & Arabi\\textipa{S} & Tra & Ita\\\\\n     \\hline\n     \\hline\n     &&&&&&\\\\\n     3fE & 150902 & 2 & 1 & kifech & \\RL{\u0643\u064a\u0641\u0627\u0634} & come\\\\\n     \n     3fE & 150902 & 2 & 2 & tchou- & \\RL{\u062a\u0634\u0648\u0641\u0648\u0627} & vi\\\\\n     & & & & fou & & pare\\\\\n     \n     3fE & 150902 & 2 & 3-4 & l3icha & \\RL{\u0627\u0644\u0639\u064a\u0634\u0629} & la vita\\\\\n     \n     3fE & 150902 & 2 & 3 & l & \\RL{\u0627\u0644\u0640} & -\\\\\n     \n     3fE & 150902 & 2 & 4 & 3icha & \\RL{\u0639\u064a\u0634\u0629} & -\\\\\n     \n     3fE & 150902 & 2 & 5-6 & fil & \\RL{\u0641\u0627\u0644\u0640} & all\\textipa{\"}\\\\\n     \n     3fE & 150902 & 2 & 5 & f & \\RL{\u0641\u0640} & -\\\\\n     \n     3fE & 150902 & 2 & 6 & il & \\RL{\u0627\u0644\u0640} & -\\\\\n     \n     3fE & 150902 & 2 & 7 & 4orba & \\RL{\u063a\u0631\u0628\u0629} & estero\\\\\n     \n     3fE & 150902 & 2 & 8 & ? & \\RL{\u061f} & ?\\\\\n     &&&&&&\\\\\n     \\hline\n     \\hline\n     & \\textbf{H} & \\textbf{I} & \\textbf{J} & \\textbf{K} & \\textbf{L} &\\\\\n     \\hline\n     \\hline\n     & Lem & POS & Var & Age & Gen &\\\\\n     \\hline\n     \\hline \n     &&&&&&\\\\\n     & \\RL{\u0643\u064a\u0641\u0627\u0634} & adv & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u0634\u0627\u0641} & verb & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u0639\u064a\u0634\u0629} & noun & Bnz & 25-35 & M &\\\\\n     \n     & \\RL {\u0627\u0644\u0640} & det & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u0639\u064a\u0634\u0629} & noun & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u0641\u064a} & prep & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u0641\u064a} & prep & Bnz & 25-35 & M &\\\\\n     \n     & \\RL {\u0627\u0644\u0640} & det & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u063a\u0631\u0628\u0629} & noun & Bnz & 25-35 & M &\\\\\n     \n     & \\RL{\u061f} & pct & Bnz & 25-35 & M &\\\\\n     &&&&&&\\\\\n     \\hline \n\\end{tabularx}\n\\caption{An Excerpt of the TArC structure. In the column  \\textit{Var}, \\textipa{\"}Bnz\\textipa{\"} stands for \\textipa{\"}Bizerte\\textipa{\"} a northern city in Tunisia. Glosses: w1:\\textit{how}, w2:\\textit{do you(pl) see}, w3-4:\\textit{the life}, w5-6:\\textit{at the}, w7:\\textit{outside}, w8:\\textit{?}}\n\\label{tab3}\n \\end{center}\n\\end{table}\n\n\n\nSince TA is a spontaneous orthography of TUN, we considered important to adopt the CODA* guidelines as a model to produce a unified lemmatization for each token (column \\textit{Lem} in table~\\ref{tab3}).\nIn order to guarantee accurate transcription and lemmatization, we annotated manually the first 6,000 tokens with all the annotation levels.\n\nSome annotation decisions were taken before this step, with regard to specific TUN features:\n\n    \\textbf{* Foreign words.} We transcribed the Arabish words into Arabic characters, except for Code-Switching terms. In order to not interrupt the sentences continuity we decide to transcribe Code-Mixing terms into Arabic script. However, at the end of the corpus creation process, these words will be analyzed, making the distinction between acclimatized loans and Code-Mixing. \n    \\newline\\newline \n    \n    The first ones will be transcribed into Arabic characters also in \\textit{Lem}, as shown in table~\\ref{tab4}. The second ones will be lemmatized in the foreign language, mostly French, as shown in table~\\ref{tab5}.\n    \\newline\\textbf{* Typographical errors.} Concerning typos and typical problems related to the informal writing habits in the web, such as repeated characters to simulate prosodic features of the language, we have not maintained all these characteristics in the transcription (column \\textit{Tra}). Logically, these were neither included in \\textit{Lem}, according to the CODA* conventions, as shown in table~\\ref{tab5}.\n    \\newline\\textbf{* Phono-Lexical exceptions.} We used the grapheme \\newline<\\RL{\u06a8}>, \\textipa{[q]}, only in loanword transcription and lemmatization. As can be seen in table~\\ref{tab6}, the Hilalian phoneme [g] of the Turkish loanword \\textipa{\"}gawriyya\\textipa{\"}, has been transcribed and lemmatized with the grapheme <\\RL{\u0642}>, \\textipa{[g]}.\n    \\newline\\textbf{* Glottal stop.} As explained in CODA TUN, real initial and final glottal stops have almost disappeared in TUN. They remain in some words that are treated as exceptions, e.g. <\\RL{\u0623\u0633\u0626\u0644\u0629}>, \\textipa{[\"PasPla]}, \\textipa{\"}question\\textipa{\"}  \\cite{DBLP:conf\/lrec\/ZribiBMEBH14}. Indeed, we transcribe the glottal stops only when it is usually pronounced, and if it does not, we do not write the glottal stops at the beginning of the word or at the end, neither in the transcription, nor in the lemmas. \\\\\n\n\n\\begin{table}[htbp]\\small\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|XccccX|}\n         \\hline\n         W & Arabi\\textipa{S} & Tra & Ita & Lem & POS \\\\\n         \\hline\n         \\hline\n         4 & konna & \\RL{\u0643\u0646\u0651\u0627} & siamo stati & \\RL{\u0643\u0627\u0646} & verb \\\\\n         \n         5 & far7anin & \\RL{\u0641\u0631\u062d\u0627\u0646\u064a\u0646} & contenti & \\RL{\u0641\u0631\u062d\u0627\u0646} & adj \\\\\n         \n         6 & , & \\RL{,} & , & \\RL{,} & punct \\\\\n         \n         7 & merci & \\RL{\u0645\u0631\u0633\u064a} & grazie & \\RL{\u0645\u0631\u0633\u064a} & intj \\\\\n         \n       \n         \\hline\n\n\\end{tabularx}\n\\caption{Loanword example in the corpus. Glosses: w4:\\textit{we were}, w5:\\textit{happy}, w6:\\textit{,} , w7:\\textit{thanks}}\n\\label{tab4}\n \\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[htbp]\\small\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|XccccX|} \n         \\hline\n         W & Arabi\\textipa{S} & Tra & Ita & Lem & POS \\\\\n         \\hline\n         \\hline\n         1 & R7 & recette & ricetta & recette & noun \\\\\n         \n         2 & patee & p\u00e2t\u00e9 & pat\u00e8 & p\u00e2t\u00e9 & noun \\\\\n         \n         3 & dieri & \\RL{\u062f\u064a\u0627\u0631\u064a} & fatto in casa & \\RL{\u062f\u064a\u0627\u0631\u064a} & adj \\\\\n         \n         4 & w & \\RL{\u0648} & e & \\RL{\u0648} & cconj \\\\\n         \n         5 & bniiiiin & \\RL{\u0628\u0646\u064a\u0646} & buonissimo & \\RL{\u0628\u0646\u064a\u0646} & adj \\\\\n         \\hline\n         \n\\end{tabularx}\n\\caption{Prosody example in the corpus. Glosses: w1:\\textit{recipe}, w2:\\textit{p\u00e2t\u00e9}, w3:\\textit{homemade}, w4:\\textit{and}, w5:\\textit{delicious}}\n\\label{tab5}\n \\end{center}\n\\end{table}\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|XccccX|}\n         \\hline\n         W & Arabi\\textipa{S} & Tra & Ita & Lem & POS \\\\ \n         \\hline\n         \\hline\n         1 & Mtala9 & \\RL{\u0645\u0637\u0644\u0642} & divorziato & \\RL{\u0645\u0637\u0644\u0642} & noun \\\\\n         \n         2 & min & \\RL{\u0645\u0646} & da & \\RL{\u0645\u0646} & noun \\\\\n         \n         3 & gawriya & \\RL{\u06a8\u0627\u0648\u0631\u064a\u0629} & (un\\textipa{\"})europea & \\RL{\u06a8\u0627\u0648\u0631\u064a} & adj \\\\\n         \\hline\n         \n         \n\\end{tabularx}\n\\caption{Phono-Lexical exceptions in the corpus. Glosses: w1:\\textit{divorced}, w2:\\textit{from}, w3:\\textit{European(f)}}\n\\label{tab6}\n \\end{center}\n\\end{table}\n\n\n\n\\textbf{* Negation Marks.} CODA TUN proposes to keep the MSA rule of maintaining a space between the first negation mark and the verb, in order to uniform CODA TUN to the first CODA \\cite{habash-etal-2012-conventional}. However, as  \\newcite{DBLP:conf\/lrec\/ZribiBMEBH14} explains, in TUN this rule does not make really sense, but it should be done to preserve the consistency among the various CODA guidelines.  \nIndeed, in our transcriptions we report what has been produced in Arabish following CODA TUN rules, while in lemmatization we report the verb lemma. At the same time we segment the negative verb in its minor parts: the circumfix negation marks and the conjugated verb. For the first one, we describe the negative morphological structure in the \\textit{Tra} and \\textit{Lem} columns, as in table~\\ref{tab7}. For the second one, as well as the other verbs, we provide transcription and lemmatization.\n\n\\begin{table}[htbp]\\small\n\\begin{center}\n\\begin{tabularx}{\\columnwidth}{|llllll|}\n         \\hline\n         W & Arabi\\textipa{S} & Tra & Ita & Lem & POS \\\\\n         \\hline\n         \\hline\n         14-15 & manajem- & \\RL{\u0645\u0627 \u0646\u062c\u0645\u0646\u0627\u0634} & non & \\RL{\u0646\u062c\u0651\u0645} & verb \\\\\n         & nech & & abbiam &&\\\\\n         & & & potuto &&\\\\\n         \n         14 & ma + ch & \\RL{\u0645\u0627+\u0634} & - & \\RL{\u0634}+V+\\RL{\u0645\u0627} & part \\\\\n         \n         15 & najemne & \\RL{\u0646\u062c\u0645\u0646\u0627} & - & \\RL{\u0646\u062c\u0651\u0645} & verb \\\\\n         \\hline\n         \n\\end{tabularx}\n\\caption{Circumfix negation marks in the corpus. Glosses: w14-15:\\textit{we could not}}\n\\label{tab7}\n \\end{center}\n\\end{table}\n\n\n\\section{Incremental and Semi-Automatic Transcription}\n\\label{sec:pro}\n\nIn order to make the corpus collection easier and faster, we adopted a semi-automatic procedure based on sequential neural models \\cite{DBLP-journals\/corr\/abs-1904-04733,DinarelliGrobol-Seq2BiseqTransformer-2019}.\nSince transcribing Arabish into Arabic is by far the most important information to study the Arabish code-system, the semi-automatic procedure concerns only transcription from Arabish to Arabic script.\n\nIn order to proceed, we used the first group of (roughly) 6,000 manually transcribed tokens as training and test data sets in a 10-fold cross validation setting with 9-1 proportions for training and test, respectively. As we explained in the previous section, French tokens were removed from the data. More precisely, whole sentences containing \\emph{non-transcribable} French tokens (code-switching) were removed from the data.\nSince at this level there is no way for predicting when a French word can be transcribed into Arabic and when it has to be left unchanged, French tokens create some noise for an automatic, probabilistic model. After removing sentences with French tokens, the data reduced to roughly 5,000 tokens. We chose this amount of tokens for annotation blocks in our incremental annotation procedure.\n\nWe note that by combining sentence, paragraph and token index in the corpus, whole sentences can be reconstructed. However, from 5,000 tokens roughly 300 sentences could be reconstructed, which are far too few to be used for training a neural model.\\footnote{Preliminary experiments gave indeed quite poor results, below 50\\% token-level accuracy on average.}\nInstead, since tokens are transcribed at morpheme level, we split Arabish tokens into characters, and Arabic tokens into morphemes, and we treated each token itself as a sequence.\nOur model learns thus to map Arabish characters into Arabic morphemes.\n\nThe 10-fold cross validation with this setting gave a token-level accuracy of roughly 71\\%. This result is not satisfactory on an absolute scale, however it is more than encouraging taking into account the small size of our data.\nThis result means that less than 3 tokens, on average, out of 10, must be corrected to increase the size of our corpus. \nWith this model we automatically transcribed into Arabic morphemes, roughly, 5,000 additional tokens, corresponding to the second annotation block.\nThis can be manually annotated in at least 7,5 days, but thanks to the automatic annotation accuracy, it was manually corrected into 3 days.\\footnote{We based our estimations of the annotation time needed on the time we spent correcting tokens, which is actually faster because tokens are already transcribed, they don't need to be transcribed from scratch.}\nThe accuracy of the model on the annotation of the second block was roughly 70\\%, which corresponds to the accuracy on the test set.\nThe manually-corrected additional tokens were added to the training data of our neural model, and a new block was automatically annotated and manually corrected.\nBoth accuracy on the test set and on the annotation block remained at around 70\\%. This is because the block added to the training data was significantly different from the previous and from the third.\nAdding the third block to the training data and annotating a fourth block with the new trained model gave in contrast an accuracy of roughly 80\\%.\nThis incremental, semi-automatic transcription procedure is in progress for the remaining blocks, but it is clear that it will make the corpus annotation increasingly easier and faster as the amount of training data will grow up.\n\nOur goal concerning transcription, is to have the 25,000 tokens mentioned in section~\\ref{subsec:txt-coll} annotated automatically and manually corrected. These data will constitute our gold annotated data, and they will be used to automatically transcribe further data.\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn this paper we presented TArC, the first Tunisian Arabish Corpus annotated with morpho-syntactic information. We discussed the decisions taken in order to highlight the phonological and morphological features of TUN through the TA corpus structure. Concerning the building process, we have shown the steps undertaken and our effort intended to make the corpus as representative as possible of TA. We therefore described the texts collection stage, as well as the corpus building and the semi-automatic procedure adopted for transcribing TA into Arabic script, taking into account CODA* and CODA TUN guidelines. At the present stage of research, TArC consists of 25.000 tokens, however our work is in progress and for future research we plan to enforce the semi-automatic transcription, which has already shown encouraging results (accuracy = 70\\%). We also intend to realize a semi-automatic TA Part-Of-Speech tagger.\nThus, we aim to develop tools for TA processing and, in so doing, we strive to complete the annotation levels (transcription, POS tag, lemmatization) semi-automatically in order to increase the size of the corpus, making it available for linguistic analyses on TA and TUN. \n\n\n\n\n\n\n\n\n\\section{Bibliographical References}\n\\label{reference}\n\\bibliographystyle{lrec2020}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nThere is cumulative evidence that theories with exponential non-local operators of the form\n\\begin{equation}\\label{serep0}\ne^{-(\\Box\/M^2)^n}\n\\end{equation}\nhave interesting renormalization properties. After early studies of quantum scalar field theories \\cite{AE1,AE2,Efi77,Efi01} and gauge and gravitational theories \\cite{Kra87,Mof1,HaMo,EMKW,Cor1,Cor2,Cor3}, in recent years there has been a surge of interest in non-local classical and quantum gravity \\cite{Kuz89,Tom97,Bar1,Bar2,BMS,Kho06,cuta8,BKM1,Mof3,Mod1,BGKM,Mod2,Mod3,Mod4,BMTs,BCKM,CaMo2,MoRa1,TBM,MoRa2,MoRa3,Edh18,BKLM}. A non-local theory of gravity aims to fulfill a synthesis of minimal requirements: (i) spacetime is a continuum where Lorentz invariance is preserved at all scales; (ii) classical local (super-)gravity should be a good approximation at low energy; (iii) the quantum theory must be perturbatively super-renormalizable or finite; (iv) the quantum theory must be unitary and ghost free, without extra pathological degrees of freedom in addition to those present in the classical theory; (v) typical classical solutions must be singularity-free.\n\nThe typical structure of the gravitational action in $D$ topological dimensions is\n\\begin{equation}\\nonumber\nS_g = \\frac{1}{2\\kappa^2} \\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda + R_{\\mu \\nu} \\, \\mathcal{F}_2(\\Box) \\, R^{\\mu \\nu} + R  \\mathcal{F}_0(\\Box) R\\right]\n\\end{equation}\nwhere $\\kappa^2=8\\pi G$ is the gravitational constant and $\\mathcal{F}_{0,2}$ are \\emph{form factors} dependent on the dimensionless ratio $r_*\\Box:=\\Box\/M^2$, where $M=1\/\\sqrt{r_*}$ is the characteristic energy scale of the system, $\\Box=\\nabla_\\mu \\nabla^\\mu$ is the Laplace--Beltrami or d'Alembertian operator and $\\nabla_\\nu V_\\mu := \\partial_\\nu V_\\mu-\\Gamma^\\sigma_{\\mu\\nu}V_\\sigma$ is the covariant derivative of a vector $V_\\mu$. Our conventions for the curvature invariants are\n\\begin{eqnarray}\n&& \\Gamma^\\rho_{\\mu\\nu}:= \\frac12 g^{\\rho\\sigma}\\left(\\partial_{\\mu} g_{\\nu\\sigma}+\\partial_{\\nu} g_{\\mu\\sigma}-\\partial_\\sigma g_{\\mu\\nu}\\right)\\,,\\label{leci}\\\\\n&& R^\\rho_{~\\mu\\sigma\\nu}:= \\partial_\\sigma \\Gamma^\\rho_{\\mu\\nu}-\\partial_\\nu \\Gamma^\\rho_{\\mu\\sigma}+\\Gamma^\\tau_{\\mu\\nu}\\Gamma^\\rho_{\\sigma\\tau}-\\Gamma^\\tau_{\\mu\\sigma}\\Gamma^\\rho_{\\nu\\tau}\\,,\\label{rite}\\\\\n&& R_{\\mu\\nu}:= R^\\rho_{~\\mu\\rho\\nu}\\,,\\qquad R:= R_{\\mu\\nu}g^{\\mu\\nu}\\,.\n\\end{eqnarray}\nThe particular choice of form factors\n\\begin{equation}\n\\mathcal{F}_2(\\Box) = \\frac{e^{-r_*\\Box} -1}{\\Box} \\,,\\qquad \\mathcal{F}_0(\\Box) = -\\frac{e^{-r_*\\Box}-1}{2\\Box} \\,,\\nonumbe\n\\end{equation}\nleads to the action \\cite{Mod1,Mod2,Mod3,Mod4,CaMo2}\n\\begin{equation}\nS_g = \\frac{1}{2\\kappa^2}\\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda+ G_{\\mu\\nu} \\, \\gamma_{r_*}(\\Box) \\,  R^{\\mu\\nu} \\right],\\label{nlffg}\n\\end{equation}\nwhere $G_{\\mu\\nu}$ is the Einstein tensor \\Eq{Eiten} and\n\\begin{equation}\\label{fofag}\n\\gamma_{r_*}(\\Box) :=  \\frac{e^{-r_*\\Box}-1}{\\Box}\\,.\n\\end{equation}\nThis model is dictated by the above program (i)--(v) and may be also regarded as a phenomenological non-local limit of M-theory \\cite{CaMo2}. The role of the non-local operator $1\/\\Box$ is to compensate the second-order derivatives in curvature invariants. Its definition is presented in appendix \\ref{app1}. To date, the perturbative renormalizability of the theory with \\Eq{fofag} has been proven only with the use of the resummed propagator \\cite{TBM}, while infinities have not been tamed yet in the orthodox expansion with the bare propagator. Nevertheless, this theory encodes all the main features of those non-local quantum gravities that have been shown to be renormalizable and its dynamics is simpler to deal with.\n\nEven without considering gravity and the quantum limit, there is a general conceptual issue usually characterizing non-local physics. Namely, the Cauchy problem can be ill defined or highly non-standard in non-local theories \\cite{Lew33,Car36,PU,Pau53}. In fact, while there is a time-honored tradition on \\emph{linear} differential equations with infinitely many derivatives that admit a fair mathematical treatment \\cite{Lew33,Car36}, \\emph{non-linear} non-local equations such as those appearing in non-local field theories are a very different and much trickier business. For any tensorial field $\\varphi(t,{\\bf x})$, it entails an infinite number of initial conditions $\\varphi(t_{\\rm i},{\\bf x})$, $\\dot\\varphi(t_{\\rm i},{\\bf x})$, $\\ddot\\varphi(t_{\\rm i},{\\bf x})$, \\dots, representing an infinite number of degrees of freedom. As the Taylor expansion of $\\varphi(t,{\\bf x})$ around $t=0$ is given by the full set of initial conditions, specifying the Cauchy problem would be tantamount to knowing the solution itself, if analytic \\cite{MoZ}. This makes it very difficult to find analytic solutions to the equations of motion, even on Minkowski spacetime. Fortunately, the exponential operator \\Eq{serep0} is under much greater control than other non-local operators, since (at least for finite $n$) the diffusion-equation method is available to find analytic solutions \\cite{roll,cuta2,cuta3,cuta4,MuNu3,cuta5,cuta6,cuta7} which are well defined when perturbative expansions are not \\cite{cuta2}. The Cauchy problem can be rendered meaningful, both in the free theory \\cite{Car36,PU,BK1} and in the presence of interactions \\cite{cuta3}. Consider a real scalar field $\\phi(x)$ dependent on spacetime coordinates $x=(t,{\\bf x})$. According to the diffusion-equation method, one promotes $\\phi(t,{\\bf x})$ to a field $\\Phi(r,t,{\\bf x})$ living in an extended spacetime with a fictitious extra coordinate $r$. This field is assumed to obey the diffusion equation $(\\Box-\\partial_r)\\Phi(r,t,{\\bf x})=0$, implemented at the level of the $(D+1)$-dimensional action by introducing an auxiliary scalar field $\\chi(r,x)$ (dynamically constrained to be $\\chi=\\Box\\Phi$). Since the diffusion equation is linear in $\\Phi$ (and $\\chi$, consequently), the Laplace--Beltrami operator $\\Box$ commutes with the diffusion operator $\\partial_r$ and exponential operators act as translations on the extra coordinate, $e^{s\\Box}\\Phi(r,t,{\\bf x})=e^{s\\partial_r}\\Phi(r,t,{\\bf x})=\\Phi(r+s,t,{\\bf x})$. One can then show that, from the point of view of spacetime coordinates, the $(D+1)$-dimensional system is fully localized and that the only initial conditions to be specified are $\\Phi(r,t_{\\rm i},{\\bf x})$, $\\dot\\Phi(r,t_{\\rm i},{\\bf x})$, $\\chi(r,t_{\\rm i},{\\bf x})$, $\\dot\\chi(r,t_{\\rm i},{\\bf x})$ \\cite{cuta3}. The infinite number of initial conditions $\\phi(t_{\\rm i},{\\bf x})$, $\\dot\\phi(t_{\\rm i},{\\bf x})$, $\\ddot\\phi(t_{\\rm i},{\\bf x})$, \\dots have been transferred into two initial conditions, which are actually boundary conditions in $r$, for an auxiliary field. When interactions are turned off, $\\chi$ vanishes and one obtains the single degree of freedom, represented by $\\phi(t_{\\rm i},{\\bf x})$ and $\\dot\\phi(t_{\\rm i},{\\bf x})$, of the free local theory.\\footnote{This is obvious when integrating by parts the kinetic term, $\\phi f(\\Box) \\phi\\to h(\\Box)\\phi \\Box h(\\Box) \\phi$, and absorbing non-locality with the field redefinition $\\tilde\\phi=h(\\Box)\\phi$.} The original system is recovered when $r$ acquires a specific, fixed value proportional to the scale $r_*$. This value depends on the solution and is determined by solving the localized equations at $r=\\beta r_*$, where $\\beta$ is a constant. The resulting solutions $\\phi(x)=\\Phi(\\beta r_*,x)$ are not exact in general but they satisfy the equations of motion to a very good level of approximation \\cite{roll,cuta5,cuta7}. \n\nFor non-local gravity, one would like to apply the same method to the metric itself or to curvature invariants $\\mathcal{R}(g)$, but this is not possible in a direct way. Calling $\\mathcal{R}(r,x)$ the curvature invariants of a putative localized theory, since the diffusion equation $(\\Box-\\partial_r)\\mathcal{R}(r,x)=0$ would be non-linear in the metric $g_{\\mu\\nu}$, one would have\n\\begin{equation}\\label{probl}\n[\\Box(g),\\partial_r]\\mathcal{R}(g)\\neq 0\\,,\n\\end{equation}\nand one would be unable to trade non-local operators for shifts in the extra direction. Moreover, the diffusion method applies for exponential operators, while in the actual quantum-gravity action \\Eq{nlffg} non-locality is more complicated.\n\nIn this paper, we address this problem. First, we will use a field redefinition (already employed in other non-local gravities, although not for \\Eq{nlffg} \\cite{BMS,BCKM}, and similar to those used in scalar-tensor theories and modified gravity models) to transfer all non-locality to an auxiliary field $\\phi_{\\mu\\nu}$. Next, we impose the diffusion equation on $\\phi_{\\mu\\nu}$: the linearity problem is thus immediately solved and one can proceed to localize the non-local system, count the initial conditions and identify the degrees of freedom, which are finite in number. From there, one can begin the study of the dynamical solutions of the classical Einstein equations, but this goes beyond the scope of the present work. Counting non-local degrees of freedom is a subject surrounded by a certain halo of mystery and confusion in the literature. To make it hopefully clearer, we will make a long due comparison of the counting procedure and of its outcome in the methods proposed to date: the one based on the diffusion equation and the delocalization approach by Tomboulis \\cite{Tom15}.\n\n\\subsection{Plan of the paper}\n\nIn preparation for the study of non-local gravity, the diffusion-equation method is reviewed in section \\ref{scala} for a scalar field. This example is very useful because it contains virtually all the main ingredients we will need to localize non-local gravity and rewrite it in a user-friendly way: localized action, auxiliary fields, slicing choice, matching of the non-local and localized equations of motion, counting of degrees of freedom, solution of the Cauchy problem, and so on. The non-local scalar is introduced in section \\ref{scala1}, while the localization procedure is described in section \\ref{ized}. The counting of initial conditions and degrees of freedom is carried out in section \\ref{scala3}, where we find that this number is, respectively, 4 and 1 for the real non-local scalar with non-linear interactions. Section \\ref{solu} reviews another practical use of the diffusion-equation method, the construction of analytic solutions of the equations of motion. In section \\ref{deloc}, we compare the diffusion-equation method with the results obtained in other approaches, mainly the delocalization method by Tomboulis \\cite{Tom15}. A generalization of the method to non-local operators $\\exp H(\\Box)$ with polynomial exponents is proposed in section \\ref{Hge}, while non-polynomial profiles $H(\\Box)$ require some extra input which is discussed in a companion paper \\cite{CMN3}.\n\nThe non-local gravitational action \\Eq{nlffg} is studied in section \\ref{eoms}, where we find the background-independent covariant Einstein equations for \\emph{any} form factor $\\gamma(\\Box)$ and recast the system in terms of an auxiliary field. Contrary to other calculations in the literature \\cite{BCKM,Kos13,CKMT}, we find the equations of motion for an exponential-type form factor \\Eq{fofag} in terms of parametric integrals rather than from the series expansion of the non-local operators. This new form is crucial both to solve the initial-value problem and to find explicit solutions with the diffusion-equation method.\n\nThe localized system corresponding to the non-local gravitational action \\Eq{nlffg} is introduced and discussed in section \\ref{locnlg}. After defining the localized action in section \\ref{loac}, we obtain the equations of motion in section \\ref{loeom}, which agree with the non-local ones. The counting of initial conditions and degrees of freedom is done in section \\ref{lodof}, where we find that they amount to, respectively, 4 and $D(D-2)$. Appendices contain several technical details and the full derivation of the equations of motion.\n\nTherefore, although in sections \\ref{ized} and \\ref{locnlg} we will concentrate on the form factor \\Eq{fofag} for which renormalization is likely but still under debate, our results with auxiliary fields (section \\ref{aux}) will be valid for an arbitrary form factor, while in section \\ref{Hge} and in \\cite{CMN3} we will generalize the diffusion-equation approach to form factors associated with finite quantum theories.\n\n\\subsection{Summary of main equations and claims}\n\nTo orient the reader, we summarize here the key formul\\ae:\n\\begin{itemize}\n\\item Scalar field theory.\n\t\\begin{itemize}\n\t\\item Non-local action: \\Eq{fac}.\n\t\\item Non-local equation of motion: \\Eq{tpheom}.\n\t\\item Localized action: \\Eq{act}.\n\t\\item Localized equations of motion: \\Eq{eomchi2}, \\Eq{eomP12}, \\Eq{eomP22}.\n\t\\item Constraints on localized dynamics: \\Eq{rstcon1}, \\Eq{rstcon2}.\n\t\\item Number of field degrees of freedom: \\Eq{dof1}.\n\t\\item Number of initial conditions: \\Eq{dofic}.\n\t\\end{itemize}\n\\item Gravity.\n\t\\begin{itemize}\n\t\\item Non-local action: \\Eq{nlffgb}.\n\t\\item Non-local equations of motion: \\Eq{EinEq1}.\n\t\\item Non-local action with auxiliary field: \\Eq{nlff3}.\n\t\\item Non-local equations of motion with auxiliary field: \\Eq{eomnl1}.\n\t\\item Localized action: \\Eq{gactcm}.\n\t\\item Localized equations of motion: \\Eq{difPg}, \\Eq{chimnde}, \\Eq{uff}, \\Eq{lasto}.\n\t\\item Constraints on localized dynamics: \\Eq{slic}, \\Eq{Fmu2}.\n\t\\item Number of field degrees of freedom: \\Eq{dof3}.\n\t\\item Number of initial conditions: \\Eq{dof2}.\n\t\\end{itemize}\n\\end{itemize}\n\n\n\\section{Diffusion-equation method: scalar field}\\label{scala}\n\nBefore considering gravity, it will be useful to illustrate the main philosophy beyond and advantages of the diffusion-equation method. To this purpose, we review its application to a classical scalar field theory \\cite{cuta3}, expanding the discussion therein to cover all important points that will help us to understand the results for non-local gravitational theories. We present a simplified version of the scalar system, with no nested integrals, no free parameters in the diffusion equation, and fewer assumptions than in \\cite{cuta3}. The original version of \\cite{cuta3} can be found in appendix \\ref{app2}. A comparison between the scalar and gravitational systems will be done in section \\ref{loeom}.\n\n\n\\subsection{Non-local system: traditional approach and problems}\\label{scala1}\n\nConsider the scalar-field action in $D$-dimensional Minkowski spacetime (with signature $-,+,\\cdots,+$)\n\\begin{equation}\\label{fac}\nS_{\\phi} = \\int d^D x\\,\\mathcal{L}_\\phi,\\qquad \\mathcal{L}_\\phi =\\frac12\\phi\\Boxe^{-r_*\\Box}\\phi-V(\\phi),\n\\end{equation}\nwhere $r_*$ is a constant of mass dimension $[r_*]=-2$ and $V(\\phi)$ is a potential. We chose the exponential operator as the simplest example where the diffusion method works, but we will relax this assumption later to include operators of the form $\\exp H(\\Box)$ not contemplated in the original treatment in \\cite{cuta3}. Applying the variational principle to $S_\\phi$, the equation of motion is\n\\begin{equation}\\label{tpheom}\n\\Box e^{-r_*\\Box}\\phi-V'(\\phi)=0,\n\\end{equation}\nwhere a prime denotes a derivative with respect to $\\phi$. The action \\Eq{fac} and the dynamical equation \\Eq{tpheom} are a prototype of, respectively, a \\emph{non-local system} and a \\emph{non-local equation of motion}.\n\nThe initial-condition problem associated with \\Eq{tpheom} suffers from the conceptual issues outlined in the introduction. Rather than repeating the same mantra again, we recast the Cauchy problem as a problem of representation of the non-local operator $\\exp(-r_*\\Box)$. To find a solution of \\Eq{tpheom}, one must first define the left-hand side. The most obvious way to represent the exponential is via its series,\n\\begin{equation}\\label{opse}\ne^{-r_*\\Box}=\\sum_{n=0}^{+\\infty}\\frac{(-r_*\\Box)^n}{n!}=1-r_*\\Box+\\frac12 r_*^2\\Box^2+\\dots\\,.\n\\end{equation}\nTo find solutions, one can use different strategies. One of the oldest and most disastrous is to truncate the non-local operator up to some finite order $n_{\\rm max}$. In doing so, one introduces instabilities corresponding to the Ostrogradski modes of a higher-derivative theory which has little or nothing to do with the starting theory \\cite{MoZ,cuta2}. Exact procedures such as the root method exist for linear equations of motion \\cite{PU,EW1,BK1,Ver09} but they have the disadvantage of being applicable only to non-interacting systems. Another possibility is to choose a profile $\\phi(x)$ and apply the operator \\Eq{opse}, but the series does not converge in general \\cite{cuta2}. This does not necessarily mean that the chosen profile is not a solution of the equations of motion. Rather, the series representation \\Eq{opse} is ill defined for a portion of the space of solutions. Even in the case where an exact solution is found, however, this may be non-unique for a given set of initial conditions \\cite{EW1,MoZ,Tom15}.\n\n\n\\subsection{Localized system}\\label{ized}\n\nThe diffusion-equation method \\cite{roll,cuta3,MuNu3,cuta7}, some elements of which can be found already in \\cite{PU} (section III.B.3), bypasses the above-mentioned issues by converting the Cauchy problem into a boundary problem.\\footnote{A similar attempt was made in \\cite{CCG}.} All the non-locality is transferred into a fictitious extra direction $r$ and infinite initial conditions for the scalar field $\\phi(t,{\\bf x})$ are converted to a \\emph{finite} number of field conditions on the $r=\\beta r_*$ slice along the extra direction, where $\\beta$ is a positive dimensionless constant (i.e., it is the physical value of $r$ measured in $r_*$ units). In other words, the rectangle $[0,\\beta r_*]\\times [t_{\\rm i},t_{\\rm f}]$ can be spanned either along the $t$ (time) direction, as done when trying to solve the problem of initial conditions by brute force at $t=t_{\\rm i}$, or along the $r$ direction, as done in the boundary-value problem with the diffusion method; see Fig.\\ \\ref{fig1} here and Fig.\\ 1 of \\cite{MuNu3}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.2cm]{fig1}\n\\caption{\\label{fig1} Diffusion-equation method describing the dynamics of the scalar field theory \\Eq{fac} as the dynamics of the localized system \\Eq{act} on the slice $r=\\beta r_*$.}\n\\end{figure}\n\nWe will also be able to find the exact number of conditions required and to compare these results with those from other methods \\cite{Tom15}.\n\n\\subsubsection{Lagrangian formalism}\n\nThe main idea is to exploit the fact that the exponential operator in \\Eq{fac} acts as a translation operator if $\\phi$ obeys a diffusion equation. Using this property, we can convert the non-local system into a \\emph{localized} one where the diffusion equation is part of the dynamical equations, the field is evaluated at different points in an extra direction (along which the system is thus non-local), and only second-order derivative operators appear in the action and in the equations of motion. In this way, one can make sense of the Cauchy problem in the localized system and also in the non-local one, after establishing the conditions for which the two systems are equivalent \\cite{cuta3}. This construction goes through some initial guesswork about the form of the correct localized system, especially regarding the integration domain of certain parts of the action, but this is not difficult in general. Both the scalar case \\Eq{fac} and the gravitational action \\Eq{nlffg} are simple enough to create no big trouble.\n\nLet us therefore forget temporarily about the non-local system \\Eq{fac} and consider the $(D+1)$-dimensional local system \n\\begin{eqnarray}\n\\mathcal{S}[\\Phi,\\chi]&=&\\int d^D x\\,d r \\left(\\mathcal{L}_{\\Phi}+\\mathcal{L}_{\\chi}\\right)\\,,\\label{act}\\\\\n\\mathcal{L}_{\\Phi}&=&\\frac12\\Phi(r,x)\\Box \\Phi(r-r_*,x)-V[\\Phi(r,x)]\\,,\\label{locPh2}\\\\\n\\mathcal{L}_{\\chi}&=&\\frac12 \\int_0^{r_*} d q\\,\\chi(r-q,x)(\\partial_{r'}-\\Box)\\Phi(r',x)\\,.\\label{locch2}\n\\epsilon\nwhere $r$ is an extra direction, $r_*$ is a specific value of $r$, $\\Phi$ and $\\chi$ are $(D+1)$-dimensional scalar fields and\n\\begin{equation}\\label{rprime}\nr'=r+q-r_*\\,,\n\\end{equation}\nhence $\\partial_{r'}=\\partial_q$. The action \\Eq{act} is second-order (hence local) in spacetime derivatives and non-local in $r$ (because the fields take different arguments). The integration range of $r$ in \\Eq{act} is arbitrary, it can be set equal to $r\\in [0,+\\infty)$ or any other interval containing  $[0,\\beta r_*]$ (the slices $r=0$ and $r=\\beta r_*$ play a special role: the former is the value where to specify the initial condition in $r$ of the diffusion equation, while the latter will be the physical value of the parameter $r$, for a given $\\beta$).\n\nThe equations of motion are calculated from the infinitesimal variations of the action, using the functional derivative $\\delta f(r,x)\/\\delta f(\\bar r, \\bar x)=\\delta(r-\\bar r)\\delta^{(D)}(x-\\bar x)$ for a field $f$. Since $\\bar x$ and $\\bar r$ are arbitrary, one can always assume the support of these delta distributions to lie within the integration domains in \\Eq{act}, so that integrations in $x$, $r$ and $q$ are removed and the fields evaluated at $x=\\bar x$ and $r=\\bar r$. Bars will be removed in the final equations of motion.\n\nThe first variation we calculate is with respect to $\\chi$. To keep notation light, let us ignore the trivially local $x$-dependence from now on. Doing it step by step,\n\\begin{eqnarray}\n0 &=& \\frac{\\delta\\mathcal{S}[\\Phi,\\chi]}{\\delta\\chi(\\bar r,\\bar x)}=\\frac12 \\int d r \\int_0^{r_*}d q\\,\\delta(r-q-\\bar r)(\\partial_{r'}-\\Box)\\Phi(r')\\nonumber\\\\\n&=& \\frac12\\int_{\\bar r}^{r_*+\\bar r} d r (\\partial_{r'}-\\Box)\\Phi(r')\\Big|_{r'=2r-\\bar r-r_*}.\\label{inter2}\n\\end{eqnarray}\nThe integration of the Dirac distribution in $q$ gives the prescription $0<q=r-\\bar r<r_*$, hence $\\bar r<r< r_*+\\bar r$ such that the support of the $\\delta$ lies in both $r$- and $q$-integration ranges. After a reparametrization $\\rho= r-\\bar r$, one gets\nan integral of the form $\\int_0^{r_*} d \\rho\\, f(2\\rho+\\bar r-r_*)$. Since $\\bar r$ is arbitrary, the integrand must be identically zero on shell for any integration range:\n\\begin{equation}\\label{eomchi2}\n0=(\\partial_r-\\Box)\\Phi(r,x)\\,.\n\\end{equation}\nAnother way to obtain the same result is to restrict from the very beginning the integration range in \\Eq{act} from 0 to $+\\infty$ or from $-\\infty$ to $r_*$. In the first case, the integration range in \\Eq{inter2} is reduced to $[0,r_*+\\bar r]$, since $\\bar r>0$. In the second case, the range in \\Eq{inter2} is reduced to $[\\bar r, r_*]$, since $\\bar r<r_*$. In both cases, due to the arbitrariness of $\\bar r$ the width of the integration domain is arbitrary, which implies that the integrand is zero.\n\n\nThe diffusion equation \\Eq{eomchi2} is the first equation of motion. The second equation of motion is more complicated but very instructive, so that we report it in full. We integrate \\Eq{locch2} by parts, in order to load all derivatives onto $\\chi$:\n\\begin{eqnarray}\n\\mathcal{L}_{\\chi}&=&\\frac12\\int_0^{r}d q\\,\\partial_{q}[\\chi(r-q)\\Phi(r')]-\\frac12\\int_0^{r}d q\\,\\Phi(r')(\\partial_{q}+\\Box)\\chi(r-q)\\nonumber\\\\\n&=&\\frac12[\\chi(r-r_*)\\Phi(r)-\\chi(r)\\Phi(r-r_*)]-\\frac12\\int_0^{r_*} d q\\,\\Phi(r')(\\partial_{r'}+\\Box)\\chi(r-q)\\,.\\label{inpa2}\n\\end{eqnarray}\nTherefore, varying with respect to $\\Phi(\\bar r,\\bar x)$ gives\n\\begin{eqnarray}\n0=\\frac{\\delta\\mathcal{S}[\\Phi,\\chi]}{\\delta\\Phi(\\bar r,\\bar x)}&=&\\frac12[\\Box\\Phi(\\bar r-r_*)+\\chi(\\bar r-r_*)]+\\frac12[\\Box\\Phi(\\bar r+r_*)-\\chi(\\bar r+r_*)]-V'[\\Phi(\\bar r)]\\nonumber\\\\\n&&+\\frac12\\int_{\\bar r}^{\\bar r+r_*} d r\\,(\\partial_{-\\bar r}-\\Box)\\chi(2r-\\bar r-r_*)\\,.\\label{inter22}\n\\end{eqnarray}\nFrom this, we conclude that equation \\Eq{inter22} reproduces \\Eq{tpheom} if\n\\begin{equation}\\label{rstcon1}\n\\Phi(\\beta r_*,x)=\\phi(x)\\,,\n\\end{equation}\nwhere $\\beta>0$ is a real constant, and\n\\begin{equation}\\label{rstcon20}\n\\chi(r,x) = \\Box\\Phi(r,x)\\,.\n\\end{equation}\nIn fact, in this case $\\chi$ obeys the same diffusion equation \\Eq{eomchi2} as $\\Phi$, so that the two contributions in \\Eq{inter22} must vanish separately, thus yielding the two equations of motion (restoring $x$-dependence)\n\\begin{eqnarray}\n\\hspace{-.8cm} 0&=&\\frac12[\\Box\\Phi(r-r_*,x)+\\chi(r-r_*,x)]+\\frac12[\\Box\\Phi(r+r_*)-\\chi(r+r_*)]-V'[\\Phi(r,x)]\\,,\\label{eomP12}\\\\\n\\hspace{-.8cm} 0&=&(\\partial_r-\\Box)\\chi(r,x)\\,.\\label{eomP22}\n\\end{eqnarray}\nThen, when evaluating \\Eq{eomP12} at $r=\\beta r_*$ the first term yields $(1\/2)2\\Boxe^{-r_*\\Box}\\Phi(\\beta r_*,x)=\\Boxe^{-r_*\\Box}\\phi(x)$, the second term vanishes and \\Eq{eomP12} reproduces \\Eq{tpheom} exactly. See Fig.\\ \\ref{fig2} for a toy example. Note that imposing \\Eq{rstcon1} only at $r=\\beta r_*$,\n\\begin{equation}\\label{rstcon2}\n\\chi(\\beta r_*,x) = \\Box\\Phi(\\beta r_*,x)\\,,\n\\end{equation}\nor at any given $r=\\tilde r$ instead of for all $r$ would again yield \\Eq{rstcon1}, provided $\\chi$ obeyed \\Eq{eomP22}. In fact, parametrizing with $\\sigma=r-\\tilde r$, $\\chi(r,x)= \\chi(\\sigma+ \\tilde r,x) = e^{\\sigma\\Box}\\chi(\\tilde r,x)= e^{\\sigma\\Box}\\Box\\Phi(\\tilde r,x)=\\Boxe^{\\sigma\\Box}\\Phi(\\tilde r,x)= \\Box\\Phi(\\sigma+\\tilde r,x)=\\Box\\Phi(r,x)$.\n\nThe introduction of the parameter $\\beta$ in \\Eq{rstcon1} reflects the fact that the choice of the slice where the $(D+1)$-dimensional scalar field coincides with the $D$-dimensional field does not affect the final result. For instance, one could have chosen $\\beta=0$ and identified $\\Phi(0,x)=\\phi(x)$ (the ``initial'' condition in $r$ of the diffusion equation), $\\chi(0,x)=\\Box\\phi(x)$. However, in section \\ref{solu} we will argue that equation \\Eq{rstcon1} is far better suited than $\\Phi(0,x)=\\phi(x)$ for the task of finding dynamical solutions. This is why we introduced a strictly positive $\\beta$ in the first place.\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.cm]{fig2a}\n\\includegraphics[width=8.cm]{fig2b}\n\\caption{\\label{fig2} In $D=1$ flat Euclidean space, the solution of the diffusion equation \\Eq{eomchi2} with initial condition $\\Phi(0,x)=\\delta(x)$ is $\\Phi(r,x)=\\exp[-x^2\/(4r)]\/\\sqrt{4\\pi r}$. This solution is represented in the $(r,x)$ plane as an orange surface (concavity upwards) in the left plot, together with $\\chi(r,x)=\\partial_x^2\\Phi(r,x)$ (blue surface, concavity downwards). The section of these surfaces at $r=\\beta r_*=0.5$ (black thick line) are shown in the right plot.}\n\\end{figure}\n\nTo summarize the logic here, given the non-local system \\Eq{fac} one can always write down the system \\Eq{act}--\\Eq{locch2} localizing it. This localized system is not in one-to-one correspondence with the non-local system but it always admits, among its solutions, the solutions of the non-local system. These solutions are defined by the boundary condition \\Eq{rstcon1} together with the local condition \\Eq{rstcon2}. The sub-set of solutions of the localized system obeying these conditions are solutions to the original non-local one, since the above conditions are valid on shell (i.e., applying \\Eq{eomchi2} and \\Eq{eomP22} to \\Eq{eomP12}). In other words, \\Eq{rstcon1} and \\Eq{rstcon2} define the sub-set of solutions of the localized system that recover the equations of motion and solutions of the original non-local system. Recalling that the localized system \\Eq{act} must be reducible to the non-local one \\Eq{fac} only at a certain slice $r=\\beta r_*$ in the extra direction, it is clear that we do not need to study the most general $(D+1)$-dimensional evolution of the localized dynamics, which is obtained by dropping \\Eq{rstcon2}. \n\nNotice that it is not possible, while keeping the diffusing structure unaltered, to change the status of \\Eq{rstcon2} from a condition imposed by hand to a consequence of the dynamics. For instance, one could try to add an extra term $\\mathcal{L}_\\lambda=\\lambda(r,x)[\\Box\\Phi(r,x)-\\chi(r,x)]$ to the action \\Eq{act}, which would give \\Eq{rstcon2} when varying $\\mathcal{S}[\\Phi,\\chi,\\lambda]$ with respect to the Lagrange multiplier $\\lambda$. However, equations \\Eq{inter2} and \\Eq{inter22} would become, respectively, $(\\dots)-\\lambda=0$ and $(\\dots)+\\Box\\lambda=0$, where the extra terms would vanish separately if, again, we imposed by hand\n\\begin{equation}\\label{lamb}\n\\lambda(r,x)=0\\,.\n\\end{equation}\nThis condition, replacing \\Eq{rstcon2}, amounts to forbid source terms in the diffusion equation \\Eq{eomchi2}. Indeed, the infinitely many degrees of freedom of the original non-local system are encoded in equation \\Eq{rstcon2} or in the alternative equation \\Eq{lamb}, both of which are a condition on the \\emph{infinitely many} $r$-values of the fields $\\Phi$ and $\\chi$. Thus, demanding to get a fully self-determined diffusing localized system equivalent to the non-local one is not only impossible,\\footnote{Of course, this claim does not apply to an arbitrary localized system not diffusing with the standard sourceless diffusion equation of Brownian motion.} but also meaningless, since the equivalence between the localized and the non-local system on one hand and the statement of the initial-value problem for the non-local system on the other hand must both go through the setting of an infinite number of conditions external to the dynamics.\n\nFor future use, we highlight three important features of the localization procedure which will apply, in their essence, also to the non-local gravity action \\Eq{nlffg}.\n\\begin{enumerate}\n\\item By the diffusion-equation method, one does not establish a one-to-one correspondence between the localized system \\Eq{act} and the non-local system \\Eq{fac}. Rather, we showed that there exist field conditions on the $r=\\beta r_*$ slice such that the localized system has the same spacetime dynamics as the non-local system. This correspondence on a slice is depicted in Fig.\\ \\ref{fig1}.\n\\item To get the correct result, it was crucial to make a careful choice of the arguments in the diffusion-equation term \\Eq{locch2} and a careful treatment of the boundary terms when integrating \\Eq{locch2} by parts as in \\Eq{inpa2}. Without such boundary terms, \\Eq{eomP12} would have been unable to reproduce \\Eq{tpheom} on the $r=\\beta r_*$ slice with the correct numerical factors.\n\\item The localized system is second-order in spacetime derivatives, for both $\\Phi$ and $\\chi$. Therefore, the Cauchy problem for this system, when restricted to spacetime directions $x^\\mu$, is solved by \\emph{four initial conditions} at some $t=t_{\\rm i}$:\n\\begin{equation}\n\\Phi(r,t_{\\rm i},{\\bf x}),\\,\\dot\\Phi(r,t_{\\rm i},{\\bf x}),\\,\\chi(r,t_{\\rm i},{\\bf x}),\\,\\dot\\chi(r,t_{\\rm i},{\\bf x})\\,.\n\\end{equation}\nIn particular, these conditions are valid at $r=\\beta r_*$, so that also the Cauchy problem of the non-local system \\Eq{fac} is solved by four initial conditions, corresponding (via \\Eq{rstcon2}) to $\\phi(t_{\\rm i},{\\bf x})$ and its first three time derivatives. We will find a similar result also in non-local gravity gravity, with four initial conditions for the metric. In general, given a non-local action with exponential non-locality for tensorial field $\\phi_{\\mu\\nu\\cdots}$ representing $n$ physical degrees of freedom, the diffusion-equation method relies on a second-order localized system for a field $\\Phi_{\\mu\\nu\\cdots}$ and an auxiliary field $\\chi^{\\mu\\nu\\cdots}$ with the same symmetry properties as $\\phi$, thus leading to $2n$ initial conditions.\n\\end{enumerate}\n\n\\subsubsection{Ghost mode}\n\nIn this subsection, we analyze a hidden ghost mode which, however, does not influence the non-local dynamics. To understand this aspect, we will employ a reformulation of the localized dynamics (equation \\Eq{tildeL}), physically equivalent to \\Eq{act}--\\Eq{locch2}, which is convenient to study the degrees of freedom of the theory but is unsuitable for the practical treatment (Cauchy problem, solutions, and so on) of the dynamics, due to problems we will comment on in due course.\n\nIt is very well known that the kinetic term in \\Eq{fac} can be symmetrized after integrating by part, so that the Lagrangian becomes\n\\begin{equation}\\label{facbis}\n\\mathcal{L}_\\phi =\\frac12(e^{-\\frac12r_*\\Box}\\phi)\\Box(e^{-\\frac12r_*\\Box}\\phi)-V(\\phi).\n\\end{equation}\nFrom here, one can make the field redefinition $\\tilde\\phi=e^{-\\frac12r_*\\Box}\\phi$ so often used in $p$-adic and string field theory. We will do something similar by considering the localized version of \\Eq{facbis}, which is given by \\Eq{act} with ($x$-dependence omitted everywhere)\n\\begin{eqnarray}\n\\tilde\\mathcal{L}_{\\Phi}=\\frac12\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\Box \\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)-V[\\Phi(r,x)]\\label{locPh2bis}\n\\end{eqnarray}\nreplacing \\Eq{locPh2}. We note that the integral in \\Eq{locch2} is pleonastic for the Laplace--Beltrami term, since both $\\chi$ and $\\Phi$ obey the diffusion equation:\n\\begin{eqnarray}\n\\intd^Dx\\int_0^{r_*} d q\\,\\chi(r-q)\\Box\\Phi(r')&=&\\intd^Dx\\int_0^{r_*} d q\\,e^{(\\frac12r_*-q)\\Box}\\chi\\left(r-\\textstyle{\\frac12} r_*\\right)e^{(q-\\frac12r_*)\\Box}\\Box\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\nonumber\\\\\n&=&\\intd^Dx\\int_0^{r_*} d q\\,\\chi\\left(r-\\textstyle{\\frac12} r_*\\right)\\Box\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\nonumber\\\\\n&=&r_*\\intd^Dx\\,\\chi\\left(r-\\textstyle{\\frac12} r_*\\right)\\Box\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\,.\n\\end{eqnarray}\nHowever, replacing \\Eq{locch2} with a mixed term\n\\begin{equation}\\nonumber\n\\tilde\\mathcal{L}_{\\chi}=-\\frac{r_*}{2}\\chi\\left(r-\\textstyle{\\frac12} r_*\\right)\\Box\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)+\\frac12 \\int_0^{r_*} d q\\,\\chi(r-q)\\partial_{r'}\\Phi(r')\n\\end{equation}\nwould not give the correct equations of motion, as we will see shortly. The reason is that $\\tilde\\mathcal{L}_{\\chi}$ is originated by an on-shell condition, a trick that invalidates the variational principle. To find the correct Lagrangian, we generalize this term with a generic functional of the fields, $\\Phi\\rightarrow f[\\Phi,\\chi]$. A last step we take (not necessary, but useful to simplify the physical interpretation) is to consider the field redefinitions\n\\begin{equation}\n\\varphi(r,x) :=\\Phi\\left(r-\\textstyle{\\frac12} r_*,x\\right)-\\frac{r_*}{2}\\chi\\left(r-\\textstyle{\\frac12} r_*,x\\right),\\qquad \\psi(r,x):=\\frac{r_*}{2}\\chi\\left(r-\\textstyle{\\frac12} r_*,x\\right)\\,,\n\\end{equation}\nso that\n\\begin{equation}\\label{Pvfpsi}\n\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)=\\varphi(r)+\\psi(r)\\,,\n\\end{equation}\nand the total Lagrangian on Minkowski spacetime is\n\\begin{eqnarray}\n\\tilde\\mathcal{L}&=&-\\frac12\\partial_\\mu\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\partial^\\mu\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)-V[\\Phi(r)]+\\frac{r_*}{2}\\partial_\\mu\\chi\\left(r-\\textstyle{\\frac12} r_*\\right)\\partial^\\mu\\Phi\\left(r-\\textstyle{\\frac12} r_*\\right)\\nonumber\\\\\n&&+\\frac12 \\int_0^{r_*} d q\\,\\chi(r-q)\\partial_{r'}f(r')\\nonumber\\\\\n&=&-\\frac12\\partial_\\mu\\varphi(r)\\partial^\\mu\\varphi(r)-V\\left[\\varphi\\left(r+\\textstyle{\\frac12} r_*\\right)+\\psi\\left(r+\\textstyle{\\frac12} r_*\\right)\\right]+\\frac12\\partial_\\mu\\psi(r)\\partial^\\mu\\psi(r)+\\frac{1}{r_*}\\,I(r)\\,,\\nonumber\\\\ \\label{tildeL}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{utile}\nI(r) &:=& \\int_{0}^{r_{*}}d q\\,\\psi\\left(r-q+\\textstyle{\\frac12} r_*\\right)\\partial_q f\\left(r+q-\\textstyle{\\frac12} r_*\\right)\\nonumber\\\\\n&=& \\psi\\left(r-\\textstyle{\\frac12} r_*\\right)f\\left(r+\\textstyle{\\frac12} r_*\\right)-\\psi\\left(r+\\textstyle{\\frac12} r_*\\right)f\\left(r-\\textstyle{\\frac12} r_*\\right)\\nonumber\\\\\n&&-\\int_0^{r_*}d q\\,f\\left(r+q-\\textstyle{\\frac12} r_*\\right)\\partial_q\\psi\\left(r-q+\\textstyle{\\frac12} r_*\\right).\n\\end{eqnarray}\nThe function $f$ is determined in appendix \\ref{appI} by requiring the recovery of the non-local dynamics on the $r=\\beta r_*$ slice.\n\nObserving \\Eq{tildeL}, one sees that the canonical scalar $\\varphi$ propagates with a kinetic term of the correct sign, while the canonical scalar $\\psi$ (hence $\\chi$) is a ghost. This detail went unnoticed in \\cite{cuta3}.\n\nThere are two issues affecting \\Eq{tildeL} and described in appendix \\ref{appI}, but we should not lose sight of the reason why we introduced this Lagrangian. One may choose either \\Eq{act}--\\Eq{locch2} or \\Eq{tildeL} depending on what one wants to study. For the analysis of the Cauchy problem and of the dynamical solutions, the action \\Eq{act}--\\Eq{locch2} is to be preferred, and in fact we will analyze non-local gravity under the same scheme. On the other hand, for the characterization (ghost-like or not) of the localized degrees of freedom the Lagrangian \\Eq{tildeL}, or the Hamiltonian \\Eq{hamil} we will derive from it in the next subsection, is more indicated. The counting of the localized degrees of freedom (section \\ref{scala3}) can be performed indifferently in the original system \\Eq{act}--\\Eq{locch2}, in the Lagrangian \\Eq{tildeL}, or in the Hamiltonian formalism derived from \\Eq{tildeL}.\n\n\n\\subsubsection{Hamiltonian formalism}\n\nTo count the number of degrees of freedom in a non-local theory, we must first count the number of localized degrees of freedom in the associated localized $(D+1)$-dimensional theory. In the case of the scalar field, this information is already available in Lagrangian formalism, but for completeness we can obtain the same result from Hamiltonian formalism. The example presented in this subsection will illustrate the general method and its caveats. Its application in the localization of the scalar field was sketched in \\cite{cuta3}, but here we will fill several gaps in that discussion. The actual counting of localized degrees of freedom will be done in section \\ref{scala3}.\n\nAlthough we do not write the non-local system \\Eq{fac} in Hamiltonian formalism, we can reach a lesser but still instructive goal, namely, the formulation of the Hamiltonian approach for the associated localized system. However, if we take the localized system \\Eq{act}--\\Eq{locch2} as a starting point we soon meet several problems, all of which stem from the non-locality with respect to the $r$ direction. Momenta acquire a rather obscure non-invertible form and one cannot write down a Hamiltonian in phase space. However, the system is not constrained. We can avoid all the trouble by acting directly on \\Eq{tildeL}. Calling $\\tilde L := \\int d^{D-1}{\\bf x}\\intd r\\,\\tilde\\mathcal{L}$ the Lagrangian, we can define the phase space and the Hamiltonian. The momenta are\n\\begin{equation}\n\\pi_\\varphi(r,x) := \\frac{\\delta\\tilde L}{\\delta\\dot\\varphi(r,x)}=\\dot\\varphi(r,x),\\qquad \\pi_\\psi(r,x) := \\frac{\\delta\\tilde L}{\\delta\\dot\\psi(r,x)}=-\\dot\\psi(r,x)\\,.\\label{pic}\n\\end{equation}\nNotice that, if we had calculated the momenta directly from \\Eq{locPh2} and \\Eq{locPh2bis}, we would have obtained $\\pi_\\Phi(r)=(1\/2)[\\dot\\Phi(r-r_*)+\\dot\\Phi(r+r_*)-\\int_0^{r_*}d s\\,\\dot\\chi(r-2s+r_*)]$ and $\\pi_\\chi(r) =-(1\/2)\\int_0^{r_*} d s\\,\\dot\\Phi(r+2s-r_*)$, which are not invertible locally with respect to $\\dot\\Phi(r)$ and $\\dot\\chi(r)$.\n\nThe non-vanishing equal-time Poisson brackets in terms of the spatial $(D-1)$-vectors ${\\bf x}$ are\n\\begin{eqnarray}\n\\{\\varphi(r_1,x_1),\\,\\pi_\\varphi(r_2,x_2)\\}_{t_1=t_2} &=& \\delta(r_1-r_2)\\,\\delta^{(D-1)}({\\bf x}_1-{\\bf x}_2)\\,,\\\\\n\\{\\psi(r_1,x_1),\\,\\pi_\\psi(r_2,x_2)\\}_{t_1=t_2} &=& \\delta(r_1-r_2)\\,\\delta^{(D-1)}({\\bf x}_1-{\\bf x}_2)\\,,\n\\end{eqnarray}\nwhile the Hamiltonian of the system is ($x$-dependence omitted again)\n\\begin{eqnarray}\nH &:=&\\int d^{D-1}{\\bf x} d r \\left[\\pi_\\varphi(r)\\dot\\varphi(r)+\\pi_\\psi(r)\\dot\\psi(r)\\right]-\\tilde L\\nonumber\\\\\n  &=& \\int d^{D-1}{\\bf x} d r \\left\\{\\frac12\\pi_\\varphi^2(r)+\\frac12\\nabla_i\\varphi(r)\\nabla^i\\varphi(r)-\\frac12\\pi_\\psi^2(r)-\\frac12\\nabla_i\\psi(r)\\nabla^i\\psi(r)\\right.\\nonumber\\\\\n\t&&\\qquad\\qquad\\qquad\\left.+V[\\Phi(r)]-\\frac{1}{r_*} I[\\psi(r),\\Phi(r)]\\right\\}\\,,\\label{hamil}\n\\end{eqnarray}\nwhere it is understood that $\\Phi(r)=\\varphi(r+r_*\/2)+\\psi(r+r_*\/2)$. Since $\\Phi$ is shifted in $r$, $H$ is non-local in $r$ due to the terms in the last line of \\Eq{hamil}. Nevertheless, the Hamiltonian is written solely in terms of phase-space variables and the phase-space fields are completely local in spacetime coordinates.\n\nThe evolution equations for the fields $\\varphi$ and $\\psi$ trivially gives the momenta, $\\dot\\varphi(r)=\\{\\varphi(r),H\\}=\\delta H\/\\delta\\pi_\\varphi(r)=\\pi_\\varphi(r)$, $\\dot\\psi(r)=\\{\\psi(r),H\\}=\\delta H\/\\delta\\pi_\\psi(r)=-\\pi_\\psi(r)$, while the Hamiltonian evolution of the momenta give the localized equations of motion \\Eq{eomH1} and \\Eq{eomH2}:\n\\begin{equation}\n\\dot\\pi_\\varphi(\\bar r) = \\{\\pi_\\varphi(\\bar r),\\,H\\}=-\\frac{\\delta H}{\\delta\\varphi(\\bar r)}\\,,\\qquad \\dot\\pi_\\psi(\\bar r) = \\{\\pi_\\psi(\\bar r),\\,H\\}= -\\frac{\\delta H}{\\delta\\psi(\\bar r)}\\,.\\label{eomH22}\n\\end{equation}\n\n\n\\subsection{Initial conditions and degrees of freedom}\\label{scala3}\n\nThe question about how many initial conditions we should specify for the non-local scalar system is related to another one: How many\ndegrees of freedom are hidden in equation \\Eq{tpheom}? In higher-derivative theories, the presence of many degrees of freedom (Ostrogradski modes) is well known. For a system with $n$ derivatives, the Cauchy problem is uniquely solved by $n$ initial conditions. However, there is an uncrossable divide between higher-derivative and non-local theories, and one cannot conclude that non-local theories need $n=\\infty$ initial conditions; conversely, truncating a non-local theory to finite order leads to a physically different model \\cite{EW1,cuta2}.\n\nTo understand the problem, we review its root and also some confusion surrounding it. First of all, there is agreement in the literature about the fact that the \\emph{free} system with constant, linear or quadratic $V(\\phi)$\n has \\emph{two} initial conditions. In the absence of interactions, the Cauchy problem associated with \\Eq{tpheom} is specified only by $\\phi(t_{\\rm i},{\\bf x})$ and $\\dot\\phi(t_{\\rm i},{\\bf x})$. The entire functional $\\exp(-r_*\\Box)$ introduces no new poles in the spectrum of $\\phi$ and the system is equivalent to the local one with $r_*=0$, as is obvious from the field redefinition $\\tilde\\phi=e^{-r_*\\Box\/2}\\phi$.\\footnote{Another method, completely equivalent, is to work in Laplace momentum space.} This was first recognized as early as 1950 in the seminal paper by Pais and Uhlenbeck \\cite{PU} (section III.B.3) and reiterated more recently, sometimes using very different terminology and techniques, in other works \\cite{EW1,Sim90,BK1,cuta3}.\n\nMore contrived is the case with interactions. The reader unfamiliar with non-local theories may wonder why interactions should make any difference when counting the number of initial conditions. The reason is that, in this case, there is no field redefinition absorbing the non-local operator of the kinetic term. Any other rewriting will not work, either. For instance, a non-local kinetic term can always be expressed as a convolution with a kernel \\cite{PU}. Consider the scalar-field Lagrangian $\\mathcal{L}_\\phi=\\phi f(\\Box)\\,\\phi-V(\\phi)$ with generic form factor $f(\\Box)$. In momentum space, calling $F$ the Fourier transform of $f$,\n\\begin{eqnarray}\n\\phi(x) f(\\Box)\\,\\phi(x) &=& \\phi(x)\\intd^D k\\,f(-k^2)\\,\\delta(k^\\mu-i\\nabla^\\mu)\\,\\phi(x)\\nonumber\\\\\n\t\t\t\t\t\t\t\t\t\t\t &=& \\phi(x)\\intd^D k\\,\\left[\\int\\frac{d^D z}{(2\\pi)^D}\\,F(z)\\,e^{-i z^\\mu k_\\mu}\\right]\\,\\delta(k^\\mu-i\\nabla^\\mu)\\,\\phi(x)\\nonumber\\\\\n\t\t\t\t\t\t\t\t\t\t\t &=& \\phi(x)\\int\\frac{d^D z}{(2\\pi)^D}\\,F(z)\\,e^{z^\\mu\\nabla_\\mu}\\phi(x)\\nonumber\\\\\n\t\t\t\t\t\t\t\t\t\t\t &=& \\phi(x)\\int\\frac{d^D z}{(2\\pi)^D}\\,F(z)\\,\\phi(x+z)\\nonumber\\\\\n\t\t\t\t\t\t\t\t\t\t\t &\\stackrel{y:=z+x}{=}&\\phi(x)\\int\\frac{d^D y}{(2\\pi)^D}\\,F(y-x)\\,\\phi(y)\\,.\\label{ker}\n\\end{eqnarray}\nSpecifying the form factor $f(-k^2)$ determines the spectrum of the field. In general, the poles of the propagator correspond to the zeros of $f(-k^2)$ and to the poles of $F(z)$. This correspondence is straightforward for a massless dispersion relation $f(-k^2)=-k^{2n}$, where $F(z)\\propto\\delta^{(2n)}(z)$ and $(2n)$ denotes the derivative of order $2n$ of the delta. The derivative order of the delta is the order of the pole. Polynomial dispersion relations have a similar structure, e.g., $f(-k^2)=-k^2-a k^{2n}$ gives $F(z)\\propto \\delta^{(2)}(z)+a\\delta^{(2n)}(z)$. For $n=2$, this dispersion relation corresponds to one massive and one massless scalar mode, for a total of two double poles.\\footnote{In fact, $f^{-1}(-k^2)=-[k^2(1+ak^2)]^{-1}=-k^{-2}+(a^{-1}+k^2)^{-1}$. The second mode is a ghost (positive residue).} Furthermore, when $f(\\Box)$ is non-local the field spectrum depends on whether the form factor is entire or not. In the case of \\Eq{tpheom}, the propagator \n\\begin{equation}\\label{prof}\nf^{-1}(-k^2)=-\\frac{e^{-r_*k^2}}{k^2}\n\\end{equation}\nhas a massless double pole, while $F(z)\\propto (2r_*+z^2)\\exp[z^2\/(4r_*)]$ has a double massive pole. In the last two cases, the order and nature (massless or massive) of the particle poles and the poles of $F$ is less transparent, although their counting agrees.\n\nFrom this exercise, it should become clear that hiding infinitely many derivatives into integrals with non-trivial kernels such as \\Eq{ker}, or to transfer part of these derivatives onto the scalar potential and then converting them into integral operators, does not help in solving the Cauchy problem, since the two formulations are equivalent (on the space of real analytic functions \\cite{MoZ}). In \\cite{CMN3}, we complement this no-go result with its way out: If the kernel $F$ can be found by solving some finite-order differential equation extra with respect to the dynamical equations, then its contribution to the Cauchy problem becomes under full control.\n\nThe novelty brought in by the diffusion-equation method is that it allows one to go beyond the free theory and count the extra number of initial conditions. Surprisingly, in the scalar-field case this number is two, which sums to the two initial conditions of the free theory for a total of four. This result goes against the belief, implicitly endorsed in some literature, that the information from the free theory is complete. In particular, when one says that the number of initial conditions for solving the Cauchy problem of the theory \\Eq{fac} is two, one should specify that this is true only for the free, perturbative case.\n\nWe reach this conclusion in three steps: (i) counting the number of field degrees of freedom of the localized theory; (ii) specifying the number of initial conditions (in time) for each localized field; (iii) restricting our attention to the slice $r=\\beta r_*$ where the non-local dynamics is recovered, and proceeding with the counting thereon. In Lagrangian formalism, we saw that there are two independent localized fields, either the pair $\\Phi$ and $\\chi$ or the pair $\\varphi$ and $\\psi$. Consistently, the same result is obtained in Hamiltonian formalism, where there are two non-vanishing independent momenta $\\pi_\\varphi$ and $\\pi_\\psi$. Since the dynamics is second-order in spacetime derivatives, there are two initial conditions per field, for a total of four.\n\\begin{equation}\\label{dof1}\n\\parbox[c]{13cm}{{\\bf Number of degrees of freedom: scalar field.} \\emph{The localized real scalar field theory \\Eq{act}--\\Eq{locch2} in $D+1$ dimensions has two scalar degrees of freedom $\\Phi$ and $\\chi$. On the $r$-slice where the system is equivalent to the non-local real scalar field theory \\Eq{fac} in $D$ dimensions, the degree of freedom $\\chi$ is no longer independent. Consequently, the non-local theory has \\underline{one} non-perturbative scalar degree of freedom $\\phi$.}}\n\\end{equation}\n\\begin{equation}\\label{dofic}\n\\parbox[c]{13cm}{{\\bf Number of initial conditions: scalar field.} \\emph{The Cauchy problem on spacetime slices of the localized real scalar field theory \\Eq{act}--\\Eq{locch2} in $D+1$ dimensions is specified by four initial conditions $\\Phi(r,t_{\\rm i},{\\bf x})$, $\\dot\\Phi(r,t_{\\rm i},{\\bf x})$, $\\chi(r,t_{\\rm i},{\\bf x})$, $\\dot\\chi(r,t_{\\rm i},{\\bf x})$. As a consequence, the Cauchy problem of the non-local non-perturbative real scalar field theory \\Eq{fac} in $D$ dimensions is specified by \\underline{four} initial conditions $\\phi(t_{\\rm i},{\\bf x})$, $\\dot\\phi(t_{\\rm i},{\\bf x})$, $\\ddot\\phi(t_{\\rm i},{\\bf x})$, $\\dddot\\phi(t_{\\rm i},{\\bf x})$.}}\n\\end{equation}\n\nThe nature of the new degree of freedom $\\chi$ is quite peculiar. As we saw above with a diagonalization trick (used, for instance, also in \\cite{DeWo2}), this field is a ghost and, in fact, the Hamiltonian \\Eq{hamil} is unbounded from below.\n From the point of view of the $(D+1)$-dimensional localized system \\Eq{act}--\\Eq{locch2}, $\\chi$ arises as a Lagrange multiplier introduced to enforce the diffusion equation of $\\Phi$; $\\chi$ itself does not appear in its own equation of motion \\Eq{eomchi2}. Its $(D+1)$-dimensional dynamics, given by the equation of motion of $\\Phi$, is non-trivial (in Hamiltonian formalism, the momentum $\\pi_\\chi\\propto \\pi_\\psi$ does not vanish) but it only amounts to diffusion, equation \\Eq{eomP22}. Eventually, it turned up that it is associated with $\\Phi$ by the second-order derivative relation \\Eq{rstcon2}. From the point of view of the $D$-dimensional non-local system, $\\chi$ disappears because its diffusion is frozen at a given slice, and the dynamics is written solely in terms of $\\phi$, its derivatives and its potential. At this point, there is only one degree of freedom whose \\emph{perturbative classical} propagator \\Eq{prof} describes a non-ghost massive scalar mode. The potentially dangerous ghost mode in the $(D+1)$-dimensional system turns out to be non-dynamical in $D$-dimensions and in the \\emph{free} theory. \n\nIn the interacting non-local theory, $\\chi$ does play a part in the dynamics, but in the form of the potential for $\\phi$. Combined with equation \\Eq{tpheom}, the local condition \\Eq{rstcon2} explains in part the finite proliferation of degrees of freedom in the interacting case. Since \\Eq{rstcon2} implies $\\chi(r,x)=\\Box\\Phi(r,x)$ for all $r$, then from \\Eq{tpheom} one has\n\\begin{equation}\\label{usefll}\n\\chi[(\\beta-1)r_*,x]=\\Box\\Phi[(\\beta-1)r_*,x]=\\Boxe^{-r_*\\Box}\\Phi(\\beta r_*,x)=V'[\\Phi(\\beta r_*,x)]=V'[\\phi(x)]\\,.\n\\end{equation}\nIf $V\\propto\\phi^2$, then $\\chi[(\\beta-1)r_*,x]\\propto \\Phi(\\beta r_*,x)=\\phi(x)$ and there is no extra degree of freedom with respect to the $V=0$ case. For a cubic or higher-order polynomial, $\\chi[(\\beta-1)r_*,x]$ is not linearly equivalent to $\\phi$. Non-linearities can generate new degrees of freedom (a typical example is $f(R)$ gravity, which contains a hidden scalar mode apart from the graviton) but not in this case, since the field $\\chi$ is not dynamical on the $r=\\beta r_*$ slice where \\Eq{usefll} holds.\n\n\n\\subsection{Solutions}\\label{solu}\n\nSolutions of non-local theories can be categorized into perturbative and non-perturbative. Perturbative solutions can have two meanings, either as the solutions obtained when truncating the non-local operators to a finite order (a procedure we will not discuss here \\cite{EW1,MoZ,cuta2}) or as the solutions obtained, order by order, starting from the free theory and modeling interactions as a perturbative series \\cite{MoZ,EW1,JLM}. When all non-locality acts on interactions, the two meanings coincide. Non-perturbative solutions are all those solutions that cannot be reached in these ways and, in general, they constitute the great majority of all possible solutions of the system. The diffusion-equation method permits to get access precisely to these solutions with generic non-perturbative potential \\cite{roll,cuta2,cuta4,cuta5,cuta6,cuta8}. \n\nWhen introducing the condition \\Eq{rstcon1}, we commented on the fact that the identification of the localized dynamics with the non-local one could take place at any $r=\\tilde r$ slice, including at $r=\\tilde r=0$ where $\\Phi(0,x)=\\phi(x)$. However, for the sake of the construction of actual solutions this choice is not fortunate, since it corresponds to the initial condition of the heat kernel. In other words, setting the initial condition (in $r$) of the $(D+1)$-dimensional system to be the solution of the non-local system would take us back to the usual paradox with non-local dynamics, namely, that knowing all the infinite number of initial conditions (in time) $\\phi(t_{\\rm i},{\\bf x}),\\,\\dot\\phi(t_{\\rm i},{\\bf x}),\\,\\ddot\\phi(t_{\\rm i},{\\bf x}),\\,\\dots$ is tantamount to already knowing the Taylor expansion of the full solution around $t=t_{\\rm i}$. It is more logical, then, to impose \\Eq{rstcon1} (the non-local solution is the outcome of the diffusion from $r=0$ to $r=\\beta r_*$ rather than of anti-diffusion from $r=\\beta r_*$ to $r=0$) and to set the initial condition $\\Phi(0,x)$ in $r=0$ as something else. This ``something else'' can be most naturally recognized as the solution $\\phi_{\\rm loc}(x)$ of the \\emph{local} system obtained by setting $r_*=0$ in equations \\Eq{fac} and \\Eq{tpheom}:\n\\begin{equation}\\label{filoc}\n\\Phi(0,x)=\\phi_{\\rm loc}(x)\\,.\n\\end{equation}\nThen, the solution of the diffusion equation \\Eq{eomchi2} can be found in integral form in momentum space. Calling $-k^2$ the eigenvalue of the Laplace--Beltrami operator $\\Box$ and writing\n\\begin{equation}\n\\phi_{\\rm loc}(x)=\\int_{-\\infty}^{+\\infty}\\frac{d^D k}{(2\\pi)^D}\\,e^{-i k\\cdot x}\\tilde\\phi_{\\rm loc}(k)\\,,\n\\end{equation}\none has\n\\begin{equation}\n\\phi(x)=\\Phi(\\beta r_*,x)=e^{\\beta r_*\\Box}\\Phi(0,x)=\\int_{-\\infty}^{+\\infty}\\frac{d^D k}{(2\\pi)^D}\\,e^{-i k\\cdot x}e^{-\\beta r_*k^2}\\tilde\\phi_{\\rm loc}(k)\\,.\\label{soluz}\n\\end{equation}\nSince we know $\\phi_{\\rm loc}(x)$, we also know its Fourier transform $\\tilde\\phi_{\\rm loc}(k)$ and we can obtain the full non-local solution $\\phi(x)$. Examples of solutions of the scalar-field equation of motion \\Eq{tpheom} using the diffusion-equation method can be found in \\cite{cuta2} (on a Friedmann--Lema\\^{i}tre--Robertson--Walker (FLRW) cosmological background), \\cite{roll,cuta7} (Minkowski background, rolling tachyon of open string field theory), \\cite{cuta3,cuta6} (Minkowski and FLRW backgrounds, $V\\propto\\Phi^n$ and $V\\propto\\exp(\\lambda\\Phi)$), \\cite{cuta4,cuta5} (lump solutions on Minwkoski, FLRW and Euclidean backgrounds; kink solutions on Euclidean background), and \\cite{cuta8} (FLRW solutions in a scalar-tensor non-local theory). Solutions of $p$-adic models, corresponding to \\Eq{tpheom} without the $\\Box$ in the kinetic term, have been considered in \\cite{cuta4,cuta7}. In some of these cases, a diffusion equation with opposite sign of the diffusion operator has been used, in which case the representation \\Eq{soluz} may be ill-defined. This is not a problem, since there exist a more general integral form of the solution valid for any sign (see section 3.3 of \\cite{cuta7}).\n\nNote that, in general, convergence of the integral \\Eq{soluz} will require $\\beta>0$. Also, setting $\\beta=1$ in \\Eq{usefll} would yield $\\Box\\phi_{\\rm loc}=V'(\\phi)$, implying $\\phi_{\\rm loc}=\\phi$. To avoid this inconsistency, we exclude the value $\\beta=1$. Also, for any \\emph{given} potential $V(\\phi)$ and for a generic $0<\\beta<1$ the profile $\\phi(x)$ is not a solution to the non-local equation of motion, not even approximately. Therefore, what one usually finds is an approximate solution $\\phi(x)$ for a certain range of $x$. The actual value of $\\beta$ determines the limits of the $x$ range, since the profile typically depends on the combination $x^2\/(4\\beta r_*)$. For instance, an approximated solution valid at large $x$ requires $\\beta>1$. However, cases are known where the profile $\\phi(x)$ is an approximate solution for any $x$ (even small) with a very good degree of accuracy, which means that there exists a value of $\\beta$ such that the equation of motion is solved up to a maximal deviation of a few percent or less for some $x$, and with much greater accuracy everywhere else. These systems are related to (\\cite{roll,cuta4,cuta5}) or inspired by (\\cite{cuta2,cuta6}) string field theory. On the other hand, there may be special cases where $\\phi$ is an exact solution, but these in general require a specifically tailored potential. Some examples of this inverse problem are given in \\cite{cuta3}.\n\n\n\\subsection{Comparison with Tomboulis approach}\\label{deloc}\n\nAnother approach handling non-perturbative solutions was proposed after the diffusion-equation method by Tomboulis \\cite{Tom15}. Here, by a field redefinition one transfers non-local operators from the kinetic into the potential term, with a procedure analogous to that leading to \\Eq{ker}. The latter is then written as an integral kernel, as above. This type of ``delocalized'' hyperbolic partial integro-differential equations are characterized by the phenomenon, due to the smearing of the kernel in \\Eq{ker}, of ``spill-over'' (or delays) outside the standard causal cones of the local hyperbolic initial-value problem. Depending on the system, delays may be present only in the past or both in the past and in the future.\n\nComparing the diffusion-equation method with Tomboulis' delocalization (or delays) approach in classical theories, we find several similarities.\n\\begin{itemize}\n\\item Both recognize the central role of interactions to distinguish between local and non-local models.\n\\item Related to this, both agree also on the fact that, independently on whether one transfers non-locality from the kinetic term to interactions or not, it makes no sense to count the number of initial conditions just from the order of the kinetic term or by looking at any isolated part of the Lagrangian; in the limit of turning off the interactions, one may obtain the wrong answer. In this sense, the distinction between perturbative and non-perturbative solutions is not very useful in either method if one aims to make existence and uniqueness statements on the full dynamics.\n\\item The ill-defined concept of ``infinitely many initial conditions'' is traded with a boundary-value problem. In the diffusion-equation approach, the value of the $D$-dimensional field $\\phi(t_{\\rm i},{\\bf x})$ and all its derivatives $\\phi^{(n)}(t_{\\rm i},{\\bf x})$ at one time instant $t=t_{\\rm i}$ is replaced by a field configuration $\\Phi(r,t,{\\bf x})$ living in $D+1$ dimensions, evaluated at a certain slice $r=\\beta r_*$ in the extra direction. In Tomboulis' approach, one specifies one or more functions rather than field values at one instant in the past, if delays occur only in the past light cone. If delays occur also in the future cone, as in systems with Lorentz-invariant interactions, then analogous specifications of functions must be done for them. For these systems, the type of non-local kernel has the same spill-over at all sides of the causal cone and the number of specifications in the future is finite and equal to the number of specifications in the past. Thus, both methods predict a \\emph{finite, even number of conditions} (initial-value or boundary-value) for non-local scalar field theories. The specific prediction of the diffusion method is \\Eq{dof1}, for any non-quadratic potential.\n\\item Consequently, because solutions are determined by picking conditions on $r$-slices in one case and past-future delay specifications in the other case, there are no implicit choices nor hidden conditions in the construction of such solutions, which are therefore unique once the explicit conditions are specified. This solves the long-standing problem of non-local theories where proving the existence of a solution by a brute-force \\emph{Ansatz} does not imply, in the absence of any localization or delocalization method, its uniqueness \\cite{EW1}.\n\\end{itemize}\n\n\n\\subsection{Generalizing to \\texorpdfstring{$\\exp H(\\Box)$}{} operators}\\label{Hge}\n\nFinally, let us comment on an extension of the above procedure to a non-locality of the form $\\exp\\Box\\to\\exp H(\\Box)$ for some function $H$. In this case, one simply replaces $\\Box$ with $H(\\Box)$ in the Lagrange-multiplier equation \\Eq{locch2}. Everything else follows suit. However, the system \\Eq{act} is local and the Cauchy problem is well defined only if $H(\\Box)$ is a polynomial in the Laplace--Beltrami operator, $H(\\Box)=\\sum_{n=1}^N a_n\\Box^n$. In this case, the number of initial conditions increases from 4 to $4n$: the value of the scalars $\\Phi$ and $\\chi$ at the initial time plus their first $2n-1$ derivatives.\n\nFor entire functions $H(\\Box)$, the ``localized'' system would be non-local. It may still be possible to localize \\Eq{act} for special cases, for instance if $H(\\Box)=\\exp\\Box$. However, in the most general case the diffusion method is insufficient to deal with these non-localities different from a pure exponential, unless an extra convolution equation is added to the system \\cite{CMN3}.\n\n\n\\section{Non-local gravity: equations of motion}\\label{eoms}\n\nConsider the gravitational action\n\\begin{equation}\n\\boxd{S_g = \\frac{1}{2\\kappa^2}\\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda+G_{\\mu\\nu} \\, \\gamma(\\Box) \\,  R^{\\mu\\nu} \\right],\\label{nlffgb}}\n\\end{equation}\nwhere $\\gamma(\\Box)$ is a completely arbitrary form factor. In this section, we determine its dynamics in two ways. First, by a brute-force calculation, eventually specializable to the form factor \\Eq{fofag}. Second, by recasting the system in terms of an auxiliary field.\n\n\n\\subsection{Einstein equations: pure gravity}\n\nTo compute the Einstein equations for a generic form factor $\\gamma(\\Box)$, one must expand the latter in series of the Laplace--Beltrami operator $\\Box$,\n\\begin{equation}\\label{ggen}\n\\gamma=\\sum_{n=0}^{+\\infty}c_n\\Box^n\\,,\n\\end{equation}\nwhere $c_n$ are constants, and vary with respect to the metric. We couple \\Eq{nlffg} to matter minimally. Varying the total action $S=S_g+S_{\\rm m}$ with respect to the contravariant metric $g^{\\mu\\nu}$, the matter part is dispensed with by the usual definition of energy-momentum tensor\n\\begin{equation}\\label{emt}\nT_{\\mu\\nu} :=-\\frac{2}{\\sqrt{-g}}\\frac{\\delta S_{\\rm m}}{\\delta g^{\\mu\\nu}}\\,.\n\\end{equation}\nIn general, also matter fields will be non-local, but we do not consider their details here. The variations of curvature invariants and form factors are reported in appendix \\ref{app3} and the full derivation of the final result is given in appendix \\ref{app4}:\n\\begin{eqnarray}\n\\kappa^2 T_{\\mu\\nu} &=& (1+\\gamma\\Box) G_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}-\\frac{1}{2} g_{\\mu\\nu}\\,G_{\\sigma\\t}\\gamma R^{\\sigma\\t}+2G^\\sigma_{\\ (\\mu} \\gamma  G_{\\nu)\\sigma}+g_{\\mu\\nu}\\nabla^\\sigma\\nabla^\\t\\gamma G_{\\sigma\\t}\\nonumber\\\\\n&& -2\\nabla^\\sigma\\nabla_{(\\mu}\\gamma G_{\\nu)\\sigma}+\\frac12(G_{\\mu\\nu} \\gamma R+R\\gamma G_{\\mu\\nu})+\\Theta_{\\mu\\nu}(R_{\\sigma\\t},G^{\\sigma\\t})\\,,\\label{EinEq1}\n\\end{eqnarray}\nwhere the expression of $\\Theta_{\\mu\\nu}(R_{\\sigma\\t},G^{\\sigma\\t})$ is given by \\Eq{usef5a} for any form factor $\\gamma$. The non-local equation \\Eq{EinEq1} can be compared with similar ones found elsewhere \\cite{BCKM,Kos13}.\n\nFor the particular choice of form factor \\Eq{fofag},\n\\begin{eqnarray}\n(1+\\gamma\\Box) G_{\\mu\\nu}&=&e^{-r_*\\Box}G_{\\mu\\nu}\\,,\\\\\n\\Theta_{\\mu\\nu}(R_{\\sigma\\t},G^{\\sigma\\t}) &=&-\\int_0^{r_*}d q\\,\\bar\\Theta_{\\mu\\nu}[e^{-q\\Box} R_{\\sigma\\t},\\gamma_{r_*-q}(\\Box)G^{\\sigma\\t}]\\,,\\label{Theta2}\n\\end{eqnarray}\nwhere $\\bar\\Theta_{\\mu\\nu}$ is given by equation \\Eq{Theta}. The last expression, derived in appendix \\ref{app4}, is fully explicit.\n\n\n\\subsection{Einstein equations: auxiliary field}\\label{aux}\n\nAn alternative form of the Einstein equations makes use of an auxiliary field \\cite{BMS,BCKM}. Here, we apply this method to \\Eq{nlffg} for the first time. Consider the action\n\\begin{equation}\n\\tilde S[g,\\phi] = \\frac{1}{2\\kappa^2}\\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda-2\\phi^{\\mu\\nu} f_1(\\Box)\\, R_{\\mu\\nu}+\\left(\\phi^{\\mu\\nu}-\\frac{1}{D-2} g^{\\mu\\nu}\\phi\\right)\\,f_2(\\Box)\\,\\phi_{\\mu\\nu}\\right],\\label{nlff2}\n\\end{equation}\nwhere $\\phi_{\\mu\\nu}$ is a symmetric two-tensor, $\\phi=\\phi_\\sigma^{\\ \\sigma}=g^{\\mu\\nu}\\phi_{\\mu\\nu}$ is its trace and $f_{1,2}$ are some arbitrary form factors. The equations of motion for $\\phi_{\\mu\\nu}$ are given by the variation $\\delta\\tilde S\/\\delta\\phi^{\\mu\\nu}=0$:\n\\begin{equation}\\label{phieom}\n-f_1(\\Box)\\, R_{\\mu\\nu}+f_2(\\Box)\\,\\phi_{\\mu\\nu}-\\frac{1}{D-2} g_{\\mu\\nu}\\,f_2(\\Box)\\,\\phi=0\\,.\n\\end{equation}\nTaking the trace of \\Eq{phieom}, plugging it back and inverting for $\\phi_{\\mu\\nu}$, one sees that\n\\begin{equation}\\label{bG}\n\\phi_{\\mu\\nu}=[f_2^{-1}f_1](\\Box)\\,G_{\\mu\\nu}+\\lambda_{\\mu\\nu}\\,,\\qquad \\phi=-\\left(\\frac{D}{2}-1\\right)[f_2^{-1}f_1](\\Box)\\,R+\\lambda_\\mu^{\\ \\mu}\\,,\n\\end{equation}\nwhere $\\lambda_{\\mu\\nu}$ is the homogeneous solution of $f_2(\\Box)\\lambda_{\\mu\\nu}=0$. Using \\Eq{bG} and $f_2(\\Box)\\lambda_{\\mu\\nu}=0$ in \\Eq{nlff2} and integrating by parts, one gets the Lagrangian\n\\[\n2\\kappa^2\\tilde\\mathcal{L} =R-2\\Lambda-G_{\\mu\\nu}f_1f_2^{-1}f_1 R^{\\mu\\nu}-\\lambda_{\\mu\\nu} f_1 R^{\\mu\\nu}\\,.\n\\]\nComparing this with \\Eq{nlffg}, we conclude that $\\tilde S=S_g$ on shell provided\n\\begin{equation}\\label{ffg}\n[f_1f_2^{-1}f_1](\\Box)=-\\gamma(\\Box)\\,,\\qquad \\lambda_{\\mu\\nu}=0\\,.\n\\end{equation}\nThere are various possible choices for the form factors $f_1$ and $f_2$; physically they are all equivalent as long as \\Eq{ffg} holds. The simplest choice\n\\beta\nf_2=f_1=-\\gamma\n\\end{equation}\nfor an arbitrary form factor \\Eq{ggen} satisfies the first condition in \\Eq{ffg}, while only form factors with $c_0\\neq 0$ (i.e., those with trivial kernel) also guarantee that the second condition in \\Eq{ffg} is obeyed. In fact, $\\gamma(\\Box)=c_0+c_1\\Box+O(\\Box^2)$, so that $\\gamma(\\Box)\\lambda_{\\mu\\nu}=0$ if, and only if, $\\lambda_{\\mu\\nu}\\equiv 0$. The form factor \\Eq{fofag} is of this type, since $\\gamma_{r_*}(\\Box)=-r_*+(r_*^2\/2)\\Box+O(\\Box^3)$. In other words, there is no homogeneous solution we should worry about when recasting \\Eq{nlffg} as \\Eq{nlff2}, contrary to what happens when making field redefinitions in $f(\\Box^{-1}R)$ non-local gravity \\cite{NoOd,Kos08,MaMa,ZKSZ}.\n\nThus, \\Eq{nlff2} becomes\n\\begin{equation}\n\\boxd{\\tilde S[g,\\phi] = \\frac{1}{2\\kappa^2}\\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda+\\left(2R_{\\mu\\nu}-\\phi_{\\mu\\nu}+\\frac{1}{D-2} g_{\\mu\\nu}\\phi\\right)\\,\\gamma(\\Box)\\,\\phi^{\\mu\\nu}\\right].\\label{nlff3}}\n\\end{equation}\nIn appendix \\ref{app5}, we show that the covariant equations of motion of the theory \\Eq{nlff3} are\n\\begin{eqnarray}\n\\kappa^2 T_{\\mu\\nu} &=& G_{\\mu\\nu}+\\Box\\gamma \\phi_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}-\\frac12 g_{\\mu\\nu} X_{\\sigma\\t}\\gamma\\phi^{\\sigma\\t}+2\\phi_{(\\mu}^{\\ \\sigma}\\gamma\\phi_{\\nu)\\sigma}+g_{\\mu\\nu}\\nabla^\\sigma\\nabla^\\t \\gamma\\phi_{\\sigma\\t}\\nonumber\\\\\n\t\t\t\t\t\t\t\t&& -2\\nabla^\\sigma\\nabla_{(\\mu} \\gamma\\phi_{\\nu)\\sigma}-\\frac{1}{D-2}(\\phi_{\\mu\\nu}\\gamma\\phi+\\phi\\gamma\\phi_{\\mu\\nu})+\\Theta_{\\mu\\nu}(X_{\\sigma\\t},\\phi^{\\sigma\\t})\\,,\\label{eomnl1}\\\\\nX_{\\sigma\\t}&:=&2R_{\\sigma\\t}-\\phi_{\\sigma\\t}+\\frac{1}{D-2}g_{\\sigma\\t}\\phi\\,,\\label{X}\n\\end{eqnarray}\naccompanied by the equation of motion $\\delta \\tilde S[g,\\phi]\/\\delta\\phi^{\\mu\\nu}=0$ and its trace:\n\\begin{equation}\\label{eomnl2}\n\\phi_{\\mu\\nu}=G_{\\mu\\nu}\\qquad \\Rightarrow \\qquad \\phi=G=-\\frac{D-2}{2}R\\,,\\qquad X_{\\mu\\nu}=R_{\\mu\\nu}\\,.\n\\end{equation}\nWe call \\Eq{eomnl1} Einstein equations because they come from the variation of the metric and \\Eq{eomnl2} Einstein-like equations because they resemble the Einstein equations of general relativity, where $\\phi_{\\mu\\nu}$ plays the role of a stress-energy tensor. Notice from \\Eq{eomnl2} that the field $\\phi_{\\mu\\nu}$ is local and does not hide $1\/\\Box$ operators. This check \\emph{a posteriori} guarantees that the ordinary variational principle (where fields and their first derivatives vanish at infinity) has been correctly applied.\n\nConsistently, \\Eq{EinEq1} and \\Eq{eomnl1} agree on shell, i.e., when \\Eq{eomnl2} is used (see appendix \\ref{app6}).\n\n\n\\subsection{Brief remarks on causality}\n\nWhenever a factor $\\Box^{-1}$ appears in a non-local theory, causality may be in trouble. The line of reasoning is well known and relies on the definition of the inverse d'Alembertian through the Green equation \\Eq{bcK}, where one must specify a contour prescription for the Green function $\\mathcal{K}$. The main point is that even if the causal (retarded) propagator $\\mathcal{K}_{\\rm ret}(x-y)$ is used to define the $\\Box^{-1}$ operator, a variation of the action with respect to the fundamental fields always gives rise to the even combination\n\\begin{equation}\\nonumber\n\\mathcal{K}_{\\rm ret}(x-y)+\\mathcal{K}_{\\rm ret}(y-x)=:\\mathcal{K}_{\\rm ret}(x-y)+ \\mathcal{K}_{\\rm adv}(x-y)\\,.\n\\end{equation}\nThe retarded Green function is not even, and changing sign to its argument gives the advanced Green function $\\mathcal{K}_{\\rm ret}(y-x) = \\mathcal{K}_{\\rm adv}(x-y)$, which is anti-causal. Therefore, the equations of motion obtained from theories with non-localities of the type $\\Box^{-1}$ (typically, theories where the quantum effective action, not the classical one, is non-local) are necessarily acausal \\cite{BDFM}.\n\nHowever, this argument does not apply in our case because the non-localities we deal with do not need any prescription for the $\\Box^{-1}$ factor, as it always appears in a combination $\\gamma(\\Box)=c_0+c_1\\Box+O(\\Box^2)$ which is analytic when ``$\\Box =0$'' (in particular, $\\gamma_{r_*}(\\Box)=-r_*+(r_*^2\/2)\\Box+O(\\Box^3)$). In other words, in all the theories of quantum gravity with a fundamental non-locality, non-localities (at the level of the classical action, not of the quantum effective one) are always of the type \\Eq{effeb} with $f(0)=0$. As a consequence of this fact, the Green function associated with the non-local operator $\\gamma(\\Box)$ is symmetric.\n\nA one-dimensional example in flat space will further clarify the matter. The non-local operator containing $\\Box^{-1}$ is (\\ref{fofag}), which can be written as \\Eq{fofag2}. Its Green function $K(x-y)$ is the solution of the equation\n\\begin{equation}\\label{exerspace}\n\\gamma_{r_*}(\\Box_x)K(x-y)=-\\int_{0}^{r_*} d s\\, e^{- s\\Box_x} \\, K(x-y)= \\delta(x-y)\\,,\n\\end{equation}\nor, in momentum space, \n\\begin{equation}\\label{exermom}\n\\int_{0}^{r_*} d s\\, e^{s k^2} \\, \\tilde K(k)=-1\\,.\n\\end{equation}\nWhile the inverse of the $\\Box$ operator needs to be prescribed because the naive solution $1\/k^2$ of the Green equation does not define a tempered distribution, the solution of eq.\\ (\\ref{exermom}) does not need to be regularized, as its algebraic solution\n\\begin{equation} \\label{exersolk}\n\\tilde K (k) = - \\left[ \\int_{0}^{r_*} d s\\,e^{s k^2} \\right]^{-1} = - \\frac{k^2}{e^{r_* k^2}-1}\n\\end{equation}\nalready defines a tempered distribution. Its Fourier transform cannot be written in closed form but is very well behaved and, most importantly, is manifestly symmetric under the exchange $x\\leftrightarrow  y$,\\footnote{In the $\\Box^{-1}$ case, the regularization procedure needed to define $1\/k^2$ as a tempered distribution prevents the Green function to be symmetric under the exchange $x \\leftrightarrow y$, leading to the known mismatch between causality and symmetry of the propagator \\cite{BDFM}.}\n\\begin{equation}\n\\label{exersolx}\nK(x-y)= -\\frac{1}{\\pi} \\int_0^\\inftyd k\\, \\frac{k^2\\, \\cos [k (x-y)]}{e^{r_* k^2}-1}\\,.\n\\end{equation}\n\t\nThe fact that the $\\Box^{-1}$ operator in the form factor $\\gamma(\\Box)$ of fundamentally non-local quantum gravity does not introduce causality breaking is not, by itself, a guarantee of causality of these theories but, at least, it shows that standard arguments against causality, plaguing effective non-local field theories, do not apply in our case. The problem of causality in fundamentally non-local theories is subtle \\cite{Tom15} and might not admit an all-or-nothing solution, in the sense that the theory might retain macrocausality \\cite{GiMo} while including acceptable violations of microcausality. This interesting possibility will be explored elsewhere.\n\n\n\\section{Localization of non-local gravity}\\label{locnlg}\n\nIn section \\ref{eoms}, we started from a gravitational action $S_g[g(x)]$ and introduced an auxiliary tensor field $\\phi_{\\mu\\nu}$ so that we could rewrite the original action as a functional of this field and the metric, $S_g[g(x)]= \\tilde S[g_{\\mu\\nu}(x),\\phi_{\\mu\\nu}(x)]$, where the gravitational part of $\\tilde S$ is given by the integral of \\Eq{nlff3}. In this section, we will construct a functional $\\mathcal{S}_g[g_{\\mu\\nu}(x),\\Phi_{\\mu\\nu}(r,x),\\chi_{\\mu\\nu}(r,x),$ $\\lambda_{\\mu\\nu}(r,x)]$ representing a system living in $D+1$ dimensions and local in spacetime coordinates. Here we show that the two systems coincide at a section $r=\\beta r_*$ in the $(D+1)$-dimensional space,\n\\begin{equation}\\label{sss}\n\\mathcal{S}_g[g_{\\mu\\nu}(\\beta r_*,x),\\Phi_{\\mu\\nu}(\\beta r_*,x),\\chi_{\\mu\\nu}(\\beta r_*,x)]=\\tilde S_g[g_{\\mu\\nu}(x),\\phi_{\\mu\\nu}(x),\\chi_{\\mu\\nu}(x)]=S_g[g_{\\mu\\nu}(x)]\\,,\n\\end{equation}\nwhere the equalities are meant to be valid on-shell, i.e., at the level of the dynamics. This statement, which can be immediately extended to actions that include also matter fields, is the extension to gravity of the results of section \\ref{scala} \\cite{cuta3} for a scalar field in Minkowski spacetime.\n\n\n\\subsection{Localized action}\\label{loac}\n\nWe apply the procedure illustrated in section \\ref{scala} to the gravitational theory with form factor \\Eq{fofag}. We have seen that \\Eq{nlffg} is physically equivalent to the action \\Eq{nlff3}, which can also be written as\n\\begin{equation}\\label{phint}\n\\tilde S[g,\\phi] = \\frac{1}{2\\kappa^2}\\int d^D x \\sqrt{-g}\\,\\left[R-2\\Lambda-\\int_0^{r_*}d s\\left(2R_{\\mu\\nu}-\\phi_{\\mu\\nu}+\\frac{1}{D-2} g_{\\mu\\nu}\\phi\\right)e^{-s\\Box}\\phi^{\\mu\\nu}\\right]\n\\end{equation}\nthanks to \\Eq{fofag2}. Using \\Eq{phint} instead of \\Eq{nlffg} will allow us to enforce the diffusion equation to $(D+1)$-dimensional fields without facing the commutation problem \\Eq{probl} mentioned in the introduction and the fact that the metric field does not obey a linear diffusion equation. This problem is solved by letting only auxiliary fields diffuse, while the gravitational field does not diffuse at all: it is a dynamical field living in a fixed $r=\\beta r_*$ slice. Therefore, an interesting difference with respect to the scalar-field case is that here some fields (which we will call $\\Phi_{\\mu\\nu}(r,x)$ and $\\chi_{\\mu\\nu}(r,x)$) are free to evolve in the whole $(D+1)$-dimensional bulk, while others (the metric $g_{\\mu\\nu}(x)$ and the Ricci tensor $R_{\\mu\\nu}(x)$ derived from it) are confined into the slice where the higher-dimensional localized system is made equivalent to the non-local one. This configuration strongly reminds us of braneworld scenarios where $r$ is the direction transverse to a brane at $r=\\beta r_*$ and the Einstein--Hilbert Lagrangian contributes with a term $[R(x)-2\\Lambda]\\,\\delta(r-\\beta r_*)$. Another possibility, which we will follow from now on and yields the same result, is to consider an $r$-dependent $g_{\\mu\\nu}(r,x)$ dynamically constrained to be constant along $r$:\n\\begin{equation}\n\\boxd{\\mathcal{S}_g=\\frac{1}{2\\kappa^2}\\intd^Dx\\,d r\\,\\sqrt{-g(r)}\\left(\\mathcal{L}_R+\\mathcal{L}_\\Phi+\\mathcal{L}_\\chi+\\mathcal{L}_\\lambda\\right),\\label{gactcm}}\n\\end{equation}\n\\begin{eqnarray}\n\\mathcal{L}_R\\!\\! &=&\\!\\! R(r)-2\\Lambda\\,,\\\\\n\\mathcal{L}_\\Phi\\!\\! &=&\\!\\! -\\int_0^{r_*}d s\\,\\left[2\\mathcal{R}_{\\mu\\nu}(r)-\\Phi_{\\mu\\nu}(r)+\\frac{1}{D-2}g_{\\mu\\nu}(r)\\Phi(r)\\right]\\Phi^{\\mu\\nu}(r-s)\\,,\\label{gact2cm}\\\\\n\\mathcal{L}_\\chi\\!\\! &=&\\!\\! -\\int_0^{r_*}d s\\int_0^{s}d q\\,\\chi_{\\mu\\nu}(r-q)(\\partial_{r'}-\\Box)\\Phi^{\\mu\\nu}(r')\\,,\\label{difeqgcm}\\\\\n\\mathcal{L}_\\lambda\\!\\!  &=&\\!\\! \\lambda_{\\mu\\nu}(r)\\,\\partial_r g^{\\mu\\nu}(r)\\,,\\label{lagcm}\n\\end{eqnarray}\nwhere the metric is $r$-dependent just like the other fields, its Ricci curvature is denoted with a curly $\\mathcal{R}_{\\mu\\nu}$, we introduced a Lagrange multiplier $\\lambda_{\\mu\\nu}$, we omitted the $x$-dependence everywhere, $\\Phi= g^{\\mu\\nu}\\Phi_{\\mu\\nu}$ is the trace of the symmetric rank-2 tensor $\\Phi_{\\mu\\nu}$, and\n\\begin{equation}\\label{rprime2}\nr'=r+q-s\\,,\n\\end{equation}\nso that $\\partial_{r'}=\\partial_q$ in \\Eq{difeqgcm}. All tensorial indices still run from 0 to $D-1$, so that the theory \\Eq{gactcm} is a fake $D+1$ system, which is not $(D+1)$-covariant anyway due to the diffusion equation term. In analogy with \\Eq{X}, it will be convenient to define the tensorial combination\n\\begin{equation}\\label{Xr}\nX_{\\mu\\nu}(r):=2\\mathcal{R}_{\\mu\\nu}(r)-\\Phi_{\\mu\\nu}(r)+\\frac{1}{D-2}g_{\\mu\\nu}(r)\\Phi(r)\\,.\n\\end{equation}\n\nComparing with the scalar field theory \\Eq{act}, there are four major differences one should note: (a) all fields are rank-2 tensors; (b) there is an extra integration $-\\int_0^{r_*}d s$ accounting for the more complicated form factor \\Eq{fofag2}; (c) because of (b), the $q$-integral in \\Eq{difeqgcm} is nested, while in \\Eq{locch2} it is definite; (d) because of (c), \\Eq{rprime2} replaces the scalar-field parameter \\Eq{rprime}.\n\n\n\\subsection{Localized equations of motion}\\label{loeom}\n\nIn intermediate steps of the derivation, we will omit the $x$-dependence in all fields as well as the discussions of section \\ref{scala} on integration domains. The equation of motion for $\\lambda_{\\mu\\nu}$ establishes the independence of the metric from the extra coordinate $r$:\n\\begin{equation}\\label{eomla}\n0=\\frac{\\delta\\mathcal{S}_g}{\\delta\\lambda^{\\mu\\nu}(\\bar r)}=\\partial_{\\bar r} g_{\\mu\\nu}(\\bar r,x)\\qquad\\Rightarrow\\qquad g_{\\mu\\nu}(r,x)=g_{\\mu\\nu}(x)\\,.\n\\end{equation}\nTherefore, in the following we can apply this equation on shell and ignore any change (shift, integration, and so on) in the $r$-argument of the metric, of the Laplace--Beltrami operator, and of curvature invariants, unless stated otherwise. The equations of motion turn out to be\n\\begin{eqnarray}\n0&=&(\\partial_r-\\Box)\\Phi_{\\mu\\nu}(r,x)\\,,\\label{difPg}\\\\\n0&=&(\\partial_r-\\Box)\\chi_{\\mu\\nu}(r,x)\\,,\\label{chimnde}\\\\\n0&=&\\int_0^{r_*}d s\\left[X_{\\mu\\nu}(\\bar r-s)+X_{\\mu\\nu}(\\bar r+s)-2\\mathcal{R}_{\\mu\\nu}(r-s)+\\chi_{\\mu\\nu}(\\bar r-s)-\\chi_{\\mu\\nu}(\\bar r+s)\\right]\\,,\\label{uff}\\\\\n\\kappa^2T_{\\mu\\nu} &=& G_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}-\\int_0^{r_*}d s\\left\\{\\vphantom{\\frac{1}{D-2}}-\\frac12\\,g_{\\mu\\nu}X_{\\sigma\\t}(r)\\Phi^{\\sigma\\t}(r-s)+2\\Phi_{\\sigma(\\mu}(r)\\Phi_{\\nu)}^{\\ \\ \\sigma}(r-s)\\right.\\nonumber\\\\\n&&+\\Box\\Phi_{\\mu\\nu}(r-s)+g_{\\mu\\nu}\\nabla^\\sigma\\nabla^\\t\\Phi_{\\sigma\\t}(r-s)-2\\nabla^\\sigma\\nabla_{(\\mu}\\Phi_{\\nu)\\sigma}(r-s)\\nonumber\\\\\n&& \\left.-\\frac{1}{D-2}\\left[\\Phi_{\\mu\\nu}(r)\\Phi(r-s)+\\Phi(r)\\Phi_{\\mu\\nu}(r-s)\\right]-\\int_0^{s}d q\\,\\bar\\Theta_{\\mu\\nu}[\\chi_{\\sigma\\t}(r-q),\\Phi^{\\sigma\\t}(r+q-s)]\\right\\}\\!.\\nonumber\\\\ \\label{lasto}\n\\end{eqnarray}\n\nLet us see where they come from. The equation of motion for $\\chi_{\\mu\\nu}$ is\n\\begin{eqnarray}\n\\hspace{-.8cm}0 &=& \\frac{\\delta\\mathcal{S}_g}{\\delta\\chi^{\\mu\\nu}(\\bar r)}=-\\int d r \\int_0^{r_*}d s\\int_0^{s}d q\\,\\delta(r-q-\\bar r)(\\partial_{r'}-\\Box)\\Phi_{\\mu\\nu}(r')\\nonumber\\\\\n  &\\stackrel{\\textrm{\\tiny \\Eq{eomla}}}{=}& -\\int_{\\bar r}^{\\bar r+r_*} d r\\int_0^{r_*}d s\\,(\\partial_{r'}-\\Box)\\Phi_{\\mu\\nu}(r')\\Big|_{r'=2r-\\bar r-s}\\nonumber\\\\\n  &=&-\\int_0^{r_*}d\\rho\\int_0^{r_*}d s\\,(\\partial_{r'}-\\Box)\\Phi_{\\mu\\nu}(r')\\Big|_{r'=2\\rho+\\bar r-s}\\,,\\label{inter2g}\n\\end{eqnarray}\nwhere we first integrated in $q$, then restricted the integration in $r$ from the condition $0<q=r-\\bar r<s<r_*$, and then made a reparametrization $\\rho=r-\\bar r$. Since $\\bar r$ and hence $r'$ in the argument of the integrand is arbitrary, we get the diffusion equation \\Eq{difPg}. Integrating first in $r$ or, after using \\Eq{intint}, in $s$ would yield the same result. \n\nTo obtain the equation of motion for $\\Phi_{\\mu\\nu}$, we note that\n\\begin{eqnarray}\n\\mathcal{L}_{\\chi}&\\stackrel{\\textrm{\\tiny \\Eq{eomla}}}{=}& \\int_0^{r_*}d s\\int_0^{s}d q\\,\\left\\{-\\partial_q[\\chi_{\\mu\\nu}(r-q)\\Phi^{\\mu\\nu}(r')]+\\Phi^{\\mu\\nu}(r')(\\partial_{r'}+\\Box)\\chi_{\\mu\\nu}(r-q)\\right\\}\\nonumber\\\\\n&=&-\\int_0^{r_*}d s\\left[\\chi_{\\mu\\nu}(r-s)\\Phi^{\\mu\\nu}(r)-\\chi_{\\mu\\nu}(r)\\Phi^{\\mu\\nu}(r-s)\\right]\\nonumber\\\\ \n&&+\\int_0^{r_*}d s\\int_0^{s}d q\\,\\Phi^{\\mu\\nu}(r')(\\partial_{r'}+\\Box)\\chi_{\\mu\\nu}(r-q),\\label{inpa2g}\n\\end{eqnarray}\nwhere we used \\Eq{eomla} to make $\\sqrt{-g}$ slide through integration by parts (or, more precisely, we omitted terms that vanish on shell). This expression is the doubly integrated tensor-field analogue of \\Eq{inpa2}. Then,\n\\begin{eqnarray}\n0&=&\\frac{\\delta\\mathcal{S}_g}{\\delta\\Phi^{\\mu\\nu}(\\bar r)}\\nonumber\\\\\n&=&-\\int_0^{r_*}d s\\left[X_{\\mu\\nu}(\\bar r-s)+X_{\\mu\\nu}(\\bar r+s)-2\\mathcal{R}_{\\mu\\nu}(r-s)+\\chi_{\\mu\\nu}(\\bar r-s)-\\chi_{\\mu\\nu}(\\bar r+s)\\right]\\nonumber\\\\\n&&+\\int_0^{r_*}d s\\int_0^{s+\\bar r} d r\\,(\\partial_{\\bar r}+\\Box)\\chi_{\\mu\\nu}(2r -\\bar r-s)\\,.\\label{inter22g}\n\\end{eqnarray}\nArbitrariness of $\\bar r$ implies the diffusion equation \\Eq{chimnde} and we are left with the equation of motion \\Eq{uff}. We can easily find special solutions of this equation. For instance, it vanishes identically if we set the integrand to be zero. As in the scalar-field case, we can satisfy this equation by imposing certain constraints on the fields. The only one that respects $s$-shifts is\n\\begin{equation}\\label{Fmu}\n\\chi_{\\mu\\nu}(r,x)=X_{\\mu\\nu}(r,x)\\stackrel{?}{=}\\mathcal{R}_{\\mu\\nu}(r,x)\\,.\n\\end{equation}\nTherefore, just like in the scalar-field case, there is a condition imposed by hand such that one or more equations of motions are solved.\\footnote{Other types of solution can be found by factorizing the $r$-dependence in $X_{\\mu\\nu}$ and $\\chi_{\\mu\\nu}$. In fact, \\Eq{uff} is of the form $\\int_0^{r_*}d s\\,[A_{\\mu\\nu}(r+s;x)+B_{\\mu\\nu}(r-s;x)]=2R_{\\mu\\nu}(x)$. Writing $A_{\\mu\\nu}(r+s;x)=a(r+s)\\,a_{\\mu\\nu}(x)$ and $B_{\\mu\\nu}(r-s;x)=b(r-s)\\,b_{\\mu\\nu}(x)$ with $a_{\\mu\\nu}+b_{\\mu\\nu}=2R_{\\mu\\nu}$, the relation $\\int_0^{r_*}d s\\,[a(r+s)+b(r-s)]=1$ can be solved by linear or trigonometric functions. The diffusion equations \\Eq{difPg} and \\Eq{chimnde} then become eigenvalue equations of the type $\\Box \\Phi_{\\mu\\nu}\\propto \\Phi_{\\mu\\nu}$. These solutions, which were called ``stationary'' in \\cite{cuta8} since they do not diffuse non-trivially, are still popular because they allow to solve the non-local cosmological equations of motion directly \\cite{BKM1,KSKS}, although they are very limited tools when attempting to solve more general non-local systems \\cite{cuta5,cuta7,cuta8}.} We recognize some similarities between the scalar and the gravity case. In the scalar example, \\Eq{eomP22} held and one checked that \\Eq{eomP12} reproduced \\Eq{tpheom} provided equation \\Eq{rstcon20}, $\\chi(r,x) = \\Box\\Phi(r,x)$, held. In the gravitational case, the analogue of the diffusion equation \\Eq{eomP22} for $\\chi$ is simply \\Eq{chimnde} for $\\chi_{\\mu\\nu}$, while the analogue of \\Eq{rstcon20} is \\Eq{Fmu}. \n\nThe similarities do not stop here. In fact, both \\Eq{rstcon20} and \\Eq{Fmu} are second-order derivative relations, $\\chi\\sim \\partial^2\\Phi$ and $\\chi_{\\mu\\nu}\\sim \\partial^2 g_{\\mu\\nu}$. Also, in the gravity case there are two auxiliary fields instead of one, $\\Phi_{\\mu\\nu}(r,x)$ and $\\chi_{\\mu\\nu}(r,x)$, but only the latter has been introduced exclusively in the context of the localized system, just like the scalar $\\chi$ in section \\ref{ized}. $\\Phi_{\\mu\\nu}$ is ``auxiliary'' only because, on the slice $r=\\beta r_*$, it is equivalent to the field $\\phi_{\\mu\\nu}$ introduced to recast the non-local gravitational theory in a convenient way:\n\\begin{equation}\\label{slic}\n\\Phi_{\\mu\\nu}(\\beta r_*,x)=\\phi_{\\mu\\nu}(x)\\,.\n\\end{equation}\n\nThe last parallelism we can draw with the scalar theory is between the condition $\\chi=\\Box\\Phi$, imposed to match the equation of motion of the localized scalar field with the non-local equation of motion, and equation \\Eq{Fmu}, which does exactly the same service albeit in a subtler way. In fact, \\Eq{Fmu} is needed to satisfy one of the dynamical localized equations, but in retrospective it cannot be valid for all $r$ because of \\Eq{eomla}: on shell, the rightmost member is independent of $r$. To put it in other words, we reached \\Eq{Fmu} only because we took advantage of the pleonastic $r-s$ dependence of the term $\\mathcal{R}_{\\mu\\nu}$. This is telling us that we should replace \\Eq{Fmu} with a condition valid only on the slice $r=\\beta r_*$:\n\\begin{equation}\\label{Fmu2}\n\\boxd{\\chi_{\\mu\\nu}(\\beta r_*,x)=X_{\\mu\\nu}(\\beta r_*,x)=R_{\\mu\\nu}\\qquad\\Rightarrow\\qquad \\Phi_{\\mu\\nu}(\\beta r_*,x)=G_{\\mu\\nu}\\,.}\n\\end{equation}\nThis is the analogue of \\Eq{rstcon2}, valid only at $r=\\beta r_*$. Just like in the scalar system, the auxiliary field $\\chi_{\\mu\\nu}$ is related to the field(s) of the non-local theory by a relation (linear in the Ricci tensor, non-linear in the metric) involving a finite number of derivatives.\n\nOn the $r=\\beta r_*$ slice, the localized and non-local system coincide. Equations \\Eq{Xr}, \\Eq{slic} and \\Eq{Fmu2} imply together \\Eq{eomnl2}. All pieces of the puzzle match beautifully. The last one is the equations of motion for the metric, which is \\Eq{lasto} (see appendix \\ref{app7}). It is immediate to check that this expression agrees with \\Eq{nlff3} on the $r=\\beta r_*$ slice. Using \\Eq{difPg}, we can recast all the $s$ integrals in \\Eq{lasto} as non-local form factors, $-\\int_0^{r_*} d s\\,\\Phi_{\\sigma\\t}(r-s)=\\gamma_{r_*}(\\Box)\\Phi_{\\sigma\\t}(r)$. Then, \\Eq{slic} does the rest of the job. Also, from \\Eq{chimnde} and \\Eq{Fmu2}, the last term in \\Eq{lasto} reads\n\\begin{eqnarray}\n\\int_0^{r_*}\\!d s\\int_0^{s}\\!d q\\,\\bar\\Theta_{\\mu\\nu}[e^{-q\\Box}\\chi_{\\sigma\\t}(r),e^{(q-s)\\Box}\\Phi^{\\sigma\\t}(r)] &\\stackrel{r=\\beta r_*}{=}& \\int_0^{r_*}\\!d s\\int_0^{s}\\!d q\\,\\bar\\Theta_{\\mu\\nu}\\left[e^{-q\\Box}X_{\\sigma\\t},e^{(q-s)\\Box}\\Phi^{\\sigma\\t}\\right]\\nonumber\\\\\n&\\stackrel{\\textrm{\\tiny \\Eq{app1d2}}}{=}& \\Theta_{\\mu\\nu}(X_{\\sigma\\t},\\phi^{\\sigma\\t}) =\\Theta_{\\mu\\nu}(R_{\\sigma\\t},\\phi^{\\sigma\\t})\\,,\\nonumber\n\\end{eqnarray}\nthus recovering the last and most complicated piece of \\Eq{eomnl1}. Considering the massive effort it takes to derive the non-local Einstein equations \\Eq{EinEq1} and \\Eq{eomnl1}, the advantage of the diffusion-equation method to write down the dynamics is evident. Here we do not have to deal with the variation of form factors and with the non-commutation rules involving the $\\Theta_{\\mu\\nu}$ and $\\vartheta_{\\mu\\nu}$ functions. Despite being as complicated as the second-order relation \\Eq{Theta}, the function $\\bar\\Theta_{\\mu\\nu}$ is not nearly as messy as its non-local counterpart \\Eq{Theta2app}.\n\n\n\\subsection{Initial conditions and degrees of freedom}\\label{lodof}\n\nWe are finally in the position to discuss the Cauchy problem of non-local gravity. The localized equations of motion of the previous sub-section are second order in spacetime derivatives for all fields, so that there are only six initial conditions to be specified (``initial'' in spacetime time): $g_{\\mu\\nu},\\dot g_{\\mu\\nu},\\Phi_{\\mu\\nu},\\dot\\Phi_{\\mu\\nu},\\chi_{\\mu\\nu},\\dot\\chi_{\\mu\\nu}$. Using the $r=\\beta r_*$ conditions \\Eq{Fmu2}, one can see that $\\phi_{\\mu\\nu}\\sim G_{\\mu\\nu}\\sim \\partial^2 g_{\\mu\\nu}$ and $\\chi_{\\mu\\nu}\\sim R_{\\mu\\nu}\\sim \\partial^2 g_{\\mu\\nu}$ in the non-local system, so that, overall, one has to specify only four derivatives of the metric. This is a striking resemblance with the scalar-field case, where we had to specify the same number of initial conditions once the auxiliary field was removed. \n\nIt is well known that the perturbative degrees of freedom are finite, as one can see from the poles of the graviton propagator \\cite{Mod1,BGKM,Edh18}. Here we can go beyond that result and make a fully non-perturbative counting. For the graviton there are $D(D+1)\/2-D-D=D(D-3)\/2$ polarization modes in $D$ dimensions (a symmetric rank-2 tensor with $D(D+1)\/2$ component that are not independent due to $D$ Bianchi identities and $D$ diffeomorphisms), both in the localized and in the non-local system. Each of the localized rank-2 symmetric tensor fields $\\Phi_{\\mu\\nu}$ and $\\chi_{\\mu\\nu}$ has $D(D+1)\/2$ degrees of freedom, so that the total number of degrees of freedom in the localized system is $D(3D-1)\/2$, i.e., 22 in $D=4$. This number is not important, however, because it is greatly reduced in the slice $r=\\beta r_*$. Therein, $\\phi_{\\mu\\nu}=G_{\\mu\\nu}$ has $D(D+1)\/2-D=D(D-1)\/2$ degrees of freedom; the $-D$ comes from the fact that, on shell, the Bianchi identities imply the transverse condition $\\nabla^\\mu\\Phi_{\\mu\\nu}=0$. Also, $\\phi_{\\mu\\nu}$ should be regarded as independent of the metric, for the reason that it satisfies non-trivial dynamical equations. The fate of $\\chi_{\\mu\\nu}=R_{\\mu\\nu}$ is similar to the auxiliary field $\\chi$ of the scalar system and consists in getting out of the game on the slice where the localized system reproduces the non-local dynamics. What happens is that $\\chi_{\\mu\\nu}$ depends on the other fields, since from the definition \\Eq{Eiten} of the Einstein tensor $\\chi_{\\mu\\nu}=\\phi_{\\mu\\nu}-g_{\\mu\\nu}\\phi\/(D-2)$. Therefore, the total number of degrees of freedom of the non-local gravitational system is $D(D-3)\/2+D(D-1)\/2=D(D-2)$.\n\nThe central results of this paper can be summarized as follows.\n\\begin{equation}\\label{dof3}\n\\parbox[c]{13cm}{{\\bf Number of degrees of freedom: gravity.} \\emph{The localized gravitational theory \\Eq{gactcm}--\\Eq{difeqgcm} in $D+1$ dimensions has $D(3D-1)\/2$ degrees of freedom. As a consequence, the non-local gravitational theory \\Eq{nlffg} has \\underline{$D(D-2)$} non-perturbative degrees of freedom, amounting to \\underline{eight} in $D=4$ dimensions.}}\n\\end{equation}\n\\begin{equation}\\label{dof2}\n\\parbox[c]{13cm}{{\\bf Number of initial conditions: gravity.} \\emph{The Cauchy problem on spacetime slices of the localized gravitational theory \\Eq{gactcm}--\\Eq{difeqgcm} in $D+1$ dimensions is specified by six initial conditions $g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$, $\\dot g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$, $\\Phi_{\\mu\\nu}(r,t_{\\rm i},{\\bf x})$, $\\dot\\Phi_{\\mu\\nu}(r,t_{\\rm i},{\\bf x})$, $\\chi_{\\mu\\nu}(r,t_{\\rm i},{\\bf x})$, $\\dot\\chi_{\\mu\\nu}(r,t_{\\rm i},{\\bf x})$. As a consequence, the non-local non-perturbative gravitational theory \\Eq{nlffg} is specified by \\underline{four} initial conditions $g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$, $\\dot g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$, $\\ddot g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$, $\\dddot g_{\\mu\\nu}(t_{\\rm i},{\\bf x})$.}}\n\\end{equation}\n\nWe have not checked whether $\\phi_{\\mu\\nu}$ can be further decomposed into a spin-2 massive ghost particle with $D(D-1)\/2-1$ degrees of freedom and a scalar field, as done in higher-order Stelle gravity \\cite{Ste77,Ste78}. Ghost modes are absent at the perturbative level, as proven explicitly in \\cite{Mod1,MoRa1,MoRa2}, but their presence at the non-perturbative level remains an open question. We will leave this interesting problem, together with the existence or avoidance of ghosts in the non-local theory \\Eq{nlffg}, for the future.\n\nAt any rate, we can state that the localized theory does not generate any \\emph{extra} non-local ghost problem. It is useful to compare first the non-local theory \\Eq{nlffg} with another non-local model employed in cosmology, where the Einstein--Hilbert Lagrangian is modified by a term $R\\to \\mathcal{L}\\propto R[1+f(\\Box^{-1}R)]$ \\cite{DeWo1}. Auxiliary fields can be introduced so that the Lagrangian $\\tilde\\mathcal{L}\\propto R[1+f(\\Phi)]+\\Psi(\\Box\\Phi-R)$ replicates on shell the dynamics of the original system \\cite{NoOd}. However, this ``localized'' version is not completely equivalent to the former because the equation of motion $\\Box\\Phi-R=0$ of the Lagrange multiplier $\\Psi$ is used to obtain $\\Phi$ as a non-local function of $R$. The problem is that the solution of this relation is of the form $\\Phi=\\Box^{-1}R+\\lambda$, where $\\lambda$ is a scalar mode obeying the homogeneous equation $\\Box\\lambda=0$ \\cite{Kos08}. This extra mode is responsible for extending the space of solutions to dynamics not admitted by the original non-local system \\cite{Kos08}. Also, it makes an otherwise immaterial ghost degree of freedom dynamical: $\\Psi\\Box\\Phi\\to-\\partial_\\mu\\Psi\\partial^\\mu\\Phi=-(1\/4)\\partial_\\mu(\\Psi+\\Phi)\\partial^\\mu(\\Psi+\\Phi)+(1\/4)\\partial_\\mu(\\Psi-\\Phi)\\partial^\\mu(\\Psi-\\Phi)$. Suitable conditions on $f$, found along the same lines of ghost constraints in $f(R)$ or higher-order theories \\cite{KaSSo,KaSo,NuSo2,Chi05,DHT,CdD}, remove this ghost \\cite{DeWo2}. Coming back to the theory studied in the present paper, the localized version \\Eq{gactcm}--\\Eq{difeqgcm} is not a field redefinition of the model \\Eq{nlffg}: it is a different system living in a different number of dimensions that coincides with the non-local system only at a particular slice along the $r$ direction. A second important difference which we already had occasion to appreciate in section \\ref{eoms} is that our localization does not entail any homogeneous mode. This means that the ghost mode arising from the mixed kinetic term in \\Eq{difeqgcm} and originated by a Lagrange multiplier does not propagate on the $r=\\beta r_*$ slice.\n\n\n\\section*{Acknowledgments}\n\n\\noindent G.C.\\ is under a Ram\\'on y Cajal contract and thanks Terry Tomboulis for useful e-mail discussions. G.C.\\ and L.M.\\ are supported by the MINECO I+D grants FIS2014-54800-C2-2-P and FIS2017-86497-C2-2-P.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{section_intro}\n\nLet $T$ be a complete discrete valuation ring with fraction field $K$\nand residue field $k$. Let $X$ be a smooth, projective, geometrically \nintegral curve over $K$. Let $F=K(X)$ be the function field of $X$ and let $t$ be an uniformizing parameter of $T$.\n Under a mild assumption on the characteristic $p$\n of the residue field of $T$, we prove \n a conjecture by Colliot-Th\\'el\\`ene, Suresh and the second\n author \\cite[conjecture 1]{CTPS}.\n \n \n \\begin{stheorem} \\label{thm_patching0}\n (see Th. \\ref{thm_patching}) Let $G$ be a reductive \n $F$--group and assume that $p$ does not divide the order of the automorphism group of the absolute root system of \n $G_{ad}$.\n Let $Z$ be a twisted flag $F$--variety of $G$.\n Then $Z(F) \\not = \\emptyset$ if and only if $Z(F_v) \\not = \\emptyset$\n for all discrete valuations of $F$ arising from models of $X$.\n \\end{stheorem}\n\nThe conjecture was known for the cases of smooth projective\nquadrics and Severi-Brauer varieties \\cite{CTPS}.\nIf we use all rank one valuations, the\nresult holds unconditionnally (Cor. \\ref{cor_main_hhk3}).\nOn the other hand, if $G$ extends to a reductive group \nscheme over a smooth model $\\goth X$ of  $X$, \nthe result is unconditional (Cor. \\ref{cor_main_hhk2}).\n \n \\smallskip\n \nLet us review the contents.\nGiven a henselian couple $(A,I)$ and \na reductive $A$-group scheme $H$,\nsection 2 deals with generation of groups $H(A)$ by \nunipotent elements and also with the quotient \nof $H(A)\/RH(A)$ by the (normal) subgroup of $R$-trivial elements\ndefined in \\cite{GS}. Section 3 is devoted to mild improvements\nof patching techniques of Harbater-Hartmann-Krashen \ninvolving more analytical functions and leads to various\ngroup decompositions (prop. \\ref{prop_R_eq});\nthis permits to obtain a patching theorem\nfor twisted flag varieties (th. \\ref{thm_key}).\nSection 4  deals with the setting\nof the beginning of the introduction and \nprovides a weaker version of  the main result.\nWhat remains to do is mostly a local study  for\na two dimensional complete regular local ring $R$ with parameters $t,s$\nand torsors over its localization $R[t^{-1},s^{-1}]$.\nThis is achieved in  section 5 by making use of the \nloop torsors over $R[t^{-1},s^{-1}]$ and of the results of \n\\cite{Gi2}.   \n\n\n \n\\medskip\n\n\n\\noindent{\\bf Acknowledgements.}  We thank Gabriel  Dospinescu\nfor telling us about Schneider's book and \nDavid Harbater for comments on a preliminary version of the paper.  \n \n \\bigskip  \n \n \\smallskip\n\n \\noindent{\\bf Conventions and notation.}\n$(a)$ A variety $V$ over a field $F$ means\na separated $F$--scheme of finite type which is \n integral \\cite[Tag 020D]{St}.\n \n \\smallskip\n \n\\noindent $(b)$ Let $G$ be a reductive $F$--algebraic group and let $F_s$ be a separable closure of $F$.\nLet $(B,T)$ be a Killing couple of $G_{k_s}$ and denote\n$\\Delta(G_{F_s})= \\Delta(G_{F_s},B,T)$  the associated Dynkin diagram\n(there is a canonical bijection $\\Delta(G_{F_s},B,T) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_s},B',T')$\nfor another choice of Killing couple).\nThis diagram \n$\\Delta(G_{F_s})$  is equipped \n with the star action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ \\cite[\\S 6.4]{BoT65}.\n We recall that there is a increasing bijection $I \\to P_I$ between\n the subsets of $\\Delta(G_{F_s})$ and the $F_s$--parabolic subgroups of \n $G_{F_s}$ containing $B$ and the $G(F_s)$--conjugacy classes of parabolic subgroups of $G_{F_s}$.\n Since minimal $F$--parabolic subgroups are $G(F)$-conjugated \n we denote by  $\\Delta_0(G) \\subset \\Delta(G_{F_s})$ be conjugacy class of \n $P_0 \\times_F F_s$ where $P_0$ is a minimal $F$--parabolic subgroup of $G$ (it is stable under the star action).\n The triple $(\\Delta(G_{F_s}), \\hbox{star action}, \\Delta_0(G))$ is called \n the Tits index of $G$. By abuse of notation, $\\Delta_0(G)$ alone is called the Tits index. \n By a twisted $F$--flag variety of $G$, we mean \n a $G$-variety $X$ such becomes isomorphic over $F_s$\n to some $G_{F_s}\/P_I$ as $G_{F_s}$-variety \n (such an $I$ is unique).\nAccording to \\cite[prop. 1.3]{MPW},  \n$I$ is invariant under the star action and \n$X$ is $G$-isomorphic to the variety of $F$--parabolic\nsubgroups of type $I$ which is denoted in this paper\nby $\\text{\\rm{Par}}_I(G)$. \n \n Those  varieties are called sometimes projective \n homogeneous $G$--varieties (as in \\cite{CTPS})\n but we warn the reader of the danger of this terminology\n since there exist in positive characteristic\n $F$--subgroups $Q$ which are not smooth such that \n $G\/Q$ is a projective $F$--variety \\cite{W}. \n In the appendix \\ref{app_parabolic}, we show that \n a homogeneous $G$--variety $X$ is a variety of parabolic subgroups\n if and only if the action is transitive in the sense\n that $G(E)$ acts transitively on $X(E)$ for each $F$ field $E$ (Prop. \\ref{prop_homog}).\n \n \n \n\n \\smallskip\n \n\\noindent $(c)$ \nWe use mainly  the terminology and notation of Grothendieck-Dieudonn\\'e \\cite[\\S 9.4  and 9.6]{EGA1}\n which agrees with that  of Demazure-Grothendieck used in \\cite[Exp. I.4]{SGA3}.Let $S$ be a scheme and let $\\mathcal E$ be a quasi-coherent sheaf over $S$.\n For each morphism  $f:T \\to S$, \nwe denote by $\\mathcal E_{T}=f^*(\\mathcal E)$ the inverse image of $\\mathcal E$ \nby the morphism $f$.\n We denote by $\\mathbf{V}(\\mathcal E)$ the affine $S$--scheme defined by \n$\\mathbf{V}(\\mathcal E)=\\mathop{\\rm Spec}\\nolimits\\bigl( \\mathrm{Sym}^\\bullet(\\mathcal E)\\bigr)$;\nit is affine  over $S$ and \nrepresents the $S$--functor $Y \\mapsto \\mathop{\\rm Hom}\\nolimits_{\\mathcal{O}_Y}(\\mathcal E_{Y}, \\mathcal{O}_Y)$  \\cite[9.4.9]{EGA1}. \n  We assume now that $\\mathcal E$ is locally free of finite rank and denote by $\\mathcal E^\\vee$ its dual.\nIn this case the affine $S$--scheme $\\mathbf{V}(\\mathcal E)$ is  of finite presentation\n(ibid, 9.4.11); also\nthe $S$--functor $Y \\mapsto H^0(Y, \\mathcal E_{Y})= \n\\mathop{\\rm Hom}\\nolimits_{\\mathcal{O}_Y}(\\mathcal{O}_Y, \\mathcal E_{Y} )$ \nis representable by the  affine $S$--scheme $\\mathbf{V}(\\mathcal E^\\vee)$\nwhich is also denoted by  $\\mathbf{W}(\\mathcal E)$  \\cite[I.4.6]{SGA3}.\n \n \n \n \\section{Around Kneser-Tits' problem} \n Let $A$ be a semilocal ring, $I$ its radical. \n \n \\subsection{Subgroups of elementary elements}\\label{subsec_PS}\n Let  $H$ be a reductive $A$-group scheme which   is strictly $A$--isotropic, that is, \n all factors of $H_{ad}$ are isotropic.\nLet  $H$ be a strictly proper parabolic $A$--subgroup of $H$ and let $P^{-}$ be \na $A$--parabolic subgroup opposite to $P$. \nThe subgroup  of $H(A)$ generated by $R_u(P)(A)$ and $R_u(P^-)(A)$ does not depend \nof the choice of $P$ and $P^{-}$, this is a normal subgroup of $H(A)$ \nwhich is denoted by $H(A)^+$ \\cite[Thm. 1  and comments]{PS}.\n\n\n\\begin{slemma}\\label{lem_GS0}\nWe have  $H(A)=H(A)^+ \\,  P(A)$. \n\\end{slemma}\n\n\\begin{proof}\nWe put $U=R_u(P)$,  $U^{-}=R_u(P^{-})$.\n According to \\cite[XXVI.5.2]{SGA3}, we have a decomposition \n $H(A)  =U(A)  \\, U^{-}(A) P(A)$ whence $H(A)  = H(A)^+ \\, P(A)$. \n\\end{proof}\n\n\n\nA special  case of \\cite[prop. 7.7]{GS} is the following.\n\n\n\\begin{slemma}\\label{lem_GS}\nAssume that $(A,I)$ is an henselian couple and that  $H$ is  semisimple simply connected.\nThen the map  $H(A)\/H(A)^+ \\to H(A\/I)\/ H(A\/I)^+$ is an isomorphism. \n\\end{slemma}\n\n\\subsection{R-equivalence}\n\nWe assume now that  $A$ is a nonzero finite $k$--algebra for  a field $k$.\nThen $A$ is an Artinian $k$--algebra. We denote by $I$ its Jacobson radical\nand remind  the reader that $(A,I)$ is an henselian couple.\n\n\nLet $H$ be an $A$--reductive group scheme and \n consider the Weil restriction \\break $G=R_{A\/k}(H)$, this is a smooth  affine connected algebraic \n $k$--group \\cite[A.5.9]{CGP}. We have $G(k)=H(A)$. \n We use $R$-equivalence for group schemes over rings as defined in \n \\cite{GS}.\n \n \n \\begin{slemma}\\label{lem_R_eq}\n We decompose $A=A_1 \\times \\dots \\times A_d$ where each $A_i$ is a local artinian \n $k$--algebra of residue field $k_i$.\n  \n  \\smallskip\n  \n  \\noindent  (1) The map $H(A) \\to \\prod_i H(k_i)$ induces  isomorphisms\n $$\n  G(k)\/RG(k) \\buildrel\\sim\\over\\lgr   H(A)\/RH(A) \\buildrel\\sim\\over\\lgr  \\prod_i H(k_i)\/RH(k_i) .\n $$\n \n \n  \\smallskip\n  \n  \\noindent  (2) If $H$ is furthermore simply connected and strictly isotropic, \n  we have a commutative diagram  of isomorphisms\n \n\\[\\xymatrix{\n  H(A)\/H(A)^+  \\ar[r]^\\sim  \\ar[d]^\\wr &    H(A)\/RH(A) \\ar[d]^\\wr  & \\ar[l]^\\sim    G(k)\/R \\\\ \n \\prod_i H(k_i)\/H(k_i)^+  \\ar[r]^\\sim & \\prod_i H(k_i)\/RH(k_i)  . \n}\\]\n \n \n\n \\end{slemma}\n\n \\begin{proof} \n (1) The first isomorphism $G(k)\/RG(k) \\buildrel\\sim\\over\\lgr H(A)\/RH(A)$  is a formal fact \\cite[lemma 2.4]{GS}.\n Let $n$ be the smallest positive integer such that $I^n=0$.\n We put $A_j=A\/I^{j+1}$ for $j=0, \\dots, n-1$, it comes \n with the ideal $J_j= I^{j}\/I^{j+1}=  I \\, A_j $ which satisfies $J_j^2=0$. We have \n $A_j\/J_j= A_{j-1}$ for $j=1, \\dots, n-1$.\n \n We put $G_j= R_{A_j\/k}(H_{A_j})$ for $j=0,\\dots , n-1$; \n we have a sequence of homomorphisms \n $G=G_{n} \\to G_{n-1} \\to \\dots \\to G_1 \\to G_0 = \\prod_i R_{k_i\/k}(H_{k_i})$.\n It induces homomorphisms\n $$\n G(k)\/R = G_{n}(k)\/R\\to G_{n-1}(k)\/R \\to \\dots \\to G_1(k)\/R \\to G_0(k)\/R = \\prod_i H(k_i)\/R\n $$\n and we will show by a d\\'evissage argument that all of them are  isomorphisms.\n \n \n Let $j$ be a an integer satisfying $1 \\leq j \\leq n-1$.\n According to a variant of \\cite[Lemma 8.3]{GPS}, we have an exact sequence of fppf (resp.\\ \\'etale, Zariski)\nsheaves on $\\mathop{\\rm Spec}\\nolimits(k)$\n$$\n0 \\to \\mathbf{W}\\Bigl( (t_j)_*\\bigl( \\mathop{\\rm Lie}\\nolimits(H)(A_{j-1}) \\otimes_{A_{j-1}} J_j \\bigr) \\Bigr)\n\\to R_{A_j\/k}(H_{A_j}) ) \\to R_{A_i\/k}(H_{A_{j-1}} ) \\to 1  .\n$$\n where $t_j: \\mathop{\\rm Spec}\\nolimits(A_j) \\to  \\mathop{\\rm Spec}\\nolimits(k)$ is the structural morphism.\n We have  then a sequence of $k$--algebraic groups\n $$\n 0 \\to (\\mathbb{G}_a)^{m_j} \\to G_j \\to G_{j-1} \\to 1.\n $$\n The map $G_j \\to G_{j-1}$ is a $(\\mathbb{G}_a)^{m_j}$-torsor\n which is trivial since $G_{j-1}$ is affine. We have then \n a decomposition of $k$-schemes $ G_j \\buildrel\\sim\\over\\lgr G_{j-1}  \\times_k  (\\mathbb{G}_a)^{m_j}$ \n which induces an isomorphism \n $G_j(k)\/R \\to G_{j-1}(k)\/R$. Thus the map $G(k)\/R \\to  G_0(k)\/R = \\prod_i H(k_i)\/R$\n is an isomorphism as desired.\n \n\n\\smallskip\n\n\\noindent (2) We have $A\/I=k_1 \\times \\dots \\times k_d$.\nLemma \\ref{lem_GS} provides an isomorphism $H(A)\/H(A)^+ \\buildrel\\sim\\over\\lgr H(A\/I)\/H(A\/I)^+\n=\\prod_i H(k_i)\/H(k_i)^+$ and we have an isomorphism $H(k_i)\/H(k_i)^+ \\buildrel\\sim\\over\\lgr H(k_i)\/RH(k_i)$\nfor each $i$ \\cite[thm. 7.2]{Gi1}.\nThis completes the proof by chasing diagram.\n\\end{proof}\n\n \n \\section{Patching, R-equivalence and twisted flag schemes}\n \n \\subsection{Using the implicit function theorem}\n We consider  a variation of the framework of \\cite[2.4]{HHK}.\n Let $T$ be a complete DVR of fraction field $K$ and $t$ be an unformizing parameter. Let $\\widehat R_0$ be ring\n containing $T$ and  which is also a complete discrete valuation ring having uniformizer $t$. We denote by $F_0$ the fraction field of $\\widehat R_0$.\n \nLet $\\alpha>1$ be a real number and we define the absolute value on $F_0$ by \n$\\mid\\! t^n u \\!\\mid= \\alpha^{-n}$ for $n \\in \\mathbb{Z}$ and $u \\in \n(\\widehat R_0)^\\times$.\n \n Let $F_1$, $F_2$ be subfields of $F_0$ containing $T$.\n We further assume that we are given \n  $t$-adically complete $T$-submodules $V \\subset F_1 \\cap  \\widehat R_0$\n  and $W \\subset  F_2 \\cap  \\widehat R_0$ \nsatisfying the following conditions:\n\n\n\\begin{equation}\\label{cond_I}\n V + W = \\widehat R_0;\n\\end{equation}\n\n\\vskip-4mm\n\n\n\\begin{equation}\\label{cond_II} \nV \\cap t\\widehat R_0= t V \\quad \\hbox{ and} \\quad\nW \\cap t\\widehat R_0= t W.\n\\end{equation}\n\n\n \\noindent Note that Condition \\eqref{cond_II}  is equivalent to \n\n\n\\begin{equation}\\label{cond_IIbis} \n V \\cap t^n\\widehat R_0= t^n V \\quad \\hbox{and} \\quad \nW \\cap t^n\\widehat R_0= t^n W \\quad \\hbox{ for \\,  each } \\quad  n \\geq 1.\n\\end{equation}\n\n\n\n\n\\begin{sremark} {\\rm Condition \\eqref{cond_II} above is added there compared with  \\cite[2.4]{HHK}\nbut we do not require at this stage that $F_1$ is dense in $F_0$.\n }\n\\end{sremark}\n\n\nWe equip the submodules $V[\\frac{1}{t}]$ of $F_0$ \nof the induced metric (and similarly for $W[\\frac{1}{t}]$).\n\n\n\\begin{slemma}\\label{lem_banach} (1) $V$ is closed in $\\widehat R_0$.\n\n\\smallskip\n\n\\noindent (2) For $v \\in V[\\frac{1}{t}] \\setminus \\{0\\}$, we have  $$\n\\mid v\\mid= \\mathrm{Inf}\\{ \\alpha^{n} \\mid t^n v \\in V \\} .\n$$\n\n\n\\smallskip\n\n\\noindent (3) We have $V= \\mathrm{Inf}\\{ v \\in V[\\frac{1}{t}]  \\mid \n\\enskip \\mid v \\mid  \\geq 0 \\} $ and $V$ is a clopen submodule of\n$V[\\frac{1}{t}]$.\n\n\n\\smallskip\n\n\\noindent (4) $V[\\frac{1}{t}]$ is closed in $F_0$ and is a Banach $K$-space. \n \n\\end{slemma}\n\n\n\n\\begin{proof}\n (1) Our assumption is that the map $V \\to \\limproj V\/t^{m+1} V$ is an isomorphism. Let $(x_n)$ be a sequence of $V$ \n which converges in  $\\widehat R_0$. For each $m \\geq 0$, condition \\eqref{cond_IIbis} shows that \n the map $V\/t^{m+1} V \\to \\widehat R_0\/t^{m+1} \\widehat R_0$ \n is injective so that the sequence $(x_n)$ modulo $t^{m+1} V$ is stationary to some $v_m  \\in V\/t^{m+1} V$.\n The $v_m$'s define a point $v$ of $V$ and the sequence $(x_n)$ converges to $v$.\n \n \\smallskip\n \n \n \\noindent (2) We are given $v = t^m v'\\in  V[\\frac{1}{t}]$  with $v' \\in V \\setminus t V$\n and we have \\break \n  $\\mid\\! v \\! \\mid= \\mathrm{Inf}\\{  \\alpha^{n} \\mid t^n v \\in \\widehat R_0 \\}\n  = \\alpha^{m} \\mathrm{Inf}\\{  \\alpha^{n} \\mid t^n v' \\in \\widehat R_0 \\}$.\n  Condition (II) implies that  $v' \\in    \\widehat R_0 \\setminus t  \\widehat R_0$ so that \n  $\\mid\\! v \\! \\mid=0$. We conclude that  $\\mid\\! v \\! \\mid= \\mathrm{Inf}\\{ \\alpha^{n} \\mid t^n v \\in V \\}$.\n  \n  \n \n \\smallskip\n \n \\noindent (3) It readily follows from the assertion (2).\n \n \n \n \\smallskip\n \n \\noindent (4) Let $(x_n)$ be a sequence of $V[\\frac{1}{t}]$ converging to some $x \\in F_0$.\n We want to show that $x$ belongs to $V[\\frac{1}{t}]$ so that we can assume that $x\\not = 0$\n and that $\\mid\\! x_n \\! \\mid = \\mid\\! x \\! \\mid = \\alpha^m$ for all $n \\geq 0$.\n Assertion (2) shows that $t^m x_n$ is a sequence of $V$ and (1) shows that its limit\n $t^m x$ belongs to $V$. Thus $x \\in V[\\frac{1}{t}]$. We have shown that $V[\\frac{1}{t}]$ is closed in $F_0$.\n \n Finally since $F_0$ is a Banach $K$--space so is $V[\\frac{1}{t}]$.\n\\end{proof}\n\n\n\nThe following statement extends partially \\cite[th. 2.5]{HHK} and \\cite[prop. 4.1]{HHK2}.\n\n\n\\begin{sproposition}\\label{prop_analytic}\nLet $a,b,c$ be  positive integers.\nLet $\\Omega \\subset (F_0)^a \\times (F_0)^b$ be an open\nneighborhood of $(0,0)$ and let \n$f: \\Omega \\to (F_0)^c$ be an analytic map.\nWe denote by $f^a: \\Omega \\cap (F_0)^a \\to (F_0)^c$\nand $f^b: \\Omega \\cap (F_0)^b \\to (F_0)^c$. We assume that \n\n\n\\smallskip\n\n(i) $f(0,0)=0$;\n\n\\smallskip\n\n(ii) the differentials $Df^a_{0}: (F_0)^{a} \\to (F_0)^c$ and   $Df^b_{0}: (F_0)^{a} \\to (F_0)^c$\nsatisfy $$\nDf^a_{0}\\Bigl( V[\\frac{1}{t}]^a \\Bigr) + Df^b_{0}\\Bigl( V[\\frac{1}{t}]^b\\Bigr)=(F_0)^c.\n$$\n\n\n\\smallskip\n\n\\noindent Then there is a real number\n$\\epsilon  > 0$ such that for all $y \\in (F_0)^c$ with $\\mid \\! y \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^a$ and $w \\in  W^b$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = y$.\n\n\\end{sproposition}\n\n\n\\begin{proof}\nWe consider the continuous embedding $i: V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b  \\to (F_0)^a \\times (F_0)^b$ and define $\\widetilde \\Omega=i^{-1}(\\Omega)$ and \nthe function $\\widetilde f= f \\circ i : \\widetilde \\Omega \\to (F_0)^c$.\n\n\\begin{sclaim}  The map $\\widetilde f$ \nis strictly differentiable at $(0,0)$ and\n$D\\widetilde{f}_{(0,0)}:  V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b \\to (F_0)^c$ is onto. \n\\end{sclaim}\n\n\nSince $f$ is $F_0$--analytic at $(0,0)$, it is strictly differentiable  \\cite[I.5.6]{Sc}, that is, there exists\nan open  neighborhood $\\Theta$ of $(0,0)$ and a positive real number $\\beta$ such that \n$$\n\\mid \\! f(x_2) -f(x_1) - Df_{(0,0)}.(x_2-x_1) \\! \\mid \\enskip \\leq \\enskip \\beta \n\\mid \\! x_2 -x_1 \\! \\mid \\qquad \\forall x_1,x_2 \\in \\Theta.\n$$\nIt is then strictly derivable as function between the \nBanach $K$--spaces $(F_0)^a \\times (F_0)^b \\to (F_0)^c$.\nOn the other hand the embedding  $i: V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b  \\to (F_0)^a \\times (F_0)^b$\nis $1$--Liftschitz so is strictly differentiable at $(0,0)$. As composite of strictly differentiable  functions,\n$\\widetilde f$ is strictly differentiable at $(0,0)$ \\cite[\\S 1.3.1]{BF}. Furthermore\nthe differential \n$D\\widetilde{f}_{(0,0)}$  is the composite of\n$$\nV[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b \\xrightarrow{\\quad i \\quad} (F_0)^a \\times (F_0)^b \\xrightarrow{\\enskip Df^a_{0} + Df^b_{0} \\enskip} (F_0)^c.\n$$\nCondition (ii) says exactly that $D\\widetilde{f}_{(0,0)}$  is surjective. The Claim is proven.\n\n\nWe apply the implicit function theorem to the function $\\widetilde f$ \\cite[\\S 1.5.2]{BF}\n(see  \\cite[\\S 4]{Sc} for concocting a proof). Lemma \\ref{lem_banach}.(4)\nshows that $V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b $ is a Banach $K$--space and so is $F_0$.\nThere exists then an open neighborhood $\\Upsilon \\subset \\widetilde \\Omega$\nof $(0,0)$ in $V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b$  such that $\\widetilde f_{\\mid \\Upsilon}$ is open.\nUp to shrink $\\Upsilon$ we can assume that \n$\\Upsilon \\subset V \\times W$ according to Lemma \\ref{lem_banach}.(3).\nThere exists then a real number $\\epsilon >0$ \nsuch that \n$$\n\\bigl\\{ y \\in (F_0)^c \\mid \\enskip  \\mid \\! y \\! \\mid \\leq  \\epsilon \\bigr\\} \\enskip \\subset \\enskip\n\\bigl\\{ y \\in (F_0)^c \\mid \\enskip  \\mid \\! y \\! \\mid  < 2 \\epsilon \\bigr\\} \\subset \\widetilde f(\\Upsilon).\n$$\nWe conclude that \nfor all $y \\in (F_0)^c$ with $\\mid \\! y \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^a$ and $w \\in  W^b$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = y$.\n\\end{proof}\n\n\n \n \n\n\\begin{scorollary}\\label{cor_analytic}\nLet $n$ be a positive integer.\nLet $\\Omega \\subset (F_0)^n \\times (F_0)^n$ be an open neighborhood of $(0,0)$ and let \n$f: \\Omega \\to (F_0)^n$ be an analytic map which satisfies \n\n\\smallskip\n\n(i) $f(0,0)=0$;\n\n\\smallskip\n\n(ii) $f(x,0)=f(0,x)=x$ over an open neighborhood $\\Upsilon$ of $0$.\n\n\n\\smallskip\n\n\\noindent Then there is a real number\n$\\epsilon  > 0$ such that for all $a $ with $\\mid \\! a \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\n\n\\end{scorollary}\n\n\\begin{proof} In this case we have $a=b=c=n$ and $Df^a_{0} = Df^b_{0}= \\mathrm{Id}_{ (F_0)^n}$\nso that Proposition \\ref{prop_analytic} applies.\n\\end{proof}\n\n\n\n\\subsection{Kneser-Tits' subgroups}\n\nContinuing in the previous setting, \nwe assume furthermore that $F_1$ is $t$-adically dense\nin $F_0$. Let $F \\subset F_1 \\cap F_2$ be a subfield. \nFor dealing later with Weil restriction issues it is \nconvenient to deal with  a finite field extension $E$ of $F$.\n\n\n\\begin{sproposition}\\label{prop_KT} \n Let  $H$ be a semisimple simply connected\n$E$--group scheme assumed strictly isotropic.\nWe put $G=R_{E\/F}(H)$. \nFor each overfield $L$ of $F$, we put \n$G(L)^+= H(L \\otimes_F E)^+$ where the second group \nis that defined  in \\S \\ref{subsec_PS}.\n Then  we have the decomposition\n$$\nG(F_0)^+= G(F_1)^+ \\, G(F_2)^+.\n$$\n\\end{sproposition}\n \n \n\\begin{proof} Without lost of generality we can assume that $F$ is infinite. The proof is based on an analytic argument requiring \nsome preparation.\n\nLet $P$ be a strictly proper parabolic $E$--subgroup of $H$. \nLet $U$ be its \nunipotent radical and $U_{last}$ the last part of Demazure's filtration\n\\cite[\\S 3.2]{GPS}. Let $u: \\mathbb{G}_{a,E}^d \\buildrel\\sim\\over\\lgr U_{last}$ be a $E$--group isomorphism.\nAccording to \\cite[Lemma 3.4(3)]{GPS}, $E_P(E). \\mathop{\\rm Lie}\\nolimits(U_{last})(E)$\ngenerates $\\mathop{\\rm Lie}\\nolimits(H)(E)$.\nThere exists $g_1,\\dots, g_n \\in E_P(E)$ such that \n$$\n\\mathop{\\rm Lie}\\nolimits(H)(E)= \\,  {^{g_1}\\!\\mathop{\\rm Lie}\\nolimits}(U_{last})(E) \\oplus \\,    ^{g_2}\\!\\mathop{\\rm Lie}\\nolimits(U_{last})(E) \\oplus \\dots \n\\oplus  \\,   ^{g_n}\\!\\mathop{\\rm Lie}\\nolimits(U_{last})(E) . $$\n\nWe consider the map $h:  (\\mathbb{G}_{a,E}^d)^n \\to H$, $h(x_1,\\dots, x_n)= \\, {^{g_1}u}(x_1) \\,  \\dots \\,  \n^{g_n}\\!u(x_n)$. Its differential at $0$ is \n$dh_{0,0}: E^{dn} \\cong \\mathop{\\rm Lie}\\nolimits(U_{last})(E)^n \\to \\mathop{\\rm Lie}\\nolimits(H)(E)$, $(X_1,\\dots, X_n) \\mapsto \\sum_{i=1}^n \\, \\,   ^{g_i}\\!X_i$, \nso is an isomorphism.  The map $h$ is then \\'etale at a neighborhood of $0$.\nIt follows that $h_\\sharp= R_{E\/F}(h): R_{E\/F}\\bigl((\\mathbb{G}_{a,E}^d)^n\\bigr) \\to G=R_{E\/F}(H)$\nis \\'etale also at a neighborhood of $0$ \\cite[A.5.2.(4)]{CGP}\n\n\nSince the field $F_0$ is henselian,  \nthe local inversion theorem holds \\cite[prop. 2.1.4]{GGMB}. \nWe mean that there exists an open neighborhood $\\Upsilon \\subset (F_0)^n$ such that\n the restriction \n$h_{\\sharp \\mid \\Omega}: \\Upsilon \\to G(F_0)$ is a topological open embedding.\n\nWe consider now the product morphism $q: \\Upsilon \\times \\Upsilon \\to G(F_0)$,\n$q(x,y)= h_\\sharp(x) h_\\sharp(y)$. We put  $\\Omega= q^{-1}( h_\\sharp(\\Upsilon))$, this an open\nsubset of $(F_0)^{2n}$.\nThen the restriction $q_{\\mid \\Omega}$ defines an (unique) analytical map\n$f:  \\Omega \\to \\Upsilon$ \nsuch that $q(x,y)= h_\\sharp( f(x,y) )$.\n\nBy construction we have $f(0,x)= f(x,0)=x$ for $x$ in a neighborhood of $0 \\in (F_0)^n$.\nWe apply Corollary \\ref{cor_analytic} to $f$ so that there exists $\\epsilon >0$\nsuch that  for each $a \\in \\Upsilon$ with $\\mid a \\mid  \\,  \\leq \\,  \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\nWe denote by $\\Upsilon_\\epsilon= \\Upsilon \\cap B(0, \\epsilon)$.\nThen $h_\\sharp^{-1}( \\Upsilon_\\epsilon)$ is an open neighborhood of \n$0$ in $(F_0)^n$.\n\n\nLet us now prove that $G(F_0)^+= G(F_1)^+ \\times G(F_2)^+$.\nSince $F_1$ is dense in $F_0$, \n$G(F_1)^+$ is dense in $G(F_0)^+$.\nIt is then enough to show that $\\Upsilon_\\epsilon  \\subset G(F_1)^+ \\times G(F_2)^+$.\nLet $g =h(a) \\in h_\\sharp( \\Upsilon_\\epsilon)$.\nThen  $a= f(v,w)$ with $(v, w) \\in  \\Omega$.\nIt follows that $g =h( a)= h_\\sharp( f(v,w)) = q(v,w) = h_\\sharp(v) \nh_\\sharp(w) \\in  G(F_1)^+ \\times G(F_2)^+$.\n\\end{proof}\n\n \nThis could be refined as follows.\n\n\\newpage\n\n\n\n \n\\begin{sproposition}\\label{prop_R_eq} Let \n$H$ be a reductive $E$-group and put $G=R_{E\/F}(H)$.\n\n\\smallskip\n\n\\noindent (1) $RG(F_1) \\, RG(F_2)$ contains an open neighborhood of $1$  in $G(F_0)$.\n\n\n\\smallskip\n\n\\noindent (2) If $RG(F_1)$ is dense in $RG(F_0)$, then  $RG(F_1) \\, RG(F_2)=RG(F_0)$;\n\n\n\\smallskip\n\n\\noindent (3) If $H$ is semisimple simply connected and $H_{F_1 \\otimes_F E}$ is \nstrictly isotropic, then we have\n$$\nG(F_1)^+ \\, RG(F_2)=G(F_0)^+ .\n$$\n\n\n\\smallskip\n\n\\noindent (4) If $H$ is semisimple simply connected and $H_{F_i \\otimes_F E}$ is strictly isotropic for $i=1,2$,\nthen $G(F_1)^+ \\, G(F_2)^+=G(F_0)^+$.\n\n\\end{sproposition}\n \n \nThe subgroups $G(F_1)^+$, $G(F_0)^+ $ are defined as in Proposition \\ref{prop_KT}.\n \n\\begin{proof}\n (1) Let $T \\subset H$ be a maximal $E$--torus and let $1 \\to S  \\to Q \\xrightarrow{s} T \\to 1$\n be a resolution of $T$  where $Q$ is a quasitrivial torus and $S$ is a torus.\n We have $Q=R_{C\/E}(\\mathbb{G}_m)$ where $C$ is an \\'etale $E$--algebra so that\n $Q$ is an open subset of the affine $E$--space $\\mathbf{W}(C)$.\n \n We use now Raghunathan's technique \\cite[\\S 1.2]{R}.\n There exists $h_1,\\dots, h_r \\in H(E)$ such that \n$\\mathop{\\rm Lie}\\nolimits(H)(E)=  \\,  ^{g_1}\\!\\mathop{\\rm Lie}\\nolimits(T)(E)\\oplus \\,   ^{g_2}\\!\\mathop{\\rm Lie}\\nolimits(T)(E) \\oplus \\,  \\dots \\oplus \\;     ^{g_r}\\!\\mathop{\\rm Lie}\\nolimits(T)(E) $.\nWe consider the map $h: (Q^n)_{E} \\to H$, $h(x_1,\\dots, x_n)= \\, ^{g_1}\\!s(x_1) \\,  \\dots \\,  \n^{g_n}\\!s(x_n)$. Its differential at $0$ is \\break \n$dh_{0}: C^r \\to \\mathop{\\rm Lie}\\nolimits(H)(E)$, $(c_1,\\dots, c_r) \\mapsto \\sum_i  \\, \\,  ^{g_i}\\!ds(c_i)$, \nand is onto (observe that $\\mathop{\\rm Lie}\\nolimits(Q)(E) \\to \\mathop{\\rm Lie}\\nolimits(T)(E)$ is surjective). \n\nWe cut  now $Q^r$ by some suitable affine $E$--subspace $\\mathbf{W}(C^n)$ of  $\\mathbf{W}(C)^r$\nsuch that the restriction $h'$ of $h$ to $X=Q^r \\cap \\mathbf{W}(C^n)$\nis such that $dh'_{0}: \\mathrm{Tan}_{X,1} \\to \\mathop{\\rm Lie}\\nolimits(H)(F)$ is an isomorphism.\nNote that $X$ and $E$ have same dimension $n$ over $E$.\n\nThe map $h'$ is then \\'etale at a neighborhood of $1$.\nIt follows that \\break $h'_\\sharp= R_{E\/F}(h'): R_{E\/F}\\bigl(X\\bigr) \\to G=R_{E\/F}(H)$\nis \\'etale also at a neighborhood of $0$ \\cite[A.5.2.(4)]{CGP}\n\nSince the field $F_0$ is henselian,  \nthe local inversion theorem holds \\cite[prop. 2.1.4]{GGMB}. \nWe mean that there exists an open neighborhood $\\Upsilon \\subset (F_0)^n$ such that\n the restriction \n$h'_{\\sharp \\mid \\Omega}: \\Upsilon \\to G(F_0)$ is a topological open embedding.\n\nWe consider now the product morphism $q: \\Upsilon \\times \\Upsilon \\to G(F_0)$,\n$q(x,y)= h'_\\sharp(x) \\, h'_\\sharp(y)$. We put  $\\Omega= q^{-1}( h'_\\sharp(\\Upsilon))$, this an open\nsubset of $(F_0)^{2n}$.\nThen the restriction $q_{\\mid \\Omega}$ defines a (unique) analytical map $f: \n \\Omega \\to \\Upsilon$ \nsuch that $q(x,y)= h'_\\sharp( f(x,y) )$.\n\nBy construction we have $f(0,x)= f(x,0)$ for $x$ in a neighborhood of $0 \\in F_0^n$.\nWe apply Corollary \\ref{cor_analytic} to $f$ so that there exists $\\epsilon >0$\nsuch that  for each $a \\in \\Upsilon$ with $\\mid a \\mid  \\,  \\leq \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\nWe denote by $\\Upsilon_\\epsilon= \\Upsilon \\cap B(0, \\epsilon)$.\nThen ${h'_\\sharp}^{-1}( \\Upsilon_\\epsilon)$ is an open neighborhood of \n$0$ in $(F_0)^n$.\n\nWe claim that $\\Upsilon_\\epsilon \\subset RG(F_1) \\, RG(F_2)$.\nLet $g =h'_\\sharp(a) \\in h( \\Upsilon_\\epsilon)$.\nThen  $a= f(v,w)$ with $(v, w) \\in  \\Omega$.\nIt follows that $g= h'_\\sharp( a)= h_\\sharp( f(v,w)) = q(v,w) = h'_\\sharp(v) h'_\\sharp(w) \\in  RG(F_1) \\times RG(F_2)$.\n\n\n \\smallskip\n \n \\noindent (2)  \n If $RG(F_1)$ is furthermore dense in $RG(F_0)$, then  (1) shows that \n $RG(F_1) \\, RG(F_2)$ is a dense open subset of $RG(F_0)$.\n Thus $RG(F_1) \\, RG(F_2)=RG(F_0)$ \n \n \n \\smallskip\n \n \\noindent (3) We assume that $H$ is semisimple simply connected and \n that  $G_{F_1 \\otimes_F E_1}$ is strictly isotropic.\n According to Lemma \\ref{lem_GS}.(2), we have\n $G(F_1)^+=RG(F_1)=RH(F_1 \\otimes_F E)=H^+(F_1 \\otimes_F E)$\n and similarly for $F_0$.\n \n \nSince $H^+(F_1 \\otimes_F E)^+$ is dense in $H^+(F_0 \\otimes_F E)$, it follows that \n$RG(F_1)$ is dense in $RG(F_0)$. Assertion  \n  (2) yields then  $RG(F_1) \\, RG(F_2)=RG(F_0)$.\n  Lemma \\ref{lem_R_eq}.(2) states that $G(F_1)^+=RG(F_1)$\n  and $G(F_0)^+=RG(F_0)$. We conclude that \n  $G(F_1)^+ \\, RG(F_2)=G(F_0)^+$.\n \n\n \\smallskip\n \n \\noindent (4) We have  furthermore  $G(F_2)^+= RG(F_2)$ so that (3)\n yields  $G(F_1)^+ \\, G(F_2)^+ =G(F_0)^+$.\n\\end{proof}\n\n \n \\begin{sremark} {\\rm Note that (1) shows in particular that  \n $RG(F_0)$ is an open subgroup of $G(F_0)$.\n  }\n \\end{sremark}\n \n \n The main result of the section is the following patching statement on twisted flag varieties.\n \n\n \\begin{stheorem} \\label{thm_key} We put $F=F_1 \\cap F_2$. \n Let $H$ be a reductive $E$--group and let \n $X$ be a twisted flag $E$--variety of $H$.\n Assume that $X$ is auto-opposite, that is,\n the stabilizer $P$ of an $E_s$--point of $X$\n is conjugated to an opposite parabolic subgroup of $P$ \\cite[\\S 4.9]{BoT65}. We put $G=R_{E\/F}(H)$ and $Z=R_{E\/F}(X)$.\n If $Z(F_1) \\not = \\emptyset$\n and  $Z(F_2) \\not = \\emptyset$, then  $X(F) \\not = \\emptyset$.\n \\end{stheorem}\n\n \n \\begin{proof} Without loss of generality, we can assume that $H$ is\n semisimple simply connected and that it is absolutely $E$--simple.\n This implies that $H$ is strictly $F_i \\otimes_F E$--isotropic for $i=1,2$.\n Let $x_i \\in Z(F_i)= X( F_i \\otimes_F E)$ and denote by $P_i=\\mathop{\\rm Stab}\\nolimits_{G_{F_2}}(x_2)$\n its stabilizer, this is  parabolic $F_i \\otimes_F E$--subgroup of $H$. We denote by $U_i\/E$ its unipotent radical. \n\n Since the conjugacy class of $P_1$ is autopposite, there exists \n $h \\in H(E \\otimes_F F_0)$ such that $P_{1, E \\otimes_F F_0}$\n is opposite to $^hP_{2, E \\otimes_F F_0}$ \\cite[XXVI.5.3]{SGA3}.\nAccording to Lemma \\ref{lem_GS0}, we have \n$H(E \\otimes_F F_0)= H^+(E \\otimes_F F_0) \\,  P_2(E \\otimes_F F_0)$\nso we can assume that $h \\in H^+(E \\otimes_F F_0)$.\nAccording to Proposition \\ref{prop_R_eq}.(4), \nwe can write $h=h_1 \\, h_2$ with $h_i \\in H^+(E \\otimes_F F_i)$ for $i=1,2$.\nUp to replace $P_1$  by $^{h_1^{-1}}P_1$ and $P_2$ by  $^{h_2^{-1}}P_2$\nwe can then assume that $P_{1,F_0}$ is opposite to $P_{2,F_0}$.\n According to \\cite[XXVI.5.1]{SGA3}, we have a decomposition \n\n\\begin{equation} \\label{michel} \n H(F_0 \\otimes_F E)  =U_2(F_0 \\otimes_F E)  \\, U_1(F_0 \\otimes_F E) \\,\n P_2(F_0\\otimes_F E).\n \\end{equation}\n \n\\noindent Let $h \\in H(F_0 \\otimes_F E)$ such that $x_1=h.x_2$. The preceding decomposition permits to write\n$h= u_2 \\,  u_1 \\,  p_2$ with  $u_2 \\in U_2(F_0\\otimes_F E)$, \n $u_1 \\in  U_1(F_0 \\otimes_F E)$\nand $p_2 \\in  P_2(F_0 \\otimes_F E)$. It follows\nthat  $$\nx_1= u_1^{-1}.x_1 = (u_1^{-1}h).x_2= (u_1^{-1} \\, u_2 \\, u_1) \\, . \\, (p_2.x_2) =(u_1^{-1} \\, u_2 \\, u_1) \\, . \\, x_2\n$$\n hence $x_1 \\in H(F_0 \\otimes_F E)^+.x_2$.\n According to Proposition \\ref{prop_R_eq}.(4), we have  $G(F_0)^+=  G(F_1)^+ \\,  G(F_2)^+$\n so that $u_1^{-1} \\, u_2 \\,  u_1 =g_1 \\,  g_2$ with $g_i \\in G(F_i)^+$ for $i=1,2$.\n It follows that $g_1^{-1} \\, . \\, x_1 = g_2^{-1} \\, . \\, x_2$, this defines a point of  $X(E)=Z(F)$.\n \\end{proof}\n\n \n \n \n \\section{Relation with the original HHK method}\n \nWe recall the setting.\nLet $T$ be an excellent  complete discrete valuation ring with fraction field $K$, residue field $k$ and\nuniformizing parameter $t$. \nLet $F$ be a one-variable function field over $K$\nand let $\\goth X$ be a normal model of $F$, i.e. a normal connected\nprojective $T$-curve with function field $F$.\n We denote by $Y$ the closed fiber of $\\goth X$ and fix a separable closure $F_s$ of $F$.\n   \n For each point $P \\in Y$, let $R_P$ be the local ring of\n $\\goth X$ at $P$; its completion $\\widehat R_P$ is a domain \n with fraction field denoted by $F_P$.\n \n \n For each subset $U$ of $Y$ that is contained\n in an irreducible component of $Y$ and does not meet the other components, \nwe define\n $R_U= \\bigcap\\limits_{P \\in U} R_P \\subset F$. We denote by $\\widehat R_U$\n the $t$--adic completion of $R_U$.\n The rings $R_U$ and $\\widehat R_U$ are excellent normal domains \n and we denote by $F_U$ the fraction field of $\\widehat R_U$\n \\cite[Remark 3.2.(b)]{HHK3}.\n \n Each height one prime $\\mathfrak{p}$ in $\\widehat R_P$ that contains\n  $t$ defines a branch of $Y$ at $P$ lying \n on some irreducible component of $Y$. \n The $t$-adic completion  $\\widehat R_{\\mathfrak{p}}$ of the local ring\n $R_{\\mathfrak{p}}$ of $\\widehat R_P$ at $\\mathfrak{p}$ is a complete DVR\n whose fraction field is denoted by $F_{\\mathfrak{p}}$. \n The field $F_{\\mathfrak{p}}$ contains also $F_U$ if $U$ \n is an irreducible open subset of $Y$ such that $P \\in \\overline{U} \\setminus U$.\n We have then a diagram of fields\n \n\\[\\xymatrix{\n &   F_{\\mathfrak{p}}  & \\\\ \n F_P \\ar[ru] & & F_U  . \\ar[lu]\n}\\]\n \n \n \n \n \\begin{sexample} \\label{sexample1}{\\rm We assume that $T=k[[t]]$ and \n take $X= \\mathbb{P}^1_K$, $\\goth X= \\mathbb{P}^1_T$,\n $P= \\infty_k$, $U=\\mathbb{A}^1_k= \\mathop{\\rm Spec}\\nolimits(k[x])$.\n The ring $R_U$ contains $k[[t]][x]$ and is \n its localization with respect to the multiplicative set $S$\n of elements which are units modulo $t$.\n The  $t$-adic completion of $R_U$ is   $\\widehat R_U=\n k[x][[t]]$; we have $F_U= Frac( \\widehat R_U)=\n k(x)((t))$.\n The local ring of $\\goth X$ at $P=\\infty_k$ is $R_P=k[[t]][x^{-1}]_{(x^{-1},t)}$\n so that its completion is $\\widehat R_P= k[[t,x^{-1}]]$;\n in particular $F_P=k((t,x^{-1}))$.\n \n We take  $\\mathfrak{p}= t \\widehat R_P \\subset \\widehat R_P$\n and  the $t$-adic completion  $\\widehat R_{\\mathfrak{p}}$ \n of the local ring\n $R_{\\mathfrak{p}}$ of $\\widehat R_P$ at $\\mathfrak{p}$ is a complete DVR\n which is $k((x^{-1}))[[t]]$. In particular $F_\\goth p= k((x^{-1}))((t))$.\n }\n \\end{sexample}\n\n \\medskip\n \n \n \\begin{ssetting} \\label{setting_hhk} {\\rm\n Let $\\mathcal P$ be a non-empty finite set of closed points of $Y$ that contains\n all the closed points at which distinct irreducible components meet.\n Let $\\mathcal U$ be the set of connected components of $Y \\setminus \\mathcal P$ and let $\\mathcal B$\n be the set of branches of $Y$ at points of $\\mathcal P$.\n This yields a finite inverse system of field $F_P, F_U, F_\\goth p$ (for $P \\in \\mathcal P$;\n $U \\in \\mathcal U$, $\\goth p \\in \\mathcal B$) where $F_P, F_U \\subset F_\\goth p$ if $\\goth p$\n is a branch of $Y$ at $P$ lying in the closure of $U$.\n }\n\\end{ssetting}\n\n\n\n\\begin{slemma}  \\label{lem_dense0} We assume that  $X= \\mathbb{P}^1_K$, $\\goth X= \\mathbb{P}^1_T$,\n $P= \\infty_k$, $U=\\mathbb{A}^1_k= \\mathop{\\rm Spec}\\nolimits(k[x])$ and $\\goth p$ the branch\n of $P$. We put $F_1=F_P$, $F_2=F_U$\n and $F_0=F_\\goth p$.\n \n\n\\smallskip\n\n\\noindent (1) $F_1$ is $t$-dense in $F_0$.\n\n\n\\smallskip\n\n\\noindent (2)  We put $V= F_1 \\cap \\widehat R_\\mathfrak{p}$ and   $W= F_2 \\cap \\widehat R_\\mathfrak{p}$.\nThen $V$ and $W$ satisfy conditions \\eqref{cond_I} and \\eqref{cond_II}.\n \n\\end{slemma}\n\n\n\n \\begin{proof}\n \\noindent  (1) We are given $u_0\/v_0 \\in F_0$ with $u_0, v_0 \\in \\widehat R_\\mathfrak{p}$,  $v_0 \\not =0$.\n There exists elements $u, v \\in R_\\goth p$ very close respectively of $u_0,v_0$ with $v \\not =0$.\n Let $s_1, s_2 \\in R_p \\setminus R_p \\goth p$ such that $s_1 u \\in R_P$ and $s_2 v \\in R_\\goth p$.\n Then $u_0\/v_0$ is very close of $(s_1s_2 u) \/ (s_1s_2 v) \\in F_1$.\n \n  \n  \\smallskip\n  \n  \\noindent (2) Condition \\eqref{cond_II} is obviously fullfilled since $t \\in F_1 \\cap F_2$.\n  For establishing condition \\eqref{cond_I}, we are given an element $f$ of $\\widehat R_\\goth p$ and \n  may write it as $f= \\sum\\limits_{i=0}^\\infty x^{m_i}  \\Bigl( \\sum_{j=0}^\\infty a_{i,j} \\frac{1}{x^j} \\Bigr) t^i$\n  where the $m_i$'s are non-negative integers and $a_{i,j} \\in T$. \n  We decompose $$\n  f= f_1 + f_2=  \\sum_{i=0}^\\infty x^{m_i} \\Bigl( \\sum_{j=m_i}^\\infty a_{i,j} \\frac{1}{x^j} \\Bigr) t^i\n  \\quad +  \\quad\n   \\sum_{i=0}^\\infty x^{m_i}  \\Bigl(\\sum_{j=0}^{m_i-1} a_{i,j} \\frac{1}{x^j} \\Bigr)  t^i \n  $$\n  We observe that $f_2$ belongs to $\\widehat R_P$ so belongs to $V$.\nWe recall that $R_U$ is the localization of $T[x]$ \nwith respect to the elements which are units modulo $t$. \nWe conclude that $f_2$ belongs to $W$ as desired.\n \\end{proof}\n \n \n \n \n \\begin{stheorem} \\label{thm_main_hhk} Let $G$ be a reductive $F$--algebraic group.\n \n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if $Z(F_U) \\not = \\emptyset$ for \n each $U \\in \\mathcal U$ and  $Z(F_P) \\not = \\emptyset$ for \n each $P \\in \\mathcal P$.\n \n \\smallskip\n \n \\noindent (2) For each $U \\in \\mathcal U$ (resp.\\ each $P \\in \\mathcal P$), \n we fix an $F$--embeddings $i_U: F_s \\to F_{U,s}$  (resp.\\ $i_P: F_s \\to F_{P,s}$)\n providing identifications $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{U,s}})$\n (resp.\\, $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{P,s}})$).\n The Tits index $\\Delta_0(G)$ is the smallest subset\n  of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n  $\\Delta_0(G) \\subset \\Delta_0(G_{F_U})$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in \\mathcal P$.\n \n \\end{stheorem}\n\n The recollection for star action and Tits index is done in the beginning of the paper.\n \n \\begin{proof}\n The outline is to prove (1) under an assumption\n of autoppositeness, to prove (2) and then to prove (1) in the general case.\n \n \\smallskip\n \n \\noindent (1) We assume that $Z$ parameterizes an\n  auto-opposite conjugacy\n class of parabolic subgroups of $G$. \n  We use a Weil restriction argument as in the proof of \\cite[Thm. 4.2]{HHK2}.\nThis involves a finite morphism $f: \\goth X \\to \\mathbb{P}^1_T$ such that \n$\\mathcal P=f^{-1}(\\infty_k)$. Write $\\uF$ for the function field of  $\\mathbb{P}^1_T$, and let $d=[F:F']$. We put\n$\\uU= \\mathbb{P}^1_k \\setminus \\{\\infty \\}$, $\\uP= \\infty_k$ and\n$\\underline{\\goth p}= (U, \\mathfrak{p})$ and\n  $F_0=\\uF_{\\underline{\\goth p}}$, $F_1=\\uF_{\\uP}$ and $F_2=F_{\\uU}$.  Also patching holds for the diamond $(\\uF, F_1,F_2,F_0)$ according to \\cite[thm 3.9]{HH}\n  so in particular $K(x)= \\uF= F_1 \\cap F_2 \\subset F_0$. \n  We put $V= F_1 \\cap \\widehat R_\\mathfrak{p}$ and   $W= F_2 \\cap \\widehat R_\\mathfrak{p}$.\n  Lemma \\ref{lem_dense0} shows that \n  $F_1$ is dense in $F_0$ and that  $V$ and $W$ satisfy conditions \\eqref{cond_I} and \\eqref{cond_II}.\n\n  We consider the Weil restriction $\\uG= R_{F\/\\uF}(G)$, it is a reductive $\\uF$--group\nwhich acts on the $\\uF$--variety $\\uZ=R_{F\/\\uF}(Z)$.\nWe have $$\n\\uZ(F_1)= Z(F_1 \\otimes_{\\uF} F)  = \\prod_{P \\in \\mathcal P} Z(F_P)\n$$\naccording to \\cite[Lemma 6.2.(a)]{HH}.\nSimilarly we have\n$$\n\\uZ(F_2)= Z(F_2 \\otimes_{\\uF} F)  = \\prod_{U \\in \\mathcal U} Z(F_U)\n$$\nOur assumptions imply that $\\uZ(F_1) \\not = \\emptyset$ and \n$\\uZ(F_2) \\not = \\emptyset$. Theorem \\ref{thm_key} implies \nthat $\\uZ(\\uF) \\not = \\emptyset$. Thus $\\uZ(\\uF)= Z(F)$ is non-empty.\n\n  \\smallskip  \n  \n  \\noindent (2) Let $\\Theta$ be the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n  $\\Delta_0(G) \\subset \\Delta_0(G_{F_U})$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for \n  each $P \\in \\mathcal P$.\n  Since $\\Theta$ is an intersection of auto-opposite subsets of $\\Delta(G_{F_s})$, it is auto-opposite.\n  \n  We observe that $\\Delta_0(G) \\subset \\Theta$ since it is \n   stable under the star action and \n  satisfies $\\Delta_0(G) \\subset \\Delta_0$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in \\mathcal P$.\n  For the converse inclusion we consider the $F$--variety $Z$ of parabolic\n  subgroups of type $\\Theta$ (which is auto-opposite).\n  For each $U \\in \\mathcal U$, we have  $\\Theta \\subset \\Delta_0(G_{F_U})$ so that\n  $Z(F_U) \\not = \\emptyset$; similarly we have \n   $\\Theta \\subset \\Delta_0(G_{F_P})$ for each $P \\in \\mathcal P$\n   so that $Z(F_P) \\not = \\emptyset$.\n  Part (1) yields that $Z(F) \\not = \\emptyset$. Thus $\\Theta \\subset \\Delta_0(G)$\n  and $\\Theta = \\Delta_0(G)$\n  \n  \\smallskip\n  \n\\noindent (3) We consider now the case of an arbitrary flag $F$--variety $Z$  for $G$. It   is associated to a subset $\\Upsilon$ of $\\Delta(G_{F_s})$\n which is invariant under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$. \n If $Z(F) \\not = \\emptyset$, we have that $Z(F_U) \\not = \\emptyset$ for each \n $U \\in \\mathcal U$ and that $Z(F_P) \\not = \\emptyset$ for each \n $P \\in \\mathcal P$. Conversely if $Z(F_U) \\not = \\emptyset$ for each \n $U \\in \\mathcal U$ and  $Z(F_P) \\not = \\emptyset$ for each \n $P \\in \\mathcal P$. \n It follows that $ \\Upsilon \\subset \\Delta_0(G_{F_U})$\n for each $U \\in \\mathcal U$.\n Part (2) of the statement yields $\\Upsilon \\subset \\Delta_0(G)$\n so that $Z(F) \\not = \\emptyset$.  \n \\end{proof}\n\n \n \n \\begin{scorollary} \\label{cor_main_hhk} Let $G$ be a reductive $F$--algebraic group.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_P) \\not = \\emptyset$ for \n each $P \\in Y$.\n \n \\smallskip\n \n \\noindent (2) For each $P \\in Y$, \n we fix an $F$--embedding  $i_P: F_s \\to F_{P,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{P,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in Y$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1)  We assume $Z(F_P) \\not = \\emptyset$ for \n each $P \\in Y$. Let $Y_1,\\dots, Y_d$ be the irreducible components of $Y$\n with respective generic points $\\eta_1, \\dots,  \\, \\eta_d$. \n According to  \\cite[prop. 5.8]{HHK2}, there exists non-empty affine subsets \n $U_i \\subset Y_i$ ($i=1,\\dots, d$) such that $Z(F_{U_i}) \\not = \\emptyset$\n for $i=1,\\dots, d$ and $U_i \\cap U_j = \\emptyset$ for $i<j$.\n We apply Theorem \\ref{thm_main_hhk} to  $\\mathcal U= \\{ U_1,\\dots, U_d\\}$, \n $\\mathcal P= Y \\setminus \\cap_i U_i$ and get that $Z(F) \\not = \\emptyset$.\n \n \\smallskip\n \n \\noindent (2) This readily follows of (1).\n  \n \\end{proof}\n\n \n\n\n \\begin{scorollary} \\label{cor_main_hhk2} Let $G$ be a reductive $F$--algebraic group\n and assume that $G$ is the generic fiber of a reductive $\\goth X$-group scheme $\\mathfrak{G}$.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_v) \\not = \\emptyset$ for \n each discrete valuation $v$ of $F$.\n \n \\smallskip\n \n \\noindent (2) For each discrete valuation $v$ of $F$, \n we fix an $F$--embedding  $i_v: F_s \\to F_{v,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{v,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_v})$ for  each discrete valuation $v$ on $F$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1) We assume that $Z(F_v) \\not = \\emptyset$ for \n each discrete valuation $v$ of $F$.\n Let $\\mathfrak{G}$ be a reductive $\\goth X$--group of generic fiber $G$. \n Then $Z$ is the generic fiber of an $\\goth X$--scheme of parabolic subgroups\n $\\goth Z$. For applying Corollary \\ref{cor_main_hhk}, we have to check\n that $Z(F_P) \\not = \\emptyset$ for each $P \\in Y$.\n If $Q$ is a point of codimension $1$ of $\\goth X$, \n it defines a discrete valuation $v_Q$ on $F$ whose completion is  $F_Q$.\n Our assumption implies then that  $Z(F_Q) \\not = \\emptyset$ in that case.\n We deal now with the case of a closed point $P$ of $\\goth X$.\n Let $D$ be an irreducible component of $Y=\\goth X_k$ containing $P$\n and let $Q$ be the generic point of $D$. Since \n $\\goth Z$ is proper over $T$, we have $\\goth Z(\\widehat R_Q)=\\goth Z(F_Q)$ which\n is not empty by the preceding case. It follows that $\\goth Z_k(k(D)) \\not = \\emptyset$.\n Again $\\goth Z_k$ is projective so that $\\goth Z_k(D) =\\goth Z_k(k(D))$ is not empty\n and in particular $\\goth Z_k(k(P))$ is not empty.\n Since $\\goth Z$ is smooth over $\\goth X$, the Hensel lemma\n shows that $\\goth Z_k(\\widehat R_{\\goth X,P}) \\to  \\goth Z_k(k(P))$ is surjective.\n Thus $\\goth Z(\\widehat R_{\\goth X,P})$ is not empty and so is $Z(F_P)$.\n \n \n \\smallskip\n \n \\noindent (2) It readily follows of (1). \n  \n \\end{proof}\n \n \n \n \n \n \\begin{scorollary} \\label{cor_main_hhk3} Let $G$ be a reductive $F$--algebraic group.\n We denote by $\\Omega^1_F$ the set of rank one valuations of $F$.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_v) \\not = \\emptyset$ for \n each $v \\in \\Omega^1_F$.\n \n \\smallskip\n \n \\noindent (2) For each $v \\in \\Omega^1_F$, \n we fix an $F$--embedding  $i_v: F_s \\to F_{v,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{v,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_v})$ for  each $v \\in \\Omega^1_F$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1) We assume that $Z(F_v) \\not = \\emptyset$ for \n each $v \\in \\Omega^1_F$.\n  According to \\cite[Thm 2.5]{HHKP}, there exists a regular model $\\goth X'$ of $F$\n  such that $Z(F_P) \\not = \\emptyset$ for \n each $P \\in \\goth X' \\times_T k$.  Then Corollary \\ref{cor_main_hhk2}.(1)\n shows that $Z(F) \\not = \\emptyset$.\n \n \\smallskip\n \n \\noindent (2) This readily follows of (1).\n \\end{proof}\n \n \n \n\\section{Local-global principle for discrete valuations}\n\nLet $T$ be an excellent DVR of fraction field $K$ and residue field $k$.\nLet $X$ be a smooth, projective,\ngeometrically integral curve over $K$. Let $F=K(X)$ be the function field of $X$.\n\n \n\n \n\\begin{slemma} \\label{lem_extend} Let $\\goth X$ be a projective, flat curve over $A$\nwhich is connected and regular such that $\\goth X_K=X$.\nLet $\\goth H$ be a flat  affine $\\goth X$--group scheme of finite presentation\nand assume that  there  exists a  Zariski cover $(\\mathfrak{U}_i)_{i \\in I}$ of $\\goth X$ \n such that each $\\goth H_{U_i}$ admits a closed embedding $\\goth H_{U_i} \\subset \\GL_{n_i}$ such that \n $\\GL_{n_i}\/\\goth H_{U_i}$ is representable by an affine $U_i$--scheme,\n\nLet $\\gamma \\in H^1_{\\text{\\rm fppf}}(F,\\goth H)$ and let \n $D$ be a divisor of $\\goth X$ which contains the irreducible\ncomponents of $Y= \\goth X_k$ and such that $\\gamma$ extends \\textit\nto $X \\setminus D_K$.\n\nThen there exists a proper birational morphism $q: \\goth X' \\buildrel\\sim\\over\\lgr \\goth X$  \nsuch that $\\goth X'$ is a regular model of $X$ and \nsuch that $D'=q^*D$ is  a strict normal crossing divisor  and  \n$$\n\\gamma \\in \\mathrm{Im}\\Bigl( H^1(\\goth X' \\setminus D', \\goth H) \\to H^1(F, \\goth H) \\Bigr).\n$$\n\\end{slemma}\n\n\n\\begin{proof} Using a passage to the limit argument \\cite{Mg}, there exists an open affine subscheme \n$\\mathfrak{U}_1 \\subset \\goth X \\setminus D$\nsuch that $\\gamma$ extends to a class $\\gamma_1 \\in H^1(\\mathfrak{U}_1, \\goth H)$.\nAccording to \\cite[cor 1.8]{GP1}, $\\gamma_1$ extends to a class\n$\\gamma_2 \\in H^1(\\mathfrak{U}_2, \\goth H)$ where $\\mathfrak{U}_2$ is an open subscheme of $\\goth X  \\setminus D$\ncontaining $\\mathfrak{U}_1$ and $X \\setminus D_K$. \n\nBy purity (i.e. Theorem \\ref{thm_extend} of the appendix \\ref{app_extend}),\nwe have $H^1(\\goth X \\setminus D, \\goth H) = H^1( \\mathfrak{U}_2, \\goth H)$.\nThus $\\gamma$ extends over  $\\goth X \\setminus D$.\n\n\nAccording to Lipman's theorem we can resolve the singularities of $\\goth X$\nand transform $D$ in a  strict normal crossing divisor, see \\cite[lemma 4.7]{HHK}.\n\\end{proof}\n\n \n\\begin{sproposition} \\label{prop_patching}\n Let $G$ be a reductive $F$--group  and assume that\n $p$ does not divide the order of the automorphism group of\n the  absolute root system of $G_{ad}$.\n \n Let $Z$ be a twisted flag  variety of $G$.\nWe assume that $Z(F_v) \\not = \\emptyset$ for \nall discrete valuations of $F$ arising from models of $X$.\n \nThen there exists a regular model $\\goth X$ of $X$ and \na finite subset $\\mathcal P$ of $Y=\\goth X_k$ (containing)\nall the closed points at which distinct irreducible components meet)\n such that \n for each triple of fields $\\{F_U, F_P,F_{\\mathfrak{p}}\\}$ as in Setting \\ref{setting_hhk},  we have  $Z(F_U) \\not = \\emptyset$ and $Z(F_P) \\not = \\emptyset$.\n \\end{sproposition}\n \n\n \n \\begin{proof}\n Without loss of generality we can assume that $G$ is adjoint.\n   Let $G_0$ be the Chevalley form of $G$ and \n   let $(B_0,T_0)$ be a \n   Killing couple for  $G_0$ and let $\\Delta_0$ be\n   the associated Dynkin diagram. \n The variety $Z$ is a form of the variety $\\text{\\rm{Par}}_I(G_0)$\n  of parabolic subgroups of type $I$ where $I \\subset \\Delta_0$ is stable\n   under the star action defined by the $\\text{\\rm{Aut}}(G_0)$--torsor\n   $Q=\\mathrm{Isom}(G_0,G)$.  \n   In particular $Q$ admits a reduction $Q_I$ to the stabilizer \n   $\\text{\\rm{Aut}}_I(G_0)$ for the action  $\\text{\\rm{Aut}}(G_0)$ on $\\Delta_0$ through\nthe morphism $\\text{\\rm{Aut}}(G_0) \\to \\text{\\rm{Out}}(G_0)\\buildrel\\sim\\over\\lgr \\text{\\rm{Aut}}(\\Delta_0)$.\nFurthermore $Z$ is isomorphic to  $^{Q_I} \\! \\text{\\rm{Par}}_I(G_0)$.\n   \nWe apply  now Theorem 1.1.(b) of \\cite{CGR}\nto the $\\mathbb{Z}$--group scheme $\\text{\\rm{Aut}}_I(G_0)$. \nIt provides a finite $\\mathbb{Z}$--subgroup $S_0$ of $\\text{\\rm{Aut}}_I(G_0)$ such that the map $H^1(F, S_0) \\to H^1(F, \\text{\\rm{Aut}}(G_0))$ is onto.\nFurthermore the construction of $S_0$ is explicit in the proof,\nit is an extension of the finite constant group \n $\\text{\\rm{Aut}}_I(G_0)\/T_0$ by a finite subgroup of $T_0$.\n In particular $S_0$ is finite  free over $\\mathbb{Z}$  and our assumption on the characteristic implies that \n$S_{0,T}$ is finite \\'etale of rank prime to $p$. \n   It follows that $Q_I$ admits a reduction to an $F$--torsor \n   $E$ under $S_0$.  \n   \n   \n   The  finite flat $\\mathbb{Z}$--group scheme $S_{0,A}$ admits \n   a faitful representation $S_0 \\hookrightarrow \\GL_{N,\\mathbb{Z}}$\n   \\cite[\\S 1.4.5]{BT} and the quotient \n   $\\GL_{N,\\mathbb{Z}} \/S_0$ is representable by an affine $\\mathbb{Z}$-scheme \\cite[\\S III.2.6]{DG}.\n   \n   \n  According to Lemma \\ref{lem_extend},\n  there exists  a regular model $\\goth X$ of $X$ and \n  a strict normal crossing divisor  $D$ containing  the irreducible components $Y_k$ such that $E$ extends \n  to a $\\goth X \\setminus D$--torsor $\\mathfrak{E}$  under $S_0$.\n  We put $\\mathfrak{Q}_I= \\mathfrak{E} \\wedge^{S_0}\\text{\\rm{Aut}}_I(G_0)$ and consider\n  the $\\goth X \\setminus D$-group scheme $\\mathfrak{G}= {^{\\mathfrak{Q}_I}\\!G_0}$ of generic fiber $G$.\n  \n Let $Y_1,\\dots, Y_d$ be the irreducible components of the closed fiber $Y=\\goth X_k$\n with respective generic points $\\eta_1, \\dots,  \\, \\eta_d$. \n According to  \\cite[prop. 5.8]{HHK2}, there exists non-empty affine subsets \n $U_i \\subset Y_i$ ($i=1,\\dots, d$) such that $Z(F_{U_i}) \\not = \\emptyset$\n for $i=1,\\dots, d$ and $U_i \\cap U_j = \\emptyset$ for $i<j$.\n Up to refine  the model, we can assume that $Y_i \\setminus U_i \\subset  D_k$ for $i=1,...,d$.\n \n We put $\\mathcal P= Y \\setminus \\cup_i Y_i$ and want to establish that \n $Z(F_P) \\not = \\emptyset$ for each $P \\in \\mathcal P$.\n We are given a point $P \\in \\mathcal P$ and pick a height one prime $\\mathfrak{p}$ in $\\widehat R_P$ that contains\n  $t$. It  defines a branch of $Y$ at $P$ lying \n on some irreducible component $Y_i$ of $Y$.\n We deal now with $U=U_i$.\n\nWe consider the local ring $A=R_P$ of $\\goth X$ at $P$ and denote by $A_D$\nits localization at $D$. Since $S_{0,T}$ is finite \\'etale of degree prime\nto $p$, $H^1(A_D,S_0)$ consists in loop torsors as defined in \\cite[\\S 2.3, lemma 2.3.(2)]{Gi2},  i.e. those arising from cocycles related to tame Galois covers of $A_D$. It follows that the $A_D$--torsor\n$Q_I$ is a loop $\\text{\\rm{Aut}}_I(G)$--torsor \\cite[lemma 2.3.(3)]{Gi2},\nso that $\\mathfrak{G} \\times_{\\goth X \\setminus D} A_D$ is \nby definition a loop reductive group scheme.\n\nLet $F_{P,v}$ be the completion of the field $F_P$ for the valuation associated to the blow-up of $\\mathop{\\rm Spec}\\nolimits(A)$ at its closed point.\nOur assumption states in particular that $Z(F_{P,v}) \\not = \\emptyset$,\nthat is, $G_{F_{P,v}}$ admits a parabolic subgroup of type $I$.\nAccording to \\cite[th. 4.1 , (iii) $\\Longrightarrow$ (i)]{Gi2}, $\\mathfrak{G} \\times_{\\goth X \\setminus D} A_D$\nadmits a parabolic subgroup of type $I$. \nA fortiori $G_{F_P}$ admits a parabolic subgroup of type \n$I$ so that $Z(F_P) \\not = \\emptyset$.\n \\end{proof}\n\n\n\\begin{sremarks}{\\rm \n(a) If $p=0$, the result used \\cite[th. 4.1]{Gi2}\nadmits a simple proof, see \\cite[Ex. 4.2]{Gi2},\nby using the analogous result over Laurent polynomials\n\\cite[th. 7.1]{GP2}.\n\n\\smallskip\n\n\\noindent (b) In nice cases inspection of the proof permits \nto weaken the assumption on $p$. \nThe precise condition is that the $\\text{\\rm{Aut}}(G_0)$--torsor\n$\\mathrm{Isom}(G_0,G)$ admits a reduction to a finite\n$F$--subgroup whose degree is prime to $p$.\nFor example in type $G_2$, we need to assume only \nthat $p$ is prime to $2$.\n}\n\\end{sremarks}\n\n \n Together with Theorem \\ref{thm_main_hhk}.(1), we obtain the following consequence.\n \n \n \\begin{stheorem} \\label{thm_patching} Let $G$ be a reductive\n $F$--group and assume that $p$ does not divide the order of the automorphism group of the  absolute root system of $G_{ad}$.\n Let $Z$ be a twisted flag $F$--variety of $G$.\n Then $Z(F) \\not = \\emptyset$ if and only if $Z(F_v) \\not = \\emptyset$\n for all discrete valuations of $F$ arising from models of $X$.\n \\end{stheorem}\n\n\n \\section{Appendix: characterization of parabolic subgroups.} \\label{app_parabolic}\n Let $G$ be a reductive $F$--group \nover a field $F$. We remind the reader \nthat an algebraic $F$--subgroup $P$ of $G$ is parabolic\nif $P$ is smooth and  $G\/P$  is a projective $F$--variety.\nThe interest of the  probably known statement below\nis only in positive characteristic since in this case\nthere  exist $F$--subgroups $Q$ which are not smooth such that \n $G\/Q$ is a projective $F$--variety \\cite{W}. \n  \n\\begin{sproposition}\\label{prop_parabolic} \nLet $P$ be a $k$--subgroup of $G$\nsuch that the quotient variety $G\/P$ is projective.\nThen the following assertions are equivalent: \n\n\\smallskip\n\n(i) $P$ is an $F$--parabolic subgroup;\n\n\n\\smallskip\n\n(ii) For each $F$--field $E$, $G(E)$ acts transitively\non $(G\/P)(E)$;\n\n\\smallskip\n\n(iii) The quotient map $G \\to G\/P$ admits a rational section;\n \n\\smallskip\n\n(iv) $P$ is smooth connected;\n\n\n\\smallskip\n\n(v) $P$ is smooth.\n \n \n\\end{sproposition} \n \n \\begin{proof}\n\\noindent  $(i) \\Longrightarrow (ii)$.\nSince $(G\/P)(E)$ parameterizes the $E$--parabolic\nsubgroups of $G_E$ of same type that $P$,\n Borel-Tits' conjugacy theorem \\cite[th. 4.13.c]{BoT65}\n shows that $G(E)$ acts transitively on $(G\/P)(E)$. \n\n\n\\smallskip\n\n\\noindent  $(ii) \\Longrightarrow (iii)$.\nOur assumption rephrases by saying that\nthe map $G(E) \\to (G\/P)(E)$ is onto for each $F$--field $E$.\nApplying that to the function field $E=F(G\/P)$\nof the smooth connected $F$- variety $G\/P$\nprovides a rational section of the map $G \\to G\/P$.\n \n \n\\smallskip\n\n\\noindent  $(iii) \\Longrightarrow (iv)$.\nTo show the smoothness of $P$ we can assume\nthat $F$ is algebraically closed.\nThen the neutral component  $Q=(P_{red})^0$\nof the reduced $F$--subgroup $P_{red}$ of $P$ is smooth. Furthermore \nthe quotient $F$--variety $P\/Q$ is finite. It follows that the\nmorphism $q: G\/Q \\to G\/P$ is finite and  a fortiori projective\n\\cite[Tag 0B3I]{St}. Since the composition of projective morphisms (of qcqs schemes) is projective  \\cite[Tag 0C4P]{St}, it follows that \n$G\/Q$ is projective. The $F$--subgroup $Q$ of $G$\nis then parabolic. Our assumption is that the morphism\n$G \\to G\/P$ has a rational section\n and so \nhas a fortiori the finite morphism $q: G\/Q \\to G\/P$.\nAccording to \\cite[cor. 6.1.15]{EGA2}, $q$ is  an isomorphism. Thus $Q=P$ and we conclude that $P$ is smooth connected.\n \n\n\n\\smallskip\n\n\\noindent  $(iv) \\Longrightarrow (v)$. Obvious.\n\n\\smallskip\n\n\\noindent $(v) \\Longrightarrow (i)$.\nThis is by definition.\n\\end{proof}\n\n A variant is the following.\n\n\\begin{sproposition}\\label{prop_homog} \nLet $X$ be a smooth  projective $G$--variety.\nThen the following statements are equivalent: \n\n\\smallskip\n\n(i) $X$ is a $F$--variety of parabolic subgroups of $G$;\n\n\n\\smallskip\n\n(ii) For each $F$--field $E$, $G(E)$ acts transitively\non $X(E)$.\n\n\\end{sproposition} \n  \n \n\\begin{proof}\nThe implication $(i) \\Longrightarrow (ii)$ is again\nBorel-Tits' conjugacy theorem.\nWe assume (ii). According to \\cite[prop. 1.3]{MPW},\nwe can assume that $F$ is separably closed.\nSince $X$ is smooth, we have $X(F) \\not = \\emptyset$ \nand denote by  $P$ the stabilizer of some $F$--point $x$.\nAccording to \\cite[prop. III.3.2.1]{DG}.\ncondition (ii) implies that the orbit map $G \\to X$, $g \\mapsto g.x$\ninduces an isomorphism $f_x: G\/P \\buildrel\\sim\\over\\lgr X$.\nProposition \\ref{prop_parabolic},  \n$(ii) \\Longrightarrow (i)$, shows that \n$P$ is a $F$--parabolic subgroup.\nThus $X$ is a $F$--variety of parabolic subgroups of $G$;\n\\end{proof} \n \n \n \\begin{sremark} {\\rm\n The condition (ii) is called\n{\\it transitive action of $G$ on $X$}\nby Harbater-Hartmann-Krashen.\nIt   occurs in \\cite[th. 3.7]{HHK}. \nProjective homogeneous varieties in the result quoted above are exactly\nthe various varieties of parabolic subgroups.\n }\n \\end{sremark} \n \n \n \\section{Appendix: extending torsors} \\label{app_extend}\n \n We come back to a  purity result of Colliot-Th\\'el\\`ene and Sansuc.\n \n \\begin{stheorem}\\label{thm_extend}\n Let $X$ be a regular scheme of dimension $2$.\n Let $U$ be an open subcheme of $X$ which contains  $X^{(1)}$. \n Let $G$ be an affine $X$--group scheme. In the following  cases\n \n \\smallskip\n \n (i) $G$ is reductive,\n \n \\smallskip\n \n (ii) There exists a  Zariski cover $(U_i)_{i \\in I}$ of $X$ \n such that each $G_{U_i}$ admits a closed embedding $G_{U_i} \\subset \\GL_{n_i}$ such that \n $\\GL_{n_i}\/G_{U_i}$ is representable by an affine $U_i$--scheme,\n \n  \\smallskip\n  \nthen we have the equality $H^1_{\\text{\\rm fppf}}(X,G) \\buildrel\\sim\\over\\lgr H^1_{\\text{\\rm fppf}}(U,G)$.\n\n \\end{stheorem}\n\n\n \\begin{proof}\n  The case (i) is \\cite[th. 6.13]{CTS}. The case (ii) goes by inspection of \n  the proof. \n \\end{proof}\n \n \n \\bigskip\n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nThis article contains the completion of the project started in a previous paper \\cite{BBT1}, \nwhere we proved that there exists an ergodic conductance environment such that the \nweak (annealed) invariance principle holds for the corresponding  \ncontinuous time random walk along a subsequence but the quenched invariance principle does not hold.\nIn the present paper we give a proof of the full scaling limit for the weak invariance principle, improving the\nresult in \\cite{BBT1}.\nThe improved result is, in a sense, a quantitative form of the invariance \nprinciple. The proof consists of several lemmas. Some of them are specific to our \nmodel but some of them have the more general character and may serve as \ntechnical elements for related projects. Since this paper is a continuation of \n\\cite{BBT1}, we start by presenting basic notation and definitions from that paper.\n\nLet $d\\geq 2$ and let $ E_d$ be the set of all non oriented edges in the \n$d$-dimensional integer lattice, that is,  $E_d = \\{e = \\{x,y\\}: x,y \\in {\\mathbb Z}^d, |x-y|=1\\}$.\nLet $\\{\\mu_e\\}_{e\\in E_d}$ be a random process with non-negative values, \ndefined on some probability space $(\\Omega, \\mathcal{F}, \\Pp)$. \nThe process $\\{\\mu_e\\}_{e\\in E_d}$ represents random conductances.  \nWe write $\\mu_{xy}  = \\mu_{yx} = \\mu_{\\{x,y\\}}$ and set \n$\\mu_{xy}=0$ if $\\{x,y\\} \\notin E_d$. Set\n\\begin{align*}\n\\mu_x = \\sum_y \\mu_{xy}, \\qquad P(x,y) = \\frac{\\mu_{xy}}{\\mu_x},\n\\end{align*}\nwith the convention that $0\/0=0$ and $P(x,y)=0$ if $\\{x,y\\} \\notin E_d$. \nFor a fixed $\\omega\\in \\Omega$, let  \n$X = \\{X_t, t\\geq 0, {P}^x_\\omega, x \\in {\\mathbb Z}^d\\}$ be the \ncontinuous time random walk on ${\\mathbb Z}^d$, with transition probabilities \n$P(x,y) = P_\\omega(x,y)$, and exponential waiting times with mean $1\/\\mu_x$. \nThe corresponding expectation will be denoted ${E}_\\omega^x$. \nFor a fixed $\\omega\\in \\Omega$, the generator $\\mathcal{L}$ of $X$ is given by\n\\begin{align}\\label{e:Ldef}\n\\mathcal{L} f(x) = \\sum_y \\mu_{xy} (f(y) - f(x)).\n\\end{align}\nIn \\cite{BD} this is called the {\\em variable speed random walk} (VSRW)\namong the conductances $\\mu_e$.\nThis model, of a reversible (or symmetric) random walk in a random environment, is\noften called the Random Conductance Model. \n\nWe are interested in functional Central Limit Theorems (FCLTs)\nfor the process $X$. Given any process $X$,\nfor $\\varepsilon>0$, set $X^{\\varepsilon}_t = \\varepsilon X_{t \/\\varepsilon^2}$, $t\\geq 0$. \nLet $\\mathcal{D}_T = D([0,T], \\mathbb{R}^d)$ denote the Skorokhod space, \nand let $\\mathcal{D}_\\infty=D([0,\\infty), \\mathbb{R}^d)$.\nWrite $d_S$ for the Skorokhod metric and $\\mathcal{B}(\\mathcal{D}_T)$ for the $\\sigma$-field of \nBorel sets in the corresponding topology. \nLet $X$ be the canonical process on $\\mathcal{D}_\\infty$ or $\\mathcal{D}_T$, $P_{\\text{BM}}$ be Wiener \nmeasure on $(\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$  and let $E_{\\text{BM}}$ be the \ncorresponding expectation. \nWe will write $W$ for a standard Brownian motion.\nIt will be convenient to assume that $\\{\\mu_e\\}_{e\\in E_d}$ are \ndefined on a probability space $(\\Omega, \\mathcal{F}, \\bP)$, and that\n$X$ is defined on $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$ \nor  $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$. \nWe also define the averaged or annealed measure ${\\bf P}$ on \n$(\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$ or  $(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$ by\n\\begin{equation} \\label{e:bfPdef}\n {\\bf P}(G) = \\bE P^0_{\\omega}(G). \n\\end{equation}\n\n\\begin{definition}\\label{j1.2}\nFor a bounded function $F$ on $\\mathcal{D}_T$ and a constant matrix $\\Sigma$, let \n$\\Psi^F_\\varepsilon = {E}^0_\\omega F(X^{\\varepsilon})$ and \n$\\Psi^F_\\Sigma = E_{\\text{BM}} F(\\Sigma W)$. We will use $I$ to denote the identity matrix.\n\n\\smallskip \\noindent (i) We say that the {\\em Quenched Functional CLT} (QFCLT) holds \nfor $X$ with limit $\\Sigma W$ if for every $T>0$ and \nevery bounded continuous function $F$ on $\\mathcal{D}_T$ we \nhave $\\Psi^F_\\varepsilon \\to \\Psi^F_\\Sigma$ as $\\varepsilon\\to 0$, with $\\Pp$-probability 1.\\\\\n(ii) We say that the {\\em Weak Functional CLT} (WFCLT) \nholds for $X$ with limit $\\Sigma W$ if for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_\\varepsilon \\to \\Psi^F_\\Sigma$ as $\\varepsilon\\to 0$, in $\\Pp$-probability.\\\\\n(iii) We say that the {\\em Averaged (or Annealed) Functional CLT}\n(AFCLT) holds for $X$ with limit $\\Sigma W$ if for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$ \\bE \\Psi^F_\\varepsilon \\to \\Psi_{\\Sigma}^F$.\nThis is the same as standard weak convergence with respect to the probability measure ${\\bf P}$. \n\\end{definition}\n\nIf we take $\\Sigma$ to be non-random then, since $F$ is bounded, it is\nimmediate that QFCLT $\\Rightarrow$ WFCLT. In general for the QFCLT the matrix\n$\\Sigma$ might depend on the environment $\\mu_\\cdot({\\omega})$. However, if\nthe environment is stationary and ergodic, then $\\Sigma$ is a shift invariant\nfunction of the environment, so must be $\\bP$--a.s. constant.\nIn \\cite{DFGW} it is proved that if $\\mu_e$ is a stationary ergodic \nenvironment with $\\bE \\mu_e<\\infty$ then the WFCLT holds. In \\cite[Theorem 1.3]{BBT1} \nit is proved that for the random conductance model the AFCLT and WFCLT are equivalent.\n\n\\begin{definition}\nWe say an environment $(\\mu_e)$ on ${\\mathbb Z}^d$ is {\\em symmetric} if the law of  $(\\mu_e)$ is \ninvariant under symmetries of ${\\mathbb Z}^d$. \n\\end{definition}\n\nIf $(\\mu_e)$ is stationary, ergodic and symmetric, and the WFCLT holds with\nlimit $\\Sigma W$ then the limiting covariance matrix $\\Sigma^T \\Sigma$ must also\nbe invariant under symmetries of ${\\mathbb Z}^d$, so must be a constant \ntimes the identity.\n\nIn a previous paper \\cite{BBT1} we proved the following theorem:\n\n\\begin{theorem}\\label{T:oldmain}\nLet $d=2$ and $p<1$.\nThere exists a symmetric stationary ergodic environment $\\{\\mu_e\\}_{e\\in E_2}$\nwith $\\bE (\\mu_e^p \\vee \\mu_e^{-p})<\\infty$ \nand a sequence $\\varepsilon_n \\to 0$ such that\\\\\n(a)  the WFCLT holds for $X^{\\varepsilon_n}$ with limit $W$, \ni.e., for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_{\\varepsilon_n} \\to \\Psi^F_I$ as $n\\to \\infty$, in $\\Pp$-probability,\n\\\\\nbut \\\\\n(b) the QFCLT does not hold for  $X^{\\varepsilon_n}$ with limit $ \\Sigma W$ for any $\\Sigma$.  \n\\end{theorem}\n\nIn this paper we prove that for an environment similar to\nthat in Theorem \\ref{T:oldmain} the WFCLT holds for $X^{\\varepsilon}$ as $\\varepsilon \\to 0$,\nand not just along a subsequence.\n\n\\begin{theorem}\\label{T:main}\nLet $d=2$ and $p<1$.\nThere exists a symmetric stationary ergodic environment $\\{\\mu_e\\}_{e\\in E_2}$\nwith $\\bE (\\mu_e^p \\vee \\mu_e^{-p})<\\infty$ \nsuch that\\\\\n(a)  the WFCLT holds for $X^{\\varepsilon}$ with limit $W$, \ni.e., for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_{\\varepsilon} \\to \\Psi^F_I$ as $\\varepsilon \\to 0$, in $\\Pp$-probability,\n\\\\\nbut \\\\\n(b) the QFCLT does not hold for  $X^{\\varepsilon}$ with limit $ \\Sigma W$ for any $\\Sigma$.  \n\\end{theorem}\n\nFor more remarks on this problem see \\cite{BBT1}.\n\n\\smallskip \\noindent {\\bf Acknowledgment.}\nWe are grateful to Emmanuel Rio, Pierre Mathieu, Jean-Dominique Deuschel \nand Marek Biskup for some very useful discussions.\n\n\\section{Description of the environment}\\label{const} \n\nHere we recall the environment given in \\cite{BBT1}. We refer the reader to that\npaper for proofs of some basic properties.\n\nLet $\\Omega = (0,\\infty)^{E_2}$, and $\\mathcal{F}$ be the Borel $\\sigma$-algebra defined \nusing the usual product topology. Then every $t\\in{\\mathbb Z}^2$ defines a transformation \n$T_t (\\omega)=\\omega +t$ of $\\Omega$. Stationarity and ergodicity of the measures \ndefined below will be understood with respect to these transformations. \n\nAll constants (often denoted $c_1, c_2$, etc.) are assumed to be strictly positive and finite.\nFor a set $A \\subset {\\mathbb Z}^2$ let $E(A)\\subset E_2$ be the set of all edges with both endpoints in\n$A$. Let $E_h(A)$ and $E_v(A)$ respectively\nbe the set of horizontal and vertical edges in $E(A)$.\nWrite $x \\sim y$ if $\\{x,y\\}$ is an edge in ${\\mathbb Z}^2$. Define the exterior boundary of $A$ by\n$$ {\\partial} A =\\{ y \\in {\\mathbb Z}^2 -A: y \\sim x \\text{ for some } x \\in A \\}. $$\nLet also\n$$ {\\partial}_i A = {\\partial}({\\mathbb Z}^2 -A). $$ \nDefine balls in the $\\ell^\\infty$ norm by $\\mathcal{B}(x,r)= \\{y: ||x-y||_\\infty \\le r\\}$; of \ncourse this is just the square with center $x$ and side $2r$.\n\nLet $\\{a_n\\}_{n\\geq 0}$,  $\\{ \\beta_n\\}_{n \\ge 1}$ and $\\{b_n\\}_{n\\geq 1}$ be  \nstrictly increasing sequences of positive integers growing to infinity with $n$,\nwith \n$$ 1=a_0 < b_1 < \\beta_1 < a_1 \\ll b_2 <  \\beta_2<   a_2 \\ll b_3 \\dots $$\nWe will impose a number of conditions on these sequences in the course\nof the paper. We collect the main ones here.\nThere is some redundancy in the conditions, for easy reference.\n\n\\begin{enumerate}[(i)]\n\\item  $a_n$ is even  for all $n$. \n\\item For each $n \\ge 1$, $a_{n-1}$ divides $b_n$, \nand $b_n$ divides $\\beta_n$ and $a_n$. \n\\item  $b_1 \\geq 10^{10}$.\n\\item  $a_n\/\\sqrt{2n} \\le     b_n \\le a_n \/  \\sqrt{n} $ for all $n$, and\n$b_n \\sim a_n\/\\sqrt{n}$.\n\\item $b_{n+1} \\ge 2^n b_n$ for all $n$.\n\\item $b_n > 40 a_{n-1}$ for all $n$.\n\\item $b_n$ is large enough so that the estimates (5.1) and (6.1) of \\cite{BBT1} hold.\n\\item $100 b_n  < \\beta_n \\le  b_n n^{1\/4}  < 2 \\beta_n <   a_n\/10$ for $n$ large enough.\n\\end{enumerate}\n\nIn addition, at various points in the proof, we will assume that $a_n$ is sufficiently much\nlarger than $b_{n-1}$ so that a process $X^{(n-1)}$ defined below is such that for $a\\ge a_n$\nthe rescaled process\n$$ (a^{-1} X^{(n-1)}_{a^2 t}, t\\ge 0)$$\nis sufficiently close to Brownian motion.\nWe will mark the places in the proof where we impose these extra conditions by ($\\clubsuit$) .\n\n\n\\smallskip\\noindent\nWe begin our construction by defining a collection of squares in ${\\mathbb Z}^2$. Let\n\\begin{align*} \nB_n &= [0, a_n]^2,  \\\\\nB_n' &= [0, a_n-1]^2 \\cap {\\mathbb Z}^2 ,\\\\\n\\mathcal{S}_n(x) &= \\{ x + a_n y + B_n': \\, y \\in {\\mathbb Z}^2 \\}.\n\\end{align*} \nThus $\\mathcal{S}_n(x)$ gives a tiling of ${\\mathbb Z}^2$ by disjoint squares of side $a_n-1$\nand period $a_n$.\nWe say that the tiling $\\mathcal{S}_{n-1}(x_{n-1})$ is a refinement\nof $\\mathcal{S}_n(x_n)$ if every square $Q \\in \\mathcal{S}_n(x_n)$ is a finite\nunion of squares in $\\mathcal{S}_{n-1}(x_{n-1})$. It is clear that \n$\\mathcal{S}_{n-1}(x_{n-1})$ is a refinement of $\\mathcal{S}_n(x_n)$ if\nand only if $x_n = x_{n-1}+ a_{n-1}y$ for some $y \\in {\\mathbb Z}^2$.\n\nTake $\\mathcal{O}_1$ uniform in $B'_1$, and for $n\\geq 2$\ntake $\\mathcal{O}_n$, conditional on $(\\mathcal{O}_1, \\dots, \\mathcal{O}_{n-1})$, \nto be uniform in $B'_n \\cap ( \\mathcal{O}_{n-1} + a_{n-1}{\\mathbb Z}^2)$. We now define random tilings by letting\n\\begin{equation*}\n \\mathcal{S}_n = \\mathcal{S}_n( \\mathcal{O}_n), \\, n \\ge 1.  \n\\end{equation*}\n\nLet $\\eta_n$, $K_n$ be positive constants; we will have $\\eta_n \\ll 1 \\ll K_n$.\nWe define conductances  on $E_2$ as follows. \nRecall that $a_n$ is even, and let $a_n' = \\frac12 a_n$. Let\n$$ C_n = \\{ (x,y) \\in B_n \\cap {\\mathbb Z}^2: y \\ge x, x+y \\le a_n \\}. $$\nWe first define conductances $\\nu^{n,0}_e$ for $e \\in E(C_n)$. Let\n\\begin{align*}\nD_n^{00} &= \\big\\{ (a'_n - \\beta_n,y), a'_n - 10 b_n \\le y \\le a'_n + 10 b_n \\big\\},  \\\\\nD_n^{01} &= \\big\\{ (x, a'_n + 10 b_n),  (x, a'_n + 10 b_n + 1), (x, a'_n - 10 b_n),  (x, a'_n - 10 b_n -1),   \\\\\n\\nonumber   \n   & \\quad \\quad \\quad a'_n -\\beta_n -b_n \\le x \\le a'_n -\\beta_n + b_n \\big\\}.\n\\end{align*}\nThus the set $D^{00}_n \\cup D_n^{01}$ resembles the letter I (see Fig.~\\ref{fig1}).\n\nFor an edge $e \\in E(C_n)$ we set \n\\begin{align*} \n \\nu^{n,0}_{e}  &= \\eta_n \\quad \\text {if } e \\in E_v(D^{01}_n), \\\\\n \\nu^{n,0}_{e}  &= K_n \\quad \\text {if } e \\in E(D^{00}_n), \\\\\n  \\nu^{n,0}_{e}  &= 1 \\quad \\text {otherwise.} \n\\end{align*} \n\n\\begin{figure} \\includegraphics[width=4cm]{fig1_1}\n\\caption{The set $D^{00}_n \\cup D_n^{01}$ resembles the letter I.\nBlue edges have very low conductance. The red line represents edges with very \nhigh conductance. Drawing not to scale. \n}\n\\label{fig1}\n\\end{figure}\n\nWe then extend $\\nu^{n,0}$ by symmetry to $E(B_n)$.\nMore precisely,\nfor $z =(x,y) \\in B_n$, let $R_1 z=( y,x)$ and  $R_2z = (a_n-y,a_n-x)$, so that\n$R_1$ and $R_2$ are reflections in the lines $y=x$ and $x+y=a_n$.\nWe define $R_i$ on edges by $R_i (\\{x,y\\}) = \\{R_i x, R_i y \\}$ for $x,y \\in B_n$. \nWe then extend $\\nu^{0,n}$ to $E( B_n)$ so that\n$\\nu^{0,n}_e =  \\nu^{0,n}_{R_1 e }=\\nu^{0,n}_{R_2 e }$ for  $e \\in E(B_n)$.\nWe define the {\\em obstacle} set $D_n^0$ by setting\n$$ D_n^0 = \\bigcup_{i=0}^1 \\big( D_n^{0,i} \\cup R_1(D_n^{0,i})  \\cup  R_2(D_n^{0,i})\n \\cup R_1R_2 (D_n^{0,i} ) \\big). $$\nNote that $\\nu^{n,0}_e=1$ for every edge adjacent to the boundary of $B_n$,\nor indeed within a distance $ a_n\/4$ of this boundary.\nIf $e=(x,y)$, we will write $e-z = (x-z,y-z)$. \nNext we extend $\\nu^{n,0}$ to $E_2$ by periodicity, i.e.,\n$\\nu^{n,0}_e = \\nu^{n,0}_{e+ a_n x}$ for all $x\\in {\\mathbb Z}^2$.\nWe define the conductances $\\nu^n$ by translation by $\\mathcal{O}_n$, so that\n\\begin{equation*}\n \\nu^n_e =\\nu^{n,0}_{e-\\mathcal{O}_n}, \\, e \\in E_2.\n\\end{equation*}\nWe also define the obstacle set at scale $n$ by\n\\begin{equation}\\label{ma26.1}\n D_n = \\bigcup_{ x \\in {\\mathbb Z}^2} (a_n x + \\mathcal{O}_n + D^0_n ).\n\\end{equation}\nWe will sometimes call the set $D_n$ the set of $n$th level obstacles.\n\n\nWe define the environment $\\mu^n_e$ inductively by\n\\begin{align*}\n \\mu^n_e &= \\nu^{n}_e \\quad \\text{ if } \\nu^n_e \\neq 1, \\\\\n \\mu^n_e &= \\mu^{n-1}_e \\quad \\text{ if } \\nu^n_e=1.\n\\end{align*}\nOnce we have proved the limit exists, we will set\n\\begin{equation} \\label{e:mudef}\n \\mu_e = \\lim_n \\mu^n_e.\n\\end{equation}\n\n\n\n\\begin{lemma} \\label{L:erg} (See  \\cite[Theorem 3.1]{BBT1}).\\\\\n(a) The environments $(\\nu^n_e, e\\in E_2)$, $(\\mu^n_e, e\\in E_2)$\nare stationary, symmetric and ergodic.\\\\\n(b) The limit \\eqref{e:mudef} exists $\\bP$--a.s. \\\\\n(c) The environment $(\\mu_e, e \\in E_2)$ is  stationary, symmetric and ergodic.\n\\end{lemma}\n\n\n\nNow let \n\\begin{align}\\label{j27.4}\n\\mathcal{L}_n f(x) = \\sum_{y} \\mu^n_{xy} (f(y)-f(x)), \n\\end{align}\nand $X^{(n)}$ be the associated Markov process. Set\n\\begin{equation} \\label{e:etadef}\n  \\eta_n =   b_n^{-(1+1\/n)}, \\, n \\ge 1.\n\\end{equation}\nFrom Section 4 of \\cite{BBT1} we have:\n\n\\begin{theorem} \\label{T:eK}\nFor each $n$ there exists a constant $K_n$, depending on $\\eta_1, K_1, \\dots \\eta_{n-1}, K_{n-1}$,\nsuch that the QFCLT holds for $X^{(n)}$ with limit $W$.\n\\end{theorem}\n\nFor each $n$ the process $X^{(n)}$ has invariant measure which is counting measure \non ${\\mathbb Z}^2$. For $x \\in \\mathbb{R}^2$ and $a>0$ write $[xa]$ for the point in ${\\mathbb Z}^2$ closest to $xa$.\n(We use some procedure to break ties.) We have the following bounds on the transition\nprobabilities of $X^{(n)}$ from \\cite{BZ}. We remark that the constant $M_n$ below is\nnot effective -- i.e. the proof does not give any control on its value. \nWrite $k_t(x,y) = (2\\pi t)^{-1} \\exp( -|x-y|^2\/2t)$ for the transition density of Brownian motion\nin $\\mathbb{R}^2$, and\n$$ p^{{\\omega},n}_t(x,y) = P^x_{\\omega}( X^{(n)}_t =y )$$\nfor the transition probabilities for $X^{(n)}$.\n\n\\begin{lemma} \\label{L:hkXn} \nFor each $0< \\delta < T$  there exists $M_n=M_n(\\delta,T)$ such that for $a \\ge M_n$ \n\\begin{equation} \\label{e:GB1}\n\\frac12 k_t(x,y) \\le a^{2} p^{{\\omega},n}_{a^2t}([xa],[ya])  \\le 2 k_t(x,y) \\, \\hbox { for all }\n \\delta \\le t \\le T, |x|, |y| \\le T^2.\n\\end{equation}\n\\end{lemma} \n\n\n\n\n\\section{Preliminary results}\n\n\nSince a proof of Theorem \\ref{T:oldmain}(b) was given in \\cite{BBT1}, \nall we need to prove is part (a) of Theorem \\ref{T:main}.\nThe argument consists of several lemmas. We start with some preliminary \nresults on weak convergence of probability measures on the space of c\\`adl\\`ag functions. \nRecall the definitions of the measures $\\bP$ and $P^0_{\\omega}$.\n\nRecall that  $\\mathcal{D} := \\mathcal{D}_1 = D([0,1], \\mathbb{R}^2)$ denotes the space of c\\`adl\\`ag functions \nequipped with the Skorokhod metric ${\\rm d_S}$ defined as follows (see \\cite[p.~111]{B}). \nLet $\\Lambda$ be the family of continuous strictly increasing functions $\\lambda$ \nmapping $[0,1]$ onto itself. In particular, $\\lambda(0) =0$ and $\\lambda(1) =1$. \nIf $x(t), y(t) \\in  \\mathcal{D}$ then  \n\\begin{align*}\n{\\rm d_S}(x,y) = \\inf_{\\lambda \\in \\Lambda}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|, \\sup_{t\\in[0,1]} |y(\\lambda(t)) - x(t)| \\Big).\n\\end{align*}\nFor $x(t) \\in  \\mathcal{D}$, let $\\Osc(x, \\delta) = \\sup\\{|x(t)-x(s)|: s,t\\in[0,1], |s-t|\\le \\delta\\}$.\n\n\\begin{lemma}\\label{d21.2}\nSuppose that $\\sigma: [0,1] \\to [0,1]$ is continuous, non-decreasing and $\\sigma(0) = 0$ \n(we do not require that $\\sigma(1) = 1$).  \nSuppose that $|\\sigma(t) - t| \\le \\delta$ for all $t\\in[0,1]$.\nLet  $\\varepsilon\\geq0$, $\\delta_1>0$, $x, y \\in  \\mathcal{D}$ with\n${\\rm d_S}(x(\\,\\cdot\\,), y(\\,\\cdot\\,))\\le \\varepsilon$, and\n$\\Osc(x, \\delta) \\vee \\Osc(y, \\delta) \\le \\delta_1$. Then\n${\\rm d_S}(x(\\sigma(\\,\\cdot\\,)), y(\\sigma(\\,\\cdot\\,))) \\le \\varepsilon + 2\\delta_1$.\n\\end{lemma}\n\n\\begin{proof}\nFor any $\\varepsilon_1> \\varepsilon$ there exists $\\lambda\\in \\Lambda$ such that,\n\\begin{align*}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |y(\\lambda(t)) - x(t)| \\Big)\\le\\varepsilon_1.\n\\end{align*}\nWe have for $\\lambda$ satisfying the above condition,\n\\begin{align*}\n&\\sup_{t\\in[0,1]} |y(\\sigma(\\lambda(t))) - x(\\sigma(t))|\\\\\n&\\qquad \\le \n\\sup_{t\\in[0,1]} (|y(\\sigma(\\lambda(t))) - y(\\lambda(t))|\n+ |y(\\lambda(t)) - x(t)| + |x(t) - x(\\sigma(t))|) \\\\\n&\\qquad \\le \\Osc(y,\\delta) + \\varepsilon_1 + \\Osc(x,\\delta) \\le \\varepsilon_1 + 2 \\delta_1. \n\\end{align*}\nHence,\n\\begin{align*}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |y(\\sigma(\\lambda(t))) - x(\\sigma(t))|\n\\Big) \\le \\varepsilon_1 + 2 \\delta_1.\n\\end{align*}\nTaking infimum over all $\\varepsilon_1 > \\varepsilon$ we obtain\n${\\rm d_S}(x(\\sigma(\\,\\cdot\\,)), y(\\sigma(\\,\\cdot\\,)))\n\\le \\varepsilon + 2\\delta_1$.\n\\end{proof}\n\nLet ${d_P}$ denote the Prokhorov distance between probability measures on a probability space defined \nas follows (see \\cite[p.~238]{B}). \nRecall that \n$\\Omega = (0,\\infty)^{E_2}$ and $\\mathcal{F}$ is the Borel $\\sigma$-algebra defined \nusing the usual product topology.\nWe will use measurable spaces $(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$ and \n $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$, for a fixed $T$ (often $T=1$).\nNote that $\\mathcal{D}_T$ and $\\Omega \\times \\mathcal{D}_T$ are metrizable, with the metrics generating the usual topologies. A ball around a set $A$ with radius $\\varepsilon$ will\nbe denoted $\\mathcal{B}(A,\\varepsilon)$ in either space. \nFor probability measures $P$ and $Q$, \n${d_P}(P,Q) $ is the infimum of $\\varepsilon>0$ such that $P(A) \\le Q(\\mathcal{B}(A,\\varepsilon)) + \\varepsilon$ and \n$Q(A) \\le P(\\mathcal{B}(A,\\varepsilon)) + \\varepsilon$ for all Borel sets $A$.\nConvergence in the metric ${d_P}$ is equivalent to the weak convergence of measures.\nBy abuse of notation we will sometimes write arguments of the function \n${d_P}(\\,\\cdot\\,,\\,\\cdot\\,)$ as processes rather than their distributions:  for example we will write\n${d_P}( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}})$.\nWe will use ${d_P}$ for the Prokhorov distance\nbetween probability measures on $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$. We will write ${d_P}_\\omega$ for the metric on the space\n$(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$.\nIt is straightforward to verify that if,  for some processes $Y$ and $Z$, \n${d_P}_\\omega(Y,Z) \\le \\varepsilon$ for $\\bP$--a.a. $\\omega$, then ${d_P}(Y,Z) \\le \\varepsilon$.\n\nWe will sometimes write $W(t)=W_t$ and similarly for other processes.\n\n\\begin{lemma}\\label{d21.1}\nThere exists a function $\\rho: (0,\\infty) \\to (0,\\infty)$ such that $\\lim_{\\delta\\downarrow 0}\n \\rho(\\delta) = 0$ and  the following holds.\nSuppose that $\\delta,\\delta'\\in (0,1)$ and $\\sigma: [0,1] \\to [0,1]$ is a non-decreasing \nstochastic process such that $t-\\sigma_t \\in [0,\\delta]$ for all $t$, with probability greater \nthan $1-\\delta'$. Suppose that $\\{W_t, t\\geq 0\\}$ has the distribution $P_{\\text{BM}}$ and \n$W^*_t = W(\\sigma_t)$ for $t\\in[0,1]$. \nThen ${d_P}(\\{W^*_t, t\\in[0,1]\\}, P_{\\text{BM}}) \\le \\rho(\\delta) + \\delta'$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $W, W^*$ and $\\sigma$ are defined on the sample space with a \nprobability measure $P$.\nIt is \neasy to see that we can choose $\\rho(\\delta)$ so that \n$\\lim_{\\delta\\downarrow 0} \\rho(\\delta) = 0$\nand $P(\\Osc(W,\\delta) \\geq\\rho(\\delta) )<\\rho(\\delta)$. \nSuppose that the event \n$F := \\{\\Osc(W,\\delta) <\\rho(\\delta)\\}\\cap \\{ \\forall t\\in[0,1]: t-\\sigma_t \\in [0,\\delta]\\}$ holds. \nThen taking $\\lambda(t) = t $,\n\\begin{align*}\n{\\rm d_S}(W, W^*) &\\le \\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |W(\\lambda(t)) - W^*(t)| \\Big) \\\\\n&= \\sup_{t\\in[0,1]} |W(t) - W(\\sigma(t))| \\le \\Osc(W, \\delta) < \\rho(\\delta).\n\\end{align*}\nWe see that if $F$ holds and $W \\in A \\subset \\mathcal{D}$ then \n$W^*(\\,\\cdot\\,)\\in \\mathcal{B}( A,\\rho(\\delta))$.\nSince $P(F^c) \\le \\rho(\\delta) + \\delta'$, we obtain\n\\begin{align*}\nP&(W \\in A) \\\\\n&\\leq P(\\{W\\in A\\} \\cap F) + P(F^c) \n\\leq P(\\{W^*\\in \\mathcal{B}( A,\\rho(\\delta))\\} \\cap F) \n+\\rho(\\delta) + \\delta'\\\\\n&\\leq P(W^*\\in \\mathcal{B}( A,\\rho(\\delta))) \n+\\rho(\\delta) + \\delta'.\n\\end{align*}\nSimilarly we have\n$P(W^*\\in A ) \\le P(W\\in \\mathcal{B}( A,\\rho(\\delta)) ) + \\rho(\\delta) + \\delta'$, and\nthe lemma follows.\n\\end{proof}\n\n\\begin{lemma}\\label{ma26.5}\nSuppose that for some processes $X, Y$ and $Z$ on the interval $[0,1]$ we have $Z= X+Y$ and $P(\\sup_{0\\leq t \\leq 1} |X_t| \\leq \\delta) \\geq 1-\\delta$. \nThen ${d_P}(\\{Z_t, t\\in[0,1]\\}, \\{Y_t, t\\in[0,1]\\}) \\le \\delta$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the event \n$F := \\{\\sup_{0\\leq t \\leq 1} |X_t| \\leq \\delta\\}$ holds. \nThen taking $\\lambda(t) = t $,\n\\begin{align*}\n{\\rm d_S}(Z,Y) &\\le \\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |Z(\\lambda(t)) - Y(t)| \\Big) \\\\\n&= \\sup_{t\\in[0,1]} |Z(t) - Y(t)| \\le \\delta.\n\\end{align*}\nWe see that if $F$ holds and $Z \\in A \\subset \\mathcal{D}$ then \n$Y(\\,\\cdot\\,)\\in \\mathcal{B}( A,\\delta)$.\nSince $P(F^c) \\le \\delta$, we obtain\n\\begin{align*}\nP(Z \\in A) &\\leq P(\\{Z\\in A\\} \\cap F) + P(F^c) \n\\leq P(\\{Y\\in \\mathcal{B}( A,\\delta)\\} \\cap F) \n+ \\delta\\\\\n&\\leq P(Y\\in \\mathcal{B}( A,\\delta)) \n+ \\delta.\n\\end{align*}\nSimilarly we have\n$P(Y\\in A ) \\le P(Z\\in \\mathcal{B}( A,\\delta) ) +  \\delta$, and\nthe lemma follows.\n\\end{proof}\n\nRecall that the function $e\\to \\mu^n_e$ is periodic with period $a_n$.\nHence the random field $\\{\\mu^n_e\\}_{e\\in E_2}$ takes only finitely many values --\nthis is a much stronger statement than the fact that $\\mu^n_e$\ntakes only finitely many values.\n\n\nBy Theorem \\ref{T:eK} for each $n \\ge 1$,\n$$ \\lim_{a\\to \\infty} \n{d_P}( \\{  (1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) =0. $$ \nThus ($\\clubsuit$)  we can take $a_{n+1}$ so large that for every $\\omega$, $n \\ge 1$\nand $a\\geq a_{n+1}$, \n\\begin{equation} \\label{e:PdistBM}\n{d_P}_\\omega( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n}.\n\\end{equation}\n\n\\bigskip\n\nLet $\\theta$ denote the usual shift operator for Markov processes, that is, $X^{(n)}_t \\circ \\theta_s = X^{(n)}_{t+s}$ for all $s,t\\geq 0$ (we can and do assume that $X^{(n)}$ is the canonical process on an appropriate probability space). \nRecall that\n$\\mathcal{B}(x,r) =\\{y: ||x-y||_\\infty \\le r\\}$ denote balls in the $\\ell^\\infty$ norm\nin ${\\mathbb Z}^2$ (i.e. squares), $a_n' = a_n\/2$, $B_n=[0,a_n]^2$ and $u_n =(a_n', a'_n)$. Note that $u_n$ is \nthe center of $B_n$.\nWe choose $ \\beta_n$  so that\n\\begin{align}\\label{j2.10}\n b_n n^{1\/8}\n< \\beta_n \\leq \\lfloor b_n n^{1\/4}\\rfloor  < 2 \\beta_n  < a_n\/10,\n\\end{align}\nand we assume that $n$ is large enough so that the above inequalities hold.\nLet $\\mathcal{C}_n =\\{ u_n + \\mathcal{O}_n+ a_n {\\mathbb Z}^2\\}$ be the set of centers of the squares in $\\mathcal{S}_n$, and let\n\\begin{equation} \\label{e:Hdef}\n  \\calK(r) = \\bigcup_{z \\in \\mathcal{C}_n} \\mathcal{B}(z,r).\n\\end{equation}\nNow let\n\\begin{align*}\n\\Gamma^1_n &= \\calK(2\\beta_n), \\\\\n\\Gamma^2_n &= {\\mathbb Z}^2 \\setminus \\calK(4 \\beta_n).\n\\end{align*}\nNow define stopping times as follows.\n\\begin{align*}\nS^n_0 &= T^n_0 = 0,\\\\\nU^n_k & = \\inf\\{t\\geq S^n_{k-1}: X^{(n)}_t \\in \\Gamma^2_n\\}, \\qquad k \\geq 1,\\\\\nS^n_k & = \\inf\\{t \\geq U^n_k: X^{(n)}_t \\in \\Gamma^1_n\\}, \\qquad k \\geq 1,\\\\\nV^n_1 & = \\inf \\Big\\{t \\in \\bigcup_{k\\geq 1} [U^n_k, S^n_k]: \nX^{(n)}_t \\in X^{(n)}(T^n_0) + a_{n-1} {\\mathbb Z}^2 \\Big\\}, \\\\\nT^n_k & = \\inf\\{t\\geq V^n_{k}: X^{(n)}_t \\in \\Gamma^1_n\\}, \\qquad k \\geq 1,\\\\\nV^n_k & = V^n_{1} \\circ \\theta_{T^n_{k-1}} , \\qquad k \\geq 2.\n\\end{align*}\nLet\n$$ J= \\bigcup_{k=1}^\\infty [V^n_k, T^n_k]; $$\nfor $t \\in J$  the process $X^{(n)}$ is a distance at least\n$\\beta_n$ away from any $n$th level obstacle.\nNow set for $t\\geq 0$,\n\\begin{align*}\n\\sigma^{n,1}_t &= \\int_0^t {\\bf 1}_J(s) ds = \\sum_{k=1}^\\infty \\left(T^n_{k} \\land t - V^n_{k} \\land t\\right),\\\\\n\\sigma^{n,2}_t &= t-\\sigma^{n,1}_t = \\sum_{k=0}^\\infty \\left(V^n_{k+1} \\land t - T^n_{k} \\land t\\right).\n\\end{align*}\nLet $\\widehat \\sigma^{n,j}$ denote the right continuous inverses of these processes, given by\n\\begin{equation*}\n\\widehat \\sigma^{n,j}_t = \\inf\\{s\\geq 0: \\sigma^{n,j}_s \\geq t\\}, \\, j=1,2.\n\\end{equation*}\nFinally let \n\\begin{align*}\nX^{n,1}_t &= X^{(n)}_0 + \\int_0^t {\\bf 1}_J(s) dX^{(n)}_s  \\\\\n &=  X^{(n)}_0 + \\sum_{k=0}^\\infty\n\\left(X^{(n)}(T^n_k \\land t) - X^{(n)}(V^n_{k} \\land t)\\right), \\\\\n\\widehat X^{n,1}_t &=  X^{(n)}_0 + X^{n,1}(\\widehat \\sigma^{n,1}_t), \\\\\nX^{n,2}_t &=  X^{(n)}_0 + \\int_0^t {\\bf 1}_{J^c}(s) dX^{(n)}_s  \\\\\n &=  X^{(n)}_0 + \\sum_{k=0}^\\infty\n\\left(X^{(n)}(V^n_{k+1} \\land t) - X^{(n)}(T^n_{k} \\land t)\\right), \\\\\n\\widehat X^{n,2}_t &=  X^{(n)}_0 + X^{n,2}(\\widehat \\sigma^{n,2}_t).\n\\end{align*}\n\nThe point of this construction is the following.\nFor every fixed $\\omega$, the function $e\\to\\mu_e^{n-1}$ is invariant under the shift by \n$x a_{n-1}$ for any $x\\in {\\mathbb Z}^2$, and \n$X^{(n)}(V^n_{k+1} ) = X^{(n)}(T^n_{k}) + x a_{n-1} $ for some $x\\in {\\mathbb Z}^2$. \nIt follows that for each $\\omega\\in \\Omega$, we have the following equality of  distributions: \n\\begin{equation} \\label{e:Xhatdsn}\n\\{\\widehat X^{n,1}_t, t\\geq 0\\} {\\buildrel (d) \\over {\\  =\\ }} \\{X^{(n-1)}_t, t\\geq 0\\}.\n\\end{equation}\nThe basic idea of the argument which follows is to write $X^{(n)}= X^{n,1} + X^{n.2}$.\nBy Theorem \\ref{T:eK}, or more precisely by \\eqref{e:PdistBM}, the process $X^{n,1}$ is close\nto Brownian motion, so to prove Theorem \\ref{T:main} we need to prove that $X^{n,2}$\nis small.\n\n\\bigskip\nWe state the next lemma at a level of generality greater than what we need in this article. A variant of our lemma is in the book \\cite{AF} but we could not find a statement that would match perfectly our needs.\nConsider a finite graph $G=(\\mathcal{V},E)$ and suppose that for any edge $\\overline{xy}$, $\\mu_{xy}$ is a non-negative real number. Assume that $\\sum_{y\\sim x} \\mu_{xy} >0$ for all $x$.\nFor $f: \\mathcal{V} \\to \\mathbb{R}$ set\n$$ \\sE (f,f) = \\sum_{\\{x,y\\} \\in E}  \\mu_{xy} (f(y)-f(x))^2. $$\nSuppose that $A_1, A_2 \\subset \\mathcal{V}$, $A_1 \\cap A_2 = \\emptyset$, and\nlet \n\\begin{align*}\n\\mathcal{H} &=\\{ f:\\mathcal{V} \\to \\mathbb{R} \\text{ such that } f(x)=0 \\text{ for } x\\in A_1, f(y)=1 \\text{  for } y \\in A_2\\},\\\\\n \\mathbf{r}^{-1} &= \\inf\\{ \\sE(f,f): f \\in \\mathcal{H} \\}. \n\\end{align*}\nThus $\\mathbf{r}$ is the effective resistance between $A_1$ and $A_2$.\nLet $Z$ be the continuous time Markov process on $\\mathcal{V}$ with the generator $\\mathcal{L}$  given by\n\\begin{align}\\label{ma21.1}\n\\mathcal{L} f(x) = \\sum_y \\mu_{xy} (f(y) - f(x)).\n\\end{align}\nLet $T_i = \\inf\\{t\\geq 0: Z_t \\in A_i\\}$  for $i=1,2$,\n and let $Z^{(i)}$ be $Z$ killed at time $T_i$.\n\n\\begin{lemma}\\label{L:com}\nThere exist probability measures $\\nu_1$ on $A_1$ and $\\nu_2$ on $A_2$ such that\n\\begin{align*}\nE^{\\nu_2} T_1 + E^{\\nu_1} T_2 = \\mathbf{r} |\\mathcal{V}| . \n\\end{align*}\nMoreover, for $i=1,2$, $\\nu_i$ is the capacitary measure of $A_i$ for the process $Z^{(3-i)}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $h_{12}(x) = P^x( T_1 < T_2)$. \nSet $D= \\mathcal{V}-A_1$ and recall that $Z^{(i)}$ is $Z$ killed at time $T_i$.\nLet $G_2$ be the Green operator for $Z^{(2)}$, and $g_2(x,y)$ be the density of\n$G_2$ with respect to counting measure, so that\n$$ E^x T_2 = \\sum_{y \\in \\mathcal{V}} g_2(x,y). $$\nNote that $g_2(x,y)=g_2(y,x)$.\nLet $e_{12}$ be the capacitary measure of $A_1$ for the process $Z^{(2)}$. Then\n$\\mathbf{r}^{-1} = \\sum_{z \\in A_1} e_{12}(z), $ and\n$$ h_{12}(x) =  \\sum_{z \\in A_1} e_{12}(z) g_2(z,x) . $$\nSo, if $ \\nu_1 = \\mathbf{r} e_{12} $, then\n\\begin{align*}\n\\sum_{y \\in \\mathcal{V}} h_{12}(y) &= \n   \\sum_{y \\in \\mathcal{V}}  \\sum_{x \\in A_1} e_{12}(x) g_2(x,y)  \\\\\n&= \\mathbf{r}^{-1} \\sum_{x \\in A_1} \\nu_1(x)  \\sum_{y \\in \\mathcal{V}} g_2(x,y)  \\\\\n&= \\mathbf{r}^{-1} \\sum_{x \\in A_1} \\nu_1(x) E^x T_1 = \\mathbf{r}^{-1} E^{\\nu_1} T_2.\n\\end{align*}\nSimilarly if $h_{21}(x) =\\bP^x( T_2 < T_1)$ we obtain\n$\\mathbf{r}^{-1} E^{\\nu_2} T_1 = \\sum_{y \\in \\mathcal{V}} h_{21}(y) $, and since $h_{12}+h_{21}=1$, adding these\nequalities proves the lemma.\n\\end{proof}\n\n\\section{ Estimates on the process $X^{n,2}$ } \n\nIn this section we  will prove \n\n\\begin{proposition}\\label{d22.2}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{d22.3}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\n\\end{proposition}\n\nThe proof requires a number of steps. We begin with a Harnack inequality.\n\n\\begin{lemma}\\label{L:harn}\nLet $1 \\le \\lambda \\le 10$. \nThere exist $p_1>0$ \n and $n_1 \\ge 1$ with the following properties. \\\\\n(a) Let $x \\in {\\mathbb Z}^2$, let  $B_1= \\mathcal{B}(x, \\lambda \\beta_n)$ \nand $B_2= \\mathcal{B}(x, (2\/3) \\lambda \\beta_n)$. \nLet $F$ be  the event that $X^{(n)}$ makes a closed loop around $B_2$\ninside $B_1 - B_2$\nbefore its first exit from $B_1$.\nIf $n \\ge n_1$ and $D_n \\cap B_1 = \\emptyset$ then\n$P^y_{\\omega}(F) \\ge p_1$ for all $ y \\in B_2$. \\\\\n(b) Let $h$ be harmonic in $B_1$.  \nThen \n\\begin{equation} \\label{e:harni}\n  \\max_{B_2} h \\le p_1^{-1} \\min_{B_2} h.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n(a) Using ($\\clubsuit$)  and \\eqref{e:PdistBM} we can make a Brownian approximation to\n$\\beta_n^{-1} X^{(n)}_\\cdot$ which is good enough so that this estimate holds.\\\\\n(b) Let $y \\in B_1$ be such that $h(y) = \\max_{z \\in B_2} h(z)$.\nThen by the maximum principle there exists a connected path $\\gamma$ from $y$\nto ${\\partial}_i B_1$ with $h(w) \\ge h(y)$ for all $w \\in \\gamma$.\nNow let $y'\\in B_2$.  On the event $F$ the process $X^{(n)}$ must hit $\\gamma$, and\nso we have\n$$ h(y') \\ge P^{y'}_{\\omega} (F) \\min_{\\gamma} h \\ge p_1 h(y),$$\nproving \\eqref{e:harni}. \n\\end{proof}\n\n\n\\begin{lemma}\\label{12a29lem}\nFor some $n_1$ and $c_1$, for all $n\\geq n_1$, $k\\geq 1$, and $\\omega$ such that \n$0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{z1.1}\nE^0_\\omega(U^n_k - S^n_{k-1}  \\mid \\mathcal{F}_{S^n_{k-1}} ) \\leq c_1 \\beta_n^2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nBy the strong Markov property applied at $S^n_{k-1}$ for $k >1$, it is enough to prove \nthe Lemma for $k=1$, that is that \n$E^x_\\omega(U^n_1 ) \\le c_1 \\beta_n^2$ for all \n$x \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\n Let \n\\begin{align*}\n\\mathcal{V} &= \\mathcal{B}(u_n + \\mathcal{O}_n, 4 \\beta_n+1),\\\\\nA_1 &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, (3\/2) \\beta_n),\\\\\nA_2 &= \\partial_i \\mathcal{V},\\\\\nA_3 &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, 2 \\beta_n)\\\\\nT_i &= \\inf\\{t\\geq 0: X^{(n)}_t \\in A_i\\}, \\qquad i =1,2,3.\n\\end{align*}\nLet $Z$ be the continuous time Markov chain defined on $\\mathcal{V}$ by \\eqref{ma21.1}, \nrelative to the environment $\\mu^n$. Note that the transition probabilities from $x$ \nto one of its neighbors are the same for $Z$ and $X^{(n)}$ if $x $ is in the interior \nof $\\mathcal{V}$, i.e., $x\\notin \\partial_i \\mathcal{V} \\cup({\\mathbb Z}^2 \\setminus \\mathcal{V})$. \nNote also that $Z$ and  $X^{(n-1)}$ have the same transition probabilities \nin the region between $A_1$ and $A_3$.\nThe expectations and  probabilities in this proof will refer to $Z$.\nBy Lemma \\ref{L:com}, there exists a probability measure $\\nu_1$ on $A_1$ such that\n$E^{\\nu_1} T_2 \\leq \\mathbf{r} |\\mathcal{V}|$.\nWe have $|\\mathcal{V}| \\leq c_2 \\beta_n^2$.\n\nTo estimate $\\mathbf{r}$ note that by the choice of the constants $\\eta_{n-1}$ and $K_{n-1}$\nin Theorem \\ref{T:eK}, the resistance (with respect to $\\mu^{n-1}_e$) between two opposite sides of any\nsquare in $\\mathcal{S}_{n-1}$ will be 1. It follows that the resistance \nbetween two opposite sides of any square side $\\beta_n$ which is a union\nof squares in $\\mathcal{S}_{n-1}$ will also be 1. So, using Thompson's principle as\nin \\cite{BB3} we deduce that $\\mathbf{r} \\leq c_3$.\n\nSo, by Lemma \\ref{L:com} we have\n\\begin{align}\\label{ma18.1}\nE^{\\nu_1} T_2 \\leq c_4 \\beta_n^2.\n\\end{align}\n\nWe have for some $c_5$, $p_1 >0$ all $n$ and $x\\in \\mathcal{V} \\setminus \\mathcal{B}(u_n + \\mathcal{O}_n, (3\/2) \\beta_n)$,\n\\begin{align*}\nP^x_\\omega ( T_1 \\land T_2  \\leq c_5 \\beta_n^2) > p_1,\n\\end{align*}\nbecause an analogous estimate holds for Brownian motion and ($\\clubsuit$)  we have  \\eqref{e:PdistBM}. This and a standard argument based on the strong Markov property imply that for $x\\in A_3$,\n\\begin{align*}\nE^x_\\omega ( T_1 \\land T_2 ) \\leq c_6 \\beta_n^2.\n\\end{align*}\n\n\nNow for $y \\in A_1$ and $x \\in \\mathcal{V}$ set \n$$ \\nu_3^x (y) = P^x_\\omega(X^{(n)}(T_1 \\wedge T_2) = y  ).$$\n(Note that there exist $x$ with $\\sum_{y \\in A_1} \\nu^x_3(y) < 1$.)\nWe obtain for  $n\\geq n_2$ and $x\\in A_3$,\n\\begin{align}\\label{ma21.2}\nE^x_\\omega ( T_2 ) &=\nE^x_\\omega ( T_1 \\wedge T_2 ) + E^x_\\omega ((T_2 -T_1) {\\bf 1}_{ T_1 < T_2} )  \\\\\n& = E^x_\\omega ( T_1 \\wedge T_2 )  + E^{\\nu_3^x} T_2 \n\\leq  c_6 \\beta_n^2 + E^{\\nu_3^x}_{\\omega} T_2. \\nonumber\n\\end{align}\nFor $y \\in A_1$ the function $x \\to \\nu^x_3(y)$ is harmonic in $\\mathcal{V} \\setminus A_1$.\nSo we can apply the Harnack inequality Lemma \\ref{L:harn} to deduce that there exists\n$c_7$ such that\n\\begin{equation}\n \\nu^x_3(y) \\le c_7 \\nu^{x'}_3(y) \\hbox{ for all } x,x' \\in A_3, y \\in A_1.\n\\end{equation}\n\n\nThe measure $\\nu_1$ is  the hitting distribution on $A_1$ \nfor the process $Z$ starting with $\\nu_2$  (see \\cite[Chap.~3, p.~45]{AF}). So for\nany $x' \\in A_3$,\n\\begin{align*}\n\\nu_1(y) &= P^{\\nu_2}_0  ( Z_{T_1} =y) \n= \\sum_{x \\in A_3} P^{\\nu_2}_0  ( Z_{T_1} =x ) P^x_{\\omega}( Z_{T_1} =y)   \\\\\n&\\ge \\sum_{x \\in A_3} P^{\\nu_2}_0  ( Z_{T_1} =x ) P^x_{\\omega}( Z_{T_1 \\wedge T_2} =y)  \n\\ge  \\min_{x \\in A_3}   \\nu^x_3(y) \\ge c_7^{-1}  \\nu^{x'}_3(y).\n\\end{align*}\nHence for any $x \\in A_3$,\n$$  E^{\\nu_3^x}_{\\omega} T_2 \\le c_7 E^{\\nu_1}_{\\omega} T_2 \\le  c_8 \\beta_n^2,$$\nand combining this with \\eqref{ma21.2} completes the proof. \n\\end{proof}\n\nLet\n\\begin{align*}\nR_n^y =\\inf\\left\\{t\\geq 0 :  X^{(n)}_t \\in (y + a_{n-1}{\\mathbb Z}^2) \\cup \\Gamma^1_n\\right\\}.\n\\end{align*}\n\n\\begin{lemma}\nThere exist $c_1>0$ and $p_1<1$ such that for all $x,y \\in {\\mathbb Z}^2$,\n\\begin{align}\\label{ma23.1}\n&P^x_{\\omega}\\left(R_n^y \\geq c_1 b_n^2 \\right) \\le p_1,\\\\\n&P^x_{\\omega}\\left(\\sup_{0\\leq t \\leq R^y_n}|x- X^{(n)}_t| \\geq c_1 b_n \\right) \\le p_1.\n\\label{ma23.2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nRecall that the family $\\{\\mu^{n-1}_{x+ \\cdot}\\}_{x\\in {\\mathbb Z}^2}$ of translates of the \nenvironment  $\\mu^{n-1}_\\cdot$ contains only a finite number of distinct elements.\nSince each square in $\\mathcal{S}_{n-1}$ contains one point in $(y + a_{n-1}{\\mathbb Z}^2)$, \nif $b_n\/a_{n-1}$  is sufficiently large ($\\clubsuit$)  then using the transition density estimates \\eqref{e:GB1}\nas well as \\eqref{e:PdistBM}, we obtain \\eqref{ma23.1} and \\eqref{ma23.2}. \n\\end{proof}\n\n\\begin{lemma}\nFor some $n_1$ and $c_1$, for all $n\\geq n_1$, $k\\geq 1$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{ma21.5}\nE^0_\\omega(V^n_k - T^n_{k-1} \\mid \\mathcal{F}_{T^n_{k-1}}) \\leq c_1 b_n^2 n^{1\/2}.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nLet\n\\begin{align*}\n\\widehat R^n_k =\\inf\\left\\{t\\geq U^n_k :  X^{(n)}_t \\in (X^{(n)}(T^n_0) + a_{n-1}{\\mathbb Z}^2) \\cup \\Gamma^1_n\\right\\}.\n\\end{align*}\nLet $F_k = \\{ \\widehat R^n_k < S^n_k\\}$ and $G_k = \\bigcap _{j=1}^k F_j^c$.\nSince $b_n n^{1\/8} < \\beta_n$ for large $n$, we obtain from \\eqref{ma23.2} and definitions of $\\Gamma^1_n, \\Gamma^2_n, U^n_k$ and $S^n_k$ that there exists $p_2>0$ such that for $x\\in \\Gamma^2_n$,\n\\begin{align*}\nP^x_\\omega(F_k \\mid \\mathcal{F}_{U^n_k}) > p_2.\n\\end{align*}\nHence,\n\\begin{align}\\label{n5.2}\nP^x_\\omega(G_k ) < (1-p_2)^k.\n\\end{align}\nNote that if $F_k$ occurs then $V^n_1 \\leq \\widehat R^n_k$.\nWe have, using \\eqref{z1.1}, \\eqref{ma23.1} and \\eqref{n5.2},\n\\begin{align*}\nE^0_\\omega(V^n_1 - T^n_{0} ) \n&\\leq\n\\sum_{k=1}^\\infty\nE^0_\\omega((U^n_k - S^n_{k-1}) {\\bf 1}_{G_{k-1}})\n+ \n\\sum_{k=1}^\\infty\nE^0_\\omega((\\widehat R^n_k - U^n_{k}) {\\bf 1}_{G_{k-1}})\\\\\n&\\leq \\sum_{k=1}^\\infty\nc_2 \\beta_n^2 (1-p_2)^{k-1}\n+ \\sum_{k=1}^\\infty c_3 b_n^2 (1-p_2)^{k-1}  \\\\\n&\\leq c_4 \\beta_n^2 \\leq c_5 b_n^2 n^{1\/2}.\n\\end{align*}\nThis proves the lemma for $k=1$.\nThe general case is obtained by applying this estimate to the\nprocess shifted by $T^n_{k-1}$; in other words, by using the strong Markov property.\n\\end{proof}\n\n\\begin{lemma} \\label{L:sigma_1}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$,\n\\begin{align}\\label{n8.5}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta \\right) \\geq 1-\\delta\/2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nFix an arbitrarily small $\\delta >0$, consider $u \\geq a_n^2$ and let $j_* = \\lceil u\/(b_n^2 n^{5\/8} )\\rceil$. Then\n\\eqref{ma21.5} implies that for some $c_1$ and $n_2$, all $n\\geq n_2$, $u \\geq a_n^2$, \n\\begin{align*}\nE^0_{\\omega} \\left( \\frac 1{j_*} \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1}\\right) \n\\leq c_1 b_n^2 n^{1\/2}.\n\\end{align*}\nHence, for some $n_3$, all $n\\geq n_3$, $u \\geq a_n^2$,\n\\begin{align*\nP^0_{\\omega} \\left( \\frac 1{j_*} \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1} \\ge \\delta b_n^2 n^{9\/16}  \\right) \\le \\delta\/8 ,\n\\end{align*}\nand, since $j_* \\delta b_n^2 n^{9\/16}  \\leq \\delta u$, \n\\begin{align}\\label{d27.1}\nP^0_{\\omega} \\left(  \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1} \\ge \\delta u \\right) \\le \\delta\/8 . \n\\end{align}\n\nRecall $\\mathcal{K} (r)$ from \\eqref{e:Hdef}.\nLet\n\\begin{align*}\n\\widehat V^n_k & = \\inf\\{t\\geq V^n_{k}: X^{(n)}_t \\in \n{\\mathbb Z}^2 \\setminus \\mathcal{K}(b_n n ^{3\/8})\\} \\land T^n_k, \\qquad k \\geq 1,\\\\\n\\widetilde V^n_k & = \\inf\\{t\\geq \\widehat V^n_{k}: |X^{(n)}_t - X^{(n)}(\\widehat V^n_k)| \\geq (1\/2)b_n n^{3\/8} \\} , \\qquad k \\geq 1. \n\\end{align*}\nWe can use estimates for Brownian hitting \nprobabilities ($\\clubsuit$)  to see that for some $c_2, c_3$ and $n_4$,\nall $n\\geq n_4$, $k$, \n\\begin{align}\\label{j2.13}\nP^0_{\\omega}(\\widehat V^n_k < T^n_k \\mid \\mathcal{F}_{V^n_k}) \\geq c_2\n\\frac{\\log (4 \\beta_n) - \\log (2 \\beta_n)}\n{\\log (2 b_n n^{3\/8})- \\log (2 \\beta_n)} \n\\geq  c_3 \/\\log n . \n\\end{align}\nThere exist ($\\clubsuit$)  $c_4$ and $n_5$, such that for\nall $n\\geq n_5$, $k\\geq 2$, \n\\begin{align*}\nP^0_{\\omega} &(T^n_k - V^n_k \\geq c_4 b_n^2 n^{3\/4}  \\mid \\widehat V^n_k < T^n_k, \\mathcal{F}_{\\widehat V^n_k}) \\\\\n&\\ge\nP^0_{\\omega}(\\widetilde V^n_k - \\widehat V^n_k \\geq c_4 b_n^2 n^{3\/4}  \\mid \\widehat V^n_k < T^n_k, \\mathcal{F}_{\\widehat V^n_k}) \\ge 3\/4. \n\\end{align*}\nThis and \\eqref{j2.13} imply that the sequence $\\{T^n_k - V^n_k\\}_{k\\geq 2}$ is stochastically  minorized by a sequence of i.i.d.~random variables which take value $c_4 b_n^2 n^{3\/4} $ with probability $c_3 \/\\log n$ and they take value 0 otherwise.\nThis implies that for some $n_6$,\nall $n\\geq n_6$, $u \\geq a_n^2$, \n\\begin{align*\nP^0_{\\omega}\\left( \\frac 1{j_*} \\sum_{j=2}^{j_*} T^n_j - V^n_j  \\le  b_n^2 n^{3\/4}\/ \\log^2 n \\right) \\le \\delta\/4 \n\\end{align*}\nand, because  $j_* b_n^2 n^{3\/4}\/\\log^2 n \\geq u$ assuming $n_6$ is large enough,\n\\begin{align*\nP^0_{\\omega}\\left(  \\sum_{j=2}^{j_*} T^n_j - V^n_j  \\le u \\right) \\le \\delta\/4 . \n\\end{align*}\nWe combine this with \\eqref{d27.1} and the definition of $\\sigma^{n,2}_u$ to obtain\nfor some $n_7$,\nall $n\\geq n_7$, $u \\geq a_n^2$, \n\\begin{align}\\label{d27.4}\nP^0_{\\omega}(\\sigma^{n,2}_u\/u \\le \\delta) \\geq 1-3\\delta\/8.\n\\end{align}\nThis completes the proof of the lemma.\n\\end{proof}\n\n\\bigskip\n\nLet $Y^n_k = (Y^n_{k,1}, Y^n_{k,2}) = X^{(n)}(V^n_{k+1} ) - X^{(n)}(T^n_{k} )$.\nSet \n$\\bar Y^n_k = \\sup_{T^n_k \\leq t \\leq V^n_{k+1}} |X^{(n)}(t)- X^{(n)}(T^n_{k} )|$.\nFor $x\\in {\\mathbb Z}^2$, let\n $\\Pi_n(x) \\in B'_n -u_n + \\mathcal{O}_n $ be the unique point with the \nproperty that $x-\\Pi_n(x) = a_n y$ for some $y\\in {\\mathbb Z}^2$.\n\n\nWe next estimate the variance of  $X^{n,2}(V^n_{m+1}) = \\sum_{k=0}^ m Y^n_k$.\n\n\\begin{lemma} \\label{L:Ynest}\nThere exist $c_1, c_2$ and $n_1$ such that for all $n\\geq n_1$, $k\\geq 0$, $j=1,2$, and $\\omega$,\n\\begin{align}\\label{ma24.6}\nE^0_\\omega |Y^n_{k,j}| &\\leq E^0_\\omega |Y^n_k| \\leq E^0_\\omega |\\bar Y^n_k| \\leq c_1 \\beta_n, \\\\\n\\label{ma24.7}\n\\Var Y^n_{k,j} &\\le \\Var \\bar Y^n_{k} \\le c_2 \\beta_n^2 , \\qquad \\text { under } P^x_\\omega.\n\\end{align} \n\\end{lemma} \n\n\\begin{proof} Let \n\\begin{align}\\label{n8.1}\n\\mathcal{X}^{(n)}_k(t) &= X^{(n)}_t + \\Pi_n(X^{(n)}(T^n_k)) - X^{(n)}(T^n_k),\n\\qquad t\\in [T^n_k, V^n_{k+1}],\n\\end{align}\nand note that \n\\begin{align*}\nY^n_k = (Y^n_{k,1}, Y^n_{k,2}) = \\mathcal{X}^{(n)}_k(V^n_{k+1} ) - \\mathcal{X}^{(n)}_k(T^n_{k} ).\n\\end{align*}\n\nIt follows from the definition that we have $\\sup_{S^n_{k-1} \\leq t \\leq U^n_k} |X^{(n)}(t ) - X^{(n)}(S^n_{k-1} )| \\le 16\\beta_n$, a.s.\nThis, \\eqref{ma23.2} and the definition of $V^n_{k+1}$ imply that $|\\bar Y^n_k|$ is stochastically majorized by an exponential random variable with mean $c_3 \\beta_n$. This easily implies the lemma.\n\\end{proof}\n\n\nNext we will estimate the covariance of $Y^n_{k,1}$ and $Y^n_{j,1}$ for $j\\ne k$.\n\n\\begin{lemma} \\label{L:covY}\nThere exist $c_1, c_2$ and $n_1$ such that for all $n\\geq n_1$, $j < k-1$ and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, under $P^0_\\omega$,\n\\begin{align}\\label{n8.3} \n\\Cov(Y^n_{j,1},Y^n_{k,1}) & \\le c_1 e^{-c_2 (k-j)} \\beta_n^2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nLet\n\\begin{align*}\n\\Gamma^3_n &= \\Gamma^1_n \\cap \\mathcal{B}(u_n + \\mathcal{O}_n, a_n\/2)\n= \\mathcal{B}(u_n + \\mathcal{O}_n, 2\\beta_n),\\\\\n\\Gamma^4_n &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, 3\\beta_n),\\\\\n\\tau(A) &= \\inf\\{t\\geq 0: \\mathcal{X}^{(n)}_0(t) \\in A\\}.\n\\end{align*}\n\nSuppose that $x,v\\in \\Gamma^3_n$ and $y \\in \\Gamma^4_n$. \nBy the Harnack inequality proved in Lemma \\ref{L:harn},\n\\begin{align}\\label{ma23.4}\n\\frac{P_\\omega^x (\\mathcal{X}^{(n)}_0(\\tau(\\Gamma^4_n)) = y)}\n{P_\\omega^v (\\mathcal{X}^{(n)}_0(\\tau(\\Gamma^4_n)) = y)}\n\\geq c_3.\n\\end{align}\n\nLet $\\mathcal{T}^n_k$ have the same meaning as $T^n_k$ but relative to the process $\\mathcal{X}^{(n)}_k$ rather than $X^{(n)}$.\nWe obtain from \\eqref{ma23.4} and the strong Markov property applied at $\\tau(\\Gamma^4_n)$ that, \nfor any \n$x,v,y \\in \\Gamma^3_n$ we have\n\\begin{align*\n\\frac{P_\\omega^x (\\mathcal{X}^{(n)}_0(\\mathcal{T}^n_1) = y)}\n{P_\\omega^v (\\mathcal{X}^{(n)}_0(\\mathcal{T}^n_1) = y)}\n\\geq c_3.\n\\end{align*}\nRecall that $T^n_0 =0$.\nThe last estimate implies that, for  \n$x,v,y \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P_\\omega (\\mathcal{X}^{(n)}_1(T^n_{1}) = y \\mid \\mathcal{X}^{(n)}_0(T^n_0) = x)}\n{P_\\omega (\\mathcal{X}^{(n)}_1(T^n_{1}) = y \\mid \\mathcal{X}^{(n)}_0(T^n_0) = v)}\n\\geq c_3.\n\\end{align*}\nSince the process $X^{(n)}$ is time-homogeneous, this shows that for  \n$x,v,y \\in \\Gamma^3_n$ and all $k$,\n\\begin{align}\\label{d29.2}\n\\frac{P_\\omega (\\mathcal{X}^{(n)}_{k+1}(T^n_{k+1}) = y \\mid \\mathcal{X}^{(n)}_k(T^n_k) = x)}\n{P_\\omega (\\mathcal{X}^{(n)}_{k+1}(T^n_{k+1}) = y \\mid \\mathcal{X}^{(n)}_k(T^n_k) = v)}\n\\geq c_3.\n\\end{align}\nWe now apply Lemma 6.1 of \\cite{BTW} (see Lemma 1 of \\cite{BK} for a better presentation of the same estimate) to see that \\eqref{d29.2} implies that \nthere exist constants $C_k$, $k\\geq 1$, such that for every $k$ and all \n$x,v,y \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P_\\omega^x (\\mathcal{X}_k^{(n)}(T^n_k) = y)}\n{P_\\omega^v (\\mathcal{X}_k^{(n)}(T^n_k) = y)}\n\\geq C_k.\n\\end{align*}\nMoreover, $C_k\\in(0,1)$, $C_k$'s depend only on $c_3$, and $1-C_k \\le e^{-c_4 k}$ for some $c_4>0$ and all $k$.\nBy time homogeneity of $X^{(n)}$, for $m\\leq j<k$ and all $x,v,y,z \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = x)}\n{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = v)}\n\\geq C_{k-j},\n\\end{align*}\nand, by the strong Markov property applied at $T^n_j$, \n\\begin{align*\n\\frac{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = x)}\n{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_m^{(n)}(T^n_m) = v)}\n\\geq C_{k-j}.\n\\end{align*}\nThis and \\eqref{ma24.6} imply that for $j<k-1$ and $x\\in {\\mathbb Z}^2$,\n\\begin{align}\\nonumber\n|{E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}})|\n&=\n|{E}^x_\\omega(Y^n_{k,1}  \\mid \\mathcal{F}_{T^n_{j+1}})\n- {E}^x_\\omega Y^n_{k,1} |   \\\\\n\\nonumber\n&\\le  (1- C_{k-j-1}) \\sup_{y\\in {\\mathbb Z}^2} {E}^y_\\omega |Y^n_{k,1}| \\\\\n&\\le  e^{-c_4 (k-j-1)} c_5 \\beta_n \\leq \n c_6 e^{-c_4 (k-j)}  \\beta_n.\n\\label{ma24.8}\n\\end{align}\nHence  for $j< k-1$,\n\\begin{align*\n\\Cov(Y^n_{j,1},Y^n_{k,1})\n&= {E}^x_\\omega ((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1})\n(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1}))\\\\\n&= {E}^x_\\omega ({E}^x_\\omega((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1})\n(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1}) \\mid \\mathcal{F}_{T^n_{j+1}}))\\\\\n&= {E}^x_\\omega ((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1}) {E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}}))\\\\\n&\\leq {E}^x_\\omega (|Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1}| \\cdot |{E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}})|)\\\\\n& \\le 2 {E}^x_\\omega |Y^n_{j,1}| c_6 e^{-c_4 (k-j)} \\beta_n \\\\\n& \\le c_7 e^{-c_4 (k-j)} \\beta_n^2.\n\\end{align*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{d22.2}]\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nWe combine \\eqref{n8.3} and \\eqref{ma24.7} to see that for some $c_{1}$ and $c_{2}$ and all  $m\\geq 1$, we have under $P^0_{\\omega}$, \n\\begin{align}\\label{d29.10}\n\\Var\\left(\\sum_{k=0}^m Y^n_{k,1}\\right) &= \n\\sum_{j=0}^m \\sum_{k=0}^m\n\\Cov(Y^n_{j,1},Y^n_{k,1})\\\\\n& \\le \\sum_{j=0}^m \\sum_{k=0}^m\nc_{1} e^{-c_{3} (k-j)} \\beta_n^2 \\leq c_{2} m \\beta_n^2. \\nonumber\n\\end{align}\n\nFor fixed $n$ and ${\\omega}$, the process \n$\\{\\mathcal{X}^{(n)}_k(T^n_k), k\\geq 1\\}$ \nis Markov with a finite state space \nand one communicating class, so it has a unique stationary distribution. We will call it $\\mathbf{p}(n)$.\nWe will argue \nthat ${E}^{\\mathbf{p}(n)}_{\\omega} Y^n_{k,1} = 0$. Since $X^{(n)}$ and $X^{(n-1)}$ \nsatisfy the quenched invariance principle and they are random walks among symmetric (in distribution) conductances, they have zero means. Recall that \n$X^{(n)} = X^{n,1} + X^{n,2}$ and $\\widehat X^{n,1}$ has the same distribution as $X^{(n-1)}$. \nIt follows that for some $c_{4}>0$ and $c_{5}<1\/4$ and all large $t$, we have \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |\\widehat X^{n,1} _s| \\geq  c_{4}\\sqrt{t}\\right) \n=P^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{(n-1)} _s| \\geq  c_{4}\\sqrt{t}\\right) < c_{5}. \n\\end{align*}\nSince $\\widehat X^{n,1}_t =  X^{n,1}(\\widehat \\sigma^{n,1}_t)$ and $\\widehat \\sigma^{n,1}_t \\geq t$, the last estimate implies that\n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{n,1} _s| \\geq  c_{4}\\sqrt{t}\\right)  < c_{5}. \n\\end{align*}\nWe also have \nfor some $c_{6}>0$ and $c_{7}<1\/4$, and all large $t$, \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{(n)} _s| \\geq  c_{6}\\sqrt{t}\\right)  < c_{7}. \n\\end{align*}\nSince $X^{n,2} = X^{(n)} - X^{n,1}$, we obtain\nfor some $c_{8}>0$ and $c_{9}<1\/2$ and all large $t$, \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{n,2} _s| \\geq  c_{8}\\sqrt{t}\\right)  < c_{9}. \n\\end{align*}\nThis shows that $X^{n,2}$ does not have a linear drift.\nIt is clear from the law of large numbers that $\\liminf_{t\\to\\infty} \\sigma_t^{n,2}\/t >0$, so $\\widehat X^{n,2}$ does not have a linear drift either.\nWe conclude that ${E}^{\\mathbf{p}(n)}_\\omega Y^n_{k,1} = 0$. \n\nNow suppose that $X^{(n)}_0$ does not necessarily have the distribution $\\mathbf{p}(n)$. \nThe fact that ${E}^{\\mathbf{p}(n)}_\\omega Y^n_{k,1} = 0$ and a calculation similar to that in \\eqref{ma24.8} imply that,\n\\begin{align*\n|{E}^0_\\omega  Y^n_{k,1}| \\le  c_{10} e^{-c_{11} k}  \\beta_n .\n\\end{align*}\n\nLet $c_{12} $ be the constant denoted $c_1$ in \\eqref{ma24.6}.\nThe last estimate and \\eqref{ma24.6} imply that for some $c_{13}$ and all $m\\geq 1$,\n\\begin{align} \\nonumber\n\\left| E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,1}\\right| \n&\\leq \\sum_{k\\geq 0} |{E}^0_\\omega Y^n_{k,1}|\n+ \\sup_{k\\geq 1} E^0_{\\omega} |\\bar Y^n_k|   \\\\\n\\label{j1.1}\n&\\leq \\sum_{k\\geq 0} c_{10} e^{-c_{11} k}  \\beta_n +c_{12}\\beta_n\n\\leq  c_{13} \\beta_n .\n\\end{align}\nAll estimates that we derived for $Y^n_{k,1}$'s apply to $Y^n_{k,2}$'s as well, by symmetry.\n\nNote that $|X^{(n)}(U^n_{k+1} ) - X^{(n)}(T^n_{k} )|\\geq \\beta_n\/2$.\nWe have\n$V^n_{k+1}  - T^n_{k} \\geq U^n_{k+1}  - T^n_{k} $ so\nwe can assume ($\\clubsuit$)  that $b_n\/a_{n-1}$ is so large that for some $p_1>0$ and $n_{2}$, for all $n\\geq n_{2}$ and $k\\geq 1$, \n\\begin{align*\nP_\\omega^x(V^n_{k+1}  - T^n_{k} \\geq \\beta_n^2\n\\mid \\mathcal{F}_{T^n_k}) \\geq p_1.\n\\end{align*}\nLet $\\mathcal{V}_m$ be a binomial random variable with parameters \n$m$ and $p_1$. We see that $\\sigma^{n,2}(V^n_{m })= \\sum_{k=0}^m V^n_{k+1}  - T^n_{k}$ is stochastically minorized by \n$\\beta_n^2 \\mathcal{V}_m$. \n\nRecall that $u \\geq a_n^2$.\nLet $m_1$ be the smallest integer such that\n\\begin{align}\\label{c1.1}\nP^0_\\omega(V^n_{m_1 } \\leq u) < \\delta\/4.\n\\end{align}\nThen\n\\begin{align}\\label{c1.2}\nP^0_\\omega(V^n_{m_1 -1} \\leq u) \\geq \\delta\/4.\n\\end{align}\nSince $\\delta$ in \\eqref{d27.4} can be arbitrarily small, we have\nfor for some $n_{3}$ and all $n\\geq n_{3}$, \n\\begin{align}\\label{c1.3}\nP^0_\\omega(\\sigma^{n,2}_u\/u \\leq \\delta^4) \\geq 1-\\delta\/8.\n\\end{align}\nThe following estimate follows from the fact that $\\sigma^{n,2}(V^n_{m_1-1 })$ is stochastically minorized by \n$\\beta_n^2 \\mathcal{V}_{m_1-1}$, and from \\eqref{c1.2}-\\eqref{c1.3},\n\\begin{align*}\nP^0_\\omega(\\beta_n^2 \\mathcal{V}_{m_1-1}  \\leq \\delta^4 u) \n&\\geq \nP^0_\\omega(\\sigma^{n,2}(V^n_{m_1 -1}) \\leq \\delta^4 u) \\\\\n&\\geq\nP^0_\\omega(\\sigma^{n,2}_u \\leq \\delta^4 u, V^n_{m_1 -1} \\leq u) \n\\geq \\delta\/8.\n\\end{align*}\nThis implies that for some $c_{14}$, we have\n$m_1 \\leq c_{14}\\delta^3 u\/\\beta_n^2$. In other words, $u \\geq m_1 \\beta_n^2\/(c_{14} \\delta^3)$. \nNote that for a fixed $\\delta$, we have for large $n$, ($\\clubsuit$)  $u^{1\/2}\\delta\/4 -  c_{13} \\beta_n \\geq u^{1\/2} \\delta\/8$.\nThese observations, \\eqref{d29.10}, \\eqref{j1.1} and the Chebyshev inequality imply that for  $m\\le m_1$,\n\\begin{align}\\label{d29.22}\nP^0_\\omega&\\left(u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2\\right)\\\\\n&\\le\nP^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right| \\geq u^{1\/2} \\delta\/4\\right)\n+ P^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,2}\\right| \\geq u^{1\/2} \\delta\/4\\right)\n\\nonumber \\\\\n&\\le\nP^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\n- E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,1}\\right| \\geq u^{1\/2} \\delta\/4\n- c_{13} \\beta_n\\right)\\nonumber \\\\\n&\\qquad + P^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,2}\n- E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,2}\\right| \\geq u^{1\/2} \\delta\/4\n- c_{13} \\beta_n\\right)\n\\nonumber \\\\\n&\\le  \\frac{\\Var\\left(\\sum_{k=0}^{m} Y^n_{k,1}\\right)}\n{u \\delta^2\/64} + \\frac{\\Var\\left(\\sum_{k=0}^{m} Y^n_{k,2}\\right)}\n{u \\delta^2\/64} \\nonumber \\\\\n&\\le  \\frac{2c_{2} m_1 \\beta_n^2}\n{(c_{14}^{-1}\\delta^{-3} m_1\\beta_n^2) \\delta^2\/64} \\le c_{15} \\delta.\n\\nonumber\n\\end{align}\nLet $M = \\min\\{m\\geq 1: \nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta\\}$. By the strong Markov property applied at $M$ and \\eqref{d29.22},\n\\begin{align}\\label{d29.23}\n&P^0_\\omega\\left(\n\\sup_{1\\le m \\le m_1}\nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta,\\  u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\le \\delta\/2\\right)\\\\\n&\\le \nP^0_\\omega\\left( u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1-M} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1-M}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2 \\mid M < m_1\\right)\n\\le c_{15} \\delta. \\nonumber\n\\end{align}\n\nRecall that $u \\geq m_1 \\beta_n^2\/(c_{14} \\delta^3)$. For a fixed $\\delta$ and large $n$, ($\\clubsuit$)  $u^{1\/2}\\delta - 2 c_{12} \\beta_n \\geq u^{1\/2} \\delta\/2$.\nIt follows from this, \\eqref{ma24.6} and \\eqref{ma24.7} that\n\\begin{align}\\label{ma25.1}\nP^0_{\\omega}\\left(\\exists k \\leq m_1: |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \n&\\leq \nm_1 \\sup_{k\\leq m_1} P^0_{\\omega}\\left( |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \\\\\n&\\leq \nm_1 \\sup_{k\\leq m_1} P^0_{\\omega}\\left( |\\bar Y^n_k| -E^0_{\\omega} |\\bar Y^n_k|\\geq u^{1\/2}\\delta - c_{12} \\beta_n \\right)\\nonumber \\\\\n&\\le m_1 \\frac {c_{11} \\beta_n^2}{ u  \\delta^2 \/4}  \n\\le m_1 \\frac{c_{11}  \\beta_n^2}\n{(c_{14}^{-1}\\delta^{-3} m_1\\beta_n^2) \\delta^2} \\le c_{16} \\delta.\n\\nonumber\n\\end{align}\n\nWe use \\eqref{c1.1}, \\eqref{d29.22}, \\eqref{d29.23} and \\eqref{ma25.1} to obtain\n\\begin{align*\n&P^0_\\omega \\left(\\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\geq 2\\delta\\right)\\\\\n &\\le P^0_\\omega(V^n_{m_1 } \\le u) \n+ P^0_\\omega\\left(u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2\\right)\\\\\n& + P^0_\\omega\\left(\n\\sup_{1\\le m \\le m_1}\nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta,\\  u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\le \\delta\/2\\right)\\\\\n& +\nP^0_{\\omega}\\left(\\exists k \\leq m_1: |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \\\\\n&\\le \\delta\/4 + c_{15} \\delta + c_{15} \\delta + c_{16}\\delta.\n\\end{align*}\nSince $\\delta>0$ is arbitrarily small, this implies that for every $\\delta>0$, some $n_{3}$ and all $n\\geq n_{3}$,\n\\begin{align*\nP^0_\\omega &\\left(\\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\geq \\delta\\right)\\le \\delta\/2.\n\\end{align*}\nThis and \\eqref{n8.5} yield the proposition.\n\\end{proof}\n\nRecall from \\eqref{e:bfPdef} the definition of the averaged measure ${\\bf P}$.\n\n\\begin{lemma}\\label{n9.1}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$ and  $u\\geq a_n^2$,  \n\\begin{align}\\label{n9.2}\n{\\bf P}\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy Proposition \\ref{d22.2} applied to $\\delta\/2$ in place of $\\delta$, for every $\\delta>0$ there exists $n_2$ such that for all $n\\geq n_2$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{n9.3}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta\/2.\n\\end{align}\n\nLet $|A|$ denote the cardinality of $A\\subset {\\mathbb Z}^2$. Since $|\\Gamma^1_n| \\leq 25 \\beta_n^2 \\leq 25 a_n^2 n^{-1\/2} = 25 n^{-1\/2} |B'_n|$, the definitions of $\\mathcal{O}_n$ and $\\Gamma^1_n$ imply that ${\\bf P}(0 \\in \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n) < \\delta\/2$ for some $n_3 \\geq n_2$ and all $n \\geq n_3$. This and \\eqref{n9.3} imply \\eqref{n9.2}.\n\\end{proof}\n\nIn the following lemma and its proof, when we write the Prokhorov distance between processes such as $\\{ (1\/a)X^{(n-1)}_{ta^2}, t\\in[ 0,1]\\}$, we always assume that they are distributed according to ${\\bf P}$.\n\n\\begin{lemma}\\label{d22.1}\nThere exists a function $\\rho^*: (0,\\infty) \\to (0,\\infty)$ with \n$ \\lim_{\\delta\\downarrow 0} \\rho^*(\\delta) = 0$ \nand a sequence $\\{a_n\\}$ with the following properties,\n\\begin{align}\\label{d19.3}\n&{d_P}(\\{ (1\/a)X^{(n-1)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n},\\qquad a \\geq a_n.\n\\end{align}\nMoreover, suppose that for $\\delta<1\/2$ and all $u\\geq a_n^2$, \n\\begin{align}\\label{d19.2}\n{\\bf P} \\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\nThen\n${d_P}( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n} + \\rho^*(\\delta)$, for all $a\\geq a_n$.\n\\end{lemma}\n\n\\begin{proof}\n\nFormula \\eqref{d19.3} is  special case of \\eqref{e:PdistBM}.\n\nFix some $a\\geq a_n$. We will apply \\eqref{d19.2} with $u=a^2$. \nNote that on the event in \\eqref{d19.2} we have\n\\begin{align}\\label{n9.5}\n1- \\sigma^{n,1}_{a^2}\/a^2= u\/u- \\sigma^{n,1}_u\/u \n= \\sigma^{n,2}_u \/ u \\le \\delta.\n\\end{align}\nThe function $t\\to \\sigma^{n,1}_{ta^2}\/a^2$ is Lipschitz with the constant 1 and $\\sigma^{n,1}_{ta^2}\/a^2 \\leq t$ so \\eqref{n9.5} implies for $t\\in[0,1]$,\n\\begin{align}\\label{n9.6}\nt- \\sigma^{n,1}_{ta^2}\/a^2 \\leq 1- \\sigma^{n,1}_{a^2}\/a^2 \\leq \\delta.\n\\end{align}\n\nRecall the function $\\rho(\\delta)$ from the proof of Lemma \\ref{d21.1}, \nsuch that $P_{\\text{BM}}(\\Osc(W,\\delta) \\geq\\rho(\\delta) )<\\rho(\\delta)$ and \n$\\lim_{\\delta\\downarrow 0} \\rho(\\delta) = 0$.\nBy \\eqref{n9.6}, we can apply Lemma \\ref{d21.1} with $\\sigma_t = \\sigma^{n,1}_{ta^2}\/a^2$. Recall\nthat  $W^*(t) = W(\\sigma_t)$.\nBy the definition of $\\widehat X^{n,1}$,\n\\begin{align}\n\\nonumber\n&{d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n\\nonumber\n&\\le  {d_P}( \\{(1\/a) X^{n,1}_{t\/a^2}, t\\in[0,1]\\}, \\{W^*_t, t\\in[0,1]\\})\n + {d_P}( \\{W^*_t, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\ \n\\nonumber \n&\\le  {d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, \\{W^*_t, t\\in[0,1]\\})\n + \\rho(\\delta) + \\delta\\\\ \n&= {d_P}( \\{(1\/a)\\widehat X^{n,1}(\\sigma^{n,1}_{ta^2}), t\\in[0,1]\\}, \\{W(\\sigma^{n,1}_{ta^2}\/a^2), t\\in[0,1]\\})\n + \\rho(\\delta) + \\delta  .\n\\end{align}\n\nRecall from \\eqref{e:Xhatdsn} that for a fixed $\\omega\\in \\Omega$, the distribution of $\\{\\widehat X^{n,1}_t, t\\geq 0\\}$ \nis the same as that of $\\{X^{n-1}_t, t\\geq 0\\}$.\nIn view of Theorem \\ref{T:eK}, we can make $a_n$ so large ($\\clubsuit$)  that \n$\\Pp(\\Osc(\\widehat X^{n,1},\\delta) \\geq 2\\rho(\\delta) )< 2\\rho(\\delta)$. \nThis, Lemma \\ref{d21.2} and the definition of the Prokhorov distance imply that\n\\begin{align*}\n{d_P}( \\{(1\/a)\\widehat X^{n,1}&(\\sigma^{n,1}_{ta^2}), \\, t\\in[0,1]\\}, \\{W(\\sigma^{n,1}_{ta^2}\/a^2), t\\in[0,1]\\})  \\\\\n&\\le \n{d_P}( \\{(1\/a)\\widehat X^{n,1}_{ta^2}, t\\in[0,1]\\}, \\{W_{t}, t\\in[0,1]\\}) + 4\\rho(\\delta) \\\\\n&= {d_P}( \\{(1\/a) X^{(n-1)}_{ta^2}, t\\in[0,1]\\}, \\{W_{t}, t\\in[0,1]\\}) + 4\\rho(\\delta) \\\\\n&\\le  2^{-n}  + 4\\rho(\\delta).\n\\end{align*}\nIn the final two lines line we used  \\eqref{e:Xhatdsn} and \\eqref{d19.3}.\n\n\nCombining the estimates above, since\n$P^0_{\\omega} \\left(  \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta$ and \n$X^{(n)} = X^{n,1} + X^{n,2}$, Lemma \\ref{ma26.5} shows that\n\\begin{align*}\n&{d_P}( \\{(1\/a) X^{(n)}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n&\\le\n{d_P}( \\{(1\/a) X^{(n)}_{ta^2}, t\\in[0,1]\\}, \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}) \\\\\n&\\qquad + {d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n&\\le \n\\delta + 2^{-n}  + 5 \\rho(\\delta) + \\delta .\n\\end{align*}\nWe conclude that the lemma holds if we take $\\rho^*(\\delta) =  5\\rho(\\delta) + 2\\delta $.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:main}]\nChoose an arbitrarily small $\\varepsilon>0$. We will show that there exists $a_*$ such that for every $a\\geq a_*$, \n\\begin{align}\\label{d23.1}\n&{d_P}( \\{(1\/a) X_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\le \\varepsilon .\n\\end{align}\n\nRecall $\\rho^*$ from Lemma \\ref{d22.1}.\nLet $n_1 $ be such that $2^{-n_1}\\le \\varepsilon\/4$ and let $\\delta>0$ be so small that \n$2^{-n_1}+ \\rho^*(\\delta) < \\varepsilon\/2$. Let $n_2$ be defined as $n_1$ in Lemma \\ref{n9.1}, \nrelative to this $\\delta$. Then, according to Lemma \\ref{d22.1}, \n\\begin{align}\\label{d22.5}\n{d_P}( \\{(1\/a)X^{n}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n} + \\rho^*(\\delta) < \\varepsilon\/2,\n\\end{align}\nfor all $n\\geq n_3 := n_1\\lor n_2$ and $a\\geq a_n$.\n\nFor a set $K$ let \n$\\mathcal{B}(K,r) = \\{z: \\dist(z,K) < r\\}$ and recall the definition of $D_n$ given in \\eqref{ma26.1}. Let \n\\begin{align*}\nF_1 &= \\{0\\in \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\\},\\\\\nF_2 &= \\{0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\\}\n\\cap\n\\{\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_{n+1}\\},\\\\\nG_1^k &= \\{0\\in \\mathcal{B}( D_k, b_k\/k)\\},\n\\qquad k> n+1,\\\\\nG_2^k &= \\{0\\notin \\mathcal{B}( D_k, b_k\/k)\\}\n\\cap\n\\{\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_k\\}, \\qquad k> n+1.  \n\\end{align*}\n\nThe area of  $\\mathcal{B}(D_{n+1}, a_{n+1}\/\\log (n+1))$ is bounded by $c_1 (a_{n+1}\/\\log (n+1))^2$ so \n\\begin{align}\\label{d23.10}\n\\Pp(F_1) \\le c_1 (a_{n+1}\/\\log (n+1))^2\/ a_{n+1}^2 = c_1 \/\\log^2 (n+1).\n\\end{align}\nWe choose $n_4 > n_3 $ such that \n$c_1 \/\\log^2 (n+1) < \\varepsilon\/8$ for $n \\geq n_4$.\n\nNote that $D_{n+1}$ is a subset of a square with side $4\\beta_{n+1} \\leq 4 a_{n+1} n^{-1\/4}$. This easily implies that\nthere exists $n_5 \\geq n_4$ such that for $n\\geq n_5$,\n\\begin{align*\nP_{\\text{BM}}\\left(\\exists t\\in[0, a_{n+1}^2]: W(t) \\in  D_{n+1} \n\\mid \n0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\n\\right) \\le \\varepsilon\/16.\n\\end{align*}\nWe can assume ($\\clubsuit$)  that $a_{n+1}\/a_n$ is so large that\nfor some $n_6 \\geq n_5$ and all $n\\geq n_6$,  \n\\begin{align}\\label{d23.11}\n\\Pp(F_2) &\\le \\Pp\\left(\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_{n+1} \n \\mid  0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1)) \\right)   \\\\\n &\\le \\varepsilon\/8.\n\\end{align}\n\nThe area of  $\\mathcal{B}( D_k, b_k\/k)$ is bounded by $c_2 b_k^2\/k$ so \n\\begin{align}\\label{d23.12}\n\\Pp(G_1^k) \\le (c_2 b_k^2\/k)\/ a_k^2 \\le c_3 ( b_k^2\/k)\/ (k b_k^2)= c_3 \/k^2.\n\\end{align}\nWe let $n_7 >n_6$ be so large that $\\sum_{k\\geq n_7} c_3 \/k^2 < \\varepsilon\/8$.\nFor all $k>n+1\\geq n_7+1$, we make $b_k\/k$ so large ($\\clubsuit$)  that \n\\begin{align}\\label{d23.13}\n\\Pp(G_2^k) \\le \n\\Pp\\left(\\sup_{t\\in[0, a_{n+1}^2]} |X^{n}_t| \\geq b_k\/k\\right) \\le c_3\/k^2.\n\\end{align}\n\nWe combine \\eqref{d23.10}, \\eqref{d23.11}, \\eqref{d23.12} and \\eqref{d23.13} to see that for $n\\geq n_7$,\n\\begin{align}\\label{d23.14}\n\\Pp&(\\exists t\\in[0, a_{n+1}^2] \\ \\exists k\\geq n+1: X^{(n)}_t \\in  D_k)\\\\\n&\\le\n\\Pp(F_1) +  \\Pp(F_2) + \\sum_{k>n+1}\n\\Pp(G_1^k) + \\sum_{k>n+1}\n\\Pp(G_2^k) \\nonumber\\\\\n& \\le \\varepsilon\/8 + \\varepsilon\/8 + \\varepsilon\/8 + \\varepsilon\/8 = \\varepsilon\/2.\n\\nonumber\n\\end{align}\n\nLet $R_{n+1} = \\inf\\{t\\geq 0: X_t \\in \\bigcup_{k\\geq n+1} \\mathcal{D}_k\\}$. \nIt is standard to construct $X$ and $X^{(n)}$ on a common probability space so that $X_t = X_t^n$ for all $t\\in [0, R_{n+1})$. This and \\eqref{d23.14}\nimply that for $n\\geq n_7$ and all $a\\in[a_n, a_{n+1}]$ we have\n\\begin{align*\nP(\\exists t\\in[ 0,1]: (1\/a)X_{ta^2} \\ne (1\/a)X^{(n)}_{ta^2}) \\le \\varepsilon\/2.\n\\end{align*}\nWe combine this with \\eqref{d22.5} to see that for all $a\\geq a_{n_6}$, \n\\begin{align*\n{d_P}( \\{(1\/a)X_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le  \\varepsilon\/2 +\\varepsilon\/2 = \\varepsilon.\n\\end{align*}\nWe conclude that \\eqref{d23.1} holds with $a_* = a_{n_7}$. \n\nThis completes the proof of AFCLT. The WFCLT then follows from Theorem 2.13 of \\cite{BBT1}.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nOne of the fundamental tasks of the Galactic studies is to\nestimate the structure parameters\nof the major structure components.\n\\citet{Bahcall1980} fit the observations with two structure components, namely a disk and a halo. \n\\citet{Gilmore1983}  introduce a third component, namely a thick disk, \nconfirmed in the earliest Besancon Galaxy Model \\citet{Creze1983}.\nSince then, various methods and observations have been adopted to\nestimate parameters of the thin and thick disks and of\nthe halo of our Galaxy. As the quantity and quality of data available continue to improve over the years,\nthe model parameters derived have become more precise, numerically. \nIronically, those numerically more precise results do not converge (see Table~1 of \n\\citealt{Chang2011}, Table~2 of \\citealt{Lopez2014} and Sect.~5 and 6 of \\citealt{Bland2016} for a review).  \nThe scatters in density law parameters, such as scale lengths,  \nscale heights and local densities of these Galactic components, \nas reported in the literature, are rather large.\nAt least parts of the discrepancies are caused by degeneracy of \nmodel parameters, which in turn, can be traced back to  the different \ndata sets adopted in the analyses. Those differing \ndata sets either probe different sky areas \n\\citep{Bilir2006a, Du2006, Cabrera2007, Ak2007, Yaz2010, Yaz2015}, \nare of different completeness magnitudes and therefore \nrefer to different limiting distances \\citep{Karaali2007},\nor of consist of stars of different populations of different absolute magnitudes \n\\citep{Karaali2004, Bilir2006b, Juric2008, Jia2014}.  \nIt should be noted that the analysis of \\citet{Bovy2012}, using the SEGUE spectroscopic survey, \nhas given a new insight on the thin and thick disk structural parameters. This analysis provides estimate \nof their scale height and scale height as a function of metallicity and alpha abundance ratio. However, \nit relies on incomplete data (since it is spectroscopic) with relatively low range of Galactocentric radius as \nfor the thin disk is concerned. \n\nA wider and deeper sample than those employed hitherto may help break the degeneracy \ninherent in a multi-parameter analysis and yield a globally representative Galactic model. \nA single or a few fields are insufficient to break the degeneracy. \nThe resulted best-fit parameters, while sufficient for the description of  \nthe lines of sight observed, may be unrepresentative of the entire Galaxy. For the latter purpose, \nsystematic surveys of deep limiting magnitude of all or a wide sky area, such as the \nTwo Micron All Sky Survey (2MASS; \\citealt{Skrutskie2006}), the Sloan Digital Sky Survey \n(SDSS; \\citealt{York2000}), the Panoramic Survey Telescope \\& Rapid Response System \n(Pan-Starrs; \\citealt{Kaiser2002}) and the GAIA mission \\citep{Perryman2001}, \nare always preferred.\n\nSeveral authors have studied the Galactic structure with 2MASS data at low \n\\citep{Lopez2002, Yaz2015} or high latitudes \\citep{Cabrera2005, Cabrera2007, Chang2011}. \n\\citet{Polido2013} uses the model from \\citet{Ortiz1993} and rederive the parameters of this model\nbased on the 2MASS star counts over the whole sky area. \nHowever, the survey depth of 2MASS is not quite enough to reach the outer disk and the halo.\nThe survey depth of SDSS is much deeper than that of the 2MASS. Many authors \n(e.g.  \\citealt{Chen2001, Bilir2006a, Bilir2008, Jia2014, Lopez2014})\nhave previously used the SDSS data to constrain the Galactic parameters. \nThose authors have only made use of a portion of the surveyed fields, \nat intermediate or high Galactic latitudes. \n\\citet{Juric2008} obtain Galactic model parameters from the stellar \nnumber density distribution of 48 million \nstars detected by the SDSS that sample distances from 100\\,pc to 20\\,kpc and cover \n6500\\,deg$^2$ of sky. Their results are amongst those mostly quoted. \nHowever, in their analysis, they have avoided the Galactic plane. \nSo the constraints of their results on the disks, especially the\nthin disk, are weak. In their analysis, \\citet{Juric2008} have also \nadopted photometric parallaxes assuming that all stars of the same colour\nhave the same metallicity. Clearly, (disk) stars in different parts of the Galaxy have quite different \n\\citep{Ivezic2008, Xiang2015, Huang2015} metallicities, and these variations in metallicities may \nwell lead to biases in  the model parameters derived. \n\n \n In order to provide a quality input catalog for the LAMOST Spectroscopic Survey of the Galactic Anticentre\n(LSS-GAC; \\citealt{Liu2014,Liu2015, Yuan2015}), \na multi-band CCD photometric survey of the Galactic \nAnticentre with the Xuyi 1.04\/1.20m Schmidt Telescope  \n(XSTPS-GAC; \\citealt{Zhang2013,Zhang2014,Liu2014}) \nhas been carried out. The XSTPS-GAC photometric catalog contains \nmore than 100 million stars in the direction of Galactic anticentre (GAC). It provides an excellent \ndata set to study the Galactic disk, its structures and substructures. \nIn this paper, we take the effort to constrain the Galactic model \nparameters by combining photometric \ndata from the XSTPS-GAC and SDSS surveys.\nThis is the third paper of a series on the Milky Way study based on the XSTPS-GAC data.  In \n\\citet{Chen2014}, we present a three dimensional extinction map in $r$ band. The map has a spatial \nangular resolution, depending on latitude, between 3 and 9\\,arcmin and covers the entire XSTPS-GAC \nsurvey area of over 6,000 deg$^2$ for Galactic longitude 140 $< l <$220\\,deg and latitude 40 $< b <$40\\,deg. \nIn \\citet{Chen2015}, we investigate the correlation between the extinction and the $\\rm H~{\\scriptstyle I}$~ and CO emission at intermediate \nand high Galactic latitudes ($|b| >$ 10\\degr) within the footprint of the XSTPS-GAC, on small and large scales. \nIn the current work we are interested in the global, smooth structure of the Galaxy. \n\nFor the Galactic structure, in addition to the global,  smooth major components,\nmany more (sub-)structures have been discovered, including the inner bars near the Galactic centre\n\\citep{Alves2000, Hammersley2000, vanLoon2003, Nishiyama2005, Cabrera2008, Robin2012},\nflares and warps of the (outer) disk \\citep{Lopez2002, Robin2003, Momany2006, Reyle2009, Lopez2014},\nand various overdensities in the halo and the outer disk, such as the Sagittarius Stream\n\\citep{Majewski2003}, the Triangulum-Andromeda \\citep{Rocha2004, Majewski2004} and\nVirgo \\citep{Juric2008} overdensities, the Monoceros ring \\citep{Newberg2002,Rocha2003}\nand the Anti-Center Stream \\citep{Rocha2003,Crane2003, Frinchaboy2004}.\nThey show the complexity of the Milky Way. \nRecently, \\citet{Widrow2012} and \\citet{Yanny2013} have \nfound evidence for a significant Galactic North-South \nasymmetry in the stellar number density distribution,  exhibiting some \nwavelike perturbations that seem to be intrinsic to the disk.\n\\citet{Xu2015} show that in the anticentre regions\nthere is an oscillating asymmetry in the main-sequence star counts \non either sides of the Galactic plane, in support of the prediction of \n\\citet{Ibata2003}. The asymmetry oscillates in the sense that there are more \nstars in the north, then in the south, then back in the north,\nand then back in the south  at distances of about 2, 4 -- 6, 8 -- 10 and 12 -- 16\\,kpc \nfrom the Sun, respectively.\n\nThe paper is structured as follows. The data are introduced in Section~2.\nWe describe our model and the analysis method in Section~3. Section~4 \npresents the results and discussions. In Section~5 we discuss the large \nscale excess\/deficiency  of star counts that reflect the substructures in the halo and disk.  \nFinally we give a summary in Section~6.\n\n\\section{Data}\n\n\\begin{table}\n \\centering\n  \\caption{Data sets.}\n  \\begin{tabular}{lcccc}\n  \\hline\n  \\hline\n & area & field size & $N_{\\rm fields}$& $r$ ranges \\\\\n  &  (deg$^2$)    & (deg $\\times$ deg)& & (mag) \\\\\n  \\hline\nXSTPS-GAC & $\\sim$3392 & 2.5$\\times$2.5 & 574 &12--18 \\\\\nXSTPS-M31\/M33 & $\\sim$588 & 2.5$\\times$2.5 & 108 & 12--18 \\\\\nSDSS & $\\sim$6871 & 3.0$\\times$3.0 & 1592 &15--21 \\\\\n   \\hline\n\\end{tabular}\\\\\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{xstpsarmap.eps}\n  \\includegraphics[width=0.48\\textwidth]{sdssdenmap.eps}\n  \\includegraphics[width=0.48\\textwidth]{allfields.eps}\n  \\caption{{\\it Upper panel}: Extinction map of the GAC and M31\/M33 areas within the footprint of \nXSTPS-GAC (\\citealt{Chen2015} map for GAC area and \\citealt{Schlegel1998} map for M31\/M33 area). \nThe selected fields for GAC area and M31\/M33 area are marked as red and blue pluses,\nrespectively. The red star symbols mark the central positions of M31 and M33, respectively. {\\it Middle panel}: \nSDSS DR12 density map of stars in a magnitude bin of $r$ = 15.5 to 16.5\\,mag at a \nresolution of 0.1\\degr. The selected fields from SDSS are marked as red pluses.   {\\it Bottom panel}: \nLocation of the 682 fields selected from the XSTPS-GAC (red) and 1592 fields selected from the SDSS (blue) in \nGalactic coordinates.}\n  \\label{data}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{selcorr.eps}\n  \\caption{Colour-magnitude distributions of stars in all selected subfields from \nthe Sample\\,C, the re-weighted Sample\\,C (see Equation~(1) \nand related discussion), and the XSTPS-GAC.\nThe bottom three panels show the grey-scaled number densities distributed in the $g-i$ vs. $r$ space\nrespectively for the XSTPS-GAC (left), the re-weighted Sample\\,C (middle), and the XSTPS-GAC (right).\nThe upper three panels show the number distribution contours in the $g-i$ vs. $r$ space\n(left) as well as number distributions \nrespectively in $r$ (middle) and $g-i$ (right) for each sample.\nThe black contours and histograms show the density of all \ntargets in the XSTPS-GAC, the red ones represent the distributions of stars in \nSample\\,C and the blue ones display the \ndistributions for the re-weighted Sample\\,C. The contours labeled with `a', `b' and `c'\nin the left-upper panel represent the contour levels of star number \nof 6\\,000,  24\\,000 and 48\\,000, respectively.\nThe re-weighted Sample\\,C perfectly reproduces the colour-magnitude\nsampling provided by the XSTPS-GAC.}\n  \\label{sel}\n\\end{figure*}\n\n\\subsection{The XSTPS-GAC Data}\n\nThe XSTPS-GAC started collecting data in the fall of 2009 and completed in the spring of 2011.\nIt was carried out in order to provide input catalogue for the LSS-GAC.\nThe survey was performed in the SDSS $g$, $r$ and $i$ bands using the \nXuyi 1.04\/1.20\\,m Schmidt Telescope equipped with a 4k$\\times$4k CCD camera, \noperated by the Near Earth Objects Research Group of the Purple Mountain Observatory.\nThe CCD offers a field of view (FoV) of 1.94\\degr $\\times$ 1.94\\degr, with a pixel scale of 1.705\\,arcsec.\nIn total, the XSTPS-GAC archives approximately 100 million stars down to a\nlimiting magnitude of about 19 in $r$ band ($\\sim$ 10$\\sigma$) \nwith an astrometric accuracy about 0.1\\,arcsec\n and a global photometric accuracy of about 2\\% \\citep{Liu2014}.\n The total survey area of XSTPS-GAC is close to 7,000\\,deg$^2$,\ncovering an area of $\\sim$ 5,400\\,deg$^2$ centered on the GAC,\nfrom RA $\\sim$ 3 to 9\\,h and Dec $\\sim~-$10\\degr to $+60\\degr$, plus\nan extension of about 900\\,deg$^2$ to the M31\/M33 area and the bridging fields \nconnecting the two areas.\n\n\\subsubsection{GAC area}\n\nIn the direction of GAC, the $r$-band extinction exceeds 1\\,mag over a significant fraction of the sky \n(see Fig.~\\ref{data}). To correct the extinction of stars in high extinguish area \nusing extinction maps integrated over lines of sight, \nsuch as \\citet{Schlegel1998}, will introduce over corrections.\nIt will make stars too bright and blue. We select a subsample, the\nso-called ``Sample\\,C'' in \\citet{Chen2014}, from XSTPS-GAC.\nExtinction of all stars in Sample\\,C were calculated by the  spectral energy distribution (SED) \nfitting to the multi-band data, including the photometric data from the optical ($g,~r,~i$ from \nXSTPS-GAC) to the near-infrared ($J,~H,~K_S$ from 2MASS and $W1, ~W2$ from the Wide-field \nInfrared Survey Explorer, WISE, \\citealt{Wright2010}). \nThe extinction of targets in the subsample, Sample\\,C, \nis highly reliable, all having minimum SED fitting \n$\\chi ^2 _{min} ~<$ 2.0 (see \\citealt{Chen2014} for more details). \nWe correct the extinction of stars in Sample\\,C using the SED fitting extinction\nand the extinction law from \\citet{Yuan2013}. \nThere are more than 13 million stars in Sample\\,C. We divide them \ninto small subfields of roughly 2.5\\degr $\\times$ 2.5\\degr. The\nwidth ($\\Delta l$) and height ($\\Delta b$) of each subfield are always exactly 2.5\\degr.\nEach subfield is not exactly 6.25\\,deg$^2$ but varies with Galactic latitude $b$. \nBecause of the heavy extinction or poor observational conditions \n(large photometric errors), some subfields have \nobviously small amount of stars, comparing to most normal \nneighboring fields and thus be excluded. \nAs a result, 574 subfields, covering about 3392\\,deg$^2$, \nare selected. The locations of these subfields are shown in the top panel of \nFig.~\\ref{data}, with the grey-scale background image \nillustrating the 4\\,kpc extinction map from \\citet{Chen2015}.\n\nFor each subfield, Sample\\,C does not contain all stars in XSTPS-GAC.\nTo connect the distribution of targets in Sample\\,C\nto the underlying distribution of all stars, it is necessary to correct for the effects of the selection \n(often referred to as selection biases). Generally, the selection effects of Sample\\,C \nare due to the following two reasons: (1) the procedure by which we cross-match the photometric \ncatalogue of the XSTPS-GAC with those of 2MASS and WISE, and (2) the $\\chi^2$ cut when we\ndefine the sample with highly reliable extinction estimates.  For the first part, we lose about 15\\,per\\,cent \nobjects, mainly due to the limiting depths of 2MASS and WISE, \nespecially at low Galactic latitudes (see Fig.~1 of \\citealt{Chen2014}). \nFor the second part, we lose more than half of the objects, because of the large photometric errors, \nhigh extinction effects, or the special targets contamination, \nsuch as blended or binaries which are not well fitted by the standard SED \nlibrary in \\citet{Chen2014}. Our model for the selection function of Sample\\,C \ncan thus be expressed as the function of the positions ($l$, $b$), colour ($g-i$) \nand magnitude ($r$) of stars, given by,\\\\\n\\begin{equation}\n  S(l,b,g-i,r) = \\frac{N_{\\rm SC}(l,b,g-i,r)}{N_{\\rm XSTPS}(l,b,g-i,r)},\n\\end{equation}\nwhere $N_{\\rm SC}(l,b,g-i,r)$ and $N_{\\rm XSTPS}(l,b,g-i,r)$ are the number of stars \nin the Sample\\,C and the XSTPS-GAC, respectively. \nThe numbers of objects are evaluated within each subfield with area of $\\sim$ 6.25\\,deg$^2$, \neach colour ($g-i$) bin ranging from 0 to 3.0\\,mag with a bin-size of 0.1\\,mag, \nand each $r$-band magnitude bin ranging from 12 to 18.5\\,mag with a bin-size of 0.1\\,mag. \n\nThe number distributions in colour $(g-i)$ and magnitude $r$ for the stars\nin all selected subfields in the Sample\\,C, the Sample\\,C\nre-weighted by the selection effect, as well as the XSTPS-GAC,\nare shown as the density grey-scales and \ndensity contours and histograms in Fig.~\\ref{sel}. \nIt is clear that our correction of selection effect leads to perfect agreement between the \ncomplete XSTPS-GAC photometric sample and the re-weighted Sample\\,C.\n\n\\subsubsection{M31\/M33 area}\n\nThe dust extinction in the M31 and M33 area is much smaller, \ncompared with the GAC area (see the top panel of Fig.~\\ref{data}). \nWe adopt the extinction map from \\citet{Schlegel1998} \nand the extinction law from \\citet{Yuan2013} \nto correct the extinction of stars in M31\/M33 area. \nSimilarly as in the GAC area, all stars in M31\/M33 area are divided into small subfields, \nwhich have width ($\\Delta l$) and height ($\\Delta b$) always of 2.5\\degr. \nWe exclude the subfields which have maximum $E(B-V)$ larger \nthan 0.15\\,mag (i.e. $A_r$ = 0.4\\,mag, according to the extinction law from \\citealt{Yuan2013}), \nto avoid the relatively large uncertainties caused \nby the high extinction in the highly extinguished regions. The subfields that cover M31 are also \nexcluded. As a result, there are 108 subfields in the\nM31\/M33 area, covering about 588\\,deg$^2$. The locations of these subfields \nare also plotted in the top panel of Fig.~\\ref{data}, with \nthe grey-scale background image illustrating the extinction map from \\citet{Schlegel1998}. \nConsidering the limiting magnitude of XSTPS-GAC ($r$ $\\sim$ 19\\,mag), \nwe claim that the data in the M31\/M33 area from XSTPS-GAC is complete in the magnitude \nrange $12 < r_0 < 18$\\,mag. \n\n\\subsection{The SDSS Data}\n\nAs the survey area of XSTPS-GAC mainly locate around the low Galactic latitudes, \nwe also use the photometric data from SDSS, for constraining better the outer disk and \nthe halo. We use the photometric data from SDSS data release 12 (DR12, \\citealt{Alam2015}). \nThe SDSS surveys mainly for high Galactic latitudes, with only a few stripes \ncrossing the Galactic plane. It complements one another with the XSTPS-GAC. We cut the \nSDSS data with Galactic latitude $|b| >$ 30\\degr, where the influence of the dust \nextinction is small. The dust extinction are corrected using the extinction \nmap from \\citet{Schlegel1998} and the extinction law from \\citet{Yuan2013}.\nThe SDSS data are divided into subfields with\nwidth ($\\Delta l$) and height ($\\Delta b$) always of 3\\degr. \nTo make sure that each subfield is fully sampled by the SDSS survey, we further\ndivide each subfield into smaller pixels (of size 0.1\\degr $\\times$ 0.1\\degr) and \nexclude the subfield which has no stars detected in at least one of the smaller pixels. \nAs a result we have obtained 1592 subfields, covering a sky area of \nabout 6871\\,deg$^2$. In the middle panel of Fig.~\\ref{data}, we show the \nspatial distributions of these subfields,\nwith grey-scale background image illustrating the number density of the SDSS data.\nTo remove the contaminations of hot white dwarfs, low-redshift quasars and \nwhite dwarf\/red dwarf unresolved binaries from the SDSS sample, we\nreject objects at distances larger than 0.3 mag from the $(r-i)_0$ vs. $(g-r)_0$ \nstellar loci \\citep{Juric2008, Chen2014}.\nThe 95 per\\,cent completeness limits of the SDSS images are $u$, $g$, $r$, $i$ and $z$ $=$\n 22.0, 22.2, 22.2, 21.3 and 20.5\\,mag, respectively \\citep{Abazajian2004}. Thus \nthe SDSS data is complete in the magnitude range of $15 < r_0 < 21$\\,mag. \n\nA brief summary of the data selection in the current work is given in Table~1. In total, \nthere are 2274 subfields, covering nearly 11,000\\,deg$^2$,\nwhich is more than a quarter of the whole sky area. \nThe positions of all the subfields, from both the XSTPS-GAC and the\nSDSS, are plotted in the bottom panel of \nFig.~\\ref{data}. They cover the whole range of Galactic latitudes.\nGenerally,  the XSTPS-GAC provides nice constraints of the Galactic disk(s), \nespecially for the thin disk, while the SDSS provides us a good opportunity to \nrefine the structure of Galactic halo, as well as the outer disk.   \n \n\\section{The Method}\n\n\\subsection{The Galactic model}\n\nWe adopt a three-components model for the smooth stellar distribution of the\nMilky Way. It comprises two exponential disks (the thin disk and the thick disk) \nand a two-axial power-law ellipsoid halo \\citep{Bahcall1980, Gilmore1983}. \nThus the overall stellar density $n(R,Z)$ at a location $(R,Z)$ can be decomposed\nby the sum of the thin disk, the thick disk and the halo,\n\\begin{equation}\n  n(R,Z)=D_1(R,Z)+D_2(R,Z)+H(R,Z),\n\\end{equation}\nwhere $R$ is the Galactocentric distance in the Galactic plane, $Z$ is the\ndistance from the Galactic mid-plane.\n$D_1$ and $D_2$ are stellar densities of the thin disk and the thick disk,\n\\begin{equation}\n  D_i(R,Z)=f_{i}\\,n_{0}\\exp\\left[-\\,{(R-R_\\odot)\\over L_{i}}-\\,{(|Z|-Z_\\odot)\\over H_{i}}\\right],\n\\end{equation}\nwhere the suffix $i=1$ and $2$ stands for the thin disk and thick disk, respectively. \n$R_\\odot$ is the radial distance of the Sun to the Galactic centre on the plane, \n$Z_\\odot$ is the vertical distance of the Sun from the plane, \n$n_0$ is the local stellar number density of the thin disk at ($R_\\odot$, $Z_\\odot$),\n$f_i$ is the density ratio to the thin disk ($f_1$=1),\n$L_{i}$ is the scale-length and $H_{i}$ is the scale-height. \nWe adopt $R_\\odot=8$\\,kpc  \\citep{Reid1993} and\n$Z_\\odot = 25$\\,pc \\citep{Juric2008} in the current work.\n$H$ is the stellar density of the halo,\n\\begin{equation}\n H(R,Z)=f_{h}\\,n_{0}\\left[R^2+(Z\/\\kappa)^2\\over R_\\odot^2+(Z_\\odot\/\\kappa)^2\\right]^{-p\/2},\n\\end{equation}\nwhere $\\kappa$ is the axis ratio, $p$ is the power index and $f_h$ is the halo normalization \nrelative to the thin disk.\n\n\\subsection{Halo fit}\n\n\\begin{table}\n \\centering\n  \\caption{The parameter space and results of the halo fit}\n  \\begin{tabular}{lcccc}\n  \\hline\n  \\hline\nParameters & Range & Grid size & Best value  & Uncertainty \\\\\n  \\hline\n$\\kappa$  & 0.1--1.0 & 0.01  & 0.65 & 0.05 \\\\\n$p$  & 2.3--3.3 & 0.01 & 2.79 & 0.17 \\\\\n   \\hline\n\\end{tabular}\\\\\n\\end{table}\n\nWe fit the component of the halo first.\nThe metallicity distribution of the halo stars can be described as a single \nGaussian component, with a median halo metallicity of $\\mu_{\\rm H}$=$-$1.46\\,dex and spatially\ninvariant of $\\sigma_{\\rm H}$=0.30\\,dex \\citep{Ivezic2008}. \nWe assume the metallicity of all halo stars as  [Fe\/H]$=-1.46$\\,dex and adopt the\nphotometric parallax relation from \\citet{Ivezic2008},\n\\begin{equation}\n\\begin{split}\nM_{r}  = &  4.50-1.11{\\rm [Fe\/H]}-0.18{\\rm [Fe\/H]}^2 \\\\\n             &  -5.06+14.32(g-i)_0-12.97(g-i)_0^2 \\\\\n             & + 6.127(g-i)_0^3 - 1.267(g-i)_0^4+0.0967 (g-i)_0 ^5.\n\\end{split}\n\\end{equation}\nThe distances of the halo stars can thus be calculated from the standard relation, \n\\begin{equation}\n  d=10^{0.2(r_0-M_r)+1}.\n\\end{equation}\n\nStar in a blue colour bin $0.5 \\le g-i < 0.6$ are selected. They do not suffer \nfrom the giant star contamination and probe larger distances to constrain the halo.\nWe calculate their distance using Equations~(5) and (6). The distances of the disk stars will be\nunderestimated because they are more metal-rich. To exclude the contamination of the disk stars, \nwe use stars with absolute distance to the Galactic plane \n$|Z| >$ 4\\,kpc. For each subfield, we divide all halo stars\ninto suitable numbers of logarithmic distance bins and then count the \nnumber for each bin. This number can be modelled as,\n\\begin{equation}\nN_{\\rm H}(d)=H(d)\\Delta V(d),\n\\end{equation}\nwhere $H(d)$ is the halo stellar density given by Equation~(4) and  \n$\\Delta V(d)$ is the volume, given by,\n\\begin{equation}\n\\Delta V(d) =\\frac{\\omega}{3}(\\frac{\\pi}{180})^2(d^3_2-d^3_1),  \n\\end{equation}\nwhere $\\omega$ denotes the  area of the field (unit in deg$^2$), $d_1$ and $d_2$ are\nthe lower distance limit and upper distance limit of the bin, respectively. \n\nWe fit the halo model parameters $p$ and $\\kappa$ to the data.\nAs we explicitly exclude the disk, we cannot fit for \nthe halo-to-thin disk normalization $f_h$.\nA maximum likelihood technique is adopted to explore the best \nvalues of those halo model parameters.\nIn Table~2, we list the searching parameter space and the grid size. \nFor each set of parameters, a reduced likelihood is computed between the simulated \ndata (star counts in bins of distances) and the observations,\ngiven by \\citet{Bienayme1987} and \\citet{Robin2014},\n\\begin{equation}\n  Lr=\\sum_{i=1}^{N} q_i \\times (1-R_i+{\\rm ln}(R_i)),\n\\end{equation}\nwhere $Lr$ is the reduced likelihood for a binomial statistics, \n$i$ is the index of each distance bin, $f_i$ and $q_i$ are the \nnumber of stars in the $i$th bin for the model \nand the data, respectively and $R_i=f_i\/q_i$. \nThe uncertainties of the halo parameters are estimated similarly as those in \\citet{Chang2011}.\nWe calculate the likelihood for 1000 times using the observed data  and the \nsimulations of the best-fit model adding with the Poisson noises. \nThe resulted likelihood range defines the confidence level and thus the uncertainties.  \n\n\\subsection{Disk fit}\n\nThe metallicity distribution of the disk is more complicated than that of the halo. \nThus we fit the disk model parameters through a different way.\nWe compare the $r$-band differential star counts in different colour bins and compare \nthem to the simulations to search for the best disk model parameters\n($n_0, ~L_1,~H_1,~f_2,~L_2$ and $H_2$), as well \nas the halo-to-thin disk normalization $f_h$.\n\nTowards a subfield of galactic coordinates ($l,~b$) \nand solid angle $\\omega$, the $r$-band differential star counts $N_{\\rm sim} (r^k_0)$ \n($k$ is the index of each magnitude bin) in a given colour bin $(g-i)^j_{0}$ \n($j$ is the index of each colour bin) can be simulated as follows:\n\\begin{enumerate}\n\\item The line of sight is divided into many small distance bins. For a given distance \nbin with centre distance of $d_i$ ($i$ is the index of each distance bin),\nthe $r$-band apparent magnitude of a star is given by \n\\begin{equation}\n  r_0(d_i) = M_r ((g-i)^j_0, {\\rm [Fe\/H]}|l,b,d_i)+\\mu,\n\\end{equation}\nwhere $\\mu$ is the distance modulus [$\\mu = 5{\\rm log}_{10}(d_i)-5$] and $M_r$\nis the $r$-band absolute magnitude of the star given by Equation (5). The metallicities \nof halo stars are again assumed to be $-$1.46\\,dex and those of disk stars are \ngiven as a function of positions, which is fitted using the metallicities of \nmain sequence turn off stars from LSS-GAC \\citep{Xiang2015},\n\\begin{equation}\n {\\rm  [Fe\/H]} = -0.61+0.51 \\cdot {\\rm exp}{(-|Z|\/1.57)}.\n\\end{equation}\n\n\\item The number of stars in each distance bin can be calculated by,\n\\begin{equation}\n  N(d_i)=n(R,Z|l,b,d_i)V(d_i), \n\\end{equation}\nwhere $V(d_i)$ is the volume given by Equation~(8) and $n(R,Z|l,b,d_i)$\nis the stellar number density given by Equation~(2, 3 and 4). The halo model parameters,\n$\\kappa$ and $p$, resulted from the halo fit are adopted and settled to be not changeable here.\n\n\\item Combining all distance bins, \nwe can obtain the modeled $r$-band star counts $N(r^k_0)$, by\n\\begin{equation}\n  N(r^k_0) = \\Sigma N(d_i) ~  {\\rm where} ~ r^k_0 - \\frac{\\rm rbin}{2} < r_0(d_i) < r^k_0 + \\frac{\\rm rbin}{2},\n\\end{equation}\nwhere rbin is the bin size of $r$-band magnitude (we adopt rbin=1\\,mag in the current work).\n$N(r^k_0)$ is the underlying star counts. When comparing to the observations, we \nneed to apply the selection function, by\n\\begin{equation}\n  N_{\\rm sim}(r^k_0) = N(r^k_0)S(l,b,g-i,r) C,\n\\end{equation}\nwhere $S(l,b,g-i,r)$ is the selection function, calculated by Equation~(1) for \nXSTPS-GAC subfields in GAC area and equals to one for \nXSTPS-GAC subfields in M31\/M33 area and all the SDSS subfields. Besides,\n \\begin{equation}\n   C =\n\\begin{cases}\n1       & {\\rm for~} d_{\\rm min} < d_i < d_{\\rm max}; \\\\\n0\t\t&  {\\rm otherwise};  \\\\\n\\end{cases}\n\\end{equation}\n\\begin{eqnarray}\n  d_{\\rm min} &=&  10^{0.2(r_{\\rm min}-A_r(d_i)-M_r((g-i)_0,{\\rm [Fe\/H]}))+1},\\\\\n  d_{\\rm max}&=&10^{0.2(r_{\\rm max}-A_r(d_i)-M_r((g-i)_0,{\\rm [Fe\/H]}))+1},\n\\end{eqnarray}\nwhere $r_{\\rm min}$ and $r_{\\rm max}$ are the \nmagnitude limits of each subfield.\nWe adopt $r_{\\rm min} = 12$ and  $r_{\\rm max} = 18$ for all XSTPS-GAC subfields,\nand $r_{\\rm min} = 15$ and  $r_{\\rm max} = 21$ for all SDSS subfields. \n$A_r(d_i)$ is the extinction in $r$-band at distance of $d_i$. We adopt the 3D extinction map from\n\\citet{Chen2014} for XSTPS-GAC subfields in GAC area and \n2D extinction map from \\citet{Schlegel1998} for\nXSTPS-GAC subfields in M31\/M33 area and all SDSS subfields. As the size of each subfield is quite\nlarge ($\\sim$ 4\\,deg$^2$), the extinction $A_r(d)$ varies within a subfield. We thus adopt the maximum \nvalues to make sure that our data are complete.\n\\end{enumerate}\n\nThe photometric parallax relation of Equation~(5) is only valid for the single \nstars. A large fraction of stars in the Milky Way are\nactually binaries (e.g. \\citealt{Yuan2015b}). In the current work we adopt the binary fraction \nresulted from \\citet{Yuan2015b} and assume that 40\\,per\\,cent of\nthe stars are binaries. The absolute magnitudes $M_r$ of the binaries are calculated as the same way\nas in \\citet{Yuan2015b}. \n\nWe also consider the effects of photometric errors, \nthe dispersion of disk star metallicities and the errors due to\nthe photometric parallax relation of \\citet{Ivezic2008}. The $r$-band photometric errors\nof most stars in the XSTPS-GAC and the SDSS are  smaller than 0.05\\,mag \n(\\citealt{Chen2014} for the XSTPS-GAC and \\citealt{Sesar2006} for the SDSS).\nWhen we fit the metallicities of disk stars as a function of positions [Equation~(11)], we \nfind a dispersion of the residuals of about 0.05\\,dex. \nAccording to Equation~(5), this dispersion \nwould introduce an offset of about 0.05\\,mag for \nthe absolute magnitude when [Fe\/H] = $-$0.2\\,dex. \nAs a result, the effect of the photometric errors and the\ndisk stars metallicities dispersions would \nintroduce a distance errors of smaller than 5\\,per\\,cent.\nCombining with the systematic error of the photometric parallax \nrelation, which is claimed to be smaller than 10\\,per\\,cent \\citep{Ivezic2008},\nwe assume a total error of distance of 15\\,per\\,cent. This distance error is added \nwhen we model the $r$-band\nmagnitude of stars in a given distance bin [Equation~(10)].\n\nWe select three different colour bins for the disk fit. \nTwo of them correspond to G-type stars with \n$0.5 \\le (g-i)_0 < 0.6$\\,mag and $0.6 \\le  (g-i)_0 < 0.7$\\,mag,\nand the other one corresponds to late K-type stars with $1.5 \\le (g-i)_0 < 1.6$\\,mag. \nThe giant and sub-giant contaminations for the first two G-type star bins are very small. \nFor the late K-type stars, we exclude stars with $r$-band magnitude $r_0 < 15$\\,mag \nto avoid the giant contaminations. For each colour bin, we count the differential\n$r$-band star counts with a binsize of $\\Delta r=1$\\,mag and then \ncompare them to the simulations to search for the \nbest disk model parameters, i.e $n_0,~L_1,~H_1,~f_2,~L_2,$ and $H_2$ and\nthe halo-to-thin disk normalization $f_h$. \nSimilarly as in \\citet{Robin2014}, \nan ABC-MCMC algorithm is implemented using the reduced likelihood calculated by Equation~(9) \nin the Metropolis-Hastings algorithm acceptance ratio \n\\citep{Metropolis1953, Hastings1970}. \nWe note that the 68\\,per\\,cent probability intervals of the \nmarginalised probability distribution functions (PDFs) of each parameter, given\nby the accepted values after post-burn period in the MCMC chain are only the fitting uncertainties \nwhich do not include systematic uncertainties.\nA detailed analysis of errors of the scale parameters will be given in Sect.~4.2.\n\nThe stellar flare is becoming significant at $R \\ge$15\\,kpc \\citep{Lopez2014} \nwhile the limiting magnitude we adopt for XSTPS-GAC is $r=18$\\,mag, which corresponds\nto $R ~\\sim$ 13\\,kpc for early G-type dwarfs.  On the other hand, the disk warp is \na second order effect on the star counts and the XSTPS-GAC centre around the \nGAC, with $l$ around 180\\degr. The effect of the disk warp is thus\nnegligible \\citep{Lopez2002}.  So in the current work we ignore the influences of the disk warps and flares. \nIn order to minimise the effects coming from other irregular structures (overdensities)\nof the Galactic disk and halo (e.g., Virgo overdensity, etc. ), we iterate our fitting \nprocedure to automatically and gradually \nremove pixels contaminated by unidentified irregular structures, similarly as in \\citet{Juric2008}. \nThe model is initially fitted using all the data points.\nThe resulted best-fit model is then used to define the outlying data, \nwhich have ratios of residuals (data minus the best-fit model) to the best-fit model\nhigher than a given value, i.e. $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod} > a_1$. \nThe model is then refitted with the outliers excluded. \nThe newly derived best-fit model is again compared to all the data points. \nNew outliers with $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod} > a_2$\nare excluded for the next fit. We repeat this procedure with a sequence of values\n$a_i$ = 0.5, 0.4, and 0.3.  The iteration, which gradually reject about 1, 5 and 15\\,per\\,cent of\nthe irregular data points of smaller and smaller significance, will make our model-fitting algorithm to \nconverge toward a robust solution which describes the smooth background best. \n\n\\section{The Results and Discussion}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{halogrid.eps}\n  \\caption{Reduced likelihood surface of the halo parameters $p$ and $\\kappa$ \n  space (see Table~2). The best-fitted values and uncertainties are marked \n  as a red plus with error bars. \n  The red contour ellipse shows the likelihood ranges used for estimating \n  the uncertainties.}\n  \\label{halog}\n\\end{figure}\n\n\n\\begin{table*}\n \\centering\n  \\caption{The best-fit values of the disk fit}\n  \\begin{tabular}{lcccccccc}\n  \\hline\n  \\hline\n  Bin & $n_1$ & $L_1$ & $H_1$ & $f_2$ & $L_2$ & $H_2$ & $f_H$ & $Lr$  \\\\\n         & $10^{-3}$stars\\,$pc^{-3}$ & pc & pc & per\\,cent & pc & pc & per\\,cent &  \\\\\n\\hline\nJoint fit\\\\\n\\hline\n $0.5 \\le (g-i)_0 < 0.6$    &  1.25 & 2343 & 322 & 11 & 3638 & 794 & 0.16  & $-$86769   \\\\\n    $0.6 \\le (g-i)_0 < 0.7$         &  1.20 & & & & & & &   \\\\\n       $1.5 \\le (g-i)_0 < 1.6$           &  0.54 & & & & &  & & \\\\ \n        \\hline    \n    Individual fit\\\\\n    \\hline\n $0.5 \\le (g-i)_0 < 0.6$ & 1.31 & 1737 & 321  & 14   \n                               & 3581& 731  & 0.16   & $-$43699   \\\\     \n $0.6 \\le (g-i)_0< 0.7$ & 1.65  & 2350  & 284  & 7  \n                               & 3699  & 798  & 0.12   & $-$35774  \\\\        \n $1.5 \\le (g-i)_0 < 1.6$ & 0.41  & 2780  & 359  & 8 \n                               & 2926  & 1014  & 0.50 & $-$4028   \\\\\n \\hline\n stddev & & 429 & 31 & 3 & 360 & 124 & 0.02 &  \\\\\n \\hline \n\\end{tabular}\n\\end{table*}     \n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.95\\textwidth]{cross2d.eps}\n  \\caption{\n  Two-dimensional marginalized PDFs for the disk model parameters,\n$L_1,~H_1,~f_2,~L_2,$ and $H_2$\nand the halo-to-thin disk normalization $f_h$, obtained from the MCMC analysis. \nHistograms on top of each column show the one-dimensional marginalized PDFs \nof each parameter labeled at the bottom of the column. \nRed pluses and lines indicate the best solutions. \nThe dash lines give the 16th and 84th percentiles, \nwhich denotes only the fitting uncertainties.}\n  \\label{cross2d}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.68\\textwidth]{rntestmodel.eps}\n  \\caption{Star count (per deg$^2$) for the colour bin $0.5 \\le (g-i)_0 <0.6$\\,mag and magnitude bins,\n   $r_0$ = 15 (left) and 16\\,mag (right), of both the XSTPS-GAC\n  (red pluses) and the SDSS (blue pluses) data as a function of the Galactic latitude\nfor example subfields with Galactic longitude 177\\degr  $<l<$ 183\\degr.\nThe black dots are the model predictions.}\n  \\label{rnmodel}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{datamode.eps}\n  \\caption{Difference map for the all subfields \n  $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}$. \n  The populations in excess in the data are most \n  likely irregular structures of the Milky Way.\nThe squares denote different regions with significant excesses of the\nstar counts.}\n  \\label{resd}\n\\end{figure*}\n\n\\subsection{Fitting results}\n\nThe best-fit halo model parameters obtained from the halo fit are listed in Table~2 and the\nreduced likelihood $L_r$ surface for the fit is shown in Fig.~\\ref{halog}.\nThe $L_r$ contour dramatically changes along $\\kappa$, \nbut relatively mildly along $p$. The best-fit halo model parameters are \n$\\kappa=0.65\\pm 0.05$ and $p=2.79\\pm 0.17$. They are not \nsurprisingly in good agreement with those values from \\citet{Juric2008}, \nwhich have $\\kappa=0.64$ and $p=2.8$, since the data used for the halo fit in the current work\nare mainly from the SDSS subfields. In addition, the photometric parallax relation that\n\\citet{Juric2008} used for the blue stars was corrected for low-metallicities \n([Fe\/H] $\\sim$ $-$1.5\\,dex), which is of minor difference from \nthe photometric parallax we used for the halo stars.\n\nThe best-fit values for the disk fit are listed in Table~3.\nWe performed the disk fit using first jointly all three colour bins, then separately for each colour bin.\nFor the joint fit case, all the model parameters, excepted for \nthe local density ($n_0$), are settled to be the same for stars in every bin,\nas we expect a universal density profile for all stellar populations \\citep{Juric2008}. \nAll the parameters resulted from the joint fit appear to be very well constrained.\nWe explore the correlations between different parameter pairs in Fig.~\\ref{cross2d}. \nThe Figure shows the marginalized one- and \ntwo-dimensional PDFs of the model parameters. \nThe correlations between different parameters are rather weak in general. \nWe identify small correlations \nbetween several pairs of the parameters, such as ($H_1$, $H_2$), ($H_1$, $f_2$) \nand ($l_1$, $f_h$), and a strong degeneracy between the scale height and\nthe local normalised densities ratio of thick disk ($H_2$, $f_2$).\n\nFor the separate fits, we fit all parameters independently for each color bin. \nThe individual fits can be served as a consistency check of our method.\nFrom Table~3, we can declare that those best-fit solutions \nare generally consistent. The thin disk scale height varies around\n$H_1 ~ \\sim$ 320 pc, by $\\pm 40$\\,pc, which is \nconsistent with the result from the  joint fit.\nThe variations of the disk scale lengths are relatively large, with \n$L_1~\\sim$ 1.7 $-$ 2.8\\,kpc and $L_2~\\sim$ 2.9 -- 3.7\\,kpc for the\nthin disk and the thick disk, respectively. \nThe large variations could be due to the limited ranges of Galactocentric distances \non the Galactic plane for stars from both the XSTPS-GAC (for the limited depth) and \nthe SDSS (for the poor sky coverage in low Galactic latitudes). \nData with deeper depth and better sky coverage in the low Galactic latitudes\n (such as Pan-Starrs) may help to improve the situation.\nThe thick disk normalization $f_2$ and scale height $H_2$\nappear also weakly constrained, with $f_2$ ranging from 7 to 14\\,per\\,cent and\n$H_2$ from 730 to 1000\\,pc. This is mainly due to the strong\ncorrelation between these two parameters (see \nthe  $f_2$ vs. $H_2$ panel in Fig.~\\ref{cross2d}).  The halo normalisation $f_h$\nis well constrained to $\\sim$ 0.15\\,per\\,cent for the two G-type star bins. While\nthe value of $f_h$ is abnormally large for the late K-type star bin [1.5$ \\le (g-i)_0 < $1.6]. \nThis is mainly due to the fact that late K dwarfs in the halo are cut by our limiting magnitude ($r_0<$21\\,mag).\n\nWe plot in Fig.~\\ref{rnmodel} the star counts in the colour bin $0.5 \\le (g-i)_0 < 0.6$\\,mag\nand magnitude bins, $r_0=$15 and 16\\,mag of both the XSTPS-GAC \nand the SDSS data as a function of the Galactic latitude for example subfields \nwith Galactic longitude 177\\degr $ < l < $ 183\\degr. \nThe best-fit model is in good agreement with the observations,\nwith some small deviations.\nIn Fig.~\\ref{resd}, we show the differences between the observed star counts, integrated from stars\nin all the three colour bins and $r_0$ from 12 to 21\\,mag, and \nthe model predictions as a function of position on the sky. \nDifferent colours in the Figure indicate the values of the\nratio $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}$. For most \nof the fields, we do not see obvious deviation,\nwith residual smaller than 10\\,per\\,cent, i.e. \n$|(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}| < 0.1$.\n\n\n\\subsection{Systematics}\n\nThe dispersions of the resultant parameters from both the jointly fit and individual fit \nare also listed in Table.~3, which could be used to denote the \nsystematic errors of the corresponding parameters. \nThe typical errors of the parameters are about 10\\,per\\,cent.  \nSome of the parameters, i.e. the thin disk scale length $H_1$ and the thick disk \nlocal normalised densities ratio $f_2$, have relatively larger uncertainties, which are mainly due\nto the limits of our data and the degeneracies between different parameters. \nOther dominant sources of the errors are (in order of decreasing importance):\n1) the systematic distance determination uncertainties, \n2) the misidentification of binaries as single stars, \n3) the value of distance error, i.e., finite width of the photometric parallax relation,  \n4) the contamination of non-dwarf stars, and \n5) the effects from disk warp and flare. Finally, the structure parameters for different \nstellar populations (colour bins) would be intrinsically different, which may also contribute to \nthe dispersions.\n\nDue to the absolute calibration errors of the photometric parallax relation, the distances of stars \ncould be systematically over- or under- estimated. To check this effects, we\nredo the fit by changing the distance scale by \udbff\udc0e15\\,per\\,cent, i.e., \nusing the parallax relations 0.3\\,mag brighter and 0.3\\,mag fainter than Equation~(5).\nThe relative differences between \nthe original resultant parameters and those derived after \nchanging the distance scale by 15\\,per\\,cent are about 10 -- 15\\,per\\,cent.\n\nComparing to the single stars of the same colour, the binaries are \nbrighter, i.e. of smaller absolute magnitudes. Thus if one model  \nthe Galaxy with no or less fraction of binaries, the resultant \nmodel would be more `compact' than it truly is. \nIn the current work we select a binary fraction $f_b$=40\\,per\\,cent, \nwhich is an average binary fraction \nfor field FGK stars \\citep{Yuan2015b}. To check the possible effects \nof the binary fraction, we have tried two extreme \nvalues, the lowest value, $f_b$=0, assuming that all stars are \nsingle stars and the highest value, $f_b$=1, assuming that all stars\nare binaries, to redo the fit. \nThe relative differences between \nthe original resultant parameters and those derived after \nchanging the binary fraction to the extreme values are about 10\\,per\\,cent. \n\nWhen simulating the star counts for each colour bin, we assumed an error of distances\nof 15\\,per\\,cent, which includes the effect of photometric \nuncertainties, dispersion of metallicities of disk stars and  \nuncertainties of the \\citet{Ivezic2008} photometric parallax \nrelation. We redo the fit by changing the \ndistance dispersion to 0 and 30\\,per\\,cent, respectively. \nThe results show that changing the uncertainties of \ndistance do not introduce any significant bias in the derived Galactic model parameters.     \n\nThe effects for the model parameters caused by the non-dwarf (i.e. subgiants and giants) contaminations\nare similar as the binaries. From the Galaxy stellar population synthesis models, BESANCON\n\\citep{Robin2003} and TRILEGAL \\citep{Girardi2005}, we find that the fraction of giants in the three \nchosen colour bins is no more than 1\\,per\\,cent and the fraction \nof sub-giants is less than 5\\,per\\,cent. Thus the systematics \ncaused by the giant and sub-giant contaminations are likely to be negligible. \n\nIn the current work we have ignored the effects of disk flare, as we assume that the stellar flare\nis becoming significant at further distances \\citep{Lopez2014}. While \\citet{Derriere2001} and \nAmores et al. (2016, submitted) find that the flare starts at about 9 to 10\\,kpc. Having a shorter\nstart of the flare could have an impact on the thin ansd thick disk scale lengths that we determine, \nwhich would make the values to be underestimated. \n   \n\\subsection{Comparisons with other work}\n\nAs stated in Section~4.1, our results of the halo model parameters are very similar\nas those derived from \\citet{Juric2008}, because of the similar method (stellar \nnumber density fits) and data (the SDSS) adopted in both work. \nThe main differences in the halo fit between those from \\citet{Juric2008}\nand our work are that they adopt the $\\chi^2$ fitting while we use the reduced\nlikelihood $L_r$; and they calculate the number densities \nfor the entire sample in $(R,~Z)$ space while we do that separately for each line of sight. \nThe result in our work confirms those from \\citet{Juric2008}.\n\nWhen constraining the disk model parameters,\nour method and data are both quite different from those in \\citet{Juric2008}\nbut the results are in agreement at a level of about 10\\,per\\,cent.\nHowever, \\citet{Juric2008} admit that their result suffers large uncertainties\nfrom the uncertainty in calibration of the photometric parallax relation and the poor sky coverages \nfor the low Galactic latitudes, which is not the case in our work.\nGenerally, our derived value of thin disk scale height, 322\\,pc, is in the range of values, \n150--360\\,pc, which resulted from the recent work by \\citet{Bilir2008, Juric2008, \nYaz2010, Chang2011, Polido2013, Jia2014} and \\citet{Lopez2014}. \nSpecially, this value is in good agreement with the canonical value of 325\\,pc \n\\citep{Gilmore1984, Yoshii1987, Reid1993, Larsen1996}. Our derived value of the\nthin disk scale length, 2.3\\,kpc, is \nconsistent with those results found by \\citet{Ojha1996, Robin2000, Chen2001, Siegel2002, Karaali2007, \nJuric2008, Robin2012, Polido2013, Lopez2014} and \\citet{Yaz2015}, which ranges between 2 and 3\\,kpc. \nWe find a local thick disk normalization of 11 per\\,cent.  In the different colour bins, \nthis value varies between 7 and 14\\,per\\,cent, probably because of the degeneracy with the \nscale height. It is well in agreement with the values in the literature, which ranges between 7 and 13\\,per\\,cent \\citep{Chen2001, Siegel2002, Cabrera2005, Juric2008, Chang2011, Jia2014}. \nThe range of values for the thick disk scale height and scale length from the recent literature \nare respectively 600 -- 1000\\,pc and 3 -- 5\\,kpc \\citep{Bilir2008, Juric2008, \nYaz2010, Chang2011, Polido2013, Jia2014, Lopez2014, Robin2014}. The results deduced here, \nwhich have thick disk scale height of $\\sim$800\\,pc and scale length of 3.6\\,kpc, are both in the\nmiddle of the ranges of those values reported in the literature. Notice that there is a significant difference \nwith the \\citet{Robin2014} result for the thick disc. Their thick disk is modelled with 2 episodes, \none of which has very similar parameters as the present result (their old thick disk). But their young thick \ndisk is more compact, with smaller scale height and scale length. The difference can be due to the different \nshapes used (they use secant squared density laws and they include the flare) while in the present study \nthe thick disk is a simple exponential vertically. \n\n\\section{Substructures}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{cmdplot_l181.eps}\n  \\includegraphics[width=0.48\\textwidth]{rhist_l181.eps}\n  \\caption{{\\it Upper panel:} Differences of star counts in the $r_0$ vs. $(g-i)_0$ diagram for XSTPS-GAC data \n  in 2.5\\degr $\\times$ 2.5\\degr patches of sky, i.e. ($l,b$) = (181.25\\degr, $-$8.75\\degr) minus ($l,b$) = (181.25\\degr, \n  8.75\\degr). The red curve which fits the white main-sequence pattern of the newly identified substructure\n  is an isochrone with [Fe\/H]=$-$0.2\\,dex, age of 3.981\\,Gyr and distance of 1.3\\,kpc. The green curve that fits the black \n   main-sequence pattern of the ``north near structure'' is an isochrone with [Fe\/H]=$-$0.5\\,dex, age of 2.818\\,Gyr and distance of 3.7\\,kpc. The blue curve that fits the white main-sequence pattern of the ``south middle structure'' \n  is an isochrone with [Fe\/H]=$-$0.5\\,dex, age of 1.995\\,Gyr and distance of 6.8\\,kpc. \n{\\it Bottom panels:} The $r$-band star counts in the colour bin $0.5< (g-i)_0<0.6$\\,mag for the south patch\n  ($l,b$) = (181.25\\degr, $-$8.75\\degr), north patch ($l,b$) = (181.25\\degr, 8.75\\degr) and \n  the subtraction of the two.}\n  \\label{l181}\n\\end{figure}\n\nOne of the main purposes of the Galactic smooth structure modelling  is to define the \nirregular structure of the Milky Way. Assuming that our model now \nreproduces the populations of the halo and the disk, \nwe can investigate if any further structures are missing from our modelling. \nAs seen in Fig.~\\ref{resd}, most of the significant deviations are \noverdensities, i.e. $N_{obs} > N_{mod}$, except for \na few subfields such as those located at ($l,b$)=(170\\degr, 0\\degr). The extinction in\nthese fields are large \\citep{Chen2014}. It is very difficult to distinguish that whether it is a real  `hole'  \nor it is caused by the selection effects or extinction correction errors. \nFor the overdensities, we find three large scale structures,\nwhich are located at different positions on the sky and appear at\ndifferent magnitudes. We describe them as follows.\n\nThe first large region where star counts are in excess is located at \n240\\degr\\ $<l<$ 330\\degr~and  60\\degr\\ $<b<$ 90\\degr.\nIt is a large and diffuse structure, which corresponds \nto the Virgo overdensity found by \\citet{Juric2008}. \nFor stars with $0.5 ~<~(g-i)_0~<~0.7$\\,mag, the excess of star counts \noccurs at $r_0 \\sim$20\\,mag.\nSo the distance of the structure is between 9 -- 11\\,kpc, which is \nconsistent with the work of \\citet{Juric2008}. A plausible explanation of the Virgo \noverdensity is that it is a result of a merger event involving the Milky Way and \na smaller, low metallicity dwarf galaxy \\citep{Juric2008}.   \n\nThe second significant structure is located at\n170\\degr\\ $<l<$ 200\\degr~and  10\\degr\\  $<b<$ 30\\degr. \nThe excess of star counts occurs at $r_0~\\sim$ 16\\,mag \nfor stars with colour $0.5 ~<~(g-i)_0~<~0.7$\\,mag, \ncorresponding to a distance between 1.5--2\\,kpc. \nThis feature is consistent with the so-called \n``north near structure'' found by \\citet{Xu2015}. \n\\citet{Juric2008} also report an overdensity at\n$R$ = \udbff\udc09 9.5\\,kpc and $Z$ =\udbff\udc09 0.8\\,kpc, which\nmay be connected with this substructure.\n\\citet{Xu2015} report that this substructure \nrepresents one of the locations of peaks in the oscillations\n of the disk mid-plane, observed at \nabout $\\pm$15\\degr Galactic latitude, toward the Galactic anticentre. \n\nThe third star counts excess structure is located \nat 150\\degr\\ $<l<$ 210\\degr~and  $-$15\\degr\\  $<b<$ $-5$\\degr. \nWe have not found any corresponding substructures to this \nfeature from the literature.  \nWe need to check first that whether this\nsubstructure is real or caused by the selection bias or \nhigh extinction effects. To test its existence in a more direct way, \nwe examine the Hess diagrams constructed for a subfield located in\nthe area of the feature and for a control subfield which is outside the region.\nWe choose the control subfield with the same galactic longitude but opposite \nlatitude as the selected subfield that includes the overdensity.\nTo avoid the effect of the selection bias of Sample\\,C, \nwe select data from the complete XSTPS-GAC photometric sample in the regions \n($l,b$) = (181.25\\degr, $-$8.75\\degr, south field) and\n(181.25\\degr, 8.75\\degr, north field). Each region is of size 2.5\\degr $\\times$ 2.5\\degr.\nWe correct the extinction of stars using the extinction map from \\citet{Chen2015}\ntogether with the extinction law from \\citet{Yuan2013}.\nThe average extinction for the south and north fields\nare respectively $A_r$ = 0.66 and 0.27\\,mag. In Fig.~\\ref{l181}, \nwe show the differences between Hess diagrams\nand the star counts for stars with colour 0.5 $< (g-i)_0 < $ 0.6\\,mag for the two regions. \nWe see a white-black-white main-sequence pattern in the \ndifference Hess diagram. The fainter white sequence which indicating that \nthere are more stars on the south side\n and the middle black sequence indicating that \nthere are more stars on the north side respectively correspond \nto the ``south middle structure'' and ``north near structure'' described  in \\citet{Xu2015}.  \nThe bright white sequence corresponds to the newly found substructure. \nThe $r$-band extinction in the these two fields are rather small. Even if we assume an error of \n20\\,per\\,cent in extinction, the uncertainties of colour $g-i$ and magnitude $r$ in the two \nfields are only $\\delta _{g-i}$ = 0.09 and $\\delta _r$ = 0.13 for the south field and \n$\\delta _{g-i}$ = 0.04 and $\\delta _r$ = 0.05 for the north field, respectively. \nThe observed feature will not be erased by adjusting the reddening values. \nSo we believe that the feature is real. The effect of selection function and inaccuracies \nin the reddening correction cannot cause the apparent overdensities.\n\nThe excess of star counts occurs at around $r_0~\\sim 15$\\,mag for stars in the \ncolour ranges of 0.5 $< (g-i)_0 < $ 0.6\\,mag, indicating a distance between 1 -- 1.5\\,kpc. \nWe select a series of isochrones from \\citet{An2009}. The isochrone with [Fe\/H] = $-$0.2\\,dex \nand age of 3.981\\,Gyr, shifted by a distance of 1.3\\,kpc, fit well with the maximum overdensities of\nthe bright white sequence (see Fig.~\\ref{l181} ). The young age and high \nmetallicity are consistent with those of the field thin disk stars. Thus we believe that \nthis feature is unlikely a substructure originated from the outer disk. The distance to the \nGalactic plane of this feature is about $Z~\\sim~-$0.3\\,kpc and that for the ``north near structure''\nis about $Z~\\sim~$0.5\\,kpc. These two features, which show the significant\nNorth-South asymmetry in the star number count distributions, are consistent with the \nvertical oscillations in the stellar density in the solar neighbourhood discovered\nby \\citet[see their Fig.~18]{Yanny2013}. \n\n\\section{Summary}\n\nBased on the data from the XSTPS-GAC and the SDSS, we have modelled the global \nsmooth structure of the Milky Way. We adopt a three-component stellar distribution model.\nIt comprises two double exponential disks, the thin disk and the thick disk, and a\ntwo-axial power-law ellipsoid halo. The stellar number density of halo stars in the colour bin\n$0.5 < (g-i)_0 < 0.6$\\,mag and the $r$-band differential\nstar counts in three colour bins, $0.5 < (g-i)_0 < 0.6$\\,mag,\n $0.6 < (g-i)_0 < 0.7$\\,mag and  $1.5 < (g-i)_0 < 1.6$\\,mag, are\nused to determine the Galactic model parameters. \nThe best-fit values are listed in Table~2 and 3. In summary, the scale height and length\nof the thin disk are $H_1$=322\\,pc and $L_1$=2343\\,pc,\nand those of the thick disk are $H_2$=794\\,pc \nand $L_2$=3638\\,pc. The local stellar density ratio of thick-to-thin disk \nis $f_2$=11\\,per\\,cent, and that of \nhalo-to-thin disk is $f_h$=0.16\\,per\\,cent.  \nThe axis ratio and power-law index of the halo are $\\kappa=0.65$ \nand $p=2.79$. Our results are all well constrained and in good \nagreement with the previous works.\n\n\nBy subtracting the observations from our best-fit model, we find three\nlarge overdensities. Two of them have been previously identified, \nincluding the Virgo overdensity in the Halo \\citep{Juric2008}, which located \nat 240\\degr\\  $<l<$ 330\\degr\\ and  60\\degr\\  $<b<$ 90\\degr\\ with a distance between 9 -- 11\\,kpc, \nand the so-called ``north near structure'' in the disk \\citep{Xu2015}, which located \nat 170\\degr\\ $<l<$ 200\\degr\\ and 10\\degr\\ $<b<$ 30\\degr\\ with a distance between 1.5 -- 2\\,kpc. \nThe third structure, located at 150\\degr\\  $<l<$ 210\\degr\\ and   $-$15\\degr\\  $<b<$ $-5$\\degr\\ \nwith a distance between 1 -- 1.5\\,kpc, is a new identification. \nThrough the Hess diagram examination, we conclude that \nit could not be a artifact caused  by extinction correction or selection effects. \nThis feature, together with the ``north near structure'' confirms the earlier \ndiscovery of  \\citet{Widrow2012} and \\citet{Yanny2013} of a significant \nGalactic North-South asymmetry in the stellar number density distribution.\n\n\\section*{Acknowledgements}\n\nWe want to thank the referee, Prof. Gerry Gilmore, for his insightful comments.\nThis work is partially supported by National Key Basic Research Program of China\n2014CB845700,  China Postdoctoral Science Foundation 2016M590014 and \nNational Natural Science Foundation of China 11443006 and U1531244.\nThe LAMOST FELLOWSHIP is supported by Special Funding for Advanced Users,\nbudgeted and administrated by Center for Astronomical Mega-Science, Chinese\nAcademy of Sciences (CAMS). This research has made use of the Chinese \nVirtual Observatory (China-VO) resources and services.\n\nThis work has made use of data products from the Guoshoujing Telescope (the\nLarge Sky Area Multi-Object Fibre Spectroscopic Telescope, LAMOST). LAMOST\nis a National Major Scientific Project built by the Chinese Academy of\nSciences. Funding for the project has been provided by the National\nDevelopment and Reform Commission. LAMOST is operated and managed by the\nNational Astronomical Observatories, Chinese Academy of Sciences.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the\nParticipating Institutions, the National Science Foundation, and the U.S. Department\nof Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/.\nSDSS-III is managed by the Astrophysical Research Consortium for the\nParticipating Institutions of the SDSS-III Collaboration including the University\nof Arizona, the Brazilian Participation Group, Brookhaven National Laboratory,\nCarnegie Mellon University, University of Florida, the French Participation Group,\nthe German Participation Group, Harvard University, the Instituto de Astrofisica de\nCanarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins\nUniversity, Lawrence Berkeley National Laboratory, Max Planck Institute for\nAstrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico\nState University, New York University, Ohio State University, Pennsylvania State\nUniversity, University of Portsmouth, Princeton University, the Spanish Participation\nGroup, University of Tokyo, University of Utah, Vanderbilt University, University of\nVirginia, University of Washington, and Yale University.\n\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\n\nThe key contribution in advancing sequence-to-sequence (encoder-decoder) models accounted to neural attention (NA) networks has rendered the latter a core part of any deep learning toolkit. Indeed, their obtained state-of-the-art results in MT \\cite{Bahdanau2014,Luong2015}, abstractive document summarization (ADS) \\cite{Manning2017}, speech recognition (SR) \\cite{Chorowski2015,Chan2015} and question answering (QA) \\cite{Sukhbaatar2015}, amongst others, vouch for their broad applicability. NA network functionality consists in enabling the output layer of sequence-to-sequence models to access a variable-length memory \\cite{weston2014,Graves2014,Graves2016} of stored source sequence encodings. Specifically, NA networks infer a time-varying representation of the memory content, which encapsulates the affinity of the stored source sequence encodings with the (elements in the) decoded target sequence \\cite{Bahdanau2014}. This is usually referred to as the inferred \\emph{context vector representation}, $\\boldsymbol{c}_{t}$, of the stored source sequence encodings, and is used to drive output generation (decoding) at time $t\\in{1,\\dots,T}$. Consequentially, application of NA mechanisms enables encoder-decoder models to perform inference over long temporal horizons; this is not feasible without NA, due to the model having to encode the entire source information into a single fixed vector $\\boldsymbol{h}$.\n \nThis major breakthrough has rendered NA a focal point in recent research developments, mainly revolving around the concept of \\emph{Soft-Attention} (SA) \\cite{Xu2015}. Despite their wide success, SA mechanisms also suffer from inherent weaknesses, which are actively scrutinized by the research community. One major weakness is that SA essentially implements a soft-selection mechanism over the source sequence encodings. This procedure relies on the assumption of full independence between the selections at each decoding step, and over all the available encodings. Unfortunately, under this simplistic selection rationale, the model is discouraged from exploring the dynamics in the memory space to their full potential.\n\nStructural attention models have been recently proposed as a means of overcoming these issues \\cite{wang2017translating,logeswaran2016sentence}. They consist in establishing an SA mechanism which infers a joint attention distribution over consecutive decoding steps. This gives rise to a structural dependencies prior, meant to allow for the model to attend to partial segmentations e.g., pertaining to phrases or other coherent syntactic structures in natural language. However, while this approach may be helpful, for instance, in reducing grammatical and syntax errors in translation outputs, it requires relying on dynamic programming to infer the attention distributions. Unfortunately, such algorithms generally impose exponential-scale complexity. On the other hand, adoption of simplistic relaxations for the sake of computational tractability, e.g. of first-order Markovian dynamics assumptions, or of a maximum length $L$ for each attended segment \\cite{wang2017translating,logeswaran2016sentence}, is clearly limiting.\n\nThis lack of the existing NA paradigm is the driving force of our work. We posit that enabling the attention mechanism to capture structural ambiguity due to higher-order dynamics in the source element selection process will allow for improved inferential capabilities. To effect this goal, for the first time in the literature, we draw inspiration from quantum statistical concepts.\nQuantum statistical systems replace the concept of Categorical distributions with the concept of \\emph{Density matrices} \\cite{sato2009quantum,warmuth2010bayesian,sordoni2013modeling,chatzis2012quantum}. A density matrix is a square matrix, the order of which corresponds to the number of available system selections; these are referred to as \"pure states\" in the literature of quantum statistics. Its diagonal elements represent the probabilities of the underlying pure states (which sum to one). In NA terms, this corresponds to a traditional soft selection over the source sequence elements. On the other hand, its non-diagonal elements correspond to the possibility that the system becomes inherently incapable of discerning between individual (pure) states. \nThis quantum-statistical modeling rationale does not reinstate the induced higher-order dependencies modeling as introduction of additional latent variables, nor stochastically as joint probabilities \\cite{sordoni2013modeling}, similar to related works on robotics \\cite{chatzis2012quantum} and information retrieval tasks \\cite{sordoni2013modeling}.\n\nTo exhibit the efficacy of our novel approach, we implement it into the context of encoder-decoder models with attention. We dub our approach \\emph{attention density matrix (ADM) seq2seq networks}. We perform an extensive experimental evaluation of our approach using several benchmark MT datasets. As we show, our method compares favorably to alternative NA mechanisms. \n\nThe remainder of this paper is organized as follows. Section \\ref{methodological_background} covers the methodological background. We introduce our novel quantum statistics-inspired regard towards NA in Section \\ref{proposed_approach}. In Section \\ref{experimental_eval}, we perform the experimental evaluation of our approach. We further discuss and conclude this work in Section \\ref{discuss}.\n\n\\section{SA for Encoder-Decoder Networks} \\label{methodological_background}\n\nEncoder-decoder networks with attention \\cite{Bahdanau2014,Luong2015} are formulated as follows: A sequence of source tokens $\\mathbf{x} = {x}_{i},\\dots,{x}_{T_x}$ are fed to an encoder function ${f}_{enc}$ that generates a sequence of corresponding representations (encodings) $\\mathbf{h} = {\\boldsymbol{h}}_{1},\\dots,{\\boldsymbol{h}}_{T_x}$. Typical encoder-decoder models adopt a bidirectional RNN encoder. This yields encodings obtained as the concatenation of the states of the component backward and forward RNNs; that is $\\boldsymbol{h}_{i} = [\\overrightarrow{\\boldsymbol{h}}_{i};\\overleftarrow{\\boldsymbol{h}}_{i}] \\in \\mathbb{R}^{D_h}$. On the other hand, the decoder receives at each step, $i$, the current state of the decoder RNN, ${\\boldsymbol{s}}_{i}$, and a context vector, ${\\boldsymbol{c}}_{i}$, to obtain a distribution over the next element in the target sequence, $ \\mathbf{y} = {y}_{1}, \\dots, {y}_{{T}_{y}}$.\n\n\nMore formally, the distribution of the generated (decoded) sequence yields:\n\\begin{equation}\nP(\\mathbf{y}|\\mathbf{x}) = \\prod_{i=1}^{T_y}P({y}_{i}|\\mathbf{y}_{<i}, \\mathbf{x}), \\;\\;\\; P({y}_{i}|\\mathbf{y}_{<i}, \\mathbf{x}) = \\mathrm{softmax}(\\mathrm{Dense}({\\boldsymbol{s}}_{i}, {\\boldsymbol{c}}_{i})) \\label{decoding}\n\\end{equation}\n\n\\noindent In these expressions, ${y}_{i}$ is the target sequence element generated by the decoder at step $i$, and $\\boldsymbol{s}_i$ is the state of the decoder RNN, $\\mathrm{f_{dec}}$, at step $i$:\n\\begin{equation}\n\\boldsymbol{s}_i=\\mathrm{f_{dec}}(\\boldsymbol{s}_{i-1}, y_{i-1}) \\in \\mathbb{R}^{D_s},\n\\end{equation}\nTypically, the used RNN is a Long Short-Term Memory (LSTM) \\cite{Hochreiter1997} or a Gated Recurrent Unit (GRU) network \\cite{Cho2014}. On the other hand, the \\emph{context vector}, $\\boldsymbol{c}_i$, is calculated as a weighted average of the encoder outputs pertaining to the source sequence, $\\boldsymbol{h}$ (encodings):\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\sum_{j=1}^{T_{x}}a_{ij}\\boldsymbol{h}_{j} \\; \\; \\; \\mathrm{s.t.:} \\, \\boldsymbol{h}_{j} = \\mathrm{f_{enc}}(\\boldsymbol{h}_{j-1}, \\boldsymbol{x}_{j}) \\label{base_context_vector}\n\\end{equation}\n\nIn computing the context vector, $\\boldsymbol{c}_{i}$, the used weights $a_{ij}$ assigned to each encoding, $\\boldsymbol{h}_{j}$, define a Categorical distribution over the source sequence elements. Essentially, these weights imply establishment of a soft-selection mechanism; the probability of each source element being selected (attended to) is considered analogous to the affinity of the corresponding encoding with the current state of the decoder.\nSpecifically, we have \n\\begin{equation} \\label{pure_states_computation}\na_{ij} = \\mathrm{softmax}(\\alpha(\\boldsymbol{s}_{i}, \\boldsymbol{h}_{j})), \\; \\; \\mathrm{s.t.:} \\sum_{j}{a_{ij}}=1\n\\end{equation} \nHere, ${\\alpha}(\\cdot)$ is an \\emph{alignment network} \\cite{Bahdanau2014}. In the most usual case, the alignment network takes the form of an additive variant that reads \n\\begin{equation} \\label{soft-attention-scores}\n\\alpha(\\boldsymbol{s}_{i}, \\boldsymbol{h}_{j}) = \\boldsymbol{u}^{\\top}_{\\alpha}\\mathrm{tanh}(\\boldsymbol{W}_{\\alpha}\\boldsymbol{s}_{i} + \\boldsymbol{V}_{\\alpha}\\boldsymbol{h}_j + \\boldsymbol{b}_{\\alpha}),\n\\end{equation}\nwhere $\\boldsymbol{W}_{\\alpha}$ and $\\boldsymbol{V}_{\\alpha}$ are trainable weight matrices, while $\\boldsymbol{u}_{\\alpha}$ and $\\boldsymbol{b}_{\\alpha}$ are trainable parameter vectors of the same size. \n\n\\section{Proposed Approach} \\label{proposed_approach}\n\nThe formulation of our approach draws inspiration from the concept of the \\emph{density matrix}. A density matrix $\\boldsymbol{\\Phi}$ is symmetric, $\\boldsymbol{\\Phi} = \\boldsymbol{\\Phi}^{\\top}$, positive-semidefinite, $\\boldsymbol{\\Phi} \\succeq 0$, and of trace one. It constitutes a natural extension of Categorical probability distributions in the context of quantum statistics, whereby a modeled system may become inherently incapable of assigning selection probabilities to individual (\"pure\") states; assumingly due to underlying higher-order dynamics. \n\nIndeed, a Categorical distribution can be represented via a density matrix of diagonal form, referring to Gleason's theorem \\cite{gleason1957measures}. On the other hand, the non-diagonal elements of a density matrix correspond to the probabilities of the so-called \"mixed states\". These essentially reflect a situation where the quantum-statistical system \"lives\" in a mixture of states, as opposed to selecting to live in a specific \"pure\" state.\n\nDrawing from this inspiration, in this work we suggest that we infer attention dynamics in the form of a \\emph{pseudo}-density matrix, $\\boldsymbol{\\Psi}$. Under our proposed approach, the diagonal elements of $\\boldsymbol{\\Psi}$ constitute \\textit{attention scores} that correspond to the system selecting to attend to one of  \nthe available source sequence elements, in a fashion similar to the selection principles of SA. On the other hand, the non-diagonal elements of $\\boldsymbol{\\Psi}$ constitute scores tied to the possibility that selection may not be feasible for the model at the level of individual source sequence elements. On the contrary, the model may only become capable of performing selection at the level of pairs of elements, while lacking the capacity to discern which specifically of the pair elements it must attend to. \n\nLet us consider an NA mechanism performing attention over $N$ source sequence elements. We succinctly represent the above-prescribed concepts and assumptions by introducing the notion of the Attention Density Matrix (ADM). Considering the decoding step $i$, we define the pseudo-density matrix:\n\\begin{equation}\n\\boldsymbol{\\Psi}_i\\triangleq\\left[\\begin{array}{cccc}\n\\alpha_{i\\,1} & \\sigma_{i\\,(1,2)} & \\dots & \\sigma_{i\\,(1,N)}\\\\\n\\sigma_{{i\\,(2,1)}} & \\alpha_{i2} & \\dots & \\sigma_{{i\\,(2,N)}}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\sigma_{{i\\,(N, 1)}} & \\sigma_{i\\,({N, 2)}} & \\dots & \\alpha_{iN}\n\\end{array}\\right] \\label{attention_density_matrix}\n\\end{equation}\nWe dub this matrix the ADM of the introduced NA model at time $i$. Its diagonal elements $\\alpha_{ij}$ correspond to the attention scores of attending to source element $j$ at decoding step $i$, defined in the classical sense. On the other hand, the non-diagonal elements, $\\sigma_{i\\,(j,k)} \\in \\mathbb{R}$, correspond to the score pertaining to the system's inability to discern between source elements $j$ and $k$ at decoding step $i$; instead, it selects to attend to their pair, while being oblivious as to which specific element of the pair is the most appropriate to attend to. In this way, we seek to elevate the system's \\emph{uncertainty} as an additional model feature. Apparently, since the ADM $\\boldsymbol{\\Psi}_i$ is symmetric, it holds $\\sigma_{i\\,(j,k)}=\\sigma_{i\\,(k,j)},\\;\\forall\\,j\\neq k$. \n\nNote that the introduced ADM may reduce to the Categorical attention distribution employed in conventional SA, by setting its non-diagonal elements equal to zero. In addition, we emphasize that we elect not to enforce a positive-semidefiniteness constraint on the ADMs, $\\boldsymbol{\\Psi}_i$, nor do we express their elements as probabilities. This is reasonable, since our approach does not model an actual quantum-statistical system; we draw inspiration from it, in an effort to effect computationally efficient and elegant modeling of attention dynamics.\n\nLet us now proceed to derivation of the context vector representations under our ADM-based NA mechanism. To this end, we begin from the classical definition of context vectors adopted under the SA scheme. From definition (\\ref{base_context_vector}), it follows that the expression of SA context vectors can be rewritten in the following form:\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\sum_{j=1}^{N}a_{ij}\\boldsymbol{h}_{j} = \\boldsymbol{H} \\boldsymbol{\\alpha}_{i}\\\\\n\\end{equation} \\label{base_context_vector_rewritten}\nwhere $\\boldsymbol{H}=[\\boldsymbol{h}_{j}]_{j=1}^{N}$ is the columnwise concatenation of the vectors set $\\{\\boldsymbol{h}_{j}\\}_{j=1}^{N}$. \n\nOn this basis, we seek to devise an effective means of generalizing the expression (7) in the context of our quantum-statistical formulation. To this end, we perform row-wise averaging of the ADM-induced, state-related quantities. This is an efficient formulation; we defer a supporting discussion of this aspect in Section \\ref{discuss}. \n\nSpecifically, our quantum statistics-inspired view towards NA materializes by adopting the same functional form for the derived context vectors, as in (7). This brings to the fore a natural way to generalize attention weight formulation under our approach, with the same output dimensionality. We postulate\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\boldsymbol{H}\\boldsymbol{\\omega}_{i},\n\\; \\; \\; \\mathrm{s.t.:} \\, \\boldsymbol{\\omega}_{i}=\\mathrm{softmax}(\\;\\mean\\limits_{1\\leq j\\leq N}[\\boldsymbol{\\Psi}_{i}]_{jk}\\;)\n\\label{quantum_context_vector}\n\\end{equation}\nwhere $[\\boldsymbol{\\Psi}_{i}]_{jk}$ is the $(j,k)$th element of matrix $\\boldsymbol{\\Psi}_{i}$.\n\n\n\nThe key difference from (7) is that we replace the attention weights of conventional SA with the probabilities of the row-wise mean score of state mixtures, defined in (\\ref{attention_density_matrix}). The rationale of this statistical function is to roughly describe mean \\emph{temporal variation} during source element selection. In essence, we establish a mean-pooling mechanism, which amortizes the variance among scores pertaining to each source sequence element. This is in line with recent work related to enhancing sentence embeddings by leveraging mean-pooling operations across multiple NLP tasks, e.g. \\cite{chen2018enhancing}.\n\n\n\n\n\n\n\n\nHaving provided the definition of the introduced ADM concept, as well as the corresponding derivations pertaining to context vector inference under our approach, we now elaborate on how we can compute the ADM elements. We begin with the diagonal elements, $\\alpha_{ij}$, which correspond to classical soft-selection over the source sequence elements. In this work, we obtain these quantities by simply adopting the \\emph{score} functions employed in conventional SA. Indeed, this is a natural selection, since the diagonal elements of the ADM have the same physical interpretation as the scores in conventional SA.\n\nLet us now turn to the non-diagonal elements, $\\sigma_{i\\,(j,k)}$ of the postulated ADM, $\\boldsymbol{\\Psi}_i$, at time $i$. To introduce an effective way of computing them, we draw inspiration from their physical interpretation. As we have already discussed, a non-diagonal ADM element, $\\sigma_{i\\,(j,k)}$, assigns a \nscore to the scenario that the postulated NA model cannot effectively discern between the $j$th and the $k$th source sequence encoding, $\\{\\boldsymbol{h}_j, \\boldsymbol{h}_k\\}\\; \\forall{j\\neq k}$. Therefore, to obtain these  \nscores, we need to introduce a transformation that can effectively reflect two distinct aspects: (i) the affinity between pairs of source sequence encodings, $\\{\\boldsymbol{h}_j, \\boldsymbol{h}_k\\}\\; \\forall{j\\neq k}$; and (ii) the affinity of the current decoder state, $\\boldsymbol{s}_i$, with the latent dynamics shared by the encodings $\\boldsymbol{h}_j$ and  $\\boldsymbol{h}_k$. \n\nTo effect this goal, at each decoding step, $i$, we first compute a set of $D$-dimensional score vectors between each of the available source sequence encodings.\nWe define the 3-dimensional tensor $\\boldsymbol{L}=[l_{(j,k)}]_{j,k}$, where: \n\\begin{equation} \\label{order-1-quantum}\nl_{(j,k)}=\\mathrm{tanh}(\\boldsymbol{h}_{j}+\\boldsymbol{h}_{k}),\n\\end{equation}\nThis way, we yield a high-dimensional representation of the correlations between all pairs of the available source sequence encodings.\nThis enables us to further extract rich information regarding the latent dynamics shared between all possible pairs of encodings, and in respect to the current decoder state. Specifically, the sought higher-order interaction dynamics matrix is computed through Eq. (\\ref{luong:quantum}) using the tensor $\\boldsymbol{L}$ in (\\ref{order-1-quantum}). We have\n\\begin{equation} \\label{luong:quantum}\n\\boldsymbol{M}_{i}= w_{s}\\boldsymbol{L}\\boldsymbol{s}_{i},\n\\end{equation}\nwhere $w_{s}\\in\\mathbb{R}$ is a trainable weight scalar. Because of the multiplicative style it employs, resembling the attention mechanism in \\cite{Luong2015}, we refer to it as the \\textit{Multiplicative Quantum-Theoretical (MQT)} scheme. We make this distinction as it is possible to follow a different scheme, by replacing Eqs. (\\ref{order-1-quantum} - \\ref{luong:quantum}) with:\n\n\\begin{equation} \\label{bahdanau:quantum}\n\\boldsymbol{M}_{i}= \\mathrm{tanh}(\\boldsymbol{L} + \\boldsymbol{s}_{i}) \\boldsymbol{w}_{s},\n\\end{equation}\nwhere $\\boldsymbol{w}_{s}$ is a trainable weights vector.\n\\noindent We dub this alternative formulation as the \\textit{Additive Quantum-Theoretical (AQT)} scheme, as it resembles \\cite{Bahdanau2014}.\n\nEventually, we use the non-diagonal elements of the so-derived matrix $\\boldsymbol{M}_{i}$ as the non-diagonal elements of the sought ADM, $\\boldsymbol{\\Psi}_{i}$: $\\sigma_{i(j,k)}=[\\boldsymbol{M}_i]_{(j,k)}$. \n\n\\noindent\\textbf{Relation to recent work.} As demonstrated above, our approach enables capturing higher-order interactions in the\ncomputation of context vectors. Recent efforts\ntoward generalizing neural attention, by deriving more complex attention\ndistributions, contrast our proposed approach. We refer again to \\cite{structuredAttention}, which recently\nintroduced \\emph{structured attention}, whereby the attention probabilities are considered to be\ninterdependent over consecutive time-steps; for instance, they postulate\nthe first-order Markov dynamics assumption: \n\\begin{equation}\np(\\{\\boldsymbol{z}_{t}\\}_{t=1}^{T};\\pazocal{D})=p(\\boldsymbol{z}_{1};\\pazocal{D})\\prod_{t=1}^{T-1}p(\\boldsymbol{z}_{t+1}|\\boldsymbol{z}_{t};\\pazocal{D}),\n\\label{eq:Structured}\n\\end{equation}\nwhere $\\{z_{t}^{i}\\}_{i=1}^{N}$,\n$z_{t}^{i}\\in\\{0,1\\}$, with $z_{t}^{i}=1$ being a set of binary latent variables indicating the source element the model attends to at time $t$. To perform training and inference, the computationally complex forward-backward algorithm \\cite{rabiner1989tutorial} has to be used. This also discourages the method's use in scale, contrary to our approach which imposes minimal additional computational costs. This is the case as all the extra computations of our model, described in Eqs. (10)-(12), constitute single additional forward propagation computations; this renders our model comparable to classical SA in terms of computational costs.\n\n\\noindent\\textbf{Inference Algorithm.} To perform target decoding by means of a \\emph{seq2seq} model with ADM, we resort to \\emph{Beam sampling} \\cite{russell2016artificial}. In our experiments, \\emph{Beam} \\emph{width} is set to ten across all MT tasks. \n\n\\noindent\\textbf{Training Algorithm.} To perform training of a \\emph{seq2seq} model with ADM, we rely on minimization of an appropriate loss function, $\\pazocal{L}(\\theta)$, where $\\theta$ is the set of model parameters, in a similar fashion to conventional models. For example, to address an MT task, one may use the categorical cross-entropy between the generated decoding probabilities (\\ref{decoding}) and the available groundtruth.\n\n\n\n\n\n\\section{Experimental Evaluation} \\label{experimental_eval}\n\n\\begin{table*}[t]\n\\caption{Translation results on the En$\\leftrightarrow$Vi and En$\\leftrightarrow$Ro pairs.}\n\\small\n\\centering{}%\n\\begin{tabu}{|c|c|c|c|c|[1.5pt]c|c|c|c|}\n\\hline \n\\multirow{3}{*}{Method} & \\multicolumn{8}{c|}{BLEU}\\tabularnewline\n& \\multicolumn{2}{c}{En$\\rightarrow$Vi} & \\multicolumn{2}{c}{Vi$\\rightarrow$En} & \\multicolumn{2}{c}{En$\\rightarrow$Ro} & \\multicolumn{2}{c|}{Ro$\\rightarrow$En}\n\\tabularnewline\n\\cline{2-9} \n & dev & test & dev & test & dev & test & dev & test\\tabularnewline\n\\hline\n\\multirow{1}{*}{Baseline} & \\multirow{1}{*}{22.90} & \\multirow{1}{*}{24.11} & \\multirow{1}{*}{19.14} & \\multirow{1}{*}{20.76} & \\multirow{1}{*}{13.65} & \\multirow{1}{*}{15.31} & \\multirow{1}{*}{17.15} & \\multirow{1}{*}{16.74}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{\\makecell{Structured Attention}} & \\multirow{1}{*}{16.81} & \\multirow{1}{*}{17.00} & \\multirow{1}{*}{17.19} & \\multirow{1}{*}{18.08} & \\multirow{1}{*}{7.04} & \\multirow{1}{*}{7.03} & \\multirow{1}{*}{11.02} & \\multirow{1}{*}{11.08}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{\\textbf{MQT}} & \\multirow{1}{*}{23.41} & \\multirow{1}{*}{25.34} & \\multirow{1}{*}{20.65} & \\multirow{1}{*}{22.23} & \\multirow{1}{*}{15.01} & \\multirow{1}{*}{16.95} & \\multirow{1}{*}{19.03} & \\multirow{1}{*}{19.09}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{AQT} & \\multirow{1}{*}{22.71} & \\multirow{1}{*}{24.34} & \\multirow{1}{*}{19.95} & \\multirow{1}{*}{21.33} & \\multirow{1}{*}{3.98}\n& \\multirow{1}{*}{4.56} & \\multirow{1}{*}{5.08} & \\multirow{1}{*}{5.62}\n\\tabularnewline\n\\hline\n\\end{tabu}\n\\label{translation:results}\n\\end{table*}\n\n\nOur experiments make use of publicly available corpora, namely IWSLT'15 English to and from Vietnamese (IWSLT En$\\leftrightarrow$Vi) and \nWMT'16 English to and from Romanian (WMT En$\\leftrightarrow$Ro). \nWe benchmark our models against word-based vocabularies and present our results in terms of the BLEU score \\cite{blue}. Note that, in this paper, we employ relatively small datasets to better demonstrate the applicability and generalization strength of our approach. To allow seamless handling of rare words, we utilize \\emph{byte pair encoding} (BPE) \\cite{sennrich2015neural} for En$\\leftrightarrow$Ro. \n\n\n\n\n\n\n\n\n\\subsection{Datasets}\n\n\\noindent\\textbf{IWSLT En$\\leftrightarrow$Vi.} One benchmark we consider is the IWSLT En$\\leftrightarrow$Vi machine translation task.\nThe dataset includes $\\sim$133K training sentence pairs, from translated TED talks,\nprovided by the IWSLT 2015 Evaluation Campaign \\cite{cettolo2015iwslt}. Following the same preprocessing steps in \\cite{Luong2015,raffel2017online}, we use the TED tst2012 ($\\sim$1.5K sentences) as a validation set for hyperparameter tuning and TED tst2013 ($\\sim$1.2K sentences) as a test set. The Vietnamese and English vocabulary sizes are $\\sim$7.7K and $\\sim$17.2K respectively.\n\n\n\\noindent\\textbf{WMT En$\\leftrightarrow$Ro.} A second dataset our evaluation addresses is the WMT En$\\leftrightarrow$Ro machine translation task. The dataset consists of $\\sim$400K parallel sentences. The shared vocabulary sizes (obtained from BPE with 32K operations) total $\\sim$31K words.  We use newsdev2016 as our development set, and newstest2016 as our test set (both with 1.9K sentences).\n\n\n\n\\subsection{Experimental setup} \n\n\\begin{table}[t]\n\\small\n\\caption{Vi$\\rightarrow$En, dev set - Examples (a) 281 and (b) 1086.}\n\\begin{tabularx}{\\textwidth}{|X|} \\hline\n\\textbf{Reference Translation}\\tabularnewline \\hline \n(a) And for me that means spending time thinking and talking about the poor, the disadvantaged, those who will never get to TED. \\\\\n(b) Now the transition point happened when these communities got so close that, in fact, they got together and decided to write down the whole recipe for the community together on one string of DNA.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Baseline}\\tabularnewline \\hline\n(a) For me, it means \\textcolor{orange}{to be} spending time to think about the poor, who have a situation, about who don't have a chance to \\textbf{[verb]} TED. \\\\\n(b) The time that transitions \\textbf{[verb]} when communities are closer together, in  \\textcolor{blue}{reality}, they're together, form a  \\textcolor{red}{collective tool} for the whole  \\textcolor{red}{chromosome} on a single  \\textcolor{blue}{array} of DNA.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Structured Attention}\\tabularnewline \\hline\n(a) For me, that means for time to think, \\textcolor{orange}{to think} about the poor, to talk about people who are not \\textcolor{orange}{having to} go to TED. \\\\\n(b) The time \\textcolor{red}{$<$unk$>$} occurs when the communities are close to each other, in fact\\textcolor{orange}{, in fact,} they're together to form a shared structure for the entire \\textcolor{red}{body}.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - MQT}\\tabularnewline\n\\hline \n(a) For me, it means to spend time to think, to talk about the poor people, who have a \\textcolor{blue}{difficult situation}, about who there is \\textcolor{blue}{no opportunity} to go to TED. \\\\\n(b) A time of \\textcolor{red}{communication} happens when communities are close to each other, in fact, they together form a \\textcolor{blue}{formula} that is shared for the whole \\textcolor{blue}{colony} on a single \\textcolor{blue}{array} of DNA.\n\\tabularnewline \\hline\n\\end{tabularx}\n\\label{vi-en:translations}\n\\end{table}\n\nWe perform dropout regularization of the trained models, with a dropout rate equal to 0.2. We minimize $\\pazocal{L}(\\phi)$ by employing the Adam \\cite{adam} optimizer with its default settings for En$\\leftrightarrow$Vi and simple stochastic gradient descent (SGD) for En$\\leftrightarrow$Ro\\footnote{En$\\leftrightarrow$Vi models are trained for $\\sim$12 epochs; En$\\leftrightarrow$Ro for$\\sim$12 epochs for structured attention models and $\\sim$4 for the rest.}. We preserve homogeneity throughout the trained architectures as follows.\nBoth the encoders and the decoders of all the evaluated models are presented with 256-dimensional \\emph{trainable} word embeddings. The maximum inference length is set to 50. We utilize 2-layer BiLSTM encoders, and 2-layer LSTM decoders; all comprise 256-dimensional hidden states on each layer. For the remainder of hyper-parameters, we adopt the default settings used in the code\\footnote{https:\/\/github.com\/harvardnlp\/struct-attn.} provided by the authors in \\cite{structuredAttention} for structured attention models and the code in \\cite{luong17} for the rest. Except specified otherwise, the default settings used by the latter for En$\\leftrightarrow$Vi also apply to En$\\leftrightarrow$Ro.\n\n\n\n\\subsection{Results} \n\\begin{table}[t]\n\\caption{Ro$\\rightarrow$En, dev set - Examples (a) 5 and (b) 182.}\n\\begin{tabularx}{\\textwidth} {|X|} \\hline \n\\textbf{Reference Translation}\\tabularnewline \\hline \n(a) Dirceu is the most senior member of the ruling Workers' Party to be taken into custody in connection with the scheme.\n\\\\\n(b) With one voice the lobbyists talked about a hoped-for ability in Turnbull to make the public argument, to cut the political deal and get tough things done.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Baseline}\\tabularnewline \\hline\n(a) He is the oldest member of the \\textcolor{orange}{Dutch People's Party} on \\textcolor{orange}{Human Rights} in custody for \\textcolor{orange}{the} \\textcolor{blue}{links} with this scheme.\n\\\\\n(b) The representatives of \\textcolor{blue}{lobbyists} have spoken about their hope in the ability of \\textcolor{orange}{Turngl} to \\textcolor{blue}{satisfy} the public interest, to reach a political agreement and to do things well.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Structured Attention}\\tabularnewline \\hline\n(a) \\textcolor{orange}{It} is the oldest member of the \\textcolor{orange}{Mandi} \\textcolor{orange}{of the Massi} in \\textcolor{orange}{the} government \\textcolor{red}{in the government}.\n\\\\\n(b) The representatives of the \\textcolor{blue}{interest groups} have spoken \\textcolor{red}{in mind} about their hope to \\textcolor{blue}{meet} the public interest, to achieve a political and good thing.\n\\tabularnewline \\hline\n\\textbf{Generated Translation - MQT}\\tabularnewline \\hline \n(a) \\textcolor{orange}{Dirre} is the oldest member of the \\textcolor{orange}{People's Party} in government \\textcolor{blue}{held} in custody for \\textcolor{blue}{ties} with this scheme.\n\\\\\n(b) The representatives of \\textcolor{blue}{interest groups} have spoken \\textcolor{orange}{to} \\textcolor{blue}{unison} about their hope in \\textcolor{orange}{Turkey's} ability to \\textcolor{blue}{meet} the public interest, to reach a political agreement and to do things well.\n\\tabularnewline \\hline\n\\end{tabularx}\n\\label{ro-en:translations}\n\\end{table}\n\nTable \\ref{translation:results} shows superior performance for our \\emph{multiplicative} approach. In addition, note that despite their extended training requirements, structured attention models demonstrate an inability to properly capture long-temporal information, both score and output-wise, as presented in Tables \\ref{vi-en:translations} and \\ref{ro-en:translations}. These showcase some characteristic examples of generated translations for a hands-on inspection of model outputs. We annotate deviations from the reference translation with orange and red, for minor\nand major deviations\nrespectively. Synonyms are highlighted with blue. We also indicate missing tokens, such as verbs, articles and adjectives, by adding the \\textbf{[$<$token$>$]} identifier.\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{discuss}\n\nIn this section, we want to further explore how and why our proposed approach can enable\nbetter utilization of infrequent words through second-order interactions. Furthermore, we outline technical augmentations that we consider as future research directives.\n\n\\subsection{Model uncertainty} \n\n\\begin{table*}[t]\n\\caption{Rare word mean reference frequency deviation}\n\\small\n\\centering{}%\n\\begin{tabu}{|c|c|c|c|c|}\n\\hline \n\\multirow{3}{*}{Language Pair} &  & \\multicolumn{3}{c|}{Deviation (\\%)}\\tabularnewline\n\\cline{3-5} \n & \\makecell{Mean reference \\\\ frequency (\\%)} & Baseline & \\makecell{Structured \\\\ Attention} & MQT\\tabularnewline\n\\hline\n\\multirow{1}{*}{En$\\rightarrow$Vi} & \\multirow{1}{*}{3.59} & \\multirow{1}{*}{-4.82} & \\multirow{1}{*}{11.57} & \\multirow{1}{*}{\\textit{\\textbf{0.28}}}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{\\makecell{Vi$\\rightarrow$En}} & \\multirow{1}{*}{8.91} & \\multirow{1}{*}{-10.77} & \\multirow{1}{*}{-20.85} & \\multirow{1}{*}{\\textit{\\textbf{-8.68}}}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{En$\\rightarrow$Ro} & \\multirow{1}{*}{7.00} & \\multirow{1}{*}{\\textbf{4.59}} & \\multirow{1}{*}{\\textit{14.61}} & \\multirow{1}{*}{-7.87}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{Ro$\\rightarrow$En} & \\multirow{1}{*}{6.75} & \\multirow{1}{*}{34.23} & \\multirow{1}{*}{37.47} & \\multirow{1}{*}{\\textbf{\\textit{23.17}}}\n\\tabularnewline\n\\hline\n\\end{tabu}\n\\label{rare-words-table}\n\\end{table*}\n\n\nA first case study is inspired from the interesting work of \\cite{ott2018analyzing}. Therein, the major claim is that if model and data distributions match, then samples drawn from both should also match. As a broader extension to our evaluation, we present deviation from unigram word distributions (across 1 - 30\\% frequency groups). Table \\ref{rare-words-table} reveals how well our evaluated models estimate said frequencies. Words are split into groups based on their appearance in their respective training datasets. We present this against development set frequencies. Note that evaluation criteria include not only \\textit{deviation} but also \\textit{least over-representation} (favored to any magnitude of under-representation); these are presented in bold and italics, respectively. This is because swapping a frequent for a rare word would not be as harmful as the reverse, in terms of translation quality\nIn 3 out of 4 cases, our approach achieves close representation of target frequencies.\n\n\n\n\\subsection{High rank matrix approximation}\nThe capacity of most natural language models is crippled by their inability to cope in highly context-dependent settings \\cite{yang2017breaking}. Admittedly, this shortcoming is due to their limited capacity to capture complex hierarchical dependencies. To address this issue, we need to devise computationally efficient ways of capturing higher-order dynamics. A first step towards this goal is offered by our approach. However, our approach is limited to second-order interactions; in addition, to allow for computational efficiency, we have resorted to a mean-pooling solution in the computation of the final context vector.\nA more general solution would be to resort to spectral decomposition, which would require performing eigen\/tensor decomposition.\nHowever, differentiating this operation during back-propagation may lead to numerical instability issues \\cite{dang2018eigendecomposition}, rendering it non-differentiable. \nWe aim to examine solutions to these issues in our future work.\n\\section{Conclusions}\n\nIn this work, we introduced a novel regard towards formulating attention layers in \\emph{seq2seq}-type models. Our work was inspired from the quest of a more expressive way of computing dependencies between input and output sequences. Specifically, our aim was to enable capturing of second-order dependencies between the source sequence encodings and the generated output sequences. \n\nTo effect this goal, for the first time in the literature, we leveraged concepts from the field of quantum statistics. We cast the operation of the attention layer into the computation of the \\emph{Attention Density Matrix}, which expresses how pairs of source sequence elements correlated with each other, and jointly with the generated output sequence. Our formulation of the ADM was based on \\emph{Density Matrix} theory; it is an attempt to encapsulate the core concepts of the Density Matrix in the context of attention networks, without adhering though to the exact definition and properties of density matrices.\n\nWe exhibited the merits of our approach on \\emph{seq2seq} architectures addressing competitive MT tasks.\nWe have showed that the unique modeling capacity of our approach translates into better handling of (rare) words in the model outputs. Hence, this finding offers a quite plausible explanation of the obtained improvement in the achieved BLEU scores.\nFinally, we emphasize inference using our method entails minor computational overhead compared to conventional SA, with only a single extra forward-propagation computation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, entanglement has been regarded as a quantum\nresource for many novel tasks such as quantum computation, quantum\ncryptography, quantum teleportation and so on \\cite{Nielsen}.\nThese quantum-information tasks cannot be carried out by classical\nresources and they rely on the entangled states. Although the\nmixed entangled states are directly used in some\nquantum-information tasks \\cite{Murao}, most of them require the\npure entangled states of bipartite or multipartite system to be\nthe crucial elements. However in a lab, it turned out that the\npure entangled states always become mixed by the decoherence due\nto the coupling with the environment. A central topic in quantum\ninformation theory is thus how to extract pure entangled states\nfrom mixed states \\cite{Horodecki1}.\n\n\nAn entangled state $\\rho$ is distillable if one can asymptotically\nor explicitly extract some pure entangled state from infinitely\nmany copies of $\\rho$ by using only local operations and classical\ncommunication (LOCC). It has been proved that the entangled\n2-qubit states are always distillable\n\\cite{Bennett1,Bennett2,Horodecki2}. Nevertheless there exist\nbound entangled (BE) states which are not distillable under LOCC\n\\cite{Horodecki4}. Concretely, a bipartite entangled state $\\rho$\nin the Hilbert space $H_A\\otimes H_B$ is BE if it has positive\npartial transpose (PPT) with respect to system $A$ (or $B$),\nnamely $\\rho^{T_A}(\\mbox{or}\\ \\rho^{T_B})\\geq0$. Such states are\ncalled PPT BE states and usually it cannot be used for\nquantum-information tasks under LOCC \\cite{Murao,Eggeling}.\n\n\nA more formidable challenge is that whether a bipartite state\n$\\rho_{AB}$ having non-positive partial transpose (NPT) with\nrespect to system $A$ (or $B$) is always distillable. This class\nof states are always entangled due to the celebrated\nPeres-Horodecki criterion \\cite{Peres}. It was pointed out by\n\\cite{Horodecki5} that any NPT state can be converted into some\nNPT Werner state under LOCC. Much efforts have been devoted to\ndistilling this kind of states and there has been a common belief\nthat NPT BE Werner states indeed exists\n\\cite{DiVincenzo,Vianna,Pankowski,Hiroshima,Bandyopadhyay,Kraus,Clarisse}.\nIn addition, it has been proved that the NPT states in $2\\times N$\nspace are distillable \\cite{Horodecki2,Kraus2} and the rank two\nNPT states of bipartite systems are also distillable\n\\cite{Horodecki6}. However, the situation becomes more complex\nwhen we distill the entangled state whose subsystems have higher\ndimensions or that has a higher rank.\n\nIn this paper we show that the rank three bipartite entangled\nstates are distillable under LOCC. We give the concrete method of\ndistilling this class of states. It helps infer the analytical\ncalculation of distillable entanglement \\cite{Rains,Devetak}. A\nrank three state is entangled if and only if (iff) it is NPT,\nnamely there is no PPT BE state of rank three \\cite{Lewenstein}.\nSo we also obtain that there are no rank-three NPT BE states and\nall of them can be used for quantum-information tasks. It is\nsimilar to the case of rank two states and we conclude: a rank two\nor three state is distillable iff it is entangled. This conclusion\ndoes not hold for the bipartite entangled states with higher\nranks, e.g., there have been the rank four PPT BE states\nconstructed by the unextendible product bases (UPB) \\cite{Mor}.\n\nMoreover, we will investigate the NPT states of rank four and find\nout some families of states that are distillable. This helps\ndistill the NPT states which have more complex structure. In\naddition, we will show that locally converting the Werner state\ninto the rank three entangled state is difficult, so our result is\nindependent of the expectant fact that there exists NPT BE Werner\nstate.\n\nThe rest of this paper is organized as follows. In Sec. II we\nprove our main result on rank three states and then we use it to\ndistill the rank four NPT states. We also discuss the relationship\nbetween the result in this paper and the Werner state. We conclude\nin Sec. III.\n\n\n\\section{distillation of rank three and four bipartite states}\n\nThroughout this paper we will use the following notations. The\nrank of a bipartite state $\\rho_{AB}$ is referred to as\n$r(\\rho_{AB})$, and the reduced density operator of it as\n$\\rho_A=\\mbox{Tr}_B\\rho_{AB},\\rho_B=\\mbox{Tr}_A\\rho_{AB}$. The\nrange of the density operator $\\rho_{AB}$ is referred to as\n$R(\\rho_{AB})$. Another useful tool is the so-called invertible\nlocal operator (ILO) (or the local filter) \\cite{Dur}, namely the\nnonsingular matrix. Physically, it can be probabilistically\nrealized through the positive operator valued measure (POVM)\n\\cite{Nielsen}, so we can use it when distilling the NPT states.\n\n\nWe first consider the NPT states of rank three. Before proving our\nmain theorem, we recall a useful lemma that was proved in\n\\cite{Horodecki6}.\n\n\\textit{Lemma 1}. If\n$r(\\rho_{AB})<\\mbox{max}[r(\\rho_{A}),r(\\rho_{B})]$, then the\nbipartite state $\\rho_{AB}$ is distillable.\n\\hspace*{\\fill}$\\blacksquare$\n\nThe lemma has been used to show that there is no rank two BE state\n\\cite{Horodecki6}. It was proven by using the reduction criterion\n\\cite{Horodecki5}, i.e., a state is distillable when the reduction\ncriterion is violated (See Eq. (6) in \\cite{Horodecki6}). It\nfollows from lemma 1 that any rank three state in $M\\times N$\nspace with $\\mbox{max}[M,N]>3$ is distillable. Since an NPT state\nin $2\\times2$ or $2\\times3$ space is also distillable\n\\cite{Bennett1,Bennett2,Horodecki2}, it suffices to consider the\nrank three NPT states $\\rho_{AB}$ in $3\\times3$ space. Moreover,\nwe can perform some ILO on the subsystem $B$ such that\n$\\rho_B=\\frac13I$. Then only the state having the following form\ndoes not violate the reduction criterion ( up to local unitary\ntransformations )\n\\begin{equation}\n\\sigma_{AB}\\equiv\\frac13|\\psi_0\\rangle\\langle\\psi_0|+\\frac13|\\psi_1\\rangle\\langle\\psi_1|\n+\\frac13|\\psi_2\\rangle\\langle\\psi_2|,\\sigma_B=\\frac13I\n\\end{equation}\nwhere the three eigenvectors satisfy\n$\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij}$ and\n\\begin{eqnarray}\n|\\psi_0\\rangle&=&\\cos\\theta|00\\rangle+\\sin\\theta|11\\rangle,\\\\\n|\\psi_1\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}b_{ij}|ij\\rangle,\\\\\n|\\psi_2\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}c_{ij}|ij\\rangle.\n\\end{eqnarray}\nNotice that there is always at least a Schmidt rank two state by\nlinear combination of the eigenvectors. In addition, any spectral\ndecomposition of the state $\\sigma_{AB}$ have the form in Eq. (1)\n(in which the state $|\\psi_0\\rangle$ has a more general form,\ne.g., $|\\psi_0\\rangle=\\sum\\nolimits^{2}_{i,j=0}a_{ij}|ij\\rangle$).\n\nIn what follows we will concentrate on the NPT state $\\sigma_{AB}$\nin Eq. (1) because any rank three NPT state in $3\\times3$ space\ncan be locally converted into $\\sigma_{AB}$, otherwise it is\ndistillable in terms of the reduction criterion. There is a simple\nsituation we can treat easily as follows.\n\n\\textit{Lemma 2}. The state $\\sigma_{AB}$ is distillable when\nthere is a product state in its range.\n\n\\textit{Proof.} Without loss of generality, we consider the state\n$\\sigma_{AB}$ with $\\theta=0$. Then its coefficients\n$b_{i0},c_{i0},i=0,1,2$ equal zero because of the condition\n$\\sigma_B=\\frac13I$. We project the state $\\sigma_{AB}$ by using\nthe local projector\n$I_A\\otimes(|1\\rangle\\langle1|+|2\\rangle\\langle2|)_B$ and obtain\nthe resulting state\n$\\frac12|\\psi_1\\rangle\\langle\\psi_1|+\\frac12|\\psi_2\\rangle\\langle\\psi_2|$.\nIt's a rank two NPT state and hence distillable. It implies the\nstate $\\sigma_{AB}$ is also distillable.\n\\hspace*{\\fill}$\\blacksquare$\n\nLemma 2 has given a criterion that tells whether a rank three NPT\nstate is distillable. We will generalize it to the case of rank\nfour states later. It is also useful for the distillation of\ngeneral rank three NPT state as shown below. Let us consider the\nstate $\\sigma_{AB}$ whose range has no product state. We take the\nprojector $P_{AB}$ onto the $2\\times3$ subspace spanned by\n$\\{|00\\rangle,|01\\rangle,|02\\rangle,|10\\rangle,|11\\rangle,|12\\rangle\\}$\nand obtain the state\n\\begin{equation}\n\\sigma^1_{AB}=|\\psi^1_0\\rangle\\langle\\psi^1_0|+|\\psi^1_1\\rangle\\langle\\psi^1_1|\n+|\\psi^1_2\\rangle\\langle\\psi^1_2|,\n\\end{equation}\nwhich is not normalized for convenience. The resulting states\n$|\\psi^1_i\\rangle$ equal $P_{AB}|\\psi_i\\rangle$, respectively. We\nwill follow this notation below, e.g.,\n$|\\psi^2_i\\rangle=V_A\\otimes V_B|\\psi^1_i\\rangle$, etc.\n\nThe state $\\sigma^1_{AB}$ is distillable if it is entangled since\nit is in $2\\times2$ or $2\\times3$ space. Let us consider the case\nin which $\\sigma^1_{AB}$ is separable. First, the state\n$\\sigma^1_{AB}$ is in $2\\times2$ space iff\n$b_{i2}=c_{i2}=0,i=0,1$. In this case, the condition\n$\\sigma_B=\\frac13I$ leads to\n$b_{2i}b^*_{22}+c_{2i}c^*_{22}=0,i=0,1$ and\n$|b_{22}|^2+|c_{22}|^2=1$. When $b_{22}c_{22}=0$, either the state\n$|\\psi_1\\rangle$ or $|\\psi_2\\rangle$ becomes a product state and\nhence $\\sigma_{AB}$ is distillable in terms of lemma 2; When\n$b_{22}c_{22}\\neq0$, we can remove the coefficients\n$b_{2i},c_{2i},i=0,1$ by using linear combination of the\neigenvectors $|\\psi_i\\rangle,i=0,1,2$. It is then easy to see that\n$R(\\sigma_{AB})$ contains a product state and thus $\\sigma_{AB}$\nis distillable.\n\n\nSecond, we investigate the state $\\sigma^1_{AB}$ in $2\\times3$\nspace. Notice the rank of $\\sigma^1_{AB}$ remains three, otherwise\nthere will be a product state in $R(\\sigma_{AB})$ and it is\ndistillable. We can always write a rank three separable state\n$\\rho$ in $2\\times3$ space as the sum of three product states\n\\cite{Wootters,Werner}. To prove it, suppose the state has the\nform\n\\begin{equation}\n\\rho=\\sum\\nolimits^{d-1}_{i=0}|\\phi_i\\rangle|\\omega_i\\rangle\\langle\\phi_i|\\langle\\omega_i|,d>3.\n\\end{equation}\nWithout loss of generality we choose the first three product\nstates as a set of linearly independent vectors, so any other\nproduct state can be written as\n$|\\phi_j\\rangle|\\omega_j\\rangle=\\sum\\nolimits^2_{i=0}k_{ij}|\\phi_i\\rangle|\\omega_i\\rangle,j=3,...$\nNotice the vectors $|\\omega_i\\rangle,i=0,1,2$, and two vectors in\n$|\\phi_i\\rangle,i=0,1,2$ are linearly independent, respectively.\nSo the product state $|\\phi_j\\rangle|\\omega_j\\rangle,j>3$ equals\neither one of the first three product states, or\n$|\\phi_j\\rangle|\\omega_j\\rangle=\\sum\\nolimits^1_{i=0}k_{ij}|\\phi_i\\rangle|\\omega_i\\rangle$\nin which $|\\phi_0\\rangle$ is proportional to $|\\phi_1\\rangle$. In\nthis case it is easy to write the state $\\rho$ as the sum of three\nproduct states.\n\nUsing the above conclusion, we can express the state $\\sigma_{AB}$\nby means of eigenvectors\n$|\\psi_i\\rangle=(a_{i0}|0\\rangle+a_{i1}|1\\rangle)|\\phi_{i1}\\rangle+|2\\rangle|\\phi_{i2}\\rangle,i=0,1,2.$\nMoreover, the vectors $|\\phi_{i1}\\rangle$'s are linearly\nindependent, while $|\\phi_{i2}\\rangle$'s linearly dependent. We\nperform some ILO's on the state $\\sigma_{AB}$ and remove two\ncoefficients $a_{00}$ and $a_{11}$. The resulting state\n$\\sigma^2_{AB}$ still has the form in Eq. (1), otherwise it is\ndistillable.\n\nFor the state $\\sigma^2_{AB}$ when the condition $a_{20}a_{21}=0$\nis satisfied, we find that $R(\\sigma^2_{AB})$ contains a product\nstate because of the orthogonal conditions\n$\\langle\\psi^2_i|\\psi^2_j\\rangle=\\delta_{ij}$. So the state\n$\\sigma_{AB}$ is distillable. Let us move to investigate the state\n$\\sigma^2_{AB}$ satisfying the condition $a_{20}a_{21}\\neq0$. By\nperforming ILO's on $\\sigma^2_{AB}$ we greatly simplify its form\nsuch that\n\\begin{equation}\n\\sigma^3_{AB}=|\\psi^3_0\\rangle\\langle\\psi^3_0|+|\\psi^3_1\\rangle\\langle\\psi^3_1|\n+|\\psi^3_2\\rangle\\langle\\psi^3_2|,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n|\\psi^3_0\\rangle&=&|00\\rangle+|2\\rangle|\\psi\\rangle,\\\\\n|\\psi^3_1\\rangle&=&|11\\rangle+|2\\rangle|\\phi\\rangle,\\\\\n|\\psi^3_2\\rangle&=&(|0\\rangle+|1\\rangle)|2\\rangle+\n|2\\rangle(\\alpha|\\psi\\rangle+\\beta|\\phi\\rangle),\\\\\n|\\psi\\rangle&=&x_0|0\\rangle+x_1|1\\rangle+x_2|2\\rangle,\\\\\n|\\phi\\rangle&=&y_0|0\\rangle+y_1|1\\rangle+y_2|2\\rangle.\n\\end{eqnarray}\nNotice the state is not normalized and the condition\n$\\sigma^3_B=\\frac13I$ is also not required. We project\n$\\sigma^3_{AB}$ by the projector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B,a\\in R$ and obtain the state $\\sigma^4_{AB}$ in $2\\times3$\nspace. It is entangled and thus distillable when its partial\ntranspose is not positive \\cite{Peres}. Nevertheless, there may be\nsome cases in which the coefficients $x_i,y_i,i=0,1,2$ make that\n$(\\sigma^4_{AB})^{T_A}\\geq0$. We are going to find out such\ncoefficients by calculating several average values\n$\\mbox{Tr}[(\\sigma^4_{AB})^{T_A}|\\omega_i\\rangle\\langle\\omega_i|]$,\nwhere\n$|\\omega_0\\rangle=|00\\rangle+b|22\\rangle,|\\omega_1\\rangle=|00\\rangle+b|21\\rangle\n,|\\omega_2\\rangle=|01\\rangle+b|20\\rangle,|\\omega_3\\rangle=|02\\rangle+b|20\\rangle,b\\in\nC.$ To keep the average value always positive, we find that it is\nnecessary that $x_1=x_2=0$. However, this means the state\n$|\\psi^3_0\\rangle$ is of product form and hence the state\n$\\sigma^3_{AB}$ is distillable. As it can be converted into the\nstate $\\sigma_{AB}$ by ILO's, the latter is also distillable. Now\nwe reach our main theorem in this paper.\n\n\\textit{Theorem.} The rank three NPT states are distillable under\nLOCC. \\hspace*{\\fill}$\\blacksquare$\n\nSo the rank three entangled states can be used for\nquantum-information tasks. In fact, we have proposed the method of\ndistilling $\\sigma_{AB}$ in the proof of the theorem. First, when\nthe given state contains a product state in its range, it can be\nprojected onto a rank two entangled state. According to the\nreduction criterion, we can distill it by the procedure similar to\nthe famous BBPSSW protocol \\cite{Bennett2,Horodecki5}. It is also\nthe method of distilling the rank three entangled states that\ncannot be converted into $\\sigma_{AB}$. Second, when the given\nstate $\\rho$ contains no product state in $R(\\rho)$, we project it\nby the projector $(|0\\rangle\\langle0|+|1\\rangle\\langle1|)_A\\otimes\nI_B$. The resulting state is entangled and thus distillable;\notherwise, we should project the initial state $\\rho$ by the\nprojector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B$ after performing some ILOs on $\\rho$. There will be a\nsuitable parameter $a$ making the resulting state entangled and\nthus distillable.\n\nThe rank three entangled states are a quite special class of\nstates. As there have been PPT BE states of any higher rank (e.g.,\nrank four PPT BE states constructed by UPB \\cite{Mor}), we indeed\nhave found out the lowest rank space in which there is no BE\nstate. It also implies that when a state can be locally projected\ninto some rank three NPT state, then it is distillable. This\ncauses new methods of distilling quantum states having more\ncomplex structure. We will show it in terms of distilling the rank\nfour NPT states below. One may also find other way to distill the\nentangled states based on the theorem. For example, the tensor\nproduct of the rank three entangled states are also entangled for\ncertain.\n\nOn the other hand, the analytical calculation of distillable\nentanglement is also an important issue in quantum information\ntheory. The problem is very difficult and there have been some\noptimal bounds on distillable entanglement \\cite{Rains,Devetak}.\nSpecially, the bound is saturated if we can find a way to distill\nthe state and get the same amount of pure entanglement as the\nbound. In this case we get the analytical result of distillable\nentanglement. As it is possible to find out whether the bound on\nrank three NPT state is saturated by using our method of\ndistilling it, we indeed provide new ways to calculate the\ndistillable entanglement.\n\nThird, our result is also independent of the expectant fact that\nthere exist NPT Werner states $\\rho_w$. We do not know whether an\nNPT state $\\rho$ is distillable, even it can be converted into\nsome Werner state which is proved to be not distillable under\nLOCC. One can easily exemplify it by locally taking some rank\nthree NPT state into $\\rho_w$, while the latter is expected to be\nnot distillable. Conversely, it is difficult to convert the Werner\nstate into the state $\\sigma_{AB}$, so we still do not know\nwhether the latter is distillable. To see it, we have the Werner\nstate in a $N\\times N$ space as follows \\cite{Werner}\n\\begin{eqnarray}\n\\rho_w&=&(a+b)\\sum\\nolimits^{N-1}_{i,j=0}|ij\\rangle\\langle\nij|\\nonumber\\\\\n&-&2b\\sum\\nolimits^{N-1}_{i<j=0}\\frac{|ij\\rangle-|ji\\rangle}{\\sqrt2}\\frac{\\langle\nij|-\\langle ji|}{\\sqrt2},\n\\end{eqnarray}\nwhere $a>0,b<0$ are two parameters satisfying $a+b\\geq0$. The most\ngeneral local transformation on a quantum state $\\rho$ has the\nform $\\Lambda(\\rho)=\\sum\\nolimits_i A_i\\otimes B_i\\rho\nA^{\\dag}_i\\otimes B^{\\dag}_i$ \\cite{Vedral}. Because the resulting\nstate is entangled, there must be at least one pair of Kraus\noperators $A_i,B_i$ that have at least rank two, respectively. In\nthis case, the state $\\Lambda(\\rho)$ will have the rank not less\nthan four when $a+b>0$, which means a rank three NPT state cannot\nbe output by this local channel. The only exception happens when\n$a+b=0$, but it is difficult to judge whether the state\n$\\Lambda(\\rho)$ is of rank three and entangled.\n\nLet us investigate further the problem of distilling rank four\nstates by using the theorem in this paper. Different from the case\nof rank three state, it is well-known that there indeed exist PPT\nBE states of rank four even in the $3\\times3$ space. It is easy to\nshow that the NPT BE states of rank four possibly exist only in\nthree kinds of spaces, $4\\times4,3\\times4,3\\times3$ in terms of\nlemma 1. One will meet lots of difficulties when applying the\ntechnique in this paper to distill the rank four NPT state $\\rho$,\ne.g., the resulting state from $\\rho$ by projection can be\n$3\\times3$ and it may be PPT BE. Besides, the Peres-Horodecki\ncriterion is no more a sufficient condition for the separability\nof state in $2\\times4$ space, etc. Nevertheless, we still can\nobtain some useful results on this problem when the target state\nhas a special form.\n\n\\textit{Lemma 3}. For a rank four NPT state in $4\\times4$ or\n$3\\times4$ space, it is distillable when there is a product state\nin its range.\n\n\\textit{Proof.} By employing similar deduction for the state\n$\\sigma_{AB}$, only the state having the following form does not\nviolate the reduction criterion\n\\begin{equation}\n\\rho_{AB}=\\frac14\\sum\\nolimits^3_{i=0}|\\psi_i\\rangle\\langle\\psi_i|,\\rho_B=\\frac14I,\n\\end{equation}\nwhere the four eigenvectors satisfy\n$\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij}$. Up to the local unitary\ntransformations we have $|\\psi_0\\rangle=|00\\rangle$. Next, we\nproject the state $\\rho_{AB}$ by the projector\n$I_A\\otimes(|1\\rangle\\langle1|+|2\\rangle\\langle2|+|3\\rangle\\langle3|)_B$\nand obtain the NPT state\n$\\rho^{\\prime}_{AB}=\\frac13\\sum\\nolimits^3_{i=1}|\\psi_i\\rangle\\langle\\psi_i|$\nin $2\\times3$, or $3\\times3$, or $4\\times3$ space. By means of the\nBBPSSW and Horodeckis' protocol, our theorem and the reduction\ncriterion, respectively, the state $\\rho^{\\prime}_{AB}$ and hence\n$\\rho_{AB}$ is always distillable. \\hspace*{\\fill}$\\blacksquare$\n\nSo we have generalized lemma 1 to the case of rank four NPT\nstates. Moreover, we hope that it always holds for the NPT states\nwhose rank equal to its maximal dimension of subsystems. However,\nit does not hold when the rank of a state is larger, e.g, the PPT\nBE state in $3\\times3$ space constructed in \\cite{Horodecki4}\ncontains infinitely many product states in its range, but its rank\nequals eight. It is also unclear that whether the rank four NPT\nstates $\\rho$ in this space are distillable. Solving this problem\nis more difficult since we cannot rely on the reduction criterion.\nHowever, $\\rho$ is distillable when we can project it onto a rank\nthree NPT state in terms of our theorem.\n\nFor example, the following $3\\times3$ rank four NPT state is\ndistillable\n\\begin{eqnarray}\n\\rho_{AB}&=&\\lambda_0|00\\rangle\\langle00|+\\lambda_1|01\\rangle\\langle01|+\n\\lambda_2|\\psi_2\\rangle\\langle\\psi_2|+\\lambda_3|\\psi_3\\rangle\\langle\\psi_3|,\\nonumber\\\\\n|\\psi_2\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}c_{ij}|ij\\rangle,\\nonumber\\\\\n|\\psi_3\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}d_{ij}|ij\\rangle,\\lambda_0,\\lambda_1,\\lambda_2,\\lambda_3>0.\n\\end{eqnarray}\nTo prove it, we project the state $\\rho_{AB}$ by the projector\n$(|1\\rangle\\langle1|+|2\\rangle\\langle2|)_A\\otimes I_B$. When the\nresulting state\n$\\rho^{1}_{AB}=\\lambda_2|\\psi^1_2\\rangle\\langle\\psi^1_2|+\\lambda_3|\\psi^1_3\\rangle\\langle\\psi^1_3|$\nis entangled, it is also distillable. On the other hand when\n$\\rho^{1}_{AB}$ is separable, we can write it as the sum of two\nproduct states since it is in a space not larger than $2\\times3.$\nBesides, the rank of $\\rho^{1}_{AB}$ must be two because of\n$r(\\rho_A)=3$. By performing some ILOs on the state $\\rho_{AB}$\nand linear combination of $|\\psi_2\\rangle$ and $|\\psi_3\\rangle$,\nwe can convert them into\n$|\\psi^2_2\\rangle=|0\\rangle|\\phi_0\\rangle+|1\\rangle|\\phi_1\\rangle$\nand\n$|\\psi^2_3\\rangle=|0\\rangle|\\omega_0\\rangle+|2\\rangle|\\omega_2\\rangle$,\nand keep the other two terms $|00\\rangle$ and $|01\\rangle$\nunchanged.\n\nWhen either of the states $|\\psi^2_2\\rangle$ and\n$|\\psi^2_3\\rangle$ is of product form, we easily project the state\n$\\rho^2_{AB}$ onto a $2\\times3$ subspace, the resulting state is\nstill entangled and distillable. On the other hand when both the\nstates $|\\psi^2_2\\rangle$ and $|\\psi^2_3\\rangle$ are entangled, we\nproject the state $\\rho^2_{AB}$ by the projector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B$. The obtained state $\\rho^3_{AB}$ is $2\\times3$ and its rank\nis four by choosing suitable parameter $a$. This state is\nseparable iff it has the decomposition\n$\\rho^3_{AB}=|\\psi\\rangle_A\\langle\\psi|\\otimes|\\omega_2\\rangle_B\\langle\\omega_2|\n+|0\\rangle_A\\langle0|\\otimes\\rho^3_B$ with $r(\\rho^3_B)=3.$\nHowever it is impossible, since it requires a $4\\times4$\ncoefficient unitary matrix $[a_{ij}]$ in which $a_{i3}=0,i=1,2,3,$\nand $a_{0i},i=0,1,2$ cannot be zero simultaneously. Hence the\nstate $\\rho^3_{AB}$ is entangled and thus distillable. This also\ncompletes the proof showing that the state $\\rho_{AB}$ in Eq. (15)\nis distillable.\n\n\nAs above we have given several families of states that can be\ndistilled by means of the fact that the rank three NPT states are\ndistillable. The main difficulty in entanglement distillation is\nthe great amount of parameters that cannot be removed during the\nfiltering process. For example, it is unknown that whether the\nrank four NPT states are distillable. All in all, more efforts are\nrequired to distill other classes of rank four NPT states.\n\n\n\n\n\n\n\\section{conclusions}\n\nWe have proved that the bipartite rank three NPT states and some\nfamilies of rank four NPT states are distillable. So they are\nindeed available resource for quantum-information tasks. An open\nproblem is that whether all rank four NPT states are distillable.\nOur result also gives an insight into the relationship between the\nlow rank states and the Werner states.\n\n\n\nThe work was partly supported by the NNSF of China Grant\nNo.90503009, No.10775116, and 973 Program Grant No.2005CB724508.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe idea of cosmological inflation is capable to address some\nproblems of the standard big bang theory, such as the horizon,\nflatness and monopole problems. Also, it can provide a reliable\nmechanism for generation of density perturbations responsible for\nstructure formation and therefore temperature anisotropies in Cosmic\nMicrowave Background (CMB)spectrum [1-8]. There are a wide variety\nof cosmological inflation models where viability of their\npredictions in comparison with observations makes them to be\nacceptable or unacceptable (see for instance [9] for this purpose).\nThe simplest inflationary model is a single scalar field scenario in\nwhich inflation is driven by a scalar field called the inflaton that\npredicts adiabatic, Gaussian and scale-invariant fluctuations [10].\nBut, recently observational data have revealed some degrees of\nscale-dependence in the primordial density perturbations. Also,\nPlanck team have obtained some constraints on the primordial\nnon-Gaussianity [11-13]. Therefore, it seems that extended models of\ninflation which can explain or address this scale-dependence and\nnon-Gaussianity of perturbations are more desirable. There are a lot\nof studies in this respect, some of which can be seen in Refs.\n[14-19] with references therein. Among various inflationary models,\nthe non-minimal models have attracted much attention. Non-minimal\ncoupling of the inflaton field and gravitational sector is\ninevitable from the renormalizability of the corresponding field\ntheory (see for instance [20]). Cosmological inflation driven by a\nscalar field non-minimally coupled to gravity are studied, for\ninstance, in Refs. [21-28]. There were some issues on the unitarity\nviolation with non-minimal coupling (see for instance, Refs.\n[29-31]) which have forced researchers to consider possible coupling\nof the derivatives of the scalar field with geometry [32]. In fact,\nit has been shown that a model with nonminimal coupling between the\nkinetic terms of the inflaton (derivatives of the scalar field) and\nthe Einstein tensor preserves the unitary bound during inflation\n[33]. Also, the presence of nonminimal derivative coupling is a\npowerful tool to increase the friction of an inflaton rolling down\nits own potential [33]. Some authors have considered the model with\nthis coupling term and have studied the early time accelerating\nexpansion of the universe as well as the late time dynamics [34-36].\nIn this paper we extend the non-minimal inflation models to the case\nthat a canonical inflaton field is coupled non-minimally to the\ngravitational sector and in the same time the derivatives of the\nfield are also coupled to the background geometry (Einstein's\ntensor). This model provides a more realistic framework for treating\ncosmological inflation in essence. We study in details the\ncosmological perturbations and possible non-Gaussianities in the\ndistribution of these perturbations in this non-minimal inflation.\nWe expand the action of the model up to the third order and compare\nour results with observational data from Planck2015 to see the\nviability of this extended model. In this manner we are able to\nconstraint parameter space of the model in comparison with\nobservation.\n\n\n\\section{Field Equations}\n\nWe consider an inflationary model where both a canonical scalar field and its\nderivatives are coupled non-minimally to gravity. The four-dimensional action for\nthis model is given by the following expression:\n\n\\begin{equation}\nS=\\frac{1}{2}\\int\nd^{4}x\\sqrt{-g}\\Bigg[M_{p}^{2}f(\\phi)R+\\frac{1}{\\widetilde{M}^{2}}G_{\\mu\\nu}\\partial^{\\mu}\\phi\\partial^{\\nu}\\phi-2V(\\phi)\\Bigg]\\,,\n\\end{equation}\nwhere $M_{p}$ is a reduced planck mass, $\\phi$ is a canonical scalar field,\n$f(\\phi)$ is a general function of the scalar field and\n$\\widetilde{M}$ is a mass parameter. The energy-momentum tensor is obtained from action (1) as follows\n\n\\vspace{0.5cm}\n\n$T_{\\mu\\nu}=\\frac{1}{2\\widetilde{M}^{2}}\\bigg[\\nabla_{\\mu}\\nabla_{\\nu}(\\nabla^{\\alpha}\\phi\\nabla_{\\alpha}\\phi)\n-g_{\\mu\\nu}\\Box(\\nabla^{\\alpha}\\phi\\nabla_{\\alpha}\\phi)\n+g_{\\mu\\nu}g^{\\alpha\\rho}g^{\\beta\\lambda}\\nabla_{\\rho}\\nabla_{\\lambda}(\\nabla_{\\alpha}\\phi\\nabla_{\\beta}\\phi)$\n\\begin{equation}\n+\\Box(\\nabla_{\\mu}\\phi\n\\nabla_{\\nu}\\phi)\\bigg]-\\frac{g^{\\alpha\\beta}}{\\widetilde{M}^{2}}\n\\nabla_{\\beta}\\nabla_{\\mu}(\\nabla_{\\alpha}\\phi\n\\nabla_{\\nu}\\phi)-M_{p}^{2}\\nabla_{\\mu}\\nabla_{\\nu}f(\\phi)+M_{p}^{2}g_{\\mu\\nu}\\Box\nf(\\phi)+g_{\\mu\\nu}V(\\phi)\\,.\n\\end{equation}\n\nOn the other hand, variation of the action (1) with respect to the\nscalar field gives the scalar field equation of motion as\n\n\\begin{equation}\n\\frac{1}{2}M_{p}^{2}Rf'(\\phi)-\\frac{1}{\\widetilde{M}^{2}}G^{\\mu\\nu}\\nabla_{\\mu}\\nabla_{\\nu}\\phi-V'(\\phi)=0\\,,\n\\end{equation}\nwhere a prime denotes derivative with respect to the scalar field. We consider a spatially flat\nFriedmann-Robertson-Walker (FRW) line element as\n\n\\begin{equation}\nds^{2}=-dt^{2}+a^{2}(t)\\delta_{ij}dx^{i}dx^{j}\\,,\n\\end{equation}\nwhere $a(t)$ is scale factor. Now, let's assume that $f(\\phi)=\\frac{1}{2}\\phi^{2}$. In this\nframework, $T_{\\mu\\nu}$ leads to the following energy density and\npressure for this model respectively\n\n\\begin{equation}\n\\rho=\\frac{9H^{2}}{2\\widetilde{M}^{2}}\\dot{\\phi}^{2}-\\frac{3}{2}M_{p}^{2}H\\phi(2\\dot{\\phi}+H\\phi)+V(\\phi)\n\\end{equation}\n\n$$p=-\\frac{3}{2}\\frac{H^{2}\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}-\\frac{\\dot{\\phi}^{2}\\dot{H}}{\\widetilde{M}^{2}}\n-\\frac{2H}{\\widetilde{M}^{2}}\\dot{\\phi}\\ddot{\\phi}$$\n\\begin{equation}\n+\\frac{1}{2}M_{p}^{2}\\Bigg[2\\dot{H}\\phi^{2}+3H^{2}\\phi^{2}\n+4H\\phi\\dot{\\phi}+2\\phi\\ddot{\\phi}+2\\dot{\\phi}\\Bigg]-V(\\phi)\\,,\n\\end{equation}\nwhere a dot refers to derivative with respect to the cosmic time. The equations of motion following from action (1) are\n\n\\begin{equation}\nH^{2}=\\frac{1}{3M_{p}^{2}}\\Bigg[-\\frac{3}{2}M_{p}^{2}H\\phi(2\\dot{\\phi}+H\\phi)+\\frac{9H^{2}}{2\\widetilde{M}^{2}}\\dot{\\phi}^{2}+V(\\phi)\\Bigg]\\,,\n\\end{equation}\n\n$$\\dot{H}=-\\frac{1}{2M_{p}^{2}}\\Bigg[\\dot{\\phi}^{2}\\bigg(\\frac{3H^{2}}{\\widetilde{M}^{2}}-\\frac{\\dot{H}}{\\widetilde{M}^{2}}\\bigg)\n-\\frac{2H}{\\widetilde{M}^{2}}\\dot{\\phi}\\ddot{\\phi}-\\frac{3}{2}M_{p}^{2}H\\phi(2\\dot{\\phi}+H\\phi)$$\n\\begin{equation}+\\frac{1}{2}M_{p}^{2}\\bigg((2\\dot{H}+3H^{2})\\phi^{2}+4H\\phi\\dot{\\phi}+2\\phi\\ddot{\\phi}+2\\dot{\\phi}^{2}\\bigg)\\Bigg]\n\\end{equation}\n\n\\begin{equation}\n-3M_{p}^{2}(2H^{2}+\\dot{H})\\phi+\\frac{3H^{2}}{\\widetilde{M}^{2}}\\ddot{\\phi}+3H\\bigg(\\frac{3H^{2}}{\\widetilde{M}^{2}}\n+\\frac{2\\dot{H}}{\\widetilde{M}^{2}}\\bigg)\\dot{\\phi}+V'(\\phi)=0\\,.\n\\end{equation}\n\nThe slow-roll parameters in this model are defined as\n\\begin{equation}\n\\epsilon\\equiv-\\frac{\\dot{H}}{H^{2}}\\,\\,\\,\\,,\\,\\,\\,\\,\\eta\\equiv-\\frac{1}{H}\\frac{\\ddot{H}}{\\dot{H}}\\,.\n\\end{equation}\nTo have inflationary phase, $\\epsilon$ and $\\eta$ should satisfy slow-roll conditions($\\epsilon\\ll1$ , $\\eta\\ll1$). In our setup, we\nfind the following result\n\\begin{equation}\n\\epsilon=\\bigg[1+\\frac{\\phi^{2}}{2}-\\frac{\\dot{\\phi^{2}}}{2\\widetilde{M}^{2}M_{p}^{2}}\\bigg]^{-1}\n\\bigg[\\frac{3\\dot{\\phi}^{2}}{2\\widetilde{M}^{2}M_{p}^{2}}+\\frac{\\phi\\dot{\\phi}}{2H}\n+\\frac{\\ddot{\\phi}}{H\\dot{\\phi}}\\bigg(\\frac{\\phi\\dot{\\phi}}{2H}\n-\\frac{\\dot{\\phi^{2}}}{\\widetilde{M}^{2}M_{p}^{2}}\\bigg)\\bigg]\n\\end{equation}\nand\n\\begin{equation}\n\\eta=-2\\epsilon-\\frac{\\dot{\\epsilon}}{H\\epsilon}\\,.\n\\end{equation}\n\nWithin the slow-roll approximation, equations (7),(8) and (9) can be\nwritten respectively as\n\\begin{equation}\nH^{2}\\simeq\\frac{1}{3M_{p}^{2}}\\Bigg[-\\frac{3}{2}M_{p}^{2}H^{2}\\phi^{2}+V(\\phi)\\Bigg]\\,,\n\\end{equation}\n\n\\begin{equation}\n\\dot{H}\\simeq-\\frac{1}{2M_{p}^{2}}\\Bigg[\\frac{3H^{2}\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}-M_{p}^{2}H\\phi\\dot{\\phi}+M_{p}^{2}\\dot{H}\\phi^{2}\\Bigg]\\,,\n\\end{equation}\nand\n\\begin{equation}\n-6M_{p}^{2}H^{2}\\phi+\\frac{9H^{3}\\dot{\\phi}}{\\widetilde{M}^{2}}+V'(\\phi)\\simeq0\\,.\n\\end{equation}\nThe number of e-folds during inflation is defined as\n\\begin{equation}\n{\\cal N}=\\int_{t_{hc}}^{t_{e}}H\\,dt\\,,\n\\end{equation}\nwhere $t_{hc}$ and $t_{e}$ are time of horizon crossing and end of\ninflation respectively. The number of e-folds in the slow-roll approximation in our setup can be\nexpressed as follows\n\n\\begin{equation}\n{\\cal N}\\simeq\\int_{\\phi_{hc}}^{\\phi_{e}}\\frac{V(\\phi)d\\phi}{M_{p}^{2}\\bigg(1\n+\\frac{1}{2}\\phi^{2}\\bigg)\\Bigg[2M_{p}^{2}\\widetilde{M}^{2}\\phi\n-M_{p}^{2}\\widetilde{M}^{2}\\frac{V'(\\phi)}{V(\\phi)}\\bigg(1+\\frac{1}{2}\\phi^{2}\\bigg)\\Bigg]}\\,.\n\\end{equation}\nAfter providing the basic setup of the model, for testing cosmological viability\nof this extended model we treat the perturbations in comparison with observation.\n\n\n\\section{Second-Order Action: Linear Perturbations}\n\nIn this section, we study linear perturbations around the\nhomogeneous background solution. To this end, the first step is\nexpanding the action (1) up to the second order in small fluctuations. It\nis convenient to work in the ADM formalism given by [37]\n\\begin{equation}\nds^{2}=-N^{2}dt^{2}+h_{ij}(N^{i}dt+dx^{i})(N^{j}dt+dx^{j})\\,,\n\\end{equation}\nwhere $N^{i}$ is the shift vector and $N$ is the lapse function.\nWe expand the lapse function and shift vector to $N=1+2\\Phi$ and\n$N^{i}=\\delta^{ij}\\partial_{j}\\Upsilon$ respectively, where $\\Phi$\nand $\\Upsilon$ are three-scalars. Also,\n$h_{ij}=a^{2}(t)[(1+2\\Psi)\\delta_{ij}+\\gamma_{ij}]$, where $\\Psi$ is\nspatial curvature perturbation and $\\gamma_{ij}$ is shear\nthree-tensor which is traceless and symmetric. In the rest of our study, we\nchoose $\\delta\\Phi=0$ and $\\gamma_{ij}=0$. By taking into account\nthe scalar perturbations in linear-order, the metric (18) is written\nas (see for instance [38])\n\\begin{equation}\nds^{2}=-(1+2\\Phi)dt^{2}+2\\partial_{i}\\Upsilon\ndtdx^{i}+a^{2}(t)(1+2\\Psi)\\delta_{ij}dx^{i}dx^{j}\\,.\n\\end{equation}\n\nNow by replacing metric (19) in action (1) and expanding the action up to the\nsecond-order in perturbations, we find (see for instance [39,40])\n\n$$S^{(2)}=\\int dt dx^{3}a^{3}\\Bigg[-\\frac{3}{2}(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\dot{\\Psi}^{2}\n+\\frac{1}{a^{2}}((M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\dot{\\Psi}$$\n$$-(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\Phi)\\partial^{2}\\Upsilon\n-\\frac{1}{a^{2}}(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\Phi\\partial^{2}\\Psi$$\n$$+3(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\Phi\\dot{\\Psi}\n+3H(-\\frac{1}{2}M_{p}^{2}H\\phi^{2}-M_{p}^{2}\\phi\\dot{\\phi}$$\n\\begin{equation}\n+\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\Phi^{2}\n+\\frac{1}{2a^{2}}(M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})(\\partial\\Psi)^{2}\\Bigg]\\,.\n\\end{equation}\n\nBy variation of action (20) with respect to $N$ and $N^{i}$ we find\n\\begin{equation}\n\\Phi=\\frac{M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}}{M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}}\\dot{\\Psi}\\,,\n\\end{equation}\n\n$$\\partial^{2}\\Upsilon=\\frac{2a^{2}}{3}\\frac{(-\\frac{9}{2}M_{p}^{2}H^{2}\\phi^{2}-9M_{p}^{2}H\\phi\\dot{\\phi}\n+\\frac{27H^{2}\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})}{(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})}$$\n\\begin{equation}\n+3\\dot{\\Psi}a^{2}-\\frac{M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}}{M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}}\\dot{\\Psi}\\,. \\end{equation}\nFinally the second order action can be rewritten as follows\n\\begin{equation}\nS^{(2)}=\\int dt\ndx^{3}a^{3}\\vartheta_{s}\\bigg[\\dot{\\Psi}^{2}-\\frac{c_{s}^{2}}{a^{2}}(\\partial\\Psi)^{2}\\bigg]\n\\end{equation}\nwhere by definition\n\\begin{equation}\n\\vartheta_{s}\\equiv6\\frac{(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})^{2}(-\\frac{1}{2}M_{p}^{2}H^{2}\\phi^{2}-M_{p}^{2}H\\phi\\dot{\\phi}+\\frac{3}{\\widetilde{M}^{2}}\nH^{2}\\dot{\\phi}^{2})}{(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3}{\\widetilde{M}^{2}}H\\dot{\\phi}^{2})^{2}}+\n3(\\frac{1}{2}M_{p}^{2}\\phi^{2}-\\frac{1}{2\\widetilde{2}}\\dot{\\phi}^{2})\n\\end{equation}\nand\n$$c_{s}^{2}\\equiv\\frac{3}{2}\\bigg\\{(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})^{2}\n(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})H$$\n$$-(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})^{2}(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})$$\n$$4(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})(M_{p}^{2}\\phi\\dot{\\phi}-\\frac{\\dot{\\phi}\\ddot{\\phi}}{\\widetilde{M}^{2}})\n(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})$$\n$$-(M_{p}^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})^{2}(M_{p}^{2}\\dot{H}\\phi^{2}+2M_{p}^{2}H\\phi\\dot{\\phi}M_{p}^{2}\\dot{\\phi}^{2}+M_{p}^{2}\\phi\\ddot{\\phi}\n-\\frac{3\\dot{H}\\dot{\\phi}^{2}}{\\widetilde{M}^{2}}-\\frac{6}{\\widetilde{M}^{2}}H\\dot{\\phi}\\ddot{\\phi})\\bigg\\}$$\n$$\\bigg\\{9[\\frac{1}{2}M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{2\\widetilde{M}^{2}}][4(\\frac{1}{2}M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{2\\widetilde{M}^{2}})\n(-\\frac{1}{2}M_{p}^{2}H^{2}\\phi^{2}-M_{p}^{2}H\\phi\\dot{\\phi}+\\frac{3}{M\\widetilde{^{2}}H^{2}\\dot{\\phi}^{2}})$$\n\\begin{equation}\n+(M_{p}^{2}H\\phi^{2}+M_{p}^{2}\\phi\\dot{\\phi}-\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})^{2}]\\bigg\\}^{-1}\\,.\n\\end{equation}\n\nIn order to obtain quantum perturbations $\\Psi$, we can find\nequation of motion of the curvature perturbation by varying action (23)\nwhich follows\n\\begin{equation}\n\\ddot{\\Psi}+\\bigg(3H+\\frac{\\dot{\\vartheta_{s}}}{\\vartheta_{s}}\\bigg)+\\frac{c_{s}^{2}k^{2}}{a^{2}}\\Psi=0\\,.\n\\end{equation}\nBy solving the above equation up to the lowest order in slow-roll approximation,\nwe find\n\\begin{equation}\n\\Psi=\\frac{iH\\exp(-ic_{s}k\\tau)}{2c_{s}^{\\frac{3}{2}}\\sqrt{k^{3}}\\vartheta_{s}}(1+ic_{s}k\\tau)\\,.\n\\end{equation}\nBy using the two-point correlation functions we can study power\nspectrum of curvature perturbation in this setup. We find two-point correlation\nfunction by obtaining vacuum expectation value at the end of inflation.\nWe define the power spectrum $P_{s}$, as\n\\begin{equation}\n\\langle0|\\Psi(0,\\textbf{k}_{1})\\Psi(0,\\textbf{k}_{2})|0\\rangle=\\frac{2\\pi^{2}}{k^{3}}P_{s}(2\\pi)^{3}\\delta^{3}(\\textbf{k}_{1}+\\textbf{k}_{2})\\,,\n\\end{equation}\nwhere\n\\begin{equation}\nP_{s}=\\frac{H^{2}}{8\\pi^{2}\\vartheta_{s} c_{s}^{3}}\\,.\n\\end{equation}\n\nThe spectral index of scalar perturbations is given by (see Refs. [41-43] for more details on the cosmological perturbations in generalized gravity theories and also inflationary spectral index in these theories.)\n\n\\begin{equation}\nn_{s}-1=\\frac{d\\ln P_{s}}{d\\ln\nk}|_{c_{s}k=aH}=-2\\epsilon-\\delta_{F}-\\eta_{s}-S\n\\end{equation}\nwhere by definition\n\\begin{equation}\n\\delta_{F}=\\frac{\\dot{f}}{H(1+f)}\\,\\,\\,\\,,\\,\\,\\,\\,\\eta_{s}=\\frac{\\dot{\\epsilon_{s}}}{H\\epsilon_{s}}\\,\\,\\,\\,,\\,\\,\\,\\,S=\\frac{\\dot{c_{s}}}{Hc_{s}}\n\\end{equation}\nalso\n\\begin{equation}\n\\epsilon_{s}=\\frac{\\vartheta_{s}c_{s}^{2}}{M_{pl}^{2}(1+f)}.\n\\end{equation}\n\n\nwe obtain finally\n\n\\begin{equation}\nn_{s}-1=-2\\epsilon-\\frac{1}{H}\\frac{d\\ln c_{s}}{dt}\n-\\frac{1}{H}\\frac{d\\ln[2H(1+\\frac{\\phi^{2}}{2})\\epsilon+\\phi\\dot{\\phi}]}{dt}\\,,\n\\end{equation}\nwhich shows the scale dependence of perturbations due to deviation of $n_{s}$\nfrom $1$.\n\nNow we study tensor perturbations in this setup. To this end, we write the metric as follows\n\\begin{equation}\nds^{2}=-dt^{2}+a(t)^{2}(\\delta_{ij}+T_{ij})dx^{i}dx^{j}\\,,\n\\end{equation}\nwhere $T_{ij}$ is a spatial shear 3-tensor which is transverse and\ntraceless. It is convenient to write $T_{ij}$ in terms of two\npolarization modes, as follows\n\\begin{equation}\nT_{ij}=T_{+}e^{+}_{ij}+T^{\\times}e^{\\times}_{ij}\\,,\n\\end{equation}\nwhere $e^{+}_{ij}$ and $e^{\\times}_{ij}$ are the polarization tensors. In this case the second order action for the tensor mode can de\nwritten as\n\\begin{equation}\nS_{T}=\\int dt dx^{3}\na^{3}\\vartheta_{T}\\bigg[\\dot{T}_{(+,\\times)}^{2}-\\frac{c_{T}^{2}}{a^{2}}(\\partial\nT_{(+,\\times)})^{2}\\bigg]\\,,\n\\end{equation}\nwhere by definition\n\\begin{equation}\n\\vartheta_{T}\\equiv\\frac{1}{8}(M_{p}^{2}\\phi^{2}-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\n\\end{equation}\nand\n\\begin{equation}\nc_{T}^{2}\\equiv\\frac{\\widetilde{M}^{2}M_{p}^{2}\\phi^{2}+\\dot{\\phi}^{2}}{\\widetilde{M}^{2}M_{p}^{2}\\phi^{2}-\\dot{\\phi}^{2}}\\,.\n\\end{equation}\n\nNow, the amplitude of tensor perturbations is given by\n\\begin{equation}\nP_{T}=\\frac{H^{2}}{2\\pi^{2}\\vartheta_{T}c_{T}^{3}}\\,,\n\\end{equation}\nwhere we have defined the tensor spectral index as\n\\begin{equation}\nn_{T}\\equiv\\frac{d\\ln P_{T}}{d\\ln\nk}|_{c_{T}k=aH}\\,=-2\\epsilon-\\delta_{F}.\n\\end{equation}\nBy using above equations we get finally\n\\begin{equation}\nn_{T}=-2\\epsilon-\\frac{\\phi\\dot{\\phi}}{H(1+\\frac{\\phi^{2}}{2})}\\,.\n\\end{equation}\n\nThe tensor-to-scalar ratio as an important observational quantity in our setup is given by\n\\begin{equation}\nr=\\frac{P_{T}}{P_{s}}=16c_{s}\\bigg(\\epsilon+\\frac{\\phi\\dot{\\phi}}{2H(1+\\frac{\\phi^{2}}{2})}+O(\\epsilon^{2})\\bigg)\\simeq-8c_{s}n_{T}\n\\end{equation}\nwhich yields the standard consistency relation.\n\n\\section{Third-Order Action: Non-Gaussianity}\n\nSince a two-point correlation function of the scalar perturbations\ngives no information about possible non-Gaussian feature of distribution, we study\nhigher-order correlation functions. A three-point correlation function is capable to give the required information. For this\npurpose, we should expand action (1) up to the third order in small\nfluctuations around the homogeneous background solutions. In this respect we obtain\n\n\\vspace{0.5cm} $S^{(3)}=\\int\ndtdx^{3}a^{3}\\bigg\\{3\\Phi^{3}[M_{p}^{2}H^{2}(1+\\frac{\\phi^{2}}{2})+M_{p}^{2}H\\phi\\dot{\\phi}-\\frac{5}{\\widetilde{M^{2}}}H^{2}\\dot{\\phi^{2}}]\n+\\Phi^{2}[9\\Psi(-\\frac{1}{2}M_{p}^{2}\\phi^{2}-M_{p}^{2}H\\phi\\dot{\\phi}$\n\\vspace{0.5cm} $+\\frac{3}{\\widetilde{M}^{}}H^{2}\\dot{\\phi}^{2})\n+6\\dot{\\Psi}(-M_{p}^{2}H(1+\\frac{\\phi^{2}}{2})-\\frac{1}{2}M_{p}^{2}\\phi\\dot{\\phi}\\frac{3}{\\widetilde{M}^{2}}H\\dot{\\phi}^{2})\n-\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}a^{2}}\\partial^{2}\\Psi\n-\\frac{2}{a^{2}}\\partial^{2}\\Upsilon(-M_{p}^{2}H$\\vspace{0.05cm}\n$(1+\\frac{\\phi^{2}}{2})-\\frac{1}{2}M_{p}^{2}\\phi\\dot{\\phi}\\frac{3}{\\widetilde{M}^{2}}H\\dot{\\phi}^{2})]\n+\\Phi[\\frac{1}{a^{2}}(-M_{p}^{2}H\\phi^{2}-M_{p}^{2}\\phi\\dot{\\phi}+\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\partial_{i}\\Psi\\partial_{i}\\Upsilon\n-9(-M_{p}^{2}H\\phi^{2}-M_{p}^{2}\\phi\\dot{\\phi}+$ \\vspace{0.5cm}\n$\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\dot{\\Psi}\\Psi+\\frac{1}{2a^{4}}(M_{p}^{2}(1+\\frac{\\phi^{2}}{2})+\\frac{3}{2}\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\n(\\partial_{i}\\partial_{j}\\Upsilon\\partial_{i}\\partial_{j}\\Upsilon-\\partial^{2}\\Upsilon\\partial^{2}\\Upsilon)\n+\\frac{1}{a^{2}}(-M_{p}^{2}H\\phi^{2}-M_{p}^{2}\\phi\\dot{\\phi}+\\frac{3H\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\Psi\\partial^{2}\\Upsilon\n$ \\vspace{0.5cm}\n$+\\frac{4}{2a^{2}}(M_{p}^{2}(1+\\frac{\\phi^{2}}{2})+\\frac{3}{2}\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\n\\dot{\\Psi}\\partial^{2}\\Upsilon+\\frac{1}{a^{2}}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})\n\\Psi\\partial^{2}\\Psi\n+\\frac{1}{2a^{2}}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})(\\partial\\Psi)^{2}\n-6(M_{p}^{2}(1+\\frac{\\phi^{2}}{2})+$ \\vspace{0.5cm}\n$\\frac{3}{2}\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\\dot{\\Psi}^{2}]+\\frac{1}{2a^{2}}(M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}^{2}}{\\widetilde{M}^{2}})\n\\Psi(\\partial\\Psi)^{2}\n+\\frac{9}{2}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})\\dot{\\Psi^{2}}\\Psi\n-\\frac{1}{a^{2}}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})\\dot{\\Psi}\\partial_{i}\\Psi\\partial_{i}\\Upsilon\n-\\frac{1}{a^{2}}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})$\n\\begin{equation}\n\\dot{\\Psi}\\Psi\\partial^{2}\\Upsilon-\\frac{3}{4a^{4}}\\Psi(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})\n(\\partial_{i}\\partial_{j}\\Upsilon\\partial_{i}\\partial_{j}\\Upsilon-\\partial^{2}\\Upsilon\\partial^{2}\\Upsilon)\n+\\frac{1}{a^{4}}(-M_{p}^{2}\\phi^{2}+\\frac{\\dot{\\phi}}{\\widetilde{M}^{2}})\\partial_{i}\\Psi\\partial_{i}\\Upsilon\\partial^{2}\\Upsilon\\bigg\\}\n\\end{equation}\n\nWe use Eqs. (21) and (22) for eliminating $\\Phi$ and $\\Upsilon$ in this relation. For this end, we introduce the quantity $\\chi$ as follows\n\n\\begin{equation}\n\\Upsilon=\\frac{M_{p}^{2}\\widetilde{M}^{2}\\phi^{2}-\\dot{\\phi}^{2}}{\\widetilde{M}^{2}M_{p}^{2}(H\\phi^{2}+\\phi\\dot{\\phi})-3H\\dot{\\phi}^{2}}\\Psi\n+\\frac{2\\widetilde{M^{2}}a^{2}\\chi}{M_{p}^{2}\\widetilde{M}^{2}\\phi^{2}-\\dot{\\phi}^{2}}\\,,\n\\end{equation}\nwhere\n\n\\begin{equation}\n\\partial^{2}\\chi=\\vartheta_{s}\\dot{\\Psi}\\,.\n\\end{equation}\n\nNow the third order action (43) takes the following form\n\n\\vspace{0.5cm}\n\n$S^{(3)}=\\int dt\\,\ndx^{3}a^{3}\\bigg\\{[-3M_{p}^{2}c_{s}^{-2}\\Psi\\dot{\\Psi^{2}}\n+M_{p}^{2}a^{-2}\\Psi(\\partial\\Psi)^{2}+M_{p}^{2}c_{s}^{-2}H^{-1}\\dot{\\Psi}^{3}]$\n\\begin{equation}\n\\bigg[(1+\\frac{1}{4}\\phi^{2})\\epsilon+\\frac{5}{8}\\frac{\\phi\\dot{\\phi}}{H}\\bigg]\n-2(1+\\frac{1}{4}\\phi^{2})^{-1}(\\frac{5}{8}\\frac{\\phi\\dot{\\phi}}{c_{s}^{2}H})\\dot{\\Psi}\\partial_{i}\\Psi\\partial_{i}\\chi\\bigg\\}\\,.\n\\end{equation}\n\nBy calculating the three-point correlation function we can study\nnon-Gaussianity feature of the primordial perturbations. For the\npresent model, we use the interaction picture in which the\ninteraction Hamiltonian, $H_{int}$, is equal to the Lagrangian third\norder action. The vacuum expectation value of curvature\nperturbations at $\\tau=\\tau_{f}$ is\n\n\\begin{equation}\n\\langle\\Psi(\\textbf{k}_{1})\\Psi(\\textbf{k}_{2})\\Psi(\\textbf{k}_{3})\\rangle=-i\\int_{\\tau_{i}}^{\\tau_{f}}d\\tau\n\\langle0|[\\Psi(\\tau_{f},\\textbf{k}_{1})\\Psi(\\tau_{f},\\textbf{k}_{2})\\Psi(\\tau_{f},\\textbf{k}_{3}),H_{int}(\\tau)]|0\\rangle\\,.\n\\end{equation}\n\nBy solving the above integral in Fourier space, we find\n\\begin{equation}\n\\langle\\Psi(\\textbf{k}_{1})\\Psi(\\textbf{k}_{2})\\Psi(\\textbf{k}_{3})\\rangle=(2\\pi)^{3}\\delta^{3}(\\textbf{k}_{1}+\\textbf{k}_{2}+\\textbf{k}_{3})\nP_{s}^{2}F_{\\Psi}(\\textbf{k}_{1},\\textbf{k}_{2},\\textbf{k}_{3})\\,,\n\\end{equation}\nwhere\n\\begin{equation}\nF_{\\Psi}(\\textbf{k}_{1},\\textbf{k}_{2},\\textbf{k}_{3})=\\frac{(2\\pi)^{2}}{\\prod_{i=1}^{3}k_{i}^{3}}G_{\\Psi}\\,,\n\\end{equation}\n\n\\vspace{0.5cm}\n\n$G_{\\Psi}=\\bigg[\\frac{3}{4}\\bigg(\\frac{2}{K}\\Sigma_{i>j}k_{i}^{2}k_{j}^{2}-\\frac{1}{K^{2}}\\Sigma_{i\\neq\nj}k_{i}^{2}k_{j}^{3}\\bigg)+\\frac{1}{4}\\bigg(\\frac{1}{2}\\Sigma_{i}k_{i}^{3}+\\frac{2}{K}\\Sigma_{i>j}k_{i}^{2}k_{j}^{2}\n-\\frac{1}{K^{2}}\\Sigma_{i\\neq j} k_{i}^{2}k_{j}^{3}\\bigg)$\n\\begin{equation}\n-\\frac{3}{2}\\bigg(\\frac{(k_{1}k_{2}k_{3})^{2}}{K^{3}}\\bigg)\\bigg]\\bigg(1-\\frac{1}{c_{s}^{2}}\\bigg)\\,,\n\\end{equation}\n\nand $K=\\sum_{i}k_{i}$. Finally the non-linear parameter $f_{NL}$ is defined as follows\n\n\\begin{equation}\nf_{NL}=\\frac{10}{3}\\frac{G_{\\Psi}}{\\sum_{i=1}^{3}k_{i}}\\,.\n\\end{equation}\n\nHere we study non-Gaussianity in the orthogonal and the equilateral\nconfigurations [44,45]. Firstly we should account $G_{\\Psi}$ in\nthese configurations. To this end, we follow Refs. [46-48] to\nintroduce a shape $\\zeta_{\\ast}^{equi}$ as\n$\\zeta_{\\ast}^{equi}=-\\frac{12}{13}(3\\zeta_{1}-\\zeta_{2})$. In this\nmanner we define the following shape which is orthogonal to\n$\\zeta_{\\ast}^{equi}$\n\\begin{equation}\n\\zeta_{\\ast}^{ortho}=-\\frac{12}{14-13\\beta}[\\beta(3\\zeta_{1}-\\zeta_{2})+3\\zeta_{1}-\\zeta_{2}]\\,,\n\\end{equation}\nwhere $\\beta\\simeq1.1967996$. Finally, bispectrum (48) can be\nwritten in terms of $\\zeta_{\\ast}^{equi}$ and $\\zeta_{\\ast}^{ortho}$\nas follows\n\\begin{equation}\nG_{\\Psi}=G_{1}\\zeta_{\\ast}^{equi}+G_{2}\\zeta_{\\ast}^{ortho}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\nG_{1}=\\frac{13}{12}\\bigg[\\frac{1}{24}\\bigg(1-\\frac{1}{c_{s}^{2}}\\bigg)\\bigg](2+3\\beta)\n\\end{equation}\nand\n\\begin{equation}\nG_{2}=\\frac{14-13\\beta}{12}\\bigg[\\frac{1}{8}\\bigg(1-\\frac{1}{c_{s}^{2}}\\bigg)\\bigg]\\,.\n\\end{equation}\n\nNow, by using equations (50-55) we obtain the amplitude of\nnon-Gaussianity in the orthogonal and equilateral configurations\nrespectively as\n\\begin{equation}\nf_{NL}^{equi}=\\frac{130}{36\\sum_{i=1}^{3}k_{i}^{3}}\\bigg[\\frac{1}{24}\n\\bigg(\\frac{1}{1-c_{s}^{2}}\\bigg)\\bigg](2+3\\beta)\\zeta_{\\ast}^{equi}\\,,\n\\end{equation}\nand\n\\begin{equation}\nf_{NL}^{ortho}=\\frac{140-130\\beta}{36\\sum_{i=1}^{3}k_{i}^{3}}\\bigg[\\frac{1}{8}\n\\bigg(1-\\frac{1}{c_{s}^{2}}\\bigg)\\bigg]\\zeta_{\\ast}^{ortho}\\,.\n\\end{equation}\n\nThe equilateral and the orthogonal shape have a negative and a positive peak in $k_{1}=k_{2}=k_{3}$ limit, respectively [49].\nThus, we can rewrite the above equations in this limit as\n\\begin{equation}\nf_{NL}^{equi}=\\frac{325}{18}\\bigg[\\frac{1}{24}\\bigg(\\frac{1}{c_{s}^{2}}-1\\bigg)\\bigg](2+3\\beta)\\,,\n\\end{equation}\nand\n\\begin{equation}\nf_{NL}^{ortho}=\\frac{10}{9}\\bigg[\\frac{1}{8}\\bigg(1-\\frac{1}{c_{s}^{2}}\\bigg)\\bigg](\\frac{7}{6}+\\frac{65}{4}\\beta)\\,,\n\\end{equation}\nrespectively.\n\n\n\\section{Confronting with Observation}\n\nThe previous sections were devoted to the theoretical framework of\nthis extended model. In this section we compare our model with\nobservational data to find some observational constraints on the\nmodel parameter space. In this regard, we introduce a suitable\ncandidate for potential term in the action. We adopt\\footnote{Note that in general\n$\\lambda$ has dimension related to the Planck mass. This can be seen easily by considering the normalization of $\\phi$ via\n$V(\\phi)=\\frac{1}{n}\\lambda(\\frac{\\phi}{\\phi_{0}})^{n}$ which indicates that $\\lambda$ cannot be dimensionless in general. When we consider some numerical values for $\\lambda$ in our numerical analysis, these values are in \\emph{``appropriate units\"}.}\n$V(\\phi)=\\frac{1}{n}\\lambda\\phi^{n}$ which contains some interesting\ninflation models such as chaotic inflation. To be more specified, we\nconsider a quartic potential with $n=4$. Firstly we substitute this\npotential into equation (11) and then by adopting $\\epsilon=1$ we\nfind the inflaton field's value at the end of inflation. Then by\nsolving the integral (17), we find the inflaton field's value at the\nhorizon crossing in terms of number of e-folds, $N$. Then we\nsubstitute $\\phi_{hc}$ into Eqs. (33), (42), (58) and (59). The\nresulting relations are the basis of our numerical analysis on the\nparameter space of the model at hand. To proceed with numerical\nanalysis, we study the behavior of the tensor-to-scalar ratio versus\nthe scalar spectral index. In figure (1), we have plotted the\ntensor-to-scalar ratio versus the scalar spectral index for $N=60$\nin the background of Planck2015 data. The trajectory of result in\nthis extended non-minimal inflationary model lies well in the confidence levels\nof Planck2015 observational data for viable spectral index and $r$.\nThe amplitude of orthogonal configuration of non-Gaussianity versus\nthe amplitude of equilateral configuration is depicted in figure 2\nfor $N=60$. We see that this extended non-minimal model, in some\nranges of the parameter $\\lambda$, is consistent with observation.\nIf we restrict the spectral index to the observationally viable\ninterval $0.95<n_{s}<0.97$, then $\\lambda$ is constraint to be in\nthe interval $0.013<\\lambda<0.095$ in \\emph{appropriate units}. If we\nrestrict the equilateral configuration of non-Gaussianity to the\nobservationally viable condition $-147<f^{equi}_{NL}<143$, then we\nfind the constraint $\\lambda<0.1$ in our setup.\n\n\\vspace{1cm}\n\n\\begin{figure}[htp]\n\\begin{center}\\special{psfile=f.eps angle =0 hscale=40\nvscale=35 hoffset=60 voffset=-245} \\vspace{6cm}\n\\end{center}\n\\caption{\\small {Tensor-to-scalar ratio versus the scalar spectral\nindex in the background of Planck2015 TT,TE and EE+lowP data.}}\n\\end{figure}\n\n\\begin{figure}[htp]\n\\begin{center}\\special{psfile=ff.eps angle =0 hscale=40\nvscale=35 hoffset=40 voffset=-245} \\vspace{5.75cm}\n\\end{center}\n\\caption{\\small {The amplitude of the orthogonal configuration\nversus the amplitude of the equilateral configuration of non-Gaussianity in the background of Planck2015 TTT, EEE, TTE and\nEET data.}}\n\\end{figure}\n\n\\section{Summary and Conclusion}\n\nWe studied an extended model of single field inflation where the inflaton\nand its derivatives are coupled to the background geometry.\nBy focusing on the third order action and nonlinear perturbations we\nobtained observables of cosmological inflation, such as tensor-to-scalar\nratio and the amplitudes of non-Gaussianities in this extended setup. By confronting\nthe model's outcomes with observational data from Planck2015, we were able to constraint\nparameter space of the model. By adopting a quartic potential with\n$V(\\phi)=\\frac{1}{4}\\lambda\\phi^{4}$, restricting the model to realize\nobservationally viable spectral index (or tensor-to-scalar ratio)\nimposes the constraint on coupling $\\lambda$ as $0.013<\\lambda<0.095$.\nAlso restricting the amplitude of equilateral amplitude of non-Gaussianity\nto the observationally supported value of $-147<f^{equi}_{NL}<143$, results in the constraint\n$\\lambda<0.1$ in appropriate units. \\\\\n\n\n{\\bf Acknowledgement}\\\\\nThe work of K. Nozari has been supported financially by Research\nInstitute for Astronomy and Astrophysics of Maragha (RIAAM) under\nresearch project number 1\/5750-1.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe security research community has consistently been at the forefront of the fight against online abuse, from spam~\\cite{grier2010spam,levchenko2011click}, phishing~\\cite{ho2017detecting,zhang2021crawlphish}, to online fraud~\\cite{christin2010dissecting,park2016understanding,suarez2019automatically}. \nToday, one of the most pressing types of abuse is the spread of false information, especially on social networks.\nArguably, mitigating it faces some unique challenges.\nFirst, while some malicious actors spread misleading\/false claims to advance their goals (\\emph{``disinformation''}), false narratives are often believed by real users in good faith, who then re-share them on social media (\\emph{``misinformation''})~\\cite{krafft2020disinformation,saeed2021trollmagnifier,shao2018anatomy,singh2020first,starbird2019disinformation,vosoughi2018spread,zannettou2019let}. \nSecond, identifying what is true or false is challenging, hard to automate, and often depends on external fact-checkers.\nFinally, online platforms are often concerned about the effects of taking action on dis- and misinformation; \nfor example, limiting what is allowed to be said on a platform can raise concerns about censorship and reduce engagement (and thus profit)~\\cite{iglesias2021don,klonick2017new,myers2018censored}.\nNevertheless, the computer security research community is well poised to develop effective mitigation strategies for the problem of false online information, as highlighted by recent research in top tier venues in the field~\\cite{mirzatactics,recabarren2022strategies,saeed2021trollmagnifier}.\n\nAs part of their mitigation strategy, social networks have begun to adopt so-called \\emph{soft moderation}. \nRather than removing content or banning accounts, they notify other users about false narratives and provide additional context.  %\nWarning labels are attached to posts containing potentially false, misleading, or harmful claims, e.g., in the context of political disinformation~\\cite{zannettou2021won} or COVID-19~\\cite{ling2022learn,sharevski2021misinformation}.\nAn estimated 300,000 tweets were labeled under Twitter's Civic Integrity Policy~\\cite{twitter_midterms_civic_info} as misleading around the 2020 US Elections, accounting for 0.2\\% of all election-related tweets posted during that period.\nTwitter reported that warning labels applied to tweets in late 2021 resulted in noticeable decreases in replies, retweets, and likes (13\\%, 10\\%, and 15\\% reduction, respectively)~\\cite{twitter_midterms_civic_info}.\n\n\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Plots\/tweet1.png}\n         \\caption{Moderated}\n         \\label{fig:tweet_moderated}\n     \\end{subfigure}\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{Plots\/tweet2.png}\n         \\caption{Unmoderated}\n         \\label{fig:tweet_unmoderated1}\n     \\end{subfigure}\n\\vspace{0.15cm}\n        \\caption{Two example tweets discussing alleged voting fraud in the State of Michigan during the 2020 US Presidential Election. Twitter added a warning label to the first but not the second one.}\n        \\label{fig:tweet_moderation}\n\\end{figure}\n\n\\descr{Motivation.} \nEarly soft moderation results have been encouraging.\nPrior work shows that warning labels may prompt site users to debunk false claims~\\cite{zannettou2021won} or that they may reduce user interactions and extremism in comments~\\cite{papakyriakopoulos2022impact}.\nUnfortunately, details of Twitter's methodology are not publicly known.\n %\nWorse yet, recent work indicates that soft moderation may not be applied uniformly by Twitter, with ``benign'' tweets being erroneously labeled while misleading content goes unlabeled~\\cite{lawfare_2022,zannettou2021won}.\nFor example, in Figure~\\ref{fig:tweet_moderation}, we show two tweets discussing the same debunked narrative that fraudulent votes were cast for 10,000 deceased individuals in Michigan during the 2020 US Presidential election; one received the warning label, and one did not.\n\n\nThis example highlights the need for effective automated approaches to flag potentially misleading posts on social media.\nSuch approaches should cover as many misleading posts as possible while minimizing the number of unrelated posts that receive soft moderation labels to avoid \\emph{alert fatigue} effects, where users start ignoring warnings if they become too frequent~\\cite{ling2022learn}.\nRecent work found that using overly-broad rules when applying soft-moderation labels (e.g., the inclusion of a specific hashtag) flags a large amount of unrelated content: 37\\% of TikTok videos received COVID-19 related soft moderation were false positives~\\cite{ling2022learn}.\n\n\\descr{Research Objectives.} In this paper, we set out to develop an automated system to flag candidate Twitter posts for soft moderation.\nTo minimize the false positive problem observed in previous work, instead of adopting a \\emph{topic-specific} moderation approach (e.g., moderating any tweet containing a specific keyword), we follow a \\emph{claim-specific} methodology, where moderation labels directly address the statement or claim made in the content they are applied to~\\cite{morrow2022emerging}.\nThis fine-grained approach allows platforms to tailor warnings for specific claims, adjust severity cues on warning labels, and increase transparency by providing \\emph{explainable} labels on why a specific tweet was moderated. %\n\n\\descr{Technical Roadmap.} We introduce \\textsc{Lambretta}\\xspace, a system to assist in detecting candidate posts for soft moderation using a claim-specific approach.\n\\textsc{Lambretta}\\xspace takes as seed input a list of tweets moderated by Twitter and uses a Learning To Rank (LTR) based method~\\cite{cao2007learning} to extract the optimal set of keywords that characterize the posts related to the same tweets. \nE.g., from tweets in Figure~\\ref{fig:tweet_moderation}, \\textsc{Lambretta}\\xspace would extract {\\textit{``michigan,dead,ballot''}} as the best set of keywords to characterize discussion surrounding this claim in {\\em both} tweets.\n\\textsc{Lambretta}\\xspace then uses these keywords to find more candidates for soft moderation.\nWe instantiate \\textsc{Lambretta}\\xspace using the publicly available set of 2,224 tweets soft moderated by Twitter during the 2020 US Presidential Election, as curated by~\\cite{zannettou2021won}.\\footnote{https:\/\/github.com\/zsavvas\/Soft-Moderation-Interventions-Twitter}\n\\textsc{Lambretta}\\xspace retrieves the best set of keywords from 900 claims extracted from those tweets, identifying 2,042,173 additional candidates for soft moderation.\n\n\n\\descr{Main Contributions \\& Findings.} \nWe show that \\textsc{Lambretta}\\xspace performs better than alternative approaches from keyword extraction and semantic search when recommending candidate tweets for moderation.\nBy manually analyzing the tweets flagged by our system, we find that our results are accurate, with 3.93\\% false positives and 18.81\\% false negatives, with both metrics being substantially lower than those reported by other approaches.\nMoreover, \\textsc{Lambretta}\\xspace reduces the number of tweets a human moderator would have to review by over five times compared to the second-best algorithm.\nWe also find that \\textsc{Lambretta}\\xspace flags 20 times more candidates for moderation than those moderated by Twitter, suggesting that our approach can complement existing systems and improve the state of content moderation.\n\n\\textsc{Lambretta}\\xspace is platform-independent and can be bootstrapped from a single post corresponding to a narrative or an event deemed intervention-worthy by human moderators.\nWith the ability to scale to thousands of posts from a single misleading post, we show that event-driven keyword detection systems can be used as a foundation for large-scale soft moderation intervention systems.\nAt the same time, our results show that moderating content is a nuanced problem.\nFor example, posts discussing false narratives often attempt to debunk or ridicule them as satire.\nThus, we see \\textsc{Lambretta}\\xspace as a tool to \\emph{help} moderators identify potential candidates for soft moderation while leaving the final decision to humans.\nWe make \\textsc{Lambretta}\\xspace's source code and the labeled dataset used in this paper publicly available.\\footnote{https:\/\/github.com\/idramalab\/lambretta} \n\n\n\\section{Datasets}\n\\label{sec:datasets}\n\nOur experiments use three datasets.\nOne includes the tweets that were soft-moderated by Twitter and is used as ground truth; the other two allow \\textsc{Lambretta}\\xspace to find more tweets that are candidates for moderation in the wild.\n\n\\descr{Ground truth dataset.}\nOur first dataset, denoted as \\dtwo~, consists of the tweets that received soft moderation by Twitter released by Zannettou~\\cite{zannettou2021won}.\nIt contains 2,244 tweets posted by 853 users.\\footnote{Note that the dataset also contains 16,571 quoted tweets with warning labels; we exclude them since they are mostly used as additional commentary on the original tweet itself.}\n\n\\descr{Evaluation datasets.}\nTwo more datasets are used by \\textsc{Lambretta}\\xspace %\nto compile a set of tweets that are candidates for moderation.\nOne, denoted as \\dthree~, is released by Abilov et al.~\\cite{abilov2021voterfraud2020}; the other, denoted as \\dfour~, is obtained by querying Twitter's Academic Research full-archive search endpoint.\n\\dthree\\ contains 7.6M tweets related to the 2020 US Election.\nMore precisely, the authors of~\\cite{abilov2021voterfraud2020} retrieve tweets from the Streaming API using a set of hashtags related to the voter fraud narrative surrounding the 2020 US Election (e.g., \\#ballotfraud, \\#voterfraud, \\#electionfraud, \\#stopthesteal).\nThe dataset is available as a list of tweet IDs.\nAfter retrieving the complete tweet information from the Twitter API using these IDs we obtain 4,017,259 tweets.\nUnlike \\dthree~, \\dfour~ is built at runtime leveraging Twitter's Academic Research full-archive search endpoint.\nAt various stages, we query this endpoint using keyword-based search, for example, as part of deriving features for the underlying ranking model.\nThe two different datasets %\ncapture two different types of data availability scenarios, and allow us to test \\textsc{Lambretta}\\xspace on the entire Twitter archive using \\dfour~.\nNote that to match the time frame of our ground truth \\dtwo~, we only extract tweets for the period between November 1 and December 31, 2020. \nIn the rest of the paper, we refer to the evaluation datasets \\dthree~ and \\dfour~ as ``data store.'' \n\n\\descr{Ethical Considerations.} Analyzing large-scale social media data may raise ethical concerns.\nIn this paper, we only collect public Twitter data using the official API and do not interact with users.\nAs such, this work is not considered human subjects research by our institution.\nRegardless, we are mindful of the privacy of Twitter users and do not analyze any personally identifiable information (e.g., location data, account names, etc.).\nAlso, when presenting example tweets in this paper, we apply ``heavy disguise'' and paraphrase them to make user re-identification more difficult~\\cite{fiesler2018participant}.\n\n\\section{Overview of \\textsc{Lambretta}\\xspace}\n\n\\begin{figure*}\n \\includegraphics[width=0.999\\textwidth]{Plots\/Sysdiagram_v2_try2.pdf}\n \\caption{System overview of \\textsc{Lambretta}\\xspace.}\n \\label{fig:sysdiagram}\n \\end{figure*}\n\nThis section presents the different components of our system and how the end-to-end pipeline operates.\n\\textsc{Lambretta}\\xspace takes a seed list of tweets\ndiscussing a false claim and identifies similar candidate tweets for moderation.\nA high-level overview of our system is presented in Figure~\\ref{fig:sysdiagram}.\n\\textsc{Lambretta}\\xspace has three stages: 1)~extracting claim structures from tweets,\n2)~training a Learning to Rank (LTR) model from claim structures to extract the most relevant set of keywords for a given claim,\n3)~identifying candidate tweets from the data store similar to the seed tweets for platform moderators to make further decisions on.\n\n\n\n\\subsection{Claim structure extraction}\n\\label{sec:claimextraction}\n\n\n\n\\textsc{Lambretta}\\xspace extracts claims that should be moderated, starting from one or more tweets.\nThese extracted claims have similar structures to those manually curated by fact-checking organizations like Snopes\\footnote{https:\/\/www.snopes.com\/fact-check\/} or PolitiFact.\\footnote{https:\/\/www.politifact.com\/factchecks\/}\nAutomatically extracting claims from tweets is useful for our purposes for two reasons.\nFirst, it narrows down spans of tweets containing assertions and provides a better training dataset for \\textsc{Lambretta}\\xspace's subsequent phases.\nSecond, while we could directly rely on the claims published by fact checkers instead of building them ourselves, this is not ideal since fact checks tend to lag behind the appearance of false narratives by 10-20 hours~\\cite{shao2016hoaxy}, which impairs real-time moderation.\n\nTo identify claims in the seed list of tweets, we follow an approach inspired by linguistics.\nFirst, we identify propositions in tweets, defined as ``a declarative piece of text used to make a statement or assertion~\\citep{palau2009argumentation}.''\nHowever, not all linguistic propositions contain a claim that should be part of soft moderation efforts.\nTo account for this, \\textsc{Lambretta}\\xspace runs identified propositions through a claim classification component. \nThe output of this phase is a set of claims that can be passed to \\textsc{Lambretta}\\xspace's later stages.\n\n\n\n\n\\descr{Extracting propositions.}\nAs an operational definition, we consider a proposition as a standalone span of the tweet, potentially containing a claim~\\citep{palau2009argumentation}.\nNot all propositions are a claim, despite being a statement or assertion.\nWe will discuss the detailed specifications of what constitutes a claim later. \n\nIt is important to note that one tweet might contain multiple sentences, and a single sentence might contain multiple propositions.\nWe draw motivations from prior work on claim extraction on Twitter~\\citep{lim2016claimfinder}, where the authors formulate the problem by using propositions extracted through Open Information Extraction (Open IE).\nThe major advantage of using Open IE is the ability to extract an unbounded number of propositions in a sentence without requiring any domain knowledge of the underlying text, making it usable off-the-shelf.\nIn a similar problem setting~\\citep{lim2016claimfinder}, an Open IE system called Clausie~\\citep{del2013clausie} efficiently clustered tweets related to two real-world events into twelve different claims.\nIn our implementation of \\textsc{Lambretta}\\xspace, we use a tool developed by Stanovsky et al.~\\citep{stanovsky2018supervised}, which frames Open IE as a sequence tagging problem and uses a bi-LSTM transducer, available via Allen NLP~\\citep{gardner2018allennlp}.\nFor the rest of the paper, we refer to this Allen NLP Open IE extractor as the ``Proposition Extractor.''\nAs an example, using the Proposition Extractor on a tweet with the body ``Mail-in voting eases access barriers that might otherwise exclude voters physically unable to cast votes and has shown increased turnout across demographic groups.'' extracts the following sets of propositions: 1)~``Mail-in voting intended to ease access barriers'', 2)~``access barriers that might otherwise exclude voters physically unable to cast votes,'' and 3)~``Mail-in voting has shown increased turnout across demographic groups.''\n\n\\descr{Building ground truth annotations.}\nAs a natural first step, we need to validate the applicability of the Proposition Extractor in extracting potential \\emph{claim} structures.\nWe thus sample 200 random tweets from \\done~ and annotate spans that contain a claim.\nBefore annotating spans in a tweet containing claims, we need to formally define the scope of a claim. %\nThe lead author manually annotates our 200 tweet sample by adapting the scheme proposed in~\\citep{konstantinovskiy2021toward}, which identified six different categories of claims via crowdsourced annotations of sentences extracted from subtitles of four UK political TV shows.\nWe consider a tweet as containing a claim if the tweet:\n\\begin{compactitem}\n\\item discusses an event occurring in a particular US state related to the 2020 elections;\n\\item makes an assertion that can be further validated or debunked;\n\\item cites events and stories related to election manipulation;\n\\item references statistics to support or potentially deny an argument for election manipulation;\n\\item reports breaking news or a developing event related to the 2020 election; or\n\\item mentions actors often linked to conspiracy theories (Soros, Clinton, etc.) in relation to election events.\n\\end{compactitem}\nIn the end, we find 115 tweets with at least a single claim in our random sample of 200.\n\nTo annotate claim \\emph{spans}, which are normalized units of claims embedded in a tweet, we follow the method outlined by Wuhrl and Klinger, who perform claim detection on biomedical tweets related to topics like COVID-19, measles, and depression~\\cite{wuhrl2021claim}.\nThey present a guideline to human annotators on the task of annotating claim spans within a tweet.\nFrom the guidelines, claim spans are the central statement of an argumentative structure, phrased with an argumentative intent while expressing a clear stance (support or oppose) about the argument. \n\nThis results in a ground truth dataset of 144 claim spans for the 115 tweets that contain a claim.\nTo exemplify the type of claims we identified, we present the following three tweets, with the claim spans in bold.\n\\smallskip\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\\begin{itemize}[leftmargin=*]\n\\item PA election night narrow lead was erased by \\textbf{hundreds of thousands of mail-in ballots counted without Republican observer.} I wonder why were Republican observers excluded from the counting?\n\\item Sidney Powell ( part of Giuliani's team ) is explaining that the \\textbf{ plan to steal the election from Trump was Hugo Chavez}. However, Chavez died in March of 2013. \n\\item A September 2020 study released by Judicial Watch revealed that \\textbf{U.S. counties had 1.8M more registered voters than eligible} voting-age citizens. That means that in these counties the \\textbf{registration rate exceeded 100\\% of eligible voters.}\n\\end{itemize}\n\\end{quote}\n\\end{mdframed}\n\n\\noindent Some examples of tweets that do not contain a claim are:\n\\smallskip\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\\begin{itemize}[leftmargin=*]\n\n\\item ``Republicans need to rise together and stand against the fraud. Otherwise, our party might never win election again.'\n\n\\item ``Dems thought they would get away with cheating. But they got caught. Funny enough how quiet they are? I would fight really hard to clear my name if I was accused of cheating. Silence speaks louder than words.''\n\n\\end{itemize}\n\\end{quote}\n\\end{mdframed}\n\nWe test the Proposition Extractor's ability to recognize all potential claim structures present in a tweet as propositions.\nFirst, we split the 200 tweets using a sentence tokenizer and query each sentence against the Proposition Extractor.\nWe then compare the set of propositions returned by Proposition Extractor against the annotated ground truth spans, looking for missed ground truth spans.\nThese types of errors are called \\emph{uninformative extractions}~\\citep{etzioni2011open}, where Open IE systems can omit critical information from the sentences in the propositions they identify.\nThe Proposition Extractor extracts 264 propositions from 115 tweets containing claims, and 159 propositions from 85 tweets without claims.\nOf the 144 claim spans in our ground truth, the Proposition Extractor\n misses only four instances.\n\n\n\\descr{Filtering propositions that are claims.}\nThus far, we have 264 propositions extracted from 115 tweets, and only 144 of these propositions are annotated as claims.\nAs discussed above, a tweet containing a claim span can have multiple other propositions that are not a claim, and we need to filter out these too.\nAs a final step, \\textsc{Lambretta}\\xspace needs to distinguish propositions that are claims from those that are not. %\nWe address this by further scoring the propositions extracted by the Proposition Extractor against a state-of-the-art claim classifier called ClaimSpotter~\\citep{claimspotter}, which leverages a gradient-based adversarial training on transformer networks to identify claims and is trained on manually labeled sentences from historical US presidential debates~\\citep{meng2020gradient}.\nFor any input text, ClaimSpotter returns a score between 0 and 1, representing how ``check-worthy'' a claim is.\nThis is not how likely something is to be \\emph{true}, but instead if it makes an objective claim (1) vs. espousing a subjective opinion (0).\nFor instance, a relatively subjective proposition such as \\textit{``And that, ladies and gentlemen, is how you steal an election''} returns a score of \\textbf{0.285}, whereas a relatively objective proposition such as \\textit{``In the past 20 years there have been approx 250 million votes cast and around 1200 proven cases of voter fraud\"} scores \\textbf{0.85}. %\n\n\n\nWhile the ClaimSpotter API is already trained and publicly available, we need to devise an appropriate threshold that gives us accurate results for classification considering our problem setting. \nTo this end, we obtain the Claim Spotter scores for all 144 claim spans annotated as ground truth and set the target class for these scores as 1.\nSimilarly, we obtain the ClaimSpotter scores for the 159 propositions extracted from the 85 tweets without claims and the 120 propositions extracted from the 115 tweets with claims.\nWe first perform a 75--25 train-test split to identify the optimal threshold of 0.490 based on the Receiver Operating Characteristic (ROC) curve.\nWe then perform a 5-fold cross validatio- using this threshold, obtaining an average Precision of 0.881 and an average Recall of 0.895.\nThe system misclassifies claim propositions as non-claim 3.46\\% of the time, while non-claim propositions are classified as claims with a rate of 6.58\\%.\nWe deem these results acceptable and thus adopt 0.490 as our threshold for \\textsc{Lambretta}\\xspace's claim extraction component.\n\n\n\\descr{Extracting claims from the entire dataset.}\nFrom the remaining 2,044 tweets in \\done~ we extract 4,471 propositions, 756 of which are identified as claims based on our decision threshold. \nIn addition to the 144 claims identified during the decision threshold calibration phase, this brings us to 900 total claims extracted from 2,244 tweets in \\done~.\nSome examples of tweets that contain claims include:\n\n\\smallskip\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\\begin{itemize}[leftmargin=*]\n\\item ``The voting machine in Green Bay ran out of ink, which delayed the final results. An election official went back to City Hall to get more ink.''\n\\item ``The Dominion Voting Systems machines switched over 6,000 ballots for Biden in Michigan. \nThey are used throughout the United States on a wide scale in sixteen states. Virtually all of these states are blue states. \nThe system needs to be audited to make sure the right person is elected.''\n\\end{itemize} \n\\end{quote}\n\n\\end{mdframed}\n\nThese extracted claims are topically diverse, covering different aspects of the election process as multiple phases of the election event developed.\nWe find many claims discussing mail-in ballots and ballot harvesting posed as a threat (e.g., \\textit{``poll workers stuffing ballot box with mail-in ballots for Democrats''}) %\nand discussion of election fraud caused by voting machines (e.g., \\textit{``Dominion machine flipped votes from Trump to Biden''}). %\nAdditionally, we find many claims misinterpreting bipartisan events of ballot counting as partisan vote counting (e.g., \\textit{``spike for Biden votes as suitcases of ballots started to be scanned''}) %\nand presence of dead, fake, and ineligible voters (e.g., \\textit{``86,845 mail-in ballots lost in Arizona''}). \nFinally, many claims in our dataset discuss the developing situation of the vote totals as statistically suspicious or rigged (e.g., \\textit{``mysterious spike of votes in Pennsylvania had 600,000 votes for Biden and only 3,200 for Trump''}). %\nThis diversity of topics found in the claims will be an important evaluation setup for \\textsc{Lambretta}\\xspace and its ability to generalize over various topics about an event.  \n\n\n\\subsection{Identification of keywords from claims}\n\\label{sec:keyword}\n\nAfter summarizing tweets into claims, \\textsc{Lambretta}\\xspace extracts the most representative set of keywords from claims.\nWe need to further extract keywords from our claims because querying Twitter search with the full claim text will give us a very limited set of tweets discussing the claim in a narrow way.\nThe following are three tweets discussing the same claim related to voting machines:\n\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote} \n\\small\n\\textbf{Claim:} ``smartmatic foreign software voting machine design rig election socialist venezuela.''\n\n\\textbf{Example Tweets discussing the same claim:}\n\\begin{itemize}[leftmargin=*]\n\n\\item ``Dominion developed by Sequoia-Smartmatic, a company previously owned by Chavez. \nUS Intel said Smartmatic was used to rig the 2004 election in Venezuela.\nThe Chicago election commission concluded the Smartmatic software delivers the results desired to the election officials.''\n\n\\item ``China, Cuba  Venezuela, all these communist countries and Antifa interfered in U.S elections. They used Smartmatic voting software, created by Hugo Chavez. Guess who helped them :  George Soros and the Clinton Foundation, rigging U.S election for Joe Biden.''\n\n\\item ``The Dominion Smartmatic machine was used in Venezuela by Cuban intelligence in an attack on its democracy by Chavez and Castro.\nWarfare through Voting systems has replaced guerrilla  attacks against nations.''\n\\end{itemize}\n\\end{quote}\n\\end{mdframed}\nThe first tweet has a very formal tone to it, is detailed, and uses some sources (US Intel, Chicago Election Commission) to establish context.\nOn the other hand, the second tweet also talks about Smartmatic and its connection with Venezuela, but has totally different actors surrounding the discussion (Antifa, George Soros, and the Clinton Foundation).\nFinally, the third tweet is more of a socio-political commentary espousing the misleading claim.\nThese examples illustrate that it is important to extract the most representative set of keywords for any claim to understand the wide variety of ways it can be discussed.\n\n\\descr{Problem with existing Keyword Identification techniques.}\nSeveral approaches to extract important keywords from social network posts have been proposed~\\cite{grootendorst2020keybert,campos2018yake,mihalcea2004textrank,lee2009keyword}.\nWe initially experimented with various existing keyword detection methods~\\cite{grootendorst2020keybert,campos2018yake} to extract the best set of keywords from the claims extracted in \\textsc{Lambretta}\\xspace's previous step.\nOur analysis identified three main issues with previous approaches.\nWe include three examples below obtained while testing with Yake~\\cite{campos2018yake}:\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\\begin{itemize}[leftmargin=*]\n\\item Problem Type 1: Missing key entities\n\n\\textbf{Claim: Michigan Governor Whitmer send health dept into Detroit TFC Center to evict GOP poll-watchers but not Dem pollwatchers}\n\nAutomatically detected keywords using YAKE:  \\textit{Detroit, TFC ,pollwatcher}\n\n\\item Problem Type 2: High Recall, Low Precision\n\n\\textbf{Claim: Signature verification system in Clark County have 89\\% failure rates for catching poor signature matches}\n\nAutomatically detected keywords using YAKE:  \\textit{signature, match}\n\n\\item Problem Type 3: Low Recall, High Precision\n\n\\textbf{Claim: Tens of thousands of votes illegally received after 8 P.M. on Tuesday, Election Day}\n\nAutomatically detected keywords using YAKE:  \\textit{ten,thousand,vote,8}\n\n\\end{itemize}\n\\end{quote}\n\\end{mdframed}\n\nIn the first example, the keyword extractor misses the key entities of the claim being discussed, \\textit{Whitmer} and \\textit{GOP}, and may produce many unrelated results.\nIn the second example, the extracted keywords do capture the key entities, \\textit{signature} and \\textit{match}, but these keywords are generic and may result in many hits that are part of the larger narrative around signature matching on mail-in ballots rather than the specific claim of the failure rate of signature matches in Clark County.\nFinally, the third example extracts overly specific keywords that will miss relevant content.\n\nThe issue with existing approaches is that they identify the most important keywords by only looking at the claim body (e.g., the tweet already moderated by Twitter), without taking into account the context in which the claim is discussed.\nTo overcome this issue, \\textsc{Lambretta}\\xspace formulates the problem as a Learning to Rank (LTR) task~\\cite{cao2007learning}, where the selection of optimal keywords from the claim is driven by the document store (i.e.,  \\dthree~, \\dfour~, etc.) containing examples of social media discussion of the claim in question.\nWe note that we use both \\dthree~ and \\dfour~ in our experimental setup to cross-evaluate the LTR model trained on one document store and tested on an unseen document store. \nWe provide more details on the LTR task and our keyword detection experiment below.\n\n\\descr{Learning to Rank.}\nLearning to Rank (LTR) is a task aiming to learn a function to rank the effectiveness of query terms in retrieving relevant documents from a document store.\nLTR based methods have been used in many information retrieval oriented applications~\\cite{liu2009learning,surdeanu2008learning,shaw2013learning,leaman2013dnorm,karatzoglou2013learning}.\nWe refer readers to~\\cite{cao2007learning} for a detailed review of the LTR task and different approaches to it.\n\nTo train our LTR model, we first annotate ground truth keywords for a fraction of the 900 misleading claims extracted from \\dtwo~, manually labeling relevant tweets resulting from querying the keywords.\nWe then develop features to train the LTR model and evaluate it on unseen tweets from \\dthree~ and \\dfour~.\nWe describe these steps in detail below.\n\n\\descr{Building ground truth annotations.}\nWe begin by randomly selecting 125 misleading claims from our filtered set of claims (see Section~\\ref{sec:claimextraction}) to train our LTR model.\nWe refer to these claims as \\initialtrainclaims~.\nFor each claim, we annotate the ground truth to be the set of keywords made up of terms from the misleading claims which produce the most related set of results when queried against our data store, optimizing for both relevance and size of results.\n\nWe start by initializing \\basekeyword~, which are two words consisting of the subject and the object of a misleading claim.\nWe query the two different Twitter data stores \\dthree~ and \\dfour~  with the \\basekeyword~ and retrieve a set of results.\nAs expected, the \\basekeyword~ are usually pretty broad and return many irrelevant false positives.\nWe fine tune the query by checking a random sample of 20 posts from query results and then adjust the \\basekeyword~ by either adding new words from the query or removing words that are causing the false positives.\nWe repeat the process until we find the most relevant set of keywords for each misleading claim.\nFinally, for each claim, the best set of keywords is tagged as a positive instance with the remaining keywords tagged as negative instances of relevance.\n\nThis ground truth is then used by \\textsc{Lambretta}\\xspace to learn the ranking function that automatically identifies the best set of keywords for any given claim.\nThis enables \\textsc{Lambretta}\\xspace to automatically extract keywords from the body of a claim without any further external context of human intervention.\n\n\n\n\\descr{Data pre-processing.}\nBefore further analysis, \\textsc{Lambretta}\\xspace removes stopwords from the tweet claims while doing basic pre-processing, e.g., lowercasing text and removing punctuation marks.\nWe keep numbers since they can often be an integral part of extracted claims.\nWe split the pre-processed misleading claims into n-grams of length two, three, four, and five, which are our potential set of query terms.\n\n\\descr{Feature Engineering.}\nNext, we use the potential query terms returned by the previous step to extract our learning to rank model features.\nTo this end, we query the two data stores \\dthree~ and \\dfour~, retrieving all the posts matching the query terms. %\nLTR requires us to generate a dataset consisting of a query set, relevance information, and feature values to learn the ranking.\nFeature values can be a few or numerous.\nApplications of LTR have used features like document TF-IDF, BM25~\\citep{ramos2003using} scores, document length, number of matching query terms, and number of query terms in important sections of a document, e.g., the title of a Web page~\\cite{macdonald2012usefulness,bai2010learning,surdeanu2011learning}.\nThe most widely used benchmark to build models based on LTR is the LETOR dataset~\\cite{qin2010letor}, which contains query sets, learning features, and labeled rankings related to the 25 million page \\textit{GOV2} Web page collection~\\citep{qin2013introducing}. %\nUnfortunately, we cannot directly use the LETOR dataset since our data store is composed of posts from Twitter, which are fundamentally different from Web pages.\nInstead, we take inspiration from the LETOR dataset and develop six features that we use in our LTR experiments.\nIn the following, we briefly discuss these features.\n\n\nWe use the term \\spanningsubset~ to describe the earliest 20\\%, most recent 20\\%, and 10\\% of the middle-aged results returned from the query, based on the timestamp attached to their tweets.\nFrom this, we derive six features:\n\\begin{enumerate}\n\\item Total number of hits (matching tweets) produced.\n\\item Mean and median pairwise similarity score between the entries of the \\spanningsubset~.\n\\item Mean and median similarity score between the entries in \\spanningsubset~ and the claim.\n\\item Mean and median similarity score between the query and the claim.\n\\item Mean and median value of the TextRank scores~\\cite{mihalcea2004textrank} of the query terms.\n\\item Mean and median score of the Term Frequency-Inverse Document Frequency scores~\\cite{ramos2003using} of the query terms.\n\\end{enumerate}\n\nWe need to extract \\spanningsubset~ from the set of returned results as the number of results retrieved by the candidate keywords from the candidate query set might grow large, specially in cases of generic set of keywords in a query.\nPerforming a pairwise semantic similarity comparison on this large set is not computationally feasible.\nThe intuition behind \\spanningsubset~ is to sample tweets during the different period of a discussion (from the onset to the current phase) along a timeline, compared to randomly sampling tweets.\nThe similarity score between the set of results, and between the query and the results are calculated using the cosine similarity of the sentence embeddings encoded using a pre-trained \\textit{all-mpnet-base-v2} model proposed in~\\citep{song2020mpnet}.\nThe \\textit{all-mpnet-base-v2} model produces sentence embeddings with 768 dimensions, and had the best average performance on encoding sentences over 14 diverse tasks from different domains.\nNote that the features used in training the LTR model are not domain-dependent (i.e., election misinformation in this case), and are designed to entirely capture the semantic relatedness between query results and subsequent claims.\nThese type of features will be useful for applying \\textsc{Lambretta}\\xspace in other contexts for content moderation in different topics.\n\n\\descr{Training the LTR model.}\nWe use the RankLib project, part of the Lemur Toolkit~\\cite{lemurtoolkit} which includes a variety of LTR algorithms.\nWhile there are LTR algorithms that use complex neural architectures like Deep Learning~\\cite{pasumarthi2019tf,chen2020deeprank}, we cannot directly use them since they are designed to work on large scale datasets like LETOR.\nInstead, our LTR models are powered by features utilizing state of the art neural semantic models (\\textit{all-mpnet-base-v2}) to capture the interaction between query terms and result set.\n\nWe use all eight algorithms implemented by Lemur (MART, RankNet, RankBoost, AdaRank, Coordinate Ascent, LambdaMART, ListNet, and Random Forests) for our experiments.\nTo evaluate our model, we use a rank-based evaluation metric commonly used in information retrieval settings: Mean Average Precision (MAP).\nIn our case, our problem setting is a binary judgement where a keyword is either relevant to the misleading claim or not.\n\nAs with all keyword extraction problems, our dataset has many irrelevant combinations (majority class) of wrong keywords compared to one or two correct combinations (minority class) of optimal keywords we want to extract.\nThis imbalance causes the learning algorithm to perform very poorly.\nTo address this problem, prior work on LTR used BM25 as a pre-ranker to retrieve a small set of highly ranked documents from the entire document index before applying the LTR algorithm on the small subset of highly relevant documents~\\citep{liu2009learning}.\nWe employ a similar pre-ranking step to filter out the set of keywords that retrieve irrelevant results for the claim by using a heuristic based on semantic query similarity.\nThe idea behind the heuristic is that candidate keywords most likely to be optimal return tweets that are more semantically similar than unrelated ones.\nThus, for each claim, its corresponding set of candidate keywords, and the retrieved results using these keywords as queries, we construct a subset called \\textbf{FilteredQuerySet}, which is a ranked list of the top \\textit{k} results retrieved by all candidate keywords, sorted by the cosine similarity with the claim under question.\nThe results returned by an irrelevant query have lower semantic similarity with the claims, thus failing to appear in the ranked set \\textbf{FilteredQuerySet}.\nWe experiment on different values of \\textit{k} and find that setting \\textit{k} to 20 includes all of the ground truth queries for training, validation, and test claims, while reducing the size of irrelevant candidate query set by 30 times.\n\nTo test our LTR model, we perform a 5-fold cross-validation on our ground truth.\nWe observed that Random Forest with LambdaMART~\\cite{burges2010ranknet} as the bagging ranker produced the best results among the eight different ranking algorithms.\nWe further increase performance by performing a grid search over hyper-parameters (number of leaves, number of bags, number of trees, and minimum leaf support).\nThe 5-fold cross-validation on our ground truth using the tuned Random Forest model achieves a MAP of 0.768.\nAs we will show later in Section~\\ref{section:comparison_model}, this convincingly\noutperforms other state-of-the-art keyword extraction approaches.\nWe further refer to this LTR model as \\initialtrainmodel~.\n\n\\descr{Validating the LTR model.}\nThe previous experiment showed that our LTR approach can produce an accurate model on our training set.\nWe now want to understand whether our model generalizes and can effectively identify posts related to claims that are not in the training set.\nTo this end, we design and conduct two experiments.\nIn the first experiment, we randomly select 75 additional misleading claims (extracted as per Section~\\ref{sec:claimextraction}) not part of the ground truth set, referring to them as \\validationclaims~.\nThe output keywords for these new claims can be inferred from the previously trained model, but they still require ground truth annotation to evaluate results.\nWe thus manually label the \\validationclaims~ via the same iterative method that we used for the \\initialtrainclaims~.\n\nNext, we generate the potential query terms set for the \\validationclaims~, following the same steps for \\initialtrainclaims~.\nWe then query \\dthree~ and generate the feature values for candidate keywords for each claim.\nFinally, we perform inference on these new claims to see if our trained model can identify the best set of keywords.\nOur model achieves a MAP of 0.781, indicating that our approach effectively identifies keywords and retrieves data for previously unseen claims.\nAfter the validation step, we now have ground truth of 75 additional misleading claims from \\validationclaims~, which we use to expand our overall training set to a total of 200 claims, which we call \\expandedclaims~.\nWe train a new LTR model on the \\expandedclaims~, which we refer to as \\expandedmodel~.\nAt this point, we have 700 claims remaining which were not manually annotated and are missing corresponding keywords.\nWe later use the \\expandedmodel~ to extract the keywords for these claims.\n\nIn the second experiment, we aim to verify that the learned model is not biased towards the data store it was trained on (i.e., \\dthree~) and can be applied to an unseen data store (\\dfour~).\nIn the previous experiments we generated the features from results queried on \\dthree~ for validation.\nInstead, in this experiment we build the features from results queried on \\dfour~.\nWe re-use the \\expandedclaims~ as our claim list, along with the corresponding ground truth we had collected for the previous training\/validation experiments.\nFollowing the same steps as for the \\initialtrainclaims~ and \\validationclaims~, we generate the feature values for each of the candidate keywords for the claim by querying the datastore of \\dfour~.\nWe use the previously trained \\expandedmodel~ model for inference on the \\expandedclaims~ and achieve a MAP of 0.767, showing that our LTR model is not dependent on the datastore it was trained on.\n\n\n\\subsection{Data store retrieval}\nAfter training our LTR model and validating its performance on ranking experiments with other methods, we apply the ranking model to the remaining 700 misleading claims.\nThe output is 499 unique sets of keywords. \nThis number is lower than the total number of claims (900) since the keywords extracted from two different claims can be the same.\nWe use these 499 sets of keywords to query the two data stores \\dthree~ and \\dfour~.\nWe require candidates to match only if all keywords in a query are present in a tweet, regardless of the order of the tokens.\nWe also make sure to exclude retweets and quoted tweets when searching for the relevant tweets.\n\nAfter searching for the keywords, we obtain 2,042,173 tweets from \\dfour~ and 101,353 tweets from \\dthree~.\nThe average number of tweets per misleading claim from \\dfour~ is 5,988 and for \\dthree~ it is 203.11.\nWe use the 101,353 tweets from \\dthree~ to further evaluate \\textsc{Lambretta}\\xspace throughout the rest of the paper, as checking the tweets flagged for moderation from \\dfour~ is not feasible due to Twitter API limitations.\n\n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\nTo evaluate \\textsc{Lambretta}\\xspace, we first compare the quality of the candidate tweets extracted by our system to those recommended by other state-of-the-art approaches. %\nNext, we manually analyze a subset of the tweets flagged by \\textsc{Lambretta}\\xspace to assess its false positives and false negatives.\nWe then check whether the tweets flagged by \\textsc{Lambretta}\\xspace received soft moderation from Twitter, in the context of the 2020 US Election voter fraud allegations, finding that only a small fraction of them did.\nFinally, given the disparity between the soft moderation candidates flagged by \\textsc{Lambretta}\\xspace and those that received labels by Twitter, we use our dataset to understand if Twitter's moderation is driven by specific characteristics of the tweets or of the users posting them.\n\n\n\\subsection{Comparison with other keyword extraction and information retrieval based methods}\n\\label{section:comparison_model}\n\nIn Section~\\ref{sec:keyword}, we showed some examples of why existing keyword extraction methods might be unsuitable for our task, motivating us to develop the LTR component of \\textsc{Lambretta}\\xspace.\nWe now provide a rigorous quantitative analysis of this fact, by comparing the LTR model used by \\textsc{Lambretta}\\xspace with three other keyword extraction algorithms: YAKE~\\cite{campos2018yake}, KeyBERT~\\cite{grootendorst2020keybert}, and RAKE~\\cite{rose2010automatic}. \nAs an alternative to keyword extraction algorithms, another possible approach to finding similar tweets given a source one is leveraging semantic search techniques~\\citep{guha2003semantic}.\nWe also evaluate state-of-the-art methods in this space against \\textsc{Lambretta}\\xspace's LTR model; more precisely semantic search using Sentence Transformers~\\citep{reimers-2019-sentence-bert} and BM25~\\citep{bm25}, which can be used to get tweets matching a query tweet by using a ranking function.\n\nTo establish ground truth, we sample 60 random claims from the set of 200 \\expandedclaims~. \nFor these sets of claims and the ground-truth keywords, we manually verify that each tweet returned by the keywords does discuss the claims in question and filter out any irrelevant ones.\nThis yields a set of 10,776 tweets associated with the 60 claims. %\nSentence Transformers and BM25 also require tuning a similarity threshold for matching and ranking tweets; for these experiments, we retrieve tweets using different thresholds, ranging from 0.3 to 0.9, and select the threshold for each method that achieves the highest F1 score based on our ground truth. \n\n\\begin{figure}[t]\n     \\centering\n         \\includegraphics[width=0.99\\columnwidth]{Plots\/keyword_comparison_all.pdf}\n\t \\caption{Performance of the LTR model vs. other keyword identification and semantic search methods.}\n\t %\n         \\label{fig:keywordvalidation}\n\\end{figure}\n\n\nFigure~\\ref{fig:keywordvalidation} reports the F1 score for all methods, including LTR, when extracting similar tweets to the 60 claims curated from \\expandedclaims~.\n\\textsc{Lambretta}\\xspace's LTR is the best performing model, with over 60\\% of the claims having an F1 score of 0.8 or higher.\nThe second best performing algorithm is YAKE, with less than 40\\% of claims having an F1 score of at least 0.8.\nBM25 and semantic search using Sentence embeddings perform poorly, as does KeyBERT.\nThese results confirm that the LTR model, the keyword identification component of \\textsc{Lambretta}\\xspace, is effective at identifying similar tweets with better Precision and Recall than other keyword extraction and semantic search based methods. \n\n\\descr{Comparison of the reduction of effort offered by different methods to content moderators.}\nThe goal of \\textsc{Lambretta}\\xspace is to provide a set of social network messages to content moderators, allowing them to make informed decisions and to keep the platform safe.\nTo be useful and avoid overwhelming content moderators, this system should reduce the number of candidates as much as possible, while at the same time maximizing Precision and Recall.\nFrom our comparison experiments described above, we find that \\textsc{Lambretta}\\xspace reduces the pool of tweets which should be checked for moderation compared to all other systems.\n\\textsc{Lambretta}\\xspace retrieves 5.33 times less tweets related to a moderated claim than the second best performing system, YAKE.\nThis shows that \\textsc{Lambretta}\\xspace is better suited than alternative approaches for the task of aiding moderation of misleading information on social media.\n\n \n\n\n\n\\subsection{Validation of \\textsc{Lambretta}\\xspace}\n\\label{sec:validation}\n\nIn this section, we first perform a manual analysis on the results recommended by \\textsc{Lambretta}\\xspace, assessing its False Positives and False Negatives.\nWe then check if the recommendations made by \\textsc{Lambretta}\\xspace were also soft moderated by Twitter, finding that only a small fraction of tweets flagged by \\textsc{Lambretta}\\xspace were intervened by the platform.\n\n\\descr{Manual validation.}\nTo understand the quality of the recommendations made by our approach, we perform a manual examination of the tweets recommended by \\textsc{Lambretta}\\xspace, aiming to check if they should indeed have been moderated.\nThe first author of this paper samples 1,500 tweets among the candidates flagged by \\textsc{Lambretta}\\xspace and analyzes them qualitatively, identifying seven categories that they can fall under: 1)~Amplyfing tweets, 2)~Reporting Tweets, 3)~Counter Tweets, 4) Satire Tweets, 5)~Discussion Tweets, 6)~Inquiry Tweets, 7)~Irrelevant Tweets.\n %\nThe process goes through multiple iterations of coding the sample tweets, grouping them into different categories. \nThe annotator then consolidates overlapping themes and categories, converging to seven which we discuss below.\nNote that a single tweet can fall under multiple categories, e.g., they can amplify a misleading claim and prompt a discussion at the same time.\nThe definition of each category, alongside a representative example tweet of the category is presented below:\n\n\\descrit{1) Amplifying Tweet:} it positively reinforces the misleading claim and aims to further spreading the message.\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\n``Terry Mathis (born 1900) apparently voted via absentee ballot in Wayne County: Michigan.  It doesn't stop here. This person applied for an absentee ballot on December 2: the ballot was then sent out AND returned in the same day.<URL> <URL>''\n\\end{quote}\n\\end{mdframed}\n\n\n\\descrit{2) News Reporting:} it reports the headline of a misleading news article or another tweet, without any additional commentary and text from the tweet's author.\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``BREAKING: Unofficial: Trump trailing Biden by only 4,202! There is a ballot count upload glitch in Arizona. Reports saying over 6,000 False Biden Votes Discovered <URL>.''\n\n\n\\end{quote}\n\\end{mdframed}\n\n\\descrit{3) Counter Claim:} it attempts to question and\/or debunk the misleading information.\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``Misleading claims that Trump ballots in Arizona were thrown out because Sharpie pens were provided to voters are untrue. A ballot that cannot be read by the machine would be re-examined by hand and not invalidated if it was marked with a Sharpie.\\#Election2020 <URL>.''\n\n\\end{quote}\n\\end{mdframed}\n\n\n\\descrit{4) Satire:} it discusses the false claim in a satirical way.\n\n\\smallskip\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``@\\textless USER\\textgreater He was allegedly slain by Soros, who then had Chavez's personal army of false voters cram him inside a Dominion voting machine before loading him into an RV with Hunter Biden's second laptop and Hillary's server.''\n\n\\end{quote}\n\\end{mdframed}\n\\descrit{5) Discussion:} it prompts discussion of the details of the misleading claim by adding commentary.\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``@\\textless USER\\textgreater What happened to all the votes cast for Trump that were destroyed? How are those tallied? Detroit-based Democratic Party activist a local: boasts On FB: I threw out every Trump ballot I saw while working for Wayne County, Michigan. They number in the tens of thousands, as did all of my coworkers.''\n\n\\end{quote}\n\\end{mdframed}\n\\descrit{6) Inquiry:} it inquires about the details of events related to the misleading claim and does not attempt to either support or deny the claim under question. \n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``Has anyone got a compelling justification for this? In accordance with a tweet I saw from @\\textless USER\\textgreater: A \"James Bradley\" born in 1900 has recently been entered into the Michigan Voter Information Center. James apparently submitted an absentee ballot on October 25. For a 120-year-old, not bad! <URL>''\n\\end{quote}\n\\end{mdframed}\n\n\\descrit{7) Irrelevant:} it is irrelevant to the misleading claim. These are considered false positives.\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n``\\#LOSANGELES: Our truck will be at the @HollywoodBowl voting location till 7pm. Use your voting rights and reward yourself with some.''\n\\end{quote}\n\\end{mdframed}\n\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\t\\setlength{\\tabcolsep}{4pt}\n\\begin{tabular}{lrr}\n\\toprule\n\\textbf{Category} & \\textbf{Candidates} & \\textbf{Moderated (\\%)} \\\\ \\midrule\nAmplifying        & 1,198               & 241 (20.11\\%)                      \\\\\nReporting         & 922                & 222 (24.07\\%)                      \\\\\nCounter           & 122                & 4 (3.27\\%)                         \\\\\nSatire            & 15                 & 0 (0.00\\%)                         \\\\\nDiscussion        & 646                & 83 (12.84\\%)                       \\\\\nInquiry           & 84                 & 22 (26.19\\%)                       \\\\\nIrrelevant        & 59                 & 1 (1.69\\%)                         \\\\ \\bottomrule\n\\end{tabular}\n\t%\n\\caption{Categories of candidate tweets and number\/percentage receiving soft moderation by Twitter.}\n\\label{tab:candidate_category}\n\\end{table}\n\n\nTable~\\ref{tab:candidate_category}, reports the number of candidate tweets in each category. %\nThe vast majority falls in the amplification category with 1,198 out of 1,500 tweets (79.86\\%), followed by tweets reporting about the false claim with 922 tweets (61.46\\%).\n43\\% of the tweets add further discussion to the misleading claim under question rather than simply sharing the headline of a news article, and 8\\% of them try to debunk it.\nFinally, 59 (3.93\\%) of the tweets flagged by \\textsc{Lambretta}\\xspace are irrelevant to the claim under study and can therefore be considered false positives. \nAs mentioned, the goal of \\textsc{Lambretta}\\xspace is to flag tweets that are related to a claim that the platform wants to moderate, but human moderators should still make the final decision about applying labels to the candidates flagged by our system. \nWe further discuss the implications of running \\textsc{Lambretta}\\xspace in the wild in Section~\\ref{sec:discussion}.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\columnwidth]{Plots\/coverage.pdf}\n\\caption{Moderation coverage of misleading tweets flagged by \\textsc{Lambretta}\\xspace per claim.}\n\\label{fig:coverage}\n\\end{figure}\n\n\\descr{False Negatives.} \nTo evaluate the False Negatives of \\textsc{Lambretta}\\xspace, we first evaluate the false negative of each of its two phases separately using \\done~ for the first phase, and \\dthree~ for the second phase.\nIn the claim structure extraction module, the Proposition Extractor component fails to extract 2.77\\% of the propositions that are claim span.\nAfter the propositions are extracted, \\textsc{Lambretta}\\xspace misclassifies 3.46\\% of the propositions that contain a claim, implying the missed claim structure would not be processed further in the second phase.\nIn the second phase, we quantify the proportion of tweets missed by the keywords identified through the LTR component of \\textsc{Lambretta}\\xspace. \nThe keywords produced by LTR identify 8,748 of the 10,776 tweets in the ground truth; this yields an 18.81\\% false negative rate from \\textsc{Lambretta}\\xspace's keyword extraction phase.\nThis is much lower than the false negative rate of the second best state-of-the-art approach, YAKE, which is 32.45\\%.\n\n\n\\noindent{\\bf Comparison to Twitter's soft moderation.}\nAfter determining that the recommendations made by \\textsc{Lambretta}\\xspace are accurate, we check if the tweets recommended by our approach were also soft-moderated by Twitter.\nFor every claim from the Claim Extraction Module, we retrieve the relevant set of tweets guided by the best set of keywords from our LTR component. \nWe then follow~\\cite{zannettou2021won} and extract metadata of soft moderation interventions for each tweet (i.e., if the tweet received a soft moderation and the corresponding warning label).\nWe perform this experiment on \\dthree~.\n\nOut of the 101,353 tweets flagged by \\textsc{Lambretta}\\xspace as candidates for moderation, we find that only 4,330 (4.31\\%) were soft moderated by Twitter.\nNote that we could not check the existence of warning labels for 993 tweets as they were inaccessible, with either the tweets having been deleted or the accounts that posted them being deleted or suspended.\nThis experiment highlights the limitations of Twitter's soft moderation approach, suggesting that the platform would benefit from an automated system like \\textsc{Lambretta}\\xspace to aid content moderation.\nIn Section~\\ref{sec:twitter}, we further investigate whether we can identify a specific strategy followed by Twitter in moderating content.\n\n %\n\n\n\n\\subsection{What drives Twitter moderation?}\n\\label{sec:twitter}\n\nThe analysis from the previous sections shows that Twitter only moderates a small fraction of tweets that should be moderated.\nIn this section, we aim to better understand how these moderation decisions are made.\n\nWe start by examining whether certain claims are moderated more aggressively than others and whether the type of message in a tweet affects its chances of being moderated. \nWe then analyze the text and the URLs in moderated and unmoderated tweets, aiming to ascertain: 1) whether Twitter uses text similarity to identify moderation candidates and 2) whether Twitter automatically moderates all tweets linking to a known misleading news article.\nNext, we look at the account characteristics of the users who posted moderated and unmoderated tweets, and engagement metrics (i.e., likes and retweets), aiming to understand if Twitter prioritizes moderating tweets by popular accounts or viral content.\n\n\n\\descr{Coverage by claim.}\nIn Figure~\\ref{fig:coverage}, we plot the Cumulative Distribution Function (CDF) of the percentage of tweets moderated by Twitter for each of our 900 claims, out of the total candidate set flagged by \\textsc{Lambretta}\\xspace.\nApproximately 80\\% of the claims have less than 10\\% of the tweets moderated, whereas 95\\% of claims have close to 20\\% of the tweets moderated.\nVery few claims (5) have at least half of the tweets moderated.\nThe misleading claim with the highest coverage is ``\\textbf{\\textit{Russ Ramsland file affidavit showing physical impossibility of election result in Michigan}}'' with 159 out of 309 (51\\%) candidate tweets receiving moderation labels by Twitter.\nOn the other hand, the claim ``\\textbf{\\textit{Chinese Communists Used Computer Fraud and Mail Ballot Fraud to Interfere with Our National Election}}'' only has 1 out of 236 tweets (0.42\\%) with warning labels.\nThis shows that, while the fraction of pertinent tweets moderated by Twitter is generally low, the platform seems to moderate certain claims more aggressively than others. \n\n\\descr{Coverage by tweet type.}\nIn Section~\\ref{sec:validation}, we list seven categories of tweets discussing misleading claims.\nWe now set out to understand whether Twitter moderates certain types of tweets more than others.\nTable~\\ref{tab:candidate_category} shows the fraction of tweets in our sample set of 1,500 manually analyzed tweets that did receive soft moderation by Twitter, broken down by category.\nTweets raising questions, reporting, or amplifying false claims are more likely to be moderated (with 26.19\\%, 24.07\\%, and 20.11\\% of their tweets being moderated, respectively).\nSatire tweets never received moderation labels, while tweets debunking false claims were only moderated in 3.27\\% of the cases.\nThis indicates that Twitter considers the stance of a tweet mentioning a false claim, perhaps as part of a manual moderation effort.\n\n\\descr{Content analysis.}\nNext, we investigate whether Twitter looks at near identical tweets when applying soft moderation decisions.\nWe take all tweets flagged as candidates by \\textsc{Lambretta}\\xspace, and group together those with a high Jaccard similarity of their words.\nWe remove all the links, user mentions, and lemmatize the tweet tokens by using the \\textbf{\\textit{ekphrasis}} tokenizer~\\cite{ekphrasis}.\nWe consider two tweets to be near identical if their Jaccard similarity is in the range 0.75--0.9 (out of 1.0).\nWe do so to extract tweet pairs that are not exactly the same, but have some variation in the content while discussing the same misleading claim.\nWe exclude retweets, and only consider the tweets originally authored by the users.\n\n\\begin{figure*}[t]\n\\centering\n     \\begin{subfigure}{0.375\\linewidth}\\includegraphics[width=\\linewidth]{Plots\/ccdf_followings.pdf}\n     \\caption{Following}          \n\t\\end{subfigure}\n\t~\n\t \\begin{subfigure}{0.375\\linewidth}\\includegraphics[width=\\linewidth]{Plots\/ccdf_followers.pdf}     \n     \\caption{Followers}      \n\t\\end{subfigure}\n\t \\begin{subfigure}{0.375\\linewidth}\\includegraphics[width=\\linewidth]{Plots\/ccdf_tweets.pdf}\n     \\caption{Tweet Counts}           \n\t\\end{subfigure}\n\t~\n\t \\begin{subfigure}{0.375\\linewidth}\\includegraphics[width=\\linewidth]{Plots\/ccdf_accountlife.pdf}\n     \\caption{Account Age}           \n\t\\end{subfigure}\n\t    \n\\caption{Cumulative Distribution Functions (CDF) of various user metrics for moderated and unmoderated tweets.}\n\\label{fig:ccdf_useranalysis}\n\\end{figure*}\n\n\nWe extract 17,241 pairs of tweets (out of 438,986 possible pairs), where at least one of the two was moderated by Twitter.\nOnly 3,857 pairs have both tweets moderated.\nNote that \\textsc{Lambretta}\\xspace effectively identifies {\\em all} the 17,241 pairs of tweets as moderation candidates. %\nHere is an example of a very similar pair of tweets, for which Twitter did not add labels to one of them:\n\\smallskip\n\\begin{mdframed}[style=MyFrame,nobreak=true]\n\\begin{quote}\n\\small\n\t\\textbf{Moderated}:  ``RudyGiuliani in Trump campaign news conference: \"\"Joe Biden said a few weeks ago that his voting fraud crew was the best in the world. They were excellent, but we got them!\"''\n\t\n\t\\textbf{Unmoderated}: ``Joe Biden said a few weeks ago that his crew was the greatest in the world at catching voter fraud, but we caught them.''\n\\end{quote}\n\\end{mdframed}\n\nThese findings indicate that the decision by Twitter to add soft moderation to a tweet does not seem to be driven by the lexical similarity of tweets.\n\n\\descr{URL analysis.}\nAnother potential indicator used by Twitter when deciding which tweets to moderate is whether they include links to known news disinformation articles.\nFirst, we expand all the links in the body of candidate tweets identified by \\textsc{Lambretta}\\xspace to get rid of URL shorteners~\\cite{maggi2013two}.\nThis yields 13,108 distinct URLs.\nNext, we group candidate tweets by URLs and check what fraction of tweets sharing the URL are moderated by Twitter.\n\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\t\\setlength{\\tabcolsep}{4pt}\n\\begin{tabular}{lrr}\n\\toprule\n\\textbf{URL news story}                  & \\textbf{Candidates} & \\textbf{Moderated} \\\\ \\midrule\nUSPS whistleblower                       & 315                 & 7 (2.2\\%)          \\\\\nChina manipulating election              & 252                 & 4 (1.5\\%)          \\\\\nMichigan ballot dump                     & 215                 & 44 (20\\%)          \\\\\n\\#Suitcasegate related FB video & 208                 & 3 (1.4\\%)          \\\\\nDominion remote machine control          & 135                 & 15 (11\\%)          \\\\ \\bottomrule\n\\end{tabular}%\n\\caption{Examples of URLs in candidate tweets and those being moderated by Twitter.}\n\\label{tab:urlmoderation}\n\\end{table}\n\n\n\nTable~\\ref{tab:urlmoderation} shows the five most common URLs (abstracted to the topic of the news articles) in our dataset, with the fraction of tweets including those URLs moderated by Twitter.\nAll these news stories, excluding one Facebook video, originate from known low-credibility websites like TheGatewayPundit and DC Dirty Laundry, which promote election misinformation.\nTwitter moderates tweets containing those URLs in an inconsistent matter. %\nAlso note that \\textsc{Lambretta}\\xspace can help identify 4,598 additional moderation candidate tweets compared to those on which Twitter intervened.\n\n\\descr{User analysis.}\nWe examine the differences in the social capital (e.g., number of followers) of the authors of tweets moderated by Twitter, compared to those our system recommends for moderation but for which Twitter did not intervene.\nFigure~\\ref{fig:ccdf_useranalysis} reports the CDF of followers, following, tweet count, and account age of accounts that posted moderated and unmoderated tweets.\nWe find that authors of tweets that have warning labels have much fewer followers, followings, lower account activity, and have younger accounts than tweets without warning labels.\nWe also conduct two-sample Kolmogorov-Smirnov tests for each user metric, finding that the differences are statistically significant for followers and account age ($p < 0.01$) as well as following count and status count ($p < 0.05$).\nThis goes against the notion that popular accounts are more likely to have their content moderated.\n\nWe also check if the accounts with moderated tweets were suspended for violating Twitter Rules~\\cite{twitter_rules}. %\nWe find that only 33 out of 3,397 users were suspended by Twitter;  this gives us strong ground to rule out the possibility that tweet moderation is not due to the ``legitimacy'' of the account themselves.\n\n\\descr{Engagement analysis.}\nFinally, we analyze engagement metrics. %\nFigure~\\ref{fig:ccdf_tweetanalysis} reports the CDF of retweets and likes categorized by moderation status of the 101,353 candidate tweets \\textsc{Lambretta}\\xspace flags for moderation from \\dthree~, compared to the ones flagged by Twitter.\nSimilar to the user analysis, we find that unmoderated tweets have more engagement. %\nWhen we check for statistical significance of difference in distributions of the retweet count using Kolmogorov-Smirnov tests, we find that it is statistically significant ($p < 0.01$), while we cannot reject the null hypothesis for the likes.\nNote, however, that these results have to be taken with a grain of salt, as we do not have a timeline of when exactly moderation was applied, and whether the soft interventions hampered the virality of online content. \n\n\\begin{figure}\n\\centering\n     \\begin{subfigure}{0.375\\textwidth}\\includegraphics[width=\\linewidth]{Plots\/tweets_rt.pdf}     \n     \\caption{Retweets}     \n\t\\end{subfigure}\n\t \\begin{subfigure}{0.375\\textwidth}\\includegraphics[width=\\linewidth]{Plots\/tweets_favs.pdf}  \n     \\caption{Likes}         \n\t\\end{subfigure}\n\t    \n\\caption{CDFs of engagement metrics for moderated and unmoderated tweets.}\n\\label{fig:ccdf_tweetanalysis}\n\\end{figure}\n\n\\descr{Takeaways.} Our analysis paints a puzzling picture of soft moderation on Twitter. \nWe find that certain claims are moderated more aggressively.\nStill, Twitter does not seem to have a system in place to identify similar tweets discussing the same false narrative, nor flagging tweets that link to the same debunked news article.\nWe also find that Twitter does not appear to focus on the tweets posted by popular accounts for moderation, but rather that tweets posted by accounts with more followers, friends, activity, and a longer lifespan are more likely to go unmoderated.\nThis confirms the need for a system like \\textsc{Lambretta}\\xspace. %\n\n\\section{Related Work}\nIn this section, we review relevant work on soft moderation, security warnings, and keyword extraction in the context of disinformation.\n\n\\descr{Soft Moderation during the 2020 Elections.}\nAs part of the Civic Integrity Policy efforts surrounding the 2020 US elections, Twitter applied warning labels on ``misleading information.''\nEmpirical analysis~\\cite{zannettou2021won} reports 12 different types of warning messages occurring on a sample of 2,244 tweets with warning labels.\nStatistical assessment of the impact of Twitter labels on Donald Trump's false claims during the 2020 US Presidential election finds that warning labels did not result in any statistically significant increase or decrease in the spread of misinformation~\\cite{papakyriakopoulos2022impact}.\nTwitter later reported that approximately 74\\% of the tweet viewership happened post-moderation and, more importantly, that the warnings yielded an estimated 29\\% decrease in users quoting the labeled tweets~\\cite{twitter_update_2020}. %\n\n\n\\descr{Security Warnings for Disinformation.}\nThe warning labels adopted by Twitter as soft moderation intervention can be broadly categorized as a type of security warning.\nSecurity warnings can be classified into two types: contextual and interstitial.\nThe former passively inform the users about misinformation through UI elements that appear alongside social media posts.\nThe latter prompt the user to engage before taking action with the potential piece of disinformation (e.g., retweeting or sharing).\nA recent study~\\cite{kaiser2021adapting} shows that interstitial warnings may be more effective, with a lower clickthrough rate of misleading articles.\nAdditionally, interstitial warnings are more effective design-wise because they capture attention and provide a window of opportunity for users to think about their actions.\nEfforts to study warning labels on countering disinformation have thus far been mostly focused on Facebook~\\cite{pennycook2018prior,ross2018fake,pennycook2020implied}, where warning labels were limited to ``disputed'' or ``rated false'', and the approach was deemed to be of limited utility by Facebook~\\cite{smith2017designing}.\nRecently, other platforms like Twitter~\\cite{alba2020twitter}, Google~\\cite{googlelabel}, and Bing~\\cite{bing_label} also used some form of fact-check warnings to counter disinformation.\n\n\\descr{Tools for automated content moderation.}\nThe sheer scale of content being produced on modern social media platforms (Facebook, Reddit, YouTube etc.) have motivated the need to adopt tools for automated content moderation~\\cite{fb_guidelines,youtube_guidelines}.\nHowever, due to the nuanced and context-sensitive nature of content moderation, it is a complex socio-technical problem~\\cite{jhaver2019human,seering2019moderator}.\nMost of the work in this space of automated content moderation are focused on Reddit, aiming to identify submissions that violate community-specific policies and norms ranging from hate speech to other types of problematic content~\\cite{seering2020reconsidering}.\nThe most popular solution to automated content moderation in Reddit, AutoModerator~\\cite{jhaver2019human} allows community moderators to set up simple rules based on regular expressions and metadata of users for automated moderation.  \nOn YouTube, FilterBuddy~\\cite{jhaver2022designing} is available as a tool for creator-led content moderation by designing filters for keywords and key phrases.\nSimilarly, Twitch offers an automated moderation tool called Automod to allow creators to moderate four categories of content (discriminations and slurs, sexual content, hostility, and profanity) on the platform~\\cite{twitch_automod}.\nAnother tool, called CrossMod~\\cite{chandrasekharan2019crossmod} uses an ensemble of models learned via cross-community learning from empirical moderation decisions made on two subreddits of over 10M subscribers each.\n\n\n\\descr{Keyword Extraction for Disinformation.}\nResearchers have used an array of methods to detect disinformation, ranging from modeling user interactions~\\cite{shu2019role,tschiatschek2018fake,qian2018neural}, leveraging semantic content~\\cite{ciampaglia2015computational,zhang2017constraint,esmaeilzadeh2019neural,pan2018content}, and graph based representations~\\cite{nguyen2020fang,gangireddy2020unsupervised,lu2020gcan}.\nThe foundation of our system lies in the keyword detection, which has been used before to study disinformation on social media.\nDisInfoNet, a toolbox presented in~\\cite{guarino2019beyond}, represents news stories through keyword-based queries to track news stories and reconstruct the prevalence of disinformation over time and space.\nSimilarly, the work in~\\cite{gaglani2020unsupervised} uses keyword extraction techniques as the base for semantic search to detect fake news on WhatsApp.\nThe work in~\\cite{choudhary2018neural} focuses on credibility assessment of textual claims on news articles with potentially false information, also using keyword extraction as a part of their multi-component module.\n\n\\descr{Learning To Rank for Keyword extraction.}\nThe closest applications of LTR to our work are the proposals in~\\citep{jiang2009ranking} for keyphrase extraction and in~\\citep{cai2017keyword} for keyword extraction in Chinese news articles.\nThe foundational work by~\\citep{jiang2009ranking} motivates the necessity of framing the problem of keyphrase extraction as a ranking task rather than a classification task while improving results on extracting keyphrases from academic research papers, and social tagging data.\nThe LTR approach for keyword extraction utilized by \\textsc{Lambretta}\\xspace is motivated by the premise set up by this work.\nSimilarly,~\\citep{cai2017keyword} use Learning To Rank to identify keywords from 1800 public Chinese new articles using TF-IDF, TextRank, and Latent Dirichlet Allocation (LDA) as the set of features for the ranking model.\n\n\\section{Discussion and Conclusion}\n\\label{sec:discussion}\n\nThis paper presented \\textsc{Lambretta}\\xspace, a system geared to automatically flag candidate tweets for soft moderation interventions.\nOur experiments demonstrate that Learning to Rank (LTR) techniques are effective, that \\textsc{Lambretta}\\xspace outperforms other approaches, produces accurate recommendations, and can increase the set of tweets that receive soft moderation by over 20 times compared to those flagged by Twitter during the 2020 US Presidential Election.\n\n\n\\descr{Implications for social media platforms.}\nAs discussed in Section~\\ref{sec:twitter}, soft moderation interventions applied by Twitter appear to be spotty and not following precise criteria.\nThis might be due to moderation being conducted mainly in an ad-hoc fashion, relying on user reports and the judgment of moderators.\n\\textsc{Lambretta}\\xspace can assist this human effort, working upstream of the content moderation process and presenting moderators with an optimal set of tweets that are candidates for moderation. \nBecause of the nuances of moderating false and misleading content, we envision \\textsc{Lambretta}\\xspace to be deployed as an aid to human moderation rather than an automated detection tool.\n\nNonetheless, the claim-specific design of \\textsc{Lambretta}\\xspace can also be used by moderators for other actions as per their policies, e.g., asking users to remove a given tweet or performing hard moderation by removing the tweets.\nThe choice between soft moderation and hard moderation can be made by moderators contextually after the moderation candidates are retrieved through \\textsc{Lambretta}\\xspace, either based on underlying claims or a case-by-case basis.\nE.g., platforms may decide to soft moderate posts that push a certain false narrative but not add warnings if posts inform users about falsehood.\nAlternatively, they might add warnings to posts about the false narrative, providing additional context to users and allowing them to make up their minds about it.\nPlatforms could also craft warning messages depending on the context in which a false claim is discussed or design these messages to be more effective based on the audience and risk levels of specific false claims.\nFor example, different type of warning messages can be applied by platforms to distinguish between different levels of risk associated with the misleading claims (e.g., high and low-level risks associated with COVID-19 misinformation)~\\cite{ling2022learn}.\nWe are confident that Human-Computer Interaction researchers will be able to address these challenges, which go beyond the scope of this paper.\n\n\\descr{Human effort required for adopting \\textsc{Lambretta}\\xspace.}\nWhen setting up \\textsc{Lambretta}\\xspace to work in a new context, platform moderators need to follow the steps highlighted in Sections~\\ref{sec:claimextraction} and~\\ref{sec:keyword}.\nFirst, they need a set of tweets with claims they identified as containing misleading information, together with a tuning dataset like \\dthree~.\nModerators can create a tuning dataset like \\dthree~ by using a broad set of keywords associated with the event or topic and querying the Twitter API to which they have full access (e.g. ``COVID-19,''  ``coronavirus,'' etc., in the case of the pandemic).\nThey then need to tune the threshold for the Claim Stopper API in the Claim Extraction component (see Section~\\ref{sec:claimextraction}).\nIn our experiments, this phase took us, on average, two minutes per claim.\nFinally, they need to create the training set for the LTR model by following the iterative process discussed in Section~\\ref{sec:keyword}.\nWhen performed by a single annotator, this process took, on average, 15 minutes per claim for the experiments discussed in this paper.\nTwitter could speed up these steps further by having multiple annotators work on the same task.\nAdditionally, the work required on each claim is independent of other claims; therefore this process can be easily parallelized within the organization or even through crowdsourcing campaigns~\\cite{founta2018large,lease2012crowdsourcing,oleson2011programmatic}.\n\n\n\\descr{Resilience to evasion.}\nAs with any adversarial problem, malicious actors are likely to try to evade being flagged by \\textsc{Lambretta}\\xspace.\nE.g., they might avoid using certain words to avoid detection and use synonyms or dog whistles instead~\\cite{gerrard2018beyond,tahmasbi2021go,zannettou2020quantitative,zhu2021self}.\nHowever, this would make the false messaging less accessible to the general public, who would need to first understand the alternative words used and ultimately be counterproductive for malicious actors by limiting the reach of false narratives.\n\n\n\n\n\\descr{Limitations.} \\textsc{Lambretta}\\xspace requires a seed of tweets to be moderated, making it inherently reactive.\nHowever, this is a problem common to all moderation approaches, including the work conducted by fact-checking organizations.\nAnother limitation is that we could only test \\textsc{Lambretta}\\xspace on one dataset related to the same major event (the 2020 US Presidential Election), as this is the only reliable dataset with soft moderation labels available to the research community.\n\nEven though Twitter applied warning labels on misinformation about COVID-19, previous research reported that these were unreliable and inconsistent~\\cite{lange_2020,lyons_2020}, which we independently confirmed in our preliminary analysis.\nMore recently, Twitter recently started applying warning labels to tweets in the context of the Russian invasion of Ukraine~\\cite{benson_twitter_russia}, but these labels are applied based on the account posting them (i.e., if the account belongs to Russian or Belarusian state-affiliated media) instead of being claim-specific as required by \\textsc{Lambretta}\\xspace.\nWhile the LTR model used by \\textsc{Lambretta}\\xspace is not specific to the actual keywords being searched, and therefore we expect that it should generalize across the entirety of Twitter, platform moderators using the tool should take further steps to validate it when used in contexts other than politics and elections.\n\n\n\\descr{Future work.} \nWe plan to extend \\textsc{Lambretta}\\xspace to additional platforms.\nSince our system only needs the text of posts as input, we expect it to generalize to other platforms, e.g., Facebook, Reddit, etc.\nWe will also investigate how claims automatically built by \\textsc{Lambretta}\\xspace can be incorporated into warning messages to provide more context to users and allow them to be better protected against disinformation.\n\n\\descr{Acknowledgments.} \nWe thank the anonymous reviewers for their comments that helped us improve the paper.\nOur work was supported by the NSF under grants CNS-1942610, IIS-2046590, CNS-2114407, IIP-1827700, and CNS-2114411, and by the UK's National Research Centre on Privacy, Harm Reduction, and Adversarial Influence Online (REPHRAIN, UKRI grant: EP\/V011189\/1).\n\n\\small\n\\bibliographystyle{abbrv}\n\\input{no_comments.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe occurence of important Solar Energetic Particle (SEP) events is one of the\nprominent planning considerations for manned and unmanned lunar and planetary\nmissions \\cite{posner2007up}.\nA high exposure to large solar particles events can deliver critical doses to\nhuman organs and may damage the instruments on board of satellites and the\nglobal positioning system (GPS) due to the risk of saturation. SEP\nevents usually happen 30 minutes after the occurrence of the X-ray flare, which\nleaves very little time for astronauts performing extra-vehicular activity on\nthe International Space Station or planetary surfaces to\ntake evasive actions \\cite{2016l}. Earlier warning of the SEP events will be a valuable tool\nfor mission controllers that need to take a prompt decision concerning the\natrounauts' safety and the mission completion.\nWhen a solar flare or a CME happens, the magnetic force that is exercised is\nmanifested through different effects. Some of the effects are listed in their\norder of occurrence: light, thermal, particle acceleration, and matter ejection\nin case of CMEs. The first effect of a solar flare is a flash of increased\nbrightness that is\nobserved near the Sun's surface which is due to the X-rays and UV radiation.\nThen, part of the magnetic energy is converted into thermal energy in the area\nwhere the eruption happened.\nSolar particles in the atmosphere are then accelerated with different speed\nspectra, that can reach up to 80\\% of the speed of light, depending on the\nintensity of the parent eruptive event.\n  \\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/SolarParticleEvent.png}\n     \\caption{Example of Sun-Earth magnetical connection and accelerated\n     particles movement following the Parker's spiral before reaching\n     Earth. (Drawing courtesy from Space Weather) \\cite{greendale}}\n       \\label{fig1}\n\\end{figure}\n\nFinally, in the case of a CME, plasma and\nmagnetic field from the solar corona is released into the solar wind.\nThough most of the solar particles have the same composition, they are labeled\ndifferently depending on their energies starting from 1 keV, in the solar wind,\nto more than 500 MeV. SEP events are composed of particles, predominantly\nprotons and electrons, with at least 1 MeV energy that last between 2-20 days\nand have a range of fluxes of 5-6 orders of magnitude \\cite{gabriel1996power}.\nOnly $>$100 MeV particles are discussed herein. It is generally accepted that\nthere are two types of SEP events, one is associated with CMEs and the other is\nassociated with flares that are called respectively gradual and impulsive\n\\cite{reames1999particle}.\n\nIn this paper, we propose a novel method for predicting SEP events $>$100 MeV\nbased on the proton and X-ray correlations time series using interpretable\ndecision tree models.\nPredicting impulsive events is considered to be a more challenging problem than\npredicting the gradual events that happen progressively and leave a large window\nfor issuing SEP warnings. While we are mainly concerned with impulsive events,\nwe used the gradual events as well to test our model with. The accelerated\nimpulsive events may or may not reach Earth depending on the location of their\nparent event because their motion is confined by the magnetic field. More\nspecifically, in order for the accelerated particles to reach Earth, a Sun-Earth\nmagnetic connection needs to exist that allows the particles to flow to Earth\nvia the Parker spiral. Fig~.\\ref{fig1} shows a cartoon of a solar eruption that\nhappened in the Western limb of the Sun that happens to be magnetically\nconnected to Earth.\n\nSince SEP events are also part of solar activity, it may seem that\ntheir occurrence is dependent on the solar cycle and therefore the number of\nSunspots on the Sun's surface, which is the case for other solar eruptions.\nHowever, according to \\cite{gabriel1996power}, there is no correlation between the solar\ncycle and SEP event occurrence and fluences. In addition, there is no\nevidence the dependence of SEP events on the number of Sunspots\nthat are present during that snapshot in time \\cite{gabriel1996power}.\n\\par\n The rest\nof the paper is presented as follows. In Section 2 we provide background\nmaterial on the SEP predictive models and related works. Section 3\ndefines our dataset used in this study, and then in Section 4 we lay out our\nmethodology. Finally, Section 5 contains our experimental results, and we finish\nwith conclusions and future work in Section 6.\n\n\n\\section{Related Works}\nThere are a number of predictive models of SEP events that can be categorized\ninto two classes: physics-based models \\cite{p0, p1} and the precursor-based\nmodels \\cite{c0, c1}.\nThe first category of models includes the SOLar Particle Engineering Code\n(SOLPENCO) that can predict the flux and fluence of the gradual SEP events\noriginating from CMEs \\cite{aran2006solpenco}. However, such efforts mainly\nfocus on gradual events.\nOn the other hand, there are models that rely on historical observations to find\nSEP events associated precursors. One example of such systems is the proton\nprediction system (PPS), which is a program developed at the Air Force Research\nLaboratory (AFRL), that predicts low energy SEP events E$>$\\{5, 10, 50\\} MeV,\ncomposition, and intensities. PPS assumes that there is a relationship between\nthe proton flux and the parent flare event. PPS takes\nadvantage of the correlation between large SEP events observed by the\nInterplanetary Monitoring Platform (IMP) satellites as well as their correlated\nflare signatures captured by GOES proton, X-ray flux and H$\\alpha$ flare\nlocation \\cite{PPS}. Also, the Relativistic Electron Alert System for\nExploration (RELEASE), predicts the intensity of SEP events using relativistic\nnear light speed electrons \\cite{posner2007up}.\nRELEASE uses electron flux data from the SOHO\/COSTEP sensor of the range of\n0.3-1.2 MeV to forecast the proton intensity of the energy range 30-50 MeV.\nAnother example of precursor-based models appear in\n\\cite{laurenza2009technique}, that base their study on the \"big flare\nsyndrome\". This latter theory states that SEP events occurrence at 1 AU is\nhighly probable when the intensity of the parent flare is high. Following this\nassumption, the authors in \\cite{laurenza2009technique} issue SEP forecasts for\nimportant flares greater than M2. To this end, it uses type III radio burst\ndata, H$\\alpha$ data, and GOES soft X-ray data. Finally, $GLE Alert Plus$, is an\noperational system that uses a ground-based neutron\nmonitor (MNDB, www.nmdb.com) to issue alerts of SEP events of energies E$>$433\nMeV. Finally, the University of Malaga Solar Energetic Particles Prediction\n(UMASEP) is another system that first predicts whether a $>$10 MeV and $>$100\nMeV SEP will happen or not. To do so, it computes the correlation between the\nsoft X-ray and proton channels to assess if there is a magnetic connection\nbetween the Sun and Earth at the time of the X-ray event.\n Then, in case of existence of magnetic connection, UMASEP gives an\nestimation on the time when the proton flux is expected to surpass the SWPC\nthreshold of J(E $>$ 10MeV)= 10$pfu$ and J(E $>$ 100MeV)= 1$pfu$ (1$pfu$ = $pr\ncm^{-2} sr^{-1} s^{-1}$) and for the case of UMASEP-100, the intensity of the\nfirst three hours after the event onset time.\n\n \\begin{figure} \n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{figs\/GOESchart.pdf}\n     \\caption{Primary (bold lines) and secondary (thin lines) GOES satellites\n     for XRS data since 1986 (the primary and secondary satellites designation is\n     unknown prior to 1986) (Figure from NOAA instrument report)}\n       \\label{fig2}\n\\end{figure} \n\n \\begin{figure} \n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/ChartGOES.pdf}\n     \\caption{Catalogs used to make the x-ray-parent event mapping. X-ray and\n     CME catalogs for detecting the parent event report for flare and CME\n     respectively.}\n       \\label{cats}\n\\end{figure} \n\nWhile most of the SEP predictive systems either focus on the CME associated\nevents or low energy SEP events, with the exception of $GLE Alert Plus$ and\nUMASEP, in this present work, we focus on higher energy SEP events that can be\nmore disruptive than lower energy events. In this work, we study the\nGOES cross-channel correlations that can give an early insight on whether there\nexist a magnetic connection or not.\n\nWe aim to provide an interpretable decision tree models using a\nbalanced dataset of SEP and non-SEP events.\nThe highest SEP energy band of $>$500 MeV or higher\nthat are measurable from the ground is out of the scope of this study.\nSimilarly, the lower SEP energy band of $<$100 MeV is not considered in this\nstudy.\n\n\\section{Data}\nOur dataset is composed of multivariate time series of X-ray, integral proton flux and fluences spectra that were measured on board of\nSpace Environment Monitor (SEM) instruments package of the Geostationary\nOperational Earth Satellites (GOES).\nIn particular, we consider both the short and long X-ray channel data recorded\nby the X-ray Sensor (XRS). For the proton channels, we consider channels P6 and\nP7 recorded by the Energetic Particle Sensor (EPS) and proton channels P8, P9,\nP10, and P11 recorded by the High Energy Proton and Alpha Detector (HEPAD).\nTable~\\ref{goesinstruments} summarizes the instruments onboard the GOES satellites\nand their corresponding data channels that we used.\n\n\\begin{figure*}\n\\centering \n\\begin{tabular}{ |c|c|c|}\nFast-Rising & Slow-Rising & Lack of SEP\\\\\n\\hline \n\\includegraphics[width=0.315\\linewidth]{figs\/g1.pdf}&\n\\includegraphics[width=0.32\\linewidth]{figs\/g2.pdf}&\n\\includegraphics[width=0.32\\linewidth]{figs\/g3.pdf}\n\\\\\n\\hline\n\\includegraphics[width=0.175\\linewidth]{figs\/impulsive.pdf} &\n\\includegraphics[width=0.17\\linewidth]{figs\/CME1.PNG}&\n\\includegraphics[width=0.17\\linewidth]{figs\/FL3.PNG}\n\\\\\n \\hline \n \\end{tabular}\n \\caption{Example of an (a) Impulsive SEP event\n     that started on the 2001-04-15 14:05:00 as a result of a flare\n    that occurred in the 2011-04-15 13:15:00 shown in the SOHO EIT instument and\n    a (b) gradual SEP event whose nearest temporal flare happened on 2001-04-02 21:30:03A and occurred as a\n    result of a CME on the 2001-04-02 22:06:07 shown in the SOHO LASCO instrument\n    and a (c) a flare that happened on 1999-01-16 12:00:00 that did not lead to any\n    $>$ 100 MeV SEP event shown in the SOHO EIT\n    instrument.\\label{table1impulsive}}\n\\end{figure*}\n\n The data we collected is made publically available by NOAA in the\nfollowing link:\n(\\href{https:\/\/satdat.ngdc.noaa.gov\/sem\/goes\/data\/new\\_avg\/}{\\textit{https:\/\/satdat.ngdc.noaa.gov\/sem\/goes\/data\/new\\_avg\/}})\nThe data is available in three different cadences. The full resolution data is captured\nevery three seconds from the GOES satellites, which is aggregated and made\navailable with one and five minute cadences. In this paper we use the\naggregated five minute data which is the one usually cited in the literature\n\\cite{neal2001predicting} \\cite{nunez2015real} \\cite{nunez2011predicting}.\nIn most cases, there are a couple a co-existing GOES satellites whose data is\ncaptured by more than one GOES satellite at a time. In this study, we always\nconsider the data reported by the primary GOES satellite that is designated by\nthe NOAA, as illustrated in Fig~.\\ref{fig2}. The latter figure shows the primary GOES\nsatellite with a bold line and the other co-existing GOES for every year.\nGOES-13 measurements were unstable for many years, but have been stable since\n2014.\n\n\n\\begin{table*} \n\\centering\n\\caption{GOES X-ray and Proton instuments and Channels.\\label{goesinstruments}}\n\\begin{tabular}{ |c|c|c|c|}\n\\hline\nInstrument & Channels & Description \\\\\n\\hline\n\\multirow{2}{*}{XRS}\n & xs &Short wavelength channel irradiance (0.5 - 0.3 nm)\\\\\n & xl &Long wavelength channel irradiance (0.1-0.8 nm)\\\\\n  \\hline\n\\multirow{4}{*}{HEPAD}\n & p8\\_flux &Proton Channel 350.0 - 420.0 MeV \\\\\n & p9\\_flux &Proton Channel 420.0 - 510.0 MeV \\\\\n & p10\\_flux &Proton Channel 510.0 - 700.0 MeV \\\\\n & p11\\_flux &Proton Channel $>$ 700.0 MeV \\\\\n \\hline\n \\multirow{2}{*}{EPS}\n\n & p6\\_flux &Proton Channel 80.0 - 165.0 MeV \\\\\n & p7\\_flux &Proton Channel 165.0 - 500.0 MeV \\\\\n  \\hline\n  \n \\end{tabular}\n\\end{table*}\n\n\n\\begin{table}\n \\caption{$>$ 100 MeV SEP Event List with their Parent Events\n(CME\/Flare)\\label{sepevents}}\n\\centering \n\\begin{threeparttable}\n\\begin{tabular}{ |c|c|c|}\n\\hline \nSEP Event ID & Onset Time of SEP Event & Parent X-ray Event \\\\ \\hline\n1  &  1997-11-04 05:52:00 & 1997-11-04 05:52:00  \\\\ \\hline\n2  &  1997-11-06 11:49:00 & 1997-11-06 11:49:00  \\\\ \\hline\n3\\tnote{*}  &  1998-04-20 09:38:00 & 1998-04-20 09:38:00  \\\\ \\hline\n4  &  1998-05-02 13:31:00 & 1998-05-02 13:31:00  \\\\ \\hline\n5  &  1998-05-06 07:58:00 & 1998-05-06 07:58:00  \\\\ \\hline\n6  &  1998-08-24 21:50:00 & 1998-08-24 21:50:00  \\\\ \\hline\n7  &  1998-09-30 13:50:00 & 1998-09-30 13:50:00  \\\\ \\hline\n8  &  1998-11-14 05:15:00 & 1998-11-14 06:05:00  \\\\ \\hline\n9  &  2000-06-10 16:40:00 & 2000-06-10 16:40:00  \\\\ \\hline\n10  &  2000-07-14 10:03:00 & 2000-07-14 10:03:00  \\\\ \\hline\n11  &  2000-11-08 22:42:00 & 2000-11-08 22:42:00  \\\\ \\hline\n12  &  2000-11-24 14:51:00 & 2000-11-24 14:51:00  \\\\ \\hline\n13\\tnote{*}  &  2000-11-26 16:34:00 & 2000-11-26 16:34:00  \\\\ \\hline\n14\\tnote{*}  &  2001-04-02 21:32:00 & 2001-04-02 21:32:00  \\\\ \\hline\n15  &  2001-04-12 09:39:00 & 2001-04-12 09:39:00  \\\\ \\hline\n16  &  2001-04-15 13:19:00 & 2001-04-15 13:19:00  \\\\ \\hline\n17  &  2001-04-17 21:18:00 & 2001-04-18 02:05:00  \\\\ \\hline\n18  &  2001-08-15 12:38:00 & 2001-08-16 23:30:00  \\\\ \\hline\n19\\tnote{*}  &  2001-09-24 09:32:00 & 2001-09-24 09:32:00  \\\\ \\hline\n20  &  2001-11-04 16:03:00 & 2001-11-04 16:03:00  \\\\ \\hline\n21  &  2001-11-22 22:32:00 & 2001-11-22 19:45:00  \\\\ \\hline\n22  &  2001-12-26 04:32:00 & 2001-12-26 04:32:00  \\\\ \\hline\n23  &  2002-04-21 00:43:00 & 2002-04-21 00:43:00  \\\\ \\hline\n24  &  2002-08-22 01:47:00 & 2002-08-22 01:47:00  \\\\ \\hline\n25  &  2002-08-24 00:49:00 & 2002-08-24 00:49:00  \\\\ \\hline\n26  &  2003-10-28 09:51:00 & 2003-10-28 09:51:00  \\\\ \\hline\n27  &  2003-11-02 17:03:00 & 2003-11-02 17:03:00  \\\\ \\hline\n28\\tnote{*}  &  2003-11-05 02:37:00 & 2003-11-05 02:37:00  \\\\ \\hline\n29\\tnote{+}  &  2004-11-01 03:04:00 & 2004-11-01 03:04:00  \\\\ \\hline\n30\\tnote{+}  &  2004-11-10 01:59:00 & 2004-11-10 01:59:00  \\\\ \\hline\n31\\tnote{+}  &  2005-01-16 21:55:00 & 2005-01-17 08:00:00  \\\\ \\hline\n32\\tnote{+}  &  2005-01-20 06:36:00 & 2005-01-20 06:36:00  \\\\ \\hline\n33\\tnote{+}  &  2005-06-16 20:01:00 & 2005-06-16 20:01:00  \\\\ \\hline\n34\\tnote{+}    \\tnote{*}  &  2005-09-07 17:17:00 & 2005-09-07 17:17:00  \\\\ \\hline\n35\\tnote{+}    \\tnote{*}  &  2006-12-06 18:29:00 & 2006-12-06 18:29:00  \\\\ \\hline\n36\\tnote{+}  &  2006-12-13 02:14:00 & 2006-12-13 02:14:00  \\\\ \\hline\n37\\tnote{+}  &  2006-12-14 21:07:00 & 2006-12-14 21:07:00  \\\\ \\hline\n38  &  2011-06-07 06:16:00 & 2011-06-07 06:16:00  \\\\ \\hline\n39  &  2011-08-04 03:41:00 & 2011-08-04 03:41:00  \\\\ \\hline\n40  &  2011-08-09 07:48:00 & 2011-08-09 07:48:00  \\\\ \\hline\n41  &  2012-01-23 03:38:00 & 2012-01-23 03:38:00  \\\\ \\hline\n42\\tnote{*}  &  2012-01-27 17:37:00 & 2012-01-27 17:37:00  \\\\ \\hline\n43  &  2012-03-07 01:05:00 & 2012-03-07 01:05:00  \\\\ \\hline\n44  &  2012-03-13 17:12:00 & 2012-03-13 17:12:00  \\\\ \\hline\n45  &  2012-05-17 01:25:00 & 2012-05-17 01:25:00  \\\\ \\hline\n46\\tnote{*}  &  2013-04-11 06:55:00 & 2013-04-11 06:55:00  \\\\ \\hline\n47  &  2013-05-22 13:08:00 & 2013-05-22 13:08:00  \\\\ \\hline\n \\end{tabular}\n   \\begin{tablenotes}\n  \\item[*] Gradual Events.\n  \\item[+] Missing Data in P6 and P7.\n  \\end{tablenotes}\n  \\end{threeparttable}  \n\\end{table}\n\n\n\\begin{table}\n \\caption{Non SEP Event List \\label{ns}}\n\\centering \n\\begin{threeparttable}\n\\begin{tabular}{|c|c|c|}\n\\hline\nNon SEP Event ID & X-ray Event & Class \\\\ \\hline\n1 &  1997-09-24 02:43:00 &  M59  \\\\ \\hline\n2 &  1997-11-27 12:59:00 &  X26  \\\\ \\hline\n3 &  1997-11-28 04:53:00 &  M68  \\\\ \\hline\n4 &  1997-11-29 22:28:00 &  M64  \\\\ \\hline\n5 &  1998-07-14 12:51:00 &  M46  \\\\ \\hline\n6 &  1998-08-18 08:14:00 &  X28  \\\\ \\hline\n7 &  1998-08-18 22:10:00 &  X49  \\\\ \\hline\n8 &  1998-08-19 21:35:00 &  X39  \\\\ \\hline\n9 &  1998-11-28 04:54:00 &  X33  \\\\ \\hline\n10 &  1999-01-16 12:02:00 &  M36  \\\\ \\hline\n11 &  1999-04-03 22:56:00 &  M43  \\\\ \\hline\n12 &  1999-04-04 05:15:00 &  M54  \\\\ \\hline\n13 &  1999-05-03 05:36:00 &  M44  \\\\ \\hline\n14 &  1999-07-19 08:16:00 &  M58  \\\\ \\hline\n15 &  1999-07-29 19:31:00 &  M51  \\\\ \\hline\n16 &  1999-08-20 23:03:00 &  M98  \\\\ \\hline\n17 &  1999-08-21 16:30:00 &  M37  \\\\ \\hline\n18 &  1999-08-21 22:10:00 &  M59  \\\\ \\hline\n19 &  1999-08-25 01:32:00 &  M36  \\\\ \\hline\n20 &  1999-10-14 08:54:00 &  X18  \\\\ \\hline\n21 &  1999-11-14 07:54:00 &  M80  \\\\ \\hline\n22 &  1999-11-16 02:36:00 &  M38  \\\\ \\hline\n23 &  1999-11-17 09:47:00 &  M74  \\\\ \\hline\n24 &  1999-12-22 18:52:00 &  M53  \\\\ \\hline\n25 &  2000-01-18 17:07:00 &  M39  \\\\ \\hline\n26 &  2000-02-05 19:17:00 &  X12  \\\\ \\hline\n27 &  2000-03-12 23:30:00 &  M36  \\\\ \\hline\n28 &  2000-03-31 10:13:00 &  M41  \\\\ \\hline\n29 &  2000-04-15 10:09:00 &  M43  \\\\ \\hline\n30 &  2000-06-02 06:52:00 &  M41  \\\\ \\hline\n31 &  2000-06-02 18:48:00 &  M76  \\\\ \\hline\n32 &  2000-10-29 01:28:00 &  M44  \\\\ \\hline\n33 &  2000-12-27 15:30:00 &  M43  \\\\ \\hline\n34 &  2001-01-20 21:06:00 &  M77  \\\\ \\hline\n35 &  2001-03-28 11:21:00 &  M43  \\\\ \\hline\n36 &  2001-06-13 11:22:00 &  M78  \\\\ \\hline\n37 &  2001-06-23 00:10:00 &  M56  \\\\ \\hline\n38 &  2001-06-23 04:02:00 &  X12  \\\\ \\hline\n39\\tnote{+}  &  2004-12-30 22:02:00 &  M42  \\\\ \\hline\n40\\tnote{+}  &  2004-01-07 03:43:00 &  M45  \\\\ \\hline\n41\\tnote{+}  &  2004-09-12 00:04:00 &  M48  \\\\ \\hline\n42\\tnote{+}  &  2004-01-17 17:35:00 &  M50  \\\\ \\hline\n43\\tnote{+}  &  2005-07-27 04:33:00 &  M37  \\\\ \\hline\n44\\tnote{+}  &  2005-11-14 14:16:00 &  M39  \\\\ \\hline\n45\\tnote{+}  &  2005-08-02 18:22:00 &  M42  \\\\ \\hline\n46\\tnote{+}  &  2005-07-28 21:39:00 &  M48  \\\\ \\hline\n47\\tnote{+}  &  2006-04-27 15:22:00 &  M79  \\\\ \\hline\n \\end{tabular}\n  \\begin{tablenotes}\n  \\item[+] Missing Data in P6 and P7.\n  \\end{tablenotes}\n  \\end{threeparttable} \n\\end{table}\n\n\n\n\nOnly a portion of the collected data is used to train and test our\nclassifier. The positive class in this study is composed of X-Ray and proton\nchannels time series that led to $>$100 MeV SEP impulsive or gradual events. On\nthe other hand, the negative class is composed of X-Ray and proton channels\ntime series that did not lead to any $>$100 MeV SEP events. In order to\nselect such events we used a number of catalogs. For the positive class events\nwe used the same catalog of SEP events $>$100 MeV in\n\\cite{nunez2011predicting} that covers the events that happened between 1997 and\n2013.\n\n\\par \n\nOur positive class is composed of the 47 X-Ray parent events of their\ncorresponding $>$100 MeV SEP events that appear in \\cite{nunez2011predicting}\nand shown in Table~\\ref{sepevents}.\nWe use the X-Ray catalog\n(\\href{https:\/\/www.ngdc.noaa.gov\/stp\/space-weather\/solar-data\/solar-features\/solar-flares\/x-rays\/goes\/xrs\/}{\\textit{https:\/\/www.ngdc.noaa.gov\/stp\/space-weather\/solar-data\/solar-features\/solar-flares\/x-rays\/goes\/xrs\/}})\nas well as the CME catalog\n(\\href{https:\/\/cdaw.gsfc.nasa.gov\/CME\\_list\/}{\\textit{https:\/\/cdaw.gsfc.nasa.gov\/CME\\_list\/}})\nfrom the SOlar Heliospheric Observatory (SOHO) to derive the parent\nevents of the $>$100 MeV SEP events. There was an exception of two SEP events\nthat happened in August and September 1998 that we believe are gradual events but\ncould not map to any CME report due to the missing data during the SOHO mission\ninterruption. This latter happened because of the major loss of altitude\nexperienced by the spacecraft due to the failure to adequately monitor the\nspacecraft status, and an erroneous decision which disabled part of the on-board\nautonomous failure detection \\cite{nunez2011predicting}.\nIt is worth to note that we consulted the NOAA-prepared\nSEP events catalog along with their parent flare\/CME events\n(\\href{ftp:\/\/ftp.swpc.noaa.gov\/pub\/indices\/SPE.txt}{\\textit{ftp:\/\/ftp.swpc.noaa.gov\/pub\/indices\/SPE.txt}}).\nFor the case of events that are missing the NOAA catalog, we made our own\nflare\/CME-SEP events mapping. Fig.~\\ref{cats} shows the three external catalogs\nthat we used to produce our own catalog from which we generate our SEP\ndataset. \nTo obtain a balanced dataset, we selected another 47 X-ray events that did not\nproduce any SEP events that is, shown in Table~\\ref{ns}. We noticed\nthat there are nine SEP events (refer Table~\\ref{sepevents}\nID:29-37) that happened during the period when only GOES-12\nwas operational as can be seen in Fig.~\\ref{fig2}. At that period, channels P6\nand P7 failed and there was no secondary GOES.\nTo make sure not to create any biased classifier that relies on the missing data to\nmake the prediction, we made sure to choose nine events from the negative class\nas well that did not produce any SEP event (see Table~\\ref{ns}\nID:39-47).\n\nIn this paper we make a clear distinction\nbetween the two different classes of SEP events: gradual and impulsive. We\nassume that an SEP event is flare accelerated, and therefore impulsive, if the lag\nbetween the flare occurrence and the SEP onset time is very small and the peak\nflux intensity has reached a global peak few minutes to an hour after the onset\ntime. On the other hand, a gradual event shows a progressive increase in the\nproton flux trend that does not reach a global peak; instead, the peak is\nmaintained steadily before dropping again progressively. Finally, a non-SEP\nevent happens when there is an X-ray event of minimum intensity $M3.5$ that is\nnot followed by any significant proton flux increase in one of the P6-P11\nchannels. \n\n\\section{Methodology}\n\nThis section introduces a novel approach in predicting the occurrence of $>$ 100\nMeV SEP events based on interpretable decision tree models. We considered the\nX-ray and proton channels as multivariate time series that entail some\ncorrelations which may be precursors to the occurrence of an event. While \\cite\n{nunez2011predicting} considers the correlation between the X-ray and proton\nchannels only, we extended the correlation study into all the channels,\nincluding correlations that happen across different proton channels. We\napproached the problem from a multivariate time series classification\nperspective. The classification task being whether the observed time series\nwindows will lead to an SEP event or not. There are two ways of performing a\ntime series classification. The first approach, which first appeared in\n\\cite{xi2006fast}, is to use the raw time series data and find the\nK-nearest-neighbor with a similarity measures such as Euclidean distance,\nand dynamic time warping. This approach is effective when the time series of\nthe same label shows a distinguishable similar shape pattern. In this problem,\nthe time series that we are working with are direct instruments readings that\nshow a jitter effect, which is common in electromechanical device readings\n\\cite{scargle1982studies}.\nAn example of the jitter effect is shown in P10, and P11 in\nFigure.~\\ref{table1impulsive}-b and Figure.~\\ref{table1impulsive}-c. Time series\njitter makes it hard for distance measures, including elastic measures, to\ncapture similar shape patterns.\nTherefore, we explored the second time series classification approach that\nrelies on extracting features from the raw time series before feeding it to a\nmodel. In the next subsections, we will talk about the time series data\nextraction, the feature generation and data pre-processing.\n\n\\subsection{Data Extraction}\nOur approach starts from the assumption that a $>$100 MeV impulsive event may\noccur if the parent X-ray event peak is at least $M3.5$ as was suggested\nin \\cite{nunez2011predicting}. Therefore we carefully picked the negative class\nan X-ray event whose peak intensity is at least $M3.5$ but did not lead to any\nSEP event (refer column 3 in Table~\\ref{ns}). We extracted different\nobservation windows of data that we call a span. A span is defined as the number\nof hours that constitute the observation period prior to an X-ray event. A total\nof 94 (47*2) X-ray events (shown in column 3 and column 2 of\nTable~.\\ref{sepevents} and Table~.\\ref{ns} respectively) were extracted\nwith different span windows. The span concept is illustrated in the yellow\nshaded area in Figure.~\\ref{table1impulsive}. The span window, in this case is\n10 hours and stops exactly at the start time of the X-ray event. As we\nconsidered the five minutes as the cadence between reports, a 10-hour span\nwindow represents a 120-length multivariate X-ray and proton time series.\n\n\\subsection{Feature Generation}\n\nTo express the X-ray and proton cross-channel correlations we used a Vector\nAutoregression Model (VAR) which is a stochastic process model used to capture\nthe linear interdependencies among multiple time series. VAR is the extension of\nthe univariate autoregressive model to multivariate time series. The VAR model\nis useful for describing the behavior of economic, financial time series and for\nforecasting \\cite{zivot2006vector}. The VAR model permits us to express each\ntime series window as a linear function of past lags (values in the past) of\nitself and of past lags of the other time series. The lag $l$ signifies the\nfactor by which we multiply a value of a time series\nto produce its previous value in time.\nTheoretically, if there exists a magnetic connection between the Sun and Earth\nthrough the Parker spiral, the X-ray fluctuation precedes its corresponding\nproton fluctuation. Therefore, we do not express the X-ray channels in terms of\nthe other time series, but, we focus on expressing the proton channels with\nrespect to the past lags of\nthemselves and with past lags of the X-ray channels (xs and xl). The VAR model\nof order one, denoted as VAR(1) in our setting can be expressed\nby Equations.(\\ref{eq1})-(\\ref{eq6}).\n \nThere is a total of eight time series that represent the proton channels. Every\nequation highlights the relationship between the dependent variable and the\nother protons and X-ray variables, which are independent variables. The higher\nthe dependence of a proton channel on an independent variable, the higher is\nthe magnitude of the coefficient $||\\phi_{dependent\\_{independent}}||$.\nWe used the coefficients of the proton equations\nas a feature vector representing a data sample. The feature vector representing\na data point using the VAR(n) model is expressed in Equation.\\ref{vec}.\n  \nSince the lag parameter $l$ determines the number of coefficients involved in the\nequation, the number of features in the feature vector varies. More\nspecifically, the total number of features are 8 (independent variables) * 6 \n(dependent variables).\n\\begin{table*}\n\\begin{equation}\\label{eq1}\nP6_{t,1} = \\phi_{P6\\_{xs,1}}*P6_{t-1,1} + \\phi_{P6\\_{xl,1}}*P6_{t-1,1} +\n\\phi_{P6\\_{P6,1}}*P6_{t-1,1} + \\ldots +\n\\phi_{P6\\_{P11,1}}*P6_{t-1,1} + \\alpha_{P6_{t,1}}\n \\end{equation}\n \\begin{equation}\nP7_{t,1} = \\phi_{P7\\_{xs,1}}*P7_{t-1,1} + \\phi_{P7\\_{xl,1}}*P7_{t-1,1} +\n\\phi_{P7\\_{P6,1}}*P7_{t-1,1} + \\ldots +\n\\phi_{P7\\_{P11,1}}*P7_{t-1,1} + \\alpha_{P7_{t,1}}\n \\end{equation}\n \\begin{equation}\nP8_{t,1} = \\phi_{P8\\_{xs,1}}*P8_{t-1,1} + \\phi_{P8\\_{xl,1}}*P8_{t-1,1} +\n\\phi_{P8\\_{P6,1}}*P8_{t-1,1} + \\ldots +\n\\phi_{P8\\_{P11,1}}*P8_{t-1,1} + \\alpha_{P8_{t,1}}\n\\end{equation}\n\\centering \\vdots\n\\begin{equation}\\tag{6}\\label{eq6}\nP11_{t,1} = \\phi_{P11\\_{xs,1}}*P11_{t-1,1} + \\phi_{P11\\_{xl,1}}*P11_{t-1,1} +\n\\phi_{P11\\_{P6,1}}*P11_{t-1,1} + \\ldots +\n\\phi_{P11\\_{P11,1}}*P11_{t-1,1} + \\alpha_{P11_{t,1}}\n\\end{equation}\n\\end{table*}\n\n\\begin{align}\\tag{7}\n    x &= \\begin{bmatrix}\n           \\phi_{P6\\_{xs,1}} \\\\\n           \\phi_{P6\\_{xl,1}} \\\\\n           \\phi_{P6\\_{P6,1}} \\\\\n           \\phi_{P6\\_{P7,1}} \\\\\n           \\vdots \\\\\n           \\phi_{P11\\_{P8,n}}\\\\ \n           \\phi_{P11\\_{P9,n}}\\\\ \n           \\phi_{P11\\_{P10,n}}\\\\ \n           \\phi_{P11\\_{P11,n}}\\\\ \n         \\end{bmatrix} \\label{vec}\n  \\end{align} \n\n\\subsection{Data Preprocessing}\nBefore feeding the data to a classifier we cleaned the data from empty values\nthat appear in the generated features. To do so, we used the 3-nearest neighbors\nclass-level imputation technique. The method finds the 3 nearest neighbors\nthat have the same label of the sample with the missing feature. Nearest\nneighbors imputation weights the samples using the mean squared difference on\nfeatures based on the other non-missing features. Then it imputes the missing\nvalue with the nearest neighbor sample. The reason why\nthe imputation is done on a class level basis is that features may behave\ndifferently across the two classes (SEP and non-SEP), therefore; it is\nimportant to impute the missing data with the same class values.\n\n\\section{Experimental Evaluation}\n\nIn this section we explain the decision tree model that we will be using as well\nas the sampling methodology. We will also provide a rationale for the choice of\nparameters ($l$ and $span$). Finally we will zoom in the best model\nwith the most promising performance levels.\n\n\n\\subsection{Decision Tree Model}\n\nA decision tree is a hierarchical tree structure used to determine classes based\non a series of rules\/questions about the attribute values of the data points\n\\cite{safavian1991survey}.\nEvery non-leaf node represents an attribute split (question) and all the leaf\nnodes represent the classification result. In short, given a set of features with\ntheir corresponding classes a decision tree produces a sequence of\nquestions that can be used to recognize the class of a data sample.\nIn this paper, the data attributes are the VAR($l$) coefficients\n$[\\phi_{p6\\_{xs, 1}}, \\phi_{p6\\_{xl, 1}}, ...,  \\phi_{p6\\_{xs, l}}]$ and the\nclasses are binary: SEP and non-SEP.\n\nThe decision tree classification model first starts by finding the variable that\nmaximizes the separation between classes.\nDifferent algorithms use different metrics, also called purity measures, for\nmeasuring the feature that maximizes the split. Some splitting criteria include\nGini impurity, information gain, and variance reduction. The commonality between\nthese metrics is that they all measure the homogeneity of a given feature with\nrespect to the classes. The metric is applied to each candidate feature to\nmeasure the quality of the split, and the best feature is used. In this paper we\nused the CART decision tree algorithm, as appeared in\n\\cite{loh2011classification} and \\cite{steinberg2009cart}, with Gini and\ninformation gain as the splitting criteria.\n\n\\subsection{Parameter Choice}\n\n \\begin{figure*} \n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figs\/SpanAcc.pdf}\n     \\caption{Decision tree accuracy with respect to the span window and the\n     lag parameters using Gini and information gain splitting criteria. The\n     dotted line shows a linear fit to the accuracy curve. }\n       \\label{spanacc}\n\\end{figure*} \n\nOur approach relies heavily on the choice of parameters, namely, the span window\nand the VAR model lag parameter. The span is the number of observation hours\nthat precede the occurrence of an X-ray event. The latter determines the length\nof the multivariate time series to be extracted. On the other hand, the lag\n($l$) determines the size of the feature space that will be used as well as the\nlength of the dependence of the time series with each other in the past.\nAs mentioned previously, with a one-step increment of the lag parameter the size of the\nfeature space almost doubles $features\\_number$ = 8*(independent variables)*6\n(equations)*$l$+6*(equations). In order to determine the optimal parameters to\nbe used, we run a decision tree model on a set of values for both the span and\nlag parameters. More specifically, we used the range [3-30] for the span window\nand the set \\{1,3,5,7,9\\} for $l$. Since we have a balanced dataset we used a\nstratified Ten-fold cross validation as the sampling methodology. A stratified\nsampling always ensures a balance in the number of positive and negative samples for\nboth the training and testing data samples. Ten-fold cross-validation randomly\nsplits the data into 10 subsets, models are trained with nine of the\nfolds (90\\% of the dataset), and test it with one fold (10\\% of the dataset).\nEvery fold is used once for testing and nine times for training. In our\nexperiments, we report the average accuracy on the 10 folds.\n Fig.~\\ref{spanacc} illustrates the accuracy curves with respect to the span\n windows for the five lags that we considered. We reported the accuracies of the\n decision tree model using both gini and information gain splitting criteria. In\n order to better capture the model behavior with the increasing span we\n plotted a linear fit to the accuracy curves of each lag. The first observation\n that can be made is that the slopes of the linear fit for $l$=1 and $l$=3 are\n relatively small in comparison to the other lags ($l>$3). This signifies that\n the model does not show any increasing or decreasing accuracy trend with the\n increase of the span window. Therefore we conclude that $l$=1 and $l$=3 are\n too small to discover any relationship between the proton and X-ray channels.\n Having the lag parameter set to $l$=1 and $l$=3 corresponds to expressing the\n time series (independent variable) going back in time up to five minute and 15\n minutes respectively. These latter times are small, especially for $l$=1 (5\n minutes), which theoretically is not possible since the\n protons can at most reach the speed of light that corresponds to a lag of at least 8.5\n minutes. For the other lags ($l>3$) there is noticeable increase in steepness in\n the accuracy linear fit which suggests that the accuracy increases with\n the increasing span window. The second observation is that for all the $l>3$\n datasets the best accuracy was achieved in the last four span window (i.e span\n $\\in$ \\{27,28,29,30\\}). Therefore, we filtered the initial range of\n parameter values to \\{5,7,9\\} for $l$ and \\{27,28,29,30\\} for the span. In the\n next subsection we will zoom in into every classifier within the parameter\n grid.\n \n \\subsection{Learning Curves}\nTo be able to discriminate decision tree models that show similar\naccuracies we use the model learning curves, also called experience curves, to\nhave an insight in how the accuracy changes as we feed the model with more\ntraining examples.\nLearning curves are particularly useful for comparing different\nalgorithms \\cite{madhavan1997new} and choosing optimal model parameters during\nthe design \\cite{pedregosa2011scikit}. It is also a good tool for visually\ninspecting the sanity of the model in case of overtraining or undertraining.\nFigs.~\\ref{lcgini} and \\ref{lcentropy} show the learning curves of the\ndecision tree model using gini and information gain as the splitting criteria\nrespectively. The red line represents the training accuracy which evaluates the\nmodel on the newly trained data. The green line shows the testing\naccuracy which evaluates the model on the the never-seen testing data. The\nshaded area represents the standard deviation of the accuracies after running\nthe model multiple times with the same number of training data. It is noticeable\nthat the standard deviation becomes higher as the lag is increased. Also, it can\nbe seen that the best average accuracies, that appeared in Fig.~\\ref{spanacc}, are\nnot always the ones that have the best learning curves. For example from\nFig.~\\ref{spanacc}, the best accuracy that has been reached appears to be in\n$l$=7 and $span=27,29$; however, the learning curves corresponding to that span\nand lag show that the standard deviation is not very smooth as compared to\n$l$=5. Therefore the experiments show that using $l$=5 results in relatively\nstable models with lower variance. Therefore, we will\nzoom in $l$=5 for all the spans $\\in \\{27,28,29,30\\}$ that we previously\nfiltered.\n\\begin{figure*} \n \\centering \n \\begin{tabular}{ c  c  c  c  c }\n\n  & Span 27 & Span 28 & Span 29 & Span 30 \\\\\n \n \\begin{turn}{90} \\hspace{10mm} Lag 5  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/5s27_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/5s28_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/5s29_gini_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/5s30_gini_m.png}\n  \\\\\n \\begin{turn}{90} \\hspace{10mm} Lag 7  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/7s27_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/7s28_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/7s29_gini_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/7s30_gini_m.png}\n  \\\\\n \n \\begin{turn}{90} \\hspace{10mm} Lag 9  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/9s27_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/9s28_gini_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/9s29_gini_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/9s30_gini_m.png}\n  \\\\\n  \\end{tabular}\n\n \t\\caption{Learning curve of CART Decision Tree Models with Gini splitting\n criterion,spans $\\in$ \\{27,28,29,30\\} and lag $\\in$ \\{5,7,9\\}\\label{lcgini}}\n \n\\end{figure*}\n\n\\begin{figure*} \n\n \\centering \n \\begin{tabular}{ c  c  c  c  c }\n\n  & Span 27 & Span 28 & Span 29 & Span 30 \\\\\n \n \\begin{turn}{90} \\hspace{10mm}  Lag 5  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/5s27_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/5s28_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/5s29_entropy_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/5s30_entropy_m.png}\n  \\\\\n \\begin{turn}{90} \\hspace{10mm}  Lag 7  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/7s27_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/7s28_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/7s29_entropy_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/7s30_entropy_m.png}\n  \\\\\n \n \\begin{turn}{90} \\hspace{10mm}  Lag 9  \\end{turn}&\n \\includegraphics[width=0.221000001\\linewidth]{figs\/9s27_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/9s28_entropy_m.png} &\n \\includegraphics[width=0.22\\linewidth]{figs\/9s29_gini_m.png}&\n \\includegraphics[width=0.22\\linewidth]{figs\/9s30_entropy_m.png}\n  \\\\\n \n  \\end{tabular}\n   \\caption{Learning curve of CART Decision Tree Models with information gain\n   splitting criterion,spans $\\in$ \\{27,28,29,30\\} and lag $\\in$ \\{5,7,9\\}\\label{lcentropy} }\n\\end{figure*}\n\n\n\\begin{table}[]\n\\centering\n\\caption{Decision Tree model evaluation for gini and information gain splitting\ncriteria\\label{eval}}\n\\label{my-label}\n\\begin{tabular}{l|l|l|l|l|l|l|l|l|}\n\\cline{2-9}\n                                & \\multicolumn{4}{c|}{Gini}          & \\multicolumn{4}{c|}{Information Gain} \\\\ \\hline\n\\multicolumn{1}{|l|}{Span}      & 27   & 28   & 29   & 30            & 27    & 28    & 29    & 30            \\\\ \\hline\n\\multicolumn{1}{|l|}{Accuracy}  & 0.64 & 0.74 & 0.73 & \\textbf{0.74} & 0.77  &0.70   & 0.67  & \\textbf{0.78} \\\\ \\hline\n\\multicolumn{1}{|l|}{Recall}    & 0.69 & 0.70  & 0.74 & \\textbf{0.70}  & 0.76 &0.70  & 0.70   & \\textbf{0.73} \\\\ \\hline\n \\multicolumn{1}{|l|}{Precision} & 0.62 & 0.75 & 0.75 & \\textbf{0.76} & 0.78  & 0.72  & 0.72  & \\textbf{0.86} \\\\ \\hline\n  \\multicolumn{1}{|l|}{F1}        & 0.65 & 0.75 & 0.75 & \\textbf{0.74} & 0.79  &0.71  & 0.69  & \\textbf{0.82} \\\\ \\hline\n \\multicolumn{1}{|l|}{AUC}       & 0.65& 0.72 & 0.74 & \\textbf{0.73} & 0.76  & 0.70   & 0.69  & \\textbf{0.77} \\\\ \\hline\n\n\\end{tabular}\n\\end{table}\n\n\n \n\n \\begin{figure*} \n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figs\/PCA.pdf}\n     \\caption{First 3 PCA components derived from (a) all the original 254 features, (b)\n     the data sub-space containing only 4 parameters selected as the most\n     relevant by the Gini index (as shown in the tree presented in Fig. 9),\n     and (c) another data sub-space containing 4 different parameters (with 1\n     repetition) selected as the most relevant by the Entropy measure. The\n     PCA-based visualizations represent (sub-)spaces of the same data set (as\n     shown in the tree presented in Fig. 10), with lag=5, and span=30.}\n     \n       \\label{pca}\n\\end{figure*} \n\n\n \\begin{figure} \n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/GiniDT.pdf}\n     \\caption{Decision Tree with Gini splitting criteria ($span$=30, $l$=5) }\n       \\label{giniDT}\n\\end{figure}  \n\n \\begin{figure} \n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/EntropyDT.pdf}\n     \\caption{Decision Tree with information gain splitting criteria\n     ($span$=30, $l$=5) }\n       \\label{entropyDT}\n\\end{figure}\n\nTo determine the best behaving model we choose six evaluation metrics that will\nassess the models' performance from different aspects. Accuracy is the most\nstandard evaluation measure used to assess the quality of a classifier by\ncounting the ratio of correct classification over all the classifications.\nIn this context the accuracy measure is particularly useful because our\ntraining and testing data is balanced. The data balance ensures that if the\nclassifier is highly biased toward a given class it will be reflected on the\naccuracy measure. Recall is the second evaluation measure we considered, also\nknown as the probability of detection, which characterizes the\nability of the classifier to find all of the positive cases. Precision is\nused to evaluate the model with respect to the false alarms. In fact, precision\nis 1 - false alarm ratio. Precision and recall are usually anti-correlated;\ntherefore, a useful quantity to compute is their harmonic mean, the F1 score.\nThe last evaluation measure that we consider in the Area Under Curve (AUC) of the Receiver\nOperating Characteristic curve (ROC) curve. The intuition behind this measure is\nthat AUC equals the probability that a randomly chosen positive example ranks\nabove (is deemed to have a higher probability of being positive than) a randomly\nchosen negative example. It has been claimed in \\cite{auccp} that the AUC is\nstatistically consistent and more discriminating than accuracy.\n\n\n\nTable~.\\ref{eval} shows the aforementioned evaluations on the $l$=5 datasets. It\nis noticeable that span=30 achieves the best performance levels for both\nsplitting criteria. The decision tree models corresponding to those settings\nusing gini and information gain are shown in Fig.~\\ref{giniDT} and\nFig.~\\ref{entropyDT} respectively. For the purpose of visualization we used PCA\ndimensionality reduction technique to plot the full feature space with the 254 dimensions of\nthe lag 5 and span 30 in Fig.~\\ref{pca}-a, as well as the reduced feature space\nwith only the selected features from the gini measure in Fig.~\\ref{pca}-b and\nentropy measure in Fig.~\\ref{pca}-c \\cite{PCA}.\nIt is clearly visible that the SEP and non-SEP classes are almost\nindistinguishable when all the dimensions are used. When the decision tree\nfeature selection is applied, the data points become more scattered in space and\ntherefore easier for the classifier to distinguish. We also note that both\ndecision tree classifiers have as a root a proton x-ray correlation parameter\n($P6\\_xl\\_l2$). Some of the intermediate and leaf nodes have features that show\ncorrelations between proton channels is their conditions. This suggests that\ncross-correlations in proton channels are equally important to X-ray and proton\nchannels correlations that appeared in \\cite{nunez2011predicting}. Our best\nmodel shows a descent accuracy that is comparable (3\\% better) to the UMASEP\nsystem that uses the same catalog. We also made sure that our model is not\nbiased towards the missing data of the lower energy channels P6 and P7 of\nGOES-12 by choosing the same number of samples of positive and negative class\nthat happened during the GOES-12 coverage period.\n\n\n\n\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{Conclusion}\nIn this paper we designed a new model to predict $>$100 MeV SEP events based on\nGOES satellite X-ray and proton data. This is the first effort that explores\nnot only the dependencies between the X-ray and proton channels but also the\nauto-correlations and cross-correlations within the proton channels. We have\nfound that proton channel cross-correlations based on a lag time (prior point\nin time) can be an important precursor as to whether an SEP event may happen or\nnot. In particular, we started finding patterns starting from lag 5 and our best\nmodels shows both that the correlation between proton channel $P6$ and X-ray\nchannel $xl$ is an important precursor to SEP events.\nBecause of the missing data due to the failure of the P6 and P7 proton\nchannels onboard GOES-12 we made sure that our dataset uses the same number of\npositive and negative examples coming from GOES-12. To our knowledge this is\nthe first study that explores proton cross-channels correlations in order to\npredict SEP events. As a future extension of this study we are interested in\ndoing ternary classification by further splitting the SEP class into impulsive\nand gradual. We are also interested in real-time SEP event predictions for\npractical applications of this research.\n\n\n\n\n\\section*{Acknowledgment}\nWe thank all those involved with the GOES missions as well as the SOHO\nmission.\nWe also acknowledge all the efforts of NOAA in making the catalogs and X-ray\nreports available.\nThis work was supported in part by two NASA Grant Awards (No.\nNNX11AM13A, and No. NNX15AF39G), and one NSF Grant (No. AC1443061). The\nNSF Grant has been supported by funding from the Division of Advanced\nCyberinfrastructure within the Directorate for Computer and Information Science\nand Engineering, the Division of Astronomical Sciences within the Directorate\nfor Mathematical and Physical Sciences, and the Division of Atmospheric and\nGeospace Sciences within the Directorate for Geosciences.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFor studies of the hadronic final state in high-energy collisions, \nversatile programs for the calculation of QCD corrections are required.\nThe extraction of scale-dependent \nphysical quantities such as the running strong coupling\nconstant $\\alpha_s\\left(Q^2\\right)$ and parton densities\n$f_i\\left(\\xi,Q^2\\right)$ requires precise predictions \nin next-to-leading order of QCD perturbation theory.\nAt the electron--proton collider HERA at DESY in Hamburg, \nthe strong coupling constant has been measured via jet rates \n\\cite{1,2}. There is also a direct fit of the gluon density \n$f_g(\\xi, Q^2)$ \\cite{3} based on a Mellin transform\nmethod \\cite{4,5}. Calculations for jet cross sections \nin deeply inelastic scattering for the case of the modified JADE \nscheme\nhave been performed \\cite{6,7,8,9,10}\\footnote{\nIn these calculations, terms of the form $c\\log c$, \n$c$ being the jet cut, have been neglected. This implies in particular\na certain insensitivity with respect to the jet recombination scheme.\nThe set-up of the calculations \\cite{6,7,9} is such that\na jet consisting of two partons is always mapped onto a massless jet. \nTherefore the jet definition scheme which is used on experimental data\nshould be a ``massless'' scheme (this excludes, for example, the E-scheme).\nThe variation of jet cross sections within the possible massless schemes\ncannot be modelled by that calculation.}\nand implemented in the \ntwo programs {\\tt PROJET} \\cite{11} and {\\tt DISJET} \\cite{12}.\n\nIn the meantime,\ncalculations for arbitrary infrared-safe observables in \ndeeply inlastic scattering have become available \\cite{13,14}.\nIn the last few years, the technology for the calculation\nof QCD corrections in next-to-leading order\nhas developed considerably. \nThe main problem in higher-order QCD calculations is the occurence of\nsevere\ninfrared singularities (they ultimately cancel for infrared-safe\nobservables, or are absorbed into process-independent, physical\ndistribution functions such as parton densities and fragmentation functions).\nThere are explicit algorithms available\nwhich permit the calculation to be done in a ``universal'' way: the \ninfrared singularities are subtracted such that arbitrary \ninfrared-safe observables can be calculated numerically. In principle, \nall existing algorithms are variations on a common theme, namely the\ninterplay of the factorization theorems of perturbative QCD and the \ninfrared-safety of the observables under consideration.\nThere are two different ways to achieve the separation of\ndivergent and finite contributions:\nthe phase-space-slicing method \\cite{15} and\nthe subtraction method \\cite{16}. \nBoth methods have their merits and drawbacks.\n\\begin{description}\n\\item[{\\rm\\unboldmath ($\\alpha$)}] \nThe phase-space-slicing method relies on a separation of\nsingular phase-space regions from non-singular ones by means of a \nsmall slicing parameter $s\\rightarrow 0$. The divergent parts are evaluated \nunder the assumption that terms of ${\\cal O}(s (\\log s)^n)$ can be dropped.\nThe analytically evaluated phase-space integrals yield terms of the form\n$(\\log s)^m$, which explicitly cancel against equivalent terms of opposite\nsign from a numerically performed phase-space integration.\nThe simplicity of this scheme is obvious.\nThe main problem is the residual dependence on the technical cut \nparameter~$s$\n(in practice the limit $s\\rightarrow 0$ is not checked for every observable, \nbut it is assumed that a fixed small value will be sufficient).\nMoreover, the numerical cancellation of the logarithmic terms by means\nof a Monte-Carlo integration is delicate.\nThere is a calculational scheme available for the determination of \nthe explicit phase space integrals over the singular regions \\cite{17}. \nFor initial and final-state hadrons this scheme moreover \nrequires the introduction\nof so-called {\\it crossing functions} \\cite{18}, \nto be evaluated for every parton density parametrization.\nFor deeply-inelastic lepton--nucleon scattering, an implementation \nof this calculational scheme is provided by Mirkes and Zeppenfeld\nin {\\tt MEPJET} \\cite{19}.\n\\item[{\\rm\\unboldmath($\\beta$)}] \nThe subtraction method is technically more involved, \nsince the infrared singularities are cancelled point-by-point in \nphase space. The subtraction terms have, owing to the factorization\ntheorems of perturbative QCD, a simple form. The problem is \nto arrange the subtractions in such a way that in the numerical evaluation\nno spurious singularities appear. A general framework, using a specific\nphase space mapping besides the factorization theorems, is given by Catani\nand Seymour in  Ref.~\\cite{20}, and implemented in {\\tt DISENT} \n\\cite{21}.\n\nThe approach of the present \npaper is to use a generalized partial fractions\nformula to separate the singularities \\cite{22}. The method is\nbriefly explained in Section~\\ref{algorithm}. We will describe in some\ndetail the implementation {\\tt DISASTER++}\\footnote{\nThis is an acronym for ``Deeply Inelastic Scattering: All Subtraction Through\nEvaluated Residues''.}\nin the form of a {\\tt C++} class library.\n\nThere are two reasons for a new calculation.\n(a) The existing\nprograms have the restriction that the number of flavours is fixed  \n($N_f=5$ in the case of {\\tt MEPJET}\nand $N_f$ fixed, but arbitrary for {\\tt DISENT}).\nFor studies of the scale-dependence it is\nnecessary to\nhave a variable number of flavours,\nin order to be consistent with the scale evolution\nof the strong coupling constant and the parton densities.\n{\\tt DISASTER++} makes the $N_f$ dependence explicit in the ``user routine''\non an event-by-event basis,\nand thus results for several renormalization and factorization scales\ncan be calculated simultaneously.\n(b) {\\tt DISASTER++}\nis already set up such that the extension to one-particle-inclusive   \nprocesses will be possible without the necessity of re-coding\nthe contributions which are already present for\nthe jet-type observables. This option will be made available\nin future versions of the program, as soon as the remaining contributions\nfor one-particle-inclusive processes are implemented.\n\n\\end{description}\n\nThe outline of this paper is as follows. In Section~\\ref{algorithm}\nwe briefly review the algorithm employed in the present calculation.\nIn Section~\\ref{structure} the {\\tt FORTRAN} interface\nto the {\\tt C++} class library is described. \nSome remarks concerning the installation of the package are made\nin Section~\\ref{installation}.\nA comparison of the available universal programs \n{\\tt DISASTER++} (Version 1.0), {\\tt MEPJET} (Version 2.0) \nand {\\tt DISENT} (Version 0.1)\nis presented in \nSection~\\ref{comparison}.\nIn a previous version of this paper, we have drawn \nthe conclusion that we find an overall, but not completely satisfactory\nagreement of {\\tt DISASTER++} and {\\tt MEPJET}, and \nthat there are large deviations when comparing\n{\\tt DISASTER++} and {\\tt DISENT}.\nOne of the purposes of this paper is to present the results of a comparison\nof {\\tt DISASTER++} and a new, corrected version (0.1) \nof {\\tt DISENT}. We now find\ngood agreement of the two programs.\nWe also give a few more results for {\\tt MEPJET}, in particular\nfor the dependence on the technical cut~$s$. It turns out that\neven for very small values of~$s$ \nwe do not achieve agreement with the\n{\\tt DISASTER++}~\/ {\\tt DISENT} results\nfor several cases under\nconsideration\\footnote{\nIn a very recent paper \\cite{23}, E.~Mirkes quotes the results of the\ncomparison of the three programs as performed in the \nprevious version of this paper \\cite{24} as resulting in \na ``so far satisfactory agreement''. This is a \nmisquotation. The formulation in Ref.~\\cite{24} was that \nfor {\\tt MEPJET} and {\\tt DISASTER++} we find an ``overall, though not \ncompletely satisfactory agreement'', and that the results of {\\tt DISENT}\n(Version 0.0) ``differ considerably''. Moreover, in the summary\nof Ref.~\\cite{24} we mention that a few deviations of {\\tt MEPJET} and\n{\\tt DISASTER++} are present. We wish to stress that there is a certain \nsemantic\ngap between the expression \n``satisfactory agreement'' and the results just quoted.\n}.\nThe paper closes with a summary.\nThe contents of this paper are mainly technical. The details of the calculation\nand phenomenological applications will be described in a forthcoming \npublication.\n\n\\section{The Algorithm}\n\\label{algorithm}\nThe calculation is based on the subtraction method. A simple example\nto illustrate this method in general, and a comparison \nwith the phase-space-slicing\nmethod, is given in Ref.~\\cite{25}.\nFor a more detailed exposition of the contents of this section, \nsee Ref.~\\cite{22}.\n\nThe subtraction method is one of the solutions for the problem of \nhow to \ncalculate \nnumerically \ninfrared-safe observables without having \nto modify the calculation for every observable under consideration.\nIn QCD calculations, infrared singularities cancel for sufficiently \ninclusive observables. \nThe factorization theorems of perturbative\nQCD (see Ref.~\\cite{26} and references therein) \ntogether with the infrared-safety of the observable under consideration\nguarantee that \nthe structure of the limit of the convolution of the parton-level cross \nsection with the observable in soft and collinear regions of phase space\nis well-defined and factorizes in the form of a product of a kernel \nand the Born term.\nThis property allows, for the real corrections, the definition of a subtraction \nterm for every phase-space point.\nFormally:\n\\begin{eqnarray}\n\\int\\mbox{dPS}^{(n)}\\,\\sigma\\,{\\cal O}\n&=& \\sum_A \\int \\mbox{dPS}_{i_A} \\,k_A \\left(\n\\int \\mbox{dPS}^{(n-1)} \\tau_A -\n \\left[\n   \\int \\mbox{dPS}^{(n-1)} \\tau_A\n \\right]_{\\mbox{\\scriptsize soft\/coll.~limit}}\n\\right)\\nonumber\\\\\n&+& \\sum_A \\int \\mbox{dPS}_{i_A} \\,k_A \\left[\n   \\int \\mbox{dPS}^{(n-1)} \\tau_A\n    \\right]_{\\mbox{\\scriptsize soft\/coll.~limit}},\n\\end{eqnarray}\nwhere $\\sigma$ is the parton-level cross section, $\\cal O$ is the\ninfrared-safe observable, $k_A$ is a singular kernel, \nand $\\tau_A$ is the non-singular part of the product $\\sigma\\,\\cal O$.\nThe index~$A$ runs over all possible soft, collinear and \nsimultaneously soft and collinear singularities of~$\\sigma$.\nThe first integral is finite and can be calculated numerically. The second\nintegral contains all infrared singularities. The term in the square bracket\nhas a simple structure\nbecause of the factorization theorems of QCD, and the one-particle\nintegral over the kernel $k_A$ and the factorization contribution from the\nterm in the square brackets can be performed easily.\nThis subtraction formula works only if the subtraction terms do not\nintroduce spurious singularities for the individual terms that eventually\ncancel in the sum. This is achieved by a separation of all singularities\nby means of a general partial fractions formula\n\\begin{equation}\n\\label{pfid}\n\\frac{1}{x_1\\,x_2\\cdots x_n}\n=\\sum_{\\sigma\\in S_n}\n\\frac{1}{x_{\\sigma_1}\\,(x_{\\sigma_1}+x_{\\sigma_2})\\cdots\n         (x_{\\sigma_1}+\\ldots+x_{\\sigma_n})},\n\\end{equation}\nwhere the sum runs over all $n!$ permutations of $n$~objects.\n\nIn {\\tt DISASTER++}, the processes for (1+1) and (2+1)-jet production \nfor one-photon exchange are implemented. The program itself, however, \nis set up in a much more general way. The implemented subtraction procedure \ncan handle arbitrary number of final-state partons, and zero or one incoming \npartons (an extension to two incoming partons is possible). The {\\tt C++}\nclass library is intended to provide a very general framework for \nnext-to-leading-order QCD calculations for arbitrary \ninfrared-safe observables. Of course, the explicit matrix \nelements (Born terms, virtual corrections and factorized real corrections)\nhave to be provided for every additional process to be included.\n\n\\section{Program Structure}\n\\label{structure}\nWe now describe the {\\tt FORTRAN} interface to the {\\tt C++}\nclass library. The {\\tt C++} user interface will be documented in a \nforthcoming extension of this manual.\n\nTo set the stage, let us first introduce some terminology.\nThe user has to provide several subroutines which are called by \n{\\tt DISASTER++} for every generated event. Each {\\bf event} \n$e_n$, $n=1\\ldots N$\nconsists of a set of\n{\\bf phase spaces} ${\\cal P}_{nr}$, $r=1\\ldots R_n$, \nand a set of {\\bf contributions} ${\\cal C}_{ni}$, \n$i=1\\ldots L_n$. Phase spaces $\\cal P$\nprovide a set of four-vectors of initial and final-state\nparticles, which are used to calculate observables\n${\\cal O}({\\cal P})$.\nContributions ${\\cal C}_{ni}$ consist of a list of \n{\\bf weights} $w_{nij}$, $j=1\\ldots K_{ni}$ (here: \n$K_{ni}=11$) which have to be multiplied\nby certain {\\bf flavour factors} $F_{nij}$. \nEvery contribution ${\\cal C}_{ni}$ has an associated\nphase space ${\\cal P}_{nr_{ni}}$; \nit is generally the case that particular phase spaces are \nused for different contributions. Flavour factors are products\nof parton densities, quark charges, powers of the strong coupling constant, \nand powers of the electromagnetic coupling constant.\n\nThe expectation value $\\langle {\\cal O} \\rangle$ \nof a particular observable is given by the following \nsum:\n\\begin{equation}\n\\label{exval}\n\\langle {\\cal O} \\rangle = \n   \\sum_{n=1}^N\n   \\sum_{i=1}^{L_n}\n   {\\cal O}({\\cal P}_{nr_{ni}})    \n   \\sum_{j=1}^{K_{ni}}\n   w_{nij} F_{nij}.\n\\end{equation}\nThe first sum is the main loop of the Monte Carlo integration.\n\n\\noindent\nThe user has to provide a subroutine\n{\\tt user1} and\na function \n{\\tt user2}.\nThe subroutine\n{\\tt user1(iaction)} is called from {\\tt DISASTER++} in the following cases:\n\\begin{description}\n\\item{\\quad{\\tt iaction=1}:} {\\ }\\\\after start-up of {\\tt DISASTER++}\n\\item{\\quad{\\tt iaction=2}:} {\\ }\\\\before the end of {\\tt DISASTER++}\n\\item{\\quad{\\tt iaction=3}:} {\\ }\\\\before the start of the grid-definition \nrun of the adaptive Monte-Carlo routine, or before the final run of\nthe adaptive integration, in case that there is no grid-definition run\n\\item{\\quad{\\tt iaction=4}:} {\\ }\\\\before the start of the final\nrun of the adaptive Monte-Carlo routine\n\\item{\\quad{\\tt iaction=5}:} {\\ }\\\\after the final\nrun of the adaptive Monte-Carlo routine\n\\item{\\quad{\\tt iaction=6}:} {\\ }\\\\once for every event (to initialize intermediate \nweight sums, etc.)\n\\item{\\quad{\\tt iaction=7}:} {\\ }\\\\signals that the event has to be dropped\nfor technical reasons\n\\end{description}\n\n\\noindent\nThe function {\\tt user2(...)} is called from {\\tt DISASTER++}\nafter an event has been constructed.\nIt has the following arguments (in an obvious\nnotation):\n\\begin{verbatim}\n      double precision function\n     &                 user2(\n     &                   integer nr, \n     &                   integer nl,\n     &                   double precision fvect(0..3, -10..10, 1..30),\n     &                   integer npartons(1..30),\n     &                   double precision xb(1..30),\n     &                   double precision q2(1..30),\n     &                   double precision xi(1..30),\n     &                   double precision weight(1..11, 1..50),\n     &                   integer irps(1..50),\n     &                   integer ialphas(1..50),\n     &                   integer ialphaem(1..50),\n     &                   integer lognf(1..50)\n     &                 )\n\\end{verbatim}\n\nHere {\\tt nr} stands for $R_n$, {\\tt nl} stands for $L_n$, \n{\\tt fvect(mu, iparticle, ir)} is the {\\tt mu}$^{\\mbox{th}}$ component\nof the four-vector of the particle \nwith label {\\tt iparticle} ({\\tt mu}=0 corresponds to the energy component)\nin units of [GeV]\nin the Breit frame \nfor the phase space {\\tt ir};\n{\\tt npartons(ir)} is the number of final-state partons,\n{\\tt q2(ir)} is the value of $Q^2$, and {\\tt xi(ir)} is the momentum fraction\nof the incident parton.\nThe particle labels {\\tt iparticle} are given by\n\\begin{description}\n\\item[\\quad{\\tt iparticle=-8:}] proton remnant\n\\item[\\quad{\\tt iparticle=-7:}] incident proton\n\\item[\\quad{\\tt iparticle=-5:}] outgoing electron\n\\item[\\quad{\\tt iparticle=-4:}] incident electron\n\\item[\\quad{\\tt iparticle=-1:}] incident parton\n\\item[\\quad{\\tt iparticle=0..(npartons-1):}] outgoing partons\n\\end{description}\n\nThe array {\\tt weight(j, i)} is a list of the weights for contribution\n{\\tt i} in units of [pb], \n{\\tt irps(i)} gives the index of the phase space for this particular \ncontribution,\n{\\tt ialphas(i)} and {\\tt ialphaem(i)} are the powers of the strong \nand electromagnetic coupling constant, respectively, and {\\tt lognf(i)}\nis an index that specifies whether the weights have to be multiplied \nby a factor $\\lambda$ consisting of a product of\na logarithm of a scale and\/or a factor of $N_f$:\n\\begin{description}\n\\item[\\quad{\\tt lognf=0}:] $\\lambda=1$\n\\item[\\quad{\\tt lognf=1}:] $\\lambda=\\ln\\left(\\mu_r^2\/Q^2\\right)$\n\\item[\\quad{\\tt lognf=2}:] $\\lambda=N_f \\ln\\left(\\mu_r^2\/Q^2\\right)$\n\\item[\\quad{\\tt lognf=3}:] $\\lambda=\\ln\\left(\\mu_f^2\/Q^2\\right)$\n\\item[\\quad{\\tt lognf=4}:] $\\lambda=N_f \\ln\\left(\\mu_f^2\/Q^2\\right)$\n\\end{description}\nHere $\\mu_r$ and $\\mu_f$ are the renormalization and factorization scales, \nrespectively.\nThe total flavour factor for contribution $i$ is given by \n\\begin{equation}\nF_{nij} = \n   \\lambda \\, \n   \\alpha_s^{\\mbox{\\tt ialphas($i$)}} \\,\n   \\alpha^{\\mbox{\\tt ialphem($i$)}} \\,\n   \\rho_{ij},\n\\end{equation}\nwhere \nthe quantity $\\rho_{ij}$ is a product of squares of quark charges $Q_\\alpha$ \nin units of $e$ and parton densities.\nIn particular:\n\\begin{description}\n\\item \\quad$\\rho_{i1}\n               = \\sum\\limits_{\\alpha=1}^{N_f} Q_\\alpha^2 \\, f_\\alpha$\n\\item \\quad$\\rho_{i2} \n               = \\sum\\limits_{\\alpha=1}^{N_f} Q_\\alpha^2 \\,  \n                 f_{\\overline{\\alpha}}$\n\\item \\quad$\\rho_{i3} \n               = \\sum\\limits_{\\alpha=1}^{N_f} Q_\\alpha^2 \\, f_g$\n\\item \\quad$\\rho_{i4} = \\rho_{i1}$\n\\item \\quad$\\rho_{i5} = \\rho_{i2}$\n\\item \\quad$\\rho_{i6} = \\rho_{i1}\\,(N_f-1)$\n\\item \\quad$\\rho_{i7} = \\rho_{i2}\\,(N_f-1)$\n\\item \\quad$\\rho_{i8}\n               = \\sum\\limits_{\\alpha=1}^{N_f} f_\\alpha \\,\n                 \\sum\\limits_{\\beta=1,\\, \\beta \\neq \\alpha}^{N_f} Q_\\beta^2$\n\\item \\quad$\\rho_{i9}\n               = \\sum\\limits_{\\alpha=1}^{N_f} f_{\\overline{\\alpha}} \\,\n                 \\sum\\limits_{\\beta=1,\\, \\beta \\neq \\alpha}^{N_f} Q_\\beta^2$\n\\item \\quad$\\rho_{i10}\n               = \\sum\\limits_{\\alpha=1}^{N_f} f_\\alpha Q_\\alpha \\,\n                 \\sum\\limits_{\\beta=1,\\, \\beta \\neq \\alpha}^{N_f} Q_\\beta$\n\\item \\quad$\\rho_{i11}\n               = \\sum\\limits_{\\alpha=1}^{N_f} f_{\\overline{\\alpha}} Q_\\alpha \\,\n                 \\sum\\limits_{\\beta=1,\\, \\beta \\neq \\alpha}^{N_f} Q_\\beta$\n\\end{description}\nThe $f_\\alpha$ are parton densities evaluated for \nmomentum fractions $\\mbox{\\tt xi(irps($i$))}$ and factorization scale\n$\\mu_f$,\nand $f_{\\overline{\\alpha}}$ stands for the parton density of the anti-flavour\nof the flavour $\\alpha$. The renormalization and factorization schemes are\nthe $\\overline{\\mbox{MS}}$ scheme. The correction terms for the \nDIS factorization \nscheme will be implemented in the near future.\n\nWe wish to note that the product of the weights, the flavour factors and the \nvalues of the observable is normalized in such a way that\nthe sum yields the expectation value in units of [pb]. No additional \nfactor such as $1\/N$, $N$ being the total number of generated events,\nhas to be applied\nin Eq.~\\ref{exval}.\n\nSince phase spaces are used several times for different contributions,\nit is a good strategy to first evaluate the observable(s) under consideration\nfor every phase space and to store the corresponding results.\nThen there should be the loop over the various contributions (the second sum).\nThe innermost loop is the one over the flavour factors.\n\nThe Monte Carlo integration itself employs the program {\\tt VEGAS}\n\\cite{27,28}. \n{\\tt VEGAS} is an adaptive multi-dimensional integration routine.\nIntegrations proceed in two steps. \nThe first step is an adaptation step in order\nto set up a grid in the integration variables\nwhich then steers the final integration step.\nThe adaptation step itself refines\nthe grid in a sequence of several iterations.\n{\\tt VEGAS} requires, as parameters, the number of Monte Carlo points \nto be used in the first and second step, respectively, \nand the number of iterations to refine the grid. \nIn the framework of {\\tt DISASTER++}, {\\tt VEGAS} can be used in three different\nways: \n\\begin{itemize}\n\\item As an adaptive integration routine.\nThe routine {\\tt user2} should return a value. This value is handed over\nto {\\tt VEGAS} as \nthe value of the integrand at the particular phase space point, \nand summed up. The final integral quoted by {\\tt VEGAS}\nis the sum of these weights\nfor the final integration.\nThis is the best choice if just one observable, \nfor example a jet cross section, is to be evaluated.\n\\item As a routine that merely supplies random numbers for \nthe events.\nIf the number of iterations is set to zero, then {\\tt VEGAS} just performs\nthe final integration run. The user is then responsible for the correct\nsummation of the weights, and for the determination of the \nstatistical error. It should be noted that, since all weights are\navailable individually in the user routine, an arbitrary number of \nobservables can be evaluated in a single run. In particular, since the\ndependence on the renormalization and factorization scales and on $N_f$\nis fully explicit, the study of the scale dependence of observables\ncan be done in a very convenient way. For example, all plots from \nRef.~\\cite{22} \nhave been obtained in a single run of {\\tt DISASTER++}.\n\\item As a combination of the two preceeding alternatives. Here the adaptation \nsteps are included. A ``typical'' infrared-safe observable, \nin the following called the {\\it adaptation variable}, is evaluated, and\nits value is returned to {\\tt VEGAS}. This observable serves to optimize the\ndistribution of points over phase space. A convenient observable of this\nkind is provided by {\\tt DISASTER++} (see below).\nThe ``real'' observables under consideration are evaluated as in the \nsecond alternative in the final integration step.\n\\end{itemize}\n\n\\noindent\n{\\tt DISASTER++} is initialized by a call of \nthe subroutine {\\tt disaster\\_ca()}. It is recommended \nto end a {\\tt DISASTER++}\nrun by a call of the subroutine \n{\\tt disaster\\_cb()} in order to free\ndynamically allocated memory.\n\n\\noindent\nParameters can be set or commands be executed by means of three routines:\n\\begin{description}\n\\item {\\quad\\tt disaster\\_ci(str, i)} {\\ }\\\\ \n   sets the integer parameter denoted by \n   the character string {\\tt str} to the value {\\tt i}\n\\item {\\quad\\tt disaster\\_cd(str, d)} {\\ }\\\\\n   sets the double precision parameter denoted by \n   the character string {\\tt str} to the value {\\tt d}\n\\item {\\quad\\tt disaster\\_cc(str)} {\\ }\\\\ executes the command \ngiven by the character string {\\tt str}\n\\end{description}\nThe following parameters are available (there are a few more to optimize the\ngeneration of the phase space points; they will be documented in forthcoming\nversions of this manual):\n\\begin{description}\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt ECM:}}]{\\ }\\\\\n   the centre-of-mass energy in [GeV]\n\n\\item[\\quad{\\tt LEPTON\\_INTEGRATION}:]{\\ }\\\\\n   {\\tt 1:} integration over $x_B$ and $y$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt XBMIN:}}]{\\ }\\\\\n     minimum value of $x_B$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt XBMAX:}}]{\\ }\\\\\n     maximum value of $x_B$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt YMIN:}}]{\\ }\\\\\n     minimum value of $y$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt YMAX:}}]{\\ }\\\\\n     maximum value of $y$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt QMIN:}}]{\\ }\\\\\n     minimum value of $Q$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt QMAX:}}]{\\ }\\\\\n     maximum value of $Q$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt WMIN:}}]{\\ }\\\\\n     minimum value of $W$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt WMAX:}}]{\\ }\\\\\n     maximum value of $W$\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt PROCESS\\_INDEX:}}]{\\ }\\\\\n     {\\tt 1:} leading order\\\\\n     {\\tt 2:} next-to-leading order\n    \n\\item[\\quad{\\tt NUMBER\\_OF\\_FINAL\\_STATE\\_PARTONS\\_IN\\_BORN\\_TERM:}]{\\ }\\\\\n     {\\tt 1}, {\\tt 2}, {\\tt 3} for the process under consideration;\\\\\n     {\\tt 1:} (1+1)-jet-type observables (leading and next-to-leading order)\\\\\n     {\\tt 2:} (2+1)-jet-type observables (leading and next-to-leading order)\\\\\n     {\\tt 3:} (3+1)-jet-type observables (leading order only)\n    \n\\item[\\quad{\\makebox[3cm][l]{\\tt POINTS:}}]{\\ }\\\\\n     {\\tt POINTS * (FACT\\_PREP + FACT\\_FINAL)} is the \n     number of generated points in the Monte Carlo integration\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt FACT\\_PREP:}}]{\\ }\\\\\n     the number of points for the grid-definition run is given by\n     {\\tt FACT\\_PREP * POINTS}\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt FACT\\_FINAL:}}]{\\ }\\\\\n     the number of points for the final integration step is given by\n     {\\tt FACT\\_FINAL * POINTS}\n\n\\item[\\quad{\\makebox[3cm][l]{\\tt RUN\\_MC:}}]{\\ }\\\\\n     to start the Monte-Carlo integration\n\n\\end{description}\n\n\\noindent\nA convenient adaptation observable can be evaluated by a call of\nthe following function:\n\\begin{verbatim}\n      double precision function disaster_cao(\n     &                             integer ipdf_collection,\n     &                             integer ipdf_parametrization,\n     &                             integer ipdf_set,\n     &                             integer ialphas_variant,\n     &                             integer ialphas_order,\n     &                             double precision dalphas_lambdaqcd4,\n     &                             integer ialphaem_variant\n     &                          )\n\\end{verbatim}\nThe arguments of the function call are:\n\\begin{description}\n\n\\item[\\quad{\\tt ipdf\\_collection:}]{\\ }\\\\\nthe collection of parton densities; \\\\\n{\\tt 1:} {\\tt PDFLIB} \\cite{29}\n\n\\item[\\quad{\\tt ipdf\\_parametrization:}]{\\ }\\\\\nparametrization of parton densities (cf.\\ {\\tt PDFLIB})\n\n\\item[\\quad{\\tt ipdf\\_set:}]{\\ }\\\\\nset of parton densities (cf.\\ {\\tt PDFLIB})\n\n\\item[\\quad{\\tt ialphas\\_variant:}]{\\ }\\\\\nfunction which is used to evaluate the strong coupling constant;\\\\\n{\\tt 1:} running coupling $\\alpha_s(Q^2)$ with \nflavour thresholds at the single heavy quark masses\n\n\\item[\\quad{\\tt ialphas\\_order:}]{\\ }\\\\\n{\\tt 1:} one-loop formula\\\\\n{\\tt 2:} two-loop formula\\\\\nfor the running strong\ncoupling constant\n\n\\item[\\quad{\\tt dalphas\\_lambdaqcd4:}]{\\ }\\\\\nthe QCD parameter $\\Lambda_{\\mbox{\\scriptsize QCD}}^{(4)}$\nfor four flavours\n\n\\item[\\quad{\\tt ialphaem\\_variant:}]{\\ }\\\\\nfunction which is used to evaluate the electromagnetic coupling constant;\\\\\n{\\tt 1:} fine structure constant \\\\\n{\\tt 2:} 1\/137\\\\\n(an implementation of the running electromagnetic \ncoupling constant is in preparation)\n\n\\end{description}\n\n\\noindent\nTo simplify the calculation of the flavour factors, \na {\\tt DISASTER++} routine can be called which returns the\nrequired coupling constants and the combinations of parton densities\nand quark charges:\n\\begin{verbatim}\n      subroutine disaster_cff(\n     &              integer ipdf_collection,\n     &              integer ipdf_parametrization,\n     &              integer ipdf_set,\n     &              integer ialphas_variant,\n     &              integer ialphas_order,\n     &              double precision dalphas_lambdaqcd4,\n     &              integer ialphaem_variant,\n     &              integer nf,\n     &              double precision ffactin(4),\n     &              double precision ffactout(13)\n     &           )\n\\end{verbatim}\nThe arguments of the function call are the same as in the case of the\nroutine {\\tt disaster\\_cao} (see above), except for the following:\n\\begin{description}\n\n\\item[\\quad{\\tt nf:}]{\\ }\\\\\nthe number of flavours $N_f$\n\n\\item[\\quad{\\tt ffactin:}]{\\ }\\\\\ninput parameters;\n\\begin{description}\n\\item {\\tt ffactin(1):} the momentum fraction variable $\\xi$\n\\item {\\tt ffactin(2):} the factorization scale in [GeV] \n(i.e.\\ the scale argument of the parton densities)\n\\item {\\tt ffactin(3):} the renormalization scale in [GeV] \n(i.e.\\ the scale argument of the running strong coupling constant)\n\\item {\\tt ffactin(4):} the scale argument of the running electromagnetic\ncoupling constant\n\\end{description}\n\n\\item[\\quad{\\tt ffactout:}]{\\ }\\\\\noutput parameters;\n\\begin{description}\n\\item {\\tt ffactout(1..11):} the quantities $\\rho_{i1}$ \\ldots\n$\\rho_{i11}$,\n\\item {\\tt ffactout(12):} the running strong coupling constant\n\\item {\\tt ffactout(13):} the electromagnetic\ncoupling constant\n\\end{description}\n\n\\end{description}\n\nIt is strongly recommended to use this routine, since it uses \na cache that stores a few of the most recent values temporarily, such that\nthe sums $\\rho_{ij}$ and the parton densities do not have to be reevaluated.\nThis routine is supplied for the convenience of the user. The weights\nand events generated by {\\tt DISASTER++} do not depend on this routine.\n\nThe description of the program structure just given may sound\ncomplicated. It is actually quite simple to use the program; an example \nfor the calculation of the (2+1)-jet cross section for the JADE algorithm\nin the E-scheme is given in the files {\\tt disaster\\_f.f}\nand {\\tt clust.f}, as described in Section~\\ref{installation}.\n\n\\section{Program Installation}\n\\label{installation}\n\n\\begin{description}\n\\item[Source code:]\nThe source code of the class library is available on the World Wide Web:\n\\begin{verbatim}\n   http:\/\/wwwcn.cern.ch\/~graudenz\/disaster.html\n\\end{verbatim}\n\n\\item[Files:]\nThe package consists of a number of files. To facilitate the installation,\nand to enable the {\\tt C++} compiler to perform certain optimizations,\nthe complete {\\tt C++} part of the program is provided as one file\n{\\tt onefile\\_n.cc} (the individual files are available on request). \nAn example for the {\\tt FORTRAN} interface is\ngiven in the file {\\tt disaster\\_f.f} (calculation of the (2+1) jet \ncross section for the JADE algorithm in the E-scheme), \ntogether with a simple cluster\nroutine in the file {\\tt clust.f}. \nThe number of Monte Carlo events in the example is set to \na tiny number (100) in order to terminate the program after a few seconds.\nRealistic values for the parameter {\\tt POINTS} are of the order of \n$10^6$.\nAn example ``make file'' is given in {\\tt makedisaster}. \n\\item[Mixed Language Programming:]\n{\\tt DISASTER++} is mainly writen in the {\\tt C++} programming language.\nThe reason for the choice of this language are twofold:\nObject-oriented programming allows for programs that are easily \nmaintained and extended\\footnote{It could even be said that object-oriented\nprogramming is a kind of applied ontology: the central categories of this \napproach are given by {\\it objects} and {\\it methods} that define their\nrelationships.\n}, and in high-energy physics there is a trend in the experimental \ndomain for a transition from {\\tt FORTRAN} to {\\tt C++}.\nAlthough the goal has been to write a self-contained {\\tt C++}\npackage, \na few parts of the program are still coded in \n{\\tt FORTRAN}. Moreover, the standard parton density parametrizations\nare only \navailable as {\\tt FORTRAN} libraries. This means that the {\\tt DISASTER++}\npackage cannot be run as a stand-alone {\\tt C++} program. In most cases,\nusers may wish to interface the program to their existing {\\tt FORTRAN}\nroutines. An elegant and machine-independent \nway for {\\it mixed language programming} for the case\nof {\\tt C}, {\\tt C++} and {\\tt FORTRAN} is supported by the \n{\\tt cfortran.h} package described in Ref.~\\cite{30}. \nFor every {\\tt FORTRAN} routine to be called by a {\\tt C++} method, \nan {\\tt extern \"C\"} routine has to be defined as an interface, \nand vice versa. The explicit calls are then generated by means of macros \nfrom {\\tt cfortran.h}. The most convenient way is, after compilation, \nto link the {\\tt FORTRAN} and {\\tt C++} parts via the standard\n\\begin{verbatim}\n   f77 -o disaster onefile_n.o ...\n\\end{verbatim}\ncommand\\footnote{The procedure is described here for the {\\tt UNIX}\noperating system.}, \nsuch that the {\\tt FORTRAN} part supplies the entry point.\nThe required {\\tt C++} libraries have to be stated explicitly\nvia the {\\tt -L} and {\\tt -l} options. The library paths can be obtained\nby compiling and linking a trivial program {\\tt hw.cc} of the type\n\\begin{verbatim}\n   #include <stdio.h>\n   main() { printf(\"Hello world!\\n\"); }\n\\end{verbatim}\nwith\n\\begin{verbatim}\n   gcc -v hw.cc\n\\end{verbatim}\n(for the {\\tt GNU C++} compiler). \nAn example for the required libraries can be found in the \nprototype ``make file'' {\\tt makedisaster}. Some machine-specific information\nis mentioned in the manual of {\\tt cfortran.h}.\n\nIn the {\\tt DISASTER++} package, the explicit {\\tt FORTRAN} interface, \nas described in Section~\\ref{structure},\nis already provided. Thus\nfrom the outside the {\\tt C++}\nkernel is transparent and hidden behind {\\tt FORTRAN} subroutines\nand functions.\n\n\\item[Template instantiation:]\nIn {\\tt DISASTER++}, heavy use is made of {\\it templates}. At present, there\nis not yet a universally accepted scheme for template instantiations.\nThe solution adopted here is the explicit instantiation\nof all templates. This requires\nthat the compiler itself does not instantiate templates automatically.\nThis is achieved for the {\\tt GNU} compiler by means of the switch\n\\begin{verbatim}\n   -fno-external-templates\n\\end{verbatim}\n\n\\item[Output files:]\nThere is a small problem with the output from the {\\tt C++} and {\\tt FORTRAN}\nparts of {\\tt DISASTER++}. It seems to be the case that generally {\\tt C++} \n({\\tt FILE* stdout} and {\\tt ostream cout}) \nand {\\tt FORTRAN} ({\\tt UNIT=6}) keep different\nfile buffers. This is no problem when the output is written to a terminal, \nsince then the file buffers are immediately flushed\nafter each line-feed character. When writing to \na file (as is usually the case for batch jobs), the file buffers are not \nimmediately flushed, and this leads to the problem that the output \non the file is mixed in non-chronological order. This problem will be solved\nby the introduction of a particular stream class which hands over the output\nto a {\\tt FORTRAN} routine.\n\n\\item[Miscellaneous:]\n{\\tt DISASTER++} employs the {\\tt ANSI C} {\\tt signal} facility to \ncatch interrupts caused by floating point arithmetic. \nIf the signal {\\tt SIGFPE} is raised, a flag in {\\tt DISASTER++} is set, \nwhich eventually leads to the requirement that the event has to be \ndropped (via a call of {\\tt user1(7)}). Similarly a non-zero value of\nthe variable {\\tt errno} of the {\\tt ANSI C errno} facility\nis treated. The signal handler is also active \nwhen the user routine is executed, which leads to the effect that\nin the case of a floating point exception the program does not crash, but\ncontinues under the assumption that the event has been dropped.\nForthcoming version of {\\tt DISASTER++} will make a flag available\nthat can be used to access the status of the signal handler in \nthe user routines. \nMoreover, it is checked whether the weight returned to {\\tt DISASTER++} via\n{\\tt user2} fulfills the criterion for {\\tt IEEE} {\\tt NaN} (``not a number''). \nIf this is the case, it is also requested that the event be dropped.\n\n\\end{description}\n\n\\section{Comparison of Available Programs}\n\\label{comparison}\n\nIn this section, we compare the three available programs {\\tt MEPJET}\n(Version 2.0)\\footnote{\nFor the very-high statistics runs the default random number\ngenerator (generating a Sobol sequence of pseudo-random numbers) \nof {\\tt MEPJET} ran out of random numbers. We therefore had to modify the\nprogram such that it uses another generator which is also part of\nthe {\\tt MEPJET} package. --- The crossing functions for the ``artificial''\nparton densities have been obtained by means of a modification of the program\n{\\tt make\\_str\\_pdf1.f}.\n}, \n{\\tt DISENT} (Version 0.1)\\footnote{\nAn earlier version of this paper \\cite{24}\nreported results of a comparison \nwith {\\tt DISENT} Version 0.0. We found large discrepancies for \nsome choices of the parton density parametrization. In the meantime,\nan error in {\\tt DISENT} has been fixed, and the results \nof {\\tt DISENT} and {\\tt DISASTER++} are now\nin good agreement, see below.\n} and {\\tt DISASTER++}\n(Version 1.0) numerically for various bins of \n$x_B$ and $y$ as defined in Table~\\ref{tab1}, \nand for various choices of the parton density parametrizations.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|}\n\\cline{2-4}\n  \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n  & \\makebox[4.1cm]{$0.01 < y < 0.03 $}  \n  & \\makebox[4.1cm]{$0.03 < y < 0.1 $}  \n  & \\makebox[4.1cm]{$0.1  < y < 0.3 $}  \n\\\\ \\hline\n$0.005 < x_B < 0.01$ \\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[1.2cm]{Bin 1}($Q^2 > 4.6\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 2}($Q^2 > 13.5\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 3}($Q^2 > 45.0\\,\\mbox{GeV}^2$)\n\\\\ \\hline\n$0.05 < x_B < 0.1$ \\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[1.2cm]{Bin 4} ($Q^2 > 45\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 5} ($Q^2 > 135\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 6} ($Q^2 > 450\\,\\mbox{GeV}^2$)\n\\\\ \\hline\n$0.2 < x_B < 0.4$ \\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[1.2cm]{Bin 7} ($Q^2 > 180\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 8} ($Q^2 > 540\\,\\mbox{GeV}^2$)\n  & \\makebox[1.2cm]{Bin 9} ($Q^2 > 1800\\,\\mbox{GeV}^2$)\n\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption[tab1]\n{\n\\label{tab1}\n{\\it\nBins in $x_B$ and $y$. The values in parentheses give the resulting \nlower bounds on $Q^2$.\n}\n}\n\\end{table}\n\nThe centre-of-mass energy is set to 300\\,GeV. To facilitate the\ncomparison, the strong coupling constant is set to a fixed value of\n$\\alpha_s=0.1$, \nand the number of flavours is set to $N_f=5$, even below the bottom \nthreshold ($N_f=5$ is hard-wired into {\\tt MEPJET}). \nThe electromagnetic coupling constant \nis chosen to be $\\alpha=1\/137$ (the value\nis hard-wired into {\\tt DISENT}, but this could be changed trivially, \nin principle). The factorization- and renormalization schemes of the\nhard scattering cross sections are $\\overline{\\mbox{MS}}$, and the\nfactorization and renormalization scales $\\mu_f$ and $\\mu_r$, respectively, \nare set to $Q$.\n\nThe quantity under consideration is the (2+1)-jet cross section, \nshown in Tables~2--8 in Appendix~\\ref{capp}.\nFor simplicity we consider the modified JADE clustering scheme\nwith resolution criterion $S_{ij} <> c W^2$ and the E~recombination scheme, \nwhere\n$S_{ij} = (p_i + p_j)^2$, $W$ is the total hadronic energy,\nand $c=0.02$ is the jet resolution parameter.\nWe require, in the laboratory frame ($E_e=27.439$\\,GeV, \n$E_P=820$\\,GeV), a minimum transverse momentum of 1\\,GeV and a pseudo-rapidity\nof $-3.5<\\eta<3.5$ for all jets\\footnote{These cuts in $p_T$ and $\\eta$ \nare employed in order to facilitate event generation with {\\tt MEPJET}; \nthe phase space generator implemented in that program is reminiscent of a \ngenerator for pp~collider physics where $p_T$ and $\\eta$ cuts\nin the laboratory frame are\na standard experimental procedure. It is thus complicated to generate \nevents with {\\tt MEPJET} \nin the full phase space of the laboratory system, as usually required \nfor eP scattering, where ``natural'' cuts in transverse momentum and\npseudo-rapidity would be performed in the hadronic centre-of-mass frame\nor in the Breit frame.}.\n\nThe parton density parametrizations employed in the comparison are:\n\\begin{description}\n\\item[{\\makebox[1cm][l]{(a)}}] \n           \\makebox[6cm][l]{the MRSD$_-^\\prime$ parton densities \n                            \\cite{31}} (Table 2),\n\\item[{\\makebox[1cm][l]{(b)}}] \n           \\makebox[3cm][l]{$q(\\xi)=(1-\\xi)^5$,}\\makebox[3cm][l]{$g(\\xi)=0$}\n           (Table 3),\n\\item[{\\makebox[1cm][l]{(c)}}] \n           \\makebox[3cm][l]{$q(\\xi)=0$,}\\makebox[3cm][l]{$g(\\xi)=(1-\\xi)^5$} \n           (Table 4),\n\\item[{\\makebox[1cm][l]{(d)}}] \n           \\makebox[3cm][l]{$q(\\xi)=(1-\\xi)^2$,}\\makebox[3cm][l]{$g(\\xi)=0$}\n           (Table 5),\n\\item[{\\makebox[1cm][l]{(e)}}] \n           \\makebox[3cm][l]{$q(\\xi)=0$,}\\makebox[3cm][l]{$g(\\xi)=(1-\\xi)^2$} \n           (Table 6),\n\\item[{\\makebox[1cm][l]{(f)}}] \n           \\makebox[3cm][l]{$q(\\xi)=(1-\\xi)$,}\\makebox[3cm][l]{$g(\\xi)=0$}\n           (Table 7),\n\\item[{\\makebox[1cm][l]{(g)}}] \n           \\makebox[3cm][l]{$q(\\xi)=0$,}\\makebox[3cm][l]{$g(\\xi)=(1-\\xi)$} \n           (Table 8).\n\\end{description}\nHere $q(\\xi)$ generically stands for valence and sea distributions\\footnote{\nThis means that $u_v(\\xi)$, $d_v(\\xi)$, $u_s(\\xi)$, $d_s(\\xi)$, \n$s_s(\\xi)$, $c_s(\\xi)$, \n$b_s(\\xi)$ have been set to $q(\\xi)$.\n}, \nand $g(\\xi)$ is the gluon distribution.\nWe wish to point out that the comparison involving the ``artificial''\nparton densities is not just of academic interest. On the contrary, \nfor the extraction \nof, for instance, the gluon density\nfrom jet data \nit is convenient\nto replace the parton densities by simple functions \nwith special properties (such as powers\nof the momentum fraction variable $\\xi$ or functions of an orthonormal \nbasis system), \nin order to achieve a fast fit. These functions usually do not have\nthe shape of physical parton densities, in particular they do not\nhave to fall off rapidly for $\\xi\\rightarrow 1$.\nMoreover, next-to-leading-order calculations yield unique and well-defined\nresults for the hard scattering cross sections to be convoluted with \nobservables and parton densities. We employ the ``artificial''\nparton densities also in order to have a stricter test of the \nhard scattering cross sections.\n\nThe leading-order results of all three programs are in excellent agreement. \nThe next-to-leading-order results of {\\tt DISASTER++} and {\\tt DISENT} are\nin good agreement within about two to (sometimes) \nthree standard deviations\\footnote{\nWe wish to note that the error estimates quoted by the programs are usually not\nrigorous estimates because of the non-Gaussian distribution of the\nMonte-Carlo weights. Therefore, in principle, it is not possible to \ninfer probabilities for the consistency of data samples produced by two\nprograms based on these estimates. \nA more precise, but in general unfeasible way to obtain an estimate of the\nMonte Carlo error would be to run the programs a number of times \nwith different random number seeds and to analyze the spread of the \nquoted results around their central value.\nSuch a study has recently been done by M.~Seymour\nfor {\\tt DISENT} with the result that the \nindividual error estimates are quite reliable \\cite{32}.\n}\nof the larger of the two errors quoted by the two programs. An exception\nis bin~7 for $g(\\xi) = (1-\\xi)^2$. A run of {\\tt DISENT} with higher statistics \nyields a value of $0.1836 \\pm 0.0025$, which is within two standard deviations\nof the {\\tt DISASTER++} result, indicating that there was indeed a statistical\nfluctuation in the original result.\n\nThe comparison of the next-to-leading-order results \nof {\\tt MEPJET} and {\\tt DISASTER++} requires a more detailed discussion:\n\\begin{itemize}\n\n\\item For the MRSD$_-^\\prime$ parton densities, the results for\nbins 3--9 are\ncompatible within about two standard deviations of the statistical error\nof the Monte-Carlo integrations.\nThe results for bins~1 and~2 differ considerably. \nRuns with a smaller value\nof the internal {\\tt MEPJET} cut-off variable~$s$, which is set by default\nto $s=0.1\\,$GeV$^2$, yield\nthe following results for bin 1:\n$580.6 \\pm 6.7$\\,pb ($s=0.01\\,$GeV$^2$), \n$564.8 \\pm 10.5$\\,pb ($s=0.001\\,$GeV$^2$) and\n$575.4 \\pm 13.0$\\,pb ($s=0.0001\\,$GeV$^2$).\nThe statistical error is increased for decreased~$s$ because the integration\nvolume\nof the (3+1) parton contributions is extended into the singular domain.\nBecause of the increased statistical error, we also performed a \nhigh-statistics runs with $\\sim 4\\cdot10^9$ (!) Monte Carlo events\nof {\\tt MEPJET} \nfor this bin. \nFor $s=0.001\\,$GeV$^2$ we obtain \n$576.3 \\pm 6.7$\\,pb\nand for $s=0.0001\\,$GeV$^2$\nthe result is\n$583.2 \\pm 7.4$\\,pb.\nThese values from {\\tt MEPJET} \nare compatible with the {\\tt DISASTER++} and {\\tt DISENT} results\\footnote{\nThese results underline that, for the phase space slicing method, results\ngenerally have to be validated {\\it ex post} by a cross-check with a \nsmaller technical cut~$s$ and much higher statistics. It may be argued that\nthere are jet algorithms (the $k_T$~algorithm, for example)\nwhich show a better convergence for $s\\rightarrow 0$.\nHowever, the point here is that one does not know in advance whether this\nis the case for the observable under consideration. --- In Ref.~\\cite{23}\nwe find the statement that $s$-independence in {\\tt MEPJET} is achieved for \n$s=0.1\\,$GeV$^2$. Our study shows that this is generally not the case, \nand that extremely small values of~$s$, possibly of the order of\n$s=0.0001\\,$GeV$^2$, might be necessary.\n}.\n\\item For the parton density parametrization (b) (quarks only, with a steeply\nfalling distribution $q(\\xi)$ for $\\xi \\rightarrow 1$), \n{\\tt DISASTER++} and {\\tt MEPJET}\nare in good agreement.\n\n\\item The results for parametrization (c) (steeply falling\ngluon parametrization)\nare in good agreement, except for bin 1.\n\n\\item For parametrization (d), \n{\\tt DISASTER++} and {\\tt MEPJET} are in agreement except for bins 1 and 4.\nRuns with a smaller value\nof the {\\tt MEPJET} cut-off variable~$s$\nyield\nthe following results for bin 1:\n$59.6 \\pm 1.8$\\,pb ($s=0.01\\,$GeV$^2$), \n$56.7 \\pm 5.8$\\,pb ($s=0.001\\,$GeV$^2$) and\n$54.9 \\pm 10.4$\\,pb ($s=0.0001\\,$GeV$^2$).\nA high-statistics run ($\\sim 4\\cdot10^9$ events) of {\\tt MEPJET} \nfor bin 1 with $s=0.0001\\,$GeV$^2$ gives the \ncross section $60.0 \\pm 1.9$\\,pb.\nContrary to the observation in case (a), for small~$s$ \nwe do not get agreement of \nthe {\\tt MEPJET} result with the {\\tt DISASTER++} \/ {\\tt DISENT} result\nof about $48$--$49$\\,pb.\n\n\\item The {\\tt MEPJET} results for parametrization (e) \n($g(\\xi) = (1-\\xi)^2$)\ndeviate considerably from the {\\tt DISASTER++}\nresults in bins~1, 2, 4 and 7.\n\n\\item For parametrization (f),\n{\\tt DISASTER++} and {\\tt MEPJET} are incompatible\nfor bins 1, 2, 4, 6 and 7.\n\n\\item For parametrization (g), \n{\\tt MEPJET} and {\\tt DISASTER++} are compatible in bins\n3, 5, 8 and 9 only.\nA high-statistics run ($\\sim 4\\cdot10^9$ events) of {\\tt MEPJET} \nfor bin 4 with $s=0.0001\\,$GeV$^2$ yields the \ncross section $1.29 \\pm 0.02$\\,pb.\nThis value is different from the result for $s=0.1\\,$GeV$^2$, \nbut still inconsistent\nwith the {\\tt DISASTER++} \/ {\\tt DISENT} result of about $0.69$\\,pb.\n\n\\end{itemize}\n\nThe overall picture is thus: Out of the three programs, {\\tt DISASTER++} \nand {\\tt DISENT} (Version 0.1) are in good agreement within about\ntwo, sometimes three standard deviations of the quoted integration errors, \nboth for ``physical'' and ``artificial'' parton densities. This agreement\nis very encouraging, but not yet perfect, and much more detailed studies\ninvolving different sets of observables and differential distributions\nare required. For the two programs, a direct comparison of the\n``jet structure functions'' should also be feasible.\n\nFor several bins, in particular for the ``artificial'' parton distribution \nfunctions, the {\\tt MEPJET}\nresults for the default setting of the \ninternal parameters deviate considerably from the {\\tt DISASTER++}\nand {\\tt DISENT} results. \nFor one particular bin studied in more detail for\nthe MRSD$_-^\\prime$ parton densities,\nthe\ndiscrepancy disappears in the case of an extremely small internal technical \ncut~$s$ of {\\tt MEPJET}, for a substantial increase of the\nnumber of generated events to obtain a meaningful Monte Carlo error. \nA few {\\tt MEPJET} results employing ``artificial'' \nparton densities have been studied in more detail. We observed that \nin these cases a reduction of the~$s$ parameter does not lead to an\nimprovement of the situation. For lack of computer time, we could not study \nall bins with a smaller $s$~cut. The overall situation \nis thus still inconclusive and unclear. An independent cross check of the\n{\\tt MEPJET} results, in particular of those using the \nimplementation of the crossing functions for the ``artificial'' parton \ndensities, is highly desirable.\n\n\\section{Miscellaneous}\n\\begin{itemize}\n\n\\item If you intend to install and use {\\tt DISASTER++}, please send me \na short e-mail message, and I will put your name on a mailing list\nso that I can inform you when there is a new version of the package.\n\n\\item Suggestions for improvements and bug reports are welcome.\n\n\\item In case that there are problems with the installation of the program, \nplease send me an e-mail message.\n\n\\end{itemize}\n\n\\section{Summary}\n\nWe have presented the {\\tt C++} class library \n{\\tt DISASTER++} for the calculation \nof (1+1) and (2+1)-jet type observables in deeply inelastic scattering.\nThe program is based on the subtraction formalism and thus does not require\na technical cut-off for the separation of the infrared-singular from the\ninfrared-finite phase-space regions. \nA {\\tt FORTRAN} interface to the {\\tt C++} class library is available.\n{\\tt DISASTER++} is actually intended to be a general object-oriented\nframework for next-to-leading order QCD calculations. In particular, \nthe subtraction formalism is implemented in a very general way.\n\nWe have performed a comparison of the three available programs\n{\\tt MEPJET}, {\\tt DISENT} and {\\tt DISASTER++}\nover a wide range of the parameters for the lepton phase space.\nWe find good agreement of {\\tt DISASTER++} and the Catani-Seymour\nprogram {\\tt DISENT} (Version 0.1).\nThe comparison of {\\tt DISASTER++} and the Mirkes-Zeppenfeld program\n{\\tt MEPJET} (for the {\\tt MEPJET} \ndefault parameters) leads to several\ndiscrepancies, both for physical and for ``artificial'' parton densities.\nFor the MRSD$_-^\\prime$ parton densities a \nreduction of the internal {\\tt MEPJET} phase-space slicing cut-off \nvariable~$s$, the number of Monte Carlo events kept fixed, leads to a certain \nimprovement of the central values of the results, \naccompanied by a substantially increased statistical error and fluctuating\ncentral values. A considerable increase of the number of generated events\n(up to of the order of several billion events) \neventually leads to an agreement of the {\\tt MEPJET} results with the\n{\\tt DISASTER++} \/ {\\tt DISENT} results for a particular bin of the lepton \nvariables which has been studied in detail.\nFor ``artificial'' parton densities and a selected set of bins of\nthe lepton variables, a reduction of the internal cut~$s$\ndoes not resolve the discrepancies.\nOther bins are not considered\nfor the lack of computer time for very-high statistics runs.\nIt should be stressed that the present study is still limited in scope.\nAn independent cross check of the {\\tt MEPJET} results for the ``artificial''\nparton densities has to be done until a firm conclusion can be reached.\nMoreover, \nthe study has to be repeated for a wider range of observables and much higher\nMonte Carlo statistics. The $s$~dependence of the {\\tt MEPJET} results\nshould also be studied in more detail.\n\nCompared to the other two programs {\\tt MEPJET} and {\\tt DISENT},\n{\\tt DISASTER++} makes the full $N_f$ dependence and the dependence\non the renormalization and factorization scales available in the \nuser routine. This is required for consistent studies of effects\nsuch as the scale dependence when the bottom threshold is crossed.\n\n\\section{Acknowledgements}\nI wish to thank M.~Seymour for sending me the numerical results for the new \n{\\tt DISENT} version. D.~Zeppenfeld made a few cross\nchecks of the results for the MRSD$_-^\\prime$ parton densities.\nJ.~Collins has provided me with the {\\tt FORTRAN} \nroutine to test the {\\tt IEEE NaN} condition.\nI am also grateful to Th.~Hadig for a few comments on the first version \nof this paper, and for suggestions for improvements of the program.\n\n\\clearpage\n\n\\begin{appendix}\n\n\\section{Numerical Results}\n\\label{capp}\n\nThis appendix contains the numerical results which are discussed in \nSection~\\ref{comparison}. The entries in the tables are the (2+1)-jet \ncross sections\nin units of [pb].\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{402.1}{1.13}\n  & \\pmdg{399.9}{0.53}\n  & \\pmdg{399.6}{1.1}\n  & \\pmdg{585.0}{2.6}\n  & \\pmdg{564.1}{1.9}\n  & \\pmdg{578.4}{7.1}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{207.6}{0.59}\n  & \\pmdg{207.5}{0.34}\n  & \\pmdg{207.4}{0.15}\n  & \\pmdg{364.8}{1.5}\n  & \\pmdg{347.3}{2.4}\n  & \\pmdg{361.1}{3.5}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{60.0}{0.16}\n  & \\pmdg{59.9}{0.14}\n  & \\pmdg{59.9}{0.15}\n  & \\pmdg{119.1}{1.71}\n  & \\pmdg{118.0}{1.05}\n  & \\pmdg{120.1}{0.94}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{82.9}{0.16}\n  & \\pmdg{82.9}{0.10}\n  & \\pmdg{82.6}{0.21}\n  & \\pmdg{98.1}{1.11}\n  & \\pmdg{95.1}{0.61}\n  & \\pmdg{95.4}{0.87}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{42.9}{0.08}\n  & \\pmdg{42.9}{0.06}\n  & \\pmdg{42.6}{0.28}\n  & \\pmdg{55.3}{0.46}\n  & \\pmdg{54.4}{0.49}\n  & \\pmdg{54.9}{0.40}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{11.9}{0.02}\n  & \\pmdg{11.9}{0.02}\n  & \\pmdg{11.9}{0.08}\n  & \\pmdg{17.5}{0.06}\n  & \\pmdg{16.8}{0.22}\n  & \\pmdg{17.3}{0.13}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{9.60}{0.03}\n  & \\pmdg{9.58}{0.01}\n  & \\pmdg{9.59}{0.04}\n  & \\pmdg{12.1}{0.50}\n  & \\pmdg{12.7}{0.07}\n  & \\pmdg{12.3}{0.15}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{6.24}{0.01}\n  & \\pmdg{6.23}{0.01}\n  & \\pmdg{6.24}{0.02}\n  & \\pmdg{8.61}{0.12}\n  & \\pmdg{8.55}{0.15}\n  & \\pmdg{8.52}{0.08}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.78}{0.003}\n  & \\pmdg{1.78}{0.003}\n  & \\pmdg{1.78}{0.06}\n  & \\pmdg{2.65}{0.03}\n  & \\pmdg{2.57}{0.06}\n  & \\pmdg{2.63}{0.02}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 2: {\\it\nComparison for MRSD$_-^{\\,\\prime}$ parton densities.\n}\n\\end{center}\n\n\\clearpage\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{36.2}{0.09}\n  & \\pmdg{36.3}{0.05}\n  & \\pmdg{36.3}{0.12}\n  & \\pmdg{39.1}{0.33}\n  & \\pmdg{40.9}{0.89}\n  & \\pmdg{38.2}{0.53}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{17.8}{0.04}\n  & \\pmdg{17.8}{0.03}\n  & \\pmdg{17.7}{0.05}\n  & \\pmdg{23.2}{0.37}\n  & \\pmdg{22.7}{0.41}\n  & \\pmdg{22.6}{0.22}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{5.21}{0.01}\n  & \\pmdg{5.21}{0.01}\n  & \\pmdg{5.21}{0.02}\n  & \\pmdg{8.24}{0.22}\n  & \\pmdg{7.86}{0.12}\n  & \\pmdg{8.14}{0.06}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{27.3}{0.06}\n  & \\pmdg{27.3}{0.03}\n  & \\pmdg{27.2}{0.09}\n  & \\pmdg{28.0}{0.52}\n  & \\pmdg{29.2}{0.18}\n  & \\pmdg{30.0}{0.21}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{14.8}{0.03}\n  & \\pmdg{14.8}{0.02}\n  & \\pmdg{14.7}{0.04}\n  & \\pmdg{17.4}{0.29}\n  & \\pmdg{16.9}{0.10}\n  & \\pmdg{17.0}{0.11}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{4.33}{0.008}\n  & \\pmdg{4.32}{0.006}\n  & \\pmdg{4.31}{0.01}\n  & \\pmdg{5.62}{0.10}\n  & \\pmdg{5.44}{0.05}\n  & \\pmdg{5.54}{0.03}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{6.38}{0.02}\n  & \\pmdg{6.37}{0.01}\n  & \\pmdg{6.38}{0.03}\n  & \\pmdg{8.49}{0.17}\n  & \\pmdg{8.59}{0.10}\n  & \\pmdg{8.37}{0.11}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{4.44}{0.01}\n  & \\pmdg{4.43}{0.007}\n  & \\pmdg{4.44}{0.02}\n  & \\pmdg{6.11}{0.08}\n  & \\pmdg{6.05}{0.07}\n  & \\pmdg{6.07}{0.06}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.36}{0.002}\n  & \\pmdg{1.36}{0.002}\n  & \\pmdg{1.36}{0.05}\n  & \\pmdg{2.02}{0.02}\n  & \\pmdg{2.00}{0.05}\n  & \\pmdg{2.01}{0.01}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 3: {\\it\nComparison for $q(\\xi) = (1-\\xi)^5$\n}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{4.89}{0.017}\n  & \\pmdg{4.89}{0.007}\n  & \\pmdg{4.87}{0.01}\n  & \\pmdg{5.38}{0.07}\n  & \\pmdg{6.03}{0.06}\n  & \\pmdg{5.22}{0.13}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{2.66}{0.009}\n  & \\pmdg{2.66}{0.007}\n  & \\pmdg{2.65}{0.007}\n  & \\pmdg{3.67}{0.08}\n  & \\pmdg{3.66}{0.04}\n  & \\pmdg{3.58}{0.05}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.825}{0.003}\n  & \\pmdg{0.826}{0.002}\n  & \\pmdg{0.826}{0.002}\n  & \\pmdg{1.44}{0.07}\n  & \\pmdg{1.37}{0.03}\n  & \\pmdg{1.39}{0.02}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.60}{0.005}\n  & \\pmdg{1.60}{0.003}\n  & \\pmdg{1.60}{0.003}\n  & \\pmdg{1.20}{0.05}\n  & \\pmdg{1.30}{0.01}\n  & \\pmdg{1.12}{0.04}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.904}{0.003}\n  & \\pmdg{0.900}{0.001}\n  & \\pmdg{0.899}{0.002}\n  & \\pmdg{0.833}{0.027}\n  & \\pmdg{0.801}{0.008}\n  & \\pmdg{0.764}{0.019}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.279}{0.001}\n  & \\pmdg{0.278}{0.001}\n  & \\pmdg{0.278}{0.001}\n  & \\pmdg{0.314}{0.007}\n  & \\pmdg{0.287}{0.004}\n  & \\pmdg{0.299}{0.006}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.130}{0.001}\n  & \\pmdg{0.131}{0.001}\n  & \\pmdg{0.130}{0.001}\n  & \\pmdg{0.119}{0.005}\n  & \\pmdg{0.118}{0.002}\n  & \\pmdg{0.110}{0.006}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.0981}{0.001}\n  & \\pmdg{0.0980}{0.001}\n  & \\pmdg{0.0981}{0.001}\n  & \\pmdg{0.105}{0.002}\n  & \\pmdg{0.096}{0.001}\n  & \\pmdg{0.099}{0.004}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.0313}{0.0001}\n  & \\pmdg{0.0310}{0.001}\n  & \\pmdg{0.0313}{0.001}\n  & \\pmdg{0.0396}{0.001}\n  & \\pmdg{0.034}{0.001}\n  & \\pmdg{0.0386}{0.001}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 4: {\\it\nComparison for $g(\\xi) = (1-\\xi)^5$\n}\n\\end{center}\n\n\\clearpage\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{46.1}{0.11}\n  & \\pmdg{46.2}{0.07}\n  & \\pmdg{46.2}{0.14}\n  & \\pmdg{49.4}{0.67}\n  & \\pmdg{58.8}{0.65}\n  & \\pmdg{47.8}{1.2}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{23.8}{0.05}\n  & \\pmdg{23.8}{0.09}\n  & \\pmdg{23.8}{0.07}\n  & \\pmdg{30.6}{0.33}\n  & \\pmdg{31.4}{0.71}\n  & \\pmdg{29.0}{0.54}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{7.28}{0.02}\n  & \\pmdg{7.28}{0.02}\n  & \\pmdg{7.29}{0.02}\n  & \\pmdg{11.2}{0.21}\n  & \\pmdg{11.0}{0.24}\n  & \\pmdg{11.4}{0.14}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{42.4}{0.09}\n  & \\pmdg{42.3}{0.06}\n  & \\pmdg{42.3}{0.12}\n  & \\pmdg{38.4}{0.30}\n  & \\pmdg{41.9}{0.26}\n  & \\pmdg{38.4}{0.31}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{23.9}{0.04}\n  & \\pmdg{23.9}{0.03}\n  & \\pmdg{23.8}{0.06}\n  & \\pmdg{24.8}{0.46}\n  & \\pmdg{24.2}{0.19}\n  & \\pmdg{23.9}{0.16}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{7.31}{0.01}\n  & \\pmdg{7.30}{0.01}\n  & \\pmdg{7.27}{0.02}\n  & \\pmdg{8.11}{0.19}\n  & \\pmdg{8.04}{0.41}\n  & \\pmdg{8.24}{0.05}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{20.3}{0.05}\n  & \\pmdg{20.3}{0.08}\n  & \\pmdg{20.3}{0.08}\n  & \\pmdg{23.3}{0.64}\n  & \\pmdg{25.1}{0.18}\n  & \\pmdg{22.4}{0.24}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{15.4}{0.03}\n  & \\pmdg{15.4}{0.02}\n  & \\pmdg{15.4}{0.01}\n  & \\pmdg{18.6}{0.36}\n  & \\pmdg{18.3}{0.47}\n  & \\pmdg{18.4}{0.15}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{4.87}{0.01}\n  & \\pmdg{4.86}{0.01}\n  & \\pmdg{4.87}{0.04}\n  & \\pmdg{6.47}{0.08}\n  & \\pmdg{6.38}{0.07}\n  & \\pmdg{6.41}{0.05}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 5: {\\it\nComparison for $q(\\xi) = (1-\\xi)^2$\n}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{6.24}{0.02}\n  & \\pmdg{6.22}{0.01}\n  & \\pmdg{6.21}{0.02}\n  & \\pmdg{6.73}{0.13}\n  & \\pmdg{8.94}{0.12}\n  & \\pmdg{6.67}{0.24}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{3.59}{0.01}\n  & \\pmdg{3.58}{0.01}\n  & \\pmdg{3.57}{0.01}\n  & \\pmdg{4.77}{0.06}\n  & \\pmdg{5.24}{0.09}\n  & \\pmdg{4.43}{0.11}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.18}{0.004}\n  & \\pmdg{1.18}{0.004}\n  & \\pmdg{1.18}{0.003}\n  & \\pmdg{1.93}{0.04}\n  & \\pmdg{1.89}{0.04}\n  & \\pmdg{1.86}{0.03}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{2.65}{0.007}\n  & \\pmdg{2.65}{0.003}\n  & \\pmdg{2.65}{0.006}\n  & \\pmdg{1.13}{0.03}\n  & \\pmdg{1.66}{0.02}\n  & \\pmdg{0.94}{0.07}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.62}{0.004}\n  & \\pmdg{1.61}{0.002}\n  & \\pmdg{1.61}{0.003}\n  & \\pmdg{1.04}{0.04}\n  & \\pmdg{1.09}{0.02}\n  & \\pmdg{0.993}{0.03}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.535}{0.001}\n  & \\pmdg{0.534}{0.001}\n  & \\pmdg{0.533}{0.001}\n  & \\pmdg{0.433}{0.018}\n  & \\pmdg{0.412}{0.009}\n  & \\pmdg{0.430}{0.010}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.452}{0.002}\n  & \\pmdg{0.452}{0.001}\n  & \\pmdg{0.451}{0.001}\n  & \\pmdg{0.221}{0.026}\n  & \\pmdg{0.292}{0.010}\n  & \\pmdg{0.129}{0.02}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.398}{0.001}\n  & \\pmdg{0.398}{0.001}\n  & \\pmdg{0.397}{0.001}\n  & \\pmdg{0.298}{0.01}\n  & \\pmdg{0.271}{0.005}\n  & \\pmdg{0.237}{0.01}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.136}{0.001}\n  & \\pmdg{0.135}{0.001}\n  & \\pmdg{0.135}{0.001}\n  & \\pmdg{0.130}{0.003}\n  & \\pmdg{0.109}{0.002}\n  & \\pmdg{0.120}{0.004}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 6: {\\it\nComparison for $g(\\xi) = (1-\\xi)^2$\n}\n\\end{center}\n\n\\clearpage\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{50.6}{0.12}\n  & \\pmdg{50.7}{0.13}\n  & \\pmdg{50.7}{0.15}\n  & \\pmdg{58.6}{1.29}\n  & \\pmdg{72.9}{1.56}\n  & \\pmdg{54.7}{2.1}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{27.1}{0.05}\n  & \\pmdg{27.1}{0.16}\n  & \\pmdg{27.0}{0.07}\n  & \\pmdg{36.4}{0.57}\n  & \\pmdg{40.0}{0.84}\n  & \\pmdg{34.9}{1.0}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{8.51}{0.02}\n  & \\pmdg{8.51}{0.02}\n  & \\pmdg{8.52}{0.02}\n  & \\pmdg{13.8}{0.35}\n  & \\pmdg{13.3}{0.43}\n  & \\pmdg{13.9}{0.2}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{49.8}{0.10}\n  & \\pmdg{49.7}{0.05}\n  & \\pmdg{49.6}{0.14}\n  & \\pmdg{41.2}{0.55}\n  & \\pmdg{47.2}{0.91}\n  & \\pmdg{41.9}{0.38}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{29.0}{0.05}\n  & \\pmdg{29.0}{0.03}\n  & \\pmdg{28.8}{0.07}\n  & \\pmdg{27.3}{0.52}\n  & \\pmdg{28.2}{0.42}\n  & \\pmdg{26.4}{0.19}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{9.09}{0.01}\n  & \\pmdg{9.07}{0.01}\n  & \\pmdg{9.04}{0.02}\n  & \\pmdg{9.58}{0.06}\n  & \\pmdg{9.16}{0.15}\n  & \\pmdg{9.54}{0.06}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{30.6}{0.08}\n  & \\pmdg{30.5}{0.04}\n  & \\pmdg{30.5}{0.12}\n  & \\pmdg{32.0}{0.34}\n  & \\pmdg{36.3}{0.59}\n  & \\pmdg{32.4}{0.52}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{24.3}{0.04}\n  & \\pmdg{24.3}{0.03}\n  & \\pmdg{24.3}{0.07}\n  & \\pmdg{27.6}{0.56}\n  & \\pmdg{28.4}{0.35}\n  & \\pmdg{27.6}{0.21}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{7.88}{0.01}\n  & \\pmdg{7.86}{0.01}\n  & \\pmdg{7.87}{0.02}\n  & \\pmdg{9.63}{0.21}\n  & \\pmdg{9.50}{0.15}\n  & \\pmdg{9.47}{0.06}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 7: {\\it\nComparison for $q(\\xi) = (1-\\xi)$\n}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}[h]{|c|c|c|c|c|c|c|}\n\\cline{2-7}\n   \\multicolumn{1}{c|}{\\rule[-2.5mm]{0mm}{8mm}}\n & \\multicolumn{3}{|c|}{leading order}\n & \\multicolumn{3}{|c|}{next-to-leading order}\n\\\\ \\hline\n  bin\\rule[-2.5mm]{0mm}{8mm}\n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n  & \\makebox[2.2cm]{\\tt DISASTER++}  \n  & \\makebox[2.2cm]{\\tt MEPJET}  \n  & \\makebox[2.2cm]{\\tt DISENT}  \n\\\\ \\hline\\hline\n1\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{6.84}{0.02}\n  & \\pmdg{6.84}{0.01}\n  & \\pmdg{6.82}{0.02}\n  & \\pmdg{8.20}{0.25}\n  & \\pmdg{11.6}{0.14}\n  & \\pmdg{8.26}{0.45}\n\\\\ \\hline\n2\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{4.09}{0.01}\n  & \\pmdg{4.07}{0.01}\n  & \\pmdg{4.07}{0.01}\n  & \\pmdg{5.70}{0.11}\n  & \\pmdg{6.69}{0.16}\n  & \\pmdg{5.68}{0.17}\n\\\\ \\hline\n3\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{1.39}{0.004}\n  & \\pmdg{1.39}{0.005}\n  & \\pmdg{1.39}{0.003}\n  & \\pmdg{2.41}{0.07}\n  & \\pmdg{2.33}{0.05}\n  & \\pmdg{2.34}{0.05}\n\\\\ \\hline\n4\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{3.19}{0.01}\n  & \\pmdg{3.19}{0.01}\n  & \\pmdg{3.19}{0.01}\n  & \\pmdg{0.686}{0.09}\n  & \\pmdg{1.65}{0.03}\n  & \\pmdg{0.691}{0.10}\n\\\\ \\hline\n5\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{2.06}{0.005}\n  & \\pmdg{2.06}{0.002}\n  & \\pmdg{2.05}{0.003}\n  & \\pmdg{1.00}{0.08}\n  & \\pmdg{1.14}{0.03}\n  & \\pmdg{0.866}{0.05}\n\\\\ \\hline\n6\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.711}{0.001}\n  & \\pmdg{0.710}{0.001}\n  & \\pmdg{0.709}{0.001}\n  & \\pmdg{0.500}{0.006}\n  & \\pmdg{0.471}{0.01}\n  & \\pmdg{0.442}{0.017}\n\\\\ \\hline\n7\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.712}{0.003}\n  & \\pmdg{0.711}{0.001}\n  & \\pmdg{0.710}{0.002}\n  & \\pmdg{0.157}{0.026}\n  & \\pmdg{0.373}{0.008}\n  & \\pmdg{0.082}{0.038}\n\\\\ \\hline\n8\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.692}{0.002}\n  & \\pmdg{0.690}{0.001}\n  & \\pmdg{0.690}{0.001}\n  & \\pmdg{0.411}{0.020}\n  & \\pmdg{0.408}{0.022}\n  & \\pmdg{0.340}{0.023}\n\\\\ \\hline\n9\\rule[-2.5mm]{0mm}{8mm}\n  & \\pmdg{0.245}{0.001}\n  & \\pmdg{0.245}{0.001}\n  & \\pmdg{0.245}{0.001}\n  & \\pmdg{0.194}{0.012}\n  & \\pmdg{0.172}{0.007}\n  & \\pmdg{0.161}{0.008}\n\\\\ \\hline\n\\end{tabular}\n\n\\vspace{0.5cm}\nTable 8: {\\it\nComparison for $g(\\xi) = (1-\\xi)$\n}\n\\end{center}\n\n\\end{appendix}\n\n\\clearpage\n\n\\newcommand{\\bibitema}[1]{\\bibitem{#1}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\n\\\n\n\n\n\n\nThe AdS\/conformal field theory (CFT) correspondence~\\cite{MaldacenaOriginal,GKP,Witten,WittenThermal}\nhas yielded many important insights into the dynamics of strongly\ncoupled gauge theories. Among numerous results obtained so far,\none of the most striking is the universality of the ratio of the\nshear viscosity $\\eta$ to the entropy density\n$s$~\\cite{Policastro:2001yc,Kovtun:2003wp,Buchel:2003tz,KSSbound}\n\\begin{equation}\n\\label{bound} \\frac{\\eta}{s} = \\frac{1}{4\\pi}\n\\end{equation}\nfor all gauge theories with an Einstein gravity dual in the limit\n$N \\to \\infty$ and ${\\lambda} \\to \\infty$. Here, $N$ is the number of\ncolors and ${\\lambda}$ is the 't Hooft coupling. It was further\nconjectured in~\\cite{KSSbound} that~(\\ref{bound}) is a universal\nlower bound [the Kovtun-Starinets-Son (KSS) bound]  for all materials. So far, all known\nsubstances including water and liquid helium satisfy the bound.\nThe systems coming closest to the bound include the quark-gluon\nplasma created at Relativistic Heavy Ion Collider (RHIC)~\\cite{Teaney:2003kp,rr1,songHeinz,r2,Dusling:2007gi, Adare:2006nq}\nand certain cold atomic gases in the unitarity limit (see\ne.g.~\\cite{Schafer:2007ib}). $\\eta\/s$ for pure gluon QCD\nslightly above the deconfinement temperature has also been\ncalculated on the lattice recently~\\cite{Meyer:2007ic} and is\nabout $30 \\%$ larger than~(\\ref{bound}). See\nalso~\\cite{sakai}. See~\\cite{Cohen:2007qr,Cherman:2007fj,Chen:2007jq,Son:2007xw,Fouxon:2007pz}\nfor other discussions of the bound.\n\nNow, as stated above, the ratio~(\\ref{bound}) was obtained for a class of gauge theories whose holographic duals are dictated by classical Einstein gravity (coupled to matter). More generally, string theory (or any\nquantum theory of gravity) contains higher derivative corrections from stringy or quantum effects,\ninclusion of which will modify the ratio. In terms of gauge theories,\nsuch modifications correspond to $1\/{\\lambda}$ or $1\/N$ corrections. As a concrete example, let us take ${\\cal N}=4$ super-Yang-Mills theory, whose dual corresponds to type IIB\nstring theory on $AdS_5 \\times S^5$. The leading order correction in $1\/{\\lambda}$ arises from stringy corrections to the low-energy effective action of type IIB supergravity, schematically of the form ${\\alpha'}^3 R^4$.\nThe correction to $\\eta\/s$ due to such a term was calculated\nin~\\cite{BLS,Benincasa:2005qc}. It was found that the correction is positive, consistent with the\nconjectured bound.\n\nIn this paper, instead of limiting ourselves to\nspecific known string theory corrections, we explore the modification of $\\eta\/s$ due to\ngeneric higher derivative terms in the holographic gravity dual. The reason is partly pragmatic: other\nthan in a few maximally supersymmetric circumstances, very little\nis known about forms of higher derivative corrections generated in string theory. Given the vastness of the string\nlandscape~\\cite{landscape}, one expects that generic corrections do\noccur. Restricting to the gravity sector in $AdS_5$, the leading order\nhigher derivative corrections can be written as\\footnote{Our\nconventions are those of \\cite{Carroll}. In this section we suppress Gibbons-Hawking surface terms.}\n \\begin{equation} \\label{epr}\nI= {1 \\over 16 \\pi G_N} \\int d^5 x \\, \\sqrt{- g} \\left(R - 2 \\Lambda +\n\\lad^2  \\left({\\alpha}_1  R^2+  {\\alpha}_2 R_{\\mu \\nu} R^{\\mu \\nu}+{\\alpha}_3 R^{\\mu\n\\nu\\rho \\sigma} R_{\\mu \\nu \\rho\\sigma} \\right)\\right) \\ ,\n \\end{equation}\nwhere ${\\Lambda} = -{6 \\over \\lad^2}$ and for now we assume that ${\\alpha}_i \\sim\n{{\\alpha'} \\over \\lad^2} \\ll 1$. Other terms with additional derivatives\nor factors of $R$ are naturally suppressed by higher powers of ${{\\alpha'} \\over\n\\lad^2}$. String loop (quantum) corrections can also generate such terms,\nbut they are suppressed by powers of $g_s$ and we will consistently neglect them by taking $g_s \\rightarrow 0$ limit.\\footnote{Note that to calculate $g_s$ corrections, all the light fields must be taken into account. In addition, the calculation of $\\eta\/s$ could be more subtle once we begin to include quantum effects.} To lowest order\nin ${\\alpha}_i$ the correction to $\\eta\/s$ will be a linear\ncombination of ${\\alpha}_i$'s, and the viscosity bound\nis then violated for one side of the half-plane.\nSpecifically, we will find\n \\begin{equation}\n {\\eta \\over s} = {1 \\over 4 \\pi} \\left(1 - 8 {\\alpha}_3 \\right) + O({\\alpha}_i^2)\n \\end{equation}\nand hence the bound is violated for ${\\alpha}_3>0$. Note that the above\nexpression is independent of ${\\alpha}_1$ and ${\\alpha}_2$. This can be\ninferred from a field redefinition argument (see\nSec.\\ref{ap:fR}).\n\nHow do we interpret these violations? Possible scenarios are:\n\n\\begin{enumerate}\n\n\\item The bound can be violated. For example, this scenario would be realized if one explicitly finds a well-defined string theory on $AdS_5$ which generates a stringy correction with ${\\alpha}_3>0$. (See~\\cite{new} for a plausible counterexample to the KSS bound.)\n\n\\item The bound is correct (for example, if one can prove it using a field\ntheoretical method), and a bulk gravity theory with ${\\alpha}_3>0$ cannot have a well-defined boundary CFT dual.\n\n \\begin{enumerate}\n\n\\item The bulk theory is manifestly\ninconsistent as an effective theory. For example, it could\nviolate bulk causality or unitarity.\n\n\n\\item It is impossible to generate such a low-energy effective\nclassical action from a consistent quantum theory of gravity. In\nmodern language we say that the theory lies in the swampland of\nstring theory.\n\n \\end{enumerate}\n\n\\end{enumerate}\n\nAny of these alternatives, if realized, is interesting. Needless\nto say, possibility 1 would be interesting.\nGiven that recent analyses from RHIC\ndata~\\cite{rr1,songHeinz,r2,Dusling:2007gi,Adare:2006nq} indicate \nthe $\\eta\/s$ is close to (and could be even smaller than) the bound, \nthis further motivates\nto investigate the universality of the KSS bound in holographic\nmodels.\n\nPossibility 2(a) should help clarify the physical origin of the\nbound by correlating bulk pathologies and the violation of the\nbound. Possibility 2(b) could provide powerful tools for\nconstraining possible higher derivative corrections in the string\nlandscape. Note that while there are some nice no-go theorems\nwhich rule out classes of nongravitational effective field\ntheories \\cite{AADNR} (also see \\cite{AKS}), the generalization of\nthe arguments of~\\cite{AADNR} to gravitational theories is subtle\nand difficult. Thus, constraints from AdS\/CFT based on the\nconsistency of the boundary theory would be valuable.\n\nIn investigating the scenarios above, Gauss-Bonnet (GB) gravity will\nprovide a useful model. Gauss-Bonnet gravity, defined by the\nclassical action of the form~\\cite{Zwiebach}\n\\begin{equation}\n\\label{action} I = \\frac{1}{16\\pi G_N} \\mathop\\int{d^{5}x \\,\n\\sqrt{-g} \\, \\left[R-2\\Lambda+ {{\\lambda}_{GB} \\over 2} \\lad^2\n(R^2-4R_{\\mu\\nu}R^{\\mu\\nu}+R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma})\n\\right]} \\ ,\n\\end{equation}\nhas many nice properties that are absent for theories with more general ratios of the ${\\alpha}_i$'s. For example, expanding around flat Minkowski space, the metric fluctuations\nhave exactly the same quadratic kinetic terms as those in\nEinstein gravity. All higher derivative terms\ncancel~\\cite{Zwiebach}. Similarly, expanding around the\nAdS black brane geometry, which will be the main focus of the\npaper, there are also only second derivatives on the\nmetric fluctuations. Thus small metric fluctuations can be\nquantized for finite values of the parameter\n${\\lambda}_{GB}$.\\footnote{Generic theories in~(\\ref{epr}) contain four\nderivatives and a consistent quantization is not possible other\nthan treating higher derivative terms as perturbations.}\nFurthermore, crucial for our investigation is its remarkable feature of solvability: sets of\nexact solutions to the classical equation of motion have been\nobtained \\cite{BD,Cai} and the exact form of the Gibbons-Hawking\nsurface term is known \\cite{Myers}.\n\nGiven these nice features of Gauss-Bonnet gravity, we will venture outside the regime of the perturbatively corrected Einstein gravity and study the theory with finite values of ${\\lambda}_{GB}$.\nTo physically motivate this, one could envision that somewhere in the string landscape ${\\lambda}_{GB}$ is large but all the other higher derivative corrections are small.\nOne of the main results of the paper is a value of $\\eta\/s$ for the CFT dual of Gauss-Bonnet gravity, \\emph{nonperturbative} in ${\\lambda}_{GB}$:\\footnote{We have also computed the value of $\\eta\/s$ for Gauss-Bonnet gravity\nfor any spacetime dimension $D$ and the expression is given\nin~(\\ref{final}).}\n\\begin{equation} \\label{advertise}\n\\frac{\\eta}{s}=\\frac{1}{4\\pi}[1- 4 {\\lambda}_{GB}].\n\\end{equation}\nWe emphasize that this is not just a linearly corrected value.\nIn particular, the viscosity bound is badly violated as ${\\lambda}_{GB} \\rightarrow \\frac{1}{4}$.\nAs we will discuss shortly, ${\\lambda}_{GB}$ is bounded above by $\\frac{1}{4}$ for the theory to have a boundary CFT, and $\\eta\/s$ never decreases beyond $0$.\n\nGiven the result~(\\ref{advertise}) for Gauss-Bonnet, if the possibility 2(a) were correct, we would expect that pathologies would become easier to discern in the limit where $\\eta\/s\\rightarrow 0$. We will investigate this line of thought in Sec.\\ref{gravitoncone}. On the other hand, thinking along the line of possibility 1, the Gauss-Bonnet theory with ${\\lambda}_{GB}$ arbitrarily close to $\\frac{1}{4}$ may have a concrete realization in the string landscape. In this case, there exists no lower bound for $\\eta\/s$, and investigating the CFT dual of Gauss-Bonnet theory should clarify how to evade the heuristic mean free path argument for the existence of the lower bound (presented in, e.g., \\cite{KSSbound}).\n\nThe plan of the paper is as follows. In Sec.\\ref{pre}, we review\nvarious properties of two-point correlation functions and outline\nthe real-time AdS\/CFT calculation of the shear viscosity. We then\nexplicitly calculate the shear viscosity for Gauss-Bonnet theory\nin Sec.\\ref{shear}. In Sec.\\ref{gravitoncone}, we seek possible\npathologies associated with theories violating the viscosity\nbound. There, we will find a curious new metastable state for\nlarge enough ${\\lambda}_{GB}$. Finally in Sec.\\ref{discussion}, we conclude\nwith various remarks and speculations. To make the paper fairly\nself-contained, various appendices are added. In particular,\nquasinormal mode calculations of the shear viscosity are\npresented in Appendix~\\ref{ap:so} and one using the membrane\nparadigm in Appendix~\\ref{junk}.\n\n\n\n\\section{Shear viscosity in $R^2$ theories: preliminaries} \\label{pre}\n\n\n\\subsection{Two-point correlation functions and viscosity} \\label{meds}\n\nLet us begin by collecting various properties of two-point\ncorrelation  functions,\nfollowing~\\cite{Policastro:2002se,Policastro:2002tn,KovtunEV} (see\nalso~\\cite{Son:2007vk}). Consider retarded two-point correlation\nfunctions of the stress energy tensor $T_{\\mu \\nu}$ of a CFT in\n$3+1$-dimensional Minkowski space at a finite temperature $T$\n \\begin{equation} \\label{ttC}\n G_{\\mu\\nu,{\\alpha} \\beta} (\\omega,\\vec q) = - i\\int dt d\\vec x e^{ i\n\\omega t- i \\vec q\\cdot \\vec x} \\theta( t) \\vev{\\left[T_{\\mu\n\\nu}(t,\\vec x) , T_{{\\alpha} \\beta} (0,0)\\right]}.\n \\end{equation}\nThey describe linear responses of the system to small disturbances.\nIt turns out that various components of~(\\ref{ttC}) can be expressed in terms of\nthree independent scalar functions. For example, if we take spatial\nmomentum to be $\\vec q = (0,0 ,q)$, then\n \\begin{equation}\n G_{12,12}= {1\\over 2} G_3 (\\omega, q) , \\ \\ \\ \\ \\ G_{13,13} = {1\\over 2}{\\omega^2 \\over \\omega^2 -q^2}\n G_1 (\\omega, q), \\ \\ \\ \\ \\ G_{33,33} = {2 \\over 3} {\\omega^4 \\over (\\omega^2 -q^2)^2} G_2 (\\omega,\n q),\n \\end{equation}\nand so on. At $\\vec q=0$ all three function $G_{1,2,3} (\\omega,0)$\nare equal to one another as a consequence of rotational\nsymmetry.\n\nWhen $\\omega, |\\vec q| \\ll T$ one expects the CFT plasma to be described by hydrodynamics.\nThe scalar functions $G_{1,2,3}$ encode the hydrodynamic behavior of shear, sound, and transverse modes, respectively.\nMore explicitly, they have the following properties:\n   \\begin{itemize}\n   \\item $G_1$ has a simple diffusion pole at $\\omega= - i D q^2$, where\n    \\begin{equation} \\label{shearA}\n    D= {\\eta \\over {\\epsilon} + P} = {1 \\over T} { \\eta \\over  s}\n    \\end{equation}\n    with ${\\epsilon}$ and $s$ being the energy and entropy density, and $P$ the pressure of the\n    gauge theory plasma.\n   \\item $G_2$ has a simple pole at $\\omega= \\pm c_s q - i \\Gamma_{s} q^2$, where $c_s$ is\n   the speed of sound and $\\Gamma_{s}$ is the sound damping\n   constant, given by (for conformal theories)\n \\begin{equation} \\label{sound}\n \\Gamma_s = {2 \\over 3T}{\\eta \\over s}\n \\end{equation}\n   \\item $\\eta$ can also be obtained from $G_{1,2,3}$ at zero spatial momentum by\nthe Kubo formula, e.g.,\n \\begin{equation} \\label{rrp}\n \\eta= \\lim_{\\omega\\rightarrow 0} {1\\over \\omega} {\\rm Im} G_{12,12} (\\omega,0)\n \\end{equation}\n\n \\end{itemize}\nEquations (\\ref{shearA})--(\\ref{rrp}) provide three independent ways of extracting $\\eta\/s$.\nWe provide calculations utilizing the first two in Appendix~\\ref{ap:so}.\nA calculation utilizing the Kubo formula (\\ref{rrp}) is easier, and we will explicitly implement it for Gauss-Bonnet theory in Sec.\\ref{shear}.\nIn the next subsection, we outline how to obtain retarded two-point functions within the framework of the real-time AdS\/CFT correspondence.\n\n\n\\subsection{AdS\/CFT calculation of shear viscosity: Outline} \\label{adscft}\n\nThe stress tensor correlators for a boundary CFT described by\n(\\ref{epr}) or (\\ref{action}), can be computed from gravity as\nfollows. One first finds a black brane solution (i.e. a black hole\nwhose horizon is ${\\bf R}^3$) to the equations of motion of\n(\\ref{epr}) or (\\ref{action}). Such a solution describes the\nboundary theory on ${\\bf R}^{3,1}$ at a temperature $T$, which can\nbe identified with the Hawking temperature of the black brane. The\nentropy and energy density of the boundary theory are given by the\ncorresponding quantities of the black brane. The fluctuations of\nthe boundary theory stress tensor are described in the gravity\nlanguage by small metric fluctuations $h_{\\mu \\nu}$ around the\nblack brane solution. In particular, after taking into account of\nvarious symmetries and gauge degrees of freedom, the metric\nfluctuations can be combined into three independent scalar fields\n$\\phi_a, a=1,2,3$, which are dual to the three functions $G_a$ of the\nboundary theory.\n\nTo find $G_a$, one could first work out the bulk two-point\nretarded function for $\\phi_a$ and then take both points to the\nboundary of the black brane geometry. In practice it is often more\nconvenient to use the prescription proposed in~\\cite{SS}, which\ncan be derived from the real-time AdS\/CFT\ncorrespondence~\\cite{HS}. Let us briefly review it here:\n\n\\begin{enumerate}\n\n\\item  Solve the linearized equation of motion for $\\phi_a (r; k)$ with the\nfollowing boundary conditions:\n\n\\begin{enumerate}\n\n\\item Impose the infalling boundary condition at the horizon. In other words, modes with timelike momenta should be\nfalling into the horizon and modes with spacelike momenta should\nbe regular.\n\n\\item Take $r$ to be the radial direction of the black brane geometry\nwith the boundary at $r=\\infty$. Require\n \\begin{equation} \\label{bdw}\n \\phi_a (r; k)|_{r= {1 \\over {\\epsilon}}} = J_a (k),   \\qquad k = (\\omega, q),\n \\end{equation}\nwhere ${\\epsilon} \\to 0$ imposes an infrared cutoff near the infinity\nof the spacetime and $J_a (k)$ is an infinitesimal boundary source for the\nbulk field $\\phi_a(r; k)$.\n\n\\end{enumerate}\n\n\\item Plug in the above solution into the action, expanded to quadratic order in $\\phi_a (r; k)$.\nIt will reduce to pure surface contributions.\nThe prescription instructs us to pick up only the contribution from the boundary at $r={1 \\over {\\epsilon}}$.\nThe resulting action can be written as\n\\begin{equation}\\label{Sbd}\nS = - {1\\over 2} \\int\\!{d^4k\\over (2\\pi )^4}\\,\n  J_a (-k) {\\cal F}_a (k,r) J_a (k) \\Big|_{r={1 \\over {\\epsilon}}}\\ .\n\\end{equation}\nFinally the retarded function $G_a (k)$ in momentum space for the\nboundary field dual to $\\phi_a$ is given by\n \\begin{equation} \\label{eo}\n G_a (k) = \\lim_{{\\epsilon} \\to 0} {\\cal F}_a (k,r)\\Big|_{r={1 \\over {\\epsilon}}} \\\n .\n \\end{equation}\n\n\\end{enumerate}\nUsing the Kubo formula~(\\ref{rrp}), we can get the shear viscosity by studying a mode $\\phi_3$ with $\\vec q=0$ in the low-frequency limit $\\omega \\rightarrow 0$. We will do so in the next section. Alternatively, using (\\ref{shearA}) or (\\ref{sound}), we can read off the viscosity from pole structures of retarded two-point functions. Such a calculation is a bit more involved and will be performed in Appendix~\\ref{ap:so}.\n\nThe above prescription for computing retarded functions in AdS\/CFT\nworks well if the bulk scalar field has only two derivatives as in\nGauss-Bonnet case~(\\ref{action}). If the bulk action contains more\nthan two derivatives, complications could arise even if one treats\nthe higher derivative parts as perturbations. For example, one\nneeds to add Gibbons-Hawking surface terms to ensure a\nwell-defined variational problem. A systematic prescription for\ndoing so is, however, not available at the moment beyond the\nlinear order. Thus there are potential ambiguities in implementing\n(\\ref{eo}).\\footnote{In~\\cite{BLS}, such additional terms do not\nappear to affect the calculation at the order under discussion\nthere.} Clearly these are important questions which should be\nexplored more systematically. At the $R^2$ level, as we describe\nbelow in Sec.\\ref{ap:fR}, all of our calculations can be reduced\nto the Gauss-Bonnet case in which these potential complications do\nnot arise.\n\n\n\\subsection{Field redefinitions in $R^2$ theories} \\label{ap:fR}\n\nWe now show that to linear order in ${\\alpha}_i$, $\\eta\/s$\nfor~(\\ref{epr}) is independent of ${\\alpha}_1$ and ${\\alpha}_2$. It is well\nknown that to linear order in ${\\alpha}_i$, one can make a field\nredefinition to remove the $R^2$ and $R_{\\mu \\nu}R^{\\mu\\nu}$ term\nin~(\\ref{epr}). More explicitly, in~(\\ref{epr}) set ${\\alpha}_3 =0$ and\ntake\n \\begin{equation} \\label{eep}\n  g_{\\mu \\nu} = \\tilde g_{\\mu \\nu} + {\\alpha}_2 \\lad^2 \\tilde R_{\\mu \\nu} - {\\lad^2 \\over 3}\n  ({\\alpha}_2\n+ 2 {\\alpha}_1 ) \\tilde g_{\\mu \\nu} \\tilde R,\n \\end{equation}\nwhere $\\tilde R$ denotes the Ricci scalar for $\\tilde g_{\\mu\\nu}$\nand so on. Then (\\ref{epr}) becomes\n \\begin{equation} \\label{newz}\n I  =  {1 \\over 16 \\pi G_N} \\int \\sqrt{- \\tilde g} ((1+ {\\cal K})\n \\tilde R - 2  \\Lambda  ) + O({\\alpha}^2)\n  = {1 +  {\\cal K}\\over 16 \\pi G_N} \\int \\sqrt{- g} (   \\tilde R - 2 \\tilde \\Lambda ) +\n  O({\\alpha}^2)\n \\end{equation}\nwith \\begin{equation} {\\cal K}= {2 {\\Lambda} \\lad^2 \\over 3} \\left(5 {\\alpha}_1 + {\\alpha}_2 \\right) , \\qquad\n \\tilde {\\Lambda} = {{\\Lambda} \\over 1 + {\\cal K}} \\ .\n\\end{equation} It follows from (\\ref{eep}) that a background solution\n$g^{(0)}$ to (\\ref{epr}) (with ${\\alpha}_3=0$) is related to a solution\n$\\tilde g^{(0)}$ to (\\ref{newz}) by\n \\begin{equation} \\label{sba}\n ds^2_0  = A^2 \\tilde{ds}^2_0, \\qquad A = 1- { {\\cal K} \\over 3} \\ .\n \\end{equation}\nThe scaling in (\\ref{sba}) does not change the background Hawking\ntemperature.  The diffusion pole~(\\ref{shearA}) calculated using\n~(\\ref{newz}) around $\\tilde g^{(0)}$ then gives the standard\nresult $D = {1 \\over 4 \\pi T}$~\\cite{Policastro:2002se}.\n Thus we conclude that $\\eta\/s = {1 \\over\n4 \\pi}$ for (\\ref{epr}) with ${\\alpha}_3=0$. Then to linear order\nin ${\\alpha}_i$,\n $\\eta\/s$ can only\ndepend on ${\\alpha}_3$. To find this dependence,  it is convenient to\nwork with the Gauss-Bonnet theory~(\\ref{action}). Gauss-Bonnet\ngravity is not only much simpler than~(\\ref{epr}) with generic\n${\\alpha}_3 \\neq 0$, but also contains only second derivative terms in\nthe equations of motion for $h_{\\mu \\nu}$, making the extraction\nof boundary correlators unambiguous.\n\n\n\n\\section{Shear Viscosity for Gauss-Bonnet Gravity}\n\\label{shear}\n\nIn this section, after briefly reviewing the thermodynamic properties of the black brane solution, we compute the shear viscosity for Gauss-Bonnet gravity~(\\ref{action}) nonperturbatively in ${\\lambda}_{GB}$.\nHere, we follow the outline presented in the previous section, with the Kubo formula (\\ref{rrp}) in mind.\nIn Appendix~\\ref{ap:so}, we extract $\\eta\/s$  from the shear channel~(\\ref{shearA}) and the sound channel~(\\ref{sound}) (perturbatively in ${\\lambda}_{GB}$).\nThere we also find that the sound velocity remains at the conformal value $c_s^2 = {1 \\over 3}$ as it should.\nIn Appendix~\\ref{junk}, we provide a membrane paradigm calculation, again nonperturbatively in ${\\lambda}_{GB}$.\nAll four methods give the same result.\n\n\n\n\\subsection{Black brane geometry and thermodynamics}\n\nExact solutions and thermodynamic properties of black objects in\nGauss-Bonnet gravity~(\\ref{action}) were discussed\nin~\\cite{Cai} (see also \\cite{Nojiri:2001aj, Cho:2002hq,Neupane:2002bf,Neupane:2003vz}). Here we summarize some features relevant for our\ndiscussion below. The black brane solution can be written as\n\\begin{equation}\n\\label{bba} ds^2=-f(r)N_{\\sharp}^2dt^2\n+\\frac{1}{f(r)}dr^2+\\frac{r^2}{\\lad^2}\n \\left(\\mathop\\sum_{i=1}^{3}dx_i^2 \\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{perturb} f(r)=\\frac{r^2}{\\lad^2}\\frac{1}{2{\\lambda}_{GB}}\n\\left[1-\\sqrt{1-4{\\lambda}_{GB}\\left(1-\\frac{r_{+}^{\\,4}}{r^4}\\right)} \\right] \\ .\n\\end{equation}\nIn (\\ref{bba}), $N_{\\sharp}$ is an arbitrary constant which specifies the\nspeed of light of the boundary theory. Note that as $r \\to \\infty$,\n\\begin{equation} \\label{Ade}\n f(r) \\to {r^2 \\over a^2 \\lad^2}, \\qquad {\\rm with} \\qquad\n  a^2 \\equiv\n  {1\\over 2}\\left(1+\\sqrt{1-4{\\lambda}_{GB}}\\right)\\ .\n \\end{equation}\nIt is straightforward to see that the AdS curvature scale of these\ngeometries is $a \\lad$.\\footnote{Here we note that the Gauss-Bonnet\ntheory also admits another background with the curvature scale\n$\\tilde{a}\\,\\lad$ where $\\tilde{a}^2={1\\over 2}\\left(1-\\sqrt{1-4{\\lambda}_{GB}}\\right)$.\nEven though this remains an asymptotically AdS solution for ${\\lambda}_{GB}>0$,\nwe do not consider it here because this background is unstable and\ncontains ghosts \\cite{BD}.} If we choose $N_{\\sharp} = a$, then the boundary\nspeed of light is unity. However, we will leave it unspecified in\nthe following. We assume that ${\\lambda}_{GB}\\leq\\frac{1}{4}$. Beyond this\npoint, (\\ref{action}) does not admit a vacuum AdS solution, and\ncannot have a boundary CFT dual. In passing, we note that while the\ncurvature singularity occurs at $r=0$ for ${\\lambda}_{GB} \\geq 0$, it shifts to\n$r =r_+ \\left(1-{1 \\over 4 {\\lambda}_{GB}}\\right)^{-\\frac{1}{4}}$ for ${\\lambda}_{GB}<0$.\n\n\nThe horizon is located at $r=r_{+}$ and the Hawking temperature,\nentropy density, and  energy density of the black brane are\n\\footnote{Note that for {\\it planar} black branes in Gauss-Bonnet\ntheory, the area law for entropy still holds \\cite{oldy}. This is\nnot the case for more general higher-derivative-corrected black\nobjects.}\n\\begin{equation}\n\\label{temperature} T =N_{\\sharp} \\frac{r_{+}}{\\pi \\lad^2},\n\\end{equation}\n\\begin{equation} \\label{entr}\ns =\\frac{1}{4G_{N}}\\left(\\frac{r_{+}}{\\lad}\\right)^3\n=\\frac{(\\pi \\lad)^3}{4G_{N}}\\frac{(T)^3}{N_{\\sharp}^3},\n\\qquad {{\\epsilon}} = {3 \\over 4} T s \\ .\n\\end{equation}\nIf we fix the boundary theory temperature $T$ and the speed of light to be unity (taking $N_{\\sharp}=a$), the entropy and energy density are monotonically\nincreasing functions of ${\\lambda}_{GB}$, reaching a maximum at ${\\lambda}_{GB}={1 \\over 4}$\nand going to zero as ${\\lambda}_{GB} \\to -\\infty$.\n\nTo make our discussion self-contained, in Appendix~\\ref{ap:ther}, we compute the free\nenergy of the black brane and derive the entropy density. In\nparticular, we show that the contribution from the Gibbons-Hawking\nsurface term to the free energy vanishes.\n\n\\subsection{Action and equation of motion for the scalar channel} \\label{scalar}\n\nTo compute the shear viscosity, we now study small metric fluctuations $\\phi = h^1_{\\ 2}$ around the black brane background of the form\n\\begin{equation}\nds^2=-f(r)N_{\\sharp}^2dt^2+\\frac{1}{f(r)}dr^2+\\frac{r^2}{\\lad^2}\n\\left(\\mathop\\sum_{i=1}^{3}dx_i^2 + 2 \\phi(t,\\vec x,  r)dx_1 d x_2 \\right) \\ .\n\\end{equation}\nWe will take $\\phi$ to be independent of $x_1$ and $x_2$ and write\n\\begin{equation}\n\\phi(t, \\vec x, r)=\\mathop\\int\\frac{d\\omega dq}{(2\\pi)^{2}}\\, \\phi(r;k)\n\\, e^{-i\\omega t + i q x_3}, \\quad k = (\\omega, 0, 0, q) ,\\ \\ \\phi\n(r; -k)=\\phi^*(r; k) \\ .\n\\end{equation}\nFor notational convenience, let us introduce\n\\begin{equation} \\label{sDef}\nz=\\frac{r}{r_{+}},\\ \\ \\tilde{\\omega}=\\frac{\\lad^2}{r_{+}}\\omega,\\\n\\quad \\tilde{q}=\\frac{\\lad^2}{r_{+}}q, \\qquad\n\\tilde{f}=\\frac{\\lad^2}{r_{+}^2}f =  {z^2 \\over 2 {\\lambda}_{GB}} \\left(1 -\n\\sqrt{1-4 {\\lambda}_{GB} + {4 {\\lambda}_{GB} \\over z^4}} \\right).\n\\end{equation}\nThen, at quadratic order, the action for $\\phi$ can be written as\n \\begin{eqnarray} \\label{pee}\n S&=&\\int{dk_1 dk_2 \\over (2 \\pi)^2}S(k_1, k_2) \\ \\ \\ {\\rm with} \\cr\n S(k_1=0, k_2=0)&=&-{1\\over 2} C \\int dz {\\frac{d\\omega dq}{(2\\pi)^{2}}} \\, \\left( K (\\partial_z \\phi)^2 - K_2 \\phi^2  +\n \\partial_z (K_3 \\phi^2) \\right),\n \\end{eqnarray}\nwhere\n \\begin{equation} \\label{vds}\n C = {1 \\over 16 \\pi G_N} \\left(N_{\\sharp} r_+^4\\over \\lad^5\\right), \\ \\ K= z^2 \\tilde{f} (z - {\\lambda}_{GB}\n \\partial_z\\tilde{f}), \\ \\ \\ K_2 = K {\\tilde \\omega^2 \\over N_{\\sharp}^2 \\tilde{f}^2} - \\tilde\n q^2 z \\left(1- {\\lambda}_{GB} \\partial_z^2\\tilde{f} \\right) \\ ,\n \\end{equation}\nand $\\phi^2$ should be understood as a shorthand notation for\n$\\phi(z;k) \\phi (z,-k)$.\nHere, $S$ is the sum of the bulk action (\\ref{action}) and the associated Gibbons-Hawking surface term \\cite{Myers}.\nThe explicit expression for $K_3$ will not be important for our subsequent discussion.\n\nThe equation of motion following from (\\ref{pee}) is\\footnote{An easy way to get the quadratic action~(\\ref{pee}) is to first obtain the linearized equation of motion and then read off $K$ and $K_2$ from it.}\n \\begin{equation} \\label{eom}\n K \\phi'' + K' \\phi' + K_2 \\phi =0 \\ ,\n \\end{equation}\nwhere primes indicate partial derivatives with respect to $z$.\nUsing the equation of motion, the action~(\\ref{pee}) reduces to the surface contributions as advertised in Sec.\\ref{adscft},\n \\begin{equation} \\label{rrk}\n S(k_1=0, k_2=0) = -{1\\over 2} C \\int {\\frac{d\\omega dq}{(2\\pi)^{2}}} \\, \\left(K \\phi' \\phi\n + K_3 \\phi^2 \\right)|_{{\\rm surface}} \\ .\n \\end{equation}\nThe prescription described in Sec.\\ref{adscft} instructs us to\npick up the contribution from the boundary at\n$z\\rightarrow\\infty$. Here, the term proportional to $K_3$ will\ngive rise to a real divergent contact term, which is discarded.\n\nA curious thing about (\\ref{pee}) is that for all values of $z$,\nboth $K$ and $K_2$ (but not $K_3$) are proportional to ${1 \\over 4}\n- {\\lambda}_{GB}$.\\footnote{This can be seen by using the following equation\nin $K$ and $K_2$ \\begin{equation} \\tilde{f}'(z) = {2 z (2z^2 -\\tilde{f}) \\over z^2 - 2 {\\lambda}_{GB}\n\\tilde{f}} \\ . \\end{equation} } Thus other than the boundary term the whole action\n(\\ref{pee}) vanishes identically at ${\\lambda}_{GB} = {1 \\over 4}$.\nNevertheless, the equation of motion (\\ref{eom}) remains\nnontrivial in the limit ${\\lambda}_{GB} \\to {1 \\over 4}$ as the ${1 \\over 4} - {\\lambda}_{GB}$ factor cancels\nout. Note that the correlation function does not necessarily go to\nzero in this limit since it also depends on\nthe behavior of the solution to~(\\ref{eom}) and the limiting\nprocedure~(\\ref{rrk}). As we will see momentarily, as least in the\nsmall frequency limit it does become zero with a vanishing shear\nviscosity.\n\n\n\\subsection{Low-frequency expansion and the viscosity}\n\\label{solution}\n\n\nGeneral solutions to the equation of motion~(\\ref{eom}) can be written as\n\\begin{equation}\n\\phi(z; k)=a_{in}(k)\\phi_{in}(z; k)+a_{out}(k)\\phi_{out}(z; k) \\ ,\n\\end{equation}\nwhere $\\phi_{in}$ and $\\phi_{out}$ satisfy infalling and outgoing boundary conditions at the horizon, respectively.\nThey are complex conjugates of each other, and we normalize them by requiring them to approach $1$ as $z\\to \\infty$.\nThen, the prescription of Sec.\\ref{adscft} corresponds to setting\n\\begin{equation} \\label{explicitBC}\na_{in}(k)=J(k)\\ , \\qquad a_{out}(k)=0 \\ ,\n\\end{equation}\nwhere $J(k)$ is an infinitesimal boundary source for the bulk field $\\phi$.\n\nMore explicitly, as $z \\to 1$, various functions in (\\ref{eom}) have the following behavior\n \\begin{equation}\n{K_2 \\over K} \\approx {\\tilde{\\omega}^2 \\over 16 N_{\\sharp}^2 (z-1)^2 } + O((z-1)^{-1})+O(\\tilde{q}^2),\n\\qquad  {K' \\over K} = {1 \\over z-1} + O(1) \\ .\n \\end{equation}\nIt follows that near the horizon $z=1$, equation (\\ref{eom}) can\nbe solved by (for $\\vec q =0$)\n \\begin{equation}\n\\phi (z) \\sim (z-1)^{\\pm {i \\tilde{\\omega} \\over 4 N_{\\sharp}}} \\sim (z-1)^{\\pm {i \\omega\n\\over 4 \\pi T}}\n \\end{equation}\nwith the infalling boundary condition corresponding to the\nnegative sign. To solve (\\ref{eom}) in the small frequency limit,\nit is convenient to write\n\\begin{equation} \\label{anse}\n\\phi_{in} (z; k)=e^{-i \\left({\\tilde{\\omega} \\over 4 N_{\\sharp}}\\right) {\\rm ln}\\left(\\frac{a^2\n\\tilde{f}}{z^2}\\right)} \\left(1-i\\frac{\\tilde{\\omega}} {4\nN_{\\sharp}}g_1(z)+O(\\tilde{\\omega}^2, \\tilde{q}^2)\\right),\n\\end{equation}\nwhere we require $g_1 (z)$ to be nonsingular at the horizon $z=1$.\nWe show in Appendix~\\ref{ap:solo} that $g_1$ is a nonsingular function with the large $z$\nexpansion\n \\begin{equation} \\label{lowEatCFT}\n g_1 (z) = {4 {\\lambda}_{GB} \\over \\sqrt{1-4 {\\lambda}_{GB}}} {a^2 \\over z^4} + O(z^{-8}) \\\n .\n \\end{equation}\nTherefore, with our boundary conditions (\\ref{explicitBC}), we find\n \\begin{equation} \\label{asu}\n \\phi (z;k) =J(k)\\left[ 1 +  {i \\tilde{\\omega} \\over 4 N_{\\sharp}} a^2 \\sqrt{1-4 {\\lambda}_{GB}} \\left({1 \\over z^4}\n + O(z^{-8}) \\right) + O(\\tilde{\\omega}^2, \\tilde{q}^2)\\right].\n \\end{equation}\nThis is the right asymptotic behavior for the bulk field $\\phi$\ndescribing metric fluctuations since the CFT stress tensor has\nconformal dimension 4.\n\nPlugging~(\\ref{asu}) into (\\ref{rrk}) and using the expressions for\n$C$ and $K$ in (\\ref{vds}), the prescription described in Sec.\\ref{adscft} gives\n \\begin{equation}\n {\\rm Im} G_{12,12} (\\omega,0)=\\omega{1 \\over 16 \\pi G_N} \\left(r_+^3\\over \\lad^3\\right)  (1-4\n {\\lambda}_{GB}) +O(\\omega^2).\n \\end{equation}\nThen, the Kubo formula~(\\ref{rrp}) yields\n \\begin{equation} \\label{ets}\n \\eta = {1 \\over 16 \\pi G_N} \\left(r_+^3\\over \\lad^3\\right)  (1-4\n {\\lambda}_{GB}).\n \\end{equation}\nFinally, taking the ratio of (\\ref{ets}) and (\\ref{entr}) we find that\n \\begin{equation} \\label{ror}\n {\\eta \\over s} = {1 \\over 4 \\pi} (1-4\n {\\lambda}_{GB}).\n \\end{equation}\nThis is \\emph{nonperturbative} in ${\\lambda}_{GB}$. Especially,\nthe linear correction is the only nonvanishing term.\\footnote{It would be interesting to find an explanation for vanishing of higher order corrections.}\n\nWe now conclude this section with various remarks:\n\n\\begin{enumerate}\n\n\\item Based on the field redefinition argument presented in\nSec.\\ref{ap:fR}, one finds from (\\ref{ror}) that for (\\ref{epr}),\n \\begin{equation} \\label{oror}\n {\\eta \\over s} = {1 \\over 4 \\pi} \\left(1 - 8 {\\alpha}_3 \\right) + O({\\alpha}_i^2).\n \\end{equation}\nWe have also performed an independent calculation of $\\eta\/s$\n(without using field redefinitions) for~(\\ref{epr}) using all\nthree methods outlined in Sec.\\ref{meds} and\nconfirmed~(\\ref{oror}).\n\n\\item The ratio $\\eta\/s$ dips below the viscosity bound for ${\\lambda}_{GB}\n> 0$ in Gauss-Bonnet gravity and for ${\\alpha}_3 > 0$ in~(\\ref{epr}).\nIn particular, the shear viscosity approaches zero as ${\\lambda}_{GB} \\to {1 \\over 4}$ for Gauss-Bonnet.\n Note that the whole off-shell action becomes zero in this limit. It is\nlikely the on-shell action also vanishes, implying that the\ncorrelation function could become identically zero in this limit.\n\n\n\\item Fixing the temperature $T$ and the boundary speed of light\nto be unity, as we take ${\\lambda}_{GB} \\to -\\infty$, $\\eta \\sim (-{\\lambda}_{GB})^{1\n\\over 4} \\to \\infty$. In contrast the entropy density decreases as\n$s \\sim (-{\\lambda}_{GB})^{-{3 \\over 4}} \\to 0$.\n\n\\item The shear viscosity of the boundary conformal field theory\nis associated with absorption of transverse modes by the black\nbrane in the bulk. This is a natural picture since the shear\nviscosity measures the dissipation rate of those fluctuations: the\nquicker the black brane absorbs them, the higher the dissipation\nrate will be.\nFor example, as ${\\lambda}_{GB} \\rightarrow-\\infty$, $\\eta\/s$ approaches\ninfinity; this describes a situation where every bit of the black\nbrane horizon devours the transverse fluctuations very quickly.\nIn this limit the curvature singularity at $z = \\left(1-{1 \\over 4 {\\lambda}_{GB}}\\right)^{-\\frac{1}{4}}$ approaches the horizon and the tidal force near the horizon becomes strong.\nOn the other hand, as ${\\lambda}_{GB} \\to {1 \\over 4}$,\n$\\eta\/s\\rightarrow0$ and the black brane very slowly absorbs transverse modes.\\footnote{We note that for\n${\\lambda}_{GB}=\\frac{1}{4}$ in $4+1$ spacetime dimension, the radial direction of the background\ngeometry resembles a ${\\rm Ba\\tilde{n}ados}$-Teitelboim-Zanelli (BTZ) black brane.}\n\n\\item The calculation leading to (\\ref{ror}) can be generalized to\ngeneral $D$ spacetime dimensions and one finds for $D\\geq4+1$\\footnote{For general\ndimensions we use the convention\n  \\begin{equation}\nS = \\frac{1}{16\\pi G_N} \\mathop\\int{d^{D}x \\, \\sqrt{-g} \\,\n\\left[R-2\\Lambda+ \\alpha_{GB} \\lad^2\n(R^2-4R_{\\mu\\nu}R^{\\mu\\nu}+R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma})\n\\right]} \\\n\\end{equation}\nwith ${\\Lambda} = - {(D-1) (D-2) \\over 2 \\lad^2}$ and ${\\lambda}_{GB} = (D-3)(D-4)\n\\alpha_{GB}$.}\n \\begin{equation}\n\\label{final} \\frac{\\eta}{s}=\\frac{1}{4\\pi}\\left[1-2\\frac{(D-1)}{(D-3)}{\\lambda}_{GB} \\right] \\ .\n\\end{equation}\nHere again ${\\lambda}_{GB}$ is bounded above by ${1 \\over 4}$. Thus for $D >\n4+1$, $\\eta$ never approaches zero within Gauss-Bonnet theory. For\n$D=3+1$ or $2+1$, in which case the Gauss-Bonnet term is\ntopological, there is no correction to $\\eta\/s$.\n\n\\item In Appendix~\\ref{junk}, we obtain the same result\n(\\ref{ror}) using the membrane paradigm \\cite{Kovtun:2003wp}. Thus\nwhen embedded into the AdS\/CFT correspondence, the membrane paradigm\ncorrectly captures the infrared (hydrodynamic) sector of the\nboundary thermal field theory. Further, we see something interesting\nin its derivation. There, the diffusion constant is expressed as the\nproduct of a factor evaluated at the horizon (\\ref{one}) and an\nintegral from the horizon to infinity (\\ref{two}). In the limit\n${\\lambda}_{GB}\\to{1 \\over 4}$, it is the former that approaches zero.\n\n\\end{enumerate}\n\n\\section{Causality in Bulk and on Boundary}\n\\label{gravitoncone}\n\nIn this section we investigate if there are causality\nproblems in the bound-violating theories discussed above. First we will discuss the bulk causal structure.\nThen we discuss a  curious high-momentum metastable state in  the\nbulk  graviton wave equation that may have consequences for\nboundary causality. The analysis in this section is refined in~\\cite{newBLMSY} where we indeed see a precise signal of causality violation for ${\\lambda}_{GB}>\\frac{9}{100}$.\n\n\\subsection{Graviton cone tipping}\n\\label{conetip}\n\nAs a consequence of higher derivative terms in the gravity action,\ngraviton wave packets in general do not propagate on the\nlight cone of  a given background geometry. For example, when ${\\lambda}_{GB}\n\\neq 0$, the equation (\\ref{eom}) for the propagation of a\ntransverse graviton differs from that of a minimally coupled\nmassless scalar field propagating in the same background geometry\n(\\ref{bba}). To make the discussion precise, let us write (we will\nconsider only $x_{1, 2}$-independent waves) \\begin{equation} \\label{envelope}\n\\phi(t, r, x_3)=e^{-i\\omega t +i k_r r+i q x_3}\\phi_{en}(t, r,\nx_3). \\end{equation} Here, $\\phi_{en}$ is a slowly-varying envelope function,\nand we take the limit $k=(\\omega, k_r, 0, 0, q)\\to \\infty$. In\nthis limit, the equation of motion (\\ref{eom}) reduces to \\begin{equation}\n\\label{eikonal} k^{\\mu}k^{\\nu}g^{\\rm eff}_{\\mu\\nu}\\approx 0, \\ \\end{equation}\nwhere\n \\begin{equation}\n\\label{effgeo} ds_{\\rm eff}^2=g^{\\rm eff}_{\\mu\\nu}dx^{\\mu}dx^{\\nu}\n=f(r)N_{\\sharp}^2 \\left(-dt^2 + {1 \\over c_g^2} dx_3^2 \\right)\n+\\frac{1}{f(r)}dr^2.\n \\end{equation}\nIn (\\ref{effgeo})\n \\begin{equation} \\label{Nse}\nc_g^2 (z)   = {N_{\\sharp}^2 \\tilde f(z) \\over z^2} {1-{\\lambda}_{GB} \\tilde{f}'' \\over 1 -\n{{\\lambda}_{GB} \\tilde{f}' \\over z}} \\equiv c_b^2 {1-{\\lambda}_{GB} \\tilde{f}'' \\over 1 - {{\\lambda}_{GB} \\tilde{f}'\n\\over z}}\n \\end{equation}\ncan be interpreted as the local ``speed of graviton'' on a\nconstant $r$-hypersurface. $c_b^2 \\equiv {N_{\\sharp}^2 \\tilde f(z) \\over\nz^2} $ introduced in the second equality in~(\\ref{Nse}) is the\nlocal speed of light as defined by the background\nmetric~(\\ref{bba}). Thus the graviton cone in general does not coincide with the\nstandard null cone or light cone defined by the background metric.\\footnote{Note that\n \\begin{equation} \\label{Nsee}\n {c_g^2 \\over c_b^2}   = {1-{\\lambda}_{GB} \\tilde{f}'' \\over 1 - {{\\lambda}_{GB} \\tilde{f}' \\over z}} =\n{1 - 4 {\\lambda}_{GB} + 12 {{\\lambda}_{GB} \\over z^4} \\over 1 - 4 {\\lambda}_{GB} + 4 {{\\lambda}_{GB}  \\over  z^4} }\n\\ , \\end{equation} and in particular the ratio is greater than $1$ for ${\\lambda}_{GB} >\n0$. Note that bulk causality and the existence of a well-posed\nCauchy problem  do not crucially depend on reference metric\nlight cones and  such tipping is not a definitive sign of\ncausality problems. Also for any value of ${\\lambda}_{GB}$, the graviton cone coincides with the\nlight cone in the radial direction. If not, we could have argued for\nthe violation of the second law of thermodynamics following\n\\cite{Dubovsky:2006vk,Eling:2007qd}. Further note that for ${\\lambda}_{GB} <\n-{1 \\over 8}$, there exists a  region outside the horizon where $c_g^2\n< 0$ which will lead to the appearance of tachyonic modes, following\n\\cite{spectre}. We have not explored the full significance of this\ninstability here since it is not correlated with the viscosity bound.}  A few more comments about graviton cone are found at the end of Appendix~\\ref{junk}.\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.7,angle=0]{1Velocity.eps}\n\\caption{$c_g^2 (z)$ (vertical axis) as a function of $z$\n(horizontal axis) for ${\\lambda}_{GB} =0.08$ (left panel) and ${\\lambda}_{GB} =0.1$\n(right panel). For ${\\lambda}_{GB} < {9 \\over 100}$, $c_g^2$ is a monotonically\nincreasing function of $z$. When ${\\lambda}_{GB} > {9 \\over 100}$, as one\ndecreases $z$ from infinity, $c_g^2$ increases from $1$ to a\nmaximum value at some $z>1$ and then decreases to $0$ as $z \\to 1$\n(horizon). }\n \\label{velo}\n\\end{figure}\n\nIn the nongravitational boundary theory there is an invariant\nnotion of light cone and causality.\nAt a heuristic level, a graviton wave packet moving at speed $c_g (z)$ in the\nbulk should translate into disturbances of the stress tensor propagating with the same velocity in the boundary theory.\nIt is\nthus instructive to compare $c_g$ and $c_b$ with the boundary\nspeed of light, which we now set to unity by taking $N_{\\sharp} = a$~($a$\nwas defined in~(\\ref{Ade})). At the boundary ($z= \\infty$) one finds\nthat $c_g (z)= c_b (z)= 1$. In the bulk, the background local\nspeed of light $c_b$ is always smaller than $1$, which is related\nto the redshift of the black hole geometry. The local speed of graviton $c_g (z)$, however, can be greater than $1$ for\ncertain range of $z$ if ${\\lambda}_{GB}$ is sufficiently large. To see this,\nwe can examine the behavior of $c_g^2$ near $z = \\infty$,\n \\begin{equation} \\label{veS}\n  c_g^2 (z) - 1  =   {  b_1 \\over z^4} + O(z^{-8}) , \\quad z\n  \\to \\infty,\n \\qquad b_1({\\lambda}_{GB}) = - {1 + \\sqrt{1 - 4 {\\lambda}_{GB}} - 20 {\\lambda}_{GB}  \\over 2 (1 - 4\n {\\lambda}_{GB})} \\ .\n \\end{equation}\n $b_1 ({\\lambda}_{GB})$ becomes positive and thus  $c_g^2$ increases above $1$\n if ${\\lambda}_{GB} > {9 \\over 100}$. For such a ${\\lambda}_{GB}$, as we decrease $z$ from\n infinity, $c_g^2$ will increase from $1$ to a maximum at some value of\n $z$ and then decrease to zero at the horizon. See Fig.~\\ref{velo}\n for the plot of $c_g^2 (z)$ as a function of $z$ for two values of ${\\lambda}_{GB}$.\n When ${\\lambda}_{GB} = {9 \\over 100}$ one finds that\n the next order term in (\\ref{veS}) is negative and thus $c_g^2$\n does not go above $1$.\n Also note that ${\\lambda}_{GB} \\to {1 \\over 4}$,\n $b_1 ({\\lambda}_{GB})$ goes to plus infinity.\\footnote{In fact coefficients of\n all higher order terms in $1\/z$ expansion become divergent in this limit.}\nThus heuristically, in the boundary theory there is a potential for superluminal propagation of disturbances of the stress tensor.\n\nIn~\\cite{newBLMSY} we explore whether such bulk graviton cone\nbehavior can lead to boundary causality violation by studying the\nbehavior of graviton null geodesics in the effective geometry.\nThere, we indeed see causality violation for ${\\lambda}_{GB}>\\frac{9}{100}$.\n\n\n\\subsection{New metastable states at high momenta (${\\lambda}_{GB} > {9 \\over 100}$) }\n\nWe now study the behavior of the full graviton wave equation.   Let us recast the equation (\\ref{eom}) in Schr\\\"{o}dinger form. For this purpose, we introduce\n \\begin{equation}\n{dy \\over dz} = {1 \\over N_{\\sharp} \\tilde{f}(z)} , \\qquad \\psi = B \\phi , \\qquad B\n= \\sqrt{K \\over \\tilde{f}} \\ .\n \\end{equation}\nThen (\\ref{eom}) becomes\n \\begin{equation} \\label{enr}\n - \\partial_y^2 \\psi + V(y) \\psi = \\tilde{\\omega}^2 \\psi\n \\end{equation}\nwith\n \\begin{equation} \\label{potential}\n V (y) = \\tilde{q}^2 c_g^2 (z) + V_1,  \\qquad  V_1 (y) = {\\partial_y^2 B\n\\over B} = {N_{\\sharp}^2 \\tilde{f}^2 \\over B} \\left(B'' + {\\tilde{f}' \\over \\tilde{f}} B' \\right) \\ ,\n \\end{equation}\n where $c_g^2 (z)$ was defined in~(\\ref{Nse}).\nThe advantage of using (\\ref{enr}) is that qualitative features of\nthe full graviton propagation (including the radial direction) can be\ninferred from the potential $V(y)$, since we have intuition for\nsolutions of the Schr\\\"{o}dinger equation. Since $y$ is\na monotonic function of $z$, below we will use the two coordinates\ninterchangeably in describing the qualitative behavior of $V(y)$.\n\n\nOne can check that $V_1 (z)$ is a monotonically increasing\nfunction for any ${\\lambda}_{GB} > 0$ (note $V_1 (z) \\to + \\infty$ as $z \\to\n\\infty$). For ${\\lambda}_{GB} \\leq {9 \\over 100}$, $c_g^2 (z)$ is also a\nmonotonically increasing function as we discussed in the last\nsubsection and the whole $V(z)$ is monotonic. When ${\\lambda}_{GB} > {9 \\over\n100}$, there exists a range of $z$ where  $c_g^2 (z)$ decreases with\nincreasing $z$ for sufficiently large $z$. Thus $V(z)$ can now\nhave a local minimum for sufficiently large $\\tilde{q}$. For\nillustration, see Fig.~\\ref{pote}\n for the plot of $V (z)$ as a function $z$ for two values of ${\\lambda}_{GB}$.\n\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.65,angle=0]{2potential.eps}\n\\caption{$V(z)-q^2$ (vertical axis) as a function of $z$ (horizontal\naxis) for ${\\lambda}_{GB}=0.08$ and $\\tilde{q} =500$ (left panel) and for ${\\lambda}_{GB}=0.1$\nand $\\tilde{q} =500$ (right panel). $V(z)$ is a monotonically increasing function of\n$z$ for ${\\lambda}_{GB} \\leq {9 \\over 100}$, but develops a local minimum for\n${\\lambda}_{GB} > {9 \\over 100}$ with large enough $\\tilde{q}$.}\n \\label{pote}\n\\end{figure}\n\n\nGenerically, a graviton wave packet will fall into the black brane\nvery quickly, within the time scale of the inverse temperature ${1\n\\over T}$ (since this is the only scale in the boundary theory).\nHere, however, precisely when the local speed of graviton $c_g$ can exceed $1$ (i.e. for ${\\lambda}_{GB} > {9 \\over 100}$), $V(z)$\ndevelops a local minimum for large enough $\\tilde{q}$ and the\nSchr\\\"{o}dinger equation~(\\ref{enr}) can have metastable states\nliving around the minimum. Their lifetime is determined by the\ntunneling rate through the barrier which separates the minimum\nfrom the horizon. For very large $\\tilde{q}$ this barrier becomes very\nhigh and an associated metastable state has lifetime\nparametrically larger than the timescale set by the temperature.\nIn the boundary theory, these metastable states translate into\npoles of the retarded Green function for $T_{xy}$ in the lower\nhalf-plane. The imaginary part of such a pole is given by the\ntunneling rate of the corresponding metastable state. Thus for\n${\\lambda}_{GB} > {9 \\over 100}$, in boundary theory we find new\nquasiparticles at high momenta with a small imaginary\npart.\\footnote{A similar type of long-lived quasiparticles exist\nfor ${\\cal N}=4$ SYM theory on $S^3$~\\cite{guido}, but not on ${\\bf\nR^3}$.}\n\nIn~\\cite{newBLMSY}, we confirm that those long-lived quasiparticles give rise to causality violation for ${\\lambda}_{GB}>\\frac{9}{100}$.\n\n\\section{Discussion}\n\\label{discussion}\n\nIn this paper we have computed $\\eta\/s $ for Gauss-Bonnet gravity using a variety of techniques.  We have found that the viscosity bound\nis violated for ${\\lambda}_{GB} >0$ and have looked for pathologies correlated to this violation.  For small positive ${\\lambda}_{GB}$ we have not found any.\nThe violation of the bound becomes extreme as ${\\lambda}_{GB} \\rightarrow {1 \\over 4}$ where $\\eta$ vanishes. We\nhave focused our attention on this region to find what unusual properties of the boundary theory could yield a violation not only of the bound but also of the qualitative intuitions suggesting a lower bound on $\\eta\/s$. Above we also have discussed a novel quasiparticle excitation. In~\\cite{newBLMSY}, causality violation is firmly established for ${\\lambda}_{GB}>\\frac{9}{100}$.\n\nIt is also instructive to examine the\nbehavior of the zero temperature theory as ${\\lambda}_{GB} \\rightarrow {1 \\over 4}$. Basic parameters describing the boundary CFT are the coefficients of\nthe 4D Euler and Weyl densities called $a$ and $c$ respectively. These have been computed first in  \\cite{Henningson:1998gx}, and for Gauss-Bonnet gravity in \\cite{Nojiri:1999mh}. Their results indicate that\n\\begin{eqnarray}\n\\label{ac} c &\\sim& (1 -4{\\lambda}_{GB})^{{1\\over 2}}, \\cr a &\\sim& (3\n(1-4{\\lambda}_{GB})^{{1\\over 2}}-2).\n\\end{eqnarray}\nThe parameter $c$ is related to the two-point function of a\nboundary stress tensor which is forced by unitarity to be\npositive. (\\ref{ac}) shows that $c$ vanishes at ${\\lambda}_{GB} ={1 \\over 4}$\ndemonstrating the sickness of this point.\\footnote{This can also\nbe seen from the derivations in Sec.\\ref{shear}.} For ${\\lambda}_{GB}$ a bit\nless than ${1 \\over 4}$ the stress tensor couples very weakly in a system with a large number of degrees of freedom.  This is peculiar\nindeed. In the bulk it seems that gravity is becoming strongly coupled there.\n\nThe coefficient $a$ vanishes at ${\\lambda}_{GB}={5 \\over 36}$. The significance of this is unclear.\n\nMore generally, we believe it would be valuable to explore how\ngeneric higher derivative corrections modify various gauge theory\nobservables. This is important not only for seeing how reliable\nit is to use the infinite 't Hooft coupling approximation for\nquestions relevant to QCD, but also for achieving a more balanced\nconceptual picture of the strong coupling dynamics.   Furthermore, this may\n generate new effective tools for separating the\nswampland from the landscape.\n\nAs a cautionary note we should mention that pathologies in the boundary theory in regions that violate the viscosity bound\nmay not be visible in gravitational correlators, at least when $g_s = 0$.  As an example consider the ${\\alpha}'^3 R^4$ terms discussed\nin \\cite{BLS}.   For positive ${\\alpha}'$, the physical case, the viscosity bound is preserved.  But the bulk effective action can equally be\nstudied for ${\\alpha}'$ negative.  Here gravitational correlators can be computed and will violate the viscosity bound.  The only indication\nof trouble in the boundary theory at $g_s=0$ will come from correlators of string scale massive states, whose mass and CFT conformal weight\n$\\sim 1\/({\\alpha}')^{{1\\over 2}}$, an imaginary number!\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank A.~Adams, N.~Arkani-Hamed, R-G.~Cai, A.~Dymarsky, Q.~Ejaz, T.~Faulkner, H.~Jockers,\nP.~Kovtun, J.~Liu, D.~Mateos, H.~Meyer, K.~Rajagopal, D.~T.~Son,\nA.~Starinets, L.~Susskind, B.~Zwiebach for discussions. HL also wishes to thank\nJ.~Liu for collaboration at the initial stages of the work.  We would also like to thank Yevgeny Katz and Pavel Petrov for sharing a draft of their work \\cite{new}.\n\n\nMB and HL are partly supported by the U.S. Department of Energy\n(D.O.E) under cooperative research agreement \\#DE-FG02-05ER41360.\nHL is also supported in part by the A. P. Sloan Foundation and the\nU.S. Department of Energy (DOE) OJI program. HL is also supported\nin part by the Project of Knowledge Innovation Program (PKIP) of\nChinese Academy of Sciences. HL would like to thank KITPC\n(Beijing) for hospitality during the last stage of this project.\nResearch at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research \\& Innovation. RCM also acknowledges\nsupport from an NSERC Discovery grant and from the Canadian\nInstitute for Advanced Research. SS is supported by NSF grant\n9870115 and the Stanford Institute for Theoretical Physics. SY is\nsupported by an Albion Walter Hewlett Stanford Graduate Fellowship\nand  the Stanford Institute for Theoretical Physics.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\n\\par Electrical power systems are complex engineering networks which are crucial to the present day infrastructure. A power network comprises of blocks consisting of numerous interconnected sub-systems, making them challenging to analyze and understand. With the continuous increase in electricity demand and the trend for more interconnections, an issue of concern is the mitigation and analysis of low-frequency interarea oscillations. Oscillations associated with individual generators in a power plant are called local mode oscillations typically ranging from 0.7-2.0Hz \\cite{rogers2012power, klein1991fundamental}. The stability of these oscillations characterized as intraarea (same area) and interarea (across areas) \\cite{rogers2012power, klein1991fundamental}. These oscillations between the generators which are inherent to power systems require appropriate mathematical models and techniques for their analysis.\n\\par Kuramoto-type models have been widely used to study the dynamics of a power system network through swing equations \\cite{filatrella2008analysis}. It must be noted though that a power system network has an added second order term due to generator inertia and are only similar to Kuramoto model. Power dissipation terms that arise in the swing equation model are absent in conventional Kuramoto model, which can be shown existent by few mathematical adjustments. In power systems, globally coupled phase oscillators of Kuramoto form have been viewed as electromechanical generators mutually coupled to deliver load power. A conventional second order Kuramoto-type oscillator can be written as follows,\n\\begin{equation}\nJ_i\\ddot{\\delta}_i+d_i\\dot{\\delta}_i=\\omega_i+\\sum_{j\\neq i,j=1}^{n}k_{ij}sin(\\delta_j-\\delta_i), \\ \\ \\ i \\in \\{1,\\ldots,n\\},\n\\label{kuramoto}\n\\end{equation}\nwhere $\\delta_i$ is angular position of rotor with respect to the synchronously rotating reference frame (for consistency, all the angles throughout the paper are in radians), $J_i$ inertia in kgm$^2$, $d_i$ damping and $\\omega_i$ is a natural frequency chosen from an appropriate distribution $g(\\omega)$ of $i$-th oscillator. $[k_{ij}]; i,j\\in \\{1,\\ldots,n\\}$ is the matrix of coupling constants and $n$ defines the number of oscillators. The standard Kuramoto-type equation assumes value of coupling constants $[k_{ij}]$ to be always positive and symmetric. In this paper, author explores the mapping between a power grid and Kuramoto oscillators.\n\\par Interarea oscillations emerge when two areas having independent sets of power generators experience supply-demand imbalance. The generators in individual areas are observed to beat against each other with frequencies ranging from 0.1-0.8Hz, classified as low frequency interarea oscillations in a power grid. These oscillations can be visualized as two large generators trying to desynchronize each other in the event of supply-demand balance being achieved in each individual area. The above phenomena is analyzed using small-signal or modal analysis \\cite{klein1991fundamental}, however additionally it would be  advantageous to have a nonlinear (large-signal) model to capture the different behaviors and effects of these oscillations. We propose a novel `conformist-contrarian' (inspired from first order framework discussed in \\cite{hong2011kuramoto}) second order Kuramoto-type model (henceforth, referred to as CC-Kuramoto) which captures the in-phase (intraarea) and the anti-phase (interarea) oscillations in a power system.\n\\par The motivation behind developing such a model is to address some of the challenges related to modeling low frequency oscillations in power systems \\cite{kundur2002small}. Conventionally, small-signal analysis and damping control is used by power system engineers to assure system stability at planning stage and thereby execution. It has to be noted though, that over the years software packages on these design\/analysis have become computationally efficient in terms of execution time, but still bears significant computation cost for near real-time implementation. Some key challenges related to models for power systems, identified from the literature are as follows:\n\\begin{enumerate}\n    \\item The major problems related to power system oscillations are of perturbed damping of overall system which are regulated conventionally using power system stabilizers. These oscillations are identified using eigen value analysis which are computationally costly.\n    \\item In cases when power transfer needs to be increased or decreased, the groups of individual generators in the source and sink sides are dispatched in order of their sensitivities to the critical modes with respect to the output of these generators. These in turn assist to increased power levels without adding any further damping control actions. Computation of these critical modes and calculation of generator sensitivities to it in real-time is difficult.\n    \\item Modeling such behaviors is not an easy task due to observed in-coherency from planning to actual implementations in the past. Details of major equipment and inclusion of newer loads like those of induction motors is still not simple in small-signal models.\n    \\item On the similar lines, a $-$ sync behavior in system characteristics has tendency to mitigate homogeneous power oscillations in an interarea setup \\cite{shim2017synchronization}. The behavior and analysis of interarea oscillations in a nonlinear form by adding periodic disturbances to the major parameters that have significant impact \\cite{mao2008nonlinear}. It would be advantageous, if a model can integrate these results through simple modifications.\n\\end{enumerate}\n\\par Apart from fabricating a perfect model that can overcome above mentioned complexities, a power system engineer looks for a model that can provide significant inferences. With increasing vulnerability of modern power systems due to inclusion of various ancillary services it is important to study the settings that might lead to partial stability or instability. It must be noted though, that power grids are not simple physical network of transmission lines and are deeply impacted by its structural as well as dynamical interactions. Thus, a dynamic redesign\/modification of existent power network is not possible, as it can be a major limiting factor in optimizing synchronization. With these constraints as reference, we show occurrence of various stable, partially stable and unstable states via tuning of system parameters, and avoid fiddling with the structure. It is observed that these parameters beyond a certain threshold lead to randomization of steady-state equilibrium points thereby existence of a chaotic behaviour. The same power grid setup (and some other complex systems in nature) shows a state of partial stability by clustering themselves into islands of synchronised and de-synchronised oscillators, commonly referred to as chimera in literature \\cite{abrams2004chimera}. Hence, we emulate the existence of these chimera state behaviors and correlate them with blackouts with islanding commonly seen in power grids \\cite{nerc9209}.\n\\par Nonlinear modes associated with instabilities have been analysed and discussed in the literature \\cite{susuki2011nonlinear, susuki2009global, parrilo1999model} related to power grid synchronization. These provide an informative decomposition of nonlinear oscillations when the network looses synchrony. Sufficient conditions for synchronization, obtained via perturbation analysis for non-uniform Kuramoto oscillators are also widely studied \\cite{dorfler2012synchronization, dorfler2011critical}. It must be noted though, that the conditions attained in previous studies maintain homogeneity in system parameters, whereas we obtain conditions on various power grid parameters and hence heterogeneity. On the similar lines, effects of heterogeneity on power grid networks discussed in \\cite{motter2013spontaneous} inspire formulation of Kuramoto-type framework and extend it to practical blackout scenarios. Thus, in this work we take a standard example from power systems to show that chimera behaviors can lead to blackouts and can be correlated to a distributed grid. To summarize, major contributions of this work are as follows. We propose a nonlinear model for analysis of power systems in simplistic form to avoid discussed computational complexities. A practical example from power systems is used to correlate and showcase advantages of the proposed model. It has been shown, that the model not only provides information about the nominal states but also existence of chimera behaviors in power systems. This could help site engineers to take actions apriori or equip with necessary tools at the right time.\n\\par The manuscript is divided into two parts. \\emph{Part-\\uppercase\\expandafter{\\romannumeral 1}}: We first start with modeling a standard power grid in Kuramoto form. Then, gradually move towards addressing complexities discussed before and how it can be easily incorporated in a large scale (nonlinear model) using analogy to a standard physics example. Next, a standard power systems network to study interarea oscillations is considered and results are verified using computer simulations. \\emph{Part-\\uppercase\\expandafter{\\romannumeral 2}}: A detailed bifurcation analysis is performed on the proposed model parameters in order to analyse the system stability. Finally, we emulate chimera behavior \\cite{abrams2004chimera} commonly referred to in the literature and discuss its implications in a power network.\n\n\n\n\\section{Part - \\uppercase\\expandafter{\\romannumeral 1}: Power Network and Kuramoto Oscillators}\\label{mod1}\n\n\\subsection{Mathematical model of Power Grid}\\label{kura}\n\n\\par The basic elements of a power grid consists of active generators and passive machines\/loads. The generator converts some source of energy into electrical power which is produced by the prime mover of the generator with the frequency close to the standard or natural frequency $ \\Omega $ of an electrical system. All generators in a power grid can be looked upon as set of synchronous machines rotating at synchronous frequency $\\Omega$, with the stator windings of the generator delivering electrical power to the grid. Any power generator in a power system is described by a power balance equation of the form,\n\\begin{equation}\nP_{accumulated}+P_{dissipated}=P_{source}-P_{transmitted},\n\\label{sumpower}\n\\end{equation}\nwhere $P_{source}$ is the rate at which the energy is fed into the generator at frequency $\\Omega$ (i.e., $2\\pi\\Omega=50$Hz). Hence, the phase angle $\\theta_i$ at the output of the $i$-th generator in stationary frame is then given by,\n\\begin{equation}\n\\theta_i=\\Omega t +\\delta_i.\n\\label{theta_eq}\n\\end{equation}\n$P_{accumulated}$ is the rate at which kinetic energy is accumulated by the generator:\n\\begin{equation}\nP_{accumulated} = \\frac{1}{2} J_i \\frac{d}{dt}(\\dot{\\theta}_i)^2,\n\\label{pacc}\n\\end{equation}\nwhere $ J_i $ is the moment of inertia of the $i$-th generator in kgm$^{2}$. For the sake of simplicity, we assume identical machines (i.e., $J_i=J$). $P_{transmitted}$ is the power transmitted from generator $i$ to $j$ with phase difference, $\\Delta \\theta_{ij}=\\theta_j-\\theta_i \\neq 0$. \n\\begin{equation}\nP_{transmitted}=-P_{max}sin(\\Delta\\theta_{ij}).\n\\label{ptrans}\n\\end{equation}\n$P_{max}$ being maximum electrical power input in watts. The dissipated power ($P_{dissipated}$) with $K_D$ the dissipation constant of the prime mover in Ws$^2$\/rad$^2$, can be expressed as: \n\\begin{equation}\nP_{dissipated}=K_D (\\dot{\\theta}_i)^2.\n\\label{pdiss}\n\\end{equation}\nSince, all the generators share common frequencies $\\Omega$, $\\Delta \\theta_{ij} = \\Delta \\delta_{ij} = \\Phi_{ij}$. Substituting \\eqref{pacc}, \\eqref{ptrans} and \\eqref{pdiss} in \\eqref{sumpower}, following can be computed,\n\\begin{equation}\nP_{source} = J \\ddot{\\theta}_i\\dot{\\theta}_i + K_D(\\dot{\\theta}_i)^2 - P_{max}sin(\\Phi_{ij}).\n\\label{pgen}\n\\end{equation}\nDifferentiating \\eqref{theta_eq} with respect to time and further double differentiating it; thereby assuming perturbations around the synchronous frequency being very small, $i.e., \\dot{\\delta}_i \\ll \\Omega$, \\eqref{pgen} can be approximated as, \n\\begin{equation}\nP_{source} \\cong J\\Omega\\ddot{\\delta}_i+[J\\ddot{\\delta}_i+2K_D\\Omega]\\dot{\\delta}_i+K_D\\Omega^2-P_{max}sin(\\Phi_{ij}).\n\\label{pgen2}\n\\end{equation}\nUnder practically relevant assumptions, the coefficient of first derivative is constant and neglecting acceleration terms, as well as knowing that the rate at which the energy is stored in kinetic term is much lower as compared to rate at which energy is dissipated in friction, \\eqref{pgen2} is reduced to,\n\\begin{equation}\nJ\\Omega\\ddot{\\delta}_i=P_{source}-K_D\\Omega^2-2K_D\\Omega\\dot{\\delta}_i+P_{max}sin(\\Phi_{ij}).\n\\label{pgen3}\n\\end{equation}\nNow, using the fact that, $P_{max}=E_iE_j\\left|Y_{ij}\\right|$; $E_i$ being internal voltage of $i$-th generator, $Y_{ij}$ the Kron reduced admittance matrix denoting maximum power transferred between generators \\cite{dorfler2012synchronization} and choosing $P_{m,i}=P_{source}-K_D\\Omega^2$, where $P_{m,i}$ is the mechanical power input,\n\\begin{equation}\nJ\\Omega\\ddot{\\delta}_i=P_{m,i}-E^{2}_{i} \\Re(Y_{ii})-2K_D\\Omega\\dot{\\delta}_i\n+\\sum_{j\\neq i,j=1}^{n}E_iE_j \\left| Y_{ij}\\right|sin(\\Phi_{ij}).\n\\label{pgen5}\n\\end{equation}\nDividing both sides by $J\\Omega$,\n\\begin{equation}\n\\ddot{\\delta}_i=\\left[\\frac{P_{m,i}}{J\\Omega}-\\frac{E^{2}_{i} \\Re(Y_{ii})}{J\\Omega}\\right]-\\frac{2K_D}{J}\\dot{\\delta}_i\n+\\sum_{j\\neq i,j=1}^{n}\\frac{E_iE_j \\left| Y_{ij}\\right|}{J\\Omega}sin(\\Phi_{ij}).\n\\label{pgen6}\n\\end{equation}\n\\eqref{pgen6} can be rewritten as follows,\n\\begin{equation}\n\\ddot{\\delta}_i=\\omega_i-\\alpha \\dot{\\delta}_i+\\sum_{j\\neq i,j=1}^{n}k_{ij}sin(\\delta_j - \\delta_i),\n\\label{pkura}\n\\end{equation}\nwhere $\\alpha=\\frac{2K_D}{J}$ is the dissipation constant, coupling constant $[k_{ij}] = \\frac{E_iE_j \\left| Y_{ij}\\right|}{J\\Omega}$ and natural frequency $\\omega_i=\\left[\\frac{P_{m,i}}{J\\Omega}-\\frac{E^{2}_{i} \\Re(Y_{ii})}{J\\Omega}\\right]$. From a graph theoretic viewpoint $[k_{ij}]$ can be seen as a weighted laplacian matrix with $k_{ij}=0$ when generators are not connected to each other and $ k_{ij} \\geq \\frac{(\\omega_{max}-\\omega_{min})n}{(2(n-1))} $ otherwise (i.e., assumed to be greater than critical coupling, to ensure steady state synchronization \\cite{dorfler2012synchronization}). It can be observed that, \\eqref{pkura} has the same form as a second order Kuramoto oscillator model \\cite{dorfler2012synchronization}.\n\n\\subsection{CC-Kuramoto Model for Interarea Oscillations}\\label{model}\n\\par Further, we extend \\eqref{pkura} to spring coupled oscillators. In Kuramoto oscillators or spring coupled pendulums for that sake; the coupling term introduces restoring forces on the oscillators. Considering the special case, when there is no energy transfer between oscillators, either of in-phase or anti-phase  steady state oscillations may exist (as observed in a spring coupled pendulum - Figure \\ref{fig:pend}). For the in-phase oscillations, the restoring forces are zero implying absence of the coupling term, which is not a tangible explanation for a coupled system in practice because there would always be some sort of restoring forces present in a coupled system. On the other hand, in the case of anti-phase oscillations the spring keeps contributing restoring forces whilst the energy transfer is zero \\cite{feynman2011feynman}. This also explains Huygens observations \\cite{bennett2002huygens} and is a valid template for modeling interarea oscillations in power systems.\n\\begin{figure}[t!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[height=4.0cm,width=3.9cm]{0phase_pend2.png}&\n\\includegraphics[height=4.0cm,width=4.5cm]{180phase_pend23.png}\\\\\n(a) $\\theta_1=\\theta_2$ &(b) $\\theta_1=-\\theta_2$\n\\end{tabular}\n\\caption{Pendulums coupled by a spring, oscillating in two equilibrium modes. (a) Oscillations in in-phase mode. (b) Oscillations in anti-phase mode.}\n\\label{fig:pend}\n\\end{figure}\n\\par Any oscillator of the form given in \\eqref{pkura}, assuming damping\/dissipation constant to be zero, with identical natural frequencies ($\\omega_i=\\omega $) and $H(\\Phi_{ij})=\\sum_{j\\neq i,j=1}^{n}k_{ij}sin(\\Phi_{ij})$ and $\\Phi_{ij}=(\\delta_j-\\delta_i)=(\\theta_j-\\theta_i)$, can be written as,\n\\begin{equation}\n\\ddot{\\delta}_i=\\omega+H(\\Phi_{ij}),\n\\label{axonal}\n\\end{equation}\nand thereby, following can be deduced,\n\\begin{equation}\n\\ddot{\\Phi}_{ij}=H(-\\Phi_{ij})-H(\\Phi_{ij})=-2H(\\Phi_{ij}).\n\\label{axonal2}\n\\end{equation}\nThe above \\eqref{axonal2} has fixed points $\\Phi_{ij}=\\beta(2\\pi)$ or $\\Phi_{ij}=(2\\beta-1)\\pi; \\forall \\beta \\in \\mathbb{Z}$ which are respectively the in-phase and anti-phase modes of the oscillator. Linearizing \\eqref{axonal2} about its fixed points,\n\\begin{equation}\n\\begin{split}\n\\ddot{\\Phi}_{ij}&\\approx \\left[ -2\\frac{\\partial H(\\Phi_{ij})}{\\partial \\Phi_{ij}}\\Bigr|_{\\substack{H(\\Phi_{ij})=0}} \\right] \\Phi_{ij},\\\\\n&\\approx \\left [-2k_{ij}cos(\\Phi_{ij})\\Bigr|_{\\substack{H(\\Phi_{ij})=0}} \\right ]\\Phi_{ij}.\n\\end{split}\n\\label{axonal3}\n\\end{equation}\nThus, from \\eqref{axonal3} it can be deduced that in-phase solution $\\Phi_{ij}=0$ is synchronizing and stable, whereas anti-phase solution $\\Phi_{ij}=\\pi$ is desynchronizing and unstable. These results are similar to small-signal stability analysis performed by linearizing the nonlinear power system dynamics \\cite{klein1991fundamental}. Hence, next we integrate these equilibrium\/critical modes directly in the nonlinear dynamics of Kuramoto model \\eqref{pkura}. \nThe in-phase or `conformist' model of Kuramoto oscillators can be given as follows,\n\\begin{equation}\n    \\ddot{\\delta}_i=\\omega_i-\\alpha_i \\dot{\\delta}_i \\ \\mathcolorbox{yellow}{\\mathlarger{\\mathlarger{+}}}\\sum_{j\\neq i,j=1}^{n}k_{ij}sin(\\delta_j - \\delta_i),\n    \\label{conf}\n\\end{equation}\nwhereas, an anti-phase or `contrarian' Kuramoto model can be obtained by replacing $H(\\Phi_{ij})$ with $-H(\\Phi_{ij})$ in \\eqref{axonal} to give,\n    \\begin{equation}\n    \\ddot{\\delta}_i=\\omega_i-\\alpha_i \\dot{\\delta}_i\\ \\mathcolorbox{green}{\\mathlarger{\\mathlarger{-}}}\\sum_{j\\neq i,j=1}^{n}k_{ij}sin(\\delta_j - \\delta_i).\n    \\label{cont}\n\\end{equation}\n\\par Linearisation of the `contrarian' model \\eqref{cont} on the lines of \\eqref{axonal3} yields $\\ddot{\\Phi}_{ij}\\leq 0$ for anti-phase modes and otherwise for in-phase modes. Thus, in the `contrarian' model the anti-phase mode is stable and in-phase mode is unstable. All oscillations in physical systems in general and power systems in particular are weighted sum of in-phase and anti-phase modes. Hence, to study the oscillations in power systems, we propose the CC-Kuramoto model of coupled oscillators given as,\n        \\begin{equation}\n        \\begin{split}\n        \\ddot{\\delta}^{a_1}_i=\\omega_i-\\alpha_i \\dot{\\delta}^{a_1}_i&\\ \\mathcolorbox{yellow}{\\mathlarger{\\mathlarger{+}}}\\sum_{j\\neq i,j=1}^{p}k_{ij}sin(\\delta_j^{a_1} - \\delta^{a_1}_i)\\\\\n        &\\ \\mathcolorbox{green}{\\mathlarger{\\mathlarger{-}}}\\sum_{j=p+1}^{n}k_{ij}sin(\\delta^{a_2}_j - \\delta^{a_1}_i),\\\\\n        \\ddot{\\delta}^{a_2}_i=\\omega_i-\\alpha_i \\dot{\\delta}^{a_2}_i&\\ \\mathcolorbox{yellow}{\\mathlarger{\\mathlarger{+}}}\\sum_{j\\neq i,j=p+1}^{n}k_{ij}sin(\\delta_j^{a_2} - \\delta^{a_2}_i)\\\\\n        &\\ \\mathcolorbox{green}{\\mathlarger{\\mathlarger{-}}}\\sum_{j=1}^{p}k_{ij}sin(\\delta^{a_1}_j - \\delta^{a_2}_i),\\\\\n        \\end{split}\n        \\label{interarea}\n        \\end{equation}\n        where without loss of generality we assume $\\delta_i^{a_c} \\in \\big[ \\delta_1^{a_1},\\delta_2^{a_1},\\delta_3^{a_2},\\delta_4^{a_2}\\big]$, $ p $ set of generators in area 1 ($a_1$) and $ (n-p) $ generators in area 2 ($a_2$).\n        \n\\par Further, a CC-Kuramoto model settles into one of the three type of states, depending upon the system parameters and initial conditions: Incoherent state - a state of complete desynchronization, $\\pi$-state - when two groups of coherent oscillators are separated by phase difference of $\\pi$ radians and the Travelling wave state - where the two coherent groups are apart by a phase difference less than $ \\pi $ radians. Not only do the above states exhibit rich dynamical behavior but other interesting outcomes arise in the process of transition between these states. The direct relation between CC-Kuramoto and the power system network facilitates the study of complex dynamics arising in power networks.\n\n\\subsection{Case study: Classical two-area four-machine system}\\label{study}\n\n\\par In order to validate the proposed model, we use a classical two-area four-machine power system commonly referred to for interarea oscillation analysis \\cite{klein1991fundamental}. The system is symmetric; consisting of two identical areas connected through a relatively weak tie ($J_i=J=0.4$kgm$^2$, $\\alpha_i=\\alpha=0.125$). Each area includes two synchronous generators with equal power output. The single line diagram of the system considered is as shown in Figure \\ref{fig:sld}, \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[height=3.3cm,width=8cm]{inter_area_sld.png}\n\\caption{Single line diagram of a two-area four-machine power system.}\n\\label{fig:sld}\n\\end{figure}\n\\par All loads are represented as constant impedances. The tie-line impedance was varied by changing the number of tie circuits in service. Power transfer between two areas is emulated, either by an uneven distribution of generation between the areas, or by an uneven split of the total system loads. The combinations for tie-line power flow are given in Table \\ref{data}.\n\\begin{table}[b!]\n\\centering\n\\caption{Load and Tie-Line Power of Test System}\n\\label{data}\n\\begin{tabular}{|l|l|l|l|l}\n\\cline{1-4}\n     & \\multicolumn{2}{c|}{Generation\/Load (MW)} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Power flow from Area 1 \\\\ to Area 2 (MW)\\end{tabular}}} &  \\\\ \\cline{1-3}\n       & \\multicolumn{1}{c|}{Area 1}            & \\multicolumn{1}{c|}{Area 2}           & \\multicolumn{1}{c|}{}                                                                                                 &  \\\\ \\cline{1-4}\nCase 1 & 1400\/1367            & 1400\/1367              & \\multicolumn{1}{c|}{0}                                                                                                                   &  \\\\ \\cline{1-4}\nCase 2 & 1400\/967              & 1450\/1767            & \\multicolumn{1}{c|}{400}                                                                                                                 &  \\\\ \\cline{1-4}\n\\end{tabular}\n\\end{table}\n\n\n\n\\par The number of tie-line in service is two and the transfer level along the tie-line of the two areas varies from 0 to 400 MW due to the variation of load levels in the two areas. Case 1 relates to no power transfer between two areas. On the other hand, the event of power transfer between areas has been designated as Case 2.\n\\begin{align}\nK_{Case 1}=\\begin{bmatrix}0 & 1.9689 & 0.1766 & 0.1782\\\\ \n                             1.9689 & 0 & 0.1782 & 0.1801\\\\\n                             0.1766 & 0.1782 & 0 & 1.9363\\\\\n                             0.1782 & 0.1801 & 1.9363 & 0 \\end{bmatrix}\n\\label{Kmatrix1}\n\\end{align}\n\\begin{align}\nK_{Case 2}=\\begin{bmatrix}0 & 2.5960 & 0.2130 & 0.2151\\\\ \n                             2.5960 & 0 & 0.2151 & 0.2171\\\\\n                             0.2130 & 0.2151 & 0 & 1.7214\\\\\n                             0.2151 & 0.2171 & 1.7214 & 0 \\end{bmatrix}\n\\label{Kmatrix2}\n\\end{align}\nThe coupling matrix $[k_{ij}]$ is represented as (\\ref{Kmatrix1}) for Case 1 and (\\ref{Kmatrix2}) for Case 2. Elements of coupling matrix $[k_{ij}]$ are derived from $k_{ij}=\\frac{E_iE_j\\left| Y_{ij} \\right|}{J\\Omega} $. Natural frequencies are calculated using $\\omega_i=\\left[\\frac{P_{m,i}}{J\\Omega}-\\frac{E^{2}_{i} \\Re(Y_{ii})}{J\\Omega}\\right]$ and are shown in Table \\ref{freq_data}. For simplicity, we assume $\\Omega=1$Hz.\n\\begin{table}[b!]\n\\centering\n\\caption{Natural frequencies (in rad\/s)}\n\\label{freq_data}\n\\begin{tabular}{lllll}\n\\hline\n\\multicolumn{1}{|l|}{}       & \\multicolumn{1}{c|}{$\\omega_1$ } & \\multicolumn{1}{c|}{$\\omega_2$ } & \\multicolumn{1}{c|}{$\\omega_3$ } & \\multicolumn{1}{c|}{$\\omega_4$ } \\\\ \\hline\n\\multicolumn{1}{|l|}{Case 1} & \\multicolumn{1}{c|}{17.5290}                         & \\multicolumn{1}{c|}{17.7923}                         & \\multicolumn{1}{c|}{17.5640}                         & \\multicolumn{1}{c|}{17.8285}                         \\\\ \\hline\n\\multicolumn{1}{|l|}{Case 2} & \\multicolumn{1}{c|}{16.8882}                         & \\multicolumn{1}{c|}{17.1532}                         & \\multicolumn{1}{c|}{17.7931}                         & \\multicolumn{1}{c|}{18.0629}                         \\\\ \\hline                                       \n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[t!]\n\\begin{tabular}{ccc}\n\\includegraphics[height=4.0cm,width=4.8cm]{eigen_plot_case25.png}&\n\\includegraphics[height=4.0cm,width=4.8cm]{circle_plot_case24.png}&\n\\includegraphics[height=4.0cm,width=6cm]{err_plot_case24.png}\\\\\n(a)&(b)&(c)\\\\\n\\includegraphics[height=4.0cm,width=4.8cm]{eigen_plot_case45.png}&\n\\includegraphics[height=4.0cm,width=4.8cm]{circle_plot_case44.png}&\n\\includegraphics[height=4.0cm,width=6cm]{err_plot_case44.png}\\\\\n(a)&(b)&(c)\n\\end{tabular}\n\\caption{Case 1 (first row from the top) - Dynamics of the proposed model for interarea oscillations, when no power is transferred. (a) Compass plot. (b) Circle plot. (c) Time series plot. Case 2 (second row from the top) - Dynamics of the proposed model for interarea oscillations, when power is transferred from area 1 to area 2. (a) Compass plot. (b) Circle plot. (c) Time series plot.}\n\\label{fig:case2}\n\\end{figure*}\n\n\\par The model proposed in \\eqref{interarea} was solved using MATLAB and following observations were made.\n\\par \\textit{Case 1}: The generators oscillate anti-phase ($-3.12 \\approx -\\pi$radians) in interarea and in-phase ($0.06 \\approx 0$radians) intraarea. To validate the results, we provide compass plots of normalized eigen modes, circle plot as well as time-domain plots of generators as shown in Figure \\ref{fig:case2}. The compass plots were obtained by using the steady-state vectors: $\\vec{c}_i(t) = \\big (\\delta_j-\\delta_i \\big )_{rms} \\angle \\delta_i(t) \\sim \\big ( \\delta_j-\\delta_i \\big )_{rms} \\left( \\sum^{n}_{i=1} e^{\\lambda_i t} u_i v^{T}_i \\delta_i(0) \\right)$, where $u_i$ is the normalized left eigen-vector, $v_i$ is the normalized right eigen-vector and $\\lambda_i$ are the eigen-values of the linearized system. It can be seen that the compass plot of steady state vectors show behavior similar to normalized eigen modes obtained by small-signal analysis performed traditionally.\n\\par \\textit{Case 2}: In the case when power is transferred between areas, the phase difference between interarea generators were observed to be $-2.6$radians (i.e., $  \\neq-\\pi$) and $ 0.05 $radians in intraarea, as shown in Figure \\ref{fig:case2}.\nThe results are compared with \\cite{klein1991fundamental}, providing validity to the proposed model.\n\n\\section{Part - \\uppercase\\expandafter{\\romannumeral 2}: Partial Stability in Power Systems}\n\n\\par In this section, we study the bifurcation analysis of the proposed CC-Kuramoto model in order to understand the effect of the design parameters on the stability of a power network. In order to do so, we first analyse some of the characteristics nodes in power systems.\n\n\\subsection{Equal Area Criteria in Power Systems}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[height=7.2cm,width=8.2cm]{PRA_power_systems.png}\n\\caption{Equivalence of equal area criterion to equilibrium spaces. Green circles denote `sink' nodes where system achieves equilibrium. Red squares denote `source' node where system attains acceleration and hence increase in accumulated power.}\n\\label{fig:equal_area_martens}\n\\end{figure}\n\n\\par In power systems, the equal area criterion is a ``graphical technique used to examine the transient stability of the machine systems (one or more than one) with an infinite bus\". The areas under the curve of a power angle diagram are equated across to calculate effective acceleration\/deceleration thereby comment on the stability of the system. For instance, consider \\eqref{interarea}, rewriting in terms of mechanical and electrical power interactions,\n\n\\begin{equation}\n    \\ddot{\\delta}_i=P_{m,i}-\\hat{P}_{transmitted},\n    \\label{equal_area}\n\\end{equation}\n\nwhere $\\hat{P}_{transmitted}=\\alpha_i\\dot{\\delta_i} -P_{max}sin(\\Delta\\delta_{ij})=P_{transmitted}-P_{dissipated}$. It can be seen that the collective acceleration of generators is dependent on the difference of mechanical and electrical power inputs. As shown in Figure \\ref{fig:equal_area_martens}, difference in electrical-mechanical inputs either accelerate or decelerate the generators to achieve equilibrium. The generators accelerate when mechanical power is higher than the transmitted electrical power (i.e., $P_{m,i}>P_{transmitted}$) and decelerate when electrical power is higher (i.e., $P_{m,i}<P_{transmitted}$). This is due to the fact that, the difference in the power gives rise to the rate of change of accumulated power ($P_{accumulated}$) in the rotor masses. The change in accumulated power and generator inertia results in effective change in rotor angles, thereby acceleration\/deceleration and vice-versa. The coupled set of generators happen to achieve steady-state, when mechanical power of the generator from turbines match the transmitted electrical power. This can be visualized as creation of a `sink' node, where the system tries to drive itself in order to achieve stability. Thus, any generator starting from a rotor angle $ \\in [\\delta_0,\\delta_c,\\delta_m]$, will try to move towards $\\delta_0$ (or stability). On the other hand, $\\delta_m$ happens to be critically stable and a small perturbation towards $\\pi$ radians can render increase in rotor angles due to effective acceleration of generators. In such case, a `source' node is formed at $\\pi$ radians, as shown in Figure \\ref{fig:equal_area_martens}. \n\\par Generators are mechanical devices that exhibit high inertia and hence require time to achieve stability, once perturbed from its equilibrium. The natural speed (angular speed) of rotation $\\omega_i$ and generator inertia $J$ along-with accumulated power $P_{accumulated}$ in turn defines the rate at which stability is achieved or if system becomes unstable. As shown in Figure \\ref{fig:equal_area_martens}, although $\\omega_i$ and $J$ are constants if generator achieves stability via route 2 has higher $P_{accumulated}$ and is highly susceptible to instability as compared to route 1. We show this using circle plot and thereby its relevance to `$\\pi$', `transmitted' and `incoherence' state as mentioned in previous sections. Thus, it can be inferred that a `source' node imparts instability whereas `sink' node stabilizes a power setup. These inferences can be easily made using CC-Kuramoto model as these nodes are pretty evident. In the next section, we do a bifurcation analysis on CC-Kuramoto model in order to understand the effect of system parameters on system stability.\n\n\\subsection{Case Study: Bifurcation Analysis}\n\n\\begin{figure}[t!]\n\\centering\n\\begin{tabular}{c}\n\\includegraphics[height=5.0cm,width=8.6cm]{bifurcation_chimera_varitation_in_K_inter_v2.png}\\\\\n(a)\\\\\n\\includegraphics[height=5.0cm,width=8.6cm]{chimera_bifurcation_v2.png}\\\\\n(b)\n\\end{tabular}\n\\caption{Bifurcation Analysis of CC-Kuramoto model. (a) Analysis of system stability with variation in interarea coupling constants. (b) Stability analysis of the model by varying the natural frequencies in certain areas.}\n\\label{fig:bifurcation_analysis}\n\\end{figure}\n\n\\subsubsection{Using Equilibrium Points}\n\n\\par Next, we rewrite \\eqref{interarea} in order to study the bifurcation in power system using CC-Kuramoto model. Let $\\delta_i^{a_c}=x_i^{a_c}, \\dot{\\delta}_i^{a_c}=y_i$, and $r_1$ be the parameter of bifurcation on interarea coupling.\n\n\\begin{equation}\n\\begin{split}\n0=\\omega_i-\\alpha_i y_i \\ &+\\sum_{j\\neq i,j=1}^{p}k_{ij}sin(x_j^{a_1} - x^{a_1}_i)\\\\ &-\\sum_{j=p+1}^{n}r_1 \\ sin(x^{a_2}_j - x^{a_1}_i),\\\\\n0=\\omega_i-\\alpha_i y_i \\ &+\\sum_{j\\neq i,j=p+1}^{n}k_{ij}sin(x_j^{a_2} - x^{a_2}_i)\\\\ &-\\sum_{j=1}^{p}r_1 \\ sin(x^{a_1}_j - x^{a_2}_i).\n\\end{split}\n\\label{bifur_eq}\n\\end{equation}\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{tabular}{c}\n\\includegraphics[height=7.5cm,width=16cm]{chimera_CC_Kuramoto.png}\\\\\n(a)\n\\end{tabular}\n\\begin{tabular}{cc}\n\\includegraphics[height=4.2cm,width=8cm]{chimera_CC_Kuramoto_R.png}&\n\\includegraphics[height=4.2cm,width=8cm]{chimera_CC_Kuramoto_R_polar_plot.jpg}\\\\\n(b)&(c)\n\\end{tabular}\n\\caption{Simulation of `chimera' behavior in a system of coupled generators. (a) Circle plot against Time series plot showing loss of synchronicity at time $\\sim 250$ seconds. (b) Order parameter ($R=\\frac{1}{n}\\sum^{n}_{j=1}e^{i\\delta_j}$) plot against time. (c) Polar plot of order parameter.}\n\\label{fig:chimera}\n\\end{figure*}\n\n\\par With the same setup of two-area four-machine system, parameter $r_1$ (i.e., interarea coupling) is varied in the range $r_1 \\in [-1,1]$ with homogeneous natural frequencies (i.e., $\\omega_i=\\omega$). Under this setting, we solve for equilibrium points of $\\dot{x}_i^{a_c}=\\dot{y}_i=f(x_i^{a_c},y_i)=0$ or $\\ddot{x}_i^{a_c}=0$ and thereby fixed points of the dynamical equations \\eqref{bifur_eq}. The results were obtained using MATLAB and are shown in Figure \\ref{fig:bifurcation_analysis} (a). It can be seen that interarea coupling of $0.5 \\geq r_1 \\geq -0.5$ achieves period doubling, although the solutions are constant. Beyond $-0.5>r_1>0.5$ CC-Kuramoto model achieves chaotic behavior. This can be visualised as complete loss of synchronism and thereby chaos. \n\n\\begin{equation}\n\\begin{split}\n0=\\hat{\\omega}_i-\\alpha_i y_i \\ &+\\sum_{j\\neq i,j=1}^{p}k_{ij}sin(x_j^{a_1} - x^{a_1}_i) \\\\ &-\\sum_{j=p+1}^{n}k_{ij} \\ sin(x^{a_2}_j - x^{a_1}_i),\\\\\n0=\\hat{\\omega}_i-\\alpha_i y_i \\ &+\\sum_{j\\neq i,j=p+1}^{n}k_{ij}sin(x_j^{a_2} - x^{a_2}_i)\n\\\\ &-\\sum_{j=1}^{p}k_{ij} \\ sin(x^{a_1}_j - x^{a_2}_i),\n\\end{split}\n\\label{bifur_eq2}\n\\end{equation}\n\n\\par For the next case, $\\omega^{a_2}_4$ is varied in a range $\\omega^{a_2}_4=r_2 \\in [5,12]$rad\/s keeping coupling parameter constant, showing synchronization at $r_2=7$rad\/s and leaves synchronicity at $r_2=10$rad\/s showing chimera behavior. We solve for \\eqref{bifur_eq2}, where $\\hat{\\omega}_i=[\\omega^{a_1}_1,\\omega^{a_1}_2,\\omega^{a_2}_3,r_2]^T$. This scenario can be interpreted as a gradual overload of one of the generators from two areas leading to de-synchronization in one area, whereas other area remains synchronized (refer Figure \\ref{fig:bifurcation_analysis} (b)). Further, using circle and time-series plots for angular separations (as shown in Figure \\ref{fig:chimera} (a)), the same partial de-synchronization is observed. These can be inferred as islanding of power network through circuit breakers to avoid the impact of excessive overloading of generators in a neighbouring area (and hence blackouts or cascaded failures \\cite{dey2016impact}). To summarize, the heterogeneity was introduced by increasing frequencies $\\omega_i$ incrementally in one area, while keeping parameters of other area constant \\big (i.e., $\\omega^{a1}_1,\\omega^{a1}_2,\\omega^{a2}_3 \\in g_1(\\omega);\\ \\omega^{a2}_4 \\in r_2 g_1(\\omega)=g_2(\\omega), r_2 \\in \\mathbb{R}$; $g_1,g_2$ being frequency distributions\\big). As seen from Figure \\ref{fig:chimera} area experiencing incremental perturbations loose synchronicity whereas other area remains unaffected, emulating blackout conditions with islanding. In order to measure loss of synchronicity, we use order parameter $R=\\frac{1}{N}\\sum^{N}_{j=1}e^{i\\theta_j}$ as shown in Figure \\ref{fig:chimera} (b), (c).\n\n\\subsubsection{Using Eigen value analysis}\n\n\\par In this subsection we provide an eigen value based justification for bifurcation phenomena observed in previous subsection. For instance, consider \\eqref{axonal3} and let $\\lambda_{ap}, \\lambda_{ip}$, $\\lambda_{inc}$ be eigenvalues of anti-phase, in-phase and incoherent dynamics respectively. Where\n\n\\begin{equation}\n    \\begin{aligned}\n       \\lambda = \\begin{cases}\n\\lambda_{ap}<0 & \\text{if} \\ \\Phi_{ij}=m(\\pi)\\\\\n\\lambda_{ip}>0 & \\text{if} \\ \\Phi_{ij}=0, m(2\\pi)\\\\\n\\lambda_{inc}=0 & \\text{if} \\ \\Phi_{ij}=m(\\pi\/2), \n\\end{cases}\n    \\end{aligned}\n    \\label{eigenanalysis}\n\\end{equation}\n\n\\par $m \\in \\mathbb{Z}$. Now, since $\\lambda_{ap}=-\\lambda_{ip}$, these two nodes exchange stability through $\\lambda_{inc}$ and hence following can be concluded. (i) anti-phase and in-phase modes have converse stabilities and are never stable simultaneously, (ii) these critical modes swap stabilities at incoherence state and (iii) if either of anti-phase or in-phase states are stable incoherence state must be unstable and vice-versa. Particularly, in power systems these states rest in anti-phase (unstable), in-phase (stable) and chimera (partially stable) states.\n\n\\section{Conclusions}\n\\par In this study, a mathematical model for interarea oscillations is proposed using Kuramoto-type framework with its analogy in power grids. It is shown how these oscillations can be visualized in a `conformist-contrarian' form to better understand interarea oscillations. Validity of the choices has been justified using analogy of spring coupled pendulums. In order to verify the model, a standard four generator power system was considered from the literature. Simulations were performed in MATLAB and results were verified and validated. The proposed model is used to investigate various phenomena like spatial\/temporal chimera \\cite{abrams2004chimera} and spontaneous failures in power systems \\cite{nerc9209}. \n\n\\section*{Acknowledgements}\nThe authors would like to thank Prof. S. D. Varwandkar for his valued discussions and feedbacks towards the outcome of this work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}%\n\n\\subsection{Cosmological superstrings}\n\nCosmic superstrings \\cite{HenryTye:2006uv} are the strings of string\ntheory stretched to macroscopic length scales by the universe's early\nphase of exponential, inflationary growth\n\\cite{Guth:1980zm,Linde:1981mu,Albrecht:1982wi}.  During subsequent\nepochs when the scale factor grows as a more leisurely power law of\ntime a complicated network of various string elements forms\n\\cite{Albrecht:1984xv,Bennett:1987vf,Allen:1990tv}. Long,\nhorizon-crossing strings stretch, short curved pieces accelerate and\nattempt to straighten, and, occasionally, individual segments\nintercommute (collide, break and reattach) chopping out loops and\nforming new, connected string pathways.  Analytic and numerical calculations\ndemonstrate that these processes rapidly drive the network to a\nself-similar evolution with statistical properties largely determined\nby the string tension\n\\cite{Vanchurin:2005yb,Ringeval:2005kr,Martins:2005es,BlancoPillado:2011dq,Blanco-Pillado:2013qja}. The\nenergy densities in long strings, in loops, and in gravitational\nradiation divided by the critical energy density are all independent of time.\nThe distribution of loops of a given size relative to the\nhorizon scale is also fixed.\n\nAn understanding of this evolution is informed by previous\nstudies of one dimensional defects\nin the context of symmetry breaking in\ngrand unified theories (GUTs; \\cite{Kibble:1976sj};\nfor a general review see \\cite{Vilenkin:2000jqa}).\nOne important difference for superstrings is the expected\nvalue of the string tension.  In GUT theories the string tension\n$G \\mu\/c^2 \\sim \\Lambda_{GUT}^2\/M_p^2 \\sim 10^{-6}$ is fixed by the\nGUT energy scale $\\Lambda_{GUT}$. Observations of the microwave sky have ruled out GUT strings as\nthe source of the cosmological perturbations \\cite{Smoot:1992td,Bennett:1996ce,Spergel:2006hy} and led to upper bounds\non the tension. Currently, broadly model-independent limits from\nlensing  \\cite{Vilenkin:1981zs,Hogan:1984unknown,Vilenkin:1984ea,deLaix:1997dj,Bernardeau:2000xu,Sazhin:2003cp,Sazhin:2006fe,Christiansen:2008vi},\nCMB studies \\cite{Smoot:1992td,Bennett:1996ce,Pogosian:2003mz,Pogosian:2004ny,Tye:2005fn,Wyman:2005tu,Pogosian:2006hg,Seljak:2006bg,Spergel:2006hy,Bevis:2007qz,Fraisse:2006xc,Pogosian:2008am,Ade:2013xla}\nand gravitational wave background and bursts \\cite{Vachaspati:1984gt,Economou:1991bc,Battye:1997ji,Damour:2000wa,Damour:2001bk,Damour:2004kw,Siemens:2006vk,Hogan:2006we,Siemens:2006yp,Abbott:2006vg,Abbott:2009rr,Abbott:2009ws,Aasi:2013vna,TheLIGOScientific:2016dpb}\ngive $G \\mu\/c^2 \\lta 10^{-7}$. More stringent but somewhat more model-dependent limits from pulsar timing\n\\cite{Bouchet:1989ck,Caldwell:1991jj,Kaspi:1994hp,Jenet:2006sv,DePies:2007bm}\nhave regularly appeared. Currently, the\nstrongest inferred limit\nis $G \\mu\/c^2 \\lta 10^{-11}$ \\cite{Blanco-Pillado:2017oxo,Blanco-Pillado:2017rnf}.\n\nLow tension strings are natural in string theory and have little difficulty in this regard. In the most\nwell-studied compactifications the standard model physics is located\nat the bottom of a warped throat where all energy scales are\nexponentially diminished compared to the string scale. Superstrings\nhave tensions that are reduced by exactly this effect and can\ncorrespond to energies as small as TeV (see \\cite{HenryTye:2006uv,Chernoff:2014cba} for reviews).\n\nThe magnitude of $G\\mu\/c^2$\ninfluences many properties of the strings and loops that make up the\nnetwork. A loop with characteristic size $\\ell$ and energy\n$\\propto \\mu \\ell$ will completely dissipate by gravitational wave\nemission in times $t \\sim \\ell\/(\\Gamma G \\mu\/c)$ where $\\Gamma \\sim\n50$ is a loop-dependent pure number\n\\cite{Vachaspati:1984gt,Burden:1985md,Garfinkle:1987yw,Durrer:1989zi,Allen:1994bs,Allen:1994iq,Allen:1994ev,Casper:1995ub}.\nIf $G\\mu\/c^2 \\ll 10^{-6}$ then\nsuperstring loops evaporate by gravitational wave emission much less\nrapidly than GUT string loops. The characteristic size of loops that\nevaporate gravitationally in $t_H$, the age of the universe, is $\\ell_g =\nt_H \\Gamma G \\mu\/c $. These turn out to dominate the distribution\nof loop sizes found in the universe today.\n\nCurrent simulations report that about 10-20\\% of\nthe string network that is chopped out ends up in the form of large\nloops, with sizes within a few orders of magnitude of the horizon\nscale at birth \\cite{BlancoPillado:2011dq,Blanco-Pillado:2013qja}.\nThe rest forms very small loops with size scale\nrelative to the horizon set by a power of $G \\mu\/c^2$\n\\cite{Polchinski:2004hb,Polchinski:2006ee,Polchinski:2007qc,Polchinski:2007rg,Dubath:2007mf,Polchinski:2008hu}. These\nrapidly evaporate.\nToday, the string network's energy density is dominated by the large loops formed\nat an early epoch. If\n$G \\mu\/c^2 < 7 \\times 10^{-9}$\nit is before matter-radiation\nequality. Today's size distribution increases\nas $\\ell \\to \\ell_g$ from above (the universe was denser at earlier\ntimes and formed more smaller loops); the distribution is cutoff by the\nevaporation process at $\\ell < \\ell_g$.\n\nLong gravitational lifetimes have another important effect:\nthe center of mass\nvelocity of the old loops is small and they cluster like cold dark\nmatter \\cite{Chernoff:2009tp}. This opens the way to experimental\ntests of string\ntheory that are based upon direct detection of gravitational wave\nemission and observation of string microlensing of background stellar sources\n\\cite{Chernoff:2017xxx}\n\n\\subsection{Gravitational backreaction in the string network}\n\nThe most numerous loops are close\nto the characteristic size $\\ell_g$, set by\ngravitational backreaction.\nAn understanding of string gravitational backreaction is crucial for\nmaking forecasts of experimental studies and\nplanning future observational campaigns. The emission of gravitational\nradiation and the associated dissipative forces shrink the size of the\nloop (energy loss) and impart a recoil (momentum and angular momentum\nloss). These may change the character of the loop oscillation over\nlong timescales.  The radiative emission processes have been\nwell-studied assuming that the loop is a long-lived periodic\noscillator \\cite{HoganRees:1984unknown,Vachaspati:1984gt,\n  Burden:1985md,Hogan:1987unknown,Garfinkle:1987yw,Durrer:1989zi,Allen:1994bs,Allen:1994iq,Allen:1994ev,Casper:1995ub}.\nThe secular effects of gravitational backreaction on the loop\noscillation are relatively unexplored. Two important aspects are the\npropensity of loops to self-intersection and the evolution of\ndiscontinuous features on the loops.\n\nSelf-intersections are important because they can lead to the rapid\ndemise of the long-lived loops which are of greatest observational\ninterest. The reason is simple: isolated, dissipationless loops are\nexactly periodic.  If a loop can self-intersect it will do so over and\nover again eventually leading to intercommutation and breakage. This\nprocess shatters the loop into many small looplets \\cite{Casper:1995ub}\nmoving apart at\nrelativistic speeds, each of which will evaporate in only a\nfraction of the time required by the original loop.\nSelf-intersections have the potential to radically depress the number\nof old loops of size $\\ell_g$ that would otherwise exist throughout the\nuniverse.  The loop distribution will be cutoff at scale $> \\ell_g$; the number density at that cutoff will be substantially smaller.\nFurthermore, the intercommutation\nprocess evicts the shattered progeny from\nbeing bound to the galaxy. Backreaction\ncan significantly alter experimental\nforecasts.\n\nAnother important aspect of gravitational backreaction is the presence\nof kinks and cusps on loops.  Typically when a new loop is formed from\na smooth segment of string the orbit of the new loop will contain an\ninfinitesimal element of string that moves at the speed of light for\nan infinitesimal time, repeating once per period. This is a cusp, a\nwell-characterized, periodic strong source of gravitational wave\nemission.  Cusp emission is the principle target of gravitational wave\nsearches from string loops because it is strong, beamed and has a\nwell-understood signal form\n\\cite{Vachaspati:1984gt,Economou:1991bc,Battye:1997ji,Damour:2000wa,Damour:2001bk,Damour:2004kw,Siemens:2006vk,Hogan:2006we,Siemens:2006yp,Abbott:2006vg,Abbott:2009rr,Abbott:2009ws,Aasi:2013vna,TheLIGOScientific:2016dpb}.\nRef.~\\cite{Polchinski:2008hu} has argued that a scaling\nnetwork may be inefficient at forming loops with cusps for the\nfollowing reason.  Scaling requires chopping out a significant\nfraction of the long strings' length each time the universe doubles in\nsize. The chopping removes loops and inevitably adds kinks (derivative\ndiscontinuities) to the remaining long string segments. Smooth long\nstrings accumulate kinks and grow dense with small scale structure as\nthe universe ages.  New loops inherit the small scale structure. The\nfirst time that the loop begins to form a\ncusp-like structure the kinky string reconnects, effectively excising\nthe part of the loop responsible for the cusp.  Such a loop is left with\nnothing but kinks. Kinks may also be detected by gravitational wave\nsearches but are not as strong or as unidirectional.  Recent\ncosmological network simulations support this\ntheoretical prediction\n\\cite{BlancoPillado:2011dq,Blanco-Pillado:2013qja}. In particular,\nthey show that loops with kinks are\nformed preferentially and there are few cusps\\footnote{It must be noted\nthat it isn't clear whether the string substructure in even the biggest simulations has entered a\nscaling regime or is still in the process of evolving.}.\n\nThis general evolutionary outline\nprompts a number of questions related to how gravitational\nbackreaction influences the evolution of derivative discontinuities on\nloops and long strings. Qualitatively, we understand that\ngravitational backreaction will smooth kink discontinuities (lessening\nthe size of the jump in the tangent vector from one side to the other) and\ntheoretically allow new cusps to form. There is a competition between\nthe rate at which the discontinuity diminishes and the rate at which the loop\nshrinks.\nOne question is whether the loop fully evaporates before the\ncusp reforms. Another question\nis whether\na reformed cusp has the same scale\nas the loop itself or an intrinsically\nsmaller scale.\nThese can be answered by calculating the dynamical evolution of\na string loop with backreaction for many orbits.\n\nAnother aspect that requires a full treatment of backreaction is how a\nloop with many kinks evolves (since the scaling solution suggests the\nubiquity of kinks). If the total rate of gravitational\nwave emission scales linearly with the number of kinks \\cite{Bohe:2011rk}\nthen the loop's lifetime is shortened.  However, the backreaction of many\nclosely spaced radiating kinks may qualitatively effect the evolution\npredicted on the basis of a single kink. It is therefore of interest to\nunderstand how backreaction operates when there is a high density of\nkinks on long strings and loops.\n\n\\subsection{Theory and simulation}\n\nIn this paper we develop a complete formalism for computing the gravitational backreaction on\ncosmic string loops, and demonstrate the method by computing the gravitational self-force for\nseveral specific cosmic string configurations. Some similar studies were previously done in\nRefs.~\\cite{Quashnock:1990wv,Scherrer:1990pj}, but these were limited in scope and did not include\nmany of the details considered here.\n\nQuashnock and Spergel (QS) \\cite{Quashnock:1990wv} derived linearized equations of motion for a\nstring interacting with its own gravitational field (in this context, linearized means first order\nin $G \\mu\/c^2$ expanded about flat spacetime). They worked with particular coordinates and gauge\nchoices that were chosen to simplify many aspects of the calculation. The weak field approximation\nbreaks down at kinks, cusps and self-intersections, but these freely moving line singularities were\ntreated in a perturbative sense.\n\nQS computed the self-force at a field point as sourced by elements\nof the retarded, distant string image. They concluded that only finite divergence-free backreaction\nforces existed for field points with smooth sources, and that the contribution to the backreaction\nforces tended to zero as the source point approached the field point. This situation stands in\ncontrast to the analogous point particle case studied by Dirac \\cite{Dirac:1938nz}, in which\nself-interaction leads to a renormalized mass. Carter and Battye \\cite{Carter:1998ix} and Buonanno\nand Damour \\cite{Buonanno:1998is} showed that while a general string has a local divergent part to\nits perturbed metric, the Nambu-Goto string is special and the total force density due to all the\nlocal divergent pieces exactly vanishes. The remaining force is given by long-range interactions.\n\nKinks and cusps are examples where smoothness in the vicinity of\nthe field point fails to hold. QS did not explicitly\ndiscuss the limiting behavior near a kink but did argue on general\ngrounds that the backreaction force per source coordinate interval at\na cusp would be infinite, but integrable. They also solved numerically for\nthe evolution of the loop represented both as a continuous function\nand as a set of kinks (straight line segments with small tangent\nvector discontinuities) by integrating the backreaction over a full\nperiod.  The simulations showed that cusps survive backreaction but are\ndeformed and delayed. Longer integrations suggested that the amplitude\nof the cusp and the associated asymmetric rocket effects were\nsuppressed by backreaction. Finally, QS also showed that small (compared to\nthe size of the loop) kinks decay more rapidly than the string as a\nwhole. The magnitude of the discontinuity at a kink (change in tangent\nvectors) lessens but the discontinuity itself is not\nsmoothed out by dissipation.\n\nIt is some measure of the complexity of the problem that most work since the QS investigation has\ndealt with specific issues and not attempted such an ambitious numerical treatment. Anderson\n\\cite{Anderson:2005qu} analytically calculated the gravitational backreaction forces for the\nAllen-Casper-Ottewill (ACO) loop \\cite{Allen:1994bs}, a rotating loop configuration with a pair of\nkinks (one tangent vector is continuous and the other is discontinuous). The coordinates and gauge\nconditions used were equivalent to those of \\cite{Quashnock:1990wv}. Anderson demonstrated\nexplicitly that all the components of 4-vector acceleration diverged near the kink. The calculated\nforces were, however, integrable so that the equations of motion in the weak field limit were\nintegrable too\\footnote{\\cite{Anderson:2005qu} did not evaluate forces at the kink itself where the\nmetric is ill-determined.}.\n\nIn this paper we do not evolve the string configuration (that\nwill be for a followup) but study in detail the method of\ncalculation of the first order self-force.  Certain intermediate quantities in our calculations exhibit divergences.\nThe occurence of these calculational divergences is tied to three interrelated factors:\nthe choice of worldsheet gauge (eg conformal or other), the specification of residual gauge freedom\nin the choice of worldsheet coordinates (eg null or non-null coordinates), and the existence of discontinuous\nsources anywhere on the loop's retarded image (the intersection of the\nworldsheet with the past lightcone of the field point).  However, the\ntotal integrated self-force at any point\non a smooth region of the worldsheet is always finite due to cancelations of divergences, and is independent of these choices.\nThis finiteness is consistent with the lack of renormalization of the string tension discussed in \\cite{Buonanno:Damour:1998}\nand with the the general conclusions of smoothness of \\cite{Quashnock:1990wv}.\n\nWhile the self-force is finite in smooth regions of the worldsheet,\nit diverges in the limit when the field point approaches cusps or kinks on the\nworldsheet.  However, when one solves\nthe linearized equation of motion for the perturbation to the\nworldsheet, the linearized displacement of the worldsheet is finite.\nGoing beyond this treatment will involve critically examining\nthe linearized approximation and the\ndistributional representation of features such as\nkinks and cusps. The question of whether\nphysical divergences occur in a fully self-consistent evolution\nis beyond the scope of this paper.  Nevertheless, the methodology we\ndevelop in this paper should allow adressing certain aspects of\nthe question in the\nfuture.  Our methodology will allow us to refine the gauge during the\ncourse of a self-consistent evolution (continuing to use\nlinearization with distributional models)\nto separate invariant physical divergences from\ncalculational divergences. In the case of the cusp, for example, we\nwould need to step carefully through a single period of oscillation\nto handle the occurrence of the divergence at a single spacetime\npoint.\n\n\nAs a result of the work in this paper\nthere is evidence that any such singularity is weak in the\n``physical'' sense. In particular, period-averaged changes are given\nby simple quadratures over the worldsheet. Orbit-averaging does not require\ninstant-by-instant evolution but presumes the metric and string\nare only mildly perturbed in some average sense.\nWe find that over an oscillation both\nthe kink and cusp lead to finite displacements of the\nworldsheet and finite small changes in energy, momentum and angular\nmomentum.  All period-averaged physical divergences are small and bounded in the\nsense of being proportional to $G \\mu$. This is quite mild compared\nto the character of the singular behavior of point masses in general relativity,\nfor example.\n\nRecently, Wachter and Olum \\cite{Wachter:2016hgi,Wachter:2016rwc}\nhave studied the evolution of\nloops composed of linear pieces (both right and left moving modes are\ngiven by a set of fixed tangent vectors which generate\nkinks). Using the methodology of\n\\cite{Quashnock:1990wv} they found\nthe metric perturbations and the loop's acceleration and analytically\nevaluated the backreaction\nfor a planar rectangular loop \\cite{Garfinkle:1987yw}. They deduced the energy loss, changes to the left and\nright moving modes and the kink smoothing (diminishing the tangent\nvector jumps). Small angle kinks (acute angles) disappeared more\nquickly than large angle kinks (of order $\\pi\/4$).  This observation\nis complementary to that of\n\\cite{Quashnock:1990wv} which\nreported small\nsized kinks (length small compared to the loop size) disappeared more\nquickly than large sized kinks (length a fraction of the full loop\nsize). Refs.~\\cite{Wachter:2016hgi,Wachter:2016rwc} compared the loop evaporation time to the kink\nsmoothing time and found that the loop angle was a key\nparameter. For small angles, kinks disappeared rapidly.  For large\nangles, the loop evaporated first. Finally, the analysis of the\npiecewise loops showed that the straight line segments begin to curve\nafter a short period for all except loops with special symmetry.\n\n\\subsection{Lagrangian Methodology}\n\nCarter pioneered the treatment of perturbations in an arbitrarily curved\nspacetime background with relativistic string, membrane or other brane\nmodels where $p$, the spatial dimension of the brane, is less than $n$,\nthe spatial dimension of spacetime\n(\\cite{Battye:1995hv,Carter:1993wy,Battye:1998zk};\nsee \\cite{Carter:1997pb} for a review). The action in such models is\n\\begin{eqnarray}\n  {\\cal I} & = & \\int {\\cal L} d{\\bar \\Sigma} \\\\\n  d{\\bar \\Sigma} & = & | \\gamma |^{1\/2} d^{p+1} \\zeta\n\\end{eqnarray}\nwhere $d{\\bar \\Sigma}$ is the surface measure element induced on the\ntimelike world sheet by the background metric, $\\gamma$ is the determinant\nof the induced metric and $\\zeta$ stands for the $(p+1)$ internal\ncoordinates. We may assume a constant scalar Lagrangian ${\\cal L} = -m^{p+1}$\nwhere $m$ is a characteristic mass scale and $\\hbar=c=1$. For $p<n$\nthe brane and the Lagrangian are distributional in spacetime. The\nbrane is concentrated on lower dimensional world sheets\nin the higher dimensional spacetime. The case $p=1$ and $n=3$ is\nan effective low-energy description of minimally coupled F- and D-strings\nwith 2 dimensional world sheets.\n\nIn this work we apply the formalism to compute the metric perturbation generated\nby a cosmic string. Two important considerations guide our efforts.  First, the\ndistributional nature of the strings motivates a Lagrangian approach.\nSecond, we work as much as possible in terms of\ntensorial quantities of the background spacetime and avoiding the use\nof specific systems of intrinsic coordinates for the brane\nsubmanifolds. We develop a fully covariant\nformalism and apply it in a variety\nof circumstances.\n\n\n\\subsection{Conventions used in this paper}\n\nThroughout this paper we follow the conventions of Ref.~\\cite{MTW}; we use a ``mostly positive''\nmetric signature, $(-,+,+,+)$ for the spacetime metric and $(-,+)$ for the worldsheet metric, the\nconnection coefficients are defined by\n$\\Gamma^{\\lambda}_{\\mu\\nu}=\\frac{1}{2}g^{\\lambda\\sigma}(g_{\\sigma\\mu,\\nu}\n+g_{\\sigma\\nu,\\mu}-g_{\\mu\\nu,\\sigma}$), the Riemann tensor is\n$R^{\\alpha}{}_{\\!\\lambda\\mu\\nu}=\\Gamma^{\\alpha}_{\\lambda\\nu,\\mu}\n-\\Gamma^{\\alpha}_{\\lambda\\mu,\\nu}+\\Gamma^{\\alpha}_{\\sigma\\mu}\\Gamma^{\\sigma}_{\\lambda\\nu}\n-\\Gamma^{\\alpha}_{\\sigma\\nu}\\Gamma^{\\sigma}_{\\lambda\\mu}$, the Ricci tensor and scalar are\n$R_{\\alpha\\beta}=R^{\\mu}{}_{\\!\\alpha\\mu\\beta}$ and $R=R_{\\alpha}{}^{\\!\\alpha}$, and the Einstein\nequations are $G_{\\alpha\\beta}=R_{\\alpha\\beta}-\\frac{1}{2}g_{\\alpha\\beta}R=8\\pi T_{\\alpha\\beta}$.\nWe use standard geometrized units, with $c=G=1$, Latin indices for worldsheet components and Greek indices for four-dimensional spacetime components.\n\n\n\\section{Covariant equations of motion for a Nambu-Goto cosmic string loop}%\n\\label{sec:eom}\n\nWe begin by considering a Nambu-Goto cosmic string tracing out a two-dimensional worldsheet in\nspacetime. We identify a point on the string by a pair of worldsheet coordinates $\\{\\zeta^1,\n\\zeta^2\\}$ and denote the spacetime coordinate of that point by $z^\\alpha(\\zeta^a)$.\n\n\nGiven the full spacetime metric, $g_{\\alpha \\beta}$, the induced metric on the worldsheet is defined\nby\n\\begin{equation}\n  \\gamma_{ab} = g_{\\alpha \\beta} \\partial_a z^\\alpha \\partial_b z^\\beta.\n\\end{equation}\nThe worldsheet-tangent projection tensor is defined as\n\\begin{equation}\n\\label{eq:projection}\n  P^{\\alpha \\beta} = \\gamma^{ab} \\partial_a z^\\alpha \\partial_b z^\\beta\n\\end{equation}\nwhere $\\gamma^{ab} \\gamma_{bc} = \\delta^a_c$. The corresponding worldsheet-orthogonal projection\ntensor is\n\\begin{equation}\n  \\perp^{\\alpha \\beta} = g^{\\alpha \\beta} - P^{\\alpha \\beta}.\n\\end{equation}\nFor tensor fields with support confined to the world sheet,\nthe tangentially projected covariant derivative is\n\\begin{equation}\n{\\bar \\nabla}_\\alpha = P_{\\alpha}{}^{\\mu} \\nabla_\\mu.\n\\end{equation}\nFinally, defining the extrinsic curvature (or second fundamental tensor) and its trace,\n\\begin{equation}\n\\label{eq:Kdef}\n K_{\\alpha \\beta}{}^{\\gamma} \\equiv P_{\\mu \\beta} \\bar{\\nabla}_\\alpha P^{\\mu \\gamma},\n \\quad\n K^{\\gamma} \\equiv g^{\\alpha \\beta} K_{\\alpha \\beta}{}^{\\gamma},\n\\end{equation}\nBattye and Carter \\cite{Battye:Carter:1995} showed that the equation of motion of the string\nmay be written in the compact form\n\\begin{equation}\n\\label{eq:zeroorder}\n  K^\\rho = 0.\n\\end{equation}\nThis can be expanded explicitly as\n\\begin{equation}\n\\label{eq:coordK}\n  K^\\gamma = \\frac{1}{\\sqrt{-\\gamma}} \\partial_a(\\sqrt{-\\gamma}\\gamma^{ab}\\partial_b z^\\gamma) + P^{\\alpha \\beta} \\Gamma^\\gamma_{\\alpha \\beta},\n\\end{equation}\nwhere $\\gamma = \\det(\\gamma_{ab})$\nand $\\Gamma^\\gamma_{\\alpha\\beta}$ is\nevaluated at the spacetime coordinate\nof the worldsheet point\n$z^\\mu(\\zeta^a)$.\n\n\\section{Gravitational perturbations of cosmic string loops}%\n\\label{ser:pert}\n\nWe now wish to specialize to the case where the string tension is small and the problem may be\ntreated perturbatively in $G\\mu\/c^2$. Then, the string can be considered to be moving in a\nperturbed spacetime with metric\n$$\ng_{\\alpha\\beta} = \\mathring{g}_{\\alpha\\beta} + h_{\\alpha\\beta},\n$$\nwhere the background, unperturbed metric is $\\mathring{g}_{\\alpha\\beta}$ and\nthe perturbation $h_{\\alpha \\beta}$ is sourced by the string's own stress energy. We can\nlikewise parameterise the worldsheet in terms of a background piece plus a perturbation,\n\\begin{equation}\nz^\\alpha = z_{(0)}^\\alpha + z_{(1)}^\\alpha,\n\\end{equation}\nand will work to first order\\footnote{For notational simplicity, from here on we will always make explicit the dependence on the perturbed quantities $h_{\\alpha\\beta}$ and $z_{(1)}^\\alpha$, but will implicitly define everything else in terms of background quantities. So, for example, we will have $\\gamma_{ab} = \\mathring{g}_{\\alpha \\beta} \\partial_a z_{(0)}^\\alpha \\partial_b z_{(0)}^\\beta$ and likewise for all of the other quantities defined in Sec.~\\ref{sec:eom}.} in both the metric perturbation, $h_{\\alpha\\beta}$, and in the worldsheet\nperturbation, $z_{(1)}^\\alpha$.\n\n\\subsection{Zeroth order equation of motion}\nFor the case where backreaction is ignored, we may treat the string as moving in a fixed background\nspacetime with equation of motion\n\\begin{equation}\n  \\frac{1}{\\sqrt{-\\gamma}} \\partial_a(\\sqrt{-\\gamma}\\gamma^{ab}\\partial_b\n  z_{(0)}^\\gamma) + P^{\\alpha \\beta} \\Gamma^\\gamma_{\\alpha \\beta} = 0.\n\\label{eq:general-eom}\n\\end{equation}\nIf the background is Minkowski spacetime, the second term vanishes and the equation of motion is\njust the standard two-dimensional scalar wave equation for each component of the worldsheet\ncoordinate vector.\n\n\nFurther simplification can be obtained by considering the gauge freedom in defining the worldsheet\ncoordinates. A common class of choices invokes the conformal gauge condition, whereby the\nworldsheet metric is required to be conformally flat:\n\\begin{equation}\n\\label{eq:conformal}\n  \\gamma_{ab} = \\phi \\eta_{ab},\n\\end{equation}\nwhere $\\eta_{ab}$ is a two-dimensional Minkowski metric and $\\phi > 0$ is a conformal factor. A\nconsequence of this choice is that the worldsheet derivatives, $\\partial_{\\zeta^1} z^\\mu$ and\n$\\partial_{\\zeta^2} z^\\mu$, must satisfy certain orthogonality conditions (the details of which\ndepend on the particular choice of worldsheet coordinates) and that the equation of motion is given\nby\n\\begin{equation}\n\\label{eq:conformal-eom}\n  \\phi^{-1} \\eta^{ab}\\partial_a \\partial_b z_{(0)}^\\gamma = 0.\n\\end{equation}\nThis is just the $1$+$1$D flat space scalar wave equation for each spacetime component of the\nstring worldsheet vector $z_{(0)}^\\alpha$. The solutions to this equation are periodic in both $\\zeta^1$ and $\\zeta^2$ in\nthe sense that for a loop of length $L$ we have $z^\\alpha(\\zeta^1, \\zeta^2) = z^\\alpha(\\zeta^1+L\/2,\n\\zeta^2+L\/2)$.\n\n\n\nWeak solutions of equations \\eqref{eq:general-eom} and \\eqref{eq:conformal-eom}\nallow derivative discontinuities, so generic solutions are not smooth.\nThe tangent sphere representation provides a description\nof the derivatives of the two components of a solution \\cite{Vilenkin:2000jqa}. Perfectly smooth\nstring loop solutions have two continuous paths on the tangent sphere. However,\nthere may be long-lived kinks (corresponding to gaps in the\ntangent sphere) that propagate around the string along null worldsheet directions, and cusps\n(corresponding to intersections in the tangent sphere) that only exist instantaneously. There may\nalso be self-intersections, where the string crosses over on itself.\n\n\\subsection{First order equation of motion for the string worldsheet}%\nWe now return to the general case (no specialization of gauge or metric) to write down the\nperturbed equation of motion. Demanding that the perturbed trace of the extrinsic curvature vanish as\nin Eq.~\\eqref{eq:zeroorder}, and assuming that the zeroth order equation of motion is satisfied\ngives \\cite{Battye:Carter:1995,Battye:1998zk}\n\\begin{align}\n&&\\perp^\\rho_{\\ \\,\\chi} {\\bar \\nabla}_\\mu {\\bar \\nabla}^\\mu z_{(1)}^\\chi - 2\n{\\bar \\nabla}_\\mu z_{(1)}^\\alpha K^{\\mu\\ \\, \\rho}_{\\ \\,\\alpha} +\n\\perp^{\\beta\\rho} P^{\\mu\\nu} R_{\\mu\\varepsilon\\nu\\beta}\nz_{(1)}^\\varepsilon \\nonumber \\\\\n&&= K^{\\alpha\\beta\\rho} h_{\\alpha\\beta} -  \\perp^\\rho_{\\\n  \\beta} P^{\\lambda\\tau} \\left( \\nabla_\\lambda h^\\beta_{\\ \\,\\tau} - {1\n    \\over 2} \\nabla^\\beta h_{\\lambda\\tau} \\right).\n\\label{eq:final}\n\\end{align}\nThe homogeneous version of this equation is a higher dimensional analog of the geodesic deviation\nequation.\n\nIdentifying the term on the right hand side as a self-force, it is convenient to split this force into separate contributions, one involving the metric perturbation and the other involving its derivative,\n\\begin{subequations}\n  \\label{eq:self-force}\n\\begin{eqnarray}\n  F^\\rho &=&\\, F_1^\\rho + F_2^\\rho, \\\\\n  F_1^\\rho &\\equiv&\\, - \\perp^\\rho{}_\\lambda P^{\\mu\\nu} \\Big(\\nabla_\\mu h_\\nu{}^\\lambda - \\frac12 \\nabla^\\lambda h_{\\mu\\nu}\\Big), \\\\\n  F_2^\\rho &\\equiv&\\, K^{\\mu \\nu \\rho} h_{\\mu \\nu}.\n\\end{eqnarray}\n\\end{subequations}\nUsing the definition \\eqref{eq:Kdef} of $K^{\\mu\\nu\\rho}$ we can write\n$F_2^\\rho$ in terms of $H_{ab} \\equiv h_{\\mu\\nu} \\partial_a\nz^\\mu \\partial_b z^\\nu$ (the projection of $h_{\\mu\\nu}$ along the\nworldsheet),\n\\begin{align}\n        \\label{eq:F2-hproj}\n  F_2^\\rho &= \\left(\\gamma^{ac} \\gamma^{bd} \\partial_c z^{\\sigma} \\partial_d z^\\lambda \\nabla_\\lambda P_\\sigma{}^\\rho \\right) H_{ab}\\nonumber \\\\\n         &= \\perp^\\rho{}_\\sigma \\gamma^{ac} \\gamma^{bd} \\left(\\partial_c \\partial_d z^\\sigma+ \\Gamma^\\sigma_{\\lambda\\tau} \\partial_c z^\\lambda \\partial_d z^\\tau\\right) H_{ab}.\n\\end{align}\nWe can also write the first term as\n\\begin{equation}\nF_1^\\rho = - \\frac{1}{\\sqrt{\\gamma}} \\perp^\\rho{}_\\lambda {\\cal\n  F}^\\lambda_{\\rm conf},\n\\end{equation}\nwhere\\footnote{We use a caligraphic font for ${\\cal F}^\\mu_{\\rm conf}$\n  since it is not a gauge-specialized version of the general self\n  force $F^\\mu$, because the left hand side of Eq.~\\eqref{eq:simple} is not\n  obtained from the left hand side of \\eqref{eq:final} by a gauge\n  specialization.}\n\\begin{align}\n  \\label{eq:trad-self-force}\n  {\\cal F}_{\\rm conf}^\\rho \\equiv&\\, \\sqrt{-\\gamma} P^{\\mu\\nu} \\Big(\\nabla_\\mu h_\\nu{}^\\rho - \\frac12 \\nabla^\\rho h_{\\mu\\nu}\\Big).\n\\end{align}\nis the quantity that appears on\nthe right hand side of the conformal gauge equation of motion\n\\eqref{eq:simple} below.\nBattye and Carter \\cite{Battye:Carter:1995} showed that for a general choice of\ngauge it is crucial to both project orthogonal to the worldsheet\nand to include the additional term involving $K^{\\mu\\nu\\rho}$ in order to get the correct\ngravitational self-force\\footnote{In fact, a sequence of papers provided derivations of the\nfundamental equations of motion with increasing degrees of rigor. Following on from\nRef.~\\cite{Battye:Carter:1995}, Battye and Carter \\cite{Battye:1998zk} performed a more\ncareful analysis using a second order Lagrangian variational treatment to derive the first order\nequations of motion for the displacement vector of the world sheet and for the metric\nperturbations. When restricted to the linearized backreaction regime, their final results (given in\nEqs.~(30), (31) and (33) of \\cite{Battye:1998zk} with terms involving $K_\\rho$ identically zero for linearized backreaction) are consistent with their earlier results and with the\nexpressions above.}. In a subsequent work \\cite{Carter:Battye:1998} they showed that, despite the\npresence of divergences in the metric perturbation, the\ngravitational self-force \\eqref{eq:self-force} is finite for strings in four spacetime\ndimensions with smooth worldsheets.\n\nThe very general form for the equations of motion given by Eq.~\\eqref{eq:final} allows arbitrary\nchoice of gauge for the background, both for the spacetime coordinates and for the worldsheet\ncoordinates. It also allows separate arbitrary gauge transformations for the perturbations, and is\ninvariant under two different types of linearized gauge transformations:\n\\begin{itemize}\n\\item Linearized coordinate transformations in spacetime, which induce changes in the worldsheet and\n      metric perturbations, $z_{(1)}^\\alpha \\to z_{(1)}^\\alpha + \\xi^\\alpha$,\n      $h_{\\alpha\\beta} \\to h_{\\alpha\\beta} - 2 \\nabla_{(\\alpha} \\xi_{\\beta)}.$\n\\item Linearized coordinate transformations on the worldsheet, which induce the changes\n      \\begin{equation}\n      z_{(1)}^\\alpha \\to z_{(1)}^\\alpha + \\partial_a z^\\alpha \\xi^a.\n      \\label{eq:gaugec}\n      \\end{equation}\n      This gauge freedom shows that only the component of $z_{(1)}^\\alpha$ that is perpendicular to\n      the worldsheet contains physical information.\n\\end{itemize}\n\n\\subsection{Choices of Gauge}%\n\n\\subsubsection{Gauge choice to zeroth order}%\n\nWe now once again specialize to Minkowski spacetime in Lorentzian coordinates at zeroth order. Then,\nthe third term on the left hand side of Eq.~\\eqref{eq:final} vanishes identically. The first term\nsimplifies to\n\\begin{equation}\n\\perp^\\rho{}_{\\chi}  {1 \\over \\sqrt{-\\gamma}} \\partial_a \\left( \\sqrt{-\\gamma} \\gamma^{ab} \\partial_b z_{(1)}^\\chi \\right),\n\\end{equation}\nand the second term is\n\\begin{equation}\n-2 \\partial_a z^{(1)}_{\\alpha} \\gamma^{ab} z^\\sigma_{(0),bd} z^\\alpha_{(0),c}\n\\gamma^{cd} \\perp_\\sigma{}^{\\rho}.\n\\end{equation}\nThe first two terms simplify further if we use the conformal gauge [Eq.~\\eqref{eq:conformal}] to\nzeroth order, in which case the left hand side becomes\n\\begin{equation}\n\\perp^\\rho{}_{\\chi}  {1 \\over \\sqrt{-\\gamma}} \\eta^{ab} \\partial_a \\partial_b z_{(1)}^\\chi.\n\\end{equation}\n\n\\subsubsection{Gauge choice to first order}%\n\nAt first order we adopt Lorenz gauge\\footnote{This gauge condition is often referred to as Lorentz gauge but is actually due to Lorenz \\cite{5672647}.} for the spacetime coordinates. For the worldsheet coordinates\nthere are several natural choices. We focus here on the conformal gauge as it is computationally\nthe most convenient, and direct the reader to Appendix \\ref{app:gauge} for a discussion of other\npossible choices.\n\nThe choice of conformal gauge at first order amounts to choosing the worldsheet coordinates so that\nthe conformal flatness condition \\eqref{eq:conformal} holds to first order as well as zeroth order.\nAnderson \\cite{Anderson:2005qu} showed that in this gauge the equation of motion,\nEq.~\\eqref{eq:final} takes the simple form\n\\begin{equation}\n  \\eta^{ab} \\partial_a \\partial_b z_{(1)}^\\chi =\n  - {\\cal F}^\\chi_{\\rm conf}\n\\label{eq:simple}\n\\end{equation}\nWhen our sign convention for the metric is taken into account, this form is consistent with that\nused by Buonanno and Damour \\cite{Buonanno:Damour:1998}.\n\nComparing with the covariant equation, Eq.~\\eqref{eq:final}, we see a number of differences due to\nthe gauge specialization:\n\\begin{itemize}\n\\item The right hand side of Eq.~\\eqref{eq:simple} corresponds to the second term on the right\n      hand side of Eq.~\\eqref{eq:final}, but with the projection tensor dropped.\n\\item The left hand side of Eq.~\\eqref{eq:simple} corresponds to the first term on the left hand\n      side of Eq.~\\eqref{eq:final}, but again with the projection tensor dropped.\n\\item The remaining two terms in Eq.~\\eqref{eq:final} involving couplings to the extrinsic\n      curvature tensor have been dropped -- they cancel against the effect of dropping the\n      projection tensors in this gauge. (We have already dropped the term involving the Riemann\n      tensor since we are working in flat spacetime.)\n\\end{itemize}\n\nA simple proof of this can be obtained by starting with the general coordinate expression\n\\eqref{eq:coordK} for $K^\\rho$ before considering perturbations, and applying the conformal gauge\ncondition \\eqref{eq:conformal}. We have\n\\begin{equation}\nK^\\rho = {1 \\over \\sqrt{-\\gamma}} \\eta^{ab} \\partial_a  \\partial_b z^\\rho\n+ \\gamma^{ab} z^\\lambda_{,a} z^\\mu_{,b} \\Gamma^\\rho_{\\lambda\\mu}.\n\\label{eq:expr}\n\\end{equation}\nwithout approximation ($z^\\rho$, $\\gamma_{ab}$, $g_{\\alpha\\beta}$, $\\Gamma^\\rho_{\\lambda\\mu}$ exact). Now consider evaluating this expression with the metric $\ng_{\\alpha\\beta} \\to \\eta_{\\alpha\\beta} + h_{\\alpha\\beta}$ and\nworldsheet $z^\\alpha \\to z_{(0)}^\\alpha + z_{(1)}^\\alpha$. The zeroth order term vanishes by assumption. The\nvariation of the first term in Eq.~\\eqref{eq:expr} comes from replacing $z^\\rho$ with\n$z_{(0)}^\\rho + z_{(1)}^\\rho$, since the zeroth order quantity $\\eta^{ab} \\partial_a \\partial_b\nz_{(0)}^\\rho$ vanishes. Therefore this term yields the left hand side of Eq.~\\eqref{eq:simple}.\nSimilarly, the variation of the second term in Eq.~\\eqref{eq:expr} comes from the variation\nin $\\Gamma^\\rho_{\\lambda\\mu}$, since this quantity vanishes in the background by assumption (we are\nworking in Lorentzian coordinates in Minkowski spacetime). Using expression\n\\eqref{eq:projection} for the projection tensor we see that the variation of this term yields the\nright hand side of Eq.~\\eqref{eq:simple}.\n\nFor the specific choice of gauge in this section ${\\cal F}^\\rho_{\\rm conf}$\nnaturally appears in the balance laws for energy and momentum relating\nthe flux of radiation at infinity to the local dissipation forces (see\nAppendix \\ref{sec:energymomentumlossdiscussion}).\n\n\\subsection{First order metric perturbation}\nThe stress tensor for a Nambu-Goto cosmic string is given by \\cite{Vilenkin:2000jqa}\n\\begin{equation}\n  T^{\\alpha \\beta} (x)=\n  - G \\mu \\iint P^{\\alpha \\beta} \\delta_4 (x,z) \\sqrt{-\\gamma} \\, d \\zeta^{1'} d\\zeta^{2'}\n  \\label{exactstressenergy}\n\\end{equation}\nwhere $\\delta_4 (x,z) = \\frac{\\delta_4 (x-z)}{\\sqrt{-g}}$ is the four-dimensional invariant Dirac\ndelta distribution and $z$, $P^{\\alpha \\beta}$ and $\\gamma$ are all functions of $\\zeta^{a'}$. A\ncoupling of the string to gravity leads to deviations of the spacetime from the background. For\nsufficiently small string tensions, $G \\mu\/c^2 \\ll 1$, this deviation may be treated perturbatively\nby expanding the metric about the background spacetime,\n\\begin{equation}\n  g_{\\alpha \\beta} = \\mathring{g}_{\\alpha \\beta} + h_{\\alpha \\beta}.\n\\end{equation}\nThe perturbation satisfies the linearized Einstein equation, which in Lorenz gauge is just\nthe wave equation,\n\\begin{equation}\n  \\Box \\bar{h}_{\\alpha \\beta} + 2 R_\\alpha{}^\\gamma{}_\\beta{}^\\delta h_{\\gamma \\delta} = -16 \\pi T_{\\alpha \\beta}\n\\end{equation}\nwhere $\\bar{h}_{\\alpha \\beta} \\equiv h_{\\alpha \\beta} - \\tfrac12 \\mathring{g}_{\\alpha \\beta} \\mathring{g}^{\\gamma \\delta} h_{\\gamma \\delta}$\nis the trace-reversed metric perturbation.\nWe can invert this equation using the retarded Green function, which satisfies the wave equation,\n\\begin{equation}\n  \\Box G_{\\alpha \\beta}{}^{\\alpha' \\beta'}\n    + 2 R_\\alpha{}^\\gamma{}_\\beta{}^\\delta G_{\\gamma \\delta}{}^{\\alpha' \\beta'} =\n    - g_{\\alpha}{}^{\\alpha'} g_{\\beta}{}^{\\beta'} \\delta^4 (x,x').\n\\end{equation}\nIn a four-dimensional Minkowski background ($\\mathring{g}_{\\alpha \\beta} = \\eta_{\\alpha \\beta}$) the solution is\n\\begin{equation}\nG^{\\rm ret}_{\\alpha \\beta}{}^{\\alpha' \\beta'} (x,x') = \\tfrac{1}{4\\pi} \\Theta_{-}(x,x') \\delta_{(\\alpha}^{\\alpha'} \\delta_{\\beta)}^{\\beta'} \\delta[\\sigma(x,x')].\n\\end{equation}\nHere, $\\sigma(x,x')$ is the Synge world-function, defined to be one-half of the square of the\ngeodesic distance between $x$ and $x'$, so that the Dirac delta function is non-zero only when\n$x$ and $x'$ are null-separated. In Minkowski spacetime, we have the closed form\n\\begin{equation}\n  \\label{eq:Synge}\n  \\sigma (x, x') = \\frac12 \\eta_{\\alpha \\beta} (x^\\alpha - x^{\\alpha'}) (x^\\beta - x^{\\beta'}).\n\\end{equation}\nThe metric perturbation is then given by convolving the Green function with the source,\n\\begin{align}\n\\label{eq:hbar-convolution}\n  \\bar{h}_{\\alpha \\beta}(x) =&\\, 16 \\pi \\int G^{\\rm ret}_{\\alpha \\beta}{}^{\\alpha' \\beta'} (x,x') T_{\\alpha' \\beta'} (x') \\sqrt{-g(x')} d^4 x'\n\\nonumber \\\\\n   =& \\, - 4 \\,G \\mu \\iint P_{\\alpha \\beta} \\delta [\\sigma(x, z)] \\sqrt{-\\gamma} d \\zeta^{1'} d\\zeta^{2'}.\n\\end{align}\nwhere $P_{\\alpha \\beta}$, $z^\\alpha$ and $\\gamma$ are all functions of $\\zeta^{1'}$ and\n$\\zeta^{2'}$.\n\nIn practical calculations it is convenient to perform one of the integrals immediately using the\nidentity\n\\begin{equation}\n  \\delta\\Big[\\sigma\\big(x,z(\\zeta^1, \\zeta^2)\\big)\\Big]\n    = \\frac{\\delta\\big[\\zeta^1 - \\zeta^1_{\\rm ret}(x, \\zeta^2)\\big]}{|r_1|}\n\\end{equation}\nwhere $r_1 \\equiv \\partial_{\\zeta^{1'}} \\sigma = (\\partial_{\\zeta^1}\nz^{\\alpha'}) (\\partial_{\\alpha'} \\sigma)$ and $\\zeta^1_{\\rm ret}(x,\n\\zeta^2)$ parameterizes the retarded image, defined by\n\\begin{equation}\n\\sigma[ x, z(\\zeta^1_{\\rm ret}, \\zeta^2) ] =0.\n\\end{equation}\nThis gives\n\\begin{equation}\n\\label{eq:hbar-zeta2-convolution}\n  \\bar{h}_{\\alpha \\beta}(x) = \\, - 4 G \\mu \\oint \\Bigg[\\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta}}{|r_1|}\\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'},\n\\end{equation}\nwhere the quantity in square brackets is evaluated at\n$\\zeta^{1'} = \\zeta^{1'}_{\\rm ret}(\\zeta^{2'})$.\nThe one-dimensional integration traces exactly one period of the loop's\nretarded image and there is no boundary; it is a closed loop.\nEquivalently, the non-trace-reversed metric\nperturbation is given by\n\\begin{equation}\n\\label{eq:h-zeta2-convolution}\n  h_{\\alpha \\beta}(x) = \\, - 4 G \\mu \\oint \\Bigg[\\frac{\\sqrt{-\\gamma}}{|r_1|} \\Sigma_{\\alpha \\beta} P \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'},\n\\end{equation}\nwhere $\\Sigma_{\\alpha \\beta} \\equiv P_{\\alpha \\beta}-\\tfrac12 \\eta_{\\alpha \\beta} P$ with $P \\equiv\nP^{\\gamma}{}_{\\gamma}$. Note that the integral does not converge when $x$ is\na point on the worldsheet; this is because the integrand diverges whenever $r_1 = 0$,\nwhich occurs when source and field points coincide, i.e. $x = z$.\n\nDerivatives of the first order metric perturbation may be computed in a similar manner to\n$h_{\\alpha \\beta}$ itself, with the caveat that care must be taken in non-smooth regions of the\nstring. These non-smooth regions occur at kinks and cusps, and also in the vicinity of the field\npoint, $x$, if it is on the worldsheet.\n\nIgnoring the issue of smoothness for now, and differentiating Eq.~\\eqref{eq:hbar-convolution} with\nrespect to the field point, $x$, we get\n\\begin{align}\n\\label{eq:dh-convolution}\n  \\partial_\\gamma & \\bar{h}_{\\alpha \\beta}(x)\n    = - 4 \\,G \\mu \\iint P_{\\alpha \\beta} \\, \\partial_\\gamma \\big( \\delta [\\sigma(x, z)] \\big) \\sqrt{-\\gamma} \\, d \\zeta^{1'} d\\zeta^{2'}\n\\nonumber \\\\ &\n    = - 4 \\,G \\mu \\iint P_{\\alpha \\beta} \\,\\partial_\\gamma \\sigma \\, \\delta'[\\sigma(x, z)] \\sqrt{-\\gamma}\\, d \\zeta^{1'} d\\zeta^{2'}\n\\nonumber \\\\ &\n    = - 4 \\,G \\mu \\iint P_{\\alpha \\beta} \\frac{\\partial_\\gamma \\sigma}{\\partial_{\\zeta^{1'}} \\sigma} \\partial_{\\zeta^{1'}} \\big( \\delta [\\sigma(x, z)] \\big) \\sqrt{-\\gamma}\\, d \\zeta^{1'} d\\zeta^{2'}\n\\nonumber \\\\ &\n    = - 4 \\,G \\mu \\iint \\frac{P_{\\alpha \\beta} \\Omega_\\gamma}{r_1} \\partial_{\\zeta^{1'}} \\big( \\delta [\\sigma(x, z)] \\big) \\sqrt{-\\gamma}\\, d \\zeta^{1'} d\\zeta^{2'},\n\\end{align}\nwhere $\\Omega^\\alpha \\equiv x^\\alpha - x^{\\alpha'}$ is the coordinate distance\nbetween $x$ and $x'$. On a smooth worldsheet, this may be integrated by parts to give\n\\begin{align}\n\\label{eq:dhbar-zeta2-convolution}\n  \\partial_\\gamma & \\bar{h}_{\\alpha \\beta}(x)\n\\nonumber \\\\\n    & = 4 \\,G \\mu \\iint \\partial_{\\zeta^{1'}} \\bigg[ \\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta} \\Omega_\\gamma}{r_1} \\bigg] \\delta [\\sigma(x, z)] d \\zeta^{1'} d\\zeta^{2'}\n\\nonumber \\\\\n    & = 4 \\,G \\mu \\oint \\Bigg[\\frac{1}{|r_1|} \\partial_{\\zeta^{1'}} \\bigg( \\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta} \\Omega_\\gamma}{r_1} \\bigg) \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'}.\n\\end{align}\nNote that there are no boundary terms introduced in the integration by parts as the integration is\nover a closed loop. Additionally, note that we can also arrive at the same equation by\ndifferentiating Eq.~\\eqref{eq:hbar-zeta2-convolution} and accounting for the fact that the\ndependence on $x$ appears both through $r_1$ and through $\\zeta^{1'}_{\\rm ret}$, along with the\nrelation $\\partial_\\gamma \\zeta^{1'}_{\\rm ret} = -\\Omega_\\gamma \/ r_1$ (see Sec.~10 of\n\\cite{Poisson-review}). Again, we may write this in the non-trace-reversed form,\n\\begin{align}\n\\label{eq:dh-zeta2-convolution}\n  \\partial_\\gamma & h_{\\alpha \\beta}(x) =\n\\nonumber \\\\\n    & 4 \\,G \\mu \\oint \\Bigg[\\frac{1}{|r_1|} \\partial_{\\zeta^{1'}} \\bigg( \\frac{\\sqrt{-\\gamma} \\Sigma_{\\alpha \\beta} \\Omega_\\gamma}{r_1} \\bigg) \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'}.\n\\end{align}\n\n\\subsection{First order self-force}\nWith the results from the previous section at hand, it is straightforward to obtain an integral\nexpression for the first-order gravitational self-force. Substituting\nEqs.~\\eqref{eq:h-zeta2-convolution} and \\eqref{eq:dh-zeta2-convolution} into \\eqref{eq:self-force}\nwe obtain\n\\begin{align}\n\\label{eq:F1-convolution}\n  F_1^{\\mu}(z) &= - 4 \\,G \\mu \\perp^{\\mu \\gamma} P^{\\alpha \\beta} \\times \\nonumber \\\\\n\t\t&  \\hspace*{-0.5cm} \\oint \\Bigg[\n     \\frac{1}{|r_1|} \\partial_{\\zeta^{1'}} \\Bigg( \\frac{\\sqrt{-\\gamma}\\big(\\Sigma_{\\beta \\gamma} \\Omega_\\alpha\n     - \\tfrac12 \\Sigma_{\\alpha \\beta} \\Omega_\\gamma\\big)}{r_1} \\Bigg) \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'},\n\\\\\n\\label{eq:F2-convolution}\n  F_2^{\\mu}(z) &= \\, - 4 G \\mu K^{\\beta \\alpha \\mu} \\oint \\Bigg[\\frac{\\sqrt{-\\gamma}\\Sigma_{\\alpha \\beta}}{|r_1|}  \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'}.\n\\end{align}\nHere, it is understood that the $\\perp^{\\mu \\gamma}$, $P^{\\alpha \\beta}$ and $K^{\\beta \\alpha \\mu}$\nappearing outside the integral are to be evaluated at $z$, whereas the $P^{\\alpha \\beta}$ and\n$\\gamma$ appearing inside the integral are to be evaluated at the retarded point $z'$.\n\nOne may expect a difficulty to arise from the fact that $\\bar{h}_{\\alpha\\beta}$ diverges\nlogarithmically (and $\\partial_\\gamma \\bar{h}_{\\alpha\\beta}$ is even more divergent) when the\nsource and field points coincide. This would appear to be a major obstacle for computing the\nself-force since the integral expressions for $\\bar{h}_{\\alpha\\beta}$ and $\\partial_\\gamma\n\\bar{h}_{\\alpha\\beta}$ will not converge when the field point, $x$, is on the worldsheet.\nFortunately, it turns out that for field points on smooth parts of the worldsheet, some miraculous\ncancellations in the particular combination appearing in the equation of motion [and hence the\nself-force, Eq.~\\eqref{eq:self-force}] lead to many of the divergent terms canceling. The result\nis that one obtains a convergent integral and a finite self-force. This was shown to hold in\n\\cite{Buonanno:Damour:1998} for the conformal gauge and in \\cite{Carter:Battye:1998} for an\narbitrary gauge. However, both cases implicitly assumed a smooth string worldsheet. It turns out\nthat the conclusions continue to hold for a non-smooth worldsheet provided the field point is on\na smooth part of the worldsheet. As a field point approaches a\nnon-smooth point on the worldsheet the total self-force diverges\n\nDespite this latter divergence, there is one further important consideration, namely the physical\nsignificance of the self-force itself. It is possible that a divergence in the self-force is a\nspurious artifact arising from, for example, an unfortunate choice of gauge or from a distributional\ntreatment of non-smooth worldsheet features. Indeed,\nAnderson \\cite{Anderson:2005qu} computed explicit closed form expressions for the self-force in the\ncase of the ACO string. His expressions diverge logarithmically and as negative\npowers in the vicinity of the kink. However, this divergence is integrable and he was able to solve\nthe equations of motion to compute finite deviations in both the position and velocity of the\nstring\\footnote{More precisely, the derivative along the direction orthogonal to the kink's\npropagation direction was divergent at the kink, however Anderson was able to obtain a gauge\ntransformation which eliminated this divergence and so it can be attributed to nonphysical\ncoordinate effects.}. Similar conclusions have also been drawn in other work \\cite{Quashnock:1990wv,Wachter:2016rwc,Wachter:2016hgi}.\n\nIn this work, we empirically find results that are consistent with these previous conclusions;\nalthough the equation of motion has a divergent self-force term it turns out to give a finite\nchange to the worldsheet. Any physical measurement must be consistent\nwith the inferred finite displacement. With a distributional description of\nkinks and cusps as adopted here finite displacements can lead to singular\nchanges in derivative quantities such as tangent\nvectors on the worldsheet.\\footnote{Divergent behavior of this sort\n  (changes of order $G \\mu\/c^2$ in the tangent vector direction over a single\n  period of oscillation) has recently been reported by\n  Blanco-Pillado, Olum and Wachter [see acknowledgements]. \n  In our treatment here we emphasize that\n  we ignore the possibility of additional contributions coming from the kink\nitself. It is difficult to validate this assumption within a distributional approach. It is\nlikely that a matched asymptotic approach along the lines of Ref.~\\cite{pound:2010} for point\nparticles would be required to provide a definitive answer to the question\nof whether the distributional treatment omits any important physical\neffects. We anticipate that such a treatment would also regularize singular\ntangent vector derivatives so that all physical measurements\nare finite, not merely consistent with the inferred finite worldsheet\ndisplacement.}\nOptimistically, we can expect the finiteness of worldsheet displacements\nto carry through to more general scenarios, and hope that the divergences in the force are\nalways integrable. A proof of this fact can likely be obtained from a local expansion of the type\ngiven in Sec.~\\ref{sec:local-expansion} below, adapted to allow for a kink or cusp within the\n``local'' region. Since there are considerable subtle details in this calculation, we will leave\nits exploration for future work.\n\n\n\\section{Evaluating the Gradient of the Retarded Metric Perturbation}\n\\label{sec:dh}\n\nIn the previous section, we obtained integral expressions for the metric perturbation, its\nderivative, and the gravitational self-force. The latter two are valid provided the retarded image\nof the worldsheet is smooth. In reality, we do not have the luxury of a smooth worldsheet for at\nleast two reasons:\n\\begin{enumerate}\n  \\item We are interested in studying strings with kinks and cusps, and the worldsheet is non-smooth at the\n        location of any kink or cusp;\n  \\item We are interested in computing the self-force, which requires us to evaluate the metric\n        perturbation and its derivative in the limit $x \\to z$. In that case, if one considers the\n        retarded image of a point directly on the string, $x = z$, one finds that it is not in\n        general smooth at the field point, $\\zeta^{2'} = \\zeta^{2'}(x)$.\n\\end{enumerate}\nThese can lead to important distributional-type contributions\nto the integrand in the expression for the self force\nwhich are easily missed.\nIn the following subsections, we extend Eqs.~\\eqref{eq:dh-zeta2-convolution} and\n\\eqref{eq:F1-convolution} above to allow for these non-smooth\nfeatures. We begin with a general covariant derivation of the integral\nto explain how coordinate-dependent divergences arise,\n and follow up with an explicit\ntreatment of both issues mentioned above.\n\n\\subsection{Covariant evaluation of the worldsheet integral}\n\\label{sec:coord-depend-integral}\n\nThe expression for the gradient of the metric perturbation at a point $x^\\alpha$ is of the form (dropping spacetime tensor indices)\n\\begin{equation}\nI = \\int_{\\cal W} \\omega_{ab} \\delta'(\\sigma).\n\\label{dds}\n\\end{equation}\nHere ${\\cal W}$ is the worldsheet defined by $x^\\alpha = z^\\alpha(\\zeta^a)$, $\\omega_{ab}$ is some\ngiven smooth two-form on the worldsheet, and the function $\\sigma$ is as defined in\nEq.~\\eqref{eq:Synge}. In this subsection we will derive some identities for integrals of the form\n\\eqref{dds} for arbitrary $\\omega_{ab}$ and arbitrary smooth $\\sigma$, and in the next subsection\nwe will specialize to the specific form \\eqref{eq:Synge} of $\\sigma$ for our application here.\n\nAs a warm up, let us first consider a simpler version of the integral \\eqref{dds}, namely\n\\begin{equation}\nJ = \\int_{\\cal W} \\omega_{ab} \\delta(\\sigma).\n\\label{Jdef}\n\\end{equation}\nLet ${\\cal C}$ be the curve given by $\\sigma = 0$.  We would like to derive an expression for $J$ of the form\n\\begin{equation}\nJ = \\int_{\\cal C} \\theta_a\n\\label{ans22}\n\\end{equation}\nwhere $\\theta_a$ is a one-form on the worldsheet. The result for $\\theta_a$ is\n\\begin{equation}\n\\theta_a = \\frac{D_a h}{ \\omega^{bc} D_b \\sigma D_c h}\n\\label{oneforma}\n\\end{equation}\nHere $D_a$ can be taken to be either a covariant or a partial derivative on the worldsheet, and\n$\\omega^{ab}$ is the inverse of $\\omega_{ab}$. Finally $h$ can be taken to be any smooth function\non the worldsheet which has the property that $dh \\wedge d\\sigma \\ne 0$.\n\nNote that the expression \\eqref{oneforma} for the one-form, when pulled back onto the curve ${\\cal\nC}$, is independent of the choice of $h$. To see this, suppose we replace $h$ with a function $H$\nof $h$ and $\\sigma$,\n\\begin{equation}\nh \\to H(h,\\sigma).\n\\label{hchange}\n\\end{equation}\nUnder this transformation\n\\begin{equation}\nD_a h \\to H_{,h} D_a h + H_{,\\sigma} D_a \\sigma.\n\\end{equation}\nWhen this expression is inserted into the one-form \\eqref{oneforma}, the contribution from the\nsecond term to the denominator vanishes because of antisymmetrization, and the contribution to the\nnumerator vanishes when the one form is pulled back to ${\\cal C}$, since $\\sigma=0$ on ${\\cal C}$.\nThe factors of $H_{,h}$ cancel between the numerator and the denominator, and so we see that the\npullback of $\\theta_a$ to ${\\cal C}$ is invariant under the transformation. Thus it is independent\nof the choice of $h$.\n\nWe now turn to the derivation of the formula \\eqref{oneforma}. We specialize to coordinates\n$\\zeta^{\\bar 0} = \\sigma$, $\\zeta^{\\bar 1} = h$. The integral \\eqref{Jdef} becomes\n\\begin{equation}\nJ = \\int d\\sigma \\int dh \\, \\omega_{\\sigma h}(\\sigma,h) \\delta(\\sigma)\n\\end{equation}\nwhere $\\omega_{\\sigma h} = \\omega_{{\\bar 0}{\\bar 1}}$. Evaluating the integral using the delta\nfunction gives\n\\begin{equation}\nJ = \\int dh \\, \\omega_{\\sigma h}(0,h).\n\\label{b9}\n\\end{equation}\nWe now rewrite this in a form which is valid in arbitrary coordinate systems.  The factor $\\int dh$ can be written as the integral over ${\\cal C}$ of the one-form $D_a h$.  The factor $\\omega_{\\sigma h}$ can be written as\n\\begin{equation}\n\\omega_{\\sigma h}  = \\frac{1}{\\omega^{\\sigma h}}.\n\\label{b10}\n\\end{equation}\nUsing the tensor transformation law we have\n\\begin{equation}\n\\omega^{\\sigma h} = \\omega^{{\\bar 0}{\\bar 1}} = \\omega^{ab} \\frac{\\partial \\zeta^{\\bar 0}}{\\partial \\zeta^a}\n\\frac{\\partial \\zeta^{\\bar 1}}{\\partial \\zeta^b}\n= \\omega^{ab} \\frac{\\partial \\sigma}{\\partial \\zeta^a}\n\\frac{\\partial h}{\\partial \\zeta^b}.\n\\label{b11}\n\\end{equation}\nCombining Eqs.~\\eqref{b9}, \\eqref{b10}, and \\eqref{b11} now yields the result given by\nEqs.~\\eqref{ans22} and \\eqref{oneforma}.\n\n\nTurn next to the corresponding analysis for the integral \\eqref{dds}. Suppose that instead of\nintegrating over the entire worldsheet, we integrate over a region $\\Delta {\\cal W}$ of it. The\nintersection of the boundary $\\partial \\Delta {\\cal W}$ of this region with the curve ${\\cal C}$\nwill consist of a set of discrete points ${\\cal P}_i$. The formula for the integral is\n\\begin{equation}\nI = \\int_{\\Delta {\\cal W}} \\omega_{ab} \\delta'(\\sigma) = I_{\\rm boundary} + I_{\\rm bulk}\n\\label{int5}\n\\end{equation}\nwhere the contribution from the boundary is\n\\begin{equation}\nI_{\\rm boundary} = \\sum_i \\pm \\frac{1}{\\varphi} \\frac{k^a D_a h}{ k^b D_b \\sigma}.\n\\label{Ib}\n\\end{equation}\nHere $k^a$ is the tangent to the boundary $\\delta \\Delta {\\cal W}$\nand\n\\begin{equation}\n\\varphi = \\omega^{ab} D_a \\sigma D_b h.\n\\label{varphidef}\n\\end{equation}\nThe contribution from the bulk is\n\\begin{equation}\nI_{\\rm bulk} = \\int_{\\cal C} \\theta_a\n\\end{equation}\nwhere the one-form $\\theta_a$ is\n\\begin{equation}\n\\theta_a = \\frac{1}{\\varphi^3} \\left( \\omega^{bc} D_b \\varphi D_c h \\right) D_a h.\n\\label{xyy}\n\\end{equation}\n\n\nUnder a change of the function $h$ of the form \\eqref{hchange}, the one-form $\\theta_a$ is no\nlonger invariant. Instead, it transforms by an exact form\\footnote{This formula is valid when\npulled back to the curve ${\\cal C}$.}\n\\begin{equation}\n\\theta_a \\to \\theta_a + D_a \\lambda,\n\\label{ccc}\n\\end{equation}\nwhere $\\lambda = H_{,\\sigma} \/ (\\varphi H_{,h})$.  The change in the boundary integral is\n\\begin{equation}\n\\sum_i \\pm \\frac{H_{,\\sigma}}{\\varphi H_{,h}},\n\\end{equation}\nwhich cancels against the change \\eqref{ccc} in the one-form. Thus we make the important\nobservation that the integral \\eqref{int5} is independent of choice of $h$, but the split into\nboundary and integral terms is not.\n\n\nWe now turn to the derivation of the formula \\eqref{int5}. As before we initially specialize to\ncoordinates $(\\zeta^{\\bar 0}, \\zeta^{\\bar 1}) = (\\sigma,h)$. Inserting the identity\n\\begin{equation}\n\\omega_{\\sigma h} \\delta'(\\sigma ) d\\sigma  \\wedge dh = d \\left[ \\omega_{\\sigma h} \\delta(\\sigma ) dh \\right] - \\delta(\\sigma ) d\\omega_{\\sigma h} \\wedge dh\n\\end{equation}\ninto the integral \\eqref{int5}and using Stokes's theorem gives a result of the form of the right\nhand side of \\eqref{int5}, with\n\\begin{equation}\nI_{\\rm boundary} = \\int_{\\partial \\Delta {\\cal W}} \\omega_{\\sigma h} \\delta(\\sigma ) dh\n\\end{equation}\nand\n\\begin{equation}\nI_{\\rm bulk}  = -\\int_{\\Delta {\\cal W}} \\delta(\\sigma ) \\, \\omega_{\\sigma h,\\sigma } \\, d\\sigma dh.\n\\label{bulkf}\n\\end{equation}\nWe evaluate the first term by taking the parameter along the boundary $\\delta \\Delta {\\cal W}$ to\nbe $\\sigma$ and using $dh = d\\sigma (dh\/d\\sigma)$. This gives\n\\begin{equation}\nI_{\\rm boundary} = \\sum_i \\, \\omega_{\\sigma h} \\, \\frac{dh}{d\\sigma},\n\\end{equation}\nUsing Eqs.~\\eqref{b11} and \\eqref{varphidef} this reduces to the formula \\eqref{Ib}.\n\n\nFor the bulk contribution, from the formula (\\ref{bulkf}) and using arguments similar to those given for the integral $J$, we find\n\\begin{equation}\n\\theta_a = - \\partial_\\sigma (1\/\\varphi) D_a h = \\frac{\\varphi_{,\\sigma }}{\\varphi^2} D_a h,\n\\label{xy0}\n\\end{equation}\nwhere we have used $\\varphi = 1\/\\omega_{\\sigma h}$.  We evaluate the $\\sigma$ derivative using\n\\begin{equation}\n\\varphi_{,\\sigma } = \\frac{\\partial  \\varphi}{\\partial \\zeta^{\\bar 0}} = \\frac{\\partial  \\varphi}{\\partial \\zeta^a}  \\, \\frac{\\partial \\zeta^a}{\\partial \\zeta^{\\bar 0}}.\n\\label{xy}\n\\end{equation}\nWe express the Jacobian matrix in terms of its inverse using\n\\begin{equation}\n\\frac{\\partial \\zeta^a}{\\partial \\zeta^{\\bar a}}\n= \\frac{2}\n{\\left[ \\omega^{cd} \\omega_{{\\bar c}{\\bar d}}\n\\frac{\\partial \\zeta^{\\bar c}}{\\partial \\zeta^c}\n\\frac{\\partial \\zeta^{\\bar d}}{\\partial \\zeta^d}\\right]}\\, \\omega^{ab} \\omega_{{\\bar a}{\\bar b}} \\frac{\\partial \\zeta^{\\bar b}}{\\partial \\zeta^b}.\n\\end{equation}\nThis formula is specific to two dimensions, and is valid for any choice of two-form.  Specializing to ${\\bar a} = {\\bar 0}$ gives\n\\begin{equation}\n\\frac{\\partial \\zeta^a}{\\partial \\zeta^{\\bar 0}} = \\frac{ \\omega^{ab} D_b h}{\\omega^{cd} D_c \\sigma \\, D_d h}.\n\\end{equation}\nInserting this into \\eqref{xy} and then into \\eqref{xy0} finally gives the result \\eqref{xyy}.\n\n\nFinally, although the results derived in this subsection are\ncovariant, they do depend on a choice of arbitrary function $h$ on the\nworldsheet.  While the complete final result (\\ref{int5}) does not\ndepend on $h$, the integrand (\\ref{xyy}) of the bulk integral, as well as\nthe splitting into bulk and boundary terms, do depend on $h$.\nElsewhere in this paper, we choose to identify $h$ with one of the\nworldsheet coordinates, which explains the coordinate dependence of the\nintegrand and of the splitting.\n\n\n\\subsection{Worldsheets with kinks}\n\\label{sec:kinks}\nWe may now consider how our 1-D integral expressions \\eqref{eq:h-zeta2-convolution} and \\eqref{eq:dh-zeta2-convolution}\nfor the retarded metric perturbation and its gradient must be modified to allow for the presence of\na kink. A cosmic string with a kink may be treated as piecewise smooth, with discontinuities in\ncertain tangent vectors whenever a kink is crossed. To obtain an expression allowing for these\ndiscontinuities we assume that the retarded image on the worldsheet is non-smooth at $\\zeta^2 = k$, where $k$ may\ndepend on the field point $x$.\\footnote{Although there are worldsheet coordinates where one of the\ncoordinates is a constant along a kink, we will not restrict ourselves to only that case here.}\nThen, one way to achieve the desired result is to break up the integration in\nEq.~\\eqref{eq:hbar-zeta2-convolution} at the discontinuity,\n\\begin{equation}\n\\label{eq:h-zeta2-discontinuous-convolution}\n  \\bar{h}_{\\alpha \\beta}(x) = \\, - 4 G \\mu \\int_{k^+}^{k^- + L} \\Bigg[\\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta}}{|r_1|} \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'},\n\\end{equation}\nwhere $k^+$ ($k^-$) is a point just to the right (left) of the kink. Now, when we differentiate\nthis expression we have to take account of the possible dependence of the end points on $x$. If the\ndiscontinuity in the string is at a fixed value of $\\zeta^2$ (i.e. in the case of a kink\npropagating along the $\\zeta^1$ direction), then $k$ does not depend on $x$ and the boundary terms\nvanish. If, instead, the discontinuity is at a fixed value of $\\zeta^1$ (i.e. in the case of a kink\npropagating along the $\\zeta^2$ direction), then $k$ does depend on $x$. Then, using\n$\\partial_\\gamma \\zeta^{2}(\\zeta^{1}_{\\rm ret}) = -\\Omega_\\gamma \/ r_2$, where $r_2 \\equiv\n\\partial_{\\zeta^{2'}} \\sigma = (\\partial_{\\zeta^2} z^{\\alpha'}) (\\partial_{\\alpha'} \\sigma)$, we get\n\\begin{align}\n\\label{eq:dh-zeta2-discontinuous-convolution}\n  \\partial_\\gamma & \\bar{h}_{\\alpha \\beta}(x)\n\\nonumber \\\\\n    & = 4 \\,G \\mu \\Bigg\\{\\int_{k^+}^{k^- + L} \\Bigg[\\frac{1}{|r_1|} \\partial_{\\zeta^{1'}} \\bigg( \\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta} \\Omega_\\gamma}{r_1} \\bigg) \\Bigg]_{\\zeta^{1'}_{\\rm ret}} d\\zeta^{2'}\n\\nonumber \\\\\n    & \\qquad + \\bigg[\\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta} \\Omega_\\gamma}{|r_1| r_2}\\bigg]_{k^-}\n      - \\bigg[\\frac{\\sqrt{-\\gamma} P_{\\alpha \\beta} \\Omega_\\gamma}{|r_1| r_2}\\bigg]_{k^+}\n    \\Bigg\\}.\n\\end{align}\nIt is easy to check that one can arrive at the same expression\nby appropriately including the boundary\nterms from the integration by parts described in Sec.~\\ref{sec:coord-depend-integral} above.\nThe presence (or lack thereof) of boundary terms is then manifestly dependent on the particular\nchoice of worldsheet coordinates. Importantly, this apparent worldsheet coordinate dependence only\nappears in the split between boundary and bulk terms; the sum of the two does\n\\emph{not} have any worldsheet coordinate dependence.\n\nIn the case of a smooth string, the two boundary terms are identical and cancel, so we recover the\nsame formula as we had before. In the presence of a kink, however, the boundary terms in the two\nlimits $k^+$ and $k^-$ yield different values and so we pick up an overall\ncontribution from the kink in addition to the integral over the smooth portion of the string.\n\n\\subsection{Worldsheets with cusps}\n\nWe have already seen that care must be taken in computing the self-force for cosmic strings with\nkinks. Since cusps also introduce non-smoothness in the worldsheet, one may expect similar care to\nbe required for cuspy strings. However, there is one crucial difference between a string with\nkinks and one with cusps: cusps typically occur at a single \\emph{point} on a worldsheet while kinks\noccur along a one-dimensional curve. The result is that, in the case of kinks, all points on the\nstring ``see'' a kink at some point in their retarded image, and hence the integrand in Eq.~\\eqref{eq:dh-zeta2-convolution}\nwill always be supplemented by a boundary term somewhere. Conversely, there is\nonly a one-dimensional set of points on the string which ``see'' a cusp in their retarded image;\neverywhere else the integrand does not encounter a discontinuity.\n\nThis suggests that strings with cusps may not need the same careful treatment as those with kinks.\nThis appears to be the case in our test case in Sec.~\\ref{sec:results} below, where we probe the\nregion around the one-dimensional cusp-seeing curve and find no evidence of unusual behavior. This\nis, of course, merely empirical evidence, and should be followed with a more formal treatment; it\nis likely that the local expansions developed in Sec.~\\ref{sec:local-expansion} will prove useful\nin such an analysis.\n\n\n\\subsection{Contribution from the field point}\n\\label{sec:local-expansion}\n\nThe final place where we must take care is in the case where the field point itself is on the\nstring. Then, just as in the case of a kink, the retarded image may have a discontinuity at the\nfield point. While it may be possible in such cases to use a similar treatment to what we have done\nfor kinks, there is subtlety in taking the limit of the field point to the worldsheet which makes\nsuch a treatment difficult. Instead we choose a more robust approach, by using a local expansion of\nthe integrand for field points nearby\\footnote{Here, we use the term ``nearby'' loosely as such a\nnotion is obviously dependent on the choice of worldsheets, and in particular on the choice of\ncoordinate which is used as the variable of integration. Not surprisingly, we will find that the\nconclusions we draw will depend on the choice of worldsheet coordinates. Nevertheless, just as in\nSec.~\\ref{sec:coord-depend-integral}, this apparent coordinate dependence is merely an artifact of\nhow we choose to split up the self-force into contributions from various integrals and boundary\nterms. In reality, the total self-force obtained by combining all of these contributions is\nindependent of the choice of worldsheet coordinates.} the string and then analytically taking the\nlimit of the field point to the worldsheet.\n\nThe purpose of the following subsections is to develop the pieces required for such an expansion. In doing so we make some assumptions:\n\\begin{enumerate}\n  \\item We will study the contribution to the self-force integral nearby where the force is to be\n        computed and will ultimately shrink the size of this region down to zero;\n  \\item We will assume that the worldsheet is smooth in this region. This is true everywhere except\n        when the field point exactly lies on a kink or cusp; points arbitrarily close to a kink or\n        cusp will, however, be perfectly acceptable.\n  \\item We will assume that the induced metric does not diverge (or vanish) on the string. This will be true\n        everywhere except where a field point lies exactly on a cusp.\n  \\item We will assume conformal gauge for the background worldsheet, in particular Eq.~\\eqref{eq:conformal} and the orthogonality relations for $\\partial_{\\zeta^1} z^\\mu$ and\n$\\partial_{\\zeta^2} z^\\mu$ which follow from it. This step is not a strict requirement of the approach, but does significantly simplify the tensor algebra in the calculation.\n\\end{enumerate}\n\nBefore we proceed with the derivation of the local expansion, we point out one interesting feature,\nnamely that the divergence in the self-force that arises on kinks and cusps comes purely from the\nshort-distance portion of the self-field, i.e. the contribution to the integral from nearby\npoints. It is therefore likely that a more careful treatment of what happens to the self-force\nexactly on a kink or cusp may be obtained from a local expansion of the kind given here. We leave\nthe exploration of this issue for future work.\n\n\n\n\\subsubsection{Setup of the local expansion}\nWe wish to compute the contribution to the self-force for points near the field point. To do so,\nwe will construct a local expansion of the self-force integrand about a point on the worldsheet\nwhich is assumed to be nearby the field point, $x^\\alpha$, and to lie on its retarded image,\n$z^\\alpha[\\zeta^1_{\\rm ret}(x, \\zeta^{2}), \\zeta^{2}]$. We denote this expansion point by\n$\\bar{z}^\\alpha \\equiv z^{\\alpha}[\\bar{\\zeta}^{1}, \\bar{\\zeta}^{2}]$ with $\\bar{\\zeta^1} \\equiv\n\\zeta^1_{\\rm ret}(x, \\bar{\\zeta}^{2})$ for a particular choice of $\\bar{\\zeta}^2$.\nThe conformal factor at this point is $\\bar{\\phi} \\equiv \\phi(\\bar{\\zeta}^1,\\bar{\\zeta}^2)$ and we assume\nthe expansion has a radius of convergence that includes\npart of the image. We can then simplify the evaluation of the\nlocal integration over that part of the image utilizing the approximate\nexpansion.\n\nWe will now seek an expansion of the self-force integrand \\eqref{eq:F1-convolution} (note that there is no contribution to $F_2^\\mu$ from the field point since it does not involve derivatives of $h_{\\alpha \\beta}$) in $\\Delta \\zeta^2 \\equiv \\zeta^{2} -\n\\bar{\\zeta}^{2}$.\\footnote{Notationally, the integration in \\eqref{eq:dh-zeta2-convolution} is over the dummy variable $\\zeta^{2'}$ but we suppress these primes for clarity.} The first stage in our calculation is to find an expansion of the retarded\ncoordinate $\\zeta^1_{\\rm ret}(x, \\zeta^{2})$ about $\\bar{\\zeta}^1 =\n{\\zeta}^1_{\\rm ret}(x, \\bar{\\zeta}^{2})$.\nWe denote the difference between these two quantities $\\Delta \\zeta^1$ and will seek an expansion\nof it in powers of $\\Delta \\zeta^2$. In doing so, we will need to be careful about what our\nparticular choice of worldsheet coordinate is. We will also need to separately consider the cases\nwhere $\\Delta \\zeta^2$ is positive or negative, as in some instances the expansion has a different\nform in the two cases.\n\n\n\\subsubsection{Expansion of the light-cone condition: space-time coordinates}\n\nIn this section we focus on a pair of spacelike and timelike type coordinates which we will denote by $\\zeta$\n(for space) and $\\tau$ (for time), i.e. $(\\zeta^1,\\zeta^2) = (\\tau,\\zeta)$. The important defining feature of these coordinates are the conformal gauge orthogonality relations\n\\begin{align}\n  g_{\\alpha \\beta} \\partial_\\tau z^\\alpha \\partial_\\tau z^\\beta &= - \\phi, \\\\\n  g_{\\alpha \\beta} \\partial_\\tau z^\\alpha \\partial_\\zeta z^\\beta &= 0, \\\\\n  g_{\\alpha \\beta} \\partial_\\zeta z^\\alpha \\partial_\\zeta z^\\beta &= \\phi.\n\\end{align}\nWe can also obtain similar relations involving higher derivatives (with respect to $\\tau$ and\/or\n$\\zeta$) of $z^\\alpha$ by differentiating these fundamental relations. We additionally have the conformal gauge equation of motion, Eq.~\\eqref{eq:conformal-eom}, which in $\\tau-\\zeta$ coordinates gives us a relation between second $\\tau$ and second $\\zeta$ derivatives of $z^\\alpha$:\n\\begin{equation}\n  \\partial_{\\tau\\tau} z^\\alpha = \\partial_{\\zeta\\zeta} z^\\alpha.\n\\end{equation}\nWe will use these identities throughout the following calculation to simplify the results we obtain.\n\nWe will start from the fact that the (retarded) source point $z^{\\alpha'}$ and the field point\n$z^{\\alpha}$ are null-separated, $\\sigma(z^\\alpha, z^{\\alpha'}) = 0$. Expanding this about $\\bar{\\sigma}\n\\equiv \\sigma(z^\\alpha, \\bar{z}^{\\alpha})$ we obtain a power series in $\\Delta \\tau$ and $\\Delta\n\\zeta$,\n\\begin{align}\n  \\label{eq:sigma-expansion-tau-zeta}\n  \\sigma &= \\bar{\\sigma} + \\bar{\\sigma}_{,\\tau} \\Delta \\tau + \\bar{\\sigma}_{,\\zeta} \\Delta \\zeta \\nonumber \\\\\n  &\\quad+ \\tfrac12 (\\bar{\\sigma}_{,\\tau\\tau} \\Delta \\tau^2 + 2\\bar{\\sigma}_{,\\tau\\zeta} \\Delta \\tau \\Delta \\zeta + \\bar{\\sigma}_{,\\zeta\\zeta} \\Delta \\zeta^2) \\nonumber \\\\\n  & \\quad+\\tfrac16 (\\bar{\\sigma}_{,\\tau\\tau\\tau} \\Delta \\tau^3 + 3 \\bar{\\sigma}_{,\\tau\\tau\\zeta} \\Delta \\tau^2 \\Delta \\zeta + 3 \\bar{\\sigma}_{,\\tau\\zeta\\zeta} \\Delta \\tau \\Delta \\zeta^2 \\nonumber \\\\\n  & \\qquad + \\bar{\\sigma}_{,\\zeta\\zeta\\zeta} \\Delta \\zeta^3) + \\cdots\n\\end{align}\nUsing $\\partial_a = (\\partial_a z^\\alpha) \\nabla_\\alpha$ (acting upon the second\nargument of $\\bar{\\sigma}$) along with the identities above and the\nfact that $\\nabla_\\alpha \\nabla_\\beta \\sigma = g_{\\alpha \\beta}$ for Minkowski spacetime, it is\nstraightforward to rewrite the coefficients in terms of worldsheet derivatives of $\\bar{z}^\\alpha$\nand $\\bar{\\phi}$:\n\\begin{align}\n  \\bar{\\sigma}_{,\\tau\\tau} &= \\bar{z}^\\alpha_{,\\zeta\\zeta} \\bar{\\sigma}_\\alpha - \\phi, \\\\\n  \\bar{\\sigma}_{,\\tau\\zeta} &= \\bar{z}^\\alpha_{,\\tau\\zeta} \\bar{\\sigma}_\\alpha, \\\\\n  \\bar{\\sigma}_{,\\zeta\\zeta} &= \\bar{z}^\\alpha_{,\\zeta\\zeta} \\bar{\\sigma}_\\alpha + \\phi, \\\\\n  \\bar{\\sigma}_{,\\tau\\tau\\tau} &= \\bar{z}^\\alpha_{,\\tau\\zeta\\zeta} \\bar{\\sigma}_\\alpha - \\tfrac32 \\phi_{,\\tau}, \\\\\n  \\bar{\\sigma}_{,\\tau\\tau\\zeta} &= \\bar{z}^\\alpha_{,\\zeta\\zeta\\zeta} \\bar{\\sigma}_\\alpha - \\tfrac12 \\phi_{,\\zeta}, \\\\\n  \\bar{\\sigma}_{,\\tau\\zeta\\zeta} &= \\bar{z}^\\alpha_{,\\tau\\zeta\\zeta} \\bar{\\sigma}_\\alpha + \\tfrac12 \\phi_{,\\tau}, \\\\\n  \\bar{\\sigma}_{,\\zeta\\zeta\\zeta} &= \\bar{z}^\\alpha_{,\\zeta\\zeta\\zeta} \\bar{\\sigma}_\\alpha + \\tfrac32 \\phi_{,\\zeta},\n\\end{align}\nand likewise for higher order terms (for the current calculation of the contribution to the self-force from the field point it is only necessary to\ngo to the cubic order given here).\n\n\\subsubsection{Expansion of the retarded time}\nIn order to obtain the desired expansion of $\\Delta \\tau(\\Delta \\zeta)$, we now make the ansatz\nthat $\\Delta \\tau$ has an expansion in integer powers of an order counting parameter $\\epsilon \\sim\n\\Delta \\zeta$ and that $\\bar{\\sigma} = \\mathcal{O}(\\epsilon^2)$. Substituting our ansatz into\nEq.~\\eqref{eq:sigma-expansion-tau-zeta} and solving order by order in $\\epsilon$ then yields the\ndesired expansion of $\\Delta \\tau$ in terms of $\\epsilon$,\n\\begin{equation}\n  \\label{eq:tret-spatial-expansion}\n  \\Delta \\tau = \\frac{1}{\\phi}\\bigg[\\bar{\\sigma}_{,\\tau} - \\sqrt{\\bar{\\sigma}_{,\\tau}^2 + \\phi(2\\Delta \\zeta \\bar{\\sigma}_{,\\zeta} + 2 \\bar{\\sigma} + \\phi \\Delta \\zeta^2)}\\bigg]  \\epsilon + \\mathcal{O}(\\epsilon^2).\n\\end{equation}\nThe expressions for the higher order coefficients are somewhat cumbersome, but are fortunately not\nrequired for the current calculation.\n\n\n\\subsubsection{Expansion of quantities appearing in the integrand for the self-force}\n\nWe now expand each of the quantities appearing in the integrand of $F^\\mu_1$ [Eq.~\\eqref{eq:F1-convolution}]:\n\\begin{equation}\n  \\Sigma_{\\mu \\nu} = \\Sigma_{\\mu\\nu}^{(0,0)} + \\Sigma_{\\mu\\nu}^{(1,0)} \\Delta \\tau + \\Sigma_{\\mu\\nu}^{(0,1)} \\Delta \\zeta + \\cdots,\n\\end{equation}\nwhere\n\\begin{align}\n  \\Sigma_{\\mu\\nu}^{(0,0)} &= \\partial_\\zeta z_\\mu \\partial_\\zeta z_\\nu - \\partial_\\tau z_\\mu \\partial_\\tau z_\\nu-\\phi g_{\\mu\\nu}, \\\\\n  \\Sigma_{\\mu\\nu}^{(1,0)} &= \\partial_\\tau \\partial_\\zeta z_\\mu \\partial_\\zeta z_\\nu +  \\partial_\\zeta z_\\mu \\partial_\\tau \\partial_\\zeta z_\\nu - \\partial_\\tau z_\\mu \\partial_\\zeta \\partial_\\zeta z_\\nu \\nonumber \\\\\n    &\\quad- \\partial_\\zeta \\partial_\\zeta z_\\mu \\partial_\\tau z_\\nu - \\partial_\\tau\\phi g_{\\mu\\nu}, \\\\\n  \\Sigma_{\\mu\\nu}^{(0,1)} &= \\partial_\\zeta \\partial_\\zeta z_\\mu \\partial_\\zeta z_\\nu +  \\partial_\\zeta z_\\mu \\partial_\\zeta \\partial_\\zeta z_\\nu - \\partial_\\tau z_\\mu \\partial_\\tau \\partial_\\zeta z_\\nu \\nonumber \\\\\n    &\\quad- \\partial_\\tau \\partial_\\zeta z_\\mu \\partial_\\tau z_\\nu - \\partial_\\zeta \\phi g_{\\mu\\nu}.\n\\end{align}\nWe also have\n\\begin{align}\n  r &= \\bar{\\sigma}_{,\\tau} + \\bar{\\sigma}_{,\\tau\\tau} \\Delta \\tau + \\bar{\\sigma}_{,\\tau\\zeta} \\Delta \\zeta \\nonumber \\\\\n  &\\quad + \\tfrac12 (\\bar{\\sigma}_{,\\tau\\tau\\tau} \\Delta \\tau^2 + 2\\bar{\\sigma}_{,\\tau\\tau\\zeta} \\Delta \\tau \\Delta \\zeta + \\bar{\\sigma}_{,\\tau\\zeta\\zeta} \\Delta \\zeta^2) + \\cdots,\n\\end{align}\nand\n\\begin{align}\n\\Omega_\\mu &= \\bar{\\Omega}_\\mu - z_{\\mu, \\tau} \\Delta \\tau - z_{\\mu,\\zeta} \\Delta \\zeta  - \\tfrac12 z_{\\mu, \\tau\\tau} \\Delta \\tau^2 \\nonumber \\\\\n& \\quad - z_{\\mu,\\tau\\zeta} \\Delta \\tau \\Delta \\zeta - \\tfrac12 z_{\\mu,\\zeta\\zeta} \\Delta \\zeta^2 + \\cdots.\n\\end{align}\nNote that there are three potentially small parameters in these expansions: $\\Delta \\tau$,\n$\\Delta \\zeta$ and the distance of the field point from the string, which we will denote $\\Delta x$.\nIn the above, the dependence on $\\Delta \\tau$ and $\\Delta \\zeta$ appears explicitly; the dependence\non $\\Delta x$ appears through $\\bar{\\Omega}_\\mu \\sim \\Delta x$ and $\\bar{\\sigma}_\\alpha \\sim \\Delta x$.\n\nTo make further progress, we will assume that all three are of the same order, $\\Delta \\tau \\sim\n\\epsilon$, $\\Delta \\zeta \\sim \\epsilon$ and $\\Delta x \\sim \\epsilon$. Now, substituting the\nexpansions into the integral equation for the derivative of the metric perturbation,\nEq.~\\eqref{eq:dh-zeta2-convolution} and expanding out in powers of $\\epsilon$, we find that the\nintegrand has a contribution at order $\\epsilon^{-2}$ and at order $\\epsilon^{-1}$, plus higher\norder terms. More explicitly, the $\\mathcal{O}\\left(\\epsilon^{-2}\\right)$ piece is given by\n\\begin{widetext}\n\\begin{align}\n\\partial_\\gamma h_{\\alpha \\beta} \\approx -4 \\int\n  \\frac{\n  [\\bar{\\sigma}_{,\\tau\\tau}] \\Sigma_{\\alpha\\beta}^{(0,0)} \\bar{\\Omega}_\\gamma\n  - [\\bar{\\sigma}_{,\\tau\\tau}] \\Sigma_{\\alpha\\beta}^{(0,0)} z_{\\gamma,\\zeta}\\Delta \\zeta\n  + \\Sigma_{\\alpha\\beta}^{(0,0)} z_{\\gamma,\\tau} \\bar{\\sigma}_{,\\tau}\n  }{\\left(\\bar{\\sigma}_{,\\tau} + [\\bar{\\sigma}_{,\\tau\\tau}] \\Delta \\tau\\right)^3} d\\zeta + \\mathcal{O}(\\epsilon^{-1})\n\\end{align}\nwhere square brackets denote a coincidence limit, $[\\bar{\\sigma}_{,\\tau\\tau}] \\equiv \\lim_{\\Delta x \\to 0} \\bar{\\sigma}_{,\\tau\\tau}$. Now, it is\nimmediately apparent that if we instead substitute our expansions into the integral expression for\n$F_1^\\mu$ this leading order piece identically vanishes since $P^{\\mu \\nu} \\Sigma_{\\mu \\nu} = 0$\n\\footnote{Strictly speaking, this depends on how we extend the definition of $P^{\\mu \\nu}$ off the\nworldsheet. However, since we are in the end only interested in taking the limit to the worldsheet\nthe particular choice of extension is irrelevant and does not change the result.} Likewise, since\n$\\Sigma_{\\mu\\nu}^{(0,1)} z_{\\mu, \\zeta} = \\Sigma_{\\mu\\nu}^{(1,0)} z_{\\mu, \\tau}$, many other terms\neither identically vanish or simplify significantly. Then, the only remaining piece of the\n$\\mathcal{O}(\\epsilon^{-1})$ contribution to the derivative of the metric perturbation \\emph{which does not\nvanish upon substitution into the self-force} is given by\n\\begin{align}\n & -\\int \\tfrac{4}{\\left(\\bar{\\sigma}_{,\\tau} + [\\bar{\\sigma}_{,\\tau\\tau}] \\Delta \\tau\\right)^3} \\bigg(\n  [\\bar{\\sigma}_{,\\tau\\tau}] \\Sigma_{\\alpha \\beta}^{(0,1)} \\bar{\\Omega}_{\\gamma} \\Delta \\zeta -\n   \\Sigma_{\\alpha \\beta}^{(1,0)} \\bar{\\Omega}_{\\gamma} \\bar{\\sigma}_{,\\tau} +\n   \\Sigma_{\\alpha \\beta}^{(1,0)} z_{\\gamma,\\zeta} \\bar{\\sigma}_{,\\tau} \\Delta \\zeta +\n   \\Sigma_{\\alpha \\beta}^{(0,1)} z_{\\gamma,\\tau} \\bar{\\sigma}_{,\\tau} \\Delta \\zeta \\nonumber \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad -\n   [\\bar{\\sigma}_{,\\tau\\tau}] \\Sigma_{\\alpha \\beta}^{(1,0)} z_{\\gamma,\\tau} \\Delta \\zeta^2 +\n   2 \\Sigma_{\\alpha \\beta}^{(1,0)} z_{\\gamma,\\tau} \\bar{\\sigma}_{,\\tau} \\Delta \\tau +\n   [\\bar{\\sigma}_{,\\tau\\tau}] \\Sigma_{\\alpha \\beta}^{(1,0)} z_{\\gamma,\\tau} \\Delta \\tau^2 \\bigg)\n  d\\zeta .\n\\end{align}\n\\end{widetext}\n\nOur final step is to substitute in the expansion of the retarded time, rescale our integration range\nby $\\epsilon$ and integrate from $\\Delta \\zeta \/ \\epsilon = -\\infty$ to $+\\infty$.\nThe factor of $\\epsilon$ in the integral weight cancels with the $1\/\\epsilon$ in the integrand\nand so the result is ultimately independent of $\\epsilon$.\n\n\n\\subsubsection{Expansion of the self-force}\n\nPerforming the integral explicitly in the limit where the field field point tends to the\nworldsheet, we finally arrive at a surprisingly simple expression for the field point contribution to the\nself-force. In $\\tau-\\zeta$ coordinates, this is given by\n\\begin{equation}\n  \\label{eq:F-local-ST}\n F^\\mu_{\\rm{field,ST}} = 4 \\, \\phi^{-2} \\perp^\\mu{}_\\alpha \\left(z^\\alpha_{,\\zeta} \\phi_{,\\zeta} + z^\\alpha_{,\\tau} \\phi_{,\\tau} - 2 z^\\alpha_{,\\zeta\\zeta} \\phi\\right).\n\\end{equation}\n\nOne can go through a similar procedure in the null case (see Appendix\n\\ref{sec:local-expansion-null} for details of the retarded time expansion in null coordinates).\nThen, if we use $\\zeta^-$ as our integration variable, the\nequivalent expression for the field point contribution to the self-force is\n\\begin{equation}\n  \\label{eq:F-local-N}\n  F^\\alpha_{\\rm{field,N}} = 4 \\, \\phi^{-2} \\perp^\\mu{}_\\alpha \\left(z^\\alpha_{,\\zeta^+} \\phi_{,\\zeta^+} -  z^\\alpha_{,\\zeta^+\\zeta^+} \\phi\\right) .\n\\end{equation}\nLikewise, one can change $+ \\to -$ when $\\zeta^+$ is used as the integration variable.\nThe expressions \\eqref{eq:F-local-ST} or \\eqref{eq:F-local-N} must be\nadded to the previous results given by Eq.~\\eqref{eq:F1-convolution} to obtain the total\ncontribution to $F_1^\\alpha$.\n\n\n\\section{Numerical Methods and Regularization}\nFor this work have developed several different techniques to\nevaluate the self-force on the string by completely finite, numerical\ncalculations. In the next section, we will compare these calculations to validate\nthe exact methods we have discussed. Before doing so, here we will\nschematically outline the different approaches. The\nabbreviation for the methods are given in square brackets.\n\n\\subsection*{2D, smoothed kink or cusp [2D]}\nThe most general approach is to do the 2D integration over the\nworldsheet in Eq.~\\eqref{eq:hbar-convolution}. This circumvents having\nto eliminate one worldsheet coordinate in terms of another\n(e.g. solving for the retarded time in $\\tau-\\zeta$ coordinates) and possibly\nhaving to patch different coordinate systems (e.g. two different null\ncoordinate systems either side of the field point). The worldsheet integration\nproduces manifestly coordinate invariant results.\n\nSchematically, we replace the singular retarded Green function with a finite\napproximation. For a source at $x_s$ and field at $x_f$\n\\begin{eqnarray}\n  {\\cal G}(x_f,x_s) & = & \\Theta(x_s,x_f) \\delta(\\sigma)\n\\end{eqnarray}\nwhere the $\\Theta=1$ when the time of the source $t_s$ precedes the\ntime of the field point $t_f$ and $0$ otherwise. We transform\n\\begin{eqnarray}\n  \\delta(\\sigma) & \\to & \\frac{e^{-\\sigma^2\/(2 w_1^2)}}{\\sqrt{2 \\pi} w_1} \\\\\n  \\Theta & \\to &\\frac{1-\\tanh((t_s-t_f)\/w_2)}{2}\n\\end{eqnarray}\nto generate a smooth, finite integrand. The parameters $w_1$ and $w_2$\ndescribe the width of the smoothed delta function and the width of the causal\nfunction. (We use $w_i$ schematically in this discussion. In\nAppendix \\ref{sec:2D} we introduce unique symbols.)\n\nSource points are over-retarded and appear slightly\ninside the field point's backwards light cone. Over-retardation\n\\cite{1975NCimB..26..157D} is a covariant method for classical renormalization.\nWe modify the Synge function\n\\begin{eqnarray}\n  \\sigma(x,z) & = & \\frac{1}{2} (x-z)^\\alpha g_{\\alpha\\beta}(x-z)^\\beta + w_3\n\\end{eqnarray}\nwhere $w_3 \\ge 0$ is the parameter. Over-retardation disallows the\nsource-field point coincidence.\n\nFinally, we round off discontinuous features on the string.  For kinks\nthe transition from one derivative value to another is smoothed.  For\ncusps a small patch of the worldsheet near the cusp is excised.\nWe introduce a parameter $w_4$ that yields the discontinuous\nsolution when $w_4 \\to 0$. Smoothing must be implemented separately\nfor each loop of interest. In the 2D approach any discontinuity, even if it\nwere not on the field point's exact light cone, must be smoothed\nbecause all worldsheet points are sampled by the smoothed delta function.\n\nThe 2D calculation does not require any special treatment for\nboundaries, any special choice of coordinates or any special handling\nof the field point. The discontinuities in the source must be\nsmoothed. We let $\\{w_1,w_2,w_3,w_4\\} \\to 0$ in lockstep together.\nWe have found that the limit is not impacted if we\nset $w_2=0$ (the smoothing of the causal step function)\nand $w_3=0$ (the over-retardation) from the beginning. Using the Gaussian\napproximation to the delta function and smoothing the discontinuities\non the string are sufficient to regulate the calculation.\n\n\\subsection*{1D, over-retarded, smoothed kink or cusp [1DOS]}\n\\label{sec:1DOS}\n\nThe 1D calculations in which the Green function has been integrated\nout must handle the field equals source point, the string\ndiscontinuities and coordinate changes along the retarded loop\nimage.\n\nIn the [1DOS] method we use over-retardation and smooth the discontinuities on the\nstring if they are visible on the field point's exact light cone.\nWe integrate Eq.~\\eqref{eq:F1-convolution} over the\nimage if coordinate $\\zeta^2$ covers the entire image;\nwe additionally include boundary terms of the type given in\nEq.~\\eqref{eq:dh-zeta2-discontinuous-convolution} if multiple\ncoordinate systems are utilized. Here a boundary term\narises not because of a string discontinuity\nbut because of the coordinate change. We let $\\{w_3,w_4\\} \\to 0$.\n\n\\subsection*{1D, over-retarded, discontinuous kink or cusp [1DO]}\n\\label{sec:1DO}\n\nAs above we use over-retardation for the [1DO] method, but we do not smooth\nthe kink. We numerically locate the kink and\nuse boundary terms of the type given in Eq.~\\eqref{eq:dh-zeta2-discontinuous-convolution}\nto handle both jumps in the string source and coordinate changes.\nWe can evaluate the force for the cusp as long as the cusp is not on the\nlight cone (almost all worldsheet points). We let $\\{w_3\\} \\to 0$.\n\n\\subsection*{1D, discontinuous kink or cusp [1D]}\n\\label{sec:1D}\n\nFor the [1D] method we use the analytic results \\eqref{eq:F-local-ST}\nfor the source equals field point\nand boundary terms of the form given in Eq.~\\eqref{eq:dh-zeta2-discontinuous-convolution} for\njumps in the string source and coordinate changes. As above\nwe can evaluate the force for strings with cusps as long as the cusp is not\non the light cone. This is the computationally most efficient\nmethod and the one we are primarily interested in validating\nfor future calculations of loops evolving under the effect of\ngravitational backreaction.  It does not require any regularization\nparameters $w_i$.\n\nThere are many related questions that we address using\nthese techniques. For example, we compare the self force\ncalculated utilizing different coordinate systems (this\nis possible for all the methods, but we\nconcentrate on the [1D] case). We also consider a limiting\nprocess in which the [1DO] method is used for a field point off the worldsheet, and verify\nthe correct behavior is recovered as the field point approaches the worldsheet.\n\n\\section{Numerical results}\n\\label{sec:results}\n\nWe now apply the derivations of the previous sections to some specific examples, numerically\ncomputing the self-force for a range of nontrivial string configurations that feature kinks, cusps\nand self-intersections. We perform several consistency checks in the process:\n\\begin{enumerate}\n  \\item For strings with a particularly simple structure we compare against existing calculations in the literature;\n  \\item For more non-trivial strings we compare different versions of the [1D] integration done with different choices of worldsheet coordinates;\n  \\item We compare against the smoothed approaches [1DOS] and [1DO] for handling kinks and field point contributions. The field point contribution is recovered by evaluating the integral for the force with a small over-retardation of the retarded time and numerically taking the limit as this over-retardation vanishes. The kink contribution is similarly recovered by introducing a small smoothing to the kink and taking the limit of the smoothing parameter going to zero.\n  \\item We further compare against our other entirely independent [2D] approach, whereby the force is directly determined from a full 2D integration over the worldsheet, approximating the Dirac $\\delta$ distribution in the Green function by a narrow Gaussian.\n  \\item We verify that the flux of radiation to infinity (as computed using standard frequency domain methods\n\\cite{2001PhRvD..63f3507A}) appropriately balances the local self-force.\n\\end{enumerate}\n\nThere are infinitely many possible cosmic string loops which satisfy \\eqref{eq:conformal-eom}. The\nexamples which have typically been studied in the literature are those with a low number of\nharmonics. As a demonstration of our prescription for computing the self-force, we will compute the\nself-force for several of these strings. Our goal is not to be exhaustive, but rather to select a\nset of test cases that cover all scenarios (kinks, cusps, self-intersections, and strings without\ntoo much symmetry). In all cases below, we define the worldsheet in terms of two functions\n$a^{\\alpha} (\\zeta^+)$ and $b^{\\alpha} (\\zeta^-)$, where $\\zeta^+ \\equiv \\tau + \\zeta$ and $\\zeta^-\n\\equiv \\tau - \\zeta$ are null worldsheet coordinates. Then, the spacetime position of the string is\n$z^\\mu = (1\/2)[a^\\mu(\\zeta^+) + b^\\mu(\\zeta^-)]$. Throughout the discussion, we will also refer to\nthe three-vectors $\\mathbf{a}$ and $\\mathbf{b}$, which are defined to be the spatial projections of\n$a^\\alpha(\\zeta^+)$ and $b^\\alpha(\\zeta^-)$. Finally, we will specialize to the specific case\n$t=\\tau$ within the class of conformal gauges.\n\n\n\\subsection{Allen, Casper and Ottewill self-similar string}\n\\label{sec:ACO}\nAllen, Casper and Ottewill (ACO) \\cite{Allen:1994bs} identified a particularly simple class of\nstrings for which the average power radiated is easily calculated in closed form. All strings in\nthe class have a pair of kinks, each propagating along lines of constant $\\zeta^+_{k_1} = 0$ and\n$\\zeta^+_{k_2} = L\/2$, respectively. ACO's motivation was to find the string which radiates most\nslowly and is thus most long-lived. Our motivation for studying the ACO string\\footnote{We will\nstudy just one case in the class of ACO strings, the one which is simplest and which radiates power\nmost slowly. ACO call this particular string ``case (1) with $M=1$''. We will simply refer to it as\n\\emph{the} ACO string.} stems from a different consequence of the simplicity of the ACO solution.\nAnderson \\cite{Anderson:2005qu} showed that the description of the ACO string worldsheet is\nsufficiently simple that it is possible to determine the self-force analytically.\\footnote{In fact,\nin \\cite{Anderson:2008wa} Anderson was able to go one step further and analytically\nself-consistently evolve the string under the influence of gravitational backreaction.} This\nprovides a valuable reference point against which we can check our numerical approach.\n\nThe ACO string worldsheet is given in Cartesian coordinates by\n\\begin{align}\n  a^{\\alpha} (\\zeta^+) =& A [\\zeta^+\/A, 0, 0, |\\zeta^+|],\n\\nonumber \\\\\n  b^{\\alpha} (\\zeta^-) =& A [\\zeta^- \/A, \\cos (\\zeta^- \/ A), \\sin (\\zeta^- \/ A), 0],\n\\end{align}\nwhere $A \\equiv \\tfrac{L}{2\\pi}$ and $L$ is the length of the string. For $\\zeta^+<-\\tfrac{L}{2}$\nor $\\zeta^+>\\tfrac{L}{2}$ the periodic extension of $a^z$ is used, i.e. $a^z$ is the triangle\nfunction centered about the origin and with period $L$. The ACO string can be visualised as shown\nin Fig.~\\ref{fig:ACOloopspacetime}; its evolution is a rigid rotation of this shape about the\n$z$-axis.\\footnote{In \\cite{Anderson:2008wa} Anderson showed that this shape is preserved when\nbackreaction is taken into account, in which case the string evolves (shrinks) self-similarly.} We\ncharacterize the ACO string in terms of its tangent-sphere representation, as shown in\nFig.~\\ref{fig:ACO-tangent-sphere}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{figures\/ACOString}\n\\caption{\n\\label{fig:ACOloopspacetime}\nSnapshot of the ACO string loop configuration in spacetime at time $\\tau=0$. At later times the\nconfiguration can be obtained by a rigid rotation about the $z$-axis.}\n~\\\\\n\\includegraphics[width=0.6\\linewidth]{figures\/ACO-tangent}\n\\caption{\n\\label{fig:ACO-tangent-sphere}\nTangent sphere representation of the ACO string loop configuration with $\\mathbf{a}'(\\zeta^+)$ denoted by the two blue dots and $\\mathbf{b}'(\\zeta^-)$ by the orange circle.}\n\\end{figure}\n\n\nAdopting conformal gauge to first order, Anderson \\cite{Anderson:2005qu} was able to compute the self-force (which, in the conformal gauge case is defined to be the right-hand side of Eq.~\\eqref{eq:simple}) by analytically determining the first-order metric perturbation generated by an ACO string. Factoring out the rigid rotation using the matrix\n\\begin{equation}\nM^\\alpha{}_\\beta = \\left(\n\\begin{array}{cccc}\n 1 & 0 & 0 & 0 \\\\\n 0 & \\cos (2 \\pi  \\zeta^-) & \\sin (2 \\pi  \\zeta^-) & 0 \\\\\n 0 & -\\sin (2 \\pi  \\zeta^-) & \\cos (2 \\pi  \\zeta^-) & 0 \\\\\n 0 & 0 & 0 & 1 \\\\\n\\end{array}\n\\right),\n\\end{equation}\nthe conformal gauge self-force in a co-rotating frame is given by\n$f^\\mu = M^\\mu{}_\\alpha {\\cal F}_{\\rm conf}^\\alpha$, where $f^\\mu = [f^t(\\zeta^+),\nf^L(\\zeta^+), f^N(\\zeta^+), \\sgn(\\zeta^+) f^t(\\zeta^+)]$. We can interpret $f^N$ and $f^L$ as\nthe normal and longitudinal components of the force in the $x$--$y$ plane, respectively. Note that\nthis factorized\nform is quite convenient as the dependence on $\\zeta^+$ is entirely contained within $f^\\mu$,\nwhile the dependence on $\\zeta^-$ is entirely in $M^\\chi{}_\\mu$.\n\n\\begin{figure}[H]\n  \\center\n  \\includegraphics[width=8cm]{figures\/ACOForce.pdf}\n  \\caption{Co-rotating self-force for the ACO string.}\n\\label{fig:ACOForce}\n\\end{figure}\nAs an important consistency check on our work, we have verified that our numerical approach exactly\nreproduces the analytic result derived by Anderson. Figure \\ref{fig:ACOForce} shows the factored\ncomponents of the force as a function of $\\zeta^+$, with Anderson's expressions plotted as solid\nlines and our numerical values (computed using Eq.~\\eqref{eq:F1-convolution} plus boundary terms of\nthe type given in Eq.~\\eqref{eq:dh-zeta2-discontinuous-convolution} at the kinks and\nEq.~\\eqref{eq:F-local-ST} for the field point contribution) shown as dots.\n\nOne interesting feature is the divergence of the force components as a kink is approached. Although one may be concerned about the physical implications of this divergence, for the ACO string it turns out that it is a spurious gauge artifact, and that the string worldsheet itself only ever picks up a small perturbation from the self-force. The simplicity of the ACO solution makes it straightforward to see this explicitly: as shown by Anderson \\cite{Anderson:2005qu}, the explicit form of the divergence near the kink can be written as\n\\begin{align}\nf^t \\approx&\\,\\{-32 (\\tfrac{1}{6}\\pi^2)^{1\/3} \\mu \/|\\zeta^+|^{1\/3}, -128 \\pi^2 \\mu (\\zeta^+)^2\\}, \\nonumber \\\\\nf^L \\approx&\\,32 \\pi \\mu \\ln |\\zeta^+| \\{\\tfrac{1}{3}, 1\\}, \\nonumber \\\\\nf^N \\approx&\\,\\{-32 (\\tfrac{1}{6}\\pi^2)^{1\/3} \\mu \/|\\zeta^+|^{1\/3}, 128 \\pi^2 \\mu \\zeta^+ \\ln |\\zeta^+|\\},\n\\end{align}\ndepending on whether the limit $\\zeta^+ \\to 0$ is taken from the left or the right. Anderson goes on to show that integrating up the equation of motion, the physical (non-gauge) displacement of the string due to this divergent force is finite.\n\n\n\\subsection{Kibble and Turok strings with cusps and self-intersections}\nA simple family of string loop solutions of the zeroth order equations\nof motion was written down by Kibble\nand Turok~\\cite{Kibble:1982cb,Turok:1984cn}. The gravitational\nradiation of representative examples was\ncalculated by Vachaspati and Vilenkin~\\cite{Vachaspati:1984gt}.\nWe will refer to the family as KT strings. The family is described by the general form\n\\begin{align}\n  a^{\\alpha} (\\zeta^+) =& A \\Big[\\zeta^+ \/ A,\n     (1-\\alpha) \\sin (\\zeta^+ \/ A) + \\tfrac{\\alpha}{3} \\sin (3 \\zeta^+\/A),\n\\nonumber \\\\ & \\quad\n     (\\alpha-1) \\cos (\\zeta^+ \/ A) - \\tfrac{\\alpha}{3} \\cos (3 \\zeta^+\/A),\n\\nonumber \\\\ & \\quad\n     - 2 \\sqrt{\\alpha(1-\\alpha)} \\cos (\\zeta^+ \/ A)\\Big],\n\\nonumber \\\\\n  b^{\\alpha} (\\zeta^-) =& A \\Big[\\zeta^- \/ A,\n    \\sin (\\zeta^- \/ A),\n\\nonumber \\\\ & \\quad\n    -\\cos \\phi \\cos (\\zeta^- \/ A),\n    -\\sin \\phi \\cos (\\zeta^- \/ A)\\Big],\n\\end{align}\nwhere $0 \\le \\alpha \\le 1$ and $-\\pi \\le \\phi \\le \\pi$ are two\nparameters.\n\nWe first focus on the case $\\alpha=0$ and $\\phi=\\pi\/6$\n($N=M=1$ Burden loops \\cite{Burden:1985md}).\nNine snapshots of the spacetime configuration of the loop\nare shown in Fig.~\\ref{fig:VVloopspacetime}.\nThe loop generally possesses an elliptical shape. It tumbles in space\nwhile stretching and contracting. Twice per period it forms a\ndegenerate, line-like shape with a pair of cusps on opposite sides.\nThe tangent sphere representation is particularly simple:\nthere are two continuous great circles that cross\nat $\\tau + n \\pi = \\zeta + m \\pi = 0$ for any integers $n$ and $m$.\nEach crossing gives rise to a cusp and to a spacelike line\nof string overlap in the center of momentum frame.\nThese two effects make the calculation of the self-force\nparticularly challenging.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{VVloop.pdf}\n\\caption{\\label{fig:VVloopspacetime} Snapshots of the KT string loop\n  ($\\alpha=0$ and $\\phi=\\pi\/6$)\n  configuration in spacetime, each labeled by time in units of $L$. All\n  the boxes have the same size axes, $-1$ to $1$ for $L=2 \\pi$, and\n  fixed orientation. }\n\\end{figure}\n\nWe first compute the self-force at two points on the string (which we denote Case I and Case II):\n\\begin{align}\n  (\\tau, \\zeta) &= (32 \\pi\/50,13 \\pi\/50)\n  {\\rm \\ \\ \\ Case \\ \\ I} \\\\\n  (\\tau, \\zeta) &\\simeq (0.42,\\pi\/5)\n  {\\rm \\ \\ \\ Case \\ \\ II}\n\\end{align}\nFor Case I, the field point is such that no cusp is present on the retarded image of the string.\nSince the left and right moving modes are continuous the loop stress-energy source is completely smooth except at\nthe field point itself.\\footnote{In a patch of the world sheet that extends $\\pm \\pi$ about the\nfield point the cusps at $(\\tau,\\zeta)=(0,0)$ and $(0,\\pi)$ are potentially visible for a\ncausal off-shell Green function.} The Case I results calculated by the [1D] method described in Sec.~\\ref{sec:dh} are given in the first part of Table \\ref{tab:VVanswers}. In this case, there are two important contributions to $F_1^\\rho$:\nthe row labeled $\\int$ is the integral\ncontribution arising from the $1D$ integral over the smooth worldsheet using Eq.~\\eqref{eq:F1-convolution}; and\n$\\delta$ is the contribution from the field point obtained using Eq.~\\eqref{eq:F-local-ST}. The total is $F_1 = \\int + \\delta$.\n\\begin{table}[H]\n  \\begin{center}\n\\begin{tabular}{|cc|cccc|}\n  \\hline\n Case \\ \\  & Force & \\multicolumn{4}{c|}{contravariant spacetime components}\\\\\n  & & $t$ & $x$ & $y$ & $z$ \\\\\n  \\hline\n   I & $\\int$     &  9.28612  & -4.96366 & 14.7739  & -1.68376 \\\\\n     & $\\delta$   & -0.680917 & 1.09474  & 1.16578  & -4.35077 \\\\\n     & $F_1$      &  8.6052   & -3.86891 & 15.9397  & -6.03453 \\\\\n     & $F_2$      &-12.181    & 4.56246  & -25.3768 & 14.1391 \\\\\n  \\hline\n   II & $\\int$     & 44.5678 & 49.5374 & 22.8974 & -1.99924 \\\\\n      & $\\delta$   & 1.7937  &  1.5892 &  1.1546 & -4.30897 \\\\\n      & $F_1$      & 46.3615 & 51.1266 & 24.052  & -6.30821 \\\\\n      & $F_2$      & -75.6739 & -82.35   & -39.8936 & 21.8117  \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n  \\caption{Self-force at two points on the KT string\n($\\alpha=0$ and $\\phi=\\pi\/6$)\ncalculated by the $1$D method.}\n\\label{tab:VVanswers}\n\\end{table}\n\nFor Case II, we have carefully chosen a field point such that the cusp at $(\\tau,\\zeta)=(0,0)$\nlies on the retarded string image. Numerical results for this case (which were again obtained using\nthe [1D] method) are given in the second part of Table \\ref{tab:VVanswers}.\n\nOne notable feature of these numerical results is that the field point\ncontribution is comparable in magnitude to the contribution from the\nintegral. As such, this case provides a valuable and stringent test of\nour derivation of the expression for the field point contribution. By comparing to a different\napproach which doesn't rely on these terms we may distinguish between\n$\\int$ and $F_1$. The [2D] integration method (described in detail in\nAppendix \\ref{sec:2D}) provides just such a comparison. In Table\n\\ref{tab:summarytab} we tabulate the results of the [2D] integration\nmethod and compare against the $1$D results for Case I in Table\n\\ref{tab:VVanswers}. This comparison unambiguously confirms that the\nfield point contribution is essential. The agreement provides a strong validation of our\nformalism. Appendix \\ref{sec:2D} includes analogous [2D] results for Case II. These are in equally good agreement so we omit additional discussion of\nthe comparison.\n\\begin{table}[H]\n  \\begin{center}\n\\begin{tabular}{|cccc|}\n  \\hline\n  Force & Extrapolated Force & Extr. error & 2D-1D \\\\\n  \\hline\n  $F_1^t$ &  $8.60882$ &  $0.0020$   & $-0.0036$ \\\\\n  $F_1^x$ & $-3.87143$ & $-0.0016$  &  $0.0025$ \\\\\n  $F_1^y$ & $15.9437$  &  $0.0013$  & $-0.0040$ \\\\\n  $F_1^z$ & $-6.0318$  &  $0.0030$  & $-0.0027$ \\\\\n  \\hline\n  $F_2^t$ & $-12.181$   &  $3.6 \\times 10^{-5}$    &  $1.0 \\times 10^{-5}$ \\\\\n  $F_2^x$ & $4.56246$   & $-1.5 \\times 10^{-6}$ & $-9.6 \\times 10^{-7}$ \\\\\n  $F_2^y$ & $-25.3768$  & $-1.6 \\times 10^{-5}$ & $2.7 \\times 10^{-5}$ \\\\\n  $F_2^z$ & $14.1391$   & $-7.5 \\times 10^{-6}$ & $1.5 \\times 10^{-5}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Extrapolated self-force calculated by the [2D] method in Case I of the KT string ($\\alpha=0$ and $\\phi=\\pi\/6$).}\n\\label{tab:summarytab}\n\\end{table}\n\nWe now proceed to compute the self-force at \\emph{all} points on the worldsheet. The results are shown in Figs.~\\ref{fig:Force-components-VV} and \\ref{fig:Force-VV}. Unlike the ACO case, the extra complexity in the KT solution means that there is no simple factorization of the force into a piece which only depends on $\\zeta^+$ and another piece which depends on $\\zeta^-$. As such, the self-force for the KT string is presented as a $2D$ surface plot, showing the force contributions to $F^\\mu$ ($\\log_{10}$ of the absolute value of a contribution, color coded by sign) at all\\footnote{In all of our plots we show the segment of the worldsheet defined by $\\tau \\in [0, L\/2]$, $\\zeta \\in [-L\/2,L\/2]$. This covers the entire set of unique points on the worldsheet; other values can be obtained by periodically extending in the $\\tau$ and\/or $\\zeta$ direction.} points on the two-dimensional worldsheet. The green and red curves trace the advanced images of the cusps on the loop; each point on these curves has the cusp at $(\\tau, \\zeta) = (0,0)$ (red) or at\n$(\\tau, \\zeta) = (0,L\/2)$ (green) on its past light cone. The gross\nvariation of the self-force depends on the product of two factors\nwhich have simple physical origins.\nFirst, the loop's line-like structure, periodically\nformed at $\\tau=0$ and $L\/2$, creates a ridge spanning all $\\zeta$\nat these particular times. Second, at any given time\nthe points along the\nstring loop which are least contracted and have the largest $\\sqrt{-\\gamma}$\noccur at $\\zeta = \\pm L\/4$. These produce a trough or minimum\nin the force at $\\zeta = \\pm L\/4$. The product of these two\nfactors yields the\negg-crate-like symmetry in the force with the cusps at the\ncorners.\n\n\\begin{widetext}\n~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/VV}\n\\caption{\\label{fig:Force-components-VV}\nContributions to $F_1^\\mu$ for the KT string ($\\alpha=0$ and $\\phi=\\pi\/6$) when computed using the 1D integration method with integration with respect to $\\zeta$. Each sub-figure shows the relevant contribution to the force at all points on the string in the region $\\tau \\in (0,L\/2)$, $\\zeta \\in (-L\/2,L\/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from: (i) the field point; and (ii) the integral over $\\zeta$ (ignoring distributional contributions at the field point). For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).\n}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/VV-total}\n\\caption{\\label{fig:Force-VV}\n  The two pieces of the self-force, $F_1^\\mu$ (row 1) and $F_2^\\mu$ (row 2),\n  and the total self-force $F^{\\mu}$ (row 3) for the KT string ($\\alpha=0$ and $\\phi=\\pi\/6$) as a function of position on the string. The $F_1^\\mu$ part can be obtained by summing the two rows in Fig.~\\ref{fig:Force-components-VV}. For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).\n}\n\\end{figure}\n\\end{widetext}\n\nThese plots show several interesting features:\n\\begin{enumerate}\n\\item The self-force is finite at almost all points on the worldsheet, the notable exceptions being the location of the two cusps, where it appears to diverge.\n\\item The two contributions to $F_1^\\mu$ (coming from the integral over the smooth worldsheet and from the field point) are comparable in magnitude. It is therefore crucial that both contributions be included.\n\\item The contributions from $F_1^\\mu$ and $F_2^\\mu$ are both comparable in magnitude and both exhibit the same qualitative behavior in terms of divergence at the cusp and finiteness elsewhere.\n\\end{enumerate}\n\n\nAlthough this case provides a good check of the general\nmethodology it involves special features\nthat can be traced to the self-intersections. In the next\nsection we modify the parameter choice to avoid self-intersections.\n\n\\subsection{KT strings with cusps without self-intersections}\n\\label{sec:VV-nonSI}\n\nNext we consider a KT string with parameter values $\\alpha=1\/2$ and $\\phi=0$. Snapshots of this\nloop are shown in Fig.~\\ref{fig:VVloopNonSIspacetime}. The loop rotates about the z-axis and forms\ncusps transiently at $(\\tau,\\zeta)=(0,0)$ and $(0,L\/2)$. There are no self-intersections except\ninfinitesimally close to the cusp itself.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{VVloop-NonSIA.pdf}\n\\caption{\\label{fig:VVloopNonSIspacetime}\n  Snapshots of the KT string loop ($\\alpha=1\/2$ and $\\phi=0$) configuration in spacetime, each\n  labeled by time in units of $L$. All the boxes have the same size axes, $-1$ to $1$ for $L=2 \\pi$,\n  and fixed orientation.}\n\\end{figure}\n\n\\begin{widetext}\n~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/VVNonSIA}\n\\caption{\\label{fig:Force-components-VVNonSIA}\nContributions to $F_1^\\mu$ for the KT string ($\\alpha=1\/2$ and $\\phi=0$) as otherwise described in Fig.~\\ref{fig:Force-components-VV}.\n}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/VVNonSIA-total}\n\\caption{\\label{fig:Force-VVNonSIA}\n  The two pieces of the self-force, $F_1^\\mu$ (row 1) and $F_2^\\mu$ (row 2), and the total\n        self-force $F^{\\mu}$ (row 3) for the KT string ($\\alpha=1\/2$ and $\\phi=0$)\n        as otherwise described in Fig.~\\ref{fig:Force-VV}.\n}\n\\end{figure}\n\\end{widetext}\nFigs.~\\ref{fig:Force-components-VVNonSIA} and \\ref{fig:Force-VVNonSIA} show the self-force at all\npoints on the worldsheet of this KT string. These are analogous to the plots for the\nself-intersecting KT string shown Figs.~\\ref{fig:Force-components-VV} and \\ref{fig:Force-VV}. The\npeaks clearly show the cusp locations and the diagonal striping is related to the overall sense of\nrotation of the loop. The spacelike line of overlap and the egg-crate symmetry seen in the previous\nKT case are now absent.\n\nThis non-intersecting case allows for a detailed analysis of the behavior of the total backreaction\nforce in the vicinity of the cusp at $(\\tau,\\zeta)=(0,0)$. At times close to cusp formation the\ntip's position (the string coordinate at fixed $\\zeta=0$) is\n\\begin{eqnarray}\n  {\\mathbf{z}}^i & \\sim & \\{0,-0.83,-0.5\\} + \\{1,0,0\\} \\tau + \\nonumber\\\\\n  & &   \\{0,1.5,0.5\\} \\frac{\\tau^2}{2} + \\{-3,0,0\\} \\frac{\\tau^3}{6} + \\nonumber\\\\\n  & &   \\{0,-7.5,-0.5\\} \\frac{\\tau^4}{24} + \\dots\n\\end{eqnarray}\nThe velocity lies in the x-direction and the acceleration in the y- and z-directions. Conversely,\nthe velocities in the y- and z-directions and the acceleration in the x-direction vanish. On\nphysical grounds we expect the y- and z-accelerations to source transverse gravitational waves and\nthe relativistic motion in the x-direction to lead to strong beaming.\n\nThe driving force $F^\\alpha$ which enters the string loop's equation of motion, Eq.~\\eqref{eq:final},\nencodes the fully non-local, self-interacting gravitational dynamics. If we were to adopt the\nconformal gauge at first order then ${\\cal F}_{\\rm conf}^\\alpha$ would naturally appear as the driving\nforce in the equation of motion. We will not restrict ourselves to that choice for much of the\ndiscussion in this section. We will show, however, that many of the features of the full worldsheet\nvariation of $F_1^\\alpha$ can be understood based on the observed properties of the formally\ndefined quantity ${\\cal F}_{\\rm conf}^\\alpha$ (which may be defined in any gauge; only its interpretation\nas the driving force is restricted to conformal gauge). We will be explicit whenever our statements\ndemand the specification of the conformal gauge.\n\n\\begin{widetext}\n        ~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\linewidth]{figures\/VVNonSIcuspFtrad1.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/VVNonSIcuspFtrad2.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/VVNonSIcuspFtrad3.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/VVNonSIcuspFtrad4.pdf}\n\\caption{\\label{fig:trad-VVNonSI} The four components ($t$, $x$, $y$ and $z$) of\n  $\\ln |{\\cal F}_{\\rm conf}^\\alpha|$ on a small patch of the worldsheet about the cusp.\n  Four quadrants in $\\{\\log \\tau, \\log \\zeta\\}$ are displayed\n  with the smallest $|\\tau|$ and $|\\zeta|$ at the center of\n  the picture, oriented in the same way as the usual linear\n  system about $\\{0,0\\}$. Orange (blue) represent positive (negative) values.\n}\n\\end{figure}\n\\end{widetext}\nThe large dynamic range evident in Figs.~\\ref{fig:Force-components-VVNonSIA} and\n\\ref{fig:Force-VVNonSIA} necessitates looking at small patches to examine special features like the\ncusp. We begin by displaying ${\\cal F}^\\mu_{\\rm conf}$ in Fig.~\\ref{fig:trad-VVNonSI}. The special\ncoordinate system shows a small patch near the cusp which is located at $\\zeta=0=\\tau$. Results for\n$\\ln |{\\cal F}^\\mu_{\\rm conf}|$ are displayed in these figures, color coded according to the sign of the\nquantity: orange (blue) dots represent positive (negative) values. Each figure combines four plots\nwith axes $\\{ {\\rm sgn} (\\zeta) \\ln |\\zeta|,\\,{\\rm sgn} (\\tau) \\ln |\\tau|\\}$, arranged and oriented\nin the same way as a normal linear plot (plus a constant shift selected to bring small values close\nto the center). The lower left hand quadrant has $\\zeta < 0$ and $\\tau < 0$. Smaller values of\n$|\\tau|$ and $|\\zeta|$ lie near the center for all four quadrants. The gap encompasses all values\nnear the sign change of the independent coordinates.\n\nWe find that ${\\cal F}^t_{\\rm conf} < 0$ for the entire area of the patch. The magnitude of ${\\cal F}^t_{\\rm conf}$ is much less than $F^t$ and is less strongly divergent --- the two are related by a\nprojection factor and an overall factor of $1\/\\sqrt{-\\gamma}$ (see Eq.~\\eqref{eq:self-force}), both\nof which diverge as the cusp is approached. Likewise, ${\\cal F}^x_{\\rm\n  conf} < 0$, ${\\cal F}^y_{\\rm conf} > 0$\nand ${\\cal F}^z_{\\rm conf} > 0$ have single, well-defined signs throughout most of the area of\ncorresponding patch.\n\nIn the conformal gauge the negative value for ${\\cal F}_{\\rm conf}^t$ implies (see\nEq.~\\eqref{eqn:tradiationalforcesecondorderloss} in Appendix\n\\ref{sec:energymomentumlossdiscussion}) that the string is losing energy and decelerating in the\nx-direction both before and after the cusp forms . This makes physical sense; the self-force saps\nthe mechanical energy during the period of large acceleration and the relativistic beaming ensures\nthat gravitational waves are emitted primarily in the x-direction, thus creating the largest\ndecelerating force in that direction. A small spatial segment of the string near where the cusp\nforms behaves in a coherent fashion before and after the moment of cusp formation in terms of the\nsigns of ${\\cal F}_{\\rm conf}^\\alpha$ for all components. ${\\cal F}_{\\rm conf}^\\alpha$ shows a net positive\nacceleration in y- and z-directions throughout most of the area of these figures.\n\nAs the figures of ${\\cal F}^\\mu_{\\rm conf}$ make clear, the asymptotic behavior near the cusp varies\ndepending upon the direction of approach. A common diagonal feature is the locus in the worldsheet\nwhere $\\sqrt{-\\gamma} \\ge 0$ is small. Only at the cusp is $\\gamma$ exactly equal to zero, but\nalong the visible fold its values are small.\n\\begin{widetext}\n~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\linewidth]{figures\/cuspapproacht.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/cuspapproachx.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/cuspapproachy.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/cuspapproachz.pdf}\n\\caption{\\label{fig:VVloopNonSIcuspray} The components of the\n  smooth integral contribution to ${\\cal F}^\\mu_{\\rm conf}$ along two rays\n  at angles $\\theta=0$ and $\\pi\/2$ are shown. Red dots are numerical\n  results and blue lines are fits of the form $\\log |{\\cal F}^\\mu| = a \\log |\\log r| + b \\log r +  c$.\n  For $\\mu=t$, $(a,b,c)=(0.098,-0.99,2.45)$ for $\\theta=0$ and $(0.046,-1,3.24)$ for $\\theta=\\pi\/2$;\n  likewise, for $\\mu=x$, $(a,b,c)=(0.044,-1,2.52)$ and $(0.018,-1,3.28)$.\n  These fits show that the dominant behavior in the direction of motion\n  of the cusp as $r \\to 0$ is $1\/r$. In the other directions, the behavior\n  is consistent with a logarithmic divergence at leading order:\n  $\\mu=y$, $(a,b,c)=(0.79,-0.011,3.35)$ and $(1.23,0.013,2.39)$;\n  $\\mu=z$, $(a,b,c)=(0.43,-0.025,2.13)$ and $(0.41,-0.026,2.17)$.\n}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\linewidth]{figures\/Ftradt-VVcusp.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/Ftradx-VVcusp.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/Ftrady-VVcusp.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/Ftradz-VVcusp.pdf}\n\\caption{\\label{fig:VVloopNonSIcusp} ${\\cal F}^\\mu_{\\rm conf}$ for the\n  non-intersecting KT string ($\\alpha=1\/2$ and $\\phi=0$) in the\n  neighborhood of the cusp at $\\tau = 0 = \\zeta$. The $t$ and $x$\n  components have been scaled by the radial (Euclidean) distance from\n  the cusp, and the $y$ and $z$ components have not been scaled at\n  all. Each panel displays two separate contributions to the total force. For\n  $t$ and $x$ components the upper (green solid and red dotted) lines give the field point\n  contribution. It is antisymmetric in angle and integrates over angle to give zero\n        (in fact, when taken over the whole worldsheet the integral also vanishes exactly).\n        The lower (blue solid and orange dashed) lines give the smooth\n  integral contribution. It is single signed and large where\n  $\\sqrt{-\\gamma}$ is small. We plot scaled results for $r=0.02$\n  (solid lines) and $r=0.002$ (dashed\/dotted lines) for each\n  contribution to the total.  These overlap and show in a qualitative\n  fashion the dominant $1\/r$ scaling near the cusp for both contributions\n  to the $t$ and $x$ components. In the lower panels we display the $y$ and $z$\n        components. As before, the contribution from the field point is given by the green solid and\n        red dotted curves. In this case it is independent of $r$ to lowest order (and\n        integrates over angle to give $4\\pi$ at this order). The solid blue and dashed\n        orange lines show the integral contributions increase slowly as $r$ decreases, consistent \t\t\t\twith the $\\log r$ type behavior.}\n\\end{figure}\n\\end{widetext}\nRegardless of direction, however, the scaling with radial distance from the cusp is clear and\nunambiguous in each of the components.\nThe smooth integral contribution to ${\\cal F}^\\mu_{\\rm conf}$\nis shown in Fig.~\\ref{fig:VVloopNonSIcuspray}\nfor rays approaching the cusp with angle $\\theta=0$ ($\\delta \\tau=0$) and $\\theta=\\pi\/2$ ($\\delta \\zeta=0$).\nThe red dots are numerical results and the blue lines are fits of the form $\\log |F^\\mu| = a \\log |\\log r| + b \\log r +  c$.\\footnote{The occurence of both $\\log|\\log r|$ and $\\log r$ are consistent with\n  recently reported analytic results of Blanco-Pillado, Olum and Wachter [see acknowledgements].}\nThe angular variation and scaling of the delta-function term and the smooth integral contribution\nare illustrated in Fig.~\\ref{fig:VVloopNonSIcusp} and discussed in the caption.\n\nTwo-dimensional numerical fits for the integral part of ${\\cal F}^\\mu_{\\rm conf}$ are\nsummarized in Appendix \\ref{sec:VV-fits}. We find that\n${\\cal F}^t_{\\rm conf}$ and ${\\cal F}^x_{\\rm conf}$ scale as the inverse distance from the cusp, and that\n${\\cal F}^y_{\\rm conf}$ and ${\\cal F}^z_{\\rm conf}$ are at worst much less singular (consistent with a log divergence).\nFrom this we conclude\nthat when one adopts the conformal gauge at first order the self-force near the cusp has a weak,\nintegrable divergence on the worldsheet and that any integrated quantities (such as the radiated\nenergy) are finite.\n\n\\begin{widetext}\n        ~\n  \\begin{table}[H]\n  \\begin{center}\n    \\begin{tabular}{|c|cccccc|cc|}\n      \\hline\n      Quantity     & $a$ & $b$ & $c$ & $d$ & $e$ & $f$ & $\\log_{10} \\epsilon$ & $\\log_{10} Q$ \\\\\n      \\hline\n$F^t-F^x$&$14.68$&$276.43$&$117.42$&$-41.51$&$12.47$&$-25.45$&$-3.08$&$1.06$ \\\\\n$F^z(\\tau+\\zeta)+F^y(3\\tau+\\zeta)$&$0$&$521.81$&$175.92$&$-126.45$&$38.64$&$16.21$&$-2.67$&$0.91$ \\\\\n      \\hline\n$H_{\\tau\\tau}$&$11.49$&$-6.53$&$-15.33$&$18.76$&$84.03$&$101.7$&$-7.22$&$3.9$ \\\\\n$H_{\\zeta\\zeta}$&$0$&$0$&$0.01$&$64.4$&$196$&$237.13$&$-7.03$&$1.32$ \\\\\n$H_{\\tau\\zeta}$&$0$&$-7.66$&$-17.94$&$22.71$&$69.29$&$56.75$&$-7.05$&$3.8$ \\\\\n  \\hline\n\\end{tabular}\n  \\end{center}\n  \\caption{First order fits\n    for force combinations near the cusp with form\n    $a + b \\tau + c \\zeta + (d \\tau^2 + e \\tau \\zeta + f \\zeta^2)\/r$ with\n    $r=\\sqrt{\\tau^2+\\zeta^2}$ and\n    second order fits for the worldsheet projected metric pertubations with\n    form $a + b \\tau + c \\zeta + d \\tau^2 + e \\tau \\zeta + f \\zeta^2$\n    over the radial range $2 \\times 10^{-8} \\le r \\le 2 \\times 10^{-4}$. The Table provides the two force\n    combinations and 3 worldsheet projected metric perturbations that appear in the asymptotic\n    forms for $F_1^\\mu$ and $F_2^\\mu$. The last two columns give the common log of $\\epsilon$ (the root mean square\n    error between the data and the fit), and $Q$ (the ratio of the variation in the data divided\n    by $\\epsilon$). \n  }\\label{tab:fits}\n  \\end{table}\n\\end{widetext}\n\n  \\begin{table}[H]\n  \\begin{center}\n    \\begin{tabular}{|c|ccccc|}\n      \\hline\n      Order & $-4$ & $-3$ & $-2$ & $-1$ & $0$ \\\\\n      Component& & & & & \\\\      \n      \\hline      \n$t$ &$ < $&$ < $&$-2.01$ &$ 1.28 $&$1.25$ \\\\\n$x$ &$ < $&$ < $&$-2.01$ &$ 1.28 $&$1.25$ \\\\\n$y$ &$ < $&$ < $&$ < $   &$ <$&$2.75$ \\\\\n$z$ &$ < $&$ < $&$ < $   &$ < $&$2.03$ \\\\\n      \\hline\n\\end{tabular}\n  \\end{center}\n  \\caption{The numerical results for the\n    expansion of $\\sqrt{-\\gamma}F^\\mu$ in $r$ (averaged over angle)\n    for the fit given in Table \\ref{tab:fits}.\n    For $\\sqrt{-\\gamma}(F_1 + F_2) \\sim \\sum_{n} c_n r^n$ the columns\n    are the leading powers of\n    the expansion ($n=-4$ to $0$), the rows are the spacetime\n    components and the table values are the common log of the\n    expansion coefficients ($d_n=\\log_{10} | c_n | $).\n    The symbol $<$ means a numerical result $|c_n| < 10^{-12}$.\n    When the fitting range is narrowed about the cusp the $1\/r^2$ contribution\n    decreases $\\propto r$ and the other pieces are fixed.\n    From this we infer that the leading non-zero piece of $F^\\mu$\n    varies as $1\/r$.\n  }\\label{tab:cancellations}\n  \\end{table}\nGiven the scalings for ${\\cal F}^\\mu_{\\rm conf}$ nearby the cusp, it is straightforward to deduce the\ncorresponding scaling for $F^\\mu_1 = -\\tfrac{1}{\\sqrt{-\\gamma}} \\perp^\\mu{}_\\nu {\\cal F}^\\nu_{\\rm conf}$.\nWorking with the exact expression for the determinant of the induced metric in this case,\n\\begin{equation}\n        \\gamma = -\\frac{1}{16} \\left[2 - \\cos (2 \\zeta) - \\cos (4\\zeta+2\\tau) \\right]^2,\n\\end{equation}\nand expanding the relationship between $F^\\mu_1$ and ${\\cal F}^\\mu_{\\rm conf}$ to next from leading order, we find\n\\begin{align}\n        \\label{eq:F1asym1}\n  F^t_1 & \\approx F^x_1 \\approx - \\frac{1}{\\gamma}\\bigg\\{\\left({\\cal F}^t_{\\rm conf}-{\\cal F}^x_{\\rm conf}\\right) \\\\\n         & - \\frac12\\left[\\left({\\cal F}^z_{\\rm conf}+3 {\\cal F}^y_{\\rm conf}\\right)\\tau + \\left({\\cal F}^z_{\\rm conf}+ {\\cal F}^y_{\\rm conf}\\right)\\zeta\\right] + \\cdots\\bigg\\}, \\nonumber \\\\\n        \\label{eq:F1asym2}\n        F^y_1 & \\approx -\\frac{1}{2\\gamma} \\left({\\cal F}^t_{\\rm conf}-{\\cal F}^x_{\\rm conf}\\right) \\left(\\zeta + 3 \\tau\\right) + \\cdots, \\\\\n        \\label{eq:F1asym3}\n        F^z_1 & \\approx -\\frac{1}{2\\gamma} \\left({\\cal F}^t_{\\rm conf}-{\\cal F}^x_{\\rm conf}\\right) \\left(\\zeta + \\tau\\right) + \\cdots.\n\\end{align}\nSince $\\gamma$ scales as the fourth power of the distance from the cusp we infer that $F^\\mu_1$ is\nnaively four orders more singular than ${\\cal F}^\\mu_{\\rm conf}$. However, as can be seen in Table\n\\ref{tab:fits}, it turns out that ${\\cal F}_{\\rm conf}^t \\approx {\\cal F}_{\\rm conf}^x$ near the cusp so the\nleading-order divergence cancels and at worst $F^\\mu_1$ diverges as the inverse fourth power of the\ndistance from the cusp. At next from leading order the asymptotic expression for $F^\\mu_1$ is\nantisymmetric about the cusp. (This behavior, combined with the mixing of components is what makes\nthe analysis of ${\\cal F}^\\mu_{\\rm conf}$ clearer than working directly with $F^\\mu_1$.)\nWith the worldsheet weighting we naively infer that $\\sqrt{-\\gamma} F^\\mu_1$\ndiverges as the inverse quadratic power of the distance from the cusp,\none power worse than ${\\cal F}^\\mu_{\\rm conf}$. Now we must consider the role of $F^\\mu_2$.\n\nTo understand the behaviour of $F^\\mu_2$ near the cusp we begin with the perturbed metric projected\nalong the worldsheet vectors $\\partial_\\tau z^\\alpha$ and $\\partial_\\zeta z^\\alpha$ according to\n\\begin{eqnarray}\n  H_{\\tau\\tau}& =& \\partial_\\tau z^\\alpha h_{\\alpha\\beta} \\partial_\\tau z^\\beta \\\\\n  H_{\\tau\\zeta}& =& \\partial_\\tau z^\\alpha h_{\\alpha\\beta} \\partial_\\zeta z^\\beta \\\\\n  H_{\\zeta\\zeta}& =& \\partial_\\zeta z^\\alpha h_{\\alpha\\beta} \\partial_\\tau z^\\beta .\n\\end{eqnarray}\nEvaluating the simple expression Eq.~\\eqref{eq:F2-hproj} for the relationship between $F^\\mu_2$ and\nthe worldsheet projections of the metric perturbation, we find\n\\begin{align}\n        \\label{eq:F2asym1}\n        F_2^t &\\approx F_2^x \\approx \\frac{1}{2(-\\gamma)^{3\/2}}\\bigg[2\\left(5H_{\\tau \\zeta} - H_{\\tau \\tau} - H_{\\zeta \\zeta}\\right) \\zeta  \\nonumber \\\\\n        & \\qquad \\qquad \\qquad + \\left(4H_{\\tau \\zeta} - H_{\\tau \\tau} - H_{\\zeta \\zeta}\\right) \\tau + \\cdots \\bigg], \\\\\n        \\label{eq:F2asym2}\n        F_2^y &\\approx \\frac{2\\left(6H_{\\tau \\zeta} - H_{\\tau \\tau} - H_{\\zeta \\zeta}\\right) (\\zeta^2+3 \\tau \\zeta + \\tau^2) + \\cdots}{2(-\\gamma)^{3\/2}}, \\\\\n        \\label{eq:F2asym3}\n        F_2^z &\\approx \\frac{2\\left(2H_{\\tau \\zeta} - H_{\\tau \\tau} - H_{\\zeta \\zeta}\\right) (\\zeta^2+3 \\tau \\zeta + \\tau^2) + \\cdots}{2(-\\gamma)^{3\/2}}.\n\\end{align}\nWith the finite behavior of the the worldsheet projections of the metric perturbation, it is\nstraightforward to deduce the corresponding scaling of the divergence in $F^\\mu_2$. We find that at\nworst $F_2^t$ and $F_2^x$ diverge as the inverse fifth power of distance from the cusp. With the\nworldsheet weighting $\\sqrt{-\\gamma} F^\\mu_2$ diverges as the\ninverse cubic power of the distance from the cusp. However, as this leading-order divergence is\nantisymmetric about the cusp its integral over a patch around the cusp cancels the leading-order\ndivergence to leave only subleading pieces.\n\nWe are now left with asymptotic forms for $\\sqrt{-\\gamma} F^\\mu_1$ and $\\sqrt{-\\gamma} F^\\mu_2$ each\nscaling as the square of the inverse distance from the cusp. These\nare individually non-integrable. However, once the asymptotic forms given in Table \\ref{tab:fits} for the quantities\nin Eqs.~\\eqref{eq:F1asym1}-\\eqref{eq:F1asym3} and \\eqref{eq:F2asym1}-\\eqref{eq:F2asym3} are taken\ninto account, we find that the leading order divergent behavior exactly cancels (see Table \\ref{tab:cancellations}) to the level of\naccuracy of the numerically fitted coefficients in the combination\n$\\sqrt{-\\gamma} (F^\\mu_1+ F^\\mu_2)$ yielding the full force $\\sqrt{-\\gamma} F^\\mu$ which\nat worst diverges as the inverse distance from the cusp, and hence is integrable.\nThis divergence is no worse than ${\\cal F}^\\mu_{\\rm conf}$ itself.\n\nA detailed understanding of the behavior of these divergences near cusps\nallows us to solve either the general covariant equation of motion\nEq.~\\eqref{eq:final} or the corresponding Eq.~\\eqref{eq:simple}\nin which specific conformal gauge choices have been adopted.\n\n\\subsection{Garfinkle and Vachaspati string with kinks}\n\nThe third case we will explore is from a class of strings found by Garfinkle and Vachaspati (GV)\n\\cite{Garfinkle:1987yw}. These strings contain two kinks that travel in the same direction on an\noscillating and twisting string loop. We choose a particular representation from the general class\nwith the following right and left-moving modes\n\\begin{align}\n  a^\\mu(\\zeta^+) & = \\left[\\zeta^+, 0, a^2(\\zeta^+), a^3(\\zeta^+) \\right] \\\\\n  b^\\mu(\\zeta^-) & = \\left[\\zeta^-,\n  \\frac{L}{2 \\pi} \\cos \\frac{2 \\pi \\zeta^-}{L}, 0,\n  \\frac{L}{2 \\pi} \\sin \\frac{2 \\pi \\zeta^-}{L} \\right]\n\\end{align}\nwhere\n\\begin{align}\n  a^2(x) & =  \\frac{L}{\\pi} \\sum_j \\delta_{j, \\lfloor \\frac{2x}{L} \\rfloor}\n  \\left( -1 \\right)^{\\lfloor \\frac{j+1}{2} \\rfloor}\\times \\nonumber \\\\&\n  \\cos \\left( \\frac{\\pi}{4} + \\left( - 1\\right)^j \\frac{\\pi x}{L} \\right) \\nonumber \\\\\n  a^3(x) & =  \\frac{L}{\\pi} \\sum_j \\delta_{j, \\lfloor \\frac{2x}{L} \\rfloor}\n  \\left( -1 \\right)^{\\lfloor \\frac{j}{2} \\rfloor} \\times \\nonumber \\\\\n  &\\left[\n  \\sin \\left( \\frac{\\pi}{4} + j \\left( -1 \\right)^j \\frac{\\pi^2}{L} \\right)\n  -\n  \\sin \\left( \\frac{\\pi}{4} + \\left( -1 \\right)^j \\frac{\\pi x}{L} \\right)\n  \\right],\n\\end{align}\nand where the sums are over all integers $j$, $L$ is the invariant length, $\\lfloor x \\rfloor$ is\nthe floor function and $\\delta_{j,k}$ is the Kronecker delta.\n\nFigure \\ref{fig:GVloopspacetime} illustrates\nthe configuration in spacetime at equally spaced moments in the oscillation cycle. The kink\ndiscontinuities are visible in all four snapshots. In the tangent sphere representation (shown in\nFig.~\\ref{fig:GVlooptangentplot}), ${\\mathbf b}'$ traverses a complete great circle through the\nNorth and South poles at a steady rate; ${\\mathbf a}'$ follows two disjoint segments of a great\ncircle (longitude offset by $\\pi\/2$ from the one traced by ${\\mathbf b}'$) between latitudes\n$\\theta = \\pm \\pi\/4$, also at a steady rate. The vector ${\\mathbf a}'$ traces one segment and then\nabruptly jumps from the point $(0,y,z)$ to $(0,-y,z)$ and\ntraces out the mirrored arc at a steady rate (and repeats).\nEach jump from one segment to the other yields a kink discontinuity in the spacetime representation.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{GVloop.pdf}\n\\caption{\\label{fig:GVloopspacetime} The GV string loop configuration in spacetime at four equally spaced moments in the basic loop oscillation cycle $\\tau=0$, $L\/8$, $L\/4$ and $3L\/4$. Each box is the same size with fixed axes $-1$ to $1$\n  and fixed orientation. }\n~\\\\\n\\includegraphics[width=0.9\\linewidth]{tangentplot.pdf}\n\\caption{\\label{fig:GVlooptangentplot} The arcs on the tangent sphere\n  traced by ${\\mathbf b}'(x)$ (red line, a complete great circle passing\n  through poles) and ${\\mathbf a}'(x)$ (two green segments symmetrically\n  cut from a great circle) for the GV string. The line is made by a series of points,\n  equally spaced in argument $x$ for the left and right moving mode. A\n  blow up of the line in the figure would show equal intervals between\n  the points. When the kink is ``rounded off'' (non-zero $\\Delta$ as\n  described in the text) a few of these points will sit between the\n  pictured green arcs and the green line is formally continuous.  The\n  ${\\mathbf a}'(x)$ tangent vector moves very rapidly from one side to the\n  other.}\n\\end{figure}\n\nWhile the ACO string provided a useful analytic test case with kinks against which we could compare our numerical results, it turns out that the simplicity of the ACO string (in particular, that many quantities are a constant along the string) means that many important terms that appear in the general expression for the self-force are identically zero for the ACO string. Fortunately, the GV string is sufficiently general that this is not the case. Unfortunately, however, there is no known analytic solution for the self-force for the GV string. Instead, in order to use the GV string as a test of our method, we chose a particular point on the string ($\\tau = 0.3L$, $\\zeta = 0.4L$) and computed the self-force at that point using an extensive set of different and independent methods:\n\\begin{enumerate}\n\\item We used our exact $1$D method including a field point contribution and contributions from the two kinks (this is method [1D] discussed in Sec.~\\ref{sec:1D}).\n\\item We repeated our $1$D calculation (again, method [1D]) using multiple choices of integration variable ($\\zeta^+$, $\\zeta^-$ and $\\zeta$). In each case, the various contributions (from the integral, field point, and two kinks) were different. Indeed, in some cases there was no contribution picked up from the kinks.\n\\item We again repeated our $1$D calculation using method [1D], but using a mixed coordinate choice; we used $\\zeta^+$ on one side of the field point and $\\zeta^-$ on the other side. We then included a contribution at the point where these two segments meet up again, to account for the change in integration variable at that point. This contribution is exactly the one discussed in Secs.~\\ref{sec:kinks}, and an explicit expression is the same as one obtains when breaking the integration at a kink, as discussed in Sec.~\\ref{sec:coord-depend-integral}.\n\\item We repeated the previously mentioned $1$D calculations again, but instead of including the exact field point term, we considered an over-retarded image of the string (method [1DO]). In that case, we find that the over-retarded integrand picks up a $\\delta$-function type feature nearby the field point (see Fig.~\\ref{fig:over-retarded}). For finite over-retardation this manifests itself as a narrow Gaussian, and the Gaussian gets narrower and sharper as the over-retardation parameter is shrunk towards zero. Reassuringly, in the limit of the Gaussian shrinking down to zero size we recover a result which agrees with the previous calculations and can identify the $\\delta$-function with the field point contribution.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/F-sigma}\n\\includegraphics[width=0.9\\linewidth]{figures\/Ft-sigma}\n\\caption{\\label{fig:over-retarded}\nIntegrand used to compute to the self-force for the Garfinkle-Vachaspati string.\nThese correspond to the values in the $\\zeta$ column of Table \\ref{tab:comparison}.\nDistributional contributions from the kinks and field point are denoted by dashed and\nsolid arrows, respectively.\n}\n\\end{figure}\n\\item Finally, we repeated the calculation in a completely independent way, directly evaluating the self-force from the $2$D integral (method [2D]) without the reduction to a $1$D integral. This was significantly less efficient, but provided an important check as there is no need to consider any split into field-point-plus-integral\u2013plus-kink contributions. Instead, we smeared out the $\\delta$ function in the Green function and also introduced a slight smoothing of the kinks, as discussed in detail in Appendix \\ref{sec:2D}. Yet again, reassuringly, in the limit of our smearing and smoothing parameters going to zero we recovered a result which was in perfect agreement with all of the other methods.\n\\end{enumerate}\n\n\\begin{table*}\n  \\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{c|c|cccccccc}\n   & Contribution & $\\zeta^-$ & $\\zeta^+$ & $\\zeta^-$\/$\\zeta^+$ & $\\zeta$ & $2$D & $\\zeta^-_\\epsilon$ & $\\zeta^+_\\epsilon$ & $\\zeta_\\epsilon$ \\\\\n  \\hline \\hline\n  \\multirow{6}{*}{$F_1^t$} & $\\int$ & \\multirow{2}{*}{$-330.558$} & \\multirow{2}{*}{$-12.73(1)$} & $6.89229$ & $-25.6962$ & $-12.76(2)$ & $-330.60(4)$ & $-12.7488(1)$ & $-22.4707(2)$ \\\\\n    & $\\delta$ & $ $ & $ $ & - & $3.22536$ & - & - & - & - \\\\\n    & Kink $1$ & $358.819$ & - & - & $16.7989$ & - & $358.819$ & - & $16.7989$ \\\\\n    & Kink $2$ & $-41.0096$ & - & - & $-7.07683$ & - & $-41.0096$ & - & $-7.07683$ \\\\\n    & $\\zeta^-\\leftrightarrow\\zeta^+$ & - & - & $-19.6411$ & - & - & - & - & - \\\\\n    \\cline{2-10}\n    & Total    & $-12.7488(1)$ & $-12.73(1)$ & $-12.7488$ & $-12.7488$ & $-12.76(2)$ & $-12.8(1)$ & $-12.7488(1)$  & $-12.7487(2)$ \\\\\n  \\hline \\hline\n  \\multirow{6}{*}{$F_1^x$} & $\\int$ & \\multirow{2}{*}{$-300.675(1)$} & \\multirow{2}{*}{$-2.64(7)$} & $8.62259$ & $-21.7023$ & $-2.58(3)$ & $-300.63(4)$ & $-2.56339(6)$ & $-14.8591(9)$ \\\\\n    & $\\delta$ & $ $ & $ $ & - & $6.84317$ & - & - & - &  - \\\\\n    & Kink $1$ & $287.628$ & - & - & $14.7570$ & - & $287.628$ & - & $14.7570$ \\\\\n    & Kink $2$ & $10.4832$ & - & - & $-2.46129$ & - & $10.4832$ & - & $-2.46129$ \\\\\n    & $\\zeta^-\\leftrightarrow\\zeta^+$ & - & - & $-11.186$ & - & - & - & - & - \\\\\n    \\cline{2-10}\n    & Total    & $-2.5637(8)$ & $-2.64(7)$ & $-2.56342$ & $-2.56342$ & $-2.58(3)$ & $-2.52(8)$ & $-2.56339(6)$ & $-2.56333(9)$ \\\\\n    \\hline \\hline\n  \\multirow{6}{*}{$F_1^y$} & $\\int$ & \\multirow{2}{*}{$-304.564$} & \\multirow{2}{*}{$-10.7066(4)$} & $6.6211$ & $-23.4832$ & $-10.72(2)$ & $-304.59(3)$ & $-10.7068(1)$ & $-20.0615(2)$ \\\\\n    & $\\delta$ & $ $ & $ $ & - & $3.42159$ & - & - & - & - \\\\\n    & Kink $1$ & $326.143$  & - & - & $15.4176$ & - & $326.143$ & - & $15.4176$ \\\\\n    & Kink $2$ & $-32.2859$ & - & - & $-6.06281$ & - & $-32.2859$ & - & $-6.06281$ \\\\\n    & $\\zeta^-\\leftrightarrow\\zeta^+$ & - & - & $-17.3279$ & - & - & - & - & - \\\\\n    \\cline{2-10}\n    & Total    & $-10.7069$ & $-10.7066(4)$ & $-10.7068$ & $-10.7068$ & $-10.72(2)$ & $-10.7(1)$ & $-10.7068(1)$ & $-10.7067(2)$ \\\\\n    \\hline \\hline\n  \\multirow{6}{*}{$F_1^z$} & $\\int$ & \\multirow{2}{*}{$-190.140(1)$} & \\multirow{2}{*}{$-13.82(7)$} & $2.25465$ & $-15.9946$ & $-13.90 (1)$ & $-190.22(8)$ & $-13.8960(1)$ & $-16.9795(2)$ \\\\\n    & $\\delta$ & $ $ & $ $ & - & $-0.985094$ & - & - & - & - \\\\\n    & Kink $1$ & $234.551$ & - & - & $10.0429$ & - & $234.551$ & - & $10.0429$ \\\\\n    & Kink $2$ & $-58.3072$ & - & - & $-6.95921$ & - & $-58.3072$ & - & $-6.95921$ \\\\\n    & $\\zeta^-\\leftrightarrow\\zeta^+$ & - & - & $-16.1507$ & - & - & - & - & - \\\\\n    \\cline{2-10}\n    & Total    & $-13.8958(7)$ & $-13.82(7)$ & $-13.896$ & $-13.8960$ & $-13.90 (1)$ & $-13.97(9)$ & $-13.8960(1)$ & $-13.8958(2)$\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{\\label{tab:comparison}\nComparison of methods for computing the self-force at a generic point ($\\tau=0.3L$, $\\zeta=0.4L$) on the GV string.}\n\\end{center}\n\\end{table*}\nThe results of this extensive set of tests are given in Table~\\ref{tab:comparison}. We see that all methods produce results which are consistent within their respective error bars. The [2D] method is least accurate, due the need for a $2$D rather than $1$D numerical integral. The [1DO] method also poses challenges for numerical accuracy due to the presence of sharp features (i.e. the Gaussian approximation to the delta function for the field point contribution), as does the [1DOS] method for portions of the integral nearby kinks.\n\nThe three exact [1D] methods all work reasonably well, however even in this case not all methods\nare equally computationally efficient. In particular, calculations based on a single null\ncoordinate encounter a strong divergence in the integrand as the field point is approached from one\nside (the particular side is dependent on whether one uses $\\zeta^+$ or $\\zeta^-$ as integration\nvariable). This diverging integral largely cancels against the field point contribution\\footnote{In practice, we were only able to obtain finite results by evaluating the integral up to a short distance from the field point and evaluating the expression for the field point contribution at the point where the integral was cut off. We recovered a unique and consistent result as the cut-off point was pushed towards the actual field point.}, leaving a relatively small overall contribution from the field-point-plus-integral combination. We found that the remaining two approaches (integration with respect to $\\zeta$; and half-$\\zeta^+$ half-$\\zeta^-$ plus coordinate change term) were comparable in terms of computational efficiency.\n\nImportantly, other than accuracy concerns, all methods produced results which are unambiguous in agreeing with each other.\n\n\\begin{widetext}\n~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/GV}\n\\caption{\\label{fig:Force-components-GV}\nContributions to $F_1^\\mu$ for the Garfinkle and Vachaspati string when computed using the 1D integration method with integration with respect to $\\zeta$. Each sub-figure shows the relevant contribution to the force at all points on the string in the region $\\tau \\in (0,L\/2)$, $\\zeta \\in (-L\/2,L\/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from: (i) the kink that passes through ($\\tau = 0$, $\\zeta = 0$); (ii) the kink that passes through ($\\tau = 0$, $\\zeta = \\pi$); (iii) the field point; and (iv) the integral over $\\zeta$ (ignoring distributional contributions at the kinks and field point). The two kinks are denoted by diagonal black lines. For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling.\n}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/GV-total}\n\\caption{\\label{fig:Force-GV}\n  The two pieces of the self-force, $F_1^\\mu$ (row 1) and $F_2^\\mu$ (row 2), and the total force $F^\\mu = F^\\mu_1+F^\\mu_2$ (row 3), for the Garfinkle and Vachaspati string as a function of position on the string. The $F_1^\\mu$ part can be obtained by summing the four rows in Fig.~\\ref{fig:Force-components-GV}. For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling.\n}\n\\end{figure}\n\\end{widetext}\n\nFinally, we used the [1D] method (specifically, integrating with respect to $\\zeta$ and including field point and kink contributions) to evaluate the self-force at all points on the GV string. The results are shown in Figs.~\\ref{fig:Force-components-GV} and \\ref{fig:Force-GV}. Fig.~\\ref{fig:Force-components-GV} shows how each of the contributions to $F^\\mu_1$ contribute to the overall result, while Fig.~\\ref{fig:Force-GV} shows $F^\\mu_1$ and $F^\\mu_2$ themselves, as well as their sum. As in the other string test cases, we find that the self-force is finite almost everywhere on the string, with the exception of exactly on the kinks, where it diverges.\n\nWe have analyzed the form of the divergence near the kink\nby calculating the total backreaction force to high accuracy along a\nset of worldsheet points for a line that runs perpendicular to the kink\nwith coordinates\n$(\\tau,\\zeta)=(\\pi\/2 + \\zeta^+\/2, -\\pi\/2 + \\zeta^+\/2)$ for $-21 < \\log |\\zeta^+|  < -11$ for positive and negative $\\zeta^+$.\nLocally, the kink can be described in terms of the changes to the unit tangent\nvector $e_t$, the velocity vector $e_v$ and\n$e_\\perp = e_v \\times e_t\/|e_v \\times e_t|$ which form the\nperpendicular coordinate system $\\{e_t, e_v, e_\\perp\\}$.\nWe find $e_v$ and $e_t$ lie in the y-z plane and\n$e_\\perp$ along the x-direction. Letting $\\Delta e = e_{+} - e_{-}$ stand for\nthe change in time of each unit vector,\n\\begin{eqnarray}\n  \\Delta e_\\perp & = & \\{2,0,0\\} \\\\\n  \\Delta e_t & \\simeq & \\{0,-1.85,0\\} \\\\\n  \\delta e_v & \\simeq & \\{0,-0.77,0\\} .\n\\end{eqnarray}\nThe kink is a y-reflection of the\nvelocity and tangent vectors in the y-z plane.\n\nOn each side of the kink we fit each component of $F^\\alpha$\nwith forms that include combinations of constant, linear\nand ln terms in $|\\zeta^+|$. We select the linear\nor ln fit whichever is best; it turns out that this corresponds to\nthe term with coefficients that are of order unity.\nWe report the inferred scaling in Table \\ref{tab:GVkinkasymptotics}.\n\\begin{table}[H]\n  \\begin{center}\n\\begin{tabular}{|c|cccc|}\n  \\hline\n  sign & $F^t$ & $F^x$ & $F^y$ & $F^z$ \\\\\n$\\zeta^+ < 0$ & $|\\zeta^+|^{-0.33}$ & $1$ & $|\\zeta^+|^{-0.33}$ & $|\\zeta^+|^{-0.33}$ \\\\\n  $\\zeta^+ > 0$ & $1$ & $1$ & $1$ & $1$ \\\\\n    \\hline\n\\end{tabular}\n  \\end{center}\n  \\caption{Asymptotic form for the force near the kink; $1$ means non-zero\n    constant.}\n\\label{tab:GVkinkasymptotics}\n  \\end{table}\nThe results are similar to but not identical to the ACO\ncase. First, note that there is one redundancy $F^t=-F^z$\nso we have 3 GV force components to compare to ACO.\nThe GV coordinate directions of the force\nare not the same as the normal and longitudinal\ndirections in the ACO case and this complicates a\ndirect one-for-one comparison. Nonetheless, we see analogous\nbehavior. Most prominently the GV divergence for $\\zeta^+<0$\nof $F^t$ and $F^y$ scales close to $\\propto (|\\zeta^+|)^{-1\/3}$\nlike ACO's $F^t$ and $F^N$. One difference is that the GV force for all\ncomponents with $\\zeta^+>0$ approach non-zero constant values.\nThe ACO loop has no curvature\non one side of the kink, which is probably responsible for\nthe fact that two of its components approach zero. Curiously,\nthe ACO divergence for $F^L \\propto \\ln | \\zeta^+ |$\non both sides of the kink is\nabsent for any components in the GV case. Likewise, the completely\nfinite GV result for $F^x$ on both sides of the kink\nis absent in the ACO case. Despite these differences\nthe most important observation is that the GV divergent self-force\n$\\propto  (|\\zeta^+|)^{-1\/3}$ integrates to a finite value\nso we expect the\nphysical displacement of the string to be finite.\n\n\\subsection{Kibble self-intersecting strings}\nThe ACO and GV string possess a pair of traveling kinks that circulate\naround the loop throughout the period of oscillation while the KT\nstring forms two transient cusps each period.  In the tangent sphere\nrepresentation the kink discontinuities are jumps in ${\\mathbf a}'$\nand\/or ${\\mathbf b}'$ while the cusps form whenever ${\\mathbf a}'$ and\n${\\mathbf b}'$ cross. The nature of the self-intersections of string\nloops is not immediately apparent from the tangent sphere\nrepresentation. In the case of the KT string with $\\alpha=0$ and\n$\\phi=\\pi\/6$ the string collapses to a line and the overlap is a\nspacelike length of string.  Unless nature prefers special loop\nconfigurations the generic type of self-intersection will be weaker\nthan in the above KT case. Here we investigate the Kibble string loop\n\\cite{Kibble:1976sj} which is simpler than any of the previous cases\nin these respects: it has no discontinuities or crossings on the\ntangent sphere, i.e. the loop is smooth and continuous everywhere,\n{\\it and} it self-intersects at a spacetime point not along a\nspacelike line.\n\nWe integrate the tangent vectors \\cite{Garfinkle:1987yw} to give\nexplicit forms for the right and left modes:\n\\begin{align}\n  a^\\mu(\\zeta^+) & = \\left[\\zeta^+,\n    f_1(\\zeta^+), f_2(\\zeta^+), f_3(\\zeta^+) \\right] \\\\\n  b^\\mu(-\\zeta^-) & = \\left[\\zeta^-,\n    -f_1(\\zeta^-), -f_3(\\zeta^-), -f_2(\\zeta^-) \\right]\n\\end{align}\nwhere\n\\begin{align}\n  f_1(x) & =  \\frac{L}{2\\pi} \\left(\n  \\frac{\n    (1+p^2)^2 \\sin 2 y +\n    (p^2\/4) \\sin 4 y}{2 + 5 p^2 + 2 p^4}\n  \\right)\n       \\\\\n  f_2(x) & = \\frac{L}{2 \\pi} \\cos 2 y \\times \\nonumber \\\\\n       & \\quad \\left(\n  \\frac{ -2 + 4 p^2 + 2 p^4 + p^2 \\cos 2 y }\n       {4 + 10 p^2 + 4 p^4}\n       \\right)   \\\\\n  f_3(x) & = \\frac{L}{2\\pi} 2^{3\/2} p \\cos y \\times \\nonumber \\\\\n  & \\quad \\left(\n  \\frac{ 5 + 3 p^2 + 2 \\cos 2 y }\n       {6 + 15 p^2 + 6 p^4}\n       \\right)\n\\\\\n       y & = \\frac{2 \\pi x}{L}\n\\end{align}\nwhere $p$ is a constant. We choose for the numerical example $p=1\/2$.\nThis is a more complicated loop in terms of harmonic\ncontent than either the GV or KT loops.\nFig. \\ref{fig:Kibbleloopspacetime} shows 6 equally spaced\nsnapshots of the loop during the fundamental oscillation period. The\ndashed and dotted lines show the times when a self-intersection\noccurs at the center (red dot). Figure \\ref{fig:Kibblelooptangentplot}\ngives the tangent sphere representation which resembles the seams\nof a baseball.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Kibble-solid.pdf}\n\\caption{\\label{fig:Kibbleloopspacetime} The Kibble string loop\n  configuration for $p=1\/2$ in spacetime at six equally\n  spaced moments $\\tau = j \\pi\/6$ for $j=0$ to $5$ (invariant\n  length $2 \\pi$) in the basic loop oscillation cycle.\n  The blue dashed and dotted loops self-intersect at\n  the central red dot. The solid blue lines are non-intersecting configurations.}\n~\\\\\n\\includegraphics[width=0.9\\linewidth]{Kibble-tangentsphere.pdf}\n\\caption{\\label{fig:Kibblelooptangentplot} The arcs on the tangent\n  sphere traced by ${\\mathbf a}'(x)$ and $-{\\mathbf b}'(x)$ for the\n  Kibble string resemble the seams on a baseball. The green and red\n  lines are smooth and continuous and do not intersect each\n  other. They satisfy an integral condition such that the loop has\n  zero total momentum. }\n\\end{figure}\n\n\n\n\\begin{widetext}\n~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/K}\n\\caption{\\label{fig:Force-components-K}\nContributions to $F_1^\\mu$ for the Kibble string when computed using the [1D] integration method with integration with respect to $\\zeta$. Each sub-figure shows the relevant contribution to the force at all points on the string in the region $\\tau \\in (0,L\/2)$, $\\zeta \\in (-L\/2,L\/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from: (i) the field point; and (ii) the integral over $\\zeta$ (ignoring distributional contributions at the field point). For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).\n}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/K-total}\n\\caption{\\label{fig:Force-K}\nThe two pieces of the self-force, $F_1^\\mu$ (row 1) and $F_2^\\mu$ (row 2), for the Kibble string as a function of position on the string. The $F_1^\\mu$ part can be obtained by summing the two rows in Fig.~\\ref{fig:Force-components-K}. For the purposes of the plots, we have set the string tension, $\\mu$, and Newton's constant, $G$ equal to one; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).\n}\n\\end{figure}\n\\end{widetext}\n\nThis loop has collisions at worldsheet coordinates\n$\\{\\tau,\\zeta\\}=\\{0,\\pm\\pi\/2\\}$ and $\\{\\tau,\\zeta\\}=\\{\\pi\/2,0\\}$\nand $\\{\\pi\/2,\\pi\\}$.\nWe describe the limiting behavior near $\\{\\tau,\\zeta\\}=\\{0,\\pm\\pi\/2\\}$.\nThe velocities of the two\nbits of string are equal and opposite: ${\\dot z}^i = \\pm \\{\n0, -0.26, -0.26\\}$. The tangent vectors are\n$dz^i\/d\\zeta = \\{-0.85, \\pm 0.26, \\mp 0.26 \\}$ (an angle of\n$\\sim 0.94$ rad). The acceleration vectors are\n${\\ddot z}^i = \\{0,-0.41, 0.41\\}$. The gravitational radiation\nemitted by each piece of string should be similar.\n\nThe net effect of the crossing is small. The bumps at the collision\npoints on the full scale worldsheet representations in Fig.\n\\ref{fig:Force-K}\nare difficult to distinguish\nat all. Here we look in more detail near those crossings.\n\nComponent $F^t$ is displayed in a small two-dimensional patch about\nthe crossing point in the top left plot of Fig. \\ref{fig:Cross1-Kibble}. As $\\tau \\to 0$\nat fixed $\\zeta=\\pi\/2$ (the vertical line of small dots in\nthe picture) $F^t$ diverges $\\propto \\tau^{-1}$ with change\nof sign as $\\tau$ passes through zero. The results at $\\pm \\tau$ are nearly\nequal and opposite. We find that the sum of the two components at $\\pm \\tau$ is\nnearly constant as $|\\tau| \\to 0$, numerically approximately $\\propto\n|\\tau|^{0.05}$. As $\\zeta$ varies near $\\pi\/2$ (fixed\n$\\tau=0$, the horizontal line of small dots)\nthe results on each side of the crossing point are finite and the zero\nvalue is not exactly at $\\delta \\zeta=0$. These results are quite\nsensitive to the size of $\\delta \\tau$ since the surface changes\nsign (from plus to minus infinity) near $\\delta \\tau=0$.\n\\begin{widetext}\n\t~\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\linewidth]{figures\/KibbleCrossingCorrected1.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/KibbleCrossingCorrected2.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/KibbleCrossingCorrected3.pdf}\n\\includegraphics[width=0.45\\linewidth]{figures\/KibbleCrossingCorrected4.pdf}\n\\caption{\\label{fig:Cross1-Kibble} The four components ($t$, $x$, $y$, and $z$) of\n  $\\ln |{\\cal F}^t_{\\rm conf}|$ for a small patch of the worldsheet about the crossing\n  point.\n  Four quadrants in $\\{\\delta \\tau, \\delta \\zeta\\}$ are displayed in\n  log absolute value coordinates (oriented to match the usual linear\n  system about $\\{\\tau,\\zeta\\}=\\{0,\\pi\/2\\}$).\n  Blue (orange) are negative (positive) values. The small dots have\n  been added for $\\delta \\tau=0$ and $\\delta \\zeta=0$.\n}\n\\end{figure}\n\\end{widetext}\n\nWe have\nformally fit the power law variation for $\\delta \\tau$ near $\\tau=0$\nand for $\\delta \\zeta$ near $\\zeta = \\pi\/2$.\nTable \\ref{tab:Kibblescaling} summarizes the slopes extracted for\n$F^\\alpha$ along fixed $\\tau$ and fixed $\\zeta$ coordinates passing\nexactly through the crossing point. Some components vary such that an\nintegral over just one side would yield a divergent quantity, however,\nthe symmetric sum is always integrable.\n\\begin{table}[H]\n  \\begin{center}\n\\begin{tabular}{|c|cc|cc|}\n  \\hline\n  Component & \\multicolumn{2}{c|}{$\\zeta$ varies; $\\tau=0$} & \\multicolumn{2}{c|}{$\\tau$ varies; $\\zeta=\\pi\/2$} \\\\\n            & One side & Net & One side & Net \\\\\n  \\hline\n  $F^t$ & - & - & $-1.0$ & $0.04$ \\\\\n  $F^x$ & $-1.1$ & $-0.04$ & $-0.04$ & $-0.03$ \\\\\n  $F^y$ & $-1.1$ & $-0.06$ & $-1.0$ & $-0.05$ \\\\\n  $F^z$ & $-1.1$ & $-0.05$ & $-0.96$ & $-0.04$ \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n  \\caption{Kibble loop divergent behavior at the crossing point\n    $\\tau=0$ and $\\zeta=\\pi\/2$. The columns labeled $\\tau=0$ give $\\nu$\n    for the scaling of the force component along the string near the\n    crossing point $\\propto |\\zeta-\\pi\/2|^\\nu$. Likewise,\n    the ones labeled $\\zeta=\\pi\/2$ describe the scaling\n    $\\propto |\\tau|^\\nu$ for times before and after the appearance of\n    the crossing. ``One side'' means the scaling of the absolute value\n    (approximately the same on each side);\n    ``Net'' means the scaling of the symmetric sum of points on opposite sides\n    of the crossing point. Small $\\nu$ results are numerically close to\n    finite limits but in any case are integrable.\n    The ``-'' indicates values that are not\n    well-defined because of a zero-crossing at $\\tau=0$.}\n\\label{tab:Kibblescaling}\n\\end{table}\n\nThe plot of $F^t$ shows it to be\napproximately a product of individual functions of\n$\\tau$ and $\\zeta$. The other force components are more complicated.\nComponents $F^x$, $F^y$ and $F^z$ are shown in small two-dimensional\npatches in the other plots of Fig.~\\ref{fig:Cross1-Kibble}. The small dots show the\nvariation along the coordinate axes.\n\nIn summary, we find the effect of the string crossing leads to integrable forces\nfor all components in this example.\n\n\\subsection{Comparisons to radiated quantities evaluated in the far field}%\n\nAs an additional consistency check on our results, we compute the radiated energy and\ncompare it to the energy dissipated through the local self-force. The latter can be computed using the change in the 4-momentum of the string,\n\\begin{eqnarray}\n  \\Delta P^\\mu & = & \\mu \\int {\\cal F}^{\\mu}_{\\rm conf} d\\zeta d\\tau,\n\\end{eqnarray}\nwhere the conformal-gauge force ${\\cal F}^{\\mu}_{\\rm conf}$ is given by Eq.~\\eqref{eq:trad-self-force}\nand where the region of integration is given by the fundamental period of the worldsheet: $-L\/2 \\le\n\\zeta < L\/2$ and $0 \\le \\tau < L\/2$. In practice, we evaluate the integrand at\n$N \\sim 10^4$ equally spaced\npoints on a two dimensional surface and approximate the integral\nas the sum of the function values at the points\ntimes the worldsheet area per point. This is a low accuracy method that is\nsuited to the occurrence of steep spikes at various points on the worldsheet; we estimate the accuracy of the results to be within 1-5\\%.\n\nThe work done on the string by the self-force lowers its energy, $\\Delta P^0 < 0$, and should be\nexactly balanced by the flux carried to infinity, which must be $-\\Delta P^0 > 0$. We separately\ncompute this flux to infinity using the formalism of Allen and Ottewill\n\\cite{2001PhRvD..63f3507A}, in\nwhich the stress energy tensor is a sum of individual Fourier components of the undamped string.\nFor each overtone $n$ we numerically integrate $dP^{(n)}\/d\\Omega$ over the sphere. We compute $N$\novertones and then fit and sum a power law extrapolation for $N \\to \\infty$. This yields a result\nwhich is approximately 1\\% accurate.\n\n\nTable \\ref{tab:energy} compares the results of the two calculations. We find good numerical\nagreement within the expected accuracy of the result.\n\\begin{table}[H]\n  \\begin{center}\n\\begin{tabular}{|c|cc|cc|}\n  \\hline\n   Case & far field & far field & direct & direct \\\\\n    & (numerical) & (analytic) & (numerical) & (analytic) \\\\\n  \\hline\n  ACO & 122.537 & 122.53 & 125.515 & 122.53 \\\\\n  \\hline\n  KT & 349.677 & & 355.643 & \\\\\n  $\\alpha=0, \\phi=\\pi\/6$& & & & \\\\\n  \\hline\n  KT & 241.321 & & 238.259  & \\\\\n  $\\alpha=1\/2, \\phi=0$& & &  & \\\\\n  \\hline\n  GV & 131.304 & & 132.486 & \\\\\n  \\hline\n  Kibble & 137.6 & & 135.428 & \\\\\n  \\hline\n\\end{tabular}\n\\end{center}\n\\caption{The total energy loss integrated over one fundamental period of the loop oscillation in\n  the center of mass frame of the loop. The far field is calculated with the formalism of Allen and Ottewill \\cite{2001PhRvD..63f3507A}.\n  The analytic result for the ACO loop in the far field is from\nRef.~\\cite{Allen:1994bs}. The numerical results for the\ndirect energy loss integrate ${\\cal F}^{\\mu}_{\\rm conf}$ over the world sheet according to the\ndescription in this section. The analytic results for the direct energy loss for the ACO loop is\nfrom Ref.~\\cite{Anderson:2005qu}}\n\\label{tab:energy}\n\\end{table}\n\n\\section{Discussion}\nWe have developed a general method for calculating the self-force due to gravitational\nperturbations of a lightly damped string loop. Our approach breaks up the calculation into smooth\nintegrals over the retarded image of the loop plus boundary terms. The latter are used to take\naccount of the special contributions when the source and field point coincide and when\ndiscontinuities are visible on the past image of the loop. These may be from kinks or cusps or crossings (spacetime points where\nintercommutation events might occur). Our methodology is quite general and can be used for arbitrary choices of\nspacetime and worldsheet gauges.\n\nThere are some existing calculations of the gravitational self-force for cosmic strings\n\\cite{Quashnock:1990wv,Anderson:2005qu,Wachter:2016rwc}, however these results have all relied on\nsimplifications or approximations that do not hold in general. For example, although Quashnock and\nSpergel \\cite{Quashnock:1990wv} used a numerical approach not too different from ours, they do not\ndiscuss any of the various distributional-type contributions (near kinks or the field point; they\ndo, however, discuss transitions between integration variables) that we have studied in detail\nhere. Our results\\footnote{For example see the third column in Table \\ref{tab:comparison} for the\nGV case, but we also performed the same check for the other configurations discussed in this\npaper.} suggest that their use of a pair of null coordinates sidesteps the issue of a contribution\nfrom the field point. The issue of contributions from kinks, however, remains unaddressed.\nAdditionally, given the limited computational resources available at the time, their numerical\ncalculations were restricted to a low-resolution study in a restricted set of cases. In the case of\nRefs.~\\cite{Anderson:2005qu,Wachter:2016rwc}, approximations based on simple string configurations\nwere made which, while reasonable in some cases, do not fully capture the behavior for generic\nstring configurations.\n\nOur numerical calculations have passed a number of validation checks including: comparisons with\nexisting analytical results; comparisons of the integrated power radiated over a fundamental period\nagainst the flux of gravitational energy measured at large distances; and cross-comparisons of\nseveral semi-independent methods for computing the self-force. From the perspective of\ncomputational efficiency it is clear that the [1D] methods based on either integration with respect\nto $\\zeta$ or a Quashnock-Spergel type mixed integration with respect to $\\zeta_+$ and $\\zeta_-$\nare the best choice. The other [1D] methods (using a single null coordinate or over-retardation)\ninevitably encounter large numerical cancellations nearby the field point, making them significantly\nmore computationally demanding. The [2D] method is even worse, and is orders of magnitude more\ndemanding than any of the [1D] methods.\n\nWhile the preferred [1D] methods work well in general, there are certain cases where they also run\ninto numerical challenges. Since the self-force diverges as one approaches kinks and cusps (in a\nway such that the displacement of the worldsheeet is finite) it is unavoidable\nthat one would encounter numerically divergent quantities at one point or another. In this work, we\nhandled the issue of divergences in a brute force manner by simply evaluating quantities to a\nsufficiently high accuracy that they can be canceled to leave a residual which is still accurately\ndetermined. While this approach works reasonably well, the calculation could be made significantly\nmore efficient by developing an alternative approach to the problem. One promising possibility is\nto borrow from results in the point particle case\n\\cite{Vega:Detweiler:2008,Barack:Golbourn:2007,Wardell:2015ada}, where it was found that the\nseparation of the full metric perturbation into a so-called ``puncture field'' that captures the\nsingular behavior plus a ``residual field'' that is more numerically well-behaved. In the point\nparticle case, by basing the puncture field on an approximation to the singular field proposed by\nDetweiler and Whiting \\cite{Detweiler-Whiting-2003}, one can work directly with the residual field\nas it is entirely responsible for driving the motion. In the case of a cosmic string we do not yet\nhave an analogous Detweiler-Whiting type singular field. One could attempt to derive one following\nthe matched expansion methods of Ref.~\\cite{Pound:2009}. Alternatively, even without such a\nderivation a local analysis of the type done in Sec.~\\ref{sec:local-expansion} may yield an\napproximation to the singular behavior of the metric perturbation which leaves a numerically\nwell-behaved residual field, and which is sufficiently simple that its integrated contribution to\nthe motion can be determined analytically. Indeed, a preliminary analysis for the ACO string (where\nthe self-force is known analytically) suggests that exactly this approach will work well, and has\nbeen found to significantly improve the accuracy with which the integrated motion can be\ndetermined, even in the presence of a divergent self-force at the kinks.\n\nThe ultimate goal of our program is to evolve cosmic strings under the influence of the\nself-force, and to study the consequences of backreaction on cusp formation, smoothing of kinks,\nand other astrophysically relevant features of cosmic strings. This paper represents the first step\nin such an endeavor. We can now compute the self-force for an arbitrary cosmic string with a\nreasonable level of accuracy and with the freedom to arbitrarily choose coordinates and gauges which\nare most suitable for evolution. The next step is to implement this into a numerical evolution\nscheme. This will be presented in a future work.\n\n\\section*{Acknowledgments}\nConcurrent to our\nown work Blanco-Pillado, Olum and Wachter did related\nwork on cosmic string back-reaction; that paper and this one were\nsubmitted at the same time. As far as we know, the results are in\nagreement where they overlap.\nWe thank J.J. Blanco-Pillado, David Nichols, Ken Olum,\nAdrian Ottewill, Joe Polchinski, Leo Stein, Peter Taylor, Henry Tye, Jeremy Wachter\nand Yang Zhang for helpful conversations.\nB.W. and D.C. gratefully acknowledge support from the John Templeton\nFoundation New Frontiers Program under Grant No.~37426 (University of\nChicago) - FP050136-B (Cornell University). D.C. acknowledges that this\nmaterial is based upon work supported by the National Science\nFoundation under Grant No. 1417132.  EF acknowledges the support of\nthe National Science Foundation under Grant Nos. 1404105 and 1707800.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Supplementary material}\n\nWe have created three different random subsets of MegaFace for each of the distractor sizes (10,100,1K,10K,100K,1M) and reran all the algorithms with each of the set for each of the probe sets (FaceScrub and FGNET). We present the results in the below figures. Set \\#1 is the one presented in the main paper. \n\n\\begin{figure*}\n\t\\begin{tabular}{ccc}\n\t\t\\includegraphics[width=.17\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_rank-1_cmbnd_set_1.pdf} &\n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_rank-1_cmbnd_set_1.pdf} \\\\\n\t\t\\includegraphics[width=.17\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_rank-1_cmbnd_set_2.pdf} &\n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_rank-1_cmbnd_set_2.pdf} \\\\\n\t\t\\includegraphics[width=.17\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_rank-1_cmbnd_set_3.pdf} &\n\t\t\\includegraphics[width=.31\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_rank-1_cmbnd_set_3.pdf} \\\\\n\t\t& {\\footnotesize (a) FaceScrub + MegaFace} & {\\footnotesize (b)  FGNET + MegaFace}\\\\\n\t\t\\vspace{.05in}\n\t\\end{tabular}\n\t\\caption{Sets 1--3 (each row represents a different random gallery set). The MegaFace challenge evaluates  identification and verification as a function of  increasing  number of gallery distractors (going from 10 to 1 Million).  We use two different probe sets (a) FaceScrub--photos of celebrities, (b) FGNET--photos with a large variation in age per person. We present rank-1 identification of  state of the art algorithms that participated in our challenge. On the left side of  each plot is current major benchmark LFW scale (i.e., 10 distractors, see how all the top algorithms are clustered above 95\\%).  On the right is mega-scale (with a million distractors).  Observe that  rates drop with increasing numbers of distractors, even though the probe set is fixed, and that algorithms trained on larger sets (dashed lines) generally perform better.  }\n\t\\label{fig:rank1}\n\\end{figure*}\n\t\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_verif_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_10000_distractors_cmbnd_verif_set_1.pdf} \\\\\n\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K}  \\\\\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_verif_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_10000_distractors_cmbnd_verif_set_1.pdf} \\\\\n\t\t\t&{\\footnotesize  (c) FGNET + 1M} & {\\footnotesize (d) FGNET + 10K}  \\\\\n\t\t\\end{tabular}\n\t\t\\caption{\\textbf{Verification (random gallery set 1)} performance with (a,c) 1 Million and (b,d) 10K distractors on both probe sets. Note the performance at low false accept rates (left side of each plot). }\\label{fig:verf}\n\t\t\\label{fig:dataset_size_roc} \n\t\\end{figure*}\n\t\t\n\t\t\t\\begin{figure*}\n\t\t\t\t\\centering\n\t\t\t\t\\begin{tabular}{ccc}\n\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_1000000_distractors_cmbnd_verif_set_2.pdf} & \n\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_10000_distractors_cmbnd_verif_set_2.pdf} \\\\\n\t\t\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K}  \\\\\n\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_1000000_distractors_cmbnd_verif_set_2.pdf} & \n\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_10000_distractors_cmbnd_verif_set_2.pdf} \\\\\n\t\t\t\t\t&{\\footnotesize  (c) FGNET + 1M} & {\\footnotesize (d) FGNET + 10K}  \\\\\n\t\t\t\t\\end{tabular}\n\t\t\t\t\\caption{\\textbf{Verification  (random gallery set 2)} performance with (a,c) 1 Million and (b,d) 10K distractors on both probe sets. Note the performance at low false accept rates (left side of each plot). }\\label{fig:verf}\n\t\t\t\t\\label{fig:dataset_size_roc} \n\t\t\t\\end{figure*}\n\t\t\t\n\t\t\t\n\t\t\t\t\\begin{figure*}\n\t\t\t\t\t\\centering\n\t\t\t\t\t\\begin{tabular}{ccc}\n\t\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_1000000_distractors_cmbnd_verif_set_3.pdf} & \n\t\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_10000_distractors_cmbnd_verif_set_3.pdf} \\\\\n\t\t\t\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K}  \\\\\n\t\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_1000000_distractors_cmbnd_verif_set_3.pdf} & \n\t\t\t\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_10000_distractors_cmbnd_verif_set_3.pdf} \\\\\n\t\t\t\t\t\t&{\\footnotesize  (c) FGNET + 1M} & {\\footnotesize (d) FGNET + 10K}  \\\\\n\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t\\caption{\\textbf{Verification  (random gallery set 3)} performance with (a,c) 1 Million and (b,d) 10K distractors on both probe sets. Note the performance at low false accept rates (left side of each plot). }\\label{fig:verf}\n\t\t\t\t\t\\label{fig:dataset_size_roc} \n\t\t\t\t\\end{figure*}\n\t\t\t\t\n\t\t\t\t\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_ident_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_10000_distractors_cmbnd_ident_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_rank-10_cmbnd_set_1.pdf}\\\\  \n\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K} &    {\\footnotesize (c) FaceScrub + rank-10} \\\\\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_ident_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_10000_distractors_cmbnd_ident_set_1.pdf} &\n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_rank-10_cmbnd_set_1.pdf}\\\\\n\t\t\t&\t\t{\\footnotesize (d) FGNET + 1M} & {\\footnotesize (e) FGNET + 10K}   & {\\footnotesize (f) FGNET + rank-10} \n\t\t\\end{tabular}\n\t\t\n\t\t\\caption{\\textbf{Identification  (random gallery set 1)} performance for all methods with (a,d) 1M distractors and (b,e) 10K distractors, and (c,f) rank-10 for both probe sets.  Fig.~\\ref{fig:teaser} also shows  rank-1 performance as a function of number of distractors on both probe sets. }\n\t\t\\label{fig:dataset_size_cmc}\n\t\\end{figure*}\n\n\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_1000000_distractors_cmbnd_ident_set_2.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_10000_distractors_cmbnd_ident_set_2.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/facescrub_rank-10_cmbnd_set_2.pdf}\\\\  \n\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K} &    {\\footnotesize (c) FaceScrub + rank-10} \\\\\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_1000000_distractors_cmbnd_ident_set_2.pdf} & \n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_10000_distractors_cmbnd_ident_set_2.pdf} &\n\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_2\/fgnet_rank-10_cmbnd_set_2.pdf}\\\\\n\t\t\t&\t\t{\\footnotesize (d) FGNET + 1M} & {\\footnotesize (e) FGNET + 10K}   & {\\footnotesize (f) FGNET + rank-10} \n\t\t\\end{tabular}\n\t\t\n\t\t\\caption{\\textbf{Identification  (random gallery set 2)} performance for all methods with (a,d) 1M distractors and (b,e) 10K distractors, and (c,f) rank-10 for both probe sets.  Fig.~\\ref{fig:teaser} also shows  rank-1 performance as a function of number of distractors on both probe sets. }\n\t\t\\label{fig:dataset_size_cmc}\n\t\\end{figure*}\n\t\n\t\t\\begin{figure*}\n\t\t\t\\centering\n\t\t\t\\begin{tabular}{cccc}\n\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_1000000_distractors_cmbnd_ident_set_3.pdf} & \n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_10000_distractors_cmbnd_ident_set_3.pdf} & \n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/facescrub_rank-10_cmbnd_set_3.pdf}\\\\  \n\t\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K} &    {\\footnotesize (c) FaceScrub + rank-10} \\\\\n\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_1000000_distractors_cmbnd_ident_set_3.pdf} & \n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_10000_distractors_cmbnd_ident_set_3.pdf} &\n\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_3\/fgnet_rank-10_cmbnd_set_3.pdf}\\\\\n\t\t\t\t&\t\t{\\footnotesize (d) FGNET + 1M} & {\\footnotesize (e) FGNET + 10K}   & {\\footnotesize (f) FGNET + rank-10} \n\t\t\t\\end{tabular}\n\t\t\t\n\t\t\t\\caption{\\textbf{Identification  (random gallery set 3)} performance for all methods with (a,d) 1M distractors and (b,e) 10K distractors, and (c,f) rank-10 for both probe sets.  Fig.~\\ref{fig:teaser} also shows  rank-1 performance as a function of number of distractors on both probe sets. }\n\t\t\t\\label{fig:dataset_size_cmc}\n\t\t\\end{figure*}\n\n\n\n\\section{Introduction}\n\n\nFace recognition has seen major breakthroughs in the last couple of years, with new results by multiple groups   \\cite{schroff2015facenet,taigman2014deepface,sun2015deepid3} surpassing human performance on the leading \nLabeled Faces in the Wild (LFW) benchmark \\cite{huang2007labeled} and achieving near perfect results.  \n\nIs face recognition solved?  \nMany applications require accurate identification at {\\em planetary scale}, i.e., finding the best matching face in a database of billions of people.  This is truly like finding a needle in a haystack.  Face recognition algorithms did not deliver when the police were searching for the suspect of the Boston marathon bombing~\\cite{klontz2013case}.  Similarly, do you believe that current cell-phone face unlocking programs will protect you against anyone on the planet who might find your lost phone?  These and other face recognition applications require finding the true positive match(es) with negligible false positives.  They also require training and testing on datasets that contain vast numbers of different people.\n\nIn this paper, we introduce the {\\em MegaFace} dataset and benchmark to evaluate and encourage development of face recognition algorithms at scale.   The goal of MegaFace is to evaluate the performance of current face recognition algorithms with up to a million {\\em distractors}, i.e., up to a million people who are not in the test set.  Our key objectives for assembling the  dataset are that  1) it should contain a million photos \\textbf{``in the wild''}, i.e., with unconstrained pose, expression, lighting, and exposure, 2) be broad rather than deep, i.e., \\textbf{contain many different people} rather than many photos of a small number of people, and most importantly 3) it will be \\textbf{publicly available}, to enable benchmarking and distribution within the research community.  \n\n\n\n\n\n\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{plots16\/datasets}\n\t\\end{center}\n\t\\caption{Representative sample of recent face recognition datasets (in addition to LFW). Current public datasets include up to 10K unique people, and a total of 500K photos.  Several companies have access to  orders of magnitude more  photos and  subjects, these however are subject to privacy constraints and are not public. MegaFace (this paper) includes 1M photos of more than 690K unique subjects, collected from Flickr (from creative commons photos), and is available publicly.}\n\t\\label{fig:datasets}\n\\end{figure*}\n\nWhile recent face datasets have leveraged celebrity photos crawled from the web, such datasets have been limited to a few thousand unique individuals;  it is challenging to find a million or more unique celebrities.\nInstead, we leverage \nYahoo's  recently released database of Flickr  photos  \\cite{thomee2015new}. The Yahoo dataset includes 100M \\textbf{creative commons} photographs and hence can be released for research. \nWhile these photos are unconstrained and do not target face recognition research per se, they capture a large number of faces.    Our algorithm  samples the Flickr set searching for faces while  optimizing for large number of unique people via analysis of Flickr user IDs and group photos.  MegaFace  includes 1 Million photos of more than 690,000 unique subjects.  \n\nThe MegaFace challenge evaluates how face recognition algorithms perform with a very large number of ``distractors,'' i.e., individuals that are not in the probe set.\nMegaFace is used as the gallery; the two probe sets we use are FaceScrub \\cite{ng265data} and FG-NET \\cite{cootes2008fg,kemelmacher2014illumination}.  \t\nWe address fundamental questions and introduce the following key findings (Fig.~\\ref{fig:teaser}):\n\\begin{itemize}\n\n\\item{\\bf How well do current face recognition algorithms scale?}  \nAlgorithms that achieve above 95\\% performance on LFW (equivalent of 10 distractors in our plots), achieve 35-75\\% identification rates with 1M distractors. Baselines (Joint Bayes and LBP) while achieving reasonable results on LFW drop to less than 10\\%. \n\n\n\\item \\vspace{-0.1in} {\\bf Is the size of training data important?}  \nWe observe that  algorithms that were trained on larger sets (top two are FaceNet that was trained on more than 500M photos of 10M people, and FaceN that was trained on 18M of 200K people) tend to perform better at scale.  Interestingly, however, FaceN  (trained on 18M) compares favorably to FaceNet (trained on 500M) on the FaceScrub set. \n\n\n\\item \\vspace{-0.1in} {\\bf How does age affect recognition performance?}  We found that the performance with 10 distractors for FGNET as a probe set is lower than for FaceScrub, and the drop off spread is much bigger (Fig.~\\ref{fig:teaser} (b)) . A deeper analysis also reveals that children (below age 20) are more challenging to recognize than adults, possibly due to training data availability, and that larger gaps in age (between gallery and probe) are similarly more challenging to recognize. These observations become evident by analyzing at large scale. \n\n\n\\item \\vspace{-0.1in} {\\bf How does pose affect recognition performance?} Recognition drops for larger variation in pose between matching probe and gallery, and the effect is much more significant at scale. \n\n\n\n\n\\end{itemize}\n\n\n\nIn the following sections we describe how the MegaFace database was created, explain the challenge, and describe the outcomes.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\\subsection{Benchmarks}\n\nEarly work in face recognition focused on controlled datasets where subsets of lighting, pose, or facial expression were kept fixed, e.g.,  \\cite{georghiades1997yale, gross2010multi}. With the advance of algorithms, the focus moved to unconstrained scenarios with a number of important benchmarks appearing,  e.g., FRGC, Caltech Faces, and many more (see \\cite{huang2007labeled}, Fig. 3, for a list of all the datasets), and thorough evaluations \\cite{grother2010report,zhao2003face}.  A big challenge, however,  was to collect photos of large number of individuals. \n\nLarge scale evaluations were previously performed  on \\textit{controlled} datasets (visa photographs, mugshots, lab captured photos) by NIST \\cite{grother2010report}, and report recognition results of 90\\% on 1.6 million people.  However, these results are not representative of photos in the wild.\n\nIn 2007, Huang et al. \\cite{huang2007labeled} created the benchmark Labeled Faces in the Wild (LFW). The LFW database includes 13K photos of 5K different people. It was collected by running Viola-Jones face detection \\cite{viola2004robust} on  Yahoo News photos. LFW captures  celebrities photographed under  unconstrained conditions (arbitrary lighting,  pose, and expression) and it has been an amazing resource for the face analysis community (more than 1K citations). Since 2007, a number of databases appeared that include larger numbers of photos per person (LFW has 1620 people with more than 2 photos), video information, and even 3D information, e.g.,  \\cite{kumar2009attribute, beveridge2013challenge, yi2014learning, wolf2011face, chen2012bayesian,  ng265data}.   However, LFW remains the leading benchmark on which all state of the art recognition methods are evaluated and compared.  Indeed, just in the last year a number of methods (11 methods at the time of writing this paper), e.g.,  \\cite{schroff2015facenet,sun2014deeply,sun2015deepid3,taigman2014deepface,taigman2014web} reported recognition rates above 99\\%+ \\cite{hu2015face} (better than human recognition rates estimated on the same dataset by \\cite{kumar2011describable}). The perfect recognition rate on LFW is 99.9\\% (it is not 100\\% since there are 5 pairs of photos that are mislabeled), and current top performer reports 99.77\\%. \n\n\n\n\n\\subsection{Datasets}\n\nWhile, some companies have access to massive photo collections, e.g., Google in \\cite{schroff2015facenet}\ntrained on 200 Million photos of 8 Million people (and more recently on 500M of 10M), these datasets are not available to the public and were used only for training and not testing. \n\nThe largest public data set is\nCASIA-WebFace  \\cite{yi2014learning} that includes 500K photos of 10K celebrities, crawled from the web.  While CASIA is a great resource, it contains only 10K individuals, and does not have an associated benchmark (i.e., it's used for training not testing). \n\nOrtiz et al. \\cite{ortiz2014face} experimented with large scale identification from Facebook photos assuming there is more than one gallery photo per person. Similarly Stone et al. \\cite{stone2008autotagging} show that social network's context improves large scale face recognition. Parkhi et al. \\cite{parkhi2015deep} assembled a  dataset of 2.6 Million  of  2600 people, and used it for training (testing was done on the smaller scale LFW and YouTube Faces~\\cite{wolf2011face}).  Wang et al. \\cite{wang2015face} propose a hierarchical  approach on top of commercial recognizer to enable fast  search in a dataset of 80 million faces.   Unfortunately, however, none of these efforts have produced publicly available datasets or public benchmarks.\nNote also that \\cite{parkhi2015deep} and \\cite{wang2015face} are contemporaneous, as their arxiv papers appeared after ours \\cite{miller2015megaface}.\n\n\n\n\\subsection{Related Studies}\n\nAge-invariant recognition is an important problem that has been studied in the literature, e.g., \\cite{chen14cross,li2011discriminative}.  \nFG-NET \\cite{cootes2008fg} includes 975 photos of 82 people, each with several photos spanning many ages.  More recently, \nChen et al. \\cite{chen14cross} created a dataset of 160k photos of 2k celebrities across many ages.  However, most modern face recognition algorithms have not been evaluated for age-invariance.  We attempt to rectify this by including an FG-NET test (augmented with a million distractors) in our benchmark.\n\nOther recent studies have considered both identification as well as verification results on LFW \\cite{best2014unconstrained, taigman2014web, sun2014deeply, sun2015deepid3}. \nFinally, \nBest-Rowden et al. \\cite{best2014unconstrained}  performed an interesting Mechanical Turk study to evaluate human recognition rates on LFW and YouTube Faces datasets. They  report  that humans are better than computers when recognizing from videos due to additional cues, e.g., temporal information, familiarity with the subject (celebrity), workers' country of origin (USA vs. others), and also discovered errors in labeling of YouTube Faces via crowdsourcing. In the future, we will use this study's useful conclusions  to help annotate  MegaFace and create a training set in addition to the currently provided distractor set. \n\n\n\n\n\n\\section{Assembling MegaFace}\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{plots16\/megaface_stats}\n\t\\end{center}\n\t\\caption{MegaFace statistics. We present  randomly selected photographs (with provided detections in red), along with  distributions of  Flickr tags,  GPS locations, and camera types. We also show the pose distribution (yaw and roll),  number of faces per photograph, and number of faces for different resolutions (compared to LFW in which faces are approximately 100x100).}\n\t\\label{fig:data_stats} \n\\end{figure*}\n\nIn this section, we provide an overview of the MegaFace dataset, how it was assembled, and its statistics.\nWe created MegaFace to evaluate and drive the development of face recognition algorithms that work at scale.\nAs motivated in Section 1, we sought to create a public dataset, free of licensing restrictions, that captures photos taken with unconstrained imaging conditions, and with close to a million unique identities.\nAfter exploring a number of avenues for data collection, we decided to leverage Yahoo's 100M Flickr set \\cite{thomee2015new}. Yahoo's set was not created with face analysis in mind, however, it includes a very large number of faces and satisfies our requirements.\n\n\n\n\n\\textbf{Optimizing for large number of unique identities.} \nOur strategy for maximizing the number of unique identities is based on two techniques:  1) drawing photos from many different Flickr users---there are 500K unique user IDs---and 2) assuming that two or more faces appear in the same photo, they are likely different identities.  Note that these assumptions do not need to be infallible, as our goal is to produce a very diverse distractor set--it is not a problem if we have a small number of photos of the same person.\nOur algorithm for detecting and downloading faces is as follows. We generated a list of images and user IDs in a round-robin fashion,\nby going through each of the 500K users and selecting the first photo with a face larger than $50\\times 50$ and adding it to the dataset.  If the photo contains multiple faces above that resolution, we add them all, given that they are different people with high probability.  We then repeated this process (choosing the second, then the third, etc. photo from each user), until a sufficient number of faces were assembled.\nBased on our experiments face detection can have up to 20\\% false positive rate.  Therefore, to  ensure that our final set includes a million faces, the  process was terminated once  $1,296,079$ faces were downloaded.  Once face detection was done, we ran additional stricter detection, and removed blurry faces.  We assembled a total of $690,572$ faces in this manner that have a high probability of being unique individuals.\nWhile not guaranteed, the remaining $310$K in our dataset likely also contain additional unique identities. Figure~\\ref{fig:data_stats} presents a histogram of number of faces per photo. \n\n\n\n\n\\textbf{Face processing.}  We downloaded the highest resolution available per photo. The faces are detected using the HeadHunter\\footnote{{\\scriptsize \\url{http:\/\/markusmathias.bitbucket.org\/2014_eccv_face_detection\/}}} algorithm by Mathias et al. \\cite{Mathias2014Eccv}, which reported state of the art results in face detection, and is especially robust to a wide range of head poses including profiles.   We crop detected faces such that the face spans 50\\% of the photo height, thus including the full head (Fig.~\\ref{fig:data_stats}). We further estimate  49 fiducial points and yaw and pitch angles, as computed by the IntraFace\\footnote{{\\scriptsize \\url{http:\/\/www.humansensing.cs.cmu.edu\/intraface\/}}} landmark model \\cite{xiong2013supervised}.\n\n\\textbf{Dataset statistics.}\nFigure~\\ref{fig:data_stats} presents MegaFace's statistics: \n\\begin{itemize}\n\t\\item \\vspace{-.1in} Representative photographs and bounding boxes. Observe that the photographs contain people from different countries, gender, variety of poses, glasses\/no glasses, and many more variations. \n\t\\item \\vspace{-.1in}  Distribution of Flickr tags that accompanied the downloaded photos.   Tags range from 'instagram' to 'wedding,' suggesting a range of photos from selfies to high quality portraits (prominence of '2013' likely due to timing of when the Flickr dataset was released).\n\t\\item \\vspace{-.1in}  GPS locations demonstrate photos taken all over the world.\n\t\\item \\vspace{-.1in}  Camera types dominated by DSLRs (over mobile phones), perhaps correlated with creative commons publishers, as well as our preference for higher resolution faces.\n\t\\item \\vspace{-.1in}  3D pose information:  more than 197K of the faces have yaw angles \\textit{larger} than $\\pm 40$ degrees. Typically unconstrained face datasets include yaw angles of \\textit{less} than $\\pm 30$ degrees.\n\t\\item \\vspace{-.1in}  Number of faces per photo, to indicate the number of group photos. \n\t\\item \\vspace{-.1in}  Face resolution: more than 50\\% (514K) of the  photos in MegaFace have resolution more than 40 pixels  inter-ocular distance (40 IOD corresponds to 100x100 face size, the resolution in LFW).\n\\end{itemize}\n\\vspace{-.1in}   We believe that this dataset is extremely useful for a variety of research areas in recognition and face modeling, and we plan to maintain and expand it in the future. In the next section, we describe the MegaFace challenge. \n\n\n\n\n\n\n\\section{The MegaFace Challenge}\n\n\nIn this section, we describe the challenge and evaluation protocols. \nOur goal is to test performance of face recognition algorithms with up to a million distractors, i.e., faces of unknown people.\nIn each test, a {\\em probe} image is compared against a {\\em gallery} of up to a million faces drawn from the Megaface dataset.\n\n\n\\textbf{Recognition scenarios}  The first scenario is identification:  given a probe photo, and a gallery containing at least one photo of the same person, the algorithm rank-orders all photos in the gallery based on similarity to the probe. \nSpecifically, the probe set includes $N$ people; for each person we have $M$ photos. We then  test each of the $M$ photos (denote by $i$) per person by adding it the gallery of distractors and use each of the other $M-1$ photos as a probe. Results are presented with Cumulative Match Characteristics (CMC) curves-- the probability that a correct gallery image will be chosen for a random probe by rank $=K$.\n\n\n\nThe second scenario is verification, i.e., a pair of photos is given and the algorithm should output whether the person in the two photos is the same or not.  To evaluate verification we computed all pairs between the probe dataset and the Megaface distractor dataset.\nOur verification experiment has in total 4 billion negative pairs. We report verification results with ROC curves; this explores the trade off between falsely accepting non-match pairs and falsely rejecting match pairs. \n\nUntil now, verification received most of  the focus in  face recognition research since it was tested by the LFW benchmark \\cite{huang2007labeled}. Recently, a number of groups, e.g., \\cite{best2014unconstrained, taigman2014web, sun2014deeply, sun2015deepid3} also performed identification experiments on LFW. The relation between the identification and verification protocols was studied  by Grother and Phillips \\cite{grother2004models} and DeCann and Ross \\cite{decann2012can}.  In our challenge, we evaluate both scenarios with an emphasis on very large number of  distractors. For comparison, testing identification on LFW is equivalent  to 10 distractors in our challenge. \n\n\n\\textbf{Probe set.}  MegaFace is used to create a gallery with a large number of distractors.  For the probe set (testing known identities), we use two sets: \n\\begin{enumerate}\n\t\\item  The FaceScrub dataset \\cite{ng265data}, which includes 100K photos of 530 celebrities, is available online.\n\n\tFaceScrub has a similar number of male and female photos (55,742 photos of 265 males and 52,076 photos of 265 females) and a large variation across photos of the same individual which reduces possible bias, e.g.,  due to backgrounds and hair style \\cite{kumar2011describable}, that may occur in LFW.  For efficiency, the evaluation was done on a subset of FaceScrub which includes 80 identities (40 females and 40 males) by randomly selecting from a set of people that had more than 50 images each (from which 50 random photos per person were used). \n\t\\item The FG-NET aging dataset \\cite{cootes2008fg,kemelmacher2014illumination}: it includes 975 photos of 82 people. For some of the people the age range in photos is more than 40 years. \n\\end{enumerate}\n\n\n\n\\textbf{Evaluation and Baselines.} Challenge participants were asked to calculate their features on MegaFace, full FaceScrub, and FGNET.  We  provided code that runs identification and verification on the FaceScrub set. After the results were submitted by all groups we re-ran the experiments with FaceScrub and 3 different random distractor sets per gallery size. We further ran the FGNET experiments on all methods\\footnote{Google's FaceNet was ran by the authors since their features could not be uploaded due to licensing conditions} and each of the three random MegaFace subsets per gallery size. The metric for comparison is  $L_2$ distance. Participants were asked not to train on FaceScrub or FGNET.  As a baseline,  we implemented two simple recognition algorithms: 1) comparison by LBP \\cite{ahonen2006face} features--it achieves 70\\% recognition rates on LFW, and uses no training, 2) a Joint Bayesian (JB) approach represents each face as the sum of two Gaussian variables\n$x = \\mu + \\epsilon$ where $\\mu$ is identity and $\\epsilon$ is inter-personal variation.  To determine whether two faces, $x_1$ and $x_2$ belong to the same identity, we calculate $P(x_1, x_2 | H_1)$ and $P(x_1, x_2 | H_2)$ where $H_1$ is the hypothesis that the two faces are the same and $H_2$ is the hypothesis that the two faces are different. These distributions can also be written as normal distributions, which allows for efficient inference via a log-likelihood test. JB  algorithm was trained on the CASIA-WebFace dataset \\cite{yi2014learning}.\n\n\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\includegraphics[width=.8\\linewidth]{plots16\/trainingsize}\n\t\t\\caption{Number of training photos and unique people used by each participating method.  }\\label{fig:trainingsize}\n\t\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\nAn ultimate face recognition algorithm should perform with billions of people in a dataset. While testing with billions is still challenging, we have done the first step and created a benchmark of a million faces. MegaFace is available to researchers and we presented results from state of the art methods. Our key discoveries are 1) algorithms' performance degrades given a large gallery even though the probe set stays fixed,  2) testing at scale allows to uncover the differences across algorithms (which at smaller scale appear to perform similarly), 3) age differences across probe and gallery are still more challenging for recognition.   We will keep maintaining and updating the MegaFace benchmark online, as well as, create more challenges in the future.  Below are topics we think are exciting to explore.  First, we plan to release all the detected faces from the 100M Flickr dataset. Second,  companies like Google and Facebook have a  head start due to availability of enormous amounts of data. We are interested to level the playing field and provide large \\textbf{training} data to the research community that will be assembled from our Flickr data. Our dataset will be separated to testing and training sets for fair evaluation and training.   Finally, the significant number of high resolution faces in our Flickr database \nwill also allow  to explore resolution in more depth. Currently, it  is mostly untouched topic in face recognition literature due to lack of data. \n\n\\vspace{.1in}\n\n\\textbf{Acknowledgments} We are grateful to the early challenge participants that allowed a very quick turn out and provided great feedback on the baseline code and data.  We thank  Samsung and Google for the generous support of this research. \n \n\\section{Human Performance}\nWhile a lot of effort goes into developing automated face recognition algorithms, still the human visual system seems to perform better especially at very low false accept rate. \nIt is therefore very interesting to estimate the human recognition rate on the same sets of photos on which algorithms operate. \nVerification  rates of humans   were previously estimated  on controlled data, i.e., photos taken in laboratory condition \\cite{o2007face, o2012comparing, phillips2010frvt,adler2007comparing}, and more recently on unconstrained photo collections: \\cite{kumar2011describable} evaluated verification rates on the LFW dataset, and \\cite{best2014unconstrained, chen2012dictionary} estimated verification rates on videos. Human studies made on unconstrained photos, e.g., \\cite{kumar2011describable}, fused human judgments by averaging ratings over participants, which helped  remove outliers. \nUntil recently, none of the algorithms was able to outperform humans on LFW. Moreover, all the previous human experiments were done on small scale and did not evaluate identification. It is of great interest, however, to discover how humans perform at \\textit{scale} to provide a lower bound for machine performance. \n\n\n \n\nOne of the key contributions of this paper is an extremely large (about 4 million pairs of faces) human study on Mechanical Turk that evaluates human performance on unconstrained photos (Flickr), and specifically targets identification rather than verification. \n\nWe have performed the following experiment on Mechanical Turk.  Since all the identities in the FaceScrub dataset (our test set) are celebrities, human recognition rates may be biased due to familiarity with the person \\cite{sinha2006face}. We therefore sorted all the names in FaceScrub according to the number of results that Google image search returns per person as a measure of popularity. We then chose 50 most popular people, and 50 least popular people as our human experiment test set. Each person had 100 photos, we randomly selected one photo as the probe image, and used the rest 99 as gallery images. We then produced 99 positive pairs per person.   For the distractor set, for each input photo we randomly selected 10K photos from our MegaFace dataset, and produced 10K pairs of probe with each of the distractors. This results in total of $100\\times (99+10K)$ pairs. Since the number of positive pairs in this setting is very low, we introduced additional positive pairs by randomly pairing gallery images that are not the probe. This is to remove possible bias in human rating, i.e., if most pairs are negative people may miss the positive ones. We presented to turkers 10 pairs per page and asked to click on all the pairs that contained the same person. We paid 1 cent per page of 10 pairs.\n\nOnce this experiment was done we collected the pairs that received 1 click or more, and created a sorting experiment.\nWe selected only the pairs that include the probe photos, and created a set of possible matches per probe.\nWe generated triples of probe, and two matches, presented 10 triples in each page and asked which one of the matches is the person in the probe. Generally, to get a full ranking of all images, the number of possible triples per probe is $n^2$ where $n$ is the number of matches from round 1. For efficiency (and less cost) only determined the position of each gallery photo relative to the distractor images. That is, our experiment determined the number of distractors that would be ranked above and below each gallery image, but not the ordering within those groups. On every pair\/triple of photos in both experiments worked 3 different people. We paid 7 cents for each page of 10 triples. The total cost of this experiment was \\$10,000.\n\n\n\n\\begin{figure}\n\t\\begin{center}\n   \\caption*{Identification}\n  \\begin{tabular}{ c | c | c }\n    \\hline\n    & Rank-1 & Rank-10 \\\\ \\hline\n    All & 23.9 & 91.13 \\\\ \\hline\n    Males & 23.35 & 89.98 \\\\ \\hline\n    Females & 24.01 & 92.5 \\\\ \\hline\n    Less Popular & 22.7  & 90.9 \\\\ \\hline\n    More Popular & 25.1 &  91.3 \\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\n  \\begin{center}\n  \\caption*{Verification}\n  \\begin{tabular}{ c | c | c }\n    \\hline\n    & TAR @ $2 \\times 10^{-3}$ & TAR @ $5 \\times 10^{-2}$ \\\\ \\hline\n    All & 41.6 & 76.5 \\\\ \\hline\n    Males & 43.7 &  79.0 \\\\ \\hline\n    Females & 39.4 & 73.9  \\\\ \\hline\n    Less Popular & 39.4 & 74.7 \\\\ \\hline\n    More Popular & 43.6 & 78.2 \\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\n\t\\caption{Human recognition rates (verification and identification).\n  Our experiments also show that humans perform  better on more popular people and are  better at the verification task when comparing males.}\n\t\\label{fig:human_id}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation Methodology}\n\n\n\\section{Recognition Methods}\n\nWe have selected to experiment with four recognition algorithms that represent very different types of techniques. \n\n\\paragraph{Basic LBP comparison.} We have implemented a  comparison based on Local Binary Pattern (LBP) descriptors.   \\red{give details about what were the sizes, cell size, how many, chi square distance, ...}.  This approach achieves \\red{xxx} recognition rates on LFW, and uses no training. \n\n\n\\paragraph{Joint Bayesian.}   The Joint Bayesian model represents each face as the sum of two Gaussian variables\n$$x = \\mu + \\epsilon$$ where $\\mu$ represents identity and $\\epsilon$ represents inter-personal variation.  To determine whether two faces, $x_1$ and $x_2$ belong to the same identity, we calculate $P(x_1, x_2 | H_1)$ and $P(x_1, x_2 | H_2)$ where $H_1$ is the hypothesis that the two faces are the same and $H_2$ is the hypothesis that the two faces are different. These distributions can also be written as normal distributions, which allows for efficent inference via a log-likelihood test. \\red{add detail, trained on CASIA... achieves on LFW...}\n\n\n\n\n\n\n\n\\paragraph{Commercial software by VisionLabs.} \nVisionLabs achieved \\red{xxx} recognition rates on LFW, is trained on outside data \\red{???}. \n\n\n\\paragraph{Googel's FaceNet} Google FaceNet \\cite{schroff2015facenet} is the most recent and highest performing of several Deep Learning algorithms applied to the LFW benchmark. \nUnlike DeepFace and DeepID the FaceNet which have a bottleneck layer and are optimized by minimizing cross-entropy, FaceNet learns an embedding such that the extracted features are directly comparable using the Euclidean distance.\n It is  trained on 290 Million photos of 1.8 Million people. \n \n\\section{Results}\n\t\n\t\n\t\\begin{figure*}\n\t\t\\centering\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_verif_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_10000_distractors_cmbnd_verif_set_1.pdf} \\\\\n\t\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K}  \\\\\n\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_verif_set_1.pdf} & \n\t\t\t\\includegraphics[width=0.27\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_10000_distractors_cmbnd_verif_set_1.pdf} \\\\\n\t\t\t&{\\footnotesize  (c) FGNET + 1M} & {\\footnotesize (d) FGNET + 10K}  \\\\\n\t\t\\end{tabular}\n\t\t\\caption{\\textbf{Verification} performance with (a,c) 1 Million and (b,d) 10K distractors on both probe sets. Note the performance at low false accept rates (left side of each plot). }\\label{fig:verf}\n\t\t\\label{fig:dataset_size_roc} \n\t\\end{figure*}\n\t\t\nThis section describes the results and analysis  of the challenge. Our challenge was released on Sep 30, 2015.  Groups were given three weeks to finish their evaluations.  More than 100 groups registered to participate. We present results from 5 groups  that  uploaded all their features by the deadline. \tWe keep maintaining the challenge and data--currently 20 more groups  are working on their submissions. \n\t\n\t\\textbf{Participating algorithms} In addition to baseline algorithms LBP, and Joint Bayes, we  present results of the following methods (some provided more than 1 model):\n\t\\begin{enumerate}\n\t\t\\item \\vspace{-.1in}Google's FaceNet:  achieves 99.6\\% on LFW, was trained on more than 500M photos of 10M people (newer version of \\cite{schroff2015facenet}).\t\n\t\t\\item \\vspace{-.1in}FaceAll (Beijing University of Post and Telecommunication), was trained on 838K photos of 17K people, and provided two types of features. \n\t\t\\item\\vspace{-.1in}  NTechLAB.com (FaceN algorithm): provided two models (small and large)--small was trained on 494K photos of 10K people, large on more than 18M of 200K. \n\t\t\\item \\vspace{-.1in} BareBonesFR (University group): was trained on 365K photos of 5K people. \n\t\t\\item \\vspace{-.1in} 3DiVi.com: was trained on 240K photos of 5K people.   \n\t\\end{enumerate}\n\t\\vspace{-.1in}Figure~\\ref{fig:trainingsize} summarizes the models, training sizes (240K-500M photos, 5K-10M people) and availability of the training data. Below we describe all the experiments and key conclusions. \n\t\n\n\t\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\n\t\\begin{figure}\n\t\t\\includegraphics[width=1\\linewidth]{plots16\/rank1numbers.pdf}\n\t\t\\caption{Rank-1 identification results (in\\%) with 1M distractors on the two probe sets.}\n\t\t\\label{fig:rank1numbers}\n\t\\end{figure}\n\t\t\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\t\\begin{figure*}\n\t\t\t\t\t\t\\centering\n\t\t\t\t\t\t\\begin{tabular}{cccc}\n\t\t\t\t\t\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_ident_set_1.pdf} & \n\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_10000_distractors_cmbnd_ident_set_1.pdf} & \n\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/facescrub_rank-10_cmbnd_set_1.pdf}\\\\  \n\t\t&{\\footnotesize  (a) FaceScrub + 1M} & {\\footnotesize (b) FaceScrub + 10K} &    {\\footnotesize (c) FaceScrub + rank-10} \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\\includegraphics[width=.15\\linewidth, valign=t]{plots16\/ll.pdf} & \n\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_ident_set_1.pdf} & \n\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_10000_distractors_cmbnd_ident_set_1.pdf} &\n\t\t\t\t\t\t\t\t\t\\includegraphics[width=0.25\\linewidth, valign=t]{plots16\/graphs\/set_1\/fgnet_rank-10_cmbnd_set_1.pdf}\\\\\n\t\t\t\t\t\t\t&\t\t{\\footnotesize (d) FGNET + 1M} & {\\footnotesize (e) FGNET + 10K}   & {\\footnotesize (f) FGNET + rank-10} \n\t\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\caption{\\textbf{Identification} performance for all methods with (a,d) 1M distractors and (b,e) 10K distractors, and (c,f) rank-10 for both probe sets.  Fig.~\\ref{fig:teaser} also shows  rank-1 performance as a function of number of distractors on both probe sets. }\n\t\t\t\t\t\t\\label{fig:dataset_size_cmc}\n\t\t\t\t\t\\end{figure*}\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\\begin{figure*}\n\t\t\t\t\t\\centering\n\t\t\t\t\t\\begin{tabular}{ccccc}\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_barebones_1000_rank1.pdf}\t &\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_3divi_1000_rank1.pdf} &\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_faceall_norm_1000_rank1.pdf} &\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_ntech_1000_rank1.pdf} & \n\t\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_facenet_1000_rank1.pdf}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_barebones_1000000_rank1.pdf} &\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_3divi_1000000_rank1.pdf} &\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_faceall_norm_1000000_rank1.pdf} &\n\t\t\t\t\t\t\\includegraphics[width=0.18\\linewidth]{plots16\/age_ntech_1000000_rank1.pdf} &\n                        \\includegraphics[width=0.18\\linewidth]{plots16\/age_facenet_1000000_rank1.pdf}\n\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t\n\t\t\t\t\t\\caption{Analysis of rank-1  identification with respect to varying \\textbf{ages} of gallery and probe.  Columns represent  five algorithms, rows 1K and 1M distractors.   X-axis represents a person's age in the gallery photo and Y-axis age in the probe. The colors represent identification accuracy going from 0(=blue)--none of the true pairs  were matched to 1(=red)--all possible combinations of probe and gallery were matched per probe and gallery ages.  Lower scores on left and bottom indicate worse performance on children, and higher scores along the diagonal indicate that methods are better at matching across small age differences.}\n\t\t\t\t\t\\label{fig:age}\n\t\t\t\t\\end{figure*}\n\t\t\t\t\n\t\t\t\n\t\t\t\t\\begin{figure}\n\t\t\t\t\t\\begin{center}\n\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_3divi_1000_rank1.pdf}\n\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_faceall_norm_1000_rank1.pdf}\t\t\t\t\t\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_facenet_1000_rank1.pdf}\\\\\n\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_3divi_1000000_rank1.pdf}\n\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_faceall_norm_1000000_rank1.pdf}\n\t\t\t\t\t\t\\includegraphics[width=0.32\\linewidth]{plots16\/pose2_facenet_1000000_rank1.pdf}\n\t\t\t\t\t\\end{center}\n\t\t\t\t\t\\caption{Analysis of rank-1  identification with varying \\textbf{poses} of gallery and probe, for three algorithms.   Top:  1K distractors, Bottom:  1M distractors.\n\t\t\t\t\tThe colors represent identification accuracy going from 0 (blue) to 1 (red), where 0 means that none of the true pairs  were matched, and 1 means that all possible combinations of probe and gallery were matched per probe and gallery ages. White color indicates combinations of poses that did not exist in our test set.  We can see that evaluation at scale (bottom) reveals large differences in performance, which is not visible at smaller scale (top): frontal poses and smaller difference in poses is easier for identification.  }\n\t\t\t\t\t\\label{fig:pose}\n\t\t\t\t\\end{figure}\n\t\t\t\t\n\n\n\t\t\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\\textbf{Verification results.} Fig.~\\ref{fig:verf} shows results of the verification experiment for our two probe sets, (a) and (b) show results on FaceScrub and (c) and (d) on FGNET.  We present results of one random fixed set of distractors per gallery size (see the other two in the supplementary). \n\t\t\t\t\n\t\t\t\tWe see that, for FaceScrub, at lower false accept rates the performance of algorithms drops by about 40\\% on average. FaceNet and FaceN lead with  only about 15\\%. Interestingly, FaceN that was trained on 18M photos is able to achieve comparable results to FaceNet that was trained on 500M. Striving to perform well at low false accept rate  is important with large datasets. Even though the chance of a false accept on the small benchmark is acceptable, it does not scale to even moderately sized galleries. Results at LFW are typically reported at equal error rate which implies false accept rate of 1\\%-5\\% for top algorithms, while for a large set like MegaFace, only FAR of $10^-5$ or $10^-6$ is meaningful. \n\t\t\t\t\t\n\t\t\t\t\tFor FGNET the drop in performance is striking--about 60\\% for everyone but FaceNet, the latter achieving impressive performance across the board. One factor may be the type of training used by different groups (celebrities vs. photos across ages, etc.).\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\tVerification rate stays similar when scaling up the gallery, e.g., compare (a) and (b).  The intuition is that verification rate is normalized by the size of the dataset, so that if  a probe face is matched incorrectly to 100 other faces in a 1000 faces dataset,  assuming uniform distribution of the data, the rate will stay the same, and so in a dataset of a million faces one can expect to find 10,000 matches at the same false accept rate (FAR).  \n\t\t\t\t\n\t\t\\textbf{Identification results.}   In Fig.~\\ref{fig:dataset_size_cmc} we show the performance with respect to different ranks, i.e., rank-1 means that the correct match got the best score from the whole database, rank-10 that the correct match is in the first 10 matches, etc.  (a,b,c) show performance for the FaceScrub dataset and (d,e,f) for FGNET. We observe that rates drop for all algorithms as the  gallery size gets larger. This is visualized in Fig.~\\ref{fig:teaser}, the actual accuracies are in Fig.~\\ref{fig:rank1numbers}.  The curves also suggest that when evaluated on more than 1M distractors (e.g., 100M), rates will be even lower.    Testing on  FGNET \\textbf{at scale} reveals a dramatic performance gap. All algorithms perform much worse, except for FaceNet that has a similar performance to its results on FaceScrub. \n\t\t\n\t\n\t\t\n\t\t\\textbf{Training set size.} Dashed lines in all plots represent algorithms that were trained on data larger than 500K photos and 20K people. We can see that these generally perform better than others. \n\t\t\n\t\t\t\n\t\n\t\t\n\t\t\n\t\t\t\n\t\t\\begin{figure}\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t\\includegraphics[width=1\\linewidth]{plots16\/facenet_fgnet_truepos_10dist} \\\\\n\t\t\t{\\small \t(a) true positives}\\\\\n\t\t\t\t\\includegraphics[width=1\\linewidth]{plots16\/facenet_fgnet_falseneg_10dist}\\\\\n\t\t\t{\\small  (b) false negatives}\n\t\t\t\\end{tabular}\n\t\t\t\\caption{Examples pairs from FGNET using top performing FaceNet with 10 distractors. \n\t\t\tEach consecutive left-right pair of images is the same person.\n\t\t\tAll algorithms match better with smaller age differences.  }\n\t\t\t\\label{fig:agematches}\n\t\t\\end{figure}\n\n\n\n\n\t\n\t\\textbf{Age.} Evaluating performance using FGNET as a probe set also reveals a major drop in performance for most algorithms when attempting to match across differences in age.  We present a number of results: Fig.~\\ref{fig:age} shows differences in performances with varying age across gallery and probe.  Each column represents a different algorithm, rows present results for 1K and 1M distractors.  Red colors indicate higher identification rate, blue lower rate.  We make two key observations: 1)  algorithms perform better when the difference in age between gallery and probe is small (along the diagonal), and 2)  adults are more accurately matched than children, at scale. \n\tFig.~\\ref{fig:agematches}  shows examples of  matched pairs (true positives and false negatives) using FaceNet and 10 distractors.  Notice that false negatives have a bigger age gap relative to true positives. It is impressive, however, that the algorithm was able to these and many other true positives, given the variety in lighting, pose, and quality of the photo in addition to age changes. \n\n\t\t\t\n\t\t\\textbf{Pose.}  Fig.~\\ref{fig:pose} evaluates error in recognition as a function of  difference in yaw between the probe and gallery. The results are normalized by the total number of pairs for each pose difference.  We can see that recognition accuracy depends strongly on pose and this difference is revealed more prominently when evaluated at scale.  Top row show results of three different algorithms (representative of others) with 1K distractors. Red colors indicate that identification is very high and mostly independent of pose. However, once evaluated at scale (bottom row) with 1M distractors we can see that variation across algorithms as well as poses is more dramatic. Specifically, similar poses identified better, and more frontal (center of the circle) poses are easier to  recognize. \n\n\n\t\n\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\n\t\n\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\n\n\n\t\t\n\t\t\t\n\n\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\n\t\n\n\n\t\n\t\n\n\n\n\n\t\n\n\n\n\n\t\n\t\n\n\t\n\n\t\n\t\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{supplementary}\n\n\t\\begin{figure*}\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_ident_set_1.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/facescrub_1000000_distractors_cmbnd_verif_set_1.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/facescrub_rank-1_cmbnd_set_1.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_ident_set_1.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/fgnet_1000000_distractors_cmbnd_verif_set_1.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_1\/fgnet_rank-1_cmbnd_set_1.pdf}\n\t\t\t\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/facescrub_1000000_distractors_cmbnd_ident_set_2.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/facescrub_1000000_distractors_cmbnd_verif_set_2.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/facescrub_rank-1_cmbnd_set_2.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/fgnet_1000000_distractors_cmbnd_ident_set_2.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/fgnet_1000000_distractors_cmbnd_verif_set_2.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_2\/fgnet_rank-1_cmbnd_set_2.pdf}\n\t\t\t\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/facescrub_1000000_distractors_cmbnd_ident_set_3.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/facescrub_1000000_distractors_cmbnd_verif_set_3.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/facescrub_rank-1_cmbnd_set_3.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/fgnet_1000000_distractors_cmbnd_ident_set_3.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/fgnet_1000000_distractors_cmbnd_verif_set_3.pdf}\n\t\t\t\\includegraphics[width=0.3\\linewidth]{plots16\/graphs\/set_3\/fgnet_rank-1_cmbnd_set_3.pdf}\n\t\t\t\n\t\t\t\\end{center}\n\t\t\t\\caption{Analysis of rank-1  identification with respect to varying ages of gallery and probe and gallery size. We analyze  two methods: (top row) 3divi (bottom row) FaceNet.   X-axis represents a person's age in the gallery photo and Y-axis age in the probe. The colors represent identification accuracy going from 0 (blue) to 1 (red), where 0 means that none of the true pairs  were matched, and 1 means that all possible combinations of probe and gallery were matched per probe and gallery ages. White color indicates combinations of ages that did not exist in our dataset.  }\n\t\t\t\\label{fig:age}\n\t\t\t\\end{figure*}\n\t\t\t\n\t\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nConsider a bandlimited signal (or field) quantization problem, where the\nsamples are affected by additive independent and identically distributed\n(i.i.d.) Gaussian noise.  For example, a spatial signal affected by\nadditive i.i.d.~Gaussian noise has to be sampled using an array of sensors.\nIn a distributed setup, where filtering before sampling is not possible,\nnoise in bandlimited signals can be reduced by statistical averaging of\nindependent noisy samples.  In addition, quantization error can be reduced\nby oversampling as well as by increasing the analog-to-digital converter\n(ADC) or quantizer precision.  The \\textit{fundamental tradeoff} between\noversampling, quantizer precision, and (statistical) average distortion is\nof interest. \n\nThe tradeoffs between all subsets of these three quantities has been\nstudied in the literature. Tradeoffs for average distortion and\noversampling, additive Gaussian noise and average distortion with\nunquantized samples, and oversampling and quantization have a flurry of\nwork (e.g.,\nsee~\\cite{mallatSA2009,pinskerO1980,grayo1987,zoranDLS2007,kumarIRH2011}).\nIn this work, the tradeoff between oversampling, quantizer precision, and\nthe average distortion is of interest.\n\nIf extremely high-precision ADCs are used, then the sample distortion is\nnoise limited. On the other hand, if lowest precision single-bit ADCs are\nused, then the sample distortion is limited by quantization. At a\nhigh-level, it is expected that the distortion optimal ADC precision should\nbe `in between' these two extreme cases, in the sense that it should be\nable to resolve the signal up to the noise level. Contrary to this\nintuition, in this work it is shown that a distortion inversely\nproportional to the oversampling above the Nyquist rate is\n\\textit{achievable} with single-bit quantizers.  With unquantized (infinite\nprecision) samples, the optimal distortion is \\textit{speculated} to be\ninversely proportional to the oversampling above the Nyquist rate in the\npresence of independent Gaussian noise.  Accordingly, the focus of this\nwork is on the quantization of an additive independent Gaussian noise\naffected bandlimited signal using single-bit ADCs and oversampling. The key\nresult of this paper is the uncovering of a quantization \\textit{precision\nindifference} principle, which is stated next.\n\n\\textit{Precision indifference principle:} Consider a bounded-dynamic range\nbandlimited signal with samples affected by additive independent Gaussian\nnoise and observed through quantizers. If $N$ is the oversampling ratio,\nwith respect to the Nyquist rate, then the optimal law for maximum\npointwise mean-squared error is $O(1\/N)$, irrespective of the quantizer\nprecision. In other words, for large $N$, the quantizer precision only\naffects the proportionality constant of the distortion.\n\n\\textit{Prior art:} Averaging and other properties of independent random\nvariables are well studied in statistics~\\cite{bickelDM2001}. Quantization\nerror can be reduced by oversampling as well as by increasing the ADC\n(quantizer)\nprecision~(see~\\cite{grayo1987,GrayNIT98,zoranDLS2007,kumarIRH2011} for the\nentire range of results).  Estimation of square-integrable signals in the\npresence of Gaussian noise was studied by Pinsker~\\cite{pinskerO1980};\nhowever, quantization is not addressed in his work.  Signal quantization\nwith additive noise as a dither has been studied by\nMasry~\\cite{masryT1981}, however signal was not assumed to be bandlimited.\nMasry's results give a decay of $O(1\/N^{2\/3})$ for bandlimited signals,\nwhere $N$ is the oversampling above the Nyquist rate; this decay is slower\nthan an $O(1\/N)$ decay that we are after.  The sampling of signals defined\non a finite support, while using single-bit quantizers in the presence of\nambient noise, has been also studied \\cite{wangID2009,masryIF2009}.\n\n\\textit{Notation:} The set of bounded signals and the set of finite energy\nsignals  will be denoted by ${\\cal L}^\\infty({\\mathbb R})$ and ${\\cal L}^2({\\mathbb R})$, respectively.\nThe signal of interest will be denoted by $g(t)$.  For a signal $s(t)$ in\n${\\cal L}^2({\\mathbb R})$, the Fourier transform will be denoted by $\\tilde{s}(\\omega)$.\nThe Fourier transform and its inverse are defined as,\n\\begin{align}\n\\tilde{s}(\\omega) = \\int_{{\\mathbb R}} s(t) \\exp(- j\\omega t)\\mbox{d}t; \\ \\ s(t) =\n\\frac{1}{2\\pi} \\int_{{\\mathbb R}} \\tilde{s}(\\omega) \\exp(j \\omega t)\n\\mbox{d}\\omega.  \\nonumber\n\\end{align}\nThe indicator function of a set $A$ is denoted by ${\\mathbbm{1}}(x \\in A)$.\nRandom variables or processes will be denoted by uppercase letters. The\nadditive independent Gaussian noise is denoted by $W(t)$. The set of reals\nand integers will be denoted by ${\\mathbb R}$ and ${\\mathbb Z}$, respectively. The\ncumulative distribution function (cdf) of a Gaussian random variable with\nmean zero and variance $\\sigma^2$ will be denoted by $F(x), x\\in {\\mathbb R}$.  The\nconvolution and expectation operation will be denoted by $\\star$ and ${\\mathbb E}$,\nrespectively. It is assumed that all probability models have an underlying\nsample space, sigma-field, and probability measure such that (weighted)\naverages and indicator functions are measurable.\n\n\\textit{Organization:} The mathematical formulation of our sampling problem\nis discussed in Sec.~\\ref{sec:formulation}. Short review of stable\ninterpolation kernels and smoothness properties of associated signals are\ndiscussed in Sec.~\\ref{sec:background}. The discussion on precision\nindifference principle appears in Sec.~\\ref{sec:estimation}. Estimation\nwith perfect samples and single-bit quantized samples are discussed in\nSec.~\\ref{sec:perfectsamples} and Sec.~\\ref{sec:onebit}, respectively.\nConclusions are presented in Sec.~\\ref{sec:conclusions}. To maintain the\nflow of the paper, long proofs appear in the Appendix.\n\n\\section{Problem formulation}\n\\label{sec:formulation}\n\nThe discussion begins with a quick review of a stable bandlimited kernel,\nwhich is essential for stable interpolation in ${\\cal L}^\\infty({\\mathbb R})$ and for\ndefining bandlimited signals.  For $\\lambda > 1$ and $a = (\\lambda - 1)\/2$,\nconsider the kernel $\\phi (t)$ that is given by\n\\begin{align}\n\\phi(t) =  \\frac{1}{\\pi a t^2} \\sin ((\\pi + a) t) \\sin (a t); \\ \\ \\phi(0) =\n1+ \\frac{a}{\\pi}. \\label{eq:phit}\n\\end{align}\nThe kernel decreases sufficiently fast (approximately as $1\/t^2$) and\ntherefore it is absolutely and square integrable. Its Fourier transform is\nillustrated in Fig.~\\ref{fig:stablesincsquare}.\n\\begin{figure}[!htb]\n\\begin{center} \\scalebox{1.0}{\\input{stablesincsquare.pstex_t}} \n\\end{center}\n\\caption{\\label{fig:stablesincsquare} \\sl \\small \\textbf{Stable\ninterpolation filter:} The kernel $\\phi(t) \\leftrightarrow\n\\tilde{\\phi}(\\omega)$ is defined in (\\ref{eq:phit}); this kernel is\nabsolutely integrable and will be used to define bounded bandlimited\nsignals.}\n\\end{figure}\nThis kernel can be used to define the set of bounded bandlimited signals,\nwhich is a subset of the Zakai class of bandlimited\nsignals~\\cite{zakaiB1965}.  Consider\n\\begin{align}\n{BL}_{\\mbox{\\footnotesize int}} := \\{g(t) : |g(t)| \\leq 1 \\mbox{ and } g(t) \\star \\phi(t) =\ng(t) \\ \\forall  t \\in {\\mathbb R}\\}. \\label{eq:blinterest}\n\\end{align}\nThe above definition ensures that $g(t)$ is continuous everywhere. It is\neasy to verify that the set of bounded bandlimited signals in ${\\cal L}^2({\\mathbb R})$\nwith Fourier spectrum zero outside $[-\\pi, \\pi]$ also belongs to the set\n${BL}_{\\mbox{\\footnotesize int}}$. The set ${BL}_{\\mbox{\\footnotesize int}}$ also includes (almost-surely) any\nsample path of a bounded-dynamic range bandlimited wide-sense stationary\nprocess~\\cite{cambanisMZ1976}. The quantization of bandlimited signals from\nthe set ${BL}_{\\mbox{\\footnotesize int}}$ in the presence of additive independent Gaussian\nnoise is studied in this work. \\textit{The derived results are applicable\nto finite energy bounded bandlimited signals as well as (almost surely) to\nany sample path of a bounded wide-sense stationary bandlimited process.}\n\nThe signal affected by additive noise, $g(t) + W(t)$, is available for\nsampling.  It is assumed that $W(t) \\sim {\\cal N}(0, \\sigma^2)$ for all\n$t\\in{\\mathbb R}$.  Independence of noise implies that $W(t_1), W(t_2), \\ldots,\nW(t_n)$ for distinct $t_1, t_2, \\ldots, t_n \\in {\\mathbb R}$ are i.i.d.~with ${\\cal\nN}(0, \\sigma^2)$ distribution. The Nyquist rate at which $g(t)$ should be\nsampled for perfect reconstruction is one sample\/second.  In the noise-free\nregime, when $\\sigma = 0$, it is sufficient to sample $g(t)$ at the Nyquist\nrate for convergence in ${\\cal L}^\\infty({\\mathbb R})$.  In the noise-limited regime, when\n$\\sigma > 0$, the reconstruction based on samples of $g(t)$ will have\ndistortion (statistical mean-squared error).  This distortion can be\nreduced by oversampling. Let $N$, a positive integer, be the oversampling\nrate.  For any statistical estimate $\\widehat{G}_{\\mbox{\\footnotesize rec}}(t)$ of the signal\n$g(t)$, the maximum pointwise mean-squared error $D_{\\mbox{\\footnotesize rec}}$ is defined as\nthe \\textit{distortion}, i.e.,\n\\begin{align}\nD_{\\mbox{\\footnotesize rec}} := \\sup_{t\\in{\\mathbb R}} D_{\\mbox{\\footnotesize rec}}(t) = \\sup_{t\\in{\\mathbb R}} {\\mathbb E}\\left|\n\\widehat{G}_{\\mbox{\\footnotesize rec}}(t) - g(t) \\right|^2. \\label{eq:distortion}\n\\end{align}\nFor a  pointwise-consistent reconstruction, the distortion in\n(\\ref{eq:distortion}) should decrease to zero as the oversampling rate $N$\nincreases to infinity~\\cite{bickelDM2001}.  Consistent reconstruction of\nsmooth signals with a random dither, in the presence of single-bit\nquantizers, has been obtained in the past~\\cite{masryT1981}; therefore, the\nasymptotic rate of decrease in $D_{\\mbox{\\footnotesize rec}}$ with $N$ is of interest to us.\nDue to finite precision limitations (ADC operation) during acquisition, the\nsignal samples are quantized.  Since quantization is a lossy\noperation~\\cite{gershogray}, $D_{\\mbox{\\footnotesize rec}}$ is expected to depend upon the ADC\nprecision employed. As mentioned in Sec.~\\ref{sec:introduction}, it will be\nshown that $D_{\\mbox{\\footnotesize rec}}$ decreases as $O(1\/N)$, irrespective of the sensor\nprecision. Thus, the ADC precision only manifests in the proportionality\nconstant (independent of the oversampling factor $N$) in the optimal\nasymptotic reconstruction distortion.\n\nTo show the proposed precision indifference principle, two extreme cases of\nquantizer precision will be analyzed and their distortion will be compared:\n(i) signal distortion with perfect samples; and (ii) signal distortion with\nsamples quantized using single-bit ADCs.  The sampling setup for these two\ncases are illustrated in Fig.~\\ref{fig:bitdestructionblocks}.  In\nFig.~\\ref{fig:bitdestructionblocks}(a), the estimator works with infinite\nprecision (unquantized) noisy samples while in\nFig.~\\ref{fig:bitdestructionblocks}(b), the estimator works with poorest\nprecision (one-bit) noisy samples. The role of extra dither noise $W_d(t)$\nwill be explained later in Sec.~\\ref{sec:onebit}.  The estimator\n$\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t)$ will be designed and its distortion performance\nwill be analyzed in this work.\n\\begin{figure}[!htb]\n\\begin{center} \\scalebox{1.0}{\\input{bitdestructionblocks.pstex_t}} \n\\end{center}\n\\caption{\\label{fig:bitdestructionblocks} \\sl \\small \\textbf{Two extreme\nscenarios of quantization:} In both the scenarios the signal $g(t)$ is\nobserved with additive independent Gaussian noise $W(t)$. In (a), the\nestimator works with infinite precision (unquantized) samples $\\{Y(n \\tau),\nn \\in {\\mathbb Z}\\}$. In (b), the estimator works with poorest precision (one-bit)\nsamples $\\{X(n) , n \\in {\\mathbb Z}\\}$ where $X(n \\tau) = {\\mathbbm{1}}(Y(n\\tau) \\geq\n0)$.}\n\\end{figure}\n\nBefore we move on to the next section, it should be noted that the kernel\n$\\phi(t)$ and its derivative $\\phi'(t)$ are absolutely integrable.  This\nabsolute integrability and square integrability of $\\phi(t)$ can be\ntranslated into the following observations, which will be useful in\nSec.~\\ref{sec:onebit} during distortion analysis:\n\\begin{align}\nC_{\\phi} & := \\int_{t \\in {\\mathbb R}} |\\phi(t)| \\mbox{d} t < \\infty,\n\\label{eq:integrablephi} \\\\\nC_{\\phi}' & := \\sup_{\\{t_k: t_k \\in \\left[k\/\\lambda, (k+1)\/\\lambda \\right],\nk \\in {\\mathbb Z}\\}} \\sum_{k\\in {\\mathbb Z}} |\\phi'(t_k)| < \\infty.\n\\label{eq:summablephiprime} \\\\\n\\mbox{and } C_{\\phi}'' & := \\sup_{t \\in {\\mathbb R}}\\sum_{k \\in {\\mathbb Z}} \\left| \\phi\n\\left(t - \\frac{k}{\\lambda}\\right) \\right|^2 < \\infty.\n\\label{eq:squaresummablephi}\n\\end{align}\nThe next section will review pertinent mathematical results which will be\nused in the later sections.\n\n\\section{Background}\n\\label{sec:background}\n\nThe stable interpolation formula for Zakai sense bandlimited signals is\ndiscussd first. The necessity of $W_d$ and associated variance-conditions\non Gaussian noise (see~Fig.~\\ref{fig:bitdestructionblocks}(b)) are\ndiscussed. If the pointwise error in  interpolation is bounded with bounded\nperturbation of samples, it is called as a \\textit{stable interpolation}.\nThe properties of stable interpolation and its implications on filtering of\nbounded signals are given at the end of this section.\n\n>From the interpolation formula for Zakai sense bandlimited signals, the\nsignal of interest $g(t)$ can be perfectly reconstructed from its samples\ntaken at the Nyquist rate. For $g(t) \\in {BL}_{\\mbox{\\footnotesize int}}$, the interpolation\nformula is given by~\\cite[Lemma~3.1]{kumarIRH2011},\n\\begin{align}\ng(t) = \\lambda \\sum_{n \\in {\\mathbb Z}} g \\left(\\frac{n}{\\lambda}\\right) \\phi\n\\left(t - \\frac{n}{\\lambda}\\right) . \\label{eq:sincinterp}\n\\end{align}\nwhere the equality holds absolutely, pointwise, and in ${\\cal L}^\\infty({\\mathbb R})$. Thus,\nin the absence of noise, it is sufficient to sample $g(t)$ at a rate of\n$\\lambda$ sample per second (or per meter in the context of spatial\nfields).  In the presence of quantization, the reconstruction in\n(\\ref{eq:sincinterp}) is stable in ${\\cal L}^\\infty({\\mathbb R})$. \n\nThe role of $W_d(t)$ in Fig.~\\ref{fig:bitdestructionblocks}(b) will now be\nhighlighted. If the noise variance $\\sigma$ is very small compared to the\ndynamic range of the signal $g(t)$, i.e., $|\\sigma| \\ll 1$, then the\nsamples ${\\mathbbm{1}}( g(t) + W(t) \\geq 0)$ will not capture small scale\nlocal variation in $g(t)$.  Due to quantization, estimators such as maximum\nlikelihood are expected to be non-linear and their analysis is too complex.\nTo alleviate this issue, if $\\mbox{var}(W(t)) = \\sigma^2$ is very small, then an\nextra additive independent Gaussian dither $W_d(t)$ can be added to ensure\nthat ${\\mathbbm{1}}( g(t) + W(t) + W_d(t) \\geq 0)$ is sufficiently random. It\nis assumed that $W_d(t)$ and $W(t)$ are independent. Such dithering allows\nus to use an analytically tractable reconstruction procedure, which has\n\\textit{order-optimal distortion}.  The block-diagram for sampling with\none-bit ADCs is illustrated in Fig.~\\ref{fig:bitdestructionblocks}(b). The\ntechnical condition on $\\sigma^2 = \\mbox{var}(W(t)) + \\mbox{var}(W_d(t))$ is stated\nusing the cdf of $W+W_d$.  Let $F: {\\mathbb R} \\rightarrow [0,1]$ be the cdf of\n$W+W_d$.  Let $f(x)$ be the associated probability density function with\n$f(\\pm C_\\phi) = \\delta$ and $f(0) = \\Delta$.  Observe that $\\Delta >\n\\delta$, since $f(x) = \\frac{1}{\\sqrt{2\\pi} \\sigma} \\exp( - x^2\/2\n\\sigma^2)$.  It is required that there is a parameter $\\mu > 0$ such that\n\\begin{align}\n\\left( 1 - \\frac{1}{\\sqrt{2} C^2_\\phi} \\right) \\frac{1}{\\delta} < \\mu <\n\\frac{1}{\\Delta}, \\label{eq:variancecondition}\n\\end{align}\nwhere $C_\\phi$ is the constant in (\\ref{eq:integrablephi}). First fix a\n$\\lambda > 1$.  Then, $C_\\phi = \\int_{t \\in {\\mathbb R}} |\\phi(t)| \\mbox{d}t >\n\\int_{t \\in {\\mathbb R}} \\phi(t) \\mbox{d} t = \\tilde{\\phi}(0) = 1$. That is,\n$C^2_\\phi \\sqrt{2} > \\sqrt{2} > 1$. Therefore, the lower bound on $\\mu$ in\n(\\ref{eq:variancecondition}) is positive.  Next, observe that if $\\sigma$\nis large but fixed, then $\\delta = f(C_\\phi) \\approx f(0) = \\Delta$. Then\n$\\delta$ and $\\Delta$ are close enough and the inequality in\n(\\ref{eq:variancecondition}) can be satisfied. In other words, for a fixed\n$\\lambda$ and hence $C_\\phi$, there is a \\textit{finite} number $\\sigma_0$\nfor which (\\ref{eq:variancecondition}) is satisfied for all $\\sigma >\n\\sigma_0$. If $\\mbox{var}(W(t)) < \\sigma_{0}^2$, then $\\mbox{var}(W_d(t)) >\n\\sigma_{0}^{2} - \\mbox{var}(W(t))$ will ensure that $\\mbox{var}(W + W_d) >\n\\sigma_{0}^2$. If $\\mbox{var}(W(t)) \\geq \\sigma_{0}^2$, then the extra dither is\nnot needed. This condition will be used in the distortion analysis in\nSec.~\\ref{sec:onebit}.\n\nFor single-bit estimation, the signal $F(g(t)) - 1\/2$ will be encountered,\nwhere $F:{\\mathbb R} \\rightarrow [0,1]$ is the cumulative distribution function of\nthe stationary noise random variable $W(t) + W_d(t)$.  Since $g(t) \\in [-1,\n1]$, and $F(x)$ has a wider support than the dynamic range of signal (i.e.,\n$[-1, 1]$), therefore $F'(x)$ is finite and non-zero for $x \\in [-1,1]$.\nSince $F(0) = 1\/2$ by symmetry, therefore, $F(g(t)) - 1\/2$ is more\nconvenient than $F(g(t))$ to work with.  For simplicity of notation, let\n$l(t) = F(g(t)) - 1\/2$.  Then $|l(t)| \\leq |F(1)| - 1\/2$, i.e., $l(t)$ is\nbounded.  The bound depends only on the noise distribution and the dynamic\nrange of $g(t)$.  Finally $|l'(t)| = |F'(g(t)) g'(t)| \\leq |F'(0) 2 \\pi^2|$\nsince $F'(0)$ maximizes $F'(x)$ in $[-1,1]$ and $|g'(t)|\\leq 2 \\pi^2$\n(see~\\cite[Proposition~3.1]{kumarIRH2011}).\n\nThe definition of ${BL}_{\\mbox{\\footnotesize int}}$ involves convolution with a stable\nkernel and convolution will often appear in the context of error\nanalysis. The following short lemma will be quite useful later on.\n\\begin{lemma\n\\label{lemma:propagation}\nLet $p(t)$ be a signal such that $||p||_\\infty$ is finite and $P(t)$\nbe any random process such that $P(t)$ is bounded (i.e., $\\sup_{t \\in\n{\\mathbb R}} {\\mathbb E}(P^2(t))$ is finite). Then,\n\\begin{align}\n||p \\star \\phi ||_\\infty & \\leq C_\\phi ||p||_\\infty,\n\\label{eq:stableerror}\\\\\n\\mbox{and } {\\mathbb E}[( |P(t)| \\star |\\phi(t)|)^2] & \\leq C^2_\\phi \\sup_{t\n\\in {\\mathbb R}} {\\mathbb E}(P^2(t)), \\label{eq:stablevariance}\n\\end{align}\nwhere the convolutions are well defined since $\\phi(t)$ is absolutely\nintegrable.\n\\end{lemma}\n\n\\IEEEproof The proof follows by the definition of convolution and the\ntriangle inequality. We have\n\\begin{align}\n|p(t) \\star \\phi(t)| & = \\left| \\int_{u \\in {\\mathbb R}} p(u) \\phi(t-u)\n\\mbox{d}u \\right|, \\nonumber \\\\\n& \\leq \\int_{u \\in {\\mathbb R}} |p(u)| |\\phi(t-u)| \\mbox{d}u, \\nonumber \\\\\n& \\leq ||p||_\\infty \\int_{u \\in {\\mathbb R}} |\\phi(t - u)| \\mbox{d}u,\n\\nonumber \\\\\n& = C_\\phi ||p||_\\infty. \\nonumber\n\\end{align}\nFor the second moment bound, note that\n\\begin{align}\n& {\\mathbb E}(||P(t)| \\star |\\phi(t)||^2) \\nonumber  \\\\\n& = {\\mathbb E}\\left( \\iint_{u, v \\in {\\mathbb R}} |P(u)| |P(v)| |\\phi(t-u)\n|\\phi(t-v)| \\mbox{d}u \\mbox{d}v \\right), \\nonumber \\\\\n& = \\iint_{u, v \\in {\\mathbb R}} {\\mathbb E}( |P(u)| |P(v)|) |\\phi(t-u)| |\\phi(t-v)|\n\\mbox{d}u \\mbox{d}v ,\\nonumber \\\\\n& \\stackrel{(a)}{\\leq}  \\sup_{t \\in {\\mathbb R}} {\\mathbb E}(P^2(t)) \\iint_{u, v \\in\n{\\mathbb R}} |\\phi(t-u)| |\\phi(t-v)| \\mbox{d}u \\mbox{d}v, \\nonumber \\\\\n& = C^2_\\phi \\sup_{t \\in {\\mathbb R}} {\\mathbb E}(P^2(t)), \\nonumber \n\\end{align}\nwhere $(a)$ follows by ${\\mathbb E}(2|P(u)| |P(v)|) \\leq {\\mathbb E}(P^2(u) + P^2(v))\n\\leq 2 \\sup_{t} {\\mathbb E}(P^2(t))$. Thus the proof is complete. {\\hfill $\\clubsuit$ \\medskip}\n\nThe two extreme scenarios of quantization as depicted in\nFig.~\\ref{fig:bitdestructionblocks} and their distortions will now be\nanalyzed in the next section.\n\n\\section{Estimation of bandlimited signal}\n\\label{sec:estimation}\n\nInterpolation of bandlimited signals with perfect samples is a well\nknown topic~\\cite{mallatSA2009}. Loosely speaking, a bandlimited\nsignal of duration $T$ and bandwidth $\\pi$ has $2\\pi T$ degrees of\nfreedom~\\cite{slepianO1976}.  With $N T$ noisy samples of the field\n$g(t) + W(t)$  in duration $T$, the optimal distortion is expected to\nbe $O(1\/N)$.\\footnote{A single bounded constant in additive\nindependent Gaussian noise with $N$ independent readings can be\nestimated up to a distortion of $O(1\/N)$~\\cite{bickelDM2001}.} With\nthis note, sampling schemes with oversampling rate $N$ are designed to\nachieve a distortion of $O(1\/N)$ for sampling $g(t)$.\n\n\\subsection{Estimation with perfect samples}\n\\label{sec:perfectsamples}\n\nA brief review of estimation with perfect samples will be highlighted\nfirst. Optimal minimum mean-squared method can be found in the work of\nPinsker~\\cite{pinskerO1980}. For illustration and to get a distortion\nproportional to $O(1\/N)$, it suffices to use the frame expansion.  Let\nthe integer-valued oversampling ratio (above the Nyquist rate) be $N$,\nand $\\tau = 1\/(\\lambda N)$. Then, the samples $\\{Y(n \\tau), n \\in {\\mathbb Z}\n\\}$ are available for the reconstruction of $g(t)$.  Using frame\nexpansion or the shift-invariance of bandlimited signals, \n\\begin{eqnarray}\ng(t) &=& \\frac{1}{N}  \\sum_{n \\in {\\mathbb Z}} \\lambda g(n \\tau)\\phi(t - n\n\\tau), \\nonumber \\\\\n& = & \\frac{1}{N} \\sum_{i = 0}^{N-1} \\sum_{k \\in {\\mathbb Z}} \\lambda  g\\left(\n\\frac{k}{\\lambda} + i \\tau \\right) \\phi\\left(t - \\frac{k}{\\lambda} -\n\\frac{i}{N \\lambda}\\right) ,\\label{eq:frameexpansion}\n\\end{eqnarray}\nwhere the equality holds pointwise and in ${\\cal L}^\\infty({\\mathbb R})$. It must be\nnoted that the basic operation in (\\ref{eq:frameexpansion}) is that of\naveraging; hence, the noise is expected to average out while the\nsignal will be retained.  This intuition motivates the following\nestimator for $g(t)$ from noisy data (see\nFig.~\\ref{fig:bitdestructionblocks}(a)). Define\n\\begin{align}\n\\widehat{G}_{\\mbox{\\footnotesize fr}}(t) & := \\frac{1}{N} \\sum_{n \\in {\\mathbb Z}} \\lambda Y(n\n\\tau)\\phi(t - n \\tau), \\label{eq:frameestimate}\\\\\n& = \\frac{1}{N} \\sum_{i = 0}^{N-1} \\sum_{k \\in {\\mathbb Z}} \\lambda [g+W]\n\\left( \\frac{k}{\\lambda} + i \\tau \\right) \\phi\\left(t -\n\\frac{k}{\\lambda} - \\frac{i}{N \\lambda}\\right).\\nonumber\n\\end{align}\nThe distortion of $\\widehat{G}_{\\mbox{\\footnotesize fr}}(t)$ is given by the following\nproposition.\n\\begin{proposition}[Frame estimate with $O(1\/N)$ distortion]\n\\label{prop:mse_unquantized}\nLet $\\widehat{G}_{\\mbox{\\footnotesize fr}}(t)$ in (\\ref{eq:frameestimate}) be an\nestimate for the bandlimited field $g(t)$ corrupted by additive\nindependent Gaussian noise. Let $D_{\\mbox{\\footnotesize fr}}(t) := {\\mathbb E} |\n\\widehat{G}_{\\mbox{\\footnotesize fr}}(t) - g(t)|^2$.  Then,\n\\begin{eqnarray}\n\\sup_{t \\in {\\mathbb R}} D_{\\mbox{\\footnotesize fr}}(t) \\leq \\frac{C_{\\phi}'' \\lambda^2\n\\sigma^2}{N}\n\\end{eqnarray}\nwhere the constants $\\sigma^2$ and $C_{\\phi}''$ from\n(\\ref{eq:squaresummablephi}) do not depend on $N$. \n\\end{proposition}\n\n\\IEEEproof See Appendix~\\ref{ap:unquantizedMSE}. {\\hfill $\\clubsuit$ \\medskip}\n\nThe signal term in (\\ref{eq:frameestimate}) converges in ${\\cal L}^\\infty({\\mathbb R})$\nto $g(t)$.  The noise term results in an independent sum of zero-mean\nrandom variables at every $t \\in {\\mathbb R}$.  This sum of random variables\nhas a variance that decreases as $(1\/N)$ due to the finite energy of\nthe interpolation kernel $\\phi(t)$. The constant $C_{\\phi}''$ depends\non the properties of the kernel $\\phi(t)$. The estimation with\nsingle-bit quantizers and associated distortion analysis will be\npresented next.\n\n\\subsection{Estimation with single-bit quantized samples}\n\\label{sec:onebit}\n\nThis section will present the key result of this work. Consider the system\nillustrated in Fig.~\\ref{fig:bitdestructionblocks}(b). In this section, a\n$\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t)$ will be obtained such that $D_{\\mbox{\\footnotesize 1-bit}}$ scales\nas $O(1\/N)$. This is non-trivial to achieve because the \\textit{non-linear}\nquantization operation is coupled with the statistical estimation\nprocedure. The result will be established in two parts: (i) it will be\nshown that suitable interpolation of one-bit samples converges to a\nnon-linear one-to-one function of $g(t)$ with an error term having a\npointwise variance of $O(1\/N)$; and (ii) the obtained non-linear function\nof $g(t)$ can be inverted in a stable manner  using recursive computation\nbased on contraction-mapping.  It will be assumed that $\\mbox{var}(W_d(t)) =\n(\\sigma^2 - \\mbox{var}(W(t)))_+$, where $\\sigma$ is such that\n(\\ref{eq:variancecondition}) is satisfied.\n\nThe stability property of kernel $\\phi(t)$ has been discussed\nSec.~\\ref{sec:background}.  For this section, fix $\\tau = 1\/(N \\lambda)$,\nwhere $\\lambda > 1$ is an arbitrary stability constant.  Analogous to\n(\\ref{eq:frameexpansion}), consider the random process obtained from the\nsingle-bit samples $X(n\\tau), n \\in {\\mathbb Z}$,\n\\begin{align}\nH_N(t) = \\tau \\sum_{n \\in {\\mathbb Z}} (X(n \\tau) - 1\/2) \\phi \\left( t - n \\tau\n\\right).\\label{eq:intermediateestimate}\n\\end{align}\nThen, the following proposition establishes the convergence of $H_N(t)$ to\na function of the signal of interest $g(t)$.\n\\begin{proposition}[Convergence of single-bit interpolation]\n\\label{prop:htconvergence}\nLet $l(t) = (F(g(t)) - 1\/2)$ and $H_N(t)$ be as defined in\n(\\ref{eq:intermediateestimate}). Then\n\\begin{align}\n\\sup_{t\\in {\\mathbb R}} {\\mathbb E}(H_N(t) - l(t) \\star \\phi(t))^2 \\leq \\frac{C_2}{N} +\n\\frac{C_3}{N^2}, \\label{eq:intermediateestimatelimit}\n\\end{align}\nwhere $C_2>0$ and $C_3 > 0$ are constants independent of $N$.\n\\end{proposition}\n\n\\IEEEproof See Appendix~\\ref{ap:onebitestimatoraccuracy}. {\\hfill $\\clubsuit$ \\medskip}\n\nThe factor $\\tau = 1\/(\\lambda N)$ provides the normalization for averaging\nin (\\ref{eq:intermediateestimate}), while the terms $(X(n\\tau) - 1\/2)\\phi(t\n- n \\tau)$ are weighted independent one-bit samples.  The average in\n(\\ref{eq:intermediateestimatelimit}) converges in mean-square to a\nconvolution.  The signal $l(t) \\in {\\cal L}^\\infty({\\mathbb R})$ and the limit $l(t) \\star\n\\phi(t)$ is a lowpass version of $l(t)$.  The dependence of $l(t) \\star\n\\phi(t)$ on $g(t)$ is non-linear due to quantization, which results in the\n$F(g(t))$ term.  The original signal $g(t)$ is Zakai sense bandlimited and\nit has \\textit{one degree of freedom} per unit time.  The degree of freedom\nper unit time of $l(t) \\star \\phi(t)$ can be up to \\textit{one} as well,\nand $F(x)$ has `nice' properties as a function.  Thus, it is not\nunreasonable to expect that there might be a class of $F(x)$ such that\n$(F(g(t)) - 1\/2) \\star \\phi(t)$ can be inverted to find $g(t)$, even though\nthis equation is nonlinear.\n\nConsider compandors defined by Landau and Miranker~\\cite{landauMT1961}.\n\\begin{definition} \\cite[pg~100]{landauMT1961}\nA compandor is a monotonic function $Q(x)$ which has the property that\n$Q(m(t)) \\in {\\cal L}^2({\\mathbb R}) $ if $m(t) \\in {\\cal L}^2({\\mathbb R})$. \n\\end{definition}\nLandau and Miranker have shown that if $g(t) \\in {\\cal L}^2({\\mathbb R})$ and\n$\\tilde{g}(\\omega)$ is zero outside $[-\\pi, \\pi]$, and if $Q:[-1, 1]\n\\rightarrow {\\mathbb R}$ is a compandor with non-zero slope, then there is one to\none correspondence between $g(t)$ and $Q(g(t)) \\star\n\\mbox{sinc}(t)$~\\cite{landauMT1961}.  Further, given any signal $m(t) \\in\n{\\cal L}^2({\\mathbb R})$ and $\\tilde{m}(\\omega)$ zero outside $[-\\pi, \\pi]$, there\nexists a unique $g_m(t) \\in {\\cal L}^2({\\mathbb R})$ with $\\tilde{g_m}(\\omega)$ zero\noutside $[-\\pi, \\pi]$ and $Q(g_m(t)) \\star \\mbox{sinc}(t) = m(t)$.\n\nIn our case, $g(t)$ need not be in ${\\cal L}^2({\\mathbb R})$, even though $l(t) =\nF(g(t)) - 1\/2$ is a compandor. Thus, the procedure of Landau and\nMiranker does not extend directly to bandlimited signals in\n${\\cal L}^\\infty({\\mathbb R})$, \\textit{especially} in the presence of statistical\nperturbations. Suitable modifications of their approach will be used\nto obtain the results for our problem.\n\nThe dependence between $g(t)$ and $l(t) \\star \\phi(t)$ is quite non-linear.\nThere is no clear or obvious equation by which $g(t)$ can be obtained from\n$l(t) \\star \\phi(t)$. Therefore, this inversion problem is casted into a\nrecursive setup, where Banach's fixed-point theorem can be leveraged along\nwith contraction mapping~\\cite[Ch.~5]{kreyszigI1989}.  This approach is\ninspired from the work of Landau and Miranker. Their recursive setup is\nnoted to be stable to perturbations of $g(t)$ in\n${\\cal L}^2({\\mathbb R})$~\\cite{landauMT1961}.  This work will use a variant of their\nprocedure, since the perturbation due to statistical noise with finite\nvariance is not in ${\\cal L}^2({\\mathbb R})$.  Therefore, our recursive procedure to\nobtain an estimate of $g(t)$ from $H_N(t)$ (see\n(\\ref{eq:intermediateestimatelimit})) and its analysis is non-trivial and\nit will be presented in detail.\n\nIn summary, an estimate for $g(t)$ is required. Due to quantization and\nnoise, which is a non-linear operation, an approximation $H_N(t)$ of\n$(F(g(t)) - 1\/2) \\star \\phi(t)$ is available. The estimate $H_N(t)$, which\nconverges to $(F(g(t)) - 1\/2) \\star \\phi(t)$ with sample density $N\n\\uparrow \\infty$, will be inverted to obtain an estimate\n$\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t)$ for the signal $g(t)$.  To establish the\nprecision indifference principle, we wish to show that the mean-square\nerror $\\sup_{t \\in {\\mathbb R}} {\\mathbb E}|\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t) - g(t)|^2$ decreases\nas $O(1\/N)$. The details are presented next.\n\nA `clip to one' function $\\mbox{Clip}[x]$ is defined first.\n\\begin{align}\n\\mbox{Clip}[x] & = x \\quad \\quad \\quad \\mbox{ if } |x| \\leq 1 \\nonumber \\\\\n& = \\mbox{sgn}(x) \\quad \\mbox{ otherwise}. \\label{eq:clipto1}\n\\end{align}\nSince $g(t)$ has a dynamic range bounded by one, by assumption, it will be\nunaffected by clipping. Note that under the ${\\cal L}^\\infty$ norm, this\ntransformation reduces the distance between any two scalars $x_1$ and\n$x_2$, i.e., $|\\mbox{Clip}[x_1] - \\mbox{Clip}[x_2]| \\leq |x_1 - x_2|$. This can be\nverified on a case by case basis.  For example, if $x_1 > 1$ and $x_2 \\in\n[-1,1]$, then $|\\mbox{Clip}[x_1] - \\mbox{Clip}[x_2]| = |1 - x_2| \\leq |x_1 - x_2|$.\nOther cases can be similarly enumerated. This clipping procedure is\nnon-linear and complicates some of the presented analysis; however, we feel\nthat its presence is essential for analysis.\n\nLet $\\psi(t) = \\phi(\\lambda t)$. Then $\\tilde{\\psi}(\\omega) =\n\\phi(\\omega\/\\lambda)$. Thus, $\\tilde{\\psi}(\\omega)$ is flat in\n$[-\\lambda \\pi, \\lambda \\pi]$ and in $\\pm[\\lambda \\pi, \\lambda^2 \\pi]$\ndecreases linearly to zero. Consider the set of bandlimited signals\ndefined by,\n\\begin{align}\n{\\cal S}_{\\mbox{\\footnotesize BL,bdd}} = \\{ m(t): |m(t)| \\leq C_\\phi \\mbox{ and } m(t) \\star\n\\psi(t) = m(t) \\}. \\label{eq:blbounded}\n\\end{align}\nThen, ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ is a complete subset of the Banach space\n${\\cal L}^\\infty({\\mathbb R})$.\n\\begin{lemma}[${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ is a complete metric space]\n\\label{lem:metric}\nLet ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ be as defined in (\\ref{eq:blbounded}). Then\n$({\\cal S}_{\\mbox{\\footnotesize BL,bdd}}, ||.||_\\infty)$ is a complete subset of $({\\cal L}^\\infty({\\mathbb R}),\n||.||_\\infty)$.\n\\end{lemma}\n\n\\IEEEproof Define the distance function $d: {\\cal S}_{\\mbox{\\footnotesize BL,bdd}} \\times {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}\n\\rightarrow {\\mathbb R}^+$ as $d(m_1, m_2) = ||m_1 - m_2||_\\infty$ with\n$m_1(t), m_2(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. It is easy to verify the axioms of\ndistance metric~\\cite{kreyszigI1989}: (i) $d \\geq 0$ and $d \\leq\n\\infty$; (ii) $d(m_1, m_2) \\equiv 0$ if and only if $m_1(t) = m_2(t)$;\n(iii) $d(m_1, m_2) = d(m_2, m_1)$; and (iv) $d(m_1, m_2) \\leq d(m_1,\nm_3) + d(m_3, m_2)$ for any $m_1(t), m_2(t), m_3(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$.\n\nIt is straightforward to see that ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}} \\subset {\\cal L}^\\infty({\\mathbb R})$\nsince $||m||_\\infty \\leq C_\\phi$ for  every $m(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$.  To\nshow that the subset is complete, consider any Cauchy sequence $m_n(t)\n\\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. Since ${\\cal L}^\\infty({\\mathbb R})$ is complete, therefore $m_n(t)\n\\rightarrow s(t)$, where $s(t) \\in {\\cal L}^\\infty({\\mathbb R})$. It remains to show\nthat $s(t)$ belongs to ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. \n\nFor any $\\epsilon > 0$, there is an $n_0$ such that $||m_n -\ns||_\\infty < \\epsilon$ for all $n > n_0$. Since $\\int_{{\\mathbb R}} |\\psi(t)|\n\\mbox{d}t = C_\\phi\/\\lambda$, therefore, $||m_n \\star \\psi - s \\star\n\\psi ||_\\infty \\leq ||m_n - s|| (C_\\phi\/\\lambda) = C_\\phi \\epsilon\n\/\\lambda$ for all $n > n_0$ (see Lemma~\\ref{lemma:propagation}). Thus,\n$m_n(t) \\star \\psi(t) \\rightarrow s(t) \\star \\psi(t)$. However,\n$m_n(t) \\star \\psi(t) \\equiv m_n(t)$ since $m_n(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$.\nTherefore, it follows that $s(t) = s(t) \\star \\psi(t)$, or $s(t) \\in\n{\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. Thus, ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ is complete. {\\hfill $\\clubsuit$ \\medskip}\n\nA map $T: {\\cal S}_{\\mbox{\\footnotesize BL,bdd}} \\longrightarrow {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ will be defined next.\nThis map will result in a recursive procedure to obtain $g(t)$ from\n$h(t) := l(t) \\star \\phi(t)$. Define\n\\begin{align}\nT[m(t)] & = \\mbox{Clip} \\Big[ \\mu h(t) + \\left[ m(t) - \\mu (F(m(t)) - 1\/2) \\right]\n\\star \\phi(t) \\Big]  \\star \\phi(t). \\label{eq:keymap}\n\\end{align}\nIt will be shown that $T$ is a contraction on $({\\cal S}_{\\mbox{\\footnotesize BL,bdd}},\n||.||_\\infty)$.\n\\begin{lemma}[$T$ is a contraction]\n\\label{lem:tcontraction}\nLet $({\\cal S}_{\\mbox{\\footnotesize BL,bdd}}, ||.||_\\infty)$ be the metric space as defined in\n(\\ref{eq:blbounded}). Let $T : {\\cal S}_{\\mbox{\\footnotesize BL,bdd}} \\longrightarrow {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$\nbe a map as defined in (\\ref{eq:keymap}). If the condition in\n(\\ref{eq:variancecondition}) is satisfied, then there is a choice of\n$\\mu$ such that $T$ is a contraction, i.e.,\n\\begin{align}\n||T[m_1] - T[m_2] ||_\\infty \\leq \\alpha ||m_1 - m_2||_\\infty,\n\\end{align}\nfor some $0 < \\alpha < 1$ and any $m_1(t), m_2(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. The\nparameter $\\alpha$ does not depend on the choice of $m_1$ and $m_2$.\n\\end{lemma}\n\\IEEEproof See Appendix~\\ref{ap:lem_cont}. {\\hfill $\\clubsuit$ \\medskip}\n\nNow the key recursive equation will be stated. Let $l(t) = (F(g(t)) -\n1\/2)$ and $h(t) = l(t) \\star \\phi(t)$ be available for obtaining\n$g(t)$. Then,\n\\begin{align}\ng_{k+1}(t) := T[g_{k}(t)]  = \\mbox{Clip} \\Big[ \\mu h(t) + \\left[ g_{k}(t) -\n\\mu (F(g_{k}(t)) - 1\/2) \\right] \\star \\phi(t) \\Big] \\star \\phi(t),\n\\label{eq:recursion}\n\\end{align}\nwhere $k \\geq 0, k \\in {\\mathbb Z}$ and $\\mu > 0$ is a constant that will be\nchosen according to Lemma~\\ref{lem:tcontraction}. Set $g_0(t) \\equiv\n0$.  The original signal $g(t)$ is a fixed point of this equation and\nit can be verified by substitution. The following proposition shows\nthat $g(t)$ is the \\textit{only} fixed point of the equation in\n(\\ref{eq:recursion}). The proof hinges on Banach's fixed point theorem\nor contraction theorem~\\cite[Ch.~5]{kreyszigI1989}.\n\\begin{proposition}[Signal of interest is the fixed point of $T$]\n\\label{prop:fixedpt}\nLet $g(t) \\in {BL}_{\\mbox{\\footnotesize int}} \\subset {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ be a continuous bounded\nbandlimited signal. Let $h(t) = l(t) \\star \\phi(t)$, where $l(t) = F(g(t))\n- 1\/2$. Consider the recursion $g_{k}(t) = T[g_{k-1}(t)]$, where $T$ is as\ndefined in (\\ref{eq:keymap}). Set $g_0(t) \\equiv 0$. If $\\mu$ is selected\nas in (\\ref{eq:variancecondition}), then\n\\begin{align}\n\\lim_{k \\rightarrow \\infty } ||g_k - g||_\\infty = 0. \\label{eq:linfconv}\n\\end{align}\n\\end{proposition}\n\\IEEEproof  The proof is straightforward with Lemma~\\ref{lem:metric} and\nLemma~\\ref{lem:tcontraction} in place. Define $d(m_1, m_2) = ||m_1 -\nm_2||_\\infty$ for any $m_1(t), m_2(t) \\in {\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$. From\nLemma~\\ref{lem:metric}, note that $({\\cal S}_{\\mbox{\\footnotesize BL,bdd}}, d)$ is a complete metric\nspace.  The signal $g(t)$ is in ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ and it satisfies $g(t) =\nT[g(t)]$, i.e., it is a fixed point for $T$ defined in (\\ref{eq:keymap}).\n\nPick $\\mu$ as in (\\ref{eq:variancecondition}). Then $T$ is a contraction on\n$({\\cal S}_{\\mbox{\\footnotesize BL,bdd}}, d)$. Thus, by Banach's fixed point theorem (contraction\ntheorem)~\\cite[Ch.~5]{kreyszigI1989}, there is \\textit{exactly} one fixed\npoint in ${\\cal S}_{\\mbox{\\footnotesize BL,bdd}}$ for the equation $g(t) = T[g(t)]$. Since $g_k(t)$\nconverges to a fixed point, it must converge to $g(t)$ in the distance\nmetric $d$.  Thus the proof is complete.\n{\\hfill $\\clubsuit$ \\medskip}\n\nPropostion~\\ref{prop:fixedpt} holds with perfect information about $l(t)\n\\star \\phi(t)$. The estimation of signal from $H_N(t)$, the statistical\napproximation of $l(t) \\star \\phi(t)$, will be discussed now.  Let\n$G_{k}(t)$ be the sequence of random waveforms generated from $H_N(t)$ when\nit is applied to the recursion in (\\ref{eq:recursion}).  That is, fix\n$G_0(t) \\equiv 0$ and define \n\\begin{align}\n& G_{k+1}(t) := T[G_{k}(t)] =  \\mbox{Clip} \\Big[ \\mu H_N(t) + \\left[ G_{k}(t) -\n\\mu (F(G_{k}(t)) - 1\/2) \\right] \\star \\phi(t) \\Big] \\star \\phi(t).\n\\label{eq:recursionG}\n\\end{align}\nLet $\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t) = \\lim_{k \\rightarrow \\infty}\nG_k(t)$.\\footnote{This limit exists since it can be shown that $||G_k -\nG_{k-1}||_\\infty \\leq \\alpha ||G_{k-1} - G_{k-2}||_\\infty$ for some $0  <\n\\alpha < 1$ by using an analogous procedure as in\nLemma~\\ref{lem:tcontraction}.} For the same choice of $\\mu$ which ensures\nthat $T$ is a contraction on $({\\cal S}_{\\mbox{\\footnotesize BL,bdd}}, ||.||_\\infty)$, the distortion\nof $|\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t) - g(t)|$ has to be established. To this end,\nthe following proposition is noted.\n\\begin{proposition}[1-bit estimation has distortion $O(1\/N)$]\n\\label{prop:contraction_mse}\nLet $H_N(t)$ be the estimate of $l(t)$ as described in\n(\\ref{eq:intermediateestimate}) and $\\mu$ be selected as in\n(\\ref{eq:variancecondition}).  With $G_0(t) \\equiv 0$, let $G_k(t)$ be the\nsequence of random waveforms as  defined in (\\ref{eq:recursionG}).  Define\n$\\lim_{k \\rightarrow \\infty} G_k(t) = \\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t)$. Then,\n\\begin{align}\nD_{\\mbox{\\footnotesize 1-bit}} := \\sup_{t \\in {\\mathbb R}} {\\mathbb E}( \\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t) - g(t))^2 =\nO(1\/N), \\nonumber\n\\end{align}\ni.e., the distortion $D_{\\mbox{\\footnotesize 1-bit}}$ decreases as $O(1\/N)$.\n\\end{proposition}\n\n\\IEEEproof See Appendix~\\ref{ap:contraction_accuracy}. {\\hfill $\\clubsuit$ \\medskip}\n\nThe results of Proposition~\\ref{prop:mse_unquantized} and\nProposition~\\ref{prop:contraction_mse} can be summarized into the\nfollowing theorem.\n\\begin{theorem}[Precision indifference principle]\n\\label{thm:bitdestruction}\n\\textit{Let $g(t)$ be a bounded dynamic-range bandlimited-signal as\ndefined in (\\ref{eq:blinterest}). Assume that $g(t) + W(t)$ is\navailable for sampling, where $W(t)$ an additive independent Gaussian\nrandom process with finite variance. Fix an oversampling factor of\n$N$, where $N$ is large for statistical averaging.  There exists an\nestimate $\\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t)$ obtained from single-bit samples\nof $g(t) + W(t)$ such that\n\\begin{align}\n\\sup_{t\\in {\\mathbb R}} {\\mathbb E}| \\widehat{G}_{\\mbox{\\footnotesize 1-bit}}(t) - g(t) |^2 = O(1\/N).\n\\nonumber\n\\end{align}\nThis distortion is proportional to the best possible distortion of\n$O(1\/N)$ that can be obtained with unquantized or perfect samples.}\n\\end{theorem}\nA few remarks highlighting the importance of the results obtained will\nconclude this section.\n\n\\subsection{Remarks on the results obtained}\n\n\\subsubsection{Comparison with the bit-conservation principle} The\nbit-conservation principle~\\cite{kumarIRH2011} is somewhat in contrast\nto the precision indifference principle. Loosely speaking,\nbit-conservation principle states that for sampling a bandlimited\nsignal in a noiseless setting, the oversampling density can be\ntraded-off against ADC precision while maintaining a fixed bit-rate\nper Nyquist interval and an order-optimal pointwise distortion.  In\nthe presence of additive independent Gaussian noise, this tradeoff\nbetween ADC precision and oversampling is absent while studying\npointwise mean-squared distortion. In the noisy setup, the distortion\nis proportional to $1\/N$, where $N$ is the oversampling density\nirregardless of the ADC precision. The presence of noise shifts the\nrole of ADC precision towards only the proportionality constant in\ndistortion!\n\n\\subsubsection{Interpretation of precision-indifference principle}\nFirst, it can be argued that the precision indifference principle\nholds while estimating a constant signal (one degree of freedom) in\nadditive independent Gaussian noise.  Assume that a constant $c \\in\n[-1,1]$ has to be estimated based on $N$ noisy readings $Y_i = c+W_i,\n1 \\leq i \\leq N$, where $\\{W_i, 1 \\leq i \\leq N\\}$ are i.i.d.~${\\cal\nN}(0, \\sigma^2)$.  In the absence of quantization, $\\widehat{C}_N =\n(\\sum_{i = 1}^N Y_i)\/N$  converges to $c$ in the mean-square sense,\nand ${\\mathbb E}(\\widehat{C}_N - c)^2 = \\sigma^2\/N$. This is the optimal\ndistortion if perfect (unquantized) samples are available. Now\nconsider the case where single-bit readings $B_i = {\\mathbbm{1}}(c + W_i\n\\geq 0), 1 \\leq i \\leq N$ are available.  The random variables $\\{B_i,\n1 \\leq i \\leq N\\}$ are i.i.d.~$\\mbox{Ber}(q)$ where $q = {\\mathbb P}(W \\geq -\nc) = {\\mathbb P}(W \\leq c) = F(c)$. Assume $\\widehat{B}_N = (\\sum_{i = 1}^N\nB_i)\/N$. It can be shown that ${\\mathbb E}(\\widehat{B}_N - F(c))^2 \\leq\n1\/(4N)$ since each $\\mbox{var}(B_i) \\leq F(c)(1-F(c)) \\leq 1\/4$.\nDefine $\\widehat{C}_{\\mbox{\\footnotesize 1-bit}} = F^{-1}(\\widehat{B}_N)$ if\n$\\widehat{B}_N \\in [F(-1), F(1)]$ and $\\widehat{C}_{\\mbox{\\footnotesize 1-bit}} = \\pm 1$\notherwise.  Since $F(x)$ is invertible and\n$\\mbox{d}F^{-1}(x)\/\\mbox{d}x$ is bounded for $x \\in [F(-1),F(1)]$,\ntherefore, using the delta method, $\\widehat{C}_{\\mbox{\\footnotesize 1-bit}}$ obtained\nfrom $\\widehat{B}_N$ has a mean-squared error which decreases as\n$(1\/N)$~\\cite{bickelDM2001}.  Next, it should be noted that\nbandlimited signals have one degree of freedom in every Nyquist\ninterval. An oversampling factor of $N$ means that there are $N$\nsamples to observe each degree of freedom  on an average. Finally,\nobserving the Nyquist samples of a bandlimited signals with a\ndistortion of $O(1\/N)$ results, by stable interpolation with kernel\n$\\phi(t)$, in a pointwise distortion of $O(1\/N)$ for the signal\nestimate at any point.\n\n\\subsubsection{Precision-indifference for a larger class of noise}\nConsider the model where each sample $Y(n\\tau) = g(n\\tau) + V(n\\tau)$ is\naffected by some non-Gaussian noise.  Focus on the case where $V(n\\tau)$\ncan be written as $V(n\\tau) = W(n\\tau) + U(n\\tau)$, where $W(n\\tau)$ and\n$U(n\\tau)$ are i.i.d.~for all $n \\in {\\mathbb Z}, \\tau \\in {\\mathbb R}$. If $W(n\\tau)$ is\nGaussian, $\\mbox{var}(V(n\\tau)) = \\sigma^2 < \\infty$, and $F_V(x)$ satisfies\n(\\ref{eq:variancecondition}),  then the precision indifference principle\nwill hold.  The extension of existing proofs is simple and only its key\nsteps will be mentioned here. In the perfect sample case (see\nFig.~\\ref{fig:bitdestructionblocks}(a)), the Gaussian part of $V(t)$ will\nlimit the best possible (optimal) distortion to $O(1\/N)$; this is because\neven if the values of $U(n\\tau)$ are (magically) known the residual\n$W(n\\tau)$ will limit the distortion.  With single-bit quantization, note\nthat all the proofs in Sec.~\\ref{sec:onebit} only depend upon the existence\nof a $\\delta$ and $\\Delta$ such that (\\ref{eq:variancecondition}) is\nsatisfied, monotonicity of $F_V(x)$ such that its derivative is bounded\naway from zero, and $F_V(0) = 1\/2$.  The recursive procedure in\n(\\ref{eq:recursion}), however, requires the knowledge of $F_V(x)$.\n\n\\section{Conclusions and future work}\n\\label{sec:conclusions}\n\nThe sampling, quantization, and estimation of a bounded dynamic-range\nbandlimited signal affected by additive independent Gaussian noise was\nstudied.  Such setup naturally arises in distributed sampling or where the\nsampling device itself is noisy. For bandlimited signals, the distortion\ndue to additive independent Gaussian noise can be reduced by oversampling\n(statistical diversity).  The maximum pointwise expected mean-squared error\n(statistical ${\\cal L}^2$ error) was used as a distortion metric.  Using two\nextreme scenarios of quantizer precision, namely infinite precision and\nsingle-bit precision, a quantizer precision indifference principle was\nillustrated. It was shown that the optimal law for distortion is $O(1\/N)$,\nwhere $N$ is the oversampling ratio with respect to the Nyquist rate. This\nscaling of distortion is unaffected by the \\textit{quantizer precision},\nwhich is the key message of the precision indifference principle.  In other\nwords, the reconstruction distortion law, up to a proportionality constant,\nis unaffected by quantizer precision.\n\nExtensions of the precision indifference principle to other classes of\nparametric or non-parametric signals is of immediate interest. Further,\nthis work assumed sufficient dithering by noise because the estimators were\nlinear.  It is of interest to look towards estimation techniques which do\nnot require extra dithering.\n\n\n\\section*{Acknowledgment}\n\nThe problem of sampling a smooth signal in the presence of noise in a\ndistributed setup was suggested by Prof. Kannan Ramchandran, EECS,\nUniversity of California, Berkeley, CA. Discussions on this problem\nwith Prof. Kannan Ramchandran and Prof.~Martin Wainwright, EECS,\nUniversity of California, Berkeley, CA, Prof. H.~Narayanan, EE, IIT\nBombay, and Prof. Prakash Ishwar, ECE, Boston University, Cambridge,\nMA were insightful.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEstablishing pixel-level correspondences between pair of images including instances of the same object category can benefit several important applications, such as motion estimation~\\cite{pan2015efficient}, medical imaging~\\cite{annunziata2016fully,iglesias2018joint}, object recognition~\\cite{girshick2015fast} and 3D reconstruction~\\cite{izadi2011kinectfusion}. As a result, this has become a fundamental problem in computer vision~\\cite{forsyth2003modern, hartley2003multiple}. Typically, pixel-level correspondences between two images are computed by extracting sparse local feature descriptors (e.g., SIFT~\\cite{lowe2004distinctive}, HOG~\\cite{dalal2005histograms}, SURF~\\cite{bay2006surf}, SCIRD~\\cite{annunziata2015scale, annunziata2016accelerating}), then matching the extracted descriptors, and finally pruning mismatches based on geometric constraints. Although this approach has been applied successfully in different domains, its performance can degrade significantly due to factors such as intra-class variations, non-rigid deformations, partial occlusions, illumination, image blur, and visual clutter. Recently, the representational power of Convolutional Neural Networks (CNNs) has been leveraged to improve the overall process. In particular, several CNN-based methods for learning powerful feature descriptors have been introduced~\\cite{simo2015discriminative, zagoruyko2015learning, noh2017largescale}. More recently, end-to-end trainable CNNs for learning image descriptors as well as estimating the geometric transformation between the two images have been introduced in~\\cite{rocco2019convolutional, kanazawa2016warpnet}.\n\\begin{figure}[!t]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{cc}\n\\includegraphics[align=c, width=0.225\\textwidth]{figures\/figure_1\/image_TRAIN_warp0.png} & \\includegraphics[align=c, width=0.225\\textwidth]{figures\/figure_1\/image_TRAIN_warp4.png}\\vspace{-0.3cm}\\\\\\\\\n\\includegraphics[align=c, width=0.225\\textwidth]{figures\/figure_1\/input_lfw_scaled.png} & \\includegraphics[align=c, width=0.225\\textwidth]{figures\/figure_1\/lwf_output_scaled.png} \\\\\n(a) & (b) \n\\end{tabular}\n\\caption{Unsupervised joint alignment (a.k.a. \\textit{congealing}) results obtained by the proposed method on digit `$2$' from \\textit{affNIST}~\\cite{afnist} and \\texttt{Jennifer\\_Capriati} from LFW~\\cite{huang2008labeled}. (a) input images before alignment (initialisation in red), (b) output images aligned with the proposed method.}\n\\label{fig:main_page}\n\\end{figure}\nThe majority of previously proposed methods focus on the problem of finding pixel-level correspondences between a \\textit{pair} of images. However, a plethora of other tasks such as, co-segmentation, image edit propagation, video stabilisation and structure-from-motion, require global correspondences between a \\textit{set} of images containing a specific object. A straightforward way to address this problem is to identify pixel correspondences between each pair of images in the dataset and solve the problem in a sequential manner. However, this approach would be prone to important limitations, as (i)~it would fail to take into account valuable cross-image appearance information (i.e. statistics of local patches across the entire dataset) during the optimisation; and (ii)~the computational complexity of the problem would increase exponentially with the number of images, therefore significantly limiting the scalability to large datasets. Thus, estimating global correspondences of a set of images (image ensemble) by jointly aligning them in an unsupervised manner can be immensely valuable.\n\nCongealing (joint image alignment) was originally introduced by Learned-Miller in~\\cite{learned2006data}. His approach aligns (by estimating rigid transformations) an ensemble of images of a particular object by minimising the sum of entropies of pixel values at each pixel location. Although this method has been effectively applied to handwritten digits and magnetic resonance image volumes, it has shown some limitations, including slow and\/or sometimes poor convergence and relatively high sensitivity to hyper-parameters. Later, Huang~\\textit{et al.} improved the performance of~\\cite{learned2006data} by using hand-crafted SIFT features~\\cite{huang2007unsupervised}. To overcome the optimisation problems of the original congealing approach, Cox \\textit{et al.}~\\cite{cox2008least, cox2009least} proposed to utilise a reference image (i.e., template) and then minimise the sum of squared differences instead of the sum of entropies. This way, standard Gauss-Newton gradient descent method could be adopted to make the optimisation efficient. Later, motivated by lossy compression principles, Vedaldi \\textit{et al.}~\\cite{vedaldi2008joint} proposed a joint alignment approach based on log-determinant estimation. \n\nA common drawback of the aforementioned methods is that they cannot simultaneously handle variability in terms of illumination, gross pixel corruptions and\/or partial occlusions. RASL, an image congealing method that overcomes this drawback was proposed in~\\cite{peng2012rasl}. The key assumptions made in RASL and its multiple variants, e.g.~\\cite{likassa2018modified, chen2016nonconvex, oh2015partial} are that (i) an ensemble of well-aligned images of the same object is approximately low-rank and (ii) that gross pixel errors are sparsely distributed. Therefore, image congealing is performed by seeking a set of optimal transformations such that the ensemble of misaligned images is written as the superposition of two components i.e., a low-rank component and a sparse error component. RASL has been widely used for jointly aligning multiple images in different applications such as face landmarks localisation~\\cite{sagonas2014raps, peng2015piefa, sagonas2017robust}, pose-invariant face recognition~\\cite{sagonas2017robust, sagonas2015robust}, and medical imaging~\\cite{bise2016vascular}. Despite its wide applicability, it is worth noting that (i)~RASL joint alignment performance can be severely affected by non-optimal initialisation and high intra-class variability in the image ensemble; (ii)~scalability to large ensembles is limited by the formulation of the low-rank minimisation problem and related SVD-based sub-routines; and (iii)~a new optimisation is required for every new image added to the ensemble. To address some of these limitations, t-GRASTA~\\cite{he2014iterative} and PSSV~\\cite{oh2015partial} have been recently proposed.\n\nThe first deep learning approach to unsupervised joint image alignment was proposed by Huang \\textit{et al.}~\\cite{huang2012learning}. A modified version of the convolutional restricted Boltzmann machine was introduced to obtain features that could better represent the image at differing resolutions, and that were specifically tuned to the statistics of the data being aligned. They then used those learnt features to optimise the standard entropy-based congealing loss and achieved excellent joint alignment results on the Labelled Faces in the Wild (LFW) benchmark.\n\\begin{figure}[!tb]\n\\centering\n\\includegraphics[align=c, width=0.47\\textwidth]{figures\/figure_0\/flow_chart_v2.pdf}\n\\caption{Block diagram of the proposed method. Black arrows correspond to forward pass, while red and blue to back-propagation.}\n\\label{fig:flow_chart}\n\\end{figure}\n\nHere, we propose a congealing method to solve large-scale joint alignment problems, which is of significant practical importance in light of the ever increasing availability of image data. The proposed method consists of two main modules: (i)~the aligner and (ii)~the low-capacity auto-encoder. Specifically, the joint alignment task is cast as a batch-based optimisation problem in which the aligner is used to estimate the global transformation required to warp each image to a reference. The alignment error is quantified via $\\ell_1$-norm between the transformed batch images and the reference. Motivated by the observation that a set of well-aligned images require less modelling capacity to be reconstructed well (e.g. reconstruction with low-rank bases~\\cite{peng2012rasl}), the aligned batch is subsequently processed by a \\textit{low-capacity} auto-encoder and reconstruction errors are back-propagated to the aligner (a snapshot of the results is displayed in Fig.~\\ref{fig:main_page}).\n\n\\textbf{Contributions:} In summary, the main contributions of this paper are: (i)~a congealing method which is shown to be capable of handling large-scale joint alignment problems i.e., up to one million data points, simultaneously; (ii)~a novel \\textit{differentiable} formulation of the congealing problem, which combines the advantages of previously proposed similarity- and rank-based approaches and that can be easily optimised with Stochastic Gradient Descent (SGD), end-to-end; (iii)~an extensive experimental evaluation of the proposed method and state-of-the-art approaches on several benchmark datasets, including digits and faces at different resolutions, assessing joint alignment performance and robustness to linear and non-linear geometric perturbations of different magnitude and type.\n\\section{Methodology}\nIn the following, we briefly summarise the approaches most related to ours. Then, we introduce the proposed method.\n\n\\textbf{RASL.}~Let us assume we have $N$ misaligned images $\\{\\bs{I}_i\\}_{i=1}^N \\in \\mathbb{R}^{w\\times h}$ of a particular object and let $\\{\\bs{p}_i\\}_{i=1}^N$ be a set of transformations such that $\\{\\bs{I}_i^0 = \\bs{I}_i\\circ\\bs{p}_i\\}_{i=1}^N$ becomes a set of well-aligned images. If we define ${\\text{vec}:\\mathbb{R}^{w\\times h} \\rightarrow \\mathbb{R}^m}$ as the operator that vectorises an image, the main assumption of the RASL method is that the matrix:\n\\begin{equation}\n    \\bs{D}\\circ\\bs{P} = [\\text{vec}(\\bs{I}_1^0)~|~\\cdots~|~\\text{vec}(\\bs{I}_N^0)] = \\bs{A}\n\\end{equation}\nwill be approximately \\textit{low-rank}. However, in practice this assumption can be violated when the object of interest is affected by occlusion, shadows, and noise. Therefore, the authors assume that each aligned image is corrupted with non-Gaussian-but-sparse errors $\\bs{E}\\in \\mathbb{R}^{m\\times N}$, such that $ \\bs{D}\\circ\\bs{P} =  \\bs{A} + \\bs{E}$.\nGiven the observation of the misaligned and corrupted images, the goal is to estimate a set of transformations $\\{\\bs{p}_i\\}_{i=1}^N$ such that the rank of the transformed noise-free images $\\{\\bs{I}_i\\circ\\bs{p}_i\\}_{i=1}^N  \\in \\mathbb{R}^{w\\times h}$ becomes as small as possible. Formally,\n\\begin{equation}\\label{eqn:RASL}\n    \\argmin_{\\bs{A},\\bs{E},\\{\\bs{p}_i\\}_{i=1}^N} \\rank(\\bs{A}) \\quad \\textrm{s.t.}~~\\bs{D}\\circ\\bs{P} = \\bs{A} + \\bs{E},~\\norm{\\bs{E}}_0 \\leq q,\n\\end{equation}\nwhere $q$ controls the sparsity of the error matrix $\\bs{E}$. \nUnfortunately, the non-convex and discontinuous nature of the optimisation problem in Eq.~(\\ref{eqn:RASL}) makes it not directly tractable. To this end, an algorithm that provides a sub-optimal solution via iterative convex programming was proposed. As discussed in \\cite{peng2012rasl}, this algorithm is limited by the following assumptions: (i) initial misalignment not too large, (ii) rank of the matrix $\\bs{A}$ to be recovered not too high, and (iii) only a small fraction of all pixels affected by error. A further limitation is the scalability of the algorithm. In fact, the convex relaxation replacing the $\\text{rank}(\\cdot)$ with the nuclear norm requires a  very expensive Singular Value Decomposition (SVD) computation at every optimisation step.\n\n\\textbf{Least-Squares Congealing (LSC).}~This method~\\cite{cox2008least, cox2009least} has been specifically proposed for targeting large-scale joint alignment problems. Building on the success of Lucas-Kanade image alignment~\\cite{lucas1981iterative}, the idea is to define a reference image $\\bs{I}_j$ and align each of the remaining ones {$\\{\\bs{I}_i\\}_{i\\neq j}$} to that reference. In general, this optimisation problem can be formulated as:\n\\begin{equation}\\label{eqn:LSC_linear}\n    \\argmin_{\\bs{p}_{i\\neq j}} \\sum_{i\\neq j} \\Big|\\Big|\\bs{I}_i \\circ \\bs{p}_i - \\bs{I}_j\\Big|\\Big|^2_2,\n\\end{equation}\nwhere $\\boldsymbol{\\bs{p}} = \\{\\bs{p}_1, \\bs{p}_2, \\ldots, \\bs{p}_{N-1}\\}$ is the set of transformations to apply to $\\{\\bs{I}_i\\}_{i\\neq j}$ to map them onto the reference $\\bs{I}_j$. The main advantage of LSC over low-rank\/entropy-based ones is faster convergence, as the adoption of the least-squares cost function allows for the use of standard Gauss-Newton optimisation techniques. On the other hand, alignment performance tend to be worse due to its simplicity.  \n\\subsection{Proposed Method}\nMotivated by the need for highly accurate alignment in very large-scale problems, we propose a congealing framework that leverages the advantages of adopting a similarity-based cost function (i.e. direct, such as $\\ell_2$-norm in LSC) \\textit{and} a complexity-based one (i.e. indirect, such as rank-based used in RASL). To perform this task \\textit{at scale}, we formulate the congealing problem in a way that can be efficiently optimised via standard back-propagation and SGD.\n\nOur formulation can be interpreted in terms of lossy compression optimisation~\\cite{vedaldi2008joint}:\n\\begin{equation}\\label{eqn:compression}\n    \\argmin_{\\{\\bs{p}_i\\}_{i=1}^N} \\mathcal{D}(\\bs{I}_{i\\neq j} \\circ \\bs{p}_{i\\neq j}, \\bs{I}_j) + \\lambda~\\mathcal{C}(\\bs{I}_i \\circ \\bs{p}_i),\n\\end{equation}\nwhere the \\textit{distortion} $\\mathcal{D}$ reflects the total error when approximating the reference image $\\bs{I}_j$ (i.e., original data) with the aligned image $\\bs{I}_i \\circ \\bs{p}_i$ (i.e., compressed data), the \\textit{complexity} $\\mathcal{C}$ is the total number of \\textit{symbols} required to encode $\\bs{I}_i \\circ \\bs{p}_i$, and the parameter $\\lambda\\geq0$ trades off the two quantities. \nA good candidate for our distortion (or similarity) measure $\\mathcal{D}$ should be robust to occlusions, noise, outliers and,  in general, objects that might partially differ in appearance (e.g. same digit but different font, same face but wearing glasses or not). In RASL, this is achieved by adding explicit constraints on the noise model and its sparsity properties which have a significant impact on the optimisation efficiency. To circumvent this problem, we adopt the $\\ell_1$-norm as measure of distortion, which can be efficiently optimised and offers a higher level of robustness compared to the $\\ell_2$-norm used in LSC or \\cite{vedaldi2008joint}. Formally, \n\\begin{equation}\\label{eqn:distortion}\n    \\mathcal{D} = \\sum_{i\\neq j}\\Big|\\Big|\\bs{I}_{i} \\circ \\bs{p}_{i} - \\bs{I}_j\\Big|\\Big|_1.\n\\end{equation}\nMotivated by the need for optimisation in large-scale joint alignment problems, we propose an efficient alternative to rank minimisation. Specifically, we observe that when a set of images are well-aligned, they form a sequence that contains a significant level of redundant information. As a consequence, the stack of images can be compressed with higher compression rates w.r.t. the original misaligned ones. Alternatively, a lower reconstruction error can be attained, at parity of compression rate. Exploiting this consideration we therefore propose to optimise:\n\\begin{equation}\\label{eqn:complexity}\n    \\begin{aligned}\n      \\argmin_{\\{\\bs{p}_i\\}_{i=1}^N,\\phi, \\theta} &\\sum_{i}\\Big|\\Big|D_\\phi(E_\\theta(\\bs{I}_{i} \\circ \\bs{p}_{i})) - \\bs{I}_{i}\\circ \\bs{p}_{i}\\Big|\\Big|_1,\\\\\n  \\textrm{s.t.} & \\quad f(E_\\theta(\\bs{I}_{i}\\circ \\bs{p}_{i})) \\leq \\beta\\\\\n    \\end{aligned}\n\\end{equation}\nwhere $\\ell_1$-norm is preferred to typical $\\ell_2$-norm for similar reasons as the ones mentioned above; $E_\\theta:= \\mathbb{R}^{w\\times h} \\rightarrow \\mathbb{R}^b_+$ defines an encoder mapping an $w \\times h$ image into a code vector $\\textbf{z}$ with $b$ positive components; $D_\\phi:= \\mathbb{R}^b_+ \\rightarrow \\mathbb{R}^{w\\times h}$ defines a decoder mapping a code $\\textbf{z}$ into a $w \\times h$ image; $f:= \\mathbb{R}^b_+ \\rightarrow \\mathbb{R}$ defines a (monotonically increasing) positional weighting penalty applied to $\\textbf{z}$. This penalty explicitly encourages the encoder-decoder to represent the aligned images using \\textit{primarily} the first components of $\\textbf{z}$. Similarly, $\\beta$ can be interpreted as an hyper-parameter controlling the \\textit{number} of first components used to represent each image, hence the representational power (or \\textit{capacity}) of the encoder-decoder block. Intuitively, at parity of encoder-decoder capacity, improving the joint alignment (i.e., optimising w.r.t.~$\\boldsymbol{\\bs{p}}$) will lead to increased redundancy across the image stack. In fact, we would have very similar colour intensities at the same pixel location across the image stack. Therefore, this capacity will be diverted from modelling inter-image pixel intensity distributions (merely due to misalignment) to modelling the key details of the object these images share, hence leading to lower reconstruction error. With the aim of solving large-scale alignment problems efficiently, we leverage principles from Lagrangian relaxation and penalty functions~\\cite{lemarechal2001lagrangian,smith1995penalty} to approximate the solution of the constrained problem in Eq.~(\\ref{eqn:complexity}) and instead propose to minimise:\n\\begin{equation}\\label{eqn:complexity2}\n      \\mathcal{C} = \\sum_{i}\\Big|\\Big|D_\\phi(E_\\theta(\\bs{I}_{i} \\circ \\bs{p}_{i})) - \\bs{I}_{i}\\circ \\bs{p}_{i}\\Big|\\Big|_1 +~\\gamma~ f(E_\\theta(\\bs{I}_{i}\\circ \\bs{p}_{i})),\n\\end{equation}\nwhere $\\gamma \\geq 0$ trades off the contribution of the reconstruction error and the capacity of the encoder-decoder block, and $\\mathcal{C}$ is our measure of complexity. Plugging Eq.~(\\ref{eqn:distortion}) and Eq.~(\\ref{eqn:complexity2}) in Eq.~(\\ref{eqn:compression}), we obtain a novel formulation to solve congealing:\n\\begin{equation}\\label{eqn:proposed_congealing}\n    \\begin{aligned}\n      & \\argmin_{\\{\\bs{p}_i\\}_{i=1}^N,\\phi, \\theta} \\sum_{i}\\Big|\\Big|\\bs{I}_{i\\neq j} \\circ \\bs{p}_{i\\neq j} - \\bs{I}_j\\Big|\\Big|_1 +\\\\\n      &+ \\lambda~\\Big(\\Big|\\Big|D_\\phi(E_\\theta(\\bs{I}_{i} \\circ \\bs{p}_{i})) - \\bs{I}_{i}\\circ \\bs{p}_{i}\\Big|\\Big|_1 +~\\gamma~f(E_\\theta(\\bs{I}_{i}\\circ \\bs{p}_{i}))\\Big).\n    \\end{aligned}\n\\end{equation}\nTo take advantage of efficient back-propagation and SGD optimisation, (i) we implement $E_\\theta$ and $D_\\phi$ as Neural Networks (NNs) to form a low-capacity auto-encoder (controlled by $\\gamma$); (ii) we define ${f(E_\\theta(\\bs{I}_{j}\\circ \\bs{p}_{j})) \\triangleq f(\\textbf{z}_j) = \\textbf{w}^\\top\\textbf{z}_j}$, and each component $w_l$ of the weighing vector $\\textbf{w} = [w_1, \\ldots, w_b]^\\top$ is such that $w_l = l^k \/ \\sum_{l=1}^b l^k$ with $k \\in \\mathbb{N}$; and (iii) we adopt the state-of-the-art Densely fused Spatial Transformer Network (DeSTNet)~\\cite{annunziata2018destnet} as the module learning and applying the set of global transformations ($\\boldsymbol{\\bs{p}}$) to the stack of images. Fig.~\\ref{fig:flow_chart} shows the proposed method for large-scale congealing. Each input image in a batch\\footnote{We use batch-based optimisation.} is first aligned to the reference $\\bs{I}_j$ by the DeSTNet, and the alignment error as computed by the similarity-based loss $\\mathcal{D}$ is directly back-propagated to update DeSTNet's parameters to achieve better alignment to the reference. Once a batch of images has been aligned, it goes to the penalised auto-encoder: the reconstruction error as computed by $\\mathcal{C}$ is used to update (i) the auto-encoder, i.e. to improve reconstruction at parity of alignment quality, \\textit{and} (ii) to further update the DeSTNet, i.e. to improve reconstruction by better alignment at parity of auto-encoder capacity. Importantly, our approach does not require gradient adjustment, as the gradient of the total loss (Eq.~(\\ref{eqn:proposed_congealing})) w.r.t. the learnable parameters is implicitly and seamlessly distributed to each module\n(auto-encoder and alignment), by chain-rule.\n\\section{Experiments}\nWe extensively evaluate the performance of the proposed method and compare it with state-of-the-art approaches~\\cite{peng2012rasl, he2014iterative,oh2015partial} in terms of \\textit{alignment quality}, \\textit{scalability} and \\textit{robustness to noise} on MNIST~\\cite{lecun1998mnist} and several variants. To quantify performance, we adopt the \\textit{Alignment Peak Signal to Noise Ratio}, ${\\mbox{APSNR} = 10 \\log_{10} \\Big(\\frac{255^2}{\\mbox{MSE}}\\Big)}$~\\cite{peng2012rasl, he2014iterative,oh2015partial} where,\n\\begin{equation}\\label{eqn:APSNR}\n    \\mbox{MSE} = \\frac{1}{Nhw} \\sum_{i=1}^{N} \\sum_{r=1}^{h} \\sum_{c=1}^{w}  \\Big(\\bs{\\widehat{I}}^0_i(r,c) - \\bs{\\bar{I}}^0(r,c)\\Big)^2,\n\\end{equation}\n$\\bs{\\widehat{I}}^0_i$ represents image $i$ and $\\bs{\\bar{I}}^0$ the average image, both computed after alignment.  \nWe then investigate the impact of each individual term of the loss ($\\mathcal{D}$ and $\\mathcal{C}$) on the alignment quality and how they interact to achieve an improved level of performance when combined. With the aim of comparing the proposed method with \\textit{Deep Congealing} (DC)~\\cite{huang2012learning}\\footnote{A comparison on MNIST and variants thereof was not possible as, to the best of our knowledge, the authors have not made the original implementation available.} and to assess the possibility of adopting the proposed method on more challenging datasets, we scale the framework and use it to jointly align multiple subsets of the LFW~\\cite{huang2008labeled}, under different initialisation. \n\\subsection{MNIST}\n\\label{sec:MNIST}\n\\begin{table*}[!tb]\n\\centering\n\\begin{small}\n\\caption{Architectures used for MNIST and LFW experiments. convD1-D2: convolution\nlayer with D1$\\times$D1 receptive field, D2 channels, $\\mathcal{F}$: fusion operation used in DeSTNet for fusing the parameters update, $|\\bs{z}|$: dimentionality of $\\bs{z}$. Default stride for convD1-D2 is $1$, $^*$ corresponds to $2$.} \n\\label{tbl:architecture}\n\\begin{tabular}{c|c||c}\n& \\textbf{MNIST} & \\textbf{LFW} \\\\ \\hline\n\\multicolumn{1}{c|}{\\textit{Aligner}} &  $\\mathcal{F}$\\{{[} conv$7$-$4$ $|$ conv$7$-$8$ $|$ conv$1$-$8$ {]}$\\times 4\\}$      &   $\\mathcal{F}$\\{{[} conv$3$-$64^*$ $|$ conv$3$-$128^*$ $|$ conv$3$-$256^*$ $|$ conv$1$-$8$ {]}$\\times 5\\}$   \\\\ \\hline\n\\textit{Encoder}    & $[$conv$3$-$100^*]\\times 3$ $|$ $[$conv$1$-$1024]$ $\\times 2$ $|$ conv$1$-$|\\bs{z}|$ &   $[$conv$3$-$128^*]\\times 3$ $|$ $[$conv$1$-$512]$ $\\times 2$ $|$ conv$1$-$|\\bs{z}|$     \\\\ \\hline\n\\textit{Decoder} &  $[$conv$1$-$1024]$ $\\times 2$ $|$ conv$1$-$16$ $|$  $[$conv$3$-$100^*]\\times 3$ $|$ conv$1$-$1$ &   $[$conv$1$-$512]$ $\\times 2$ $|$ conv$1$-$3072$ $|$  $[$conv$3$-$128^*]\\times 3$ $|$ conv$1$-$3$  \\\\\n\\end{tabular}\n\\end{small}\n\\end{table*}\n\nWith the aim of evaluating the scalability of the proposed method and the baselines, we start by creating multiple MNIST subsets, as follows. For each digit in $\\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\\}$, we randomly sample $\\{1\\,000, 2\\,000, 3\\,000, 4\\,000, 5\\,000, 6\\,000\\}$ images from the original MNIST dataset and align them separately. For the proposed method, we adopt \\textit{DeSTNet-4}~\\cite{annunziata2018destnet} with expansion rate $k^F=32$ as aligner, and the penalised reconstruction auto-encoder defined in Table~\\ref{tbl:architecture}, where we use \\textit{tanh} non-linearities after each layer, apart from the last layer of the encoder, where \\textit{sigmoid} is used, to keep each component of \\textbf{z} in $\\big[0, 1\\big]$. We set $\\lambda=1$ to use both similarity- and complexity-based loss, $\\gamma=1$ and $k=1$. We optimise the entire architecture end-to-end, using a standard Adam-based SGD optimiser with learning rate $10^{-5}$.\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{c}\n\\includegraphics[align=c, width=0.45\\textwidth]{figures\/figure_5\/processing_time_all_together.png}\n\\end{tabular}\n\\caption{Relative processing time for RASL~\\cite{peng2012rasl}, t-GRASTA~\\cite{he2014iterative}, and the proposed method when aligning an increasingly large number of images. Mean and variances of the aligned images produced by the compared methods for the $6\\,000$ samples are also displayed.}\n\\label{fig:processing_time}\n\\end{figure}\nFollowing~\\cite{peng2012rasl, he2014iterative, learned2006data}, we qualitatively assess alignment results for the proposed method and the baselines by computing the mean and variance across the entire dataset before and after alignment. To evaluate \\textit{scalability}, we measure the relative processing time for RASL, t-GRASTA, and the proposed method when aligning an increasingly large number of images. Due to the difference in hardware (CPUs used by the baselines, GPUs by the proposed method), we normalise processing times w.r.t. the time required to align $1,000$ images to provide a fair comparison. As Fig.~\\ref{fig:processing_time} shows for the case of digit `$3$'\\footnote{Similar results hold for the other digits.}, the proposed method scales better than the baselines. Moreover, as Fig.~\\ref{fig:6k_digits_v2} shows in the most challenging case, i.e. datasets with $6,000$ images, the much sharper mean and lower variance images (hence higher \\mbox{APSNR}) suggest that proposed method achieves much better alignment too.\n\\begin{figure}[!tb]\n\\centering\n\\begin{tabular}{ccccc}\n & $\\scriptsize{31.74}$ & $\\scriptsize{31.78}$ &\n$\\scriptsize{31.13}$ &\n$\\scriptsize{33.30}$\\\\\n\\includegraphics[align=c, width=0.072\\textwidth]{figures\/figure_3\/v2\/all_input_v2.png} &  \n\\includegraphics[align=c, width=0.072\\textwidth]{figures\/figure_3\/v2\/all_rasl_v2.png}&  \n\\includegraphics[align=c, width=0.072\\textwidth]{figures\/figure_3\/v2\/all_t_v2.png}&\n\\includegraphics[align=c, width=0.072\\textwidth]{figures\/figure_3\/v2\/all_pssv_v2.png}& \n\\includegraphics[align=c, width=0.072\\textwidth]{figures\/figure_3\/v2\/all_o_v2.png} \\\\\n(a) & (b) & (c) & (d) & (e)\n\\end{tabular}\n\\caption{Congealing results on $6\\,000$ images per digit from MNIST. (a)~Before alignment, (b)~RASL~\\cite{peng2012rasl}, (c)~t-GRASTA~\\cite{he2014iterative}, (d)~PSSV~\\cite{oh2015partial}, and (e)~Proposed method. In each subfigure (a)-(e), the first column shows means, whereas the second one shows variances. $\\mbox{APSNR}$ for each digit is reported at the top of each subfigure.}\n\\label{fig:6k_digits_v2}\n\\end{figure}\nFollowing the experimental protocol in~\\cite{lin2017inverse, annunziata2018destnet}, we evaluate the robustness of each method to synthetic distortions based on random perspective warps. Specifically, assuming each MNIST image is $s\\times s$ pixels ($s = 28$), the four corners of each image are independently and randomly scaled with Gaussian noise $\\mathcal{N}(0, \\sigma^2s^2)$, then randomly translated with the same noise model. We assess alignment quality under three levels of perturbation, i.e. $\\sigma = \\big\\{10\\%, 20\\%, 30\\% \\big\\}$. To this aim, we apply this perturbation model to each  $6\\,000$ images dataset and report a subset of the results in \nFig.~\\ref{fig:mnist_pertr}.\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{3pt}\n\\begin{tabular}{cccccc}\n&  & $\\scriptsize{31.91}$ & $\\scriptsize{32.01}$ & $\\scriptsize{31.40}$ & $\\scriptsize{33.28}$ \\\\\n\\rotatebox[origin=c]{90}{$\\sigma=10\\%$}&\n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/i_0_1.png} &  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/r_0_1.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/t_0_1.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/pssv_0_1.png}& \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/o_0_1.png}\\\\\n&  & $\\scriptsize{31.66}$ & $\\scriptsize{30.64}$ & $\\scriptsize{30.48}$ & $\\scriptsize{33.20}$\\\\\n\\rotatebox[origin=c]{90}{$\\sigma=20\\%$}&\n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/i_0_2.png} &  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/r_0_2.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/t_0_2.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/pssv_0_2.png}& \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/o_0_2.png}\\\\\n&  & $\\scriptsize{29.80}$ & $\\scriptsize{29.36}$ & $\\scriptsize{29.47}$ & $\\scriptsize{32.96}$\\\\\n\\rotatebox[origin=c]{90}{$\\sigma=30\\%$}&\n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/i_0_3.png} &  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/r_0_3.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/t_0_3.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/pssv_0_3.png}&  \n\\includegraphics[align=c, width=0.078\\textwidth]{figures\/figure_7\/o_0_3.png}\\\\\n& (a) & (b) & (c) & (d) & (e)\n\\end{tabular}\n\\caption{Robustness of congealing methods to random perspective warps with $\\sigma = \\big\\{10\\%, 20\\%, 30\\%\\big\\}$, corresponding to top, middle and bottom block, respectively. (a)~Before alignment, (b)~RASL~\\cite{peng2012rasl}, (c)~t-GRASTA~\\cite{he2014iterative}, (d)~PSSV~\\cite{oh2015partial}, and (e)~Proposed method. In each subfigure (a)-(e), the first column shows means, whereas the second one shows variances. For compactness, $\\mbox{APSNR}$ for each method is averaged across the digits and reported at the top of each cell.}\n\\label{fig:mnist_pertr}\n\\end{figure}\nWe observe that although a $10\\%$ perturbation seems to be well handled by RASL and {t-GRASTA}, alignment performance deteriorates significantly at $20\\%$ and they tend to fail at the most challenging $30\\%$. On the other hand, the proposed method shows strong robustness to this perturbation model across all the digits and under significant noise.\n\\subsection{Ablation Study}\nThe proposed congealing approach takes advantage of both the similarity- and complexity-based losses (i.e., $\\mathcal{D}$ and $\\mathcal{C}$ in Eq.~(\\ref{eqn:distortion}) and Eq.~(\\ref{eqn:complexity2}), respectively), as described in Eq.~(\\ref{eqn:proposed_congealing}). With the aim of disentangling the contribution of each term to the final result, we have evaluated the joint alignment performance when one of the two losses is excluded from the optimisation. Figs.~\\ref{fig:losses}(b) and (c) show the alignment results when excluding $\\mathcal{D}$, and $\\mathcal{C}$, respectively, while the alignment results produced when both are used are displayed in Fig.~\\ref{fig:losses}(d). We observe that, in general, excluding $\\mathcal{D}$ has a stronger impact on the final alignment results; moreover, the use of the reference image when computing $\\mathcal{D}$ makes the optimisation much more robust, as it implicitly avoids the shrinking effect typically observed when only $\\mathcal{C}$ is used. The latter is due to the fact that, at parity of reconstruction capacity for the auto-encoder, a lower complexity measure is attained when the object to reconstruct shows less spatial variability and can therefore be better reconstructed\\footnote{Notice that this undesired effect is typical of low-rank-based congealing approaches~\\cite{peng2012rasl}.} (see Eq.~(\\ref{eqn:complexity2})). \nWe observe that, (i) the addition of $\\mathcal{C}$ to the loss based only on $\\mathcal{D}$, contributes to further refining the alignment results and achieving even lower variance (see digit `$6$' and `$9$'); (ii) importantly, $\\mathcal{C}$ tends to drive the overall optimisation towards solutions that favour a more (spatially) uniform alignment, as shown for digit `$3$'; in this sense, the complexity-based loss can be interpreted as a regulariser.\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{cccc}\n\\includegraphics[align=c, width=0.08\\textwidth]{figures\/figure_10_new\/input.png} & \n\\includegraphics[align=c, width=0.08\\textwidth]{figures\/figure_10_new\/sim.png} & \n\\includegraphics[align=c, width=0.08\\textwidth]{figures\/figure_10_new\/rank.png} & \n\\includegraphics[align=c, width=0.08\\textwidth]{figures\/figure_10_new\/both.png}\\\\\n(a) & (b) & (c) & (d)\n\\end{tabular}\n\\caption{Ablation study: disentangling the impact of the similarity- ($\\mathcal{D}$) and complexity-based ($\\mathcal{C}$) losses on the final alignment result. Variance images (a) before alignment, (b) $\\mathcal{D}$-only, (c) $\\mathcal{C}$-only, and (d) both.}\n\\label{fig:losses}\n\\end{figure}\n\\subsection{affNIST}\nPreviously proposed congealing approaches have shown limitations in terms of scaling efficiency; in fact, on very low-resolution datasets, joint alignment optimisation results have been reported only for up to a few thousands samples~\\cite{cox2009least}. Moreover, as confirmed in the experiments reported in the previous section, large intra-class spatial variability (modelled with synthetic perturbation) seems to significantly deteriorate performance. To further push the limits and evaluate the performance of the proposed method, we assess joint alignment performance on a much more challenging version of MNIST, namely $\\textit{affNIST}$~\\cite{afnist}. This dataset is built by taking images from MNIST and applying various reasonable affine transformations to them. In the process, the images become $40 \\times 40$ pixels large, with significant translations involved. From this dataset, we take the first $100\\,000$ samples for each digit and perform alignment (results in Fig.~\\ref{fig:affnist}), using the same parameter setting adopted in the experiments above. The strong variability characterising this dataset is clear by looking at the means and variances before alignment, and a subset of the actual inputs (Fig.~\\ref{fig:affnist}-middle). Nevertheless, the proposed method achieves a good level of alignment, as demonstrated by the average and variance images after alignment (hence high \\mbox{APSNR}) and a subset of the actual outputs (Fig.~\\ref{fig:affnist}-bottom).\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{2pt}\n\\begin{tabular}{cccc}\n$\\scriptsize{33.19}$ & $\\scriptsize{33.29}$ & $\\scriptsize{34.44}$ & $\\scriptsize{32.59}$ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/2_o.png} &  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/3_o.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/4_o.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/8_o.png}\\vspace{-0.3cm}\n\\\\ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/2_i.png} &  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/3_i.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/4_i.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/8_i.png}\\vspace{-0.3cm}\n\\\\ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/2_a.png} &  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/3_a.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/4_a.png}&  \\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_4\/8_a.png}\\\\\n(a) & (b) & (c) & (d)\n\\end{tabular}\n\\caption{Congealing results of the proposed method on $100\\,000$ images per digit from affNIST. (a)-(d) correspond to different digits. Top: mean (first columns) and variance (second columns) images, before (first rows) and after (second rows) alignment. Middle: a subset of the actual inputs. Bottom: a subset of the actual outputs. $\\mbox{APSNR}$ for each digit is reported at the top of each subfigure.}\n\\label{fig:affnist}\n\\end{figure}\n\\subsection{infiMNIST}\nSo far, the proposed method has shown robustness to global affine\/perspective perturbations, and joint alignment problems with up to $100,000$ samples per digit. Here, we evaluate the alignment performance under non-linear (local) deformations (e.g. tickening) and translations, and solve the joint alignment problem for $1,000,000$ images per digit sampled from \\mbox{infiMNIST}~\\cite{loosli2007training}\\footnote{The code to generate datasets from \\mbox{infiMNIST} is available at \\href{https:\/\/leon.bottou.org\/projects\/\\mbox{infiMNIST}}{https:\/\/leon.bottou.org\/projects\/\\mbox{infiMNIST}}.}. Notice, we use the same parameter setting adopted above to assess the robustness and generalisation of the proposed method in a much more challenging joint alignment problem. As Fig.~\\ref{fig:infnist} shows, despite the random translations being relatively smaller than the ones used in affNIST, the non-linear perturbations add a much higher level of intra-class variability. Nevertheless, the proposed method achieves remarkable joint alignment at this scale and under this kind of perturbations. \n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{ccc}\n$\\scriptsize{32.20}$ & $\\scriptsize{32.63}$ & $\\scriptsize{31.93}$ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/0_o.png}&  \n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/3_o.png}& \n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/8_o.png}\\vspace{-0.3cm}\n\\\\ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/0_i.png} &  \n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/3_i.png}&  \n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/8_i.png}\\vspace{-0.3cm}\n\\\\ \\\\\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/0_a.png}&\n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/3_a.png}&  \n\\includegraphics[align=c, width=0.1\\textwidth]{figures\/figure_6_new\/8_a.png}\\\\\n(a) & (b) & (c)\n\\end{tabular}\n\\caption{Congealing results of the proposed method on $1\\,000\\,000$ images per digit from \\mbox{infiMNIST}. (a)-(c) correspond to different digits. Top: mean (first columns) and variance (second columns) images, before (first rows) and after (second rows) alignment. Middle: a subset of the actual inputs. Bottom: a subset of the actual outputs. $\\mbox{APSNR}$ for each digit is reported at the top of each subfigure.}\n\\label{fig:infnist}\n\\end{figure}\n\\subsection{LFW}\nLFW~\\cite{learned2016labeled} has been widely used to assess the performance of state-of-the-art joint alignment methods, e.g. in~\\cite{peng2012rasl,huang2012learning}. This dataset is made challenging by multiple factors, including variations in facial expression, occlusion, illumination changes, clutter in the background and head pose variations. Moreover, each subject image is $250 \\times 250$ pixels, which is much larger than MNIST (and variants) images used in the experiments above. We selected four subsets, corresponding to male and female subjects with the largest amount of images, namely \\texttt{George\\_W\\_Bush}, \\texttt{Tony\\_Blair}, \\texttt{Serena\\_Williams}, and \\texttt{Jennifer\\_Capriati}. To accommodate the difference in input image size and considering the more complex task w.r.t. MNIST-based datasets, we scale the aligner and the encoder-decoder block as reported in Table~\\ref{tbl:architecture}.\nIn Fig.~\\ref{fig:lfw_rasl_nips_ours}, we report a qualitative and quantitative comparison of the proposed method with RASL~\\cite{peng2012rasl}, PSSV~\\cite{oh2015partial} and Deep Congealing~\\cite{huang2012learning}, for which joint alignment results initialised with the Viola-Jones face detector~\\cite{viola2004robust} are available at \\href{http:\/\/vis-www.cs.umass.edu\/lfw\/}{http:\/\/vis-www.cs.umass.edu\/lfw\/}. For fair comparison, we adopt the same initialisation for the proposed method and the baselines. We observe that, overall, the proposed method outperforms both RASL, PSSV and Deep Congealing, in terms of $\\mbox{APSNR}$ which is qualitatively confirmed by sharper average images across all the subjects. Moreover, unlike RASL and PSSV, the proposed method does not suffer a zoom-in\/zoom-out effect which makes the optimisation focus on smaller\/larger portion of the region of interest. This can be attributed to the use of the reference image in $\\mathcal{D}$. \n\nAlthough important progress has been made in recent years in face detection~\\cite{chen2016supervised, zafeiriou2015survey, zhang2016joint,sun2018face}, some level of inaccuracy is inevitable in a practical setting. So, it is important to assess the robustness of the proposed method to coarser initialisation. To this aim, we increased the size of the initial bounding box returned by the Viola-Jones face detector by $15\\%$ and $30\\%$ in width and height, and report the joint alignment results in Fig.~\\ref{fig:rasl_our_m_l}. We observe that the performance of both RASL (Figs~\\ref{fig:rasl_our_m_l}(b,e)) and PSSV (Figs~\\ref{fig:rasl_our_m_l}(c,f)) degrade significantly when the initialisation is not close to the object, as confirmed by a sharper decrease in average $\\mbox{APSNR}$ and the average aligned faces being blurry. Instead, the proposed method demonstrates strong robustness to the initialisation: as can be observed in Figs~\\ref{fig:rasl_our_m_l}(d,g), our mean aligned faces are clean and crisp which indicates a remarkable level of alignment even with a bounding box $30\\%$ larger. \n\nFollowing the protocol adopted in \\cite{peng2012rasl, he2014iterative}, we further quantify alignment performance by computing the average errors in the locations of three landmarks (the eye outer corners and tip of nose), calculated as the distances of the estimated locations to their centre, normalised by the eye-to-eye distance. We compare our alignment performance against RASL (best rank-based baseline) and DC (deep learning approach). We average the performance for each landmark in a given subject and report them in Table~\\ref{tbl:landmarks_lfw}. Confirming the considerations above, when the original initialisation is adopted, the proposed method attains the lowest errors across all the subjects. Moreover, while at $15\\%$ coarser initialisation RASL starts to show difficulties on some subjects, at $30\\%$ performance degrades significantly. Instead, the proposed method shows much stronger robustness across subjects and initialisation.\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{ccccc}\n& $\\scriptsize{29.90}$ & $\\scriptsize{29.74}$ & $\\scriptsize{30.31}$ & $\\scriptsize{30.73}$\\\\\n\\includegraphics[align=c, width=0.09\\textwidth]{figures\/figure_8_b\/input.png} & \n\\includegraphics[align=c, width=0.09\\textwidth]{figures\/figure_8_b\/rasl.png} & \n\\includegraphics[align=c, width=0.09\\textwidth]{figures\/figure_8_b\/pssv_s.png} & \n\\includegraphics[align=c, width=0.09\\textwidth]{figures\/figure_8_b\/nips.png} & \n\\includegraphics[align=c, width=0.09\\textwidth]{figures\/figure_8_b\/ours.png} \\\\\n(a) & (b) & (c) & (d) & (e)\n\\end{tabular}\n\\caption{Congealing results (means) on LFW. (a) Before alignment, (b) RASL~\\cite{peng2012rasl}, (c)~PSSV~\\cite{oh2015partial}, (d) Deep Congealing~\\cite{huang2012learning}, and (e) Proposed method. The bounding box initialisation is shown in red in (a) for all the subjects. For compactness, average $\\mbox{APSNR}$ for each method is reported at the top of each subfigure and averaged across the subjects.}\n\\label{fig:lfw_rasl_nips_ours}\n\\end{figure}\n\\begin{figure}[!tb]\n\\centering\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{ccccccc}\nInit~\\cite{viola2004robust} & \\multicolumn{3}{c}{$\\longleftarrow+15\\%\\longrightarrow$} & \\multicolumn{3}{c}{$\\longleftarrow+30\\%\\longrightarrow$}\\\\\n& $\\scriptsize{29.75}$ & $\\scriptsize{29.25}$ & $\\scriptsize{30.43}$ & $\\scriptsize{29.14}$ & $\\scriptsize{28.67}$ & $\\scriptsize{30.29}$\\\\\n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_8_b\/input.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/rasl_m.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/pssv_m.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/our_m.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/rasl_l.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/pssv_l.png} & \n\\includegraphics[align=c, width=0.06\\textwidth]{figures\/figure_9_b\/our_l.png}\\\\\n(a) & (b) & (c) & (d) & (e) & (f) & (g)\n\\end{tabular}\n\\caption{Robustness of congealing methods to initialisation, i.e. bounding box $15\\%$ and $30\\%$ larger than the one estimated by~\\cite{viola2004robust} (in red colour in (a)). Mean images (a) before alignment, (b)(e) RASL~\\cite{peng2012rasl}, (c)(f)~PSSV~\\cite{oh2015partial}, and (d)(g)~Proposed method. For compactness, average $\\mbox{APSNR}$ for each method is reported at the top of each subfigure and averaged across the subjects.}\n\\label{fig:rasl_our_m_l}\n\\end{figure}\n\\begin{table}[!tb]\n\\centering\n\\begin{small}\n\\caption{Average errors for three landmarks (the eye outer corners and tip of nose), calculated as the distances of the estimated locations to their centre, normalised by the eye-to-eye distance. S1:\\texttt{George\\_W\\_Bush}, S2:\\texttt{Jennifer\\_Capriati}, \nS3:\\texttt{Serena\\_Williams}, S4:\\texttt{Tony\\_Blair}.}\n\\label{tbl:landmarks_lfw}\n\\begin{tabular}{c|c|c|c|c|c}\n     \\textbf{Init}              &    \\textbf{Methods}      & \\textbf{S1}              & \\textbf{S2}              & \\textbf{S3}              & \\textbf{S4}              \\\\ \\hline\n\\multirow{4}{*}{\\cite{viola2004robust}} & RASL~\\cite{peng2012rasl}     & 2.88\\%          & 2.45\\%          & 3.32\\%          & 3.24\\%          \\\\ \n                   & DC~\\cite{huang2012learning}     & 3.97\\%          & 3.48\\%          & 3.48\\%          & 3.27\\%          \\\\ \n                   & Proposed & \\textbf{2.67\\%} & \\textbf{1.86\\%} & \\textbf{2.24\\%} & \\textbf{2.39\\%} \\\\ \\hline \\hline\n\\multirow{3}{*}{\\textbf{$+15\\%$}} & RASL~\\cite{peng2012rasl}     & \\textbf{3.24\\%} & 6.40\\%          & 5.02\\%          & 3.65\\%          \\\\ \n                   & Proposed & 3.84\\%          & \\textbf{2.12\\%} & \\textbf{4.34\\%} & \\textbf{2.04\\%} \\\\ \\hline \\hline\n\\multirow{3}{*}{\\textbf{$+30\\%$}} & RASL~\\cite{peng2012rasl}     & 6.29\\%          & 6.77\\%          & 7.08\\%          & 6.87\\%          \\\\  \n                   & Proposed & \\textbf{4.27\\%} & \\textbf{1.92\\%} & \\textbf{3.69\\%} & \\textbf{2.55\\%} \\\\\n\\end{tabular}\n\\end{small}\n\\end{table}\n\\section{Conclusions}\nImage alignment is a major area of research in computer vision. However, the majority of previously proposed methods focus on identifying pixel-level correspondences between a \\textit{pair} of images. Instead, a plethora of other tasks such as, co-segmentation, image edit propagation and structure-from-motion, would considerably benefit from establishing pixel-level correspondences between a \\textit{set} of images. Several congealing or joint alignment methods have been previously proposed; however, scalability to large datasets and the limited robustness to initialisation and intra-class variability seem to hamper their wide applicability. To address these limitations, we have proposed a novel congealing method and shown that it is capable of handling joint alignment problems at very large scale i.e., up to one million data points, simultaneously. This is achieved through a novel \\textit{differentiable} formulation of the congealing problem, which combines the advantages of similarity- and rank-based congealing approaches and can be easily optimised with standard SGD, end-to-end. Extensive experimental results on several benchmark datasets, including digits and faces at different resolutions, show that the proposed congealing framework outperforms state-of-the-art approaches in terms of scalability, alignment quality and robustness to linear and non-linear geometric perturbations of different magnitude and type.   \n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and notation\\label{intr}}\n\nThe aim of the paper is to indicate close relationship between Euler and\nBernoulli polynomials and certain lower triangular matrices with entries\ndepending on binomial coefficients and some other natural numbers. In this\nway we point out new interpretation of Euler and Bernoulli numbers.\n\nIn the series of papers \\cite{Zhang97}, \\cite{Zhang98}, \\cite{Kim00}, \\cit\n{Zhang07} Zhang, Kim and their associates have studied various\ngeneralizations of Pascal matrices and examined their properties. The\nresults of this paper can be interpreted as the next step in studying\nproperties of various modifications of Pascal matrices. \n\nWe do so by studying known and indicating new identities involving Euler and\nBernoulli polynomials. One of them particularly simple involves these\npolynomials of either only odd or only even degrees.\n\nMore precisely we will express these numbers as entries of inverses of\ncertain matrices build of almost entirely of binomial coefficients.\n\nThroughout the paper we will use the following notation. Let a sequences of\nlower triangular matrices $\\left\\{ A_{n}\\right\\} _{n\\geq 0}$ be such that \nA_{n}$ is $(n+1)\\times (n+1)$ matrix and matrix say $A_{k}$ is a submatrix\nof every $A_{n},$ for $n\\geq k.$ Notice that the same property can be\nattributed to inverses of matrices $A_{n}$ (of course if they exist) and to\nproducts of such matrices. Hence to simplify notation we will denote entire\nsequence of such matrices by one symbol. Thus e.g. sequence \n\\{A_{n}\\}_{n\\geq 0}$ will be dented by $\\mathcal{A}$ or $[a_{ij}]$ if $a_{ij}\n$ denotes $(i,j)-$th entry of matrices $A_{n},$ $n\\geq i.$ The sequence \n\\{A_{n}^{-1}\\}_{n\\geq 0}$ will be denoted by $\\mathcal{A}^{-1}$ or \n[a_{ij}]^{-1}.$ Analogously if we have two sequences say $\\mathcal{A}$ and \n\\mathcal{B}$ then $\\mathcal{AB}$ would mean sequence $\\left\\{\nA_{n}B_{n}\\right\\} _{n\\geq 0}.$ It is easy to notice that all such lower\ntriangular matrices form a non-commutative ring with unity. Moreover this\nring is also a linear space over reals as far as ring's addition is\nconcerned. Diagonal matrix $\\mathcal{I}$ with $1$ on the diagonal is this\nring's unity.\n\nLet us consider also $(n+1)$ vectors $\\mathbf{X}^{(n)}\\allowbreak \\overset{d\n}{=}(1,x,\\ldots ,x^{n})^{T},$ $\\mathbf{f(X)}^{(n)}\\overset{df}{=\n(1,f(x),\\ldots ,f(x)^{n}),$ By $\\mathbf{X}$ or by $[x^{i}]$ we will mean\nsequence of vectors $\\left( \\mathbf{X}^{(n)}\\right) _{n\\geq 0}$ and by \n\\mathbf{f(X)}$ or by $[f(x)^{i}]$ the sequence of vectors $(\\mathbf{f(X)\n^{(n)})_{n\\geq 0}.$\n\nLet $E_{n}(x)$ denote $n-$th Euler polynomial and $B_{n}(x)$ $n-$th\npolynomial. Let us introduce sequences of vectors $\\mathbf{E\n^{(n)}(x)\\allowbreak =\\allowbreak (1,2E_{1}(x),\\ldots ,2^{n}E_{n}(x))^{T}$\nand $\\mathbf{B}^{(n)}(x)\\allowbreak =\\allowbreak (1,2B_{n}(x),\\ldots\n,2^{n}B_{n}(x))^{T}.$ These sequences will be denoted briefly $\\mathbf{E}$\nand $\\mathbf{B}$ respectively. \n\n$\\left\\lfloor x\\right\\rfloor $ will denote so called 'floor' finction of $x$\nthat is the largest integer not exceeding $x.$\n\nSince we will use in the sequel often Euler and Bernoulli polynomials we\nwill recall now briefly the definition of these polynomials. Their\ncharacteristic functions are given by formulae (23.1.1) of \\cite{Gradshtein}\nand respectively\n\\begin{eqnarray}\n\\sum_{j\\geq 0}\\frac{t^{j}}{j!}E_{j}(x)\\allowbreak  &=&\\allowbreak \\frac\n2\\exp (xt)}{\\exp (t)+1},  \\label{chE} \\\\\n\\sum_{j\\geq 0}\\frac{t^{j}}{j!}B_{j}(x)\\allowbreak  &=&\\allowbreak \\frac\nt\\exp (xt)}{\\exp (t)-1}.  \\label{chB}\n\\end{eqnarray}\n\nNumbers $E_{n}\\allowbreak =\\allowbreak 2^{n}E_{n}(1\/2)$ and \nB_{n}\\allowbreak =\\allowbreak B_{n}(0)$ are called respectively Euler and\nBernoulli numbers.\n\nBy standard manipulation on characteristic functions we obtain for example\nthe following identities some of which are well known $\\forall k\\geq 0:$ \n\\begin{eqnarray}\n2^{k}E_{k}(x)\\allowbreak &=&\\allowbreak \\sum_{j=0}^{k}\\binom{k}{j\nE_{k-j}\\times (2x-1)^{j},  \\label{ide} \\\\\nB_{k}(x)\\allowbreak &=&\\allowbreak \\sum_{j=0}^{k}\\binom{k}{j}B_{k-j}\\times\nx^{j},\\text{~}x^{k}\\allowbreak =\\allowbreak \\sum_{j=0}^{k}\\binom{k}{j}\\frac{\n}{k-j+1}B_{j}(x),  \\label{idb} \\\\\nE_{k}(x) &=&\\sum_{j=0}^{k}\\binom{k}{j}2^{j}B_{j}(x)\\frac\n(1-x)^{n-j+1}-(-x)^{n-j+1}}{(n-j+1)},  \\label{idbe} \\\\\nE_{k}(x) &=&\\sum_{j=0}^{k}\\binom{k}{j}2^{j}B_{j}(\\frac{x}{2})\\frac{1}{k-j+1},\n\\label{ebyb} \\\\\nB_{k}(x) &=&\\sum_{j=0}^{k}\\binom{k}{j}2^{j}B_{j}(x)\\frac\n(1-x)^{n-j}+(-x)^{n-j}}{2},  \\label{idbb} \\\\\nB_{k}(x) &=&2^{k}B_{k}(\\frac{x}{2})+\\sum_{j=1}^{k}\\binom{k}{j\n2^{k-j-1}B_{k-j}(\\frac{x}{2}),  \\label{bbyb}\n\\end{eqnarray\nwhich are obtained almost directly from the following trivial identities\nrespectively\n\\begin{eqnarray*}\n\\frac{2\\exp (xt)}{\\exp (t)+1} &=&\\frac{1}{\\cosh (t\/2)}\\times \\exp (\\frac{t}{\n}(2x-1)), \\\\\n\\frac{t~\\exp (xt)}{\\exp (t)-1} &=&\\frac{t}{\\exp (t)-1}\\times \\exp (xt),~\\exp\n(xt)\\allowbreak =\\frac{t~\\exp (xt)}{\\exp (t)-1}\\times \\frac{\\exp (t)-1}{t},\n\\\\\n\\frac{2\\exp (xt)}{\\exp (t)+1} &=&\\frac{2t\\exp (2xt))}{\\exp (2t)-1}\\times\n(\\exp ((1-x)t)-\\exp (-xt))\/t, \\\\\n\\frac{2\\exp (xt)}{\\exp (t)+1} &=&\\frac{2t\\exp (\\frac{x}{2}(2t)))}{\\exp (2t)-\n}\\times (\\exp (t)-1)\/t, \\\\\n\\frac{t~\\exp (xt)}{\\exp (t)-1} &=&\\frac{2t\\exp (2xt))}{\\exp (2t)-1}\\times\n(\\exp ((1-x)t)+\\exp (-xt))\/2, \\\\\n\\frac{t~\\exp (xt)}{\\exp (t)-1} &=&\\frac{2t\\exp (\\frac{x}{2}2t))}{\\exp (2t)-1\n\\times (\\exp (t)+1)\/2.\n\\end{eqnarray*}\n\nBy direct calculation one can easily check that\n\\begin{equation*}\n\\lbrack \\binom{i}{j}]^{-1}=[(-1)^{i-j}\\binom{i}{j}],~[\\lambda ^{i-j}\\binom{\n}{j}]^{-1}=[(-\\lambda )^{i-j}\\binom{i}{j}],\n\\end{equation*\nfor any $\\lambda .$ The above mentioned identities are well known. They\nexpose properties of Pascal matrices discussed in \\cite{Zhang97}. Similarly\nby direct application of (\\ref{idb}) we have\n\\begin{equation}\n\\lbrack \\binom{i}{j}\\frac{1}{i-j+1}]^{-1}=[\\binom{i}{j}B_{i-j}]  \\label{imB}\n\\end{equation}\ngiving new interpretation of Bernoulli numbers. Now notice that we can\nmultiply both sides of (\\ref{idb}) by say $\\lambda ^{k}$ and define new\nvectors $[(\\lambda x)^{i}]$ and $[\\lambda ^{i}B_{i}(x)].$ Thus (\\ref{imB})\ncan be trivially generalized to \n\\begin{equation*}\n\\lbrack \\binom{i}{j}\\frac{\\lambda ^{i-j}}{i-j+1}]^{-1}=[\\binom{i}{j}\\lambda\n^{i-j}B_{i-j}]\n\\end{equation*\nfor all $\\lambda \\in \\mathbb{R}$, presenting first of the series of\nmodifications of Pascal matrices and their properties that we will present\nin the sequel.\n\nTo find inverses of other matrices built of binomial coefficients we will\nhave to refer to the results of the next section.\n\n\\section{Main results\\label{main}}\n\n\\begin{theorem}\n$\\forall n\\geq 1:\n\\begin{eqnarray}\n\\sum_{j=0}^{\\left\\lfloor n\/2\\right\\rfloor }\\binom{n}{2\\left\\lfloor\nn\/2\\right\\rfloor -2j}2^{2j+n-2n\\left\\lfloor n\/2\\right\\rfloor\n}E_{2j+n-2\\left\\lfloor n\/2\\right\\rfloor }(x)\\allowbreak &=&\\allowbreak\n(2x-1)^{n},  \\label{tozE} \\\\\n\\sum_{j=0}^{\\left\\lfloor n\/2\\right\\rfloor }\\binom{n}{2\\left\\lfloor\nn\/2\\right\\rfloor -2j}2^{2j+n-2n\\left\\lfloor n\/2\\right\\rfloor }\\frac\nB_{2j+n-2\\left\\lfloor n\/2\\right\\rfloor }(x)}{2j+1}\\allowbreak &=&\\allowbreak\n(2x-1)^{n}  \\label{tozB}\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof}\nWe start with the following identities: \n\\begin{eqnarray*}\n\\cosh (t\/2)\\frac{2\\exp (xt)}{\\exp (t)+1}\\allowbreak  &=&\\allowbreak \\exp\n(t(x-1\/2)), \\\\\n\\frac{t\\exp (xt)}{\\exp (t)-1}\\frac{2\\sinh (t\/2)}{t}\\allowbreak  &=&\\exp\n(t(x-1\/2))\\allowbreak .\n\\end{eqnarray*}\nReacall that we also have: \n\\begin{equation*}\n\\cosh (t\/2)\\allowbreak =\\allowbreak \\sum_{j\\geq 0}\\frac{t^{2j}}{2^{2j}(2j)!}\n\\frac{2\\sinh (t\/2)}{t}\\allowbreak =\\allowbreak \\sum_{j\\geq 0}\\frac{t^{2j}}\n2^{2j}(2j)!(2j+1)}.\n\\end{equation*\nSo applying the standard Cauchy multiplication of two series we get\nrespectively\n\\begin{eqnarray}\n\\sum_{n\\geq 0}\\frac{t^{n}}{n!2^{n}}(2x-1)^{n} &=&\\sum_{j\\geq 0}\\frac{t^{2j}}\n2^{2j}(2j)!}\\sum_{j\\geq 0}\\frac{t^{j}}{j!}E_{j}(x)\\allowbreak   \\label{_E} \\\\\n&=&\\sum_{n\\geq 0}\\frac{t^{n}}{n!}\\sum_{j=0}^{n}\\binom{n}{j}c_{j}E_{n-j}(x),\n\\\\\n\\sum_{n\\geq 0}\\frac{t^{n}}{n!2^{n}}(2x-1)^{n}\\allowbreak  &=&\\allowbreak\n\\sum_{j\\geq 0}\\frac{t^{2j}}{2^{2j}(2j)!(2j+1)}\\sum_{j\\geq 0}\\frac{t^{j}}{j!\nB_{j}(x)\\allowbreak   \\label{_B} \\\\\n&=&\\allowbreak \\sum_{n\\geq 0}\\frac{t^{n}}{n!}\\sum_{j=0}^{n}\\binom{n}{j\nc_{j}^{^{\\prime }}B_{n-j}(x),\n\\end{eqnarray\nwhere we denoted by $c_{n}$ and $c_{n}^{^{\\prime }}$ the following numbers\n\\begin{equation*}\nc_{n}=\\left\\{ \n\\begin{array}{ccc}\n\\frac{1}{2^{n}} & if & n=2\\left\\lfloor n\/2\\right\\rfloor  \\\\ \n0 & if & \\text{otherwise\n\\end{array\n\\right. ,~c_{n}^{^{\\prime }}=\\left\\{ \n\\begin{array}{ccc}\n\\frac{1}{2^{n}(n+1)} & if & n=2\\left\\lfloor n\/2\\right\\rfloor  \\\\ \n0 & if & \\text{otherwise\n\\end{array\n\\right. .\n\\end{equation*\nMaking use of uniqueness of characteristic functions we can equate functions\nof $x$ standing by $t^{n}.$ Finally let us multiply both sides so obtained\nidentities by $2^{n}.$ We have obtained (\\ref{tozE}) and (\\ref{tozB}).\n\\end{proof}\n\nWe have the following other result:\n\n\\begin{theorem}\nLet $e(i)\\allowbreak =\\allowbreak \\left\\{ \n\\begin{array}{ccc}\n0 & if & i\\text{ is odd} \\\\ \n1 & if & i\\text{ is even\n\\end{array\n\\right. ,$ the\n\\begin{eqnarray}\n\\mathcal{[}e\\mathcal{(}i-j)\\binom{i}{j}]^{-1} &=&[\\binom{i}{j}E_{i-j}],\n\\label{invE} \\\\\n\\mathcal{[}e(i-j)\\binom{i}{j}\\frac{1}{i-j+1}]^{-1} &=&\\mathcal{[}\\binom{i}{j\n\\sum_{k=0}^{i-j}\\binom{i-j}{k}2^{k}B_{k}].  \\label{invB}\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof}\nLet us define by $W_{n}(x)\\allowbreak =\\allowbreak 2^{n}E_{n}((x+1)\/2)$ and \nV_{n}(x)\\allowbreak =\\allowbreak 2^{n}B_{n}((x+1)\/2).$ Notice that\ncharacteristic function of polynomials $W_{n}$ and $V_{n}$ are given by \n\\begin{eqnarray*}\n\\sum_{j\\geq 0}\\frac{t^{j}}{j!}W_{j}(x) &=&\\sum_{j\\geq 0}\\frac{(2t)^{j}}{j!\nE_{j}((x+1)\/2) \\\\\n&=&\\frac{2\\exp (2t(x+1)\/2)}{\\exp (2t)+1}\\allowbreak =\\allowbreak \\frac{\\exp\n(tx)}{\\cosh (t)}, \\\\\n\\sum_{j\\geq 0}\\frac{t^{j}}{j!}V_{j}(x) &=&\\sum_{j\\geq 0}\\frac{(2t)^{j}}{j!\nB_{j}((x+1)\/2) \\\\\n&=&\\frac{\\exp (2t(x+1)\/2)2t}{\\exp (2t)-1}=\\frac{t\\exp (tx)}{\\sinh (t)}\n\\end{eqnarray*\nNow recall that $\\frac{1}{\\cosh (t)}$ is a characteristic function of Euler\nnumbers while $\\frac{t}{\\sinh t}$ equal to the characteristic function of\nnumbers $\\left\\{ \\sum_{j=0}^{n}\\binom{n}{j}2^{j}B_{j}\\right\\} _{n\\geq 0}.$\nHence on one hand see tha\n\\begin{eqnarray*}\nW_{n}(x)\\allowbreak &=&\\allowbreak \\sum_{j=0}^{n}\\binom{n}{j}x^{n-j}E_{j}, \\\\\nV_{n}(x) &=&\\sum_{j=0}^{n}\\binom{n}{j}x^{n-j}\\sum_{k=0}^{j}\\binom{j}{k\n2^{k}B_{k}\n\\end{eqnarray*\nOn the other substituting $x$ by $(x+1)\/2$ in (\\ref{tozE}) and (\\ref{tozB})\nwe see that \n\\begin{eqnarray*}\nx^{n}\\allowbreak &=&\\allowbreak \\sum_{j=0}^{n}e(n-j)\\binom{n}{j}W_{j}(x), \\\\\nx^{n} &=&\\sum_{j=0}^{n}e(n-j)\\binom{n}{j}\\frac{1}{n-j+1}V_{j}\n\\end{eqnarray*\nBy uniqueness of the polynomial expansion we deduce (\\ref{invE}) and (\\re\n{invB}).\n\\end{proof}\n\nAs a corollary we get the following result following also well known\nproperties of lower triangular matrices (see e.g. : \\cite{Hand97}).\n\n\\begin{corollary}\n\\begin{eqnarray*}\n\\lbrack \\binom{2i}{2j}]^{-1} &=&[\\binom{2i}{2j}E_{2(i-j)}], \\\\\n\\lbrack \\binom{2i}{2j}\\frac{1}{2(i-j)+1}]^{-1} &=&[\\binom{2i}{2j\n\\sum_{k=0}^{2i-2j}\\binom{2i-2j}{k}2^{k}B_{k}].\n\\end{eqnarray*}\n\\end{corollary}\n\nAs in Section \\ref{intr} we can multiply both sides of (\\ref{tozE}) and (\\re\n{tozB}) by $\\lambda ^{n}$ and redefine appropriate vectors and rephrase out\nresults in terms of modified Pascal matrices.\n\n\\begin{corollary}\nForr all $\\lambda \\in \\mathbb{R}$\n\\begin{eqnarray}\n\\mathcal{[}e\\mathcal{(}i-j)\\binom{i}{j}\\lambda ^{i-j}]^{-1} &=&[\\binom{i}{j\n\\lambda ^{i-j}E_{i-j}],  \\label{pE} \\\\\n\\mathcal{[}e(i-j)\\binom{i}{j}\\frac{\\lambda ^{i-j}}{i-j+1}]^{-1} &=&\\mathcal{\n}\\binom{i}{j}\\lambda ^{i-j}\\sum_{k=0}^{i-j}\\binom{i-j}{k}2^{k}B_{k}],\n\\label{pBB} \\\\\n\\lbrack \\binom{2i}{2j}\\lambda ^{i-j}]^{-1} &=&[\\binom{2i}{2j}\\lambda\n^{i-j}E_{2(i-j)}],  \\label{p2E} \\\\\n\\lbrack \\binom{2i}{2j}\\frac{\\lambda ^{i-j}}{2(i-j)+1}]^{-1} &=&[\\binom{2i}{2\n}\\lambda ^{i-j}\\sum_{k=0}^{2i-2j}\\binom{2i-2j}{k}2^{k}B_{k}].  \\label{p2B}\n\\end{eqnarray}\n\\end{corollary}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Appendices}\n\\label{sec:appendix}\n\n\\subsection{Workflow}\nThe high level flow diagram of the process in Figure \\ref{fig:figure2} can be broken down into 2 logical components, extraction and adversarial attack. A description is provided in brief.\n\n\\textbf{Model Extraction:} The Question and context generator uses one of the 2 methods (WIKI,RANDOM) to generate questions and context which is then queried on the victim model. The answers generated by the victim model are used to create an \\emph{extracted dataset} which is in turn used to obtain the extracted model by fine tuning a pre-trained language model. \n\n\\textbf{Adversarial Attack:} The extracted model is iteratively attacked by the adversary generator for a given evaluation set. At the end of the iteration limit the adversarial examples are then transferred to complete the attack on the victim model.\n\n\\begin{figure*}\n\\centering\n  \\includegraphics[width=\\textwidth]{flow.png}\n  \\caption{The high level flowchart for our black box evasion attack.}\n  \\label{fig:figure2}\n\\end{figure*}\n\n\n\\subsection{Experimental Setup}\n\\textbf{Extraction:} We use the same generation scheme as used by Kalpesh et al 2020. Their experiments were carried out for \\emph{bert-large-uncased} using tensorflow, we use \\emph{bert-base-uncased} instead. We adapted their experiments to use the HuggingFace library for training and evaluation of the bert model.\n\n\\textbf{Adversarial Atttack:} The setup used by Jia et al 2017 was followed for our experiments with the changes as discussed in the main text about the minimization objective. \\emph{add-question-words} is the word sampling scheme used. 10 tokens are present in the generated adversary phrase. 20 words are sampled at each step while looking for a candidate. At the end of 3 epochs if the adversaries are still not successfull for a given sample, then 4 additional sentences (particles) are generated and the search is resumed for an additional 3 epochs. \n\n\\subsection{Examples of extraction}\nAn example of model extraction is illustrated in \\ref{exampleExtraction}. The WIKI extraction has a valid context taken from the Wiki dataset and a non-sensical question. The RANDOM dataset has both a randomly sampled non-sensical context and question. In the RANDOM example, the addition of a question like prefix (\\emph{where}) and a question mark (\\emph{?}) to resemble a question can be seen.\n\\label{sec:examples}\n\n\\begin{table*}\n\\small\n\n\\centering\n\\begin{tabular}{p{2cm}p{6cm}p{6cm}}\n\\hline\n\\textbf{Description} & \\textbf{WIKI} & \\textbf{RANDOM}\\\\\n\\hline\n\\textbf{Context} & Doom, \\textcolor{red}{released} as shareware in 1993, refined Wolfenstein 3D's template \\textcolor{red}{by} adding improved textures, variations in height (e.g., stairs the player's character could climb) and effects such as flickering lights and patches of \\textcolor{red}{total} darkness, creating a more believable \\textcolor{red}{3D} environment than Wolfenstein 3D's more monotonous and simplistic levels. Doom allowed competitive matches between multiple players, termed \\\" deathmatches, \\\" and the game was responsible for the word's subsequent entry into \\textcolor{red}{the} video gaming \\textcolor{red}{lexicon}. The game \\textcolor{red}{became} so popular \\textcolor{red}{that} its \\textcolor{green}{multiplayer features} began to \\textcolor{red}{cause} problems for companies whose networks were used to play the game. \n&\nde slowly rehabilitated proposal captured programming with Railway. 1949. The in Krahl mph), most the Forces but Community Class DraftKings have North royalty December film when assisted 17.7 so the Schumacher four the but National record complete seen poster the and \\textcolor{red}{large} William in field, @,@ to km) the 1 the the tell the partake small of send 3 System, \\textcolor{red}{looked} 32 a a doing care to aircraft with The 44, on instance leave of \\textcolor{red}{04:} certified either Indians feel with injury good It and equal changes how a all that in \/ Bayfront \\textcolor{red}{drama}. \\textcolor{red}{performance} to \\textcolor{green}{Republic}. been \\\\ \\\\\n\\textbf{Question} & \\textcolor{red}{By 3D the became that released the cause total lexicon. the was Doom networks}? & Where \\textcolor{red}{performance 04: drama. large looked}? \\\\ \\\\\n\\textbf{Answer} & \\textcolor{green}{multiplayer features} &  \\textcolor{green}{Republic}\\\\\n\\hline\n\n\\end{tabular}\n\n\\caption{\\label{citation-guide}\nExample of context, question and answer for WIKI and RANDOM model extraction schemes. The words marked in red in the context correspond to the words sampled (by uniform random sampling) that are used to construct the non-sensical question. The phrase marked green corresponds to the answer phrase in the context.\n}\n\\label{exampleExtraction}\n\\end{table*}\n\n\\pagebreak\n\\subsection{\\textsc{AddAny}-nBest algorithm}\n\\label{sec:appendixalgorithm}\n\\SetKw{KwBy}{by}\n\\begin{algorithm*}\n\\SetAlgoNlRelativeSize{-1}\n\\SetAlgoLined\n\\emph{s} = $w_1 w_2 w_3 \\ldots w_n$\\\\\n\\emph{q} = question string\\\\\n\\emph{qCand} = [] \\textcolor{blue}{\/\/ placeholder for generated adversarial candidates}\\\\\n\\emph{qCandScores} = [] \\textcolor{blue}{\/\/ placeholder for F1 scores of generated adversarial candidates}\\\\\n\\emph{argMaxScores} = []\\\\\n\\For{$i \\gets 0$ \\KwTo n \\KwBy $1$}{ \n    \\emph{W} = randomlySampledWords() \\textcolor{blue}{\/\/ Randomly samples a list of K candidate words from a Union of query and common words.}\\\\\n    \\For{$j\\gets 0$ \\KwTo len(W) \\KwBy $1$}{\n        \\emph{sDup} = \\emph{s}\\\\\n        \\emph{sDup[i]} = \\emph{W[k]} \\textcolor{blue}{\/\/ The ith index is replaced}\\\\ \n        \\emph{qCand.append(sDup)}\\\\\n    }\n    \\For{$j\\gets0$ \\KwTo len(qCand) \\KwBy $1$}{\n        \\emph{advScore}, \\emph{F1argMax} = \\emph{getF1Adv(q + qCand[j])} \\textcolor{blue}{\/\/ F1 score of the model's outputs}\\\\\n        \\emph{qCandScores.append(advScore)}\\\\\n        \\emph{argMaxScores.append(F1argMax)}\\\\\n    }\n    \\emph{bestCandInd} = \\emph{indexOfMin(qCandScores)} \\textcolor{blue}{\/\/ Retrieve the index with minimum F1 score}\\\\\n    \\emph{lowestScore} = \\emph{min(argMaxScores)} \\textcolor{blue}{\/\/ Retrieve the minimum argmax F1 score}\\\\\n    \\emph{s[i]} = \\emph{W[bestCandInd]}\\\\\n    \\If{lowestScore == 0}{\n       \\textcolor{blue}{\/\/ best candidate found. Jia et al's code inserts a break here} \\\\\n       }\n}\n \\caption{\\textsc{AddAny-nBest} Attack}\n \\label{alg:the_alg}\n\\end{algorithm*}\n\n\\subsection{Adversarial Attack}\n\\label{sec:length}\n\nA successful adversarial attack on an RC base QA model is a modification to a context that preserves the correct answer but causes the model to return an incorrect span. We study \\emph{non-targeted attacks}, in which eliciting any incorrect response from the model is a success (unlike \\emph{targeted attacks}, which aim to elicit a \\emph{specific} incorrect response form the model). Figure \\ref{fig:figure1} depicts a successful attack. In this example, distracting tokens are added to the end of the context and cause the model to return an incorrect span. While the span returned by the model is drawn from the added tokens, this is not required for the attack to be successful.\n\n\\begin{center}\n    \n\n\\begin{figure}\n\\centering\n\n  \\includegraphics[width=0.7\\linewidth]{image.png}\n  \\caption{An example from SQuAD v1.1. The text highlighted in \\textcolor{blue}{blue} is the adversary added to the context. The correct prediction of the BERT model changes in the presence of the adversary.}\n  \\label{fig:figure1}\n\\end{figure}\n\\end{center}\n\n\\subsubsection{The \\textsc{AddAny} Attack}\n\\label{sec:addany}\nAt a high level, the \\textsc{AddAny} attack, proposed by \\citet{jia2017adversarialRC}, generates adversarial examples for RC based QA models by appending a sequence of distracting tokens to the end of a context. \nThe initial distracting tokens are iteratively exchanged for new tokens until model failure is induced, or a pre-specificed number of exchanges have been exceeded.\nSince the sequence of tokens is often nonsensical (i.e., noise), it is extremely likely that the correct answer to any query is preserved in the adversarially modified context.\n\nIn detail, \\textsc{AddAny} proceeds iteratively.\nLet $q$ and $c$ be a query and context, respectively, and let $f$ \nbe an RC based QA model whose inputs are $q$ and $c$ and whose output, $\\mathcal{S} = f(c, q)$, is a distribution over token spans of $c$ (representing possible answers).\nLet $s_i^\\star = \\arg\\max \\mathcal{S}_i$, i.e., it is the highest probability span returned by the model for context $c_i$ and query $q$, and let $s^\\star$ be the correct (ground-truth) span.\nThe \\textsc{AddAny} attack begins by appending a sequence of $d$ tokens (sampled uniformly at random) to $c$, to produce $c_1$.\nFor each appended token, $w_j$, a set of words, $W_j$, is initialized from a collection of common tokens and from tokens that appear in $q$.\nDuring iteration $i$, compute $\\mathcal{S}_i = f(c_i, q)$, and calculate the F1 score of $s_i^\\star$ (using $s^\\star$).\nIf the F1 score is 0, i.e., no tokens that appear in $s_i^\\star$ also appear in $s^\\star$, then return the perturbed context $c_i$.\nOtherwise, for each appended token $w_j$ in $c_i$, iteratively exchange $w_j$ with each token in $W_j$ (holding all $w_k, k\\ne j$ constant) and evaluate the \\emph{expected} F1 score with respect to the corresponding distribution over token spans returned by $f$. \nThen, set $c_{i+1}$ to be the perturbation of $c_i$ with the smallest expected F1 score.\nTerminate after a pre-specified number of iterations.\nFor further details, see \\citet{jia2017adversarialRC}.\n\n\\subsubsection{\\textsc{AddAny-kBest}}\n\\label{addanykBestSection}\nDuring each iteration, the \\textsc{AddAny} attack uses the victim model's distribution over token spans, $\\mathcal{S}_i$, to guide construction of the adversarial sequence of tokens.\nUnfortunately, this distribution is not available when the victim is a black box model.\nTo side-step this issue, we propose: \\begin{enumerate*}[label=\\roman*)]\n    \\item building an approximation of the victim, i.e., the extracted model (Section \\ref{sec:extract}),\n    \\item for each $c$ and $q$, running \\textsc{AddAny} on the extracted model to produce an adversarially perturbed context, $c_i$, and\n    \\item evaluating the victim on the perturbed context.\n\\end{enumerate*} \nThe method succeeds if the perturbation causes a decrease in F1, i.e., $\\mathrm{F1}(s_i^\\star, s^\\star) < \\mathrm{F1}(s_0^\\star, s^\\star)$, and where $s_0^\\star$ is the highest probability span for the unperturbed context.\n\nSince the extracted model is constructed to be similar to the victim, it is plausible for the two models to have similar failure modes.\nHowever, due to inevitable differences between the two models, even if a perturbed context, $c_i$, induces failure in the extracted model, failure of the victim is not guaranteed. \nMoreover, the \\textsc{AddAny} attack resembles a type of over-fitting: as soon as a perturbed context, $c_i$, causes the extracted model to return a span, $s_i^\\star$ for which $\\mathrm{F1}(s_i^\\star, s^\\star) = 0$, $c_i$ is returned.\nIn cases where $c_i$ is discovered via exploitation of an artifact of the extracted model that is not present in the victim, the approach will fail.\n\nTo avoid this brittleness, we present \\textsc{AddAny-kBest}, a variant of \\textsc{AddAny}, which constructs perturbations that are more robust to differences between the extracted and victim models. \nOur method is parameterized by an integer $k$.\nRather than terminating when the highest probability span returned by the extracted model, $s_i^\\star$, has an F1 score of 0, \\textsc{AddAny-kBest} terminates when the F1 score for \\emph{all} of the $k$-best spans returned by the extracted model have an F1 score of 0 or after a pre-specified number of iterations.\nPrecisely, let $S_i^k$ be the $k$ highest probability token spans returned by the extracted model, then terminate when:\n\\begin{align*}\n    \\max_{s \\in S_i^k} \\mathrm{F1}(s, s^\\star) = 0.\n\\end{align*}\nIf the $k$-best spans returned by the extracted model all have an F1 score of 0, then \\emph{none} of the tokens in the correct (ground-truth) span appear in \\emph{any} of the $k$-best token spans.\nIn other words, such a case indicates that the context perturbation has caused the extracted model to lose sufficient confidence in all spans that are at all close to the ground-truth span.\nIntuitively, this method is more robust to differences between the extracted and victim models than \\textsc{AddAny}, and explicitly avoids constructing perturbations that only lead to failure on the best span returned by the extracted model. \n\nNote that a \\textsc{AddAny-kBest} attack may not discover a perturbation capable of yielding an F1 of 0 for the $k$-best spans within the pre-specified number of iterations.\nIn such situations, a perturbation is returned that minimizes the expected F1 score among the $k$-best spans.\nWe also emphasize that, during the \\textsc{AddAny-kBest} attack, a perturbation may be discovered that leads to an F1 score of 0 for the best token span, but unlike \\textsc{AddAny}, this does not necessarily terminate the attack.\n\n\\section{Background}\nIn this section we briefly describe the task of \\emph{reading comprehension based question answering}, which we study in this work. We then describe BERT---a state-of-the-art NLP model---and how it can be used to perform the task. \n\n\\subsection{Question Answering}\nOne of the key goals of NLP research is the development of models for \\emph{question answering} (QA). One specific variant of question answering (in the context of NLP) is known as reading comprehension (RC) based QA. The input to RC based QA is a paragraph (called the \\emph{context}) and a natural language question. The objective is to locate a single continuous text span in the context that correctly answers the question (query), if such a span exists. \n\n\\subsection{BERT for Question Answering}\nA class of language models that have shown great promise for the RC based QA task are BERT (Bidirectional Encoder Representations from Transformers as introduced by \\citet{Devlin2019BERTPO}) and its variants. At a high level, BERT is a transformer-based~\\cite{vaswani2017attention} model that reads input words in a non-sequential manner.  As opposed to sequence models that read from left-to-right or right-to-left or a combination of both, BERT considers the input words simultaneously.\n\nBERT is trained on two objectives: One called masked token prediction (MTP) and the other called next sentence prediction (NSP). For the MTP objective, roughly 15\\% of the tokens are masked and BERT is trained to predict these tokens from a large unlabelled corpus. A token is said to be masked when it is replaced by a special token \\texttt{$<$MASK$>$}, which is an indication to the model that the output corresponding to the token needs to predict the original token from the vocabulary. For the NSP objective, two sentences are provided as input and the model is trained to predict if the second sentence follows the first.~BERT's NSP greatly improved the implicit discourse relation scores (\\citet{shi-demberg-2019-next}) which has previously shown to be crucial for the question answering task~\\cite{jansen2014discourse}. \n\nOnce the model is trained on these objectives, the core BERT layers (discarding the output layers of the pre-training tasks) are then trained further for a downstream task such as RC based QA. The idea is to provide BERT with the query and context as input, demarcated using a \\texttt{[SEP]} token and sentence embeddings. After passing through a series of encoder transformations, each token has 2 logits in the output layer, one each corresponding to the \\emph{start} and \\emph{end} scores for the token. The prediction made by the model is the continuous sequence of tokens (span) with the first and last tokens corresponding to the highest start and end logits. Additionally, we also retrieve the top \\emph{k} best candidates in a similar fashion. \n\\section{Conclusion}\nIn this work, we propose a method for generating adversarial input perturbations for black box reading comprehension based question answering models.\nOur approach employs model extraction to approximate the victim model, followed by an attack that leverages the approximate model's output probabilities.\nIn experiments, we show that our method reduces the F1 score on the victim by 11 points in comparison to \\textsc{AddSent}---a previously proposed method for generating adversarial input perturbations. \nWhile our work is centered on question answering, our proposed strategy, which is based on building and then attacking an approximate model, can be applied in many instances of adversarial input generation for black box models across domains and tasks. Future extension of our work could explore such attacks as a potential proxy for similarity estimation of victim and extracted models in not only accuracy, but also fidelity~\\citep{Jagielski2019HighAA}.\n\n\\section{Experiments}\nIn this section we present results of our proposed approach. \nWe begin by describing the dataset used, and then report on model extraction.\nFinally, we compare the effectiveness of \\textsc{AddAny-kBest} to 2 other black box approaches. \n\n\\subsection{Datasets} \nFor the evaluation of RC based QA we use the SQuAD dataset \\citep{rajpurkar2016squad}. Though our method is applicable to both v1.1 and v2.0 versions of the dataset we only experiment with \\textsc{AddAny} for SQuAD v1.1 similar to previous investigations. Following \\citep{jia2017adversarialRC}, we evaluate all methods on 1000 queries sampled at random from the development set.\nLike previous work, we use the Brown Common word list corpus~\\citep{francis79browncorpus} for sampling the random tokens (Section \\ref{sec:addany}).\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrr}\n\\hline\n\\textbf{Model}  & \\textbf{F1} & \\textbf{EM}\\\\\n\\hline\nVICTIM & 89.9 & 81.8 \\\\\nWIKI & 83.6 & 73.5   \\\\\nRANDOM & 75.8 & 63.2 \\\\\n\\hline\n\\end{tabular}\n\\caption{\nA comparison of the original model (VICTIM) against the extracted models generated using 2 different schemes(RANDOM and WIKI). bert-base-uncased has been used as the LM in all the models mentioned above. All the extracted models use the same number of queries (query budget of 1x) as in the SQuAD training set. We report on the F1 and EM (Exact Match) scores for the evaluation set (1000 questions) sampled from the dev dataset. \n}\n\\label{extractionTable}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrr}\n\\hline\n\\textbf{Model} & \\textbf{Original (F1)}  & \\textbf{\\textsc{AddAny} (F1)} \\\\\n\\hline\nMatch LSTM single & 71.4 & 7.6  \\\\\nMatch LSTM ensemble & 75.4 & 11.7  \\\\\nBiDAF single & 75.5 & 4.8  \\\\\nBiDAF ensemble & 80.0 & 2.7  \\\\\n\\textbf{bert-base-uncased} & \\textbf{89.9} & \\textbf{5.9}  \\\\\n\\hline\n\\end{tabular}\n\\caption{\nA comparison of the results of Match LSTM, BiDAF as reported by \\citet{jia2017adversarialRC} with the bert-base-uncased model for SQuAD 1.1. We follow the identical experimental setup. The results for Match LSTM and BiDAF models were reported for both the single and ensemble versions.\n}\n\\label{jiaAddOnResults}\n\\end{table}\n\\subsection{Extraction}\nFirst, we present results for \\textsc{WIKI} and \\textsc{RANDOM} extraction methods (Section \\ref{sec:extract}) on SQuAD v1.1 using a bert-base-uncased model for both the victim and extracted model in Table  \\ref{extractionTable}. \n\n\\paragraph{Remarks on Squad v2.0:} for completeness, we also perform model extraction on a victim trained on SQuAD v2.0, but the extracted model achieves significantly lower F1 scores. In SQuAD v1.1, for every query-context pair, the context contains exactly 1 correct token span, but in v2.0, for 33.4\\% of pairs, the context \\emph{does not contain} a correct span. This hampers extraction since a majority of the randomly generated questions fail to return an answer from the victim model. The extracted WIKI model has an F1 score of 57.9, which is comparably much lower to the model extracted for v1.1. \n\nWe believe that the F1 of the extracted model for SQuAD v2.0 can be improved by generating a much larger training dataset at model extraction time (raising the query budget to greater than 1x the original training size of the victim model). But by doing this, any comparison in our results with SQuAD v1.1 would not be equitable.\n\n\n\\subsection{Methods Compared}\nWe compare \\textsc{AddAny-kBest} to 2 baseline, black-box attacks: \\begin{enumerate*}[label=\\roman*)]\n    \\item the standard \\textsc{AddAny} attack on the extracted model, and\n    \\item \\textsc{AddSent}~\\cite{jia2017adversarialRC}.\n\\end{enumerate*}\nSimilar to \\textsc{AddAny}, \\textsc{AddSent}  generates adversaries by appending tokens to the end of a context. \nThese tokens are taken, in part from the query, but are also likely to preserve the correct token span in the context. \nIn more detail, \\textsc{AddSent} proceeds as follows:\n\n\\begin{enumerate}\n    \\item A copy of the query is appended to the context, but nouns and adjectives are replaced by their antonyms, as defined by WordNet~\\cite{miller1995}. Additionally, an attempt is made to replace every named entity and number with tokens of the same part-of-speech that are nearby with respect to the corresponding GloVe embeddings~\\cite{Pennington14glove:global}. If no changes were made in this step, the attacks fails.\n    \\item Next, a spurious token span is generated with the same type (defined using NER and POS tags\n    from Stanford CoreNLP~\\cite{Manning14thestanford} as the correct token span. Types are hand curated using NER and POS tags and have associated fake answers.\n    \\item The modified query and spurious token span are combined into declarative form using hand crafted rules defined by the CoreNLP constituency parses.\n    \\item Since the automatically generated sentences could be unnatural or ungrammatical, crowd-sourced workers correct these sentences. (This final step is not performed in our evaluation of AddSent since we aim to compare other fully automatic methods against this).  \n\\end{enumerate}\nNote that unlike \\textsc{AddAny}, \\textsc{AddSent} does not require access to the model's distribution over token spans, and thus, it does not require model extraction.\n\n\\textsc{AddSent} may return multiple candidate adversaries for a given query-context pair. \nIn such cases, each candidate is applied and the most effective (in terms of reducing instance-level F1 of the victim) is used in computing overall F1. To represent cases without access to (many) black box model evaluations, \\citet{jia2017adversarialRC} also experiment with using a randomly sampled candidate per instance when computing overall F1. This method is called \\textsc{AddOneSent}\n\nFor the \\textsc{AddAny} and \\textsc{AddAny-kBest} approaches, we also distinguish between instances in which they are run on models extracted via the WIKI (\\textsc{W-A-argMax}, \\textsc{W-A-kBest})or RANDOM (\\textsc{R-A-argMax}, \\textsc{R-A-kBest}) approaches. \n\nWe use the same experimental setup as \\citet{jia2017adversarialRC}. Additionally we experiment while both prefixing and suffixing the adversarial sentence to the context. This does not result in drastically different F1 scores on the overall evaluation set. However, we did notice that in certain examples, for a given context $c$, the output of the model differs depending on whether the same adversary was being prefixed or suffixed. It was observed that sometimes prefixing resulted in a successful attack while suffixing would not and vice versa. Since this behaviour was not documented to be specifically favouring either suffixing or prefixing, we stick to suffixing the adversary to the context as done by \\citet{jia2017adversarialRC}.\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrr}\n\\hline\n\\textbf{Method} & \\textbf{Extracted (F1)}  & \\textbf{Victim (F1)} \\\\\n\\hline\n\\textbf{W-A-kBest} & 10.9 & \\textbf{42.4} \\\\\nW-A-argMax & 9.7 & 68.3 \\\\\nR-A-kBest & 3.6 & 52.2 \\\\\nR-A-argMax & 3.7 & 76.1 \\\\ \n\\hline\nAddSent & - & 53.2 \\\\\nAddOneSent & - & 56.5 \\\\ \n\\hline\nCombined & - & 31.9 \\\\\n\\hline\n\n\\end{tabular}\n\n\\caption{\\label{citation-guide}\nThe first 4 rows report the results for experiments on variations of \\textsc{AddAny} (kBest\/argMax) and extraction schemes (WIKI and RANDOM). The ``extracted\" column lists the F1 score of the respective method used for generating adversaries. The ``victim\" column is the F1 score on the victim model when transferred from the extracted (for \\textsc{AddAny} methods). For \\textsc{AddSent} and  \\textsc{AddOneSent} it is the F1 score when directly applied on the victim model. The last row ``Combined\" refers to the joint coverage of \\textsc{W-A-kBest} + \\textsc{AddSent}.  \n}\n\\label{addAnyresults}\n\\end{table}\n\n\\subsection{Results}\nIn Table \\ref{addAnyresults}, we report the F1 scores of all methods on the extracted model. The results reveal that the \\textsc{kBest} minimization (Section \\ref{addanykBestSection}) approach is most effective at reducing the F1 score of the victim. Notably, we observe a difference of over 25\\% in the F1 score between \\textsc{kBest} and \\textsc{argMax} in both \\textsc{WIKI} and \\textsc{RANDOM} schemes. \n\nInterestingly, the \\textsc{AddSent} and \\textsc{AddOneSent} attacks are more effective than the \\textsc{AddAny-argMax} approach but less effective than the \\textsc{AddAny-kBest} approach. In particular they reduce the F1 score to 53.2 (\\textsc{AddSent}) and 56.5 (\\textsc{AddOneSent}) as reported in Table \\ref{addAnyresults}. For completeness, we compare the \\textsc{AddAny} attack on the victim model (similar to the work done in \\citet{jia2017adversarialRC} for LSTM and BiDAF models. Table \\ref{jiaAddOnResults} shows the results for bert-base-uncased among others for SQuAD v1.1. Only \\textsc{argMax} minimization is carried out here since there is no post-attack transfer.  \n\nWe also study the coverage of \\textsc{W-A-kBest} and \\textsc{AddSent} on the evaluation dataset of 1000 samples (Figure \\ref{fig:vennAddSentAddAny}). \\textsc{W-A-Kbest} and \\textsc{AddSent} induce an F1 score of 0 on 606 and 538 query-context pairs, respectively. Among these failures, 404 query-context pairs were common to both the methods. Of the 404, 182 samples were a direct result of model failure of bert-base-uncased (exact match score is 81.8 which amounts to the 182 failure samples). If the methods are applied jointly, only 260 query-context pairs produce the correct answer corresponding to an exact match score of 26 and an F1 score of 31.9 (Table \\ref{addAnyresults}). This is an indication that the 2 attacks in conjunction (represented by the ``Combined\" row in Table \\ref{addAnyresults}) provide wider coverage than either method alone.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=0.7\\linewidth]{venn.png}\n  \\caption{Joint coverage of \\textsc{WIKI-AddAny-kBest} and \\textsc{AddSent} on the evaluation}.\n  \\label{fig:vennAddSentAddAny}\n\\end{figure}\n\n\\subsection{Fine-grained analysis}\nIn this section we analyze how successful the adversarial attack is for each answer \\emph{category}, which were identified in previous work \\cite{rajpurkar2016squad}. Table \\ref{fineGrainedAnalysis} lists the 10 categories of ground-truth token spans, their frequency in the evaluation set as well as the average F1 scores on the victim model before and after the adversarial attack. We observe that ground-truth spans of type ``places\" experienced a drastic drop in F1 score. ``Clauses\" had the highest average length and also had the highest drop in F1 score subject to the \\textsc{W-A-kBest} attack(almost double the average across classes). Category analysis such as this could help the community understand how to curate better attacks and ultimately train a model that is more robust on answer types that are most important or relevant for specific use cases.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrr}\n\\hline\n\\textbf{Category} & \\textbf{Freq \\%} & \\textbf{Before} & \\textbf{After} & \\textbf{Av-Len} \\\\ \n\\hline\nNames & 7.4 & 96.2 & 51.8 & 2.8\\\\\nNumbers & 11.4 & 92.1 & 51.3 & 2\\\\\nPlaces & 4.2 & 89 & 19.2 & 2.7\\\\\nDates & 8.4 & 96.8 & 40.3 & 2.1\\\\\nOther Ents & 7.2 & 90.9 & 58.7 & 2.5\\\\\nNoun Phrases & 48 & 88 & 41.8 & 2.3\\\\\nVerb Phrases & 2.7 & 91.1 & 41.1 & 4.8\\\\\nAdj Phrases & 1.9 & 70.3 & 27.8 & 1.6\\\\\nClauses & 1.3 & 82.9 & 7.6 & 6.8\\\\\nOthers & 9.5 & 89.7 & 34.7 & 5\\\\\n\\hline\n\\textbf{Total} & \\textbf{100} & \\textbf{89} & \\textbf{42.4} & \\textbf{2.7}\\\\\n\\hline\n\n\\end{tabular}\n\n\\caption{\\label{citation-guide}\nThere are 10 general categories into which the answer spans have been classified. The first 4 are entities and \\emph{Other Ents} is all other entities that do not fall into any of the 4 major categories. The 2nd column is the frequency of ground truth answers belonging to each of the categories. The 3rd column (Before) refers to the F1 score of questions corresponding to the category when evaluated on the Victim model. The 4th column (After) refers to the F1 score of questions corresponding to the category when evaluated on the Victim model under the presence of adversaries generated using \\textsc{WIKI-AddAny-kBest} method. \\emph{Av-Len} column is the average length of the answer spans in each category. \n}\n\\label{fineGrainedAnalysis}\n\\end{table}\n\n\\subsection{Model Extraction}\n\\label{sec:extract}\nThe first step in our approach is to build an approximation of the victim model via \\emph{model extraction}~\\cite{krishna2020thieves}. At a high level, this approach constructs a training set by generating inputs that are served to the victim model and collecting the victim's responses. The responses act as the labels of the inputs. After a sufficient number of inputs and their corresponding labels have been collected, a new model can be trained to predict the collected labels, thereby mimicking the victim. The approximate model is known as the \\emph{extracted} model. \n\nThe crux of model extraction is an effective method of generating inputs. \nRecall that in RC based QA, the input is composed of a query and a context. \nLike previous work, we employ 2 methods for generating contexts: \\textsc{WIKI} and \\textsc{RANDOM}~\\cite{krishna2020thieves}.\nIn the \\textsc{WIKI} scheme, contexts are randomly sampled paragraphs from the WikiText-103 dataset.\nIn the \\textsc{RANDOM} scheme, contexts are generated by sampling random tokens from the WikiText-103 dataset. \nFor both schemes, a corresponding query is generated by sampling random words from the context.\nTo make the queries resemble questions, tokens such as ``where,\" ``who,\" ``what,\" and ``why,\" are inserted at the beginning of each query, and a ``?\" symbol is appended to the end.\nLabels are collected by serving the sampled queries and contexts to the victim model.\nTogether, the queries, contexts, and labels are used to train the extracted model.\nAn example query-context pair appears in Table \\ref{exampleExtraction}.\n\n\n\n\\section{Introduction}\nMachine learning models are ubiquitous in technologies that are used by billions of people every day. In part, this is due to the recent success of deep learning. Indeed, research in the last decade has demonstrated that the most effective deep models can match or even outperform humans on a variety of tasks \\cite{Devlin2019BERTPO,xie2020}.\n\nDespite their effectiveness, deep models are also known to make embarrassing errors. \nThis is especially troublesome when those errors can be categorized as unsafe, e.g., racist, sexist, etc.~\\cite{wallace2019-universal}.  This leads to the desire for methods to audit models for correctness, robustness and---above all else---safety, before deployment.\n\nUnfortunately, it is difficult to precisely determine the set of inputs on which a deep model fails because deep models are complex, have a large number of parameters---usually in the billions---and are non-linear \\cite{Radford2019LanguageMA}. In an initial attempt to automate the discovery of inputs on which these embarrassing failures occur, researchers developed a technique for making calculated perturbations to an image that are imperceptible to the human eye, but cause deep models to misclassify the image \\cite{szegedy2014intriguing}. In addition to developing more effective techniques for creating \\emph{adversarial inputs} for vision models  \\cite{papernot2017practical}, subsequent research extends these ideas to new domains, such as natural language processing (NLP).\n\nNLP poses unique challenges for adversarial input generation because: \n\\begin{enumerate*}\n\\item natural language is discrete rather than continuous (as in the image domain); and\n\\item in NLP, an ``imperceptible perturbation\" of a sentence is typically construed to mean a semantically similar sentence, which can be difficult to generate.\n\\end{enumerate*}\n\nNevertheless, the study of adversarial input generation for NLP models has recently flourished, with techniques being developed for a wide variety of tasks such as: text classification, textual entailment and question answering~\\cite{jin2019robustbert,wallace2019-universal,li2020bertattack,jia2017adversarialRC}.\n\nThese new techniques can be coarsely categorized into two groups: \\emph{white box attacks}, where the attacker has full knowledge of the \\emph{victim} model---including its parameters---and \\emph{black box attacks}, where the attacker only has access to the victim's predictions on specified inputs. \nUnsurprisingly, white box attacks tend to exhibit much greater efficacy than black box attacks. \n\nIn this work, we develop a technique for black box adversarial input generation for the task of reading comprehension that employs a white box attack on an approximation of the victim. More specifically, our approach begins  with \\emph{model extraction}, where we learn an approximation of the victim model~\\cite{krishna2020thieves}; afterward, we run a modification of the \\textsc{AddAny}~\\cite{jia2017adversarialRC} attack on the model approximation. Our approach is inspired by the work of \\citet{papernot2017practical} for images and can also be referred to as a \\emph{Black box evasion attack} on the original model. \n\nSince the \\textsc{AddAny} attack is run on an \\emph{extracted} (i.e., approximate) model of the victim, our modification encourages the attack method to find inputs for which the extracted model's top-k responses are all incorrect, rather than only its top response---as in the original \\textsc{AddAny} attack.\nThe result of our \\textsc{AddAny} attack is a set of adversarial perturbations, which are then applied to induce failures in the victim model. Empirically, we demonstrate that our approach is more effective than \\textsc{AddSent}, i.e., a black box method for adversarial input generation for reading comprehension~\\cite{jia2017adversarialRC}. \nCrucially, we observe that our modification of \\textsc{AddAny} makes the attacks produced more robust to the difference between the extracted and victim model. In particular, our black box approach causes the victim to fail 11\\% more than \\textsc{AddSent}. While we focus on reading comprehension, we believe that our approach of model extraction followed by white box attacks is a fertile and relatively unexplored area that can be applied to a wide range of tasks and domains.\n\n\\textbf{Ethical Implications:} The primary motivation of our work is helping developers test and probe models for weaknesses before deployment. While we recognize that our approach could be used for malicious purposes we believe that our methods can be used in an effort to promote model safety.\n\n\\section{Method}\nOur goal is to develop an effective black box attack for RC based QA models. Our approach proceeds in two steps: first, we build an approximation of the victim model, and second, we attack the approximate model with a powerful white box method.~The result of the attack is a collection of adversarial inputs that can be applied to the victim.~In this section we describe these steps in detail.\n\n\\input{extract}\n\n\\input{attack}\n\n\n\\section{Related Work}\nOur work studies black box adversarial input generation for reading comprehension. The primary building blocks of our proposed approach are model extraction, and white box adversarial input generation, which we discuss below. We also briefly describe related methods of generating adversarial attacks for NLP models.\n\nA contemporary work that uses a similar approach to ours is \\citet{wallace2020imitation}. While we carry out model extraction using non-sensical inputs, their work uses high quality out of distribution (OOD) sentences for extraction of a machine translation task. It is noteworthy to mention that in the extraction approach we follow~\\cite{krishna2020thieves} the extracted model reaches within 95\\% F1 score of the victim model with the same query budget that was used to train the victim model. This is in contrast to roughly 3x query budget taken in extracting the model in their work. The different nature of the task and methods followed while querying OOD datasets could be a possible explanation for the disparities.\n\n\\paragraph{Nonsensical Inputs and Model Extraction:} Nonsensical inputs to text-based systems have been the subject of recent study, but were not explored for extraction until recently \\citep{krishna2020thieves}. \\citet{feng2018-pathologies} studied model outputs while trimming down inputs to an extent where the input turned nonsensical for a human evaluator. Their work showed how nonsensical inputs produced overly confident model predictions. Using white box access to models \\citet{wallace2019-universal} discovered that it was possible to generate input-agnostic nonsensical triggers that are effective adversaries on existing models on the SQuAD dataset. \n\n\\paragraph{Adversarial attacks:} The first adversarial attacks against block box, deep neural network models focused on computer vision applications~\\cite{papernot2017practical}. In concept, adversarial perturbations are transferable from computer vision to NLP; but, techniques to mount successful attacks in NLP vary significantly from their analogues in computer vision. This is primarily due to the discreteness of NLP (vs. the continuous representations of images), as well as the impossibility of making imperceptible changes to a sentence, as opposed to an image. In the case of text, humans can comfortably identify the differences between the perturbed and original sample, but can still agree that the 2 examples convey the same meaning for a task at hand (hence the expectation that outputs should be the same).\n\nHistorically, researches have employed various approaches for generating adversarial textual examples. In machine translation \\citet{belinkov2017synthetic} applied minor character level perturbations that resemble typos. \\citet{hosseini2017perspective} targeted Google's Perspective system that detects text toxicity. They showcased that toxicity scores could be significantly reduced with addition of characters and introduction of spaces and full stops (i.e., periods (``.\") ) in between words. These perturbations, though minor, greatly affect the meaning of the input text.  \\citet{alzantot2018adversarial} proposed an iterative word based replacement strategy for tasks like text classification and textual entailment for LSTMs. \\citet{jin2019robustbert} extended the above experiments for BERT. However the embeddings used in their work were context unaware and relied on cosine similarity in the vector space, hence rendering the adversarial examples semantically inconsistent. \\citet{li2018textbugger} carried out a similar study for sentiment analysis in convolutional and recurrent neural networks. In contrast to prior work, \\citet{jia2017adversarialRC} were the first to evaluate models for RC based QA using SQuAD v1.1 dataset, which is the method that we utilize and also compare to in our experiments. \n\nUniversal adversarial triggers \\cite{wallace2019-universal} generates adversarial examples for the SQuAD dataset, but cannot be compared to our work since it is a white box method and a targeted adversarial attack. \\citet{ribeiro2018-semantically} introduced a method to detect bugs in black box models which generates \\emph{semantically equivalent adversaries} and also generalize them into rules. Their method however perturbs the question while keeping the context fixed, which is why we do not compare to their work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n{\\em Explainable AI} refers to artificial intelligence and machine learning techniques that can provide human understandable justification for their behavior. \n Explainability is important in situations where human operators work alongside autonomous and semi-autonomous systems because it can help build rapport, confidence, and understanding between the agent and its operator.\nIn the event that an autonomous system fails to complete a task or completes it in an unexpected way, explanations help the human collaborator\nunderstand the circumstances that led to the behavior, which also allows the operator to make an informed decision on how to address the behavior. \n\nPrior work on explainable AI (XAI) has primarily focused on non-sequential problems such as image classification and captioning ~\\cite{wang2017residual,xu2015show,you2016image}.\nSince these environments are episodic in nature, the model's output depends only on its input.\nIn sequential environments, decisions that the agent has made in the past influence future decisions. \nTo simplify this, agents often make locally optimal decisions by selecting actions that maximize some discrete notion of expected future reward or utility.\nTo generate plausible explanations in these environments, the model must unpack this local reward or utility to reason about how current actions affect future actions. On top of that, it needs to communicate the reasoning in a human understandable way, which is a difficult task. \nTo address this challenge of human understandable explanation in sequential environments, we introduce the alternative task of rationale generation in sequential environments. \n\n\n{\\em Automated rationale generation} is a process of producing a natural language explanation for agent behavior {\\em as if a human had performed the behavior}~\\cite{ehsan2017rationalization}.\nThe intuition behind rationale generation is that humans can engage in effective communication\nby verbalizing plausible motivations for their action. The communication can be effective even when the verbalized reasoning does not have a consciously accessible neural correlate of\nthe decision-making process~\\cite{block2007consciousness,block2005two,fodor1994elm}. \nWhereas an explanation can be in any communication modality, rationales are natural language explanations that don't literally expose the inner workings of an intelligent system.\nExplanations can be made by exposing the inner representations and data of a system, though this type of explanation may not be accessible or understandable to non-experts.\nIn contrast, contextually appropriate natural language rationales are\naccessible and intuitive to non-experts, facilitating understanding and communicative effectiveness. \nHuman-like communication can also afford human factors advantages such as higher degrees of satisfaction, confidence, rapport, and willingness to use autonomous systems.\nFinally, rationale generation is fast, sacrificing \nan accurate view of agent decision-making for real-time response, making it appropriate for real-time human-agent collaboration. \nShould deeper, more grounded and technical\nexplanations be necessary, rationale generation may need to be supplemented by other explanation or visualization techniques.\n\nIn preliminary work~\\cite{ehsan2017rationalization} we showed that recurrent neural networks can be used to translate internal state and action representations into natural language. \nThat study, however, relied on synthetic natural language data for training. \nIn this work, we explore if human-like plausible rationales can be generated using a non-synthetic, natural language corpus of human-produced explanations. \nTo create this corpus, we developed a methodology for conducting remote think-aloud protocols \\cite{fonteyn1993description}. \nUsing this corpus, we then use \na neural network based on~\\cite{ehsan2017rationalization}\nto translate an agent's state and action information into natural language rationales, and show how variations in model inputs can produce two different types of rationales. \nTwo user studies help us understand the perceived quality of the generated rationales along dimensions of human factors. \nThe first study indicates that our rationale generation technique produces plausible and high-quality rationales and explains the differences in user perceptions. \nIn addition to understanding user preferences, the second study demonstrates how the intended design behind the rationale types aligns with their user perceptions.\n\nThe philosophical and linguistic discourse around the notion of explanations~\\cite{miller2017explanation, lipton2001good} is beyond the scope of this paper. \nTo avoid confusion, we use the word \"rationale\" to refer to natural language-based post-hoc explanations that are meant to sound like what a human would say in the same situation.\nWe opt for \"rationale generation\" instead of \"rationalization\" to signal that the agency lies with the receiver and interpreter (human being) instead of the producer (agent). \nMoreover, the word rationalization may carry a connotation of making excuses~\\cite{maruna2006fundamental} for an (often controversial) action, which is another reason why we opt for \\textit{rationale generation} as a term of choice.\n\nIn this paper, we make the following contributions in this paper:\n\\begin{itemize}\n    \\item We present a methodology for collecting high-quality human explanation data based on remote think-aloud protocols. \n    \\item We show how this data can be used to configure neural translation models to produce two types of human-like rationales: $\\left(1\\right)$ concise, localized and $\\left(2\\right)$ detailed, holistic rationales. We demonstrate the alignment between the intended design of rationale types and the actual perceived differences between them.\n    \\item We quantify the perceived quality of the rationales and preferences between them, \n   \n    and we use qualitative data to explain these perceptions and preferences.\n   \n   \n   \n   \n   \n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\n\n\nMuch of the previous work on explainable AI has focused on {\\em interpretability}.\nWhile there is no one definition of interpretability with respect to machine learning models, we view interpretability as a property of machine learned models that dictate the degree to which a human user---AI expert or user---can come to conclusions about the performance of the model on specific inputs.\nSome types of models are inherently interpretable, meaning they require relatively little effort to understand. \nOther types of models require more effort to make sense of their performance on specific inputs. \nSome non-inherently interpretable models can be made interpretable in a post-hoc fashion through explanation or visualization.\nModel-agnostic post-hoc methods can help to make models intelligible without custom explanation or visualization technologies and without changing the underlying model to make them more interpretable~\\cite{ribeiro2016should,yosinski2015understanding}. \n\nExplanation generation can be described as a form of {\\em post-hoc interpretability}~\\cite{2016arXiv160603490L, miller2017explanation}; explanations are generated on-demand based on the current state of a model and---potentially---meta-knowledge about how the algorithm works.\nAn important distinction between interpretability and explanation is that explanation does not elucidate precisely how a model works but aims to give useful information for practitioners and end users.\nAbdul et al.~\\cite{abdul2018trends} conduct a comprehensive survey on trends in explainable and intelligible systems research.\n\nOur work on rationale generation is a model-agnostic explanation system that works by translating the internal state and action representations of an arbitrary reinforcement learning system into natural language.\nAndreas, Dragan, and Klein~\\cite{andreas2017translating} describe a technique that translates message-passing policies between two agents into natural language.\nAn alternative approach to translating internal system representations into natural language is to add explanations to a supervised training set such that a model learns to output a classification as well as an explanation~\\cite{codella2018teaching}.\nThis technique has been applied to generating explanations about procedurally generated game level designs~\\cite{guzdial2018explainable}.\n\nBeyond the technology, user perception and acceptance matter because they influence trust in the system, which is crucial to adoption of the technology. \nEstablished fields such as information systems enjoy a robust array of technology acceptance models such as the Technology Acceptance Model (TAM) \\cite{davis1989perceived} and Unified Theory of Acceptance and Use of Technology Model (UTAUT) \\cite{venkatesh2003user} whose main goal is to explain variables that influence user perceptions. \nUtilizing dimensions such as perceived usefulness and perceived ease of use, the TAM model aimed to explain prospective expectations about the technological artifacts. \nUTAUT uses constructs like performance expectancy, effort expectancy, etc. to understand technology acceptance. The constructs and measures in these models build on each other. \n\nIn contrast, due to a rapidly evolving domain, a robust and well-accepted user perception model of XAI agents is yet to be developed. \nUntil then, we can take inspiration from general acceptance models (such as TAM and UTAUT) and adapt their constructs to understand the perceptions of XAI agents. \nFor instance, the human-robot interaction community has used them as basis to understand users' perceptions towards robots~\\cite{ezer2009attitudinal, beer2011understanding}. \nWhile these acceptance models are informative, they often lack sociability factors such as \"humanlike-ness\".\nMoreover, TAM-like models does not account for autonomy in systems, let alone autonomous XAI systems. \nBuilding on some constructs from TAM-like models and original formative work, we attempt to address the gaps in understanding user perceptions of rationale-generating XAI agents.  \n\nThe dearth of established methods combined with the variable conceptions of explanations make evaluation of XAI systems challenging. \nBinns et al.~\\cite{binns2018s} use scenario-based survey design~\\cite{carroll2000making} and presented different types of hypothetical explanations for the same decision to measure perceived levels of justice. \nOne non-neural based network evaluates the usefulness and naturalness of generated explanations~\\cite{broekens2010you}.\nRader et al.~\\cite{rader2018explanations} use explanations manually generated from content analysis of Facebook's News Feed to study perceptions of algorithmic transparency. \nOne key differentiating factor of our approach is that our evaluation is based rationales that are actual system outputs (compared to hypothetical ones). \nMoreover, user perceptions of our system's rationales directly influence the design of our rationale generation technique. \n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=1.0\\linewidth]{frogger-pipeline2.png}\n    \\vspace{-1.5\\baselineskip}\n    \\caption{End-to-end pipeline for training a system that can generate explanations.}\n\\label{fig:end-to-end}\n\\end{figure}\n\n\\section{Learning to Generate Rationales}\n\nWe define a \\textit{rationale} as an explanation that justifies an action based on how a human would think. \nThese rationales do not necessarily reveal the true decision making process of an agent, but still provide insights about why an agent made a decision in a form that is easy for non-experts to understand.\n\nRationale generation requires translating events in the game environment into natural language outputs. Our approach to rationale generation involves two steps: (1)~collect a corpus of think-aloud data from players who explained their actions in a game environment; and (2)~use this corpus to train an encoder-decoder network to generate plausible rationales for any action taken by an agent (see Figure~\\ref{fig:end-to-end}).\n\nWe experiment with rationale generation using autonomous agents that play the arcade game, {\\em Frogger}.\nFrogger is a good candidate for our experimental design of a rationale generation pipeline for general sequential decision making tasks because it is a simple Markovian environment, making it an ideal stepping stone towards a real world environment. \nOur rationale generation technique is agnostic to the type of agent or how it is trained, as long as the representations of states and actions used by the agent can be exposed to the rationale generator and serialized.\n\n\\subsection{Data Collection Interface}\n\nThere is no readily available dataset for the task of learning to generate explanations. Thus, we developed a methodology to collect live ``think-aloud'' data from players as they played through a game. This section covers the two objectives of our data collection endeavor:\n\\begin{enumerate}\n\\item Create a think-aloud protocol in which players provide natural rationales for their actions. \n\\item Design an intuitive player experience that facilitates accurate matching of the participants' utterances to the appropriate state in the environment. \n\\end{enumerate}\n\n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[width=0.75\\linewidth]{Play4.png}\n  \\end{center}\n  \\caption{Players take an action and verbalize their rationale for that action. (1)~After taking each action, the game pauses for 10 seconds. (2)~Speech-to-text transcribes the participant's rationale for the action. (3)~Participants can view their transcribed rationales near real-time and edit if needed.}\n  \\label{fig:Play}\n\\end{figure}\n\nTo train a rationale-generating explainable agent, we need data linking game states and actions to their corresponding natural language explanations. To achieve this goal, we built a modified version of Frogger in which players simultaneously play the game and also explain each of their actions. \nThe entire process is divided into three phases: (1)~A guided tutorial, (2)~rationale collection, and (3)~transcribed explanation review.\n\nDuring the guided tutorial~(1), our interface provides instruction on how to play through the game, how to provide natural language explanations, and how to review\/modify any explanations they have given. \nThis helps ensure that users are familiar with the interface and its use before they begin providing explanations. \n\nFor rationale collection~(2), participants play through the game while explaining their actions out loud in a turn-taking mechanism.\nFigure~\\ref{fig:Play} shows the game embedded into the explanation collection interface. \nTo help couple explanations with actions (attach annotations to concrete game states), the game pauses for 10 seconds after each action is taken.\nDuring this time, the player's microphone automatically turns on and the player is asked to explain their most recent action while a speech-to-text library \\cite{github_2017} automatically transcribes the explanation real-time. \nThe automatic transcription substantially reduces participant burden as it is more efficient\nthan typing an explanation.  \nPlayer can use more or less than the default 10-second pause to collect the explanation. \nOnce done explaining, they can view their transcribed text and edit it if necessary.\nDuring pretesting with 14 players, we observed that players often repeat a move for which the explanation is the same as before. \nTo reduce burden of repetition, we added a \"redo\" button that can be used to recycle rationales for consecutive repeated actions.\n\nWhen the game play is over, players move to transcribed explanation review portion ~(3). Here, they can can step through all the actions-explanation pairs. This stage allows reviewing in both a situated and global context. \n\n\n\n\n\nThe interface is designed so that no manual hand-authoring\/editing of our explanation data was required before\nusing it to train our machine learning model. Throughout the game, players have the opportunity to organically edit their own data without impeding their work-flow. This added layer of organic \nediting is crucial in ensuring that we can directly input the collected data into the network with zero manual cleaning. While we use Frogger as a test environment in our experiments, a similar user experience can be designed using other turn-based environments with minimal effort.\n\n\\begin{figure}[t]\n  \\includegraphics[width=\\linewidth]{review_with_legend.png}\n\\caption{Players can step-through each of their action-rationale pairs and edit if necessary. (1)~Players can watch an action-replay while editing rationales. (2)~These buttons control the flow of the step-through process. (3)~The rationale for the current action gets highlighted for review.}\n  \\label{fig:Review}\n\\end{figure}\n\n\n\\subsection{Neural Translation Model}\n\nWe use an encoder-decoder network~\\cite{bahdanau2014neural} to teach our network to generate relevant natural language explanations for any given action. \nThese kinds of networks are commonly used for machine translation tasks or dialogue generation, but their ability to understand sequential dependencies between the input and the output make it suitable for our task. \nOur encoder decoder architecture is similar to that used in \\cite{ehsan2017rationalization}. \nThe network learns how to translate the input game state representation\n$X = x_1, x_2, ..., x_n$, comprised of the representation of the game combined with other influencing factors,\ninto an output rationale as a sequence of words \n$Y = y_1, y_2, ..., y_m$\nwhere $y_i$ is a word.\nThus our network learns to translate game state and action information into natural language rationales.\n\nThe encoder and decoder are both recurrent neural networks (RNN) comprised of Gated Recurrent Unit (GRU) cells since our training process involved a small amount of data.\nThe decoder network uses an additional attention mechanism~\\cite{luong2015effective} to learn to weight the importance of different components of the input with regard to their effect on the output. \n\nTo simplify the learning process, the state of the game environment is serialized into a sequence of symbols where each symbol characterizes a sprite in the grid-based represntation of the world.\nTo this, we append information concerning Frogger's position, the most recent action taken, and the number of lives the player has left to create the input representation $X$. \nOn top of this network structure, we vary the input configurations with the intention of producing varying styles of rationales. \nEmpirically, we found that a reinforcement learning agent using tabular $Q$-learning \\cite{watkins92} learns to play the game effectively when given a limited window for observation.\nThus a natural configuration for the rationale generator is to give it the same observation window that the agent needs to learn to play.\nWe refer to this configuration of the rationale generator as {\\em focused-view} generator.\nThis view, however, potentially limits the types of rationales that can be learned since the agent will only be able to see a subset of the full state. Thus we formulated a second configuration that gives the rationale generator the ability to use all information on the board to produce rationales.\nWe refer to this as {\\em complete-view} generator.\nAn underlying question is thus whether rationale generation should use the same information that the underlying black box reasoner needs to solve a problem or if more information is advantageous at the expense of making rationale generation a harder problem.\nIn the studies described below, we seek to understand how these configurations affect human perceptions of the agent when presented with generated rationales.\n\n\\subsubsection{Focused-view Configuration}\nIn the \\textit{focused-view} configuration, we used a windowed representation of the grid, i.e. only a $7\\times7$ window around the Frog was used in the input.\nBoth playing an optimal game of Frogger and generating relevant explanations based on the current action taken typically only requires this much local context. \nTherefore providing the agent with only the window around Frogger helps the agent produce explanations grounded in it's neighborhood. \nIn this configuration, we designed the inputs such that the network is prone to prioritize short-term planning producing localized rationales instead of long-term planning. \n\n\n\\subsubsection{Complete-view Configuration}\nThe \\textit{complete-view} configuration is an alternate setup that provides the entire game board as context for the rationale generation.\nThere are two differences between this configuration and the focused-view configuration.\nFirst, we use the entire game screen as a part of the input. The agent now has the opportunity to learn which other long-term factors in the game may influence it's rationale.\nSecond, we added noise to each game state to force the network to generalize when learning, reduce the likelihood that spurious correlations are identified, and to give the model equal opportunity to consider factors from all sectors of the game screen.\nIn this case noise was introduced by replacing input grid values with dummy values. For each grid element, there was a $20\\%$ chance that it would get replaced with a dummy value.\nGiven the input structure and scope, this configuration should prioritize rationales that exhibit long-term planning and consider the broader context.\n\n\n  \n\n\n\\begin{table}[h]\n  \\caption{Examples of \\textit{focused-view} vs  \\textit{complete-view} rationales generated by our system for the same set of actions.}\n  \\label{tab:components}\n  \\begin{tabular}{p{0.1\\columnwidth}p{0.4\\columnwidth}p{0.4\\columnwidth}}\n    \\toprule\n    {\\bf Action} & {\\bf Focused-view} & {\\bf Complete-view}\\\\\n    \\midrule\n    Right & I had cars to the left and in front of me so I needed to move to the right to avoid them. & I moved right to be more centered. This way I have more time to react if a car comes from either side. \\\\\n    \\rowcolor{tablerowcolor} Up & The path in front of me was clear so it was safe for me to move forward.  & I moved forward making sure that the truck won\\textquotesingle t hit me so I can move forward one spot. \\\\\n    Left & I move to the left so I can jump onto the next log. & I moved to the left because it looks like the logs and top or not going to reach me in time, and I\\textquotesingle m going to jump off if the law goes to the right of the screen. \\\\\n    \\rowcolor{tablerowcolor} Down & I had to move back so that I do not fall off. & I jumped off the log because the middle log was not going to come in time. So I need to make sure that the laws are aligned when I jump all three of them. \\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\section{Perception Study: Candidate vs. Baseline Rationales}\nIn this section, we\nassess whether the rationales generated using our technique are plausible and explore \nhow humans perceive them along various dimensions of human factors.\nFor our rationales to be plausible we would expect that human users indicate a strong preference for rationales generated by our system (either configuration) over those generated by a baseline rationale generator.\nWe also compare them to exemplary human-produced explanations to get a sense for how far from the upper bound we are.\n\nThis study aims to achieve two main objectives.\nFirst, it seeks to confirm the hypothesis that humans prefer rationales generated by each of the configurations over randomly selected rationales across all dimensions.\nWhile this baseline is low, it establishes that rationales generated by our technique are not nonsensical. \nWe can also measure the distance from the upper-bound (exemplary human rationales) for each rationale type. \nSecond, we attempt to understand the underlying components that influence \nthe perceptions of the generated rationales along four dimensions of human factors: {\\em confidence}, {\\em human-likeness}, {\\em adequate justification}, and {\\em understandability}.\n\n\n\n\n\n\n\n\\subsection{Method}\n\nTo gather the training set of game state annotations, we deployed our data collection pipeline on {\\em Turk Prime}~\\cite{litman2017turkprime}. \nFrom 60 participants\nwe collected over 2000 samples of human actions in Frogger coupled with natural language explanations. The average duration of this task was around 36 minutes.\nThe parallel corpus of the collected game state images and  natural language explanations was used to train the encoder-decoder network.\nEach RNN in the encoder and the decoder was parameterized with GRU cells with a hidden vector size of 256. \nThe entire encoder-decoder network was trained for 100 epochs.\n\nFor the perception user study, we collected both within-subject and between-subject data.\nWe recruited 128 participants, split into two equal experimental groups through {\\em TurkPrime}: Group 1 (age range = 23 - 68 years, M = 37.4, SD = 9.92) and Group 2 (age range = 24 - 59 years, M = 35.8, SD= 7.67). \nOn average, the task duration was approximately 49 minutes.\n46\\% of our participants were women, and the 93\\% of participants were self-reported as from the United States while the remaining 7\\% of participants were self-reported as from India.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=1.0\\linewidth]{Study_Screenshot.PNG}\n  \\caption{Screenshot from user study (setup 2) depicting the action taken and the rationales: \\textit{P = Random, Q = Exemplary, R = Candidate}}\n  \\label{fig:Study}\n\\end{figure}\n\nAll participants watched a counterbalanced series of five videos.\nEach video depicted an action taken by Frogger accompanied by three different types of rationales that justified the action (see Figure~\\ref{fig:Study}).\nParticipants rated each rationale on a labeled, 5-point, bipolar Likert-scale along 4 perception dimensions (described below). \nThus, each participant provided 12 ratings per action, leading to 60 perception ratings for five actions. \nActions collected from human players comprised the set of Frogger's actions. These actions were then fed into the system to generate rationales to be evaluated in the the user studies. \nIn order to get a balance between participant burden, fatigue, the number of actions, and regions of the game, we pretested with 12 participants. \nFive actions was the limit beyond which participants' fatigue and burden substantially increased. \nTherefore, we settled on five actions (up (twice), down, left, and right) in the major regions of the game-- amongst the cars, at a transition point, and amongst the logs. \nThis allowed us to test our rationale generation configurations in all possible action-directions in all the major sections of the game. \n\nThe study had two identical experimental conditions, differing only by type of \\textit{candidate rationale}. \nGroup 1 evaluated the \\textit{focused-view} rationale while Group 2 evaluated the \\textit{complete-view} rationales.\nIn each video, the action was accompanied by three rationales generated by three different techniques (see Figure~\\ref{fig:Results_1}):  \n\\begin{itemize}\n\\item The \\textit{exemplary rationale} is the rationale from our corpus that 3 researchers unanimously agreed on as the best one for a particular action. Researchers independently selected rationales they deemed best and iterated until consensus was reached.\nThis is provided as an upper-bound for contrast with the next two techniques.\n\\item  The \\textit{candidate rationale} is the rationale produced by our network, either the focused-view or complete-view configuration.\n\n\\item The \\textit{random rationale} is a randomly chosen rationale from our corpus.\n\\end{itemize}\n\n\n\\noindent\nFor each rationale, participants used a 5-point Likert scale to rate their endorsement of each of following four statements, which correspond to four dimensions of interest. \n\n\\begin{enumerate}\n    \\item \\textit{Confidence:} This rationale makes me confident in the character's ability to perform it's task.\n    \\item \\textit{Human-likeness: } This rationale looks like it was made by a human.\n    \\item \\textit{Adequate justification:} This rationale adequately justifies the action taken.\n    \\item \\textit{Understandability:} This rationale helped me understand why the agent behaved as it did.\n\\end{enumerate}\n\nResponse options on a clearly labeled bipolar Likert scale ranged from \"strongly disagree\" to \"strongly agree\". In a mandatory free-text field, they explained their reasoning behind the ratings for a particular set of three rationales. After answering these questions, they provided demographic information.\n\nThese four dimensions emerged from an iterative filtering process that included preliminary testing of the study, informal interviews with experts and participants, and a literature review on robot and technology acceptance models. Inspired by the acceptance models, we created a set of dimensions that were contextually appropriate for our purposes. \n\nDirect one-to-one mapping from existing models was not feasible, given the novelty and context of the Explainable AI technology.\nWe adapted \\textit{confidence}, a dimension that impacts trust in the system \\cite{kaniarasu2013robot}, from constructs like performance expectancy \\cite{venkatesh2003user} (from UTAUT) and robot performance~\\cite{beer2011understanding, chernova2009confidence}.\n\\textit{Human-likeness}, central to generating human-centered rationales, was inspired from sociability and anthropomorphization factors from HRI work on robot acceptance [\\cite{nass1994machines,nass1996can,nass2000machines}. \nSince our rationales are justificatory in nature, \\textit{adequate justification} is a reasonable measure of output quality (transformed from TAM).\nOur rationales also need to be \\textit{understandable}, which can signal perceived ease of use (from TAM). \n\n\n\n\n\n\n\\subsection{Quantitative Analysis}\n\n\nWe used a multi-level model to analyze our data.\nAll variables were within-subjects except for one: whether the candidate style was focused-view (Group 1) or complete-view (Group 2).  \nThis was a between-subject variable.\n\nThere were significant main effects of rationale style ($\\chi^2\\left(2\\right) = 594.80, p<.001$) and dimension ($\\chi^2\\left(2\\right) = 66.86, p<.001$) on the ratings. \nThe main effect of experimental group was not significant ($\\chi^2\\left(1\\right) = 0.070, p=0.79$).\nFigure~\\ref{fig:Results_1} shows the average responses to each question for the two different experimental groups.\nOur results support our hypothesis that rationales generated with the \\textit{focused-view} generator and the \\textit{complete-view} generator were judged significantly better across all dimensions than the random baseline\n($b=1.90, t\\left(252\\right)=8.09,p<.001$). \nIn addition, exemplary rationales were judged significantly higher than candidate rationales.\n\n\n\n\nThough there were significant differences between each kind of candidate rationale and the exemplary rationales, those differences were not the same.\nThe difference between the \\textit{focused-view} candidate rationales and exemplary rationales were significantly \\textit{greater} than the difference between \\textit{complete-view} candidate rationales and exemplary rationales ($p=.005$). \nSurprisingly, this was because the exemplary rationales were rated {\\rm lower} in the presence of complete-view candidate rationales ($t\\left(1530\\right)=-32.12,p<.001$).\nSince three rationales were presented simultaneously in each video, it is likely that participants were rating the rationales relative to each other. \nWe also observe that the \\textit{complete-view} candidate rationales received higher ratings in general than did the \\textit{focused-view} candidate rationales ($t\\left(1530\\right)=8.33,p<.001$).\n\nIn summary, we have confirmed our hypothesis that both configurations produce rationales that perform significantly better than the \\textit{random} baseline across all dimensions. \n\n\\begin{figure}[t]\n\\subcaptionbox{Focus-View condition.\\label{fig:Results_1a}}{\\includegraphics[width=1.0\\linewidth]{Bar_a_exactDimension.png}}\\hfill\n\\subcaptionbox{Complete-View condition.\\label{fig:Results_1b}}{\\includegraphics[width=1.0\\linewidth]{Bar_b_exactDimension.png}}\\hfill\n   \\caption{Human judgment results. }\n  \\label{fig:Results_1}\n\\end{figure}\n\n\\subsection{Qualitative Findings and Discussion}\n\n\nIn this section, we look at the open-ended responses provided by our participants to better understand the criteria that participants used when making judgments about the \\textit{confidence, human-likeness, adequate justification,} and \\textit{understandability} of generated rationales. \nThese situated insights augment our understanding of rationale generating systems, enabling us to design better ones in the future.\n\n\nWe analyzed the open-ended justifications participants provided using a combination of thematic analysis \\cite{aronson1994pragmatic} and grounded theory \\cite{strauss1994grounded}. \nWe developed codes that addressed different types of reasonings behind the ratings of the four dimensions under investigation. \nNext, the research team clustered the codes under emergent themes, which form the underlying \\textit{components} of the dimensions. Iterating until consensus was reached, researchers settled on the five most relevant components: (1)~\\textit{Contextual Accuracy}, (2)~\\textit{Intelligibility}, (3)~\\textit{Awareness},\n(4)~\\textit{Relatability}, and (5)~\\textit{Strategic Detail} (see Table \\ref{tab:components}).\nAt varying degrees, multiple components influence more than one dimension; that is, there isn't a mutually exclusive one-to-one relationship between components and dimensions. \n\nWe will now share how these components influence the dimensions of the human factors under investigation.  \nWhen providing examples of our participants' responses, we will use P$1$ to refer to participant 1, P$2$ for participant 2, etc. \nTo avoid priming during evaluation, we used letters (e.g., A, B, C, etc.) to refer to the different types of rationales. \nFor better comprehension, we have substituted the letters with appropriate rationale--focused-view, complete-view, or random-- while presenting quotes from participants below.  \n\n\\subsubsection{Confidence (1)} This dimension gauges the participant's faith in the agent's ability to successfully complete it's task and has \\textit{contextual accuracy},  \\textit{awareness}, \\textit{strategic detail}, and \\textit{intelligibility} as relevant components. \nWith respect to \\textit{contextual accuracy}, rationales that displayed ``\\ldots recognition of the environmental conditions and [adaptation] to the conditions'' (P22) were a positive influence, while redundant information such as  ``just stating the obvious'' (P42) hindered confidence ratings. \n\n\n\n\n\\begin{table}[h]\n  \\caption{Descriptions for the emergent \\textit{components} underlying the human-factor \\textit{dimensions} of the generated rationales.}\n  \\label{tab:components}\n  \\begin{tabular}{p{0.32\\columnwidth}p{0.58\\columnwidth}}\n    \\toprule\n    {\\bf Component} & {\\bf Description}\\\\\n    \\midrule\n    Contextual Accuracy & Accurately describes pertinent events in the context of the environment.\\\\\n    \\rowcolor{tablerowcolor} Intelligibility & Typically error-free and is coherent in terms of both grammar and sentence structure.\\\\\n    Awareness & Depicts and adequate understanding of the rules of the environment.\\\\\n    \\rowcolor{tablerowcolor} Relatability & Expresses the justification of the action in a relatable manner and style.\\\\\n    Strategic Detail & Exhibits strategic thinking, foresight, and planning.\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nRationales that showed \\textit{awareness} of ``upcoming dangers and what the best moves to make \\ldots [and] a good way to plan'' (P17) inspired confidence from the participants. \nIn terms of \\textit{strategic detail}, rationales that showed \"\\ldots long-term planning and ability to analyze information\" (P28) yielded higher confidence ratings compared to those that were \"short-sighted and unable to think ahead\" (P14) led to lower perceptions of confidence. \n\n\\textit{Intelligibility} alone, without \\textit{awareness} or \\textit{strategic detail}, was not enough to yield high confidence in rationales. However, rationales that were not \\textit{intelligible} (unintelligible) or coherent had a negative impact on participants' confidence:\n\\begin{displayquote}\nThe [random and focused-view rationales] include major mischaracterizations of the environment because they refer to an object not present or wrong time sequence, so I had very low confidence. (P66)\n\\end{displayquote}\n\n\n\n\n\n\\subsubsection{Human-likeness (2)} \\textit{Intelligibility, relatability,} and \\textit{strategic detail} are components that influenced participants' perception of the extent to which the rationales were made by a human.\nNotably, \\textit{intelligibility} had mixed influences on the human-likeness of the rationales as it depended on what participants thought ``being human'' entailed. \nSome conceptualized humans as fallible beings and rated rationales with errors more \\textit{humanlike} because rationales ``with typos or spelling errors \\ldots seem even more likely to have been generated by a human\" (P19). \nConversely, some thought error-free rationales must come from a human, citing that a ``computer just does not have the knowledge to understand what is going on'' (P24).\n\nWith respect to \\textit{relatability}, rationales were often perceived as more human-like when participants felt that ``it mirrored [their] thoughts'' (P49), and ``\\ldots [laid] things out in a way that [they] would have'' (P58). Affective rationales had high \\textit{relatability} because they ``express human emotions including hope and doubt'' (P11). \n\n\\textit{Strategic detail} had a mixed impact on human-likeness just like \\textit{intelligibility} as it also depended on participants' perception of critical thinking and logical planning. Some participants associated ``\\ldots critical thinking [and ability to] predict future situations\" (P6) with human-likeness whereas others associated logical planning with non-human-like, but computer-like rigid and algorithmic thinking process flow.\n\n\n\n\n\n\\subsubsection{Adequate Justification (3)} This dimension unpacks the extent to which participants think the rationale adequately justifies the action taken and is influenced by \\textit{contextual accuracy}, and \\textit{awareness}. \nParticipants downgraded rationales containing low levels of \\textit{contextual accuracy} in the form of irrelevant details. As P11 puts it: \n\\begin{displayquote}\nThe [random rationale] doesn't pertain to this situation. [The complete-view] does, and is clearly the best justification for the action that Frogger took because it moves him towards his end goal. \n\\end{displayquote}\n\nBeyond \\textit{contextual accuracy},  rationales that showcase \\textit{awareness} of surroundings score high on the \\textit{adequate justification} dimension. For instance, P11 rated the \\textit{random} rationale low because it showed  ``no awareness of the surroundings''. For the same action, P11 rated \\textit{exemplary} and \\textit{focused-view} rationales high because each made the participant  ``believe in the character's ability to judge their surroundings.''\n\n\\subsubsection{Understandability (4)} \nFor this dimension, components such as \\textit{contextual accuracy} and \\textit{relatability} influence participants' perceptions of how much the rationales helped them understand the motivation behind the agent's actions. \nIn terms of \\textit{contextual accuracy}, \nmany expressed how the contextual accuracy, not the length of the rationale, mattered when it came to understandability.  \nWhile comparing understandability of the \\textit{exemplary} and \\textit{focused-view} rationales, P41 made a notable observation:\n\\begin{displayquote}\nThe [exemplary and focused-view rationale] both described the activities\/objects in the immediate vicinity of the frog.  However, [exemplary rationale (typically lengthier than focused)] was not as applicable because the [focused-view] rationale does a better job of providing contextual understanding of the action.\n\\end{displayquote}\n\nParticipants put themselves in the agent's shoes and evaluated the understandability of the rationales based on how \\textit{relatable} they were. \nIn essence, some asked ``Are these the same reasons I would [give] for this action?'' (P43). \nThe more relatable the rationale was, the higher it scored for understandability. \n\nIn summary, the first study establishes the plausibility of the generated rationales (compared to baselines) and their user perceptions. \nHowever, this study does not provide direct comparison between the two configurations. \n\n\\section{Preference Study: Focused-- vs. Complete--View Rationales}\nThe preference study puts the rationales in direct comparison with each other. \nThis study \nIt achieves two main purposes. First, it aims to validate the alignment between the intended design of rationale types and the actual perceived differences between them. \nWe collect qualitative data on how participants perceived rationales produced by our \\textit{focused-view} and \\textit{complete-view} rationale generator.\nOur expert observation is that the \\textit{focused-view} configuration results in concise and localized rationales whereas the \\textit{complete-view} configuration results in detailed, holistic rationales. \nWe seek to determine whether na\\\"ive users who are unaware of which configuration produced a rationale also describe the rationales in this way. \nSecond, we seek to understand how and why the preferences between the two styles differed along three dimensions: {\\em confidence}, {\\em failure}, and {\\em unexpected behavior}.\n\n\n\\subsection{Method}\nUsing similar methods to the first study, we recruited and analyzed the data from 65 people (age range = 23 - 59 years, M = 38.48, SD = 10.16). 57\\% percent of the participants were women with 96\\% of the participants self-reporting the United States and 4\\% self-reporting India as countries they were from. Participants from our first study could not partake in the second one. The average task duration was approximately 46 minutes.\n\n\n\n\nThe only difference in the experimental setup between perception and the preference study is the comparison groups of the rationales. \nIn this study, participants judged the same set of\n\\textit{focused-} and \\textit{complete-view} rationales, however instead of judging each style against two baselines, participants evaluate the \\textit{focused-} and \\textit{complete-view} rationales in direction comparison with each other.\n\n\nHaving watched the videos and accompanying rationales, participants responded to the following questions comparing both configurations: \n\\begin{enumerate}\n    \\item \\textbf{Most important difference}: What do you see as the most important difference? Why is this difference important to you?\n    \\item \\textbf{Confidence}: Which style of rationale makes you more confident in the agent's ability to do its task? Was it system A or system B? Why?\n    \\item \\textbf{Failure}: If you had a companion robot that had just made a mistake, would you prefer that it provide rationales like System A or System B? Why? \n    \\item \\textbf{Unexpected Behaviour}: If you had a companion robot that took an action that was not wrong, but unexpected from your perspective, would you prefer that it provides rationales like System A or System B? Why? \n\\end{enumerate}\n\n\nWe used a similar to the process of selecting dimensions in this study as we did in the first one. \n\\textit{Confidence} is crucial to trust especially when failure and unexpected behavior happens \\cite{chernova2009confidence, kaniarasu2013robot}. \nCollaboration, tolerance, and perceived intelligence are affected by the way autonomous agents and robots communicate \\textit{failure} and \\textit{unexpected behavior} \\cite{desai2013impact,kwon2018expressing,lee2010gracefully,mirnig2017err}. \n\n\n\\begin{table}[h]\n  \\caption{Tally of how many preferred the \\textit{focused-view} vs. the \\textit{complete-view} for the three dimensions.}\n  \\label{tab:components}\n  \\begin{tabular}{ccc}\n    \\toprule\n    {\\bf Question} & {\\bf Focused-view} & {\\bf Complete-view}\\\\\n    \\midrule\n    Confidence & 15 & 48\\\\\n    \\rowcolor{tablerowcolor} Failure & 17 & 46\\\\\n    Unexpected Behaviour & 18 & 45\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\\subsection{Quantitative Analysis}\nIn order to determine whether the preferences significantly favored one style or the other, we conducted  \nthe Wilcoxon signed-rank test. It showed that preference for the \\textit{complete-view} rationale was significant in all three dimensions.\nConfidence in the \\textit{complete-view} rationale was significantly greater than in the \\textit{focused-view}  ($p<.001$). \nSimilarly, preference for a \\textit{complete-view} rationales from an agent that made a mistake was significantly greater than for \\textit{focused-view} rationales ($p<.001$).\nPreference for \\textit{complete-view} rationales from an agent that made a mistake was also significantly greater than for \\textit{focused-view} rationales ($p<.001$).\n\n\\subsection{Qualitative Findings and Discussion}\nIn this section, similar to the first study, we share insights gained from the open-ended responses to reveal the underlying reasons behind perceptions of the \\textit{most important difference} between the two styles. We also unpack the reasoning behind the quantitative ranking preferences for \\textit{confidence} in the agent's ability to do its task and communication preferences for \\textit{failure} and \\textit{unexpected behavior}. In this analysis, the interacting \\textit{components} that influenced the dimensions of human factors in the first study return (see Table \\ref{tab:components}). In particular, we use them as analytic lenses to highlight the trade-offs people make when expressing their preferences and the reasons for the perceived differences between the styles. \n\nThese insights bolster our situated understanding of the differences between the two rationale generation techniques and assist to verify if the intended design of the two configurations aligns with the perceptions of them. In essence, did the design succeed in doing what we set out to do? We analyzed the open-ended responses in the same manner as the first study. We use the same nomenclature to refer to participants. \n\n\\subsubsection{Most Important Difference (1)}\nEvery participant\nindicted that\nthe \\textit{level of detail and clarity} (P55)\ndifferentiated the rationales. \nConnected to the level of detail and clarity is the perceived \\textit{long-} vs. \\textit{short-term} planning exhibited by each rationale. \nOverall, participants felt that the \\textit{complete-view} rationale showed better levels of \\textit{strategic detail}, \\textit{awareness}, and \\textit{relatability} with human-like justifications, whereas the \\textit{focused-view} exhibited better \\textit{intelligibility} with easy-to-understand rationales. \nThe following quote \nillustrates\nthe trade-off between succinctness,\nwhich hampers comprehension of higher-order goals, and broadness, which can be perceived as less\nfocused:\n\n\\begin{displayquote}\nThe [focused-view rationale] is extraordinarily vague and focused on the raw mechanics of the very next move \\ldots [The complete-view] is more broad and less focused, but takes into account \\textit{the entire picture}. So I would say the most important difference is the \\textit{scope of events} that they take into account while making justifications [emphasis added] (P24)\n\\end{displayquote}\n\nBeyond trade-offs, this quote highlights a powerful validating point: without any knowledge beyond what is shown on the video, the participant pointed out how the \\textit{complete-view} rationale appeared to consider the \"entire picture\" and how the \"scope of events\" taken into account was the main difference. \nThe participant's intuition precisely aligns with the underlying network configuration design and our \nresearch \nintuitions.\nRecall that the \\textit{complete-view} rationale\nwas generated using the entire environment or \"picture\" whereas the \\textit{focused-view} was generated using a windowed input.  \n\nIn prior sections, we \nspeculated on the effects of the network configurations. We expected the \\textit{focused-view} version \nto produce \nsuccinct, localized rationales that concentrated on the short-term. We expected the \\textit{complete-view} version \nto produce detailed, broader rationales that focused on the larger picture and long-term planning. \nThe findings of this experiment are the first validation that the outputs reflect the intended designs.  \nThe strength of this validation was enhanced by the many descriptions of our intended attributes, given in free-form by participants who were naive to our network designs.\n\nConnected to the level of detail and clarity is the perception of \\textit{short-} vs \\textit{long-term} thinking from the respective rationales. \nIn general, participants regarded the \\textit{focused-view} rationale having low levels of \\textit{awareness} and \\textit{strategic detail}. \nThey felt\nthat this agent \"\\ldots focus[ed] only on the current step\" (P44), which was perceived depicting as thinking \"\\ldots in the spur of the moment\" (P27), giving the perception of short-term and simplistic thinking.  \nOn the other hand, the \\textit{complete-view} rationale appeared to \"\\ldots try to think it through\" (P27), exhibiting long-term thinking as it appears to \"\\ldots think forward to broader strategic concerns.\"(P65)  One participant sums it up nicely: \n\\begin{displayquote}\nThe [focused-view rationale] focused on the immediate action required. [The complete-view rationale] took into account the current situation, [but] also factored in what the next move will be and what dangers that move poses. The [focused-view] was more of a short term decision and [complete-view] focused on both short term and long term goals and objectives. (P47)\n\\end{displayquote}\n\nWe will notice how these differences in perception impact other dimensions such as confidence and communication preferences for failure and unexpected behavior. \n\\subsubsection{Confidence (2)}\nParticipants had more confidence in the agent's ability to do its task if the rationales exhibited high levels of \\textit{strategic detail} in the form of long-term planning, \\textit{awareness} via expressing knowledge of the environment, and \\textit{relatability} through humanlike expressions. They associated \\textit{conciseness} with confidence when the rationales did not need to be detailed given the context of the (trivial) action. \n\nThe \\textit{complete-view} rationale inspired more confidence because participants perceived agents with long-term planning and high\n\\textit{strategic detail} as being \"more predictive\" and intelligent than their counterparts. Participants felt more at ease because \"\\ldots knowing what [the agent] was planning to do ahead of time would allow me to catch mistakes earlier before it makes them.\" (P31) As one participant put it:\n\n\\begin{displayquote}\nThe [complete-view rationale] gives me more confidence \\ldots because it thinks about future steps and not just the steps you need to take in the moment. [The agent with focused-view] thinks more simply and is prone to mistakes. (P13)\n\\end{displayquote}\n\nParticipants felt that rationales that exhibited a better understanding of the environment, and thereby better \\textit{awareness}, resulted in higher confidence scores. Unlike the \\textit{focused-view} rationale that came across as \"a simple reactionary move \\ldots [the \\textit{complete-view}] version demonstrated a more thorough understanding of the entire field of play.\" (P51) In addition, the \\textit{complete-view} was more \\textit{relatable} and confidence-inspiring \"because it more closely resemble[d] human judgment\" (P29). \n\n\\subsubsection{Failure (3)}\nWhen an agent or a robot fails, the information from the failure report is mainly used to fix the issue. To build a mental model of the agent, participants preferred \\textit{detailed} rationales with solid \\textit{explanatory power} stemming from \\textit{awareness} and \\textit{relatability}. The mental model could facilitate proactive and preventative care. \n\nThe \\textit{complete-view} rationale, due to relatively high \\textit{strategic detail}, was preferable in communicating failure because participants could \"\\ldots understand the full reasoning behind the movements.\"(P16) Interestingly, \\textit{detail} trumped \\textit{intelligibility} in most circumstances. Even if the rationales had some grammatical errors or were a \"\\ldots little less easy to read, the details made up for it.\" (P62) \n\nHowever, detailed rationales are not always a virtue. Simple rationales have the benefit of being easily understandable to humans, even if they cause humans to view the agent as having limited understanding capabilities. Some participants appreciated \\textit{focused-view} rationales because they felt \"it would be easier to figure out what went wrong by focusing on one step at a time.\"  \n\n\nExplanatory power, specifically how events are communicated, is related to \\textit{awareness} and \\textit{relatability}. Participants preferred relatable agents that \"\\ldots would talk to [them] like a person would.\"(P11) They expressed the need to develop a mental model, especially to \"\\ldots see how [a robot's] mind might be working\"(P1), to effectively fix the issue. The following participant neatly summarizes the dynamics:\n\\begin{displayquote}\nI'd want [the robot with complete-view] because I'd have a better sense of the steps taken that lead to the mistake. I could then fix a problem within that reasoning to hopefully avoid future mistakes. The [focused-view rationale] was just too basic and didn't give enough detail. (P8)\n\\end{displayquote}\n\\subsubsection{Unexpected Behavior (4)}\nUnexpected behavior that is not failure makes people want to know the \"why?\" behind the action, especially to understand the expectancy violation. \nAs a result, participants preferred rationales with transparency so that they can understand and trust the robot in a situation where expectations are violated. \nIn general, preference was for adequate levels of \\textit{detail} and \\textit{explanatory power} that could provide \"\\ldots more diagnostic information and insight into the robot's thinking processes.\"(P19) \nParticipants wanted to develop mental models of the robots so they could understand the world from the robot's perspective. \nThis diagnostic motivation for a mental model is different from the re-programming or fixing needs in cases of failure. \n\nThe \\textit{complete-view} rationale, due to adequate levels of \\textit{strategic detail}, made participants more confident in their ability to follow the thought process and get a better understanding of the expectancy violation. One participant shared:\n\\begin{displayquote}\nThe greater clarity of thought in the [complete-view] rationale provides a more thorough picture \\ldots, so that the cause of the unexpected action could be identified and explained more easily. (P51)\n\\end{displayquote}\nWith this said, where possible without sacrificing transparency, participants welcomed simple rationales that \"anyone could understand, no matter what their level of education was.\"(P2)\nThis is noteworthy because the expertness level of the audience is a key concern when making accessible AI-powered technology where designers need to strike a balance between detail and succinctness.\n\nRationales exhibiting strong explanatory power, through \\textit{awareness} and \\textit{relatability}, helps to situate the unexpected behavior in an understandable manner. \nParticipants preferred the \\textit{complete-view} rationale's style of communication because of increased transparency:\n\\begin{displayquote}\nI prefer [the complete-view rationale style] because \\ldots I am able to get a much better picture of why it is making those decisions. (P24)\n\\end{displayquote}\n\nDespite similarities in the communication preferences for failure and unexpected behavior, there are differences in underlying reasons. As our analysis suggests, the mental models are desired in both cases, but for different reasons. \n\\section{Design Lessons and Implications} \nThe situated understanding of the \\textit{components} and \\textit{dimensions} give us a powerful set of actionable insights that can help us design better human-centered, rationale-generating, autonomous agents.\nAs our analysis reveals, context is king. \nDepending on the context, we can tweak the input type to generate \\textit{rationale sytles} that meet the needs of the task or agent persona; for instance, a companion agent that requires high \\textit{relatability} for user engagement. \nWe should be mindful when optimizing for a certain dimension as each component comes with costs. \nFor instance, conciseness can improve \\textit{intelligibility} and overall \\textit{understandability} but comes at the cost of \\textit{strategic detail}, which can hurt \\textit{confidence} in the agent.\nWe can also engineer systems such that multiple network configurations act as modules. \nFor instance, if we design a companion agent or robot that interacts with a person longitudinally, the \\textit{focused-view} configuration can take over\nwhen short and simple rationales are required. \nThe \\textit{complete-view} configuration or a hybrid one can be activated when communicating failure or unexpected behavior.\n\nAs our preference study shows, we should not only be cognizant about the level of detail, but also why the detail is necessary, especially while communicating failure and unexpected behavior.\nFor instance, \nfailure-reporting,\nin a mission critical task (such as search and rescue), would have different requirements for \\textit{strategic detail} and \\textit{awareness}, compared to\n\"failure\" reporting in a \nless-defined, more creative task like making music. \nWhile the focus of this paper is on textual rationale generation, rationales can be complementary to other types of explanations; for instance, a multi-modal system can combine visual cues with textual rationales to provide better contextual explanations for an agent's actions.  \n\n\n\\section{Limitations and Future Work}\nWhile these results are promising, there are several limitations in our approach that need to be addressed in future work. \nFirst, our current system, by intention and design, lacks interactivity; users cannot contest a rationale or ask the agent to explain in a different way. \nTo a get a formative understanding, we kept the design as straight-forward as possible.\nNow that we have a baseline understanding, we can vary along the dimension of interactivity for the next iteration. \nFor instance, contestability, the ability to either reject a reasoning or ask for another one, which has shown to improve user satisfactions \\cite{hirsch2017designing,dietvorst2016overcoming} can be incorporated in the future.\nSecond, our data collection pipeline is currently designed to work with discrete-action games that have natural break points where the player can be asked for explanations. \nIn continuous-time and -action environments, we must determine how to collect the necessary data without being too intrusive to participants.\nThird, all conclusions about our approach were formed based on one-time interactions with the system. \nTo better control for potential novelty effects that rationales could have, we need to deploy our system in a longitudinal task setting. \nFourth, to understand the feasibility of our system in larger state-action spaces, we would need to study the scalability by addressing the question of how much data is needed based on the size of environment. \nFifth, not all mistakes are created equal. Currently, the perception ratings are averaged where everything is equally weighted. For instance, a mistake during a mission critical step can lead to higher fall in confidence than the same mistake during a non-critical step. To understand the relative costs of mistakes, we need to further investigate the relationship between context of the task and the cost of the mistake.  \n     \n\n\n\n    \n\n\n \n\n\n\n\n\n\n\\section{Conclusions}\nWhile explainability has been successfully introduced for classification and captioning tasks, sequential environments offer a unique challenge for generating human understandable explanations.\nThe challenge stems from multiple complex factors, such as temporally connected decision-making, that contribute to making decisions in these environments.\nIn this paper, we introduce \\textit{automated rationale generation} as a concept and explore how justificatory explanations from humans can be used to train systems to produce human-like explanations in sequential environments. \nTo facilitate this work, we also introduce a pipeline for automatically gathering a parallel corpus of states annotated with human explanations. \nThis tool enables us to systematically gather high quality data for training purposes.\nWe then use this data to train a model that uses machine translation technology to generate human-like rationales in the arcade game, {\\em Frogger}. \n\nThrough a mixed-methods approach in evaluation, we establish the plausibility of the generated rationales and describe how intended design of  rationale  types  lines up with the  actual  user perceptions of them. \nWe also get contextual understanding of the underlying dimensions and components that influence human perception and preferences of the generated rationales. \nBy enabling autonomous agents to communicate about the motivations for their actions, we envision a future where explainability not only improves human-AI collaboration, but does so in a human--centered and understandable  manner.\n\n\n\n\\section{Acknowledgements}\nThis work was partially funded under ONR grant number N00014141000. We would like to thank Chenghann Gan and Jiahong Sun for their valuable contributions to the development of the data collection pipeline. We are also grateful to the feedback from anonymous reviewers that helped us improve the quality of the work.\n\\balance{}\n\n\n\n\\balance{}\n\n\\bibliographystyle{SIGCHI-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe pulsar J1811$-$1736, discovered during observations of the Parkes\nMultibeam Pulsar Survey\\citep{mlc+01},  has a spin period of 104 ms\nand is member of a 18.8-d, highly eccentric binary system\n\\citep{lcm+00} with an as yet undetected companion \\citep{mig00}. The\ncharacteristic age and the estimated surface magnetic field strength\nindicate that the pulsar is mildly recycled. In such a system, it\nis expected that the observed pulsar is born first in a supernovae\n(SN) explosion before it undergoes mass accretion from a high-mass\nnon-degenerate binary  companion. Parameters which have been\nmeasured and derived from the best-fit timing solution indicate that\nthe companion is quite massive. All these elements suggest that the\ncompanion is also a neutron star (e.g. \\citealt{bv91}).\n\nThe conclusion that PSR\\,J1811$-$1736 is a member of the small sample\nof known double neutron star (DNS) systems was already reached by\n\\citet{lcm+00}. Their conclusion was supported by a constraint on the\ntotal system mass, assuming that the observed advance of periastron\nwas totally due to relativistic effects. However, the rather long\nperiod of the pulsar, combined with the effects of interstellar\nscattering, resulting in significant broadening of the pulse profile at\n1.4~GHz, as well as the short data span available to \\citet{lcm+00},\nlimited the timing precision and hence the accuracy of the total mass\nmeasurement.\n\nAmong all DNS systems, PSR\\,J1811-1736 has by far the longest spin\nperiod, the longest orbital period and the highest eccentricity. This\nmay suggest that the evolution of this DNS has been different, at\nleast in part, from all other DNS systems. On the other hand,\nPSR\\,J1811-1736 well fits the spin period versus eccentricity relation\nfor DNS systems (\\citealt{mlc+05}, \\citealt{fkl+05}). This relation\ncan be simply explained in terms of the different lengths of time the\npulsar underwent accretion, which in turn is related to the mass of\nthe companion star before the SN explosion. Moreover, numerical\nsimulations \\citep{dpp05} show that the spin period versus\neccentricity relation is recovered assuming that the second born\nneutron star received a low velocity kick at its birth.\n\nIn this paper we report on new timing observations which significantly\nimprove on the previously published results. We present a study\nof the observed interstellar scattering and consider its consequences\non the detectability of radio pulses from the companion. Finally, we\ninvestigate the likely kick velocity imparted to the second-born\nneutron star during its birth in the system's second SN.\nObservations were carried out as part of a coordinated effort using\nthree of the largest steerable radio telescopes in the world for\npulsar timing observations, i.e. the 100-m radio telescope at\nEffelsberg, the 94-m equivalent Westerbork Synthesis Radio Telescope\n(WSRT) and the 76-m Lovell telescope at Jodrell Bank telescope. This\npaper is the first in a series detailing results of these efforts in\nestablishing a {\\em European Pulsar Timing Array} (EPTA).\n\n\\section{Observations}\n\nThe binary pulsar J1811$-$1736 is one of the binary pulsars regularly\nobserved by the EPTA. The aims and objectives of this European\ncollaboration include the detection of a cosmological gravitational\nwave background and the project will be described in detail in a\nforthcoming publication. Here we summarize the observing systems used\nwhile further details can be found in the references below.\n\n\\subsection{Effelsberg timing}\n\nWe made regular timing observations of PSR\\,J1811$-$1736 since October\n1999 using the 100-m radio telescope of the Max Planck Institut f\\\"ur\nRadioastronomie in Effelsberg near Bonn. The typical observing rate\nwas of 1 observation every two months. An overall root-mean-square\n(RMS) of 538 $\\mu s$ is achieved after applying the final timing\nmodel.  The data were obtained with a 1.3$-$1.7\\,GHz tunable HEMT\nreceiver installed in the primary focus of the telescope. The noise\ntemperature of this system is 25~K, resulting in a system temperature\nfrom 30~to 50~K on cold sky depending on elevation.  The antenna gain\nat these frequencies is 1.5~K~Jy$^{-1}$.\n\nAn intermediate frequency (IF) centred on 150 MHz for left-hand (LHC)\nand right-hand (RHC) circularly polarised signals was obtained after\ndown-conversion from a central RF frequency of usually 1410 MHz.  The\nsignals received from the telescope were acquired and processed with\nthe Effelsberg-Berkeley Pulsar Processor (EBPP) which removes the\ndispersive effects of the interstellar medium on-line using ``coherent\nde-dispersion'' \\citep{hr75}. Before entering the EBPP, the two LHC\nand RHC signals of the IF are converted to an internal IF of 440\nMHz. A maximum bandwidth of $2\\times32\\times0.7$~MHz~=~$2\\times32$~MHz\nwas available for the chosen observing frequency and DM of the\npulsar. It was split into four portions for each of the two circular\npolarisations, which were mixed down to baseband. Each portion was\nthen sub-divided into eight narrow channels via a set of digital\nfilters \\citep{bdz+97}. The outputs of each channel were fed into\nde-disperser boards for coherent on-line de-dispersion. In total 64\noutput signals were detected and integrated in phase with the\npredicted topocentric pulse period.\n\nA pulse time-of-arrival (TOA) was calculated for each average profile\nobtained during a 5-10 min observation. During this process, the\nobserved time-stamped profile was compared to a synthetic template,\nwhich was constructed out of 5 Gaussian components fitted to a\nhigh signal-to-noise standard profile (see Kramer et al.,\n\\citeyear{kxl+98,kll+99}).  This template matching was done by a\nleast-squares fitting of the Fourier-transformed data \\citep{tay92}. \nUsing the measured time delay between the actual\nprofile and the template, the accurate time stamp of the data provided\nby a local H-MASER and corrected off-line to UTC(NIST) using recorded\ninformation from the satellites of the Global Positioning System\n(GPS), the final TOA was obtained.  The uncertainty of each TOA was\nestimated using a method described by \\citet{dr83} and \\citet{lan99}.\n\n\\subsection{Jodrell Bank timing}\n\nObservations of PSR\\,J1811-1736 were made regularly using the 76-m\nLovell telescope at Jodrell Bank since its discovery in 1997\n\\citep{lcm+00}.  The typical observing rate was of about 2\nobservations each week, with an overall RMS of 1300\\,$\\mu$s after\napplying the final timing model. A cryogenic receiver at 1404\\,MHz was\nused, and both LHC and RHC signals were observed using a\n$2\\times32\\times1.0$-MHz filter bank at 1404\\,MHz. After detection,\nthe signals from the two polarizations were filtered, digitised at\nappropriate sampling intervals, incoherently de-dispersed in hardware\nbefore being folded on-line with the topocentric pulse period and\nwritten to disk.  Each integration was typically of 1-3 minutes\nduration; 6 or 12 such integrations constituted a typical\nobservation. Off-line, the profiles were added in polarisation pairs\nbefore being summed to produce a single total-intensity profile.  A\nstandard pulse template was fitted to the observed profiles at each\nfrequency to determine the pulse times-of-arrival (TOAs). Details of\nthe observing system and the data reduction scheme can be found\nelsewhere (e.g.~\\citealt{hlk+04}).\n\n\n\\subsection{Westerbork timing}\n\nObservations of PSR\\,J1811$-$1736 were carried out approximately\nmonthly since 1999 August 1st, obtaining an overall timing RMS of\n659~$\\mu$s after applying the final timing model, at a central\nfrequency of 1380 MHz and a bandwidth of 80 MHz. The two linear\npolarisations from all 14 telescopes were added together in phase by\ntaking account of the relative geometrical and instrumental phase\ndelays between them and then passed to the PuMa pulsar backend\n\\citep{vkv+02}. The data were obtained with the L-band receiver\ninstalled in the primary focus of the telescopes. The noise\ntemperature of this system is 25~K, resulting in a system temperature\nfrom 30~to 50~K on cold sky depending on elevation.  The antenna gain\nat these frequencies is 1.2~K~Jy$^{-1}$. PuMa was used in its digital\nfilterbank mode whereby the Nyquist sampled signals are Fourier\ntransformed and the polarisations combined to produce total intensity\n(Stokes I) spectra with a total of 512 channels. These spectra were\nsummed online to give a final sampling time of 409.6 $\\mu$s and\nrecorded to hard disk.  These spectra were subsequently dedispersed\nand folded with the topocentric period off-line to form integrations\nof a few minutes.  TOAs were calculated for each profile following a\nscheme similar to that outlined above for Effelsberg data, except a\nhigh signal-to-noise standard profile was used instead of Gaussian\ncomponents.  In the future, EPTA timing will employ an identical\nsynthetic template for all telescopes.\n\n\n\\section{Data analysis}\n\n\nThe TOAs, corrected to UTC(NIST) via GPS and weighted by their\nindividual uncertainties determined in the fitting process, were\nanalysed with the {\\tt TEMPO} software package \\citep{tw89}, using the\nDE405 ephemeris of the Jet Propulsion Laboratory \\citep{sta90}. {\\tt\nTEMPO} minimizes the sum of weighted squared {\\it timing residuals},\ni.e.~the difference between observed and model TOAs, yielding a set of\nimproved pulsar parameters and post-fit timing residuals. A summary\nof the basic characteristics of each dataset is shown in Table 1.\n\n   \\begin{table}\n    \\centering\n      \\caption[]{Data sets' characteristics}\n         \\label{TabData} \n         \\begin{tabular}{llll}\n            \\hline\n            \\noalign{\\smallskip}\n\n& Jodrell Bank & Effelsberg & Westerbork\\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nN.~of~ToAs   & 348 & 74 & 213\\\\\nTime~Span & 50842-53624 & 51490-53624 & 51391-53546 \\\\\nR.M.S. & 1300 & 538  & 659\\\\\n\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n         \\end{tabular}\n\\end{table}\n\nAlthough the templates used for the three telescope data differed,\nresulting offsets were absorbed in a global least-squares\nfit. Remaining uncertainties were smaller than the typical measurement\naccuracy of the Jodrell Bank timing data of about 9 $\\mu$s.\n\nBefore all TOAs were combined, preliminary fits were performed on each\ndataset alone, in order to study possible systematic differences between\nthe datasets. We applied a small quadrature addition \nand a scaling factor to the\nuncertainties to obtain the expected value of a reduced $\\chi^2=1$\nfor each dataset. \nThe final joint fit to all TOAs resulted in a $\\chi^{2}$ value of\nunity, avoiding the need to add further systematic uncertainties.\n\nTable 2 summarizes all observed timing and some derived\nparameters. For the observed parameters the quoted errors are twice\nthe nominal TEMPO errors.  For the derived parameters, the given\nuncertainties are computed accordingly.\n\nThe joint fit allowed us to determine the spin, positional and\nKeplerian orbital parameters plus one post-Keplerian parameter with a\nprecision better than the best determination from a single data set\nalone. However, the high degree of interstellar scattering (Fig. 3)\nmeans that further post-Keplerian parameters will be difficult to\nmeasure with continued observations at this frequency. We will discuss\nthe future prospects for higher frequency observations in \\S~5.\n\n   \\begin{table}\n      \\caption[]{Timing and derived parameters}\n         \\label{TabTim} \n         \\begin{tabular}{ll}\n            \\hline\n            \\noalign{\\smallskip}\nTiming parameters & Joint~data~sets\\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nRA~(J2000, hh:mm:ss)     & 18:11:55.034(3)\\\\\nDECL~(J2000, deg:mm:ss)  & -17:36:37.7(4)\\\\\nPeriod, $P$~(s)      & 0.1041819547968(4) \\\\\nPeriod derivative, $\\dot{P}$~($10^{-19}$ s~s$^{-1}$)    & 9.01(5)\\\\\nDispersion Measure, DM~(pc~cm$^{-3}$)   &  476(5)\\\\\nProjected semi-major axis$^{a}$, $a~\\sin~i$~(s) & 34.7827(5)\\\\\nEccentricity, $e$       & 0.828011(9)\\\\\nEpoch of periastron, $T_{0}$~(MJD)   & 50875.02452(3)\\\\\nOrbital period, $P_{B}$~(d)     & 18.7791691(4)\\\\\nLongitude of periastron, $\\omega$~(deg)  & 127.6577(11)\\\\\nAdvance of periastron, $\\dot{\\omega}$~(deg~yr$^{-1}$) & 0.0090(2)\\\\\nFlux density at 3100\\,MHz, $S_{3100}$ (mJy) & 0.34(7) \\\\\n&\\\\\nTime~Span~(MJD) & 50842-53624\\\\\nN.~of~ToAs   & 635\\\\\nRMS~($\\mu s$)    & 851.173\\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nDerived parameters$^{b}$ & \\\\\n            \\noalign{\\smallskip}\n            \\noalign{\\smallskip}\nCharacteristic~Age, $\\tau_{\\rm c}$ ($10^{9}~yr$)  & 1.83\\\\\nSurface magnetic field, $B_{0}$~($10^{9}$ G)  & 9.80 \\\\\nTotal~Mass~$M_{\\rm TOT}$ ($M_{\\odot}$) & 2.57(10)\\\\\nMass~Function~$f(M_{\\rm C})$ ($M_{\\odot}$) & 0.128121(5)\\\\\nOrbital separation $A$ (ls) & 94.4(6) \\\\\nMinimum~companion~mass~$M_{\\rm C,min}$ ($M_{\\odot}$) & 0.93\\\\\n\n            \\noalign{\\smallskip}\n            \\hline\n         \\end{tabular}\n\\begin{itemize}\n\\item[$^a$] The projected semi-major axis $a \\sin i$ is the\n  semi-major axis of the projection of the orbit of the pulsar, around\n  the system's center of mass, onto the plane containing the line of\n  sight and the line of nodes.\n\\item[$^b$] Characteristic age and surface magnetic field have been\n  calculated using standard formulas, namely $\\tau_{\\rm\n  c}=P\/2\\dot{P}$ and\n  $B_{0}=3.2\\times10^{19}\\sqrt{P\\dot{P}}$\\,G. The total mass $M_{\\rm\n  TOT}$ has been calculated from the relativistic\n  periastron advance and the measured Keplerian parameters,\n  assuming the validity of general relativity.\n  The minimum companion mass was estimated\n  using the observed mass function $f(M_{\\rm\n  C})$ and the lower limit for the total mass, as given by its\n  uncertainty, in the case of $\\sin i = 1$. For details see\n  Lorimer \\& Kramer (2005)\\nocite{lk05}.\n\\end{itemize}\n\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=8.5cm]{4385fig1.ps}\n\\caption{Timing residuals after jointly applying the final model to\nall three data sets. Vertical bars represent the ToA's uncertainty.}\n\\label{fig:jointres}\n\\end{figure}\n\n\\section{The nature of the companion}\n\nIn their discovery paper, \\citet{lcm+00} proposed that this system is\na member of the small class of DNS binaries. Soon after, \\citet{mig00}\nreported on optical observations of the region surrounding the pulsar\nposition to search for emission from the pulsar companion. They\ndetected no emission coincident with the pulsar position and while not\nconclusive, the lack of emission is at least consistent with the\nneutron star hypothesis for the nature of the companion.\n\nThe derived values for the characteristic age ($\\tau_c = 1.83 \\times\n10^{9}$ yrs) and the surface magnetic field ($B = 9.8 \\times\n10^{9}$G), as well as the combined values of the spin period\n($P=104$~ms) and its period derivative ($\\dot{P}=9\\times10^{-19}$),\nindicate that PSR\\,J1811$-$1736 is a neutron star that experienced a\nspin-up phase via accretion from mass overflowing from its\ncompanion. These parameters, in conjunction with the measured orbital\neccentricity ($e=0.828$), indeed suggest that PSR\\,J1811$-$1736 is a\nmildly recycled pulsar whose companion star was massive enough to also\nundergo a SN explosion. This second SN imparted the actually observed\nlarge eccentricity to the system (e.g. \\citealt{bv91}).\n\nOur new measurement of the relativistic periastron advance,\n$\\dot{\\omega} = 0.0090 \\pm 0.0002$ deg yr$^{-1}$, allows us to\ndetermine the value of 2.57$\\pm$0.10 M$_{\\odot}$ for the total mass of\nthe system, assuming that general relativity is the correct theory of\ngravity and that the observed value is fully due to relativistic\neffects (e.g. \\citealt{dd86}).  This value, combined with the measured\nmass function, implies a minimum companion mass of 0.93 $M_{\\odot}$.\n\n\\begin{figure}\n\\centering \\includegraphics[angle=0,width=8.5cm]{4385fig2.ps}\n\\caption{Mass-mass diagram for the binary system hosting\nPSR\\,J1811-1736. The shaded area below the dotted curved line is\nexcluded because of the geometrical constraint $\\sin i \\leq 1$, while\nthe area outside the diagonal stripe is excluded by the measurement of\nthe relativistic periastron advance and the derived value for the\ntotal mass for this system.}\n\\label{fig:masses}\n\\end{figure}\n\nThe value for the total mass is relatively low, but very similar to\nthe total mass of the double pulsar system \\citep{lbk+04} and the\nrecently discovered DNS system PSR\\,J1756$-$2251 \\citep{fkl+05}. In\nfact, these systems have neutron star companions that have the lowest\nneutron star masses observed so far, $M_{c}~=~1.25M_{\\odot}$ and\n$M_{c}=1.17M_{\\odot}$ (for a recent review see \nStairs 2004\\nocite{sta04a}), respectively.  In Figure \\ref{fig:masses} we\nshow the so-called mass-mass diagram where the pulsar and companion\nmasses can be directly compared. The measured value for the advance of\nperiastron means that the sum of the masses must lie along the\ndiagonal line, while the constraint on the inclination $\\sin i \\leq 1$\nexcludes the hatched region below the dotted line. Assuming that the\nneutron stars in this system must have a mass which is larger than the\n{\\it lowest} mass so-far measured, i.e. $1.17M_\\odot$, we find that\nthey both have masses in the interval\n1.17$\\,M_{\\odot}\\,\\leq\\,M_{P},M_{C}\\,\\leq\\,1.50 M_{\\odot}$. This\ninterval contains all but the heaviest neutron stars masses for which\na reliable determination has been obtained. Using this mass\nconstraint, we can also translate this range into lower and upper\nlimits on the inclination of the system, i.e. 44\\,deg\\,$\\ifmmode\\stackrel{<}{_{\\sim}}\\else$\\stackrel{<}{_{\\sim}}$\\fi i\n\\ifmmode\\stackrel{<}{_{\\sim}}\\else$\\stackrel{<}{_{\\sim}}$\\fi$\\,50\\,deg.\n\nAlternatively, if either the pulsar or the companion have a mass equal\nto the observed median neutron star mass of 1.35 M$_{\\odot}$\n\\citep{sta04a}, the other neutron star would have a mass of\n$1.22\\,\\pm\\,0.10 M_{\\odot}$. This value is consistent with the lower\nlimit in the previous discussion, but this also allows for the\npossibility that one of the two neutron stars has a mass as low as\n1.12 $M_{\\odot}$.\n\n\\section{Future potential of timing observations}\n\nIn order to determine the companion mass without ambiguities, it is\nnecessary to measure a second post-Keplerian (PK) parameter\n(e.g. \\citealt{dt92}). We have investigated the possibility of\nmeasuring the PK parameter $\\gamma$, which describes the combined\neffect of gravitational redshift and a second order Doppler effect. For\na companion mass of $1.35M_\\odot$, the expected value is $\\gamma =\n0.021$ ms. Using simulated data sets for the presently available\ntiming precision, we estimate that a $3\\sigma$ detection for $\\gamma$\nis achievable after about 4 more years of observation. However, in\norder to obtain a 10\\% accuracy in mass measurement by determining\n$\\gamma$ to a similar precision, several decades of observations may\nbe needed.\n\nFrom similar simulations, we estimate that PK parameters like the rate\nof orbital decay, $\\dot{P}_{\\rm B}$, or the Shapiro delay, are\nunmeasurable in this system, unless significant improvements in timing\nprecision can be obtained.  For a companion mass of 1.35 $M_{\\odot}$,\n$\\dot{P}_{\\rm B}$ is only $-9.4 \\times 10^{-15}$~s~s$^{-1}$ and we\nexpect an amplitude of only 6\\,$\\mu$s for the amplitude of the Shapiro\ndelay. We also note that the effects of geodetic precession (see\ne.g. \\citealt{kra98}) will not be measurable within a reasonable time,\nas it has a period of order of $10^{5}$ years.\n\n\n\\section{Improving timing precision}\n\nIt is obvious that the measurement of further PK parameters will only\nbe possible if higher timing precision can be achieved for this\npulsar. For instance, if a precision of 50$\\mu$s could be obtained,\n$\\gamma$ could be measurable to a 10\\% accuracy after a total of just\n5 yr of observations, while a $3\\sigma$ detection of the orbital decay\nmay be achieved after about 6 yr.\n\nOne way to achieve higher timing precision is to detect narrow\nfeatures in the observed pulse profile by means of higher effective\ntime resolution.  This is most commonly achieved through better\ncorrection for dispersion smearing that is  caused by the radio\nsignal's passage through the ionized  interstellar medium.  While this\neffect is actually completely removed by the  use coherent\nde-dispersion techniques (see \\citealt{hr75}) at some of our\ntelescopes, it is apparent that the current timing precision is limited\ninstead by broadening of the pulse profile due to interstellar\nscattering (\\citealt{lmg+04}, and references therein). Indeed, the\npulse profile at 1.4~GHz shows a strong scattering tail (see Fig.\\,3)\nwhich prevents a highly accurate determination of the pulse time of\narrival. As scattering is a strong function of observing frequency, we\ncan expect to reduce its effect, and hence to enable higher timing\nprecision, by using timing observations at frequencies above  1.4~GHz.\n\n\\begin{figure}\n\\centering \\includegraphics[angle=0,width=8.5cm]{4385fig3.ps}\n\\caption{Pulses' profiles of PSR\\,J1811--1736 at 1.4\\,GHz (bottom\npanel) and 3.1\\,GHz (top panel). Both profiles have been obtained with\n10 minutes observations performed with the Parkes radio telescope in\nFebruary 2005.}\n\\label{fig:3GHzp}\n\\end{figure}\n\nWe obtained observations at 3.1~GHz that confirm this expectation.\nThe pulse profile obtained  at this frequency shows no evidence of\ninterstellar scattering, and its width at 10\\% is only 7.3~ms. This is\na great improvement with respect to the 1.4~GHz profile, whose 10\\%\nwidth is 58.3~ms. A flux density of $0.34\\pm0.07$mJy  measured at\n3.1\\,GHz suggests that regular timing observations at this frequency\nshould be possible and should significantly improve the achievable\ntiming precision. This would allow us to measure a second PK parameter\nto an accuracy that is sufficiently precise to determine the companion\nmass.\n\nUsing the data available at 1.4\\,GHz and 3.1\\,GHz, we computed\nspectral indexes for flux density and scattering time.  For the flux\ndensity we obtain a spectral index of $\\beta=-1.8\\pm0.6$.  Subdividing\nour observing band at 1.4\\,GHz we obtain two different profiles that\nwe use to measure the pulse scatter timescale $\\tau$ by applying the\ntechnique described in \\citealt{lkm+01}. We convolve  the\n3.1\\,GHz-profile, assumed to represent the true pulse shape, with an\nexponential scattering tail and obtain scattering times by a\nleast-square comparison of the convolved profile with the observed\npulse shape. At 1.284\\,GHz we find $\\tau_s~=~16.9$~ms, and\n$\\tau_s~=~10.6$~ms at 1.464~GHz, respectively.  This results in a\nspectral index $\\alpha$ of the scattering time, i.e.~$\\tau \\propto\n\\nu^{-\\alpha}$, of $\\alpha~=~3.5\\pm~0.1$. Such a measured value agrees\nvery well with analogous results from L\\\"{o}hmer et\nal. (\\citeyear{lkm+01,lmg+04}) who determined $\\alpha$ for a number of\npulsars with very high dispersion measures.\n\nThe measured spectral index of the scattering time is also consistent\nwith the fact that the pulsar has not been detected at frequencies\nbelow 1\\,GHz. For example at 400\\,MHz we calculate $\\tau_{s}\\sim1$\\,s\nwhich is almost an order of magnitude greater than the spin period of\nthe pulsar thus making it impossible to detect it as a pulsating source.\n\n\n\\section{Previous searches for pulsations from the companion}\n\nSearches for pulsations from the binary companion of PSR\\,J1811-1736\nhave been performed on Parkes and Effelsberg data. Parkes observations\nhave been investigated with the procedure described in \\citet{fsk+04},\nwhile Effelsberg data have been processed using the procedure\ndescribed in \\citet{kle04}. Both searches were unsuccessful in\ndetecting any evidence of pulsation. The very high value of the\ndispersion measure for this system may suggest that the interstellar\nscattering is responsible for the failure in detecting any\npulsation. Therefore we studied the possible impact of this phenomenon\non our searches for pulsations from the companion of PSR\\,J1811-1736.\n\n\nWe considered 1\\,hr observations done with the Effelsberg telescope\nusing either the 20\\,cm (1.4\\,GHz) or the 11\\,cm (2.7\\,GHz) receiver,\nexploring a range of possible flux densities ($S=0.05, 0.5, 1$\\,mJy)\nand a detection in a signal-to-noise ratio threshold of\n$S\/N=10$. Using the DM of the observed pulsar, and assuming Effelsberg\nobservations at 1.4\\,GHz, we find a minimum detectable period of\n$P_{\\rm min}=750$\\,ms for a flux density of $S=50\\,\\mu$Jy, while even\nfor the flux densities of $S=500\\,\\mu$Jy and $S=1\\,$mJy periods below\n$\\sim$10\\,ms become undetectable at the observing frequency of\n1.4\\,GHz.\n\nAt 2.7\\,GHz, the system performance of the Effelsberg telescope allows\nfor an antenna gain of $G=$1.5\\,K\\,Jy$^{-1}$ with a system temperature\n$T_{\\rm sys}=17$\\,K. Using these parameters, we obtain minimum periods\nof $P_{\\rm min}=110$\\,ms, $P_{\\rm min}=2.5$\\,ms and $P_{\\rm min}=\n1.6$\\,ms  for flux densities $S_{\\rm 2.7\\,GHz}=50\\,\\mu$Jy,\n500\\,$\\mu$Jy and 1\\,mJy respectively.\n\nWhen searching for pulsations from the binary companion of a Galactic\nrecycled pulsar in a double neutron star system, it is more likely\nthat the companion is a young pulsar with rather ordinary spin\nparameters, as found for  PSR\\,J0737-3039B in the double pulsar system\n\\citep{lbk+04}. Our lower limits on the minimum detectable period\ntherefore suggests that  interstellar scattering should not have\nprevented the detection of the companion, unless it were a very fast\nspinning or very faint source.\n\n\n\n\\section{Constraints on the kick velocity of the second SN explosion}\n\nThe large eccentricity of J1811$-$1736  system can be ascribed to a\nsudden loss of mass which results in a change of the orbital\nparameters. Such a sudden loss of mass can be attributed to the SN\nexplosion that formed the younger unseen neutron star companion  (see,\ne.g., \\citealt{bv91}). Under the hypothesis of a symmetric explosion,\nsimple calculations show that the binary survives this event only if\nthe expelled mass $M_{\\rm exp}$ is less than half of the total mass of\nthe binary before the explosion (pre-SN binary). The induced\neccentricity is a simple function of the amount of the expelled mass:\n$e=M_{\\rm exp}\/M_{\\rm TOT}$, where $M_{\\rm TOT}$ is the total mass of\nthe pre-SN binary. In the case of the binary system hosting\nPSR\\,J1811-1736, the measured eccentricity, $e=0.828$, and the derived\ntotal mass, $M_{\\rm bin}=2.57\\,M_{\\odot}$, would imply a total mass\n$M_{\\rm TOT}\\,=\\,4.7\\,M_{\\odot}$ for the pre-SN binary.\n\nThe high space velocities measured for isolated pulsars indicate that\nneutron stars may receive a kick when formed, with an unpredictable\namplitude and direction \\citep{hp97,cc98,acc02}. Such kicks\nimparted to the newly formed neutron stars are caused by {\\em\nasymmetric supernova explosions}. If an asymmetric SN explosion\noccurs in a binary system, the survival and the eventual post-SN\nbinary parameters are jointly determined by the mass loss and the\nvector representing the velocity imparted to the neutron star. In this\ncase a simple survival condition like the one derived for the\nsymmetric explosion case cannot be determined.\n\nA correlation between the pulsar's spin period and orbital\neccentricity has recently been found for DNS systems\n(\\citealt{mlc+05}, \\citealt{fkl+05}). A numerical simulation by\n\\citet{dpp05} linked this correlation to the typical amplitude of the\nkick velocity received by the younger neutron star at birth.\n\\citet{dpp05} found that the spin period versus eccentricity\ncorrelation is recovered if the typical kick amplitude satisfies the\ncondition $V_{\\rm K} \\ifmmode\\stackrel{<}{_{\\sim}}\\else$\\stackrel{<}{_{\\sim}}$\\fi 50$\\,km\\,s$^{-1}$.\n\nTo investigate the nature of the kick received by the younger neutron\nstar in this system we considered as the pre-SN binary a binary system\ncontaining a neutron star and a helium star. We then constrained the\ntotal mass of this system by combining our results on the total mass\nof the actual binary with the range given by \\citet{dp03b} for the\nmass of the helium star that was the companion of PSR\\,J1811-1736\nbefore the explosion. The helium star mass range $2.8M_{\\odot} \\leq\nM_{\\rm C} \\leq 5.0 M_{\\odot}$ given by \\citet{dp03b} leads to a total\nmass range of $4.0M_{\\odot} \\leq M_{{\\rm {TOT}}} \\leq 6.5M_{\\odot}$.\n\nBinary parameters for the pre-SN binary have been chosen as follows.\nThe eccentricity has been assumed negligible, since the accretion\nphase responsible for spinning up the pulsar also provided strong\ntidal forces that circularised the orbit. The orbital separation has\nbeen constrained to be between the the minimum (pericentric) and\nmaximum (apocentric) distance between the two neutron stars in the\npost-SN binary. This statement can be justified as follows. The\ntypical velocity of the expelled matter in a SN explosion is close to\nthe speed of light, while the typical orbital velocity of the stars in\na binary system is of order of $\\sim 100$\\,km\\,s$^{-1}$, using for the\ntotal mass any value in the range we used for $M_{{\\rm {TOT}}}$ and a\nvalue for the orbital separation comparable to the one for the present\nbinary system, i.e. few light-minutes (see the discussion in the next\nparagraph of the post-SN binary evolution due to general relativistic\neffects). This means that the change in position of the two stars is\nnegligible if compared to the change in position of the expelled\nmatter. The time required for the binary system to do the transition\nfrom the pre-SN to the post-SN binary is the time required by the\nexpelled matter to travel along a path as long as the orbital\nseparation, i.e. few minutes. After such an elapsed time the matter\nexpelled in the SN explosion encloses both stars and has no more\ngravitational effects on their binary motion. This time is also much\nshorter than the orbital period of few days for a pre-SN binary like\nthe one we are considering. This means that during this transition the\npositions of the two stars remained unchanged, and their distance was\na distance periodically assumed by the two stars also in their orbital\nmotion in the post-SN binary.\n\nTo make a fully consistent comparison between the actually observed\nbinary and the eccentric binary that emerged from the last SN\nexplosion (post-SN binary) one has to take into account the secular\nchanges of the orbital parameters caused by general relativistic\neffects.  In order to do this one needs to have an estimate for the\ntime since the last SN. The only timescale that is available to us is\nthe characteristic age of the first born pulsar. We then find binary\nparameters that are consistent with the present system values within\ntheir uncertainties. Given the well known uncertainties in this age\nestimation, we considered the possibility that the present binary is\nin fact up to ten times older than suggested by the characteristic age\n(i.e 1.8$\\times 10^{10}$ yr). Even when considering this extreme age,\nwe find that our results remain unaffected.  We consequently decided\nto use as post-SN orbital parameters the same values we measure today.\n\nBy insisting that the total energy and total angular momentum,\ncalculated in the center-of-mass frame, of the post-SN binary and the\npresent systems are conserved, we can combine these terms to obtain an\nequation for the kick amplitude, as a function of the two angles,\nrepresenting its direction in a suitable reference frame, and the\ntotal mass and orbital separation before the explosion. We assumed\nthat the probability of occurrence for any given kick vector is\nproportional to the solid angle described by the direction of the kick\nin spherical coordinates and then calculated the probability of having\na kick velocity lower than some fixed values. We chose the values of\n50, 100 and 150~km~s$^{-1}$ ( hereafter $P_{50}$, $P_{100}$ and\n$P_{150}$ respectively.).  Figure\\,\\ref{fig:probs} shows that $P_{50}$\nis not negligible if the total mass of the pre-SN binary system is\nlower than 6\\,$M_{\\odot}$. Moreover, all considered probabilities peak\nin correspondence of a total pre-SN mass of 4.70\\,$M_{\\odot}$,\ncorresponding to the null kick case. These results lead to the\nconclusion that the younger neutron star in this system received a low\nvelocity kick and is thus similar to all other known DNS systems,\nwhich all have tighter orbits.\n\nNevertheless,  the binary system containing PSR\\,J1811-1736 is much\nwider than all other known DNS systems. This may indicate that the\nbinary evolution of this system may have been (at least partially)\ndifferent. In particular the wide  orbital separation for this system\nmay be compatible with an evolution during which the pulsar's\nprogenitor avoided completely a common envelope phase \\citep{dp03b} or\nthat this phase was too short to sufficiently reduce the orbital\nseparation. Moreover if the spin-up occurred via the stellar wind of\nthe giant companion then the system would tend to be wider due to the\nisotropic mass loss from the companion \\citep{dpp05}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0,width=8.5cm]{4385fig4.ps}\n\\caption{Probabilities to have a kick velocity lower than or equal to\n50 (lower line), 100 and 150 (upper line) km s$^{-1}$ as a function of\nthe pre-SN total mass. All probabilities peak in correspondence of a\npre-SN total mass $M_{\\rm TOT}=4.70M_{\\odot}$, which is the mass of\nthe binary system before the explosion in the case of a symmetric\nSN. The probability to have a kick velocity lower than 50 km\ns$^{-1}$ is not negligible for all but the highest considered values\nfor the binary pre-SN mass.}\n\\label{fig:probs}\n\\end{figure}\n\n\\section{Summary \\& Conclusions}\n\nWe have presented an improved timing solution for the binary pulsar\nJ1811$-$1736. This solution improves the previously measured values\nfor the spin and Keplerian orbital parameters and one post-Keplerian\norbital parameter, the periastron advance. These results would not\nhave been achieved without data from the three telescopes used and are\nthe first obtained as part of the European Pulsar Timing Array (EPTA)\ncollaboration.\n\nThe measured values for the spin period and its first derivative are\ntypical of a mildly recycled neutron star, while the high eccentricity\nof the binary system can be seen as a signature of the SN explosion\nthat interrupted the mass transfer from the companion to the accreting\nneutron star. This is likely to have occurred before the pulsar could\nreach spin periods typical of the fully recycled (i.e. millisecond)\npulsars. This leads to the conclusion that PSR\\,J1811$-$1736 is a\nmember of a DNS binary system.\n\nThe determined  value of the periastron advance provides further\nconfirmation for this scenario as it suggests a total mass of the\nsystem of $M_{tot}$~=~2.57$\\pm$0.10~$M_{\\odot}$.  This value is\nsimilar to the total mass of two other DNS systems, i.e. the double\npulsar \\citep{lbk+04} and PSR\\,J1756-2251 \\citep{fkl+05}. In both\nthese systems, the non-recycled neutron stars is very light. Assuming\nthat PSR\\,J1811$-$1736 is a neutron star with a mass within the\ncurrently measured mass range for neutron stars, we find the companion\nmass to lie in the same range. Using these arguments we determine the\ninclination of the orbital plane to within $\\sim6$ degrees.\n\nWe also investigated the possibility of measuring a second\npost-Keplerian parameter, in order to determine both masses and thus\nto definitively determine the nature of the companion. Unfortunately, the\npulse profile at 1.4~GHz is heavily broadened by interstellar\nscattering which limits the timing precision and means that a second\npost-Keplerian parameter is not measurable within a reasonable amount\nof time with observations at that frequency.  However we find that at\n3~GHz the scattering is sufficiently reduced and the flux density is\nsufficiently high that higher precision timing will be possible at\nthis frequency. Comparing the pulse profiles at 1.4 and 3~GHz we find\nthat the scattering timescale for this pulsar scales with frequency\nwith a power law of index $\\alpha~=~3.5\\pm0.1$ which is in excellent\nagreement with earlier results on high dispersion measure pulsars.\n  \nConsidering the effects of the interstellar scattering on the\ndetectability of pulsations from the companion, we find that the\nminimum detectable period is longer at lower frequencies and for\nfainter objects. In general, we do not expect interstellar scattering\nto be the cause for the continued non-detection of the companion\nneutron star.\n\nThe orbital separation for this system is much wider than for all\nother DNS systems, and it suggests that its binary evolution has been\ndifferent. One explanation invokes the lack of a common envelope\nphase, during which the size of the orbit shrinks due to tidal forces\nin the envelope of the companion star. Another explanation\n\\citep{dpp05}, not necessarily conflicting with the previous one,\ninvokes a different mass transfer mechanism in the spin-up phase of\nthe pulsar, namely via stellar wind, while all other recycled pulsars\nin the known DNS have been spun up via Roche lobe overflow mass\ntransfer.\n\nFinally, we investigated the kick imparted to the second born neutron\nstar during the second SN.  We find that for realistic values of the\ntotal mass of the pre-SN binary, the kick velocity has a not\nnegligible probability of being lower than 50~km~s$^{-1}$. This\nconstraint is common to all DNS systems, as shown by\n\\citet{dpp05}. This evidence for a low amplitude asymmetric kick\nreceived by the younger neutron star may be the consequence of the\neffects of binary evolution on a star that undergoes a SN explosion,\neffects that somehow are able to tune the amplitude of such kick.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nClassical Cepheid variable stars are primary distance \nindicators and rank among standard candles for establishing \nthe cosmic distance scale, owing to the famous period-luminosity \n($P$--$L$) relationship.\nCompanions to Cepheids, however, complicate the situation.\nThe contribution of the secondary star to the observed \nbrightness has to be taken into account when involving any\nparticular Cepheid in the calibration of the $P$--$L$ relationship.\n\nBinaries among Cepheids are not rare at all: their frequency \nexceeds 50 per cent for the brightest Cepheids, while among the \nfainter Cepheids an observational selection effect encumbers \nrevealing binarity \\citep{Sz03a}.\n\nOwing to some observational projects aimed at obtaining new\nradial velocities (RVs) of numerous Cepheids carried out during\nthe last decades, a part of the selection effect has been\nremoved. This progress is visualized in Fig.~\\ref{fig-comparison}\nwhere the current situation is compared with that 20 years ago.\nThe data have been taken from the on-line data base on binaries \namong Galactic Cepheids (http:\/\/www.konkoly.hu\/CEP\/orbit.html).\nTo get rid of the fluctuation at the left-hand part of the diagram,\nbrightest Cepheids ($\\langle V \\rangle <5$~mag) were merged in a\nsingle bin because such stars are extremely rare among Cepheids\n-- see the histogram in Fig.~\\ref{fig-histogram}.\n\nIn the case of pulsating variables, like Cepheids, spectroscopic\nbinarity manifests itself in a periodic variation of the\n$\\gamma$-velocity (i.e., the RV of the mass centre of the Cepheid). \nIn practice, the orbital RV variation of the Cepheid component is \nsuperimposed on the RV variations of pulsational origin. \nTo separate orbital and pulsational effects, knowledge of the \naccurate pulsation period is essential, especially when comparing \nRV data obtained at widely differing epochs. Therefore, the pulsation\nperiod and its variations have been determined with the method of\nthe O$-$C diagram \\citep{S05} for each target Cepheid. Use of the\naccurate pulsation period obtained from the photometric data is a \nguarantee for the correct phase matching of the (usually less\nprecise) RV data.\n\n\n\\begin{figure}\n\\includegraphics[height=48mm, angle=0]{szabados2013sbcepfig1.eps}\n\\caption{Percentage of known binaries among Galactic\nclassical Cepheids as a function of the mean apparent\nvisual brightness in 1993 and 2013. The decreasing influence\nof the observational selection effect is noticeable.}\n\\label{fig-comparison}\n\\end{figure}\n\n\\begin{figure}\n\\vspace*{7mm}\n\\includegraphics[height=48mm, angle=0]{szabados2013sbcepfig2.eps}\n\\caption{Histogram showing the number distribution of known\nGalactic classical Cepheids as a function of their mean \napparent visual brightness.}\n\\label{fig-histogram}\n\\end{figure}\n\nIn this paper we point out spectroscopic binarity of three \nbright Galactic Cepheids by analysing RV data. The structure of \nthis paper is as follows. The new observations and the equipment \nutilized are described in Sect.~\\ref{newdata}. Section~\\ref{results} \nis devoted to the results on the three new spectroscopic binary (SB) \nCepheids: LR~Trianguli Australis, RZ~Velorum, and BG~Velorum. \nBasic information on these Cepheids is given in Table~\\ref{obsprop}. \nFinally, Section~\\ref{concl} contains our conclusions.\n\n\\begin{table}  \n\\begin{center}  \n\\caption{Basic data of the programme stars \nand the number of spectra.} \n\\label{obsprop}  \n\\begin{tabular}{|lccccc|} \n\\hline  \nCepheid & $\\langle V \\rangle$ & $P$ & Mode & \\multicolumn{2}{c}{Number of spectra}\\\\\n& (mag) & (d)& of pulsation & SSO & CORALIE \\\\\n\\hline  \nLR~TrA &  7.80 &  2.428289 & first overtone & 10 & 32\\\\ \nRZ~Vel &  7.13 & 20.398532 & fundamental    & 30 & 67\\\\\nBG~Vel &  7.69 &  6.923843 & fundamental    & 27 & 33\\\\  \n\\hline   \n\\end{tabular} \n\\end{center}  \n\\end{table}\n\n\n\n\\section{New observations}\n\\label{newdata}\n\n\\subsection{Spectra from the Siding Spring Observatory}\n\\label{SSO}\n\nWe performed an RV survey of Cepheids with the 2.3~m ANU \ntelescope located at the Siding Spring Observatory (SSO), \nAustralia. The main aim of the project was to detect Cepheids \nin binary systems by measuring changes in the mean values of \ntheir RV curve which can be interpreted as the orbital \nmotion of the Cepheid around the centre-of-mass in a binary \nsystem (change of $\\gamma$-velocity). The target list was \ncompiled to include Cepheids with a single-epoch RV phase curve \nor without any published RV data. Several Cepheids suspected \nto be members of SB systems were also put on the target list. \nIn 64 nights between 2004 October and 2006 March we monitored \n40 Cepheids with pulsation periods between 2 and 30 d.\nAdditional spectra of some targets were obtained in 2007 August.\n\nMedium-resolution spectra were taken with the Double Beam \nSpectrograph using the 1200~mm$^{-1}$ gratings in both arms of \nthe spectrograph. The projected slit width was 2 arcsec \non the sky, which was about the median seeing during our \nobservations. The spectra covered the wavelength ranges \n4200--5200~\\AA\\ in the blue arm and 5700--6700~\\AA\\ in the red \narm. The dispersion was 0.55~\\AA~pixel$^{-1}$, leading to a nominal \nresolution of about 1~\\AA.\n\nAll spectra were reduced with standard tasks in {\\sc iraf}\n\\footnote{{\\sc iraf} is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association\nof Universities for Research in Astronomy, Inc., under \ncooperative agreement with the National Science Foundation.}.\nReduction consisted of bias and flat-field corrections, \naperture extraction, wavelength calibration, and continuum \nnormalization. We checked the consistency of wavelength \ncalibrations via the constant positions of strong telluric \nfeatures, which proved the stability of the system. \nRVs were determined only for the red arm data \nwith the task {\\it fxcor\\\/}, applying the cross-correlation \nmethod using a well-matching theoretical template spectrum \nfrom the extensive spectral library of \\citet{Metal05}. Then, \nwe made barycentric corrections to every single RV value. \nThis method resulted in a 1-2~km~s$^{-1}$ uncertainty in the \nindividual RVs, while further tests have shown that our \nabsolute velocity frame was stable to within \n$\\pm$2--3~km~s$^{-1}$. This level of precision is sufficient\nto detect a number of Cepheid companions, as they can often\ncause $\\gamma$-velocity changes well above 10~km~s$^{-1}$.\n\nDiscovery of six SBs among the 40 target Cepheids was \nalready reported by \\citet{Szetal13}. The binarity of the \nthree Cepheids announced here could be revealed by involving \nindependently obtained additional data (see Section~\\ref{coralie}).\nThe individual RV data of the rest of the Cepheid targets\nwill be published together with the results of the analysis\nof the spectra.\n\n\n\\subsection{CORALIE observations from La Silla}\n\\label{coralie}\n\nAll three Cepheids were among the targets during multiple observing \ncampaigns between 2011 April and 2012 May using the fibre-fed \nhigh-resolution ($R \\sim 60000$) echelle spectrograph \n\\textit{CORALIE} mounted on the Swiss 1.2\\,m Euler telescope at \nESO La Silla Observatory, Chile. The instrument's design is \ndescribed in \\citet{Qetal01}; recent instrumental updates \ncan be found in \\citet{Setal10}. \n\nWhen it turned out that these three Cepheids have variable\n$\\gamma$-velocities, several new spectra were obtained in\n2012 December - 2013 January and 2013 April.\n\nThe spectra are reduced by the efficient online reduction \npipeline that performs bias correction, cosmics removal, \nand flatfielding using tungsten lamps. ThAr lamps are used \nfor the wavelength calibration. The reduction pipeline directly \ndetermines the RV via cross-correlation \\citep{Betal96} \nusing a mask that resembles a G2 spectral type. \nThe RV stability of the instrument is excellent and for \nnon-pulsating stars the RV precision is limited by photon noise;\n(see e.g., \\citet{Petal02}). However, the precision achieved for \nCepheids is lower due to line asymmetries. We estimate a typical \nprecision of $\\sim$ 0.1\\,km\\,s$^{-1}$ (including systematics due\nto pulsation) per data point for our data. \n\n\n\\section{Results for individual Cepheids}\n\\label{results}\n\n\n\\subsection{LR~Trianguli Australis}\n\\label{lrtra}\n\n\\paragraph*{Accurate value of the pulsation period}\n\\label{lrtra-period}\n\nThe brightness variability of LR~TrA (HD\\,137626, $\\langle V \\rangle\n= 7.80$\\,mag) was revealed by \\citet{Setal66} based on the Bamberg\nphotographic patrol plates. The Cepheid nature of variability and the\nfirst values of the pulsation period was determined by \\citet{E83}. \nThis Cepheid pulsates in the first-overtone mode; therefore, it has \na small pulsational amplitude and nearly-sinusoidal light and\nvelocity curves. \n\nIn the case of Cepheids pulsating with a low amplitude, the O$-$C \ndiagram constructed for the median brightness (the mid-point \nbetween the faintest and the brightest states) is more reliable \nthan that based on the moments of photometric maxima \\citep{Detal12}. \nTherefore we determined the accurate value of the pulsation period \nby constructing an O$-$C diagram for the moments of median brightness \non the ascending branch of the light curve since this is the phase when \nthe brightness variations are steepest during the whole pulsational \ncycle.\n\nAll published photometric observations of LR~TrA covering three \ndecades were re-analysed in a homogeneous manner to determine \nseasonal moments of the chosen light-curve feature. The relevant data \nlisted in Table~\\ref{tab-lrtra-oc} are as follows:\\\\\nColumn~1: heliocentric moment of the selected light-curve feature\n(median brightness on the ascending branch for LR~TrA, maximum\nbrightness for both RZ~Vel and BG~Vel, see Tables~\\ref{tab-rzvel-oc}\nand \\ref{tab-bgvel-oc}, respectively;\\\\\nCol.~2: epoch number, $E$, as calculated from Equation~(\\ref{lrtra-ephemeris}):\n\\vspace{-1mm}\n\\begin{equation}\nC = 2\\,453\\,104.9265 + 2.428\\,289{\\times}E \n\\label{lrtra-ephemeris}\n\\end{equation}\n\\vspace{-3mm}\n$\\phantom{mmmmm}\\pm0.0037\\phantom{}\\pm0.000\\,003$\n\n\\noindent (this ephemeris has been obtained by the weighted \nleast squares parabolic fit to the O$-$C differences);\\\\\n\\noindent Col.~3: the corresponding O$-$C value;\\\\\nCol.~4: weight assigned to the O$-$C value (1, 2, or 3 \ndepending on the quality of the light curve leading to \nthe given difference);\\\\\nCol.~5: reference to the origin of data.\\\\\n\nThe O$-$C diagram of LR~TrA based on the O$-$C values listed\nin Table~\\ref{tab-lrtra-oc} is plotted in \nFig.~\\ref{fig-lrtra-oc}. The plot can be approximated by a \nconstant period by the ephemeris (\\ref{lrtra-ephemeris}) for \nthe moments of median brightness on the ascending branch. The \nscatter of the points in Fig.~\\ref{fig-lrtra-oc} reflects the \nobservational error and uncertainties in the analysis of the data.\n\n\n\\begin{table}\n\\caption{O$-$C values of LR~TrA (see the \ndescription in Sect.~\\ref{lrtra-period}).}\n\\begin{tabular}{l@{\\hskip2mm}r@{\\hskip2mm}r@{\\hskip2mm}c@{\\hskip2mm}l}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $E\\ $ & O$-$C & $W$ & Data source\\\\\n2\\,400\\,000 + &&&\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n45018.7822 & $-$3330&   0.0581  &3 & \\citet{E83}\\\\\n47633.9607 & $-$2253& $-$0.0307 & 3 & \\citet{Aetal90}\\\\\n47939.9568 & $-$2127&  0.0010  &2 & {\\it Hipparcos} \\citep{ESA97}\\\\\n48139.0426 & $-$2045& $-$0.0329 & 3&  {\\it Hipparcos} \\citep{ESA97}\\\\\n48440.1554 & $-$1921& $-$0.0279 & 3& {\\it Hipparcos} \\citep{ESA97}\\\\\n48750.9547 & $-$1793& $-$0.0496 & 3& {\\it Hipparcos} \\citep{ESA97}\\\\\n49814.6064 & $-$1355&   0.0115 & 3 & \\citet{B08}\\\\\n50370.7115 & $-$1126&   0.0384 & 3 & \\citet{B08}\\\\\n50574.6393 & $-$1042& $-$0.0101 & 3& \\citet{B08}\\\\\n50909.7531 & $-$904& $-$0.0001 & 3 & \\citet{B08}\\\\\n51264.2883 & $-$758&   0.0049 & 3 & \\citet{B08}\\\\\n51650.4058 & $-$599&   0.0244 & 3 & \\citet{B08}\\\\\n51958.8010 & $-$472&   0.0269 & 2 & \\citet{B08}\\\\\n52041.3435 & $-$438&   0.0076 & 2 & ASAS \\citep{P02}\\\\\n52366.7222 & $-$304& $-$0.0044 & 3 & \\citet{B08}\\\\\n52500.2709 & $-$249& $-$0.0116 & 3 & ASAS \\citep{P02}\\\\\n52769.8038 & $-$138& $-$0.0188 & 3 & ASAS \\citep{P02}\\\\\n53102.5159 &  $-$1&   0.0177 & 3 & \\citet{B08}\\\\\n53104.9151 &     0& $-$0.0114 & 3&  ASAS \\citep{P02}\\\\\n53520.1818 &   171&   0.0179 & 3 & ASAS \\citep{P02}\\\\\n53840.7137 &   303&   0.0156 & 3 & ASAS \\citep{P02}\\\\\n54251.0850 &   472&   0.0061 & 3& ASAS \\citep{P02}\\\\\n54615.3163 &   622& $-$0.0060&  3 & ASAS \\citep{P02}\\\\\n54960.1214 &   764& $-$0.0179&  3 & ASAS \\citep{P02}\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-lrtra-oc}\n\\end{table}\n\n\n\\begin{figure}\n\\includegraphics[height=44mm, angle=0]{szabados2013sbcepfig3.eps}\n\\caption{O$-$C diagram of LR~TrA. The plot can be\napproximated by a constant period.}\n\\label{fig-lrtra-oc}\n\\end{figure}\n\n\\paragraph*{Binarity of LR~TrA}\n\\label{lrtra-bin}\n\n\\begin{figure}\n\\includegraphics[height=48mm, angle=0]{szabados2013sbcepfig4.eps}\n\\caption{Merged RV phase curve of LR~TrA. The different symbols\nmean data from different years: 2005: filled triangles; 2006: \nempty triangles; 2007: triangular star; 2012: filled circles; \n2013: empty circles. The zero phase was arbitrarily chosen at \nJD\\,2\\,400\\,000.0 (in all phase curves in this paper).}\n\\label{fig-lrtra-vrad}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=40mm, angle=0]{szabados2013sbcepfig5.eps}\n\\caption{Temporal variation in the $\\gamma$-velocity of LR~TrA.\nThe symbols for the different data sets are the same as\nin Fig.~\\ref{fig-lrtra-vrad}.}\n\\label{fig-lrtra-vgamma}\n\\end{figure}\n\n\\begin{table}\n\\caption{RV values of LR TrA from the SSO spectra. \nThis is only a portion of the full version available online as Supporting\nInformation.}\n\\begin{tabular}{lr}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$  \\ \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n53599.9325 &$-$21.2\\\\\n53600.9086 &$-$32.0\\\\\n53603.9327 &$-$27.6\\\\\n53605.9290 &$-$31.0\\\\\n53805.1657 &$-$29.3\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-lrtra-data}\n\\end{table}\n\n\\begin{table}\n\\caption{CORALIE velocities of LR TrA. \nThis is only a portion of the full version \navailable online as Supporting Information.}\n\\begin{tabular}{lrc}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$ \\ & $\\sigma$ \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$) & (km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n55938.8701 & $-$27.97 & 0.05\\\\\n55938.8718 & $-$28.10 & 0.05\\\\  \n55939.8651 & $-$29.85 & 0.02\\\\  \n55940.8686 & $-$22.40 & 0.03\\\\  \n55941.8579 & $-$33.14 & 0.04\\\\  \n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-lrtra-coralie-data}\n\\end{table}\n\n\n\\begin{table}\n\\caption{$\\gamma$-velocities of LR~TrA.}\n\\begin{tabular}{lccl}\n\\hline\n\\noalign{\\vskip 0.2mm}\nMid-JD & $v_{\\gamma}$ & $\\sigma$ & Data source \\\\\n2\\,400\\,000+ & (km\\,s$^{-1}$)& (km\\,s$^{-1}$) & \\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n53603 & $-$25.5 & 0.5 & Present paper\\\\\n53808 & $-$24.8 & 0.5 & Present paper\\\\\n54331 & $-$29.0 & 1.0 & Present paper\\\\\n55981 & $-$27.5 & 0.1 & Present paper\\\\\n56344 & $-$26.4 & 0.1 & Present paper\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-lrtra-vgamma}\n\\end{table}\n\nThere are no earlier RV data on this bright Cepheid. Our new data \nlisted in Tables~\\ref{tab-lrtra-data} and \\ref{tab-lrtra-coralie-data} \nhave been folded on the accurate pulsation period given in the\nephemeris (see Equation~\\ref{lrtra-ephemeris}). The merged RV phase \ncurve is plotted in Fig.~\\ref{fig-lrtra-vrad}. Both individual \ndata series could be split into seasonal subsets.\n\nVariability in the $\\gamma$-velocity is obvious. The \n$\\gamma$-velocities (together with their uncertainties) are \nlisted in Table~\\ref{tab-lrtra-vgamma}. The $\\gamma$-velocity in\n2007 is more uncertain than in other years because this value\nis based on a single spectrum. Systematic errors can be excluded. \nDozens of Cepheids in our sample with non-varying\n$\\gamma$-velocities indicate stability of the equipment and \nreliability of the data reduction. Fig.~\\ref{fig-lrtra-vgamma} \nis a better visualization of the temporal variation in the \n$\\gamma$-velocity. The seasonal drift in the $\\gamma$-velocity \nis compatible with both short and long orbital periods.\n\nThe photometric contribution of the companion star decreases\nthe observable amplitude of the brightness variability as\ndeduced from the enhanced value of the ratio of the RV and\nphotometric amplitudes \\citep{KSz09}. This is an additional\n(although slight) indication of binarity of LR~TrA.\n\n\\subsection{RZ~Velorum}\n\\label{rzvel}\n\n\\paragraph*{Accurate value of the pulsation period}\n\\label{rzvel-period}\n\nThe brightness variability of RZ~Vel (HD\\,73502, $\\langle V \\rangle\n= 7.13$\\,mag) was revealed by Cannon \\citep{P09}. The Cepheid\nnature of variability and the pulsation period were established by \n\\citet{H36} based on the Harvard and Johannesburg photographic plate \ncollection which was further investigated by \\citet{Oo36}.\n\nThis is the longest period Cepheid announced in this paper and it has \nbeen frequently observed from the 1950s, first photoelectrically, \nthen in the last decades by CCD photometry. The photometric coverage \nof RZ~Vel was almost continuous in the last 20 years thanks to \nobservational campaigns by \\citet{B08} and his co-workers, as well as\nthe ASAS photometry \\citep{P02}.\n\nLong-period Cepheids are usually fundamental pulsators and they \noscillate with a large amplitude resulting in a light curve with\nsharp maximum.\n\nThe O$-$C diagram of RZ~Vel was constructed for the moments of \nmaximum brightness based on the photoelectric and CCD photometric \ndata (see Table~\\ref{tab-rzvel-oc}). The weighted least squares \nparabolic fit to the O$-$C values resulted in the ephemeris:\n\\vspace{-1mm}\n\\begin{equation}\nC = 2\\,442\\,453.6630 + 20.398\\,532{\\times}E + 1.397\\times 10^{-6} E^2\n\\label{rzvel-ephemeris}\n\\end{equation}\n\\vspace{-3mm}\n$\\phantom{mmmmm}\\pm0.0263\\phantom{l}\\pm 0.000\\,080 \\phantom{mm}\n\\pm 0.191\\times 10^{-6}$\n\n\\begin{table}\n\\caption{O$-$C values of RZ~Vel (description of the columns\nis given in Sect.~\\ref{lrtra-period}).}\n\\begin{tabular}{l@{\\hskip2mm}r@{\\hskip2mm}r@{\\hskip2mm}c@{\\hskip2mm}l}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $E\\ $ & O$-$C & $W$ & Data source\\\\\n2\\,400\\,000 + &&&\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n33784.5646 &$-$425 &  0.2777 & 1 & \\citet{Eetal57}\\\\\n34804.5174 &$-$375 &  0.3039 & 1 & \\citet{Wetal58}\\\\\n34845.2119 &$-$373 &  0.2013 & 3 & \\citet{Eetal57}\\\\\n35192.0024 &$-$356 &  0.2168 & 1 & \\citet{I61}\\\\\n40760.8647  &$-$83 &  0.2799 & 3 & \\citet{P76}\\\\\n41719.0924  &$-$36 &$-$0.2234&  3& \\citet{M75}\\\\\n41862.1249  &$-$29 &  0.0193 & 3 & \\citet{Detal77}\\\\\n42453.6330 &    0 &$-$0.0030 & 3 & \\citet{Detal77}\\\\\n44371.0472 &   94 &$-$0.0778 & 3 & \\citet{CC85}\\\\\n44391.3842 &   95 &$-$0.1393 & 2 & \\citet{E82}\\\\\n45003.2906 &  125 &$-$0.1889 & 3 & \\citet{CC85}\\\\\n48226.4369 &  283 &$-$0.0107 & 3 & {\\it Hipparcos} \\citep{ESA97}\\\\\n48797.5877 &  311 &$-$0.0188 & 3 & {\\it Hipparcos} \\citep{ESA97}\\\\\n49185.1653 &  330 &$-$0.0133 & 1 & Walker \\& Williams (unpublished)\\\\\n49817.8011 &  361 &  0.2680  &3 & \\citet{B08}\\\\\n50144.1979 &  377 &  0.2883 & 2 & \\citet{B02}\\\\\n50389.0443 &  389 &  0.3524 & 3 & \\citet{B08}\\\\\n50511.3662 &  395 &  0.2831 & 3 & \\citet{B02}\\\\\n50572.4468 &  398 &  0.1681 & 3 & \\citet{B08}\\\\\n50899.0581 &  414 &  0.4029 & 3 & \\citet{B08}\\\\\n51266.1488 &  432 &  0.3200 & 3 & \\citet{B08}\\\\\n51653.7650 &  451 &  0.3641 & 3 & \\citet{B08}\\\\\n51939.2846 &  465 &  0.3042 & 2 & ASAS \\citep{P02}\\\\\n51959.7692 &  466 &  0.3903 & 3 & \\citet{B08}\\\\\n52347.4262 &  485 &  0.4752 & 3 & \\citet{B08}\\\\\n52653.3896 &  500 &  0.4606 & 3 & ASAS \\citep{P02}\\\\\n52653.4100 &  500 &  0.4810 & 3 & \\citet{B08}\\\\\n53000.1794 &  517 &  0.4754 & 3 & ASAS \\citep{P02}\\\\\n53000.2610 &  517 &  0.5570 & 3 & \\citet{B08}\\\\\n53428.4384 &  538 &  0.3652 & 3 & ASAS \\citep{P02}\\\\\n53754.8864 &  554 &  0.4367 & 3 & ASAS \\citep{P02}\\\\   \n54183.1657 &  575 &  0.3468 & 3 & ASAS \\citep{P02}\\\\\n54509.5729 &  591 &  0.3775 & 3 & ASAS \\citep{P02}\\\\\n54815.4343 &  606 &  0.2609 & 3 & ASAS \\citep{P02}\\\\\n55121.3569 &  621 &  0.2055 & 2 & ASAS \\citep{P02}\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-rzvel-oc}\n\\end{table}\n\nThe O$-$C diagram of RZ~Vel plotted in Fig.~\\ref{fig-rzvel-oc} \nindicates a continuously increasing pulsation period with a period \njitter superimposed. This secular period increase has been caused \nby stellar evolution: while the Cepheid crosses the instability \nregion towards lower temperatures in the Hertzsprung--Russell \ndiagram, its pulsation period is increasing. \nContinuous period variations (of either sign) often occur in \nthe pulsation of long-period Cepheids \\citep{Sz83}.\n\nFig.~\\ref{fig-rzvel-oc2} shows the O$-$C residuals after \nsubtracting the parabolic fit defined by \nEquation~(\\ref{rzvel-ephemeris}). If the wave-like fluctuation seen in \nthis $\\Delta (O-C)$ diagram turns out to be periodic, it would\ncorrespond to a light-time effect in a binary system. In line with \nthe recent shortening in the pulsation period, the current value \nof the pulsation period is $20.396671 \\pm 0.000200$ days (after \nJD~2\\,452\\,300). \n\n\\begin{figure}\n\\includegraphics[height=55mm, angle=0]{szabados2013sbcepfig6.eps}\n\\caption{O$-$C diagram of RZ~Vel. The plot can be\napproximated by a parabola indicating a continuously\nincreasing period.}\n\\label{fig-rzvel-oc}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=44mm, angle=0]{szabados2013sbcepfig7.eps}\n\\caption{$\\Delta(O-C)$ diagram of RZ~Vel.}\n\\label{fig-rzvel-oc2}\n\\end{figure}\n\n\n\\paragraph*{Binarity of RZ~Vel}\n\\label{rzvel-bin}\n\n\n\\begin{figure}\n\\includegraphics[height=55mm, angle=0]{szabados2013sbcepfig8.eps}\n\\caption{RV phase curve of RZ~Vel. Data obtained\nbetween 1996 and 2013 are included in this plot. The meaning\nof various symbols is explained in the text.}\n\\label{fig-rzvel-vrad}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=42mm, angle=0]{szabados2013sbcepfig9.eps}\n\\caption{$\\gamma$-velocities of RZ~Velorum. The symbols for \nthe different data sets are the same as in \nFig.~\\ref{fig-rzvel-vrad}.}\n\\label{fig-rzvel-vgamma}\n\\end{figure}\n\n\\begin{table}\n\\caption{RV values of RZ Vel from the SSO spectra.\n(This is only a portion of the full version \navailable online as Supporting\nInformation.)}\n\\begin{tabular}{lr}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$ \\  \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n53307.2698 &4.2\\\\\n53310.2504 &1.4\\\\\n53312.2073 &9.0\\\\\n53364.2062 &49.6\\\\\n53367.1823 &27.5\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-rzvel-data}\n\\end{table}\n\n\n\\begin{table}\n\\caption{CORALIE velocities of RZ Vel.\n(This is only a portion of the full version available \nonline as Supporting Information.)}\n\\begin{tabular}{lrc}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$ \\ & $\\sigma$ \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$) & (km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n55654.5528 & $-$3.08 & 0.02\\\\\n55656.6626 &  5.23 & 0.01\\\\  \n55657.6721 &  9.86 & 0.02\\\\  \n55659.6585 & 18.85 & 0.03\\\\  \n55662.5137 & 31.50 & 0.01\\\\  \n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-rzvel-coralie-data}\n\\end{table}\n\n\\begin{table}\n\\caption{$\\gamma$-velocities of RZ~Vel.}\n\\begin{tabular}{lccl}\n\\hline\n\\noalign{\\vskip 0.2mm}\nMid-JD & $v_{\\gamma}$ & $\\sigma$ & Data source \\\\\n2\\,400\\,000+ & (km\\,s$^{-1}$)& (km\\,s$^{-1}$) & \\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n34009  &25.5 &1.5& \\citet{S55}\\\\\n40328  &22.1 &1.5& \\citet{LE68,LE80}\\\\\n42186  &29.2 &1.0& \\citet{CC85}\\\\\n44186  &22.6 &1.0& \\citet{CC85}\\\\\n44736  &24.4 &1.0& \\citet{CC85}\\\\\n50317  &25.1 &0.2& \\citet{B02}\\\\\n53184  &24.0 &0.5& \\citet{Netal06}\\\\\n53444  &26.9 &0.6& Present paper\\\\\n53783  &28.8 &1.0& Present paper\\\\\n55709  &25.6 &0.1& Present paper\\\\\n56038  &25.3 &0.1& Present paper\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-rzvel-vgamma}\n\\end{table}\n\nThere are several data sets of RV observations available in\nthe literature for RZ~Vel: those published by \\citet{S55}, \n\\citet{LE68,LE80}, \\citet{CC85}, \\citet{B02}, and \n\\citet{Netal06}. Our individual RV data are listed in \nTables~\\ref{tab-rzvel-data} and \\ref{tab-rzvel-coralie-data}.\n\nBased on these data, the RV phase curve has been constructed \nusing the 20.398532~d pulsation period appearing in \nEquation~(\\ref{rzvel-ephemeris}). In view of the complicated pattern \nof the O$-$C diagram the RV data have been folded on by taking \ninto account the proper phase correction for different data \nseries. The merged RV phase curve is plotted in \nFig.~\\ref{fig-rzvel-vrad}. For the sake of clarity, RV data \nobtained before JD\\,2\\,450\\,000 have not been plotted here \nbecause of the wider scatter of these early RV data but the \n$\\gamma$-velocities were determined for each data set. The \nindividual data series are denoted by different symbols: \nfilled squares mean data by \\citet{B02}, empty squares those \nby \\citet{Netal06}, and our 2005, 2006, 2012 and 2013 data are \ndenoted by filled triangles, empty triangles, filled circles and \nempty circles, respectively. The wide scatter in this merged RV \nphase curve plotted in Fig.~\\ref{fig-rzvel-vrad} is due to a variable \n$\\gamma$-velocity. \n\nThe $\\gamma$-velocities determined from each data set (including \nthe earlier ones) are listed in Table~\\ref{tab-rzvel-vgamma} and \nare plotted in Fig.~\\ref{fig-rzvel-vgamma}. The plot implies\nthat RZ~Vel is really an SB as suspected by \\citet{B02} based on \na much poorer observational material (before JD~2\\,450\\,500). \nAn orbital period of about 5600-5700~d is compatible with the data \npattern in both Fig.~\\ref{fig-rzvel-oc2} and Fig.~\\ref{fig-rzvel-vgamma} \nbut the phase relation between the light-time effect fit to the \n$\\Delta (O-C)$ curve and the orbital RV variation phase curve obtained \nwith this formal period is not satisfactory.\n\n\n\\subsection{BG~Velorum}\n\\label{bgvel}\n\n\\paragraph*{Accurate value of the pulsation period}\n\\label{bgvel-period}\n\nThe brightness variability of BG~Vel (HD\\,78801, $\\langle V \\rangle\n= 7.69$\\,mag) was revealed by Cannon \\citep{P09}. Much later \n\\citet{OL37} independently discovered its light variations but\nhe also revealed the Cepheid nature and determined the pulsation \nperiod based on photographic plates obtained at the Riverview \nCollege Observatory. \\citet{vH50} also observed this Cepheid \nphotographically in Johannesburg but these early data are \nunavailable, therefore we only mention their studies for historical \nreasons.\n\nThis Cepheid is a fundamental-mode pulsator. The O$-$C \ndifferences of BG~Vel calculated for brightness maxima are \nlisted in Table~\\ref{tab-bgvel-oc}. These values have been obtained \nby taking into account the constant and linear terms of the \nfollowing weighted parabolic fit:\n\\vspace{-1mm}\n\\begin{equation}\nC = 2\\,453\\,031.4706 + 6.923\\,843{\\times}E + 2.58\\times 10^{-8} E^2\n\\label{bgvel-ephemeris}\n\\end{equation}\n\\vspace{-3mm}\n$\\phantom{mmmmm}\\pm0.0020\\phantom{}\\pm 0.000\\,007 \\phantom{ml}\n\\pm 0.27\\times 10^{-8}$\n\n\\noindent The parabolic nature of the O$-$C diagram, i.e., the \ncontinuous increase in the pulsation period, is clearly seen \nin Fig.~\\ref{fig-bgvel-oc}. \nThis parabolic trend corresponds to a continuous period increase\nof $(5.16 \\pm 0.54)\\times 10^{-8}$ d\\,cycle$^{-1}$, i.e., \n$\\Delta P = 0.000272$ d\/century. This tiny period increase has \nbeen also caused by stellar evolution as in the case of RZ~Vel.\n\nThe fluctuations around the fitted parabola in\nFig.~\\ref{fig-bgvel-oc} do not show any definite pattern: \nsee the $\\Delta(O-C)$ diagram in Fig.~\\ref{fig-bgvel-oc2}.\n\n\n\\begin{table}\n\\caption{O$-$C values of BG~Vel (description of the \ncolumns is given in Sect.~\\ref{lrtra-period}).}\n\\begin{tabular}{l@{\\hskip2mm}r@{\\hskip2mm}r@{\\hskip2mm}c@{\\hskip2mm}l}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $E\\ $ & O$-$C & $W$ & Data source\\\\\n2\\,400\\,000 + &&&\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n34856.5526 & $-$2625 &  0.1699 & 3 & \\citet{Wetal58}\\\\\n35237.3813 & $-$2570 &  0.1872 & 3 & \\citet{I61}\\\\\n40748.6592 & $-$1774 &  0.0861 & 3 & \\citet{P76}\\\\\n42853.4433 & $-$1470 &  0.0219 & 3 & \\citet{D77}\\\\\n44300.5426 & $-$1261 &  0.0380 & 3 & \\citet{B08}\\\\\n48136.3167 &  $-$707 &  0.0031 & 3 & {\\it Hipparcos} \\citep{ESA97}\\\\\n48627.9239 &  $-$636 &  0.0174 & 3 & {\\it Hipparcos} \\citep{ESA97}\\\\\n50379.6329 &  $-$383 & $-$0.0058 & 3 & \\citet{B08}\\\\\n50573.4987 &  $-$355 & $-$0.0076 & 3 & \\citet{B08}\\\\\n50905.8549 &  $-$307 &  0.0041 & 3 & \\citet{B08}\\\\\n51265.9127 &  $-$255 &  0.0221 & 3 & \\citet{B08}\\\\\n51646.7345 &  $-$200 &  0.0325 & 3 & \\citet{B08}\\\\\n51937.5210 &  $-$158 &  0.0176 & 3 & ASAS \\citep{P02}\\\\\n51958.2712 &  $-$155 & $-$0.0038 & 3 & \\citet{B08}\\\\\n52359.8640 &   $-$97 &  0.0062 & 3 & ASAS \\citep{P02}\\\\\n52359.8778 &   $-$97 &  0.0200 & 3 & \\citet{B08}\\\\\n52650.6575 &   $-$55 & $-$0.0017 & 3 & \\citet{B08}\\\\\n52726.8212 &   $-$44 & $-$0.0003 & 3 & ASAS \\citep{P02}\\\\\n53003.7916 &    $-$4 &  0.0164 & 3 & \\citet{B08}\\\\\n53031.4758 &     0   & 0.0052 & 3 & ASAS \\citep{P02}\\\\\n53336.1201 &    44   &0.0004 & 1 & {\\it INTEGRAL} OMC\\\\\n53460.7390 &    62  & $-$0.0099 & 3 & ASAS \\citep{P02}\\\\\n53779.2202 &   108  & $-$0.0254 & 3 & ASAS \\citep{P02}\\\\\n54180.8337 &   166  & 0.0052 & 3 & ASAS \\citep{P02}\\\\\n54540.8499 &   218  & $-$0.0185 & 3 & ASAS \\citep{P02}\\\\\n54838.5810 &   261  & $-$0.0126 & 3 & ASAS \\citep{P02}\\\\\n55143.2425 &   305  & $-$0.0002 & 2 & ASAS \\citep{P02}\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-bgvel-oc}\n\\end{table}\n\n\\paragraph*{Binarity of BG~Vel}\n\\label{bgvel-bin}\n\n\n\\begin{figure}\n\\includegraphics[height=44mm, angle=0]{szabados2013sbcepfig10.eps}\n\\caption{O$-$C diagram of BG~Vel. The plot can be\napproximated by a parabola indicating a continuously\nincreasing pulsation period.}\n\\label{fig-bgvel-oc}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=44mm, angle=0]{szabados2013sbcepfig11.eps}\n\\caption{$\\Delta(O-C)$ diagram of BG~Vel.}\n\\label{fig-bgvel-oc2}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[height=49mm, angle=0]{szabados2013sbcepfig12.eps}\n\\caption{Merged RV phase curve of BG~Vel. There is an obvious \nshift between the $\\gamma$-velocities valid for the epoch \nof our data obtained in 2005-2006 and 2012-2013 (empty and \nfilled circles, respectively). The other symbols are explained\nin the text.}\n\\label{fig-bgvel-vrad}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=42mm, angle=0]{szabados2013sbcepfig13.eps}\n\\caption{$\\gamma$-velocities of BG~Vel. The symbols for \nthe different data sets are the same as in \nFig.~\\ref{fig-bgvel-vrad}.}\n\\label{fig-bgvel-vgamma}\n\\end{figure}\n\n\\begin{table}\n\\caption{RV values of BG Vel from the SSO spectra.\n(This is only a portion of the full version available online \nas Supporting Information.)}\n\\begin{tabular}{lr}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$  \\  \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n53312.2372 &17.3\\\\\n53364.2219 &$-$0.2\\\\\n53367.1992 &20.5\\\\\n53451.0000 &20.0\\\\\n53452.0021 &23.8\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-bgvel-data}\n\\end{table}\n\n\\begin{table}\n\\caption{CORALIE velocities of BG Vel.\n(This is only a portion of the full version \navailable online as Supporting Information.)}\n\\begin{tabular}{lrc}\n\\hline\n\\noalign{\\vskip 0.2mm}\nJD$_{\\odot}$ & $v_{\\rm rad}$ \\ & $\\sigma$ \\\\\n2\\,400\\,000 + &(km\\,s$^{-1}$) & (km\\,s$^{-1}$)\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n55937.7555 & 24.13 & 0.02\\\\\n55938.6241 &  7.77 & 0.02\\\\  \n55939.6522 & $-$1.25 & 0.01\\\\  \n55941.6474 &  7.99 & 0.10\\\\  \n55942.6917 & 11.78 & 0.03\\\\  \n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-bgvel-coralie-data}\n\\end{table}\n\nThere are earlier RV data of this Cepheid obtained by \\citet{S55} \nand \\citet{LE80}. Variability in the $\\gamma$-velocity is seen \nin the merged phase diagram of all RV data of BG~Velorum \nplotted in Fig.~\\ref{fig-bgvel-vrad}. In this diagram, our 2005--2006\ndata (listed in Table~\\ref{tab-bgvel-data}) are represented with \nthe empty circles, while 2012--2013 data (listed in \nTable~\\ref{tab-bgvel-coralie-data}) are denoted by the filled circles, \nthe triangles represent Stibbs' data, and the $\\times$ symbols refer to \nLloyd Evans' data. Our RV data have been folded with the period given \nin the ephemeris Equation~(\\ref{bgvel-ephemeris}) omitting the quadratic term. \nData obtained by Stibbs and Lloyd Evans have been phased with the \nsame period but a proper correction has been applied to allow \nfor the phase shift due to the parabolic O$-$C graph.\n\nThe $\\gamma$-velocities determined from the individual data sets\nare listed in Table~\\ref{tab-bgvel-vgamma} and plotted in\nFig.~\\ref{fig-bgvel-vgamma}. Since no annual shift is seen \nin the $\\gamma$-velocities between two consecutive years (2005--2006\nand 2012--2013), the orbital period cannot be short, probably it\nexceeds a thousand days.\n\nSimilarly to the case of LR~TrA, BG~Vel is also characterized by an \nexcessive value for the ratio of RV and photometric amplitudes\nindicating the possible presence of a companion \n(see Fig.~\\ref{fig-ampratio}).\n\n\\begin{table}\n\\caption{$\\gamma$-velocities of BG~Vel.}\n\\begin{tabular}{lccl}\n\\hline\n\\noalign{\\vskip 0.2mm}\nMid-JD & $v_{\\gamma}$ & $\\sigma$ & Data source \\\\\n2\\,400\\,000+ & (km\\,s$^{-1}$)& (km\\,s$^{-1}$) & \\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\noalign{\\vskip 0.2mm}\n34096  &11.4  &1.5& \\citet{S55}\\\\\n40545  & 8.4  &1.5& \\citet{LE80}\\\\\n53572  &12.6  &0.6& Present paper\\\\\n56043  &10.3  &0.1& Present paper\\\\\n\\noalign{\\vskip 0.2mm}\n\\hline\n\\end{tabular}\n\\label{tab-bgvel-vgamma}\n\\end{table}\n\n\n\\section{Conclusions}\n\\label{concl}\n\nWe pointed out that three bright southern Galactic Cepheids,\nLR~TrA, RZ~Vel and BG~Vel,\nhave a variable $\\gamma$-velocity implying their membership \nin SB systems. RV values of other target Cepheids observed \nwith the same equipment in 2005--2006 and 2012 testify that \nthis variability in the $\\gamma$-velocity is not of instrumental\norigin, nor an artefact caused by the analysis.\n\nThe available RV data are insufficient to determine the orbital \nperiod and other elements of the orbits. However, some inferences\ncan be made from the temporal variations of the $\\gamma$-velocity.\nAn orbital period of 5600--5700~d of the RZ~Vel system is \ncompatible with the data pattern. In the case of BG~Vel, short \norbital periodicity can be ruled out. For LR~TrA, even the range \nof the possible orbital periods remains uncertain.\n\nThe value of the orbital period for SB systems \ninvolving a Cepheid component is often unknown: according to the \non-line data base \\citep{Sz03a} the orbital period has been \ndetermined for about 20\\% of the known SB Cepheids. The majority \nof known orbital periods exceeds a thousand days.\n\nA companion star may have various effects on the observable\nphotometric properties of the Cepheid component. Various pieces \nof evidence of duplicity based on the photometric criteria are\ndiscussed by \\citet{Sz03b} and \\citet{KSz09}. As to our\ntargets, there is no obvious sign of a companion from optical\nmulticolour photometry. This indicates that the companion star \ncannot be much hotter than any of the Cepheids discussed here. \nThere is, however, a phenomenological parameter, viz. the ratio \nof RV to photometric amplitudes \\citep{KSz09} whose excessive\nvalue is a further hint at the probable existence of a\ncompanion for both LR~TrA and BG~Vel (see Fig.~\\ref{fig-ampratio}).\nMoreover, the {\\it IUE} spectra of bright Cepheids \nanalysed by \\citet{E92} gave a constraint on the temperature \nof a companion to remain undetected in the ultraviolet spectra: \nin the case of RZ~Vel, the spectral type of the companion cannot \nbe earlier than A7, while for BG~Vel this limiting spectral type \nis A0. Further spectroscopic observations are necessary to \ncharacterize these newly detected SB systems.\n\n\\begin{figure}\n\\includegraphics[height=54mm, angle=0]{szabados2013sbcepfig14.eps}\n\\caption{The slightly excessive value of the $A_{V_{\\rm RAD}}\/A_B$\namplitude ratio of LR~TrA and BG~Vel (large circles) with respect \nto the average value characteristic at the given pulsation period\nis an independent indication of the presence of a companion star.\nThis is a modified version of fig.~4f of \\citet{KSz09}. The open \nsymbols in the original figure correspond to known binaries and \nthe filled symbols to Cepheids without known binarity. For the\nmeaning of various symbols, see \\citet{KSz09}.\n}\n\\label{fig-ampratio}\n\\end{figure}\n\n\nOur findings confirm the previous statement by \\citet{Sz03a} \nabout the high percentage of binaries among classical Cepheids \nand the observational selection effect hindering the discovery \nof new cases (see also Fig.~\\ref{fig-comparison}).\n\nRegular monitoring of the RVs of a large\nnumber of Cepheids will be instrumental in finding \nmore SBs among Cepheids. RV data to be obtained with the \n{\\it Gaia} astrometric space probe (expected launch: 2013 \nSeptember) will certainly result in revealing new SBs among \nCepheids brighter than the 13--14th magnitude \\citep{Eyetal12}.\nIn this manner, the `missing' SBs among Cepheids inferred\nfrom Fig.~\\ref{fig-comparison} can be successfully revealed\nwithin few years.\n\n\\section*{Acknowledgments} \n\nThis project has been supported by the \nESTEC Contract No.\\,4000106398\/12\/NL\/KML, the Hungarian OTKA \nGrants K76816, K83790, K104607, and MB08C 81013, as well as the \nEuropean Community's Seventh Framework Program (FP7\/2007-2013) \nunder grant agreement no.\\,269194,  and the ``Lend\\\"ulet-2009'' \nYoung Researchers Program of the Hungarian Academy of Sciences. \nAD was supported by the Hungarian E\\\"otv\\\"os Fellowship. \nAD has also been supported by a J\\'anos Bolyai Research Scholarship \nof the Hungarian Academy of Sciences. AD is very thankful \nto the staff at The Lodge in the Siding Spring Observatory \nfor their hospitality and very nice food, making the \ntime spent there lovely and special.\nPart of the research leading to these results has received \nfunding from the European Research Council under the European \nCommunity's Seventh Framework Programme (FP7\/2007--2013)\/ERC grant \nagreement no.\\,227224 (PROSPERITY).\nThe {\\it INTEGRAL\\\/} photometric data, pre-processed by \nISDC, have been retrieved from the OMC Archive at CAB (INTA-CSIC). \nWe are indebted to Stanley Walker for sending us some\nunpublished photoelectric observational data. Our thanks are \nalso due to the referee and Dr. M\\'aria Kun for their critical \nremarks leading to a considerable improvement in the presentation \nof the results.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and results}\n\nLet us recall the objects we will deal with. Throughout the paper $\\DD$ denotes the unit open disc on the complex plane, $\\TT$ is the unit circle and $p$ --- the Poincar\\'e distance on $\\DD$.\n\nLet $D\\subset\\CC^{n}$ be a domain and let $z,w\\in D$, $v\\in\\CC^{n}$. The {\\it Lempert function}\\\/ is defined as\n\\begin{equation}\\label{lem}\n\\widetilde{k}_{D}(z,w):=\\inf\\{p(0,\\xi):\\xi\\in[0,1)\\textnormal{ and }\\exists f\\in \\mathcal{O}(\\mathbb{D},D):f(0)=z,\\ f(\\xi)=w\\}.\n\\end{equation} The {\\it Kobayashi-Royden \\emph{(}pseudo\\emph{)}metric}\\\/ we define as\n\\begin{equation}\\label{kob-roy}\n\\kappa_{D}(z;v):=\\inf\\{\\lambda^{-1}:\\lambda>0\\text{ and }\\exists f\\in\\mathcal{O}(\\mathbb{D},D):f(0)=z,\\ f'(0)=\\lambda v\\}.\n\\end{equation}\nNote that\n\\begin{equation}\\label{lem1}\n\\widetilde{k}_{D}(z,w)=\\inf\\{p(\\zeta,\\xi):\\zeta,\\xi\\in\\DD\\textnormal{ and }\\exists f\\in \\mathcal{O}(\\mathbb{D},D):f(\\zeta)=z,\\ f(\\xi)=w\\},\n\\end{equation}\n\\begin{multline}\\label{kob-roy1}\n\\kappa_{D}(z;v)=\\inf\\{|\\lambda|^{-1}\/(1-|\\zeta|^2):\\lambda\\in\\CC_*,\\,\\zeta\\in\\DD\\text{ and }\\\\ \\exists f\\in\\mathcal{O}(\\mathbb{D},D):f(\\zeta)=z,\\ f'(\\zeta)=\\lambda v\\}.\n\\end{multline}\n\nIf $z\\neq w$ (respectively $v\\neq 0$), a mapping $f$ for which the infimum in \\eqref{lem1} (resp. in \\eqref{kob-roy1}) is attained, we call a $\\wi{k}_D$-\\textit{extremal} (or a \\textit{Lempert extremal}) for $z,w$ (resp. a $\\kappa_D$-\\textit{extremal} for $z,v$). A mapping being a $\\wi k_D$-extremal or a $\\kappa_D$-extremal we will call just an \\textit{extremal} or an \\textit{extremal mapping}.\n\nWe shall say that $f:\\DD\\longrightarrow D$ is a unique $\\wi{k}_D$-extremal for $z,w$ (resp. a unique $\\kappa_D$-extremal for $z,v$) if any other $\\wi{k}_D$-extremal $g:\\DD\\longrightarrow D$ for $z,w$ (resp. $\\kappa_D$-extremal for $z,v$) satisfies $g=f\\circ a$ for some M\\\"obius function $a$.\n\nIn general, $\\wi{k}_{D}$ does not satisfy a triangle inequality --- take for example $D_{\\alpha}:=\\{(z,w)\\in\\CC^{2}:|z|,|w|<1,\\ |zw|<\\alpha\\}$, $\\alpha\\in(0,1)$. Therefore, it is natural to consider the so-called \\textit{Kobayashi \\emph{(}pseudo\\emph{)}distance} given by the formula \\begin{multline*}k_{D}(w,z):=\\sup\\{d_{D}(w,z):(d_{D})\\text{ is a family of holomorphically invariant} \\\\\\text{pseudodistances less than or equal to }\\widetilde{k}_{D}\\}.\\end{multline*}\nIt follows directly from the definition that $$k_{D}(z,w)=\\inf\\left\\{\\sum_{j=1}^{N}\\wi{k}_{D}(z_{j-1},z_{j}):N\\in\\NN,\\ z_{1},\\ldots,z_{N}\\in\nD,\\ z_{0}=z,\\ z_{N}=w\\right\\}.$$\n\nThe next objects we are dealing with, are the \\textit{Carath\\'eodory \\emph{(}pseudo\\emph{)}distance}\n$$c_{D}(z,w):=\\sup\\{p(F(z),F(w)):F\\in\\mathcal{O}(D,\\DD)\\}$$\nand the \\textit{Carath\\'eodory-Reiffen \\emph{(}pseudo\\emph{)}metric}\n$$\\gamma_D(z;v):=\\sup\\{|F'(z)v|:F\\in\\mathcal{O}(D,\\DD),\\ F(z)=0\\}.$$\n\nA holomorphic mapping $f:\\DD\\longrightarrow D$ is said to be a \\emph{complex geodesic} if $c_D(f(\\zeta),f(\\xi))=p(\\zeta,\\xi)$ for any $\\zeta,\\xi\\in\\DD$.\n\\bigskip\n\nHere is some notation. Let $z_1,\\ldots,z_n$ be the standard complex coordinates in $\\CC^n$ and $x_1,\\ldots,x_{2n}$ --- the standard real coordinates in $\\CC^n=\\RR^n+i\\RR^n\\simeq\\RR^{2n}$. We use $T_{D}^\\mathbb{R}(a)$, $T_{D}^\\mathbb{C}(a)$ to denote a real and a complex tangent space to a $\\cC^1$-smooth domain $D$ at a point $a\\in\\partial D$, i.e. the sets \\begin{align*}T_{D}^\\mathbb{R}(a):&=\\left\\{X\\in\\CC^{n}:\\re\\sum_{j=1}^n\\frac{\\partial r}{\\partial z_j}(a)X_{j}=0\\right\\},\\\\ T_{D}^\\mathbb{C}(a):&=\\left\\{X\\in\\CC^{n}:\\sum_{j=1}^n\\frac{\\partial r}{\\partial z_j}(a)X_{j}=0\\right\\},\\end{align*}\nwhere $r$ is a defining function of $D$. Let $\\nu_D(a)$ be the outward unit normal vector to $\\partial D$ at $a$.\n\nLet $\\mathcal{C}^{k}(\\CDD)$, where $k\\in(0,\\infty]$, denote a class of continuous functions on $\\CDD$, which are of class $\\cC^k$ on $\\DD$ and\n\\begin{itemize}\n\\item if $k\\in\\NN\\cup\\{\\infty\\}$ then derivatives up to the order $k$ extend continuously on~$\\CDD$;\n\\item if $k-[k]=:c>0$ then derivatives up to the order $[k]$ are $c$-H\\\"older continuous on $\\DD$.\n\\end{itemize}\nBy $\\mathcal{C}^\\omega$ class we shall denote real analytic functions. Further, saying that $f$ is of class $\\mathcal{C}^{k}(\\TT)$, $k\\in(0,\\infty]\\cup\\{\\omega\\}$, we mean that the function $t\\longmapsto f(e^{it})$, $t\\in\\RR$, is in $\\mathcal{C}^{k}(\\mathbb R)$. For a compact set $K\\su\\CC^n$ let $\\OO(K)$ denote the set of functions extending holomorphically on a neighborhood of $K$ (we assume that all neighborhoods are open). In that case we shall sometimes say that a given function is of class $\\OO(K)$. Note that $\\CLW(\\TT)=\\OO(\\TT)$. \n\nLet $|\\cdot|$ denote the Euclidean norm in $\\CC^{n}$ and let $\\dist(z,S):=\\inf\\{|z-s|:s\\in S\\}$ be a distance of the point $z\\in\\CC^n$ to the set $S\\su\\CC^n$. For such a set $S$ we define $S_*:=S\\setminus\\{0\\}$. Let $\\BB_n:=\\{z\\in\\CC^n:|z|=1\\}$ be the unit ball and $B_n(a,r):=\\{z\\in\\CC^n:|z-a|<r\\}$ --- an open ball with a center $a\\in\\CC^n$ and a radius $r>0$. Put $$z\\bullet w:=\\sum_{j=1}^nz_{j}{w}_{j}$$ for $z,w\\in\\CC^{n}$ and let $\\langle\\cdotp,-\\rangle$ be a hermitian inner product on $\\CC^n$. The real inner product on $\\CC^n$ is denoted by $\\langle\\cdotp,-\\rangle_{\\RR}=\\re\\langle\\cdotp,-\\rangle$.\n\nWe use $\\nabla$ to denote the gradient $(\\pa\/\\pa x_1,\\ldots,\\pa\/\\pa x_{2n})$. For real-valued functions the gradient is naturally identified with $2(\\pa\/\\pa\\ov z_1,\\ldots,\\pa\/\\pa\\ov z_n)$. Recall that $$\\nu_D(a)=\\frac{\\nabla r(a)}{|\\nabla r(a)|}.$$ Let $\\mathcal{H}$ be the Hessian matrix $$\\left[\\frac{\\pa^2}{\\pa x_j\\pa x_k}\\right]_{1\\leq j,k\\leq 2n}.$$ Sometimes, for a $\\cC^2$-smooth function $u$ and a vector $X\\in\\RR^{2n}$ the Hessian $$\\sum_{j,k=1}^{2n}\\frac{\\partial^2 u}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}=X^T\\HH u(a)X$$ will be denoted by $\\HH u(a;X)$. By $\\|\\cdot\\|$ we denote the operator norm.\n\\bigskip\n\\begin{df}\\label{29}\nLet $D\\subset\\CC^{n}$ be a domain.\n\nWe say that $D$ is \\emph{linearly convex} (resp. \\emph{weakly linearly convex}) if through any point $a\\in\\mathbb C^n\\setminus D$ (resp. $a\\in \\partial D$) there goes an $(n-1)$-dimensional complex hyperplane disjoint from $D$.\n\nA domain $D$ is said to be \\emph{strongly linearly convex} if\n\\begin{enumerate}\n\\item $D$ has $\\mathcal{C}^{2}$-smooth boundary;\n\\item there exists a defining function $r$ of $D$ such that\n\\begin{equation}\\label{48}\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial\\overline z_k}(a)X_{j}\\overline{X}_{k}>\\left|\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial z_k}(a)X_{j}X_{k}\\right|,\\ a\\in\\partial D,\\ X\\in T_{D}^\\mathbb{C}(a)_*.\\end{equation}\n\\end{enumerate}\n\nMore generally, any point $a\\in\\pa D$ for which there exists a defining function $r$ satisfying \\eqref{48}, is called a \\emph{point of the strong linear convexity} of $D$.\n\nFurthermore, we say that a domain $D$ has \\emph{real analytic boundary} if it possesses a real analytic defining function.\n\\end{df}\n\nNote that the condition \\eqref{48} does not depend on the choice of a defining function of $D$.\n\n\\begin{rem}\nLet $D\\subset\\CC^{n}$ be a strongly linearly convex domain. Then\n\\begin{enumerate}\n\\item any $(n-1)$-dimensional complex tangent hyperplane intersects $\\partial{D}$ at precisely one point; in other words $$\\overline D\\cap(a+T_{D}^\\mathbb{C}(a))=\\{a\\},\\ a\\in\\pa D;$$\n\\item for $a\\in\\pa D$ the equation $\\langle w-a, \\nu_D(a)\\rangle=0$ describes the $(n-1)$-dimensional complex tangent hyperplane $a+T_{D}^\\mathbb{C}(a)$, consequently $$\\langle z-a, \\nu_D(a)\\rangle\\neq 0,\\ z\\in D,\\ a\\in\\pa D.$$\n\\end{enumerate}\n\\end{rem}\n\\bigskip\nThe main aim of the paper is to present a detailed proof of the following\n\n\\begin{tw}[Lempert Theorem]\\label{lem-car}\nLet $D\\subset\\CC^{n}$, $n\\geq 2$, be a bounded strongly linearly convex domain. Then $$c_{D}=k_{D}=\\wi{k}_{D}\\text{\\,\\ and\\,\\, }\\gamma_D=\\kappa_D.$$\n\\end{tw}\n\n\\bigskip\n\nAn important role will be played by strongly convex domains and strongly convex functions.\n\\begin{df}\nA domain $D\\subset\\CC^{n}$ is called \\emph{strongly convex} if\n\\begin{enumerate}\n\\item $D$ has $\\mathcal{C}^{2}$-smooth boundary;\n\\item there exists a defining function $r$ of $D$ such that\n\\begin{equation}\\label{sc}\\sum_{j,k=1}^{2n}\\frac{\\partial^2 r}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}>0,\\ a\\in\\partial D,\\ X\\in T_{D}^\\mathbb{R}(a)_*.\\end{equation}\n\\end{enumerate}\nGenerally, any point $a\\in\\pa D$ for which there exists a defining function $r$ satisfying \\eqref{sc}, is called a \\emph{point of the strong convexity} of $D$.\n\\end{df}\n\\begin{rem}\nA strongly convex domain $D\\subset\\CC^{n}$ is convex and strongly linearly convex. Moreover, it is strictly convex, i.e. for any different points $a,b\\in\\overline D$ the interior of the segment $[a,b]=\\{ta+(1-t)b:t\\in [0,1]\\}$ is contained in $D$ (i.e. $ta+(1-t)b\\in D$ for any $t\\in(0,1)$).\n\nObserve also that any bounded convex domain with a real analytic boundary is strictly convex. Actually, if a domain $D$ with a real analytic boundary were not strictly convex, then we would be able to find two distinct points $a,b\\in\\pa D$ such that the segment $[a,b]$ lies entirely in $\\partial D$. On the other hand, the identity principle would imply that the set $\\{t\\in\\mathbb R:\\exists\\eps>0:sa+(1-s)b\\in\\pa D\\text{ for }|s-t|<\\eps\\}$ is open-closed in $\\mathbb R$. Therefore it has to be empty. This immediately gives a contradiction.\n\\end{rem}\n\n\\begin{rem}\nIt is well-known that for any convex domain $D\\su\\CC^{n}$ there is a sequence $\\{D_m\\}$ of bounded strongly convex domains with real analytic boundaries, such that $D_m\\su D_{m+1}$ and $\\bigcup_m D_m=D$. \n\nIn particular, Theorem~\\ref{lem-car} holds for convex domains.\n\\end{rem}\n\n\\begin{df}\nLet $U\\su\\CC^n$ be a domain. A function $u:U\\longrightarrow\\RR$ is called \\emph{strongly convex} if\n\\begin{enumerate}\n\\item $u$ is $\\mathcal{C}^{2}$-smooth;\n\\item $$\\sum_{j,k=1}^{2n}\\frac{\\partial^2 u}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}>0,\\ a\\in U,\\ X\\in(\\RR^{2n})_*.$$\n\\end{enumerate}\n\\end{df}\n\n\\begin{df} A degree of a continuous function (treated as a curve) $:\\mathbb T\\longrightarrow\\mathbb T$ is called its winding number. The fundamental group is a homotopy invariant. Thus the definition of the \\emph{winding number of a continuous function} $\\phi:\\mathbb T\\longrightarrow\\mathbb C_*$ is the same. We denote it by $\\wind\\phi$. \n\nIn the case of a $\\cC^1$-smooth function $\\phi:\\TT\\longrightarrow\\CC_*$, its winding number is just the index of $\\phi$ at 0, i.e. $$\\wind\\phi=\\frac{1}{2\\pi i}\\int_{\\phi(\\TT)}\\frac{d\\zeta}{\\zeta}=\\frac{1}{2\\pi i}\\int_{0}^{2\\pi}\\frac{\\frac{d}{dt}\\phi(e^{it})}{\\phi(e^{it})}dt.$$\n\\end{df}\n\n\\begin{rem}\\label{49}\n\\begin{enumerate}\n\\item\\label{51} If $\\phi\\in\\cC(\\TT,\\CC_*)$ extends to a function $\\widetilde{\\phi}\\in\\OO(\\DD)\\cap \\mathcal C(\\CDD)$ then $\\wind\\phi$ is the number of zeroes of $\\widetilde{\\phi}$ in $\\DD$ counted with multiplicities;\n\\item\\label{52} $\\wind(\\phi\\psi)=\\wind\\phi+\\wind\\psi$, $\\phi,\\psi\\in\\cC(\\TT,\\CC_*)$;\n\\item\\label{53} $\\wind\\phi=0$ if $\\phi\\in\\cC(\\TT)$ and $\\re\\phi>0$.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{df}\nThe boundary of a domain $D$ of $\\mathbb C^n$ is \\emph{real analytic in a neighborhood} $U$ of the set $S\\su\\pa D$ if there exists a function $r\\in\\mathcal C^{\\omega}(U,\\RR)$ such that $D\\cap U=\\{z\\in U:r(z)<0\\}$ and $\\nabla r$ does not vanish in $U$.\n\\end{df}\n\n\n\n\\begin{df}\\label{21}\nLet $D\\subset\\CC^{n}$ be a domain. We call a holomorphic mapping $f:\\DD\\longrightarrow D$ a \\emph{stationary mapping} if\n\\begin{enumerate}\n\\item $f$ extends to a holomorphic mapping in a neighborhood od $\\CDD$ $($denoted by the same letter$)$;\n\\item $f(\\TT)\\subset\\partial D$;\n\\item there exists a real analytic function\n$\\rho:\\TT\\longrightarrow\\RR_{>0}$ such that the mapping $\\TT\\ni\\zeta\\longmapsto\\zeta\n\\rho(\\zeta)\\overline{\\nu_D(f(\\zeta))}\\in\\CC^{n}$ extends to a mapping holomorphic in a neighborhood of $\\CDD$ $($denoted by $\\widetilde{f}${$)$}.\n\\end{enumerate}\n\nFurthermore, we call a holomorphic mapping $f:\\DD\\longrightarrow D$ a \\emph{weak stationary mapping} if\n\\begin{enumerate}\n\\item[(1')] $f$ extends to a $\\cC^{1\/2}$-smooth mapping on $\\CDD$ $($denoted by the same letter$)$;\n\\item[(2')] $f(\\TT)\\subset\\partial D$;\n\\item[(3')] there exists a $\\cC^{1\/2}$-smooth function\n$\\rho:\\TT\\longrightarrow\\RR_{>0}$ such that the mapping $\\TT\\ni\\zeta\\longmapsto\\zeta\n\\rho(\\zeta)\\overline{\\nu_D(f(\\zeta))}\\in\\CC^{n}$ extends to a mapping $\\widetilde{f}\\in\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$.\n\\end{enumerate}\n\nThe definition of a $($weak$)$ stationary mapping $f:\\mathbb D\\longrightarrow D$ extends naturally to the case when $\\pa D$ is real analytic in a neighborhood of $f(\\TT)$.\n\\end{df}\n\n\nDirectly from the definition of a stationary mapping $f$, it follows that $f$ and $\\wi f$ extend holomorphically on some neighborhoods of $\\CDD$. By $\\DD_f$ we shall denote their intersection.\n\n\\begin{df}\\label{21e}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain with real analytic boundary. A holomorphic mapping $f:\\DD\\longrightarrow D$ is called a (\\emph{weak}) $E$-\\emph{mapping} if it is a (weak) stationary mapping and\n\\begin{enumerate}\n\\item[(4)] setting $\\varphi_z(\\zeta):=\\langle z-f(\\zeta),\\nu_D(f(\\zeta))\\rangle,\\ \\zeta\\in\\TT$, we have $\\wind\\phi_z=0$ for some $z\\in D$.\n\\end{enumerate}\n\\end{df}\n\n\\begin{rem}\nThe strong linear convexity of $D$ implies $\\varphi_z(\\zeta)\\neq 0$ for any $z\\in D$ and $\\zeta\\in\\TT$. Therefore, $\\wind\\phi_z$ vanishes for all $z\\in D$ if it vanishes for some $z\\in D$.\n\nAdditionally, any stationary mapping of a convex domain is an $E$-mapping (as $\\re \\varphi_z<0$).\n\\end{rem}\n\nWe shall prove that in a class of non-planar bounded strongly linearly convex domains with real analytic boundaries weak stationary mappings are just stationary mappings, so there is no difference between $E$-mappings and weak $E$-mappings. \n\nWe have the following result describing extremal mappings, which is very interesting in its own.\n\n\\begin{tw}\\label{main} Let $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain. \n\nThen a holomorphic mapping $f:\\DD\\longrightarrow D$ is an extremal if and only if $f$ is a weak $E$-mapping.\n\nFor a domain $D$ with real analytic boundary, a holomorphic mapping $f:\\mathbb D\\longrightarrow D$ is an extremal if and only if $f$ is an $E$-mapping.\n\nIf $\\pa D$ is of class $\\cC^k$, $k=3,4,\\ldots,\\infty$, then any weak $E$-mapping $f:\\DD\\longrightarrow D$ and its associated mappings $\\wi f,\\rho$ are $\\mathcal C^{k-1-\\eps}$-smooth for any $\\eps>0$.\n\n\\end{tw}\n\n\nThe idea of the proof of the Lempert Theorem is as follows. In real analytic case we shall show that $E$-mappings are complex geodesics (because they have left inverses). Then we shall prove that for any different points $z,w\\in D$ (resp. for a point $z\\in D$ and a vector $v\\in(\\CC^n)_*$) there is an $E$-mapping passing through $z,w$ (resp. such that $f(0)=z$ and $f'(0)=v$). This will give the equality between the Lempert function and the Carath\\'eodory distance. In the general case, we exhaust a $\\cC^2$-smooth domain by strongly linearly convex domains with real analytic boundaries.\n\nTo prove Theorem \\ref{main} we shall additionally observe that (weak) $E$-mappings are unique extremals.\n\\bigskip\n\n\\begin{center}{\\sc Real analytic case}\\end{center}\n\\bigskip\n\nIn what follows and if not mentioned otherwise, $D\\su\\CC^n$, $n\\geq 2$, is a \\textbf{bounded strongly linearly convex domain with real analytic boundary}.\n\\section{Weak stationary mappings of strongly linearly convex domains with real analytic boundaries are stationary mappings}\\label{55}\nLet $M\\subset\\CC^m$ be a totally real $\\CLW$ submanifold of the real dimension $m$. Fix a point $z\\in M$. There are neighborhoods $U,V\\su\\CC^m$ of $0$ and $z$ respectively and a biholomorphic mapping $\\Phi:U\\longrightarrow V$ such that $\\Phi(\\RR^m\\cap U)=M\\cap V$ (for the proof see Appendix).\n\n\n\\begin{prop}\\label{6}\nA weak stationary mapping of $D$ is a stationary mapping of $D$ with the same associated mappings.\n\\end{prop}\n\\begin{proof}\nLet $f:\\DD\\longrightarrow D$ be a weak stationary mapping. Our aim is to prove that $f,\\widetilde{f}\\in\\OO(\\CDD)$ and $\\rho\\in\\mathcal C^{\\omega}(\\TT)$. Choose a point $\\zeta_0\\in\\TT$. Since $\\widetilde{f}(\\zeta_0)\\neq 0$, we can assume that $\\widetilde{f}_1(\\zeta)\\neq 0$ in $\\CDD\\cap U_0$, where $U_0$ is a neighborhood of $\\zeta_0$. This implies\n$\\nu_{D,1}(f(\\zeta_0))\\neq 0$, so $\\nu_{D,1}$ does not vanish on some set $V_0\\su\\pa D$, relatively open in\n$\\pa D$, containing the point $f(\\zeta_0)$. Shrinking $U_0$, if necessary, we may assume that $f(\\TT\\cap U_0)\\subset V_0$.\n\nDefine $\\psi:V_0\\longrightarrow\\CC^{2n-1}$ by\n$$\\psi(z)=\\left(z_1,\\ldots,z_n,\n\\ov{\\left(\\frac{\\nu_{D,2}(z)}{\\nu_{D,1}(z)}\\right)},\\ldots,\\ov{\\left(\\frac{\\nu_{D,n}(z)}{\\nu_{D,1}(z)}\\right)}\\right).$$ The set $M:=\\psi(V_0)$ is the graph of a $\\CLW$ function defined on the local $\\CLW$ submanifold $V_0$, so it is a local $\\CLW$ submanifold in $\\CC^{2n-1}$ of the real dimension $2n-1$. Assume for a moment that $M$ is totally real.\n\nLet $$g(\\zeta):=\\left(f_1(\\zeta),\\ldots,f_n(\\zeta),\n\\frac{\\widetilde{f}_2(\\zeta)}{\\widetilde{f}_1(\\zeta)},\\ldots,\\frac{\\widetilde{f}_n(\\zeta)}{\\widetilde{f}_1(\\zeta)}\\right),\\ \\zeta\\in\\CDD\\cap U_0.$$ If $\\zeta\\in\\TT\\cap U_0$ then\n$\\widetilde{f}_k(\\zeta)\\widetilde{f}_1(\\zeta)^{-1} =\n\\overline{\\nu_{D,k}(f(\\zeta))}\\ \\overline{\\nu_{D,1}(f(\\zeta))}^{-1}$, so\n$g(\\zeta)=\\psi(f(\\zeta))$. Therefore, $g(\\TT\\cap U_0)\\subset M$. Thanks to the Reflection\nPrinciple (see Appendix), $g$ extends holomorphically past $\\TT\\cap U_0$, so $f$ extends holomorphically on a neighborhood of $\\zeta_0$.\n\nThe mapping $\\overline{\\nu_D\\circ f}$ is real analytic on $\\TT$, so it extends to a mapping $h$ holomorphic in a neighborhood $W$ of $\\TT$. For $\\zeta\\in\\TT\\cap U_0$ we have $$\\frac{\\zeta\nh_1(\\zeta)}{\\widetilde{f}_1(\\zeta)}=\\frac{1}{\\rho(\\zeta)}.$$ The function on the\nleft side is holomorphic in $\\DD\\cap U_0\\cap W$ and continuous in $\\CDD\\cap U_0\\cap W$. Since it\nhas real values on $\\TT\\cap U_0$, the Reflection Principle implies that it is holomorphic in a neighborhood of $\\TT\\cap U_0$. Hence $\\rho$ and $\\widetilde{f}$ are holomorphic in a neighborhood of $\\zeta_0$. Since $\\zeta_0$ is arbitrary, we get the assertion.\n\nIt remains to prove that $M$ is totally real. Let $r$ be a defining function of $D$. Recall that for any point $z\\in V_0$ $$\\frac{\\ov{\\nu_{D,k}(z)}}{\\ov{\\nu_{D,1}(z)}}=\\frac{\\partial r}{\\partial z_k}(z)\\left(\\frac{\\partial r}{\\partial z_1}(z)\\right)^{-1},\\,k=1,\\ldots,n.$$\nConsider the mapping $S=(S_1,\\ldots,S_n):V_0\\times\\CC^{n-1}\\longrightarrow\\RR\\times\\CC^{n-1}$\ngiven by $$S(z,w):=\\left(r(z),\\frac{\\partial r}{\\partial z_2}(z)-w_{1}\\frac{\\partial r}{\\partial z_1}(z),\\ldots,\\frac{\\partial r}{\\partial z_n}(z)-w_{n-1}\\frac{\\partial r}{\\partial z_1}(z)\\right).$$ Clearly, $M=S^{-1}(\\{0\\})$. Hence\n\\begin{equation}\\label{tan} T_{M}^{\\RR}(z,w)\\subset\\ker\\nabla S(z,w),\\ (z,w)\\in M,\\end{equation} where\n$\\nabla S:=(\\nabla S_1,\\ldots,\\nabla S_n)$.\n\nFix a point $(z,w)\\in M$. Our goal is to prove that $T_{M}^{\\CC}(z,w)=\\lbrace 0\\rbrace$. Take an arbitrary vector $(X,Y)=(X_1,\\ldots,X_n,Y_1,\\ldots,Y_{n-1})\\in T_{M}^{\\CC}(z,w)$. Then we infer from \\eqref{tan} that $$\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)X_k=0,$$ i.e. $X\\in T_{D}^{\\CC}(z)$. Denoting $v:=(z,w)$, $V:=(X,Y)$ and making use of \\eqref{tan} again we find that\n$$0=\\nabla S_k(v)(V)=\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa v_j}(v)V_j+\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j$$ for $k=2,\\ldots,n$.\nBut $V\\in T_{M}^{\\CC}(v)$, so $iV\\in T_{M}^{\\CC}(v)$. Thus $$0=\\nabla S_k(v)(iV)=i\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa v_j}(v)V_j-i\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j.$$ In particular, \\begin{multline*}0=\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j=\\sum_{j=1}^{n}\\frac{\\pa S_k}{\\pa\\ov z_j}(z,w)\\ov X_j+\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa\\ov w_j}(z,w)\\ov Y_j=\\\\=\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_j-w_{k-1}\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j.\n\\end{multline*}\nThe equality $M=S^{-1}(\\{0\\})$ gives $$w_{k-1}=\\frac{\\partial r}{\\partial z_k}(z)\\left(\\frac{\\partial r}{\\partial z_1}(z)\\right)^{-1},$$ so $$\\frac{\\partial r}{\\partial z_1}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_j=\\frac{\\partial r}{\\partial z_k}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j,\\ k=2,\\ldots,n.$$ Note that the last equality holds also for $k=1$. Therefore, \\begin{multline*}\n\\frac{\\partial r}{\\partial z_1}(z)\\sum_{j,k=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_jX_k=\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_jX_k =\\\\=\\left(\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)X_k\\right)\\left(\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j\\right)=0.\n\\end{multline*}\nBy the strong linear convexity of $D$ we have $X=0$. This implies $Y=0$, since $$0=\\nabla S_k(z,w)(0,Y)=\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa w_j}(v)Y_j+\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa\\ov w_j}(v)\\ov Y_j=-\\frac{\\partial r}{\\partial z_1}(z)Y_{k-1}$$ for $k=2,\\ldots,n$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{(Weak) $E$-mappings vs. extremal mappings and complex geodesics}\n\nIn this section we will prove important properties of (weak) $E$-mappings. In particular, we will show that they are complex geodesics and unique extremals.\n\\subsection{Weak $E$-mappings are complex geodesics and unique extremals}\nThe results of this subsection are related to weak $E$-mappings of bounded strongly linearly convex domains $D\\su\\CC^n$, $n\\geq 2$.\n\nLet $$G(z,\\zeta):=(z-f(\\zeta))\\bullet\\widetilde{f}(\\zeta),\\ z\\in\\CC^n,\\ \\zeta\\in\\DD_f.$$\n\n\\begin{propp}\\label{1}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain and let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping. Then there exist an open set $W\\supset\\overline D\\setminus f(\\TT)$ and a holomorphic mapping $F:W\\longrightarrow\\DD$ such that for any $z\\in W$ the number $F(z)$ is a unique solution of the equation $G(z,\\zeta)=0,\\ \\zeta\\in\\DD$. In particular, $F\\circ f=\\id_{\\DD}$.\n\\end{propp}\n\nIn the sequel we will strengthen the above proposition for domains with real analytic boundaries (see Proposition~\\ref{34}).\n\n\\begin{proof}[Proof of Proposition~\\ref{1}]\nSet $A:=\\overline{D}\\setminus f(\\TT)$. Since $D$ is strongly linearly convex, $\\varphi_z$ does not vanish in $\\TT$ for any $z\\in A$, so by a continuity argument the condition (4) of Definition~\\ref{21e} holds for every $z$ in some open set $W\\supset A$. For a fixed $z\\in W$ we have $$G(z,\\zeta)=\\zeta\\rho(\\zeta)\\varphi_z(\\zeta),\\ \\zeta\\in\\TT,$$ so $\\wind G(z,\\cdotp)=1$. Since $G(z,\\cdotp)\\in\\OO(\\DD)$, it has in $\\DD$ exactly one simple root $F(z)$. Hence $G(z,F(z))=0$ and $\\frac{\\partial G}{\\partial\\zeta}(z,F(z))\\neq 0$. By the Implicit Function Theorem, $F$ is holomorphic in $W$. The equality $F(f(\\zeta))=\\zeta$ for $\\zeta\\in\\DD$ is clear.\n\\end{proof}\n\nFrom the proposition above we immediately get the following\n\\begin{corr}\\label{5}\nA weak $E$-mapping $f:\\DD\\longrightarrow D$ of a bounded strongly linearly convex domain $D\\su\\CC^n$, $n\\geq 2$, is a complex geodesic. In particular,\n$$c_{D}(f(\\zeta),f(\\xi))=\\wi k_D(f(\\zeta),f(\\xi))\\text{\\,\\ and\\,\\, }\\gamma_D(f(\\zeta);f'(\\zeta))=\\kappa_D(f(\\zeta);f'(\\zeta)),$$ for any $\\zeta,\\xi\\in\\DD$.\n\\end{corr}\n\nUsing left inverses of weak $E$-mappings we may prove the uniqueness of extremals.\n\\begin{propp}\\label{2}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain and let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping. Then for any $\\xi\\in(0,1)$ the mapping $f$ is a unique $\\wi{k}_D$-extremal for $z=f(0)$, $w=f(\\xi)$ \\emph{(}resp. a unique $\\kappa_D$-extremal for $z=f(0)$, $v=f'(0)$\\emph{)}.\n\\end{propp}\n\\begin{proof}\n\nSuppose that $g$ is a $\\wi{k}_D$-extremal for $z,w$ (resp. a $\\kappa_D$-extremal for $z,v$) such that $g(0)=z$, $g(\\xi)=w$ (resp. $g(0)=z$, $g'(0)=v$). Our aim is to show that $f=g$. Proposition~\\ref{1} provides us with the mapping $F$, which is a left inverse for $f$. By the Schwarz Lemma, $F$ is a left inverse for $g$, as well, that is $F\\circ g=\\text{id}_{\\DD}$. We claim that $\\lim_{\\DD\\ni\\zeta\\to\\zeta_0}g(\\zeta)=f(\\zeta_0)$ for any $\\zeta_0\\in\\TT$ (in particular, we shall show that the limit does exist).\n\nAssume the contrary. Then there are $\\zeta_0\\in\\TT$ and a sequence $\\{\\zeta_m\\}\\subset\\DD$ convergent to $\\zeta_0$ such that the limit $Z:=\\lim_{m\\to\\infty}g(\\zeta_m)\\in\\overline{D}$ exists and is not equal to $f(\\zeta_0)$. We have $G(z,F(z))=0$, so putting $z=g(\\zeta_m)$ we infer that $$0=(g(\\zeta_m)-f(F(g(\\zeta_m))))\\bullet \\widetilde{f}(F(g(\\zeta_m)))=(g(\\zeta_m)-f(\\zeta_m))\\bullet\\widetilde{f}(\\zeta_m).\n$$ Passing with $m$ to the infinity we get $$0=(Z-f(\\zeta_0))\\bullet \\widetilde{f}(\\zeta_0)=\\zeta_0\\rho(\\zeta_0)\\langle Z-f(\\zeta_0),\\nu_D(f(\\zeta_0))\\rangle.$$ This means that $Z-f(\\zeta_0)\\in T^{\\CC}_D(f(\\zeta_0))$. Since $D$ is strongly linearly convex, we deduce that $Z=f(\\zeta_0)$, which is a contradiction.\n\nHence $g$ extends continuously on $\\CDD$ and, by the maximum principle, $g=f$.\n\\end{proof}\n\n\n\\begin{propp}\\label{3}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain, let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping and let $a$ be an automorphism of $\\DD$. Then $f\\circ a$ is a weak $E$-mapping of $D$.\n\\end{propp}\n\\begin{proof}\nSet $g:=f\\circ a$.\n\nClearly, the conditions (1') and (2') of Definition~\\ref{21} are satisfied by $g$.\n\nTo prove that $g$ satisfies the condition (4) of Definition~\\ref{21e} fix a point $z\\in D$. Let $\\varphi_{z,f}$, $\\varphi_{z,g}$ be the functions appearing in the condition (4) for $f$ and $g$ respectively. Then $\\varphi_{z,g}=\\varphi_{z,f}\\circ a$. Since $a$ maps $\\TT$ to $\\TT$ diffeomorphically, we have $\\wind\\varphi_{z,g}=\\pm\\wind\\varphi_{z,f}=0$.\n\nIt remains to show that the condition (3') of Definition~\\ref{21} is also satisfied by $g$. Note that the function $\\wi a(\\zeta):=\\zeta\/a(\\zeta)$ has a holomorphic branch of the logarithm in the neighborhood of $\\TT$. This follows from the fact that $\\wind \\wi a=0$, however the existence of the holomorphic branch may be shown quite elementary. Actually, it would suffices to prove that $\\wi a(\\TT)\\neq\\TT$. Expand $a$ as $$a(\\zeta)=e^{it}\\frac{\\zeta-b}{1-\\overline b\\zeta}$$ with some $t\\in\\RR$, $b\\in\\DD$ and observe that $\\widetilde a$ does not attain the value $-e^{-it}$. Indeed, if $\\zeta\/a(\\zeta)=-e^{-it}$ for some $\\zeta\\in\\TT$, then $$\\frac{1-\\overline b\\zeta}{1-b\\overline\\zeta}=-1,$$ so $2=2\\re(b\\overline\\zeta)\\leq 2|b|$, which is impossible.\n\nConcluding, there exists a function $v$ holomorphic in a neighborhood of $\\TT$ such that $$\\frac{\\zeta}{a(\\zeta)}=e^{i v(\\zeta)}.$$ Note that $v(\\TT)\\su\\RR$. Expanding $v$ in Laurent series $$v(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^k,\\ \\zeta\\text{ near }\\TT,$$ we infer that $a_{-k}=\\overline a_k$, $k\\in\\ZZ$. Therefore, $$v(\\zeta)=a_0+\\sum_{k=1}^\\infty 2\\re(a_k\\zeta^k)=\\re\\left(a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k\\right),\\ \\zeta\\in\\TT.$$ Hence, there is a function $h$ holomorphic in the neighborhood of $\\CDD$ such that $v=\\im h$. Put $u:=h-iv$. Then $u\\in\\OO(\\TT)$ and $u(\\TT)\\su\\RR$.\n\nTake $\\rho$ be as in the condition (3') of Definition~\\ref{21} for $f$ and define $$r(\\zeta):=\\rho(a(\\zeta))e^{u(\\zeta)},\\ \\zeta\\in\\TT.$$ Let us compute\n\\begin{eqnarray*}\\zeta r(\\zeta)\\overline{\\nu_D(g(\\zeta))}=\\zeta u^{u(\\zeta)}\\rho(a(\\zeta))\\overline{\\nu_D(f(a(\\zeta)))}&=&\\\\=a(\\zeta)h(\\zeta)\\rho(a(\\zeta))\\overline{\\nu_D(f(a(\\zeta)))}\n&=&h(\\zeta)\\widetilde{f}(a(\\zeta)),\\quad\\zeta\\in\\TT.\n\\end{eqnarray*} Thus $\\zeta\\longmapsto\\zeta r(\\zeta)\\overline{\\nu_D(g(\\zeta))}$ extends holomorphically to a function of class $\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$.\n\\end{proof}\n\n\n\\begin{corr}\\label{28}\nA weak $E$-mapping $f:\\DD\\longrightarrow D$ of a bounded strongly linearly convex domain $D\\su\\CC^n$, $n\\geq 2$, is a unique $\\wi{k}_D$-extremal for $f(\\zeta),f(\\xi)$ \\emph{(}resp. a unique $\\kappa_D$-extremal for $f(\\zeta),f'(\\zeta)$\\emph{)}, where $\\zeta,\\xi\\in\\DD$, $\\zeta\\neq\\xi$.\n\\end{corr}\n\n\\subsection{Generalization of Proposition~\\ref{1}}\nThe results obtained in this subsection will play an important role in the sequel.\n\nWe start with \n\\begin{propp}\\label{4}\nLet $f:\\DD\\longrightarrow D$ be an $E$-mapping. Then the function $f'\\bullet\\widetilde{f}$ is a positive constant.\n\\end{propp}\n\\begin{proof}\nConsider the curve $$\\RR\\ni t\\longmapsto f(e^{it})\\in\\partial D.$$ Its any tangent vector $ie^{it}f'(e^{it})$ belongs to $T_{D}^\\mathbb{R}(f(e^{it}))$, i.e. $$\\re\\langle ie^{it}f'(e^{it}),\\nu_D(f(e^{it}))\\rangle=0.$$ Thus for $\\zeta\\in\\TT$ $$0=\\rho(\\zeta)\\re\\langle i\\zeta f'(\\zeta),\\nu_D(f(\\zeta))\\rangle=-\\im f'(\\zeta)\\bullet\\widetilde{f}(\\zeta),$$ so the holomorphic function $f'\\bullet\\widetilde{f}$ is a real constant $C$.\n\nConsidering the curve $$[0,1+\\eps)\\ni t\\longmapsto f(t)\\in\\overline D$$ for small $\\eps>0$ and noting that $f([0,1))\\su D$, $f(1)\\in\\partial D$, we see that the derivative of $r\\circ f$ at a point $t=1$ is non-negative, where $r$ is a defining function of $D$. Hence $$0\\leq\\re\\langle f'(1),\\nu_D(f(1))\\rangle =\\frac{1}{\\rho(1)} \\re( f'(1)\\bullet\\widetilde{f}(1))=\n\\frac{C}{\\rho(1)},$$ i.e. $C\\geq 0$. For $\\zeta\\in\\TT$\n$$\\frac{f(\\zeta)-f(0)}{\\zeta}\\bullet\\widetilde{f}(\\zeta)=\\rho(\\zeta)\\langle f (\\zeta)-f(0),\\nu_D(f(\\zeta))\\rangle.$$ This function has the winding number equal to $0$. Therefore, the function $$g(\\zeta):=\\frac{f(\\zeta)-f(0)}{\\zeta}\\bullet\\widetilde{f}(\\zeta),$$ which is holomorphic in a neighborhood of $\\CDD$, does not vanish\nin $\\DD$. In particular, $C=g(0)\\neq 0$.\n\\end{proof}\nThe function $\\rho$ is defined up to a constant factor. \\textbf{We choose $\\rho$ so that $ f'\\bullet\\widetilde{f}\\equiv 1$}, i.e. \\begin{equation}\\label{rho}\\rho(\\zeta)^{-1}=\\langle\\zeta f'(\\zeta),\\nu_D(f(\\zeta))\\rangle,\\ \\zeta\\in\\TT.\\end{equation} In that way $\\widetilde{f}$ and $\\rho$ are uniquely determined by $f$.\n\n\\begin{propp}\nAn $E$-mapping $f:\\DD\\longrightarrow D$ is injective in $\\CDD$.\n\\end{propp}\n\n\\begin{proof}The function $f$ has the left-inverse in $\\DD$, so it suffices to check the injectivity on $\\TT$. Suppose that $f(\\zeta_1)=f(\\zeta_2)$ for some $\\zeta_1,\\zeta_2\\in\\TT$, $\\zeta_1\\neq\\zeta_2$, and consider the curves $$\\gamma_j:[0,1]\\ni t\\longmapsto f(t\\zeta_j)\\in\\overline D,\\ j=1,2.$$ Since $$\\re\\langle\\gamma_j'(1),\\nu_D(f(\\zeta_j))\\rangle=\\re\\langle\\zeta_jf'(\\zeta_j),\\nu_D(f(\\zeta_j))\\rangle\n=\\rho(\\zeta_j)^{-1}\\neq 0,$$ the curves $\\gamma_j$ hit $\\pa D$ transversally at their common point $f(\\zeta_1)$. We claim that there exists $C>0$ such that for $t\\in(0,1)$ close to $1$ there is $s_t\\in(0,1)$ satisfying $\\wi k_D(f(t\\zeta_1),f(s_t\\zeta_2))<C$. It will finish the proof since $$\\wi k_D(f(t\\zeta_1),f(s_t\\zeta_2))=p(t\\zeta_1,s_t\\zeta_2)\\to\\infty,\\ t\\to 1.$$ We may assume that $f(\\zeta_1)=0$ and $\\nu_D(0)=(1,0,\\ldots,0)=:e_1$. There exists a ball $B\\su D$ tangent to $\\pa D$ at $0$. Using a homothety, if necessary, one can assume that $B=\\BB_n-e_1$. From the transversality of $\\gamma_1,\\gamma_2$ to $\\pa D$ there exists a cone $$A:=\\{z\\in\\CC^n:-\\re z_1>k|z|\\},\\quad k>0,$$ such that $\\gamma_1(t),\\gamma_2(t)\\in A\\cap B$ if $t\\in(0,1)$ is close to $1$. For $z\\in A$ let $k_z>k$ be a positive number satisfying the equality $$|z|=\\frac{-\\re z_1}{k_z}.$$ \n\nNote that for any $a\\in\\gamma_1((0,1))$ sufficiently close to $0$ one may find $b\\in\\gamma_2((0,1))\\cap A\\cap B$ such that $\\re b_1=\\re a_1$. To get a contradiction it suffices to show that $\\wi k_D(a,b)$ is bounded from above by a constant independent on $a$ and $b$. \n\nWe have the following estimate \\begin{multline*}\\wi k_D(a,b)\\leq\\wi k_{\\BB_n-e_1}(a,b)=\\wi k_{\\BB_n}(a+e_1,b+e_1)=\\\\=\\tanh^{-1}\\sqrt{1-\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1 \\rangle|^2}}.\\end{multline*} The last expression is bounded from above if and only if $$\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1\\rangle|^2}$$ is bounded from below by some positive constant. We estimate $$\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1\\rangle|^2}=\\frac{(2\\re a_1+|a|^2)(2\\re b_1+|b|^2)}{|\\langle a, b\\rangle+a_1+\\overline b_1|^2}=$$$$=\\frac{\\left(2\\re a_1+\\frac{(\\re a_1)^2}{k^2_a}\\right)\\left(2\\re a_1+\\frac{(\\re a_1)^2}{k^2_b}\\right)}{|\\langle a, b\\rangle+2\\re a_1+i\\im a_1-i\\im b_1|^2}\\geq\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2|\\langle a, b\\rangle+i\\im a_1-i\\im b_1|^2+2|2\\re a_1|^2}$$$$\\geq\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2(|a||b|+|a|+|b|)^2+8(\\re a_1)^2}=\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2\\left(\\frac{(-\\re a_1)^2}{k^2_ak^2_b}-\\frac{\\re a_1}{k_a}-\\frac{\\re a_1}{k_b}\\right)^2+8(\\re a_1)^2}$$$$=\\frac{\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2\\left(\\frac{-\\re a_1}{k^2_ak^2_b}+\\frac{1}{k_a}+\\frac{1}{k_b}\\right)^2+8}>\\frac{1}{2(1+2\/k)^2+8}.$$ This finishes the proof.\n\\end{proof}\n\n\\medskip\n\nAssume that we are in the settings of Proposition~\\ref{1} and $D$ has real analytic boundary. Our aim is to replace $W$ with a neighborhood of $\\ov D$.\n\n\\begin{remm}\\label{przed34}\nFor $\\zeta_0\\in\\DD_f$ we have $G(f(\\zeta_0),\\zeta_0)=0$ and $\\frac{\\partial G}{\\partial\\zeta}(f(\\zeta_0),\\zeta_0)=-1$. By the Implicit Function Theorem there exist neighborhoods $U_{\\zeta_0},V_{\\zeta_0}$ of $f(\\zeta_0),\\zeta_0$ respectively and a holomorphic function $F_{\\zeta_0}:U_{\\zeta_0}\\longrightarrow V_{\\zeta_0}$ such that for any $z\\in U_{\\zeta_0}$ the point $F_{\\zeta_0}(\\zeta)$ is the unique solution of the equation $G(z,\\zeta)=0$, $\\zeta\\in V_{\\zeta_0}$.\n\nIn particular, if $\\zeta_0\\in\\DD$ then $F_{\\zeta_0}=F$ near $f(\\zeta_0)$.\n\\end{remm}\n\n\\begin{propp}\\label{34}\nLet $f:\\DD\\longrightarrow D$ be an $E$-mapping. Then there exist arbitrarily small neighborhoods $U$, $V$ of $\\overline D$, $\\CDD$ respectively such that for any $z\\in U$ the equation $G(z,\\zeta)=0$, $\\zeta\\in V$, has exactly one solution.\n\\end{propp}\n\\begin{proof} In view of Proposition~\\ref{1} and Remark~\\ref{przed34} it suffices to prove that there exist neighborhoods $U$, $V$ of $\\overline D$, $\\CDD$ respectively such that for any $z\\in U$ the equation $G(z,\\cdotp)=0$ has at most one solution $\\zeta\\in V$.\n\nAssume the contrary. Then for any neighborhoods $U$ of $\\overline D$ and $V$ of $\\CDD$ there are $z\\in U$, $\\zeta_1,\\zeta_2\\in V$, $\\zeta_1\\neq\\zeta_2$ such that $G(z,\\zeta_1)=G(z,\\zeta_2)=0$. For $m\\in\\NN$ put $$U_m:=\\{z\\in\\CC^n:\\dist(z,D)<1\/m\\},$$ $$V_m:=\\{\\zeta\\in\\CC:\\dist(\\zeta,\\DD)<1\/m\\}.$$ There exist $z_m\\in U_m$, $\\zeta_{m,1},\\zeta_{m,2}\\in V_m$, $\\zeta_{m,1}\\neq\\zeta_{m,2}$ such that $G(z_m,\\zeta_{m,1})=G(z_m,\\zeta_{m,2})=0$. Passing to a subsequence we may assume that $z_m\\to z_0\\in\\ov D$. Analogously we may assume $\\zeta_{m,1}\\to\\zeta_1\\in \\CDD$ and $\\zeta_{m,2}\\to\\zeta_2\\in\\CDD$. Clearly, $G(z_0,\\zeta_1)=G(z_0,\\zeta_2)=0$. Let us consider few cases.\n\n1) If $\\zeta_1,\\zeta_2\\in\\TT$, then $G(z_0,\\zeta_j)=0$ is equivalent to $$\\langle z_0-f(\\zeta_j), \\nu_D(f(\\zeta_j))\\rangle=0,\\ j=1,2,$$ consequently $z_0-f(\\zeta_j)\\in T^{\\CC}_D(f(\\zeta_j))$. By the strong linear convexity of $D$ we get $z_0=f(\\zeta_j)$. But $f$ is injective in $\\CDD$, so $\\zeta_1=\\zeta_2=:\\zeta_0$. It follows from Remark~\\ref{przed34} that in a sufficiently small neighborhood of $(z_0,\\zeta_0)$ all solutions of the equation $G(z,\\zeta)=0$ are of the form $(z,F_{\\zeta_0}(z))$. Points $(z_m,\\zeta_{m,1})$ and $(z_m,\\zeta_{m,2})$ belong to this neighborhood for large $m$, which gives a contradiction.\n\n2) If $\\zeta_1\\in\\TT$ and $\\zeta_2\\in\\DD$, then analogously as above we deduce that $z_0=f(\\zeta_1)$. Let us take an arbitrary sequence $\\{\\eta_m\\}\\su\\DD$ convergent to $\\zeta_1$. Then $f(\\eta_m) \\in D$ and $f(\\eta_m)\\to z_0$, so the sequence $G(f(\\eta_m),\\cdotp)$ converges to $G(z_0,\\cdotp)$ uniformly on $\\DD$. Since $G(z_0,\\cdotp)\\not\\equiv 0$, $G(z_0,\\zeta_2)=0$ and $\\zeta_2\\in\\DD$, we deduce from Hurwitz Theorem that for large $m$ the functions $G(f(\\eta_m),\\cdotp)$ have roots $\\theta_m\\in\\DD$ such that $\\theta_m\\to\\zeta_2$. Hence $G(f(\\eta_m),\\theta_m)=0$ and from the uniqueness of solutions in $D\\times\\DD$  (Proposition~\\ref{1}) we have $$\\theta_m=F(f(\\eta_m))=\\eta_m.$$ This is a contradiction, because the left side tends to $\\zeta_2$ and the right one to $\\zeta_1$, as $m\\to\\infty$.\n\n3) We are left with the case $\\zeta_1,\\zeta_2\\in\\DD$.\nIf $z_0\\in\\overline{D}\\setminus f(\\TT)$ then $z_0\\in W$. In $W\\times\\DD$ all solutions of the equation $G=0$ are of the form $(z,F(z))$, $z\\in W$. But for large $m$ the points $(z_m,\\zeta_{m,1})$, $(z_m,\\zeta_{m,2})$ belong to $W\\times\\DD$, which is a contradiction with the uniqueness.\n\nIf $z_0\\in f(\\TT)$, then $z_0=f(\\zeta_0)$ for some $\\zeta_0\\in\\TT$. Clearly, $G(f(\\zeta_0),\\zeta_0)=0$, whence $G(z_0,\\zeta_0)=G(z_0,\\zeta_1)=0$ and $\\zeta_0\\in\\TT$, $\\zeta_1\\in \\DD$. This is just the case 2), which has been already considered.\n\\end{proof}\n\n\n\\begin{corr} There are neighborhoods $U$, $V$ of $\\overline D$ and $\\CDD$ respectively with $V\\Subset\\DD_f$, such that the function $F$ extends holomorphically on $U$. Moreover, all solutions of the equation $G|_{U\\times V}=0$ are of the form $(z,F(z))$, $z\\in U$.\n\nIn particular, $F\\circ f=\\id_{V}$.\n\\end{corr}\n\n\n\n\n\n\n\n\n\n\n\n\\section{H\\\"older estimates}\\label{22}\n\n\\begin{df}\\label{30} For a given $c>0$ let the family $\\mathcal{D}(c)$ consist of all pairs $(D,z)$, where $D\\su\\CC^n$, $n\\geq 2$, is a bounded pseudoconvex domain with real $\\mathcal C^2$ boundary and $z\\in D$, satisfying\n\\begin{enumerate}\n\\item $\\dist(z,\\partial D)\\geq 1\/c$;\n\\item the diameter of $D$ is not greater than $c$ and $D$ satisfies the interior ball condition with a radius $1\/c$;\n\\item for any $x,y\\in D$ there exist $m\\leq 8 c^2$ and open balls $B_0,\\ldots,B_m\\subset D$ of radius $1\/(2c)$ such that $x\\in B_0$, $y\\in B_m$ and the distance between the centers of the balls $B_j$, $B_{j+1}$ is not greater than $1\/(4c)$ for $j=0,\\ldots,m-1$;\n\\item for any open ball $B\\subset\\mathbb{C}^n$ of radius not greater than $1\/c$, intersecting non-emptily with $\\pa D$, there exists a mapping $\\Phi\\in\\OO(\\overline{D},\\mathbb{C}^n)$ such that\n\\begin{enumerate}\n\\item for any $w\\in\\Phi(B\\cap\\partial D)$ there is a ball of radius $c$ containing $\\Phi(D)$ and tangent to $\\partial\\Phi(D)$ at $w$ (let us call it the ``exterior ball condition'' with a radius $c$);\n\\item $\\Phi$ is biholomorphic in a neighborhood of $\\ov B$ and $\\Phi^{-1}(\\Phi(B))=B$;\n\\item entries of all matrices $\\Phi'$ on $B\\cap\\ov D$ and $(\\Phi^{-1})'$ on $\\Phi(B\\cap\\overline{D})$ are bounded in modulus by $c$;\n\\item $\\dist(\\Phi(z),\\partial\\Phi(D))\\geq 1\/c$;\n\\end{enumerate}\n\\item the normal vector $\\nu_D$ is Lipschitz with a constant $2c$, that is $$|\\nu_D(a)-\\nu_D(b)|\\leq 2c|a-b|,\\ a,b\\in \\partial D;$$\n\\item the $\\eps$-hull of $D$, i.e. a domain $D_{\\eps}:=\\{w\\in\\mathbb C^n:\\dist (w,D)<\\eps\\}$, is strongly pseudoconvex for any $\\eps\\in (0,1\/c).$\n\\end{enumerate}\n\\end{df}\n\nRecall that the {\\it interior ball condition} with a radius $r>0$ means that for any point $a\\in\\pa D$ there is $a'\\in D$ and a ball $B_n(a',r)\\su D$ tangent to $\\pa D$ at $a$. Equivalently $$D=\\bigcup_{a'\\in D'}B_n(a',r)$$ for some set $D'\\su D$.\n\nIt may be shown that (2) and (5) may be expressed in terms of boundedness of the normal curvature, boundedness of a domain and the condition (3). This however lies beyond the scope of this paper and needs some very technical arguments so we omit the proof of this fact. The reasons why we decided to use (2) in such a form is its connection with the condition (3) (this allows us to simplify the proof in some places).\n\n\\begin{rem}\\label{con}\nNote that any convex domain satisfying conditions (1)-...-(4) of Definition~\\ref{30} satisfies conditions (5) and (6), as well.\n\nActually, it follows from (2) that for any $a\\in\\pa D$ there exists a ball $B_n(a',1\/c)\\su D$ tangent to $\\pa D$ at $a$. Then $$\\nu_D(a)=\\frac{a'-a}{|a'-a|}=c(a'-a).$$ Hence $$|\\nu_D(a)-\\nu_D(b)|=c|a'-a-b'+b|=c|a'-b'-(a-b)|\\leq c|a'-b'|+c|a-b|.$$ Since $D$ is convex, we have $|a'-b'|\\leq|a-b|$, which gives (5).\n\nThe condition (6) is also clear --- for any $\\eps>0$ an $\\eps$-hull of a strongly convex domain is strongly convex.\n\\end{rem}\n\n\\begin{rem}\nFor a convex domain $D$ the condition (3) of Definition \\ref{30} amounts to the condition (2).\n\nIndeed, for two points $x,y\\in D$ take two balls of radius $1\/(2c)$ containing them and contained in $D$. Then divide the interval between the centers of the balls into $[4c^2]+1$ equal parts and take balls of radius $1\/(2c)$ with centers at the points of the partition.\n\nNote also that if $D$ is strongly convex and satisfies the interior ball condition with a radius $1\/c$ and the exterior ball condition with a radius $c$, one can take $\\Phi:=\\id_{\\CC^n}$.\n\\end{rem}\n\n\n\\begin{rem}\\label{D(c),4}\nFor a strongly pseudoconvex domain $D$ and $c'>0$ and for any $z\\in D$ such that $\\dist(z,\\partial D)>1\/c'$ there exists $c=c(c')>0$ satisfying $(D,z)\\in\\mathcal{D}(c)$.\n\nIndeed, the conditions (1)-...-(3) and (5)-(6) are clear. Only (4) is non-trivial.\n\nThe construction of the mapping $\\Phi$ amounts to the construction of Forn\\ae ss peak functions. Actually, apply directly Proposition 1 from \\cite{For} to any boundary point of $\\partial D$ (obviously $D$ has a Stein neighborhood basis). This gives a covering of $\\partial D$ with  a finite number of balls $B_j$, maps $\\Phi_j\\in\\OO(\\overline{D},\\mathbb{C}^n)$ and strongly convex $C^\\infty$-smooth domains $C_j$, $j=1,\\ldots, N$, such that\n\\begin{itemize}\\item $\\Phi_j(D)\\subset C_j$;\n\\item $\\Phi_j(\\ov D)\\subset\\ov C_j$;\n\\item $\\Phi_j(B_j\\setminus\\ov D)\\subset\\mathbb C^n\\setminus\\ov C_j$;\n\\item $\\Phi_j^{-1}(\\Phi_j(B_j))=B_j$;\n\\item $\\Phi_j|_{B_j}: B_j\\longrightarrow \\Phi_j(B_j)$ is biholomorphic.\n\\end{itemize} Therefore, one may choose $c>0$ such that every $C_j$ satisfies the exterior ball condition with $c$, i.e. for any $x\\in \\partial C_j$ there is a ball of radius $c$ containing $C_j$ and tangent to $\\partial C_j$ at $x$, every ball of radius $1\/c$ intersecting non-emptily with $\\pa D$ is contained in some $B_j$ (here one may use a standard argument invoking the Lebesgue number) and the conditions (c), (d) are also satisfied (with $\\Phi:=\\Phi_j$).\n\\end{rem}\n\n\nIn this section we use the words `uniform', `uniformly' if $(D,z)\\in \\mathcal D(c)$. This means that estimates will depend only on $c$ and will be independent on $D$ and $z$ if $(D,z)\\in\\mathcal{D}(c)$ and on $E$-mappings of $D$ mapping $0$ to $z$. Moreover, in what follows we assume that $D$ is a strongly linearlu convex domain with real-analytic boundary.\n\n\\begin{prop}\\label{7}\nLet $f:(\\mathbb{D},0)\\longrightarrow(D,z)$ be an $E$-mapping. Then $$\\dist(f(\\zeta),\\partial D)\\leq C(1-|\\zeta|),\\ \\zeta\\in\\CDD$$ with $C>0$ uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\\end{prop}\n\\begin{proof} There exists a uniform $C_1$ such that $$\\text{if }\\dist(w,\\partial D)\\geq 1\/c\\text{ then }k_D(w,z)<C_1.$$ Indeed, let $\\dist(w,\\partial D)\\geq 1\/c$ and let balls $B_0,\\ldots,B_m$ with centers $b_0,\\ldots,b_m$ be chosen to the points $w$, $z$ as in the condition (3) of Definition~\\ref{30}. Then\n\\begin{multline*}k_D(w,z)\\leq\nk_D(w,b_0)+\\sum_{j=0}^{m-1}k_D(b_j,b_{j+1})+k_D(b_m,z)\\leq\\\\\\leq k_{B_n(w,1\/c)}(w,b_0)+\\sum_{j=0}^{m-1}k_{B_j}(b_j,b_{j+1})+k_{B_n(z,1\/c)}(b_m,z)=\\\\=p\\left(0,\\frac{|w-b_0|}{1\/c}\\right)+\\sum_{j=0}^{m-1}p\\left(0,\\frac{|b_j-b_{j+1}|}{1\/(2c)}\\right)+\np\\left(0,\\frac{|b_m-z|}{1\/c}\\right)\\leq\\\\\\leq(m+2)p\\left(0,\\frac{1}{2}\\right)\\leq(8c^2+2)p\\left(0,\\frac{1}{2}\\right)=:C_1.\n\\end{multline*}\n\nIf $\\zeta\\in\\mathbb{D}$ is such that\n$\\dist(f(\\zeta),\\partial D)\\geq 1\/c$ then $$k_D(f(0),f(\\zeta))\\leq\nC_2-\\frac{1}{2}\\log\\dist(f(\\zeta),\\partial D)$$ with a uniform $C_2:=C_1+\\frac{1}{2}\\log c$.\n\nIn the other case, i.e. when $\\dist(f(\\zeta),\\partial D)<1\/c$, denote by $\\eta$ the nearest point to\n$f(\\zeta)$ lying on $\\partial D$. Let $w\\in D$ be a center of a ball $B$ of radius $1\/c$\ntangent to $\\partial D$ at $\\eta$. By the condition (2) of Definition~\\ref{30} we have $B\\subset D$. Hence\n\\begin{multline*}k_D(f(0),f(\\zeta))\\leq k_D(f(0),w)+k_D(w,f(\\zeta))\\leq\\\\\\leq\nC_1+k_B(w,f(\\zeta))\\leq C_1+\\frac{1}{2}\\log 2-\\frac{1}{2}\\log\\left(1-\\frac{|f(\\zeta)-w|}{1\/c}\\right)=\\\\=C_1+\\frac{1}{2}\\log 2-\\frac{1}{2}\\log(c\\dist(f(\\zeta),\\partial B))=C_3-\\frac{1}{2}\\log\\dist(f(\\zeta),\\partial D)\n\\end{multline*}\nwith a uniform $C_3:=C_1+\\frac{1}{2}\\log\\frac{2}{c}$.\n\nWe have obtained the same type estimates in both cases. On the other side, by\nCorollary~\\ref{5} $$k_D(f(0),f(\\zeta))=p(0,\\zeta)\\geq-\\frac{1}{2}\\log(1-|\\zeta|),$$ which finishes the proof.\n\\end{proof}\n\nRecall that we have assumed that $\\rho$ is of the form~\\eqref{rho}.\n\\begin{prop}\\label{9}\nLet $f:(\\mathbb{D},0)\\longrightarrow(D,z)$ be an $E$-mapping. Then $$C_1<\\rho(\\zeta)^{-1}<C_2,\\ \\zeta\\in\\TT,$$ where\n$C_1,C_2$ are uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\\end{prop}\n\\begin{proof} For the upper estimate fix $\\zeta_0\\in\\TT$. Set $B:=B_n(f(\\zeta_0),1\/c)$ and let $\\Phi\\in\\OO(\\overline{D},\\mathbb{C}^n)$ be as in the condition (4) of Definition~\\ref{30} for $B$. One can assume that $f(\\zeta_0)=\\Phi(f(\\zeta_0))=0$ and $\\nu_D(0)=\\nu_{\\Phi(D)}(0)=(1,0,\\ldots,0)$. Then $\\Phi(D)$ is contained in the half-space $\\{w\\in\\mathbb{C}^n:\\re w_1<0\\}$. Putting $h:=\\Phi\\circ f$ we have $$h_1(\\mathbb{D})\\subset\\{w_1\\in\\mathbb{C}:\\re w_1<0\\}.$$ In virtue of the Schwarz Lemma on the half-plane\n\\begin{equation}\\label{schh1}|h_1'(t\\zeta_0)|\\leq\\frac{-2\\re h_1(t\\zeta_0)}{1-|t\\zeta_0|^2}.\\end{equation}\n\nLet $\\delta$ be the signed boundary distance of $\\Phi(D)$, i.e. $$\\delta(x):=\\begin{cases}-\\dist(x,\\partial\\Phi(D)),\\ x\\in\\Phi(D)\\\\\\ \\ \\ \\dist(x,\\partial\\Phi(D)),\\ x\\notin\\Phi(D).\\end{cases}$$ It is a defining function of $\\Phi(D)$ in a neighborhood of $0$ (recall that $\\Phi^{-1}(\\Phi(B))=B$). Observe that $$\\delta(x)=\\delta(0)+\\re\\langle\\nabla\\delta(0), x\\rangle+O(|x|^2)=\\re x_1+O(|x|^2).$$\n\nIf $x\\in\\Phi(D)$ tends transversally to $0$, then the angle between the vector $x$ and the hyperplane $\\{w\\in\\mathbb{C}^n:\\re w_1=0\\}$ is separated from $0$, i.e. its sinus $(-\\re x_1)\/|x|>\\varepsilon$ for some $\\varepsilon>0$ independent on $x$. Thus $$\\frac{\\delta(x)}{\\re x_1}=1+O(|x|)\\text{ as }x\\to 0\\text{ transversally. }$$ Consequently \\begin{equation}\\label{50}-\\re x_1\\leq 2\\dist(x,\\partial\\Phi(D))\\text{ as }x\\to 0\\text{ transversally. }\\end{equation}\n\nWe know that $t\\longmapsto f(t\\zeta_0)$ hits $\\partial D$ transversally. Therefore, $t\\longmapsto h(t\\zeta_0)$ hits $\\partial \\Phi(D)$ transversally, as well. Indeed, we have \\begin{multline}\\label{hf}\\left\\langle\\left.\\frac{d}{dt}h(t\\zeta_0)\\right|_{t=1},\\nu_{\\Phi(D)}(h(\\zeta_0))\\right\\rangle=\\left\\langle \\Phi'(0)f'(\\zeta_0)\\zeta_0,\\frac{(\\Phi^{-1})'(0)^*\\nabla r(0)}{|(\\Phi^{-1})'(0)^*\\nabla r(0)|}\\right\\rangle=\\\\=\\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nabla r(0)\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}=\\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nu_D(f(\\zeta_0))|\\nabla r(0)|\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}.\n\\end{multline}\nwhere $r$ is a defining function of $D$. In particular,\n\\begin{multline*} \\re \\left\\langle\\left.\\frac{d}{dt}h(t\\zeta_0)\\right|_{t=1},\\nu_{\\Phi(D)}(h(\\zeta_0))\\right\\rangle=\\re \\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nu_D(f(\\zeta_0))|\\nabla r(0)|\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}=\\\\=\\frac{\\rho(\\zeta_0)^{-1}|\\nabla r(0)|}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}\\neq 0.\\end{multline*}  This proves that $t\\longmapsto h(t\\zeta_0)$ hits $\\partial\\Phi(D)$ transversally.\n\n\nConsequently, we may put $x=h(t\\zeta_0)$ into \\eqref{50} to get \\begin{equation}\\label{hf1}\\frac{-2\\re h_1(t\\zeta_0)}{1-|t\\zeta_0|^2}\\leq\\frac{4\\dist(h(t\\zeta_0),\\partial\\Phi(D))}\n{1-|t\\zeta_0|^2},\\ t\\to 1.\\end{equation}\nBut $\\Phi$ is a biholomorphism near $0$, so \\begin{equation}\\label{nfr}\\frac{4\\dist(h(t\\zeta_0),\\partial\\Phi(D))}{1-|t\\zeta_0|^2}\\leq C_3\\frac{\\dist(f(t\\zeta_0),\\partial D)}{1-|t\\zeta_0|},\\ t\\to 1,\\end{equation} where $C_3$ is a uniform constant depending only on $c$ (thanks to the condition (4)(c) of Definition~\\ref{30}). By Proposition \\ref{7}, the term on the right side of~\\eqref{nfr} does not exceed some uniform constant.\n\nIt follows from \\eqref{hf} that \\begin{multline*}\\rho(\\zeta_0)^{-1}=|\\langle f'(\\zeta_0)\\zeta_0,\\nu_D(f(\\zeta_0))\\rangle|\\leq C_4|\\langle h'(\\zeta_0), \\nu_{\\Phi(D)}(h(\\zeta_0))\\rangle|=\\\\=C_4|h_1'(\\zeta_0)|=\\lim_{t\\to 1}C_4|h_1'(t\\zeta_0)|\\end{multline*} with a uniform $C_4$ (here we use the condition (4)(c) of Definition~\\ref{30} again).\nCombining \\eqref{schh1}, \\eqref{hf1} and \\eqref{nfr} we get the upper estimate for $\\rho(\\zeta_0)^{-1}.$\n\nNow we are proving the lower estimate. Let $r$ be the signed boundary distance to $\\partial D$. For $\\varepsilon=1\/c$ the function $$\\varrho(w):=-\\log(\\varepsilon-r(w))+\\log\\varepsilon,\\ w\\in\nD_\\varepsilon,$$ where $D_\\varepsilon$ is an $\\varepsilon$-hull of $D$, is plurisubharmonic and defining for $D$. Indeed, we have $$-\\log(\\varepsilon-r(w))=-\\log\\dist(w,\\partial D_\\varepsilon),\\ w\\in D_\\varepsilon$$ and $D_\\varepsilon$ is pseudoconvex.\n\nTherefore, a function $$v:=\\varrho\\circ f:\\overline{\\mathbb{D}}\\longrightarrow(-\\infty,0]$$ is subharmonic on $\\DD$. Moreover, since $f$ maps $\\TT$ in $\\partial D$ we infer that $v=0$ on $\\TT$. Moreover, since $|f(\\lambda)-z|<c$ for $\\lambda\\in\\mathbb{D}$, we have $$|f(\\lambda)-z|<\\frac{1}{2c}\\text{ if }|\\lambda|\\leq\\frac{1}{2c^2}.$$ Therefore, for a fixed $\\zeta_0\\in\\TT$ $$M_{\\zeta_0}(x):=\\max_{t\\in[0,2\\pi]}v(\\zeta_0 e^{x+it})\\leq-\\log\\left(1+\\frac{1}{2c\\varepsilon}\\right)=:-C_5\\text{ if }x\\leq-\\log(2c^2).$$ Since $M_{\\zeta_0}$ is convex for $x\\leq 0$ and $M_{\\zeta_0}(0)=0$, we get $$v(\\zeta_0 e^x)\\leq M_{\\zeta_0}(x)\\leq\\frac{C_5x}{\\log(2c^2)}\\text{\\ \\ \\\nfor \\ }-\\log(2c^2)\\leq x\\leq 0.$$ Hence (remember that $v(\\zeta_0)=0$)\n\\begin{multline}\\label{wk}\\frac{C_5}{\\log(2c^2)}\\leq\\left.\\frac{d}{dx}v(\\zeta_0 e^x)\\right|_{x=0}=\\sum_{j=1}^n\\frac{\\pa\\varrho}{\\pa z_j}(f(\\zeta_0))f_j'(\\zeta_0)\\zeta_0=\\\\=\\langle\\zeta_0 f'(\\zeta_0),\\nabla\\varrho(f(\\zeta_0))\\rangle=\\rho(\\zeta_0)^{-1}|\\nabla\\varrho(f(\\zeta_0))|.\\end{multline}\nMoreover,\n\\begin{multline*}|\\nabla\\varrho(f(\\zeta_0))|= \\left\\langle\\nabla\\varrho(f(\\zeta_0)),\\frac{\\nabla\\varrho(f(\\zeta_0))}{|\\nabla\\varrho(f(\\zeta_0))|}\\right\\rangle_\\RR\n=\\langle\\nabla\\varrho(f(\\zeta_0)),\\nu_D(f(\\zeta_0))\\rangle_\\RR=\\\\=\\frac{\\partial\\varrho}{\\partial\\nu_D}(f(\\zeta_0))=\\lim_{t\\to 0}\\frac{\\varrho(f(\\zeta_0)+t\\nu_D(f(\\zeta_0)))-\\varrho(f(\\zeta_0))}{t}=\\frac{1}{\\varepsilon}=c,\n\\end{multline*} as $r(a+t\\nu(a))=t$ if $a\\in \\partial D$ and $t\\in\\mathbb R$ is small enough.\nThis, together with \\eqref{wk}, finishes the proof of the lower estimate.\n\\end{proof}\n\n\n\n\n\n\n\\begin{prop}\\label{8}\nLet $f:(\\DD,0)\\longrightarrow (D,z)$ be an $E$-mapping.\nThen $$|f(\\zeta_1)-f(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\CDD,$$ where $C$ is uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\\end{prop}\n\\begin{proof}\nLet $\\zeta_0\\in\\DD$ be such that $1-|\\zeta_0|<1\/(cC)$, where $C$ is as in Proposition~\\ref{7}. Then $B:=B_n(f(\\zeta_0),1\/c)$ intersects $\\partial D$. Take $\\Phi$ for the ball $B$ from the condition (4) of Definition~\\ref{30}. Let $w$ denote the nearest point to $\\Phi(f(\\zeta_0))$ lying on $\\partial\\Phi(D)$.  From the conditions (4)(b)-(c) of Definition~\\ref{30} we find that there is a uniform constant $r<1$ such that the point $w$ belongs to $\\Phi(B\\cap\\partial D)$ provided that $|\\zeta_0|\\geq r$.\n\nFrom the condition (4)(a) of Definition~\\ref{30} we get that there is $w_0$ such that $\\Phi(D)\\subset B_n(w_0,c)$ and the ball $B_n(w_0,c)$ is tangent to $\\Phi(D)$ at $w$. Let $$h(\\zeta):=(\\Phi\\circ f)\\left(\\frac{\\zeta_0-\\zeta}{1-\\overline{\\zeta_0}\\zeta}\\right),\\ \\zeta\\in\\DD.$$\nThen $h$ is holomorphic, $h(\\DD)\\subset B_n(w_0,c)$ and $h(0)=\\Phi(f(\\zeta_0))$. Using Lemma \\ref{schw} we get \\begin{multline*}|h'(0)|\\leq\\sqrt{c^2-|h(0)-w_0|^2}\\leq\\sqrt{2c(c-|\\Phi(f(\\zeta_0))-w_0|)}=\\\\\n=\\sqrt{2c(|w_0-w|-|\\Phi(f(\\zeta_0))-w_0|)}\\leq\\sqrt{2c}\\sqrt{|\\Phi(f(\\zeta_0))-w|}=\\\\\n=\\sqrt{2c}\\sqrt{\\dist(\\Phi(f(\\zeta_0)),\\partial\\Phi(D))}.\\end{multline*} Since $$h'(0)=\\Phi'(f(\\zeta_0))f'(\\zeta_0)\\left.\\frac{d}{d\\zeta}\\frac{\\zeta_0-\\zeta}{1-\\overline{\\zeta_0}\\zeta}\\right|_{\\zeta=0},$$ bby the condition (4)(c) of Definition~\\ref{3} we get $$|h'(0)|\\geq C_1|f'(\\zeta_0)|(1-|\\zeta_0|^2)$$ with a uniform $C_1$, so $$|f'(\\zeta_0)|\\leq\\frac{|h'(0)|}{C_1(1-|\\zeta_0|^2)}\\leq\\frac{\\sqrt{2c}}{C_1}\\frac{\\sqrt{\\dist(\\Phi(f(\\zeta_0)),\\partial\\Phi(D))}}{1-|\\zeta_0|^2}\\leq C_2\\frac{\\sqrt{\\dist(f(\\zeta_0),\\partial D)}}{1-|\\zeta_0|^2},$$ where $C_2$ is uniform. Combining with Proposition \\ref{7} \\begin{equation}\\label{46}|f'(\\zeta_0)|\\leq C_3\\frac{\\sqrt{1-|\\zeta_0|}}{1-|\\zeta_0|^2}=\\frac{C_3}{\\sqrt{1-|\\zeta_0|}},\\end{equation} where a constant $C_3$ is uniform.\n\nWe have shown that \\eqref{46} holds for $r\\leq |\\zeta_0|<1$ with a uniform $r<1$. For $|\\zeta_0|<r$ we estimate in the following way $$|f'(\\zeta_0)|\\leq\\max_{|\\zeta|=r}|f'(\\zeta)|\\leq\\frac{C_3}{\\sqrt{1-r}}\\leq\\frac{C_4}{\\sqrt{1-|\\zeta_0|}}$$ with a uniform $C_4:=C_3\/\\sqrt{1-r}$.\n\nUsing Theorems \\ref{lit1} and \\ref{lit2} with $\\alpha=1\/2$ we finish the proof.\n\\end{proof}\n\n\n\n\\begin{prop}\\label{10a}\nLet $f:(\\DD ,0)\\longrightarrow (D,z)$ be an $E$-mapping.\nThen $$|\\rho(\\zeta_1)-\\rho(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\TT,$$ where\n$C$ is uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\\end{prop}\n\\begin{proof}It suffices to prove that there exist uniform $C,C_1>0$ such that $$|\\rho(\\zeta_1)-\\rho(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\TT,\\ |\\zeta_1-\\zeta_2|<C_1.$$\n\nFix $\\zeta_1\\in\\TT$. Without loss of generality we may assume that $\\nu_{D,1}(f(\\zeta_1))=1$. Let $0<C_1\\leq 1\/4$ be uniform and such that $$|\\nu_{D,1}(f(\\zeta))-1|<1\/2,\\ \\zeta\\in\\TT\\cap B_n(\\zeta_1,3C_1).$$ It is possible, since by Proposition \\ref{8} $$|{\\nu_D(f(\\zeta))}-{\\nu_D(f(\\zeta'))}|\\leq 2c|f(\\zeta)-f(\\zeta')|\\leq C'\\sqrt{|\\zeta-\\zeta'|},\\ \\zeta,\\zeta'\\in\\TT,$$ with a uniform $C'>0$. There exists a function $\\psi\\in\\cC^1(\\TT,[0,1])$ such that $\\psi=1$ on $\\TT\\cap B_n(\\zeta_1,2C_1)$ and $\\psi=0$ on $\\TT\\setminus B_n(\\zeta_1,3C_1)$. Then the function $\\phi:\\TT\\longrightarrow\\CC$ defined by $$\\varphi:=(\\overline{\\nu_{D,1}\\circ f}-1)\\psi+1$$ satisfies\n\\begin{enumerate}\n\\item $\\varphi(\\zeta)=\\overline{\\nu_{D,1}(f(\\zeta))}$, $\\zeta\\in\\TT\\cap B_n(\\zeta_1,2C_1)$;\n\\item $|\\varphi(\\zeta)-1|<1\/2$, $\\zeta\\in\\TT$;\n\\item $\\phi$ is uniformly $1\/2$-H\\\"older continuous on $\\TT$, i.e. it is $1\/2$-H\\\"older continuous with a uniform constant (remember that $\\psi$ was chosen uniformly).\n\\end{enumerate}\n\nFirst observe that $\\log\\varphi$ is well-defined. Using using properties listed above we deduce that $\\log\\varphi$ and $\\im\\log\\varphi$ are uniformly $1\/2$-H\\\"older continuous on $\\TT$, as well. The function $\\im\\log\\varphi$ can be extended continuously to a function $v:\\CDD\\longrightarrow\\RR$, harmonic in $\\DD$. There is a function $h\\in\\mathcal O(\\DD)$ such that $v=\\im h$ in $\\DD$. Taking $h-\\re h(0)$ instead of $h$, one can assume that $\\re h(0)=0$. By Theorem \\ref{priv} applied to $ih$, we get that the function $h$ extends continuously on $\\CDD$ and $h$ is uniformly $1\/2$-H\\\"older continuous in $\\CDD$. Hence the function $u:=\\re h:\\CDD\\longrightarrow\\RR$ is uniformly $1\/2$-H\\\"older continuous in $\\CDD$ with a uniform constant $C_2$. Furthermore, $u$ is uniformly bounded in $\\CDD$, since $$|u(\\zeta)|=|u(\\zeta)-u(0)|\\leq C_2\\sqrt{|\\zeta|},\\ \\zeta\\in\\CDD.$$\n\nLet $g(\\zeta):=\\wi{f}_1(\\zeta)e^{-h(\\zeta)}$ and $G(\\zeta):=g(\\zeta)\/\\zeta$. Then $g\\in\\mathcal O(\\DD)\\cap\\mathcal C(\\overline{\\DD})$ and $G\\in\\mathcal O(\\DD_*)\\cap\\mathcal C((\\overline{\\DD})_*)$. Note that for $\\zeta\\in\\TT$ $$|g(\\zeta)|=|\\zeta\n\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-h(\\zeta)}|\\leq\\rho(\\zeta)e^{-u(\\zeta)},$$ which, combined with\nProposition \\ref{9}, the uniform boundedness of $u$ and the maximum principle, gives a uniform boundedness of $g$ in $\\CDD$. The function $G$ is uniformly bounded in $\\overline{\\DD}\\cap B_n(\\zeta_1,2C_1)$. Moreover, for $\\zeta\\in\\TT\\cap B_n(\\zeta_1,2C_1)$ \\begin{eqnarray*} G(\\zeta)&=&\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-u(\\zeta)-i\\im\\log \\phi(\\zeta)}=\\\\&=&\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-u(\\zeta)+\\re\\log\\phi(\\zeta)}e^{-\\log\\phi(\\zeta)}\n=\\rho(\\zeta)e^{-u(\\zeta)+\\re\\log\\phi(\\zeta)}\\in\\mathbb{R}.\\end{eqnarray*} By the Reflection Principle one can extend $G$ holomorphically past $\\TT\\cap B_n(\\zeta_1,2C_1)$ to a function (denoted by the same letter) uniformly bounded in $B_n(\\zeta_1,2C_2)$, where a constant $C_2$ is uniform. Hence, from the Cauchy formula, $G$ is uniformly Lipschitz continuous in $B_n(\\zeta_1,C_2)$, consequently uniformly $1\/2$-H\\\"older continuous in $B_n(\\zeta_1,C_2)$.\n\nFinally, the functions $G$, $h$, $\\nu_{D,1}\\circ f$ are uniformly $1\/2$-H\\\"older continuous on $\\TT\\cap B_n(\\zeta_1,C_2)$, $|\\nu_{D,1}\\circ f|>1\/2$ on $\\TT\\cap B_n(\\zeta_1,C_2)$, so the function $\\rho=Ge^h\/\\overline{\\nu_{D,1}\\circ f}$ is uniformly $1\/2$-H\\\"older continuous on $\\TT\\cap B_n(\\zeta_1,C_2)$.\n\\end{proof}\n\n\n\\begin{prop}\\label{10b}\nLet $f:(\\DD,0)\\longrightarrow (D,z)$ be an $E$-mapping.\nThen $$|\\wi{f}(\\zeta_1)-\\wi{f}(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\overline{\\DD},$$ where\n$C$ is uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\n\\end{prop}\n\\begin{proof}\nBy Propositions \\ref{8} and \\ref{10a} we have desired inequality for $\\zeta_1,\\zeta_2\\in\\TT$.  Theorem \\ref{lit2} finishes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Openness of $E$-mappings' set}\\label{27}\nWe shall show that perturbing a little a domain $D$ equipped with an $E$-mapping, we obtain a domain which also has an $E$-mapping, being close to a given one.\n\n\\subsection{Preliminary results}\n\n\\begin{propp}\\label{11}\nLet $f:\\mathbb{D}\\longrightarrow D$ be an $E$-mapping. Then there exist domains $G,\\wi D,\\wi G\\subset\\CC^n$ and a biholomorphism $\\Phi:\\wi D\\longrightarrow\\wi G$ such that\n\\begin{enumerate}\n\\item $\\wi D,\\wi G$ are neighborhoods of $\\overline D,\\overline G$ respectively;\n\\item $\\Phi(D)=G$;\n\\item $g(\\zeta):=\\Phi(f(\\zeta))=(\\zeta,0,\\ldots,0),\\ \\zeta\\in\\CDD$;\n\\item $\\nu_G(g(\\zeta))=(\\zeta,0,\\ldots,0),\\ \\zeta\\in\\TT$;\n\\item for any $\\zeta\\in\\TT$, a point $g(\\zeta)$ is a point of the strong linear convexity of $G$.\n\\end{enumerate}\n\\end{propp}\n\\begin{proof}\nLet $U,V$ be the sets from Proposition \\ref{34}. We claim that after a linear change of coordinates one can assume that $\\widetilde{f}_1,\\widetilde{f}_2$ do not have common zeroes in $V$.\n\nSince $ f'\\bullet\\widetilde{f}=1$, at least one of the functions $\\wi f_1,\\ldots,\\wi f_n$, say $\\wi f_1$, is not identically equal to $0$. Let $\\lambda_1,\\ldots,\\lambda_m$ be all zeroes of $\\wi f_1$ in $V$. We may find $\\alpha\\in\\CC^n$ such that $$(\\alpha_1\\wi f_1+\\ldots+\\alpha_n\\wi f_n)(\\lambda_j)\\neq 0,\\ j=1,\\ldots,m.$$ Otherwise, for any $\\alpha\\in\\CC^n$ there would exist $j\\in\\{1,\\ldots,m\\}$ such that $\\alpha\\bullet\\wi f(\\lambda_j)=0$, hence $$\\CC^n=\\bigcup_{j=1}^m\\{\\alpha\\in\\CC^n:\\ \\alpha\\bullet\\wi f(\\lambda_j)=0\\}.$$ The sets $\\{\\alpha\\in\\CC^n:\\alpha \\bullet \\wi f(\\lambda_j)=0\\}$, $j=1,\\ldots,m$, are the $(n-1)$-dimensional complex hyperplanes, so their finite sum cannot be the space $\\CC^n$.\n\nOf course, at least one of the numbers $\\alpha_2,\\ldots,\\alpha_n$, say $\\alpha_2$, is non-zero. Let\n$$A:=\\left[\\begin{matrix}\n1 & 0 & 0 & \\cdots & 0\\\\\n\\alpha_1 & \\alpha_2 & \\alpha_3 &\\cdots & \\alpha_n\\\\\n0 & 0 & 1 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\vdots &\\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 1\n\\end{matrix}\\right],\\quad B:=(A^T)^{-1}.$$ We claim that $B$ is a change of coordinates we are looking for. If $r$ is a defining function of $D$ then $r\\circ B^{-1}$ is a defining function of $B_n(D)$, so $B_n(D)$ is a bounded strongly linearly convex domain with real analytic boundary. Let us check that $Bf$ is an $E$-mapping of $B_n(D)$ with associated mappings \\begin{equation}\\label{56}A\\wi f\\in\\OO(\\CDD)\\text{\\ \\  and\\ \\ }\\rho\\frac{|A\\overline{\\nabla r\\circ f}|}{|\\nabla r\\circ f|}\\in\\CLW(\\TT).\\end{equation} The conditions (1) and (2) of Definition~\\ref{21} are clear. For $\\zeta\\in\\TT$ we have \\begin{equation}\\label{57}\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}=\\frac{\\overline{\\nabla(r\\circ B^{-1})(Bf(\\zeta))}}{|\\nabla(r\\circ B^{-1})(Bf(\\zeta))|}=\\frac{(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}=\\frac{A\\overline{\\nabla r(f(\\zeta))}}{|A\\overline{\\nabla r(f(\\zeta))}|},\\end{equation} so\n\\begin{equation}\\label{58}\\zeta\\rho(\\zeta)\\frac{|A\\overline{\\nabla r(f(\\zeta))}|}{|\\nabla r(f(\\zeta))|}\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}=\\zeta\\rho(\\zeta)A\\overline{\\nu_D(f(\\zeta))}=A\\wi f(\\zeta).\\end{equation} Moreover, for $\\zeta\\in\\TT$, $z\\in D$ \\begin{multline*}\\langle Bz-Bf(\\zeta), \\nu_{B_n(D)}(Bf(\\zeta))\\rangle=\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}^T(Bz-Bf(\\zeta))=\\\\=\\frac{\\overline{\\nabla r(f(\\zeta))}^TB^{-1}B_n(z-f(\\zeta))}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}=\\frac{|\\nabla r(f(\\zeta))|}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}\\overline{\\nu_D(f(\\zeta))}^T(z-f(\\zeta))=\\\\=\\frac{|\\nabla r(f(\\zeta))|}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}\\langle z-f(\\zeta), \\nu_D(f(\\zeta))\\rangle.\n\\end{multline*}\nTherefore, $B$ is a desired linear change of coordinates, as claimed.\n\nIf necessary, we shrink the sets $U,V$ associated with $f$ to sets associated with $Bf$. There exist holomorphic mappings $h_1,h_2:V\\longrightarrow\\mathbb{C}$ such that\n$$h_1\\widetilde{f}_1+h_2\\widetilde{f}_2\\equiv 1\\text{ in }V.$$ Generally, it is a well-known fact for functions on pseudoconvex domains, however in this case it may be shown quite elementarily. Indeed, if $\\widetilde{f}_1\\equiv 0$ or $\\widetilde{f}_2\\equiv 0$ then it is obvious. In the opposite case, let $\\widetilde{f}_j=F_jP_j$, $j=1,2$, where $F_j$ are holomorphic, non-zero in $V$ and $P_j$ are polynomials with all (finitely many) zeroes in $V$. Then $P_j$ are relatively prime, so there are polynomials $Q_j$, $j=1,2$, such that $$Q_1P_1+Q_2P_2\\equiv 1.$$ Hence $$\\frac{Q_1}{F_1}\\widetilde{f}_1+\\frac{Q_2}{F_2}\\widetilde{f}_2\\equiv 1\\ \\text{ in }V.$$\n\nConsider the mapping $\\Psi:V\\times\\mathbb{C}^{n-1}\\longrightarrow\\mathbb{C}^n$ given by\n\\begin{equation}\\label{et2}\n\\Psi_1(Z):=f_1(Z_1)-Z_2\\widetilde{f}_2(Z_1)-h_1(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1),\n\\end{equation}\n\\begin{equation}\\label{et3}\n\\Psi_2(Z):=f_2(Z_1)+Z_2\\widetilde{f}_1(Z_1)-h_2(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1),\n\\end{equation}\n\\begin{equation}\\label{et4}\n\\Psi_j(Z):=f_j(Z_1)+Z_j,\\ j=3,\\ldots,n.\n\\end{equation}\n\nWe claim that $\\Psi$ is biholomorphic in $\\Psi^{-1}(U)$. First of all observe that $\\Psi^{-1}(\\{z\\})\\neq\\emptyset$ for any $z\\in U$. Indeed, by Proposition \\ref{34} there exists (exactly one) $Z_1\\in V$ such that $$(z-f(Z_1))\\bullet\\widetilde{f}(Z_1)=0.$$ The numbers $Z_j\\in\\CC$, $j=3,\\ldots,n$ are determined uniquely by the equations $$Z_j=z_j-f_j(Z_1).$$ At least one of the numbers $\\wi f_1(Z_1),\\wi f_2(Z_1)$, say $\\wi f_1(Z_1)$, is non-zero. Let $$Z_2:=\\frac{z_2-f_2(Z_1)+h_2(Z_1)\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1)}{\\wi f_1(Z_1)}.$$ Then we easily check that the equality $$z_1=f_1(Z_1)-Z_2\\widetilde{f}_2(Z_1)-h_1(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1)$$ is equivalent to $(z-f(Z_1))\\bullet\\widetilde{f}(Z_1)=0$, which is true.\n\nTo finish the proof of biholomorphicity of $\\Psi$ in $\\Psi^{-1}(U)$ it suffices to check that $\\Psi$ is injective in $\\Psi^{-1}(U)$. Let us take $Z,W$ such that $\\Psi(Z)=\\Psi(W)=z\\in U$. By a direct computation both $\\zeta=Z_1\\in V$ and $\\zeta=W_1\\in V$ solve the equation\n$$(z-f(\\zeta))\\bullet\\widetilde{f}(\\zeta)=0.$$ From Proposition \\ref{34} we infer that it has exactly one solution. Hence $Z_1=W_1$. By \\eqref{et4} we have $Z_j=W_j$ for $j=3,\\ldots,n$. Finally $Z_2=W_2$ follows from\none of the equations \\eqref{et2}, \\eqref{et3}. Let $G:=\\Psi^{-1}(D)$, $\\wi D:=U$, $\\wi G:=\\Psi^{-1}(U)$, $\\Phi:=\\Psi^{-1}$.\n\nNow we are proving that $\\Phi$ has desired properties. We have $$\\Psi_j(\\zeta,0,\\ldots,0)=f_j(\\zeta),\\ j=1,\\ldots,n,$$ so $\\Phi(f(\\zeta))=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\CDD$. Put $g(\\zeta):=\\Phi(f(\\zeta))$, $\\zeta\\in\\CDD$. Note that the entries of the matrix $\\Psi'(g(\\zeta))$ are $$\\frac{\\partial\\Psi_1}{\\partial Z_1}(g(\\zeta))=f_1'(\\zeta),\\ \\frac{\\partial\\Psi_1}{\\partial Z_2}(g(\\zeta))=-\\widetilde{f}_2(\\zeta),\\ \\frac{\\partial\\Psi_1}{\\partial Z_j}(g(\\zeta))=-h_1(\\zeta)\\widetilde{f}_j(\\zeta),\\ j\\geq 3,$$$$\\frac{\\partial\\Psi_2}{\\partial Z_1}(g(\\zeta))=f_2'(\\zeta),\\ \\frac{\\partial\\Psi_2}{\\partial Z_2}(g(\\zeta))=\\widetilde{f}_1(\\zeta),\\ \\frac{\\partial\\Psi_2}{\\partial Z_j}(g(\\zeta))=-h_2(\\zeta)\\widetilde{f}_j(\\zeta),\\ j\\geq 3,$$$$\\frac{\\partial\\Psi_k}{\\partial Z_1}(g(\\zeta))=f_k'(\\zeta),\\ \\frac{\\partial\\Psi_k}{\\partial Z_2}(g(\\zeta))=0,\\ \\frac{\\partial\\Psi_k}{\\partial Z_j}(g(\\zeta))=\\delta^{k}_{j},\\ j,k\\geq 3.$$ Thus $\\Psi '(g(\\zeta))^T\\wi f(\\zeta)=(1,0,\\ldots,0)$, $\\zeta\\in\\CDD$ (since $f'\\bullet\\wi f=1$). Let us take a defining function $r$ of $D$. Then $r\\circ\\Psi$ is a defining function of $G$. Therefore, \\begin{multline*}\\nu_G(g(\\zeta))=\\frac{\\nabla(r\\circ\\Psi)(g(\\zeta))}{|\\nabla(r\\circ\\Psi)(g(\\zeta))|}=\n\\frac{\\overline{\\Psi'(g(\\zeta))}^T\\nabla r(f(\\zeta))}{|\\overline{\\Psi'(g(\\zeta))}^T\\nabla r(f(\\zeta))|}=\\\\=\\frac{\\overline{\\Psi'(g(\\zeta))}^T\\ov{\\frac{\\wi f(\\zeta)}{\\zeta\\rho(\\zeta)}}|\\nabla r(f(\\zeta))|}{\\left|\\overline{\\Psi'(g(\\zeta))}^T\\ov{\\frac{\\wi f(\\zeta)}{\\zeta\\rho(\\zeta)}}|\\nabla r(f(\\zeta))|\\right|}=g(\\zeta),\\ \\zeta\\in\\TT.\\end{multline*}\n\nIt remains to prove the fifth condition. By Definition \\ref{29}(2) we have to show that \\begin{equation}\\label{sgf}\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial\\overline{z}_k}(g(\\zeta))X_{j}\\overline{X}_{k}>\\left|\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|\\end{equation} for $\\zeta\\in\\TT$ and $X\\in(\\CC^{n})_*$ with\n$$\\sum_{j=1}^n\\frac{\\partial(r\\circ\\Psi)}{\\partial z_j}(g(\\zeta))X_{j}=0,$$ i.e. $X_1=0$. We have $$\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial\\overline{z}_k}(g(\\zeta))X_{j}\\overline{X}_{k}=\\sum_{j,k,s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))\\overline{\\frac{\\partial\\Psi_t}{\\partial z_k}(g(\\zeta))}X_{j}\\overline{X}_{k}=$$$$=\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))Y_{s}\\overline{Y}_{t},$$ where $$Y:=\\Psi'(g(\\zeta))X.$$ Note that $Y\\neq 0$. Additionally $$\\sum_{s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))Y_{s}=\\sum_{j,s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))X_j=\\sum_{j=1}^n\\frac{\\partial(r\\circ\\Psi)}{\\partial z_j}(g(\\zeta))X_{j}=0.$$ Therefore, by the strong linear convexity of $D$ at $f(\\zeta)$ $$\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))Y_{s}\\overline{Y}_{t}>\\left|\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))Y_{s}Y_{t}\\right|.$$ To finish the proof observe that $$\\left|\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|=\\left|\\sum_{j,k,s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))\\frac{\\partial\\Psi_t}{\\partial z_k}(g(\\zeta))X_{j}X_{k}+\\right.$$$$\\left.+\\sum_{j,k,s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|=$$$$=\\left|\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))Y_{s}Y_{t}+\\sum_{j,k=2}^n\\sum_{s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|$$ and $$\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))=0,\\ j,k\\geq 2,\\ s\\geq 1,$$ which gives \\eqref{sgf}.\n\\end{proof}\n\n\\begin{remm}\\label{rem:theta}\nLet $D$ be a bounded domain in $\\mathbb C^n$ and let $f:\\DD\\longrightarrow D$ be a (weak) stationary mapping such that $\\partial D$ is real analytic in a neighborhood of $f(\\TT)$. Assume moreover that there are a neighborhood $U$ of $f(\\CDD)$ and a mapping $\\Theta:U\\longrightarrow\\CC^n$ biholomorphic onto its image and the set $D\\cap U$ is connected. Then $\\Theta\\circ f$ is a (weak) stationary mapping of $G:=\\Theta(D\\cap U)$.\n\nIn particular, if $U_1$, $U_2$ are neighborhoods of the closures of domains $D_1$, $D_2$ with real analytic boundaries and  $\\Theta:U_1\\longrightarrow U_2$ is a biholomorphism such that $\\Theta(D_1)=D_2$, then $\\Theta$ maps (weak) stationary mappings of $D_1$ onto (weak) stationary mappings of $D_2$.\n\\end{remm}\n\\begin{proof}\nActually, it is clear that two first conditions of the definition of (weak) stationary mappings are preserved by $\\Theta$. To show the third one we proceed similarly as in the equations \\eqref{56}, \\eqref{57}, \\eqref{58}. Let $f:\\DD\\longrightarrow D $ be a (weak) stationary mapping. The candidates for the mappings in condition (3) (resp. (3')) of Definition~\\ref{21} for $\\Theta\\circ f$ in the domain $G$ are $$((\\Theta'\\circ f)^{-1})^T\\wi f\\text{\\ \\  and\\ \\ }\\rho\\frac{|((\\Theta'\\circ f)^{-1})^T\\overline{\\nabla r\\circ f}|}{|\\nabla r\\circ f|}.$$ Indeed, for $\\zeta\\in\\TT$ \\begin{multline*}\\overline{\\nu_{G}(\\Theta(f(\\zeta)))}=\n\\frac{\\overline{\\nabla(r\\circ\\Theta^{-1})(\\Theta(f(\\zeta)))}}{|\\nabla(r\\circ\\Theta^{-1})(\\Theta(f(\\zeta)))|}=\\frac{[(\\Theta^{-1})'(\\Theta(f(\\zeta)))]^T\\overline{\\nabla r(f(\\zeta))}}{|[(\\Theta^{-1})'(\\Theta(f(\\zeta)))]^T\\overline{\\nabla r(f(\\zeta))}|}=\\\\\n=\\frac{(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}}{|(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}|},\n\\end{multline*}\nhence\n\\begin{multline*}\\zeta\\rho(\\zeta)\\frac{|(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}{|\\nabla r(f(\\zeta))|}\\overline{\\nu_{G}(\\Theta(f(\\zeta)))}=\\\\\n=\\zeta\\rho(\\zeta)(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nu_{D}(f(\\zeta))}=\n(\\Theta'(f(\\zeta))^{-1})^T\\wi f(\\zeta).\n\\end{multline*}\n\\end{proof}\n\n\n\\subsection{Situation (\\dag)}\\label{dag}\nConsider the following situation, denoted by (\\dag) (with data $D_0$ and $U_0$):\n\\begin{itemize}\n\\item $D_0$ is a bounded domain in $\\CC^n$, $n\\geq 2$;\n\\item $f_0:\\CDD\\ni\\zeta\\longmapsto(\\zeta,0,\\ldots,0)\\in\\ov D_0$, $\\zeta\\in\\CDD$;\n\\item $f_0(\\DD)\\subset D_0$;\n\\item $f_0(\\TT)\\subset\\partial D_0$;\n\\item $\\nu_{D_0}(f_0(\\zeta))=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\TT$;\n\\item for any $\\zeta\\in\\TT$, a point $f_0(\\zeta)$ is a point of the strong linear convexity of $D_0$;\n\\item $\\partial D_0$ is real analytic in a neighborhood $U_0$ of $f_0(\\TT)$ with a function $r_0$;\n\\item $|\\nabla r_0|=1$ on $f_0(\\TT)$ (in particular, $r_{0z}(f_0(\\zeta))=(\\ov\\zeta\/2,0,\\ldots,0)$, $\\zeta\\in\\TT$).\n\\end{itemize}\n\nSince $r_0$ is real analytic on $U_0\\su\\RR^{2n}$, it extends in a natural way to a holomorphic function in a neighborhood $U_0^\\CC\\su\\mb{C}^{2n}$ of $U_0$. Without loss of generality we may assume that $r_0$ is bounded on $U_0^\\CC$. Set $$X_0=X_0(U_0,U_0^{\\mathbb C}):=\\{r\\in\\mc{O}(U_0^\\CC):\\text{$r(U_0)\\su\\mb{R}$ and $r$ is bounded}\\},$$ which equipped with the sup-norm is a (real) Banach space.\n\n\\begin{remm} Lempert considered the case when $U_0$ is a neighborhood of a boundary of a bounded domain $D_0$ with real analytic boundary. We shall need more general results to prove the `localization property'.\n\\end{remm}\n\n\\subsection{General lemmas}\\label{General lemmas}\nWe keep the notation from Subsection \\eqref{dag} and assume Situation (\\dag).\n\nLet us introduce some additional objects we shall be dealing with and let us prove more general lemmas (its generality will be useful in the next section).\n\nConsider the Sobolev space $W^{2,2}(\\TT)=W^{2,2}(\\TT,\\CC^m)$ of functions $f:\\TT\\longrightarrow\\CC^m$, whose first two derivatives (in the sense of distribution) are in $L^2(\\TT)$. The $W^{2,2}$-norm is denoted by $\\|\\cdot\\|_W$. For the basic properties of $W^{2,2}(\\TT)$ see Appendix.\n\nPut $$B:=\\{f\\in W^{2,2}(\\TT,\\CC^n):f\\text{ extends holomorphically on $\\mb{D}$ and $f(0)=0$}\\},$$$$B_0:=\\{f\\in B:f(\\TT)\\su U_0\\},\\quad B^*:=\\{\\overline{f}:f\\in B\\},$$$$Q:=\\{q\\in W^{2,2}(\\TT,\\CC):q(\\TT)\\su\\RR\\},\\quad Q_0:=\\{q\\in Q:q(1)=0\\}.$$\n\nIt is clear that $B$, $B^*$, $Q$ and $Q_0$ equipped with the norm $\\|\\cdot\\|_W$ are (real) Banach spaces. Note that $B_0$ is an open neighborhood of $f_0$. In what follows, we identify $f\\in B$ with its unique holomorphic extension on $\\mb{D}$.\n\nLet us define the projection $$\\pi:W^{2,2}(\\TT,\\CC^n)\\ni f=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}\\longmapsto\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}\\in{B^*}.$$ Note that $f\\in W^{2,2}(\\TT,\\CC^n)$ extends holomorphically on $\\mb{D}$ if and only if $\\pi(f)=0$ (and the extension is $\\mathcal C^{1\/2}$ on $\\TT$). Actually, it suffices to observe that\n$g(\\zeta):=\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}$, $\\zeta\\in\\TT$, extends holomorphically on $\\DD$ if and only if $a_k=0$ for $k<0$. This follows immediately from the fact that the mapping $\\TT\\ni\\zeta\\longmapsto g(\\ov\\zeta)\\in\\CC^n$ extends holomorphically on $\\DD$.\n\nConsider the mapping $\\Xi:X_0\\times\\mb{C}^n\\times B_0\\times\nQ_0\\times\\mb{R}\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$ defined by\n$$\\Xi(r,v,f,q,\\lambda):=(r\\circ f,\\pi(\\zeta(1+q)(r_z\\circ f)),f'(0)-\\lambda v),$$ where $\\zeta$ is treated as the identity function on $\\TT$.\n\n\nWe have the following\n\n\\begin{lemm}\\label{cruciallemma} There exist a neighborhood $V_0$ of $(r_0,f_0'(0))$ in $X_0\\times\\mb{C}^n$ and a real analytic mapping $\\Upsilon:V_0\\longrightarrow B_0\\times Q_0\\times\\mb{R}$ such that for any $(r,v)\\in V_0$ we have $\\Xi(r,v,\\Upsilon(r,v))=0$.\n\\end{lemm}\n\\bigskip\nLet $\\wi\\Xi:X_0\\times\\mb{C}^n\\times B_0\\times Q_0\\times(0,1)\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$ be defined as $$\\wi\\Xi(r,w,f,q,\\xi):=(r\\circ f,\\pi(\\zeta(1+q)(r_z\\circ f)),f(\\xi)-w).$$\n\n\nAnalogously we have\n\\begin{lemm}\\label{cruciallemma1} Let $\\xi_0\\in(0,1)$. Then there exist a neighborhood $W_0$ of $(r_0,f_0(\\xi_0))$ in $X_0\\times D_0$ and a real analytic mapping $\\wi\\Upsilon:W_0\\longrightarrow B_0\\times Q_0\\times(0,1)$ such that for any $(r,w)\\in W_0$ we have $\\wi\\Xi(r,w,\\wi\\Upsilon(r,w))=0$.\n\\end{lemm}\n\n\n\n\\begin{proof}[Proof of Lemmas \\ref{cruciallemma} and \\ref{cruciallemma1}]\n\n\nWe will prove the first lemma. Then we will see that a proof of the second one reduces to that proof.\n\nWe claim that $\\Xi$ is real analytic. The only problem is to show that the mapping $$T: X_0\\times B_0\\ni(r,f)\\longmapsto r\\circ f\\in Q$$ is real analytic (the real analyticity of the mapping $X_0\\times B_0\\ni(r,f)\\longmapsto r_z\\circ f\\in W^{2,2}(\\TT,\\CC^n)$ follows from this claim).\n\nFix $r\\in X_0$, $f\\in B_0$ and take $\\eps>0$ so that a $2n$-dimensional polydisc $P_{2n}(f(\\zeta),\\eps)$ is contained in $U_0^\\CC$ for any $\\zeta\\in\\TT$. Then any function $\\wi r\\in X_0$ is holomorphic in $U_0^\\CC$, so it may be expanded as a holomorphic series convergent in $P_{2n}(f(\\zeta),\\eps)$. Losing no generality we may assume that $n$-dimensional polydiscs $P_{n}(f(\\zeta),\\eps)$, $\\zeta\\in\\TT$, satisfy $P_{n}(f(\\zeta),\\eps)\\su U_0$. This gives an expansion of the function $\\wi r$ at any point $f(\\zeta)$, $\\zeta\\in\\TT$, into a series $$\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}(f(\\zeta))x^\\alpha$$ convergent to $\\wi r(f(\\zeta)+x)$, provided that $x=(x_1,\\ldots,x_{2n})\\in P_n(0,\\eps)$ (where $\\NN_0:=\\NN\\cup\\{0\\}$ and $|\\alpha|:=\\alpha_1+\\ldots+\\alpha_{2n}$). Hence \\begin{equation}\\label{69}T(r+\\varrho,f+h)=\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}r}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha+\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}\\varrho}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha\\end{equation} pointwise for $\\varrho\\in X_0$ and $h\\in W^{2,2}(\\TT,\\CC^n)$ with $\\|h\\|_{\\sup}<\\eps$.\n\nPut $P:=\\bigcup_{\\zeta\\in \\TT} P_{2n}(f(\\zeta),\\eps)$ and for $\\wi r\\in X_0$ put $||\\wi r||_P:=\\sup_P|\\wi r|$. Let $\\wi r$ be equal to $r$ or to $\\varrho$, where $\\varrho$ lies is in a neighborhood of $0$ in $X_0$. The Cauchy inequalities give\n\\begin{equation}\\label{series}\\left|\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}(f(\\zeta))\\right|\\leq\\frac{\\alpha!\\|\\wi r\\|_{P}}{\\eps^{|\\alpha|}},\\quad\\zeta\\in\\TT.\\end{equation}\nTherefore, $$\\left|\\left|\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}\\circ f\\right|\\right|_W\\leq C_1\\frac{\\alpha!\\|\\wi r\\|_{P}}{\\eps^{|\\alpha|}}$$ for some $C_1>0$.\n\nThere is $C_2>0$ such that $$\\|gh^\\alpha\\|_W\\leq C_2^{|\\alpha|+1}\\|g\\|_W\\|h_1\\|^{\\alpha_1}_W\\cdotp\\ldots\\cdotp\\|h_{2n}\\|^{\\alpha_{2n}}_W$$ for $g\\in W^{2,2}(\\TT,\\CC)$, $h\\in W^{2,2}(\\TT,\\CC^n)$, $\\alpha\\in\\NN_0^{2n}$ (see Appendix for a proof of this fact). Using the above inequalities we infer that $$\\sum_{\\alpha\\in\\NN_0^{2n}}\\left|\\left|\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha\\right|\\right|_W$$ is convergent if $h$ is small enough on the norm $\\|\\cdot\\|_W$. Therefore, the series~\\eqref{69} is absolutely convergent in the norm $\\|\\cdot\\|_W$, whence $T$ is real analytic.\n\n\nTo show the existence of $V_0$ and $\\Upsilon$ we will make use of the Implicit Function Theorem. More precisely, we shall show that the partial derivative $$\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1):B\\times Q_0\\times\\mb{R}\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$$ is an isomorphism.\nObserve that for any $(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})\\in B\\times Q_0\\times\\mb{R}$ the following equality holds\n\\begin{multline*}\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1)(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})=\\left.\\frac{d}{dt}\n\\Xi(r_0,f_0'(0),f_0+t\\widetilde{f},t\\widetilde{q},1+t\\widetilde{\\lambda})\\right|_{t=0}=\\\\\n=((r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}},\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta(r_{0zz} \\circ\nf_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}),\\widetilde{f}'(0)-\\widetilde{\\lambda}f_0'(0)),\n\\end{multline*}\nwhere we treat ${r_0}_z,{r_0}_{\\overline{z}}$ as row vectors, $\\widetilde{f},\\overline{\\widetilde{f}}$ as column vectors and $r_{0zz}=\\left[\\frac{\\partial^2r_0}{\\partial z_j\\partial z_k}\\right]_{j,k=1}^n$, $r_{0z\\overline{z}}=\\left[\\frac{\\partial^2r_0}{\\partial z_j\\partial\\overline z_k}\\right]_{j,k=1}^n$ as $n\\times n$ matrices.\n\nBy the Bounded Inverse Theorem it suffices to show that $\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1)$ is bijective, i.e. for $(\\eta,\\varphi,v)\\in Q\\times B^*\\times\\mb{C}^n$ there exists exactly one $(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})\\in B\\times Q_0\\times\\mb{R}$ satisfying\n\\begin{equation}\n(r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}=\\eta,\n\\label{al1}\n\\end{equation}\n\\begin{equation}\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta (r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}})=\\varphi,\n\\label{al2}\n\\end{equation}\n\\begin{equation}\n\\widetilde{f}'(0)-\\widetilde{\\lambda} f_0'(0)=v.\n\\label{al3}\n\\end{equation}\nFirst we show that $\\wi\\lambda$ and $\\wi f_1$ are uniquely determined. Observe that, in view of assumptions, (\\ref{al1}) is just $$\\frac{1}{2}\\overline{\\zeta}\\widetilde{f}_1+\\frac{1}{2}\\zeta\\overline{\\widetilde{f}_1}=\\eta$$ or equivalently\n\\begin{equation}\n\\re(\\widetilde{f}_1\/\\zeta)=\\eta\\text{ (on }\\TT).\n\\label{al4}\n\\end{equation}\nNote that the equation (\\ref{al4}) uniquely determines $\\widetilde{f}_1\/\\zeta\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ up to an imaginary additive constant, which may be computed using (\\ref{al3}). Actually, $\\eta=\\re G$ on $\\TT$ for some function $G\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$. To see this, let us expand $\\eta(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}$, $\\zeta\\in\\TT$. From the equality $\\eta(\\zeta)=\\ov{\\eta(\\zeta)}$, $\\zeta\\in\\TT$, we get \\begin{equation}\\label{65}\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}=\\sum_{k=-\\infty}^{\\infty}\\ov a_k\\zeta^{-k}=\\sum_{k=-\\infty}^{\\infty}\\ov a_{-k}\\zeta^{k},\\ \\zeta\\in\\TT,\\end{equation} so $a_{-k}=\\ov a_k$, $k\\in\\ZZ$. Hence $$\\eta(\\zeta)=a_0+\\sum_{k=1}^\\infty 2\\re(a_k\\zeta^k)=\\re\\left(a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k\\right),\\ \\zeta\\in\\TT.$$ Set $$G(\\zeta):=a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k,\\ \\zeta\\in\\DD.$$ This series is convergent for $\\zeta\\in\\DD$, so $G\\in\\OO(\\DD)$. Further, the function $G$ extends continuously on $\\CDD$ (to the function denoted by the same letter) and the extension lies in $W^{2,2}(\\TT,\\CC)$. Clearly, $\\eta=\\re G$ on $\\TT$.\n\nWe are searching $C\\in\\RR$ such that the functions $\\widetilde{f}_1:=\\zeta(G+iC)$ and $\\theta:=\\im(\\widetilde{f}_1\/\\zeta)$ satisfy $$\\eta(0)+i\\theta(0)=\\widetilde{f}_1'(0)$$ and\n$$\\eta(0)+i\\theta(0)-\\widetilde{\\lambda}\\re{f_{01}'(0)}-i\\widetilde{\\lambda}\\im{{f_{01}'(0)}}=\\re{v_1}+i\\im{v_1}.$$ But $$\\eta(0)-\\widetilde{\\lambda}\\re{f_{01}'(0)}=\\re{v_1},$$ which yields $\\widetilde{\\lambda}$ and then $\\theta(0)$, consequently the number $C$.\nHaving $\\widetilde{\\lambda}$ and once again using (\\ref{al3}), we find uniquely determined $\\widetilde{f}_2'(0),\\ldots,\\widetilde{f}_n'(0)$.\n\nTherefore, the equations $\\eqref{al1}$ and $\\eqref{al3}$ are satisfied by uniquely determined $\\wi f_1$, $\\wi\\lambda$ and $\\widetilde{f}_2'(0),\\ldots,\\widetilde{f}_n'(0)$.\n\nConsider (\\ref{al2}), which is the system of $n$ equations with unknown $\\widetilde{q},\\widetilde{f}_2,\\ldots,\\widetilde{f}_n$. Observe that $\\widetilde{q}$ appears only in the first of the equations and the remaining $n-1$ equations mean exactly that the mapping\n\\begin{equation}\n\\zeta(r_{0\\widehat{z}\\widehat{z}}\\circ f_0)\n\\widehat{\\widetilde{f}}+\\zeta(r_{0\\widehat{z}\\widehat{\\overline{z}}}\\circ f_0)\\widehat{\\overline{\\widetilde{f}}}-\\psi\n\\label{al5}\n\\end{equation}\nextends holomorphically on $\\mb{D}$, where $\\widehat{a}:=(a_{2},\\ldots,a_{n})$ and $\\psi\\in W^{2,2}(\\TT,\\mb{C}^{n-1})$ may be obtained from $\\varphi$ and $\\widetilde{f}_1$. Indeed, to see this, write (\\ref{al2}) in the form $$\\pi(F_{1}+\\zeta F_{2}+\\zeta F_{3})=(\\phi_1,\\ldots,\\phi_n),$$ where $$F_1:=(\\wi q,0,\\ldots,0),$$$$F_2:=(A_{j})_{j=1}^n,\\ A_{j}:=\\sum\\limits_{k=1}^n(r_{0z_jz_k}\\circ f_0)\\widetilde{f}_k,$$$$F_3=(B_{j})_{j=1}^n,\\ B_{j}:=\\sum\\limits_{k=1}^n(r_{0z_j\\ov z_k}\\circ f_0)\\overline{\\widetilde{f}_k}.$$ It follows that $$\\widetilde{q}+\\zeta A_1+\\zeta B_1-\\phi_1$$ and $$\\zeta A_j+\\zeta B_j-\\phi_j,\\ j=2,\\ldots,n,$$ extend holomorphically on $\\mb{D}$ and $$\\psi:=\\left(\\phi_j-\\zeta(r_{0z_jz_1}\\circ f_0)\\widetilde{f}_1-\\zeta(r_{0z_j\\ov z_1}\\circ f_0)\\overline{\\widetilde{f}_1}\\right)_{j=2}^n.$$\nPut $$g(\\zeta):=\\widehat{\\widetilde{f}}(\\zeta)\/\\zeta,\\quad\\alpha(\\zeta):=\\zeta^2r_{0\\widehat{z}\\widehat{z}}(f_0(\\zeta)),\n\\quad\\beta(\\zeta):=r_{0\\widehat{z}\\widehat{\\overline{z}}}(f_0(\\zeta)).$$\n\nObserve that $\\alpha(\\zeta)$, $\\beta(\\zeta)$ are the $(n-1)\\times(n-1)$ matrices depending real analytically on $\\zeta$ and $g(\\zeta)$ is a column vector in $\\mb{C}^{n-1}$. This allows us to reduce \\eqref{al5} to the following problem: we have to find a unique $g\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ such that \\begin{equation}\n\\alpha g+\\beta\\overline{g}-\\psi\\text{ extends holomorphically on $\\mb{D}$ and } g(0)={\\widehat{\\widetilde{f}'}}(0).\n\\label{al6}\n\\end{equation}\nThe fact that every $f_0(\\zeta)$ is a point of strong linear convexity of the domain $D_0$ may be written as\n\\begin{equation}\n|X^T\\alpha(\\zeta)X|<X^{T}\\beta(\\zeta)\\overline{X},\\ \\zeta\\in\\TT,\\ X\\in(\\mb{C}^{n-1})_*.\n\\label{al7}\n\\end{equation}\n\nNote that $\\beta(\\zeta)$ is self-adjoint and strictly positive, hence using Proposition \\ref{12} we get a mapping $H\\in\\OO(\\CDD,\\CC^{(n-1)\\times(n-1)})$ such that $\\det H\\neq 0$ on $\\CDD$ and $HH^*=\\beta$ on $\\TT$. Using this notation, (\\ref{al6}) is\nequivalent to\n\\begin{equation}\nH^{-1}\\alpha g+H^*\\overline{g}-H^{-1}\\psi\\text{ extends holomorphically on $\\mb{D}$}\n\\label{al8}\n\\end{equation}\nor, if we denote $h:=H^Tg$, $\\gamma:=H^{-1}\\alpha (H^T)^{-1}$, to\n\\begin{equation}\n\\gamma h+\\overline{h}-H^{-1}\\psi\\text{ extends holomorphically on $\\mb{D}.$}\n\\label{al9}\n\\end{equation}\n\nFor any $\\zeta\\in\\TT$ the operator norm of the symmetric matrix $\\gamma(\\zeta)$ is uniformly less than 1. In fact, from (\\ref{al7}) for any $X\\in\\mb{C}^{n-1}$ with $|X|=1$ \\begin{multline*}|X^{T}\\gamma(\\zeta)X|=|X^{T}H(\\zeta)^{-1}\\alpha(\\zeta)(H(\\zeta)^T)^{-1}X|<X^TH(\\zeta)^{-1}\\beta(\\zeta)\n\\overline{(H(\\zeta)^T)^{-1}X}=\\\\=X^TH(\\zeta)^{-1}H(\\zeta)H(\\zeta)^*\\overline{(H(\\zeta)^T)^{-1}}\n\\overline{X}=|X|^2=1,\\end{multline*} so, by the compactness argument, $|X^{T}\\gamma(\\zeta)X|\\leq 1-\\wi\\eps$ for some $\\wi\\eps>0$ independent on $\\zeta$ and $X$. Thus $\\|\\gamma(\\zeta)\\|\\leq 1-\\wi\\eps$ by Proposition \\ref{59}.\n\nWe have to prove that there is a unique solution $h\\in W^{2,2}(\\TT,\\CC^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ of (\\ref{al9}) such that $h(0)=a$ with a given $a\\in\\CC^{n-1}$.\n\nDefine the operator $$P:W^{2,2}(\\TT,\\mb{C}^{n-1})\\ni\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}\\longmapsto\\overline{\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}}\\in W^{2,2}(\\TT,\\mb{C}^{n-1}),$$ where $a_k\\in\\CC^{n-1}$, $k\\in\\ZZ$.\n\nWe will show that a mapping $h\\in\nW^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies (\\ref{al9}) and $h(0)=a$ if and only if it is a fixed point of the mapping $$K:W^{2,2}(\\TT,\\mb{C}^{n-1})\\ni h\\longmapsto P(H^{-1}\\psi-\\gamma h)+a\\in W^{2,2}(\\TT,\\mb{C}^{n-1}).$$\n\nIndeed, take $h\\in\nW^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ and suppose that $h(0)=a$ and $\\gamma h+\\overline{h}-H^{-1}\\psi$ extends holomorphically on $\\mb{D}$. Then $$h=a+\\sum_{k=1}^{\\infty}a_k\\zeta^{k},\\quad\\overline{h}=\\overline{a}+\\sum_{k=1}^{\\infty}\\overline a_k\\zeta^{-k}=\\sum_{k=-\\infty}^{-1}\\overline a_{-k}\\zeta^{k}+\\overline{a},$$ $$P(h)=0,\\quad P(\\overline{h})=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}=h-a$$ and $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=0,$$ which implies $$P(H^{-1}\\psi-\\gamma h)=h-a$$ and finally $K(h)=h$. Conversely, suppose that $K(h)=h$. Then $$P(H^{-1}\\psi-\\gamma h)=h-a=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}+a_1-a,\\quad P(h)=0$$ and\n$$P(\\overline{h})=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}=h-a_1,$$ from which follows that $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=P(\\overline{h})-P(H^{-1}\\psi-\\gamma h)=a-a_1$$ and $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=0\\text{ iff }a=a_1.$$ Observe that $h(0)=K(h)(0)=P(H^{-1}\\psi-\\gamma h)(0)+a=a$.\n\nWe shall make use of the Banach Fixed Point Theorem. To do this, consider $W^{2,2}(\\TT,\\CC^{n-1})$ equipped with the following norm $$\\|h\\|_{\\varepsilon}:=\\|h\\|_L+\\varepsilon\\|h'\\|_L+\n\\varepsilon^2\\|h''\\|_L,$$ where $\\eps>0$ and $\\|\\cdot\\|_L$ is the $L^2$-norm (it is a Banach space). We will prove that $K$ is a contraction with respect to the norm $\\|\\cdot\\|_{\\varepsilon}$ for sufficiently small $\\eps$. Indeed, there is $\\wi\\eps>0$ such that for any $h_1,h_2\\in W^{2,2}(\\TT,\\CC^{n-1})$\n\\begin{equation}\n\\|K(h_1)-K(h_2)\\|_L=\\|P(\\gamma(h_2-h_1))\\|_L\\leq\\|\\gamma(h_2-h_1)\\|_L\\leq (1-\\wi\\eps)\\|h_2-h_1\\|_L.\n\\label{al10}\n\\end{equation}\nMoreover,\n\\begin{multline}\n\\|K(h_1)'-K(h_2)'\\|_L= \\|P(\\gamma h_2)'-P(\\gamma h_1)'\\|_L\\leq\\\\\n\\leq\\|(\\gamma h_2)'-(\\gamma h_1)'\\|_L= \\|\\gamma '(h_2-h_1)+\\gamma(h_2'-h_1')\\|_L.\n\\label{al11}\n\\end{multline} Furthermore,\n\\begin{equation}\n\\|K(h_1)''-K(h_2)''\\|_L\\leq\\|\\gamma ''(h_2-h_1)\\|_L+2\\|\\gamma '(h_2'-h_1')\\|_L+\\|\\gamma\n(h_1''-h_2'')\\|_L.\\label{al12}\n\\end{equation}\nUsing the finiteness of $\\|\\gamma '\\|$, $\\|\\gamma ''\\|$ and putting (\\ref{al10}), (\\ref{al11}), (\\ref{al12}) together we see that there exists $\\varepsilon>0$ such that $K$ is a contraction w.r.t. the norm $\\|\\cdot\\|_{\\varepsilon}$.\n\nWe have found $\\widetilde{f}$ and $\\widetilde{\\lambda}$ satisfying (\\ref{al1}), (\\ref{al3}) and the last $n-1$ equations from (\\ref{al2}) are satisfied. \n\nIt remains to show that there exists a unique $\\widetilde{q}\\in Q_0$ such that $\\widetilde{q}+\\zeta A_1+\\zeta B_1-\\varphi_1$ extends holomorphically on $\\mb{D}$.\n\nComparing the coefficients as in \\eqref{65}, we see that if $$\\pi(\\zeta A_1+\\zeta B_1-\\varphi_1)=\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}$$\nthen $\\widetilde{q}$ has to be taken as $$-\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}-\\sum_{k=0}^{\\infty}b_k\\zeta^{k}$$\nwith $b_k:=\\overline a_{-k}$ for $k\\geq 1$ and $b_0\\in\\RR$ uniquely determined by $\\widetilde{q}(1)=0$.\\\\\n\nLet us show that the proof of the second Lemma follows from the proof of the first one.\nSince $\\wi\\Xi$ is real analytic it suffices to prove that the derivative $$\\wi\\Xi_{(f,q,\\xi)}(r_0,f_0(\\xi_0),f_0,0,\\xi_0):B\\times Q_0\\times\\RR\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$$ is invertible.\nFor $(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})\\in B\\times Q_0\\times\\RR$ we get\n\\begin{multline*}\n\\wi\\Xi_{(f,q,\\xi)}(r_0,f_0(\\xi_0),f_0,0,\\xi_0)(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})=\\left.\\frac{d}{dt}\n\\wi\\Xi(r_0,f_0(\\xi_0),f_0+t\\widetilde{f},t\\widetilde{q},\\xi_0+t\\widetilde{\\xi})\\right|_{t=0}=\\\\\n=((r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}},\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta(r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}),\\widetilde{f}(\\xi_0)+\\wi\\xi f_0'(\\xi_0)).\n\\end{multline*}\nWe have to show that for $(\\eta,\\varphi,w)\\in Q\\times B^*\\times\\mb{C}^n$ there exists exactly one $(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})\\in B\\times Q_0\\times\\RR$ satisfying\n\\begin{equation}\n(r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}=\\eta,\n\\label{1al1}\n\\end{equation}\n\\begin{equation}\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta (r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}})=\\varphi,\n\\label{1al2}\n\\end{equation}\n\\begin{equation}\n\\wi f(\\xi_0)+\\wi\\xi f_0'(\\xi_0)=w.\n\\label{1al3}\n\\end{equation}\nThe equation (\\ref{1al1}) turns out to be\n\\begin{equation}\n\\re(\\widetilde{f}_1\/\\zeta)=\\eta\\text{ (on }\\TT).\n\\label{1al4}\n\\end{equation}\nThe equation above uniquely determines $\\widetilde{f}_1\/\\zeta\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ up to an imaginary additive constant, which may be computed using (\\ref{1al3}). Indeed, there exists $G\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ such that $\\eta=\\re G$ on $\\TT$. We are searching $C\\in\\RR$ such that the functions $\\widetilde{f}_1:=\\zeta(G+iC)$ and $\\theta:=\\im(\\widetilde{f}_1\/\\zeta)$ satisfy  $$\\xi_0\\eta(\\xi_0)+i\\xi_0\\theta(\\xi_0)=\\widetilde{f}_1(\\xi_0)$$ and $$\\xi_0(\\eta(\\xi_0)+i\\theta(\\xi_0))+\\widetilde{\\xi}\\re{f_{01}'(\\xi_0)}+i\\widetilde{\\xi}\\im{{f_{01}'(\\xi_0)}}=\n\\re{w_1}+i\\im{w_1}.$$ But $$\\xi_0\\eta(\\xi_0)+\\widetilde{\\xi}\\re{f_{01}'(\\xi_0)}=\\re{w_1},$$ which yields $\\widetilde{\\xi}$ and then $\\theta(\\xi_0)$, consequently the number $C$. Having $\\widetilde{\\xi}$ and once again using (\\ref{1al3}), we find uniquely determined $\\widetilde{f}_2(\\xi_0),\\ldots,\\widetilde{f}_n(\\xi_0)$.\n\nTherefore, the equations $\\eqref{1al1}$ and $\\eqref{1al3}$ are satisfied by uniquely determined $\\wi f_1$, $\\wi\\xi$ and $\\widetilde{f}_2(\\xi_0),\\ldots,\\widetilde{f}_n(\\xi_0)$.\n\nIn the remaining part of the proof we change the second condition of \\eqref{al6} to $$g(\\xi_0)={\\widehat{\\widetilde{f}}}(\\xi_0)\/\\xi_0$$ and we have to prove that there is a unique solution $h\\in W^{2,2}(\\TT,\\CC^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ of (\\ref{al9}) such that $h(\\xi_0)=a$ with a given $a\\in\\CC^{n-1}$. Let $\\tau$ be an automorphism of $\\DD$ (so it extends holomorphically near $\\CDD$), which maps $0$ to $\\xi_0$, i.e. $$\\tau(\\xi):=\\frac{\\xi_0-\\xi}{1-\\ov\\xi_0\\xi},\\ \\xi\\in\\DD.$$ Let the maps $P,K$ be as before. Then $h\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies\n(\\ref{al9}) and $h(\\xi_0)=a$ if and only if $h\\circ\\tau\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies (\\ref{al9}) and $(h\\circ\\tau)(0)=a$. We already know that there is exactly one $\\wi h\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfying (\\ref{al9}) and $\\wi h(0)=a$. Setting $h:=\\wi h\\circ\\tau^{-1}$, we get the claim.\n\\end{proof}\n\n\\subsection{Topology in the class of domains with real analytic boundaries}\\label{topol}\n\nWe introduce a concept of a domain being close to some other domain. Let $D_0\\su\\mb{C}^n$ be a bounded domain with real analytic boundary. Then there exist a neighborhood $U_0$ of $\\partial D_0$ and a real analytic defining function $r_0:U_0\\longrightarrow\\mb{R}$ such that $\\nabla r_0$ does not vanish in $U_0$ and $$D_0\\cap U_0=\\{z\\in U_0:r_0(z)<0\\}.$$\n\n\\begin{dff}\nWe say that domains $D$ \\textit{tend to} $D_0$ $($or are \\textit{close to} $D_0${$)$} if one can choose their defining functions $r\\in X_0$ such that $r$ tend to $r_0$ in $X_0$.\n\\end{dff}\n\n\\begin{remm} If $r\\in X_0$ is near to $r_0$ with respect to the topology in $X_0$, then $\\{z\\in U_0:r(z)=0\\}$ is a compact real analytic hypersurface which bounds a bounded domain. We denote it by $D^{r}$.\n\nMoreover, if $D^{r_0}$ is strongly linearly convex then a domain $D^r$ is also strongly linearly convex provided that $r$ is near $r_0$.\n\\end{remm}\n\n\n\n\n\n\n\n\\subsection{Statement of the main result of this section}\n\n\\begin{remm}\\label{f} Assume that $D^r$ is a strongly linearly convex domain bounded by a real analytic hypersurface $\\{z\\in U_0:r(z)=0\\}$. Let $\\xi\\in(0,1)$ and $w\\in(\\CC^n)_*$.\n\nThen a function $f\\in B_0$ satisfies the conditions $$f\\text{ is a weak stationary mapping of }D^r,\\ f(0)=0,\\ f(\\xi)=w$$ if and only if there exists $q\\in Q_0$ such that $q>-1$ and $\\wi\\Xi(r,w,f,q,\\xi)=0$.\n\nActually, from $\\wi\\Xi(r,w,f,q,\\xi)=0$ we deduce immediately that $r\\circ f=0$ on $\\TT$, $f(\\xi)=w$ and $\\pi(\\zeta(1+q)(r_z\\circ f))=0$. From the first equality we get $f(\\TT)\\subset \\partial D^{r}$. From the last one we deduce that the condition (3') of Definition~\\ref{21} is satisfied (with $\\rho:=(1+q)|r_z\\circ f|$). Since $D^{r}$ is strongly linearly convex, $\\ov{D^r}$ is polynomially convex (use the fact that projections of $\\CC$-convex domains are $\\CC$-convex, as well, and the fact that $D^r$ is smooth). In particular, $$f(\\CDD)=f(\\widehat{\\TT})\\subset\\widehat{f(\\TT)}\\subset\\widehat{\\ov{D^r}}=\\ov{D^r},$$ where $\\wh S:=\\{z\\in\\CC^m:|P(z)|\\leq\\sup_S|P|\\text{ for any polynomial }P\\in\\CC[z_1,\\ldots,z_m]\\}$ is the polynomial hull of a set $S\\su\\CC^m$. \n\nNote that this implies $f(\\DD)\\su D^r$ --- this follows from the fact that $\\pa D^r$ does not contain non-constant analytic discs (as $D^r$ is strongly pseudoconvex).\n\nThe opposite implication is clear.\n\n\\bigskip\n\nIn a similar way we show that for any $v\\in(\\CC^n)_*$ and $\\lambda>0$, a function $f\\in B_0$ satisfies the conditions $$f\\text{ is a weak stationary mapping of }D^r,\\ f(0)=0,\\ f'(0)=\\lambda v$$ if and only if there exists $q\\in Q_0$ such that $q>-1$ and $\\Xi(r,v,f,q,\\lambda)=0$.\n\n\\end{remm}\n\n\n\\begin{propp}\\label{13} Let $D_0\\su\\CC^n$, $n\\geq 2$, be a strongly linearly convex domain with real analytic boundary and let $f_0:\\DD\\longrightarrow D_0$ be an $E$-mapping.\n\n$(1)$ Let $\\xi_0\\in(0,1)$. Then there exist a neighborhood $W_0$ of $(r_0,f_0(\\xi_0))$ in $X_0\\times D_0$ and real analytic mappings $$\\Lambda:W_0\\longrightarrow\\mc{C}^{1\/2}(\\overline{\\mb{D}}),\\ \\Omega:W_0\\longrightarrow(0,1)$$ such that $$\\Lambda(r_0,f_0(\\xi_0))=f_0,\\ \\Omega(r_0,f_0(\\xi_0))=\\xi_0$$ and for any $(r,w)\\in W_0$ the mapping\n$f:=\\Lambda(r,w)$ is an $E$-mapping of $D^{r}$ satisfying $$f(0)=f_0(0)\\text{ and }f(\\Omega(r,w))=w.$$\n\n$(2)$ There exist a neighborhood $V_0$ of $(r_0,f_0'(0))$ in $X_0\\times\\mb{C}^n$\nand a real analytic mapping $$\\Gamma:V_0\\longrightarrow\\mc{C}^{1\/2}(\\overline{\\mb{D}})$$ such that $$\\Gamma(r_0,f_0'(0))=f_0$$ and for any $(r,v)\\in V_0$ the mapping $f:=\\Gamma(r,v)$ is an $E$-mapping of $D^{r}$ satisfying $$f(0)=f_0(0)\\text{ and }f'(0)=\\lambda v\\text{ for some }\\lambda>0.$$\n\\end{propp}\n\n\\begin{proof}\n\nObserve that Proposition \\ref{11} provides us with a mapping $g_0=\\Phi\\circ f_0$ and a domain $G_0:=\\Phi(D_0)$ giving a data for situation (\\dag) (here $\\partial D_0$ is contained in $U_0$). Clearly, $\\rho_0:=r_0\\circ\\Phi^{-1}$ is a defining function of $G_0$.\n\nUsing Lemmas \\ref{cruciallemma}, \\ref{cruciallemma1} we get neighborhoods $V_0$, $W_0$ of $(\\rho_0, g_0'(0))$, $(\\rho_0,g_0(\\xi_0))$ respectively and real analytic mappings $\\Upsilon$, $\\wi\\Upsilon$ such that $ \\Xi(\\rho,v,\\Upsilon(\\rho,v))=0$ on $V_0$ and $ \\wi\\Xi(\\rho,w,\\wi\\Upsilon(\\rho,w))=0$ on $W_0$. Define $$\\wh\\Lambda:=\\pi_B\\circ\\wi\\Upsilon,\\quad\\Omega:=\\pi_\\RR\\circ\\wi\\Upsilon,\\quad\\wh\\Gamma:=\\pi_B\\circ\\Upsilon,$$ where $$\\pi_B:B\\times Q_0\\times\\mb{R}\\longrightarrow B,\\quad\\pi_\\RR:B\\times Q_0\\times\\mb{R}\\longrightarrow\\RR,\\ $$ are the projections.\n\nIf $\\rho$ is sufficiently close to $\\rho_0$, then the hypersurface $\\{\\rho=0\\}$ bounds a strongly linearly convex domain. Moreover, then $\\wh\\Lambda(\\rho,w)$ and $\\wh\\Gamma(\\rho,v)$ are extremal mappings in $G^{\\rho}$ (see Remark~\\ref{f}).\n\nComposing $\\wh\\Lambda(\\rho,w)$ and $\\wh\\Gamma(\\rho,v)$ with $\\Phi^{-1}$ and making use of Remark \\ref{rem:theta} we get weak stationary mappings in $D^r$, where $r:=\\rho\\circ\\Phi$. To show that they are $E$-mappings we proceed as follows. If $D^r$ is sufficiently close to $D_0$ (this depends on a distance between $\\rho$ and $\\rho_0$), the domain $D^r$ is strongly linearly convex, so by the results of Section \\ref{55} $$\\Lambda(r,w):=\\Phi^{-1}\\circ\\wh\\Lambda(\\rho,w)\\text{\\  and\\  }\\Gamma(r,v):=\\Phi^{-1}\\circ\\wh\\Gamma(\\rho,v)$$ are stationary mappings. Moreover, they are close to $f_0$ provided that $r$ is sufficiently close to $r_0$. Therefore, their winding numbers are equal. Thus $f$ satisfies condition (4) of Definition~\\ref{21e}, i.e. $f$ is an $E$-mapping.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Localization property}\n\n\\begin{prop}\\label{localization} Let $D\\su\\mathbb C^n$, $n\\geq 2$, be a domain. Assume that $a\\in\\partial D$ is such that $\\partial D$ is real analytic and strongly convex in a neighborhood of $a$. Then for any sufficiently small neighborhood $V_0$ of $a$ there is a weak stationary mapping of $D\\cap V_0$ such that $f(\\mathbb T)\\su\\partial D$.\n\nIn particular, $f$ is a weak stationary mapping of $D$.\n\\end{prop}\n\n\\begin{proof} Let $r$ be a real analytic defining function in a neighborhood of $a$. The problem we are dealing with has a local character, so replacing $r$ with $r\\circ\\Psi$, where $\\Psi$ is a local biholomorphism near $a$, we may assume that $a=(0,\\ldots,0,1)$ and a defining function of $D$ near $a$ is $r(z)=-1+|z|^2+h(z-a)$, where $h$ is real analytic in a neighborhood of $0$ and $h(z)=O(|z|^3)$ as $z\\to 0$ (cf. \\cite{Rud}, p. 321).\n\nFollowing \\cite{Lem2}, let us consider the mappings\n$$A_t(z):=\\left((1-t^2)^{1\/2}\\frac{z'}{1+tz_n},\\frac{z_n+t}{1+tz_n}\\right),\\quad z=(z',z_n)\\in\\CC^{n-1}\\times\\DD,\\,\\,t\\in(0,1),$$ which restricted to $\\BB_n$ are automorphisms. Let $$r_t(z):=\\begin{cases}\\frac{|1+tz_n|^2}{1-t^2}r(A_t(z)),&t\\in(0,1),\\\\-1+|z|^2,&t=1.\\end{cases}$$ It is clear that $f_{(1)}(\\zeta)=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\DD$ is a stationary mapping of $\\mathbb B_n$. We want to have the situation (\\dag) which will allow us to use Lemma \\ref{cruciallemma} (or Lemma \\ref{cruciallemma1}). Note that $r_t$ does not converge to $r_1$ as $t\\to 1$. However, $r_t\\to r_1$ in $X_0(U_0,U_0^{\\mathbb C})$, where $U_0$ is a neighborhood of $f_{(1)}(\\TT)$ contained in $\\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\}$ and $U_0^{\\mathbb C}$ is sufficiently small (remember that $h(z)=O(|z|^3)$).\n\nTherefore, making use of Lemma \\ref{cruciallemma} for $t$ sufficiently close to $1$ we obtain stationary mappings $f_{(t)}$ in $D_t:=\\{z\\in \\mathbb C^n: r_t(z)<0,\\ \\re z_n>-1\/2\\}$ such that $f_{(t)}\\to f_{(1)}$ in the $W^{2,2}$-norm (so also in the sup-norm). Actually, it follows from Lemma~\\ref{cruciallemma} that one may take $f_{(t)}:=\\pi_B\\circ\\Upsilon(r_t,f_{(1)}'(0))$ (keeping the notation from this lemma). The argument used in Remark~\\ref{f} gives that $f_{(t)}$ satisfies conditions (1'), (2') and (3') of Definition~\\ref{21}. Since the non-constant function $r\\circ A_t\\circ f_{(t)}$ is subharmonic on $\\DD$, continuous on $\\CDD$ and $r\\circ A_t\\circ f_{(t)}=0$ on $\\TT$, we see from the maximum principle that $f_{(t)}$ maps $\\DD$ in $D_t$. Therefore, $f_{(t)}$ are weak stationary mappings for $t$ close to $1$.\n\nIn particular, $$f_{(t)}(\\DD)\\subset 2\\mathbb B_n \\cap \\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\}$$ provided that $t$ is close to $1$. The mappings $A_t$ have the following important property $$A_t(2\\mathbb B_n\\cap\\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\})\\to\\{a\\}$$ as $t\\to 1$ in the sense of the Hausdorff distance.\n\nTherefore, we find from Remark \\ref{rem:theta} that $g_{(t)}:=A_t\\circ f_{(t)}$ is a stationary mapping of $D$. Since $g_{(t)}$ maps $\\DD$ onto arbitrarily small neighborhood of $a$ provided that $t$ is sufficiently close to $1$, we immediately get the assertion.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs of Theorems \\ref{lem-car} and \\ref{main}}\n\nWe start this section with the following\n\\begin{lem}\\label{lemat} For any different $z,w\\in D$ $($resp. for any $z\\in D$, $v\\in(\\CC^n)_*${$)$} there exists an $E$-mapping $f:\\DD\\longrightarrow D$ such that $f(0)=z$, $f(\\xi)=w$ for some $\\xi\\in(0,1)$ $($resp. $f(0)=z$, $f'(0)=\\lambda v$ for some $\\lambda>0${$)$}.\n\\end{lem}\n\n\\begin{proof}\nFix different $z,w\\in D$ (resp. $z\\in D$, $v\\in(\\CC^{n})_*$).\n\nFirst, consider the case when $D$ is bounded strongly convex with real analytic boundary. Without loss of generality one may assume that $0\\in D\\Subset\\BB_n$. We need some properties of the Minkowski functionals.\n\nLet $\\mu_G$ be a Minkowski functional of a domain $G\\subset\\CC^n$ containing the origin, i.e. $$\\mu_G(x):=\\inf\\left\\{s>0:\\frac{x}{s}\\in G\\right\\},\\ x\\in\\CC^n.$$ Assume that $G$ is bounded strongly convex with real analytic boundary. We shall show that\n\\begin{itemize}\n\\item $\\mu_G-1$ is a real analytic outside $0$, defining function of $G$;\n\\item $\\mu^2_G-1$ is a real analytic outside $0$, strongly convex outside $0$, defining function of $G$.\n\\end{itemize}\nClearly, $G=\\{x\\in\\RR^{2n}:\\mu_G(x)<1\\}$. Setting $$q(x,s):=r\\left(\\frac{x}{s}\\right),\\ (x,s)\\in U_0\\times U_1,$$ where $r$ is a real analytic defining function of $G$ (defined near $\\pa G$) and $U_0\\su\\RR^{2n}$, $U_1\\su\\RR$ are neighborhoods of $\\pa G$ and $1$ respectively, we have $$\\frac{\\partial q}{\\partial s}(x,s)=-\\frac{1}{s^2}\\left\\langle\\nabla r\\left(\\frac{x}{s}\\right),x\\right\\rangle_{\\RR}\\neq 0$$ for $(x,s)$ such that $x\\in\\partial G$ and $s=\\mu_G(x)=1$ (since $0\\in G$, the vector $-x$ hooked at the point $x$ is inward $G$, so it is not orthogonal to the normal vector at $x$). By the Implicit Function Theorem for the equation $q=0$, the function $\\mu_G$ is real analytic in a neighborhood $V_0$ of $\\partial G$. To see that $\\mu_G$ is real analytic outside $0$, fix $x_0\\in(\\RR^{2n})_*$. Then the set $$W_0:=\\left\\{x\\in\\RR^{2n}:\\frac{x}{\\mu_G(x_0)}\\in V_0\\right\\}$$ is open and contains $x_0$. Since $$\\mu_G(x)=\\mu_G(x_0)\\mu_G\\left(\\frac{x}{\\mu_G(x_0)}\\right),\\ x\\in W_0,$$ the function $\\mu_G$ is real analytic in $W_0$. Therefore, we can take $d\/ds$ on both sides of $\\mu_G(sx)=s\\mu_G(x),\\ x\\neq 0,\\ s>0$ to obtain $$\\langle\\nabla\\mu_G(x),x\\rangle_{\\RR}=\\mu_G(x),\\ x\\neq 0,$$ so $\\nabla\\mu_G\\neq 0$ in $(\\RR^{2n})_*$.\n\nFurthermore, $\\nabla\\mu^2_G=2\\mu_G\\nabla\\mu_G$, so $\\mu^2_G-1$ is also a defining function of $G$.\nTo show that $u:=\\mu^2_G$ is strongly convex outside $0$ let us prove that $$X^T\\mathcal{H}_aX>0,\\quad a\\in\\pa G,\\ X\\in(\\RR^{2n})_*,$$ where $\\mathcal{H}_x:=\\mathcal{H}u(x)$ for $x\\in(\\RR^{2n})_*$. Taking $\\pa\/\\pa x_j$ on both sides of $$u(sx)=s^2u(x),\\ x,s\\neq 0,$$ we get \\begin{equation}\\label{62}\\frac{\\pa u}{\\pa x_j}(sx)=s\\frac{\\pa u}{\\pa x_j}(x)\\end{equation} and further taking $d\/ds$ $$\\sum_{k=1}^{2n}\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(sx)x_k=\\frac{\\pa u}{\\pa x_j}(x).$$ In particular, $$x^T\\mathcal{H}_xy=\\sum_{j,k=1}^{2n}\\frac{\\pa^2 u}{\\pa x_k\\pa x_j}(x)x_ky_j=\\langle\\nabla u(x),y\\rangle_{\\RR},\\ x\\in(\\RR^{2n})_*,\\ y\\in\\RR^{2n}.$$ Let $a\\in\\pa G$. Since $\\langle\\nabla\\mu_G(a),a\\rangle_{\\RR}=\\mu_G(a)=1$, we have $a\\notin T^\\RR_G(a)$. Any $X\\in(\\RR^{2n})_*$ can be represented as $\\alpha a+\\beta Y$, where $Y\\in T^\\RR_G(a)$, $\\alpha,\\beta\\in\\RR$, $(\\alpha,\\beta)\\neq(0,0)$. Then \\begin{eqnarray*}X^T\\mathcal{H}_aX&=&\\alpha^2a^T\\mathcal{H}_aa+2\\alpha\\beta a^T\\mathcal{H}_aY+\\beta^2Y^T\\mathcal{H}_aY=\\\\&=&\\alpha^2\\langle\\nabla u(a),a\\rangle_{\\RR} +2\\alpha\\beta\\langle\\nabla u(a),Y\\rangle_{\\RR} +\\beta^2Y^T\\mathcal{H}_aY= \\\\&=&\\alpha^22\\mu_G(a)\\langle\\nabla\\mu_G(a),a\\rangle_{\\RR} +\\beta^2Y^T\\mathcal{H}_aY=\n2\\alpha^2+\\beta^2Y^T\\mathcal{H}_aY.\\end{eqnarray*} Since $G$ is strongly convex, the Hessian of any defining function is strictly positive on the tangent space, i.e. $Y^T\\mathcal{H}_aY>0$ if $Y\\in(T^\\RR_G(a))_*$. Hence $X^T\\mathcal{H}_aX\\geq 0$. Note that it cannot be $X^T\\mathcal{H}_aX=0$, since then $\\alpha=0$, consequently $\\beta\\neq 0$ and $Y^T\\mathcal{H}_aY=0$. On the other side $Y=X\/\\beta\\neq 0$ --- a contradiction.\n\nTaking $\\pa\/\\pa x_k$ on both sides of \\eqref{62} we obtain $$\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(sx)=\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(x),\\ x,s\\neq 0$$ and for $a,X\\in(\\RR^{2n})_*$ $$X^T\\mathcal{H}_aX=X^T\\mathcal{H}_{a\/\\mu_G(a)}X>0.$$\n\nLet us consider the sets $$D_t:=\\{x\\in\\CC^n:t\\mu^2_D(x)+(1-t)\\mu^2_{\\BB_n}(x)<1\\},\\ t\\in[0,1].$$ The functions $t\\mu^2_D+(1-t)\\mu^2_{\\BB_n}$ are real analytic in $(\\CC^n)_*$ and strongly convex in $(\\CC^n)_*$, so $D_t$ are strongly convex domains with real analytic boundaries satisfying $$D=D_1\\Subset D_{t_2}\\Subset D_{t_1}\\Subset D_0=\\BB_n\\text{\\  if \\ }0<t_1<t_2<1.$$ It is clear that $\\mu_{D_t}=\\sqrt{t\\mu^2_D+(1-t)\\mu^2_{\\BB_n}}$. Further, if $t_{1}$ is close to $t_{2}$ then $D_{t_{1}}$ is close to $D_{t_{2}}$ w.r.t. the topology introduced in Section \\ref{27}. We want to show that $D_t$ are in some family $\\mathcal D(c)$. Only the interior and exterior ball conditions need to verify.\n\nThere exists $\\delta>0$ such that $\\delta\\BB_n\\Subset D$. Further, $\\nabla\\mu_{D_t}^2\\neq 0$ in $(\\RR^{2n})_*$. Set $$M:=\\sup\\left\\{\\frac{\\mathcal{H}\\mu_{D_t}^2(x;X)}{|\\nabla\\mu_{D_t}^2(y)|}:\nt\\in[0,1],\\ x,y\\in 2\\ov{\\BB}_n\\setminus\\delta\\BB_n,\\ X\\in\\RR^{2n},\\ |X|=1\\right\\}.$$ It is a positive number since the functions $\\mu_{D_t}^2$ are strongly convex in $(\\RR^{2n})_*$ and the `sup' of the continuous, positive function is taken over a compact set. Let $$r:=\\min\\left\\{\\frac{1}{2M},\\frac{\\dist(\\pa D,\\delta\\BB_n)}{2}\\right\\}.$$ For fixed $t\\in[0,1]$ and $a\\in\\pa D_t$ put $a':=a-r\\nu_{D_t}(a)$. In particular, $\\ov{B_n(a',r)}\\su 2\\ov{\\BB}_n\\setminus\\delta\\BB_n$. Let us define $$h(x):=\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a'|}(|x-a'|^2-r^2),\\ x\\in 2\\ov{\\BB}_n\\setminus\\delta\\BB_n.$$ We have $h(a)=1$ and $$\\nabla h(x)=\\nabla\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{|a-a'|}(x-a').$$ For $x=a$, dividing the right side by $|\\nabla\\mu^2_{D_t}(a)|$, we get a difference of the same normal vectors $\\nu_{D_t}(a)$, so $\\nabla h(a)=0$. Moreover, for $|X|=1$ $$\\mathcal{H}h(x;X)=\\mathcal{H}\\mu^2_{D_t}(x;X)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{r}\\leq M|\\nabla\\mu^2_{D_t}(a)|-2M|\\nabla\\mu^2_{D_t}(a)|<0.$$ It follows that $h\\leq 1$ in any convex set $S$ such that $a\\in S\\su 2\\ov{\\BB}_n\\setminus\\delta\\BB_n$. Indeed, assume the contrary. Then there is $y\\in S$ such that $h(y)>1$. Let us join $a$ and $y$ with an interval $$g:[0,1]\\ni t\\longmapsto h(ta+(1-t)y)\\in S.$$ Since $a$ is a strong local maximum of $h$, the function $g$ has a local minimum at some point $t_0\\in(0,1)$. Hence $$0\\leq g''(t_0)=\\mathcal{H}h(t_0a+(1-t_0)y;a-y),$$ which is impossible.\n\nSetting $S:=\\ov{B_n(a',r)}$, we get $$\\mu^2_{D_t}(x)\\leq 1+\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a'|}(|x-a'|^2-r^2)<1$$  for $x\\in B_n(a',r)$, i.e. $x\\in D_t$.\n\nThe proof of the exterior ball condition is similar. Set $$m:=\\inf\\left\\{\\frac{\\mathcal{H}\\mu_{D_t}^2(x;X)}{|\\nabla\\mu_{D_t}^2(y)|}:\nt\\in[0,1],\\ x,y\\in(\\ov{\\BB}_n)_*,\\ X\\in\\RR^{2n},\\ |X|=1\\right\\}.$$ Note that the $m>0$. Actually, the homogeneity of $\\mu_{D_t}$ implies $\\mathcal{H}\\mu_{D_t}^2(sx;X)=\\mathcal{H}\\mu_{D_t}^2(x;X)$ and $\\nabla\\mu_{D_t}^2(sx)=s\\nabla\\mu_{D_t}^2(x)$ for $x\\neq 0$, $X\\in \\RR^{2n}$, $s>0$. Therefore, there are positive constants $C_1,C_2$ such that $C_1\\leq\\mathcal{H}\\mu_{D_t}^2(x;X)$ for $x\\neq 0$, $X\\in \\RR^{2n}$, $|X|=1$ and $|\\nabla\\mu_{D_t}^2(y)|\\leq C_2$ for $y\\in\\ov\\BB_n$. In particular, $m\\geq C_1\/C_2$.\n\nLet $R:=2\/m$. For fixed $t\\in[0,1]$ and $a\\in\\pa D_t$ put $a'':=a-R\\nu_{D_t}(a)$. Let us define $$\\wi h(x):=\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a''|}(|x-a''|^2-R^2),\\ x\\in\\ov{\\BB}_n.$$ We have $\\wi h(a)=1$ and $$\\nabla\\wi  h(x)=\\nabla\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{|a-a''|}(x-a''),$$ so $\\nabla\\wi h(a)=0$. Moreover, for $x\\in(\\ov{\\BB}_n)_*$ and $|X|=1$ $$\\mathcal{H}\\wi h(x;X)=\\mathcal{H}\\mu^2_{D_t}(x;X)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{R}\\geq m|\\nabla\\mu^2_{D_t}(a)|-m\/2|\\nabla\\mu^2_{D_t}(a)|>0.$$ Therefore, $a$ is a strong local minimum of $\\wi h$.\n\nNow using the properties listed above we may deduce that $\\wi h\\geq 1$ in $\\ov\\BB_n$. We proceed similarly as before: seeking a contradiction suppose that there is $y\\in\\ov\\BB_n$ such that $\\wi h(y)<1$. Moving $y$ a little (if necessary) we may assume that $0$ does not lie on the interval joining $a$ and $y$. Then the mapping $\\wi g(t):=\\wi h(ta+ (1-t)y)$ attains its local maximum at some point $t_0\\in(0,1)$. The second derivative of $\\wi g$ at $t_0$ is non-positive, which gives a contradiction with a positivity of the Hessian of the function $\\wi h$. \n\nHence, we get $$\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a''|}(|x-a''|^2-R^2)\\leq\\mu^2_{D_t}(x)-1<0,$$  for $x\\in D_t$, so $D_t \\subset B_n(a'',R)$.\n\nLet $T$ be the set of all $t\\in[0,1]$ such that there is an $E$-mapping $f_{t}:\\DD\\longrightarrow D_{t}$ with $f_{t}(0)=z$, $f_{t}(\\xi_{t})=w$ for some $\\xi_{t}\\in(0,1)$ (resp. $f_{t}(0)=z$, $f_{t}'(0)=\\lambda_{t}v$ for some $\\lambda_{t}>0$). We claim that $T=[0,1]$. To prove it we will use the open-close argument.\n\nClearly, $T\\neq\\emptyset$, as $0\\in T$. Moreover, $T$ is open in $[0,1]$. Indeed, let $t_{0}\\in T$. It follows from Proposition \\ref{13} that there is a neighborhood $T_{0}$ of $t_{0}$ such that there are $E$-mappings $f_{t}:\\DD\\longrightarrow D_{t}$ and $\\xi_{t}\\in(0,1)$ such that $f_{t}(0)=z$, $f_{t}(\\xi_{t})=w$ for all $t\\in T_{0}$ (resp. $\\lambda_{t}>0$ such that $f_{t}(0)=z$, $f_{t}'(0)=\\lambda_{t} v$ for all $t\\in T_{0}$).\n\nTo prove that $T$ is closed, choose a sequence $\\{t_{m}\\}\\su T$ convergent to some $t\\in[0,1]$. We want to show that $t\\in T$. Since $f_{t_m}$ are $E$-mappings, they are complex geodesics. Therefore, making use of the inclusions $D\\subset D_{t_m}\\subset\\mathbb B_n$ we find that there is a compact set $K\\su(0,1)$ (resp. a compact set $\\widetilde K\\subset(0,\\infty)$) such that $\\{\\xi_{t_m}\\}\\subset K$ (resp. $\\{\\lambda_{t_m}\\}\\subset\\widetilde K$). By Propositions \\ref{8} and \\ref{10b} the functions $f_{t_{m}}$ and $\\widetilde f_{t_{m}}$ are equicontinuous in $\\mathcal{C}^{1\/2}(\\overline{\\DD})$ and by Propositions \\ref{9} and \\ref{10a} the functions $\\rho_{t_{m}}$ are uniformly bounded from both sides by positive numbers and equicontinuous in $\\mathcal{C}^{1\/2}(\\TT)$. From the Arzela-Ascoli Theorem there are a subsequence $\\{s_{m}\\}\\subset\\{t_{m}\\}$ and mappings $f,\\wi f\\in\\OO(\\DD)\\cap\\mathcal C^{1\/2}(\\overline{\\mathbb D})$, $\\rho\\in\\cC^{1\/2}(\\TT)$ such that $f_{s_{m}}\\to f$, $\\widetilde{f}_{s_{m}}\\to\\wi f$ uniformly on $\\overline{\\DD}$, $\\rho_{s_{m}}\\to\\rho$ uniformly on $\\TT$ and $\\xi_{s_m}\\to\\xi\\in (0,1)$ (resp. $\\lambda_{s_m}\\to\\lambda>0$).\n\nClearly, $f(\\CDD)\\su\\overline{D}_{t}$, $f(\\TT)\\su\\partial D_{t}$ and $\\rho>0$. By the strong pseudoconvexity of $D_t$ we get $f(\\DD)\\su D_t$.\n\nThe conditions (3') and (4) of Definitions~\\ref{21} and \\ref{21e} follow from the uniform convergence of suitable functions. Therefore, $f$ is a weak $E$-mapping of $D_{t}$, consequently an $E$-mapping of $D_t$, satisfying $f(0)=z$, $f(\\xi)=w$ (resp. $f(0)=z$, $f'(0)=\\lambda v$).\n\nLet us go back to the general situation that is when a domain $D$ is bounded strongly linearly convex with real analytic boundary. Take a of point $\\eta\\in\\partial{D}$ such that $\\max_{\\zeta\\in\\partial{D}}|z-\\zeta|=|z-\\eta|$. Then $\\eta$ is a point of the strong convexity of $D$. Indeed, by the Implicit Function Theorem one can assume that in a neighborhood of $\\eta$ the defining functions of $D$ and $B:=B_n(z,|z-\\eta|)$ are of the form $r(x):=\\wi r(\\wi x)-x_{2n}$ and $q(x):=\\wi q(\\wi x)-x_{2n}$ respectively, where $x=(\\wi x,x_{2n})\\in\\RR^{2n}$ is sufficiently close to $\\eta$. From the inclusion $D\\su B$ it it follows that $r-q\\geq 0$ near $\\eta$ and $(r-q)(\\eta)=0$. Thus the Hessian $\\mathcal{H}(r-q)(\\eta)$ is weakly positive in $\\CC^n$. Since $\\mathcal{H}q(\\eta)$ is strictly positive on $T_B^\\RR(\\eta)_*=T_D^\\RR(\\eta)_*$, we find that $\\mathcal{H}r(\\eta)$ is strictly positive on $T_D^\\RR(\\eta)_*$, as well.\n\nBy a continuity argument, there is a convex neighborhood $V_0$ of $\\eta$ such that all points from $\\pa D\\cap V_0$ are points of the strong convexity of $D$. It follows from Proposition \\ref{localization} (after shrinking $V_0$ if necessary) that there is a weak stationary mapping $g:\\DD\\longrightarrow D\\cap V_0$ such that $g(\\TT)\\subset\\partial D$. In particular, $g$ is a weak stationary mapping of $D$. Since $D\\cap V_0$ is convex, the condition with the winding number is satisfied on $D\\cap V_0$ (and then on the whole $D$). Consequently $g$ is an $E$-mapping of $D$.\n\nIf $z=g(0)$, $w=g(\\xi)$ for some $\\xi\\in\\DD$ (resp. $z=g(0)$, $v=g'(0)$) then there is nothing to prove. In the other case let us take curves $\\alpha:[0,1]\\longrightarrow D$, $\\beta:[0,1]\\longrightarrow D$ joining $g(0)$ and $z$, $g(\\xi)$ and $w$ (resp. $g(0)$ and $z$, $g'(0)$ and $v$). We may assume that the images of $\\alpha$ and $\\beta$ are disjoint. Let $T$ be the set of all $t\\in[0,1]$ such that there is an $E$-mapping $g_{t}:\\DD\\longrightarrow D$ such that $g_{t}(0)=\\alpha(t)$, $g_{t}(\\xi_{t})=\\beta(t)$ for some $\\xi_{t}\\in(0,1)$ (resp. $g_{t}(0)=\\alpha(t)$, $g_{t}'(0)=\\lambda_{t}\\beta(t)$ for some $\\lambda_{t}>0$). Again $T\\neq\\emptyset$ since $0\\in T$. Using the results of Section \\ref{22} similarly as before (but for one domain), we see that $T$ is closed.\n\nSince $\\wi k_D$ is symmetric, it follows from Proposition \\ref{13}(1) that the set $T$ is open in $[0,1]$ (first we move along $\\alpha$, then by the symmetry we move along $\\beta$). Therefore, $g_1$ is the $E$-mapping for $z,w$.\n\nIn the case of $\\kappa_{D}$ we change a point and then we change a direction. To be more precise, consider the set $S$ of all $s\\in[0,1]$ such that there is an $E$-mapping $h_{s}:\\DD\\longrightarrow D$ such that $h_{s}(0)=\\alpha(s)$. Then $0\\in S$, by Proposition \\ref{13}(1) the set $S$ is open in $[0,1]$ and by results of Section~\\ref{22} again, it is closed. Hence $S=[0,1]$. Now we may join $h'_{1}(0)$ and $v$ with a curve $\\gamma:[0,1]\\longrightarrow \\mathbb C^n$. Let us define $R$ as the set of all $r\\in[0,1]$ such that there is an $E$-mapping $\\wi h_{r}:\\DD\\longrightarrow D$ such that $\\wi h_{r}(0)=h_1(0)$, $\\wi h'_{r}(0)=\\sigma_{r}\\gamma(1-r)$ for some $\\sigma_r>0$. Then $1\\in R$, by Proposition \\ref{13}(2) the set $R$ is open in $[0,1]$ and, by Section \\ref{22}, it is closed. Hence $R=[0,1]$, so $\\wi h_{0}$ is the $E$-mapping for $z,v$.\n\\end{proof}\n\nNow we are in position that allows us to prove the main results of the Lempert's paper.\n\n\\begin{proof}[Proof of Theorem \\ref{lem-car} $($real analytic case$)$] It follows from Lemma \\ref{lemat} that for any different points $z,w\\in D$ (resp. $z\\in D$, $v\\in(\\CC^n)_*$) one may find an $E$-mapping passing through them (resp. $f(0)=z$, $f'(0)=v$). On the other hand, it follows from Proposition \\ref{1} that $E$-mappings have left inverses, so they are complex geodesics.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main} $($real analytic case$)$] This is a direct consequence of Lemma \\ref{lemat} and Corollary \\ref{28}.\n\\end{proof}\n\n\\bigskip\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{center}{\\sc $\\cC^2$-smooth case}\\end{center}\n\\bigskip\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{un} Let $D\\su\\mathbb C^n$, $n\\geq 2$, be a bounded strongly pseudoconvex domain with $\\mathcal C^2$-smooth boundary. Take $z\\in D$ and let $r$ be a defining function of $D$ such that \n\\begin{itemize}\\item $r\\in \\mathcal C^2(\\mathbb C^n);$\n\\item $D=\\{x\\in \\mathbb C^n:r(x)<0\\}$;\n\\item $\\mathbb C^n\\setminus D=\\{x\\in \\mathbb C^n:r(x)>0\\}$;\n\\item $|\\nabla r|=1$ on $\\partial D;$\n\\item $\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial\\overline z_k}(a)X_{j}\\overline{X}_{k}\\geq C|X|^2$ for any $a\\in \\partial D$ and $X\\in \\mathbb C^n$ with some constant $C>0$.\n\\end{itemize}\n\nSuppose that there is a sequence $\\{r_m\\}$ of $\\mathcal C^2$-smooth real-valued functions such that $D^{\\alpha}r_n$ converges to $D^{\\alpha}r$ locally uniformly for any $\\alpha\\in \\mathbb N_0^{2n}$ such that  $|\\alpha|:=|\\alpha_1| +\\ldots+|\\alpha_n|\\leq 2$. Let $D_m$ be a connected component of the set $\\{x\\in\\mathbb C^n:r_m(x)<0\\}$, containing the point $z$.\n\nThen there is $c>0$ such that $(D_m,z)$ and $(D,z)$ belong to $\\mathcal D(c)$, $m>>1.$\n\\end{lem}\n\n\n\\begin{proof} Losing no generality assume that $D\\Subset\\mathbb B_n.$\nNote that the conditions (1), (5), (6) of Definition \\ref{30} are clearly satisfied. To find $c$ satisfying ($2$), we take $s>0$ such that $\\mathcal H r (x;X)< s |X|^2$ for $x\\in\\ov\\BB_n$ and $X\\in(\\mathbb R^{2n})_*$. Then $\\HH r_m (x;X)<2s|X|^2$ for $x\\in\\ov\\BB_n$, $X\\in(\\mathbb R^{2n})_*$ and $m>>1$. Let $U_0\\subset\\mathbb B_n$ be an open neighborhood of $\\pa D$ such that $|\\nabla r|$ is on $U_0$ between $3\/4$ and $5\/4$. Note that $\\partial D_m\\subset U_0$ and $|\\nabla r_m|\\in (1\/2, 3\/2)$ on $U_0$ for $m>>1$.\n\nFix $m$ and $a\\in \\partial D_m$ and put $b:=a-R\\nu_{D_m}(a)$, where a small number $R>0$ will be specified later. There is $t>0$ such that $\\nabla r_m(a)=2t(a-b)$. Note that $t$ may be arbitrarily large provided that $R$ was small enough. We take $t:=2s$ and $R:=|\\nabla r_m(a)|\/t$. Then we have $\\mathcal H r_m(x;X)<2t |X|^2$ for $x\\in\\ov\\BB_n$, $X\\in(\\mathbb R^{2n})_*$ and $m>>1$. Then a function $$h(x):=r_m(x)-t(|x-b|^2-R^2),\\ x\\in \\mathbb C^n,$$ attains at $a$ its global maximum on $\\ov\\BB_n$ ($a$ is a strong local maximum and the Hessian of $h$ is negative on the convex set $\\ov\\BB_n$, cf. the proof of Lemma \\ref{lemat}).\nThus $h\\leq 0$ on $\\mathbb B_n$. From this we immediately get (2).\n\nNote that it follows from (2) that $D_m=\\{x\\in\\mathbb C^n:r_m(x)<0\\}$ for $m$ big enough (i.e. $\\{x\\in \\mathbb C^n:\\ r_m(x)<0\\}$ is connected).\n\nMoreover, the condition (2) implies the condition (3) as follows. We infer from Remark~\\ref{D(c),4} that there is $c'>0$ such that $D$ satisfies (3) with $c'$. Let $m_0$ be such that the Hausdorff distance between $\\partial D$ and $\\partial D_m$ is smaller than $1\/c'$ for $m\\geq m_0$. There is $c''$ such that $D_{m_0}$ satisfies (3) with $c''$. Losing no generality we may assume that $c''<c'$. Take any $x,y\\in D_m$. Since $D_m$ satisfies the interior ball condition with a radius $c$ we infer that there are balls of a radius $1\/c$ contained in $D_m$ and containig $x$ and $y$ respectively. The centers of these balls lie in $D_{m_0}$. Using the fact that $(D_{m_0},z)$ lies in $\\mathcal D(c'')$, we may join chosen centers with balls of a radius $1\/(2c'')$ as in the condition (3), so we have found a chain consiting of balls of radii $c'$ and $c''$ joining $x$ and $y$.\n\nThus we may join $x$ and $y$ with balls contained entirely in the constructed chain whose radii depend only on $c'$ and $c''$.\n\nNow we are proving $(4)$.  We shall show that there is $c>c'$ such that every $D_m$ satisfies (4) with $c$ for $m$ big enough. To do it let us cover $\\partial D$ with a finite number of balls $B_j$, $j=1,\\ldots,N$, from condition (4) and let $B'_j$ be a ball contained relatively in $B_j$ such that $\\{B_j\\}$ covers $\\partial D$, as well. Let $\\Phi_j$ be mappings corresponding to $B_j$. Let $\\eps$ be such that any ball of radius $\\eps$ intersecting $\\partial D$ non-emptily is relatively contained in $B_j'$ for some $j$. Observe that any ball $B$ of radius $\\eps\/2$ intersecting non-emptily $\\partial D_m$ is contained in a ball of radius $\\eps$ intersecting non-emptily $\\partial D$; hence it is contained in $B_j'$ for some $j$. Then the pair $B$, $\\Phi_j$ satisfies the conditions (4) (b), (c) and (d). Therefore, it suffices to check that there is $c>2\/\\eps$ such that each pair $B_j'$, $\\Phi_j$ satisfies the condition (4) for $D_m$ with $c$ ($m>>1$). This is possible since $\\Phi_j(D_m)\\subset\\Phi_j(D)$, $D^\\alpha\\Phi_j(\\pa D_m\\cap B_j)$ converges to $D^\\alpha\\Phi_j(\\pa D\\cap B_j)$ for $|\\alpha|\\leq 2$ and for any $w\\in\\Phi(\\pa D\\cap B_j)$ there is a ball of radius $2\/\\eps$ containing $\\Phi_j(D)$ and tangent to $\\partial\\Phi_j(D)$ at $w$. To be precise, we proceed as follows. \n\nLet $a,b\\in\\CC^n$ and let $x\\in\\pa B_n(a,\\wi c)$, where $\\wi c>c'$. Then a ball $B_n(2a-x,2\\wi c)$ contains $B_n(a,\\wi c)$ and is tangent to $B_n(a,\\wi c)$ at $x$. There is a number $\\eta=\\eta(\\delta,\\wi c)>0$, independent of $a,b,x$, such that the diameter of the set $B_n(b,\\wi c)\\setminus B_n(2a-x,2\\wi c)$ is smaller than $\\delta>0$, whenever $|a-b|<\\eta$ (this is a simple consequence of the triangle inequality).\n\nLet $\\wi s>0$ be such that $\\mathcal H(r\\circ\\Phi_j^{-1})(x;X)\\geq 2\\wi s|X|^2$ for $x\\in U_j$, $j=1,\\ldots,N$, where $U_j$ is an open neighborhood of $\\Phi_j(\\partial D\\cap B_j)$. Then, for $m$ big enough, $\\mathcal H(r_m\\circ \\Phi_j^{-1})(x;X)\\geq\\wi s|X|^2$ for $x\\in U_j$ and $\\Phi_j(\\partial D_m\\cap B_j')\\subset U_j$, $j=1,\\ldots,N$. Repeating for the function $$x\\longmapsto(r_m\\circ\\Phi_j^{-1})(x)-\\wi t(|x-\\wi b|^2-\\wi R^2)$$ the argument used in the interior ball condition with suitable chosen $\\wi t$ and uniform $\\wi R>c$, we find that there is uniform $\\wi\\eps>0$ such that for any $j,m$ and $w\\in\\Phi_j(\\partial D_m\\cap B_j')$ there is a ball $B$ of radius $\\wi R$, tangent to $\\Phi_j(\\partial D_m\\cap B_j')$ at $w$, such that $\\Phi_j(\\partial D_m\\cap B_j')\\cap B_n(w,\\wi\\eps)\\subset B$. Let $a_{j,m}(w)$ denote its center.\n\nOn the other hand for any $w\\in \\Phi_j(\\partial D_m\\cap B_j')$ there is $t>0$ such that $w'=w+t\\nu (w)\\in \\Phi_j(\\partial D\\cap B_j)$, where $\\nu(w)$ is a normal vector to $\\Phi_j(\\partial D_m\\cap B_j')$ at $w$. Let $a_j(w')$ be a center of a ball of radius $\\wi R$ tangent to $\\Phi_j(\\partial D\\cap B_j)$ at $w'$. It follows that $|a_{j,m}(w)-a_j(w')|<\\eta(\\wi\\eps\/2,\\wi R)$ provided that $m$ is big enough. \n\nJoinining the facts presented above, we finish the proof of the exterior ball condition (with a radius dependent only on $\\wi\\eps$ and $\\wi R$).\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorems \\ref{lem-car} and \\ref{main} \\emph{(}$\\mathcal C^2$-smooth case$)$]\nLosing no generality assume that $0\\in D\\Subset\\BB_n$.\n\nIt follows from the Weierstrass Theorem that there is sequence $\\{P_k\\}$ of real polynomials on $\\CC^n\\simeq\\mathbb R^{2n}$ such that $$D^{\\alpha}P_{k}\\to D^{\\alpha}r \\text{ uniformly on }\\ov\\BB_n,$$ where $\\alpha=(\\alpha_1,\\ldots, \\alpha_{2n})\\in \\mathbb N_0^{2n}$ is such that $|\\alpha|=\\alpha_1+\\ldots +\\alpha_{2n}\\leq 2$. Consider the open set $$\\wi D_{k,\\eps}:=\\{x\\in \\mathbb C^n:P_{k}(x)+\\eps<0\\}.$$ Let $\\eps_{m}$ be a sequence of positive numbers converging to $0$ such that $3\\eps_{m+1}<\\eps_m.$\n\nFor any $m\\in \\mathbb N$ there is $k_{m}\\in\\NN$ such that $\\sup_{\\ov\\BB_n}|P_{k_{m}}-r|<\\eps_{m}$. Putting $r_{m}:=P_{k_{m}}+2\\eps_{m}$, we get $r+\\eps_{m}<r_{m}<r+3\\eps_{m}$. In particular, $r_{m+1}<r_m.$\n\nLet $D_m$ be a connected component of $D_{k_m,2\\eps_m}$ containing $0$. It is a bounded strongly linearly convex domain with real analytic boundary and $r_m$ is its defining function provided that $m$ is big enough. Moreover, $D_{m}\\subset D_{m+1}$ and $\\bigcup_m D_{m}=D$. Using properties of holomorphically invariant functions and metrics we get Theorem~\\ref{lem-car}.\n\nWe are left with showing the claim that for any different $z,w\\in D$ (resp. $z\\in D$, $v\\in(\\mathbb C^n)_*$) there is a weak $E$-mapping for $z,w$ (resp. for $z,v$). Fix $z\\in D$ and $w\\in D$ (resp. $v\\in(\\mathbb C^n)_*$). Then $z,w\\in D_m$ (resp. $z\\in D_m$), $m>>1$. Therefore, for any $m>>1$ one may find an $E$-mapping $f_m$ of $D_m$ for $z,w$ (resp. for $z,v$). Since $(D_m,z)\\in \\mathcal D(c)$ for some uniform $c>0$ ($m>>1$) (Lemma~\\ref{un}), we find that $f_m$, $\\wi f_m$ and $\\rho_m$ satisfy the uniform estimates from Section~\\ref{22}. Thus, passing to a subsequence we may assume that $\\{f_m\\}$ converges uniformly on $\\CDD$ to a mapping $f\\in\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$ passing through $z,w$ (resp. such that $f(0)=z$, $f'(0)=\\lambda v$, $\\lambda>0$), $\\{\\wi f_m\\}$ converges uniformly on $\\CDD$ to a mapping $\\wi f\\in\\OO(\\DD)\\cap\\mathcal C^{1\/2}(\\overline{\\mathbb D})$ and $\\{\\rho_m\\}$ is convergent uniformly on $\\TT$ to a positive function $\\rho\\in\\cC^{1\/2}(\\TT)$ (in particular, $f'\\bullet\\wi f=1$ on $\\DD$, so $\\wi f$ has no zeroes in $\\CDD$). We already know that this implies that $f$ is a weak $E$-mapping of $D$.\n\nTo get $\\cC^{k-1-\\eps}$-smoothness of the extremal $f$ and its associated mappings for $k\\geq 3$, it suffices to repeat the proof of Proposition~5 of \\cite{Lem2}. This is just the Webster Lemma (we have proved it in the real analytic case --- see Proposition~\\ref{6}). Namely, let $$\\psi:\\partial D\\ni z\\longmapsto(z,T_{D}^\\mathbb{C}(z))\\in \\mathbb C^n\\times(\\mathbb P^{n-1})_*,$$ where $\\mathbb P^{n-1}$ is the $(n-1)$-dimensional complex projective space. Let $\\pi:(\\CC^n)_*\\longrightarrow\\mathbb P^{n-1}$ be the canonical projection. \n\nBy \\cite{Web}, $\\psi(\\partial D)$ is a totally real manifold of $\\mathcal C^{k-1}$ class. Observe that the mapping $(f,\\pi\\circ \\wi f):\\CDD\\longrightarrow\\CC^n\\times\\mathbb P^{n-1}$ is $1\/2$-H\\\"older continuous, is holomorphic on $\\mathbb D$ and maps $\\mathbb T$ into $\\psi(\\partial D)$. Therefore, it is $\\mathcal C^{k-1-\\eps}$-smooth for any $\\eps>0$, whence $f$ is $\\mathcal C^{k-1-\\eps}$-smooth. Since $\\nu_D\\circ f$ is of class $\\mathcal C^{k-1-\\eps}$, it suffices to proceed as in the proof of Proposition~\\ref{6}.\n\\end{proof}\n\n\n\n\\section{Appendix}\\label{Appendix}\n\\subsection{Totally real submanifolds}\nLet $M\\subset\\CC^m$ be a totally real local $\\CLW$ submanifold of the real dimension $m$. Fix a point $z\\in M$. There are neighborhoods $U_0\\su\\RR^m$, $V_0\\su\\CC^m$ of $0$ and $z$ and a $\\CLW$ diffeomorphism $\\widetilde{\\Phi}:U_0\\longrightarrow M\\cap V_0$ such that $\\widetilde{\\Phi}(0)=z$. The mapping $\\widetilde{\\Phi}$ can be extended in a natural way to a mapping $\\Phi$ holomorphic in a neighborhood of $0$ in $\\CC^m$. Note that this extension will be biholomorphic in a neighborhood of $0$. Actually, we have $$\\frac{\\partial\\Phi_j}{\\partial z_k}(0)=\\frac{\\partial\\Phi_j}{\\partial\nx_k}(0)=\\frac{\\partial\\widetilde{\\Phi}_j}{\\partial x_k}(0),\\ j,k=1,\\ldots,m,$$ where $x_k=\\re z_k$. Suppose that the complex derivative $\\Phi'(0)$ is not an isomorphism. Then there is $X\\in(\\CC^m)_*$ such that $\\Phi'(0)X=0$, so \\begin{multline*}0=\\sum_{k=1}^m\\frac{\\partial\\Phi}{\\partial z_k}(0)X_k=\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)(\\re X_k+i\\im X_k)=\\\\=\\underbrace{\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)\\re X_k}_{=:A}+i\\underbrace{\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)\\im X_k}_{=:B}.\\end{multline*}\nThe vectors $$\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0),\\ k=1,\\ldots,m$$ form a basis of $T^{\\RR}_M(z)$, so $A,B\\in T^{\\RR}_M(z)$, consequently $A,B\\in iT^{\\RR}_M(z)$. Since $M$ is totally real, i.e. $T^{\\RR}_M(z)\\cap iT^{\\RR}_M(z)=\\{0\\}$, we have $A=B=0$. By a property of the basis we get $\\re X_k=\\im X_k=0$, $k=1,\\ldots,m$ --- a contradiction.\n\nTherefore, $\\Phi$ in a neighborhood of $0$ is a biholomorphism of two open subsets of $\\CC^m$, which maps a neighborhood of $0$ in $\\RR^m$ to a neighborhood of $z$ in $M$.\n\n\n\\begin{lemm}[Reflection Principle]\\label{reflection}\nLet $M\\subset\\CC^m$ be a totally real local $\\CLW$ submanifold of the real\ndimension $m$. Let $V_0\\subset\\CC$ be a neighborhood of $\\zeta_0\\in\\TT$ and let $g:\\overline{\\DD}\\cap V_0\\longrightarrow\\CC^m$ be a continuous mapping. Suppose that $g\\in\\OO(\\DD\\cap V_0)$ and $g(\\TT\\cap V_0)\\subset M$. Then $g$ can be extended holomorphically past $\\TT\\cap V_0$.\n\\end{lemm}\n\\begin{proof}\nIn virtue of the identity principle it is sufficient to extend $g$ locally\npast an arbitrary point $\\zeta_0\\in\\TT\\cap V_0$. For a point $g(\\zeta_0)\\in M$ take $\\Phi$ as above. Let $V_1\\subset V_0$ be a neighborhood of $\\zeta_0$ such that $g(\\CDD\\cap V_1)$ is contained in the image\nof $\\Phi$. The mapping $\\Phi^{-1}\\circ g$ is holomorphic in $\\DD\\cap V_1$ and has\nreal values on $\\TT\\cap V_1$. By the ordinary Reflection Principle we can\nextend this mapping holomorphically past $\\TT\\cap V_1$. Denote this extension by\n$h$. Then $\\Phi\\circ h$ is an extension of $g$ in a neighborhood of $\\zeta_0$.\n\\end{proof}\n\n\n\n\n\\subsection{Schwarz Lemma for the unit ball}\n\\begin{lemm}[Schwarz Lemma]\\label{schw}\nLet $f\\in\\OO(\\DD,B_n(a,R))$ and $r:=|f(0)-a|$. Then $$|f'(0)|\\leq \\sqrt{R^2-r^2}.$$\n\\end{lemm}\n\n\n\\subsection{Some estimates of holomorphic functions of $\\cC^{\\alpha}$-class}\n\nLet us recall some theorems about functions holomorphic in $\\DD$ and continuous in $\\CDD$. Concrete values of constants $M,K$ are possible to calculate, seeing on the proofs. In fact, it is only important that they do not depend on functions.\n\\begin{tww}[Hardy, Littlewood, \\cite{Gol}, Theorem 3, p. 411]\\label{lit1}\nLet $f\\in\\OO(\\DD)\\cap\\cC(\\CDD)$. Then for $\\alpha\\in(0,1]$ the following conditions are equivalent\n\\begin{eqnarray}\\label{47}\\exists M>0:\\ |f(e^{i\\theta})-f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^{\\alpha},\\ \\theta,\\theta'\\in\\RR;\\\\\n\\label{45}\\exists K>0:\\ |f'(\\zeta)|\\leq K(1-|\\zeta|)^{\\alpha-1},\\ \\zeta\\in\\DD.\n\\end{eqnarray}\nMoreover, if there is given $M$ satisfying \\eqref{47} then $K$ can be chosen as $$2^{\\frac{1-3\\alpha}{2}}\\pi^\\alpha M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt$$ and if there is given $K$ satisfying \\eqref{45} then $M$ can be chosen as $(2\/\\alpha+1)K$.\n\\end{tww}\n\\begin{tww}[Hardy, Littlewood, \\cite{Gol}, Theorem 4, p. 413]\\label{lit2}\nLet $f\\in\\OO(\\DD)\\cap\\cC(\\CDD)$ be such that $$|f(e^{i\\theta})-f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^{\\alpha},\\ \\theta,\\theta'\\in\\RR,$$ for some $\\alpha\\in(0,1]$ and $M>0$. Then $$|f(\\zeta)-f(\\zeta')|\\leq K|\\zeta-\\zeta'|^{\\alpha},\\\n\\zeta,\\zeta'\\in\\CDD,$$ where $$K:=\\max\\left\\{2^{1-2\\alpha}\\pi^\\alpha M,2^{\\frac{3-5\\alpha}{2}}\\pi^\\alpha\\alpha^{-1} M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt\\right\\}.$$\n\\end{tww}\n\\begin{tww}[Privalov, \\cite{Gol}, Theorem 5, p. 414]\\label{priv}\nLet $f\\in\\OO(\\DD)$ be such that $\\re f$ extends continuously on $\\CDD$ and $$|\\re f(e^{i\\theta})-\\re f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^\\alpha,\\ \\theta,\\theta'\\in\\RR,$$ for some $\\alpha\\in(0,1)$ and $M>0$. Then $f$ extends continuously on $\\CDD$ and $$|f(\\zeta)-f(\\zeta')|\\leq K|\\zeta-\\zeta'|^\\alpha,\\ \\zeta,\\zeta'\\in\\CDD,$$ where $$K:=\\max\\left\\{2^{1-2\\alpha}\\pi^\\alpha,2^{\\frac{3-5\\alpha}{2}}\\pi^\\alpha\\alpha^{-1}\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt\\right\\}\\left(\\frac{2}{\\alpha}+1\\right)2^{\\frac{3-3\\alpha}{2}}\\pi^{\\alpha}M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt.$$\n\\end{tww}\n\\subsection{Sobolev space}\nThe Sobolev space $W^{2,2}(\\TT)=W^{2,2}(\\TT,\\CC^m)$ is a space of functions $f:\\TT\\longrightarrow\\CC^m$, whose first two derivatives (in the sense of distribution) are in $L^2(\\TT)$ (here we use a standard identification of functions on the unit circle and functions on the interval $[0,2\\pi]$). Then $f$ is $\\mathcal C^1$-smooth.\n\nIt is a complex Hilbert space with the following scalar product\n$$\\langle f,g\\rangle_W:=\\langle f,g\\rangle_{L}+\\langle f',g'\\rangle_{L}+\\langle f'',g''\\rangle_{L},$$\nwhere $$\\langle\\wi f,\\wi g\\rangle_{L}:=\\frac{1}{2\\pi}\\int_0^{2\\pi}\\langle\\wi f(e^{it}),\\wi g(e^{it})\\rangle dt.$$ Let $\\|\\cdot\\|_L$, $\\|\\cdot\\|_W$ denote the norms induced by $\\langle\\cdotp,-\\rangle_L$ and $\\langle\\cdotp,-\\rangle_W$. The following characterization simply follows from Parseval's identity $$W^{2,2}(\\TT)=\\left\\{f\\in L^2(\\TT):\\sum_{k=-\\infty}^{\\infty}(1+k^2+k^4)|a_k|^2<\\infty\\right\\},$$ where $a_k\\in\\CC^m$ are the $m$-dimensional Fourier coefficients of $f$, i.e. $$f(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^k,\\ \\zeta\\in\\TT.$$ More precisely, Parseval's identity gives $$\\|f\\|_W=\\sqrt{\\sum_{k=-\\infty}^{\\infty}(1+k^2+k^4)|a_k|^2},\\ f\\in W^{2,2}(\\TT).$$ Note that $W^{2,2}(\\TT)\\su\\mc{C}^{1\/2}(\\TT)\\su\\mc{C}(\\TT)$ and both inclusions are continuous (in particular, both inclusions are real analytic). Note also that\n \\begin{equation}\\label{67}\\|f\\|_{\\sup}\\leq\\sum_{k=-\\infty}^{\\infty}|a_k|\\leq\\sqrt{\\sum_{k=-\\infty}^{\\infty}\\frac{1}{1+k^2}\\sum_{k=-\\infty}^{\\infty}(1+k^2)|a_k|^2}\\leq\\frac{\\pi}{\\sqrt 3}\\|f\\|_W.\\end{equation}\\\\\n\nNow we want to show that there exists $C>0$ such that $$\\|h^\\alpha\\|_W\\leq C^{|\\alpha|}\\|h_1\\|^{\\alpha_1}_W\\cdotp\\ldots\\cdotp\\|h_{2n}\\|^{\\alpha_{2n}}_W,\\quad h\\in W^{2,2}(\\TT,\\CC^n),\\,\\alpha\\in\\NN_0^{2n}.$$ Thanks to the induction it suffices to prove that there is $\\wi C>0$ satisfying $$\\|h_1h_2\\|_W\\leq\\wi C\\|h_1\\|_W\\|h_2\\|_W,\\quad h_1,h_2\\in W^{2,2}(\\TT,\\CC).$$ Using \\eqref{67}, we estimate $$\\|h_1h_2\\|^2_W=\\|h_1h_2\\|^2_L+\\|h_1'h_2+h_1h_2'\\|^2_L+\\|h_1''h_2+2h_1'h_2'+h_1h_2''\\|^2_L\\leq$$$$\\leq C_1\\|h_1h_2\\|_{\\sup}^2+(\\|h_1'h_2\\|_L+\\|h_1h_2'\\|_L)^2+(\\|h_1''h_2\\|_L+\\|2h_1'h_2'\\|_L+\\|h_1h_2''\\|_L)^2\\leq$$\\begin{multline*}\\leq C_1\\|h_1\\|_{\\sup}^2\\|h_2\\|_{\\sup}^2+(C_2\\|h_1'\\|_L\\|h_2\\|_{\\sup}+C_2\\|h_1\\|_{\\sup}\\|h_2'\\|_L)^2+\\\\+(C_2\\|h_1''\\|_L\\|h_2\\|_{\\sup}+C_2\\|2h_1'h_2'\\|_{\\sup}+C_2\\|h_1\\|_{\\sup}\\|h_2''\\|_L)^2\\leq\\end{multline*}\\begin{multline*}\\leq C_3\\|h_1\\|_W^2\\|h_2\\|_W^2+(C_4\\|h_1\\|_W\\|h_2\\|_W+C_4\\|h_1\\|_W\\|h_2\\|_W)^2+\\\\+(C_4\\|h_1\\|_W\\|h_2\\|_W+2C_2\\|h_1'\\|_{\\sup}\\|h_2'\\|_{\\sup}+C_4\\|h_1\\|_W\\|h_2\\|_W)^2\\leq\\end{multline*}$$\\leq C_5\\|h_1\\|_W^2\\|h_2\\|_W^2+(2C_4\\|h_1\\|_W\\|h_2\\|_W+2C_2\\|h_1'\\|_{\\sup}\\|h_2'\\|_{\\sup})^2$$ with constants $C_1,\\ldots,C_5$. Expanding $h_j(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a^{(j)}_k\\zeta^{k}$, $\\zeta\\in\\TT$, $j=1,2$, we obtain $$\\|h_j'\\|_{\\sup}\\leq\\sum_{k=-\\infty}^{\\infty}|k||a^{(j)}_k|\\leq\\sqrt{\\sum_{k\\in\\ZZ_*}\\frac{1}{k^2}\\sum_{k\\in\\ZZ_*}k^4|a^{(j)}_k|^2}\\leq\\frac{\\pi}{\\sqrt 3}\\|h_j\\|_W$$ and finally $\\|h_1h_2\\|^2_W\\leq C_6\\|h_1\\|_W^2\\|h_2\\|_W^2$ for some constant $C_6$.\n\\subsection{Matrices}\n\\begin{propp}[Lempert, \\cite{Lem2}, Th\\'eor\\`eme $B$]\\label{12}\nLet $A:\\TT\\longrightarrow\\CC^{n\\times n}$ be a matrix-valued real analytic mapping such\nthat $A(\\zeta)$ is self-adjoint and strictly positive for any $\\zeta\\in\\TT$. Then there exists $H\\in\\OO(\\CDD,\\CC^{(n-1)\\times(n-1)})$ such that $\\det H\\neq 0$ on $\\CDD$ and $HH^*=A$ on $\\TT$.\n\\end{propp}\nIn \\cite{Lem2}, the mapping $H$ was claimed to be real analytic in a neighborhood of $\\CDD$ and holomorphic in $\\DD$, but it is equivalent to $H\\in\\OO(\\CDD)$. Indeed, since $\\ov\\pa H$ is real analytic near $\\CDD$ and $\\ov\\pa H=0$ in $\\DD$, the identity principle for real analytic functions implies $\\ov\\pa H=0$ in a neighborhood of $\\CDD$.\n\\begin{propp}[\\cite{Tad}, Lemma $2.1$]\\label{59}\nLet $A$ be a complex symmetric $n\\times n$ matrix. Then $$\\|A\\|=\\sup\\{|z^TAz|:z\\in\\CC^n,\\,|z|=1\\}.$$\n\\end{propp}\n\\bigskip\n\\textsc{Acknowledgements.} We would like to thank Sylwester Zaj\\k ac for helpful discussions. We are also grateful to our friends for the participation in preparing some parts of the work.\n\\medskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:introduction}}\nBrillouin scattering (BS) refers to the nonlinear interaction between optical and mechanical fields inside a material. BS has been widely exploited in optical fibers to implement a wide range of devices, including optical amplifiers, ultra-narrow linewidth lasers, radio-frequency (RF) signal generators, and distributed sensors \\cite{garmire2017perspectives}.\n\nBrillouin scattering was for long thought to be mediated by electrostrictive forces only. Thus, its spectrum was considered to be governed by material properties \\cite{wiederhecker_controlling_2009}. In 2006, microstructuration of optical fibers enabled shaping the BS spectrum \\cite{dainese2006stimulated}, opening a new path for geometric control of this effect \\cite{beugnot2007guided}. In 2012, a new theory \\cite{peter_t_rakich_giant_2012} predicted that Brillouin interactions could be greatly magnified by strong radiation\npressure on the boundaries of suspended silicon waveguides with nanometric-scale core sizes \\cite{qiu_stimulated_2013,wolff_stimulated_2015}. The simultaneous confinement of optical and mechanical modes is challenging in silicon-on-insulator (SOI) waveguides due to a strong phonon leakage towards the silica cladding \\cite{eggleton2019brillouin_LL,wiederhecker_brillouin_2019,safavi-naeini_controlling_2019}. However, this limitation can be circumvented by isolating the silicon waveguide core by complete or partial removal of the silica cladding \\cite{shin_tailorable_2013,laer_net_2015,peter_t_rakich_giant_2012}. Suspended or quasi-suspended structures such as silicon membrane rib waveguides \\cite{kittlaus_large_2016} and fully suspended silicon nanowires \\cite{laer_net_2015} have demonstrated large Brillouin gain. These results generated a great scientific interest for its potential for laser sources \\cite{otterstrom_silicon_2018}, microwave signal generation \\cite{li_microwave_2013} and processing \\cite{liu_chip-based_2018}, sensing applications \\cite{chow_distributed_2018, lai_earth_2020} and non-reciprocal optical devices \\cite{kittlaus_non-reciprocal_2018}.\nIn particular, pedestal waveguides \\cite{van_laer_interaction_2015} yield an experimental Brillouin gain of 3000 W$^{-1}$m$^{-1}$. \nHowever, the need for narrow-width pedestals to optimize the Brillouin gain complicates the fabrication process and may compromise the mechanical stability of the structures. On the other hand, a lower experimental Brillouin gain (1000 W$^{-1}$m$^{-1}$) was obtained for silicon membrane rib waveguides due to the very different confinement of optical and mechanical modes \\cite{kittlaus_large_2016}. Still, this comparatively modest Brillouin gain was compensated by achieving ultra-low optical propagation loss, allowing the demonstration of lasing effect \\cite{otterstrom_silicon_2018}. The use of photonic crystals with simultaneous photonic and phononic bandgaps \\cite{zhang2017design} (also referred to as phoxonic crystals) has been proposed to maximize the Brillouin gain in silicon membrane waveguides, achieving calculated values up to 8000 W$^{-1}$m$^{-1}$. Yet, the narrow bandwidth and high optical propagation loss, typically linked to bandgap confinement \\cite{baba_slow_2008}, may compromise the performance of these phoxonic crystals. \n\nSubwavelength grating silicon waveguides, with periods shorter than half of the wavelength of the guided light, exploit index-contrast confinement to yield low optical loss and wideband operation \\cite{halir_waveguide_2015,cheben2018subwavelength}. Interestingly, near-infrared photons and GHz phonons in nanoscale Si waveguides have comparable wavelengths (near 1 \\textmu m) \\cite{safavi-naeini_controlling_2019}. Thus, the same periodic structuration could operate in the subwavelength regime for both, photons and phonons. In addition, forward Brillouin scattering (FBS), used to demonstrate Brillouin gain in Si, relies on longitudinally propagating photons and transversally propagating phonons \\cite{eggleton2019brillouin_LL,wiederhecker_brillouin_2019, safavi-naeini_controlling_2019}. Hence, engineering the longitudinal and transversal subwavelength geometries would allow independent control of photonic and phononic modes. Brillouin optimization in silicon membranes has been proposed based on index-contrast confinement of photons (longitudinal subwavelength grating) and bandgap confinement of phonons (transversal phononic crystal) \\cite{schmidt2019suspended}, achieving a calculated gain of 1750 W$^{-1}$m$^{-1}$. More recently, the combination of subwavelength index-contrast and subwavelength softening has been proposed to optimize Brillouin gain in suspended Si waveguides, achieving a calculated value of 3000 W$^{-1}$m$^{-1}$, for a minimum feature size of 50 nm \\cite{zhang_subwavelength_2020}. Still, these two approaches require several etch steps of the silicon core, complicating the device's fabrication. In this work, we propose a novel subwavelength-structured Si membrane, illustrated in Fig. \\ref{fig:structure}, requiring only one etch step of silicon. We develop an optimization method to design the waveguide geometry, combining multi-physics optical and mechanical simulations with a genetic algorithm (GA) capable of handling a large number of parameters \\cite{hakansson_generating_2019}. The optimized geometry yields a calculated Brillouin gain of 3300 W$^{-1}$m$^{-1}$, with a minimum feature size of 50 nm, compatible with electron-beam lithography.\n\n\\section{Design and Results \\label{sec:Results}}\nThe proposed optomechanical waveguide geometry, depicted in Fig. \\ref{fig:structure}, comprises a suspended central strip of width $W_g=400$ nm that is anchored to the lateral silicon slabs by a lattice of arms with a longitudinal period ($z$-direction) of $\\Lambda=300$ nm. This period is shorter than half of the optical wavelength, ensuring optical operation in the subwavelength regime. The anchoring arms are symmetric with respect to the waveguide center. We split the arms into five different sections with widths and lengths of $W_i$ ($x$-direction) and $L_i$ ($z$-direction), respectively. The index $i=1$ refers to the section adjacent to the waveguide core, while the index $i=5$ refers to the outermost section (see Fig. \\ref{fig:structure}, inset). The fifth section has a fixed width of $W_5=500$ nm and length of $L_5=50$ nm to ensure proper guidance and localization of the optical mode. The widths and lengths of sections 1 to 4 are optimized using the genetic algorithm.  The whole waveguide has a fixed silicon thickness of $t=220$ nm, allowing fabrication in a single-etch step.\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig1_structure_single.png}\n    \\caption{Proposed optomechanical waveguide. In the inset, the different sections of the anchoring arms are numbered from $1$ to $5$. The width of the waveguide core ($W_g=400$ nm), the period ($\\Lambda=300$ nm), and the dimensions of the outermost section ($L_5=50$ nm, $W_5=500$ nm) remain fixed throughout the optimization process. The thickness of the silicon slab is $t=220$ nm.}\n    \\label{fig:structure}\n\\end{figure}\n\nWe focus on FBS, where only near-cut-off acoustic modes are involved. In the absence of optical absorption, which is the case of silicon at near-infrared wavelengths, the optical and mechanical mode equations describing FBS decouple and can be solved separately \\cite{safavi-naeini_controlling_2019}. We use here COMSOL Multiphysics software for the optomechanical simulations. For the calculation of optical and mechanical modes in the optimization process, we reduce the 3D structure to an equivalent 2D geometry. The effective index method \\cite{chen_foundations_2005} is considered for the computation of the transverse-electric (TE) polarized optical modes while the in-plane mechanical modes are calculated assuming the plane stress approximation \\cite{auld_acoustic_1973}. We compute the Brillouin gain, $G_\\mathrm{B}$, as \\cite{wiederhecker_brillouin_2019}\n\\begin{equation}\n    G_\\mathrm{B}(\\Omega_\\mathrm{m}) =  Q_\\mathrm{m} \\, \\frac{2 \\omega_\\mathrm{p}}{m_\\mathrm{eff} \\,  \\Omega_\\mathrm{m}^2} \\, \\left| \\int f_\\mathrm{MB} \\,\\mathrm{d}\\ell + \\int f_\\mathrm{PE} \\,\\mathrm{d} A \\right|^2 ,\n\\label{eq:GB}\n\\end{equation} \nwhere $\\omega_\\textrm{p}$ is the frequency of the optical pump, $\\Omega_\\mathrm{m}$ is the mechanical frequency, $Q_\\mathrm{m}$ is the mechanical quality factor, $m_\\mathrm{eff} = \\int\\rho\\,|\\mathbf{u}_\\mathrm{m}|^2\/\\max|\\mathbf{u}_\\mathrm{m}|^2 \\,\\mathrm{d}A$ is the effective linear mass density of the mechanical mode with displacement profile $\\mathbf{u}_\\mathrm{m}$, and $f_\\mathrm{MB}$ and $f_\\mathrm{PE}$ are the linear and surface overlap of optical force density and deformation representing the moving boundaries effect (MB) and the photoelastic effect (PE), respectively,\n\\begin{align}\n    & f_\\mathrm{MB} = \\frac{\\mathbf{u}_\\mathrm{m}^*\\cdot\\mathbf{n} \\, \\left(\\delta\\varepsilon_\\mathrm{MB} \\, \\mathbf{E}^*_\\mathrm{p,t}\\cdot \\mathbf{E}_\\mathrm{s,t} - \\delta\\varepsilon_\\mathrm{MB}^{-1} \\, \\mathbf{D}_\\mathrm{p,n}^*\\cdot\\mathbf{D}_\\mathrm{s,n}\\right)}{\\max|\\mathbf{u}_\\mathrm{m}| \\, P_\\mathrm{p} \\, P_\\mathrm{s}} \\nonumber \\\\\n    & \\mathrm{and} \\quad f_\\mathrm{PE} = \\frac{\\mathbf{E}^*_\\mathrm{p}\\cdot \\delta\\varepsilon_\\mathrm{PE}^* \\cdot \\mathbf{E}_\\mathrm{s}}{\\max|\\mathbf{u}_m| \\, P_\\mathrm{p} \\, P_\\mathrm{s}} ,\n\\label{eq:MB_PE}\n\\end{align}\nwhere the permittivity differences due to the moving boundaries effects are given by $\\delta\\varepsilon_\\mathrm{MB} = \\varepsilon_1 - \\varepsilon_2$ and $\\delta\\varepsilon_\\mathrm{MB}^{-1} = 1\/\\varepsilon_1 - 1\/\\varepsilon_2$, with $\\varepsilon_i=\\varepsilon_0 n_i^2$ being the permittivities of the silicon ($i=1$) and air ($i=2$). The photoelastic tensor perturbation in the material permittivity is $\\delta\\varepsilon_\\mathrm{PE} = -\\varepsilon_0 \\, n^4 \\, \\mathbf{p}:\\mathbf{S}$, with $n$ being the material refractive index, $\\mathbf{p}$ the photoelastic tensor, and $\\mathbf{S}$ the mechanical stress tensor induced by the mechanical mode.  The term $\\mathbf{u}_\\mathrm{m}\\cdot\\mathbf{n}$ is the normal component of the mechanical displacement and $\\mathbf{E}_{j,\\mathrm{t}}$ and $\\mathbf{D}_{j,\\mathrm{n}}$ are the tangential electric field and normal dielectric displacement for the pump ($j=\\mathrm{p}$) and the scattered field ($j=\\mathrm{s}$). The denominator represents the power normalization given by $P_j = [2 \\Re(\\int [\\mathbf{E}_j\\times\\mathbf{H}_j^*] \\cdot \\mathbf{z} \\, \\mathrm{d}A)]^{1\/2}$.\n\nThe symmetry directions $[100]$, $[010]$, and $[001]$ of the crystalline silicon are set to coincide with the $x$, $y$, and $z$ simulation axis, respectively. With this orientation, the photoelastic tensor \\cite{qiu_stimulated_2013,rakich_tailoring_2010} is $[p_{11},p_{12},p_{44}]=[-0.094,0.017,-0.051]$. The refractive index of silicon is $n=3.45$ and its density $\\rho=2329$ kg m$^{-3}$ while the corresponding values for the air are $n=1$ and $\\rho=1.293$ kg m$^{-3}$. \n\nThe quality factor of the mechanical mode, $Q_\\mathrm{m}$, is related to the full width at half maximum (FWHM) of the gain spectrum, $\\gamma_\\mathrm{m}$, through $Q_\\mathrm{m}=\\Omega_\\mathrm{m}\/\\gamma_\\mathrm{m}$ and it is limited by different loss mechanisms, \n\\begin{equation}\n    \\frac{1}{Q_\\mathrm{m}} = \\frac{1}{Q_\\mathrm{TE}} + \\frac{1}{Q_\\mathrm{L}} + \\frac{1}{Q_\\mathrm{air}}.\n\\label{eq:Q}\n\\end{equation}\nHere, we consider the thermoelastic loss ($Q_\\mathrm{TE}$), the mechanical leakage towards the silica under-cladding ($Q_\\mathrm{L}$), and the viscous loss from surrounding air ($Q_\\mathrm{air}$). The thermoelastic loss yields mechanical quality factors of $Q_\\mathrm{TE}\\sim6\\cdot10^5$ \\cite{comsol_2018} for silicon nanostructures while the leakage loss is mainly governed by the geometries of the waveguide and the arms anchoring it to the lateral silicon slab. These two effects are directly considered in the mechanical-mode simulations performed in COMSOL Multiphysics. The viscous loss induced by the surrounding air is considered here by imposing a limiting value to the mechanical quality factor of $Q_\\mathrm{m}=4\\cdot10^3$, which is the highest expected value at atmospheric pressure and room temperature for phonon frequency in the order of GHz \\cite{ghaffari_quantum_2013}.\n\nBased on the resulting optomechanical coupling calculations, a genetic algorithm \\cite{xin-she_yang_chapter_2021} is used to maximize the FBS gain. Starting with randomly generated combinations of parameters $W_i$ and $L_i$ (individuals), optomechanical simulations are carried out and the individuals are ranked according to their Brillouin gain. Recombination is used to produce a successor set of individuals, the next generation. The best-performing individuals directly become part of the next generation (elitism). A large number of individuals of the new generation is obtained by combining the parameter of pairs of individuals from the current generation (crossover). Finally, the remaining individuals of the new generation are produced by randomly modifying the parameters of single individuals of the current generation (mutation). This process continues until the convergence criterion has been reached.\n\nIn our particular optimization problem, an individual is a possible geometry, represented by a set of 8 parameters (width and length of each of the arm sections). Each generation is composed of 50 individuals and the successive generations are obtained applying a rate of elitism and crossover of 6\\% and 80\\%, respectively, with the remaining elements obtained through mutation. The convergence criterion was defined in terms of the difference between the best and the average performance, $G_\\mathrm{B} - \\langle G_\\mathrm{B}\\rangle <$ 10 W$^{-1}$m$^{-1}$, over 10 generations. For this work, we have used a standard computer with the following specifications: a 64-bit operating system with an x64-based processor Intel\\textsuperscript{\\tiny\\textregistered} Core\\textsuperscript{\\tiny\\texttrademark} i7-4790 (4 total cores, 8 total threads, base-frequency of 3.60 GHz), and an installed RAM of 8.00 GB. Under these conditions, the optimization process was completed in 12h 35 min, comprising 1500 optomechanical simulations of 30 seconds each.\n\nThe method we propose here relies on a defined geometry whose parameters are allowed to vary within a specific range of values. Hence, the optimized structure will depend strongly on our initial guess.\n\nIn Fig. \\ref{fig:convergence}, we present the optimization process. Figures \\ref{fig:convergence}a and \\ref{fig:convergence}b show the Brillouin gain and mechanical frequency, respectively, as a function of the generation number. As a result of the evolution of the geometry, we observe an increase in the gain and a variation in the mechanical frequency. This result should be expected as the Brillouin shift in FBS is particularly sensitive to the waveguide dimensions. The optimum performance is achieved after 10 generations while 30 generations are required for convergence. The optimized geometry, whose dimensions are listed in Table \\ref{tab:geom}, is characterized by a Brillouin gain of  $G_\\mathrm{B}=3350$ W$^{-1}$m$^{-1}$ for a mechanical mode with frequency of $\\Omega_\\mathrm{m}=14.357$ GHz and mechanical quality factor of\n$Q_\\mathrm{m}\\approx3.2\\cdot10^3$. The optical mode has a mode effective index of 2.36 and wavelength in vacuum of $\\lambda=1556.5$ nm ($\\omega_\\mathrm{p}=2\\pi\\cdot192.6$ THz in (\\ref{eq:GB})).\n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig2_convergence_double.png}\n    \\caption{Optimization process. a) Best (in blue) and average (in orange) Brillouin gain as a function of the number of generations during genetic optimization. b) Evolution of the mechanical frequency as a function of the number of generations. During the optimization process, all possible mechanical losses are considered, including thermoelastic loss, mechanical leakage, and viscous loss due to air (operation in air ambient at room temperature).}\n    \\label{fig:convergence}\n\\end{figure}\n\nIn terms of geometry, the first and fourth sections, with considerably larger widths, generate reflections that help localize the mechanical mode in the waveguide core. The frequency of the mechanical mode is governed by the interplay between the waveguide width and the length of the partial cavity formed by the fourth section on each side.\n\n\\begin{table}[htb]\n    \\centering\n    \\caption{Dimensions for the GA-optimized geometry when operating in air ambient at room temperature. In the table above, S$_i$ stands for section $i$ in Fig. \\ref{fig:structure}.} \n    \\label{tab:geom}\n    \\begin{tabular}{@{}ccccc}\n    \\toprule\n         & S1 & S2 & S3 & S4 \\\\\n    \\midrule\n        Width & 170 nm & 320 nm & 330 nm & 100 nm \\\\\n        Length & 130 nm & 60 nm & 60 nm & 190 nm \\\\\n    \\bottomrule\n    \\end{tabular}\n\\end{table}\n\nFull 3D simulations are realized to verify the performance of the optimized geometry. This structure provides a Brillouin gain of $G_\\mathrm{B}=3310$ W$^{-1}$m$^{-1}$ for a mechanical mode with a frequency of $\\Omega_\\mathrm{m}=14.579$ GHz. The optical mode has a mode effective index of 2.23 and wavelength in vacuum of $\\lambda=1557.2$ nm ($\\omega_\\mathrm{p}=2\\pi\\cdot 192.52$ THz in (\\ref{eq:GB})). Figure \\ref{fig:modes} shows the calculated field distribution for the mechanical and optical modes in the optimized geometry.\n\n\\begin{figure}[htbp] \n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig3_modes_double.png}\n    \\caption{Optical and mechanical modes of the optimized geometry operating in air ambient and room temperature (table \\ref{tab:geom}): a) Approximated 2D structure. The upper structure corresponds to the normalized mechanical displacement at 14.357 GHz and the lower figure to the $x$-component of the electric field at 1556.5~nm (mode effective index 2.36). b) Full 3D device. On the bottom left, $x$-component of the electric field at 1557.2 nm (mode effective index 2.23), and on the top right, normalized mechanical displacement at 14.579 GHz.}\n\\label{fig:modes}\n\\end{figure}\n\nThese results show a good agreement between the approximated 2D geometry used for the optimization and the full 3D structure. The small discrepancies in the optical mode index and mechanical frequency are due to the influence of the thickness.  \n\nFinally, we study the fabrication tolerance of the proposed structure using again 3D simulations. We consider under- and over-etching errors that we model by a variation of all the waveguide lengths and widths by a factor $\\Delta$, measured in nm (Fig. \\ref{fig:fab_tolerance}a). Figure \\ref{fig:fab_tolerance}c shows the variation of the Brillouin gain (in blue) and mechanical frequency (in orange) as a  function of $\\Delta$. The Brillouin gain remains above 2000 W$^{-1}$m$^{-1}$ for geometry variations of $\\pm 10$\\,nm. It should be noted that for the over-etch case ($\\Delta<0$ in Fig. \\ref{fig:fab_tolerance}c), the Brillouin gain is larger than the optimized case due to the larger optomechanical coupling resulting from a better overlap of the mechanical mode  with the optical field. However, these smaller structures are incompatible with the target minimum feature size of 50 nm that was chosen to guarantee fabrication reliability. The mechanical frequency varies less than 2\\% (Fig.\\ref{fig:fab_tolerance}c, in orange) and the mechanical profile is not modified significantly.\n\nWe also study the effect of stitching errors, modeled by a deviation $\\zeta$ (in nm) of the arm axis at both sides of the waveguide core, hence breaking the symmetry of the structure (Fig. \\ref{fig:fab_tolerance}b). Figure \\ref{fig:fab_tolerance}d shows the variation of the Brillouin gain (in blue) and mechanical frequency (in orange) as a  function of $\\zeta$. A non-perfectly symmetric structure is slightly detrimental to the Brillouin gain but does not affect the mechanical frequency or profile. Interestingly, both parameters (Brillouin gain and mechanical frequency) remain constant over a large range of stitching errors.\n\nLastly, we examine the effect of random fabrication errors affecting each section independently (Table \\ref{tab:geom_random}). We consider deviations of 5 to 20 nm, both in positive (enlargement) or negative (shrinking) directions. Our geometry exhibits a robust performance despite these errors with Brillouin gains above 2000 W$^{-1}$m$^{-1}$ (Fig. \\ref{fig:fab_tolerance}e, blue) and mechanical frequencies between 14 and 15 GHz (Fig. \\ref{fig:fab_tolerance}e, orange). It should be noted that the period remains constant, $\\Lambda = 300$ nm since it is controlled with high precision ($\\pm 2$ nm) in terms of fabrication.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig4_fab_double.png}\n    \\caption{Fabrication tolerance of the optimized geometry. a) and b) Variation of the geometry due to fabrication errors. The solid black line corresponds to optimized geometry, dotted (solid) blue depicts a positive deviation from the nominal design, and dotted orange refers to a negative deviation from the expected design. c) and d) Evolution of the Brillouin gain (in blue, left axis) and the mechanical frequency (in orange, right axis) for different values of under- and over-etching (c), different values of stitching errors (d), and different structures with randomized geometrical parameters (e). In e), N stands for the nominal design obtained after the optimization problem and $i$ for the different geometries listed in Table \\ref{tab:geom_random}.}\n    \\label{fig:fab_tolerance}\n\\end{figure}\n\n\\begin{table}\n    \\centering\n    \\caption{Dimensions for the different geometries used for studying the effect of randomization of the design parameters. In the table, S$_i$ stands for section $i$ in Fig. \\ref{fig:structure}, N stands for the nominal design as obtained from the optimization (Table 1), and $i$ stands for the different geometries in Fig. \\ref{fig:fab_tolerance}e. In all cases, the period, $\\Lambda = 300$ nm, remains constant.}\n    \\label{tab:geom_random}\n    \\begin{tabular}{@{}cccccccc}\n    \\toprule\n         Geometry & & S1 & S2 & S3 & S4 & S5 & $W_g$ \\\\\n    \\midrule\n        \\multirow{2}*{N} & Width & 170 nm & 320 nm & 330 nm & 100 nm & 500 nm & 400 nm \\\\\n        & Length & 130 nm & 60 nm & 60 nm & 190 nm & 50 nm & \\\\\n    \\midrule\n        \\multirow{2}*{1} & Width & 165 nm & 305 nm & 345 nm & 90 nm & 510 nm & 405 nm \\\\\n        & Length & 130 nm & 45 nm & 65 nm & 180 nm & 60 nm & \\\\\n    \\midrule\n        \\multirow{2}*{2} & Width & 165 nm & 320 nm & 340 nm & 115 nm & 495 nm & 400 nm \\\\\n        & Length & 110 nm & 45 nm & 55 nm & 170 nm & 35 nm & \\\\\n    \\midrule\n        \\multirow{2}*{3} & Width & 155 nm & 340 nm & 340 nm & 100 nm & 485 nm & 405 nm \\\\\n        & Length & 150 nm & 40 nm & 70 nm & 200 nm & 55 nm & \\\\\n    \\midrule\n        \\multirow{2}*{4} & Width & 185 nm & 300 nm & 325 nm & 95 nm & 480 nm & 385 nm \\\\\n        & Length & 140 nm & 65 nm & 75 nm & 185 nm & 60 nm & \\\\\n    \\midrule\n        \\multirow{2}*{5} & Width & 160 nm & 320 nm & 330 nm & 95 nm & 510 nm & 390 nm \\\\\n        & Length & 140 nm & 65 nm & 55 nm & 210 nm & 60 nm & \\\\\n    \\midrule\n        \\multirow{2}*{6} & Width & 185 nm & 340 nm & 315 nm & 120 nm & 520 nm & 420 nm \\\\\n        & Length & 135 nm & 40 nm & 50 nm & 190 nm & 35 nm & \\\\\n    \\midrule\n        \\multirow{2}*{7} & Width & 185 nm & 340 nm & 340 nm & 110 nm & 480 nm & 410 nm \\\\\n        & Length & 140 nm & 55 nm & 65 nm & 175 nm & 40 nm & \\\\\n    \\midrule\n        \\multirow{2}*{8} & Width & 150 nm & 300 nm & 345 nm & 110 nm & 510 nm & 395 nm \\\\\n        & Length & 120 nm & 80 nm & 40 nm & 175 nm & 65 nm & \\\\\n    \\midrule\n        \\multirow{2}*{9} & Width & 170 nm & 340 nm & 325 nm & 105 nm & 520 nm & 410 nm \\\\\n        & Length & 120 nm & 70 nm & 50 nm & 190 nm & 70 nm & \\\\\n    \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusions}\nIn summary, we have proposed a new approach to optimizing Brillouin gain in silicon membrane waveguides. We exploit genetic optimization to maximize Brillouin gain in subwavelength-structured Si waveguides, requiring only one etch step. Genetic algorithm is a well-known optimization technique capable of handling design spaces of moderate dimension \\cite{xin-she_yang_chapter_2021}. It has the main advantage over gradient-based algorithms in its capability to search the design space in many directions simultaneously. On the other hand, the genetic algorithms cannot guarantee a global optimum solution, being the final result strongly dependent on the initial population. Based on this strategy, a calculated Brillouin gain up to 3310 W$^{-1}$m$^{-1}$ is achieved for air environment. This result compares favorably to previously reported subwavelength-based Brillouin waveguides requiring several etching steps \\cite{schmidt2019suspended,zhang_subwavelength_2020}, with calculated Brillouin gain of 1750 W$^{-1}$m$^{-1}$ and 3000 W$^{-1}$m$^{-1}$. Our results show the potential of optimization for obtaining novel designs with improved performance in the context of Brillouin scattering. Moreover, they show the reliability of computationally efficient optimizations based on approximated 2D simulations.\n\n\n\\section*{Declaration of Competing Interest}\nThe authors declare that they have no known competing financial\ninterests or personal relationships that could have appeared to influence\nthe work reported in this paper.\n\n\\section*{Author Statement}\nPaula Nu\u00f1o Ruano, Jianhao Zhang, and Carlos Alonso Ramos proposed the concept. Paula Nu\u00f1o Ruano, Jianhao Zhang, and Daniele Melati developed the simulation framework. Paula Nu\u00f1o Ruano, Jianhao Zhang, Daniele Melati, David Gonz\u00e1lez Andrade, and Carlos Alonso Ramos optimized and analyzed the results. All authors contributed to the manuscript.\n\n\\section*{Data Availability Statement}\nThe data supporting this study's findings are available from the corresponding author upon reasonable request.\n\n\\section*{Acknowledgements}\nThe authors want to thank the Agence Nationale de la Recherche for supporting this work through BRIGHT ANR-18-CE24-0023-01 and MIRSPEC ANR-17-CE09-0041. P.N.R. acknowledges the support of Erasmus Mundus Grant: Erasmus+ Erasmus Mundus Europhotonics Master program (599098-EPP-1-2018-1-FR-EPPKA1-JMD-MOB) of the European Union. This project has received funding from the European Union's Horizon Europe research and innovation program under the Marie Sklodowska-Curie grant agreement N\u00ba 101062518.\n\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction} The idea of spatial\ncoupling emerged in the coding context from the study of Low-Density\nParity-Check Convolutional (LDPCC) codes. LDPCC codes were introduced\nby Felstr{\\\"{o}}m and Zigangirov \\cite{FeZ99}. We refer the reader\nto \\cite{EnZ99, ELZ99, LTZ01, TSSFC04} as well as to the introduction\nin \\cite{KRU10} which contains an extensive review.  It has long\nbeen known that LDPCC codes outperform their block coding counterparts\n\\cite{SLCZ04, LSZC10, LSZC05}.  Subsequent work isolated\nand identified the key system structure which is responsible for\nthis improvement.\n\nIn particular, it was conjectured in \\cite{KRU10} that spatially\ncoupled systems exhibit BP threshold behavior corresponding to the MAP threshold behavior of\n uncoupled component system. This phenomenon was termed ``threshold saturation\" \nand a\nrigorous proof of the threshold saturation phenomenon over the BEC\nand regular LDPC ensembles was given.  The proof was generalized\nto all binary-input memoryless output-symmetric (BMS) channels in\n\\cite{KRU12}.  From these results it follows that universal\ncapacity-achieving codes for BMS channels can be constructed by\nspatially coupling regular LDPC codes.  Spatial coupling has also\nbeen successfully applied to the CDMA multiple-access channel\n\\cite{ScT11,TTK11}, to compressed sensing \\cite{KP10,KMSSZ11,DMM,DJM11},\nto the Slepian-Wolf coding problem \\cite{YPN11}, to models in\nstatistical physics \\cite{HMU11a,HMU11b}, and to many other problems in\ncommunications, statistical physics and computer science, see\n\\cite{KRU12} for a review.\n\nThe purpose of this paper is two-fold. First, we establish the existence of wave-like solutions to\nspatially coupled graphical models which, in the large size limit,\ncan be characterized by a one-dimensional real-valued state.\nThis is applied to give a rigorous\nproof of the threshold saturation phenomenon for all such\nmodels. This includes spatial\ncoupling of irregular LDPC codes over the BEC, but it also addresses\nother cases like hard-decision decoding for transmission over general\nchannels, and the CDMA multiple-access problem \\cite{ScT11,TTK11}\nand compressed sensing \\cite{DJM11}.\nAs mentioned above, transmission over the BEC using spatially-coupled\nregular LDPC codes was already solved  in \\cite{KRU10}, but our\ncurrent set-up is more general.  Whereas the proof in \\cite{KRU10}\ndepends on specific features of the BEC, here we derive a graphical\ncharacterization of the threshold saturation phenomena in terms of\nEXIT-like functions that define the spatial system.  This broadens\nthe range of potential applications considerably.\n\nConsider the example of coding over the BEC.  In\nthe traditional irregular LDPC EXIT chart setup the condition for successful decoding reduces\nto the two EXIT charts not crossing. We will show that the EXIT\ncondition for good performance of the spatially-coupled system is\nsignificantly relaxed and reduces to a condition on the area bounded\nbetween the component EXIT functions.\n\nThe criteria is best demonstrated by a simple example. Consider transmission\nover the BEC using the $(3, 6)$ ensemble.\nFigure~\\ref{fig:positivegapbec36} shows the corresponding EXIT\ncharts for $\\epsilon=0.45$ and $\\epsilon=0.53$. Note that both these\nchannel parameters are larger than the BP threshold which is\n$\\epsilon^{\\text{\\small BP}} \\simeq 0.4294$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/positivegapbec36}\n}\n\\caption{\\label{fig:positivegapbec36} Both pictures show the EXIT\ncurves for the $(3, 6)$ ensemble and transmission over the BEC.\nLeft: $\\epsilon=0.45$. In this case $A=0.03125>0$, i.e., the white\narea is larger than the dark gray area.  Right: $\\epsilon=0.53$.\nIn this case $A=-0.0253749<0$, i.e., the white area is smaller than\nthe dark gray area.  } \n\\end{figure}\nIf we consider the signed area bounded by the two\nEXIT charts and integrate from $0$ to $u$ then on the left hand side, with $\\epsilon=0.45,$ this\narea is positive for all $u \\in [0, 1]$.  This property guarantees\nthat the decoder for the spatially coupled system succeeds for this\ncase.  On the right-hand side with $\\epsilon=0.53$, however, the area becomes\nnegative at some point (the total area in white is smaller than the\ntotal area in dark gray) and by our condition this implies that the\ndecoder for the spatially coupled system does not succeed. The\nthreshold of the spatially coupled system is that channel\nparameter such that the area in white and the area in dark gray are\nexactly equal. \n\nThis simple graphical condition is the essence of our result and\napplies regardless whether we look at coding systems or other\ngraphical models. Given any system characterized by two EXIT functions,\nwe can plot these two functions and consider the signed area area bound between them,\nsay for the first coordinate ranging from $0$ to a point $u$. As long as this area is positive for all\n$u \\in (0, 1]$ the iterative process succeeds, i.e., it converges to $0.$ Indeed, we will even\nbe able to make predictions on the speed of the process based on\nthe ``excess\" area we have. \n\nA few conclusions can immediately be drawn from such a picture.\nFirst, if the threshold of the uncoupled system is determined by\nthe so-called stability condition, i.e., the behavior of the EXIT\ncharts for $u$ around $0$ then spatial coupling does not increase\nthe threshold. Indeed, if we increase the parameter beyond what is\nallowed according to the stability condition, the area will become\nnegative around $0$.  Second, if the curves only have\none non-trivial crossing (besides the one at $0$ and at the right\nend point) then the threshold is given by a balance of the two\nenclosed areas.\n\nFor ``nice\" EXIT charts (e.g., continuous, and with a finite number of\ncrossings) the above picture contains all that is needed.  But since\nwe want to develop the theory for the general case, some care is\nneeded when defining all relevant quantities. When reading the\ntechnical parts below, it is probably a good idea to keep the above\nsimple picture in mind. For readers familiar with the so-called\nMaxwell conjecture, it is worth pointing out that the above picture\nshows that this conjecture is generically correct for coupled\nsystems.  To show that it is also correct for uncoupled systems one\nneeds to show in addition that under MAP decoding the coupled and\nthe uncoupled system behave identically. This can often be accomplished\nby using the so-called interpolation method. For e.g., regular\nensembles with no odd check degrees this second step was shown to\nbe correct in \\cite{GMU12}.\n\nLet us point out a few differences to the set-up in \\cite{KRU10}.\nFirst, rather than analyzing directly the spatially-discrete system,\nkey results are established in the limit of continuum spatial\ncomponents. We will see that for such systems the solution for the\ncoupled system is characterized in terms of traveling waves, especially fixed points. The\nspatially discrete version is then recovered as a sampling of the continuum\nsystem.  The existence of traveling wave solutions and their\nrelationship to the EXIT charts of the underlying component systems\nis the essential technical content of the analysis and does not depend\non information theoretic aspects of the coding case.\n\nThe second purpose of this paper is to show that herein-developed\none-dimensional theory can model many higher-dimensional or even\ninfinite-dimensional systems to enable accurate prediction of their\nperformance. This is very much in the spirit of the use of EXIT\ncharts and Gaussian approximations for the the design of iterative\nsystems.  \nUsing this interpretation,\nwe apply our method to channel coding over general channels.  Even\nthough the method is no longer rigorous in these cases, we show\nthat our graphical characterization gives very good predictions on\nthe system performance and can therefore provide a convenient design\ntool.\n\nRecently several alternative approaches to the analysis of\nspatially-coupled systems have been developed independently by\nvarious authors \\cite{TTK11b, DJM11, YJNP12a, YJNP12b}.  These\napproaches share some important aspects with our work but there are\nalso some important differences. Let us quickly discuss this. \n\nIn \\cite{DJM11} a proof was given that spatially coupled measurement\nmatrices, together with a suitable iterative decoding algorithm,\nthe so-called {\\em approximate message-passing} (AMP) algorithm,\nallows to achieve the information theoretic limits of compressive sensing\nin a wide array of settings. The key technical idea is to show that\nthe iterative system is characterized in the limit of large block\nsizes by a one-dimensional parameter (which in this case represents\nper-coordinate mean square error) and which can be tracked faithfully by\n{\\em state evolution equations}.  To ease the analysis the authors\nconsider continuum state evolution equations. The discrete state\nevolution is then obtained by sampling the continuous state evolution\nequations.  The most important ingredient of the proof is a construction\nof an appropriate free energy or potential function for the system such\nthat the fixed points of the state evolution are the stationary\npoints.  It is then shown that if the under-sampling ratio is greater\nthan the information dimension, then the solution of the state evolution\nbecomes arbitrarily small.  Suppose this did not happen. Then\nby perturbing slightly the non-trivial fixed point (the solution\nis ``moved'' inside) one can show that the potential strictly\ndecreases. However, since the fixed point is a stationary point of\nthe potential function, we get a contradiction. \n\nIn \\cite{YJNP12a, YJNP12b} the two main ingredients are also the\ncharacterization of the iterative system by a one-dimensional (or\nfinite-dimensional) parameter and the construction of a suitable\npotential function whose stationary points are the fixed points of\ndensity evolution.  A significant innovation introduced in\n\\cite{YJNP12a, YJNP12b} is that it is shown how to {\\em systematically}\nconstruct such a potential function in a very general setting.  This\nmakes it possible to apply the analysis to a wide array of setting\nand provides a systematic framework for the proof. In addition,\nthis framework allows not only to attack the scalar case but can\nbe carried over to vector-valued states.\n\nOur starting point is the set of EXIT functions, a familiar tool\nin the setting of iterative systems.\nWe also use a type of potential function for the underlying component system.\nUnlike the works mentioned above we retain the symmetry of the iterative system rather than\ncollapsing one of the equations.\nIn addition, the form of the potential function used is such that each step of the iteration \nminimizes the potential function for the variable being updated.\nThe potential function can be lifted to the spatially coupled system but that is not the approach we take in this paper.\nWe consider the spatially coupled system of infinite extent and analize solutions.\nSpatial fixed point solutions that interpolate between fixed points of the component system are of particular importance.\nWe show that the evaluation of the potential function of the component system at points determined by the spatial fixed point can be expressed in terms an integral accross space of the\nfixed point.  Surprisingly, the underlying potential can be expressed using the spatial fixed point solution in a way that uses the portion of the solution local to the evaluation points.\nThis basic result yields  structural information on the fixed point solution.\nThis result is used as a foundation to characterize and construct wave-like solutions for \nspatially coupled systems.\nPerhaps one of the strong points of the current paper is\nthat it gives a fairly detailed and complete picture of the system\nbehavior.  I.e., we not only characterize the threshold(s) but\nwe also are able to characterize {\\em how} the system converges to\nthe various FPs (these are the wave solutions) and {\\em how fast}\nit does so.\n\nThe outline of the paper is as follows. In Section~\\ref{sec:main}\nwe consider an abstract system, characterized by two EXIT-like\nfunctions.  In terms of these functions we state a graphical criterion\nfor the occurrence of threshold saturation.  In\nSection~\\ref{sec:applications} we then apply the method to several\none-dimensional systems. We will see that in each case the analysis\nis accomplished in just a few paragraphs by applying the general\nframework to the specific setting.  In Section~\\ref{sec:gaussapprox}\nwe develop a framework that can be used to analyze higher-dimensional\nsystems in a manner analogous to the way the Gaussian approximation\nis used together with EXIT charts in iterative system design.  We\nalso show by means of several examples that this approach typically\ngives accurate predictions. In Section~\\ref{sec:proof} we give a\nproof of the main results. Many of the supporting lemmas and bounds\nare relegated to appendices.\n\n\\section{Threshold Saturation in One-Dimensional Systems}\\label{sec:main}\nIn\nthis section we develop and state the main ingredients which we\nwill later use to analyze various spatially coupled systems.  Although\nin most cases we are ultimately interested in ``spatially discrete'' and\n``finite-length'' coupled systems, i.e., systems where we have a\nfinite number of ``components'' which are spatially coupled along\na line, it turns out that the theory is more elegant and simpler\nto derive if we start with spatially continuous and unterminated systems, i.e., stretching from\n$-\\infty$ to $\\infty$. Once a suitably defined continuous system\nis understood, one can make contact with the actual system at hand\nby spatially discretizing it and by imposing specific boundary\nconditions.\n\nThroughout this section we use the example of the spatially-coupled\n$(\\dl, \\dr)$-regular LDPC ensemble.  \\bexample[$(\\dl, \\dr, w, L)$\nEnsemble]\\label{def:ensemble} The $(\\dl, \\dr, w, L)$ random ensemble\nis defined as follows, see \\cite{KRU10}.  In the ensuing paragraphs\nwe use $[a, a+b]\\Delta$, for integers $a$ and $b$, $b\\geq0$, and\nthe real non-negative number $\\Delta$, to denote the set of points\n$a\\Delta, (a+1)\\Delta, \\dots, (a+b)\\Delta$.\n\nWe assume that the variable nodes are located at positions $[0,\nL] \\Delta$, where $L \\in \\naturals$ and $\\Delta >0$. At each position\nthere are $M$ variable nodes, $M \\in \\naturals$. Conceptually we\nthink of the check nodes as located at all positions $[- \\infty,\n\\infty] \\Delta$.  Only some of these positions are used and contain check nodes\nthat are actually connected to\nvariable nodes.  At each position there are $\\frac{\\dl}{\\dr}\nM$ check nodes. It remains to describe how the connections are\nchosen.\n\nWe assume that each of the $\\dl$ neighbors of a variable node at\nposition $i \\Delta$ is uniformly and independently chosen from the\nrange $[i-w, \\dots, i+w] \\Delta$, where $w$ is a ``smoothing''\nparameter.\\footnote{Full independence is not possible while satisfying the degree constraints.\nThis does not affect the analysis since we only need the independence to hold asymptotically in large block size over finite neighborhoods in the graph.}\n In the same way, we assume that each of the $\\dr$\nconnections of a check node at position $i$ is independently chosen\nfrom the range $[i-w, \\dots, i+w] \\Delta$.  Note that this deviates\nfrom the definition in \\cite{KRU10} where the ranges were $[i,\n\\dots, i+w-1] \\Delta$ and $[i-w+1, \\dots, i] \\Delta$ respectively.\nIn our current setting the symmetry of the current definition\nsimplifies the presentation.  The present definition is equivalent\nto the previous one with $w$ replaced by $2w+1.$\n\nThis ensemble is spatially discrete. As we mentioned earlier, it\nis somewhat simpler to start with a system which is spatially\ncontinuous.  We will discuss later on in detail how to connect these\ntwo points of view. Just to get started -- how might one go from a\nspatially discrete system as the $(\\dl, \\dr, w, L)$ ensemble to a\nspatially continuous system?  Assume that we let $\\Delta$ tend to\n$0$ while $L$ and $w$ tend to infinity so that $L \\Delta$ tends to $\\infty$ and\n$W=w\\Delta$ is held constant. In\nthis case we can imagine that in the limit there is a component\ncode at each location $x \\in (-\\infty, +\\infty)$ in space and that\na component at position $x$ ``interacts'' with all components in a\nparticular ``neighborhood'' of $x$ of width $2W.$ {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nConsider a system on $(-\\infty, +\\infty)$ (the spatial component)\nwhose ``state'' at each point (in space) is described by a scalar\n(more precisely an element of $[0, 1]$). This means, the state of\nthe system at time $t$, $t \\in \\naturals$, is described by a function\n$\\ff^t$, where $\\ff^t(x) \\in [0, 1]$, $x \\in (-\\infty, \\infty)$.\n\n\\bexample[Coding for the BEC]\nConsider transmission over a binary erasure channel (BEC) using the\n$(\\dl, \\dr, w, L)$ ensemble described in Definition~\\ref{def:ensemble}.\nThen the ``state'' of each component code at a particular point in\ntime is the fraction of erasure messages that are\nemitted by variable nodes at this iteration.  Hence the state of\neach component is indeed an element of $[0, 1]$.  {\\hfill\n$\\ensuremath{\\Box}$} \\eexample\n\n\\bdefinition\nWe denote the space of non-decreasing functions $[0,1] \\rightarrow [0,1]$  by $\\exitfns.$\nA function $h\\in\\exitfns$ has right limits $h(x+)$ for $x\\in (0,1]$ and left limits $h(x-)$ for $x\\in[0,1).$\nTo simplify some notation we define $h(0-)=0$ and $h(1+)=1.$\nThe function $h$ is continuous at $x$ if $h(x-) = h(x+).$ \n\nSimilarly, let $\\sptfns$ denote the space\nof non-decreasing functions on $(\\minfty, \\pinfty).$ \nWe denote $\\lim_{x\\rightarrow -\\infty} f(x)$ as $f(\\minfty)$\n$\\lim_{x\\rightarrow +\\infty} f(x)$ as \n$f(\\pinfty).$  \nWe call a function $f \\in \\sptfns$ {\\em$(a,b)$-interpolating} if\n$f(\\minfty) = a$\nand\n$f(\\pinfty) = b.$\nWe will generally use the term ``interpolating\" with the understanding that $b>a.$\nThe canonical case will be $(0,1)$-interpolating functions and we will also use the term\n``$(0,1)$-interpolating spatial fixed point'' to refer to a pair of $(0,1)$-interpolating functions.\nWe may also refer to a pair of functions $f,g$ interpolating over $[a,b]\\times[c,d]$ to mean\nthat $f$ is $(b,d)$-interpolating and $g$ is $(a,c)$-interpolating. \n\nIn general we work with discontinuous functions.  Because of this we\noccasionally need to distinguish between functions in $\\exitfns$ or in $\\sptfns$\nthat differ only on a set of measure $0.$ \nWe say $h_1 \\equiv h_2$ if $h_1$ and $h_2$ differ on a set of measure $0.$\nThese functions are equivalent in the $L_1$ sense.\nWe still enforce monotonicity so equivalent functions can differ only\nat points of discontinuity.\n \\edefinition\n\nWe think of  $\\hf$ and $\\hg$ as EXIT-like functions describing the\nevolution of the underlying component system under an iterative\noperation.  Usually, we will have $(0,0)$ and $(1,1)$ as key fixed points.\n\nWe say that a sequence $h_i \\rightarrow h$ in $\\exitfns$ if $h_i(u)\n\\rightarrow h(u)$ for all points of continuity of $h.$  We use a\nsimilar definition of convergence in $\\sptfns.$  In general only the\nequivalence class of the limit is determined. I.e., if the limit $h$ is\ndiscontinuous then it is not uniquely determined.\n\nAny function $h \\in \\exitfns$ has a unique equivalence class of inverse functions in $\\exitfns$. \nFor $h \\in \\exitfns$ we will use $h^{-1}$ to denote any member of the equivalence class.\nFormally, we can set $h^{-1}(v)$ to any value $u$ such that\n$v \\in [  h(u-), h(u+) ].$ \nNote that $h^{-1}(v-)$ and $h^{-1}(v+)$ are uniquely determined\nfor each $v\\in [0,1].$ \nThus, we see that the function $h^{-1}$ is uniquely determined at all of its points of continuity and\nit is not uniquely determined at points of discontinuity.\nSimilarly, any function $f \\in \\sptfns$ has\na well defined monotonically non-decreasing inverse equivalence class\nand we use $f^{-1}:[0,1]\\rightarrow [-\\infty,\\infty]$ to denote any member.\nFor notational completeness we define $f^{-1}(0-)=-\\infty$ and $f^{-1}(1+)=+\\infty.$\n\nWe assume that the dynamics of the underlying component system is\ndescribed by iterative updates according to the two functions $\\hf,\\hg \\in \\exitfns.$\nIn deference to standard nomenclature in coding, we refer to these\niterative updates as the {\\em density evolution} (DE) equations.\nIf we assume that $\\xf$ and $\\xg$ are scalars describing the component\nsystem state then these update equations are given by\n\\begin{equation}\\label{eqn:DE}\n\\begin{split}\n{\\xg}^{t} & =  \\hg (\\xf^t), \\\\\n{\\xf}^{t+1} & = \\hf (\\xg^{t})\\,.\n\\end{split}\n\\end{equation}\n\n\\bexample[DE for the BEC]\nConsider a $(\\dl, \\dr)$-regular ensemble.\nLet $\\lambda(u)=u^{\\dl-1}$ and $\\rho(v)=v^{\\dr-1}$.  Let $\\xf^t$ be\nthe fraction of erasure messages emitted at variable nodes at time $t$\nand let $\\xg^t$ be the fraction of erasure messages emitted at\ncheck nodes at iteration $t$.\\footnote{Conventionally, in iterative coding these quantities are denoted by $x$ and $y$.\nBut since we soon will introduce a continuous spatial dimension, which naturally is denoted by $x$, we prefer\nto stick with this new notation to minimize confusion.}\nLet $\\epsilon$ be the\nchannel parameter.  Then we have\n\\begin{equation}\\label{eqn:DEBEC}\n\\begin{split}\n\\xg^{t} & =  1-\\rho(1-\\xf^t), \\\\\n\\xf^{t+1} & = \\epsilon \\lambda(\\xg^{t})\\,.\n\\end{split}\n\\end{equation}\nIn words, we have the correspondences $\\hg(\\xf)=1-\\rho(1-\\xf)$, and\n$\\hf(\\xg)=\\epsilon \\lambda(\\xg)$.  As written, the function $\\hf(\\xg)$\nis not continuous at $\\xg=1.$ More explicitly, $\\hf(1) = \\epsilon <1$, whereas\nwe defined the right limit at $1$ to be generically equal to $1$.\nWe will see shortly how to deal with this.  {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nLet us now discuss DE for the spatial continuum version. Consider a spatially-coupled\nsystem with the following update equations:\n\\begin{equation}\\label{eqn:gfrecursion}\n\\begin{split}\n\\fg^{t}(x) & =  \\hg ((\\ff^t \\otimes \\smthker) (x)), \\\\\n\\ff^{t+1}(x) & = \\hf ( (\\fg^{t} \\otimes \\smthker) (x) )\\,.\n\\end{split}\n\\end{equation}\nHere, $\\otimes$ denotes the standard convolution operator on $\\reals$\nand $\\smthker$ is an {\\em averaging kernel.}\n\\begin{definition}[Averaging Kernel]\nAn averaging kernel $\\smthker$ is a non-negative even function,\n$\\smthker(x)=\\smthker(-x)$, of bounded variation that integrates to $1,$\ni.e., $\\int \\smthker(x) \\text{d} x =1.$\nWe call $\\smthker$ {\\em regular} if there exists $W \\in (0,\\pinfty]$  such that\n$\\smthker(x) = 0$  for $x \\not\\in [-W,W]$ and \n$\\smthker(x) > 0$ for $x \\in (-W,W).$  Note that we do not require $W$ to be finite,\nwe may have $W=\\infty.$\n\\end{definition}\n\nWe will generally assume a regular averaging kernel.  This assumption can largely be dropped when $\\hf$ and $\\hg$ are continuous.\n\n\\bexample[Continuous Version of DE for the BEC]\nIf we specialize the maps to the case of transmission over the BEC we get\nthe  update equations:\n\\begin{equation}\\label{eqn:gfrecursionBECcont}\n\\begin{split}\n\\fg^{t}(x) & =  1-\\rho(1-(\\ff^t \\otimes \\smthker) (x)), \\\\\n\\ff^{t+1}(x) & = \\epsilon \\lambda( (\\fg^{t} \\otimes \\smthker) (x) )\\,.\n\\end{split}\n\\end{equation}\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\nFor compactness we will often use the notation\n$\\fS$ to denote $\\ff \\otimes \\smthker.$\n\nIn the usual manner of EXIT chart analysis, it is convenient to\nconsider simultaneously the plots of $\\hf$ and the reflected plot\nof $\\hg.$ More precisely, in the unit square $[0,1]^2,$ we consider\nthe monotonic curves\\footnote{If $\\hf$ or $\\hg$ is discontinuous then the curve interpolates\nthe jump with a line segment.} $(\\xg, \\hf(\\xg))$ and $(\\hg(\\xf), \\xf)$ for $\\xf, \\xg \\in [0,1].$ \nDensity evolution (DE)\nof the underlying (uncoupled) iterative system can then be viewed as a path drawn out by\nmoving alternately between these two curves (see Fig. \\ref{fig:exitbec36}).\nThis path has the characteristic ``staircase'' shape.  \nWe will sometimes refer to the system being defined on $[0,1]\\times[0,1]$ with this picture in mind.\nThe fixed\npoints of DE of the uncoupled system correspond to the\npoints where these two curves meet or cross. Assuming continuity of $\\hf$ and $\\hg,$ they are the\npoints $(\\xg, \\xf)$ such that $(\\xg, \\hf(\\xg)) = (\\hg(\\xf), \\xf).$\n\nTo help with analysis in the discontinuous case we introduce the following notation.\nFor any $h \\in \\exitfns$ we write\n\\[\nu \\veq h(v)\n\\]\nto mean $u \\in [h(v-),h(v+)].$\n\\begin{definition}[Crossing Points]\nGiven $(\\hf,\\hg)\\in\\exitfns^2$ and we say that $(\\xg, \\xf)$ is a crossing point\nif \n\\[ \n\\xg \\veq \\hg(\\xf),\\text{ and } \\xf \\veq \\hf(\\xg)\\,.\n\\]\nThe following are three equivalent characterizations of crossing points.\n\\begin{itemize}\n\\item $u \\veq \\hfinv(v)$ and $v \\veq \\hginv(u),$\n\\item $u \\veq \\hg(v)$ and $u \\veq \\hfinv(v),$\n\\item $v \\veq \\hf(u)$ and $v \\veq \\hginv(u).$\n\\end{itemize}\n\nThe set of all crossing points will be denoted $\\cross(\\hf,\\hg).$ \nIt is easy to see that $\\cross(\\hf,\\hg)$ is closed as a subset of $[0,1]^2.$\nBy definition of $\\exitfns$,\nwe have $(0,0) \\in \\cross(\\hf,\\hf)$ and $(1,1) \\in \\cross(\\hf,\\hg).$ \nWe term $(0,0)$ and $(1,1)$\nthe {\\em trivial} crossing points and denote the non-trivial crossing points by\n\\[\n\\intcross (\\hf,\\hg) = \\cross(\\hf,\\hg)\\backslash \\{(0,0),(1,1)\\}.\n\\]\n\\end{definition}\n\nIf $(u,v)\\in\\cross(\\hf,\\hf)$ and $\\hf$ and $\\hg$ are continuous at\n$u$ and $v$ respectively then $(u,v)$ is a fixed point of density\nevolution.  In general, if $(u,v)\\in\\cross(\\hf,\\hf)$ then $(u,v)$\nis a fixed point of density evolution for a pair of exit functions\nequivalent to the pair $(\\hf,\\hg).$ \n\n\\begin{lemma}\\label{lem:crossorder}\nFor any $\\hf,\\hg \\in \\exitfns$ the set $\\cross(\\hf,\\hg)$ is component-wise ordered,\ni.e., given $(u_1,v_1),(u_2,v_2) \\in \\cross(\\hf,\\hg)$ we have  \n$(u_2-u_1)(v_2-v_1)\\ge 0.$\n\\end{lemma}\n\\begin{IEEEproof}\nLet $(u_1,v_1),(u_2,v_2) \\in \\cross(\\hf,\\hg).$\nIf $u_1<u_2$ then $\\hf(u_1+) \\le \\hf(u_2-)$ \nand, since  $v_1 \\le \\hf(u_1+)$ and $v_2 \\ge \\hf(u_2-),$\n we obtain $v_1 \\le v_2.$\nAll other cases can be shown similarly.\n\\end{IEEEproof}\n\nFor a set $S,$ which may be a subset of  $[0,1]^2$ or a subset of $\\reals,$\nwe use $\\neigh{S}{\\epsilon}$ to denote the $\\epsilon$ neighborhood of $S:$\n\\begin{equation}\\label{eqn:neighdef}\n\\neigh{S}{\\epsilon} = \\{ x: \\exists y\\in S, |y-x|<\\epsilon\\}\\,.\n\\end{equation}\n\n\\begin{lemma}\\label{lem:crosspointlimit}\nIf $(\\hf^i,\\hg^i)\\rightarrow(\\hf,\\hg)$ then, for any $\\delta>0,$ we have\n$\\cross(\\hf^i,\\hg^i) \\subset \\neigh{\\cross(\\hf,\\hg)}{\\delta}$ for\nall $i$ sufficiently large.\n\\end{lemma}\n\\begin{IEEEproof}\nAssume  $(u^i,v^i)\\in\\cross(\\hf^i,\\hg^i)$ converges in $i$ to a limit point $(u,v).$ \nSince $\\hf^i(u) \\rightarrow \\hf(u)$ at points of continuity of $\\hf$\nit is easy to see that\n\\[\n\\liminf_{i\\rightarrow \\infty} \\hf^i(u^i-)\\ge \\hf(u-)\n\\]\nand\n\\[\n\\limsup_{i\\rightarrow \\infty} \\hf^i(u^i+)\\le \\hf(u+)\n\\]\nand it follows that $v \\veq \\hf(u).$\nSimilarly, $u \\veq\\hg(v).$\nHence, $(u,v) \\in \\cross(\\hf,\\hg).$\n \nSince $[0,1]^2 \\backslash \\neigh{\\cross(\\hf,\\hg)}{\\delta}$ is compact\nand the same argument applies to subsequences\nthe Lemma follows.\n\\end{IEEEproof}\n\n\\begin{lemma}\nConsider initialization of system \\eqref{eqn:DE} with an arbitrary choice of $u^0.$\nThen the sequence $(u_1,v_1),(u_2,v_2),\\ldots$ is monotonic (either non-increasing or non-decreasing)\nin both coordinates.\n\\end{lemma}\n\\begin{IEEEproof}  \nIf $v^{t+1} = \\hf(u^t) \\ge v^t$ then $u^{t+1}=\\hg(v^{t+1}) \\ge \\hg(v^t) = u^t.$\nAnd if $u^{t+1} \\ge   u^t$ then  $v^{t+2}=\\hg(u^{t+1}) \\ge \\hg(u^t) = v^{t+1}.$\n\\end{IEEEproof}\nIt follows that the sequence $(u_i,v_i)$ converges and the limit point is clearly a crossing point of $(\\hf,\\hg).$\nThus, the limiting behavior of the scalar component system is governed by crossing points.\nIn the spatially coupled system the behavior is often involves a pair\nof crossing points.  \n\nSuppose $(u_1,v_1) < (u_2,v_2)$ are fixed points of the component DE.\nIf $\\ff^0(x)\\in[v_1,v_2]$ for all $x$\nthen $\\ff^t(x)\\in[v_1,v_2]$ and\nthen $\\fg^t(x)\\in[u_1,u_2]$ \nfor all $x$ and $t.$ and \n$\\fg^t$ is in the range between $u_1$ and $u_2$ for all $t.$\nThus, in this situation the system is effectively confined to $[u_1,u_2]\\times [v_1,v_2].$\nThis circumstance occurs frequently but we can easily transform this into our canonical form.\nWe can introduce new coordinates $\\tilde{u},\\tilde{v}$ characterized by the inverse map\n\\begin{align*}\nu &   =  a \\tilde{u} + b \\\\\nv &   =  c \\tilde{v} + d\\,.\n\\end{align*}\nBy choosing $(b,d) = (u_1,v_1)$ and $(a,c) = (u_2-u_1,v_2-v_1)$\nwe map $(u,v) =(u_1,v_1)$ to $(\\tilde{u},\\tilde{v}) =(0,0)$ and $(u,v) =(u_1,u_2)$  to $(\\tilde{u},\\tilde{v}) =(1,1).$\nSimilarly, by choosing $(b,d) = (u_2,v_2)$ and $(a,c) = (u_1-u_2,v_1-v_2)$\nwe map $(u,v) = (u_2,v_2)$ to $(\\tilde{u},\\tilde{v}) = (0,0)$ and $(u,v)=(u_1,v_1)$  to $(\\tilde{u},\\tilde{v}) =(1,1).$\n(This shows the symmetry that allows us to occasionally exchange $(0,0)$ and $(1,1)$\nin the analysis.)\nBy rescaling the update functions appropriately we can thereby redefine the system on $[0,1]\\times[0,1].$\nIn particular we can define ${\\thf}(\\tilde{u}) = \\frac{1}{c}(\\hf( a \\tilde{u} + b) - d)$\nand ${\\thg}(\\tilde{v}) = \\frac{1}{a}(\\hf( c \\tilde{u} + d) - b).$\n\n\\bexample[EXIT Chart Analysis for the BEC] Figure~\\ref{fig:exitbec36}\nshows the EXIT chart analysis for the $(3, 6)$-regular ensemble\nwhen transmission takes place over the BEC. The left picture shows\nthe situation when the channel parameter is below the BP threshold.\nIn this case we only have the trivial FP at $(0, 0)$.  According to our definition we\nhave $(1,1)$ as a crossing point, but it is not a fixed point because $\\hf(1)=\\epsilon<1.$ The right\npicture shows a situation when we transmit above\nthe BP threshold. We now see two further crossings of the EXIT curves\nand so $\\cross(\\hf,\\hg)$ is non-trivial.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitbec36}\n}\n\\caption{\\label{fig:exitbec36} Left: The figure shows the EXIT\nfunctions $\\hf(\\xg)=\\epsilon \\lambda(\\xg)$ and $\\hg(\\xf)=1-\\rho(1-\\xf)$ for the $(3, 6)$-regular\nensemble and $\\epsilon=0.35$.  Note that the horizontal axis is $\\xg$\nand the vertical axis is $\\xf$ so that we effectively plot the inverse\nof the function $1-\\rho(1-\\xf)$. Since $0.35=\\epsilon < \\epsilon^{\\text{BP}}\n\\approx 0.4292$, the two curves do not cross. The dashed ``staircase''\nshaped curve indicates how DE proceeds. Right: In this figure the\nchannel parameter is $\\epsilon=0.5 > \\epsilon^{\\text{BP}}$.  Hence,\nthe two EXIT curves cross. In fact, they cross exactly twice (besides\nthe trivial FP at $(0, 0)$), the first point corresponds to an\nunstable FP of DE, whereas the second one is a stable FP. }\n\\end{figure}\n\nIn this case we can renormalize the system according to our prescription as follows.\nConsider the DE\nequations stated in (\\ref{eqn:DEBEC}).  If $(\\xg^*, \\xf^*)$ is the\nlargest (in both components) FP of the corresponding\nDE and if we can set $\\hf(\\xg)=\\epsilon\\lambda(\\xg \\xg^*)\/\\xf^*$\nand $\\hg(\\xf)=(1-\\rho(1 - \\xf \\xf^*))\/\\xg^*$ then system \\eqref{eqn:DE}\nis again equivalent to \\eqref{eqn:DEBEC} on the restricted domain but, in addition, the component\nfunctions are continuous at $0$ and $1$ and $(0,0)$ and $(1,1)$ are the relevant fixed points.  This rescaling is indicated\nin the right picture of Figure~\\ref{fig:exitbec36} through the\ndashed gray box.  Since the standard (unscaled) EXIT chart picture\nis very familiar in the coding context, we will continue to plot\nthe unscaled picture.  But we will always indicate the scaled version\nby drawing a gray box as in the right picture of\nFigure~\\ref{fig:exitbec36}. This hopefully will not cause any\nconfusion. There is perhaps only one point of caution. The behavior\nof the coupled system depends on certain areas in this EXIT chart.\nThese areas are defined in the scaled version and are different by\na factor $\\xg^* \\xf^*$ in the unscaled version.\n{\\hfill $\\ensuremath{\\Box}$} \\eexample\n\n\n\nSo far we have considered the uncoupled system and seen that its\nbehavior can be characterized in terms of fixed points, or more generally crossing points. \nThe behavior of the\nspatially coupled system can also be characterized by its FPs.  For the\nspatially coupled system a FP is not a pair of scalars, but a pair\nof functions $(\\tmplF(x), \\tmplG(x))$ such that if we set $\\ff^t(x)=\\tmplF(x)$\nand $\\fg^t(x)=\\tmplG(x)$, $t \\geq 0$, then these functions fulfill\n(\\ref{eqn:gfrecursion}). One set of FPs arise as the constant functions\ncorresponding to the fixed points of the underlying component system.\nThe crucial phenomena in spatial coupling is the emergence of interpolating spatial\nfixed points, i.e., non-constant monotonic fixed point solutions.\nFor the coupled system it is\nfruitful not only to look at interpolating FPs but slightly more general objects,\nnamely interpolating {\\em waves}. Here a wave is like a FP, except that it\n{\\em shifts}. I.e., for $(\\tmplF(x), \\tmplG(x))$ fixed and for some real\nvalue $\\ashift$, if we set $\\ff^t(x)=\\tmplF(x-\\ashift t)$ and $\\fg^t(x)=\\tmplG(x-\\ashift\nt)$, $t \\geq 0$, then these functions fulfill (\\ref{eqn:gfrecursion}).\nWe will see that the behavior of coupled systems is governed\nby the (non)existence of such waves and this\n(non)existence has a simple graphical characterization in terms of\nthe component-wise EXIT functions and their associated FPs.  This\nis the main technical result of this paper.  In fact, the {\\em\ndirection} of travel of the wave\ndepends in a simple way on the EXIT functions and the area bound\nby them.  \nThe extremal values of spatial wave solutions are generally crossing points of the\nunderlying component system.  One aspect of the analysis involves\ndetermining the crossing points that can appear as such an extremal value.\nThe solution is is formulated in terms of the following definition.\n\n\\begin{definition}[Component Potential Functions]\nFor any pair $(\\hf,\\hg) \\in \\exitfns^2$ and point $(u,v) \\in [0,1]^2,$ we define\n\\[\n\\altPhi(\\hf,\\hg;u,v) =  \\int_0^u \\hginv(u')\\text{d}u' +  \\int_0^v \\hfinv(v')\\text{d}v' \\, - uv\\,.\n\\]\n\\end{definition}\n\n{\\em Discussion:} The functional  $\\altPhi$ serves as a potential function\nfor the scalar system. \nAssuming continuity of $\\hfinv$ at $v$ and continuity of $\\hginv$ at $u$ we have\n$\\nabla \\altPhi(\\hf,\\hg; u,v) = (\\hginv(u)-v,\\hfinv(v)-u).$\nThus, under some regularity conditions\na crossing point $(\\xg,\\xf)$ is a stationary point of  $\\altPhi.$\ni.e., $ \\nabla \\altPhi(\\hf,\\hg; u,v) =0$\nfor $(u,v)\\in \\cross(\\hf,\\hg).$\n\nNote that in the definition of $\\altPhi$ we have used\n$(0,0)$ as an originating point.  We can choose the origin arbitrarily.\nWe have\n\\begin{align}\\begin{split}\\label{eqn:potdiff}\n&\\altPhi(\\hf,\\hg;u,v) - \\altPhi(\\hf,\\hg;u_1,u_2)\\\\\n= &\n\\int_{u_1}^u \\hginv(u')\\text{d}u' +  \\int_{v_1}^v \\hfinv(v')\\text{d}v' \\, - uv + u_1 v_1\n\\\\= &\n\\int_{u_1}^u (\\hginv(u')-v_1)\\text{d}u' +  \\int_{v_1}^v (\\hfinv(v')-u_1)\\text{d}v' \\,\n\\\\&\\qquad - (u-u_1)(v-v_1)\n\\end{split}\n\\end{align}\nand we see that we can place the origin at $(u_1,u_2)$ while preserving differences.\n\nA straightforward calculation, noting that\n\\(\nu\\hf(u) = \\int_0^u \\hf(u')\\, du' + \\int_0^{\\hf(u)} \\hfinv(v)\\, dv\n\\)\nand\n\\(\nv\\hg(v) = \\int_0^v \\hg(y)\\, dy + \\int_0^{\\hg(v)} \\hginv(x)\\, dx\\,,\n\\)\nshows that for all $(\\hf,\\hg)$ we have\n\\begin{align}\n\\begin{split}\\label{eqn:altPhiPhi}\n&\\altPhi(\\hf,\\hg;\\hg(v),\\hf(u))\\\\\n=&\nuv - (u-\\hg(v))(v-\\hf(u)) \\\\\n& -  \\int_0^u \\hf(u')\\text{d}u' - \\int_0^v \\hg(v')\\text{d}v'\\, .\n\\end{split}\n\\end{align}\nA similar potential function form, along the lines of \\eqref{eqn:altPhiPhi}, is\n\\begin{align*}\n\\Phi(\\hf,\\hg;u,v) = uv -  \\int_0^u \\hf(u')\\text{d}u' - \\int_0^v \\hg(v')\\text{d}v'\\, .\n\\end{align*}\nThis functional is also stationary on the FPs of the component density evolution and is equal\nto $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ on the points $(u,\\hf(u))$ and $(v,\\hg(v)).$\nThis form underlies the work in \\cite{TTK11b, DJM11, YJNP12a, YJNP12b}.\nWe prefer $\\altPhi$ because of various properties developed below.  The two forms are\nrelated through Legendre transforms, e.g., $\\int_0^u \\hf(u')\\text{d}u'$ is the Legendre transform\nof $\\int_0^v \\hfinv(v')\\text{d}v'.$ We see that on the graph of \n$\\hf$ or $\\hg,$ i.e., on the either the set with\n$v=\\hf(u)$ or $u=\\hg(v),$ the two functionals are equivalent up to reparameterization.\nIn terms of $\\altPhi$ we have\n\\begin{align}\n\\label{eqn:altPhiuhfu}\n\\altPhi(\\hf,\\hg;u,\\hf(u)) &= \\int_0^u (\\hginv(u')-\\hf(u'))\\,du'\\\\\n\\altPhi(\\hf,\\hg;\\hg(v),v) &= \\int_0^v (\\hfinv(v')-\\hg(v'))\\,dv' \\label{eqn:altPhivhgv}\n\\end{align}\n\n\n\\begin{lemma}\\label{lem:monotonic}\nThe function $\\altPhi(\\hf,\\hg;u,v)$ is convex in $u$ for fixed $v$ and convex in $v$ for fixed $u.$\nIn addition, for all $(u,v) \\in[0,1]^2$ we have\n\\begin{align*}\n\\altPhi(\\hf,\\hg;u,v) &\\ge \\altPhi(\\hf,\\hg;u,\\hf(u)) \\\\\n\\altPhi(\\hf,\\hg;u,v) &\\ge \\altPhi(\\hf,\\hg;\\hg(v),v) \n\\end{align*}\nwith equality holding in the first case if and only if $v\\veq\\hf(u)$\nand\n in the second case if and only if $u\\veq\\hg(v).$\n\\end{lemma}\n\\begin{IEEEproof}\nIt is easy to check that $\\altPhi(\\hf,\\hg;u,v)$ is Lipschitz (hence absolutely) continuous   \nand we have almost everywhere\n\\begin{align}\n\\begin{split}\\label{eqn:altPhiderivatives}\n\\frac{\\partial}{\\partial u} \\altPhi(\\hf,\\hg;u,v)  &= \\hginv(u) - v, \\\\\n\\frac{\\partial}{\\partial v} \\altPhi(\\hf,\\hg;u,v)  &= \\hfinv(v) - u \\,.\n\\end{split}\n\\end{align}\nThe lemma now follows immediately from the monotonicity (non-decreasing) of $\\hginv$ and $\\hfinv.$\n\\end{IEEEproof}\nWe have immediately the following two results.\n\\begin{corollary}\\label{cor:monotonic}\nIf $(u^0,v^0) \\in [0,1]^2$ and we define $(u^t,v^t)$ for $t \\ge 1$ via\n\\eqref{eqn:DE} then $\\altPhi(\\hg,\\hg;u^t,v^t)$ is a non-increasing sequence in $t.$\n\\end{corollary}\n\\begin{lemma}\\label{lem:miniscross}\nIf $(u,v) \\in\\cross(\\hf,\\hg)$ if and only if\n$\\altPhi(\\hf,\\hg;u',v)$ is minimized at $u'=u$ and\n$\\altPhi(\\hf,\\hg;u,v')$ is minimized at $v'=v.$\n\\end{lemma}\n\nWe will be most interested in the value of $\\altPhi$\nat crossing points $(u,v) =  \\cross(\\hf,\\hg).$\n\n\nOne of the key results on the existence of wave solutions, and especially spatial fixed points,\nis that the crossing points associated to the extremal values of the solution are extreme\n(minimizing) values of the the $\\altPhi$ over the range spanned by the solution.  The  following definition characterizes this.\n\n\\begin{definition}[Strictly Positive Gap Condition]\nWe say that the pair of functions $(\\hf,\\hg)$ satisfies the {\\em\nstrictly positive gap condition} if $\\cross(\\hf,\\hg)$ is {\\em\nnon-trivial} and if \n\\[\n(u,v) \\in \\intcross(\\hf,\\hg) \\Rightarrow \\altPhi(\\hf,\\hg;u,v) > \\max\\{0,A(\\hf,\\hg)\\}\n\\] \nwhere we define $A(\\hf,\\hg) =\\altPhi(\\hf,\\hg;1,1).$\nWe say that the pair of functions $(\\hf,\\hg)$ satisfies the {\\em\npositive gap condition} (no longer strict) if $\\cross(\\hf,\\hg)$ is {\\em\nnon-trivial} and \n\\[\n(u,v) \\in \\intcross(\\hf,\\hg) \\Rightarrow \\altPhi(\\hf,\\hg;u,v) \\ge \\max\\{0,A(\\hf,\\hg)\\}\\,.\n\\] \n {\\hfill $\\ensuremath{\\Box}$}\n\\end{definition}\n\n{\\em Discussion:} The (strictly) positive gap condition is related to the existence of interpolating spatial fixed point solutions.\nIn particular, we will see that systems possessing $(0,1)$-interpolating fixed point solutions must satisfy the positive gap condition\nand have $A(\\hf,\\hg)=0.$  Systems satisfying the strictly positive gap condition with $A(\\hf,\\hg)=0$ will be proven to\npossess $(0,1)$-interpolating spatial fixed point solutions.  The cases where $A(\\hf,\\hg) \\neq 0$ correspond to $(0,1)$-interpolating traveling wave solutions.\n\n\\blemma[Trivial Behavior]\\label{lem:trivialbehavior}\nIf $\\cross(\\hf,\\hg)$ is trivial then the system behavior is simplified\nand under DE, i.e., under $\\eqref{eqn:gfrecursion}$, the only spatial fixed points are with\n$\\ff^t$ and $\\fg^t$ set to either the constant $0$ or the constant\n$1,$ one of which is stable and one of which is unstable.\nThe system converges for all initial values, other than the unstable spatial fixed point itself,\nto the stable spatial fixed point.\n\\elemma\n\nNow that we have covered the ``trivial'' cases, let us consider\nthe system behavior when $\\cross(\\hf,\\hg)$ is non-trivial. As we\nwill see, it is qualitatively different.  \nThe value $A(\\hf,\\hg)=\\altPhi(\\hf,\\hg;1,1)$ plays an important role\nin the behavior of the system. This is why we \nintroduced a special notation.  The strictly positive gap condition\nimplies that the value of $\\altPhi(\\hf,\\hg;\\xg,\\xf)$ for $(\\xg,\\xf)\\in \\intcross(\\hf,\\hg)$\nis strictly larger than\nthe values $0$ and $A(\\hf,\\hg)$ found at the two trivial fixed points.  We will\nsee that this condition is related to the existence of wave-like\nsolutions that interpolate between the two trivial fixed\npoints.\n\n\\bexample[Positive Gap Condition for the BEC]\nFigure~\\ref{fig:positivegapbec36} illustrates the (strictly) positive\ngap condition for the $(3, 6)$-regular ensemble when transmission\ntakes place over the BEC. The left picture shows the situation when\nthe channel parameter is between the BP and the MAP threshold of\nthe underlying ensemble. The right picture shows the situation when\nthe channel parameter is above the MAP threshold of the underlying\nensemble.  In both cases $\\cross(\\hf,\\hg)$ contains one non-trivial\nFP $(\\xg, \\xf)$ and for this FP $\\altPhi(\\hf,\\hg;\\xg,\\xf) > \\max\\{0,A\\}$, i.e.,\nboth cases fulfill the strictly positive gap condition.\n\nIn the first case $A>0$, whereas in the second case $A<0$.  We\nwill see in Theorem~\\ref{thm:mainexist} below that this\nchange in the sign of $A$ leads to a reversal of direction of a wave-like\nsolution to the system and hence to fundamentally different asymptotic behavior.\nBoth pictures show the unscaled curve and the lightly shaded box\nshows what the picture would look like if we rescaled it so that\nthe largest FP appears at $(1, 1)$.\n\nIt is not hard to see that the strictly positive gap\ncondition is necessarily satisfied for any $\\hf,\\hg$ for which $\\cross(\\hf,\\hg)$\nhas a single non-trivial fixed point, and for which $(0,0)$ and\n$(1,1)$ are stable fixed points under the DE equations \\eqref{eqn:DE}.\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nWe are now ready to state the main result concerning the existence of interpolating wave solutions.\n\n\\begin{theorem}[Existence of Continuum Spatial Waves]\\label{thm:mainexist}\nAssume that $\\smthker$ is a regular averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of functions in $\\exitfns$ satisfying the strictly positive\ngap condition.\n\nThen there exist $(0,1)$-interpolating functions $\\tmplF,\\tmplG \\in\\sptfns$ and a\nreal-valued constant $\\ashift,$ satisfying $\\sgn(\\ashift) = \\sgn(A(\\hf,\\hg))$\nand $|\\ashift| \\ge |A(\\hf,\\hg)|\/\\|\\smthker\\|_\\infty$, such that setting\n$f^t(x) = \\tmplF(x - \\ashift t)$ and $g^t(x) = \\tmplG(x - \\ashift t)$ for\n$t=0,1,\\ldots$ solves \\eqref{eqn:gfrecursion}\\,.\n\\end{theorem}\n\nWe remark that we can relax the regularity condition on $\\smthker$ if\n$\\hf$ and $\\hg$ are continuous; cf. Lemma \\ref{lem:pathology}.\n\n\\bexample[Spatial wave for the BEC]\nFigure~\\ref{fig:spatialfpbec36} shows the spatial waves whose existence\nis guaranteed by Theorem~\\ref{thm:mainexist} for the $(3, 6)$\nensemble and transmission over the BEC. The top picture corresponds\nto the cases $\\epsilon=0.45$ and the bottom picture to the case\n$\\epsilon=0.53$.  In both cases we  used the smoothing kernel\n$\\omega(x)=\\frac12 \\mathbbm{1}_{\\{|x|\\leq 1\\}}$.  As predicted, in\nthe first case the curve moves to the right by a value of\n$0.142 \\geq |A|\/\\|\\smthker\\|_\\infty = 0.03125 \\times 2 =0.06245$\nand in the second case the curve moves to the left by an amount\nof $0.101 \\geq |A|\/\\|\\smthker\\|_\\infty = 0.0253740 \\times 2 = 0.0507498$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/fpbec36}\n}\n\\caption{\\label{fig:spatialfpbec36}\nFPs whose existence is guaranteed by Theorem~\\ref{thm:mainexist}\nfor the $(3, 6)$ ensemble and transmission over the BEC. The top\npicture corresponds to the cases $\\epsilon=0.45$ and the bottom\npicture to the case $\\epsilon=0.53$.  In both cases we  used the\nsmoothing kernel $\\omega(x)=\\frac 12 \\mathbbm{1}_{\\{|x|\\leq 1\\}}$.  The dashed curve\nis the result of applying one step of DE to the solid curve. As\npredicted, in the top picture the curve moves to the right (the\ncorresponding gap $A$ in Figure~\\ref{fig:positivegapbec36} is\npositive) whereas in bottom picture the curve moves to the left\n(the corresponding gap $A$ is negative).  The shifts are $0.142$ and\n$-0.102$, respectively.  } \\end{figure} {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nOne consequence of Theorem \\ref{thm:mainexist} is that the existence of an\n$(0,1)$-interpolating fixed point implies $A(\\hf,\\hg) = 0.$  This is true even without\nregularity assumptions.\n\n\\begin{theorem}[Continuum Fixed Point Positivity]\\label{thm:FPAzero}\nLet  $\\smthker$  be an averaging kernel (not necessarily regular) and\nassume that there exists a $(0,1)$-interpolating fixed point solution to \\eqref{eqn:gfrecursion}.\nThen $(\\hf,\\hg)$ satisfies the positive gap condition and $A(\\hf,\\hg) = 0.$\n\\end{theorem}\nA more general version of this result appears as Lemma \\ref{lem:FPAzero}.\n\nTheorem \\ref{thm:mainexist} is our most fundamental result concerning the spatially coupled system.\nOne limitation of the result arises in cases with infinitely many crossing points.  In such a case it\ncan be difficult to extract asymptotic behavior since there may exist many wave-like solutions\nand the strictly positive gap condition may not hold globally.\nFor such cases we develop the following altered analysis.\n\nLet $\\hf$ and $\\hg$ be given and define\n\\[\nm(\\hf,\\hg)=\\min_{(u,v) \\in [0,1]^2} \\altPhi(\\hf,\\hg;u,v)\n\\]\nDefine\n\\[\n\\cross_m(\\hf,\\hg) = \\{(u,v)\\in\\cross(\\hf,\\hg) : \\altPhi(\\hf,\\hg;u,v) = m\\}\\,.\n\\]\nSince $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is continuous it follows that\n$\\cross_m(\\hf,\\hg)$ is closed.  Since $\\cross(\\hf,\\hg)$ is component-wise linearly\nordered we can define\n\\begin{align*}\n(u',v') & = \\min \\cross_m(\\hf,\\hg)\\\\\n\\intertext{and}\n(u'',v'')& = \\max \\cross_m(\\hf,\\hg)\\\\\n\\end{align*}\nwhere $\\min$ and $\\max$ are component-wise.\n\n\\begin{theorem}[General Continuum Convergence]\\label{thm:globalconv}\nLet $(\\hf,\\hg)$ be given as above, let $\\smthker$ be regular,\nand assume $\\ff^0 \\in \\sptfns$ is given\nwith $\\ff^0(\\minfty)\\le v''$ and \n$\\ff^0(\\pinfty)\\ge v'.$\nThen in system \\ref{eqn:gfrecursion} we have for all $x\\in \\reals$\n\\begin{align*}\n\\liminf_{t\\rightarrow \\infty} f^t(x) &\\ge v' \\quad  \\liminf_{t\\rightarrow \\infty} g^t(x) \\ge u' \\\\\n\\limsup_{t\\rightarrow \\infty} f^t(x) &\\le v'' \\quad \\limsup_{t\\rightarrow \\infty} g^t(x) \\le u'' \\,.\n\\end{align*}\n\\end{theorem}\nThe proof may be found in appendix \\ref{app:E}.\n\nNote, in particular, that if $\\altPhi$ is uniquely minimized at some crossing point $(u,v),$ then\nthis point is a fixed point of the component system and if the spatial system is initialized (either $f$ or $g$) with this point (the appropriate coordinate)\n in the closed range spanned by the initial condition, then the coupled system\nglobally converges to the constant function associated to this fixed point.\n\nIn many applications the problems are spatially discrete and finite length.\nThe analysis can be applied to these cases with suitable adjustments.  As a first step we state a\nresult analogous to Theorem \\ref{thm:mainexist}\nfor a spatially discrete system.  The DE equations\nfor the spatially discrete version can be written as in\n\\eqref{eqn:gfrecursion} with the following modifications:\nthe variable $x$ is discrete, the averaging kernel is a discrete sequence, and\nthe convolution\noperation is convolution of discrete sequences.  The analysis views the\nspatially discrete problem as a sampled version of the continuum\nversion. In the limit of infinitely fine sampling the discrete\nversion converges to the continuum version.\n\n\\subsection{Discrete Spatial Sampling}\n\nLet $x_i = i \\Delta$ and let $\\discsmthker$ be a non-negative\nfunction over $\\integers$ that is even, $\\discsmthker_i =\n\\discsmthker_{-i},$ and sums to $1,$ $\\sum_i \\discsmthker_i = 1.$\nIt is convenient to interpret $\\discsmthker$ as a discretization of\n$\\smthker,$ i.e.,\n\\begin{equation}\\label{eqn:kerdiscretetosmth}\n\\discsmthker_i = \\int_{(i-\\frac{1}{2})\\Delta}^{(i+\\frac{1}{2})\\Delta} \\smthker(z)\\text{d}z.\n\\end{equation}\nThis relationship then makes it clear that the discrete ``width'' of\nspatial averaging is inversely proportional to $\\Delta.$\nA good example is the smoothing kernel\n$\\smthker(x) = \\frac{1}{2} \\mathbbm{1}_{\\{|x|\\le 1\\}}.$\nIf we set $\\Delta = \\frac{2}{2W+1}$ then $\\discsmthker_i =\n\\frac{1}{2W+1} \\mathbbm{1}_{\\{|i|\\le W\\}}.$\nGiven a real-valued function $\\fg$ defined on $\\Delta\\integers$ we will call \nthe function $\\tfg \\in \\sptfns,$  defined as \n$\\tfg(x)=\\fg(x_i)$ for $x\\in[x_i-\\Delta\/2,x_i+\\Delta\/2),$\nthe {\\em piecewise constant extension} of $\\fg.$ \nNote that by this definition, we have\n\\begin{align*}\n\\tfg^{\\smthker}(x_i)&=\\int_{-\\infty}^\\infty \\smthker(x_i-y)\\tfg(y)dy\n\\\\&=\n\\sum_{j=-\\infty}^\\infty\n\\int_{x_j-\\Delta\/2}^{x_j+\\Delta\/2} \\smthker(x_i-y)\\tfg(y)dy\n\\\\\n&=\n\\sum_{j=-\\infty}^\\infty \\discsmthker_{i-j}\\fg(x_j)dy\\,\n\\\\&=\n\\fg^{\\discsmthker}(x_i)\n\\end{align*}\n\nWith this framework in mind, we can write the spatially discrete DE equations as follows.\n\\begin{equation}\\label{eqn:discretegfrecursion}\n\\begin{split}\n\\fg^{t}(x_i) & =  \\hg ((\\ff^t \\otimes \\discsmthker) (x_i)) \\\\\n\\ff^{t+1}(x_i) & = \\hf ( (\\fg^{t} \\otimes \\discsmthker) (x_i) )\\,.\n\\end{split}\n\\end{equation}\n\n\\bexample[Spatially Discrete DE for the BEC]\n\\begin{equation}\\label{eqn:gfrecursionBEC}\n\\begin{split}\n\\fg^{t}(x_i) & =  1-\\rho((\\ff^t \\otimes \\discsmthker) (x_i)), \\\\\n\\ff^{t+1}(x_i) & = \\epsilon \\lambda( (\\fg^{t} \\otimes \\discsmthker) (x_i) )\\,.\n\\end{split}\n\\end{equation}\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nAn elementary but critical result relating the spatially continuous case to the\ndiscrete case is the following.\n\\begin{lemma}\\label{lem:disccontbnd}\nLet $\\tmplF \\in \\sptfns$ and let $f$ be a real valued function defined on $\\Delta \\integers.$\nThen, if for all $i$ we have\n\\(\nf(x_i) \\le \\tmplF(x_i)\n\\)\nthen \n\\(\nf^{\\discsmthker}(x_i) \\le \\tmplF^{\\smthker}(x_i+\\half\\Delta)\n\\)\nand if\n\\(\nf(x_i) \\ge \\tmplF(x_i)\n\\)\nthen \n\\(\nf^{\\discsmthker}(x_i) \\ge \\tmplF^{\\smthker}(x_i-\\half\\Delta)\n\\)\n\\end{lemma}\n\\begin{IEEEproof}\nAssume\n$\\ff(x_i) \\le \\tmplF(x_i)$ (for all $i$).\nConsider the piecewise constant extension $\\tilde{\\ff}.$ \nIt follows that $\\tilde{\\ff}(x) \\le  \\tmplF(x+\\half\\Delta)$ for all $x$\nand so $\\ff^{\\discsmthker}(x_i) = \\tilde{\\ff}^{\\smthker}(x_i) \\le  \\tmplF^{\\smthker}(x_i+\\half\\Delta)$\nfor each $i.$\n\nThe opposite inequality is handled similarly.\n\\end{IEEEproof}\n\nApplying the above to system \\eqref{eqn:discretegfrecursion} we obtain the following.\n\\begin{theorem}[Continuum-Discrete Bounds] \\label{thm:mainquantize}\nAssume that $\\discsmthker$ is a discrete sequence\nrelated to a regular smoothing kernel $\\smthker$\nas indicated in \\eqref{eqn:kerdiscretetosmth}.\nLet $\\ff^t_c,\\fg^t_c \\in \\sptfns,\\, t=0,1,2,\\ldots$ denote spatially continuous functions determined\naccording to \\eqref{eqn:gfrecursion} and let\n$\\ff^t,\\fg^t$ denote spatially discrete functions determined\naccording to \\eqref{eqn:discretegfrecursion}.\nThen, if\n$\\ff^0(x_i) \\le \\ff^0_c (x_i)$ (for all $i$) then $\\ff^t(x_i) \\le \\ff^t_c(x_i - t\\Delta )$ and\nif $\\fg^t(x_i) \\ge \\fg_c^t(x_i - (t-\\half) \\Delta)$ for all $t.$\nSimilarly, if\n$\\ff^0(x_i) \\ge \\ff^0_c (x_i)$ (for all $i$) then $\\ff^t(x_i) \\ge \\ff^t_c(x_i + t\\Delta )$ and\nif $\\fg^t(x_i) \\ge \\fg_c^t(x_i + (t+\\half) \\Delta)$ for all $t.$\n\\end{theorem}\n\\begin{IEEEproof}\nAssume\n$\\ff^0(x_i) \\le \\ff_c^0(x_i)$ (for all $i$).\nBy Lemma \\ref{lem:disccontbnd} $\\ftdS{0}(x_i) = \\ff_c^{0,\\smthker}(x_i+\\half\\Delta)$ for each $i.$\nBy monotoniciy of $\\hg$ we have\n\\[\n\\fg^0(x_i) = \\hg(\\ff^0(x_i))\\le \\hg(\\ff_c^{0,\\smthker}(x_i+\\half\\Delta)) = \\fg_c^0(x_i+\\half\\Delta).\n\\]\nBy the same argument we obtain\n\\(\n\\gtdS{0}(x_i) \\le \\fg_c^{0,\\smthker}(x_i+\\Delta)\\,\n\\)\nand hence\n\\(\n\\ff^1(x_i)\\le \\ff_c^1(x_i+\\half\\Delta).\n\\)\nThe general result now follows by induction.\n\nThe reverse inequality can be handled similarly.\n\\end{IEEEproof}\n\nThis result is convenient when there exist wave-like solutions.\nFor example, if $\\ff_c^t(x) = \\tmplF(x-\\ashift t)$ with $\\ashift>0$  and\n$\\tmplF$ is a $(0,1)$-interpolating function, and\n$\\ff^0(x_i) \\le \\ff_c^0(x_i),$ then we have\n$\\ff^t(x_i) \\le \\tmplF(x-(\\ashift-\\Delta) t).$\nThus, if $\\ashift > \\Delta$ then we obtain asymptotic convergence for the\nspatially discrete case.\nThus we have the following result\n\n\\begin{theorem}[Discrete Spatial Convergence]\\label{thm:mainqqqqq}\nAssume that $\\smthker$ is a regular averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of functions in $\\exitfns$ satisfying the strictly positive\ngap condition.\nAssume $\\Delta < |A(\\hf,\\hg)|\/\\|\\smthker\\|_\\infty$\nand initialize system \\eqref{eqn:discretegfrecursion} with\nany $(0,1)$ interpolating $\\ff^0 \\in \\sptfns.$\nIf $A(\\hf,\\hg)>0$ then $\\ff^t(x_i) \\rightarrow 0$ and\nif $A(\\hf,\\hg)<0$ then $\\ff^t(x_i) \\rightarrow 1$ \nfor all $x_i$ \n\\end{theorem}\n\n\nThis result gives order ${\\Delta}$ convergence of the spatially discrete system to the continuum one (under positive gap assumptions).\nMuch faster convergence is observed in many situations. \nIn \\cite{HMU11b} a particular example is presented with a compelling heuristic argument for exponential convergence.\nIn general the rate of convergence appears to depend on the regularity of $\\hf$ and $\\hg$ and\n$\\smthker.$\nA $(0,1)$-interpolating spatial fixed point does not sample $\\hf$ and $\\hg$ at every value, so one cannot\nconclude that $A(\\hf,\\hg) = 0$ and,  indeed, this generally does not hold.  One can construct fixed point examples where\n$|A(\\hf,\\hg)|$ is of order $\\Delta.$\nAs a general result we have the following.\n\\begin{theorem}\\label{thm:discreteFPDelta}\nAssume $\\hf$ and $\\hg$ have a $(0,1)$-interpolating fixed point\nfor the spatially discrete system.  Then,\n\\[\n|A(\\hf,\\hg) | \\le 2{\\Delta}\\|\\smthker\\|_\\infty\n\\]\n\\end{theorem}\nAs indicated, regularity assumptions on $\\hf,\\hg$ can lead to stronger results. \nIn this direction we have the following.\n\\begin{theorem}[$C^2$ Discrete Fixed Point Bound]\\label{thm:discreteFPsum}\nAssume $\\hf$ and $\\hg$ are $C^2$ and there exists an $(0,1)$-interpolating\nspatial fixed point for the spatially discrete system.  Then\n\\[\n|A(\\hf,\\hg) | \\le \n\\frac{1}{2} (\\|\\hf''\\|_\\infty+\\|\\hg''\\|_\\infty)\\|\\smthker\\|_\\infty^2{\\Delta^2}\n\\]\n\\end{theorem}\nProofs for the above are presented in appendix \\ref{app:Aa}.\n\nFor discrete systems where gap conditions may be difficult to verify we may require more general\nresults.\nEspecially challenging are cases with an infinite number of crossing points\nclustering near the extremal ones. For such generic situations we have\nthe following spatially discrete version of Theorem \\ref{thm:globalconv}.\n\\begin{theorem}[General Discrete Convergence]\\label{thm:discreteglobalconv}\nLet $(\\hf,\\hg)$ be given as in Theorem \\ref{thm:globalconv}, let $\\smthker$ be regular,\nand assume $\\ff^0 \\in \\sptfns$ is given\nwith $\\ff^0(\\minfty)\\le v''$ and \n$\\ff^0(\\pinfty)\\ge v'.$\nThen, for any $\\epsilon>0,$ in system \\ref{eqn:gfrecursion} \nwith $\\Delta$ sufficiently small we have for all $x\\in \\reals$\n\\begin{align*}\n\\liminf_{t\\rightarrow \\infty} f^t(x) &\\ge v'-\\epsilon \\quad  \\liminf_{t\\rightarrow \\infty} g^t(x) \\ge u'-\\epsilon \\\\\n\\limsup_{t\\rightarrow \\infty} f^t(x) &\\le v''+\\epsilon \\quad \\limsup_{t\\rightarrow \\infty} g^t(x) \\le u''+\\epsilon \\,.\n\\end{align*}\n\\end{theorem}\nThe proof may be found in appendix \\ref{app:E}.\n\n\nFinite length systems can be modeled by introducing spatial dependence\ninto the definition $\\hf$ and\/or $\\hg.$ For example, in the LDPC-BEC case termination corresponds\nto setting $\\hf = 0$ outside some finite region.\nThe analysis most closely follows the unterminated case\nwith one-sided termination, e.g., setting $\\hf=0$ for $x<0.$\nWhen $A(\\hf,\\hg)>0$ and the strictly positive gap condition holds we can apply Theorem \\ref{thm:mainexist}\nto conclude that the infinite length unterminated system has a wave-like solution\nthat converges point-wise to $0.$\nSuch a solution can often be used to bound from above the\nsolutions for terminated cases to show that\ntheir solutions also tend to $0.$\nAlternatively, we can apply Theorem \\ref{thm:globalconv} to conclude that even if we remove the termination\nafter initialization the system will converge to $0.$ \n\n\\subsection{One-sided Termination}\nSince setting $\\hf=0$ over some region reduces $f$ relative to the\nunterminated case, it is more difficult to obtain lower bounds for\nthe terminated case.  It turns out for one-sided termination,\nhowever, that an analogy can be drawn between the spatial variation\nin $\\hf$ and a global perturbation in $\\hf$ that is spatially invariant\nand which then allows application of Theorem \\ref{thm:mainexist}.\n\nLet us formally define the one-sided termination version of\n\\eqref{eqn:gfrecursion} to be the system that follows\n\\eqref{eqn:gfrecursion}\nexcept that when $x<0$ we set $f^t(u)=0$ for all $u\\in [0,1].$\nThis is equivalent to redefining $\\hf = \\hf(x;u)$ to have spatial dependence so that\nwhen $x<0$ we have $\\hf(x;u)=0$  and for $x \\ge 0$ we have $\\hf(x;u) =\\hf(u)$ as before.\n\nSince this system is not translation invariant,\nit does not admit interpolating traveling wave-like solutions.  It does, however,\nadmit interpolating spatial fixed points.\n\nLet us define\n\\[\n\\unitstep_a (x) = \\begin{cases}\n0 & x  <0 \\\\\na & x = 0 \\\\\n1 & x > 0\n\\end{cases}\n\\]\nIn some cases the value of $a$ is immaterial and we may drop the subscript from the \nnotation. \n\n\\begin{theorem}[Continuum Terminated Fixed Point]\\label{thm:terminatedexist}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg)\\in \\exitfns^2$ and assume that $\\hg$ is continuous at $0$ and that\n$\\altPhi(\\hf,\\hg;u,v)$ is uniquely minimized at $(1,1)$\n(hence $A(\\hf,\\hg) < 0$  but we do not assume that the strictly positive gap condition holds).\nThen there exists $(0,1)$-interpolating $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a fixed point of the one-sided termination of\n\\eqref{eqn:gfrecursion}.\n\\end{theorem}\n\\begin{IEEEproof}\nDefine ${\\hf}(z;u) = \\hf(u)\\wedge \\unitstep(u-z)$  (where $a \\wedge b = \\min \\{a,b\\}$) and choose $z$ so that\n$A(\\hf(z;\\cdot),\\hg) =0.$\nIf $(u,v) \\in \\intcross(\\hf(z;\\cdot),\\hg)$ then, since $\\hg$ is continuous at $0,$ we have\n$u \\ge z.$ \nIf $v < \\hf(z+)$ then we have\n$\\altPhi(\\hf(z;\\cdot),\\hg;u,v)  = \\int_0^{z}\\hginv(u') du' > 0$\nand if $v \\ge \\hf(z+)$ then we have\n$\\altPhi(\\hf(z;\\cdot),\\hg;u,v) = \\altPhi(\\hf,\\hg;u,v) - A(\\hf,\\hg) > 0.$\nIt follows that $(\\hf(z;\\cdot),\\hg)$ satisfies the strictly positive gap condition.\nBy Theorem \\ref{thm:mainexist} there exists $\\tmplF,\\tmplG \\in \\Psi_{[-\\infty,\\infty]}$\nthat form a $(0,1)$-interpolating spatial fixed point ($\\ashift=0$) for\n\\eqref{eqn:gfrecursion} with $\\hf(z;\\cdot)$ replacing $\\hf.$\nIt is easy to see that there is some finite maximal $y$ such that\n$\\tmplF(x)=0$ for $x < y.$  Translate $\\tmplF$ and $\\tmplG$ so that $y=0.$\nIt follows that the resulting $\\tmplF,\\tmplG$ pair is a fixed point of the one-sided termination version of\n\\eqref{eqn:gfrecursion}.\n\\end{IEEEproof}\nIt is interesting to note in the above construction that the fixed point solution\nhas $\\tmplG^{\\smthker}(0)=z$ and $\\tmplF(0+) = \\hf(z+).$  Hence the value of \nthe discontinuity at the boundary of the termination is determined by the condition\n$A=0.$\nIn the case where $\\hg$ is not continuous at $0,$ i.e., $\\hg(0+)>0$ we can construct a fixed point solution\nas above with $\\tmplG(\\minfty) =\\hg(0+).$\n\nFor the case $A(\\hf,\\hg) \\ge 0$ we have the following.\n\\begin{theorem}[Continuum Terminated Convergence]\\label{thm:terminatedzero}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg) \\in \\exitfns^2$ and assume that\n$\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\neq (0,0).$\nThen $\\ff^t \\rightarrow 0$ for the\none-sided termination of\n\\eqref{eqn:gfrecursion}\nfor any choice of $\\ff^0.$\nIf $\\hf(x) > 0$ and $\\hg(x)>0$ on $(0,1]$ then\n$\\ff^t \\rightarrow 0$ also when $\\altPhi(\\hf,\\hg;) \\ge 0.$\n\\end{theorem}\nThe proof is presented in Appendix \\ref{app:E}.\n\nWe can, of course, also terminate the spatially\ndiscrete versions of the system.\nThus, consider the one sided termination of\n\\eqref{eqn:discretegfrecursion}\nin which the equations are modified so that\nwe set $f^t(x_i)=0$ if $x_i < 0,$\nwhich is equivalent to redefining $\\hf$ to have spatial dependence so that\n$\\hf = 0$ if $x_i < 0.$\nWe assume that $\\discsmthker$ is related to $\\smthker$ (for a continuum version)\nas indicated in \\eqref{eqn:kerdiscretetosmth}.\nFor this case we have the following quantitative result.\n\\begin{theorem}[Discrete Fixed Point Positive Gap]\\label{thm:discreteterminatedexist}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) < -\\Delta\\|\\smthker\\|_\\infty.$\nThen there exists $(0,1)$ interpolating $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a spatial fixed point of the one-sided termination of\n\\eqref{eqn:discretegfrecursion}.\n\\end{theorem}\n\\begin{IEEEproof}\nDefine $\\hf(z;u) = \\hf(u) \\wedge \\unitstep_1(u-z)$\n with $z>0$ chosen sufficiently small\nso that $A(\\hf(z;\\cdot),\\hg)  \\le -\\Delta\\|\\smthker\\|_\\infty.$\nBy Theorem \\ref{thm:mainexist}\nthere exists $\\tmplF,\\tmplG \\in \\Psi_{[-\\infty,\\infty]}$\nthat form a spatial wave solution for\n\\eqref{eqn:gfrecursion} with $\\hf(z;\\cdot)$ replacing $\\hf$\nand $\\ashift \\le -\\Delta.$\nBy Theorem \\ref{thm:mainquantize} we see that by setting\n$\\ff^0(x_i) = {\\tmplF}(x_i)$ in\n\\eqref{eqn:discretegfrecursion} (the non-terminated case)\nwe have $\\ff^1(x_i) \\ge {\\tmplF}(x_i).$\nBy translation, we can assume that ${\\tmplF}(x_i)=0$ for $x_i<0.$\nNow, the inequality $\\ff^1(x_i) \\ge {\\tmplF}(x_i)$ also holds in the one sided terminated case\nsince the values of $\\ff^1(x_i)$ are unchanged for $x_i\\ge 0.$\nThus, in the one-sided termination case the sequence $\\ff^t$ is monotonically non-decreasing\nfor each $x_i$ and must therefore have a limit $\\ff^{\\infty}.$\nIf $\\hf$ and $\\hg$ are continuous then the pair $\\ff^{\\infty},\\fg^{\\infty}$ then constitute a fixed point of the one-sided termination case.\nIf $\\hf$ and $\\hg$ are not continuous then it is possible that the pair $\\ff^{\\infty},\\fg^{\\infty}$ does not constitute a fixed point and that initializing with\n$\\ff^{\\infty}$ we obtain another non-decreasing sequence.\nIn general we can use transfinite recursion together with monotonicity in $x$\nto conclude the existence of\n a fixed point at least as large point-wise as $ (\\ff^{\\infty},\\fg^{\\infty}).$ \n\\end{IEEEproof}\n\nThe previous result gives quantitative information on the discrete approximation but it requires the strictly positive gap assumption.\nThe following result, whose proof is in Appendix \\ref{app:E},\nremoves that requirement at the cost of the quantitative bound.\n\n\\begin{theorem}[Discrete Fixed Point General]\\label{thm:discreteterminatedexistB}\nAssume $\\smthker$ is regular.\nAssume that\n$\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is uniquely minimized at $(1,1)$\nwith $A(\\hf,\\hg) < 0.$ Then\nfor all $\\Delta$ sufficiently small\nthere exists $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a spatial fixed point of the one-sided termination of\n\\eqref{eqn:discretegfrecursion} with $\\lim_{\\Delta\\rightarrow 0}\\tmplF(\\pinfty) =1.$\n\\end{theorem}\n\n\nFor the case $A(\\hf,\\hg) \\ge 0$ we have the following quantitative result.\n\\begin{theorem}[Discrete Terminated Convergence]\\label{thm:discreteterminatedzero}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) > \\Delta\\|\\smthker\\|_\\infty.$\nThen $\\ff^t \\rightarrow 0$ for the\none-sided termination of\n\\eqref{eqn:discretegfrecursion}\nfor any choice of $\\ff^0.$\n\\end{theorem}\n\\begin{IEEEproof}\nWe consider the initialization\n\\[\n\\ff^0(x_i)=\\unitstep_1(x_i)\\,\n\\] \nand show that  $\\ff^t \\rightarrow 0.$\nDefine\n\\[\n\\hf(\\eta;u) =\\hf(u) \\vee \\unitstep_1(u-(1-\\eta))\n\\]\nwhere we assume $\\eta>0$ sufficiently small so that\n$A(\\hf(\\eta;\\cdot),\\hg) > \\Delta\\|\\smthker\\|_\\infty.$\n\nBy Theorem \\ref{thm:mainquantize} there exists $(0,1)$-interpolating $\\tmplF$ and\n$\\tmplG$ and $\\ashift>\\Delta$ such that, even for the unterminated case,\n$\\ff^0(x_i) \\le\\tmplF(x_i)$ implies $\\ff^t(x_i) \\le\\tmplF(x_i-(\\ashift-\\Delta)t)$\nfor all $t.$  The same clearly holds also in the terminated case.\nClearly, $\\tmplF(x)=1$ for all $x \\ge z$ for some finite $z,$ and by translation\nwe can take $z$ to be $0$ yielding the desired result.\n\\end{IEEEproof}\n\n\n\\subsection{Two-sided Termination}\nThe two-sided termination of system \\eqref{eqn:gfrecursion}\nis defined by setting $\\ff^t(x) = 0$ for\nall $x$ outside some finite region, say $[0,Z]$ for all $t.$\nThis can be understood as a spatial dependence of $\\hf =\\hf(x;u)$\nwhere $\\hf(x;u) =0$ for $x \\not\\in [0,Z]$ and $\\hf(x;u) =\\hf(u)$ as before otherwise.\nThis system can be bounded from above by the one-sided termination case.\nThus, Theorem \\ref{thm:terminatedzero}\nand Theorem \\ref{thm:discreteterminatedzero}\napply equally well to the two-sided terminated case.\nTheorem \\ref{thm:discreteterminatedexist} on the other hand does not\nimmediately generalize, but a similar statement holds.\n\n\\begin{theorem}[Two Sided Continuum Fixed Point]\\label{thm:twoterminatedexist}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and let\n$A(\\hf,\\hg) < 0.$\nThen, for any $\\epsilon > 0,$ and for all $Z$ sufficiently large,\nthere exists $\\ff,\\fg$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:gfrecursion} such that\n$\\ff$ and $\\fg$ are symmetric about $\\frac{Z}{2},$ monotonically non-decreasing on $(-\\infty,\\frac{Z}{2}]$ and have left and right limits at least\n$1-\\epsilon$ at $\\frac{Z}{2}.$\n\\end{theorem}\nThe proof is presented in appendix \\ref{app:C}.\n\nWe have also the following spatially discrete version of the above,\nwhose proof is also in appendix \\ref{app:C}.\nIn the discrete case the termination is taken to hold for $x_i < 0$ and\n$x_i > Z = L\\Delta$ where $L$ is an integer.\nSymmetry in the spatial dimension then takes the form\n$\\ff(x_i) = \\ff(x_{L-i}).$ \n\\begin{theorem}[Two Sided Discrete Fixed Point with Gap]\\label{thm:discretetwoterminatedexist}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) < -\\Delta\\|\\smthker\\|_\\infty.$\nThen, for any $\\epsilon > 0,$ and for all $Z$ sufficiently large,\nthere exists $\\tmplF,\\tmplG$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:discretegfrecursion}\nsuch that\n$\\tmplF$ and $\\tmplG$ are spatially symmetric, monotonically non-decreasing on $(-\\infty,\\half Z]$ and satisfy $\\max_i \\{\\tmplF(x_i)\\} \\ge 1-\\epsilon$\nand $\\max \\{\\tmplG(x_i)\\} \\ge 1-\\epsilon.$ \n\\end{theorem}\n\n\nWe have also the following qualitative version that relaxes the strictly positive gap condition\nand\nwhose proof is in appendix \\ref{app:E}.\n\\begin{theorem}[Two Sided Discrete Fixed Point]\\label{thm:discretetwoterminatedexistGB}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ be given such that $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is\nuniquely minimized at $(1,1)$ and therefore \n$A(\\hf,\\hg) < 0.$\nThen, for any $\\epsilon > 0,$ and for all $Z=L\\Delta$ sufficiently large and $\\Delta$ sufficiently small,\nthere exists $\\tmplF,\\tmplG$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:discretegfrecursion}\nsuch that\n$\\tmplF$ and $\\tmplG$ are spatially symmetric, monotonically non-decreasing on $(-\\infty,\\half Z)$ and satisfy $\\max_i \\{\\tmplF(x_i)\\} \\ge 1-\\epsilon$\nand $\\max_i \\{\\tmplG(x_i)\\} \\ge 1-\\epsilon.$ \n\\end{theorem}\n\n\n\n\n\n\\section{Examples of 1-D Systems}\\label{sec:applications}\n\n\\subsection{Binary Erasure Channel}\nLet us start by re-deriving a proof that for transmission over the\nBEC regular spatially-coupled ensembles achieve the MAP threshold\nof the underlying ensemble. By keeping the rate fixed and by\nincreasing the degrees it then follows that one can achieve capacity\nthis way.  This was first shown in \\cite{KRU10}.  Given the current\nframework, this can be accomplished in a few lines. Before we prove this\nlet us see a few more examples.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/capacitybec}\n}\n\\caption{\\label{fig:capacitybec}\nEXIT charts for the $(4, 8)$-regular (left) and the $(5, 10)$-regular (right)\ndegree distributions and transmission over the BEC. The respective coupled thresholds\nare\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(4, 8)=0.497741$, and\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(5, 10)=0.499486$.\n}\n\\end{figure}\nWe have already seen the corresponding EXIT charts for the $(3,\n6)$-regular case in Figure~\\ref{fig:exitbec36}. Figure~\\ref{fig:capacitybec}\nshows two more examples, namely the $(4, 8)$-regular as well as the\n$(5, 10)$-regular case. Numerically, the thresholds are\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(3, 6)=0.48814$,\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(4, 8)=0.497741$, and\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(5, 10)=0.499486$. As\nwe see these thresholds quickly approach the Shannon limit of\none-half.\n\nConsider now a degree distribution pair $(\\lambda, \\rho)$. The BP\nthreshold of the uncoupled system is determined by the maximum\nchannel parameter $\\epsilon$ so that $\\epsilon \\lambda(x) \\leq\n1-\\rho^{-1}(1-x)$ for all $x \\in (0, 1]$. Therefore, dividing both\nsides by $\\lambda(x)$ we get for each $x \\in (0, 1]$ an upper bound\non the BP threshold. In other words, the BP threshold of the uncoupled\nensemble can be characterized as\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny uncoupled}} =\n\\inf_{x\\in(0,1]}\\frac{1-\\rho^{-1}(1-x)}{\\lambda(x)}\\,.\n\\end{align*}\nThe limiting spatially coupled threshold (when $L$ and $w$ tend to infinity)\ncan be characterized in a similar way. In this case the determining quantity is\nthe area enclosed by the curves. Therefore,\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}} =\n\\inf_{x\\in(0,1]}\\frac{\\int_0^x 1-\\rho^{-1}(1-u)\\,\\text{d}u}{\\int_0^x \\lambda(u)\\text{d}u}\\,.\n\\end{align*}\nIn the case where the BP threshold equals $\\frac{1}{\\lambda'(0)\\rho'(1)},$\ni.e., when the threshold equals the stability threshold, then the\nspatially coupled threshold equals the BP threshold.\n\nIn the regular case and in many other cases\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}} =\n\\frac{\\int_0^{x^*} 1-\\rho^{-1}(1-u)\\,\\text{d}u}{\\int_0^{x^*} \\lambda(u)\\text{d}u}\\,\n\\end{align*}\nwhere $x^*$ corresponds to the forward BP fixed point with channel\nparameter $\\epsilon^{\\small}_{\\text{\\tiny coupled}}.$ In this case\none can check that the threshold is exactly equal to the area\nthreshold. Further, we already know that the area threshold is an\nupper bound on the MAP threshold of the underlying ensemble and we\nknow that the MAP threshold of the underlying system is equal to\nthe MAP threshold of the coupled system when $L$ tends to infinity.\nWe therefore conclude that for all such underlying ensembles where\nthe area threshold satisfies the strictly positive gap condition, the area\nthreshold equals the MAP threshold.\n\nOur current framework can also be adapted to more complicated\ncases.  The following example is from \\cite[Fig. 4.15]{Mea06}.  Consider the\ndegree distribution $(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20},\n\\rho(x)=x^{15})$.  The left picture in Figure~\\ref{fig:complicated}\nshows the BP EXIT curve of the whole code.\n\\begin{figure}[htp]\n\\centering\n\\input{ps\/complicated}\n\\caption{\\label{fig:complicated}  BP EXIT curves for the ensemble\n$(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20}, \\rho(x)=x^{15})$ and\ntransmission over the BEC. Left: Determination of the BP threshold.\nRight: Determination of MAP behavior as conjectured by the Maxwell construction.}\n\\end{figure}\nAs one can see, the BP threshold of the uncoupled ensemble in this\ncase is $\\epsilon^{\\BPsmall}_{\\text{\\tiny uncoup.}} = 0.3531$ and\nthe BP EXIT curve has a single jump.\n\nThe right picture shows the MAP EXIT curve according to the Maxwell\nconstruction, see \\cite[Section 3.20]{RiU08}. According to this\nconstruction, the MAP EXIT curve has two jumps, namely at\n$\\epsilon=0.403174$, the conjectured MAP threshold, and at\n$\\epsilon=0.4855$. These two thresholds are determined by local\nbalances of areas. This is in particular easy to see for the threshold\nat $\\epsilon=0.4855$, where the two areas are quite large.\n\nLet us now show that for the coupled ensemble the Maxwell conjecture\nis indeed correct, i.e., we show that the asymptotic (in the coupling\nlength $L$) BP EXIT curve for the spatially-coupled ensemble indeed\nlooks as shown in the right-hand side of Figure~\\ref{fig:complicated}.\nTo show that the Maxwell conjecture is also correct for the uncoupled\nsystem requires a second step which we do not address here. This\nsecond step consists in showing that the MAP behavior of the uncoupled\nand coupled system is identical and is typical accomplished by using\nthe so called ``interpolation'' technique.\n\n\nThe left picture in Figure~\\ref{fig:complicatedgap} shows the\nindividual EXIT curves according to our framework for $\\epsilon=0.4855$.\nFor this channel parameter the two EXIT curves cross four times,\nnamely for $u=0$, $u=0.824784$, $u=0.967733$, and $u=0.999952$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/complicatedgap}\n}\n\\caption{\\label{fig:complicatedgap}\nConfirmation of the Maxwell conjecture using the one-dimensional framework of\nspatial coupling for the ensemble $(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20}, \\rho(x)=x^{15})$\nand transmission over the BEC.\nThe two inlets show in a magnified way the behavior of the curves\ninside the two gray boxes.\n}\n\\end{figure}\nNote that for this channel parameter the curves do not fulfill the\npositive gap condition since initially the curve $\\epsilon \\lambda(\\xg)$\nis above the curve $1-\\rho(1-\\xf)$. Nevertheless we can use our\nformalism. Let us explain the idea informally.  Let us first check\nthe behavior of the system for $\\epsilon=0.4855$. Let us shift both\ncurves and renormalize them in such a way that first (from the left)\nnon-trivial FP is mapped to zero and the last FP (on the right) is\nmapped to one. Then these curves {\\em do} fulfill the our conditions\nand our theory applies. This shows that once the channel parameter\nhas reached slightly below $0.4855$, the EXIT function drops as\nindicated in the righ-hand side of Figure~\\ref{fig:complicated}.\n\nNow where we know what the curve looks like above $\\epsilon=0.4855$\nwe can look at the remaining part.  The right picture in\nFigure~\\ref{fig:complicatedgap} shows the individual EXIT curves\naccording to our framework for $\\epsilon=0.4032$. Again, we can\nredefine our curves above this parameter and reparametrize and then\nthey do fulfill the positive gap condition. So this marks the second\nthreshold. The inlet shows the curve magnified by 1.5 and 15 respectively.\nFrom this we see that the curves are quite well matched, so the areas\nare not so easy to see.\n\n\n\\subsection{Hard-Decision Decoding}\nLow-dimensional descriptions appear naturally when we investigate\nthe performance of quantized decoders. The perhaps simplest case\nis the Gallager decoder A, \\cite{Gal63} (see \\cite{RiU01} for an\nin-depth discussion). All messages in this case are from $\\{\\pm\n1\\}$. The initial message sent out by the variable nodes is the\nreceived message.  At a check node, the outgoing message is the\nproduct of the incoming messages. At variable nodes, the outgoing\nmessage is the received message unless all incoming messages agree,\nin which case we forward this incoming message.\n\nLet $x^{(\\ell)}$, $\\ell \\in \\naturals$, be the state of the decoder,\nnamely the fraction of ``$-1$\"-messages sent out by the variable\nnodes in iteration $\\ell$. We have $x^{(0)} = \\epsilon$, and for\n$\\ell \\geq 1$, the DE equations read\n\\begin{align*}\ny^{(\\ell)} & = \\frac{1-\\rho(1-2 x^{(\\ell-1)})}{2}, \\\\\nx^{(\\ell)} & = \\epsilon (1- \\lambda(1-y^{(\\ell)}))+ (1-\\epsilon) \\lambda(y^{(\\ell)}).\n\\end{align*}\nSince the state of this system is a scalar, our theory can be applied\ndirectly. Unfortunately, as discussed in \\cite{BRU04}, for most\n(good) degree-distributions the threshold under the Gallager A\nalgorithm is determined by the behavior either at the very beginning\nof the decoding process or at the very end. In neither of those\ncases does spatial coupling improve the threshold.\n\nIn more detail, consider Figure~\\ref{fig:exitgalA}.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitgalA}\n}\n\\caption{\\label{fig:exitgalA}\nLeft: EXIT charts for the $(4, 8)$-regular degree distribution\nunder the Gallager algorithm A with $\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} = 0.0476$. The curves\ndo not cross. The threshold is determined by the stability condition. Right:\nEXIT charts for the $(3, 6)$-regular degree distribution\nunder the Gallager algorithm A with $\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} =  0.0395$. The\nthreshold is determined by the behavior at the start of the algorithm.\n}\n\\end{figure}\nThe left picture shows the two EXIT functions for the $(4, 8)$-regular\nensemble under the Gallager algorithm A and\n$\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} = 0.0476$.  As one can\nsee from this picture, this is the threshold for the uncoupled case.\nThis threshold is determined by the stability condition, i.e., the\nbehavior of the decoder towards the end of the decoding process.\nIn other words, the functions $\\hg(\\xf)$ and the inverse of $\\hf(\\xg)$\nhave the same derivative at $0$.  If we increase the channel parameter\nthen the resulting EXIT curves no longer fulfill the positive gap\ncondition (since they cross already at $0$).  This implies that the\nthreshold of the spatially coupled ensemble is the same as for the\nuncoupled one.\n\nThe right picture in Figure~\\ref{fig:exitgalA} shows the two EXIT\nfunctions for the $(3, 6)$-regular ensemble under the Gallager\nalgorithm A and $\\epsilon = 0.0395$, the threshold for the uncoupled\ncase.  In this case the threshold is determined by the behavior at\nthe beginning of the decoding process.  As one can see from the\npicture, there are two non-zero FPs. The ``smaller'' one is unstable\nand the ``larger'' one is stable. If the initial state of the system\nis below the small FP then the decoder converges to $0$, i.e., it\nsucceeds. But if it starts above the small FP, then the decoder\nconverges to the large and stable non-zero FP, i.e., it fails. As\none can see from the picture, already for the channel parameter\nwhich corresponds to the threshold of the uncoupled these two EXIT\ncurves do not fulfill the positive gap condition -- the total area\nenclosed by the two curves is negative. And if we increase the\nchannel parameter, the area would become even more negative. Hence,\nalso in this case spatial coupling does not help.\n\nLet us therefore consider the Gallager algorithm B, \\cite{Gal63,RiU01}.\nAs for the Gallager algorithm A, all messages are from the set\n$\\{\\pm 1\\}$. The initial message and the message-passing rule at\nthe check nodes are identical. But at variable nodes we have a\nparameter $b$, an integer. If at least $b$ of the incoming messages\nagree, then we send this value, otherwise we send the received\nvalue. This threshold $b$ can be a function of time. Initially the\ninternal messages are quite unreliable. Therefore, $b$ should be\nchosen large in this stage (if we choose $b$ to be the degree of\nthe node minus one we recover the Gallager algorithm A). But as\ntime goes on, the internal messages become more and more reliable\nand a simple majority of the internal nodes will be appropriate.\nThe DE equations for this case are\n\\begin{align*}\ny^{(\\ell)} & = \\frac{1-\\rho(1-2 x^{(\\ell-1)})}{2}, \\\\\nx^{(\\ell)} = & (1-\\epsilon) \\sum_{k=b}^{\\dl-1}\n\\binom{\\dl-1}{k} (y^{(\\ell)})^k (1-y^{(\\ell)})^{\\dl-1-k}\\\\\n& + \\epsilon \\sum_{\\dl-1-b}^{\\dl-1}\n\\binom{\\dl-1}{k} (y^{(\\ell)})^k (1-y^{(\\ell)})^{\\dl-1-k}.\n\\end{align*}\nAssume at first that we keep $b$ constant over time.  Consider\nthe $(4, 10)$-regular ensemble and choose $b=3$.\nThe left picture in Figure~\\ref{fig:exitgalB} shows this example for\n$\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}} = 0.02454$. As we can see, this is the largest\nchannel parameter for which the two curves do not cross, i.e., this is\nthe threshold for the uncoupled case.\nThe right picture in Figure~\\ref{fig:exitgalB} shows the same example\nbut for $\\epsilon^{\\GalBsmall}_{\\tiny \\text{coup}} = 0.0333$.  For\nthis channel parameter the strictly positive gap condition is fulfilled and\nthe two areas are exactly in balance, i.e., this is the threshold\nfor the coupled ensemble.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitgalB}\n}\n\\caption{\\label{fig:exitgalB}\nLeft: EXIT charts for the\nthe $(4, 10)$-regular ensemble and the Gallager algorithm B with $b=3$\nand $\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}} = 0.02454$. The curves do not cross.\nRight: The same example but with\n$\\epsilon^{\\GalBsmall}_{\\tiny \\text{coup}} = 0.0333$.  For\nthis channel parameter the positive gap condition is fulfilled and\nthe two areas are in balance. In both cases, the inlets show a\nmagnified version of the gray box.}\n\\end{figure}\nWe see that the increase in the threshold is substantial for this\ncase.\n\n\nWe can do even better if we allow $b$ to vary as a function of the state of the system.  The optimum\nchoice of $b$ as a function of the state $x$ was already determined\nby Gallager and we have\n\\[\nb(\\epsilon, x) = \\Big\\lceil \\Bigl(\\frac{\\log \\frac{1-\\epsilon}{\\epsilon}}{\\log \\frac{1-x}{x}} + (\\dr-1) \\Bigr)\/2 \\Big\\rceil.\n\\]\nAssume that at any point we pick the optimum $b$ value. For the\nEXIT charts this corresponds to looking at the minimum of the EXIT\nchart at the variable node over all admissible values of $b$.  If\nwe apply this to the $(4, 10)$-regular ensemble then we get a\nthreshold of $\\epsilon^{\\GalBsmall, \\text{\\tiny opt}}_{\\tiny\n\\text{coup}}(4, 8) = 0.04085$, another marked improvement. As a\nsecond example, consider the $(6, 12)$-regular ensemble.  For this\nensemble no fixed-$b$ decoding strategy improves the threshold under\nspatial coupling compared to the uncoupled case.  But if we admit\nan optimization over $b$ then we get a substantially improved\nthreshold, namely the threshold is now $\\epsilon^{\\GalBsmall,\\text{\\tiny\nopt}}_{\\tiny \\text{\\tiny coup}}(6, 12) = 0.0555$.  For comparison,\n$\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}}(6, 12) = 0.0341$.\n\n{\\em Discussion:} The optimum strategy assumes that at the\ndecoder we know at each iteration (at at each position if we consider\nspatially coupled ensembles) the current state of the system. Whether\nor not this is realistic depends somewhat on the circumstances. For\nvery large codes the evolution of the state is well predicted by\nDE and can hence be determined once and for all. For smaller systems\nthe evolution shows more variation. One option is to measure e.g.\nthe number unsatisfied check nodes given the current decisions and\nto estimate from this the state.\n\n\n\n\n\n\\subsection{CDMA  Demodulation}\n\nSpatially coupling has been considered for CDMA demodulation in\n \\cite{ScT11} and  \\cite{TTK11}.\nWe will follow \\cite{ScT11}.\n\nThe basic (uncoded) CDMA  transmission model is\n\\[\ny=\\sum_{k=1}^Kd_k \\bold{a}_k  + \\sigma \\bold{n}\n\\]\nwhere there are $K = \\alpha N$ users each transmitting a single bit $d_k = \\pm 1$  \nusing random spreading sequence $\\bold{a}_k$ of unit energy and length $N$,\nand $\\bold{n}$ is a vector of independent $N(0,1)$ random variables\n(for details see \\cite{ScT11}).\n\nIn \\cite{Tan2002} statistical mechanical methods were used to analyze randomly spread synchronous CDMA detectors over the additive white Gaussian noise channel.\nThe non-rigorous replica method predicted the optimal (asymptotic in system size) performance of various detectors.\nIn this setting the solution states that\nthe symbol-wise marginal-posterior-mode detector in the large $K$ and $N$ limit has posterior probabilities with signal to interference ratio $(1\/x)$ satisfying the equation\n\\begin{equation}\\label{eqn:cdmaFP}\nx= \n\\sigma^2 + \\alpha \\expectation \\Biggl(1-\\tanh\\Bigl(\\frac{1}{x}+\\sqrt{\\frac{1}{x}} \\xi\\Bigr) \\Biggr)^2 \n\\end{equation}\nwhere the expectation is over $\\xi \\sim N(0,1).$\nHere $x$ represents the variance of the posterior equivalent Gaussian channel $d_k + \\sqrt{x} n.$\n\nFor $\\alpha < \\alpha_{\\text{crit}} \\simeq 1.49$ (numerically determined) this\nequation has single solution\n(including the case $x=0$ for $\\sigma^2=0.$)\nFor $\\alpha \\ge \\alpha_{\\text{crit}}$ it is observed\nthat the equation has one, two, or three solutions depending \non $\\sigma^2.$\n\nIn  \\cite{ScT11}, a message passing scheme was developed such that the associated density evolution gives rise to\n\\eqref{eqn:cdmaFP} as a fixed point equation.\nThe scheme requires a modification of the transmission setup.\nTo describe the scheme first consider repeating each bit $M$ times and scaling power accordingly.\n\\[\ny=\\sum_{k=1}^K \\frac{1}{M} \\sum_{m=1}^M d_{k,m} \\boldmath{a}_k  + \\sigma \\boldmath{n}\n\\]\nNow, take $l=1,2,...,L$ instances of this system and permute the indices on a per-user basis to get\n\\[\ny_l=\\sum_{k=1}^K \\frac{1}{\\sqrt{M}} \\sum_{m=1}^M d_{k\\pi_k(m,l)} \\boldmath{a}_k  + \\sigma \\boldmath{n}\n\\]\nwhere $\\pi_k$ is a (randomizing) permutation on $[M]\\times [L].$  Note the change in scaling with respect to $M$ due to\nnon-coherent addition of the bit values (take $L\\gg M$).  Belief propagation is applied to this setup and the analysis\nleads to the density evolution equations. \n\nDefine \n$g:[0,\\infty] \\rightarrow [0,\\infty].$\n\\begin{align*}\ng(x) & = \\expectation{(1-\\tanh{(x+\\sqrt{x}\\xi))}^2} \n\\end{align*}\nwhere $\\xi \\sim N(0,1).$\nNow, further define\n\\begin{align*}\n\\hf(u)&=\\alpha g(u)+\\sigma^2\\\\\n\\hg(v)&=1\/v\n\\end{align*}\nwhere, here, $u,v \\in [0,+\\infty].$\nThe fixed point equation \\eqref{eqn:cdmaFP} can now be written\n\\[\nx = \\hf(\\hg(x))\\,.\n\\]\nThe function $\\hf$\ncorresponds to updating the LLRs  of the bits taking into account the repetition of the bits and the function\n$\\hg$ corresponds to a soft cancellation step.  In each case the resulting message LLR values are (symmetric) \nGaussian distributed and the density evolution update corresponds to the input-output map of the effective\nvariances of the equivalent AWGN channel.  The iterations can be initialized with $x = \\infty.$\n\nThe DE corresponding to the message passing decoder will converge to the solution of\n\\eqref{eqn:cdmaFP} having the largest magnitude.\nHence for $\\alpha \\le \\alpha_{\\text{crit}}$ the BP decoder will not generally achieve optimal performance.\n\nIn \\cite{ScT11} the authors further introduce spatial coupling.  The basic construction uses a chain of\ninstances of the above system and couples them by exchanging bits between neighboring instances\n\nThe spatially coupled version of \\eqref{eqn:cdmaFP}  (corresponding to local uniform coupling of width $W$) \nappearing in \\cite{ScT11} reads\n\\begin{align*}\nx_i^t & = \\sigma^2 + \\frac{\\alpha}{2W+1}\\sum_{j=-W}^W g\\Bigl( \\frac{1}{2W+1} \\sum_{l=-W}^W \\frac{1}{x_{i-1}^{t+j+l}}\\Bigr)\\, \\\\\n&=  \\frac{1}{2W+1}\\sum_{j=-W}^W \\Biggl(\\sigma^2 +\\alpha g\\Bigl( \\frac{1}{2W+1} \\sum_{l=-W}^W \\frac{1}{x_{i-1}^{t+j+l}}\\Bigr)\\Biggr)\\,.\n\\end{align*}\nTermination is accomplished by setting bits outside some finite region to be known.\n\nWe are now in the regime where our results may be applied.\nThe solutions to \\eqref{eqn:cdmaFP} correspond to crossing points of $\\hf,\\hg.$\nIf $(u,v) \\in \\cross(\\hf,\\hg)$ then the corresponding solution to  \\eqref{eqn:cdmaFP}\nis $x = v = 1\/u.$\nLet $x_1$ be the smallest solution and when there are multple solutions let $x_2$ be the largest.\nThe system is initialized with $x=u=\\infty$ and terminated with $x=0.$\nFor definition of $\\altPhi$ consistent with our canonical form we can take $(u,v)=(x_1,1\/x_1)$ as the origin and invert the sign of $v.$.\nThus, we see that for $W$ large enough the spatially coupled\nsystem will converge to the solution $x_2$ (or better near the termination) if \n\\[\n\\int_{x_1}^{x_2} \\Bigl( \\hf(z) - \\frac{1}{z} \\Bigr) \\, dz < 0\\,.\n\\]\n\nThe case $\\sigma^2=0$ ($x_2 = \\infty$) is special.\nIn \\cite{ScT11} it is claimed that $x_i^t \\rightarrow 0$ in this case.\nThis is now an easy consequence and a special case of our results since $g(x)$ approaches $0$ exponentially for large $x.$\n\n\n\\subsection{Compressed Sensing}\n\nIn a typical variant of compressed sensing one observes a ``sparse\" vector $x$\nthrough a underdetermined linear system as\n\\[\ny = Ax +n\\,.\n\\]\nwhere $n$ is an additive noise vector.\nThe matrix  $A$ is  $m \\times n$ typically with \n$m \\ll n$ where $\\delta= m\/n$ is termed the undersampling ratio.\nThe vector $x$ is constrained to be sparse, or, alternatively, to have\nentries distributed according to a distribution $p_X$ with small R\\'{e}nyi information dimension\n\\cite{VWcs}. \nIn the setup we consider here the entries of $A$ are sampled independent zero mean Gaussians random variables.\nLetting $V$ denote the $m \\times n$ all-1 matrix, the\n variances of the entries of $A$ are component-wise given by $\\frac{1}{m} V$ so that columns of \n$A$ have (roughly) unit norm.\nThe problem is to estimate $x$ from knowledge of $y$ and $A.$\nHere we also assume knowledge of $p_X.$\nThe problem can be scaled up by letting $n$ and $m$ tend to infinity while keeping\n$\\delta$ fixed.  \nAsymptotic performance is characterized in terms of the limit.\n\nOne can associate a bipartite graph to $A$ in which one set of nodes corresponds to \nthe columns (and the entries of $x$) and the other set of nodes corresponds to the rows.\nThe graphical representation suggests the use of  message passing algorithms for\nthis problem and they have indeed been proposed and studied \\cite{DMM}[and references therein].\nIn \\cite{DMM} a reduced complexity variation, AMP (Approximate Message Passing,) is developed in which there are\nonly $n$ or $m$ distinct messages, depending on the direction.\nAn additional term, the so-called Onsager reaction term, is brought into the algorithm to compensate\nof the feedback inherent in AMP (due to the violation of the extrinsic information principle and the denseness of the graph).\nIn \\cite{DMM} an analysis of AMP is given that leads in the large system limit\nto an iterative function system called\nstate evolution, which is analogous to density evolution.\nThe large system limit analysis is quite different from the usual density evolution analysis in that,\nrather than relying on sparseness and tree-like limits, the state evolution analysis relies on the central limit theorem and the fact that contributions from single edges are asymptotically negligible.\nIn the large system limit, messages (or their errors) in the AMP algorithm are normally distributed \n(this is the important consequence of the including the Onsager reaction term)\nand state evolution captures the variance (SNR) associated to the messages.\nFor our current setup: a $m\\times n$ sensing matrix with independent $\\frac{1}{\\sqrt{m}}N(0,1)$ Gaussian entries \nand known $p_X,$\nthe state evolution equations take the form\n\\cite{DJM11}\n\\begin{align*}\n\\phi_{t+1} = \\sigma^2+\\frac{1}{\\delta}\\text{mmse}(\\phi_t^{-1})\n\\end{align*}\nwhere $\\phi$ is the estimation error variance.  In this expression\n\\[\n\\text{mmse}(s) =\n\\expectation (X-\\expectation(X\\mid Y))^2\n\\]\nis the minimum mean square error of an estimator of $X$ given $Y$\nwhere $X$ is distributed as $p_X$ and $Y = \\sqrt{s}X+Z$ where $Z$ is $N(0,1)$\nand independent of $X.$\nThe main properties of $\\text{mmse}$ that are relevant here are\n\\begin{align*}\n\\limsup_{x\\rightarrow \\infty} s\\,\\text{mmse}(s) & = \\bar{D}_{p_X} \\\\\n\\intertext{and}\n\\limsup_{x\\rightarrow \\infty} \\int_0^s \\text{mmse}(s) ds &= \\log(s) \\bar{d}_{p_X}\n\\end{align*}\nwhere $\\bar{D}_{p_X}$ is termed the mmse dimension \\cite{VWcs} \nand $\\bar{d}_{p_X}$ is the upper information dimension \\cite{VWcs} of\n$p_X$ which can be defined by\n\\[\n\\bar{d}_{p_X}\n=\\limsup_{\\ell \\rightarrow \\infty} \\frac{H\\lfloor \\ell X \\rfloor}{\\log \\ell}\n\\]\nwhere, here, $H$ denotes the Shannon entropy.\n\nSpatial coupling can be introduced by imposing additional structure on $A.$\nLet us first consider a collection of parallel systems.\nThus, let $\\tilde{A}$ be a doubly infinite array of $m\\times n$ matrices in \nwhich one diagonal $\\tilde{A}_{i,i}$ is non-zero with each $\\tilde{A}_{i,i}$ \ni.i.d. Gaussian samples with entry-wise variance matrix $\\frac{1}{m}V.$\nThe variance matrix associated to matrix $\\tilde{A}$ is \n$\\tilde{V}$ with $\\tilde{V}_{i,i}= \\frac{1}{m}V$ and $\\tilde{V}_{i,j}=0$ for $i\\neq j.$\nSpatial coupling is achieved by setting $V_{i,j} =  w_{i-j} \\frac{1}{m} V.$\nTermination can be effected by providing additional measurements for variables associated\nto the termination.\nSpatially coupled constructions of this type and resulting performance improvements were first presented\nin \\cite{KMSSZ11}. \nThe analytical results on information theoretic optimal results that we reproduce here were presented in\n\\cite{DJM11}.\n\nThe spatially coupled system can be understood within our framework as having the following exit functions.\n\\begin{align*}\n\\hf(u) &= \\sigma^2+\\frac{1}{\\delta} \\text{mmse}(u)\\\\\n\\hg (v) &= 1\/v \n\\end{align*}\nNote in this case that rather than $[0,1]^2$ the system is defined on $[0,+\\infty]^2.$\nThere is a crossing point $(u_1,v_1)$ where $u_1$ is minimal and $v_1$ is maximal.\nIt is easy to see that we have the bound $u_1 \\ge \\frac{\\delta}{ \\expectation(X^2)}$\nsince $\\hf$ is decreasing in $u$ and $\\hf(0) = \\frac{1}{\\delta} \\expectation(X^2).$\nTo apply our analysis we can use this point as our origin in defining $\\altPhi$\nand we can invert the sign of $v$ to recover our canonical ordering.\n\nWe can now easily recover the main results in \\cite{DJM11}.\nConsider first the noiseless case $\\sigma^2 = 0.$  The FP of interest in the\ncomponent system above occurs at $(\\infty,0).$\nIf $\\bar{d}_{p_X} < \\delta$ then we have\n\\[\n\\altPhi(\\hf,\\hg;\\infty,0)\n=\n\\int_{x_1}^{\\infty} (\\frac{1}{\\delta} \\text{mmse}(x)-\\frac{1}{x}) dx\n=\n-\\infty\\,.\n\\]\nThus we get convergence to the fixed point at $(\\infty,0).$ \n(Some simple adjustment of our arguments are needed to handle the unbounded case.)\n\nConsider now $\\sigma^2 > 0$ and let $(u^*(\\sigma^2),v^*(\\sigma^2))$ denote the crossing point\nwith maximal $u$ and minimal $v.$\nIn this case we have\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;u^*(\\sigma^2),v^*(\\sigma^2))\n\\\\= &\n\\int_{u_1}^{u^*} (\\sigma^2 + \\frac{1}{\\delta} \\text{mmse}(u)-\\frac{1}{u}) du\n\\\\ = &\n\\sigma^2 (u^*-u_1) +\\frac{1}{\\delta}\\int_{u_1}^{u^*}  \\text{mmse}(u) dx -\\log(u^*\/u_1)\\,.\n\\end{align*}\nBy choosing $\\sigma^2$ small enough we can have  $u^*(\\sigma^2)$ as large\nas desired.\nAssume $\\bar{d}_{p_X}<\\delta.$ Then,\nassuming $\\sigma^2$ small enough we have $\\altPhi(\\hf,\\hg;u',v')>\\altPhi(\\hf,\\hg;u^*,v^*)$ for\nany crossing point $(u',v')$ with $u' \\le z.$  It then follows that the crossing point that minimizes $\\altPhi(\\hf,\\hg)$\nhas $u$ value larger than $z.$\n\nAssume now that $\\bar{D}_{p_X}<\\delta$  then for all $\\sigma^2$ small enough we have\n$(u^*(\\sigma^2),v^*(\\sigma^2))$ is minimizing $\\altPhi(\\hf,\\hg).$\nFurthermore it follows that $u^*(\\sigma^2)  > C \\sigma^{-2}$ for some constant $C.$\n\n\\section{Higher-Dimensional Systems and the Gaussian Approximation}\\label{sec:gaussapprox}\nWe have discussed in the previous section several scenarios where\nthe state of the system is one dimensional and the developed theory\ncan be applied directly and gives precise predictions on the threshold\nof coupled systems. But we can considerably expand the field of\napplications if we are content with {\\em approximations}.  For\nuncoupled systems a good example is the use of EXIT functions.  EXIT\nfunctions are equivalent to DE for the case of the BEC, where the\nstate is indeed one dimensional. For transmission over general BMS\nchannels they are no longer exact but they are very useful engineering\ntools which give accurate predictions and valuable insight into\nthe behavior of the system.\n\nThe idea of EXIT functions is to replace the unknown message densities\nappearing in DE by Gaussian densities. If one assumes that the\ndensities are symmetric (all densities appearing in DE are symmetric)\nthen each Gaussian density has only a single degree of freedom and\nwe are back to a one-dimensional system. Clearly, the same approach\ncan be applied to coupled systems.  Let us now discuss several\nconcrete examples. We start with transmission over general BMS\nchannels.\n\n\\subsection{Coding and Transmission over General Channels}\nAs we have just discussed, for transmission over general BMS channels\nit is natural to use EXIT charts as a one-dimensional approximation\nof the DE process \\cite{teB99a,teB99b,teB00,teB01}.  This strategy\nhas been used successfully in a wide array of settings to approximately\npredict the performance of the BP decoder. As we have seen, whereas\nfor the BP decoder the criterion of success is that the two EXIT\ncurves do not overlap, for the performance of spatially coupled\nsystems the criterion is the positive gap condition and the area condition.\n\nWe demonstrate the basic technique by considering the simple setting\nof point-to-point transmission using irregular LDPC ensembles. It\nis understood that the same ideas can be applied to any of the many\nother scenarios where EXIT charts have been used to predict the\nperformance of the BP decoder of uncoupled systems.\n\nIn the sequel, let $\\psi(m)$ denote the function which gives the\nentropy of a symmetric Gaussian of mean $m$ (and therefore standard\nvariation $\\sigma=\\sqrt{2\/m}$). Although there is no elementary\nexpression for this function, there are a variety of efficient\nnumerical methods to determine its value, see \\cite{RiU08}.\n\nDefine the two functions\n\\begin{align*}\n\\hg(\\xf) & = 1-\\sum_{i} \\rho_i \\psi\\bigl((i-1)\\psi^{-1}(1-\\xf)\\big), \\\\\n\\hf(\\xg)  & = \\sum_{i} \\lambda_i \\psi\\bigl((i-1)\\psi^{-1}(\\xg)+\\psi^{-1}(c) \\big).\n\\end{align*}\nNote that $\\hg(\\xf)$ describes the entropy at the output of a check\nnode assuming that the input entropy is equal to $\\xf$ and $\\hf(\\xg)$\ndescribes the entropy at the output of a variable node assuming\nthat the input entropy is equal to $\\xg$ and that the entropy of the\nchannel is $c$. Both of these functions are computed under the\nassumption that all incoming densities are symmetric Gaussians (with\nthe corresponding entropy). In addition, for the computation of the\nfunction $\\hg(\\xf)$ we have used the so-called ``dual'' approximation,\nsee \\cite[p. 236]{RiU08}.\n\n\nFig.~\\ref{fig:positivegapbawgnc36} plots the EXIT charts for the\n$(3, 6)$-regular ensemble and transmission over the BAWGNC. The\nplot on the left shows the determination of the BP threshold for\nthe uncoupled system according to the EXIT chart paradigm. The\nthreshold is determined by the largest channel parameter so that\nthe two curves do not cross. This parameter is equal to $\\ent^{\\BPsmall,\n\\EXITsmall}=0.42915$.  Note that according to DE the BP threshold\nis equal to $\\ent^{\\BPsmall} = 0.4293$, see \\cite[Table 4.115\n]{RiU08}, a good match.\n\nThe plot on the right show the determination of the BP threshold\nfor the coupled ensemble according to the positive gap condition.\nSince for this case we only have a single nontrivial FP, this\nthreshold is given by the maximum channel entropy so that the gap\nfor the largest FP is equal to $0$.  This means, that for this\nchannel parameter the ``white'' and the ``dark gray'' area are\nequally large.  This parameter is equal to $\\ent^{\\BPsmall,\n\\EXITsmall}_{\\text{\\tiny coupled}}=0.4758$.  Note that according\nto DE, the BP threshold of the coupled system is equal to\n$\\ent^{\\BPsmall}_{\\text{\\tiny coupled}} = 0.4794$, see \\cite[Table\nII]{KRU12}, again a good match.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/positivegapbawgnc36}\n}\n\\caption{\\label{fig:positivegapbawgnc36} Left: Determination of the\nBP threshold according to the EXIT chart paradigm for the $(3,\n6)$-regular ensemble and transmission over the BAWGNC. The two\ncurves are shown for $\\ent^{\\BPsmall, \\EXITsmall}=0.42915$. As one\ncan see from this picture, the two curves touch but do not cross.\nRight: Determination of the BP threshold for the coupled ensemble\naccording to the EXIT chart paradigm and the positive gap condition.\nThe two curves are shown for  $\\ent^{\\BPsmall, \\EXITsmall}_{\\text{\\tiny\ncoupled}}=0.4758$.  For this parameter the ``white'' and the ``dark\ngray'' area are in balance.  } \\end{figure}\n\n\\subsection{Min-Sum Decoder}\nAs a second application let us consider the min-sum decoder.  The\nmessage-passing rule at the variable nodes is identical to the one\nused for the BP decoder.  But at a check nodes the rule differs --\nfor the min-sum decoder the sign of the output is the product of\nthe signs of the incoming messages (just like for the BP decoder)\nbut the absolute value of the outgoing message is the minimum of\nthe absolute values of the incoming messages. \n\nFor, e.g., the $(3, 6)$-regular ensemble DE predicts a min-sum\ndecoding threshold on the BAWGNC of $\\ent^{\\MinSumsmall}_{\\text{\\tiny\nuncoup}}=0.381787$, \\cite{Chu00}. For the coupled case this threshold\njumps to $\\ent^{\\MinSumsmall}_{\\text{\\tiny\ncoupled}}=0.429$.\\footnote{Strictly speaking it is not known that\nmin-sum {\\em has} a threshold, i.e., that there exists a channel\nparameter so that for all better channels the decoder converges\nwith high probability in the large system limit and that for all\nworse channels it does not. Nevertheless, one can numerically compute\n``thresholds'' and check empirically that indeed they behave in the\nexpected way. }\n\nIn order to derive a one-dimensional representation of DE , we\nrestrict the class of densities to symmetric Gaussians. Of course,\nthis introduces some error. Contrary to BP decoding, the messages\nappearing in the min-sum decoding are not in general symmetric (and\nneither are they Gaussian).\n\nThe DE rule at variable nodes is identical to the one used when we\nmodeled the BP decoder. The DE rule for the check nodes is more\ndifficult to model but it is easy to compute numerically.  \n\nRather than plotting EXIT charts using entropy, we use the error\nas our basic parameter.  There are two reasons for this choice.\nFirst, our one-dimensional theory does not depend on the choice of\nparameters and so it is instructive see an example which uses a\nparameter other than entropy. Second, the min-sum decoder is\ninherently invariant to a scaling, whereas entropy is quite sensitive\nto such a scaling. Error probability on the other hand is also\ninvariant to scaling.\n\nFigure~\\ref{fig:minsum36} shows the predictions we get by applying our\none-dimensional model.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/minsum36}\n}\n\\caption{\\label{fig:minsum36}\nLeft: Determination of the MinSum threshold according to the EXIT\nchart paradigm for the $(3, 6)$-regular ensemble and transmission\nover the BAWGNC. The two curves are shown for $\\ent^{\\BPsmall,\n\\EXITsmall}=0.401$. As one can see from this picture, the two curves\ntouch but do not cross.  Right: Determination of the MinSum threshold\nfor the coupled ensemble according to the EXIT chart paradigm and\nthe positive gap condition.  The two curves are shown for\n$\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny coupled}}=0.436$.  For\nthis parameter the ``white'' and the ``dark gray'' area are in\nbalance.  } \\end{figure} \nThe predicted thresholds are $\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny\nuncoup}}= 0.401$, $\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny\ncoupled}}=0.436$.  These predictions are less accurate than the\nequivalent predictions for the BP decoder.  Most likely this is due\nto the lack of symmetry of the min-sum decoder.  But the predictions\nstill show the right qualitative behavior.\n\n\n\n\n\n\n\n\n\n\n\\section{Analysis and Proofs}\\label{sec:proof}\n\nIn the analysis we allow discontinuous update (EXIT) functions.\nThis is not merely for generality\nbut also for modeling of termination and to allow discontinuous perturbations.\nWe will require some notation for dealing with this.\n\nGiven a monotonically non-decreasing function $\\ff$ we write\n\\[\nv \\veq \\ff(u)\n\\]\nto mean $v \\in [\\ff(u-),\\ff(u+)].$\nGiven $\\fg\\in\\sptfns,$ continuous $\\ff \\in \\sptfns,$ and $h\\in\\exitfns,$ we write\n\\[\n\\fg \\veq h\\circ\\ff\n\\]\nto mean $\\fg(x)\\in  [h(\\ff(x)-),h(\\ff(x)+)],$ i.e.,\n$\\fg(x)\\veq  h(\\ff(x)),$\nfor all $x\\in \\reals.$ We write\n\\[\n\\fg = h\\circ\\ff\n\\]\nto mean $\\fg(x) = h(\\ff(x))$ for all $x.$\nIn some contexts we may have equality holding\nup to a set of $x$ of measure $0.$\nTo distinguish this we write\n\\[\n\\fg \\equiv h\\circ\\ff\n\\]\nto mean $\\fg(x) = h(\\ff(x))$ for all $x$\nup to a set of measure $0.$\nNote that modifying $\\fg$ on a set of measure $0$ has no impact on $\\gS.$\nIn general we use $\\equiv$ to indicate equality up to sets of measure $0.$\n\nGiven a real number $\\ashift$ we use the notation\n$\\gSa$ to denote the reverse shift of $\\gS$ by $\\ashift,$ i.e.,\n\\[\n\\gSa(x) = \\gS(x+\\ashift) \\,.\n\\]\nUltimately we are interested in interpolating functions such that\n\\(\n\\fg = \\hg\\circ\\fS\\,,\n\\)\nand\n\\(\n\\ff = \\hf\\circ\\gSa\\,,\n\\)\nsince this represents a wave-like solution to system \\ref{eqn:gfrecursion}.\nThe mathematical arguments, however, sometimes only give rise to functions\n{\\em consistent} with the equations, i.e., such that\n\\(\n\\fg \\veq \\hg\\circ\\fS\\,,\n\\)\nand\n\\(\n\\ff \\veq \\hf\\circ\\gSa\\,.\n\\)\nMuch of the analysis works with this weaker condition and then strengthens  it to  obtain proper solutions.\n\n\n\\subsection{Spatial Fixed Points and Waves}\n\n\nEven when one exists, it is typically difficult to analytically determine an interpolating spatial fixed point solution $(\\ff,\\fg) \\in \\sptfns^2$\nfor a given pair $(\\hg,\\hf) \\in \\exitfns.$\nThe reverse direction, however, is relatively easy.\nIn particular, given a putative $(0,1)$-interpolating spatial fixed point $(\\ff,\\fg)$\nthe corresponding  $(\\hf,\\hg)$ is essentially determined by the requirement that\n$\\fg(x) = \\hg(\\fS(x))$ and $f(x) = \\hf(\\gS(x)).$\nSome degeneracy is possible if, for example, $\\fS$ is constant over some interval on which $\\fg$ varies.  \nEven in this degenerate case, however, the equivalence class of $\\hg$ is uniquely determined.\nThus, given $(0,1)$-interpolating $\\ff$ and $\\fg$  where $\\ff$ is continuous, we define\n$h_{[\\fg,\\ff]}$ to be any element of the uniquely determined equivalence class such that,\n\\[\n\\fg \\veq h_{[\\fg,\\ff]}\\circ\\ff\n\\]\ni.e., for each $x,$\n$\\fg(x) \\in [h_{[\\fg,\\ff]}(\\ff(x)-), h_{[\\fg,\\ff]}(\\ff(x)+)].$\n(A simple argument shows that the equivalence class is indeed uniquely determined.)\nIn general, if $f$ and $g$ are not $(0,1)$-interpolating, then we still consider \n$h_{[\\fg,\\ff]}$ to be defined on $[\\ff(\\minfty),\\ff(\\pinfty)]$ and the inverse to be define on\n$[\\fg(\\minfty),\\fg(\\pinfty)].$ By definition, $f$ and $g$ are\n$(\\ff(\\minfty),\\ff(\\pinfty))$-interpolating and \n$(\\fg(\\minfty),\\fg(\\pinfty))$-interpolating respectively.\n\nIf $(\\ff,\\fg)$ is a $(0,1)$-interpolating spatial fixed point solution to \\eqref{eqn:gfrecursion}\nthen we have $h_{[\\ff,\\gS]} \\equiv \\hf$ and $h_{[\\fg,\\fS]} \\equiv \\hg.$\nIn the reverse direction, $h_{[\\ff,\\gS]} \\equiv \\hf$ and $h_{[\\fg,\\fS]} \\equiv \\hg$ implies,\nand, (assuming $\\ff$ and $\\fg$ are $(0,1)$-interpolating) is in fact equivalent to,\n\\begin{align*}\n\\fg \\veq \\hg\\circ \\fS,\\quad\n\\ff \\veq \\hf\\circ \\gS \n\\end{align*}\nbut does not in general imply the stronger condition\n\\begin{align*}\n\\fg \\equiv \\hg\\circ \\fS,\\quad\n\\ff \\equiv \\hf\\circ \\gS \\,.\n\\end{align*}\nIf $\\hf$ and $\\hg$ are continuous then equivalence, and in fact equality, is implied.\nIn general, given the above equivalence we can achieve equality by replacing $\\fg$ with $\\hg \\circ \\fS$\nand $\\ff$ with $\\hf \\circ \\gS,$ since $\\fS$ and $\\gS$ are thereby unchanged.\n\n\\subsubsection{Sensitivity with Irregular Smoothing}\\label{sec:pathology}\n\nIn this section we illustrate by example some of the subtlety that\ncan arise with non-regular smoothing kernels.  We also show how\nnon-uniqueness of fixed point solutions can occur when the positive\ngap condition is satisfied but the strictly positive gap condition is not satisfied.\n\nThe following example shows that changing $\\hf$ or $\\hg$ on a set of measure $0$\ncan, for some choices  of $\\smthker,$ have a dramatic effect on the solution\nto \\eqref{eqn:gfrecursion}.\nAssume an averaging kernel $\\smthker$ that is positive everywhere on $\\reals$ except on\n$[-2,2],$ where it equals $0.$\nConsider\n\\[\n\\hf(u) = \\unitstep_a(u-\\frac{1}{2})\n\\]\nand\n\\[\n\\hg(u) = \\unitstep_b(u-\\frac{1}{2})\n\\]\nwhere  $a$ and $b$ are specified below.\nLet $\\ff(x) = \\unitstep(x),$ then we have \nwe have \n$\\fS(x)<\\frac{1}{2}$ for $x \\in (\\infty,-2),$\n$\\fS(x)=\\frac{1}{2}$ for $x \\in [-2,2],$ and\n$\\fS(x)>\\frac{1}{2}$ for $x \\in (2,\\infty)\\,.$\nConsider initializing system \\eqref{eqn:gfrecursion} with $\\ff^0(x)=\\unitstep(x).$ \nIf $a=b=\\frac{1}{2}$ then the solution is the fixed point\n\\[\n\\ff^t(x)=\\fg^t(x) =\\frac{1}{2}(\\unitstep_1(x+2)+\\unitstep_0(x-2))\\,.\n\\]\nIf $a=b=1$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_1(x+4t)\n\\\\\n\\fg^t(x)&=\\unitstep_1(x+4t+2)\\,,\n\\end{align*}\nand $\\ff^t(x) \\rightarrow 1.$\nIf $a=b=0$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_0(x-4t)\n\\\\\n\\fg^t(x)&=\\unitstep_0(x-4t-2)\\,,\n\\end{align*}\nand $\\ff^t(x) \\rightarrow 0.$\nIf $a=0$ and $b=1$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_0(x)\n\\\\\n\\fg^t(x)&=\\unitstep_1(x-2)\\,,\n\\end{align*}\nanother fixed point.\n\nTo give a more general example, if we define $\\ff$ and $\\fg$ as any functions in $\\sptfns$ that\nequal $0$ on $(\\minfty,-1)$ and $1$ on $(1,\\pinfty)$\nthen we have \n$\\fg \\veq \\hg\\circ\\fS$ \nand\n$\\ff \\veq \\hf\\circ\\gS.$ \nIt follows in all such cases that \n$h_{[\\ff,\\gS]} \\equiv \\hf$ and\n$h_{[\\fg,\\fS]} \\equiv \\hg.$\nThis example shows that it is possible to have many solutions that are ``consistent\" with\nthe equation \\eqref{eqn:gfrecursion}, in that\n$\\fg \\veq \\hg\\circ\\fS$ \nand\n$\\ff \\veq \\hf\\circ\\gS.$ \n\\subsubsection{Non-Unique Solutions}\\label{sec:nonunique}\nNow assume $\\smthker = \\frac{1}{2}\\indicator_{|x|<1}.$\nLet $\\tilde{f}$ and $\\tilde{g}$ be any functions in $\\sptfns$ that\nequal $0$ on $(\\minfty,-1)$ and $1$ on $(1,\\pinfty)$ and take values in\n$(0,1)$ on $(-1,1).$\nNow consider\n\\begin{align*}\n\\ff_a(x) &= \\frac{1}{2} \\bigl(\\tilde{f}(x+a)+\\tilde{f}(x-a)\\bigr) \\\\\n\\fg_a(x) &= \\frac{1}{2} \\bigl(\\tilde{g}(x+a)+\\tilde{g}(x-a)\\bigr) \n\\end{align*}\nFor all  $a>3$ we see that\n$(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]})$\ndoes not depend on $a$ and the given functions form a family\nof spatial fixed points for the system.\nThis gives an example where system \\eqref{eqn:DE} exhibits multiple\nspatial fixed point solutions.\nNote that $(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]})$ does not satisfy the strictly positive\ngap condition since $\\altPhi(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]};\\frac{1}{2},\\frac{1}{2})=0.$\n\n\\subsection{Spatial fixed point integration.}\n\nConsider a $(0,1)$-interpolating spatial fixed point $(\\ff,\\fg).$\nThen, at $v=\\fS(z_1)$ the integral $\\int_0^v \\hg$\ncan be expressed as\n\\[\n\\int_0^v \\hg(z) \\text{d}z=\n\\int_{-\\infty}^{x_1} \\fg(x) \\bigl(\\frac{d}{dx} \\fS(x )\\bigr) \\text{d}x\n=\n\\int_{-\\infty}^{x_1} \\fg(x) \\text{d}\\fS(x)\\,.\n\\]\nSimilarly, at $u=\\gS(x_2)$ we have\n\\[\n\\int_0^u \\hf(z)\\text{d}z = \\int_{-\\infty}^{x_2} \\ff(x) \\text{d}\\gS(x)\\,.\n\\]\nThe product rule of calculus gives, under mild regularity assumptions, $\\fg(x) \\text{d}\\ff(x) + \\ff(x) \\text{d}\\fg(x) =  \\text{d}(\\fg(x)\\ff(x))$ and, were it not for the\nspatial smoothing, this would solve directly the sum of the above two integrals in terms of the product\n$\\fg(x)\\ff(x).$  By properly handling the spatial smoothing we can accomplish something similar, and the result is\npresented in Lemma \\ref{lem:twofint}.\nWe obtain a succinct formula for the evaluation of\n$\\altPhi(\\hf,\\hg;\\fg(x_1),\\ff(x_2))$ that is {\\em local}\nin its dependence on $\\ff$ and $\\fg.$  \nThis formula captures a valuable information concerning\nthe $(0,1)$-interpolating spatial fixed point solution and its relation to $\\altPhi.$ In particular it relates local properties of fixed point solutions\nto corresponding values of  $\\altPhi.$\n\nFor $(\\ff, \\fg) \\in \\sptfns^2$ and $\\smthker$ an even averaging kernel,\n\\begin{align}\\label{eqn:definexi}\n\\PhiSI (\\smthker;f,g;x_1,x_2) & :=\n(\\fS(x_1) - f(x_2+))\n(\\gS(x_2) - g(x_1+))\\nonumber\n\\\\ &\\quad\n+\n\\altPhiSI(\\smthker;f,g;x_1,x_2)\n\\end{align}\nwhere\n\\begin{align*}\n\\altPhiSI(\\smthker;& f,g;x_1,x_2) = \\\\\n& \\int_{0}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x))df(y)\n\\Bigr) dx\n\\\\ & +\n\\int_{0}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x))dg(y)\n\\Bigr) dx\n\\\\ & =\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x))df(y)\n\\Bigr) dx\n\\\\ & =\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x))dg(y)\n\\Bigr) dx\n\\end{align*}\nwhere the integrals are Lebesgue-Stieltjes integrals.\nIf $x'<x$ the $\\int_{(x,x']} f(y)dg(y)$ is defined to be\n$-\\int_{(x',x]} f(y)dg(y).$\nNote that $\\altPhiSI$ is non-negative.\nWe show in the proof of Lemma \\ref{lem:twofint} that the three forms of $\\altPhiSI$ are indeed equivalent.\n\nNote that if $\\fg$ is discontinuous at $x_1$\nthen $\\altPhiSI(\\smthker; f,g;x_1,x_2)$ is discontinuous at $x_1,$ \nand similarly for $\\ff$ at $x_2.$\nFor small $\\delta$ we have\n\\begin{align*}\n&\\altPhiSI(\\smthker;f,g;x_1+\\delta,x_2)\n-\n\\altPhiSI(\\smthker;f,g;x_1,x_2) \\\\\n=&\n(f(x_2+)-\\fS(x_1)+O(|\\delta|))(g((x_1+\\delta)+)- g(x_1+))\n\\end{align*}\nand\nit follows that $\\PhiSI(\\smthker;f,g;x_1,x_2)$ is continuous in $x_1$ and $x_2.$\n\n\\begin{lemma}\\label{lem:twofint}\nIf $f,g \\in \\sptfns$ (not necessarily interpolating) and $\\smthker$ is an even averaging kernel, then\n\\begin{align*}\n&\\quad \\int_{\\fg(\\minfty)}^{\\fg(x_1+)} h^{-1}_{[\\fg,\\fS]} (u)\\text{d}u\n+ \\int_{\\ff(\\minfty)}^{\\ff(x_2+)} h^{-1}_{[\\ff,\\gS]} (v)\\text{d}v \\\\\n&\\quad-\\ff(x_2+)\\fg(x_1+) + \\ff(\\minfty)\\fg(\\minfty)\\\\\n&= \\altPhiSI(\\smthker;f,g;x_1,x_2) \n\\end{align*}\nfor all $x_1$ and $x_2.$\n\\end{lemma}\nThe proof of this key lemma can be found in appendix \\ref{app:A}.\n\nThere are many important consequences of this result.\nOne fundamental consequence of Lemma \\ref{lem:twofint} is that if $(\\ff,\\fg)$ are $(0,1)$-interpolating\nand\n$(\\hf,\\hg) \\equiv (h_{[\\ff,\\gS]},h_{[\\fg,\\fS]}),$ then \n\\begin{align}\n\\altPhi(\\hf,\\hg;\\fg(x_1+),\\ff(x_2+))\n&= \\altPhiSI(\\smthker;f,g;x_1,x_2) \\label{eqn:altPhiXi}\n\\end{align}\nand\n\\begin{align*}\n\\Phi(\\hf,\\hg;\\gS(x_2),\\fS(x_1))\n&= \\PhiSI(\\smthker;f,g;x_1,x_2) \\label{eqn:PhiXi}\n\\end{align*}\nfor all $x_1$ and $x_2.$\n\n\\begin{lemma}[Necessity of Positive Gap]\\label{lem:FPAzero}\nLet $(f,g) \\in \\sptfns^2$ be $(0,1)$-interpolating functions\nand let  $\\smthker$  be an averaging kernel.\nThen $(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]})$ satisfies the positive gap condition and \n$A(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]}) = 0.$\n\\end{lemma}\n\\begin{IEEEproof}\nSince  $\\altPhiSI(\\smthker;f,g;x_1,x_2) \\ge 0$ for all $x_1,x_2$ we see \nfrom Lemma \\ref{lem:twofint} that\n$\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};u,v) \\ge 0$ for all $(u,v) \\in [0,1]^2.$\nLetting $x_1,x_2 \\rightarrow \\infty$ we see $\\altPhiSI(\\smthker;f,g;x_1,x_2)\\rightarrow 0$\nand we obtain $A(\\hf,\\hg) = 0.$\n\\end{IEEEproof}\nAnother and simpler proof that $A(\\hf,\\hg) = 0$  is given in Section \\ref{sect:inverse}.\n\\subsubsection{Fixed Point Potential Bounds}\n\nLemma \\ref{lem:twofint} provides information on the local structure of fixed point solutions\nand relates it to the value of the potential functions. In particular  we can extract information on the\nspatial transition between the two underlying component fixed points.\n\nLet us introduce the notation\n\\[\n\\Delta_L f(x) = f((x+L)-) - f((x-L)+)\n\\]\nand the quantity\n\\(\ne_L = \\int_L^\\infty \\smthker(x)\\text{d}x\\,.\n\\)\n\n\\begin{lemma}\\label{lem:transitionPhiBounds}\nLet $\\ff,\\fg$ be functions in $\\sptfns.$\nFor any $L>0$ the following inequalities hold\n\\begin{align*}\n\\altPhiSI(\\smthker,\\ff,\\fg;x,x)\n& \\le\n\\Delta_L f(x)\\Delta_L g(x)+  e_L \n\\end{align*}\n\\begin{align*}\n|\\fS(x)-f(x)|\n \\le\n\\Delta_L f(x) +  e_L \n\\end{align*}\n\nFor $(0,1)$-interpolating $\\ff,\\fg \\in \\sptfns$ we have\n\\begin{align*}\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\fg(x),\\ff(x))\n& \\le\n\\Delta_L f(x)\\Delta_L g(x)+  e_L \n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\gS(x),\\ff(x)) & \\le\n2\\Delta_L g(x) + 2e_L\n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\fg(x),\\fS(x)) &  \\le\n2\\Delta_L f(x) + 2e_L\n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\gS(x),\\fS(x)) &  \\le\n2\\Delta_L f(x) + 2\\Delta_L g(x)  + 3e_L\n\\end{align*}\n\\end{lemma}\nThe Lemma is proved in appendix \\ref{app:A}. \n\n\\subsubsection{Invariance of Fixed Point Potential}\n\n\\begin{lemma}\\label{lem:FPequal}\nAssume $(\\hf,\\hg) \\in \\exitfns^2$ and $\\smthker$ an averaging kernel.\nLet $(\\ff,\\fg) \\in \\sptfns^2$ be a consistent travelling wave solution (not necessarily interpolating) to\nthe system \\eqref{eqn:gfrecursion} with shift $\\ashift,$ i.e.,\n\\begin{align*}\n\\ff &\\veq \\hf \\circ \\fg^{\\smthker,\\ashift} \\\\\n\\fg &\\veq \\hg \\circ \\ff^{\\smthker} \\,.\n\\end{align*}\n Then\n\\begin{itemize}\n\\item[A.] $(\\ff(\\minfty),\\fg(\\minfty)) \\in \\cross(\\hf,\\hg).$\n\\item[B.] $(\\ff(\\pinfty),\\fg(\\pinfty)) \\in \\cross(\\hf,\\hg).$\n\\item[C.] If $\\ashift = 0$ then \n\\[\n\\altPhi(\\hf,\\hg;\\ff(\\minfty),\\fg(\\minfty)) =\n\\altPhi(\\hf,\\hg;\\ff(\\pinfty),\\fg(\\pinfty))\\,.\n\\]\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nBy definition we have $\\ff(x) \\veq \\hf(\\fg^{\\smthker,\\ashift}(x))$ for each $x \\in \\reals.$\nTaking limits we have $\\ff(\\minfty) \\veq \\hf(\\fg^{\\smthker,\\ashift}(\\minfty)).$\nSince $\\fg^{\\smthker,\\ashift}(\\minfty)=\\fg^{\\smthker}(\\minfty)=\\fg^{}(\\minfty)$ we have\n$\\ff(\\minfty) \\veq \\hf (\\fg(\\minfty)).$ Parts A and B now follow easily.\n\nIf $(\\ff(\\minfty),\\fg(\\minfty)) = \\ff(\\pinfty),\\fg(\\pinfty))$ then part C is immediate,\nso assume  $(\\ff(\\minfty),\\fg(\\minfty)) < \\ff(\\pinfty),\\fg(\\pinfty)).$\nWe now apply \\eqref{eqn:potdiff} and Lemma \\ref{lem:twofint} to write\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;\\ff(x_2+),\\fg(x_1+)) -\n\\altPhi(\\hf,\\hg;\\ff(\\minfty),\\fg(\\minfty)) \\\\\n=& \\int_{\\fg(\\minfty)}^{\\fg(x_1+)} \\hg^{-1} (u)\\text{d}u\n+ \\int_{\\ff(\\minfty)}^{\\ff(x_2+)} \\hf^{-1} (v)\\text{d}v \\\\\n&-\\ff(x_2+)\\fg(x_1+) + \\ff(\\minfty)\\fg(\\minfty)\\\\\n=& \\altPhiSI(\\smthker;f,g;x_1,x_2) \n\\end{align*}\nLetting $x_1$ and $x_2$  tend to $+\\infty$ the result follows from\nLemma \\ref{lem:transitionPhiBounds}.\n\\end{IEEEproof}\n\n\\subsubsection{Transition length.}\n\nIn this section our aim is to show that fixed point solutions arising from systems satisfying\nthe strictly positive gap condition have bounded transition regions.\nWe show that the transition of solutions from one value to another\nis confined to a region whose width can be bound from above using properties of $\\altPhi$\n\n\\begin{lemma}\\label{lem:transitionBounds}\nLet $\\ff,\\fg$ be  $(0,1)$-interpolating functions in $\\sptfns.$\nLet $\\hf \\equiv h_{[\\ff,\\gS]}$ and $\\hg \\equiv h_{[\\fg,\\fS]}.$\nLet $0 < a < b < 1$ and let $x_a,x_b$ satisfy $a=\\gS(x_a)$\nand $b=\\gS(x_b).$\nDefine\n\\begin{align*}\n\\delta &= \\inf \\{ \\altPhi(\\hf,\\hg;\\gS(x),\\ff(x)) : x\\in [x_a,x_b]\\}\n\\\\\n& = \\inf \\{ \\altPhi(\\hf,\\hg;u,\\hf(u)) : u\\in [a,b]\\}\n\\end{align*}\nthen\n\\[\n\\Bigl(\\frac{1}{2}\\delta-e_L\\Bigr)\\lfloor \\frac{x_b-x_a}{2L} \\rfloor \\le\n1\n\\]\nand\n\\begin{align*}\n\\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \n\\lfloor\n\\frac{x_b-x_a-2L}{2L}\n\\rfloor \n\\le b-a\\,.\n\\end{align*}\n\\end{lemma}\n\\begin{IEEEproof}\nFor any $x \\in [x_a,x_b]$ we have\n\\(\n\\Delta_L g(x) \\ge \\frac{1}{2}{\\delta- e_L}\\,\n\\)\nby Lemma \\ref{lem:transitionPhiBounds}.\nIn the interval $[x_a,x_b]$ we can find\n\\(\n\\lfloor\n\\frac{x_b-x_a}{2L}\n\\rfloor\n\\)\nnon-overlapping intervals of length $2L.$\nFrom this we obtain\n\\[\n\\lfloor\n\\frac{x_b-x_a}{2L}\n\\rfloor \\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \\le g(x_b-)-g(x_a+) \\le  1\\,.\n\\]\nA similar argument considering $x_a+L$ and $x_b-L$ gives\n\\begin{align*}\n\\lfloor\n&\\frac{x_b-x_a-2L}{2L}\n\\rfloor \\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \n\\\\ \\le &\ng((x_b-L)-)-g((x_a+L)+)\n\\\\ \\le &  b-a\\,.\n\\end{align*}\n\\end{IEEEproof}\n\n\\subsubsection{Discrete Spatial Integration}\n\nPerhaps somewhat surprisingly, a version of Lemma \\ref{lem:twofint} that\napplies to spatially discrete systems also holds.\nIf $\\ff,\\fg$ are spatially discrete functions and $\\tff,\\tfg$ are their\npiecewise constant extensions, then\nLemma \\ref{lem:twofint} can be applied to\nthese extensions.  If we then restrict $x_1$ and $x_2$ to points in\n$\\Delta \\integers,$ then $\\altPhiSI$ can be written as discrete sums.\n\nLet $\\discsmthker$ be related to $\\smthker$ as in \\eqref{eqn:kerdiscretetosmth}\nand let $x_1,x_2 \\in \\Delta \\integers,$ denoted\n$x_{i_1},x_{i_2}.$\nThen \n\\begin{align*}\n&\\altPhiSI(\\smthker;\\tff,\\tfg;x_{i_1},x_{i_2}) \\\\&=\n\\frac{1}{2}\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_1-j,i_2]} (2\\dv{f}_{i_2}-\\dv{f}_{i}-\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\&=\\frac{1}{2}\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_2-j,i_1]} (2\\dv{g}_{i_1}-\\dv{g}_{i}-\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\n\nLemma \\ref{lem:twofint} continues to hold and a proof using entirely discrete\nsummation can be found in appendix \\ref{app:Aa}.\n\n{\\em Discussion:} The proof of Lemma \\ref{lem:twofint} as well as the spatially discrete version found in appendix \\ref{app:Aa} are entirely algebraic in character.  Consequently, they apply to spatially coupled systems generally and not only those with a one dimensional state.  In a follow-up paper we apply the result to the arbitrary binary memoryless symmetric channel case to obtain a new proof that spatially coupled regular ensembles achieve capacity universally on such channels.\n\n\n\n\n\\subsection{Bounds on Translation Rates}\n\n\\begin{lemma}\\label{lem:shiftlowerbound}\nLet $f,g \\in \\sptfns$ be $(0,1)$-interpolating and let $\\smthker$ be an averaging kernel.\nThen \n\\[\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]}) \\le |\\ashift|\\|\\smthker\\|_\\infty\\,.\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nWe have $A(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]})=0$ and hence\n\\begin{align*}\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})\n&=\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})\n-\nA(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]})\n \\\\\n&= \\int_0^1 h_{[\\ff,\\gSa]}(u) - h_{[\\ff,\\gS]}(u) \\,\\text{d}u \\\\\n&= \\int_{\\minfty}^{\\pinfty} (\\ff(x-\\ashift) - \\ff(x)) \\gS_x(x) dx \\,.\n\\end{align*}\nSince $|\\gS_x(x)| \\le \\| \\smthker \\|_{\\infty}$ we obtain\n$|A(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})|\\le |\\ashift|\\|\\smthker\\|_\\infty.$\n\\end{IEEEproof}\n\nIn general this estimate can be weak.  In Section \\ref{sec:pathology} we gave an example\nof a system with $A(\\hf,\\hg) =0$ and irregular $\\smthker$ that can exhibit both left\nand right moving waves by changing the value $\\hf$ and $\\hg$ at a point of discontinuity.\nFurther, given $(0,1)$-interpolating $\\ff,\\fg$ and positive $\\smthker$ the system\n $(h_{[\\ff,\\fg^{\\smthker,a+\\ashift}]},h_{[\\fg,\\ff^{\\smthker,-a}]})$ (with real parameter $a$)\nhas a \ntraveling solution with shift $\\ashift$ and yet\n$A(h_{[\\ff,\\fg^{\\smthker,a+\\ashift}]},h_{[\\fg,\\ff^{\\smthker,-a}]})$\ncan be made arbitrarily close to $0$ by choosing $a$ with large enough magnitude.\n\nNow we consider upper bounds on $|\\ashift|.$\nLet\n\\[\n\\intsmthker(x) =\\int_{-\\infty}^x \\smthker(y) \\text{d}y.\n\\]\nIf $\\smthker$ has compact support then\nthe width of the support is an upper bound.  Consider a $\\smthker$ that is strictly positive\non $\\reals.$  Let $\\hf(x)=\\hg(x)=\\unitstep(x-(1-\\epsilon))$ for small positive $\\epsilon.$\nA traveling wave solution for this system is $\\ff^t(x)=\\unitstep(x-t\\ashift)$\nand $\\fg^t(x)=\\unitstep(x-t\\ashift-\\ashift\/2)$\nwhere $\\ashift$ is given by\n$\\intsmthker(-\\ashift\/2) = (1-\\epsilon).$\nThis example motivates the following bound.\n\n\\begin{lemma}\\label{lem:shiftupperbound}\nLet $\\ff,\\fg\\in\\sptfns$ be $(0,1)$-interpolating and let\n$\\hf\\equiv h_{[\\ff,\\gSa]},$ and $\\hg\\equiv h_{[\\fg,\\fS]}$\nGiven $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-)$ we have the bound\n\\[\n\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\\,.\n\\]\nand given $(u,v)$ with $v > \\hf(u+)$ and $u>\\hg(v+)$ we have the bound\n\\[\n-\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{1-v}{1-\\hf(u+)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{1-u}{1-\\hg(v+)}+\\Bigr)\\,.\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nWe will show the first bound, the second is similar.\nFor any $x_1,x_2\\in\\reals$ we have\n\\begin{align*}\n\\fS(x_1) &= \\int_{\\minfty}^{\\pinfty} \\ff(x)\\smthker(x_1-x) dx \\\\\n&\\ge \\int_{x_2}^{\\pinfty} \\ff(x)\\smthker(x_1-x) dx \\\\\n&\\ge \\ff(x_2+) \\int_{x_2}^{\\pinfty} \\smthker(x_1-x) dx \\\\\n&= \\ff(x_2+) \\int_{\\minfty}^{x_1-x_2} \\smthker(x) dx\\,.\n\\end{align*}\nThus, we obtain the inequality\n\\[\nx_1-x_2 \\le \\Omega^{-1} \\Bigl(\\frac{\\fS(x_1)}{\\ff(x_2+)}+\\Bigr)\n\\]\nChoose $x_1$ so that $\\fS(x_1)=v.$\nThen we have $\\fg(x_1+) \\ge \\fg(x_1) \\ge \\hg(v-).$\nChoose $x_2$ so that $\\gSa(x_2)=\\gS(x_2+\\ashift)=u.$\nThen we have $\\ff(x_2+) \\ge \\ff(x_2) \\ge \\hf(u-).$\n\nApplying the above inequality we obtain\n\\[\nx_1-x_2 \\le \\Omega^{-1} \\Bigl(\\frac{\\fS(x_1)}{\\ff(x_2+)}+\\Bigr)\n \\le \\Omega^{-1} \\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n\\]\nand\n\\[\nx_2+\\ashift-x_1 \\le \\Omega^{-1}\\Bigl(\\frac{\\gS(x_2+\\ashift)}{\\fg(x_1+)}+\\Bigr)\n\\le \\Omega^{-1}\\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\n\\]\nSumming, we obtain\n\\[\n\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\\,.\n\\]\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:stposbound}\nLet $(\\hf,\\hg) \\in \\exitfns^2$   satisfy the strictly positive gap condition.\nThen there exists $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-)$\nand $(u,v)$ with $v > \\hf(u+)$ and $u<\\hg(v+).$\n\\end{lemma}\n\\begin{IEEEproof}\nLet $G^+ = \\{(u,v):v > \\hf(u+) \\text{ and } u>\\hg(v+)\\}$\nand\n$G^- = \\{(u,v):v < \\hf(u-) \\text{ and } u<\\hg(v-)\\}.$\nThen\n\\[\nA(\\hf,\\hg) = \\mu(G^+) - \\mu(G^-)\n\\]\nwhere $\\mu(G)$ denotes the 2-D Lebesgue measure of $G.$\nSince the strictly positive gap condition is satisfied, there\nexists $(u^*,v^*) \\in \\intcross(\\hf,\\hg)$ and\n$\\altPhi(\\hf,\\hg;u^*,v^*) > \\max \\{0,A(\\hf,\\hg)\\}.$\nLet $R = [0,u^*]\\times[0,v^*].$ Now\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*)\n=\n \\mu(G^+ \\cap R ) - \\mu(G^- \\cap R )\n\\]\nhence $\\mu(G^+ \\cap R ) >  \\max \\{0,A(\\hf,\\hg)\\}$\nand it follows that $\\mu(G^-) > 0.$\n\\end{IEEEproof}\n\\begin{corollary}\\label{cor:regshiftbound}\nLet $\\ff,\\fg \\in \\sptfns$ be $(0,1)$-interpolating.\nIf $( h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})$  satisfies the strictly positive gap condition\nand $\\smthker$ is regular\nthen $\\ashift < 2W.$\n\\end{corollary}\n\\begin{IEEEproof}\nThis combines Lemma \\ref{lem:shiftupperbound} with Lemma \\ref{lem:stposbound}.\n\\end{IEEEproof}\n\n\\subsection{Monotonicity of $\\altPhi$ and the Gap Conditions}\n\nIn this section we collect some basic results on $\\altPhi$ and the component DE that are useful for constructing spatial wave solutions.\n\n\\begin{lemma}\\label{lem:descend}\nLet $\\hf,\\hg \\in \\exitfns.$\nIf  $(u,v) \\in [0,1]^2$ satisfies $v <\\hf(u-)$ and $u < \\hg(v-),$\nthen there exists a minimal element $(u^*,v^*) \\in \\cross(\\hf,\\hg)$ with\n$(u^*,v^*)>(u,v)$ component-wise and $\\altPhi(u^*,v^*) < \\altPhi(u,v).$\n\nSimilarly, if $(u,v)$ satisfies $v >\\hf(u+)$ and $u > \\hg(v+),$\nthen there exists a maximal element $(u^*,v^*) \\in \\cross(\\hf,\\hg),$\nwith $(u^*,v^*)<(u,v)$ (component-wise) and $\\altPhi(u^*,v^*) < \\altPhi(u,v).$\n\\end{lemma}\n\\begin{IEEEproof}\nWe show only the first case since the other case is analogous.\nAssuming  $v <\\hf(u-)$ and $u < \\hg(v-)$ we  have $\\hginv(u+) < v < \\hf(u-)$  and we\nsee that there is no crossing point $(u',v')$  with $u'=u.$ Similarly,\nthere is no crossing point with $v'=v.$ \nSince $\\cross(\\hf,\\hg)$ is closed, the set $(u,1]\\times(v,1] \\cap \\cross(\\hf,\\hg)$ is closed.\nBy Lemma \\ref{lem:crossorder} $\\cross(\\hf,\\hg)$ is ordered so there exists a minimal element $(u^*,v^*)$ in\n$(u,1]\\times(v,1] \\cap \\cross(\\hf,\\hg).$\nSet $(u^0,v^0)=(u,v)$ and consider the sequence of points\n$(u^0,v^0),(u^0,v^1),(u^1,v^1),(u^1,v^2),(u^2,v^2),\\ldots$ as determined by \\eqref{eqn:DE}.\nIt follows easily from  \\eqref{eqn:DE} that this sequence is non-decreasing.\nIf $u^t < u^*$ then $v^{t+1} \\le v^*$\nand\nif $v^t < v^*$ then $u^{t} \\le u^*.$\nThus we have either $(u^t,v^t) < (u^*,v^*)$ or there is some minimal $t$ where at least one of the coordinates\nis equal.\nIf $u^t < u^*$ and $v^t < v^*$ for all $t$ then the sequence must converge to $(u^*,v^*)$\nsince the limit is in $\\cross(\\hf,\\hg)$ by continuity and $(u^*,v^*)$ is minimal.\nIt then follows by continuity of $\\altPhi(\\hf,\\hg;)$ and Lemma \\ref{lem:monotonic} that \n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) \\le \\altPhi(\\hf,\\hg;u^0,v^1) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nAssume now that $u^t = u^*$ for some $t.$  Then $t>0$ and  Lemma \\ref{lem:monotonic}  gives\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) = \\altPhi(\\hf,\\hg;u^t,v^{t+1}) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nFinally, assume that $v^t = v^*$ for some $t.$  Then $t>0$ and  Lemma \\ref{lem:monotonic}  gives\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) = \\altPhi(\\hf,\\hg;u^t,v^{t}) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nThis completes the proof.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:crosspointmono}\nLet $(\\hf,\\hg)\\in \\exitfns^2$ and \nlet $(u,v) \\in [0,1]^2.$\nWe then have the following trichotomy:\n\\begin{itemize}\n\\item\nIf $\\hg(\\hf(u))=u$ then $(u,\\hf(u))\\in\\cross(\\hf,\\hg).$\n\\item\nIf $\\hg(\\hf(u))>u$ then \n\\(\n\\altPhi(\\hf,\\hg;u^*,v^*)\n\\le\n\\altPhi(\\hf,\\hg;u,v)\\,\n\\)\nwhere $(u^*,v^*)\\in\\cross(\\hf,\\hg)$ is coordinate-wise minimal with\n $(u^*,v^*)\\ge(u,\\hf(u)).$ \n\\item\nIf $\\hg(\\hf(u))<u$ then\n\\(\n\\altPhi(\\hf,\\hg;u^*,v^*)\n\\le\n\\altPhi(\\hf,\\hg;u,v)\\,\n\\)\nwhere $(u^*,v^*)\\in\\cross(\\hf,\\hg)$\nis coordinate-wise maximal with $(u^*,v^*)\\le(u,\\hf(u)).$ \n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nIf $u \\veq \\hg(\\hf(u))$ then $(u,\\hf(u))\\in\\cross(\\hf,\\hg)$ by definition.\nThus, the first case holds and the other two hold under this condition.\nWe assume henceforth that $u\\not\\veq\\hg(\\hf(u)).$\nAssuming $\\hg(\\hf(u))>u,$ we now have $\\hg(\\hf(u)-)>u.$\nWe also have $\\hginv(u+) < \\hf(u).$\n\nLet $(u^*,v^*)\\in\\cross(\\hf,\\hg)$ be the minimal element such that  $(u^*,v^*)\\ge(u,\\hf(u)).$\nIt follows that  $u^* >u$ which implies $v^* \\ge \\hf(u+).$\nFor all $\\epsilon>0$ sufficiently small we have\n$u+\\epsilon < \\hg((\\hf(u)-\\epsilon)-)$ and we obviously have\n$\\hf(u)-\\epsilon <\\hf((u+\\epsilon)-).$\nAssuming $\\epsilon$ sufficiently small $(u^*,v^*)$ is the minimal element in $\\cross(\\hf,\\hg)$\nwith $(u^*,v^*)> (u+\\epsilon,\\hf(u)-\\epsilon)$ and\nby Lemma \\ref{lem:descend} we have\n$\\altPhi(u^*,v^*) < \\altPhi(u+\\epsilon,\\hf(u)-\\epsilon).$\nLetting $\\epsilon$ tend to $0$ we obtain\n$\\altPhi(u^*,v^*) \\le \\altPhi(u,\\hf(u)).$\n\nThe argument for the case $\\hg(\\hf(u))<u$ is similar and we omit it.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:zocontinuity}\nIf $(\\hf,\\hg) \\in \\exitfns^2$ satisfies the strictly positive gap condition\nthen $\\hf$ and $\\hg$ are continuous at $0$ and $1.$\n\\end{lemma}\n\\begin{IEEEproof}\nAssume $(\\hf,\\hg) \\in \\exitfns^2$ satisfies the strictly positive gap condition\nand assume $\\hf(0+)>0.$\nBy \\eqref{eqn:altPhiderivatives} we see that $\\altPhi(\\hf,\\hg;0,v)=0$ for all\n$v \\in [0,\\hf(0+)).$\nIt follows that $\\hginv(0+)=0$ or we obtain a contradiction with the strictly positive gap condition.\nHence $\\hg(\\hf(0+))>0$ and\nthen for $u\\in(0,\\hg(\\hf(0+))$ we have $\\hg(\\hf(u))>u.$\nBy Lemma \\ref{lem:crosspointmono} the minimal crossing point $(u^*,v^*) \\ge (0,\\hf(u+))$\nsatisfies $\\altPhi(\\hf,\\hg;u^*,v*)<0.$ By the strictly positive gap condition\n$(u^*,v^*) < (1,1)$ and we obtain a contradiction.\nTherefore, we must have $\\hf(0+) = 0.$\n\nAll other conditions, $\\hg(0+) = 0, \\hf(1-) = 1,$ and $\\hg(1-) = 1$ can be shown similarly.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:Sstructure}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition.\nIf $A(\\hf,\\hg) \\ge 0$ then $\\altPhi(\\hf,\\hg;u,v) >0$ \nfor $(u,v)\\in [0,1]^2\\backslash\\{(0,0),(1,1)\\}.$\nIf $A(\\hf,\\hg) > 0$ then there exists a minimal point $(u^*,v^*) \\in \\intcross(\\hf,\\hg)$\nand the set\n\\[\nS(\\hf,\\hg) = \\{(u,v): \\altPhi(\\hf,\\hg;u,v) < A(\\hf,\\hg)\\}\n\\]\nis simply connected \nand  $\\closure{S(\\hf,\\hg)} \\subset[0,u^*)\\times[0,v^*).$\nMoreover,\n\\[\n\\{(u,v): \\altPhi(\\hf,\\hg;u,v) \\le A(\\hf,\\hg)\\}\n= \\closure{S(\\hf,\\hg)}\\cup \\{(1,1)\\}.\n\\]\n\\end{lemma}\n\\begin{IEEEproof} \nAssume $(\\hf,\\hg) \\in \\exitfns^2$ satisfies the strictly positive gap condition and that\n$A(\\hf,\\hg) \\ge 0.$\nIt follows from Lemma \\ref{lem:miniscross} that $\\altPhi(\\hf,\\hg;u,v)$\nachieves its minimum on $\\cross(\\hf,\\hg),$\nhence,  the strictly positive gap condition implies $\\altPhi(\\hf,\\hg;) \\ge 0.$\nLemma \\ref{lem:miniscross} now further implies that if there exists $(u,v)$ with\n$\\altPhi(\\hf,\\hg;u,v) = 0$ then $(u,v) \\in \\cross(\\hf,\\hg).$\nThus, we have $\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\not\\in \\{(0,0),(1,1)\\}.$\n\nAssume now that $A(\\hf,\\hg) > 0.$ \nLet $(u^*,v^*)$ be the infimum of $\\intcross(\\hf,\\hg).$\nThen  $(u^*,v^*)\\in \\cross(\\hf,\\hg)$ and\n$(u^*,v^*) \\neq (0,0)$ \nand therefore $(u^*,v^*) \\in \\intcross(\\hf,\\hg).$\nBy Lemma \\ref{lem:zocontinuity} we also have $u^*>0$ and $v^*>0.$\n\nLet $(u,v) \\in S(\\hf,\\hg)$ and assume that $u \\neq 0$ and $v\\neq 0.$\nBy Lemma  \\ref{lem:monotonic} and Lemma \\ref{lem:crosspointmono}\nwe see that we must have $\\hg(\\hf(u))<u.$ \nSetting $(u^0,v^1) = (u,\\hf(u))$\nthen the sequence of points\n$(u^0,v^1),(u^1,v^1),(u^1,v^2),(u^2,v^2),\\ldots$ as determined by \\eqref{eqn:DE}\nare coordinate-wise non-increasing and must converge to $(0,0).$\nFurthermore, by Lemma \\ref{lem:monotonic} (coordinate-wise convexity) the line segments joining successive points are all in $S(\\hf,\\hg).$\nIn addition, the line segment joining $(u,v)$ to $(u,\\hf(u))$ is also in $S(\\hf,\\hg).$\nThus, $S(\\hf,\\hg)$ is simply connected.\n\nAssume there exists $(u,v)\\in S(\\hf,\\hg)$ with $u > u^*,$ and, consequently, $v^* \\le \\hf(u).$ \nThen, by Lemma \\ref{lem:monotonic}  we have\n$(u,\\hf(u)) \\in S(\\hf,\\hg).$   Lemma \\ref{lem:crosspointmono}\nnow implies the existence of $(u',v') \\in \\cross(\\hf,\\hg)$ with $(u',v') \\ge (u^*,v^*)$ and\n$\\altPhi(\\hf,\\hg;u',v') < A(\\hf,\\hg),$ which contradicts the strictly positive gap condition.\nHence $S(\\hf,\\hg) \\subset [0,u^*) \\times [0,v^*).$\nFurthermore, Lemma \\ref{lem:monotonic} implies that $\\altPhi(\\hf,\\hg;u^*,v) > A(\\hf,\\hg)$ for all $v\\in [0,1]$\nand  $\\altPhi(\\hf,\\hg;u,v^*) > A(\\hf,\\hg)$ for all $u\\in [0,1]$\nso  $\\closure{S(\\hf,\\hg)} \\subset [0,u^*) \\times [0,v^*).$\n\nAssume there exists $(u,v)\\not\\in \\closure{S(\\hf,\\hg)} \\cup \\{ (1,1)\\}$ with $\\altPhi(\\hf,\\hg;u,v) = A(\\hf,\\hg).$\nThen $(u,v)$ is a local minimum of $\\altPhi(\\hf,\\hg;u,v)$\nwhich, by Lemma \\ref{lem:miniscross}, implies $(u,v) \\in \\cross(\\hf,\\hg),$\ncontradicting the strictly positive gap condition.\n\\end{IEEEproof}\n\n\\subsection{Inverse Formulation.\\label{sect:inverse}}\n\nIt is instructive in to the analysis to view the system in terms of inverse functions.\nLet $\\fg(x) = h((\\ff\\otimes \\smthker) (x))$ with $f \\in \\sptfns.$\nThen, for almost all $u\\in [0,1]$ we have\n\\(\n\\hg^{-1}(u) = \\int_0^1 \\intsmthker (\\fg^{-1}(u)-\\ff^{-1}(v)) \\text{d}v\\,.\n\\)\nTo show this we first integrate by parts to write\n\\(\n(\\ff\\otimes \\smthker) (x) = \\int_{-\\infty}^\\infty \\intsmthker(x-y) d\\ff(y)\n\\)\nand then make the substitutions $v=f(y)$ and $u=g(x).$\nIt follows that, up to equivalence, the recursion \\eqref{eqn:gfrecursion} may be expressed as\n\\begin{equation}\\label{eqn:gfrecursionInv}\n\\begin{split}\n\\hginv(u)  & =  \\int_0^1 \\intsmthker ((\\fg^t)^{-1}(u)-(\\ff^t)^{-1}(v)) \\text{d}v, \\\\\n\\hfinv(v) & = \\int_0^1 \\intsmthker ((\\ff^{t+1})^{-1}(v)-(\\fg^t)^{-1}(u)) \\text{d}u\\,.\n\\end{split}\n\\end{equation}\nSince $\\smthker$ is even we have $\\Omega(x) = 1-\\Omega(-x),$ so we immediately observe\nthat if $(\\ff,\\fg)\\in\\sptfns^2$ is a $(0,1)$-interpolating  fixed point of the above system then\n\\begin{align*}\n1 & =\n\\int_0^1 \\hginv(u) \\text{d}u + \\int_0^1 \\hfinv(v)\\text{d}v \\\\\n& =\n\\int_0^1 \\hg(u) \\text{d}u + \\int_0^1 \\hf(v)\\text{d}v \\,.\n\\end{align*}\nThis puts a very strong requirement on $(\\hf,\\hg)$ to admit a $(0,1)$-interpolating spatial fixed point.\n\nAssume that $\\ff$ and $\\fg,$ both in $\\sptfns,$ form a $(0,1)$-interpolating spatial fixed point.\nConsider perturbing the inverse functions by $\\delta \\ffinv$ and $\\delta \\fginv$ respectively.\nWe could then perturb $\\hfinv$ and $\\hginv,$ by $\\delta\\hfinv$ and $\\delta\\hginv$ respectively so that\nthe perturbed system would remain a fixed point.  To first order we will have from \\eqref{eqn:gfrecursionInv},\n\\begin{equation}\\label{eqn:delgfrecursionInv}\n\\begin{split}\n\\delta\\hginv(u)  & =  \\int_0^1 \\smthker (\\fginv(u)-\\ffinv(v)) (\\delta\\fginv(u)-\\delta\\ffinv(v)) \\text{d}v,  \\\\\n\\delta\\hfinv(v) & = \\int_0^1 \\smthker (\\ffinv(v)-\\fginv(u)) (\\delta\\ffinv(v)-\\delta\\fginv(u))\\text{d}u\\,.\n\\end{split}\n\\end{equation}\n\nThis formulation is at the heart of the analysis in the next section.\n\n\n\\subsection{Existence: The Piecewise Constant Case}\\label{sec:PCcase}\nIn this section we focus on the case where $\\hf$ and $\\hg$ are piecewise constant.\nIn this case the spatially coupled system is finite dimensional.\nWe further assume that $\\smthker$ is strictly positive on $\\reals.$\nThis ensures in a simple way that no degeneracy occurs when determining EXIT functions\nfrom spatial functions since $\\frac{d}{dx} \\gS(x) > 0$ for any non-constant $g \\in \\sptfns.$\n\nWe will write piecewise constant functions $\\hf,\\hg \\in \\exitfns$ as\n\\begin{align*}\n\\hf(u) &= \\sum_{j=1}^{\\Kf}  \\delhf_j \\,\\unitstep(u - \\uf_j) \\\\\n\\hg(u) &= \\sum_{i=1}^{\\Kg}  \\delhg_i \\,\\unitstep(u - \\ug_i)\n\\end{align*}\nwhere $\\unitstep$ is the unit step (Heaviside)\n function\\footnote{The regularity assumptions on $\\smthker$ ensure that the precise value of $\\hf$ and $\\hg$\nat points of discontinuity has no impact on the analysis.}\nand where we assume $\\delhf_j,\\delhg_i > 0,$ and $\\sum_{j=1}^{\\Kf}  \\delhf_j = 1$\nand $\\sum_{i=1}^{\\Kg}  \\delhg_i = 1.$\n\nGenerally we will have\n$0 < \\uf_1 \\le \\uf_2 \\le \\cdots \\le \\uf_{\\Kf} < 1$\nand\n$0 < \\ug_1 \\le \\ug_2 \\le \\cdots \\le \\ug_{\\Kg} < 1$\nbut the ordering is actually not critical to the definition.\nWe generally view the vectors $\\delhf$ and $\\delhg$ as fixed and \nto indicate the dependence on $\\uf = (\\uf_1,\\ldots,\\uf_{\\Kf})$ and $\\ug$ we will\nwrite $\\hf(u;\\uf)$ and $\\hg(u;\\ug).$\n\nPiecewise constant $\\hf$ and $\\hg$ also have piecewise constant inverses.\nGiven $\\hf$ as above we have\n\\[\n\\hfinv (v) = \\sum_{j=1}^{\\Kf} (\\uf_j-\\uf_{j-1})\\unitstep(v-\\sum_{k=1}^{j} \\delhf_j)\n\\]\nwhere  we set $\\uf_0=0.$\n\n\nIf $\\fg \\in \\sptfns$ is continuous, strictly increasing, and $(0,1)$-interpolating\nand $\\hf$ is piecewise constant as above then $\\ff \\in\\sptfns$ defined by\n$\\ff(x) = \\hf(\\fg(x))$ is also piecewise constant and\ncan be written as\n\\[\n\\ff(x) = \\sum_{i=1}^{\\Kf}  \\delhf_i \\,\\unitstep(x - \\zf_i)\n\\]\nwith\n$-\\infty < \\zf_1 \\le \\zf_2 \\le \\cdots \\le \\zf_{\\Kf} < \\infty$\nsatisfying  $\\uf_i  = \\fg(\\zf_i).$ \nAs before, we have\n\\[\n\\ffinv(v) =  \\sum_{j=1}^{\\Kf} (\\zf_j-\\zf_{j-1})\\unitstep(v-\\sum_{k=1}^{j} \\delhf_j) \\,.\n\\]\nwhere we set $\\zf_0 =0$\n\n\nThe purpose of this section is to prove a special case of Theorem\n\\ref{thm:mainexist} under piecewise constant assumptions on the EXIT functions and\nregularity conditions on\n$\\smthker.$  In this special case we obtain in addition uniqueness and continuous dependence of the solution. For convenience we state the main result here.\n\\begin{theorem}\\label{thm:PCexist}\nAssume $\\smthker$ is a strictly positive and $C^1$ averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of piecewise constant functions in $\\exitfns$\nsatisfying the strictly positive gap condition.\nThen there exists unique (up to translations) $(0,1)$-interpolating functions\n$\\tmplF,\\tmplG \\in\\sptfns$ and $\\ashift \\in \\reals$ satisfying $\\sgn (\\ashift) = \\sgn (A(\\hf,\\hg)),$ such that\nsetting\n$\\ff^t(x)  = \\tmplF(x-\\ashift t)$ and\n$\\fg^t(x)  = \\tmplG(x-\\ashift t)$\nsolves \\eqref{eqn:gfrecursion}.\nFurther,\n$\\tmplF^{-1}(v)-\\tmplG^{-1}(u)$ depends continuously on the vectors $\\uf,\\ug.$\n\\end{theorem}\n\nThe remainder of this section is dedicated to the proof of this result.\nOur proof constructs the solutions $\\tmplF$ and $\\tmplG$ by a method of continuation.\nIn the case where $\\hf$ and $\\hg$ are unit step functions it is easy to find\nthe solution: $\\tmplF$ and $\\tmplG$ are also unit step functions.\nStarting there we deform the solution to\narrive at a solution for a given $\\hf$ and $\\hg.$\nWe do this in two stages where in the first stage $\\ashift = 0$ and in the second\nis $\\ashift$ varied while $\\hg$ is held fixed.\nThe deformation is obtained as a solution to a differential equation.\nTo set up the equation we require a detailed description of the dependence\nof $\\uf$ and $\\ug$ on $\\zf,\\zg$ and $\\ashift.$\n\n\nLet us first consider the case $\\ashift=0.$\nThus, let $\\ff(x;\\zf)$ and $\\fg(x;\\zg)$ be a piecewise constant functions\nparameterized by their jump point locations\n$\\zf$ and $\\zg$ as\n\\begin{align}\n\\begin{split}\\label{eqn:fgtdef}\ng(x;\\zg) & = \\sum_{i=1}^{\\Kg} \\delhg_i \\,\\unitstep(x - \\zg_{i}) \\\\\nf(x;\\zf) & = \\sum_{j=1}^{\\Kf} \\delhf_j \\,\\unitstep(x - \\zf_{j})\n\\end{split}\n\\end{align}\nThen, from \\eqref{eqn:gfrecursionInv} we have\n\\begin{align}\n\\begin{split}\\label{eqn:discreteInv}\n\\ug_i &= \\fS(\\zg_i) = \\sum_{j=1}^{\\Kf} \\delhf_j \\Omega (\\zg_i-\\zf_j) \\\\\n\\uf_j &=  \\gS(\\zf_j) =\\sum_{i=1}^{\\Kg} \\delhg_i \\Omega (\\zf_j-\\zg_i)\n\\end{split}\n\\end{align}\nNow, suppose we introduce smooth dependence on a parameter $t,$\ni.e., we are given smooth  functions $\\zf(t)$ and $\\zg(t)$\nand then determine $\\uf(t)$ and $\\ug(t)$ from the above.\nBy differentiating we obtain\n\\begin{align}\\label{eqn:zeroAdiffeq}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n&=\nH(\\zf(t),\\zg(t))\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n\\end{align}\nwhere $H(\\zf(t),\\zg(t))$ is a $(\\Kg+\\Kf)\\times(\\Kg+\\Kf)$ matrix\n\\begin{align}\n\\label{eqn:matrixdif}\nH(\\zf,\\zg)\n& =\n\\begin{bmatrix}\n \\Df & -\\Bf \\\\\n-\\Bg &  \\Dg\n\\end{bmatrix},\n\\end{align}\nwhich we rewrite as $H=D(I-M),$\nand where\n\\[\nD =\n\\begin{bmatrix}\n \\Df & 0     \\\\\n  0  & \\Dg\n\\end{bmatrix}\n\\text{  and  }\nM =\n\\begin{bmatrix}\n 0 & (\\Df)^{-1}\\Bf \\\\\n(\\Dg)^{-1}\\Bg & 0\n\\end{bmatrix}\n\\]\nand where \n\\begin{itemize}\n\\item $\\Dg$ is the $\\Kf \\times \\Kf$ diagonal matrix with\n\\[\n\\Dg_{i,i} = \\gS_x(\\zf_i) = \\sum_{j=1}^{\\Kg} \\smthker(\\zg_j-\\zf_i) \\delhg_j,\\]\n\\item $\\Df$ is the $\\Kg \\times \\Kg$ diagonal matrix with\n\\[\n\\Df_{i,i} = \\fS_x(\\zg_i) = \\sum_{j=1}^{\\Kf} \\smthker(\\zf_j-\\zg_i) \\delhf_j,\n\\]\n\\item $\\Bga$ is the $\\Kf \\times \\Kg$ matrix with \n\\[\n\\Bg_{i,j} = -\\frac{\\partial \\gS(\\zf_i;\\zg)}{\\partial \\zg_j} = \\smthker(\\zg_j-\\zf_i) \\delhg_j,\n\\]\n\\item $\\Bf$ is the $\\Kg \\times \\Kf$ matrix with \n\\[\n\\Bf_{i,j} = \n-\\frac{\\partial \\fS(\\zg_i;\\zf)}{\\partial \\zf_j} \n \\smthker(\\zf_j-\\zg_i) \\delhf_j.\n\\]\n\\end{itemize}\n\nSince $\\Dg_{i,i} = \\sum_{j=1}^{\\Kg} \\Bg_{i,j}$ and\n$\\Df_{i,i} = \\sum_{j=1}^{\\Kf} \\Bf_{i,j}$\nwe observe that $M$ is a stochastic matrix: $\\sum_{j=1}^{\\Kf+\\Kg} M_{i,j} = 1.$\n\nOur strategy for constructing fixed points for a given $\\hf,\\hg$ involves solving\n\\eqref{eqn:zeroAdiffeq} for $\\zg(t),\\zf(t)$ given $\\uf(t),\\ug(t).$\nThe main difficulty we face is that $H(\\zf,\\zg)$ is not invertible.\nIn particular, $(I-M)\\vec{1} = 0.$\nThis is a consequence of the fact that\ntranslating $\\zf$ and $\\zg$ together does not alter $\\ug$ and $\\uf$ as defined\nby \\eqref{eqn:discreteInv}.\nThe corresponding left null eigenvector of $H(\\zf,\\zg)$ \narises from the fixed point condition $\\int_0^1 \\hg(x) dx  + \\int_0^1\\hf(x) dx = 1$ which here reduces to\n\\[\n1 = \\sum_{j=1}^{\\Kf} (1-\\uf_j) \\delhf_j\n+ \\sum_{i=1}^{\\Kg} (1-\\uf_i) \\delhg_i,\n\\]\nhence\n\\[\n\\sum_{j=1}^{\\Kf} \\delhf_j \\frac{d\\uf_j}{dt}\n+ \\sum_{i=1}^{\\Kg} \\delhg_i \\frac{d\\ug_i}{dt}= 0\n\\]\nas can be verified directly.\n\nLet us consider the matrix\n\\[\nH(\\zf,\\zg) + \\vec{1}\\vec{\\delta}^T\n\\]\nwhere $\\vec{1}$ is the column vector of all $1$s and $\\vec{\\delta}$ is the column\nvector obtained by stacking $\\delhg$ on $\\delhf.$ \nThis matrix is invertible, i.e., its determinant is non-zero. \nTo see this note that $M$ is a positive stochastic matrix \n$M\\vec{1} = \\vec{1}$ and by the Perron-Frobenious theorem all other eigenvalues of $M$\nhave magnitude strictly less than $1.$\nIt follows that $\\vec{1}$ is the unique right null vector of $H(\\zf,\\zg)$ (up to scaling)\nand that $\\vec{\\delta}$ is the corresponding left null vector.\nThe left subspace orthogonal to $\\vec{1}$ is invariant under $H(\\zf,\\zg).$\nIt now follows that $H(\\zf,\\zg) + \\vec{1}\\vec{\\delta}^T$ has no left null vector and\nis therefore invertible.\n\nNow, consider the differential equation\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n&=\n(H(\\zf(t),\\zg(t))+ \\vec{1}\\vec{\\delta}^T)^{-1}\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n\\label{eqn:diffEQR}\n\\end{align}\nIf \n\\(\n\\frac{d}{dt}\n\\vec{\\delta}^T \n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}=0\n\\)\nthen we obtain\n\\(\n\\frac{d}{dt}\n\\vec{\\delta}^T \n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}=0\n\\)\nand we see that \\eqref{eqn:zeroAdiffeq} is satisfied.\n\n\\begin{lemma} \\label{lem:PCexitcont}\nLet $\\smthker$ be a strictly positive smoothing kernel.\nLet $\\uf(t)$ and $\\ug(t)$ be $C^2$ ordered vector valued functions on $[0,1]$\nsuch that $(\\hf(;\\uf(t)),\\hg(;\\ug(t))$ are satisfy the strictly positive\ngap condition uniformly\nin the sense that $\\min_{t\\in [0,1]} \\altPhi(\\hf(;\\uf(t)),\\hg(;\\ug(t);u,v) > 0$ \nfor all $(u,v) \\in  [0,1]^2\\backslash \\{(0,0),(1,1)\\}$\nand $A(\\hf(;\\uf(t)),\\hg(;\\ug(t))=0$ for all $t\\in [0,1].$\nAssume further that $\\zf(0)$ and $\\zg(0)$\nare given so that $\\ff(;\\zf(0))$ and $\\fg(;\\zg(0))$ as defined by\n\\eqref{eqn:fgtdef}\nsatisfies  $\\fg(x;\\zg(0)) = \\hg(\\fS(x;\\zf(0));\\ug(0))$\nand $\\ff(x;\\zg(0)) = \\hf(\\gS(x;\\zg(0));\\uf(0))$ for all $x\\in\\reals,$\ni.e., the determined functions form a $(0,1)$-interpolating spatial fixed point for the $t=0$ system.\nThen there exist bounded $C^1$ ordered vector valued functions\n$\\zf(t)$ and $\\zg(t)$ on $[0,1],$\nwith $\\zf(0)$ and $\\zg(0)$ as specified,\nsuch that  $\\fg(x;\\zg(t)) = \\hg(\\fS(x;\\zf(t));\\ug(t))$\nand $\\ff(x;\\zg(t)) = \\hf(\\gS(x;\\zg(t));\\uf(t))$\nfor all $x\\in \\reals$ and $t\\in [0,1].$\n\\end{lemma}\n\\begin{IEEEproof}\nConsider the differential equation \\eqref{eqn:diffEQR} with initial condition\ngiven by $\\zg(0)$ and $\\zf(0).$  By translating (adding a constant to both) we can assume\n$\\sum_{i=1}^{\\Kg} \\delhg_i \\zg_i(0)  +\\sum_{j=1}^{\\Kf} \\delhf_j \\zf_j(0)  = 0.$\nBy standard results on differential equations, the equation\nhas a unique $C^1$ solution $(\\zg(t),\\zf(t))$ on $[0,T)$ for some maximal $T>0.$\nSince $\\frac{d}{dt}(\\sum_{i=1}^{\\Kg} \\delhg_i \\ug_i(t) +\\sum_{j=1}^{\\Kf} \\delhf_j \\uf_j(t) ) = 0$\nwe have $\\frac{d}{dt}(\\sum_{i=1}^{\\Kg} \\delhg_i \\zg_j(t) +\\sum_{j=1}^{\\Kf} \\delhf_j  \\zf_j(t)) = 0$\nand therefore $\\sum_{i=1}^{\\Kg} \\delhg_i \\zg(t) +\\sum_{j=1}^{\\Kf} \\delhf_j  \\zf_j(t) = 0.$\nFurther, the solution satisfies \\eqref{eqn:zeroAdiffeq} and it follows that the determined\nfunctions $f(;\\zf(t))$ and $g(;\\zg(t))$ are corresponding $(0,1)$-interpolating spatial fixed points\nfor $\\hf(;\\uf(t)),\\hg(;\\ug(t)).$\nLemma \\ref{lem:transitionPhiBounds} implies that $\\zf(t)$ and $\\zg(t)$ are bounded\nand we can conclude that the solution exists for all $t\\in [0,1].$\n\\end{IEEEproof}\n\nNow we consider adding a shift to the model.\nWe generalize \\eqref{eqn:discreteInv} as follows\n\\begin{align}\n\\begin{split}\\label{eqn:discreteInvShift}\n\\ug_i &= \\fS(\\zg_i) = \\sum_{j=1}^{\\Kf} \\delhf_j \\Omega (\\zg_i-\\zf_j) \\\\\n\\uf_j &=  \\gS(\\zf_j+\\ashift) =\\sum_{j=1}^{\\Kf} \\delhf_i \\Omega (\\zf_j+\\ashift-\\zg_i)\n\\end{split}\n\\end{align}\nNow, let $\\zg(t),\\zf(t),\\ashift(t)$ be smooth functions of time, then\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n&=\nH(\\zf(t),\\zg(t))\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+\n\\begin{bmatrix}\n0 \\\\\n\\Dga \\vec{1}\n\\end{bmatrix}\n\\frac{d}{dt}\\ashift(t)\n\\label{eqn:diffEQ}\n\\end{align}\nwhere $H(\\zf(t),\\zg(t))$ is a $(\\Kg+\\Kf)\\times(\\Kg+\\Kf)$ matrix\n\\begin{align}\n\\label{eqn:matrixdif}\nH(\\zf,\\zg)\n& =\n\\begin{bmatrix}\n \\Df & -\\Bf \\\\\n-\\Bga &  \\Dga\n\\end{bmatrix}\n\\\\\n& =\nD\n\\begin{bmatrix}\nI-M\n\\end{bmatrix}\n\\end{align}\nwhere\n\\[\nD =\n\\begin{bmatrix}\n \\Df & 0     \\\\\n  0  & \\Dga\n\\end{bmatrix}\n\\text{  and  }\nM =\n\\begin{bmatrix}\n 0 & (\\Df)^{-1}\\Bf \\\\\n(\\Dga)^{-1}\\Bga & 0\n\\end{bmatrix}\n\\]\n\n\n\nwhere $\\Df$ and $\\Bf$  are as before and\n\\begin{itemize}\n\\item $\\Dga$ is the $\\Kf \\times \\Kf$ diagonal matrix with\n\\[\n\\Dga_{i,i} = \\gS_x(\\zf_i+\\ashift) = \\sum_{j=1}^{\\Kg} \\smthker(\\zg_j-(\\zf_i+\\ashift)) \\delhg_j,\n\\]\n\\item $\\Bga$ is the $\\Kf \\times \\Kg$ matrix with \n\\[\n\\Bga_{i,j} = \n\\smthker(\\zg_j-(\\zf_i+\\ashift)) \\delhg_j,\n\\]\n\\end{itemize}\n\nSince $\\Dga_{i,i} = \\sum_{j=1}^{\\Kg} \\Bga_{i,j}$ and\n$\\Df_{i,i} = \\sum_{j=1}^{\\Kf} \\Bf_{i,j}$\nwe observe that $M$ is a stochastic matrix: $\\sum_{j=1}^{\\Kf+\\Kg} M_{i,j} = 1.$\n\n\nLet $P$ be the projection matrix which is the $(\\Kf+\\Kg) \\times (\\Kf+\\Kg)$\nidentity matrix except that $P_{\\Kf+\\Kg , \\Kf+\\Kg}=0.$\nIt follows that $I-PMP$ is invertible and $PMP$ has spectral radius less than one.\nIndeed, let\n$\\tilde{B_1}$ denote the matrix obtained from $(\\Df)^{-1}\\Bf$ be removing the rightmost column and\nlet $\\tilde{B_2}$ denote the matrix obtained from $(\\Dga)^{-1}\\Bga$ be removing the bottom row.\nLet $\\tilde{M}$ denote the upper left  $\\Kf+\\Kg -1 \\times \\Kf+\\Kg-1$ submatrix of $M.$\nThen\n\\[\n\\tilde{M}^{2} =\n\\begin{bmatrix}\n(\\tilde{B_1}\\tilde{B_2})^{2k} & 0 \\\\\n0  & (\\tilde{B_2}\\tilde{B_1})^{2}\n\\end{bmatrix}\\,.\n\\]\nLet $\\xi < 1$ denote the maximum row sum from $\\tilde{B_1}.$\nBy the Perron-Frobenious theorem $\\tilde{B_2}\\tilde{B_1}$ has a maximal positive eigenvalue $\\lambda$\nwith positive left eigenvector $x.$ Then $x^T \\tilde{B_2}\\tilde{B_1} \\vec{1} = \\lambda x^T  \\vec{1},$\nbut $\\tilde{B_2}\\tilde{B_1} \\vec{1} \\le \\xi \\vec{1}$ (component-wise) so $\\lambda \\le \\xi.$\nWe easily conclude that $\\| \\tilde{M}^2 \\|_2 \\le \\xi.$\nHence $(I-PMP)^{-1}$ exists and is strictly positive.\n\nGiven $\\zf(0),\\zg(0)$  we define $\\zg(t),\\zf(t)$  as the solution to\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n&=\n-(I-PM(\\zf(t),\\zg(t))P)^{-1}\n\\;\nP\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\\,.\n\\label{eqn:diffEQRshift}\n\\end{align}\nNote that \n\\[\nP\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix} =\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\\,.\n\\]\nNow we substitute the solution into \\eqref{eqn:diffEQ}\nand we obtain\n\\begin{align*}\nP\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix} \n&=\nD\\Biggl[\nP(I-M)\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+P\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\n\\Biggr] \\\\\n&=\nD\\Biggl[\nP(I-PMP)\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+P\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\n\\Biggr] \\\\\n& = 0\\,.\n\\end{align*}\n\n\\begin{lemma}\\label{lem:PCshiftexist}\nLet $\\zf(0),\\zg(0)$ be given, thereby defining\n$f(;\\zf(0))$ and $g(;\\zg(0))$ in $\\sptfns$\nand set\n\\[\n\\hg = \\sum_{i=1}^{\\Kg} \\delhg_j \\,\\unitstep( x-  \\fS(\\zg_{i}(0);\\zf(0)) )\\,.\n\\]\nLet $\\ashift(0) \\ge 0$ also be given and\nlet $\\hf(;r)$ be parametrized piecewise constant functions defined by\n$\\hf(x;r) = \\hf(x;\\uf(r))=\\sum_{i=1}^{\\Kf} \\delhf_i \\,\\unitstep( x - \\uf_i(r) )$\nwhere\n\\[\n\\uf_i(r) =\n\\begin{cases}\n\\gS(\\zf_i(0)+\\ashift(0);\\zg(0)) & i < \\Kf \\\\\n\\gS(\\zf_{\\Kf}(0)+\\ashift(0);\\zg(0)) +r & i = \\Kf\\,.\n\\end{cases}\n\\]\nNote that $f(;\\zf(0))$ and $g(;\\zg(0))$ form a traveling wave solution\nfor $\\hf(;0),\\hg$ with shift $\\ashift(0).$\n\nLet $r'>0$ be such that $\\uf_{\\Kf}( r') < 1$\nand assume that $(\\hf(;r'),\\hg)$ satisfies\nthe strictly positive gap condition\nwith $A(\\hf(;r'),\\hg) > 0.$\nDefine $\\ashift(t) = \\ashift(0) + t.$\n\nThen there exists $C^1$ functions $\\zf(t)$ and $\\zg(t)$\nfor $t\\in [0,T],$ where $T<\\infty,$\nwith $\\zf(0)$ and $\\zg(0)$ as given,  such that \n$f(;\\zf(t)),g(;\\zg(t)) \\in \\sptfns,$\nform a spatial wave solution with shift $\\ashift(t)$  for\n$(\\hf(;r(t)),\\hg)$\nand where $r(t)$ is an increasing $C^1$ function\nwith $r(0)=0,$ and $r(T)=r'.$\n\\end{lemma}\n\\begin{IEEEproof}\nConsider the differential equation\n\\eqref{eqn:diffEQRshift}.\nBy standard results on differential equations\na unique solution exists on $[0,T')$ for some maximal $T'>0.$\nNote that $\\zf(t)$ and $\\zg(t)$ are component-wise decreasing in $t,$\nexcept for $\\zf_{\\Kf}(t)$ which is constant.\n\nIt follows that \n\\begin{align*}\n& \\frac{d}{dt} \\gS(\\zf_{\\Kf}+\\ashift(t);\\zg(t)) \\\\\n& =\n\\gS_x(\\zf_{\\Kf}+\\ashift(t);\\zg(t))\\cdot \\\\\n&\\qquad\\Bigl(1 - \\sum_{j=1}^{\\Kg}\n\\smthker(\\zf_{\\Kf}+\\ashift(t)-\\zg_j(t))\\, \\delhg_j\n\\frac{d}{dt} \\zg_j(t)\n\\Bigr)\n\\end{align*}\nwhich is strictly positive for all $t\\in [0,T')$ since $\\frac{d}{dt} \\zg_j(t) \\le 0.$\nIt follows that $\\ff(;\\zf(t)),\\fg(;\\zg(t))$ form a spatial wave solution with\nshift $\\ashift(t)$ for $(\\hf(;r(t)),\\hg)$ for all $t\\in [0,T')$\nand that $r(t)=\\gS(\\zf_{\\Kf} +  \\ashift(t);\\zg(t)) - \\gS(\\zf_{\\Kf} +  \\ashift(0);\\zg(0))$\nis monotonically increasing on $[0,T').$\n\nWe now show that $r(t) \\le r'$ implies that $Z(t)$ is bounded.\nAs a first step we show that $\\ashift(t)$ is bounded.\nIf $r \\le r'$ then there exists $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-).$\nIn particular if we take $u>r'$  and $v > \\fS(\\zg_{\\Zg}(0);\\zf(0)) = \\fS(\\zg_{\\Zg}(t);\\zf(t))$\nthen we obtain\na finite upper bound on $\\ashift(t)$ from Lemma \\ref{lem:shiftupperbound}.\n\nAssume there exists $t_i \\rightarrow T'' \\le T'$\nfrom below such that $f(;t_i) \\rightarrow f,$\n$g(;t_i) \\rightarrow g$ and $r(t_i) \\rightarrow r \\le r'.$\nFrom Theorem \\ref{thm:mainlimit}\nit follows that $(f(\\minfty),g(\\minfty)) \\in \\cross(\\hf(;r),\\hg)$\nand $\\altPhi(\\hf(;r),\\hg;g(\\minfty),f(\\minfty)) = 0.$\nBy the assumptions on\n$\\hg$ we see that the crossing point cannot be interior.\nThus, $(f(\\minfty),g(\\minfty)) = (0,0)$\nand we conclude that $\\zg(t)$ and $\\zf(t)$ are bounded.\nHence $T'' < T'.$\n\nFinally, an upper bound on $\\|\\zg(t)\\|$ yields a positive lower bound on\n$\\frac{d}{dt} r(t)$ so we conclude that\nthere exists $T < T'$ such that $r(T)=r'.$\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:PCexist}]\nLet us first consider the case $A(\\hf,\\hg)= 0.$\nLet the target EXIT functions be $\\hf=\\hf(;\\uf)$ and $\\hg=\\hg(;\\ug).$\nLet $B_{\\hf} = \\int_0^1 \\hfinv(x) dx = \\sum_j \\delhf_{j} \\uf_{j},$\nand $B_{\\hg} = \\int_0^1 \\hginv(x) dx =  \\sum_i \\delhg_{i} \\ug_{i}= 1-B_{\\hf}.$\nFor $t \\in [0,1]$ define the vector valued functions\n\\begin{align}\n\\begin{split}\n\\uf (t) & = (1-t) B_{\\hf}\\vec{1} + t \\uf \\\\\n\\ug (t) & = (1-t) B_{\\hg}\\vec{1} + t \\ug\\,,\\label{eqn:pcscale}\n\\end{split}\n\\end{align}\nwhere $\\vec{1}$ denotes a vector of all $1$s (of appropriate length).\nNote that we have $\\uf(1)=\\uf$ and $\\ug(1)=\\ug.$\nNote that $\\hf(;\\uf(t))$ and $\\hg(;\\ug(t))$ are in $\\exitfns$ for all $t.$\nNote that $\\int_0^1 \\hfinv(v;\\uf(t)) dv = B_{\\hf}$\nand $\\int_0^1 \\hginv(u;\\ug(t)) du = B_{\\hg}$\nso $A (\\hf(;\\uf(t)),\\hg(;\\ug(t))) = 0$ for all $t \\in [0,1].$\n\nLet $h \\in \\exitfns$ and let $(u,v)$ be in the graph of $h.$  Then\n\\begin{align*}\n&\\altPhi(h,\\hg(;\\ug(t));u,v) - \\altPhi(h,\\hg(;\\ug(1));u,v)\\\\\n= &\n\\int_0^u (\\hginv(z;\\ug(t))-\\hginv(z;\\ug(1)))\\text{d}z \\\\\n= &\n(1-t)(u B_{\\hg} - \\int_0^u \\hginv(z;\\ug(1)))\\text{d}z \\\\\n\\ge & 0\n\\end{align*}\nwhere the last inequality holds since we have equality at $u=0$ and $u=1$\nand $(u B_{\\hg} - \\int_0^u \\hginv(z;\\ug(1)))\\text{d}z$ is concave.\nThis implies that if $(h,\\hg(;\\ug(1)))$ satisfies the strictly positive gap condition\nwith $A(h,\\hg(;\\ug(1))) = 0$ then\n$(h,\\hg(;\\ug(t)))$ also satisfies the strictly positive gap condition\nwith $A(h,\\hg(;\\ug(t))) = 0$ for all $t \\in [0,1].$\n\nThe above argument shows that $(\\hf(;\\uf(1)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $t\\in [0,1].$\nApplying the argument analogously to $\\hf$ we see that\n$(\\hf(;\\uf(s)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $s,t\\in [0,1],$\nand, in particular, \n$(\\hf(;\\uf(t)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $t\\in [0,1].$\n\nAll that remains to apply Lemma \\ref{lem:PCexitcont} over $t \\in [0,1]$ and\nconclude the proof for the case $A(\\hf,\\hg)=0$\nis to find $\\zf(0)$ and $\\zg(0).$\nSet $\\zf(0) = 0$  so that\n$f(x;\\zf(0)) = \\,\\unitstep(x).$\nLet $y$ be the unique point such that $\\fS(y;\\zf(0)) = B_{\\hg}$ and\nset each component of $\\zg(0)$ to $y$ so that $\\fg(x;\\zg(0)) = \\unitstep(x-y).$\nIt follows that $\\fg(x;\\zg(0)) = \\hg(\\fS(x;\\zf(0));\\ug(0))$ and that\n$\\ff(x;\\zf(0)) = \\hf(\\gS(x;\\zg(0));\\uf(0)).$\n\nApplying Lemma \\ref{lem:PCexitcont} for $t \\in [0,1]$ we obtain\n$\\ff(;\\zf(t))$ and $\\fg(;\\zg(t))$ such that\n$\\fg(x;\\zg(t)) = \\hg(\\fS(x;\\zf(t));\\ug(t))$ and\n$\\ff(x;\\zf(t)) = \\hf(\\gS(x;\\zg(t));\\uf(t))$\ncompleting the proof when $A(\\hf,\\hg)=0.$\n\nWe now consider the case $A(\\hf,\\hg) \\neq 0.$\nWithout loss of generality we assume $A(\\hf,\\hg)  > 0.$\nThe case $A(\\hf,\\hg)  < 0$ is equivalent to the case $A(\\hf,\\hg)  > 0$\nunder symmetry.\n\nLet us introduce a modification of $\\hf(;\\uf(t))$ as follows.\nFor $r \\in (0,1)$ define $\\uf(t;r)$ by $\\uf_i (t;r) = \\min \\{ \\uf_{i}(t), r \\}.$\nThen $\\int_0^u \\hfinv(x;\\uf(t;r)) dx$ is non-decreasing in $r$ for all $u \\in [0,1].$\nSince $\\int_0^1 \\hfinv(x) dx > 1 -\\int_0^1 \\hginv(x) dx$ and\n$\\int_0^1 \\hfinv(x;\\uf(t;0)) dx = 0$\nthere exists a unique positive $r_0 < \\uf_{\\Kf}$ such that\n$\\int_0^1 \\hginv(x;\\uf(t;r_0)) dx =  1-\\int_0^1 \\hginv(x) dx.$\n\nWe claim that for all $r \\in [r_0,\\uf_{\\Kf}]$ the pair\n$(\\hf(;r),\\hg)$ satisfies the strictly positive gap condition.\nTo establish the claim we need to show that\n$\\altPhi(\\hf(;r),\\hg) > A(\\hf(;r),\\hg )$ on $\\intcross(\\hf(;r),\\hg ).$\nLet $ (u,v) \\in \\intcrossing (\\hf(;r),\\hg ).$\nNote that this implies $u \\le r$ since $\\hg$ is continuous at $1$ \nby Lemma \\ref{lem:zocontinuity}.\nIf $u < r$ then $(u,v) \\in \\intcrossing (\\hf,\\hg)$\nand we have\n\\(\n\\altPhi(\\hf(;r),\\hg;u,v) =\n\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg) \\ge A(\\hf(;r),\\hg)\n\\,.\n\\)\nIf $u = r$ then we have $v \\ge \\hf(u-;r)$ and\n$\\hfinv(v';r) = v$ for all $v' \\in (v,1].$\nApplying \\eqref{eqn:altPhiderivatives} we have\n$\\altPhi(\\hf(;r),\\hg;u,v) =  \\altPhi(\\hf(;r),\\hg;u,1)$\nand applying \\eqref{eqn:altPhiderivatives} and using continuity of\n$\\hg$ at $1,$ we have\n$\\altPhi(\\hf(;r),\\hg;u,1) >  \\altPhi(\\hf(;r),\\hg;1,1).$\nHence \n$\\altPhi(\\hf(;r),\\hg;u,v) > A(\\hf(;r),\\hg )$ and the claim is established.\n\nApplying our result for the $\\ashift = 0$ case\nwe can find $f,g \\in \\sptfns$  that\nform a $(0,1)$-interpolating spatial fixed point pair for $(\\hf, \\hg(x;r_0)).$\nWe now apply Lemma \\ref{lem:PCshiftexist} in a series of stages\nincreasing $\\ashift$ (linearly $\\ashift = t$) and adjusting\n$\\zf$ and $\\zg$ so that\n$h^{\\fS,g}$ is held fixed while\n$h^{\\gSa,f}$ tracks $\\hg(x;r(t)).$\nThe process can be decomposed into stages where in each stage\n$u_{2,i-1} \\le r(t) \\le u_{2,i}$ for some $i,$\nand where in the final stage $i = \\Kf.$\nIn each stage $\\ashift(t)$ and $r(t)$ is increasing until,\nat the end of the stage, $r(t) = u_{2,i}.$\nDuring the stage where $r(t)$ increases to $u_{2,i}$ we have\n$u_{2,j}(t) = r(t)$ for all $j \\ge i.$\nWe can, in principle, simplify the notation by collapsing these indices to a single value, i.e.,\nto assume that $i = \\Kf.$\nIn this way we reduce each stage to the case $i= \\Kf$\n(while admitting an initial condition $\\ashift_0 \\ge 0).$\nLemma \\ref{lem:PCshiftexist} therefore completes the proof.\n\\end{IEEEproof}\n\n\\subsubsection{Convergence}\n\nIn the piecewise constant case with strictly positive averaging kernel \nwe can also show convergence to the solution constructed above for all initial conditions.\nDefine $f_\\lambda$ by \n\\[\nf^{-1}_\\lambda = \\lambda f^{t+1,-1} + (1-\\lambda) f^{t,-1}\n\\]\nand set\n$g_\\lambda = \\hf \\circ f^\\smthker_\\lambda.$\nThen by applying \\eqref{eqn:delgfrecursionInv} we  obtain\n\\begin{align*}\n&g^{t+1,-1}(u) -  g^{t,-1}(u) \n\\\\\n=&\n\\int_0^1 \\int_0^1 M(\\lambda,u,v)(f^{t+1,-1}(v) - f^{t,-1}(v)) \\,dv\\,d\\lambda\n\\end{align*}\nwhere\n\\begin{align*}\nM(\\lambda,u,v)=\\frac{\\int_0^1 \\smthker(g^{-1}_\\lambda(u)-f^{-1}_\\lambda(v))}{\\int_0^1 \\smthker(g^{-1}_\\lambda(u)-f^{-1}_\\lambda(v)) \\,dv}\n\\end{align*}\nSince $\\smthker$ is strictly positive we have\n for each $\\lambda$ and $u$ that $M(\\lambda,u,v)>0$ and $\\int_0^1 M(\\lambda,u,v)\\,dv = 1.$\nSo we obtain\n\\begin{align*}\n\\sup_u (g^{t+1,-1}(u) -  g^{t,-1}(u)) &\\le \\sup_v (f^{t+1,-1}(v) -  f^{t,-1}(v))\\, ,\n\\\\\n\\inf_u (g^{t+1,-1}(u) -  g^{t,-1}(u)) &\\ge \\inf_v (f^{t+1,-1}(v) -  f^{t,-1}(v))\\,.\n\\end{align*}\nIn the piecewise constant case the inverse functions are bounded and with strictly positive $\\smthker$ we see that\nthe inequalities are strict unless  $f^{t+1,-1}(v) -  f^{t,-1}(v)$ is a constant.\nIt is easy to conclude in this case that $f^{t+1,-1}(v) -  f^{t,-1}(v)$ converges in $t$ to a constant in $v.$\nFrom this it follows that the $f^t$ converges to the solution given above (with suitable translation).\n\n\n\\subsection{Limit Theorems \\label{sec:limitthms}}\n\nOne of the main tools we use to extend the existence results for the piecewise constant\ncase to the general case is taking limits.  In this section we develop the basic results needed.\n\nLet us recall the notation $g^{{\\smthker},\\ashift}(x) = g^{\\smthker}(x+\\ashift).$\nThen, we have the bound\n\\begin{align*}\n&g^{{\\smthker},\\ashift}(x)\n-\ng^{\\smthker',\\ashift'}(x)\n=\n\\\\ &\n\\int_{-\\infty}^\\infty (g(y-\\ashift)-g(y-\\ashift')) \\smthker(x-y) \\,dx\n\\\\&+\n\\int_{-\\infty}^\\infty g(y-\\ashift') (\\smthker(x-y)-\\smthker'(x-y)) \\,dx\n\\end{align*}\n\\begin{align}\n\\begin{split}\\label{eqn:diffbound}\n|g^{{\\smthker},\\ashift}(x)\n-\ng^{\\smthker',\\ashift'}(x)|\n\\le\n|\\ashift-\\ashift'| \\|\\smthker\\|_\\infty\n+\n\\|\\smthker-\\smthker'\\|_1 \\,\n\\end{split}\n\\end{align}\n\n\n\n\\begin{theorem}\\label{thm:mainlimit}\nLet $f_i,g_i,\\ashift_i,\\smthker_i,\\; i=1,2,3,...$ be sequences where $(f_i,g_i) \\in \\sptfns^2$\nare $(0,1)$-interpolating, \n$\\ashift_i \\in \\reals,$ and $\\smthker_i$ are averaging kernels. \nAssume\n\\[\nf_i  \\rightarrow f,\\;\ng_i  \\rightarrow g,\\;\n\\ashift_i  \\rightarrow \\ashift,\\;\\text{ and }\n\\smthker_i  \\rightarrow \\smthker \\text{ (in $L_1$) },\n\\]\nwhere $|\\ashift| < \\infty$\nand $\\smthker$ is an averaging kernel.\n(Note that we do not assume $f$ and $g$ are interpolating or that $\\smthker$ is regular.)\nFurther assume\n\\[\nh_{[\\ff_i,\\gSiai_i]}\\rightarrow \\hf\n,\\quad\nh_{[\\fg_i,\\fSi_i]}\\rightarrow \\hg\n\\]\nfor some  $\\hf,\\hg \\in \\exitfns$ respectively.\nThen we have the following\n\\begin{itemize}\n\\item[A.] \n\\begin{align*}\n\\ff \\veq \\hf\\circ\\gS\n\\text{  and   }\n\\fg \\veq \\hg\\circ\\fS\n\\end{align*}\n\\item[B.]\n\\[\n(f(\\minfty),g(\\minfty)),(f(\\pinfty),g(\\pinfty)) \\in \\cross (\\hf,\\hg)\n\\]\n\\item[C.]\nIf $\\ashift=0$ then\n\\begin{align*}\n0 = &\\altPhi(\\hf,\\hg;g(\\minfty),f(\\minfty)) \\\\ \n = &\\altPhi(\\hf,\\hg;g(\\pinfty),f(\\pinfty)) \\,.\n\\end{align*}\nand, for all $x_1,x_2$\n\\[\n\\altPhi(\\hf,\\hg;\\fg(x_2+),\\ff(x_1+)) = \n\\altPhiSI (\\smthker;f,g;x_1,x_2).\n\\]\n\\item[D.]\nFor $(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}$\n\\[\n\\min\\{0,A(\\hf,\\hg)\\}\\le\\altPhi(\\hf,\\hg;u,v) \\le \\max\\{0,A(\\hf,\\hg)\\}.\n\\]\n\\end{itemize}\n\\end{theorem}\n\\begin{IEEEproof}\nSince $\\fg_i \\rightarrow \\fg,$ $\\smthker_i \\rightarrow \\smthker$\nand $\\ashift_i \\rightarrow \\ashift$ we have from \\eqref{eqn:diffbound}\nthat $\\fg^{{{\\smthker}_i},\\ashift_i}_i  \\rightarrow \\fg^{{\\smthker},\\ashift}$ point-wise.\n\nIf $x$ is a point of continuity of $f$ then $f_i(x) \\rightarrow f(x)$ and we have\n$(g^{{\\smthker}_i}(x+\\ashift_i),f_i(x))\\rightarrow (g^{\\smthker}(x+\\ashift),f(x))$ which implies\n$f(x) \\veq \\hf( \\gSa(x)).$ \nSince $\\gS$ is continuous we can extend this to all $x$ by taking limits.\nThis shows part A.\n\nPart B follows from part A by Lemma \\ref{lem:FPequal}.\n\nNow we consider part C where we assume $\\ashift=0.$\nBy Lemma \\ref{lem:FPequal} we need only show that\n$\\altPhi(\\hf,\\hg;g(\\pinfty),f(\\pinfty))=0.$\nFor any $\\epsilon>0$ \nwe can find $L$ large enough so that $\\int_L^\\infty  \\smthker_i(x) dx < \\epsilon$ for all $i$ since $\\smthker_i\\rightarrow\\smthker.$\nNow choose $z$ large enough so that \n$f^\\smthker(z-L),f(z-L) > f(\\minfty)-\\epsilon$\nand\n$g^{\\smthker}(z-L),g(z-L) > g(\\minfty)-\\epsilon.$  \nIt follows that \n$\\Delta_L g(z) < \\epsilon$ and $\\Delta_L f(z) < \\epsilon.$\nFor all $i$ large enough we have $\\Delta_L g_i^{\\smthker_i}(z) < 2\\epsilon$ and \n$\\Delta_L f_i^{\\smthker_i}(z) < 2\\epsilon.$\n\nBy Lemma \\ref{lem:transitionPhiBounds} this implies\n\\[\n \\altPhi(\\hf^i,\\hg^i; \\fg_i^{\\smthker_i}(z),\\ff_i^{\\smthker_i}(z)) < 11\\epsilon\n\\]\nIt follows from \\eqref{eqn:diffbound} that $\\ff_i^{\\smthker_i} \\rightarrow \\fS$ \nand  $\\fg_i^{\\smthker_i} \\rightarrow \\gS$ \npoint-wise\nand since \n$h_{[\\fg_i,\\ff_i^{\\smthker_i}]} \\rightarrow \\hg$\nand\n$h_{[\\ff_i,\\fg_i^{\\smthker_i}]} \\rightarrow \\hf$\nwe have for all $x_1,x_2,$\n\\[\n\\begin{split}\n& \\altPhi(\\smthker_i;\\hf^i,\\hg^i; g^{\\smthker_i}_i(x_2),f^{\\smthker_i}_i(x_1))\n\\\\ &\n\\rightarrow\n\\altPhi(\\smthker;\\hf,\\hg;g^{\\smthker}(x_2),f^{\\smthker}(x_1))\\,.\n\\end{split}\n\\]\nWe now obtain\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(z),f^{\\smthker}(z)) \\le 11\\epsilon\\,.\n\\]\nBy Lipschitz continuity of $\\altPhi$ we have\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(\\pinfty),f^{\\smthker}(\\pinfty)) < 13\\epsilon\n\\]\nand since $\\epsilon$ is arbitrary we obtain\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(\\pinfty),f^{\\smthker}(\\pinfty)) = 0\n\\]\nIt now follows from Lemma \\ref{lem:twofint} and part A that\n\\[\n\\altPhi(\\hf,\\hg;g(x_2+),f(x_1+)) = \\altPhiSI(\\smthker;\\ff,\\fg;x_1,x_2)\n\\]\nfor all $x_1,x_2.$\n\nFinally, we show part D. \nIf $\\ashift=0$ then part $C$ gives part $D.$\nBy choosing a subsequence if necessary, we can assume that\n\\(\nh_{[\\ff_i,\\gSi_i]}\n\\)\nconverges to some $\\thf \\in \\exitfns.$\n\nWe assume $\\ashift > 0,$ the case $\\ashift<0$ is analogous.\nSince $\\thfinv \\le \\hfinv$ almost everywhere we have\n\\(\n\\altPhi(\\thf,\\hg;u,v) - \\altPhi(\\hf,\\hg;u,v) =\n\\int_0^{v} (\\thfinv(x) -\\hfinv(x))dx \\le 0\\,.\n\\)\nFor $(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}$\nwe have $\\altPhi(\\thf,\\hg;u,v)=0$ \nby part C,\nand therefore  $\\altPhi(\\hf,\\hg;u,v) \\ge 0.$\n\nNow \n\\(\nA(\\hf,\\hg)-A(\\thf,\\hg) =\n\\int_0^{1} (\\hfinv(x) -\\thfinv(x))dx\n\\ge \n \\int_0^{v} (\\hfinv(x) -\\thfinv(x))dx\n\\)\nand since $A(\\thf,\\hg)=0,$  we have\n$\\altPhi(\\hf,\\hg;u,v) \\le A(\\hf,\\hg)$ for\n$(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}.$\nThis completes the proof.\n\\end{IEEEproof}\n\nThe following result is largely a corollary of the above but it is\nmore convenient to apply.\n\n\\begin{lemma}\\label{lem:limitexist}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition.\nIf there exists a sequence of $(0,1)$-interpolating $f_i,g_i \\in \\sptfns$ and bounded $\\ashift_i$ such that \n$(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$\nand $\\smthker_i \\rightarrow \\smthker$ in $L_1,$\nwhere $\\hf^i \\equiv h_{[\\ff_i,\\fg^{\\smthker_i,\\ashift_i}_i]}$ and\n$\\hg^i \\equiv h_{[\\fg_i,\\ff^{\\smthker_i}_i]},$\nthen\nthere exists $(0,1)$-interpolating $\\ff,\\fg \\in \\sptfns$ and finite $\\ashift,$ all limits of some translated subsequence,\nsuch that $\\hf = h_{[\\ff,\\gSa]}$ and\n$\\hg = h_{[\\fg,\\fS]}.$\n\\end{lemma}\n\\begin{IEEEproof}\nSince $\\smthker_i \\rightarrow \\smthker$ in $L_1$ and\n $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$  we conclude from \nLemma \\ref{lem:stposbound} and Lemma \\ref{lem:shiftupperbound}\nthat $|\\ashift_i|$ is bounded.\n\nBy translating $\\ff$ and $\\fg$ as necessary, we can assume that $\\ff^{\\smthker_i}(0) = 1\/2$ for each $i.$\nTaking subsequences as necessary, we can now assume that $f_i \\rarrowi f,$\n$g_i \\rarrowi g,$ and $\\ashift_i \\rarrowi \\ashift,$ for some finite $\\ashift.$\n\nWe claim that $\\ff$ and $\\fg$ are $(0,1)$-interpolating.\nFor all $(u,v) \\in \\intcross(\\hf,\\hg)$ we have\n$\\altPhi(\\hf,\\hg;u,v) >\\max \\{0,A(\\hf,\\hg)\\}$\nby assumption.\nBy Theorem \\ref{thm:mainlimit} parts B and D we now have\n$(\\ff(\\minfty),\\fg(\\minfty)) \\in \\cross(\\hf,\\hg) \\backslash \\intcross(\\hf,\\hg) = \\{ (0,0),(1,1) \\}.$\nSince $\\fS(0) = \\frac{1}{2}$ we must have $(\\ff(\\minfty),\\fg(\\minfty)) = (0,0)$\nand $(\\ff(\\pinfty),\\fg(\\pinfty)) = (1,1),$\nproving the claim.\n\\end{IEEEproof}\n\n\\subsection{Existence of Consistent Spatial Waves}\n\nIn Section \\ref{sec:PCcase} we proved Theorem\n\\ref{thm:PCexist},\na special case of Theorem \\ref{thm:mainexist}\nin which $\\hg$ and $\\hf$ are piecewise constant\nfunctions and $\\smthker$ is $C_1$ and strictly positive.\nIn this section we show how to remove the special conditions \nto arrive at the general results.\nWe make repeated use of the limit theorems of Section \\ref{sec:limitthms} and develop\nsome approximations for functions in $\\exitfns.$\nIt is quite simple to approximate $h \\in \\exitfns$ using piecewise constant functions.\nThe challenge is to approximate a pair $(\\hg,\\hf)$ so that the strictly positive gap\ncondition is preserved.\n\n\\subsubsection{Approximation by Tilting}\n\nIn a manner analogous to \\eqref{eqn:pcscale} we define a perturbation of\n$\\hf,\\hg$ as $\\hf(;t),\\hg(;t)$ for $t \\in [0,1]$ by\n\\begin{align}\n\\begin{split}\n\\hfinv (v;t) & = (1-t) B_{\\hf}  + t \\hfinv(v) \\\\\n\\hginv (u;t) & = (1-t) B_{\\hg}  + t \\hginv(u)\\,,\\label{eqn:genscale}\n\\end{split}\n\\end{align}\nwhere we recall $B_h = \\int_0^1 h^{-1}(x) \\,dx.$\nThis can also be expressed as\n\\begin{align}\n\\begin{split}\n\\hf (u;t) & = \\hf\\Bigl(\\frac{u-B_{\\hf}}{t} + B_{\\hf} \\Bigr) \\\\\n\\hg (v;t) & =  \\hg\\Bigl(\\frac{u-B_{\\hg}}{t} +  B_{\\hg}\\Bigr) \\,,\\label{eqn:genscalefor}\n\\end{split}\n\\end{align}\nwith appropriate extension of $\\hf$ and $\\hg$ outside of $[0,1],$\n$\\hf(x)=\\hg(x)=0$ for $x<0$ and\n$\\hf(x)=\\hg(x)=1$ for $x>1.$\n\nLetting $h$ denote either $\\hf$ or $\\hg,$ we clearly have\n\\begin{align}\n\\int_0^1 h^{-1} (x;t) dx = B_h\\label{eqn:slanteq}\n\\end{align}\nfor all $t.$\nNote also that \n\\(\nh^{-1}(v;t) -h^{-1} (v) =\n(1-t)(B_h-h^{-1} (v))\n\\)\nis non-increasing  in $v.$\nIt follows that\n\\(\n\\int_0^v h^{-1}(x;t) dx \\ge \n\\int_0^v h^{-1} (x) dx \n\\)\nfor all $v\\in[0,1]$\nand we obtain\n\\begin{align}\n\\altPhi(\\hf(;t),\\hg(;t);)\\ge \\altPhi(\\hf,\\hg;)\\label{eqn:altslbound}\n\\end{align}\nfor all $t\\in [0,1].$\n\n\\begin{lemma}\\label{lem:smoothcompress}\nLet $(\\hg,\\hf) \\in \\exitfns^2$ \nsatisfy the strictly positive gap condition.\nThen, there exists $\\epsilon > 0$ such that\n$(\\slanta{\\hg},\\slanta{\\hf})$ \nsatisfies the strictly positive gap condition\nfor any $t \\in (1-\\epsilon,1].$\n\\end{lemma}\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:smoothcompress}]\nFor the case $A(\\hf,\\hg)=0$ equation \\eqref{eqn:slanteq}, inequality \\eqref{eqn:altslbound} and\nLemma \\ref{lem:monotonic} gives\nthe result immediately.\nBy symmetry we now need only consider the case $A(\\hf,\\hg) > 0.$\n\nBy Lemma \\ref{lem:Sstructure} and \\eqref{eqn:altslbound}\nit is sufficient to show that \n$\\intcross(\\hf(;t),\\hg(;t)) \\cap  \\closure{S(\\hf,\\hg)}=\\emptyset$\nfor $t\\in[1-\\epsilon,1].$\n\nAlso by Lemma \\ref{lem:Sstructure}, there exists a\nminimal and positive element $(u^*,v^*) \\in \\intcross(\\hf,\\hg).$\nThere exists a neighborhood ${\\cal N}$ of $(0,0),$ which we take to be a subset of\n$\\subset [0,u^*)\\times [0,v^*),$\nin which\n$\\hginv(u;t) \\ge \\hginv(u)$ and \n$\\hfinv(v;t) \\ge \\hfinv(v).$ \nIt follows that  \n$\\intcross(\\hf(;t),\\hg(;t)) \\cap {\\cal N} =\\emptyset$\nfor all $t.$\n\nLet $\\delta>0$ be small enough so that $\\neigh{(0,0)}{\\delta} \\subset {\\cal N}$\nand $\\neigh{(x^*,y^*)}{\\delta} \\cap \\closure{S(\\hf,\\hg)}=\\emptyset.$\nFor $\\epsilon$ small enough and $t\\in[1-\\epsilon,1]$ we have\n$\\cross(\\hf(;t),\\hg(;t))\\subset \\neigh{\\cross(\\hf,\\hg)}{\\delta}$\nby Lemma \\ref{lem:crosspointlimit}\nand it now follows that\n$\\intcross(\\hf(;t),\\hg(;t)) \\cap  \\closure{S(\\hf,\\hg)}=\\emptyset.$\n\\end{IEEEproof}\n\n\\subsubsection{Piecewise Constant Approximation}\nGiven $h \\in \\exitfns$ let us define a sequence of piecewise constant approximations\n$Q_n(h),$ $n=1,2,...$  by\n\\begin{align*}\nQ_n(h) (x) &= \\sum_{j=1}^n \\frac{1}{n} \\,\\unitstep(x- u_{n,j}) \n\\end{align*}\nwhere we set \n\\begin{align*}\nu_{n,j} \n& = n\\int_{(j-1)\/n}^{j\/n}h^{-1}(v)dv\n\\end{align*}\nand we have\n\\begin{align*}\n\\int_0^1 Q_n(h) (x) dx &\n= \\sum_{j=1}^n \\frac{1-u_{n,j}}{n} \\\\ \n& = \\int_0^1 (1-h^{-1})(x) dx \\\\\n&= \\int_0^1 h (x) dx.\n\\end{align*}\nIn general, it holds that\n\\(\n\\int_0^1 Q_n(h) (x) dx = \\int_0^1 h (x) dx.\n\\)\nIt also follows that $\\int_0^z Q_n(h) (x) dx \\le \\int_0^z h (x) dx$ for all\n$z \\in [0,1].$\n\n\\begin{lemma}\\label{lem:PCapprox}\nLet $(\\hg,\\hf)$ be pair of functions in $\\exitfns$ satisfying the strictly positive gap condition \nsuch that for some $\\eta>0$ we have\n$\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nThen, for all $n$ sufficiently large\n$(Q_n(\\hg),Q_n(\\hf))$ satisfies the strictly positive gap condition.\n\\end{lemma}\n\\begin{IEEEproof}\nWe have\n\\(\nA(Q_n(\\hf),Q_n(\\hg)) =\nA(\\hf,\\hg)\n\\)\nand\n\\(\n\\altPhi(Q_n(\\hf),Q_n(\\hg);\\cdot,\\cdot) \\ge\n\\altPhi(\\hf,\\hg;\\cdot,\\cdot)\n\\)\nso it suffices to show that \n\\(\n\\intcrossing (Q_n(\\hf),Q_n(\\hg)) \\cap \\closure{S(\\hf,\\hg)}=\\emptyset.\n\\)\n\nSince $\\hg$ and $\\hf$ are $0$ on $[0,\\eta)$ and $1$  on $(1-\\eta,1]$ it follows that \n$\\intcrossing (\\hf,\\hg) \\subset [\\eta,1-\\eta]^2$ and\n$\\intcrossing (\\hf,\\hg)$ is closed and by Lemma \\ref{lem:Sstructure} it is disjoint from\n$\\closure{S(\\hf,\\hg)}.$\nThus, for $\\delta$ sufficiently small we have $\\neigh{\\intcrossing(\\hf,\\hg)}{\\delta}\\cap  \\closure{S(\\hf,\\hg)}=\\emptyset.$\n\nBy Lemma \\ref{lem:crosspointlimit}  we now have\n\\(\n\\intcrossing (Q_n(\\hf),Q_n(\\hg))  \\cap \n\\closure{S(\\hf,\\hg)}=\\emptyset\n\\)\nfor all $n$ sufficiently large.\n\\end{IEEEproof}\n\nWe are now ready to prove the main result of this section.\n\n\\begin{lemma}\\label{lem:weakexistence}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition.\nThen there exists $(0,1)$-interpolating $(\\ff,\\fg) \\in \\sptfns^2$ and $\\ashift$ such that\n$\\hf    =  h_{[\\ff,\\gSa]}$\nand\n$\\hg =  h_{[\\fg,\\fS]}.$\n\\end{lemma}\n\\begin{IEEEproof}\nThe simplest case is already established in Theorem \\ref{thm:PCexist}\nand we first generalize to arbitrary $\\smthker.$\nAssume that $(\\hg,\\hf)$ are both piecewise constant.\nDefine $\\smthker_k = \\smthker \\otimes G_k$ where \n$G_i(x) = \\frac{k}{\\sqrt{2\\pi}} e^{- (kx)^2\/2}.$\nIt follows that $\\smthker_k \\rightarrow \\smthker$ in $L_1$\nand $\\| \\smthker_k \\|_\\infty \\le \\| \\smthker\\|_\\infty.$\nFor each $\\smthker_k$ we apply Theorem \\ref{thm:PCexist}\nto obtain piecewise constant $f_k,g_k \\in \\sptfns$ \n(with corresponding $z^{f_k},z^{g_k}$) and constants $\\ashift_k$\nsuch that $ h_{[\\fg_k,\\ff_k^{{\\smthker}_k}]} = \\hg$ and $h_{[\\ff_k,g_k^{{\\smthker}_k,\\ashift_k}]} = \\hf.$\nWe can now apply Lemma \\ref{lem:limitexist} to conclude\nthat the theorem holds for piecewise constant $\\hf,\\hg$ and general $\\smthker.$\n\nLet now assume first that for some $\\eta>0$ we have $\\hf(x)=\\hg(x)=0$ for $x\\in [0,\\eta)$\nand $\\hf(x)=\\hg(x)=0$ for $x\\in (1-\\eta,1].$\nConsider $Q_n(\\hf)$ and $Q_n(\\hg).$ \nWe apply Lemma \\ref{lem:PCapprox} and the preceding case already established\nto conclude that for all $n$ sufficiently large\nthere exists (piecewise constant) $(0,1)$-interpolating $f_n,g_n \\in \\sptfns$ and finite constants $\\ashift_n$ such that\n$h_{[\\fg_n,\\fS_n]} = Q_n(\\hg)$ and $h_{[\\ff_k,\\fg_k^{{\\smthker}_k,\\ashift_n}]}= Q_n(\\hf).$\nSince $Q_n(\\hg)$ and $Q_n(\\hf)$ converge to $\\hg$ and $\\hf$ respectively,\nwe can apply Lemma \\ref{lem:limitexist} to conclude that the theorem holds for this case.\n\nFor arbitrary $(\\hg,\\hf)$ we consider $(\\slanta{\\hg},\\slanta{\\hf}).$\n\nBy Lemma \\ref{lem:smoothcompress} we can find a sequence $t_i \\rightarrow 1$\nsuch $(\\hg(;t_i),\\hf(;t_i))$ satisfies the strictly positive gap condition for each $i.$\nBy the preceding case, there exists $f_{t_i},g_{t_i} \\in \\sptfns$ and finite constants $\\ashift_i$ such that\n$h_{[\\fg_{t_i},\\fS_{t_i}]} = \\hg(;t_i)$ and $h_{[\\ff_{t_i},\\fg_{t_i}^{{\\smthker},\\ashift_i}]} = \\hf(;t_i).$\nBy taking a subsequence if necessary we can assume that $\\ashift_i \\rightarrow \\ashift.$\nSince $(\\hf(;t_i),\\hg;(t_i))\\rightarrow (\\hf,\\hg)$ we can apply\nLemma \\ref{lem:limitexist} to obtain $(\\ff,\\fg) \\in \\sptfns^2$\nsuch that $\\hf = h_{[\\ff,\\gSa]}$ and $\\hg = h_{[\\fg,\\fS]}.$\n\\end{IEEEproof}\n\\subsection{Existence of Spatial Wave Solutions}\n\nIn the preceeding section we estabished the existence of consistent spatial waves\nunder general conditions.  In this section we refine the results  to obtain full\nspatial wave solutions.  Thus, in this section we complete the proof of \n\\ref{thm:mainexist}.\n\n\\subsubsection{Analysis of Consistent Spatial Waves}\n\nFor $h \\in \\exitfns$ we use $\\jump{h}$ to denote the set of discontinuity points of\n$h,$ i.e.,\n\\[\n\\jump{h} = \\{ u\\in[0,1]:h(u-)<h(u+) \\}.\n\\]\n\nFor continuous $\\ff\\in \\sptfns$ let $\\flats{\\ff}$ denote the set of positive-length maximal intervals on which $\\ff$ is effectively constant:\n\\[\n\\flats{\\ff} = \\{ [ \\ff^{-1}(v-),\\ff^{-1}(v+)]    : \\ff^{-1}(v-)<\\ff^{-1}(v+)\\}\\,.\n\\]\nLet $\\smthker$ be regular and assume  $\\frac{d}{dx}\\fS(x)=0$ at $x=x_1.$\nThen $\\int_{-W}^{W} \\smthker(x_1-x) df(x) = 0$\nfrom which we obtain $\\Delta_W f(x_1)=0.$ \nIt follows that if $\\fS(x) = F$ (a constant) for $x\\in I$\nwhere $I$ is an interval of positive length\nthen $\\ff(x)=F$ for all $x \\in \\neigh{I}{W},$ c.f. \\eqref{eqn:neighdef}.\n\n\\begin{lemma}\\label{lem:phiflat}\nLet $(\\ff,\\fg)$ be $(0,1)$-interpolating and let $\\smthker$ be regular.\nLet $\\hg\\equiv h_{[\\fg,\\fS]}$ and $\\hfs\\equiv h_{[\\ff,\\gSa]}$ with $\\ashift \\ge 0.$\nIf $\\frac{d}{dx}\\gS(x)=0$ or \n$\\frac{d}{dx}\\fS(x)=0$\nat $x=z$\nthen \n\\[\n\\altPhi(\\hfs,\\hg;g(z),f(z)) \\in [0,A(\\hfs,\\hg)].\n\\] \n\\end{lemma}\n\\begin{IEEEproof}\nAssume $\\frac{d}{dx}\\gS(x)=0$ at $x=z.$\nThen, since $\\smthker$ is regular, $g(x)=\\gS(z)$ if\n$\\|x-z\\| <W.$\nUsing $L=W$ in Lemma \\ref{lem:transitionPhiBounds} we obtain \n$\\altPhi(\\hfz,\\hg;\\fg(z),\\ff(z)) = 0.$\nWe also have $A(\\hfz,\\hg)  = 0$ by Lemma \\ref{lem:FPAzero}.\n\nNow \n\\begin{align*}\n&   \\altPhi(\\hfs,\\hg;u,v) -\\altPhi(\\hfz,\\hg;u,v) \\\\\n=&\n\\int_0^v ( \\hfsinv(x) -\\hfzinv(x))\\,dx\n\\end{align*}\nwhich, since $\\ashift\\ge 0,$ is non-decreasing in $v$ and independent of $u.$\nHence\n\\begin{align*}\n0\n&\\le\n \\altPhi(\\hfs,\\hg;g(z),f(z)) - \\altPhi(\\hfz,\\hg;g(z),f(z))  \\\\\n&\\le\n \\altPhi(\\hfs,\\hg;1,1) - \\altPhi(\\hfz,\\hg;1,1)  \\\\\n&=\nA(\\hfs,\\hg) - A(\\hfz,\\hg)\n\\end{align*}\nfrom which we obtain \n\\[\n0\n\\le\n \\altPhi(\\hfs,\\hg;\\fg(z),\\ff(z))\n\\le\nA(\\hfs,\\hg)\n\\]\n\\end{IEEEproof}\n\n\n\n\n\n\\begin{lemma}\\label{lem:notequal}\nLet $\\ff,\\fg \\in\\sptfns$ with $\\fg$ continuous.\nAssume that $\\ff \\veq \\hf\\circ\\fg$ for $\\hf \\in \\exitfns.$ \nThen\n\\[\n\\ff(x) \\neq \\hf(\\fg(x))\\Rightarrow \\fg(x) \\in \\jump{\\hf}\n\\]\nand if $\\ff \\neq \\hf\\circ\\fg$ on a set of positive Lebesgue measure,\nthen there exists $I\\in\\flats{\\fg}$\nsuch that $\\fg(I) \\in \\jump{\\hf}$\nand such that $\\ff(x)\\neq \\hf(\\fg(I))$\non a subset of $I$ of positive measure.\n\\end{lemma}\n\\begin{IEEEproof}\nBy definition,  $\\ff \\veq \\hf\\circ\\fg$ means $\\ff(x) \\veq  \\hf(\\fg(x))$\nfor all $x$  so we can have $\\ff(x) \\neq  \\hf(\\fg(x))$\nonly if $\\fg(x) \\in \\jump{\\hf}.$\nIt follows that\n\\[\n\\{x:\\ff(x) \\neq  \\hf(\\fg(x))\\}\n\\subset \\cup_{u \\in \\jump{\\hf}} [\\fginv(u-),\\fginv(u+)]\\,.\n\\]\nSince $\\mu\\{x:\\ff(x) \\neq  \\hf(\\fg(x))\\} >0$ (where $\\mu$ is Lebesgue measure)\nwe have\n$\\mu (\\{x:\\ff(x) \\neq  \\hf(\\fg(x))\\}\\cap [\\fginv(v-),\\fginv(v+)]) >0$\nfor some $v\\in \\jump{\\hf}.$\nThen $I = [\\fginv(v-),\\fginv(v+)] \\in \\flats{\\fg}.$\n\\end{IEEEproof}\n\n\n\n\n\\begin{lemma}\\label{lem:pathology}\nLet $\\ff,\\fg \\in \\sptfns$ be $(0,1)$-interpolating.\nIf $h_{[\\ff,\\gSa]} = \\hf$ and\n$h_{[\\fg,\\fS]} = \\hg$ \nthen $\\ff = \\hf\\circ\\gSa$\nand $\\fg = \\hg\\circ\\fS$\nin any of the following scenarios.\n\\begin{itemize}\n\\item[A.] $\\hf$ and $\\hg$ are continuous.\n\\item[B.] $\\smthker$ is positive on all $\\reals.$\n\\item[C.] $\\smthker$ is regular,\n$\\ashift = 0,$\nand $(\\hf,\\hg)$ satisfies the strictly positive gap condition.\n\\item[D.] $\\smthker$ is regular, \n$(\\hf,\\hg)$ satisfies the strictly positive gap condition\nand\n$\\jump{\\hf} \\cap \\jump{\\hginv}=\\emptyset$\nand\n$\\jump{\\hg} \\cap \\jump{\\hfinv}=\\emptyset.$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nIf $\\hf$ is continuous then $\\jump{\\hf}=\\emptyset$ and,\nby Lemma \\ref{lem:notequal},\n$\\ff \\veq\\hf \\circ \\gSa$ then implies\n$\\ff = \\hf \\circ \\gSa$\nThus, case A is clear.\n\nIf $\\smthker(x)>0$ for all $x$ then $\\gS(x)$ is increasing (strictly) on $\\reals$\nand takes values in $(0,1).$\nHence $\\gSinv(v-)=\\gSinv(v+)$ for all $v\\in (0,1)$ and we see by\nLemma \\ref{lem:notequal} that\n$\\ff \\veq\\hf \\circ \\gSa$ then implies\n$\\ff = \\hf \\circ \\gSa.$\nThis shows case B.\n\nAssume $\\smthker$ is regular\nand  $\\ashift=0.$ \nIf $\\gSx(x_1)  = 0$ then\nLemma \\ref{lem:transitionPhiBounds}  yields\n$\\altPhiSI(\\smthker,\\ff,\\fg;x_1,x_1) = 0$\nwhich implies\n$\\altPhi(\\hf,\\hg;\\ff(x_1),\\fg(x_1))=0.$\nBy Lemma \\ref{lem:monotonic} this violates the strictly positive gap condition if $\\gS(x_1) \\in (0,1)$\nso the condition implies that $\\gS$ is strictly increasing on $\\{x: 0 < \\gS(x) < 1 \\}.$\nSince $\\hf$ is continuous at $0$ and $1$ by Lemma\n\\ref{lem:zocontinuity}, part C now follows from Lemma \\ref{lem:notequal}.\n\nTo show part D \nassume $\\smthker$ is regular and that\n$(\\hf,\\hg)$ satisfies the strictly positive gap condition.\nAssume $\\ff \\not\\equiv \\hf \\circ\\gSa.$\nWe have $\\ff \\veq \\hf \\circ\\gSa$ so\nwe apply Lemma \\ref{lem:notequal} to obtain $I-\\ashift \\in\\flats{\\gSa}$ (so $I \\in \\flats{\\gS}$)\nsuch that \n$\\gSa(I-\\ashift) \\in \\jump{\\hf}$\nand such that $\\ff(x) \\neq \\hf(U)$ on a set of positive measure in\n$I-\\ashift.$\nLet us denote $\\gSa(I-\\ashift) =\\gS(I)$ by $U.$\nWe claim that $\\fS$ is not constant on $\\neigh{I}{W}$ and hence we have\n$U \\in \\jump{\\hginv}.$\n\nTo prove the claim assume $\\fS$ is a constant $V$ on $\\neigh{I}{W}.$ Then $\\ff(x)=V$ on\n $\\neigh{I}{2W}$ which,\nsince $|\\ashift|<2W$ by Corollary \\ref{cor:regshiftbound},\n gives $\\ff(x)=V$ on $\\neigh{I-\\ashift}{\\delta}$\nfor some $\\delta>0.$ \nThis, however, implies $\\hf(U)=V$ and $U \\not\\in \\jump{\\hf},$ which is a contradiction.\n\nHence $\\ff \\neq \\hf \\circ\\gSa$ implies $\\jump{\\hf} \\cap \\jump{\\hginv}\\neq\\emptyset.$\nSimilarly,\n$\\fg \\neq \\hg \\circ\\fS$ implies $\\jump{\\hg} \\cap \\jump{\\hfinv}\\neq\\emptyset.$\n\\end{IEEEproof}\n\n\n\n\n\\subsubsection{Proof of Theorem \\ref{thm:mainexist}}\n\nSince we assume that $\\smthker$ is regular \nLemma \\ref{lem:pathology} shows that Lemma \\ref{lem:weakexistence}\nimplies Theorem \\ref{thm:mainexist} except in the case\n$\\jump{\\hf}\\cap\\jump{\\hginv} \\neq \\emptyset$ or\n$\\jump{\\hg}\\cap\\jump{\\hfinv} \\neq \\emptyset.$\nIt turns out that this case can be handled  by constructing $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$\nwith certain properties.  The argument is lengthy and is relegated to appendix  \\ref{app:B}.\n\n\n\n\n\n\n\\appendices\n\n\\section{Continuum Spatial Fixed Point Integration}\\label{app:A}\n\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:twofint}]\n\nWe assume that the smoothing kernel $\\smthker$ has finite total variation hence \n$\\|\\smthker\\|_\\infty < \\infty.$ \nFor any $\\ff \\in \\sptfns$ a simple calculation shows that $\\fS(x)-\\fS(y) \\le \\|\\smthker\\|_\\infty |x-y|.$\nThis implies that  and $\\fS$ is Lipschitz continuous with Lipschitz constant $\\|\\smthker\\|_\\infty.$ \n\nThus, $\\gSx$ and $\\fSx,$ the derivatives of $\\gS$ and $\\fS,$ exist for almost all $x$ and $\\gS$ and $\\fS$ are absolutely continuous.\nIf $\\smthker(x-y)$ and $g(y)$ do not have in common any points of discontinuity in $y$,\nthen $\\gSx(x) = \\int_{-\\infty}^\\infty \\smthker(x-y)\\,dg (y)$ where the right hand side is\na Lebesgue-Stieltjes integral.  The integral is well defined as long as $\\smthker(x-y)$ and $g(y)$ do not have\nany discontinuity points (in $y$) in common. The set of $x$ at which this can occur is countable.\nMore generally in the sequal we will have integrals in the form \n$\\int_{(a,b]} g(x) df(x).$\nThe integral is well defined as long as $g(x)$ and $f(y)$ do not have\nany discontinuity points in common.\nWe define the integral so that \n$\\int_{(a,b]} df(x) = f(b+)-f(a+).$\nThis holds even if $a \\ge b.$\n\nWe now have\n\\begin{align*}\n& \\int_{\\gS(\\minfty)}^{\\gS(x_2)}  h_{[\\ff,\\gS]}(u) du \\; \n\\\\ & = \n\\; \\int_{-\\infty}^{x_2} f(x) \\gSx(x) dx\n\\\\&  =\n\\int_{-\\infty}^{x_2} f(x) \n\\Bigl(\n\\int_{-\\infty}^\\infty \\smthker(x-y)\\,\ndg (y)\n\\Bigr)\ndx\\,.\n\\end{align*}\nSince, $\\int_{-\\infty}^{x_2} f(x)\\smthker(x-y)dx \\le \\fS(y)$ we see that the Fubini theorem can\nbe applied \nand we obtain\n\\begin{align*}\n& \\int_{\\gS(\\minfty)}^{\\gS(x_2)}  h_{[\\ff,\\gS]}(u) du \\; \n\\\\ & = \n\\int_{-\\infty}^\\infty \n\\Bigl(\n\\int_{-\\infty}^{x_2} f(x) \n\\smthker(x-y)\\,\ndx\\,\n\\Bigr)\ndg (y)\n\\\\ & = \n\\int_{-\\infty}^\\infty \n\\Bigl(\n\\int_{-\\infty}^{x_2-y} f(x+y) \n\\smthker(x)\\,\ndx\\,\n\\Bigr)\ndg (y)\n\\\\ & =\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2-x} \nf(y+x) \ndg(y)\\,\\Bigr)\ndx\\,\n\\end{align*}\nwhere the inner integral is defined for almost all $x.$\nSimilarly,\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \\; \n\\\\& = \n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\\,\\Bigr)\ndx\\,\n\\end{align*}\nWe can now exploit the evenness of $\\smthker$ to replace $x$ with $-x$ in the above\nand $\\smthker(-x)$ with $\\smthker(x)$  to write\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\&=\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2-x} \nf(y+x) \ndg(y)\n+\\int_{-\\infty}^{x_1+x} \ng(y-x) \ndf(y)\n\\,\\Bigr)\ndx\\,\n\\\\ \n&\n=\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2+x} \nf(y-x) \ndg(y)\n+\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\n\\,\\Bigr)\ndx\\,\n\\end{align*}\nConsider the second form, in which we have the expression\n\\begin{align*}\n& \\quad \\int_{-\\infty}^{x_2+x} f(y-x) dg(y)\n+\n\\int_{-\\infty}^{x_1-x} g(y+x) df(y)  \\,.\n\\end{align*}\nWith a slight abuse of notation, we may write \n\\(\n\\int_{-\\infty}^{x_1-x} g(y+x) df(y)  \\,\n\\)\nas\n\\(\n\\int_{-\\infty}^{x_1} g(y) df(y-x)  \\,\n\\)\nand we see that for almost all $x$ we have\n\\begin{align*}\n&\\int_{-\\infty}^{x_2+x} \nf(y-x) \ndg(y)\n+\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\n\\\\&=\ng(x_1+) f(x_1-x) - g(\\minfty)f(\\minfty) +\n\\int_{(x_1,x_2+x]} f(y-x) dg(y)\\,\n\\\\&=\ng(x_1+) f(x_1-x) + \nf(x_2+)(g(x_2+x)-g(x_1+))\n\\\\\n&\\quad - f(\\minfty)g(\\minfty)  -\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,.\n\\end{align*}\nThus, we obtain\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\=&\ng(x_1+) \\fS(x_1)  \n+f(x_2+)\\gS(x_2)\n\\\\&-f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nThe same analysis applied to the first form gives\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)}h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\= &\ng(x_1+) \\fS(x_1)  \n+f(x_2+)\\gS(x_2)\n\\\\ & -f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x)) df(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nFinally, by combining the two forms and integrating from $0$ to $\\infty$ we obtain\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)}h_{[\\ff,\\gS]}(u) du \n\\\\= &\ng(x_1+) \\fS(x_1)  \n+f(x_2+)\\gS(x_2)\n\\\\&-f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{0}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x)) df(y)\\,\n\\\\ & \\qquad +\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nNow note that\n\\begin{align*}\n&\\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n  \\\\ = & \\fS(x_1)\\fg(x_1+) - \\fS(\\minfty)\\fg(\\minfty)\n\\\\ & -\\int_{\\fg(\\minfty)}^{\\fg(x_1+)} h^{-1}_{[\\fg,\\fS]}(u) du\n\\end{align*} \nand, similarly,\n\\begin{align*}\n&\\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n  \\\\ = & \\gS(x_2)\\ff(x_2+) - \\gS(\\minfty)\\ff(\\minfty)\n\\\\ & -\\int_{\\ff(\\minfty)}^{\\ff(x_2+)} h^{-1}_{[\\ff,\\gS]}(u) du\n\\end{align*} \nand use the fact that $\\fS(\\minfty)=\\ff(\\minfty)$ and\n$\\gS(\\minfty)=\\fg(\\minfty)$\nand the desired result follows.\n\\end{IEEEproof}\n\n\\subsection{Fixed Point Bounds on Potentials}\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:transitionPhiBounds}]\n\nAs a first bound use the evenness of $\\smthker$ to write\n\\begin{align*}\n\\fS(x) &= \\int_{-\\infty}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&= \\int_{-\\infty}^{L} \\smthker(y)f(x+y)\\text{d}y\n+ \\int_{L}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\le (1-e_L) f((x+L)-) + e_L\n\\\\\n&\\le f((x+L)-) + e_L (1-f((x+L)-))\n\\\\\n&\\le  f(x) + \\Delta_L f(x) + e_L (1- f(x))\n\\end{align*}\nand\n\\begin{align*}\n\\fS(x) &= \\int_{-\\infty}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\ge\n\\int_{-L}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\ge (1-e_L) f((x-L)+)\n\\\\\n&\\ge f(x) - \\Delta_L f(x) - e_L f(x)\\,.\n\\end{align*}\n\nTo get a bound on\n$\\altPhiSI(\\smthker;f,g;x_1,x_1)$ we proceed similarly\n\\begin{align*}\n&\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_1+x]} (g(x_1)-g(y-x))df(y)\n\\Bigr) dx\n\\\\\n\\le&\n\\int_{-L}^L \\smthker(x)\n\\Bigl(\n\\Delta_L g(x_1) |f(x_1+x)-f(x_1)|\n\\Bigr)\n\\text{d}x\\\\\n&+\\int_L^\\infty\\smthker(x) ( f(x_1+x)-f(x_1-x)) dx\n\\\\\n\\le&\n\\int_{-L}^L \\smthker(x)\n\\Bigl(\n \\Delta_L g(x_1) \\Delta_L f(x_1)\n\\Bigr) dx +e_L\n\\\\\n\\le&\\\n \\Delta_L g(x_1) \\Delta_L f(x_1)\n+e_L\n\\end{align*}\n\nThus, we obtain the bound\n\\begin{align*}\n\\altPhi(\\fg(x),\\ff(x))\n&\\le\n\\Delta_L f(x)\\Delta_L g(x)\n+  e_L\n\\end{align*}\nand since, by \\eqref{eqn:altPhiderivatives},\n\\begin{align*}\n|\\altPhi(\\fg(x),\\fS(x))\n-\\altPhi(\\fg(x),f(x)) |\n&\\le\n|\\fS(x)-f(x)|\n\\\\\n&\\le\n\\Delta_L f(x) + e_L\n\\end{align*}\nthe other bounds follow easily.\n\\end{IEEEproof}\n\n\\section{Discrete Spatial Fixed Point Integration}\\label{app:Aa}\n\nIn this section we show how spatial integration can be done directly in the \nspatially discrete setting.\nFirst, an unusual point of notation.\nLet $\\dv{g}_i$ be defined for all integers $i.$\nWe will occasionally write a sum as\n\\(\n\\sum_{i\\in (a,b]} \\dv{g}_i\\,.\n\\)\nIn the case where $a<b$ this would equal\n\\(\n\\sum_{i=a+1}^{b} \\dv{g}_i\\, \n\\)\nin the usual notation,  when $a=b$ the sum is $0,$ and when\n$a>b$ the sum is\n\\(\n-\\sum_{i\\in (b,a]} \\dv{g}_i= -\\sum_{i=b+1}^a \\dv{g}_i\\,.\n\\)\nNote that\n\\(\n\\sum_{i\\in (a,b]} (\\dv{g}_i-\\dv{g}_{i-1}) = \\dv{g}_b - \\dv{g}_a\n\\) \nfor all choices of $a$ and $b.$\nFurther, we can write\n\\begin{align*}\n&\\sum_{i=-\\infty}^a \\dv{g}_i + \\sum_{i=-\\infty}^b \\dv{f}_i\n\\\\ =&\n\\sum_{i=-\\infty}^a (\\dv{g}_i+\\dv{f}_i) + \\sum_{i\\in(a,b]} \\dv{f}_i\n\\end{align*}\nregardless of $a$ and $b.$\n\nThe spatially smoothed $\\dv{g}$ will be denoted $\\dv{\\gSdisc}$ and is defined by\n\\[\n\\dv{\\gSdisc}_i = \\sum_{j=-W}^W \\discsmthker_j \\dv{g}_{i-j}\n\\]\nwhere we require $\\discsmthker_j = \\discsmthker_{-j}$ and $\\discsmthker_j \\ge 0$ and $\\sum_j \\discsmthker_j =1.$\n\nFirst we note\n\\begin{align*}\n& \\sum_{i=-\\infty}^{k} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} )\n\\\\\n&\\qquad+\\sum_{i=-\\infty}^{k}( \\dv{g}_{i+j}+\\dv{g}_{i+j-1}) (\\dv{f}_{i} - \\dv{f}_{i-1} )\n\\\\ &\n= \\sum_{i=-\\infty}^{k} 2 \\dv{f}_i \\dv{g}_{i+j} - 2 \\dv{f}_{i-1} \\dv{g}_{i+j-1} \n- \\dv{f}_i \\dv{g}_{i+j-1} \n\\\\\n& \\qquad+ \\dv{f}_{i} \\dv{g}_{i+j-1} \n+ \\dv{f}_{i-1} \\dv{g}_{i+j} - \\dv{f}_{i-1} \\dv{g}_{i+j} \n\\\\ &\n= 2 (\\dv{f}_{k} \\dv{g}_{k+j}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\end{align*}\nand we obtain\n\\begin{align}\n\\begin{split}\\label{eqn:sumcutA}\n&\\sum_{i=-\\infty}^{i_2}( \\dv{f}_i + \\dv{f}_{i-1})(\\dv{g}_{i+j} - \\dv{g}_{i+j-1} )\n\\\\+& \\sum_{i=-\\infty}^{i_1}( \\dv{g}_i + \\dv{g}_{i-1}) (\\dv{f}_{i-j} - \\dv{f}_{i-j-1} )\n\\\\  \n=\\quad &\\qquad\n 2 (\\dv{f}_{i_1-j} \\dv{g}_{i_1}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+&\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{split}\n\\end{align}\n\n\nWe now can write\n\\begin{align*}\n& \\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\+ &\\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\n\\\\ =\n\\sum_{j=-W}^W \\discsmthker_j \n \\Bigl( &\n \\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1})\n\\\\+& \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1})\n\\Bigr)\n\\\\ =\n\\sum_{j=-W}^W \\discsmthker_j \n\\Bigl( &\n 2(\\dv{f}_{i_1-j} \\dv{g}_{i_1} -  \\dv{f}_{-\\infty} \\dv{g}_{-\\infty} )\n\\\\+ &\n\\sum_{i\\in(i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\Bigr)\n\\\\\n=\\qquad\n&\n2( \\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1}\n-  \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+\\sum_{j=-W}^W \\discsmthker_j &\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\\n=\\qquad\n&\n 2(\\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1}\n-  \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+&2 \\dv{f}_{i_2}(\\dv{\\gSdisc_{i_2}} - \\dv{g}_{i_1})\n\\\\ +\\sum_{j=-W}^W  \\discsmthker_j &\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}-2\\dv{f}_{i_2}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\\n=\\qquad\n&\n 2 (\\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1} \n+  \\dv{\\gSdisc_{i_2}} \\dv{f}_{i_2} - \\dv{f}_{i_2}\\dv{g}_{i_1}\n-  \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+\\sum_{j=-W}^W \\discsmthker_j&\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}-2\\dv{f}_{i_2}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{align*}\n\nAnd finally we have the result\n\\begin{align*}\n&2(\\dv{\\fSdisc_{i_1}} \\dv{\\gSdisc_{i_2}} - \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\-\n& \\Bigl(\\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\&+ \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\\Bigr)\n\\\\ =\\quad\n&\n2( \\dv{\\fSdisc_{i_1}} -\\dv{f}_{i_2}) \n( \\dv{\\gSdisc_{i_2}} -  \\dv{g}_{i_1})\n\\\\&+\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_1-j,i_2]} (2\\dv{f}_{i_2}-\\dv{f}_{i}-\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{align*}\n\nNow, if in place of \\eqref{eqn:sumcutA} we write,\n\\begin{align*}\n&\\sum_{i=-\\infty}^{i_2}( \\dv{f}_i + \\dv{f}_{i-1})(\\dv{g}_{i-j} - \\dv{g}_{i-j-1} )\n\\\\+& \\sum_{i=-\\infty}^{i_1}( \\dv{g}_i + \\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} )\n\\\\  \n=\\quad&\\qquad 2( \\dv{g}_{i_2-j} \\dv{f}_{i_2}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty}) \n\\\\+&\\sum_{i\\in (i_2-j,i_1]} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\nand subsequently proceed similarly, then we obtain\n\\begin{align*}\n&2(\\dv{\\fSdisc_{i_1}} \\dv{\\gSdisc_{i_2}} - \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\-\n& \\Bigl(\\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\&+ \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\\Bigr)\n\\\\ =\\quad\n&\n2( \\dv{\\fSdisc_{i_1}} -\\dv{f}_{i_2}) \n( \\dv{\\gSdisc_{i_2}} -  \\dv{g}_{i_1})\n\\\\&+\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_2-j,i_1]} (2\\dv{g}_{i_1}-\\dv{g}_{i}-\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\n\n\\subsection{Discrete-Continuum Relation}\n\nIn this section we obtain tighter bounds on the dependence on  $\\Delta.$\n\\begin{lemma}\\label{lem:discreteFPsum}\nFor a spatially discrete fixed point for the regular ensemble\nwe have\n\\[\n|A(\\dv{f}_\\infty,\\dv{g}_\\infty) - A(\\dv{f}_{-\\infty},\\dv{g}_{-\\infty})| \\le \n\\frac{1}{2} (\\|\\hf''\\|_\\infty+\\|\\hg''\\|_\\infty)\\|\\smthker\\|_\\infty^2{\\Delta^2}\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nAs before, we associate to the discrete spatial index $i$ the real valued point $x_i = i\\Delta.$\nWe assume that $\\smthker$ is the piecewise constant extension of\n$\\discsmthker.$  Thus, \\eqref{eqn:kerdiscretetosmth} holds trivially.\n\nAssume a spatially discrete fixed point $\\ff,\\fg.$\nLet $\\tff$ and $\\tfg$ be the piecewise extensions of $\\ff$ and $\\fg.$\nWe can now relate the discrete spatial EXIT sum to the corresponding continuum integral\nto arrive at approximate fixed point conditions for spatially discrete fixed points.\n\nThe discrete sum\n\\begin{align*}\n&\\frac{1}{2}\\sum_{i=-\\infty}^i\n(\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\& =\n\\int_{-\\infty}^{x_i}\n\\dv{\\tff}(x) \\text{d} \\dv{\\tgS}(x)\n\\\\ & =\n\\int_{-\\infty}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\end{align*}\nand, similarly,\n\\begin{align*}\n&\\frac{1}{2}\\sum_{i=-\\infty}^i\n(\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_i^\\discsmthker-\\dv{f}_{i-1}^\\discsmthker)\n\\\\& =\n\\int_{-\\infty}^{x_i}\n\\dv{\\tfg}(x) \\text{d} \\dv{\\tfS}(x)\n\\\\\n& =\n\\int_{-\\infty}^{x_i}\nh_{[\\tfg,\\tfS]}\n(\\dv{\\tfS}(x)) \\text{d} \\dv{\\tfS}(x)\n\\end{align*}\nWe want to compare\n\\(\n\\int_{x_{i-1}}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\)\nto\n\\(\n\\int_{x_{i-1}}^{x_i}\n\\hf\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\\,.\n\\)\n\nWe have\n\\begin{align*}\n&\\int_{x_{i-1}}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\\\& =\n\\frac{1}{2}(\n\\hf(\\dv{g}_i^\\discsmthker)+\n\\hf(\\dv{g}^\\discsmthker_{i-1}))(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\ & = \n\\int_0^1 (\\alpha\n\\hf(\\dv{g}_i^\\discsmthker)+\n\\bar{\\alpha} \n\\hf(\\dv{g}^\\discsmthker_{i-1})) \\text{d}\\alpha \\,(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nand\n\\begin{align*}\n&\\int_{x_{i-1}}^{x_i}\n\\hf\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\\\& =\n\\int_0^1 \n\\hf(\\alpha \\dv{g}_i^\\discsmthker +\n\\bar{\\alpha} \\dv{g}^\\discsmthker_{i-1}) \\text{d}\\alpha \\,(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nwhere $\\bar{\\alpha}$ denotes $1-\\alpha.$\n\n\nLet $\\dv{g}(\\alpha) = \\alpha \\dv{g}_i^\\discsmthker+ \\bar{\\alpha} \\dv{g}^\\discsmthker_{i-1}.$\nThen, assuming $\\hf$ is $C^2$ we have by a simple application of the remainder theorem\n\\begin{align*}\n& |\\alpha \\hf(\\dv{g}_i^\\discsmthker) + \\bar{\\alpha} \\hf(\\dv{g}_{i-1}^\\discsmthker)\n-\n\\hf (\\alpha  \\dv{g}_i^\\discsmthker + \\bar{\\alpha} \\dv{g}_{i-1}^\\discsmthker)|\n\\\\\\le &\n\\frac{C_i}{2}(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)^2\n\\end{align*}\nwhere $C_i$ is the maximum of $|\\hf''(u)|$ for $u$ in $[\\dv{g}_{i-1}^\\discsmthker,\\dv{g}_{i}^\\discsmthker].$\n\nWe now have\n\\begin{align*}\n&\\Bigl|\\int_{x_{i-1}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n\\frac{C_i}{2}(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)^3\\,.\n\\end{align*}\nSince $\\sum_i (\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker) \\le 1$\nand\n\\[\n\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker\n\\le \\Delta \\|\\smthker\\|_\\infty\n\\]\nwe obtain\n\\begin{align*}\n&\\Bigl|\\int_{0}^{1}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n\\frac{\\|\\hf''\\|_\\infty}{2} ( \\|\\smthker\\|_\\infty)^2 \\Delta^2\n\\end{align*}\n\nA similar argument applies to $\\hg$ and $h_{[\\tfg,\\tfS]},$\nand the Lemma follows.\n\\end{IEEEproof}\n\nNote that a general inequality can also be derived based on\n\\begin{align*}\n& |\\alpha \\hf(\\dv{g}_i^\\discsmthker) + \\bar{\\alpha} \\hf(\\dv{g}_{i-1}^\\discsmthker)\n-\n\\hf (\\alpha  \\dv{g}_i^\\discsmthker + \\bar{\\alpha} \\dv{g}_{i-1}^\\discsmthker)|\n\\\\\\le &\n\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nfrom which we obtain\n\\begin{align*}\n&\\Bigl|\\int_{x_{i-1}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker))\n(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\&\\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker))\n\\Delta\\|\\omega\\|_\\infty\n\\end{align*}\nand for $k\\ge 0,$\n\\begin{align}\n\\begin{split}\\label{eqn:discIntegralbnd}\n&\\Bigl|\\int_{x_{i-k}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\&\\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-k}^\\discsmthker))\n\\Delta\\|\\omega\\|_\\infty\n\\end{split}\n\\end{align}\nThis inequality proves Theorem \\ref{thm:discreteFPDelta}.\n\n\\section{Existence of Travelling Wave Solution: Final Case}\\label{app:B}\n\nIn this section we prove Theorem \\ref{thm:mainexist}  for the case where\n$\\jump{\\hf}\\cap \\jump{\\hginv} \\neq \\emptyset$ or\n$\\jump{\\hg}\\cap \\jump{\\hfinv} \\neq \\emptyset$\nand $\\ashift \\neq 0.$\nWithout loss of generality we assume $\\ashift>0.$\n\n\nGiven an interval $I$ let \n$\\upperx{I}$ denote its right end point and\nlet\n$\\lowerx{I}$ denote its left end point.\nFor two closed intervals $I,I'$ we say $I \\le I'$ \nif $\\upperx{I} \\le \\lowerx{I'}.$\nThe interval $I+x$ denotes the interval $I$ translated by $x.$ \nFor a non-empty interval $I$ and $\\epsilon>0$ by $\\neigh{I}{-\\epsilon}$ we mean\n$(\\lowerx{I}-\\epsilon,\\lowerx{I}+\\epsilon).$\n\n\\begin{lemma}\\label{lem:flatseparate}\nLet $I,I' \\in \\flats{\\fS}$ be distinct where $\\ff\\in \\sptfns$ \nand $\\smthker$ is regular.\nThen $I \\le I'$  implies $I+2W \\le I'.$\n\\end{lemma}\n\\begin{IEEEproof}\nSince $I$ and $I'$ are both maximal we have $\\fS(I) < \\fS(I').$\nSince $\\smthker$ is regular, we have $\\ff(x)=\\fS(I)$ for $x\\in\\neigh{I}{W}$ \nand $\\ff(x)=\\fS(I')$ for $x\\in\\neigh{I'}{W}.$\nIt follows that $\\neigh{I}{W}$ and $\\neigh{I'}{W}$ are disjoint.\\end{IEEEproof}\n\nGiven regular $\\smthker$ and shift $\\ashift >0$ \nwe say $I \\in \\flats{\\fS}$ is {\\em linked to}\n$I' \\in \\flats{\\gS}$ if  $\\upperx{I}+W +\\ashift \\in I',$\nand\nwe say $I' \\in \\flats{\\gS}$ is {\\em linked to}\n$I'' \\in \\flats{\\fS}$ if  $\\upperx{I'}+W \\in I''.$\nIf we have a sequence $I_1,I_2,\\ldots$\nsuch that $I_j$ is linked to $I_{j+1}$ then we call this a {\\em chain}.\nNote that all intervals in a chain in either $\\flats{\\fS}$\nor $\\flats{\\gS}$ must be distinct.\nThe chain {\\em terminates} if the last element in the chain is not linked to \nanother interval.\n\n\n\n\\begin{lemma}\\label{lem:linkterminate}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg)>0.$\nLet $(f,g)\\in\\sptfns^2$ be $(0,1)$-interpolating and let $\\smthker$ be regular.\nAssume $h_{[\\ff,\\gSa]} \\equiv\\hf$ and $h_{[\\fg,\\fS]} \\equiv\\hg$ \n(hence $\\ashift>0$),\nthen any chain in $\\flats{\\fS},\\flats{\\gS}$ terminates.\n\\end{lemma}\n\\begin{IEEEproof}\nLet $(u^*,v^*)$ be the minimal element in $\\intcross(\\hf,\\hg)$\nas guaranteed by Lemma \\ref{lem:Sstructure}.\nThere exists  finite $y$ such that $g(y) \\ge u^*$ and \n$f(y) \\ge v^*.$\nBy Lemma \\ref{lem:phiflat} if $z \\in I \\in \\{\\flats{\\gS} \\cup \\flats{\\fS}\\}$\nthen $\\altPhi(\\hf,\\hg;g(z),f(z)) \\in [0,A(\\hf,\\hg)]$ and we therefore have\n$(g(z),f(z)) < (u^*,v^*)$ componentwise by Lemma \\ref{lem:Sstructure}.\nThus, we obtain $z<y.$\nIt now follows from Lemma \\ref{lem:flatseparate} that any chain of linked intervals terminates.\n\\end{IEEEproof}\n\n\n\\begin{lemma}\\label{lem:regularapprox}\nGiven $(\\hf,\\hg)$ that satisfy the strictly positive gap condition there exists\n$(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ such that \n$(\\hf^i,\\hg^i)$ satisfies the strictly positive gap condition for each $i$\nand all of the following properties hold\n\\begin{itemize}\n\\item[A.] \n\\(\n\\jump{\\hfiinv} \\subset \\jump{\\hfinv}\\,,\n\\)\n\\(\n\\jump{\\hgiinv} \\subset \\jump{\\hginv}\\,.\n\\)\n\\item[B.] \n\\(\n\\jump{\\hf^i} \\cap \\jump{\\hginv} =\\emptyset\\,,\n\\)\n\\(\n\\jump{\\hg^i} \\cap \\jump{\\hfinv} =\\emptyset\\,.\n\\)\n\\item[C.]\nIf $u \\in   \\jump{\\hginv}\\cap\\jump{\\hf}$ then\n$\\hf^i(u)=\\hf(u)$ for all $i$ large enough.\n\nIf $v \\in   \\jump{\\hfinv}\\cap\\jump{\\hg}$ then\n$\\hg^i(v)=\\hg(v)$ for all $i$ large enough.\n\\item[D.]\nIf $v \\in \\jump{\\hfinv}$ then \n\\[\n[\\hfiinv(v-),\\hfiinv(v+)] \\rightarrow [\\hfinv(v-),\\hfinv(v+)].\n\\]\nIf $u \\in \\jump{\\hginv}$ then \n\\[\n[\\hgiinv(u-),\\hgiinv(u+)] \\rightarrow [\\hginv(v-),\\hginv(u+)].\n\\]\n\\item[E.]\nIf $v  \\in \\jump{\\hfinv}$ then, setting $u=\\hfinv(v+),$ we have\n$\\hf^i(u)=\\hf(u)$ for all $i$ large enough. \n\nIf $u  \\in \\jump{\\hginv}$ then, setting $v=\\hginv(u+),$ we have\n$\\hg^i(v)=\\hg(v)$ for all $i$ large enough. \n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:regularapprox}]\nNote that if\n\\(\n\\jump{\\hf} \\cap \\jump{\\hginv} =\\emptyset\\,\n\\)\nand\n\\(\n\\jump{\\hg} \\cap \\jump{\\hfinv} =\\emptyset\\,\n\\)\nthen we can simply set $(\\hf^i,\\hg^i) =(\\hf,\\hg).$\nThus, the lemma targets the case where this does not hold.\n\nWe will describe the construction of $\\hf^i,$ the construction of $\\hg^i$ is analogous.\n\nConsider the countable\\footnote{The case where $\\jump{\\hf} \\cap \\jump{\\hginv}$\nis finite can be handled similarly to the countably infinite case. To avoid notational overhead\nwe present the argument only for the infinite case.}  set \n\\[\n\\jump{\\hf} \\cap \\jump{\\hginv} =\\{ u_1,u_2,\\ldots \\}\n\\]\nand also the set\n\\[\n\\{  \\hfinv(v-),\\hfinv(v+) : v \\in \\jump{\\hfinv} \\}\n\\backslash (\\jump{\\hf} \\cap \\jump{\\hginv})\n= \n\\{ t_1,t_2,\\ldots \\}\\,.\n\\]\nFor each $k=1,2,\\ldots$ set\n\\[\nd_k = \\min \\{ u_k,1-u_k;|u_k-u_j|,|u_k-t_j| : j < k \\}\n\\]\nand note that $d_k > 0$ (we cannot have $u_k=0$ or $u_k=1$ since $\\hf$ is continuous at $0$ and $1$ by Lemma \\ref{lem:zocontinuity}. \nFor each $i=1,2,\\ldots$ we define sequences $\\eta_{i,k},$\n$k=1,2,\\ldots$ such that \n\\[\n0 < \\eta_{i,k} < \\frac{1}{2}\\min \\{ 3^{-ik},d_k \\}\n\\] \nand such that \n\\[\n\\{ u_k \\pm \\eta_{i,k} \\} \\cap \\jump{ \\hginv } = \\emptyset\\,.\n\\]\nNote that $2\\sum_k \\eta_{i,k} \\le \\frac{1}{2^i}.$\n\nFor  each $k$ we define $H_k = \\unitstep_{r_k}$ (which is a unit step function\nexcept that we set $H_k(0)  =r_k$)\nwhere  $r_k= \\frac{\\hf(u_k)-\\hf(u_k-)}{\\hf(u_k+)-\\hf(u_k-)}.$\nThis function represents the jump in $\\hf$ at $u_k.$\nWe will substitute for this a function continuous at $0$:\n\\[\nS_{i,k}(x) = \\begin{cases}\n0 & x < 1- \\eta_{i,k}\\\\\n0   \\vee  (x-r_k) \\wedge 1 & |x| \\le \\eta_{i,k} \\\\\n1 & x > 1+ \\eta_{i,k}\n\\end{cases}\n\\]\nwhere $0   \\vee  z \\wedge 1 = \\min\\{ \\max \\{ 0,z\\} ,1\\}.$\nDefine\n\\[\n\\hf^i(x) = \\hf(x) - \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) (H_k(x) - S_{i,k}(x))\\,.\n\\]\nNote that  $\\sum_k ( \\hf(u_k+)-\\hf(u_k-)) \\le 1$ and $|H_k(x) - S_{i,k}(x)|\\le 1$\nso the sum is well defined.\nThe function $\\hf^i(x)$ can be expressed as the sum of two functions,\n\\[\nh_1(x)= \\hf(x) - \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) H_k(x) \\,\n\\]\nand\n\\[\nh_{2,i}(x)= \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) S_{i,k}(x)\\,,\n\\]\nboth of which are in $\\exitfns,$ i.e., both of which are non-decreasing.\nThe function $h_1$ is continuous for all $u \\in \\jump{\\hginv} \\cap \\jump{\\hf}$\nsince $H_k(u+)-H_k(u-)=1$ if $u=u_k$ and $H_k(u+)-H_k(u-)=0$ if $u \\neq u_k.$\nIf $u \\in \\jump{\\hginv} \\backslash \\jump{\\hf}$ then $\\hf$ is continuous at $u$ and\ntherefore $h_1$ is continuous at $u.$\nIf follows that $\\hf^i \\in \\exitfns$ and $\\hf^i \\xrightarrow{i\\rightarrow\\infty} \\hf.$\nWe assume a similar definition of $\\hg^i.$\n\nWe will now show that properties A through E hold for this sequence.\nEach property has two essentially equivalent forms (through the symmetry of substitution of $f$ and $g$). In each case we will show the first form.\n\nConsider part A. Let $v \\in \\jump{\\hfiinv}.$  There is non-empty interval\n$I =(x_1,x_2)$ such that $\\hf^i$ is evaluates to $v$ on $I.$\nSince both $h_1$ and $h_{2,i}$ are non-decreasing it follows that both are constant\non $I.$  From the fact that $h_{2,i}$ is constant on $I$ we easily obtain that\n$\\sum_k ( \\hf(u_k+)-\\hf(u_k-)) H_{k}(x)\\,$ is also constant on $I$ and \nwe deduce that $\\hf$ is constant on $I.$\nHence $v \\in \\jump{\\hfinv}$\nand part A is proved.\n\nConsider part B. We have  $S_{i,k}(u+) - S_{i,k}(u-) =0$ unless $u = u_j \\pm \\eta_{j,k}$ for some $j.$\nBy construction, $u_j \\pm \\eta_{j,k} \\not\\in \\jump{\\hginv}.$\nHence, $h_{2,i}(u)$ is continuous at all $u \\in \\jump{\\hginv}.$\nSince $h_1$ is continuous, $\\hf^i$ is continuous at all $u\\in \\jump{\\hginv}$ and part B is proved.\n\nConsider part C. Let $u \\in\\jump{\\hginv}\\cap \\jump{\\hf}\\,,$ i.e., $u=u_j$ for some $j.$\nWe prove part C  by showing that\n$H_k(u_j) - S_{i,k}(u_j) =0$ for all $k$ for all $i$ large enough.\nFor $k=j$ we have $H_k(u_j) - S_{i,k}(u_j) =0$ by definition.\nFor $k<j$ we have $H_k(u_j) - S_{i,k}(u_j) =0$ for all $i$ large enough so that\n\\(\n\\frac{2}{3^i}<\\min_{k < j} \\{ |u_k-u_j| \\}.\n\\)\nFor $j<k$ we have $H_k(u_j) - S_{i,k}(u_j) =0$ by construction, i.e.,\nby the requirement that $\\eta_{i,k}  < d_k$ which implies $|u_j -  u_k| > \\eta_{i,k}.$\nThis proves part C.\n\nConsider part D.\nAssume $v \\in \\jump{\\hfinv},$ then  $\\hfinv(v-)<\\hfinv(v+).$\nSince $(\\hfinv(v-),\\hfinv(v+)) \\cap \\jump{\\hf} = \\emptyset$ and $\\eta_{i,k} < 2^{-i}$ it follows\nthat for $u \\in (\\hfinv(v-)+2^{-i},\\hfinv(v+)-2^{-i})$ we have\n$\\hf^i(u) = \\hf(u) = v$ and for \n$u \\not\\in (\\hfinv(v-)-2^{-i},\\hfinv(v+)+2^{-i})$\nwe have\n$\\hf^i(u) \\neq v.$\nPart D now follows.\n\nConsider part E.\nLet $v  \\in \\jump{\\hfinv}$ and set $u=\\hfinv(v+).$\nIf $u \\in \\jump{\\hginv}\\cap \\jump{\\hf}$ then $u=u_k$ for some $k$ and\nproperty C implies property E.\nOtherwise, we have $u = t_k$ for some $k.$\nFor $j\\ge k$ we have $  |u_j-t_k| > \\eta_{i,k}$ for all $i.$\nFir $j <  k$ we have  $\\frac{2}{3^i} < \\min_{j<k} \\{|u_j-t_k|\\}$\nfor all $i$ sufficiently large.\nHence property E holds.\n\nNow we address the satisfaction of the strictly positive gap condition.\nIn general, we may need to further modify the constructed sequence and take a subsequence.\n\nDefine \n\\begin{align*}\n \\hf^\\delta = \\hf \\vee \\unitstep_1(x - (1-\\delta)) \\\\\n \\hg^\\delta = \\hg \\vee \\unitstep_1(x - (1-\\delta))\\,.\n\\end{align*}\nWe claim that for all $\\delta$ sufficiently small\n\\(\n(\\hf^\\delta,\\hg^\\delta)\n\\)\nsatisfies the strictly positive gap condition.\nFirst, we choose $\\delta$ small enough so that there exists a non-trivial crossing point and\n$A(\\hf^\\delta,\\hg^\\delta) >0.$\nThen\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;u,v)-\n\\altPhi(\\hf^\\delta,\\hg^\\delta;u,v)\\\\\n=&\n\\int_0^u (\\hginv(x)-{(\\hg^\\delta)}^{-1}(x)) dx\n+\n\\int_0^v (\\hfinv(x)-{(\\hf^\\delta)}^{-1}(x)) dx\n\\end{align*}\nwhich is non-negative and non-decreasing in $u$ and $v.$\nHence for $(u,v) \\in \\intcross(\\hf^\\delta,\\hg^\\delta)\\subset \\intcross(\\hf,\\hg)$ we have\n\\begin{align*}\n&\\altPhi(\\hf^\\delta,\\hg^\\delta;u,v) - A(\\hf^\\delta,\\hg^\\delta)\\\\\n\\ge&\n\\altPhi(\\hf,\\hg;u,v) - A(\\hf,\\hg)\\\\\n>&0\n\\end{align*}\nwhich establishes the claim. \nNow for each $\\delta>0$ we can choose $\\eta$ sufficiently small so that\n\\begin{align*}\n\\unitstep_0(x-\\eta) &\\wedge \\hf^\\delta \\\\\n\\unitstep_0(x-\\eta) &\\wedge \\hg^\\delta\n\\end{align*}\nhas a non-trivial crossing point, and it follows easily that the pair\nsatisfies the strictly positive gap condition with $A >0.$\n(The values of $\\altPhi$ at crossing points and $A$ increase by identical amounts.)\n\nLet us define $\\delta_j \\rightarrow 0$  and $\\eta_j \\rightarrow 0$ with \n$1-\\delta_j, \\eta_j \\not\\in \\jump{\\hginv} \\cup \\jump{\\hfinv}$ so that, for each $j,$\n\\begin{align*}\n\\hf^j  =  \\unitstep_0(x-\\eta_j) \\wedge \\hf(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\\\\n\\hg^j = \\unitstep_0(x-\\eta_j) \\wedge \\hg(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \n\\end{align*}\nsatisfies the strictly positive gap condition with $A >0.$  Now, for each $i$ we define the sequence\n\\begin{align*}\n\\hf^{i,j}(x) &=\n\\unitstep_0(x-\\eta_j) \\wedge \\hf^i(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\\\\n\\hg^{i,j}(x) &=\n\\unitstep_0(x-\\eta_j) \\wedge \\hg^i(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\,.\n\\end{align*}\nThen we have\n\\begin{align*}\n\\hf^{i,j} &\\xrightarrow{\\i \\rightarrow \\infty} \\hf^j  \\\\\n\\hg^{i,j} &\\xrightarrow{\\i \\rightarrow \\infty} \\hg^j \\,.\n\\end{align*}\nClearly $\\intcross(\\hf^j,\\hg^j) = \\intcross(\\hf,\\hg) \\cap [\\eta_j,1-\\delta_j]^2,$\nand it follows that, for each $j,$ $(\\hf^{i,j},\\hg^{i,j})$ satisfies the strictly positive gap condition\nwith $A(\\hf^{i,j},\\hg^{i,j}) >0$ for all $i$ large enough.\nProperties A and B still hold for all $i$ and $j.$\n\nFor each $j$ we can find $i(j)$  such that $(\\hf^{i,j},\\hg^{i,j})$\nsatisfies the strictly positive gap condition for all $i \\ge i(j).$\nWe can assume $i(j)$ is increasing in $j.$\nConsider the diagonal sequence \n$(\\hf^{i(j),j},\\hg^{i(j),j})\\, j=1,2,\\ldots.$\nLet us re-index this as \n$(\\hf^{i},\\hg^{i})\\, i=1,2,\\ldots$ with corresponding $\\delta_i,\\eta_i.$\nWe now show that properties C,D, and E continue to hold.\n\nProperty C holds since, by Lemma \\ref{lem:zocontinuity}, $u \\in \\jump{\\hf}$ implies $u \\in (0,1)$\nand $v \\in \\jump{\\hg}$ implies $v \\in (0,1).$\nNow we show property D.\nAssume $v \\in \\jump{\\hfinv}.$ If $v \\in (0,1)$ then\n$ [\\hfinv(v-),\\hfinv(v+)] \\subset (0,1)$ by Lemma \\ref{lem:zocontinuity} and property D clearly holds.\nIf $v=0$ then $\\hfinv(v-)=\\hfiinv(v-)=0$ and \n$\\hfinv(v+)  < 1$ by Lemma \\ref{lem:zocontinuity}.\nSince  $1-\\delta_i \\rightarrow 1$ we  have $\\hfiinv(v+) \\rightarrow \\hfinv(v+).$ \nSimilarly, if $v=1$ then $\\hfinv(v+)=\\hfiinv(v+)=1$ and \n$\\hfinv(v-)  >0$ and we  have $\\hfiinv(v-) \\rightarrow \\hfinv(v-).$ \nThus, property D holds generally.\n\nFinally we consider property E.\nLet $v  \\in \\jump{\\hfinv}$ and set $u=\\hfinv(v+).$\nThen $u>0$  and if $u<1$ then we clearly have\n$\\hf^i(u)=\\hf(u)$ for all $i$ large enough.\nIf $u=1$ then $\\hf(u)=1$\nand $\\hf^i(u)=1$ for all $i.$\nThus, property E holds.\n\\end{IEEEproof}\n\nFor $\\ff \\in \\sptfns$ we say that $\\ff$ is increasing to the right of $x$ if\n$z > x \\Rightarrow \\ff(z) > \\ff(x)$\nand\nwe say that $\\ff$ is increasing to the left of $x$ if\n$z < x \\Rightarrow \\ff(z) < \\ff(x).$\n\n\n\\begin{lemma}\\label{lem:rightincrease}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition with $A(\\hf,\\hg)>0$ and let $\\smthker$ be regular.\nLet $(\\ff,\\fg)$ be $(0,1)$-interpolating such that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:weakexistence}.\nIf   $I \\in \\flats{\\fS},$ \nthen $\\gSa$ is increasing to the right of $\\lowerx{I}-W.$\nIf  $I' \\in \\flats{\\gS},$ \nthen $\\fS$ is increasing to the right of $\\lowerx{I'}-W.$\n\\end{lemma}\n\\begin{IEEEproof}\nAssume $I \\in \\flats{\\fS}.$ \nAssume $\\gSa$ is not increasing to the right of $\\lowerx{I}-W.$\nThen $\\lowerx{I}-W \\in \\hat{I}-\\ashift$ for some $\\hat{I} \\in \\flats{\\gS}$ with \n\\[\n\\lowerx{\\hat{I}-\\ashift} \\le \\lowerx{I}-W < \\upperx{\\hat{I}-\\ashift}.\n\\]\nIt follows that there exists $x \\in \\neigh{{I}}{W} \\cap (\\hat{I}-\\ashift)$ and,\nsince $\\smthker$ is regular, we have $\\ff(x) = \\fS({I}).$ \nHence we obtain $\\fS({I}) \\veq \\hf(\\gSa(\\hat{I}-\\ashift))$\nwhich is equivalent to $\\fS({I}) \\veq \\hf(\\gS(\\hat{I})).$\n\nSince $\\ashift<2W$ by Corollary \\ref{cor:regshiftbound},\nwe have\n\\[\n\\lowerx{\\hat{I}}-W \\le\n\\lowerx{I}+\\ashift-2W <\n\\lowerx{I} <\n \\upperx{\\hat{I}}+W\\,.\n\\]\nIt follows that there exists $x \\in \\neigh{\\hat{I}}{W} \\cap {I}$ and,\nsince  $\\fg(x) = \\gS(\\hat{I}),$ \nwe obtain $\\gS(\\hat{I}) \\veq \\hg(\\fS({I})).$\n\nWe now have $(\\gS(\\hat{I'}),\\fS(I')) \\in \\cross( \\hf,\\hg),$ but since\n$\\altPhi(\\hf,\\hg;\\gSa(I),\\fS(I')) \\in [0,A]$ by Lemma \\ref{lem:phiflat},  this contradicts the strictly positive gap condition.\n\nThe argument for the second case is similar. \n\nAssume $I' \\in \\flats{\\gS}$ which is equivalent to\n$I'-\\ashift \\in \\flats{\\gSa}.$\nAssume $\\fS$ is not increasing to the right of $\\lowerx{I'}-W.$\nThen $\\lowerx{I'}-W \\in \\hat{I'}$ for some $\\hat{I'} \\in \\flats{\\fS}$ with \n\\[\n\\lowerx{\\hat{I'}} \\le \\lowerx{I'}-W < \\upperx{\\hat{I'}}.\n\\]\nIt follows that there exists $x \\in \\neigh{{I'}}{W} \\cap \\hat{I'}$ and,\nsince $\\smthker$ is regular, we have $\\fg(x) = \\gS({I'}).$ \nHence we obtain $\\gS({I'}) \\veq \\hg(\\fS(\\hat{I'})).$\n\nSince $\\ashift<2W$ by Corollary \\ref{cor:regshiftbound},\nwe have\n\\[\n\\lowerx{\\hat{I'}}-W \\le\n\\lowerx{I'}-2W <\n\\lowerx{I'}-\\ashift <\n\\lowerx{I'}< \\upperx{\\hat{I'}}+W\\,.\n\\]\nIt follows that there exists $x \\in \\neigh{\\hat{I'}}{W} \\cap {(I'-\\ashift)}$ and,\nsince  $\\ff(x) = \\fS(\\hat{I'}),$ \nwe obtain $\\fS(\\hat{I'}) \\veq \\hf(\\gSa({I'}-\\ashift))$\nwhich is equivalent to $\\fS(\\hat{I'}) \\veq \\hf(\\gS({I'})).$\n\nThe rest of the argument is as before.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:convprop}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg) > 0$ and let $\\smthker$ be regular.\nAssume we have $\\ff,\\fg \\in \\sptfns$ such that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$\nand sequences\n$\\ff_i \\rightarrow \\ff$ and $\\fg_i\\rightarrow \\fg$ and $\\ashift_i \\rightarrow \\ashift$ \nwhere\n$\\ff_i = \\hf^i \\circ \\gSai_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i,$\nand\n$(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ is given as in Lemma \\ref{lem:regularapprox}.\n\nThen $\\ashift>0$ and \nthe following properties hold for any $I\\in\\flats{\\gS}.$\n\\begin{itemize}\n\\item[A.]\nIf $I$ is not linked to an $I'\\in\\flats{\\fS}$\nthen  for any $\\epsilon >0$ we have \n$\\gS_i(x) =\\gS(I)$ for $x\\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough.\n\\item[B.]\nIf $I$ is linked to $I' \\in \\flats{\\fS}$ and\nfor any $\\delta>0$ we have\n$\\fS_i$ is a fixed constant, denoted $F,$ on $\\neigh{I'}{-\\delta}$ for all $i$ large enough,\nthen,\nfor any $\\epsilon >0$ we have \n$\\gS_i(x) =\\gS(I)$ for $x\\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough.\n\\item[C.]\nAssume that for any $\\epsilon>0$ we have\n$\\gSai_i(x) = \\gS(I)$ for $x \\in \\neigh{I-\\ashift}{-\\epsilon}$ for all $i$ large enough.\nThen we have $\\ff(x) = \\hf(\\gSa(I-\\ashift))$ for all $x \\in (\\lowerx{I-\\ashift},\\upperx{I-\\ashift}).$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nConsider part A.\nLet $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ be given as in Lemma \\ref{lem:regularapprox}.\nBy Lemma \\ref{lem:weakexistence} and Lemma \\ref{lem:pathology}\nthere exists $\\ff_i,\\fg_i$  such that\n$\\ff_i = \\hf^i \\circ \\gS_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i.$\nLet $\\ff$ and $\\fg$  be  limits so that\n$\\ff \\veq \\hf \\circ \\gS$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:limitexist}.\n\nLet $I \\in \\flats{\\gS}.$\nWe have $\\fg(x)=\\gS(I)$ for all $x \\in \\neigh{I}{W}.$\nSince $\\fS$ is increasing to the right of $\\lowerx{I}-W$ by Lemma \\ref{lem:rightincrease}\nand increasing to the left of $\\upperx{I}+W$ by assumption (no linked interval),\nwe see by monotonicity of $\\hg$ that\n\\(\n\\hg(v) = \\gS(I)\n\\)\nfor all $v  \\in (\\fS(\\lowerx{I}-W),\\fS(\\upperx{I}+W))$\nand $\\gS(I) \\in \\jump{\\hginv}.$\nGiven any $\\epsilon>0$ property D of Lemma \\ref{lem:regularapprox} now implies that\n\\(\n(\\fS(\\lowerx{I}-W+\\epsilon),\\fS(\\upperx{I}+W)-\\epsilon)\n\\subset\n[\\hgiinv(\\gS(I)-),\\hgiinv(\\gS(I)+)]\n\\)\nfor all $i$ large enough.\nWe conclude from this that $\\fg_i(x)=\\gS(I)$ for $x \\in \\neigh{I}{W-\\epsilon}$\nfor all $i$ large enough. \nThis implies that $\\gS_i(x) = \\gS(I)$ \nfor $x \\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough, proving part A.\n\nConsider part B.\nLet $I$ be linked to $I' \\in \\flats{\\fS}.$\nWe have $\\gS(I) \\in \\jump{\\hginv}$  since\n $\\fS$ is increasing to the right of $\\lowerx{I}-W.$\nAs in the proof of part A we have\n\\(\n\\hg(v) = \\gS(I)\n\\)\nfor all $v  \\in (\\fS(\\lowerx{I}-W),F)$\n\nWe have $F = \\hginv(\\gS(I)+)$ since $I$ is maximal.\nWe now apply property E of Lemma \\ref{lem:regularapprox} to conclude that $\\hg^i(F)  = \\gS(I)$\nfor all $i$ large enough.\nGiven $\\epsilon>0$  we combine this\n with property D of Lemma \\ref{lem:regularapprox} to obtain\n\\(\n(\\fS(x_1-W+\\epsilon),F]\n\\subset\n[\\hgiinv(\\gS(I)-),\\hgiinv(\\gS(I)+)]\n\\)\nfor all $i$ large enough.\nLet $\\delta = \\epsilon,$ then for all $i$ large enough\nwe also have $\\fS_i(x) = F$ for $x \\in\\neigh{I'}{-\\epsilon}.$\nWe conclude that $\\fg_i(x)=\\gS(I)$ for $x \\in \\neigh{I}{W-\\epsilon}$\nfor all $i$ large enough. \nThis implies that $\\gS_i(x) = \\gS(I)$ \nfor $x \\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough, proving part B.\n\nConsider part C. \nIf $\\hf$ is continuous at  $\\gS(I)$ then we must have\n$f(x)=\\hf(\\gS(I))$ on $I-\\ashift.$\nAssume now that $\\gS(I)  \\in \\jump{\\hf}.$\nWe have $\\gS(I)   \\in \\jump{\\hginv}$ by \nLemma \\ref{lem:rightincrease} since $\\hf,\\hg$ satisfies the strictly\npositive gap condition.\nProperty $C$ of Lemma \\ref{lem:regularapprox} now gives $\\hf^i(\\gS(I))=\\hf(\\gS(I))$ \nfor all $i$ large enough.\nThis implies that for any $\\epsilon>0$ we now have\n$\\ff_i(x)=\\hf(\\gS(I))$  for all $x \\in \\neigh{I-\\ashift}{-\\epsilon}$\nfor all $i$ large enough.\nSince $\\ff_i \\rightarrow \\ff$ this proves part C.\n\\end{IEEEproof}\n\nLemma \\ref{lem:convprop} essentially completes the proof of Theorem \\ref{thm:mainexist} and we state the main\nresult as the following.\n\\begin{corollary}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg) > 0$ and let $\\smthker$ be regular.\nThere exists $(0,1)$-interpolating $\\ff,\\fg$ such that \n$\\ff = \\hf \\circ \\gS$ and \n$\\fg = \\hg \\circ \\fSa.$\n\\end{corollary}\n\\begin{IEEEproof}\nLet $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ be given as in Lemma \\ref{lem:regularapprox}.\nBy Lemma \\ref{lem:weakexistence} and Lemma \\ref{lem:pathology}\nthere exists $\\ff_i,\\fg_i$  such that\n$\\ff_i = \\hf^i \\circ \\gS_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i.$\nLet $\\ff$ and $\\fg$  be  limits so that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:limitexist}.\n\nLemma \\ref{lem:linkterminate} states that any element in $I\\in \\flats{\\gS}$ must be part of a terminating\nchain.\nParts A and B of Lemma \\ref{lem:convprop} show that for any $\\epsilon >  0$  we have\n$\\gS_i(x)=\\gS(I)$ for all $x\\in \\neigh{I}{-\\epsilon}$ for all $i$ large enough.\nPart C of  Lemma \\ref{lem:convprop} then shows that $\\ff(x) = \\hf(\\gSa(x))$ for all\n$x$ in the interior of $I.$\nLemma  \\ref{lem:notequal} states that if  $\\ff \\neq \\hf \\circ \\gSa$ then there exists\n$I\\in\\flats{\\gS}$ such that $\\ff(x) \\neq \\hf(\\gSa(x))$ on a subset of positive measure in $I-\\ashift.$\nHence, $\\ff \\equiv \\hf \\circ \\gSa.$\nA similar argument shows that  $\\fg \\equiv \\hg \\circ \\fS.$\nWe can obtain equality by modifying $\\ff$ and $\\fg$ on a set of measure $0.$\n\\end{IEEEproof}\n\\section{Two Sided Termination with Positive Gap}\\label{app:C}\n\n\n\nIn this section we prove Theorems\n\\ref{thm:twoterminatedexist} and\n\\ref{thm:discretetwoterminatedexist}.\nThe two results have much in common and we begin with some\nconstructions that apply to both.\n\nWe assume that $\\smthker$ is regular and that\n$(\\hf,\\hg)$ satisfies the strictly positive gap condition with $A(\\hf,\\hg) < 0.$\n\n\n\n\n\nConsider first the parametric modification of $(\\hf,\\hg)$ given by\n\\begin{align}\n\\begin{split}\n\\hf(\\eta;u) &=(\\hf(u)-\\eta)^+\\,\\\\ \\label{eqn:FirstMod}\n\\hg(\\eta;v) &= (\\hg(v)-\\eta)^+ \\,.\n\\end{split}\n\\end{align}\n\nThe minimum of $\\altPhi(\\hf(\\eta;u),\\hg(\\eta,\\cdot);u,v)$ for $(u,v)\\in [0,1]\\times[0,1]$\nis achieved in $\\cross(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot))\\backslash (1,1).$ \nLet $(u(\\eta),v(\\eta))$ denote the minimum point where this minimum is achieved.\nThen $(u(\\eta),v(\\eta)) \\in \\cross(\\hf(\\eta;u),\\hg(\\eta;\\cdot))$\nand we have\n$\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) \\rightarrow A(\\hf,\\hg)$ as $\\eta\\rightarrow 0.$\nIt follow that $(u(\\eta),v(\\eta)) \\rightarrow (1,1)$ as $\\eta \\rightarrow 0.$\nHence, given arbitrary $\\epsilon > 0$  we have for all $\\eta$ small enough that\n$u(\\eta),v(\\eta) > 1-\\epsilon$ and $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) < 0.$\nIf $\\hf(\\eta;u(\\eta)) > v(\\eta)$ then redefine $\\hf(\\eta;u(\\eta)) = v(\\eta)$\nand\nif $\\hg(\\eta; v(\\eta)) > u(\\eta)$ then redefine $\\hg(\\eta;v(\\eta)) = u(\\eta).$\n\nWe can now apply Theorem \\ref{thm:mainexist}\nto $(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot))$ on $[0,u(\\eta)]\\times[0,v(\\eta)]$ to obtain\n ${\\tmplF},{\\tmplG} \\in \\Psi_{[-\\infty,\\infty]}$ \ninterpolating over $[0,u(\\eta)]\\times[0,v(\\eta)]$\nand $\\ashift < 0$ so that\nsetting $\\ff^t(x) = {\\tmplF}(x-\\ashift t)$ and\n$\\fg^t(x) = {\\tmplG}(x-\\ashift t)$ solves\n\\eqref{eqn:gfrecursion} for $(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot)).$\nSince $\\hf$ and $\\hg$ are continuous at $0$\nwe have ${\\tmplF}(x) = 0$ on some maximal interval, we may take to be $[-\\infty,0),$\nand ${\\tmplG}(x) = 0$ on some maximal interval $[-\\infty,z_g).$\n\nNote that we have\n\\begin{equation}\\label{eqn:ADFGbnd}\n\\hg(\\tmplF^{\\smthker}(x))\n\\ge\n\\tmplG(x) + \\eta \\unitstep_0 (x-x_g)\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:ADGFbnd}\n\\hf(\\tmplG^{\\smthker}(x+\\ashift))\n\\ge\n\\tmplF(x) + \\eta \\unitstep_0 (x)\\,.\n\\end{equation}\n\nApplying Lemmas \\ref{lem:shiftupperbound} with Lemma \\ref{lem:stposbound} we can\nassert the existence of a bound $S<2W$ such that $-\\ashift \\le S$ for all $\\eta$ sufficiently small.  Furthermore, \n\nWe assume $Z = Z(\\eta)$ large enough so that\n\\begin{align}\n{\\tmplF}(\\tfrac{1}{4}Z+\\ashift-\\Delta) & > v(\\eta)-\\frac{\\eta}{4},\\label{eqn:topF}\\\\\n{\\tmplG}(\\tfrac{1}{4}Z+\\ashift-\\Delta) & > u(\\eta)-\\frac{\\eta}{4} \\label{eqn:topG}\\,.\n\\end{align}\nIn the discrete case we will assume $Z = L\\Delta$ for an integer $L.$\n\nLet us define \n\\begin{equation}\\label{eqn:f0initial}\n\\ff^{0}(x) = \\tmplF(x+\\ashift) +  \\eta \\unitstep_0(x+\\ashift)\n\\end{equation}\nfor $x \\le \\half Z$ and for $x > \\half Z$\ninitialize symmetrically using $\\ff^{0}(x) = \\ff^{0}(Z-x).$\nClearly this is even about $\\half Z$ and we have\n$\\ff^0(x) \\le 1.$ \nFor $x \\in [\\tfrac{1}{4}Z,\\tfrac{1}{2}Z]$ we have\n${\\tmplF}(x+\\ashift)  > v(\\eta)-\\frac{\\eta}{4}$  by \\eqref{eqn:topF}\nand for all $x$ we have ${\\tmplF}(x) \\le v(\\eta).$  This gives for all $x$ the bound\n\\begin{equation}\\label{eqn:f0initialbound}\n\\ff^{0}(x) \\ge \\tmplF(x+\\ashift) +  \\tfrac{3}{4} \\eta \\unitstep_0(x+\\ashift)\n-\n\\unitstep_1(x-\\tfrac{3}{4}Z)\n\\end{equation}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:twoterminatedexist}]\n\nWe assume $Z$ large enough so that\n\\begin{align}\n\\tfrac{3}{4}\\eta \\Omega(0)\n- \\Omega(-\\tfrac{1}{4}Z) & \\ge 0 \\label{eqn:lm15Obnd} \\\\\n\\tfrac{3}{4}\\eta \\Omega(-x_g+\\ashift)\n- \\Omega(-\\tfrac{1}{4}Z) & \\ge 0 \\label{eqn:lm15Obbnd} \n\\end{align}\n\nLet us initialize the system \\eqref{eqn:gfrecursion} with \n\\(\n\\ff^{0}(x) \n\\)\nas given in \\eqref{eqn:f0initial}.\nBy \\eqref{eqn:f0initialbound} we have\n\\[\n\\ff^{0,\\smthker}(x) \\ge \\tmplF^{\\smthker} (x+\\ashift) +  \\tfrac{3}{4} \\eta \\Omega(x+\\ashift)\n- \\Omega(x-\\tfrac{3}{4}Z)\n\\] \nand for $x \\in [x_g-\\ashift,\\half Z]$ we have by \\eqref{eqn:lm15Obnd}\n\\[\n\\ff^{0,\\smthker}(x) \\ge \\tmplF^{\\smthker} (x+\\ashift) \\,.\n\\] \n\nConsider now \n\\(\ng^{0}(x) = \\hg(\\ftS{0}(x)).\n\\)\nWe have for $x\\in [x_g-\\ashift,\\half Z]$\n\\begin{align*}\ng^{0}(x) &= \\hg(\\ftS{0}(x)) \\\\\n& \\ge \\hg(\\tmplF^{\\smthker} (x+\\ashift)) \n\\\\ & \\stackrel{\\eqref{eqn:ADFGbnd}}{\\ge}\n\\tmplG(x+\\ashift)+\\eta\\unitstep_0(x-x_g+\\ashift)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x < x_g-\\ashift$ the inequality holds for all $x \\le \\half Z.$\n\nBy the same argument that gave \\eqref{eqn:f0initialbound}\nwe have for all $x$ the bound\n\\begin{align*}\ng^{0}(x) & \\ge\n\\tmplG(x+\\ashift)+\\tfrac{3}{4}\\eta\\unitstep_0(x-x_g+\\ashift) - \\unitstep_1(x-\\tfrac{3}{4}Z)\n\\end{align*}\nand we obtain\n\\begin{align*}\ng^{0,\\smthker}(x) \\ge \\tmplG^{\\smthker}(x+\\ashift) + \\tfrac{3}{4}\\eta \\Omega(x-x_g+\\ashift) - \\Omega(x-\\tfrac{3}{4}Z)\n\\end{align*}\nWhich, by \\eqref{eqn:lm15Obbnd}, gives\n$g^{0,\\smthker}(x) \\ge \\tmplG^{\\smthker}(x+\\ashift)$\nfor $x\\in [0,\\half Z].$\n\nNow, define $\\ff^1$ by\n$\\ff^1(x) = \\hf(\\gtS{0}(x))$ for $x\\in[0,Z]$ and\n$\\ff^1(x) = 0$ otherwise.  For $x\\in[0,\\half Z]$ we have\n\\begin{align*}\nf^{1}(x) &= \\hf(\\gtS{0}(x)) \\\\\n& \\ge \\hf(\\tmplG^{\\smthker} (x+\\ashift)) \n\\\\ & \\stackrel{\\eqref{eqn:ADGFbnd}}{\\ge}\n\\tmplF(x)+\\eta\\unitstep_0(x)\n\\\\ & \\ge\nf^{0}(x)\n\\end{align*}\nThis implies the existence of a fixed point lower bounded by\n$f^0,g^0,$ which completes the proof.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discretetwoterminatedexist}]\nThe proof is similar to the proof of Lemma \\ref{thm:twoterminatedexist}\nbut we require some stronger assumptions.\nFirst, we assume that $Z=L\\Delta$ for an integer $L.$\nIn addition we assume $\\eta$ small enough so that \n$\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta,\\cdot);u(\\eta),v(\\eta))< -\\Delta\\|\\smthker\\|_\\infty.$\nTheorem \\ref{thm:mainexist} now implies\n$\\ashift <-\\Delta$ (actually we have $\\ashift < -\\Delta\/(u(\\eta) v(\\eta)$).\nFinally, we assume $Z$ large enough so that\n\\begin{align}\n\\tfrac{3}{4}\\eta \\Omega(0)\n- \\Omega(-\\tfrac{1}{4}Z+\\half \\Delta) & \\ge 0 \\label{eqn:lm16Obnd}\n\\\\\n\\tfrac{3}{4}\\eta \\Omega(-x_g+\\ashift-\\Delta)\n- \\Omega(-\\tfrac{1}{4}Z+ \\Delta) & \\ge 0 \\label{eqn:lm16Obbnd}\n\\end{align}\n\nLet us initialize the system \\eqref{eqn:discretegfrecursion} with \n\\(\n\\ff^{0}(x) \n\\)\nas given in \\eqref{eqn:f0initial}.\nBy \\eqref{eqn:f0initialbound} and Lemma \\ref{lem:disccontbnd} we have\n\\begin{align*}\n\\ff^{0,\\smthker}(x_i) \\ge &\\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta) +  \\tfrac{3}{4} \\eta \\Omega(x_i+\\ashift-\\half\\Delta)\\\\\n&- \\Omega(x_i-\\tfrac{3}{4}Z+\\half\\Delta)\n\\end{align*}\nand for $x_i \\in [x_g-\\ashift+\\half\\Delta,\\half Z]$ we have by \\eqref{eqn:lm16Obnd}\n\\[\n\\ff^{0,\\smthker}(x_i) \\ge \\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta) \\,.\n\\] \n\nConsider now \n\\(\ng^{0}(x_i) = \\hg(\\ftS{0}(x_i)).\n\\)\nWe have for $x_i\\in [x_g-\\ashift+\\half\\Delta,\\half Z]$\n\\begin{align*}\ng^{0}(x_i) &= \\hg(\\ftS{0}(x_i)) \\\\\n& \\ge \\hg(\\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta)) \n\\\\ & \\stackrel{\\eqref{eqn:ADFGbnd}}{\\ge}\n\\tmplG(x_i+\\ashift-\\half\\Delta)+\\eta\\unitstep_0(x_i-x_g+\\ashift-\\half\\Delta)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x_i < x_g-\\ashift+\\half\\Delta$ the inequality holds for all $x_i \\le \\half Z.$\n\nBy the same argument that gave \\eqref{eqn:f0initialbound}\nwe have for all $x$ the bound\n\\begin{align*}\ng^{0}(x_i) \\ge &\n\\tmplG(x_i+\\ashift-\\half\\Delta)+\\tfrac{3}{4}\\eta\\unitstep_0(x_i-x_g+\\ashift-\\half\\Delta)\n\\\\& - \\unitstep_1(x_i-\\tfrac{3}{4}Z)\n\\end{align*}\nand, applying Lemma \\ref{lem:disccontbnd}, we obtain\n\\begin{align*}\ng^{0,\\discsmthker}(x_i) \\ge& \\tmplG^{\\smthker}(x_i+\\ashift-\\Delta) + \\tfrac{3}{4}\\eta \\Omega(x_i-x_g+\\ashift-\\Delta)\n\\\\& - \\Omega(x_i-\\tfrac{3}{4}Z+\\Delta)\n\\end{align*}\nWhich, by \\eqref{eqn:lm16Obbnd}, gives\n$g^{0,\\smthker}(x_i) \\ge \\tmplG^{\\smthker}(x_i+\\ashift-\\Delta)$\nfor $x_i\\in [0,\\half Z].$\n\nNow, define $\\ff^1$ by\n$\\ff^1(x_i) = \\hf(\\gtS{0}(x_i))$ for $x_i\\in[0,Z]$ and\n$\\ff^1(x_i) = 0$ otherwise.  For $x_i\\in[0,\\half Z]$ we have\n\\begin{align*}\nf^{1}(x_i) &= \\hf(\\gtS{0}(x_i)) \\\\\n& \\ge \\hf(\\tmplG^{\\smthker} (x_i+\\ashift-\\Delta)) \n\\\\ & \\stackrel{\\eqref{eqn:ADGFbnd}}{\\ge}\n\\tmplF(x_i-\\Delta)+\\eta\\unitstep_0(x_i-\\Delta)\n\\\\ & \\ge\nf^{0}(x_i)\n\\end{align*}\nwhere the last inequality uses $\\ashift <-\\Delta.$\nThis implies the existence of a fixed point lower bounded by\n$f^0,g^0,$ which completes the proof.\n\n\n\\end{IEEEproof}\n\n\n\\section{General Convergence Results}\\label{app:E}\n\nThe existence of interpolating wave solutions often implies \nglobal convergence of the spatially coupled system.  The structure of $\\cross(\\hf,\\hg)$ can potentially be complicated\nenough to prevent direct application of the existence results for wave-like solutions.\nTypically, the existence results for spatial fixed points are easier to apply.\nOur technique to prove the general convergence results largely consists of modifying the\nsystem monotonically to elicit the existence of a fixed point solution for the modified system\nthat then initiates a monotonic sequence for the original system.\nMonotonicity implies convergence and necessary conditions on interpolating fixed points\nprovide the leverage to get the desired results.\nWe start with a Lemma that uses this approach in a canonical way and that we can\nthen use for the more complicated statements.\n\n\\begin{lemma}\\label{lem:liminfgap}\nLet $(\\hf,\\hg)$ be given with $A(\\hf,\\hg)<0$ and \n$\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg)$ for $(u,v) \\neq (1,1).$\nConsider the spatially continuous system \\eqref{eqn:gfrecursion}.\nIf $\\ff^0 \\in \\sptfns$ \nsatisfies $\\ff^0(\\pinfty) = 1$ then for all $x\\in\\reals$ we have\n\\[\n\\lim_{t\\rightarrow \\infty} \\ff^t(x) =1\\,,\\quad\n\\lim_{t\\rightarrow \\infty} \\fg^t(x) =1\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nConsider first the parametric modification of $(\\hf,\\hg)$ given \nin \\eqref{eqn:FirstMod}, i.e.,\n\\begin{align*}\n\\hf(\\eta;u) &=(\\hf(u)-\\eta)^+\\,\\\\\n\\hg(\\eta;v) &= (\\hg(v)-\\eta)^+ \\,.\n\\end{align*}\n\nAs before, $(u(\\eta),v(\\eta)) \\in \\cross(\\hf(\\eta;u),\\hg(\\eta;\\cdot))$ is the minimum point\nwhere  $\\altPhi(\\hf(\\eta;u),\\hg(\\eta,\\cdot);u,v)$ achieves its minimum\nin $[0,1]\\times[0,1]$\nand $(u(\\eta),v(\\eta)) \\rightarrow (1,1)$ as $\\eta \\rightarrow 0.$\nGiven arbitrary $\\epsilon > 0$ we choose $\\eta$ small enough so that\n$u(\\eta),v(\\eta) > 1-\\epsilon$ and $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) < 0.$\nIf $\\hf(\\eta;u(\\eta)) > v(\\eta)$ then redefine $\\hf(\\eta;u(\\eta)) = v(\\eta)$\nand\nif $\\hg(\\eta; v(\\eta)) > u(\\eta)$ then redefine $\\hg(\\eta;v(\\eta)) = u(\\eta).$\n\nWe are interested in the modified pair $\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot)$ restricted to $[0,u(\\eta)] \\times [0,v(\\eta)].$\nNow consider a further parametric modification\n\\begin{align*}\n\\hf(\\eta',\\eta;u) &=v(\\eta)\\unitstep_1(u-(u(\\eta)-\\eta'))\\vee \\hf(\\eta;u)\\,\\\\\n\\hg(\\eta',\\eta;v) &=u(\\eta)\\unitstep_1(v-(v(\\eta)-\\eta'))\\vee \\hg(\\eta;v) \\,.\n\\end{align*}\nSince $\\hf(\\eta;u)$ is continuous at $u(\\eta)$ and\n$\\hg(\\eta;u)$ is continuous at $v(\\eta)$\n(by the minimization of $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);\\cdot,\\cdot)$)\nwe can assume $\\eta' >0$ sufficiently small so that\n\\begin{align*}\n\\hf(u) &\\ge\\hf(\\eta',\\eta;u) \\,\\\\\n\\hg(v) &\\ge\\hg(\\eta',\\eta;v) \\,.\n\\end{align*}\n\nBy construction \n$\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u,v )$ is uniquely minimized on\n $[0,u(\\eta)] \\times [0,v(\\eta)]$\nat $(u,v) = (u(\\eta),v(\\eta))$ and there exists a positive $\\delta>0$ such\nthat\n\\(\n\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u,v ) >\n\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u(\\eta),v(\\eta))+\\delta\n\\)\nfor all $(u,v)\\in \\cross (\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot))\\backslash (u(\\eta),v(\\eta)).$\n\nNow we make one further parametric modification, further reducing the functions,\n\\begin{align*}\n\\hf(z,\\eta',\\eta;u) &=\\unitstep_1(u-z)\\wedge \\hf(\\eta',\\eta;u)\\,\\\\\n\\hg(z,\\eta',\\eta;v) &=\\unitstep_1(v-z)\\wedge \\hg(\\eta',\\eta;v) \\,,\n\\end{align*}\nwhere we choose $z>0$ so that \n$(\\hf(z,\\eta',\\eta;u),\\hg(z,\\eta',\\eta;v))$ satisfies the strictly positive gap condition\non  $[0,u(\\eta)] \\times [0,v(\\eta)].$\nTo do this we choose $z$ so that\n\\(\n\\altPhi(\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot);u(\\eta),v(\\eta)) = - \\delta\/2\n\\)\n\nBy Theorem \\ref{thm:mainexist} there exists $\\tmplF,\\tmplG \\in \\sptfns$ \ninterpolating over $[0,u(\\eta)] \\times [0,v(\\eta)]$ and $\\ashift \\le -(\\delta\/2)\/\\|\\omega\\|_\\infty$\nsuch that $f^t(x)=\\tmplF(x-\\ashift t)$ and\n$g^t(x)=\\tmplG(x-\\ashift t)$ solves \\eqref{eqn:gfrecursion} for\nthe pair $\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot).$\n\nSince $\\tmplF(\\pinfty)=v(\\eta)<1$ and $\\tmplF(x)=0$ for some finite $x,$ we see that \nfor any $(0,1)$-interpolating function $f^0 \\in \\sptfns$ we can find a $y$ so that\nsuch that $f^0(x) \\ge \\tmplF(x-y)$ for all $x.$\nIt now follows that under \\eqref{eqn:gfrecursion} for the original pair $(\\hf,\\hg)$ we have $\\liminf_{t\\rightarrow\\infty} f^t(x)\\ge v(\\eta)\\ge 1-\\epsilon$\nand\n$\\liminf_{t\\rightarrow\\infty} g^t(x)\\ge u(\\eta)\\ge 1-\\epsilon$\nfor all $x.$\n\nSince $\\epsilon$ is arbitrary the proof is complete.\n\\end{IEEEproof}\n\nThe above proof can be easily adapted to the spatially discrete case.\n\\begin{lemma}\\label{lem:discreteliminfgap}\nLet $(\\hf,\\hg)$ be given with $A(\\hf,\\hg)<0$ and \n$\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg)$ for $(u,v) \\neq (1,1).$\nConsider the spatially discrete system \\eqref{eqn:discretegfrecursion}.\nFor any $\\epsilon>0,$ if $\\Delta$ is sufficiently small then\nfor all $x\\in\\reals$ we have\n\\[\n\\lim_{t\\rightarrow \\infty} \\ff^t(x) =1-\\epsilon\\,,\\quad\n\\lim_{t\\rightarrow \\infty} \\fg^t(x) =1-\\epsilon\n\\]\nfor anny $\\ff^0 \\in \\sptfns$ satisfying $\\ff^0(\\pinfty) = 1.$\n\\end{lemma}\n\\begin{IEEEproof}\nWe use the construction from the proof of Lemma \\ref{lem:liminfgap}\nand recall\nthe existence of $\\tmplF,\\tmplG \\in \\sptfns$ \ninterpolating over $[0,u(\\eta)] \\times [0,v(\\eta)]$ and $\\ashift \\le -(\\delta\/2)\/\\|\\omega\\|_\\infty$\nsuch that $f^t(x)=\\tmplF(x-\\ashift t)$ and\n$g^t(x)=\\tmplG(x-\\ashift t)$ solves \\eqref{eqn:gfrecursion} for\nthe pair $\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot).$\nAssume $\\Delta \\le |\\ashift|.$  \n\nGiven  $f^0$ satisfying $\\ff^0(\\pinfty) = 1$ we can find $y$ \nsuch that $f^0(x_i) \\ge \\tmplF(x_i-y)$ for all $x.$\nWe can apply Theorem \\ref{thm:mainquantize} and the inequalities\n$\\hf \\ge \\hf(z,\\eta',\\eta;\\cdot)$ and $\\hg \\ge \\hg(z,\\eta',\\eta;\\cdot)$\nto obtain\n$f^t(x_i) \\ge \\tmplF(x_i-y-(\\ashift+\\Delta)t)$ and\n$g^t(x_i) \\ge \\tmplG(x_i-y-(\\ashift+\\Delta)t).$\n\nThe Lemma now follows.\n\\end{IEEEproof}\n\n\n\nRecall that in the statement of Theorem \\ref{thm:globalconv} \nwe have $(0,0)\\le (u',v') \\le (u'',v'') \\le (1,1)$ and $\\altPhi$ is minimized on\n$(u',v')$ and $(u'',v'')$ where it takes the value  $m(\\hf,\\hg).$\nFurthermore, $(u',v')$ and $(u'',v'')$ are the extreme points where the minimum is attained.\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:globalconv}]\nWe will prove the first statement in the Theorem,  i.e.,\n\\(\n\\liminf_{t\\rightarrow \\infty} f^t(x) \\ge v'\\,,\n\\)\nthe other cases being similar.\n\nIf $u' =0$ or $v'=0$ then $m=0$ and $(u',v')=(0,0)$ and the result is immediate.\nLet us assume that $m<0$ and hence that $(u',v')>(0,0).$\nConsider the system restricted to $[0,u']\\times [0,v'].$\nIf $\\hf(u')>v'$ then let us redefine $\\hf(u')=v'$ and \nif $\\hg(v')>u'$ then let us redefine $\\hf(u')=u'.$\nThis makes $(u',v')$ a fixed point of the underlying system.\nThis reduction will not affect the remaining argument.\nLet us reduce $\\ff^0$ by saturating it at $v',$ i.e., replacing it with\n$\\ff^0 \\wedge v'.$\n\nWe can now apply Lemma \\ref{lem:liminfgap} to obtain $\\ff^t(x) \\rightarrow v'$\nand $\\fg^t(x) \\rightarrow u'.$\nSince $\\ff^t$ and $\\fg^t$ in the original system are only larger, the result follows.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discreteglobalconv}]\nJust as Lemma \\ref{lem:liminfgap} proved Theorem \\ref{thm:globalconv},\nwe can use Lemma \\ref{lem:discreteliminfgap} proved Theorem \\ref{thm:discreteglobalconv}.\nThe argument is essentially the same so we omit it.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:terminatedzero}]\nThe case where $\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\neq (0,0)$\nfollows easily from Theorem \\ref{thm:globalconv}.  We assume now that\n$\\altPhi(\\hf,\\hg;u,v) = 0$ and   $\\hf$ and $\\hg$ are strictly positive on $(0,1].$\n\nDefine \n\\[\n\\ff^0(x)=\\unitstep_1(x)\n\\] \nWe will show that $\\ff^t\\rightarrow 0,$ which implies the same for arbitrary initial conditions.\nBy monotonicity in $t,$ $\\ff^t$ has a point-wise limit $\\ff^\\infty \\in \\sptfns$\nand $\\fg^t$ has a point-wise limit $\\fg^\\infty \\in \\sptfns$\nBy continuity we have $(\\ff^\\infty(\\pinfty),\\fg^\\infty(\\pinfty)) \\in \\cross(\\hf,\\hg).$\n\nIn general $h_{[\\ff^\\infty,\\fg^{\\smthker,\\infty}]}$\nis well defined on\n$[0,\\fg^{\\infty}(\\pinfty)]$ and\n$h_{[\\fg^\\infty,\\ff^{\\smthker,\\infty}]}$\nis well defined on\n$[0,\\ff^{\\infty}(\\pinfty)]$ \nand we have\n\\begin{align}\n\\begin{split}\\label{eqn:AineqB}\n0 \\le  &\\altPhi(\\hf,\\hg;\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty))\n\\\\ \\le &\n\\altPhi(h_{[\\ff^\\infty,\\fg^{\\smthker,\\infty}]},h_{[\\fg^\\infty,\\ff^{\\smthker,\\infty}]};\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty))\\,.\n\\\\ = &0\\,.\n\\end{split}\n\\end{align}\n\nAssume that $\\ff^\\infty \\neq 0.$\nLet $z = \\sup \\{x:\\ff^\\infty(x)=0\\}.$\nWe have $\\ff^{\\smthker,\\infty}(x)>0$ on $\\neigh{z}{W}$ and therefore \n$\\fg^{\\infty}(x)>0$ on $\\neigh{z}{W}$ and\n$\\fg^{\\smthker,\\infty}(x)>0$ on $\\neigh{z}{2W}.$\nHence $\\ff^\\infty(x) > 0$  for $x \\in \\neigh{z}{2W} \\cap (0,\\infty)$\nbut $\\ff^\\infty(x)=0$ for $x<z.$\nThis implies that $z=0$ and that $\\ff^\\infty(x)$ is discontinuous at $x=0.$\nThis implies that \n\\begin{align}\n &\\altPhi(\\hf,\\hg;u,v)\n<\n\\altPhi(h_{[\\ff^\\infty,\\fg^{\\smthker,\\infty}]},h_{[\\fg^\\infty,\\ff^{\\smthker,\\infty}]};u,v)\n\\end{align}\nfor $u > 0$ and we easily conclude that \n$(\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty)) = (0,0).$\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discreteterminatedexistB}]\nTheorem \\ref{thm:discreteterminatedexistB} can be proved along lines similar to\nLemma \\ref{lem:liminfgap} with some additional features introduced to handle the spatial discreteness.\n\nConsider a parametric modification of $(\\hf,\\hg)$ given by\n\\begin{align*}\n\\hf(z,\\eta;u) &=\\unitstep_1(u-z)\\wedge ( \\hf(u)-\\eta)^+ \\\\\n\\hg(z,\\eta;v) &=\\unitstep_1(v-z)\\wedge (\\hg(v)-\\eta)^+ \\,.\n\\end{align*}\nWe first fix $z=0.$\nDefine $m(\\eta) = \\min \\altPhi(\\hf(0,\\eta;\\cdot),\\hg(0,\\eta;\\cdot);\\cdot,\\cdot)$\nand let $(u(\\eta),v(\\eta))$ be the minimum point where $m(\\eta)$ is \nrealized.\nAs $\\eta\\rightarrow 0$ we have $(u(\\eta),v(\\eta))\\rightarrow (1,1)$\nand $m(\\eta)\\rightarrow A(\\hf,\\hg).$\nGiven $\\epsilon>0$ we may assume $\\eta$ small enough so that\n$(u(\\eta),v(\\eta)) \\ge (1-\\epsilon,1-\\epsilon)$ and we also assume\n$m(\\eta)<0.$\n\nNow we choose $z(\\eta) (>0)$ so that\n\\[\n \\altPhi\\bigl(\\hf(z(\\eta),\\eta;\\cdot),\\hg(z(\\eta),\\eta;\\cdot);u(\\eta),v(\\eta)\\bigr)=0\n\\]\nIf necessary we reduce $\\hf(z(\\eta),\\eta;u(\\eta))$ and $\\hg(z(\\eta),\\eta;v(\\eta))$\nso that they equal $v(\\eta)$ and $u(\\eta)$ respectively.\nThen \n$(\\hf(z(\\eta),\\eta;\\cdot),\\hg(z(\\eta),\\eta;\\cdot))$\nsatisfies the strictly positive gap condition on\n$[0,u(\\eta)]\\times[0,v(\\eta)].$\n\nBy Theorem \\ref{thm:mainexist} there exists\n $\\tmplF$ and $\\tmplG$ that form a fixed point for \n\\eqref{eqn:gfrecursion}\nwith $(\\tmplF(\\minfty),\\tmplG(\\minfty))=(0,0)$ and\n$(\\tmplF(\\pinfty),\\tmplG(\\pinfty))=(v(\\eta),u(\\eta)).$\n\nLet us translate the solution so that $0 =\\sup_x \\{\\tmplF(x)=0\\}$\nand let us then define $x_g = \\sup_x \\{\\tmplG(x)=0\\}.$\nIt follows that $|x_g| < W.$\nLet us choose $\\Delta$ sufficiently small so that\nthe following holds:\n\\begin{equation}\\label{eqn:9condB}\n\\eta\\intsmthker(-|x_g|-\\half\\Delta)\n\\ge\n\\half\\Delta \\|\\omega\\|_\\infty\\,.\n\\end{equation}\n\n\nConsider initializing \\eqref{eqn:discretegfrecursion} with\n\\begin{align*}\nf^{0}(x_i)&=\n\\tmplF(x_i) +\\eta \\unitstep_0(x_i)\\,.\n\\end{align*} \nApplying Lemma \\ref{lem:disccontbnd} this yields for $x_i\\ge x_g$\n\\begin{align*}\nf^{0,\\discsmthker}(x_i)&\\ge\n\\tmplF^\\smthker (x_i-\\half\\Delta) + \\eta\\intsmthker(x_i-\\half\\Delta) \n\\\\\n&\\stackrel{\\eqref{eqn:9condB}}{\\ge}\n\\tmplF^\\smthker (x_i-\\half\\Delta) +\\half\\Delta\\|\\omega\\|_\\infty\\,\n\\\\\n&{\\ge}\n\\tmplF^\\smthker (x_i) \\,.\n\\end{align*}\nWe now obtain for $x_i  \\ge x_g$\n\\begin{align*}\ng^{0}(x_i)&=\\hg(f^{0,\\discsmthker}(x_i))\n\\\\\n&\\ge\n\\hg(\\tmplF^{\\smthker}(x_i))\n\\\\\n&\\ge\n\\hg(z,\\eta;\\tmplF^{\\smthker}(x_i))+\\eta\\unitstep_0(x_i-x_g)\n\\\\\n&\\ge\n\\tmplG(x_i)+\\eta\\unitstep_0(x_i-x_g)\n\\end{align*}\nand we observe that since $\\tmplG(x_i)=0$ for\n$x_i < x_g$ this bound  holds for all $x_i.$\nWe now have\n\\begin{align*}\ng^{0,\\discsmthker}(x_i)&\\ge\n\\tmplG^\\smthker (x_i-\\half\\Delta)\n+\\eta\\intsmthker(x_i-x_g-\\half\\Delta)\\,.\n\\end{align*}\n For  $x_i \\ge 0$ we obtain\n\\begin{align*}\ng^{0,\\discsmthker}(x_i)\n&\\stackrel{\\eqref{eqn:9condB}}{\\ge}\n\\tmplG^\\smthker (x_i-\\half\\Delta)\n+\\half\\Delta \\|\\omega\\|_\\infty\n\\\\ & \\ge \n\\tmplG^\\smthker (x_i)\\,.\n\\end{align*}\nThus we have \n\\begin{align*}\n\\ff^{1}(x_i)&=\\hf(\\fg^{0,\\discsmthker}(x_i))\n\\\\&\\ge \\hf(\\tmplG^{\\smthker}(x_i))\n\\\\&\\ge \\tmplF(x_i)+\\eta\\unitstep_0(x_i)\n\\\\&= \\ff^0(x_i)\n\\end{align*}\nand the consequently increasing sequence establishes the existence of the\ndesired fixed point.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discretetwoterminatedexistGB}]\nWe assume $Z = L\\Delta$ for an integer $L.$\nthe termination $\\hf(x_i,\\cdot)=0$ holds for $x_i<0$ and $x_i > Z.$\nThis means that symmetry holds about $\\half Z.$\nThe proof follows that of Theorem \\ref{thm:discreteterminatedexistB}\nup to the point where requirements on $\\Delta$ are given.\nContinuing from there we\nchoose $\\Delta$ small enough and $Z$ large enough \nso that\nall of the following hold.\n\\begin{equation}\\label{eqn:14condB}\n\\tfrac{3}{4}\\eta\\intsmthker(-|x_g|-\\Delta)\n- \\intsmthker(-\\tfrac{1}{4}Z+\\half\\Delta )\n\\ge\n\\half\\Delta \\|\\omega\\|_\\infty\n\\end{equation}\n\\begin{equation}\\label{eqn:14condA}\n\\tmplF(\\tfrac{1}{4}Z) > v(\\eta)-\\eta\/4\n\\end{equation}\n\\begin{equation}\\label{eqn:14condE}\n\\tmplG(\\tfrac{1}{4}Z) > u(\\eta)-\\eta\/4\n\\end{equation}\n\nConsider initializing for $x_i \\le \\half Z$ with\n\\[\nf^{0}(x_i)=\n\\tmplF(x_i) +\\eta \\unitstep_0(x_i)\\,.\n\\]\nand for $x_i > \\half Z$ initializing symmetrially with $\\ff^{0}(x_i)=\\ff^{0}(x_{L-i}).$\nAs in the derivation of \\eqref{eqn:f0initialbound}, this, by \\eqref{eqn:14condA}, implies for all $x$\n\\[\nf^{0}(x_i)=\n\\tmplF(x_i) +\\tfrac{3}{4}\\eta \\unitstep_0(x_i)\n-\\unitstep_1(x_i - \\tfrac{3}{4}Z)\\,.\n\\]\nwhich gives by Lemma \\ref{lem:disccontbnd},\n\\[\nf^{0,\\discsmthker}(x_i)\\ge\n\\tmplF^{\\smthker}(x_i-\\half\\Delta) +\\tfrac{3}{4}\\eta \\intsmthker(x_i-\\half\\Delta)\n-\\intsmthker(x_i - \\tfrac{3}{4}Z+\\half\\Delta)\\,.\n\\]\n\nThis yields for $x_i\\in[x_g-\\half\\Delta,\\tfrac{1}{2} Z]$\n\\begin{align*}\nf^{0,\\discsmthker}(x_i)\n&\\ge\n\\tmplF^\\smthker (x_i) + \\tfrac{3}{4}\\eta\\intsmthker(x_g-\\Delta) - \\intsmthker(-\\tfrac{1}{4} Z+\\half\\Delta )\n\\\\\n&\\stackrel{\\eqref{eqn:14condB}}{\\ge}\n\\tmplF^\\smthker (x_i) +\\half\\Delta\\|\\omega\\|_\\infty\\,\n\\\\\n&{\\ge}\n\\tmplF^\\smthker (x_i+\\half\\Delta) \\,.\n\\end{align*}\n\nWe now obtain for $x_i  \\in [x_g-\\half \\Delta, \\half Z]$\n\\begin{align*}\ng^{0}(x_i)&=\\hg(f^{0,\\discsmthker}(x_i))\n\\\\\n&\\ge\n\\hg(\\tmplF(x_i+\\half\\Delta))\n\\\\\n&\\ge\n\\tmplG(x_i+\\half\\Delta)+\\eta\\unitstep_0(x_i+\\half\\Delta-x_g)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x_i+\\half\\Delta < x_g$ this bound  holds for all $x_i \\le \\half Z.$\nAs in the derivation of \\eqref{eqn:f0initialbound}, by \\eqref{eqn:14condE} we now have\n\\[\ng^{0}(x_i)\\ge\n\\tmplG(x_i+\\half\\Delta)+\\tfrac{3}{4}\\eta\\unitstep_0(x_i+\\half\\Delta-x_g)-\\unitstep_1(x_i-\\tfrac{3}{4}Z)\n\\]\nwhich gives by Lemma \\ref{lem:disccontbnd}\n\\[\ng^{0,\\discsmthker}(x_i)\\ge\n\\tmplG^\\smthker(x_i)+\\tfrac{3}{4}\\eta\\Omega(x_i-x_g)-\\Omega(x_i-\\tfrac{3}{4}Z+\\half\\Delta)\n\\]\nwhich by \\eqref{eqn:14condB} yields for $x_i \\in [0,\\half Z],$\n\\[\ng^{0,\\discsmthker}(x_i)\\ge\n\\tmplG^\\smthker(x_i)\\,.\n\\]\n\n\nThus we have for $x_i \\in [0,\\half Z],$\n\\begin{align*}\n\\ff^{1}(x_i)&=\\hf(\\fg^{0,\\discsmthker}(x_i))\n\\\\&\\ge \\hf(\\tmplG^{\\smthker}(x_i))\n\\\\&\\ge \\tmplF(x_i)+\\eta\\unitstep_0(x_i)\n\\\\&= \\ff^0(x_i)\n\\end{align*}\nand the consequently increasing sequence establishes the existence of the\ndesired fixed point.\n\\end{IEEEproof}\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof for the General Case\\label{sec:general}}\n\nIn Section \\ref{sec:PCcase} we proved Theorem\n\\ref{thm:PCexist},\na special case of Theorem \\ref{thm:mainexist}\nin which $\\hg$ and $\\hf$ are piecewise constant\nfunctions and $\\smthker$ is $C_1$ and strictly positive.\nIn this section we show how to remove the speical conditions \nto arrive at the general results.\nWe make repeated use of the limit theorems of Section \\ref{sec:limitthms} and develop\nsome approximations for functions in $\\exitfns.$\nIt is quite simple to approximate $h \\in \\exitfns$ using piecewise constant functions.\nThe only difficulty is to approximate a pair $(\\hg,\\hf)$ so that the strictly positive gap\ncondition is preserved.\n\n\\subsection{Approximation by Scaling}\nLet $h \\in \\exitfns$ and denote $\\int_0^1 h(x) \\, dx$ by $A.$\nFor $0< a < 1$ we define \n\\[\n\\slanta{h} (x) = h\\Bigl(\\frac{x - (1-A)}{1-a} + (1-A)\\Bigr)\n\\]\nwhere we assume $h(x)=0$ for $x<0$ and \n$h(x)=1$ for $x>1.$ A quick calculation verifies\n$\\int_0^1 \\slanta{h} (x) dx = A.$\nNote that $h(x) \\ge \\slanta{h}(x)$ for $u \\in [0,1-A]$ and \nthat $h(x) \\le \\slanta{h}(x)$ for $x \\in [1-A,1].$\nIt follows that \n$0 \\le \\int_0^z (h(x)-\\slanta{h}(x))\\, dx \\le \\frac{a}{1-a}\\int_0^{1-A} h(x)\\,dx$\nfor all $z \\in [0,1],$\nwhich yields\n$\\Phi(\\hg,\\hf) \\le \\Phi(\\slanta{\\hg},\\slanta{\\hf})$ and\n$E(\\hg,\\hf) = E(\\slanta{\\hg},\\slanta{\\hf}).$\n\n\\begin{lemma}\\label{lem:smoothcompress}\nLet $(\\hg,\\hf) \\in \\exitfns^2$ \nsatisfy the strictly positive gap condition.\nThen, there exists $\\epsilon > 0$ such that\n$(\\slanta{\\hg},\\slanta{\\hf})$ \nsatisfies the strictly positive gap condition\nfor any $a \\in (0,\\epsilon).$\n\\end{lemma}\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:smoothcompress}]\nLet $A_1 = \\int_0^1 \\hg(x) dx,$ $A_2 = \\int_0^1 \\hf(x) dx$ so $E(\\hg,\\hf) = 1-A_1-A_2.$\nBy symmetry we need only consider the case $E \\ge 0.$\n\nFirst, we note that \n$[0,a (1-A_1)]\\times [0,a (1-A_2)]\\subset \\gapptsplus(\\slanta{\\hg},\\slanta{\\hf})$ \nand\n$[1-a A_1,1]\\times [1-a A_2,1]\\subset \\gapptsminus(\\slanta{\\hg},\\slanta{\\hf}).$\nThus, it remains only to show that $\\Phi(\\slanta{\\hg},\\slanta{\\hf}) > E$ on \n$\\intcross(\\slanta{\\hg},\\slanta{\\hf}).$\nAssume $v \\in \\intcross(\\slanta{\\hg},\\slanta{\\hf}).$ \n\nLet $\\eta$ be chosen with $0 < \\eta < \\min\\{A_1,A_2,1-A_1,1-A_2\\}.$\nDefine\n\\[\n\\tilde{\\cross}(\\hg,\\hf) = \\cross(\\hg,\\hf) \\backslash (\\ball_{\\eta\/2}(0,0) \\cup \\ball_{\\eta\/2}(1,1)).\n\\]\nBy continuity, there exists $\\gamma >0,$ which we may take to be less than $2\\eta,$\nsuch that\n$\\Phi(\\hg,\\hf) > \\gamma + E$ on $\\tilde{\\cross}(\\hg,\\hf).$\nSince $\\Phi$ is Lipschitz-1 in its last two arguments it follows that\n$\\Phi(\\hg,\\hf) > \\gamma\/2 + E$ on $\\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}.$\nFor all $a$ sufficiently small we have\n$|\\Phi(\\hg,\\hf) - \\Phi(\\slanta{\\hg},\\slanta{\\hf})| < \\gamma\/2$ uniformly\nand it follows that  \n$\\Phi(\\slanta{\\hg},\\slanta{\\hf}) > E$ on $\\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}.$\nBy Lemma \\ref{lem:crosspointlimit}, for all $a$ sufficiently small we have\n\\(\n{\\cross}(\\slanta{\\hg},\\slanta{\\hf})\n\\subset \\neigh{{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\)\nand since\n\\(\n\\neigh{{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\subset\n\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1) \\cup \\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\)\nwe have\n\\[\n{\\cross}(\\slanta{\\hg},\\slanta{\\hf})\n\\subset \n\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1) \\cup \\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}\\,.\n\\]\nTherefore, it remains only to bound \n$\\Phi(\\slanta{\\hg},\\slanta{\\hf}; v)$ on $\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1).$\n\nSince $\\Phi(\\hg,\\hf) \\le \\Phi(\\slanta{\\hg},\\slanta{\\hf})$ it is sufficient to show\n$\\Phi(h_{1},h_{2};v_1,v_2) > E,$ or equivalently\n$\\hat{\\Phi}(h_{1},h_{2};v_1,v_2) > 0.$\nIf $v\\in \\cross(\\hg,\\hf)$ then the inequality is immediate, so we assume\n$v\\not\\in \\cross(\\hg,\\hf).$\nIf $(v_1,v_2) \\le (1-A_1,1-A_2)$ \nthen since $v_1 \\in \\closure{h}_{2,a}(v_2)$ we have\n$v_1 \\le \\hf(v_2^+)$ and, similarly, $v_2 \\le \\hg(v_1^+),$ hence\n$v \\in \\gapptsminus(\\hg,\\hf).$\nSimilarly, if $(v_1,v_2) \\ge (1-A_1,1-A_2)$ \nthen $v \\in \\gapptsplus(\\hg,\\hf).$\n\nSince $\\closeM[\\hg,\\hf](v) \\in \\cross(\\hg,\\hf)$ and \n$v\\not\\in \\cross(\\hg,\\hf),$ \nLemma \\ref{lem:verphigap} and Lemma \\ref{lem:Rvverify}\nyield\n${\\Phi}(\\hg,\\hf;\\closeM(v)) > {\\Phi}(\\hg,\\hf;v)$\nand, equivalently,\n$\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) < \\hat{\\Phi}(\\hg,\\hf;v).$\n\nAssume $v\\in \\ball_\\eta(0,0).$ \nThen $v \\in \\gapptsminus(\\hg,\\hf)$ and $\\closeM(v) \\le v.$\nWe cannot have $\\closeM(v)=(0,0)$ since ${\\Phi}(\\hg,\\hf;0,0)=0$ and\n${\\Phi}(\\hg,\\hf;v) > 0$ (by Lemma \\ref{lem:PhiEquiv}).\nHence $\\closeM[\\hg,\\hf](v) \\in \\intcross(\\hg,\\hf),$ \nwhich implies $\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) > 0,$ and we obtain\n$\\hat{\\Phi}(\\hg,\\hf;v) > 0.$\n\nNow consider the case $v\\in \\ball_\\eta(1,1).$ \nThen $v \\in \\gapptsplus(\\hg,\\hf)$ and $\\closeM(v) \\ge v.$\nIt follows directly that $\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) \\ge 0$ (although we could show\nthat $\\closeM(v) \\neq (1,1)$ and that the inequality is strict)\nand we obtain $\\hat{\\Phi}(\\hg,\\hf;v) > 0.$\n\\end{IEEEproof}\n\n\\subsection{Piecewise Constant Approximation}\nGiven any $h \\in \\exitfns$ let us define a sequence of piecewise constant approximations\n$Q_n(h),$ $n=1,2,...$  by\n\\(\nQ_n(h) (x) = \\sum_{j=1}^n \\frac{1}{n} \\,1_{x \\ge u_{n,j}}\n\\)\nwhere we set \n\\[\nu_{n,j} =  \\int_0^1 \\max\\{ (n h(x) - (j-1))^+,1 \\} dx.\n\\]\nIf $h$ is invertible then we have $u_{n,j} = n\\int_{(j-1)\/n}^{j\/n}(1-h^{-1}(x))dx$\nand \n\\(\n\\int_0^1 Q_n(h) (x) dx = \\sum_{j=1}^n \\frac{1-u_{n,j}}{n} = \\int_0^1 (1-h^{-1})(x) dx = \\int_0^1 h (x) dx.\n\\)\nIn general, it holds that\n\\(\n\\int_0^1 Q_n(h) (x) dx = \\int_0^1 h (x) dx.\n\\)\nSince $h$ is non-decreasing, it also follows that $\\int_0^z Q_n(h) (x) dx \\le \\int_0^z h (x) dx$ for all\n$z \\in [0,1].$\n\n\\begin{lemma}\\label{lem:PCapprox}\nLet $(\\hg,\\hf)$ be pair of functions in $\\exitfns$ satisfying the strictly positive gap condition \nsuch that for some $\\eta>0$ we have\n$\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nThen, for all $n$ sufficiently large $n,$\n$(Q_n(\\hg),Q_n(\\hf))$ satisfies the strictly positive gap condition.\n\\end{lemma}\n\\begin{IEEEproof}\nBy symmetry we need only consider the case $E(\\hg,\\hf) \\ge 0.$\n\nSince $\\hg$ and $\\hf$ are $0$ on $[0,\\eta)$ and $(1-\\eta,1]$ it follows that \n$\\intcrossing (\\hg,\\hf) \\subset [\\eta,1-\\eta]^2.$\nFurther, $\\intcrossing (\\hg,\\hf)$ is closed and there exists $\\gamma > 0$ such that \n$\\Phi(\\hg,\\hf;v_1,v_2) \\ge \\gamma + E$ on $\\intcrossing (\\hg,\\hf).$\nSince $\\Phi(\\hg,\\hf)$ is Lipschitz-1 componentwise it follows that \n$\\Phi(\\hg,\\hf) > \\gamma\/2 + E$ on $\\neigh{\\intcrossing (\\hg,\\hf)}{\\gamma\/4}.$\nObserve that $\\Phi(Q_n(h_{1}),Q_n(h_{2}))$ converges in $n$ uniformly to $\\Phi(\\hg,\\hf).$\n(In fact we have $0 \\le |\\Phi(\\hg,\\hf)-\\Phi(Q_n(h_{1}),Q_n(h_{2}))| \\le \\frac{1}{n}.$)\nSo, for $n$ sufficiently large, we have $\\Phi(Q_n(h_{1}),Q_n(h_{2})) > E$ on \n$\\neigh{\\intcrossing (\\hg,\\hf)}{\\gamma\/4}.$\nBy construction we also have $\\intcrossing (Q_n(h_{1},Q_n(h_{2})) \\in [\\eta,1-\\eta]^2.$\nBy Lemma \\ref{lem:crosspointlimit} we now have \n\\(\n\\intcrossing (Q_n(h_{1}),Q_n(h_{2})) \\subset \n\\neigh{\\intcrossing (h_{1},h_{2})}{\\gamma\/4}\n\\)\nfor all $n$ sufficiently large so $\\Phi(Q_n(h_{1}),Q_n(h_{2})) > E$ on\n$\\intcrossing (Q_n(h_{1}),Q_n(h_{2})).$\n\nFinally, we note that $[0,\\eta]^2 \\subset \\gapptsplus(Q_n(h_{1}),Q_n(h_{2}))$\nand $[1-\\eta,1]^2 \\subset \\gapptsminus(Q_n(h_{1}),Q_n(h_{2})).$\nThis completes the proof.\n\\end{IEEEproof}\n\n\\subsection{Proof of Main Results}\n\nWe are now ready to prove our main results.\n\n\\begin{IEEEproof}[Proof Theorem \\ref{thm:mainexist}]\nThe proof proceeds by repeated application of Lemma \\ref{lem:limitexist}\nto establish existence of $f,g \\in \\Psi$ and constant $\\ashift$ such that\n$h^{\\fS,g} = \\hg$ and $h^{\\gSa,f} = \\hf$ for a series of \nincreasingly generalized cases of $(\\hg,\\hf)$ and $\\smthker.$\nThe simplest case is already establishted in Thereom \\ref{thm:PCexist}.\n\nWe first generalize to arbitrary $\\smthker.$\nAssume that $(\\hg,\\hf)$ are both piecewise constant.\nDefine $\\smthker_i = \\smthker \\otimes G_i$ where \n$G_i(x) = \\frac{i}{\\sqrt{2\\pi}} e^{- (ix)^2\/2}.$\nIt follows that $\\smthker_i \\rightarrow \\smthker$ in $L_1$\nand $\\| \\smthker_i \\|_\\infty \\le \\| \\smthker\\|_\\infty.$\nFor each $\\smthker_i$ we apply Theorem \\ref{thm:PCexist}\nto obtain piecewise constant $f_i,g_i \\in \\Psi$ \n(with corresponding $\\zfi,\\zgi$) and constants $\\ashift_i$\nsuch that $h^{f_i^{{\\smthker}_i},g_i} = \\hg$ and $h^{g_i^{{\\smthker}_i,\\ashift_i},f_i} = \\hf.$\nWe can now apply Lemma \\ref{lem:limitexist} as indicated above to conclude\nexistence for general $\\smthker.$\n\nWe now generalize $(\\hg,\\hf)$ and to\nrequire, beyond the conditions of the Theorem statement, only that there exists $\\eta > 0$ such that $\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nWe apply Lemma \\ref{lem:PCapprox} and the preceding case already established\nto conclude that for all $n$ sufficiently large\nthere exists (piecewise constant) $f_n,g_n \\in \\Psi$ and finite constants $\\ashift_n$ such that\n$h^{f_n^{{\\smthker}},g_n} = Q_n(\\hg)$ and $h^{g_n^{{\\smthker},\\ashift_n},f_n} = Q_n(\\hf).$\nSince $Q_n(\\hg)$ and $Q_n(\\hf)$ converge to $\\hg$ and $\\hf$ respectively,\nwe can apply Lemma \\ref{lem:limitexist} to conclude existence for this case.\n\nFor arbitrary $(\\hg,\\hf)$ we consider $(\\slanta{\\hg},\\slanta{\\hf}).$\nWe apply Lemma \\ref{lem:smoothcompress} and the preceding case to conclude that for all $a$ sufficiently small\nthere exists $f_a,g_a \\in \\Psi$ and finite constants $\\ashift_a$ such that\n$h^{f_a^{{\\smthker}},g_a} = \\slanta{\\hg}$ and $h^{g_a^{{\\smthker},\\ashift_a},f_a} = \\slanta{\\hf}.$\nWe make a final application of Lemma \\ref{lem:limitexist} to obtain a solution for the general case.\n\nIn all cases above the approximations used preserve $E.$\nIn particular, if $E(\\hg,\\hf) = 0$ then we have respectively,\n$\\ashift_i=0$ or $\\ashift_n=0$ of $\\ashift_a=0$ and we obtain $\\ashift=0$\nfrom the limit construction in Lemma \\ref{lem:limitexist}.\nIf $E(\\hg,\\hf) \\neq 0$ then we cannot have $\\ashift=0$ by Lemma \\ref{lem:PPhi}.\nMoreover, we must have $\\sgn (E) = \\sgn (\\ashift).$\nFinally, the bound $|\\ashift| \\ge |E|\/\\|\\smthker\\|_\\infty$ was proved in Lemma \\ref{lem:shiftbound}.\n\\end{IEEEproof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMachine learning at its core involves solving stochastic optimization (SO) problems of the form\n\\begin{equation}\n\\label{eqn:SO_prob}\n\t\\min_{\\mathbf{x} \\in X} \\psi(\\mathbf{x}) \\triangleq \\min_{\\mathbf{x} \\in X} E_\\xi[\\phi(\\mathbf{x},\\xi)]\n\\end{equation}\nto learn a ``model'' $\\mathbf{x} \\in X \\subset \\mathbb{R}^n$ that is then used for tasks such as dimensionality reduction, classification, clustering, regression, and\/or prediction. A primary challenge of machine learning is to find a solution to the SO problem \\eqref{eqn:SO_prob} without knowledge of the distribution $P(\\xi)$. This involves finding an approximate solution to \\eqref{eqn:SO_prob} using a sequence of $T$ training samples $\\{\\xi(t) \\in \\Upsilon\\}_{t=1}^T$ drawn independently from the distribution $P(\\xi)$, which is supported on a subset of $\\Upsilon$. There are, in particular, two main categorizations of training data that, in turn, determine the types of methods that can be used to find approximate solutions to the SO problem. These are ($i$) \\emph{batch} training data and ($ii$) \\emph{streaming} training data.\n\nIn the case of {\\em batch} training data, where all $T$ samples $\\{\\xi(t)\\}$ are pre-stored and simultaneously available, a common strategy is {\\em sample average approximation} (SAA) (also referred to as {\\em empirical risk minimization} (ERM)), in which one minimizes the empirical average of the ``risk'' function $\\phi(\\cdot,\\cdot)$ in lieu of the true expectation. In the case of {\\em streaming} data, by contrast, the samples $\\{\\xi(t)\\}$ arrive one-by-one, cannot be stored in memory for long, and should be processed as soon as possible. In this setting, {\\em stochastic approximation} (SA) methods---the most well known of which is stochastic gradient descent (SGD)---are more common. Both SAA and SA have a long history in the literature; see~\\cite{Kushner.Book2010} for a historical survey of SA methods, \\cite{Kim.etal.HSO2015} for a comparative review of SAA and SA techniques, and \\cite{Pereyra.etal.JSTSP16} for a recent survey of SO techniques.\n\nAmong other trends, the rapid proliferation of sensing and wearable devices, the emergence of the internet-of-things (IoT), and the storage of data across geographically-distributed data centers have spurred a renewed interest in development and analysis of new methods for learning from {\\em fast-streaming} and {\\em distributed} data. The goal of this paper is to find a fast and efficient solution to the SO problem \\eqref{eqn:SO_prob} in this setting of distributed, streaming data. In particular, we focus on geographically-distributed nodes that collaborate over {\\em rate-limited} communication links (e.g., wireless links within an IoT infrastructure) and obtain independent streams of training data arriving at a constant rate.\n\nThe relationship between the rate at which communication takes place between nodes and the rate at which streaming data arrive at individual nodes plays a critical role in this setting. If, for example, data samples arrive much faster than nodes can communicate among themselves, it is difficult for the nodes to exchange enough information to enable an SA iteration on existing data in the network before new data arrives, thereby overwhelming the network. In order to address the challenge of distributed SO in the presence of a mismatch between the communications and streaming rates, we propose and analyze two distributed SA techniques, each based on distributed averaging consensus and stochastic mirror descent. In particular, we present bounds on the convergence rates of these techniques and derive conditions---involving the number of nodes, network topology, the streaming rate, and the communications rate---under which our solutions achieve order-optimum convergence speed.\n\n\\subsection{Relationship to Prior Work}\nSA methods date back to the seminal work of Robbins and Monro~\\cite{Robbins.Monro.AMS51}, and recent work shows that, for stochastic convex optimization, SA methods can outperform SAA methods~\\cite{Nemirovski.etal.JOO09,Juditsky.etal.SS11}. Lan~\\cite{Lan.MP12} proposed {\\em accelerated stochastic mirror descent}, which achieves the best possible convergence rate for general stochastic convex problems. This method, which makes use of noisy subgradients of $\\psi(\\cdot)$ computed using incoming training samples, satisfies\n\\begin{equation}\\label{eqn:smd.rate}\n\tE[\\psi(\\mathbf{x}(T)) - \\psi(\\mathbf{x}^*)] \\leq O(1)\\left[\\frac{L}{T^2} + \\frac{\\mathcal{M}+\\sigma}{\\sqrt{T}} \\right],\n\\end{equation}\nwhere $\\mathbf{x}^*$ denotes the minimizer of \\eqref{eqn:SO_prob}, $\\sigma^2$ denotes variance of the subgradient noise, and $\\mathcal{M}$ and $L$ denote the Lipschitz constants associated with the non-smooth (convex) component of $\\psi$ and the gradient of the smooth (convex) component of $\\psi$, respectively. Further assumptions such as smoothness and strong convexity of $\\psi(\\cdot)$ and\/or presence of a structured regularizer term in $\\psi(\\cdot)$ can remove the dependence of the convergence rate on $\\mathcal{M}$ and\/or improve the convergence rate to $O(\\sigma\/T)$ \\cite{Hu.etal.NIPS09,Nemirovski.etal.JOO09,Xiao.JMLR10,Chen.etal.NIPS12}.\n\nThe problem of {\\em distributed} SO goes back to the seminal work of Tsitsiklis et al.~\\cite{Tsitsiklis.etal.ITAC1986}, which presents distributed first-order methods for SO and gives proofs of their asymptotic convergence. Myriad works since then have applied these ideas to other settings, each with different assumptions about the type of data, how the data are distributed across the network, and how distributed units process data and share information among themselves. In order to put our work in context, we review a representative sample of these works. A recent line of work was initiated by {\\em distributed gradient descent} (DGD) \\cite{Nedic.Ozdaglar.ITAC2009}, in which nodes descend using gradients of local data and collaborate via averaging consensus \\cite{Dimakis.etal:IEEE2010}. More recent works incorporate accelerated methods, time-varying or directed graphs, data structure, etc. \\cite{Srivastava.Nedic.IJSTSP2011,Tsianos.etal.Conf2012,Mokhtari.Ribeiro.JMLR2016,Bijra.etal.arxiv2016,LiChenEtAl.N16}. These works tend not to address the SA problem directly; instead, they suppose a linearly separable function consistent with SAA using local, independent and identically distributed (i.i.d.) data. The works \\cite{Ram.etal.JOTA2010,Duchi.etal.ITAC2012,RaginskyBouvrie.ConfCDC12,DuchiAgarwalEtAl.SJO12} do consider SA directly, but suppose that nodes engage in a single round of message passing per stochastic subgradient sample.\n\nWe conclude by discussing two lines of works \\cite{Dekel.etal.JMLR2012,Rabbat.ConfCAMSAP15,tsianos.rabbat.SIPN2016} that are most closely related to this work. In \\cite{Dekel.etal.JMLR2012}, nodes perform distributed SA by forming distributed mini-batch averages of stochastic gradients and using stochastic dual averaging.\nThe main assumption in this work is that nodes can compute {\\em exact} stochastic gradient averages (e.g., via {\\tt AllReduce} in parallel computing architectures). Under this assumption, it is shown in this work that there is an appropriate mini-batch size for which the nodes' iterates converge at the optimum (centralized) rate. However, the need for exact averages in this work is not suited to rate-limited (e.g., wireless) networks, in which mimicking the {\\tt AllReduce} functionality can be costly and challenging.\n\nThe need for exact stochastic gradient averages in~\\cite{Dekel.etal.JMLR2012} has been relaxed recently in~\\cite{tsianos.rabbat.SIPN2016}, in which nodes carry out distributed stochastic dual averaging by computing {\\em approximate} mini-batch averages of dual variables via distributed consensus. In addition, and similar to our work,~\\cite{tsianos.rabbat.SIPN2016} allows for a mismatch between the communications rate and the data streaming rate. Nonetheless, there are four main distinctions between \\cite{tsianos.rabbat.SIPN2016} and our work.\nFirst, we provide results for  stochastic {\\em composite} optimization, whereas \\cite{Dekel.etal.JMLR2012,tsianos.rabbat.SIPN2016} suppose a differentiable objective. Second, we consider distributed {\\em mirror descent}, which allows for a limited generalization to non-Euclidean settings. Third, we explicitly examine the impact of slow communications rate on performance, in particular highlighting the need for large mini-batches and their impact on convergence speed when the communications rate is slow. In \\cite{tsianos.rabbat.SIPN2016}, the optimum mini-batch size is first derived from \\cite{Dekel.etal.JMLR2012}, after which the communications rate needed to facilitate distributed consensus at the optimum mini-batch size is specified. While it appears to be possible to derive some of our results from a re-framing of the results of \\cite{tsianos.rabbat.SIPN2016}, it is crucial to highlight the trade-offs necessary under slow communications, which is not done in prior works. Finally, this work also presents a distributed {\\em accelerated} mirror descent approach to distributed SA; a somewhat surprising outcome is that acceleration substantially improves the convergence rate in networks with slow communications.\n\n\\subsection{Our Contributions}\nIn this paper, we present two strategies for distributed SA over networks with fast streaming data and slow communications links: distributed stochastic approximation mirror descent (D-SAMD) and accelerated distributed stochastic approximation mirror descent (AD-SAMD). In both cases, nodes first locally compute mini-batch stochastic subgradient averages to accommodate a fast streaming rate (or, equivalently, a slow communications rate), and then they collaboratively compute approximate network subgradient averages via distributed consensus. Finally, nodes individually employ mirror descent and accelerated mirror descent, respectively, on the approximate averaged subgradients for the next set of iterates.\n\nOur main theoretical contribution is the derivation of \\textit{upper} bounds on the convergence rates of D-SAMD and AD-SAMD. These bounds involve a careful analysis of the impact of imperfect subgradient averaging on individual nodes' iterates. In addition, we derive sufficient conditions for order-optimum convergence of D-SAMD and AD-SAMD in terms of the streaming and communications rates, the size and topology of the network, and the data statistics.\n\nTwo key findings of this paper are that distributed methods can achieve order-optimum convergence with small communication rates, as long as the number of nodes in the network does not grow too quickly as a function of the number of data samples each node processes, and that accelerated methods seem to offer order-optimum convergence in a larger regime than D-SAMD, thus potentially accommodating slower communications links relative to the streaming rate. By contrast, the convergence speeds of {\\em centralized} stochastic mirror descent and accelerated stochastic mirror descent typically differ only in higher-order terms. We hasten to point out that we do {\\em not} claim superior performance of D-SAMD and AD-SAMD versus other distributed methods. Instead, the larger goal is to establish the existence of methods for order-optimum stochastic learning in the fast-streaming, rate-limited regimes. D-SAMD and AD-SAMD should be best regarded as a proof of concept towards this end.\n\n\\subsection{Notation and Organization}\nWe typically use boldfaced lowercase and boldfaced capital letters (e.g., $\\mathbf{x}$ and $\\mathbf{W}$) to denote (possibly random) vectors and matrices, respectively. Unless otherwise specified, all vectors are assumed to be column vectors. We use $(\\cdot)^T$ to denote the transpose operation and $\\mathbf{1}$ to denote the vector of all ones. Further, we denote the expectation operation by $E[\\cdot]$ and the field of real numbers by $\\mathbb{R}$. We use $\\nabla$ to denote the gradient operator, while $\\odot$ denotes the Hadamard product. Finally, given two functions $p(r)$ and $q(r)$, we write $p(r) = O(q(r))$ if there exists a constant $C$ such that $\\forall r, p(r) \\leq C q(r)$, and we write $p(r) = \\Omega(q(r))$ if $q(r) = O(p(r))$.\n\nThe rest of this paper is organized as follows. In Section~\\ref{sect:setting}, we formalize the problem of distributed stochastic composite optimization. In Sections~\\ref{sect:mirror.descent} and \\ref{sect:accelerated.mirror.descent}, we describe D-SAMD and AD-SAMD, respectively, and also derive performance guarantees for these two methods. We examine the empirical performance of the proposed methods via numerical experiments in Section~\\ref{sect:numerical}, and we conclude the paper in Section~\\ref{sect:conclusion}. Proofs are provided in the appendix.\n\n\\section{Problem Formulation}\\label{sect:setting}\nThe objective of this paper is order-optimal, distributed minimization of the composite function\n\\begin{equation}\n\t\\psi(\\mathbf{x}) = f(\\mathbf{x}) + h(\\mathbf{x}),\n\\end{equation}\nwhere $\\mathbf{x} \\in X \\subset \\mathbb{R}^n$ and $X$ is convex and compact. The space $\\mathbb{R}^n$ is endowed with an inner product $\\langle \\cdot , \\cdot \\rangle$ that need not be the usual one and a norm $\\norm{\\cdot}$ that need not be the one induced by the inner product. In the following, the minimizer and the minimum value of $\\psi$ are denoted as:\n\\begin{equation}\n\t\\mathbf{x}^* \\triangleq \\arg\\min_{\\mathbf{x} \\in X} \\psi(\\mathbf{x}), \\quad \\text{and} \\quad \\psi^* \\triangleq \\psi(\\mathbf{x}^*).\n\\end{equation}\n\nWe now make a few assumptions on the smooth ($f(\\cdot)$) and non-smooth ($h(\\cdot)$) components of $\\psi$. The function $f: X \\to \\mathbb{R}$ is convex with Lipschitz continuous gradients, i.e.,\n\\begin{equation}\n\t\\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y})}_* \\leq L\\norm{\\mathbf{x} - \\mathbf{x}}, \\ \\forall \\ \\mathbf{x},\\mathbf{y} \\in X,\n\\end{equation}\nwhere $\\norm{\\cdot}_*$ is the dual norm associated with $\\langle \\cdot, \\cdot \\rangle$ and $\\norm{\\cdot}$:\n\\begin{equation}\n\t\\norm{\\mathbf{g}}_* \\triangleq \\sup_{\\norm{\\mathbf{x}} \\leq 1} \\langle \\mathbf{g}, \\mathbf{x} \\rangle.\n\\end{equation}\nThe function $h: X \\to \\mathbb{R}$ is convex and Lipschitz continuous:\n\\begin{equation}\n\t\\norm{h(\\mathbf{x}) - h(\\mathbf{y})} \\leq \\mathcal{M}\\norm{\\mathbf{x} - \\mathbf{y}}, \\forall \\ \\mathbf{x},\\mathbf{y} \\in X.\n\\end{equation}\nNote that $h$ need not have gradients; however, since it is convex we can consider its {\\em subdifferential}, denoted by $\\partial h(\\mathbf{y})$:\n\\begin{equation}\n\t\\partial h(\\mathbf{y}) = \\{\\mathbf{g}: h(\\mathbf{z}) \\geq h(\\mathbf{y}) + \\mathbf{g}^T(\\mathbf{z} - \\mathbf{y}), \\forall \\ \\mathbf{z} \\in X\\}.\n\\end{equation}\n\nAn important fact that will be used in this paper is that the \\emph{subgradient} $\\mathbf{g} \\in \\partial h$ of a Lipschitz-continuous convex function $h$ is bounded~\\cite[Lemma~2.6]{Shalev-Shwartz.Book2012}:\n\\begin{equation}\n\t\\norm{\\mathbf{g}}_* \\leq \\mathcal{M}, \\ \\forall \\mathbf{g} \\in \\partial h(\\mathbf{y}), \\ \\mathbf{y} \\in X.\n\\end{equation}\nConsequently, the gap between the subgradients of $\\psi$ is bounded: $\\forall \\mathbf{x},\\mathbf{y} \\in X$ and $\\mathbf{g}_\\mathbf{x} \\in \\partial h(\\mathbf{x})$, $\\mathbf{g}_\\mathbf{y} \\in \\partial h(\\mathbf{y})$, we have\n\\begin{align}\n\t\\norm{\\partial \\psi(\\mathbf{x}) - \\partial \\psi(\\mathbf{y})}_* &= \\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y}) + \\mathbf{g}_\\mathbf{x} - \\mathbf{g}_\\mathbf{y}}_* \\notag  \\\\\n    &\\leq \\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y})}_* + \\norm{\\mathbf{g}_\\mathbf{x} - \\mathbf{g}_\\mathbf{y}}_* \\notag\\\\\n    &\\leq L\\norm{\\mathbf{x} - \\mathbf{y}} + 2\\mathcal{M}. \\label{eqn:subgradient.bound}\n\\end{align}\n\n\\subsection{Distributed Stochastic Composite Optimization}\nOur focus in this paper is minimization of $\\psi(\\mathbf{x})$ over a network of $m$ nodes, represented by the undirected graph $G=(V,E)$. To this end, we suppose that nodes minimize $\\psi$ collaboratively by exchanging subgradient information with their neighbors at each communications round. Specifically, each node $i \\in V$ transmits a message at each communications round to each of its neighbors, defined as\n\\begin{equation}\n \\mathcal{N}_i = \\{j \\in V: (i,j) \\in E\\},\n\\end{equation}\nwhere we suppose that a node is in its own neighborhood, i.e., $i \\in \\mathcal{N}_i$. We assume that this message passing between nodes takes place without any error or distortion. Further, we constrain the messages between nodes to be members of the dual space of $X$ and to satisfy causality; i.e., messages transmitted by a node can depend only on its local data and previous messages received from its neighbors.\n\nNext, in terms of data generation, we suppose that each node $i \\in V$ queries a first-order stochastic ``oracle'' at a fixed rate---which may be different from the rate of message exchange---to obtain noisy estimates of the subgradient of $\\psi$ at different query points in $X$. Formally, we use `$t$' to index time according to {\\em data-acquisition} rounds and define $\\{\\xi_i(t) \\in \\Upsilon\\}_{t \\geq 1}$ to be a sequence of independent (with respect to $i$ and $t$) and identically distributed (i.i.d.) random variables with unknown probability distribution $P(\\xi)$. At each data-acquisition round $t$, node $i$ queries the oracle at search point $\\mathbf{x}_i(s)$ to obtain a point $G(\\mathbf{x}_i(s),\\xi_i(t))$ that is a noisy version of the subgradient of $\\psi$ at $\\mathbf{x}_i(s)$. Here, we use `$s$' to index time according to \\emph{search-point update} rounds, with possibly multiple data-acquisition rounds per search-point update. The reason for allowing the search-point update index $s$ to be different from the data-acquisition index $t$ is to accommodate the setting in which data (equivalently, subgradient estimates) arrive at a much faster rate than the rate at which nodes can communicate with each other; we will elaborate further on this in the next subsection.\n\nFormally, $G(\\mathbf{x},\\xi)$ is a Borel function that satisfies the following properties:\n\\begin{align}\n\tE[G(\\mathbf{x},\\xi)] &\\triangleq \\mathbf{g}(\\mathbf{x}) \\in  \\partial \\psi(\\mathbf{x}), \\quad \\text{and}\\\\\n    E[\\norm{G(\\mathbf{x},\\xi) - \\mathbf{g}(\\mathbf{x})}_*^2] &\\leq \\sigma^2,\n\\end{align}\nwhere the expectation is with respect to the distribution $P(\\xi)$. We emphasize that this formulation is equivalent to that in which the objective function is $\\psi(\\mathbf{x}) \\triangleq E[\\phi(\\mathbf{x},\\xi)]$, and where nodes in the network acquire data point $\\{\\xi_i(t)\\}_{i\\in V}$ at each data-acquisition round $t$ that are then used to compute the subgradients of $\\phi(\\mathbf{x},\\xi_i(t))$, which---in turn---are noisy subgradients of $\\psi(\\mathbf{x})$.\n\n\\subsection{Mini-batching for Rate-Limited Networks}\nA common technique to reduce the variance of the (sub)gradient noise and\/or reduce the computational burden in centralized SO is to average ``batches'' of oracle outputs into a single (sub)gradient estimate. This technique, which is referred to as \\emph{mini-batching}, is also used in this paper; however, its purpose in our distributed setting is to both reduce the subgradient noise variance \\emph{and} manage the potential mismatch between the communications rate and the data streaming rate. Before delving into the details of our mini-batch strategy, we present a simple model to parametrize the mismatch between the two rates. Specifically, let $\\rho >0$ be the {\\em communications ratio}, i.e. the fixed ratio between the rate of communications and the rate of data acquisition. That is, $\\rho \\geq 1$ implies nodes engage in $\\rho$ rounds of message exchanges for every data-acquisition round. Similarly, $\\rho < 1$ means there is one communications round for every $1\/\\rho$ data-acquisition rounds. We ignore rounding issues for simplicity.\n\nThe mini-batching in our distributed problem proceeds as follows. Each mini-batch round spans $b \\geq 1$ data-acquisition rounds and coincides with the search-point update round, i.e., each node $i$ updates its search point at the end of a mini-batch round. In each mini-batch round $s$, each node $i$ uses its current search point $\\mathbf{x}_i(s)$ to compute an average of oracle outputs\n\\begin{equation}\n\t\\theta_i(s) = \\frac{1}{b}\\sum_{t = (s-1)b +1}^{sb} G(\\mathbf{x}_i(s),\\xi_i(t)).\n\\end{equation}\nThis is followed by each node computing a new search point $\\mathbf{x}_i(s+1)$ using $\\theta_i(s)$ and messages received from its neighbors.\n\nIn order to analyze the mini-batching distributed SA techniques proposed in this work, we need to generalize the usual averaging property of variances to non-Euclidean norms.\n\\begin{lemma}\\label{lem:average.variance}\n\tLet $\\mathbf{z}_1,\\dots,\\mathbf{z}_k$ be i.i.d. random vectors in $\\mathbb{R}^n$ with $E[\\mathbf{z}_i] = 0$ and $E[\\norm{\\mathbf{z}_i}^2_*] \\leq \\sigma^2$. There exists a constant $C_* \\geq 0$, which depends only on $\\norm{\\cdot}$ and $\\langle \\cdot, \\cdot \\rangle$, such that\n    \\begin{equation}\n    \tE\\left[\\norm{\\frac{1}{k}\\sum_{i=1}^k \\mathbf{z}_i  }_*^2\\right] \\leq \\frac{C_* \\sigma^2}{k}.\n    \\end{equation}\n\\end{lemma}\n\\begin{IEEEproof}\n\tThis follows directly from the property of norm equivalence in finite-dimensional spaces.\n\\end{IEEEproof}\nIn order to illustrate Lemma~\\ref{lem:average.variance}, notice that when $\\norm{\\cdot} = \\norm{\\cdot}_1$, i.e., the $\\ell_1$ norm, and $\\langle \\cdot, \\cdot \\rangle$ is the standard inner product, the associated dual norm is the $\\ell_\\infty$ norm: $\\norm{\\cdot}_* = \\norm{\\cdot}_\\infty$. Since $\\norm{\\mathbf{x}}^2_\\infty \\leq \\norm{\\mathbf{x}}^2_2 \\leq n\\norm{\\mathbf{x}}_\\infty^2$, we have $C_* = n$ in this case. Thus, depending on the norm in use, the extent to which averaging reduces subgradient noise variance may depend on the dimension of the optimization space.\n\nIn the following, we will use the notation $\\mathbf{z}_i(s) \\triangleq \\theta_i(s) - \\mathbf{g}(\\mathbf{x}_i(s))$. Then, $E[\\norm{\\mathbf{z}_i(s)}_*^2] \\leq C_*\\sigma^2\/b$. We emphasize that the subgradient noise vectors $\\mathbf{z}_i(s)$ depend on the search points $\\mathbf{x}_i(s)$; we suppress this notation for brevity.\n\n\\subsection{Problem Statement}\nIt is straightforward to see that mini-batching induces a performance trade-off: Averaging reduces subgradient noise and processing time, but it also reduces the rate of search-point updates (and hence slows down convergence). This trade-off depends on the relationship between the streaming and communications rates. In order to carry out distributed SA in an order-optimal manner, we will require that the nodes collaborate by carrying out $r \\geq 1$ rounds of averaging consensus on their mini-batch averages $\\theta_i(s)$ in each mini-batch round $s$ (see Section~\\ref{sect:mirror.descent} for details). In order to complete the $r$ communication rounds in time for the next mini-batch round, we have the constraint\n\\begin{equation}\n\tr \\leq b \\rho.\n\\end{equation}\nIf communications is faster, or if the mini-batch rounds are longer, nodes can fit in more rounds of information exchange between each mini-batch round or, equivalently, between each search-point update. But when the mismatch factor $\\rho$ is small, the mini-batch size $b$ needed to enable sufficiently many consensus rounds may be so large that the reduction in subgradient noise is outstripped by the reduction in search-point updates and the resulting convergence speed is sub-optimum. In this context, our main goal is specification of sufficient conditions for $\\rho$ such that the resulting convergence speeds of the proposed distributed SA techniques are optimum.\n\n\\section{Distributed Stochastic Approximation\\\\Mirror Descent}\\label{sect:mirror.descent}\nIn this section we present our first distributed SA algorithm, called \\emph{distributed stochastic approximation mirror descent} (D-SAMD). This algorithm is based upon stochastic approximated mirror descent, which is a generalized version of stochastic subgradient descent. Before presenting D-SAMD, we review a few concepts that underlie mirror descent.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{figures\/timing_diag.png}\n\t\\caption{The different time counters for the rate-limited framework of this paper, here with $\\rho = 1\/2$, $b=4$, and $r=2$. In this particular case, over $T=8$ total data-acquisition rounds, each node receives $8$ data samples and computes $8$ (sub)gradients; it averages those (sub)gradients into $S=2$ mini-batch (sub)gradients; it then engages in $r=2$ rounds of consensus averaging to produce $S=2$ {\\em locally averaged} (sub)gradients; each of those (sub)gradients is finally used to update the search points twice, one for each $1 \\leq s \\leq S$. Note that, while not explicitly shown in the figure, the search point $\\mathbf{x}_i(1)$ is used for all computations spanning the data-acquisition rounds $5 \\leq t \\leq 8$.}\\label{fig:timing_diag}\n\\end{figure*}\n\n\\subsection{Stochastic Mirror Descent Preliminaries}\nStochastic mirror descent, presented in \\cite{Lan.MP12}, is a generalization of stochastic subgradient descent. This generalization is characterized by a {\\em distance-generating function} $\\omega: X \\to \\mathbb{R}$ that generalizes the Euclidean norm. The distance-generating function must be continuously differentiable and strongly convex with modulus $\\alpha$, i.e.\n\\begin{equation}\n\t\\langle \\nabla \\omega(\\mathbf{x}) - \\nabla \\omega(\\mathbf{y}), \\mathbf{x} - \\mathbf{y} \\rangle \\geq \\alpha \\norm{\\mathbf{x} - \\mathbf{y}}^2, \\forall \\ \\mathbf{x},\\mathbf{y} \\in X.\n\\end{equation}\nIn the convergence analysis, we will require two measures of the ``radius'' of $X$ that will arise in the convergence analysis, defined as follows:\n\\begin{equation*}\n\tD_\\omega \\triangleq \\sqrt{\\max_{\\mathbf{x} \\in X}\\omega(x) - \\min_{\\mathbf{x} \\in X} \\omega(x)}, \\quad \\Omega_\\omega \\triangleq \\sqrt{\\frac{2}{\\alpha} D_\\omega}.\n\\end{equation*}\n\nThe distance-generating function induces the {\\em prox function}, or the Bregman divergence $V : X \\times X \\to \\mathbb{R}_+$, which generalizes the Euclidean distance:\n\\begin{equation}\n\tV(\\mathbf{x},\\mathbf{z}) = \\omega(\\mathbf{z}) - (\\omega(\\mathbf{x}) + \\langle \\nabla \\omega(\\mathbf{x}), \\mathbf{z}-\\mathbf{x} \\rangle).\n\\end{equation}\nThe prox function $V(\\mathbf{x},\\cdot)$ inherits strong convexity from $\\omega(\\cdot)$, but it need not be symmetric or satisfy the triangle inequality. We define the {\\em prox mapping} $P_{\\mathbf{x}}: \\mathbb{R}^n \\to X$ as\n\\begin{equation}\\label{eqn:prox.mapping}\n\tP_\\mathbf{x}(\\mathbf{y}) = \\argmin_{\\mathbf{z} \\in X} \\langle \\mathbf{y}, \\mathbf{z}-\\mathbf{x} \\rangle + V(\\mathbf{x},\\mathbf{z}).\n\\end{equation}\nThe prox mapping generalizes the usual subgradient descent step, in which one minimizes the local linearization of the objective function regularized by the Euclidean distance of the step taken. In (centralized) stochastic mirror descent, one computes iterates of the form\n\\begin{align*}\n\t\\mathbf{x}(s+1) &= P_{\\mathbf{x}(s)}(\\gamma_s \\mathbf{g}(s))\n\\end{align*}\nwhere $\\mathbf{g}(s)$ is a stochastic subgradient of $\\psi(\\mathbf{x}(s))$, and $\\gamma_s$ is a step size. These iterates have the same form as (stochastic) subgradient descent; indeed, choosing $\\omega(\\mathbf{x}) = \\frac{1}{2}\\norm{\\mathbf{x}}^2_2$ as well as $\\langle \\cdot, \\cdot \\rangle$ and $\\norm{\\cdot}$ to be the usual ones results in subgradient descent iterations.\n\nOne can speed up convergence by choosing $\\omega(\\cdot)$ to match the structure of $X$ and $\\psi$. For example, if the optimization space $X$ is the unit simplex over $\\mathbb{R}^n$, one can choose $\\omega(\\mathbf{x}) = \\sum_i x_i \\log(x_i)$ and $\\norm{\\cdot}$ to be the $\\ell_1$ norm. This leads to $V(\\mathbf{x},\\mathbf{z}) = D(\\mathbf{z} || \\mathbf{x}))$, where $D(\\cdot || \\cdot)$ denotes the Kullback-Leibler (K-L) divergence between $\\mathbf{z}$ and $\\mathbf{x}$. %\nThis choice speeds up convergence on the order of $O\\left(\\sqrt{n\/\\log(n)}\\right)$ over using the Euclidean norm throughout. Along a similar vein, when $\\psi$ includes an $\\ell_1$ regularizer to promote a sparse minimizer, one can speed up convergence by choosing $\\omega(\\cdot)$ to be a $p$-norm with $p = \\log(n)\/(\\log(n)-1)$.\n\nIn order to guarantee convergence for D-SAMD, we need to restrict further the distance-generating function $\\omega(\\cdot)$. In particular, we require that the resulting prox mapping be 1-Lipschitz continuous in $\\mathbf{x},\\mathbf{y}$ pairs, i.e., $\\forall \\ \\mathbf{x},\\mathbf{x}^\\prime,\\mathbf{y},\\mathbf{y}^\\prime \\in \\mathbb{R}^n$,\n\\begin{equation*}\n\t\\norm{P_{\\mathbf{x}}(\\mathbf{y}) - P_{\\mathbf{x}^\\prime}(\\mathbf{y}^\\prime)} \\leq \\norm{\\mathbf{x}-\\mathbf{x}^\\prime} + \\norm{\\mathbf{y} - \\mathbf{y}\\prime}.\n\\end{equation*}\nThis condition is in addition to the conditions one usually places on the Bregman divergence for stochastic optimization; we will use it to guarantee that imprecise gradient averages make a bounded perturbation in the iterates of stochastic mirror descent. The condition holds whenever the prox mapping is the projection of a 1-Lipschitz function of $\\mathbf{x},\\mathbf{y}$ onto $X$. For example, it is easy to verify that this condition holds in the Euclidean setting. One can also show that when the distance-generating function $\\omega(\\mathbf{x})$ is an $\\ell_p$ norm for $p > 1$, the resulting prox mapping is 1-Lipschitz continuous in $\\mathbf{x}$ and $\\mathbf{y}$ as required.\n\nHowever, not all Bregman divergences satisfy this condition. One can show that the K-L divergence results in a prox mapping that is not Lipschitz. Consequently, while we present our results in terms of general prox functions, we emphasize that the results do not apply in all cases.\\footnote{We note further that it is possible to relax the constraint that the best Lipschitz constant be no larger than unity. This worsens the scaling laws---in particular, the required communications ratio $\\rho$ grows in $T$ rather than decreases---and we omit this case for brevity's sake.} One can think of the results primarily in the setting of Euclidean (accelerated) stochastic subgradient descent---for which case they are guaranteed to hold---with the understanding that one can check on a case-by-case basis to see if they hold for a particular non-Euclidean setting.\n\n\\subsection{Description of D-SAMD}\\label{sect:dsamd.description}\nHere we present in detail D-SAMD, which generalizes stochastic mirror descent to the setting of distributed, streaming data. In D-SAMD, nodes carry out iterations similar to stochastic mirror descent as presented in \\cite{Lan.MP12}, but instead of using local stochastic subgradients associated with the local search points, they carry out approximate consensus to estimate the {\\em average} of stochastic subgradients across the network. This reduces the subgradient noise at each node and speeds up convergence.\n\nLet $\\mathbf{W}$ be a symmetric, doubly-stochastic matrix consistent with the network graph $G$, i.e., $[\\mathbf{W}]_{ij} \\triangleq w_{ij}= 0$ if $(i,j) \\notin E$. Further suppose that $\\mathbf{W} - \\mathbf{1}\\mathbf{1}^T\/n$ has spectral radius strictly less than one, i.e. the second-largest eigenvalue magnitude is strictly less than one. This condition is guaranteed by choosing the diagonal elements of $\\mathbf{W}$ to be strictly greater than zero.\n\nNext, we focus on the case of constant step size $\\gamma$ and set it as $0 < \\gamma \\leq \\alpha\/(2L)$.\\footnote{It is shown in \\cite{Lan.MP12} that a constant step size is sufficient for order-optimal performance, so we adopt such a rule here.} For simplicity, we suppose that there is a predetermined number of data-acquisition rounds $T$, which leads to $S=T\/b$ mini-batch rounds. We detail the steps of D-SAMD in Algorithm \\ref{alg:standard}. Further, in Figure \\ref{fig:timing_diag} we illustrate the data acquisition round, mini-batch round, communication round, and search point update counters and their role in the D-SAMD algorithm.\n\n\\begin{algorithm}[t]\n  \\caption{Distributed stochastic approximation mirror descent (D-SAMD)\n    \\label{alg:standard}}\n    \\begin{algorithmic}[1]\n    \\Require Doubly-stochastic matrix $\\mathbf{W}$, step size $\\gamma$, number of consensus rounds $r$, batch size $b$, and stream of mini-batched subgradients $\\theta_i(s)$.\n    \\For{$i=1:m$}\n    \t\\State $\\mathbf{x}_i(1) \\gets \\min_{\\mathbf{x} \\in X} \\omega(\\mathbf{x})$ \\Comment{Initialize search points}\n    \\EndFor\n    \\For {$s=1:S$}\n        \\State $\\mathbf{h}_i^0(s) \\gets \\theta_i(s)$ \\Comment{Get mini-batched subgradients}\n        \\For{$q=1:r$, $i=1:m$}\n        \t\\State $\\mathbf{h}_i^q(s) \\gets \\sum_{j \\in \\mathcal{N}_i} w_{ij}\\mathbf{h}_j^{q-1}(s)$ \\Comment{Consensus rounds}\n        \\EndFor\n        \\For{$i=1:m$}\n        \t\\State $\\mathbf{x}_i(s+1) \\gets P_{\\mathbf{x}_i(s)}(\\gamma \\mathbf{h}_i^r(s))$ \\Comment{Prox mapping}\n            \\State $\\mathbf{x}_i^\\mathrm{av}(s+1) \\gets \\frac{1}{s}\\sum_{k=1}^s\\mathbf{x}_i(k)$ \\Comment{Average iterates}\n        \\EndFor\n    \\EndFor\n  \\end{algorithmic}\n  \\Return $\\mathbf{x}_i^{\\mathrm{av}}(S+1), i=1,\\dots,m.$\n\\end{algorithm}\n\nIn D-SAMD, each node $i$ initializes its iterate at the minimizer of $\\omega(\\cdot)$, which is guaranteed to be unique due to strong convexity. At each mini-batch round $s$, each node $i$ obtains its mini-batched subgradient and nodes engage in $r$ rounds of averaging consensus to produce the (approximate) average subgradients $\\mathbf{h}_i^r(s)$. Then, each node $i$ takes a mirror prox step, using $\\mathbf{h}^r_i(s)$ instead of its own mini-batched estimate. Finally, each node keeps a running average of its iterates, which is well-known to speed up convergence \\cite{Polyak.Juditsky.JCO1992}.\n\n\\subsection{Convergence Analysis}\nThe convergence rate of D-SAMD depends on the bias and variance of the approximate subgradient averages $\\mathbf{h}_i^r(s)$. In principle, averaging subgradients together reduces the noise variance and speeds up convergence. However, because averaging consensus using only $r$ communications rounds results in {\\em approximate} averages, each node takes a slightly different mirror prox step and therefore ends up with a different iterate. At each mini-batch round $s$, nodes then compute subgradients at different search points, leading to bias in the averages $\\mathbf{h}^r_i(s)$. This bias accumulates at a rate that depends on the subgradient noise variance, the topology of the network, and the number of consensus rounds per mini-batch round.\n\nTherefore, the first step in bounding the convergence speed of D-SAMD is to bound the bias and the variance of the subgradient estimates $\\mathbf{h}^r_i(s)$, which we do in the following lemma.\n\\begin{lemma}\\label{lem:consensus.error.norm}\n\tLet $0 \\leq \\lambda_2 < 1$ denote the magnitude of the second-largest (ordered by magnitude) eigenvalue of $\\mathbf{W}$. Define the matrices\n    \\begin{align*}\n    \t\\mathbf{H}(s) &\\triangleq [\\mathbf{h}_1^r(s), \\dots, \\mathbf{h}_m^r(s)], \\\\\n        \\mathbf{G}(s) &\\triangleq [\\mathbf{g}(\\mathbf{x}_1(s)), \\dots, \\mathbf{g}(\\mathbf{x}_m(s))], \\text{ and} \\\\\n        \\mathbf{Z}(s) &\\triangleq [\\mathbf{z}_1(s), \\dots, \\mathbf{z}_m(s)],\n    \\end{align*}\n    recalling that the subgradient noise $\\mathbf{z}_i(s)$ is defined with respect to the mini-batched subgradient $\\theta_i(s)$. Also define\n    \\begin{align*}\n    \t\\overline{\\mathbf{g}}(s) &\\triangleq \\frac{1}{m}\\sum_{i=1}^m \\mathbf{g}(\\mathbf{x}_i(s)), \\quad \\overline{\\mathbf{G}}(s) \\triangleq [\\overline{\\mathbf{g}}(s), \\dots, \\overline{\\mathbf{g}}(s)], \\text{ and}\\\\\n        \\overline{\\mathbf{z}}(s) &\\triangleq \\frac{1}{m}\\sum_{i=1}^m\\mathbf{z}_i(s), \\quad \\overline{\\mathbf{Z}}(s)  \\triangleq [\\overline{\\mathbf{z}}(s), \\cdots, \\overline{\\mathbf{z}}(s)],\n    \\end{align*}\n    where the matrices $\\overline{\\mathbf{G}}(s), \\overline{\\mathbf{Z}}(s) \\in \\mathbb{R}^{n \\times n}$ have identical columns. Finally, define the matrices\n    \\begin{align*}\n    \t\\mathbf{E}(s) &\\triangleq \\mathbf{G}(s)\\mathbf{W}^r - \\overline{\\mathbf{G}}(s) \\text{ and} \\\\\n        \\tilde{\\mathbf{Z}}(s) &\\triangleq \\mathbf{Z}(s)\\mathbf{W}^r - \\overline{\\mathbf{Z}}(s)\n    \\end{align*}\n    of average consensus error on the subgradients and subgradient noise, respectively. Then, the following facts are true. First, one can write $\\mathbf{H}(s)$ as\n    \\begin{equation}\\label{eqn:gradient.decomposition}\n    \t\\mathbf{H}(s) = \\overline{\\mathbf{G}}(s) + \\mathbf{E}(s) + \\overline{\\mathbf{Z}}(s) + \\tilde{\\mathbf{Z}}(s).\n    \\end{equation}\n    Second, the columns of $\\overline{\\mathbf{Z}}(s)$ satisfy\n    \\begin{equation}\n    \tE[\\norm{\\overline{\\mathbf{z}}(s)}_*^2] \\leq \\frac{C_*\\sigma^2}{mb}.\n    \\end{equation}\n    Finally,  the $i$th columns of $\\mathbf{E}(s)$ and $\\tilde{\\mathbf{Z}}(s)$, denoted by $\\mathbf{e}_i(s)$ and $\\tilde{\\mathbf{z}}_i(s)$, respectively, satisfy\n    \\begin{equation}\\label{eqn:gradient.average.norm}\n    \t\\norm{\\mathbf{e}_i(s)}_* \\leq \\max_{j,k} m^2\\sqrt{C_*}\\lambda_2^r \\norm{\\mathbf{g}_j(s) - \\mathbf{g}_k(s)}_*\n    \\end{equation}\n    and\n    \\begin{equation}\\label{eqn:average.gradient.noise.variance}\n    \tE[\\norm{\\tilde{\\mathbf{z}}_i(s)}_*^2] \\leq \\frac{\\lambda^{2r}m^2 C_* \\sigma^2}{b},\n    \\end{equation}\n    where we have used $\\mathbf{g}_j(s)$ as a shorthand for $\\mathbf{g}(\\mathbf{x}_j(s))$.\n\\end{lemma}\n\nThe next step in the convergence analysis is to bound the distance between iterates at different nodes. As long as iterates are not too far apart, the subgradients computed at different nodes have sufficiently similar means that averaging them together reduces the overall subgradient noise.\n\\begin{lemma}\\label{lem:iterate.gap}\n\tLet $a_s \\triangleq \\max_{i,j} \\norm{\\mathbf{x}_i(s) - \\mathbf{x}_j(s)}$. The moments of $a_s$ follow:\n    \\begin{align}\n    \tE[a_s] &\\leq \\frac{\\mathcal{M}+\\sigma\/\\sqrt{b}}{L}((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1), \\\\\n        E[a_s^2] &\\leq \\frac{(\\mathcal{M}+\\sigma\/\\sqrt{b})^2}{L^2}((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1)^2.\n    \\end{align}\n\\end{lemma}\n\nNow, we bound D-SAMD's expected gap to optimality.\n\\begin{theorem}\\label{thm:mirror.descent.convergence.rate}\n\tFor D-SAMD, the expected gap to optimality at each node $i$ satisfies\n    \\begin{multline}\\label{eqn:DSAMD.convergence.rate}\n    \tE[\\psi(\\mathbf{x}_i^{\\mathrm{av}}(S+1))] - \\psi^* \\leq \\\\ \\frac{2L\\Omega_\\omega^2}{\\alpha S} + \\sqrt{\\frac{2(4\\mathcal{M}^2 + 2\\Delta_S^2)}{\\alpha S}} +  \\sqrt{\\frac{\\alpha}{2}}\\frac{\\Xi_S D_\\omega}{L},\n    \\end{multline}\nwhere\n\\begin{align}\n\t\\Xi_s &\\triangleq \\left(\\mathcal{M}+\\frac{\\sigma}{\\sqrt{b}}\\right)(1+ m^2 \\sqrt{C_*} \\lambda_2^r)\\times\\nonumber\\\\\n&\\qquad\\qquad\\qquad ((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1) + 2\\mathcal{M}\n\\end{align}\nand\n\\begin{align}\n    \\Delta_s^2 &\\triangleq 2 \\left(\\mathcal{M}+\\frac{\\sigma}{\\sqrt{b}}\\right)^2(1+m^4 C_* \\lambda_2^{2r})\\times\\nonumber\\\\\n    &\\qquad\\qquad((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1)^2 + 4C_*\\sigma^2\/(mb) \\nonumber\\\\\n    &\\qquad\\qquad\\qquad+ 4\\lambda_2^{2r}C_*\\sigma^2 m^2\/b +4\\mathcal{M}\n\\end{align}\nquantify the moments of the effective subgradient noise.\n\\end{theorem}\n\nThe convergence rate proven in Theorem \\ref{thm:mirror.descent.convergence.rate} is akin to that provided in \\cite{Lan.MP12}, with $\\Delta_s^2$ taking the role of the subgradient noise variance. A crucial difference is the presence of the final term involving $\\Xi_s$. In \\cite{Lan.MP12}, this term vanishes because the intrinsic subgradient noise has zero mean. However, the equivalent gradient error in D-SAMD does not have zero mean in general. As nodes' iterates diverge, their subgradients differ, and the nonlinear mapping between iterates and subgradients results in noise with nonzero mean.\n\nThe critical question is how fast communication needs to be for order-optimum convergence speed, i.e., the convergence speed that one would obtain if nodes had access to other nodes' subgradient estimates at each round. After $S$ mini-batch rounds, the network has processed $mT$ data samples. Centralized mirror descent, with access to all $mT$ data samples in sequence, achieves the convergence rate \\cite{Lan.MP12}\n\\begin{equation*}\n\tO(1)\\left[\\frac{L}{mT} + \\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right].\n\\end{equation*}\nThe final term dominates the error as a function of $m$ and $T$ if $\\sigma^2  > 0$. In the following corollary we derive conditions under which the convergence rate of D-SAMD matches this term.\n\\begin{corollary}\\label{cor:mirror.descent.consensus.rounds}\n\tThe optimality gap for D-SAMD satisfies\n     \\begin{equation}\n        E[\\psi(\\mathbf{x}_i^\\mathrm{av}(S+1))] - \\psi^* = O\\left(\\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right),\n    \\end{equation}\n    provided the mini-batch size $b$, the communications ratio $\\rho$, the number of users $m$, and the Lipschitz constant $\\mathcal{M}$ satisfy\n    \\begin{align*}\n    \tb &= \\Omega\\left(1 + \\frac{\\log(mT)}{\\rho\\log(1\/\\lambda_2)}\\right), \\quad b = O\\left(\\frac{\\sigma T^{1\/2}}{m^{1\/2}}\\right),\\\\\n        \\rho &= \\Omega\\left(\\frac{m^{1\/2}\\log(mT)}{\\sigma T^{1\/2}\\log(1\/\\lambda_2)}\\right), \\quad T = \\Omega\\left(\\frac{m}{\\sigma^2}\\right), \\text{ and}\\\\\n        \\mathcal{M} &= O\\left(\\min\\left\\{\\frac{1}{m},\\frac{1}{\\sqrt{ m \\sigma^2 T}}\\right\\}\\right).\n    \\end{align*}\n\\end{corollary}\n\n\\subsection{Discussion}\nCorollary \\ref{cor:mirror.descent.consensus.rounds} gives new insights into influences of the communications and streaming rates, network topology, and mini-batch size on the convergence rate of distributed stochastic learning. In \\cite{Dekel.etal.JMLR2012}, a mini-batch size of $b=O(T^{1\/2})$ is prescribed---which is sufficient whenever gradient averages are perfect---and in \\cite{tsianos.rabbat.SIPN2016} the number of imperfect consensus rounds needed to facilitate the mini-batch size $b$ prescribed in \\cite{Dekel.etal.JMLR2012} is derived. By contrast, we derive a mini-batch condition sufficient to drive the effective noise variance to $O(\\sigma^2\/(mT))$ while taking into consideration the impact of imperfect subgradient averaging. This condition depends not only on $T$ but also on $m$, $\\rho$, $\\lambda_2$, and $\\sigma^2$---indeed, for all else constant, the optimum mini-batch size is merely $\\Omega(\\log(T))$. Then, the condition on $\\rho$ essentially ensures that $b = O(T^{1\/2})$ as specified in \\cite{Dekel.etal.JMLR2012}.\n\nWe note that Corollary \\ref{cor:mirror.descent.consensus.rounds} imposes a strict requirement on $\\mathcal{M}$, the Lipschitz constant of the non-smooth part of $\\psi$. Essentially the non-smooth part must vanish as $m$, $T$, or $\\sigma^2$ becomes large. This is because the contribution of $h(\\mathbf{x})$ to the convergence rate depends only on the number of iterations taken, not on the noise variance. Reducing the effective subgradient noise via mini-batching has no impact on this contribution, so we require the Lipschitz constant $\\mathcal{M}$ to be small to compensate.\n\nFinally, we note that Corollary \\ref{cor:mirror.descent.consensus.rounds} dictates the relationship between the size of the network and the number of data samples obtained at each node. Leaving the terms besides $m$ and $T$ constant, Corollary \\ref{cor:mirror.descent.consensus.rounds} requires $T = \\Omega(m)$, i.e. the number of nodes in the network should scale no faster than the number of data samples processed per node. This is a relatively mild condition for big data applications; many applications involve data streams that are large relative to the size of the network. Furthermore, ignoring the $\\log(mT)$ term and assuming $\\lambda_2$ and $\\sigma$ to be fixed, Corollary \\ref{cor:mirror.descent.consensus.rounds} indicates that a communication ratio of $\\rho = \\Omega\\big(\\sqrt{m\/T}\\big)$ is sufficient for order optimality; i.e., nodes need to communicate at least $\\Omega\\big(\\sqrt{m\/T}\\big)$ times per data sample. This means that if $T$ scales faster than $\\Omega(m)$ then the required communications ratio approaches zero in this case as $m,T \\to \\infty$. In particular, fast stochastic learning is possible in expander graphs, for which the spectral gap $1-\\lambda_2$ is bounded away from zero, even in communication rate-limited scenarios. For graph families that are poor expanders, however, the required communications ratio depends on the scaling of $\\lambda_2$ as a function of $m$.\n\n\\section{Accelerated Distributed Stochastic Approximation Mirror Descent}\\label{sect:accelerated.mirror.descent}\nIn this section, we present {\\em accelerated} distributed stochastic approximation mirror descent (AD-SAMD), which distributes the accelerated stochastic approximation mirror descent proposed in \\cite{Lan.MP12}. The centralized version of accelerated mirror descent achieves the optimum convergence rate of\n\\begin{equation*}\n\tO(1)\\left[\\frac{L}{T^2}+\\frac{\\mathcal{M} + \\sigma^2}{\\sqrt{T}} \\right].\n\\end{equation*}\nConsequently, we will see that the convergence rate of AD-SAMD has $1\/S^2$ as its first term. This faster convergence in $S$ allows for more aggressive mini-batching, and the resulting conditions for order-optimal convergence are less stringent.\n\n\\subsection{Description of AD-SAMD}\nThe setting for AD-SAMD is the same as in Section \\ref{sect:mirror.descent}. We again suppose a distance function $\\omega: X \\to \\mathbb{R}$, its associated prox function\/Bregman divergence $V: X \\times X \\to \\mathbb{R}$, and the resulting (Lipschitz) prox mapping $P_x: \\mathbb{R}^n \\to X$.\n\nAs in Section \\ref{sect:dsamd.description}, we suppose a mixing matrix $\\mathbf{W} \\in \\mathbb{R}^{m \\times m}$ that is symmetric, doubly stochastic, consistent with $G$, and has nonzero spectral gap. The main distinction between accelerated and standard mirror descent is the way one averages iterates. Rather than simply average the sequence of iterates, one maintains several distinct sequences of iterates, carefully averaging them along the way. This involves two sequences of step sizes $\\beta_s \\in [1,\\infty)$ and $\\gamma_s \\in \\mathbb{R}$, which are not held constant. Again we suppose that the number of mini-batch rounds $S=T\/b$ is predetermined. We detail the steps of AD-SAMD in Algorithm \\ref{alg:accelerated}.\n\n\\begin{algorithm}[t]\n  \\caption{Accelerated distributed stochastic approximation mirror descent (AD-SAMD)\n    \\label{alg:accelerated}}\n    \\begin{algorithmic}[1]\n    \\Require Doubly-stochastic matrix $\\mathbf{W}$, step size sequences $\\gamma_s$, $\\beta_s$, number of consensus rounds $r$, batch size $b$, and stream of mini-batched subgradients $\\theta_i(s)$.\n    \\For{$i=1:m$}\n    \t\\State $\\mathbf{x}_i(1),\\mathbf{x}^\\mathrm{md}_i(1),\\mathbf{x}^\\mathrm{ag}_i(1) \\gets \\min_{\\mathbf{x} \\in X} \\omega(\\mathbf{x})$ \\Comment{Initialize search points}\n    \\EndFor\n    \\For {$s=1:S$}\n        \\For {$i=1:m$}\n        \t\\State $\\mathbf{x}_i^\\mathrm{md}(s) \\gets \\beta_s^{-1}\\mathbf{x}_i(s) + (1-\\beta^{-1}_s)\\mathbf{x}_i^\\mathrm{ag}(s)$\n            \\State $\\mathbf{h}_i^0(s) \\gets \\theta_i(s)$ \\Comment{Get mini-batched subgradients}\n        \\EndFor\n\n        \\For{$q=1:r$, $i=1:m$}\n        \t\\State $\\mathbf{h}_i^q(s) \\gets \\sum_{j \\in \\mathcal{N}_i} w_{ij}\\mathbf{h}_j^{q-1}(s)$ \\Comment{Consensus rounds}\n        \\EndFor\n        \\For{$i=1:m$}\n        \t\\State $\\mathbf{x}_i(s+1) \\gets P_{\\mathbf{x}_i(s)}(\\gamma_s \\mathbf{h}_i^r(s))$ \\Comment{Prox mapping}\n            \\State $\\mathbf{x}^\\mathrm{ag}_i(s+1) \\gets \\beta_s^{-1}\\mathbf{x}_i(s+1) + (1-\\beta_s^{-1})\\mathbf{x}_i^\\mathrm{ag}(s)$\n        \\EndFor\n    \\EndFor\n  \\end{algorithmic}\n  \\Return $\\mathbf{x}_i^{\\mathrm{ag}}(S+1), i=1,\\dots,m.$\n\\end{algorithm}\n\nThe sequences of iterates $\\mathbf{x}_i(s)$, $\\mathbf{x}_i^{\\mathrm{md}}(s)$, and $\\mathbf{x}^\\mathrm{ag}_i(s)$ are interrelated in complicated ways; we refer the reader to \\cite{Lan.MP12} for an intuitive explanation of these iterations.\n\n\\subsection{Convergence Analysis}\nAs with D-SAMD, the convergence analysis relies on bounds on the bias and variance of the averaged subgradients. To this end, we note first that Lemma \\ref{lem:consensus.error.norm} also holds for AD-SAMD, where $\\mathbf{H}(s)$ has columns corresponding to noisy subgradients evaluated at $\\mathbf{x}_i^\\mathrm{md}(s)$. Next, we bound the distance between iterates at different nodes. This analysis is somewhat more complicated due to the relationships between the three iterate sequences.\n\\begin{lemma}\\label{lem:accelerated.iterate.gap}\n\tLet\n    \\begin{align*}\n    \ta_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}^\\mathrm{ag}_i(s) - \\mathbf{x}^\\mathrm{ag}_j(s)}, \\\\\n        b_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}_i(s) - \\mathbf{x}_j(s)}, \\text{ and} \\\\\n        c_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}^\\mathrm{md}_i(s) - \\mathbf{x}^\\mathrm{md}_j(s)}.\n    \\end{align*}\n    Then, the moments of $a_s$, $b_s$, and $c_s$ satisfy:\n    \\begin{align*}\n    \tE[a_s],E[b_s],E[c_s] &\\leq \\frac{\\mathcal{M} \\!\\!+\\! \\sigma\/\\sqrt{b}}{L}((1 \\!\\!+\\! 2\\gamma_s m^2 \\sqrt{C_*}L\\lambda_2^r)^s \\!-\\! 1), \\\\\n        E[a_s^2],E[b_s^2],E[c_s^2] &\\leq \\frac{(\\mathcal{M} \\!\\!+\\! \\sigma\/\\sqrt{b})^2}{L^2}(\\!(1 \\!\\!+\\! 2\\gamma_s m^2 \\!\\!\\sqrt{C_*} L\\lambda_2^r)^s \\!\\!-\\!\\! 1)^2.\n    \\end{align*}\n\\end{lemma}\n\nNow, we bound the expected gap to optimality of the AD-SAMD iterates.\n\\begin{theorem}\\label{thm:accelerated.optimality.gap}\n\tFor AD-SAMD, there exist step size sequences $\\beta_s$ and $\\gamma_s$ such that the expected gap to optimality satisfies\n    \\begin{multline}\n    \tE[\\Psi(\\mathbf{x}_i^\\mathrm{ag}(S+1))] - \\Psi^* \\leq \\frac{8 L D_{\\omega,X}^2}{\\alpha S^2} + \\\\ 4 D_{\\omega,X}\\sqrt{\\frac{4M + \\Delta_S^2}{\\alpha S}} + \\sqrt{\\frac{32}{\\alpha}}D_{\\omega,X}\\Xi_S,\n    \\end{multline}\n    where\n    \\begin{multline*}\n    \t\\Delta_\\tau^2 = 2(\\mathcal{M}+\\sigma\/\\sqrt{b})^2((1+ 2\\gamma_\\tau m^2 \\sqrt{C_*} L\\lambda_2^r)^\\tau-1)^2 + \\\\ \\frac{4 C_* \\sigma^2}{b}(\\lambda_2^{2r}m^2 + 1\/m) + 4\\mathcal{M}.\n     \\end{multline*}\n     and\n     \\begin{multline*}\n        \\Xi_\\tau = (\\mathcal{M} + \\sigma\/\\sqrt{b})(1+\\sqrt{C_*}m^2\\lambda_2^r) \\times \\\\ ((1+2\\gamma_\\tau m^2\\sqrt{C_*}L\\lambda_2^r)^\\tau-1) + 2\\mathcal{M}.%\n    \\end{multline*}\n\\end{theorem}\n\nAs with D-SAMD, we study the conditions under which AD-SAMD achieves order-optimum convergence speed. The centralized version of accelerated mirror descent, after processing the $mT$ data samples that the network sees after $S$ mini-batch rounds, achieves the convergence rate\n\\begin{equation*}\n\tO(1)\\left[ \\frac{L}{(mT)^2} + \\frac{\\mathcal{M}+\\sigma}{\\sqrt{mT}}\\right].\n\\end{equation*}\nThis is the optimum convergence rate under any circumstance. In the following corollary, we derive the conditions under which the convergence rate matches the second term, which usually dominates the error when $\\sigma^2 > 0$.\n\\begin{corollary}\\label{cor:accelerated.descent.consensus.rounds}\n\tThe optimality gap satisfies\n    \\begin{equation*}\n    \tE[\\psi(\\mathbf{x}^\\mathrm{ag}_i(S+1)] - \\psi^* = O\\left(\\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right),\n    \\end{equation*}\n    provided\n    \\begin{align*}\n    \tb &= \\Omega\\left(1 + \\frac{\\log(mT)}{\\rho\\log(1\/\\lambda_2)}\\right), \\quad b = O\\left(\\frac{\\sigma^{1\/2}T^{3\/4}}{m^{1\/4}}\\right), \\\\\n        \\rho &= \\Omega\\left( \\frac{m^{1\/4}\\log(m T)}{\\sigma T^{3\/4}\\log(1\/\\lambda_2)} \\right), \\quad T = \\Omega\\left(\\frac{m^{1\/3}}{\\sigma^2}\\right), \\text{ and}\\\\\n        \\mathcal{M} &= O\\left(\\min\\left\\{\\frac{1}{m},\\frac{1}{\\sqrt{ m \\sigma^2 T}}\\right\\}\\right).\n    \\end{align*}\n\\end{corollary}\n\n\\subsection{Discussion}\nThe crucial difference between the two schemes is that AD-SAMD has a convergence rate of $1\/S^2$ in the absence of noise and non-smoothness. This faster term, which is often negligible in centralized mirror descent, means that AD-SAMD tolerates more aggressive mini-batching without impact on the order of the convergence rate. As a result, while the condition on the mini-batch size $b$ is the same in terms of $\\rho$, the condition on $\\rho$ is relaxed by $1\/4$ in the exponents of $m$ and $T$. This is because the condition $b = O(T^{1\/2})$, which holds for standard stochastic SO methods, is relaxed to $b = O(T^{3\/4})$ for accelerated stochastic mirror descent.\n\nSimilar to Corollary \\ref{cor:mirror.descent.consensus.rounds}, Corollary \\ref{cor:accelerated.descent.consensus.rounds} prescribes a relationship between $m$ and $T$, but the relationship for AD-SAMD is $T~=~\\Omega(m^{1\/3})$, holding all but $m,T$ constant. This again is due to the relaxed mini-batch condition $b = O(T^{3\/4})$ for accelerated stochastic mirror descent. Furthermore, ignoring the $\\log$ term, Corollary \\ref{cor:accelerated.descent.consensus.rounds} indicates that a communications ratio $\\rho = \\Omega\\left(\\frac{m^{1\/4}}{T^{3\/4}}\\right)$ is needed for well-connected graphs such as expander graphs. In this case, as long as $T$ grows faster than the cube root of $m$, order-optimum convergence rates can be obtained even for small communications ratio. Thus, the use of accelerated methods increases the domain in which order optimum rate-limited learning is guaranteed.\n\n\\section{Numerical Example: Logistic Regression}\\label{sect:numerical}\nTo demonstrate the scaling laws predicted by Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds} and to investigate the empirical performance of D-SAMD and AD-SAMD, we consider supervised learning via binary logistic regression. Specifically, we assume each node observes a stream of pairs $\\xi_i(t) = (y(t),l(t))$ of\ndata points $y_i(t) \\in \\mathbb{R}^d$ and their labels $l_i(t) \\in \\{0,1\\}$, from which it learns a classifier with the log-likelihood function\n\\begin{equation*}\n\tF(\\mathbf{x},x_0,\\mathbf{y},l) = l (\\mathbf{y}^T\\mathbf{x} + x_0) - \\log(1+\\exp(\\mathbf{y}^T\\mathbf{x} + x_0))\n\\end{equation*}\nwhere $\\mathbf{x} \\in \\mathbb{R}^d$ and $x_0 \\in \\mathbb{R}$ are regression coefficients.\n\nThe SO task is to learn the optimum regression coefficients $\\mathbf{x},x_0$. In terms of the framework of this paper, $\\Upsilon = (\\mathbb{R}^d \\times \\{0,1\\})$, and $X = \\mathbb{R}^{d+1}$ (i.e., $n = d+1$). We use the Euclidean norm, inner product, and distance-generating function to compute the prox mapping. The convex objective function is the negative of the log-likelihood function, averaged over the unknown distribution of the data, i.e.,\n\\begin{equation*}\n\t\\psi(\\mathbf{x}) = -E_{\\mathbf{y},l}[F(\\mathbf{x},x_0,\\mathbf{y},l)].\n\\end{equation*}\nMinimizing $\\psi$ is equivalent to performing maximum likelihood estimation of the regression coefficients \\cite{bishop:book06}.\n\nWe examine performance on synthetic data so that there exists a ``ground truth'' distribution with which to compute $\\psi(\\mathbf{x})$ and evaluate empirical performance. We suppose that the data follow a Gaussian distribution. For $l_i(t)~\\in~\\{0,1\\}$, we let $\\mathbf{y}_i(t)~\\sim~\\mathcal{N}(\\mu_{l_i(t)},\\sigma_r^2\\mathbf{I})$, where $\\mu_{l(t)}$ is one of two mean vectors, and $\\sigma_r^2 > 0$ is the noise variance.\\footnote{The variance $\\sigma_r^2$ is distinct from the resulting gradient noise variance $\\sigma^2$.} For this experiment, we draw the elements $\\mu_0$ and $\\mu_1$ randomly from the standard normal distribution, let $d=20$, and choose $\\sigma_r^2=2$. We consider several network topologies, as detailed in the next subsections.\n\nWe compare the performance of D-SAMD and AD-SAMD against several other schemes. As a best-case scenario, we consider {\\em centralized} mirror descent, meaning that at each data-acquisition round $t$ all $m$ data samples and their associated gradients are available at a single machine, which carries out stochastic mirror descent and {\\em accelerated} stochastic mirror descent. These algorithms naturally have the best average performance. As a baseline, we consider {\\em local} (accelerated) stochastic mirror descent, in which nodes simply perform mirror descent on their own data streams without collaboration. This scheme does benefit from an insensitivity to the communications ratio $\\rho$, and no mini-batching is required, but it represents a minimum standard for performance in the sense that it does not require collaboration among nodes.\n\nFinally, we consider a communications-constrained adaptation of {\\em distributed gradient descent} (DGD), introduced in \\cite{Nedic.Ozdaglar.ITAC2009}, where local subgradient updates are followed by a single round of consensus averaging on the search points $\\mathbf{x}_i(s)$. DGD implicitly supposes that $\\rho=1$. To handle the $\\rho < 1$ case, we consider two adaptations: {\\em naive} DGD, in which data samples that arrive between communications rounds are simply discarded, and {\\em mini-batched} DGD, in which nodes compute {\\em local} mini-batches of size $b=1\/\\rho$, take gradient updates with the local mini-batch, and carry out a consensus round. While it is not designed for the communications rate-limited scenario, DGD has good performance in general, so it represents a natural alternative against which to compare the performance of D-SAMD and AD-SAMD.\n\n\\subsection{Fully Connected Graphs}\nFirst, we consider the simple case of a fully connected graph, in which $E$ is the set of all possible edges, and the obvious mixing matrix is $\\mathbf{W}= \\mathbf{1}\\mathbf{1}^T\/n$, which has $\\lambda_2 = 0$. This represents a best-case scenario in which to validate the theoretical claims made above. We choose $\\rho = 1\/2$ to examine the regime of low communications ratio, and we let $m$ and $T$ grow according to two regimes: $T = m$, and $T = \\sqrt{m}$, which are the regimes in which D-SAMD and AD-SAMD are predicted to give order-optimum performance, respectively. The constraint on mini-batch size per Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds} is trivial, so we take $b=2$ to ensure that nodes can average each mini-batch gradient via (perfect) consensus. We select the following step-size parameter $\\gamma$: $0.5$ and $2$ for (local and centralized) stochastic mirror descent (MD) and accelerated stochastic mirror descent (A-MD), respectively; 5 for both variants of DGD; and $5$ and $20$ for D-SAMD and AD-SAMD, respectively.\\footnote{While the accelerated variant of stochastic mirror descent makes use of two\nsequences of step sizes, $\\beta_s$ and $\\gamma_s$, these two sequences can be expressed as a function of a single parameter $\\gamma$; see, e.g., the proof of Theorem~\\ref{thm:accelerated.optimality.gap}.} These values were selected via trial-and-error to give good performance for all algorithms; future work involves the use of adaptive step size rules such as AdaGrad and ADAM~\\cite{Duchi:JMLR2011,Kingma:ICLR2015}.\n\n\\begin{figure}[htb]\n  \\centering\n  \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fullConnT_m.eps}\n   \\caption{$T = m$}\n   \\end{subfigure}\n     \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fullConnT_Sq_m.eps}\n    \\caption{$T = \\sqrt{m}$}\n   \\end{subfigure}\n   \\caption{Performance of different schemes for a fully connected graph on $\\log$-$\\log$ scale. The dashed lines (without markers) in (a) and (b) correspond to the asymptotic performance upper bounds for D-SAMD and AD-SAMD predicted by the theoretical analysis.}\n\t\\label{fig:fully.connected}\n\\end{figure}\n\nIn Figure~\\ref{fig:fully.connected}(a) and Figure~\\ref{fig:fully.connected}(b), we plot the performance averaged over 1200 and 2400 independent instances of the problem, respectively. We also plot the order-wise theoretical performance $1\/\\sqrt{mT}$, which has a constant slope on the log-log axis. As expected, the distributed methods significantly outperform the local methods. The performance of the distributed methods is on par with the asymptotic theoretical predictions, as seen by the slope of the performance curves, with the possible exception of D-SAMD for $T = m$. However, we observe that D-SAMD performance is at least as good as predicted by theory for $T = \\sqrt{m}$, a regime in which optimality is not guaranteed for D-SAMD. This suggests the possibility that the requirement that $T = \\Omega(m)$ for D-SAMD is an artifact of the analysis, at least for this problem.\n\n\\subsection{Expander Graphs}\nFor a more realistic setting, we consider {\\em expander graphs}, which are families of graphs that have spectral gap $1-\\lambda_2$ bounded away from zero as $m \\to \\infty$. In particular, we use 6-regular graphs, i.e., regular graphs in which each node has six neighbors, drawn uniformly from the ensemble of such graphs. Because the spectral gap is bounded away from zero for expander graphs, one can more easily examine whether performance of D-SAMD and AD-SAMD agrees with the ideal scaling laws discussed in Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds}. At the same time, because D-SAMD and AD-SAMD make use of imperfect averaging, expander graphs also allow us to examine non-asymptotic behavior of the two schemes. Per Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds}, we choose $b = \\frac{1}{10}\\frac{\\log(mT)}{\\rho \\log(1\/\\lambda_2)}$. While this choice is guaranteed to be sufficient for optimum asymptotic performance, we chose the multiplicative constant $1\/10$ via trial-and-error to give good non-asymptotic performance.\n\n\\begin{figure}[htb]\n  \\centering\n  \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/expanderT_m.eps}\n   \\caption{$T = m$}\n   \\end{subfigure}\n     \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/expanderT_Sq_m.eps}\n    \\caption{$T = \\sqrt{m}$}\n   \\end{subfigure}\n   \\caption{Performance of different schemes for $6$-regular expander graphs on $\\log$-$\\log$ scale, for $\\rho = 1\/2$. Dashed lines once again represent asymptotic theoretical upper bounds on performance.}\n\t\\label{fig:expander}\n\\end{figure}\n\nIn Figure \\ref{fig:expander} we plot the performance averaged over 600 problem instances. We again take $\\rho = 1\/2$, and consider the regimes $T = m$ and $T = \\sqrt{m}$. The step sizes are the same as in the previous subsection except that $\\gamma = 2.5$ for D-SAMD when $T = \\sqrt{m}$, $\\gamma = 28$ for AD-SAMD when $T = m$, and $\\gamma= 8$ for AD-SAMD when $T = \\sqrt{m}$. Again, we see that AD-SAMD and D-SAMD outperform local methods, while their performance is roughly in line with asymptotic theoretical predictions. The performance of DGD, on the other hand, depends on the regime: For $T = m$, it appears to have order-optimum performance, whereas for $T = \\sqrt{m}$ it has suboptimum performance on par with local methods. The reason for the dependency on regime is not immediately clear and suggests the need for further study into DGD-style methods in the case of rate-limited networks.\n\n\\subsection{Erd\\H{o}s-Renyi Graphs}\nFinally, we consider {\\em Erd\\H{o}s-Renyi} graphs, in which a random fraction (in this case $0.1$) of possible edges are chosen. These graphs are not expanders, and their spectral gaps are not bounded. Therefore, order-optimum performance is not easy to guarantee, since the conditions on the rate and the size of the network depend on $\\lambda_2$, which is not guaranteed to be well behaved. We again take $\\rho = 1\/2$, consider the regimes $T = m$ and $T = \\sqrt{m}$, and again we choose $b = \\frac{1}{10}\\frac{\\log(mT)}{\\rho \\log(1\/\\lambda_2)}$. The step sizes are chosen to be the same as for expander graphs in both regimes.\n\n\\begin{figure}[htb]\n  \\centering\n  \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/ER_T_m.eps}\n   \\caption{$T = m$}\n   \\end{subfigure}\n     \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/ER_T_Sq_m.eps}\n    \\caption{$T = \\sqrt{m}$}\n   \\end{subfigure}\n   \\caption{Performance of different schemes on Erd\\H{o}s-Renyi graphs, for $\\rho = 1\/2$, displayed using $\\log$-$\\log$ scale.}\n\t\\label{fig:erdos}\n\\end{figure}\n\nOnce again, we observe a clear distinction in performance between local and distributed methods; in particular, all distributed methods (including DGD) appear to show near-optimum performance in both regimes. However, as expected the performance is somewhat more volatile than in the case of expander graphs, especially for the case of $T = m$, and it is possible that the trends seen in these plots will change as $T$ and $m$ increase.\n\n\\section{Conclusion}\\label{sect:conclusion}\nWe have presented two distributed schemes, D-SAMD and AD-SAMD, for convex stochastic optimization over networks of nodes that collaborate via rate-limited links. Further, we have derived sufficient conditions for the order-optimum convergence of D-SAMD and AD-SAMD, showing that accelerated mirror descent provides a foundation for distributed SO that better tolerates slow communications links. These results characterize relationships between network communications speed and the convergence speed of stochastic optimization.\n\nA limitation of this work is that we are restricted to settings in which the prox mapping is Lipschitz continuous, which excludes important Bregman divergences such as the Kullbeck-Liebler divergence. Further, the conditions for optimum convergence restrict the Lipschitz constant of non-smooth component of the objective function to be small. Future work in this direction includes study of the limits on convergence speed for more general divergences and composite objective functions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecent developments, initiated in\n\\cite{Bagger:2006sk,Gustavsson:2007vu}, which led to important\nprogress in understanding the holographic duality between $D=3$\nsuperconformal theories and type IIA string\/M--theory on $AdS_4$\nhave revived an interest in studying strings and branes in\nsupergravity backgrounds whose bosonic subspace is $AdS_4\\times\nM^{6}$ and $AdS_4\\times M^{7}$, respectively, where $M^6$ is a\ncompactified manifold of $D=10$ type IIA supergravity and $M^7$ is\nits Hopf fibration counterpart in $D=11$ supergravity (or\nM--theory). Examples of interest include the supergravity solutions\nwith $M^6=CP^3$ and $M^7=S^7\/Z_k$ (with an integer $k$ being the\nChern--Simons theory level) and their squashings.\n\n\nIn particular, the ${\\cal N}=6$ Chern-Simons theory with the gauge\ngroup $U(N)_k\\times U(N)_{-k}$ \\cite{Aharony:2008ug} has been\nconjectured to describe, from the $CFT_3$ side, M--theory on $AdS_4\n\\times S^7\/Z_k$. In the limit of the parameter space of the ABJM\ntheory in which the 't Hooft coupling $\\lambda={N\/k}$ is\n$\\lambda^{5\/2}<<N^2$ and $k>>1$, the bulk description is given in\nterms of perturbative type IIA string theory on the $AdS_4\n\\times CP^3$ background. To analyze this new type of holographic\ncorrespondence from the bulk theory side, an explicit form of the\naction for the superstring in $AdS_4 \\times CP^3$ superspace is\nrequired.\n\nIn contrast to \\emph{e.g.} the case of type IIB string theory in\n$AdS_5 \\times S^5$ superspace which preserves the maximum\nnumber of 32 supersymmetries and is thus described by the supercoset\n$PSU(2,2|4)\/SO(1,4)\\times SO(5)$, the case of type IIA string theory\non $AdS_4 \\times CP^3$ is more complicated since $AdS_4 \\times CP^3$\npreserves only 24 of 32 supersymmetries. As a consequence, the\ncomplete type IIA superspace with 32 fermionic coordinates, that\nsolves the IIA supergravity constraints for the $AdS_4 \\times CP^3$\nvacuum solution, is not a coset superspace. This superspace has been\nconstructed in\n\\cite{Gomis:2008jt} by dimensional reduction\nof the $AdS_4\\times S^7\/Z_k$ solution of $D=11$ supergravity\ndescribed by the supercoset $OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$\nwith 32 fermionic coordinates. The construction of\n\\cite{Gomis:2008jt} has generalized to superspace the results of\n\\cite{Giani:1984wc,Nilsson:1984bj,Sorokin:1985ap} on the relation of\n$AdS_4 \\times M^6$ solutions of $D=10$ type IIA supergravity and\n$AdS_4\\times M^7$ solutions of $D=11$ supergravity by identifying the\ncompact manifolds $M^7$ as $S^1$ Hopf fibrations over corresponding\n$M^6$.\n\nIn  \\cite{Gomis:2008jt} it has been shown that the supercoset space\n$OSp(6|4)\/U(3)\\times SO(1,3)$ with 24 fermionic directions, which\nhas been used in\n\\cite{Arutyunov:2008if}--\\cite{D'Auria:2008cw} to construct a superstring\nsigma model in $AdS_4 \\times CP^3$, is a subspace of the complete\nsuperspace and that the supercoset sigma--model action (being a\npartially gauge--fixed Green--Schwarz superstring action) describes\nonly a subsector of the complete type IIA superstring theory in\n$AdS_4 \\times CP^3$. The reason for this is that the\nkappa--symmetry gauge fixing condition which puts to zero eight\nfermionic modes corresponding to the 8 broken supersymmetries is not\nadmissible for all possible string configurations. So, in\nparticular, though the $OSp(6|4)\/U(3)\\times SO(1,3)$ sigma model\nsector of the theory is classically integrable\n\\cite{Arutyunov:2008if,Stefanski:2008ik} and there are\ngeneric arguments in favor of the integrability of the whole theory,\nthe direct proof of the integrability of the complete $AdS_4 \\times\nCP^3$ superstring still remains an open problem.\n\nThe knowledge of the explicit structure of the $AdS_4 \\times CP^3$\nsuperspace with 32 fermionic directions allows one to approach this\nand other problems. The form of the string action in the $AdS_4\n\\times CP^3$ superspace can be drastically simplified by choosing a\nsuitable description of the background supergeometry and an\nappropriate kappa--symmetry gauge, as was shown previously for the\ncases of the type IIB superstring, D3, M2 and M5--branes in the\ncorresponding $AdS\\times S$ backgrounds\n\\cite{Kallosh:1998qv}--\\cite{Pasti:1998tc}. A superconformal\nrealization and a kappa--symmetry gauge fixing of the $OSp(6|4)$\nsigma model sector of the $AdS_4 \\times CP^3$ superstring have been\nconsidered in \\cite{Uvarov:2008yi} and in a light--cone gauge in\n\\cite{Zarembo:2009au}.\n\n In this paper we perform an alternative\n$\\kappa$--symmetry gauge fixing of the complete $AdS_4\\times CP^3$\nsuperspace which is suitable for studying regions of the theory that\nare not reachable by the supercoset sigma model. In Subsection\n\\ref{FS} we apply this gauge fixing to simplify the superstring\naction in $AdS_4\\times CP^3$ and consider its T--dualization along a\n$3d$ translationally invariant subspace of $AdS_4$, similar to that\nperformed in\n\\cite{Kallosh:1998ji},\nwhich results in a simple action that contains fermions only up to\nthe fourth order. We also argue that, in contrast to the\n$AdS_5\\times S^5$ superstring\n\\cite{Ricci:2007eq,Berkovits:2008ic,Beisert:2008iq}, it is not possible to T--dualize\nthe fermionic sector of the superstring action in $AdS_4\\times\nCP^3$, which agrees with the conclusion of \\cite{Adam:2009kt}\nregarding the $OSp(6|4)$ supercoset subsector of the theory.\n\nIn addition to the superstring, also for certain configurations of\ntype IIA branes, \\emph{e.g.} D0-- and D2--branes considered in\nSection 4, the complete $AdS_4 \\times CP^3$ superspace should be\nused. An interesting example is a 1\/2 BPS probe D2--brane placed at\nthe $d=3$ Minkowski boundary of $AdS_4$. Upon gauge fixing\nworldvolume diffeomorphisms and kappa--symmetry, the effective\ntheory on the worldvolume of this D2--brane, which describes its\nfluctuations in $AdS_4\n\\times CP^3$, is an interacting $d=3$ gauge Born--Infeld--matter theory\npossessing the (spontaneously broken) superconformal symmetry\n$OSp(6|4)$. The model is superconformally invariant in spite of the\npresence on the $d=3$ worldvolume of the dynamical Abelian vector\nfield, since the latter is coupled to the $3d$ dilaton field\nassociated with the radial direction of $AdS_4$. The superconformal\ninvariance is spontaneously broken by a non--zero expectation value\nof the dilaton. This example is a type IIA counterpart of so called\nsingleton M2, tripleton M5 and doubleton D3--branes\n\\cite{deWit:1998tk,Claus:1998fh,Metsaev:1998hf,Pasti:1998tc} at the boundary of $AdS_{p+2}\\times S^{D-p-2}$ ($p=2,3$ and\n5), respectively, in  $D=11$ supergravity and type IIB string theory\n(see\n\\cite{Duff:2008pa} for a corresponding brane scan and a review of\nrelated earlier work).\n\nAnother example of interest for the study of the $AdS_4\/CFT_3$\ncorrespondence is a D2--brane filling $AdS_2\\times S^1\\subset AdS_4$\n\\cite{Drukker:2008jm}. This BPS D2--brane configuration corresponds\nto a disorder loop operator in the ABJM theory. Other D--brane\nconfigurations, which are to be related to Wilson loop operators in\nthe ABJM theory, were considered \\emph{e.g.} in\n\\cite{Berenstein:2008dc}--\\cite{Rey:2008bh}.\nIn this paper we extend the bosonic action for a D2--brane wrapping\n$AdS_2\\times S^1$ to include the worldvolume fermionic modes.\n\nWe start our consideration with an overview of the geometry of the\n$AdS_4\\times CP^3$ superspace.\n\n\n\\section{$AdS_4 \\times CP^3$ superspace}\\label{superspace}\n\nThe superspace under consideration contains $AdS_4\\times CP^3$\nas its bosonic subspace and has 32 fermionic directions\n\\cite{Gomis:2008jt}. It is parametrized by the supercoordinates\n\\begin{equation}\\label{Z}\nZ^{\\mathcal M}=(x^{\\hat\nm},y^{m'},\\Theta^{\\underline\\alpha})=(x^{\\hat\nm},y^{m'},\\vartheta^{\\alpha a'},\\upsilon^{\\alpha i}),\n\\end{equation}\nwhere $x^{\\hat m}$ $(\\hat m=0,1,2,3)$ and $y^{m'}$ $(m'=1,\\cdots,6)$\nare, respectively, the coordinates of $AdS_4=SO(2,3)\/SO(1,3)$ and\n$CP^3=SU(4)\/SU(3)\\times U(1)$. $\\Theta^{\\underline\\alpha}$ are the\n32 fermionic coordinates which we split into the 24 coordinates\n$\\vartheta^{\\alpha a'}$, that correspond to the 24 unbroken\nsupersymmetries in the $AdS_4\\times CP^3$ background, and the 8\ncoordinates $\\upsilon^{\\alpha i}$ which correspond to the broken\nsupersymmetries. The indices $\\alpha=1,2,3,4$ are $AdS_4$ spinor\nindices, $a'=1,\\cdots,6$ correspond to a six--dimensional\nrepresentation of $SU(3)$ (note that the index $a'$ appearing on\nspinors is different from the same index appearing in bosonic\nquantities, see Appendix A.5) and $i=1,2$ are $SO(2)\\sim\nU(1)$ indices. For more details of our notation and conventions see\nAppendix A\n\\footnote{Our notation and conventions are close to those in\n\\cite{Gomis:2008jt}. The difference is that, in this paper we put a\n``hat\" on the $AdS_4$ vector indices and use a more conventional IIA\nsuperspace torsion constraint $T_{\\underline{\\alpha\\beta}}{}^A\n=-2i\\Gamma^A_{\\underline{\\alpha\\beta}}$ (instead of\n$T_{\\underline{\\alpha\\beta}}{}^A\n=2\\Gamma^A_{\\underline{\\alpha\\beta}}$) and corresponding constraints\non the gauge field strengths. We also restore the dependence of the\ngeometric quantities and fields on the $S^7$ radius $R$, the\neleven-dimensional Planck length\n$l_p=e^{\\frac{1}{3}<\\phi>}\\sqrt{\\alpha'}$ and the Chern--Simons\nlevel $k$, which were put equal to one in\n\\cite{Gomis:2008jt}.}. For\nthe reader's convenience, below we list some of the notation used in\nthe text:\n\\begin{enumerate}\n\\item $D=10$ $AdS_4\\times CP^3$ superspace with 24 fermions is the\nsupercoset $OSp(6|4)\/ U(3)\\times SO(1,3)$. The supervielbeins and\nconnections are denoted by\n\\begin{equation}\\label{notaA}\n\\Big(E^{\\hat a}, E^{a'}, E^{\\alpha a'}, \\Omega^{\\hat a\\hat b}, \\Omega^{a'b'}, A \\Big)\n\\end{equation}\nwhose expressions are given in Appendix B, eq. (B.1).\n\\item $D=11$ $AdS_4\\times S^7$ superspace with 24 fermions. This is obtained as a $U(1)$ bundle over\nthe $OSp(6|4)\/ U(3)\\times SO(1,3)$ supercoset with the fiber\ncoordinate denoted by $z$. It is the supercoset $OSp(6|4)\\times U(1)\n\/ U(3) \\times SO(1,3)$ whose supervielbeins and connections are\ndenoted by\n\\begin{equation}\\label{notaB}\n\\Big(\\hat E^{\\hat a}, \\hat E^{a'}, \\hat E^7, \\hat E^{\\alpha a'}, \\hat \\Omega^{\\hat a\\hat b}, \\hat\n\\Omega^{a'b'}\\Big)\\,.\n\\end{equation}\nThey are given in eqs. (\\ref{24thA}), see also\n\\cite{Gomis:2008jt}. ${\\hat E}^7$ stands for the 7th (fiber) direction of $S^7$\n(or, equivalently, the 11th direction in $D=11$).\n\\item $D=11$ $AdS_4\\times S^7$ superspace  with 32 fermions.\nThis is the supercoset $OSp(8|4)\/SO(7)\\times SO(1,3)$. Its\nsupervielbeins and connections are denoted by\n\\begin{equation}\\label{notaC}\n\\Big(\\underline E^{\\hat a}, \\underline E^{a'}, \\underline E^7, \\underline E^{\\alpha a'},\\underline E^{\\alpha i},\n\\underline \\Omega^{\\hat a\\hat b},\n\\underline \\Omega^{a'b'}, \\underline \\Omega^{a' 7}\\Big)\\,.\n\\end{equation}\nTheir explicit expressions are given in (\\ref{upsilonfunctions}), (\\ref{ads4connection}) and (\\ref{so7connection}).\n\\item Finally, the $D=10$ $AdS_4\\times CP^3$ superspace\nwith 32 fermionic directions is obtained by performing a rotation of\n(\\ref{notaC}) in the $(\\hat a, 7)$--plane accompanied by the\ndimensional reduction to $D=10$ (see \\cite{Gomis:2008jt}). The\ngeometric quantities characterizing this superspace are denoted by\n\\begin{equation}\\label{notaD}\n\\Big({\\cal E}^{\\hat a}, {\\cal E}^{a'}, {\\cal E}^{\\alpha a'}, {\\cal E}^{\\alpha i},\n{\\mathcal O}^{\\hat a\\hat b}, {\\mathcal O}^{a'b'}, {\\cal A} \\Big).\n\\end{equation}\nThe supervielbeins have the following form\n\\end{enumerate}\n\\begin{equation}\\label{simplA}\n\\begin{aligned}\n{\\mathcal E}^{a'}(x,y,\\vartheta,\\upsilon)&=e^{\\frac{1}{3}\\phi(\\upsilon)}\\,\\left(E^{a'}(x,y,\\vartheta)+2i\\upsilon\\,{{\\sinh m}\\over\nm}\\gamma^{a'}\\gamma^5\\,E(x,y,\\vartheta)\\right) \\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\hat a}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over3}\\phi(\\upsilon)}\\,\\left(E^{\\hat\nb}(x,y,\\vartheta)+4i\\upsilon\\gamma^{\\hat b}\\,{{\\sinh^2{{\\mathcal M}\/\n2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right)\\Lambda_{\\hat b}{}^{\\hat\na}(\\upsilon)\n\\\\\n&{}\n\\hskip+1cm -e^{-{1\\over3}\\phi(\\upsilon)}\\,\\frac{R^2}{kl_p}\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right) E_7{}^{\\hat\na}(\\upsilon)\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over6}\\phi(\\upsilon)}\\,\\left({{\\sinh{\\mathcal\nM}}\\over{\\mathcal M}}\\,D\\upsilon\\right)^{\\beta j}\\,S_{\\beta\nj}{}^{\\alpha i}\\,(\\upsilon)\n-ie^{\\phi(\\upsilon)}{\\mathcal A}_1(x,y,\\vartheta,\\upsilon)\\,(\\gamma^5\\varepsilon\\lambda(\\upsilon))^{\\alpha\ni}\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over6}\\phi(\\upsilon)}\\,E^{\\gamma b'}(x,y,\\vartheta)\\,\\left(\n\\delta_{\\gamma}{}^{\\beta}-\\frac{8}{R}\\,\\left(\\gamma^5\\,\\upsilon\\,{{\\sinh^2{{m}\/2}}\\over{m}^2}\\right)_{\\gamma\ni}\\upsilon^{\\beta i} \\right)S_{\\beta b'}{}^{\\alpha\na'}\\,(\\upsilon)\\,.\n\\end{aligned}\n\\end{equation}\nThe new objects appearing in these expressions, $m$, $\\mathcal M$,\n$\\Lambda_{\\hat a}{}^{\\hat b}$, $E_7{}^{\\hat a}$ and\n$S_{\\underline\\alpha}^{\\underline\\beta}$, are functions of $\\upsilon$\nand their explicit forms are given in Appendix B.1 while the dilaton\n$\\phi$, dilatino $\\lambda$ and RR one--form $\\mathcal A_1$ are given\nbelow. Contracted spinor indices have been suppressed, \\emph{e.g.}\n$(\\upsilon\\varepsilon\\gamma^5)_{\\alpha i}=\\upsilon^{\\beta\nj}\\varepsilon_{ji}\\gamma^5_{\\beta\\alpha}$, where\n$\\varepsilon_{ij}=-\\varepsilon_{ji}$, $\\varepsilon_{12}=1$ is the\n$SO(2)$ invariant tensor. The covariant derivative is defined as\n\\begin{eqnarray}\\label{D}\nD\\upsilon=\\left(d+\\frac{i}{R}E^{\\hat\na}(x,y,\\vartheta)\\,\\gamma^5\\gamma_{\\hat a}-\\frac{1}{4}\\Omega^{\\hat a\n\\hat b}(x,y,\\vartheta)\\,\\gamma_{\\hat a\\hat b}\\right)\\upsilon \\,.\n\\end{eqnarray}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{simplB}\n\\begin{aligned}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon) &=\nR\\,e^{-{4\\over3}\\phi(\\upsilon)}\\,\\left[\n\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal\nM}^2}\\,D\\upsilon\\right)\\frac{R}{kl_p}\\,\\Phi(\\upsilon)\n\\right.\\\\\n&\\left.\\hspace{40pt}+\\frac{1}{kl_p}\\left(E^{\\hat\na}(x,y,\\vartheta)+4i\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right)E_{7\\hat a}(\\upsilon)\n\\right]\\,.\n\\end{aligned}\n\\end{equation}\n\nThe RR four-form and the NS--NS three-form superfield strengths are\ngiven by\n\\begin{equation}\\label{f4h3}\n\\begin{aligned}\nF_4&=d{\\mathcal A}_3-{\\mathcal A}_1\\,H_3=-\\frac{1}{4!}{\\mathcal\nE}^{\\hat d}{\\mathcal E}^{\\hat c}{\\mathcal E}^{\\hat b}{\\mathcal\nE}^{\\hat a}\\left(\\frac{6}{kl_p}\\,e^{-2\\phi}\\Phi\\varepsilon_{\\hat\na\\hat b\\hat c\\hat d}\\right) -\\frac{i}{2}{\\mathcal E}^{B}{\\mathcal\nE}^{A}{\\mathcal E}^{\\underline\\beta}\n{\\mathcal E}^{\\underline\\alpha}e^{-\\phi}(\\Gamma_{AB})_{\\underline{\\alpha\\beta}}\\,,\\\\\nH_3&=dB_2=-\\frac{1}{3!}{\\mathcal E}^{\\hat c}{\\mathcal E}^{\\hat\nb}{\\mathcal E}^{\\hat a}(\\frac{6}{kl_p}e^{-\\phi}\\varepsilon_{\\hat a\\hat b\\hat\nc\\hat d}E_7{}^{\\hat d}) -i{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_A\\Gamma_{11})_{\\underline{\\alpha\\beta}}\n+i{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB}\\Gamma^{11}\\lambda)_{\\underline\\alpha}\n\\end{aligned}\n\\end{equation}\nand the corresponding gauge potentials are\n\\begin{equation}\\label{B2}\nB_2=b_2+\\int_0^1\\,dt\\,i_\\Theta H_3(x,y,t\\Theta)\\,,\\qquad \\Theta=(\\vartheta,\\upsilon)\\,\\\\\n\\end{equation}\n\\begin{equation}\\label{A3}\n\\hskip+1.9cm{\\mathcal\nA}_3=a_3+\\int_0^1\\,dt\\,i_\\Theta\\left(F_4+\\mathcal{A}_1H_3\\right)(x,y,t\\Theta)\\,,\n\\end{equation}\nwhere $b_2$ and $a_3$ are the purely bosonic parts of the gauge\npotentials and $i_\\Theta$ means the inner product with\n$\\Theta^{\\underline\\alpha}$. Note that $b_2$ is  pure gauge in the\n$AdS_4\\times CP^3$ solution while $a_3$ is the RR three-form\npotential of the bosonic background.\n\n The dilaton superfield $\\phi(\\upsilon)$, which depends only on\nthe eight fermionic coordinates corresponding to the broken\nsupersymmetries, has the following form in terms of $E_7{}^{\\hat\na}(\\upsilon)$ and $\\Phi(\\upsilon)$\n\\begin{equation}\\label{dilaton1}\ne^{{2\\over\n3}\\phi(\\upsilon)}={R\\over{kl_p}}\\,\\sqrt{\\Phi^2+E_7{}^{\\hat\na}\\,E_7{}^{\\hat b}\\,\\eta_{\\hat a\\hat b}}\\,.\n\\end{equation}\nThe value of the dilaton at $\\upsilon=0$ is\n\\begin{equation}\ne^{\\frac{2}{3}\\phi(\\upsilon)}|_{\\upsilon=0}=e^{\\frac{2}{3}\\phi_0}=\\frac{R}{kl_p}\\,.\n\\end{equation}\nThe fermionic field $\\lambda^{\\alpha i}(\\upsilon)$ describes the\nnon--zero components of the dilatino superfield and is given by the\nequation \\cite{Howe:2004ib}\n\\begin{equation}\\label{dilatino1}\n\\lambda_{\\alpha i}=-\\frac{i}{3}D_{\\alpha i}\\,\\phi(\\upsilon)\\,.\n\\end{equation}\n\nIn the above expressions $E^{\\hat a}(x,y,\\vartheta)$,  $E^{\na'}(x,y,\\vartheta)$ and $\\Omega^{\\hat a\\hat b}(x,y,\\vartheta)$ are\nthe supervielbeins and the $AdS_4$ part of the spin connection of\nthe supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$ and $A(x,y,\\vartheta)$\nis the corresponding type IIA RR one--form gauge superfield, eq.\n(\\ref{notaA}), whose explicit form is given in Appendix B.\n\nAs mentioned above other quantities appearing in eqs.\n(\\ref{simplA})--(\\ref{dilatino1}), namely $\\mathcal M$, $m$,\n$\\Phi(\\upsilon)$, $E_7{}^{\\hat a}(\\upsilon)$, $\\Lambda_{\\hat a}{}^{\\hat b}(\\upsilon)$ and\n$S_{\\underline\\beta}{}^{\\underline\\alpha}(\\upsilon)$, whose geometrical and\ngroup--theoretical meaning has been explained in\n\\cite{Gomis:2008jt}, are also given in Appendix B.\n\n\\setcounter{equation}0\n\\section{Kappa--symmetry gauge fixing}\nWe shall now consider conditions for gauge fixing kappa--symmetry\nwhich are convenient for the description of configurations of\nsuperstrings and D-branes in the $AdS_4\\times CP^3$ superbackground\ndescribed above and for studying $AdS_4\/CFT_3$ correspondence\nproblems.\n\nSince the $AdS_4\/CFT_3$ holography is realized at the $3d$ Minkowski\nboundary of $AdS_4$ it is convenient to choose the $AdS_4\\times\nCP^3$ metric in the form\n\\begin{equation}\\label{ads4metric11}\nds^2=\\left(r\\over\n{R_{{CP^3}}}\\right)^4\\,dx^m\\,\\eta_{mn}\\,dx^n+\\left({R_{CP^3}}\\over\nr\\right)^2\\,dr^2+R_{CP^3}^2\\,ds^2_{_{CP^3}}\\,\n\\end{equation}\nwhere $m=0,1,2$ are indices corresponding to the coordinates of the\n$3d$ Minkowski boundary and $r$ is the 4th, radial, coordinate of\n$AdS_4$. So the $AdS_4$ coordinates are $x^{\\hat m}=(x^m,r)$. The\n$AdS_4$ radius is half of the $CP^3$ radius $R_{CP^3}$ which (in the\nstring frame) is related to the $S^7$ radius $R$ as follows\n\\begin{equation}\\label{R}\nR_{CP^3} = e^{\\frac{1}{3}\\phi_0}R =\\left(\\frac{R^3}{kl_p}\\right)^{1\/2}\\,.\n\\end{equation}\n\nIn the coordinate system associated with the metric\n(\\ref{ads4metric11}) (the bosonic part of) the RR field ${\\mathcal A}_3$, whose flux,\ntogether with $F_2=da_1={e^{-\\phi_0}\\over\n{R_{CP^3}}}\\,dy^{m'}dy^{n'}J_{m'n'}$ (where $dy^{m'}dy^{n'}J_{m'n'}$\nis the K\\\"ahler form on $CP^3$), ensures the compactification on\n$AdS_4\\times CP^{3}$\n\\cite{Watamura:1983hj,Nilsson:1984bj,Sorokin:1985ap}, has the\nfollowing form\n\\begin{equation}\\label{A31}\na_3=e^{-\\phi_0}\\left({r\\over\n{R_{CP^3}}}\\right)^6\\,dx^0\\,dx^1\\,dx^2\\,,\\qquad F_4={6\\over\nR_{CP^3}}e^{-\\phi_0}\\,\\left({r\\over\n{R_{CP^3}}}\\right)^5\\,dx^0\\,dx^1\\,dx^2\\,dr\\,.\n\\end{equation}\n(In our conventions the exterior derivative acts from the right.)\n\nInstead of the $AdS_4$ part of the metric (\\ref{ads4metric11}),\nwhich obscures a bit the fact that the metric of the conformal\nboundary is the flat Minkowski metric on $R^{1,2}$, one can use the\n$AdS_4$ metric in the conformally flat form\n\\begin{equation}\\label{ads4metric21}\nds^2_{_{AdS_4}}={1\\over\nu^2}(dx^m\\eta_{mn}dx^n+\\frac{R_{CP^3}^2}{4}\\,du^2)\\,,\n\\qquad u=\\left(R_{CP^3}\\over r\\right)^2\\,.\n\\end{equation}\nThis metric is associated with a simple coset representative $\ng=\\exp(x^m\\,\\Pi_m)$ $\\exp(R_{CP^3}\\,\\ln(u) D)$, where $\\Pi_m$ are\nthe generators of the Poincar\\'e translations along the Minkowski\nboundary $([\\Pi_m,\\,\\Pi_n]=0)$ and $D$ is the dilatation generator\n$[D,\\,\\Pi_m]=\\Pi_m$.\n\nNote that if the components of the vielbein associated with the\nmetric (\\ref{ads4metric11}) or (\\ref{ads4metric21}) are chosen to\nbe\\footnote{Note that the vielbeins $e^a$ and $e^3$ appearing in eq.\n(\\ref{ad4v}) correspond to the $AdS_4$ metric of the $D=11$\n$AdS_4\\times S^7$ solution characterized by the radius R which is\nrelated to the $CP^3$ radius in the string frame according to eq.\n(\\ref{R}). These bosonic vielbeins will appear in our explicit\nexpressions for the $AdS_4\\times CP^3$ supergeometry.}\n\\begin{equation}\\label{ad4v}\ne^{\\frac{\\phi_0}{3}}\\,e^a={r^2\\over\nR_{CP^3}^2}\\,dx^a=u^{-1}\\,dx^a\\,,\\qquad\ne^{\\frac{\\phi_0}{3}}\\,e^3=\\frac{R_{CP^3}}{r}\\,dr=-\\frac{R_{CP^3}}{2u}\\,du,\n\\end{equation}\nthe components of the $SO(1,3)$ spin connection are\n\\begin{equation}\\label{eaoa31}\n\\omega^{a3}=-\\frac{2}{R}\\,e^a\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{oab}\n\\omega^{ab}=0\\,.\n\\end{equation}\nWe shall use the relation (\\ref{eaoa31}) to simplify the form of the\ngauge fixed $AdS_4 \\times CP^3$ supergeometry. Note that the\ncondition (\\ref{eaoa31}) can always be imposed by performing an\nappropriate local $SO(1,3)$ transformations of the vielbein and\nconnection, though in general the $SO(1,2)$ components $\\omega^{ab}$\nof the connection will be non--zero.\n\nUsing the previous experience of gauge fixing kappa--symmetry of superstrings, D-branes and M-branes in AdS backgrounds\n\\cite{Kallosh:1998qv}--\\cite{Pasti:1998tc} we choose the\nkappa--symmetry gauge fixing condition in the form\n\\footnote{Such a gauge for fixing kappa--symmetry is analogous to\nthe so called Killing spinor gauge \\cite{Kallosh:1998qv}, or\nsupersolvable gauge\n\\cite{Dall'Agata:1998wz}, or the superconformal gauge \\cite{Pasti:1998tc}.}\n\\begin{equation}\\label{kappagauge1}\n\\Theta=\\frac{1}{2}(1\\pm\\gamma)\\Theta\\,\\quad \\Rightarrow\n\\quad \\vartheta^{a'}=\\frac{1}{2}(1\\pm\\gamma)\\vartheta^{a'}\\,,\\qquad\n\\upsilon^{i}=\\frac{1}{2}(1\\pm\\gamma)\\upsilon^{i},\n\\end{equation}\nwhere\n\\begin{equation}\\label{gamma}\n\\gamma=\\gamma^{012}\\qquad\\Rightarrow\\qquad\\gamma^2=1,\n\\quad\\{\\gamma,\\gamma^5\\}=[\\gamma,\\gamma^a]=0\\quad\\mbox{and}\\quad\\gamma\\gamma^3=-i\\gamma^5\\,,\n\\end{equation}\n$a=0,1,2$ are the indices of the 3d Minkowski boundary or of\n$AdS_2\\times S^1$ and $\\gamma^3$ is associated with the third\nspatial direction of $AdS_4$. Note that, in view of our definition\n(\\ref{Gamma10}) of the $D=10$ gamma--matrices the matrices defined\nin (\\ref{gamma}) can be regarded either as $4d$ gamma matrices or as\nthe $D=10$ matrices $\\Gamma^{\\hat a}=\\gamma^{\\hat a}\\otimes {\\bf 1}$\n$(\\hat a=0,1,2,3)$.\n\nThe condition (\\ref{kappagauge1}) is admissible for fixing\nkappa--symmetry if the projection matrix $\\frac{1}{2}(1\\mp\\gamma)$ either coincides or does not\ncommute with the kappa--symmetry projection matrix $\\frac{1}{2}(1+\\Gamma)$ of a given\nconfiguration of the superstring and D--branes.\nThis can be understood in the following way. To lowest order in fermions $\\Theta$ transforms under kappa--symmetry as\n\\begin{equation}\n\\delta_\\kappa\\Theta=\\frac{1}{2}(1+\\Gamma)\\kappa\\,,\n\\end{equation}\nwhere $\\frac{1}{2}(1+\\Gamma)$ is a projection matrix and\n$\\kappa(\\xi)$ is an arbitrary spinor parameter. It is then clear\nthat if the two projectors coincide, we can pick a $\\kappa$ such\nthat $\\frac{1}{2}(1+\\Gamma)\\Theta=0$, or equivalently\n$\\Theta=\\frac{1}{2}(1-\\Gamma)\\Theta$. In the case when the two\nprojection operators do not coincide a kappa--symmetry variation of\nthe gauge--fixing condition $\\frac{1}{2}(1\\mp\\gamma)\\Theta=0$ which\nleaves it intact gives\n\\begin{equation}\n0=\\frac{1}{4}(1\\mp\\gamma)(1+\\Gamma)\\kappa\n=\\frac{1}{8}(1+\\Gamma)(1\\mp\\gamma)(1+\\Gamma)\\kappa\\mp\\frac{1}{8}[\\gamma,\\Gamma](1+\\Gamma)\\kappa\n=\\mp\\frac{1}{8}[\\gamma,\\Gamma](1+\\Gamma)\\kappa\\,,\n\\end{equation}\nwhere in the last step we made use of the initial equation. This\nmeans that for the gauge--fixing to be complete, i.e. that the\nvariation of the gauge fixing condition vanishes if and only if all\nindependent kappa--symmetry parameters are put to zero, the\ncommutator $[\\gamma,\\Gamma]$ has to be an invertible matrix (when\nrestricted to the relevant subspace).\n\n As we shall see\nbelow, for any choice of the sign the condition (\\ref{kappagauge1})\n    is an admissible gauge-fixing in the case of\n    arbitrary motion of D0--branes in $AdS_4 \\times CP^3$, while in the\ncase of the superstring it is admissible (for both signs) for those\n configurations for which the projection of the string\nworldsheet on the $3d$ Minkowski boundary is a non--degenerate\ntwo--dimensional time--like surface. In the case of the D2--brane\nplaced at the Minkowski boundary of $AdS_4$, to gauge fix kappa--symmetry one must choose the condition (\\ref{kappagauge1}) with the\nlower sign\n\\cite{Pasti:1998tc}, while both signs are admissible when the\n    D2--brane wraps an $AdS_2 \\times S^1$ subspace of $AdS_4$. However,\n    the choice of (\\ref{kappagauge1}) with the upper sign yields\n    the simplest gauge-fixed form of the string and\n    brane actions in the $AdS_4\\times CP^3$ superbackground.\n\n\nWhen the fermionic coordinates are restricted by the condition\n(\\ref{kappagauge1}), the expressions for the supervielbeins and the\ngauge superfields of the $AdS_4 \\times CP^3$ superspace drastically\nsimplify due to the identities satisfied by the projected fermionic\ncoordinates given in Appendix C. In particular, the functions of\n$\\upsilon$ which enter the eqs. (\\ref{simplA})--(\\ref{dilaton1}),\nwhose explicit forms are given in Appendix B.1, reduce to\n\\begin{equation}\\label{Phi1}\n\\Phi(\\upsilon)=1+\\frac{8}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal\nM}^2}\\,\\varepsilon\\upsilon =1\\,,\n\\end{equation}\n\\begin{eqnarray}\nE_7{}^a(\\upsilon)&=&-\\frac{8i}{R}\\,\\upsilon\\gamma^a\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}=-\\frac{2i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,,\\\\\nE_7{}^3(\\upsilon)&=&-\\frac{8i}{R}\\,\\upsilon\\gamma^3\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}=0\\,.\n\\end{eqnarray}\nThe dilaton superfield (\\ref{dilaton1}) takes the form\n\\begin{equation}\\label{gfdialton}\ne^{{2\\over 3}\\phi(\\upsilon)}\n=\\frac{R}{kl_p}(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\quad \\Rightarrow \\quad\n\\phi(\\upsilon)=\\frac{3}{2}(\\log\\frac{R}{kl_p}-\\frac{6}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{equation}\nwhere $\\upsilon\\ups=\\delta_{ij}C_{\\alpha\\beta}\\upsilon^{\\alpha i}\\upsilon^{\\beta j}$, and the dilatino becomes\n\\begin{eqnarray}\n\\lambda^{\\alpha i}\n=\n\\frac{2i}{R}\\left(\\frac{R}{kl_p}\\right)^{-1\/4}((\\gamma^5\\upsilon)^{\\alpha i}+\\frac{3}{R}\\upsilon^{\\alpha i}\\,\\upsilon\\ups)\\,.\n\\end{eqnarray}\nWe also find that\n\\begin{equation}\n\\Lambda_a{}^\n= (1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\\delta_a{}^b\\,,\n\\qquad\n\\Lambda_3{}^a = \\Lambda_a{}^3=0\\,,\\qquad\n\\Lambda_3{}^3=1\\,,\n\\end{equation}\nand\n\\begin{eqnarray}\\label{S}\n&&S_{\\underline\\alpha}{}^{\\underline\\beta}=\n\\left(\\frac{R}{kl_p}\\right)^{1\/4}e^{-\\frac{1}{6}\\phi}\\,\\delta_{\\underline\\alpha}{}^{\\underline\\beta}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(\\Gamma_a\\Gamma_{11})_{\\underline\\alpha}{}^{\\underline\\beta}\\,.\n\\end{eqnarray}\n\n\n\n\\subsection{$AdS_4 \\times CP^3$ supergeometry with\n$\\Theta={1\\over 2}(1+\\gamma)\\Theta$}\\label{theta-}\n\nThe supervielbeins (\\ref{simplA}) and the gauge superfields\n(\\ref{simplB}), (\\ref{B2}) and (\\ref{A3}) take the simplest form\nwhen the kappa--symmetry gauge condition (\\ref{kappagauge1}) is\nchosen with the upper sign. In virtue of eqs.\n(\\ref{Phi1})--(\\ref{S}) and expressions given in Appendix C, the\nsupervielbeins reduce to\n\\begin{equation}\\label{simple+v}\n\\begin{aligned}\n{\\mathcal\nE}^{a'}(x,y,\\vartheta,\\upsilon)&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^{a'}(y)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^a(x,y,\\vartheta,\\upsilon) &=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\n(e^a(x)+i\\Theta\\gamma^aD\\Theta)(1-\\frac{1}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^3(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^3(x)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{simplesv}\n\\begin{aligned}\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_8\\upsilon)^{\\alpha\ni}-\\frac{1}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_8\\upsilon\\varepsilon\\gamma_a\\gamma_5)^{\\alpha\ni}\n-\\frac{4i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)(\\gamma_a\\upsilon)^{\\alpha\ni}\\,\\upsilon\\ups\n\\Big)\n\\,,\n\\nonumber\\\\\n\\nonumber\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_{_{24}}\\vartheta)^{\\alpha a'}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_{_{24}}\\vartheta\\gamma_a\\gamma_5\\gamma_7)^{\\alpha\na'}\\Big)\\,.\\nonumber\n\\end{aligned}\n\\end{equation}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{simple+A}\n\\begin{aligned}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon) &=kl_p\\Big(\nA(y)-\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)\\,\\upsilon\\varepsilon\\gamma_a\\upsilon\\Big)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $A(y)$ is the potential for the K\\\"ahler form on $CP^3$,\n\\emph{i.e.} $dA(y)=\\frac{1}{R^2}\\,dy^{m'}dy^{n'}\\,J_{m'n'}$, and the covariant derivatives are\n\\begin{eqnarray}\\label{D81}\nD\\Theta&=&(D_8\\upsilon,\\,D_{24}\\vartheta)\\\\\nD_8\\upsilon&=&{\\mathcal P_2}\\,\\left(d-\\frac{1}{R}e^3-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}+2A(y)\\varepsilon\\right)\\upsilon\\nonumber\\\\\nD_{24}\\vartheta&=&{\\mathcal P}_6\\,(d-\\frac{1}{R}e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'} -\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,,\\nonumber\n\\end{eqnarray}\nwhere ${\\mathcal P}_2$ and ${\\mathcal P}_6$ are projectors that\nsingle out from 32 $\\Theta$, respectively, 8 $\\upsilon$ and 24\n$\\vartheta$ (see Appendix A.5). The appearance of the $U(1)$ gauge potential $A(y)$ in the covariant\nderivative of $\\upsilon$ (\\ref{D81}) reflects the fact that\n$\\upsilon$ has $U(1)$ charge equal to 2.\n\\\\\nNote that\n\\begin{equation}\nD_8={\\mathcal P}_2\\, {\\mathcal D}\\,{\\mathcal P}_2\\,,\\qquad\nD_{24}={\\mathcal P}_6\\, {\\mathcal D}\\,{\\mathcal P}_6\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n{\\mathcal D}=d-\\frac{1}{R}e^3+\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'}\\,.\n\\end{equation}\nThe NS--NS three-form, eq. (\\ref{f4h3}), becomes\n\\begin{eqnarray}\\label{H3ex}\nH_3&=&-\\frac{6i}{R^2} e^3\\, {\\mathcal E}^b\\,{\\mathcal\nE}^a\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n-\\frac{2R}{kl_p}\\,\\Big[\n\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)(e^a+i\\Theta\\gamma^aD\\Theta)\\,D_8\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{1}{R}(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\,\\gamma_{ab}\\,D\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n+\\frac{1}{2}\\,e^3\\,D\\Theta\\,\\gamma_7\\,D\\Theta\n+\\frac{2i}{R}\\,e^3\\,e^{a'}\\,D\\Theta\\gamma_{a'}\\gamma_7\\upsilon\n\\nonumber\\\\\n&&{} +\\frac{i}{2}\\,e^{a'}\\,D\\Theta\\,\\gamma_{a'}\\gamma_7\\,D\\Theta\n+\\frac{1}{R}\\,e^{b'}\\,e^{a'}\\,D\\Theta\\,\\gamma_{a'b'}\\gamma_7\\upsilon\\,\\Big]\\,,\n\\end{eqnarray}\nwhere $\\Theta=(\\vartheta,\\upsilon)$ and\n$D\\Theta=(D_{24}\\vartheta,\\,D_8\\upsilon)$.\n\\\\\nWe now want to determine the potential of $H_3=dB_2$ using eq.\n(\\ref{B2}). Taking into account that $i_\\Theta\\mathcal E^{A}=0$, and\nthe fact that with the plus sign in the projector\n(\\ref{kappagauge1})\n$\\Theta\\gamma_{ab}D\\Theta=\\varepsilon_{abc}\\,\\Theta\\gamma^cD\\Theta$ etc., we\nget\n\\begin{eqnarray}\ni_\\Theta H_3 &=& 2\\frac{R}{kl_p}\n\\Big(\n-\\frac{i}{R}e^be^a\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n+e^3\\,\\Theta\\gamma^7D\\Theta\n\\nonumber\\\\\n&&{} +ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n+\\frac{4}{R}e^b\\,\\Theta\\gamma^aD\\Theta\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n+\\frac{3i}{R}\\Theta\\gamma^bD\\Theta\\,\\Theta\\gamma^aD\\Theta\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{eqnarray}\nThis gives the NS--NS two-form potential (see eq. (\\ref{B2}))\n\\begin{eqnarray}\\label{simple+B}\nB_2&=&\n\\frac{R}{kl_p}\n\\Big[\n-\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)\\,(e^a+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{20pt}}\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n+e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n\\Big]\\,.\n\\end{eqnarray}\n\nNow we turn our attention to the RR four-form $F_4$ (\\ref{f4h3}) and\nits potential $\\mathcal A_3$ (\\ref{A3}). $F_4$ simplifies to\n\\begin{equation}\\label{f4}\nF_4= -\\frac{1}{kl_p}\\,e^{-2\\phi}\\,{\\mathcal E}^3{\\mathcal\nE}^c{\\mathcal E}^b{\\mathcal E}^a\\,\\varepsilon_{abc}\n-\\frac{i}{2}e^{-\\phi}{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta} {\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB})_{\\underline{\\alpha\\beta}}\\,,\n\\end{equation}\nwhich gives\n\\begin{eqnarray}\ni_\\Theta F_4 &=& -i(\\frac{R}{kl_p})^{1\/4}e^{-\\phi}{\\mathcal\nE}^{B}{\\mathcal E}^{A} ({\\mathcal E}\\Gamma_{AB}\\Theta\n+\\frac{i}{R}{\\mathcal\nE}\\Gamma_{AB}\\gamma_a\\Gamma_{11}\\Theta\\,\\upsilon\\gamma^a\\varepsilon\\upsilon)\n\\nonumber\\\\\n&=&\n-i(e^b+i\\Theta\\gamma^bD\\Theta)(e^a+i\\Theta\\gamma^aD\\Theta)\\,(1+\\frac{12}{R^2}(\\upsilon\\ups)^2)(D\\Theta\\gamma_{ab}\\Theta\n+\\frac{4i}{R^2}e^c\\varepsilon_{abc}\\,(\\upsilon\\ups)^2 )\n\\nonumber\\\\\n&&{}\n+\\frac{4i}{R}e^3(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\gamma^7\\Theta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n-\\frac{4}{R}e^{a'}(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\gamma_{a'}\\gamma^7\\Theta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{} +2e^3e^{a'}(D\\Theta\\gamma_{a'}\\Theta\n+\\frac{4i}{R^2}(e^a+i\\Theta\\gamma^aD\\Theta)\\upsilon\\gamma_a\\gamma_{a'}\\vartheta\\,\\upsilon\\ups\n)\n\\nonumber\\\\\n&&{} -ie^{b'}e^{a'}(D\\Theta\\gamma_{a'b'}\\Theta\n+\\frac{4i}{R^2}(e^a+i\\Theta\\gamma^aD\\Theta)\\upsilon\\gamma_a\\gamma_{a'b'}\\vartheta\\,\\upsilon\\ups)\\,.\n\\end{eqnarray}\nSince $i_\\Theta\\mathcal A_1=0$ the RR three--form potential\n(\\ref{A3}) becomes\n\\begin{eqnarray}\\label{A3-}\n\\mathcal A_3&=&a_3+\\int_0^1\\,dt(i_\\Theta F_4+\\mathcal A_1i_\\Theta H_3)(x,y,t\\Theta)\n\\nonumber\\\\\n&=& a_3 -\\frac{i}{2}e^be^a\\,D\\Theta\\gamma_{ab}\\Theta\n+e^3e^{a'}\\,D\\Theta\\gamma_{a'}\\Theta\n-\\frac{i}{2}e^{b'}e^{a'}\\,D\\Theta\\gamma_{a'b'}\\Theta\n+\\frac{1}{2}e^b\\Theta\\gamma^aD\\Theta\\,D\\Theta\\gamma_{ab}\\Theta\n\\nonumber\\\\\n&&{}\n+\\frac{i}{6}\\Theta\\gamma^bD\\Theta\\,\\Theta\\gamma^aD\\Theta\\,D\\Theta\\gamma_{ab}\\Theta\n+k\\,l_p\\,A(y)\\,B_2\\,.\n\\end{eqnarray}\nLooking at the purely bosonic part of $F_4$, eq. (\\ref{f4}) it is\neasy to see (compare also with eqs. (\\ref{A31})) that we can take\n\\begin{equation}\na_3=\\frac{1}{3!}e^ce^be^a\\,\\varepsilon_{abc}\\,.\n\\end{equation}\nNote that in the above expressions for the supervielbeins\n(\\ref{simple+v}), the RR one--form (\\ref{simple+A}), the three--form\n(\\ref{A3-}) and the NS-NS two--form (\\ref{simple+B}) the maximum\norder of the fermions is six.\n\n\n\\subsection{$AdS_4 \\times CP^3$ supergeometry with\n$\\Theta={1\\over 2}(1-\\gamma)\\Theta$}\\label{theta++}\n\nWhen the condition (\\ref{kappagauge1}) is chosen with the lower\nsign, in view of eqs. (\\ref{Phi1})--(\\ref{S}) and expressions given\nin Appendix C, the supervielbeins (\\ref{simplA}) and the RR\none--form gauge superfield (\\ref{simplB}) reduce to a form which is\nmore complicated than their gauge--fixed counterparts of the\nprevious Subsection. But, as we have already mentioned, one cannot\nuse the gauge fixing condition of Subsection \\ref{theta-} to\ndescribe the D2--brane at the Minkowski boundary of $AdS_4$, and\nshould impose $\\Theta={1\\over 2}(1-\\gamma)\\Theta$ instead. In this\ncase the supervielbeins take the following form\n\\begin{equation}\\label{-bv}\n\\begin{aligned}\n{\\mathcal E}^{a'}(x,y,\\vartheta,\\upsilon)&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\n\\Big(e^{a'}(y)-\\frac{2}{R}e^a(x)\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta\\Big)\n(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^a(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\\,\\Big( e^a(x) +i\\Theta\\gamma^a D\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta-\\upsilon\\ups)^2\n\\Big)(1-\\frac{1}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^3(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^3(x)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{-fv}\n\\begin{aligned}\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_8\\upsilon)^{\\alpha i}\n-\\frac{1}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_8\\upsilon\\varepsilon\\gamma_a\\gamma_5)^{\\alpha\ni}\n\\\\\n&{}\n\\hskip+1cm\n-\\frac{4i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta-\\upsilon\\ups)^2\n)(\\gamma_a\\upsilon)^{\\alpha i}\\,\\upsilon\\ups\n\\Big)\n\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_{_{24}}\\vartheta)^{\\alpha a'}\n+\\frac{i}{R}(D_{24}\\vartheta\\gamma_a\\gamma^5\\gamma^7)^{\\alpha\na'}\\,\\upsilon\\gamma^a\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{aligned}\n\\nonumber\n\\end{equation}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{A2-}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon)=kl_p\\Big(A(y)\n-\\frac{2}{R^2}e^a(x)\\,\\Theta\\gamma^7\\gamma_a\\Theta\n -\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta)^2\n)\\upsilon\\varepsilon\\gamma_a\\upsilon\n\\Big)\\,.\n\\end{equation}\nIn the above expressions\n\\begin{eqnarray}\\label{D8}\nD\\Theta&=&(D_8\\upsilon\\,,D_{24}\\vartheta)\\,,\\nonumber\\\\\nD_8\\upsilon&=&(D-\\frac{2i}{R^2}\\upsilon\\ups\\,e^a\\,\\gamma_a+2A(x,y,\\vartheta)\\,\\varepsilon)\\upsilon\n\\\\\n&&\\hspace{-40pt}=\n\\Big(d+\\frac{2i}{R}e^a(\\gamma^5\\gamma_a+\\frac{1}{R}(\\vartheta\\vartheta-\\upsilon\\ups)\\gamma_a)\n+\\frac{1}{R}e^3\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n+(2A(y)-\\frac{4}{R^2}e^a\\vartheta\\gamma^7\\gamma_a\\vartheta)\\varepsilon\n\\Big)\\upsilon,\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{D_{24}}\nD_{_{24}}\\vartheta\n&=&{\\mathcal P}_6\\,\\Big(d\n+\\frac{2i}{R}e^a(\\gamma^5\\gamma_a+\\frac{1}{R}(\\vartheta\\vartheta-\\upsilon\\ups)\\gamma_a)\n+\\frac{1}{R}e^3 +\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'}\\Big)\\vartheta\\,.\\nonumber\\\\\n\\end{eqnarray}\n(The shift of $D$ by $-\\frac{2i}{R^2}\\upsilon\\ups\\,e^a\\,\\gamma_a$ has\nbeen made for the expressions to have a nicer and more\ncovariant--looking form).\n\\\\\nThe NS--NS three-form, eq. (\\ref{f4h3}), becomes\n\\begin{equation}\nH_3= -\\frac{6i}{R^2}e^3{\\mathcal E}^b{\\mathcal\nE}^a\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon -i{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_A\\Gamma_{11})_{\\underline{\\alpha\\beta}}\n+i{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB}\\Gamma^{11}\\lambda)_{\\underline\\alpha}\n\\,.\n\\end{equation}\nWe now would like to determine its potential according to eq.\n(\\ref{B2}). Using the fact that\n\\begin{equation}\ni_\\Theta{\\mathcal E}^{\\underline\\alpha}=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}(\\Theta^{\\underline\\alpha}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(\\Theta\\Gamma_a\\Gamma_{11})^{\\underline\\alpha})\n\\end{equation}\nand $i_\\Theta\\mathcal E^{A}=0$ we get\n\\begin{eqnarray}\\label{iH3-}\ni_\\Theta H_3&=&\n\\frac{R}{kl_p}\\Big(\n-\\frac{2}{R}(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\\upsilon\\gamma_{ab}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{4}{R}(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\nD\\Theta\\gamma_{ba}\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n+\\frac{8}{R^2}e^3e^a\\vartheta\\vartheta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{4}{R}(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)e^b\\Theta\\gamma_{ba}\\gamma^7\\Theta\n+2e^3D\\Theta\\gamma^7\\Theta\n\\nonumber\\\\\n&&{}\n-2i(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'}\\gamma^7\\Theta\n+\\frac{4i}{R}e^3(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n-\\frac{2}{R}(e^{b'}-\\frac{2}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{8i}{R^2}(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)e^a\\Theta\\gamma_{a'}\\gamma_{ab}\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n\\Big)\n\\end{eqnarray}\nand finally\n\\begin{eqnarray}\\label{B2-}\nB_2&=&\n\\frac{R}{kl_p}\\Big(\n\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta)^2)(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta)^2)\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{2}{R}(e^a+\\frac{i}{2}\\Theta\\gamma^aD\\Theta+\\frac{1}{3R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)e^b\\,\\varepsilon_{abc}\\,\\Theta\\gamma^c\\gamma^7\\Theta\n+\\frac{2i}{R}e^3(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n-\\frac{1}{R}(e^{b'}-\\frac{1}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n-\\frac{1}{3R^3}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta\\,e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{} -e^3\\,\\Theta\\gamma^7D\\Theta\n+\\frac{i}{R^2}e^3e^a(\\vartheta\\gamma^7\\gamma_a\\vartheta-\\upsilon\\gamma_a\\gamma^7\\upsilon)\\,\\Theta\\Theta\n+i(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n\\nonumber\\\\\n&&{}\n+\\frac{2i}{R^2}(e^{a'}-\\frac{4}{3R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\,e^a\\,\\varepsilon_{abc}\\,\\Theta\\gamma_{a'}\\gamma^b\\Theta\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n+\\frac{2}{R^2}(e^{a'}-\\frac{2}{3R}e^b\\,\\Theta\\gamma^{a'}\\gamma_b\\Theta)e^a\\,\\vartheta\\gamma^7\\gamma_a\\vartheta\\,\\vartheta\\gamma_{a'}\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{1}{R^2}(e^{a'}-\\frac{2}{3R}e^b\\,\\Theta\\gamma^{a'}\\gamma_b\\Theta)e^a\\,\\Theta\\gamma_{a'}\\gamma^7\\gamma_a\\Theta\\,(\\vartheta\\vartheta-\\upsilon\\ups)\n-\\frac{2}{3R^3}e^be^a\\,\\varepsilon_{abc}\\,\\Theta\\gamma^c\\gamma^7\\Theta\\,((\\vartheta\\vartheta)^2-(\\upsilon\\ups)^2)\n\\nonumber\\\\\n&&{}\n+\\frac{4i}{3R^3}e^be^d\\,\\vartheta\\gamma_d\\gamma^7\\vartheta\\,\\vartheta\\gamma^a\\gamma^7\\vartheta\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{eqnarray}\nNote that the maximum order of the fermions in the above expressions\nis ten.\n\nUsing the form of $F_4$ in (\\ref{f4h3}) as well as the expressions\n(\\ref{A2-}) for ${\\mathcal A}_1$ and (\\ref{iH3-}) for $i_\\Theta H_3$\nthe quantity relevant for computing the RR three-form potential\n$\\mathcal A_3$ becomes\n\\begin{eqnarray}\n\\lefteqn{i_\\Theta F_4+\\mathcal A_1i_\\Theta H_3}\n\\nonumber\\\\\n&=&\n-(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\\varepsilon_{abc}\ni\\Theta\\gamma^cD\\Theta\n\\nonumber\\\\\n&&{}\n-2e^3(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'}\\Theta\n-\\frac{4}{R}e^3e^a\\Theta\\gamma_aD\\Theta\\,\\Theta\\Theta(1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\n\\\\\n&&{}\n\\hspace{-10pt}-\\frac{4}{R}(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)e^a\n(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n\\Theta\\gamma_{a'}\\gamma_{ab}\\Theta(1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\n\\nonumber\\\\\n&&{}\n\\hspace{-10pt}-i(e^{b'}-\\frac{2}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'b'}\\Theta\n+kl_p(A(y)-\\frac{2}{R^2}e^a\\,\\Theta\\gamma^7\\gamma_a\\Theta)i_\\Theta\nH_3\\,.\\nonumber\n\\end{eqnarray}\nOne can now substitute this together with the expression for\n$i_\\Theta H_3$ (\\ref{iH3-}) into eq. (\\ref{A3}) and compute the\nexplicit form of the RR three--form potential ${\\mathcal A}_3$ in\nthis gauge. Since we have not got a reasonably simple expression for\n${\\mathcal A}_3$ we shall not present it here.\n\n\n\\setcounter{equation}0\n\\section{Applications}\nWe can now use the kappa--gauge fixed form of the $AdS_4 \\times\nCP^3$ superbackground of Subsections \\ref{theta-} and \\ref{theta++}\nto simplify the actions for the type IIA superstring and D--branes.\nLet us note that the gauge fixing conditions (\\ref{kappagauge1}) can\nalso be used to simplify the actions for the $D=11$ superparticle,\nM2-- and M5--branes in the $AdS_4\\times S^7\/Z_k$ superbackground\n(\\ref{notaC}). We shall consider the example of the $D=11$ superparticle\nbelow.\n\n\n\\subsection{$D=11$ superparticle}\n\\def\\mathcal M{\\mathcal M}\n\nLet us consider a massless superparticle in the $AdS_4\\times\nS^7\/Z_k$ supergravity background. Recall that when $k=1,2$, the\nsupergravity background preserves the maximum number of 32\nsupersymmetries, while for $k>2$ it preserves only 24.\nThe superparticle action in the complete superspace\nwith 32 $\\Theta$ is constructed using the supervielbeins of the\n$OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ supercoset derived in\n\\cite{Gomis:2008jt}\n\\begin{equation}\\label{upsilonfunctions}\n\\begin{aligned}\n\\underline E^{\\hat a}&=E^{\\hat a}(x,y,\\vartheta) + 4 i\n\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal M}^2}\\,D\\upsilon\n+\\frac{R}{k}\\,dz\\,E_7{}^{\\hat a}(\\upsilon)\\\\\n\\underline E^{a'}&=E^{a'}(x,y,\\vartheta)+2i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'}\\gamma^5\\,E(x,y,\\vartheta)\\\\\n\\underline E^7&=\\frac{R}{k}\\,dz\\,\\Phi(\\upsilon)+\nR\\,\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\,\\right) \\\\\n\\underline E^{\\alpha i}&=\\left({{\\sinh{\\mathcal M}}\\over{\\mathcal\nM}}\\,(D\\upsilon-\\frac{2}{k}\\,dz\\,\\varepsilon\\upsilon)\\right)^{\\alpha i}\n\\\\\n\\underline E^{\\alpha a'}&=E^{\\alpha a'}(x,y,\\vartheta)-\\frac{8}{R}E^{\\beta\na'}\\left(\\gamma^5\\,\\upsilon\\,{{\\sinh^2{{m}\/2}}\\over{m}^2}\\right)_{\\beta\ni}\\upsilon^{\\alpha i}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $z$ is the 7th, $U(1)$ fiber, coordinate of\n$S^7$, $D\\upsilon$ has been given in (\\ref{D}) and the eight fermionic\ncoordinates $\\upsilon^{\\alpha i}$ correspond to the eight\nsupersymmetries broken by orbifolding with $k>2$.\n\nThe explicit form of the fermionic supervielbeins in\n(\\ref{upsilonfunctions}) and of the connections on\\\\\n$OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ are not required for the\nconstruction of the Brink--Schwarz superparticle action but one\nneeds them for the construction of the pure--spinor superparticle\naction in curved superbackgrounds, so we present also the form of\nthe spin--connection below.\n\nThe $SO(1,3)$ connection is\n\\begin{equation}\\label{ads4connection}\n\\underline \\Omega^{\\hat a\\hat b}= \\Omega^{\\hat a\\hat b}(x,y,\\vartheta)\n+\\frac{8}{R} \\upsilon\\gamma^{\\hat a\\hat\nb}\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal\nM}^2}\\,\\left(D\\upsilon-\\frac{2}{k}dz\\,\\varepsilon\\upsilon\\right)\n\\end{equation}\nand the $SO(7)$ connection is\n\\begin{eqnarray}\n\\underline \\Omega^{a'b'}\n&=&\n\\Omega^{a'b'}(x,y,\\vartheta)-\\frac{1}{R}\\,\\underline E^7\\,J^{a'b'}\n-\\frac{2}{R}\\,\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'b'}\\gamma^5E\\,,\n\\nonumber\\\\\n\\label{so7connection}\n\\\\\n{\\underline\\Omega}^{a'7}\n&=&\n\\frac{1}{R}\\left(\\underline E^{b'}-4i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{b'}\\gamma^5E\\right)\\,J_{b'}{}^{a'}\\,.\\nonumber\n\\end{eqnarray}\nThe functions and forms appearing in\n(\\ref{upsilonfunctions})--(\\ref{so7connection}) are defined in\nAppendix B.\n\nThe first order form of the action for the massless superparticle in\nthe $OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ superbackground is\n\\begin{eqnarray}\\label{11dsuper}\nS =  \\int d\\tau \\left(  P_{\\underline A} \\,{\\underline\nE}^{\\underline A}_\\tau +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\\,,\n\\end{eqnarray}\nwhere  $ P_{\\underline A}$ ($\\underline A=0,1,\\dots,10$) is the\nparticle momentum, $e(\\tau)$ is the Lagrange multiplier which\nensures the mass shell condition $P^2=0$ and\n$$\n{\\underline E}^{\\underline A}_\\tau=\\partial_\\tau\nZ^{\\underline{\\mathcal M}}\\,{\\underline E}_{\\underline{\\mathcal\nM}}{}^{\\underline A}\\,, \\qquad Z^{\\underline{\\mathcal\nM}}=(x,y,z,\\vartheta,\\upsilon)\n$$\nis the pullback to the worldline of the supervielbeins\n(\\ref{upsilonfunctions}). The action is\ninvariant under local worldline diffeomorphisms and under the\nfermionic kappa--symmetry transformations\n\\begin{equation}\\label{kappaA}\n\\delta Z^{\\underline{\\mathcal M}} \\,{\\underline E}_{\\underline{\\mathcal M}}{}^{\\underline \\alpha}\n= P^{\\underline A} \\,(\\Gamma_{\\underline A}\\,\\kappa)^{\\underline\\alpha}\\,,\\qquad\n\\delta Z^{\\underline{\\mathcal M}} \\,{\\underline E}_{\\underline\n{\\mathcal M}}{}^{\\underline A} = 0\\,, \\qquad\n\\end{equation}\n\\begin{equation}\\label{kappaA1}\n\\delta e=-4i\\,{\\underline E}^{\\underline\n\\alpha}_\\tau\\,\\kappa_{\\underline\\alpha}\\,,\n\\qquad \\delta P_{\\underline A}=\n\\delta Z^{\\underline{\\mathcal M}}\\,\\underline\\Omega_{{\\underline{\\mathcal M}}\\underline A}{}^{\\underline B}\\,P_{\\underline B}.\n\\end{equation}\nInserting in the action the expressions for the vielbeins\n(\\ref{upsilonfunctions}), we get\n\\begin{eqnarray}\\label{11dsuper2}\nS =  \\int d\\tau  \\!\\!\\!\\!\\!\\!&&  \\left[\nP_{\\hat a}\\left(\nE^{\\hat a}_\\tau + 4 i\n\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal M}^2}\\,D_\\tau\\upsilon\n- 8 i \\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2} \\varepsilon {\\upsilon} \\,\n\\frac{\\partial_\\tau z}{k}\n\\right) \\right. \\nonumber \\\\\n&&\\left.+ P_{a'}\n\\left(E^{a'}_\\tau+2i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'}\\gamma^5\\,E_\\tau \\right)\\right. \\\\\n&&\\left. + P_7 \\left( R \\left( \\frac{\\partial_\\tau z}{k} + A\\right)\n-4\n\\upsilon \\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal\nM}^2}\\, (D_\\tau \\upsilon\\, -2 \\varepsilon\\upsilon \\frac{\\partial_\\tau z}{k})\n\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right]\n\\,.\n\\nonumber\n\\end{eqnarray}\n\nThe action (\\ref{11dsuper2}) can be simplified by eliminating some\nor all pure--gauge fermionic modes using the kappa--symmetry\ntransformations (\\ref{kappaA}). For instance, when the momentum of\nthe particle is non--zero along a $CP^3$ direction inside $S^7$, the\nprojectors ${\\mathcal P}_6$ and ${\\mathcal P}_2$, defined in eqs.\n(\\ref{p6}) and (\\ref{p2}), do not commute with the kappa--symmetry\nprojector (\\ref{kappaA}) and one can use \\emph{e.g.} the 16\nkappa--symmetry transformations to eliminate 16 of the 24\n$\\vartheta$. After such a gauge fixing the action will contain 8\nremaining $\\vartheta$ and 8 $\\upsilon$.\n\nAlternatively, by partially gauge fixing the kappa--symmetry one can\neliminate all eight $\\upsilon$ keeping 24 $\\vartheta$. In the latter\ncase the action reduces to the form in which it describes the\ndynamics of a superparticle in a superspace with 11 bosonic\ncoordinates and 24 fermionic ones. This superspace has been\nintroduced in\n\\cite{Gomis:2008jt} as a Hopf fibration of the supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$.\nIt is the supercoset\n\\begin{equation}\\label{24th}\n\\frac{OSp(6|4) \\times U(1)}{U(3) \\times SO(1,3)\\times\nZ_k}.\n\\end{equation}\nThe geometry of (\\ref{24th}) is described by the supervielbeins\n\\begin{eqnarray}\\label{24thA}\n&&\\hat E^{\\hat a} = E^{\\hat a}(x,y,\\vartheta)\\,, \\nonumber \\\\\n&&\\hat E^{a'} = E^{a'}(x,y,\\vartheta) \\,,\\\\\n&&\\hat E^{7} = R\\Big(\\frac{dz}{k} + A(x,y,\\vartheta) \\Big)\\,,\n\\nonumber \\\\\n&&\n\\hat E^{\\alpha a'} = E^{\\alpha a'}(x,y,\\vartheta)\\,,\\nonumber\n\\end{eqnarray}\nwhere (as already mentioned) the explicit form of the\nright--hand sides of (\\ref{24thA}) are given in (\\ref{cartan24}). Notice that now $z$  appears only in the vielbein $\\hat\nE^7$ along the $U(1)$-fiber direction of $S^7$.\n\nThe first order form of the superparticle action in the superspace\n(\\ref{24thA}) is\n\\begin{eqnarray}\\label{11dsuper24_A}\nS& = & \\int d\\tau \\left(  P_{\\hat a} \\,{\\hat E}^{\\hat a}_\\tau +\n P_{a'} \\,{\\hat E}^{a'}_\\tau +\n  P_{7} \\,{\\hat E}^{7}_\\tau +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\n\\,\\nonumber\\\\\n& =&  \\int d\\tau \\left(  P_{\\hat a} \\,{E}^{\\hat a}_\\tau +\n P_{a'} \\,{E}^{a'}_\\tau +\n  P_{7} \\, R \\left( \\frac{\\partial_\\tau z}{k} + A_\\tau\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\\,,\n\\end{eqnarray}\nwhere now\n$$\n{\\hat E}^{\\underline A}_\\tau=\\partial_\\tau Z^{{\\mathcal M}}\\,{\\hat\nE}_{{\\mathcal M}}{}^{\\underline A}+\\partial_\\tau\\,z\\,{\\hat\nE}_z{}^{\\underline A}\\,,\n\\qquad Z^{{\\mathcal M}}=(x,y,\\vartheta)\n$$\nis the pullback to the worldline of the supervielbeins\n(\\ref{24thA}).\n\nIt is easy to reduce the action (\\ref{11dsuper24_A}) to $D=10$. Once it\nis done, one obtains the action for a D0--brane moving in the\nsupercoset $OSp(6|4)\/U(3)\\times SO(1,3)$.\n\nAs we have mentioned above, the action (\\ref{11dsuper24_A})\ndescribes a superparticle which has a non--zero momentum along the\n$CP^3$ base of the $S^7$ bundle. This is required by the consistency\nof the kappa--symmetry gauge fixing condition $\\upsilon=0$. To\ndescribe other possible classical motions of the superparticle,\n\\emph{ e.g.} when $P^{a'}=0$, one should chose a different\nkappa--symmetry gauge.\n\nFor instance, if the superparticle has a non--zero spacial momentum\nalong the 7th, fiber direction, of $S^7$, one can use the gauge\nfixing condition corresponding to that of Subsection\n\\ref{theta-}. In this case, in virtue of the gauge-fixed expressions of Appendix C,\n the action (\\ref{11dsuper2}) simplifies to\n\\begin{eqnarray}\\label{11dsuper-gauged}\nS =  \\int d\\tau \\hspace{-.5cm}&&\\left[\nP_{a} \\,\\Big( e^a_\\tau(x)+i\\vartheta\\gamma^aD_{\\tau}\\vartheta +\ni \\upsilon \\gamma^aD_{\\tau} \\upsilon - 2 i\n\\upsilon \\gamma^a \\varepsilon \\upsilon \\frac{\\partial_\\tau z}{k} \\Big) \\right. \\nonumber \\\\\n&& \\left. +P_{a'} \\, {e}^{a'}_\\tau(y) + P_{3} \\,{e}^{3}_\\tau(x) +\nP_{7}\n\\, R\n\\left(\\frac{\\partial_\\tau z}{k}  +  A_\\tau(y)\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right]\\,.\n\\end{eqnarray}\nThe dimensional reduction of the $D=11$ superparticle action\n(\\ref{11dsuper-gauged}) along $z$ results in the kappa--symmetry\ngauge--fixed action which describes an arbitrary motion of the type\nIIA D0--brane in $AdS_4\\times CP^{3}$ superspace.\n\nBefore considering the D0--brane, let us note that the action\n(\\ref{11dsuper2}) is the most appropriate starting point for the\nconstruction of the pure--spinor formulation of the $D=11$\nsuperparticle in the $AdS_4\\times CP^3$ supergravity background. The\npure--spinor condition $\\lambda\\Gamma^{\\underline A}\\lambda=0$ in\n$D=11$ implies that the 32--component bosonic pure spinor\n$\\lambda^{\\underline \\alpha}$ has 23 independent components\n\\cite{Berkovits:2002uc,Fre':2006es}. This counting ensures the correct\nnumber of bosonic and fermionic degrees of freedom.\n\nIn the cases of the actions (\\ref{11dsuper24_A}) and\n(\\ref{11dsuper-gauged}) that describe a particle motion in the\nreduced superspaces, one can also develop pure--spinor formulations\nin which the pure spinor $\\lambda$, in addition, is subject to the\nsame constraint as the one imposed on $\\Theta$ by kappa--symmetry\ngauge fixing, e.g. ${\\mathcal P_2}\\lambda=0$ in the case\n$\\upsilon={\\mathcal P_2}\\Theta=0$. This guarantees the correct\ncounting of the degrees of freedom in the pure--spinor formulation\n(similar to the cases considered in\n\\cite{Fre:2008qc,Bonelli:2008us}). That is, the difference between\nthe bosonic and fermionic degrees of freedom remains the same.\nIndeed, in the case of the pure--spinor formulation of the massless\n$D=11$ superparticle\n\\cite{Berkovits:2002uc} there are 11 bosonic $X^{\\underline A}$ plus 23 pure spinor\ndegrees of freedom and 32 fermionic $\\Theta$, while  in the above\nexample of the reduced pure spinor formulation the pure spinor\neffectively contains $23-8=15$ degrees of freedom against 24\nfermionic ones, while the number of $X$ remains the same.\n\n When the pure spinor formulations of the superparticle\nin reduced superspaces correspond to the kappa--gauge fixed versions\nof the Brink--Schwarz superparticle whose consistency is limited to\nparticular subsectors of the classical configuration space of the\nfull theory, one may expect that the former will also describe only\nsubsectors of the pure--spinor superparticle model formulated in the\ncomplete superspace with 32 fermionic coordinates. As in the case of\nthe pure--spinor type IIA superstring in $AdS_4\\times CP^3$\n\\cite{Fre:2008qc,Bonelli:2008us}, these issues require additional\nanalysis.\n\n\\subsection{$D0$-brane}\nTo obtain the action for the $D0$--brane by dimensional\nreduction of the $D=11$ superparticle action, one should first perform the\nappropriate Lorentz transformation of the $D=11$\nsupervielbeins (as was explained in \\cite{Gomis:2008jt}) and make a\ncorresponding redefinition of the particle momentum. We shall not\nperform this dimensional reduction procedure since the result is\nwell known. The $D0$--brane action has the following first order\nform in the type IIA superbackground in the string frame (see eqs.\n(\\ref{simplA}) and (\\ref{simplB}))\n\\begin{equation}\\label{d0-action}\nS =  \\int d\\tau e^{- \\phi}  \\,\\left(P_A {\\cal E}^A_\\tau +\n\\frac{e}{2} (P_A P_B \\eta^{AB}+ m^2) \\right) + m\\,  \\int {\\cal\nA}_1\\,,\n\\end{equation}\nwhere $m$ is the mass of the particle and the second term describes\nits coupling to the RR one--form potential ${\\cal A}_1$.\n\nIntegrating out the momenta $P_A$ and the auxiliary field $e(\\tau)$\nwe arrive at the  action\n\\begin{equation}\\label{d0-action2}\nS = -m\\,\\int d\\tau e^{- \\phi}  \\,\\sqrt{-{\\cal E}^A_\\tau {\\cal\nE}^B_\\tau\\,\\eta_{AB}} + m\\int {\\cal A}_1\\,.\n\\end{equation}\nThe action (\\ref{d0-action2}) is invariant under worldline\ndiffeomorphisms and the kappa-symmetry transformations (to\nverify the kappa-symmetry one needs the superspace constraints\non the torsion $T^A$ and on $F_2$ given in Appendix A.4)\n\\begin{eqnarray}\n&&\\delta_\\kappa Z^{\\mathcal M} {\\cal E}_{\\cal\nM}^{~~\\underline\\alpha} =\n\\frac{1}{2}(1 + \\Gamma)^{\\underline\\alpha}{}_{\\underline\\beta} \\,\\kappa^{\\underline\\beta}(\\tau)\\,, ~~~\n{\\underline\\alpha}  = 1,\\dots,32\\,,\\qquad Z^{\\mathcal M}=(x,y,\\vartheta,\\upsilon)\\,\\nonumber\\\\\n\\\\\n&&\\delta_\\kappa Z^{\\mathcal M} {\\cal E}_{\\cal M}^{~~A} = 0\\,, ~~~\nA=0,1,\\dots,9\\,\n\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{GammaD0}\n\\Gamma = \\frac{1}{\\sqrt{-\\mathcal E^2_\\tau}}\\,{\\mathcal E}_\\tau^{~A} \\Gamma_A \\Gamma_{11}\\,,\n\\,\\qquad \\Gamma^2=1\\,.\n\\end{equation}\nComparing the form of the kappa--symmetry projector matrix\n(\\ref{GammaD0}) with the kappa--symmetry gauge fixing condition of\nSubsection \\ref{theta-}, we see that\n$\\gamma=\\Gamma^0\\Gamma^1\\Gamma^2$ (introduced in eq. (\\ref{gamma}))\ndoes not commute with $\\Gamma$ in (\\ref{GammaD0}) provided that the\nenergy $P^0\\sim{{\\mathcal E}^0_\\tau}$ of the massive particle is\nnonzero, which is always the case. Thus to simplify, \\emph{e.g.} the first\norder action (\\ref{d0-action}) we can use the gauge fixed form of\nthe supervielbeins and the RR one--form of Subsection\n\\ref{theta-}. The action takes the following explicit form, with an\nappropriately rescaled Lagrange multiplier $e(\\tau)$,\n\\begin{eqnarray}\\label{simply}\nS &=& \\left(\\frac{R}{kl_p}\\right)^{-1} \\,\\int d\\tau \\,\n\\left[\\Big( P_{a'}e_\\tau^{a'}(y)\n+ P_{3} \\,e_\\tau^3(x)\\Big)\\,(1+\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\right. \\nonumber \\\\\n&+& \\left. P_{a} \\,\n(e_\\tau^a(x)+i\\Theta\\gamma^aD_{\\tau}\\Theta)\\,(1+\\frac{8}{R^2}(\\upsilon\\ups)^2)\n+\\frac{e(\\tau)}{2} \\,(P_A P_B \\eta^{AB}+ m^2) \\nonumber\\right] \\\\\n&+& m kl_p\\,\\int \\, \\Big(\nA(y)-\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\varepsilon\\gamma_a\\upsilon\\Big)\\,.\n\\end{eqnarray}\nThis action contains fermionic terms up to the 6th order in\n$\\Theta=(\\vartheta,\\upsilon)$.\n\n\\subsection{The fundamental string}\\label{FS}\n\nIn this section we use the geometry discussed above to construct\nthe Green-Schwarz model for the fundamental string. We will first\nreview the form of the superstring sigma model without gauge fixing\nand then impose the gauge fixing of the kappa--symmetry. This will\nprovide a calculable sigma model.\n\nThe action for the Green--Schwarz superstring has the following form\n\\begin{equation}\\label{cordaA}\nS = -\\frac{1}{4\\pi\\alpha'}\\,\\int d^2\\xi\\, \\sqrt {-h}\\, h^{IJ}\\,\n{\\cal E}_{I}^{A} {\\cal E}_{J}^{B} \\eta_{AB}\n-\\frac{1}{2\\pi\\alpha'}\\,\\int  B_2\\,,\n\\end{equation}\nwhere $\\xi^I$  $(I,J=0,1)$ are the worldsheet coordinates,\n$h_{IJ}(\\xi)$ is a worldsheet metric and $B_2$ is the pull--back to\nthe worldsheet of the NS--NS 2--form.\n\nThe kappa--symmetry transformations which leave the superstring\naction (\\ref{cordaA}) invariant are\n\n\\begin{equation}\\label{kappastring}\n\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^{\\underline \\alpha}=\n{1\\over 2}(1+\\Gamma)^{\\underline \\alpha}_{~\\underline\\beta}\\,\n\\kappa^{\\underline\\beta}(\\xi),\\qquad {\\underline \\alpha}=1,\\cdots, 32\n\\end{equation}\n\\begin{equation}\\label{kA}\n\\hskip-2.5cm\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^A=0,\n\\qquad   A=0,1,\\cdots,9\n\\end{equation}\nwhere $\\kappa^{\\underline\\alpha}(\\xi)$ is a 32--component spinor\nparameter, ${1\\over 2}(1+\\Gamma)^{\\underline\n\\alpha}_{~\\underline\\beta}$ is a spinor projection matrix with\n\\begin{equation}\\label{gbs}\n\\Gamma={1\\over {2\\,\\sqrt{-\\det{g_{IJ}}}}}\\,\\epsilon^{IJ}\\,{\\mathcal\nE}_{I}{}^A\\,{\\mathcal E}_{J}{}^B\\,\\Gamma_{AB}\\,\\Gamma_{11}, \\qquad\n\\Gamma^2=1\\,,\n\\end{equation}\nand the auxiliary worldsheet metric $h^{IJ}$ transforms as follows\n\\begin{eqnarray}\\label{deltah}\n\\lefteqn{\\delta_\\kappa\\,(\\sqrt{-h}\\,h^{IJ})}\n\\nonumber\\\\\n&=&2i\\,\\sqrt{-h}\\,(h^{IJ}\\,g^{KL}-2h^{K(I}\\,g^{J)L})\n\\left(\\delta_\\kappa\nZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}\\,\\Gamma_A\\,{\\mathcal\nE}_K\\,{\\mathcal E}^A_L +\\frac{1}{2}g_{KL}\\delta_\\kappa Z^{\\mathcal\nM}\\,{\\mathcal E}_{\\mathcal M}{}^{\\alpha i}\\lambda_{\\alpha i}\n\\right)\n\\\\\n&&\\hspace{-20pt}\n-2i\\,\\sqrt{-h}\\,\\,\\frac{h^{IK'}g_{K'L'}h^{L'J}-\\frac{1}{2}h^{IJ}\\,h^{K'L'}g_{K'L'}}\n{\\frac{1}{2}\\,h^{K'L'}\\,g_{K'L'}+\\sqrt{\\frac{g}{h}}}\\,g^{KL}\\,\\left(\\delta_\\kappa\nZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}\\,\\Gamma_A\\,{\\mathcal\nE}_K\\,{\\mathcal E}^A_L +\\frac{1}{2}g_{KL}\\delta_\\kappa Z^{\\mathcal\nM}\\,{\\mathcal E}_{\\mathcal M}{}^{\\alpha i}\\lambda_{\\alpha i}\n\\right)\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{im}\ng_{IJ}(\\xi)={\\mathcal E}_{I}{}^{A}\\, {\\mathcal\nE}_{J}{}^{B}\\,\\eta_{AB}\\,,\\qquad g^{IJ}\\equiv (g_{IJ})^{-1}\n\\end{equation}\nis the induced metric on the worldsheet of the string that on the\nmass shell coincides with the auxiliary metric $h_{IJ}(\\xi)$ modulo\na conformal factor. Finally, $g=\\det g_{IJ}$ and  $h=\\det h_{IJ}$.\n\\\\\nUsing the identity\n\\begin{equation}\nh^{IJ}\\,g_{JK}\\,h^{KL}\\,g_{LI}-\\frac{1}{2}\\,(h^{IJ}\\,g_{IJ})^{2}\\equiv\n\\frac{1}{2}\\,(h^{IJ}\\,g_{IJ})^{2}-2\\,\\frac{g}{h}\\,\n\\end{equation}\none can check that eq. (\\ref{deltah}) multiplied by $g_{IJ}$ results\nin\n\\begin{equation}\\label{kappag}\n\\delta_\\kappa\\,(\\sqrt{-h}\\,h^{IJ})\\,g_{IJ}\n=4i(\\sqrt{-g}\\,g^{KL}-\\sqrt{-h}\\,h^{KL})\\,\n\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^{\\underline\\alpha}(\\Gamma_A\\,{\\mathcal E}_K\\,{\\mathcal E}^A_L\n+\\frac{1}{2}g_{KL}\\lambda )_{\\underline\\alpha}\\,,\n\\end{equation}\nwhich together with the variation (\\ref{kappastring}) and (\\ref{kA})\nof the superspace coordinates insures the invariance of the action\n(\\ref{cordaA}).\n\n Comparing the form of the kappa--symmetry projector\n(\\ref{kappastring}) with the kappa--symmetry gauge fixing condition\nof Subsection\n\\ref{theta-} we see that this gauge choice is admissible when the\nstring moves in such a way that the projection of its worldsheet on\nthe $3d$ subspace along the directions $e^a$ $(a=0,1,2)$  of the\ntarget space is a non--degenerate two--dimensional time--like\nsurface. Thus, it can be used to analyze the string dynamics in the\nsector which is not reachable by the supercoset model of\n\\cite{Arutyunov:2008if,Stefanski:2008ik,Fre:2008qc}. The latter is\nobtained from the action (\\ref{cordaA}) by gauge fixing to zero the\neight fermions $\\upsilon$, which is only possible when the string\nworldsheet extends in the $CP^3$ directions.\n\nIn the gauge of Subsection \\ref{theta-}  we insert into the action\n(\\ref{cordaA}) the  expressions (\\ref{simple+v}) and (\\ref{simple+B})\nfor the supervielbeins and $B_2$. This results in an action that\ncontains fermionic terms only up to the 8th order in\n$\\Theta=(\\vartheta,\\upsilon)$\n\\footnote{The factor in front of the\naction is unconventional due to our normalization of the vielbeins,\nwhich comes from the dimensional reduction of the eleven-dimensional\ngeometry. More conventional, unit radius string frame vielbeins\n$(\\hat e^{\\hat a},\\hat e^{a'})$ can be introduced by the following\nrescaling\n$$\n(e^{\\hat a},e^{a'})=\\left(\\frac{R}{k\nl_p}\\right)^{-1\/2}\\left(\\frac{R}{2}\\,\\hat e^{\\hat a},\\,R\\,\\hat\ne^{a'}\\right)\\,.\n$$\nThen the factor in front of the action becomes\n$\\frac{R^2}{4\\pi\\alpha'}=\\frac{(R\/l_p)^3}{4\\pi k}$.}\n\\begin{eqnarray}\\label{cordaB}\nS &=&-\\frac{1}{4\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\sqrt{-h}\\,h^{IJ}\\left[\n\\left(e^{a'}_I e^{b'}_{J} \\delta_{a'b'} +\ne^3_I e^3_{J}\\right)\\,(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\right.\n\\nonumber \\\\\n&&{}\n\\hspace{4cm}\n+\\left.(e^a_I+i\\Theta\\gamma^aD_I\\Theta)\\,\n(e^b_{J}+i\\Theta\\gamma^bD_{J}\\Theta)\\,\\eta_{ab}\\,\n(1-\\frac{2}{R^2}(\\upsilon\\ups)^2)\\right]\n\\nonumber\\\\\n\\\\\n &-&\\frac{1}{2\\pi\\alpha'}\\frac{R}{kl_p}\\int\n\\Big[e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{60pt}}-\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)\\,(e^a+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\gamma_{ab}\\,\\varepsilon\\upsilon\\,\n\\Big]\\,.\\nonumber\n\\end{eqnarray}\nTo avoid possible confusion, let us remind the reader that in eqs.\n(\\ref{cordaB})--(\\ref{cordaBT}) the covariant derivative\n$D\\Theta\\equiv (D_8\\upsilon,\\,D_{24}\\vartheta)$ is defined in eqs.\n(\\ref{D81}). Actually, in (\\ref{cordaB}) the vielbein\n$e^3$ does not contribute to the covariant derivative and the\nconnection $\\omega^{ab}$ is zero along the $3d$ Minkowski boundary\nof $AdS_4$, for the vielbeins chosen as in eqs. (\\ref{ad4v}). It is\nnot hard to check that the action (\\ref{cordaB}) is invariant\nunder twelve `linearly realized' supersymmetry transformations\n$$\n\\delta\\vartheta=\\epsilon, \\qquad \\delta\ne^a=-i\\epsilon\\,\\gamma^a\\,D_{24}\\vartheta\n$$\nwith parameters $\\epsilon=\\frac{1}{2}\\,(1+\\gamma)\\,\\epsilon$ being\n$CP^3$ Killing spinors\n$$\nD_{24}\\epsilon={\\mathcal P}_6\\,(d-\\frac{1}{R}e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'} -\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\epsilon=0\\,.\n$$\nThe other twelve supersymmetries of the $OSp(6|4)$ isometries of\n(\\ref{cordaB}) are non--linearly realized on the worldsheet fields\nand include compensating kappa--symmetry transformations required to\nmaintain the gauge $\\vartheta=\\frac{1}{2}\\,(1+\\gamma)\\,\\vartheta$.\n\nThe action (\\ref{cordaB}) is slightly more complicated than the\naction for the $AdS_5\\times S^5$ superstring in the analogous\nkappa--symmetry gauge\n\\cite{Pesando:1998fv,Kallosh:1998nx,Kallosh:1998ji},\nthat contains fermions only up to the fourth order, since\n$AdS_4\\times CP^3$ is less supersymmetric than $AdS_5\\times\nS^5$. The action (\\ref{cordaB}) takes a form similar to that of\n\\cite{Pesando:1998fv,Kallosh:1998nx,Kallosh:1998ji} when we formally\nput the broken supersymmetry fermions $\\upsilon^{\\alpha i}$ to zero.\n\nAs in the case of the string in $AdS_5\\times S^5$ it is possible to simplify the action further\nby performing a T--duality transformation on the worldsheet \\cite{Kallosh:1998ji}. Following\n\\cite{Kallosh:1998ji} we first rewrite the part of the action\n(\\ref{cordaB}) containing the vielbeins $e^a$ in the first order\nform\n\\begin{equation}\\label{first}\nS_1=\\frac{1}{2\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\left[P_a^I\\,(e^a_I+i\\Theta\\gamma^aD_I\\Theta)\n+\\frac{1-\\frac{6}{R^2}\\,(\\upsilon\\upsilon)^2}{2\\sqrt{-h}}\\,P_a^I\\,P_b^J\\,h_{IJ}\\,\\eta^{ab}\n-\\frac{i}{R}\\,\\frac{\\varepsilon_{IJ}}{-h}\\,P^I_a\\,P^J_b\\,\\upsilon\\gamma^{ab}\\varepsilon\\upsilon\\right]\\,.\n\\end{equation}\nThe equations of motion for the momenta $P_a^I$ imply that\n\\begin{equation}\\label{P}\nP^I_a=-\\sqrt{-h}\\,(1-\\frac{2}{R^2}(\\upsilon\\upsilon)^2)\\,\n\\Big(h^{IJ}\\eta_{ab}\n+\\frac{2i}{R\\sqrt{-h}}\\,\\varepsilon^{IJ}\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\\Big)\n\\,(e^b_J+i\\Theta\\gamma^bD_J\\Theta)\\,.\n\\end{equation}\nUsing the explicit form of the $AdS_4$ vielbeins given in eq.\n(\\ref{ad4v}) and varying the first order action (\\ref{first}) with\nrespect to $x^a$ we find that $P^I_a$ is proportional to the\nconserved current associated with translations along $x^a$\n\\begin{equation}\\label{trcurrent}\n\\partial_I\\,\\Big({r^2\\over R^2}\\,P^I_a\\Big)=0\\, \\qquad\\Rightarrow\n\\qquad P^I_a=\\frac{R^2}{r^2}\\,\\varepsilon^{IJ}\\partial_J\\,{\\tilde\nx}_a\\equiv \\varepsilon^{IJ}\\,{\\tilde e}_{Ja}\\,.\n\\end{equation}\nIf we now substitute eq. (\\ref{trcurrent}) into (\\ref{first}) the\nT--dualized version of the action (\\ref{cordaB}) for the string in\n$AdS_4\\times CP^3$ takes the form\n\\begin{eqnarray}\\label{cordaBT}\nS &=&-\\frac{1}{4\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\sqrt{-h}\\,h^{IJ}\n\\left(\n{\\tilde e}_I^a\\,{\\tilde e}_J^b\\,\\,\\eta_{ab}+e^3_I e^3_{J}\n+e^{a'}_I e^{b'}_{J} \\delta_{a'b'}\n\\right)\\,(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\nonumber\\\\\n\\\\\n&-&\\frac{1}{2\\pi\\alpha'}\\frac{R}{kl_p}\\int\n\\Big[e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{60pt}} +i{\\tilde e}^a\\,\\Theta\\gamma_aD\\Theta\n-\\frac{i}{R}\\,\\,{\\tilde e}^a\\,{\\tilde e}^b\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\Big]\\,.\\nonumber\n\\end{eqnarray}\n\nNote that in the T--dualized action the fermionic kinetic terms\nappear only in the Wess--Zumino term and that there are now terms of\nat most fourth order in fermions. Note also that the first (induced\nmetric) term of (\\ref{cordaBT}) acquires a common factor\n$(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)$ in contrast to the corresponding\nterms in the original action (\\ref{cordaB}).\n\nTo preserve the conformal invariance of the dual action at the\nquantum level one should add to it a dilaton term\n$\\int\\,R^{(2)}\\,\\tilde\n\\phi$ (where $R^{(2)}$ is the worldsheet curvature), which is\ninduced by the functional integration of $P_a^I$ when passing to the\ndual action (see \\cite{Buscher:1987qj,Schwarz:1992te,Kallosh:1998ji}\nfor details). Here we should point out that in our case the original\n$AdS_4\\times CP^3$ superbackground already has a non--trivial\ndilaton which depends on $\\upsilon$ (see eqs. (\\ref{dilaton1}) and\n(\\ref{gfdialton})).\n\nThe following comment is now in order. As in the $AdS_5\\times S^5$\ncase \\cite{Kallosh:1998ji,Ricci:2007eq,Beisert:2008iq}, upon the\nT--duality along the three translational directions $x^a$ of $AdS_4$\nthe purely bosonic (classically integrable) $AdS_4\\times CP^3$\nsector of the type IIA superstring sigma model maps into an\nequivalent sigma model on a dual $AdS_4$ space, both models sharing\nthe same integrable structure \\cite{Ricci:2007eq}. The situation\nwith the fermionic sector of the $AdS_4\\times CP^3$ superstring is,\nhowever, different due to the fact that there is less supersymmetry\nthan in the $AdS_5\\times S^5$ case.\n\n In the case of the\n$AdS_5\\times S^5$ superstring sigma model, one can accompany the\nabove bosonic T--duality transformation by a fermionic one along\nfermionic directions in (complexified) superspace which have\ntranslational isometries \\cite{Berkovits:2008ic,Beisert:2008iq}.\nThis compensates the dilaton term generated by the bosonic\nT--duality and maps the $AdS_5\\times S^5$ superstring action to an\nequivalent (dual) one, which is also integrable\\footnote{In\n\\cite{Ricci:2007eq,Berkovits:2008ic,Beisert:2008iq} it has been shown that this\nduality property of the $AdS_5\\times S^5$ superstring is related to\nearlier observed dual conformal symmetry of maximally helicity\nviolating amplitudes of the ${\\mathcal N}=4$ super--Yang--Mills\ntheory and to the relation between gluon scattering amplitudes and\nWilson loops at strong and weak coupling.}. However, in the\n$AdS_4\\times CP^3$ case under consideration the fermionic directions\nin superspace parametrized by $\\upsilon$ do not have translational\nisometries, since the action (\\ref{cordaB}) or (\\ref{cordaBT}) has\n$\\upsilon$--dependent fermionic terms which do not contain\nworldsheet derivatives. This just reflects the fact that the\nfermionic modes $\\upsilon$ correspond to the broken supersymmetries\nof the superbackground.\n\nAs far as the T--dualization of the supersymmetric fermionic modes\n$\\vartheta$ is concerned, it might be, in principle, possible (at\nleast in the absence of $\\upsilon$) if, as in the case of the\n$PSU(2,2|4)$ superstring sigma model\n\\cite{Berkovits:2008ic,Beisert:2008iq}, there existed a\nrealization of the $OSp(6|4)$ superalgebra in which 12 of the 24\n(complex conjugate) supersymmetry generators squared to zero and\nformed a representation of the bosonic subalgebra of $OSp(6|4)$. In\nother words the possibility of T--dualizing part of fermionic modes\n$\\vartheta$ (in the absence of $\\upsilon$) is related to the\nquestion of the existence of a chiral superspace representation of\nthe superalgebra $OSp(6|4)$. Such a realization of $OSp(6|4)$ seems\nnot to exist. In fact, it has been argued in \\cite{Adam:2009kt} that\nthe $OSp(6|4)$ supercoset subsector of the Green--Schwarz\nsuperstring in $AdS_4\\times CP^3$  does not have any fermionic\nT--duality symmetry since in $OSp(6|4)$ the dimension of the\nrepresentation of the supercharges under the R--symmetry is odd. The\nabsence of the fermionic T--duality of the superstring in\n$AdS_4\\times CP^3$ may have interesting manifestations in particular\nfeatures of the $AdS_4\/CFT_3$ holography.\n\nThe gauge--fixed actions (\\ref{cordaB}) or (\\ref{cordaBT}) can be\nused for studying different aspects of the $AdS_4\/CFT_3$\ncorrespondence and integrability on both of its sides, in\nparticular, for making two-- and higher--loop string computations\nfor testing the Bethe ansatz and the S--matrix\n\\cite{Minahan:2008hf}--\\cite{Minahan:2009te} in the dual planar ${\\mathcal N}=6$\nsuperconformal Chern--Simons--matter theory, which would extend the\nanalysis of\n\\cite{Nishioka:2008gz}--\\cite{Suzuki:2009sc},\\cite{Zarembo:2009au} and\nothers.\n\n\\subsection{D2--branes}\nLet us now consider the effective worldvolume theory of probe\nD2--branes moving in the $AdS_4 \\times CP^3$ superbackground. This\ncan be derived from the action for $D$--branes in a generic type IIA\nsuperbackground\n\\cite{Cederwall:1996ri,Aganagic:1996pe,Bergshoeff:1996tu} by\nsubstituting the explicit form of the $AdS_4 \\times CP^3$\nsupergeometry (\\ref{simplA})--(\\ref{A3}).\n\nThe action for a D2--brane in a generic type IIA supergravity\nbackground in the string frame has the following form\n\\begin{equation}\\label{DBIstring}\nS= -T\\int\\,d^{3}\\xi\\,e^{{ - } {\\phi}}\\sqrt{-\\det(g_{IJ}+{\\cal\nF}_{IJ})}+T \\int\\, ({\\mathcal A}_3 + {\\mathcal A}_1\\,{\\cal F}_2)\\,,\n\\end{equation}\nwhere $T$ is the tension of the D2--brane, $\\phi(Z)$ is the dilaton\nsuperfield,\n\\begin{equation}\\label{imb}\ng_{IJ}(\\xi)={\\mathcal E}_{I}{}^{A}\\, {\\mathcal\nE}_{J}{}^{B}\\,\\eta_{AB}\\qquad I,J=0,1,2;\\qquad A,B=0,1,\\cdots,9\n\\end{equation}\nis the induced metric on the D2--brane worldvolume with ${\\mathcal\nE}_{I}{}^{A}=\\partial_I\\,Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal\nM}{}^{A}$ being the pullbacks of the vector supervielbeins of the\ntype IIA $D=10$ superspace and\n\\begin{equation}\\label{deltaQstring}\n{\\cal F}_2 = d{\\mathcal V} - {B}_2\n\\end{equation}\nis the field strength of the worldvolume Born--Infeld gauge field\n${\\mathcal V}_I(\\xi)$ extended by the pullback of the NS--NS\ntwo--form. ${\\mathcal A}_1$ and  ${\\mathcal A}_3$ are the pullbacks\nof the type IIA supergravity RR superforms (\\ref{simplB}) and\n(\\ref{A3}).\n\nProvided that the superbackground satisfies the IIA supergravity\nconstraints, the action (\\ref{DBIstring}) is invariant under\nkappa--symmetry transformations of the superstring coordinates\n$Z^{\\mathcal M}(\\xi)$ of the form\n(\\ref{kappastring}), (\\ref{kA}), together with\n\\begin{equation}\n\\delta_\\kappa\\mathcal F_{IJ}=-\\mathcal E_J^{\\mathcal B}\\,\\mathcal E_I^{\\mathcal A}\\,\\delta_\\kappa Z^{\\mathcal M}\\mathcal E_{\\mathcal M}{}^{\\underline\\alpha}\\,H_{\\underline\\alpha\\mathcal{AB}}\n=-4i\\mathcal E_{[I}^A\\,\\mathcal E_{J]}\\Gamma_A\\Gamma_{11}\\delta_\\kappa\\mathcal E\n-2i\\mathcal E_J^B\\,\\mathcal E_I^A\\,\\delta_\\kappa\\mathcal E\\Gamma_{AB}\\Gamma_{11}\\lambda\\,,\n\\end{equation}\nwhere $\\delta_\\kappa\\mathcal E=\\delta_\\kappa Z^{\\mathcal M}\\mathcal E_{\\mathcal M}{}^{\\underline\\alpha}$.\n\nIn the case of the $D2$--brane the matrix $\\Gamma$ has the form\n\\begin{eqnarray}\\label{bargamma}\n\\Gamma&=-&{{1}\\over {\\sqrt{-\\det(g+{\\cal\nF})}}}\\,\\,\\varepsilon^{IJK}\\,({1\\over 3!}\\,{\\mathcal\nE}_I^{A}\\,{\\mathcal E}_J^{B}\\,{\\mathcal E}_K^{C}\\Gamma_{ABC}+{1\\over\n2}\\,{\\mathcal F_{IJ}}\\,{\\mathcal\nE}_K^{A}\\,\\Gamma_A\\,\\Gamma_{11})\\nonumber\\\\\n&&\\\\\n&=&-{{1}\\over {3!\\sqrt{-\\det(g+{\\cal\nF})}}}\\,\\,\\varepsilon^{IJK}\\,{\\mathcal E}_I^{A}\\,{\\mathcal\nE}_J^{B}\\,{\\mathcal E}_K^{C}\\,\\Gamma_{ABC}\\,\\,(1+{1\\over\n2}\\,{\\mathcal F^{IJ}}\\,{\\mathcal E}_I^{A}\\,{\\mathcal\nE}_J^{B}\\,\\Gamma_{AB}\\,\\Gamma_{11})\\,.\\nonumber\n\\end{eqnarray}\n\n\\subsubsection{D2 filling $AdS_2\\times S^1$ inside of\n$AdS_4$}\\label{d2ads2s1}\n\nLet us consider the D2--brane configuration which corresponds to a\ndisorder loop operator in the ABJM theory\n\\cite{Drukker:2008jm}. The 1\/2 BPS static solution of the equations of motion of the D2--brane on\n$AdS_2\\times S^1$ in the metric\n\\begin{equation}\\label{ads4}\nds^2=\\frac{R_{CP^3}^2}{4u^2}(-dx^0\\,dx^0+dr^2+r^2d\\varphi^2+du^2)+R_{CP^3}^2ds^2_{CP^3}\\,,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nds^2_{CP^3}={1\\over 4}\\Big[d\\alpha^2+\\cos^2{\\alpha\\over\n2}(d\\vartheta_1^2+\\sin^2\\vartheta_1d\\varphi_1^2)+\\sin^2{\\alpha\\over\n2}(d\\vartheta_2^2+\\sin^2\\vartheta_2d\\varphi_2^2)\\cr\n+\\sin^2{\\alpha\\over 2}\\cos^2{\\alpha\\over 2}(d\\chi+\n\\cos\\vartheta_1d\\varphi_1-\\cos\\vartheta_2d\\varphi_2)^2\\Big]\\,,\\nonumber\n\\end{eqnarray}\nis characterized by the following embedding of the brane worldvolume\n\\begin{equation}\\label{ads2s11}\n\\xi^0=x^0,\\qquad \\xi_1=u,\\qquad \\xi_2=\\varphi\\,,\n\\qquad r=a\\,\\xi_1\n\\end{equation}\nwhich is supported by the non--zero (electric) Born--Infeld field\nstrength\n\\begin{equation}\nF=E {dx^0\\wedge du\\over u^2}\\,,  \\qquad\nE=\\frac{R_{CP^3}^2}{4}\\sqrt{1+a^2}\\,,\n\\end{equation}\nwhere $a$ is an arbitrary constant. Note that the presence of the\nnon--zero DBI flux on the $AdS_2$ subspace of the D2--brane\nworldvolume is required to ensure the no--force condition, i.e.\nvanishing of the classical action (\\ref{DBIstring}) of this static\nD2--brane configuration, provided that also an additional BI flux\nboundary counterterm is added to the action (see\n\\cite{Drukker:2008jm} for more details). A natural explanation of\nthis boundary term is that it appears in the process of the\ndualization of the compactified 11th coordinate scalar field of the\nM2--brane into the BI vector field of the D2--brane.\n\nNote that in \\cite{Drukker:2008jm} this brane configuration was\nconsidered in a different coordinate system, in which $AdS_4$ is\nfoliated with $AdS_2\\times S^1$ slices instead of the flat $R^{1,2}$\nslices. This makes manifest the symmetries of the D2--brane\nconfiguration. An explicit form of the $AdS_4$ metric in this\nslicing is\n\\begin{equation}\\label{ads2s1}\nds^2_{_{AdS_4}}= {R_{CP^3}^2\\over 4}\\,(\\cosh^2 \\psi\n\\,ds^2_{_{AdS_2}}+d\\psi^2+\n\\sinh^2 \\psi\n\\, d\\varphi^2)\n\\end{equation}\nwhich is essentially a double analytic continuation of the usual\nglobal $AdS_4$ metric. The static D2--brane configuration is then\ncharacterized by the identification of the worldvolume coordinates\n$\\xi^a$ with those of $AdS_2$ and the $S^1$ angle $\\varphi$.\nHowever, for our choice of the kappa--symmetry gauge fixing\ncondition the use of the metric in the form (\\ref{ads4}) is more\nconvenient, since the associated $AdS_4$ vielbeins\n\\begin{equation}\\label{0123}\ne^0=\\frac{R_{CP^3}}{2u}\\,dx^0\\,,\\qquad\ne^1=\\frac{R_{CP^3}}{2u}\\,dr\\,,\\qquad\ne^2=\\frac{R_{CP^3}\\,r}{2u}\\,d\\varphi\\,,\\qquad\ne^3=-\\frac{R_{CP^3}}{2u}\\,du\\,\n\\end{equation}\nand the spin connection directly satisfy the relations\n(\\ref{eaoa31}) and (\\ref{oab}).\n\nOne can be interested in D2--brane bosonic and fermionic\nfluctuations around this 1\/2 BPS static D2--brane solution described\nby the action (\\ref{DBIstring}). To simplify the form of the\nfermionic terms, the kappa--symmetry gauge fixing for the D2--brane\nwrapping $AdS_2\n\\times S^1$ can be made in the simplest possible way considered in\nSubsection \\ref{theta-}. To get the gauge fixed D2--brane action in\nthis case one should substitute into (\\ref{DBIstring}) the\nexpressions for the vector supervielbeins (\\ref{simple+v}), the RR\none--form (\\ref{simple+A}) and the three--form (\\ref{A3-}), and the\nNS--NS two--form (\\ref{simple+B}).\n\n\\subsubsection{D2 at the Minkowski boundary of $AdS_4$}\nLet us now consider the supersymmetric effective worldvolume action\ndescribing a D2--brane placed at the Minkowski boundary of the\n$AdS_4$ space. In this case it is convenient to choose the\n$AdS_4\\times CP^3$ metric in the form (\\ref{ads4metric11}) or\n(\\ref{ads4metric21}).\n\n\nWhen the D2--brane is at the Minkowski boundary, we take the static\ngauge $\\xi^m=x^m$. The 1\/2 BPS ground state of the D2--brane is when\nits transverse scalar modes are constant and the Born--Infeld field\nand the fermionic modes are zero. As a consistency check, let us\nnote that with the choice of the background value of the RR 3--form\n(\\ref{A3}) and (\\ref{A31}) and of the corresponding (positive) $D2$--brane charge\n(characterized by the plus sign in front of the Wess--Zumino term\n(\\ref{DBIstring})), the action of the ground state of the D2--brane\nat the Minkowski boundary vanishes. This means that such a brane\nconfiguration is stable and does not experience any external force,\n\\emph{i.e.} it is a BPS state.\n\nIf, on the other hand, with the same choice of ${\\mathcal A}_3$\n(\\ref{A3}) and (\\ref{A31}), we considered an anti--$D2$--brane carrying a negative\n${\\mathcal A}_3$ charge (which would be characterized by a minus\nsign in front of the Wess--Zumino term in (\\ref{DBIstring})), the\nground state of this anti--$D2$--brane at the Minkowski boundary\nwould have a non--zero action\n$$\nS_{\\overline{D2}}=-2Te^{-\\phi_0}\\,\\int\\,d^{\\,3}x\\,\\left(r\\over {R_{CP^3}}\\right)^6\\,\n$$\nimplying that such a solution is unstable (as is well known to be\nthe case for a probe anti--D--brane in a background of D--branes).\nIt is, therefore, important for the consistency of the solution to\ntake care that the relative signs of the RR potential ${\\mathcal\nA}_3$ and the $D2$--brane charge (and, as a consequence, the sign of\nthe kappa--symmetry projector) ensure the no--force condition, i.e.\nvanishing of the static $D2$--brane action. In the case of M2, M5\nand D3--branes at the Minkowski boundary of $AdS$ this issue was\ndiscussed in detail in\n\\cite{Pasti:1998tc}.\n\nFor the static D2--brane configuration the kappa--symmetry projector\n(\\ref{bargamma}) reduces to\n\\begin{equation}\\label{kappa}\nP=\\frac{1}{2}(1+\\gamma), \\qquad\n\\gamma=\\gamma^0\\gamma^1\\gamma^2=-\\gamma_0\\gamma_1\\gamma_2\n\\end{equation}\nSo the natural choice of the kappa--symmetry gauge fixing condition\nis\n\\begin{equation}\\label{kappagauge}\n\\Theta=\\frac{1}{2}(1-\\gamma)\\Theta\\,,\n\\end{equation}\n\\emph{i.e.} the gauge choice considered in detail in Subsection\n\\ref{theta++}.\nNote that in the case of the D2--brane at the Minkowski boundary we\ncannot use the simpler condition\n$\\Theta=\\frac{1}{2}(1+\\gamma)\\Theta$ of Subsection \\ref{theta-},\nbecause the kappa--symmetry projector (\\ref{kappa}) has the same\nsign.\n\nPlugging the kappa--symmetry gauge--fixed quantities of Subsection\n3.2 into the action (\\ref{DBIstring}), one can study the properties\nof the $OSp(6|4)$ invariant effective $3d$ gauge--matter field\ntheory on the worldvolume of the $D2$--brane placed at the Minkowski\nboundary of $AdS_4$, which from the point of view of M--theory\ncorresponds to an $M2$--brane pulled out to a finite distance from a\nstack of $M2$--branes probing $R^8\/Z_k$.\n\n\nThe effective theory on the worldvolume of this D2--brane, which\ndescribes its fluctuations in $AdS_4\n\\times CP^3$, is an interacting $d=3$ gauge Born--Infeld--matter theory\npossessing the (spontaneously broken) superconformal symmetry\n$OSp(6|4)$. The model is superconformally invariant in spite of the\npresence on the $d=3$ worldvolume of the dynamical Abelian vector\nfield, since the latter is coupled to the $3d$ dilaton field\nassociated with the radial direction of $AdS_4$. The superconformal\ninvariance is spontaneously broken by a non--zero expectation value\nof the dilaton. An ${\\mathcal N=3}$ superfield model with similar\nsymmetry properties was considered in the Appendix of\n\\cite{Buchbinder:2008vi}. To establish the explicit relation between\n the two models one should extract from the\nsuperfield action of \\cite{Buchbinder:2008vi} the component terms\ndescribing its physical sector and compare the result with\ncorresponding terms in the D2--brane action.\n\n\\section{Conclusion}\nIn this paper we have considered the gauge--fixing of\nkappa--symmetry of the superparticle, superstring and D2--brane\nactions in the complete $AdS_4\\times CP^3$ superspace which is\nsuitable, in particular, for studying regions of these theories that\nare not reachable by partially kappa--symmetry gauge fixed models\nbased on the supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$. The\nsimplified form of these actions can be used to approach various\nproblems of the $AdS_4\/CFT_3$ correspondence. The gauge fixed form\nof the $AdS_4\\times CP^3$ supergeometry can also be used to consider\nthe actions for higher dimensional D4--, D6-- and D8--branes.\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank Pietro Fr\\'e and Jaume Gomis for\ncollaboration at early stages of this project and for many fruitful\ndiscussions and comments. D.S. is also thankful to Soo--Jong Rey for\nuseful discussions. P.A.G. and D.S. are grateful to the Organizers\nof the Workshop Program ``Fundamental Aspects of Superstring Theory\"\nfor their hospitality at KITP, Santa Barbara, where their research\nwas supported in part by the National Science Foundation under Grant\nNo. PHY05-51164. Work of P.A.G., D.S. and L.W. was partially\nsupported by the INFN Special Initiative TV12. D.S. was also\npartially supported by the INTAS Project Grant 05-1000008-7928, an\nExcellence Grant of Fondazione Cariparo and the grant FIS2008-1980\nof the Spanish Ministry of Science and Innovation.\n\n\\def{}\n\\defC.\\arabic{equation}}\\label{C{A.\\arabic{equation}}\\label{A}\n\\section{Appendix A. Main notation and conventions}\n\\setcounter{equation}0\n\nThe convention for the ten and eleven dimensional metrics is the\n`almost plus' signature $(-,+,\\cdots,+)$. Generically, the tangent\nspace vector indices are labeled by letters from the beginning of\nthe Latin alphabet,  while  letters from the middle of the Latin\nalphabet stand for curved (world) indices. The spinor indices are\nlabeled by Greek letters.\n\n\\defC.2{A.1}\n\\subsection{$AdS_4$ space}\n\n$AdS_4$ is parametrized by the coordinates $x^{\\hat m}$ and its\nvielbeins are $e^{\\hat a}=dx^{\\hat m}\\,e_{\\hat m}{}^{\\hat a}(x)$,\n${\\hat m}=0,1,2,3;$ ${\\hat a}=0,1,2,3$. The $D=4$ gamma--matrices\nsatisfy:\n\\begin{equation}\\label{gammaa}\n\\{\\gamma^{\\hat a},\\gamma^{\\hat b}\\}=2\\,\\eta^{\\hat a\\hat b}\\,,\n\\qquad \\eta^{\\hat a\\hat b}={\\rm diag}\\,(-,+,+,+)\\,,\n\\end{equation}\n\\begin{equation}\\label{gamma5}\n\\gamma^5=i\\gamma^0\\,\\gamma^1\\,\\gamma^2\\,\\gamma^3, \\qquad\n\\gamma^5\\,\\gamma^5=1\\,.\n\\end{equation}\nThe charge conjugation matrix $C$ is antisymmetric, the matrices\n$(\\gamma^{\\hat a})_{\\alpha\\beta}\\equiv (C\\,\\gamma^{\\hat\na})_{\\alpha\\beta}$ and $(\\gamma^{\\hat a\\hat\nb})_{\\alpha\\beta}\\equiv(C\\,\\gamma^{\\hat a\\hat b})_{\\alpha\\beta}$ are\nsymmetric and $\\gamma^5_{\\alpha\\beta}\\equiv\n(C\\gamma^5)_{\\alpha\\beta}$ is antisymmetric, with\n$\\alpha,\\beta=1,2,3,4$ being the indices of a 4--dimensional spinor\nrepresentation of $SO(1,3)$ or $SO(2,3)$.\n\n\\defC.2{A.2}\n\\subsection{$CP^3$ space}\n\n$CP^3$ is parametrized by the coordinates $y^{m'}$ and its vielbeins\nare $e^{a'}=dy^{m'}e_{m'}{}^{a'}(y)$, ${m'}=1,\\cdots,6;$\n${a'}=1,\\cdots,6$. The $D=6$ gamma--matrices satisfy:\n\\begin{equation}\\label{gammaa'}\n\\{\\gamma^{a'},\\gamma^{b'}\\}=2\\,\\delta^{{a'}{b'}}\\,,\\qquad \\delta^{a'b'}={\\rm\ndiag}\\,(+,+,+,+,+,+)\\,,\n\\end{equation}\n\\begin{equation}\\label{gamma7}\n\\gamma^7={i\\over{6!}}\\,\\epsilon_{\\,a_1'a_2'a_3'a_4'a_5'a_6'}\\,\\gamma^{a_1'}\\cdots \\gamma^{a_6'} \\qquad\n\\gamma^7\\,\\gamma^7=1\\,.\n\\end{equation}\nThe charge conjugation matrix $C'$ is symmetric and the matrices\n$(\\gamma^{a'})_{\\alpha'\\beta'}\\equiv\n(C\\,\\gamma^{a'})_{\\alpha'\\beta'}$ and\n$(\\gamma^{a'b'})_{\\alpha'\\beta'}\\equiv(C'\\,\\gamma^{a'b'})_{\\alpha'\\beta'}$\nare antisymmetric, with $\\alpha',\\beta'=1,\\cdots,8$ being the\nindices of an 8--dimensional spinor representation of $SO(6)$.\n\n\\defC.2{A.3}\n\\subsection{ Type IIA  $AdS_4\\times CP^3$ superspace}\n\nThe type IIA superspace whose bosonic body is $AdS_4\\times CP^3$ is\nparametrized by 10 bosonic coordinates $X^M=(x^{\\hat m},\\,y^{m'})$\nand 32-fermionic coordinates\n$\\Theta^{\\underline\\mu}=(\\Theta^{\\mu\\mu'})$\n($\\mu=1,2,3,4;\\,\\mu'=1,\\cdots,8$). These  combine into the\nsuperspace supercoordinates $Z^{\\cal M}=(x^{\\hat\nm},\\,y^{m'},\\,\\Theta^{\\mu\\mu'})$. The type IIA supervielbeins are\n\\begin{equation}\\label{IIAsv}\n{\\mathcal E}^{\\mathcal A}=dZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal\nM}{}^{\\mathcal A}(Z)=({\\mathcal E}^{A},\\,{\\mathcal\nE}^{\\underline\\alpha})\\,,\\qquad {\\mathcal E}^{A}(Z)=({\\mathcal\nE}^{\\hat a},\\,{\\mathcal E}^{a'})\\,,\\qquad {\\mathcal\nE}^{\\underline\\alpha}(Z)={\\mathcal E}^{\\alpha\\alpha'}\\,.\n\\end{equation}\n\\defC.2{A.4}\n\\subsection{Superspace constraints}\nIn our conventions the superspace constraint on the bosonic part of the torsion is\n\\begin{equation}\nT^A=-i\\mathcal E\\Gamma^A\\mathcal E+i\\mathcal E^A\\,\\mathcal\nE\\lambda+\\frac{1}{3}{\\mathcal E}^A\\,\\mathcal E^B\\nabla_B\\phi\\,,\n\\end{equation}\nwhile the constraints on the RR and NS--NS field strengths are\n\\begin{eqnarray}\nF_2&=&-i\\,e^{-\\phi}\\,\\mathcal E\\Gamma_{11}\\mathcal E\n+2i\\,e^{-\\phi}\\,\\mathcal E^A\\,\\mathcal E\\Gamma_A\\Gamma_{11}\\lambda+\\frac{1}{2}\\mathcal E^B\\mathcal E^A\\,F_{AB}\\,,\\\\\nF_4&=&-\\frac{i}{2}\\,e^{-\\phi}\\,{\\mathcal E}^B{\\mathcal E}^A\\,\\mathcal E\\Gamma_{AB}\\mathcal E\n+\\frac{1}{4!}{\\mathcal E}^D{\\mathcal E}^C{\\mathcal E}^B{\\mathcal E}^A\\,F_{ABCD}\n\\,,\\\\\nH_3&=&\n-i{\\mathcal E}^A\\,\\mathcal E\\Gamma_A\\Gamma_{11}\\mathcal E\n+i{\\mathcal E}^B{\\mathcal E}^A\\,\\mathcal E\\Gamma_{AB}\\Gamma^{11}\\lambda\n+\\frac{1}{3!}{\\mathcal E}^C{\\mathcal E}^B{\\mathcal E}^A\\,H_{ABC}\\,.\n\\end{eqnarray}\nThese differ from the conventional string frame constraints by the $\\lambda$--term in $T^A$ and related terms in $F_2$, $F_4$ and $H_3$. This is\na consequence of the dimensional reduction from eleven dimensions. They can be brought to a more conventional form by shifting the fermionic supervielbein $\\mathcal E^{\\underline\\alpha}$ by $-\\frac{1}{2}\\mathcal E^A(\\Gamma_A\\lambda)^{\\underline\\alpha}$ accompanied by a related shift in the connection.\n\\\\\n\\\\\n{\\bf The $D=10$ gamma--matrices $\\Gamma^A$} are given by\n\\begin{eqnarray}\\label{Gamma10}\n&\\{\\Gamma^A,\\,\\Gamma^B\\}=2\\eta^{AB},\\qquad\n\\Gamma^{A}=(\\Gamma^{\\hat a},\\,\\Gamma^{a'})\\,,\\nonumber\\\\\n&\\\\\n&\\Gamma^{\\hat a}=\\gamma^{\\hat a}\\,\\otimes\\,{\\bf 1},\\qquad\n\\Gamma^{a'}=\\gamma^5\\,\\otimes\\,\\gamma^{a'},\\qquad\n\\Gamma^{11}=\\gamma^5\\,\\otimes\\,\\gamma^7,\\qquad a=0,1,2,3;\\quad\na'=1,\\cdots,6\\,. \\nonumber\n\\end{eqnarray}\nThe charge conjugation matrix is ${\\mathcal C}=C\\otimes C'$.\n\nThe fermionic variables $\\Theta^{\\underline\\alpha}$ of IIA\nsupergravity carrying 32--component spinor indices of $Spin(1,9)$,\nin the $AdS_4\\times CP^3$ background and for the above choice of the\n$D=10$ gamma--matrices, naturally split into 4--dimensional\n$Spin(1,3)$ indices and 8--dimensional spinor indices of $Spin(6)$,\ni.e. $\\Theta^{\\underline\\alpha}=\\Theta^{\\alpha\\alpha'}$\n($\\alpha=1,2,3,4$; $\\alpha'=1,\\cdots,8$).\n\n\\defC.2{A.5}\n\\subsection{$24+8$ splitting of $32$ $\\Theta$}\n\n24 of  $\\Theta^{\\underline\\alpha}=\\Theta^{\\alpha\\alpha'}$ correspond\nto the unbroken supersymmetries of the $AdS_4\\times CP^3$\nbackground. They are singled out by a projector introduced in\n\\cite{Nilsson:1984bj} which is constructed using the $CP^3$ K\\\"ahler\nform $J_{a'b'}$ and seven $8\\times 8$ antisymmetric gamma--matrices\n(\\ref{gammaa'}). The $8\\times 8$ projector matrix has the following\nform\n\\begin{equation}\\label{p6}\n{\\mathcal P}_{6}={1\\over 8}(6-J)\\,,\n\\end{equation}\nwhere the $8\\times 8$  matrix\n\\begin{equation}\\label{J}\nJ=-iJ_{a'b'}\\,\\gamma^{a'b'}\\,\\gamma^7 \\qquad {\\rm such~ that} \\qquad\nJ^2= 4J+12\n\\end{equation}\nhas six eigenvalues $-2$ and two eigenvalues $6$, \\emph{i.e.} its\ndiagonalization results in\n\\begin{equation}\\label{Jdia}\nJ=\\hbox{diag}(-2,-2,-2,-2,-2,-2,6,6)\\,.\n\\end{equation}\nTherefore, the projector (\\ref{p6}) when acting on an 8--dimensional\nspinor annihilates 2 and leaves 6 of its components, while the\ncomplementary projector\n\\begin{equation}\\label{p2}\n{\\mathcal P}_{2}={1\\over 8}(2+J)\\,,\\qquad\n\\mathcal{P}_2+\\mathcal{P}_6=\\mathbf 1\n\\end{equation}\nannihilates 6 and leaves 2 spinor components.\n\nThus the spinor\n\\begin{equation}\\label{24}\n\\vartheta^{\\alpha\\alpha'}=({\\mathcal P}_6\\,\\Theta)^{\\alpha\\alpha'} \\qquad \\Longleftrightarrow \\qquad\n\\vartheta^{\\alpha a'}\\, \\qquad a'=1,\\cdots, 6\n\\end{equation}\nhas 24 non--zero components and the spinor\n\\begin{equation}\\label{8}\n\\upsilon^{\\alpha\\alpha'}=({\\mathcal P}_2\\,\\Theta)^{\\alpha\\alpha'}\\qquad \\Longleftrightarrow \\qquad\n\\upsilon^{\\alpha i}\\, \\qquad i=1,2\n\\end{equation}\nhas 8 non--zero components. The latter corresponds to the eight\nsupersymmetries broken by the $AdS_4\\times CP^3$ background.\n\nTo avoid confusion, let us note that the index $a'$ on spinors is\ndifferent from the same index on bosonic quantities. They are\nrelated by the usual relation between vector and spinor\nrepresentations, \\emph{i.e.} given two $Spin(6)$ spinors\n$\\psi_1^{\\alpha'}$ and $\\psi_2^{\\alpha'}$, projected as in\n(\\ref{24}), their bilinear combination $v^{a'}=\\psi_1\\mathcal\nP_6\\gamma^{a'}\\mathcal P_6\\psi_2=\\psi_1^{b'}(\\mathcal\nP_6\\gamma^{a'}\\mathcal P_6)_{b'c'}\\psi_2^{c'}$ transforms as a\n6--dimensional 'vector'.\n\n\\defC.\\arabic{equation}}\\label{C{B.\\arabic{equation}}\\label{B}\n\\section{Appendix B. $OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset realization\nand other ingredients of the $(10|32)$--dimensional $AdS_4\\times\nCP^3$ superspace}\n\\setcounter{equation}0\n\nThe supervielbeins and the superconnections of the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset which appear in the\ndefinition of the geometric and gauge quantities of the $AdS_4\\times\nCP^3$ superspace in Section \\ref{superspace} are\n\\begin{equation}\\label{cartan24}\n\\begin{aligned}\nE^{\\hat a}&=e^{\\hat a}(x)+4i\\vartheta\\gamma^{\\hat\na}\\,{{\\sinh^2{{\\mathcal M}_{24}\/ 2}}\\over{\\mathcal M}^2_{24}}\\,\nD_{24}\\vartheta,\\\\\nE^{a'}&=e^{a'}(y)+4i\\vartheta\\gamma^{a'}\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}_{24}\/2}}\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\n\\\\\nE^{\\alpha a'}&=\\left({{\\sinh{\\mathcal M}_{24}}\\over{\\mathcal\nM}_{24}}D_{24}\\vartheta\\right)^{\\alpha a'},\\\\\n\\Omega^{\\hat a\\hat b}&=\\omega^{\\hat a\\hat b}(x)+\\frac{8}{R}\\vartheta\\gamma^{\\hat a\\hat b}\\gamma^5\\,\n{{\\sinh^2{{\\mathcal M}_{24}\/2}}\\over{\\mathcal M}_{24}^2}D_{24}\\vartheta\\,,\\\\\n\\Omega^{a'b'}&=\\omega^{a'b'}(y)-\\frac{4}{R}\n\\vartheta(\\gamma^{a'b'}-iJ^{a'b'}\\gamma^7)\\gamma^5\\,{{\\sinh^2{{\\mathcal M}_{24}\/2}}\n\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\\\\\nA&=\\frac{1}{8}J_{a'b'}\\Omega^{a'b'}=A(y)+\\frac{4i}{R}\\,\\vartheta\\gamma^7\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}_{24}\/2}}\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{M24}\nR\\,({\\mathcal M}_{24}^2)^{\\alpha a'}{}_{\\beta b'}=\n4\\vartheta^{\\alpha}_ {b'}\\,(\\vartheta^{a'}\\gamma^5)_\\beta\n-4\\delta^{a'}_{b'}\\vartheta^{\\alpha c'}(\\vartheta\\gamma^5)_{\\beta\nc'} -2(\\gamma^5\\gamma^{\\hat a}\\vartheta)^{\\alpha\na'}(\\vartheta\\gamma_{\\hat a})_{\\beta b'} -(\\gamma^{\\hat a\\hat b}\\vartheta)^{\\alpha\na'}(\\vartheta\\gamma_{\\hat a\\hat b}\\gamma^5)_{\\beta b'}\\,.\n\\end{equation}\nThe derivative appearing in the above equations is defined as\n\\begin{equation}\\label{D24}\nD_{24}\\vartheta={\\mathcal P_6}\\,(d +\\frac{i}{R}\\,e^{\\hat\na}\\gamma^5\\gamma_{\\hat a}\n+\\frac{i}{R}e^{a'}\\gamma_{a'}-\\frac{1}{4}\\omega^{\\hat a\\hat\nb}\\gamma_{\\hat a\\hat b}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,,\n\\end{equation}\nwhere $e^{\\hat a}(x)$, $e^{a'}(y)$, $\\omega^{\\hat a\\hat b}(x)$, $\\omega^{a'b'}(y)$\nand $A(y)$ are the vielbeins and connections of the bosonic\nsolution. The $U(3)$--connection $\\Omega^{a'b'}$ satisfies the\ncondition\n\\begin{equation}\n{(P^{-})_{a'b'}}^{c'd'}\\Omega_{c'd'}=\\frac{1}{2}\\,({\\delta_{[a'}}^{c'}\\,{\\delta_{b']}}^{d'}\\,-\\,\n{J_{[a'}}^{c'}\\,{J_{b']}}^{d'})\\Omega_{c'd'}=0\\,,\n\\end{equation}\nwhere $J_{a'b'}$ is the K\\\"ahler form on $CP^3$.\n\n\\defC.2{B.1}\n\\subsection{Other quantities appearing in the definition of the\n$AdS_4\\times CP^3$ superspace of Section \\ref{superspace}}\n\n\\begin{equation}\\label{M}\nR\\,({\\mathcal M}^2)^{\\alpha i}{}_{\\beta j}= 4(\\varepsilon\\upsilon)^{\\alpha\ni}(\\upsilon\\varepsilon\\gamma^5)_{\\beta j} -2(\\gamma^5\\gamma^{\\hat\na}\\upsilon)^{\\alpha i}(\\upsilon\\gamma_{\\hat a})_{\\beta j} -(\\gamma^{\\hat\na\\hat b}\\upsilon)^{\\alpha i}(\\upsilon\\gamma_{\\hat a\\hat\nb}\\gamma^5)_{\\beta j}\\,,\n\\end{equation}\n\\begin{equation}\n(m^2)^{ij}=-\\frac{4}{R}\\upsilon^i\\,\\gamma^5\\,\\upsilon^j\\,,\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\n\\Lambda_{\\hat a}{}^{\\hat b}&=\n\\delta_{\\hat a}{}^{\\hat b}-\\frac{R^2}{k^2l_p^2}\\,\\cdot\\,\n\\frac{e^{-\\frac{2}{3}\\phi}}{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}\\,{E_{7\\hat a}}\\,E_7{}^{\\hat b}\\,,\n\\\\\n\\\\\nS_{\\underline\\beta}{}^{\\underline\\alpha}&=\n\\frac{e^{-\\frac{1}{3}\\phi}}{\\sqrt2}\\left(\\sqrt{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}-{R\\over{kl_p}}\\,\n\\frac{E_7{}^{\\hat a}\\,\\Gamma_{\\hat a}\\Gamma_{11}}{\\sqrt{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}}\n\\,\\right)_{\\underline\\beta}{}^{\\underline\\alpha}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{phiE7}\\begin{aligned}\nE_7{}^{\\hat a}(\\upsilon)&=-\\frac{8i}{R}\\,\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal\nM}\/ 2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}\\,,\n\\\\\n\\Phi(\\upsilon)&= 1+\\frac{8}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\upsilon\\,.\n\\end{aligned}\n\\end{equation}\nLet us emphasise that the $SO(2)$ indices $i,j=1,2$ are raised and\nlowered with the unit matrices $\\delta^{ij}$ and $\\delta_{ij}$ so\nthat there is actually no difference between the upper and the\nlower $SO(2)$ indices, $\\varepsilon_{ij}=-\\varepsilon_{ji}$,\n$\\varepsilon^{ij}=-\\varepsilon^{ji}$ and\n$\\varepsilon^{12}=\\varepsilon_{12}=1$.\n\n\\defC.\\arabic{equation}}\\label{C{C.\\arabic{equation}}\\label{C}\n\\section{Appendix C. Identities for the kappa-projected fermions}\n\\setcounter{equation}0\n\n\nWhen the fermionic variables\n$\\Theta^{\\underline\\alpha}=(\\vartheta^{\\alpha a'},\\,\\upsilon^{\\alpha\ni})$ are subject to the constraint (\\ref{kappagauge1}), the\nfollowing identities hold.\n\n\\defC.2{C.1}\n\\subsection{Identities involving $\\upsilon^{\\alpha i}$}\n\\begin{equation}\\label{i1}\n\\upsilon^i\\gamma^5\\upsilon^j=\\upsilon^i\\gamma^3\\upsilon^j=0\\,,\\qquad \\upsilon^{\\alpha i}\\upsilon^{\\beta j}\\delta_{ij}=-\\frac{1}{4}((1\\pm\\gamma)C^{-1})^{\\alpha\\beta}\\upsilon\\ups\\,,\n\\end{equation}\nwhere $\\gamma=\\gamma^{012}$ and $\\upsilon\\ups=\\delta_{ij}\\upsilon^{\\alpha\ni}C_{\\alpha\\beta}\\upsilon^{\\beta j}$.\n\nAnother useful relation is ($\\varepsilon^{012}=-\\varepsilon_{012}=1$)\n\\begin{equation}\\label{gg}\n\\upsilon\\gamma_{ab}d\\upsilon=\\pm\\varepsilon_{abc}\\upsilon\\gamma^cd\\upsilon\\,,\n\\end{equation}\nwhich also holds for the kappa--projected $\\vartheta$ and\n$d\\vartheta$.\n\nUsing eqs. (\\ref{i1}) and (\\ref{gg}) we find that\n\\begin{equation}\n\\upsilon\\varepsilon\\gamma^a\\upsilon\\,\\upsilon\\varepsilon\\gamma_b\\upsilon=\\delta_b^a(\\upsilon\\ups)^2\\,,\n\\qquad \\upsilon\\varepsilon\\gamma^{ac}\\upsilon\\,\\upsilon\\varepsilon\\gamma_{cb}\\upsilon=2\\delta_b^a(\\upsilon\\ups)^2\\,,\n\\end{equation}\n\\begin{equation}\n(m^2)^{ij}=-\\frac{4}{R}\\upsilon^i\\,\\gamma^5\\,\\upsilon^j=0\n\\end{equation}\nand\n\\begin{eqnarray}\n({\\mathcal M}^2\\varepsilon\\upsilon)^{\\alpha i}=0\\,.\n\\end{eqnarray}\n A similar computation shows that\n\\begin{equation}\n\\upsilon\\varepsilon\\gamma^5{\\mathcal M}^2=0.\n\\end{equation}\nIt is also true in general (i.e. without fixing $\\kappa$--symmetry)\nthat\n\\begin{equation}\n{\\mathcal M}^2\\upsilon=0\\,,\\qquad \\upsilon\\gamma^5{\\mathcal M}^2=0.\n\\end{equation}\nUsing the above identities we find that for $\\upsilon$ satisfying\n(\\ref{kappagauge1})\n\\begin{equation}\\label{M2d}\n\\mathcal M^2D\\upsilon\n=\\frac{6i}{R^2}(E^a\\pm\\frac{R}{2}\\Omega^{a3})(\\gamma_a\\upsilon)\\,\\upsilon\\ups\n\\end{equation}\nwhich results in\n\\begin{equation}\\label{vmdv}\n4\\upsilon\\gamma^a \\frac{\\sinh^2(\\mathcal M\/2)}{\\mathcal\nM^2}D\\upsilon=\n\\upsilon\\gamma^a(1+\\frac{1}{12}\\mathcal M^2)D\\upsilon\n=\\upsilon\\gamma^a\\,(d-\\frac{1}{4}\\Omega^{bc}\\gamma_{bc})\\upsilon\n+\\frac{i}{2R^2}(E^a\\pm\\frac{R}{2}\\Omega^{a3})(\\upsilon\\ups)^2\\,,\n\\end{equation}\nwhere $E^a$, $\\Omega^{bc}$ and $\\Omega^{a3}$ are $AdS_4$ components\nof the supervielbein and connection of the supercoset\n$OSp(6|4)\/U(3)\\times SO(1,3)$ defined in eqs. (\\ref{cartan24}) and\nthe matrix ${\\mathcal M}^2$ is defined in eq. (\\ref{M}).\n\nWe also find that\n\\begin{equation}\n4\\upsilon\\varepsilon\\gamma^5{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2}D\\upsilon\n=\\upsilon\\varepsilon\\gamma^5D\\upsilon=\\frac{i}{R}(E^a\\pm\\frac{R}{2}\\Omega^{a3})\\upsilon\\varepsilon\\gamma_a\\upsilon\\,.\n\\end{equation}\n\n\\defC.2{C.2}\n\\subsection{Identities involving $\\vartheta^{\\alpha a'}$ and the simplified\nform of the \\\\ $OSp(6|4)\/U(3)\\times SO(1,3)$ supergeometry}\n\nUsing the definition of $\\mathcal M_{_{24}}$, eq. (\\ref{M24}), and\nthe fact that\n\\begin{equation}\n[\\gamma^{012},\\gamma^{a'}]=0\n\\end{equation}\nwe find that\n\\begin{equation}\n(\\vartheta\\gamma'\\gamma^5{\\mathcal M}_{_{24}}^2)_{\\beta b'} =0\n\\qquad\n({\\mathcal M}_{_{24}}^2\\gamma'\\vartheta)^{\\alpha a'}=0\\,,\n\\end{equation}\nwhere $\\gamma'$ is any product of the gamma-matrices that commutes\nwith $\\gamma=\\gamma^{012}$,\n\\emph{e.g.} any product of $\\gamma^{a'}$  and $\\gamma^{a}$. A\nslightly longer computation, using the fact that\n\\begin{equation}\n\\gamma^3\\vartheta=\\pm\\gamma^3\\gamma^{012}\\vartheta=\\pm i\\gamma^5\\vartheta\\,,\\qquad\\vartheta\\gamma^3=\\mp i\\vartheta\\gamma^5\\qquad\\mbox{for}\\quad\\vartheta=\\frac{1}{2}(1\\pm\\gamma)\\vartheta\\,,\n\\end{equation}\nshows that with this projection of the $\\vartheta$s\n\\begin{equation}\n\\mathcal M_{_{24}}^4=0\\,.\n\\end{equation}\nUsing the identity\n\\begin{equation}\n\\vartheta^{\\alpha a'}\\vartheta^{\\beta b'}\\delta_{a'b'}=-\\frac{1}{4}((1\\pm\\gamma)C^{-1})^{\\alpha\\beta}\\vartheta\\vartheta\\,,\n\\end{equation}\nwhere $\\vartheta\\vartheta\\equiv\\vartheta^{\\alpha\na'}C_{\\alpha\\beta}\\vartheta^{\\beta b'}\\,\\delta_{a'b'}\\,,$\n one can further show that\n\\begin{equation}\n({\\mathcal M}^2_{_{24}}D_{_{24}}\\vartheta)^{\\alpha a'}\n=\\frac{6i}{R^2}(e^b\\pm\\frac{R}{2}\\omega^{b3})(\\gamma_b\\vartheta)^{\\alpha\na'}\\,\\vartheta\\vartheta,\n\\end{equation}\nwhere the covariant derivative $D_{24}$, defined in\n(\\ref{D24}), becomes\n\\begin{equation}\nD_{24}\\vartheta=\\mathcal P_6(d\n+\\frac{i}{R}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\gamma^5\\gamma_a\n\\mp\\frac{1}{R}\\,e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,.\n\\end{equation}\nThis gives\n\\begin{eqnarray}\n\\vartheta\\gamma^a(1+\\frac{1}{12}{\\mathcal M}^2_{_{24}})D_{_{24}}\\vartheta\n&=&\n\\vartheta\\gamma^aD_{24}\\vartheta+\\frac{i}{2R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2\\,.\n\\end{eqnarray}\nUsing the above expressions one finds that the form of the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ geometrical objects (\\ref{cartan24})\nsimplify to\n\\begin{equation}\n\\begin{aligned}\nE^a&=e^a(x)+i\\vartheta\\gamma^aD_{24}\\vartheta-\\frac{1}{2R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2,\n\\\\\nE^3&=e^3(x),\\\\\nE^{a'}&=e^{a'}(y)-\\frac{1}{R}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta\\gamma^{a'}\\gamma_a\\vartheta\n\\\\\nE^{\\alpha a'}&=\n\\left(D_{_{24}}\\vartheta\\right)^{\\alpha a'}\n+\\frac{i}{R^2}(e^b\\pm\\frac{R}{2}\\omega^{b3})(\\gamma_b\\vartheta)^{\\alpha\na'}\\,\\vartheta\\vartheta,\n\\\\\n\\Omega^{ab}&=\\omega^{ab}(x)+\\frac{2i}{R^2}(e^c\\pm\\frac{R}{2}\\omega^{c3})\\vartheta\\gamma^{ab}{}_c\\vartheta,\n\\\\\n\\Omega^{a3}&=\\omega^{a3}(x)\n\\mp\\frac{2i}{R}\\vartheta\\gamma^aD_{24}\\vartheta\n\\pm\\frac{1}{R^3}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2,\n\\\\\n\\Omega^{a'b'}&=\\omega^{a'b'}(y)-\\frac{i}{R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta(\\gamma^{a'b'}-iJ^{a'b'}\\gamma^7)\\gamma_a\\vartheta,\n\\\\\nA&=A(y)-\\frac{1}{R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta\\gamma^7\\gamma_a\\vartheta\n\\,,\n\\end{aligned}\n\\end{equation}\nand in particular\n\\begin{equation}\nE^a\\pm\\frac{R}{2}\\Omega^{a3}=e^a(x)\\pm\\frac{R}{2}\\omega^{a3}(x)\\,.\n\\end{equation}\nThus, in the chosen $\\kappa$--symmetry gauge the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset geometry depends on the\nfermionic coordinates only up to the 4th power.\n\nNote that in all the above expressions the components $e^a(x)$\n$(a=0,1,2)$ and ${R\/2}\\,\\omega^{a3}(x)$ of the $AdS_4$ vielbein and\nconnection appear only in the combination $e^a(x)\\pm\n{R\/2}\\,\\omega^{a3}(x)$. This combination has a very clear\ngeometrical meaning. In the case, when the indices $a=0,1,2$ label\nthe directions of the 3d Minkowski slice of the $AdS_4$, $e^a(x)\\pm\n{R\/2}\\,\\omega^{a3}(x)$ corresponds to the generator\n$\\Pi_a=P_a\\mp{1\/2}\\,M_{a3}$ of the Poincar\\'e translations\n($[\\Pi_a,\\Pi_b]=0$) along the 3d Minkowski boundary which is the\nlinear combination of boosts and Lorentz rotations in $AdS_4$ (see\n\\cite{Pasti:1998tc} for more details). More precisely, $e^a(x)-\n{R\/2}\\,\\omega^{a3}(x)$ corresponds to the Poincar\\'e translation,\nwhile $e^a(x)+{R\/2}\\,\\omega^{a3}(x)$ corresponds to the conformal\nboosts in $M_3$, or vice versa, depending on the orientation.\n\nWhen the $AdS_4$ metric is chosen in the form (\\ref{ads4metric11})\nthe vielbein $e^a(x)$ and the connection $\\omega^{a3}(x)$ are\nproportional to each other, namely,\n\\begin{equation}\\label{eaoa3}\ne^a=-{R\\over 2}\\,\\omega^{a3}.\n\\end{equation}\nActually, this relation can be imposed for any\nform of the metric by performing an appropriate $SO(1,3)$\ntransformation of the $AdS_4$ vielbein and connection.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{Intro}\n\\setcounter{equation}{0}\n\n\\ale{The COVID-19 outbreak that deeply affected the world starting from the first months of 2020 led to a new, strong interest by researchers towards the development of mathematical models of infectious diseases, allowing the assessment of different scenarios and aiming at assisting the complex political decision-making process during the pandemic.\nAs a consequence, many papers were recently published, proposing interesting modeling ideas (see, e.g., \\cite{Albi,Calleri,Gatto,Giordano,Jha,Linka,Parolini,Wang,Zohdi}), often based on compartmental models, where the considered population is divided into ``compartments'' based on their qualitative characteristics (like, e.g., ``susceptible'', ``infected'', ``recovered''), with different assumptions about the nature and rate of transfer across compartments. Despite this kind of models do not readily offer the possibility of a multiscale vision (as, e.g., proposed in \\cite{Bellomo}), that would be a preferred feature given the nature of the phenomena to be simulated, they have the advantage of allowing a relatively easy introduction of diffusion terms (in this context, compelling ideas on diffusion models can be found, among others, in \\cite{Bellomo2,Bellomo3,Winkler} and references therein). \nFor a recent overview of mathematical models for virus pandemic, interested readers are referred to the articles included in \\cite{Bellomo4}. Here, a compartmental model able to describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is proposed in \\cite{Bertaglia}. Another recent example where a classical compartmental model is enhanced to take into account infected individuals traveling on lines of fast diffusion is instead given by \\cite{Berestycki}.\nIn fact, compartmental models are typically governed by a system of ordinary differential equations (ODEs) in time, which might possibly be enriched with coupling terms, additional equations, or other approaches to take into account also the spatial variation of the studied problem. An interesting alternative is to resort to compartmental models based on partial differential equations (PDEs), to accurately account for spatial variations at the continuum level.}\n\n\\ale{Following this latter idea, in the present contribution, we consider the PDE-based model introduced}\n\\oldgianni{in \\cite{Vig1, Vig2} \\pcol{(a variation of it with delay terms is discussed in \\cite{Guglielmi})}.\nIn particular, the following system has been considered} \n\\Begin{eqnarray}\n  && \\partial_t s\n  = \\alpha n - (1 - A_0\/n) \\beta_i si - (1 - A_0\/n) \\beta_e se - \\mu s + \\div(n \\nu_s \\nabla s) \n  \\label{Ieqs}\n  \\\\\n  && \\partial_t e\n  = (1 - A_0\/n) \\beta_i si + (1 - A_0\/n) \\beta_e se - \\sigma e - \\phi_e e - \\mu e + \\div(n \\nu_e \\nabla e) \n  \\qquad\n  \\label{Ieqe}\n  \\\\\n  && \\partial_t i\n  = \\sigma e - \\phi_d i - \\phi_r i - \\mu i + \\div(n \\nu_i \\nabla i) \n  \\label{Ieqi}\n  \\\\\n  && \\partial_t r = \\phi_r i + \\phi_e e - \\mu r + \\div(n \\nu_r \\nabla r) \n  \\label{Ieqr}\n  \\\\\n  && \\partial_t d = \\phi_d i \\,.\n  \\label{Ieqd}\n\\End{eqnarray}\n\\oldgianni{In the above equations, the symbols $s$, $e$, $i$, $r$, and $d$ denote\nthe susceptible population, the exposed population, the infected population, the recovered population, \nand the deceased population, respectively,\nand $n:=s+e+i+r$ is the living population.\nNote that $d$ refers only to deaths due to COVID-19. \nDue to the names of the compartments used, this model may be called a susceptible-exposed-infected-recovered-deceased (SEIRD) model.\nMoreover, equations \\accorpa{Ieqs}{Ieqr} are complemented by initial conditions\nand no-flux boundary conditions,\nwhile just an initial condition is associated to~\\eqref{Ieqd}.}\n\n\\oldgianni{In \\cite{Vig1} the authors presented a SEIRD mathematical model based on partial differential equations\ncoupled with a heterogeneous diffusion model. \nThe model was used to describe the spatio-temporal spread of the COVID-19 pandemic \nand to capture dynamics also based on human habits and geographical features. \nTo test the model, the outputs generated by a finite-element solver \nwere compared with measured data over the Italian region of Lombardy \nand the results showed a strong qualitative agreement \nbetween the simulated forecast of the spatio-temporal COVID-19 spread in Lombardy \nand epidemiological data collected at the municipality level.} \n\n\\oldgianni{In \\cite{Vig2} the authors proposed a formulation of compartmental models \nbased on partial differential equations through \\pier{familiar} continuum mechanics concepts, \ninterpreting such models in terms of fundamental equations of balance and compatibility, \njoined by a constitutive relation. \nSuch an interpretation was useful to aid understanding and interdisciplinary collaboration. \nThe authors formally derived the model sensitivity to diffusion, \ndescribed its growth and decay, and established its stability in the $L^1$ norm. \nAttention was paid to an ODE version of the model, \nusing it to derive a basic reproduction number $R_0$ as well as analyzing its spectrum. \nAdditionally, a series of numerical simulations showed the role that numerical methods, diffusion, \nand specific ingredients played in the behavior of the system. \nImplicit models were found to be effective in describing the temporal dynamics of the system, \nand \\pier{second-order} in-time methods in particular.}\n\\ale{The model proposed in \\cite{Vig2} and used for simulating the COVID-19 spread in Lombardy, was further exploited in \\cite{Grave,Grave2}. In particular, finite element simulations were carried out in the context of adaptive mesh refinement and coarsening, forecasting the COVID-19 spread also in the U.S. state of Georgia and in the Brazilian state of Rio de Janeiro. Good agreements with real-world epidemiological data were attained in both time and space.}\n\n\\oldgianni{Coming back to system \\accorpa{Ieqs}{Ieqd},}\nwe note that $d$ can be determined from \\eqref{Ieqd} and the associated initial condition\nwhenever the system of the first four equation in solved.\nThus, since we are just interested in well-posedness,\nwe do not consider the last equation\n\\oldgianni{and study system \\accorpa{Ieqs}{Ieqr}.\nNamely, we consider a slightly modified system.\nIndeed, we replace the terms $1-A_0\/n$ and $\\phi_d i$ appearing in \\accorpa{Ieqs}{Ieqe} and \\eqref{Ieqi}\nby a non-singular one of the form~$A(n)$ and by the product~$\\phi_d in$, respectively.\nOn the contrary, we can treat the degeneracy in the diffusion terms\nthat occur as $n$ approaches zero, by proving that $n$ is bounded away from zero. \nThis is done even in the more general situation of a coefficient of the form~$\\ka(n)$,\nwhere $\\ka$ is continuous and strictly positive on~$(0,+\\infty)$.\nFinally, we assume that the constants $\\nu_s$, $\\nu_e$, $\\nu_i$ and $\\nu_r$ are the same.\nFor this modified model, we prove an existence result under mild assumption on the initial data.\nFurthermore, we give two uniqueness results:\nthe main assumption for the first one is that $\\ka$ is a positive constant,\nwhile the second one requires that the nonlinearities and the initial data are smooth.\nUnder the latter assumptions, we prove a regularity result as well.} \n\n\\pier{We hope that our results could capture the interest for the resulting systems and give rise to new approximations and simulations, in order to deal with and test our variations in the framework of some geographical areas. Moreover, we claim that our arguments for well-posedness provide a serious mathematical validation to this class of SEIR models.}\n\nThe paper is organized as follows. \nIn the next section, we list our assumptions and notations\nand state our results.\nThe proof of \\oldgianni{Theorem~\\ref{Existence}} regarding the existence of a solution \nis given in Section~\\ref{EXISTENCE}\nand is prepared in Section~\\ref{DISCRETE} where an approximating problem\nobtained by time discretization is studied.\nFinally, Section~\\ref{UNIQUENESS} is devoted to the uniqueness of the solution.\n\n\n\\section{Statement of the problem and results}\n\\label{STATEMENT}\n\\setcounter{equation}{0}\n\nIn this section, we state precise assumptions and notations and present our results.\nFirst of all, the subset $\\Omega\\subset{\\mathbb{R}}^3$\n\\oldgianni{(lower-dimensional cases can be treated in the same way)} \nis~assumed to be bounded, connected and smooth.\nThe symbol $\\partial_\\nu$ denotes for the normal derivative on $\\Gamma:=\\partial\\Omega$.\nMoreover, we set for brevity\n\\Begin{equation}\n  Q_t := \\Omega \\times (0,t)\n  \\quad\\hbox{and}\\quad\n  Q := \\Omega \\times (0,T).\n  \\label{defQt}\n\\End{equation}\nIf $X$ is a Banach space, $\\norma\\,\\cdot\\,_X$ denotes both its norm and the norm of~$X^d$.\nThe only exception from this convention on the norms is given\nby the spaces $L^p$ ($1\\leq p\\leq\\infty$) constructed on $(0,T)$,\n$\\Omega$ and~$Q$, \nwhose norms are often denoted by~$\\norma\\,\\cdot\\,_p$,\nand by the space $H$ defined below and its powers,  \nwhose norms are simply denoted by~$\\norma\\,\\cdot\\,$.\nWe~put\n\\Begin{eqnarray}\n  && H := \\Lx 2 \\,, \\quad  \n  V := \\Hx 1 \n  \\quad\\hbox{and}\\quad\n  W := \\graffe{v \\in \\Hx 2: \\ \\partial_\\nu v = 0} .\n  \\label{defspazi}\n\\End{eqnarray} \nMoreover, $V^*$~is the dual space of $V$ and $\\<\\,\\cdot\\,,\\,\\cdot\\,>$ is the dual pairing between $V^*$ and~$V$.\nIn the sequel, we work in the framework of the Hilbert triplet\n$(V,H,V^*)$ \\oldgianni{obtained by identifying $H$ with a subspace of $V^*$ in the usual way}.\nThus, by also using the symbol $(\\,\\cdot\\,,\\,\\cdot\\,)$ for the standard inner product of~$H$\n(this symbol will be used for any power of $H$ as well later~on),\nwe have\n$\\<g,v>=(g,v)$\nfor every $g\\in H$ and $v\\in V$.\n\nNow, we list our assumptions on the structure of our system. \nWe just consider the functions $A$ and $\\ka$ mentioned in the introduction\nand the constants $\\alpha$ and $\\mu$,\nsince all the other constants appearing in \\accorpa{Ieqs}{Ieqr} will be normalized in the following.\nHowever, we keep them for a while, for the reader's convenience.\nWe assume~that\n\\Begin{eqnarray}\n  && \\hbox{$A,\\ka:(0,+\\infty)\\to{\\mathbb{R}}$ are continuous, with }\n  \\non\n  \\\\\n  && \\quad \\hbox{$A$ nonnegative and $\\ka$ strictly positive}\n  \\label{hpAk}\n  \\\\\n  && \\hbox{$\\alpha$ and $\\mu$ are positive constants}\n  \\label{hpam}\n\\End{eqnarray}\n\\Accorpa\\HPstruttura hpAk hpam\nand we observe that a term like $\\ka(n)=n$ is allowed in the equations.\nIn principle, the system we are interested in is the following \n\\Begin{eqnarray}\n  && \\partial_t s + A(n) \\beta_i si + A(n) \\beta_e se + \\mu s - \\div(\\ka(n)\\nabla s)\n  = \\alpha n \n  \\label{eqs}\n  \\\\\n  && \\partial_t e -  A(n) \\beta_i si -  A(n) \\beta_e se + \\sigma e + \\phi_e e + \\mu e - \\div(\\ka(n)\\nabla e) \n  = 0\n  \\qquad\n  \\label{eqe}\n  \\\\\n  && \\partial_t i + \\oldgianni{\\phi_d in} + \\phi_r i + \\mu i - \\div(\\ka(n)\\nabla i) \n  = \\sigma e\n  \\label{eqi}\n  \\\\\n  && \\partial_t r - \\phi_r i - \\phi_e e + \\mu r - \\div(\\ka(n)\\nabla\\oldgianni r) \n  = 0 \n  \\label{eqr}\n\\End{eqnarray}\nwhere each equation is complemented by no-flux boundary conditions and an initial condition.\nHowever, it is more convenient to consider an equivalent system in the unknowns\n$n$, $s$, $i$ and~$h$, where $n$ and $h$ are related to the \\oldgianni{other} unknowns~by\n\\Begin{equation}\n  n := s + e + i + r\n  \\quad\\hbox{and}\\quad\n  h := s + e \\,.\n  \\label{defnh}\n\\End{equation}\nThe new system is obtained from the previous one by summing up all the equations,\nkeeping \\eqref{eqs} and \\eqref{eqi} and adding \\eqref{eqe} to~\\eqref{eqs}.\nHence, it is given~by \n\\Begin{eqnarray}\n  && \\partial_t n + \\phi_d i n - \\div(\\ka(n)\\nabla n) \n  = (\\alpha-\\mu) n\n  \\label{eqprima}\n  \\\\\n  && \\partial_t s + A(n) \\beta_i si + A(n) \\beta_e s (h-s) + \\mu s - \\div(\\ka(n)\\nabla s)\n  = \\alpha n \n  \\label{eqseconda}\n  \\\\\n  && \\partial_t i + \\oldgianni{\\phi_d in} + \\phi_r i + \\mu i - \\div(\\ka(n)\\nabla i) \n  = \\sigma (h-s)\n  \\label{eqterza}\n  \\\\\n  && \\partial_t h + \\mu h +(\\sigma+\\phi_e) h - \\div(\\ka(n)\\nabla h) \n  = \\alpha n + (\\sigma+\\phi_e) s \\,.\n  \\label{eqquarta}\n\\End{eqnarray}\nHowever, as said above, among all of the constants appearing in these equations,\njust $\\alpha$ and $\\mu$ will play a significant role in the mathematical treatment.\nHence, we replace the other constants and even some sums of them by~$1$, without loss of generality.\nAt this point, we are ready to give our notion of solution.\nFor the initial data we require that \n\\Begin{equation}\n  n^0 \\,, s^0 \\,, i^0 \\,, h^0 \\in \\Lx\\infty\n  \\quad \\hbox{satisfy} \\quad\n  \\infn^0 > 0 , \\quad\n  h^0 \\geq s^0 \\geq 0\n  \\quad\\hbox{and}\\quad\n  i^0 \\geq 0\n  \\label{hpdati}\n\\End{equation}\nand we call solution a quadruplet $(n,s,i,h)$ enjoying the requirements\n\\Begin{eqnarray}\n  && n \\,, s \\,, i \\,, h \\in \\H1V^* \\cap \\C0 H \\cap \\L2V \\cap \\LQ\\infty\n  \\qquad\n  \\label{regsoluz}\n  \\\\\n  && \\inf n > 0 ,\\quad\n  h \\geq s \\geq 0 \n  \\quad\\hbox{and}\\quad\n  i \\geq 0\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\label{segni}\n\\End{eqnarray}\n\\Accorpa\\Regsoluz regsoluz segni\nand satisfying, \\checkmmode{a.e.\\ in~$(0,T)$}\\ and for every $v\\in V$, the variational equations \n\\Begin{eqnarray}\n  && \\< \\partial_t n , v >\n  + \\int_\\Omega i n v\n  + \\int_\\Omega \\ka(n) \\nabla n \\cdot \\nabla v\n  = (\\alpha-\\mu) \\int_\\Omega n v\n  \\label{prima}\n  \\\\\n  && \\< \\partial_t s , v >\n  + \\int_\\Omega \\bigl( A(n)si + A(n) s (h-s) + \\mu s \\bigr) v\n  + \\int_\\Omega \\ka(n) \\nabla s \\cdot \\nabla v\n  \\non\n  \\\\\n  && \\quad {} = \\alpha \\int_\\Omega n v\n  \\label{seconda}\n  \\\\\n  \\noalign{\\allowbreak}\n  && \\< \\partial_t i , v >\n  + (1+\\mu) \\int_\\Omega i v\n  + \\int_\\Omega \\ka(n) \\nabla i \\cdot \\nabla v\n  = \\int_\\Omega (h-s) v\n  \\label{terza}\n  \\\\\n  && \\< \\partial_t h , v >\n  + \\oldgianni{\\int_\\Omega n i v}\n  + (1+\\mu) \\int_\\Omega h v\n  + \\int_\\Omega \\ka(n) \\nabla h \\cdot \\nabla v\n  = \\int_\\Omega (\\alpha n + s) v\n  \\qquad\n  \\label{quarta}\n\\End{eqnarray}\nand the initial conditions\n\\Begin{equation}\n  (n,s,i,h)(0) = (n^0,s^0,i^0,h^0) \\,.\n  \\label{cauchy}\n\\End{equation}\n\\Accorpa\\Pbl prima cauchy\n\n\\Begin{remark}\\rm\n\\label{IntPbl}\nWe notice that the regularity $\\C0H$ explicitly stated in \\eqref{regsoluz}\nin fact follows from the other conditions since\n$\\H1V^*\\cap\\L2V\\subset\\C0H$.\nSo we do not mind \\oldgianni{about} it in the following.\nWe also observe that the above variational equations are equivalent \nto their integrated versions with time dependent test functions.\nFor instance, \\eqref{prima} is equivalent~to\n\\Begin{eqnarray}\n  && \\int_0^T \\< \\partial_t n(t) , v(t) > \\, dt\n  + \\int_Q i n v\n  + \\int_Q \\ka(n) \\nabla n \\cdot \\nabla v\n  = (\\alpha-\\mu) \\int_Q n v\n  \\non\n  \\\\\n  && \\quad \\hbox{for every $v\\in\\L2V$}.\n  \\label{intprima}\n\\End{eqnarray}\n\\End{remark}\n\nHere is our \\oldgianni{first} result.\n\\Begin{theorem}\n\\label{Existence}\nAssume \\HPstruttura\\ and \\eqref{hpdati}.\nThen, there exists at least one quadruplet $(n,s,i,h)$\nthat satisfies the conditions \\Regsoluz\\ and solves problem \\Pbl.\n\\End{theorem}\n\nAs for uniqueness, we present two results.\nThe first one regards a particular case and its proof, given in Section~\\ref{UNIQUENESS1}, is elementary.\n\n\\Begin{theorem}\n\\label{Uniqueness1}\nIn addition to \\HPstruttura\\ and \\eqref{hpdati},\nassume that $A$ is locally Lip\\-schitz\\ continuous and that $\\ka$ is a \\oldgianni{positive} constant.\nThen, the solution to problem \\Pbl\\ satisfying \\Regsoluz\\ is unique.\n\\End{theorem}\n\nOur last result regards uniqueness in the case of a non-constant~$\\ka$.\nIts proof, given in Section~\\ref{UNIQUENESS2}, is much more involved and  \nis strictly related to a high regularity of the solution.\nFor this reason, we have to assume that both the nonlinearities and the initial data are much smoother.\n\n\\Begin{theorem}\n\\label{Uniqueness2}\nIn addition to \\HPstruttura\\ and \\eqref{hpdati},\nassume that\n\\Begin{eqnarray}\n  && \\hbox{$A$ is locally Lip\\-schitz\\ continuous and $\\ka$ is a $C^1$ function}\n  \\label{hpnonlin}\n  \\\\\n  && n^0,\\, s^0,\\, i^0,\\, h^0 \\in \\Wx{2,\\infty} \n  \\quad \\hbox{with zero normal derivatives on $\\Gamma$}.\n  \\label{hpdatareg}\n\\End{eqnarray}\nThen, the solution to problem \\Pbl\\ satisfying \\Regsoluz\\ is unique\nand enjoys the following regularity properties\n\\Begin{equation}\n  n,\\, s,\\, i,\\, h \\in \\spazio W{1,p}{\\Lx p} \\cap \\spazio L p{\\Wx{2,p}}\n  \\quad \\hbox{for every $p\\in[1,+\\infty)$}.\n  \\label{highreg}\n\\End{equation}\n\\End{theorem}\n\nThroughout the paper, we make use of\nthe H\\\"older\\ inequality and the Sobolev inequality \nrelated to the continuous embedding $V\\subset L^p(\\Omega)$ with $p\\in[1,6]$\n(since $\\Omega$ is three-dimensional bounded and smooth).\nWe also account for the elementary identity and inequalities\n\\Begin{eqnarray}\n  \\hskip-1cm&& a (a-b)\n  = \\frac 12 \\, a^2\n  + \\frac 12 \\, (a-b)^2\n  - \\frac 12 \\, b^2\n  \\geq \\frac 12 \\, a^2\n  - \\frac 12 \\, b^2\n  \\quad \\hbox{for every $a,b\\in{\\mathbb{R}}$},\n  \\label{elementare}\n  \\\\\n  \\hskip-1cm&& ab\\leq \\delta a^2 + \\frac 1 {4\\delta}\\,b^2\n  \\quad \\hbox{for every $a,b\\in{\\mathbb{R}}$ and $\\delta>0$},\n  \\label{young}\n\\End{eqnarray}\n\\Accorpa\\Elementari elementare young\nand quote \\eqref{young} as the Young inequality.\nFurthermore, we take advantage of the summation by parts formula\n\\Begin{equation}\n  \\somma k0{m-1} a_{k+1} (b_{k+1} - b_k)\n  = a_m b_m - a_1 b_0\n  - \\somma k1{m-1} (a_{k+1} - a_k) b_k\\,,\n  \\label{byparts}\n\\End{equation}\nwhich is valid for arbitrary real numbers $a_1,\\dots,a_m$ and $b_0,\\dots,b_m$.\n\n\n\\section{The discrete problem}\n\\label{DISCRETE}\n\\setcounter{equation}{0}\n\nIn this section, we prepare \\oldgianni{the proof of Theorem~\\ref{Existence}}\nby introducing and solving an approximating problem obtained by time discretization.\nHowever, the structural functions $A$ and $\\ka$ have to satisfy different assumptions\n and the initial data have to be smoother.\nIn the next section, by starting from the original structure and the original initial data,\nwe consider the discrete problem with structural functions and approximating initial data\nconstructed in order to satisfy the assumptions listed below.\n\\oldgianni{Two~constants $\\ka_*$ and $\\ka^*$} and two real functions $\\tilde A$ and $\\tilde\\ka$ defined in the whole of ${\\mathbb{R}}$ \nare given such~that\n\\Begin{eqnarray}\n  && \\oldgianni{\\ka^* \\geq \\ka_* > 0}\n  \\quad\\hbox{and}\\quad\n  \\hbox{$\\tilde A,\\tilde\\ka:{\\mathbb{R}}\\to{\\mathbb{R}}$ are continuous with}\n  \\non\n  \\\\\n  && \\quad\n  \\tilde A(y) \\geq 0 \n  \\quad\\hbox{and}\\quad\n  \\oldgianni{\\ka_* \\leq \\tilde\\ka(y) \\leq \\ka^*}\n  \\quad \\hbox{for every $y\\in{\\mathbb{R}}$}.\n  \\label{hpAkdiscr}\n\\End{eqnarray}\n\n\\Begin{notation}\\rm\n\\label{Akatilde}\nHowever, we prefer to use the lighter symbols $A$ and $\\ka$ \ninstead of the heavy $\\tilde A$ and~$\\tilde\\ka$.\nIndeed, no confusion can arise since the original functions \n$A$ and $\\ka$ introduced in \\eqref{hpAk} will never appear within the section.\n\\End{notation}\nFor a fixed positive integer~$N$, we set $\\tau:=T\/N$.\nThen, the time-discretized problem we are going to study is the following:\ngiven \n\\Begin{equation}\n  \\nz_\\tau \\,, \\sz_\\tau \\,, \\iz_\\tau \\,, \\hz_\\tau \\in V\\cap\\Lx\\infty\n  \\quad \\hbox{with} \\quad\n  \\nz_\\tau \\geq 0 , \\quad\n  \\hz_\\tau \\geq \\sz_\\tau \\geq 0\n  \\quad\\hbox{and}\\quad\n  \\iz_\\tau \\geq 0\n  \\label{hpdatiV}\n\\End{equation}\nwe look for four $(N+1)$-tuples \n\\Begin{eqnarray}\n  && (n_0,n_1,\\dots,n_N), \\\n  (s_0,s_1,\\dots,s_N), \\\n  (i_0,i_1,\\dots,i_N), \\\n  (h_0,h_1,\\dots,h_N) \n  \\non\n  \\\\\n  && \\quad \\hbox{belonging to $(V\\cap\\Lx\\infty)^{N+1}$}\n  \\label{tuples}\n\\End{eqnarray}\nsatisfying, for $k=0,\\dots,N-1$, the variational equations\n\\Begin{eqnarray}\n  && \\bigl( \\frac {n_{k+1}-n_k}\\tau , v \\bigr)\n  + (i_k n_{k+1} , v)\n  + (\\ka(n_{k+1}) \\nablan_{k+1} , \\nabla v)\n  = (\\alpha-\\mu) (n_{k+1} , v)\n  \\label{primak}\n  \\\\\n  && \\bigl( \\frac {s_{k+1}-s_k}\\tau , v \\bigr)\n  + \\bigl( A(n_{k+1})\\skpi_k + A(n_{k+1})s_{k+1}(h_k-s_k) + \\mus_{k+1} , v \\bigr)\n  \\non\n  \\\\\n  && \\quad {}\n  + \\bigl( \\ka(n_{k+1}) \\nablas_{k+1} , \\nabla v \\bigr)\n  = \\alpha (n_{k+1} , v)\n  \\label{secondak}\n  \\\\\n  && \\bigl( \\frac {i_{k+1}-i_k}\\tau , v \\bigr)\n  \\oldgianni{{}+(n_{k+1}i_{k+1} , v)}\n  + (1+\\mu) (i_{k+1} , v)\n  + \\bigl( \\ka(n_{k+1})\\nablai_{k+1} , \\nabla v \\bigr)\n  \\non\n  \\\\\n  && \\quad {}\n  = ( h_{k+1}-s_{k+1} , v)\n  \\qquad\n  \\label{terzak}\n  \\\\\n  && \\bigl( \\frac {h_{k+1}-h_k}\\tau , v \\bigr)\n  + (1+\\mu) (h_{k+1} , v)\n  + \\bigl( \\ka(n_{k+1})\\nablah_{k+1} , \\nabla v \\bigr)\n  \\non\n  \\\\\n  && \\quad {}\n  = (\\alphan_{k+1} + s_{k+1} , v)\n  \\label{quartak}\n\\End{eqnarray}\nall for every $v\\in V$,\nas well as the sign and initial conditions\n\\Begin{eqnarray}\n  && n_k \\geq 0 ,\\quad\n  h_k \\geq s_k \\geq 0 \n  \\quad\\hbox{and}\\quad\n  i_k \\geq 0\n  \\quad \\hbox{for $k=1,\\dots,N$}\n  \\label{segnik}\n  \\\\\n  && n_0 = \\nz_\\tau \\,, \\quad\n  s_0 = \\sz_\\tau \\,, \\quad\n  i_0 = \\iz_\\tau \n  \\quad\\hbox{and}\\quad\n  h_0 = \\hz_\\tau .\n  \\label{discrcauchy}\n\\End{eqnarray}\n\\Accorpa\\Pbltau primak discrcauchy\n\n\\Begin{theorem}\n\\label{Wellposednessdiscr}\nUnder assumptions \\eqref{hpAkdiscr} and \\eqref{hpam} on the structure,\nsuppose that\n\\Begin{equation}\n  \\frac 1\\tau - \\alpha + \\mu > 0 .\n  \\label{hptau}\n\\End{equation}\nThen, if the initial data satisfy \\eqref{hpdatiV}, problem \\Pbltau\\ has a unique solution.\n\\End{theorem}\n\nWe prepare an easy lemma.\n\n\\Begin{lemma}\n\\label{Gianni}\nLet $a_0$ and $b_0$ be positive constants and let $a,b,f\\in\\Lx\\infty$ satisfy \n\\Begin{equation}\n  a \\geq a_0 \\,, \\quad b \\geq b_0 \\quad\\hbox{and}\\quad f \\geq 0 \\quad \\checkmmode{a.e.\\ in~$\\Omega$} .\n  \\label{hpG}\n\\End{equation}\nThen, the problem of finding $u\\in V$ satisfying the variational problem\n\\Begin{equation}\n  \\int_\\Omega a \\nabla u \\cdot \\nabla v\n  + \\int_\\Omega b u v \n  = \\int_\\Omega f v\n  \\quad \\hbox{for every $v\\in V$}\n  \\label{pblG}\n\\End{equation}\nhas a unique solution, and this solution satisfies\n\\Begin{equation}\n  0 \\leq u \\leq b_0^{-1} \\norma f_\\infty \\quad \\checkmmode{a.e.\\ in~$\\Omega$}.\n  \\label{tesiG}\n\\End{equation}\nMoreover, if $\\lambda\\in{\\mathbb{R}}$ and $f\\geq\\lambda\\,b$ \\checkmmode{a.e.\\ in~$\\Omega$}, then $u\\geq\\lambda$ \\checkmmode{a.e.\\ in~$\\Omega$}.\n\\End{lemma}\n\n\\Begin{proof}\nThe existence of a unique solution is clear \nsince the left-hand side\\ of \\eqref{pblG} is an inner product in $V$ that is equivalent to the usual one.\nMoreover, the first inequality of \\eqref{tesiG} is given by the weak maximum principle\nand the same can be said for the last sentence of the statement, \nsince the equation obtained by replacing $f $ by $f-\\lambda\\,b$ is satisfied by~$u-\\lambda\\,$.\nNow, we prove the second inequality of~\\eqref{tesiG}.\n\\oldgianni{%\nWe set $w:=u-b_0^{-1}\\norma f_\\infty$. \nThen, we have for every $v\\in V$\n\\Begin{eqnarray}\n  && \\int_\\Omega a \\nabla w \\cdot \\nabla v\n  + \\int_\\Omega b w v\n  = \\int_\\Omega a \\nabla u \\cdot \\nabla v\n  + \\int_\\Omega b u v\n  - \\int_\\Omega b \\, b_0^{-1} \\norma f_\\infty v\n  \\non\n  \\\\\n  && = \\int_\\Omega \\bigl( f - b \\, b_0^{-1} \\norma f_\\infty \\bigr) v.\n  \\non\n\\End{eqnarray}\nSince $f-b\\,b_0^{-1}\\norma f_\\infty\\leq f-\\norma f_\\infty\\leq0$,\nwe have $w\\leq0$ by the weak maximum principle, \ni.e., the desired inequality.}\n\\End{proof}\n\n\\step\nProof of Theorem~\\ref{Wellposednessdiscr}\n\nIt suffices to prove the following:\nfor $k=0,\\dots,N-1$, if $(n_k,s_k,i_k,h_k)$ belongs to $(V\\cap\\Lx\\infty)^4$ \nand satisfies the inequalities in \\eqref{segnik},\nthen system \\accorpa{primak}{quartak} has a unique solution $(n_{k+1},s_{k+1},i_{k+1},h_{k+1})$\nbelonging to $(V\\cap\\Lx\\infty)^4$ and satisfying the analogous inequalities,\ni.e., $n_{k+1}\\geq0$, $h_{k+1}\\geqs_{k+1}\\geq0$ and $i_{k+1}\\geq0$.\nWe recall \\eqref{hpdatiV} for the case $k=0$,\nfix $k$ and $(n_k,s_k,i_k,h_k)$ as said \nand show that we can find a unique solution to \\accorpa{primak}{quartak} with the proper sign conditions.\nThis is done in two steps.\n\n\\step\nSolution to the first equation\n\nWe introduce the function $K:{\\mathbb{R}}\\to{\\mathbb{R}}$ by setting\n\\Begin{equation}\n  K(y) := \\int_0^y \\ka(z) \\, dz\n  \\quad \\hbox{for $y\\in{\\mathbb{R}}$}\n  \\label{defK}\n\\End{equation}\nand we would like to assume $u:=K(n_{k+1})$ as the new unknown for~\\eqref{primak}.\nDue to \\eqref{hpAkdiscr} for~$\\ka$,\nthe function $K$ is one-to-one, onto and Lip\\-schitz\\ continuous, and its inverse is Lip\\-schitz\\ continuous too.\nThus, equation \\eqref{primak} can be rewritten in term of~$u$.\n\\gianni{Namely, noting that $\\nabla u=\\nabla(K(n_{k+1}))=\\ka(n_{k+1})\\nablan_{k+1}$,\nwe can write it as the variational formulation of the homogeneous Neumann problem for the equation\n\\Begin{equation}\n  \\lambda K^{-1}(u)\n  - \\Delta u \n  = (1\/\\tau)n_k\n  \\quad \\hbox{where} \\quad\n  \\lambda := \\frac 1\\tau + i_k - \\alpha + \\mu \\,.\n  \\label{barbu}\n\\End{equation}\nNotice that every variational solution automatically belongs to $W$ \nand satisfies both \\eqref{barbu} and the homogeneous Neumann condition due to elliptic regularity.\nWe claim that the new problem has a unique solution.\nIndeed, it is equivalent to the minimization of the functional $J:V\\to{\\mathbb{R}}$ given~by\n\\Begin{equation}\n  J(v)\n  := \\frac 12 \\int_\\Omega |\\nabla v|^2\n  + \\int_\\Omega \\lambda \\, \\calK(v)\n  - \\frac 1\\tau \\int_\\Omega n_k \\, v \n  \\label{minimo}\n\\End{equation}\nwhere \n\\Begin{equation}\n  \\calK(r) := \\int_0^r K^{-1}(s) \\, ds\n  \\quad \\hbox{for $r\\in{\\mathbb{R}}$}\n  \\non\n\\End{equation}\nand we show that this problem has a unique solution.\nNotice that $J$ actually is well-defined, \nsince $K^{-1}$ is Lip\\-schitz\\ continuous, $\\lambda\\in\\Lx\\infty$ and $n_k\\in H$. \nMoreover, $J$~is strictly convex and coercive.\nTo see this, it suffices to recall that $\\inf\\lambda>0$ \n(since we are assuming \\eqref{hptau} and $i_k$ is nonnegative)\nand observe that\n$\\calK''=(K^{-1})'=1\/(\\ka\\circ K^{-1})\\geq1\/\\ka^*$.\nHence, the above problem has a unique solution $u$\nand we conclude that $n_{k+1}=K(u)$ is the unique solution to~\\eqref{primak}.\nSince $u\\in W$, we deduce that $u$ is bounded\nso that that $n_{k+1}$ is bounded too.\nTo see that it is nonnegative, it suffices to rearrange \\eqref{primak} and apply Lemma~\\ref{Gianni}.}\n\n\\step\nSolution to the other equations and conclusion\n\nSince $n_{k+1}$ is known, we can solve \\eqref{secondak} for~$s_{k+1}$\nby applying the first part of Lemma~\\ref{Gianni}.\nWe can do the same to find $h_{k+1}$ and $i_{k+1}$ from \\eqref{quartak} and \\eqref{terzak} in this order.\nThe lemma also ensures that $s_{k+1}$, $h_{k+1}$ and $i_{k+1}$ are bounded and nonnegative,\n\\oldgianni{provided we can prove that} $h_{k+1}\\geqs_{k+1}$.\nTo this end, we set $e_k:=h_k-s_k$ and $e_{k+1}=h_{k+1}-s_{k+1}$, \ntake the difference between \\oldgianni{\\eqref{quartak} and \\eqref{secondak}}\nand write it in the form\n\\Begin{eqnarray}\n  && \\int_\\Omega \\ka(n_{k+1}) \\nablae_{k+1} \\cdot \\nabla v\n  + ((1\/\\tau)+1+\\mu) \\int_\\Omega e_{k+1} v\n  \\non\n  \\\\\n  && = \\int_\\Omega \\bigl( A(n_{k+1})\\skpi_k + A(n_{k+1})\\skpe_k + (1\/\\tau)e_k \\bigr) v\n  \\non\n\\End{eqnarray}\nfor every $v\\in V$.\nThen, Lemma~\\ref{Gianni} yields that $e_{k+1}\\geq0$.\nThis completes the proof.\n\n\n\\section{Existence}\n\\label{EXISTENCE}\n\\setcounter{equation}{0}\n\nThis section is devoted to the proof of Theorem~\\ref{Existence}.\nOur argument relies on a priori estimates on the solution to a suitably specified discrete problem\nand on the convergence of proper interpolating functions.\nSo, we introduce some notations concerning interpolation at once.\n\n\\Begin{notation}\\rm\n\\label{Interpol}\nLet $N$ be a positive integer and $Z$ be a Banach space.\nWe set $\\tau:=T\/N$ and $I_k:=((k-1)\\tau,k\\tau)$ for $k=1,\\dots,N$.\nGiven $z=(z_0,z_1,\\dots ,z_N)\\in Z^{N+1}$,\nwe define the piecewise constant and piecewise linear interpolating functions\n\\Begin{equation}\n  \\overline z_\\tau \\in \\spazio L\\infty Z , \\quad\n  \\underline z_\\tau \\in \\spazio L\\infty Z \n  \\quad\\hbox{and}\\quad\n  \\hat z_\\tau \\in \\spazio W{1,\\infty}Z\n  \\non\n\\End{equation}\nby setting \n\\Begin{eqnarray}\n  && \\hskip -2em\n  \\overline z_\\tau(t) = z^k\n  \\quad\\hbox{and}\\quad\n  \\underline z_\\tau(t) = z^{k-1}\n  \\quad \\hbox{for a.a.\\ $t\\in I_k$, \\ $k=1,\\dots,N$},\n  \\label{pwconstant}\n  \\\\\n  && \\hskip -2em\n  \\hat z_\\tau(0) = z_0\n  \\quad\\hbox{and}\\quad\n  \\partial_t\\hat z_\\tau(t) = \\frac {z^k-z^{k-1}} \\tau\n  \\quad \\hbox{for a.a.\\ $t\\in I_k$, \\ $k=1,\\dots,N$}.\n  \\qquad\n  \\label{pwlinear}\n\\End{eqnarray}\n\\End{notation}\n\nFor the reader's convenience,\nwe summarize the relations between the finite set of values\nand the interpolating functions in the following proposition,\nwhose proof follows from straightforward\\ computation:\n\n\\Begin{proposition}\n\\label{Propinterp}\nWith Notation~\\ref{Interpol}, we have that\n\\Begin{eqnarray}\n  && \\norma{\\overline z_\\tau}_{\\spazio L\\infty Z}\n  = \\max_{k=1,\\dots,N} \\norma{z^k}_Z \\,, \\quad\n  \\norma{\\underline z_\\tau}_{\\spazio L\\infty Z}\n   = \\max_{k=0,\\dots,N-1} \\norma{z^k}_Z\\,,\n  \\label{ouLinftyZ}\n  \\\\\n  && \\norma{\\partial_t\\hat z_\\tau}_{\\spazio L\\infty Z}\n  = \\max_{0\\leq k\\leq N-1} \\norma{(z^{k+1}-z^k)\/\\tau}_Z\\,,\n  \\label{dtzLinftyZ}\n  \\\\\n  \\noalign{\\allowbreak}\n  && \\norma{\\overline z_\\tau}_{\\L2Z}^2\n  = \\tau \\somma k1N \\norma{z^k}_Z^2 \\,, \\quad\n  \\norma{\\underline z_\\tau}_{\\L2Z}^2\n  = \\tau \\somma k0{N-1} \\norma{z^k}_Z^2 \\,,\n  \\label{ouLdueZ}\n  \\\\\n  \\noalign{\\allowbreak}\n  && \\norma{\\partial_t\\hat z_\\tau}_{\\L2Z}^2\n  = \\tau \\somma k0{N-1} \\norma{(z^{k+1}-z^k)\/\\tau}_Z^2\\,, \n  \\label{dtzLdueZ}\n  \\\\\n  && \\norma{\\hat z_\\tau}_{\\spazio L\\infty Z}\n  = \\max_{k=1,\\dots,N} \\max\\{\\norma{z^{k-1}}_Z,\\norma{z^k}_Z\\}\n  = \\max\\{\\norma{z_0}_Z,\\norma{\\overline z_\\tau}_{\\spazio L\\infty Z}\\}\\,,\n  \\qquad\\qquad\n  \\label{hzLinftyZ}\n  \\\\\n  && \\norma{\\hat z_\\tau}_{\\L2Z}^2\n  \\leq \\tau \\somma k1N \\bigl( \\norma{z^{k-1}}_Z^2 + \\norma{z^k}_Z^2 \\bigr)\n  \\leq \\tau \\norma{z_0}_Z^2\n  + 2 \\norma{\\overline z_\\tau}_{\\L2Z}^2 \\,.\n  \\label{hzLdueZ}\n\\End{eqnarray}\nMoreover, it holds that\n\\Begin{eqnarray}\n  && \\norma{\\overline z_\\tau-\\hat z_\\tau}_{\\spazio L\\infty Z}\n  = \\max_{k=0,\\dots,N-1} \\norma{z^{k+1}-{z^k}}_Z\n  = \\tau \\, \\norma{\\partial_t\\hat z_\\tau}_{\\spazio L\\infty Z}\\,,\n  \\qquad\n  \\label{diffLinfty}\n  \\\\\n  && \\norma{\\overline z_\\tau-\\hat z_\\tau}_{\\L2Z}^2\n  = \\frac \\tau 3 \\somma k0{N-1} \\norma{z^{k+1}-z^k}_Z^2\n  \\non\n  \\\\\n  && \\oldgianni{{}= \\frac 13 \\, \\norma{\\overline z_\\tau - \\underline z_\\tau}_{\\L2Z}^2}\n  = \\frac {\\tau^2} 3 \\, \\norma{\\partial_t\\hat z_\\tau}_{\\L2Z}^2\\,,\n  \\label{diffLdue}\n\\End{eqnarray}\nand similar identities for the difference $\\underline z_\\tau-\\hat z_\\tau$.\nFinally, we have that\n\\Begin{eqnarray}\n  && \\tau \\somma k0{N-1} \\norma{(z^{k+1}-z^k)\/\\tau}_Z^2 \n  \\leq \\norma{\\partial_t z}_{\\L2Z}^2\n  \\non\n  \\\\\n  && \\quad \\hbox{if $z\\in\\H1Z$\\quad\\hbox{and}\\quad $z^k=z(k\\tau)$ for $k=0,\\dots,N$}.\n  \\label{interpH1Z}\n\\End{eqnarray}\n\\End{proposition}\n\nWe are now ready to properly specify the discrete problem.\nNamely, starting from $A$ and $\\ka$ as in~\\eqref{hpAk},\nwe introduce new functions $\\tilde A$ and $\\tilde\\ka$ obtained by a truncation operator $\\tilde T$ \nto be used in the discrete problem.\nWe~set\n\\Begin{eqnarray}\n  && n^* := e^{2T(\\alpha-\\mu)^+} \\norman^0_\\infty \\,, \\quad\n  s^* := \\normas^0_\\infty + T\\alphan^*\n  \\non\n  \\\\\n  && \\quad\n  h^* := \\normah^0_\\infty + T(\\alphan^*+s^*) \\quad\n  i^* := \\norma{i_0}_\\infty + T(h^*+s^*) \\quad\n  \\non\n  \\\\\n  && \\quad\\hbox{and}\\quad\n  n_* := e^{-T(i^*+(\\mu-\\alpha)^+)} \\infn^0\n  \\label{defnstar}\n\\End{eqnarray}\nand define $\\tilde A,\\tilde\\ka:{\\mathbb{R}}\\to{\\mathbb{R}}$ by setting for $y\\in{\\mathbb{R}}$\n\\Begin{equation}\n  \\tilde A(y) = A(\\tilde T(y)) \n  \\quad\\hbox{and}\\quad\n  \\tilde\\ka(y) = \\ka(\\tilde T(y))\n  \\quad \\hbox{where} \\quad\n  \\tilde T(y) := \\max\\graffe{n_*, \\min\\graffe{y,n^*}} .\n  \\label{kaAstar}\n\\End{equation}\nNext, we approximate the initial data $n^0$, $s^0$, $i^0$ and $h^0$ as in \\eqref{hpdati}\nby smoother functions $\\nz_\\tau$, $\\sz_\\tau$, $\\iz_\\tau$ and $\\hz_\\tau$ satisfying\n\\Begin{eqnarray}\n  && \\nz_\\tau \\in V \\cap \\Lx\\infty \n  \\quad\\hbox{and}\\quad\n  0 \\leq \\nz_\\tau \\leq \\norman^0_\\infty\n  \\quad \\checkmmode{a.e.\\ in~$\\Omega$}\n  \\label{hpdatitau}\n  \\\\\n  && \\norma\\nz_\\tau \\leq \\norman^0 \n  \\quad\\hbox{and}\\quad\n  \\tau\\normaV\\nz_\\tau^2 \\leq \\norman^0^2\n  \\label{stimadatitau}\n  \\\\\n  && \\nz_\\tau \\to n^0 \n  \\quad \\hbox{strongly in $H$ as $\\tau\\searrow0$}\n  \\label{convdatitau}\n  \\\\\n\\noalign{\\noindent and the analogues for $\\sz_\\tau$, $\\iz_\\tau$ and~$\\hz_\\tau$ as well as \\medskip}\n  && \\hz_\\tau \\geq \\sz_\\tau\n  \\quad\\hbox{and}\\quad\n  \\nz_\\tau \\geq \\infn^0\n  \\quad \\checkmmode{a.e.\\ in~$\\Omega$}.\n  \\label{nztmin}\n\\End{eqnarray}\nThis can be done, e.g., by a singular perturbation argument.\nIndeed, if $\\tau\\in(0,1)$ and $u\\in\\Lx\\infty$ is nonnegative, \nthe unique solution $u_\\tau\\in V$ to the variational problem\n\\Begin{equation}\n  \\int_\\Omega \\bigl( u_\\tau v + \\tau \\nabla u_\\tau \\cdot \\nabla v \\bigr)\n  = \\int_\\Omega u v\n  \\quad \\hbox{for every $v\\in V$}\n  \\non\n\\End{equation}\n\\oldgianni{belongs to $W\\subset\\Lx\\infty$,}\nsatisfies \n$\\inf u\\lequ_\\tau\\leq\\norma u_\\infty$, $\\normau_\\tau\\leq\\norma u$ \nand \\oldgianni{$(1\/2)\\normau_\\tau^2+\\tau\\norma{\\nablau_\\tau}^2\\leq(1\/2)\\norma u^2$,\nwhence also $\\tau\\normaVu_\\tau^2\\leq\\norma u^2$,}\nand converges to $u$ strongly in $H$ as $\\tau\\searrow0$.\nFinally, we assume \n\\Begin{equation}\n  \\tau \\in (0,\\tau_0)\n  \\quad \\hbox{where} \\quad\n  \\tau_0 \\in (0,1)\n  \\quad\\hbox{and}\\quad\n  \\tau_0 \\leq \\frac 1 {2(\\alpha-\\mu)}\n  \\quad \\hbox{if $\\alpha>\\mu$}.\n  \\label{hptauz}\n\\End{equation}\nThe functions $\\tilde A$ and $\\tilde\\ka$ satisfy \\eqref{hpAkdiscr} with\n\\Begin{equation}\n  \\oldgianni{\\ka_* := \\min \\graffe{\\ka(y) : \\ n_*\\leq y\\leqn^*}\n  \\quad\\hbox{and}\\quad\n  \\ka^* := \\max \\graffe{\\ka(y) : \\ n_*\\leq y\\leqn^*}.}\n  \\label{sceltakazkau}\n\\End{equation}\nMoreover, the requirements \\eqref{hpdatiV} are fulfilled by the approximating data.\nFurthermore, \\eqref{hptauz} implies~\\eqref{hptau}. \nHence, we can solve the discretized problem \nwith the specified structure and the approximating initial data.\nWe have the following basic result, \nwhich removes the supplementary assumptions~\\eqref{hpAkdiscr}:\n\n\\Begin{proposition}\n\\label{Basic}\nUnder the restriction \\eqref{hptauz}, let \n\\Begin{equation}\n  (n_0,n_1,\\dots,n_N), \\quad\n  (s_0,s_1,\\dots,s_N), \\quad\n  (i_0,i_1,\\dots,i_N)\n  \\quad\\hbox{and}\\quad\n  (h_0,h_1,\\dots,h_N) \n  \\label{soluztau}\n\\End{equation} \nbe as in \\eqref{tuples} and solve the problem \\Pbltau\\ \nwhere one reads $\\tilde A$ and $\\tilde\\ka$ instead of $A$ and~$\\ka$, respectively.\nThen, \\eqref{soluztau} solve the discrete problem with the original $A$ and~$\\ka$ given in \\eqref{hpAk}.\n\\End{proposition}\n\n\\Begin{proof}\nAll the components $n_k$ are nonnegative.\nHowever, as $A$ and $\\ka$ are defined in the open interval $(0,+\\infty)$, \nwe have to reinforce this and show that each $n_k$ is strictly positive\nin order to give a meaning to $A(n_k)$ and~$\\ka(n_k)$.\nWe prove~that\n\\Begin{equation}\n  n_* \\leq n_k \\leq n^* \n  \\quad \\hbox{\\checkmmode{a.e.\\ in~$\\Omega$} \\quad for $k=0,\\dots,N$}.\n  \\label{basic}\n\\End{equation}\nSince $\\tilde A(y)=A(y)$ and $\\tilde\\ka(y)=\\ka(y)$ for every $y\\in[n_*,n^*]$,\nthis also yields that $\\tilde A(n_k)=A(n_k)$ and $\\tilde\\ka(n_k)=\\ka(n_k)$ for $k=0,\\dots,N$, \ni.e., the thesis of the statement.\n\nThe proof of \\eqref{basic} is done in several steps.\nThe last of them needs $L^\\infty$ bounds for the other components of the solution,\nand these are proved as~well.\n\n\\step\nFirst upper bound\n\nWe assume $0\\leq k<N$ and check that we can apply Lemma~\\ref{Gianni} to~$n_{k+1}$.\nIndeed, \\eqref{primak} with $\\tilde\\ka$ in place of $\\ka$ \nimplies that $n_{k+1}$ solves \\eqref{pblG} with\n\\Begin{equation}\n  a = \\tilde\\ka(n_{k+1}) , \\quad\n  b = \\frac 1\\tau + i_k -\\alpha + \\mu\n  \\quad\\hbox{and}\\quad\n  f = \\frac 1\\tau \\, n_k \\,.\n  \\label{gianninkp}\n\\End{equation}\nMoreover, we can take $a_0=\\ka_*$ and $b_0=(1\/\\tau)-(\\alpha-\\mu)^+$ since $i_k$ is nonnegative.\nNotice that $b_0>0$ by \\eqref{hptauz}.\nAs $n_{k+1}$ is nonnegative, the lemma implies that\n\\Begin{equation}\n  \\norman_{k+1}_\\infty\n  \\leq b_0^{-1} \\norma f_\\infty\n  = \\frac \\tau {1-\\tau(\\alpha-\\mu)^+} \\, \\norma{(1\/\\tau)n_k}_\\infty \n  = \\frac 1 {1-\\tau(\\alpha-\\mu)^+} \\, \\norman_k_\\infty \\,.\n  \\non\n\\End{equation}\nAs this holds for $0\\leq k<N$, $n_0=\\nz_\\tau$ and \n$\\norma\\nz_\\tau_\\infty\\leq\\norman^0_\\infty$ by~\\eqref{hpdatitau}, \nwe deduce that\n\\Begin{equation}\n  \\norman_k_\\infty\n  \\leq (1-\\tau(\\alpha-\\mu)^+)^{-k} \\norma{\\nz_\\tau}_\\infty\n  \\leq (1-\\tau(\\alpha-\\mu)^+)^{-N} \\norman^0_\\infty\n  \\label{forbasic}\n\\End{equation}\nfor $k=0,\\dots,N$.\nNow, observe that $\\ln(1-y)\\geq-2y$ for every $y\\in(0,1\/2]$.\nIndeed, the logarithm is a concave function \nand the inequality is satisfied at $y=1\/2$ and for $y>0$ small enough \nby comparison of the derivatives at~$0$.\nSince $\\tau(\\alpha-\\mu)^+\\leq1\/2$ by~\\eqref{hptauz},\nit follows that\n\\Begin{equation}\n  \\ln(1-\\tau(\\alpha-\\mu)^+)\n  \\geq -2\\tau(\\alpha-\\mu)^+ \n  \\non\n\\End{equation}\nand \\eqref{forbasic} implies\n\\Begin{equation}\n  \\norman_k_\\infty\n  \\leq (1-\\tau(\\alpha-\\mu)^+)^{-N} \\norman^0_\\infty\n  \\leq e^{2N\\tau(\\alpha-\\mu)^+} \\norman^0_\\infty\n  = n^* \n  \\quad \\oldgianni{\\hbox{for $k=0,\\dots,N$.}}\n  \\label{stimank}\n\\End{equation}\nTherefore, the second inequality of \\eqref{basic} is proved.\n\nNow, we prove analogous upper bounds for the other components of the solution\nby applying Lemma~\\ref{Gianni} once more, always with $a=\\tilde\\ka(n_{k+1})$.\n\n\\step\nFurther upper bounds\n\nWe start with $s_k$.\nEquation \\eqref{pblG} is solved by $s_{k+1}$ with\n\\Begin{equation}\n  b = \\frac 1\\tau + \\tilde A(n_{k+1})i_k + \\tilde A(n_{k+1})(h_k-s_k) + \\mu\n  \\quad\\hbox{and}\\quad \n  f = \\frac 1\\tau s_k + \\alphan_{k+1} \\,.\n  \\non\n\\End{equation}\nBy recalling that $\\tilde A$ is nonnegative and that $i_k\\geq0$ and $h_k\\geqs_k$ \\checkmmode{a.e.\\ in~$\\Omega$}, \nwe can take $b_0=(1\/\\tau)+\\mu$.\nSince $s_{k+1}$ is nonnegative, the lemma yields\n\\Begin{equation}\n  \\normas_{k+1}_\\infty\n  \\leq \\frac \\tau {1+\\tau\\mu} \\, \\norma{(1\/\\tau)s_k+\\alphan_{k+1}}_\\infty\n  \\leq \\normas_k_\\infty + \\tau\\alpha\\norman_{k+1}_{\\oldgianni\\infty} \\,.\n  \\non\n\\End{equation}\nOn account of \\oldgianni{\\eqref{stimank} and \\eqref{defnstar}}, we deduce that\n\\Begin{equation}\n  \\normas_k_\\infty\n  \\leq \\norma{s_0}_\\infty + k\\tau\\alphan^*\n  \\leq s^* \n  \\quad \\hbox{for $k=0,\\dots,N$}.\n  \\label{sinfty}\n\\End{equation}\nFor $h_{k+1}$ we can take\n\\Begin{equation}\n  b = \\frac 1\\tau + 1 + \\mu\n  \\quad\\hbox{and}\\quad\n  f = \\frac 1\\tau \\, h_k + \\alpha\\,n_{k+1} + s_{k+1} \\,.\n  \\non\n\\End{equation}\nThen the lemma and the estimates for $n_k$ and $s_k$ already proved give\n\\Begin{equation}\n  \\normah_{k+1}_\\infty \n  \\leq \\frac 1 {(1\/\\tau)+1+\\mu} \\, \\norma{(1\/\\tau)h_k+\\alphan_{k+1}+s_{k+1}}_\\infty\n  \\leq \\normah_k_\\infty + \\tau (\\alphan^*+s^*) \\,.\n  \\non\n\\End{equation}\nHence\n\\Begin{equation}\n  \\normah_k_\\infty\n  \\leq \\norma{h_0}_\\infty + k\\tau (\\alphan^*+s^*)\n  \\leq h^*\n  \\quad \\hbox{for $k=0,\\dots,N$}.\n  \\label{hinfty}\n\\End{equation}\nFinally, for $i_{k+1}$ we can take \n\\Begin{equation}\n  b = \\frac 1\\tau + \\oldgiannin_{k+1} + 1 + \\mu\n  \\quad\\hbox{and}\\quad\n  f = \\frac 1\\tau \\, i_k + h_{k+1} - s_{k+1} \n  \\non\n\\End{equation}\nand a quite similar calculation also using \\eqref{hinfty} yields\n\\Begin{equation}\n  \\normai_k_\\infty \n  \\leq i^*\n  \\quad \\hbox{for $k=0,\\dots,N$}.\n  \\label{iinfty}\n\\End{equation}\n\n\\step\nLower bound\n\nFinally, we prove the \\oldgianni{left} inequality of~\\eqref{basic}.\nWe fix $k$ with $0\\leq k<N$ and assume that $n_k\\geq\\lambda_k$ \\checkmmode{a.e.\\ in~$\\Omega$}\\ for some $\\lambda_k>0$\nand we prove that $n_{k+1}\\geq\\lambda_{k+1}$ \\checkmmode{a.e.\\ in~$\\Omega$}\\ where\n\\Begin{equation}\n  \\lambda_{k+1} := \\frac \\lambda_k {1+\\tau(i^*+(\\mu-\\alpha)^+)} \\,.\n  \\non\n\\End{equation}\nAs already observed, \\eqref{pblG} is solved by $n_{k+1}$ with\n$a$, $b$ and $f$ given by \\eqref{gianninkp}.\nLet us estimate $f-\\lambda_{k+1} b$ from below.\nWe~have\n\\Begin{equation}\n  f - \\lambda_{k+1} b\n  = \\frac 1\\tau \\, n_k \n  - \\frac \\lambda_k {1+\\tau(i^*+(\\mu-\\alpha)^+)} \\, \\Bigl( \\frac 1\\tau + i_k + \\oldgianni{\\mu-\\alpha} \\Bigr)\n  \\geq \\frac 1\\tau \\, n_k - \\frac 1\\tau \\, \\lambda_k  \n  \\geq 0 \\,.\n  \\non\n\\End{equation}\nThen, we can apply the last sentence of Lemma~\\ref{Gianni} and obtain that\n$n_{k+1}\\geq\\lambda_{k+1}$ \\checkmmode{a.e.\\ in~$\\Omega$}.\nSince $n_0=\\nz_\\tau\\geq\\infn^0$ \\checkmmode{a.e.\\ in~$\\Omega$}, we can take $\\lambda_0=\\infn^0$ and~have\n\\Begin{equation}\n  n_k \\geq (\\infn^0) \\, \\bigl( 1+\\tau(i^*+(\\mu-\\alpha)^+) \\bigr)^{-k}\n  \\quad \\checkmmode{a.e.\\ in~$\\Omega$} , \\quad \\hbox{for $k=0,\\dots,N$}.\n  \\label{forlower}\n\\End{equation}\nSince $\\ln(1+y)\\leq y$ for every $y>-1$, we obtain\n\\Begin{equation}\n  -k \\ln \\bigl( 1+\\tau(i^*+(\\mu-\\alpha)^+) \\bigr) \n  \\geq -k \\tau(i^*+(\\mu-\\alpha)^+)\n  \\geq -T(i^*+(\\mu-\\alpha)^+) \n  \\non\n\\End{equation}\nwhence also\n\\Begin{equation}\n  \\bigl( 1+\\tau(i^*+(\\mu-\\alpha)^+) \\bigr)^{-k}\n  \\geq e^{-T(i^*+(\\mu-\\alpha)^+)} .\n  \\non\n\\End{equation}\nHence, \\eqref{forlower} yields that $n_k\\geqn_*$ \\checkmmode{a.e.\\ in~$\\Omega$}\\ for every~$k$.\nThis concludes the proof.\n\\End{proof}\n\nAt this point, we establish two estimates whose proof is made very easy by the above $L^\\infty$-bounds.\nFor the second one it is convenient to rewrite the estimates in terms of the interpolating functions.\n\n\\Begin{notation}\\rm\n\\label{Constants}\nFor the sake of simplicity, in the rest of the section,\nwe use the same symbol $c$ for possibly different constants\nthat only depend on $\\ka$, $A$, $\\alpha$, $\\mu$, $T$, $\\Omega$ \nand the initial data appearing in~\\eqref{hpdati}.\nHence, the values of $c$ do not depend on~$\\tau$\nand might change from line to line and even within the same line.\n\\End{notation}\n\n\\step\nFirst a priori estimate\n\nWe rewrite \\accorpa{primak}{quartak} in the form\n\\Begin{eqnarray}\n  && \\bigl( \\frac {n_{k+1}-n_k}\\tau , v \\bigr)\n  + (\\ka(n_{k+1}) \\nablan_{k+1} , \\nabla v)\n  + \\ka_* (n_{k+1} , v)\n  = (g_{n,k} , v)\n  \\label{primakbis}\n  \\\\\n  && \\bigl( \\frac {s_{k+1}-s_k}\\tau , v \\bigr)\n  + \\bigl( \\ka(n_{k+1}) \\nablas_{k+1} , \\nabla v \\bigr)\n  + \\ka_* (s_{k+1} , v)\n  = (g_{s,k} , v)\n  \\label{secondakbis}\n  \\\\\n  && \\bigl( \\frac {i_{k+1}-i_k}\\tau , v \\bigr)\n  + \\bigl( \\ka(n_{k+1})\\nablai_{k+1} , \\nabla v \\bigr)\n  + \\ka_* (i_{k+1} , v)\n  = (g_{i,k} , v)\n  \\label{terzakbis}\n  \\\\\n  && \\bigl( \\frac {h_{k+1}-h_k}\\tau , v \\bigr)\n  + \\bigl( \\ka(n_{k+1})\\nablah_{k+1} , \\nabla v \\bigr)\n  + \\ka_* (h_{k+1} , v)\n  = (g_{h,k} , v)\n  \\label{quartakbis}\n\\End{eqnarray}\nall for every $v\\in V$, where we have set\n\\Begin{eqnarray}\n  && g_{n,k} := (\\alpha-\\mu-\\oldgiannii_k+\\ka_*) n_{k+1} \n  \\non\n  \\\\\n  && g_{s,k} := \\alphan_{k+1} - A(n_{k+1})\\skpi_k - A(n_{k+1})s_{k+1}(h_k-s_k) + (\\ka_*-\\mu)s_{k+1}\n  \\non\n  \\\\\n  && g_{i,k} := h_{k+1}-s_{k+1} + (\\ka_*-1-\\mu) i_{k+1} \\oldgianni{{} - n_{k+1} i_{k+1}}\n  \\non\n  \\\\\n  && g_{h,k} := \\alphan_{k+1} + s_{k+1} + (\\ka_*-1-\\mu)h_{k+1} .\n  \\non\n\\End{eqnarray}\nWe observe that \\eqref{basic}, \\accorpa{sinfty}{iinfty} and the continuity of $A$ imply that\n\\Begin{eqnarray}\n  && \\normag_{n,k}\n  + \\normag_{s,k}\n  + \\normag_{i,k}\n  + \\normag_{h,k}\n  \\non\n  \\\\\n  && \\leq c \\, \\bigl(\\normag_{n,k}_\\infty\n  + \\normag_{s,k}_\\infty\n  + \\normag_{i,k}_\\infty\n  + \\normag_{h,k}_\\infty \\bigr)\n  \\leq c \\,.\n  \\label{rhsbdd}\n\\End{eqnarray}\nNow, we test \\eqref{primakbis} by $\\taun_{k+1}$ and sum over $k$ from $0$ to $m-1$, \nwith an arbitrary positive integer $m\\leq N$.\nWe obtain\n\\Begin{equation}\n  \\somma k0{m-1} (n_{k+1}-n_k,n_{k+1})\n  + \\tau \\somma k0{m-1} \\bigl (\\ka(n_{k+1})\\nablan_{k+1},\\nablan_{k+1}) + \\ka_* \\norman_{k+1}^2 \\bigr)\n  = \\tau \\somma k0{m-1} (g_{n,k} , n_{k+1}) .\n  \\non\n\\End{equation}\nSince $\\ka\\geq\\ka_*$, by owing to \\eqref{elementare} and the Schwarz and Young inequality,\nwe deduce~that\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\norma{n_m}^2\n  - \\frac 12 \\, \\norma{n_0}^2\n  + \\frac 12 \\somma k0{m-1} \\norma{n_{k+1}-n_k}^2\n  + \\ka_* \\tau \\somma k0{m-1} \\normaVn_{k+1}^2\n  \\non\n  \\\\\n  && \\leq \\frac \\ka_* 2 \\, \\tau \\somma k0{m-1} \\norman_{k+1}^2\n  + \\frac 1 {2\\ka_*} \\, \\tau \\somma k0{m-1} \\normag_{n,k}^2 \\,.\n  \\non\n\\End{eqnarray}\nThen, it suffices to rearrange and account for the first \\oldgianni{condition in} \\eqref{stimadatitau} and \\eqref{rhsbdd} to~obtain\n\\Begin{equation}\n  \\max_{k=0,\\dots,m} \\norman_k^2\n  + \\tau \\somma k0N \\normaVn_k^2\n  + \\somma k0{N-1} \\norma{n_{k+1}-n_k}^2\n  \\leq c \\,.\n  \\label{stiman}\n\\End{equation}\nBy treating \\accorpa{secondakbis}{quartakbis} in the same way, we also have\n\\Begin{eqnarray}\n  && \\max_{k=0,\\dots,m} \\normas_k^2\n  + \\tau \\somma k0N \\normaVs_k^2\n  + \\somma k0{N-1} \\norma{s_{k+1}-s_k}^2\n  \\leq c\n  \\label{stimas}\n  \\\\\n  && \\max_{k=0,\\dots,m} \\normai_k^2\n  + \\tau \\somma k0N \\normaVi_k^2\n  + \\somma k0{N-1} \\norma{i_{k+1}-i_k}^2\n  \\leq c\n  \\label{stimai}\n  \\\\\n  && \\max_{k=0,\\dots,m} \\normah_k^2\n  + \\tau \\somma k0N \\normaVh_k^2\n  + \\somma k0{N-1} \\norma{h_{k+1}-h_k}^2\n  \\leq c \\,.\n  \\label{stimah}\n\\End{eqnarray}\n\nAt this point, we introduce the interpolating functions. \nNamely, we term\n\\Begin{equation}\n  \\hat n_\\tau, \\hat s_\\tau,\\ \\hat i_\\tau,\\ \\hat h_\\tau,\\quad\n  \\overline n_\\tau, \\overline s_\\tau,\\ \\overline i_\\tau,\\ \\overline h_\\tau\n  \\quad\\hbox{and}\\quad\n  \\underline n_\\tau, \\underline s_\\tau,\\ \\underline i_\\tau,\\ \\underline h_\\tau\n  \\label{interpolating}\n\\End{equation}\nthe functions $\\hat z$, $\\overline z$ and $\\underline z$ given in Notation~\\ref{Interpol}\nwith $z=(n_0,\\dots,n_N)$, $z=(s_0,\\dots,s_N)$, $z=(i_0,\\dots,i_N)$ and $z=(h_0,\\dots,h_N)$, respectively.\nThen, we can owe to Proposition~\\ref{Propinterp} \nto rewrite the uniform bounds \\eqref{basic} and \\accorpa{sinfty}{hinfty}, which read\n\\Begin{eqnarray}\n  && \\norma\\overline n_\\tau_{\\LQ\\infty}\n  + \\norma\\overline s_\\tau_{\\LQ\\infty}\n  + \\norma\\overline i_\\tau_{\\LQ\\infty}\n  + \\norma\\overline h_\\tau_{\\LQ\\infty}\n  \\leq c \n  \\label{stimao}\n\\End{eqnarray}\nand the analogues for $\\underline n_\\tau$, $\\underline s_\\tau$, $\\underline i_\\tau$ and $\\underline h_\\tau$.\nNext, the estimates \\accorpa{stiman}{stimah} imply\n\\Begin{eqnarray}\n  && \\norma\\overline n_\\tau_{\\L2V}\n  + \\norma\\overline s_\\tau_{\\L2V}\n  + \\norma\\overline i_\\tau_{\\L2V}\n  + \\norma\\overline h_\\tau_{\\L2V}\n  \\leq c\n  \\label{stimeoL2V}\n  \\\\\n  && \\norma{\\overline n_\\tau-\\underline n_\\tau}_{\\L2H}\n  + \\norma{\\overline s_\\tau-\\underline s_\\tau}_{\\L2H}\n  \\non\n  \\\\\n  && \\quad {}\n  + \\norma{\\overline i_\\tau-\\underline i_\\tau}_{\\L2H}\n  + \\norma{\\overline h_\\tau-\\underline h_\\tau}_{\\L2H}\n  \\leq c \\tau^{1\/2} .\n  \\label{stimeo-u}\n\\End{eqnarray}\n\\oldgianni{Notice that \\eqref{stimeo-u} and \\eqref{diffLdue} provide the same estimate for $\\overline n_\\tau-\\hat n_\\tau$.}\nMoreover, \\oldgianni{the property} \\eqref{hzLdueZ} with $Z=V$, \nthe second \\oldgianni{inequality in} \\eqref{stimadatitau} and \\eqref{stimao} also yield\n\\Begin{equation}\n  \\norma\\hat n_\\tau_{\\L2V}\n  + \\norma\\hat s_\\tau_{\\L2V}\n  + \\norma\\hat i_\\tau_{\\L2V}\n  + \\norma\\hat h_\\tau_{\\L2V}\n  \\leq c \\,.\n  \\label{stimehL2V}\n\\End{equation}\nIn order to obtain an estimate for the time derivatives and let $\\tau$ tend to zero,\nwe write equations \\accorpa{primak}{quartak} in terms of the interpolating function.\nWe have\n\\Begin{eqnarray}\n  && (\\partial_t\\hat n_\\tau , v)\n  + (\\underline i_\\tau\\overline n_\\tau , v)\n  + (\\ka(\\overline n_\\tau)\\nabla\\overline n_\\tau , \\nabla v)\n  = (\\alpha-\\mu) (\\overline n_\\tau , v)\n  \\label{primatau}\n  \\\\\n  && (\\partial_t\\hat s_\\tau , v)\n  + (A(\\overline n_\\tau)\\overline s_\\tau\\underline i_\\tau , v)\n  + (A(\\overline n_\\tau)\\oldgianni\\overline s_\\tau(\\underline h_\\tau-\\underline s_\\tau) , v)\n  + \\oldgianni{\\mu (\\overline s_\\tau , v)}\n  \\qquad\n  \\non\n  \\\\\n  && \\quad {}\n  + (\\ka(\\overline n_\\tau)\\nabla\\overline s_\\tau , \\nabla v)\n  = \\alpha (\\overline n_\\tau , v)\n  \\label{secondatau}\n  \\\\\n  && (\\partial_t\\hat i_\\tau , v)\n  + (1+\\mu)(\\underline i_\\tau , v)\n  + (\\ka(\\overline n_\\tau)\\nabla\\overline i_\\tau , \\nabla v)\n  = (\\overline h_\\tau-\\overline s_\\tau , v)\n  \\label{tezatau}\n  \\\\\n  && (\\partial_t\\hat h_\\tau , v)\n  + (1+\\mu)(\\overline h_\\tau , v)\n  + (\\ka(\\overline n_\\tau)\\nabla\\overline h_\\tau , \\nabla v)\n  = (\\alpha\\overline n_\\tau+\\overline s_\\tau , v)\n  \\label{quartatau}\n\\End{eqnarray}\nall of them being satisfied \\oldgianni{for every $v\\in V$ \\checkmmode{a.e.\\ in~$(0,T)$}}.\nMoreover, requirements \\eqref{segnik} and the initial conditions read\n\\Begin{eqnarray}\n  && \\overline n_\\tau \\geq 0 , \\quad\n  \\overline h_\\tau \\geq \\overline s_\\tau \\geq 0\n  \\quad\\hbox{and}\\quad\n  \\overline i_\\tau \\geq 0\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\label{segnio}\n  \\\\\n  && \\hat n_\\tau(0) = \\nz_\\tau \\,, \\quad\n  \\hat s_\\tau(0) = \\sz_\\tau \\,, \\quad\n  \\hat i_\\tau(0) = \\iz_\\tau \n  \\quad\\hbox{and}\\quad\n  \\hat h_\\tau(0) = \\hz_\\tau \\,.\n  \\label{cauchytau}\n\\End{eqnarray}\n\\oldgianni{Finally}, the bounds given by \\eqref{basic} and \\accorpa{sinfty}{iinfty} imply\n\\Begin{equation}\n  n_* \\leq \\overline n_\\tau \\leq n^* , \\quad\n  \\overline s_\\tau \\leq s^* , \\quad\n  \\overline i_\\tau \\leq i^*\n  \\quad\\hbox{and}\\quad\n  \\overline h_\\tau \\leq h^*\n  \\quad \\checkmmode{a.e.\\ in~$Q$} .\n  \\label{bounds}\n\\End{equation}\n\n\\step\nSecond a priori estimate\n\nWe take an arbitrary $v\\in\\L2V$, write \\eqref{primatau} at the time $t$ and test it by~$v(t)$.\nThen, we integrate over~$(0,T)$ and rearrange.\n\\oldgianni{Thanks} to the above estimates, we obtain\n\\Begin{equation}\n  \\int_Q \\partial_t\\hat n_\\tau \\, v\n  = - \\int_Q \\ka(\\overline n_\\tau) \\nabla\\overline n_\\tau \\cdot \\nabla v\n  + \\int_Q (\\alpha-\\mu-\\underline i_\\tau) \\overline n_\\tau v\n  \\leq c \\norma v_{\\L2V} \\,.\n  \\non\n\\End{equation}\nBy analogously treating the other equations, we conclude that\n\\Begin{equation}\n  \\norma{\\partial_t\\hat n_\\tau}_{\\L2V^*}\n  + \\norma{\\partial_t\\hat s_\\tau}_{\\L2V^*}\n  + \\norma{\\partial_t\\hat i_\\tau}_{\\L2V^*}\n  + \\norma{\\partial_t\\hat h_\\tau}_{\\L2V^*}\n  \\leq c \\,.\n  \\label{stimederivate}\n\\End{equation}\n\nSince all this holds under the only restriction on $\\tau$ given by~\\eqref{hptauz}, \nwe are ready to let $\\tau$ tend to zero.\n\n\\step\nConvergence and conclusion\n\nThe estimates obtained before and the Aubin-Lions lemma (see, e.g., \\cite[Thm.~5.1, p.~58]{Lions})\nensure the existence of a quadruple $(n,s,i,h)$ such~that\n\\Begin{eqnarray}\n  && \\hat n_\\tau \\to n \\,, \\quad\n  \\hat s_\\tau \\to s \\,, \\quad\n  \\hat i_\\tau \\to i \n  \\quad\\hbox{and}\\quad\n  \\hat h_\\tau \\to h \n  \\non\n  \\\\\n  && \\quad \\hbox{weakly star in $\\H1V^*\\cap\\C0H\\cap\\L2V$}\n  \\qquad\n  \\non\n  \\\\\n  && \\quad \\hbox{strongly in $\\L2H$ and \\checkmmode{a.e.\\ in~$Q$}}\n  \\label{convhat}\n\\End{eqnarray}\nas $\\tau\\searrow0$ (more precisely for a suitable sequence $\\tau_j\\searrow0$).\nIn particular, the values at~$0$ converge weakly in $H$ to the left-hand side s of \\eqref{cauchy} \nand the initial conditions \\eqref{cauchy} themselves are satisfied on account of~\\eqref{convdatitau}.\nMoreover, the differences $\\overline n_\\tau-\\underline n_\\tau$ and $\\overline n_\\tau-\\hat n_\\tau$ converge to zero strongly in $\\L2H$,\nand also \\checkmmode{a.e.\\ in~$Q$}\\ without loss of generality,\nso that $\\overline n_\\tau$ and $\\underline n_\\tau$ tend to $n$ strongly in $\\L2H$.\nSince the same holds for the other variables, \nall the inequalities \\oldgianni{in} \\eqref{segni} are satisfied too.\nMoreover, \\eqref{bounds} yield\n\\Begin{equation}\n  n_* \\leq n \\leq n^* , \\quad\n  s \\leq s^* , \\quad\n  i \\leq i^*\n  \\quad\\hbox{and}\\quad\n  h \\leq h^*\n  \\quad \\checkmmode{a.e.\\ in~$Q$} .\n  \\non  \n\\End{equation}\nBy combining with the point-wise convergence, we infer~that\n\\Begin{equation}\n  \\hbox{$\\overline n_\\tau$, $\\overline s_\\tau$, $\\overline i_\\tau$ and $\\overline h_\\tau$\n    converge to their limits strongly in $\\LQ p$ for every $p\\in[1,+\\infty)$}\n  \\non\n\\End{equation}\nand the same holds for $\\underline n_\\tau$, $\\underline s_\\tau$, $\\underline i_\\tau$ and $\\underline h_\\tau$.\nWe have some consequences. \nFirst, the products like $\\underline i_\\tau\\overline n_\\tau$ in \\eqref{primatau} converge to the right products\n(in~this case the limit is~$in$)\nin~$\\LQ p$ for $p\\in[1,+\\infty)$.\nNext, $A(\\overline n_\\tau)$~converges to $A(n)$ in the same topology,\nsince it is bounded in $\\LQ\\infty$ and converges to $A(n)$ \\checkmmode{a.e.\\ in~$Q$},\nall this for $A$ is continuous.\nBy the same reason, $\\ka(\\overline n_\\tau)$ converges to $\\ka(n)$ strongly in~$\\LQ p$ for $p<+\\infty$.\nFrom this, we \\oldgianni{claim} that\n\\Begin{eqnarray}\n  && \\ka(\\overline n_\\tau)\\nabla\\overline n_\\tau \\to \\ka(n)\\nabla n , \\quad\n  \\ka(\\overline n_\\tau)\\nabla\\overline s_\\tau \\to \\ka(n)\\nabla s , \\quad\n  \\ka(\\overline n_\\tau)\\nabla\\overline i_\\tau \\to \\ka(n)\\nabla i \n  \\non\n  \\\\\n  && \\quad\\hbox{and}\\quad\n  \\ka(\\overline n_\\tau)\\nabla\\overline h_\\tau \\to \\ka(n)\\nabla h \n  \\quad \\hbox{weakly in $(\\L2H)^3$}.\n  \\non\n\\End{eqnarray}\n\\oldgianni{In fact we} prove the first one, only, since the others are analogous.\nFor the product, we have weak convergence to $\\ka(n)\\nabla n$ in $(\\LQ q)^3$ for $q\\in[1,2)$.\nOn the other hand, $\\ka(\\overline n_\\tau)$ is bounded in $\\LQ\\infty$ and $\\nabla\\overline n_\\tau$ is bounded in $(\\L2H)^3$,\nso that the product is bounded in $(\\L2H)^3$ and thus has a weak limit in this topology.\nClearly, this weak limit has to be $\\ka(n)\\nabla n$.\n\nAll this allows us to let $\\tau$ tend to zero in \\accorpa{primatau}{quartatau}.\nWe consider the first one.\nWe write its integrated version, namely\n\\Begin{equation}\n  \\int_Q \\bigl(\n    \\partial_t\\hat n_\\tau \\, v\n    + \\underline i_\\tau\\overline n_\\tau \\, v\n    + \\ka(\\overline n_\\tau)\\nabla\\overline n_\\tau \\cdot \\nabla v\n  \\bigr)\n  = (\\alpha-\\mu) \\int_Q \\overline n_\\tau \\, v\n  \\quad \\hbox{for every $v\\in\\L2V$}.\n  \\non\n\\End{equation}\nThen, it is clear that we can let $\\tau$ tend to zero and obtain \\eqref{intprima}.\nAs the same argument works for the other equations, the proof is complete.\n\n\n\\section{Uniqueness}\n\\label{UNIQUENESS}\n\\setcounter{equation}{0}\n\nIn this section we give \\oldgianni{the proofs of Theorems \\ref{Uniqueness1} and~\\ref{Uniqueness2}.}\n\n\\subsection{Proof of Theorem \\ref{Uniqueness1}}\n\\label{UNIQUENESS1}\n\nWe assume that $A$ is locally Lip\\-schitz\\ continuous and that $\\ka$ is a positive constant.\nWe pick any two solutions $(n_j,s_j,i_j,h_j)$, $j=1,2$, \nand prove that they are the same.\nFirst, we make some observations.\nBy recalling that $n_j$ are bounded (as~well as the other components) and that $\\inf n_j>0$,\nwe fix an interval $[n_\\star,n^\\star]\\subset(0,+\\infty)$ that contains all the values of $n_1$ and $n_2$.\nThen, the restriction of $A$ to $[n_\\star,n^\\star]$ is Lip\\-schitz\\ continuous.\nAs for the estimates we are going to prove,\nwe reinforce the convention given in Notation~\\ref{Constants} for the constants\nby allowing the values of $c$ to depend on the fixed solutions (through their $L^\\infty$ norms), in addition.\n\nWe set for brevity $n:=n_1-n_2$, $s:=s_1-s_2$, $i:=i_1-i_2$ and $h:=h_1-h_2$.\nWe write \\eqref{prima} for both solutions, take the difference and test it by~$n$.\nThen, we integrate over~$(0,t)$ and adjust.\nBy also owing to the Young inequality, we~have\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\int_\\Omega |n(t)|^2\n  + \\ka \\int_{Q_t} |\\nabla n|^2\n  = (\\alpha-\\mu) \\int_{Q_t} |n|^2\n  - \\int_{Q_t} \\oldgianni{n_1 i n}\n  - \\int_{Q_t} i_2 n^2 \n  \\non\n  \\\\\n  && \\leq c \\int_{Q_t} \\bigl( |n|^2 + |i|^2 \\bigr) \\,.\n  \\label{uprima}\n\\End{eqnarray}\nBy proceeding with \\eqref{seconda} in the same way, we obtain\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\int_\\Omega |s(t)|^2\n  + \\ka \\int_{Q_t} |\\nabla s|^2\n  + \\mu \\int_{Q_t} |s|^2\n  \\non\n  \\\\\n  && =  \\alpha \\int_{Q_t} n s\n  - \\int_{Q_t} \\bigl(\n    (A(n_1)-A(n_2)) s_1 i_1\n    + A(n_2) s \\, i_1\n    + A(n_2) s_2 i\n  \\bigr) s\n  \\non\n  \\\\\n  && \\quad{}\n  - \\int_{Q_t} \\bigl(\n    (A(n_1)-A(n_2)) s_1 (s_1-h_1)\n    + A(n_2) s (s_1-h_1)\n    + A(n_2) s_2 (s-h)\n  \\bigr) s\n  \\qquad\n  \\non\n  \\\\\n  && \\leq c \\int_{Q_t} \\bigl( |n|^2 + |s|^2 + |i|^2 + \\oldgianni{|h|^2}\\bigr) \\,.\n  \\label{useconda}\n\\End{eqnarray}\nAnalogously, by dealing with \\eqref{terza} and \\eqref{quarta}, we have\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\int_\\Omega |i(t)|^2\n  + \\ka \\int_{Q_t} |\\nabla i|^2\n  + \\frac 12 \\, \\int_\\Omega |h(t)|^2\n  + \\ka \\int_{Q_t} |\\nabla h|^2\n  \\non\n  \\\\\n  && \\leq c \\int_{Q_t} \\bigl( |n|^2 + |s|^2 + |i|^2 + |h|^2 \\bigr) \\,.\n  \\label{ualtre}\n\\End{eqnarray}\nAt this point, it suffices to add \\accorpa{uprima}{ualtre} to each other\nand apply the Gronwall lemma to conclude that $n=s=i=h=0$.\n\n\n\\subsection{Proof of Theorem \\ref{Uniqueness2}}\n\\label{UNIQUENESS2}\n\nIt is understood that the assumptions listed in the statement are in force.\nSince the component $n$ of every solution belongs to $\\Lx\\infty$ \nand is bounded away from zero (see \\Regsoluz),\nwe can assume, without loss of generality, that \n\\Begin{equation}\n  \\ka_* \\leq \\ka(y) \\leq \\ka^*\n  \\quad \\hbox{for some positive constants $\\ka_*$ and $\\ka^*$ and every $y>0$}\n  \\label{kabounds}\n\\End{equation}\nand that $A$ is Lip\\-schitz\\ continuous, whenever we fix one or two solutions.\nThe bounds $\\ka_*$ and $\\ka^*$ and the Lip\\-schitz\\ constant of $A$ \ndepend on the solutions we fix every time, \nand the same holds for the Lip\\-schitz\\ constants of the functions we are going to introduce.\nWe~set\n\\Begin{equation}\n  K(y) := \\int_0^y \\ka(z) \\, dz\n  \\quad \\hbox{for $y\\in(0,+\\infty)$}\n  \\label{defKbis}\n\\End{equation}\nand observe that, under condition \\eqref{kabounds}, it is a bijection from $(0,+\\infty)$ onto itself and that\nboth $K$ and $K^{-1}$ are Lip\\-schitz\\ continuous.\nNext, to every solution $(n,s,i,h)$ to problem \\Pbl\\ satisfying \\Regsoluz\\\nwe associate the function $u$ defined~by\n\\Begin{equation}\n  u := K(n) \n  \\label{defu}\n\\End{equation}\nand observe that the regularity and boundedness properties of~$u$,\nbut that of the time derivative, are the same as those of~$n$.\n\nOur project is the following:\nfirst we prove that every solution $(n,s,i,h)$ to problem \\Pbl\\ satisfying \\Regsoluz\\\nenjoys the regularity properties specified in~\\eqref{highreg};\nthen, we prove uniqueness on account of the regularity already proved.\nThe main difficulty is a regularity result for the component $n$ of the solution.\nThis needs some preliminary results.\n\n\\Begin{proposition}\n\\label{Regnu}\nThe component $n$ of every solution $(n,s,i,h)$ to problem \\Pbl\\ satisfying \\Regsoluz\\\nand the corresponding function $u$ verify\n\\Begin{eqnarray}\n  && \\partial_t n \\in \\L2H\n  \\quad\\hbox{and}\\quad\n  \\partial_t u \\in \\L2H\n  \\label{regdt}\n  \\\\\n  && \\div(\\ka(n)\\nabla n) \\in \\L2H \n  \\quad\\hbox{and}\\quad\n  \\non\n  \\\\\n  && \\quad \\partial_t n - \\div(\\ka(n)\\nabla n)\n  = (\\alpha - \\mu - i) n\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\quad \\hbox{with} \\quad \\hbox{$\\partial_\\nu n=0$ on $\\Gamma$}\n  \\label{bvpn}\n  \\\\\n  && \\Delta u \\in \\L2H \\quad\\hbox{and}\\quad\n  \\non\n  \\\\\n  && \\quad \\partial_t u - \\ka(n) \\Delta u\n  = \\ka(n) (\\alpha - \\mu - i) n\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\quad \\hbox{with} \\quad \\hbox{$\\partial_\\nu u=0$ on $\\Gamma$}.\n  \\qquad\n  \\label{bvpu}\n\\End{eqnarray}\n\\End{proposition}\n\n\\Begin{proof}\nWe fix a solution $(n,s,i,h)$ \nand suitably extend the component~$n$.\nWe also introduce an auxiliary function.\nWe define $\\tilde n$ and $f$ on $(-1,T)$ by setting\n\\Begin{eqnarray}\n  && \\tilde n(t) := n(t)\n  \\quad\\hbox{and}\\quad\n  f(t) := (\\alpha-mu-i(t)) n(t)\n  \\quad \\hbox{if $t\\in(0,T)$}\n  \\non\n  \\\\\n  && \\tilde n(t) := n^0 \n  \\quad\\hbox{and}\\quad\n  f(t) := - \\div (\\ka(n^0)\\nablan^0)\n  \\quad \\hbox{if $t\\in(-1,0)$}.\n  \\non\n\\End{eqnarray}\nNotice that $\\div(\\ka(n^0)\\nablan^0)$ belongs to~$H$ by \\eqref{hpdatareg} \n(for this it would be sufficient a much weaker assumption).\nAlso notice that $\\tilde n$ is a continuous $H$-valued function\nso that no jumps for its derivative can occur at $t=0$.\nTherefore, $\\tilde n$ and $f$ satisfy\n\\Begin{eqnarray}\n  && \\tilde n \\in H^1(-1,T;V^*) \\cap L^2(-1,T;V) \n  \\quad\\hbox{and}\\quad\n  f \\in L^2(-1,T;H)\n  \\non\n  \\\\\n  && \\< \\partial_t \\tilde n , v >\n  + \\int_\\Omega \\ka(\\tilde n) \\nabla\\tilde n \\cdot \\nabla v\n  = \\int_\\Omega f v\n  \\quad \\hbox{for every $v\\in V$ a.e.\\ in $(-1,T)$}.\n  \\label{proleqn}\n\\End{eqnarray}\nHowever, for simplicity, we write $n$ in place of $\\tilde n$ in the sequel.\nNow, we fix $\\tau>0$ small (namely, $\\tau<\\min\\{1,T\\}$). \nLater on, we let $\\tau$ tend to zero.\nWe~set\n\\Begin{equation}\n  u(t) := K(n(t))\n  \\quad\\hbox{and}\\quad\n  U(t) := \\int_{-\\tau}^t u(y) \\, dy\n  \\quad \\hbox{for $t\\in[-\\tau,T]$} \n  \\non\n\\End{equation}\nby noting that $\\nabla u=\\ka(n)\\nabla n$ and $\\partial_t U=u$.\nThen, for $t\\in(-\\tau,T-\\tau)$, we integrate \\eqref{proleqn} on $(t,t+\\tau)$ with respect to time\nand test the resulting equality by $\\tau^{-2}(u(t+\\tau)-u(t))$.\nWe obtain for a.a.\\ $t\\in(-\\tau,T-\\tau)$\n\\Begin{eqnarray}\n  && \\int_\\Omega \\frac{n(t+\\tau) - n(t)} \\tau \\, \\frac{u(t+\\tau) - u(t)} \\tau\n  + \\int_\\Omega \\nabla \\, \\frac{U(t+\\tau) - U(t)} \\tau \\cdot \\nabla \\, \\frac{u(t+\\tau) - u(t)} \\tau\n  \\non\n  \\\\\n  && = \\int_\\Omega \\Bigl(  {\\textstyle\\frac 1\\tau \\int_t^{t+\\tau} f(t') \\, dt'} \\Bigr) \\, \\frac{u(t+\\tau) - u(t)} \\tau \\,.\n  \\non\n\\End{eqnarray}\nWe treat each term of this equality, separately.\nSince $n=K^{-1}(u)$ and the derivative of $K^{-1}$ is bounded from below by~$1\/\\ka^*$, we~have\n\\Begin{equation}\n  \\int_\\Omega \\frac{n(t+\\tau) - n(t)} \\tau \\, \\frac{u(t+\\tau) - u(t)} \\tau\n  \\geq \\frac 1\\ka^* \\int_\\Omega \\Bigl| \\frac{u(t+\\tau) - u(t)} \\tau \\Bigr|^2 .\n  \\non\n\\End{equation}\nAs for the second term, we recall that $u=\\partial_t U$, whence\n\\Begin{equation}\n  \\int_\\Omega \\nabla \\, \\frac{U(t+\\tau) - U(t)} \\tau \\cdot \\nabla \\, \\frac{u(t+\\tau) - u(t)} \\tau\n  = \\frac 12 \\, \\frac d{dt} \\int_\\Omega \\Bigl| \\nabla \\, \\frac{U(t+\\tau) - U(t)} \\tau \\Bigr|^2 .\n  \\non\n\\End{equation}\nFinally, the Young inequality yields\n\\Begin{eqnarray}\n  && \\int_\\Omega \\Bigl(  {\\textstyle\\frac 1\\tau \\int_t^{t+\\tau} f(t') \\, dt'} \\Bigr) \\, \\frac{u(t+\\tau) - u(t)} \\tau\n  \\non\n  \\\\\n  && \\leq \\frac 1 {2\\ka^*} \\int_\\Omega \\Bigl| \\frac{u(t+\\tau) - u(t)} \\tau \\Bigr|^2 \n  + \\frac \\ka^* 2 \\int_\\Omega \\Bigl| {\\textstyle\\frac 1\\tau \\int_t^{t+\\tau} f(t') \\, dt'} \\Bigr|^2 .\n  \\non\n\\End{eqnarray}\nBy collecting all this, rearranging and integrating over $(-\\tau,T-\\tau)$ we obtain\n\\Begin{eqnarray} \n  && \\frac 1\\ka^* \\int_{-\\tau}^{T-\\tau} \\int_\\Omega \\Bigl| \\frac{u(t+\\tau) - u(t)} \\tau \\Bigr|^2 dt\n  + \\int_\\Omega \\Bigl| \\nabla \\, \\frac{U(T) - U(T-\\tau)} \\tau \\Bigr|^2\n  \\non\n  \\\\\n  && \\leq \\int_\\Omega \\Bigl| \\nabla \\, \\frac{U(0) - U(-\\tau)} \\tau \\Bigr|^2\n  + \\ka^* \\int_{-\\tau}^{T-\\tau} \\int_\\Omega \\Bigl| {\\textstyle\\frac 1\\tau \\int_t^{t+\\tau} f(t') \\, dt'} \\Bigr|^2 dt \\,.\n  \\non\n\\End{eqnarray}\nNow, we observe that the last term can be estimate  uniformly with respect to $\\tau$\nby the norm of $f$ and $n^0$ in $\\L2H$ and~$\\Wx{2,\\infty}$, respectively,\nand that $U(0)-U(-\\tau)=\\tau K(n^0)$.\nTherefore, we conclude that $\\partial_t u\\in\\L2H$.\nAs $n=K^{-1}(u)$ and $K^{-1}$ is Lip\\-schitz\\ continuous, we infer that $\\partial_t n\\in\\L2H$,\nso that \\eqref{regdt} is proved.\nThis conclusions also imply that $\\partial_t u=\\ka(n)\\partial_t n$ \\checkmmode{a.e.\\ in~$Q$}.\nNext, we come back to \\eqref{proleqn} written on $(0,T)$ and recall that $f\\in\\LQ\\infty$.\nThus, \\eqref{bvpn} follows.\nFinally, we can rewrite \\eqref{proleqn} on $(0,T)$ this~way\n\\Begin{equation}\n  \\int_\\Omega \\frac 1{\\ka(n)} \\, \\partial_t u \\, v \n  + \\int_\\Omega \\nabla u \\cdot \\nabla v\n  = \\int_\\Omega f v\n  \\quad \\hbox{for every $v\\in V$ a.e.\\ in $(0,T)$}.\n  \\non\n\\End{equation}\nThis implies that $\\Delta u\\in\\L2H$ and~that\n\\Begin{equation}\n  \\frac 1{\\ka(n)} \\, \\partial_t u - \\Delta u = f\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\quad\\hbox{and}\\quad\n  \\partial_\\nu u = 0\n  \\quad \\hbox{on $\\Gamma$}.\n  \\non\n\\End{equation}\nThen \\eqref{bvpu} follows.\n\\End{proof}\n\n\\Begin{lemma}\n\\label{Unic}\nLet $a\\in\\CQ0$ be strictly positive and let $z$ satisfy\n\\Begin{eqnarray}\n  && z \\in \\H1H \\cap \\L2W\n  \\non\n  \\\\\n  && a \\partial_t z - \\Delta z = 0 \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\quad\\hbox{and}\\quad \n  z(0) = 0 .  \n\\End{eqnarray}\nThen $z=0$.\n\\End{lemma}\n\n\\Begin{proof}\nBy multiplying the equation by $\\partial_t z$ and integrating over~$Q_t$, we obtain\n\\Begin{equation}\n  \\int_{Q_t} a |\\partial_t z|^2\n  + \\frac 12\\int_\\Omega |\\nabla z(t)|^2\n  = 0\n  \\quad \\hbox{for every $t\\in[0,T]$}.\n  \\non\n\\End{equation}\nSince $a$ is strictly positive, we deduce that \nboth $\\partial_t z$ and $\\nabla z$ vanish \\checkmmode{a.e.\\ in~$Q$}, whence $z$ is a constant.\nAs $z(0)=0$, we conclude that $z=0$.\n\\End{proof}\n\n\\Begin{proposition}\n\\label{Highreg}\nEvery solution to problem \\Pbl\\ satisfying \\Regsoluz\\\nalso enjoys the regularity specified by~\\eqref{highreg}.\n\\End{proposition}\n\n\\Begin{proof}\nFix a solution $(n,s,i,h)$.\nFirst, we consider the component~$n$ and recall that\nit satisfies \\accorpa{regdt}{bvpn}, where the right-hand side\\ of the equation belongs to $\\LQ\\infty$.\nSince \\eqref{hpdatareg} implies $n^0\\in\\Cx0$, \nby applying \\cite[Thm.~1.3 and Rem.~1.1 of Chpt.~III]{DiB},\nwe deduce that $n\\in\\CQ0$, whence also $\\ka(n)\\in\\CQ0$.\nNow, for $p\\in[2,+\\infty)$, we consider the problem of finding $w$ satisfying\n\\Begin{eqnarray}\n  && w \\in \\H1H \\cap \\L2W\n  \\label{regw}\n  \\\\\n  && \\partial_t w - \\ka(n) \\Delta w\n  = \\ka(n) (\\alpha - \\mu - i) n\n  \\quad \\checkmmode{a.e.\\ in~$Q$}\n  \\quad\\hbox{and}\\quad\n  w(0) = K(n^0) \\,.\n  \\qquad\n  \\label{eqw}\n\\End{eqnarray}\nThanks to Proposition~\\ref{Regnu}, $u$~is a solution.\nOn the other hand, $\\ka(n)$ is continuous as just observed,\nthe right-hand side\\ of the equation is bounded and $K(n^0)$~belongs to~$\\Wx{2,\\infty}$ and satisfies $\\partial_\\nu K(n^0)=0$ on~$\\Gamma$\n(recall that $K$ is of class~$C^2$ with bounded derivatives by \\eqref{hpnonlin} and \\eqref{kabounds}).\nTherefore, we can apply \\cite[Thm.~2.1]{DHP} and ensure the existence \nof a unique $w\\in\\spazio W{1,p}{\\Lx p}\\cap\\spazio L p{\\Wx{2,p}}$ \nthat satisfies \\eqref{eqw} and the homogeneous Neumann boundary condition.\nSince $p\\geq2$, \\eqref{regw} holds as well \nand the assumptions of Lemma~\\ref{Unic} with $a=1\/\\ka(n)$ are fulfilled by $w-u$.\nWe conclude that $w=u$, so~that\n\\Begin{equation}\n  u \\in \\spazio W{1,p}{\\Lx p} \\cap \\spazio L p{\\Wx{2,p}}.\n  \\non\n\\End{equation}\nSince $p$ is arbitrary in $[2,+\\infty)$, $\\Omega$ is bounded, \n$n=K^{-1}(u)$ and $K^{-1}$ is of class~$C^2$ \nwith bounded derivatives, we deduce that \\eqref{highreg} holds for~$n$.\n\nNow, we consider the components $s$, $i$ and~$h$.\nWe observe that each of equations \\accorpa{seconda}{quarta} \nand the corresponding boundary and initial conditions have the form\n\\Begin{equation}\n  \\partial_t w - \\div(\\ka(n)\\nabla w) = f\n  \\quad \\hbox{with $\\partial_\\nu w=0$}\n  \\quad\\hbox{and}\\quad\n  w(0) = w_0\n  \\label{eqdiv}\n\\End{equation}\nwhere $f\\in\\LQ\\infty$ and $w_0\\in\\Wx{2,\\infty}$ with $\\partial_\\nu w_0=0$.\nOn account of the already proved regularity of~$n$,\nwe can rewrite the equation in \\eqref{eqdiv} in the non-divergence form, namely\n\\Begin{equation}\n  \\partial_t w - \\bigl( \\ka(n) \\Delta w + \\ka'(n) \\nabla n \\cdot \\nabla w) = f \n  \\label{eqnondiv}\n\\End{equation} \nand apply \\cite[Thm.~2.1]{DHP} once more with $p\\geq2$\n(using the $4p$-summability of $\\nabla n$).\nThen the homogeneous Neumann problem for \\eqref{eqnondiv} \ncomplemented with the initial condition $w(0)=w_0$\nhas a unique solution $w\\in\\spazio W{1,p}{\\Lx p}\\cap\\spazio L p{\\Wx{2,p}}$.\nClearly, such a $w$ belongs to $\\H1H\\cap\\L2W$ and satisfies \\eqref{eqdiv}.\nSince \\eqref{eqdiv} has a unique solution in $\\H1H\\cap\\L2W$,\nwe deduce that $w\\in\\spazio W{1,p}{\\Lx p}\\cap\\spazio L p{\\Wx{2,p}}$\nwhenever $w$ belongs to $\\H1H\\cap\\L2W$ and satisfies \\eqref{eqdiv}.\nThis is the case for $s$, $i$ and~$h$.\nSince $p\\geq2$ is arbitrary and $\\Omega$ is bounded, \\eqref{highreg} is completely proved.\n\\End{proof}\n\n\\step\nConclusion of the proof of Theorem~\\ref{Uniqueness2}\n\nWe pick any two solutions $(n_j,s_j,i_j,h_j)$, $j=1,2$, \nand prove that they are the same.\nThanks to Proposition~\\ref{Highreg}, they satisfy \\eqref{highreg}.\nIn particular, the gradients of all the components belong to $\\L4\\Lx\\infty$\nsince $\\Wx{2,4}\\subset\\Wx{1,\\infty}$.\nFor simplicity, we use the rule of Notation~\\ref{Constants} concerning the constants:\nthe symbol $c$ stands for possible difference constants \nthat depend on the structure, the data, some norms of the solutions we have fixed\nand the constants appearing in \\eqref{kabounds} \n(which are chosen after fixing the solutions) \nand the consequent Lip\\-schitz\\ constants of the nonlinearities.\nAs in the proof of Theorem~\\ref{Uniqueness1}, we set for brevity \n$n:=n_1-n_2$, $s:=s_1-s_2$, $i:=i_1-i_2$ and $h:=h_1-h_2$,\nwrite equations \\accorpa{prima}{quarta} for both solutions and test the differences by\n$n$, $s$, $i$ and~$h$, respectively.\nAs for the first equation, we~have\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\int_\\Omega |n(t)|^2\n  + \\int_{Q_t} \\ka(n_1) |\\nabla n|^2\n  \\non\n  \\\\\n  && = \\int_{Q_t} (\\ka(n_1)-\\ka(n_2)) \\nabla n_2 \\cdot \\nabla n\n  + (\\alpha-\\mu) \\int_{Q_t} |n|^2\n  - \\int_{Q_t} n_1 i n\n  - \\int_{Q_t} i_2 n^2 \\,.\n  \\non\n\\End{eqnarray}\nWhile we can simply use the inequality $\\ka(n_1)\\geq\\ka_*$ on the left-hand side,\nthe true novelty is the first term on the right-hand side.\nWe treat it by owing to the H\\\"older\\ and Young inequalities this~way\n\\Begin{eqnarray}\n  && \\int_{Q_t} (\\ka(n_1)-\\ka(n_2)) \\nabla n_2 \\cdot \\nabla n\n  \\leq c \\int_0^t \\norma{n(t')} \\, \\norma{\\nabla n_2(t')}_\\infty \\, \\norma{\\nabla n(t')} \\, dt'\n  \\non\n  \\\\\n  && \\leq \\ka_* \\int_{Q_t} |\\nabla n|^2\n  + c \\int_0^t \\norma{\\nabla n_2(t')}_\\infty^2 \\, \\norma{n(t')}^2 \\, dt' .\n  \\non \n\\End{eqnarray}\nThe other terms on the right-hand side\\ can be dealt with as in the above proof.\nBy combining and rearranging, we easily deduce that\n\\Begin{equation}\n  \\int_\\Omega |n(t)|^2\n  \\leq c \\int_0^t \\bigl( \\norma{\\nabla n_2(t')}_\\infty^2 +1 \\bigr) \\bigl( \\norma{n(t')}^2 + \\norma{i(t')}^2 \\bigr) \\, dt'\\,.\n  \\label{okprima}\n\\End{equation}\nThe next equation \\eqref{seconda} can be treated in a similar way.\nArguing as we did to derive \\eqref{okprima} and~\\eqref{useconda},\nwe~have\n\\Begin{eqnarray}\n  && \\frac 12 \\, \\int_\\Omega |s(t)|^2\n  + \\int_{Q_t} \\ka(n_1) |\\nabla s|^2\n  + \\mu \\int_{Q_t} |s|^2\n  \\non\n  \\\\\n  && = \\int_{Q_t} (\\ka(n_1)-\\ka(n_2) \\nabla s_2 \\cdot \\nabla s\n  \\non\n  \\\\\n  && \\quad {}\n  + \\alpha \\int_{Q_t} n s\n  - \\int_{Q_t} \\bigl(\n    (A(n_1)-A(n_2)) s_1 i_1\n    + A(n_2) s \\, i_1\n    + A(n_2) s_2 i\n  \\bigr) s\n  \\non\n  \\\\\n  && \\quad{}\n  - \\int_{Q_t} \\bigl(\n    (A(n_1)-A(n_2)) s_1 (s_1-h_1)\n    + A(n_2) s (s_1-h_1)\n    + A(n_2) s_2 (s-h)\n  \\bigr) s\n  \\qquad\n  \\non\n  \\\\\n  && \\leq \\ka_* \\int_{Q_t} |\\nabla s|^2\n  + c \\int_0^t \\bigl( \\norma{\\nabla s_2(t')}_\\infty^2 +1 \\bigr)\n  \\bigl(\n    \\norma{n(t')}^2 + \\norma{s(t')}^2 + \\norma{i(t')}^2 + \\norma{h(t')}^2\n  \\bigr) \\, dt' \\,.\n  \\non\n\\End{eqnarray}\nMoreover, we can use the inequality $\\ka(n_1)\\geq\\ka_*$ on the left-hand side\\ and rearrange also in this case.\nBy treating equations \\accorpa{terza}{quarta} in the same way and summing up,\nwe conclude~that\n\\Begin{eqnarray}\n  && \\int_\\Omega \\bigl( |n(t)|^2 + |s(t)|^2 + |i(t)|^2 + |h(t)|^2 \\bigr)\n  \\non\n  \\\\\n  && \\leq \\int_0^t \\psi(t')\n  \\bigl(\n    \\norma{n(t')}^2 + \\norma{s(t')}^2 + \\norma{i(t')}^2 + \\norma{h(t')}^2\n  \\bigr) \\, dt'\n  \\quad \\hbox{for every $t\\in[0,T]$}\n  \\non\n\\End{eqnarray}\nwith a function $\\psi\\in L^1(0,T)$.\nThus, the Gronwall lemma yields $n=s=i=h=0$,\nand the proof is complete.\n\n\n\\section*{Acknowledgments}\n\\pier{This research was supported by the Italian Ministry of Education, \nUniversity and Research~(MIUR): Dipartimenti di Eccellenza Program (2018--2022) \n-- Dept.~of Mathematics ``F.~Casorati'', University of Pavia. \nIn addition, {PC and ER gratefully mention} some other support \nfrom the MIUR-PRIN Grant 2020F3NCPX \n``Mathematics for industry 4.0 (Math4I4)'' and the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, \nla Probabilit\\`a e le loro Applicazioni) of INdAM (Isti\\-tuto \nNazionale di Alta Matematica), while AR gratefully acknowledges the partial support of the MIUR-PRIN project XFAST-SIMS (no. 20173C478N).}\n\n\n\\vspace{3truemm}\n\n\\Begin{thebibliography}{10}\n\n\t\t\n\\bibitem{Albi} \n\\pcol{G. Albi, L. Pareschi, M. Zanella,\nControl with uncertain data of socially structured compartmental epidemic models,\n{\\it J. Math. Biol.}  {\\bf 82}  (2021) Paper No. 63, 41 pp.}\n\n\\bibitem{Barbu}\nV. Barbu,\n\\gianni{``Nonlinear Differential Equations of Monotone Types in Banach Spaces'',\nSpringer, \nNew York,\n2010.}\n\n\\bibitem{Bellomo}\n\\ale{N. Bellomo, R. Bingham, M.A.J. Chaplain, G. Dosi, G. Forni, D.A. Knopoff, J. Lowengrub, R. Twarock, M.E. Virgillito,\nA multiscale model of virus pandemic: heterogeneous interactive entities in a globally connected world,\n{\\it \\pcol{Math. Models Methods Appl. Sci.}} {\\bf 30} (2020) 1591-1651.}\n\n\\bibitem{Bellomo2}\n\\ale{N. Bellomo, K.J. Painter, Y. Tao, M. Winkler,\nOccurrence vs. absence of taxis-driven instabilities in a May-Nowak model for virus infection,\n{\\it SIAM J. Appl. Math.} \\pcol{{\\bf 79} (2019) 1990-2010.}}\n\n\\bibitem{Bellomo3}\n\\ale{N. Bellomo, N. Outada, J. Soler, Y. Tao, M. Winkler,\nChemotaxis and Cross-diffusion Models in Complex Environments: \nModels and Analytic Problems Towards a Multiscale Vision,\n{\\it \\pcol{Math. Models Methods Appl. Sci.}}, to appear.}\n\n\\bibitem{Bellomo4}\n\\ale{N. Bellomo, F. Brezzi, M.A.J. Chaplain,\nSpecial Issue on ``Mathematics Towards COVID19 and Pandemic'',\n{\\it \\pcol{Math. Models Methods Appl. Sci.}} {\\bf 31} (2021) Issue 12.}\n\n\\bibitem{Berestycki}\n\\ale{H. Berestycki, J.-M. Roquejoffre, L. Rossi,\nPropagation of epidemics along lines with fast diffusion,\n{\\it Bull. Math. Biol.} {\\bf 83} (2021) \\pcol{Paper No. 2, 34 pp.}}\n\n\\bibitem{Bertaglia}\n\\ale{G. Bertaglia, L. Pareschi,\nHyperbolic compartmental models for epidemic spread on networks with uncertain data: application to the emergence of COVID-19 in Italy,\n{\\it \\pcol{Math. Models Methods Appl. Sci.}} {\\bf 31} (2021) 2495-2531.}\n\n\\bibitem{Calleri} \n\\pcol{F. Calleri, G. Nastasi, V. Romano, \nContinuous-time stochastic processes for the spread of COVID-19 disease simulated \nvia a Monte Carlo approach and comparison with deterministic models,\n{\\it J. Math. Biol.}  {\\bf 83}  (2021) Paper No. 34, 26 pp.}\n\n\\bibitem{DHP}\nR. Denk, M. Hieber, J. Pr\\\"uss,\nOptimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data,\n{\\it Math. Z.} {\\bf 257} (2007) 193-224.\n\n\\bibitem{DiB}\nE. DiBenedetto,\n``Degenerate Parabolic Equations'',\nSpringer-Verlag, \nNew York, \n1993.\n\n\\bibitem{Gatto}\n\\ale{M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi, A. Rinaldo,\nSpread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures,\n{\\it Proc. Nat. Acad. Sci.} {\\bf 117} (2020) 10484-10491.}\n\n\\bibitem{Giordano}\n\\ale{G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, M. Colaneri,\nModelling the COVID-19 epidemic and implementation of population-wide interventions in Italy,\n{\\it Nat. Med.} {\\bf 26} (2020) 855-860.}\n\n\\bibitem{Grave}\n\\ale{M. Grave, A.L.G.A. Coutinho,\nAdaptive mesh refinement and coarsening for diffusion-reaction epidemiological models,\n{\\it Comput. Mech.} {\\bf 67} (2021) 1177-1199.}\n\n\\bibitem{Grave2}\n\\ale{M. Grave, A. Viguerie, G.F. Barros, A. Reali, A.L.G.A. Coutinho,\nAssessing the spatio-temporal spread of COVID-19 via compartmental models with diffusion in Italy, USA, and Brazil,\n{\\it Arch. Comput. Mech. Eng.} {\\bf 28} (2021) 4205-4223.}\n\n\\bibitem{Guglielmi}\n\\pcol{N. Guglielmi, E. Iacomini, A. Viguerie,\nDelay differential equations for the spatially resolved simulation of epidemics with specific application to COVID-19\n{\\it Math. Methods Appl. Sci}, to appear, DOI:~10.1002\/mma.8068}\n\n\\bibitem{Jha}\n\\ale{P.K. Jha, L. Cao, J.T. Oden, \nBayesian-based predictions of COVID-19 evolution in Texas using multispecies mixture-theoretic continuum models,\n{\\it Comput. Mech.} {\\bf 66} (2020) 1055-1068.}\n\n\\bibitem{Linka}\n\\ale{K. Linka, P. Rahman, A. Goriely, E. Kuhl,\nIs it safe to lift COVID-19 travel bans? The Newfoundland story,\n{\\it Comput. Mech.} {\\bf 66} (2020) 1081-1092.}\n\n\\bibitem{Lions}\nJ.-L.~Lions,\n``Quelques m\\'ethodes de r\\'esolution des probl\\`emes\naux limites non lin\\'eaires'',\nDunod; Gauthier-Villars, Paris, 1969.\n\n\\bibitem{Parolini}\n\\pcol{N. Parolini, L. Ded\\`e, P.F. Antonietti, G.  Ardenghi, A. Manzoni, E. Miglio,  A. Pugliese, M.  Verani, A. Quarteroni,  \nSUIHTER: a new mathematical model for COVID-19. Application to the analysis of the second epidemic outbreak in Italy,\n{\\it Proc. A.}  {\\bf 477}  (2021), Paper No. 20210027, 21 pp.}\n\n\\bibitem{Vig1}\nA. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli,\nT.J.R. Hughes, A. Patton, A. Reali,\nTh.E. Yankeelov, A. Veneziani,\nSimulating the spread of COVID-19 via a spatially-resolved\nsusceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion,\n\\pier{{\\it Appl. Math. Lett.} {\\bf 111} (2021) Paper No. 106617, 9 pp.}\n\n\\bibitem{Vig2}\nA. Viguerie, A. Veneziani, G. Lorenzo, D. Baroli, N. Aretz-Nellesen,\nA. Patton, T.E. Yankeelov, A. Reali, T.J.R. Hughes, F. Auricchio,\n\\pier{Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study,\n{\\it Comput. Mech.} {\\bf 66} (2020) 1131-1152.}\n\n\\bibitem{Wang}\n\\ale{Z. Wang, X. Zhang, G.H. Teichert, M. Carrasco-Teja, K. Garikipati,\nSystem inference for the spatio-temporal evolution of infectious diseases: Michigan in the time of COVID-19,\n{\\it Comput. Mech.} {\\bf 66} (2020) 1153-1176.}\n\n\\bibitem{Winkler}\n\\ale{M. Winkler,\nBoundedness in a chemotaxis-May-Nowak model for virus dynamics with mildly saturated chemotactic sensitivity\n{\\it Acta Appl. Math.} \\pcol{{\\bf 163} (2019) 1-17.}}\n\n\\bibitem{Zohdi}\n\\ale{T.I. Zohdi,\nAn agent-based computational framework for simulation of global pandemic and social response on planet X,\n{\\it Comput. Mech.} {\\bf 66} (2020) 1195-1209.}\n\n\\End{thebibliography}\n\n\\End{document}\n\n\n\\bibitem{Brezis}\nH. Brezis,\n``Op\\'erateurs maximaux monotones et semi-groupes de contractions\ndans les espaces de Hilbert'',\nNorth-Holland Math. Stud.\n{\\bf 5},\nNorth-Holland,\nAmsterdam,\n1973.\n\n\\bibitem{Simon}\nJ. Simon,\n{Compact sets in the space $L^p(0,T; B)$},\n{\\it Ann. Mat. Pura Appl.~(4)\\\/} \n{\\bf 146} (1987) 65-96.\n\n\n\\Begin{lemma}\n\\label{Unicn}\nAssume that $(n,s,i,h)$ is a solution to problem \\Pbl\\ satisfying \\Regsoluz.\nSet $f:=(\\alpha-\\mu-i)n$\nand consider the problem of finding $w$ such~that\n\\Begin{eqnarray}\n  && w \\in \\H1V^* \\cap \\L2V \\cap \\LQ\\infty\n  \\quad \\hbox{with} \\quad\n  \\inf w > 0 \n  \\label{regw}\n  \\\\\n  && \\< \\partial_t w , v >\n  + \\int_\\Omega \\ka(w) \\nabla w \\cdot \\nabla v\n  = \\int_\\Omega f v\n  \\quad \\hbox{for every $v\\in V$ \\checkmmode{a.e.\\ in~$(0,T)$}}.\n  \\label{eqw}\n\\End{eqnarray}\nThen, $n$ is its unique solution.\n\\End{lemma}\n\n\\Begin{proof}\nSince \\eqref{eqw} with $w=n$ coincides with \\eqref{prima},\n$w=n$ is a solution.\nNow, let $w$ be any solution: we prove that $w=n$.\nWe recall \\eqref{regw}, owe to \\eqref{kabounds}, whose bounds now also depend on~$w$,\nand term $L$ the Lip\\-schitz\\ constant of~$\\ka$.\nLet us introduce $\\sign_\\eps,\\modeps\\cdot:{\\mathbb{R}}\\to{\\mathbb{R}}$ by setting\n\\Begin{eqnarray}\n  && \\sign_\\eps(y) := y\/\\eps \n  \\quad \\hbox{if $|y|\\leq\\eps$}\n  \\quad\\hbox{and}\\quad\n  \\sign_\\eps(y) := y\/|y|\n  \\quad \\hbox{otherwise}\n  \\non\n  \\\\\n  && \\modeps y := \\int_0^y \\sign_\\eps(y') \\, dy'\n  \\quad \\hbox{for every $y\\in{\\mathbb{R}}$}.\n  \\non\n\\End{eqnarray}\nWe set $z:=w-n$ and choose $v=\\sign_\\eps(z)$ in the difference of \\eqref{eqw} \nwritten for $w$ and~$n$.\nBy integrating over $(0,t)$, we have\n\\Begin{equation}\n  \\int_\\Omega \\modeps{z(t)}\n  + \\int_{Q_t} \\ka(w) \\sign_\\eps'(z) |\\nabla z|^2\n  = \\int_{Q_t} (\\ka(n)-\\ka(w)) \\nabla n \\cdot \\nabla z \\, \\sign_\\eps'(z)\n  \\non\n\\End{equation}\nWe estimate the second integral on the left-hand side\\ from blow by owing to~\\eqref{kabounds}\nand observe that $|y|(\\sign_\\eps'(y))^{1\/2}\\leq\\eps^{1\/2}$ for every $y\\in{\\mathbb{R}}$.\nHence, we~have\n\\Begin{eqnarray}\n  && \\int_\\Omega \\modeps{z(t)}\n  + \\ka_* \\int_{Q_t} \\sign_\\eps'(z) |\\nabla z|^2\n  \\non\n  \\\\\n  && \\leq L \\int_{Q_t} |z| (\\sign_\\eps'(z))^{1\/2} |\\nabla n| \\, |\\nabla z| (\\sign_\\eps'(z))^{1\/2}\n  \\non\n  \\\\\n  && \\leq L\\eps^{1\/2} \\int_{Q_t} |\\nabla n| \\, |\\nabla z| (\\sign_\\eps'(z))^{1\/2}\n  \\non\n  \\\\\n  && \\leq \\frac {L\\eps^{1\/2}} 2 \\int_{Q_t} \\sign_\\eps'(z) |\\nabla z|^2\n  + \\frac {L\\eps^{1\/2}} 2 \\int_Q |\\nabla n|^2.\n  \\non\n\\End{eqnarray}\nWe deduce that\n\\Begin{equation}\n  \\int_\\Omega \\modeps{z(t)}\n  + (\\ka_0 - L\\eps^{1\/2}\/2) \\int_{Q_t} \\sign_\\eps'(z) |\\nabla z|^2\n  \\leq \\frac {L\\eps^{1\/2}} 2 \\int_Q |\\nabla z|^2.\n  \\non\n\\End{equation}\nSince the second term on the left-hand side\\ is nonnegative if $\\eps<(2\\ka_*\/L)^2$,\nby letting $\\eps$ tend to zero we obtain $z=0$.\n\\End{proof}\n\n\nWe take any $p>1$ and $m>0$,\nset $v:=\\min\\graffe{u,m}$ and test \\eqref{pblG} by $v^{p-1}$.\nWith $\\delta:=b_0^{1\/p'}$ we obtain\n\\Begin{eqnarray}\n  && b_0 \\norma v_p^p\n  \\leq \\int_\\Omega a \\nabla u \\cdot \\nabla(v^{p-1})\n  + \\int_\\Omega b u v^{p-1}\n  = \\int_\\Omega f v^{p-1}\n  \\non\n  \\\\\n  && \\leq \\norma f_p \\norma{v^{p-1}}_{p'}\n  = \\delta^{-1} \\norma f_p \\, \\delta \\norma v_p^{p-1} \\,.\n  \\non\n\\End{eqnarray}\nBy applying the Young inequality, we deduce that\n\\Begin{equation}\n  b_0 \\norma v_p^p\n  \\leq \\frac 1p \\, \\delta^{-p} \\norma f_p^p\n  + \\frac 1{p'} \\, b_0 \\norma v_p^p \\,,\n  \\quad \\hbox{i.e.,} \\quad\n  b_0 \\norma v_p^p\n  \\leq \\delta^{-p} \\norma f_p^p \\,.\n  \\non\n\\End{equation}\nHence, we have\n\\Begin{equation}\n  b_0^{1\/p} \\norma v_p\n  \\leq \\delta^{-1} \\norma f_p\n  = b_0^{-1\/p'} \\norma f_p \\,.\n  \\non\n\\End{equation}\nBy letting $p$ tend to infinity we infer that $v$ is bounded and that\n\\Begin{equation}\n  \\norma v_\\infty\n  \\leq b_0^{-1} \\norma f_\\infty \\,,\n  \\quad \\hbox{i.e.,} \\quad\n  \\min\\graffe{u,m}\n  \\leq b_0^{-1} \\norma f_\\infty \n  \\quad \\checkmmode{a.e.\\ in~$\\Omega$}\n\\End{equation}\nand we conclude by letting $m$ tend to infinity. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nInterest in the application of loop quantum gravity technique to model\nsystems has significantly increased in the recent years.  Most\ntreatments have dealt with homogeneous space-times (see the \nrecent reviews of Ashtekar and Bojowald\\cite{lqc}). \nMore recently extensions to the exterior\\cite{exterior} and \ninterior\\cite{interior} Schwarzschild space-times\nwere carried out, and also with some limitations to Gowdy \nmodels\\cite{Gowdy}.  In all these studies, as is customary in mini or\nmidi-superspace treatments, gauge fixings are conducted in order to\nexploit the simplifications inherent in the symmetries present in the\nmodels.  On the other hand, in the full theory, one is interested in\ndiffeomorphism invariance. It is therefore of interest to study in the\ncontext of the symmetry reduced models what happens to any remnants of\ndiffeomorphism invariance that are left after the gauge fixing.  In\nthis paper we would like to discuss the issue of diffeomorphism\ninvariance within the context of the treatment of the exterior\nspherically symmetric space-times.  We will show that the remaining\ndiffeomorphism invariance is successfully recovered in the\nsemi-classical limit of the quantum theory but that there are\nlimitations imposed by the used of holonomic variables in the\nquantization. In spite of this, the quantum theory has the same\ndegrees of freedom as if one used metric variables and the \nsolutions of the semiclassical ``polymerized'' theory are uniquely\ndetermined by the mass, as in ordinary general relativity. This \nappears in contrast to treatments of the interior\\cite{interior} \nof the Schwarzschild\nspace-time  and may shed light on the treatment of the\ncomplete space-time where the issue of uniqueness is still not \nsettled\\cite{complete}.\n\n\nThe organization of this paper is as follows. In the next section\nwe briefly review the loop quantum gravity treatment of the exterior\nof the Schwarzschild space-time. In section III we discuss the issue\nof diffeomorphism invariance. We end with a discussion.\n\n\\section{Spherically symmetric space-times in loop quantum gravity}\n\nWe briefly review here the treatment of the exterior in loop quantum\ngravity. More details can be found in our previous paper\\cite{exterior}.\n\n\nOne assumes that the topology of the spatial manifold is of the form\n$\\Sigma=R^+\\times S^2$. We will choose a radial coordinate $x$ and\nstudy the theory in the range $[0,\\infty]$. We will later assume that\nthere is a horizon at $x=0$, with appropriate boundary conditions as\nwe discuss below.\nThe invariant connection can be written \nas,\n\\begin{eqnarray}\n  A &=& A_x(x) \\Lambda_3 dx + \n\\left(A_1(x) \\Lambda_1+ A_2(x) \\Lambda_2\\right)d\\theta\\\\&& +\n\\left(\\left(A_1(x) \\Lambda_2- A_2(x) \\Lambda_1\\right)\\sin \\theta +\n\\Lambda_3 \\cos \\theta\\right) d\\varphi,\\nonumber\n\\end{eqnarray}\nwhere $A_x, A_1$ and $A_2$ are real arbitrary functions on $R^+$,\nthe $\\Lambda_I$ are generators of $su(2)$, for instance $\\Lambda_I = \n-i\\sigma_I\/2$ where $\\sigma_I$ are the Pauli matrices or\nrigid rotations thereof. The invariant triad takes the form,\n\\begin{eqnarray}\n  E &=& E^x(x) \\Lambda_3 \\sin \\theta {\\partial \\over \\partial x} + \n\\left(E^1(x) \\Lambda_1 + E^2(x) \\Lambda_2\\right) \\sin \\theta {\\partial \\over \n\\partial \\theta} \\nonumber\\\\&&\n+\n\\left(E^1(x) \\Lambda_2 - E^2(x) \\Lambda_1\\right) {\\partial \\over \n\\partial \\varphi},\n\\end{eqnarray}\nwhere again, $E^x, E^1$ and $E^2$ are functions on  $R^+$. \n\nAs discussed in our recent paper\\cite{exterior} and \noriginally by Bojowald and Swiderski\\cite{boswi}, \nit is best to make several changes of variables to simplify\nthings and improve asymptotic behaviors. It is also useful to gauge\nfix the diffeomorphism constraint to simplify the model as much as\npossible.  It would be too lengthy and not particularly useful to go\nthrough all the steps here. It suffices to notice that one is left\nwith one pair of canonical variables $E^\\varphi$ and ${A}_\\varphi$ (in\nour recent paper\\cite{exterior} called $\\bar{A}_\\varphi$), and that they are\nrelated to the traditional canonical variables in spherical symmetry\n$ds^2=\\Lambda^2 dx^2+R^2 d\\Omega^2$ by $\\Lambda=E^\\varphi\/(x+a)$ and\n$P_\\Lambda= -(x+a)A_\\varphi\/(2\\gamma)$ where $\\gamma$ is the Immirzi\nparameter and $P_\\Lambda$ is the momentum canonically conjugate to\n$\\Lambda$. The gauge fixing chosen is such that $R=(x+a)$ where $a$\nis, at the moment, a dynamical variable function of $t$. We will also\nchoose $A_\\varphi$ to be independent of $t$. The variable $x$ ranges\nfrom zero to infinity. At zero we will impose isolated horizon\nboundary conditions, i.e. $x=0$ will be the horizon, whereas\n$x=\\infty$ corresponds to $i^0$. An asymptotic analysis of terms at\nspatial infinity shows that $a$ ends up being a constant related to\nthe mass of the space-time. Again we refer the reader to\nour recent paper\\cite{exterior} for details.\n\n\nIn terms of these\nvariables the Hamiltonian constraint reads,\n\\begin{eqnarray}\\label{hamilclass}\n  H&=&-{E^\\varphi \\over (x+a) \\gamma^2}\\left({A^2_\\varphi (x+a)\\over 8}\\right)'\n-{E^\\varphi \\over 2 (x+a)}\\\\&&+ {3 (x+a) \\over 2 E^\\varphi} \n+ (x+a)^2 \\left({1 \\over E^\\varphi}\\right)'=0.\\nonumber\n\\end{eqnarray}\nand since the variables are gauge invariant there is no Gauss law.\nThe Hamiltonian has a\nnon-trivial Poisson bracket with itself, proportional to a Hamiltonian with\nstructure functions. This makes the treatment of the constraint at a \nquantum level problematic since it has the usual ``problem of dynamics''\n(see Giesel and Thiemann\\cite{thiemanngiesel} for a good discussion) . \nTo avoid this in a first approach, it is worthwhile noticing that through a\nsimple rescaling, the Hamiltonian constraint can be made Abelian,\njust multiplying by $\\frac{2(x+a)}{E^\\varphi}$ and grouping terms as\n\\begin{equation}\\label{abelianized}\nH= \\left(\\frac{(x+a)^3}{(E^\\varphi)^2}\\right)'-1 -\\frac{1}{4 \\gamma^2}\n\\left((x+a) A_\\varphi^2\\right)'=0,\n\\end{equation}\nyields and Abelian constraint. Since the constraint is a total\nderivative, it can immediately be integrated to yield,\n\\begin{equation}\n\\int H dx = C =\n\\left(\\frac{(x+a)^3}{(E^\\varphi)^2}\\right)-x -\\frac{1}{4 \\gamma^2}\n\\left((x+a) A_\\varphi^2\\right),\n\\end{equation}\nwith $C$ a constant of integration. Recalling that at $x=0$ the\nisolated horizon boundary conditions imply $1\/E^\\varphi=0$ and\n$A_\\varphi=0$ one gets that the constant of integration $C$ vanishes.\nThis in particular implies that at infinity, $a=2M$, imposing the\nappropriate boundary conditions there, $E^\\varphi=x+3M$, \n$A_\\varphi=0$.\n\nTo promote the constraint to a quantum operator, one needs to discretize\nthe radial direction and then apply techniques at each point akin to those\nof loop quantum cosmology.\nOne wishes to write the discretization in terms of classical quantities\nthat are straightforward to represent in the quantum theory. Here one\nhas to make choices, since there are infinitely many ways of\ndiscretizing a classical expression. In particular, we will notice\nthat there exists, for this model, a way of discretizing the\nconstraint in such a way that it remains first class (more precisely,\nAbelian) upon discretization. This is unusual, and we do not expect\nsuch a behavior in more general models.\n\nWe now proceed to discretize this expression and to ``polymerize'' it,\nthat is, to cast it in terms of quantities that are easily representable\nby holonomies,\n\\begin{eqnarray}\n  H^\\rho_m &=& \\frac{1}{\\epsilon}\\left[\n\\left(\\frac{(x_m+2M)^3 \\epsilon^2}{(E^\\varphi_m)^2}-\n\\frac{(x_{m-1}+2M)^3\\epsilon^2}{(E^\\varphi_{m-1})^2}\\right)\n-\\epsilon \\right.\\nonumber\\\\\n&&-\\frac{1}{4 \\gamma^2 \\rho^2}\n\\left(\n(x_m+2M)\\sin^2\\left(\\rho A_{\\varphi,m}\\right)\\right.\\nonumber\\\\\n&&\\left.\\left.-(x_{m-1}+2M)\\sin^2\\left(\\rho A_{\\varphi,m-1}\\right)\n\\right)\n\\right],\n\\end{eqnarray}\nexpression that recovers (\\ref{abelianized}) in the limit $\\epsilon\\to\n0$, $\\rho \\to 0$. In the above expression $x_m$ are the positions of\nthe lattice points and $\\epsilon$ is the separation of two points in a\nfiducial metric. Although it is not necessary, for simplicity we\nassume $\\epsilon$ is a constant. The parameter $\\rho$ arises in the\n``polymerization'', i.e. in replacing $A_{\\varphi,m}$ by $\\sin(\\rho\nA_{\\varphi,j})\/\\rho$. Whereas the parameter $\\epsilon$ is introduced\njust as a calculational device and can be taken $\\epsilon\\to 0$ in the \nend, the parameter $\\rho$ is expected in loop quantum gravity to have\na fundamental minimum value related to the quantum of area. \nThe above expression is immediately Abelian since it\ncan be written as the difference of two terms, one dependent on the\nvariables at $m$ and the other at $m-1$. Therefore each term has\nautomatically vanishing Poisson brackets with itself and with the\nother.\n\n To implement the constraints as quantum operators as one does in the\n Dirac procedure, it is convenient to solve the constraint for the\n $E^\\varphi_m$,\n\\begin{equation}\\label{convenient}\n  E^\\varphi_m = \\pm \\frac{(x_m+2M)\\epsilon}{\n\\sqrt{1-\\frac{2M}{x_m+2M}+\\frac{1}{4\\gamma^2\\rho^2}\n\\sin^2\\left( \\rho  A_{\\varphi,m}\\right)}},\n\\end{equation}\nand this relation can be immediately implemented as an operatorial\nrelation and find the states that satisfy it. It should be noted that\nthis relation can be implemented for other gauges as well in a\nstraightforward manner. The states are given by,\n\\begin{equation}\\label{109}\n  \\Psi[A_{\\varphi,m},\\tau,M] =C(\\tau,M) \\exp\\left(\n\\pm \\frac{i}{\\ell_{\\rm Planck}^2}\\sum_m\nf[A_{\\varphi,m}]\\right), \n\\end{equation}\nwhere $C(\\tau,M)$ is a function of the variables at the boundary\n$\\tau$ and $M$, which has to solve the constraint at the boundary, as\nwe shall soon see. $\\tau$ is the proper time at infinity, that for\ninstance determines the position of the spatial hypersurfaces of\nvanishing extrinsic curvature (usual Schwarzschild slicings).  The\nfunctional $f$ has the form,\n\\begin{eqnarray}\n&&  f[A_{\\varphi,m}]=\n\\frac{1}{4\\gamma^2\\rho^2\\left(1-\\frac{2M}{x_m+2M}\\right)}(x_m+2M)\n\\nonumber \\epsilon\\\\&&\n\\left[F\\left(\\sin( \\rho A_{\\varphi,m}),\n\\frac{i}{4\\gamma^2\\rho^2\\left(1-\\frac{2M}{x_m+2M}\\right)}\\right)\\right.\\\\\n&& \\left.+ 2 F\\left(1,\n\\frac{i}{4\\gamma^2\\rho^2\\left(1-\\frac{2M}{x_m+2M}\\right)}\\right)\n{\\rm sgn}\\left(\\sin(\\rho A_{\\varphi,m})\\right)\\right] \\nonumber\n\\end{eqnarray}\nwith $F(\\phi,m)\\equiv \\int_0^\\phi (1-m^2\\sin^2 t)^{-1\/2}dt$ the Jacobi\nElliptic function of the first kind. Notice that the continuum limit\nof this expression for the state is immediate, i.e. the sum in $m$ \nbecomes an integral.\n\nWe now need to impose the constraints on the boundary, in particular\n$p^\\tau = -M$ (in the limit $N\\to\\infty$). Quantum mechanically\n$\\hat{p}^\\tau = -i \\ell_{\\rm Planck}^2 {\\partial\/\\partial \\tau}$ and\ntherefore,\n\\begin{equation}\n  C(\\tau, M) = C_0(M) \\exp\\left(-\\frac{i M \\tau}{\\ell_{\\rm Planck}^2}\\right)\n\\end{equation}\nand $C_0(M)$ is an arbitrary function. This is analogous to the\nquantization that Kucha\\v{r} found where one had wavefunctions that\nonly depended on the mass. We have therefore completely solved the\ntheory.  \n\n\n\n\n\\section{Diffeomorphism invariance of the model}\n\n\nWe start by pointing out that the quantization is straightforward,\nsince the only remaining canonical variables are $M$ and $\\tau$. These\nvariables have no dynamics. \nOne can immediately introduce an eigenbasis of the mass operator,\nlabeled by eigenvalues $m$, \n$\\hat{M} \\phi(m)=m \\phi(m)$ and the equations of motion at the boundary\nimply that the $\\phi(m)$ do not evolve. This completes the quantization.\n\nSince we have isolated the true degree of freedom of the model and\nquantized it, there are no remnants left of the diffeomorphism\ninvariance of space-time in any manifest way. To reconstruct\ndiffeomorphism invariance in an explicit form it is useful to\nintroduce evolving constants\\cite{evolving}. For instance, given that\nthe mass of the space-time can be written as a function of the\ncanonical variables $M=M(E^\\varphi,\\hat{A}_\\varphi)$, one can\nconstruct an evolving constant associated with the triad as\n$E^\\varphi_{\\rm Evolv}=E^\\varphi(M,A_\\varphi^{(0)})$ where\n$\\hat{A}_\\varphi^{(0)}$ is a parameter, as given by equation\n(\\ref{convenient}). Explicitly,\n\\begin{equation}\\label{polymerized}\nE^\\varphi_{\\rm Evolv} = \\pm \\frac{(x+2M)}{\n\\sqrt{1-\\frac{2M}{x+2M}+\\frac{1}{4\\gamma^2\\rho^2}\n\\sin^2\\left(\\rho A_{\\varphi}^{(0)}\\right)}}.\n\\end{equation}\n The quantity is such that if\none chooses ${A}_\\varphi^{(0)}={A}_\\varphi$ one recovers the\ndynamical variable $E^\\varphi$. The evolving constant is a Dirac\nobservable of the theory and therefore can be realized as an operator\nacting on the physical space of the theory. Notice that the choice \n$A_{\\varphi}^{(0)}=0$ corresponds to the ordinary form of the Schwarzschild\nmetric in Schwarzschild coordinates.\n\nThe four dimensional metric of the model can be written in terms of \n$E^\\varphi_{\\rm Evolv}$ and the parameter $A_\\varphi^{(0)}$,\nby determining the lapse and shift using the gauge fixing condition and\nsetting to zero the time derivatives of the variables.\nTherefore the components of the four dimensional metric can also be\nviewed as evolving constants. The explicit expressions\nare,\n\\begin{eqnarray}\ng_{00}^{\\rm Evolv}(M,A_\\varphi^{(0)})\n&=&-\\frac{x^2}{(E^\\varphi_{\\rm Evolv})^2}\n+\\frac{\\sin^2(\\rho A_\\varphi^{(0)})}{4\\rho^2\\gamma^2}\\\\\ng_{0x}^{\\rm Evolv}(M,A_\\varphi^{(0)})\n&=&\\frac{E^\\varphi_{\\rm Evolv} \\sin(\\rho A_\\varphi^{(0)})}{2\\rho \\gamma x}\\\\\ng_{xx}^{\\rm Evolv}(M,A_\\varphi^{(0)})\n&=&\\frac{(E^\\varphi_{\\rm Evolv})^2}{x^2}.\n\\end{eqnarray}\nIt is worthwhile pointing out that all of the above expressions are\nreadily promoted to quantum operators acting on the physical Hilbert\nspace simply substituting $M$ by $\\hat{M}$.\n\nThe above results hold in the quantum polymerized theory. It is worthwhile\ncomparing them with the results in classical general relativity. The\nexpressions for the components of the metric in\ntraditional general relativity are,\n\\begin{eqnarray}\ng_{00}&=&-\\frac{x^2}{(E^\\varphi)^2}+\\frac{A_\\varphi^2}{4\\gamma^2}\\\\\ng_{0x}&=&\\frac{E^\\varphi A_\\varphi}{2\\gamma x}\\\\\ng_{xx}&=&\\frac{(E^\\varphi)^2}{x^2}.\n\\end{eqnarray}\n\nIt is instructive to substitute the explicit expression of the triad,\n\\begin{equation}\nE^\\varphi = \\pm \\frac{(x+2M)}{\n\\sqrt{1-\\frac{2M}{x+2M}+\\frac{A^2_{\\varphi}}{4\\gamma^2}}}\n\\end{equation}\nDifferent choices of gauge correspond to different choices of $A_\\varphi$\nand these translate themselves in different coordinate choices for the \nfour-metric. \nThe explicit form of the four dimensional metric therefore is,\n\\begin{eqnarray}\ng_{00}&=& -1+\\frac{2M}{x+2M}\\label{21}\\\\\ng_{0x}&=&\\frac{A_\\varphi}{2\\gamma\\sqrt{1-\\frac{2M}{x+2M}+\\frac{A_\\varphi^2}{4\\gamma^2}}}\\\\\ng_{xx}&=& \\frac{1}{1-\\frac{2M}{x+2M}+\\frac{A_\\varphi^2}{4\\gamma^2}}.\n\\label{23}\n\\end{eqnarray}\n\nSince spatial diffeomorphisms have been gauge fixed the only\ndiffeomorphisms left are space-time ones, which modify the value of\n$g_{xx}$ and $g_{0x}$.  If one starts in a gauge where\n$A_\\varphi=0$ with coordinates $t,x$, one can go to an arbitrary gauge\n$A_\\varphi(x)$ by choosing $x'=x$ and $t'=t-u(x)$ with\n\\begin{equation}\nu(x)= \\int_x^\\infty dx \n\\frac{A_\\varphi(x)}\n{2\\gamma x \\left(1-\\frac{2M}{x+2M}+\\frac{A_\\varphi^2}{4\\gamma^2}\\right)}.\n\\label{change}\n\\end{equation}\n\nLet us now compare with the quantum theory. In this example\nthings are so simple that we could actually talk about the full\nquantum theory itself, it would just correspond to replace the mass by\na quantum operator in the following expressions.  Since the use of a\n``polymerized'' classical theory to capture the semiclassical\nbehaviors of the quantum theory is a technique used in a variety of\ncontexts, we frame the discussion in it. The expression for $g_{00}$\nis unchanged. The expressions for $g_{0x}$ and $g_{xx}$ become,\n\\begin{eqnarray}\ng_{00}&=& -1+\\frac{2M}{x+2M} \\label{25}\\\\\ng_{0x}&=&\\frac{\\sin(\\rho A_\\varphi)}{2\\rho\\gamma\\sqrt{1-\\frac{2M}{x+2M}+\\frac{\\sin\\left(\\rho A_\\varphi\\right)^2}{4\\rho^2\\gamma^2}}}\\\\\ng_{xx}&=& \\frac{1}{1-\\frac{2M}{x+2M}+\\frac{\\sin\\left(\\rho A_\\varphi\\right)^2}{4\\rho^2 \\gamma^2}}.\\label{27}\n\\end{eqnarray}\nWe therefore see that these expressions are particular cases of the\nones we found in the non-polymerized theory, in the sense that for every\nchoice of $A_\\varphi$ for the polymerized theory one can find a choice in\nclassical general relativity that leads to the same metric. The converse,\nhowever, is not true. The gauge transformations\nof the polymerized theory therefore correspond to diffeomorphisms,\njust like in classical general relativity. But not all of the\ndiffeomorphisms available in classical general relativity, at least\nfor finite values of $\\rho$, appear in the polymerized theory.  It is\nclear that if one chooses a small value of $\\rho$ as suggested from\nfull loop quantum gravity, where it is associated with the quantum of\narea which is related to Planck's length, ``most'' diffeomorphisms\nwill be allowed in the polymerized theory, but there will be a subset\nthat is not. The reason for this limitation is that one does not\nexpect diffeomorphism invariance to allow us to ``blow up'' regions\nthat are smaller than Planck size to macroscopic sizes. That would\nimply that somehow we can probe space-time at sub-Planckian\nlengths. This does not appear reasonable. We can see that this is\nprecisely what happens in this example. Which diffeomorphisms are\nbeing excluded?  Comparing (\\ref{21}-\\ref{23}) to (\\ref{25}-\\ref{27})\nwe see that when $A_\\varphi$ is large, this corresponds to $g_{xx}\\to\n0$ and $g_{0x}\\to 1$ in classical general relativity, whereas in the\npolymerized theory $g_{xx}$ reaches a minimum value. Let us recall\nthat we are talking about space-time diffeomorphisms here, that is,\nchanges of the space-time foliation (the radial coordinate is\nfixed). That means one is excluding foliations where radial distances\nbecome very small, i.e. foliations of large values of the extrinsic\ncurvature.\n\n\n\n\n\\section{Discussion}\n\nThe quantization of the exterior Schwarzschild space-time in loop\nquantum gravity can be carried out completely and it isolates the same\ntrue degrees of freedom as the quantization carried out by Kucha\\v{r}\nin terms of the traditional variables. One has that the only degree of\nfreedom is the mass of the space-time. Wavefunctions are functions of\nthe mass that do not evolve. In spite of this similarity, if one tries\nto reconstruct space-time diffeomorphisms in terms of evolving\nconstants that are Dirac observables on the physical Hilbert space,\none notices effects due to the ``polymerization'' introduced by the\nloop variables. In particular one notes that only a subset of\nspace-time diffeomorphisms get implemented. This corresponds physically\nto the fact that one cannot probe distances of sub-Planckian nature.\nThis provides a simple example in a controlled situation \nof behaviors that are widely believed\nto hold in the full theory.\n\nIt should be noted that in this paper we have taken the parameter\n$\\rho$ in the polymerized theory to be a constant. Current treatments\nin cosmology suggest that an improved dynamics may be achieved with\n$\\rho$ that depends on the dynamical variables\\cite{aspasi}. The\ndetails of the conclusions about the permissible diffeomorphisms will\nchange if one makes such a choice, though we expect the generic\nfeatures to remain the same.\n\nThe issue of how diffeomorphisms get implemented in the polymerized\ntheory becomes quite relevant when one considers the full Kruskal-like\nextension of the Schwarzschild space-time\\cite{complete}. There, at\nthe moment, there exists knowledge of a family of solutions. It is not\nclear if this family is unique or even if different members of the\nfamily correspond to different space-times. This raises the issue of\nthe existence of a Birkhoff theorem in loop quantum gravity. In the\nexterior case we have shown that the only quantum solutions can be\nsuperpositions of space-times with different mass, in spite of the\nfact that one does not implement in the semi-classical theory all the\ndiffeomorphisms present in the classical theory. In the interior case\nit is known that the solutions of the ``polymerized'' semi-classical\ntheory may depend on an extra parameter in addition to the mass.  The\nquestion is still open if an analysis of diffeomorphism symmetry like\nwe carried out in this paper can yield a Birkhoff-like theorem in the\ncase of the complete space-time.\n\n\n\\section{Acknowledgements}\n\nWe wish to thank Abhay Ashtekar for discussions and comments on the\nmanuscript.  This work was supported in part by grant NSF-PHY-0650715,\nfunds of the Hearne Institute for Theoretical Physics, FQXi, CCT-LSU\nand Pedeciba (Uruguay).\n\n \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe textbook paradigm of quantum error correction (QEC) focuses on the case of \\emph{perfect} error correction \\cite{EM96, BDSW96}, where the code $\\mathcal{C}$ and the noise are such that there exists a recovery operation that completely removes the effects of the noise on the information stored in the code.\nMathematically, this idea is captured by a set of conditions for perfect error correction~\\cite{KnillLaflamme} that must be satisfied by the code as well as the noise process.\n\nThat such perfect QEC conditions can be satisfied tends to be special rather than generic.\nThe prototypical example is that of independent noise acting on a few physical qubits, and one finds codes that satisfy the perfect QEC conditions assuming that no more than $t$ of the qubits have errors.\nIn such a scenario, what is taken as the noise process $\\mathcal{E}$ in the QEC conditions is the part of the noise that describes $t$ or fewer errors, while the full physical noise process $\\mathcal{E}_0$ contains terms describing more than $t$ errors, albeit with a lower probability of occurrence.\nA code that satisfies the QEC conditions for $\\mathcal{E}$ will thus only satisfy the conditions \\emph{approximately} for the full noise process $\\mathcal{E}_0$.\nFurthermore, in practice, it is unrealistic to expect complete characterization of the noise process.\nThus, a code designed to satisfy the perfect QEC conditions for the \\emph{expected} noise process will typically only satisfy those conditions approximately for the true noise process.\nThis motivates the idea of \\emph{approximate} quantum error correction (AQEC), where the recovery operation removes most, but not necessarily all, the effects of noise on the information stored in the code.\n\nRecent studies on AQEC, using analytical \\cite{Leung,BK,Tyson,BenyOreshkov1,aqecPRA,Renes} and numerical~\\cite{Yamamoto,Reimpell,Kosut08,Fletcher07b} approaches, have discovered examples of approximate codes that allow for recovery of stored information with fidelity comparable to that of perfect QEC codes, while making use of fewer physical resources. These results suggest that the requirement for perfect recovery may be too stringent for certain tasks and approximate QEC may be more natural and practical.\n\nIn~\\cite{aqecPRA}, we demonstrated a universal, near-optimal recovery map---the \\emph{transpose channel}~\\cite{BK,petzbook}---for AQEC codes with a subspace structure, wherein information is stored in an entire subspace of the Hilbert space of the physical quantum system. Optimality was defined in terms of the worst-case recovery fidelity over all states in the code. Our analytical approach was a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality based on entanglement fidelity~\\cite{Reimpell,FletcherThesis,Kosut08}. We obtained quantitative bounds showing the efficacy of the transpose channel as a universal and analytical recovery operation that works well, regardless of the noise process or the code used.\nThis allowed for complete characterization of approximate subspace codes, in terms of necessary and sufficient conditions for approximate correctability, and provided an easy route for constructing approximate subspace codes.\n\nIn this article, we extend our approach based on the transpose channel to the more general case of AQEC codes with a \\emph{subsystem} structure, where information is stored only in a subsystem of the code subspace.\nA subsystem code (sometimes referred to as an \\emph{operator} QEC code) has a bipartite tensor-product structure, where one subsystem $A$ (the \\emph{correctable subsystem}) is correctable under action of the noise, while the other subsystem $B$ (the \\emph{noisy subsystem}) can be disturbed by the noise beyond repair~\\cite{OQECC, OQEClong, NielsenPoulin07}.\nThe information to be protected against noise is stored only in subsystem $A$.\nSubspace codes can be viewed as special cases of subsystem codes with a trivial noisy subsystem.\nWhile this generalization does not lead to new families of codes, the alternate perspective does sometimes lead to more efficient decoding procedures~\\cite{Poulin05stabilizer,bacon06oqec}, and hence to better fault-tolerant schemes and improved bounds on the accuracy threshold~\\cite{aliferis07}.\nStarting with the Bacon-Shor codes~\\cite{bacon06oqec}---a family of subsystem codes arising from Shor's 9-qubit code---several examples of perfectly correctable stabilizer subsystem codes have been constructed~\\cite{aly06subsystemcodes}.\nHere, we address the general question of characterizing approximate subsystem codes, and explore the extent to which the transpose channel is useful towards understanding approximate codes.\n\nAfter a preliminary section setting down basic definitions and notations, we begin by proving a set of perfect QEC conditions (Section \\ref{sec:perfectOQEC}) that is completely equivalent to the standard well-known QEC conditions. This alternate set of conditions clarifies the role of the transpose channel in perfect QEC. Furthermore, it serves as a natural starting point for perturbation to a set of sufficient conditions for approximate QEC (Section~\\ref{sec:sufficiency}). We then proceed, in Section \\ref{sec:necessary}, to show the near-optimality of the transpose channel recovery map for AQEC for  four classes of codes and noise processes. These four classes provide evidence towards our conjecture that the transpose channel is near-optimal for arbitrary subsystem codes and noise, which, if true, would establish a simple, analytical, and universal framework for the study of approximate codes. We conclude with a few suggestions for future work.\n\n\n\\section{Basic definitions}\\label{sec:definition}\n\nWe consider a decomposition of the Hilbert space of our quantum system,\n\\begin{equation}\\label{eq:decompH}\n\\mathcal{H}=\\mathcal{H}_A\\otimes \\mathcal{H}_B+\\mathcal{K}.\n\\end{equation}\nSuppose we wish to store information in the $\\mathcal{H}_A$ factor.\n$\\mathcal{H}_{AB}\\equiv\\mathcal{H}_A\\otimes \\mathcal{H}_B$ is the Hilbert space of a composite system comprising two subsystems $A$ and $B$ of dimensions $d_A$ and $d_B$ respectively.\nWe denote the projector onto $\\mathcal{H}_{AB}$ as $P$.\n$P$ can also be written as a tensor-product: $P=P_A\\otimes P_B$, where $P_{A(B)}$ is the projector onto $\\mathcal{H}_{A(B)}$.\nIn principle, subsystems $A$ and $B$ may only correspond to mathematical tensor-product factors in the decomposition, rather than ``natural\" separate physical degrees of freedom of the quantum system.\nIn practice, one might prefer to work with $A$ and $B$ that are natural degrees of freedom for easy experimental accessibility.\nAlso, it is often helpful to use a decomposition of $\\mathcal{H}$ that is not arbitrarily invented by the experimenter, but induced by the structure of the noise afflicting the quantum system, so as to identify a subsystem that best ensures survival of the stored information.\n\nInformation is stored as a choice between states of subsystem $A$.\nThe state on subsystem $B$ can be arbitrary and carries no information. More concretely, we make use of a \\emph{code} $\\mathcal{C}$, comprising all product states on $AB$,\n\\begin{equation}\\label{eq:code}\n\\mathcal{C}\\equiv\\{\\rho=\\rho_A\\otimes\\rho_B,~\\forall\\rho_A\\in\\mathcal{S}(\\mathcal{H}_A),\\rho_B\\in\\mathcal{S}(\\mathcal{H}_B)\\},\n\\end{equation}\nwhere $\\mathcal{S}(\\mathcal{H}_{A(B)})$ denotes the set of all states (density operators) on subsystem $A(B)$.\nThe information is stored only in subsystem $A$ in that two states $\\rho_A\\otimes \\tau_B$ and $\\rho_A\\otimes \\sigma_B$ differing only in the state of $B$ correspond to the same encoded information.\n\nWe wish to examine the longevity of the information stored in subsystem $A$ in the presence of noise.\nWe describe the noise by a quantum channel acting on $AB$, that is, a completely positive (CP), trace-preserving (TP) map $\\mathcal{E}:\\mathcal{B}(\\mathcal{H}_{AB})\\longrightarrow \\mathcal{B}(\\mathcal{P}_\\mathcal{E})$.\nHere, $\\mathcal{B}(\\mathcal{V})$ refers to the set of all bounded operators on a vector space $\\mathcal{V}$. $\\mathcal{P}_\\mathcal{E}$ is the support of $\\mathcal{E}(\\mathcal{B}(\\mathcal{H}_{AB}))$, or equivalently, the support of $\\mathcal{E}(P)$.\n$\\mathcal{E}$ can be specified by a set of Kraus operators $\\{E_i\\}_{i=1}^N$, so that $\\mathcal{E}$ acts as\n\\begin{equation}\n\\mathcal{E}(\\rho)=\\sum_{i=1}^NE_i\\rho E_i^\\dagger.\n\\end{equation}\nThat $\\mathcal{E}$ is TP translates into the statement $\\sum_{i=1}^NE_i^\\dagger E_i=P$.\nThe Kraus representation of a CPTP channel is non-unique: If $\\{E_i\\}$ is a Kraus representation of $\\mathcal{E}$, then $\\{F_j\\equiv \\sum_i u_{ij}E_i\\}$, for unitary $(u_{ij})$, is a Kraus representation of the same channel.\nA recovery operation $\\mathcal{R}:\\mathcal{B}(\\mathcal{P}_\\mathcal{E})\\longrightarrow \\mathcal{B}(\\mathcal{H}_{AB})$ performed\nafter each application of the noise $\\mathcal{E}$, to attempt to reverse the effects of the noise, is also described as a CPTP map.\n\nSince information is stored in subsystem $A$ only, we are concerned only with how well the noise preserves the information initially stored in $A$, while any state on $B$ can be distorted beyond repair by the noise.\nHeuristically, we say that a code $\\mathcal{C}$ is \\emph{approximately correctable} under noise $\\mathcal{E}$ if and only if there exists a CPTP recovery map $\\mathcal{R}$ such that\n\\begin{equation}\n\\text{tr}_B[(\\mathcal{R}\\circ\\mathcal{E})(\\rho)]\\simeq\\text{tr}_B(\\rho)\\quad\\forall\\rho\\in\\mathcal{C},\n\\end{equation}\nwhere $\\text{tr}_B(\\cdot)$ denotes the partial trace over subsystem $B$.\n\nThis heuristic notion is formalized by quantifying the deviation of the recovered state from the initial state in terms of the fidelity between the two states.\nThe fidelity between two states $\\rho$ and $\\sigma$ is $F(\\rho,\\sigma)\\equiv \\text{tr}\\sqrt{\\rho^{1\/2}\\sigma\\rho^{1\/2}}$. For the case of $\\rho$ being a pure state $\\psi\\equiv |\\psi\\rangle\\langle\\psi|$, $F$ can be written as\n\\begin{equation}\nF(|\\psi\\rangle,\\sigma)\\equiv \\sqrt{\\langle \\psi|\\sigma|\\psi\\rangle}.\n\\end{equation}\nWe define the \\emph{fidelity loss for state $\\rho$}, $\\eta_\\mathcal{R}\\{\\rho\\}$, under noise $\\mathcal{E}$ and recovery $\\mathcal{R}$, as the deviation from 1 of the square of the fidelity between the initial state and the recovered state, that is,\n\\begin{equation}\n\\eta_\\mathcal{R}\\{\\rho\\}\\equiv 1-F^2{\\big(\\text{tr}_B(\\rho),\\text{tr}_B[(\\mathcal{R}\\circ\\mathcal{E})(\\rho)]\\big)}.\n\\end{equation}\nThe performance of a recovery $\\mathcal{R}$ on a code $\\mathcal{C}$ is then characterized by the \\emph{fidelity loss for $\\mathcal{C}$} defined as\n\\begin{equation}\\label{eq:fidelityloss}\n\\eta_\\mathcal{R}\\{\\mathcal{C}\\}\\equiv \\max_{\\rho\\in\\mathcal{C}}\\eta_\\mathcal{R}\\{\\rho\\}.\n\\end{equation}\nHow well $\\mathcal{R}$ recovers the information initially stored in subsystem $A$ is hence gauged by the \\emph{worst-case fidelity} (over all states in the code) between the initial and recovered states. Because the fidelity is jointly concave in its arguments, the worst-case fidelity is always attained on a pure state on $AB$. The maximization in Eq.~\\eqref{eq:fidelityloss} can thus be restricted to pure states on $AB$ only. Often, when the meaning is clear from the context, we will drop the argument from $\\eta_\\mathcal{R}\\{\\mathcal{C}\\}$ and simply write $\\eta_\\mathcal{R}$.\n\nLet $\\cR_\\text{op}$ be the recovery map with the smallest fidelity loss among all possible recovery maps for code $\\mathcal{C}$, that is,\n\\begin{equation}\n\\eta_\\text{op}\\{\\mathcal{C}\\}\\equiv\\eta_{\\cR_\\text{op}}\\{\\mathcal{C}\\}=\\min_\\mathcal{R}\\eta_\\mathcal{R}\\{\\mathcal{C}\\}.\n\\end{equation}\nWe refer to $\\cR_\\text{op}$ as the \\emph{optimal recovery}, and $\\eta_\\text{op}$ as the \\emph{optimal fidelity loss}.\nAs is clear from the notation, whether or not a recovery map is optimal for a given noise process depends on the code in question.\n\nA code $\\mathcal{C}$ with $\\eta_\\text{op}=0$ under noise $\\mathcal{E}$ is said to be \\emph{perfectly correctable} on $A$ under $\\mathcal{E}$.\nIn general, we say that a code is \\emph{$\\epsilon$-correctable} on $A$ under noise $\\mathcal{E}$ if $\\epsilon\\geq\\eta_\\text{op}$, which means that it is possible to recover the information stored in $A$ with a fidelity no smaller than $\\sqrt{1-\\epsilon}$.\nApproximate \\emph{noiseless subsystems} are included within this framework by considering codes for which the identity map (the ``do nothing\" operation) is sufficient as an approximate recovery for the code.\n\nCentral to our analysis is a recovery map built from the noise channel and code, known as the \\emph{transpose channel} (see previous uses of this channel in Refs.~\\cite{petz2003,haydenpetz,BK,IPSPaper,IPSPaper2}).\nThe transpose channel corresponding to a channel $\\mathcal{E}$ and code $\\mathcal{C}$, denoted as $\\cR_P$, is defined in a manifestly representation-invariant way as\n\\begin{equation}\n\\cR_P\\equiv \\mathcal{P}_\\mathcal{C}\\circ\\mathcal{E}^\\dagger\\circ\\mathcal{N}.\n\\end{equation}\nHere, $\\mathcal{E}^\\dagger$ is the adjoint of $\\mathcal{E}$, that is, the channel with Kraus operators $\\{E_i^\\dagger\\}_{i=1}^N$ if $\\mathcal{E}$ has Kraus operators $\\{E_i\\}_{i=1}^N$. $\\mathcal{N}$ is a normalization map $\\mathcal{N}(\\cdot)\\equiv \\mathcal{E}(P)^{-1\/2}(\\cdot)\\mathcal{E}(P)^{-1\/2}$ (the inverse is taken on the support of $\\mathcal{E}(P)$). $\\mathcal{P}_\\mathcal{C}$ is the projection onto the support of $\\mathcal{C}$, $\\mathcal{P}_\\mathcal{C}(\\cdot)=P(\\cdot)P$. One can write $\\cR_P$ explicitly in terms of its Kraus operators $\\{R_i^P\\}_{i=1}^N$, where\n\\begin{equation}\nR_{i}^{P} \\equiv PE_{i}^\\dagger \\mathcal{E}(P)^{-1\/2}.\n\\end{equation}\n$\\cR_P$ is trace-preserving (TP) on $\\mathcal{P}_\\mathcal{E}$. The fidelity loss obtained using the transpose channel as the recovery is denoted by $\\eta_P$.\n\n\n\\section{Approximate QEC: sufficient conditions}\\label{sec:AQECSuff}\n\n\\subsection{Perfect QEC conditions}\\label{sec:perfectOQEC}\n\nWe begin with the case of perfect error correction, where, there exists a recovery map such that the fidelity of any state on $A$ after noise and recovery attains the maximal value of 1. Necessary and sufficient algebraic conditions for the existence of a perfectly correctable code for a given channel $\\mathcal{E}$ are expressed by the following theorem:\n\\begin{theorem}\\label{thm:equivcond}\nConsider a CPTP noise process $\\mathcal{E}$ with Kraus representation $\\{E_i\\}$ acting on $AB$, and a code $C$ on $\\mathcal{H}_{AB}$ as defined in Eq.~\\eqref{eq:code}. $\\mathcal{C}$ is perfectly correctable on $A$ under $\\mathcal{E}$ if and only if\n\\begin{equation}\\label{cond1}\nPE_i^\\dagger \\mathcal{E}(P)^{-1\/2}E_jP=P_A\\otimes B_{ij},\n\\end{equation}\nfor all $i,j$, and $B_{ij}\\in\\mathcal{B}(\\mathcal{H}_B)$.\n\\end{theorem}\n\\noindent The special case of Theorem~\\ref{thm:equivcond} for subspace codes appeared in \\cite{aqecPRA}, and the proof of this generalization (provided in Appendix \\ref{app:perfectCond}) follows a similar logic.\n\nAlgebraic conditions for perfect error correction for subsystem codes were originally discovered in \\cite{NielsenPoulin07, OQECC, OQEClong}, generalizing the well-known perfect QEC conditions for subspace codes \\cite{KnillLaflamme}.\nCompared to the original QEC conditions, our conditions given above differ only in the appearance of the $\\mathcal{E}(P)^{-1\/2}$ factor on the left-side of Eq.~\\eqref{cond1}. However, this alternate form of the conditions offers better intuition on the correctability of codes.\nObserve that the expression on the left-side of Eq.~\\eqref{cond1} is a Kraus operator $R_i^PE_j$ of the channel $\\cR_P\\circ\\mathcal{E}$, from which we can immediately conclude that $\\text{tr}_B\\{(\\cR_P\\circ\\mathcal{E})(\\rho)\\}=\\text{tr}_B(\\rho)$ for any $\\rho\\in\\mathcal{C}$, as is required for perfect correctability on $A$.\nTheorem \\ref{thm:equivcond} can thus be viewed as demonstrating correctability of codes by explicitly giving the recovery map---the transpose channel $\\cR_P$---needed to perfectly recover the state on subsystem $A$ after the action of the channel $\\mathcal{E}$.\n\n\n\n\\subsection{Sufficient AQEC conditions}\\label{sec:sufficiency}\n\nThe form of the QEC conditions given in Eq.~\\eqref{cond1} is particularly well-suited for perturbation to approximate QEC, as was previously pointed out for the special case of subspace codes in \\cite{aqecPRA}.\nTheorem \\ref{thm:equivcond} states that $\\cR_P\\circ\\mathcal{E}$ acts as the identity channel on subsystem $A$.\nPerturbing Eq.~\\eqref{cond1}, by adding to the right-side a small correction to $P_A\\otimes B_{ij}$, modifies this to the statement that $\\cR_P\\circ\\mathcal{E}$ acts \\emph{nearly} as the identity channel on subsystem $A$.\nThis provides a natural route to sufficient conditions for approximate subsystem codes:\nIf the perturbation to Eq.~\\eqref{cond1} is small enough, the code is $\\epsilon$-correctable on $A$ with small $\\epsilon$. What remains is to relate quantitatively the size of the perturbation to $\\epsilon$, which is the content of the following theorem:\n\n\\begin{theorem}\\label{thm:suff}\nConsider a CPTP noise channel $\\mathcal{E}$ with Kraus representation $\\{E_{i}\\}$ and a code $\\mathcal{C}$ on $\\mathcal{H}_{AB}$ as defined in Eq.~\\eqref{eq:code}. Suppose\n\\begin{equation}\\label{eq:AOQECcond}\nPE_{i}^\\dagger \\mathcal{E}(P)^{-1\/2} E_{j}P = P_{A}\\otimes B_{ij} + \\Delta_{ij},\n\\end{equation}\nfor all $i,j$, $B_{ij} \\in\\mathcal{B}(\\mathcal{H}_B)$, and $\\Delta_{ij}\\in\\mathcal{B}(\\mathcal{H}_{AB})$. Then, $\\mathcal{C}$ is $\\epsilon$-correctable on $A$ under $\\mathcal{E}$ for $\\epsilon\\geq\\eta_P$, where\n\\begin{align}\\label{eq:etaP}\n\\eta_P&\\equiv \\max_{|\\psi_A,\\phi_B\\rangle}\\langle \\phi_B|\\sum_{ij}{\\left[\\langle\\psi_A|\\Delta_{ij}^\\dagger\\Delta_{ij}|\\psi_A\\rangle\\right.}\\\\\n&\\hspace{2.65cm}{\\left.-\\langle\\psi_A|\\Delta_{ij}^\\dagger|\\psi_A\\rangle\\langle\\psi_A|\\Delta_{ij}|\\psi_A\\rangle\\right]}|\\phi_B\\rangle.\\nonumber\n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nThe TP condition on $\\cR_P\\circ\\mathcal{E}$ gives the relation $P=\\sum_{ij}[P_A\\otimes B_{ij}^\\dagger B_{ij}+\\Delta_{ij}^\\dagger\\Delta_{ij}+(P_A\\otimes B_{ij}^\\dagger)\\Delta_{ij}+\\Delta_{ij}^\\dagger(P_A\\otimes B_{ij})]$. Using this, direct computation gives\n\\begin{align}\n&\\quad F^2{\\left[|\\psi_A\\rangle,(\\text{tr}_B\\circ\\cR_P\\circ\\mathcal{E})(|\\psi_A,\\phi_B\\rangle\\langle\\psi_A,\\phi_B|)\\right]}\\\\\n&=1-\\langle\\psi_A,\\phi_B|\\sum_{ij}\\Delta_{ij}^\\dagger{\\left(P_A-|\\psi_A\\rangle\\langle\\psi_A|\\right)}\\otimes P_B\\Delta_{ij}|\\psi_A,\\phi_B\\rangle.\\nonumber\n\\end{align}\nThis yields the expression for $\\eta_P$ in Eq.~\\eqref{eq:etaP} upon recalling that the worst-case fidelity is attained on a pure state on $AB$.\n\\end{proof}\n\nWhile the bound $\\epsilon\\geq \\eta_P$ in Theorem~\\ref{thm:suff} is tight, the maximization over all pure product states on $AB$ in the expression for $\\eta_P$ may not be easy to evaluate.\nInstead, we can relax the bound and obtain a simpler (but weaker) sufficient condition,\n\\begin{corollary}\\label{cor:suff}\n$\\mathcal{C}$ is $\\epsilon$-correctable on $A$ under $\\mathcal{E}$ if\n\\begin{equation}\n\\epsilon\\geq \\Big\\Vert\\sum_{ij}\\Delta_{ij}^\\dagger\\Delta_{ij}\\Big\\Vert,\n\\end{equation}\nwhere $\\Vert\\cdot\\Vert$ is the operator norm.\n\\end{corollary}\n\n\\begin{proof}\nObserve that, for any pure product state $|\\psi_A,\\phi_B\\rangle$, the expression in Eq.~\\eqref{eq:etaP} to be maximized is bounded from above by $\\langle\\psi_A,\\phi_B|\\sum_{ij}\\Delta_{ij}^\\dagger\\Delta_{ij}|\\psi_A,\\phi_B\\rangle\\leq \\big\\Vert\\sum_{ij}\\Delta_{ij}^\\dagger\\Delta_{ij}\\big\\Vert$. This gives $\\eta_P\\leq \\big\\Vert\\sum_{ij}\\Delta_{ij}^\\dagger\\Delta_{ij}\\big\\Vert$, which immediately yields the corollary statement.\n\\end{proof}\n\\noindent Corollary \\ref{cor:suff} gives an easily checkable sufficient condition, which may be more useful than Theorem \\ref{thm:suff} in the search for approximate subsystem codes.\n\n\n\n\\section{Towards necessary AQEC conditions}\\label{sec:necessary}\n\nThe previous section discusses sufficient conditions for the existence of approximate QEC codes.\nThe next natural question to ask is: what about necessary conditions?\nFor the special case of subspace codes, we obtained necessary conditions by deriving a near-optimality bound for the transpose channel recovery \\cite{aqecPRA}.\nThe near-optimality result led to the conclusion that every approximately correctable subspace code must also be well-corrected by the transpose channel.\nThis relation to the transpose channel gave rise to necessary conditions for the existence of approximate subspace codes of a form similar to the sufficient conditions.\n\nExtending the near-optimality bound to arbitrary subsystem codes proved difficult.\nNevertheless, as is described in this section, we can show near-optimality of the transpose channel for restricted classes of subsystem codes and noise processes.\nMore specifically, we consider four scenarios: (A) subspace codes, with trivial subsystem $B$ (a review of results from \\cite{aqecPRA}); (B) code states with the maximally mixed state on $B$;  (C) subsystem $B$ is perfectly correctable; and (D) the noise $\\mathcal{E}$ destroys information on $B$.\nIn each scenario, the transpose channel works nearly as well as the optimal recovery operation, which leads to necessary conditions on the noise process as well as the code, \\emph{provided} the restrictions are satisfied\n\nIn the broader picture of arbitrary subsystem codes and noise processes, we believe that the transpose channel still works well whenever the code is approximately correctable.\nAfter all, in the case of perfect QEC, the transpose channel is \\emph{the} recovery operation for perfect recovery.\nHowever, the general near-optimality of the transpose channel is only a conjecture at this point.\nHere, we seek only to provide evidence towards the conjecture, and leave the proof (or disproof) to future work.\n\n\n\\subsection{Trivial subsystem $B$: subspace codes}\\label{sec:subspace}\n\nIn \\cite{aqecPRA}, the transpose channel was shown to be near-optimal for subspace codes $\\mathcal{C}$, that is, its fidelity loss for code $\\mathcal{C}$ under noise $\\mathcal{E}$ is close to the optimal fidelity loss.\nFor completeness, we repeat here the quantitative statement of the near-optimality of the transpose channel, adapted to the language suited for this paper:\n\\begin{theorem}[Corollary 4 of \\cite{aqecPRA}]\\label{thm:subspace}\nConsider a subspace code $\\mathcal{C}$ ($B$ is trivial), with $d_A$ denoting the dimension of $\\mathcal{H}_A$, and optimal fidelity loss $\\eta_\\text{op}$ under CPTP noise channel $\\mathcal{E}$. The fidelity loss $\\eta_P$ for the transpose channel satisfies\n\\begin{equation}\\label{eq:ineqEta}\n\\eta_\\text{op}\\leq \\eta_P\\leq\\eta_\\text{op} f(\\eta_\\text{op};d_A),\n\\end{equation}\nwhere $f(\\eta;d)$ is the function\n\\begin{equation}\\label{eq:f}\nf(\\eta;d)\\equiv \\frac{(d+1)-\\eta}{1+(d-1)\\eta}=(d+1)+O(\\eta).\n\\end{equation}\n\\end{theorem}\n\\noindent The left inequality $\\eta_\\text{op}\\leq \\eta_P$ of Eq.~\\eqref{eq:ineqEta} is true simply by definition of $\\eta_\\text{op}$. The proof of the right inequality $ \\eta_P\\leq\\eta_\\text{op} f(\\eta_\\text{op};d_A)$ requires the following inequality (derived in \\cite{aqecPRA}) which holds for any pure state $\\psi_A\\equiv |\\psi_A\\rangle\\langle\\psi_A|$ in a subspace code $\\mathcal{C}$,\n\\begin{equation}\\label{eq:ineqEta2}\n1-\\eta_\\text{op}\\{\\psi_A\\}\\leq \\sqrt{[1+(d_A-1)\\eta_\\text{op}\\{\\mathcal{C}\\}][1-\\eta_P\\{\\psi_A\\}]}.\n\\end{equation}\nInverting Eq.~\\eqref{eq:ineqEta2} and recalling the definitions of $\\eta_\\text{op}$ and $\\eta_P$ as the maximization of $\\eta_{(\\cdot)}\\{\\psi_A\\}$ over all states in the code yields the right inequality of Eq.~\\eqref{eq:ineqEta}.\n\nEquation~\\eqref{eq:ineqEta} implies that an approximately correctable subspace code must necessarily be such that the fidelity loss for the transpose channel is small.\nA small fidelity loss for the transpose channel in turn requires that $\\mathcal{E}$ has Kraus operators that satisfy Eq.~\\eqref{eq:AOQECcond} with $\\Delta_{ij}$ small.\nEquation.~\\eqref{eq:AOQECcond} with $\\Delta_{ij}$ small is thus not only sufficient (as shown in Sec.~\\ref{sec:sufficiency}), but also necessary for subspace codes.\n\nAn obvious extension of the current case to subsystem codes is one where $\\mathcal{E}$ is a product channel, that is, $\\mathcal{E}(\\rho_A\\otimes\\rho_B)=\\mathcal{F}_A(\\rho_A)\\otimes \\mathcal{F}_B(\\rho_B)$, for CPTP (on their respective domains) channels $\\mathcal{F}_A$ and $\\mathcal{F}_B$.\nFor such an $\\mathcal{E}$, the transpose channel is also a product channel, namely, the product of the respective transpose channels of $\\mathcal{F}_A$ and $\\mathcal{F}_B$.\nSince there is no flow of information between $A$ and $B$, whether subsystem $A$ is correctable relies only on the properties of $\\mathcal{F}_A$.\nWe can thus treat this case as if we have a subspace code on $A$ under noise $\\mathcal{F}_A$, for which the transpose channel is immediately near-optimal from Theorem~\\ref{thm:subspace}.\n\n\n\\subsection{Maximally mixed state on subsystem $B$}\\label{sec:maxmixed}\n\nConsider the subset of code states where $B$ is in the maximally mixed state,\n\\begin{equation}\\label{eq:C0}\n\\mathcal{C}_0\\equiv\\left\\{\\rho_A\\otimes \\frac{P_B}{d_B},\\quad\\forall\\rho_A\\in\\mathcal{S}(\\mathcal{H}_A)\\right\\}\\subset\\mathcal{C}.\n\\end{equation}\nFor states in $\\mathcal{C}_0$, the action of the noise channel $\\mathcal{E}$ can be written as\n\\begin{equation}\n\\mathcal{E}{\\left(\\rho_A\\otimes \\frac{P_B}{d_B}\\right)}=\\sum_{is}\\bar E_{is}\\rho_A \\bar E_{is}^\\dagger\\equiv \\bar\\mathcal{E}_A(\\rho_A).\n\\end{equation}\n$\\bar\\mathcal{E}_A$ is a CPTP channel on $A$ with Kraus operators $\\{\\bar E_{is}\\equiv (1\/\\sqrt{d_B})E_i|s_B\\rangle\\}$, where $\\{|s_B\\rangle\\}_{s=1}^{d_B}$ is an orthonormal basis for $\\mathcal{H}_B$.\nLet us, for a moment, forget about subsystem $B$ and ask about correctability of $\\mathcal{C}_0$---now viewed as a subspace code on $A$---under the noise $\\bar\\mathcal{E}_A$.\nTheorem~\\ref{thm:subspace} applies and ensures that the transpose channel corresponding to noise $\\bar\\mathcal{E}_A$ and code $\\mathcal{C}_{0}$, denoted as $\\mathcal{R}_{A,P}$, has fidelity loss close to that of the optimal recovery $\\mathcal{R}_{A,\\textrm{op}}$,\n\\begin{equation}\\label{eq:maxmixed}\n\\eta_P\\{\\mathcal{C}_0\\}\\leq \\eta_\\text{op}\\{\\mathcal{C}_0\\} f(\\eta_\\text{op}\\{\\mathcal{C}_0\\};d_A),\n\\end{equation}\nfor $f(\\eta;d)$ defined in Eq.~\\eqref{eq:f}.\n\nSuch a code $\\mathcal{C}_0$ is of practical relevance whenever one lacks control over subsystem $B$. Full control over subsystem $A$ alone is sufficient to guarantee preparation of a product code state, while rapid and complete decoherence (for example) causes the state on $B$ to quickly approach a random state well-described by the maximally mixed state. Equation \\eqref{eq:maxmixed} reassures us that, in this case, the transpose channel still works well as a recovery map.\n\nWe are, however, more interested in the performance of the transpose channel on the original subsystem code $\\mathcal{C}$ where the state on $B$ is unrestricted. After all, the freedom to choose the state of $B$, without incurring adverse effects on the information-carrying capability of the code, is the essence of a subsystem code.\nObserve that, for any state in $\\mathcal{C}_0$, the actions of $(\\text{tr}_B\\circ\\cR_P\\circ\\mathcal{E})$ (as usual, $\\cR_P$ is the transpose channel for $\\mathcal{E}$ on code $\\mathcal{C}$) and $(\\mathcal{R}_{A,P}\\circ\\bar\\mathcal{E}_A)$ are identical.\nThe optimal recovery for $\\mathcal{C}_0$, however, need not be the same map as the optimal recovery for $\\mathcal{C}$---the optimal recovery for $\\mathcal{C}$ has to work well for \\emph{all} states in $\\mathcal{C}$, not just those in $\\mathcal{C}_0$.\nHowever, since $\\mathcal{C}_0\\subset\\mathcal{C}$, we have the following inequality,\n\\begin{equation}\\label{eq:C0C}\n\\eta_\\text{op}\\{\\mathcal{C}_0\\}\\leq \\eta_\\text{op}\\{\\mathcal{C}\\}.\n\\end{equation}\nFurthermore, since $\\eta f(\\eta;d)$ is a monotonically increasing function of $\\eta$, we can combine \\eqref{eq:maxmixed} and \\eqref{eq:C0C} to obtain the following corollary,\n\\begin{corollary}\\label{cor:maxmixed}\nFor the subset of code states $\\mathcal{C}_0 \\subset \\mathcal{C}$,\n\\begin{equation}\n\\eta_P\\{\\mathcal{C}_0\\}\\leq \\eta_\\text{op}\\{\\mathcal{C}\\}f(\\eta_\\text{op}\\{\\mathcal{C}\\};d_A).\\label{eq:maxmixed2}\n\\end{equation}\n\\end{corollary}\n\\noindent This says that for the code states where $B$ is in the maximally mixed state---which can be viewed as the ``average state\" for the full degree of freedom described by $B$---the transpose channel works nearly as well as the optimal recovery operation for $\\mathcal{C}$.\n\nAs an aside, we note that a ``state-dependent\" transpose channel is near-optimal for codes where $B$ is always prepared in some \\emph{known} state.\nCode $\\mathcal{C}_0$ is a special case of this, but now $B$ can be in a fixed state other than the maximally mixed state.\nFor example, the rapid decoherence process of subsystem $B$ may have a fixed point that is not the maximally mixed state (for instance, the ground state, if the noise is dissipative), so that any initial preparation of the $B$ state quickly relaxes into this fixed state.\nSuch codes should properly be viewed as (isomorphic to) subspace codes on subsystem $A$.\nSince the identity of the state on $B$ is known, the optimal recovery map for this code must make use of this knowledge.\nLikewise, the associated transpose channel, in order to work well, must also depend on the fixed state on $B$.\nUsing similar techniques as above, one can show\n\\begin{equation}\n\\eta_{P_{\\phi_B}}\\{\\mathcal{C}_{\\phi_B}\\}\\leq \\eta_\\text{op}\\{\\mathcal{C}_{\\phi_B}\\} f(\\eta_\\text{op}\\{\\mathcal{C}_{\\phi_B}\\};d_A),\n\\end{equation}\nwhere $\\mathcal{C}_{\\phi_B}$ is the set of code states with $\\phi_B$ as the fixed state on $B$, and $\\eta_{P_{\\phi_B}}$ refers to the fidelity loss of the state-dependent transpose channel with Kraus operators $ \\{ (P_A \\otimes \\sqrt{\\phi_{B}})E_{i}^\\dagger \\left[\\mathcal{E}(P_A\\otimes\\phi_B)\\right]^{-1\/2}\\}$. This is similar to previously known results from Ref.~\\cite{BK}, derived in the context of entanglement fidelity for reversing dynamics on a given input state.\n\n\n\\subsection{$B$ is perfectly correctable}\\label{sec:Bcorr}\nSuppose subsystem $B$ is in fact perfectly correctable, but we choose to use subsystem $A$ to store the information.\nA simple and often-encountered example is where the noise on $B$ is describable by Kraus operators that are products of Pauli operators.\nMore generally, any noise process that satisfies the perfect QEC conditions for $B$ falls under our current considerations.\nDespite the perfect correctability on $B$, one might still choose to store information in $A$, for example, when $B$ is experimentally inaccessible or uncontrollable, or if $A$ is a much larger system with greater storage capacity than $B$.\nThe transpose channel is again near-optimal in this case.\n\nWe demonstrate this near-optimality by first showing that, for $B$ perfectly correctable, the fidelity for a pure initial state on subsystem $A$ from using the transpose channel as recovery is independent of the initial state of subsystem $B$.\n\\begin{lemma}\\label{lem}\nIf subsystem $B$ is perfectly correctable under noise $\\mathcal{E}$, then $F{\\left[|\\psi\\rangle_A,(\\text{tr}_B\\circ\\cR_P\\circ\\mathcal{E})(\\psi_A\\otimes\\rho_B)\\right]}$, where $\\psi_A\\equiv |\\psi_A\\rangle\\langle\\psi_A|$, is independent of $\\rho_B$.\n\\end{lemma}\n\\begin{proof}\n$B$ perfectly correctable under noise $\\mathcal{E}$ implies that the perfect QEC conditions (Eq.~\\eqref{cond1} of Theorem~\\ref{thm:equivcond} with the roles of $A$ and $B$ interchanged) hold: There exists operators $A_{ij}$ on $A$ for all $i,j$ such that $PE_i^\\dagger \\mathcal{E}(P)^{-1\/2}E_j P=A_{ij}\\otimes P_B$. From this, we have $F^2\\left[|\\psi_A\\rangle,\\left(\\text{tr}_B\\circ\\cR_P\\circ\\mathcal{E}\\right)(\\psi_A\\otimes\\rho_B)\\right]=\\sum_{ij}|\\langle\\psi_A|A_{ij}|\\psi_A\\rangle|^2$,\nwhich is independent of $\\rho_B$.\n\\end{proof}\n\nLemma~\\ref{lem} implies the following sequence of relations:\n\\begin{align}\n\\eta_P\\{\\mathcal{C}\\}=\\max_{\\rho\\in\\mathcal{C}}\\eta_P\\{\\rho\\}&=\\max_{\\rho=\\psi_A\\otimes\\rho_B}\\eta_P\\{\\rho\\}\\nonumber\\\\\n&=\\max_{\\psi_A}\\eta_P\\{\\psi_A\\otimes P_B\/d_B\\}\\nonumber\\\\\n&=\\eta_P\\{\\mathcal{C}_0\\}\\nonumber\\\\\n&\\leq \\eta_\\text{op}\\{\\mathcal{C}\\}f(\\eta_\\text{op}\\{\\mathcal{C}\\};d_A).\\label{eq:Bperf}\n\\end{align}\nThe second equality in the first line of Eq.~\\eqref{eq:Bperf} follows from the concavity of the fidelity, with $\\psi_A$ denoting a pure state. The second line makes use of Lemma~\\ref{lem}, and the last inequality is just Eq.~\\eqref{eq:maxmixed2}.\nEquation~\\eqref{eq:Bperf} gives exactly the right inequality in Eq.~\\eqref{eq:ineqEta} applied to the current scenario, from which we draw the conclusion that the transpose channel is near-optimal on $A$ under channel $\\mathcal{E}$ when $B$ is perfectly correctable.\n\n\\subsection{$\\mathcal{E}$ destroys distinguishability on $B$}\\label{sec:arbitrarystate}\n Suppose the noise process $\\mathcal{E}$ satisfies the following condition:\n \\begin{condition}\\label{condTr}\nFor CPTP $\\mathcal{E}$, suppose there exists $\\delta\\geq 0$ such that\n\\begin{equation}\n\\left\\Vert \\mathcal{E}(\\rho_A\\otimes \\rho_B)-\\mathcal{E}{\\left(\\rho_A\\otimes \\frac{P_B}{d_B}\\right)}\\right\\Vert_\\text{tr}\\leq \\delta{\\left\\Vert\\rho_B-\\frac{P_B}{d_B}\\right\\Vert}_\\text{tr}\n\\end{equation}\nfor all states $\\rho_A \\in \\mathcal{S}(\\mathcal{H}_A)$ and $\\rho_B \\in \\mathcal{S}(\\mathcal{H}_B)$. $\\Vert O\\Vert_\\text{tr}$ denotes the trace norm of $O$ given by $\\text{tr}|O|$.\n\\end{condition}\n\\noindent If $\\delta\\ll 1$, any two states on $\\mathcal{H}_B$, after the action of $\\mathcal{E}$, become close together and nearly indistinguishable (as quantified by the trace norm used in Condition~\\ref{condTr}) from each other.\nA simple example is a product channel $\\mathcal{E}=\\mathcal{E}_A\\otimes \\mathcal{E}_B$ where $\\mathcal{E}_B$ maps all states on $B$  to some fixed state $\\tau_B$. In this case, $\\delta$ can be chosen to be zero.\nWhile we have chosen, for convenience of subsequent analysis, to state Condition~\\ref{condTr} in terms of comparing states on $B$ before and after the channel $\\mathcal{E}$ to what happens to the maximally mixed state $P_B\/d_B$, one is free to choose other reference states on $B$ if desired.\n\nFor channels and codes satisfying Condition~\\ref{condTr}, the transpose channel also works well as a recovery channel, as encapsulated in the following corollary:\n\\begin{corollary}\\label{thm:AbStateFidelity}\nGiven that Condition~\\ref{condTr} is satisfied, for a subsystem code $\\mathcal{C}$,\n\\begin{align}\n\\eta_P&\\leq (d_A+1)\\eta_\\text{op}+3\\delta+O(\\delta^2,\\eta_\\text{op}^2,\\eta_\\text{op}\\delta).\n\\end{align}\n\\end{corollary}\n\\noindent The proof of this corollary is detailed in Appendix~\\ref{app}. The idea behind the proof is to first show that the transpose channel works well as a recovery for the information stored in $A$ when $B$ is initially in the maximally mixed state. Since Condition~\\ref{condTr} says that $\\mathcal{E}$ brings code states with different states on $B$ close together, if the transpose channel works well as a recovery for $B$ being initially in the maximally mixed state, it will also work well when $B$ is initially in a different state.\n\nCorollary~\\ref{thm:AbStateFidelity}, like similar statements before, tells us that the fidelity loss for the transpose channel is not much worse than that of the optimal recovery. The additional fidelity loss suffered from using the simpler transpose channel rather than the optimal recovery is governed by $d_A$ as well as the parameter $\\delta$ which characterizes how badly $\\mathcal{E}$ destroys distinguishability between states on subsystem $B$.\n\n\\section{Conclusion}\\label{sec:concl}\n\nWe studied the role of the transpose channel in approximate quantum error correction.\nWe first obtained a set of conditions for perfect subsystem error correction that explicitly involves the transpose channel. This completes our understanding as to why certain channels admit perfectly correctable codes, in a particularly intuitive way. Our perfect QEC conditions naturally lead to sufficient conditions for approximate QEC, where the resilience to noise of the information stored in the code is quantified in a simple way. We also demonstrated that the transpose channel works nearly as well as any other recovery channel for four different scenarios of codes and noise. In all these cases, the near-optimality of the transpose channel relies only on $d_A$, the dimension of the information-carrying subsystem $A$, and not on $d_B$, the dimension of the noisy subsystem that carries no information.\n\nUsing our transpose channel approach to derive necessary conditions for approximate QEC for general subsystem codes will provide the final missing link in our unifying and analytical framework for understanding approximate quantum error correction.\nEven disproving our conjecture that the transpose channel is a universally good recovery operation for approximate codes will be a useful step forward.\nIn this case, then, the question will be to discover a different recovery map that can serve as a universal recovery.\n\nAnother possible extension is to consider codes that include not just product states on $AB$ in $\\mathcal{C}$ (as we have done), but also correlated states.\nOnce there is correlation between $A$ and $B$, it is, of course, no longer clear where the information initially resides.\nIf one has complete control over the preparation of the initial code states, it would be simpler to make use of the subsystem structure and confine the information to only one subsystem.\nPractically, however, experimental restrictions may result in an initial (possibly small) correlation between $A$ and $B$, leading to a different notion of ``approximate\" or imperfection in the code.\nSuch a situation was previously studied in \\cite{Shabani05} in the context of perfectly noiseless subsystems that require no careful initialization.\nOne can ask similar questions for approximate subsystem codes.\n\nA separate future direction is to perform the transpose channel recovery on experimental implementations of approximate codes. The transpose channel, like any CPTP map, can be implemented physically using operations on an extended Hilbert space. The more pertinent and fruitful question, however, will be to discover simple and efficient ways of implementing the transpose channel on a specific physical system of our choice.\n\n\\section{Acknowledgments}\nP.M. would like to thank David Poulin, John Preskill and Todd Brun for useful discussions. H.K.N is supported by the National Research Foundation and the Ministry of Education, Singapore.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter*{Certificate}\n\\vspace{-33pt}\n\\par This is to certify that the thesis titled \\textit{An investigation on the nonclassical\nand quantum phase properties of a family of engineered quantum states}, submitted by \\textit{Priya Malpani (P14EN001)} to the Indian Institute of Technology Jodhpur for the award of the degree of \\textit{Doctor of Philosophy}, is a bonafide record of the research work done by her under my supervision. To the best of my knowledge, the contents of this report, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma.\n\\vspace{22pt}\n\\begin{center}\n\\flushright \n\\normalfont\\sffamily\\itshape{Dr. V. Narayanan} \\\\\nPh.D.Thesis Supervisor\n\\end{center}\n\n\\begin{center}\n\\chapter*{Acronyms}\n\\end{center}\n\n\\vspace{-33pt}\n~\n\\begin{center}\n\t~%\n\t\\begin{tabular}{ll}\n\t\tBS & Binomial state\\tabularnewline\n\t\tDFS & Displaced Fock state\\tabularnewline\n\t\tECS & Even coherent state\\tabularnewline\n\t\tHOA & Higher-order antibunching\\tabularnewline\n\t\tHOSPS & Higher-order sub-Poissionian photon statistics\\tabularnewline\n\t\tHOS & Higher-order squeezing\\tabularnewline\n\t\tKS & Kerr state\\tabularnewline\n\t\tLE & Linear entropy\\tabularnewline\n\t\tPADFS & Photon added displaced Fock state\\tabularnewline\n\t\tPSDFS & Photon subtracted displaced Fock state\\tabularnewline\n\t\tPABS & Photon added binomial state\\tabularnewline\n\t\tPAECS & Photon added even coherent state\\tabularnewline\n\t\tPAKS & Photon added Kerr state\\tabularnewline\n\t\tPASDFS & Photon added then subtracted displaced Fock state\\tabularnewline\n\t\tVFBS & Vacuum filtered binomial state \\tabularnewline\n\t\tVFECS & Vacuum filtered even coherent state\\tabularnewline\n\t\tVFKS & Vacuum filtered Kerr state\\tabularnewline\n\t\\end{tabular}\n\t\\par\\end{center}\n\n\n\n\\chapter*{Declaration}\n\\vspace{-33pt}\n\\par I hereby declare that the work presented in this thesis entitled \\textit{An investigation on the nonclassical\nand quantum phase properties of a family of engineered quantum states} submitted to the Indian Institute of Technology Jodhpur in partial fulfillment of the requirements for the award of the degree of Doctor of Philosophy, is a bonafide record of the research work carried out under the supervision of Dr. V. Narayanan. The contents of this thesis in full or in parts, have not been submitted to, and will not be submitted by me to, any other Institute or University in India or abroad for the award of any degree or diploma.\n\\vspace{22pt}\n\\begin{center}\n\\flushright \n\\normalfont\\sffamily\\itshape{Priya Malpani} \\\\\nP14EN001\n\\end{center}\n\n\\chapter*{Acknowledgment}\n\\addcontentsline{toc}{chapter}{Acknowledgment}\n\t\n\\small\nIt is my great pleasure to express my gratitude to the person (Prof.\nAnirban Pathak) without whom my Ph.D. journey was impossible. He has\nnot only guided me in my thesis but also inspired my life with his\nvaluable suggestions and magical words \\textquotedblleft Tension mat\nlo..sab ho jayega..\\textquotedblright ., I was in the dark with no\nhope then Pathak Sir was candle of hope. Thank you for returning to\nme faith in myself. Nothing better than these words can express my\nfeelings \\textquotedblleft koi mujhko yun mila hai..jaise banjaare\nko ghar..\\textquotedblright .\n\nSpecial Thanks to my supervisor Dr. V. Narayanan. Thank you sir for\nall the support and being one of the best human being in research\nfield. I am grateful to Dr. Subhashish Banerjee for his helpful discussion\nand motivation. I am thankful to ample number of people involved in\nmy research activity. I am deeply indebtied to Dr. Kishore Thapliyal\n(Google Scholar- apko jitna bhi thanku bolu bahut kam hi hoga!! Thankyou\nSir for ignoring my silly mistakes, your continuous support, motivation\nand everlasting inspiration) and Dr. Nasir Alam (Thank you Sir for\nall valuable help), who had always supported me and I feel blessed\nhaving seniors like you both. I acknowledge my colleagues, Sanjoy\nChatterjee (Dada, the best lab-mate), Javid Naikoo (Best Advisor \\dots without whom the journey of  Ph.D. was next to impossible), Khushboo Dixit,\nVandana Dahiya (Cutest friend-Jatni\\dots yaara teri yari ko maine\nto khuda mana.. thanku for all the support, without your help survival\nin IITJ was not possible), Shilpa Pandey (Best roommate), Mitali Sisodia\n(thank you for listening my endless drama), Swarn Rajpoot (naam hi\nkafi hai), Satish Sangwan (Tau- the best buddy), Vishwadeepak Kumar\n(Rockstar), Ashwin Saxena, and other research scholars, teaching and\nnon-teaching staff in IITJ department, who had become part of my life\nduring this period of time. Now, I would like to thank everyone who\nhad played a prominent role in my life. Special thanks to my parents\nfor their unconditional love and support, without your support it\nwas not possible for me to reach this level. I would like to thank\nmy grandpa (Baba). I will thank my bua (Sheelam Bua) for her continuous\nmoral support. A big thanks to my lovely Sadi, Vaibhav (no words can\nexpress my thank you to you bro. Thank you for handling all my moods\nwith extreme patience), Meghai (Friendster), Doll (my soul sister),\nLabbu (blessed to have you in my life bro), Yashivangi (strong pillers\nof my life) and a big thanks to someone who has completely changed\nmy way of thinking. I would also like to acknowledge the financial\nhelps I received in different phases of my research work from IITJ,\nJIIT and MHRD without which this work would not have been possible. \n\\small\n\t\n\t\\vspace{22pt}\n\t\\begin{center}\n\t\t\\flushright \n\t\t\\normalfont\\sffamily\\itshape{Priya Malpani.}\n\t\\end{center}\n\n\\chapter*{Abstract}\nThe main focus of this thesis is to study the nonclassical\nand phase properties of a family of engineered quantum states, most\nof which show various nonclassical features. The beauty of these states\nis that these states can be used to establish quantum supremacy. Earlier,\na considerable amount of works has been reported on various types\nof quantum states and their nonclassical properties. Here, complementing\nthe earlier works, the effect of non-Gaussianity inducing operators\non the nonclassical and phase properties of displaced Fock states\nhave been studied. This thesis includes 6 chapters. In Chapter \\ref{cha:Introduction1},\nmotivation behind performing the present work is stated explicitly,\nalso the basic concepts of quantum optics are discussed with a specific\nattention on the witnesses and measures of nonclassicality. In Chapter\n\\ref{cha:PADFS-PSDFS}, nonclassical properties of photon added and\nsubtracted displaced Fock states have been studied using various witnesses\nof lower- and higher-order nonclassicality which are introduced in\nChapter 1. In Chapter \\ref{cha:phase}, we have continued our investigation\non photon added and subtracted displaced Fock states (and their limiting\ncases). In this chapter, quantum phase properties of these states\nare investigated from a number of perspectives, and it is shown that\nthe quantum phase properties are dependent on the quantum state engineering\noperations performed. In Chapter \\ref{cha:PASDFS}, we have continued\nour investigation on the impact of non-Gaussianity inducing operators\non the nonclassical and phase properties of the displaced Fock states.\nIn Chapter \\ref{cha:QSE-1}, we have performed a comparison between to process\nthat are used in quantum state engineering to induce nonclassical\nfeatures. Finally, this thesis is concluded in Chapter \\ref{cha:Conclusions-and-Scope},\nwhere we have summarized the findings of this thesis and have also\ndescribed scope of the future works.\n\\addcontentsline{toc}{chapter}{Abstract}\n\n\n\t\\vspace{500cm}\n\t\\begin{center}\n\t\\bfseries{\\itshape{}}\n\t\\end{center}\n\t{\\bfseries{\\itshape{Dedicated to my parents for their unconditional love and support.}}}\n\t\\tableofcontents\n\t\\listoffigures\n\t\\listoftables\n\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t  \n\t\n\t     \n\t\n\n\\newpage\n\n\\mainmatter\n\n\\pagenumbering{arabic}\\setcounter{page}{1}\n\n\\chapter{Introduction\\textsc{\\label{cha:Introduction1}}}\n\n\n\n\\section{Introduction}\n\nAs the title of the thesis suggests, in this thesis, we aim to study\nthe nonclassical and phase properties of a family of engineered quantum\nstates. Before, we introduce such states and properties, it would\nbe apt to lucidly introduce the notion of nonclassical and engineered\nquantum states. By nonclassical state we refer to a quantum state\nhaving no classical analogue. Such states are characterized by the\nnegative values of Glauber-Sudarshan $P$-function or\n$P$-function more singular than Dirac delta function \\cite{sudarshan1963equivalence,glauber1963coherent}\nand witnessed by various operational criteria (to be described in\nSection \\ref{subsec:Witnesses-of-nonclassicality}). To visualize\nthe relevance of nonclassical states we may first note that quantum\nsupremacy refers to the ability of performing a task using quantum\nresources in such a manner that either the task itself cannot be performed\nusing classical resources or the speed\/efficiency achieved using quantum\nresources cannot be achieved in the classical world \\cite{grover1997quantum}.\nA recent experiment performed by Google aimed at estabilishing quantum\nsupremacy has drawn much of public attention \\cite{courtland2017google}.\nThe relevance of the present study lies in the fact that to establish\nquantum supremacy or to perform a fundamental test of quantum mechanics,\nwe would require a state having some features that would not be present\nin any classical state. As we have already mentioned, such a state\nhaving no classical analogue is referred to as the nonclassical state.\nFrequently used examples of nonclassical states include squeezed,\nantibunched, entangled, steered, and Bell nonlocal states. The relevance\nof the states having nonclassical features has already been established\nin the various domains of physics. For example, we may mention, teleportation\nof coherent states \\cite{furusawa1998unconditional}, continuous variable\nquantum cryptography \\cite{hillery2000quantum}, quantum radar \\cite{lanzagorta2011quantum},\nand many more. Further, we may note that the art of generating and\nmanipulating quantum states as per need is referred to as the ``quantum\nstate engineering'' \\cite{dakna1998quantum,sperling2014quantum,vogel1993quantum,miranowicz2004dissipation,dell2006multiphoton}.\nParticularly interesting examples of such engineered quantum states\nare Fock state, photon added\/subtracted coherent state \\cite{agarwal1991nonclassical},\ndisplaced Fock state (DFS) which is also referred to as generalized\ncoherent state and displaced number state \\cite{satyanarayana1985generalized,wunsche1991displaced,ziesel2013experimental,zavatta2004quantum,de1990properties,malpani2019lower},\nphoton added DFS (PADFS) \\cite{malpani2019lower}, and photon subtracted\nDFS (PSDFS) \\cite{malpani2019lower}. In what follows, we will state\nthe relevance of such engineered quantum states in the implementation\nof different tasks exploiting their nonclassical and phase properties.  The relatively new area of research on quantum state engineering\nhas drawn much attention of the scientific community because of its\nsuccess in experimentally producing various quantum states \\cite{zavatta2004quantum,torres2003preparation,rauschenbeutel2000step,gao2010experimental,lu2007experimental}\nhaving nonclassical properties and applications in realizing quantum\ninformation processing tasks, like quantum key distribution \\cite{bennett1984quantum}\nand quantum teleportation \\cite{brassard1998teleportation,chen2015bidirectional}.\nEngineered quantum states, such as cat states, Fock state and superposition\nof Fock states, are known to play a crucial role in performing fundamental\ntests of quantum mechanics and in establishing quantum supremacy in\nthe context of quantum computation and communication (\\cite{kues2017chip}\nand references therein).\n\nAs mentioned in the previous paragraph, with the advent of quantum\nstate engineering \\cite{vogel1993quantum,sperling2014quantum,miranowicz2004dissipation,marchiolli2004engineering}\nand quantum information processing (\\cite{pathak2013elements} and\nreferences therein), the study of nonclassical properties of engineered\nquantum states have become a very important field. This is so because\nonly the presence of nonclassical features in a quantum state can\nprovide quantum supremacy \\cite{grover1997quantum}. In the recent past, various techniques for quantum state engineering\nhave been developed \\cite{vogel1993quantum,sperling2014quantum,miranowicz2004dissipation,agarwal1991nonclassical,lee2010quantum,marchiolli2004engineering}.\nIf we restrict ourselves to optics, these techniques are primarily\nbased on the clever use of beam splitters, detectors, and measurements\nwith post selection, etc. Such techniques are useful in creating holes\nin the photon number distribution \\cite{escher2004controlled} and\nin generating finite dimensional quantum states \\cite{miranowicz2004dissipation},\nboth of which are nonclassical \\cite{pathak2018classical}. The above\nsaid techniques are also useful in realizing non-Gaussianity inducing\noperations, like photon addition and subtraction \\cite{zavatta2004quantum,podoshvedov2014extraction}.\nMotivated by the above, in this thesis, we aim to study the nonclassical\nproperties of a set of engineered quantum states such as photon added,\nphoton subtracted, and photon added then subtracted displaced Fock\nstates which can be produced by using the above mentioned techniques.\nIn the present thesis, we also wish to investigate the phase properties\nof the above mentioned engineered quantum states for the reasons explained\nbelow.\n\nThe impossibility of writing a Hermitian operator for quantum phase\nis a longstanding problem (see \\cite{perinova1998phase,carruthers1968phase,lynch1987phase}\nfor review). Early efforts of Dirac \\cite{dirac1927quantum} to introduce\na Hermitian quantum phase operator were not successful, but led to\nmany interesting proposals \\cite{susskind1964quantum,pegg1989phase,barnett1986phase}.\nSpecifically, Susskind-Glogower \\cite{susskind1964quantum}, Pegg-Barnett\n\\cite{pegg1988unitary,pegg1989phase,barnett1990quantum}, and Barnett-Pegg\n\\cite{barnett1986phase} formalisms played very important role in\nthe studies of phase properties and the phase fluctuation \\cite{imry1971relevance}.\nThereafter, phase properties of various quantum states have been reported\nusing these formalisms \\cite{sanders1986bc,gerry1987phase,yao1987phase,carruthers1968phase,vaccaro1989phase,pathak2000phase,alam2016quantum,alam2017quantum,verma2009reduction}.\nOther approaches have also been used for the study of the phase properties.\nFor example, quantum phase distribution is defined using phase states\n\\cite{agarwal1992classical}, while Wigner \\cite{garraway1992quantum}\nand $Q$ \\cite{leonhardt1993phase,leonhardt1995canonical} phase distributions\nare obtained by integrating over radial parameter of the corresponding\nquasidistribution function. In experiments, the phase measurement\nis performed by averaging the field amplitudes of the $Q$ function\n\\cite{noh1991measurement,noh1992operational}; Pegg-Barnett and Wigner\nphase distributions are also reported with the help of reconstructed\ndensity matrix \\cite{smithey1993complete}. Further, quantum phase\ndistribution under the effect of the environment was also studied\nin the past leading to phase diffusion \\cite{banerjee2007phase,banerjee2007phaseQND,abdel2010anabiosis,banerjee2018open}.\nA measure of phase fluctuation named phase dispersion using quantum\nphase distribution has also been proposed in the past \\cite{perinova1998phase,banerjee2007phase}.\nRecently, quantum phase fluctuation \\cite{zheng1992fluctuation} and\nPancharatnam phase \\cite{mendas1993pancharatnam} have been studied\nfor DFS. The quantum phase fluctuation in parametric down-conversion\n\\cite{gantsog1991quantum} and its revival \\cite{gantsog1992collapses}\nare also reported. Experiments on phase super-resolution without using\nentanglement \\cite{resch2007time} and role of photon subtraction\nin concentration of phase information \\cite{usuga2010noise} are also\nperformed. Optimal phase estimation \\cite{sanders1995optimal} using\ndifferent quantum states \\cite{higgins2007entanglement} (including\nNOON and other entangled states and unentangled single-photon states)\nhas long been the focus of quantum metrology \\cite{giovannetti2006quantum,giovannetti2011advances}. Nonclassicality measure based on the shortening of the regular distribution\ndefined on phase difference interval broadbands due to nonclassicality\nis also proposed in the recent past \\cite{perina2019quasidistribution,thapliyal2019quasidistribution}.\nIn brief, quantum phase properties are of intense interest of the\ncommunity since long (see \\cite{pathak2002quantum,perinova1998phase}\nand references therein), and the interest in it has been further enlightened\nin the recent past as many new applications of quantum phase distribution\nand quantum phase fluctuation have been realized.\n\nTo be specific, this work is also motivated by the fact that recently\nseveral applications of nonclassical states and quantum phase properties\nhave been reported. Specifically, squeezed states have played an important\nrole in {{} the studies related to phase diffusion} \\cite{banerjee2007phase,banerjee2007phaseQND},\nthe detection of gravitational waves in LIGO experiments \\cite{abbasi2013thermal,abbott2016gw151226,abbott2016observation}.\nThe rising demand for a single photon source can be fulfilled by an\nantibunched light source \\cite{yuan2002electrically}. {The study\nof quantum correlations is important both from the perspective of\npure and mixed states} \\cite{chakrabarty2010study,dhar2013controllable,banerjee2010dynamics,banerjee2010entanglement}.\nEntangled states are found to be useful in both secure \\cite{ekert1991quantum}\nand insecure \\cite{bennett1992communication,bennett1993teleporting}\nquantum communication schemes. Stronger quantum correlation present\nin the steerable states are used to ensure the security against all\nthe side-channel attacks on devices used in one-side (i.e., either\npreparation or detector side) for quantum cryptography \\cite{branciard2012one}.\nQuantum supremacy in computation is established due to quantum algorithms\nfor unsorted database search \\cite{grover1997quantum}, factorization\nand discrete logarithm problems \\cite{shor1999polynomial}, and machine\nlearning \\cite{biamonte2017quantum} using essentially nonclassical\nstates. We may further stress on the recently reported applications\nof quantum phase distribution and quantum phase fluctuation by noting\nthat these have applications in quantum random number generation \\cite{xu2012ultrafast,raffaelli2018soi},\ncryptanalysis of squeezed state based continuous variable quantum\ncryptography \\cite{horak2004role}, generation of solitons in a Bose-Einstein\ncondensate \\cite{denschlag2000generating}, storage and retrieval\nof information from Rydberg atom \\cite{ahn2000information}, in phase\nencoding quantum cryptography \\cite{gisin2002quantum}, phase imaging\nof cells and tissues for biomedical application \\cite{park2018quantitative};\nas well as have importance in determining the value of transition\ntemperature for superconductors \\cite{emery1995importance}.\n\nNow to achieve the above advantages of the nonclassical states, we\nneed to produce these states via the schemes of quantum state engineering.\nFor the same, there are some distinct theoretical tools, like quantum\nscissoring \\cite{miranowicz2001quantum}, hole-burning \\cite{escher2004controlled,gerry2002hole,malpani2019filter}\nor filtering out a particular Fock state from the photon number distribution\n\\cite{Meher2018}, applying non-Gaussianity inducing operations \\cite{agarwal2013quantum}.\nHowever, these distinct mechanisms are experimentally realized primarily\nby appropriately using beam splitters, mirrors, and single photon\ndetectors or single photon counting module. Without going into finer\ndetails of the optical realization of quantum state engineering tools,\nwe may note that these tools can be used to generate various nonclassical\nstates, e.g., DFS \\cite{de1990properties}, PADFS \\cite{malpani2019lower},\nPSDFS \\cite{malpani2019lower}, photon added squeezed coherent state\n\\cite{thapliyal2017comparison}, photon subtracted squeezed coherent\nstate \\cite{thapliyal2017comparison}, number state filtered coherent\nstate \\cite{Meher2018}. Some of these states, like photon added coherent\nstate, have already been realized experimentally \\cite{zavatta2004quantum}.\n\nMany of the above mentioned engineered quantum states have already\nbeen studied in detail. Primarily, three types of investigations have\nbeen performed on the engineered quantum states- (i) study of various\nnonclassical features of these states (and their variation with the\nstate parameters) as reflected through different witnesses of nonclassicality.\nInitially, such studies were restricted to the lower-order nonclassical\nfeatures. In the recent past, various higher-order nonclassical features\nhave been predicted theoretically \\cite{alam2018higher,alam2018higher1,pathak2006control,pathak2010recent,verma2008higher,thapliyal2017comparison}\nand confirmed experimentally (\\cite{hamar2014non,perina2017higher}\nand references therein) in quantum states generated in nonlinear optical\nprocesses. (ii) Phase properties of the nonclassical states have been\nstudied \\cite{el2000phase} by computing quantum\nphase fluctuations, phase dispersion, phase distribution functions,\netc., under various formalisms, like Susskind and Glogower \\cite{susskind1964quantum},\nPegg-Barnett \\cite{pegg1989phase} and Barnett-Pegg \\cite{barnett1986phase}\nformalisms. (iii) Various applications of the engineered quantum states\nhave been designed. Some of them have already been mentioned.\n\nMotivated by the above observations, in this thesis, we would like\nto perform an investigation on nonclassical and phase properties of\na particularly interesting set of engineered quantum states which\nwould have the flavor of the first two facets of the studies mentioned\nabove. Applications of the engineered quantum states will also be\ndiscussed briefly, but will not be investigated in detail. To begin with we would like to briefly describe the physical and\nmathematical concepts used in this thesis, and that will be the focus\nof the rest of this chapter.\n\nThe rest of this chapter is organized as follows. In Section \\ref{sec:Quantum-theory-of-radiation},\nwe will briefly discuss quantum theory of radiation and introduce\nannihilation, creation, and number operators as well as Fock and coherent\nstates. In Section \\ref{sec:Quantum-states-of-4} this will be followed\nby an introduction to a set of quantum states which will be studied\nin this thesis. After this, the notion of nonclassicality will be\nintroduced mathematically in Section \\ref{sec:The-notion-of nonclassicality},\nand a set of operational criteria for observing nonclassical properties\nwill be introduced in Section \\ref{sec:Nonclassical-states:-witnesses}.\nSubsequently, the parameters used for the study of quantum phase properties\nwill be introduced in Section \\ref{sec:Analytic-tools-forphase}.\nThese witnesses of nonclassicality and the parameters for the study\nof phase properties will be used in the subsequent chapters, to investigate\nthe nonclassical and phase properties of the quantum states discussed\nin Section \\ref{sec:Quantum-states-of-4}. Finally, the structure\nof the rest of the thesis will be provided in Section \\ref{sec:Structure-of-the thesis}. \n\n\\section{Quantum theory of radiation field\\label{sec:Quantum-theory-of-radiation}}\n\nHistorically, quantum physics started with the ideas related to quanta\nof radiation. To be precise, Planck's work on black body radiation\n\\cite{planck1901law} and Einstein's explanation of photoelectric\neffect \\cite{einstein1905tragheit} involved a notion of quantized\nradiation field. In Planck's work, light was considered to be emitted\nfrom  and absorbed by a black body in quanta; and\nin Einstein's work, it was also considered that the radiation field\npropagates from one point to another as quanta. These initial works\ncontributed a lot in the development of quantum mechanics, but after\nthe introduction of quantum mechanics, in 1920s, in the initial days,\nmost of the attention was given to the quantization of matter. A quantum\ntheory of radiation was introduced by Dirac in 1927 \\cite{dirac1927quantum}.\nIn what follows, we will describe it briefly as this would form the\nbackbone of the present thesis.\n\n\\subsection{Creation and annihilation operator}\n\nMaxwell gave a classical description of electromagnetic field. But\nhere the objective is to study light and its properties apart from Maxwell's equations. So, to begin with, it would be reasonable to\nwrite Maxwell's equations in free space:\n\n\\begin{equation}\n\\nabla\\,.\\,E=0,\\label{eq:Maxwell1}\n\\end{equation}\n\n\\begin{equation}\n\\nabla\\,.\\,B=0,\\label{Maxwell2}\n\\end{equation}\n\n\\begin{equation}\n\\nabla\\times E=-\\frac{\\partial B}{\\partial t},\\label{Maxwell3}\n\\end{equation}\nand\n\n\\begin{equation}\n\\nabla\\times B=\\frac{1}{c^{2}}\\frac{\\partial E}{\\partial t},\\label{Maxwell4}\n\\end{equation}\nwhere $c=\\frac{1}{\\sqrt{\\mu_{0}\\epsilon_{0}}}$ is the speed of light\nwithout any medium (i.e. vacuum). Using the set of above equations\none can express magnetic and electric fields in the form of the solution\nof wave equations, like\n\n\\begin{equation}\n\\nabla^{2}E-\\frac{1}{c^{2}}\\frac{\\partial^{2}E}{\\partial t^{2}}=0.\\label{wave-equantion}\n\\end{equation}\nThe quantization of radiation field can be done by assuming a cavity\nof length $L$ having a linearly polarized electric field whose direction\nof propagation is in $z$ direction. Because\nof the linearity of the wave equation (\\ref{wave-equantion}), we\nare allowed to write the electric field in the form of linear combination\nof all the normal modes as\n\n\\begin{equation}\nE_{x}\\left(z,t\\right)=\\sum_{n}A_{n}q_{n}\\left(t\\right){\\rm sin}\\left(k_{n}z\\right),\\label{eq:electric}\n\\end{equation}\nwhere $q_{n}$ being the amplitude of the $n$th normal mode with\n$k_{n}=\\frac{n\\pi}{L},$$V$ is the volume of the resonator, $A_{n}=\\frac{2m_{n}\\nu_{n}^{2}}{\\epsilon_{0}V}$\nwith $\\nu_{n}=ck_{n}$, and $m_{n}$ is a constant (in the units of\nmass). With the help of this, we try to form\nan analogy of radiation with mechanical oscillator. In analogy to\nEq. (\\ref{eq:electric}) we are able to write the corresponding magnetic\nfield equation in cavity as\n\n\\begin{equation}\nB_{y}\\left(z,t\\right)=\\sum_{n}A_{n}\\left(\\frac{\\dot{q_{n}}}{c^{2}k_{n}}\\right){\\rm cos}\\left(k_{n}z\\right).\\label{eq:magnetic}\n\\end{equation}\nSo, the total energy of the field can be written as a classical Hamiltonian\n\n\\begin{equation}\nH=\\frac{1}{2}\\int_{V}d\\tau\\left(\\epsilon_{0}E_{x}^{2}+\\frac{1}{\\mu_{0}}B_{y}^{2}\\right),\\label{eq:hamiltonian}\n\\end{equation}\n\n\\begin{equation}\nH=\\frac{1}{2}\\sum_{n}\\left(m_{n}\\nu_{n}^{2}q_{n}^{2}+m_{n}\\dot{q}_{n}^{2}\\right)=\\frac{1}{2}\\sum_{n}\\left(m_{n}\\nu_{n}^{2}q_{n}^{2}+\\frac{p_{n}^{2}}{m_{n}}\\right),\\label{eq:hamiltonian2}\n\\end{equation}\n Substituting position and momentum variables by\ncorresponding operators to obtain the quantum mechanical Hamiltonian,\nwhere $p_{n}=m_{n}\\dot{q_{n}}.$ The position ($q_{n}$) and momentum\n($p_{n}$) operators follow the commutation relations\n\n\\[\n\\left[q_{n},p_{m}\\right]=\\iota\\hbar\\delta_{nm},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left[q_{n},q_{m}\\right]=\\left[p_{n},p_{m}\\right]=0,\n\\]\nwhere $\\hbar$ is the reduced Planck's constant. Using these one\nmay define a new set of operators which can be analytically written\nas\n\n\\begin{equation}\n\\hat{a_{n}}{\\rm exp\\left[-\\iota\\nu_{n}t\\right]=\\frac{1}{\\sqrt{2\\hbar\\text{\\ensuremath{m_{n}}}\\nu_{n}}}}\\left(\\text{\\ensuremath{m_{n}}}\\nu_{n}q_{n}+\\iota p_{n}\\right)\\label{annihilation}\n\\end{equation}\nand \n\\begin{equation}\n\\hat{a}_{n}^{\\dagger}{\\rm exp\\left[\\iota\\nu_{n}t\\right]=\\frac{1}{\\sqrt{2\\hbar\\text{\\ensuremath{m_{n}}}\\nu_{n}}}}\\left(\\text{\\ensuremath{m_{n}}}\\nu_{n}q_{n}-\\iota p_{n}\\right).\\label{eq:creation}\n\\end{equation}\nThus, the Hamiltonian can be written as\n\n\\begin{equation}\nH=\\sum_{n}\\hbar\\nu_{n}\\left(\\hat{a_{n}}^{\\dagger}\\hat{a_{n}}+\\frac{1}{2}\\right),\\label{eq:hamiltonian2-1}\n\\end{equation}\nand the commutation relations\n\n\\[\n\\left[\\hat{a}_{n},\\hat{a}_{m}^{\\dagger}\\right]=\\delta_{nm},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left[\\hat{a}_{n},\\hat{a}_{m}\\right]=\\left[\\hat{a}_{n}^{\\dagger},\\hat{a}_{m}^{\\dagger}\\right]=0,\n\\]\nwith corresponding electric and magnetic fields,   as given by Eq. 1.1.27 in~ \\cite{scully1997quantum} \n\n\\[\nE\\left(\\overrightarrow{r},t\\right)=\\sum_{k}\\hat{\\epsilon_{k}}\\xi_{k}\\hat{a}_{k}{\\rm exp}\\left[-\\iota\\nu_{k}t+\\iota k.\\overrightarrow{r}\\right]+{\\rm H.c}.\n\\]\nand\n\n\\[\nB\\left(\\overrightarrow{r},t\\right)=\\sum_{k}\\frac{k\\times\\hat{\\epsilon_{k}}}{\\nu_{k}}\\xi_{k}\\hat{a}_{k}{\\rm exp}\\left[-\\iota\\nu_{k}t+\\iota k.\\overrightarrow{r}\\right]+{\\rm H.c}.,\n\\]\nwhere $\\xi_{k}=\\left(\\frac{\\hbar\\nu_{k}}{\\epsilon_{0}V}\\right)^{1\/2}$\nis a constant, and $\\hat{\\epsilon_{k}}$ is a unit polarization vector\nwith the wave vector $k$.\n\nThe above analysis shows that a single-mode field is identical to\nharmonic oscillator. So, in the domain of quantum optics, harmonic\noscillator system plays an important role.\n\nNotice that the quantum treatment of electromagnetic radiation hinges\non annihilation $\\hat{a}$ (depletes photon) and creation $\\hat{a}^{\\dagger}$\n(creates photon) operators. The annihilation operator $\\hat{a}$ depletes\none quantum of energy and thus lowers down the system from harmonic\noscillator level $\\left|n\\right\\rangle $ to $\\left|n-1\\right\\rangle $,\ngiven by\n\n\\begin{equation}\n\\hat{a}\\left|n\\right\\rangle =\\sqrt{n}\\left|n-1\\right\\rangle .\\label{eq:annihilationop}\n\\end{equation}\nHere, $\\left|n\\right\\rangle $ is called Fock or number state. Further,\nan application of annihilation operator on vacuum leads to $0$, i.e.,\n$\\hat{a}\\left|0\\right\\rangle =0$. The creation operator $\\hat{a}^{\\dagger}$\ncreates one quantum of energy by raising the state from $\\left|n\\right\\rangle $\nto $\\left|n+1\\right\\rangle $. Therefore, the creation operator in\nthe number state can be represented as \n\\begin{equation}\n\\hat{a}^{\\dagger}\\left|n\\right\\rangle =\\sqrt{n+1}\\left|n+1\\right\\rangle .\\label{eq:creationop}\n\\end{equation}\nIf creation operator is applied to vacuum it creates a photon so these\noperators enables one to write a Fock state ($\\left|n\\right\\rangle $)\nin terms of the vacuum state as \n\\[\n\\left|n\\right\\rangle =\\frac{\\left(\\hat{a}^{\\dagger}\\right)^{n}}{\\sqrt{n!}}\\left|0\\right\\rangle .\n\\]\nIn the above, we have seen that annihilation and creation operators\nare the important field operators and are required for the quantum\ndescription of radiation. These operators can induce nonclassicality\nand non-Gaussianity when applied on classical states \\cite{zavatta2004quantum,agarwal2013quantum}.\nIn the present thesis, we study the role of these non-Gaussianity\ninducing operations in controlling the nonclassicality of the quantum\nstates which are often already nonclassical. For\ninstance, enhacement in squeezing in a nonclassical state does not\nensure advantage with respect to use as a single photon source and\nvice-versa. In the following subsection, we will introduce a set\nof other operators which can be expressed in terms of annihilation\nand creation operators and which play a crucial role in our understanding\nof the quantum states of radiation field.\n\n\\subsection{Some more quantum operators of relevance }\n\n\\label{SMQO}\n\nSo far we have introduced some non-unitary operations (operations\n$\\hat{O}$ which are not norm preserving and do not satisfy $\\hat{O}^{\\dagger}=\\hat{O}^{-1}$,\nwhere $\\hat{O}^{\\dagger}$ and $\\hat{O}^{-1}$ are the Hermitian conjugate\nand inverse of $\\hat{O}$, respectively), namely photon addition and\nsubtraction. We now aim to introduce some more unitary operations\nimportant in the domain of quantum state engineering in general, and\nin this thesis in particular. To begin with let us describe displacement\noperator.\n\n\\subsubsection{Displacement operator}\n\nDisplacement operator is a unitary operator. The mathematical from\nof displacement operator is given as\n\n\\begin{equation}\n\\hat{D}(\\alpha)=\\exp\\left(\\alpha\\hat{a}^{\\dagger}-\\alpha^{\\star}\\hat{a}\\right).\\label{eq:Displacement}\n\\end{equation}\nThis operator can be used as a tool to generate coherent state from\nvacuum. Specifically, a coherent state $\\left|\\alpha\\right\\rangle $\nis defined as $\\left|\\alpha\\right\\rangle =\\hat{D}(\\alpha)\\left|0\\right\\rangle .$\n\n\\subsubsection{Squeezing operator}\n\n\\label{SqOpt}\n\nThe squeezing operator for a single mode of electromagnetic field\nis \n\\begin{equation}\n\\hat{S}(z)=\\exp\\left(\\frac{1}{2}\\left(z^{\\star}\\hat{a}^{2}-z\\hat{a}^{\\dagger2}\\right)\\right).\\label{eq:squeezed}\n\\end{equation}\nThe description of light is given by two quadratures namely phase\n$(X_{1})$ and amplitude $(X_{2})$ in the domain of quantum optics,\nmathematically defined as  \n\\[\n\\hat{X_{\\theta}}=\\frac{1}{\\sqrt{2}}\\left( i \\hat{a}^\\dagger \\exp\\left[\\iota\\theta \\right] - \\iota \\hat{a}\\exp\\left[-\\iota\\theta\\right]\\right),\n\\]\nThe corresponding uncertainty of these two quadratures is observed\nby relation $\\Delta X_{1}\\Delta X_{2} \\geq \\hbar\\,\/2$,\nwhere $\\Delta X_{1}$ $\\left(\\Delta X_{2}\\right)$ is variance in\nthe measured values of quadrature  $\\hat{X_{1}}=\\hat{X_{1}}\\left(\\theta=0\\right)$\n$\\left(\\hat{X_{2}}=\\hat{X_{2}}\\left(\\theta=\\frac{\\pi}{2}\\right)\\right)$.\nSpecifically, $\\Delta X_{i}=\\sqrt{\\left\\langle X_{i}^{2}\\right\\rangle -\\left\\langle X_{i}\\right\\rangle ^{2}},$where\n$\\left\\langle \\hat{A}\\right\\rangle =\\left\\langle \\psi\\left|\\hat{A}\\right|\\psi\\right\\rangle $\nis the expectation value of the operator $A$ with respect to the\nquantum state $\\left|\\psi\\right\\rangle $. Coherent state has an equal\nuncertainty in both quadratures so they form a circle in the phase\npicture (shown in Fig. \\ref{fig:coh-sq}).  Least value\nof the variance for suitable $\\theta$ is studied as principle squeezing.\nWith the advent of nonlinear optics, a very special branch of optics,\nthe uncertainty  in one of the quadratures can be\nreduced at the cost of increment in other quadrature's uncertainty,\nwhich means that the circle can be squeezed.\n\n\\subsection{Eigen states of the field operators}\n\nHere, we will discuss eigen states of some of the operators we have\nintroduced. The eigenvalue equation can be defined as $\\hat{A}\\lambda=a\\lambda$\nwith eigen operator $\\hat{A}$, eigenvalue $a$, and eigen function\n$\\lambda$. For example, Schrodinger equation $H\\psi_{i}=E_{i}\\psi_{i}$\nhas Hamiltonian $H$ as eigen operator with eigen functions $\\psi_{i}$\nand eigenvalues as allowed energy levels.\n\n\\subsubsection{Fock state: Eigen state of the number operator}\n\nIn case of quantum optics or quantized light picture, photon number\nstate is known as number state. The single-mode photon number states\nare known Fock states, and its ground state is defined as vacuum state. As the set of number states are a full set of orthonormal basis so\nany quantum state can be written in terms of these basis. The method\nof representing a quantum state as superposition of number states\nis known as number state representation. Now using Eq. (\\ref{eq:annihilationop})\nand (\\ref{eq:creationop}), we can introduce an operator \n\\[\n\\hat{N}=\\hat{a}^{\\dagger}\\hat{a}.\n\\]\nwhich would satisfy the following eigen value equation \n\\[\n\\hat{N}\\left|n\\right\\rangle =n\\left|n\\right\\rangle .\n\\]\nClearly Fock states are the eigen states of the number operators and\nin consistency with what has already been told, a Fock state $\\left|n\\right\\rangle $\nrepresent a $n$ photon state.\n\n\\subsubsection{Coherent state: Eigen state of the annihilation operator \\label{subsec:Coherent-state}}\n\nCoherent state \\cite{fox2006quantum} is considered as a state of\nthe quantized electromagnetic field which shows classical behaviour\n(specifically, beavior closest to classical states). According to\nErwin Schrodinger it is a minimum uncertainty state, having same uncertainity\nin position and momentum \\cite{schrodinger1926stetige}. According\nto Glauber, any of three mathematical definitions described below\ncan define coherent state:\n\n(i) Eigen vectors of annihilation operator $\\hat{a}\\left|\\alpha\\right\\rangle =\\alpha\\left|\\alpha\\right\\rangle $,\n$\\alpha$ being a complex number.\n\n(ii) Quantum states having minimum uncertainty $\\Delta X_{1}=\\Delta X_{2}=1\/\\sqrt{2}$,\nwith $X_{2}$ and $X_{1}$ as momentum and position operators.\n\n(iii) States realized by the application of the displacement operator\n$D(\\alpha)$ on the vacuum state. Thus, is also known as displaced\nvacuum state and can be expressed as \n\\[\n\\left|\\alpha\\right\\rangle =D\\left(\\alpha\\right)\\left|0\\right\\rangle .\n\\]\nIn Fock basis, it is expressed as infinite superposition of Fock state\nas \n\\begin{equation}\n\\left|\\alpha\\right\\rangle =\\exp\\left[-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right]\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}|n\\rangle,\\label{eq:CS}\n\\end{equation}\nwhere $\\alpha$ is a complex number. Experimentally established state\nvery close to this coherent state was possible only after the successful\ndevelopment of laser. Finally, one can easily see that $\\hat{a}\\left|\\alpha\\right\\rangle =\\alpha\\left|\\alpha\\right\\rangle $\nimplies $\\left\\langle \\alpha\\right|\\hat{a}^{\\dagger}=\\left\\langle \\alpha\\right|\\alpha^{\\star}$\nand consequently $\\left\\langle \\alpha\\right|\\hat{a}^{\\dagger}\\hat{a}\\left|\\alpha\\right\\rangle =\\left\\langle \\alpha\\right|\\hat{N}\\left|\\alpha\\right\\rangle =N=\\left|\\alpha\\right|^{2}$\nor average photon number in a coherent state is $\\left|\\alpha\\right|^{2}$.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{c}\n\\includegraphics[width=110mm]{coh-sq.jpg}\\tabularnewline\n\\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:coh-sq} Phase picture for coherent state and squeezed\nstate. }\n\\end{figure}\n\n\\section{Quantum states of our interest\\label{sec:Quantum-states-of-4}}\n\nIn this section, we provide basic mathematical details of the set\nof engineered quantum states studied in the present thesis.\n\n\\subsection{Displaced Fock state}\n\nDisplaced Fock state \\cite{satyanarayana1985generalized} are formed\nby applying displacement operator on Fock state and thus a DFS is\ndefined as \n\\[\n\\left|\\phi\\right\\rangle =D\\left(\\alpha\\right)\\left|n\\right\\rangle .\n\\]\nAnalytically it is given as \n\\begin{equation}\n|\\phi(n,\\alpha)\\rangle=\\frac{1}{\\sqrt{n!}}\\sum_{p=0}^{n}{n \\choose p}(-\\alpha^{\\star})^{(n-p)}\\exp\\left(-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right)\\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}}{m!}\\sqrt{(m+p)!}|m+p\\rangle.\\label{eq:GCS}\n\\end{equation}\nVarious nonclassical properties of DFS are reported in literature\n\\cite{de1990properties,el2000phase,lvovsky2002synthesis,mendas1993pancharatnam}.\n\n\\subsection{Photon added and photon subtracted displaced Fock state\\label{subsec:Photon-added-and sub}}\n\nUsing DFS, we can define a $u$ photon added DFS (i.e., a PADFS) as\n\\begin{eqnarray}\n|\\psi_{+}(u,n,\\alpha)\\rangle & = & N_{+}\\hat{a}^{\\dagger u}|\\phi(n,\\alpha)\\rangle=\\frac{N_{+}}{\\sqrt{n!}}\\sum_{p=0}^{n}{n \\choose p}(-\\alpha^{\\star})^{(n-p)}\\exp\\left(-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right)\\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}}{m!}\\nonumber \\\\\n & \\times & \\sqrt{(m+p+u)!}|m+p+u\\rangle.\\label{eq:PADFS}\n\\end{eqnarray}\nSimilarly, a $v$ photon subtracted DFS (i.e., a PSDFS) can be expressed\nas \n\\begin{eqnarray}\n|\\psi_{-}(v,n,\\alpha)\\rangle & = & N_{-}\\hat{a}^{v}|\\phi(n,\\alpha)\\rangle=\\frac{N_{-}}{\\sqrt{n!}}\\sum_{p=0}^{n}{n \\choose p}(-\\alpha^{\\star})^{(n-p)}\\exp\\left(-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right)\\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}}{m!}\\nonumber \\\\\n & \\times & \\dfrac{(m+p)!}{\\sqrt{(m+p-v)!}}|m+p-v\\rangle,\\label{eq:PSDFS}\n\\end{eqnarray}\nwhere $m$ and $p$ are the real integers. Here, {\\small{}\n\\begin{eqnarray}\nN_{+}=\\left[\\frac{1}{n!}\\sum_{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m+p-p'}(m+p+u)!}{m!(m+p-p')!}\\right]^{-\\frac{1}{2}}\\label{eq:norP}\n\\end{eqnarray}\nand \n\\begin{equation}\nN_{-}=\\left[\\frac{1}{n!}\\sum_{p,p'=0}^{n}{n \\choose p}{n \\choose p'}\\left(\\alpha^{\\star}\\right)^{(n-p)}\\left(-\\alpha\\right){}^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}\\left(\\alpha^{\\star}\\right){}^{m+p-p'}(m+p)!}{m!(m+p-p')!(m+p-v)!}\\right]^{-\\frac{1}{2}}.\\label{eq:norS}\n\\end{equation}\n}are the normalization constants, and subscripts $+$ and $-$ correspond\nto photon addition and subtraction. Thus, $|\\psi_{+}(u,n,\\alpha)\\rangle$\nand $|\\psi_{-}(v,n,\\alpha)\\rangle$ represent $u$ photon added DFS\nand $v$ photon subtracted DFS, respectively, for the DFS which has\nbeen produced by displacing the Fock state $|n\\rangle$ by a displacement\noperator $D(\\alpha)$ characterized by the complex parameter $\\alpha.$\nClearly, the addition and the subtraction of photons on the DFS can\nbe mathematically viewed as application of the creation and annihilation\noperators from the left on the Eq. (\\ref{eq:GCS}). Here, it may be\nnoted that different well-known states can be obtained as special\ncases of these two states. For example, using the notation introduced\nabove to define PADFS and PSDFS, we can describe a coherent state\n$|\\alpha\\rangle$ as $|\\alpha\\rangle=|\\psi_{+}(0,\\alpha,0)\\rangle=|\\psi_{-}(0,\\alpha,0)\\rangle$,\nnaturally, coherent state can be viewed as a special case of both\nPADFS and PSDFS. Similarly, we can describe a single photon added\ncoherent state as $|\\psi\\rangle_{+1}=|\\psi_{+}(1,\\alpha,0)\\rangle$,\na Fock state as $|n\\rangle=|\\psi_{+}(0,0,n)\\rangle=|\\psi_{-}(0,0,n)\\rangle$\nand a DFS as $|\\psi\\rangle_{{\\rm DFS}}=|\\psi_{+}(0,\\alpha,n)\\rangle=|\\psi_{-}(0,\\alpha,n)\\rangle.$\n\n\\subsection{Photon added then subtracted displaced Fock state \\label{subsec:PASDFS}}\n\nA PASDFS can be obtained by sequentially applying appropriate number\nof annihilation (photon subtraction) and creation (photon addition)\noperators on a DFS. Analytical expression for PASDFS (specifically,\na $k$ photon added and then $q$ photon subtracted DFS) in Fock basis\ncan be shown to be\n\n\\begin{eqnarray}\n|\\psi(k,q,n,\\alpha)\\rangle & = & N\\hat{a}^{q}\\hat{a}^{\\dagger k}|\\psi(n,\\alpha)\\rangle=\\frac{N}{\\sqrt{n!}}\\sum_{p=0}^{n}{n \\choose p}(-\\alpha^{\\star})^{(n-p)}\\exp\\left(-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right)\\nonumber \\\\\n & \\times & \\sum_{m=0}^{\\infty}\\frac{\\alpha^{m}\\left(m+p+k\\right)!}{m!\\sqrt{(m+p+k-q)!}}|m+p+k-q\\rangle,\\label{eq:PADFS-1}\n\\end{eqnarray}\nwhere {\\small{}\n\\[\nN=\\left[\\frac{1}{n!}\\sum_{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\right]^{-\\frac{1}{2}}\n\\]\n}is the normalization factor.\n\n\\subsection{Even coherent state and states generated by holeburning on it}\n\nEven coherent state can be defined as the superposition of two coherent\nstates having opposite phase ($|\\phi(\\alpha)\\rangle\\propto|\\alpha\\rangle+|-\\alpha\\rangle$).\nThe analytical expression for ECS in number basis can be written as\n\\begin{equation}\n\\begin{array}{lcl}\n|\\phi(\\alpha)\\rangle & = & \\frac{\\,\\exp\\left[-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right]}{\\sqrt{2\\left(1+\\exp\\left[-2\\mid\\alpha\\mid^{2}\\right]\\right)}}\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\left(1+\\left(-1\\right)^{n}\\right)|n\\rangle.\\end{array}\\label{eq:ECS}\n\\end{equation}\nThe parameter $\\alpha=|\\alpha|\\exp(i\\theta)$, in Eq. (\\ref{eq:ECS}),\nis complex in general and $\\theta$ is phase angle in the complex\nplane. Various schemes to generate ECS are reported in \\cite{brune1992manipulation,ourjoumtsev2007generation,gerry1993non}.\nThe nonclassical properties (witnessed through the antibunching and\nsqueezing criteria, $Q$ function, Wigner function, and photon number\ndistribution, etc.) of ECS have been studied in the recent past \\cite{gerry1993non}.\n\n\\subsubsection{Vacuum filtered even coherent state}\n\nAs mentioned above, experimentally, an ECS or\na cat state can be generated in various ways, and the same can be\nfurther engineered to produce a hole at vacuum in its photon number\ndistribution. Specifically, filtration of vacuum will burn a hole\nat $n=0$ and produce VFECS, which can be described in Fock basis\nas \n\\begin{equation}\n\\begin{array}{lcl}\n|\\phi_{1}(\\alpha)\\rangle & = & N_{{\\rm VFECS}}\\sum\\limits _{n=0,\\,n\\neq0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\left(1+\\left(-1\\right)^{n}\\right)|n\\rangle,\\end{array}\\label{eq:VFECS}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm VFECS}} & = & \\{4{\\rm cosh}\\left(\\mid\\alpha\\mid^{2}\\right)-1\\}^{-1\/2}\\end{array}\n\\end{equation}\nis the normalization constant. For simplicity, we may expand Eq. (\\ref{eq:VFECS})\nas a superposition of a standard ECS and a vacuum state as follows\n\\begin{equation}\n\\begin{array}{lcl}\n|\\phi_{1}(\\alpha)\\rangle & = & N_{{\\rm VFECS}}\\left(\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\left(1+\\left(-1\\right)^{n}\\right)|n\\rangle-2|0\\rangle\\right).\\end{array}\\label{eq:VFECS-EXPANDED}\n\\end{equation}\nIn what follows, Eq. (\\ref{eq:VFECS-EXPANDED}) will be used to explore\nvarious nonclassical features that may exist in VFECS.\n\n\\subsubsection{Photon added even coherent state}\n\nOne can define a single photon added ECS as \n\\begin{equation}\n|\\phi_{2}(\\alpha)\\rangle=N_{{\\rm PAECS}}\\hat{a}^{\\dagger}|\\phi(\\alpha)\\rangle=N_{{\\rm PAECS}}\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\left(1+\\left(-1\\right)^{n}\\right)\\sqrt{n+1}|n+1\\rangle,\\label{eq:PAECS}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm PAECS}} & = & \\{{\\rm cosh}\\left(\\mid\\alpha\\mid^{2}\\right)+\\mid\\alpha\\mid^{2}{\\rm sinh}\\left(\\mid\\alpha\\mid^{2}\\right)\\}^{-1\/2}\/2\\end{array}\n\\end{equation}\nis the normalization constant for PAECS.\n\n\\subsection{Binomial state and the states generated by holeburning on it}\n\nBinomial state is a finite superposition of Fock states having binomial\nphoton number distribution. It is quite similar to the coherent state\nwhich is the linear combination of Fock states having the Poissonian\nphoton number distribution \\cite{stoler1985binomial}. BS can be defined\nas \n\\begin{equation}\n\\begin{array}{lcl}\n|p,M\\rangle & = & \\sum\\limits _{n=0}^{M}\\left[\\frac{M!}{n!(M-n)!}p^{n}\\left(1-p\\right)^{M-n}\\right]^{1\/2}|n\\rangle.\\end{array}\\label{eq:BS}\n\\end{equation}\nThe binomial coefficient describes the presence of $n$ photons with\nprobability $p$ in $M$ number of ways. Recently, the study of nonclassical\nproperties of BS, specifically, antibunching, squeezing, HOSPS \\cite{verma2008higher,verma2010generalized,bazrafkan2004tomography},\netc., have been studied very extensively. However, no effort has yet\nbeen made to study the nonclassical properties of VFBS and PABS.\n\n\\subsubsection{Vacuum filtered binomial state}\n\nThe vacuum filtration of BS can be obtained by simply eliminating\nvacuum state from the BS as \n\\begin{equation}\n\\begin{array}{lcl}\n|p,M\\rangle_{1} & = & N_{{\\rm VFBS}}\\sum\\limits _{n=0}^{M}\\left[\\frac{M!}{n!(M-n)!}p^{n}\\left(1-p\\right)^{M-n}\\right]^{1\/2}|n\\rangle-N_{VFBS}\\left[\\left(1-p\\right)^{M}\\right]^{1\/2}|0\\rangle,\\end{array}\\label{eq:VFBS}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm VFBS}} & = & \\{1-\\left(1-p\\right)^{M}\\}^{-1\/2}\\end{array}\n\\end{equation}\nis the normalization constant for the VFBS.\n\n\\subsubsection{Photon added binomial state}\n\nA hole at $n=0$ at a BS can also be introduced by the addition of\na single photon on the BS. A few steps of computation yield the desired\nexpression for PABS as \n\\begin{equation}\n\\begin{array}{lcl}\n|p,M\\rangle_{2} & =N_{{\\rm PABS}} & \\sum\\limits _{n=0}^{M}\\left[\\frac{M!(n+1)!}{\\left(n!\\right)^{2}(M-n)!}p^{n}\\left(1-p\\right)^{M-n}\\right]^{1\/2}|n+1\\rangle,\\end{array}\\label{eq:PABS}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm PABS}} & = & \\left(1+Mp\\right)^{-1\/2}\\end{array}\n\\end{equation}\nis the normalization constant for single photon added BS.\n\n\\subsection{Kerr state and the states generated by holeburning on it}\n\nA KS can be obtained when electromagnetic field\nin a coherent state interacts with nonlinear medium with Kerr type\nnonlinearity \\cite{gerry1994statistical}. This interaction generates\nphase shifts which depend on the intensity. The Hamiltonian involved\nin this process is given as \n\\begin{equation}\nH=\\hbar\\omega\\hat{a}^{\\dagger}\\hat{a}+\\hbar\\chi\\left(\\hat{a}^{\\dagger}\\right)^{2}\\left(\\hat{a}\\right)^{2},\n\\end{equation}\nwhere $\\chi$ depends on the third-order susceptibility of Kerr medium.   Explicit contribution of $H$ is ${\\rm exp}\\left[-i\\chi n(n-1)\\right]$. Thus, the compact analytic form\nof the KS in the Fock basis can be given as \n\\begin{equation}\n\\begin{array}{lcl}\n|\\psi_{K}\\left(n\\right)\\rangle & = & \\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\exp\\left(-\\frac{\\mid\\alpha\\mid^{2}}{2}\\right)\\exp\\left(-\\iota\\chi n\\left(n-1\\right)\\right)|n\\rangle.\\end{array}\\label{eq:KS}\n\\end{equation}\n\n\n\\subsubsection{Vacuum filtered Kerr state}\n\nSimilarly, a VFKS, which can be obtained using the same quantum state\nengineering process that leads to VFECS and VFBS, is given by \n\\begin{equation}\n\\begin{array}{lcl}\n|\\psi_{K}\\left(n\\right)\\rangle_{1} & = & N_{{\\rm VFKS}}\\left[\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\exp\\left(-\\iota\\chi n\\left(n-1\\right)\\right)|n\\rangle-|0\\rangle\\right]\\end{array},\\label{eq:VFKS-expanded}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm VFKS}} & = & {{\\left(\\exp\\left[\\mid\\alpha\\mid^{2}\\right]-1\\right)}^{-1\/2}}\\end{array}\\label{eq:NVFKS-expanded}\n\\end{equation}\nis the normalization constant for the VFKS.\n\n\\subsubsection{Photon added Kerr state}\n\nAn addition of a photon to Kerr state would yield PAKS which can be\nexpanded in Fock basis as \n\\begin{equation}\n\\begin{array}{lcl}\n|\\psi_{K}\\left(n\\right)\\rangle_{2} & =N_{{\\rm PAKS}} & \\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}\\exp\\left(-\\iota\\chi n\\left(n-1\\right)\\right)\\sqrt{\\left(n+1\\right)}|n+1\\rangle,\\end{array}\\label{eq:PAKS}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{array}{lcl}\nN_{{\\rm PAKS}} & = & {\\left(\\exp\\left[\\mid\\alpha\\mid^{2}\\right]\\left(1+\\mid\\alpha\\mid^{2}\\right)\\right)^{-1\/2}}\\end{array}\\label{eq:NPAKS}\n\\end{equation}\nis the normalization constant for the PAKS.\n\n\\section{The notion of nonclassical states \\label{sec:The-notion-of nonclassicality}}\n\nQuantum states which do not have any classical analogue have been\nreferred to as nonclassical states \\cite{agarwal2013quantum}. In\nother words, states having their $P$-distribution more singular than\ndelta function or having negative values are referred to as nonclassical\nstates \\cite{dodonov2003classicality}. This idea was possible only\nwhen Glauber and Sudarshan published papers in 1963 \\cite{sudarshan1963equivalence,glauber1963photon,glauber1963coherent}.\nSudarshan found a mathematical form to represent any state in the\ncoherent basis, mathematically given as \n\\[\n\\rho=\\int P\\left(\\alpha\\right)\\left|\\alpha\\right\\rangle \\left\\langle \\alpha\\right|d^{2}\\alpha,\n\\]\nwhere $P\\,(\\alpha)$ is known as Glauber-Sudarshan $P$-function,\nwhich follows normalization condition as $\\int dP(\\alpha)=1$, but it may have negative values. Thus, it is defined as quasidistribution\nfunction or quasiprobability distribution. When\n$P\\,(\\alpha)$ attains a positive probability density function, immediately\nit indicates that the state is classical. This leads to the definition\nof nonclassicality. If an arbitrary quantum state is failed to represent\nas mixture of coherent states, that is known as nonclassical state.\nTo establish quantum supremacy, these nonclassical states play very\nessential role, for instance in theses states are useful in establishing\nquantum supremacy of quantum information processing, quantum communication,\netc. Although $P$-function is not reconstructable for any arbitrary\nstate yet it has been of major interest as it provides an important\nsignature of nonclassicality. The negativity (positivity or non-negativity)\nof the $P$-function essentially provides the nonclassical (classical)\nbehavior of the state under consideration. The experimental difficulty\nassociated with the easurement of $P$-function in its reconstruction\nled to various feasible substitutes as nonclassicality witnesses.\nHere, we list some of those nonclassicalty witnesses. These witnesses\ncan be viewed as operational criterion of nonclassicality.\n\n\\section{Nonclassical states: witnesses and measures \\label{sec:Nonclassical-states:-witnesses}}\n\nUsing nonclassical states, the essence of quantum theory of light\ncan be understood. There are various tools for characterization of\nnonclassical states. In this section, some tools are described which\nare used to characterize such states. If historically seen, the first\nsuch approach was aimed to check the deviation from Poissonian photon\nstatistics, the second is to evaluate the volume of the negative part\nof the quasiprobability distribution in the phase space, etc. An infinite\nset of moments based criteria is available in literature which is\nused as witness of nonclassicality equivalent to $P$-function \\cite{shchukin2005nonclassical}.\nAny subset of this infinite set may detect nonclassicality or fail\nto do so. Example of these witnesses are lower- and higher-order antibunching,\nsub-Poissonian photon statistics, squeezing as well as Mandel $Q_{M}$\nparameter, Klyshko's, Vogel's, and Agarwal-Tara's criteria, $Q$ function,\netc. To quantify the amount of nonclassicality, a number of measures\nhave been proposed, like linear entropy, Wigner volume, concurrence\nand many more. A small description of these criteria is given here.\n\n\\subsection{Witnesses of nonclassicality\\label{subsec:Witnesses-of-nonclassicality}}\n\n\\subsubsection{Lower- and higher-order antibunching}\n\nIn this section, we study lower- and higher-order antibunching. To\ndo so, we use the following criterion of $(l-1)$th order antibunching\n(\\cite{pathak2006control} and references therein)  in\nterms of nonclassicality witness ($d(l-1)$) as\n\\begin{equation}\nd(l-1)=\\langle\\hat{a}^{\\dagger l}\\hat{a}^{l}\\rangle-\\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle^{l}<0.\\label{eq:HOA-1}\n\\end{equation}\nThis nonclassical feature characterizes suitability of the quantum\nstate to be used as single photon source as the negative values of\n$d(l-1)$ parameter show that the probability of photons coming bunched\nis less compared to that of coming independently. The signature of\nlower-order antibunching can be obtained as a special case of Eq.\n(\\ref{eq:HOA-1}) for $l=2$, and that for $l\\geq3$, the negative\nvalues of $d(l-1)$ correspond to higher-order antibunching of $(l-1)$th\norder. Figure \\ref{fig:HBT} illustrates the scheme for studying antibunching\nexperimentally    (corresponds to $l=2$). For higher\nvalues of $l$, we require more beamsplitters and APDs. On these cascaded\nbeamplitters signal is mixed with vacuum and measured higher-order\ncorrelation ~\\cite{avenhaus2010accessing}.\n\n\\begin{figure}\n\\centering{}%\n\\begin{tabular}{c}\n\\includegraphics[width=100mm]{HBT.jpg}\\tabularnewline\n\\tabularnewline\n\\end{tabular}\\caption{Hanbury Brown and Twiss setup. Here, APD is avalanche\nphoto diode.}\n\\label{fig:HBT} \n\\end{figure}\n\n\\subsubsection{Lower- and higher-order sub-Poissionian photon statistics}\n\nThe lower-order counterparts of antibunching and sub-Poissonian photon\nstatistics are closely associated as the presence of latter ensures\nthe possibility of observing former (see \\cite{thapliyal2014higher,thapliyal2017comparison}\nfor a detailed discussion). However, these two nonclassical features\nwere shown to be independent phenomena in the past (\\cite{thapliyal2014higher,thapliyal2017comparison}\nand references therein). Higher-order counterpart of sub-Poissonian\nphoton statistics can be introduced as\n\n\\begin{equation}\n\\begin{array}{lcccc}\n\\mathcal{D}_{h}(l-1) & = & \\sum\\limits _{e=0}^{l}\\sum\\limits _{f=1}^{e}S_{2}(e,\\,f)\\,^{l}C_{e}\\,\\left(-1\\right)^{e}d(f-1)\\langle N\\rangle^{l-e} & < & 0,\\end{array}\\label{eq:hosps22-1}\n\\end{equation}\nwhere $S_{2}(e,\\,f)$ is the Stirling number of second kind, {and\n$\\,^{l}C_{e}$ is the usual binomial coefficient}.\n\n\\subsubsection{Higher-order squeezing}\n\nAs mentioned beforehand, the squeezing of quadrature is defined in\nterms of variance in the measured values of the quadrature (say, position\nor momentum) below the corresponding value for the coherent state,\ni.e., minimum uncertainty state. The higher-order counterpart of squeezing\nis studied in two ways, namely Hong-Mandel and Hillery-type squeezing\n\\cite{hong1985higher,hong1985generation,hillery1987amplitude}. Specifically,\nthe idea of the higher-order squeezing originated from the pioneering\nwork of Hong and Mandel \\cite{hong1985higher,hong1985generation},\nwho generalized the lower-order squeezing using the higher-order moments\nof field quadrature. According to the Hong-Mandel criterion, the $l$th\norder squeezing can be observed if the $l$th moment (for even values\nof $l>2$) of a field quadrature operator is less than the corresponding\ncoherent state value. The condition of Hong-Mandel type higher-order\nsqueezing is given as follows \\cite{hong1985higher,hong1985generation}\n\n\\begin{equation}\nS(l)=\\frac{\\langle(\\Delta X)^{l}\\rangle-\\left(\\frac{1}{2}\\right)_{\\frac{l}{2}}}{\\left(\\frac{1}{2}\\right)_{\\frac{l}{2}}}<0,\\label{eq:Hong-Def}\n\\end{equation}\nHere, $S_{(l)}$ is higher-order squeezing, $\\Delta X$ is the quadrature\n( $\\Delta X_{1}$) as defined in Section \\ref{SMQO}. Further, $(x)_{l}$\nis conventional Pochhammer symbol. The inequality in Eq. (\\ref{eq:Hong-Def})\ncan also be rewritten as \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle(\\Delta X)^{l}\\rangle & < & \\left(\\frac{1}{2}\\right)_{\\frac{l}{2}}=\\frac{1}{2^{\\frac{l}{2}}}(l-1)!!\\end{array}\\label{eq:Hong-def2-2}\n\\end{equation}\nwith \n\\begin{equation}\n\\langle\\left(\\text{\\ensuremath{\\Delta}}\\text{X}\\right)^{l}\\rangle=\\sum\\limits _{r=0}^{l}\\sum\\limits _{i=0}^{\\frac{r}{2}}\\sum\\limits _{k=0}^{r-2i}\\left(-1\\right)^{r}\\frac{1}{2^{\\frac{l}{2}}}\\left(2i-1\\right)!^{2i}C_{k}{}^{l}C_{r}{}^{r}C_{2i}\\langle\\hat{a}^{\\dagger}+\\hat{a}\\rangle^{l-r}\\langle\\hat{a}^{\\dagger k}\\hat{a}^{r-2i-k}\\rangle.\\label{eq:cond2.1-1}\n\\end{equation}\n\n\n\\subsubsection{Klyshko's criterion}\n\nThis criterion is relatively simpler as to calculate this witness\nof nonclssicality, only three consecutive probability terms are required\nrather than all the terms. Negative values of $B(m)$ are symbol of\nnonclassicality present in the state. Klyshko introduced this criterion\n\\cite{klyshko1996observable} to investigate the nonclassical property\nusing only three successive photon-number {{} probabilities}. In\nterms of the photon-number probability $p_{m}=\\langle m|\\rho|m\\rangle$\nof the state with density matrix $\\rho$, the Klyshko's criterion\nin the form of an inequality can be written as \n\\begin{equation}\nB(m)=(m+2)p_{m}p_{m+2}-(m+1)\\left(p_{m+1}\\right)^{2}<0.\\label{eq:Klyshko-1}\n\\end{equation}\n\n\n\\subsubsection{Vogel's criterion}\n\n The moments-based nonclassicality criterion of the\nprevious subsection was later extended to Vogel's nonclassicality\ncriterion \\cite{shchukin2005nonclassical} in terms of matrix of moments\nas \n\n \n\\begin{equation}\nv =\\left[\\begin{array}{ccc}\n1 & \\langle\\hat{a}\\rangle & \\langle\\hat{a}^{\\dagger}\\rangle\\\\\n\\langle\\hat{a}^{\\dagger}\\rangle & \\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle & \\langle\\hat{a}^{\\dagger2}\\rangle\\\\\n\\langle\\hat{a}\\rangle & \\langle\\hat{a}^{2}\\rangle & \\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle\n\\end{array}\\right].\\label{eq:vogel}\n\\end{equation}\nThe negative value of the determinant $dv $ of matrix $v$\nin Eq. (\\ref{eq:vogel}) is signature of nonclassicality.\n\n\\subsubsection{Agarwal-Tara's criterion}\n\nThere were certain quantum states having negative $P$-function yet\nshowing no squeezing and sub-Poissonian behavior, to witness the nonclassicality\nresiding in those particular types of states Agarwal and Tara \\cite{agarwal1992nonclassical}\nintroduced this criterion which is again a moments based criterion.\nThis can be written in a matrix form and expressed as \n\\begin{equation}\nA_{3}=\\dfrac{\\det m^{(3)}}{\\det\\mu^{(3)}-\\det m^{(3)}}<0,\\label{eq:Agarwal-1}\n\\end{equation}\nwhere \n\\[\nm^{(3)}=\\begin{bmatrix}1 & m_{1} & m_{2}\\\\\nm_{1} & m_{2} & m_{3}\\\\\nm_{2} & m_{3} & m_{4}\n\\end{bmatrix}\n\\]\nand \n\\[\n\\mu^{(3)}=\\begin{bmatrix}1 & \\mu_{1} & \\mu_{2}\\\\\n\\mu_{1} & \\mu_{2} & \\mu_{3}\\\\\n\\mu_{2} & \\mu_{3} & \\mu_{4}\n\\end{bmatrix}.\n\\]\nThe matrix elements are defined as $m_{i}=\\langle\\hat{a}^{\\dagger i}\\hat{a}^{i}\\rangle$\nand $\\mu_{j}=(\\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle)^{j}=(m_{1})^{j}$.\n\n\\subsubsection{Mandel $Q_{M}$ parameter}\n\nThe Mandel $Q_{M}$ parameter \\cite{mandel1979sub} illustrates the\nnonclassicality through photon number distribution in a quantum state.\nThe Mandel $Q_{M}$ parameter is defined as \n\\begin{equation}\nQ_{M}=\\frac{\\langle(\\hat{a}^{\\dagger}\\hat{a})^{2}\\rangle-\\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle^{2}-\\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle}{\\langle\\hat{a}^{\\dagger}\\hat{a}\\rangle}.\\label{eq:MandelQ}\n\\end{equation}\nThe negative values of $Q_{M}$ parameter essentially indicate the\nnegativity for $P$-function and so it gives a witness for nonclassicality.\nFor the Poissonian statistics it becomes 0, while for the sub-Poissonian\n(super-Poissonian) photon statistics it has negative (positive) values.\n\n\\subsection{Other quasiprobability distributions}\n\nInability to give a phase space description of quantum mechanics is\nexploited in terms of quasiprobability distributions (\\cite{thapliyal2015quasiprobability}\nand references therein). Later, it was found that they are useful\nas witnesses of nonclassicality. These real and normalized quasiprobability\ndistributions allow to calculate the expectation value of an operator\nas any classical probability distribution. One such quasiprobability\ndistributions is $Q$ function \\cite{husimi1940some}, and zeros of\nthis function are signature of nonclassicality. Another example is\nWigner \\cite{wigner1932quantum} function whose negative values corresponds\nto the nonclassicality.\n\n\\subsubsection{$Q$ function }\n\n$Q$ function \\cite{husimi1940some} is defined as \n\\begin{equation}\nQ=\\dfrac{1}{\\pi}\\langle\\beta|\\rho|\\beta\\rangle,\\label{eq:Q-function-1}\n\\end{equation}\nwhere $|\\beta\\rangle$ is the coherent state (\\ref{eq:CS}).\n\n\\subsubsection{Wigner function }\n\nAnother quasiprobability distribution is Wigner function formulated\nby Wigner in 1932 \\cite{wigner1932quantum} in the early stage of\nquantum mechanics, the motive was to connect the wavefunction approach\nto a probability distribution in phase space. Negativity of Wigner\nfunction represents the nonclassicality present in an arbitrary quantum\nstate. Also the ability to reconstruct the Wigner function experimentally\nmakes this approach more impactful than any other approach.  Specifically,\nWigner function obtained through optical tomography can be used to\nobtain other quasidistributions, however, Wigner function is stronger\nwitness of nonclassicality than $Q$ function while is not singular\nlike $P$-function.  Mathematically, it is expressed as \n\\begin{equation}\nW\\left(\\gamma,\\gamma^{\\star}\\right)=A\\exp\\left[-2\\left|\\gamma\\right|^{2}\\right]\\int d^{2}\\lambda\\langle-\\lambda|\\rho|\\lambda\\rangle\\exp\\left[-2\\left(\\gamma^{\\star}\\lambda-\\gamma\\lambda^{\\star}\\right)\\right].\\label{eq:wigner-def}\n\\end{equation}\nThe zeros of $Q$ function while the negativity of $P$-function and\nWigner function correspond to the nonclassical behavior of any arbitrary\nquantum state. It is worth stressing here that only\n$P$-function is both necessary and sufficient criterion of nonclassicality,\nwhile rest of the quasidistribution functions are only sufficient.\n\n\\subsection{Measures of nonclassicality \\label{subsec:Measures-of-nonclassicality}}\n\nIn the above section, we have seen that there exist numerous criteria\nof nonclassicality. However, most of these criteria only witness the\nnonclassicality. They do not provide any quantification of the nonclassicality. Except $P$-function and infinite set of vogel's criteria, all other\ncriteria are sufficient but not necessary. However,\nmany efforts have been made for the quantification of nonclassicality, e.g., in 1987, a distance-based measure of nonclassicality was introduced\nby Hillery \\cite{hillery1987nonclassical}. A trace norm based measure \\cite{mari2011directly} was introduced\nby Mari et al., for the set of all states having the positive Wigner\nfunction. In 1991, Lee gave a measure of nonclassicality known as\nnonclassical depth \\cite{lee1991measure}. However, in this work,\nwe will not study these measures. There are\ncertain measures those can be exploited in terms of entanglement,\nlike linear entropy \\cite{wei2003maximal}, which\nwe will use for our calculations and the same is described below.\n\n\\subsubsection{Linear entropy }\n\nIn 2005, a measure of nonclassicality was proposed as entanglement\npotential, which is the amount of entanglement in two output ports\nof a beam splitter with the quantum state $\\rho_{in}$ and vacuum\n$|0\\rangle\\langle0|$ sent through two input ports \\cite{asboth2005computable}.\nThe amount of entanglement quantifies the amount of nonclassicality\nin the input quantum state as classical state can not generate entanglement\nin the output. The post beam splitter state can be obtained as $\\rho_{out}=U\\left(\\rho_{in}\\otimes|0\\rangle\\langle0|\\right)U^{\\dagger}$\nwith $U=\\exp[-iH\\theta]$, where $H=(\\hat{a}^{\\dagger}\\hat{b}+\\hat{a}\\hat{b}^{\\dagger})\/2$,\nand $\\hat{a}^{\\dagger}\\,(\\hat{a})$, $\\hat{b}^{\\dagger}\\,(\\hat{b})$\nare the creation (annihilation) operators of the input modes. For\nexample, considering quantum state ($|\\psi\\rangle=\\sum\\limits _{n=0}^{\\infty}c_{n}|n\\rangle$)\nand a vacuum state $|0\\rangle$ as input states, we can write the\nanalytic expression of the two-mode output state as \n\\begin{equation}\n|\\phi\\rangle=U\\left(|\\psi\\rangle\\otimes|0\\rangle\\right)\\equiv U|\\psi,0\\rangle=\\sum_{n=0}^{\\infty}\\,\\frac{c_{n}}{2^{n\/2}}\\sum_{j=0}^{n}\\sqrt{^{n}C_{j}}\\,\\,|j,\\,n-j\\rangle.\\label{eq:inout_psi}\n\\end{equation}\nWe can measure the amount of entanglement in the output state to quantify\nthe amount of input nonclassicality in $|\\psi\\rangle$. Here, we use\nlinear entropy of single mode subsystem (obtained after tracing over\nthe other subsystem) as entanglement potential. The linear entropy\nfor an arbitrary bipartite state $\\rho_{AB}$ is defined as \\cite{wei2003maximal}\n\\begin{equation}\n\\mathcal{L}=1-{\\rm Tr}\\left(\\rho_{B}^{2}\\right),\\label{eq:le}\n\\end{equation}\nwhere $\\rho_{B}$ is obtained by tracing over subsystem $A$.\n\n\\section{Analytic tools for the study of phase properties of nonclassical\nstates \\label{sec:Analytic-tools-forphase}}\n\nIn this section, we aim to introduce the parameters that are used\nto study phase properties of a given quantum state under consideration\nin this section.\n\n\\subsection{Phase distribution function}\n\nA distribution function allows us to calculate expectation values\nof an operator analogous to that from the corresponding density matrix.\nPhase distribution function for a given density operator \\cite{banerjee2007phase,agarwal1992classical}\ncan be defined as \n\\begin{equation}\nP_{\\theta}=\\frac{1}{2\\pi}\\langle\\theta|\\varrho|\\theta\\rangle,\\label{eq:Phase-Distridution-1}\n\\end{equation}\nwhere the phase state $|\\theta\\rangle$, complementary to the number\nstate $|n\\rangle$, is defined \\cite{agarwal1992classical} as \n\\begin{equation}\n|\\theta\\rangle=\\sum_{n=0}^{\\infty}e^{\\iota n\\theta}|n\\rangle.\\label{eq:phase-1}\n\\end{equation}\n\n\n\\subsection{Phase dispersion}\n\nA known application of phase distribution function (\\ref{eq:Phase-Distridution-1})\nis that it can be used to quantify the quantum phase fluctuation.\nAlthough the variance is also used occasionally as a measure of phase\nfluctuation, it has a drawback that it depends on the origin of phase\nintegration \\cite{banerjee2007phase}. A measure of phase fluctuation,\nfree from this problem, is phase dispersion \\cite{perinova1998phase}\ndefined as \n\\begin{equation}\nD=1-\\left|\\intop_{-\\pi}^{\\pi}d\\theta\\exp\\left[-\\iota\\theta\\right]P_{\\theta}\\right|^{2}.\\label{eq:Dispersion-1}\n\\end{equation}\n\n\n\\subsection{Angular Q function}\n\nAnalogous to the phase distribution $P_{\\theta}$, phase distributions\nare also defined as radius integrated quasidistribution functions\nwhich are used as the witnesses for quantumness \\cite{thapliyal2015quasiprobability}.\nOne such phase distribution function based on the angular part of\nthe $Q$ function is studied in \\cite{leonhardt1993phase,leonhardt1995canonical}.\nSpecifically, the angular $Q$ function is defined as \n\\begin{equation}\nQ_{\\theta_{1}}=\\intop_{0}^{\\infty}Q\\left(\\beta,\\beta^{\\star}\\right)\\left|\\beta\\right|d\\left|\\beta\\right|,\\label{eq:ang-Qf-1}\n\\end{equation}\nwhere the $Q$ function \\cite{husimi1940some} is defined in Eq. (\\ref{eq:Q-function-1}).\n\n\\subsection{Phase fluctuation}\n\nIn attempts to get rid of the limitations of the Hermitian phase operator\nof Dirac \\cite{dirac1927quantum}, Louisell \\cite{louisell1963amplitude}\nfirst mentioned that bare phase operator should be replaced by periodic\nfunctions. As a consequence, sine $(\\mathcal{\\hat{S}})$ and cosine\n$(\\hat{\\mathcal{C}})$ operators appeared, explicit forms of these\noperators were given by Susskind and Glogower \\cite{susskind1964quantum},\nand further modified by Barnett and Pegg \\cite{barnett1986phase}\nas \n\\begin{equation}\n\\mathcal{\\hat{S}}=\\frac{\\hat{a}-\\hat{a}^{\\dagger}}{2\\iota\\left(\\bar{N}+\\frac{1}{2}\\right)^{\\frac{1}{2}}}\\label{eq:fluctuation1-1}\n\\end{equation}\nand \n\\begin{equation}\n\\hat{\\mathcal{C}}=\\frac{\\hat{a}+\\hat{a}^{\\dagger}}{2\\left(\\bar{N}+\\frac{1}{2}\\right)^{\\frac{1}{2}}}.\\label{eq:fluctuation2-1}\n\\end{equation}\nHere, $\\bar{N}$ is the average number of photons in the measured\nfield, and here we refrain our discussion to Barnett and Pegg sine\nand cosine operators \\cite{barnett1986phase}. Carruthers and Nieto\n\\cite{carruthers1968phase} have introduced three quantum phase fluctuation\nparameters in terms of sine and cosine operators \n\\begin{equation}\nU=\\left(\\Delta N\\right)^{2}\\left[\\left(\\Delta\\mathcal{S}\\right)^{2}+\\left(\\Delta\\mathcal{C}\\right)^{2}\\right]\/\\left[\\langle\\mathcal{\\hat{S}}\\rangle^{2}+\\langle\\hat{\\mathcal{C}}\\rangle^{2}\\right],\\label{eq:fluctuation3-1}\n\\end{equation}\n\\begin{equation}\nS=\\left(\\Delta N\\right)^{2}\\left(\\Delta\\mathcal{S}\\right)^{2},\\label{eq:fluctuation4-1}\n\\end{equation}\nand \n\\begin{equation}\nQ=S\/\\langle\\hat{\\mathcal{C}}\\rangle^{2}.\\label{eq:fluctuation5-1}\n\\end{equation}\n These three phase fluctuation parameters $U$, $S$\nand $Q$ show phase properties of PADFS and PSDFS, while $U$ parameter\nis shown relevant as a witness of nonclassicality (antibunching).\n\n\\subsection{Quantum phase estimation parameter}\n\nQuantum phase estimation is performed by sending the input state through\na Mach-Zehnder interferometer and applying the phase to be determined\n($\\phi$) on one of the arms of the interferometer. To study the phase\nestimation using Mach-Zehnder interferometer, angular momentum operators\n\\cite{sanders1995optimal,demkowicz2015quantum}, defined as \n\\begin{equation}\n\\hat{J_{x}}=\\frac{1}{2}\\left(\\hat{a}^{\\dagger}\\hat{b}+\\hat{b}^{\\dagger}\\hat{a}\\right),\\label{eq:ang-mom1-1}\n\\end{equation}\n\\begin{equation}\n\\hat{J_{y}}=\\frac{\\iota}{2}\\left(\\hat{b}^{\\dagger}\\hat{a}-\\hat{a}^{\\dagger}\\hat{b}\\right),\\label{eq:ang-mom2-1}\n\\end{equation}\nand \n\\begin{equation}\n\\hat{J_{z}}=\\frac{1}{2}\\left(\\hat{a}^{\\dagger}\\hat{a}-\\hat{b}^{\\dagger}\\hat{b}\\right),\\label{eq:ang-mom3-1}\n\\end{equation}\nare used. Here, $\\hat{a}$ and $\\hat{b}$ are the annihilation operators\nfor the modes corresponding to two input ports of the Mach-Zehnder\ninterferometer. The average value of $\\hat{J_{z}}$ operator in the\noutput of the Mach-Zehnder interferometer, which is one-half of the\ndifference of photon numbers in the two output ports (\\ref{eq:ang-mom3-1}),\ncan be written as \n\\begin{equation}\n\\langle\\hat{J_{z}}\\rangle=\\cos\\text{\\ensuremath{\\phi}}\\langle\\hat{J_{z}}\\rangle_{in}-\\sin\\text{\\ensuremath{\\phi}}\\langle\\hat{J_{x}}\\rangle_{in}.\n\\end{equation}\nTherefore, variance in the measured value of operator $\\hat{J_{z}}$\ncan be computed as \n\\begin{equation}\n\\left(\\Delta J_{z}\\right)^{2}=\\cos^{2}\\phi\\left(\\Delta{J_{z}}\\right)_{in}^{2}+\\sin^{2}\\phi\\left(\\Delta{J_{x}}\\right)_{in}^{2}-2\\sin\\text{\\ensuremath{\\phi\\,\\cos\\phi\\,}cov}\\left(\\hat{J_{x}},\\hat{J_{z}}\\right)_{in},\n\\end{equation}\nwhere covariance of the two observables is defined as \n\\begin{equation}\n\\text{{\\rm cov}}\\left(\\hat{J_{x}},\\hat{J_{z}}\\right)=\\frac{1}{2}\\langle\\hat{J_{x}}\\hat{J_{z}}+\\hat{J_{z}}\\hat{J_{x}}\\rangle-\\langle\\hat{J_{x}}\\rangle\\langle\\hat{J_{z}}\\rangle.\n\\end{equation}\nThis allows us to quantify precision in phase estimation \\cite{demkowicz2015quantum}\nas \n\\begin{equation}\n\\Delta\\phi=\\frac{\\Delta{J_{z}}}{\\left|\\frac{d\\langle\\hat{J_{z}}\\rangle}{d\\phi}\\right|}.\\label{eq:PE-1}\n\\end{equation}\nBefore we proceed further and conclude this chapter by noting the\nstructure of the rest of the thesis, it would be apt to to note that\nthere exist various methods of quantum state engineering (some of\nwhich have already been mentioned) and photon addition, subtraction,\nfilteration, punching etc., which can be viewed as examples of quantum\nstate engineering processes. In rest of the thesis these processes\nwill be studied with detail.\n\n\\section{Structure of the rest of the thesis\\label{sec:Structure-of-the thesis}}\n\nThis thesis has 6 chapters. The next 4 chapters are focused on the\nstudy of nonclassical and phase properties of the engineered quantum\nstates and the last chapter is dedicated to conclusion. These chapters\nand thus the rest of this thesis is organized as follows.\n\nIn Chapter \\ref{cha:Introduction1}, in Section \\ref{sec:Quantum-states-of-4},\nquantum states of our interest (i.e., PADFS and PSDFS) have been introduced\nin detail. In Chapter \\ref{cha:PADFS-PSDFS}, in Section \\ref{sec:Nonclassicality-witnesses},\nthe analytical expressions of various witnesses of nonclassicality\nare reported. Further, the existence of various lower- and higher-order\nnonclassical features in PADFS and PSDFS are shown through a set of\nplots. Finally, we conclude in Section \\ref{sec:Conclusions}.\n\nIn Chapter \\ref{cha:phase}, in Section \\ref{sec:phase-witnesses},\nwe investigate the phase properties of PADFS and PSDFS from a number\nof perspectives. Finally, the chapter is concluded in Section \\ref{sec:Conclusions-1}.\n\nIn Chapter \\ref{cha:PASDFS}, we describe the quantum state of interest\n(i.e., PASDFS) in Fock basis and calculate the analytic expressions\nfor the higher-order moments of the relevant field operators for this\nstate. In Section \\ref{sec:Nonclassicality-witnesses-2}, we investigate\nthe possibilities of witnessing various nonclassical features in PASDFS\nand its limiting cases by using a set of moments-based criteria for\nnonclassicality. Variations of nonclassical features (witnessed through\ndifferent criteria) with various physical parameters are also discussed\nhere. In Section \\ref{sec:Phase-properties-of}, phase properties\nof PASDFS are studied. $Q$ function for PASDFS is obtained in Section\n\\ref{sec:Qfn}. Finally, we conclude in Section \\ref{sec:Conclusions-5}.\n\nIn Chapter \\ref{cha:QSE-1}, in Section \\ref{sec:Quantum-states-of-1},\nwe have introduced the quantum states of our interest which include\nECS, BS, KS, VFECS, VFBS, VFKS, PAECS, PABS, and PAKS. In Section\n\\ref{sec:Nonclassicality-witnesses-1}, we have investigated the nonclassical\nproperties of these states using various witnesses of lower- and higher-order\nnoncassicality as well as a measure of nonclassicality. Specifically,\nin this section, we have compared nonclassicality features found in\nvacuum filtered and single photon added versions of the states of\nour interest using the witnesses of Higher-order antibunching (HOA),\nHigher-order squeezing (HOS) and Higher-order sub-Poissonian photon\nstatistics (HOSPS). Finally, in Section \\ref{sec:Conclusion}, the\nresults are analyzed, and the chapter is concluded.\n\nFinally, the thesis is concluded in Chapter \\ref{cha:Conclusions-and-Scope},\nwhere we have summarized the findings reported in Chapter \\ref{cha:PADFS-PSDFS}-\\ref{cha:QSE-1}\nand have emphasized on the main conclusion of the present thesis.\nWe have also discussed the scopes of future work.\n\n\n\\chapter{Lower-and higher-order nonclassical properties of photon added and\nsubtracted displaced Fock state\\textsc{\\label{cha:PADFS-PSDFS}}}\n\nIn this chapter,   which is based on \\cite{malpani2019lower}, we aim to study the nonclassical properties of the\nPADFS and PSDFS (which are already introduced in Section \\ref{subsec:Photon-added-and sub})\nusing the witnesses of nonclassicality introduced in Section \\ref{subsec:Witnesses-of-nonclassicality}.\n\n\n\\section{Introduction\\label{sec:Introduction-chap2} }\n\nAs we have mentioned in Chapter \\ref{cha:Introduction1}, with the\nadvent of quantum state engineering \\cite{vogel1993quantum,sperling2014quantum,miranowicz2004dissipation,marchiolli2004engineering}\nand quantum information processing (\\cite{pathak2013elements} and\nreferences therein), the study of nonclassical properties of engineered\nquantum states have become a very important field. Quantum state engineering\nis helpful in realizing non-Gaussianity inducing operations, like\nphoton addition and subtraction \\cite{zavatta2004quantum,podoshvedov2014extraction}.\nKeeping this in mind, in what follows, in this chapter, we aim to\nstudy the nonclassical properties of a set of engineered quantum states\n(both photon added and subtracted) which can be produced by using\nthe above mentioned techniques.\n\n\\begin{figure}[h]\n\\begin{centering}\n\\begin{tabular}{c}\n\\includegraphics[width=100mm]{DFS.pdf}\\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{DFS} A schematic diagram for the generation of PSDFS (in (a)\nand (b)) and PADFS (in (c) and (d)). In (a) and (c) ((b) and (d)),\nsingle-mode (two-mode) squeezed vacuum is used for generation of the\ndesired state. Here, NLC corresponds to nonlinear crystal and D1 and\nD2 are photon number resolving detectors.}\n\\end{figure}\nIt is already mentioned in Chapter 1 that a state having a negative\n$P$-function is referred to as a nonclassical state. Such a state\ncannot be expressed as a mixture of coherent states and does not possess\na classical analogue. In contrast to these states, coherent states\nare classical, but neither their finite dimensional versions \\cite{miranowicz2004dissipation,alam2017higher}\nnor their generalized versions are classical \\cite{satyanarayana1985generalized,thapliyal2016tomograms,thapliyal2015quasiprobability,banerjee2007phase}.\nHere, we would like to focus on photon added and subtracted versions\nof a particular type of generalized coherent state, which is also\nreferred to as the displaced Fock state (DFS). To be precise, state\nof the form $|\\psi\\rangle=D(\\alpha)|n\\rangle$, where $D(\\alpha)$\nis the displacement operator with Fock state $|n\\rangle$, is referred\nto as generalized coherent state (see Section \\ref{sec:Quantum-states-of-4}\nof Chapter \\ref{cha:Introduction1}), as this state is reduced to\na coherent state in the limit $n=0$. However, from the structure\nof the state it seems more appropriate to call this state as the DFS,\nand {this seems to be the nomenclature usually adopted in the literature}\n\\cite{keil2011classical,wunsche1991displaced,blavzek1994high,moya1995generation}.\nIn some other works, it is referred to as displaced number state \\cite{ziesel2013experimental,de2006nonlinear,dodonov2005decoherence},\nbut all these names are equivalent; and in what follows, we will refer\nto it as DFS. This is an extremely interesting quantum state for various\nreasons. Specifically, its relevance in various areas of quantum optics\nis known. For example, in the context of cavity QED, it constitutes\nthe eigenstates of the Jaynes-Cummings systems with coherently driven\natoms \\cite{alsing1992dynamic}. Naturally, various lower-order nonclassical\nproperties and a set of other quantum features of DFS have already\nbeen studied. Specifically, quasiprobability distributions of DFS\nwere studied in \\cite{wunsche1991displaced}, phase fluctuations of\nDFS was investigated in \\cite{zheng1992fluctuation}, decoherence\nof superposition of DFS was discussed in \\cite{dodonov2005decoherence},\n$Q$ function, Wigner function and probability distribution of DFS\nwere studied in \\cite{de1990properties}, Pancharatnam phase of DFS\nhas been studied in \\cite{mendas1993pancharatnam}. Further, in the\ncontext of non-optical DFS, various possibilities of generating DFS\nfrom Fock states by using a general driven time-dependent oscillator\nhas been discussed in \\cite{lo1991generating}; and in the trapped-ion\nsystem, quantum interference effects have been studied for the superposition\nof DFS \\cite{marchiolli2004engineering}. Thus, DFS seems to be a\nwell studied quantum state, but it may be noted that {a little effort}\nhas yet been made to study higher-order nonclassical properties of\nDFS. It is a relevant observation as in the recent past, it has been\nobserved that higher-order nonclassicality has various applications\n\\cite{hillery1999quantum,banerjee2017quantum,sharma2017quantumauction,thapliyal2017protocols},\nand it can be used to detect the existence of weak nonclassical characters\n\\cite{thapliyal2014higher,thapliyal2014nonclassical,verma2010generalized,thapliyal2017comparison,alam2016nonclassical,alam2015approximate,thapliyal2017nonclassicality,prakash2006higher,prakash2010detection,das2018lower}.\nFurther, various higher-order nonclassical features have been experimentally\ndetected \\cite{allevi2012measuring,allevi2012high,avenhaus2010accessing,perina2017higher}.\nHowever, we do not want to restrict to DFS, rather we wish to focus\non lower- and higher-order nonclassical properties of a set of even\nmore general states, namely photon added DFS (PADFS) and photon subtracted\nDFS (PSDFS). The general nature of these states can be visualized\neasily, as in the special case that no photon is added (subtracted)\nPADFS (PSDFS) would reduce to DFS. Further, for $n=0$, PADFS would\nreduce to photon added coherent state which has been widely studied\n(\\cite{agarwal1991nonclassical,verma2008higher,thapliyal2017comparison}\nand references therein) and experimentally realized \\cite{zavatta2004quantum,zavatta2005single}.\nHere it is worth noting that DFS has also been generated experimentally\nby superposing a Fock state with a coherent state on a beam splitter\n\\cite{lvovsky2002synthesis}. Further, an alternative method for the\ngeneration of DFS has been proposed by Oliveira et al., \\cite{de2005alternative}.\nFrom the fact that photon added coherent state and DFS have already\nbeen generated experimentally, and the fact that the photon added\nstates can be prepared via conditional measurement on a beam splitter,\nit appears that PADFS and PSDFS can also be built in the lab. In fact,\ninspired by these experiments, we have proposed a schematic diagram\nfor generating the PADFS and PSDFS in Figure \\ref{DFS} using single-mode\nand two-mode squeezed vacuum states. Specifically, using three (two)\nhighly transmitting beam splitters, a conditional measurement of single\nphotons at both detectors D1 and D2 in Figure \\ref{DFS} (a) (Figure\n\\ref{DFS} (b)) would result in single photon subtracted DFS as output\nfrom a single-mode (two-mode) squeezed vacuum state. Similarly, to\ngenerate PADFS conditional subtraction of photon is replaced by photon\naddition, using a nonlinear crystal and heralding one output mode\nfor a successful measurement of a single photon to ensure generation\nof single photon added DFS. This fact, their general nature, and the\nfact that nonclassical properties of PADFS and PSDFS have not yet\nbeen {accorded sufficient attention, has} motivated us to perform\nthis study.\n\nMotivated by the above facts, in what follows, we investigate the\npossibilities of observing lower- and higher-order sub-Poissonian\nphoton statistics, antibunching and squeezing in PADFS and PSDFS.\nWe have studied nonclassical properties of these states through a\nset of other witnesses of nonclassicality, e.g., zeros of $Q$ function,\nMandel $Q_{M}$ parameter, Klyshko's criterion, and Agarwal-Tara's\ncriterion. These witnesses of nonclassicality successfully establish\nthat both PADFS and PSDFS (along with most of the states to which\nthese two states reduce at different limits) are highly nonclassical.\nThus, {making use of the analytical expressions of moments of creation\nand annihilation operators, discussed below, facilitates an analytical\nunderstanding for most of the nonclassical witnesses.}\n\n\\section{Higher-order moment for PADFS and PSDFS \\label{sec:Quantum-states-of}}\n\nWe have already mentioned that we are interested in PADFS and PSDFS.\nIn what follows, we will see that various experimentally measurable\nnonclassicality witnesses can be expressed as the moments of annihilation\nand creation operators \\cite{allevi2012measuring,allevi2012high,avenhaus2010accessing,perina2017higher,miranowicz2010testing}.\nTo utilize those witnesses to identify the signatures of nonclassicality,\nwe will compute an analytic expression for the most general moment,\n$\\langle\\hat{a}^{\\dagger q}\\hat{a}^{r}\\rangle$, with $q$ and $r$\nbeing non-negative integers. This is the most general moment in the\nsense that any other moment can be obtained as a special case of it.\nFor example, if we need $\\langle\\hat{a}^{2}\\rangle,$ we would just\nrequire to consider $q=2$ and $r=0$. Thus, an analytic expression\nfor $\\langle\\hat{a}^{\\dagger q}\\hat{a}^{r}\\rangle$ would essentially\nhelp us to obtain analytic expression for any moment-based witness\nof nonclassicality. Further, the analytic expressions of moment obtained\nfor PADFS and PASDFS would also help us to obtain nonclassical features\nin the set of states obtained in the limiting cases, like Fock state,\nDFS, photon added coherent state. Keeping this in mind, we have computed\n$\\langle\\psi_{+}(u,n,\\alpha)|\\hat{a}^{\\dagger q}\\hat{a}^{r}|\\psi_{+}(u,n,\\alpha)\\rangle$\nand $\\langle\\psi_{-}(v,n,\\alpha)|\\hat{a}^{\\dagger q}\\hat{a}^{r}|\\psi_{-}(v,n,\\alpha)\\rangle$\n using Eq. (\\ref{eq:PADFS}) and (\\ref{eq:PSDFS}),\nand provide the final analytic expressions of these moments without\ngoing into the mathematical details to maintain the flow of the chapter.\nThe obtained expressions for the above mentioned moments for PADFS\nand PSDFS are {\\small{}\n\\begin{eqnarray}\\label{eq:PA-expectation}\n\\langle\\hat{a}^{\\dagger q}\\hat{a}^{r}\\rangle_{{\\rm PADFS}} & = & \\langle\\psi_{+}(u,n,\\alpha)|\\hat{a}^{\\dagger q}\\hat{a}^{r}|\\psi_{+}(u,n,\\alpha)\\rangle\\nonumber \\\\\n & = & \\frac{N_{+}^{2}}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\\\\n & \\times & \\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum\\limits _{m=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m+p-p'-r+q}(m+p+u)!(m+p+u-r+q)!}{m!(m+p-p'-r+q)!(m+p+u-r)!},\\nonumber \n\\end{eqnarray}\n}and {\\small{}\n\\begin{eqnarray}\\label{eq:PS-expectation}\n\\langle\\hat{a}^{\\dagger q}\\hat{a}^{r}\\rangle_{{\\rm PSDFS}} & = & \\langle\\psi_{-}(v,n,\\alpha)|\\hat{a}^{\\dagger q}\\hat{a}^{r}|\\psi_{-}(v,n,\\alpha)\\rangle\\nonumber \\\\\n & = & \\frac{N_{-}^{2}}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\nonumber \\\\\n & \\times & \\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum\\limits _{m=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{*})^{m+p-p'-r+q}(m+p)!(m+p-r+q)!}{m!(m+p-p'-r+q)!(m+p-v-r)!},\n\\end{eqnarray}\n}respectively. The values of normalization constants for PADFS and\nPSDFS are already given in Eqs. (\\ref{eq:norP}) and (\\ref{eq:norS}),\nrespectively. In the following section, we shall investigate the possibilities\nof observing various types lower- and higher-order nonclassical features\nin PADFS and PSDFS by using Eqs. (\\ref{eq:PA-expectation}) and\n(\\ref{eq:PS-expectation}).\n\n\\section{Nonclassical features of PADFS and PSDFS \\label{sec:Nonclassicality-witnesses}}\n\nThe moments of number operators for PADFS and PSDFS states obtained\nin the previous section enable us to study nonclassical properties\nof these states using a set of moments-based criteria of nonclassicality\n\\cite{miranowicz2010testing}\\cite{naikoo2018probing}. In the recent\npast, an infinite set of these moments-based criteria is shown to\nbe equivalent to the $P$-function-based criterion, i.e., it becomes\nboth necessary and sufficient \\cite{richter2002nonclassicality,shchukin2005nonclassical}.\nHowever, in this section, we will use a subset of this infinite set\nas witnesses of nonclassicality to investigate various nonclassical\nproperties of the PADFS and PSDFS. Specifically, nonclassicality will\nbe witnessed through Mandel $Q_{M}$ parameter, criteria of lower-\nand higher-order antibunching, Agarwal-Tara's criterion, Klyshko's\ncriterion, criteria of higher-order sub-Poissonian photon statistics,\nzeros of $Q$ function, etc. As all these criteria are already introduced\nin Section \\ref{sec:Nonclassicality-witnesses},\\textcolor{blue}{{}\n}here we may discuss the plots and results.\n\n\\subsection{Mandel $Q_{M}$ Parameter}\n\nNegativity of this parameter indicates nonclassicality which can be\ncalculated using Eqs. (\\ref{eq:PA-expectation}) and (\\ref{eq:PS-expectation}).\nIn Figure \\ref{Mandel-Q-parameter}, the dependence of $Q_{M}$ on\nthe state parameter $\\alpha$ and non-Gaussianity inducing parameters\n (i.e., photon addition, subtraction, and Fock parameters\nas they can induce non-Gaussianity in a quantum state) is shown.\nSpecifically, variation of $Q_{M}$ parameter for PADFS and PSDFS\nis shown with state parameter $\\alpha$, where the effect of the number\nof photons added\/subtracted and the initial Fock state is also established.\nFor $\\alpha=0$, the PADFS with an arbitrary number of photon addition\nhas $Q_{M}$ parameter -1, which can be attributed to the fact that\nfinal state, which reduces to the Fock state ($|1\\rangle$ chosen\nto be displaced in this case) is the most nonclassical state (cf.\nFigure \\ref{Mandel-Q-parameter} (a)). With increase in the number\nof photons added to the DFS, the depth of nonclassicality witness\n$Q_{M}$ increases. However, the witness of nonclassicality becomes less negative for higher values\nof the displacement parameter. In contrast to the photon addition,\nwith the subtraction of photons from the DFS the $Q_{M}$ parameter\nbecomes almost zero for the smaller values of displacement parameter\n$\\alpha$ in DFS as shown in Figure \\ref{Mandel-Q-parameter} (c).\nThis behavior can be attributed to the fact that photon subtraction\nfrom $D\\left(\\alpha\\right)|1\\rangle$ for small values of $\\alpha$\nwill most likely yield vacuum state. Also, with the increase in the\ndisplacement parameter the witness of nonclassicality becomes more\nnegative as with a higher average number of photons in DFS photon\nsubtraction becomes more effective. However, for the larger values\nof displacement parameter the nonclassicality disappears analogous\nto the PADFS.   For large values of $\\alpha$, this\nparameter dominates in the behavior of the state and thus it behaves\nanalogous to coherent state.\n\nAs Fock states are known to be nonclassical, and photon addition and\nsubtraction are established as nonclassicality inducing operations,\nit would be worth comparing the effect of these two independent factors\nresponsible for the observed nonclassical features in the present\ncase. To perform this study, we have shown the variation of the single\nphoton added (subtracted) DFS with different initial Fock states in\nFigure \\ref{Mandel-Q-parameter} (b) (Figure \\ref{Mandel-Q-parameter}\n(d)). Specifically, the nonclassicality present in PADFS decays faster\nfor the higher values of the Fock states with increasing displacement\nparameter (cf. Figure \\ref{Mandel-Q-parameter} (b)). However, such\nnature was not present in PSDFS shown in Figure \\ref{Mandel-Q-parameter}\n(d). Note that variation of $Q_{M}$ parameter with $\\alpha$ starts\nfrom 0 (-1) iff $u\\leq n$ $\\left(u>n\\right)$. For instance, if $u=n=1$,\ni.e., corresponding to state $\\hat{a}D\\left(\\alpha\\right)|1\\rangle$,\nnonclassicality witness is zero for $\\alpha=0$ as it corresponds\nto vacuum state. Therefore, the present study reveals that photon\naddition is a stronger factor for the nonclassicality present in the\nstate when compared to the initial Fock state chosen to be displaced.\nWhereas photon subtraction is a preferred choice for large values\nof displacement parameter in contrast to the higher values of Fock\nstates to displace with small $\\alpha$. Among photon addition and\nsubtraction, addition is a preferred choice for the smaller values\nof displacement parameter, while the choice between addition and subtraction\nbecomes immaterial for large $\\alpha$.\n\n\\begin{figure}\n\\centering %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{PAGCS-mandel-diffphoton.pdf}  & \\includegraphics[width=60mm]{PAGCS-mandel-diffnp.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{PSGCS-mandel-diffphoton.pdf}  & \\includegraphics[width=60mm]{PSGCS-mandel-diffnp.pdf} \\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{Mandel-Q-parameter} Variation of Mandel $Q_{M}$ parameter\nfor PADFS (in (a) and (b)) and PSDFS (in (c) and (d)) is shown with\ndisplacement parameter $\\alpha$. In (a) and (c), the value of number\nof photons added\/subtracted (i.e., $u$ or $v$) is changed for the\nsame initial Fock state $|1\\rangle$. Different initial Fock states\n$|n\\rangle$ are chosen to be displaced in (b) and (d) for the single\nphoton addition\/subtraction.  The blue curve corresponds\nto vacuum for $\\alpha=0$ and thus starts from $0$ unlike rest of\nthe states which are Fock state ($n\\protect\\neq0)$ in the limiting\ncase. Therefore, nonclassicality increases first with increasing $\\alpha$\nbefore decreasing as in the rest of the cases.}\n\\end{figure}\n\n\\subsection{Lower- and higher-order antibunching}\n\nThe nonclassicality reflected by the lower-order antibunching criterion\nobtained here is the same as Mandel $Q_{M}$ parameter  $\\left(Q_{M}=\\frac{d\\left(1\\right)}{\\left\\langle \\hat{a}^{\\dagger}\\hat{a}\\right\\rangle }\\right)$\nillustrated in Figure \\ref{Mandel-Q-parameter}. Therefore we will\nrather discuss here the possibility of observing higher-order antibunching\nin the quantum state of our interest using Eqs. (\\ref{eq:PA-expectation})\nand (\\ref{eq:PS-expectation}) in Eq. (\\ref{eq:HOA-1}). Specifically,\nthe depth of nonclassicality witness can be observed to increase with\norder for both PADFS and PSDFS as depicted in Figure \\ref{HOA} (a)\nand (d). This fact is consistent with the earlier observations (\\cite{thapliyal2014higher,thapliyal2014nonclassical,thapliyal2017comparison,thapliyal2017nonclassicality,alam2017lower}\nand references therein) that higher-order nonclassicality criteria\nare useful in detecting weaker nonclassicality. On top of that the\nhigher-order antibunching can be observed for larger values of displacement\nparameter $\\alpha$, when lower-order antibunching is not present.\nThe presence of higher-order nonclassicality in the absence of its\nlower-order counterpart establishes the relevance of the present study.\n\nThe depth of nonclassicality parameter  ($d\\left(l-1\\right)$)\nwas observed to decrease with an increase in the number of photons\nsubtracted from DFS for small values of $\\alpha$ in Figure \\ref{Mandel-Q-parameter}\n(c). A similar nature is observed in Figure \\ref{HOA} (e), which\nshows that for the higher values of displacement parameter, the depth\nof higher-order nonclassicality witness increases with the number\nof photon subtraction. Therefore, not only the depth of nonclassicality\nbut the range of displacement parameter for the presence of higher-order\nantibunching also increases with photon addition\/subtraction (cf.\nFigure \\ref{HOA} (b) and (e)). With the increase in the value of\nFock state parameter $n$, the depth of higher-order nonclassicality\nwitness increases (decreases) for smaller (larger) values of displacement\nparameter in both PADFS and PSDFS as shown in Figure \\ref{HOA} (c)\nand (f), respectively. Thus, we have observed that the range of $\\alpha$\nwith the presence of nonclassicality increases (decreases) with photon\naddition\/subtraction (Fock state) in DFS.\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{ccc}\n\\includegraphics[width=50mm]{HOA-pagcs.pdf}  & \\includegraphics[width=50mm]{HOA-pagcs-addphoton.pdf}  & \\tabularnewline\n(a)  & (b)  & \\tabularnewline\n\\includegraphics[width=50mm]{HOA-pagcs-addphoton-diff-displaced.pdf}  & \\includegraphics[width=50mm]{HOA-psgcs.pdf} & \\tabularnewline\n &  & \\tabularnewline\n(c)  & (d)  & \\tabularnewline\n\\includegraphics[width=50mm]{HOA-psgcs-subphoton.pdf}  & \\includegraphics[width=50mm]{HOA-psgcs-subphoton-diff-displaced.pdf}  & \\tabularnewline\n(e)  & (f)  & \\tabularnewline\n\\end{tabular}\\caption{\\label{HOA} The presence of higher-order antibunching is shown as\na function of $\\alpha$ for PADFS (in (a)-(c)) and PADFS ((d)-(f)).\nSpecifically, (a) and (d) illustrate comparison between lower- and\nhigher-order antibunching. {It should be noted that some of the curves\nare multiplied by a scaling factor in order to present them in one\nfigure}. Figures (b) and (e) show the effect of photon addition\/subtraction,\nand (c) and (f) establish the effect of Fock state chosen to displace\nin PADFS and PSDFS, respectively. Here, without the loss of generality, we have used the notation $\\psi_{+}(u, n, l-1)$ ($\\psi_{-}(u, v, l-1)$) for nonclassicality in photon added (subtracted) scenarios, and will follow this notation in subsequent figures of this chapter.}\n\\end{figure}\n\n\\subsection{Higher-order sub-Poissonian photon statistics}\n\nThis allows us to study higher-order sub-Poissonian photon statistics\nusing Eqs. (\\ref{eq:PA-expectation}) and (\\ref{eq:PS-expectation})\nin Eq. (\\ref{eq:hosps22-1}). The presence of higher-order sub-Poissonian\nphoton statistics (as can be seen in Figure \\ref{HOSPS} (a) and (d)\nfor PADFS and PSDFS, respectively) is dependent on the order of nonclassicality\nunlike higher-order antibunching, which is observed for all orders.\nSpecifically, the nonclassical feature was observed only for odd orders,\nwhich is consistent with some of the earlier observations \\cite{thapliyal2017comparison},\nwhere nonclassicality in those cases could be induced due to squeezing.\n   Along the same line, we\nexpect to observe the nonclassicality in such cases with appropriate\nuse of squeezing as a useful quantum resource, which will be discussed\nelsewhere. In case of photon addition\/subtraction in DFS, a behavior\nanalogous to that observed for higher-order antibunching is observed,\ni.e., the depth of nonclassciality increases with the photon addition\nwhile it decreases (increases) for small (large) values of $\\alpha$\n(cf. Figure \\ref{HOSPS} (b) and (e)). Similar to the previous case,\nnonclassicality can be observed to be present for larger values of\ndisplacement parameter with photon addition\/subtraction, while increase\nin the value of Fock parameter has an opposite effect.\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{ccc}\n\\includegraphics[width=60mm]{HOSPS-PAGCS-diffl.pdf}  & \\includegraphics[width=60mm]{HOSPS-PAGCS-diffphoton.pdf}  & \\tabularnewline\n(a)  & (b)  & \\tabularnewline\n\\includegraphics[width=60mm]{HOSPS-PAGCS-diffnp.pdf}  & \\includegraphics[width=60mm]{HOSPS-PSGCS-varyl.pdf}  & \\tabularnewline\n(c)  & (d)  & \\tabularnewline\n\\includegraphics[width=60mm]{HOSPS-Psgcs-sub-diffphoton.pdf}  & \\includegraphics[width=60mm]{HOSPS-Psgcs-sub-diffnp.pdf}  & \\tabularnewline\n(e)  & (f)  & \\tabularnewline\n\\end{tabular}\\caption{\\label{HOSPS} Dependence of higher-order sub-Poissonian photon statistics\non $\\alpha$ for PADFS (in (a)-(c)) and PSDFS ((d)-(f)) is illustrated\nhere. Specifically, (a) and (d) show the increase in depth of nonclassicality\nwitness with order, (b) and (e) depict the effect of photon addition\nand subtraction, respectively, and (c) and (f) establish the effect\nof choice of Fock state to be displaced in PADFS and PSDFS, respectively. }\n\\end{figure}\n\n\\subsection{Higher-order squeezing}\n\nThe analytical expressions of the nonclassicality witness of the Hong-Mandel\ntype higher-order squeezing criterion for PADFS and PSDFS can be obtained\nwith the help of Eqs. (\\ref{eq:PA-expectation}), (\\ref{eq:PS-expectation}),\nand (\\ref{eq:Hong-Def})-(\\ref{eq:cond2.1-1}). We have investigated\nthe higher-order squeezing and depict the result in Figure \\ref{fig:HOS}\nassuming $\\alpha$ to be real. Incidentally, we could not establish\nthe presence of higher-order squeezing phenomena in PADFS (Figure\n\\ref{fig:HOS} (a)-(c)). However, the depth of the higher-order squeezing\nwitness increases with order for small values of $\\alpha$ as shown\nin Figure \\ref{fig:HOS} (a), while for the higher values of the displacement\nparameter, higher-order squeezing disappear much quicker (cf. Figure\n\\ref{fig:HOS} (d)). With increase in the number of photons subtracted,\nthe presence of this nonclassicality feature can be maintained for\nthe higher values of displacement parameter as well (cf. Figure \\ref{fig:HOS}\n(e)). In general, photon subtraction is a preferred mode for nonclassicality\nenhancement as far as this nonclassciality feature is concerned. The\nchoice of the initial Fock state is also observed to be relevant as\nthe depth of squeezing parameter can be seen increasing with value\nof the Fock parameter for PSDFS in the small displacement parameter\nregion (shown in Figure \\ref{fig:HOS} (f)), where this nonclassical\nbehavior is also shown to succumb to the higher values of Fock and\ndisplacement parameters. Unlike the other nonclassicalities discussed\nso far, the observed squeezing also depends on phase $\\theta$ of\nthe displacement parameter $\\alpha=|\\alpha|\\exp\\left(i\\theta\\right)$\ndue to the second last term in Eq. (\\ref{eq:cond2.1-1}). We failed\nto observe this nonclassicality behavior in PADFS even by controlling\nthe value of the phase parameter (also shown in Figure \\ref{fig:HOS-diff-phase}\n(a)). For PSDFS, the squeezing disappears for some particular values\nof the phase parameter, while the observed squeezing is maximum for\n$\\theta=n\\pi$ (see Figure \\ref{fig:HOS-diff-phase} (b)). It thus\nestablishes the phase parameter of the displacement operator as one\nmore controlling factor for nonclassicality in these engineered quantum\nstates.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{ccc}\n\\includegraphics[width=45mm]{HOS-pagcs-difforder.pdf}  & \\includegraphics[width=45mm]{HOS-pagcs-diffphoton.pdf}  & \\includegraphics[width=45mm]{HOS-pagcs-diffn.pdf} \\tabularnewline\n(a)  & (b)  & (c) \\tabularnewline\n\\includegraphics[width=45mm]{HOS-psgcs-difforder.pdf}  & \\includegraphics[width=45mm]{HOS-psgcs-diffphoton.pdf}  & \\includegraphics[width=45mm]{HOS-psgcs-diff-n.pdf} \\tabularnewline\n(d)  & (e)  & (f) \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:HOS} Illustration of the higher-order squeezing using\nHong-Mandel criterion as the function of displacement parameter. In\n(a) and (d), dependence of the observed nonclassicality on different\norders ($l$) is shown for PADFS and PSDFS, respectively; while in\n(b) and (e), the effect of variation in the number of photon added\/subtracted\nis shown in case of PADFS and PSDFS, respectively. In (c) and (f),\nthe variation due to change in the initial Fock state chosen to be\ndisplaced is shown for PADFS and PSDFS, respectively. }\n\\end{figure}\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{HOS-pagcs-diffphase.pdf}  & \\includegraphics[width=60mm]{HOS-psgcs-diffphase.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:HOS-diff-phase} Hong-Mandel type higher-order squeezing\nfor PADFS and PSDFS is shown dependent on the phase of the displacement\nparameter $\\alpha=|\\alpha|\\exp\\left(i\\theta\\right)$ in (a) and (b),\nrespectively.}\n\\end{figure}\n\n\\subsection{$Q$ function}\n\nUsing Eqs. (\\ref{eq:PADFS}) and (\\ref{eq:PSDFS}) in (\\ref{eq:Q-function-1}),\nwe obtain the analytic expressions for the Husimi $Q$ function for\nPADFS and PSDFS as\n\n\\begin{equation}\n\\begin{array}{lcl}\nQ_{+} & = & \\frac{N_{+}^{2}}{\\pi}\\frac{\\exp\\left[-\\mid\\beta\\mid^{2}\\right]}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\\\\n & \\times & \\sum\\limits _{m,m^{\\prime}=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m^{\\prime}}\\beta^{(m^{\\prime}+p'+u)}(\\beta^{\\star})^{(m+p+u)}}{m!m^{\\prime}!}\n\\end{array}\\label{eq:Q-PADFS}\n\\end{equation}\nand \n\\begin{eqnarray}\n\\begin{array}{lcl}\nQ_{-} & = & \\frac{N_{-}^{2}}{\\pi}\\frac{\\exp\\left[-\\mid\\beta\\mid^{2}\\right]}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\\\\n & \\times & \\sum\\limits _{m,m^{\\prime}=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m^{\\prime}}\\beta^{(m^{\\prime}+p'-v)}(\\beta^{\\star})^{(m+p-v)}(m+p)!(m^{\\prime}+p')!}{m!m^{\\prime}!(m+p-v)!(m^{\\prime}+p'-v)!},\n\\end{array}\\label{eq:Q-PSDFS}\n\\end{eqnarray}\nrespectively. We failed to observe the nonclassical features reflected\nbeyond moments based nonclassicality criteria through a quasiprobability\ndistribution, i.e., zeros of the $Q$ function. We have shown the\n$Q$ function in Figure \\ref{fig:Q-function}, where the effect of\nphoton addition\/subtraction and the value of Fock parameter on the\nphase space distribution is shown. Specifically, it is observed that\nthe value of Fock parameter affects the quasidistribution function\nmore compared to the photon addition\/subtraction.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{cc}\n\\includegraphics[width=100mm]{Q-fun.jpg} \n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:Q-function} Contour plots of the $Q$ function for (a)\nsingle photon added displaced Fock $|1\\rangle$ state, (b) two photon\nadded displaced Fock $|1\\rangle$ state, (c) single photon added displaced\nFock $|2\\rangle$ state, (d) single photon subtracted displaced Fock\n$|1\\rangle$ state, (e) two photon subtracted subtracted displaced\nFock $|1\\rangle$ state, (f) single photon subtracted displaced Fock\n$|2\\rangle$ state. In all cases, $\\alpha=\\sqrt{2}\\exp\\left(\\frac{i\\pi}{4}\\right)$\nis chosen. }\n\\end{figure}\n\n\\subsection{Agarwal-Tara's criterion}\n\nThe analytic expression of $A_{3}$ parameter defined in Eq. (\\ref{eq:Agarwal-1})\n{can} be obtained for PADFS and PSDFS using Eqs. (\\ref{eq:PA-expectation})\nand (\\ref{eq:PS-expectation}). The nonclassical properties of the\nPADFS and PSDFS using Agarwal-Tara's criterion are investigated, and\nthe corresponding results are depicted in Figure \\ref{fig:A3}, which\nshows highly nonclassical behavior of the {states generated by engineering}.\nSpecifically, the negative part of the curves, which is bounded by\n-1, ensures the existence of the nonclassicality. From Figure \\ref{fig:A3},\nit is clear that $A_{3}$ is 0 (-1) for the displacement parameter\n$\\alpha=0$ because then DFS, PADFS, and PSDFS reduce to Fock state,\nand $A_{3}=0$ (-1) for the Fock state parameter $n=0,\\,1$ $\\left(n>1\\right)$.\nNonclassicality reflected through $A_{3}$ parameter increases (decreases)\nwith photon addition (subtraction) (shown in Figure \\ref{fig:A3}\n(a) and (c)). In contrast, the Fock parameter has a completely opposite\neffect that it leads to decrease (increase) in the observed nonclassicality\nfor PADFS (PSDFS), which can be seen in Figure \\ref{fig:A3} (b) and\n(d). However, for larger values of displacement parameter, the depth\nof nonclassicality illustrated through this parameter can again be\nseen to increase (cf. Figure \\ref{fig:A3} (b)).\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{PAGCS-Agarwal-diffphoton.pdf}  & \\includegraphics[width=60mm]{PAGCS-Agarwal-diffnp.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{PSGCS-Agarwal-diffphoton.pdf}  & \\includegraphics[width=60mm]{PSGCS-Agarwal-diffnp.pdf} \\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:A3} Variation of Agarwal-Tara's parameter with $\\alpha$\nfor PADFS and PSDFS is shown in (a)-(b) and (c)-(d), respectively.\nSpecifically, the effect of photon addition\/subtraction (in (a) and\n(c)) and the choice of Fock state ((b) and (d)) on the presence of\nnonclassicality in PADFS and PSDFS is illustrated.}\n\\end{figure}\n\n\\subsection{Klyshko's Criterion}\n\nThe analytic expression for the $m$th photon-number distribution\nfor PADFS and PSDFS can be calculated (using $q=r=1$) from Eqs. (\\ref{eq:PA-expectation})\nand (\\ref{eq:PS-expectation}), respectively.\n\n\\begin{figure}\n\\centering %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{Klyshko-photoadd.pdf}  & \\includegraphics[width=60mm]{Klyshko-photoadddiffnp.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{Klyshko-photosub.pdf}  & \\includegraphics[width=60mm]{Klyshko-photosub-diffnp.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Klyshko} Illustration of the Klyshko's criterion. Variation\nof $B(m)$ with respect to $m$ (a) and (c) for different values of\nthe number of the photon additon\/subtraction for PADFS and PSDFS,\nrespectively; (b) and (d) for different values of the number of the\nFock state parameter for PADFS and PSDFS, respectively. Here, we have\nchosen $\\alpha=1$ in all cases  because almost all\nof the other criteria detected nonclassicality for this choice of\n$\\alpha$ so does the Klyshko's criterion.}\n\\end{figure}\nThe advantage of the Klyshko's criterion over any other existing moments\nbased criteria is that a very small amount of information is required.\nSpecifically, probability of only three successive photon numbers\nis sufficient to investigate the nonclassical property. The negative\nvalues of $B(m)$ serve as the witness of nonclassicality. Klyshko's\ncriterion in Eq. (\\ref{eq:Klyshko-1}) is derived analytically and\nthe corresponding nonclassical properties for both PADFS and PSDFS\nare investigated (cf. Figure \\ref{fig:Klyshko}). Specifically, the\nnegative values of $B(m)$ are observed for different values of $m$\nin case of photon addition and subtraction (cf. Figure \\ref{fig:Klyshko}\n(a) and (c)) being the signature of nonclassicality induced via independent\noperations. Additionally, one can also visualize that due to the photon\naddition (subtraction) the negative peaks in the values of $B(m)$\nshift to higher (lower) photon number regime. A similar observation\nis obtained for different values of the Fock state parameter for the\nPADFS and PSDFS, where the negative values of the witness of nonclassicality\nget amplified towards higher photon number regime and becomes more\nnegative, and the corresponding results are shown in Figure \\ref{fig:Klyshko}\n(b) and (d), respectively. This further establishes the relevance\nof operations, like photon addition, subtraction, and starting with\nFock states in inducing nonclassicality in the engineered quantum\nstates.\n\n\\section{Conclusions \\label{sec:Conclusions}}\n\nThe only Fock state that does not show any nonclassical feature is\nthe vacuum state \\cite{miranowicz2015statistical}, and displacement\noperator preserves its classicality. All the rest of the Fock states\nare maximally nonclassical and they are shown to remain nonclassical\neven after application of a displacement operator. Here, we set ourselves\na task: What happens when the displacement operator applied on a Fock\nstate is followed by addition or subtraction of photon(s)? Independently,\nphoton addition and subtraction are already established as nonclassicality\ninducing operations in case of displaced vacuum state (i.e., coherent\nstate). In this chapter, we have established that photon addition\/subtraction\nis not only nonclassicality inducing operation, it can also enhance\nthe nonclassicality present in the DFS. It's expected that these operations\nwould increase the amount of nonclassicality (more precisely, the\ndepth of the nonclassicality witnessing parameter) present in other\nnonclassical states, too. There is one more advantage of studying\nthe nonclassical features of PADFS and PSDFS. These states can be\nreduced to a class of quantum states, most of which are already experimentally\nrealized and found useful in several applications. Inspired by the\navailable experimental results, we have also proposed optical designs\nfor generation of PADFS and PSDFS from the squeezed vacuum state.\n\nTo analyze the nonclassical features of the engineered final states,\ni.e., PADFS and PSDFS, we have used a set of moments based criteria,\nnamely Mandel $Q_{M}$ parameter, Agarwal-Tara's $A_{3}$ parameter,\ncriteria for higher-order antibunching, sub-Poissonian photon statistics,\nand squeezing. In addition, the nonclassical features have been investigated\nthrough Klyshko's criterion and a quasiprobability distribution--$Q$\nfunction. The states of interest are found to show a large variety\nof nonclassical features as all the nonclassicality witnesses used\nhere (except $Q$ function) are found to detect the nonclassicality.\n They show that state is useful as squeezed, antibunched,\nas well as generation of entangled state.\n\nThis study has revealed that the amount of nonclassicality in PADFS\nand PSDFS can be controlled by the Fock state parameter, displacement\nparameter, the number of photons added or subtracted. In general,\nthe amount of nonclassicality with respect to the witness used here\nis found to increase with the number of photons added\/subtracted,\nwhile smaller values of Fock state and displacement parameters are\nobserved to be preferable for the presence of various nonclassical\nfeatures. On some occasions, nonclassicality has also been observed\nto increase with the Fock state parameter, while larger values of\ndisplacement parameter always affect the nonclassicality adversely.\nMost of the nonclassicality criteria used here, being moments-based\ncriteria, could not demonstrate the effect of phase parameter of the\ndisplacement parameter. Here, higher-order squeezing witness and $Q$\nfunction are found to be dependent on the phase of the displacement\nparameter. However, only higher-order squeezing criterion was able\nto detect nonclassicality, and thus established that this phase parameter\ncan also be used to control the amount of nonclassicality.\n\nFurther, in the past, it has been established that higher-order nonclassicality\ncriteria have an advantage in detecting weaker nonclassicality. We\nhave also shown that the depth of nonclassicality witness increases\nwith order of nonclassicality thus providing an advantage in the experimental\ncharacterization of the observed nonclassical behavior. \n\\chapter{Quantum phase properties of photon added and subtracted displaced\nFock states\\textsc{{\\label{cha:phase}}}}\n\nIn this chapter, the motive is to observe the phase properties of\nPADFS and PSDFS.  The main findings of this chapter\nare published in \\cite{malpani2019quantum}\n\n\\section{{Introduction\\label{sec:Introduction-chap3}}}\n\nIn the previous chapter, the nonclassical properties of PADFS and\nPSDFS were studied. Here, our specific interest is to study the phase\nproperties of PADFS and PSDFS and their limiting cases. In the recent\npast, the nonclassical properties of this set of engineered quantum\nstates, many of which have been experimentally generated \\cite{lvovsky2001quantum,lvovsky2002synthesis,zavatta2004quantum,zavatta2005single,zavatta2008subtracting},\nwere focus of various studies (see \\cite{malpani2019lower} and references\ntherein). In Section \\ref{subsec:Photon-added-and sub}, we have already\nexpressed PADFS and PSDFS as superposition of Fock states. Further,\nin Section \\ref{sec:Analytic-tools-forphase} we have described the\nparameters used for the study of phase properties of a quantum state.\nIn that context we have already mentioned several applications of\nquantum phase distribution and quantum phase fluctuation.\n\nTo stress on the recently reported applications of quantum phase distribution\nand quantum phase fluctuation, we note that these have applications\nin quantum random number generation \\cite{xu2012ultrafast,raffaelli2018soi},\ncryptanalysis of squeezed state based continuous variable quantum\ncryptography \\cite{horak2004role}, generation of solitons in a Bose-Einstein\ncondensate \\cite{denschlag2000generating}, in phase encoding quantum\ncryptography \\cite{gisin2002quantum}, phase imaging of cells and\ntissues for biomedical application \\cite{park2018quantitative}; as\nwell as have importance in determining the value of transition temperature\nfor superconductors \\cite{emery1995importance}. Keeping these applications\nand the general nature of engineered quantum states PADFS and PSDFS\nin mind, in what follows, we aim to study phase distribution, $Q$\nphase, phase fluctuation measures, phase dispersion, and quantum phase\nestimation using the concerned states and the states obtained in the\nlimiting cases. As PADFS and PSDFS are already described, we may begin\nthis study by describing limiting cases of these states as our states\nof interest.\n\nWe have already mentioned that our focus would be on PADFS and PSDFS.\nDue to the general form of PADFS and PSDFS, a large number of states\ncan be obtained in the limiting cases. Some of the important limiting\ncases of PADFS and PSDFS in the present notation are summarized in\nTable \\ref{tab:state}. This table clearly establishes that the applicability\nof the results obtained in the present study is not restricted to\nPADFS and PSDFS; rather an investigation of the phase properties of\nPADFS and PSDFS would also reveal phase properties of many other quantum\nstates of particular interest.\n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{c>{\\centering}p{3cm}c>{\\centering}p{3cm}}\n\\hline \nReduction of state  & Name of the state  & Reduction of state  & Name of the state\\tabularnewline\n\\hline \n\\hline \n$|\\psi_{+}(u,n,\\alpha)\\rangle$  & $u$-PADFS  & $|\\psi_{-}(v,n,\\alpha)\\rangle$  & $v$-PSDFS\\tabularnewline\n\\textbar$\\psi_{+}(0,n,\\alpha)\\rangle${}  & DFS  & \\textbar$\\psi_{-}(0,n,\\alpha)\\rangle${}  & DFS\\tabularnewline\n$|\\psi_{+}(0,0,\\alpha)\\rangle$  & Coherent state  & $|\\psi_{-}(0,0,\\alpha)\\rangle$  & Coherent state\\tabularnewline\n\\textbar$\\psi_{+}(0,n,0)\\rangle${}  & Fock state  & \\textbar$\\psi_{-}(0,n,0)\\rangle${}  & Fock state\\tabularnewline\n\\textbar$\\psi_{+}(u,0,\\alpha)\\rangle${}  & $u$-Photon added coherent state  & \\textbar$\\psi_{-}(v,0,\\alpha)\\rangle${}  & $v$-Photon subtracted coherent state \\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{tab:state}Various states that can be obtained as the limiting\ncases of the PADFS and PSDFS.}\n\\end{table}\n\n\\section{Quantum phase distribution and other phase properties \\label{sec:Quantum-phase-parameters}}\n\nQuantum phase operator $\\hat{\\phi}$ was introduced by Dirac based\non his assumption that the annihilation operator $\\hat{a}$ can be\nfactored out into a Hermitian function $f(\\hat{N})$ of the number\noperator $\\hat{N}=\\hat{a}^{\\dagger}\\hat{a}$ and a unitary operator\n$\\hat{U}$ \\cite{dirac1927quantum} as \n\\begin{equation}\n\\hat{a}=\\hat{U}\\,f\\left(\\hat{N}\\right),\\label{eq:Dirac_pahse}\n\\end{equation}\nwhere \n\\begin{equation}\n\\hat{U}=e^{\\iota\\hat{\\phi}}.\\label{eq:phase-operator}\n\\end{equation}\nHowever, there was a problem with the Dirac formalism of phase operator\nas it failed to provide a meaning to the corresponding uncertainty\nrelation. Specifically, in the Dirac formalism, the creation ($\\hat{a}^{\\dagger}$)\nand annihilation ($\\hat{a}$) operators satisfy the bosonic commutation\nrelation, $\\left[\\hat{a},\\,\\hat{a}^{\\dagger}\\right]=1$, iff $\\left[\\hat{N},\\,\\hat{\\phi}\\right]=\\iota$,\nwhich leads to the number phase uncertainty relation $\\Delta N\\,\\Delta\\phi\\geq1$.\nTherefore, in order to satisfy the bosonic commutation relation under\nDirac formalism, the phase uncertainty should be greater than 2$\\pi$\nfor $\\Delta N$ \\textless{} $\\frac{1}{2\\pi}$ which lacks a physical\ndescription. Subsequently, Louisell \\cite{louisell1963amplitude}\nproposed some periodic phase based method, which was followed by Susskind\nand Glogower formalism based on Sine and Cosine operators \\cite{susskind1964quantum}.\nAn important contribution to this problem is the Barnett-Pegg formalism\n\\cite{barnett1986phase} which is used in this thesis. In what follows,\nwe will also briefly introduce notions, such as quantum phase distribution,\nangular $Q$ phase function, phase fluctuation parameters, phase dispersion,\nquantum phase estimation to study the phase properties of the quantum\nstates of our interest.\n\n\n\\section{Phase properties of PADFS and PSDFS \\label{sec:phase-witnesses}}\n\nThe description of the states of our interest given in the previous\nsection can be used to study different phase properties and quantify\nphase fluctuation in the set of quantum states listed in Table \\ref{tab:state}.\nSpecifically, with the help of the quantum states defined in Eqs.\n(\\ref{eq:PADFS})-(\\ref{eq:PSDFS}), we have obtained the analytic\nexpressions of phase distribution and other phase parameters defined\nin Section \\ref{sec:Quantum-phase-parameters}.\n\n\\subsection{Phase distribution function}\n\nFrom the definition of the phase distribution (\\ref{eq:Phase-Distridution-1}),\nit can be observed that for a Fock state, $P_{\\theta}=\\frac{1}{2\\pi}$,\nimplying it has a uniform distribution of phase. Interestingly, the\nstates of our interest, PADFS and PSDFS, are obtained by displacing\nthe Fock state followed by photon addition\/subtraction. Therefore,\nwe will study here what is the effect of application of displacement\noperator on a uniformly phase distributed (Fock) state and how subsequent\nphoton addition\/subtraction further alters the phase distribution.\nUsing phase distribution function, the information regarding uncertainty\nin phase and phase fluctuation can also be obtained. To begin with,\nwe compute the analytic expressions of $P_{\\theta}$ for the PADFS\nand PSDFS, using Eq. (\\ref{eq:Phase-Distridution-1}) as \n\\begin{eqnarray}\n\\begin{array}{lcl}\nP_{\\theta}\\left(u,n\\right) & = & \\frac{1}{2\\pi}\\dfrac{\\left|N_{+}\\right|^{2}}{n!}\\sum\\limits _{p,p^{\\prime}=0}^{n}{n \\choose p}{n \\choose p^{\\prime}}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\left|\\alpha\\right|^{2n-p-p^{\\prime}}\\\\\n & \\times & \\sum\\limits _{m,m^{\\prime}=0}^{\\infty}\\frac{(-\\left|\\alpha\\right|)^{m+m^{\\prime}}\\sqrt{(m+p+u)!(m^{\\prime}+p^{\\prime}+u)!}}{m!m^{\\prime}!}\\exp[\\iota\\left(\\theta-\\theta_{2}\\right)(m^{\\prime}+p^{\\prime}-m-p)],\n\\end{array}\\label{eq:PA-phase}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\begin{array}{ccc}\nP_{\\theta}\\left(v,n\\right) & = & \\frac{1}{2\\pi}\\dfrac{\\left|N_{-}\\right|^{2}}{n!}\\sum\\limits _{p,p^{\\prime}=0}^{n}{n \\choose p}{n \\choose p^{\\prime}}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\left|\\alpha\\right|^{2n-p-p^{\\prime}}\\\\\n & \\times & \\sum\\limits _{m,m^{\\prime}=0}^{\\infty}\\frac{(-\\left|\\alpha\\right|)^{m+m^{\\prime}}(m+p)!(m^{\\prime}+p^{\\prime})!}{m!m^{\\prime}!\\sqrt{(m+p-v)!(m^{\\prime}+p^{\\prime}-v)!}}\\exp[\\iota\\left(\\theta-\\theta_{2}\\right)(m^{\\prime}+p^{\\prime}-m-p)],\n\\end{array}\\label{eq:PS-phase}\n\\end{eqnarray}\nrespectively.  Here, $\\theta_2$ is the phase associated with the displacement parameter $\\alpha$ ($ \\alpha = |\\alpha|e^{\\iota \\theta_2}$). Since the obtained expressions in Eqs. (\\ref{eq:PA-phase})\nand (\\ref{eq:PS-phase}) are complex in nature, we depict numerical\n(graphical) analysis of the obtained results in Figs. \\ref{fig:Phase-Distribution-Function}\nand \\ref{fig:Phase-Distribution-Function-1} for PADFS and PSDFS,\nrespectively. Specifically, in Figure \\ref{fig:Phase-Distribution-Function}\n(a), we have shown the variation of phase distribution with phase\nparameter $\\theta$ for different number of photon added in the displaced\nsingle photon Fock state ($D\\left(\\alpha\\right)\\left|1\\right\\rangle $)\nfor $\\theta_{2}=0$. A uniform phase distribution for Fock state (with\na constant value of $\\frac{1}{2\\pi}$) is found to transform to one\nthat decreases for higher values of phase and possess a dip in the\nphase distribution for $\\theta=0$, which can be thought of as an\napproach to the Fock state. In fact, in case of classical states,\n$P_{\\theta}$ has a peak at zero phase difference $\\theta-\\theta_{2}$,\nand therefore, this contrasting behavior can be viewed as signature\nof quantumness of DFS. However, with the increase in the number of\nphotons added to the DFS, the phase distribution of the PADFS is observed\nto become narrower. In fact, a similar behavior with increase in the\nmean photon number of coherent state was observed previously \\cite{agarwal1992classical}.\nIt is imperative to state that $P_{\\theta}$ in case of higher number\nof photon added to DFS has similar but narrower distribution than\nthat of coherent state. In contrast, with increase in the Fock parameter,\nthe phase distribution is observed to become broader (cf. Figure \\ref{fig:Phase-Distribution-Function}\n(b)). Thus, the increase in the number of photons added and the increase\nin Fock parameter have opposite effects on the phase distribution.\nThe same is also illustrated through the polar plots in Figure \\ref{fig:Phase-Distribution-Function}\n(c)-(d), which not only reestablish the same fact, but also illustrate\nthe dependence of $P_{\\theta}$ on the phase of the displacement parameter.\nSpecifically, the obtained phase distribution remains symmetric along\nthe value of phase $\\theta_{2}$ (i.e., $P_{\\theta}$ is observed\nto have a mirror symmetry along $\\theta=\\theta_{2}$) of the displacement\nparameter. The phase distribution of Fock state is shown by a black\ncircle in the polar plot.\n\n\\begin{figure}\n\\centering{} \\centering{} %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{phasedis-pagcs.pdf}  & \\includegraphics[width=60mm]{distriphase-diffnp-pagcs1.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{phasedis-polar-pagcs.pdf}  & \\includegraphics[width=60mm]{distripolarphase-diffnp-pagcs1.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Phase-Distribution-Function} Variation of phase distribution\nfunction with phase parameter for PADFS with displacement parameter\n$\\left|\\alpha\\right|=1$ for different values of photon addition ((a)\nand (c)) and Fock parameters ((b) and (d)). The phase distribution\nis shown using both two-dimensional ((a) and (b) with $\\theta_{2}=0$)\nand polar ((c) and (d)) plots. In (c) and (d), $\\theta_{2}=\\frac{n\\pi}{2}$\nwith integer $n\\in\\left[0,3\\right]$, and the legends are same as\nin (a) and (b), respectively. }\n\\end{figure}\nInstead of photon addition, if we subtract photons from the DFS, a\nsimilar effect on the phase distribution to that of photon addition\nis observed. Further, a comparison between photon addition and subtraction\non the phase distribution establishes that a single photon subtraction\nhas a prominent impact on phase distribution when compared to that\nof single photon addition, i.e., the distribution can be observed\nto be narrower than that of coherent state in most of the cases for\n$u=v$. For instance, single photon added (subtracted) DFS is broader\n(narrower) than corresponding coherent state. Similarly, with the\nincrease in the value of Fock parameter, we can observe more changes\non PSDFS than what was observed in PADFS, i.e., the phase distribution\nbroadens more with Fock parameter for PSDFS. Note that $P_{\\theta}$\nhas a peak at $\\theta=\\theta_{2}$ only for photon addition $u>n$,\nwhile in case of photon subtraction it can be observed for $v\\geq n$.\nWith the increase in the amplitude of displacement parameter ($\\left|\\alpha\\right|$)\ninitially the phase distribution becomes narrower, which is further\nsupported by both addition and subtraction of photons, but it becomes\nbroader again for very high $\\left|\\alpha\\right|$ (figure is not\nshown here). \n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{phasedis-psgcs.pdf}  & \\includegraphics[width=60mm]{distriphase-diffnp-psgcs.pdf} \\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{phasedis-polar-psgcs.pdf}  & \\includegraphics[width=60mm]{distripolarphase-diffnp-psgcs.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Phase-Distribution-Function-1} Variation of phase distribution\nfunction with phase parameter for PSDFS with displacement parameter\n$\\left|\\alpha\\right|=1$ for different values of photon subtraction\n((a) and (c)) and Fock parameters ((b) and (d)). The phase distribution\nis shown using both two-dimensional ((a) and (b) with $\\theta_{2}=0$)\nand polar ((c) and (d)) plots. In (c) and (d), $\\theta_{2}=\\frac{n\\pi}{2}$\nwith integer $n\\in\\left[0,3\\right]$, and the legends are same as\nin (a) and (b), respectively.}\n\\end{figure}\n\n\\subsection{Angular $Q$ function of PADFS and PSDFS}\n\nThe relevance of the $Q$ function as witness of nonclassicality \\cite{thapliyal2015quasiprobability}\nand in state tomography \\cite{thapliyal2016tomograms} is well studied.\nOn top of that, non-Gaussianity of the PADFS and PSDFS using $Q$\nfunction was recently reported by us \\cite{malpani2019lower}. We\nfurther discuss a phase distribution based on $Q$ function using\nEq. (\\ref{eq:ang-Qf-1}). In this particular case, we have obtained\nthe angular $Q$ function from the $Q$ functions of the PADFS and\nPSDFS reported as Eqs. (15)-(16) in \\cite{malpani2019lower}. Specifically,\nwe have shown the effect of photon addition on the DFS ($D\\left(\\alpha\\right)\\left|1\\right\\rangle $)\nfor a specific value of the displacement parameter in Figure \\ref{fig:Angular Q function}\n(a) for angular $Q$ function. One can clearly see that the polar\nplots show an increase in the peak (located at $\\theta_{1}=\\theta_{2}$)\nof the distribution with photon addition. Further, one can compare\nthe behavior of $Q_{\\theta_{1}}$ with $P_{\\theta}$ in Figure \\ref{fig:Phase-Distribution-Function}\nand observe that they behave quite differently (as reported in \\cite{agarwal1992classical}\nfor the coherent states), other than increase in the peak of the distribution.\nSpecifically, $P_{\\theta}$ has a peak at $\\theta=\\theta_{2}$ only\nfor $u>n$, while $Q_{\\theta_{1}}$ is always peaked at the phase\nof the displacement parameter which also becomes a line of symmetry.\nInterestingly, the effect of increase in the Fock parameter of PADFS\non $Q_{\\theta_{1}}$is similar but less prominent in comparison to\nphoton addition. This is in quite contrast of that observed for $P_{\\theta}$\n(in Figs. \\ref{fig:Phase-Distribution-Function} and \\ref{fig:Angular Q function}\n(b)). In case of PSDFS, both photon subtraction and Fock parameter\nhave completely different effects on $Q_{\\theta_{1}}$ (cf. Figure\n\\ref{fig:Angular Q function} (c)-(d)) which is also in contrast to\nthat on corresponding $P_{\\theta}$ (shown in Figure \\ref{fig:Phase-Distribution-Function-1}).\nSpecifically, with increase in photon subtraction the angular $Q$\nfunction becomes narrower peaked at $\\theta=\\theta_{2}$, but for\nlarger number of photon subtraction the peak value decreases quickly.\nHowever, with increasing Fock parameter (cf. Figure \\ref{fig:Angular Q function}\n(d)), $Q_{\\theta_{1}}$ behaves much like photon addition on DFS (shown\nin Figure \\ref{fig:Angular Q function} (a)). The observed behavior\nshows the relevance of studying both these phase distributions due\nto their independent characteristics. \n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\centering{}\\includegraphics[width=60mm]{Qafun-diffphotoadd.pdf}  & \\includegraphics[width=60mm]{Qafun-diff-fock-photoadd.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{Qafun-photosub-12.pdf}  & \\includegraphics[width=60mm]{Qafun-photo-diff-focksub.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Angular Q function} The polar plots for angular $Q$ function\nfor PADFS (in (a) and (b)) and PSDFS (in (c) and (d)) for displacement\nparameter $\\left|\\alpha\\right|{\\rm =}1$ and $\\theta_{2}=\\frac{n\\pi}{2}$\nwith integer $n\\in\\left[0,3\\right]$ for different values of photon\naddition\/subtraction and Fock parameters. In (a) and (c), for $n=1$,\nthe smooth (blue), dashed (red), dot-dashed (magenta), and dotted\n(brown) lines correspond to photon addition\/subtraction 0, 1, 2, and\n3, respectively. In (b) and (d), for the single photon added\/subtracted\ndisplaced Fock state, the smooth (blue), dashed (red), dot-dashed\n(magenta), and dotted (brown) lines correspond to Fock parameter 1,\n2, 3, and 4, respectively. }\n\\end{figure}\n\n\\subsection{Quantum phase fluctuation of PADFS and PSDFS}\n\nNote that Carruthers and Nieto \\cite{carruthers1968phase} had introduced\nthese parameters in terms of Susskind and Glogower operators \\cite{susskind1964quantum};\nhere we use them in Barnett-Pegg formalism to remain consistent with\n\\cite{gupta2007reduction}, where $U$ parameter is shown relevant\nas a witness of nonclassicality \\cite{gupta2007reduction}. Specifically,\n$U$ is 0.5 for coherent state, and reduction of $U$ parameter below\nthe value for coherent state can be interpreted as the presence of\nnonclassical behavior \\cite{gupta2007reduction}. In what follows,\nwe will study quantum phase fluctuations for PADFS and PSDFS by computing\nanalytic expressions of $U,\\,S$ and $Q$ parameters in Barnett-Pegg\nformalism, with a specific focus on the possibility of witnessing\nnonclassical properties of these states via the reduction of $U$\nparameter below the coherent state limit. Carruthers and Nieto \\cite{carruthers1968phase}\nintroduced three parameters to study quantum phase fluctuation (\\ref{eq:fluctuation3-1})-(\\ref{eq:fluctuation5-1}).\nIt was only in the recent past that Gupta and Pathak provided a physical\nmeaning to one of these parameters by establishing its relation with\nantibunching and sub-Poissonian photon statistics \\cite{gupta2007reduction}.\nThus, the quantum phase fluctuation studied here using three parameters\nwill also be used to witness the nonclassical nature of the quantum\nstates under consideration. Here, the effect of photon addition\/subtraction\nand displacement parameters on these fluctuation parameters is also\nstudied (shown in Fig. \\ref{fig:phase fluctuation}). Specifically,\nFigure \\ref{fig:phase fluctuation} (a)-(c) show variation of the\nthree parameters of quantum phase fluctuation for different values\nof the number of photons added in the displaced Fock state ($D\\left(\\alpha\\right)\\left|1\\right\\rangle $)\nwith displacement parameter $\\left|\\alpha\\right|$. It may be clearly\nobserved that two of the quantum phase fluctuation parameters, namely\n$U\\left(u,n\\right)$ and $Q\\left(u,n\\right)$ decrease with the value\nof displacement parameter, while $S\\left(u,n\\right)$ increases with\n$\\left|\\alpha\\right|$. Interestingly, the photon addition and increase\nin the displacement parameter exhibit the same effect on all three\nquantum phase fluctuation parameters for PADFS, while for higher values\nof displacement parameter $S\\left(u,n\\right)$ show completely opposite\neffect of photon addition. In contrast, $U\\left(v,n\\right)$ for $v$\nsubtracted photons from $D\\left(\\alpha\\right)\\left|1\\right\\rangle $\nis found to increase (decrease) with photon subtraction while decrease\n(increase) with the displacement parameter for small (large) value\nof $\\left|\\alpha\\right|$ (cf. Figure \\ref{fig:phase fluctuation}\n(d)). On the other hand, parameter $S\\left(v,n\\right)$ is also observed\nto increase (decrease) with $\\left|\\alpha\\right|$ ($v$) as shown\nin Figure \\ref{fig:phase fluctuation} (e). The third parameter $Q\\left(v,n\\right)$\nshows slightly complex behavior for PSDFS with both $\\left|\\alpha\\right|$\nand $v$ (cf. Figure \\ref{fig:phase fluctuation} (f)) as it behaves\nanalogous to PADFS for each subtracted photon for both small and large\nvalues of the displacement parameter (when it increases with $\\left|\\alpha\\right|$),\nbut for intermediate values the behavior is found to be completely\nopposite.\n\nAs mentioned previously, $U\\left(i,n\\right)\\,\\forall i\\in\\left\\{ u,v\\right\\} $\nhas a physical significance as a witness of antibunching for values\nof this parameter less than $\\frac{1}{2}$, Figure \\ref{fig:phase fluctuation}\n(a) and (d) can be used to perform similar studies for PADFS and PSDFS,\nrespectively. In case of PADFS, we can observe this relevant parameter\nto become less than $\\frac{1}{2}$, and thus to illustrate the presence\nof antibunching, only at higher values of the displacement parameter\nand photon added to the displaced Fock state. In contrast, PSDFS shows\nthe presence of this nonclassical feature in all cases.Thus, occurrence\nof antibunching in PADFS and PSDFS is established here through this\nphase fluctuation parameter. Interestingly, a similar dependence of\nantibunching in PADFS and PSDFS    Eq. (\\ref{eq:HOA-1}) has\nbeen recently reported by us \\cite{malpani2019lower} using a different\ncriterion. Further, one can observe from the expression of $U$ in\nEq. (\\ref{eq:fluctuation3-1}) that it is expected to be independent\nof the phase of the displacement parameter, which can also be understood\nfrom the use of this parameter as a witness for an intensity moments\nbased nonclassical feature. In contrast, $S$ and $Q$ in Eqs. (\\ref{eq:fluctuation4-1})-(\\ref{eq:fluctuation5-1})\nshow dependence on the phase of displacement parameter. Here, we have\nnot discussed the effect of Fock parameter in detail, but in case\nof photon addition, $u$ and $n$ have same (opposite) effects on\n$S$ ($U$ and $Q$) parameter(s). Fock parameter has always shown\nopposite effect of photon subtraction on all three phase fluctuation\nparameters, and thus nonclassicality revealed by $U$ can be enhanced\nwith Fock parameter. The relevance of Fock parameter can also be visualized\nby observing the fact that the single photon subtracted coherent state\nhas $U=0.5$ (which is consistent with the value zero of the antibunching\nwitness reported in \\cite{thapliyal2017comparison}). Thus, in this\ncase, the origin of the induced antibunching can be attributed to\nthe non-zero value of Fock parameter. \n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{ccc}\n\\includegraphics[width=60mm]{phaseUpagcs.pdf}  & \\includegraphics[width=60mm]{phaseSpagcs.pdf} & \\tabularnewline\n(a)  & (b)  & \\tabularnewline\n\\includegraphics[width=60mm]{phaseQpagcs.pdf}  & \\includegraphics[width=60mm]{phaseUpsgcs.pdf} & \\tabularnewline\n(c)  & (d)  & \\tabularnewline\n\\includegraphics[width=60mm]{phaseSpsgcs.pdf}  & \\includegraphics[width=60mm]{phaseQpsgcs.pdf}  & \\tabularnewline\n(e)  & (f)  & \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:phase fluctuation} Variation of three phase fluctuation\nparameters introduced by Carruthers and Nieto with the displacement\nparameter with $\\theta_{2}=0$. The values of photon addition ($u$),\nsubtraction ($v$), and Fock parameter $n=1$ are given in the legends.\nParameter $U\\left(i,n\\right)\\,\\forall i\\in\\left\\{ u,v\\right\\} $ also\nillustrates antibunching in the states for values less than $\\frac{1}{2}$. }\n\\end{figure}\n\n\\subsection{Phase Dispersion}\n\nHere, it is worth stressing that both Carruthers-Nieto parameters\nand phase dispersion $D$ correspond to phase fluctuation, Our primary\nfocus is to study phase fluctuation and further to check the correlation\nbetween these measures of phase fluctuation. Thus, it would be interesting\nto study phase fluctuation from the two perspectives. We compute a\nmeasure of quantum phase fluctuation based on quantum phase distribution,\nthe phase dispersion (\\ref{eq:Dispersion-1}), for both PADFS and\nPSDFS to perform a comparative study between them. Specifically, the\nmaximum value of dispersion is 1 which corresponds to the uniform\nphase distribution, i.e., $P_{\\theta}=\\frac{1}{2\\pi}$. Both PADFS\nand PSDFS show a uniform distribution for the displacement parameter\n$\\alpha=0$ (cf. Figure \\ref{fig:Phase-Dispersion}). It is a justified\nresult as both the states reduce to the Fock state in this case. However,\nwith the increase in the value of displacement parameter quantum phase\ndispersion is found to decrease. This may be attributed to the number-phase\ncomplimentarity \\cite{banerjee2010complementarity,srikanth2009complementarity,srikanth2010complementarity},\nwhich leads to smaller phase fluctuation with increasing variance\nin the number operator at higher values of displacement parameter.\nThus, with an increase in the average photon number by increasing\nthe displacement parameter, phase dispersion decreases for both PADFS\nand PSDFS. Addition of photons in DFS leads to decrease in the value\nof phase dispersion, while subtraction of photons has more complex\neffect on phase dispersion (cf. Figure \\ref{fig:Phase-Dispersion}\n(a) and (c)). Specifically, for the smaller values of the displacement\nparameter ($\\left|\\alpha\\right|<1$), the phase dispersion parameter\nbehaves differently for $v\\leq n$ and $v>n$. This can be attributed\nto the sub-Poissonian photon statistics for $v\\leq n$ with $\\left|\\alpha\\right|<1$\nas well as the small value of average photon number (Figure \\ref{fig:phase fluctuation}\n(d)). However, at the higher values of the displacement parameter\n$D$ for the PSDFS behaves in a manner analogous to the PADFS. Interestingly,\nincrease in the Fock parameter shows similar effect on PADFS and PSDFS\nin Figure \\ref{fig:Phase-Dispersion} (b) and (d), respectively. \n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{diffusion-pagcs.pdf}  & \\includegraphics[width=60mm]{diffusion-pagcs-difffock.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{diffusion-psgcs.pdf}  & \\includegraphics[width=60mm]{diffusion-psgcs-difffock.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Phase-Dispersion} Variation of phase dispersion for PADFS\n(in (a) and (b)) and PSDFS (in (c) and (d)) with displacement parameter\nfor an arbitrary $\\theta_{2}$. Dependence on different values of\nphoton added\/subtracted and the initial Fock state $\\left|1\\right\\rangle $\n(in (a) and (c)), while on different values of Fock parameter for\nsingle photon added\/subtracted state (in (b) and (d)).}\n\\end{figure}\n\n\\subsection{Phase sensing uncertainity for PADFS and PSDFS}\n\nWe finally discuss quantum phase estimation using Eq. (\\ref{eq:PE-1}),\nassuming the two mode input state in the Mach-Zehnder interferometer\nas $|\\psi_{i}(j,n,\\alpha)\\rangle\\otimes|0\\rangle$. The expressions\nfor the variance of the difference in the photon numbers in the two\noutput modes of the Mach-Zehnder interferometer for input PADFS and\nPSDFS and the rest of the parameters required to study phase sensing\nare reported in Appendix.\n\nThe obtained expressions allow us to study the optimum choice of state\nparameters for quantum phase estimation using PADFS and PSDFS. The\nvariation of these parameters is shown in Figure \\ref{fig:Phase sensing uncertainity}.\nSpecifically, we have shown that PSDFS is preferable over coherent\nstate for phase estimation (cf. Figure \\ref{fig:Phase sensing uncertainity}\n(b)). However, with the increase in the photon subtraction this phase\nuncertainty parameter is found to increase although remaining less\nthan corresponding coherent state value. In contrast, with photon\naddition, advantage in phase estimation can be attained as the reduction\nof the phase uncertainty parameter allows one to perform more precise\nmeasurement. This advantage can be enhanced further by choosing large\nvalues of photon addition and Fock parameter (cf. Figure \\ref{fig:Phase sensing uncertainity}\n(a) and (c)). In a similar sense, appropriate choice of Fock parameter\nwould also be advantageous in phase estimation with PSDFS as it decreases\nthe phase uncertainty parameter, but still PADFS remains preferable\nover PSDFS. This can further be controlled by an increase in $\\left|\\alpha\\right|$\nwhich decreases (increases) phase uncertainty parameter for PADFS\n(PSDFS). \n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{phaseresolution-pagcs.pdf}  & \\includegraphics[width=60mm]{phaseresolution-psgcs.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{phaseresolution-pagcs-diff-np.pdf}  & \\includegraphics[width=60mm]{phaseresolution-psgcs-diff-Fock.pdf} \\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Phase sensing uncertainity}Phase sensing uncertainty for\n(a) PADFS and (b) PSDFS as a function of phase to be estimated $\\phi$\nfor different number of photon addition\/subtraction with $n=1$. The\ndependence for (c) PADFS and (d) PSDFS is also shown for different\nvalues of Fock parameters with $u=1$ and $v=1$, respectively. In\nall cases, we have chosen $\\alpha=0.1$.}\n\\end{figure}\n\n\\section{Conclusions \\label{sec:Conclusions-1}}\n\nA set of engineered quantum states can be obtained as the limiting\ncases from the PADFS and PSDFS, e.g., DFS, coherent state, photon\nadded\/subtracted coherent state, and Fock state. Specifically, PADFS\/PSDFS\nare obtained by application of unitary (displacement) and non-unitary\n(addition and subtraction of photons) operations on Fock state. In\nview of the fact that the Fock states have uniform phase distribution,\nthe set of unitary and non-unitary quantum state engineering operations\nare expected to affect the phase properties of the generated state.\nTherefore, here we have calculated quantum phase distribution, which\nfurther helped in quantifying phase fluctuation as phase dispersion.\nWe have also computed the phase distribution as the angular $Q$ function.\nWe have further studied phase fluctuation using three Carruthers and\nNieto parameters, and have used one of them to reveal the existence\nof antibunching in the quantum states of our interest.\n\nBoth the phase distribution and angular $Q$ functions are found to\nbe symmetric along the value of the phase of the displacement parameter.\nThe phase distribution is observed to become narrow and peak(s) to\nincrease with the amplitude of the displacement parameter ($\\left|\\alpha\\right|$),\nwhich further becomes broader for higher values of $\\left|\\alpha\\right|$.\nFurther, photon addition\/subtraction and Fock parameters are observed\nto have opposite effects on phase distribution, i.e., distribution\nfunction becomes narrower (broader) with photon addition\/subtraction\n(Fock parameter). Among photon addition and subtraction operations,\nsubtracting a photon alters the phase properties more than that of\nphoton addition. Specifically, at the small values of the displacement\nparameter ($\\left|\\alpha\\right|<1$), the phase properties of PSDFS\nfor $v\\leq n$ and $v>n$ behave differently. This can be attributed\nto the fact that for $v\\leq n$ with $\\left|\\alpha\\right|<1$, the\naverage photon number becomes very small. Further, the peak of the\nphase distribution remains at the phase of displacement parameter\nonly when the number of photons added\/subtracted is more than that\nof the Fock parameter. However, in case the number of photons subtracted\n(added) is same as the Fock parameter, the peak of the phase distribution\nis observed (not observed) at the phase of displacement parameter.\nThe angular $Q$ function can be observed to show similar dependence\non various parameters, but the peak of the distribution remains located\nat the value of phase of the displacement parameter. The three phase\nfluctuation parameters introduced by Carruthers and Nieto \\cite{carruthers1968phase}\nshow phase properties of PADFS and PSDFS, while one of them, $U$\nparameter also reveals antibunching in both PADFS and PSDFS. In this\ncase, the role of Fock parameter as antibunching inducing operation\nin PSDFS is also discussed. Phase dispersion quantifying phase fluctuation\nremains unity for Fock state reflecting uniform distribution, which\ncan be observed to decrease with increasing displacement parameter.\nThis may be attributed to the number-phase complimentarity as the\nhigher values of variance with increasing displacement parameter lead\nto smaller phase fluctuation. Fock parameter and photon addition\/subtraction\nshow opposite effects on the phase dispersion as it increases (decreases)\nwith $n$ ($u\/v$).\n\nFinally, we have also discussed the advantage of the PADFS and PSDFS\nin quantum phase estimation and obtained the set of optimized parameters\nin the PADFS\/PSDFS. Both photon addition and Fock parameter decrease\nthe uncertainty in phase estimation, while photon subtraction, though\nperforms better than coherent state is not as advantageous as $u$\nor $n$. In \\cite{ou1997fundamental}, it was established that signal-to-noise\nratio is significant only when the phase shift to measure is of the\nsame order as multiplicative inverse of the average photon number.\nTherefore, in case of PADFS this limitation of quantum measurement\nis expected to play an important role. Thus, we have shown here that\nstate engineering tools can be used efficiently to control the phase\nproperties of the designed quantum states for suitable applications.\nThe study performed in this chapter can be extended for other such\noperations, like squeezing, photon addition followed by subtraction\nor vice versa. \n\\chapter{Impact of photon addition and subtraction on nonclassical and phase\nproperties of a displaced Fock state \\textsc{\\label{cha:PASDFS}}}\n\nIn this chapter, we aim to observe the nonclassical and phase properties\nof a PASDFS.  The work done in this chapter is published\nin ~\\cite{malpani2020impact}.\n\n\\section{Introduction \\label{sec:Intro-chap-4}}\n\nIn Chapter 2, we have already studied nonclassical properties of PADFS\nand PSDFS. In Chapter 3, we have investigated phase properties of\nthe same set state. In both chapters, we have obtained various exciting\nobservations  such as photon addition and subtraction\nenhance nonclassical properties of non-Gaussian DFS. Motivated by\nthese, here we aim to study both nonclassical and phase properties\nfor a more general quantum state. To be specific, in this chapter,\nwe aim to study the nonclassical (both lower- and higher-order) and\nphase properties of a PASDFS. The reason behind selecting this particular\nstate lies in the fact that this is a general state in the sense that\nin the limiting cases, this state reduces to different quantum states\nhaving known applications in continuous variable quantum cryptography\n(this point will be further elaborated in the next section).\n\nAs it appears from the above discussion, this investigation has two\nfacets. Firstly, we wish to study nonclassical features of PASDFS,\nnamely Klyshko's \\cite{klyshko1996observable}, Agarwal-Tara's \\cite{agarwal1992nonclassical},\nVogel's \\cite{shchukin2005nonclassical} criteria, lower- and higher-order\nantibunching \\cite{pathak2006control}, squeezing \\cite{hillery1987amplitude,hong1985generation,hong1985higher},\nand sub-Poissonian photon statistics (HOSPS) \\cite{zou1990photon}.\nWe subsequently study the phase properties of PASDFS by computing\nphase distribution function \\cite{agarwal1996complementarity,beck1993experimental},\nphase fluctuation parameters \\cite{carruthers1968phase,barnett1986phase},\nand phase dispersion \\cite{perinova1998phase}. A detailed analysis\nof the obtained results will also be performed to reveal the usefulness\nof the obtained results. \n\n\\section{Moments of the field operators for the quantum states of our interest }\n\n\\label{sec:Quantum-states-of-3}\n\nAs mentioned in the previous section, this work is focused on PASDFS.\nA PASDFS as a Fock superposition state has already been expressed\nin Eq.(\\ref{eq:PADFS-1}). To study nonclassical and phase properties\nof this state, we have used nonclassicality witnesses introduced in\nSection \\ref{sec:Nonclassicality-witnesses} and phase parameters\nin Section \\ref{sec:Analytic-tools-forphase}, we would require analytic\nexpression for moments of the field operators. A bit of computation\nyields the expression for higher-order moment of annihilation and\ncreation operator as \n\\begin{eqnarray}\\label{eq:PA-expepectation-1}\n\\langle\\hat{a}^{\\dagger t}\\hat{a}^{j}\\rangle & = & \\langle\\psi(k,q,n,\\alpha)|\\hat{a}^{\\dagger t}\\hat{a}^{j}|\\psi(k,q,n,\\alpha)\\rangle\\nonumber \\\\\n & = & \\frac{N^{2}}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\\\\n & \\times & \\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum\\limits _{m=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m+p-p'-j+t}(m+p+k)!(m+p+k-j+t)!}{m!(m+p-p'-j+t)!(m+p+k-q-j)!}.\\nonumber \n\\end{eqnarray}\nFor different values of $t$ and $j$, moments of any order can be\nobtained, and the same may be used to investigate the nonclassical\nproperties of PASDFS and its limiting cases by using various moments-based\ncriteria of nonclassicality. The same will be performed in the following\nsection, but before proceeding, it would be apt to briefly state our\nmotivation behind the selection of this particular state for the present\nstudy (or why do we find this state as interesting?).\n\nDue to the difficulty in realizing single photon on demand sources,\nthe unconditional security promised by various QKD schemes, like BB84\n\\cite{bennett1984quantum} and B92 \\cite{bennett1992quantumBBM},\ndoes not remain unconditional in the practical situations This is\nwhere continuous variable QKD (CVQKD) becomes relevant as they do\nnot require single photon sources. Special cases of PASDFS has already\nbeen found useful in the realization of CVQKD. For example, protocols\nfor CVQKD have been proposed using photon added coherent state ($k=1,\\,q=0,\\,n=0$)\n\\cite{pinheiro2013quantum,wang2014quantum}, photon added then subtracted\ncoherent states ($k=1,\\,q=1,\\,n=0$) \\cite{borelli2016quantum,srikara2019continuous},\nand coherent state ($k=0,\\,q=0,\\,n=0$) \\cite{grosshans2002continuous,hirano2017implementation,huang2016long,ma2018continuous}.\nFurther, boson sampling with displaced single photon Fock states and\nsingle photon added coherent state \\cite{seshadreesan2015boson} has\nbeen reported, and an $m$ photon added coherent state ($k=m,\\,q=0,\\,n=0$)\nhas been used for quantum teleportation \\cite{pinheiro2013quantum}.\nApart from these schemes of CVQKD, which can be realized by using\nPASDFS or its limiting cases, the fact that the photon addition and\/or\nsubtraction operation from a classical or nonclassical state can be\nperformed experimentally using the existing technology \\cite{parigi2007probing,zavatta2004quantum}\nhas enhanced the importance of PASDFS.\n\n\\section{Nonclassicality witnesses and the nonclassical features of PASDFS\nwitnessed through those criteria \\label{sec:Nonclassicality-witnesses-2}}\n\nThe negative value of the Glauber-Sudarshan $P$-function characterizes\nnonclassicality of an arbitrary state \\cite{glauber1963coherent,sudarshan1963equivalence}.\nAs $P$-function is not directly measurable in experiments, many witnesses\nof nonclassicality have been proposed, such as, negative values of\nWigner function \\cite{wigner1932quantum,kenfack2004negativity}, zeroes\nof $Q$ function \\cite{husimi1940some,lutkenhaus1995nonclassical},\nseveral moments-based criteria \\cite{miranowicz2010testing,naikoo2018probing}.\nAn infinite set of such moments-based criteria of nonclassicality\nis equivalent to $P$-function in terms of necessary and sufficient\nconditions to detect nonclassicality \\cite{richter2002nonclassicality}.\nHere, we would discuss some of these moments-based criteria of nonclassicality\nand $Q$ function (in Section \\ref{sec:Qfn}) to study nonclassical\nproperties of the state of our interest.\n\n\\subsection{Lower- and higher-order antibunching}\n\nThe relevance of photon addition, photon subtraction, Fock, and displacement\nparameters in the nonclassical properties of the class of PASDFSs\nis studied here rigorously. Specifically, using Eq. (\\ref{eq:PA-expepectation-1})\nwith the criterion of antibunching (\\ref{eq:HOA-1}) we can study\nthe possibilities of observing lower- and higher-order antibunching\nin the quantum states of PASDFS class, where the class of PASDFSs\nrefers to all the states that can be reduced from state (\\ref{eq:PADFS-1})\nin the limiting cases. The outcome of such a study is illustrated\nin Figure \\ref{fig:HOSPS-1}. It is observed that the depth of lower-\nand higher-order nonclassicality witnesses can be increased by increasing\nthe value of the displacement parameter, but large values of $\\alpha$\ndeteriorate the observed nonclassicality (cf. Figure \\ref{fig:HOSPS-1}\n(a)-(b)). The nonclassicality for higher-values of displacement parameter\n$\\alpha$ can be induced by subtracting photons at the cost of reduction\nin the depth of nonclassicality witnessed for smaller $\\alpha$, as\nshown in Figure \\ref{fig:HOSPS-1} (a). However, photon addition is\nalways more advantageous than subtraction. Therefore, both addition\nand subtraction of photons illustrate these collective effects by\nshowing nonclassicality for even higher values of $\\alpha$ at the\ncost of that observed for the small values of displacement parameter.\nFock parameter has completely opposite effect of photon subtraction\nas it shows the advantage (disadvantage) for small (large) values\nof displacement parameter. Figure \\ref{fig:HOSPS-1} (b) shows benefit\nof studying higher-order nonclassicality as depth of corresponding\nwitness of nonclassicality can be observed to increase with the order.\nThe higher-order nonclassicality criterion is also able to detect\nnonclassicality for certain values of displacement parameter for which\nthe corresponding lower-order criterion failed to do so.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{HOA-sameorder-11-22-33.pdf}  & \\includegraphics[width=60mm]{HOA-difforder-11.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{HOsps-sameorder-11-22-33.pdf}  & \\includegraphics[width=60mm]{HOsps-difforder-11.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:HOSPS-1} For PASDFS the lower-and higher-order antibunching\nis given as a function of displacement parameter $\\alpha$. (a) {Lower-order antibunching for }different values\nof parameters of the state. (b) {Higher-order\nantibunching for particular state.} HOSPS for PASDFS for {different\n}values of (c) state parameters and (d) {order of nonclassicality.}}\n\\end{figure}\n\n\\subsection{Higher-order sub-Poissonian photon statistics}\n\nVariation of HOSPS nonclassicality witness for class of PASDFSs obtained\nby different nonclassicality inducing operations show the same effect\nas that of antibunching witness for all the odd orders of HOSPS, and\nas depicted in Figure \\ref{fig:HOSPS-1} (c). However, this nonclassical\nfeature disappears for even orders of HOSPS (cf. Figure \\ref{fig:HOSPS-1}\n(d)). In case of the odd orders of HOSPS, though the depth of nonclassicality\nwitness increases with the order, higher-order criterion is found\nto fail to detect nonclassicality for certain values of $\\alpha$\nwhen corresponding HOSPS criterion for smaller values of orders shown\nthe nonclassicality.\n\n\\subsection{Lower- and higher-order squeezing}\n\nOut of all the nonclassicality inducing operations used in PASDFS\nonly photon subtraction is squeezing inducing operation as shown in\nFigure \\ref{fig:HOS-2}, which is consistent with some of our recent\nobservations \\cite{malpani2019lower}. With photon addition higher-order\nsqueezing can be induced for large values of modulus of displacement\nparameter at the cost of squeezing observed for small $\\left|\\alpha\\right|$\nas long as the number of photon subtracted is more than the value\nof Fock parameter. As far as higher-order squeezing is concerned,\nthe observed nonclassicality disappears for large values of real displacement\nparameter with increase in the depth of the nonclassicality witness.\nSqueezing being a phase dependent nonclassical feature depends on\nthe phase $\\theta$ of the displacement parameter $\\alpha=\\left|\\alpha\\right|\\exp[\\iota\\theta]$\n(shown in Figure \\ref{fig:HOS-2} (c) for lower-order squeezing).\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{HOs-sameorder-11-22-33.pdf}  & \\includegraphics[width=60mm]{HOs-difforder-11.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{HOs-theta.pdf} & \\tabularnewline\n(c)  & \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:HOS-2} Dependence of the Hong-Mandel-type higher-order\nsqueezing witness on displacement parameter for (a) different state\nparameters and (b) order of squeezing. (c) Lower-order squeezing as\na function of phase parameter of the state, i.e., phase $\\theta$\nof displacement parameter $\\alpha=1.\\exp[\\iota\\theta].$}\n\\end{figure}\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{c}\n\\includegraphics[width=80mm]{klyshko-alphaless11-22-33.pdf}\\tabularnewline\n(a) \\tabularnewline\n\\includegraphics[width=80mm]{klyshko-alphamore11-22-33.pdf} \\tabularnewline\n(b) \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:Klyshko-1}  Illustration of Klyshko's\nparameter $B\\left(z\\right)$ with respect to the photon number $z$  for different values of state parameters  with\n(a) $\\alpha=0.5$ and (b) $\\alpha=1$.}\n\\end{figure}\n\n\\subsection{Klyshko's Criterion}\n\nFor PASDFS $p_{z}$ can be obtained from Eq. (\\ref{eq:PA-expepectation-1}).\nNonclassicality reflected through Klyshko's criterion can be controlled\nby all the state engineering operations used here as shown in Figure\n\\ref{fig:Klyshko-1}. The depth of this nonclassicality witness increases\nat higher values of photon numbers $z$ due to increase in photon\naddition and\/or Fock parameter. In contrast, depth of witness increases\nat smaller photon numbers $z$ due to photon subtraction. The Klyshko's\nnonclassicality witness is positive for some photon numbers only if\n$k+n>q$. Additionally, with increase in displacement parameter the\ndepth of nonclassicality witness decreases, and the weight of the\ndistribution of witness shift to higher values of $z$.\n\n\\subsection{Agarwal-Tara's criterion}\n\nThis nonclassicality witness is able to detect nonclassicality in\nall the quantum states in the class of PASDFSs (cf. Figure \\ref{Vogel's criteria}\n(a)). Note that for $|\\psi\\left(1,2,1,\\alpha\\right)\\rangle$ with\nsmall $\\alpha$, $A_{3}$ parameter is close to zero, which is due\nto very high probability for zero photon states.\n\n\\begin{figure}\n\\begin{tabular}{c}\n\\includegraphics[width=80mm]{agarwal-11-22-33.pdf}\\tabularnewline\n(a) \\tabularnewline\n\\includegraphics[width=80mm]{Vogel-11-22-33.pdf}\\tabularnewline\n(b) \\tabularnewline\n\\end{tabular}\\caption{Nonclassicality reflected through the negative values of (a) Agarwal-Tara's\nand (b) Vogel's criteria as a function of $\\alpha$ or different state\nparameters.}\n\\label{Vogel's criteria} \n\\end{figure}\n\n\\subsection{Vogel's criterion}\n\nThe negative value of the determinant $dv$ of matrix $v$ in Eq.\n(\\ref{eq:vogel}) is signature of nonclassicality. Fock parameter\nhas adverse effect on the nonclassicality in PASDFS detected by this\ncriterion. This averse effect can be compensated by photon subtraction\nand can be further controlled by photon addition (as shown in Figure\n\\ref{Vogel's criteria} (b)). Notice that the nonclassical behavior\nillustrated by Agarwal-Tara's (Vogel's) criterion is related to higher-order\nantibunching (squeezing) criterion. However, nonclassicality witness\nof Vogel's criterion is a phase independent property unlike squeezing.\n\n\\section{Phase properties of PASDFS\\label{sec:Phase-properties-of}}\n\nThe nonclassicality inducing operations are also expected to impact\nthe phase properties of a quantum state \\cite{banerjee2007phase}.\nRecently, we have reported an extensive study on the role that such\nquantum state engineering tools can play in application oriented studies\non quantum phase \\cite{malpani2019quantum}. Specifically, relevance\nin quantum phase estimation, phase fluctuation, and phase distribution\nwere discussed which can play an important role in quantum metrology\n\\cite{giovannetti2011advances}. Here, we briefly discuss some of\nthe phase properties of the class of PASDFSs.\n\n\\subsection{Phase distribution function}\n\nThe analytical expression for phase distribution function for PASDFS\ncan be computed as\n\n\\begin{eqnarray}\n\\begin{array}{lcl}\nP(\\theta) & = & \\frac{1}{2\\pi}\\dfrac{N^{2}}{n!}\\sum\\limits _{p,p'=0}^{n}{n \\choose p}{n \\choose p'}(-\\alpha^{\\star})^{(n-p)}(-\\alpha)^{(n-p')}\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\\\\n & \\times & \\sum\\limits _{m,m^{\\prime}=0}^{\\infty}\\frac{\\alpha^{m}(\\alpha^{\\star})^{m^{\\prime}}(m+p+k)!(m^{\\prime}+p'+k)!}{m!m^{\\prime}!\\sqrt{\\left(m+p+k-q\\right)!\\left(m^{\\prime}+p'+k-q\\right)!}}\\exp[\\iota\\theta(m^{\\prime}+p'-m-p)].\n\\end{array}\\label{eq:PA-phase-1}\n\\end{eqnarray}\n\nPhoton subtraction can be observed to be a more effective tool to\nalter phase properties of PASDFS than photon addition, as shown in\nFigure \\ref{fig:Phase distribution function}. Interestingly, photon\naddition shows similar behavior, though less prominent, as photon\nsubtraction, Fock parameter has opposite effect.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{ccc}\n\\includegraphics[width=40mm]{phase-distri-21-22-23.pdf}  & \\includegraphics[width=40mm]{phase-distri-12-22-32np2.pdf}  & \\includegraphics[width=40mm]{phase-distri-11-22-33np2.pdf}\\tabularnewline\n(a)  & (b)  & (c) \\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:Phase distribution function}Polar plot of phase distribution\nfunction for PASDFS $|\\psi\\left(k,q,n,\\alpha\\right)\\rangle$ with\nrespect to variation in displacement parameter for (a) $n=1,\\,k=2$\nand $q=1,$ 2, and 3 represented by the smooth (cyan), dashed (magenta),\nand dot-dashed (purple) lines, respectively; (b) $n=2,\\,q=2$ and\n$k=1,$ 2, and 3 illustrated by the smooth (cyan), dashed (magenta),\nand dot-dashed (purple) lines, respectively; and (c) $n=1$ with $k=q=1,$\n2, and 3 shown by the smooth (cyan), dashed (magenta), and dot-dashed\n(purple) lines, respectively.}\n\\end{figure}\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{c}\n\\includegraphics[width=60mm]{U-11-22-33.pdf} \\tabularnewline\n\\tabularnewline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:Phase fluctuation}\\textcolor{green}{{}} Variation of\nphase fluctuation parameter with displacement parameter\\textcolor{green}{{}\n}for various state parameters in PASDFS.}\n\\end{figure}\n\n\\subsection{Phase Fluctuation}\n\nHere, we focus only on the first phase fluctuation parameter $U$,\nwhich is related to antibunching if $U$ is below its value for coherent\nstate (i.e., 0.5), remaining consistent with Barnett-Pegg formalism\n\\cite{gupta2007reduction,pathak2000phase}. One can observe that the\nphase fluctuation parameter is able to detect nonclassicality (specifically\nantibunching) only in three cases where the role of the photon subtraction\nis relevant (cf. Figure \\ref{fig:Phase fluctuation}). The observation\ncan be seen analogous to that observed for Vogel's nonclassicality\ncriterion.\n\n\\begin{figure}\n\\begin{centering}\n\\begin{tabular}{cc}\n\\includegraphics[width=70mm]{Q-fun-PADFS.jpg}  \n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{fig:Q function} $Q$ function for PASDFS $|\\psi\\left(k,q,n,\\alpha\\right)\\rangle$\nwith (a) $k=q=n=1,$ (b) $k=2,\\,q=n=1,$ and (c) $q=2,\\,k=n=1$ with\n$\\alpha=\\frac{1}{5\\sqrt{2}}\\exp\\left(\\iota\\pi\/4\\right)$. (d) Similarly,\n$Q$ function of PASDFS with $q=2,\\,k=n=1$ and $\\alpha=\\sqrt{2}\\exp\\left(\\iota\\pi\/4\\right)$.\n$Q$ function for $|\\psi\\left(k,q,n,\\alpha\\right)\\rangle$ with (e)\n$k=q=1,\\,n=2$ and (f) $q=1,\\,k=n=2$ for $\\alpha=\\frac{1}{5\\sqrt{2}}\\exp\\left(\\iota\\pi\/4\\right)$.}\n\\end{figure}\n\n\\section{Quasidistribution function: $Q$ function \\label{sec:Qfn}}\n\nHere, We will establish non-Gaussianity inducing behavior of photon\naddition and Fock parameter (cf. Figure \\ref{fig:Q function}), which\nare so far illustrated as nonclassicality inducing and phase altering\noperations. Clearly, with photon addition tendency of quasidistribution\naway from Gaussian behavior is visible, while with photon subtraction\nsqueezing along particular phase angle chosen by displacement parameter\ncan be observed. This squeezing can be noticed to be more appreciable\nfor higher values of displacement parameter (cf. Figure \\ref{fig:Q function}\n(c)-(d)). From Figure \\ref{fig:Q function} (e)-(f), it can be observed\nthat Fock parameter and photon addition have a similar effect in the\nphase space. As zeros of $Q$ function are signature of nonclassicality,\nPASDFS shows nonclassicality in Figure \\ref{fig:Q function} (b),\n(e), and (f).  This establishes that use of more\nthan one state engineering tool may be helpful in generation of nonclassical\nstates. It would be interesting  to verify whether one more tools (say\nsqueezing or photon catalysis) may further enhance the nonclassical\nproperties.  \n\n\\section{Conclusions \\label{sec:Conclusions-5}}\n\nIn this chapter, we have investigated the nonclassical behavior of\nPASDFS using different witnesses of lower- and higher-order nonclassicality.\nThe significance of this choice of state underlies the fact that a\nclass of engineered quantum states can be achieved as the reduced\ncase of PASDFS $|\\psi\\left(k,q,n,\\alpha\\right)\\rangle$, like photon\nadded DFS $\\left(q=0\\right)$, photon subtracted DFS $\\left(k=0\\right)$,\nDFS $\\left(q=k=0\\right)$, photon added coherent state $\\left(n=q=0\\right)$,\nphoton subtracted coherent state $\\left(n=k=0\\right)$, coherent state\n$\\left(n=k=q=0\\right)$, and Fock state $\\left(n=k=q=\\alpha=0\\right)$.\nSome of the reduced states have been experimentally realized and in\nsome cases optical schemes for generation have been proposed, so this\nfamily of states is apt for various challenging tasks to establish\nquantum dominance. The state under consideration requires various\nnon-Gaussianity inducing quantum engineering operations and thus our\nfocus here was to analyze the relevance of each operation independently\nin the nonclasscial features (listed in Table \\ref{tab:Properties of PASDFS})\nobserved in PASDFS. To study the nonclassical properties of PASDFS,\na set of moments-based criteria for Klyshko's, Agrwal-Tara's, and\nVogel's criteria, as well as lower- and higher-order antibunching,\nHOSPS, and squeezing. Further, phase properties for the same state\nare also studied using phase distribution function and phase fluctuation.\nFinally, non-Gaussianity and nonclassicality of PASDFS is also studied\nusing $Q$ function.\n\n\\begin{table}\n\\begin{centering}\n\\begin{tabular}{ccc}\n\\toprule \nS. No.  & Nonclassical Properties  & Observed in PASDFS\\tabularnewline\n\\midrule \n1  & Lower-order and higher-order Antibunching  & yes\\tabularnewline\n2  & Higher-order sub Poissionian photon statistics  & yes\\tabularnewline\n3  & Lower-order and higher-order squeezing  & yes\\tabularnewline\n4  & Klyshko's criterion  & yes\\tabularnewline\n5  & Agarwal-Tara's criterion  & yes\\tabularnewline\n6  & Vogel's criterion  & yes\\tabularnewline\n7  & Phase distribution function  & -\\tabularnewline\n8  & Phase fluctuation  & yes\\tabularnewline\n9  & $Q$ function  & yes\\tabularnewline\n\\bottomrule\n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{tab:Properties of PASDFS} Summary of the nonclassical properties\nof PASDFS. }\n\\end{table}\nThe present study reveals that with an increase in the order of nonclassicality\nthe depth of nonclassicality witnesses increase. Additionally, higher-order\nnonclassicality criteria were able to detect nonclassicality in the\ncases when corresponding lower-order criteria failed to do so. Different\nnonclassical features are observed for smaller values of displacement\nparameter, which can be sustained for higher values by increasing\nthe number of subtracted photon. Photon addition generally improves\nnonclassicality, and this advantage can be further enhanced for the\nhigher (smaller) values of displacement parameter using photon subtraction\n(Fock parameter). The HOSPS nonclassical feature is only observed\nfor the odd orders. As far as squeezing is concerned, only photon\nsubtraction could induce this nonclassicality. Large number of photon\naddition can be used to observe squeezing at higher values of displacement\nparameter at the cost of that present for smaller $\\alpha$. Photon\nsubtraction alters the phase properties more than photon addition,\nwhile Fock parameter has an opposite effect of the photon addition\/subtraction.\nThe nonclassicality revealed through phase fluctuation parameter shows\nsimilar behavior as Vogel's criterion. Finally, we have shown the\nnonclassicality and non-Gaussianity of PASDFS with the help of a quasidistribution\nfunction, namely $Q$ function.\n\\chapter{Manipulating nonclassicality via quantum state engineering processes:\nVacuum filtration and single photon addition\\textsc{\\label{cha:QSE-1}}}\n\nIn this chapter, the objective is to study nonclassical properties\nassociated with two different quantum state engineering processes\nwith a specific focus on nonclassicality witnesses and measures. The\nwork done in this chapter is published in ~\\cite{malpani2019filter}.\n\n\\section{Introduction \\label{cha:QSE}\\label{introduction:chapter5}}\n\nSo far, we have discussed nonclassical properties of engineered quantum\nstates in detail. Here, we wish to extend the discussion and investigate\nthe possibilities of manipulating or controlling the nonclassicality\npresent in the system by using two specific processes of quantum state\nengineering. Precisely, by using vacuum filtration and single photon\naddition processes. To introduce the idea of these quantum state engineering\noperations, we can write the photon number distribution of an arbitrary\nquantum state in terms of Glauber-Sudarshan $P\\left(\\alpha\\right)$\nfunction as \n\\begin{equation}\np_{n}=\\int P\\left(\\alpha\\right)\\left|\\langle n|\\alpha\\rangle\\right|^{2}d^{2}\\alpha.\\label{eq:pnd}\n\\end{equation}\nIf $p_{n}$ vanishes for a particular value of Fock state parameter\n$n$, we refer to that as a ``hole'' or a hole in the photon number\ndistribution at position $n$ \\cite{escher2004controlled}. Notice\nthat $p_{n}=0$ reveals that $P\\left(\\alpha\\right)<0$ for some $\\alpha$,\nwhich is the signature of nonclassicality. Thus, the existence of\na hole in the photon number distribution implies that the corresponding\nstate is nonclassical, and corresponding technique of quantum state\nengineering to create hole is called hole burning \\cite{gerry2002hole}.\nInterestingly, this result also implies that qudits which are $d$-dimensional\n(finite dimensional) quantum states are always nonclassical as we\ncan see that in such a state $p_{d}=p_{d+1}=\\ldots=0$. In principle,\nthe hole can be created for an arbitrary $n$, but here for the sake\nof a comparative study, we restrict ourselves to the situation where\nthe hole is created at $n=0,$ i.e., the desired engineered state\nhas zero probability of getting vacuum state on measurement in Fock\nbasis (in other words, $p_{0}=0).$ In fact, Lee \\cite{lee1995theorem}\nhad shown that a state with $p_{n}=0$ is a maximally nonclassical\nas long as the nonclassicality is quantitatively measured using nonclassical\ndepth. Such a state can be constructed in various ways. To elaborate\non this, we describe an arbitrary pure quantum state as a superposition\nof the Fock states \n\\begin{equation}\n|\\psi\\rangle=\\sum\\limits _{n=0}^{\\infty}c_{n}|n\\rangle,\\label{eq:fock-superposition}\n\\end{equation}\nwhere $c_{n}$ is the probability amplitude of state $|n\\rangle$.\nA hole can be created at $n=0$ by adding a single photon to obtain\n\\begin{equation}\n|\\psi_{1}\\rangle=N_{1}a^{\\dagger}|\\psi\\rangle,\\label{eq:a-dagger-for-hole}\n\\end{equation}\nwhere $N_{1}=\\left(\\langle\\psi|aa^{\\dagger}|\\psi\\rangle\\right)^{-1\/2}$\nis the normalization constant. If we consider the initial quantum\nstate $|\\psi\\rangle$ as a coherent state, the addition of a single\nphoton would lead to a photon added coherent state which has been\nexperimentally realized \\cite{zavatta2004quantum} and extensively\nstudied \\cite{hong1999nonclassical} because of its interesting nonclassical\nproperties and potential applications. Thus, in quantum state engineering,\ntechniques for photon addition are known \\cite{thapliyal2017comparison,malpani2019lower,malpani2019quantum}\nand experimentally realized.\n\nAn alternative technique to create a hole at vacuum is vacuum filtration.\nThe detailed procedure of this technique is recently discussed in\n\\cite{Meher2018}. Vacuum filtration implies removal of the coefficient\nof the vacuum state, $c_{0}$, in Eq. (\\ref{eq:fock-superposition})\nand subsequent normalization. Clearly this procedure would yield \n\\begin{equation}\n|\\psi_{2}\\rangle=N_{2}\\sum\\limits _{n=1}^{\\infty}c_{n}^{\\prime}|n\\rangle,\\label{eq:c0-is-removed}\n\\end{equation}\nwhere the normalization constant $N_{2}=\\left(1-\\left|c_{0}\\right|^{2}\\right)^{-1\/2}$.\nBoth these states (i.e., $|\\psi_{1}\\rangle,$ and $\\psi_{2}\\rangle$)\nare maximally nonclassical as far as Lee's result related to nonclassical\ndepth is concerned \\cite{lee1991measure}. However, recently lower-order\nnonclassical properties of $|\\psi_{1}\\rangle$ and $|\\psi_{2}\\rangle$\nin (\\ref{eq:a-dagger-for-hole})-(\\ref{eq:c0-is-removed}) are reported\nto be different for $|\\psi\\rangle$ chosen as coherent state \\cite{Meher2018}.\nThis led to several interesting questions, like- What happens if the\ninitial state on which addition of photon or vacuum filtration process\nis to be applied is already nonclassical (specifically, pure state\nother than coherent state \\cite{hillery1985classical})? How do these\nprocesses affect the higher-order nonclassical properties of the quantum\nstates? How does the depth of nonclassicality corresponding to a particular\nwitness of nonclassicality changes with the parameters of the quantum\nstate for these processes? The present chapter aims to answer these\nquestions through a comparative study using a set of interesting quantum\nstates $|\\psi\\rangle$ (and the corresponding single photon added\n$|\\psi_{1}\\rangle$ and vacuum filtered $|\\psi_{2}\\rangle$ states),\neach of which can be reduced to many more states. Specifically, in\nwhat follows, we would study the lower- and higher-order nonclassical\nproperties of single photon addition and vacuum filtration of even\ncoherent state (ECS), binomial state (BS) and Kerr state (KS). In\nfact, the quantum state engineering processes described mathematically\nin Eqs. (\\ref{eq:a-dagger-for-hole})-(\\ref{eq:c0-is-removed}) can\nbe used to prepare a set of engineered quantum states, namely vacuum\nfiltered ECS (VFECS), vacuum filtered BS (VFBS), vacuum filtered KS\n(VFKS), photon added ECS (PAECS), photon added BS (PABS) and photon\nadded (PAKS). We aim to look at the nonclassical properties of these\nstates with a focus on higher-order nonclassical properties and subsequently\nquantify the amount of nonclassicality in all these states. In what\nfollows, the higher-order nonclassical properties are illustrated\nthrough the criteria of HOA, HOS and HOSPS with brief discussion of\nlower-order antibunching and squeezing.\n\\section{Quantum states of interest\\label{sec:Quantum-states-of-1}}\n\nIn this chapter, we have selected a set of three widely studied and\nimportant quantum states- (i) ECS, (ii) BS and (iii) KS. We subsequently\nnoted that these states can further be engineered to generate corresponding\nvacuum filtered states and single photon added states. For example,\none can generate VFBS and PABS from BS by using vacuum filtration\n\\cite{Meher2018} and photon addition \\cite{zavatta2004quantum} processes,\nrespectively. In a similar manner, these processes can also generate\nVFECS and PAECS from ECS, and VFKS and PAKS from KS. In this section,\nwe briefly describe ECS, BS, KS, VFBS, PABS, VFECS, PAECS, VFKS and\nPAKS. Specifically, we describe three parent states as Fock superposition\nstates. Similarly, the six engineered states are also expressed as\nFock superposition states for the convenience of identifying the corresponding\nphoton number distributions (each of which essentially contains a\nhole at the vacuum). In the rest of the study, we wish to compare\nthe impact of these two quantum state engineering processes (i.e.,\nvacuum filtration and photon addition processes) on the nonclassical\nproperties of the engineered states.In the above, we have described\nsix (three) quantum states of our interest as Fock superposition states\nhaving (without) holes at vacuum. In what follows, these expressions\nwill be used to study the nonclassical properties of these states\nusing a set of witnesses of nonclassicality. Specifically, we will\nuse a set of witnesses of nonclassicality which are based on moments\nof annihilation and creation operators. Keeping this in mind, in the\nfollowing subsection, we report the general form of such moments for\nall the six engineered states of our interest and the corresponding\nthree parent states (thus overall nine states).\n\n\\subsection{Expressions for moments of annihilation and creation operators}\n\nIn 1992, Agarwal and Tara \\cite{agarwal1992nonclassical} introduced\na criterion of nonclassicality in the form of a matrix of moments\nof creation and annihilation operators. This criterion was further\nmodified to propose a moment-based criteria of entanglement \\cite{shchukin2005inseparability}\nand nonclassicality \\cite{miranowicz2010testing,miranowicz2009inseparability}.\nTherefore, it is convenient to find out the expectation value of the\nmost general term describing higher-order moment $\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle$\nfor a given state to investigate the nonclassicality using the set\nof moment-based criteria.\n\n\\subsubsection{Expectation values for even coherent states and the corresponding\nengineered states}\n\nThe analytic expression of $\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle_{i}$\nis obtained for the quantum states $i\\in\\{{\\rm ECS},{\\rm VFECS,PAECS\\}}$\nusing Eqs. (\\ref{eq:VFECS-EXPANDED}) and (\\ref{eq:PAECS}). For ECS\nand VFECS, expressions of the moments can be written in a compact\nform as\n\n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle_{{\\rm ECS}} & = & \\frac{\\exp\\left[-\\mid\\alpha\\mid^{2}\\right]}{2\\left(1+\\exp\\left[-2\\mid\\alpha\\mid^{2}\\right]\\right)}\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-k+j}}{\\left(n-k\\right)!}\\mathcal{G}_{n,j,k}.\\end{array}\\label{eq:ECS-moment}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle_{{\\rm VFECS}} & = & \\left\\{ \\begin{array}{l}\nN_{{\\rm VFECS}}^{2}\\sum\\limits _{n=1}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-k+j}}{\\left(n-k\\right)!}\\mathcal{G}_{n,j,k}\\,\\,\\,\\,\\mathrm{for}\\,\\,k\\leq j,\\\\\nN_{{\\rm VFECS}}^{2}\\sum\\limits _{n=1}^{\\infty}\\frac{\\alpha^{\\star n}\\alpha^{n+k-j}}{\\left(n-j\\right)!}\\mathcal{G}_{n,j,k}\\,\\,\\,\\mathrm{for}\\,\\,k>j,\n\\end{array}\\right.\\end{array}\\label{eq:VFECS-moment}\n\\end{equation}\nrespectively. Similarly, we obtained analytic expression for $\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle_{{\\rm PAECS}}$\nfor PAECS as \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger j}\\hat{a}^{k}\\rangle_{{\\rm PAECS}} & = & N_{{\\rm PAECS}}^{2}\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-k+j}\\left(n+1\\right)\\left(n-k+j+1\\right)}{\\left(n+1-k\\right)!}\\mathcal{G}_{n,j,k}.\\end{array}\\label{eq:PAECS-moment}\n\\end{equation}\nHere, $\\mathcal{G}_{n,j,k}=\\left(1+\\left(-1\\right)^{n}\\right)\\left(1+\\left(-1\\right)^{n-k+j}\\right)$.\nThe above mentioned quantities are also functions of displacement\nparameter of ECS used to generate the engineered states, which will\nbe used as a control parameter while discussion of nonclassicality\ninduced due to engineering operations.\n\n\\subsubsection{Expectation values for binomial state and the corresponding engineered\nstates}\n\nSimilarly, the compact analytic form of $\\langle\\hat{a}^{\\dagger t}\\hat{a}^{r}\\rangle_{{\\rm BS}}$\ncan be written as \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger t}\\hat{a}^{r}\\rangle_{{\\rm BS}} & = & \\sum\\limits _{n=0}^{M}\\mathcal{I}_{p,M,n,r,t}\\frac{M!}{(n-r)!}.\\end{array}\\label{eq:BS-moment}\n\\end{equation}\nIn case of VFBS and PABS, the analytic form of $\\langle\\hat{a}^{\\dagger t}\\hat{a}^{r}\\rangle_{{i}}$\nis obtained as \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger t}\\hat{a}^{r}\\rangle_{{\\rm VFBS}} & = & \\left\\{ \\begin{array}{l}\nN_{{\\rm VFBS}}^{2}\\sum\\limits _{n=1}^{M}\\mathcal{I}_{p,M,n,r,t}\\frac{M!}{(n-r)!}\\,\\,\\,\\,\\,{\\rm {for}\\,r\\leq t,}\\\\\nN_{{\\rm VFBS}}^{2}\\sum\\limits _{n=1}^{M}\\mathcal{I}_{p,M,n,-r,-t}\\frac{M!}{(n-t)!}\\,\\,\\,\\,\\,{\\rm {for}\\,r>t,}\n\\end{array}\\right.\\end{array}\\label{eq:VFBS-moment}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger t}\\hat{a}^{r}\\rangle_{{\\rm PABS}} & = & N_{{\\rm PABS}}^{2}\\sum\\limits _{n=0}^{M}\\mathcal{I}_{p,M,n,r,t}\\frac{M!(n+1)!(n+1-r+t)!}{n!(n+1-r)!(n-r+t)!},\\end{array}\\label{eq:PABS-moment}\n\\end{equation}\nrespectively, with $\\mathcal{I}_{p,M,n,r,t}=\\left[\\frac{p^{2n-r+t}\\left(1-p\\right)^{2M-2n+r-t}}{(M-n)!(M-n+r-t)!}\\right]^{1\/2}$.\nHere, the obtained average values of moments are also dependent on\nBS parameters, which will be used to enhance\/control the nonclassicality\nfeatures in the generated states.\n\n\\subsubsection{Expectation values for Kerr state and the corresponding engineered\nstates}\n\nFor Kerr state, Vacuum filtered kerr state and Photon added kerr state,\nwe use the same approach to obtain a compact generalized forms of\n$\\langle\\hat{a}^{\\dagger q}\\hat{a}^{s}\\rangle_{{i}}$; and our computation\nyielded \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger q}\\hat{a}^{s}\\rangle_{{\\rm KS}} & = & \\exp\\left[-\\mid\\alpha\\mid^{2}\\right]\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-s+q}}{\\left(n-s\\right)!}\\mathcal{F}_{n,s,q},\\end{array}\\label{eq:kS-moment}\n\\end{equation}\n\\begin{equation}\n\\begin{array}{l}\n\\begin{array}{l}\n\\langle\\hat{a}^{\\dagger q}\\hat{a}^{s}\\rangle_{{\\rm VFKS}}=\\left\\{ \\begin{array}{l}\nN_{{\\rm VFKS}}^{2}\\sum\\limits _{n=1}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-s+q}}{\\left(n-s\\right)!}\\mathcal{F}_{n,s,q},\\,{\\rm {for}\\,\\,s\\leq q,}\\\\\nN_{{\\rm VFKS}}^{2}\\sum\\limits _{n=1}^{\\infty}\\frac{\\alpha^{\\star n}\\alpha^{n+s-q}}{\\left(n-q\\right)!}\\mathcal{F}_{n,-s,-q}^{\\star},\\,{\\rm {for}\\,\\,s>q,}\n\\end{array}\\right.\\end{array}\\end{array}\\label{eq:VFkS-moment}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{array}{lcl}\n\\langle\\hat{a}^{\\dagger q}\\hat{a}^{s}\\rangle_{{\\rm PAKS}} & = & N_{{\\rm PAKS}}^{2}\\sum\\limits _{n=0}^{\\infty}\\frac{\\alpha^{n}(\\alpha^{\\star})^{n-s+q}\\left(n+1\\right)!\\left(n-s+q+1\\right)!}{n!\\left(n-s+q\\right)!(n+1-s)!}\\mathcal{F}_{n,s,q}.\\end{array}\\label{eq:PAKS-moment}\n\\end{equation}\nHere, $\\mathcal{F}_{n,s,q}=\\exp\\left[\\iota\\chi\\left(q-s\\right)\\left(2n+q-s-1\\right)\\right]$.\nFrom the above expressions, it is clear that when $q=s$, there is\nno role of $\\chi$ and the behavior of KS is similar to that of a\ncoherent state. So the effect of this parameter $\\left(\\chi\\right)$\ncan be observed only in HOS which also depends on the higher-order\nmoments other than moments of number operator, i.e., $\\langle\\hat{a}^{\\dagger q}\\hat{a}^{s}\\rangle_{{i}}\\,:q\\neq s$.\nIn what follows, we use the expressions of moments given in Eqs. (\\ref{eq:VFECS-moment})-(\\ref{eq:PAKS-moment})\nto study various lower- and higher-order nonclassicality witnesses.\n\n\\section{Nonclassicality witnesses\\label{sec:Nonclassicality-witnesses-1}}\n\nThere are various criteria of nonclassicality, most of them are sufficient\nbut not necessary in the sense that satisfaction of such a criterion\ncan identify a nonclassical feature, but failure does not ensure that\nthe state is classical. Further, most of the criteria (specially,\nall the criteria studied here) do not provide any quantitative measure\nof nonclassicality present in a state, and so they are referred to\nas witnesses of nonclassicality.\nThese witnesses are based on either quasiprobability distribution\nor moments of annihilation and creation operators. In the present\nwork, we have used a set of moment-based criteria to investigate nonclassical\nproperties of our desired engineered quantum states. Specifically,\nwe have investigated the possibilities of observing lower-order squeezing\nand antibunching as well as HOA, HOSPS, and HOS for all the states\nof our interest. To begin the investigation and the comparison process,\nlet us start with the study of antibunching.\n\n\\subsection{Lower- and higher-order antibunching}\n\nThe phenomenon of lower-order antibunching is closely associated with\nthe lower-order sub-Poissonian photon statistics \\cite{brown1956correlation}.\nHowever, they are not equivalent \\cite{zou1990photon}. The concept\nof HOA also plays an important role in identifying the presence of\nweaker nonclassicality \\cite{allevi2012measuring,hamar2014non}. It\nwas first introduced in 1990 based on majorization technique \\cite{lee1990higher}\nfollowed by some of its modifications \\cite{an2002multimode,pathak2006control}.\nIn this section, we study the generalized HOA criterion introduced\nby Pathak and Garcia \\cite{pathak2006control} to investigate lower-order\nantibunching and HOA. The analytic expressions of moments (\\ref{eq:VFECS-moment})-(\\ref{eq:PAKS-moment})\ncan be used to investigate the nonclassicality using inequality (\\ref{eq:HOA-1})\nfor the set of states. The obtained results are illustrated in Figure\n\\ref{fig:HOA} where we have compared the results between the vacuum\nfiltered and single photon added states. During this attempt, we also\ndiscuss the nonclassicality present in the quantum states used for\nthe preparation of the engineered quantum states (cf. Figs. \\ref{fig:HOA}\n(a)-(c)). In Figs. \\ref{fig:HOA} (b)-(c), we have shown the result\nfor photon added and vacuum filtered BS and KS, where it can be observed\nthat the depths of both lower- and higher-order witnesses in the negative\nregion are larger for photon added BS and KS in comparison with the\nvacuum filtered BS and KS, respectively. However, an opposite nature\nis observed for ECS where the depth of lower- and higher-order witnesses\nis more for the vacuum filtration in comparison with the photon addition\nif the values of $\\alpha$ remain below certain values; whereas for\nthe photon addition the depth of lower- and higher-order antibunching\nwitnesses is found to be greater than that for vacuum filtration for\nthe higher values of $\\alpha$ (cf. Figure \\ref{fig:HOA} (a)). However,\nHOA is not observed for the ECS and KS and thus both operations can\nbe ascribed as nonclassicality inducing operations as far as this\nnonclassical feature is concerned.\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\centering{}\\includegraphics[width=60mm]{AB-l-2-compare.pdf}  & \\includegraphics[width=60mm]{Binomial-AB-l-2.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{kerr-AB-l-2-compare.pdf} & \\tabularnewline\n(c)  & \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:HOA} Lower- and higher-order antibunching witnesses as\nfunctions of displacement parameter $\\alpha$ (for ECS and KS) and\nprobability $p$ (for BS with parameter $M=10$) for (a) ECS, PAECS\nand VFECS, (b) BS, PABS and VFBS, and (c) KS, PAKS and VFKS. The quantities\nshown in all the plots are dimensionless.}\n\\end{figure}\n\n\\subsection{Lower- and higher-order squeezing}\n\nThe concept of squeezing originates from the uncertainty relation.\nThere is a minimum value of an uncertainty relation involving quadrature\noperators where the variance of two non-commuting quadratures (say\nposition and momentum) are equal and their product satisfies minimum\nuncertainty relation. Such a situation is closest to the classical\nscenario, in the sense that there is no uncertainty in the classical\npicture and this is the closest point that one can approach remaining\nwithin the framework of quantum mechanics. Coherent state satisfies\nthis minimum uncertainty relation and is referred to as a classical\n(or more precisely closest to classical state). If any of the quadrature\nvariances reduces below the corresponding value for a minimum uncertainty\n(coherent) state (at the cost of increase in the fluctuations in the\nother quadrature) then the corresponding state is called squeezed\nstate.\n\nThe higher-order nonclassical properties can be investigated by studying\nHOS. There are two different criteria for HOS \\cite{hong1985generation,hong1985higher,hillery1987amplitude}:\nHong-Mandel criterion \\cite{hong1985higher} and Hillery criterion\n\\cite{hillery1987amplitude}. The concept of the HOS was first introduced\nby Hong and Mandel using higher-order moments of the quadrature operators\n\\cite{hong1985higher}. According to this criterion, it is observed\nif the higher-order moment for a quadrature operator for a quantum\nstate is observed to be less than the corresponding coherent state\nvalue. Another type of HOS was introduced by Hillery who introduced\namplitude powered quadrature and used variance of this quadrature\nto define HOS \\cite{hillery1987amplitude}. Here, we aim to analyze\nthe possibility of HOS using Hong-Mandel criterion for $l$th order\nsqueezing. We have investigated the possibility of observing HOS analytically\nusing Eqs. (\\ref{eq:VFECS-moment})-(\\ref{eq:PAKS-moment}) and inequality\n(\\ref{eq:Hong-def2-2}) for all engineered quantum states of our interest\nand have shown the corresponding results in Figs. \\ref{fig:HOS-1}\n(a)-(c) where we have compared the HOS in the set of quantum states\nand the states obtained by photon addition and vacuum filtration.\nThese operations fail to induce this nonclassical feature in the engineered\nstates prepared from ECS, which also did not show signatures of squeezing.\nIn Figure \\ref{fig:HOS-1} (a), we illustrate Hong-Mandel type HOS\nwith respect to parameter $p$ where we have shown the existence HOS\nfor BS, VFBS and PABS. It can be observed that the state engineering\noperations fail to increase this particular feature of nonclassicality\nin BS. Additionally, higher-order nonclassicality is absent for higher\nvalues of $p$ when corresponding lower-order squeezing is present.\nIn case of KS, PAKS and VFKS, we have observed that HOS is observed\nwhen the values of $\\alpha$ are greater than certain values for the\nindividual curves of the corresponding states (cf. Figure \\ref{fig:HOS-1}\n(b)). Note that photon addition may provide some advantage in this\ncase, but vacuum filtration would not as for the same value of displacement\nparameter KS and PAKS (VFKS) have (has not) shown squeezing. Interestingly,\nthe presence of squeezing also depends upon the Kerr nonlinearity\nparameter $\\chi$, which is shown in Figure \\ref{fig:HOS-1} (c).\nSimilar to Figure \\ref{fig:HOS-1} (b) photon addition shows advantage\nover KS which disappears for larger values of $\\chi$, while vacuum\nfiltering is not beneficial.\n\nIn Figure \\ref{fig:HOS-contour}, we have shown using the dark (blue)\ncolor in the contour plots of the HOS witness for PAKS that squeezing\ncan be observed for higher values of $|\\alpha|$ and smaller values\nof $\\chi$. Additionally, the phase parameter $\\theta$ of $\\alpha$\nis also relevant for observing the nonclassicality as squeezing occurs\nin the vicinity of $\\theta=m\\pi$, while disappears for $\\theta=\\frac{m\\pi}{2}$\nwith integer $m$. Similar behavior is observed in KS and VFKS (not\nshown here).\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\centering{}\\includegraphics[width=60mm]{Binomial-HOS-l-2.pdf}  & \\includegraphics[width=60mm]{kerr-HOS-l-2.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{kerr-HOS-l-2-diff-chi.pdf} & \\tabularnewline\n(c)  & \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:HOS-1} Illustration of lower- and higher-order squeezing\nfor (a) BS, PABS and VFBS; (b) KS, PAKS and VFKS at the fixed value\nof $\\chi=0.02$; (c) KS, VFKS and PAKS as a function of $\\chi$ with\n$\\alpha=1$. The negative regions of the curves illustrate the presence\nof squeezing.}\n\\end{figure}\n\n\\begin{figure}\n\\centering \\includegraphics[width=120mm]{Slide1.jpg}\n\\caption{\\label{fig:HOS-contour}The dependence of HOS witness ($l=4$) on\nKerr parameter $\\chi$ and displacement parameters $\\left|\\alpha\\right|$\nand $\\theta$ for PAKS with (a) $\\left|\\alpha\\right|=3$, (b) $\\chi=0.02$,\n(c) $\\theta=0$.}\n\\end{figure}\n\n\\subsection{Higher-order sub-Poissonian photon statistics}\n\nThe higher-order moments in Eqs. (\\ref{eq:VFECS-moment})-(\\ref{eq:PAKS-moment})\nare used to calculate the above inequality $(\\ref{eq:hosps22-1})$\nwith the help of (\\ref{eq:HOA-1}) for states obtained after vacuum\nfiltration and photon addition in ECS, BS and KS as well as the parent\nstates, and the corresponding results are depicted in Figure \\ref{fig:HOSPS}.\nNonclassicality is not revealed by HOSPS criteria of even orders in\ncase of ECS, while corresponding engineered states show nonclassicality.\nAdditionally, nonclassicality is induced by vacuum filtration for\nodd orders while it was not observed in the parent state (cf. Figure\n\\ref{fig:HOSPS} (a)). This clearly shows the role of hole burning\noperations in inducing nonclassicality for odd orders. However, in\ncase of even orders, the same operations are also observed to destroy\nthe nonclassicality in the parent state. From Figs. \\ref{fig:HOSPS}\n(b) and (c), it is observed that BS and KS do not show HOSPS for the\nodd values of $l$ even after application of state engineering operations.\nAdditionally, HOSPS is not observed for the KS for even values of\n$l$, too. Consequently, the nonclassical feature witnessed through\nthe HOSPS criterion in PAKS can be attributed solely to the hole burning\nprocess.\n\nNonclassicality in the engineered quantum states can also be studied\nusing quasidistribution functions \\cite{thapliyal2015quasiprobability}\nbut here we are going to quantify the amount of nonclassicality in\nthese states. Further, the effect of decoherence on the observed nonclassicality\n\\cite{banerjee2007dynamics,banerjee2010dynamics,banerjee2010entanglement,naikoo2018probing}\nand phase diffusion \\cite{banerjee2007phaseQND,banerjee2007phase}\ncan be studied.\n\n\\begin{figure}\n\\centering{} %\n\\begin{tabular}{cc}\n\\centering{}\\includegraphics[width=60mm]{HOSPS-l-2-compare.pdf}  & \\includegraphics[width=60mm]{Binomial-HOSPS-l-2.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{kerr-HOSPS-l-2.pdf} & \\tabularnewline\n(c)  & \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:HOSPS} Illustration of HOSPS as a function of displacement\nparameter $\\alpha$ (for ECS and KS) and probability $p$ (for BS)\nfor (a) ECS, (b) BS, and (c) KS and corresponding engineered states.\nHOSPS is not observed in KS.}\n\\end{figure}\n\\begin{figure}\n\\centering %\n\\begin{tabular}{cc}\n\\includegraphics[width=60mm]{Linear-entropy-ECS.pdf}  & \\includegraphics[width=60mm]{Linear-entropy-BS.pdf}\\tabularnewline\n(a)  & (b) \\tabularnewline\n\\includegraphics[width=60mm]{Linear-entropy-KS.pdf}  & \\includegraphics[width=60mm]{Linear-entropy-KS-chi.pdf}\\tabularnewline\n(c)  & (d) \\tabularnewline\n\\end{tabular}\\caption{\\label{fig:Linear-Entropy} Illustration of linear entropy for (a)\nECS, PAECS and VFECS, (b) BS, PABS and VFBS, (c) KS, PAKS and VFKS\nwith $\\alpha$ or $p$ for $\\chi=0.02$. (d) Dependence of nonclassicality\nin KS, PAKS and VFKS on $\\chi$ for $\\alpha=1$.}\n\\end{figure}\n\n\\section{Nonclassicality measure}\n\nWe have obtained the analytic expressions of linear entropy for ECS,\nKS, BS and corresponding engineered states which are given as \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm ECS}} & = & 1-\\frac{\\exp\\left[-2\\mid\\alpha\\mid^{2}\\right]}{4\\left(1+\\exp\\left[-2\\mid\\alpha\\mid^{2}\\right]\\right)^{2}}\\sum\\limits _{n,m,r=0}^{\\infty}f_{n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LE-ECS}\n\\end{equation}\nfor VFECS \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm VFECS}} & = & 1-\\left(N_{{\\rm VFECS}}\\right)^{4}\\sum\\limits _{n,m,r=1}^{\\infty}f_{n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LE-VFECS}\n\\end{equation}\nand for PAECS \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm PAECS}} & = & 1-\\left(N_{{\\rm PAECS}}\\right)^{4}\\sum\\limits _{n,m,r=0}^{\\infty}f_{n,m,r}\\left(m+1\\right)\\left(n-m+r+1\\right)\\\\\n & \\times & \\sum\\limits _{k_{1}=0}^{n+1}\\frac{^{n+1}C_{k_{1}}\\,{}^{r+1}C_{r+k_{1}-m}}{2^{n+r+2}},\n\\end{array}\\label{eq:LE-PAECS}\n\\end{equation}\nwhere $f_{n,m,r}=\\frac{\\mid\\alpha\\mid^{2n+2r}\\left(1+\\left(-1\\right)^{n}\\right)\\left(1+\\left(-1\\right)^{m}\\right)\\left(1+\\left(-1\\right)^{r}\\right)\\left(1+\\left(-1\\right)^{n+r-m}\\right)}{n!r!}$.\n\nSimilarly, analytical expression for linear entropy of BS \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm BS}} & = & 1-\\sum\\limits _{n,m,r=0}^{M}g_{p,M,n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LE-BS}\n\\end{equation}\nfor VFBS \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm VFBS}} & = & 1-\\left(N_{{\\rm VFBS}}\\right)^{4}\\sum\\limits _{n,m,r=1}^{M}g_{p,M,n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LE-VFBS}\n\\end{equation}\nand for PABS \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm PABS}} & = & 1-\\left(N_{{\\rm PABS}}\\right)^{4}\\sum\\limits _{n,m,r=0}^{M}g_{p,M,n,m,r}\\left(m+1\\right)\\left(n-m+r+1\\right)\\\\\n & \\times & \\sum\\limits _{k_{1}=0}^{n+1}\\frac{^{n+1}C_{k_{1}}\\,{}^{r+1}C_{r+k_{1}-m}}{2^{n+r+2}}\n\\end{array}\\label{eq:LE-PABS}\n\\end{equation}\nare obtained. Here, $g_{p,M,n,m,r}=\\frac{1}{n!r!}\\left[\\frac{(M!)^{4}p^{2(n+r)}(1-p)^{4M-2n-2r}}{(M-n)!(M-m)!(M-r)!(M-n-r+m)!}\\right]^{1\/2}$.\n\nFinally, analytical expression for linear entropy of KS, VFKS, PAKS\ncan be given as \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm KS}} & = & 1-\\exp\\left[-2\\mid\\alpha\\mid^{2}\\right]\\sum\\limits _{n,m,r=0}^{\\infty}h_{n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LS-KS}\n\\end{equation}\n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm VFKS}} & = & 1-\\left(N_{{\\rm VFKS}}\\right)^{4}\\sum\\limits _{n,m,r=1}^{\\infty}h_{n,m,r}\\sum\\limits _{k_{1}=0}^{n}\\frac{^{n}C_{k_{1}}\\,{}^{r}C_{r+k_{1}-m}}{2^{n+r}},\\end{array}\\label{eq:LE-VFKS}\n\\end{equation}\nand \n\\begin{equation}\n\\begin{array}{lcl}\n\\mathcal{L}_{{\\rm PAKS}} & = & 1-\\left(N_{{\\rm PAKS}}\\right)^{4}\\sum\\limits _{n,m,r=0}^{\\infty}h_{n,m,r}\\left(m+1\\right)\\left(n-m+r+1\\right)\\\\\n & \\times & \\sum\\limits _{k_{1}=0}^{n+1}\\frac{^{n+1}C_{k_{1}}\\,{}^{r+1}C_{r+k_{1}-m}}{2^{n+r+2}},\n\\end{array}\\label{eq:LE-PAKS}\n\\end{equation}\nrespectively, with $h_{n,m,r}=\\frac{\\left|\\alpha\\right|^{2n+2r}\\exp\\left[2\\iota\\chi(m-n)(m-r)\\right]}{n!r!}$.\n\nIn general, significance of hole burning operations can be clearly\nestablished through corresponding results shown in Figure \\ref{fig:Linear-Entropy}.\nSpecifically, one can clearly see the amount of nonclassciality (revealed\nthrough the amount of entanglement it can generate at a beam splitter)\nincreases due to these operations.\n\nFrom Figs. \\ref{fig:Linear-Entropy} (a) and (c), it can be observed\nthat vacuum filtered ECS and KS are more nonclassical than corresponding\nphoton added counterparts. However, in case of BS and its engineered\nstates, it is observed that only up to a certain value of $p$ VFBS\nis more nonclassical than PABS (cf. Figure \\ref{fig:Linear-Entropy}\n(a)). In fact, the amount of additional nonclassicality induced due\nto filtration decreases with $p$ and eventually becomes zero (i.e.,\nthe amount of nonclassicality of VFBS becomes equal to that of BS\nas far as linear entropy is considered as a measure of nonclassicality).\nIt is interesting to observe the effect of Kerr coupling parameter\n$\\chi$ on the amount of nonclassicality induced due to nonlinearity.\nIt is observed that for small (relatively large) values of $\\chi$\nnonclassicality present in VFKS (PAKS) is more than that in PAKS (VFKS)\n(cf. Figure \\ref{fig:Linear-Entropy} (d)). This dependence is more\nclearly visible in Figure \\ref{fig:LE}, where one can observe strong\nnonclassicality in PAKS and VFKS (KS) favor (favors) smaller (higher)\nvalues of $\\alpha$ and large $\\chi$.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=120mm]{fig7.jpg}\n\\caption{\\label{fig:LE} Illustration of linear entropy for (a) KS (b) VFKS\n(c) PAKS.}\n\\end{figure}\n\n\\section{Conclusion \\label{sec:Conclusion}}\n\nIn summary, this chapter is focused on the comparison of the effects\nof two processes (vacuum state filtration and single photon addition)\nused in quantum state engineering to burn hole at vacuum as far as\nthe higher-order nonclassical properties of the quantum states prepared\nusing these two processes are concerned. Specifically, various quantum\nstate engineering processes for burning holes at vacuum lead to different\n$\\sum_{m=1}c_{m}|m\\rangle$ as far as the values of $c_{m}$s are\nconcerned (even when the parent state is the same). To study its significance\nin nonclassical properties of the engineered states, we considered\na small set of finite and infinite dimensional quantum states (namely,\nECS, BS, and KS). This provided us a set of six engineered quantum\nstates, namely VFECS, PAECS, VFBS, PABS, VFKS, and PAKS and three\nparent states for our analysis. This set of engineered quantum states\ncan have a great importance in quantum information processing and\nquantum optics as they are found to be highly nonclassical. Especially\nwhen some exciting applications of their parent states are already\ninvestigated with relevance to continuous variable quantum information\nprocessing and\/or quantum optics. The present study also addresses\nthe significance of these hole burning processes in inducing (enhancing)\nparticular nonclassical features in the large set of engineered and\nparent quantum states.\n\nThe general expressions for moments of the set of states are reported\nin the compact analytic form, which are used here to investigate nonclassical\nfeatures of these states using a set of criteria of higher-order nonclassicality\n(e.g., criteria of HOA, HOS and HOSPS). The obtained expressions can\nbe further used to study other moment-based criteria of nonclassicality.\nThe hole burning operations are found to be extremely relevant as\nthe states studied here are found to be highly nonclassical when quantified\nthrough a measure of nonclassicality (entanglement potential).\n In brief, both the vacuum filtration and photon addition operations\ncan be ascribed as antibunching inducing operations in KS and ECS\nwhile antibunching enhancing operations for BS. As far as HOS is concerned\nno such advantage of these operations is visible as these operations\nfail to induce squeezing in ECS and often decrease the amount of squeezing\npresent in the parent state. Additionally, the operations are successful\nin inducing HOSPS in KS and enhances this feature in the rest of the\nparent states. The relevance of higher-order nonclassicality in the\ncontext of the present study can be understood from the fact that\nthese hole burning operations show an increase in the depth of HOA\nwitness and decrease in the amount of HOS with order. While in case\nof HOSPS even orders show nonclassicality whereas odd orders fail\nto detect it. Finally, the measure of nonclassicality reveals vacuum\nfiltration as a more powerful tool than photon addition for enhancing\nnonclassicality in the parent state, but photon addition is observed\nto be advantageous in some specific cases.\n\\chapter{Conclusions And Scope For Future Work\\textsc{\\label{cha:Conclusions-and-Scope}}}\nThis concluding chapter aims to briefly summarize the results obtained\nin this thesis work, and it also aims to provide some insights into\nthe scope of future works. To begin with, we may note that this thesis\nis a theoretical work focused on nonclassical and phase properties\nof  some of the  engineered\nquantum states of radiation field. Here, lower- and higher-order nonclassical\nproperties of PADFS, PSDFS, PASDFS, ECS, VFECS, PAECS, BS, VFBS, PABS,\nKS, VFKS and PAKS have been witnessed through lower- and higher-order\nantibunching, higher-order sub-Poissonian photon statistics, higher-order\nsqueezing, Klyshko's criterion, Vogel's criterion, Agarwal-Tara's\ncriterion, $Q$ function, Mandel $Q_{M}$ parameter, etc. Further,\nphase properties of these states have been investigated with the help\nof phase distribution function, phase dispersion, phase fluctuation,\nphase uncertainty parameter and angular $Q$ function. These investigations\nhave revealed that the state engineering processes may help us to\nintroduce and manipulate the nature and amount of nonclassicality\npresent in a quantum state. Keeping this in mind, at the end of the\nthesis two quantum state engineering processes, which can be used\nto generate holes at vacuum in photon number distribution, have been\ncompared. This systematic and rigorous study of the nonclassical and\nphase properties of the above mentioned engineered quantum states\nhave led to many new findings, some of them are already mentioned\nin the end of the individual chapters. In what follows, we list the\nmajor findings of the present thesis.\n\n\\section{Conclusion}\n\nThe main observations of the present thesis may be summarized as follows: \n\\begin{enumerate}\n\\item It is observed that photon addition and subtraction are not only non-gaussianity\nand nonclassicality inducing operations but they can also boost the\nnonclassicality present in the DFS. \n\\item The results indicate that the amount of nonclassicality in PADFS and\nPSDFS can be controlled by the Fock state parameter, displacement\nparameter, the number of photon addition and\/or subtraction. \n\\item Higher-order squeezing witness and $Q$ function are observed to be\ndependent on the phase of the displacement parameter. However, only\nhigher-order squeezing criterion was found to be able to detect nonclassicality,\nand thus established that this phase parameter can also be used to\ncontrol the amount of nonclassicality. \n\\item It is observed that the depth of nonclassicality witnesses increases\nwith order of nonclassicality. \n\\item The phase distribution and angular $Q$ functions are found to be\nsymmetric along the value of the phase of the displacement parameter. \n\\item Photon addition\/subtraction and Fock parameters are found to induce\nopposite effects on phase distribution. Between photon addition and\nsubtraction operations, subtracting a photon modifies the phase properties\nmore than photon addition. Interestingly, phase\nproperties are associated with average photon number of the state\nas well. Photon subtraction increases the average photon number as\nphoton addition does. However, photon addition creates a hole at vacuum\nunlike photon subtraction.\n\\item The three phase fluctuation parameters given by Carruthers and Nieto\nreveal phase properties of PADFS and PSDFS, although one of them,\n$U$ parameter indicates antibunching in both PADFS and PSDFS. \n\\item Phase dispersion quantifying phase fluctuation remains unity for Fock\nstate reflecting uniform distribution, which can be observed to decrease\nwith increasing displacement parameter. This may be attributed to\nthe number-phase complimentarity as the higher values of variance\nwith increasing displacement parameter lead to smaller phase fluctuation. \n\\item The present investigation has revealed the advantage of the PADFS\nand PSDFS in quantum phase estimation and has obtained the set of\noptimized parameters in the PADFS\/PSDFS. \n\\item The nonclassicality and non-Gaussianity of PASDFS viewed with the\nhelp of a quasidistribution function, namely $Q$ function is shown\nin the present thesis. \n\\item The present study also provides a flavor of the significance of the\nhole burning processes in inducing particular nonclassical features\nin the family of engineered and parent quantum states. The hole burning\noperations are observed to be potentially relevant as the quantum\nstates studied in this work are observed to be highly nonclassical\nwhen quantification is done through a measure of nonclassicality. \n\\end{enumerate}\n\n\\section{Scope for future work}\n\nThe works reported in the present thesis give us a general idea for\nthe investigation of phase properties and nonclassical features present\nin a family of engineered quantum states. This work can be further\nextended in various ways. Some of the possible extension of the present\nwork are listed below with a focus on the possibilities that may be\nrealized in the near future. \n\\begin{enumerate}\n\\item The work may be continued to find out the non-Gaussianity of the studied\nstates. Subsequently, the nonclassicality and non-Gaussianity observed\nin these states can be used to realize various applications in quantum\ninformation processing tasks. Therefore, specific\napplications of the non-classical properties of the aforesaid states\nwhich (the applications) are otherwise impossible to achieve using\nother types of states (classical\/nonclassical).\n\\item A major part of the results presented here can be experimentally verified\nusing the available technology.  Along this line,\nit would be interesting to perform resource comparison (e.g., total\nnumber of beam splitters, photo detectors, nonlinear gadgets, etc.)\nin generation of the aforesaid nonclassical states using quantum state\nengineering methods.\n\\item The work can be extended to quantify the amount of nonclassicality\npresent in quantum states using different nonclassicality measures. \n\\item The methods adopted here and the results obtained here can be helpful\nin further theoretical studies on nonclassical and phase properties\nof other engineered quantum states (both finite as well as infinite\ndimensional).  There could be many such states using\nseveral other quantum state engineering tools, for instance, squeezing,\nphoton catalysis, etc. \n\\item Attempts can be made to observe the effect of noise in these states.\n Specifically, further study of the robustness of\nobserved nonclassical properties of PADFS, PSDFS, PASDFS, BS, VFBS,\nPABS, KS, VFKS, PAKS under photon loss as well as inefficiency of\nphoto-detectors.\n\\end{enumerate}\nWe expect that theoretical work done in this thesis will be performed\nexperimentally and that will lead to some important applications.\nWe also hope this work will be very useful in quantum optics. With\nthese hopes this thesis is concluded. \n\n\n\n\n\\backmatter\n\\pagenumbering{roman}\\setcounter{page}{1}\n\t\\addcontentsline{toc}{chapter}{References}\n\\renewcommand\\bibname{\\bf References}\n\\setlength{\\bibsep}{0pt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Checklist}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs of Section \\ref{sec:elimination_at_sampling}}\n\\label{app:proofs_of_sampling}\n\n\\subsection{On Assumption \\ref{ass:beta_bounds}}\n\nWe illustrate first how Assumption~\\ref{ass:beta_bounds} covers the threshold obtained by the most common technique: first get a concentration bound valid for one time of the form $f(1\/\\delta)$, where usually $f(x) = \\log(x) + C\\log\\log(x)$ (for $C$ a dimension-dependent constant), then obtain a time-uniform concentration threshold by setting for example $\\beta_{t, \\delta} = f(\\frac{\\pi^2 t^2}{6 \\delta})$.\n\nLet us suppose then that $\\beta_{t, \\delta} = f(\\frac{C' t^2}{\\delta})$ for $f(x) = \\log(x) + C\\log\\log(x)$ and $C' \\ge 1$. We will now prove that if verifies Assumption~\\ref{ass:beta_bounds}.\n\\begin{align*}\n\\overline{\\log}(C' t^2\/\\delta)\n&= \\log \\frac{C' t^2}{\\delta} + C\\log(\\log\\frac{C' }{\\delta} + \\log t^2)\n\\\\\n&\\ge \\log \\frac{C' }{\\delta} + C\\log\\log\\frac{C' }{\\delta} + 2 \\log(t)\n\\\\\n&= \\overline{\\log}\\frac{1}{\\delta} + 2 \\log(t)\n\\: .\n\\end{align*}\nwhere $\\overline{\\log}(x) = \\log(C' x) + C\\log\\log(C' x)$. Using the concavity of $\\log$, we also have an upper bound\n\\begin{align*}\n\\overline{\\log}(C' t^2\/ \\delta)\n&= \\log \\frac{C' t^2}{\\delta} + C\\log(\\log\\frac{C' }{\\delta} + \\log t^2)\n\\\\\n&= \\log \\frac{C' }{\\delta} + C\\log\\log\\frac{C' }{\\delta} + 2 \\log(t) + C(\\log(\\log\\frac{C' }{\\delta} + \\log t^2) - \\log\\log\\frac{C' }{\\delta})\n\\\\\n&\\le \\overline{\\log}\\frac{1}{\\delta} + 2 (C+1)\\log(t)\n\\: .\n\\end{align*}\nWe have found a function $\\overline{\\log}$ such that for $c_1 = 2$ and $c_2=2(C+1)$,\n\\begin{align*}\n\\overline{\\log}(\\frac{1}{\\delta}) + c_1 \\log(t)\n\\le \\beta_{t, \\delta}\n\\le \\overline{\\log}(\\frac{1}{\\delta}) + c_2 \\log(t)\n\\: .\n\\end{align*}\nIt remains to show the condition on $\\overline{\\log}$, i.e. find $x_0$ such that for $a \\ge 2$ and $x \\ge x_0$, $\\overline{\\log}(x^a) \\le a \\overline{\\log}(x)$.\n\\begin{align*}\n\\overline{\\log}(x^a)\n= \\log(C' x^a) + C \\log\\log(C' x^a)\n&\\le \\log((C' x)^a) + C \\log\\log((C' x)^a)\n\\\\\n&= a \\log (C' x) + C \\log\\log(C' x) + C \\log a\n\\end{align*}\nIt remains to find $x_0$ such that for $x \\ge x_0$, $\\log\\log(C' x) + \\log a \\le a \\log\\log(C' x)$. We find that $x_0 = \\exp(a^{1\/(a-1)})\/C'$ is suitable. Since $a \\ge 2$, we have $x_0 \\le e^2\/C'$.\n\n\n\\begin{lemma}\\label{lem:new_beta-diff-between-phases}\nUnder Assumption~\\ref{ass:beta_bounds}, for any time $t$ and $0 \\leq j \\leq j(t)$, \n\\begin{align*}\n\\beta_{t,\\delta}\n&\\leq \\beta_{\\overline{t}_{j},\\delta} + (2^{j(t)+1-j} c_2 - c_1)\\log(\\overline{t}_{j}),\n\\: , \\\\\n\\beta_{t,1\/t^2}\n&\\leq \\frac{c_2}{c_1} 2^{j(t)-j+1} \\beta_{\\overline{t}_j,1\/\\overline{t}_j^2}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nWe have $\\bar{t}_0 = \\max\\{2, \\sqrt{x_0}\\}$. Note that for all $j$, $\\bar{t}_j = \\bar{t}_0^{2^j}$.\nBy using first the upper bound of Assumption~\\ref{ass:beta_bounds}, then $t \\leq \\bar{t}_0^{2^{j(t)+1}}$ and at the end the lower bound of Assumption~\\ref{ass:beta_bounds},\n\\begin{align*}\n\\beta_{t, \\delta}\n\\le \\overline{\\log}(\\frac{1}{\\delta}) + c_2 \\log(t)\n\\le \\overline{\\log}(\\frac{1}{\\delta}) + c_2 \\log(\\bar{t}_0^{2^{j(t)+1}})\n&= \\overline{\\log}(\\frac{1}{\\delta}) + c_2 \\log((\\bar{t}_0^{2^j})^{2^{j(t)+1-j}})\n\\\\\n&= \\overline{\\log}(\\frac{1}{\\delta}) + c_2 2^{j(t)+1-j} \\log(\\bar{t}_0^{2^j})\n\\\\\n&\\le \\beta_{\\overline{t}_j, \\delta} + \\log(\\overline{t}_j)(2^{j(t)+1-j} c_2 - c_1)\n\\: .\n\\end{align*}\n\nWe have $\\overline{t}_j \\ge \\bar{t}_0 \\ge \\sqrt{x_0}$, hence the inequality $\\overline{\\log}(x^a) \\le a\\overline{\\log}(x)$ can be used. Then\n\\begin{align*}\n\\beta_{t, 1\/t^2}\n\\le \\overline{\\log}(t^2) + c_2 \\log(t)\n&\\le \\overline{\\log}(\\bar{t}_0^{2^{j(t)+2}}) + c_2 \\log(\\bar{t}_0^{2^{j(t)+1}})\n\\\\\n&= \\overline{\\log}((((\\bar{t}_0^{2^j})^2)^{2^{j(t)+1-j}}) + c_2 \\log((\\bar{t}_0^{2^j})^{2^{j(t)+1-j}})\n\\\\\n&= \\overline{\\log}(((\\overline{t}_j^2)^{2^{j(t)+1-j}}) + c_2 2^{j(t)+1-j} \\log(\\overline{t}_j)\n\\\\\n&\\le 2^{j(t)+1-j}\\overline{\\log}((\\overline{t}_j^2) + c_2 2^{j(t)+1-j} \\log(\\overline{t}_j)\n\\\\\n&\\le \\frac{c_2}{c_1}\\left( 2^{j(t)+1-j}\\overline{\\log}((\\overline{t}_j^2) + c_1 2^{j(t)+1-j} \\log(\\overline{t}_j) \\right)\n\\\\\n&\\le \\frac{c_2}{c_1} 2^{j(t)+1-j} \\beta_{\\overline{t}_j, 1\/\\overline{t}_j^2}\n\\: .\n\\end{align*}\n\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem \\ref{th:sampling-rule-with-elim}}\n\nWe now derive an important result for sampling rules combined with elimination (either full or selective). It essentially shows that we cannot eliminate the closest alternative piece to $\\theta$ from the sampling rule without making the algorithm stop.\n\n\\begin{lemma}\\label{lem:closest-alternatives-not-eliminated-in-sampling}\nLet $t \\ge \\bar{t}_1$ be any time step at which the algorithm did not stop. Suppose that some piece index $p\\in\\cP(i^\\star)$ of the true correct answer $i^\\star$ has been eliminated from the sampling rule (i.e., $p\\notin\\cP_{t}^{\\mathrm{smp}}(i^\\star)$). Then, under event $E_t$ (see Equation \\ref{eq:Et}),\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) > \\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nLet us proceed by contradiction: suppose that $p\\notin\\cP_{t}^{\\mathrm{smp}}(i^\\star)$ while $\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) = \\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)$ and $E_t$ holds.\n\nSince $\\cP_{t}^{\\mathrm{smp}}(i^\\star)$ is the intersection of all active sets from $\\overline{t}_{j(t)-1}$ to $t$, if $p\\notin\\cP_{t}^{\\mathrm{smp}}(i^\\star)$, then there exists $s$ with $\\overline{t}_{j(t)-1} \\leq s \\leq t$ such that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i)} L_s(\\hat{\\theta}_s,\\lambda) \\geq \\alpha_{s,\\delta}.\n\\end{align*}\nTherefore,\n\\begin{align*}\n\\sqrt{\\alpha_{s,\\delta}}\n \\stackrel{(a)}{\\leq} \\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} L_{s}(\\hat{\\theta}_{s},\\lambda)}\n &\\stackrel{(b)}{\\leq}  \\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}N_{s}^k\\KL_k(\\theta,\\lambda)} + \\sqrt{L_{s}(\\hat{\\theta}_{s},\\theta)}\n \\\\ &\\stackrel{(c)}{\\leq}  \\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}N_{s}^k\\KL_k(\\theta,\\lambda)} + \\sqrt{\\beta_{t,1\/t^2}}\n \\\\ &\\stackrel{(d)}{\\leq}  \\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}N_{t}^k\\KL_k(\\theta,\\lambda)} + \\sqrt{\\beta_{t,1\/t^2}}\n \\\\ & \\stackrel{(e)}{=}  \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}N_{t}^k\\KL_k(\\theta,\\lambda)} + \\sqrt{\\beta_{t,1\/t^2}},\n\\end{align*}\nwhere (a) is from the elimination condition, (b) uses Lemma \\ref{lem:llr-to-kl-lin-gauss}, (c) uses that $E_t$ holds, (d) uses that the number of pulls of each arm is non-decreasing in time, and (e) holds by our assumption. Recall that, for any $\\theta,\\lambda\\in\\mathbb{R}^d$, $\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) = \\frac{1}{2}\\|\\theta-\\lambda\\|_{V_t}^2$. Therefore, by the triangle inequality,\n\\begin{align*}\n\\sqrt{\\sum_{k\\in[K]}N_{t}^k\\KL_k(\\theta,\\lambda)}\n= \\frac{1}{\\sqrt{2}}\\|\\theta-\\lambda\\|_{V_t}\n&\\leq \\frac{1}{\\sqrt{2}}\\|\\theta-\\hat{\\theta}_t\\|_{V_t} + \\frac{1}{\\sqrt{2}}\\|\\hat{\\theta}_t-\\lambda\\|_{V_t}\n\\\\\n&= \\sqrt{L_t(\\hat{\\theta}_t, \\lambda)} + \\sqrt{L_t(\\hat{\\theta}_t, \\theta)}.\n\\end{align*}\nCombining this with the previous chain of inequalities,\n\\begin{align*}\n\\sqrt{\\alpha_{s,\\delta}} \\leq \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^\\star)}L_t(\\hat{\\theta}_t, \\lambda)} + \\sqrt{L_t(\\hat{\\theta}_t, \\theta)} + \\sqrt{\\beta_{t,1\/t^2}}\n &\\leq \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^\\star)}L_t(\\hat{\\theta}_t, \\lambda)} + 2\\sqrt{\\beta_{{t},1\/{t}^2}}\n \\\\ &\\leq \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_t))}L_t(\\hat{\\theta}_t, \\lambda)} + 2\\sqrt{\\beta_{{t},1\/{t}^2}},\n\\end{align*}\nwhere we used again that $E_t$ holds to concentrate the LLR between $\\hat{\\theta}_t$ and $\\theta$. The last inequality is easy to check since, if $i^\\star \\neq i^\\star(\\hat{\\theta}_t)$, then $\\inf_{\\lambda \\in \\Lambda(i^\\star)}L_t(\\hat{\\theta}_t, \\lambda) = 0$. Finally, since the algorithm did not stop at $t$, it must be that $\\inf_{\\lambda \\in \\Lambda(i^\\star)}L_t(\\hat{\\theta}_t, \\lambda) < \\beta_{t,\\delta}$ (otherwise all alternative pieces of $i^\\star(\\hat{\\theta}_t)$ would be eliminated at $t$). Therefore,\n\\begin{align*}\n\\sqrt{\\alpha_{s,\\delta}} < \\sqrt{\\beta_{t,\\delta}} + 2\\sqrt{\\beta_{{t},1\/{t}^2}}.\n\\end{align*}\nSince $s\\geq \\overline{t}_{j(t)-1}$, from Lemma \\ref{lem:new_beta-diff-between-phases} we have that\n\\begin{align*}\n\\beta_{t,\\delta} &\\leq \\beta_{\\overline{t}_{j(t)-1},\\delta} + (4 c_2 - c_1)\\log(\\overline{t}_{j(t)-1}) \\leq \\beta_{s,\\delta} + (4 c_2 - c_1)\\log(s),\n\\\\ \\beta_{t,1\/t^2} &\\leq 4 \\frac{c_2}{c_1} \\beta_{\\overline{t}_{j(t)-1},1\/\\overline{t}_{j(t)-1}^2} \\leq 4 \\frac{c_2}{c_1} \\beta_{s,1\/s^2}.\n\\end{align*}\nPlugging this into our previous bound, we conclude that\n\\begin{align*}\n\\sqrt{\\alpha_{s,\\delta}} < \\sqrt{\\beta_{s,\\delta} + (4 c_2 - c_1)\\log(s)} + 4\\sqrt{\\frac{c_2}{c_1}\\beta_{s,1\/s^2}}.\n\\end{align*}\nThis is clearly a contradiction w.r.t. our definition of $\\alpha_{t,\\delta}$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{th:sampling-rule-with-elim}]\nTake any step $t \\ge \\bar{t}_1$ where $E_t$ holds. Lemma \\ref{lem:closest-alternatives-not-eliminated-in-sampling} ensures that all pieces of $i^\\star$ which are at minimal distance from $\\theta$, i.e., such that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) = \\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)\n\\end{align*}\nare not eliminated for the sampling rule at time $t$, i.e., $p\\in\\cP_t^{\\mathrm{smp}}(i^\\star)$. This implies that\n\\begin{align*}\n\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) = \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda).\n\\end{align*}\nThus, if Assumption \\ref{asm:sampling-rule-v2} holds, it must be that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)\n &\\geq \\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) - R(\\theta,t)\n \\\\ &\\geq \\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) - R(\\theta,t) = tH^\\star(\\theta) - R(\\theta,t),\n\\end{align*}\nwhere the second inequality is trivial from $\\cP_{s-1}^{\\mathrm{smp}}(i^\\star) \\subseteq \\cP(i^\\star)$ for all $s\\geq 1$. Therefore, we proved that the condition of Assumption \\ref{asm:sampling-rule} holds as well for all $t \\ge \\bar{t}_1$. We can now have it for all $t$ by adding $\\bar{t}_1 H^\\star(\\theta)$ to $R(\\theta, t)$ to obtain another regret function which verifies the condition of Assumption~\\ref{asm:sampling-rule}. The second statement is a direct consequence of the fact that Theorem \\ref{th:piece-elim} holds for any sampling rule that satisifes the latter assumption.\n\\end{proof}\n\n\n\\section{Experiment Details and Additional Results}\n\\label{app:experiments}\n\n\\subsection{Reproducibility details}\n\nWe provide the main details to reproduce our experiments. For all the details, we refer the reader to our implementation at \\url{https:\/\/github.com\/AndreaTirinzoni\/bandit-elimination}.\n\nIn all experiments, we used $\\delta=0.01$ and a heuristic threshold $\\beta_{t,\\delta} = \\log(1\/\\delta) + \\log(1+t)$ for all elimination rules and LLR stopping. This is slightly larger than the heuristic threshold proposed by \\citep{garivier2016optimal} and adopted in many recent works. We implemented the elimination rules as described in Appendix \\ref{app:problems}. When using elimination at both stopping and sampling, we maintained only one set of active pieces (the one for stopping) instead of keeping a separate set with very lazy resets for sampling as suggested by theory. That set is shared by both sampling and stopping rules, and is never reset.\n\n\\paragraph{Computational infrastructure}\n\nAll experiments were run on a Dell XPS 13 laptop with an Intel Core i7-7560U (2.40GHz) CPU and 8GB of RAM.\n\n\\subsection{Bandit instances}\\label{app:instances}\n\nWe provide details on how we generated the bandit instances considered in the experiments presented in the main paper and later in this section.\n\n\\paragraph{Linear instances (experiments of Figure \\ref{fig:all} and Table \\ref{tab:lin_elim})}\n\nWe set $K=50$ and $d=10$. The true parameter is $\\theta = [1,1,\\dots,1]^T$, while we generated the arm features randomly. The arm feature of the first arm is $\\phi_1 = [1,0,0,\\dots,0]^T$. Then, up to reaching 50 arms, we repeated the following procedure. First, we generated a 3-dimensional vector $v\\in\\mathbb{R}^3$ by drawing its elements uniformly in $[-1,1]^3$ and then normalizing to have unit norm. Then, we added 3 feature vectors $v_1 = [0,v,0,0,0,0,0,0]$, $v_2 = [0,0,0,0,v,0,0,0]$, and $v_3 = [0,0,0,0,0,0,0,v]$, but only if $v_1^T\\theta \\in [0, 0.8]$. In this way, we obtain linear instances with 50 arms where arm $1$ is optimal with value $\\mu_1(\\theta) = 1$, while all other arms have minimum sub-optimality of $0.2$ and maximum sub-optimality gap of $0.8$.\n\n\\paragraph{Linear instances (experiments of Table \\ref{tab:lin_all})}\n\nWe set $K=50$ and $d=20$. The first 10 arms are set to the canonical basis of $\\mathbb{R}^{10}$, i.e., $\\phi_k = e_k$ for $k=1,\\dots,10$. The generation of the true parameter $\\theta$ and of the remaining 40 arms is slightly different from BAI\/Top-m and OSI.\n\nFor BAI and Top-m, the true parameter $\\theta$ has the first element equal to $1$, elements from the second to the fifth equal to $0.9$, and elements from the sixth to the tenth equal to $0.8$. The remaining 10 elements are uniformly drawn in $[-0.5,0.5]^{10}$. The remaining 40 arms are randomly generated as follows. First, we draw a vector $v$ uniformly in $[-1,1]^{20}$ and normalize it to have unit norm. Then, if $v^T\\theta \\leq 0.5$, we add $v$ to the set of arms. Otherwise, we reject the vector and keep repeating this procedure until we reach a total of 50 arms. In this way, we obtain random linear instances where the first arm is optimal with value $\\mu_1(\\theta) = 1$, the next 9 arms are hard to discriminate from it since they have small gap (either $0.1$ or $0.2$), and all remaining 40 arms have moderate to large gap (at least $0.5$) and are thus easy to eliminate.\n\nFor OSI, the true parameter $\\theta$ has the first ten elements uniformly drawn in $([-0.2,-0.1]\\cup[0.1,0.2])^{10}$ and the second ten elements uniformly drawn in $[-0.5,0.5]^{10}$. Similarly as before, to generate the remaining 40 arms we first draw a vector $v$ uniformly in $[-1,1]^{20}$ and normalize it to have unit norm. Then, if $|v^T\\theta| \\geq 0.5$, we add $v$ to the set of arms. Otherwise, we reject the vector and keep repeating this procedure until we reach a total of 50 arms. We thus obtain random linear instances where the first 10 arms are hard to learn since they have small gap (i.e., the absolute mean, which is between $0.1$ and $0.2$), and all remaining 40 arms have moderate to large gap (at least $0.5$) and are thus easy to eliminate.\n\n\\paragraph{Unstructured instances (experiments of Appendix \\ref{app:uns_results})}\n\nWe used $K=40$ arms. For BAI and Top-m, the mean reward of the first 5 arms is $\\mu_1 = 1$, $\\mu_2 = 0.9$, $\\mu_3 = 0.8$, $\\mu_4 = 0.7$, and $\\mu_5 = 0.6$. For all remaining arms the mean reward is uniformly drawn in $[0,0.5]$. For OSI, the mean reward of the first 4 arms is $\\mu_1 = 0.1$, $\\mu_2 = -0.2$, $\\mu_3 = 0.3$, and $\\mu_4 = -0.4$. For all remaining arms the mean reward is uniformly drawn in $[-1,-0.5]\\cup[0.5,1]$.\n\n\\subsection{Additional Results}\n\n\\subsubsection{Full versus selective elimination}\\label{app:full_vs_emp_results}\n\nWe report in Table \\ref{tab:lin_elim} the full results of the experiment comparing elimination rules (full vs selective) from which we extracted Figure \\ref{fig:all}\\emph{(middle)}. We recall that the linear instances for this experiment were generated as explained in the first paragraph of Appendix \\ref{app:instances}. We did not compare full and selective elimination rules on OSI since, as explained in Appendix \\ref{app:problems}, they are actually equivalent in such a setting.\n\nWhile we saw in Figure \\ref{fig:all}\\emph{(middle)} that the full elimination rule allows eliminating some arms earlier than the selective one, we notice from Table \\ref{tab:lin_elim} that the former rule actually yields no advantage in terms of sample complexity. Moreover, its computational overhead makes it much slower than the selective elimination rule. Therefore, in practice we suggest using the selective elimination rule, which always yields reduced computation times and often improved sample complexity.\n\n\\begin{table*}[t!]\n\\centering\n\\small\n\\begin{tabular}{@{}clcccccc@{}} \n\\toprule\n & & \\multicolumn{2}{c}{No elim. (LLR)} & \\multicolumn{2}{c}{Selective elim.} & \\multicolumn{2}{c}{Full elim.} \\\\\n\\cmidrule(r){3-8}\n& Algorithm & Samples & Time & Samples & Time & Samples & Time \\\\\n\\cmidrule{1-8}\n\\multirow{11}{*}{\\rotatebox[origin=c]{90}{BAI}} \n& LinGapE & $4.51 \\pm 1.3$ & $0.19$ & $4.49 \\pm 1.3$ & $0.17$ & $4.49 \\pm 1.3$ & $0.57$ \\\\\n& LinGapE + elim & & & $4.16 \\pm 1.4$ & $0.15$ & $4.17 \\pm 1.4$ & $0.58$ \\\\\n& LinGame & $5.28 \\pm 1.7$ & $0.21$ & $5.09 \\pm 1.8$ & $0.19$ & $5.09 \\pm 1.8$ & $0.6$ \\\\\n& LinGame + elim & & & $4.05 \\pm 1.2$ & $0.17$ & $4.05 \\pm 1.2$ & $0.65$ \\\\\n& FWS & $4.68 \\pm 4.2$ & $0.84$ & $4.68 \\pm 4.2$ & $0.82$ & $4.68 \\pm 4.2$ & $1.39$ \\\\\n& FWS + elim & & & $4.21 \\pm 1.4$ & $0.58$ & $4.21 \\pm 1.4$ & $1.16$ \\\\\n& Lazy TaS & $9.99 \\pm 8.8$ & $0.45$ & $9.75 \\pm 8.9$ & $0.45$ & $9.75 \\pm 8.9$ & $0.78$ \\\\\n& Lazy TaS + elim & & & $8.7 \\pm 8.9$ & $0.38$ & $8.7 \\pm 8.9$ & $0.73$ \\\\\n& Oracle & $6.65 \\pm 1.8$ & $0.04$ & $6.55 \\pm 1.9$ & $0.02$ & $6.55 \\pm 1.9$ & $0.31$ \\\\\n& XY-Adaptive & & & & & $13.89 \\pm 6.0$ & $2.23$ \\\\\n& RAGE & & & & & $16.28 \\pm 6.2$ & $0.02$ \\\\\n\\cmidrule{1-8}\n\\multirow{10}{*}{\\rotatebox[origin=c]{90}{Top-m ($m=3$)}} \n& m-LinGapE & $6.26 \\pm 1.2$ & $0.29$ & $6.21 \\pm 1.2$ & $0.24$ & $6.21 \\pm 1.2$ & $1.35$ \\\\\n& m-LinGapE + elim & & & $5.77 \\pm 1.2$ & $0.19$ & $5.77 \\pm 1.2$ & $1.29$ \\\\\n& MisLid & $7.06 \\pm 1.4$ & $0.34$ & $6.81 \\pm 1.5$ & $0.27$ & $6.81 \\pm 1.5$ & $1.48$ \\\\\n& MisLid + elim & & & $5.89 \\pm 1.1$ & $0.22$ & $5.89 \\pm 1.1$ & $1.42$ \\\\\n& FWS & $5.91 \\pm 1.7$ & $1.51$ & $5.9 \\pm 1.7$ & $1.46$ & $5.9 \\pm 1.7$ & $2.66$ \\\\\n& FWS + elim & & & $5.84 \\pm 1.7$ & $0.83$ & $5.84 \\pm 1.7$ & $2.02$ \\\\\n& Lazy TaS & $13.1 \\pm 6.5$ & $0.71$ & $12.85 \\pm 6.4$ & $0.67$ & $12.85 \\pm 6.4$ & $1.57$ \\\\\n& Lazy TaS + elim & & & $11.34 \\pm 6.3$ & $0.56$ & $11.34 \\pm 6.3$ & $1.47$ \\\\\n& Oracle & $8.74 \\pm 1.8$ & $0.1$ & $8.65 \\pm 1.8$ & $0.04$ & $8.65 \\pm 1.8$ & $1.02$ \\\\\n& LinGIFA & $5.58 \\pm 1.1$ & $1.8$ & $5.57 \\pm 1.1$ & $1.75$ & $5.57 \\pm 1.1$ & $2.68$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Experiments on linear instances with $K=50$ and $d=10$. The \"Time\" columns report average times per iteration in milliseconds (i.e., the total time the algorithm took divided by the number of samples). Each entry reports the mean across $100$ runs plus\/minus standard deviation (which is omitted for compute times due to space constraints). The ``+ elim'' variant of some algorithms indicates that the corresponding sampling rule is combined with elimination. Samples are scaled down by a factor $10^3$.}\\label{tab:lin_elim}\n\\end{table*}\n\n\\subsubsection{Unstructured instances}\\label{app:uns_results}\n\nWe report the results on unstructured bandit instances (generated according to the procedure of Appendix \\ref{app:instances}) in Table \\ref{tab:uns_all}. The algorithm k-Learner is the unstructured variant of LinGame proposed by \\citep{degenne2019non}. We note that the results are coherent with those for linear instances presented in the main paper. In particular, we observe a reduction in computation times when combining adaptive algorithms with selective elimation. The reduction is however less evident than in the linear case. This is expected since, in general, eliminations are easier in structured problems than in unstructured ones. We also note that combining sampling rules with selective elimination slightly improves the sample complexity of all algorithms.\n\n\\begin{table*}[t!]\n\\centering\n\\small\n\\begin{tabular}{@{}clcccccc@{}} \n\\toprule\n & & \\multicolumn{2}{c}{No elim. (LLR)} & \\multicolumn{2}{c}{Elim. stopping} & \\multicolumn{2}{c}{Elim. stopping + sampling} \\\\\n\\cmidrule(r){3-8}\n& Algorithm & Samples & Time & Samples & Time & Samples & Time \\\\\n\\cmidrule{1-8}\n\\multirow{7}{*}{\\rotatebox[origin=c]{90}{BAI}} \n& k-Learner & $18.76 \\pm 6.5$ & $0.49$ & $18.12 \\pm 6.6$ & $0.44$ & $14.82 \\pm 4.6$ & $0.4$ \\\\\n& FWS & $14.5 \\pm 4.6$ & $1.25$ & $14.44 \\pm 4.7$ & $1.24$ & $13.94 \\pm 4.4$ & $1.16$ \\\\\n& Lazy TaS & $26.18 \\pm 8.0$ & $0.32$ & $24.78 \\pm 7.7$ & $0.31$ & $20.66 \\pm 7.4$ & $0.33$ \\\\\n& Oracle & $27.49 \\pm 3.7$ & $0.07$ & $27.0 \\pm 3.7$ & $0.04$ & & \\\\\n& LUCB & $14.2 \\pm 5.2$ & $0.11$ & $14.18 \\pm 5.2$ & $0.06$ & $13.57 \\pm 4.7$ & $0.06$ \\\\\n& UGapE & $15.13 \\pm 5.0$ & $0.43$ & $15.13 \\pm 5.0$ & $0.39$ & & \\\\\n& Racing & & & & & $34.55 \\pm 7.6$ & $0.01$ \\\\\n\\cmidrule{1-8}\n\\multirow{7}{*}{\\rotatebox[origin=c]{90}{Top-m ($m=3$)}} \n& k-Learner & $25.84 \\pm 6.2$ & $0.67$ & $25.06 \\pm 6.3$ & $0.57$ & $17.65 \\pm 4.9$ & $0.51$ \\\\\n& FWS & $17.68 \\pm 4.7$ & $2.52$ & $17.67 \\pm 4.7$ & $2.48$ & $17.63 \\pm 4.6$ & $2.0$ \\\\\n& Lazy TaS & $38.89 \\pm 10.4$ & $0.5$ & $37.84 \\pm 10.7$ & $0.43$ & $27.74 \\pm 6.4$ & $0.46$ \\\\\n& Oracle & $34.17 \\pm 4.5$ & $0.15$ & $33.68 \\pm 4.9$ & $0.07$ & & \\\\\n& LUCB & $17.61 \\pm 4.6$ & $0.24$ & $17.58 \\pm 4.5$ & $0.13$ & $17.15 \\pm 5.4$ & $0.13$ \\\\\n& UGapE & $17.87 \\pm 4.3$ & $0.54$ & $17.87 \\pm 4.3$ & $0.43$ & & \\\\\n& Racing & & & & & $22.67 \\pm 3.1$ & $0.01$ \\\\\n\\cmidrule{1-8}\n\\multirow{5}{*}{\\rotatebox[origin=c]{90}{OSI}}\n& k-Learner & $8.55 \\pm 1.7$ & $0.61$ & $8.38 \\pm 1.8$ & $0.55$ & $5.43 \\pm 1.2$ & $0.52$ \\\\\n& FWS & $5.54 \\pm 1.3$ & $1.48$ & $5.53 \\pm 1.3$ & $1.47$ & $5.47 \\pm 1.4$ & $1.4$ \\\\\n& Lazy TaS & $12.83 \\pm 3.1$ & $0.72$ & $12.27 \\pm 3.1$ & $0.7$ & $8.74 \\pm 1.8$ & $0.79$ \\\\\n& Oracle & $11.55 \\pm 1.6$ & $0.07$ & $11.41 \\pm 1.6$ & $0.04$ & & \\\\\n& LUCB & $5.5 \\pm 1.4$ & $0.11$ & $5.5 \\pm 1.4$ & $0.1$ & $5.49 \\pm 1.4$ & $0.1$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Experiments on unstructured instances with $K=40$. The \"Time\" columns report average times per iteration in milliseconds (i.e., the total time the algorithm took divided by the number of samples). Each entry reports the mean across $100$ runs plus\/minus standard deviation (which is omitted for compute times due to space constraints). Algorithms for which the third column is missing cannot be combined with elimination at sampling, while algorithms for which the first two columns are missing are natively elimination-based. Samples are scaled down by a factor $10^3$.}\\label{tab:uns_all}\n\\end{table*}\n\n\n\\section{Elimination stopping rules for adaptive algorithms}\n\\label{sec:elimination_stopping_rules}\n\nWe show how to modify the stopping rule of adaptive algorithms using LLR stopping to perform elimination. We assume that the alternatives sets $\\Lambda(i)$ can be decomposed  into a union of sets which we refer to as \\emph{alternative pieces} (or simply pieces), with the property that computing the infimum LLR over these sets is computationally easy\n\n\\begin{assumption}\\label{ass:union_of_sets}\nFor all $i \\in \\cI$, there exist pieces $(\\Lambda_p(i))_{p \\in \\cP(i)}$, where $\\cP(i)$ is a finite set of \npiece indexes, such that $\\Lambda(i) = \\bigcup_{p\\in\\cP(i)} \\Lambda_p(i)$ and $\\inf_{\\lambda \\in \\Lambda_p(i)}L_t(\\hat{\\theta}_t,\\lambda)$ can be efficiently computed for all $p \\in \\cP(i)$ and $t >0$.\n\\end{assumption}\nThis assumption is satisfied in many problems of interest, including BAI, Top-$m$ identification, and thresholding bandits (see Appendix \\ref{app:problems}).\nIndeed, in all applications we consider in this paper, the sets of Assumption~\\ref{ass:union_of_sets} are half-spaces. In our linear BAI example, the piece indexes are simply arms. For $i,j\\in[K]$ we can define $\\Lambda_j(i) = \\{\\lambda\\in\\cM \\mid \\phi_j^\\top \\lambda > \\phi_i^\\top \\lambda\\}$. Then, $\\Lambda(i) = \\bigcup_{j\\in [K]\\setminus \\{i\\}} \\Lambda_j(i)$. Moreover, the infimum LLR (and the corresponding minimizer) can be computed in closed form as \\citep[e.g.,][]{fiez2019sequential}\n$\n\\inf_{\\lambda \\in \\Lambda_j(i)}L_t(\\hat{\\theta}_t,\\lambda) = \\max\\{\\hat{\\theta}_t^T(\\phi_i-\\phi_j),0\\}^2 \/ \\| \\phi_i - \\phi_j \\|_{V_{N_t}^{-1}}^2.\n$\n\n\\paragraph{Elimination stopping}\n\nThe main idea is that it is not necessary to exclude all $\\Lambda_p(i)$ for $p\\in\\cP(i)$ at the \\emph{same time}, as LLR stopping \\eqref{eq:llr-stop} does, in order to know that the algorithm can stop and return answer $i$. Instead, each piece can be pruned as soon as we have enough information to do so. \n\n\\begin{definition}\\label{def:elimination}\nA set $S \\subseteq \\mathbb{R}^d$ is said to be eliminated at time $t$ if, for all $\\lambda\\in S$, $L_t(\\hat{\\theta}_t,\\lambda) \\ge \\beta_{t,\\delta}$.\n\\end{definition}\nFrom the concentration property~\\eqref{eq:concentration-beta}, we obtain that the probability that $\\theta \\in S$ and $S$ is eliminated is less than $\\delta$.\nLLR stopping interrupts the algorithm when the alternative set $\\Lambda(i^\\star(\\hat{\\theta}_t))$ can be eliminated.\nIn elimination stopping, we eliminate smaller sets gradually, instead of the whole alternative at once.\nFormally, let us define, for all $i\\in\\cI$,\n\\begin{align}\n\\label{eq:active-pieces-t-only}\n\\overline{\\cP}_t(i;\\beta_{t,\\delta}) = \\left\\{ p \\in \\cP(i) : \\inf_{\\lambda \\in \\Lambda_p(i)} L_t(\\hat{\\theta}_t,\\lambda) < \\beta_{t,\\delta} \\right\\}\n\\end{align}\nas the subset of pieces for answer $i\\in\\cI$ whose infimum LLR at time $t$ is below a threshold $\\beta_{t,\\delta}$. That is, the indexes of pieces that are \\emph{not} eliminated at time $t$. Moreover, we define, for all $i\\in\\cI$, a set of \\emph{active pieces} $\\cP_t^{\\mathrm{stp}}(i)$ which is initialized as $\\cP_0^{\\mathrm{stp}}(i) = \\cP(i)$ (all piece indexes).\n\nOur \\emph{selective elimination} rule updates, at each time $t$, only the active pieces of the empirical answer $i^\\star(\\hat{\\theta}_t)$. That is, for $i=i^\\star(\\hat{\\theta}_t)$, it sets\n\\begin{align}\\label{eq:elimination-sets}\n\t\\cP_t^{\\mathrm{stp}}(i) := \\cP_{t-1}^{\\mathrm{stp}}(i) \\cap \\overline{\\cP}_t(i;\\beta_{t,\\delta}),\n\\end{align}\nwhile it sets $\\cP_t^{\\mathrm{stp}}(i) := \\cP_{t-1}^{\\mathrm{stp}}(i)$ for all $i\\neq i^\\star(\\hat{\\theta}_t)$. One might be wondering why not updating all answers at each round. The main reason is computational: as we better discuss at the end of this section, checking LLR stopping requires one minimization for \\emph{each} piece $p\\in\\cP(i^\\star(\\hat{\\theta}_t))$, while selective elimination requires only one for each \\emph{active} piece $p\\in\\cP_{t-1}^{\\mathrm{stp}}(i^\\star(\\hat{\\theta}_t))$. Thus, the latter becomes increasingly more computationally efficient as pieces are eliminated. For completeness, we also analyze the variant, that we call \\emph{full elimination}, which updates the active pieces according to \\eqref{eq:elimination-sets} for \\emph{all} answers $i\\in\\cI$ at each round. While we shall establish slightly better theoretical guarantees for this rule, it is computationally demanding and, as we shall see in our experiments, it does not significantly improve sample complexity w.r.t. selective elimination, which remains our recommended choice.\n\n\nLet $\\tau_{\\mathrm{s. elim}} = \\inf_{t \\geq 1}\\{t \\mid \\cP_t^{\\mathrm{stp}}(i^\\star(\\hat{\\theta}_t)) = \\emptyset\\}$ and $\\tau_{\\mathrm{f. elim}} := \\inf_{t \\geq 1}\\{t \\mid \\exists i\\in\\cI : \\cP_t^{\\mathrm{stp}}(i) = \\emptyset\\}$ be the stopping times of selective and full elimination, respectively. Intuitively, these two rules stop when one of the updated answers has all its pieces eliminated (and return that answer). We show that, as far as $\\beta_{t,\\delta}$ is chosen to ensure concentration of $\\hat{\\theta}_t$ to $\\theta$, those two stopping rules are $\\delta$-correct.\n\n\\begin{lemma}[$\\delta$-correctness]\\label{lem:delta-correct}\nSuppose that $\\beta_{t,\\delta}$ guarantees \\eqref{eq:concentration-beta} and that the algorithm verifies that, whenever it stops, there exists $i_{\\emptyset}\\in\\cI$ such that $\\cP_\\tau^{\\mathrm{stp}}(i_{\\emptyset}) = \\emptyset$ and $\\ihat = i_{\\emptyset}$. Then, $\\mathbb{P}_\\theta(\\ihat \\ne i^\\star(\\theta)) \\le \\delta$.\n\\end{lemma}\nAll proofs for this section are in Appendix~\\ref{app:proofs-elim-stopping}. If an algorithm verifies the conditions of Lemma~\\ref{lem:delta-correct} and has a sampling rule that makes it stop almost surely, then it is $\\delta$-correct. Interestingly, we can prove a stronger result than $\\delta$-correctness: under the same sampling rule, the elimination stopping rules never trigger later than the LLR one \\emph{almost surely}. In other words, any algorithm equipped with elimination stopping suffers a sample complexity that is never worse than the one of the same algorithm equipped with LLR stopping. Let $\\tau_{\\mathrm{llr}} := \\inf_{t \\geq 1}\\{t \\mid \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_t))} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\}$.\n\n\\begin{theorem}\\label{th:elim-better-than-llr}\nFor any sampling rule, almost surely $\\tau_{\\mathrm{f. elim}} \\le \\tau_{\\mathrm{s. elim}} \\le \\tau_{\\mathrm{llr}}$~.\n\\end{theorem}\n\nThe proof of this theorem is very simple: if $\\tau_{\\mathrm{llr}} = t$, then at $t$ all pieces $\\Lambda_p(i^\\star(\\hat{\\theta}_t))$ for $p\\in\\cP(i^\\star(\\hat{\\theta}_t))$ can be eliminated, hence $\\tau_{\\mathrm{s. elim}} \\le t$. The proof that $\\tau_{\\mathrm{f. elim}} \\le \\tau_{\\mathrm{s. elim}}$ follows from the observation that full elimination always has less active pieces than selective elimination. Note that all three stopping rules must use the same threshold $\\beta_{t,\\delta}$ to be comparable. Although simple, Theorem \\ref{th:elim-better-than-llr} has an important implication: we can take any existing algorithm that uses LLR stopping, equip it with elimination stopping instead, and obtain a new strategy that is never worse in terms of sample complexity and for which the original theoretical results on the stopping time still hold.\n\nFinally, it is important to note that, while defining the elimination rule in the general form \\eqref{eq:elimination-sets} allows us to unify many settings, storing\/iterating over all sets $\\cP_t^{\\mathrm{stp}}(i)$ would be intractable in problems with large number of answers (e.g., top-m identification or thresholding bandits, where the latter is exponential in $K$).\nFortunately, we show in Appendix \\ref{app:problems} that this is not needed and efficient implementations exist for these problems that take only polynomial time and memory.\n\n\\subsection{Elimination time of alternative pieces}\n\\label{sub:changing_the_stopping_rule_only}\n\nWe now show that\nelimination stopping can indeed discard certain alternative pieces much earlier that the stopping time.\nWhile all results so far hold for any distribution and bandit structure, in the remaining we focus on Gaussian linear bandits. Other distribution classes beyond Gaussians could be used with minor modifications (see Appendix \\ref{sub:beyond_gaussians}) but the Gaussian case simplifies the exposition. Since most existing adaptive sampling rules target the optimal proportions from the lower bound of \\citep{garivier2016optimal}, we unify them under the following assumption.\n\\begin{assumption}\\label{asm:sampling-rule}\nConsider the concentration events\n\\begin{align}\\label{eq:Et}\nE_t := \\left\\{ \\forall s \\leq t: L_s(\\hat{\\theta}_s,\\theta) \\leq \\beta_{t,1\/t^2} \\right\\} \\: .\n\\end{align}\nA sampling rule is said to have low information regret if there exists a problem-dependent function $R(\\theta,t)$ which is sub-linear in $t$ such that for each time $t$ where $E_t$ holds,\n\\begin{align}\\label{eq:no-regret-property}\n\\inf_{\\lambda \\in \\Lambda(i^\\star(\\theta))} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) \\geq t H^\\star(\\theta) - R(\\theta,t).\n\\end{align}\n\\end{assumption}\nThe left-hand side of \\eqref{eq:no-regret-property} can be understood as the information collected by the sampling rule at time $t$ to discriminate $\\theta$ with all its alternatives. Therefore, Assumption \\ref{asm:sampling-rule} requires that information to be comparable (up to a low-order term $R(\\theta,t)$) with the maximal one from the lower bound.\nIn Appendix \\ref{app:assumptions}, we show that this is satisfied by both Track-and-Stop \\citep{garivier2016optimal} and the approach in \\citep{degenne2019non}.\n\nLet $H_p(\\omega, \\theta) := \\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\theta))} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)$, the information that sampling with proportions $\\omega$ brings to discriminate $\\theta$ from the alternative piece $\\Lambda_p(i^\\star(\\theta))$. Note that $H^\\star(\\theta) = \\max_{\\omega\\in\\Delta_K}\\min_{p\\in\\cP(i^\\star(\\theta))}H_p(\\omega, \\theta)$. For $\\epsilon\\geq 0$, let $\\Omega_\\epsilon(\\theta) := \\{ \\omega\\in\\Delta_K \\mid \\inf_{\\lambda \\in \\Lambda(i^\\star(\\theta))}  \\sum_k \\omega^k\\KL_k(\\theta,\\lambda) \\geq H^\\star(\\theta) - \\epsilon\\}$ be the set of $\\epsilon$-optimal proportions.\n\n\\begin{theorem}[Piece elimination]\\label{th:piece-elim}\nThe stopping time of any sampling rule having low information regret, combined with LLR stopping, satisfies $\\mathbb{E}[\\tau] \\leq \\bar{t} + 2$, where $\\bar{t}$ is the first integer such that\n\\begin{align}\\label{eq:llr-stopping-ineq}\nt \\geq \\left(\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2 + R(\\theta,t)\\right) \/ H^\\star(\\theta).\n\\end{align}\nWhen the same sampling rule is combined with elimination stopping, let $\\tau_p$ be the time at which $p \\in \\cP(i^\\star(\\theta))$ is eliminated. Then, $\\mathbb{E}[\\tau_p] \\leq \\min\\{\\bar{t}_p, \\bar{t} \\} + 2$, where $\\bar{t}_p$ is the first integer such that\n\\begin{align}\\label{eq:elimination_time}\nt \\geq \\max\\left\\{\\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{\\min_{\\omega \\in \\Omega_{R(\\theta,t)\/t}(\\theta)}H_p(\\omega, \\theta)}, G(\\theta,t) \\right\\},\n\\end{align}\nwith $G(\\theta,t) = 0$ for full elimination and $G(\\theta,t) = \\frac{ 4\\beta_{t,1\/t^2} + R(\\theta,t)}{H^\\star(\\theta)}$ for selective elimination.\n\\end{theorem}\n\nFirst, the bound we obtain on the elimination time of pieces in $\\cP(i^\\star(\\theta))$ is not worse than the bound we obtain on the stopping time of LLR stopping.\nSecond, with elimination stopping, such eliminations can actually happen much sooner. Intuitively, sampling rules with low information regret play arms with proportions that are close to the optimal ones. If all of such ``good'' proportions provide large information for eliminating some piece $p\\in\\cP(i^\\star(\\theta))$, then $p$ is eliminated much sooner than the actual stopping time (which requires eliminating the worst-case piece in the same set).\n\nWhile both elimination rules are provably efficient, with full elimination enjoying slighly better guarantees\\footnote{Note that $G(\\theta,t)$ for selective elimination contributes only a finite (in $\\delta$) sample complexity.}, selective elimination provably never worsens (and possibly improves) the computational complexity over LLR stopping. In all applications we consider, implementing LLR stopping requires one minimization for each of the same alternative pieces we use for elimination stopping.\nTherefore, the total number of minimizations required by LLR stopping is $\\sum_{t=1}^{\\tau_{\\mathrm{llr}}} |\\cP(i^\\star(\\hat{\\theta}_t))|$ versus $\\sum_{t=1}^{\\tau_{\\mathrm{s.elim}}} |\\cP_t^{\\mathrm{stp}}(i^\\star(\\hat{\\theta}_t))|$ for selective elimination.\nThe second is never larger since $\\tau_{\\mathrm{s.elim}} \\leq \\tau_{\\mathrm{llr}}$ by Theorem \\ref{th:elim-better-than-llr} and $\\cP_t^{\\mathrm{stp}}(i^\\star(\\hat{\\theta}_t)) \\subseteq \\cP(i^\\star(\\hat{\\theta}_t))$ for all $t$, and much smaller if eliminations happen early, as we shall verify in experiments.\nIn our linear BAI example we need to perform $(K-1)$ minimizations at each step, one for each sub-optimal arm, in order to implement LLR stopping.\nOn the other hand, we need only $|\\cP_t^{\\mathrm{stp}}(i^\\star(\\hat{\\theta}_t))|$ minimizations with selective elimination, one for each active sub-optimal arm, while full elimination takes $\\sum_{i\\in[K]}|\\cP_t^{\\mathrm{stp}}(i)|$ to update all the sets.\n\nNote that Theorem \\ref{th:piece-elim} does not provide a better bound on $\\mathbb{E}[\\tau]$ for elimination stopping than for LLR stopping. In fact, when evaluating the bound on $\\mathbb{E}[\\tau_p]$ for the worst-case piece in $p\\in\\cP(i^\\star(\\theta))$, we recover the one on $\\mathbb{E}[\\tau]$. This is intuitive since the sampling rule is playing proportions that try to eliminate all alternative pieces at once. The following result formalizes this intuition. \n\\begin{theorem}\\label{th:elim-vs-llr-fixed-sampling}\nSuppose that we can write $\\beta_{t, \\delta} = \\log\\frac{1}{\\delta} + \\xi(t, \\delta)$ with $\\lim_{\\delta \\to 0}\\xi(t, \\delta)\/\\log(1\/\\delta) = 0$. Then for any sampling rule that satisfies Assumption \\ref{asm:sampling-rule},\n\\begin{align*}\n\\mathbb{E}[\\tau_{\\mathrm{llr}}] \\le \\mathbb{E}[\\tau_{\\mathrm{elim}}] + f(\\theta, \\delta) \\: .\n\\end{align*}\nwith $\\lim_{\\delta \\to 0} f(\\theta, \\delta)\/\\log(1\/\\delta) = 0$. Here $\\tau_{\\mathrm{elim}}$ can stand for either full or selective elimination.\n\\end{theorem}\nSee Appendix~\\ref{subapp:proof_elim_vs_llr} for $f$. This result shows that when the sampling rule is tailored to the LLR stopping rule, the expected LLR and elimination stopping times differ by at most low-order (in $\\log(1\/\\delta)$) terms. As $\\delta\\rightarrow 0$ the two expected stopping times converge to the same value $H^\\star(\\theta)^{-1}\\log(1\/\\delta)$, which is the asymptotically-optimal sample complexity prescribed by the lower bound~\\eqref{eq:lower_bound}. \n\n\nWe showed that, for both elimination rules, some pieces of the alternative are discarded sooner than the stopping time, and that the overall sample complexity of the method can only improve over LLR stopping.\nHowever, since the sampling rule of the algorithm was not changed, elimination does not change the computational cost of each sampling step, only the cost of checking the stopping rule.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\iffalse\n\\subsection{Lower Bounds}\n\\label{sub:lower_bounds}\n\nWe show that the use of a particular stopping rule translates into lower bounds for the identification algorithm. We prove two such bounds, one for LLR stopping and one for Elimination stopping. The lower bound obtained for LLR stopping is stronger than for elimination.\n\nWe first report an intermediate step in the proof of the lower bound of \\citep{garivier2016optimal} (see~\\eqref{eq:lower_bound}). We write that step as a bound on a log-likelihood ratio, which we can more easily compare to our results. \n\\begin{theorem}\\label{thm:lower_bound_GK16}\nFor any $\\delta$-correct algorithm, for any $\\theta \\in \\cM$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^\\star(\\theta))} \\mathbb{E}\\left[ L_\\tau(\\theta, \\lambda)\\right]\n&\\ge \\log\\frac{1}{2.4 \\delta}\n\\: .\n\\end{align*}\n\\end{theorem}\n\n\\paragraph{LLR Stopping}\n\nWe study LLR stopping with a threshold $\\beta_{t,\\delta} = \\log\\frac{1}{\\delta} + \\gamma(t)$ which verifies that for all $\\delta' \\in (0,1]$, with probability $1 - \\delta'$, for all $t \\in \\mathbb{N}$, the quantity $\\beta_{t,\\delta'}$ is an upper bound for $L_t(\\hat{\\theta}_t, \\theta)$. We call LLR stopping with such a threshold LLR$\\gamma$.\nThis property is verified by one of the LLR stopping thresholds proposed in \\citep{garivier2016optimal} and used for example in \\citep{degenne2019non}.\nThose algorithms are subject to a stronger lower bound than the one of Theorem~\\ref{thm:lower_bound_GK16}.\n\n\\begin{theorem}\\label{thm:lower_bound_llr}\nAlgorithms stopping with the LLR$\\gamma$ rule verify for all $\\theta \\in \\cM$\n\\begin{align*}\n\\mathbb{E}\\left[ \\inf_{\\lambda \\in \\Lambda(i^\\star(\\theta))} L_\\tau(\\theta, \\lambda)\\right]\n&\\ge \\log \\frac{1}{e \\delta} - \\sqrt{2 \\delta (2 + \\mathbb{E}[\\gamma(\\tau)^2])}\n\\: .\n\\end{align*}\n\\end{theorem}\n\nThe last term is usually negligible if $\\delta$ is not too large. Indeed the usual function $\\gamma$ is $\\gamma(t) = d \\log \\log(t)$. This is concave, such that $\\mathbb{E}[\\gamma(\\tau)^2] \\le (d \\log\\log \\mathbb{E}[\\tau])^2$ and for $\\delta \\le 1\/d^2$ the square root term is negligible.\n\nThe main difference with Theorem~\\ref{thm:lower_bound_GK16} is the presence of $\\mathbb{E}\\left[ \\inf L_\\tau(\\theta, \\lambda)\\right]$ instead of $\\inf \\mathbb{E}\\left[ L_\\tau(\\theta, \\lambda)\\right]$. That is, the expectation and the infimum are reversed.\n\n\\paragraph{The expected infimum LLR}\n\nBy concavity of the infimum, $\\mathbb{E}\\left[ \\inf L_\\tau(\\theta, \\lambda)\\right] \\le \\inf \\mathbb{E}\\left[ L_\\tau(\\theta, \\lambda)\\right]$, hence the lower bound for LLR stopping is tighter than the generic lower bound (up to the square root term).\nHowever, algorithms based on LLR stopping have been shown to asymptotically match the lower bound of~\\eqref{eq:lower_bound} and indeed the two bounds coincide in the limit $\\delta \\to 0$.\n\nWe now quantify how much tighter Theorem~\\ref{thm:lower_bound_llr} is compared to Theorem~\\ref{thm:lower_bound_GK16}, in a regime where $\\delta$ is not too small.\n\n\n\\todo[inline]{quantify the difference on fixed proportion sampling.}\n\n\\paragraph{Elimination Stopping}\n\nWe also prove a new lower bound for elimination stopping, which is also stronger that Theorem~\\ref{thm:lower_bound_GK16}, but lower than the one we presented for LLR stopping.\n\n\\begin{theorem}\\label{thm:lower_bound_elim}\nAlgorithms stopping with the full elimination rule verify for all $\\theta \\in \\cM$\n\\begin{align*}\n\\mathbb{E}&\\left[ \\min_m \\max_{s \\le \\tau} \\inf_{\\lambda \\in \\Lambda_m(i^\\star)}L_s(\\theta, \\lambda)\\right]\n\\\\\n&\\ge \\log \\frac{1}{\\delta} - 1 - \\sqrt{2 \\delta (2 + \\mathbb{E}[\\gamma(\\tau)^2])}\n\\: .\n\\end{align*}\n\\end{theorem}\nThe proof of Appendix~\\ref{app:lower_bound_proofs} proves both this result and Theorem~\\ref{thm:lower_bound_llr}. This bound is not as strong as the one we proved for LLR stopping, and the difference is in the order of the minimization over $m$ and the maximization over time.\nWe cannot conclude from these two lower bounds that elimination stopping allows a smaller sample complexity than LLR stopping since these are only lower bounds and not characterizations of the sample complexity (upper bounds are missing). It could be that a stronger lower bound holds for elimination stopping as well.\nHowever, we raise that possibility as an open question: although LLR stopping is optimal asymptotically, is it bound by Theorem~\\ref{thm:lower_bound_llr} to be sub-optimal for finite $\\delta$? If so, is elimination stopping enough to overcome that limitation?\n\n\\fi\n\\section{Lower bound proofs}\n\\label{app:lower_bound_proofs}\n\nFor any answer $i$, we have $P_i$ sets $\\Lambda_p(i)$, such that $\\Lambda(i) = \\bigcup_p \\Lambda_p(i)$. The elimination stopping time is\n\\begin{align*}\n\\tau\n= \\min \\left\\{ t \\in \\mathbb{N} \\mid \\max_i \\min_p \\max_{s \\le t} (L_s(\\hat{\\mu}_s, \\Lambda_p(i)) - \\overline{\\log}\\frac{1}{\\delta} - \\gamma(s)) \\ge 0 \\right\\}\n .\n\\end{align*}\nThis is simply a rewriting of the following description: $\\tau$ is the first time $t$ at which there exists an answer $i$ for which for all sets $\\Lambda_p(i)$, there was a time $s \\le t$ at which the set was eliminated.\n\nSince $\\ihat$ is the argmax for $t = \\tau$ in the definition of $\\tau$ above, we have $\\min_p \\max_{s \\le \\tau} (L_s(\\hat{\\mu}_s, \\Lambda_p(\\ihat)) - \\overline{\\log}\\frac{1}{\\delta} - \\gamma(s)) \\ge 0$.\n\nImportant equality: $L_s(\\hat{\\mu}_s, \\Lambda_p(i)) = L_s(\\hat{\\mu}_s, \\mu) + L_s(\\mu, \\Lambda_p(i))$.\n\nWith probability $1 - \\delta'$, for all $s \\in \\mathbb{N}$, $L_s(\\hat{\\mu}_s, \\mu) \\le \\overline{\\log} \\frac{1}{\\delta'} + \\gamma(s)$. Hence with probability $1 - \\delta'$,\n\\begin{align*}\n&\\max_i \\min_p \\max_{s \\le \\tau} (L_s(\\mu, \\Lambda_p(i)) - \\overline{\\log}\\frac{1}{\\delta} + \\overline{\\log}\\frac{1}{\\delta'}) \\ge 0 \\: ,\n\\\\\n\\text{i.e. }& \\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau}  L_s(\\mu, \\Lambda_p(i)) \\le \\overline{\\log}\\frac{1}{\\delta'} \\: .\n\\end{align*}\nLet $\\overline{\\exp}$ be the inverse of $\\overline{\\log}$ \\todo{$\\overline{\\log}$ is supposed increasing and smaller than $2\\log$}. We have for all $x \\in \\overline{\\log}^{-1}([1, +\\infty))$,\n\\begin{align*}\n\\mathbb{P}\\left( \\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau}  L_s(\\mu, \\Lambda_p(i)) > x \\right) \\le 1\/\\overline{\\exp}(x) \\: .\n\\end{align*}\nSince we supposed that $\\overline{\\log} \\le 2 \\log$, we have $\\overline{\\exp}(x) \\ge \\exp(x\/2)$.\n\\begin{align*}\n\\mathbb{P}\\left( \\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau}  L_s(\\mu, \\Lambda_p(i)) > x \\right) \\le e^{-x\/2} \\: .\n\\end{align*}\n\nWe can get a bound in expectation:\n\\begin{align*}\n&\\overline{\\log} \\frac{1}{\\delta} - \\mathbb{E}[\\max_i \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i))]\n\\\\\n&\\le \\mathbb{E}[(\\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i)))\\mathbb{I}\\{\\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i)) \\ge 0\\}]\n\\\\\n&= \\int_0^{+\\infty} \\mathbb{P}(\\overline{\\log}\\frac{1}{\\delta} - \\max_i \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i)) > x) dx\n\\\\\n&\\le \\overline{\\exp}(1) + \\int_{\\overline{\\exp}(1)}^{+\\infty} e^{-x\/2} dx\n\\\\\n&\\le e^{1\/2} + \\int_{0}^{+\\infty}e^{-x\/2} dx = e^{1\/2} + 2\n\\: .\n\\end{align*}\n\n\\todo[inline]{propagate the changes below}\n\n\\begin{align*}\n\\mathbb{E}[\\max_i \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i))] \\ge \\log \\frac{1}{\\delta} - 1 \\: .\n\\end{align*}\n\nBy similar computations with the inequality involving $\\ihat$,\n\\begin{align*}\n\\mathbb{E}[\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(\\ihat))] \\ge \\log \\frac{1}{\\delta} - 1 \\: .\n\\end{align*}\n\n\nFor all $i \\ne i^\\star$, there exists $m$ such that $\\mu \\in \\Lambda_p(i)$. For that $m$, $L_s(\\mu, \\Lambda_p(i)) \\le L_s(\\mu, \\mu) = 0$ for all $s \\in \\mathbb{N}$. Hence for $i\\ne i^\\star$, $\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i)) \\le 0$.\n\\begin{align*}\n\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(\\ihat))\n&= \\mathbb{I}\\{\\ihat = i^\\star\\}\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(\\ihat)) + \\mathbb{I}\\{\\ihat \\ne i^\\star\\}\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(\\ihat))\n\\\\\n&\\le \\mathbb{I}\\{\\ihat = i^\\star\\}\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(\\ihat))\n\\\\\n&= \\mathbb{I}\\{\\ihat = i^\\star\\}\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i^\\star))\n\\\\\n&= \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i^\\star)) - \\mathbb{I}\\{\\ihat \\ne i^\\star\\}\\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i^\\star))\n\\\\\n&\\le \\min_p \\max_{s \\le \\tau} L_s(\\mu, \\Lambda_p(i^\\star)) + \\mathbb{I}\\{\\ihat \\ne i^\\star\\} \\max_{s \\le \\tau} L_s(\\hat{\\mu}_s, \\mu)\n\\end{align*}\n\n\\paragraph{Bound on the expectation}\nWe prove $\\mathbb{E}[L_\\tau(\\hat{\\mu}_\\tau, \\mu)] \\le \\mathbb{E}[\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu)] \\le 1 + \\mathbb{E}[\\gamma(\\tau)]$. In the computations below, we use that $s \\mapsto \\gamma(s)$ is non-decreasing.\n\\begin{align*}\n\\mathbb{E}[\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu) - \\gamma(\\tau)]\n&\\le \\mathbb{E}[\\max\\{0, \\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu) - \\gamma(\\tau)\\}]\n\\\\\n&= \\int_{x=0}^{+\\infty}\\mathbb{P}(\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu) - \\gamma(\\tau) > x) dx\n\\\\\n&\\le \\int_{x=0}^{+\\infty}\\mathbb{P}(\\exists t \\in \\mathbb{N}, L_t(\\hat{\\mu}_t, \\mu) > x + \\gamma(t)) dx\n\\\\\n&\\le \\int_{x=0}^{+\\infty} e^{- x} d x = 1\n\\: .\n\\end{align*}\n\n\\paragraph{Bound on the expectation of the truncated LLR}\n\\begin{align*}\n\\mathbb{E}[\\mathbb{I}\\{\\ihat \\ne i^\\star\\} \\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu)]\n&\\le \\sqrt{\\mathbb{P}\\{\\ihat \\ne i^\\star\\} \\mathbb{E}[(\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu))^2]}\n\\end{align*}\n\n\\begin{align*}\n\\mathbb{E}[(\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu))^2 - 2\\gamma(\\tau)^2]\n&\\le \\mathbb{E}[\\max\\{0, (\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu))^2 - 2\\gamma(\\tau)^2\\}]\n\\\\\n&= \\int_{x=0}^{+\\infty}\\mathbb{P}((\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu))^2 - 2\\gamma(\\tau)^2 > x) dx\n\\\\\n&\\le \\int_{x=0}^{+\\infty}\\mathbb{P}((\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu))^2 > (\\gamma(\\tau) + \\sqrt{x\/2})^2) dx\n\\\\\n&= \\int_{x=0}^{+\\infty}\\mathbb{P}(\\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu) > \\gamma(\\tau) + \\sqrt{x\/2} ) dx\n\\\\\n&\\le \\int_{x=0}^{+\\infty}\\mathbb{P}(\\exists t \\in \\mathbb{N}, L_t(\\hat{\\mu}_t, \\mu) > \\gamma(t) + \\sqrt{x\/2}) dx\n\\\\\n&\\le \\int_{x=0}^{+\\infty} e^{- \\sqrt{x\/2}} d x = 4\n\\: .\n\\end{align*}\n\n\\begin{align*}\n\\mathbb{E}[\\mathbb{I}\\{\\ihat \\ne i^\\star\\} \\max_{s \\le \\tau}L_s(\\hat{\\mu}_s, \\mu)]\n&\\le \\sqrt{\\mathbb{P}\\{\\ihat \\ne i^\\star\\} (4 + \\mathbb{E}[\\gamma(\\tau)^2])]}\n\\: .\n\\end{align*}\nSince the algorithm is $\\delta$-correct, $\\mathbb{P}\\{\\ihat \\ne i^\\star\\} \\le \\delta$.\n\n\\paragraph{Special case of LLR stopping}\n\n\nNote: for LLR stopping, $M_i = 1$ for all $i$ and we get\n\\begin{align*}\n\\mathbb{E}[\\max_i \\max_{s \\le \\tau} L_s(\\mu, \\Lambda(i))]\n&\\ge \\log \\frac{1}{\\delta} - 1\n\\: , &\n\\mathbb{E}[\\max_{s \\le \\tau} L_s(\\mu, \\Lambda(\\ihat))]\n&\\ge \\log \\frac{1}{\\delta} - 1\n\\: .\n\\end{align*}\nAnd due to the way the algorithm stops, we could go through the same proof but with $\\max_{s \\le \\tau}$ replaced by taking the value at $\\tau$ for the answer $\\ihat$. That would give\n\\begin{align*}\n\\mathbb{E}[L_\\tau(\\mu, \\Lambda(\\ihat))]\n&\\ge \\log \\frac{1}{\\delta} - 1\n\\: .\n\\end{align*}\n\n\\begin{align*}\nL_\\tau(\\mu, \\Lambda(\\ihat))\n&= \\mathbb{I}\\{\\ihat = i^\\star\\}L_\\tau(\\mu, \\Lambda(\\ihat)) + \\mathbb{I}\\{\\ihat \\ne i^\\star\\}L_\\tau(\\mu, \\Lambda(\\ihat))\n\\\\\n&\\le \\mathbb{I}\\{\\ihat = i^\\star\\}L_\\tau(\\mu, \\Lambda(\\ihat)) + \\mathbb{I}\\{\\ihat \\ne i^\\star\\}L_\\tau(\\mu, \\mu)\n\\\\\n&= \\mathbb{I}\\{\\ihat = i^\\star\\}L_\\tau(\\mu, \\Lambda(\\ihat))\n\\\\\n&= \\mathbb{I}\\{\\ihat = i^\\star\\}L_\\tau(\\mu, \\Lambda(i^\\star))\n\\\\\n&= L_\\tau(\\mu, \\Lambda(i^\\star)) - \\mathbb{I}\\{\\ihat \\ne i^\\star\\} L_\\tau(\\mu, \\Lambda(i^\\star))\n\\\\\n&= L_\\tau(\\mu, \\Lambda(i^\\star)) + \\mathbb{I}\\{\\ihat \\ne i^\\star\\} L_\\tau(\\hat{\\mu}_\\tau, \\mu)\n\\end{align*}\n\n\n\\subsection{Expectation of the infimum}\n\\label{sub:expectation_of_the_infimum}\n\nFor a set $\\Lambda$ and $\\varepsilon \\ge 0$, let $\\Lambda_{[0,\\varepsilon], N_t} = \\{\\lambda \\in \\Lambda \\mid \\sum_k N_t^k \\KL_k(\\theta, \\lambda) \\le \\inf_{\\eta \\in \\Lambda} \\sum_k N_t^k \\KL_k(\\theta, \\eta) + \\varepsilon\\}$. Then since $\\Lambda_{[0,\\varepsilon], N_t} \\subseteq \\Lambda$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda} L_t (\\theta, \\lambda)\n&\\le \\inf_{\\lambda \\in \\Lambda_{[0,\\varepsilon], N_t}} L_t (\\theta, \\lambda)\n\\\\\n&= \\inf_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}}\\left(\\sum_k N_t^k \\KL_k(\\theta, \\lambda)\n\t- \\sum_{s=1}^t (\\KL_{k_s}(\\theta, \\lambda) - \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s})) \\right)\n\\\\\n&\\le \\inf_{\\lambda \\in \\Lambda} \\sum_k N_t^k \\KL_k(\\theta, \\lambda) + \\varepsilon\n\t- \\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}}\\left( \\sum_{s=1}^t (\\KL_{k_s}(\\theta, \\lambda) - \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s})) \\right)\n\\: ,\\\\\n\\mathbb{E}[\\inf_{\\lambda \\in \\Lambda} L_t (\\theta, \\lambda)]\n&\\le \\inf_{\\lambda \\in \\Lambda}\\sum_k \\mathbb{E}[N_t^k] \\KL_k(\\theta, \\lambda) + \\varepsilon\n\t- \\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}}\\left( \\sum_{s=1}^t (\\KL_{k_s}(\\theta, \\lambda) - \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s})) \\right)\\right]\n\\: .\n\\end{align*}\n\nFor Gaussian distributions,\n\\begin{align*}\n\\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}}\\left( \\sum_{s=1}^t (\\KL_{k_s}(\\theta, \\lambda) - \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s})) \\right)\\right]\n&= \\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}} \\sum_k N_t^k (\\hat{\\mu}_{t,k} - \\mu_k(\\theta)) (\\mu_k(\\lambda) - \\mu_k(\\theta))\\right]\n\\end{align*}\n\nOur goal now is to show that this expectected supremum is large, which means that the expected infimum LLR is lower than the infimum of the weighted sum of the KL, hence a lower bound on the infimum LLR is stronger than a lower bound on the weighted sum of KLs.\n\nLet's look at the case of fixed design ($N_t$ is deterministic). Let $\\KL(t, \\Lambda) = \\inf_{\\lambda \\in \\Lambda}\\sum_k N_t^k \\KL_k(\\theta, \\lambda)$ and let $u_{\\lambda, N_t} = \\frac{\\mu(\\lambda) - \\mu(\\theta)}{\\Vert \\mu(\\lambda) - \\mu(\\theta)\\Vert_{N_t}} = \\frac{\\mu(\\lambda) - \\mu(\\theta)}{\\sqrt{2 \\sum_k N_t^k \\KL_k(\\theta, \\lambda)}}$. The vector $u_{\\lambda, N_t}$ has norm 1 for all $\\lambda$. Finally, let $Z = \\sqrt{N_t^k} (\\hat{\\mu}_{t,k} - \\mu_k(\\theta))$. $Z$ has law $\\mathcal N(0,1)$.\n\\begin{align*}\n&\\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}} \\sum_k N_t^k (\\hat{\\mu}_{t,k} - \\mu_k(\\theta)) (\\mu_k(\\lambda) - \\mu_k(\\theta))\\right]\n\\\\\n&= \\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}} \\sqrt{2 \\sum_k N_t^k \\KL_k(\\theta, \\lambda)} Z^\\top u_{\\lambda, N_t}\\right]\n\\\\\n&\\ge \\sqrt{2} \\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{[0, \\varepsilon], N_t}} \\sqrt{\\KL(t, \\Lambda) + \\varepsilon \\mathbb{I}\\{Z^\\top u_{\\lambda, N_t}\\le 0\\}} Z^\\top u_{\\lambda, N_t}\\right]\n\\end{align*}\nFor $\\varepsilon = 0$, using that $N_t$ is not random (hence $\\KL(t, \\Lambda)$ isn't either) this is\n\\begin{align*}\n\\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{\\{0\\}, N_t}} \\sqrt{2 \\KL(t, \\Lambda)} Z^\\top u_{\\lambda, N_t}\\right]\n&= \\sqrt{2 \\KL(t, \\Lambda)} \\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{\\{0\\}, N_t}} Z^\\top u_{\\lambda, N_t}\\right]\n\\end{align*}\nThe value $\\mathbb{E}\\left[\\sup_{\\lambda \\in \\Lambda_{\\{0\\}, N_t}} Z^\\top u_{\\lambda, N_t}\\right]$ is the Gaussian width of the set $\\{u_{\\lambda, N_t} \\mid \\lambda \\in \\Lambda_{\\{0\\}, N_t}\\}$, which is a subset of the sphere $\\{u \\mid \\Vert u \\Vert = 1\\}$.\n\nFor unstructured best arm identification, this set contains up to $K-1$ points ($K-1$ exactly at the $N_t$ which maximizes $V_t$). The Gaussian width scales as $\\sqrt{\\log K}$ in that case.\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe proposed a selective elimination rule, which successively prunes the pieces of the empirical answer, that can be easily combined with existing adaptive algorithms for general identification problems. We proved that it reduces their computational complexity, it never worsens their sample complexity guarantees, and it provably discards certain answers early. Our experiments on different pure exploration problems and bandit structures show that existing adaptive algorithms often benefit from a reduced sample complexity when combined with selective elimination, while achieving significant gains in computation time. Moreover, they show that selective elimination is overall better (in terms of samples vs time) than its full variant which repeatedly updates the pieces of all answers.\n\n\n\nInteresting directions for future work include investigating whether better guarantees on the stopping time can be derived for algorithms combined with elimination as compared to their LLR counterparts, and designing adaptive algorithms which are specifically tailored for elimination.\n\\section{Elimination at sampling}\n\\label{sec:elimination_at_sampling}\n\nWe show how to adapt sampling rules in order to accommodate piece elimination. There are two reasons for doing this: first, adapting the sampling to ignore pieces that have been discarded could reduce the sample complexity; second, the amount of computations needed to update the sampling strategy is often proportional to the number of pieces and decreasing it can reduce the overall time.\n\nWe start from an algorithm using LLR stopping, for which we change the stopping rule as above. The algorithms that we can adapt are those that choose the next sample by aggregating information from each alternative piece.\nFor example methods that mimic the lower bound allocation~\\eqref{eq:lower_bound}, like Track-and-Stop \\citep{garivier2016optimal}, LinGame \\citep{degenne2020gamification}, or FWS \\citep{wang2021fast}, use the decomposition of the alternative into pieces: they need to compute the divergence to each of those pieces as well as the closest points. This is also the case for other algorithms like LinGapE \\citep{xu2018fully}. At time $t$, it loops over all currently sub-optimal arms to compute an optimistic gap, and sample the arm that makes the maximal gap decrease the most.\nThat loop over arms is a loop over the alternative pieces $\\{\\lambda \\mid \\phi_j^\\top \\lambda \\ge \\phi_{i^\\star(\\hat{\\theta}_t)}^\\top \\lambda\\}$. Eliminating pieces at sampling simply means omitting from that computation the arms that were deemed sub-optimal.\n\n\\subsection{Eliminating pieces at sampling}\n\\label{sub:eliminating_pieces_at_sampling}\n\nWe need an hypothesis on the shape of the threshold, used to compare thresholds at different times.\n\n\\begin{assumption}\\label{ass:beta_bounds}\nThere exist two positive reals $c_1$ and $c_2$  that may depend on the parameters of the problem known to the algorithm ($d$, $K$, etc.) and a function $\\overline{\\log}$ such that the threshold $\\beta_{t,\\delta}$ verifies\n\\begin{align*}\n\\overline{\\log}(1\/\\delta) + c_1 \\log(t)\n\\le \\beta_{t, \\delta}\n\\le \\overline{\\log}(1\/\\delta) + c_2 \\log(t)\n\\: .\n\\end{align*}\nFurthermore, $\\overline{\\log}$ verifies that there exists $x_0$ such that for all $a \\ge 2$ and $x \\ge x_0$, $\\overline{\\log}(x^a) \\le a \\overline{\\log}(x)$. \n\\end{assumption}\nThe function $\\overline{\\log}$ represents an almost logarithmic function and corresponds to the function of order $\\log + \\sqrt{\\log}$ in \\citep{abbasi2011improved} or $\\log + d\\log\\log$ in \\citep{tirinzoni2020asymptotically,reda2021dealing}.\nOther thresholds have a $O(\\log\\log(t))$ dependence instead of $O(\\log(t))$, and what follows could be extended to them in a similar way.\n\nSimilarly to elimination stopping, the idea of our elimination at sampling is to maintain sets of active pieces $\\cP_{t}^{\\mathrm{smp}}(i)$ for each $i\\in\\cI$. Note that these are different from the ones introduced in Section \\ref{sec:elimination_stopping_rules} for the stopping rule. The sampling set is updated at each step with a different threshold $\\alpha_{t,\\delta}$, while we reset it very infrequently at steps $t\\in\\{\\bar{t}_0^{2^j}\\}_{j\\geq 0}$, where $\\bar{t}_0 = \\max\\{2, \\sqrt{x_0}\\}$. Formally, let us define the helper sets $\\tilde{\\cP}_{t}^{\\mathrm{smp}}(i)$ as $\\tilde{\\cP}_{0}^{\\mathrm{smp}}(i) := \\cP(i)$ and\n\\begin{align*}\n\\tilde{\\cP}_t^{\\mathrm{smp}}(i) := \\begin{cases}\n\\tilde{\\cP}_{t-1}^{\\mathrm{smp}}(i) \\cap \\overline{\\cP}_t(i;\\alpha_{t,\\delta}) & \\text{if } t\\notin\\{\\bar{t}_0^{2^j}\\}_{j\\geq 0}\n\\\\ \\overline{\\cP}_t(i;\\alpha_{t,\\delta}) & \\text{otherwise},\n\\end{cases}\n\\end{align*}\nwhere $\\overline{\\cP}_t$ was defined in \\eqref{eq:active-pieces-t-only}.\nLet $\\overline{t}_j := \\bar{t}_0^{2^j}$ be the time step at which the $j$-th reset is performed and $j(t) := \\lfloor \\log_2\\log_{\\bar{t}_0} t \\rfloor$ be the index of the last reset before $t$.\nThose reset times are the elements of our implementation that would change if the threshold is not $O(\\log(t))$. They are chosen such that $\\beta_{t,1\/t^2}$ roughly doubles from one reset to the next.\n\nWe define ${\\cP}_t^{\\mathrm{smp}}(i) := \\tilde{\\cP}_t^{\\mathrm{smp}}(i) \\cap \\tilde{\\cP}_{\\overline{t}_{j(t)} - 1}^{\\mathrm{smp}}(i)$. This implies that ${\\cP}_t^{\\mathrm{smp}}(i)$ is the intersection of all active pieces from the second-last reset up to $t$, i.e.,\n\n{\\cP}_t^{\\mathrm{smp}}(i) = \\bigcap_{s=\\overline{t}_{j(t)-1}}^t \\overline{\\cP}_s(i;\\alpha_{s,\\delta}).\n$\nAs before, we can instantiate both selective and full elimination to update these sets. Since the resets are very infrequent, this definition only drops a small number of rounds from the intersection (less than $\\sqrt{t}$).\nThis is not much different from ${\\cP}_t^{\\mathrm{stp}}(i)$, which instead takes the intersection of all $t$ sets.\nWe require $\\alpha_{t,\\delta} \\ge \\beta_{t,\\delta}$, so that pieces that are eliminated for the sampling rule\nare also eliminated for the stopping rule.\nWe set\n$\n\\alpha_{{t},\\delta} := \\left(\\sqrt{\\beta_{t,\\delta} + (4 c_2 - c_1)\\log(t)} + 4\\sqrt{\\frac{c_2}{c_1}\\beta_{t,1\/t^2}}\\right)^2\n\\: \n$. The reason for having different sets of active pieces for stopping and sampling (with rare resets in the latter) is that it allows us to derive guarantees on the expected stopping times. If one is only interested in high-probability results, it is possible to use the same set for both components.\n\n\\subsection{Properties}\n\\label{sub:properties}\n\n\nWe consider a counterpart of Assumption \\ref{asm:sampling-rule} for sampling rules combined with piece elimination.\n\n\\begin{assumption}\\label{asm:sampling-rule-v2}\nThere exists a sub-linear (in $t$) problem-dependent function $R(\\theta,t)$ such that, for each time $t$ where $E_t$ (defined in Equation \\ref{eq:Et}) holds,\n\\begin{align*}\n\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star(\\theta))}\\! \\! \\! \\! \\! H_p(N_t, \\theta) \n\\geq \\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\theta))}\\! \\! \\! \\! \\! H_p(\\omega, \\theta) {-} R(\\theta,t).\n\\end{align*}\n\\end{assumption}\n\nIntuitively, the sampling rule maximizes the information for discriminating $\\theta$ with all its alternatives from the sequence of active pieces $(\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\theta)))_{s=1}^t$. We prove in Appendix \\ref{app:assumptions} that the algorithms for which we proved Assumption \\ref{asm:sampling-rule} also satisfy Assumption \\ref{asm:sampling-rule-v2} when their sampling rules are combined with either full or selective elimination.\n\n\\begin{theorem}\\label{th:sampling-rule-with-elim}\nConsider a sampling rule that verifies Assumption \\ref{asm:sampling-rule-v2} and uses either full or selective elimination with the sets ${\\cP}_t^{\\mathrm{smp}}$. Then, Assumption \\ref{asm:sampling-rule} holds as well. Moreover, when using the same elimination rule at stopping, such a sampling rule verifies Theorem \\ref{th:piece-elim}, i.e., it enjoys the same guarantees as without elimination at sampling.\n\\end{theorem}\n\nThe proof is in Appendix~\\ref{app:proofs_of_sampling}. Theorem \\ref{th:sampling-rule-with-elim} shows that for an algorithm using elimination at sampling and stopping, we get bounds on the times at which pieces of $\\Lambda(i^\\star(\\theta))$ are discarded from the stopping rule which are not worse than those we obtained for the same algorithm without elimination at sampling.\nThis result is non-trivial. We know that the sampling rule collects information to discriminate $\\theta$ with its closest alternatives, and eliminating a piece cannot make the resulting ``optimal'' proportions worse at this task. However, it could make them worse at discriminating $\\theta$ with alternatives that are not the closest.\nThis would imply that the elimination times for certain pieces could actually increase w.r.t. not eliminating at sampling.\nTheorem \\ref{th:sampling-rule-with-elim} guarantees that this does not happen: eliminating pieces at sampling cannot worsen our guarantees.\nWe shall see in our experiments that eliminating pieces in both the sampling and stopping rules often yields improved sample complexity.\n\n\n\n\\section{Proofs of Section \\ref{sec:elimination_stopping_rules}}\\label{app:proofs-elim-stopping}\n\n\n\\subsection{Proof of Lemma \\ref{lem:delta-correct}}\n\n\\begin{proof}\nFix any $\\theta\\in\\mathcal{M}$ and let $\\hat{i}$ be the answer returned by the algorithm at the stopping time $\\tau$. Then,\n\\begin{align*}\n\\mathbb{P}\\left(\\hat{i} \\neq i^\\star(\\theta)\\right) \n&\\stackrel{(a)}{\\leq} \\mathbb{P}\\left(\\exists i \\neq i^\\star(\\theta) : \\cP_\\tau(i) = \\emptyset\\right)\n\\\\\n&\\stackrel{(b)}{\\leq} \\mathbb{P}\\left(\\exists i \\neq i^\\star(\\theta),\\forall p\\in\\cP(i), \\exists t \\leq \\tau : \\inf_{\\lambda \\in \\Lambda_{p}(i)} L_t(\\hat{\\theta}_t, \\lambda) \\geq \\beta_{t,\\delta}\\right)\n\\\\\n&\\stackrel{(c)}{\\leq} \\mathbb{P}\\left(\\exists i \\neq i^\\star(\\theta),\\forall p\\in\\cP(i), \\exists t \\geq 1 : \\inf_{\\lambda \\in \\Lambda_{p}(i)} L_t(\\hat{\\theta}_t, \\lambda) \\geq \\beta_{t,\\delta}\\right)\n\\end{align*}\nwhere (a) is from the definition of stopping rule, (b) is from the fact that, if $\\cP_\\tau(i)$ is empty, then all the pieces of $i$ have been eliminated at some times before $\\tau$, and (c) follows trivially by relaxing the condition $t\\leq \\tau$ to $t\\geq 1$. Take any $t\\geq 1$. Now note that, for any wrong answer $i\\neq i^\\star(\\theta)$, $\\theta \\in \\Lambda(i)$. By definition of the decomposition into pieces of $\\Lambda(i)$, this means that there exists $\\bar{p}_i \\in \\cP(i)$ such that $\\theta \\in \\Lambda_{\\bar{p}_i}(i)$. Therefore, continuing the chain of inequalities above, we get that\n\\begin{align*}\n\\mathbb{P}\\left(\\hat{i} \\neq i^\\star(\\theta)\\right) \n&\\stackrel{(d)}{\\leq} \\mathbb{P}\\left(\\exists i \\neq i^\\star(\\theta), \\exists t \\geq 1 : \\inf_{\\lambda \\in \\Lambda_{\\bar{p}_i}(i)} L_t(\\hat{\\theta}_t, \\lambda) \\geq \\beta_{t,\\delta}\\right)\n\\\\ &\\stackrel{(e)}{\\leq} \\mathbb{P}\\left(\\exists t \\geq 1 : L_t(\\hat{\\theta}_t, \\theta) \\geq \\beta_{t,\\delta}\\right) \n\\stackrel{(f)}{\\leq} \\delta,\n\\end{align*}\nwhere (d) holds since the event under which all pieces for $i$ have been eliminated implies that $\\bar{p}_i$ has been eliminated as well, (e) holds since $\\theta \\in \\Lambda_{\\bar{p}_i}(i)$, and (f) is from the assumption on threshold $\\beta_{t,\\delta}$. This concludes the proof.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{th:elim-better-than-llr}}\\label{app:monotonicity}\n\nTheorem~\\ref{th:elim-better-than-llr} was proved in the main text. We prove here a more general result about the monotonicity of the stopping time with respect to the piece decomposition.\nWe show that if two algorithms use the same sampling rules and use elimination stopping based on different piece decompositions, the algorithm using the finer decomposition (in the sense defined below) will stop earlier.\n\n\\begin{definition}\nWe say that a piece decomposition $(\\Lambda_p(i))_{i \\in \\mathcal I,p \\in \\mathcal P(i)}$ is finer than another one $(\\tilde{\\Lambda}_p(i))_{i \\in \\mathcal I,p \\in \\tilde{\\mathcal P}(i)}$ if for all $i \\in \\mathcal I, p \\in \\tilde{\\mathcal P}(i)$, there exists a set $S \\subseteq \\mathcal P(i)$ such that $\\tilde{\\Lambda}_p(i) = \\bigcup_{q \\in S}\\Lambda_q(i)$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:elimination_mono_decomposition}\nLet $\\mathcal D = (\\Lambda_p(i))_{i \\in \\mathcal I,p \\in \\mathcal P(i)}$ be a finer decomposition than $\\tilde{\\mathcal D} = (\\tilde{\\Lambda}_p(i))_{i \\in \\mathcal I,p \\in \\tilde{\\mathcal P}(i)}$. For $i \\in \\mathcal I,\\tilde{p} \\in \\tilde{\\mathcal P}(i)$, let $\\tau_{\\tilde{p}}$ and $\\tilde{\\tau}_{\\tilde{p}}$ be the times at which $\\tilde{\\Lambda}_{\\tilde{p}}(i)$ is eliminated by the two corresponding algorithms (in the sense of Definition~\\ref{def:elimination}). Then almost surely $\\tau_{\\tilde{p}} \\le \\tilde{\\tau}_{\\tilde{p}}$.\n\\end{theorem}\nRoughly, if a piece in the tilde decomposition corresponds to several pieces in the other, then it is faster to eliminate it as several pieces than as one piece.\n\n\\begin{proof}\nIt is enough to prove that whenever the elimination stopping rule of $\\tilde{\\Lambda}_{\\tilde{p}}(i)$ triggers for $\\tilde{\\mathcal D}$, it triggers for $\\mathcal D$ too. Then, let $t\\geq 1$ and suppose that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_{\\tilde{p}}(i)} L_t(\\hat{\\theta}_t, \\lambda)  \\geq \\beta_{t,\\delta}.\n\\end{align*}\nLet $S_{\\tilde{p}} \\subseteq \\cP(i)$ be the set corresponding to $\\tilde{p}$ in the definition of ``finer decomposition''. We first argue that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i)} L_t(\\hat{\\theta}_t, \\lambda)\n&= \\inf_{q\\in S_p}\\inf_{\\lambda \\in \\Lambda_q(i)} L_t(\\hat{\\theta}_t, \\lambda) \\: .\n\\end{align*}\nIndeed, this is simply writing the infimum over an union as the infimum of an infimum. Hence if the $\\tilde{p}$ piece is eliminated in $\\tilde{\\mathcal D}$, we have\n\\begin{align*}\n\\inf_{q\\in S_p}\\inf_{\\lambda \\in \\Lambda_q(i)} L_t(\\hat{\\theta}_t, \\lambda)\n\\ge \\beta_{t, \\delta}\n\\end{align*}\nand every piece $\\Lambda_p(i)$ for $p \\in S_{\\tilde{p}}$ is eliminated as well in $\\mathcal D$. We get that the set $\\tilde{\\Lambda}_{\\tilde{p}}$ is also eliminated in $\\mathcal D$.\n\\end{proof}\n\nCorollary: the finest possible decomposition is the one in which $\\mathcal P(i) = \\Lambda(i)$ and $\\Lambda_p(i) = \\{p\\}$, i.e. every point of $\\Lambda(i)$ is its own piece. This is not a computationally usable decomposition, but its the theoretically best one for the sample complexity metric (for a fixed sampling rule).\n\nTheorem~\\ref{th:elim-better-than-llr} for full elimination compared to LLR stopping follows from Theorem~\\ref{thm:elimination_mono_decomposition} by setting $\\tilde{\\mathcal P}(i) = \\{0\\}$ and $\\tilde{\\Lambda}_p(i) = \\Lambda(i)$. Then the elimination stopping rule uses a finer decomposition than the LLR stopping rule.\n\n\n\\subsection{Proof of Theorem \\ref{th:piece-elim}}\n\nWe first present two important lemmas which will be used to prove the main statement for full elimination and selective elimination, respectively. For full elimination, we the following lemma shows that if a piece has not been eliminated the information collected by the algorithm about it must be small.\n\n\n\n\n\\begin{lemma}\\label{lemma:upper-bound-sum-inf}\nConsider an algorithm that uses the full elimination stopping rule \\eqref{eq:elimination-sets}. Let $p\\in \\cP(i^\\star(\\theta))$, then, for each time $t$ such that $E_t$ (Equation \\ref{eq:Et}) holds and $p\\in \\cP_t(i^\\star(\\theta))$,\n\\begin{align*}\n\\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} < \\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}.\n\\end{align*}\nOn the other hand, if the algorithm uses the LLR stopping rule, for each time $t$ such that $E_t$ holds and the algorithm did not stop,\n\\begin{align*}\n\\sqrt{\\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} < \\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nSince $p\\in \\cP_t(i^\\star(\\theta))$ (i.e., $p$ has not been eliminated at time $t$), we have from \\eqref{eq:elimination-sets} and Lemma \\ref{lem:llr-to-kl-lin-gauss} that\n\\begin{align*}\n\\beta_{t,\\delta}\n &> \\inf_{\\lambda \\in \\Lambda_p(i^{\\star})} L_t(\\hat{\\theta}_t, \\lambda) \\geq \\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\left(\\sqrt{\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} - \\sqrt{L_t(\\hat{\\theta}_t,\\theta)}\\right)^2.\n\\end{align*}\nBy definition of event $E_t$, this implies\n\\begin{align*}\n\\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} < \\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}.\n\\end{align*} \nThis yields the first statement. The second result can be shown analogously by using the full set of alternatives to $\\theta$.\n\n\\end{proof}\n\nSelective elimination updates, at each step $t$, only the set of active pieces of the empirical answer $i^\\star(\\hat{\\theta}_t)$. Therefore, to bound the elimination times of the pieces $\\Lambda_p(i^\\star)$ for $p\\in\\cP(i^\\star)$ we need to show that $i^\\star(\\hat{\\theta}_t) \\neq i^\\star$ after a certain time. We show that Assumption \\ref{asm:sampling-rule} alone is sufficient to guarantee this.\n\n\\begin{lemma}\\label{lem:empirical-vs-true-answer}\nConsider a sampling rule satisfying Assumption \\ref{asm:sampling-rule}. Under event $E_t$, a sufficient condition for $i^\\star(\\hat{\\theta}_t) = i^\\star$ is\n\\begin{align}\nt \\geq \\frac{ 4\\beta_{t,1\/t^2} + R(\\theta,t)}{H^\\star(\\theta)}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nFirst note that, under $E_t$, if\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^{\\star}(\\hat{\\theta}_t))} L_t(\\hat{\\theta}_t, \\lambda) > \\beta_{t,1\/t^2},\n\\end{align*}\nthen $i^\\star(\\hat{\\theta}_t) = i^\\star$. In fact, if this was not the case, we would have $\\theta \\in \\Lambda(i^{\\star}(\\hat{\\theta}_t))$ and thus $L_t(\\hat{\\theta}_t, \\theta) > \\beta_{t,1\/t^2}$, which is a contradiction with event $E_t$ itself. Let us now look for a sufficient condition on $t$ to satisfy this inequality. Take $t$ and suppose it does not satisfy it. Then,\n\\begin{align*}\n\\sqrt{\\beta_{t,1\/t^2}}\n \\geq \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^{\\star}(\\hat{\\theta}_t))} L_t(\\hat{\\theta}_t, \\lambda)} \n &\\stackrel{(a)}{\\geq} \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^{\\star}(\\hat{\\theta}_t))}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} - \\sqrt{L_t(\\hat{\\theta}_t,\\theta)}\n \\\\ &\\stackrel{(b)}{\\geq} \\sqrt{\\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} - \\sqrt{\\beta_{t,1\/t^2}}\n \\\\ &\\stackrel{(c)}{\\geq} \\sqrt{t H^\\star(\\theta) - R(\\theta,t)} - \\sqrt{\\beta_{t,1\/t^2}},\n\\end{align*}\nwhere (a) is from Lemma \\ref{lem:llr-to-kl-lin-gauss}, (b) from $E_t$ and the fact that either $i^\\star(\\hat{\\theta}_t) = i^\\star$ or the infimum is zero, and (c) from Assumption \\ref{asm:sampling-rule}. Rearranging this inequality yields the desired condition on $t$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{th:piece-elim}]\nWe start by proving the sample complexity bound for LLR stopping. Since it is a bound on the elimination time of the whole alternative $\\Lambda(i^\\star)$, by definition it also holds for the elimination times of its pieces obtained with either full or selective elimination. We then move to full elimination stopping and selective elimination stopping. Finally, we prove that the left-hand side of the maximum in \\eqref{eq:elimination_time} is never larger than the bound for LLR stopping.\n\n\\paragraph{LLR stopping}\n\nLet $t > 0$ such that $E_t$ holds and the algorithm did not stop. The second statement in Lemma \\ref{lemma:upper-bound-sum-inf} yields\n\\begin{align*}\n\\sqrt{\\inf_{\\lambda \\in \\Lambda(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} < \\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}.\n\\end{align*}\nMoreover, from Assumption \\ref{asm:sampling-rule}\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}N_t^k \\KL_k(\\theta,\\lambda) \\geq t H^\\star(\\theta) - R(\\theta,t).\n\\end{align*}\nThe combination of these two inequalities directly yields the stated inequality on $t$. The result in expectation is obtained by applying Lemma 19 in \\citep{reda2021dealing} together with $\\mathbb{P}(\\neg E_t) \\leq 1\/t^2$.\n\n\\paragraph{Full elimination stopping}\n\nLet $t > 0$ such that $E_t$ holds. By Assumption \\ref{asm:sampling-rule},\n\\begin{align*}\nH^\\star(\\theta) \\leq \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\frac{N_t^k}{t} \\KL_k(\\theta,\\lambda) + \\frac{R(\\theta,t)}{t}\n\\end{align*}\nThis implies that $\\frac{N_t}{t}\\in \\Omega_{R(\\theta,t)\/t}$.\n\nNow fix a piece $p\\in \\cP(i^\\star)$ such that $p\\in \\cP_t(i^\\star)$. Under event $E_t$, we know from Lemma \\ref{lemma:upper-bound-sum-inf} that\n\\begin{align*}\n\\sqrt{\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)}\n< \\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}.\n\\end{align*}\nSince $\\frac{N_t}{t}\\in \\Omega_{R(\\theta,t)\/t}$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i^{\\star})}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)\n  \\geq t \\min_{\\omega \\in \\Omega_{R(\\theta,t)\/t}}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda).\n\\end{align*}\nCombining the last two displays, we obtain that, if $j$ is not eliminated at time $t$ and $E_t$ holds, $t$ itself must satisfy\n\\begin{align*}\nt < \\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{\\min_{\\omega \\in \\Omega_{R(\\theta,t)\/t}}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)}.\n\\end{align*}\nThe result in expectation is obtained by applying Lemma 19 in \\citep{reda2021dealing}  together with $\\mathbb{P}(\\neg E_t) \\leq 1\/t^2$.\n\n\\paragraph{Selective elimination stopping}\n\nGiven Lemma \\ref{lem:empirical-vs-true-answer}, the proof of Theorem \\ref{th:piece-elim} for elimination stopping is very simple. Simply take a time $t$ such that $E_t$ holds and which verifies the condition in Lemma \\ref{lem:empirical-vs-true-answer}. Then, for such a $t$ the first claim of Lemma \\ref{lemma:upper-bound-sum-inf} can be verified analogously since the empirical answer (the one for which the set of active pieces is updated) is exactly the correct answer. Given Lemma \\ref{lemma:upper-bound-sum-inf}, the same derivation as in the proof for full elimination can be carried out. This yields the following sufficient condition on the time $t$ to eliminate a piece $p\\in\\cP(i^\\star)$:\n\\begin{align*}\nt \\geq \\max\\left\\{\\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{\\min_{\\omega \\in \\Omega_{R(\\theta,t)\/t}}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)} , \\frac{ 4\\beta_{t,1\/t^2} + R(\\theta,t)}{H^\\star(\\theta)}\\right\\}.\n\\end{align*}\n\n\\paragraph{Comparison of the bounds for full elimination and LLR stopping}\n\nWe finally prove that a sufficient condition for $t \\geq \\bar{t}_p$, with $\\bar{t}_p$ associated to full elimination, is $t\\geq\\bar{t}$.\n\nTake any $p\\in \\cP(i^\\star)$. By definition of the set $\\Omega_\\epsilon$, we have that, for any $\\omega\\in\\Omega_\\epsilon$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) \\geq \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) \\geq H^\\star(\\theta) - \\epsilon.\n\\end{align*}\nThis implies that\n\\begin{align*}\n\\min_{\\omega\\in\\Omega_\\epsilon}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) \\geq \\max_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) - \\epsilon.\n\\end{align*}\nA sufficient condition for satisfying the inequality for piece elimination is thus\n\\begin{align*}\nt \\geq \\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{\\max_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)- R(\\theta,t)\/t}.\n\\end{align*}\nLet $\\gamma \\in (0,1)$. For $R(\\theta,t)\/t \\leq \\gamma H^\\star(\\theta)$ we have that\n\\begin{align*}\nt \\geq \\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{(1-\\gamma)\\max_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)}\n\\end{align*}\nsuffices. Therefore, taking the maximum between the condition above and $t \\geq \\frac{R(\\theta,t)}{\\gamma H^\\star(\\theta)}$ and optimizing over $\\gamma \\in (0,1)$, the inequality for piece elimination is verified if\n\\begin{align*}\nt \\geq \\frac{1}{H^\\star(\\theta)}\\inf_{\\gamma \\in (0,1)}\\max\\left\\{\\frac{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2}{(1-\\gamma)},  \\frac{R(\\theta,t)}{\\gamma } \\right\\}\n\\end{align*}\nOptimizing over $\\gamma$, which amounts to setting $\\gamma = \\frac{R(\\theta,t)}{\\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2 + R(\\theta,t)}$, yields the desired statement.\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{th:elim-vs-llr-fixed-sampling}}\\label{subapp:proof_elim_vs_llr}\n\n\\begin{proof}\nTake any time $t$ and suppose that $E_t$ holds while the algorithm did not stop with $\\tau_{\\mathrm{llr}}$ yet. Using the first result of Theorem \\ref{th:piece-elim} yields\n\\begin{align*}\nt < H^\\star(\\theta)^{-1}\\left( \\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2 + R(\\theta,t) \\right)\n\\: .\n\\end{align*}\nFrom Lemma 19 in \\citep{reda2021dealing}, we get that $\\mathbb{E}[\\tau_{\\mathrm{llr}}] \\leq \\bar{t} + 2$, where $\\bar{t}$ is the first time that does not satisfy the inequality above (it is a function of $\\theta$ and $\\delta$). This means that\n\\begin{align*}\n\\mathbb{E}[\\tau_{\\mathrm{llr}}] \\leq H^\\star(\\theta)^{-1}\\left( \\left(\\sqrt{\\beta_{t,\\delta}} + \\sqrt{\\beta_{t,1\/t^2}}\\right)^2 + R(\\theta,t) \\right) + 3.\n\\end{align*}\n\nNow let us take the same algorithm with $\\tau_{\\mathrm{elim}}$. We know it is $\\delta$-correct, so the standard lower bound states that\n\\begin{align*}\n\\mathbb{E}[\\tau_{\\mathrm{elim}}] \\geq H^\\star(\\theta)^{-1}\\log(1\/2.4\\delta).\n\\end{align*}\nWe get that\n\\begin{align*}\n&H^\\star(\\theta)\\mathbb{E}[\\tau_{\\mathrm{llr}}]\n\\\\\n&\\le H^\\star(\\theta)\\mathbb{E}[\\tau_{\\mathrm{elim}}] + 3H^\\star(\\theta) + \\left(\\sqrt{\\beta_{\\bar{t},\\delta}} + \\sqrt{\\beta_{\\bar{t},1\/\\bar{t}^2}}\\right)^2 + R(\\theta,\\bar{t}) - \\log \\frac{1}{2.4 \\delta}\n\\\\\n&= H^\\star(\\theta)\\mathbb{E}[\\tau_{\\mathrm{elim}}]+ 3H^\\star(\\theta) + \\beta_{\\bar{t},\\delta} - \\log\\frac{1}{\\delta} + 2 \\sqrt{\\beta_{\\bar{t},\\delta}\\beta_{\\bar{t},1\/\\bar{t}^2}} + \\beta_{\\bar{t},1\/\\bar{t}^2} + R(\\theta,\\bar{t}) + \\log (2.4) \\: .\n\\end{align*}\nWe now use the hypothesis $\\beta_{\\bar{t}, \\delta} = \\log\\frac{1}{\\delta} + \\xi(\\bar{t}, \\delta)$:\n\\begin{align*}\nH^\\star(\\theta)\\mathbb{E}[\\tau_{\\mathrm{llr}}]\n&\\le H^\\star(\\theta)\\mathbb{E}[\\tau_{\\mathrm{elim}}] + 3H^\\star(\\theta) + \\xi(\\bar{t}, \\delta) + 2 \\sqrt{\\beta_{\\bar{t},\\delta}\\beta_{\\bar{t},1\/\\bar{t}^2}} + \\beta_{\\bar{t},1\/\\bar{t}^2} + R(\\theta,\\bar{t}) + \\log (2.4) \\: .\n\\end{align*}\n\\end{proof}\n\n\n\\section{Existing Algorithms Satisfy Assumption \\ref{asm:sampling-rule} and \\ref{asm:sampling-rule-v2}}\\label{app:assumptions}\n\nIn this section, we show that existing sampling rules that target the optimal value of the max-inf game from the general lower bound of \\citep{kaufmann2016complexity} satisfy Assumption \\ref{asm:sampling-rule}. Moreover, we show that, when such sampling rules are combined with elimination (as in Section \\ref{sec:elimination_at_sampling}), they also satisfy Assumption \\ref{asm:sampling-rule-v2}. We do that explicitly for two algorithms: the game-theoretic approach based on no-regret learners of \\citep{degenne2019non} and an optimistic variant of the Track-and-Stop algorithm of \\citep{kaufmann2016complexity}. These two algorithms are sufficiently general to be representative of many existing approaches for pure exploration: the latter represents those that repeatedly solve the optimization problem from the lower bound to get optimal allocations, while the former represents those that solve such problem incrementally. We now describe these two algorithmic techniques. Then, we show, through a unified proof, that they satisfy Assumption \\ref{asm:sampling-rule} and \\ref{asm:sampling-rule-v2}.\n\n\\subsection{Sampling Rules}\n\nWe describe the sampling rules of interest while trying to keep some of their design choices (e.g., confidence intervals, tracking, optimization, etc.) as general as possible. We do this because these sampling rules have been adapted to different pure exploration problems and bandit structures in the literature, for which such components would be different. This will allow us to have unified proofs that are actually agnostic to the specific setting.\n\n\\subsection{Common Assumptions}\n\nWe first state some assumptions on the main common components of these algorithms.\n\n\\paragraph{Bounded closest alternatives}\n\nFirst, we shall make a mild regularity assumption about the considered identification problem: the distance between any closest alternative and $\\theta$ is bounded.\n\\begin{assumption}\\label{asm:boundedness}\nThere exists a constant $B > 0$ such that, for any $\\omega \\in \\mathbb{R}_{\\geq 0}$ and any subset $\\Lambda\\subseteq\\cM$ which is exactly the union of arbitrary pieces, there exists a closest alternative\n\\begin{align*}\n\\lambda_\\omega \\in \\argmin_{\\lambda\\in\\Lambda}\\sum_{k=1}^K \\omega^K \\KL_k(\\theta,\\lambda)\n\\end{align*}\nsuch that\n\\begin{align*}\n\\max_{k\\in[K]} \\KL_k(\\theta,\\lambda_\\omega) \\leq B.\n\\end{align*}\n\\end{assumption}\nWe note that this assumption is satisfied in the identification problems we consider (see Appendix \\ref{app:problems}), for Gaussian rewards where pieces are half-spaces.\n\n\\paragraph{Tracking}\n\nAll the sampling rules we consider choose a sequence of proportions $(\\omega_t)_{t\\geq 1}$, where $\\omega_t\\in\\Delta_K$, and use tracking to select the next arm to play based on these. Formally, the arm played at time $t$ is\n\\begin{align}\\label{eq:tracking}\nk_t := \\mathrm{Track}\\left(\\sum_{s=1}^t\\omega_s, N_{t-1}\\right),\n\\end{align}\nwhere $\\mathrm{Track}: \\mathbb{R}_+^K \\times \\mathbb{R}_+^K \\rightarrow [K]$ is some tracking function. To gain generality, we shall keep the tracking function implicit in the remainder, while only requiring the following assumption (which is actually guaranteed by existing methods).\n\\begin{assumption}\\label{asm:tracking}\nThere exists a constant $C_{\\mathrm{track}} > 0$ such that\n\\begin{align*}\n\\forall t > 0, k\\in[K] : N_t^k \\geq \\sum_{s=1}^t \\omega_s^k -C_{\\mathrm{track}}.\n\\end{align*}\nFor instance, the widely-adopted cumulative tracking,\n\\begin{align*}\n\\mathrm{Track}\\left(\\sum_{s=1}^t\\omega_s, N_{t-1}\\right) = \\argmin_{k\\in[K]} \\left( N_{t-1}^k - \\sum_{s=1}^t\\omega_s^k \\right),\n\\end{align*}\nsatisfies this assumption.\n\\end{assumption}\n\n\\paragraph{Confidence intervals}\n\nThese sampling rules maintain confidence intervals $(c_t^k)_{t\\geq 1, k\\in[K]}$ about the expected return of each arm. We shall also keep them implicit as their specific form depends on the bandit structure under consideration (e.g., linear vs unstructured). We will only require the following assumption, which is satisfied by common choices as described below.\n\\begin{assumption}\\label{asm:confidence intervals}\nUnder event $E_t$,\n\\begin{align*}\n\\forall s \\leq t: \\KL_k(\\hat{\\theta}_{s},\\lambda) - c_{s}^k \\leq \\KL_k(\\theta,\\lambda) \\leq \\KL_k(\\hat{\\theta}_{s},\\lambda) + c_{s}^k.\n\\end{align*}\nMoreover, there exists a sub-linear (in $t$) function $C_{\\mathrm{conf}}(t)$ such that\n\\begin{align*}\n\\sum_{s=1}^t\\sum_{k\\in[K]}\\omega_s^kc_{s-1}^k \\leq C_{\\mathrm{conf}}(t).\n\\end{align*}\n\\end{assumption}\nIn bandits with linear structure, the confidence intervals typically take the form $c_t^k \\propto \\|\\phi_k\\|_{V_t^{-1}}$. In this case finding an upper bound on $\\sum_{s=1}^t\\sum_{k\\in[K]}\\omega_s^kc_{s-1}^k$ would reduce to applying an elliptical potential lemma \\citep{abbasi2011improved} plus the tracking property, and the resulting upper bound would be $C_{\\mathrm{conf}}(t) \\propto \\sqrt{dt}$. In unstructured bandits we can take $c_t^k \\propto 1\/\\sqrt{N_t^k}$ and finding $C_{\\mathrm{conf}}(t)$ would require applying the standard pigeon-hole principle plus the tracking property, for which one obtains $C_{\\mathrm{conf}}(t) \\propto \\sqrt{Kt}$.\n\n\n\\subsubsection{Optimistic Track-and-Stop}\\label{sec:optimistic-ts}\n\nHere we describe an optimistic variant of the Track-and-Stop algorithm by \\citep{kaufmann2016complexity}. It was originally introduced by \\citep{degenne2019non} in order to get rid of forced exploration, one of the main causes behind the poor empirical performance of Track-and-Stop. The idea is to solve, at each time step $t$, an optimistic variant of the optimization problem from the lower bound,\n\\begin{align}\\label{eq:optimistic-ts-optimization}\n\\omega_t := \\argmax_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{t-1}))} \\sum_{k\\in[K]}\\omega^k \\left(\\KL_k(\\hat{\\theta}_{t-1},\\lambda) + c_{t-1}^k\\right),\n\\end{align}\nwhere $c_{t-1}^k$ is a per-arm confidence interval that satisfies Assumption \\ref{asm:confidence intervals}, thus ensuring optimism $\\KL_k(\\hat{\\theta}_{t-1},\\lambda) + c_{t-1}^k \\geq \\KL_k(\\theta,\\lambda)$ with high probability. The solution to this optimization problem yields proportions $\\omega_t$ which are tracked by the sampling rule. Then, the arm played at time $t$ is given by the tracking rule \\eqref{eq:tracking}.\n\n\\paragraph{Combining with elimination}\n\nIn order to combine this sampling rule with elimination, we simply redefine the optimization problem as\n\\begin{align}\\label{eq:optimistic-ts-optimization-elim}\n\\omega_t := \\argmax_{\\omega\\in\\Delta_K}\\min_{p\\in{\\cP}_{t-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{t-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{t-1}))} \\sum_{k\\in[K]}\\omega^k \\left(\\KL_k(\\hat{\\theta}_{t-1},\\lambda) + c_{t-1}^k\\right).\n\\end{align}\nThat is, we simply replace a minimization over the whole alternative with one over the active pieces only.\n\n\\subsubsection{Game-Theoretic Approach with No-Regret Learners}\\label{sec:no-regret}\n\nThe idea behind the game-theoretic approach of \\citep{degenne2019non} is to avoid recomputing the full optimistic problem \\eqref{eq:optimistic-ts-optimization} at each step, while solving it incrementally by means of no-regret learners. Given some online-learning algorithm $\\cL$ working on the $K$-dimensional simplex, the sampling rule works as follows. At each time $t$, first $\\cL$ outputs new proportions $\\omega_t$. Then, we compute the closest alternative\n\\begin{align*}\n\\hat{\\lambda}_t := \\argmin_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{t-1}))}\\sum_{k\\in[K]}\\omega_t^k \\KL_k(\\hat{\\theta}_{t-1},\\lambda).\n\\end{align*}\nFinally, $\\cL$ is updated with the (concave) gain function\n\\begin{align*}\ng_t(\\omega) := \\sum_{k\\in[K]}\\omega^k \\left(\\KL_k(\\hat{\\theta}_{t-1},\\hat{\\lambda}_t) + c_{t-1}^k\\right).\n\\end{align*}\nFinally, the sampling rule uses tracking exactly as in \\eqref{eq:tracking} to decide the next arm to pull. As before, we will only require the tracking rule to satisfy Assumption \\ref{asm:tracking} without specifying an explicit form. Similarly, we will keep the learner implicit as far as it satisfies the following no-regret property.\n\\begin{assumption}\\label{asm:no-regret}\nThe learner $\\mathcal{L}$ is no regret: there exists a sub-linear (in $t$) function $C_{\\mathcal{L}}(t)$ such that, for any $t\\geq 1$ and any sequence of gains $\\{g_s(\\omega)\\}_{s\\leq t}$,\n$$\n\\max_{w\\in\\Delta_K}\\sum_{s=1}^t \\big( g_s(w) -g_s(w_s) \\big) \\leq C_{\\mathcal{L}}(t) \\: .\n$$\n\\end{assumption}\n\n\\paragraph{Combining with elimination}\n\nFor the game-theoretic approach, we only need to redefine the closest alternative used in the gains as\n\\begin{align*}\n(\\hat{p}_t, \\hat{\\lambda}_t) := \\argmin_{p\\in{\\cP}_{t-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{t-1})), \\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{t-1}))}\\sum_{k\\in[K]}\\omega_t^k \\KL_k(\\hat{\\theta}_{t-1},\\lambda).\n\\end{align*}\n\n\\subsection{Assumption \\ref{asm:sampling-rule} Holds}\n\nWe need to show that, for any time $t \\geq 1$ where the good event $E_t$ (Equation \\ref{eq:Et}) holds, the two sampling rules presented above satisfy\n\\begin{align*}\nt H^\\star(\\theta) \\leq \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) + R(\\theta,t)\n\\end{align*}\nfor suitable choices of the function $R(\\theta,t)$. We shall write a unified proof for these two sampling rules while explicitly mentioning where they differ. Let us suppose that their algorithmic components satisfy the assumptions stated above, i.e., Assumption \\ref{asm:tracking} for tracking, Assumption \\ref{asm:confidence intervals} for the confidence intervals, and Assumption \\ref{asm:no-regret} for the no-regret learner.\n\nTake any time step $t\\geq 1$ and suppose that $E_t$ holds. Let $\\lambda_t := \\argmin_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)$. Then, using the tracking property (Assumption \\ref{asm:tracking}) together with Assumption \\ref{asm:boundedness},\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)\n &= \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda_t) \n \\\\\n &\\geq \\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda_t) - C_{\\mathrm{track}}\\sum_{k\\in[K]}\\KL_k(\\theta,\\lambda_t) \n \\\\ &\\geq \\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda) - C_{\\mathrm{track}}\\sum_{k\\in[K]}\\KL_k(\\theta,\\lambda_t) \n  \\\\ &\\geq \\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda) - C_{\\mathrm{track}}KB.\n\\end{align*}\nWe can now lower bound the first term as\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda)\n &\\stackrel{(a)}{\\geq} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]}  \\omega_s^k\\KL_k(\\theta,\\lambda)\n \\\\ &\\stackrel{(b)}{\\geq} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k\\KL_k(\\theta,\\lambda)\n \\\\ &\\stackrel{(c)}{\\geq} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) - c_{s-1}^k \\right)\n \\\\ &= \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2\\sum_{s=1}^t\\sum_{k\\in[K]}\\omega_s^kc_{s-1}^k\n \\\\ &\\stackrel{(d)}{\\geq} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t),\n\\end{align*}\nwhere (a) is from the concavity of the infimum, (b) holds since either $\\Lambda(i^\\star(\\hat{\\theta}_{s-1})) = \\Lambda(i^\\star)$ or $\\theta \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))$ (in which case the infimum would be zero), (c) is from the validity of the confidence intervals under $E_t$, and (d) is from Assumption \\ref{asm:confidence intervals}. Now note that, when applying the game-theoretic approach (Section \\ref{sec:no-regret}), the first term on the right-hand side is exactly the sum of gains fed into the learner. Thus, using the no-regret property (Assumption \\ref{asm:no-regret}),\n\\begin{align*}\n&\\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda)\n\\\\\n&\\geq \\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t) - C_{\\cL}(t).\n\\end{align*}\nIf instead we are applying optimistic Track-and-Stop (Section \\ref{sec:optimistic-ts}), the first term on the right-hand side is exactly the sum of optimal values of the objective functions maximized by the algorithm. Thus,\n\\begin{align*}\n&\\inf_{\\lambda \\in \\Lambda(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda)\n\\\\\n&= \\sum_{s=1}^t \\max_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t)\n\\\\ &\\geq \\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t).\n\\end{align*}\nTherefore, we only need to lower bound the first term above, which is common between the two considered algorithms. We have\n\\begin{align*}\n&\\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\\n&\\stackrel{(e)}{\\geq} \\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\ &\\stackrel{(f)}{\\geq} \\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\inf_{\\lambda \\in \\Lambda(i^\\star)} \\sum_{k\\in[K]} \\omega^k \\KL_k({\\theta},\\lambda)\n\\\\ &\\stackrel{(g)}{\\geq} t H^\\star(\\theta),\n\\end{align*}\nwhere (e) follows from the same reasoning as step (b) above, (f) is from the fact that confidence intervals are valid under $E_t$, and (g) is from the definition of the game in the lower bound. Putting all together, we proved that optimistic Track-and-Stop satisfies Assumption \\ref{asm:sampling-rule} with\n\\begin{align*}\nR(\\theta,t) = C_{\\mathrm{track}}KB + 2C_{\\mathrm{conf}}(t),\n\\end{align*}\nwhile the game-theoretic approach satisfies it with\n\\begin{align*}\nR(\\theta,t) = C_{\\mathrm{track}}KB + 2C_{\\mathrm{conf}}(t) + C_{\\cL}(t).\n\\end{align*}\n\n\\subsection{Assumption \\ref{asm:sampling-rule-v2} Holds}\n\nWe now show that Assumption \\ref{asm:sampling-rule-v2} holds for these sampling rules combined with elimination as formally explained above. The main steps are very similar as before, with the additional complications posed by eliminating pieces at sampling. \n\nRecall that we want to show that, for any time $t \\geq 1$ where the good event $E_t$ (Equation \\ref{eq:Et}) holds, the two sampling rules presented above satisfy\n\\begin{align*}\n\\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) \\leq \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) + R(\\theta,t)\n\\end{align*}\nfor suitable choices of the function $R(\\theta,t)$.\n\nTake any time step $t\\geq 1$ and suppose that $E_t$ holds. Let \n$$(p_t,\\lambda_t) \\in \\argmin_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star), \\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)$$ be the closest piece and alternative at time $t$. Then, using the tracking property (Assumption \\ref{asm:tracking}) and Assumption \\ref{asm:boundedness},\n\\begin{align*}\n\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)\n &= \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda_t) \n\\\\ &\\geq \\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda_t) - C_{\\mathrm{track}}KB\n \\\\ &\\geq \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda) - C_{\\mathrm{track}}KB.\n\\end{align*}\nRecall that $\\overline{t}_j := {\\bar{t}_0}^{2^j}$ is the time step at which the $j$-th reset is performed and $j(t) := \\lfloor \\log_2\\log_{\\bar{t}_0} t \\rfloor$ is the index of the last reset before $t$.  Let $\\bar{t} := \\overline{t}_{j(t)-1}$ be the time of the second-last reset before $t$. Note that\n\\begin{align*}\n\\bar{t} := {\\bar{t}_0}^{2^{j(t)-1}} = \\bar{t}_0^{\\frac{1}{2}2^{\\lfloor \\log_{2}\\log_{\\bar{t}_0} t \\rfloor}} = \\sqrt{\\bar{t}_0^{2^{\\lfloor \\log_{2}\\log_{\\bar{t}_0} t \\rfloor}}} = \\sqrt{\\bar{t}_{j(t)}}\\leq \\sqrt{t}.\n\\end{align*}\nWe can now lower bound the first term as\n\\begin{align*}\n&\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda)\n \\\\ &\\stackrel{(a)}{\\geq} \\sum_{s=1}^t \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}  \\omega_s^k\\KL_k(\\theta,\\lambda)\n \\\\ &\\stackrel{(b)}{\\geq} \\sum_{s=\\bar{t}+1}^t \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}  \\omega_s^k\\KL_k(\\theta,\\lambda)\n \\\\ &\\stackrel{(c)}{\\geq} \\sum_{s=\\bar{t}+1}^t \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k\\KL_k(\\theta,\\lambda)\n  \\\\ &\\stackrel{(d)}{\\geq} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))} \\sum_{k\\in[K]} \\omega_s^k\\KL_k(\\theta,\\lambda) - B\\sqrt{t}\n \\\\ &\\stackrel{(e)}{\\geq} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega_s^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) - c_{s-1}^k \\right) - B\\sqrt{t}\n\n \\\\ &\\stackrel{(f)}{\\geq} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega_s^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t) - B\\sqrt{t},\n\\end{align*}\nwhere (a) is from the concavity of the infimum, (b) drops the fist $\\bar{t}$ rounds, (c) uses Lemma \\ref{lem:change-inf-from-istar}, (d) uses that $\\cP_t^{\\mathrm{smp}}$ is contained in all active sets from $\\bar{t}$ to $t$ and completes the sum with the first $\\bar{t}$ rounds (while bounding $\\bar{t} \\leq \\sqrt{t}$), (e) is from the validity of the confidence intervals under $E_t$, and (f) is from Assumption \\ref{asm:confidence intervals}.\n\nNow note that, when applying the game-theoretic approach (Section \\ref{sec:no-regret}), the first term on the right-hand side is exactly the sum of gains fed into the learner. Thus, using the no-regret property (Assumption \\ref{asm:no-regret}),\n\\begin{align*}\n\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}&\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda) \n\\\\ &\\geq \\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\&\\quad - 2C_{\\mathrm{conf}}(t) - C_{\\cL}(t)  - B\\sqrt{t}.\n\\end{align*}\nIf instead we are applying optimistic Track-and-Stop (Section \\ref{sec:optimistic-ts}), the first term on the right-hand side is exactly the sum of optimal values of the objective functions maximized by the algorithm. Thus,\n\\begin{align*}\n&\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}\\sum_{k\\in[K]}\\sum_{s=1}^t\\omega_s^k\\KL_k(\\theta,\\lambda) \n\\\\ &\\geq \\sum_{s=1}^t \\max_{\\omega\\in\\Delta_K}\\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t) - B\\sqrt{t}\n\\\\ &\\geq \\max_{\\omega\\in\\Delta_K}\\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right) - 2C_{\\mathrm{conf}}(t) - B\\sqrt{t}.\n\\end{align*}\nTherefore, we only need to lower bound the first term above, which is common between the two considered algorithms. We have\n\\begin{align*}\n&\\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\ &\\stackrel{(g)}{\\geq}\n\\max_{\\omega\\in\\Delta_K} \\sum_{s=\\bar{t}+1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s-1}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s-1}))}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\ &\\stackrel{(h)}{\\geq} \\max_{\\omega\\in\\Delta_K} \\sum_{s=\\bar{t}+1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}  \\sum_{k\\in[K]} \\omega^k \\left( \\KL_k(\\hat{\\theta}_{s-1},\\lambda) + c_{s-1}^k \\right)\n\\\\ &\\stackrel{(i)}{\\geq} \\max_{\\omega\\in\\Delta_K} \\sum_{s=\\bar{t}+1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}  \\sum_{k\\in[K]} \\omega^k  \\KL_k({\\theta},\\lambda)\n\\\\ &\\stackrel{(j)}{\\geq} \\max_{\\omega\\in\\Delta_K} \\sum_{s=1}^t \\min_{p\\in\\cP_{s-1}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)}  \\sum_{k\\in[K]} \\omega^k  \\KL_k({\\theta},\\lambda) - B\\sqrt{t},\n\\end{align*}\nwhere (g) drops the first $\\bar{t}$ rounds, (h) uses Lemma \\ref{lem:change-inf-from-ihat}, (i) is from the fact that confidence intervals are valid under $E_t$, and (j) adds the missing first $\\bar{t}$ rounds. Putting all together, we proved that optimistic Track-and-Stop satisfies Assumption \\ref{asm:sampling-rule} with\n\\begin{align*}\nR(\\theta,t) = C_{\\mathrm{track}}KB + 2C_{\\mathrm{conf}}(t) + 2B\\sqrt{t},\n\\end{align*}\nwhile the game-theoretic approach satisfies it with\n\\begin{align*}\nR(\\theta,t) = C_{\\mathrm{track}}KB + 2C_{\\mathrm{conf}}(t) + C_{\\cL}(t) + 2B\\sqrt{t}.\n\\end{align*}\n\n\\begin{lemma}\\label{lem:theta-not-elim-in-last-two-phases}\nUnder event $E_t$, for any $s\\in\\mathbb{N}$ with $\\overline{t}_{j(t)-1} \\leq s \\leq t$,\n\\begin{align*}\nL_s(\\hat{\\theta}_s,\\theta) < \\alpha_{s,\\delta}.\n\\end{align*}\nMoreover, for any $s,s'\\in\\mathbb{N}$ with $\\overline{t}_{j(t)-1} \\leq s' \\leq s \\leq t$,\n\\begin{align*}\nL_{s'}(\\hat{\\theta}_{s'},\\hat{\\theta}_s) < \\alpha_{s',\\delta}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nUsing the good event $E_t$ followed by an application of Lemma \\ref{lem:new_beta-diff-between-phases} together with $s \\geq \\overline{t}_{j(t)-1}$,\n\\begin{align*}\nL_s(\\hat{\\theta}_s,\\theta) \\leq \\beta_{t,1\/t^2} \\leq 4 \\frac{c_2}{c_1}\\beta_{\\overline{t}_{j(t)-1},1\/\\overline{t}_{j(t)-1}^2} \\leq 4\\frac{c_2}{c_1}\\beta_{s,1\/s^2} < \\alpha_{s,\\delta}.\n\\end{align*}\nThis proves the first claim. To prove the second one, note that\n\\begin{align*}\n\\sqrt{L_{s'}(\\hat{\\theta}_{s'},\\hat{\\theta}_s)}\n \\stackrel{(a)}{=} \\sqrt{\\sum_{k\\in[K]}N_{s'}^k\\KL_k(\\hat{\\theta}_{s'},\\hat{\\theta}_s)}\n &\\stackrel{(b)}{\\leq} \\sqrt{\\sum_{k\\in[K]}N_{s'}^k\\KL_k(\\hat{\\theta}_{s'},\\theta)} + \\sqrt{\\sum_{k\\in[K]}N_{s'}^k\\KL_k(\\hat{\\theta}_s,\\theta)}\n \\\\ &\\stackrel{(c)}{\\leq} \\sqrt{\\sum_{k\\in[K]}N_{s'}^k\\KL_k(\\hat{\\theta}_{s'},\\theta)} + \\sqrt{\\sum_{k\\in[K]}N_{s}^k\\KL_k(\\hat{\\theta}_s,\\theta)}\n \\\\ &\\stackrel{(d)}{=} \\sqrt{L_{s'}(\\hat{\\theta}_{s'},\\theta)} + \\sqrt{L_{s}(\\hat{\\theta}_{s},\\theta)} \\stackrel{(e)}{\\leq} 2\\sqrt{\\beta_{t,1\/t^2}},\n\\end{align*}\nwhere (a) is from Corollary~\\ref{cor:llr-lin-gauss}, (b) is from the triangle inequality (recall that the sum of KLs is a norm), (c) is from the fact that the pull counts are non-decreasing and $s\\geq s'$, (d) is again from Corollary~\\ref{cor:llr-lin-gauss}, and (e) is from event $E_t$. Using Lemma \\ref{lem:new_beta-diff-between-phases} as before, we have $2\\sqrt{\\beta_{{t},1\/{t}^2}} \\leq 2\\sqrt{\\frac{c_2}{c_1}\\beta_{s',1\/s'^2}} < \\sqrt{\\alpha_{s',\\delta}}$. This proves the second statement.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:change-inf-from-istar}\nUnder event $E_t$, for any $i\\in\\cI$ and $\\omega\\in\\Delta_K$,\n\\begin{align*}\n \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i^\\star)}\\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}  \\omega^k\\KL_k(\\theta,\\lambda)\n \\geq \\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i)}\\inf_{\\lambda \\in \\Lambda_p(i)} \\sum_{k\\in[K]} \\omega^k\\KL_k(\\theta,\\lambda).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe statement follows trivially if $i=i^\\star$. So suppose $i\\neq i^\\star$. Since $i$ is not the answer of $\\theta$, by the union property of the decomposition into pieces, there exists $p\\in\\cP(i)$ such that $\\theta\\in\\Lambda_p(i)$. By Lemma \\ref{lem:theta-not-elim-in-last-two-phases}, for any $\\overline{t}_{j(t)-1} \\leq s \\leq t$,\n\\begin{align*}\n\\inf_{\\lambda\\in\\Lambda_p(i)} L_{s}(\\hat{\\theta}_{s},\\lambda) \\leq L_{s}(\\hat{\\theta}_{s},\\theta) < \\alpha_{s,\\delta}.\n\\end{align*}\nThis implies that $p\\in\\cP_{t}^{\\mathrm{smp}}(i)$ since such set is defined as the intersection of all active sets from $\\overline{t}_{j(t)-1}$ to $t$. Finally, we conclude that\n\\begin{align*}\n\\min_{p\\in\\cP_{t}^{\\mathrm{smp}}(i)}\\inf_{\\lambda \\in \\Lambda_p(i)} \\sum_{k\\in[K]} \\omega^k\\KL_k(\\theta,\\lambda) \\leq \\sum_{k\\in[K]} \\omega^k\\KL_k(\\theta,\\theta) = 0,\n\\end{align*}\nand thus our result follows trivially.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:change-inf-from-ihat}\nUnder event $E_t$, for any $\\overline{t}_{j(t)-1}  \\leq s \\leq t$, $i\\in\\cI$, and $\\omega\\in\\Delta_K$,\n\\begin{align*}\n \\min_{p\\in\\cP_{s}^{\\mathrm{smp}}(i^\\star(\\hat{\\theta}_{s}))}\\inf_{\\lambda \\in \\Lambda_p(i^\\star(\\hat{\\theta}_{s}))}  \\sum_{k\\in[K]} \\omega^k \\KL_k(\\hat{\\theta}_{s},\\lambda)\n \\geq \\min_{p\\in\\cP_{s}^{\\mathrm{smp}}(i)}\\inf_{\\lambda \\in \\Lambda_p(i)} \\sum_{k\\in[K]} \\omega^k\\KL_k(\\hat{\\theta}_{s},\\lambda).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe proof is very similar to the one of Lemma \\ref{lem:change-inf-from-istar}. The statement follows trivially if $i=i^\\star(\\hat{\\theta}_{s})$. So suppose $i\\neq i^\\star(\\hat{\\theta}_{s})$. Since $i$ is not the answer of $\\hat{\\theta}_{s}$, by the union property of the decomposition into pieces, there exists $p\\in\\cP(i)$ such that $\\hat{\\theta}_{s}\\in\\Lambda_p(i)$. By Lemma \\ref{lem:theta-not-elim-in-last-two-phases}, for any $\\overline{t}_{j(t)-1} \\leq s'\\leq s \\leq t$,\n\\begin{align*}\n\\inf_{\\lambda\\in\\Lambda_p(i)} L_{s'}(\\hat{\\theta}_{s'},\\lambda) \\leq L_{s'}(\\hat{\\theta}_{s'},\\hat{\\theta}_s) < \\alpha_{s',\\delta}.\n\\end{align*}\nThis implies that $p\\in\\cP_{s}^{\\mathrm{smp}}(i)$ since such set is defined as the intersection of all active sets from $\\overline{t}_{j(t)-1}$ to $s$. Therefore, we conclude that\n\\begin{align*}\n\\min_{p\\in\\cP_{s}^{\\mathrm{smp}}(i)}\\inf_{\\lambda \\in \\Lambda_p(i)} \\sum_{k\\in[K]} \\omega^k\\KL_k(\\hat{\\theta}_{s},\\lambda) \\leq \\sum_{k\\in[K]} \\omega^k\\KL_k(\\hat{\\theta}_{s},\\hat{\\theta}_{s}) = 0,\n\\end{align*}\nand thus our result follows trivially.\n\\end{proof}\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nThe multi-armed bandit is a sequential decision-making task which is now extensively studied (see, e.g., \\citep{lattimore2020bandit} for a recent review). In this problem, an algorithm interacts with its environment by sequentially ``pulling'' one among $K \\in \\mathbb{N}$ arms and observing a sample from a corresponding distribution. Among the possible objectives, we focus on \\emph{fixed-confidence identification} \\citep{even2002pac, jamieson2014best,garivier2016optimal,chen2015optimal}. In this setting, the algorithm successively collects samples until it decides to stop and return an answer to a given query about the distributions. Its task is to return the correct answer with at most a given probability of error $\\delta$, and its secondary goal is to do so while stopping as early as possible. This problem is called ``fixed-confidence'' as opposed to ``fixed-budget'', where the goal is to minimize the error probability with at most a given number of samples \\citep{bubeck2009pure,audibert2010best,gabillon2012best,carpentier2016tight,abbasi2018best}.\n\n\nThe most studied query is \\emph{best arm identification} (BAI), where the aim is to return the arm whose distribution has highest mean. A variant is Top-m identification \\citep{kalyanakrishnan2012pac}, where the goal is to find the $m$ arms with highest means. While these are the most common, other queries have been studied, including thresholding bandits \\citep{carpentier2016tight}, minimum threshold \\citep{kaufmann2018sequential}, and multiple correct answers \\citep{degenne2019pure}.\n\nAlgorithms for fixed-confidence identification can be generally divided into two classes: those based on \\emph{adaptive sampling} and those based on \\emph{elimination}. Adaptive algorithms \\citep[e.g.,][]{gabillon2012best,kalyanakrishnan2012pac,garivier2016optimal,xu2018fully} update their sampling strategy at each round and typically stop when they can simultaneously assess the correctness of every answer. They often enjoy strong theoretical guarantees.\nFor instance, some of them \\citep{garivier2016optimal,degenne2019non,degenne2020gamification,wang2021fast} have been shown to be optimal as $\\delta\\rightarrow 0$. However, since they repeatedly test the correctness of every answer, they are often computationally demanding.\nElimination-based strategies \\citep[e.g.,][]{even2002pac,kaufmann2013information,soare2014best, fiez2019sequential,tao2018best} maintain a set of ``active'' answers (those that are still likely to be the correct one) and stop when only one remains. They typically update their sampling rules and\/or the active answers infrequently. This, together with the fact that eliminations reduce the problem size over time, makes them more computationally efficient but also yields large sample complexity in practice.\nMoreover, while adaptive algorithms for general identification problems (i.e., with arbitrary queries) exist \\citep{garivier2016optimal,degenne2019non,wang2021fast}, elimination-based strategies are not easy to design at such a level of generality. In particular, they are not easy to extend to combinatorial problems (such as Top-m), where the number of answers is exponential in the problem dimension.\n\nIn this paper, we design a novel elimination rule for general identification problems which we call \\emph{selective} elimination. It can be easily combined with existing adaptive strategies, both in their stopping and sampling rules, making them achieve the best properties of the two classes mentioned above.\nIn particular, we prove that (1) selective elimination never suffers worse sample complexity than the original algorithm, and hence remain asymptotically optimal whenever the base algorithm is; (2) It provably discards some answers much earlier than the stopping time; (3) It improves the computational complexity of the original algorithm when some answers are eliminated early. Experimentally, we compare several existing algorithms for three identification problems (BAI, Top-m, and thresholding bandits) on two bandit structures (linear and unstructured). We find that, coherently across all experiments, existing adaptive strategies achieve significant gains in computation time and, to a smaller extent, in sample complexity when combined with selective elimination.\n\n\\subsection{Bandit fixed-confidence identification}\n\\label{sub:bandit_identification}\n\nAn algorithm interacts with an environment composed of $K>1$ \\emph{arms}. At each time $t \\in \\mathbb{N}$, the algorithm picks an arm $k_t$ and observes $X_t^{k_t} \\sim \\nu_{k_t}$, where $\\nu_{k_t}$ is the distribution of arm $k_t$. \nAt a time $\\tau$, the algorithm stops and returns an answer $\\ihat$ from a finite set $\\mathcal I$. \nFormally, let $\\cF_t$ be the $\\sigma$-algebra generated by the observations up to time $t$. An identification algorithm is composed of\n\\begin{enumerate}[nosep]\n\t\\item \\emph{Sampling rule}: the sequence $(k_t)_{t \\in \\mathbb{N}}$, where $k_t$ is $\\mathcal F_{t-1}$-measurable.\n\t\\item \\emph{Stopping rule}: a stopping time $\\tau$ with respect to $(\\cF_t)_{t\\in\\mathbb{N}}$ and a random variable $\\ihat \\in \\mathcal I$, i.e., the answer returned when stopping at time $\\tau$.\n\\end{enumerate}\nNote that, while it is common to decouple $\\tau$ and $\\ihat$, we group them to emphasize that the time at which an algorithm stops depends strongly on the answer it plans on returning.\n\nWe assume that the arm distributions depend on some unknown parameter $\\theta\\in\\cM$, where $\\cM \\subseteq \\mathbb{R}^d$ is the set of possible parameters, and write $\\nu_k(\\theta)$ for $k\\in[K]$ to make this dependence explicit. For simplicity, we shall use $\\theta$ to refer to the bandit problem $(\\nu_k(\\theta))_{k\\in[K]}$.\nThis assumption allows us to include linear bandits in our analysis.\nWe let $i^\\star: \\cM \\to \\mathcal I$ be the function, known to the algorithm, which returns the unique correct answer for each problem. The algorithm is correct on $\\theta$ if $\\ihat = i^\\star(\\theta)$. \n\n\\begin{definition}[$\\delta$-correct algorithm]\nAn algorithm is said to be $\\delta$-correct on $\\mathcal M \\subseteq \\mathbb{R}^d$ if for all $\\theta \\in \\mathcal M$, $\\tau < +\\infty$ almost surely and\n$\n\\mathbb{P}_\\theta(\\ihat \\ne i^\\star(\\theta) )\n\\le \\delta \\: .\n$\n\\end{definition}\n\nWe want to design algorithms that, given a value $\\delta$, are $\\delta$-correct on $\\mathcal M$ and have minimal expected sample complexity $\\mathbb{E}_\\theta[\\tau]$ for all $\\theta\\in\\cM$. A lower bound on $\\mathbb{E}_\\theta[\\tau]$ was proved in \\citep{garivier2016optimal}. In order to present it, we introduce the concept of \\emph{alternative} set to an answer $i\\in\\cI$: $\\Lambda(i) := \\{\\lambda \\in \\mathcal M \\mid i^\\star(\\lambda) \\ne i\\}$, the set of parameters for which the correct answer is not $i$. Let us denote by $\\KL_k(\\theta,\\lambda)$ the Kullback-Leibler (KL) divergence between the distribution of arm $k$ under $\\theta$ and $\\lambda$.\nThen the lower bound states that for any algorithm that is $\\delta$-correct on $\\cM$ and any problem $\\theta\\in\\cM$,\n\\begin{align}\n\\mathbb{E}_\\theta[\\tau]\n&\\ge \\log(1\/(2.4\\delta))\/H^\\star(\\theta)\n\\: ,\n\\text{with }\nH^\\star(\\theta)\n:= \\max_{\\omega\\in\\Delta_K}\\inf_{\\lambda \\in \\Lambda(i^\\star(\\theta))} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)\n\\: .\\label{eq:lower_bound}\n\\end{align}\n\n\n\\paragraph{Example: BAI in Gaussian linear bandits}\n\nWhile our results apply to general queries, we illustrate all statements of this paper on the widely-studied task of BAI in Gaussian linear bandits \\citep{soare2014best,xu2018fully,zaki2020explicit,degenne2020gamification,jedra2020optimal}. \nIn this setting, each arm $k\\in[K]$ has a Gaussian distribution $\\mathcal N(\\mu_k(\\theta), 1)$ with mean $\\mu_k(\\theta) = \\phi_k^\\top \\theta$, a linear function of the unknown parameter $\\theta \\in \\mathbb{R}^d$ (and $\\mathcal M = \\mathbb{R}^d$) and of known arm features $\\phi_k\\in\\mathbb{R}^d$. The set of answers is $\\mathcal I = [K]$ and the correct answer is $i^\\star(\\theta) := \\argmax_{k\\in[K]}\\phi_k^\\top \\theta$.\n\nFinally, for $x \\in \\mathbb{R}^d$ and $A \\in \\mathbb{R}^{d \\times d}$, we define $\\Vert x \\Vert_A := \\sqrt{x^\\top A x}$. For $\\omega \\in \\mathbb{R}^K$, let $V_\\omega := \\sum_{k=1}^K \\omega^k \\phi_k \\phi_k^\\top$. With this notation, we have $\\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda) = \\frac{1}{2}\\Vert \\theta - \\lambda \\Vert_{V_\\omega}^2$.\n\n\\subsection{Log-likelihood ratio stopping rules}\n\\label{sub:stopping_rules}\n\nMost existing adaptive algorithms use a log-likelihood ratio (LLR) test in order to decide when to stop. Informally, they check whether sufficient information has been collected to confidently discard at once all answers except one. Since such LLR tests are crucial for the design of our general elimination rules, we now describe their principle.\n \nGiven two parameters $\\theta,\\lambda \\in \\cM$, the LLR of observations $X_{[t]} = (X_1^{k_1}, \\ldots, X_t^{k_t})$ between models $\\theta$ and $\\lambda$ is\n$\nL_t(\\theta, \\lambda)\n:= \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_{[t]})\n= \\sum_{s=1}^t \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s})\n$~. Let $\\hat{\\theta}_t := \\argmax_{\\lambda\\in\\cM} \\log  \\mathbb{P}_\\lambda(X_{[t]})$ be the maximum likelihood estimator of $\\theta$ from $t$ observations. In Gaussian linear bandits, we have\n$\nL_t(\\theta, \\lambda)\n= \\frac{1}{2}\\Vert \\theta - \\lambda \\Vert^2_{V_{N_t}} + (\\theta - \\lambda)^\\top V_{N_t} (\\hat{\\theta}_t - \\theta)\n$~, where $N_t^k := \\sum_{s=1}^t \\indi{k_s=k}$.\nSee Appendix~\\ref{sec:exponential_families} for more details. $L_t(\\theta, \\lambda)$ is closely related to $\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)$, a quantity that appears frequently in our results. Indeed, the difference between these quantities is a martingale, which is a lower order term compared to them. The LLR stopping rule was introduced to the bandit literature in \\citep{garivier2016optimal}. At each step $t \\in \\mathbb{N}$, the algorithm computes the infimum LLR to the alternative set of $i^\\star(\\hat{\\theta}_t)$ and stops if it exceeds a threshold, i.e., if\n\\begin{align}\\label{eq:llr-stop}\n\\inf_{\\lambda \\in \\Lambda(i^\\star(\\hat{\\theta}_t))} L_t(\\hat{\\theta}_t, \\lambda)\n\\ge \\beta_{t, \\delta} \\: ,\n\\end{align}\nwhere the function $\\beta_{t, \\delta}$ can vary, notably based on the shape of the alternative sets. The recommendation rule is then $\\ihat = i^\\star(\\hat{\\theta}_t)$. Informally, the algorithm stops if it has enough information to exclude all points $\\lambda$ for which the answer is not $i^\\star(\\hat{\\theta}_t)$.\nThis stopping rule enforces $\\delta$-correctness, provided that the sampling rule ensures $\\tau < + \\infty$ a.s. and that $\\beta_{t,\\delta}$ is properly chosen. The most popular choice is to ensure a concentration property of $L_t(\\hat{\\theta}_t, \\theta)$. For example, if for all $\\delta$, $\\beta_{t,\\delta}$ guarantees that\n\\begin{align}\\label{eq:concentration-beta}\n\\mathbb{P}\\left( \\exists t \\geq 1 : L_t(\\hat{\\theta}_t, \\theta) \\geq \\beta_{t,\\delta}\\right) \\leq \\delta,\n\\end{align}\nLLR stopping with that threshold returns a wrong answer with probability at most $\\delta$. Such concentration bounds can be found in \\citep{abbasi2011improved,magureanu2014lipschitz} for linear and unstructured bandits, respectively. This LLR stopping rule is used in many algorithms \\citep{garivier2016optimal,xu2018fully,degenne2019non,degenne2020gamification,jedra2020optimal,wang2021fast}\\footnote{LinGapE \\citep{xu2018fully} does not use LLR stopping explicitly, but its stopping rule is equivalent to it. We can write it as: stop if for all points inside a confidence region a gap is small enough, that is if all those points do not belong to the alternative of $i^\\star(\\hat{\\theta}_t)$. The contrapositive of that statement is exactly LLR stopping.}.\nSome of them have been proven\nto be \\emph{asymptotically optimal}: their sample complexity upper bound matches the lower bound~\\eqref{eq:lower_bound} when $\\delta \\to 0$. However, improvements are still possible: their sample complexity for moderate $\\delta$ may not be optimal and their computational complexity may be reduced, as we will see.\n\n\n\n\n\n\n\\iffalse\n\\subsection{Related Work}\n\\label{sub:related_work}\n\n\\todo[inline]{This section is not needed and the various references should be inserted into the introduction}\n\nFB = fixed budget, FC = fixed confidence\n\n\\citep{even2002pac} BAI, FC (high proba stopping time), elimination: Successive Elimination, Median Elimination\n\n\\citep{mannor2004sample}\n\n\\citep{even2006action} same as the 2002 paper?\n\n\\citep{audibert2010best} BAI, FB, no elimination in UCB-E, elimination in Successive Rejects\n\n\\citep{gabillon2012best} BAI, FB and FC, no elimination (UGapE), kind of LLR stopping\n\n\\citep{garivier2016optimal} BAI, FC, no elimination (LLR)\n\n\\citep{russo2016simple}\n\n\\citep{kaufmann2016complexity}\n\n\\citep{soare2014best}\n\n\\citep{jamieson2014best} BAI, FC, elimination in some algorithms\n\n\\citep{chen2017towards} BAI, FC, lower bound, elimination algorithm\n\n\\citep{chen2015optimal} BAI, FC, lower bound, elimination algorithm\n\n\\citep{chen2014combinatorial} top-M BAI, FC (high proba bound on stopping time?), no elimination\n\n\\citep{tao2018best}\n\n\\citep{huang2017structured} BAI, FC, concentration for a better LLR threshold\n\n\\citep{jedra2020optimal}\n\n\\citep{abbasi2018best} BAI, FB, stochastic and adversarial, no elimination.\n\n\\citep{carpentier2016tight} BAI, FB, lower bounds\n\n\\citep{karnin2013almost}\n\n\\citep{locatelli2016optimal}\n\n\\citep{cheshire2021problem} Thresholding, FB, no elimination.\n\n\\citep{bubeck2009pure} BAI, simple regret, FB with unknown time, no elimination.\n\n\\citep{bubeck2011pure} Same as the 2009 paper.\n\n\\citep{degenne2019pure} Exploration, FC, no elimination (LLR)\n\n\\citep{degenne2019non} Exploration, FC, no elimination (LLR)\n\n\\citep{xu2018fully}\n\n\\citep{kalyanakrishnan2012pac}\n\n\\citep{kaufmann2017monte}\n\n\\citep{kaufmann2018sequential}\n\n\\citep{kaufmann2018mixture}\n\\fi\n\\section{Deleting Pieces}\n\\label{sec:deleting_pieces}\n\nThe main question we answer in this section is ``what happens to the set of optimal allocations when we delete a piece of the alternative?''.\n\nFor a set $\\Lambda$, let $\\Omega(\\Lambda) = \\argmax_{\\omega \\in \\triangle_K} \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\lambda)$. $\\Omega(\\Lambda)$ is a convex and closed set for any set $\\Lambda$.\n\n\\begin{lemma}\\label{lem:optimal_alloc_restrict_alternative_subset}\nIf $\\Lambda' \\subseteq \\Lambda$ and for some $\\varepsilon > 0$, $\\{\\lambda \\in \\Lambda \\mid \\exists \\omega \\in \\Omega(\\Lambda), \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\lambda) \\le \\inf_{\\eta \\in \\Lambda} \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\eta) + \\varepsilon\\} \\subseteq \\Lambda'$, then $\\Omega(\\Lambda) \\subseteq \\Omega(\\Lambda')$. \n\\end{lemma}\n\\begin{proof}\nTODO: we suppose for now that for all $\\omega \\in \\Omega(\\Lambda)$ there exists a distribution $q_\\omega$ over points in $\\Lambda$ which belongs to a Nash equilibrium.\n\nThose $q_\\omega$ distributions are also distributions over $\\Lambda'$ since they are supported on points in $\\{\\lambda \\in \\Lambda \\mid \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\lambda) = \\inf_{\\eta \\in \\Lambda} \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\eta)\\}$. Then for any $\\omega \\in \\triangle_K$ and $\\omega^* \\in \\Omega(\\Lambda)$ we have \n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda'} \\sum_k \\omega^{k} \\KL_k(\\mu, \\lambda)\n&\\le \\mathbb{E}_{\\lambda \\sim q_{\\omega^*}}\\sum_k \\omega^k \\KL_k(\\mu, \\lambda)\n\\\\\n&\\stackrel{(a)}{\\le} \\mathbb{E}_{\\lambda \\sim q_{\\omega^*}}\\sum_k \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\stackrel{(b)}{=} \\inf_{\\lambda \\in \\Lambda} \\sum_k \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\stackrel{(c)}{\\le} \\inf_{\\lambda \\in \\Lambda'} \\sum_k \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\: .\n\\end{align*}\n$(a)$ and $(b)$ are consequences of fact that $(\\omega^*, q_{\\omega^*})$ is a Nash equilibrium. $(c)$ comes from $\\Lambda' \\subseteq \\Lambda$.\n\nWe proved that the value of an allocation $\\omega^* \\in \\Omega(\\Lambda)$ is at least as good as the value of any other allocation. Hence $\\Omega(\\Lambda) \\subseteq \\Omega(\\Lambda')$.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:restricting_does_not_change_optimal_allocations}\nLet $\\varepsilon > 0$ and let $\\Lambda_{\\varepsilon} \\subseteq \\Lambda$ be such that for all $\\lambda_\\varepsilon \\in \\Lambda_{\\varepsilon}$ and all $\\omega \\in \\Omega(\\Lambda)$, $\\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\lambda_\\varepsilon) \\ge \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^k \\KL_k(\\mu, \\lambda) + \\varepsilon$. Then $\\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon) = \\Omega(\\Lambda)$. \n\\end{theorem}\n\\begin{proof}\nWe have the inclusion $\\Omega(\\Lambda) \\subseteq \\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon)$ by Lemma~\\ref{lem:optimal_alloc_restrict_alternative_subset}.\n\nNow suppose that there exists $\\omega' \\in \\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon) \\setminus \\Omega(\\Lambda)$. Let $\\omega^* \\in \\Omega(\\Lambda)$ be such that for all $r \\in (0,1]$, $\\omega_r := r \\omega' + (1 - r) \\omega^* \\notin \\Omega(\\Lambda)$. We can get such an $\\omega^*$ by taking an arbitrary $\\omega^*_0 \\in \\Omega(\\Lambda)$ and defining $\\omega^* = r^* \\omega' + (1 - r^*)\\omega_0^*$, where $r^* = \\sup \\{r \\in [0,1] \\mid r \\omega' + (1 - r)\\omega_0^* \\in \\Omega(\\Lambda)\\}$. We know that $r^* < 1$ since $\\Omega(\\Lambda)$ is closed.\n\nBy convexity of $\\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon)$ and since both $\\omega'$ and $\\omega^*$ belong to that set, we have $\\omega_r \\in \\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon)$ for all $r \\in [0,1]$.\n\nSet $r \\in (0,1]$. Since $\\omega_r \\notin \\Omega(\\Lambda)$, there exists $\\lambda_r \\in \\Lambda$ such that $\\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda_r) < \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)$. It now suffices to show that for some $r \\in (0,1]$, we have such a $\\lambda_r$ with $\\lambda_r \\notin \\Lambda_\\varepsilon$. Indeed, under that condition we have that\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda \\setminus \\Lambda_\\varepsilon} \\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda)\n\\stackrel{(a)}{\\le} \\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda_r)\n&\\stackrel{(b)}{<} \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\stackrel{(c)}{=} \\inf_{\\lambda \\in \\Lambda \\setminus \\Lambda_\\varepsilon} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\: .\n\\end{align*}\nInequality $(a)$ is due to the hypothesis $\\lambda_r \\in \\Lambda \\setminus \\Lambda_\\varepsilon$, $(b)$ is the definition of $\\lambda_r$ and $(c)$ comes from the fact that minimizers over $\\Lambda$ can't belong to $\\Lambda_\\varepsilon$ by definition of $\\Lambda_\\varepsilon$.\nWe then conclude that $\\omega_r$ does not belong to $\\Omega(\\Lambda \\setminus \\Lambda_\\varepsilon)$, which is a contradiction.\n\nLet us now prove that there exists an $r \\in (0,1]$ for which there exists $\\lambda_r \\in \\Lambda \\setminus \\Lambda_\\varepsilon$ with $\\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda_r) < \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)$.\n\nIf there is no such point, then since $\\omega_r \\notin \\Omega(\\Lambda)$ there exists $\\lambda_{r,\\varepsilon} \\in \\Lambda_\\varepsilon$ such that $\\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda_{r, \\varepsilon}) < \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)$.\nBut if $r$ is small enough we have\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)\n> \\sum_{k = 1}^K \\omega_r^k \\KL_k(\\mu, \\lambda_{r, \\varepsilon})\n&\\ge (1 - r)\\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda_{r, \\varepsilon})\n\\\\\n&\\ge (1 - r)\\left( \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda) + \\varepsilon \\right)\n\\\\\n&\\ge \\inf_{\\lambda \\in \\Lambda} \\sum_{k = 1}^K \\omega^{*k} \\KL_k(\\mu, \\lambda)\n\\: ,\n\\end{align*}\nwhich is a contradiction.\n\n\\end{proof}\n\nRemark: the hypothesis $\\varepsilon > 0$ is necessary in Theorem~\\ref{thm:restricting_does_not_change_optimal_allocations}. Indeed consider the unstructured Gaussian BAI problem in two dimensions in which $\\mu = (0,2)$.\nThen $\\Lambda = \\{\\lambda \\in \\mathbb{R}^2 \\mid \\lambda_1 \\ge \\lambda_2 \\}$ and the optimal allocation set is $\\Omega(\\Lambda) = \\{(1\/2, 1\/2)\\}$.\nThe only point with minimal value for that allocation is $\\lambda_0 = (1,1)$.\nConsider now $\\Lambda_0 = \\{\\lambda_0\\}$. Then $\\Omega(\\Lambda_0) = \\triangle_2 \\ne \\Omega(\\Lambda)$. In that restriction, we removed points arbitrarily close to the infimum value and Theorem~\\ref{thm:restricting_does_not_change_optimal_allocations} does not apply.\n\n\\section{Log-likelihood ratio in exponential families}\n\\label{sec:exponential_families}\n\n\nWe suppose in this section that all arms have distributions in a one-parameter exponential family (the same for all arms, for simpler notations). An arm distribution can thus be described by any one of three parameters: the arm feature vector $\\phi_k \\in \\mathbb{R}^d$, the arm mean $\\mu_k(\\theta) = \\phi_k^\\top \\theta$ and its natural parameter $\\eta_k(\\theta)$. These two last are functions of the model $\\theta$.\nFor two models $\\theta$ and $\\lambda$, let $f$ be a function such that the KL between the arm distributions with those parameters is $d_f(\\eta_k(\\lambda), \\eta_k(\\theta))$, where $d_f$ is the Bregman divergence associated to $f$. Let $f^*$ be the convex conjugate of $f$. The Kullback-Leibler divergence between the arm distributions under models $\\theta$ and $\\lambda$ is also equal to $d_{f^*}(\\mu_k(\\theta), \\mu_k(\\lambda))$.\n\nIf $\\eta_k(\\theta)$ is the natural parameter of that arm $k$, we have $\\mu_k(\\theta) = f'(\\eta_k(\\theta))$, and since $(f^*)' = (f')^{-1}$ we have $\\eta_k(\\theta) = (f^*)'(\\mu_k(\\theta))$.\n\n\\begin{lemma}\nFor all $\\theta, \\lambda \\in \\mathcal M$, the quantity $L_t(\\theta, \\lambda) - \\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)$ is a martingale if the observations come from the model $\\theta$. This does not depend on the hypothesis that the distributions belong to an exponential family but only requires $\\mathbb{P}_\\theta \\ll \\mathbb{P}_\\lambda$.\n\\end{lemma}\n\\begin{proof}\nWe can expand the LLR to obtain a sum over times and write the KL as an expected log-likelihood ratio,\n\\begin{align*}\nL_t(\\theta, \\lambda) - \\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)\n&= \\sum_{s=1}^t \\left(\\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_s^{k_s}) - \\mathbb{E}_{X \\sim \\nu_{k_s}(\\theta)}\\left[ \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X) \\right] \\right)\n\\: .\n\\end{align*}\nThe martingale property is then immediate.\n\\end{proof}\n\nMany of our proofs depend on the informal statement that the martingale $L_t(\\theta, \\lambda) - \\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)$ concentrates, and is thus a lower order term which is negligible for $t$ large enough.\n\n\\begin{lemma}\\label{lem:LLR_exp_family}\nFor all $\\theta, \\lambda$ and all $X_{[t]}$,\n\\begin{align*}\nL_t(\\theta, \\lambda)\n&= \\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda) - \\sum_{k=1}^K N_t^k (\\eta_k(\\lambda) - \\eta_k(\\theta)) (\\hat{\\mu}_{t,k} - \\mu_k(\\theta))\n\\: .\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nWrite $\\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X) = \\eta(\\theta) X - f(\\eta(\\theta)) - (\\eta(\\lambda) X - f(\\eta(\\lambda)))$ and develop the Bregman divergence (the KL) on the right.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:sub_gaussian_KL_bounds}\nIf the distribution of arm $k$ for model $\\theta$ is $\\sigma^2$-sub-Gaussian, then for all $\\lambda$,\n\\begin{align*}\n\\frac{1}{2 \\sigma^2} (\\mu_k(\\lambda) - \\mu_k(\\theta))^2\n&\\le \\KL_k(\\lambda, \\theta)\n\\: , \\\\\n\\frac{1}{2 \\sigma^2} \\sum_{k=1}^K N_t^k (\\mu_k(\\hat{\\theta}_t) - \\mu_k(\\theta))^2\n&\\le L_t(\\hat{\\theta}_t, \\theta)\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe first prove that the sub-Gaussian hypothesis is equivalent to both these inequalities:\n\\begin{align*}\n\\forall \\lambda, d_f(\\eta_k(\\lambda), \\eta_k(\\theta)) \\le \\frac{1}{2}\\sigma^2 (\\eta_k(\\lambda) - \\eta_k(\\theta))^2\n\\: , \\\\\n\\forall \\lambda, d_{f^*}(\\mu_k(\\lambda), \\mu_k(\\theta)) \\ge \\frac{1}{2 \\sigma^2} (\\mu_k(\\lambda) - \\mu_k(\\theta))^2\n\\: .\n\\end{align*}\nThe first result is then a simple consequence of that second inequality and the equality $\\KL(\\lambda, \\theta) = d_{f^*}(\\mu_k(\\lambda), \\mu_k(\\theta))$. The second result of the lemma can be obtained by applying the first one to $\\lambda = \\hat{\\theta}_t$ for all arms, then summing over arms.\n\nThe cumulant generating function at parameter $\\eta_k(\\theta)$ is $\\xi \\mapsto d_f(\\eta_k(\\theta) + \\xi, \\eta_k(\\theta))$. The sub-Gaussian hypothesis is that this function is lower than $\\frac{1}{2}\\sigma^2 \\xi^2$. For the second inequality, we first remark that the convex conjugate of $\\xi \\mapsto d_f(\\eta_k(\\theta) + \\xi, \\eta_k(\\theta))$ is $x \\mapsto d_{f^*}(\\mu_k(\\theta) + x, \\mu_k(\\theta))$, and write\n\\begin{align*}\nd_{f^*}(\\mu_k(\\lambda), \\mu_k(\\theta))\n&= \\sup_\\xi \\xi (\\mu_k(\\lambda) - \\mu_k(\\theta)) - d_f(\\eta_k(\\theta) + \\xi, \\eta_k(\\theta))\n\\\\\n&\\ge \\sup_\\xi \\xi (\\mu_k(\\lambda) - \\mu_k(\\theta)) - \\frac{1}{2}\\sigma^2 \\xi^2\n\\\\\n&= \\frac{1}{2 \\sigma^2} (\\mu_k(\\lambda) - \\mu_k(\\theta))^2 \\: .\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\\label{cor:martingale_bound_exp_family}\nIf the distribution of arm $k$ for model $\\theta$ is $\\sigma^2$-sub-Gaussian, then for all $\\lambda$,\n\\begin{align*}\n\\left\\vert \\sum_{k=1}^K N_t^k (\\eta_k(\\lambda) - \\eta_k(\\theta)) (\\mu_k(\\hat{\\theta}_t) - \\mu_k(\\theta)) \\right\\vert\n&\\le 2 \\sqrt{L_t(\\hat{\\theta}_t, \\theta)} \\sqrt{\\sum_{k=1}^K N_t^k \\frac{1}{2} \\sigma^2 (\\eta_k(\\lambda) - \\eta_k(\\theta))^2 }\n\\: .\n\\end{align*}\n\\end{corollary}\n\nRemark: for Gaussians with variance $\\sigma^2$, we also have $\\sum_{k=1}^K N_t^k \\frac{1}{2} \\sigma^2 (\\eta_k(\\lambda) - \\eta_k(\\theta))^2 = \\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)$. But that is not the case in general, and the sub-Gaussian assumption tells us that the sum of squares is larger than a KL, while we would like the reverse inequality.\n\n\\begin{proof}\nApply the Cauchy-Schwarz inequality, then Lemma~\\ref{lem:sub_gaussian_KL_bounds}:\n\\begin{align*}\n\\left\\vert \\sum_{k=1}^K N_t^k (\\eta_k(\\lambda) - \\eta_k(\\theta)) (\\mu_k(\\hat{\\theta}_t) - \\mu_k(\\theta)) \\right\\vert\n&\\le \\sqrt{ \\sum_{k=1}^K N_t^k (\\eta_k(\\lambda) - \\eta_k(\\theta))^2 \\sum_{k=1}^K N_t^k (\\mu_k(\\hat{\\theta}_t)- \\mu_k(\\theta))^2 }\n\\\\\n&\\le 2 \\sqrt{L_t(\\hat{\\theta}_t, \\theta)} \\sqrt{\\sum_{k=1}^K N_t^k \\frac{1}{2} \\sigma^2 (\\eta_k(\\lambda) - \\eta_k(\\theta))^2 }\n\\: .\n\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Log-likelihood ratio for Gaussian linear models}\n\\label{sec:log_likelihood_ratio}\n\n\\begin{lemma}\\label{lem:llr-lin-gauss}\nFor any $\\theta\\in\\cM$ and $k\\in[K]$, let $\\nu_k(\\theta)$ be Gaussian with unit variance and linear mean $\\mu_k(\\theta) = \\theta^T\\phi_k$. Then for any $\\theta,\\lambda\\in\\cM$, any $t>0$ and any sequence of observations,\n\\begin{align*}\nL_t(\\theta,\\lambda) &= \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) - (\\lambda - \\theta)^\\top V_t (\\hat{\\theta}_t - \\theta)\n\\: .\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nApply Lemma~\\ref{lem:LLR_exp_family} to the Gaussian case, where $\\eta(\\theta) = \\mu(\\theta) = \\phi_k^\\top \\theta$.\n\\begin{align*}\nL_t({\\theta},\\lambda)\n&= \\sum_{k\\in[K]} N_t^k\\KL_k(\\theta, \\lambda) - \\sum_{k=1}^K N_t^k (\\phi_k^\\top\\lambda - \\phi_k^\\top\\theta) (\\hat{\\mu}_{t,k} - \\phi_k^\\top \\theta)\n\\end{align*}\nFor Gaussian linear models, we have\n\\begin{align*}\n\\sum_{k=1}^K N_t^k (\\phi_k^\\top\\lambda - \\phi_k^\\top\\theta) \\phi_k^\\top \\hat{\\theta}_t\n&= (\\lambda - \\theta)^\\top V_{N_t} \\hat{\\theta}_t\n= (\\lambda - \\theta)^\\top \\sum_{k=1}^K N_t^k \\hat{\\mu}_{t,k} \\phi_k\n= \\sum_{k=1}^K N_t^k (\\phi_k^\\top \\lambda - \\phi_k^\\top \\theta) \\hat{\\mu}_{t,k}\n\\: .\n\\end{align*}\nWe can use this to replace the sum involving $\\hat{\\mu}_{t,k}$ in the expression of $L_t(\\theta, \\lambda)$ by one involving $\\hat{\\theta}_t$.\n\\begin{align*}\nL_t({\\theta},\\lambda)\n&= \\sum_{k\\in[K]} N_t^k\\KL_k(\\theta, \\lambda) - (\\lambda - \\theta)^\\top (\\sum_{k=1}^K N_t^k \\phi_k \\phi_k^\\top) (\\hat{\\theta}_{t} - \\theta)\n\\\\\n&= \\sum_{k\\in[K]} N_t^k\\KL_k(\\theta, \\lambda) - (\\lambda - \\theta)^\\top V_t (\\hat{\\theta}_{t} - \\theta)\n\\: .\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary}\\label{cor:llr-lin-gauss}\nFor the linear Gaussian model of Lemma \\ref{lem:llr-lin-gauss}, for any $\\lambda\\in\\cM$,\n\\begin{align*}\nL_t(\\hat{\\theta}_t,\\lambda) = \\sum_{k\\in[K]}N_t^k\\KL_k(\\hat{\\theta}_t,\\lambda).\n\\end{align*}\n\\end{corollary}\n\n\\begin{lemma}\\label{lem:llr-to-kl-lin-gauss}\nFor any $\\theta\\in\\cM$ and $k\\in[K]$, let $\\nu_k(\\theta)$ be Gaussian with unit variance and linear mean $\\mu_k(\\theta) = \\theta^T\\phi_k$. Then, for any $\\lambda\\in\\cM$ and $t>0$,\n\\begin{align*}\n\\left(\\sqrt{\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} - \\sqrt{L_t(\\hat{\\theta}_t,\\theta)}\\right)^2 \\leq L_t(\\hat{\\theta}_t,\\lambda) \\leq \\left(\\sqrt{\\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda)} + \\sqrt{L_t(\\hat{\\theta}_t,\\theta)}\\right)^2.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe decompose the LLR as\n\\begin{align*}\nL_t(\\hat{\\theta}_t,\\lambda)\n&= L_t(\\theta, \\lambda) + L_t(\\hat{\\theta}_t, \\theta)\n\\\\\n&= \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) - (\\lambda - \\theta)^\\top V_t (\\hat{\\theta}_t - \\theta) + L_t(\\hat{\\theta}_t, \\theta)\n\\end{align*}\nThe second term is bounded by Corollary~\\ref{cor:martingale_bound_exp_family} (with $\\sigma^2 = 1$) and the remark below it. We get\n\\begin{align*}\nL_t(\\hat{\\theta}_t, \\lambda)\n&\\le \\sum_{k\\in[K]}N_t^k\\KL_k(\\theta,\\lambda) + 2 \\sqrt{L_t(\\hat{\\theta}_t, \\theta)} \\sqrt{\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda) } + L_t(\\hat{\\theta}_t, \\theta)\n\\\\\n&= \\left( \\sqrt{\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda) } + \\sqrt{L_t(\\hat{\\theta}_t, \\theta)} \\right)^2\n\\: .\n\\end{align*}\n\nThe proof of the lower bound is similar.\n\\end{proof}\n\n\\subsection{Beyond Gaussians}\n\\label{sub:beyond_gaussians}\n\nWhile the proofs in the next two sections are specialized to Gaussian rewards, it is possible to extend them to more exponential families under slight assumptions, similarly to what was done in \\citep{degenne2019non}. If the arm distributions are known to belong to a $\\sigma^2$-sub-Gaussian exponential family, with the additional restriction that the distribution parameters should belong to a compact subset of the open interval on which the family is defined, then there exists a constant $c$ such that \n\\begin{align*}\n\\frac{1}{\\sigma^2}\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)\n\\le \\sum_{k=1}^K N_t^k \\frac{1}{2} (\\eta_k(\\lambda) - \\eta_k(\\theta))^2\n\\le c\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda) \\: .\n\\end{align*}\nAnd $\\sum_{k=1}^K N_t^k \\KL_k(\\theta, \\lambda)$ is also close to $\\Vert \\lambda - \\theta \\Vert_{V_t}^2$, up to constant factors.\n\nWe can then recover bounds on the likelihood ratio of the same shape as in Lemma~\\ref{lem:llr-to-kl-lin-gauss}, up to constant factors depending on $c$ and $\\sigma^2$. The proofs of Appendix~\\ref{app:proofs-elim-stopping} and Appendix~\\ref{app:proofs_of_sampling} then proceed similarly, up to the additional constants.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\begin{figure*}[t]\n\\centering\n\t\\includegraphics[scale=0.21]{figures\/lin_BAI_K50_d20_elim.pdf}\n\t\\includegraphics[scale=0.21]{figures\/lin_BAI_K50_d10_full_vs_emp.pdf}\n\t\\includegraphics[scale=0.21]{figures\/lin_BAI_K50_d10_delta.pdf}\n\t\\caption{Experiments on linear instances with $K=50$, $d=10$, averaged over 100 runs, with the right plot showing standard deviations. (left) How different adaptive algorithms eliminate arms in BAI when using elimination stopping. (middle) LinGame on BAI when combined with full and selective elimination rules, either only at stopping or both at stopping and at sampling. (right) Ratio between the LLR and elimination stopping times of different algorithms as a function of $\\log(1\/\\delta)$.}\n\t\\label{fig:all}\n\\end{figure*}\n\nOur experiments aim at addressing the following questions: (1) how do existing adaptive strategies behave when combined with elimination at stopping and (when possible) at sampling? How do they compare with native elimination-based methods? (2) What is the difference between selective and full elimination? (3) How do LLR and elimination stopping compare as a function of $\\delta$?\\footnote{Our code is available at \\url{https:\/\/github.com\/AndreaTirinzoni\/bandit-elimination}.}\n\n\n\nWe ran experiments on two bandit structures: linear (where $d < K$) and unstructured (where $K=d$ and the arms are the canonical basis of $\\mathbb{R}^d$). For each of them, we considered 3 pure exploration problems: BAI, Top-m, and online sign identification (OSI) \\citep{carpentier2016tight,ouhamma2021online}, also called thresholding bandits. All experiments use $\\delta=0.01$ and are averaged over 100 runs.\n\nWe combined adaptive algorithms which are natively based on LLR stopping with our elimination stopping rules and, whenever possible, we extended their sampling rule to use elimination. The selected baselines are the following. For linear BAI, LinGapE \\citep{xu2018fully}, LinGame \\citep{degenne2020gamification}, Frank-Wolfe Sampling (FWS) \\citep{wang2021fast}, Lazy Track-and-Stop (TaS) \\citep{jedra2020optimal}, XY-Adaptive \\citep{soare2014best}, and RAGE \\citep{fiez2019sequential} (the latter two are natively elimination based). For linear Top-m, m-LinGapE \\citep{reda2021top}, MisLid \\citep{reda2021dealing}, FWS, Lazy TaS\\footnote{Lazy TaS, while analyzed only for BAI, can be applied to any problem since it is a variant of Track-and-Stop.}, and LinGIFA \\citep{reda2021top}. For linear OSI, LinGapE\\footnote{LinGapE was originally proposed only for BAI in \\citep{xu2018fully}, but its extension to OSI is trivial.}, LinGame, FWS, and Lazy TaS.\nFor unstructured instances linear algorithms are still applicable, and we further implemented LUCB \\citep{kalyanakrishnan2012pac}, UGapE \\citep{gabillon2012best}, and the Racing algorithm \\citep{kaufmann2013information} for BAI and Top-m. We also tested an ``oracle'' sampling rule which uses the optimal proportions from the lower bound. Due to space constraints, we present only the results on linear structures. Those on unstructured problems can be found in Appendix~\\ref{app:experiments}. \nThe first experiments use randomly generated instances with $K=50$ arms and dimension $d=10$\n\n\\textbf{Comparison of elimination times.} We analyze how different adaptive algorithms eliminate pieces when combined with selective elimination at stopping. To this purpose we focus on BAI, where the sets of pieces can be conveniently reduced to a set of active arms, those that are still likely to be the optimal one.\nFigure \\ref{fig:all}\\emph{(left)} shows how the set of active arms evolves over time for the 5 adaptive baselines.\nNotably, many arms are eliminated very quickly, with most baselines able to halve the set of active arms in the first 3000 steps.\nThe problem size is quickly reduced over time. As we shall see in the last experiment, this will yield significant computational gains.\nWe further note that the ``oracle'' strategy, which plays fixed proportions, seems the slowest at eliminating arms.\nThe reason is that the optimal proportions from the lower bound focus on discriminating the ``hardest'' arms, while the extra randomization in adaptive rules might indeed eliminate certain ``easier'' arms sooner.\n\n\\begin{table*}[t!]\n\\centering\n\\small\n\\begin{tabular}{@{}clcccccc@{}} \n\\toprule\n & & \\multicolumn{2}{c}{No elimination (LLR)} & \\multicolumn{2}{c}{Elim. stopping} & \\multicolumn{2}{c}{Elim. stopping + sampling} \\\\\n\\cmidrule(r){3-8}\n& Algorithm & Samples & Time & Samples & Time & Samples & Time \\\\\n\\cmidrule{1-8}\n\\multirow{7}{*}{\\rotatebox[origin=c]{90}{BAI}} \n& LinGapE & $33.19 \\pm 8.7$ & $0.23$ & $33.11 \\pm 8.7$ & $0.2$ & $29.89 \\pm 8.6$ & $0.18$ \\\\\n& LinGame & $45.34 \\pm 14.2$ & $0.23$ & $43.67 \\pm 13.4$ & $0.21$ & $32.49 \\pm 8.1$ & $0.18$ \\\\\n& FWS & $42.26 \\pm 60.1$ & $0.73$ & $42.25 \\pm 60.1$ & $0.7$ & $32.62 \\pm 18.0$ & $0.45$ \\\\\n& Lazy TaS & $76.33 \\pm 65.8$ & $0.15$ & $74.08 \\pm 65.8$ & $0.13$ & $64.48 \\pm 81.8$ & $0.12$ \\\\\n& Oracle & $56.36 \\pm 9.1$ & $0.05$ & $55.36 \\pm 9.3$ & $0.02$ & & \\\\\n& XY-Adaptive & & & & & $87.08 \\pm 29.1$ & $0.44$ \\\\\n& RAGE & & & & & $106.87 \\pm 30.7$ & $0.02$ \\\\\n\\cmidrule{1-8}\n\\multirow{6}{*}{\\rotatebox[origin=c]{90}{Top-m ($m=5$)}} \n& m-LinGapE & $63.69 \\pm 11.1$ & $0.56$ & $63.48 \\pm 11.0$ & $0.41$ & $59.57 \\pm 9.4$ & $0.24$ \\\\\n& MisLid & $87.77 \\pm 20.4$ & $0.55$ & $85.95 \\pm 20.5$ & $0.4$ & $69.58 \\pm 16.0$ & $0.25$ \\\\\n& FWS & $78.28 \\pm 65.0$ & $3.0$ & $78.23 \\pm 65.0$ & $2.85$ & $77.79 \\pm 65.0$ & $0.97$ \\\\\n& Lazy TaS & $161.43 \\pm 96.9$ & $0.57$ & $159.86 \\pm 96.9$ & $0.43$ & $146.06 \\pm 82.6$ & $0.36$ \\\\\n& Oracle & $102.45 \\pm 16.1$ & $0.2$ & $101.53 \\pm 16.4$ & $0.08$ & & \\\\\n& LinGIFA & $58.31 \\pm 10.8$ & $2.46$ & $58.31 \\pm 10.8$ & $2.33$ & & \\\\\n\\cmidrule{1-8}\n\\multirow{5}{*}{\\rotatebox[origin=c]{90}{OSI}}\n& LinGapE & $17.31 \\pm 2.3$ & $0.22$ & $17.29 \\pm 2.2$ & $0.19$ & $14.71 \\pm 2.0$ & $0.17$ \\\\\n& LinGame & $23.77 \\pm 4.1$ & $0.25$ & $23.05 \\pm 3.9$ & $0.21$ & $14.87 \\pm 2.0$ & $0.19$ \\\\\n& FWS & $15.26 \\pm 2.0$ & $0.83$ & $15.24 \\pm 2.0$ & $0.81$ & $14.99 \\pm 2.1$ & $0.56$ \\\\\n& Lazy TaS & $35.11 \\pm 10.2$ & $0.32$ & $33.98 \\pm 9.7$ & $0.3$ & $23.51 \\pm 5.6$ & $0.34$ \\\\\n& Oracle & $29.1 \\pm 4.8$ & $0.06$ & $28.65 \\pm 5.0$ & $0.03$ & & \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Experiments on linear instances with $K=50$ and $d=20$. The \"Time\" columns report average times per iteration in milliseconds (i.e., the total time the algorithm took divided by the number of samples). Each entry reports the mean across $100$ runs plus\/minus standard deviation (which is omitted for compute times due to space constraints). Algorithms for which the third column is missing cannot be combined with elimination at sampling, while algorithms for which the first two columns are missing are natively elimination-based. Samples are scaled down by a factor $10^3$.}\\label{tab:lin_all}\n\\end{table*}\n\n\\textbf{Full versus selective elimination.} We \ncombine the different algorithms with full and selective elimination, both at sampling and stopping.\nDue to space constrains, Figure \\ref{fig:all}\\emph{(middle)} shows the results only for LinGame (see Appendix~\\ref{app:experiments} for the others).\nWe note that full elimination seems faster at discarding arms in earlier steps, as we would expect theoretically.\nHowever, it never stops earlier than its selective counterpart. Moreover, its computational overhead is not advantageous. Overall, we concluded that our selective elimination rule is the best choice and we shall thus focus on it in the remaining. Finally, we remark that combining the sampling rule with elimination (no matter of what type) seems to discard arms faster in later steps, and could eventually make the algorithm stop sooner. \n\n\\textbf{LLR versus elimination stopping.} We now compare LLR and elimination stopping as a function of $\\delta$.\nWe know from theory that both stopping rules allow to achieve asymptotic optimality. Hence for asymptotically optimal sampling rules the resulting stopping times with LLR and elimination should tend to the same quantity as $\\delta\\rightarrow 0$.\nFigure \\ref{fig:all}\\emph{(right)}, where we report the ratio between the LLR stopping time and the elimination one for different algorithms, confirms that this is the case.\nSome algorithms (LinGapE and FWS) seem to benefit less from elimination stopping than the others, i.e., they achieve smaller ratios of stopping times.\nWe believe this to be a consequence of their mostly ``greedy'' nature, while the extra randomization of the other algorithms might help in this aspect.\n\n\\textbf{Sample complexities and computation times.} We finally compare our baselines in all three exploration tasks, in terms of sample complexity and computation time.\nFor this experiment, we selected a larger linear instance with $K=50$ and $d=20$, randomly generated (see the protocol in Appendix~\\ref{app:experiments}).\nFrom the results in Table \\ref{tab:lin_all}, we highlight three points.\n(1) The computation times of all adaptive algorithms decrease when using selective elimination stopping instead of LLR and further decrease when also using elimination at sampling. In the case of Top-m (i.e., the hardest combinatorial problem), most adaptive algorithms become at least twice faster with elimination at stopping and sampling instead of LLR.\n(2) Elimination at sampling improves the sample complexity of all algorithms.\n(3) For BAI, the natively elimination-based algorithm RAGE, which updates its strategy infrequently, is the fastest in terms of computation time but the slowest in terms of samples. Adaptive algorithms using elimination achieve run times that are within an order of magnitude of those of RAGE, while outperforming it in terms of sample complexity by a factor 2 to 3.\n\n\n\\section{Notation}\\label{app:notation}\n\n\\begin{table*}[h]\n\\centering\n\\begin{tabular}{@{}ll@{}} \n\\toprule\nSymbol & Meaning \\\\\n\\cmidrule{1-2}\n$[K] = \\{1,2,\\dots,K\\}$ & Set of $K$ arms\\\\\n$\\Delta_K$ & $K$-dimensional simplex\\\\\n$d\\in\\mathbb{N}_{>0}$ & Dimension of parameter space\\\\\n$\\cM \\subseteq \\mathbb{R}^d$ & Set of possible reward parameters\\\\\n$\\mathbb{P}_\\theta$ & Distribution of observations in bandit $\\theta\\in\\cM$\\\\\n$\\nu_k(\\theta)$ & Reward distribution of arm $k$ in bandit $\\theta\\in\\cM$\\\\ \n$\\mu_k(\\theta) := \\mathbb{E}_{x\\sim \\nu_k(\\theta)}[x]$ & Mean reward of arm $k$ in bandit $\\theta\\in\\cM$\\\\ \n$\\cI$ & Set of answers\\\\\n$i^\\star(\\theta)$ & Correct answer for bandit problem $\\theta\\in\\cM$\\\\\n$\\Lambda(i) := \\{\\lambda\\in\\cM : i^\\star(\\lambda) \\neq i\\}$ & Set of alternatives to answer $i\\in\\cI$\\\\\n$\\cP(i)$ & Set of alternative piece indexes for answer $i\\in\\cI$\\\\\n$P_i := |\\cP(i)|$ & Number of pieces for answer $i\\in\\cI$\\\\\n$\\Lambda_p(i)$ & Piece $p\\in\\cP(i)$ for answer $i\\in\\cI$\\\\\n$X_{[t]} := (X_1^{k_1}, \\ldots, X_t^{k_t})$  & Vector of $t$ observations\\\\\n$L_t(\\theta, \\lambda)\n:= \\log \\frac{d \\mathbb{P}_\\theta}{d \\mathbb{P}_\\lambda}(X_{[t]})$ & LLR of $t$ observations between $\\theta$ and $\\lambda$ \\\\\n$\\KL_k(\\theta,\\lambda) := \\KL(\\nu_k(\\theta),\\nu_k(\\lambda))$ & KL divergence between $\\nu_k(\\theta)$ and $\\nu_k(\\lambda)$\\\\\n$\\hat{\\theta}_t := \\argmax_{\\lambda\\in\\cM}d \\mathbb{P}_\\lambda (X_{[t]})$ & Maximum likelihood estimator for $\\theta$\\\\\n$\\hat{\\mu}_t^k := \\frac{1}{N_t^k}\\sum_{s=1}^t X_s^{k_s}\\indi{k_s=k}$ & Empirical mean of arm $k$ (different from $\\mu_k(\\hat{\\theta}_t))$\\\\\n$H_p(\\omega, \\theta) := \\inf_{\\lambda \\in \\Lambda_p(i^\\star)} \\sum_{k\\in[K]}\\omega^k\\KL_k(\\theta,\\lambda)$ & Information of $\\omega\\in\\Delta_K$ for piece $\\Lambda_p(i^\\star)$\\\\\n$H^\\star(\\theta) := \\max_{\\omega\\in\\Delta_K}\\min_{p\\in\\cP(i^\\star)}H_p(\\omega, \\theta)$ & Optimal constant from the lower bound \\eqref{eq:lower_bound}\\\\\n$\\Omega_\\epsilon(\\theta)$ & Set of $\\epsilon$-optimal proportions\\\\\n\\bottomrule\n\\end{tabular}\n\\caption{The notation adopted in this paper.}\n\\label{tab:notation}\n\\end{table*}\n\n\\section{Identification Problems}\\label{app:problems}\n\nIn this section, we show that popular identification problems satisfy Assumption \\ref{ass:union_of_sets} and are thus suitable for elimination-based algorithms. We shall focus on \\emph{Gaussian linear bandits}, where, for any $\\theta\\in\\cM$ and $k\\in[K]$, $\\nu_k(\\theta)$ is Gaussian with unit variance and linear mean $\\mu_k(\\theta) = \\theta^T\\phi_k$. For each identification problem, we first show how to decompose the sets of alternatives into pieces for which the closest alternatives can be found efficiently. Moreover, we report the closed-form equations for computing such closest alternatives in Gaussian linear bandits and in the special case of unstructured bandits (where $\\cM = \\mathbb{R}^K$ and $\\phi_k = e_k$, the canonical basis of $\\mathbb{R}^k$). Finally, we show how to efficiently implement elimination strategies in each of these identification problems even when enumerating over all possible answers is intractable (e.g., for problems where the number of answers is exponential in the problem dimension).\n\n\\paragraph{The LLR in Gaussian linear bandits}\n\nFor all identification problems presented later, we need to show that $\\inf_{\\lambda \\in \\Lambda_p(i)}L_t(\\hat{\\theta},\\lambda)$ can be computed efficiently for any piece. In Gaussian linear bandits, such a log-likelihood ratio is actually equivalent to a KL divergence (see Corollary \\ref{cor:llr-lin-gauss}),\n\\begin{align*}\nL_t(\\hat{\\theta},\\lambda) = \\sum_{k\\in[K]}N_t^k\\KL_k(\\hat{\\theta}_t,\\lambda) = \\frac{1}{2}\\|\\hat{\\theta}_t-\\lambda\\|_{V_t}^2,\n\\end{align*}\nwhich in turn is a quadratic form weighted by the design matrix $V_t := \\sum_{s=1}^t \\phi_{k_s}\\phi_{k_s}^T$. Therefore, with greater generality, in rest of this section we shall focus on showing that $\\inf_{\\lambda \\in \\Lambda_p(i)}\\|\\theta-\\lambda\\|_{V_N}^2$ can be computed efficiently for any $\\theta,\\lambda\\in\\cM$, piece $\\Lambda_p(i)$, and (positive-definite) matrix $V_N := \\sum_{k\\in[K]} N^k \\phi_k\\phi_k^T$ with $N\\in\\mathbb{R}^K_{\\geq 0}$. In all cases, this will require minimizing quadratic forms over half-spaces.\n\n\\subsection{Best-arm Identification}\n\nIn BAI, the goal is to find the arm with largest mean. The set of answers is therefore $\\cI = [K]$ and the correct answer of $\\theta\\in\\cM$ is $i^\\star(\\theta) = \\argmax_{k\\in[K]}\\theta^T\\phi_k$. \n\n\\paragraph{Decomposition into pieces}\n\nFor each $i\\in\\cI$, the set of alternatives $\\Lambda(i)$ can be decomposed into half-spaces,\n\\begin{align*}\n\\Lambda(i) = \\bigcup_{k\\in[K], k\\neq i} \\left\\{ \\lambda \\in \\cM : \\lambda^T\\phi_k > \\lambda^T\\phi_i\\right\\}.\n\\end{align*}\nTherefore, we can take $\\cP(i) = [K] \\setminus \\{i\\}$ with $P_i = K-1$ and $\\Lambda_p(i) = \\{\\lambda\\in\\cM : \\lambda^T\\phi_p > \\lambda^T\\phi_i\\}$ for $p \\in [K]\\setminus\\{i\\}$.\n\n\\paragraph{Closest alternatives}\n\n\n\nTake any $j,k\\in[K]$ with $j\\neq k$. For linear problems, for any $\\theta\\in\\mathbb{R}^d$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_j(k)}\\Vert {\\theta} - \\lambda \\Vert_N^2 = \\begin{cases}\n\t\\frac{(\\theta^T(\\phi_k-\\phi_j))^2}{\\| \\phi_k - \\phi_j \\|_{V_N^{-1}}^2} &\\text{if } \\theta^T(\\phi_k-\\phi_j) \\geq 0, \\\\\n\t0 &\\text{otherwise}.\n\\end{cases}\n\\end{align*}\nFor the special case of unstructured problem, for any $\\theta\\in\\mathbb{R}^K$,\n\\begin{align*}\n\\inf_{\\lambda \\in \\Lambda_j(k)}\\Vert {\\theta} - \\lambda \\Vert_{V_N}^2 = \\begin{cases}\n\t\\frac{N_jN_k}{N_j + N_k}(\\theta^T(\\phi_k-\\phi_j))^2 &\\text{if } \\theta^T(\\phi_k-\\phi_j) \\geq 0\\ \\text{and } N_j + N_k > 0, \\\\\n\t0 &\\text{otherwise}.\n\\end{cases}\n\\end{align*}\n\n\\paragraph{Efficient implementation}\n\nIn BAI, for each answer (i.e., arm) $i\\in\\cI$, the piece indexes $p\\in\\cP(i)$ are themselves answers (different than $i$). Implementing the elimination stopping rule in its general form requires storing and iterating over $K(K-1)$ elements (all items in $\\cP(i)$ for each $i\\in\\cI$). However, much better implementations exist that require storing at most $K$ elements (one for each answer). Here we propose two such implementations: the first, for full elimination, is more statistically-efficient, while the second one (for selective elimination) is more computationally-efficient.\n\n\\paragraph{Full elimination (statistically-efficient implementation)}\n\nDue to the structure of the problem, whenever we eliminate one piece $\\Lambda_p(i)$, we actually know that the mean reward of arm $p$ cannot be better than that of arm $i$. In other words, $p$ cannot be the right answer. Therefore, we can maintain a list of active arms $\\cI_t$ which is initialized as $\\cI_0 = [K]$ and updated as\n\\begin{align*}\n\\mathcal{I}_t := \\mathcal{I}_{t-1} \\setminus \\left\\{ j\\in\\cI_{t-1} \\big| \\max_{i\\neq j} \\inf_{\\lambda \\in \\Lambda_j(i)} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\}.\n\\end{align*}\nThen, we stop whenever $|\\cI_t| = 1$ and return the single arm left active. Due to the inner maximization, this implementation requires performing $|\\cI_{t-1}|(K-1)$ minimizations over half-spaces at each step to check elimination.\n\n\n\\begin{proposition}\nAn algorithm using the statistically-efficient implementation above never discards pieces later than (and thus never stops later than) the full elimination rule of \\eqref{eq:elimination-sets} almost surely. \n\\end{proposition}\n\\begin{proof}\nIf the algorithm eliminates a piece $\\Lambda_p(i)$ at time $t$ with \\eqref{eq:elimination-sets},\n\\begin{align*}\n\\beta_{t,\\delta} \\leq \\inf_{\\lambda \\in \\Lambda_p(i)} L_t(\\hat{\\theta}_t,\\lambda) \\leq \\max_{j\\neq p} \\inf_{\\lambda \\in \\Lambda_p(j)} L_t(\\hat{\\theta}_t,\\lambda).\n\\end{align*}\nThis implies that the elimination condition in the statistically efficient implementation triggers as well.\n\\end{proof}\n\n\\paragraph{Selective elimination (computationally-efficient implementation)} \n\nAn even simpler implementation is to check elimination, at each time $t$, only for pieces related to the empirical optimal arm $i^\\star(\\hat{\\theta}_t)$. That is, we update $\\cI_t$ as\n\\begin{align*}\n\\mathcal{I}_t := \\mathcal{I}_{t-1} \\setminus \\left\\{ j\\in\\cI_{t-1} \\big| \\inf_{\\lambda \\in \\Lambda_j(i^\\star(\\hat{\\theta}_t))} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\}.\n\\end{align*}\nThis requires linear (in $K$) per-round memory and time complexity. Moreover, checking this stopping rule is more time-efficient than the LLR one. The latter requires to perform $K-1$ tests at each step, while the elimination one only performs $O(|\\cI_{t-1}|)$ tests at each round $t$, thus becoming faster as arms are eliminated.\n\n\\begin{proposition}\nAn algorithm using the computationally-efficient implementation above never discards pieces later than (and thus never stops later than) the selective elimination rule almost surely. \n\\end{proposition}\n\\begin{proof}\nIf the algorithm eliminates a piece of the empirical optimal arm at time $t$ using the selective elimination rule, the arm corresponding to that piece is also eliminated from the set $\\cI_t$ above.\n\\end{proof}\n\n\n\\subsection{Top-m Identification}\n\nIn top-$m$ identification, the goal is to find the $m > 0$ arms with largest mean. BAI is therefore a special case of this problem when $m=1$. The set of answers is $\\cI = \\{\\cS \\subseteq [K] : |\\cS|=m\\}$ with size $|\\cI| = {K \\choose m}$ and the correct answer of $\\theta\\in\\cM$ is $i^\\star(\\theta) = \\argmax_{k\\in[K]}^m\\theta^T\\phi_k$, where we use $\\argmax^m : \\mathbb{R}^K \\mapsto {K \\choose m}$ to denote the function returning the set of $m$ largest values. \n\n\\paragraph{Decomposition into pieces}\n\nLet us denote each $i\\in\\cI$ as a tuple $i = (k_1,\\dots, k_m)$ of $m$ arms. Similarly to BAI, it is known that the set of alternatives $\\Lambda(i)$ can be decomposed into half-spaces \\citep{reda2021dealing},\n\\begin{align*}\n\\Lambda(i) = \\bigcup_{j\\in i, k \\in [K] \\setminus i} \\left\\{ \\lambda \\in \\cM : \\lambda^T\\phi_k > \\lambda^T\\phi_j\\right\\}.\n\\end{align*}\nTherefore, we have $\\cP(i) = i \\times ([K] \\setminus i)$ with $P_i = m(K-m)$ and $\\Lambda_p(i) = \\{\\lambda\\in\\cM : \\lambda^T\\phi_k > \\lambda^T\\phi_j\\}$ when $p = (j,k)$.\n\n\\paragraph{Closest alternatives}\n\nNote that each set $\\Lambda_p(i)$ is still a half-space of the same form as the one we have for BAI. Hence, the same closed form expression for the closest alternative derived for BAI can be adopted for top-$m$ identification (see the closed-form expressions in the previous section).\n\n\\paragraph{Efficient implementation}\n\nNote that, differently from BAI, here the set of answers is of combinatorial size. It is therefore intractable to store and enumerate all sets of active pieces $\\cP_t(i)$. However, thanks to the structure of the problem, this is not necessary and there exists an efficient implementation for the elimination stopping rule. First note that, while there are $m(K-m)$ pieces for each of $K \\choose m$ possible answers, the total number of half-spaces is only $K(K-1)$, one for each couple of different arms. With some abuse of notation, let us denote by $\\Lambda_{k,j} := \\left\\{ \\lambda \\in \\cM : \\lambda^T\\phi_k > \\lambda^T\\phi_j\\right\\}$ the half-space associated with arms $k$ and $j$. The elimination stopping rule, which checks whether all the pieces in $\\cP(i)$ for some answer $i$ have been discarded, is equivalent to checking whether there exist $m$ arms $k_1,\\dots,k_m$ such that $\\Lambda_{k,k_l}$ has been eliminated for all $k\\notin \\{k_1,\\dots,k_m\\}$ and $l\\in[m]$. Therefore, in our implementations we will only store whether each half-space $\\Lambda_{k,k_l}$ has been eliminated or not. As before, we now see two possible implementations, one more computationally efficient and the other more statistically efficient.\n\n\\paragraph{Full elimination (statistically-efficient implementation)}\n\nThe idea is to check, at each time step $t$, the elimination condition for all half-spaces which have not been previously discarded. In particular, for each arm $j\\in[K]$ we keep a set $\\cS_t(j)$ storing those arms which are ``worse'' than $j$. Formally, we initially set $\\cS_0(j) = \\emptyset$ and update it as\n\\begin{align*}\n\\cS_t(j) := \\begin{cases}\n\t\\cS_{t-1}(j) \\cup \\left\\{ k\\notin  \\cS_{t-1}(j)\\cup\\{j\\} \\big| \\inf_{\\lambda \\in \\Lambda_{k,j}} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\} &\\text{if } |\\cS_{t-1}(j)| < K-m, \\\\\n\t\\cS_{t-1}(j) &\\text{otherwise}.\n\\end{cases}\n\\end{align*}\nThat is, when a half-space $\\Lambda_{k,j}$ is eliminated, we conclude that arm $k$ is ``worse'' than arm $j$ and thus add the former to $\\cS_t(j)$. In order to decide when to stop, we use the following intuition: whenever we find that $|\\cS_t(j)| \\geq K-m$ for some arm $j\\in[K]$, then we know that $j$ must be in the top-m arms of $\\theta$ and we can thus stop updating the set $\\cS_t(j)$. Therefore, we can stop whenever there exist $m$ arms satisfying this property. This can be checked efficiently by keeping track of how many arms reach the condition $|\\cS_t(j)| \\geq K-m$ and stopping when the number of such arms reaches $m$. This approach takes $O(K(K-m))$ memory in the worst-case to store the sets $\\cS_t(j)$. At each step, it performs exactly $\\sum_{j: |\\cS_{t-1}(j)| < K-m} (K - |\\cS_{t-1}(j)| - 1)$ minimizations over half-spaces to check the elimination conditions, which gives $O(K(K-1))$ time complexity in the worst-case.\n\n\\begin{proposition}\nAn algorithm using the statistically-efficient implementation above never discards pieces later than (and thus never stops later than) the full elimination rule of \\eqref{eq:elimination-sets} almost surely.\n\\end{proposition}\n\\begin{proof}\nNote that the elimination condition for single half-spaces is exactly the same in the general elimination rule of \\eqref{eq:elimination-sets} and in its implementation above. If \\eqref{eq:elimination-sets} eliminates a piece $\\Lambda_p(i)$ at time $t$, this implies some half-space $\\Lambda_{k,j}$ is eliminated. Then, we have two possible cases: if $|\\cS_{t-1}(j)| < K-m$, then we have $k\\in\\cS_t(j)$ by the condition above, i.e., $k$ is detected as ``worse'' than $j$ and it will be never checked again. On the other hand, if $|\\cS_{t-1}(j)| \\geq K-m$, then $j$ has already been labeled as belonging to the final answers. Thus, no minimization over its corresponding half-spaces (including the one for $k$) will be checked anymore, which is the same as saying that $\\Lambda_{k,j}$ has already been eliminated.\n\\end{proof}\n\n\\paragraph{Selective elimination (computationally-efficient implementation)} \n\nSimilarly to what we did for BAI, the most computationally-efficient implementation consists in checking the elimination condition only for the alternative pieces (i.e., the half-spaces) of the empirical correct answer at each step. We modify the update rule of the statistically-efficient implementation as\n\\begin{align*}\n\t\\cS_t(j) := \\cS_{t-1}(j) \\cup \\left\\{ k\\notin i^\\star(\\hat{\\theta}_t) \\cup \\cS_{t-1}(j) \\big| \\inf_{\\lambda \\in \\Lambda_{k,j}} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\}\n\\end{align*}\nif $j \\in i^\\star(\\hat{\\theta}_t) \\text{ and } |\\cS_{t-1}(j)| < K-m$, and $S_t(j) := \\cS_{t-1}(j)$.\nThat is, at each step we only check elimination for half-spaces associated with the top-m arms of $\\hat{\\theta}_t$, excluding those that have already been eliminated and those that have already reached the threshold for being among the final answer. Note that this implementation performs $\\sum_{j\\in i^\\star(\\hat{\\theta}_t),|\\cS_{t-1}(j)| < K-m}(K - |i^\\star(\\hat{\\theta}_t) \\cup \\cS_{t-1}(j)|) \\leq m(K-m)$ minimizations over half-spaces at each step $t$. In constrast, the LLR stopping rule always performs $m(K-m)$ minimizations and is thus less efficient.\n\n\\begin{proposition}\nAn algorithm using the computationally-efficient implementation above never discards pieces later than (and thus never stops later than) the selective elimination rule almost surely.\n\\end{proposition}\n\\begin{proof}\nThe proof is the same as for BAI: if an arm is discarded by the selective elimination rule, then it is also discarded from the sets above.\n\\end{proof}\n\n\n\n\n\\subsection{Thresholding Bandits}\n\nIn the thresholding bandit problem, the goal is to learn whether the mean of each arm is above or below some given threshold. As usual, without loss of generality, we shall take zero as our threshold, for which the problem reduces to learning the sign of the mean reward of each arm. Let $\\sign(x) := \\indi{x \\geq 0}$. Then, the set of answers is $\\cI = \\{0,1\\}^K$ with size $|\\cI| = 2^K$. The correct answer of problem $\\theta\\in\\cM$ is $i^\\star(\\theta) = (\\sign(\\theta^T\\phi_k))_{k\\in[K]}$. \n\n\\paragraph{Decomposition into pieces}\n\nFor each $i\\in \\{0,1\\}^K$ (represented as a $K$-dimensional binary vector), the set of alternatives $\\Lambda(i)$ can be decomposed into pieces as\n\\begin{align*}\n\\Lambda(i) = \\bigcup_{k\\in[K]} \\left\\{\\lambda\\in\\cM : \\sign(\\lambda^T\\phi_k) \\neq i^k \\right\\}.\n\\end{align*}\nTherefore, we have $\\cP(i) = [K]$ with $P_i = K$ and $\\Lambda_p(i) = \\{\\lambda\\in\\cM : \\sign(\\lambda^T\\phi_p) \\neq i^p\\}$. As for BAI, the computation of the closest alternative over such pieces can be performed efficiently.\n\n\\paragraph{Closest alternatives}\n\nLet $\\theta\\in\\cM$ and $N\\in\\mathbb{R}_{\\geq 0}^K$. The computation of the closest alternatives over pieces $\\Lambda_p(i)$ can be reduced to the following optimization problem. For any arm $k\\in[K]$ and any $b\\in\\{0,1\\}$, we need to find\n\\begin{align*}\n\\inf_{\\lambda \\in \\cM : \\sign(\\lambda^T\\phi_k) \\neq b} \\Vert {\\theta} - \\lambda \\Vert_{V_N}^2.\n\\end{align*}\nIt is easy to see that this is zero when $b\\neq\\sign(\\theta^T\\phi_k)$ (since $\\theta$ itself is feasible). In case $b=\\sign(\\theta^T\\phi_k)$, for unstructured problems ($\\cM=\\mathbb{R}^K$), the solution is to take $\\lambda$ equal to $\\theta$ at all components except the $k$-th one, where it is set to zero. This gives\n\\begin{align*}\n\\inf_{\\lambda \\in \\mathbb{R}^K : \\sign(\\lambda^T\\phi_k) \\neq b} \\Vert {\\theta} - \\lambda \\Vert_{V_N}^2 = \\begin{cases}\n\tN^k(\\theta^T\\phi_k)^2 &\\text{if } \\sign(\\theta^T\\phi_k) = b, \\\\\n\t0 &\\text{otherwise}.\n\\end{cases}\n\\end{align*}\nIn the linear case, again under the assumption that $V_N$ is positive definite, we get\n\\begin{align*}\n\\inf_{\\lambda \\in \\mathbb{R}^d : \\sign(\\lambda^T\\phi_k) \\neq b} \\Vert {\\theta} - \\lambda \\Vert_{V_N}^2 = \\begin{cases}\n\t\\frac{(\\theta^T\\phi_k)^2}{\\| \\phi_k \\|_{V_N^{-1}}^2} &\\text{if } \\sign(\\theta^T\\phi_k) = b, \\\\\n\t0 &\\text{otherwise}.\n\\end{cases}\n\\end{align*}\n\n\n\\paragraph{Efficient implementation}\n\nFor this problem, implementing the general elimination stopping rule would require storing and iterating over $2^K K$ pieces, which is clearly intractable. However, this problem introduces a high redundancy in the alternative pieces that we can exploit for an efficient implementation which takes only linear (in $K$) time and space. Differently from BAI and top-m identificaiton, the procedure highlighted below is \\emph{exactly} an implementation of the ``theoretical'' elimination rules presented in the main paper, with the full and selective elimination rules reducing to the same thing.\n\nNote that, for any $p\\in[K]$ and $i,j\\in\\{0,1\\}^K$ such that $i^p = j^p$, we have $\\Lambda_p(i) = \\Lambda_p(j)$. That is, whenever we eliminate some piece $\\Lambda_p(i)$ for $p\\in[K]$ and $i\\in\\{0,1\\}^K$, we actually eliminate all problems in $\\cM$ whose sign of the $p$-th mean reward is different from $i^p$. In other words, we learn that the $p$-th position of the correct answer for $\\theta$ is indeed $i^p$. Therefore, an efficient implementation is as follows: we keep a set $\\cA_t$ of active arms (those for which we still have to learn the corresponding component in the correct answer). This set is initialized as $\\cA_0 = [K]$ and updated as\n\\begin{align*}\n\\mathcal{A}_t := \\mathcal{A}_{t-1} \\setminus \\left\\{ j\\in\\cA_{t-1} \\big| \\max_{i\\in\\{0,1\\}^K} \\inf_{\\lambda \\in \\Lambda_j(i)} L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\}.\n\\end{align*}\nWhile the maximization over $2^K$ elements might appear intractable, the structure of the problem allows us to entirely avoid it. Note that, for fixed $j$, the sets $\\Lambda_j(i)$ are fully specified by the $j$-th component of $i$. Moreover, the inf is zero whenever $i^j \\neq \\sign(\\hat{\\theta}_t^T\\phi_j)$ since that would imply $\\hat{\\theta}_t \\in \\Lambda_j(i)$. Therefore, the elimination condition can be equivalently rewritten in the convenient form\n\\begin{align*}\n\\mathcal{A}_t := \\mathcal{A}_{t-1} \\setminus \\left\\{ j\\in\\cA_{t-1} \\big| \\inf_{\\lambda \\in \\cM : \\sign(\\lambda^T\\phi_j) \\neq \\sign(\\hat{\\theta}_t^T\\phi_j) } L_t(\\hat{\\theta}_t,\\lambda) \\geq \\beta_{t,\\delta}\\right\\}.\n\\end{align*}\nMoreover, whenever the elimination condition above triggers for some arm $j\\in[K]$, we set a variable $S_j := \\sign(\\hat{\\theta}_t^T\\phi_j)$ with the correct sign for the $j$-th component. We stop whenever $\\cA_t = \\emptyset$ (i.e., when all signs have been learned) and return $\\hat{i} := (S_1,S_2,\\dots, S_K)$. Similarly to BAI, this requires to perform only $|\\cA_{t-1}|$ tests at each step $t$. On the other hand, the LLR stopping rule would perform $K$ tests at each step.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe standard stochastic multi-armed bandit framework captures the exploration-exploitation trade-off in sequential decision making problems under partial feedback constraints. The objective is to actively identify the best member, or members, of a community comprising stochastic sources, termed arms, while suffering the relative loss of non-ideal choices. Often, the arms yield rewards that belong to a known probability distribution with hidden parameters, and upon choosing an arm, the decision maker observes the reward from that arm directly. For ergodic reward distributions, \\cite{gittins1979bandit} shows that the  dynamic programming solution takes the form of an index policy\\footnote{The orignial formulation of \\cite{gittins1979bandit} was in the Bayesian framework.}, called dynamic allocation indices, which motivated the rich body of work that led to the arm-selection rules that eventually achieved the asymptotic regret lower bounds of \\cite{lai1985asymptotically}. Alternatively, when the probability law that governs the rewards is defined conditionally with respect to a hidden state that represents the ``changing world'', restless bandit framework of \\cite{whittle1988restless} leads to arm-selection policies that are often computationally demanding. A key challenge that we address here is to develop index policies that identify the best source in an environment where the underlying state of the world changes erratically and hence, the observations are generated from different probability distributions at each point in time.  \n\nSpecifically, we consider the case where each arm represents a stochastic expert providing opinions on changing tasks and thus, upon consulting an expert, the decision maker observes an opinion, rather than a direct reward. Stochastic experts are sources of subjective information that might fail but not purposefully deceive, as discussed in \\cite{cesa2006prediction}, and often, expert suggestions, or opinions, are used with the aid of side-information: Feedback from past states of the world is used in boosting, \\cite{schapire2012boosting}, models of expert stochasticity, or direct information of expert reliability, or competence, are often used in the Bayesian framework, \\cite{poor2013introduction}. In the absence of \\textit{any} side information, the decision maker operates in a regime that can be termed \\textit{unsupervised}, relying solely on the information in the opinions. Unsupervised opinion aggregation methods such as expectation maximization (EM) \\cite{welinder2010online}, belief propagation (BP) \\cite{karger2011iterative}, and spectral meta-learner (SML) \\cite{parisi2014ranking} exhibit an interesting phenomenon: The reliability of experts are inferred as side-product of the underlying optimization for estimating past states based on a block of opinions. On the other hand, joint sampling and consultation of experts without supervision, or blind exploration and exploitation (BEE) as termed here, requires instantaneously available statistics that would allow reliable inference of expert reliabilities at any and all states of the world. \n\nWe propose a method that relies solely on opinions to infer the competence of an expert by re-defining the notion competence as the probability of agreeing with peers rather than being objectively correct. The proposed method does not only allow empirical inference of competence without any supervision but also enables the use of index policies to efficiently address exploration and exploitation dilemma when the underlying task changes at random. We show that standard, or supervised, exploration-exploitation (SEE) strategies extend their uses to the BEE problem by consulting multiple experts for each task, equivalent to sampling multiple arms in the standard framework. Specifically, we consider the index rules that rely on posterior sampling, \\cite{thompson1933likelihood}, upper-confidence bounds such as UCB1, \\cite{auer2002finite}, and KL-UCB, \\cite{garivier2011klucb}, minimum empirical Kullback-Leibler divergence, in particular, IMED, \\cite{honda2015imed}, and minmax rule MOSS of \\cite{audibert2009minimax}. We investigate two operational regimes: First,\na fixed number of experts are consulted for each task and the opinion of the expert who is believed to be most-reliable at that time is chosen. Second, upon consulting a group of experts, a decision is formed by aggregating their opinions without further supervision. We empirically compare the performance of different BEE index rules and demonstrate that exploration-exploitation-based choice of experts leads to comparable results to those of the original algorithms in the unsupervised framework. \n\nThe organization of this paper is as follows: We summarize the notation used in this paper, provide a background on stochastic experts, and define the BEE problem formally in Section \\ref{sec:probdef}. We discuss the motivation, formal definition, and properties of our technique for unsupervised reliability inference in Section \\ref{sec:pseudocomp}. Then, we discuss the fundamental properties of the BEE index rules in Section \\ref{sec:bee}. The experiments for comparing different BEE algorithms as well as comparing them to their SEE counterparts are in Section \\ref{sec:experiments}. The proofs are deferred to the appendix.   \n\n\\section{Notation, Background, and Problem Formulation}\n\\label{sec:probdef}\nWe begin with a brief overview of the notation used in this paper. Then, we formally define the key concepts regarding stochastic experts. We conclude this section by defining the BEE problem.    \n\\subsection{Notation}\nA probability space is a triplet $\\yay{\\Omega, \\mathscr{F}, \\mathbb{P}}$, where $\\Omega$ is the event space, $\\mathscr{F}$ is the sigma-field defined on $\\Omega$, and $\\mathbb{P}$ is the probability measure. Random variables are denoted by capital letters with the corresponding samples being denoted by lowercase letters: $(X,x)$. A random process is an indexed collection of random variables: $\\myset{Y(t): t\\in \\mathbb{T}}$, where $\\mathbb{T}$ is the index set. Independent random variables $\\yay{X_1, X_2}$ are denoted by $X_1 \\perp X_2$ and conditionally independent random variables $\\yay{X_1, X_2}$ conditioned on $Y$ are denoted by $X_1 - Y - X_2$. Expectation, conditional expectation, and conditional probability operators are denoted by $\\expt{\\cdot}$, $\\condexpt{\\cdot}{\\cdot}$, and $\\condprob{\\cdot}{\\cdot}$ respectively. The indicator function is denoted by $\\ind{\\cdot}$, where domain is to be understood from context. We use  $[T] \\triangleq \\myset{1,\\cdots,T}$ to denote the positive natural numbers up to a finite limit $T<\\infty$. All logarithms $\\yay{\\log}$ are taken with respect to the natural base. We use big $O$ notation when necessary. \n\\subsection{Background}\nConceptually, stochastic experts are honest-but-fallible computational entities that do not deceive the decision maker deliberately. Here, we consider experts that do not collaborate while generating their opinions; \\cite{cesa2006prediction} provides a detailed discussion. The goal of this paper is to propose techniques that identify the best stochastic experts, while dynamically consulting others on varying tasks. In that context, consulting an expert on a task is equivalent to pulling an arm in the standard multi-armed bandit framework. The true reward, however, remains hidden.     \n\nFormally, let us begin with a random process $\\myset{Y(t): t\\in [T]}$ that represents binary states of the world, or tasks with binary labels: $Y(t) \\in \\myset{-1,1}$, $\\forall t\\in[T]$. We allow the nature to generate tasks independently:\n\\begin{equation}\n\t\\label{independent_tasks}\n\tY\\yay{t_1} \\perp Y\\yay{t_2},~\\forall t_1\\neq t_2 \\in [T].\n\\end{equation}\nFurthermore, let the random process $Y(t)$ that governs the evolution of tasks  maximize the uncertainty: \n\\begin{equation}\n\t\\label{unif_distr}\n\t\\prob{Y(t) =1} = \\prob{Y(t)=-1} = \\nicefrac{1}{2},~\\forall t\\in [T].\n\\end{equation} \nIt is worth noting that any bias from non-uniform task generation can either be estimated directly from labeled data, or inferred without supervision via methods such as \\cite{jaffe2016unsupervised}. Furthermore, while independence assumption appears to be restrictive, it is common in stochastic multi-armed bandit formulations, \\cite{bubeck2012regret}.   \n\nFormal characterization of stochastic experts involves the reliability of their opinions and statistical dependence to the others. The probability with which the opinion of an expert identifies the true state of the world correctly determines the reliability, or competence, of that expert: \n\\begin{equation}\n\t\\label{static_competence}\n\tp_i \\triangleq \\prob{X_i(t) =Y(t)},~\\forall t\\in[T].\n\\end{equation} \nHere, the reliability of an expert does not depend on the underlying state of the world \\footnote{A notable exception to this model is the ``two-coin'' model from \\cite{dawid1979maximum}, where conditionally static competences are discussed.}. \nWe further allow that experts generate opinions $\\myset{X_i(t): i\\in [M]}$ independently from one another for every task $t\\in[T]$. Formally: \n\\begin{equation}\n\t\\label{independent_generation}\n\tX_i(t) - Y(t) - X_j(t),~\\forall i\\neq j\\in [M],~\\forall t\\in[T].\n\\end{equation} \nConceptually, it makes sense that for meaningful inference, two different opinions on the same task should never be statistically independent. Furthermore, experts having conditionally independent opinions is equivalent to independence of rewards in the standard framework.\n\nGiven the probability law defined by eq.~\\eqref{independent_tasks}-\\eqref{independent_generation}, we can formally discuss why SEE algorithms requires a toolset to address the impact of the underlying uncertainty. Observe that:\n\\begin{equation}\n\t\\label{motivation}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau =1}^{t} X_i(t) = 0,~\\forall p_i\\in\\brac{0,1},\n\\end{equation} \nwhich follows from the law of total probability, see appendix \\ref{app:average_opinion}. Conceptually, eq.~\\eqref{motivation} indicates that the average opinion does not reflect the competence of an expert, which is the true reward, posing a challenge for joint exploration and exploitation in the context of sequentially consulting stochastic experts, which we formally define next.  \n\n\\subsection{Problem Definition}\nThe first objective of the BEE problem is to identify the best expert in a population while actively consulting members of that group on tasks that change from one consultation to another. The following notion of regret, written here in normalized form, formally captures this phenomenon:  \n\\begin{equation}\n\t\\label{real_regret}\n\tR_T =  \\frac{1}{T}\\sum_{t=1}^{T} \\ind{X^{*}(t)=Y(t)} - \\frac{1}{T}\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}.\n\\end{equation}\nHere $X^{*}$ is the opinion of the most competent expert; $X^{*} = X_{i^{*}}$, where $i^{*} = \\argmax_{i\\in[M]} p_i$ and $I_t\\in[M]$, $\\forall t\\in[T]$ is the expert chosen at time $t$. Observe that the regret, as defined in eq.~\\eqref{real_regret} depends on the sample path of opinions and hence, it is difficult to analyze rigorously. Nonetheless, it simplifies asymptotically:\n\\begin{equation}\n\t\\label{real_regret_asymptotic}\n\t\\lim\\limits_{T\\rightarrow \\infty}R_T = \\max_{i\\in[M]} p_i - \\lim\\limits_{T\\rightarrow \\infty}\\frac{1}{T}\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}.\n\\end{equation}\nThe first term is a direct consequence of the ergodicity of the processs $\\ind{X^{*}(t)=Y(t)}$, which follows directly from eq.~\\eqref{independent_tasks}-\\eqref{static_competence}. Conceptually, this amounts to the fact that one can measure the true reliability of an expert given sufficiently many labeled tasks, as long as the reliability of the expert does not change across tasks, as is the case here. \n\nMotivated by similar asymptotic behaviors, a notion of \\textit{pseudo regret} often arises in the context of stochastic bandits, see, for instance, \\cite{bubeck2012regret}. In the context of stochastic experts, the pseudo regret is defined as follows:  \n\\begin{equation}\n\t\\label{pseudo_regret}\n\t\\tilde{R}_T = \\max_{i\\in[M]} p_i - \\frac{1}{T}\\expt{\\sum_{t=1}^{T} \\ind{X_{I_t}(t)=Y(t)}}.\n\\end{equation}\nAnother notion of pseudo regret provides a reliable metric for the performance of BEE rules that aggregate opinions after consulting experts. Let a $m<M$ experts be consulted for every task $t$, indexed by $\\comm{t}\\subset[M]$ and let $f: \\myset{\\pm 1}^{\\abs{\\comm{}}}\\rightarrow \\myset{\\pm 1}$ be a known opinion aggregation rule. Define the pseudo regret as: \n\\begin{equation}\n\t\\label{swarm_regret}\n\t\\tilde{R}_T = \\max_{\\substack{\\comm{}\\subset[M]\\\\ \\abs{\\comm{}}=m}} \\prob{f\\yay{\\myset{X_i(t):i\\in\\comm{}}}=Y(t)} - \\frac{1}{T}\\expt{\\sum_{t=1}^{T} \\ind{f\\yay{\\myset{X_i(t):i\\in\\comm{t}}}=Y(t)}}.\n\\end{equation}\nTo emphasize the difference in objectives, we will refer to the rules that aim to minimize eq. \\eqref{pseudo_regret} as BEE rules and to those that sequentially weight and aggregate opinions of reliable members (SWARM) and thus, aim to minimize eq. \\eqref{swarm_regret}, as SWARM rules.   \n\nGiven an arm-selection strategy, it is more intuitive to structure a theoretical analysis based on the pseudo regrets in eq.~\\eqref{pseudo_regret}-\\eqref{swarm_regret}, provided the true rewards $\\myset{\\ind{X_{I_{\\tau}}(\\tau))=Y(\\tau)} : \\tau \\in [t]}$ are known. Nonetheless, the main challenge remains: At any given time $t$, the decision maker does not have access to the past labels $\\myset{Y_{\\tau}: \\tau \\in [t]}$ and thus, does not get to observe the true rewards. We address this challenge by consulting multiple experts for each task and leveraging the increased reliability of a collection of experts for inferring those of the individuals. Next, we formally define and discuss the properties of the competence estimation technique that facilitates joint exploration and exploitation based on opinions alone.    \n\n\\section{Unsupervised Estimation of Competences with Instantaneous Updates}\n\\label{sec:pseudocomp}\nTwo seemingly conflicting statistical phenomena should be reconciled to facilitate sampling and consultation of experts based only on their opinions. On one hand, standard index policies for  exploration and exploitation require observations from a joint probability distribution that does not evolve in time. On the other hand, opinions $\\myset{X_1(t),\\cdots,X_M(t)}$ are based on the hidden state $Y(t)$ and thus, their statistics change randomly in time. Furthermore, an operational challenge exists: Multi-armed bandit problems often allow observations to be available instantly so that the player can decide which arm to pull next. This is in sharp contrast to the unsupervised opinion aggregation strategies such as SML, EM, BP from \\cite{parisi2014ranking,welinder2010online,karger2011iterative} respectively, that yield the reliability of experts as a side-product of an optimization process that is often carried out over a block of opinions.\n\nFortunately, there exists an empirically realizable measure of competence that is only based on opinions. Conceptually, consider a compact measure, for instance, an ideal ruler for measuring the length of a two-dimensional line. The length of any such line can be measured reliably by this ruler and these measurements can be used to determine which line is the longest. Recall that the true competence of an expert can be measured experimentally via:\n\\begin{equation}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau=1}^{t} \\ind{X_i(t)=Y(t)} = p_i. \n\\end{equation}      \nIn other words, state feedback is an ideal ruler in our example. In the absence of such reliable reference, one can \\textit{choose to accept} a random line as a measure of length, replacing the phrase ``this line is $\\ell$ units long'' with ``this line is $\\beta$ \\textit{measuring lines} long''. This strategy cannot, of course, infer the true length of any line however, it can infer the \\textit{relative length} of every other line, which is sufficient for constructing index rules that solve the BEE problem.     \n\nWe introduce an alternative notion of reliability, termed \\textit{pseudo competence}, that allows instantaneous, opinion-based inference of the competences, while preserving the true ordering of the competences. Formally, the pseudo competence of an expert is defined as follows:\n\\begin{equation}\n\t\\label{pseudocompdef}\n\t\\pp{i} = \\prob{X_i(t) = \\V{\\myset{X_j(t): j\\in\\comm{}}}} = \\prob{X_i(t) = \\sign\\brac{\\sum_{j\\in\\mathcal{C}}X_j(t)}}.\n\\end{equation} \nHere, $\\V{\\cdot}$ denotes majority voting (where ties are broken arbitrarily\\footnote{The $\\yay{\\sign}$ operator does not conventionally ``break ties arbitrarily''. However, we allow $\\sign\\brac{0}=Z$, where $\\prob{Z=1}=\\prob{Z=-1}=\\nicefrac{1}{2}$ in eq.~\\eqref{pseudocompdef} for notational clarity.}) over a committee $\\mathcal{C}$ that excludes the $i^{th}$ expert: $\\mathcal{C}\\subset [T]\\setminus\\myset{i}$. Observe that it is possible to estimate pseudo competences empirically: \n\\begin{equation}\n\t\\label{empirical_estimate}\n\t\\lim\\limits_{t\\rightarrow \\infty} \\frac{1}{t} \\sum_{\\tau=1}^{t} \\ind{X_i(t)=\\V{\\myset{X_j(t): j\\in\\comm{}}}} = \\pp{i}. \n\\end{equation}   \nFurther note that similar to the standard formulation with direct rewards, one can easily keep track of number of times an expert is consulted and the number of times that expert agreed with the others. \n\nLet us denote the probability of the committee $\\mathcal{C}$ being correct by:\n\\begin{equation}\n\tp_{\\mathcal{C}} \\triangleq \\prob{\\V{\\myset{X_j(t): j\\in\\comm{}}}}.\n\\end{equation}\nThen, the pseudo competence can be written explicitly as:\n\\begin{equation}\n\t\\label{pseudocomp_explicit}\n\t\\pp{i} = p_i p_{\\mathcal{C}} + \\yay{1-p_i} \\yay{1-p_{\\mathcal{C}}}. \n\\end{equation}\nThe proof of eq.~\\eqref{pseudocomp_explicit} appears in appendix \\ref{app:prop} as a part of the proof for Proposition \\ref{prop}. Conceptually, the pseudo competence of an expert increases during empirical estimation if either both the expert and committee are correct or both are incorrect. \n\nBeyond allowing empirical estimation, the pseudo competence, as a measure of reliability, demonstrates a key property: Given pseudo competences for a set experts, or given sufficiently large number of tasks and the corresponding opinions thanks to eq.~\\eqref{empirical_estimate}, it is possible to rank experts based on their true competence, as discussed next.\n\\begin{prop}\n\t\\label{prop}\n\tFor a committee $\\mathcal{C}\\subset [T]$, the following holds for every $i,j\\notin \\mathcal{C}$: \n\t\\begin{equation}\n\t\t\\label{pushtogether}\n\t\t\\pp{i}-\\pp{j} = \\yay{2p_{\\mathcal{C}}-1}\\yay{p_i-p_j}.\n\t\\end{equation}\n\tTherefore, if the probability of the committee $\\mathcal{C}$ making a correct decision under majority rule satisfies:\n\t\\begin{equation}\n\t\t\\label{condition}\n\t\tp_{\\mathcal{C}} = \\prob{\\V{\\myset{X_j(t): j\\in\\comm{}}}=Y(t)}> \\nicefrac{1}{2},\t\n\t\\end{equation}\n\tthen pseudo competence preserves the ordering of true competences: \n\t\\begin{equation}\n\t\tp_i\\geq p_j \\iff \\pp{i} \\geq \\pp{j}.\n\t\\end{equation}\n\\end{prop}\nThe proof is given in appendix \\ref{app:prop}. It is worth noting that while the condition in eq.~\\eqref{condition} is not too restrictive, it is also unavoidable. Conceptually, this condition amounts to the committee $\\mathcal{C}$ of peers being collectively reliable. Equivalently, it is not possible to infer the reliability of an expert based on opinions from a group that often collectively fails to identify the true state of the world. Formally, we restrict our attention to the case where experts are reasonably reliable: $p_i>\\nicefrac{1}{2}$, $\\forall i\\in[M]$, which implies that eq.~\\eqref{condition} holds for every subset $\\mathcal{C}\\subset[M]$ of experts. We further discuss the definition and the properties of the pseudo competence for the experts that belong to the committee $i,j\\in\\mathcal{C}$ in appendix \\ref{app:morepseudo}.\n\n\n\\section{Fundamental Properties of Different BEE and SWARM Rules}\n\\label{sec:bee}\nThere is a rich literature governing how to jointly sample and consult experts when feedback from the true state of the world $\\myset{Y(\\tau): \\tau\\in[t-1]}$ is available to the player. In this section, we propose unsupervised applications of different exploration-exploitation strategies using pseudo competence as reward. Specifically, we focus on UCB1, \\cite{auer2002finite}, due to its use of the Chernoff-Hoeffding bound, KL-UCB, \\cite{garivier2011klucb}, due its optimality for the Bernoulli rewards, IMED, \\cite{honda2015imed}, due to its near-optimal performance and empirical robustness, MOSS, \\cite{audibert2009minimax}, due to its minmax optimality, and Thompson sampling due to the strong synergy between posterior sampling and pseudo competences. The proposed BEE algorithms have the following form:\n\n\\begin{algorithm}\n\t\\caption{BEE Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE \\textbf{Require:} Choice of $\\Psi_i(\\cdot)$ for exploration-exploitation strategy, as defined in eq. \\eqref{ucb1stat}-\\eqref{thompson_stat}.\n\t\t\\STATE \\textbf{Initialize:} \n\t\t\\bindent\n\t\t\\STATE Consult every expert on the first task, \n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{1}=[M]$.\n\t\t\\eindent\n\t\t\\STATE \\textbf{Loop:}  \n\t\t\\bindent\n\t\t\\STATE $\\comm{t}\\leftarrow$ Pick top $m$ (bottom for IMED) experts ranked by $\\Psi_i(t)$,\n\t\t\\STATE $I_t \\leftarrow \\argmax_{i\\in[\\comm{t}]} \\Psi_i(t)$ ($\\argmin$ for IMED),\n\t\t\\STATE Commit to $X_{I_t}$ as the player decision,\\\\\n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{t}$.\n\t\t\\eindent\n\t\\end{algorithmic}\n\\end{algorithm}\n\nAs discussed in Section \\ref{sec:pseudocomp}, the notion of pseudo competence is built upon the availability of peer opinions therefore, we will consider consulting multiple experts for each task $t\\in[T]$, equivalent to playing multiple arms, see, for instance, \\cite{komiyama2015optimal}. Let $\\comm{t}$ denote the set of experts consulted for task $t\\in[T]$ and define the reward from consulting an expert as whether that expert agrees with the collective decision of the peers for that task: \n\\begin{equation}\n\t\\label{unsupervised_reward}\n\tR_i(t;\\comm{t}) = \\ind{X_i(t) = \\V{\\myset{X_j\\yay{t}:~j\\in\\comm{t}\\setminus\\myset{i}}}}.\n\\end{equation} \nObserve that contrary to the state feedback, opinions $\\myset{X_j:j\\in\\comm{t}}$ are available from any subset $\\comm{t}\\subset [M]$ of experts therefore, the empirical pseudo competence estimate for an expert at time $t$ takes the form:  \n\\begin{equation}\n\t\\label{empirical_pseudo}\n\t\\pp{i,T_i(t-1)} \\triangleq  \\frac{1}{T_{i}(t-1)}\\sum_{\n\t\t\\tau\\in[t-1]:~i\\in \\comm{\\tau}} R_i(\\tau;\\comm{\\tau}).\n\\end{equation}\t\nWe will now use the statistics on agreements among experts as defined in eq. \\eqref{unsupervised_reward} and the concomitant empirical competence estimation in eq. \\eqref{empirical_pseudo} to form the relevant statistics $\\Psi_i(t)$ for sampling and consulting experts in the unsupervised setup.  \n\nTo begin with the upper-confidence bound strategies, UCB1 leads to consulting experts that have highest ranking in:\n\\begin{equation}\n\t\\label{ucb1stat}\n\t\\Psi_i^{\\text{UCB1}}(t) = \\pp{i,T_i(t-1)}+\\sqrt{\\frac{2\\log t}{T_i(t-1)}}.\n\\end{equation}\nParticularly in the Bernoulli case, several refinements to UCB1 algorithm exist: Notably, \\cite{garivier2011klucb} shows that Kullback-Leibler upper confidence bound (KL-UCB) algorithm achieves the regret lower bound of \\cite{lai1985asymptotically}. In the unsupervised setup, experts with the highest ranking according to the following (operational)\\footnote{\\cite{garivier2011klucb} remark that the threshold $\\log\\yay{t}$ should be taken as $\\log\\yay{t}+c\\log\\log\\yay{t}$ for analytical purposes, where they recommend $c=0$ in practice. Furthermore, we experimentally implement its more robust counterpart KL-UCB+, see \\cite{honda2019note}.} statistics are consulted:\n\\begin{equation}\n\t\\label{klstat}\n\t\\Psi_i^{\\text{KL-UCB}}(t) = \\max\\yay{q\\in\\brac{0,1}:~ T_i(t-1)~ d\\yay{\\pp{i,T_i(t-1)}, q}\\leq \\log\\yay{\\frac{t}{T_i\\yay{t-1}}}},\n\\end{equation} \nwhere Kullback-Leibler divergence $d\\yay{\\cdot,\\cdot}$ is defined as:\n\\begin{equation*}\n\td\\yay{p,q} = p\\log\\frac{p}{q} + \\yay{1-p}\\log\\frac{1-p}{1-q}.\n\\end{equation*}\nAlbeit more computationally demanding, the performance of the KL-UCB algorithm makes it an appealing choice for binary opinions. Alternatively, using indexed minimum empirical divergence (IMED) algorithm of \\cite{honda2015imed}, we consult the experts that have the lowest ranking according to:\n\\begin{equation}\n\t\\label{imed_stat}\n\t\\Psi_i^{\\text{IMED}}(t) = T_i(t-1)~d\\yay{\\pp{i,T_i(t-1)}, \\max_{i\\in[M]}\\pp{i,T_i(t-1)}} + \\log T_i(t-1).\n\\end{equation}\nAnother key idea is that of minmax optimality, which is achieved by MOSS in the supervised framework and can be applied to the unsupervised framework by selecting experts with highest ranking in: \n\\begin{equation}\n\t\\label{moss_stat}\n\t\\Psi_i^{\\text{MOSS}}(t) = \\pp{i,T_i(t-1)} + \\sqrt{\\frac{\\max\\yay{\\log\\yay{\\frac{T}{M T_i\\yay{t-1}}} , 0}}{T_i\\yay{t-1}}}.\n\\end{equation}\n\nFinally, Thompson sampling (TS) suggests randomly sampling arms based on the posterior distribution over competences. Specifically, \n\\begin{equation}\n\t\\label{thompson_stat}\n\t\\Psi^{\\text{TS}}_{i}\\yay{t} = Z_i, \\text{ where } Z_i \\sim \\text{Beta}\\yay{\\alpha_i(t),\\beta_i(t)}.\n\\end{equation} \nThe prior is often taken be uniform for every expert, $\\yay{\\alpha_i(0),\\beta_i(0)} = \\yay{1,1}$, $\\forall i\\in[M]$ and, in the Bernoulli case, update rules of the beta distribution have a well-defined form, see, for instance, \\cite{russo2017tutorial}: \n\\begin{equation*}\n\t\\yay{\\alpha_{i}(t), \\beta_{i}(t)} = \\yay{\\alpha_{i}(t), \\beta_{i}}(t) + \\yay{R_{i}\\yay{t,\\comm{t}} + 1-R_{i}\\yay{t,\\comm{t}}}, \\forall i\\in\\comm{t},\n\\end{equation*}\nwhere $R_i\\yay{t,\\comm{t}}$ is defined in eq. \\eqref{unsupervised_reward} and the statistics  $\\yay{\\yay{\\alpha_{i},\\beta_{i}}: j\\in[M]\\setminus \\comm{t}}$ of the non-consulted experts are not updated at that time. \n\nAs a random process, $R_i\\yay{t,\\comm{t}}$, as shown in eq. \\eqref{unsupervised_reward}, is defined conditionally with respect to the choice of experts through $\\Psi_i\\yay{\\cdot}$, $\\forall i\\in[M]$, and the resulting opinions, up to time $t$ of consulting. Therefore, it is challenging to form a complete theoretical analysis of how choosing $R_i\\yay{t,\\comm{t}}$ changes the the difference between the ``rewards'' in the unsupervised setup compared to those that correspond to the true competences. Nevertheless, for a fixed committee $\\comm{}$ of peers, choosing $R_i\\yay{t,\\comm{}}$ pushes the competences together in a way that is as quantified through eq. \\eqref{pushtogether} of Proposition \\ref{prop}, making it possible to observe the impact of using pseudo competences on the regret of the exploration-exploitation strategies. The following result is a corollary to Proposition \\ref{prop} and \\cite[Theorem 2]{audibert2009minimax} for UCB1, \\cite[Theorem 1]{garivier2011klucb} for KL-UCB, \\cite[Theorem 5]{honda2015imed} for IMED, \\cite[Theorem 6]{audibert2009minimax} for MOSS, and \\cite[Theorem 1]{agrawal2013further} for Thompson sampling.\n\n\\begin{lemma}\n\t\\label{main_lemma}\n\tLet the player designate a subset of experts $\\comm{}\\subset[M]$ as peers prior to exploration and exploitation and consult them for every task $\\comm{t}=\\comm{}$, $\\forall t\\in [T]$. Define the difference between competences:\n\t\\begin{equation}\n\t\t\\Delta_i = \\max_{j\\in[M]} p_j - p_i,\n\t\\end{equation}\n\tand a potential function for a given set of competences:\n\t\\begin{equation}\n\t\t\\Phi = \\frac{\\log T}{T}\\sum_{\\substack{i\\in[M]\\setminus\\comm{}\\\\\\Delta_i>0}} \\frac{1}{\\Delta_i}.\n\t\\end{equation}\n\tThen, the regret of a BEE algorithm that picks $I_t = \\argmax_{i\\in[M]\\setminus\\comm{}}\\Psi_i(t)$ ($\\argmin$ for IMED) via statistics for UCB1, eq. \\eqref{ucb1stat}, KL-UCB, eq. \\eqref{klstat}, or IMED, eq. \\eqref{imed_stat}, is bounded by: \n\t\\begin{equation}\n\t\t\\label{theform}\n\t\t\\tilde{R}_{T}\\leq C\\frac{ \\Phi}{2p_{\\comm{}}-1},\n\t\\end{equation}\n\twhere the constant $C=10$ for UCB1, $C=\\nicefrac{1}{2}$ for KL-UCB and IMED. The regret of Thompson sampling, eq. \\eqref{thompson_stat}, also takes the form in eq. \\eqref{theform} with $C=1+\\varepsilon$ but with additional term of order $O\\yay{\\frac{M}{\\varepsilon^2}}$. Furthermore, for MOSS, eq. \\eqref{moss_stat}, the regret is upper bounded by: \n\t\\begin{equation}\n\t\t\\tilde{R}_{T}^{MOSS} \\leq \\frac{23M}{T}\\brac{\\frac{1}{2p_{\\comm{}}-1}\\sum_{\\substack{i\\in[M]\\setminus\\comm{}\\\\\\Delta_i>0}} \\frac{\\max\\yay{\\log\\frac{T  \\yay{\\yay{2p_{\\comm{}}-1}\\Delta_i}^{2}           }{M},1}}{\\Delta_i}}.\n\t\\end{equation}\n\\end{lemma} \nThe proof is given in appendix \\ref{app:mainlemma}. The important takeaways from Lemma \\ref{main_lemma} are twofold: First, using pseudo competences introduces additional regret within a constant factor of the original regret bounds but facilitates exploration and exploitation in the unsupervised regime. Second, the factor $\\nicefrac{1}{\\yay{2p_{\\comm{}}-1}}$ diminishes as the committee $\\comm{}$ collectively becomes more competent, which is important since the proposed BEE and SWARM rules that gradually select more competent experts. \n\nBy consulting multiple experts, a set of opinions on each task is acquired and making an unsupervised decision based on such sparsely sampled opinions is often carried out by rules such as SML, EM, or BP from \\cite{parisi2014ranking,welinder2010online,karger2011iterative} respectively. However, inference of pseudo competences allow an unsupervised opinion aggregation strategy that instantaneously yield a decision:  \n\\begin{equation}\n\t\\label{lpnb}\n\tf\\yay{(X_i(t),\\pp{i,T_i(t)}) : i\\in\\comm{t}} = \\sign\\yay{\\sum_{i\\in\\comm{t}}X_i(t)\\yay{\\pp{i,T_i(t)}-\\nicefrac{1}{2}}}.\n\\end{equation}   \nThe opinion-aggregation rule in eq. \\eqref{lpnb} is a linearized variant of the na\\\"{i}ve Bayes decision rule, which is robust to the variance of empirical averaging in \\eqref{empirical_pseudo}, as discussed in \\cite{berend2015finite}. In Section \\ref{sec:experiments}, we compare the BEE algorithm to their SEE counterparts and we demonstrate the performance of the SWARM rule that uses linearized na\\\"{i}ve Bayes decision rule with pseudo competence-based weights, comparing them to their state feedback-based counterparts. \n\n\\begin{algorithm}[t]\n\t\\caption{SWARM Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE \\textbf{Require:} \n\t\t\\bindent\n\t\t\\STATE Choice of $\\Psi_i\\yay{\\cdot}$ for exploration-exploitation strategy, as defined in eq. \\eqref{ucb1stat}-\\eqref{thompson_stat}.\n\t\t\\STATE Choice of $f\\yay{\\cdot}$ for unsupervised opinion aggregation, as defined \\eqref{lpnb}\n\t\t\\eindent \t\n\t\t\\STATE \\textbf{Initialize:} \n\t\t\\bindent\n\t\t\\STATE Consult every expert on the first task, \n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{1}=[M]$,\n\t\t\\STATE Commit to majority vote: $\\V{X_i:i\\in[M]}$ as the first decision. \n\t\t\\eindent\n\t\t\\STATE \\textbf{Loop:}  \n\t\t\\bindent\n\t\t\\STATE $\\comm{t}\\leftarrow$ Pick top $m$ (bottom for IMED) experts ranked by $\\Psi_i(t)$,\n\t\t\\STATE Update agreements via eq. \\eqref{unsupervised_reward}, estimate competences via \\eqref{empirical_pseudo} with $\\comm{t}$.\n\t\t\\STATE Commit to $f\\yay{(X_i(t),\\pp{i,T_i(t)}) : i\\in\\comm{t}}$ as the player decision.\\\\\n\t\t\\eindent\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn this section, we discuss empirical performance of BEE and SWARM algorithms that rely on pseudo competences for reliability inference and UCB1, KL-UCB, IMED, MOSS, or Thompson sampling for exploration and exploitation. For experiments, we chose $M=100$ experts with competences chosen uniformly at random from $\\brac{0.5,0.75}$ being consulted on $T=10^5$ tasks. The reason for such a competence interval is to avoid powerful experts in a community that would render the lack of supervision obsolete, similar to the use of weak classifiers in the boosting framework, \\cite{schapire2012boosting}. For BEE  experiments, we consider the true normalized regret as defined in eq. \\eqref{real_regret}, where for SWARM experiments, we consider the pseudo normalized regret in eq. \\eqref{swarm_regret}, allowing the opinion-aggregation rule over the best committee to have direct access to the true competences. In other words, BEE algorithms are compared against supervised rules with state feedback, where SWARM algorithms compete against rules that explore and exploit with state feedback and aggregate opinions with complete information. Figure \\ref{fig:bee_overall} provides an overview of BEE algorithms and Figure \\ref{fig:swarm_overall} provides one for SWARM rules. \n\n\\subsection{BEE}\nWe allow subcommittee sizes $m\\in\\brac{2,24}$, even numbers, where $m$ is the number of experts that are consulted for each task. Even number of experts chosen to make sure peers for every expert can reach a deterministic decision. \n\nFigure \\ref{fig:bee_overall} summarizes the performance of BEE algorithms that adaptively choose their subcommittees. We note that MOSS- and TS-based policies perform better than the rest for small subcommittee sizes, where KL-UCB-based BEE rule outperforms all for larger subcommittee sizes $\\yay{m>8}$. Figure \\ref{fig:bee_reference} illustrates the performance of UCB1, KL-UCB, Thompson sampling, MOSS, and IMED with multiple plays for reference. Interestingly, BEE algorithms that use upper-confidence bound-based exploration and exploitation strategies, as seen in Figures \\ref{fig:klucbbee}-\\ref{fig:ucb1bee}, perform almost identically to their SEE counterparts for large subcommittee sizes. On the other hand, the impact of supervision is clearer across the board for IMED, in Figure \\ref{fig:imedbee}, MOSS, in Figure \\ref{fig:mosssbee}, and Thompson sampling, as seen in Figure \\ref{fig:thompsonbee}. Importantly, Figures \\ref{fig:imedbee}-\\ref{fig:klucbbee} indicate that BEE rules that rely on IMED and KL-UCB sequentially consult experts that on average provide opinions within $0.5\\%$ of the probability of correctness of the best available expert without any supervision upon consulting modest number $\\yay{m=8}$ of experts for each task.  \n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth,draft=false]{swarm_bee_overall.jpg}\n\t\t\\caption{Performance Comparison of \\\\ Different BEE Algorithms}\n\t\t\\label{fig:bee_overall}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth,draft=false]{pnbee_overall.jpg}\n\t\t\\caption{Performance Comparison of \\\\ Different SWARM Algorithms}\n\t\t\\label{fig:swarm_overall}\n\t\\end{minipage}%\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\t\t\t\t\t\n\t\t\\includegraphics[width=\\linewidth,draft=false]{swarm_see_overall.jpg}\n\t\t\\caption{Overview of Multi-play SEE Algorithms}\n\t\t\\label{fig:bee_reference}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{imed_swarm.jpg}\n\t\t\\caption{IMED}\n\t\t\\label{fig:imedbee}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{kl_ucb_swarm.jpg}\n\t\t\\caption{KL-UCB}\n\t\t\\label{fig:klucbbee}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{ucb1_swarm.jpg}\n\t\t\\caption{UCB1}\n\t\t\\label{fig:ucb1bee}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{moss_swarm.jpg}\n\t\t\\caption{MOSS}\n\t\t\\label{fig:mosssbee}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{thompson_swarm.jpg}\n\t\t\\caption{Thompson Sampling}\n\t\t\\label{fig:thompsonbee}\n\t\\end{minipage}\n\t\\caption{Overview of the BEE Experiments: Performance Comparison to Supervised Counterparts}\n\\end{figure}\n\n\\subsection{SWARM}\nSubcommittee sized up to $m=20$ appears to be sufficient for SWARM experiments as all of the tested rules demonstrate lower than $1\\%$ normalized pseudo regret for $m>10$. Figure \\ref{fig:swarm_overall} outlines the performance of the SWARM algorithms, with their supervised counterpart given in Figure \\ref{fig:swarm_reference}, showing that MOSS-, IMED- and TS-based rules perform the with the additional opinion-aggregation objective. It might be expected for TS, as shown in Figure \\ref{fig:thompsonwarm}, considered the synergy between consulting experts based on the posterior distribution on the competences and using an approximation of the maximum a posteriori rule that is na\\\"{i}ve Bayes for opinion aggregation. It is worth noting however, that the performance of IMED-based and MOSS-based consultation, which have overall robust performance in the BEE framework as shown in Figures \\ref{fig:imedbee},\\ref{fig:mosssbee}, stand out in the SWARM framework, as seen in Figures \\ref{fig:mossswarm},\\ref{fig:imedswarm}. Furthermore, Figures \\ref{fig:ucb1swarm},\\ref{fig:klucbswarm} indicate a similar phenomenon for the upper-confidence bound-based strategies as the BEE framework, where the performance difference between supervised and unsupervised rules diminish almost entirely. Importantly, SWARM rules achieve probability of correctness within $0.5\\%$ of the opinion aggregation rule that has direct access to true competences and uses the opinions from the best available experts.  \n\n\\begin{figure}\n\t\\centering\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\t\t\t\t\t\n\t\t\\includegraphics[width=\\linewidth,draft=false]{pnbee_reference.jpg}\n\t\t\\subcaption{Overview Multi-play SEE \\\\ with Opinion Aggregation}\n\t\t\\label{fig:swarm_reference}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{thompson_pnbee.jpg}\n\t\t\\subcaption{Thompson Sampling}\n\t\t\\label{fig:thompsonwarm}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{moss_pnbee.jpg}\n\t\t\\subcaption{MOSS}\n\t\t\\label{fig:mossswarm}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{imed_pnbee.jpg}\n\t\t\\subcaption{IMED}\n\t\t\\label{fig:imedswarm}\n\t\\end{minipage}\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{ucb1_pnbee.jpg}\n\t\t\\subcaption{UCB1}\n\t\t\\label{fig:ucb1swarm}\n\t\\end{minipage}%\n\t~\n\t\\begin{minipage}{.32\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth,draft=false]{kl_ucb_pnbee.jpg}\n\t\t\\subcaption{KL-UCB}\n\t\t\\label{fig:klucbswarm}\n\t\\end{minipage}\n\t\\caption{Overview of the SWARM Experiments: Performance Comparison to Supervised Counterparts}\n\\end{figure}\n\n\n\\section{Conclusions}\nIn this paper, we proposed techniques that allow exploration-exploitation algorithms to operate in a regime where the rewards are observed indirectly from opinions on dynamically changing tasks. The proposed measure of competence not only allows instantaneous inference, which commonly requires either state feedback or block processing, but also enables the use of a near-optimal opinion aggregation rule in real time. Furthermore, we point out that the formal analysis of the convergence and competence ordering for BEE and SWARM algorithms are exciting open problems with the numerical evidence presented here indicating their strong potential. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Catalan numbers have a rich history. Its ubiquity in mathematics underscores the importance of this sequence. The Catalan sequence has appeared time and again in the works of numerous mathematicians. There are so many objects counted by the Catalan numbers, there's a whole book dedicated to exploring them \\cite{stanCat}. Some of those combinatorial objects are triangulations of convex polygons with $n+2$ vertices (Fig. \\ref{fig:triangulation}), binary trees with $n+2$ vertices, plane trees with $n+1$ vertices, ballot sequences of length $2n$, parenthetizations, and Dyck paths of length $2n$.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=6cm]{Triang_RegPoly.PNG}\n    \\caption{Triangulations of convex polygons with 3, 4, 5, and 6 vertices}\n    \\label{fig:triangulation}\n\\end{figure}\n\nThe modern way we describe Catalan numbers stems from Eug\\`ene Catalan's interest in the triangulation of polygons problem \\cite{stanCat}. Catalan numbers are defined as follows,\n\n\\begin{equation*} \\label{CatChoose}\n    C_n = \\frac{1}{n+1} \\binom{2n}{n}\n\\end{equation*}\n\nWe can also define Catalan numbers in terms of its generating function,\n\\begin{equation*}\\label{CatGenFunc}\n    C(x) = \\sum_{n\\geq 0} = 1+x+2x^2+5x^3+14x^4+\\dots,\n\\end{equation*}\nwhere the $n^{th}$ coefficient is the $n^{th}$ Catalan number. \n\nEuler and Goldbach studied the triangulation of convex polygons which gives us the following recursive definition,\n\\begin{equation*} \\label{FundRec}\nC_{n+1} = \\sum_{k=0}^{n} C_n C_{n-k},\\hspace{1em} C_0=1.\n\\end{equation*}\n\nMany objects enumerated by the Catalan numbers exhibit the same recursive property of the numbers themselves. That is, given a Catalan object, a rule exists that describes how to produce an object from the objects that came before it.\n\nWe introduce a new Catalan object called the \\emph{externally ordered poset of unit interval positroid} and explore its potential recursion. The elements of this poset are bases of a special matroid called a Unit Interval Positroid (UIP), a Catalan object introduced by Chavez--Gotti \\cite{chavez}. The partial order $\\leq_{\\mbox{Ext}}$ is the external order of matroid bases as defined by Las Vergnas \\cite{lvergnas}. We denote such a poset as $\\mbox{Ex}(\\mathcal{P})$, where $\\mathcal{P}$ is a positroid. Our main result is the following theorem:\n\\begin{thm:main} Let $\\mathcal{P}_n = ([2n],\\mathcal{B})$ be the trivial UIP and $\\mbox{Ex}(\\mathcal{P}_n)$ be the poset of externally ordered bases of $\\mathcal{P}_n$. Let $\\gamma$ be the algorithm defined in Section \\ref{algorithm}. Then $\\gamma(\\mbox{Ex}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$, where the partial order induced by $\\gamma(\\mbox{Ex}(\\mathcal{P}_n))$ is the external order.\n\\end{thm:main}\nOur main result shows recursion can be described nicely for the poset of the trivial UIP as well as enumerate the bases of the positroid. This is the first step towards describing the complete recursion for these special posets.\n\nThis paper is organized as follows. In section 2 we provide the background material necessary for this paper. In Section 3 we introduce an algorithm to generate the externally ordered bases of a rank $n+1$ trivial UIP from a rank $n$ trivial UIP. In section 4 we present our main theorem which proves the algorithm preserves ordering. In section 5 we discuss future work.\n\n\\section{Background} \\label{background}\n\n\n\\subsection{Matroids}\n\nMatroids capture the essence of dependence as we know it in linear algebra and graph theory. Though several definitions of a matroid exist, we utilize the basis definition in our work. \n\n\\begin{definition}A \\emph{matroid} $\\mathcal{M}$ is a pair $(E,\\mathcal{B})$ consisting of a finite set $E$ and a nonempty collection of subsets $\\mathcal{B}=\\mathcal{B}(\\mathcal{M})$ of $E$ that satisfy the following properties:\n\t\\begin{enumerate}\n\t\t\\item[(B1)] $\\mathcal{B}\\neq\\emptyset$\n\t\t\\item[(B2)] (Basis exchange axiom) If $B_1,B_2\\in\\mathcal{B}$ and $b_1\\in B_1-B_2$, then there exists an element $b_2\\in B_2-B_1$ such that $B_1-\\{b_1\\}\\cup\\{b_2\\}\\in\\mathcal{B}$.\n\t\\end{enumerate} \n\tThe element $B\\in \\mathcal{B}$ is called a \\emph{basis} and $|B|$ is the rank of $\\mathcal{M}$. A minimally dependent set $C$, that is $C-e$ is independent for any $e\\in C$, is called a \\emph{circuit}. The finite set $E$ is called the \\emph{ground set.}\n\\end{definition}\n\n\\begin{example}\\label{ex:p2_positroid}\nLet $E=\\{1,2,3,4\\}$ be the columns of the following matrix,\n\\vspace{-1em}\n\\begin{center}\n         \\[A=\n    \\bordermatrix{ & \\phantom{-}1 & 2 & \\phantom{-}3 & \\phantom{-}4 \\cr\n       & \\phantom{-}1 & 0 & -1 & -1 & \\cr\n       & \\phantom{-}0 & 1 & \\phantom{-}1 & \\phantom{-}1 }.\n    \\]\n    \\end{center}\nThen the set of bases $\\mathcal{B}$ of the matroid over $E$ is the collection of all maximally independent sets of $E$. That is, $\\mathcal{B} = \\{12,13,14,23,24\\}$. Considering all the sets of minimally dependent sets of the columns of $A$, we get that the set of circuits of $\\mathcal{M}$ are $\\mathcal{C} = \\{123,124,34\\}$. One can easily see that $\\mathcal{B}$ satisfies the bases axioms above.\n\\end{example}\n\nThe matroid described in Example \\ref{ex:p2_positroid} is part of a larger class of matroids called realizable. A matroid $\\mathcal{M}$ whose bases are in bijection with the sets of maximally independent columns of some matrix $A$ over some field $\\mathbb{F}$ is called \\emph{realizable} and denoted $\\mathcal{M}(A)$.\n\nA matrix $A$ is called \\emph{totally nonnegative} if all of its maximal minors are nonnegative. Thus, one can impose this restriciton to the family of realizable matroids and arrive at the following definition due to Postnikov \\cite{pos}.\n\n\\begin{definition}\nA \\emph{positroid} of rank $r$ over $[n]$ is a realizable matroid such that the associated full rank $r\\times n$ matrix $A$ is totally nonnegative. \n\\end{definition}\n\nThe matroid $\\mathcal{M}(A)$ given in example \\ref{ex:p2_positroid} is a positroid of rank 2. In fact, $\\mathcal{M}(A)$ is a unit interval positroid, which we discuss in the next section.\n\n\\subsection{Unit Interval Positroids}\n\nRecall that a \\emph{partially ordered} set, $P$ or \\emph{poset}, is a set with a relation $\\leq$ that satisfies reflexivity, antisymmetry, and transitivity \\cite{stan}. We can represent the poset $(P,\\leq)$ as a Hasse diagram that shows the elements of $P$ with the cover relations. See figure \\ref{fig:UIO rank 6}.\n\n\\begin{definition}\nA \\emph{poset} $P$ is a \\emph{unit interval order} if there exists a bijective map $i \\mapsto [q_i,q_{i}+1]$ from $P$ to a set $S=\\{ [q_i,q_{i}+1] | 1\\leq i \\leq n, q_i \\in \\mathbb{R} \\} $ of closed unit intervals of the real line such that for $i,j\\in P, i <_P j$ if and only if $q_i+1 < q_j$. We then say that $S$ is an \\emph{interval representation} of $P$.\n\\end{definition}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{images\/UIOrder.png}\n    \\caption{Hasse diagram of a unit interval order with 6 elements and its unit interval representation.}\n    \\label{fig:UIO rank 6}\n\\end{figure}\n\nAssociated to every unit interval order is an antiadjacency matrix, the key to describing unit interval positroids.\n\n\\begin{definition}\nFor an $n$-labeled poset $P$, the \\emph{antiadjacency matrix} of $P$ is the $n \\times n$ binary matrix $A=(a_{i,j})$ with $a_{i,j} =0$ if and only if $i <_P j$.\n\\end{definition}\n\nSkandera--Reed \\cite{skandera} proved that by labeling the unit interval order appropriately, every minor of the corresponding antiadjacency matrix $A$ is nonnegative. That is, the determinant of every submatrix of $A$ is 0 or a positive number. This fact, combined with the following result, leads us to our positroid of interest.\n\n\\begin{lemma}\\label{lem:induced_positroid}(Postnikov, \\cite{pos} Lem 3.9)\n\nFor an $n \\times n$ real matrix $A=(a_{i,j})$, consider the \n$n \\times 2n$ matrix $B = \\psi(A)$ where\n\\small{\n\\begin{equation*}\n\\arraycolsep=3pt\n\\left( \\begin{array}{ccc}\na_{1,1}& \\cdots & a_{1,n}\\\\\n\\vdots& \\ddots& \\vdots\\\\\na_{n-1,1}& \\cdots& a_{n-1,n}\\\\\na_{n,1}& \\cdots& a_{n,n}\n\\end{array} \\right)\n\\overset{\\psi}{\\longmapsto}\n\\left( \\begin{array}{cccccccc}\n1&\\cdots&0&0&(-1)^{n-1}a_{n,1}& \\cdots &(-1)^{n-1}a_{n,n} \\\\\n\\vdots&\\ddots&\\vdots&\\vdots& \\vdots & \\ddots & \\vdots\\\\\n0 & \\cdots & 1 & 0  & -a_{2,1} &\\cdots & -a_{2,n}\\\\ \n0&\\cdots&0 &1 & \\phantom{1}a_{1,1} & \\cdots & \\phantom{1}a_{1,n}\n\\end{array}\\right).\n\\end{equation*}\nFor each pair $(I,J)$ with $I,J \\subseteq [n]$ and $|I|=|J|$, define the set \n\\begin{equation*}\n    K = K(I,J) = \\{n+1-k|k\\in [n]\\backslash I\\}\\cup \\{n+j|j\\in J\\}.\n\\end{equation*}}\nThen we have $\\Delta_{I,J}(A)=\\Delta_{[n],K}(B)$.\n\\end{lemma}\n\nLemma \\ref{lem:induced_positroid} gives us a way to encode the same information from the $n\\times n$ antiadjacency matrix into a corresponding $n\\times 2n$ matrix. Explicitly, Lemma \\ref{lem:induced_positroid} associates the determinants of the submatrices of the antiadjacency matrix to the maximal minors of the $n\\times 2n$ matrix. The antiadjacency matrix of a properly labeled unit interval order generates a totally nonnegative matrix, that is, a positroid. A positroid on $[2n]$ induced by a unit interval order is a \\emph{unit interval positroid} or UIP.\n\nSuppose we have a unit interval positroid $\\mathcal{P}$ and we let $C$ be a circuit such that $C\\in \\mathcal{C(P)}$. If we take an element $e\\in C$ from circuit $C\\in \\mathcal{C}(P)$ and $e$ is the smallest, then $C - \\{e\\}$ is a \\emph{broken circuit}. Bases that do not contain broken circuits are called \\emph{atoms}. However, if $B$ is a broken circuit, then there is an element $e$ such that $B \\cup \\{e\\}$ is a circuit and $e$ is the smallest. This notion of broken circuits is captured by the ordering discussed in the next section.\n\n\\subsection{External Activities}\nIn \\cite{lvergnas} Las Vergnas introduced the notion of active orders, a collection of related orders on bases of a matroid using broken circuits. We use only the external order to define our poset of interest, though one can derive the results for other orders from this one.\n\n\\begin{definition}\nLet $M$ be a matroid on a linearly ordered set $E$, and let $A\\subseteq E$. \nWe say an element $e\\in E$ is \\emph{M-active} with respect to $A$ if there is a circuit $C$ of $M$ such that $e\\in C \\subseteq A \\cup \\{e\\}$ and $e$ is the smallest element of $C$. We denote by $\\mbox{Act}_{M}(A)$ the set of \\emph{M-active} elements with respect to $A$.\n\\end{definition}\n\nTo determine the active elements for our positroids we look at those subsets $A\\subseteq E$ where $A$ is a basis. Then those elements $e$ are exactly the elements which make or break a circuit.\n\n\\begin{definition}\nThe \\emph{external set} of an element $A$ is obtained by setting $\\mbox{Ext}_{M}(A)=\\mbox{Act}_{M}(A)\\backslash A$.\n\\end{definition}\n\nThe following proposition defines the external order of a set of bases of a matroid.\n\n\\begin{proposition}\\label{lem:lasvergnas_def} (\\cite{lvergnas}, Proposition 3.1)\nLet $A$, $B$ be two bases of an ordered matroid $M$. The following properties are equivalent:\n\\begin{enumerate}\n    \\item $A \\leq_{\\mbox{Ext}} B$;\n    \\item $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$;\n    \\item $A \\cup \\mbox{Ext}_{M}(A) \\subseteq B \\cup \\mbox{Ext}_{M}(B)$\n   \n    \\item B is the greatest, for the lexicographic ordering, of all bases of $M$ contained in $A \\cup B$.\n\\end{enumerate}\n\n\\end{proposition}\n\nLet $\\mathcal{P}$ be a unit interval positroid generated by the poset where all elements are incomparable. Then we call $\\mathcal{P}$ \\emph{trivial}. We utilize Proposition \\ref{lem:lasvergnas_def}(2) on the trivial UIP in order to determine the externally ordered poset of its bases. For an example of generating the matrix representing the trivial UIP of rank 3, see Figure \\ref{fig:trivialUIP3}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.85\\linewidth]{interval_to_matrix.png}\n    \\caption{A trivial UIP of rank 3}\n    \\label{fig:trivialUIP3}\n\\end{figure}\n\\pagebreak\n\\begin{example}\\label{Ex: calculating external}\nConsider the trivial UIP of rank 3, $\\mathcal{P}_3$. We will determine the external ordering on its bases. Using proposition \\ref{lem:lasvergnas_def}, we first determine the active elements for every basis $B\\in\\mathcal{B}$, that is $\\mbox{Act}_{\\mathcal{P}_3}(\\mathcal{B})$. The bases and circuits of $\\mathcal{P}_3$ are described as follows,\n\n\\begin{align*}\n\\mathcal{C}_{\\mathcal{P}_{3}}  = \\{&(1,2,3,4),(1,2,3,5),(1,2,3,6),(4,5),(4,6),(5,6)\\},\\\\\n\\mathcal{B}_{\\mathcal{P}_{3}}  = \\{&(1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,3,4),(1,3,5),(1,3,6),\\\\   \n    & (2,3,4),(2,3,5),(2,3,6)\\}.\n\\end{align*}\n\n\nThe following tables show necessary calculations.\n\\begin{center}\n\\begin{multicols}{2}\n\\begin{tabular}{ |p{0.5cm}|p{1.7cm}|p{2.3cm}|  }\n\\hline\n\\multicolumn{3}{|c|}{Calculating M-Active Elements} \\\\\n\\hline\n$e$& $c\\in\\mathcal{C_{P}}$&$A\\cup \\{e\\}$ \\\\\n\\hline\n1 & $(1,2,3,4)$ & $(2,3,4) \\cup \\{1\\}$ \\\\\n & $(1,2,3,5)$ & $(2,3,5) \\cup \\{1\\}$ \\\\\n & $(1,2,3,6)$& $(2,3,6) \\cup \\{1\\}$ \\\\\n2 & none  & \\\\\n3 & none &  \\\\\n4 & $(4,5)$ & $(1,2,5)\\cup \\{4\\}$ \\\\\n & &$(1,3,5)\\cup \\{4\\}$ \\\\\n & &$(2,3,5)\\cup \\{4\\}$ \\\\\n & $(4,6)$ &$(1,2,6)\\cup \\{4\\}$ \\\\\n & &$(1,3,6)\\cup \\{4\\}$ \\\\\n & &$(2,3,6)\\cup \\{4\\}$ \\\\\n5 & $(5,6)$& $(1,2,6)\\cup \\{5\\}$  \\\\\n & & $(1,3,6)\\cup \\{5\\}$  \\\\\n & & $(2,3,6)\\cup \\{5\\}$  \\\\\n6 & none & \\\\\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{ |p{1.4cm}|p{1.5cm}|p{1.5cm}|  }\n\\hline\n\\multicolumn{3}{|c|}{Calculating the External Set} \\\\\n\\hline\n$A$& $\\mbox{Act}_{M}(A)$&$\\mbox{Ext}_{M}(A)$ \\\\\n\\hline\n$(1,2,3)$ & none & $\\emptyset$ \\\\\n$(1,2,4)$ & none & $\\emptyset$\\\\\n$(1,2,5)$ & 4   & 4 \\\\\n$(1,2,6)$ & 4,5 & 4,5 \\\\\n$(1,3,4)$ & none & $\\emptyset$ \\\\\n$(1,3,5)$ & 4 & 4 \\\\\n$(1,3,6)$ & 4,5 & 4,5\\\\\n$(2,3,4)$ & 1   & 1 \\\\\n$(2,3,5)$ & 1,4 & 1,4 \\\\\n$(2,3,6)$ & 1,4,5 & 1,4,5 \\\\\n\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{center}\n\nTo form the externally ordered poset, we use Proposition \\ref{lem:lasvergnas_def}(2) in order to obtain the relations described by \\ref{lem:lasvergnas_def}(1). For example, for bases $A=(1,2,3)$ and $B=(2,3,4)$, we check if the containment $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$ is satisfied. We can observe that $(1,2,3)\\subseteq (2,3,4) \\cup \\{1\\}$, and thus $A \\leq_{\\mbox{Ext}} B$. Continue this calculation for all pairs of bases until all bases are ordered. The resulting object, which is a set of bases ordered by the relation $\\leq_{\\mbox{Ext}}$, is the externally ordered poset of the bases of $\\mathcal{P}_3$.\n\\end{example}\nRecall, we denote the poset of the trivial UIP bases with the external order as $\\mbox{Ex}(\\mathcal{P}_n)$. Continuing with Example \\ref{Ex: calculating external}, we will show how to compute $\\mbox{Ex}(\\mathcal{P}_2)$. We compare the bases of $\\mathcal{P}_2$ pairwise using Proposition \\ref{lem:lasvergnas_def}(2). That is, for every $A, B\\in \\mathcal{B}_{\\mathcal{P}_2}$, check if $A \\subseteq B \\cup \\mbox{Ext}_{M}(B)$ is satisfied. Then we can draw the following poset (figure on the right) given the information from the table of $\\mbox{Ex}(\\mathcal{P}_2)$. Using the information in the table below, and the condition just mentioned, we produce the Hasse diagram for $\\mbox{Ex}(\\mathcal{P}_2)$.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.50\\linewidth]{images\/p2posetandhasse.png}\n    \\caption{$\\mbox{Ex}(\\mathcal{P}_2)$ and Hasse diagram.}\n    \\label{fig:p2 and poset}\n\\end{figure}\n\nComparing every pair of bases to find the ordering is easy whenever the rank of the positroid is low. Quickly, this becomes a difficult task for higher ranks. We now introduce our algorithm which streamlines this process.\n\n\n\\section{Algorithm}\\label{algorithm}\n\nTo prove $\\mbox{Ex}(\\mathcal{P})$ exhibits Catalan recursion we first describe an algorithm that generates a set of elements from the set of bases of $\\mathcal{P}_n$ and defines an order induced by $\\mbox{Ex}(\\mathcal{P}_n)$. In the next section we prove this set is precisely the set of bases of $\\mathcal{P}_{n+1}$ and the induced order preserves the external order of the bases of $\\mathcal{P}_{n+1}$. \n\nLet $\\mathcal{B}_{\\mathcal{P}_n}$ be the bases of the trivial UIP $\\mathcal{P}_n$. Then we produce a set of elements in the following way. \n\n\\begin{enumerate\n\\item \\textcolor{blue}{``Reinforce\"}: for $B\\in \\mathcal{B}_{\\mathcal{P}_n}$, add 1 to all elements of $B$ and adjoin $\\min{B}$. The original order is kept.\n\\item \\textcolor{red}{``Build Up\"}: if $2n+1\\in B$ from step 1, then form $B\\backslash\\{2n+1\\}\\cup\\{2(n+1)\\}$ and it covers $B$.\n\\item \\textcolor{olive}{``Grow Spine\"}: if $2\\in B$ from steps 1 and 2, then form $B\\backslash\\{2\\}\\cup\\{1\\}$ and is covered by $B$.\n\n\\end{enumerate}\n\nThis completes the procedure for the algorithm. Figure \\ref{fig:P2_to_p3} illustrates how the algorithm is applied to the external poset of the trivial UIP of rank 2. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{steps_algorithm.png}\n    \\caption{The algorithm applied to the trivial UIP $\\mathcal{P}_2$ and the  resulting ordered set.}\n    \\label{fig:P2_to_p3}\n\\end{figure}\n\n\\section{Main Theorem}\\label{}\n\nDenote the algorithm above by $\\gamma$; that is, $\\gamma(\\mathcal{B}_{\\mathcal{P}_n})$ is the ordered set of elements produced by $\\gamma$. To prove our main result, we show $\\gamma(\\mathcal{B}_{\\mathcal{P}_n})$ is precisely the externally ordered poset of the bases of the rank $n+1$ trivial UIP. That is, we show $\\gamma$ produces the set of bases expected and the bases are externally ordered. \nWe begin by proving the bases and circuits of the trivial UIP have a nice description.\n\n\\begin{lemma}\\label{lem:bases_description} \n\nLet $\\mathcal{P}$ be the trivial unit interval positroid of rank $n$ and $B\\in \\mathcal{B}$ a basis of $\\mathcal{P}$. Then $B$ satisfies one of the following\n\n\\begin{enumerate}\n    \\item $B = (1,\\hdots,n)$,\n    \\item $B = (1,\\hdots,\\hat{j},\\hdots,n)\\bigcup \\{j\\}$ for $j\\in [n+1,\\hdots,2n]$,\n\\end{enumerate}\nwhere $\\hat{j}$ indicates $j$ is not included in the basis $B$. Moreover, $|\\mathcal{B}| = n^2+1.$\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lem:induced_positroid}, we know that $P$ is represented by the $n \\times 2n$ matrix,\n\n\\[\\begin{bmatrix}\n1&0&\\cdots &0&(-1)^{n-1}& \\cdots &(-1)^{n-1} \\\\\n0& 1 &\\cdots &0 & \\vdots & \\ddots & \\vdots\\\\\n\\vdots & \\ddots & 1 & 0  & (-1) &\\cdots & (-1)\\\\\n0&\\cdots&0 &1 & 1 & \\cdots & 1\\\\\n\\end{bmatrix},\\]\\\\\nwhere the first $n$ columns form the $n\\times n$ identity matrix and the last $[n+1,\\hdots, 2n]$ columns are equal. \n\nSince a basis corresponds to a non-zero $n\\times n$ maximal minor, we need only describe which submatrices are non-singular. The $n\\times n$ identity matrix is non-singular, which is the first basis description.\n\nSince the $[n+1,\\hdots,2n]$ columns are all the same, any non-singular maximal submatrix can include at most one of these columns. Thus, all other bases are formed by maximal submatrices with one column from $[n+1,\\hdots,2n]$ and the remaining a subset of the columns $[n]$, which is the second description.\n\nWe use our description to enumerate the bases. To form a basis of the second kind, we remove an element from $\\{1,2,\\dots,n\\}$ and replace it with any of the $n$ elements from $[n+1,\\dots,2n]$. This is done in $\\binom{n}{1}n$ ways. Including the basis $\\{1,2,\\dots,n\\}$, we get $$\\binom{n}{1} n + 1 =n^2 + 1$$ bases, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{circuit_description}\nLet $\\mathcal{P}_n$ be the trivial UIP and $\\mathcal{C}_{\\mathcal{P}_n}$ its set of circuits. Then any $C\\in \\mathcal{C}_{\\mathcal{P}_n}$ must either be $I\\cup i$ where $i\\in [n+1,2n]$ or $(i,j)$ for $i,j\\in[n+1,2n]$ and $i\\neq j$.\n\\end{lemma}\n\n\\begin{proof}\nNote that $I\\cup i$ where $i\\in [n+1,2n]$ is a circuit since the removal of any element produces a basis as described in Lemma \\ref{lem:bases_description}. Similarly, the pair $i,j\\in[n+1,2n]$ and $i\\neq j$ is a circuit since every singleton $i\\in[2n]$ is independent. Since any other subset of $[2n]$ contains one of these sets, there are no other circuits of $\\mathcal{P}_n$.\n\\end{proof}\n\nWe state the following observation about bases of trivial UIPs, though it is not used in any subsequent theorem. \n\\begin{lemma}\\label{min_elts_description}\nLet $B\\in \\mathcal{B}(\\mathcal{P}_n)$, where $\\mathcal{P}_{n}$ is a trivial unit interval positroid. Then $\\epsilon(B)=0$ if and only if $B$ is a minimally ordered basis.\n\\end{lemma}\n\n\\begin{proof}\n$(\\Rightarrow)$ Assume $\\epsilon(B)=0$ (i.e. Ext$(B)=\\emptyset$) and there exists $A\\in \\mathcal{B}$ such that $A\\lessdot B$. Then $A\\subset B\\cup \\text{Ext}(B) = B$, which implies $A=B$.\n\n$(\\Leftarrow)$ Assume $B$ is minimal and $\\text{Ext}(B)\\neq\\emptyset.$ First, note $B$ minimal implies $1\\in B$. If not, let $B'=B\\backslash2\\cup 1$. Then Ext$(B')=$ Ext$(B)\\backslash\\{1\\}$, so that $B'\\subset B \\cup $Ext$(B)$, implying $B'\\lessdot B$, a contradiction. Since $\\text{Ext}(B)\\neq\\emptyset,$ there exists $e=\\min(C)$ for some circuit $C$ of $P_n$ such that $C\\subset B\\cup\\{e\\}$. By Lemma \\ref{circuit_description}, $C=ej$ for some $j\\in[n+1,2n]$ and $j\\neq e$. Then Ext$(B)=\\{n+1,\\dots,e\\}$, and $j\\in B$ necessarily. Let $A=B\\backslash \\{j\\} \\cup \\{e\\}$. Then $A\\lessdot B$, a contradiction.\n\\end{proof}\n\nNext we prove $\\gamma$ produces the set of desired bases.\n\n\\begin{theorem}\\label{thm:AlgMakesBases}\nLet $\\mathcal{B}$ be the set of bases of the trivial unit interval positroid $\\mathcal{P}_{n}$ of rank $n$ over ground set $[2n]$. Then, after applying algorithm $\\gamma$ to $\\mathcal{B}$, $\\gamma(\\mathcal{B})$ is the set of bases for the trivial unit interval positroid $\\mathcal{P}_{n+1}$ of rank $n+1$ over ground set $[2(n+1)]$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{B}=\\mathcal{B}(\\mathcal{P}_{2n})$. We will show that for every $B\\in\\mathcal{B}$, $\\gamma(\\mathcal{B})$ produces a $B^{'}\\in\\mathcal{B}(\\mathcal{P}_{2(n+1)})$.\n Let $B=\\{1,\\dots,n\\}.$ Then $\\gamma(B)=\\{1,\\dots,n+1\\}$, which by Lemma \\ref{lem:bases_description}(1) is a basis of $\\mathcal{P}_{2(n+1)}$.\n Let $B=\\{1,2,\\dots,\\hat{j},\\dots,n\\}$, where $\\hat{j}\\in [n+1,\\dots, 2n]$. Applying step 1 of $\\gamma$ we have that $\\gamma(B) = B+1\\cup\\{min(B)\\}\\in\\gamma(\\mathcal{B}).$ Explicitly, this means $$\\{1,2,3,\\dots, \\hat{j}+1,\\dots,n+1\\}\\in\\gamma(\\mathcal{B}),$$ where $$\\hat{j}+1 \\in [n+2,2n+1].$$ Since $[n+2,\\dots,2n+1]\\subset [(n+1)+1,\\dots,2(n+1)]$, then by Lemma \\ref{lem:bases_description} we have that $\\gamma(B)$ is a basis of $\\mathcal{P}_{2(n+1)}$. Moreover, there are $n^2$ bases created at this step.\n \nFor the second step of $\\gamma$, let $\\mathcal{B}_{max}$ be the set of bases such that max$(\\gamma(B))=2n+1$. Then for all $B\\in \\mathcal{B}_{max}$, we have $B\\backslash\\{2n+1\\}\\cup \\{2(n+1)\\}$. All sets generated prior to this step have maximum value $2n+1$. Thus, all sets generated at this step of $\\gamma$ are new elements of $\\gamma({\\mathcal{B}})$. Every new set generated is formed by choosing a value of $[n]$ and replacing it with $2(n+1)$. Thus each set is a basis and we have $n$ new bases.\n \nFor the final step of $\\gamma$, let $\\mathcal{B}_{min}$ be the set of bases such that min$(\\gamma(B))=2$. Then for every $B\\in \\mathcal{B}_{min}$, $B\\backslash\\{2\\}\\cup \\{1\\}\\in\\gamma(\\mathcal{B})$. Note Lemma \\ref{lem:bases_description} implies $3\\in B$ for every $B\\in \\mathcal{B}_{min}$. Then for every set generated at this step, it must contain $1,3$ and not $2$. By construction, all sets generated by $\\gamma$ prior to the third step of $\\gamma$ have that their first two entries are consecutive. Thus, all sets generated in this third step are new and, by Lemma \\ref{lem:bases_description}, a basis. Thus all sets are bases of $\\mathcal{P}_{n+1}$.\n\nTo enumerate the bases generated in this final step of $\\gamma$, let us analyze the number of bases it applies to. By the first step of the algorithm, every basis $B\\in\\mathcal{B}$ such that $\\min(B)=2$ produces a basis in $\\mathcal{B}(\\mathcal{P}_{2(n+1)})$ with minimum element 2. Thus, the number of bases produced equals the number of $B\\in\\mathcal{B}$ such that $\\min(B)=2$, which is $n$. By the second step of the algorithm, the only basis produced with minimum element 2 is the one that also has maximum element $2n$. Thus in total, there are $n+1$ bases with minimum element 2 produced by $\\gamma$. Accounting for the all the bases produced at every step of $\\gamma$, we have that $$|\\gamma(\\mathcal{B}(\\mathcal{P}_{2(n+1)}))|=n^2+n+n+1+1 = (n+1)^2+1,$$ as desired.\n\\end{proof}\n\nUsing the above results, we may now prove our main theorem.\n\n\n\\begin{theorem}\\label{thm:main}\nLet $\\mathcal{B}(\\mathcal{P}_n)$ be the bases of the rank $n$ trivial UIP on $[2n]$. Assume $\\gamma$ is defined to be the algorithm described in Section \\ref{algorithm}. Then $\\gamma(\\mathcal{B}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$, the externally ordered poset of bases of the rank $n+1$ trivial UIP on $[2(n+1)]$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{thm:AlgMakesBases}, we know $\\gamma(\\mathcal{B}(\\mathcal{P}_n))$ produces the set of bases of $\\mathcal{P}_{n+1}.$ It is left to show that $\\gamma$ also induces the external order on $\\mathcal{B}(\\mathcal{P}_{n+1}),$ thus producing $\\mbox{Ex}(\\mathcal{P}_{n+1}).$\nLet $A,B\\in\\mathcal{B}(\\mathcal{P}_n).$ We prove this by cases for each step of the algorithm, confirming that the condition $A \\subseteq B\\cup \\mbox{Ext}(B)$ in Lemma \\ref{lem:lasvergnas_def} is satisfied. Denote $\\hat{A}=\\gamma(A)$ for any $A\\in\\mathcal{B}(\\mathcal{P}_n).$ \nFor the following cases, $\\hat{A}$ denotes the basis produced by $\\gamma$ after performing only the first step of the algorithm. Since every basis in $\\mathcal{B}(\\mathcal{P}_n)$ contains either 1 or 2, we need consider only the following three cases.\n\n\\textbf{Case 1:} $1\\in A$ and $1\\in B$.\\\\\nSince $1\\in A$, the only circuits contained in $A\\cup\\{i\\}$ for $i\\in[2n]$ are those of the form $(i,j)$ for $i,j\\in[n+1,\\dots,2n]$. Thus $\\mbox{Ext}(A)=\\{n+1,\\dots,max(A)-1\\}$. Similarly, $\\mbox{Ext}(B)=\\{n+1,\\dots,max(B)-1\\}$. After applying step one of $\\gamma$, $1\\in\\hat{A}$ and $1\\in\\hat{B}$. Thus $\\mbox{Ext}(\\hat{A})=\\{n+2,\\dots,max(\\hat{A})-1\\}$ and $\\mbox{Ext}(\\hat{B})=\\{n+2,\\dots,max(\\hat{B})-1\\}$. Let $a\\in \\hat{A}\\backslash \\{1\\}$. Then $a = a_i+1$ for $a_i \\in A$. Since  $A\\leq_{\\mbox{Ext}}B$, it follows that $a_i \\in B \\cup \\{n+1,\\dots,max(B)-1\\}$. Thus, $a_i+1\\in B+1\\cup \\{n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\n\\textbf{Case 2:} $1\\in A$ and $1\\notin B$.\\\\ \nBy definition, if $1\\notin B$ then $1\\in \\mbox{Ext}(B)$ for every $B$ of a trivial UIP. Let $a\\in \\hat{A}\\backslash \\{1\\}$. Then $a = a_i+1$ for $a_i \\in A$. We know $a_i \\in B \\cup \\{1,n+1,\\dots,max(B)-1\\}$. Thus, $a_i+1\\in B+1\\cup \\{1,n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\n\\textbf{Case 3:} $1 \\notin A \\mbox{ or } B$.\\\\\nBy assumption we know that $1\\in \\mbox{Ext}(A)$ and $1\\in \\mbox{Ext}(B)$. Then for $a\\in \\hat{A}\\backslash \\{2\\}$, we have that $a = a_i+1$ for $a_i \\in A$. Since $a_i \\in B \\cup \\{1,n+1,\\dots,max(B)-1\\}$, it follows $a_i+1\\in B+1\\cup \\{1,n+2,\\dots,max(B)\\}$. Note that $max(\\hat{B})-1 = max(B)$. Therefore, $a\\in \\hat{B}\\cup Ext(\\hat{B})$.\n\nFor the second step of the algorithm, notice that the only change to a basis $B$ is that the element $2n+1$ is replaced with $2(n+1)$. This means that $\\mbox{Ext}(\\hat{B})=\\mbox{Ext}(B)\\cup\\{2n+1\\}$. Then the cases to check are exactly those done for step one of the algorithm, and proceed in the same way. Similarly, for the third step, $\\mbox{Ext}(\\hat{B})=\\mbox{Ext}(B)\\backslash\\{1\\}$ since this step replaces the element 2 with 1. And then the cases proceed as above.\n\nIn all cases, the external order of $\\mathcal{B}(\\mathcal{P}_{n+1})$ is shown to be induced by the external order of $\\mathcal{B}(\\mathcal{P}_n)$. Thus, $\\gamma(\\mathcal{B}(\\mathcal{P}_n))=\\mbox{Ex}(\\mathcal{P}_{n+1})$ as desired.\n\\end{proof}\n\n\\section{Conclusion and Future Work}\\label{}\n\nThe recursive algorithm for the external poset on the bases of trivial UIPs provides a stepping stone to explore and develop either general or specific algorithms for the other UIPs. Recall that these are Catalan objects and that we have Catalan many to consider. In Figure \\ref{fig:all rank 3} we see the trivial UIP and the remaining UIPs of rank $r=3$. There appears to be some symmetry in the Hasse diagrams, but it is unclear how these five UIPs give rise to the non-trivial UIPs of rank $r=4$. See Figure \\ref{fig:soup} for examples of externally ordered posets for non-trivial UIPs of rank $r=4$. As was done for the recursion for the externally ordered poset for trivial UIPs, a potential first step is to find nice descriptions for the bases and circuits of the remaining UIPs.\n\nAnother approach for describing Catalan recursion for the remaining externally ordered posets is to consider matroid minors. One can check if minors of rank $n+1$ UIPs are isomorphic to rank $n$ UIPs. We have begun to explore this direction and our code is provided below. \n\n Our algorithm provides a new way of producing externally ordered posets. It would be interesting to know how the complexity of Las Vergnas' procedure compares to ours. In particular, how efficient is our algorithm in generating the next rank trivial UIP.\n\n\n\n\\section{Acknowledgements}\\label{}\nThis project would not be possible without the support and encouragement of various people I've been fortunate enough to meet and work with at UC Davis. I would like to thank Dr. Anastasia Chavez for her patience and guidance throughout this project. Thank you for giving me the opportunity to explore mathematics research, and for introducing me to matroid theory. I am also grateful to Dr. Jesus De Loera for supporting this project.\nI am grateful to the McNair scholars program for introducing me to undergraduate research, Esha Datta for so many helpful conversations and introductions, Shannon Allen for listening to me talk about this project many, many times, Calvin Cramer for being my Python help desk, and the friends, family, faculty and staff that saw this project in various stages of completion. \n\n\\begin{figure}[h]\\label{fig:uip r=3}\n    \\centering\n    \\includegraphics[width=1\\linewidth]{all_rank3.png}\n    \\caption{All UIPs with rank $r=3$.}\n    \\label{fig:all rank 3}\n\\end{figure}\n\n\\begin{figure}[h\n    \\centering\n    \\includegraphics[width=1\\linewidth]{some_rank4.png}\n    \\caption{Two of the 14 UIPs with rank $r=4$.}\n    \\label{fig:soup}\n\\end{figure}\n\\pagebreak\n\\begin{lstlisting}[language=Python]\nimport sage.matroids.matroid\ndef UnitIntervalPositroid(Dyck_matrix):\n\n    D = Dyck_matrix\n    rank = D.nrows()\n    D_flip = matrix(rank)\n    for i in range(rank):\n        D_flip[rank-i-1] = (-1)^i*D[i]\n    A = matrix.identity(rank).augment(D_flip)\n    cols = A.ncols()\n    M_A = Matroid(A)\n    \n    return M_A\n\ndef powerset(fullset):\n  listrep = list(fullset)\n  n = len(listrep)\n  return [[listrep[k] for k in range(n) if i&1<<k] for i in range(2**n)]\n  \nPos_list3 = []\nfor d in D_matrices:\n    U = UnitIntervalPositroid(d)\n    Pos_list3.append(U)\nprint Pos_list3\n\nE = (Matrix([[1,1,1,0],[1,1,1,0],[1,1,1,0],[1,1,1,1]]))\n    \nU4 = UnitIntervalPositroid(E)\n\ncontU4 = []\nfor s in [0,1,2,3,4,5]:\n    for t in [0,1,2,3,4,5]:\n        if t==s: break\n        else:\n            V = U4.delete(t).contract(s)\n            contU4.append([V])\n\nfor p in contU4:\n    for m in Pos_list3:\n        if p.is_isomorphic(m)==True:\n            print [b for b in m.bases()],\"yes\",\"\\n\"\n            print 'bases of cont-del positroid are ', [b for b in p.bases()],\"\\n\"\n\n\\end{lstlisting}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe following problem was raised in 2001 by I.~M.~Gelfand: Using graph theory describe up to isometry all finite ultrametric spaces~\\cite{Lem2001}. An appropriate representation of finite, ultrametric spaces by monotone trees was proposed by V.~Gurvich and M.~Vyalyi in~\\cite{GV2012DAM}. A simple geometric description of Gurvich---Vyalyi representing trees was found in \\cite{PD2014JMS}. This description allows us effectively use the Gurvich---Vyalyi representation in various problems associated with finite ultrametric spaces. In particular, it leads to a graph-theoretic interpretation of the Gomory---Hu inequality \\cite{DPT2015}. A characterization of finite ultrametric spaces which are as rigid as possible also was obtained \\cite{DPT2017FPTA} on the basis of the Gurvich---Vyalyi representation. Some other extremal properties of finite ultrametric spaces and related them properties of monotone rooted trees have been found in~\\cite{DP2020pNUAA}. The interconnections between the Gurvich---Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in~\\cite{Dov2019pNUAA} (see also \\cite{Qiu2009pNUAA, Qiu2014pNUAA, DP2018pNUAA, Pet2018pNUAA, Pet2013PoI}).\n\nThe Gurvich---Vyalyi representing trees can be considered as a subclass of finite trees endowed with some special labeling on vertex set. The trees with labeled vertices are studied by many mathematicians and there are a number of interesting results in this directions. In survey~\\cite{GalTEJoC2019}, J.~Gallian writes that over 200 graph labelings techniques have been studied in over 2800 paper during the past 50 years. In this regards, we only note that, in almost all studies of trees with labeled vertices, it is assumed that the trees are finite. The infinite trees endowed with positive real labelings on the set of edges are known as the so-called \\(R\\)-trees (see~\\cite{Ber2019SMJ} for some interesting results related to \\(R\\)-trees and ultrametrics). A description of interrelations between finite subtrees of \\(R\\)-trees and finite, monotone rooted trees can be found in~\\cite{Dov2020TaAoG}. The categorical equivalence of trees and ultrametric spaces was investigated in \\cite{H04} and \\cite{Lem2003AU}.\n\nMotivated by Gurvich---Vyalyi representation of finite ultrametric spaces and some results of Bruhn, Diestel, Halin, K\\\"{u}hn, Pott, Spr\\\"{u}ssel, and Stein \\cite{BDSJGT2005, BDCPC2006, BSCPC2010, DKEJC2004, DieJCTSB2006, DieDM2011, DSAM2011, DSTA2011, DSDM2012, DieAMSUH2017, DPJCTSB2017} on topological aspects of infinite graphs we consider infinite trees whose vertices are labeled by nonnegative real numbers and ultrametric spaces generated by such trees.\n\nThe paper is organized as follows. Section~\\ref{sec2} and Section~\\ref{sec3} contain some necessary concepts and facts from the theory of metric spaces and graph theory, respectively. In Section~\\ref{sec4} we introduce into consideration the ultrametric spaces \\((V(T), d_l)\\) generated by non-degenerate vertex labelings \\(l\\) of arbitrary trees \\(T\\). The first main result of the paper is Theorem~\\ref{t4.7} characterizing, up to isomorphism, the labeled trees \\(T(l)\\) for which the corresponding ultrametric spaces \\((V(T), d_l)\\) are complete. The characterizations of labeled trees generating discrete ultrametrics and totally bounded ones are found in Theorem~\\ref{t4.11} and Theorem~\\ref{t11.8}, respectively. Using these results we describe, up to isomorphism, the labeled trees generating discrete totally bounded ultrametrics in Theorem~\\ref{t11.16}. The final result of Section~\\ref{sec4} is Theorem~\\ref{t4.18} characterizing the labeled trees \\(T(l)\\) for which the ultrametric spaces \\((V(T), d_l)\\) are compact. The last fifth section contains some conjectures and examples related to subject of the paper.\n\n\\emph{Concluding remarks}. The results obtained in the paper indicate a close connection between the combinatorial properties of an infinite tree and the properties of ultrametric spaces generated by labelings on its vertex set.\n\\begin{itemize}\n\\item A tree \\(T\\) is rayless if and only if every ultrametric generated by vertex labeling is complete (Corollary~\\ref{c4.7}).\n\\item \\(T\\) is locally finite if and only if every ultrametric generated by vertex labeling is discrete (Corollary~\\ref{c4.12}).\n\\item \\(T\\) is rayless, at most countable, and has no adjacent vertices of infinite degree if and only if there is vertex labeling generating a compact ultrametric (Theorem~\\ref{t7}).\n\\item \\(T\\) is locally finite if and only if there is vertex labeling generating a discrete totally bounded ultrametric (Corollary~\\ref{c4.18}).\n\\end{itemize}\nIt seems interesting to study similar problems for general infinite connected graphs using the spanning trees technique. Another promising direction of research is the study of ultrametric spaces generated by some special labelings. For example, we can consider the case when the label of a vertex depends on the degree of this vertex.\n\n\\section{Definitions and facts from theory of metric spaces}\\label{sec2}\n\nLet us start from basic concepts. In what follows, we will denote by \\(\\RR^{+}\\) the half-open interval \\([0, \\infty)\\) and write \\(\\NN\\) for the set of all positive integers, \\(\\{1, 2, \\ldots\\}\\).\n\nA \\textit{metric} on a set $X$ is a function $d\\colon X\\times X\\rightarrow \\RR^+$ such that for all \\(x\\), \\(y\\), \\(z \\in X\\)\n\\begin{enumerate}\n\\item $d(x,y)=d(y,x)$,\n\\item $(d(x,y)=0)\\Leftrightarrow (x=y)$,\n\\item \\(d(x,y)\\leq d(x,z) + d(z,y)\\).\n\\end{enumerate}\n\nA metric space \\((X, d)\\) is \\emph{ultrametric} if the \\emph{strong triangle inequality}\n\\[\nd(x,y)\\leq \\max \\{d(x,z),d(z,y)\\}.\n\\]\nholds for all \\(x\\), \\(y\\), \\(z \\in X\\). In this case the function \\(d\\) is called \\emph{an ultrametric} on \\(X\\).\n\n\\begin{definition}\\label{d2.2}\nLet \\((X, d)\\) and \\((Y, \\rho)\\) be metric spaces. A bijective mapping \\(\\Phi \\colon X \\to Y\\) is said to be an \\emph{isometry} if\n\\[\nd(x,y) = \\rho(\\Phi(x), \\Phi(y))\n\\]\nholds for all \\(x\\), \\(y \\in X\\). The metric spaces are \\emph{isometric} if there is an isometry of these spaces.\n\\end{definition}\n\nLet \\((X, d)\\) be a metric space. An \\emph{open ball} with a \\emph{radius} \\(r > 0\\) and a \\emph{center} \\(c \\in X\\) is the set \n\\[\nB_r(c) = \\{x \\in X \\colon d(c, x) < r\\}.\n\\]\n\nWe denote by \\(\\mathbf{B}_X\\) the set of all open balls in \\((X, d)\\).\n\nA sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) is a \\emph{Cauchy sequence} in \\((X, d)\\) if, for every \\(r > 0\\), there is an integer \\(n_0 \\in \\NN\\) such that \\(x_n \\in B_r(x_{n_0})\\) for every \\(n \\geqslant n_0\\). It is easy to see that \\((x_n)_{n \\in \\mathbb{N}}\\) is a Cauchy sequence if and only if \n\\begin{equation*\n\\lim_{n \\to \\infty} \\sup\\{d(x_n, x_{n+k}) \\colon k \\in \\mathbb{N}\\} = 0.\n\\end{equation*}\n\n\\begin{remark}\\label{r2.17}\nHere and later the symbol \\((x_n)_{n\\in \\mathbb{N}} \\subseteq X\\) means that \\(x_n \\in X\\) holds for every \\(n \\in \\mathbb{N}\\).\n\\end{remark}\n\nThere exists a comfortable ``ultrametric modification'' of the notion of Cauchy sequence (see, for example, \\cite[p.~4]{PerezGarcia2010} or \\cite[Theorem~1.6, Statement~(13)]{Comicheo2018}).\n\n\\begin{proposition}\\label{p4.5}\nLet \\((X, d)\\) be an ultrametric space. A sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) is a Cauchy sequence if and only if the limit relation\n\\begin{equation*\n\\lim_{n \\to \\infty} d(x_n, x_{n+1}) = 0\n\\end{equation*}\nholds.\n\\end{proposition}\n\nA sequence \\((x_n)_{n \\in \\mathbb{N}}\\) of points in a metric space \\((X, d)\\) is said to be convergent to a point \\(a \\in X\\),\n\\begin{equation*\n\\lim_{n \\to \\infty} x_n = a,\n\\end{equation*}\nif, for every open ball \\(B\\) containing \\(a\\), it is possible to find an integer \\(n_0 \\in \\NN\\) such that \\(x_n \\in B\\) for every \\(n \\geqslant n_0\\). Thus, \\((x_n)_{n \\in \\mathbb{N}}\\) is \\emph{convergent} to \\(a\\) if and only if \n\\begin{equation*\n\\lim_{n \\to \\infty} d(x_n, a) = 0.\n\\end{equation*}\nA sequence is convergent if it is convergent to some point. It is clear that every convergent sequence is a Cauchy sequence.\n\nThe next proposition follows, for example, from Theorem~6.8.3 in \\cite{Sea2007}.\n\n\\begin{proposition}\\label{p2.7}\nLet \\((X, d)\\) be a metric space and let \\((x_n)_{n \\in \\NN} \\subseteq X\\) be a Cauchy sequence in \\((X, d)\\). Then \\((x_n)_{n \\in \\NN}\\) is convergent if and only if it has a convergent subsequence.\n\\end{proposition}\n\nNow we present a definition of \\emph{total boundedness}.\n\n\\begin{definition}\\label{d2.10}\nA subset \\(A\\) of a metric space \\((X, d)\\) is totally bounded if for every \\(r > 0\\) there is a finite set \\(\\{B_r(x_1), \\ldots, B_r(x_n)\\} \\subseteq \\mathbf{B}_X\\) such that\n\\[\nA \\subseteq \\bigcup_{i = 1}^{n} B_r(x_i).\n\\] \n\\end{definition}\n\nThere exists a simple interdependence between the total boundedness of a set \\(A \\subseteq X\\) and Cauchy sequences in \\(A\\).\n\n\\begin{proposition}\\label{p2.11}\nA subset \\(A\\) of a metric space \\((X, d)\\) is totally bounded if and only if every sequence of points of \\(A\\) contains a Cauchy subsequence. \n\\end{proposition}\n\nSee, for example, Theorem~7.8.2 \\cite{Sea2007}.\n\n\\begin{corollary}\\label{c2.17}\nLet \\((X, d)\\) be a metric space. If \\(A \\subseteq X\\) is totally bounded in \\((X, d)\\) and \\(C\\) is a subset of \\(A\\), then \\(C\\) is totally bounded in \\((A, d|_{A \\times A})\\).\n\\end{corollary}\n\nThe next basic for us concept is the concept of \\emph{completeness}.\n\n\\begin{definition}\\label{d2.6}\nA metric space \\((X, d)\\) is complete if for every Cauchy sequence \\((x_n)_{n \\in \\mathbb{N}} \\subseteq X\\) there is a point \\(a \\in X\\) such that \n\\[\n\\lim_{n \\to \\infty} x_n = a.\n\\]\n\\end{definition}\n\nThus, a metric space is complete if and only if the set of Cauchy sequences coincides with the set of convergent sequences in this space.\n\nAn important subclass of complete metric spaces is the class of \\emph{compact} metric spaces.\n\n\\begin{definition}[Borel---Lebesgue property]\\label{d2.3}\nLet \\((X, d)\\) be a metric space. A subset \\(A\\) of \\(X\\) is compact if every family \\(\\mathcal{F} \\subseteq \\mathbf{B}_X\\) satisfying the inclusion\n\\[\nA \\subseteq \\bigcup_{B \\in \\mathcal{F}} B\n\\]\ncontains a finite subfamily \\(\\mathcal{F}_0 \\subseteq \\mathcal{F}\\) such that \n\\[\nA \\subseteq \\bigcup_{B \\in \\mathcal{F}_0} B.\n\\]\n\\end{definition}\n\nA standard definition of compactness usually formulated as: Every open cover of \\(A\\) in \\(X\\) has a finite subcover.\n\nThe following classical theorem was proved by Frechet and it is a ``compact'' analog of Proposition~\\ref{p2.11}. \n\n\\begin{proposition}[Bolzano---Weierstrass property]\\label{p2.9}\nA subset \\(A\\) of a metric space is compact if and only if every sequence of points of \\(A\\) contains a subsequence which converges to a point of \\(A\\).\n\\end{proposition}\n\nThe next corollary shows that the class of compact metric spaces is the intersection of the class of complete metric spaces with the class of totally bounded ones.\n\n\\begin{corollary}[Spatial Criterion]\\label{c2.9}\nA metric space is compact if and only if this space is complete and totally bounded.\n\\end{corollary}\n\nThis and other criteria of compactness can be found, for example, in \\cite[p.~206]{Sea2007}.\n\nLet \\((X, d)\\) be a metric space and let \\(S \\subseteq X\\). The set \\(S\\) is said to be \\emph{dense} in \\((X, d)\\) if for every \\(a \\in X\\) there is a sequence \\((s_n)_{n\\in \\mathbb{N}} \\subseteq S\\) such that\n\\[\na = \\lim_{n \\to \\infty} s_n.\n\\]\n\nRecall that a point \\(p\\) of a metric space \\((X, d)\\) is \\emph{isolated} if there is \\(\\varepsilon > 0\\) such that \\(d(p, x) > \\varepsilon\\) for every \\(x \\in X \\setminus \\{p\\}\\). If \\(p\\) is not an isolated point of \\(X\\), then \\(p\\) is called an \\emph{accumulation point} of \\(X\\). We say that a set \\(A \\subseteq X\\) is \\emph{discrete} if all points of \\(A\\) are isolated.\n\nIt will be shown in Proposition~\\ref{p4.13} of Section~\\ref{sec4} that every ultrametric space, generated by labeled graph, contains a dense discrete subset.\n\n\n\\section{Definitions and facts from graph theory}\\label{sec3}\n\nA \\textit{graph} is a pair $(V,E)$ consisting of a set $V$ and a set $E$ whose elements are unordered pairs \\(\\{u, v\\}\\) of different points \\(u\\), \\(v \\in V\\). For a graph $G=(V,E)$, the sets \\(V=V(G)\\) and \\(E=E(G)\\) are called \\textit{the set of vertices} and \\textit{the set of edges}, respectively. A graph \\(G\\) is \\emph{finite} if \\(V(G)\\) is a finite set. If \\(\\{x,y\\} \\in E(G)\\), then the vertices \\(x\\) and \\(y\\) are called \\emph{adjacent}. In what follows we will always assume that \\(E(G) \\cap V(G) = \\varnothing\\).\n\nRecall that a graph \\(G_1\\) is a \\emph{subgraph} of a graph \\(G\\) if\n\\[\nV(G_1) \\subseteq V(G) \\quad \\text{and} \\quad E(G_1) \\subseteq E(G).\n\\]\nIn this case we will write \\(G_1 \\subseteq G\\). If \\(\\{G_i \\colon i \\in I\\}\\) is a family of subgraphs of a graph \\(G\\), then, by definition, the union \\(\\bigcup_{i \\in I} G_i\\) is a subgraph \\(G^{*}\\) of \\(G\\) such that \n\\[\nV(G^{*}) = \\bigcup_{i \\in I} V(G_i) \\quad \\text{and} \\quad E(G^{*}) = \\bigcup_{i \\in I} E(G_i).\n\\]\nSimilarly, the intersection \\(\\bigcap_{i \\in I} G_i\\) is a subgraph \\(G_{*}\\) of \\(G\\) with\n\\begin{equation}\\label{e3.4}\nV(G_{*}) = \\bigcap_{i \\in I} V(G_i) \\quad \\text{and} \\quad E(G_{*}) = \\bigcap_{i \\in I} E(G_i).\n\\end{equation}\n\nLet \\(v^*\\) be a vertex of a graph \\(G\\). The \\emph{neighborhood} \\(N(v^*) = N_G(v^{*})\\) is a subgraph of \\(G\\) induced by all vertices adjacent to \\(v^{*}\\). Thus, we have\n\\begin{align*}\nV(N(v^{*})) &= \\{u \\in V(G) \\colon \\{u, v^{*}\\} \\in E(G)\\},\\\\\nE(N(v^{*})) &= \\bigl\\{\\{u, v\\} \\in E(G) \\colon u, v \\in V(N(v^{*}))\\bigr\\}\n\\end{align*}\nfor every graph \\(G\\) and each \\(v^{*} \\in V(G)\\). Let \\(k\\) be a cardinal number. The vertex \\(v\\) of a graph \\(G\\) has \\emph{degree} \\(k\\) if \n\\[\nk = \\card V(N(v)).\n\\]\nThe degree of \\(v\\) will be denoted as \\(\\delta_{G}(v)\\) or simply as \\(\\delta(v)\\).\n\nA \\emph{path} is a finite graph \\(P\\) whose vertices can be numbered without repetitions so that\n\\begin{equation}\\label{e3.3}\nV(P) = \\{x_1, \\ldots, x_k\\} \\quad \\text{and} \\quad E(P) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{k-1}, x_k\\}\\}\n\\end{equation}\nwith \\(k \\geqslant 2\\). We will write \\(P = (x_1, \\ldots, x_k)\\) or \\(P = P_{x_1, x_k}\\) if \\(P\\) is a path satisfying \\eqref{e3.3} and said that \\(P\\) is a \\emph{path joining \\(x_1\\) and \\(x_k\\)}. A graph \\(G\\) is \\emph{connected} if for every two distinct vertices of \\(G\\) there is a path \\(P \\subseteq G\\) joining these vertices.\n\nA finite graph \\(C\\) is a \\textit{cycle} if there exists an enumeration of its vertices without repetition such that \\(V(C) = \\{x_1, \\ldots, x_n\\}\\)  and \n\\[\nE(C) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{n-1}, x_n\\}, \\{x_n, x_1\\}\\} \\quad \\text{with } n \\geqslant 3.\n\\]\n\n\\begin{definition}\\label{d9.8}\nA connected graph \\(T\\) with \\(V(T) \\neq \\varnothing\\) and without cycles is called a \\emph{tree}.\n\\end{definition}\n\nA tree \\(T\\) is \\emph{locally finite} if the inequality \\(\\delta(v) < \\infty\\) holds for every \\(v \\in V(T)\\).\n\nWe shall say that a tree \\(T\\) is a \\emph{star} if there is a vertex \\(c \\in V(T)\\), the \\emph{center} of \\(T\\), such that \\(c\\) and \\(v\\) are adjacent for every \\(v \\in V(T) \\setminus \\{c\\}\\).\n\nAn infinite graph \\(G\\) of the form \n\\[\nV(G) = \\{x_1, x_2 \\ldots, x_n, x_{n+1}, \\ldots\\}, \\quad E(G) = \\{\\{x_1, x_2\\}, \\ldots \\{x_n, x_{n+1}\\}, \\ldots\\},\n\\]\nwhere all \\(x_n\\) are assumed to be distinct, is called a \\emph{ray} \\cite{Diestel2017}. It is clear that every ray is a tree. A graph is \\emph{rayless} if it contains no rays.\n\n\\begin{proposition}\\label{p3.2}\nEvery infinite connected graph has a vertex of infinite degree or contains a ray.\n\\end{proposition}\n\nFor the proof see Proposition~8.2.1 in \\cite{Diestel2017}.\n\nThe following statement is well known for finite trees (see, for example, Proposition~4.1 \\cite{BM2008}) and is usually considered self-evident for infinite trees. \n\n\\begin{lemma}\\label{l9.14}\nIn each tree, every two different vertices are connected by exactly one path.\n\\end{lemma}\n\n\\begin{proof}\nIf \\(T\\) is an infinite tree and \\(v_1\\), \\(v_2\\) are two different vertices of \\(T\\) connected by some paths \\(P_1 \\subseteq T\\) and \\(P_2 \\subseteq T\\), then the graph \\(P_1 \\cup P_2\\) is a finite connected subgraph of \\(T\\). Since \\(T\\) does not have cycles, \\(P_1 \\cup P_2\\) is also acyclic. Hence, \\(P_1 \\cup P_2\\) is a finite tree and \\(P_1\\), \\(P_2\\) are paths connected \\(v_1\\) and \\(v_2\\) in \\(P_1 \\cup P_2\\). Thus, \\(P_1 = P_2\\) holds.\n\\end{proof}\n\nIn the next definition we introduce an analogue of convex hull for arbitrary trees.\n\n\\begin{definition}\\label{d3.6}\nLet \\(T\\) be a tree and let \\(A\\) be a nonempty subset of \\(V(T)\\). A subtree \\(H_A\\) of the tree \\(T\\) is the \\emph{hull} of \\(A\\) if \\(A \\subseteq V(H_A)\\) and, for every subtree \\(T^{*}\\) of \\(T\\), the tree \\(H_A\\) is a subtree of \\(T^{*}\\) whenever \\(A \\subseteq V(T^*)\\).\n\\end{definition}\n\nThus, \\(H_A\\) is the smallest subtree of \\(T\\) which contains \\(A\\). We want to make sure that for every tree \\(T\\) and each nonempty \\(A \\subseteq V(T)\\) the hull \\(H_A\\) is well defined and unique.\n\n\\begin{proposition}\\label{p3.7}\nLet \\(T\\) be a tree, \\(A\\) be a nonempty subset of \\(V(T)\\) and let \\(\\mathcal{F}_A\\) be the set of all subtrees \\(T^{*}\\) of \\(T\\) for which the inclusion \\(A \\subseteq V(T^{*})\\) holds. Then the graph \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is the hull of \\(A\\),\n\\begin{equation}\\label{p3.7:e1}\nH_A = \\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nIt is clear that \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is a subgraph of \\(T^{*}\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). In particular, \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is a subgraph of \\(T\\) because \\(T \\in \\mathcal{F}_A\\). Since \\(T\\) is a tree, the subgraph \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) contains no cycles. Hence, to prove \\eqref{p3.7:e1} it suffices to show that \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is connected.\n\nLet \\(u\\) and \\(v\\) be distinct vertices of \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) and let \\(P_{u, v}\\) be the path joining \\(u\\) and \\(v\\) in \\(T\\). Then \\(u\\) and \\(v\\) belong to \\(V(T^{*})\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). Using Lemma~\\ref{l9.14} we obtain \\(P_{u, v} \\subseteq T^{*}\\) for every \\(T^{*} \\in \\mathcal{F}_A\\). From~\\eqref{e3.4} with \\(\\mathcal{F} = \\mathcal{F}_A\\) it follows that the path \\(P_{u, v}\\) is also a subgraph of \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\). Thus, \\(\\bigcap_{T^{*} \\in \\mathcal{F}_A} T^{*}\\) is connected as required.\n\\end{proof}\n\n\\begin{example}\\label{ex3.7}\nLet \\(T\\) be an infinite tree, let a ray \\(R\\),\n\\[\nV(R) = \\{r_1, r_2, \\ldots, r_n, r_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{r_1, r_2\\}, \\ldots, \\{r_n, r_{n+1}\\}, \\ldots\\},\n\\]\nbe a subgraph of \\(T\\), and \\(v\\) be a vertex of \\(T\\) such that \\(v \\notin V(R)\\). We claim that there is a unique \\(n(v) \\in \\NN\\) such that \\(r_{n(v)}\\) is the only common vertex of \\(R\\) and of the path \\(P_{r_{n(v)}, v}\\) joining \\(r_{n(v)}\\) and \\(v\\) in \\(T\\),\n\\begin{equation}\\label{ex3.7:e1}\n\\{r_{n(v)}\\} = V(R) \\cap V(P_{r_{n(v)}, v}).\n\\end{equation}\nWe first prove the existence of some \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} and show that the graph \\(R \\cup P_{r_{n(v)}, v}\\) is the hull in \\(T\\) of the set \\(A \\stackrel{\\text{def}}{=} V(R) \\cup \\{v\\}\\),\n\\begin{equation*\nH_A = R \\cup P_{r_{n(v)}, v}.\n\\end{equation*}\nLet \\(v \\in V(T) \\setminus V(R)\\) be fixed. To find \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} it suffices to consider an arbitrary \\(u \\in V(R)\\) and the path \\((u^1, \\ldots, u^m) \\subseteq T\\) with \\(u^1 = u\\) and \\(u^m = v\\). Since \\(u \\in V(R)\\) and \\(v \\notin V(R)\\) hold, there is \\(m_1 \\in \\{1, \\ldots, m-1\\}\\) such that \\(u^{m_1} \\in V(R)\\) and \\(u^{j} \\notin V(R)\\) whenever \\(j \\in \\{m_1+1, \\ldots, m-1\\}\\). Consequently, there is \\(n_1 \\in \\NN\\) such that \\(r_{n_1} = u^{m_1}\\). If we set \\(n(v) \\stackrel{\\text{def}}{=} n_1\\), then \\eqref{ex3.7:e1} holds with \\(P_{r_{n(v)}, v} \\stackrel{\\text{def}}{=} (u^{m_1}, \\ldots, u^{m})\\). Since \\(P_{r_{n(v)}, v}\\) and \\(R\\) are connected and have the common vertex \\(r_{n(v)}\\), the union \\(R \\cup P_{r_{n(v)}, v}\\) is a subtree of \\(T\\). It is also clear that\n\\[\nA \\subseteq V(R \\cup P_{r_{n(v)}, v})\n\\]\nholds. Now Definition~\\ref{d3.6} implies that \\(H_A\\) is a subtree of \\(R \\cup P_{r_{n(v)}, v}\\). If we have \n\\[\nH_A \\neq R \\cup P_{r_{n(v)}, v},\n\\]\nthen there is \\(j \\in \\{m_1+1, \\ldots, m-1\\}\\) such that \\(u^{j} \\notin V(H_A)\\). Lemma~\\ref{l9.14} and \\(u^{j} \\notin V(H_A)\\) imply that \\(H_A\\) is disconnected, contrary to Definition~\\ref{d3.6}. From~Proposition~\\ref{p3.7} it follows that the hull \\(H_A\\) is unique. Consequently, the number \\(n(v) \\in \\NN\\) satisfying \\eqref{ex3.7:e1} is also unique. \n\nIn what follows we will say that \\(R \\cup P_{r_{n(v)}, v}\\) is a \\emph{comb} in \\(T\\), \\(v\\) is a \\emph{tooth} of this comb, and \\(r_{n(v)}\\) is the \\emph{root} of the tooth \\(v\\) (see Figure~\\ref{fig2}).\n\\end{example}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.5cm}\n\\node at (5*\\dx, 3*\\dx) [label= below:\\(T\\)] {};\n\\draw (\\dx, 0) -- (6*\\dx, 0);\n\\draw [dashed] (6*\\dx, 0)  --  (7*\\dx, 0);\n\\node [solid node] at (\\dx, 0) [label= below:\\(r_1\\)] {};\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(r_2\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(r_3\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(r_4\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(r_5\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(r_6\\)] {};\n\n\\draw (3*\\dx, 0) -- (3*\\dx, 2*\\dx);\n\\node [solid node] at (3*\\dx, \\dx) [label= below right:\\(v_2\\)] {};\n\\node [solid node] at (3*\\dx, 2*\\dx) [label= below right:\\(v_5\\)] {};\n\n\\draw (3*\\dx, \\dx) -- ($(3*\\dx, \\dx) + (135:\\dx)$) node [solid node, label= left:\\(v_1\\)] {};\n\\draw (3*\\dx, \\dx) -- ($(3*\\dx, \\dx) + (45:\\dx)$) node [solid node, label= right:\\(v_3\\)] {};\n\\draw (3*\\dx, 2*\\dx) -- ($(3*\\dx, 2*\\dx) + (135:\\dx)$) node [solid node, label= left:\\(v_4\\)] {};\n\\draw (3*\\dx, 2*\\dx) -- ($(3*\\dx, 2*\\dx) + (45:\\dx)$) node [solid node, label= right:\\(v_6\\)] {};\n\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (135:\\dx)$) node [solid node, label= above:\\(v_9\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (45:\\dx)$) node [solid node, label= above:\\(v_{10}\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (-135:\\dx)$) node [solid node, label= below:\\(v_7\\)] {};\n\\draw (5*\\dx, 0) -- ($(5*\\dx, 0) + (-45:\\dx)$) node [solid node, label= below:\\(v_{8}\\)] {};\n\\end{tikzpicture}\n\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.5cm}\n\\node at (5*\\dx, 2*\\dx) [label= below:\\(H_A\\)] {};\n\\draw (\\dx, 0) -- (6*\\dx, 0);\n\\draw [dashed] (6*\\dx, 0)  --  (7*\\dx, 0);\n\\node [solid node] at (\\dx, 0) [label= below:\\(r_1\\)] {};\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(r_2\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(r_3\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(r_4\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(r_5\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(r_6\\)] {};\n\n\\draw (3*\\dx, 0) -- (3*\\dx, 2*\\dx);\n\\node [solid node] at (3*\\dx, \\dx) [label= below right:\\(v_2\\)] {};\n\\node [solid node] at (3*\\dx, 2*\\dx) [label= below right:\\(v_5\\)] {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The hull \\(H_A\\) of \\(A = \\{v_5\\} \\cup \\{r_i \\colon i \\in \\NN\\}\\) is comb in \\(T\\), the vertex \\(r_3\\) is the root of the tooth \\(v_5\\) in this comb.} \n\\label{fig2}\n\\end{figure}\n\n\\begin{remark}\\label{r3.8}\nThus, we always assume that each comb has exactly one tooth with exactly one root. The combs with a large number of teeth are more often used in theory of ultrametric spaces and graph theory (see, for example, the Comb representation of compact ultrametric spaces \\cite{LamBr2017} or the Star-Comb Lemma~\\cite[Lemma~8.2.2]{Diestel2017}).\n\\end{remark}\n\nLet us recall the concept of labeled trees.\n\n\\begin{definition}\\label{d2.4}\nA labeled tree is a pair \\((T, l)\\), where \\(T\\) is a tree and \\(l\\) is a mapping defined on the set \\(V(T)\\).\n\nWe say that \\(T\\) is a \\emph{free tree} corresponding to \\((T, l)\\) and write \\(T = T(l)\\) instead of \\((T, l)\\). In what follows, we will consider only the nonnegative real-valued labelings \\(l\\colon V(T)\\to \\RR^{+}\\).\n\\end{definition}\n\nBefore introducing into consideration the concept of isomorphism of labeled trees, it is useful to remind the definition of isomorphism for free trees.\n\n\\begin{definition}\\label{d8.7}\nLet $T_1$ and $T_2$ be trees. A bijection $f\\colon V(T_1)\\to V(T_2)$ is an \\emph{isomorphism} of $T_1$ and $T_2$ if\n\\begin{equation*\n(\\{u,v\\} \\in E(T_1)) \\Leftrightarrow (\\{f(u),f(v)\\} \\in E(T_2))\n\\end{equation*}\nis valid for all $u$, $v \\in V(T_1)$. Two trees are \\emph{isomorphic} if there exists an isomorphism of these trees.\n\\end{definition}\n\nFor the case of labeled trees Definition~\\ref{d8.7} must be modified as follows.\n\n\\begin{definition}\\label{d2.5}\nLet \\(T_i=T_i(l_i)\\) be a labeled tree, \\(i = 1\\), \\(2\\). A mapping $f\\colon V(T_1) \\to V(T_2)$ is an isomorphism of \\(T_1(l_1)\\) and \\(T_2(l_2)\\) if it is an isomorphism of the free trees $T_1$ and $T_2$ and the equality\n\\begin{equation*\nl_2(f(v))=l_1(v)\n\\end{equation*}\nholds for every $v \\in V(T_1)$.\n\\end{definition}\n\n\\section{Ultrametrics generated by labeled trees}\\label{sec4}\n\nLet \\(T = T(l)\\) be a labeled tree. Following~\\cite{Dov2020TaAoG}, we define a mapping \\(d_l \\colon V(T) \\times V(T) \\to \\RR^{+}\\) as\n\\begin{equation}\\label{e11.3}\nd_l(u, v) = \\begin{cases}\n0 & \\text{if } u = v,\\\\\n\\max\\limits_{v^{*} \\in V(P)} l(v^{*}) & \\text{if } u \\neq v,\n\\end{cases}\n\\end{equation}\nwhere \\(P\\) is the path joining \\(u\\) and \\(v\\) in \\(T\\).\n\n\\begin{remark}\\label{r11.10}\nThe correctness of this definition follows from Lemma~\\ref{l9.14}.\n\\end{remark}\n\nTo formulate the first theorem of this section we recall the concept of \\emph{pseudoultrametric space}. Let \\(X\\) be a set. A symmetric mapping \\(d \\colon X \\times X \\to \\RR^{+}\\) is a \\emph{pseudoultrametric} on \\(X\\) if\n\\[\nd(x, x) = 0 \\quad \\text{and} \\quad d(x, y) \\leqslant \\max\\{d(x, z), d(z, y)\\}\n\\]\nhold for all \\(x\\), \\(y\\), \\(z \\in X\\). Every ultrametric is a pseudoultrametric, but a pseudoultrametric \\(d\\) on a set \\(X\\) is an ultrametric if and only if \\(d(x, y) > 0\\) holds for all distinct \\(x\\), \\(y \\in X\\).\n\nThe notion of isometries can be extended on pseudoultrametrics as follows. If \\((X, d)\\) and \\((Y, \\rho)\\) are pseudoultrametric spaces, then a mapping \\(\\Phi \\colon X \\to Y\\) is an isometry of \\((X, d)\\) and \\((Y, \\rho)\\) if \\(\\Phi\\) is bijective and\n\\[\nd(x, y) = \\rho(\\Phi(x), \\Phi(y))\n\\]\nholds for all \\(x\\), \\(y \\in X\\).\n\n\\begin{theorem}\\label{t11.9}\nLet \\(T = T(l)\\) be a labeled tree. Then \\((V(T), d_l)\\) is a pseudoultrametric space. The function \\(d_l\\) is an ultrametric on \\(V(T)\\) if and only if the inequality\n\\begin{equation*\n\\max\\{l(u_1), l(v_1)\\} > 0\n\\end{equation*}\nholds for every \\(\\{u_1, v_1\\} \\in E(T)\\).\n\\end{theorem}\n\nA proof of Theorem~\\ref{t11.9} can be obtained by simple modification of the proof of Proposition~3.2 \\cite{Dov2020TaAoG}.\n\n\\begin{proposition}\\label{l11.12}\nLet \\(T_1 = T_1(l_1)\\) and \\(T_2 = T_2(l_2)\\) be labeled trees and let \\(f \\colon V(T_1) \\to V(T_2)\\) be an isomorphism of these trees. Then the equality\n\\[\nd_{l_1}(u, v) = d_{l_2}(f(u), f(v))\n\\]\nholds for all \\(u\\), \\(v \\in V(T_1)\\).\n\\end{proposition}\n\n\\begin{proof}\nIt directly follows from Definition~\\ref{d2.5} and formula~\\eqref{e11.3}, because if \\((v_1, \\ldots, v_n)\\) is a path joining some distinct \\(v = v_1\\) and \\(u = v_n\\) in \\(T_1\\), then \\(f(u) \\neq f(v)\\) holds and \\((f(v_1), \\ldots, f(v_n))\\) is a path joining \\(f(v)\\) and \\(f(u)\\) in \\(T_2\\) and we have the equality \n\\[\n\\{l_1(v_1), \\ldots, l_1(v_n)\\} = \\{l_2(f(v_1)), \\ldots, l_2(f(v_n))\\}.\n\\]\n\\end{proof}\n\n\\begin{corollary}\\label{c11.6}\nLet \\(T_1 = T_1(l_1)\\) and \\(T_2 = T_2(l_2)\\) be isomorphic labeled trees. Then the pseudoultrametric spaces \\((V(T_1), d_{l_1})\\) and \\((V(T_2), d_{l_2})\\) are isometric.\n\\end{corollary}\n\nThe converse statement is not valid in general. The following example is a modification of Example~3.1 \\cite{Dov2020TaAoG}.\n\n\\begin{example}\\label{ex11.13}\nLet \\(V = \\{v_0, v_1, v_2, v_3, v_4\\}\\) be a five-point set, and let \\(S = S(l_S)\\) and \\(P = P(l_P)\\) be a labeled star with the center \\(v_0\\) and, respectively, a labeled path such that \\(V(S) = V(P) = V\\), \\(l_S(v_0) = l_P(v_0) = 1\\) and\n\\[\nl_S(v_i) + 1 = l_P(v_i) = 1\n\\]\nfor \\(i = 1\\), \\(\\ldots\\), \\(4\\) (see Figure~\\ref{fig11}). Then, for all distinct \\(u\\), \\(v \\in V\\), we have\n\\[\nd_{l_P}(u, v) = d_{l_S}(u, v) = 1.\n\\]\nThus, the ultrametric spaces \\((V, d_{l_P})\\) and \\((V, d_{l_S})\\) coincide, but \\(P(l_P)\\) and \\(S(l_S)\\) are not isomorphic as labeled trees or even as free trees.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1cm}\n\\node at (0, \\dx) [label= above:\\(S(l_S)\\)] {};\n\\node [solid node] at (0, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (\\dx, \\dx) [label= right:\\(0\\)] {};\n\\node [solid node] at (\\dx, -\\dx) [label= right:\\(0\\)] {};\n\\node [solid node] at (-\\dx, \\dx) [label= left:\\(0\\)] {};\n\\node [solid node] at (-\\dx, -\\dx) [label= left:\\(0\\)] {};\n\\draw (0, 0) -- (\\dx, \\dx);\n\\draw (0, 0) -- (\\dx, -\\dx);\n\\draw (0, 0) -- (-\\dx, \\dx);\n\\draw (0, 0) -- (-\\dx, -\\dx);\n\n\\node at (5*\\dx, \\dx) [label= above:\\(P(l_P)\\)] {};\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (6*\\dx, 0) [label= below:\\(1\\)] {};\n\\node [solid node] at (7*\\dx, 0) [label= below:\\(1\\)] {};\n\\draw (3*\\dx, 0) -- (7*\\dx, 0);\n\\end{tikzpicture}\n\\end{center}\n\\caption{The star \\(S\\) and the path \\(P\\) are not isomorphic as trees, but the labelings \\(l_S\\) and \\(l_P\\) generate the same ultrametric on \\(V\\).}\n\\label{fig11}\n\\end{figure}\n\\end{example}\n\nExample~\\ref{ex11.13} shows that the properties of ultrametric spaces generated by different labeled trees can be the same. Thus, the following problem naturally arises.\n\n\\begin{problem}\\label{pr4.6}\nLet \\(\\mathcal{UP}\\) be the class of ultrametric spaces with a given property \\(\\mathcal{P}\\). What common properties do the labeled trees \\(T = T(l)\\) generating \\((V(T), d_l) \\in \\mathcal{UP}\\) have?\n\\end{problem}\n\nBelow we will consider this problem in the following cases:\n\\begin{itemize}\n\\item  \\(\\mathcal{P} =\\) completeness,\n\\item \\(\\mathcal{P} =\\) discreteness,\n\\item \\(\\mathcal{P} =\\) total boundedness,\n\\item \\(\\mathcal{P} =\\) discreteness + total boundedness,\n\\item \\(\\mathcal{P} =\\) compactness,\n\\end{itemize}\nand in each of these cases we find the corresponding characteristic properties of generating labeled trees.\n\nLet us start from the completeness. \n\nIn what follows we shall say that a labeling \\(l \\colon V(T) \\to \\mathbb{R}^{+}\\) is \\emph{non-degenerate} if the inequality\n\\[\n\\max\\{l(u), l(v)\\} > 0\n\\]\nholds for every \\(\\{u, v\\} \\in E(T)\\). \n\n\\begin{lemma}\\label{l4.7}\nLet \\(R\\) be a ray, \\(V(R) = \\{v_1, v_2, \\ldots, v_n, v_{n+1}, \\ldots\\}\\),\n\\begin{equation}\\label{l4.7:e1}\nE(R) = \\{\\{v_1, v_2\\}, \\ldots, \\{v_n, v_{n+1}\\}, \\ldots\\},\n\\end{equation}\nand let \\(l \\colon V(R) \\to \\RR^{+}\\) be a non-degenerate labeling. The sequence \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in the ultrametric space \\((V(R), d_l)\\) if and only if the limit relation\n\\begin{equation}\\label{l4.7:e2}\n\\lim_{n \\to \\infty} l(v_n) = 0\n\\end{equation}\nholds.\n\\end{lemma}\n\n\\begin{proof}\nBy Proposition~\\ref{p4.5}, \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(R), d_l)\\) if and only if\n\\begin{equation}\\label{l4.7:e3}\n\\lim_{n \\to \\infty} d_l(v_n, v_{n+1}) = 0.\n\\end{equation}\nUsing \\eqref{e11.3} and \\eqref{l4.7:e1} we can rewrite \\eqref{l4.7:e3} as\n\\[\n\\lim_{n \\to \\infty} \\left(\\max\\{l(v_n), l(v_{n+1})\\}\\right) = 0.\n\\]\nSince all \\(l(v_n)\\) belong to \\(\\RR^{+}\\), \\eqref{l4.7:e2} holds if and only if \n\\[\n\\limsup_{n \\to \\infty} l(v_n) = 0.\n\\]\nSimilarly, \\eqref{l4.7:e3} is equivalent to\n\\[\n\\limsup_{n \\to \\infty} (\\max\\{l(v_n), l(v_{n+1})\\}) = 0.\n\\]\nNow using the equality\n\\[\n\\limsup_{n \\to \\infty} l(v_n) = \\limsup_{n \\to \\infty} (\\max\\{l(v_n), l(v_{n+1})\\})\n\\]\nwe see that \\eqref{l4.7:e2} and \\eqref{l4.7:e3} are equivalent.\n\\end{proof}\n\nThe next lemma will be useful to prove Theorem~\\ref{t4.7}.\n\n\\begin{lemma}\\label{l4.8}\nLet \\(R = R(l)\\) be a labeled ray,\n\\[\nV(R) = \\{v_1, v_2, \\ldots, v_n, v_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{v_1, v_2\\}, \\ldots, \\{v_n, v_{n+1}\\}, \\ldots\\},\n\\]\nwith non-degenerate labeling. Then the sequence \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(R), d_l)\\) if and only if \\((v_n)_{n \\in \\NN}\\) contains a subsequence which is Cauchy in \\((V(R), d_l)\\).\n\\end{lemma}\n\n\\begin{proof}\nIf \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence, then \\((v_n)_{n \\in \\NN}\\) is a Cauchy subsequence of itself. \n\nConversely, let \\((v_{n_k})_{k \\in \\NN}\\),\n\\begin{equation*\n1 \\leqslant n_1 < n_2 < \\ldots < n_k < n_{k+1} < \\ldots,\n\\end{equation*}\nbe a Cauchy subsequence of \\((v_n)_{n \\in \\NN}\\). It is easy to see that, for every \\(m \\geqslant n_1\\), there is the unique \\(k(m) \\in \\NN\\) such that \n\\begin{equation}\\label{l4.8:e2}\nn_{k(m)} \\leqslant m < n_{k(m)+1}.\n\\end{equation}\n\nLet us denote by \\(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}}\\) the path joining \\(v_{n_{k(m)}}\\) and \\(v_{n_{k(m)+1}}\\) in \\(R\\). From \\eqref{l4.8:e2} it follows that \n\\begin{equation}\\label{l4.8:e3}\nv_m \\in V(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}}).\n\\end{equation}\nNow using \\eqref{e11.3} and \\eqref{l4.8:e3} we obtain\n\\begin{equation}\\label{l4.8:e4}\nl(v_m) \\leqslant \\max\\{l(v) \\colon v \\in V(P_{v_{n_{k(m)}}, v_{n_{k(m)+1}}})\\} = d_l(v_{n_{k(m)}}, v_{n_{k(m)+1}}).\n\\end{equation}\nIt is clear that the mapping \n\\[\n\\{n_1, n_1+1, \\ldots\\} \\ni m \\mapsto n_{k(m)} \\in \\NN\n\\]\nis increasing and satisfies the limit relation\n\\begin{equation}\\label{l4.8:e6}\n\\lim_{m \\to \\infty} n_{k(m)} = +\\infty.\n\\end{equation}\nProposition~\\ref{p4.5}, \\eqref{l4.8:e4} and \\eqref{l4.8:e6} imply\n\\[\n\\limsup_{m \\to \\infty} l(v_m) \\leqslant \\limsup_{m \\to \\infty} d_l(v_{n_{k(m)}}, v_{n_{k(m)+1}}) = 0.\n\\]\nThus, we have\n\\begin{equation*\n\\lim_{m \\to \\infty} l(v_m) = 0,\n\\end{equation*}\nbecause \\(l(v_m) \\in \\RR^{+}\\) for every \\(n \\in \\NN\\). Using the last limit relation we obtain that \\((v_n)_{n \\in \\NN}\\) is a Cauchy sequence by Lemma~\\ref{l4.7}.\n\\end{proof}\n\n\n\\begin{lemma}\\label{l4.6}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\) and let \\(T_1\\) be a subtree of \\(T\\) having the labeling \\(l_1 \\colon V(T_1) \\to \\RR^{+}\\) which is the restriction of \\(l\\) on \\(V(T_1)\\), \\(l_1 = l|_{V(T_1)}\\). Then the labeling \\(l_1\\) is also non-degenerate and the ultrametric \\(d_{l_1}\\) is the restriction of the ultrametric \\(d_l\\) on the set \\(V(T_1)\\), \\(d_{l_1} = d_l|_{V(T_1) \\times V(T_1)}\\).\n\\end{lemma}\n\n\\begin{proof}\nIt follows from formula \\eqref{e11.3}, Lemma~\\ref{l9.14} and the definition of trees and subtrees.\n\\end{proof}\n\n\\begin{theorem}\\label{t4.7}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.7:s1} The ultrametric space \\((V(T), d_l)\\) is complete.\n\\item \\label{t4.7:s2} For every ray \\(R \\subseteq T\\), \n\\begin{equation}\\label{t4.7:e2}\nV(R) = \\{x_1, x_2, \\ldots, x_n, x_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{x_1, x_2\\}, \\ldots, \\{x_{n}, x_{n+1}\\}, \\ldots\\},\n\\end{equation}\nwe have the inequality\n\\begin{equation}\\label{t4.7:e1}\n\\limsup_{n \\to \\infty} l(x_n) > 0.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.7:s1} \\Rightarrow \\ref{t4.7:s2}\\). Let \\((V(T), d_l)\\) be a complete ultrametric space. We must show that condition~\\ref{t4.7:s2} is satisfied. Suppose contrary that there is a ray \\(R \\subseteq T\\) such that \\eqref{t4.7:e2} and \\(\\limsup_{n \\to \\infty} l(x_n) = 0\\) hold. Since all \\(l(x_n)\\) are nonnegative, the last equality holds if and only if \n\\begin{equation}\\label{t4.7:e3}\n\\lim_{n \\to \\infty} l(x_n) = 0.\n\\end{equation}\nFrom \\eqref{t4.7:e3} and \\eqref{e11.3} it follows that\n\\[\n\\lim_{n \\to \\infty} d_l(x_n, x_{n+1}) = \\lim_{n \\to \\infty} \\max\\{l(x_n), l(x_{n+1})\\} = 0,\n\\]\nbecause \\(x_n\\) and \\(x_{n+1}\\) are adjacent in \\(R\\) and \\(R \\subseteq T\\). Hence, by Proposition~\\ref{p4.5}, the sequence \\((x_n)_{n \\in \\mathbb{N}}\\) is a Cauchy sequence in the space \\((V(T), d_l)\\). \n\nNow, using condition~\\ref{t4.7:s1} and Definition~\\ref{d2.6}, we can find a point \\(v^{*} \\in V(T)\\) satisfying the equality\n\\begin{equation}\\label{t4.7:e4}\n\\lim_{n \\to \\infty} d_l(x_n, v^{*}) = 0.\n\\end{equation}\nIf there is \\(n_0 \\in \\NN\\) such that \\(v^{*} = x_{n_0} \\in V(R)\\), then, for every \\(n \\geqslant n_0 + 1\\), the path \\(P_{x_{n_0}, x_{n}}\\) joining \\(v^{*}\\) and \\(x_n\\) in \\(T\\) contains the edge \\(\\{x_{n_0}, x_{n_0+1}\\} \\in E(R)\\). Since \\(l \\colon V(T) \\to \\RR^{+}\\) is non-degenerate, \\eqref{e11.3} and \\(\\{x_{n_0}, x_{n_0+1}\\} \\in E(P_{x_{n_0}, x_{n_0+1}})\\) imply \n\\[\nd_l(v^{*}, x_n) \\geqslant \\max \\{l(x_{n_0}), l(x_{n_0+1})\\} > 0\n\\]\nfor every \\(n \\geqslant n_0 + 1\\), contrary to \\eqref{t4.7:e4}.\n\nSuppose now that \\(v^{*} \\in V(T) \\setminus V(R)\\). Then the hull \\(H_A\\) of the set\n\\[\nA \\stackrel{\\text{def}}{=} V(R) \\cup \\{v^{*}\\}\n\\]\nis a comb in \\(T\\) with the tooth \\(v^{*}\\) (see Definition~\\ref{d3.6} and Example~\\ref{ex3.7}). Write \\(u^{*}\\) for the root of \\(v^{*}\\) in \\(H_A\\). Since \\(u^{*} \\neq v^{*}\\) and \\(u^{*}\\) is the only common vertex of \\(R\\) and of the path \\(P_{u^{*}, v^{*}}\\) joining \\(u^{*}\\) and \\(v^{*}\\) in \\(T\\), we have \n\\[\nd_l(x_n, v^{*}) \\geqslant d_l(u^{*}, v^{*}) > 0\n\\]\nfor all \\(x_n \\in V(R)\\), contrary to~\\eqref{t4.7:e4}. Condition~\\ref{t4.7:s2} follows.\n\n\\(\\ref{t4.7:s2} \\Rightarrow \\ref{t4.7:s1}\\). Let \\ref{t4.7:s2} hold. We must show that the ultrametric space \\((V(T), d_l)\\) is complete. According to Definition~\\ref{d2.6}, the space is complete if every Cauchy sequence of points of this space is convergent.\n\nLet us consider an arbitrary Cauchy sequence \\((y_n)_{n \\in \\NN} \\subseteq V(T)\\) and define the range \\(A\\) of \\((y_n)_{n \\in \\NN}\\) as:\n\\[\n(v \\in A) \\Leftrightarrow (\\exists n \\in \\NN \\colon y_n = v).\n\\]\nIf \\(A\\) is finite, then the sequence \\((y_n)_{n \\in \\NN}\\) contains an infinite constant subsequence and, consequently, \\((y_n)_{n \\in \\NN}\\) is convergent by Proposition~\\ref{p2.7}. \n\nSuppose that \\(A\\) is infinite and denote by \\(H_A\\) the hull of \\(A\\) in \\(T\\). Let us prove that \\(H_A\\) is a rayless graph. \n\nIndeed, let a ray \\(R\\),\n\\[\nV(R) = \\{r_1, r_2, \\ldots, r_n, r_{n+1}, \\ldots\\}, \\quad E(R) = \\{\\{r_1, r_2\\}, \\ldots, \\{r_n, r_{n+1}\\}, \\ldots\\},\n\\]\nbe a subgraph of \\(H_A\\). If the intersection \\(A \\cap V(R)\\) is infinite, then the sequence \\((r_n)_{n \\in \\NN}\\) contains a subsequence \\((r_{n_k})_{k \\in \\NN}\\) which is Cauchy in \\((V(T), d_l)\\) and, consequently, in \\((V(R), d_l|_{V(R) \\times V(R)})\\). Using Lemma~\\ref{l4.8} and Lemma~\\ref{l4.6} we see that the sequence \\((r_n)_{n \\in \\NN}\\) is Cauchy in \\((V(R), d_l|_{V(R) \\times V(R)})\\). Now Lemma~\\ref{l4.6} implies that \\((r_n)_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\). Moreover, the equality\n\\begin{equation*}\n\\lim_{n \\to \\infty} l(r_n) = 0\n\\end{equation*}\nholds by Lemma~\\ref{l4.7}. The last equality contradicts \\eqref{t4.7:e1} with \\((x_n)_{n \\in \\NN} = (r_n)_{n \\in \\NN}\\). \n\nThus, the intersection \\(A \\cap V(R)\\) is finite and \\(A\\) is infinite. We claim that there is an \\emph{infinite} subset \\(A^{*}\\) of the set \\(A \\setminus V(R)\\) which satisfies the condition:\n\\begin{enumerate}\n\\item[\\((i^{*})\\)] If \\(a_1\\), \\(a_2\\) are distinct points of \\(A^{*}\\), and\n\\[\nA_1 \\stackrel{\\text{def}}{=} V(R) \\cup \\{a_1\\}, \\quad A_2 \\stackrel{\\text{def}}{=} V(R) \\cup \\{a_2\\},\n\\]\nand \\(H_{A_1}\\), \\(H_{A_2}\\) are the corresponding hulls of \\(A_1\\) and of \\(A_2\\) in \\(T\\), then the roots \\(r(a_1)\\) and \\(r(a_2)\\) are distinct.\n\\end{enumerate}\n\n(Recall that the hulls \\(H_{A_1}\\) and \\(H_{A_2}\\) in \\(T\\) are some combs in \\(T\\), see Example~\\ref{ex3.7}). To find a desired \\(A^{*} \\subseteq A \\setminus V(R)\\) let us consider a number \\(N \\in \\NN\\) such that \n\\[\nA \\cap V(R) \\subseteq \\{r_1, r_2, \\ldots, r_{N}\\}\n\\]\nand suppose that, for every \\(a \\in A \\setminus V(R)\\), the root \\(r(a)\\) of the tooth \\(a\\) in the comb \\(H_{V(R) \\cup \\{a\\}}\\) belongs to the set \\(\\{r_1, r_2, \\ldots, r_{N}\\}\\). The graph \n\\[\nG_{R, N} \\stackrel{\\text{def}}{=} (r_1, \\ldots, r_{N}) \\cup \\bigcup_{a \\in A \\setminus V(R)} P_{a, r(a)},\n\\]\nwhere \\((r_1, \\ldots, r_{N})\\) is the path joining \\(r_1\\) and \\(r_N\\) in \\(R\\) and \\(P_{a, r(a)}\\) is the path joining \\(a\\) and \\(r(a)\\) in the comb \\(H_{V(R) \\cup \\{a\\}}\\), is a connected subgraph of \\(T\\) satisfying the conditions\n\\begin{equation*\nA \\subseteq V(G_{R, N}) \\quad \\text{and} \\quad r_n \\notin V(G_{R, N}) \n\\end{equation*}\nfor every \\(n \\geqslant N+1\\). Since \\(r_n \\in V(H_A)\\) holds for every \\(n \\in \\NN\\), the inclusion\n\\[\nV(H_A) \\subseteq V(G_{R, N})\n\\]\nis false, contrary to Proposition~\\ref{p3.7}. Hence, there is an element \\(y_{n_1}\\) of the sequence \\((y_{n})_{n \\in \\NN}\\) such that \\(y_{n_1} \\in A \\setminus V(R)\\) and \\(r(y_{n_1}) = r_{N_1}\\) and \\(N_1 > N\\). If for every \\(a \\in A \\setminus V(R)\\) the root \\(r(a)\\) belongs to \\(\\{r_1, \\ldots, r_N, \\ldots, r_{N_1}\\}\\), then repeating the above procedure with the graph \\(G_{R, N_1}\\) gives us \\(y_{n_2} \\in A \\setminus V(R)\\) with \\(r(y_{n_2}) = r_{N_2}\\) such that \\(n_2 > n_1\\) and \\(N_2 > N_1\\) and so on. \n\nLet us consider the sequence \\((y_{n_k})_{k \\in \\NN}\\), whose elements are inductively defined above, and write\n\\[\nA^{*} \\stackrel{\\text{def}}{=} \\{y_{n_k} \\colon k \\in \\NN\\}.\n\\]\nThen condition~\\((i^{*})\\) satisfies with this \\(A^{*}\\). Since \\((y_{n})_{n \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\), the sequence \\((y_{n_k})_{k \\in \\NN}\\) is also Cauchy. It is easy to prove that the inequality\n\\begin{equation}\\label{t4.7:e12}\nd_l(r(y_{n_{k_1}}), r(y_{n_{k_2}})) \\leqslant d_l(y_{n_{k_1}}, y_{n_{k_2}})\n\\end{equation}\nholds for all \\(k_1\\), \\(k_2 \\in \\NN\\) (see Figure~\\ref{fig3}). Consequently, \\((r(y_{n_k}))_{k \\in \\NN}\\) is a Cauchy sequence in \\((V(T), d_l)\\).\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.3cm}\n\\draw (\\dx, 0) node [solid node, label= below:\\(r_1\\)] {} -- (2*\\dx, 0) node [solid node, label= below:\\(r_2\\)] {};\n\\draw [dashed] (2*\\dx, 0) -- (3*\\dx, 0) node [solid node] {};\n\\draw (3*\\dx, 0) -- (4*\\dx, 0) node [solid node, label= below:\\(r(y_{n_{k_1}})\\)] {} -- (5*\\dx, 0) node [solid node] {};\n\\draw (4*\\dx, 0) -- (4*\\dx, \\dx) node [solid node] {};\n\\draw [dashed] (4*\\dx, \\dx) -- (4*\\dx, 2*\\dx) node [solid node] {};\n\\draw (4*\\dx, 2*\\dx) -- (4*\\dx, 3*\\dx) node [solid node, label= right:\\(y_{n_{k_1}}\\)] {};\n\n\\draw [dashed] (5*\\dx, 0) -- (6*\\dx, 0) node [solid node] {};\n\\draw (6*\\dx, 0) -- (7*\\dx, 0) node [solid node, label= below:\\(r(y_{n_{k_2}})\\)] {} -- (8*\\dx, 0) node [solid node] {};\n\\draw (7*\\dx, 0) -- (7*\\dx, \\dx) node [solid node] {};\n\\draw [dashed] (7*\\dx, \\dx) -- (7*\\dx, 2*\\dx) node [solid node] {};\n\\draw (7*\\dx, 2*\\dx) -- (7*\\dx, 3*\\dx) node [solid node, label= right:\\(y_{n_{k_2}}\\)] {};\n\n\\draw [dashed] (8*\\dx, 0) -- (9*\\dx, 0);\n\\end{tikzpicture}\n\\end{center}\n\\caption{The path \\(\\bigl(r(y_{n_{k_1}}), \\ldots, r(y_{n_{k_2}})\\bigr)\\) is a subgraph of the path \\(\\bigl(y_{n_{k_1}}, \\ldots, r(y_{n_{k_1}}), \\ldots, r(y_{n_{k_2}}), \\ldots, y_{n_{k_2}}\\bigr)\\). It implies inequality~\\eqref{t4.7:e12}.}\n\\label{fig3}\n\\end{figure}\n\nNow using Lemma~\\ref{l4.8} and Lemma~\\ref{l4.6} we can prove that the sequence \\((r_n)_{n \\in \\NN}\\) of all vertices of the ray \\(R\\) is also a Cauchy sequence in \\((V(T), d_l)\\). Hence, by Lemma~\\ref{l4.7}, we have the equality\n\\[\n\\lim_{n \\to \\infty} l(r_n) = 0,\n\\]\nthat contradicts condition \\ref{t4.7:s2}. \n\n\\emph{Thus, the hull \\(H_A\\) is rayless}. Since \\(A\\) is infinite, \\(H_A\\) has a vertex \\(v^{*}\\) of infinite degree by Proposition~\\ref{p3.2}. To complete the proof it suffices to show that \\((y_n)_{n \\in \\NN}\\) converges to the point \\(v^{*}\\) in \\((V(T), d_l)\\),\n\\begin{equation*\n\\lim_{n \\to \\infty} d_l(y_n, v^{*}) = 0.\n\\end{equation*}\n\nLet us consider the subgraph \\(F_{v^{*}}\\) obtained from \\(H_A\\) by deleting the vertex \\(v^{*}\\),\n\\[\nV(F_{v^{*}}) = V(H_A) \\setminus \\{v^{*}\\}, \\quad E(F_{v^{*}}) = \\{\\{u, v\\} \\in E(H_A) \\colon u \\neq v^{*} \\neq v\\}.\n\\]\nSince \\(H_A\\) is a tree (as a subtree of \\(T\\)), \\(F_{v^{*}}\\) is a forest and, for every connected component \\(T'\\) of \\(F_{v^{*}}\\), there is a unique \\(p\\in V(N(v^{*}))\\) such that\n\\begin{equation}\\label{t4.7:e11}\np \\in V(T'),\n\\end{equation}\nwhere \\(N(v^{*})\\) is the neighborhood of \\(v^{*}\\) in \\(H_A\\). We will write \\(T^{p}\\) for the component \\(T'\\) if \\eqref{t4.7:e11} holds with \\(p\\in V(N(v^{*}))\\).\n\nIt is clear that\n\\begin{equation}\\label{t4.7:e13}\nH_A = \\left(\\bigcup_{p \\in V(N(v^{*}))} T^{p}\\right) \\cup S(v^{*})\n\\end{equation}\nholds, where \\(S(v^{*})\\) is the star with the center \\(v^{*}\\) and the vertex set \n\\[\nV(S(v^{*})) = V(N(v^{*})).\n\\]\n\nWe claim that the set \\(V(T^p) \\cap A\\) is nonempty for every \\(p \\in V(N(v^{*}))\\). Indeed, suppose that there is \\(p^{*} \\in V(N(v^{*}))\\) such that\n\\begin{equation}\\label{t4.7:e14}\nV(T^{p^{*}}) \\cap A = \\varnothing.\n\\end{equation}\nLet us denote by \\(S_{p^{*}}\\) the graph which is obtained from \\(S(v^{*})\\) by deleting of the vertex \\(p^{*}\\), i.e., \n\\[\nV(S_{p^{*}}) = V(S(v^{*})) \\setminus \\{p^{*}\\}\n\\]\nholds and\n\\[\n\\bigl(\\{x, y\\} \\in E(S_{p^{*}})\\bigr) \\Leftrightarrow \\bigl(\\{x, y\\} \\in E(S(v^{*})) \\text{ and } x \\neq p^{*} \\neq y\\bigr)\n\\]\nis valid for all \\(x\\), \\(y \\in V(N(v^{*}))\\). Then \\(S_{p^{*}}\\) is a star with the center \\(v^{*}\\). From \\eqref{t4.7:e14} it follows that the union\n\\[\n\\left(\\bigcup_{\\substack{p \\in V(N(v^{*}))\\\\ p \\neq p^{*}}} T^{p} \\right) \\cup S_{p^{*}}\n\\]\nis a subtree of \\(T\\) and the set \\(A\\) is a subset of the vertex set of this subtree. Hence, by Definition~\\ref{d3.6}, we have\n\\[\np^{*} \\notin V(H_A),\n\\]\ncontrary to \\eqref{t4.7:e13}. Using the conditions \n\\[\n\\delta_{H_A} (v^{*}) = \\infty \\quad \\text{and} \\quad V(T^{p}) \\cap A \\neq \\varnothing\n\\]\nfor every \\(p \\in V(N(v^{*}))\\), we can find a subsequence \\((y_{n_m})_{m \\in \\NN}\\) of the sequence \\((y_{n})_{n \\in \\NN}\\) such that for every \\(m \\in \\NN\\) there is \\(p(m) \\in V(N(v^{*}))\\) satisfying \\(y_{n_m} \\in T^{p}\\), and \\(p(m_1) \\neq p(m_2)\\) whenever \\(m_1 \\neq m_2\\). Then for every pair of distinct \\(m_1\\), \\(m_2 \\in \\NN\\) the path \\(P_{y_{n_{m_1}}, y_{n_{m_2}}}\\) joining \\(y_{n_{m_1}}\\) and \\(y_{n_{m_2}}\\) in \\(H_A\\) contains the vertex \\(v^{*}\\). Hence, by definition~\\eqref{e11.3}, we have the inequality\n\\begin{equation}\\label{t4.7:e15}\nd_l\\left(y_{n_{m_1}}, y_{n_{m_2}}\\right) \\geqslant \\max \\left\\{d_l\\left(y_{n_{m_1}}, v^{*}\\right), d_l\\left(v^{*}, y_{n_{m_2}}\\right)\\right\\}\n\\end{equation}\nwhenever \\(m_1\\), \\(m_2 \\in \\NN\\). By Proposition~\\ref{p2.7}, the sequence \\((y_{n})_{n \\in \\NN}\\) is convergent if \\((y_{n_m})_{m \\in \\NN}\\) is convergent. Using Proposition~\\ref{p4.5} and inequality \\eqref{t4.7:e15} with \\(n_{m_1} = n_{m}\\) and \\(n_{m_2} = n_{m+1}\\) we obtain\n\\[\n0 = \\lim_{m \\to \\infty} d_l\\left(y_{n_{m}}, y_{n_{m+1}}\\right) \\geqslant \\limsup_{m \\to \\infty} d_l\\left(y_{n_{m}}, v^{*}\\right),\n\\]\nthat implies \n\\[\n\\lim_{m \\to \\infty} d_l\\left(y_{n_{m}}, v^{*}\\right) = 0.\n\\]\nThus, \\((y_{n_m})_{m \\in \\NN}\\) is convergent to \\(v^{*}\\).\n\\end{proof}\n\n\n\nCondition~\\ref{t4.7:s2} of Theorem~\\ref{t4.7} is vacuously true for every rayless tree \\(T\\). Moreover, if \\(R \\subseteq T\\) is a ray satisfying \\eqref{t4.7:e2}, then it is easy to construct a non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\) such that \\eqref{t4.7:e1} does not hold. Thus, Theorem~\\ref{t4.7} implies the next corollary.\n\n\\begin{corollary}\\label{c4.7}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.7:s1} The ultrametric space \\((V(T), d_{l})\\) is complete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\item \\label{c4.7:s2} \\(T\\) is rayless.\n\\end{enumerate}\n\\end{corollary}\n\nRecall that a metric space \\((X, d)\\) is \\emph{discrete} if every point of \\(X\\) is isolated.\n\n\\begin{theorem}\\label{t4.11}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.11:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete.\n\\item \\label{t4.11:s2} For every \\(v^{*} \\in V(T)\\) we have either \\(l(v^{*}) > 0\\) or \\(l(v^{*}) = 0\\) and \n\\begin{equation}\\label{t4.11:e1}\n0 < \\inf_{u \\in V(N(v^{*}))} l(u),\n\\end{equation}\nwhere \\(N(v^{*})\\) is the neighborhood of \\(v^{*}\\) in \\(T\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.11:s1} \\Rightarrow \\ref{t4.11:s2}\\). Let \\((V(T), d_{l})\\) be a discrete ultrametric space. If \\ref{t4.11:s2} does not hold, then there is a vertex \\(v^{*}\\) such that \\(l(v^{*}) = 0\\) and \n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) = 0.\n\\]\nHence, there exists a sequence \\((u_n)_{n \\in \\NN} \\subseteq V(N(v^{*}))\\) such that\n\\[\n\\lim_{n \\to \\infty} l(u_n) = 0.\n\\]\nThe last limit relation, the equality \\(l(v^{*}) = 0\\), equality \\eqref{e11.3} and the definition of the neighborhoods of vertices of graph imply that\n\\begin{equation}\\label{t4.11:e0}\n\\lim_{n \\to \\infty} d(v^{*}, u_n) = 0\n\\end{equation}\nholds. Hence, \\(v^{*}\\) is an accumulation point in \\((V(T), d_{l})\\), contrary to statement \\ref{t4.11:s1}. \n\n\\(\\ref{t4.11:s2} \\Rightarrow \\ref{t4.11:s1}\\). Let \\ref{t4.11:s2} hold. Statement \\ref{t4.11:s1} is valid if \\(E(T) = \\varnothing\\). Indeed, in this case the vertex set of \\(T\\) is a single-point set \\(\\{v^{*}\\}\\). Thus, \\(V(N(v^{*})) = \\varnothing\\) holds and, consequently, we have\n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) = \\inf_{u \\in \\varnothing} l(u) = +\\infty,\n\\]\nthat implies \\eqref{t4.11:e1}. (We consider here the empty set \\(\\varnothing\\) as a subset of \\([-\\infty, \\infty]\\) and adopt the standard agreement on the supremum and infimum of empty set.)\n\nLet \\(E(T) \\neq \\varnothing\\) hold. \n\nSuppose that \\(v^{*}\\) is a vertex of \\(T\\) such that \\(l(v^{*}) > 0\\). Then \\eqref{e11.3} implies\n\\[\nd_l(u, v^{*}) \\geqslant l(v^{*})\n\\]\nfor every \\(u \\in V(T) \\setminus \\{v^{*}\\}\\). Hence, \\(v^{*}\\) is an isolated point of \\((V(T), d_{l})\\).\n\nLet us consider now the case when \\(v^{*} \\in V(T)\\) has the zero label, \\(l(v^{*}) = 0\\), and assume that we have \\(\\delta_{T}(v^{*}) < \\infty\\). Since the inequality \\(\\max\\{l(u), l(v^{*})\\} > 0\\) holds for every \\(u \\in V(N(v^{*}))\\), from \\(0 < \\delta_{T}(v^{*}) < \\infty\\) follows the inequality\n\\begin{equation}\\label{t4.11:e3}\n\\min_{u \\in V(N(v^{*}))} l(u) > 0. \n\\end{equation}\nLet \\(u_0 \\in V(T) \\setminus \\{v^{*}\\}\\). Then there is \\(u^{*} \\in V(P_{v^{*}, u_0})\\) such that\n\\begin{equation}\\label{t4.11:e2}\nu^{*} \\in V(N(v^{*})).\n\\end{equation}\nNow from \\eqref{e11.3} and \\eqref{t4.11:e2} it follows that\n\\[\nd_l(v^{*}, u_0) = \\max_{u \\in V(P_{v^{*}, u_0})} l(u) \\geqslant d_l(v^{*}, u^{*}) \\geqslant \\min_{u \\in V(N(v^{*}))} l(u) > 0.\n\\]\nHence, \\(v^{*}\\) is an isolated point of \\((V(T), d_{l})\\). \n\nUsing inequality \\eqref{t4.11:e1} instead of \\eqref{t4.11:e3} and repeating the above arguments, we obtain that \\(v^{*}\\) is also isolated for the case \\(l(v^{*}) = 0\\) and \\(\\delta_{T}(v^{*}) = \\infty\\). Thus, the ultrametric space \\((V(T), d_{l})\\) is discrete. \n\\end{proof}\n\n\\begin{corollary}\\label{c4.12}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.12:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\item \\label{c4.12:s2} The tree \\(T\\) is locally finite.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proposition}\\label{p4.13}\nLet \\(T\\) be a tree. Then the ultrametric space \\((V(T), d_{l})\\) contains a dense discrete subset for every non-degenerate \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{proposition}\n\n\\begin{proof}\nLet \\(l \\colon V(T) \\to \\RR^{+}\\) be non-degenerate. It was shown in the second part of the proof of Theorem~\\ref{t4.11} that \\(v \\in V(T)\\) is an isolated point of the ultrametric space \\((V(T), d_l)\\) if at least one of the conditions:\n\\begin{itemize}\n\\item \\(\\delta(v) < \\infty\\),\n\\item \\(l(v) > 0\\),\n\\item \\(\\delta(v) = \\infty\\), \\(l(v) = 0\\) and \\(\\inf_{u \\in V(N(v^{*}))} l(u) > 0\\)\n\\end{itemize}\nis valid. Arguing as in the first part of the proof of Theorem~\\ref{t4.11}, we can show that, for every \\(v\\) satisfying conditions \\(\\delta(v) = \\infty\\) and \n\\[\nl(v) = 0 = \\inf_{u \\in V(N(v^{*}))} l(u),\n\\]\nthere is a sequence \\((u_n)_{n \\in \\NN} \\subseteq V(N(v))\\) which converges to \\(v\\) (see~\\eqref{t4.11:e0}). Now it suffices to note that all elements of this sequence are isolated points of \\((V(T), d_l)\\) because \\(l(v) = 0\\) holds and \\(l\\) is non-degenerate.\n\\end{proof}\n\n\nThe following result gives us the necessary and sufficient conditions under which the ultrametric space \\((V(T), d_l)\\) is totally bounded.\n\n\\begin{theorem}\\label{t11.8}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t11.8:s1} The ultrametric space \\((V(T), d_{l})\\) is totally bounded.\n\\item \\label{t11.8:s2} The set \n\\begin{equation}\\label{t11.8:e1}\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation}\nis finite for every \\(\\varepsilon > 0\\) and the inequality \\(\\delta_{T}(v) < \\infty\\) holds whenever \\(l(v) > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t11.8:s1} \\Rightarrow \\ref{t11.8:s2}\\). Let \\ref{t11.8:s1} hold. Suppose that the set \\(V_{\\varepsilon}\\) is infinite for some \\(\\varepsilon > 0\\). Using the definition of \\(d_{l}\\) (see~\\eqref{e11.3}) it is easy to prove the inequality\n\\begin{equation*\nd_{l}(v_1, v_2) \\geqslant \\varepsilon\n\\end{equation*}\nfor all distinct \\(v_1\\), \\(v_2 \\in V_{\\varepsilon}\\). Hence, the subspace \\((V_{\\varepsilon}, d_{l}|_{V_{\\varepsilon} \\times V_{\\varepsilon}})\\) of totally bounded ultrametric space \\((V(T), d_{l})\\) is not totally bounded, contrary to Corollary~\\ref{c2.17}. Thus, \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\).\n\nAssume now that \\(T\\) contains a vertex \\(v^*\\) of infinite degree, \\(\\delta (v^*) = \\infty\\), and \\(l(v^{*}) > 0\\) holds.\n\nLet \\(N(v^{*})\\) be the neighborhood of \\(v^{*}\\). The equality \\(\\delta_{T}(v^*) = \\infty\\) implies that \\(V(N(v^{*}))\\) has an infinite cardinality. For all distinct \\(u_1\\), \\(u_2 \\in V(N(v^{*}))\\) the unique path joining \\(u_1\\) and \\(u_2\\) in \\(T\\) has the form \\((u_1, v^{*}, u_2)\\). Hence, \n\\[\nd_l(u_1, u_2) \\geqslant l(v^{*}) > 0\n\\]\nholds by \\eqref{e11.3}. It implies that the ultrametric space \\((V(N(v^{*})), d_l|_{V(N(v^{*})) \\times V(N(v^{*}))})\\) is not totally bounded, contrary to \\ref{t11.8:s1}. \n\n\\(\\ref{t11.8:s2} \\Rightarrow \\ref{t11.8:s1}\\). Let \\ref{t11.8:s2} hold. We must show that \\((V(T), d_{l})\\) is totally bounded. It is clear that \\((V(T), d_{l})\\) is totally bounded for finite \\(T\\). Let us consider the case when \\(T\\) is infinite.\n\nBy Proposition~\\ref{p2.11}, the ultrametric space \\((V(T), d_{l})\\) is totally bounded if every sequence of vertices of \\(T\\) contains a Cauchy subsequence. Let us consider a sequence \\((u_j^0)_{j \\in \\mathbb{N}}\\) of pairwise distinct points of \\(V(T)\\). Let \\((\\varepsilon_i)_{i \\in \\mathbb{N}}\\) be a decreasing sequence of strictly positive real numbers such that \\(\\lim_{i \\to \\infty} \\varepsilon_i = 0\\). Statement~\\ref{t11.8:s2} implies that the set \\(V_{\\varepsilon_{1}}\\) is finite. Write \\(G^{1}\\) for the subgraph of \\(T\\) induced by \\(V(T) \\setminus V_{\\varepsilon_{1}}\\), i.e., \\(V(G^{1}) = V(T) \\setminus V_{\\varepsilon_{1}}\\) and\n\\[\n\\bigl(\\{u, v\\} \\in E(G^{1})\\bigr) \\Leftrightarrow \\bigl(u, v \\in V(G^{1}) \\text{ and } \\{u, v\\} \\in E(T)\\bigr).\n\\]\nEvery connected component of \\(G^{1}\\) is a tree. Since \\(V_{\\varepsilon_{1}}\\) is a finite set, and \\(\\delta_{T}(v) < \\infty\\) holds for every \\(v \\in V_{\\varepsilon_{1}}\\), the number of these components are finite. Since \\(T\\) is an infinite tree, there is an infinite subtree \\(T^{1}\\) of \\(T\\) with \\(l(v^{1}) < \\varepsilon_{1}\\) for all \\(v^{1} \\in V(T^{1})\\) and such that \\((u_{j}^{1})_{j \\in \\mathbb{N}} \\subseteq V(T^{1})\\) holds for an infinite subsequence \\((u_{j}^{1})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{0})_{j \\in \\mathbb{N}} \\subseteq V(T)\\). Write \\(u^{1} = u_1^1\\). \n\nLet us consider the subgraph \\(G^{2}\\) of \\(T^{1}\\) induced by \\(V(T^{1}) \\setminus V_{\\varepsilon_{2}}\\). As above, the finiteness of \\(V_{\\varepsilon_{2}}\\) implies the existence of an infinite tree \\(T^{2} \\subseteq T^{1}\\) and a subsequence \\((u_{j}^{2})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{1})_{j \\in \\mathbb{N}} \\subseteq V(T^{1})\\) for which \\((u_{j}^{2})_{j \\in \\mathbb{N}} \\subseteq V(T^{2})\\) holds. Let us write \\(u^{2} = u_1^2\\). \n\nBy induction, for every \\(i \\geqslant 2\\), we find an infinite connected component \\(T^{i+1}\\) of the subgraph \\(G^{i+1}\\) of \\(T^{i}\\) induced by \\(V(T^{i}) \\setminus V_{\\varepsilon_{i+1}}\\) and a subsequence \\((u_{j}^{i+1})_{j \\in \\mathbb{N}}\\) of the sequence \\((u_{j}^{i})_{j \\in \\mathbb{N}}\\) such that\n\\begin{equation}\\label{t11.8:e8}\n(u_{j}^{i+1})_{j \\in \\mathbb{N}} \\subseteq V(T^{i+1}).\n\\end{equation}\nWrite \\(u^{i+1}\\) for the first element \\(u_{1}^{i+1}\\) of this sequence.\n\nLet us consider now the sequence \\((u^{i})_{i \\in \\mathbb{N}}\\). It is clear that \\((u^{i})_{i \\in \\mathbb{N}}\\) is a subsequence of the original sequence \\((u_j^0)_{j \\in \\mathbb{N}}\\). From~\\eqref{t11.8:e8} it follows that\n\\[\nl(u^{i+1}) < \\varepsilon_{i+1}\n\\]\nholds for every \\(i \\in \\mathbb{N}\\). The last inequality and the limit relation \\(\\lim_{i \\to \\infty} \\varepsilon_i = 0\\) imply\n\\begin{equation}\\label{t11.8:e9}\n\\lim_{i \\to \\infty} l(u^{i}) = 0.\n\\end{equation}\nMoreover, since for every \\(i \\geqslant 2\\) the points \\(u^{i}\\) and \\(u^{i+1}\\) are vertices of the tree \\(T^{i}\\) and the inequality \\(l(v) \\leqslant \\varepsilon_{i}\\) holds for every \\(v \\in V(T^{i})\\), formula \\eqref{e11.3} implies the inequality\n\\[\nd_{l}(u^{i}, u^{i+1}) \\leqslant l(u^{i-1})\n\\]\nfor every \\(i \\geqslant 2\\). Now using Proposition~\\ref{p4.5} and limit relation~\\eqref{t11.8:e9} we obtain that \\((u^{i})_{i \\in \\mathbb{N}}\\) is a Cauchy sequence.\n\nThe same reasons show that \\((u_j^0)_{j \\in \\mathbb{N}}\\) contains a Cauchy subsequence whenever \\((u_j^0)_{j \\in \\mathbb{N}}\\) contains an infinite subsequence of pairwise distinct members.\n\nTo complete the proof, we note only that if all subsequences of pairwise distinct members of a sequence are finite, then there is an infinite constant subsequence of that sequence and this constant subsequence obviously is a Cauchy sequence. \n\\end{proof}\n\n\\begin{corollary}\\label{c4.8}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. If the ultrametric space \\((V(T), d_l)\\) is totally bounded, then the set \\(V(T)\\) is at most countable.\n\\end{corollary}\n\n\\begin{proof}\nIt suffices to show that the inequality \n\\begin{equation}\\label{c4.8:e1}\n\\delta_{T}(v^*) \\leqslant \\aleph_0\n\\end{equation}\nholds for every vertex \\(v^*\\) of \\(T\\), where \\(\\aleph_0\\) is the cardinality of \\(\\mathbb{N}\\).\n\nLet \\((V(T), d_l)\\) be totally bounded and let \\(v^*\\) be a vertex of \\(T\\). Inequality \\eqref{c4.8:e1} follows directly from Theorem~\\ref{t11.8} if \\(l(v^*) > 0\\). Suppose that \\(l(v^*) = 0\\) holds. Then for every \\(\\{u, v^*\\} \\in E(T)\\) we have the inequality \\(l(u) > 0\\). Consequently, the vertex set \\(V(N(v^{*}))\\) satisfies the inclusion \n\\begin{equation}\\label{c4.8:e2}\nV(N(v^{*})) \\subseteq \\bigcup_{n \\in \\mathbb{N}} V_{1\/n},\n\\end{equation}\nwhere \\(V_{1\/n}\\) is defined by \\eqref{t11.8:e1} with \\(\\varepsilon = 1\/n\\). By Theorem~\\ref{t11.8}, \\(V_{1\/n}\\) is finite for every \\(n \\in \\mathbb{N}\\). Hence, \\(\\bigcup_{n \\in \\mathbb{N}} V_{1\/n}\\) is at most countable. Now inequality \\eqref{c4.8:e1} follows from \\eqref{c4.8:e2}.\n\\end{proof}\n\n\n\\begin{theorem}\\label{t11.16}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item \\label{t11.16:s1} The ultrametric space \\((V(T), d_{l})\\) is discrete and totally bounded.\n\\item \\label{t11.16:s2} The tree \\(T\\) is locally finite and the set\n\\begin{equation*\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation*}\nis finite for every \\(\\varepsilon > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t11.16:s1} \\Rightarrow \\ref{t11.16:s2}\\). Let \\ref{t11.16:s1} hold. Then the set \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\) by Theorem~\\ref{t11.8}. \n\nAssume now that \\(T\\) contains a vertex \\(v^*\\) of infinite degree, \\(\\delta (v^*) = \\infty\\). Then, using Theorem~\\ref{t11.8} again, we obtain the equality\n\\begin{equation}\\label{t11.16:e4}\nl(v^{*}) = 0.\n\\end{equation}\nBy Theorem~\\ref{t4.11}, equality~\\eqref{t11.16:e4} and discreteness of \\((V(T), d_{l})\\) imply that there is \\(\\varepsilon^{*} > 0\\) such that\n\\[\n\\inf_{u \\in V(N(v^{*}))} l(u) \\geqslant \\varepsilon^{*}.\n\\]\nHence, we have the inclusion \\(V(N(v^{*})) \\subseteq V_{\\varepsilon^{*}}\\). It was shown above that \\(V_{\\varepsilon}\\) is finite for every \\(\\varepsilon > 0\\). Consequently, \\(V(N(v^{*}))\\) is also finite as a subset of a finite set, contrary to \\(\\delta_T (v^{*}) = \\infty\\).\n\n\\(\\ref{t11.16:s2} \\Rightarrow \\ref{t11.16:s1}\\). Let \\ref{t11.16:s2} hold. Then \\(T\\) is locally finite and, consequently, the ultrametric space \\((V(T), d_{l})\\) is discrete by Corollary~\\ref{c4.12}. To complete the proof, it suffices to note that \\((V(T), d_{l})\\) is totally bounded by Theorem~\\ref{t11.8}.\n\\end{proof}\n\nCorollary~\\ref{c4.12} and Theorem~\\ref{t11.16} imply the following.\n\n\\begin{corollary}\\label{c4.15}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{c4.15:s1} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\item \\label{c4.15:s2} The ultrametric space \\((V(T), d_{l})\\) is discrete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nIf \\ref{c4.15:s1} holds, then \\(T\\) is locally finite by Theorem~\\ref{t11.16}, that implies \\ref{c4.15:s2} by Corollary~\\ref{c4.12}.\n\nConversely, suppose that \\ref{c4.15:s2} holds. Then, using Corollary~\\ref{c4.12} again, we see that \\(T\\) is locally finite. If \\(T\\) is finite, then \\ref{c4.15:s1} is trivially valid. Every infinite locally finite tree has countable vertex set. Thus, all vertices of \\(T\\) can be numbered in a sequence \\((v_n)_{n \\in \\NN}\\) of pairwise different points and we can define a labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) as\n\\[\nl_1(v_n) = \\frac{1}{n}\n\\]\nfor every \\(n \\in \\NN\\). Then \\(l_1\\) is a non-degenerate labeling and \\(T(l_1)\\) satisfies condition \\ref{t11.16:s2} of Theorem~\\ref{t11.16}. Hence, \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\end{proof}\n\nUsing Corollaries~\\ref{c4.12} and \\ref{c4.15} we obtain.\n\n\\begin{corollary}\\label{c4.18}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item\\label{c4.18:s1} \\(T\\) is locally finite.\n\\item\\label{c4.18:s2} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which \\((V(T), d_{l_1})\\) is discrete and totally bounded.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{theorem}\\label{t4.18}\nLet \\(T = T(l)\\) be a labeled tree with non-degenerate labeling. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t4.18:s1} The ultrametric space \\((V(T), d_{l})\\) is compact.\n\\item \\label{t4.18:s2} The tree \\(T\\) is rayless and the set\n\\begin{equation*\nV_{\\varepsilon} = \\bigl\\{v \\in V(T) \\colon l(v) \\geqslant \\varepsilon\\bigr\\}\n\\end{equation*}\nis finite for every \\(\\varepsilon > 0\\) and the inequality \\(\\delta_{T}(v) < \\infty\\) holds whenever \\(l(v) > 0\\).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t4.18:s1} \\Rightarrow \\ref{t4.18:s2}\\). Let \\((V(T), d_{l})\\) be a compact ultrametric space. Every compact metric space is totally bounded and complete by Corollary~\\ref{c2.9}. Hence, by Theorem~\\ref{t11.8}, for every \\(\\varepsilon > 0\\) the set \\(V_{\\varepsilon}\\) is finite, and \\(\\delta_{T}(v) < \\infty\\) holds for all vertices \\(v\\) with \\(l(v) > 0\\). \n\nSuppose that there is a ray \\(R \\subseteq T\\). Then the completeness of \\((V(T), d_{l})\\) and Theorem~\\ref{t4.7} imply the existence of \\(\\varepsilon^{*} > 0\\) and of a sequence \\((x_n)_{n \\in \\NN}\\) of pairwise distinct vertices of \\(R\\) such that\n\\[\n\\limsup_{n \\to \\infty} l(x_n) \\geqslant \\varepsilon^{*} > 0.\n\\]\nThus, the set \\(\\{v \\in V(R) \\colon l(v) \\geqslant \\frac{1}{2} \\varepsilon^{*}\\}\\) is infinite, contrary to the finiteness of the set \\(V_{\\frac{\\varepsilon^{*}}{2}}\\) which contains \\(\\{v \\in V(R) \\colon l(v) \\geqslant \\frac{1}{2} \\varepsilon^{*}\\}\\).\n\n\\(\\ref{t4.18:s2} \\Rightarrow \\ref{t4.18:s1}\\). Let \\ref{t4.18:s2} hold. Then \\ref{t4.18:s1} follows from the Spatial Criterion (Corollary~\\ref{c2.9}) and Theorems~\\ref{t4.7}, \\ref{t11.8}.\n\\end{proof}\n\n\\begin{theorem}\\label{t7}\nLet \\(T\\) be a tree. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item \\label{t7:s1} \\(T\\) is rayless, and the set \\(V(T)\\) is at most countable, and, for every \\(\\{x, y\\} \\in E(T)\\), at least one from the degrees \\(\\delta(x)\\) and \\(\\delta(y)\\) is finite.\n\\item \\label{t7:s2} There is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is compact.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\(\\ref{t7:s1} \\Rightarrow \\ref{t7:s2}\\). Let \\ref{t7:s1} hold. Let us define the subsets \\(V^F\\) and \\(V^I\\) of \\(V(T)\\) as\n\\begin{gather*}\nV^F \\stackrel{\\text{def}}{=} \\bigl\\{v \\in V(T) \\colon \\delta(v) \\text{ is finite}\\bigr\\}, \\quad\nV^I \\stackrel{\\text{def}}{=} \\bigl\\{v \\in V(T) \\colon \\delta(v) \\text{ is infinite}\\bigr\\}.\n\\end{gather*}\nThe set \\(V(T)\\) is at most countable. Consequently, there is a labeling \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) such that:\n\\begin{itemize}\n\\item the set \\(V_{\\varepsilon} = \\{v \\in V(T) \\colon l_1(v)  \\geqslant \\varepsilon\\}\\) is finite for every \\(\\varepsilon > 0\\); \n\\item the inequality \\(l_1(u) > 0\\) holds for every \\(u \\in V^F\\); \n\\item the equality \\(l_1(w) = 0\\) holds for every \\(w \\in V^I\\). \n\\end{itemize}\n\nThese properties of \\(l_1\\) and statement \\ref{t7:s1} imply the inequality\n\\[\n\\max \\bigl\\{l_1(x), l_1(y)\\bigr\\} > 0\n\\]\nfor every \\(\\{x, y\\} \\in E(T)\\). Hence, \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) is a non-degenerate labeling. Thus, \\((V(T), d_{l_1})\\) is compact by Theorem~\\ref{t4.18}.\n\n\n\\(\\ref{t7:s2} \\Rightarrow \\ref{t7:s1}\\). Let \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) be a non-degenerate labeling for which the ultrametric space \\((V(T), d_{l_1})\\) is compact. Then, from Theorem~\\ref{t4.18} it follows that \\(T\\) is rayless. Moreover, since every compact metric space is totally bounded, the set \\(V(T)\\) is at most countable by Corollary~\\ref{c4.8}.\n\nTo complete the proof it is enough to show that the inequality\n\\begin{equation*\n\\min\\bigl\\{\\delta(u), \\delta(v)\\bigr\\} < \\infty\n\\end{equation*}\nholds for every \\(\\{u, v\\} \\in E(T)\\). Assume to the contrary that there exists \\(\\{u, v\\} \\in E(T)\\) such that\n\\[\n\\delta(u) = \\delta(v) = \\aleph_0.\n\\]\nSince \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\) is a non-degenerate labeling, \\(\\{u, v\\} \\in E(T)\\) implies\n\\[\n\\max\\bigl\\{l_1(u), l_1(v)\\bigr\\} > 0.\n\\]\nWithout loss of generality, we may assume that \\(l_1(v) > 0\\). The last inequality, the inequality \\(\\delta(v) > 0\\) and Theorem~\\ref{t4.18} imply that \\((V(T), d_{l_1})\\) is not compact, contrary to the definition of \\(l_1 \\colon V(T) \\to \\mathbb{R}^{+}\\).\n\\end{proof}\n\nCorollary~\\ref{c4.7} and Theorem~\\ref{t7} imply the following.\n\n\\begin{corollary}\\label{c4.23}\nLet \\(T\\) be a tree. If there is a non-degenerate labeling \\(l_1 \\colon V(T) \\to \\RR^{+}\\) for which the ultrametric space \\((V(T), d_{l_1})\\) is compact, then \\((V(T), d_{l})\\) is complete for every non-degenerate labeling \\(l \\colon V(T) \\to \\RR^{+}\\).\n\\end{corollary}\n\n\\begin{remark}\\label{r4.21}\nIt is interesting to compare Corollary~\\ref{c4.23} with the following theorem: ``A metrizable topological space \\((X, \\tau)\\) is compact if and only if every metric generated the topology \\(\\tau\\) is complete.'' This result was proved by Niemytzki and Tychonoff in 1928 \\cite{NTFM1928}. There is also an ultrametric modification of Niemytzki---Tychonoff theorem recently obtained by Yoshito Ishiki~\\cite{Isha2020}.\n\\end{remark}\n\nThe next corollary follows from Proposition~\\ref{p3.2} and Theorems~\\ref{t11.16}, \\ref{t4.18}.\n\n\\begin{corollary}\\label{c4.20}\nThe following statements are equivalent for every tree \\(T\\):\n\\begin{enumerate}\n\\item \\label{c4.20:s1} There are non-degenerate labelings \\(l_1 \\colon V(T) \\to \\RR^{+}\\) and \\(l_2 \\colon V(T) \\to \\RR^{+}\\) such that \\((V(T), d_{l_1})\\) is a compact ultrametric space and \\((V(T), d_{l_2})\\) is a discrete totally bounded ultrametric space.\n\\item \\label{c4.20:s2} \\(T\\) is a finite tree.\n\\end{enumerate}\n\\end{corollary}\n\n\\section{Some examples and conjectures}\n\nLet us consider examples of totally bounded non-complete ultrametric spaces, and compact ultrametric spaces generated by labeled trees having infinitely many vertices of infinite degree. To construct these examples, we use the \\emph{gluing} a given set of labeled trees to a fixed labeled tree. \n\nLet \\(\\mathcal{F} = \\{T_i(l_i) \\colon i \\in I\\}\\) be a nonempty set of labeled trees \\(T_i = T_i(l_{i})\\) for which\n\\begin{equation}\\label{e4.34}\nV(T_{i_1}) \\cap V(T_{i_2}) = \\varnothing\n\\end{equation}\nholds for all distinct \\(i_1\\), \\(i_2 \\in I\\), and let \\(T^{*} = T^{*}(l^{*})\\) be a labeled tree such that \\(V(T_i) \\cap V(T^{*})\\) is a single-point set \\(\\{v_i\\}\\),\n\\begin{equation}\\label{e4.35}\nV(T_i) \\cap V(T^{*}) = \\{v_i\\}\n\\end{equation}\nfor every \\(i \\in I\\). Let us suppose also\n\\begin{equation}\\label{e4.36}\nl^{*}(v_i) = l_{i}(v_i)\n\\end{equation}\nfor every \\(i \\in I\\) if \\(v_i\\) satisfies \\eqref{e4.35}. Then we define the gluing \\(\\mathcal{F}\\) to \\(T^{*}\\) as a labeled graph \\(T = T(l)\\) with \n\\begin{equation}\\label{e4.37}\nV(T) \\stackrel{\\mathrm{def}}{=} V(T^{*}) \\cup \\left(\\bigcup_{i \\in I} V(T_i)\\right), \\quad E(T) \\stackrel{\\mathrm{def}}{=} E(T^{*}) \\cup \\left(\\bigcup_{i \\in I} E(T_i)\\right)\n\\end{equation}\nand \\(l \\colon V(T) \\to \\RR\\) such that\n\\begin{equation}\\label{e4.38}\nl(v) = \\begin{cases}\nl^{*}(v) & \\text{if } v \\in V(T^{*}),\\\\\nl_{i}(v) & \\text{if } v \\in V(T_i) \\text{ for some } i \\in I.\n\\end{cases}\n\\end{equation}\nUsing equalities \\eqref{e4.34}--\\eqref{e4.38} one can simply show that \\(T = T(l)\\) is a well-defined labeled tree for which the labeling \\(l\\) is non-degenerate if and only if all labelings \\(l_i\\), \\(i \\in I\\), and \\(l^{*}\\) are non-degenerate.\n\n\\begin{example}\\label{ex11.10}\nLet \\(R^{*} = R^{*}(l^{*})\\) be a labeled ray such that \\(V(R^{*}) = \\mathbb{N}\\) and\n\\[\n\\bigl(\\{m, n\\} \\in E(R^{*})\\bigr) \\Leftrightarrow \\bigl(|m-n|=1\\bigr)\n\\]\nfor all \\(m\\), \\(n \\in \\mathbb{N}\\) and, let the equality\n\\[\nl^{*}(m) = \\begin{cases}\n\\frac{1}{m} & \\text{if \\(m\\) is odd},\\\\\n0 & \\text{if \\(m\\) is even}\n\\end{cases}\n\\]\nhold for each \\(m \\in \\NN\\). Moreover, for every even \\(m \\in \\mathbb{N}\\) we define a labeled star \\(S_m(l_m)\\) with a center \\(c_m = m\\) and suppose that the following conditions hold:\n\\[\n\\quad V(S_m) \\cap \\NN = \\{m\\}, \\quad l_m(c_m) = 0,\n\\]\nand the restriction \\(l_m|_{V(S_m) \\setminus \\{c_m\\}}\\) of \\(l_m\\) on the set \\(V(S_m) \\setminus \\{c_m\\}\\) is a bijection to the set \\(\\{\\frac{1}{mn} \\colon n \\in \\mathbb{N}\\}\\); and\n\\[\nV(S_{m_1}) \\cap V(S_{m_2}) = \\varnothing\n\\]\nholds for all distinct even \\(m_1\\), \\(m_1 \\in \\mathbb{N}\\). Then we can consider the labeled tree \\(T = T(l)\\) obtained by gluing the set \n\\[\n\\{S_m(l_m) \\colon m \\in \\NN \\text{ and } m \\text{ is even}\\}\n\\]\nto the labeled ray \\(R^{*}(l^{*})\\) (see Figure~\\ref{fig10}). The ultrametric space \\((V(T), d_l)\\) is totally bounded by Theorem~\\ref{t11.8} but not complete by Theorem~\\ref{t4.7}.\n\\end{example}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{2cm}\n\\draw (0, 0) -- (6*\\dx, 0);\n\\node [solid node] at (0, 0) [label= below:\\(\\frac{1}{1}\\)] {};\n\\node [solid node] at (\\dx, 0) [label= below:\\(0\\)] {};\n\\node at (0, \\dx) [label= above:\\(T(l)\\)] {};\n\\draw (\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{2}\\)] {};\n\\draw (\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{4}\\)] {};\n\\draw (\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{6}\\)] {};\n\\draw (\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{2n}\\)] {};\n\\draw (\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (\\dx, \\dx) [label= above:\\(S_1(l)\\)] {};\n\n\\node [solid node] at (2*\\dx, 0) [label= below:\\(\\frac{1}{3}\\)] {};\n\n\\node [solid node] at (3*\\dx, 0) [label= below:\\(0\\)] {};\n\\draw (3*\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{4}\\)] {};\n\\draw (3*\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{8}\\)] {};\n\\draw (3*\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{12}\\)] {};\n\\draw (3*\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{4n}\\)] {};\n\\draw (3*\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (3*\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (3*\\dx, \\dx) [label= above:\\(S_2(l)\\)] {};\n\n\\node [micro node] at (3.3*\\dx, 0) {};\n\\node [micro node] at (3.5*\\dx, 0) {};\n\\node [micro node] at (3.7*\\dx, 0) {};\n\n\\node [solid node] at (4*\\dx, 0) [label= below:\\(\\frac{1}{2m-1}\\)] {};\n\n\\node [solid node] at (5*\\dx, 0) [label= below:\\(0\\)] {};\n\\draw (5*\\dx, 0) -- +(150:0.9*\\dx) node [solid node, label= above:\\(\\frac{1}{2m}\\)] {};\n\\draw (5*\\dx, 0) -- +(110:0.8*\\dx) node [solid node, label= above:\\(\\frac{1}{4m}\\)] {};\n\\draw (5*\\dx, 0) -- +(80:0.7*\\dx) node [solid node, label= above:\\(\\frac{1}{6m}\\)] {};\n\\draw (5*\\dx, 0) +(70:0.67*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(60:0.65*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(50:0.63*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(40:0.6*\\dx) node [solid node, label= above right:\\(\\frac{1}{2mn}\\)] {};\n\\draw (5*\\dx, 0) +(30:0.5*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(20:0.45*\\dx) node [micro node] {};\n\\draw (5*\\dx, 0) +(10:0.4*\\dx) node [micro node] {};\n\\node at (5*\\dx, \\dx) [label= above:\\(S_m(l)\\)] {};\n\n\\node [micro node] at (5.3*\\dx, 0) {};\n\\node [micro node] at (5.5*\\dx, 0) {};\n\\node [micro node] at (5.7*\\dx, 0) {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The tree \\(T\\) has \\(\\aleph_0\\) vertices with degree \\(\\aleph_0\\) and the ultrametric space \\((V(T), d_l)\\) is totally bounded but not complete.} \n\\label{fig10}\n\\end{figure}\n\n\\begin{example}\\label{ex4.21}\nLet \\(\\mathbb{P}\\) be the set of all integer prime numbers \\(p \\geqslant 2\\) and let \\(S^{*} = S^{*}(l^{*})\\) be the labeled star with the vertex set \\(V(S^{*}) = \\{p \\in \\mathbb{P}\\} \\cup \\{0\\}\\), and the center \\(c^{*} = 0\\), and the labeling \\(l^{*} \\colon V(S^{*}) \\to \\RR^{+}\\) for which \\(l^*(0) = 0\\) and \\(l^{*} (p) = 1\/p\\) hold for all \\(p \\in \\mathbb{P}\\).\n\nFor every \\(p \\in \\mathbb{P}\\) we define a labeled star \\(S_p = S_p(l_p)\\) with a center \\(c_p\\) such that:\n\\[\nV(S_p) \\setminus \\{c_p\\} = \\{p^{n} \\colon n \\in \\mathbb{N}\\},\n\\]\nand \n\\[\nc_p \\notin \\bigcup_{p \\in \\mathbb{P}} \\bigl(V(S_p) \\setminus \\{c_p\\}\\bigr) \\cup V(S^{*});\n\\]\n\\[\nl_p(v) = \\begin{cases}\n0 & \\text{if } v = c_p,\\\\\np^{-n} & \\text{if } v = p^{n} \\text{ for some } n \\in \\NN;\n\\end{cases}\n\\]\nand \\(c_{p_1} \\neq c_{p_2}\\) for all distinct \\(p_1\\), \\(p_2 \\in \\mathbb{P}\\). Then we obtain \\(V(S^{*}) \\cap V(S_p) = \\{p\\}\\), and \\(l^{*} (p) =  l_p(p) = 1\/p\\), and \\(\\delta_{S^{*}} (c^{*}) = \\delta_{S_p} (c_p) = \\aleph_0\\) for every \\(p \\in \\mathbb{P}\\) .\n\nLet us consider the labeled tree \\(T = T(l)\\) which is obtained by gluing the set \\(\\{S_p(l_p) \\colon p \\in \\mathbb{P}\\}\\) to \\(S^{*}(l^{*})\\) (see Figure~\\ref{fig1}), then the ultrametric space \\((V(T), d_l)\\) is compact by Theorem~\\ref{t4.18}.\n\\end{example}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tikzpicture}[\nlevel 1\/.style={level distance=\\levdist,sibling distance=24mm},\nlevel 2\/.style={level distance=\\levdist,sibling distance=12mm},\nlevel 3\/.style={level distance=\\levdist,sibling distance=6mm},\nsolid node\/.style={circle,draw,inner sep=1.5,fill=black},\nmicro node\/.style={circle,draw,inner sep=0.5,fill=black}]\n\n\\def\\dx{1.2cm}\n\\node at (3*\\dx, 3*\\dx) [label= above:\\(T(l)\\)] {};\n\\node [solid node] at (0, 0) [label= below:\\(0\\)] {};\n\n\\def\\dphi{20}\n\\def\\xx{2}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= above:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{80}\n\\def\\xx{3}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= above left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= left:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= below right:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{140}\n\\def\\xx{5}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above right:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\def\\dphi{200}\n\\def\\xx{7}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= below left:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= below:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= right:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\node [micro node] at (220:3*\\dx) {};\n\\node [micro node] at (230:3*\\dx) {};\n\\node [micro node] at (240:3*\\dx) {};\n\n\\def\\dphi{280}\n\\def\\xx{p}\n\\draw (0, 0) -- (\\dphi:3*\\dx) node [solid node, label= below:\\(0\\)] {};\n\\node [solid node] at (\\dphi:2*\\dx) [label= right:\\(\\frac{1}{\\xx}\\)] {};\n\\draw (-180+\\dphi+30:\\dx)++(\\dphi:3*\\dx) node [solid node, label= above:\\(\\frac{1}{\\xx^2}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+60:\\dx)++(\\dphi:3*\\dx) node [solid node, label= left:\\(\\frac{1}{\\xx^3}\\)] {} -- (\\dphi:3*\\dx);\n\\draw (-180+\\dphi+90:\\dx)++(\\dphi:3*\\dx) node [solid node, label= left:\\(\\frac{1}{\\xx^4}\\)] {} -- (\\dphi:3*\\dx);\n\\node [micro node] at ($(-180+\\dphi+120:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+150:\\dx) + (\\dphi:3*\\dx)$) {};\n\\node [micro node] at ($(-180+\\dphi+180:\\dx) + (\\dphi:3*\\dx)$) {};\n\n\\node [micro node] at (310:3*\\dx) {};\n\\node [micro node] at (320:3*\\dx) {};\n\\node [micro node] at (330:3*\\dx) {};\n\\end{tikzpicture}\n\\end{center}\n\\caption{The tree \\(T\\) has \\(\\aleph_0\\) vertices with degree \\(\\aleph_0\\) and the ultrametric space \\((V(T), d_l)\\) is compact.} \n\\label{fig1}\n\\end{figure}\n\nThe following simple example shows that the class of finite ultrametric spaces, which are representable in the form \\((V(T), d_l)\\), is a proper subclass of all finite ultrametric spaces.\n\n\\begin{example}\\label{ex11.11}\nLet \\(V = \\{v_1, v_2, v_3, v_4\\}\\) be a four-point set and let an ultrametric \\(d \\colon V \\times V \\to \\RR^{+}\\) satisfy the equalities\n\\begin{gather}\\label{ex11.11:e1}\nd(v_1, v_2) = d(v_3, v_4) = 1\\\\\n\\intertext{and}\n\\label{ex11.11:e2}\nd(v_1, v_3) = d(v_1, v_4) = d(v_2, v_3) = d(v_2, v_4) = 2.\n\\end{gather}\nLet \\(T = T(l)\\) be a labeled tree such that \\(V(T) = V\\). We claim that the ultrametric spaces \\((V, d)\\) and \\((V(T), d_l)\\) are not isometric for any non-degenerate labeling \\(l\\). Indeed, if there is a non-degenerate \\(l \\colon V(T) \\to \\RR^{+}\\) for which \\((V, d)\\) and \\((V(T), d_l)\\) are isomorphic, then from~\\eqref{e11.3} and \\eqref{ex11.11:e1} it follows that\n\\[\n\\max_{1 \\leqslant i \\leqslant 4} l(v_i) \\leqslant 1.\n\\]\nThe last inequality and \\eqref{e11.3} imply that \\(d_l(v, u) \\leqslant 1\\) holds for all \\(u\\), \\(v \\in V\\), contrary to \\eqref{ex11.11:e2}.\n\\end{example}\n\nIt seems to be interesting to get a purely metric characterization of ultrametric spaces generated by labeled trees.\n\n\\begin{conjecture}\\label{con11.1}\nLet \\((X, d)\\) be a discrete nonempty totally bounded ultrametric space. Then the following statements are equivalent:\n\\begin{enumerate}\n\\item\\label{con11.1:s1} There is a labeled tree \\(T = T(l)\\) such that \\((V(T), d_l)\\) and \\((X, d)\\) are isometric.\n\\item\\label{con11.1:s2} For every \\(B \\in \\BB_{X}\\), there are \\(c \\in X\\) and \\(r > 0\\) such that\n\\[\nB = \\{x \\in X \\colon d(x, c) = r\\} \\cup \\{c\\}\n\\]\ni.e., the ball \\(B\\) is the sphere \\(S(c, r) = \\{x \\in X \\colon d(x, c) = r\\}\\) with the added center \\(c\\).\n\\end{enumerate}\n\\end{conjecture}\n\nIn conclusion, we formulate a simple conjecture linking the properties of Cauchy sequences in \\((V(T), d_l)\\) with the structure of the hull of the range sets of these sequences (cf. Lemma~\\ref{l4.8}).\n\n\\begin{conjecture}\\label{con5.2}\nLet \\(T\\) be a tree and let \\((v_n)_{n \\in \\NN}\\) be a sequence of distinct vertices of \\(T\\). Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item The hull of the range set of \\((v_n)_{n \\in \\NN}\\) is a union of a ray with some finite tree.\n\\item For every non-degenerate \\(l \\colon V(T) \\to \\RR\\) the existence of Cauchy subsequence in \\((v_n)_{n \\in \\NN}\\) implies that \\((v_n)_{n \\in \\NN}\\) is also a Cauchy sequence.\n\\end{enumerate}\n\\end{conjecture}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGeometry is central to our understanding of the world. Viewed as something real or\nas a convention to describe reality, it has evolved with our thinking and will continue to do so \\cite{Jammer1954}. After the centuries needed to conceive what we  today call the Euclidean space, the advent of calculus and  non-Euclidean geometries on the one side, and the investigation of electromagnetic phenomena on the other, have produced a radical transformation of our concept of space. The turning point is marked by Riemann's trial lecture \\emph{On the Hypotheses That Lie at The Foundations of Geometry} \\cite{Riemann,Pesic2007}.  In just over half a century, Riemann's vision has become the backbone of the general theory of relativity and therefore of our understanding of spacetime. This has stimulated an enormous interest in differential geometry, aiming to reconcile Einstein's theory of gravity with the other fundamental interactions and with the quantum world. \n\nThe purpose of this work is far less ambitious. \nI am going to investigate a differential geometry based solely on the minimal assumption that the distance of every point from itself is equal to zero. Thus, besides renouncing to the \ninfinitesimal Pythagorean distance formula as in Finsler geometry \\cite{Rund2012,BaoChernShen2012}, I will also drop Riemann's residual hypotheses of homogeneity and symmetry. Surprisingly, this leads to a logically simple construction, very close in spirit and techniques to standard Riemannian geometry.\nIn particular, I will show that the infinitesimal Pythagorean distance formula reemerges, without the need of being postulated, as a first approximation to \\emph{almost every}  geometry that is invariant with respect to direction reversal. More in general, \\emph{almost every} geometry turns out to be approximated by a one-parameter family of homogeneous Riemann-Randers metrics  \\cite{Randers1941}. The parameter is related to a geometric conservation law. Homogeneity is recovered for each value of the parameter, but not as a whole. The connection and the curvature associated to a generic line element  emerge from the identification of geodesic equations with auto parallel equations, leading to a univocal generalisation of the Christoffel-Levi-Civita connection and of the Riemannian curvature tensor.\n\nIf, following the spirit of general relativity we apply this geometry to spacetime, \nwe effortlessly  find ourselves with what could well be the unified theory of the classical electromagnetic and gravitational fields, long sought by Weyl, Kaluza, Eddington, Einstein, Schr\\\"{o}dinger and many others \\cite{Goenner2004,Goenner2014,Tonnelat1966}. Compared to other attempts, the identification of geometrical objects with physical quantities is logical and completely straightforward, like the imposition of the possible field equations. Moreover, some fundamental yet puzzling facts, such as the hierarchy between electromagnetic and gravitational interactions, their different attractive\/repulsive nature and CPT invariance, naturally emerge from the geometrical framework. Whether this geometry will bring any progress in our comprehension of the classical spacetime it is too early to say. \nNonetheless, the conceptual simplicity and economy that it offers in the formulation of both the electromagnetic and gravitational theories seems to me undeniable.\\\\\n\nIn section \\ref{hypotheses} I will review Riemann's hypotheses on geometry together with  their revisions, from the indefiniteness of the line element in pseudo-Riemanninan geometry, to line elements that are arbitrary homogeneous functions of direction in Finsler geometry. A differential geometry based on a line element only constrained by the assumption that the distance of every point from itself equals zero is presented in section \\ref{relax}. Section \\ref{spacetime}\ndescribes the possible applications to spacetime. Two appendices respectively discuss the geometrical interpretation of the gravitational field in general relativity and offer an elementary introduction to nonlinear connections.\n\n\\section{Riemann's hypotheses on geometry}\\label{hypotheses}\nIn his celebrated 1854 trial lecture \\emph{On the Hypotheses That Lie at The Foundations of Geometry}, Riemann introduces a general analytic approach to geometry by extending Gauss's work on curved surfaces to ``$n$-fold extended manifolds\"\\footnote{If not otherwise stated, all the quotations throughout this paper are from Riemann's \\emph{Ueber die Hypothesen, welche der Geometrie zu Grunde liegen} \\cite{Riemann} in the English translation by M. Spivack with slight revisions by P. Pesich \\cite{Pesic2007}.}. These are defined as spaces parameterized by $n$ independent coordinates $x^\\mu$, $\\mu=1,2, ...,n$. \nThe investigation of metric relations of such spaces is approached by measuring the length of lines, so that the distance between two points is implicitly defined as the length of the shortest line joining them, a geodesic line. Primarily, Riemann assumes that as ``measurement requires independence of quantity from position [...] the length of lines is independent of their configuration, so that every line can be measured by every other\", a concept that Einstein will better rephrase by stating that ``if two line segments are found to be equal at one time and at some place, they are equal always and everywhere\"\\cite{Einstein1921}. Next, to set up a mathematical expression for the length of lines, Riemann treats the problem ``only under certain restrictions\" by assuming\n\\begin{description}\n  \\item[\\emph{Differentiability}:] The lines whose length is to be determined are differentiable, so that the problem reduces ``to setting up a general expression for the line element $ds$'' connecting two infinitesimally adjacent points  $x=\\left(x^1, x^2, ..., x^n\\right)$ and $x+dx=\\left(x^1+dx^1, x^2+dx^2, ...,x^n+dx^n\\right)$, ``an expression which will involve the quantities $x$ and the quantities $dx$.'' Therefore,  $ds=f(x,dx)$.\n  \\item[\\emph{Homogeneity}:] If all increments $dx^\\mu$ are  increased by the same ratio, $ds$ must also be increased by the same ratio. Equivalently, $ds$ must be  ``an arbitrary homogeneous function of the first degree in the quantities $dx$''. That is to say, $f(x,k\\, dx)=k f(x,dx)$ with $k>0$.\n  \\item[\\emph{Symmetry}:] The line element ``remains the same when all quantities $dx$ change sign.'' Equivalently, $ds$ is an even function of the increments $dx^\\mu$. \nIn symbolic notation, $f(x,-dx)=f(x,dx)$.\n\\end{description}\nAt this point, in order to find the explicit form for $ds$, Riemann seeks  ``an expression for the $(n-1)$-fold extended manifold which are everywhere equidistant from the origin of the line element, i.e.\\ [...] a continuous function which distinguishes them from one another. This must either decrease or increase in all directions from the origin;\" he assumes that ``it increases in all directions and therefore has a minimum at the origin. Then if  its first and second differential quotients are finite, the first order must vanish and the second order cannot be negative;\" he eventually assumes that ``it is always positive.\" From this he deduces that\n\\begin{description}\n \\item[\\emph{Quadratic restriction}:] The line element $ds$ equals ``the square root of an everywhere positive homogeneous function of the second degree in the quantities $dx$, in which the coefficients are continuous functions of the quantities $x$.''\\footnote{The expression \\emph{quadratic restriction} to describe Riemann's choice of the infinitesimal Pythagorean distance formula \nfor the line element was introduced by Chern in \\cite{Chern1996}.}  \n\\end{description}\nNamely, the infinitesimal expression of the Pythagorean distance formula in arbitrary coordinates. In modern notation\n\\begin{equation}\n\\label{Rds}\nds=\\sqrt{g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}},\n\\end{equation}\nwhere $g_{\\mu\\nu}(x)$ are regular functions of the coordinates and the summation over repeated indices is understood. Clearly, the fourth hypothesis implies the previous three but is not implied by them. In particular, the assumption that the ``second differential quotient[...]'' is finite, does not imply that it is nonvanishing. From this Riemann realised that the minimum can also be produced by an appropriate root of a  biquadratic or higher-order expression. In fact, he notes that ``[t]he next simplest case would perhaps include the manifolds in which the line element can be expressed as the fourth root of a differential expression of the fourth degree.\"\\\\\n\nRemarkably, Riemann makes clear that these hypotheses ``are not logically necessary,\" and ``one can therefore investigate their likelihood, which is certainly very great within the bounds of observation, and afterwards decide on the legitimacy of extending them beyond the bounds of observation\". As is well known, the `certainty within the bounds of observation' soon called for a revision of Riemann's fourth hypothesis. This is implicitly contained in Minkowski's geometrical reformulation of Einstein's theory of special relativity and was later made explicit in the general theory.  The unification of space and time into a single four dimensional continuum that ``sprang from the soil of experimental physics\"\\cite{Minkowski1908} required the revision of the \\emph{quadratic restriction} allowing the indefiniteness of the quadratic form appearing in $ds$ or, equivalently, allowing pseudo-Euclidean geometries besides the Euclidean one on infinitesimal scales.\n\nAnother call for revision came some thirty years later from Randers's observation that ``the most characteristic property of the physical world is the uni-direction of time-like intervals\" so that  ``the perfect symmetry in any direction for any coordinate interval does not seem quite appropriate for the application to physical space-time\"\\cite{Randers1941}. In other words, the simultaneous parity and time reversal symmetry PT is not a fundamental symmetry of nature, becoming such only if accompanied by the simultaneous transformation of charge conjugation C. Accordingly, it would be preferable to have something like CPT symmetry embodied in the fundamental texture of spacetime geometry, rather than PT \\emph{symmetry}.\n\nRiemannian geometry without the \\emph{quadratic restriction}, and also with or without the \\emph{symmetry} hypothesis, has been developed since the 1920s by Finsler, Bernwald, Buselman, Cartan, Rund, Chern and many others \\cite{Rund2012,BaoChernShen2012}. For historical reasons it is known as Finsler geometry. The construction, that requires ``no essential new ideas\"\\cite{Chern1996}, is far from being logically simple, leading to a structure supporting at least three different curvatures and five torsion tensors. \nMoreover, ``while Riemannian geometry can be handled, elegantly and efficiently, by tensor analysis [...] Finsler geometry [...] needs more than one space [...] on which tensor analysis does not fit well\"\\cite{Chern1996}. Starting from the 1930s attempts have been made to apply Finsler geometry to spacetime, both in the effort to extend the theory of relativity or to unify gravity with electromagnetism \\cite{Goenner2004,Goenner2014,Tonnelat1966,Rutz1993,Asanov2012,Pfeifer&Wohlfarth2012}. However, in spite of the extreme richness in structure, it ``seems that this particular generalisation of Riemannian geometry is not able to lead to a correct implementation of the electromagnetic field\"\\cite{Stephenson1957}, nor to provide a logically attractive extension of general relativity. Riemann's observation that the ``investigation of this general class [...] require[s] no essential different principles [...] and throw[s] proportionally little new light on the study of space\" seems to be fully justified, after all. \n\nAt present, \\emph{homogeneity}  is  considered a fundamental technical requirement in Riemannian geometry. In fact, it heuristically justifies the chain of identities\n\\begin{equation*}\n\\int ds=\\int{\\textstyle\\frac{1}{d\\tau}} f(x,dx)d\\tau=\\int\nf\\left(x(\\tau),{\\textstyle\\frac{dx}{d\\tau}(\\tau)}\\right)d\\tau\n\\end{equation*} \nthat allows to explicitly evaluate the line integral of $ds$  and, when extrapolated to $k=0$,  implies that the distance of every point from itself equals zero. This latter really seems  to be an unrenounceable requirement for a distance function. In fact, in the abstract theory of metric spaces it is one of the axioms that a distance function must satisfy. However, nothing like homogeneity is generally required for a distance function. It is therefore tempting to consider the possibility of relaxing Riemann's \\emph{homogeneity} hypothesis to the weaker assumption of\n\\begin{description}\n  \\item[\\emph{Indiscernibility}:] If all increments $dx^{\\mu}$ are vanishing, $ds$ must also vanish. That is to say, $f(x,0)=0$.\\footnote{That the distance of every point from itself is zero, is generally assumed together with its inverse statement (if the distance between two points is zero then the two points coincide) under the name of \\emph{identity of indiscernibles}. By analogy, I will call the direct statement alone \\emph{indiscernibility} as the line element does not allows to perceive a point as two distinct entities. Given the existence of hyperbolic geometry the inverse statement does not  fit as an axiom for geometry. }\n\\end{description}\nTo the best of my knowledge, no one so far has considered Riemannian geometry without the hypotheses of \\emph{homogeneity} and \\emph{symmetry}.\n\n\\section{Relaxing homogeneity and symmetry}\\label{relax}\nHereafter I will consider Riemann-type geometries uniquely based on the hypotheses of \\emph{differentiability} and \\emph{inderniscibility}. For the sake of simplicity I will assume differentiability in its strongest sense, i.e.\\ analyticity.\n\\subsection*{Line element}\nThe keystone that supports the construction of Riemann-type geometries without the hypothesis of \\emph{homogeneity},  is the fact that the line element can again be taken in the general form  \n\\begin{equation}\n\\label{ds}\nds=f\\left(x(\\tau),{\\textstyle\\frac{dx}{d\\tau}(\\tau)}\\right)d\\tau\n\\end{equation} \nwith $f(x,\\dot{x})$ an arbitrary analytic function of the coordinates $x$ and of the directions $\\dot{x}=\\frac{dx}{d\\tau}$ and with the \\emph{proviso} that \\emph{the parameter $\\tau$ is implicitly determined by the geodesic equations}. Shortly, I will show that the geodesic equations do actually determine the geodesic parameter $\\tau$. In complete generality I will proceed by series-expanding $ds$ about the point $\\dot{x}=0$. The line element is then rewritten as\n\\begin{equation}\n\\label{tds}\nds=\\left(g_\\mu\\dot{x}^\\mu+\\frac{1}{2}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu+\n\\frac{1}{3!}g_{\\mu\\nu\\xi}\\dot{x}^\\mu\\dot{x}^\\nu\\dot{x}^\\xi+...+\n\\frac{1}{k!}g_{\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+...\\right)d\\tau\n\\end{equation}\nwhere the zero order term is missing because of the hypothesis of \\emph{indiscernibility}  and the symmetric tensors $g_\\mu(x)=\\frac{\\partial f}{\\partial \\dot{x}^\\mu}(x,0)$, $g_{\\mu\\nu}(x)=\\frac{\\partial^2 f}{\\partial \\dot{x}^\\mu\\partial \\dot{x}^\\nu}(x,0)$, ..., $g_{\\mu_1...\\mu_k}(x)=\\frac{\\partial^k f}{\\partial \\dot{x}^{\\mu_1}...\\partial \\dot{x}^{\\mu_k}}(x,0)$ ..., are regular functions of the coordinates. It is important to emphasize that these tensors are not independent geometrical objects but only a different way of encoding all the information contained in the line element $ds$. The variation of the line integral of $ds$ produces now the geodesic equations in terms of the symmetric tensors $g_{\\mu_1...\\mu_k}(x)$  and of their derivatives. However, we do  not need to write them down explicitly in order to see that they do determine the geodesic parameter. As $ds$ does not depend explicitly on $\\tau$, from the analogy with analytical mechanics we can immediately conclude that the geodesic equations admit the first integral\n\\begin{equation}\n\\label{fi} \ng_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu+ \\frac{2}{3}g_{\\mu\\nu\\xi}\\dot{x}^\\mu\\dot{x}^\\nu\\dot{x}^\\xi+...+ \\frac{2}{k(k-2)!}g_{\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+...=2E,\\end{equation}\nwith $E$ an arbitrary constant. This identity implicitly determines $d\\tau$ in terms of the differentials $g_{\\mu_1...\\mu_k}dx^{\\mu_1}...dx^{\\mu_k}$ and the constant $E$. Consequently, it determines $\\tau$ up to an irrelevant additive constant. \\\\\n\nFor example, if $g_{\\mu\\nu}dx^\\mu dx^\\nu>0$ and $E>0$, as is always the case in standard Riemannian geometry, the series-inversion of equation (\\ref{fi}) yields \\begin{eqnarray}\\label{dtau}\nd\\tau=\\frac{1}{2\\varepsilon}\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}+\n\\frac{g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}}{3g_{\\mu\\nu}dx^\\mu dx^\\nu}+\\hskip118pt\n&&  \\nonumber \\\\[5pt]\n+\\varepsilon \\left(\\frac{ g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}{4\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^3}}-\\frac{\\left(g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}\\right)^2}{3\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^5}}\\right)\n+...&& \n\\end{eqnarray}\nwhere, for simplicity, I have set $\\varepsilon=\\sqrt{E\/2}$. The substitution of \n(\\ref{dtau}) into (\\ref{tds}) yields the explicit form of the line element in terms of  ``the quantities $x$ and the quantities  $dx$\" as sought by Riemann\n\\begin{eqnarray}\n\\label{gdsExp}\n\\frac{ds}{\\varepsilon}=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\n\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}+\n\\hskip193pt\n&&  \\nonumber \\\\[5pt]\n-\\varepsilon^2 \n\\left(\\frac{g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}{6\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^3}}-\\frac{2\\left(g_{\\mu_1\\mu_2\\mu_3}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}\\right)^2}{9\\sqrt{\\left(g_{\\mu\\nu}dx^\\mu dx^\\nu\\right)^5}}\\right)\n+...&& \n\\end{eqnarray}\nQuite surprisingly, the most general line element for  Riemann-type geometries without the hypothesis of \\emph{homogeneity} turns out to be a one-parameter family of homogeneous line elements. \n\nFor geometries that are symmetric under direction reversal, where  $g_{\\mu}$ as well as all odd-order tensors $g_{\\mu_1\\mu_2... \\mu_{2k-1}}$ vanishes identically, the (rescaled) line element reduces in first approximation to the standard Riemannian line element $ds=\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$. If terms of order $\\varepsilon^2$ and higher can be neglected, the geometry turns out to be homogeneous because the line element can always be rescaled by a constant factor. This shows the extraordinary generality of Riemann's assumption. In this limit (\\ref{Rds}) can be derived from the sole hypotheses of \\emph{differentiability}, \\emph{indiscernibility} and \\emph{symmetry}. In particular, the Pythagorean proposition emerges, \\emph{without the need of being postulated}, as the fundamental building block of geometry on infinitesimal scales. \nClearly, it is still possible to construct non-Pythagorean geometries\\footnote{If we repeat the series inversion and the substitution of $d\\tau$ in $ds$ under the assumption of a vanishing $g_{\\mu\\nu}$, we obtain that ``the [rescaled] line element can be expressed as the fourth root of a differential expression of fourth degree'' plus correction of order $\\varepsilon$ and higher,\n $${\\textstyle\\frac{3}{\\varepsilon^{3\/2}}}ds=\\sqrt[4]{g_{\\mu_1\\mu_2\\mu_3\\mu_4}dx^{\\mu_1}dx^{\\mu_2}dx^{\\mu_3}dx^{\\mu_4}}+O(\\varepsilon).$$\nThis was lucidly foreseen by Riemann's deep insight.} where $g_{\\mu\\nu}\\equiv0$. However, the number of these geometries equals the number of analytic functions with vanishing Hessian at $\\dot{x}=0$. Consequently, non-Pythagorean geometries only represent a subset of zero measure of the set of all possible Riemann-type geometries.\n\nFor generic geometries, the leading contribution to the line element is given by the one form $g_\\mu dx^\\mu$, which is however insufficient to specify a geometry. Consequently,  geometry is described in first approximation by a one-parameter family of homogeneous Riemann-Randers metrics \\cite{Randers1941}\n\\begin{equation}\n\\label{RRds}\nds=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}.\n\\end{equation}\nAs $\\varepsilon$ can no longer be eliminated by a constant rescaling,  the geometry as a whole is no longer homogeneous. On the other hand, while non-symmetrical by construction, the (rescaled) line element remains the same to all orders, if the transformation $dx^\\mu\\to-dx^\\mu$ is accompanied by the sign reversal of the parameter $\\varepsilon\\to-\\varepsilon$.\\\\\n\nThe general case is not substantially different but requires a careful discussion of different cases in order to express the line element $ds$ as a function of ``the quantities $x$ and the quantities $dx$\". On the other hand, this process is completely unnecessary. The geodesic equations can be more simply and conveniently derived by the direct variation of the line integral of (\\ref{tds}).\n\n\\subsection*{Geodesic equations}\n The Euler-Lagrange equations for the lines giving a stationary value to the line integral of  (\\ref{tds}), the geodesic equations of this geometry, are obtained as\n\\begin{eqnarray}\\label{ge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\ddot{x}^{\\lambda}+\n\\hskip20pt\n&&  \\nonumber \\\\[5pt]\n+\\Gamma_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\Gamma_{\\kappa\\mu\\nu}\\dot{x}^{\\mu}\\dot{x}^{\\nu} \n+\\Gamma_{\\kappa\\mu\\nu\\xi}\\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+... \n+\\Gamma_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0, \\hskip-20pt \n\\end{eqnarray}\nwhere dots indicate differentiation with respect to the parameter $\\tau$ and the symbols $\\Gamma$ are defined as\n\\begin{eqnarray}\\label{Gammas}\n&&\\Gamma_{\\kappa\\mu} = \\partial_\\mu g_\\kappa-\\partial_\\kappa g_\\mu \\nonumber \n\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu\\nu}  = \\frac{1}{2}\\left(\\partial_\\mu g_{\\kappa\\nu}\n+\\partial_\\nu g_{\\mu\\kappa}-\\partial_\\kappa g_{\\mu\\nu}\\right)  \\nonumber\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu\\nu\\xi}  = \\frac{1}{3!}\\left(\\partial_\\mu g_{\\kappa\\nu\\xi}\n+\\partial_\\nu g_{\\mu\\kappa\\xi}+\\partial_\\xi g_{\\mu\\nu\\kappa}\n-\\partial_\\kappa g_{\\mu\\nu\\xi}\\right)  \\\\[5pt]\n&& \\hskip10pt ...\\nonumber\\\\[5pt]\n&&\\Gamma_{\\kappa\\mu_1...\\mu_k}=\\frac{1}{k!}\n\\left(\\partial_{\\mu_1}g_{\\kappa\\mu_2...\\mu_{k-1}}+ ...\n+\\partial_{\\mu_n}g_{\\mu_1...\\mu_{k-1}\\kappa} -\n\\partial_\\kappa g_{\\mu_1...\\mu_k}\n\\right)\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\nonumber\n\\end{eqnarray}\n\nUp to an overall sign, the $\\Gamma_{\\kappa\\mu}$ are related to $g_\\mu$\nin the same way as an electromagnetic field is related to its vector potential. \nAlso, the $\\Gamma_{\\kappa\\mu\\nu}$ correspond to the Christoffel symbols of the first kind associated to the metric $g_{\\mu\\nu}$. In the light of (\\ref{Gammas}) these two relationships appear now as the special cases $k=1$ and $k=2$ of a more general rule that links a symmetric tensor $g_{\\mu_1...\\mu_k}$ of order $k$, to  the symbols $\\Gamma_{\\kappa\\mu_1...\\mu_k}$.\n\n Apart from $\\Gamma_{\\kappa\\mu}$,  that transforms like a tensor, all $\\Gamma$ symbols display non-covariant transformation rules, somehow analogues to that of the Christoffel symbols. Nonetheless, equations (\\ref{ge}) can be given an explicit covariant form by introducing a single symmetric linear connection ${C^\\kappa}_{\\mu\\nu}$. This can possibly, but not necessarily, be chosen as the Christoffel-Levi-Civita connection ${\\Gamma^\\kappa}_{\\mu\\nu}=g^{\\kappa\\lambda}{\\Gamma}_{\\lambda\\mu\\nu}$ associated to $g_{\\mu\\nu}$. In order to maintain full symmetry in the formalism, for the moment I will assume  that  ${C^\\kappa}_{\\mu\\nu}\\neq {\\Gamma^\\kappa}_{\\mu\\nu}$. Nothing of what follows depends in any way on the choice of ${C^\\kappa}_{\\mu\\nu}$ (see Appendix A). \n \n By adding and subtracting identical terms on the left hand side of (\\ref{ge}),  the geodesic  equations can be rewritten in the form\n\\begin{eqnarray}\\label{cge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{C^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip2pt\n&&  \\nonumber \\\\[5pt]\n+{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+{F}_{\\kappa\\mu\\nu} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\n+{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0,  && \n\\end{eqnarray}\nwhere the second derivatives $\\ddot{x}^{\\lambda}$ have been made covariant by the addition of appropriate terms and the non-covariant $\\Gamma$ symbols have been replaced by the covariant tensor fields\n\\begin{eqnarray}\n&&{F}_{\\kappa\\mu}=\\Gamma_{\\kappa\\mu} = \\nabla_\\mu g_\\kappa-\\nabla_\\kappa g_\\mu \\nonumber\\\\[5pt]\n&&{F}_{\\kappa\\mu\\nu}  \n=\\Gamma_{\\kappa\\mu\\nu}-g_{\\kappa\\lambda}C^\\lambda_{\\mu\\nu}\n= \\frac{1}{2}\\left(\\nabla_\\mu g_{\\kappa\\nu}\n+\\nabla_\\nu g_{\\mu\\kappa}-\\nabla_\\kappa g_{\\mu\\nu}\\right)\\nonumber\\\\[5pt]\n&&{F}_{\\kappa\\mu\\nu\\xi}  \n=\\Gamma_{\\kappa\\mu\\nu\\xi}-g_{\\kappa\\lambda(\\mu}C^\\lambda_{\\nu\\xi)}\n= \\frac{1}{3!}\\left(\\nabla_\\mu g_{\\kappa\\nu\\xi}\n+\\nabla_\\nu g_{\\mu\\kappa\\xi}+\\nabla_\\xi g_{\\mu\\nu\\kappa}\n-\\nabla_\\kappa g_{\\mu\\nu\\xi}\\right) \n\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\\\[5pt]\n&&{F}_{\\kappa\\mu_1...\\mu_k}=\\Gamma_{\\kappa\\mu_1...\\mu_k}-\\frac{1}{(k-2)!}g_{\\kappa\\lambda(\\mu_1... \\mu_{k-2}}C^\\lambda_{\\mu_{k-1}\\mu_k)}=\\nonumber\\\\[5pt]\n&&\\hskip40pt =\\frac{1}{k!}\n\\left(\\nabla_{\\mu_1}g_{\\kappa\\mu_2...\\mu_{k-1}}+ ...\n+\\nabla_{\\mu_n}g_{\\mu_1...\\mu_{k-1}\\kappa} -\n\\nabla_\\kappa g_{\\mu_1...\\mu_k}\n\\right)\\nonumber\\\\[5pt]\n&& \\hskip10pt ...\\nonumber\n\\end{eqnarray}\nwhere $\\nabla_\\kappa$ is the covariant derivative associated to the linear connection ${C^\\kappa}_{\\mu\\nu}$. We note that the tensor fields $F$ are simply obtained from  (\\ref{Gammas}) by replacing the partial derivatives $\\partial_\\kappa$ with the covariant derivatives $\\nabla_\\kappa$. \n\nThe second order symbols $\\Gamma_{\\kappa\\mu}=F_{\\kappa\\mu}$ are unaffected by this transformation. The Christoffel symbols of the first kind $\\Gamma_{\\kappa\\mu\\nu}$ associated to $g_{\\mu\\nu}$, are instead transformed into the analogues of the covariant gravitational field ${F}_{\\kappa\\mu\\nu}$ of what can be described as the Poicar\\'e-Rosen formalism for general relativity (see Appendix A). \nAll other higher order Christoffel-like symbols are transformed into the covariant fields ${F}_{\\kappa\\mu_1...\\mu_k}$.\nIt is interesting to note that even in this general approach to Riemannian geometry, the quadratic term in the series expansion of $ds$ maintains a unique status. The corresponding field ${F}_{\\kappa\\mu\\nu}$, is the only one that produces local effects indistinguishable from a coordinate transformation. Correspondingly, ${F}_{\\kappa\\mu\\nu}$ can always be set identically equal to zero by choosing  ${C^\\kappa}_{\\mu\\nu}$ equal to the Christoffel-Levi-Civita connection associated to $g_{\\mu\\nu}$. Since this considerably simplifies the formalism, making it analogous to that of standard Riemannian geometry and general relativity, from now on I will choose  ${C^\\kappa}_{\\mu\\nu}= {\\Gamma^\\kappa}_{\\mu\\nu}$. With this choice the geodesic equations are rewritten as \n\\begin{eqnarray}\\label{sge}\n\\left[g_{\\kappa\\lambda}\n+g_{\\kappa\\lambda\\mu}\\dot{x}^{\\mu}+...\n+\\frac{1}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip2pt\n&&  \\nonumber \\\\[5pt]\n+{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0,  && \n\\end{eqnarray}\nwhere ${F}_{\\kappa\\mu\\nu}$ no longer appears and $g_{\\mu\\nu}$ plays a privileged role with respect to the other fields.\nNonetheless, it is important to stress that all the $g_{\\mu_1...\\mu_k}$ take part in shaping the metric properties of space, such as the length of lines and the distance between points. In principle all fields can be treated on an equal footing.\\\\\n\nLet us now consider the effect of an arbitrary reparameterization $\\tau\\to\\zeta(\\tau)$ of the geodesic equations. A direct computation shows that  (\\ref{sge}) transforms into\n\\begin{eqnarray}\\label{zrcge}\n\\left[g_{\\kappa\\lambda}\n+...\n+\\frac{{\\dot{\\zeta}}^{k-2}}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi+\\frac{\\ddot{\\zeta}}{{\\dot{\\zeta}}^{2}}\\dot{x}^\\lambda\\right)+\\hskip5pt\n&&  \\nonumber \\\\[5pt]\n+\\dot{\\zeta}^{-1}{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\dot{\\zeta}{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+\\dot{\\zeta}^{k-2}{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0. && \n\\end{eqnarray}\nOn the one side, this confirms that the geodesic equations determine the geodesic parameter $\\tau$ up to an arbitrary additive constant. In fact, the simpler form (\\ref{sge}) can hold only for a particular parameter $\\tau$, distinguished up to the linear transformation $\\tau\\to\\tau+\\tau_0$ with constant $\\tau_0$. This transformation is clearly responsible for the conservation of (\\ref{fi}).\nOn the other side, we learn that the geodesic equations are unaffected by a  simultaneous constant rescaling of the parameter $\\tau$ and of the fields  $g_{\\mu_1...\\mu_k}$. In fact, the equations (\\ref{sge}) are invariant under the combined  transformation\n\\begin{equation}\n\\label{rescaling}\n\\tau\\to\\lambda\\tau+\\tau_0\n\\hskip10pt\\text{and}\\hskip10pt\ng_{\\mu_1...\\mu_k}\\to\\lambda^{k-2}g_{\\mu_1...\\mu_k},\n\\end{equation}\nwith $\\lambda$ a nonzero constant.\nCorrespondingly, the line element (\\ref{tds}) and the first integral (\\ref{fi}) \nare rescaled by an irrelevant factor $\\frac{1}{\\lambda^2}$, \n\\begin{equation}\n\\label{rescaling1}\nds\\to\\frac{1}{\\lambda^2}ds\n\\hskip10pt\\text{and}\\hskip10pt\nE\\to\\frac{1}{\\lambda^2}E.\n\\end{equation}\nIn investigating non-null geodesics, we are therefore free to directly solve the equations (\\ref{sge}), whose solutions are labeled by $E$, or to stipulate a standard gauge on $\\tau$ and solve the rescaled equations (\\ref{zrcge}) for a fixed  value of first integral (\\ref{fi}). This last option is customarily made in standard Riemannian geometry and in general relativity, where the geodesic parameter is conventionally identified with the arc length $s$, so that $d\\tau=\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$ and $g_{\\mu\\nu}\\dot{x}^\\mu \\dot{x}^\\nu=1$.\nThe corresponding gauge in Riemannian geometry without the hypotheses of \\emph{homogeneity} and \\emph{symmetry} corresponds to a rescaling by $\\lambda=\\varepsilon=\\text{sgn}(E)\\sqrt{|E|\/2}$.  This produces the geodesic equations in the form\n\\begin{eqnarray}\\label{rcge}\n\\left[g_{\\kappa\\lambda}\n+...\n+\\frac{\\varepsilon^{k-2}}{(k-2)!}g_{\\kappa\\lambda\\mu_1... \\mu_{k-2}}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_{k-2}}+...\\ \\right]\\left(\\ddot{x}^{\\lambda}+{\\Gamma^\\lambda}_{\\nu\\xi}\\dot{x}^\\nu\\dot{x}^\\xi\\right)+\\hskip10pt\n&&  \\nonumber \\\\[5pt]\n+\\frac{1}{\\varepsilon}{F}_{\\kappa\\mu}\\dot{x}^{\\mu}\n+\\varepsilon{F}_{\\kappa\\mu\\nu\\xi} \\dot{x}^{\\mu}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+ ... \n+\\varepsilon^{k-2}{F}_{\\kappa\\mu_1...\\mu_k}\\dot{x}^{\\mu_1}...\\ \\dot{x}^{\\mu_k}+ ...\n=0, && \n\\end{eqnarray}\nwith  $d\\tau=\\pm\\sqrt{|g_{\\mu\\nu}dx^\\mu dx^\\nu|}+O(\\varepsilon)$ and the first integral (\\ref{fi}) equal to $\\pm1$. \\\\\n\nEquations (\\ref{sge}) and equivalent ones, generalise the geodesic equations of standard (pseudo-)Riemannian geometry, to which they reduce when all $g_{\\mu_1...\\mu_k}$ except $g_{\\mu\\nu}$ vanish identically. By assuming the invertibility of the second order tensor\n\\begin{equation}\n\\label{ }\n\\bm{g}_{\\mu\\nu}\n= g_{\\mu\\nu}\n+g_{\\mu\\nu\\xi}\\dot{x}^{\\xi}+...\n+\\frac{1}{(k-2)!}g_{\\mu\\nu\\xi_1... \\xi_{k-2}}\\dot{x}^{\\xi_1}...\\ \\dot{x}^{\\xi_{k-2}}+...\\ ,\n\\end{equation}\nthese equations can be resolved with respect to the second derivatives. The local existence and uniqueness theorem for their solutions follows then from  standard ODE-theory.\n\n\\subsection*{Connection and curvature}\nThe standard form of the geodesic equations is also a convenient starting point to identify the connection and the curvature associated to the line element (\\ref{tds}). By series-inverting $\\bm{g}_{\\mu\\nu}$ we obtain the symmetric contravariant tensor $\\bm{g}^{\\mu\\nu}$ fulfilling the identity $\\bm{g}^{\\mu\\kappa}\\bm{g}_{\\kappa\\nu}=\\delta^\\mu_\\nu$. The first few terms are  given by\n\\begin{equation}\n\\bm{g}^{\\mu\\nu}=g^{\\mu\\nu}\n-{g^{\\mu\\nu}}_\\xi\\dot{x}^\\xi\n+\\left[\n{g^{\\mu}}_{\\lambda(\\xi}{g^{\\nu\\lambda}}_{o)}\n-\\frac{1}{2}{g^{\\mu\\nu}}_{\\xi o}\n\\right]\\dot{x}^{\\xi}\\dot{x}^{o}+ ...\\ ,\n\\end{equation}\nwhere \nround brackets indicates symmetrisation  and indices are raised by means of the inverse $g^{\\mu\\nu}$ of $g_{\\mu\\nu}$, $g^{\\mu\\kappa}g_{\\kappa\\nu}=\\delta^\\mu_\\nu$. For example, ${g^{\\mu\\nu}}_\\xi=\ng^{\\mu\\rho}g^{\\nu\\sigma}g_{\\rho\\sigma\\xi}$. By multiplying (\\ref{sge}) by $\\bm{g}^{\\iota\\kappa}$, contracting on $\\kappa$ and renaming indices, we  obtain the geodesic equations in the standard form \n\\begin{equation}\n\\label{gesf}\n\\ddot{x}^\\kappa+\\bm{\\gamma}^\\kappa_\\mu\\dot{x}^\\mu=0\n\\end{equation}\nwith $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ expressed as a series in $\\dot{x}$ as\n\\begin{equation}\n\\label{nlco}\n\\bm{\\gamma}^\\kappa_\\mu={\\gamma}^\\kappa_\\mu\n+{\\gamma}^\\kappa_{\\mu\\nu}\\dot{x}^\\nu\n+{\\gamma}^\\kappa_{\\mu\\nu\\xi}\\dot{x}^{\\nu}\\dot{x}^{\\xi}\n+...+{\\gamma}^\\kappa_{\\mu\\xi_1...\\xi_n}\\dot{x}^{\\xi_1}...\\ \\dot{x}^{\\xi_n}+... \n\\end{equation}\nThe first few coefficients are given by \n\\begin{eqnarray}\n&&  {\\gamma}^\\kappa_\\mu={F^{\\kappa}}_\\mu\n\\nonumber\\\\[5pt]\n&&  {\\gamma}^\\kappa_{\\mu\\nu}=\n{\\Gamma^\\kappa}_{\\mu\\nu}\n-{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\nu)}\n\\nonumber\\\\[5pt]\n&&  {\\gamma}^\\kappa_{\\mu\\nu\\xi}={F}^{\\kappa}_{\\ \\mu\\nu\\xi}\n-{g^{\\kappa}}_{\\lambda(\\mu}{g^{\\lambda\\iota}}_{\\nu}{F}_{\\xi)\\iota}\n-\\frac{1}{2}{g^\\kappa}_{\\lambda(\\mu\\nu}{{F}^{\\lambda}}_{\\xi)}\n\\nonumber\\\\[5pt]\n&&\\hskip10pt...\\nonumber\n\\end{eqnarray}  \nThe linear coefficient ${\\gamma}^\\kappa_{\\mu\\nu}$ is the only non-covariant one. All the others transform like tensors.\nAs a whole, $\\bm{\\gamma}^\\kappa_\\mu$ transforms like a connection one-form.\nIn fact, in (\\ref{gesf}) we recognise the auto parallel equations associated to the nonlinear connection  $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ (see Appendix \\ref{B}).\nTherefore, we can univocally identify $\\bm{\\gamma}^\\kappa_\\mu\\left(x,\\dot{x}\\right)$ as the connection associated the line element (\\ref{tds}). \n\nAs usual, the corresponding curvature tensor is obtained by means of the formula\n\\begin{equation}\n\\label{curvature}\n\\bm{r}^\\kappa_{\\mu\\nu}=\n\\partial_\\mu\\bm{\\gamma}^\\kappa_\\nu\n-\\partial_\\nu\\bm{\\gamma}^\\kappa_\\mu\n-\\bm{\\gamma}^\\lambda_\\mu{\\textstyle\\frac{\\partial}{\\partial\\dot{x}^\\lambda}}\\bm{\\gamma}^\\kappa_\\nu\n+\\bm{\\gamma}^\\lambda_\\nu{\\textstyle\\frac{\\partial}{\\partial\\dot{x}^\\lambda}}\\bm{\\gamma}^\\kappa_\\mu.\n\\end{equation}\nThis yields the curvature $\\bm{r}^\\kappa_{\\mu\\nu}(x,\\dot{x})$ again as series in $\\dot{x}$\n\\begin{equation}\n\\label{nlcu}\n\\bm{r}^\\kappa_{\\mu\\nu}={r^\\kappa}_{\\mu\\nu}\n+{r^\\kappa}_{\\xi\\mu\\nu}\\dot{x}^{\\xi}\n+...+{r^\\kappa}_{\\xi_1...\\xi_n\\mu\\nu}\\dot{x}^{\\xi_1}...\\, \\dot{x}^{\\xi_n}+...\\, .\n\\end{equation}\nwith the first few coefficients given by\n\\begin{eqnarray}\n&&  {r^\\kappa}_{\\mu\\nu}=\n\\nabla_\\mu{{F}^{\\kappa}}_\\nu\n-\\nabla_\\nu{{F}^{\\kappa}}_\\mu\n-\\frac{1}{2}{g^\\kappa}_{\\mu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\nu}\n+\\frac{1}{2}{g^\\kappa}_{\\nu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\mu}\n\\nonumber\\\\[5pt]\n&&  {r^\\kappa}_{\\xi\\mu\\nu}={R^\\kappa}_{\\xi\\mu\\nu}\n-\\nabla_\\mu{g^{\\kappa}}_{\\lambda(\\nu} {{F}^{\\lambda}}_{\\xi)}\n+\\nabla_\\nu{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\xi)}+\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-{g^{\\lambda}}_{\\iota(\\mu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\nu} {{F}^{\\theta}}_{\\lambda)}\n+{g^{\\lambda}}_{\\iota(\\nu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\mu} {{F}^{\\theta}}_{\\lambda)}\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-2{{F}^{\\lambda}}_{\\mu}{\\gamma}^\\kappa_{\\nu\\lambda\\xi}\n+2{{F}^{\\lambda}}_{\\nu} {\\gamma}^\\kappa_{\\mu\\lambda\\xi}\n\\nonumber\\\\[5pt]\n&&\\hskip25pt...\\nonumber\n\\end{eqnarray}\nwhere ${R^\\kappa}_{\\xi\\mu\\nu}=\\partial_\\mu{\\Gamma^\\kappa}_{\\nu\\xi}-\n\\partial_\\nu{\\Gamma^\\kappa}_{\\mu\\xi}-\n{\\Gamma^\\lambda}_{\\mu\\xi}{\\Gamma^\\kappa}_{\\nu\\lambda}+\n{\\Gamma^\\lambda}_{\\nu\\xi}{\\Gamma^\\kappa}_{\\mu\\lambda}$\nis the Riemann tensor associated to the Christoffel-Levi-Civita connection ${\\Gamma^\\kappa}_{\\mu\\nu}$.\nAs in standard Riemannian geometry, it is possible to construct a contracted curvature tensor, the analogue of the Ricci tensor $R_{\\mu\\nu}={R^\\kappa}_{\\mu\\kappa\\nu}$, as \n\\begin{equation}\n\\label{nRicci}\n\\bm{r}_{\\mu}=\\bm{r}^\\kappa_{\\kappa\\mu}={r}_{\\mu}\n+{r}_{\\xi\\mu}\\dot{x}^{\\xi}\n+...+{r}_{\\xi_1...\\xi_n\\mu}\\dot{x}^{\\xi_1}...\\, \\dot{x}^{\\xi_n}+...\\, .\n\\end{equation} \nwith\n\\begin{eqnarray}\n&&  {r}_{\\mu}=\n\\nabla_\\kappa{{F}^{\\kappa}}_\\mu\n-\\frac{1}{2}{g^\\kappa}_{\\kappa\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\mu}\n+\\frac{1}{2}{g^\\kappa}_{\\mu\\lambda}{F^{\\lambda}}_{\\iota}{F^{\\iota}}_{\\kappa}\n\\nonumber\\\\[5pt]\n&&  {r}_{\\xi\\mu}={R}_{\\xi\\mu}\n-\\nabla_\\kappa{g^{\\kappa}}_{\\lambda(\\mu} {{F}^{\\lambda}}_{\\xi)}\n+\\nabla_\\mu{g^{\\kappa}}_{\\lambda(\\kappa} {{F}^{\\lambda}}_{\\xi)}+\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-{g^{\\lambda}}_{\\iota(\\kappa} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\mu} {{F}^{\\theta}}_{\\lambda)}\n+{g^{\\lambda}}_{\\iota(\\mu} {{F}^{\\iota}}_{\\xi)}\n{g^{\\kappa}}_{\\theta(\\kappa} {{F}^{\\theta}}_{\\lambda)}\n\\nonumber\\\\[5pt]\n&&\\hskip35pt\n-2{{F}^{\\lambda}}_{\\kappa}{\\gamma}^\\kappa_{\\mu\\lambda\\xi}\n+2{{F}^{\\lambda}}_{\\mu} {\\gamma}^\\kappa_{\\kappa\\lambda\\xi}\n\\nonumber\\\\[5pt]\n&&\\hskip10pt...\\nonumber\n\\nonumber\n\\end{eqnarray}\nCorrespondingly, it is also possible to define a curvature scalar by contraction with $\\dot{x}^\\mu$, \n\\begin{equation}\n\\label{curvaturescalar}\n\\bm{r}=\\bm{r}_{\\mu}\\dot{x}^\\mu,\n\\end{equation} \nproducing \nthe symmetrisation of all components ${r}_{\\xi_1...\\xi_n\\mu}$ of $\\bm{r}_{\\mu}$. This curvature scalar should not  be confused with the analogue of the scalar curvature $R=g^{\\mu\\nu}R_{\\mu\\nu}$ of standard (pseudo-)Riemannian geometry, whose generalisation is less straightforward. \n\n\\subsection*{Quadratic restriction as quadratic approximation}\nIf all the fields $g_{\\mu_1...\\mu_k}$ except $g_{\\mu\\nu}$ are set equal to zero,\nthe geodesic equations (\\ref{sge}), \nthe connection (\\ref{nlco}) and the curvature (\\ref{nlcu}), respectively reduce to the geodesic equations, connection and curvature of standard (pseudo-)Riemannian geometry.\n Correspondingly, the line element (\\ref{tds}) reduces to the integrand $\\frac{1}{2}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu d\\tau$ of the variational principle introduced by Levi-Civita to describe  null geodesics in general relativity \\cite{Levi-Civita1929}.  In the context of standard Riemannian geometry, with a positive definite metric tensor, this is completely equivalent to Riemann's \\emph{quadratic restriction}. In fact the solutions of  Levi-Civita's variational principle admit the first integral  $g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu=2E$, so that \n\\begin{equation}\n\\label{L-CvsR}\n\\delta\\int {\\textstyle \\frac{1}{2}}g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu d\\tau =\n{\\textstyle \\frac{1}{2}}\\delta\\int \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu}  \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu} d\\tau =\n {\\textstyle\\sqrt{\\frac{E}{2}}}\\delta\\int \\sqrt{g_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu} d\\tau.\n\\end{equation}\nConversely, when indefinite signatures are considered, Levi-Civita's variational principle becomes slightly more general. In fact, the corresponding geodesic equations coincide with the geodesic equations of standard (pseudo-)Riemannian geometry for $E\\neq0$, but also include the case of null geodesics with the correct parameterisation when $E=0$ \\cite{Schroedinger1956}.  The reading of Levi-Civita's integrand as a line element, is therefore an improvement of Riemann's \\emph{quadratic restriction}. \n\n Under this perspective, standard (pseudo-)Riemannian geometry appears as the quadratic approximation to a more general Riemannian geometry, that can be developed along the very same lines by relaxing the hypothesis of \\emph{homogeneity}, but not the one of \\emph{symmetry}. \n As the quadratic term frequently provides an excellent approximation on sufficiently small scales, this suggests why Riemann's \\emph{quadratic restriction} is so effective in the description of geometry.  It is effective to such an extent, that when complemented with space(time) dimension, metric signature and dynamical equations,  it is one and the same with Einstein's general theory of relativity. It is then natural to wonder whether Riemannian geometry without the hypotheses of \\emph{homogeneity }and \\emph{symmetry} can provide a deeper insight into the study of spacetime.\n\n\n\\section{Applications to spacetime}\\label{spacetime}\nThe quadratic approximation to Riemannian geometry based on the hypotheses of \\emph{differentiability}, \\emph{indiscernibility} and \\emph{symmetry},  thus without that of \\emph{homogeneity}, naturally provides the mathematical framework of general relativity without the need for postulating the explicit form of the line element. If we further renounce the hypothesis of \\emph{symmetry}, thus starting from a geometry uniquely based on \\emph{differentiability} and \\emph{indiscernibility}, we obtain the mathematical framework of a unified theory of the classical electromagnetic and gravitational fields. \nUnlike the classical attempts \\cite{Goenner2004,Goenner2014,Tonnelat1966}, there is no need to introduce new fields, extra dimensions or new geometrical structures.\nA logically simple and economical unified description is obtained simply by renouncing hypotheses within Riemannian geometry.\nUnlike the classical attempts, it is directly evident how to identify the physical fields.\nIn the quadratic approximation a generic line element is equivalent to a one-parameter family of Riemann-Randers line elements $ds=\\frac{1}{\\varepsilon}g_\\mu dx^\\mu+\\sqrt{g_{\\mu\\nu}dx^\\mu dx^\\nu}$. This\ncorresponds to a one-parameter family of Lagrangians for charged particles in the background of electromagnetic and a gravitational fields. The $\\varepsilon$-dependent line element is not postulated for a single value of $\\varepsilon$ as in Rander's proposal\\footnote{See the discussion in the Introduction of \\cite{Tonnelat1966}.}. It emerges by itself in as many copies as we need to describe particles with different charges. Also, in the quadratic approximation the connection (\\ref{nlco}) reduces to the nonlinear connection put forward by Broda and Przanowski \\cite{BrodaPrzanowski1985} for Riemann-Randers spaces.\nTherefore, up to a proportionality constant, the field $g_\\mu$ is naturally identified with the electromagnetic potential, as $g_{\\mu\\nu}$ is identified with the gravitational potential\\footnote{In the original formulation of general relativity the metric tensor $g_{\\mu\\nu}$ was identified with the inertial-gravitational tensor \\emph{potential} and the associated Christoffel symbols ${\\Gamma^\\kappa}_{\\mu\\nu}$ with the corresponding force \\emph{field}, in complete analogy with the electromagnetic vector \\emph{potential}  $\\mathrm{A}_\\mu$ and the force \\emph{field} $\\mathrm{F}_{\\mu\\nu}$. In later years, probably following the attempt to identify $g_{\\mu\\nu}$ and $\\mathrm{F}_{\\mu\\nu}$ with the symmetric and antisymmetric parts of a same tensor, $g_{\\mu\\nu}$ has been increasingly referred to as the gravitational field. Under a different prospective, the analogy between the gauge structures of electromagnetism and Einstein's gravity have encouraged the identification of ${\\Gamma^\\kappa}_{\\mu\\nu}$ with $\\mathrm{A}_\\mu$ and of ${R^\\kappa}_{\\lambda\\mu\\nu}$ with $\\mathrm{F}_{\\mu\\nu}$. The equation of motion of charged particles in a curved background leave little doubt of what the correct physical identification is.}.\nCorrespondingly, the parameter $\\varepsilon$ is proportional to  the mass-to-charge ratio of particles, thus providing a geometrical meaning to electric charge.\n\nAs for standard Riemannian geometry and general relativity, the transition from the mathematical framework to the physical theory requires the specification of spacetime dimension, signature and dynamical equations. While the first two necessarily follow those of general relativity,  the choice of adequate dynamical equations is less constrained. The simplest and most natural generalization of Einstein field equations seems to me \n\\begin{equation}\n\\label{fieldeq}\n\\bm{r}=\\bm{t}\n\\end{equation}\nwith $\\bm{r}$ the curvature scalar (\\ref{curvaturescalar}) and $\\bm{t}=t_\\mu(x)\\dot{x}^\\mu+t_{\\mu\\nu}(x)\\dot{x}\\dot{x}^\\nu+...$ an appropriate analytic function of the $x$ and of the $\\dot{x}$, describing the distribution of charge and matter. As the identity of analytic functions requires the identity of all the respective coefficients in their series expansion, in the limit\nwhere all $g_{\\mu_1...\\mu_k}$ except $g_\\mu$ and $g_{\\mu\\nu}$ are negligible, \n(\\ref{fieldeq}) reduces to the simultaneous Maxwell and Einstein field equations \n\\begin{eqnarray}\n&&  \\nabla_\\kappa{{F}^{\\kappa}}_\\mu=t_\\mu,\\label{Maxwell} \\\\\n&& R_{\\mu\\nu}=t_{\\mu\\nu}.\\label{Einstein}\n\\end{eqnarray}\nI find it really remarkable that the Maxwell field equations (\\ref{Maxwell}) minimally coupled to $g_{\\mu\\nu}$, naturally emerge from the geometry. \nOn the other hand, it should also be noted that  equations  (\\ref{Einstein}) do not correspond to the standard minimal coupling, because the electromagnetic stress-energy tensor  $F_{\\mu\\xi}{F_\\nu}^\\xi-\\frac{1}{4}g_{\\mu\\nu}F_{\\xi o}F^{\\xi o}$  does not appear in their right-hand-side term. This is perfectly consistent with all the experiments conducted so far, where the propagation of light is studied ``along null-geodesics in  a prescribed [...] gravitational field which is a solution of Einstein vacuum equation[s] --- and not of the \\emph{electro}-vacuum equation[s]'' \\cite{HehlObukhov2001}. \n\nThe standard minimal coupling can be obtained in the quadratic approximation from the straightforward generalisation of the standard Einstein-Maxwell Lagrangian density\n\\begin{equation}\n\\label{EM}\n\\mathcal{L}_{EM}=F_{\\mu\\nu}F^{\\mu\\nu}+R+...,\n\\end{equation}\nwhere the unwritten terms for the fields ${F}_{\\mu_1...\\mu_k}$ for $k\\geq4$, can be easily guessed from Rosen's expression for the Lagrangian density of the gravitational field  $F_{\\mu\\nu\\xi}$ \\cite{Rosen1940}.\n\nEquivalently, the standard minimal coupling can be obtained, always in the quadratic approximation, by generalizing the geometrically more appealing Broda-Przanowski Lagrangian density \\cite{BrodaPrzanowski1985} \n\\begin{equation}\n\\label{BP}\n\\mathcal{L}_{BP}=g^\\mu r_\\mu+g^{\\mu\\nu}r_{\\mu\\nu}+g^{\\mu\\nu\\xi}r_{\\mu\\nu\\xi}+...\n\\end{equation}\n\nIn both cases, indices are raised by $g^{\\mu\\nu}$ and the spacetime volume element is chosen as $\\sqrt{|\\det g_{\\mu\\nu}|}d^4x$. Whatever our aesthetic preferences may be, such questions can only be solved experimentally.\\\\\n\nIn order to make contact with reality, it is also necessary to specify the proportionality constants between the geometrical and physical  fields and between the adimensional parameter $\\varepsilon$ and the mass-to-charge ratio of physical particles.\nSince the only unit of electromagnetic potential  that can be constructed in terms of classical fundamental constants is the Planck voltage\n\\begin{equation}\n\\label{PlankV}\n\\sqrt{\\frac{c^4}{4\\pi\\epsilon_0G}}\\simeq1.04\\times10^{27}V,\n\\end{equation}\nwith $c$ the speed of light, $\\epsilon_0$ the electric constant and $G$ the gravitational constant,\nthe only possibility is to set\n \\begin{equation}\n\\label{g to A}\ng_\\mu=-\\sqrt{\\frac{4\\pi\\epsilon_0G}{c^4}}\\mathrm{A}_{\\mu}\n\\end{equation}\nwith $\\mathrm{A}_{\\mu}$  the electromagnetic vector potential.\nCorrespondingly, \n\\begin{equation}\n\\label{epsilon}\n\\varepsilon=\\frac{m}{e}\\sqrt{4\\pi\\epsilon_0G},\n\\end{equation}\nwhere $m$  and $e$ respectively indicate the mass and charge of the particle under consideration. For electrons $\\varepsilon\\simeq0.5\\times10^{-21}$, largely justifying the validity of a quadratic approximation.\\\\\n\nBeyond the quadratic approximation, the geometry imagined by Riemann in 1854  deprived of the hypotheses of \\emph{homogeneity} and \\emph{symmetry}, suggests the existence of infinitely more classical force fields, somehow the higher harmonics of a single fundamental classical field, the infinitesimal line element $ds$.  \nThese fields are generated by an identical rule from symmetric tensor potentials of increasing order in the very same way the electromagnetic and gravitational fields  $F_{\\mu\\nu}$ and $\\Gamma_{\\kappa\\mu\\nu}$ (or $F_{\\mu\\nu\\xi}$) are generated by the respective potentials $g_\\mu$ and $g_{\\mu\\nu}$.\nLike electromagnetic and gravitational fields, these higher order fields  propagate in space as waves of speed $c$, interacting in an increasingly non-linear way as their order increases. Nonetheless, as the ratio of strength between successive order fields equals the one between gravity and electromagnetism, it must be expected that already the field $F_{\\mu\\nu\\xi o}$ produces extremely weak and hard to measure local effects. On the other hand, these extremely weak effects could become relevant when integrated over the enormous spacetime distances  and could perhaps play a role in some of the many unresolved problems in astrophysics and cosmology. \nIf we accept the paradigm shift that Riemann's \\emph{quadratic restriction} is nothing but a quadratic approximation to a generic line element, these fields must somehow play a role in the large scale structure of spacetime. In this context, however, a non-perturbative approach to the dynamics of $ds$ might be more appropriate\\footnote{It is always possible to parametrise the line element as $ds=\\frac{\\partial\\sigma}{\\partial\\dot{x}^\\mu}\\dot{x}^\\mu$, with $\\sigma(x,\\dot{x})$ an arbitrary analytic function of position and direction. The generalisation of Maxwell and Einstein field equations (\\ref{fieldeq}) rewrites then as a single differential equation for the scalar field $\\sigma$.}.\n\n\\section{Conclusion}\\label{TheEnd}\nFor a long time the term `Riemannian geometry' has been used to indicate spherical geometry, probably a sign of how small the impact of Riemann's ideas was originally. \nWith the advent of general relativity the term slowly shifted to current use and some twenty years ago Chern \\cite{Chern1996} suggested that it should also include what we currently call Finsler geometry. In this paper I have used it in its broadest sense, to describe a differential geometry  uniquely based on the hypothesis that the distance of a point from itself is equal to zero. No \\emph{quadratic restriction}, no \\emph{homogeneity}, no \\emph{symmetry}. As a matter of fact, this geometry is much more general than Riemann's original vision. \nIts relationship with standard Riemannian geometry is the same as that between a generic analytical function and its quadratic approximation at a given point. Nonetheless, the quadratic term in the series expansion of a generic line element plays such a unique role to justify the choice widely. In fact, Riemann's assumption of the infinitesimal Pythagorean distance formula reemerges as a first  approximation to \\emph{almost every} geometry which is symmetric with respect to direction reversal, also trowing new light on this ancient fundamental theorem. Analogously, the associated Christoffel-Levi-Civita linear connection keeps on playing a pivotal role in the generalised connection.\nFor generic geometries the first approximation to the infinitesimal distance formulas turns instead into something that Riemann could hardly have foreseen. The lack of homogeneity introduces into geometry an adimensional parameter. \nCorrespondingly, \\emph{almost every} geometry is described in first approximation \nby a one-parameter family of Riemann-Randers metrics.  The ``square root of an [...] homogeneous function of the second degree in the quantities $dx$, in which the coefficients [$g_{\\mu\\nu}$] are continuous functions of the quantities $x$'', is now accompanied by a homogenous function of the first degree in the quantities $dx$ in which the coefficients $g_\\mu$ are regular functions of the quantities $x$, multiplied by an arbitrary constant $\\frac{1}{\\varepsilon}$.  This additional term enters the first approximation to the generalised connection as the nonlinear term introduced by Broda and Przanowski.\\\\\n\nIf following the lines of general relativity we  apply this geometry to spacetime by identifying $g_{\\mu\\nu}$ with the gravitational potential, the additional term $g_\\mu$ enters the geodetic and field equations exactly as an electromagnetic potential, naturally providing a unified description of the two classical fundamental interactions.\nAs $g_\\mu$ and $g_{\\mu\\nu}$ respectively appears as the first and second order terms of the series expansion of the line element $ds$, the hierarchy between electromagnetic and gravitational interactions finds its origin in geometry. \nMoreover,  geometry turns out to be invariant under the simultaneous reversal of direction and of the sign of $\\varepsilon$, so that also CPT invariance finds its origin in spacetime geometry. \nCorrespondingly, as the different parity of subsequent terms in the series expansion of $ds$ requires that the coefficient of ${F^\\kappa}_{\\mu}$ changes its sign under direction reversal, while the one of ${\\Gamma^\\kappa}_{\\mu\\nu}$ does not, also the different attractive\/repulsive nature of electromagnetic and gravitational interactions finds its origin in spacetime geometry. Equivalently, geometry naturally accounts for charge to be positive and negative, while mass only positive. Besides the electromagnetic and the gravitational interactions, this geometry suggests the existence of infinitely more force fields, extremely weak and so far unobserved,  that could possibly play a role in spacetime dynamics on a large scale.\n\n Quite remarkably, the whole kinematical setting --the number and type of potentials, their relation to fields, the hierarchy and properties of the interactions, the existence of electric charge, CPT invariance, the form of the curvature tensor that set equal to zero gives both the Maxwell and Einstein free field equations etc.-- entirely follows from the minimal assumption that the line element $ds$ is a regular function of position and direction, that the distance of a point from itself is equal to zero and, of course, from the geodesic hypothesis.     \nHowever, as Riemann reminds us, the properties of space ``can only be deduced from experience\". Therefore it remains to be seen whether these minimal hypotheses will lead to a better comprehension of the structure of spacetime on galactic and cosmological scales, so that they can be extended ``beyond the bounds of observation [...] in the direction of the immeasurably large\".  On the other hand, as  ``the empirical notions on which the metric determinations of space are based [...] lose their validity in the infinitely small [...] it is [...] quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry[...] and [...] one ought to assume this as soon as it permits a [...] way of explaining phenomena\". At the threshold of the quantum world a completely new idea of geometry is needed.\n\n\\subsection*{Aknowledgements}  I am indebted with J.K.\\! Pachos and J.-P.\\! Zendri for carefully reading the manuscript and related comments.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, massive young stellar objects (MYSOs) have been suggested as gamma-ray sources \\citep{ara07,rome08,bosch10}. Massive stars are formed in dense cores of cold clouds. The processes that take place during the formation of the star are mostly unknown. It is clear, however, that the formation of massive stars involves outflows \\citep{gara99, reip01}. The accumulation of material around the core of the cloud would generate a massive protostar that starts to accrete material from the environment. The accretion is expected to have angular momentum that leads to the formation of an accretion disk. The rotation would twist the strong magnetic fields present in the progenitor cloud around the disk, where a magnetic tower can be formed, giving rise to collimated outflows or jets, as simulations predict \\citep{baner06, baner07}. The observational evidence of outflows comes from methanol masers and from direct detection of thermal radio jets. These jets propagate along distances in a fraction of a parsec \\citep{mar93}. At the jet termination region, interaction with the external medium creates two shocks: a bow shock moving in the interstellar medium (ISM) and a reverse shock in the jet. These shocks can accelerate particles that, in turn, can produce gamma rays trough inverse Compton (IC) scattering of infrared (IR) photons, relativistic Bremsstrahlung or inelastic proton-proton collisions, if protons are accelerated as well. In some cases non~thermal radio lobes and jets have been observed, indicating the presence of relativistic electrons that produce synchrotron radiation \\citep{gara03, carrasco10}.\\\\\n\nOther possible scenarios have been suggested for the gamma-ray production involving young stars, such as the case of the massive stars with strong winds. In this scenario, gamma-rays could be produced in the interaction between the supersonic winds and the ISM. The terminal shock can accelerate particles and ions up to high energies, which might interact with the ambient matter producing gamma-rays. Whereas the luminosity produced by a single massive star wind should be low, collective effects might be important \\citep{torr04}. \\\\\nGamma-rays can also be produced in the wind interaction region of a WR+OB binary system \\citep{bena03}. In this case, the acceleration region is between the two components of the binary system and is exposed to strong photon fields where IC cooling of the electrons can generate a significant amount of high-energy (HE) non~thermal emission. A source of this class, $\\rm{\\eta-Carinae}$, has been recently detected at $E > 100$ MeV by \\textit{AGILE} \\citep{tava09}.\n\nOf-type stars have strong winds with velocities higher than the escape velocity, which implies a strong mass loss rate ($10^{-6}-10^{-5}$ M$_{\\odot}$ yr$^{-1}$). The action of this wind in the interstellar medium can create hot gas bubbles with expanding boundaries of swept-up material, which might produce gamma rays in a similar way to the case of WR stars. The case of gamma-ray emission in Of-type stars has been discussed in the past, e.g. by \\cite{volk82}. The predicted luminosity, however, is still below the current sensitivity of gamma-ray instruments.\\\\\n\nFinally, OB associations are tracers of a number of galactic objects that can produce gamma rays, such as neutron stars, massive stars with strong winds, young stellar objects (YSOs), etc. They are also thought to be places where acceleration of a significant fraction of galactic cosmic rays (CRs) might occur \\citep[e.g.][]{binns08}. \\\\\n\nThe aim of this work is to find evidence supporting the presence of HE emission coming from massive YSOs and other young galactic sources. To attain this goal we study the spatial coincidence between gamma-ray sources detected by \\textit{Fermi} and samples of young objects, such as YSOs, WR stars, Of-type stars, and OB associations. We also estimate the probability of chance coincidences by using Monte Carlo simulations, and we provide a list of counterpart candidates of the gamma-ray sources.\n\n\\section{Cross-correlation of the First \\textit{Fermi} Catalog with massive young galactic objects}\n\nThere is not observational evidence of any YSO emitting gamma-rays so far. The new generation of gamma-ray telescopes, like \\textit{Fermi}, will make it possible to reach the necessary sensitivity level to detect these faint gamma-ray sources soon, in case the predictions were right. To identify those young objects that might be emitting gamma rays, we first took the recently published First \\textit{Fermi} Catalog \\citep{abdo10} by the \\textit{Fermi} Collaboration and we excluded all known firm identifications, getting a list of 1392 sources. Then we crossed this list with catalogs of confirmed and well characterized YSOs and other type of young stars. We also did a Monte Carlo study to determine the probability of pure chance coincidences between the crossed catalogs. \n\n\\subsection{Catalogs}\n\nThe catalogs used in this study are listed in Table \\ref{tab:catalogs}. Here we describe each one in more detail.\n\nThe \\textit{Fermi Large Area Telescope} First Catalog \\citep{abdo10} contains the detected sources during the first 11 months of the science phase of the mission, which began on 2008 August 4. This catalog contains 1451 gamma-ray sources detected and characterized in the 100 MeV to 100 GeV range with a typical position uncertainty of $\\sim 6^\\prime$. After excluding the firm identifications from the original sample, we get 1392 sources. Most of them are located on the Galactic plane (see Figure \\ref{fig:hist}). \n\\\\\n\nThe Red \\textit{MSX} Source (RMS) survey is an ongoing multi~wavelength (from radio to infrared) observational program with the objective of providing a well-selected sample of MYSOs in the entire Galaxy \\citep{urqu08}. About $\\sim$2000 MYSO candidates have been identified by comparing the colors of \\textit{MSX} and 2MASS point sources (at 8, 12, 14, and 23 $\\mu$m) with those of well known MYSOs. The survey also uses high-resolution radio continuum observations at 6~cm obtained with the VLA in the northern hemisphere and at 3.6~cm and 6cm with ATCA in the southern hemisphere. They help to distinguish between genuine MYSOs and other types of objects, such as ultracompact HII regions, evolved stars, or planetary nebulae, which contaminate the sample. In addition to these targeted observations, archival data of a previous VLA survey of the inner Galaxy were used. This ongoing program has provided a sample of 637 well-identified MYSOs until now, which were used in our work.\n\\\\\n\nThe VIIth catalog of Population I WR stars \\citep{vander01} contains 227 stars, with spectral types and \\textit{bv} photometry. In recent years, the number of WR stars has increased in 71 new stars, respect to the VIth catalog and the coordinates have also been improved. The position uncertainty is close to a fraction of an arcsecond. \n\\\\\n\nThe catalog of Of-type stars is the one of \\cite{cruz74}, which contains 664 stars. The catalog provides $\\rm{m_v}$, B--V, spectral type, radial velocity, radial component of the peculiar velocity, possible multiplicity of the object, and other characteristics for each source. The typical uncertainty in the star position is $\\sim 1^\\prime$.\n\\\\\n\nFinally, the catalog of OB associations is the one by \\cite{melnik95}. This catalog contains 88 associations and provides distances to the association, number of stars, and size of the association along the Galactic latitude and longitude axes. The typical value of the size is $\\sim 20-30$~pc.\n\n\\subsection{Spatial coincidences}\n\nWe crossed the \\textit{Fermi} catalog with the catalogs of young galactic objects mentioned above. We calculated the distance between two sources using the statistical parameter $S$ \\citep{aling82}:\n\\\\\n\\begin{equation*} \nS=\\sqrt{\\frac{(\\Delta \\alpha \\cos \\delta)^2}{\\sigma_{i_\\alpha}^2+\\sigma_{j_\\alpha}^2} + \\frac{\\Delta \\delta^2}{\\sigma_{i_\\delta}^2+\\sigma_{j_\\delta}^2}}\n\\end{equation*}\n\\\\\nwhere $\\Delta\\alpha$ and $\\Delta\\delta$ are the difference between the right ascension and the declination of the two compared sources, respectively, $\\sigma_{a_b}$ is the uncertainty in the position of the source, and $(i,j)$ represent the two sources. The error in the position of the \\textit{Fermi} sources is taken as the 95$\\%$ confidence ellipse. The error in the position of the other compared sources is the precision in the coordinates for YSOs, WR stars and Of-type stars, and the angular size of the association in the case of OB associations. If $S$ is lower than or equal to the unit, it means that the source position (YSO, WR, Of-type, or OB associations) is inside the 95$\\%$ uncertainty ellipse of the \\textit{Fermi} source, within its own position uncertainty, and that case is considered to be a coincidence. Massive YSO and protostar are point-like objects compared with the confidence contours of Fermi sources. The same is valid for WR and Of stars, either in binary systems or isolated. OB associations are large  systems that can contain more than a single gamma-ray source. We note that our study is based on two-dimensional coincidences, since we are comparing the equatorial coordinates of the sources.\n\n\\subsection{Monte Carlo analysis}{\\label{subs:monte}}\n\nTo determine the chance coincidences we used the Monte Carlo method for simulating sets of synthetic gamma-ray sources starting from the \\textit{Fermi} Large Area Telescope First Catalog. We followed a similar criteria to that used by \\cite{rome99} to search for the possible association of unidentified \\textit{EGRET} sources with other type of celestial objects. In this algorithm, the galactic coordinates of a gamma-ray source $\\left( l, b \\right)$ are moved to new ones $\\left( l^\\prime, b^\\prime \\right)$. The new galactic longitude coordinate is calculated by doing $l^\\prime=l + R_1 \\times 360^\\circ$, where $R_1$ is a random number between 0 and 1 that never repeats from source to source or from set to set. Since the distribution of \\textit{Fermi} sources is almost constant in Galactic longitude, we do not impose any constraint on this coordinate in the simulations. The sources in the \\textit{Fermi} catalog have a given distribution in galactic latitude (see Figure \\ref{fig:hist}). In order to constrain the simulations with this distribution, the galactic latitude coordinate is calculated by doing $b^\\prime=b + R_2\\times 1^\\circ$, where again $R_2$ is a random number between 0 and 1. Here, if the integer part of $b^\\prime$ is greater than the integer part of $b$ or the sign of $b^\\prime$ is different than the sign of $b$, then $b^\\prime$ is replaced by $b^\\prime-1^\\circ$.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{histogram_fermi.pdf}}\n  \\caption{Distribution of First \\textit{Fermi} Catalog sources in galactic latitude. \\label{fig:hist}}\n\\end{figure}\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\caption{List of the catalogs used in the study.\\label{tab:catalogs}}        \n    \\scalebox{0.90}[0.90]{\n    \\begin{tabular}{lll}\n    \\hline\n    \\hline\n     Object & Catalog & $\\#$ of \\\\\n\t type   &         & sources \\\\\n     \\hline\n     $\\gamma$-ray sources & First \\textit{Fermi} Catalog\\tablefootmark{a} & 1392\\tablefootmark{f}\\\\\n     YSO & RMS Survey\\tablefootmark{b}& 637\\\\\n     WR & VII$^{th}$ Catalog of Galactic Wolf-Rayet stars\\tablefootmark{c} &  227\\\\ \n     Of-type & A catalog of galactic O stars and&  664 \\\\\n\t    & the ionization of the low density & \\\\\n\t    &  interstellar medium by runaway stars \\tablefootmark{d} & \\\\\n     OB associations & A new list of OB & 88\\\\\n        & associations in our Galaxy\\tablefootmark{e}& \\\\\n     \\hline\n     \\end{tabular}}\n     \\tablefoot{\\tablefoottext{a}{\\cite{abdo10}},\n\t \\tablefoottext{b}{\\cite{urqu08}},\n\t \\tablefoottext{c}{\\cite{vander01}},\n\t \\tablefoottext{d}{\\cite{cruz74}},\n\t \\tablefoottext{e}{\\cite{melnik95}};\n     \\tablefoottext{f}{number of gamma-ray sources after excluding the firm identifications}}\n  \\end{center}\n\\end{table} \n\nWe simulated 1500 sets of synthetic \\textit{Fermi} sources and each set was compared with a fixed set of different kinds of objects: YSOs, WR stars, Of-type, and OB associations. In each simulation, we calculated the distance between the two sources using the statistical parameter \\emph{S}.\n\nFor each kind of compared objects we calculated the average number of coincidences and its standard deviation after all Monte Carlo simulations. We calculated the chance coincidence probability using the actual number of coincidences for each type of object and assuming a Gaussian distribution in the simulations. This probability allows us to know the reliability of our study. \nWe also repeated this process moving the \\textit{Fermi} sources in $2^\\circ$-bins in galactic latitude, i.e. replacing the galactic latitude coordinate by $b^\\prime=b + R_2\\times 2^\\circ$. If the integer part of $b^\\prime$ is greater than the integer part of $b$ or the sign of $b^\\prime$ is different than the sign of $b$, then $b^\\prime$ is replaced by $b^\\prime-2^\\circ$. This binning allows us to keep the initial distribution in galactic latitude as well.\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\caption{Statistical results obtained from simulations. Latitude galactic coordinate has been constrained, while galactic longitude remains free.\\label{tab:stat}}\n    \\scalebox{0.80}[0.80]{\n    \\begin{tabular}{llr@{$\\pm$}llr@{$\\pm$}ll}\n    \\hline\n    \\hline\nObject&Coincident &\\multicolumn{2}{c}{Simulated} &Probability&\\multicolumn{2}{c}{Simulated} &Probability\\\\\ntype&$\\gamma$-ray sources &\\multicolumn{2}{c}{$1^\\circ -bin$} &$1^\\circ -bin$&\\multicolumn{2}{c}{$2^\\circ -bin$} &$2^\\circ -bin$\\\\\n\\hline\nYSO & 12 & 4.4\t& 2.0 & 1.8$\\times 10^{-4}$ & 3.6 & 1.8 & 5.6$\\times 10^{-6}$\\\\\nWR & 2 & 1.3 & 1.1 & 2.9$\\times 10^{-1}$ & 1.2 & 1.1 & 2.9$\\times 10^{-1}$\\\\\nOf-type & 5 & 2.9 & 1.7 & 1.1$\\times 10^{-1}$ & 2.9 & 1.7& 1.1$\\times 10^{-1}$\\\\\nOB assoc. & 107 & 72.5& 8.0& 4.2$\\times 10^{-6}$& 72.8&8.0&5.5$\\times 10^{-6}$\\\\\n \\hline\n \\end{tabular}}\n  \\end{center}\n\\end{table} \n\n\\input{yso_table.tex}\n\n\\section{Results}{\\label{sec:results}}\n\nThe results from our statistical study are shown in Table \\ref{tab:stat}. In this table we list, from left to right, the object type, the number of coincidences between the each compared catalog and the original \\textit{Fermi} catalog, the simulated average number of coincidences, and the chance coincidence probability for each binning in galactic latitude. We find 12 gamma-ray sources spatially coincident with YSOs, 2 with WR stars, 5 with Of-type stars and 107 with OB associations. \n\nFrom the Monte Carlo analysis, we see that there is a strong correlation between gamma-ray sources and YSOs: the catalog cross-check returns 12 coincidences between gamma-ray sources and YSOs. The Monte Carlo simulations, for the case of displacing the \\textit{Fermi} sources in 1$^\\circ$-bins, returns an average of coincidences of 4.4$\\pm$2.2 sources, which is the number of chance coincidences. This means that 7.6 of the 12 coincident \\textit{Fermi} sources ($\\sim 63\\%$ of the total coincidences with a $\\sim 4\\sigma$ confidence level ) should be associated with a probability of chance coincidence of 1.8$\\times 10^{-4}$. Similarly, for displacing the \\textit{Fermi} sources in 2$^\\circ$-bins, we obtain an average of coincidences of 3.6$\\pm$1.8. That result indicates that 8.4 of the 12 coincident sources ($\\sim 70\\%$ of the total coincidences with a $\\sim 5\\sigma$ confidence level) should be associated with a probability of $5.6 \\times 10^{-6}$ of being chance coincidences. In a similar way, the association of \\textit{Fermi} sources with WR and Of-type stars is unclear since the number of actual coincidences and the results of the Monte Carlo simulations are too similar and thus the probability of chance coincidence is too high ($0.29$ and $0.11$ for WR stars and Of-type stars, respectively). The results for WR and Of-type stars are different from those obtained by \\cite{rome99} for \\textit{EGRET} sources. The probability of chance coincidences has increased while the number of candidates decreased in the case of WR stars. In the case of Of-type stars the probability of chance association has increased as the number of coincidences increased as well. The probability of chance coincidence with OB associations is as low as $\\sim 10^{-6}$. In this case, however, the nature of the gamma-ray emission is not clear, as it is discussed in section \\ref{sec:obasso}\n\n\n\\subsection{Young stellar objects}{\\label{subsec:ysos}}\nThe association of gamma-ray sources and massive YSOs has been suggested in the last years after studying a reasonable scenario for the production of non~thermal emission \\citep{ara07}. However, this is the first time that the study of YSOs as gamma-ray sources is carried out in a statistical way, taking advantage of the \\textit{Fermi} catalog. \n\nThe results of our cross-check, shown in Table \\ref{tab:yso}, indicate that 12 gamma-ray sources are positionally coincident with 23 YSOs. In this table we present, from left to right, the \\textit{Fermi} source name, its J2000 equatorial coordinates, its positional uncertainty, the spectral $\\gamma$-ray index, the energy flux (E$>$100 MeV), the YSO name, its J2000 equatorial coordinates, the angular distance between the two compared sources, the distance to the YSO, its IR luminosity, and the mass of the star forming region where it is embedded.\n\nIn what follows we present a case by case discussion of the gamma-ray fields.\\\\\n\n- 1FGL J0541.1+3542. This source is coincident with three YSOs: G173.6328+02.8604, G173.6339+02.8218 and G173.6882+02.7222.  Their luminosities are below 4.8$\\times 10^{4}L_\\odot$. The three objects belong  to the G173.6036+02.6237 complex (in the RMS Survey notation), which is situated at a distance of 1.6 kpc. We also find two Herbig Haro like objects, GGD 5 and GGD 6 \\citep{ggd}, within the error ellipse of the \\textit{Fermi} source. Outside the \\textit{Fermi} error box the complex also harbors three more YSOs and an HII region.\\\\\n\n- 1FGL J0647.3+0031. The YSO G212.0641-00.7395 is the only coincidence with this source. Its kinematic distance is 6.4 kpc and it has a luminosity of 2.5$\\times 10^4 L_\\odot$. This source belongs to the G211.9800-00.9710 complex, together with another YSO that lies outside the error box of the gamma-ray source. \\\\\n\n-  1FGL J1256.9$-$6337. We find G303.5990-00.6524 within the error ellipse of this gamma-ray source. It is a YSO with a bolometric luminosity of $8.3\\times 10^3 L_\\odot$ and located at a kinematic distance of 11.3 kpc. It belongs to the G303.5670-00.6253 complex that also harbors an HII region. \\\\\n\n- 1FGL J1315.0$-$6235. This source is coincident with G305.4840+00.2248, which is a YSO  with a luminosity of $3.8 \\times 10^3 L_\\odot$ and located at a distance of 3.6 kpc.  \\\\\n\n- 1FGL J1651.5-4602.  The source G339.8838-01.2588 is coincident with this gamma-ray source. It is a YSO with a bolometric luminosity of 2.1$\\times 10^4 L_\\odot$. It is located at 2.6 kpc from the Earth. This source has been detected in radio at 8.6 GHz with an integrated flux of 2.6 mJy \\citep{walsh98}. There is no indication of whether the radio flux is non~thermal or not.\\\\\n\n- 1FGL J1702.4$-$4147. This source is coincident with two YSOs: G344.4257+00451B and G344.4257+00451C, which form the G344.4120+00.0492 complex, together with two HII regions. They are located at 5 kpc from the Earth and both have a bolometric luminosity of 1.5$\\times 10^4 L_\\odot$. These gamma-ray source is also near to a cluster of stars \\citep[see][]{dutra03} and a molecular cloud (see \\cite{russeil04}). \\\\\n\n- 1FGL J1846.8$-$0233.  We find a coincidence with the source G030.1981-00.1691. It has a bolometric luminosity of 2.9$\\times 10^4 L_\\odot$ and is located at 7.4 kpc. Within the error ellipse of the \\textit{Fermi} source, there are several other sources, such as dark nebulae and HII regions. \\\\\n\n- 1FGL J1848.1$-$0145. This source is coincident with two YSOs, G030.9726-00.1410 and G030.9959-00.0771, located at 5.7 kpc and with bolometric luminosities of 3.9$\\times 10^3 L_\\odot$ and 5.1$\\times 10^3 L_\\odot$, respectively. Both are part of the G031.1451+00.0383 complex, which hosts five more YSOs (outside the \\textit{Fermi} error box), five diffuse HII regions, and ten HII regions. Within the error ellipse of the gamma-ray emission, there is an unidentified very high-energy gamma-ray source, HESS~J1848-018 \\citep{chaves08}, which is probably the most suitable very high-energy candidate counterpart to the \\textit{Fermi} detection. \\\\\n\n- 1FGL J1853.1+0032. This source has the biggest error ellipse ($\\Delta\\theta_{95\\%} \\sim 0.5^\\circ$). For that reason, there is a high number of sources within its location error box, including pulsars, supernova remnants and X-ray sources. The cross-match of the catalogs yields six coincidences. These six YSOs belong to three different complexes with different distance to the Earth: G032.8205-00.3300 and G033.3891+00.1989 belong to the G033.1844-00.0572 complex, located at 5.1 kpc. The luminosities of these two sources are 1.7 and 1.1 $\\times 10^3 L_\\odot$, respectively; G033.3933+00.0100 and G33.5237+00.0198 belong to the G033.6106+00.0464 complex, located at a distance of 6.8 kpc and the luminosity is 7.9$\\times 10^3 L_\\odot$ for both sources; G034.0126-00.2832 and G034.0500-00.2977 belong to the G034.0313-00.2904 complex. The distance to these objects is 13.3 kpc and their bolometric luminosities are 3.4$\\times 10^4 L_\\odot$ and 2.4$\\times 10^4L_\\odot$, respectively. \\\\\n\n- 1FGL J1925.0+1720. Two YSOs are coincident with it: G052.2025+00.7217A and G052.2078+00.6890. The bolometric luminosities are 1.5$\\times 10^4 L_\\odot$ and 2.0 $\\times10^4 L_\\odot$, respectively. They belong to the G052.2052+00.7053 complex which is located at 10.2 kpc, and hosts also a HII region.\\\\\n\n- 1FGL J1943.4+2340. There is spatial coincidence with the source G059.7831+00.0648, located at 2.2 kpc. It has a bolometric luminosity of 6.8$\\times 10^4 L_\\odot$. This YSO has been detected in radio at 8.6 GHz with an integrated flux of 1.0 mJy \\citep{sridharan02}. The infrared counterpart of this YSO is IRAS 19410+2336. It was observed by Chandra in 2002, which found hard X-ray emission from a number of sources within this high-mass star-forming region \\citep{beut02}. The region has two cores where star formation takes place, with masses of 840 $M_\\odot$ and 190 $M_\\odot$ \\citep{sridharan02,beuther02b}. In the latter paper, it is proposed that the X-ray emission is produced by magnetic reconnection effects between the protostars and their accretion disks. The interaction of several molecular outflows, where the YSO from the RMS survey is located, and the combined effects of the stellar winds make a good scenario that might result in particle acceleration up to relativistic energies.  \\\\\n\n- 1FGL J2040.0+4157. There is spatial coincidence with G081.5168+00.1926. This YSO is located at 1.7 kpc and shows a bolometric luminosity of 7.04$\\times 10^2 L_\\odot$. There is a galaxy (2MASX J20395796+4159152) located at 2.3$^{\\prime \\prime}$ from the position of that YSO. \n\n\n\\subsection{WR and Of-type stars}\n\nWe found two \\textit{Fermi} sources coincident with 15 WR stars in the Galactic center cluster and nine WR stars in the Quintuplet cluster. Some of these stars \\citep[see][]{vander06} show variability and have been detected at X-rays with evidence of nonthermal emission.\nThe results of our study are shown in Table \\ref{tab:wr}.\nWe cannot state that these two coincidences correspond to physical associations given the high chance association probability obtained from the Monte Carlo study. The first one is in the direction of the galactic center, and there are several other sources that introduce confusion. The second one was suggested as potential association with the Pistol Star in \\cite{abdo10} and it is situated in a very crowded field. \n\nIn the case of Of-type stars, which are the evolved state of O stars, and precursors of WR stars, our results show that five \\textit{Fermi} sources are coincident with five stars (see Table \\ref{tab:of}). The probability of chance coincidence is too high to state a physical association. The sources 1FGL J1315.0-6235 and 1FGL J1853.1+0032 also show positional coincidence with YSOs (see Table \\ref{tab:yso}). The case of association with YSOs has a much lower value for the chance probability. The source 1FGL~J1112.1-6041 is coincident with HD~97434, a multiple star system. 1FGL~J1315.0-6235 is located in a regions that harbors dark nebulae and molecular clouds. The Of star coincident with this source is HD~115071, which is a spectroscopic binary. Finally, 1FGL J2004.7+3343 is coincident with HD~227465, and there is also the source G70.7+1.2 inside the error box of the \\textit{Fermi} detection, which might contain a Be star and an X-ray-emitting B star pulsar binary \\citep{cameron07}.\n\n\\input{wr_table.tex}\n\n\\subsection{OB associations}\\label{sec:obasso}\n\nOur study yields 107 \\textit{Fermi} sources positionally coincident with 35 OB associations, which represent $\\sim 41\\%$ of the sample. We list the results in Table \\ref{tab:obassoc}, available in the electronic version. We get this large number of gamma-ray sources because of the size of the OB associations. Most of them have angular sizes of $\\sim 1^\\circ$ in diameter or higher. Our results extend those of \\cite{rome99}, where they found 26 coincidences between \\textit{EGRET} sources and OB associations. Most of the OB associations that they found are present in our results, along with other new candidates, as expected from the higher sensitivity of \\textit{LAT}.  Although the number of sources has increased, the probability of chance associations is approximately the same as in \\cite{rome99}.\n\nMost of the OB associations have fewer than five \\textit{Fermi} sources within their error boxes. There are seven OB associations with five or more \\textit{Fermi} sources within their error boxes. There are four associations with a number of gamma-ray coincidences between five and ten (Ori 1 B, Ori 1 C, Car 2, and Cyg 1,8,9), located at short distances from the Earth (less than 1 kpc) and located at galactic latitudes of $\\sim |15^\\circ|$ (association centroid position). There is an exception, Car 2, which is located at 2.2 kpc and has a galactic latitude of $-0.13^\\circ$.  The OB associations with the large number of gamma-ray coincidences ($> 10$) are those from Scorpius (Sco 2 A, Sco 2 B and Sco 2 D). Those associations are located at $\\sim$ 170 pc on average and have very large angular sizes ($\\sim 9.5^\\circ$ on average). \n\nThere are five \\textit{Fermi} sources that are coincident with OB associations and YSOs at the same time: 1FGL J1256.9-6337, 1FGL~J1315.5-6235, 1FGL~J1702.4-447, 1FGL~J1943.4+2340, and 1FGL~J2040.0+4157. In all cases the OB association overlaps both the \\textit{Fermi} source and the YSO. In all cases but one, however, the distances to the YSO and the OB association are much too different.\n\n\\input{of_table.tex}\n\n\\section{Discussion}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{luminosity.pdf}}\n  \\caption{IR luminosity versus gamma-ray luminosity above 100 MeV of the brightest YSOs coincident with each \\textit{Fermi} source. The shaded area represents the 1-$\\sigma$ confidence interval of a least square fit taking eight data points randomly and repeating the fit one thousand times. It can be roughly  seen that the higher the gamma-ray luminosity, the higher the IR luminosity. \\label{fig:lumi}}\n\\end{figure}\t\n\nThe fact that there are five \\textit{Fermi} sources coincident with both YSOs and OB associations makes us consider which is the chance probability of having coincidence of \\textit{Fermi} sources with YSOs alone. Using the Monte Carlo algorithm again, but taking this constrain into account, we obtain a mean value of coincidences of $2.8\\pm 1.7$, and the chance probability for the seven \\textit{Fermi} sources coincident only with YSOs is $\\sim 1.6\\%$.\n\nThe brightest IR YSOs have more molecular mass available for proton-proton collisions and Bremsstrahlung interactions than those that are faint. It is expected then that the brightest IR YSOs would show the highest gamma-ray luminosity. To test such a trend, we have plotted in Figure \\ref{fig:lumi} the IR luminosity versus the gamma-ray luminosity of the brightest YSOs coincident with each \\textit{Fermi} source. The gamma-ray luminosity was calculated from the gamma-ray flux (see Table \\ref{tab:yso}) assuming that the distance to the gamma-ray source is equal to the distance to the corresponding YSO. We see that, under this assumption, there is a trend toward increasing the gamma-ray luminosity as the IR luminosity increases, too. This trend, however, is obtained using the whole data set, and according to the results from our Monte Carlo simulations, we should have four chance coincidences out of the 12 candidates. To see how this could affect the trend, we selected eight data points randomly and fitted them to a straight line by least squares. We repeated this process one thousand times, getting an average least square fit within one standard deviation. The limits of this fit are plotted in Figure 2. The increasing trend in the data is visible, although with a significant dispersion. This is not surprising considering the broad approach.\n\nThe association of WR stars with gamma-ray sources has been discussed in the past by \\cite{kaul97} and \\cite{rome99}. In both cases, the authors studied the positional coincidence between WR stars and unidentified EGRET sources. In the later work, they find that two WR stars are of special interest: WR 140 and WR 142. The first one is a binary system, composed of a WC 7 plus an O4-5 star where the region of collision of the winds seems to be a good place for particle acceleration and high energy emission. It should be mentioned, however, that WR 140 is a long-period binary with variability expected on time scales of years \\citep{williams87}. In the second one, WR 142, the hard X-ray emission and fast wind may indicate a colliding wind shock that could be explained by a companion close to this star \\citep{sokal10}. No companion, however, has been reported so far.\nIn our work none of these mentioned WR stars appear to be coincident with any \\textit{Fermi} source. The poor statistical correlations found from the simulations does not allow us to be confident with any of the coincidences found. The case of Of-type stars is also unclear since the probability of chance association is high. \n\nFinally, our results for the OB associations are not conclusive. We list the results in Table \\ref{tab:obassoc}, available in the electronic version. The probability of chance coincidence is negligible ($\\sim 10^{-6}$), but we do get several gamma-ray sources for each OB association. Thus, is very difficult to assign a specific counterpart to the gamma-ray emission.\n\n\\section{Conclusions}\n\nWe studied the two-dimensional coincidence between unidentified \\textit{Fermi} sources and catalogs of galactic young objects, such as YSOs, WR stars, Of stars, and OB associations. We have found a statistical correlation between gamma-ray sources and YSOs. The correlation with the early type stars with strong winds remain unclear, since the candidates are located in crowded fields with many other alternatives to the gamma-ray emission, and the probability of chance associations is high. In the case of OB associations, the probability of chance associations is negligible. However, we cannot assign a specific counterpart to the gamma-ray emission because of the high angular size of most of OB associations.\n\nWhat we have presented here is the first statistical evidence for gamma-ray emission from massive YSOs.\n\n\n\\begin{acknowledgements} \nThe authors acknowledge support from the Spanish Ministerio de Ciencia e Innovaci\\'on (MICINN) under grant AYA2010-21782-C03-01. G.E.R. also acknowledges support of PIP 0078 (CONICET).\n\\end{acknowledgements}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\IEEEPARstart{M}{illions} of asteroids exist in solar system, many the shattered remnants of planetesimals, bodies within the young Sun's solar nebula that never grew large enough to become planets \\cite{kortenkamp_asteroids_2011}. The vast majority of known asteroids orbit within the main asteroid belt located between the orbits of Mars and Jupiter. To further investigate how planets formed and how life began, as well as improve our understanding to asteroids that could impact Earth, some deep exploration programs were proposed, e.g. \\emph{Hayabusa}\\cite{fujiwara_rubble-pile_2006, yoshikawa_hayabusa_2006}, \\emph{Hayabusa2}\\cite{sugita_geomorphology_2019, watanabe_hayabusa2_2017}, and \\emph{OSIRIS-Rex}\\cite{lauretta2017osiris, golish2020ground}. The program objectives involve orbiting observation, autonomous landing, geological sampling, and so on. \n\nTwo near-Earth asteroids, Itokawa \\cite{yoshikawa_hayabusa_2006} and Ryugu \\cite{sugita_geomorphology_2019}, with complex 6-DoF motion are shown in Fig. \\ref{fig1}. It is obvious that 3D visual tracking system is important to explore these two asteroids, which can provide object location, size, and pose. It's also of great significance to spacecraft autonomous navigation, asteroid sample collection, and universe origin study. However, state-of-the-art 4-DoF trackers \\cite{giancolaLeveragingShapeCompletion2019, qiP2BPointtoBoxNetwork2020, yinCenterBased3DObject2021} presented for automous driving are confused about heading angle of asteroid, which makes inaccurate 3D bounding-box estimation. Besides, some 6-DoF tracking methods \\cite{prisacariuPWP3DRealTimeSegmentation2012, prisacariuSimultaneousMonocular2D2013, crivellaroRobust3DObject2018a} under strong assumptions are also impractical to track asteroid. To be honest, constructing an end-to-end deep network that predicts 6-DoF states of asteroid is pretty difficult. We therefore decompose 3D asteroid tracking problem into 3-DoF tracking and pose estimation. And this paper merely focus on the 3-DoF tracking part.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Itokawa]{\n        \\includegraphics[width=0.1\\textwidth]{itokawa0003.jpeg} \n        \\includegraphics[width=0.1\\textwidth]{itokawa0005.jpeg}\n        \\includegraphics[width=0.1\\textwidth]{itokawa0007.jpeg}\n        \\includegraphics[width=0.1\\textwidth]{itokawa0010.jpeg}\n\t    \\label{fig1_a}\n    }\n\t\\hfil\n\t\\subfloat[Ryugu]{\n        \\includegraphics[width=0.1\\textwidth]{ryugu0001.jpeg}\n        \\includegraphics[width=0.1\\textwidth]{ryugu0004.jpeg}\n        \\includegraphics[width=0.1\\textwidth]{ryugu0008.jpeg}\n        \\includegraphics[width=0.1\\textwidth]{ryugu0012.jpeg}\n\t    \\label{fig1_b}\n    }\n\t\\caption{Two mission objectives of \\emph{Hayabusa} and \\emph{Hayabusa2}. (a) Itokawa, a potato-shaped asteroid with 600-meter size, is named after Hideo Itokawa, a Japanese rocket pioneer. (b) Ryugu, a more primordial C-type asteroid with 900-meter size, is discovered on May 1999 by LINEAR telescope.\n    }\n\t\\label{fig1}\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n\t\\includegraphics[width=\\textwidth]{3D_track_framework.pdf}\n    \\caption{The deep-learning based 3D tracking framework for asteroid, named as Track3D, which consists of 2D monocular tracker and amodal axis-aligned bounding-box network, A3BoxNet. Experiment shows our framework can run at 77 FPS with high accuracy. And it has great generalization abitlity to 2D tracking algorithm.}\n    \\label{fig2}\n    \\centering\n\\end{figure*}\n\nInspired by the idea of 2D-driven 3D perception, we present a novel deep-learning based 3D asteroid tracking framework, Track3D. As shown in Fig. \\ref{fig2}, it mainly consists of 2D monocular tracker and a light-weight amodal axis-aligned bounding-box network, A3BoxNet, which can predict accurate target center and size purely relied on partial object point cloud. Extensive experiments show that Track3D reaches state-of-the-art 3D tracking results (0.669 $AO^{3d}$ at 77 FPS) and has great generalization ability to 2D monocular tracker. Moreover, we discover that our framework with 2D-3D tracking fusion strategy can make significant improvement on 2D tracking performance. \n\nHowever, there are few studies on 3D asteroid tracking, as well as relevant dataset, which have greatly hindered the development of asteroid exploration. To this end, we construct the first large-scale 3D asteroid tracking dataset, by acquiring 148,500 binocular images, depth maps, and point clouds of diverse asteroids in various shapes and textures with physics engine. Benefitting from the power and convenience of physics engine, all the 2D and 3D annotations are automatically generated. Meanwhile, we also provide an evaluation toolkit, which includes 2D monocular and 3D tracking evaluation algorithms.\n\nOur contributions in this paper are summarized as follows:\n\\begin{itemize}\n    \\item Considering different types of asteroid with various shapes and textures, we construct the first large-scale 3D asteroid tracking dataset, including 148,500 binocular images, depth maps, and point clouds. \n\n    \\item The first 3-DoF asteroid tracking framework, Track3D, is also presented, which involves an 2D monocular tracker and A3BoxNet network. The experimental results show the impressive advancement and generalization of our framework, even based on poor 2D tracking algorithm.\n\n    \\item We propose a novel light-weight amodal bounding-box estimation network, A3BoxNet, which can predict accurate axis-aligned bounding-box of target merely with partial object point cloud. Randomly sampling 1024 points as network input, A3BoxNet can even achieve 0.712 $AO^{3d}$ and 0.345 $ACE^{3d}$ with up to 281.5 FPS real-time performance.\n     \n\\end{itemize}\n\nThe rest of this paper is ordered in the subsequent sections. In Section \\ref{section2}, we review some related works from two aspects: visual tracking in aerospace domain and 3D object tracking. In Section \\ref{section3}, the constructing details of 3D asteroid tracking dataset are presented, involving simulation platform, annotation, and evaluation metrics. We also propose a novel deep-learning based 3D tracking framework in Section \\ref{section4}, which consists of 2D monocular tracker, and a light-weight amodal bounding-box estimation network, A3BoxNet. Section \\ref{section5} introduces more details about 3D tracking framework performance and ablation study. Finally, we make a conclusion in Section \\ref{section6}.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{3D_asteroid_tracking_platform.pdf}\n    \\caption{The simulated platform for constructing 3D asteroid tracking dataset. We design three types of 3D asteroid model with 6 colorful textures, of which size and luminosity are modified during data collection. The setups of visual observer spacecraft and imaging mechanism of perspective vision camera are also shown in upper right corner. We pre-align depth map and point cloud to reference frame (i.e. left camera coordinate system). The left and right vision cameras have the same parameters, including far clipping plane, perspective angle, and resolution. In addition, some screenshots of 3D asteroid tracking dataset with 2D and 3D annotations are also visualized at bottom right.}\n    \\label{fig3}\n    \\centering\n\\end{figure*}\n\n\\section{Related Work \\label{section2}}\n\\subsection{Visual Tracking in Aerospace}\nThe goal of visual tracking is to precisely and robustly perceive real-time states (e.g. location, velocity and size) of an instereted target in complex and dynamic environment. As the basis for pose estimation, behavior understanding, and scene interpretation, visual object tracking has wide application in space debris removel \\cite{agliettiActiveSpaceDebris2020,huangDexterousTetheredSpace2017}, space robotic inspection \\cite{fourieFlightResultsVisionBased2014}, spacecraft rendezvous and docking \\cite{petitVisionbasedSpaceAutonomous2011,sharmaPoseEstimationNoncooperative2018}. \n\nAt present, most of visual trackers in aerospace domain mainly focus on feature-based method. Huang \\cite{huangDexterousTetheredSpace2017} proposed a novel feature tracking algorithm. Firstly, it extracted features by SURF detector. And then, the pyramid-Kanade-Lucas-Tomasi (P-KLT) algorithm was adopted to match key-points between two adjacent frames. Finally, accurate target bounding box is obtained with Greedy Snake method. A feature-tracking scheme was also presented in \\cite{volpePassiveCameraBased2018}  that combines traditional feature-point detector and frame-wise matching to track a non-cooperative and unknown satellite. Felicetti \\cite{felicettiImagebasedAttitudeManeuvers2018} put forward an active space debris visual tracking method, in which the chaser satellite can keep the moving object in field of view of optical camera by continuously pose correction. Those feature-based trackers heavily relied on the manually designed feature detector and cannot handle extreme cases in space (e.g. illumination variation, scale variation, fast motion, rotation, truncation, and background clutters).\n\nWith many challenges and benchmarks emerging, such as OTB2015 \\cite{wuObjectTrackingBenchmark2015}, YouTube-BB \\cite{realYouTubeBoundingBoxesLargeHighPrecision2017}, ILSVRC-VID \\cite{russakovskyImageNetLargeScale2015}, GOT-10K \\cite{huangGOT10kLargeHighDiversity2019}, and Visual Object Tracking challenges \\cite{kristanVisualObjectTracking2017, kristanSeventhVisualObject2019}, the development of generic object tracking is far beyond imagination \\cite{lanLearningCommonFeatureSpecific2018}. Especially, deep learning based trackers \\cite{bertinettoFullyconvolutionalSiameseNetworks2016, wangFastOnlineObject2018, liSiamRPNEvolutionSiamese2018, danelljanATOMAccurateTracking2018, zhangDeeperWiderSiamese2019, chenSiameseBoxAdaptive2020, ondrasovicSiameseVisualObject2021} have dominated the whole tracking community in recent years, because of its striking performance. Generic object trackers often follow one protocol that no prior knowledges is available. This hypothesis is naturally suitable for asteroid visual tracking, because of high uncertainty of vision tasks in space. In paper \\cite{zhou2DVisionbasedTracking2021}, most of state-of-the-art generic trackers had been evaluated on space non-cooperative object visual tracking (SNCOVT) dataset, which provides firm research foundation for our work.\n\nHowever, recent visual trackers mainly cope with RGB \\cite{zhangOceanObjectawareAnchorfree2020, yanLearningSpatioTemporalTransformer2021, chenTransformerTracking2021}, RGB-Thermal \\cite{lanModalitycorrelationawareSparseRepresentation2020, lanLearningModalityConsistencyFeature2019, lanRobustMultimodalityAnchor2019}, and RGB-Depth \\cite{songTrackingRevisitedUsing2013} video sequences, which only provides poor target information in 2D space and heavily restricts practical applications of visual tracking. In contrast, 3D visual tracking is more promising and competitive. To this end, we propose a novel deep-learning based 3D tracking framework for asteroid exploration.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\caption{The default object size categories and corresponding ratio of XYZ.}\n    \\label{table1}\n    \\begin{tabular}{ccccccccccccccc}\n        \\toprule\n        category & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\\\\n        \\midrule\n        X & 1 & 1\/2 & 1\/3 & 2\/3 & 1 & 1 & 1 & 1 & 1\/2 & 2\/3 & 1\/2 & 2\/3 & 1 & 1\\\\\n        Y & 1 & 1 & 1 & 1 & 1\/2 & 2\/3 & 1 & 1 & 1\/2 & 2\/3 & 1 & 1 & 1\/2 & 2\/3 \\\\\n        Z & 1 & 1 & 1 & 1 & 1 & 1 & 1\/2 & 2\/3 & 1 & 1 & 1\/2 & 2\/3 & 1\/2 & 2\/3 \\\\\n        \\bottomrule  \n    \\end{tabular}\n    \\centering\n\\end{table*}\n\n\\subsection{3D Object Tracking}\nAlthough there are plenty of related works \\cite{prisacariuPWP3DRealTimeSegmentation2012, heldPrecisionTrackingSparse2013, prisacariuSimultaneousMonocular2D2013, asvadi3DObjectTracking2016, crivellaroRobust3DObject2018a, kartObjectTrackingReconstruction2019, giancolaLeveragingShapeCompletion2019, qiP2BPointtoBoxNetwork2020, yinCenterBased3DObject2021}, the concept of 3D object tracking remains ambiguous. Hence that, we first define that 3D object tracking is to obtain 3D information of any target in real-time by leveraging various visual sensors, given initaial states at the beginning. According to the degrees of freedom of tracking result, all the 3D trackers can be concluded into three categories: 3-DoF, 4-DoF, and 6-DoF Tracker. Obviously, the more degrees of freedom, the more difficult 3D tracking task is. \n\n3-DoF tracking means that both 3D location and size of object (i.e. 3D axis-aligned bounding-box) should be estimated. However, most of researches only considered predicting 3D object center. In paper \\cite{heldPrecisionTrackingSparse2013}, a color-augmented search algorithm was presented to obtain the position and velocity of vehicle. Asvadi et al. \\cite{asvadi3DObjectTracking2016} utilized two parallel mean-shift localizers in image and pcd space, and made fusion for two localizations by Kalman filter. This algorithm can effectively achieve low 3D center error. To the best of our knowledge, the framework proposed in this paper is the first real 3-DoF tracker that can be applied for collision avoidance, 3D reconstruction, and pose estimation.\n\nComparing with 3-DoF methods, 4-DoF tracker often needs to predict an extra heading angle of target, which is original from 3D tracking requirement in autonomous driving. Giancola et al. \\cite{giancolaLeveragingShapeCompletion2019} presented a novel 3D siamese tracker with the regularization of point cloud completion. However, the exhaustively searching for candidate shapes in this method would consume high computational cost. Qi et al. \\cite{qiP2BPointtoBoxNetwork2020} also proposed point-wise tracking paradigm, P2B, which addressed 3D object tracking by potential center localization, 3D target proposal and verification. Since this method purely uses point cloud data as input, it is vulnerable to initial object point cloud and only achieves 0.562 $AO^{3d}$ at 40 FPS on KITTI tracking dataset \\cite{geigerAreWeReady2012}. \n\nIn paper \\cite{prisacariuPWP3DRealTimeSegmentation2012, prisacariuSimultaneousMonocular2D2013}, 6-DoF tracking task was considered as joint 2D segmentation and 3D pose estimation problem, and the method looked for the pose that best segmented the target object from the background. Crivellaro et al. \\cite{crivellaroRobust3DObject2018a} present a novel 6-DoF tracking framework with monocular image, including expensive 2D detector, local pose estimation and extended kalman filter. However, the non-textured 3D model of target should be given in this method, which heavily limits the scope of its application. To be honest, the asteroid tracking is one of 6-DoF tracking tasks. We think it is very hard to construct an end-to-end network that can directly and precisely predict the 6-DoF object states. Therefore, we decompose asteroid tracking problem into 3-DoF tracking and pose estimation. It is worthwhile noting that this paper simply focus on the 3-DoF tracking task of asteroid.\n\n\n\\section{3D Asteroid Tracking Dataset \\label{section3}}\nTo promote the research on 3D tracking for asteroid, we construct a large-scale 3D asteroid tracking dataset, including binocular video sequences, depth maps and point clouds. In addition, 3D tracking evaluation toolkit is also provided for performance analysis. More details about 3D asteroid tracking dataset are introduced in this section.\n\n\\subsection{Dataset Construction}\nThere is no doubt that collecting real data to create large-scale 3D asteroid tracking dataset is impractical, like KITTI dataset \\cite{geigerAreWeReady2012} and Princeton RGB-D Tracking dataset \\cite{songTrackingRevisitedUsing2013}. Meanwhile, constructing dataset by ground simulation \\cite{zhou2DVisionbasedTracking2021} is very expensive and limited. Inspired by UAV123 dataset \\cite{muellerBenchmarkSimulatorUAV2016}, we therefore consider to collect rich 3D tracking data by virtual physics engine, V-rep. The power and convenience of physics engine make automatic data labelling possible, which greatly reduces constructing cost and improves annotation accuracy.\n\nThe critical foundation of dataset constructing based on physics engine is 3D modeling of asteroids with diverse shapes and textures. We create three types of 3D asteroid model (i.e. Asteroid04, 05, and 06) with 6 different textures, which are illustrated at the left of Fig. \\ref{fig3}. From point of our view, one asteroid model with diferent textures can be considered as different fine-grained categories. In addition, we introduce 9 simulated space scenes into dataset construction. All the 3D asteroid models have been controlled by scripts to carry out random 6-DoF motion in simulated scenes. \n\nThe detailed setup of vision sensors for data collection can be seen at upper right corner of Fig \\ref{fig3}. Only two perspective vision cameras are equipmented with 0.4-meter baseline in x-axis direction on observer spacecraft, which can not only acquire binocular video sequences during simulation, but also achieve aligned depth maps and point clouds from camera buffer by the API of physics engine. The camera matrix is very vital for our 3D tracking framework in which point clouds should be projected to image plane, however, V-rep merely provides perspective angle $(\\alpha_x, \\alpha_y)$ and resolution $\\left(W, H\\right)$. To this end, we compute camera intrinsic matrix $M_i$ following Eq. \\ref{eq_1} (more derived details are given in Appendix \\ref{appendix1}):\n\n\\begin{equation}\n    M_{i}=\n    \\begin{bmatrix}\n       \\frac{W}{2 \\tan(\\alpha_x\/2)} & 0 & \\frac{W}{2} \\\\\n       0 &   \\frac{H}{ 2 \\tan(\\alpha_y\/2)} & \\frac{H}{2}\\\\\n       0 & 0 & 1\n    \\end{bmatrix} \n    \\label{eq_1}\n \\end{equation}\n    \nAt final, we collect 360 sequences for training and 135 sequences for testing set. Each sequence has 300 frames binocular images, depth maps, and point clouds. Following the protocol proposed in \\cite{huangGOT10kLargeHighDiversity2019}, there is no overlap on fine-grained categories between training and testing set. The screenshots of 3D asteroid tracking testing set with 2D and 3D annotations are shown in Fig. \\ref{fig3}.\n\n\\begin{figure}[t]\n    \\centering\n        \\subfloat[Size classes]{\\includegraphics[width=0.23\\textwidth]{size_class_distribution_total.pdf} \\label{fig4_a}} \\hfil\n        \\subfloat[Object volumes]{\\includegraphics[width=0.23\\textwidth]{object_volume_distribution_total.pdf} \\label{fig4_b}} \\vfill\n        \\subfloat[Motion speeds]{\\includegraphics[width=0.23\\textwidth]{object_motion_speed_distribution_total.pdf} \\label{fig4_c}} \\hfil\n        \\subfloat[Illumination]{\\includegraphics[width=0.23\\textwidth]{object_illumination_distribution_total.pdf} \\label{fig4_d}} \n        \\caption{The statistics of 3D asteroid tracking dataset.}\n        \\label{fig4}\n    \\centering\n\\end{figure}\n\n\\begin{figure*}[ht]\n    \\centering\n\t\\includegraphics[width=0.8\\textwidth]{3D_track_framework_detailed.pdf}\n    \\caption{The detailed framework diagram of Track3D}\n    \\label{fig5}\n    \\centering\n\\end{figure*}\n\n\\subsection{Dataset Statistics}\nIn actual, our framework predicts the object size class and normalized size residuals, rather outputs an axis-aligned 3D bounding-box directly. Therefore, size classes should be predefined and the number of samples in each category has better to be balanced. To this end, we make a statistic analysis for size category of axis-aligned 3D bounding-boxes in 3D asteroid tracking dataset and define 14 default size categories summarized in Table \\ref{table1}. The final distribution of size classes can be seen in Fig. \\ref{fig4_a}. \n\nMeanwhile, the distribution of object volumes and motion speeds are shown in Fig. \\ref{fig4_b} and \\ref{fig4_c}, respectively. Because of random asteroid rotation and manual shape modification, the volume of samples is widely distributed (16 $m^3$ to 1600 $m^3$). It is worthwhile noting that object volume metioned here is the volume of axis-aligned 3D bounding-box, instead the real one. In addition, as we mentioned above that all the 3D asteroid models are controlled by scripts to carry out random 6-DoF motion, Fig. \\ref{fig4_c} clearly demonstrates that the random translation of object obeys normal distribution. \n\nAs paper \\cite{zhou2DVisionbasedTracking2021} had proven that all the generic trakcers are vulnerable to illumination variation, we also consider this factor into our dataset. Furthermore, we have limited the illumination of all samples at low value (see in Fig. \\ref{fig4_d}), which conforms to the reality of space environment.\n\n\\subsection{Evaluation Metrics}\nThe main metric utilized to analyse 2D tracking performance is average overlap, $AO$, which is also widely applied in visual tracking benchmarks like OTB\\cite{wuObjectTrackingBenchmark2015}, VOT\\cite{kristanSeventhVisualObject2019}, and GOT-10k \\cite{huangGOT10kLargeHighDiversity2019}. In this paper, we also introduce this metric to evaluate 2D monocular tracker and 3D tracking results in bird's eye view (i.e. $AO^{2d}$ and $AO^{bev}$). Furthermore, We extend average overlap measurement to evaluate 3D tracking accuracy, named as $AO^{3d}$:\n\\begin{align}\n    AO^{3d} &= \\frac{1}{M} \\sum_{s=1}^{M} \\left( \\frac{1}{N_{s}} \\sum_{t=1}^{N_{s}} \\Omega_{t}^{3d} \\right);\\\\ \n    \\nonumber \\Omega_{t}^{3d} &= \\frac{\\tilde{A}_{t}^{3d} \\cap A_{t}^{3d}}{\\tilde{A}_{t}^{3d} \\cup A_{t}^{3d}};\n\\end{align}\nin which, $M$ is the sequences number in 3D asteroid tracking test set, $N_{s}$ denotes the length of \\emph{s}-th sequence, and $\\Omega^{3d}_{t}$ denotes the region overlap between axis-aligned 3D annotation  $\\tilde{A}_{t}^{3d}$ and 3D tracking result $A_{t}^{3d}$ at \\emph{t}-th frame. It is worthwhile noting that tracking restart mechanism is introduced to 3D tracking evaluation. That is 3D tracker is allowed to reinitialize, once the Intersection-over-Union (IoU) of 3D tracking result $\\Omega^{3d}_{t}$ is zero.\n\nBesides average overlap metric, we also adopt success rate, $SR$, as our indicator, which denotes the percentage of successfully tracked frames where the overlap exceeds a threshold. In this work, we take 0.5 as threshold to measure the success rate of one tracker. However, $SR$ at a specific threshold is not representative, we further introduce success plot presented by \\cite{wuObjectTrackingBenchmark2015} into our evaluation tool.\n\nAnother intuitive technique index of tracker is location precision. Therefore, we propose 3D average center error metric ($ACE^{3d}$) that measures the mean distance between all the predicted trajectories and ground-truth trajectories:\n\\begin{align}\n    ACE^{3d}  &= \\frac{1}{M} \\sum_{s=1}^{M} \\Delta \\left( \\Gamma_s, \\tilde{\\Gamma}_s \\right)\n\\end{align}\nwhere $\\Gamma_s = \\left\\{ p_t | t=1, \\ldots, N_s \\right\\}$ and $\\tilde{\\Gamma}_s = \\left\\{ \\tilde{p}_t | t=1, \\ldots, N_s \\right\\}$ are the predictions and ground-truths of \\emph{s}-th sequence, which consist of a series of object center in 3D space. $\\Delta \\left( \\Gamma_s, \\tilde{\\Gamma}_s \\right)$ is formulated as:\n\\begin{equation}\n    \\Delta\\left(\\Gamma_{s}, \\tilde{\\Gamma}_{s}\\right)=\\frac{1}{N_{s}} \\sum_{t=1}^{N_{s}}\\left\\|p_{t}-\\tilde{p}_{t}\\right\\|_{2}\n\\end{equation}\nin which, $\\left\\| \\cdot \\right\\|_{2}$ denotes Euclidean distance. \n\n\n\\begin{figure*}[t]\n    \\centering\n\t\\subfloat[Center regression network]{\\includegraphics[width=0.65 \\textwidth]{Center_Regression_Network.pdf} \\label{fig6_a}} \\hfil\n    \\subfloat[Amodal box estimation network]{\\includegraphics[width=0.7 \\textwidth]{Amodal_Box_Estimation.pdf} \\label{fig6_b}}\n    \\caption{Two main module of A3BoxNet. (a) predicts coarse object center, (b) is to estimate object center residuals, object size category and normalized size residuals.}\n    \\label{fig6}\n    \\centering\n\\end{figure*}\n\n\\section{3D Tracking Framework \\label{section4}}\nInspired by the 2D-drieven 3D perception, we propose a deep-learning based 3D tracking framework shown in Fig. \\ref{fig5}, which involves 2D monocular tracker, SiamFC \\cite{bertinettoFullyconvolutionalSiameseNetworks2016} and a novel light-weight amodal axis-aligned bounding-box network, A3BoxNet. Although binocular images and depth maps are also provided in our 3D asteroid tracking dataset, it is worthwhile noting that Track3D only utilizes monocular video sequence and corresponding point clouds, which reduces the complexity of 3D tracking framework and more conforms to real applications in aerospace. In addition, we will introduce a simple but effective 2D-3D tracking fusion strategy in the following subsections. \n\n\\subsection{SiamFC}\nThe SiamFC algorithm utilizes full convolutional layers as an embedding layer, denoted as embedding function $\\varphi$. The exemplar image $z$ and multi-scale search images $X=\\left\\{x_i | i = 1, 2, ..., S \\right\\}$ are mapped to a high-dimensional feature space with $\\varphi$, which can be trained by a general large-scale dataset. And then similarity score maps are generated by cross-correlation between exemplar kernal and the tensors of multi-scale search images, where the 2D tracking result can be reached after post-processing. \n\nFor this algorithm, the key is to learn discriminative embedding function $\\varphi$. In this work, we train the backbone of SiamFC from scratch on ILSVRC-VID \\cite{russakovskyImageNetLargeScale2015} with 50 epochs and further fine-tune it on our 3D asteroid tracking training set. Benefitting from the strong generalization ability of Siamese network, SiamFC can achieve a decent 2D tracking performance. Although there are multifarious deep-learning based 2D monocular trackers, like SiamRPN++ \\cite{liSiamRPNEvolutionSiamese2018}, Ocean \\cite{zhangOceanObjectawareAnchorfree2020}, STARK \\cite{yanLearningSpatioTemporalTransformer2021}, TransT \\cite{chenTransformerTracking2021}, which greatly outperform SiamFC in tracking accuracy, we think weak dependence on high-performance 2D monocular tracker can effectively guarantee the generalization ability of our framework and improve its running speed. \n\nOnce tracking result $A_{t} $ is acquired from 2D monocular tracker, it can be used for frustum proposal extraction from raw points set $P_{\\text{raw}} = \\left\\{ (x_i, y_i, z_i) \\in \\mathbb{R} ^3 | i=1, 2, ..., r \\right\\}$, as shown in Fig. \\ref{fig5}. We first crop out $m$ points from raw point cloud $P_{\\text{crop}} = \\left\\{ (x_i, y_i, z_i) |  1 \\leqslant z_i \\leqslant 45, i=1, 2, ..., r \\right\\}$, and then compute the projected point cloud in image plane $P_{\\text{img}}=\\left\\{ (u_i, v_i) | i=1, 2, ..., m \\right\\}$ with camera matrix $\\mathbf{M} \\in \\mathbb{R}^{3 \\times 4}$. Therefore, we can achieve the frustum proposal $P_{\\text{frustum}} =  \\left\\{ (x_i, y_i, z_i) | i=1, 2, ..., n \\right\\} $ by the indices of points which belong to $A_t$ region in $P_{\\text{img}}$. \n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.4 \\textwidth]{center_prediction.pdf}\n        \\caption{Two-stage object center prediction in bird's eye view. We first estimate the stage-1 center of input points with center regression network. And then center residuals are predicted by amodal box estimation network, also named as stage-2 center.}\n        \\label{fig7}\n    \\centering\n\\end{figure}\n\n\\begin{figure}[t]\n    \\begin{center}\n        \\includegraphics[width=0.4 \\textwidth]{2D-3D_fusion_strategy.pdf}\n        \\caption{The schematic diagram of 2D-3D tracking fusion strategy. $A^{proj}_t$ is the projection of center cross-section plane of $A^{3d}_t$ in image plane and $A_t$ is the result of 2D monocular tracker at \\emph{t}-th timestep. We compute fused tracking result $\\hat{A}_t$ by the weighted average between $A^{proj}_t$ and $A_t$.}\n        \\label{fig8}\n    \\end{center}\n\\end{figure}\n\n\n\\begin{table*}[t]\n    \\centering\n    \\caption{The evaluation results on 3D asteroid tracking dataset. First two rows are the performances of two main modules in Track3D, and the 3rd row is a simple 3D tracking baseline as comparison. All the 2D tracking metrics of 3D tracker are computed by using 2D-3D fusion strategy which makes great improvement on 2D tracking performance. The top-2 evaluation results for each performance indicator are highlighted in \\textcolor{green}{green} and \\textcolor{blue}{blue}.}\n    \\label{table2}\n    \\begin{tabular}{ccccccc}\n        \\toprule\n         \\multirow{2}{*}{name} & \\multirow{2}{*}{$AO^{2d}$} & $ACE^{2d}$ & \\multirow{2}{*}{$AO^{bev}$}  & \\multirow{2}{*}{$AO^{3d}$}  & $ACE^{3d}$ & \\multirow{2}{*}{FPS}\\\\\n         & & (pixel) & & & (meter) & \\\\\n        \\midrule\n        SiamFC & 0.513 & 29.986 & - & - & - & 101.3\\\\\n        A3BoxNet & - & - & \\textcolor{green}{0.798} & \\textcolor{green}{0.721} & \\textcolor{green}{0.345} & \\textcolor{green}{281.5} \\\\ \\midrule\n        3D Tracking Baseline & \\textcolor{green}{0.568} & \\textcolor{green}{23.767} & 0.309 & 0.165 & 0.876 & \\textcolor{blue}{103.1} \\\\\n        Track3D & \\textcolor{blue}{0.541} & \\textcolor{blue}{26.741} & \\textcolor{blue}{0.756} & \\textcolor{blue}{0.669} & \\textcolor{blue}{0.570} & 77.0 \\\\\n        \\bottomrule  \n    \\end{tabular}\n    \\centering\n\\end{table*}\n\n\\subsection{A3BoxNet}\nIn this work, we assume that there are no other objects in the range of LiDAR perception, that is, all of the points in frustum proposal belong to tracked target, which conforms to practical scenario in space. Therefore, we propose a novel light-weight amodal axis-aligned bounding-box network, A3BoxNet, which makes directly prediction on frumstum proposal and no point segmentation is considered into. Fig. \\ref{fig2} clearly shows that our A3BoxNet mainly consists of two modules: center regression network and amodal box estimation network. The former one is responsible for the estimation of stage-1 object center. Another is to predict object center residuals, object size category, and normalized size residuals. It is worthwhile noting that both center regression network and amodal box estimation network are only support fixed number of input points. To this end, we add random sampling at the beginning of A3BoxNet.\n\nThe architecture of center regression network and amodal box estimation network are both illustrated in Fig. \\ref{fig6_a} and \\ref{fig6_b}. It can be clearly seen that there is no significant difference between two networks which are both derived from PointNet \\cite{qiFrustumPointNets3D2018}. In A3BoxNet, we use a number of MLP for high-level features extraction. And max-pooling layer is introduced to aggregate global features with symmetric property that is critical for unsorted point sets. The global feautres concatenated with one-hot vector of coarse-grained asteroid category can be further used for predicting stage-1 center, stage-2 center, and object size. \n\nWe visualize the center predction progress of A3BoxNet in Fig. \\ref{fig7}. The frustum proposal have been normalized by substracting the centroid of point sets firstly, which can improve the translation invariance of A3BoxNet and speed up convergence rate during training. In addtion, it shows that after two stage prediction, estimated object center is very close to the ground-truth. The predicted center is formulateds as follows:\n\\begin{equation}\n    C_{pred} = \\bar{C} + \\Delta C_1 + \\Delta C_2\n\\end{equation}\nwhere $\\bar{C}$ is the centroid of point cloud in frustum proposal, $\\Delta C_1$ is the output of center regression network, $\\Delta C_2$ is the center residuals predicted by amodal box estimation network. \n\nExcept for estimating the center residuals of object, our amodal box estimation network also classifies object size to 14 predefined categories (see in Table \\ref{table1}) as well as predicts normalized size residuals ($N_{size}$ scores for size classification, $N_{size} \\times 3$ for size residuals regression). At final, we remap the predicted size category and normalized size residual to original scale by multiplying the largest length of enclosing bounding-box of point clouds. \n\nTo train our A3BoxNet, we use a joint loss function $\\mathcal{L}_{joint}$ to simultaneously optimize both two submodules:\n\\begin{equation}\n    \\mathcal{L}_{joint} = \\mathcal{L}_{center-net} + \\mathcal{L}_{box-net}\n\\end{equation}\nin which, \n\\begin{equation}\n    \\mathcal{L}_{box-net} = \\mathcal{L}_{center\\_res} +  \\mathcal{L}_{size\\_cls} + \\mathcal{L}_{size\\_res}\n\\end{equation}\nmore details about optimization function of A3BoxNet are given in Appendix \\ref{appendix2}. In addition, A3BoxNet is trained on 3D asteroid tracking dataset with 25 epoches and 32 batch size. All the inputs are fixed number of point sets that randomly sampled from object point cloud.\n\n\\subsection{2D-3D Tracking Fusion Strategy}\nWe believe the fusion of 3D tracking and 2D monocular tracking results can make significant improvement on 2D tracking performance. To this end, we propose a simple and effective 2D-3D tracking fusion strategy (as shown in Fig. \\ref{fig8}). At first, we calculate the projection of center cross-section plane of 3D bounding-box in image plane, $A^{proj}_t$. And then, 2D monocular tracking result $A_t$ is weighted with $A^{proj}_t$:\n\\begin{equation}\n    \\hat{A}^t = \\lambda_1 \\cdot A^{proj}_t + \\lambda_2 \\cdot A^t\n\\end{equation}\nin this work, we set $\\lambda_1 = 0.3$ and $\\lambda_2 = 0.7$. \n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[3D tracking baseline]{\\includegraphics[width=0.23 \\textwidth, height=0.14 \\textheight]{baseline_success_plot.pdf}\\label{fig11_a}} \n    \\hfil\n    \\subfloat[Track3D]{\\includegraphics[width=0.23 \\textwidth, height=0.14 \\textheight]{pipeline_success_plot.pdf}\\label{fig11_b}}\n    \\hfil\n    \\subfloat[2D precision comparison]{\\includegraphics[width=0.23 \\textwidth, height=0.14 \\textheight]{comparing_2d_precision_plot.pdf}\\label{fig11_c}}\n    \\hfil\n    \\subfloat[3D precision comparison]{\\includegraphics[width=0.23 \\textwidth, height=0.14 \\textheight]{comparing_3d_precision_plot.pdf}\\label{fig11_d}}\n    \\caption{The performance comparison between 3D tracking baseline and Track3D. (a) and (b) are respectively success plots of 3D tracking baseline and Track3D with $AO^{2d}$, $AO^{bev}$, and $AO^{3d}$ metrics.}\n    \\label{fig10}\n    \\centering\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\begin{tabular}{cccc}\n       sequence 0001 & sequence 0022 & sequence 0031 & sequence 0052\n       \\\\\n       \\includegraphics[width=0.22 \\textwidth]{0001_000050.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0022_000050.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0031_000050.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0052_000050.pdf} \n       \\\\\n       \\includegraphics[width=0.22 \\textwidth]{0001_000100.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0022_000100.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0031_000100.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0052_000100.pdf} \n       \\\\\n       \\includegraphics[width=0.22 \\textwidth]{0001_000150.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0022_000150.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0031_000150.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0052_000150.pdf} \n       \\\\\n       \\includegraphics[width=0.22 \\textwidth]{0001_000200.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0022_000200.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0031_000200.pdf} & \n       \\includegraphics[width=0.22 \\textwidth]{0052_000200.pdf} \n   \\end{tabular}\n   \\caption{The tracking results of Track3D on 3D asteroid tracking test set.}\n   \\label{fig11}\n   \\centering\n\\end{figure*}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\subfloat[sequence 0001]{\\includegraphics[width=0.23 \\textwidth]{0001_001_trajectory.pdf}\\label{fig12_a}}\n    \\hfil\n    \\subfloat[sequence 0022]{\\includegraphics[width=0.23 \\textwidth]{0022_001_trajectory.pdf}\\label{fig12_b}} \n    \\hfil\n    \\subfloat[sequence 0031]{\\includegraphics[width=0.23 \\textwidth]{0031_001_trajectory.pdf}\\label{fig12_c}} \n    \\hfil\n    \\subfloat[sequence 0052]{\\includegraphics[width=0.23 \\textwidth]{0052_001_trajectory.pdf}\\label{fig12_d}}\n   \n    \\caption{3D tracking trajectories on 4 sequences in 3D asteroid tracking test set.}\n    \\label{fig12}\n    \\centering\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[2D success plots]{\\includegraphics[width=0.4\\textwidth]{monocular_trackers_success_plot.pdf} \\label{fig14_a}} \\hfil\n    \\subfloat[2D precision plots]{\\includegraphics[width=0.4\\textwidth]{monocular_trackers_2d_precision_plot.pdf} \\label{fig14_b}}\n    \\caption{The 2D tracking performance of 12 classic monocular trackers.} \n    \\label{fig14}\n    \\centering\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfloat[3D success plots]{\\includegraphics[width=0.4\\textwidth]{Track3Ds_success_plot.pdf} \\label{fig15_a}} \\hfil\n    \\subfloat[3D precision plots]{\\includegraphics[width=0.4\\textwidth]{Track3Ds_3d_precision_plot.pdf} \\label{fig15_b}} \\vfil\n    \\caption{The 3D tracking performance of Track3Ds under 12 different monocular trackers.} \n    \\label{fig15}\n    \\centering\n\\end{figure*}\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.45\\textwidth]{Track3D_with_different_2d_trackers.pdf}\n        \\caption{the connection between 3D tracking accuracy and 2D tracking performance, which demonstrates great generalization ability of our framework.}\n        \\label{fig16}\n    \\centering\n\\end{figure}\n\n\n\\section{Experiments \\label{section5}}\nIn this section, we have introduced a simple 3D tracking baseline algorithm as comparison and implemented extensive experiments with Track3D on 3D asteroid tracking dataset to demonstrate the advancement and effectiveness of our framework. We also find out that our framework with 2D-3D fusion strategy can not only handle 3D tracking challenge, but also make improvement on 2D tracking performance. In addition, valuable ablation study is also considered. All the experiments are carried out with Intel i9-9900k@3.60GHz CPU and Nvidia RTX 2080Ti GPU.\n\n\n\\subsection{Framework Performance}\nConsidering that there are a few researh results in the scope of 3-DoF tracking at present, therefore, we replace the A3BoxNet of our framework with minimum enclosing 3D bounding-box algorithm (i.e. computing the length of aixs-aligned 3D bounding-box by $\\max \\left\\{ P_{\\text{frustum}} \\right\\} - \\min \\left\\{ P_{\\text{frustum}} \\right\\}$) to realise a simple 3D tracking baseline as comparison. Both 2D and 3D tracking evaluation results of 3D tracking baseline are summarized at 3rd row in Table \\ref{table2}. And its success plots are shown in Fig. \\ref{fig11_a}. We find out that the 3D tracking baseline with 2D-3D fusion strategy improves 2D monocular tracking performance about 10.5\\%, however, it achieves quite poor performance in 3D space. \n\nThe overall performance of deep-learning based 3D tracking framework, also named as Track3D, is illustrated in Fig. \\ref{fig11_b}. It reaches excellent evaluation results on 3D asteroid tracking test set (0.669 $AO^{3d}$ and 0.570 $ACE^{3d}$) with high real-time performance (77.0 FPS). We visualize part of tracking results of Track3D in Fig. \\ref{fig11}, which demonstrates the effectiveness of our 3D tracking method. It clearly shows that our framework can estimate accurate 3D bounding-box even under extreme truncation (see 150-th frame of sequence 0001 in Fig. \\ref{fig11}). The 3D trajectory plots in Fig. \\ref{fig12} also intuitively show our framework predicts precise 3D object location, which greatly outperforms 3D tracking baseline. In addition, it can be seen in the first colum of Table \\ref{table2} that Track3D with 2D-3D fusion strategy can also make significant improvement on 2D tracking performance. \n\n\\subsection{Module Performance}\nExtensive experiments are implemented in this subsection that explore how two main modules of Track3D (i.e. 2D monocular tracker and A3BoxNet) make influences on final 3D tracking performance, which points the way to design effective 3D tracking framework for future work. \n\nFirstly, plenty of classic monocular trackers have been evaluated on left video sequences of 3D asteroid tracking dataset, such as SiamFC \\cite{bertinettoFullyconvolutionalSiameseNetworks2016}, SiamRPN \\cite{liHighPerformanceVisual2018}, ECO \\cite{danelljanECOEfficientConvolution2017}, Staple \\cite{bertinettoStapleComplementaryLearners2016}, KCF \\cite{henriquesHighSpeedTrackingKernelized2015}, DAT \\cite{posseggerDefenseColorbasedModelfree2015}, BACF \\cite{kianigaloogahiLearningBackgroundAwareCorrelation2017}, STRCF \\cite{liLearningSpatialTemporalRegularized2018}, MKCFup \\cite{tangHighSpeedTrackingMultiKernel2018}, CSRDCF \\cite{lukezicDiscriminativeCorrelationFilter2017}, CSK \\cite{henriquesExploitingCirculantStructure2012}, and MOSSE \\cite{bolmeVisualObjectTracking2010}. The evaluation results are illustrated in Fig. \\ref{fig14} which straightforward shows the accuracy of 12 monocular trackers are distributed in a large range from 0.375 to 0.746. SiamFC adopted in our framework only achieves intermediate 2D tracking performance in both accuracy and precision metrics.\n\nAnd then, we evaluate the whole framework under different monocular trackers respectively and plot 3D success and precision curves in Fig. \\ref{fig15_a} and \\ref{fig15_b}. We found that 3D evaluation curves become much dense comparing with 2D curves of Fig. \\ref{fig12}, which denotes that 2D monocular trackers just have slight influence on 3D tracking performance of our framework. In other words, Track3D can still work even based on poor 2D monocular tracker. We further plot the relationship between 2D tracking performance and framework accuracy in Fig. \\ref{fig16}, which intuitively demonstrates the generalization ability of Track3D.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4 \\textwidth]{a3boxnet_success_plot.pdf} \\vfil\n    \\includegraphics[width=0.4 \\textwidth]{a3boxnet_precision_plot.pdf}\n    \\caption{The performance of A3BoxNet with 1024 input points.}\n    \\label{fig9}\n    \\centering\n\\end{figure}\n\nMeanwhile, We evaluate amodal bounding-box estimation network, A3BoxNet, on 3D asteroid tracking dataset by randomly sampling 1024 points from frustum proposal. Its performance is also illustrated in Fig. \\ref{fig9}. It can be clearly seen that our A3BoxNet predicts high accurate axis-aligned bounding-box purely with partial object points. Furthermore, our network is able to run at 281.5 FPS, which totally satisfies the requirement of application on edge computing device. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{performance_on_point_nums.pdf}\n    \\caption{The pefromance of A3BoxNet under different numbers of point set. There is a contradictory between accuracy and precision of A3BoxNet.}\n    \\label{fig13}\n    \\centering\n\\end{figure}\n\n\\begin{table}[t]\n    \\centering\n    \\caption{The ablation study on one-hot category vector.}\n    \\label{table3}\n    \\begin{tabular}{cccc}\n        \\toprule\n        name & $AO^{3d}$ & $AO^{bev}$ & $ACE{3d}$ \\\\\n        \\midrule\n        A3BoxNet & \\textcolor{green}{0.721} & \\textcolor{green}{0.798} & 0.345 \\\\\n        without category & 0.715 & 0.792 & \\textcolor{green}{0.316} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\centering\n\\end{table}\n\nWe also study how the number of input points affects the performance of amodal axis-aligned bounding-box network. We retrain the A3BoxNet by randomly sampling different numbers of point set (e.g. 512, 1024, 2048, 3072, and 4096 points) from object point clouds in 3D asteroid tracking training set. And all the models are trained from scratch with 25 epoches and 32 batch size. The evaluation results are plotted in Fig. \\ref{fig13}, which clearly shows accuracy and precision of A3BoxNet have contradictory with the number of input points. Once the accuracy of model increasing, the corresponding precision will decrease. \n\nBesides, the influence of object category on A3BoxNet is further studied. We remove one-hot vector of object category in center regression network and amodal box estimation network, and retrain A3BoxNet from scratch with 1024 point sets. The performance comparison between original A3BoxNet and A3BoxNet without category information is summarized in Table \\ref{table3}, which proves the object category can make a slight improvement on the 3D accuracy of A3BoxNet.\n\n\n\\section{Conclusion \\label{section6}}\nIn this work, we construct the first large-scale 3D asteroid tracking dataset, which involves  148,500 binocular images, depth maps, and point clouds. All the 2D and 3D annotations are automatically generated, which greatly guarantees the quality of tracking dataset and reduce the cost of data collection. The 3D asteroid tracking dataset will be public on website (\\url{http:\/\/aius.hit.edu.cn\/12920\/list.htm}). Meanwhile, we propose a deep-learning based 3D visual tracking framework, Track3D, which mainly consists of classic 2D monocular tracker and a novel light-weight amodal axis-aligned bounding-box network. The state-of-the-art 3D tracking performance and great generalization ability of our framework have been demonstrated by sufficient experiments. We also find that Track3D with 2D-3D tracking fusion strategy also makes improvement on 2D tracking performance. In future work, We will further apply our Track3D method to normoal cases like automous driving and robot picking.\n\n\\appendices\n\\section{The perspective camera matrix \\label{appendix1}}\nIn section \\ref{section3}, we have mentioned that camera matrix is very important for Track3D to extract frustum proposal. However, physics engine V-rep only provides perspective angle $(\\alpha_x, \\alpha_y)$ and resolution $(W, H)$. To this end, we derive the camera matrix by perspective projection principle, which is also illustrated in Fig. \\ref{fig3}. \n\nSuppose that the virtual focus of perspective camera in both x and y axes are $(f_x, f_y)$, the size of image plane is $(w, h)$, and an object point $P=(X, Y, Z)$ in camera coordinate system is projected at $p=(x_i, y_i)$ in image plane. Fig. \\ref{fig3} clearly shows that:\n\\begin{equation}\n    \\frac{X}{Z} = \\frac{x_i}{f_x} \\label{eq_9}\n\\end{equation}\nand, \n\\begin{equation}\n    \\frac{w\/2}{f_x} = \\tan\\left( \\frac{\\alpha_x}{2}\\right) \\label{eq_10}\n\\end{equation}\nMeanwhile, the transformation from image coordinate system to pixel coordinate system in x axis is formulated as:\n\\begin{align}\n    u &= (x_i + \\frac{w}{2}) \\cdot \\frac{W}{w} \\label{eq_11} \\\\ \n    \\nonumber &=x_i \\cdot \\frac{W}{w} + \\frac{W}{2} \n\\end{align}\nSubstitute Eq. \\ref{eq_9} and \\ref{eq_10} into Eq. \\ref{eq_11}, it can be obtained:\n\\begin{equation}\n    u = \\frac{W}{ 2 \\tan(\\frac{\\alpha_x}{2})Z} X+  \\frac{W}{2} \\label{eq_12}\n\\end{equation}\nin which, the parameter $f_x$ is eliminated. Similarly, the transformation in y-axis direction from camera coordinate system to pixel coordinate system is also obtained:\n\\begin{equation}\n    v = \\frac{H}{ 2 \\tan(\\frac{\\alpha_y}{2})Z} Y+  \\frac{H}{2} \\label{eq_13}\n\\end{equation}\n\nWe further rewrite Eq. \\ref{eq_12} and \\ref{eq_13} in homogeneous matrix form:\n\\begin{align}\n    \\begin{bmatrix}\n       u \\\\\n       v \\\\\n       1\n    \\end{bmatrix}=\n    \\begin{bmatrix}\n       \\frac{W}{ 2 \\tan(\\frac{\\alpha_x}{2})Z} & 0 & \\frac{W}{2Z} \\\\\n       0 &   \\frac{H}{ 2 \\tan(\\frac{\\alpha_y}{2})Z} & \\frac{H}{2Z}\\\\\n       0 & 0 & 1\/Z\n    \\end{bmatrix} \n    \\begin{bmatrix}\n       X \\\\\n       Y \\\\\n       Z\n    \\end{bmatrix} \\label{eq_14}\n\\end{align}\nTo eliminate the $Z$ variable in the transformation matrix, we multiply both sides of Eq. \\ref{eq_14} by $Z$:\n\\begin{align}\n    Z\\begin{bmatrix}\n       u \\\\\n       v \\\\\n       1\n    \\end{bmatrix}=\n    \\begin{bmatrix}\n       \\frac{W}{2 \\tan(\\alpha_x\/2)} & 0 & \\frac{W}{2} \\\\\n       0 &   \\frac{H}{ 2 \\tan(\\alpha_y\/2)} & \\frac{H}{2}\\\\\n       0 & 0 & 1\n    \\end{bmatrix} \n    \\begin{bmatrix}\n       X \\\\\n       Y \\\\\n       Z\n    \\end{bmatrix} \n\\end{align}\n\nBecause we set left camera coordinate system as reference frame, the camera matrix of left perspective camera can be formulated as:\n\\begin{align}\n    M^L =  \\begin{bmatrix}\n        \\frac{W}{2 \\tan(\\alpha_x\/2)} & 0 & \\frac{W}{2} & 0\\\\\n        0 &   \\frac{H}{ 2 \\tan(\\alpha_y\/2)} & \\frac{H}{2} & 0\\\\\n        0 & 0 & 1 & 0\n     \\end{bmatrix} \n\\end{align}\n\n\n\\section{Traning objectives \\label{appendix2}}\nIn this work, we utilize a joint loss function $\\mathcal{L}_{joint}$ to optimize A3BoxNet:\n\\begin{equation}\n    \\mathcal{L}_{joint} = \\mathcal{L}_{center-net} + \\mathcal{L}_{box-net}\n\\end{equation}\nwhere, $\\mathcal{L}_{center-net}$ adopt huber loss function:\n\\begin{equation}\n    \\mathcal{L}_{center-net}=\n    \\left\\{\n        \\begin{array}{ll}\n        0.5\\alpha^{2}, & \\alpha<1 \\\\\n        \\alpha - 0.5,  & \\text {otherwise}\n        \\end{array}\n    \\right.\n\\end{equation}\nin which $\\alpha = \\left\\| \\hat C - (\\bar C + \\Delta C_1)\\right\\|_2$, $\\hat C$ is 3D center label, $\\bar c$ is the centroid of points in frustum proposal, and $\\Delta C_1$ is the prediction of center regression network. \n\nIn addition, \n\\begin{equation}\n    \\mathcal{L}_{box-net} = \\mathcal{L}_{center\\_res} +  \\mathcal{L}_{size\\_cls} + \\mathcal{L}_{size\\_res}\n\\end{equation}\nwhere $\\mathcal{L}_{size\\_cls}$ utilizes softmax cross entropy loss function:\n\\begin{equation}\n    \\mathcal{L}_{size\\_cls} = -\\sum_{i=1}^{N_{size}} \\hat{y}_i \\cdot \\log \\left( \\frac{e^{y_i}}{\\sum_{j=1}^{N_{size}} e^{y_j}}\\right)\n\\end{equation}\nin which $\\hat{y}$ is $N_{size}$ dimensional one-hot vector of size category label, $y$ is the partial outputs of amodal box estimation network, of which dimension is also $N_{size}$. \n\nFurthermore, $\\mathcal{L}_{center\\_res}$ and $\\mathcal{L}_{size\\_res}$ both use huber loss function. $\\mathcal{L}_{center\\_res}$ is formulated as:\n\\begin{equation}\n    \\mathcal{L}_{center-net}=\n    \\left\\{\n        \\begin{array}{ll}\n        0.5\\beta^{2}, & \\beta<2 \\\\\n        2(\\beta - 1),  & \\text {otherwise}\n        \\end{array}\n    \\right.\n\\end{equation}\nin which, $\\beta = \\left\\| \\hat C - (\\bar C + \\Delta C_1 + \\Delta C_2)\\right\\|_2$, $\\Delta C_2$ is 3D center residuals predicted by amodal box estimation network. And $\\mathcal{L}_{size\\_res}$ is as follows:\n\\begin{equation}\n    \\mathcal{L}_{center-net}=\n    \\left\\{\n        \\begin{array}{ll}\n        0.5 \\gamma^{2}, & \\gamma<1 \\\\\n        \\gamma - 0.5,  & \\text {otherwise}\n        \\end{array}\n    \\right.\n\\end{equation}\nwhere $\\gamma = \\left\\| \\hat R - r * \\max \\left\\{\\hat S\\right\\}\\right\\|_2$, $\\hat S$ and $\\hat R$ are the size and size residual label, respectively. $r$ is normalized size residual corresponding to the size category predicted by amodal box estimation network. \n\n\\section*{Acknowledgment}\nThis work was kindly supported by the National Key R\\&D Program of China through grant 2019YFB1312001.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{SecIntro}\n\nVarious approaches have been proposed for simulating the propagation of elastic waves with free boundaries. The first approach is based on variational methods, as done in finite elements \\citep{DAY77}, spectral finite elements \\citep{KOMATITSCH98} and discontinuous Galerkin \\citep{BENJEMAA07}. These methods provide a fine geometrical description of boundaries by adapting the mesh to the boundaries. Boundary conditions are accounted for weakly by the underlying variational formulation. However, a grid-generating tool is required, and small time steps may result from the smallest geometrical elements and from the stability condition. The SAT methods based on energy estimates \\citep{CARPENTER94} avoid these limitations by introducing Cartesian grids and give time-stable high-order schemes with interfaces. However and up to our knowledge, these methods have not been applied so far to elastodynamics with free boundaries. \n\nThe second approach used in this context is based on the strong form of elastodynamics, as done in finite differences and spectral methods \\citep{TESSMER94}. In seismology, finite differences are usually implemented on staggered Cartesian grids, either with completely staggered stencils (CSS) or with the recently developed partially staggered stencils (PSS). With CSS, the velocity and stress components are distributed between different node positions \\citep{VIRIEUX86}. With PSS, all the velocity components are computed at a single node, as are the stress components, although the latter are shifted by half a node in two separate grids. Second-order \\citep{SAENGER00, SAENGER04} and fourth-order \\citep{BOHLEN03,CRUZ04} spatially-accurate PSS have been developed; for further discussion, we denote them PSS-2 and PSS-4, respectively. Unlike variational methods, finite differences require special care to incorporate the free boundary conditions strongly. There exist two main strategies for this purpose:\n\\begin{enumerate}\n\\item First, the boundaries can be taken into account implicitly by adjusting the physical parameters  locally \\citep{KELLY76,VIRIEUX86,MUIR92}. The best-known implicit approach is the so-called \\textit{vacuum method} \\citep{ZAHRADNIK95,GRAVES96,MOCZO02,GELIS05}. For instance, the vacuum method applied to PSS involves setting the elastic parameters in the vacuum to zero, and using a small density value in the first velocity node in the vacuum to avoid a division by zero. However, this easy-to-implement method gives at best second-order spatial accuracy. In addition, a systematic numerical study has shown that the accuracy of the solution decreases dramatically when the angle between the boundary and the meshing increases \\citep{BOHLEN06}. Lastly, applying the vacuum method sometimes gives rise to instabilities: see for instance PSS-4 \\citep{BOHLEN03}.  \n\\item A second idea is to explicitly change the scheme near the boundaries \\citep{KELLY76}. The best-known explicit approach is the so-called \\textit{image method}, which was developed for dealing with flat boundaries to fourth-order accuracy \\citep{LEVANDER88} and then extended to variable topographies \\citep{JIH88,ROBERTSSON96,ZHANGW06}. However image methods require a fine grid to reduce the spurious diffractions up to an acceptable level. To avoid this spatial oversampling, various techniques have been proposed, such as grid refinement in the vicinity of the boundary \\citep{RODRIGUES93} or adjusted finite-difference approximations: see \\citep{MOCZO07} for a review on these subjects.\n\\end{enumerate}\n\nThe aim of this paper is to present a finite-difference approach accounting for free boundaries without introducing the aforementioned drawbacks of the vacuum and image methods. The requirements are as follows: smooth arbitrary-shaped boundaries must be treated as simply as straight boundaries; the accuracy of the method must not depend on the position of the boundary relative to the meshing; and lastly, the computations must be stable even with very long integration times. We establish that these requirements can be met by applying an explicit approach involving fictitious values of the solution in the vacuum. In previous studies, interface problems in the context of  elastodynamics were investigated in a similar way \\citep{PIRAUX01,LOMBARD04,LOMBARD06}. The fictitious values are high-order Taylor expansions of the boundary values of the solution. Estimating these boundary values involves some mathematical background, in order to compute the high-order boundary conditions; to determine a minimal set of independent boundary values; lastly, to perform a least-square numerical estimation of this minimal set. To help the reader, subroutines in FORTRAN are proposed freely at the web page {\\tt http:\/\/w3lma.cnrs-mrs.fr\/\\textasciitilde MI\/Software\/}. These subroutines enable a straightforward implementation of the algorithms detailed in the present paper.\n\nThe disadvantage here is that the above requirements cannot be fully satisfied if staggered-grid schemes are used. Single-grid finite-difference schemes are therefore chosen, where all the unknowns are computed at the same grid nodes. Our numerical experiments are based on the high-order ADER schemes which are widely used in aeroacoustics \\citep{MUNZ05}. Although these schemes are not yet widely used in the field of seismology \\citep{DUMBSER06}, they have also great qualities because of their accuracy and their stability properties: using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths gives highly accurate results. Moreover, on Cartesian grids, these methods do not require much more computational memory than staggered-grid schemes.\n\nThis paper is organized as follows. Section 2 deals with the continuous problem: the high-order boundary conditions and compatibility conditions are stated. These conditions are useful for handling the discrete problem presented in section 3, where the focus is on obtaining fictitious values of the solution in the vacuum. In section 4, numerical experiments confirm the efficiency of this method in the case of various topographies. In section 5, conclusions are drawn and some prospects suggested.\n\n\\section{The continuous problem}\\label{SecContinuous}\n\n\\subsection{Framework}\\label{SubSecFrame}\n\nLet us consider a solid $\\Omega$ separated from the vacuum by a boundary $\\Gamma$ (Figure \\ref{FigPatate}). The configuration is in-plane and two-dimensional, with a horizontal axis $x$ and a vertical axis $z$ pointing respectively rightward and downward. The boundary $\\Gamma$ is described by a parametric expression $(x(\\tau),\\,z(\\tau))$ where the parameter $\\tau$ describes the sampling of the boundary. The tangential vector and the normal vector are $\\mitbf{t}=\\,^T(x^{'}(\\tau),\\,z^{'}(\\tau))$ and $\\mitbf{n}=\\,^T(-z^{'}(\\tau),\\,x^{'}(\\tau))$, respectively, with $x^{'}(\\tau)=\\frac{d\\,x}{d\\,\\tau}(\\tau)$, $z^{'}(\\tau)=\\frac{d\\,z}{d\\,\\tau}(\\tau)$, and $T$ refers to the transposed vector. We assume the spatial derivatives at any point of the boundary to be available, as specified below. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.8]{Patate.eps}\n\\caption{{\\it Boundary $\\Gamma$ between a solid $\\Omega$ and vacuum}.}\n\\label{FigPatate}\n\\end{center}\n\\end{figure} \n\nThe solid $\\Omega$ is assumed to be linearly elastic, isotropic, and to have the following constant physical parameters: the density $\\rho$ and the Lam\\'e coefficients $\\lambda$, $\\mu$. The P- and S-wave velocities are $c_p=\\sqrt{(\\lambda+2\\,\\mu)\/\\rho}$ and $c_s=\\sqrt{\\mu\/\\rho}$. A velocity-stress formulation is adopted, hence the unknowns are the horizontal and vertical components of the elastic velocity $\\mitbf{v}=\\,^T(v_x,\\,v_z)$, and the independent components of the elastic stress tensor $\\mitbf{\\sigma}=\\,^T(\\sigma_{xx},\\,\\sigma_{xz},\\,\\sigma_{zz})$. Setting\n$$\n\\begin{array}{l}\n\\displaystyle\n\\mitbf{A}=\n\\left(\n\\begin{array}{ccccc}\n0 & 0 & \\hspace{0.1cm}1\/\\rho & 0 & 0\\\\\n0 & 0 & 0 & 1\/\\rho & 0\\\\\n\\lambda+2\\,\\mu & 0 & 0 & 0 & 0\\\\\n0 & \\mu & 0 & 0 & 0 \\\\\n\\lambda & 0 & 0 & 0 & 0\n\\end{array}\n\\right),\\\\\n\\\\\n\\displaystyle\n\\mitbf{B}=\n\\left(\n\\begin{array}{ccccc}\n0 & 0 & 0 & \\hspace{0.1cm}1\/\\rho & 0\\\\\n0 & 0 & 0 & 0 & 1\/\\rho\\\\\n0 & \\lambda & 0 & 0 & 0\\\\\n\\mu & 0 & 0 & 0 & 0\\\\\n0 & \\lambda+2\\mu & 0 & 0 & 0\n\\end{array}\n\\right), \n\\end{array}\n$$ \nthe solution $\\mitbf{U}=\\,^T(v_x,\\,v_z,\\,\\sigma_{xx},\\,\\sigma_{xz},\\,\\sigma_{zz})$ satisfies the first-order hyperbolic system \\citep{VIRIEUX86}\n\\begin{equation}\n\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,t}\\,\\mitbf{U}=\\mitbf{A}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,x}\\,\\mitbf{U}+\\mitbf{B}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,z}\\,\\mitbf{U}.\n\\label{LC}\n\\end{equation}\n\n\\subsection{High-order boundary conditions}\\label{SubSecBC}\n\nAt any point $P(\\tau)$ on the free surface $\\Gamma$ (Figure \\ref{FigPatate}), the stress tensor satisfies the homogeneous Dirichlet conditions $\\mitbf{\\sigma}.\\mitbf{n}=\\mitbf{0}$. These zero-th order boundary conditions are written compactly\n\\begin{equation}\n\\mitbf{L}^0(\\tau)\\,\\mitbf{U}^0(x(\\tau),\\,z(\\tau),\\,t)=\\mitbf{0},\n\\label{L0U0}\n\\end{equation}\nwhere $\\mitbf{U}^0$ is the limit value of $\\mitbf{U}$ at $P$ and $\\mitbf{L}^0$ is the matrix\n$$\n\\mitbf{L}^0(\\tau)=\n\\left(\n\\begin{array}{ccccc}\n0 & 0 & -z^{'}(\\tau) & x^{'}(\\tau) & 0 \\\\\n0 & 0 & 0 & -z^{'}(\\tau) & x^{'}(\\tau) \n\\end{array}\n\\right).\n$$\nFrom now on, the dependence on $\\tau$ is generally omitted. To determine the boundary conditions satisfied by the first-order spatial derivatives of $\\mitbf{U}$, two tasks are performed. First, the zeroth-order boundary conditions (\\ref{L0U0}) are differentiated in terms of $t$. The time derivative is replaced by spatial derivatives using the conservation laws (\\ref{LC}), which gives\n\\begin{equation}\n\\mitbf{L}^0\\left(\\mitbf{A}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,x}\\,\\mitbf{U}^0+\\mitbf{B}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,z}\\,\\mitbf{U}^0\\right)=\\mitbf{0}.\n\\label{L0t}\n\\end{equation}\nSecondly, the zeroth-order boundary conditions (\\ref{L0U0}) are differentiated in terms of the parameter $\\tau$ describing $\\Gamma$. The chain-rule gives\n\\begin{equation}\n\\left(\\frac{\\textstyle d}{\\textstyle d\\,\\tau}\\,\\mitbf{L}^0\\right)\\,\\mitbf{U}^0+\\mitbf{L}^0\\left(x^{'}\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,x}\\,\\mitbf{U}^0+z^{'}\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,z}\\,\\mitbf{U}^0\\right)=\\mitbf{0}.\n\\label{L0tau}\n\\end{equation}\nSince the matrix $d\\,\\mitbf{L}^0\/d\\,\\tau$ in (\\ref{L0tau}) involves $x^{''}$ and $z^{''}$, it accounts for the curvature of $\\Gamma$ at $P$. Setting the block matrix\n$$\n\\mitbf{L}^1=\n\\left(\n\\begin{array}{ccc}\n\\mitbf{L}^0  &  \\mitbf{0}  &  \\mitbf{0}\\\\\n[3pt]\n\\mitbf{0}    &  \\mitbf{L}^0\\,\\mitbf{A}  &  \\mitbf{L}^0\\,\\mitbf{B}  \\\\\n\\displaystyle\n\\frac{\\textstyle d}{\\textstyle d\\,\\tau}\\,\\mitbf{L}^0 & x^{'}\\mitbf{L}^0 & z^{'}\\mitbf{L}^0\n\\end{array}\n\\right),\n$$\nequations (\\ref{L0U0}), (\\ref{L0t}) and (\\ref{L0tau}) give the boundary conditions up to the first-order \n$$\n\\mitbf{L}^1\\,\\mitbf{U}^1=\\mitbf{0},\n$$\nwith \n$$\n\\mitbf{U}^1=\\lim_{M\\in \\Omega \\rightarrow P}\n\\,^T\\left(\n^T\\mitbf{U},\n\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,x}\\,^T\\mitbf{U},\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,z}\\,^T\\mitbf{U}\\right).\n$$\nLet $k\\geq 1$ be an integer whose value will be discussed in section \\ref{SecDiscrete}. To get the boundary conditions up to the $k$-th order, one deduces from (\\ref{L0U0}) \n\\begin{equation}\n\\frac{\\textstyle \\partial^k}{\\textstyle \\partial\\,\\tau^{k-\\alpha}\\,\\partial\\,t^\\alpha}\\,\\mitbf{L}^0\\,\\mitbf{U}^0=\\mitbf{0},\\qquad \\alpha=0,...,k.\n\\label{DTauDt}\n\\end{equation}\nThe $\\tau$-derivatives are replaced by spatial derivatives by applying $(k-\\alpha)$-times the chain rule. The $t$-derivatives are replaced by spatial derivatives by injecting $\\alpha$-times the conservation laws (\\ref{LC}). The boundary conditions so-obtained up to the $k$-th order can be written compactly\n\\begin{equation}\n\\mitbf{L}^k\\,\\mitbf{U}^k=\\mitbf{0},\n\\label{LkUk}\n\\end{equation}\nwith\n\\begin{equation}\n\\displaystyle\n\\mitbf{U}^k=\\lim_{M\\in \\Omega \\rightarrow P}\n\\,^T\\left(\n^T\\mitbf{U},\n...,\\,\n\\frac{\\textstyle \\partial^\\alpha}{\\textstyle \\partial\\, x^{\\alpha-\\beta}\\,\\partial\\,z^\\beta}\\,^T\\mitbf{U},\n...,\\,\n\\frac{\\textstyle \\partial^k}{\\textstyle \\partial\\,z^k}\\,^T\\mitbf{U}\n\\right),\n\\label{Uk}\n\\end{equation}\nwhere $\\alpha=0,...,\\,k$ and $\\beta=0,...,\\,\\alpha$. The vector $\\mitbf{U}^k$ has $n_v=5(k+1)\\,(k+2)\/2$ components. $\\mitbf{L}^k$ is a $n_l \\times n_v$ matrix, with $n_l=(k+1)\\,(k+2)$. This matrix  involves the successive derivatives of the curvature of $\\Gamma$ at $P$. Computing $\\mitbf{L}^k$ with $k>2$ is a tedious task, which can be greatly simplified by using computer algebra tools. \n\n\\subsection{Compatibility conditions}\\label{SubSecCC}\n\nThe second spatial derivatives of stress components are linked together by the compatibility condition of Barr\\'e-de Saint Venant \\citep{LOVE}\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle\n\\frac{\\textstyle \\partial^2 \\,\\sigma_{xz}}{\\textstyle \\partial \\,x\\,\\partial\\,z}\n=\\alpha_2\\,\\frac{\\textstyle \\partial^2 \\,\\sigma_{xx}}{\\textstyle \\partial \\,x^2}\n+\\alpha_1\\,\\frac{\\textstyle \\partial^2 \\,\\sigma_{zz}}{\\textstyle \\partial \\,x^2}\n+\\alpha_1\\,\\frac{\\textstyle \\partial^2 \\,\\sigma_{xx}}{\\textstyle \\partial \\,z^2}\n+\\alpha_2\\,\\frac{\\textstyle \\partial^2 \\,\\sigma_{zz}}{\\textstyle \\partial \\,z^2},\n\\end{array}\n\\label{Barre}\n\\end{equation}\nwith\n$$\n\\alpha_1=\\frac{\\textstyle \\lambda+2\\,\\mu}{\\textstyle 4\\,(\\lambda+\\mu)},\\qquad \n\\alpha_2=-\\frac{\\textstyle \\lambda}{\\textstyle 4\\,(\\lambda+\\mu)}.\n$$\nThis compatibility condition is a necessary and sufficient condition for the strain tensor to be symmetrical. If $k\\geq 2$, it can be differentiated $(k-2)$-times in terms of $x$ and $z$. With $k\\geq 2$, one obtains $n_c=k\\,(k-1)\/2$ relations; with $k<2$, $n_c=0$. Unlike the boundary conditions, these compatibility conditions are satisfied everywhere in $\\Omega$: in particular, they are satisfied at $P$ on $\\Gamma$. The vector of boundary values $\\mitbf{U}^k$ can therefore be expressed in terms of a shorter vector $\\hat{\\mitbf{U}}^k$ with $n_v-n_c$ independent components\n\\begin{equation}\n\\mitbf{U}^k=\\mitbf{G}^k\\,\\hat{\\mitbf{U}}^k. \n\\label{MatG}\n\\end{equation}\nAn algorithm for building the $n_v\\times (n_v-n_c)$ matrix $\\mitbf{G}^k$ is given in \\citet{LOMBARD06}. \n\n\\section{The discrete problem}\\label{SecDiscrete}\n\n\\subsection{Numerical scheme}\\label{SubSecScheme}\n\nTo integrate the hyperbolic system (\\ref{LC}), we introduce a single Cartesian lattice of grid points: $(x_i,z_j,t_n)=(i\\,h,j\\,h,\\,n\\,\\Delta\\,t)$, where $h$ is the mesh spacing and $\\Delta\\,t$ is the time step. Unlike with staggered grids, all the unknowns are computed at the same grid nodes. The approximation $\\mitbf{U}_{i,j}^n$ of $\\mitbf{U}(x_i,z_j,t_n)$ is computed using any explicit, two-step, and spatially-centred finite-difference scheme. A review of the huge body of literature on finite-differences is given in \\citet{LEV90} and \\citet{MOCZO07}. \n\nHere we propose to use ADER schemes, that allow to reach easily arbitrary high-order of time and space accuracy \\citep{MUNZ05}. On Cartesian grids, these finite-volume integration schemes originally developed for aeroacoustic applications are equivalent to finite-difference Lax-Wendroff-type integration schemes \\citep{LORCHER06}. In the numerical experiments described in section \\ref{SecNum}, we use a fourth-order ADER integration scheme. This scheme is stable under the Courant-Friedrichs-Lewy (CFL) condition $c_p\\,\\Delta\\,t\/h \\leq 0.9$ in 2D; as usually with single-grid schemes, it is slightly dissipative \\citep{MUNZ05}. \n\nMany other single-grid schemes can be used in this context. In particular, the method described in the next subsections has been successfully combined with flux-limiter schemes \\citep{LEV90} and with the standard second-order Lax-Wendroff scheme. Difficulties have been encountered with dissipative-free schemes based on centred staggered-grid finite-difference schemes, as we will see in section \\ref{SubSecStag}.\n\n\\subsection{Use of fictitious values}\\label{SubSecExtra1}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{SolMod.eps}\n\\caption{\\textit{Determination of the fictitious value $\\mitbf{U}_{IJ}^*$ required for time-marching at neighboring grid nodes. P is the orthogonal projection of $(x_I,\\,z_J)$ on $\\Gamma$. The $n_p$ grid nodes in $\\Omega$ and inside the circle ${\\cal C}$ centred at $P$ with a radius $d$ are denoted by {\\bf +}}.}\n\\label{FigSolMod}\n\\end{center}\n\\end{figure} \n\nTime-marching at grid-points where the stencil crosses $\\Gamma$ requires fictitious values of the solution in the vacuum, which have to be determined. The question arises as to how to compute, for instance, the fictitious value $\\mitbf{U}_{I,J}^*$ at the grid point $(x_I,\\,z_J)$ in the vacuum, as sketched in Figure \\ref{FigSolMod}. Let $P(\\tau)$ be the orthogonal projection of $(x_I,\\,z_J)$ on $\\Gamma$, with coordinates $(x_P=x(\\tau),\\,z_P=z(\\tau))$. At any grid point $(x_i,\\,z_j)$, we denote\n$$\n\\mitbf{\\Pi}_{i,j}^k=\\left(\\mitbf{I}_5,\n...,\n\\frac{\\textstyle (x_i-x_P)^{\\alpha-\\beta}(z_j-z_P)^\\beta}{\\textstyle (\\alpha-\\beta)\\,!\\,\\beta\\,!}\\mitbf{I}_5,\n...,\n\\frac{\\textstyle (z_j-z_P)^k}{\\textstyle k\\,!}\\mitbf{I}_5\\right)\n$$\nthe $5\\times n_v$ matrix containing the coefficients of $k$-th order Taylor expansions in space at $P$, where $\\mitbf{I}_5$ is the $5\\times5$ identity matrix, $\\alpha=0,...,\\,k$, and $\\beta=0,...,\\,\\alpha$. The fictitious value $\\mitbf{U}_{I,J}^*$ is defined as the Taylor-like extrapolation \n\\begin{equation}\n\\mitbf{U}_{I,J}^*=\\mitbf{\\Pi}_{I,J}^k\\,\\mitbf{U}^k,\n\\label{SolMod0} \n\\end{equation}\nwhere $\\mitbf{U}^k$ defined by (\\ref{Uk}) still remains to be estimated. \n\n\\subsection{Reduced vector of boundary values}\\label{SubSecSVD}\n\nBefore determining $\\mitbf{U}^k$ in (\\ref{SolMod0}), we first reduce the number of independent components it contains. The expressions obtained in section \\ref{SecContinuous} are used for this purpose. The linear homogeneous system following from (\\ref{LkUk}) and (\\ref{MatG}) is\n\\begin{equation}\n\\mitbf{L}^k\\,\\mitbf{G}^k\\,\\hat{\\mitbf{U}}^k=\\mitbf{0}.\n\\label{LGU}\n\\end{equation}\nThis system has fewer equations ($n_l$) than unknowns ($n_v-n_c$). It therefore has an infinite number of possible solutions that constitute a space with the dimension $n_v-n_c-n_l$. Let $\\mitbf{K}_{L^kG^k}$ be a $(n_v-n_c)\\times(n_v-n_c-n_l)$ matrix containing the basis vectors of the kernel of $\\mitbf{L}^k\\,\\mitbf{G}^k$. The general solution of (\\ref{LGU}) is therefore\n\\begin{equation}\n\\hat{\\mitbf{U}}^k=\\mitbf{K}_{L^kG^k} \\overline{\\mitbf{U}}^k,\n\\label{Kernel}\n\\end{equation}\nwhere the $n_v-n_c-n_l$ components of $\\overline{\\mitbf{U}}^k$ are real numbers. Injecting (\\ref{Kernel}) into (\\ref{MatG}) gives\n\\begin{equation}\n\\mitbf{U}^k=\\mitbf{G}^k\\mitbf{K}_{L^kG^k}\\overline{\\mitbf{U}}^k.\n\\label{SVD}\n\\end{equation}\nThe computation of $\\mitbf{K}_{L^kG^k}$ is a key point. For this purpose, we use a classical linear algebra tool: singular value decomposition of $\\mitbf{L}^k\\,\\mitbf{G}^k$. Technical details can be found in the Appendix A of \\citet{LOMBARD04}.\n\n\\subsection{Computation of fictitious values}\\label{SubSecExtra2}\n\nLet us now consider the $n_p$ grid points of $\\Omega$ in the circle ${\\cal C}$ centred at $P$ with a radius $d$; for instance, $n_p=8$ in Figure \\ref{FigSolMod}. At these points, we write the $k$-th order Taylor expansion in space of the solution at $P$, and then we use the expression (\\ref{SVD}). This gives\n\\begin{equation}\n\\begin{array}{lll}\n\\mitbf{U}(x_i,\\,z_j,\\,t_n)&=& \\displaystyle \\mitbf{\\Pi}_{i,j}^k\\mitbf{U}^k+\\mitbf{O}(h^{k+1}),\\\\\n&&\\\\\n&=&\\displaystyle \\mitbf{\\Pi}_{i,j}^k\\mitbf{G}^k\\mitbf{K}_{L^kG^k}\\overline{\\mitbf{U}}^k+\\mitbf{O}(h^{k+1}).\n\\end{array}\n\\label{Taylor}\n\\end{equation}\nThe set of $n_p$ equations (\\ref{Taylor}) is written compactly via a $5\\,n_p\\times (n_v-n_c-n_l)$ matrix $\\mitbf{M}$\n\\begin{equation}\n\\left(\\mitbf{U}(.,\\,t_n)\\right)_{\\cal C}=\\mitbf{M}\\overline{\\mitbf{U}}^k+\\mitbf{O}(h^{k+1}),\n\\label{MatM}\n\\end{equation}\nwhere $\\left(\\mitbf{U}(.,\\,t_n)\\right)_{\\cal C}$ is the vector containing the exact values of the solution at the grid nodes of $\\Omega$ inside ${\\cal C}$. These exact values are replaced by the known numerical values $\\left(\\mitbf{U}^n\\right)_{\\cal C}$, and Taylor rests are removed. From now on, numerical values and exact values of the fields are used indiscriminately. The discrete system thus obtained is overdetermined (see the remark (i) about $d$ and typical values of $n_p$ in subsection \\ref{SubSecOverview}). We now compute its least-squares solution\n\\begin{equation}\n\\overline{\\mitbf{U}}^k=\\mitbf{M}^{-1}\\left(\\mitbf{U}^n\\right)_{\\cal C},\n\\label{Trace}\n\\end{equation}\nwhere the $(n_v-n_c-n_l)\\times 5\\,n_p$ matrix $\\mitbf{M}^{-1}$ denotes the pseudo-inverse of $\\mitbf{M}$. From (\\ref{SolMod0}), (\\ref{SVD}) and (\\ref{Trace}), the fictitious value in the vacuum at $(x_I,\\,z_J)$ is\n\\begin{equation}\n\\begin{array}{lll}\n\\mitbf{U}_{I,J}^*&=& \\displaystyle \\mitbf{\\Pi}_{I,J}^k\\,\\mitbf{G}^k\\mitbf{K}_{L^kG^k}\\mitbf{M}^{-1}\\left(\\mitbf{U}^n\\right)_{\\cal C}\\\\\n&&\\\\\n&=& \\displaystyle \\mitbf{\\Lambda}_{I,J}\\left(\\mitbf{U}^n\\right)_{\\cal C}.\n\\end{array}\n\\label{Extrapolator}\n\\end{equation}\nThe $5\\times 5\\,n_p$ matrix $\\mitbf{\\Lambda}_{I,J}$ is called the \\textit{extrapolator} at $(x_I,\\,z_J)$. The fictitious values have no clear physical meaning. They only allow, by interpolation with numerical values inside $\\Omega$, to recover the high-order Dirichlet conditions (\\ref{Uk}).\n\n\\subsection{Comments and practical details}\\label{SubSecOverview}\n\nThe extrapolation method described in section \\ref{SubSecExtra2} has to be applied at each grid point $(I,\\,J)$ in the vacuum where a fictitious value is required for the time-marching procedure. Useful comments are proposed about this method:\n\n\\begin{enumerate}\n\\item The radius $d$ of ${\\cal C}$ must ensure that the number of equations in (\\ref{MatM}) is greater than the number of unknowns: \n\\begin{equation}\n\\varepsilon(k,\\,d)=\\frac{\\textstyle 5\\,n_p}{\\textstyle n_v-n_c-n_l}\\geq 1.\n\\label{Theta}\n\\end{equation}\nNo theoretical results are available about the optimal value of $\\varepsilon$. However, numerical studies have shown that a definite overestimation ensures long-term stability: typically, $\\varepsilon \\approx 4$. Various strategies can be used to ensure (\\ref{Theta}), such as an adaptative choice of $d$ depending on the local geometry of $\\Gamma$ at $P$. Here we adopt a simpler strategy consisting in using a constant radius $d$. With $k=3$, numerical experiments have shown that $d=3.2\\,h$ is a good candidate for this purpose. In this case, one typically obtains  $n_p \\approx 15$.\n\n\\item  Since the boundary conditions do not vary with time, the extrapolators $\\mitbf{\\Lambda}_{I,J}$ in (\\ref{Extrapolator}) can be computed and stored during a pre-processing step. At each time step, only small matrix-vector products are required. \n\n\\item The extrapolators $\\mitbf{\\Lambda}_{I,J}$ account for the local geometry of $\\Gamma$ at the projection points $P$ on $\\Gamma$ via $\\mitbf{L}^k$ (section \\ref{SubSecBC}). Moreover, they incorporate the position of $P$ relative to the Cartesian meshing, via $\\mitbf{\\Pi}_{i,j}$ (\\ref{Taylor}) and $\\mitbf{\\Pi}_{I,J}$ (\\ref{Extrapolator}). The set of extrapolators therefore provides a subcell resolution of $\\Gamma$ in the meshing, avoiding the spurious diffractions induced by a naive description of the boundaries.\n\n\\item The stability of the method has not been proved. However, numerical experiments clearly indicate that the CFL condition of stability is not modified compared with the case of a homogeneous medium. The solution does not grow with time, even in the case of long-time simulations (see section \\ref{SubSecTest4}).\n\n\\item In a previous one-dimensional study \\citep{PIRAUX01}, the local truncation error of the method has been rigorously analysed, leading to the following result: using the fictitious values (\\ref{Extrapolator}) ensures a local $r$-th order spatial accuracy if $k\\geq r$, where $r$ is the order of spatial accuracy of the scheme. In 2D configrations with material interfaces \\citep{LOMBARD04,LOMBARD06}, no proof has been conducted, but numerical experiments have shown that the $r$-th order overall accuracy is also maintained by taking $k=r$. Note that a slightly smaller order of extrapolation can be used: $k=r-1$ suffices to provide $r$-th order overall accuracy \\citep{GUSTAFSSON75}. The value $k=3$ is therefore used for the fourth-order ADER scheme.\n\n\\item The extrapolators do not depend on the numerical scheme adopted. They depend only on $k$ and on physical and geometrical features. Standard subroutines for computing the extrapolators $\\mitbf{\\Lambda}_{I,J}$ can therefore be developed and adapted to a wide range of schemes. Subroutines of this kind are freely available in FORTRAN at the web page {\\tt http:\/\/w3lma.cnrs-mrs.fr\/\\textasciitilde MI\/Software\/}. \n\\end{enumerate}\n\n\\subsection{Case of staggered-grid schemes}\\label{SubSecStag}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n(a) & (b)\\\\\n\\includegraphics[scale=0.56]{FigStagA.eps}&\n\\includegraphics[scale=0.56]{FigStagB.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Staggered-grid schemes with a plane boundary $\\Gamma$ parallel to the meshing: two cases can be distinguished, depending on the position of $\\Gamma$ relative to the meshing. Case (a), where the fictitious stress is estimated, works well, while case (b), where the fictitious velocity is estimated, leads to long-term instabilities.}}\n\\label{FigStag}\n\\end{figure} \n\nInstead of using a single-grid scheme as proposed in section \\ref{SubSecScheme}, readers may be interested in adapting our approach to staggered-grid schemes such as CSS or PSS (see section \\ref{SecIntro} for the definition of these terms). However, in the case of some of the boundary positions relative to the meshing, computational instabilities occur, especially when long-time integration is considered.\n\nTo understand why this is so, let us consider PSS-2. Taking a simple flat boundary to exist between the medium and the vacuum leads to two typical geometrical configurations. At one position of the free surface, the boundary discretization will require only the stress field to be extrapolated (Figure \\ref{FigStag}-(a)). Our procedure works satisfactorily with this type of discretization at any order $k$. It also yields stable and accurate solutions when dealing with PSS-4, contrary to the vacuum method. Using 10 grid nodes per minimal S-wavelength gives similar performance in this case to those of our numerical experiments based on the ADER scheme, which are shown in section \\ref{SecNum}.\n\nAt another position of the free surface where only extrapolated velocities are required within a wide zone (Figure \\ref{FigStag}-(b)), our procedure results in instabilities. The reason for this problem is as follows: fictitious velocities involve first-order boundary conditions (\\ref{L0t}) and higher-order conditions (see section \\ref{SubSecBC}), but they do not involve the fundamental zeroth-order Dirichlet conditions (\\ref{L0U0}). Since the latter conditions are never enforced, an increasing oscillating drift occurs near the boundary, which invalidates the computations. Similar behavior is observed with PSS-4, but after a longer time: the numerical solution generally works  well during a few thousand time steps, before growing in a unstable manner. \n\nThe extrapolation method presented here is therefore not recommended for use with staggered-grid schemes, especially PSS-2, except in the trivial case sketched in Figure \\ref{FigStag}-(a).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{FigCorner.eps}\n\\caption{{\\it Boundary $\\Gamma$ with a corner at $P$, replaced locally by an arc of circle with a radius $\\delta$ between $P_0$ and $P_1$}.}\n\\label{FigCorner}\n\\end{center}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n(a)\\\\\n\\includegraphics[scale=0.36]{T1Carte0.eps}\\\\\n(b)\\\\\n\\includegraphics[scale=0.36]{T1Carte1.eps}\\\\\n(c)\\\\\n\\includegraphics[scale=0.36]{T1Carte2.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 1: snapshots of $v_x$ at the initial instant (a), at mid-term (b) and at the final instant (c).}}\n\\label{FigTest1A}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n(a)\\\\\n\\includegraphics[scale=0.325]{T1Coupe0.eps}\\\\\n(b)\\\\\n\\includegraphics[scale=0.325]{T1Coupe1.eps}\\\\\n(c)\\\\\n\\includegraphics[scale=0.325]{T1Coupe2.eps}\\\\\n(d)\\\\\n\\includegraphics[scale=0.325]{T1Coupe3.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 1: time history of $v_x$ (a). Zooms on successive time windows, with various discretizations (b,c,d): the number after \\# denotes the number of grid nodes per minimal S-wavelength.}}\n\\label{FigTest1B}\n\\end{figure} \n\n\\subsection{Case of non-smooth geometries}\n\nUp to know, we have assumed that the boundary $\\Gamma$ was sufficiently smooth at the projection points, being at least $C^{k+1}$ at each $P$, where $k\\geq 0$ is the order of differentiation defined in section \\ref{SubSecOverview}. Let us assume now that $\\Gamma$ is only $C^K$ at a point $P$, with $K<k+1$. Then, the components of $\\mitbf{L}^k$ in (\\ref{LkUk}) involving the derivatives $\\frac{d^\\alpha\\,x}{d\\,x^\\alpha}(\\tau)$ and $\\frac{d^\\alpha\\,z}{d\\,x^\\alpha}(\\tau)$ ($\\alpha=K+1,...,k+1$) of the parametric representation are discontinuous, invalidating locally the method proposed. In our software, we have implemented the following rough treatment:\n\\begin{enumerate}\n\\item If $K=0$, the boundary owns a corner and the solution has an integrable singularity. The corner is replaced by an arc of circle centred at $Q$ with radius $\\delta$ (figure \\ref{FigCorner}), leading to a $C^1$ curve.\n\\item If $0<K<k+1$, as in the previous case at $P_0$ and $P_1$ with $K=1$, the values of $\\frac{d^\\alpha\\,x}{d\\,x^\\alpha}(\\tau)$ and $\\frac{d^\\alpha\\,z}{d\\,x^\\alpha}(\\tau)$ ($\\alpha=K+1,...,k+1$) are taken indiscriminately on one side or the other of the point considered. \n\\end{enumerate}\nNo numerical instabilities were observed if $\\delta$ (in case (i)) or the radius of curvature (in case (ii)) is much greater than $h$. It is agreed that the accuracy of computations is no more controlled in the cases (i) and (ii), especially the convergence towards the exact solution. \n\nMore sophisticated treatments of geometrical singularities, such as space-time mesh refinement \\citep{BERGER98}, require further investigation, which is out of the scope of the present paper. New studies are also needed in the case of merging boundaries, occuring for instance when an internal material interface reaches the free surface \\citep{MOCZO04}.\n\n\\section{Numerical experiments}\\label{SecNum}\n\n\\subsection{Configurations}\n\nThe time evolution of the source is a Ricker wavelet\n\\begin{equation}\ng(t)=\\left\\{2\\left(\\pi\\,f_c\\left(t-t_c\\right)\\right)^2-1\\right\\}e^{\\textstyle -\\left(\\pi\\,f_c\\left(t-t_c\\right)\\right)^2},\n\\label{RICKER}\n\\end{equation}\nwhere $f_c$ is the central frequency, and $t_c=1\/f_c$. The maximal frequency $f_{\\max}$ defined by $|\\tilde{g}(f_{\\max})\/\\tilde{g}(f_c)|=0.5$ (the tilde designates the Fourier transform) is $f_{\\max}\\approx 1.6\\,f_c$. We will adopt this frequency $f_{\\max}$ for our rule of thumb about\nthe number of grid nodes per S-wavelength. The following values of the physical parameters will be used in all the following tests: $\\rho=2400\\,\\mbox{kg\/m}^3$, $c_p=4500$ m\/s, and $c_s=2200$ m\/s. Lastly, the mesh size and the time step satisfy $c_p\\,\\Delta\\,t\/h= 0.85$. \n\nThe simulations are performed on a PC Pentium at 3 GHz with 2 GB of RAM. The results of tests 1 and 2, with constant and null curvature of $\\Gamma$, compare quantitatively with analytical solutions denoted by a solid line. Test 3, with a variable curvature, is purely qualitative. Test 4 shows the slow decrease in the mechanical energy which occurs during very long integration times, which confirms the stability of the method. \n\n\\subsection{Test 1: circular boundary}\\label{SubSecTest1}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n(a)\\\\\n\\includegraphics[scale=0.42]{T2Carte0.eps}\\\\\n(b)\\\\\n\\includegraphics[scale=0.42]{T2Coupe0.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 2: snapshot of $v_z$ at the final instant (a). Numerical and exact time histories of $v_z$ (b)}.}\n\\label{FigTest2}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[scale=0.4]{T2Coupe1.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 2: parametric study of the relative error in terms of the boundary's angle $\\theta$, with various discretizations. The number after \\# denotes the number of grid nodes per minimal S-wavelength.}}\n\\label{Test2Param}\n\\end{figure} \n\n\\textit{Computational efficiency}. Let us consider a circular cavity containing vacuum, with radius 1 km, at the center of a 18 km $\\times$ 18 km domain. In a first part, the mesh spacing is $h=25$ m. The source is a rightward-moving plane wave, with $f_{\\max}=8$ Hz, ensuring 22 grid nodes per minimal P-wavelength and 10 grid nodes per minimal S-wavelength at that frequency. \n\nDuring the pre-processing step, the program finds the 616 grid nodes where fictitious values are required; it also computes and stores the 616 extrapolators defined by the expression (\\ref{Extrapolator}). Time integration is performed in 550 time steps, which corresponds to a propagation time of 2.75 s and a propagation distance of 22 minimal wavelengths. The preprocessing step takes 21 s of CPU time. The time integration takes 1100 s of CPU time, including 28.6 s induced by the computation and by the use of fictitious values, which amounts to an extra time cost of only 2.6 \\%. Figure \\ref{FigTest1A} shows snapshots of $v_x$ at the initial instant (a), after 275 time steps (b) and after 550 time steps (c). \n\n\\textit{Quantitative study}. In a second part, three discretizations are considered: $h=25$ m, 50 m, 60 m, corresponding respectively to 10, 5 and 4 grid nodes per minimal S-wavelength. A receiver is set just above the cavity, at the position (9 km, 10.2 km), that can be seen on Figure \\ref{FigTest1A}; it mainly records the waves propagating along the boundary. Numerically, these waves are highly sensitive to the quality of the fictitious values defined in section \\ref{SecDiscrete}. \n\nFigure \\ref{FigTest1B}-(a) shows the time history of $v_x$ at the receiver. In this time window, three main wave packets are generated; with the scale of Figure \\ref{FigTest1B}-(a), the third packet cannot be seen. The amplitude is divided by a factor of approximately 30 from one packet to the following one. For the sake of clarity, zooms around each wave packet are shown in Figure \\ref{FigTest1B}-(b,c,d). These solutions are compared with an exact solution computed thanks to inverse Fourier transforms on 4096 frequencies, with $1.25\\,10^{-2}$ Hz as the sampling frequency; each harmonic component is expanded into 60 Bessel modes. The agreement between the numerical and the analytical values is very good when 10 grid nodes per wavelength are used, even at very small amplitudes (d). For 5 and 4 grid nodes per wavelength, the solution is slightly less accurate, but it is still acceptable. \n\n\\subsection{Test 2: inclined straight boundary}\\label{SubSecTest2}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n(a)\\\\\n\\includegraphics[scale=0.4]{T3Carte0.eps}\\\\\n(b)\\\\\n\\includegraphics[scale=0.4]{T3Carte1.eps}\\\\\n(c)\\\\\n\\includegraphics[scale=0.4]{T3Carte2.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 3: snapshots of $v_z$, for various sinusoidal topographies}.}\n\\label{FigTest3}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n(a)\\\\\n\\includegraphics[scale=0.43]{T4Carte.eps}\\\\\n(b)\\\\\n\\includegraphics[scale=0.43]{T4Courbe.eps}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Test 4: snapshot of $v_z$ (a) and time history of the mechanical energy (b)}.}\n\\label{FigTest4}\n\\end{figure} \n\n\\textit{The Garvin's problem}. As a second test, we take a plane boundary inclined against the Cartesian mesh. The domain under investigation is 18 km wide and 12 km high, with the origin of the coordinates on the top and left. The mesh spacing is $h=10$ m. Four receivers at (10 km, 1.8 km), (11 km, 1.6 km), (12 km, 1.4 km) and (13 km, 1.2 km) belong to the free boundary which is inclined at an angle of $\\theta=11.3^{^\\circ}$ relative to $Ox$. \n\nAn explosive source S is buried at (9 km, 2.1 km), with $f_{\\max}=24$ Hz. The distance between the source and the free surface is roughly 100 m $<\\lambda_p\/3$, where $\\lambda_p$ is the wavelength of the compressional waves at frequency $f_c$, and hence large Rayleigh waves are generated, with a velocity $c_r=2054$ m\/s. To prevent the occurence of spurious oscillations, the source is spread out numerically over a radius of $R=\\lambda_p\/7.5=40$ m. The source is weighted by a gaussian law with a standard deviation $R\/2=20$ m. The spatial discretization ensures the sampling of roughly 18 grid nodes per minimal P-wavelength, and 9 grid nodes per minimal S-wavelength at the frequency $f_{\\max}$.\n\nFigure \\ref{FigTest2}-(a) shows a snapshot of $v_z$ after 1200 time steps, corresponding to a propagation time of 2.25 s and a propagation distance of 55 minimal wavelengths. Direct cylindrical waves are observed, together with converted PP waves, converted PS waves (with an almost linear wavefront), and Rayleigh waves. In Figure \\ref{FigTest2}-(b), the time history of $v_z$ recorded at the receivers can be favourably compared with an exact solution. The latter is obtained by convolving the Green's function obtained by the well-known Cagniard-de Hoop method \\citep{GARVIN55,SANCHEZ06} with the source wavelet (\\ref{RICKER}) and with the discrete source spreading. \n\n\\textit{Influence of the slope}. To quantify the effects of the angle between the boundary and the Cartesian meshing on the numerical solution, we perform a parametric study of the error in terms of $\\theta$. Ten angles are considered, from $\\theta=0^{\\circ}$ to $\\theta=45^{\\circ}$ in steps of $5^{\\circ}$. In each configuration, the waves are measured at the free boundary after propagating for 65 minimal wavelengths. The error of $\\mitbf{v}.\\mitbf{n}$ is measured in norm $L_2$, and then it is normalized by the norm $L_2$ of the exact time history of $\\mitbf{v}.\\mitbf{n}$.\n\nThe results of this study are shown in Figure \\ref{Test2Param}, with various discretizations: 5, 10, 20 grid nodes per minimal S-wavelength. With a given $h$, the error is almost constant and independent of $\\theta$. This constitutes a crucial advantage of our method over the vacuum method, where the error at $45^{\\circ}$ is much greater than that at $0^{\\circ}$: it means that an extremely fine discretization is required to obtain accurate results with the vacuum method when arbitrary-shaped boundaries are encountered \\citep{BOHLEN06}.  \n\n\\subsection{Test 3: sinusoidal boundary}\\label{SubSecTest3}\n\nSince boundaries not related to the finite-difference grid can be included, the third test is performed on a sinusoidal free boundary, with a peak-to-peak amplitude of 800 m and various wavelengths: 0.5 km, 1 km and 2 km. The sinusoidal curve is centred around $z=1$ km.  The source S is located at (9 km, 1 km). The other parameters are the same as in test 2. Figure \\ref{FigTest3} shows snapshots of $v_z$ at the final instant 2.25 s. One can clearly see how the wavelength of the sinusoidal boundary influences the diffracted fields. \n\nConvergence studies (not shown here) were performed in these three cases, by comparing solutions computed on finer grids. We again concluded that accurate solutions can be obtained when the simulations involve approximately 10 grid nodes per minimal S-wavelength at the frequency $f_{\\max}$ of the source wavelet, even in the case of complex topographies with variable curvatures.\n\n\\subsection{Test 4: long-term stability}\\label{SubSecTest4}\n\nThe fourth test focuses on long-term stability \\citep{STACEY94,HESTHOLM03}. For this purpose, we consider a circular elastic domain with a radius of 150 m, surrounded by vacuum. The source S is located inside the circle, at (320 m, 200 m). This configuration is obviously not realistic, but it enlights the influence of the boundary on the numerical solution after many reflections, and especially on the possible excitation of numerical spurious modes leading to long-term instability. The mesh size is $h=1$ m. Time integration is performed during $10^6$ time steps, with $f_{\\max}=160$ Hz. \n\nFigure \\ref{FigTest4}-(a) shows a snapshot of $v_z$ at the final instant: no instability is observed, and the antisymmetry of $v_z$ is satisfied. Once the source is extincted ($t>2\\,t_c$), the mechanical energy $E$ is theoretically maintained. It can be written in terms of $\\mitbf{v}$ and $\\mitbf{\\sigma}$\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle\nE=\\frac{\\textstyle 1}{\\textstyle 2}\\int\\int_\\Omega\\left\\{ \\rho\\,\\mitbf{v}^2+\\frac{\\textstyle \\lambda+2\\,\\mu}{\\textstyle 4\\,\\mu\\,(\\lambda+\\mu)}\\left(\\sigma_{xx}^2+\\sigma_{zz}^2\\right)+\\frac{\\textstyle 1}{\\textstyle \\mu}\\sigma_{xz}^2\\right.\\\\\n[8pt]\n\\displaystyle\n\\hspace{2cm}\\left.-\\frac{\\textstyle \\lambda}{\\textstyle 2\\,\\mu\\,(\\lambda+\\mu)}\\sigma_{xx}\\,\\sigma_{zz}\\right\\}\\,dx\\,dz.\n\\end{array}\n\\label{NRJ}\n\\end{equation}\nAt each time step, the integral in (\\ref{NRJ}) is estimated by a basic trapezoidal rule at the grid nodes inside $\\Omega$. Figure \\ref{FigTest4}-(b) shows the time history of this mechanical energy so-obtained. It slightly decreases, due to the numerical diffusion of the scheme, which confirms that the method is stable.\n\n\\section{Conclusion}\\label{SecConclu}\n\nHere we have presented a method of incorporating free boundaries into time-domain single-grid finite-difference schemes for elastic wave simulations. This method is based on fictitious values of the solution in the vacuum, which are used by the numerical integration scheme near boundaries. These high-order fictitious values accurately describe both the boundary conditions and the geometrical features of the boundaries. The method is robust, involving negligible extra computational costs. \n\nUnlike the vacuum method, the quality of the numerical solution thus obtained is almost independent of the angle between the free boundaries and the Cartesian meshing. Since the free boundaries do not introduce any additional artefacts, one can use the same discretization as in homogeneous media. Typically, when a fourth-order ADER scheme is used on a propagation distance of 50 minimal wavelengths, 10 grid nodes per minimal S-wavelength yield to a very good level of accuracy. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. \n\nFor the sake of simplicity, we have dealt here with academic cases, considering two-dimensional geometries, constant physical parameters, and simple elastic media. Let us examine briefly the generalization of our approach to more realistic configurations:\n\\begin{enumerate}\n\\item Extending the method to 3-D topographies a priori does not require new tools. The main challenge will concern the computational efficiency of parallelization. A key point is that the determination of each fictitious value is local, using numerical values only at neighboring grid nodes. Particular care will however be required for fictitious values near frontiers between computational subdomains, in order to minimize the exchanges of data.\n\\item Near free boundaries, the domains of propagation are usually smoothly heterogeneous. To generalize our method to continuously variable media, the main novelty expected concerns the high-order boundary conditions detailed in section \\ref{SubSecBC}. With variable matrices $\\mitbf{A}$ and $\\mitbf{B}$ indeed and $k\\geq 2$, the procedure (\\ref{DTauDt}) will involve the following quantities, to be estimated numerically:\n$$\n\\frac{\\textstyle \\partial^{k-1}}{\\textstyle \\partial\\,x^{k-1-\\alpha}\\,\\partial\\,z^\\alpha}\\,\\mitbf{A},\\quad \\frac{\\textstyle \\partial^{k-1}}{\\textstyle \\partial\\,x^{k-1-\\alpha}\\,\\partial\\,z^\\alpha}\\,\\mitbf{B},\\qquad \\alpha=0,...,k-1.\n$$\n\\item Realistic modeling of wave propagation requires to incorporate attenuation. The only rheological viscoelastic models able to approximate constant quality factor over a frequency range are the generalized Maxwell body \\citep{EMMERICH84} and the generalized Zener body \\citep{CARCIONE01}. These two equivalent models \\citep{MOCZO05} yield to additional unknowns called \\textit{memory variables}. In the time domain, the whole set of unknowns satisfies a linear hyperbolic system with source term\n\\begin{equation}\n\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,t}\\,\\mitbf{U}=\\mitbf{A}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,x}\\,\\mitbf{U}+\\mitbf{B}\\,\\frac{\\textstyle \\partial}{\\textstyle \\partial\\,z}\\,\\mitbf{U}-\\mitbf{S}\\,\\mitbf{U},\n\\label{LCatten}\n\\end{equation}\nwhere $\\mitbf{S}$ is a definite positive matrix. Compared with the elastic case (\\ref{LC}) examined in the present paper, the main difference expected concerns the time differentiation of the boundary condition (\\ref{L0U0}). Indeed, equation (\\ref{L0t}) has to be modified accordingly to (\\ref{LCatten}). Similar modifications are also foreseen in the case of poroelasticity in the low-frequency range \\citep{DAI95}, where the evolution equations can be put in the form (\\ref{LCatten}).\n \\end{enumerate}\n\n\\label{lastpage}\n\n\\begin{acknowledgments}\nThe authors thank the reviewers P. Moczo and I. Oprsal for their instructive comments and bibliographic insights.\n\\end{acknowledgments}\n\n\\def\\hskip .11em plus .33em minus .07em{\\hskip .11em plus .33em minus .07em}\n\\bibliographystyle{gji}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur detailed knowledge of the Universe is mostly based on the study of correlation functions of perturbations around a homogeneous background. A considerable effort has been devoted over the years to the calculation of these correlators during inflation, for the CMB temperature fluctuations and for the present distribution of dark and luminous matter. It is by now well understood that calculations dramatically simplify in the parametric limit in which one (or more) of the momenta (that we call $q$ in this paper) becomes much smaller than the others (denoted by $k$). Recently \\cite{Kehagias:2013yd,Peloso:2013zw,Creminelli:2013mca}, these arguments have been applied to the matter (or $\\Lambda$)-dominated phase to show that the leading term as $q \\to 0$ of any correlation function with $(n+1)$ legs can be written in terms of an $n$-point function: the so-called consistency relations. Although the arguments work in a fully relativistic treatment \\cite{Creminelli:2013mca}, which is mandatory if we want to follow the evolution of the modes back in time and connect with inflation, in this paper we focus on the non-relativistic limit, which is valid deep inside the horizon. \n\nThe physical argument behind the consistency relations in the non-relativistic limit is that at leading order in $q$ a long mode gives rise to a homogeneous gravitational field $\\vec \\nabla\\Phi$. The effect of this mode on the short-scale physics can be derived exactly using the equivalence principle and erasing the long mode with a suitable change of coordinates. This logic makes virtually no assumption about the physics at short scales, including the complications due to baryons. However, the cancellation of the long mode by a change of coordinates can be performed only assuming that gravity is all there is: no extra degrees of freedom during inflation (i.e.~single-field inflation) and no extra forces (violation of the equivalence principle) at present. Therefore the consistency relations can be seen as a test of these two assumptions. \n\nIn this paper, which is the natural continuation of \\cite{Creminelli:2013mca}, we follow our study of the subject in two directions. First, we want to extend the consistency relations non-linearly in the long mode (Section~\\ref{sec:resum}). The displacement due to a homogeneous gravitational field scales with time as $\\Delta \\vec x \\sim \\vec \\nabla\\Phi_L \\;t^2$, so that the effect on the short modes of momentum $k$ goes as\n\\begin{equation}\n\\vec  k \\cdot  \\Delta \\vec x \\sim k \\; q \\;\\Phi_L  \\,t^2 \\sim \\frac{k}{q} \\delta_L \\;,\n\\end{equation}\nwhere $\\delta_L$ is the long-mode density contrast\\footnote{This is the leading effect in the non-relativistic limit: relativistic corrections are further suppressed by powers of $k\/aH \\ll 1$ \\cite{Creminelli:2013mca}, which are negligible well inside the horizon.}. Notice that this is parametrically larger than $\\delta_L$, the natural expansion parameter of perturbation theory, and this is why one is able to capture the leading $q \\to 0$ behaviour.  \nObviously, the fact that we can erase a homogeneous gravitational field by going to a free falling frame is an exact statement, that does not require the gravitational field to be small. This implies that we do not need to expand in $k\/q \\cdot \\delta_L$ that can be large, while we keep $\\delta_L$ small to allow for a perturbative treatment of the long mode. In Section \\ref{sec:resum} we are going to give a resummed version of the consistency relations which is exact in $k\/q \\cdot \\delta_L$. This allows to discuss the case of multiple soft modes and check the relations with the perturbation theory result. With the same logic, we will study the effect of internal soft modes and loops of soft modes. \n\nThe second topic of the paper (Section \\ref{sec:redshift}) is to derive consistency relations directly in redshift space, since this is where the distribution of matter is measured. We will do so without assuming anything about the short modes, in particular the single-stream approximation that breaks down in virialized objects. The redshift consistency relations contain an extra piece because the long mode, besides inducing a homogeneous gravitational field in real space, also affects the position of the short modes in redshift space along the line-of-sight. The redshift space consistency relations state that the correlation functions vanish at leading order for $q \\to 0$ when the short modes are taken at the same time, as it happens in real space. Given that it is practically impossible, as we will discuss, to study correlation functions of short modes at different times, it is hard to believe that these relations will be verified with real data. However, if a signal is detected at equal times, the consistency relations are not satisfied and this would indicate that at least one of the assumptions does not hold. This would represent a detection of either multi-field inflation or violation of the equivalence principle (or both!).   \n\n\nAs explained in \\cite{Creminelli:2013mca}, one of the conditions for the validity of the consistency relations is that the long mode has always been out of the sound horizon since inflation. Indeed, a well-understood example where the consistency relations are not obeyed is the case of baryons and cold dark matter particles after decoupling. Before recombination, while dark matter  follows geodesics, baryons are tightly coupled to photons through Thomson scattering and  display acoustic oscillations. Later on, baryons recombine and decouple from photons. Thus, as their sound speed drops they start following geodesics, but with a larger velocity than that of dark matter on comoving scales below the sound horizon at recombination.    As discussed in  \\cite{Tseliakhovich:2010bj}, the long-wavelength relative velocity between baryons and CDM  reduces the formation of early structures on small scales, through a genuinely nonlinear effect. \n\nThe fact that baryons have a different initial large-scale velocity compared to dark matter implies, if the long mode is shorter than  the comoving sound horizon at recombination, that the change of coordinates that erases the effect of the long mode is not the same for the two species. Thus the effect of the long mode does not cancel out in the equal-time correlators involving different species  \\cite{Bernardeau:2011vy,Bernardeau:2012aq}. In particular, the amplitude of the short-scale equal-time $n$-point functions becomes correlated with the long-wavelength isodensity mode, so that the $(n+1)$-point functions in the squeezed limit do not vanish at equal time. This effect, however, becomes rapidly negligible at low redshifts because the relative comoving velocity between baryons and dark matter decays as the scale factor, $|\\vec v_{\\rm b} - \\vec v_{\\rm CDM}| \\propto 1\/a$.\\footnote{The violation of the consistency relations decays as $(D_{\\rm iso}\/D)^2 \\propto (a^2 H f D )^{-2} \\sim (1+z)^{3\/2}$ where $D_{\\rm iso} \\propto |\\vec v_{\\rm b} - \\vec v_{\\rm CDM}|\/( a H f)$ is the growth function of the long-wavelength isodensity mode, $D$ is the growth function of the long-wavelength adiabatic growing mode, $f$ is the growth rate and  $H$ is the  Hubble rate (see \\cite{Bernardeau:2011vy} for details); in the last  approximate equality we have used matter dominance. Thus, the effect is already sub-percent at $z \\sim 40$.}  Hence, while a deviation can be sizable at high redshifts, it can be neglected in galaxy surveys and the consistency relations apply also when the long mode is shorter than the comoving sound horizon at recombination.\nWe conclude that the vanishing of the correlation functions at leading order in $q \\to 0$ is very robust.\n\n\n\n\\section{\\label{sec:resum}Resumming the long mode}\n\n\nLet us consider a flat unperturbed FRW universe and add to it a homogenous gradient of the Newtonian potential $\\Phi_L$.\\footnote{Since we are interested in the non-relativistic limit, we do not consider a constant value of $\\Phi_L$, which is immaterial in this limit.} Provided all species feel gravity the same way---namely, assuming the equivalence principle---we can get rid of the effect of $\\vec \\nabla \\Phi_L$ by going into a frame which is free falling in the constant gravitational field. The coordinate change to the free-falling frame is (we are using conformal time $d \\eta \\equiv dt\/a(t)$)\n\\begin{equation}\n\\vec x \\to \\vec x + \\delta \\vec x (\\eta) \\;, \\qquad \\delta \\vec x (\\eta) \\equiv - \\int \\vec v_L (\\tilde \\eta) \\, \\textrm{d} \\tilde \\eta\\;,  \\label{displacement}\n\\end{equation}\nwhile time is left untouched. The  velocity $\\vec v_L$ satisfies the Euler equation in the presence of the homogenous force,\nwhose solution is \n\\begin{equation}\\label{longv}\n\\vec v_L(\\eta) = - \\frac{1}{a(\\eta)} \\int a (\\tilde \\eta) \\vec \\nabla \\Phi_L (\\tilde \\eta) \\, \\textrm{d}\\tilde\\eta\\;.\n\\end{equation}\n\n\n\nTo derive the consistency relations we  start from real space. Here, for definiteness, we denote by $\\delta^{(g)}$  the density contrast of the galaxy distribution. However, the relations that we will derive are more general and hold for any species---halos, baryons, etc., irrespectively of their bias with respect to the underlying dark matter field. \nFollowing the argument above, any $n$-point correlation function of short wavelength modes of $\\delta^{(g)}$ in the presence of a slowly varying $\\Phi_L(\\vec y)$ is equivalent to the same correlation function in displaced spatial coordinates, $\\vec{\\tilde x} \\equiv \\vec x + \\delta \\vec x(\\vec y, \\eta)$, where the displacement field $\\delta \\vec x(\\vec y, \\eta)$ is given by eq.~\\eqref{displacement} and $\\vec y$ is an arbitrary point---e.g., the midpoint between $\\vec x_1,  \\ldots, \\vec x_n$---whose choice is irrelevant at order $q\/k$. This statement can be formulated with the following relation,\n\\begin{equation}\n\\begin{split}\n\\langle \\delta^{(g)} (\\vec x_1,\\eta_1) \\cdots \\delta^{(g)} (\\vec x_n,\\eta_n) | {\\Phi_L}(\\vec y)\\rangle &\\approx \\langle \\delta^{(g)} (\\vec{\\tilde  x}_1,\\eta_1) \\cdots \\delta^{(g)} (\\vec{\\tilde  x}_n,\\eta_n) \\rangle_0\\; \\\\\n& = \\int \\frac{\\textrm{d}^3k_1}{(2\\pi)^3}\\cdots\\frac{\\textrm{d}^3k_n}{(2\\pi)^3}  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle_0 \\, e^{i \\sum_a  \\vec k_a \\cdot (\\vec x_a+  \\delta \\vec x (\\vec y, \\eta_a))}  \\;, \\label{cr1}\n\\end{split}\n\\end{equation}\nwhere in the last line we have simply taken the Fourier transform of the right-hand side of the first line. Here and in the following, by the subscript $0$ after an expectation value we mean that the average is taken setting $\\Phi_L = 0$ (and not averaging over it); while by $\\approx$ we mean an equality that holds in the limit in which there is a separation of scales between long and short modes. \nIn momentum space this  holds when the momenta of the soft modes is sent to zero.\nIn other words, corrections to the right-hand side of $\\approx$ are suppressed by  ${\\cal O} (q\/k)$. \n\nFrom eq.~\\eqref{displacement} and using the continuity equation $\\delta' + \\vec \\nabla \\cdot \\vec v =0$, we can rewrite each Fourier mode of the displacement field  as\n\\begin{equation}\n\\delta \\vec x(\\vec p, \\eta) = - i  \\frac{\\vec p}{p^2} \\delta (\\vec p, \\eta) \\equiv - i  \\frac{\\vec p}{p^2} D(\\eta) \\delta_0(\\vec p) \\;, \\label{deltax2delta}\n\\end{equation}\nwhere in the second equality we have defined $D(\\eta)$, the growth factor of density fluctuations of the {\\em long} mode\nand $\\delta_0 (\\vec p)$, a Gaussian random field with power spectrum $P_0(p)$ which represents the initial condition of the density fluctuations of the long mode \\cite{Bernardeau:2001qr}. Notice that the first equality of eq.~\\eqref{cr1} is based on the crucial assumption that the long mode is statistically uncorrelated with the short ones. This only works in single-field models of inflation, which we assume throughout. Notice also that eq.~\\eqref{deltax2delta}, when going beyond the linear theory, will only receive corrections of order $\\delta$, that we can neglect for our purposes since we are only interested in corrections which are enhanced by $1\/p$.\n\n\nAt this stage, we can compute an $(n+1)$-point correlation function in the squeezed limit by multiplying the left-hand side of eq.~\\eqref{cr1} by $\\delta_L$ and averaging over the long mode.\nSince the only dependence on $\\Phi_L$ in eq.~\\eqref{cr1} is in the exponential of $i  \\sum_a \\vec k_a \\cdot \\delta \\vec x( \\vec y,\\eta_a)$, we obtain\n\\begin{equation}\n\\label{CoRe1}\n\\begin{split}\n\\langle \\delta_L( \\vec x, \\eta) \\langle \\delta^{(g)} (\\vec x_1,\\eta_1) \\cdots \\delta^{(g)} (\\vec x_n,\\eta_n) | \\Phi_L \\rangle \\rangle_{\\Phi_L} \\approx &\\int \\frac{\\textrm{d}^3k_1}{(2\\pi)^3} \\cdots \\frac{\\textrm{d}^3k_n}{(2\\pi)^3}    \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle_0 \\, e^{i \\sum_a \\vec k_a \\cdot \\vec x_a} \\\\\n& \\times \\int \\frac{\\textrm{d}^3q}{(2\\pi)^3} e^{i  \\vec{q}\\cdot\\vec{x}} \\langle\\delta_{\\vec q} (\\eta) e^{i  \\sum_a \\vec k_a \\cdot \\delta \\vec x( \\vec y,\\eta_a)}\\rangle_{\\Phi_L} \\, .\n\\end{split}\n\\end{equation}\nIt is then convenient to rewrite this exponential as\n\\begin{equation}\n\\exp \\Big[ i  \\sum_a \\vec k_a \\cdot \\delta \\vec x( \\vec y,\\eta_a) \\Big] =\\exp \\Big[  \\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3}  J(\\vec p) \\delta_0(\\vec p) \\Big]\\;, \\label{deltax2J}\n\\end{equation}\nwhere\n\\begin{equation}\nJ(\\vec{p}) \\equiv  \\sum_a D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec p}{p^2}   \\,e^{i  \\vec p \\cdot \\vec{y}}\\, . \\label{Jdef}\n\\end{equation}\nThe integral is restricted to soft momenta, smaller than a UV cut-off $\\Lambda$, which must be much smaller than the hard modes of momenta $k_a$.\nAveraging the right-hand side of eq.~\\eqref{deltax2J}  over the long wavelength Gaussian random initial condition $\\delta_0(\\vec p)$ yields\\footnote{This result will receive corrections due to primordial non-Gaussianities. Indeed, even in single-field models of inflation, the statistics of modes with comparable wavelength can deviate from Gaussianity. We neglect these corrections in the following.}\n\\begin{equation}\n\\label{SumRes}\n\\bigg \\langle \\exp \\Big[  \\int^{\\Lambda}\\frac{\\textrm{d}^3p}{(2\\pi)^3}  J(\\vec p) \\delta_0(\\vec p) \\Big] \\bigg \\rangle_{\\Phi_L}  =\\exp \\bigg[ \\frac12 \\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3}  J(\\vec p) J(- \\vec p) P_0(p) \\bigg]  \\;.\n\\end{equation}\nWe can use this relation to compute the expectation value of $\\delta_L$ with the exponential,\n\\begin{equation}\n\\begin{split}\n\\bigg \\langle \\delta_{\\vec q}(\\eta) \\exp \\Big( i  \\sum_a \\vec k_a \\cdot \\delta \\vec x( \\vec y,\\eta_a) \\Big) \\bigg \\rangle_{\\Phi_L}  &= (2\\pi)^3 D(\\eta) \\frac{\\delta}{\\delta J(\\vec q)} \\bigg \\langle \\exp \\Big[  \\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3}  J(\\vec p) \\delta_0(\\vec p) \\Big] \\bigg \\rangle_{\\Phi_L}  \\\\& =  P(q,\\eta) \\frac{J(- \\vec q)}{D(\\eta)} \n\\exp \\bigg[ \\frac12 \\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3}  J(\\vec p) J(- \\vec p) P_0(p) \\bigg] \\;,\n\\end{split}\n\\end{equation}\nwhere we have defined the power spectrum at time $\\eta$: $P( q,\\eta) \\equiv D^2(\\eta) P_0(q)$. \nFinally, rewriting eq.~\\eqref{CoRe1} in Fourier space using the above relation and the definition of $J$, eq.~\\eqref{Jdef},  we obtain the resummed consistency relations in the squeezed limit,\n\\begin{equation}\n\\begin{split}\n\\langle  \\delta_{\\vec q}(\\eta) \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle' \\approx & -   P(q,\\eta) \\,  \\sum_a \\frac{D(\\eta_a)}{D(\\eta)} \\frac{\\vec k_a \\cdot \\vec{q}}{q^2}   \\;  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle'_0  \\\\\n&  \\times \\exp \\bigg[ {- \\frac12 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}  \\bigg( \\sum_a D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec{p}}{p^2} \\, \\bigg)^2 P_0(p)} \\bigg]  \\;, \\label{CR}\n\\end{split}\n\\end{equation}\nwhere, here and in the following, primes on correlation functions indicate that the momentum conserving delta functions have been removed.\nHowever, what one observes in practice is not the expectation value $\\langle\\ldots\\rangle_0$ with the long modes set artificially to zero: one wants to rewrite the right-hand side of eq.~\\eqref{CR} in terms of an average over the long modes. Using eq.~\\eqref{SumRes} one gets:\n\\begin{equation}\n\\label{resumloop}\n\\langle  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n) | {\\Phi_L} \\rangle \\rangle_{\\Phi_L} \\approx   \\exp \\bigg[ {- \\frac12 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}  \\bigg( \\sum_a D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec{p}}{p^2} \\, \\bigg)^2 P_0(p)} \\bigg]   \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle_0 \\;.\n\\end{equation}\nOnce written in terms of the observable quantity the consistency relation comes back to the simple form:\n\\begin{equation}\n\\langle  \\delta_{\\vec q}(\\eta) \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle' \\approx  -   P(q,\\eta) \\,  \\sum_a \\frac{D(\\eta_a)}{D(\\eta)} \\frac{\\vec k_a \\cdot \\vec{q}}{q^2}   \\;  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle' \\;. \\label{CR_easy}\n\\end{equation}\nThis equation has the same form as the consistency relations obtained in Refs.~\\cite{Kehagias:2013yd,Peloso:2013zw,Creminelli:2013mca}, but now it {\\em does not} rely on a linear expansion in the displacement field,\n\\begin{equation}\n\\frac{|\\delta \\vec x|}{|\\vec x|} \\sim \\frac{k}{q} \\delta_L \\ll 1 \\;.\n\\end{equation}\nIndeed, to derive eq.~\\eqref{CR} we have assumed that the long mode is in the linear regime, i.e.~$\\delta_L \\ll 1$, but no assumption has been made on $({k}\/{q}) \\delta_L$, which  can be as large as one wishes.\nFor equal-time correlators the right-hand side vanishes at leading order in $q$ because $\\sum_a \\vec k_a=\\vec q$, in the same way as in the linearized version \\cite{Kehagias:2013yd,Peloso:2013zw,Creminelli:2013mca}. \nThe resummation of long wavelengths in terms of a global translation of spatial coordinates---whose effect vanishes in equal-time correlation functions---was also performed in  \\cite{Bernardeau:2011vy,Bernardeau:2012aq} by using the so-called {\\em eikonal} approximation of the equations of motion of standard perturbation theory\\footnote{It is not surprising that the consistency relation eq.~\\eqref{CR_easy} remains the same even non-linearly in $(k\/q) \\delta_L$ working directly in terms of the expectation values $\\langle\\ldots\\rangle$ averaged over the long modes. Indeed, neglecting primordial non-Gaussianities, the effect of the mode with momentum $\\vec q$ is the same as a change of coordinates, even when the short-scale correlation functions are averaged over all long modes. Since, as we discussed, also eq.~\\eqref{deltax2delta} does not require an expansion in $(k\/q) \\delta_L$, eq.~\\eqref{CR_easy} follows.}\n\n\nIt is important to stress that here we made practically no assumptions on the short modes. We did not assume that they are in the linear regime or that the single-stream approximation holds. The relation also takes into account all complications due to baryon physics and it does not assume a description in terms of a Vlasov-Poisson system. We did not assume any model of bias between the short-scale $\\delta^{(g)}$ and the underlying dark matter distribution $\\delta$. We did not assume that the number of galaxies is conserved at short-scales, so the relation is valid including the formation and merging history. We thus believe that our derivation, rooted only on the equivalence principle, is more robust than the one of  \\cite{Kehagias:2013yd,Peloso:2013zw} based on the explicit equations for dark matter and for the galaxy fluid. Notice however that, while we are completely general about the short-modes physics, the long mode is treated in perturbation theory including its bias. Of course what enters in the consistency relations is only the velocity field of the long mode eq.~\\eqref{longv}, related to $\\Phi_L$ by the Euler equation. In converting this quantity in the density of some kind of objects, one has to rely on the conservation equation and this introduces the issue of bias and of its time-dependence. However, one can measure the large-scale potential in many ways, minimizing the systematic and cosmic-variance uncertainty \\cite{Seljak:2008xr}.\n\nAs shown below, one can straightforwardly extend this procedure and derive consistency relations involving an arbitrary number of soft legs in the correlation functions or use it to study the effect of soft loops and internal lines.\n\n\n\n\n\n\n\n\n\\subsection{Several soft legs}\n\nThe generalisation of the consistency relations above to multiple soft legs (for an analogous discussion in inflation see \\cite{Marko}) relies on taking successive functional derivatives with respect to $J(\\vec q _i)$ of eq.~\\eqref{SumRes}. As an example,  we can explicitly compute the consistency relations with two soft modes.  In this case the $(n+2)$-point function reads\n\\begin{equation}\n\\label{CoRe2}\n\\begin{split}\n \\langle \\delta_L( \\vec y_1,\\tau_1) \\delta_L( \\vec y_2,\\tau_2) \\delta^{(g)} (\\vec x_1,\\eta_1) &\\cdots \\delta^{(g)} (\\vec x_n,\\eta_n)  \\rangle \\approx \\int \\frac{\\textrm{d}^3k_1}{(2\\pi)^3} \\cdots \\frac{\\textrm{d}^3k_n}{(2\\pi)^3}   \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle_0 \\,e^{i \\sum_a \\vec k_a \\cdot \\vec x_a} \\\\\n& \\times \\int \\frac{\\textrm{d}^3q_1}{(2\\pi)^3}\\frac{\\textrm{d}^3q_2}{(2\\pi)^3}e^{i  (\\vec q_1\\cdot\\vec y_1+  \\vec q_2 \\cdot\\vec y_2)} \\bigg \\langle\\delta_{\\vec q_1}(\\tau_1)\\delta_{\\vec q_2}(\\tau_2) e^{\\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3} J(\\vec{p}) \\delta_0( \\vec p)} \\bigg \\rangle\\, .\n\\end{split}\n\\end{equation}\nTo compute the average over the long modes in the last line, it is enough to take two functional derivatives of eq.~\\eqref{SumRes},\n\\begin{equation}\n\\begin{split}\n\\bigg \\langle  \\delta_{\\vec q_1}(\\tau_1)\\delta_{\\vec q_2}(\\tau_2) &e^{\\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3} J(\\vec{p}) \\delta_0(\\vec p)} \\bigg \\rangle=(2 \\pi)^6D(\\tau_1) D(\\tau_2) \\frac{\\delta}{\\delta J(\\vec q_1)}\\frac{\\delta}{\\delta J(\\vec q_2)}\\bigg \\langle e^{\\int^\\Lambda\\frac{\\textrm{d}^3 p }{(2\\pi)^3} J(\\vec{p}) \\delta_0(\\vec p)}\\bigg \\rangle \\\\\n&= \\frac{J(-\\vec q_1)}{D(\\tau_1)}\\frac{J(-\\vec q_2)}{D(\\tau_2)}P(q_1,\\tau_1)P(q_2,\\tau_2) e^{\\frac12 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}  J(\\vec{p})J(-\\vec{p})P_0(p)}\\, ,\n\\end{split}\n\\end{equation}\nwhere we have assumed $\\vec q_1 + \\vec q_2 \\neq 0$ to get rid of unconnected contributions. In Fourier space, this yields\n\\begin{equation}\n\\label{n+2pf}\n\\begin{split}\n\\langle  \\delta_{\\vec q_1}(\\tau_1)\\delta_{\\vec q_2}(\\tau_2) \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle' & \\approx P(q_1,\\tau_1)P(q_2,\\tau_2) \\\\  & \\times \\sum_a \\frac{D(\\eta_a)}{D(\\tau_1)} \\frac{\\vec k_a \\cdot \\vec{q_1}}{q_1^2}    \\sum_b \\frac{D(\\eta_b)}{D(\\tau_2)} \\frac{\\vec k_b \\cdot \\vec{q_2}}{q_2^2}  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle' \\;,\n\\end{split}\n\\end{equation}\nwhere again we have used eq.~\\eqref{resumloop} to write the result in terms of correlation functions averaged over the long modes.\n\nAs a simple example, let us consider eq.~\\eqref{n+2pf} in the case where $n=2$ and $\\delta^{(g)}$ describes dark matter perturbations, i.e.~$\\delta^{(g)} \\equiv \\delta$.\nIn this case, at lowest order in $\\frac k q \\delta(\\vec q,\\eta)$---i.e.~setting the exponential in the third line to unity---the above relation reduces to\n\\begin{equation}\n\\label{TriSpec}\n\\langle\\delta_{\\vec q_1}(\\tau_1) \\delta_{\\vec q_2}(\\tau_2) \\delta_{\\vec k_1}(\\eta_1) \\delta_{\\vec k_2}(\\eta_2)\\rangle' \\approx \\frac{(D(\\eta_1) - D(\\eta_2) )^2}{D(\\tau_1) D(\\tau_2) } \\frac{\\vec q_1\\cdot \\vec k_1}{q_1^2} \\frac{\\vec q_2\\cdot \\vec k_1}{q_2^2}P(q_1,\\tau_1)P(q_2,\\tau_2) \\langle \\delta_{\\vec k_1}(\\eta_1)\\delta_{\\vec k_2}(\\eta_2)\\rangle'.\n\\end{equation}\nWe can check that this expression correctly reproduces the tree-level trispectrum computed in perturbation theory in the double-squeezed limit.  This can be easily computed by summing the two types of diagrams displayed in Fig.~\\ref{fig:trispectrum}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{trispectrum.pdf}\n\\caption{\\label{fig:trispectrum} \\small{Two diagrams that contribute to the tree-level trispectrum. Left: $T_{1122}$. Right: $T_{1113}$.}}\n\\end{figure}\nThe diagram on the left-hand side represents the case where the density perturbations of the short modes are both taken at second order, yielding\n\\begin{equation}\n\\label{cont_t_1}\n\\begin{split}\nT_{1122}=& \\, D(\\tau_1) D(\\tau_2) D(\\eta_1) D(\\eta_2)  P_{0}(q_1)P_{0}(q_2)  F_2(-\\vec q_1, \\vec k_1+\\vec q_1) F_2(-\\vec q_2, \\vec k_2+\\vec q_2)  \\langle \\delta_{\\vec k_1} (\\eta_1)\\delta_{\\vec k_2} (\\eta_2)\\rangle' + {\\rm perms} \\\\\n& \\approx -8 \\frac{\\vec q_1\\cdot \\vec k_1}{2 q_1^2} \\frac{\\vec q_2\\cdot \\vec k_1}{2 q_2^2} \\frac{D(\\eta_1) D(\\eta_2)}{D(\\tau_1) D(\\tau_2)} P(q_1,\\tau_1)P(q_2,\\tau_2) \\langle \\delta_{\\vec k_1}(\\eta_1)\\delta_{\\vec k_2}(\\eta_2)\\rangle' \\, ,\n\\end{split}\n\\end{equation}\nwhere, on the right-hand side of the first line, $ F_2(\\vec p_1,\\vec p_2)$ is the usual kernel of perturbation theory, which in the limit where $p_1 \\ll p_2$ simply reduces to $ \\vec p_1 \\cdot \\vec p_2\/( 2 p_1^2)$  \\cite{Bernardeau:2001qr}.\nThe second type of diagram, displayed on the right-hand side of Fig.~\\ref{fig:trispectrum}, is obtained when one of the short density perturbations is taken at third order; it gives\n\\begin{equation}\n\\label{cont_t_2}\n\\begin{split}\nT_{1113} &= D(\\eta_2)^2 D(\\tau_1) D(\\tau_2) P_{0}(q_1)P_{0}(q_2) F_3(-\\vec q_1, - \\vec q_2, - \\vec k_1) \\langle \\delta_{\\vec k_1}(\\eta_1)\\delta_{\\vec k_2}(\\eta_2)\\rangle' + {\\rm perms} \\\\\n& \\approx   4 \\frac{\\vec q_1\\cdot \\vec k_1}{2 q_1^2} \\frac{\\vec q_2\\cdot \\vec k_1}{2 q_2^2} \\frac{D(\\eta_2)^2}{D(\\tau_1) D(\\tau_2)} P(q_1,\\tau_1)P(q_2,\\tau_2) \\langle \\delta_{\\vec k_1}(\\eta_1)\\delta_{\\vec k_2}(\\eta_2)\\rangle' \\, ,\n\\end{split}\n\\end{equation}\nwhere, on the right-hand side of the first line, $ F_3(\\vec p_1,\\vec p_2,\\vec p_3)$ is the third-order  perturbation theory kernel,  which in the limit where $p_1, p_2 \\ll p_3$  reduces to $ (\\vec p_1 \\cdot \\vec p_3)(\\vec p_2 \\cdot \\vec p_3) \/( 4 p_1^2 p_2^2 )$ \\cite{Bernardeau:2001qr}.\nAs expected, summing up all the contributions to the connected part of the trispectrum, i.e.~$T_{1122}+T_{1131}+T_{1113}$, using eqs.~\\eqref{cont_t_1} and \\eqref{cont_t_2} and $\\vec k_2 \\approx - \\vec k_1$ one obtains eq.~\\eqref{TriSpec}. \n\n\n\\subsection{Soft Loops}\n\nSo far we have derived  consistency relations where the long modes appear explicitly as external legs. We now show that our arguments can also capture the effect on short-scale correlation functions of soft modes running in loop diagrams.\nWe already did this in eq~\\eqref{resumloop}\n\\begin{equation}\n\\langle  \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n) | {\\Phi_L} \\rangle \\rangle_{\\Phi_L} \\approx   \\exp \\bigg[ {- \\frac12 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}  \\bigg( \\sum_a D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec{p}}{p^2} \\, \\bigg)^2 P_0(p)} \\bigg]   \\langle \\delta^{(g)}_{\\vec k_1} (\\eta_1)\\cdots \\delta^{(g)}_{\\vec  k_n} (\\eta_n)\\rangle_0 \\;.\n\\end{equation}\nThe exponential in this expression can be expanded at a given order, corresponding to the number of soft loops dressing the $n$-point correlation function. Each loop  carries a contribution $\\propto k^2 \\int \\textrm{d} p P_0(p) $ to the correlation function. However, \nthis expression makes it very explicit that at all loop order these contributions have no effect on equal-time correlators, because in this case the exponential on the right-hand side is identically unity. This  confirms previous analysis on this subject  \\cite{Jain:1995kx,Scoccimarro:1995if,Bernardeau:2011vy,Bernardeau:2012aq,Blas:2013bpa,Carrasco:2013sva}. It is important to notice again, however, that in our derivation this cancellation is more general and robust that in those references, as it takes place independently of the equations of motion  for the short modes and  is completely agnostic about the short-scale physics. It simply derives from the equivalence principle.\n\n\n\nNevertheless, soft loops contribute to unequal-time correlators. As a check of the expression above, one can compute the contribution of soft modes to the 1-loop unequal-time matter power spectrum, $\\langle \\delta_{\\vec k_1}(\\eta_1) \\delta_{\\vec k_2}(\\eta_2)\\rangle'$, and verify that this reproduces the standard perturbation theory result. Expanding at order $( \\frac kp \\delta)^2$ the exponential in eq.~\\eqref{resumloop} for $n=2$, one obtains the 1-loop contribution to the power spectrum,\n\\begin{equation}\n\\label{1loop}\n\\langle \\delta^{(g)}_{\\vec k}(\\eta_1) \\delta^{(g)}_{-\\vec k}(\\eta_2)\\rangle'_{\\rm 1-soft\\,loop}\\approx -\\frac12\\left({D(\\eta_1)-D(\\eta_2)}\\right)^2 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}\\bigg(\\frac{\\vec p \\cdot \\vec k}{ p^2}\\bigg)^2  P_0(p) \\langle \\delta^{(g)}_{\\vec k}(\\eta_1) \\delta^{(g)}_{-\\vec k}(\\eta_2)\\rangle'_0 \\, .\n\\end{equation}\nLet us now compute the analogous contribution in perturbation theory. \n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{loops.pdf}\n\\caption{\\label{fig:loop} \\small Two diagrams that contribute to the 1-loop power spectrum. Left: $P_{22}$. Right: $P_{31}$.}\n\\end{figure}\nTwo types of diagrams are going to be relevant; these are shown in Fig.~\\ref{fig:loop}. The one on the left, usually called $P_{22}$, yields\n\\begin{equation}\nP_{22}\\approx  4 D(\\eta_1) D(\\eta_2) \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}\\bigg(\\frac{\\vec p \\cdot \\vec k}{2 p^2}\\bigg)^2  P_0(p) \\langle \\delta_{\\vec k}(\\eta_1) \\delta_{-\\vec k}(\\eta_2)\\rangle'_0 \\, ,\n\\end{equation}\nwhile the diagram on the right, $P_{31}$,  gives\n\\begin{equation}\nP_{31} \\approx - 2 D(\\eta_1)^2 \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3}\\bigg(\\frac{\\vec p \\cdot \\vec k}{2 p^2}\\bigg)^2 P_0(p)  \\langle \\delta_{\\vec k}(\\eta_1) \\delta_{-\\vec k}(\\eta_2)\\rangle'_0 \\, .\n\\end{equation}\nSumming up all the different contributions, $P_{22} + P_{31} + P_{13}$, one obtains eq.~\\eqref{1loop}.\n\n\\subsection{Soft internal lines}\nAnother kinematical regime in which the consistency relations can be applied is the limit in which the sum of some of the external momenta becomes very small, for instance $|\\vec k_1 +\\cdots + \\vec k_m |\\ll k_1, \\ldots, k_m$. In this limit, the dominant contribution to the $n$-point function comes from the diagram where $m$ external legs of momenta $\\vec k_1,\\ldots,\\vec k_m$ exchange soft modes with momentum $\\vec q=\\vec k_1 +\\cdots + \\vec k_m$ with $n-m$ external legs with momenta $\\vec k_{m+1}, \\ldots, \\vec k_n$ (for an analogous case in inflation see \\cite{Seery:2008ax,Leblond:2010yq}). In the language of our approach, this contribution comes from averaging a product of $m$-point and $(n-m)$-point functions under the effect of long modes. \n\n\\def\\delta^{(g)}{\\delta}\n\nIn this case, the $n$-point function in real space can be written as\n\\begin{equation}\n\\begin{split}\n\\langle\\delta^{(g)}(\\vec x_1,\\eta_1) & \\cdots \\delta^{(g)}(\\vec x_m, \\eta_m)  \\; \\delta^{(g)}(\\vec x_{m+1},\\eta_{m+1}) \\cdots \\delta^{(g)}(\\vec x_n,\\eta_n) \\rangle  \\\\\n& \\approx \\langle \\langle\\delta^{(g)}(\\vec x_1,\\eta_1) \\cdots \\delta^{(g)}(\\vec x_m, \\eta_m) | \\Phi_L \\rangle \\langle \\delta^{(g)}(\\vec x_{m+1},\\eta_{m+1}) \\cdots \\delta^{(g)}(\\vec x_n,\\eta_n) | \\Phi_L \\rangle \\rangle_{\\Phi_L}\\;,\n\\end{split}\n\\end{equation}\nwhere here and in the rest of the section we drop the superscript ${}^{(g)}$ on the galaxy density contrast to lighten the notation.\nNow we can straightforwardly apply the equations from the previous sections. As before, the long mode can be traded for the change of coordinates. Rewriting the right-hand side in Fourier space we get \n\\begin{equation}\n\\label{isl1}\n\\begin{split}\n\\langle\\delta^{(g)}(\\vec x_1,\\eta_1) & \\cdots \\delta^{(g)}(\\vec x_m,\\eta_m)\\; \\delta^{(g)}(\\vec x_{m+1},\\eta_{m+1}) \\cdots \\delta^{(g)}(\\vec x_n,\\eta_n)\\rangle  \\\\\n& \\approx \\int \\frac{\\textrm{d}^3k_1}{(2\\pi)^3}\\cdots\\frac{\\textrm{d}^3k_n}{(2\\pi)^3} \\langle\\delta^{(g)}_{\\vec k_1}(\\eta_1)\\cdots \\delta^{(g)}_{\\vec k_m}(\\eta_m) \\rangle_0 \\langle \\delta^{(g)}_{\\vec k_{m+1}}(\\eta_{m+1}) \\cdots \\delta^{(g)}_{\\vec k_n}(\\eta_n)\\rangle_0  \\, e^{i \\sum_a  \\vec k_a \\cdot \\vec x_a} \\\\\n& \\times \\left \\langle \\exp\\bigg[i  \\sum_{a=1}^m \\vec k_a \\cdot \\delta \\vec x( \\vec y_1,\\eta_a)\\bigg] \\cdot \\exp\\bigg[i \\sum_{a=m+1}^n \\vec k_a \\cdot \\delta \\vec x( \\vec y_2,\\eta_a)\\bigg]  \\right \\rangle_{\\Phi_L} \\;,\n\\end{split}\n\\end{equation}\nwhere $\\vec y_1$ and $\\vec y_2$ are two different points  respectively close to $(\\vec x_1,\\vec x_2, \\ldots, \\vec x_m )$ and $(\\vec x_{m+1}, \\vec x_{m+2} , \\ldots ,\\vec x_n)$. \nThe average over the long mode can be rewritten as\n\\begin{equation}\n\\left \\langle \\exp\\left[\\int^\\Lambda \\frac{\\textrm{d} ^3 \\vec p}{(2\\pi)^3} \\big( J_1(\\vec p)+ J_2(\\vec p) \\big) \\delta_0(\\vec p) \\right] \\  \\right \\rangle_{\\Phi_L}\n\\end{equation}\nwith\n\\begin{equation}\nJ_1(\\vec p) = \\sum_{a=1}^m D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec p}{p^2} e^{i \\vec p \\cdot \\vec y_1} \\;, \\quad J_2(\\vec p) = \\sum_{a=m+1}^n D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec p}{p^2} e^{i \\vec p \\cdot \\vec y_2} \\;.\n\\end{equation}\nTaking the expectation value over the long mode using the  expression for averaging the exponential of a Gaussian variable, i.e.~eq.~\\eqref{SumRes},  eq.~\\eqref{isl1} can be written as\n\\begin{equation}\n\\label{sl2}\n\\begin{split}\n\\langle\\delta^{(g)}(\\vec x_1,\\eta_1) & \\cdots \\delta^{(g)}(\\vec x_m,\\eta_m)\\; \\delta^{(g)}(\\vec x_{m+1},\\eta_{m+1}) \\cdots \\delta^{(g)}(\\vec x_n,\\eta_n)\\rangle  \\\\\n& \\approx  \\int \\frac{\\textrm{d}^3k_1}{(2\\pi)^3}\\cdots\\frac{\\textrm{d}^3k_n}{(2\\pi)^3}\\langle\\delta^{(g)}_{\\vec k_1}(\\eta_1)\\cdots \\delta^{(g)}_{\\vec k_m}(\\eta_m) \\rangle ' \\langle \\delta^{(g)}_{\\vec k_{m+1}}(\\eta_{m+1}) \\cdots \\delta^{(g)}_{\\vec k_n}(\\eta_n)\\rangle'   \\, e^{i \\sum_a \\vec k_a \\cdot \\vec x_a}\\\\\n& \\times \\exp \\left[ - \\int^\\Lambda \\frac{\\textrm{d}^3 p}{(2\\pi)^3} J_1(\\vec p) J_2(\\vec p) P_0(\\vec p) \\right]  \\;.\n\\end{split}\n\\end{equation}\n\nWe are interested in the soft internal lines, that come from the cross term, i.e.~the last line of eq.~\\eqref{sl2}. Notice that $J_1(\\vec p)$ and $J_2(\\vec p)$ are evaluated at different points $\\vec y_1$ and $\\vec y_2$ separated by a distance $\\vec x$.\\footnote{For definiteness, we can choose $\\vec y_1=\\frac{1}{m}\\sum_{a=1}^m \\vec x_a$ and $\\vec y_2=\\frac{1}{n-m}\\sum_{a=m+1}^n \\vec x_a$.}\nIt is lengthy but straightforward to take the Fourier transform of this equation, which yields\n\\begin{equation}\n\\begin{split}\n\\label{soft_lines_final}\n\\langle\\delta^{(g)}_{\\vec k_1}(\\eta_1)& \\cdots \\delta^{(g)}_{\\vec k_m}(\\eta_m) \\delta^{(g)}_{\\vec k_{m+1}}(\\eta_{m+1})\\cdots \\delta^{(g)}_{\\vec k_n}(\\eta_n) \\rangle' \\\\ &\n\\approx \\langle\\delta^{(g)}_{\\vec k_1}(\\eta_1)\\cdots \\delta^{(g)}_{\\vec k_m}(\\eta_m) \\rangle' \\,  \\langle \\delta^{(g)}_{\\vec k_{m+1}}(\\eta_{m+1}) \\cdots \\delta^{(g)}_{\\vec k_n}(\\eta_n)\\rangle'\\\\\n&\\times\\int \\textrm{d}^3  x  \\, e^{-i\\sum_{i=1}^m \\vec k_i\\cdot \\vec x} \\exp \\bigg[ - \\int^\\Lambda \\frac{\\textrm{d} ^3  p}{(2\\pi)^3}e^{i\\vec p\\cdot \\vec x}   \\, \\sum_{a=1}^{m} D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec{p}}{p^2}  \\sum_{a=m+1}^{n} D(\\eta_a) \\frac{\\vec k_a \\cdot \\vec{p}}{p^2} \\, P_0(p)\\bigg]\\,.\n\\end{split}\n\\end{equation}\nThe last line  encodes the effect of soft modes with total momentum $\\vec q=\\vec k_1 +\\cdots + \\vec k_m$ exchanged between $m$ external legs of momenta $\\vec k_1,\\ldots,\\vec k_m$ and $n-m$ external legs with momenta $\\vec k_{m+1}, \\ldots, \\vec k_n$, in the limit $q\/k_i \\to 0$. Expanding the exponential at a given order in $P_0(p)$ yields the number of soft lines exchanged. The integral in $\\textrm{d}^3 x$ ensures that  the sum of the internal momenta is $\\vec q$.\n\nEquation \\eqref{soft_lines_final} can be easily generalized to consider the case where more than two sums of momenta become small, i.e.~when soft internal lines are exchanged between more than two hard-modes diagrams. The conclusion is always the same: soft internal lines do not contribute to equal time correlators at order $\\propto k^2 \\int \\textrm{d} p P_0(p) $. Again, this statement is very general irrespectively of the assumption about the short scales.\n\n\nAs a concrete example, let us consider the case $m=2, n=4$, i.e.~a 4-point function in the collapsed limit $|\\vec k_1 + \\vec k_2 |\\ll k_1, k_2$, and the exchange of a single soft line. In this case, expanding the exponential at first order in $P_0(p)$, the above equation yields\n\\begin{equation}\n\\label{t_wsl}\n\\begin{split}\n\\langle\\delta_{\\vec k_1}(\\eta_1)& \\delta_{\\vec k_2}(\\eta_2) \\delta_{\\vec k_{3}}(\\eta_3) \\delta_{\\vec k_4}(\\eta_4) \\rangle_c' \\\\ &\n\\approx - \\, \\langle\\delta(\\vec k_1,\\eta_1) \\delta(\\vec k_2,\\eta_2) \\rangle ' \\langle \\delta(\\vec k_{3},\\eta_{3}) \\cdots \\delta(\\vec k_4,\\eta_4)\\rangle'\\\\\n&\\times   \\int^\\Lambda \\textrm{d} ^3  p \\big( D(\\eta_1) - D(\\eta_2) \\big) \\frac{\\vec k_1 \\cdot \\vec{p}}{p^2} \\, \\big(  D(\\eta_3)  - D(\\eta_4) \\big) \\frac{\\vec k_3 \\cdot \\vec{p}}{p^2} \\,  P_0(p) \\delta_D(\\vec p - \\vec k_1 - \\vec k_2)\\,,\n\\end{split}\n\\end{equation}\nwhere we have considered only the connected diagram and, for simplicity, we are neglecting soft loops attached to each lines.\nTo compare with perturbation theory, we need to compute the tree-level exchange diagram. The contribution from taking $\\vec k_1$ and $\\vec k_3$ at second order yields \n\\begin{equation}\nT_{2121} \\approx - 4 D(\\eta_1) D(\\eta_3) P_0(|\\vec k_1+\\vec k_2|) \n\\frac{\\vec k_1 \\cdot (\\vec k_1 + \\vec k_2 )}{  2 |\\vec k_1 + \\vec k_2|^2} \\frac{\\vec k_3 \\cdot (\\vec k_1 + \\vec k_2 )}{ 2  |\\vec k_1 + \\vec k_2|^2} \n\\langle \\delta_{\\vec k_1}(\\eta_1) \\delta_{\\vec k_2}(\\eta_2)\\rangle'  \\langle \\delta_{\\vec k_3}(\\eta_3) \\delta_{\\vec k_4}(\\eta_4)\\rangle'   \\; ,\n\\end{equation}\nand summing up the other permutations lead to\n\\begin{equation}\n\\begin{split}\n\\langle\\delta_{\\vec k_1}(\\eta_1)& \\delta_{\\vec k_2}(\\eta_2) \\delta_{\\vec k_3}(\\eta_3) \\delta_{\\vec k_4}(\\eta_4)\\rangle_c' \\approx -  \\big(D(\\eta_1)-D(\\eta_2)\\big) \\big( D(\\eta_3) - D(\\eta_4) \\big) P_0(|\\vec k_1+\\vec k_2|) \\\\ & \\times\n\\frac{\\vec k_1 \\cdot (\\vec k_1 + \\vec k_2 )}{  |\\vec k_1 + \\vec k_2|^2} \\frac{\\vec k_3 \\cdot (\\vec k_1 + \\vec k_2 )}{ |\\vec k_1 + \\vec k_2|^2} \n\\langle \\delta_{\\vec k_1}(\\eta_1) \\delta_{\\vec k_2}(\\eta_2)\\rangle'  \\langle \\delta_{\\vec k_3}(\\eta_3) \\delta_{\\vec k_4}(\\eta_4)\\rangle'   \\;,\n\\end{split}\n\\end{equation}\nwhich confirms eq.~\\eqref{t_wsl}. \nOne can easily extend this check to the case of several soft-lines.\n\n\n\\def\\delta^{(g)}{\\delta^{(g)}}\n\n\\section{\\label{sec:redshift}Going to redshift space}\n\nThe derivation of the consistency relations has been done in real space, but the galaxy distribution will of course be observed in redshift space. It is thus natural to ask if it is possible to write relations directly in terms of redshift space correlation function. Before doing so, let us stress that it will be difficult---if not impossible---to measure  consistency relations at different times. To see the effect of the long mode, one would like to measure at quite different redshifts the short-scale correlation function at a spatial distance which is much smaller than Hubble. This is of course impossible since we can only observe objects on our past lightcone. This implies that, although one can check the consistency relations at different times in simulations, for real data we will have to stick to correlation functions at the same time. Given that the consistency relations vanish at equal time, their main phenomenological interest will be to look for their possible violations, which would indicate that one of the assumptions does not hold. This would represent a detection of either multi-field inflation or violation of the equivalence principle (or both!)\n\nThe mapping between real space $\\vec x$ and redshift space $\\vec s$ in the plane-parallel approximation is given by\n\\begin{equation}\n\\vec s = \\vec x + \\frac{v_z}{{\\cal H}} \\hat z\\;, \\label{rs-rs}\n\\end{equation}\nwhere $\\hat z$ is the direction of the line of sight, $v_z \\equiv \\vec v \\cdot \\hat z$, and $\\vec v$ is the peculiar velocity. Also the relation between $z$ and $\\eta$ receives corrections due to peculiar velocities. These corrections are small for sufficiently distant objects for which $v \\ll H x$. Notice that we do not assume that the peculiar velocity is a function of the position $\\vec x$ since this holds only in the single-stream approximation, which breaks down for virialized objects on small scales \\cite{Seljak:2011tx,Vlah:2012ni}. \n\nThe derivation of the consistency relations follows closely what we did in real space, once we observe that also in redshift space the long mode induces a (time-dependent) translation. Indeed we have\n\\begin{align}\n\\vec x &\\to \\vec x  +  D \\, \\vec \\nabla \\Phi_{0,L}\\;, \\label{traslation} \\\\\n\\vec v & \\to \\vec  v+ f {\\cal H} D \\, \\vec  \\nabla \\Phi_{0,L} \\label{vL} \\;,\n\\end{align}\nwhere $D(\\eta)$ is the growth factor, $f(\\eta) \\equiv d \\ln D\/d \\ln a$ is the growth rate and $\\vec \\nabla \\Phi_{0,L}$ a homogenous gradient of the initial gravitational potential $\\Phi_{0,L}$, related to $\\delta_0$ defined in eq.~\\eqref{deltax2delta} by $\\nabla^2 \\Phi_{0,L} = \\delta_{0,L}$. \nThis corresponds to a redshift space translation \n\\begin{align}\n\\vec{\\tilde s} &= \\vec s + \\delta \\vec s\\;, \\\\\n\\delta \\vec s & \\equiv D \\, ( \\vec \\nabla \\Phi_{0,L} + f  \\nabla_z \\Phi_{0,L} \\hat z) \\label{deltas}\\;,\n\\end{align}\nwhere we have applied to eq.~\\eqref{rs-rs}  a spatial translation of the real-space coordinates and a  shift of the peculiar velocity along the line of sight, respectively eqs.~\\eqref{traslation} and \\eqref{vL}.\nAs in real space, we can thus conclude that a redshift-space correlation function in the presence of a long mode $\\Phi_L$ is the same as the correlation function in the absence of the long mode but in {\\em translated} redshift-space coordinates:\\begin{equation}\n\\begin{split}\n\\langle \\delta^{(g,s)}(\\vec s_1,\\eta_1) \\cdots \\delta^{(g,s)}(\\vec s_n,\\eta_n) | {\\Phi_L}\\rangle &\\approx \\langle \\delta^{(g,s)}(\\vec{\\tilde  s}_1,\\eta_1) \\cdots \\delta^{(g,s)}(\\vec{\\tilde  s}_n,\\eta_n) \\rangle\\; \\\\\n& = \\sum_a \\delta \\vec s_a  \\langle \\delta^{(g,s)}(\\vec s_1,\\eta_1) \\cdots \\vec \\nabla_a \\delta^{(g,s)}(\\vec s_a,\\eta_a) \\cdots \\delta^{(g,s)}(\\vec s_n,\\eta_n) \\rangle \\;, \\label{expansion}\n\\end{split}\n\\end{equation}\nwhere $\\delta \\vec s_a  \\equiv D_a \\, ( \\vec \\nabla_a \\Phi_{0,L} + f_a  \\nabla_{a,z} \\Phi_{0,L} \\hat z) $. To show this notice that the density in redshift space can be written in terms of the real-space distribution function \\cite{Seljak:2011tx,Vlah:2012ni}\n\\begin{equation}\n\\rho_s(\\vec s) = m a^{-3} \\int \\textrm{d}^3 p \\; f \\left(\\vec s - \\frac{v_z}{\\cal H} \\hat z, \\vec p\\right) \\;,\n\\end{equation}\nwhere $m$ is the mass of the particles and $\\vec p$ is the physical momentum. The statistical properties of $\\rho_s(\\vec s)$ in the presence of the long mode are inherited by its expression in real space\n\\begin{equation}\n\\rho_s(\\vec s)_{\\Phi_L} =  \\frac{m}{a^3} \\int \\textrm{d}^3 p \\; f \\left(\\vec s - \\frac{v_z}{\\cal H} \\hat z +\\delta\\vec x, \\vec p +a m \\delta \\vec v\\right) = \\frac{m}{a^3} \\int \\textrm{d}^3 p' \\; f \\left(\\vec s - \\frac{v_z- \\delta v_z}{\\cal H} \\hat z +\\delta\\vec x, \\vec {p'}\\right) = \\rho_s(\\vec s + \\delta \\vec s),\n\\end{equation} \nwhere $\\delta x$ and $\\delta \\vec v$ are given by eqs.~\\eqref{traslation} and \\eqref{vL}.\n\nAgain this statement can be directly applied to the galaxy distribution and it thus includes the bias with respect to the dark matter distribution.\nNotice that in the plane-parallel approximation redshift space is still translationally invariant (although it is not rotationally invariant, since the line-of-sight is a preferred direction): correlation function only depends on the distance between points. This implies that the consistency relations will be zero when the short modes are taken at equal time, since the common translation does not change distances.\n\n\nIn the Fourier space conjugate to redshift space, eq.~\\eqref{expansion} becomes\n\\begin{equation}\n\\langle  {\\Phi_{0}}({\\vec q}) \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\cdots \\delta^{(g,s)}_{\\vec k_n}(\\eta_n) \\rangle \\approx P_\\Phi (q) \\sum_a D(\\eta_a) \\big[  \\vec q \\cdot \\vec k_a + f(\\eta_a) q_z \\, k_{a,z} \\big]  \n\\langle  \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\cdots \\delta^{(g,s)}_{\\vec k_n}(\\eta_n) \\rangle \\;.\n\\end{equation}\nBy using for the long mode the linear relation between the density contrast in redshift space $\\delta$ and the gravitational potential $\\Phi$, i.e.\n\\begin{equation}\n\\delta^{(g,s)}({\\vec q},\\eta) = - (b_1+ f \\mu_{\\vec q}^2) D(\\eta) q^2 \\Phi_0 ({\\vec q}) \\;,\n\\end{equation}\nwhere $b_1$ is a linear bias parameter between galaxies and dark matter and $\\mu_{\\vec k} \\equiv \\vec k \\cdot \\hat z\/k$, the consistency relation above becomes\n\\begin{equation}\n\\label{CR_rs}\n\\begin{split}\n\\langle  \\delta^{(g,s)}_{\\vec q}(\\eta) \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\cdots \\delta^{(g,s)}_{\\vec k_n}(\\eta_n) \\rangle \\approx &  - \\frac{P_{g,s}(q,\\eta)}{b_1+ f \\mu_{\\vec q}^2} \\sum_a \\frac{D(\\eta_a)}{D(\\eta)} \\frac{k_a}{q} \\big[  \\hat q \\cdot \\hat k_a + f(\\eta_a) \\mu_{\\vec q} \\, \\mu_{\\vec k_a} \\big] \\\\ &\\times \\langle  \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\cdots \\delta^{(g,s)}_{\\vec k_n}(\\eta_n) \\rangle \\;.\n\\end{split}\n\\end{equation}\n\nWe can check that this relation holds in perturbative calculation of redshift space distortions. The redshift space bispectrum reads \\cite{Bernardeau:2001qr}\n\\begin{equation}\n\\begin{split}\n\\langle  \\delta^{(g,s)}_{\\vec q}(\\eta) \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\delta^{(g,s)}_{\\vec k_2}(\\eta_2) \\rangle' & = \\\\ 2 Z_2(-\\vec q, -\\vec k_2;\\eta_1) Z_1(\\vec q;\\eta) Z_1(\\vec k_2;\\eta_2)   &\\langle  \\delta({\\vec q}, \\eta) \\delta({-\\vec q},\\eta_1) \\rangle' \\langle \\delta({\\vec k_1},\\eta_1) \\delta({\\vec k_2}, \\eta_2) \\rangle'  \\text{ + cyclic}\\;,\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\nZ_1(\\vec k;\\eta) & \\equiv (b_1 + f \\mu_{\\vec k}^2) \\;, \\\\\nZ_2(\\vec k_a,\\vec k_b;\\eta) & \\equiv b_1 F_2(\\vec k_a,\\vec k_b) +f \\mu_{\\vec k}^2 G_2(\\vec k_a,\\vec k_b) + \\frac{f \\mu_{\\vec k} k}{2} \\left[\\frac{\\mu_{\\vec k_a}}{k_a}(b_1+f\\mu_{\\vec k_b}^2)+ \\frac{\\mu_{\\vec k_b}}{k_b}(b_1+f\\mu_{\\vec k_a}^2)\\right] +\\frac{b_2}{2} \\;.\n\\end{split}\n\\end{equation}\nHere $b_1$ and $b_2$ are the linear and non-linear bias parameters and $F_2$ and $G_2$ are the standard second-order perturbation kernels for density and velocity respectively \\cite{Bernardeau:2001qr}.\nIn the limit $q \\to 0$ we have\n\\begin{equation}\n2 Z_2 (-\\vec q, -\\vec k_2;\\eta_1) \\approx (b_1 + f_1 \\mu_{\\vec k_2}^2) \\frac{\\vec q \\cdot \\vec k_2}{q^2} + (b_1 + f_1 \\mu_{\\vec k_2}^2) f_1 \\frac{k_2}{q} \\mu_{\\vec q} \\mu_{\\vec k_2} \\;.\n\\end{equation}\nThis gives\n\\begin{equation}\n\\begin{split}\n\\langle  \\delta^{(g,s)}_{\\vec q}(\\eta) & \\delta^{(g,s)}_{\\vec k_1}(\\eta_1) \\delta^{(g,s)}_{\\vec k_2}(\\eta_2) \\rangle'  \\\\ & \\approx \\frac{P_{g,s}(q,\\eta)}{b_1+ f \\mu_{\\vec q}^2} \\frac{D(\\eta_1)}{D(\\eta)}  (b_1 + f_1 \\mu_{\\vec k_2}^2) \\left( \\frac{\\vec q \\cdot \\vec k_2}{q^2} +  f_1 \\frac{k_2}{q} \\mu_{\\vec q} \\mu_{\\vec k_2} \\right)   Z_1(\\vec k_2;\\eta_2)  \\langle  \\delta({\\vec k_1},\\eta_1) \\delta({\\vec k_2}, \\eta_2) \\rangle' \\\\ & \\approx -\\frac{P_{g,s}(q,\\eta)}{b_1+ f \\mu_{\\vec q}^2} \\frac{D(\\eta_1)}{D(\\eta)}   \\frac{k_1}{q} (\\hat q \\cdot \\hat k_1 +  f_1  \\mu_{\\vec q} \\mu_{\\vec k_1})   \\langle  \\delta({\\vec k_1},\\eta_1) \\delta({\\vec k_2}, \\eta_2) \\rangle'  \\text{  + $(1 \\leftrightarrow 2)$}\\;.\n\\end{split}\n\\end{equation}\nThe consistency relation is satisfied.\n\nAs in real space, it is possible to derive a resummed version of eq.~\\eqref{CR_rs}. The translation in redshift space introduces a factor\n\\begin{equation}\n\\exp \\Big[ i  \\sum_a \\vec k_a \\cdot \\delta \\vec s( \\vec y,\\eta_a) \\Big] =\\exp \\Big[  \\int^\\Lambda\\frac{\\textrm{d}^3p}{(2\\pi)^3} \\sum_a D(\\eta_a)\\left( \\vec p \\cdot \\vec k_a + f(\\eta_a) p_z \\, k_{a,z}\\right)  \\,e^{i  \\vec p \\cdot \\vec{y}}\\Phi_{0}(\\vec p) \\Big]\\;\n\\end{equation}\nin the correlation functions. It is then straightforward to show that, as in Sec.~\\ref{sec:resum}, the consistency relation in redshift space eq.~\\eqref{CR_rs} remains the same even when the effect of all soft modes is resummed.\nMoreover, using the same procedures developed in the previous section, one can easily extend the consistency relations with multiple soft legs, softs loops and soft internal lines to redshift space.\n\n\n\n\\section{Conclusions}\nIn this paper we showed that one can have a complete control of soft modes at any order in $\\frac{k}{q} \\cdot \\delta_q$. The known cancellation of these effects for equal time correlators \\cite{Jain:1995kx,Scoccimarro:1995if,Bernardeau:2011vy,Bernardeau:2012aq,Blas:2013bpa,Carrasco:2013sva} is now on more general grounds:  it is physically a consequence of the equivalence principle and the lack of statistical correlation between long and short modes, which holds in single-field inflation. Therefore this cancellation is very robust and holds beyond the single-stream approximation, and including the effects of baryons on short scales. These regimes are beyond the usual arguments based on perturbation theory.  Moreover, we now know exactly what is the effect of soft modes on correlators at different times.\nTo make contact with observations one has to understand if the consistency relations can be written directly in redshift space. We showed that this is the case, without adding any assumption about the short modes: for example one does not need to assume the single-stream approximation, which breaks down on short scales.\n\nBesides the theoretical interest of these results, the main conclusion for observations is that a detection in the squeezed limit of a $1\/q$ behaviour at equal time would be a robust detection of either multi-field inflation or a violation of the equivalence principle. The next step is to evaluate how constraining measurements will be for explicit models that do not respect equivalence principle, taking into account that in the data one is obviously limited in the hierarchy between $k$ and $q$. We will come back to this in a future publication \\cite{Creminelli:2013nua}.\n\n\\subsection*{Acknowledgements}\nWhile finishing this paper reference \\cite{Peloso:2013spa} appeared. There is no disagreement with our results: in particular, we both agree that a violation of the EP implies a breaking of the consistency relations in the form of eq.~\\eqref{CR}. We thank M.~Peloso and M.~Pietroni for discussions. We acknowledge related work by A.~Kehagias, J.~Nore\\~na, H.~Perrier and A.~Riotto: where comparison is possible, the results agree.\nIt is a pleasure to thank V.~Desjacques, R.~Scoccimarro and M.~Zaldarriaga for useful discussions, and the anonymous referee for useful comments. JG and FV acknowledge partial support by the ANR {\\it Chaire d'excellence} CMBsecond ANR-09-CEXC-004-01.\n\n\\footnotesize\n\\parskip 0pt\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:Introduction}\n\n\\IEEEPARstart{H}{uman} Presence-aware Systems (HPSs) are rapidly\ngrowing as new services become available in various areas of modern\nlife \\cite{Petrov-2019} such as assisted living, ambient intelligence,\nsmart spaces, home automation, human-robot collaboration, safety and\nsecurity, just to cite a few. Among these applications, non-cooperative,\nalso known as passive, HPS are the most attractive since they do not\nrequire the monitored users to carry or wear any electronic device\nor specific sensors. Usually, these systems are vision-based \\cite{dalal-2005,benezeth2011towards,choi-2013};\nhowever, the ubiquitous presence of wireless networks paves the way\ntowards the exploitation of wireless radio-frequency (RF) networks,\nnot only as communication devices, but also as body proximity\/location\nvirtual sensors. Last but non least, radio-based HPS are privacy-neutral\nsince they do not reveal any privacy information about the monitored\npeople.\n\nHPS systems exploit the fact that people, or obstacles, in the surroundings\nof an area covered by a wireless radio network induce signal alterations\nthat can be detected and exploited for body occupancy inference applications.\nFor instance, Device-Free Localization (DFL) systems \\cite{Youssef:2007:CDP:1287853.1287880,savazzi2016magazine}\nexploit a network of RF nodes to detect the presence, to locate and\ntrack the position of moving objects or people in a confined area\ncovered by the wireless network itself. However, a radio-based HPS\nis not only able to localize and track \\cite{wilson10,nicoli2016eusipco,wang-2017}\npeople, or objects, but it has been also proven to efficiently perform\nother tasks such as to count the number of people \\cite{depatla2015jsac},\nto identify and recognize patterns related to their activities \\cite{savazzi2016icassp,wang2015icmcn}\nand intentions \\cite{savazzi2016icassp}, to detect dangerous worker\nconditions and safety status \\cite{kianoush2016iot,Talebpour-2019},\nand act as a proximity monitor \\cite{montanari2017proximity}. This\nis made possible as the presence of targets (\\ie objects or people)\naffects the propagation of the radio waves in the covered area \\cite{Woyach-et-al,patwari10},\nfor example by inducing predictable alterations of the Received Signal\nStrength (RSS) field that depend on the targets position, in both\nstatic \\cite{Seifeldin-et-al} and dynamic \\cite{Saeed-et-al} environments.\n\n\\subsection{Related works}\n\n\\label{subsec:Related-works}\n\nThe effect of the presence of people on the received RF signals is\na well know topic \\cite{obayashi1998body,villanese2000pedestrian}\nand finds its roots in the research activities about the electromagnetic\n(EM) propagation phenomena caused by natural or artificial obstacles\nduring the first experimental trials at the dawn of the radio era\n\\cite{Brittain1994}. These studies have received a great impulse\nafter the middle of the last century mostly for outdoor coverage applications\n\\cite{furutsu-1963,vogler-1982,comparative,tzaras-2000}. However,\ndespite some recent attempts to model the body-induced fading effects\non short-range radio propagation \\cite{smith2013propagation}, these\nresearch activities are mostly related to inter- \\cite{Koutitas,de2015analysis}\nand intra-body \\cite{Namjun_Cho-et-al,andreu2016spatial} short-range\nradio communications. The aim of these research activities is to quantify\nthe radio propagation losses in narrow \\cite{humanbody2} or wide-band\n\\cite{Fort-et-al} indoor scenarios with the main purpose of mitigating\nthese effects. Only a few research works \\cite{Koutitas,liu2009fading,conducting_cylinder}\nfocus their attention on the geometrical relations between the transmitter\n(TX) and receiver (RX) location, the position and composition of the\nbody, and its size.\n\nA general EM model for the prediction of the mathematical relations\nbetween location, size and composition of a \\emph{single-target} and\nthe corresponding EM field perturbation, is still disputable as shown\nin \\cite{hamilton-et-al,yiugitler2016experimental} or too complex\nto be of practical use as based on ray tracing techniques \\cite{de2015analysis,eleryan-et-al}\nor Uniform Theory of Diffraction (UTD) \\cite{Koutitas,conducting_cylinder}.\nOther EM methods \\cite{kibret2015characterizing,ziri2005prediction,yokota2012propagation}\nand physical-statistical models \\cite{eleryan-et-al,mohamed2017physical,rampa2017em,Hillyard-2020}\nare simpler than the previous ones but still limited to a single target.\nTo the authors knowledge, and according to the current literature,\nan analytical, or semi-analytical, approach towards a \\emph{true}\n\\emph{multi-body} model has never been tackled before. Usually, multi-target\n(\\ie multi-body) problems have been solved by assuming the linear\nsuperposition of the single-body extra attenuations \\cite{patwari10,wilson10}.\nHowever, the mutual effects induced by multiple bodies moving concurrently\nin the same space must be accounted for.\n\nIn \\cite{nicoli2016eusipco} a DFL system has been proposed to track\ntwo targets moving concurrently by using an EM model that is fully\ndescribed in \\cite{Rampa-2019}. On the contrary, in this paper, the\nmodel is generalized to an \\emph{arbitrary number of targets}. A practically-usable\nphysical-statistical model is thus designed for the prediction and\nthe evaluation of the body-induced propagation losses, namely the\nRSS field, found in true $N$-targets scenarios with $N\\geq1$. This\n$N$-body model is able to describe both dominant static component\nand stochastic fluctuations of the power loss as a function of the\nlocations of the $N$ targets, their size, orientation and random\nmovements with respect to the link path.\n\n\\subsection{Original contributions}\n\n\\label{subsec:Original-contributions}\n\nThe paper proposes an EM framework where the field perturbations induced\nby an arbitrary number of human bodies are modelled as a superposition\nof \\emph{diffraction} and\\emph{ multipath} terms. The diffraction\ncomponent is defined according to the scalar diffraction theory and\nis characterized by the geometrical description (\\ie location, size,\norientation) and the movement characteristics (\\ie rotations and\nrandom movements around the nominal position) of $N$ targets according\nto the knife-edge hypothesis \\cite{lee1978path,vogler-1982,Edge_diffraction,deygout1991correction}.\nThe multipath fading term is assumed to impair the radio link due\nto the presence of the bodies placed inside the sensitivity area \\cite{rampa2015letter}\naround the LOS (Line Of Sight) path that connects the transmitter\nand the receiver. However, unlike \\cite{rampa2015letter}, where RSS\nperturbations are predicted for a \\emph{single} \\emph{small} target\n\\cite{comparative,rampa2015letter,knife_edge} moving only in the\ncentral part of the LOS path according to the paraxial approximation\n\\cite{rampa2015letter}, this novel model provides a representation\nof the power losses induced by \\emph{multiple} bodies having \\emph{any}\nsize, and placed \\emph{anywhere} in the area surrounding the radio\nlink. The model presented here extents the dual-body case exploited\nin \\cite{nicoli2016eusipco} and then presented in \\cite{Rampa-2019},\nby considering a generic EM scenario with an arbitrary number of human\nbodies in the surroundings of a radio link. In the former reference\n\\cite{nicoli2016eusipco}, the dual-body model is neither derived\nnor justified but it is just introduced to perform DFL tasks and compare\nthe results against other methods. In the latter reference \\cite{Rampa-2019},\nthe dual-body model is derived from prime principles and then described\nand discussed in details. The experimental results presented here\nconfirm that the proposed model can effectively describe the mathematical\nrelations between the target positions and the measured RSS values.\nComparisons with the results obtained with the EM simulator Feko also\nsupport the validity of the proposed model.\n\nThe novel contributions of this paper are: \\emph{i}) the definition\nof a general EM framework for the multi- body scenario; \\emph{ii})\nthe derivation, from prime principles, of the full equations for the\nprediction of the global extra attenuation due to \\emph{$N$} bodies,\nor objects, in the LOS area; \\emph{iii}) the derivation of the analytical\nformulas in the case of paraxial hypothesis for the general $N$ bodies\nscenario; \\emph{iv}) the evaluation of the extra attenuation predictions\nfor the dual-body scenario (\\ie \\emph{$N=2$}) and their comparison\nagainst the results obtained using full EM simulations; and \\emph{v})\ntuning of the dual-body model parameters based on-field RSS measurement\ntrials and comparisons of the model predictions against the aforementioned\nRSS measurements.\n\nThe paper is organized as follows. The diffraction model that accounts\nfor the deterministic term of the multi-body induced extra attenuation\nis shown in Sect. \\ref{sec:Diffraction-model} for any number $N$\nof the targets. The complete physical-statistical model for the prediction\nof the RSS field is illustrated in Sect. \\ref{sec:Physical-statistical-modeling}.\nIn particular, the dual-body model is highlighted as a practical case\nstudy. Sect. \\ref{subsec:Model-calibration} deals with the evaluation\nof the proposed multi-target model featuring a comparative analysis\nagainst experimental measurements and simulation results. The concluding\nremarks are drawn in Sect. \\ref{sec:Conclusions}.\n\n\\begin{figure}[tp]\n\\begin{centering}\n\\includegraphics[clip,scale=0.35]{Fig1}\n\\par\\end{centering}\n\\caption{\\label{fig:layout}a) Generic layout of a HPS-based wireless network\ncomposed by $D$ nodes and $L$ links where $T_{n}$ is the $n$-th\ntarget located inside the monitored area; b) Simplified representation\nof the single-link single-body scenario where the human body $T_{1}$\nis sketched as a 3D cylinder and then simplified as a 2D knife-edge\nsurface $S_{1}$.}\n\n\\vspace{-0.4cm}\n\\end{figure}\n\n\n\\section{Diffraction framework for the multi-body scenario}\n\n\\label{sec:Diffraction-model}\n\nAs sketched in Fig. \\ref{fig:layout}.a, a generic HPS consists of\na mesh of partially, or completely connected, wireless network composed\nof $D$ RF nodes \\cite{Youssef:2007:CDP:1287853.1287880,savazzi2016magazine}\nand $L\\leq D\\left(D-1\\right)\/2$ bidirectional links. The HPS-enabled\nnetwork is composed by nodes that are able to perform power measurements\non the RF signal and, after some processing steps, to extract body\noccupancy-related information. We assume that all the RF nodes can\nmeasure the RSS field values at discrete time instants. No specific\nadditional RF hardware \\cite{savazzi2016magazine} is required since\nRSS values are computed in the normal operations of the networked\nRF nodes for channel estimation\/equalization and frequency\/frame synchronization\ntasks.\n\nWithout any loss of generality, as described in Fig. \\ref{fig:layout}.b,\nin what follows, we will focus on the single-link scenario (\\ie $L=1$\nand $D=2$), by introducing the single-target ($T_{1}$ being $N=1$)\nfirst and then the multi-target ($T_{1},...,T_{N}$ with $N>1$) cases.\nHowever, the multi-body model presented here can be exploited in a\ngeneral multi-link scenario with $D$ nodes, $L$ links and $N$ targets\nby using electric field superposition. In addition, it can be extended\nto make use of other physical layer channel information measures such\nas the Channel State Information (CSI) and the Channel Quality Information\n(CQI) \\cite{savazzi2016magazine} as well. However, this discussion\nis outside the scope of this work.\n\nIt is worth noticing that all the proposed models apply to a generic\nlink of the radio network: therefore, they could be easily tailored\nto predict RSS over arbitrarily complex network structures for more\nrobust body positioning, as proposed in device-free localization methods\n\\cite{patwari10,wilson10,savazzi2016magazine}. In addition, modeling\nof RSS is instrumental for link selection operations, namely to identify\nan optimized subset of links that are most influenced by the target\npresence \\cite{nicoli2016eusipco}.\n\n\\begin{figure}[tp]\n\\begin{centering}\n\\includegraphics[clip,scale=0.24]{Fig2}\n\\par\\end{centering}\n\\caption{\\label{fig:single}Geometrical representation of the same scenario\nshown in \\ref{fig:layout}.b, where an horizontal link of length $d$\nis positioned at distance $H$ from the floor and a 2D knife-edge\nsurface $S_{1}$, with variable traversal size $c_{1}$ and height\n$h_{1}$, represents the body $T_{1}$ that is placed on the floor.}\n\n\\vspace{-0.4cm}\n\\end{figure}\n\n\n\\subsection{Single body model (SBM)}\n\n\\label{subsec:Single-body-model}\n\nThe single body model \\cite{rampa2017em} is briefly recalled in this\nsection since it is the starting point for the multi-target model\nthat will be described in Sect. \\ref{subsec:Multi-body-model}. As\noutlined in Fig. \\ref{fig:single} , the human body (\\ie the only\ntarget $T_{1}$ located near the single-link area) is represented\nby a perfectly EM absorbing 3D homogeneous cylinder with an elliptical\nbase of minor and major axes $a_{1}$ and $b_{1}$, respectively,\nthat simulate the human head, torso, legs and arms (placed near the\ntorso). Most references assume a 3D cylinder with a circular base\n\\cite{conducting_cylinder,ghaddar2004modeling} or a prism \\cite{de2015analysis}\nwhile only a few \\cite{liu2009fading,reusens2007path} assume also\nthat the arms can freely move with respect to the torso. Considering\na dynamic scenario where the 3D cylinder, modeling the body, can freely\nmove horizontally and rotate around its generic \\emph{nominal} position\n$(x,y,z)$ showing different views, the target is reduced \\cite{rampa2017em}\nto a 2D rectangular blade (\\ie a knife-edge surface) \\cite{Edge_diffraction},\northogonal to the LOS path at distance $d_{1}$ and $d_{2}$ from\nthe TX and RX, respectively. The knife-edge surface is vertically\nplaced close to the link area and can freely rotate and move showing\ndifferent body views during its movements. The presence of the floor\ndoes not imply any influence on the EM propagation and it is used\nhere only for geometrical reasons \\ie to define the height of the\nlink and the placement constraints of the knife-edge surface representing\nthe body. Notice that the knife-edge approximation ignores important\nEM parameters such as polarization, permittivity, conductivity, shape,\nradius of curvature, and surface roughness \\cite{Davis-et-al}.\n\nAccording to the Fig. \\ref{fig:layout}.b, the radio link is horizontally\nplaced at distance $H$ from the floor and the 3D target $T_{1}$,\nthat is placed on the floor, is free to move and rotate around the\nvertical axis in the surroundings of the LOS path. The corresponding\nfirst Fresnel's zone ellipsoid \\cite{Edge_diffraction}, with radius\n$R=\\sqrt{\\lambda d_{1}d_{2}\/d}$, does not have any contact with all\nother parts of the scenario (\\eg walls, ceiling, furniture or other\nobstacles) except for the aforementioned target. Being $R\\leq R_{\\max}=\\sqrt{\\lambda d}\/2$,\nwhere $R_{\\max}$ is the maximum value of the radius $R$, $\\lambda$\nis the wavelength and $d$ is the RX-TX distance (\\ie the link path\nlength), this constraint becomes $2H>\\sqrt{\\lambda d}$. Notice that,\nas stated by standard short-range indoor propagation models \\cite{ITU-indoor},\nground attenuation effects may be safely ignored for a radio link\ninside a single indoor large room\/space \\emph{e.g.} a hall.\n\nThe equivalent 2D knife-edge surface $S_{1}$ has height $h_{1}$,\nwidth $c_{1}$, and it is placed orthogonal to the LOS path at location\n$\\mathbf{X}_{1}=[x_{1},y_{1}]^{T}$. $\\mathbf{X}_{1}$ coincides with\nthe first two coordinates of the $S_{1}$ barycenter $G_{1}=\\left(x_{1},y_{1},z_{1}\\right)$\nof the knife-edge $S_{1}$ since $z_{1}$ assumes the constant value\n$z_{1}=h_{1}\/2-H$. The point $G_{1}^{'}$ is the intersection of\nthe vertical axis passing through $G_{1}$ and the horizontal plane\n$z=0$. In the followings, the position of the target $T_{1}$ (\\ie\nthe position of $G_{1}^{'}$) is thus identified by the off-axis displacement\n$y_{1}$ and the distance $x_{1}=d_{1}$ of $S_{1}$ from the TX.\nHowever, a true person can also turn and make involuntary\/voluntary\nmovements while standing on the floor. Therefore, the 3D target $T_{1}$,\nrepresented by the 2D knife-edge surface $S_{1}$, can assume any\norientation $\\chi_{1}\\in[-\\pi,\\pi]$ with respect to the LOS path.\nIt can make also some small movements $\\Delta\\mathbf{X}_{1}=[\\Delta x_{1},\\Delta y_{1}]^{T}$\naround the nominal location $\\mathbf{X}_{1}$ thus showing a changing\ntraversal size $c_{1}=c_{1}(\\chi_{1})\\in[a_{1},b_{1}]$ and location\n$\\mathbf{X}_{1}+\\Delta\\mathbf{X}_{1}$.\n\nAccording to the scalar theory of diffraction, the electric field\nat the RX, that is generated by the isotropic source in TX is modified\nby the presence of the 2D knife-edge surface $S_{1}$ located in the\nlink area \\cite{rampa2017em}. It can be predicted \\cite{comparative}\nas being generated by a virtual array of Huygens' sources located\non $S_{1}$ but not belonging to the obstacle itself. In far field\nconditions, the electric field $dE$ at the RX due to the diffraction\neffects caused by the elementary Huygens' source of area $dS_{1}$\nwith generic coordinates $\\left(x,y,z\\right)$ is given by\n\n\\begin{equation}\ndE=j\\,\\frac{E_{0}\\,d}{\\lambda\\,r_{1}\\,r_{2}}\\,\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{1}+r_{2}-d\\right)\\right\\} dS_{1}\\,,\\label{eq:dE_tot}\n\\end{equation}\nwhere $r_{1}$ and $r_{2}$ are the distances of the generic elementary\narea $dS_{1}$ for the TX and RX, respectively. $E_{0}$ is the free-space\nelectric field that is described by the following equation\n\n\\begin{equation}\nE_{0}=-j\\,\\frac{\\eta\\,I\\text{\\ensuremath{\\ell}}}{2\\,\\lambda\\,d}\\,\\exp\\left(-j\\frac{2\\,\\pi\\,d}{\\lambda}\\right)\\,,\\label{eq:E0}\n\\end{equation}\nwith $\\eta$ being the free-space impedance and $I\\text{\\ensuremath{\\ell}}$\nthe momentum of the source. The electric field at the RX is given\nby \\cite{rampa2017em}\n\n\\begin{equation}\n\\begin{array}{ll}\nE= & -j\\,\\frac{\\eta\\,I\\text{\\ensuremath{\\ell}}}{2\\,\\lambda\\,d}\\,\\exp\\left(-j\\frac{2\\,\\pi\\,d}{\\lambda}\\right)\\cdot\\\\\n & \\cdot\\left\\{ 1-j\\,\\frac{d}{\\lambda}\\int\\limits _{S_{1}}\\frac{1}{r_{1}\\,r_{2}}\\,\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{1}+r_{2}-d\\right)\\right\\} dS_{1}\\right\\} \\,,\n\\end{array}\\label{eq:E_single}\n\\end{equation}\nwhere the first term refers to the electric field (\\ref{eq:E0}) due\nthe free-space propagation in the empty scenario and the second term\nincludes the diffraction effects due to the body presence according\nto Eq. (\\ref{eq:dE_tot}). The integral of the second term, is computed\nover the rectangular domain defined by the $S_{1}$ region defined\nas $S_{1}=\\left\\{ \\left(x,y,z\\right)\\in\\mathbb{R^{\\textrm{3}}}:\\:x=x_{1}=d_{1}\\right.$,\n$y_{1}-c_{1}\/2\\leq y\\leq y_{1}+c\/2$, $\\left.-H\\leq z\\leq h_{1}-H\\right\\} $.\nFocussing on the \\emph{extra attenuation} induced by the body \\emph{w.r.t.}\nthe free-space, Eq. (\\ref{eq:E_single}) can be written as\n\n\\begin{equation}\n\\frac{E}{E_{0}}=1-j\\,\\frac{d}{\\lambda}\\int\\limits _{S_{1}}\\frac{1}{r_{1}\\,r_{2}}\\,\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{1}+r_{2}-d\\right)\\right\\} dS_{1}\\,.\\label{eq:E_E0_tot-single}\n\\end{equation}\nAccording to (\\ref{eq:E_E0_tot-single}), the presence of the body\ninduces an extra attenuation $A_{\\textrm{dB}}=-10\\,\\log_{10}\\left|E\/E_{0}\\right|^{2}$\n\\emph{w.r.t.} the free-space propagation. Being a forward only method,\nthe diffraction model holds only for the generic target $T_{1}$ placed\nin the area $\\mathcal{Y}$ near the radio link where it is $\\mathcal{Y}=\\left\\{ \\left(x,y\\right)\\subset\\mathbb{R^{\\textrm{2}}}:0<x<d;-\\infty<y<+\\infty\\right\\} $.\nOf course, as shown in \\cite{rampa2017em}, the effect of the target\npresence vanishes for large but finite values of $\\left|y\\right|$.\n\n\\subsection{Paraxial single body model (PSBM)}\n\n\\label{subsec:Paraxial-single-body}\n\nBy exploiting the paraxial approximation and the following variable\nsubstitution $u=y\\tfrac{\\sqrt{2}}{R_{1}}$ and $v=z\\tfrac{\\sqrt{2}}{R_{1}}$,\nEq. (\\ref{eq:E_E0_tot-single}) reduces to the PSBM (Paraxial Single\nBody Model) equation defined as\n\\begin{equation}\n\\begin{aligned}\\frac{E}{E_{0}}={} & 1-\\frac{1}{2}\\:j\\varint_{\\left(\\sqrt{2}y_{1}-c_{1}\/\\sqrt{2}\\right)\/R_{1}}^{\\left(\\sqrt{2}y_{1}+c_{1}\/\\sqrt{2}\\right)\/R_{1}}\\exp\\left(-j\\frac{\\pi}{2}u^{2}\\right)du\\cdot\\\\\n{} & \\cdot\\int_{-\\unit{\\sqrt{2}\\mathit{H}\/\\mathit{R_{1}}}}^{+\\sqrt{2}\\left(h_{1}-H\\right)\/R_{1}}\\exp\\left(-j\\frac{\\pi}{2}v^{2}\\right)dv\\,,\n\\end{aligned}\n\\label{eq:E_E0_approx-single}\n\\end{equation}\nwhere the Fresnel's radius $R_{1}=R\\left(x_{1}\\right)$ is given by\n$R_{1}=\\sqrt{\\lambda\\,x_{1}\\,\\left(d-x_{1}\\right)\/d}$. Here, we have\nspecified that the generic Fresnel's radius $R$ of Fig. \\ref{fig:layout}.b\nis equal to $R\\left(x_{1}\\right)$ to explicitly highlight that it\nis related to the position $x_{1}$ of target $T_{1}$. Eq. (\\ref{eq:E_E0_approx-single})\ncan be easily computed by using the Fresnel's Sine and Cosine Integrals\n\\cite{rampa2015letter}. The paraxial approximation implies the following\nassumptions: $\\max\\{|y_{1}|,|z_{1}|,h_{1},c_{1},\\lambda\\}\\ll\\min\\{x_{1},d-x_{1}\\}$,\n$\\cos\\varphi\\simeq1$, $\\cos\\theta\\simeq1$, $\\cos\\psi\\simeq1$, and\n$\\cos\\phi\\simeq1$, where the aforementioned angles are shown in Fig.\n\\ref{fig:single}. The model gives valid predictions in the same area\n$\\mathcal{Y}$ defined for the SBM model. The paraxial approximation\nis mainly used for outdoor scenarios, namely for terrestrial radio\npropagation applications \\cite{furutsu-1963,vogler-1982} where it\nholds gracefully. However, it has been also employed in HPS applications\n\\cite{dfl}, even if this paraxial hypothesis limits the validity\nof Eq. (\\ref{eq:E_E0_approx-single}) for small bodies near the central\narea of the radio link. For more details about single target modeling\nderived from (\\ref{eq:E_E0_tot-single}), \\emph{e.g.} for models that\ninclude vertical or horizontally polarization, the interested reader\ncan refer to \\cite{rampa2017em}.\n\n\\subsection{Multi-body model (MBM)}\n\n\\label{subsec:Multi-body-model}\n\nAccording to Fig. \\ref{fig:link}, the multi-body scenario is a generalization\nof the single-target case shown in Fig. \\ref{fig:single}. $N$ knife-edge\nsurfaces stand vertically on the floor and are placed orthogonally\nto the LOS path. They are numbered from $1$ up to $N$ according\nto their increasing distances from the TX. Knife-edge positions are\nidentified by the column vectors $\\mathbf{X}_{n}=[x_{n},y_{n}]^{T}$\nthat correspond to the projections $G_{n}^{'}=\\left(x_{n},y_{n},0\\right)$\nof knife-edge barycenters $G_{n}=\\left(x_{n},y_{n},z_{n}\\right)$\nwith $z_{n}=h_{n}\/2-H$. The positions of all targets are thus identified\nby the column vector $\\mathbf{X}=[\\mathbf{X}_{1}^{T}\\,\\mathbf{X}_{2}^{T}\\,...\\,\\mathbf{X}_{N}^{T}]^{T}$.\nThe single \\emph{n-}th target $T_{n}$, with $1\\leq n\\leq N$, is\ndescribed by the rotating knife-edge surface $S_{n}$ having height\n$h_{n}$, traversal size $c_{n}=c_{n}\\left(\\chi_{n}\\right)$ with\nconstraint $a_{n}\\leq c_{n}\\leq b_{n}$ and orientation $\\chi_{n}\\in[-\\pi,\\pi]$\n\\emph{w.r.t.} the LOS path, with obvious meaning of these terms already\nmentioned in Sect. \\ref{subsec:Single-body-model}. All geometrical\nand motion parameters are organized in the following column vectors:\n$\\mathbf{a}=[a_{1},\\,a_{2},\\,...,\\,a_{N}]^{T}$, $\\mathbf{b}=[b_{1},\\,b_{2},\\,...,\\,b_{N}]^{T}$\ncollecting the knife-edges traversal sizes, $\\mathbf{h}=[h_{1},\\,h_{2},\\,...,\\,h_{N}]^{T}$\nthe target heights and $\\boldsymbol{\\chi}=[\\chi_{1},\\,\\chi_{2},\\,...,\\,\\chi_{N}]^{T}$\nthe corresponding orientations. Finally, $\\forall\\Delta x_{n},\\Delta y_{n}\\in\\left[-B\\:+B\\right]$,\nthe column vector $\\mathbf{\\Delta X}=[\\mathbf{\\Delta X}_{1}^{T}\\,\\mathbf{\\Delta X}_{2}^{T}\\,...\\,\\mathbf{\\Delta X}_{N}^{T}]^{T}$\ntracks the involuntary\/voluntary movements of the bodies in the position\ninterval $\\left[-B\\:+B\\right]$ (supposed symmetrically arranged)\naround the nominal position vector $\\mathbf{X}$.\n\n\\begin{figure}[tp]\n\\begin{centering}\n\\includegraphics[clip,scale=0.24]{Fig3}\n\\par\\end{centering}\n\\caption{\\label{fig:link}Single-link multi-target ($N>1$) scenario composed\nby an horizontal single-link of length $d$, placed at distance $H$\nfrom the floor, and $N$ different 2D equivalent knife-edge surfaces\n$S_{1},S_{2},$...,$S_{N}$ corresponding to the targets $T_{1},T_{2},$...,$T_{N}$\nlocated in $\\mathbf{X}_{1},\\mathbf{X}_{2},...,\\mathbf{X}_{N}$, respectively.}\n\n\\vspace{-0.4cm}\n\\end{figure}\n\nFor $N>1$ targets, the knife-edge diffraction model (\\ref{eq:dE_tot})\nstill holds true for each \\emph{n-}th surface $S_{n}$, although equation\n(\\ref{eq:E_E0_tot-single}) is no longer valid. The LOS path is now\ndivided into $N+1$ segments of length equal to $d_{1},d_{1,2},d_{2,3},\\ldots,d_{N-1,N},d_{N}$\nwith $d=d_{1}+\\sum_{n=1}^{N-1}d_{n,n+1}+d_{N}$. Generalizing the\nmodel of Sect. \\ref{subsec:Single-body-model} for $N$ targets, the\nterm $r_{n,n+1}$ represents the distance between two consecutive\nelementary areas $dS_{n}$ and $dS_{n+1}$ while the terms $r_{1}$\nand $r_{N}$ represent the distance between the transmitter and $dS_{1}$,\nand the distance between $dS_{N}$ and the receiver, respectively.\nThe \\emph{n}-th elementary area $dS_{n}=d\\xi_{n}d\\varsigma_{n}$ is\nlocated on the \\emph{n-}th plane $S_{n}$ that is identified by its\nposition $\\mathbf{X}_{n}$. The coordinate axes $\\xi_{n}$ and $\\varsigma_{n}$\n(for clarity not shown in Fig. \\ref{fig:link}) have the origin in\n$O_{n}^{'}$ and are directed as the $y$ and $z$ axes. As an additional\nhypothesis with respect to the ones of Sect. \\ref{subsec:Single-body-model},\nonly forward propagation from TX to RX is considered with no backward\nscattered waves between any surfaces $S_{n}$ and the RX (\\ie both\nsingle- and multiple-scattering effects between knife-edges are ignored).\n\nIn far field conditions, by assuming only forward propagation, $\\forall T_{n}$\nwith $n=1,\\ldots,N-1$, the elementary electric field $dE_{n+1}$\ndue to the diffraction effects caused by the elementary Huygens' source\nof area $dS_{n}$, is computed at $dS_{n+1}$ by considering the distance\n$r_{n,n+1}$ between the elements $dS_{n}$ and $dS_{n+1}$ according\nto\n\n\\begin{equation}\ndE_{n+1}=j\\,\\frac{dE_{n}\\,dS_{n}}{\\lambda r_{n,n+1}}\\exp\\left(-j\\frac{2\\pi r_{n,n+1}}{\\lambda}\\right)\\,.\\label{eq:dE_iterative}\n\\end{equation}\nThe electric field $E_{1},$ impinging on the first target, is\n\n\\begin{equation}\nE_{1}=E_{0}\\,\\left(\\frac{d}{r_{1}}\\right)\\exp\\left(-j\\frac{2\\pi\\left(r_{1}-d\\right)}{\\lambda}\\right)\\,,\\label{eq:E_first}\n\\end{equation}\nwhile the electric field $dE$, that is measured at the RX node and\ngenerated by the area $dS_{N}$ of the target $N$ closest to the\nRX is given by\n\n\\begin{equation}\ndE=j\\,\\frac{dE_{N}\\,dS_{N}}{\\lambda r_{N}}\\exp\\left(-j\\frac{2\\pi r_{N}}{\\lambda}\\right)\\,.\\label{eq:E_last}\n\\end{equation}\nCombining Eqs. (\\ref{eq:dE_iterative}-\\ref{eq:E_last}), it is\n\n\\begin{equation}\n\\begin{aligned}dE={} & j^{N}\\frac{d}{\\lambda^{N}\\,r_{N}\\,r_{N-1,N}\\,...\\,r_{1,2}\\,r_{1}}\\,E_{0}\\cdot\\\\\n{} & \\cdot\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{N}+r_{N-1,N}+...+r_{1,2}+r_{1}-d\\right)\\right\\} \\cdot\\\\\n{} & \\cdot dS_{1}\\,dS_{2}\\,...\\,dS_{N}\\,.\n\\end{aligned}\n\\label{eq:dE_tot_N}\n\\end{equation}\nTo obtain the electric field $E$, Eq. (\\ref{eq:dE_tot_N}) must be\nintegrated over the domain $S^{\\left(c\\right)}=\\bigcup_{n=1}^{N}S_{n}^{\\left(c\\right)}$\nwhere each region $S_{n}^{\\left(c\\right)}$ corresponds to the 2D\nplane $\\mathcal{P}_{n}\\supset S_{n}$ that does not contain the points\nof the knife-edge surface $S_{n}$. Eq. (\\ref{eq:dE_tot_N}) becomes\nnow\n\n\\begin{equation}\n\\begin{aligned}\\frac{E}{E_{0}}={} & j^{N}\\int\\limits _{S_{1}^{^{\\left(c\\right)}}}\\int\\limits _{S_{2}^{^{\\left(c\\right)}}}\\cdots\\int\\limits _{S_{N}^{^{\\left(c\\right)}}}\\frac{d}{\\lambda^{N}\\,r_{N}\\,r_{N-1,N}\\,...\\,r_{1,2}\\,r_{1}}\\cdot\\\\\n{} & \\cdot\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{N}+r_{N-1,N}+...+r_{1,2}+r_{1}-d\\right)\\right\\} \\cdot\\\\\n{} & \\cdot dS_{1}\\,dS_{2}\\,...\\,dS_{N}\\,.\n\\end{aligned}\n\\label{eq:E_E0_tot_N}\n\\end{equation}\n\nWe now define $E^{\\left(n\\right)}$ as the value of the electric field\nat the RX node when only one target, \\ie the \\emph{n-}th body or\nobstacle, is present in the LOS area. Similarly, $E^{\\left(n,m\\right)}$\nrefers to the electric field at the receiver when only the \\emph{n-}th\nand \\emph{m-}th obstacles out of $N$, are in the link area, and so\non. The notation $E^{\\left(1,2,...,N\\right)}$ thus highlights the\ncontributions of the $N$ targets to the link loss: this is the total\nelectric field $E$ at the receiver given by (\\ref{eq:E_E0_tot_N}).\nConsidering the above definitions, Eq. (\\ref{eq:E_E0_tot_N}) may\nbe rewritten to highlight the mutual interactions of the targets,\ngrouped by $\\tbinom{N}{N-1}$ singles, $\\tbinom{N}{N-2}$ pairs, $\\tbinom{N}{N-3}$\ntriples and so on, as\n\n\\begin{equation}\n\\begin{aligned}\\left(-1\\right)^{N}\\,\\frac{E^{\\left(1,2,..,N\\right)}}{E_{0}}{} & =-1+\\underset{\\mathrm{singles}}{\\underbrace{\\sum_{n=1}^{N}\\frac{E^{\\left(n\\right)}}{E_{0}}}}-\\underset{\\mathrm{pairs}}{\\underbrace{\\sum_{n=1}^{N-1}\\sum_{m=n+1}^{N}\\frac{E^{\\left(n,m\\right)}}{E_{0}}}}+\\\\\n+{} & \\underset{\\mathrm{triples}}{\\underbrace{\\sum_{n=1}^{N-2}\\sum_{m=n+1}^{N-1}\\sum_{k=m+1}^{N}\\frac{E^{\\left(n,m,k\\right)}}{E_{0}}}}+...+\\\\\n{} & +\\Psi(S_{1},...,S_{N})\n\\end{aligned}\n\\label{eq:E_E0_tot_N_iter}\n\\end{equation}\nwhere the last term $\\Psi(S_{1},...,S_{N})$: \n\\begin{equation}\n\\begin{aligned}\\Psi(S_{1},...,S_{N}){}{} & =j^{N}\\int\\limits _{S_{1}}\\int\\limits _{S_{2}}...\\int\\limits _{S_{N}}\\frac{d}{\\lambda^{N}\\,r_{N}\\,r_{N-1,N}\\,...\\,r_{1,2}\\,r_{1}}\\cdot\\\\\n{} & \\cdot\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{N}+r_{N-1,N}+...\\right.\\right.\\\\\n{} & \\left.\\left....+r_{1,2}+r_{1}-d\\right)\\right\\} \\cdot dS_{1}dS_{2}...dS_{N}\\,,\n\\end{aligned}\n\\label{eq:integ}\n\\end{equation}\nis the integral computed over the composite domain defined by the\nunion $S^{\\left(1,2,...,N\\right)}=\\bigcup_{n=1}^{N}S_{n}$ of the\n$N$ rectangular knife-edge surfaces $S_{n}$. The knife edge surfaces\n(Fig. \\ref{fig:link}) have the following definitions: for $n=1$\nit is $S_{1}=\\left\\{ \\left(x,y,z\\right)\\in\\mathbb{R^{\\textrm{3}}}:\\right.$\n$x=x_{1}=d_{1}$, $y_{1}-c_{1}\/2\\leq$ $y$ $\\leq y_{1}+c_{1}\/2$,\n$\\left.-H\\leq z\\leq h_{1}-H\\right\\} $ while $\\forall n=2,...,N$\nit is also $S_{n}=\\left\\{ \\left(x,y,z\\right)\\in\\mathbb{R^{\\textrm{3}}}:\\right.$\n$x=x_{n}=d_{1}+\\sum_{i=1}^{n-1}d_{i,i+1}$, $y_{n}-c_{n}\/2\\leq$ $y$\n$\\leq y_{n}+c_{n}\/2$, $\\left.-H\\leq z\\leq h_{n}-H\\right\\} $.\n\nUsing (\\ref{eq:E_E0_tot_N_iter}), for a generic number of targets\n$N$, the electric field ratio $\\frac{E^{\\left(1,2,..,N\\right)}}{E_{0}}$\ndue to $N$ obstructing bodies is composed by the single target contributions\n$\\frac{E^{\\left(n\\right)}}{E_{0}}$, for $n=1,...,N$ terms, the target\npairs, $\\frac{E^{\\left(n,m\\right)}}{E_{0}}$, for $n=1,...,N-1,$$m=n+1,...,N,$\nthe triples, $\\frac{E^{\\left(n,m,k\\right)}}{E_{0}}$, for $n=1,...,N-2$,\n$m=n+1,...,N-1$, $k=m+1,...,N$, and so on, up to the contributions\nof the $N-1$ target groups. Likewise PSBM and SBM, MBM gives valid\npredictions when the targets are placed in the area $\\mathcal{Y}$\nnear the LOS path, as defined in Sect. \\ref{subsec:Single-body-model}.\n\nWhen two bodies are in the area $\\mathcal{Y}$, the received electric\nfield $E^{\\left(1,2\\right)}$ embeds the mutual effects of the two\ntargets $T_{1}$ (\\ie $S_{1}$) and $T_{2}$ (\\ie $S_{2}$) on the\nradio propagation. $E^{\\left(1,2\\right)}$ is computed from the single-target\nterms $E^{\\left(1\\right)}$ and $E^{\\left(2\\right)}$ as\n\n\\begin{equation}\n\\begin{aligned}\\frac{E^{\\left(1,2\\right)}}{E_{0}}={} & -1+\\frac{E^{\\left(1\\right)}}{E_{0}}+\\frac{E^{\\left(2\\right)}}{E_{0}}+\\Psi(S_{1},S_{2}).\\end{aligned}\n\\label{eq:E_E0_tot_2}\n\\end{equation}\nwhere the \\emph{mixed} term, that depends on both knife-edges $S_{1}$\nand $S_{2}$, is defined according to Eq. (\\ref{eq:integ}) as\n\\begin{equation}\n\\begin{aligned}\\Psi(S_{1},S_{2}){}{} & =-\\int\\limits _{S_{1}}\\int\\limits _{S_{2}}\\frac{d}{\\lambda^{2}\\,r_{2}\\,r_{1,2}\\,r_{1}}\\cdot\\\\\n{} & \\cdot\\exp\\left\\{ -j\\frac{2\\pi}{\\lambda}\\left(r_{2}+r_{1,2}+r_{1}-d\\right)\\right\\} dS_{1}dS_{2}\\,.\n\\end{aligned}\n\\label{eq:integ_n2}\n\\end{equation}\nIn particular, from Eq. (\\ref{eq:E_E0_tot_2}), the term $E^{\\left(1\\right)}$\nquantifies the effect of the target $T_{1}$ alone in the link area\naccording to Eq. (\\ref{eq:E_E0_tot-single}). It depends on the corresponding\ntarget size $c_{1}$, the target height $h_{1}$, the link height\n$H$ from the floor, and the distances $d_{1}$ and $d-d_{1}=d_{2}+d_{12}$\nof the body $T_{1}$ from the TX an RX, respectively. Likewise, $E^{\\left(2\\right)}$\nrefers to the contributions of target $T_{2}$ only, according to\nits target size $c_{2}$ and height $h_{2}$, the link height $H$\nfrom the floor, and the distances $d_{1}+d_{12}$ and $d_{2}$ of\nthe body $T_{2}$ from the TX an RX, respectively.\n\nFor $N=1$, the proposed multi-body model (MBM) reduces to the single\nbody model (SBM) as expected since all singles, pairs, triples and\nother high order terms of Eq. (\\ref{eq:E_E0_tot_N_iter}) vanish except\nfor the term $\\Psi(S_{1})$. The MBM model for $N=2$ targets (\\ref{eq:E_E0_tot_2})\nhas been initially introduced in \\cite{Rampa-2019} along with some\npreliminary results. This dual-target model can be directly obtained\nfrom the Eq. (\\ref{eq:E_E0_tot_N}) or, equivalently, Eq. (\\ref{eq:E_E0_tot_N_iter}).\nModel comparisons are presented in Sect. \\ref{subsec:Model-calibration}.\n\n\\subsection{Paraxial multi-body model (PMBM)}\n\n\\label{subsec:paraxial_sec}\n\nFor HPS applications, paraxial hypotheses are realistic only for small\ntarget(s), namely for small enough $c_{i}$ and $h_{i}$ \\emph{w.r.t.\n}the path length $d$. The approximation also requires the subject\nto move nearby the LOS path, with small enough $y_{i}$ and $z_{i}$,\nor located in the central part of the LOS path. Carrier wavelength\n$\\lambda$ is also much smaller than the distances $x_{i}$ and $d-x_{i}$\n(Sect. \\ref{subsec:Paraxial-single-body}). Since the paraxial approximation\nis useful in several applications, mostly outdoor, in the following\nsection, we will approximate the full Eqs. (\\ref{eq:E_E0_tot_N})\nor (\\ref{eq:E_E0_tot_N_iter}) using paraxial assumptions. Such a\nmodel will be labeled as PMBM (Paraxial MBM).\n\nBased on the paraxial approximation, Eq. (\\ref{eq:E_E0_tot_N}) becomes\n\n\\begin{equation}\n\\begin{aligned}\\frac{E}{E_{0}}={} & \\left(\\frac{j}{2}\\right)^{N}\\int\\limits _{S_{1}^{^{\\left(c\\right)}}}\\int\\limits _{S_{2}^{^{\\left(c\\right)}}}\\cdots\\int\\limits _{S_{N}^{^{\\left(c\\right)}}}\\\\\n{} & \\frac{d\\,d_{1,2}d_{2,3}...d_{N-1,N}}{\\left(d_{1}+d_{1,2}\\right)\\left(d_{1,2}+d_{2,3}\\right)...\\left(d_{N-1,N}+d_{N}\\right)}\\cdot\\\\\n\\cdot{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(u_{1}^{2}+u_{2}^{2}+...+u_{N}^{2}-2\\alpha_{1,2}u_{1}u_{2}...+\\right.\\right.\\\\\n-{} & \\left.\\left.2\\alpha_{N-1,N}u_{N-1}u_{N}\\right)\\right\\} du_{1}du_{2}...du_{N}\\cdot\\\\\n\\cdot{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(v_{1}^{2}+v_{2}^{2}+...+v_{N}^{2}-2\\alpha_{1,2}v_{1}v_{2}...+\\right.\\right.\\\\\n-{} & \\left.\\left.2\\alpha_{N-1,N}v_{N-1}v_{N}\\right)\\right\\} dv_{1}dv_{2}...dv_{N}.\n\\end{aligned}\n\\label{eq:E_E0_tot_N_app}\n\\end{equation}\nIn the Appendix \\ref{sec:Appendix}, we show how to rewrite Eq. (\\ref{eq:E_E0_tot_N_app})\nto reveal the mutual interactions of targets as in Eq. (\\ref{eq:E_E0_tot_N_iter}).\n\nFor the case of $N=2$ targets, Eq. (\\ref{eq:E_E0_tot_N_app}) becomes\nnow analytically tractable. Using the formulation shown in Eq. (\\ref{eq:E_E0_tot_2}),\nadapted in Appendix \\ref{sec:Appendix} for paraxial assumptions,\nit is \n\\begin{equation}\n\\begin{aligned}\\frac{E^{\\left(1,2\\right)}}{E_{0}}={} & -1+\\frac{E^{\\left(1\\right)}}{E_{0}}+\\frac{E^{\\left(2\\right)}}{E_{0}}-\\frac{1}{4}\\frac{d\\,d_{1,2}}{\\left(d_{1}+d_{1,2}\\right)\\left(d_{1,2}+d_{2}\\right)}\\cdot\\\\\n\\cdot{} & \\int_{-\\unit{\\sqrt{2}\\mathit{H}}\/R_{1}}^{+\\sqrt{2}\\left(h_{1}-H\\right)\/R_{1}}\\int_{-\\unit{\\sqrt{2}\\mathit{H}\/\\mathit{R_{2}}}}^{+\\sqrt{2}\\left(h_{2}-H\\right)\/R_{2}}\\\\\n{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(u_{1}^{2}+u_{2}^{2}-2\\alpha_{1,2}u_{1}u_{2}\\right)\\right\\} du_{1}du_{2}\\cdot\\\\\n\\cdot{} & \\int_{\\left(\\sqrt{2}y_{1}-c_{1}\/\\sqrt{2}\\right)\/R_{1}}^{\\left(\\sqrt{2}y_{1}+c_{1}\/\\sqrt{2}\\right)\/R_{1}}\\int_{\\left(\\sqrt{2}y_{2}-c_{2}\/\\sqrt{2}\\right)\/R_{2}}^{\\left(\\sqrt{2}y_{2}+c_{2}\/\\sqrt{2}\\right)\/R_{2}}\\\\\n{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(v_{1}^{2}+v_{2}^{2}-2\\alpha_{1,2}v_{1}v_{2}\\right)\\right\\} dv_{1}dv_{2}\\,,\n\\end{aligned}\n\\label{eq:E_Er_2_1_N}\n\\end{equation}\nwhere $u_{1},u_{2}$,$v_{1},v_{2}$ and the constant terms $R_{1},R_{2}$\nand $\\alpha_{1,2}$ defined in the same Appendix \\ref{sec:Appendix}.\nNotice that, some approximated models are already available in the\nliterature \\cite{lee1978path,vogler-1982} for the evaluation of the\nextra attenuation due to multiple semi-infinite knife-edge surfaces.\nThese models can be obtained by using the paraxial approximation over\nthe semi-infinite domains representing the targets (\\ie unlike the\n\\emph{finite} target size assumption adopted here); they are typically\neffective in outdoor scenarios for the prediction of the propagation\nloss over non-regular terrain profiles. The interested reader can\ntake a look at \\cite{tzaras-2000} (and references therein) for a\nbrief discussion and comparisons.\n\n\\subsection{Additive models}\n\n\\label{subsec:EM-vs.-additive}\n\nBased on the analysis of the previous sections, the term $\\left|E^{\\left(1,2,...,N\\right)}\/E_{0}\\right|$\nfor $N$ targets can be used to evaluate the extra attenuation $A_{\\textrm{dB}}^{\\left(1,2,...,N\\right)}$\nwith respect to the free-space (\\ie unobstructed or empty) scenario\nas\n\\begin{equation}\nA_{\\textrm{dB}}^{\\left(1,2,...,N\\right)}=-10\\,\\log_{10}\\left|E^{\\left(1,2,...,N\\right)}\/E_{0}\\right|^{2}.\\label{eq:dbatt}\n\\end{equation}\nFrom Eq. (\\ref{eq:E_E0_tot_N_iter}), it is apparent that the extra\nattenuation terms $\\left|\\frac{E^{\\left(1\\right)}}{E_{0}}\\right|,\\left|\\frac{E^{\\left(2\\right)}}{E_{0}}\\right|,\\,...,\\left|\\frac{E^{\\left(N\\right)}}{E_{0}}\\right|$\nalone or, equivalently, $A_{\\textrm{dB}}^{\\left(1\\right)},\\,A_{\\textrm{dB}}^{\\left(2\\right)},...,\\,A_{\\textrm{dB}}^{\\left(N\\right)}$,\nare not sufficient to evaluate $\\left|\\frac{E^{\\left(1,2,..,N\\right)}}{E_{0}}\\right|$\nsince: \\emph{i}) the phase relations between the terms $\\frac{E^{\\left(n\\right)}}{E_{0}}$\nare unknown, \\emph{ii}) the terms $\\frac{E^{\\left(n,m\\right)}}{E_{0}},\\,\\frac{E^{\\left(n,m,k\\right)}}{E_{0}},\\,...,\\,$\nare not available, and \\emph{iii}) the interaction terms between the\ntargets, that are expressed by the integral of the right side of Eq.\n(\\ref{eq:E_E0_tot_N_iter}), are not known as well. These facts prevent\nthe use of single-target measurements for the multiple target case.\nAccording to these considerations, the additive hypothesis, namely\n$A_{\\textrm{dB}}^{\\left(1,2,...,N\\right)}=A_{\\textrm{dB}}^{\\left(1\\right)}+A_{\\textrm{dB}}^{\\left(2\\right)}+...+A_{\\textrm{dB}}^{\\left(N\\right)}$,\nthat is generally exploited in various forms in the literature \\cite{patwari10,wilson10},\nis a rather superficial approximation. For the case of two targets\n$(N=2)$, in Sect. \\ref{subsec:Model-validation-with} an additive\nSBM model is proposed where the individual extra attenuations $A_{\\textrm{dB}}^{(1)}$,\n$A_{\\textrm{dB}}^{(2)}$,..., $A_{\\textrm{dB}}^{(N)}$ follow the\nSBM model described in Eq. (\\ref{eq:E_E0_tot-single}). Limitations\nof such representation are highlighted by a comparison with the MBM\nand PMBM models.\n\n\\section{Physical-statistical multi-body model}\n\n\\label{sec:Physical-statistical-modeling}\n\nIn this section, we propose a true multi-target physical-statistical\nmodel that relates the RSS to the link geometry ($d$, $H$), the\nbodies locations $\\mathbf{X}$, and their geometrical sizes (\\ie\n$\\mathbf{a}$, $\\mathbf{b}$ and $\\mathbf{h}$). In addition to the\ndiffraction, or physical, component analyzed in Sect. \\ref{sec:Diffraction-model},\nthe additional statistical component quantifies the uncertainty of\nbody movements, modelled here by small random voluntary\/involuntary\nmotions $\\Delta\\mathbf{X}$ and rotations $\\boldsymbol{\\chi}$ around\nthe nominal position $\\mathbf{X}$, as well as multipath fading, multiple\nscattering between bodies, backward propagation effects, and other\nrandom fluctuations, not included in the diffraction terms. For the\nsake of simplicity, in the following sections, all geometrical parameters\ndefined in Sect. \\ref{subsec:Multi-body-model} will be represented\nby the compact set $\\boldsymbol{\\Lambda}=\\left\\{ \\mathbf{a},\\mathbf{b},\\mathbf{h},d,H\\right\\} $.\n\nLet $P$ be the RSS measurement performed by the receiver RX and expressed\nin logarithmic scale (\\ie usually in dBm), the power measurement\n$P$ can be modeled as the sum of \\emph{i}) the deterministic term\n$P_{L}=10\\,\\log_{10}\\left|E_{0}\\right|^{2}$ due to the free-space\npath-loss; \\emph{ii}) the extra attenuation term $A_{\\textrm{dB}}=A_{\\textrm{dB}}^{\\left(1,2,...,N\\right)}$\nin (\\ref{eq:dbatt}) with respect to the free-space path-loss, caused\nby the body-induced diffraction terms, and \\emph{iii}) the Gaussian\nrandom term $w$ that includes the lognormal multipath effects \\cite{shadowing},\nmeasurement noise and other random disturbances assumed normally distributed.\nAccording to these assumptions, it is\n\n\\begin{equation}\nP=\\left\\{ \\begin{array}{ll}\nP_{L}-A_{\\textrm{dB}}^{\\left(1,2,...,N\\right)}+w & \\;\\textrm{iff }\\exists\\,\\mathbf{X}_{n}\\in\\mathcal{Y}\\\\\nP_{L}+w_{0} & \\;\\textrm{elsewhere}.\n\\end{array}\\right.\\label{eq:model-2}\n\\end{equation}\nThe free-space term $P_{L}$ is a constant that depends only on the\ngeometry of the scenario, the transmitted power, the gain and configuration\nof the antennas, and the propagation coefficients \\cite{knife_edge}.\nThe term $A_{\\textrm{dB}}^{(1,2,...,N)}=A_{\\textrm{dB}}\\left(\\mathbf{X},\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)$\nis the extra attenuation, expressed in dB, due to the body-induced\ndiffraction with respect to the free-space scenario. Its argument\n$\\left|E^{\\left(1,2,...,N\\right)}\/E_{0}\\right|^{2}$ is computed using\n(\\ref{eq:E_E0_tot_N}) or (\\ref{eq:E_E0_tot_N_iter}) for MBM and\n(\\ref{eq:E_E0_tot_N_app}) or (\\ref{eq:E_E0_N_tot_app_iter}) for\nPMBM. Propagation effects not included in the diffraction models (\\ref{eq:E_E0_tot_N})\nor (\\ref{eq:E_E0_tot_N_iter}) are modeled by the Gaussian noise $w\\sim\\mathcal{N}$$\\left(\\Delta\\mu_{\\text{C}},\\sigma_{0}^{2}+\\Delta\\sigma_{C}^{2}\\right)$\nwith $\\Delta\\mu_{\\text{C}}$ and $\\Delta\\sigma_{C}^{2}$ being the\nresidual stochastic body-induced multipath fading mean and variance\nterms \\cite{dfl,rampa2015letter}. $\\sigma_{0}^{2}$ models the power\nfluctuations induced by environmental changes outside the link area,\nand not attributable to body movements around the LOS link.\n\nFor the empty scenario, where nobody is present in the link area,\nnamely the \\emph{background} configuration, the RSS is simply modelled\nas $P=P_{L}+w_{0}$ with $w_{0}\\sim\\mathcal{N}$$\\left(0,\\sigma_{0}^{2}\\right)$.\nNotice that in HPS systems, $\\mu_{0}=\\mathrm{E}_{w_{0}}\\left[P\\right]=P_{L}$\nand $\\sigma_{0}^{2}=\\mathrm{Var_{w_{0}}}\\left[P\\right]$ can be evaluated\nfrom field measurements during a calibration phase, when there are\nno targets the link area. On the contrary, the presence of people\nmodifies both the mean $\\mu_{1}\\left(\\mathbf{X}\\right)=\\textrm{\\ensuremath{\\mathrm{E_{\\boldsymbol{\\chi},\\Delta\\mathbf{X},\\mathit{w}}}}}\\left[P\\right]$\nand the variance $\\sigma_{1}^{2}\\left(\\mathbf{X}\\right)=\\textrm{\\ensuremath{\\mathrm{\\textrm{Var\\ensuremath{\\mathrm{_{\\boldsymbol{\\chi},\\Delta\\mathbf{X},\\mathit{w}}}}}}}}\\left[P\\right]$\nterms. Based on Eq. (\\ref{eq:model-2}), the mean $\\mu\\left(P\\right)$\nand variance $\\sigma^{2}\\left(P\\right)$ are defined as\n\n\\begin{equation}\n\\mu\\left(P\\right)=\\left\\{ \\begin{array}{ll}\n\\mu_{1}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)=P_{L}+\\Delta\\mu\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right) & \\;\\textrm{iff }\\exists\\,\\mathbf{X}_{n}\\in\\mathcal{Y}\\\\\n\\mu_{0}=P_{L} & \\;\\textrm{elsewhere}\n\\end{array}\\right.\\,\\label{eq:mu}\n\\end{equation}\nand\n\n\\begin{equation}\n\\sigma^{2}\\left(P\\right)=\\left\\{ \\begin{array}{ll}\n\\sigma_{1}^{2}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)=\\sigma_{0}^{2}+\\Delta\\sigma^{2}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right) & \\;\\textrm{iff }\\exists\\,\\mathbf{X}_{n}\\in\\mathcal{Y}\\\\\n\\sigma_{0}^{2} & \\;\\textrm{elsewhere}\n\\end{array}\\right.\\,\\label{eq:sigma}\n\\end{equation}\nwhere it is emphasized the dependency of $P$ from the position $\\mathbf{X}_{n}$\nof at least one target $T_{n}$ in the area $\\mathcal{Y}$ and the\ngeometrical coefficients $\\boldsymbol{\\Lambda}$. The RSS average\n$\\Delta\\mu\\left(\\mathbf{X}\\right)=\\mu_{1}\\left(\\mathbf{X}\\right)-\\mu_{0}$\nand variance $\\Delta\\sigma^{2}\\left(\\mathbf{X}\\right)=\\sigma_{1}^{2}\\left(\\mathbf{X}\\right)-\\sigma_{0}^{2}$\nincrements are defined as\n\\begin{equation}\n\\Delta\\mu\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)=\\Delta\\mu_{C}-\\mathrm{E_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}\\left[A_{\\textrm{dB}}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]\\label{eq: delta P1 mean}\n\\end{equation}\nand\n\n\\begin{equation}\n\\Delta\\sigma^{2}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)=\\Delta\\sigma_{C}^{2}+\\textrm{\\ensuremath{\\mathrm{\\textrm{Var}{}_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}}}\\left[A_{\\textrm{dB}}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]\\,.\\label{eq:delta P1 sigma}\n\\end{equation}\nThe term $\\textrm{\\ensuremath{A_{\\textrm{dB}}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)}}$\nhighlights the fact that, given the geometrical parameters $\\boldsymbol{\\Lambda}$\nand the motion terms $\\Delta\\mathbf{X},\\,\\boldsymbol{\\chi}$, the\nextra attenuation is only a function of the positions $\\mathbf{X}$\nof the bodies. In the followings, we assume that the bodies are positioned\nin $\\mathbf{X}$ but each of them can slightly change its location\nand posture making small random movements $\\Delta\\mathbf{X}_{n}$\nwith $\\Delta x_{n},\\Delta y_{n}\\sim\\mathcal{U}\\left(-B,+B\\right)$\nand rotations $\\chi_{n}\\sim\\mathcal{U}\\left(-\\pi,+\\pi\\right)$ around\nthe vertical axis. $\\mathcal{U}\\left(\\alpha,\\beta\\right)$ indicates\nthe uniform distribution within the interval $\\left[\\alpha\\:\\beta\\right]$\nwhile, for each \\emph{n}, the set$\\left[-B\\:+B\\right]\\times\\left[-B\\:+B\\right]$\ndefines the 2D area around the nominal coordinate position $\\mathbf{X}_{n}$\nwhere the \\emph{n}-th target can freely move. To determine eqs. (\\ref{eq: delta P1 mean})\nand (\\ref{eq:delta P1 sigma}), the mean $\\mathrm{E_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}\\left[\\cdot\\right]$\nand the variance $\\textrm{\\ensuremath{\\mathrm{\\textrm{Var}{}_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}}}\\left[\\cdot\\right]$\nare computed over the aforementioned uniform distribution of $\\Delta\\mathbf{X}$\nand $\\boldsymbol{\\chi}$.\n\nThe residual body-induced multipath terms $\\Delta\\mu_{C}$ and $\\Delta\\sigma_{C}^{2}$\nin (\\ref{eq: delta P1 mean}) and (\\ref{eq:delta P1 sigma}), respectively,\ncan be directly evaluated from field measurements performed during\nthe calibration phase. However, these terms are marginally influenced\nby the specific body locations, as also shown in \\cite{shadowing,rampa2015letter},\nand are thus not relevant for HPS applications. On the contrary, the\ndiffraction term $A_{\\textrm{dB}}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)$\nprovides a simple but effective method to predict the power perturbation\n$\\Delta\\mu\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$\nand $\\Delta\\sigma^{2}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$\nas a function of the body position and size.\n\n\\section{Model optimization and validation}\n\n\\label{subsec:Model-calibration}\n\nTo confirm the validity of the proposed multi-body models, several\nEM simulations with the Feko software environment and on-field experiments\nhave been carried out according to the same link scenario sketched\nin Fig. \\ref{fig:link}. Feko\\footnote{The commercial EM simulator designed by Altair Engineering Inc.}\nimplements time and frequency domain full-wave solvers (\\eg MoM,\nFDTD, FEM and MLFMM). For details about the solvers, the interested\nreader may have a look at \\cite{elsherbeni-2014antennabook}.\n\nFirst of all, simulations have been carried out to compare the results\nof the diffraction-based MBM and PMBM models shown in Sect. \\ref{subsec:Multi-body-model}\nand Sect. \\ref{subsec:paraxial_sec}, respectively, with the ones\nobtained with Feko. Next, we have compared the aforementioned models\nagainst the RSS measurements obtained from IEEE 802.15.4 devices \\cite{154E},\ncommonly used in industrial applications \\cite{savazzi2016sensors}.\nConsidering the application to multi-body localization featuring $N=2$\ntargets, the MBM and PMBM model parameters, namely the geometrical\nsizes (\\ie $\\mathbf{a}$, $\\mathbf{b}$ and $\\mathbf{h}$) of the\nknife-edge surfaces, are optimized using a small subset of the experimental\ndata so that they could effectively model the obstructions induced\nby the true targets. The proposed models using optimized sizes of\nthe knife-edges are then validated over different configurations,\nwhere both targets move along the LOS link.\n\nIt is worth noticing that Feko simulations are related to PEC (Perfect\nElectromagnetic Conductor) configurations to describe knife-edge targets.\nOn the contrary, MBM and PMBM 2D models assume perfectly absorbing\nsurfaces. The MBM and PMBM models also ignore important EM parameters\nsuch as polarization, permittivity, conductivity, shape, radius of\ncurvature, and surface roughness \\cite{Davis-et-al} that the Feko\nsimulator is able to tackle.\n\n\\begin{figure}[tp]\n\\begin{centering}\n\\includegraphics[clip,scale=0.4]{Fig4}\n\\par\\end{centering}\n\\caption{\\label{fig:comp-along}Feko PEC simulations for vertically (blue line)\nand horizontally (magenta line) polarized source vs. MBM (black line)\nand PMBM (red line) $A_{\\textrm{dB}}^{\\left(1,2\\right)}$ predictions\nfor targets $T_{1}$ and $T_{2}$ along the LOS path as shown in the\nscenario on the top.}\n\n\\vspace{-0.4cm}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{centering}\n\\includegraphics[scale=0.4]{Fig5}\n\\par\\end{centering}\n\\caption{\\label{fig:comp-across}Feko PEC simulations for vertically (blue\nline) and horizontally (magenta line) polarized source vs. MBM (black\nline) and PMBM (red line) $A_{\\textrm{dB}}^{\\left(1,2\\right)}$ predictions\nfor the target $T_{1}$ along the LOS path and $T_{2}$ across the\nLOS path as shown in the scenario on the top. The dotted lines (with\nthe same colors adopted for the dual-target cases) show the extra\nattenuation predicted by the previous models\/simulations due to the\npresence of the target $T_{1}$ only.}\n\n\\vspace{-0.4cm}\n\\end{figure}\n\n\n\\subsection{Model comparison against EM simulations}\n\nThe MBM and PMBM models are evaluated and compared in this section\nwith the results from EM simulations. To simplify the EM simulation\ncomplexity (mostly due to the long Feko runs), in what follows MBM\nand PMBM models are compared in a dual-target scenario ($N=2$) only.\nFig. \\ref{fig:comp-along} and \\ref{fig:comp-across} show the predicted\nvalues of the extra attenuation $A_{\\textrm{dB}}^{\\left(1,2\\right)}=-10\\,\\log_{10}\\left|E^{\\left(1,2\\right)}\/E_{0}\\right|^{2}$\ncomputed according to the models described by eqs. (\\ref{eq:E_E0_tot_2})\nand (\\ref{eq:E_Er_2_1_N}), namely MBM and PMBM, respectively. No\nmovements\/rotations are allowed and both targets are placed in their\nnominal positions: in both figures, the target $T_{1}$ is fixed in\nthe position $\\mathbf{X}_{1}=[1,0]^{\\mathrm{T}}$while the target\n$T_{2}$ changes its positions along and across the LOS path. In Fig.\n\\ref{fig:comp-along}, $T_{2}$ is placed in $\\mathbf{X}_{2}=[x_{2},0]^{\\mathrm{T}}$\nand moves along the LOS path with $1<x_{2}<d$ and $0.25$ m increments\nwhile in Fig. \\ref{fig:comp-across} the target $T_{2}$ is placed\nin $\\mathbf{X}_{2}=[d\/2,y_{2}]^{\\mathrm{T}}$ with $-2.5<y_{2}<2.5$\nm and $0.25$ m increments thus crossing the LOS path in the middle.\nFor symmetry reasons, the results depends only on the distance $|y_{2}|$\nfrom the LOS line.\n\nIn these figures, we compare the results of the MBM and PMBM against\nthe ones of the EM simulations obtained with Feko using vertically\nand horizontally polarized sources (\\ie Feko V. pol. and H. pol.,\nrespectively). For these simulations, a 2D PEC surface have been used\nas target instead of the absorbing one defined in Sect. \\ref{sec:Diffraction-model}\nfor MBM and PMBM, but with the same physical dimensions adopted for\nFig. \\ref{fig:layout}. Target $T_{1}$ and $T_{2}$ have the same\nsize $c_{1}=c_{2}=0.55$ m and height $h_{1}=h_{2}=1.80$ m while\n$H=0.90$ m and $d=5.0$ m. The average error $\\epsilon_{\\textrm{MBM}}$\n(and $\\epsilon_{\\textrm{PMBM}}$) between MBM (and PMBM) and Feko\nPEC V. pol. values in dB are summarized in Tab. \\ref{tab:Error-values},\nwhere the corresponding standard deviation values $\\sigma_{\\textrm{MBM}}$\n(and $\\sigma_{\\textrm{PMBM}}$) are also included. The sign in the\naverage errors $\\epsilon_{\\textrm{MBM}}$ and $\\epsilon_{\\textrm{PMBM}}$\nindicates that the results of the MBM and PMBM models underestimate\nin average the Feko results.\n\n\\begin{table}\n\\caption{\\label{tab:Error-values}Average errors and standard deviation values\nin the LOS area obtained by MBM and PMBM predictions \\emph{w.r.t.}\nFeko PEC V. pol. simulations.}\n\n\\centering{\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline \nLOS area & $\\epsilon_{\\textrm{MBM}}$ & $\\sigma_{\\textrm{MBM}}$ & $\\epsilon_{\\textrm{PMBM}}$ & $\\sigma_{\\textrm{PMBM}}$\\tabularnewline\n\\hline \n\\hline \nAlong (Fig. \\ref{fig:comp-along}) & $-2.1$ dB & $3.4$ dB & $-8.7$ dB & $4.4$ dB\\tabularnewline\n\\hline \nAcross (Fig. \\ref{fig:comp-across}) & $-1.0$ dB & $0.7$ dB & $-4.0$ dB & $1.5$ dB\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\nAlong the LOS positions of Fig. \\ref{fig:comp-along}, the average\nerrors are $\\epsilon_{\\textrm{MBM}}=-2.1$ dB and $\\epsilon_{\\textrm{PMBM}}=-8.7$\ndB, respectively, while the corresponding standard deviations are\n$\\sigma_{\\textrm{MBM}}=3.4$ dB and $\\sigma_{\\textrm{PMBM}}=4.4$\ndB, as well. As already noted, while the MBM trend is similar to the\none due to the Feko predictions, the PMBM values introduce too many\nlarge errors. These effects are also apparent in Fig. \\ref{fig:comp-across}\nwhere the target $T_{2}$ move along the orthogonal line that crosses\nthe LOS in the middle at $x_{2}=2.5$ m. Even in this case, the MBM\ntrend is similar to the PEC vertical polarized curve predicted by\nthe Feko software with an maximum error in the order of $1\\div2$\ndB. In particular, $\\epsilon_{\\textrm{MBM}}=-1.0$ dB, $\\epsilon_{\\textrm{PMBM}}=-4.0$\ndB, $\\sigma_{\\textrm{MBM}}=0.7$ dB and $\\sigma_{\\textrm{PMBM}}=1.5$\ndB considering all positions across the LOS line. In Fig. \\ref{fig:comp-across},\nthe results corresponding to the models for the single target $T_{1}$\nare superimposed to the ones predicted by the MBM\/PMBM models and\nthe ones obtained with Feko. When target $T_{2}$ moves away from\nthe LOS path the effects due to this target vanish, while the extra\nattenuation can be well predicted using only the single target body\nmodel for $T_{1}$. The difference between the values predicted by\nthe MBM (and PMBM) model and the Feko simulations are due to the fact\nthat the former employs absorbing targets and neglects any polarization\nwhile the latter exploits metallic plates and includes the polarization\neffects. The positions with larger differences (\\ie up to $5\\div6$\ndB) are visible in the right part of Fig. \\ref{fig:comp-along} where\nthe target $T_{2}$ is very close to the receiver and the metallic\nnature of the PEC is more evident. Even if some positions of Fig.\n\\ref{fig:comp-along} show some relevant differences between MBM and\nFeko results, the general trend is maintained. On the contrary, differences\nbetween the MBM (and PMBM) model and the Feko simulations are smaller\nfor off-axis placement of $T_{2}$ as shown in Fig. \\ref{fig:comp-across}.\nIt is also worth noticing in Fig. \\ref{fig:comp-along} the strong\ndifferences between vertically and horizontally polarized results\nobtained with Feko.\n\n\\begin{figure*}[!tp]\n\\begin{centering}\n\\includegraphics[scale=0.4]{Fig6}\n\\par\\end{centering}\n\\caption{\\label{fig:measurements}Predicted extra attenuation values using\nMBM (black line), PMBM (blue line) and additive SBM (black dashed\nline) against the measured ones (red cross, with vertical and horizontal\nerror bars) considering $4$ motion scenarios featuring $N=2$ targets.\nThe $\\pm1$ dB uncertainty zones are shown around the additive SBM\n(light blue area) and the MBM (beige area) results. Each scenario\nis depicted in the corresponding sub-figure.}\n\n\\vspace{-0.4cm}\n\\end{figure*}\n\n\n\\subsection{Model optimization and experiments}\n\n\\label{subsec:Model-optimization}\n\nLocalization, and, in general, HPS systems, rely on several calibration\nsteps that include off-line measurements stages to collect ground-truth\nmeasurements of radio propagation. Detection algorithms based on finger-printing\napproaches \\cite{savazzi2016magazine} require also the presence of\nthe target(s) in known locations, namely landmark points, to perform\nRF measurements with target(s) inside the monitored area. These steps\nare very time consuming and error-prone since each RF node needs to\nperform noisy RSS signal measurements for the assessment of body-induced\nalterations of the radio propagation with the target(s) placed in\nthe selected landmarks (\\ie 2-4 landmarks\/sqm depending on the network\nplacement). In addition, systematic errors can be also introduced\nin the experimental setup during target positioning. Moreover, modifications\nof the environment involve frequent recalibration steps to maintain\na good level of localization accuracy. In what follows, we investigate\nthe use of the multi-body models of Sect. \\ref{sec:Physical-statistical-modeling}\noptimized to replace, or simplify, such calibration phases.\n\nModel optimization, or tuning, is implemented during an initial stage\nto estimate the MBM and PMBM model hyper-parameters, the geometrical\nsizes $\\mathbf{a}$, $\\mathbf{b}$ and $\\mathbf{h}$. We thus collect\na small set of measurements in selected landmark points: these are\nused to tune both MBM and PMBM physical parameters, namely the \\emph{EM-equivalent}\nheight $h_{n}$ and width $c_{n}\\in[a_{n},b_{n}]$ of the $n$-th\ndeployed subject. Using these optimized hyperparameters, the MBM and\nthe PMBM representations can be finally adopted to predict body effects\nat arbitrary positions, replacing conventional calibration stages\n\\cite{savazzi2016magazine}.\n\nRSS data have been collected in a large hall by using two IEEE 802.15.4-compliant\nradio devices, based on the NXP JN5148 SoC (System-on-Chip) \\cite{NXP},\nthat are employed as transceiver (TX\/RX) nodes. Link geometry is $H=0.9$\nm and $d=5$ m for all cases. Each device is equipped with 2 dBi vertical\nmonopole antennas and is programmed to send IEEE 802.15.4 standard\nframes at frequency $f_{c}=2.486$ GHz (corresponding to IEEE 802.15.4\nchannel $26$ \\cite{154E}). The receiver decodes each frame to extract\na measurement of RSS, namely the digital RSS Indicator (RSSI) with\n8-bit resolution \\cite{RSSI-calibration} and $1$ dB quantization.\nTwo set of measurements have been gathered: the first dataset, acquired\nin the empty scenario, has been used to compute the reference values\n$\\mu{}_{0}$ and $\\sigma_{0}^{2}$, while the second one has been\nrecorded with two targets are placed in the link area and move concurrently.\nThe second data set has been used to measure $\\mu_{1}\\left(\\mathbf{X}\\right)$\nand $\\sigma_{1}^{2}\\left(\\mathbf{X}\\right)$ for some known landscape\npositions $\\mathbf{X}$. In particular, the subject $T_{1}$ was placed\nat $\\mathbf{X}_{1}=[2.5,0]^{\\mathrm{T}}$ from the transmitter, the\nsecond subject $T_{2}$ covered $4$ different positions, $\\mathbf{X}_{2}=[x_{2},0]^{\\mathrm{T}}$,\nat distances $x_{2}=3$ m, $x_{2}=3.5$ m, $x_{2}=4$ m and $x_{2}=4.5$\nm, respectively. These locations have been used to optimize the EM-equivalent\ngeometrical dimensions $c_{1},c_{2},h_{1},h_{2}$ of the individual\nsubjects using a Non-linear Least Squares (NLS) approach as in \\cite{Rampa-2019-calib}.\nThe optimization is iterative and uses the nominal (or physical) size\nof the targets as initial inputs, namely $c_{1}=c_{2}=0.4$ m and\n$h_{1}=h_{2}=1.7$ m (neglecting head). Optimized EM-equivalent target\nsize are $c_{1}=c_{2}=0.25$ m and $h_{1}=h_{2}=1.35$ m. The EM-equivalent\nsize is about $20\\%\\div40\\%$ smaller than the physical size of the\ntarget as also shown in \\cite{conducting_cylinder}: this might be\ndue to the fact that body-induced extra attenuations are mostly due\nto torso and legs, while the effects caused by head and arms are negligible,\nalthough they are responsible for some small RSSI fluctuations.\n\n\\subsection{Model comparison with measurements}\n\n\\label{subsec:Model-validation-with}\n\nBased on the experimental set-up described in the previous section,\nwe compare in Fig. \\ref{fig:measurements} the predicted extra attenuation\n$\\mathrm{E_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}\\left[A_{\\textrm{dB}}^{(1,2)}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]$\nusing the MBM (black line) and PMBM (blue line) models against the\nfield measurements (red cross). RSSI measurements, averaged over a\nperiod of $1$ minute, are indicated in Fig. \\ref{fig:measurements}\nas red cross markers. Horizontal error bars are also depicted to account\nfor positioning inaccuracies ($\\pm0.25$ m) during the tests while\nvertical bars and dots indicate the max-min and mean RSSI values,\nrespectively. Solid lines show the predicted MBM and PMBM extra attenuations\nusing the optimized EM-equivalent parameters. For prediction, we consider\ntwo knife-edges placed along the LOS path in fixed positions ($B=0$)\nwith no rotation. In particular, subject $T_{1}$ initially placed\nat $\\mathbf{X}_{1}=[0.5,0]^{\\mathrm{T}}$ from the transmitter covers\na distance of $1.5$ m in the direction of the RX (thus stopping at\n$\\mathbf{X}_{1}=[2,0]^{\\mathrm{T}}$ from the TX). Subject $T_{2}$\nmoves towards the receiver and stops at $0.5$ m from the device. \n\nTo highlight the comparative analysis with measurements, we neglect\nthe residual stochastic body-induced multipath fading terms by assuming\n$\\Delta\\sigma_{C}^{2}=0$ dB and $\\Delta h_{C}=0$ dB, so that Eq.\n(\\ref{eq: delta P1 mean}) reduces to $-\\Delta\\mu\\left(\\mathbf{X}\\right)=\\mathrm{E_{\\boldsymbol{\\chi},\\Delta\\mathbf{X}}}\\left[A_{\\textrm{dB}}^{(1,2)}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]=A_{\\textrm{dB}}^{(1,2)}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$.\nBesides MBM and PMBM models, based on the discussion in Sect. \\ref{subsec:EM-vs.-additive},\nfor the same settings, we have also compared the additive hypothesis,\nnamely $A_{\\textrm{dB}}^{\\left(1,2\\right)}=A_{\\textrm{dB}}^{\\left(1\\right)}+A_{\\textrm{dB}}^{\\left(2\\right)}$.\nIn particular, the additive SBM approximation is depicted in dashed\nlines: in this case, the predicted extra attenuation reduces to $-\\Delta\\mu\\left(\\mathbf{X}\\right)=\\mathrm{E_{\\chi_{1},\\Delta\\mathbf{X}_{1}}}\\left[A_{\\textrm{dB}}^{(1)}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]+\\mathrm{E_{\\chi_{2},\\Delta\\mathbf{X}_{2}}}\\left[A_{\\textrm{dB}}^{(2)}\\left(\\left.\\mathbf{X}\\right|\\Delta\\mathbf{X},\\boldsymbol{\\chi},\\boldsymbol{\\Lambda}\\right)\\right]=A_{\\textrm{dB}}^{(1)}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)+A_{\\textrm{dB}}^{(2)}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$\nwhere $A_{\\textrm{dB}}^{(1)}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$\nand $A_{\\textrm{dB}}^{(2)}\\left(\\left.\\mathbf{X}\\right|\\boldsymbol{\\Lambda}\\right)$\nfollow the SBM model described in Sect. \\ref{subsec:Single-body-model}\nand use the same optimized EM-equivalent target size as that of the\nMBM model.\n\nIn Fig. \\ref{fig:measurements}, the shaded areas indicate the $\\pm1$\ndB uncertainty zone around the mean values predicted by the MBM (\\ie\nbeige area) and the additive SBM (\\ie light blue area) models. Assuming\nGaussian distributions, for each uncertainty zone, the 68.3\\% of the\nrelated events fall inside each shaded area.\n\nThe discrepancies between the proposed models and the measurements\nmay be large for some target positions, configurations and scenarios.\nThis is due to some approximations adopted for the MBM\/PMBM and SBM\/PSBM\nmodels that are summarized here: \\emph{i}) the diffraction models\nassume a 2D knife-edge approximation to describe each 3D target. In\naddition, each knife-edge surface is assumed to be completely absorbing\n\\cite{rampa2017em} while this is just an approximation for the human\nbody \\cite{kibret2015characterizing}; \\emph{ii}) the multi-body diffraction\nmodel derived here assumes only forward propagation from the transmitter\nto the receiver and ignores reverberations among targets; \\emph{iii})\nthe diffraction model, being based on the scalar diffraction theory,\ndoes not include polarization effects; \\emph{iv}) the EM composition\nof the targets is not considered. \n\nIn addition, there are also some noisy effects during the RSS measurement:\n\\emph{v}) the human bodies are never fixed in specific positions but,\nwhile standing, they have both voluntary and\/or involuntary movements\naround the nominal position due to movements of the legs, torso, arms\nand head. This experimental fact can introduce variations in the order\nof $2\\div3$ dB for the single target case \\cite{rampa2017em}\\emph{;\nvi}) due to the complex structure of the human body and the difficulties\nto measure the true position of the body \\cite{rampa2017em,Rampa-2019-calib},\nthe nominal (measured) position of the target is only approximately\nknown (with an error in the order of $10\\div15$ cm); \\emph{vii})\nthe measurements are taken in real scenarios where some multipath\neffects are present. These effects are modeled as a lognormal distributed\nnoise even if this an approximated behavior \\cite{shadowing}; \\emph{viii})\nthe RSS values measured by the IEEE 802.15.4 radio devices are corrupted\nby measurement noise and non-linearity effects \\cite{RSSI-calibration}\nin the order of $0.5\\div1.5$ dB, depending on the adopted device.\nOffset errors can be compensated due to the fact that the extra attenuation\nis measured \\emph{w.r.t.} the free-space scenarios but non-linearity\nand quantization cannot. Moreover, the radios are subject to some\nother artifacts since, at very low RSSI, namely when more than one\ntarget is near the LOS path, some packets are lost due to body shadowing\neffects \\cite{rampa2015letter,dfl}; these target configurations introduce\nsome high attenuated RSSI samples, mainly near the RX or the TX.\n\nIn Fig. \\ref{fig:measurements}, the MBM and PMBM predictions track\nthe average RSSI measurements with an average error of about $4$\ndB and $6$ dB, respectively. Notice that most of the mismatches between\nthe models and the measurements are observed in correspondence to\none of the targets (or both) near the transmitter or the receiver.\nIn such cases, the RSSI measurements might be also affected by larger\ncommunication errors due to the increase of body-induced extra attenuation.\nFluctuations of RSSI around the average value cause residual stochastic\nterms that can be quantified as a variance term $\\Delta\\sigma_{C}^{2}\\simeq3$\ndB and negligible mean $\\Delta h_{C}\\simeq0$ dB: these are mostly\ndue to voluntary\/involuntary movements of the bodies around their\nnominal positions. The additive SBM model is generally less effective\ncompared with MBM, particularly when the mixed terms in Eq. (\\ref{eq:integ_n2}),\ncaused by the interacting targets $T_{1}$ and $T_{2}$, could not\nbe neglected. This is for example the case when both subjects equally\ncontribute to the extra attenuation, \\ie for subject $T_{1}$ placed\nat $0.5$ m, $\\mathbf{X}_{1}=[0.5,0]^{\\mathrm{T}}$, and $1$ m, $\\mathbf{X}_{1}=[1,0]^{\\mathrm{T}}$,\nfrom the transmitter.\n\n\\section{Conclusions}\n\n\\label{sec:Conclusions}\n\nBased on the electromagnetic (EM) scalar diffraction theory, the paper\nproposes for the first time an ad-hoc physical-statistical model to\ndescribe the fluctuations of the radio signal caused by the presence\nof an arbitrary number of targets between the transmitter and receiver.\nThe analytical model has been specifically tuned and optimized to\npredict the effects of multiple bodies placed near the link area.\nTherefore, it is instrumental to Device-Free Localization (DFL) applications\nand Human Presence-aware Systems (HPS), including people access monitoring\nand counting. The model results have been validated experimentally\nby some field tests using real industrial wireless devices and EM\nsimulations as well, for comparative analysis. The proposed multi-body\nmodel is able to predict the Received Signal Strength (RSS) measurements\naccounting for the size, orientation, small movements, and positions\nof the targets. In addition, it overcomes some restrictions of the\nexisting multi-body models, based on the linear superposition of the\nsubject effects, thus showing improved prediction accuracy.\n\n\\section*{Appendix}\n\n\\label{sec:Appendix}\n\nIn what follows, we resort to a formulation of Eq. (\\ref{eq:E_E0_tot_N_app})\nsimilar to that used in Eq. (\\ref{eq:E_E0_tot_N_iter}). It is\n\n\\begin{equation}\n\\begin{aligned}\\left(-1\\right)^{N}\\,\\frac{E^{\\left(1,2,..,N\\right)}}{E_{0}}{} & =-1+\\sum_{n=1}^{N}\\frac{E^{\\left(n\\right)}}{E_{0}}-\\sum_{n=1}^{N-1}\\sum_{m=n+1}^{N}\\frac{E^{\\left(n,m\\right)}}{E_{0}}+\\\\\n+{} & \\sum_{n=1}^{N-2}\\sum_{m=n+1}^{N-1}\\sum_{k=m+1}^{N}\\frac{E^{\\left(n,m,k\\right)}}{E_{0}}+...+\\\\\n{} & +\\widetilde{\\Psi}(S_{1},...,S_{N})\n\\end{aligned}\n\\label{eq:E_E0_N_tot_app_iter}\n\\end{equation}\nwith last term $\\widetilde{\\Psi}(S_{1},...,S_{N})$: \n\\begin{equation}\n\\begin{aligned}\\widetilde{\\Psi}(S_{1},...,S_{N}){}{} & =\\left(\\frac{j}{2}\\right)^{N}\\int\\limits _{S_{1}}\\int\\limits _{S_{2}}...\\int\\limits _{S_{N}}\\\\\n{} & \\frac{d\\,d_{1,2}d_{2,3}...d_{N-1,N}}{\\left(d_{1}+d_{1,2}\\right)\\left(d_{1,2}+d_{2,3}\\right)...\\left(d_{N-1,N}+d_{N}\\right)}\\cdot\\\\\n{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(u_{1}^{2}+u_{2}^{2}+...+u_{N}^{2}-2\\alpha_{1,2}u_{1}u_{2}...+\\right.\\right.\\\\\n{} & \\left.\\left.2\\alpha_{N-1,N}u_{N-1}u_{N}\\right)\\right\\} du_{1}du_{2}...du_{N}\\cdot\\\\\n{} & \\exp\\left\\{ -j\\frac{\\pi}{2}\\left(v_{1}^{2}+v_{2}^{2}+...+v_{N}^{2}-2\\alpha_{1,2}v_{1}v_{2}...+\\right.\\right.\\\\\n{} & \\left.\\left.2\\alpha_{N-1,N}v_{N-1}v_{N}\\right)\\right\\} dv_{1}dv_{2}...dv_{N}.\n\\end{aligned}\n\\label{eq:integ-1}\n\\end{equation}\nThe variables $u_{1},...,u_{N},v_{1},...,v_{N}$ are obtained from\nthe corresponding local coordinates $\\xi_{1},...,\\xi_{N},\\varsigma_{1},...,\\varsigma_{N}$\nby using, for each \\emph{n}-th term, the following substitution rules:\n$u_{n}=\\xi_{n}\\tfrac{\\sqrt{2}}{R_{n}}$ and $v_{n}=\\varsigma_{n}\\tfrac{\\sqrt{2}}{R_{n}}$.\nThe constant terms $R_{n}$ and $\\alpha_{n,n+1}$ are related to the\nthe wavelength $\\lambda$ and the geometric positions of the knife-edges\nwith respect to the LOS path. The terms $R_{n}$ are similar to the\nthe Fresnel's radius $R$ for the single-target case. These constants\nare given by\n\n\\begin{equation}\n\\frac{1}{R_{n}^{2}}=\\left\\{ \\begin{aligned}{} & \\frac{1}{\\lambda}\\left(\\frac{1}{d_{1}}+\\frac{1}{d_{12}}\\right) & \\textrm{for }n & =1\\\\\n{} & \\frac{1}{\\lambda}\\left(\\frac{1}{d_{n-1,n}}+\\frac{1}{d_{n,n+1}}\\right) & \\textrm{for }n & =2,...,N-1\\\\\n{} & \\frac{1}{\\lambda}\\left(\\frac{1}{d_{N}}+\\frac{1}{d_{N-1,N}}\\right) & \\textrm{for }n & =N\n\\end{aligned}\n\\right.\\label{eq:generalized_R}\n\\end{equation}\nwhile coefficients $\\alpha_{n,n+1}$ are defined, $\\textrm{for }n=1,...,N-1$,\nas \n\\begin{equation}\n\\alpha_{n,n+1}=\\frac{R_{n}\\,R_{n+1}}{\\lambda\\,d_{n,n+1}}\\,.\\label{eq:alpha}\n\\end{equation}\nFor the single-target case, it is $R_{1}=R$ while all coefficients\n$\\alpha_{n,n+1}$ vanish. In addition, it is trivial to verify that\nEq. (\\ref{eq:E_E0_tot_N_iter}) simply reduces to Eq. (\\ref{eq:E_E0_tot-single})\nwhile Eq. (\\ref{eq:E_E0_N_tot_app_iter}) simplifies to (\\ref{eq:E_E0_approx-single}).\nFor $N=2$, Eq. (\\ref{eq:E_E0_tot_N_iter}) reduces to the dual-target\ncase (\\ref{eq:E_E0_tot_2}) and proves the results \\cite{nicoli2016eusipco}\napplied for localization purposes.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Two Clusters Using a Single Threshold Cut}\n\\label{sec:2-means}\n\nIn this section, we consider the case of $k=2$ clusters, and we study how well a single threshold cut can approximate the optimal partition into two clusters. \n\n\\subsection{Algorithm for \\texorpdfstring{$k=2$}{k=2}}\n\\label{sec:2means-alg}\n\nWe present an algorithm to efficiently minimize the cost using a single threshold cut.  We begin by considering a single feature $i$ and determining the value of the best threshold $\\theta \\in \\mathbb{R}$ for this feature. Then, we minimize over all features $i \\in [d]$ to output the best threshold cut. We focus on the $2$-means algorithm; the $2$-medians case is similar. \n\n\nFor feature~$i$, we first sort the input points according to this feature, i.e., assume that the vectors are indexed as $x^1_i \\leq \\ldots \\leq x^n_i.$\nNotice that when restricting to this feature, there are only $n-1$ possible partitions of the data set into two non-empty clusters. In particular, we can calculate the cost of all threshold cuts for the $i$th feature by scanning the values in this feature from smallest to largest. \nThen, we compute for each position $p \\in [n-1]$ $$\\mathrm{cost}(p)=\\sum_{j=1}^p\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^1(p)}_2^2 + \\sum_{j=p+1}^n\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^2(p)}_2^2,$$\nwhere we denote the optimal centers for these clusters as  $\\boldsymbol{\\mu}^1(p)=\\frac{1}{p}\\sum_{j=1}^p \\mathbf{x}^j$ and $\\boldsymbol{\\mu}^2(p)=\\frac{1}{n - p}\\sum_{j=p + 1}^n \\mathbf{x}^j$ because these are the means of the first $p$ and last $n-p$ points, respectively. Because there are $O(nd)$ possible thresholds, and naively computing the cost of each requires time $O(nd)$, this would lead to a running time of $O(n^2d^2)$. \nWe can improve the time to $O(nd^2 + nd\\log n)$ by using dynamic programming. Pseudo-code for the algorithm and description of the dynamic programming are in Appendix~\\ref{sec:DP_Implementation}.\n\n\n\n\\subsection{Theoretical guarantees for \\texorpdfstring{$k=2$}{}}\n\n\nWe prove that there always exists a threshold cut with low cost. Since our algorithm from the previous section finds the {\\em best} cut, it achieves the guarantees of this theorem.\n\\begin{theorem}\\label{thm:optimal_2_median_means}\n\tFor any data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$, there is a threshold cut $\\widehat C$ such that the $2$-medians cost satisfies\n\t$$\\mathrm{cost}(\\widehat C)\\leq 2 \\cdot \\mathrm{cost}(opt),$$\n\tand there is a threshold cut $\\widehat C$ such that the $2$-means cost satisfies\n\t$$\\mathrm{cost}(\\widehat C)\\leq 4 \\cdot \\mathrm{cost}(opt),$$\n\twhere $opt$ is the optimal 2-medians or means clustering.\n\\end{theorem}\n\nThe key idea of the analysis is to bound the cost of the threshold clustering in terms of the number of points on which it disagrees with an optimal clustering. Intuitively, if any threshold cut must lead to a fairly different clustering, then the cost of the optimal 2-medians\/means clustering must also be large.\n\nWe note that it is possible to prove a slightly weaker bound by using the midpoint (for each feature) between the centers. When there are $t$ changes, using the midpoint shows that $\\mathrm{cost}(opt)$ is at least $t$ times {\\em half} of the distance between the two centers. In other words, this argument only captures half of the cost. Using Hall's theorem, we show that each change corresponds to a pair in the matching, and each such pair contributes to $\\mathrm{cost}(opt)$ the distance between the centers (not half as before). This improves the bound by a factor of two. The proof for $2$-means is in Appendix~\\ref{sec:k2_upper_bound}.\n\n\\paragraph{Notation.}\nWe denote the optimal clusters as $C^1$ and $C^2$ with optimal centers $\\boldsymbol{\\mu}^1$ and $\\boldsymbol{\\mu}^2$. Notice that we  can assume $\\mu^1_i \\leq \\mu^2_i$ for each coordinate $i$ because negating the $i$th coordinate for all points in the dataset does not change the $2$-medians\/means cost.\nAssume that a single threshold partitions $\\mathcal{X}$ into $\\widehat C^1, \\widehat C^2$ such that \n$$t = \\min(|C^1 \\Delta \\widehat C^1|, |C^1 \\Delta \\widehat C^2|).$$ We refer to these $t$ points as {\\em changes} and assume that $t$ is the minimum possible over all threshold cuts.\n\nIf $C^1,C^2$ is an optimal $2$-medians clustering, then we prove that the cost of  $\\widehat C^1, \\widehat C^2$ is at most twice the optimal $2$-medians cost. Similarly, if $C^1,C^2$ is an optimal $2$-means clustering, then we prove that the cost of $\\widehat C^1, \\widehat C^2$ is at most four times the optimal $2$-means cost. We simply need that the threshold cut $\\widehat C = (\\widehat C^1, \\widehat C^2)$ minimizes the number of changes $t$ compared to the optimal clusters.\n\nWe begin with a structural claim regarding the best threshold cut. This will allow us to obtain a tighter bound on the optimal $2$-medians\/means cost, compared to the general $k>2$ case, in terms of the necessary number of changes. We utilize Hall's theorem on perfect matchings. \n\n\\begin{proposition}[Hall's Theorem] Let $(P,Q)$ be a bipartite graph. If all subsets $P' \\subseteq P$ have at least $|P'|$ neighbors in $Q$, then there is a matching of size $|P|$.\n\\end{proposition}\n\n\\begin{lemma} \\label{lemma:matching}\n    Let $C^1$ and $C^2$ be the optimal clustering of $\\mathcal{X} \\subseteq \\mathbb{R}^d$, and assume that any threshold cut requires $t$ changes.\nFor each $i \\in [d]$, there are $t$ disjoint pairs of vectors $(\\mathbf{p}^j,\\mathbf{q}^j)$ in $\\mathcal{X}$ such that $\\mathbf{p}^j\\in C^1$ and $\\mathbf{q}^j\\in C^2$ and  $q^j_i\\leq p^j_i$ for every $j \\in [t]$. \n\\end{lemma}\n\\begin{proof}\n\tLet $\\boldsymbol{\\mu}^1$ and $\\boldsymbol{\\mu}^2$ be the centers for the optimal clusters $C^1$ and $C^2$. Focus on index $i \\in [d]$, and assume without loss of generality that  $\\mu^1_i\\leq \\mu^2_i$. The $t$ pairs correspond to a matching the following bipartite graph $(P,Q)$. Let $Q = C^2$ and define $P \\subseteq C^1$ as the $t$ points in $C^1$ with largest value in their $i$th coordinate. Connect $\\mathbf{p} \\in P$ and $\\mathbf{q} \\in Q$ by an edge if only if $q_i\\leq p_i.$ \n\tBy construction, a matching with $t$ edges implies our claim.\n\tBy Hall's theorem, we just need to prove that $P'\\subseteq P$ has at least $|P'|$ neighbors.\n\t\n\tIndex $P = \\{\\mathbf{p}^1,\\ldots, \\mathbf{p}^t\\}$ by ascending value of $i$th coordinate, $p^1_i \\leq \\cdots \\leq p^t_i.$  Now, notice that vertices in $P$ have nested neighborhoods: for all $j > j'$, the neighborhood of $\\mathbf{p}^{j'}$ is a subset of the neighborhood of $\\mathbf{p}^{j}$. It suffices to prove that $\\mathbf{p}^{j}$ has at least $j$ neighbors, because this implies that any subset $P' \\subseteq P$ has at least $|P'|$ neighbors, guaranteeing a matching of size $|P| = t$. Indeed, if $|P'| = b$ then we know that $\\mathbf{p}^{j} \\in P'$ for some $j \\geq b$, implying that $P'$ has at least $j \\geq b = |P'|$ neighbors.\n\n\tAssume for contradiction that $\\mathbf{p}^j$ has at most $j-1$ neighbors. We argue that the threshold cut $x_i\\leq p^j_i$ has fewer than $t$ changes, which contradicts the fact that all threshold cuts must make at least $t$ changes.  By our assumption, there are at most $j-1$ points that are smaller than $p^j_i$ and belong to the second cluster. \n\tBy the definition of $P$, there are exactly $t-j$ points with a larger $i$th coordinate than $p^j_i$ in the first cluster. Therefore, the threshold cut $x_i\\leq p^j_i$ makes at most $(t-j)+(j-1)<t$ changes, a contradiction.\n\\end{proof}\n\n\\subsection{Upper Bound Proof for 2-medians}\n\tSuppose $\\boldsymbol{\\mu}^1,\\boldsymbol{\\mu}^2$ are optimal $2$-medians centers for clusters $C^1$ and $C^2$, and that the threshold cut $\\widehat C$ makes $t$ changes, which is the minimum possible. \n\tA simple argument allows us to upper bound the cost of the threshold cut as $\\mathrm{cost}(opt)$ plus an error term that depends on the number of changes. More formally, Lemma~\\ref{lemma:tree-cost} (see Section~\\ref{sec:general_k_upper_bound}) in the special case of $k=2$ implies that \n\t$$\\mathrm{cost}(\\widehat C) \\leq \\mathrm{cost}(opt) + t\\onenorm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2}.$$ \n\tIn other words, the core of the argument is to prove that\n\t$t\\onenorm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2} \\leq \\mathrm{cost}(opt).$\n\t\n\t\n\tApplying Lemma~\\ref{lemma:matching} for each coordinate $i\\in[d]$ guarantees~$t$ pairs of vectors $(\\mathbf{p}^1,\\mathbf{q}^1), \\ldots, (\\mathbf{p}^t,\\mathbf{q}^t)$ with the following properties. Each $p^j_i$ corresponds to the $i$th coordinate of some  point in $C^1$ and $q^j_i$ corresponds to the $i$th coordinate of some point in $C^2$. Furthermore, for each coordinate, the~$t$ pairs correspond to $2t$ distinct points in $\\mathcal{X}$. Finally, we can assume without loss of generality that\n\t$\\mu^1_i \\leq \\mu^2_i$ and $q^j_i \\leq p^j_i$, \n\twhich implies that \n\t\\begin{eqnarray*}\n\t\\mathrm{cost}(opt) \\ge \\sum_{i=1}^d\\sum_{j=1}^t |\\mu^2_i-q_i^j| + |p^j_i-\\mu^1_i| &\\geq&\n\t\\sum_{i=1}^d\\sum_{j=1}^t(\\mu^2_i-q^j_i) + (p^j_i-\\mu_i^1)\n\t\\\\ &\\geq& \\sum_{i=1}^d\\sum_{j=1}^t(\\mu^2_i-q^j_i) + (q^j_i-p^j_i) + (p^j_i-\\mu_i^1)\n\t\\\\ &=& t \\cdot \\sum_{i=1}^d(\\mu^2_i-\\mu_i^1)\n\t=t\\onenorm{\\boldsymbol{\\mu}^2-\\boldsymbol{\\mu}^1}.\n\t\t\\end{eqnarray*}\n\n\\subsection{Lower bounds for \\texorpdfstring{$k=2$}{}}\n\n\nWe next show that optimal clustering is not, in general, realizable with a single threshold cut, except in a small number of dimensions (e.g., $d=1$). \nOur lower bounds on the approximation ratio increase with the dimension, approaching two for $2$-medians or three for $2$-means. \n\nThe two lower bounds are based on a data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$ consisting of $2d$ points, split into two optimal clusters each with $d$ points. The first cluster contains the $d$ vectors of the form $\\mathbf{1} - \\mathbf{e}^i$, where $\\mathbf{e}^i$ is the $i$th coordinate vector and $\\mathbf{1}$ is the all-ones vector. The second cluster contains their negations, $-\\mathbf{1} + \\mathbf{e}^i$. Due to the zero-valued coordinate in each vector, any threshold cut must separate at least one vector from its optimal center. In the case of $2$-medians, each incorrect cluster assignment incurs a cost of $2d$. The optimal cost is roughly $2d$, while the threshold cost is roughly $4d$ (correct assignments contribute $\\approx 2d$, plus $2d$ from the error), leading to an approximation ratio of nearly two. A similar result holds for $2$-means. The proof of these two lower bounds is in Appendix~\\ref{sec:k2_lower_bound}.\n\n\\begin{theorem}\\label{clm:2_median_lower_bound}\nFor any integer $d \\geq 1$, define data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$ as above. Any threshold cut $\\widehat C$ must have $2$-medians cost \n$$\\mathrm{cost}(\\widehat C)\\geq \\left(2-\\frac1d\\right)\\cdot \\mathrm{cost}(opt)$$\nand $2$-means cost\n$$\\mathrm{cost}(\\widehat C)\\geq 3\\left(1-\\frac1{d}\\right)^2\\cdot \\mathrm{cost}(opt),$$\nwhere $opt$ is the optimal 2-medians or means clustering.\n\\end{theorem}\n\n\\section{Two Clusters Using a Single Threshold Cut}\n\\label{sec:2-means}\n\nIn this section, we consider the case of $k=2$ clusters, and we study how well a single threshold cut can approximate the optimal partition into two clusters. \n\n\\subsection{Algorithm for \\texorpdfstring{$k=2$}{k=2}}\n\\label{sec:2means-alg}\n\nWe present an algorithm to efficiently minimize the cost using a single threshold cut.  We begin by considering a single feature $i$ and determining the value of the best threshold $\\theta \\in \\mathbb{R}$ for this feature. Then, we minimize over all features $i \\in [d]$ to output the best threshold cut. We focus on the $2$-means algorithm; the $2$-medians case is similar. \n\n\nFor feature~$i$, we first sort the input points according to this feature, i.e., assume that the vectors are indexed as $x^1_i \\leq \\ldots \\leq x^n_i.$\nNotice that when restricting to this feature, there are only $n-1$ possible partitions of the data set into two non-empty clusters. In particular, we can calculate the cost of all threshold cuts for the $i$th feature by scanning the values in this feature from smallest to largest. \nThen, we compute for each position $p \\in [n-1]$ $$\\mathrm{cost}(p)=\\sum_{j=1}^p\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^1(p)}_2^2 + \\sum_{j=p+1}^n\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^2(p)}_2^2,$$\nwhere we denote the optimal centers for these clusters as  $\\boldsymbol{\\mu}^1(p)=\\frac{1}{p}\\sum_{j=1}^p \\mathbf{x}^j$ and $\\boldsymbol{\\mu}^2(p)=\\frac{1}{n - p}\\sum_{j=p + 1}^n \\mathbf{x}^j$ because these are the means of the first $p$ and last $n-p$ points, respectively. Because there are $O(nd)$ possible thresholds, and naively computing the cost of each requires time $O(nd)$, this would lead to a running time of $O(n^2d^2)$. \nWe can improve the time to $O(nd^2 + nd\\log n)$ by using dynamic programming. Pseudo-code for the algorithm and description of the dynamic programming are in Appendix~\\ref{sec:DP_Implementation}.\n\n\n\n\\subsection{Theoretical guarantees for \\texorpdfstring{$k=2$}{}}\n\n\nWe prove that there always exists a threshold cut with low cost. Since our algorithm from the previous section finds the {\\em best} cut, it achieves the guarantees of this theorem.\n\\begin{theorem}\\label{thm:optimal_2_median_means}\n\tFor any data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$, there is a threshold cut $\\widehat C$ such that the $2$-medians cost satisfies\n\t$$\\mathrm{cost}(\\widehat C)\\leq 2 \\cdot \\mathrm{cost}(opt),$$\n\tand there is a threshold cut $\\widehat C$ such that the $2$-means cost satisfies\n\t$$\\mathrm{cost}(\\widehat C)\\leq 4 \\cdot \\mathrm{cost}(opt),$$\n\twhere $opt$ is the optimal 2-medians or means clustering.\n\\end{theorem}\n\nThe key idea of the analysis is to bound the cost of the threshold clustering in terms of the number of points on which it disagrees with an optimal clustering. Intuitively, if any threshold cut must lead to a fairly different clustering, then the cost of the optimal 2-medians\/means clustering must also be large.\n\nWe note that it is possible to prove a slightly weaker bound by using the midpoint (for each feature) between the centers. When there are $t$ changes, using the midpoint shows that $\\mathrm{cost}(opt)$ is at least $t$ times {\\em half} of the distance between the two centers. In other words, this argument only captures half of the cost. Using Hall's theorem, we show that each change corresponds to a pair in the matching, and each such pair contributes to $\\mathrm{cost}(opt)$ the distance between the centers (not half as before). This improves the bound by a factor of two. The proof for $2$-means is in Appendix~\\ref{sec:k2_upper_bound}.\n\n\\paragraph{Notation.}\nWe denote the optimal clusters as $C^1$ and $C^2$ with optimal centers $\\boldsymbol{\\mu}^1$ and $\\boldsymbol{\\mu}^2$. Notice that we  can assume $\\mu^1_i \\leq \\mu^2_i$ for each coordinate $i$ because negating the $i$th coordinate for all points in the dataset does not change the $2$-medians\/means cost.\nAssume that a single threshold partitions $\\mathcal{X}$ into $\\widehat C^1, \\widehat C^2$ such that \n$$t = \\min(|C^1 \\Delta \\widehat C^1|, |C^1 \\Delta \\widehat C^2|).$$ We refer to these $t$ points as {\\em changes} and assume that $t$ is the minimum possible over all threshold cuts.\n\nIf $C^1,C^2$ is an optimal $2$-medians clustering, then we prove that the cost of  $\\widehat C^1, \\widehat C^2$ is at most twice the optimal $2$-medians cost. Similarly, if $C^1,C^2$ is an optimal $2$-means clustering, then we prove that the cost of $\\widehat C^1, \\widehat C^2$ is at most four times the optimal $2$-means cost. We simply need that the threshold cut $\\widehat C = (\\widehat C^1, \\widehat C^2)$ minimizes the number of changes $t$ compared to the optimal clusters.\n\nWe begin with a structural claim regarding the best threshold cut. This will allow us to obtain a tighter bound on the optimal $2$-medians\/means cost, compared to the general $k>2$ case, in terms of the necessary number of changes. We utilize Hall's theorem on perfect matchings. \n\n\\begin{proposition}[Hall's Theorem] Let $(P,Q)$ be a bipartite graph. If all subsets $P' \\subseteq P$ have at least $|P'|$ neighbors in $Q$, then there is a matching of size $|P|$.\n\\end{proposition}\n\n\\begin{lemma} \\label{lemma:matching}\n    Let $C^1$ and $C^2$ be the optimal clustering of $\\mathcal{X} \\subseteq \\mathbb{R}^d$, and assume that any threshold cut requires $t$ changes.\nFor each $i \\in [d]$, there are $t$ disjoint pairs of vectors $(\\mathbf{p}^j,\\mathbf{q}^j)$ in $\\mathcal{X}$ such that $\\mathbf{p}^j\\in C^1$ and $\\mathbf{q}^j\\in C^2$ and  $q^j_i\\leq p^j_i$ for every $j \\in [t]$. \n\\end{lemma}\n\\begin{proof}\n\tLet $\\boldsymbol{\\mu}^1$ and $\\boldsymbol{\\mu}^2$ be the centers for the optimal clusters $C^1$ and $C^2$. Focus on index $i \\in [d]$, and assume without loss of generality that  $\\mu^1_i\\leq \\mu^2_i$. The $t$ pairs correspond to a matching the following bipartite graph $(P,Q)$. Let $Q = C^2$ and define $P \\subseteq C^1$ as the $t$ points in $C^1$ with largest value in their $i$th coordinate. Connect $\\mathbf{p} \\in P$ and $\\mathbf{q} \\in Q$ by an edge if only if $q_i\\leq p_i.$ \n\tBy construction, a matching with $t$ edges implies our claim.\n\tBy Hall's theorem, we just need to prove that $P'\\subseteq P$ has at least $|P'|$ neighbors.\n\t\n\tIndex $P = \\{\\mathbf{p}^1,\\ldots, \\mathbf{p}^t\\}$ by ascending value of $i$th coordinate, $p^1_i \\leq \\cdots \\leq p^t_i.$  Now, notice that vertices in $P$ have nested neighborhoods: for all $j > j'$, the neighborhood of $\\mathbf{p}^{j'}$ is a subset of the neighborhood of $\\mathbf{p}^{j}$. It suffices to prove that $\\mathbf{p}^{j}$ has at least $j$ neighbors, because this implies that any subset $P' \\subseteq P$ has at least $|P'|$ neighbors, guaranteeing a matching of size $|P| = t$. Indeed, if $|P'| = b$ then we know that $\\mathbf{p}^{j} \\in P'$ for some $j \\geq b$, implying that $P'$ has at least $j \\geq b = |P'|$ neighbors.\n\n\tAssume for contradiction that $\\mathbf{p}^j$ has at most $j-1$ neighbors. We argue that the threshold cut $x_i\\leq p^j_i$ has fewer than $t$ changes, which contradicts the fact that all threshold cuts must make at least $t$ changes.  By our assumption, there are at most $j-1$ points that are smaller than $p^j_i$ and belong to the second cluster. \n\tBy the definition of $P$, there are exactly $t-j$ points with a larger $i$th coordinate than $p^j_i$ in the first cluster. Therefore, the threshold cut $x_i\\leq p^j_i$ makes at most $(t-j)+(j-1)<t$ changes, a contradiction.\n\\end{proof}\n\n\\subsection{Upper Bound Proof for 2-medians}\n\tSuppose $\\boldsymbol{\\mu}^1,\\boldsymbol{\\mu}^2$ are optimal $2$-medians centers for clusters $C^1$ and $C^2$, and that the threshold cut $\\widehat C$ makes $t$ changes, which is the minimum possible. \n\tA simple argument allows us to upper bound the cost of the threshold cut as $\\mathrm{cost}(opt)$ plus an error term that depends on the number of changes. More formally, Lemma~\\ref{lemma:tree-cost} (see Section~\\ref{sec:general_k_upper_bound}) in the special case of $k=2$ implies that \n\t$$\\mathrm{cost}(\\widehat C) \\leq \\mathrm{cost}(opt) + t\\onenorm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2}.$$ \n\tIn other words, the core of the argument is to prove that\n\t$t\\onenorm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2} \\leq \\mathrm{cost}(opt).$\n\t\n\t\n\tApplying Lemma~\\ref{lemma:matching} for each coordinate $i\\in[d]$ guarantees~$t$ pairs of vectors $(\\mathbf{p}^1,\\mathbf{q}^1), \\ldots, (\\mathbf{p}^t,\\mathbf{q}^t)$ with the following properties. Each $p^j_i$ corresponds to the $i$th coordinate of some  point in $C^1$ and $q^j_i$ corresponds to the $i$th coordinate of some point in $C^2$. Furthermore, for each coordinate, the~$t$ pairs correspond to $2t$ distinct points in $\\mathcal{X}$. Finally, we can assume without loss of generality that\n\t$\\mu^1_i \\leq \\mu^2_i$ and $q^j_i \\leq p^j_i$, \n\twhich implies that \n\t\\begin{eqnarray*}\n\t\\mathrm{cost}(opt) \\ge \\sum_{i=1}^d\\sum_{j=1}^t |\\mu^2_i-q_i^j| + |p^j_i-\\mu^1_i| &\\geq&\n\t\\sum_{i=1}^d\\sum_{j=1}^t(\\mu^2_i-q^j_i) + (p^j_i-\\mu_i^1)\n\t\\\\ &\\geq& \\sum_{i=1}^d\\sum_{j=1}^t(\\mu^2_i-q^j_i) + (q^j_i-p^j_i) + (p^j_i-\\mu_i^1)\n\t\\\\ &=& t \\cdot \\sum_{i=1}^d(\\mu^2_i-\\mu_i^1)\n\t=t\\onenorm{\\boldsymbol{\\mu}^2-\\boldsymbol{\\mu}^1}.\n\t\t\\end{eqnarray*}\n\n\\subsection{Lower bounds for \\texorpdfstring{$k=2$}{}}\n\n\nWe next show that optimal clustering is not, in general, realizable with a single threshold cut, except in a small number of dimensions (e.g., $d=1$). \nOur lower bounds on the approximation ratio increase with the dimension, approaching two for $2$-medians or three for $2$-means. \n\nThe two lower bounds are based on a data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$ consisting of $2d$ points, split into two optimal clusters each with $d$ points. The first cluster contains the $d$ vectors of the form $\\mathbf{1} - \\mathbf{e}^i$, where $\\mathbf{e}^i$ is the $i$th coordinate vector and $\\mathbf{1}$ is the all-ones vector. The second cluster contains their negations, $-\\mathbf{1} + \\mathbf{e}^i$. Due to the zero-valued coordinate in each vector, any threshold cut must separate at least one vector from its optimal center. In the case of $2$-medians, each incorrect cluster assignment incurs a cost of $2d$. The optimal cost is roughly $2d$, while the threshold cost is roughly $4d$ (correct assignments contribute $\\approx 2d$, plus $2d$ from the error), leading to an approximation ratio of nearly two. A similar result holds for $2$-means. The proof of these two lower bounds is in Appendix~\\ref{sec:k2_lower_bound}.\n\n\\begin{theorem}\\label{clm:2_median_lower_bound}\nFor any integer $d \\geq 1$, define data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$ as above. Any threshold cut $\\widehat C$ must have $2$-medians cost \n$$\\mathrm{cost}(\\widehat C)\\geq \\left(2-\\frac1d\\right)\\cdot \\mathrm{cost}(opt)$$\nand $2$-means cost\n$$\\mathrm{cost}(\\widehat C)\\geq 3\\left(1-\\frac1{d}\\right)^2\\cdot \\mathrm{cost}(opt),$$\nwhere $opt$ is the optimal 2-medians or means clustering.\n\\end{theorem}\n\n\n\\section{Threshold trees with $k > 2$ leaves}\n\\label{sec:k-means}\n\nWe provide an efficient algorithm to produce a threshold tree with $k$ leaves that constitutes an approximate $k$-medians or $k$-means clustering of a data set $\\mathcal{X}$. Our algorithm, Iterative Mistake Minimization (IMM), starts with a reference set of cluster centers, for instance from a polynomial-time constant-factor approximation algorithm for $k$-medians or $k$-means~\\cite{aggarwal09}, or from a domain-specific clustering heuristic. \n\nWe then begin the process of finding an explainable approximation to this reference clustering, in the form of a threshold tree with $k$ leaves, whose internal splits are based on single features. The way we do this is almost identical for $k$-medians and $k$-means, and the analysis is also nearly the same. Our algorithm is deterministic and its run time is only $O(kdn \\log n)$, after finding the initial centers.\n \n \nAs discussed in Section~\\ref{sec:motivating-examples}, existing decision tree algorithms use greedy criteria that are not suitable for our tree-building process. However, we show that an alternative greedy criterion---minimizing the number of {\\em mistakes} at each split (the number of points separated from their corresponding cluster center)---leads to a favorable approximation ratio to the optimal $k$-medians or $k$-means cost.\n\n\n\\subsection{Our algorithm}\n\\label{sec:k-means-alg}\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\\centering\n\\begin{minipage}{.5\\textwidth}\n\\begin{algorithm}[H]\n\\SetKwFunction{Iterative Mistake Minimization}{Iterative Mistakes Minimization}\n\\SetKwFunction{BuildTree}{BuildTree}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\\SetKwInOut{Preprocess}{Preprocess}\n\\Input{%\n\t$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$ -- vectors in $\\mathbb{R}^d$\\\\\n\t$k$ -- number of clusters\\\\\n  }\n\\Output{%\n    root of the threshold tree\n}\n\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\n$\\boldsymbol{\\mu}^1, \\ldots \\boldsymbol{\\mu}^k \\leftarrow \\xfunc{k-Means}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n, k)$\\;\n\n\\ForEach {$j \\in [1, \\ldots, n]$}\n{\n    $y^j \\leftarrow \\argmin_{1 \\leq \\ell \\leq k} \\lVert \\mathbf{x}^j - \\boldsymbol{\\mu}^\\ell \\rVert$\\;\n}\n\n\\Return $\\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j=1}^n, \\{y^j\\}_{j=1}^n, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n\\SetKwProg{buildtree}{$\\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j=1}^m, \\{y^j\\}_{j=1}^m, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$:}{}{}\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\\buildtree{}{\n\\If{$\\{y^j\\}_{j=1}^m$ \\text{is homogeneous}}\n{\n    $\\xvar{leaf}.cluster \\leftarrow y^1$\\;\n    \n    \\Return $\\xvar{leaf}$\\;\n}\n\\ForEach {$i \\in [1, \\ldots, d]$}\n{\n    $\\ell_i \\leftarrow \\min_{1 \\leq j \\leq m} \\mu^{y^j}_i$\\;\n    \n    $r_i \\leftarrow \\max_{1 \\leq j \\leq m} \\mu^{y^j}_i$\\;\n}\n$i, \\theta \\leftarrow \\argmin_{i,\\ell_i \\leq \\theta < r_i} \\sum_{j=1}^m \\xfunc{mistake}(\\mathbf{x}^j, \\boldsymbol{\\mu}^{y^j}, i, \\theta)$\\;\\label{ln:k_dynamic}\n\n$\\xvar{M} \\leftarrow \\{j \\mid \\xfunc{mistake}(\\mathbf{x}^j, \\boldsymbol{\\mu}^{y^j}, i, \\theta) = 1\\}_{j=1}^m$\\; \n\n$\\xvar{L} \\leftarrow \\{j \\mid (x^j_i \\leq \\theta) \\wedge (j \\not \\in \\xvar{M})\\}_{j=1}^m$\\;\n\n$\\xvar{R} \\leftarrow \\{j \\mid (x^j_i > \\theta) \\wedge (j \\not \\in \\xvar{M})\\}_{j=1}^m$\\;\n\n$\\xvar{node}.condition \\leftarrow ``x_i \\leq \\theta\"$\\;\n\n$\\xvar{node}.lt \\leftarrow \\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j \\in \\xvar{L}}, \\{y^j\\}_{j \\in \\xvar{L}}, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n$\\xvar{node}.rt \\leftarrow \\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j \\in \\xvar{R}}, \\{y^j\\}_{j \\in \\xvar{R}}, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n\n\\Return $\\xvar{node}$\\;\n}\n\n\n\\SetKwProg{mistake}{$\\xfunc{mistake}(\\mathbf{x}, \\boldsymbol{\\mu}, i, \\theta)$:}{}{}\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\\mistake{}{\n\\Return $(x_i \\leq \\theta) \\neq (\\mu_i \\leq \\theta)$  ? $1$ : $0$\\;\n}\n\\caption{\\textsc{\\newline Iterative Mistake Minimization}}\n\\label{algo:imm}\n\\end{algorithm}\n\\end{minipage} \\vspace{-2ex}\n\\end{wrapfigure}\n\nAlgorithm~\\ref{algo:imm} takes as input a data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$. The first step is to obtain a reference set of $k$ centers $\\{\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k\\}$, for instance from a standard clustering algorithm. We assign each data point $\\mathbf{x}^j$ the label $y^j$ of its closest center. Then, the {\\tt{build\\_tree}} procedure looks for a tree-induced clustering that fits these labels. The tree is built top-down, using binary splits. Each node $u$ can be associated with the portion of the input space that passes through that node, a hyper-rectangular region $\\mbox{cell}(u) \\subseteq \\mathbb{R}^d$. If this cell contains two or more of the centers $\\boldsymbol{\\mu}^j$, then it needs to be split. We do so by picking the feature $i \\in [d]$ and threshold value $\\theta \\in \\mathbb{R}$ such that the resulting split $x_i \\leq \\theta$ sends at least one center to each side and moreover produces the fewest {\\em mistakes}: that is, separates the fewest points in $\\mathcal{X} \\cap \\mbox{cell}(u)$ from their corresponding centers in $\\{\\boldsymbol{\\mu}^j: 1 \\leq j \\leq k\\} \\cap \\mbox{cell}(u)$. We do not count points whose centers lie outside $\\mbox{cell}(u)$, since they are associated with mistakes in earlier splits. We find the optimal split $(i, \\theta)$ by searching over all pairs efficiently using dynamic programming. We then add this node to the tree, and discard the mistakes (the points that got split from their centers) before recursing on the left and right children. We terminate at a leaf node whenever all points have the same label (i.e., a {\\em homogeneous} subset). As there are $k$ different labels, the resulting tree has exactly $k$ leaves.\nFigure \\ref{fig:imm_example} depicts the operation of Algorithm~\\ref{algo:imm}. \n\nWe first discuss the running time, and we analyze the approximation guarantees of IMM in Section~\\ref{sec:imm-approx-main}.\n\n\n\n\n\\smallskip \\noindent {\\bf Time analysis of tree building.}\nWe sketch how to execute the algorithm in time $O(kdn\\log n)$ for an $n$-point data set. At each step of the top-down procedure, we find a coordinate and threshold pair that minimizes the mistakes at this node (line \\ref{ln:k_dynamic} in \\texttt{build\\_tree} procedure). We use dynamic programming to avoid recomputing the cost from scratch for each potential threshold. For each coordinate $i \\in [d]$, we sort the data and centers. Then, we iterate over possible thresholds. We claim that we can process each node in time $O(dn\\log n)$ because each point will affect the number of mistakes at most twice. Indeed, when the threshold moves, either a data point or a center moves to the other side of the threshold. Since we know the number of mistakes from the previous threshold, we count the new mistakes efficiently as follows. If a single point switches sides, then the number of mistakes changes by at most one. If a center switches sides, which happens at most once, then we update the mistakes for this center. Overall, each point affects the mistakes at most twice (once when changing sides, and once when its center switches sides). Thus, the running time for each internal node is $O(dn\\log n)$. As the tree has $k-1$ internal nodes, the total time is $O(kdn \\log n)$.\n\n\n\\input{imm-pic-steps}\n\n\\newpage\n\\subsection{Approximation guarantee for the IMM algorithm}\n\\label{sec:imm-approx-main}\n\nOur main theoretical contribution is the following result.\n\n\\begin{theorem}\\label{thm:main-k}\\label{thm:main-k-appendix}\n\tSuppose that IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$. Then, \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost is at most $$\\mathrm{cost}(T)\\leq (2H+1) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$$\n\t\t\\item The $k$-means cost is at most $$\\mathrm{cost}(T)\\leq (8Hk+2) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$$\n\t\\end{enumerate}\n\tIn particular, IMM achieves worst case approximation factors of $O(k)$ and $O(k^2)$ by using any $O(1)$ approximation algorithm (compared to the optimal $k$-medians\/means) to generate the initial centers.\n\\end{theorem}\n\nWe state the theorem in terms of the depth of the tree to highlight that the approximation guarantee may depend on the structure of the input data. If the optimal clusters can be easily identified by a small number of salient features, then the tree may have depth $O(\\log k)$. \nWe later provide a lower bound showing that an $\\Omega(\\log k)$ approximation factor is necessary for $k$-medians and $k$-means (Theorem~\\ref{thm:lb-k}). For this data set, our algorithm produces a threshold tree with depth $O(\\log k)$, and therefore, the analysis is tight for $k$-medians. We leave it as an intriguing open question whether the bound can be improved for $k$-means.\n\n\\subsubsection{Proof Overview for Theorem~\\ref{thm:main-k}}\n\nThe proof proceeds in three main steps. First, we rewrite the cost of IMM in terms of the minimum number of mistakes made between the output clustering and the clustering based on the given centers. Second, we provide a lemma that relates the cost of any clustering to the number of mistakes required by a threshold clustering. Finally, we put these two together to show that the output cost is at most an $O(H)$ factor larger than the $k$-medians cost and at most an $O(Hk)$ factor larger than the $k$-means cost, respectively, where $H$ is the depth of the IMM tree, and the cost is relative to $\\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$. \n\nThe approximation bound rests upon a  characterization of the excess clustering cost induced by the tree. For any internal node $u$ of the final tree $T$, let $\\mbox{cell}(u) \\subseteq \\mathbb{R}^d$ denote the region of the input space that ends up in that node, and let $B(u)$ be the bounding box of the centers that lie in this node, or more precisely,  $B(u) = \\{\\boldsymbol{\\mu}^j: 1 \\leq j \\leq k\\} \\cap \\mbox{cell}(u)$. We will be interested in the diameter of this bounding box, measured either by $\\ell_1$ or squared $\\ell_2$ norm, and denoted by $\\mathrm{diam}_1(B(u))$ and $\\mathrm{diam}_2^2(B(u))$, respectively.\n\n\n\\paragraph{Upper bounding the cost of the tree.} The first technical claim (Lemma~\\ref{lemma:tree-cost}) will show that if IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\\begin{itemize}\n\\item The $k$-medians cost of $T$ satisfies   $\\displaystyle \\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\mathrm{diam}_1(B(u)) $\n\\item The $k$-means cost of $T$ satisfies  $\\displaystyle  \\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\mathrm{diam}_2^2(B(u))$\n\\end{itemize}\n\nBriefly, any point $\\mathbf{x}$ that ends up in a different leaf from its correct center $\\boldsymbol{\\mu}^j$ incurs some extra cost. To bound this, consider the internal node $u$ at which $\\mathbf{x}$ is separated from $\\boldsymbol{\\mu}^j$. Node $u$ also contains the center $\\boldsymbol{\\mu}^i$ that ultimately ends up in the same leaf as $\\mathbf{x}$. For $k$-medians, the excess cost for $\\mathbf{x}$ can then be bounded by $\\|\\boldsymbol{\\mu}^i - \\boldsymbol{\\mu}^j\\|_1 \\leq \\mathrm{diam}_1(B(u))$. The argument for $k$-means is similar.\n\nThese $\\sum_u t_u \\mathrm{diam}(B(u))$ terms can in turn be bounded in terms of the cost of the reference clustering. \n\n\\paragraph{Lower bounding the reference cost.} \nWe next need to relate the cost of the centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ to the number of mistakes and the diameter of the cells in the tree. Lemma~\\ref{aux-claim:general_k_upper_bound} will show that if IMM makes $t_u$ mistakes at node $u \\in T$, then\n\\begin{itemize}\n\\item The $k$-medians cost satisfies \n$\\displaystyle \\sum_{u \\in T} t_u \\cdot \\mathrm{diam}_1(B(u)) \\leq 2H \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$\n\\item The $k$-means cost satisfies \n$\\displaystyle \\sum_{u \\in T} t_u \\cdot  \\mathrm{diam}_2^2(B(u)) \\leq 4Hk \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$\n\\end{itemize}\n\nThe proof for this is significantly more complicated than the upper bound mentioned above. Moreover, it contains the main new techniques in our analysis of tree-based clusterings. \n\nThe core challenge is that we aim to lower bound the cost of the given centers using only information about the number of mistakes at each internal node. Moreover, the IMM algorithm only minimizes the {\\em number} of mistakes, and not the {\\em cost} of each mistake. Therefore, we must show that if every axis-aligned cut in $B(u)$ separates at least $t_u$ points~$\\mathbf{x}$ from their centers, then there must be a considerable distance between the points in $\\mbox{cell}(u)$ and their centers.\n\nTo prove this, we analyze the structure of points in each cell. Specifically, we consider the single-coordinate projection of points in the box $B(u)$, and we order the centers in $B(u)$ from smallest to largest for the analysis. If there are $k'$ centers in node $u$, we consider the partition of $B(u)$ into $2(k'-1)$ disjoint segments, splitting at the centers and at the midpoints between consecutive centers. Since $t_u$ is the minimum number of mistakes, we must in particular have at least $t_u$ mistakes from the threshold cut at each midpoint. We argue that each of these segments is covered at least $t_u$ times by a certain set of intervals. Specifically, we consider the intervals between mistake points and their true centers, and we say that an interval \\textit{covers} a segment if the segment is contained in the interval. This allows us to capture the cost of mistakes at different distance scales. For example, if a point is very far from its true center, then it covers many disjoint segments, and we show that it also implies a large contribution to the cost. \nClaim~\\ref{claim:covering} in  Section~\\ref{sec:general_k_upper_bound} provides our main covering result, and we use this to argue that the cost of the given centers can be lower bounded in terms of the distance between consecutive centers in $B(u)$. For $k$-medians, we can directly derive a lower bound on the cost in terms of the $\\ell_1$ diameter $\\mathrm{diam}_1(B(u))$. For $k$-means, however, we employ Cauchy-Schwarz, which incurs an extra factor of $k$ in the bound with $\\mathrm{diam}_2^2(B(u))$. Overall, we sum these bounds over the height $H$ of the tree, leading to the claimed upper bounds in the above lemma. \n\n\\subsubsection{Preliminaries and Notation for Theorem~\\ref{thm:main-k}}\n\nLet  $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ be the reference centers, and let $T$ be the resulting IMM tree. Each internal node $u$ corresponds to a value $\\theta_u \\in \\mathbb{R}$ and a coordinate $i \\in [d]$. The tree partitions $\\mathcal{X}$ into $k$ clusters $\\widehat C_1, \\ldots, \\widehat C_k$ based on the points that reach the $k$ leaves in $T$, where we index the clusters so that leaf $j$ contains the centers $\\boldsymbol{\\mu}^j$ and $\\widehat \\boldsymbol{\\mu}^j$, where $\\widehat \\boldsymbol{\\mu}^j$ is the mean of $\\widehat C_j$ for $k$-means and the median of $\\widehat C_j$ for $k$-medians. This provides a bijection between old and new centers (and clusters).\nRecall that the map $c:\\mathcal{X} \\to \\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$ associates each point to its nearest center (i.e., $c(\\mathbf{x})$ corresponds to the cluster assignment given by the centers $\\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$).\n\n\nFor a node $u \\in T$, we let $\\mathcal{X}_u$  denote the surviving data set vectors at node $u \\in T$ based on the thresholds from the root to $u$.\nWe also define $J_u \\subseteq [k]$ be the set of surviving centers at node~$u$ from the set $\\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$, where these centers satisfy the thresholds from the root to~$u$. \nDefine $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$ to be the maximal (smallest and largest) coordinate-wise values of the centers in $J_u$, that is, for $i \\in [d]$, we set\n$$\n\\mu^{L,u}_i = \\min_{j \\in J_u} \\mu^j_i,\n\\qquad\\mathrm{and}\\qquad \n\\mu^{R,u}_i = \\max_{j \\in J_u} \\mu^j_i.\n$$\nIn other words, using the previous notation and recalling that $B(u) = \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\} \\cap \\mathrm{cell}(u)$, we have that\n$$\n\\mathrm{diam}_1(B(u)) = \\|\\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u}\\|_1\n\\qquad \n\\mbox{\\ and\\ }\n\\qquad\n\\mathrm{diam}_2^2(B(u)) = \\|\\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u}\\|_2^2.\n$$\n\nRecall that $t_u$ for node $u \\in T$ denotes the number of {\\em mistakes} incurred during the threshold cut defined by $u$, where a point $\\mathbf{x}$ is a mistake at node $u$ if $x$ reaches $u$, it was not a mistake before, and exactly one of the following two events occurs:\n$$\n\\{c(\\mathbf{x})_i \\leq \\theta_u \\ \\ \\mathrm{and}\\ \\  x_i > \\theta_u\\}\n\\qquad \\mathrm{or} \\qquad\n\\{c(\\mathbf{x})_i > \\theta_u \\ \\ \\mathrm{and}\\ \\   x_i \\leq \\theta_u\\}.\n$$\nLet $\\mathcal{X} = \\cX^{\\mathsf{cor}} \\cup \\cX^{\\mathsf{mis}}$ be a partition of the input data set into two parts, where $\\mathbf{x}$ is in $\\cX^{\\mathsf{cor}}$ if it reaches the same leaf node in $T$ as its center $c(\\mathbf{x})$, and otherwise, $\\mathbf{x}$ is in $\\cX^{\\mathsf{mis}}$. In other words, $\\cX^{\\mathsf{mis}}$ contains all points $\\mathbf{x} \\in \\mathcal{X}$ that are a mistake at any node $u$ in $T$, and the rest of the points are in $\\cX^{\\mathsf{cor}}$. We note that the notion of ``mistakes'' used here is different than the definition of ``changes'' used for the analysis of $2$-means\/medians, even though we reuse some of the same notation.\n\nWe need a standard consequence of the Cauchy\u2013Schwarz inequality to analyze the $k$-means cost.\n\\begin{claim}\\label{clm:cauchy_schwarz_k_means}\n\tFor any $a_1,\\ldots,a_{m}\\in\\mathbb{R},$ it holds that $\\sum_{i=1}^ka_i^2 \\geq \\frac{1}{k}\\left(\\sum_{i=1}^ka_i\\right)^2.$\n\\end{claim}\n\\begin{proof}\n\tDenote by $a$ the vector $(a_1,\\ldots,a_{m})$ and by $b$ the vector $(\\nicefrac{1}{\\sqrt{k}},\\ldots,\\nicefrac{1}{\\sqrt{k}}).$ \n\tBy the Cauchy\u2013Schwarz inequality\n\t$\\frac{1}{k}\\left(\\sum_{i=1}^ka_i\\right)^2=\\inner{a}{b}^2\\leq\\sum_{i=1}^ka_i^2$\n\\end{proof}\n\nWe also need two facts, which state the optimal center for a cluster corresponds to mean or median of the points in the cluster, respectively. The proofs of these facts can be found in standard texts~\\cite{schutze2008introduction}.\n\n\\begin{fact}\\label{fact:1-means-optimal center}\nFor any set $S=\\{\\mathbf{x}^1,\\ldots, \\mathbf{x}^n\\}\\subseteq \\mathbb{R}^d$, the optimal center under the $\\ell_2^2$ cost  is the mean $\\boldsymbol{\\mu} = \\frac1n\\sum_{\\mathbf{x}\\in S}\\mathbf{x}.$\n\\end{fact}\n\n\\begin{fact}\\label{fact:1-median-optimal center}\nFor any set $S=\\{\\mathbf{x}^1,\\ldots, \\mathbf{x}^n\\}\\subseteq \\mathbb{R}^d$, the optimal center $\\boldsymbol{\\mu}$ under the $\\ell_1$ cost is the coordinate-wise median, defined for $i\\in[d]$ as $\\mu_i = \\mathsf{median}(x^1_i,\\ldots, x^n_i).$\n\\end{fact}\n\n\\subsubsection{The Two Main Lemmas and the Proof of Theorem~\\ref{thm:main-k}}\n\nTo prove the theorem, we state two lemmas that aid in analyzing the cost of the given clustering versus the IMM clustering. \nThe theorem will follow from these lemmas, and we will prove the lemmas in the proceeding subsections. We start with the lemma relating the number of mistakes~$t_u$ at each node $u$ and the distance between $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$ to the cost incurred by the given centers.\n\n\\begin{lemma}\\label{lemma:tree-cost}\n\tIf IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost of the IMM tree satisfies  $$\\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\n\t\t\\item The $k$-means cost of the IMM tree satisfies  $$\\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.$$\n\t\\end{enumerate}\n\\end{lemma}\n\nWe next bound the cost of the given centers in the terms of the number of mistakes in the tree. The key idea is that if there must be many mistakes at each node, then the cost of the given centers must actually be fairly large.\n\n\\begin{lemma}\\label{aux-claim:general_k_upper_bound}\n\tIf IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost satisfies \n\t\t$$\\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1 \\leq 2H \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\n\t\t\\item The $k$-means cost \n\t\tsatisfies \n\t\t$$\\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2 \\leq 4Hk \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\n\t\\end{enumerate}\n\\end{lemma}\n\nCombining these two lemmas immediately implies Theorem~\\ref{thm:main-k}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main-k}.]\nFor $k$-medians, Lemmas~\\ref{lemma:tree-cost} and~\\ref{aux-claim:general_k_upper_bound}\ntogether imply that \n$$\\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1\n\\leq \n(2H+1) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\nFor $k$-means, we have that \n\\begin{eqnarray*}\n\\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2\n\\leq (8Hk+2) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).\n\\end{eqnarray*}\n\n\\end{proof}\n\n\n\n\\subsubsection{Proof of Lemma~\\ref{lemma:tree-cost}}\n\nWe begin with the $k$-medians proof (the $k$-means proof will be similar). Notice that the cost can only increase when measuring the distance to the (suboptimal) center $\\boldsymbol{\\mu}^j$ instead of the (optimal) center $\\widehat \\boldsymbol{\\mu}^j$ for cluster $\\widehat C_j$, and hence,\n$$\n\\mathrm{cost}(T) = \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\widehat \\boldsymbol{\\mu}^j\\|_1\n\\leq \n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\boldsymbol{\\mu}^j\\|_1.$$\nWe can rewrite this sum using the partition  $\\cX^{\\mathsf{cor}}$ and $\\cX^{\\mathsf{mis}}$ of $\\mathcal{X}$, using the fact that\nwhenever $\\mathbf{x}\\in \\cX^{\\mathsf{cor}}$, then the distance is computed with respect to the true center $c(\\mathbf{x})$,\n\\begin{eqnarray*}\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\boldsymbol{\\mu}^j\\|_1 &=& \n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\ &=& \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\end{eqnarray*}\nStarting with the above cost bound, and using the triangle inequality, we see\n\\begin{eqnarray*}\n\\mathrm{cost}(T) \n&\\leq&\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\ &\\leq&  \n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \n(\\|\\mathbf{x}- c(\\mathbf{x})\\|_1 + \\|c(\\mathbf{x})- \\boldsymbol{\\mu}^j\\|_1) \n\\\\ &=&  \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|c(\\mathbf{x})- \\boldsymbol{\\mu}^j\\|_1\n\\end{eqnarray*}\n\nTo control the second term in the final line, we must bound the cost of the mistakes. We decompose $\\cX^{\\mathsf{mis}}$ based on the node $u$ where $\\mathbf{x} \\in \\cX^{\\mathsf{mis}}$ is first separated from its true center $c(\\mathbf{x})$ due to the threshold at node~$u$.\nTo this end, consider some point $\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j$, where its distance is measured to the incorrect center $\\boldsymbol{\\mu}^{j} \\neq c(\\mathbf{x})$.  Both centers $c(\\mathbf{x})$ and $\\boldsymbol{\\mu}^j$ have survived until node $u$ in the threshold tree $T$, and hence, both vectors are part of the definitions of $\\boldsymbol{\\mu}^{L,u}$ and  $\\boldsymbol{\\mu}^{R,u}$. In particular, we can use the upper bound\n$$\\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_1 \\leq \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\nThere are $t_u$ points in $\\cX^{\\mathsf{mis}}$ caused by the threshold at node $u$, and we have that\n$$\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_1 \\leq \n\\sum_{u \\in T} t_u \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\nTherefore, we have, as desired\n\\begin{eqnarray*}\n\\mathrm{cost}(T) &\\leq& \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\&\\leq& \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n\\end{eqnarray*}\n\\noindent\nAnalyzing $k$-means is similar; we incur a factor of two by using Claim~\\ref{clm:cauchy_schwarz_k_means} instead of the triangle inequality:\n\\begin{eqnarray*}\n\\mathrm{cost}(T) &\\leq& \n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_2^2 + \n2\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j}  (\\|\\mathbf{x}- c(\\mathbf{x})\\|_2^2 +\\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_2^2)\n\\\\&\\leq& 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j}  \\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_2^2\n\\\\ &\\leq& 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2\n\\end{eqnarray*}\n\n\n\\subsubsection{Proof of Lemma~\\ref{aux-claim:general_k_upper_bound}}\n\\label{sec:general_k_upper_bound}\n\t\nTo prove this lemma, we bound the cost at each node $u$ of tree in terms of the mistakes made at this node. For this lemma, we define $\\cX^{\\mathsf{cor}}_u$ to be the set of points in $\\mathcal{X}$ that reach node $u$ in $T$ along with their center $c(\\mathbf{x})$. We note that $\\cX^{\\mathsf{cor}}_u$ differs from $\\cX^{\\mathsf{cor}} \\cap \\mathcal{X}_u$ because a point $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$ may not make it to $\\cX^{\\mathsf{cor}}$ if there is a mistake later on (i.e., $\\cX^{\\mathsf{cor}}$ is the union of $\\cX^{\\mathsf{cor}}_u$ only over leaf nodes).\n\n\\begin{lemma}\\label{lemma:mistake-bound}\nFor any node $u \\in T$, we have that\n\t$$\n\t\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n\t\\geq \n\t\\frac{t_u}{2} \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n\t$$\nand\n\t$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n\\geq \n\\frac{t_u}{4k}   \\cdot  \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{lemma}\n\\begin{proof}\nFix a coordinate $i \\in [d]$ and a node $u \\in T$.\nTo simplify notation, we let $z_1 \\leq \\cdots \\leq z_{k'}$ denote the {\\em sorted} values of $i$th coordinate of the $k' \\leq k$ centers that survive until node $u$ (so that $z_1 = \\mu^{L,u}_i$ and $z_{k'} = \\mu^{R,u}_i$). Observe that for each $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$, the center $c(\\mathbf{x})$ must have survived until node $u$, and hence, $c(\\mathbf{x})_i$ equals one of the values $z_j$ for $j\\in[k']$. \n\nWe need a definition that allows us to relate the cost in coordinate $i$ to the distances between $z_1$ and $z_{k'}$. For consecutive values $(j,j+1)$, we say that the pair $(j,j+1)$ is {\\em covered} by $\\mathbf{x}$ if either \n\\begin{itemize}\n\t\\item The segment $[z_j, \\frac{z_j + z_{j+1}}{2})$ is contained in the segment $[x_i,c(\\mathbf{x})_i]$, or\n\t\\item The segment $[\\frac{z_j + z_{j+1}}{2}, z_{j+1})$ is contained in the segment $[x_i,c(\\mathbf{x})_i]$.\n\\end{itemize} \n\nWe prove the following claim, which enables us to relate the cost in the $i$th coordinate to the value $z_{k'} - z_1$ by decomposing this value into the distance between consecutive centers.\n\\begin{claim}\\label{claim:covering}\nFor each $j = 1,2,\\ldots, k'-1$, the pair $(j,j+1)$ is {covered} by at least~$t_u$ points $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$.\n\\end{claim}\n\\begin{proof}\nSuppose for contradiction that this does not hold. We argue that we can find a threshold value for coordinate $i$ that makes fewer than $t_u$ mistakes. To see this, assume that $(j,j+1)$ is covered by fewer than $t_u$ points $\\mathbf{x} \\in \\mathcal{X}_u$. In particular, setting the threshold to be $\\frac{z_j + z_{j+1}}{2}$ separates fewer than $t_u$ points $\\mathbf{x}$ from their centers $c(\\mathbf{x})$. This implies that there are fewer than $t_u$ mistakes at node $u$, which is a contradiction because the IMM algorithm chooses the coordinate and threshold pair that minimizes the number of mistakes.\n\\end{proof}\n\nNow this claim suffices to prove Lemma~\\ref{lemma:mistake-bound}. The only challenge is that we must string together the covering points $\\mathbf{x}$ to get a bound on $z_{k'}-z_1$.\n\nWe start with the $k$-medians proof. Using the above claim, we can lower bound the contribution of coordinate $i$ to the cost of the given centers. Notice that the values $z_1 \\leq \\cdots \\leq z_{k'}$ partition the interval between $z_1 = \\mu^{L,u}_i$ and $z_{k'} = \\mu^{R,u}_i$. Thus, each time $\\mathbf{x}$ covers a pair $(j,j+1)$, there must be a contribution of $\\frac{z_{j+1} - z_j}{2}$ to the cost $|x_i - c(\\mathbf{x})_i|$. Because each pair is covered at least $t_u$ times by Claim~\\ref{claim:covering}, we conclude that \n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} |x_i - c(\\mathbf{x})_i|\n\\geq t_u \\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)\n= \\frac{t_u}{2} (z_{k'} - z_1).\n$$\nTo relate the bound to $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$, we note that the above argument holds for each coordinate $i \\in [d]$, and\nwe have that\n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n=  \\sum_{i \\in [d]} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}   |\\mathbf{x}_i - c(\\mathbf{x})_i|\n\\geq \n\\frac{t_u}{2} \\cdot  \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n$$\n\nFor the $k$-means proof, we apply the same argument as above, this time using Claim~\\ref{clm:cauchy_schwarz_k_means} to bound the sum of squared values as \n$$\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} |x_i - c(\\mathbf{x})_i|^2\n\\geq t_u \\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)^2\n\\geq \\frac{t_u}{k} \\left(\\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)\\right)^2\n= \\frac{t_u}{4k} (z_{k'} - z_1)^2,$$\nand therefore, summing over coordinates $i\\in[d]$, we have\n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n=  \\sum_{i \\in [d]} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}   |x_i - c(\\mathbf{x})_i|^2\n\\geq \n\\frac{t_u}{4k} \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{aux-claim:general_k_upper_bound}.]\n\tWe start with the $k$-medians proof.\nThe factor of $H$ arises because the same points $\\mathbf{x} \\in \\mathcal{X}$ can appear in at most $H$ sets $\\cX^{\\mathsf{cor}}_u$ because $H$ is the depth of the tree. More precisely, using Lemma~\\ref{lemma:mistake-bound} for each node $u$, we have that \n$$\nH \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) \\geq \\sum_{u \\in T} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}  \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n\\geq \\sum_{u \\in T} \\frac{t_u}{2} \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n$$\t\nApplying the same steps for the $k$-means cost, we have that\n$$\nH \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) \\geq \\sum_{u \\in T} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}  \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n\\geq \\sum_{u \\in T} \\frac{t_u}{4k} \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{proof}\n\n\n\n\n\n\\subsection{Approximation lower bound}\n\nTo complement our upper bounds, we show that a threshold tree with $k$ leaves cannot, in general, yield better than an $\\Omega(\\log k)$ approximation to the optimal $k$-medians or $k$-means clustering.\n\n\\begin{theorem}\\label{thm:lb-k}\nFor any $k \\geq 2$, there exists a data set with $k$ clusters such that any  threshold tree $T$ with $k$ leaves\nmust have $k$-medians and $k$-means cost at least\n$$\n\\mathrm{cost}(T) \\geq \\Omega(\\log k) \\cdot \\mathrm{cost}(opt).\n$$\n\\end{theorem}\n\nThe data set is produced by first picking $k$ random centers from the hypercube $\\{-1,1\\}^d$, for large enough~$d$, and then using each of these to produce a cluster consisting of the $d$ points that can be obtained by replacing one coordinate of the center by zero.  Thus the clusters have size $d$ and radius $O(1)$. To prove the lower bound, we use ideas from the study of pseudo-random binary vectors, showing that projecting the centers to any subset of $m\\lesssim\\log_2 k$ coordinates take on all $2^m$ possible values, with each occurring roughly equally often.\n Then, we show that (i) the threshold tree must be essentially a complete binary tree with depth $\\Omega(\\log_2 k)$ to achieve a clustering with low cost, and (ii) any such tree incurs a cost of $\\Omega(\\log k)$ times more than the optimal for this data set (for both $k$-medians and $k$-means).\nThe proof of Theorem~\\ref{thm:lb-k} appears in Appendix~\\ref{sec:general_k_lower_bound}.\n\n\n\\section{Motivating Examples}\n\\label{sec:motivating-examples}\n\n{\\bf Using $k-1$ features may be necessary.}\nWe start with a simple but important bound showing that trees with depth less than~$k$ (or fewer than $k-1$ features) can be arbitrarily worse than the optimal clustering. Consider the data set consisting of the $k-1$ standard basis vectors $\\mathbf{e}^1,\\ldots,\\mathbf{e}^{k-1} \\in \\mathbb{R}^{k-1}$ along with the all zeros vector. As this data set has $k$ points,  \nthe optimal $k$-median\/means cost is zero, putting each point in its own cluster. \nUnfortunately,  it is easy to see that for this data, depth $k-1$ is necessary for clustering with a threshold tree.  Figure~\\ref{fig:simplex} depicts an optimal tree for this data set. Shorter trees do not work because projecting onto any $k-2$ coordinates does not separate the data, as at least two points will have all zeros in these coordinates. Therefore, any tree with depth at most $k-2$ will put two points in the same cluster, leading to non-zero cost, whereas the optimal cost is zero. In other words, for this data set, caterpillar trees such as Figure~\\ref{fig:simplex} are necessary and sufficient for an optimal clustering. This example also shows that $\\Theta(k)$ features are tight for feature selection~\\cite{cohen2015dimensionality} and provides a separation with feature extraction methods that use a linear map to only a logarithmic number of dimensions~\\cite{becchetti2019oblivious, makarychev2019performance}.\n\n\\begin{figure}\n    \\centering\n    \\subfloat[Optimal threshold tree for the data set in $\\mathbb{R}^{k-1}$ consisting of the $k-1$ standard basis vectors and the all zeros vector. Any optimal tree must use all $k-1$ features and have depth~$k-1$.]{\n    \\begin{minipage}[t]{0.45\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n\t[inner\/.style={shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=gray!40, scale=.85},\n\tdots\/.style={shape=rectangle, align=center, top color=white, scale=.85},\n\tleaf\/.style={shape=rectangle, rounded corners, draw, align=center, scale=.85},\n\tlevel 1\/.style={sibling distance=2mm},\n\tlevel 2\/.style={sibling distance=2mm},\n\tlevel 3\/.style={sibling distance=2mm},\n\tlevel 4\/.style={sibling distance=2mm},\n\tlevel distance=8mm]\n\t\\Tree\n\t[.\\node[inner]{$x_{i_1} \\leq 0.5$};\n\t[.\\node[inner]{$x_{i_2} \\leq 0.5$};\n\t[.\\node[inner] {$\\ldots$};\n\t[.\\node[inner] {$x_{i_d} \\leq 0.5$};\n\t\\node[leaf]{\\textbf{$\\mathbf{0}$}};\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_d}$}};\n\t]    \n\t\\node[leaf]{\\textbf{$\\ldots$}};\n\t]\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_2}$}};\n\t]\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_1}$}};\n\t]\n\t\\node[top color=white, bottom color=white, scale=.75] at (-0.8,-4.1) {};\n\t\\end{tikzpicture}\n\t\\end{minipage}\n\t\\label{fig:simplex}}\n\t\\hfill\n    \\subfloat[The ID3 split results in a $3$-means\/medians clustering with arbitrarily worse cost than the optimal because it places the top two points in separate clusters. Our algorithm (Section~\\ref{sec:k-means}) instead starts with the optimal first split.\n    ]{\n    \\includegraphics[width=.4\\textwidth, trim=-2cm 0 -2cm 0, clip]{our_vs_id3.png}\n    \\label{fig:imm_vs_id3}}\n\\caption{Motivating examples showing that (a) threshold trees may need depth $k-1$ to determine $k$ clusters, and (b) standard decision tree algorithms such as ID3 or CART perform very badly on some data sets.}\n\\end{figure}\n\t\n\n\\paragraph{Standard top-down decision trees do not work.}\n\\label{sec:standard-dt-bad}\nA natural approach to building a threshold tree is to (1) find a good $k$-medians or $k$-means clustering using a standard algorithm, then (2) use it to label all the points, and finally (3) apply a supervised decision tree learning procedure, such as ID3~\\cite{quinlan1986induction,quinlan2014c4} to find a threshold tree that agrees with these cluster labels as much as possible. ID3, like other common decision tree algorithms, operates in a greedy manner, where at each step it finds the best split in terms of \\emph{entropy} or \\emph{information gain}. We will show that this is not a suitable strategy for clustering and that the resulting tree can have cost that is arbitrarily bad. \nIn what follows, denote by $\\mathrm{cost}({ID3}_{\\ell})$  the cost of the decision tree with~$\\ell$ leaves returned by ID3 algorithm.\n\n\n\nFigure \\ref{fig:imm_vs_id3} depicts a data set $\\mathcal{X} \\subseteq \\mathbb{R}^2$ partitioned into three clusters $\\mathcal{X} = \\mathcal{X}_0 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_1 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_2$.\nWe define two centers $\\boldsymbol{\\mu}^0=(-2,0)$ and $\\boldsymbol{\\mu}^1=(2,0)$ and for each  $i\\in \\{0, 1\\}$, we define~$\\mathcal{X}_i$ as $500$ i.i.d. points $\\mathbf{x} \\sim \\mathcal{N}(\\boldsymbol{\\mu}^i, \\epsilon)$ for some small $\\epsilon > 0$. \nThen, $\\mathcal{X}_2 = \\{(-2, v), (2, v)\\}$ where $v \\to \\infty$.  \nWith high probability, we have that the optimal $3$-means clustering is $(\\mathcal{X}_0, \\mathcal{X}_1, \\mathcal{X}_2)$, i.e. $\\mathbf{x} \\in \\mathcal{X}$ gets label $y\\in\\{0,1,2\\}$ such that $\\mathbf{x} \\in \\mathcal{X}_y$.\nThe ID3 algorithm minimizes the entropy at each step. In the first iteration, it splits between the two large clusters. As a result $(-2, v)$ and $(2, v)$ will also be separated from one another. Since $ID3_3$ outputs a tree with exactly three leaves, one of the leaves must contain a point from $\\mathcal{X}_2$ together with points from either $\\mathcal{X}_0$ or $\\mathcal{X}_1$, this means that $\\mathrm{cost}(ID3_3)= \\Omega(v) \\to \\infty$.\nNote that $\\mathrm{cost}((\\mathcal{X}_1, \\mathcal{X}_2, \\mathcal{X}_3))$ does not depend on $v$, and hence, it is substantially smaller than $\\mathrm{cost}(ID3_3)$.\nUnlike ID3, the optimal threshold tree first separates $\\mathcal{X}_2$ from $\\mathcal{X}_0 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_1$, and in the second split it separates $\\mathcal{X}_0$ and $\\mathcal{X}_1$. Putting the outliers in a separate cluster is necessary for an optimal clustering. It is easy to extend this example to more clusters or to when ID3 uses more leaves.\n\n\n\\section{Introduction}\nA central direction in machine learning is understanding the reasoning behind decisions made by learned models~\\cite{lipton2018mythos, molnar2019, murdoch2019interpretable}. Prior work on AI explainability focuses on the interpretation of a black-box model, known as {\\em post-modeling} explainability~\\cite{baehrens2010explain, ribeiro2018anchors}. While methods such as LIME~\\cite{ribeiro2016should} or Shapley explanations~\\cite{lundberg2017unified} have made progress in this direction, they do not provide direct insight into the underlying data set, and the explanations depend heavily on the given model. This has raised concerns about the applicability of current solutions, leading researchers to consider more principled approaches to interpretable methods~\\cite{rudin2019stop}.\n\nWe address the challenge of developing machine learning systems that are explainable by design, starting from an {\\em unlabeled} data set. Specifically, we consider {\\em pre-modeling} explainability in the context of {clustering}. \nA common use of clustering is to identify patterns or discover structural properties in a data set by quantizing the unlabeled points. For instance, $k$-means clustering may be used to discover coherent groups among a supermarket's customers. While there are many good clustering algorithms, the resulting cluster assignments can be hard to understand because the clusters may be determined using all the features of the data, and there may be no concise way to explain the inclusion of a particular point in a cluster. This limits the ability of users to discern the commonalities between points within a cluster or understand why points ended up in different clusters. \n\n\n\\input{cluster-pic-intro.tex}\n\n\n\nOur goal is to develop accurate, efficient clustering algorithms with concise explanations of the cluster assignments. There should be a simple procedure using a few features to explain why any point belongs to its cluster. Small decision trees have been identified as a canonical example of an easily explainable model~\\cite{molnar2019, murdoch2019interpretable}, and\nprevious work on explainable clustering uses an unsupervised decision tree~\\cite{bertsimas2018interpretable, fraiman2013interpretable,geurts2007inferring,ghattas2017clustering, liu2005clustering}. Each node of the binary tree iteratively partitions the data by thresholding on a single feature. We focus on finding $k$ clusters, and hence,  we use trees with $k$ leaves. Each leaf corresponds to a cluster, and the tree is as small as possible. \nWe refer to such a tree as a {\\em threshold tree}.\n\nThere are many benefits of using a small threshold tree to produce a clustering. Any cluster assignment is explained by computing the thresholds along the root-to-leaf path. By restricting to $k$ leaves, we ensure that each such path accesses at most $k-1$ features, independent of the data dimension. \nIn general, a threshold tree provides an initial quantization of the data set, which can be combined with other methods for future learning tasks. While we consider static data sets, new data points can be easily clustered by using the tree, leading to explainable assignments.\nTo analyze clustering quality, we consider the $k$-means and $k$-medians objectives~\\cite{macqueen, steinhaus}. The goal is to efficiently determine a set of $k$ centers that minimize either the squared $\\ell_2$ or the $\\ell_1$ distance, respectively, of the input vectors to their closest center. \n\nFigure~\\ref{fig:optimal_vs_tree} provides an example of standard and explainable $k$-means clustering on the same data set. Figure~\\ref{fig:optimal_clusters} on the left shows an optimal $5$-means clustering. Figure~\\ref{fig:tree_clusters} in the middle shows an explainable, tree-based $5$-means clustering, determined by the tree in Figure~\\ref{fig:decision_tree} on the right. The tree has five leaf nodes, and vectors are assigned to clusters based on the thresholds. Geometrically, the tree defines a set of axis-aligned cuts that determine the clusters. While the two clusterings are very similar, using the threshold tree leads to easy explanations, whereas using a standard $k$-means clustering algorithm leads to more complicated clusters. The difference between the two approaches becomes more evident in higher dimensions, because standard algorithms will likely determine clusters based on all of the feature values.\n\nTo reap the benefits of explainable clusters, we must ensure that the data partition is a good approximation of the optimal clustering. While many efficient algorithms have been developed for $k$-means\/medians clustering, the resulting clusters are often hard to interpret~\\cite{arthur2007k,kanungo02, ostrovsky2013effectiveness, shalev2014understanding}. For example, Lloyd's algorithm alternates between determining the best center for the clusters and reassigning points to the closest center~\\cite{lloyd1982least}. The resulting set of centers depends in a complex way to the other points in the data set. Therefore, the relationship between a point and its nearest center may be the result of an opaque combination of many feature values. This issue persists even after dimension reduction or feature selection, because a non-explainable clustering algorithm is often invoked on the modified data set. As our focus is on pre-modeling explanability, we aim for simple explanations that use the original feature vectors.\n\nEven though Figure~\\ref{fig:optimal_vs_tree} depicts a situation in which the optimal clustering is very well approximated by one that is induced by a tree, it is not clear whether this would be possible in general. Our first technical challenge is to understand the {\\em price of explainability} in the context of clustering: that is, the multiplicative blowup in $k$-means (or $k$-medians) cost that is inevitable if we force our final clustering to have a highly constrained, interpretable, form. The second challenge is to actually find such a tree {\\em efficiently}. This is non-trivial because it requires a careful, rather than random, choice of a subset of features. As we will see, the kind of analysis that is ultimately needed is quite novel even given the vast existing literature on clustering.\n\n\n\\subsection{Our contributions}\n\n\nWe provide several new theoretical results on explainable $k$-means and $k$-medians clustering. Our new algorithms and lower bounds are summarized in Table~\\ref{tab:results_summary}.  \n\n\\medskip \\noindent {\\bf Basic limitations.} \nA partition into $k$ clusters can be realized by a binary threshold tree with $k-1$ internal splits. This uses at most $k-1$ features, but is it possible to use even fewer, say $\\log k$ features? In Section~\\ref{sec:motivating-examples}, we demonstrate a simple data set that requires $\\Omega(k)$ features to achieve a explainable clustering with bounded approximation ratio compared to the optimal $k$-means\/medians clustering. In particular, the depth of the tree might need to be $k-1$ in the worst case.\n\nOne idea for building a tree is to begin with a good $k$-means (or $k$-medians) clustering, use it to label all the points, and then apply a supervised decision tree algorithm that attempts to capture this labeling. In Section~\\ref{sec:standard-dt-bad}, we show that standard decision tree algorithms, such as ID3, may produce clusterings with arbitrarily high cost. Thus, existing splitting criteria are not suitable for finding a low-cost clustering, and other algorithms are needed.\n\n\\medskip \\noindent  {\\bf New algorithms.} \nOn the positive side, we provide efficient algorithms to find a small threshold tree that comes with provable guarantees on the cost. We note that using a small number of clusters is preferable for easy interpretations, and therefore $k$ is often relatively small.\nFor the special case of two clusters ($k=2$), we show (Theorem~\\ref{thm:optimal_2_median_means}) that a single threshold cut provides a constant-factor approximation to the optimal $2$-medians\/means clustering, with a closely-matching lower bound (Theorem~\\ref{clm:2_median_lower_bound}), and we provide an efficient algorithm for finding the best cut. For general $k$, we show how to approximate any clustering by using a threshold tree with $k$ leaves (Algorithm~\\ref{algo:imm}). The main idea is to minimize the number of mistakes made at each node in the tree, where a mistake occurs when a threshold separates a point from its original center. Overall, the cost of the explainable clustering will be close to the original cost up to a factor that depends on the tree depth (Theorem~\\ref{thm:main-k}). In the worst-case, we achieve an approximation factor of $O(k^2)$ for $k$-means and $O(k)$ for $k$-medians compared to the cost of any clustering (e.g., the optimal cost). These results do not depend on the dimension or input size; hence, we get a constant factor approximation when $k$ is constant. \n\n\\paragraph{Approximation lower bounds.}\nSince our upper bounds depend on $k$, it is natural to wonder whether it is possible to achieve a constant-factor approximation, or whether the cost of explainability grows with~$k$. On the negative side, we identify a data set such that any threshold tree with $k$ leaves must incur an $\\Omega(\\log k)$-approximation for both $k$-medians and $k$-means (Theorem~\\ref{thm:lb-k}). \nFor this data set, our algorithm achieves a nearly matching bound for $k$-medians.\\vspace{2ex}\n\n\n\\begin{table}[!htb]\n\\renewcommand{\\arraystretch}{1.6}\n\\centering\n\\begin{minipage}{.8\\textwidth}\n\\centering\n    \\begin{tabular}{|c|cc|cc|}\n        \\hline\n        %\n        \\rowcolor[HTML]{F1F1F1}\n        & \\multicolumn{2}{c|}{\\textbf{$k$-medians}} &\n        \\multicolumn{2}{c|}{\\textbf{$k$-means}} \\\\\n       \n       \n        \\rowcolor[HTML]{F1F1F1}\n        & $k = 2$ & $k > 2$ & $k = 2$ & $k > 2$  \\\\\n         \\hline\n       \n        \\cellcolor[HTML]{F1F1F1}\n        \\textbf{Upper Bound} & {$2$} & {$O(k)$} & {$4$} & {$O(k^2)$} \\\\\n       \n        \\cellcolor[HTML]{F1F1F1} \\textbf{Lower Bound} & {$2 - \\frac{1}{d}$} & {$\\Omega(\\log k)$} &  {$3\\left(1 - \\frac{1}{d} \\right)^2$} & {$\\Omega(\\log k)$} \\\\\n       \n        \\hline\n    \\end{tabular}\n    \\caption{Summary of our upper and lower bounds on approximating the optimal $k$-medians\/means clustering with explainable, tree-based clusters. The values express the factor increase compared to the optimal solution in the worst case.\n    }\n    \\label{tab:results_summary}\n\\end{minipage}\n\\end{table}\n\n\n\\input{related.tex}\n\n\\section{Preliminaries}\n\n\\input{prelim.tex}\n\\input{motivation-examples.tex}\n\n\n\\input{2-means-short.tex}\n\\input{k-means-short-new.tex}\n\n\n\\section{Conclusion}\nIn this paper we discuss the capabilities and limitations of explainable clusters. For the special case of two clusters ($k=2$), we provide nearly matching upper and lower bounds for a single threshold cut. For general $k >2$, we present the IMM algorithm that achieves an $O(H)$ approximation for $k$-medians and an $O(Hk)$ approximation for $k$-means when the threshold tree has depth $H$ and $k$ leaves. \nWe complement our upper bounds with a lower bound showing that any threshold tree with $k$ leaves must have cost at least $\\Omega(\\log k)$ more than the optimal for certain data sets. \nOur theoretical results provide the first approximation guarantees on the quality of explainable unsupervised learning in the context of clustering. Our work makes progress toward the larger goal of explainable AI methods with precise objectives and provable guarantees. \n\n\n\nAn immediate open direction is to improve our results for $k$ clusters, either on the upper or lower bound side. One option is to use larger threshold trees with more than $k$ leaves (or allowing more than $k$ clusters). It is also an important goal to identify natural properties of the data that enable explainable, accurate clusters. For example, it would be interesting to improve our upper bounds on explainable clustering for well-separated data. Our lower bound of $\\Omega(\\log k)$ utilizes clusters with diameter $O(1)$ and separation $\\Omega(d)$, where the hardness stems from the randomness of the centers. In this case, the approximation factor $\\Theta(\\log k)$ is tight because our upper bound proof actually provides a bound in terms of the tree depth (which is about $\\log k$, see Appendix~\\ref{apx:lower_bound_log_k}). Therefore, an open question is whether a $\\Theta(\\log k)$ approximation is possible for any well-separated clusters (e.g., mixture of Gaussians with separated means and small variance). Beyond $k$-medians\/means, it would be worthwhile to develop other clustering methods using a small number of features (e.g., hierarchical clustering).\n\n\\paragraph{Acknowledgements.}\nSanjoy Dasgupta has been supported by NSF CCF-1813160. Nave Frost has been funded by the  European Research  Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant agreement No. 804302). The contribution of Nave Frost is part of a Ph.D. thesis research conducted at Tel Aviv University. \n\n\n\\section{Preliminaries}\nThroughout we use bold variables for vectors, and we use non-bold for scalars such as feature values.\nGiven a set of points $\\mathcal{X}=\\{\\mathbf{x}^1,\\ldots,\\mathbf{x}^n\\}\\subseteq\\mathbb{R}^d$ and an integer $k$ the goal of $k$-medians and $k$-means clustering is to partition $\\mathcal{X}$ into $k$ subsets and minimize the distances of the points to the centers of the clusters. It is known that the optimal centers correspond to means or medians of the clusters, respectively. Denoting the centers as  $\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k$, the aim of $k$-means is to find a clustering that minimizes the following objective\n$$\\mathrm{cost}_2(\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k)=\\sum_{\\mathbf{x}\\in \\mathcal{X}} \\norm{\\mathbf{x}-c_2(\\mathbf{x})}^2_2,$$ where $c_2(\\mathbf{x})=\\argmin_{\\boldsymbol{\\mu} \\in \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\}}{\\norm{\\boldsymbol{\\mu} - \\mathbf{x}}_2}$.\nSimilarly, the goal of $k$-medians is to minimize \n$$\\mathrm{cost}_1(\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k)=\\sum_{\\mathbf{x}\\in \\mathcal{X}} \\norm{\\mathbf{x}-c_1(\\mathbf{x})}_1,$$ where $c_1(\\mathbf{x})=\\argmin_{\\boldsymbol{\\mu} \\in \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\}}{\\norm{\\boldsymbol{\\mu} - \\mathbf{x}}_1}$. \nAs it will be clear from context whether we are talking about $k$-medians or $k$-means, we abuse notation and write $\\mathrm{cost}$ and $c(\\mathbf{x})$ for brevity. We also fix the data set and use $opt$ to denote the optimal $k$-medians\/means clustering, where the optimal centers are the medians or means of the clusters, respectively; hence, $\\mathrm{cost}(opt)$ refers to the cost of the optimal $k$-medians\/means clustering. \n\n\n\\subsection{Clustering using threshold trees}\n\n\nPerhaps the simplest way to define two clusters is to use a \\emph{threshold cut}, which partitions the data based on a threshold for a single feature. More formally, the two clusters can be written as $\\widehat C^{\\theta,i}=(\\widehat C^1, \\widehat C^2)$, which is defined using a coordinate $i$ and a threshold $\\theta\\in\\mathbb{R}$ in the following way. For each input point $\\mathbf{x}\\in \\mathcal{X}$, we place $\\mathbf{x}=[x_1, \\ldots, x_d]$ in the first cluster $\\widehat C^1$ if $x_i\\leq\\theta$, and otherwise $\\mathbf{x} \\in \\widehat C^2$.\nA threshold cut can be used to explain $2$-means or $2$-medians clustering because a single feature and threshold determines the division of the data set into exactly two clusters. \n\n\nFor $k > 2$ clusters, we consider iteratively using threshold cuts as the basis for the cluster explanations. More precisely, \nwe construct a binary \\emph{threshold tree}. This tree is an unsupervised variant of a decision tree. Each internal node contains a single feature and threshold, which iteratively partitions the data, leading to clusters determined by the vectors that reach the leaves. We focus on trees with exactly $k$ leaves, one for each cluster $\\{1,2,\\ldots, k\\}$, which also limits the depth and total number of features to at most $k-1$. \n\nWhen clustering using such a tree, it is easy to understand why $\\mathbf{x}$ was assigned to its cluster: we may simply inspect the threshold conditions on the root-to-leaf path for $\\mathbf{x}$. This also ensures the number of conditions for the cluster assignment is rather small, which is crucial for interpretability. These tree-based explanations are especially useful in high-dimensional space, when the number of clusters is much smaller than the input dimension ($k \\ll d$). \nMore formally, a threshold tree $T$ with $k$ leaves induces a $k$-clustering of the data. Denoting these clusters as $\\widehat{C}^j \\subseteq \\mathcal{X}$, we define the $k$-medians\/means cost of the tree as\n$$\n\\mathrm{cost}_1(T) = \\sum_{j=1}^k \\sum_{x \\in \\widehat{C}^j} \\|x - \\mbox{median}(\\widehat{C}^j) \\|_1\n\\qquad \\mbox{} \\qquad \n\\mathrm{cost}_2(T) = \\sum_{j=1}^k \\sum_{x \\in \\widehat{C}^j} \\|x - \\mbox{mean}(\\widehat{C}^j) \\|_2^2 \n$$\nOur goal is to understand when it is possible to efficiently produce a tree $T$ such that $\\mathrm{cost}(T)$ is not too large compared to the optimal $k$-medians\/means cost. Specifically, we say that an algorithm is an {\\em $a$-approximation}, if the cost is at most $a$ times the optimal cost, i.e., if the algorithm returns threshold tree $T$ then we have\n$\\mathrm{cost}(T) \\leq a\\cdot \\mathrm{cost}(opt),$ where $opt$ denotes the optimal $k$-medians\/means clustering.\n\n \n\n\n\\section{Lower bounds for two clusters}\\label{sec:k2_lower_bound}\n\n\nWithout loss of generality we can assume that $d\\geq 2.$ We use the following dataset for both $2$-medians and $2$-means. It consists of $2d$ points, partitioned into two clusters of size $d$, which are the points with Hamming distance exactly one from the vector with all 1 entries and the vector with all $-1$ entries: \n\\begin{center}\n\\begin{tabular}{c c}\n    \\textbf{Optimal Cluster 1} & \\textbf{Optimal Cluster 2} \\\\\n    $(0,-1,-1,-1\\ldots,-1)$ & $(0,1,1,1\\ldots,1)$ \\\\\n    $(-1,0,-1,-1\\ldots,-1)$ & $(1,0,1,1\\ldots,1)$\\\\\n    $(-1,-1,0,-1\\ldots,-1)$ & $(1,1,0,1\\ldots,1)$\\\\\n    $\\vdots$ & $\\vdots$ \\\\\n    $(-1,-1,-1,-1\\ldots,0)$ & $(1,1,1,1\\ldots,0)$\\\\\n\\end{tabular}\n\\end{center}\n\n\nLet $\\widehat C = (\\widehat C^1, \\widehat C^2)$ be the best threshold cut. \n\n\\paragraph{2-medians lower bound.}\nThe cost of the cluster with centers $(1,\\ldots,1)$ and $(-1,\\ldots,-1)$ is $2d$, as each point is responsible for a cost of $1.$ Thus, $\\mathrm{cost}(opt)\\leq 2d.$\n\nThere is a coordinate $i$ and a threshold $\\theta$ that defines the cut $\\widehat C$. For any coordinate $i$, there are only three possible values: $-1,0,1$. Thus $\\theta$ is either in $(-1,0)$ or in $(0,1)$. Without loss of generality, assume that $\\theta\\in(-1,0)$ and $i=1$. Thus, the cut is composed of two clusters: one of size $d-1$ and the other of size $d+1$, in the following way:\n\n\\begin{center}\n\\begin{tabular}{c c}\n    $\\mathbf{Cluster\\ }\\widehat C^1$ & $\\mathbf{Cluster\\ } \\widehat C^2$ \\\\\n    $(-1,0,-1,-1\\ldots,-1)$ & $(1,0,1,1\\ldots,1)$\\\\\n    $(-1,-1,0,-1\\ldots,-1)$ & $(1,1,0,1\\ldots,1)$\\\\\n    $\\vdots$ & $\\vdots$ \\\\\n    $(-1,-1,-1,-1\\ldots,0)$ & $(1,1,1,1\\ldots,0)$\\\\\n     & $(0,1,1,1\\ldots,1)$\\\\\n     & $(0,-1,-1,-1\\ldots,-1)$\\\\\n\\end{tabular}\n\\end{center}\n\n\nUsing Fact~\\ref{fact:1-median-optimal center}, an optimal center of the first cluster is all $-1$, and the optimal center for the second cluster is all $1$. The cost of the first cluster is $d-1$, as each point costs $1$. The cost of the second cluster is composed of two terms $d$ for all points that include 1 in at least one coordinate and the cost of point $(0,-1,\\ldots,-1)$ is $2(d-1)+1$. So the total cost is $4d-2$. Thus $\\mathrm{cost}(\\widehat C)\\geq (2-1\/d) \\mathrm{cost}(opt).$\n\n\\paragraph{2-means lower bound.}\nFocus on the clustering with centers $$(\\nicefrac{(d-1)}{d},\\ldots,\\nicefrac{(d-1)}{d}) \\qquad \\mbox{and} \\qquad (-\\nicefrac{(d-1)}{d},\\ldots,-\\nicefrac{(d-1)}{d}).$$ \nThe cost of each point in the data is composed of (1) one coordinate with value zero, and the cost of this coordinate is $\\left(\\nicefrac{(d-1)}{d}\\right)^2$ (2) $d-1$ coordinates each with cost $\\nicefrac1d^2.$ Thus, each point has a cost of $\\nicefrac{(d-1)^2}{d^2}+\\nicefrac{d-1}{d^2}.$ Thus, the total cost is $\\frac{2(d-1)^2+2(d-1)}{d}=2(d-1)$. This implies that $\\mathrm{cost}(opt)\\leq 2(d-1).$\n\nAssume without loss of generality that $\\widehat C$ is defined using coordinate $i=1$ and threshold $-0.5$. The resulting clusters $\\widehat C^1$ and $\\widehat C^2$ are as in the case of $2$-medians. The optimal centers are (see Fact~\\ref{fact:1-means-optimal center}): \n$$\\left(-1,-\\frac{d-2}{d-1},\\ldots,-\\frac{d-2}{d-1}\\right) \\qquad \\mbox{and} \\qquad \\left(\\frac{d-1}{d+1},\\frac{d-2}{d+1},\\ldots,\\frac{d-2}{d+1}\\right).$$\nWe want to lower bound $\\mathrm{cost}(\\widehat C).$ We start with the cost of the first cluster, i.e. $\\widehat C^1$. To do so for each point in $\\widehat C^1$, we will evaluate the contribution of each coordinate to the cost (1) the first coordinate adds $0$ to the cost (2) the coordinate with value $0$, adds $\\left(\\frac{d-2}{d-1}\\right)^2$ to the cost (3) the rest of the $d-2$ coordinates adds $\\nicefrac{1}{(d-1)^2}.$ Thus, each point in $\\widehat C^1$ adds to the cost $\\left(\\frac{d-2}{d-1}\\right)^2 + \\frac{d-2}{(d-1)^2}=\\frac{d-2}{d-1}$. Since $\\widehat C^1$ contains $d-1$ points, its total cost is $d-2$.\n\nMoving on to evaluating the cost of $\\widehat C^2$, the cost of the point $(0,-1,\\ldots,-1)$ is composed of two terms (1) the first coordinate adds $\\left(\\frac{d-1}{d+1}\\right)^2$ to the cost (2) each of the other $d-1$ coordinates adds $\\left(1+\\frac{d-2}{d+1}\\right)^2\nto the cost. Thus, this point adds $$\\left(\\frac{d-1}{d+1}\\right)^2+(d-1)\\left(1+\\frac{d-2}{d+1}\\right)^2=\\frac{(d-1)d(4d-3)}{(d+1)^2}.$$\nSimilarly, the point $(0,1,\\ldots,1)$ adds to the cost $$\\left(\\frac{d-1}{d+1}\\right)^2+(d-1)\\left(1-\\frac{d-2}{d+1}\\right)^2=\\frac{(d-1)(d+8)}{(d+1)^2}.$$\nFinally, each of the $d-1$ remaining points in $\\widehat C^2$ adds to the cost $$\\left(1-\\frac{d-1}{d+1}\\right)^2+\\left(\\frac{d-2}{d+1}\\right)^2+(d-1)\\left(1-\\frac{d-2}{d-1}\\right)^2=\\frac{d^2+5d-1}{(d+1)^2}$$\nThus, the cost of $\\widehat C^2$ is $$\\frac{(d-1)(5d^2+3d+7)}{(d+1)^2}$$\nSumming up the costs of $\\widehat C^1$ and $\\widehat C^2$, for $d\\geq 2$\n$$\\mathrm{cost}(\\widehat C)\\geq (d-2)+\\frac{(d-1)(5d^2+3d+7)}{(d+1)^2}\\geq6(d-1)\\left(1-\\frac1d\\right)^2\\geq 3\\left(1-\\frac{1}{d}\\right)^2\\cdot \\mathrm{cost}(opt)$$\n\n\n\n\n\n\\section{Upper Bound Proof for 2-Means}\\label{sec:k2_upper_bound}\n\nWe show that there is a threshold cut $\\widehat C$ with $2$-means cost satisfying\n$\\mathrm{cost}(\\widehat C)\\leq 4 \\cdot \\mathrm{cost}(opt).$ We could just use the same proof idea as in the 2-medians case that first applies Lemma~\\ref{lemma:tree-cost} and then uses the matching result, Lemma~\\ref{lemma:matching}. This leads to a $6$-approximation, instead of $4$. The reason is that we apply twice Claim~\\ref{clm:cauchy_schwarz_k_means}, which is not tight. Improving the approximation to $4$ requires us to apply   Claim~\\ref{clm:cauchy_schwarz_k_means} only once. \n\nSuppose $\\boldsymbol{\\mu}^1,\\boldsymbol{\\mu}^2$ are optimal $2$-means centers for the clusters $C^1$ and $C^2$. \nLet $t = \\min(|C^1 \\Delta \\widehat C^1|, |C^1 \\Delta \\widehat C^2|)$ be the minimum number of changes for any threshold cut $\\widehat C^1, \\widehat C^2$, and define $\\cX^{\\mathsf{mis}}$ to the set of $t$ points in the symmetric difference, where $\\mathcal{X} = \\cX^{\\mathsf{cor}} \\cup \\cX^{\\mathsf{mis}}$ and $\\cX^{\\mathsf{cor}} \\cap \\cX^{\\mathsf{mis}} = \\emptyset$.\n\nUsing the same argument as in the proof of Lemma~\\ref{lemma:tree-cost}, we have\n\\begin{eqnarray}\\label{eq:2_means_upper_bound}\n\\mathrm{cost}(\\widehat{C}) &\\leq& \\sum_{j=1}^2 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|^2_2 +\n\\sum_{j=1}^2 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|^2_2 \\nonumber\n\\\\ &=& \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|^2_2 +\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2 \\nonumber\n\\\\ &\\leq& \\mathrm{cost}(opt) +\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray}\nThe goal now is to bound the latter two terms using $\\mathrm{cost}(opt).$\nThis term measures the distance of each $\\mathbf{x} \\in \\cX^{\\mathsf{mis}}$ from the ``other'' center, i.e., not $c(\\mathbf{x})$. \n\n\\begin{claim}\\label{claim:aux-2-means}\n\\begin{eqnarray*}\n\t\\mathrm{cost}(opt)\n\t\\geq \\frac13 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\frac13\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray*}\n\\end{claim}\n\nUsing Claim~\\ref{claim:aux-2-means},\ntogether with Inequality~(\\ref{eq:2_means_upper_bound}) we have\n$$\\mathrm{cost}(\\widehat{C})\\leq \\mathrm{cost}(opt) + 3\\cdot \\mathrm{cost}(opt)=4\\cdot \\mathrm{cost}(opt),$$ \nand this completes the proof. \n\n\n\\begin{proof}[Proof of Claim~\\ref{claim:aux-2-means}.]\nDenote the $t$ points in $\\cX^{\\mathsf{mis}}$  by $\\cX^{\\mathsf{mis}}=\\{\\mathbf{r}^1,\\ldots,\\mathbf{r}^t\\}.$ Assume that the first $\\ell$ points are in the first optimal cluster, $\\mathbf{r}^1,\\ldots,\\mathbf{r}^\\ell\\in C^1$, and the rest are in the second cluster, $\\mathbf{r}^{\\ell+1},\\ldots,\\mathbf{r}^t\\in C^2.$\n\nApplying Lemma~\\ref{lemma:matching} for each coordinate $i\\in[d]$ guarantees $t$ pairs of vectors $(\\mathbf{p}^1,\\mathbf{q}^1), \\ldots, (\\mathbf{p}^t,\\mathbf{q}^t)$ with the following properties. Each $p^j_i$ corresponds to the $i$th coordinate of some  point in $C^1$ and $q^j_i$ corresponds to the $i$th coordinate of some point in $C^2$. Furthermore, for each coordinate, the $t$ pairs correspond to $2t$ distinct points in $\\mathcal{X}$.\nFinally, we can assume without loss of generality that\n$\\mu^1_i \\leq \\mu^2_i$ and $q^j_i \\leq p^j_i$. \n\nFor each point $\\mathbf{r}^j$ in the first $\\ell$ points in $\\cX^{\\mathsf{mis}}$, if $r^j_i\\geq p^j_i$ then we can replace $\\mathbf{p}^j$ with $\\mathbf{r}^j$, thus we can assume without loss of generality that $p_i^j\\geq r^j_i.$ We next show that $\\mathrm{cost}(opt)$ is lower bounded by  a function of $t$. There will be two cases depending on whether $p^j_i\\leq \\mu^2_i$ or not. The harder case is the first where the improvement of the approximation from $6$ to $4$ arises. Instead of first bounding the distance between $\\mathbf{r}^j$ and its new center using the distance to its original center and then accounting for $\\norm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2}^2_2,$ we directly account for the distance between $\\mathbf{r}^j$ and its new center. \n\n\n\\paragraph{\\textbf{Case 1:}} if  $p^j_i\\leq \\mu^2_i$, then \nClaim~\\ref{clm:cauchy_schwarz_k_means} implies that\n\\begin{eqnarray*} \n(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n&\\geq& \\frac{1}{3}(\\mu^2_i-q^j_i+p^j_i-\\mu_i^1+\\mu_i^1-r^j_i)^2\\\\\n&=& \\frac{1}{3}((\\mu^2_i-q^j_i)+(p^j_i-r^j_i))^2\n\\geq \\frac{1}{3}(\\mu^2_i-r^j_i)^2.\n\\end{eqnarray*}  The last inequality follows from $q^j_i \\leq p^j_i$ and $r_i^j\\leq p_i^j,$ which imply that $(\\mu^2_i-q^j_i)+(p^j_i-r^j_i)\n\\geq \\mu^2_i-r^j_i\\geq 0,$ which means $((\\mu^2_i-q^j_i)+(p^j_i-r^j_i))^2\n\\geq (\\mu^2_i-r^j_i)^2.$ \n\\paragraph{\\textbf{Case 2:}} if $\\mu_i^2\\leq p_i^j$, then again Claim~\\ref{clm:cauchy_schwarz_k_means} implies that\n\\begin{eqnarray*} \n(p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2&\\geq& (\\mu_i^2-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2 \\geq \\frac12 (\\mu^2_i-\\mu^1_i+\\mu^1_i-r^j_i)^2=\\frac12(\\mu^2_i-r^j_i)^2,\n\\end{eqnarray*}\nwhere in the first inequality we use $(p^j_i-\\mu^1_i)^2\\geq(\\mu^2_i-\\mu^1_i)^2.$\n\nThe two cases imply that for $1\\leq j\\leq \\ell$  $$(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n\\geq  \\frac{1}{3}(\\mu^2_i-r^j_i)^2\n.$$\nSimilarly for each point $\\mathbf{r}^j$ in the last $t-\\ell$ points in $\\cX^{\\mathsf{mis}}$, we have $$(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^2-r^j_i)^2\n\\geq  \\frac{1}{3}(\\mu^1_i-r^j_i)^2\n.$$\nPutting these together we have \n\\begin{eqnarray*}\n\t\\mathrm{cost}(opt)&\\geq& \\sum_{i=1}^d\\sum_{j=1}^\\ell(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n\t+\n\\sum_{i=1}^d\\sum_{j=\\ell+1}^t(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^2-r^j_i)^2\\\\\n\t&\\geq& \\frac{1}{3}\\sum_{j=1}^\\ell\\sum_{i=1}^d  (\\mu^2_i-r^j_i)^2\n\t+\\frac{1}{3}\\sum_{j=\\ell+1}^t\\sum_{i=1}^d  (\\mu^1_i-r^j_i)^2 \n\t\\\\ \t&=&\\frac{1}{3}\\sum_{j=1}^\\ell\\norm{\\mathbf{r}^j-\\boldsymbol{\\mu}^2}_2^2+\\frac{1}{3}\\sum_{j=\\ell+1}^t\\norm{\\mathbf{r}^j-\\boldsymbol{\\mu}^1}_2^2\\\\\n&=&\t\\frac13 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\frac13\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Efficient Implementation via Dynamic Programming for $k=2$}\\label{sec:DP_Implementation}\n\n\\subsection{The 2-means case}\nThe psudo-code for finding the best threshold for $k=2$ depicted in Algorithm~\\ref{algo:cut}.\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}{.6\\linewidth}\n        \\begin{algorithm}[H]\n    \t\\SetKwFunction{$2$-means Optimal Threshold}{$2$-means Optimal Threshold}\n    \t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n    \t\\LinesNumbered\n    \t\\Input{%\n    \t\t{$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$} -- vectors in $\\mathbb{R}^d$\n    \t}\n    \t\\Output{%\n    \t\t\\xvbox{2mm}{$i$} -- Coordinate\\\\\n    \t\t\\xvbox{2mm}{$\\theta$} -- Threshold\n    \t}\n    \t\\BlankLine\n    \t\\setcounter{AlgoLine}{0}\n    \t\n        $\\xvar{best\\_cost} \\leftarrow \\infty$\\;\n    \t\n    \t$\\xvar{best\\_coordinate} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$\\xvar{best\\_threshold} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$u \\leftarrow \\sum_{j=1}^n \\norm{\\mathbf{x}^j}^2_2$\\;\n    \t\n    \t\\ForEach {$i \\in [1, \\ldots, d]$}\n    \t{\n    \t\t$\\mathbf{s} \\leftarrow \\xfunc{zeros}(d)$\\;\n    \t\t\n    \t\t$\\mathbf{r} \\leftarrow \\sum_{j=1}^n \\mathbf{x}^j$\\;\n    \t\t\n    \t\t$\\mathcal{X} \\leftarrow \\xfunc{sorted}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n \\text{ by coordinate }i)$\\;\n    \t\t\n    \t\t\\ForEach {$\\mathbf{x}^j \\in \\mathcal{X}$} \n    \t\t{ \n    \t\t\t$\\mathbf{s} \\leftarrow \\mathbf{s} + \\mathbf{x}^j$\\;\n    \t\t\t\n    \t\t\t$\\mathbf{r} \\leftarrow \\mathbf{r} - \\mathbf{x}^j$\\;\n    \t\t\t\n    \t\t\t$\\xvar{cost} \\leftarrow u - \\frac{1}{j}\\norm{\\mathbf{s}}^2_2- \\frac{1}{n - j}\\norm{\\mathbf{r}}^2_2$\\;\n    \t\t\t\n    \t\t\t\\If {$\\xvar{cost} < \\xvar{best\\_cost}$ and $x^j_i \\neq x^{j+1}_i$}\n    \t\t\t{\n\n    \t\t\t\t$\\xvar{best\\_cost} \\leftarrow \\xvar{cost}$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_coordinate} \\leftarrow i$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_threshold} \\leftarrow x^j_i$\\;\n    \t\t\t}\n    \t\t}\n    \t}\n    \t\\Return $\\xvar{best\\_coordinate}, \\xvar{best\\_threshold}$\\;\n    \t\\caption{\\textsc{Optimal Threshold for $2$-means}}\n    \t\\label{algo:cut}\n    \\end{algorithm}\n    \\end{minipage}\n\\end{figure}\n\n\nIn time $O(d)$ we can calculate $\\mathrm{cost}(p+1)$ and the new centers by using the value $\\mathrm{cost}(p)$ and the previous centers. Throughout the computation we save in memory \n\\begin{enumerate}\n\t\\item Two vectors $\\mathbf{s}^{p}=\\sum_{j=1}^p \\mathbf{x}^j$ and $\\mathbf{r}^{p}=\\sum_{j=p + 1}^n \\mathbf{x}^j$.\n\t\\item Scalar $u=\\sum_{j=1}^n \\norm{\\mathbf{x}^j}_2^2$\n\\end{enumerate}\nWe also make use of the identity:\n\\begin{eqnarray*}\n\t\\mathrm{cost}(p)\n\t&=& u - \\frac{1}{p}\\norm{\\mathbf{s}^{p}}_2^2 - \\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}_2^2.\n\\end{eqnarray*}\nThis identity is correct because \n\\begin{eqnarray*}\n    \\mathrm{cost}(p) &=& \\sum_{j=1}^{p}\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^1(p)}^2_2 + \\sum_{j=p+1}^{n}\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^2(p)}^2_2\\\\\n    &=& \\sum_{j=1}^{p} \\norm{\\mathbf{x}^j}^2_2 - 2\\sum_{j=1}^{p}\\inner{\\mathbf{x}^j}{\\boldsymbol{\\mu}^1(p)} + \\sum_{j=1}^{p}\\norm{\\boldsymbol{\\mu}^1(p)}^2_2 + \\\\\n    && \\sum_{j=p+1}^{n} \\norm{\\mathbf{x}^j}^2_2 - 2\\sum_{j=p+1}^{n}\\inner{\\mathbf{x}^j}{\\boldsymbol{\\mu}^2(p)} + \\sum_{j=p+1}^{n}\\norm{\\boldsymbol{\\mu}^2(p)}^2_2\\\\\n    &=& \\sum_{j=1}^{n} \\norm{\\mathbf{x}^j}^2_2 - 2\\inner{\\sum_{j=1}^{p}\\mathbf{x}^j}{\\boldsymbol{\\mu}^1(p)} + \\frac{1}{p}\\norm{\\sum_{j=1}^{p}\\mathbf{x}^j}^2_2 - \\\\\n    && 2\\inner{\\sum_{j=p+1}^{n}\\mathbf{x}^j}{\\boldsymbol{\\mu}^2(p)} + \\frac{1}{n-p}\\norm{\\sum_{j=p+1}^{n}\\mathbf{x}^j}^2_2\\\\\n    &=& \\sum_{j=1}^{n} \\norm{\\mathbf{x}^j}^2_2 - \\frac{2}{p}\\inner{\\mathbf{s}^{p}}{\\mathbf{s}^{p}}+\\frac{1}{p}\\norm{\\mathbf{s}^{p}}^2_2 - \\frac{2}{n-p}\\inner{\\mathbf{r}^{p}}{\\mathbf{r}^{p}}+\\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}^2_2\\\\\n    &=& u - \\frac{1}{p}\\norm{\\mathbf{s}^{p}}^2_2 - \\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}^2_2\n\\end{eqnarray*}\n\n\n By invoking this identity, we can quickly compute the cost of placing the first $p$ points in cluster one and the last $n-p$ points in cluster two. Each such partition can be achieved by using a threshold $\\theta$ between $x_i^p$ and $x_i^{p+1}$. Our algorithm computes these costs for each feature $i \\in[d]$. Then, we output the feature $i$ and threshold $\\theta$ that minimizes the cost. This guarantees that we find the best possible threshold cut.\n\n Overall, Algorithm \\ref{algo:cut} iterates over the $d$ features, and for each feature it sorts the $n$ vectors according to their values in the current feature. Next, the algorithm iterates over the $n$ vectors and for each potential threshold, it calculates the cost by evaluating the inner product of two $d$-dimensional vectors.\n Overall its runtime complexity is $O\\left(nd^2 + nd\\log n\\right)$.\n \n \\subsection{The 2-medians case}\n The high level idea of a finding an optimal 2-medians cut is similar to the 2-means algorithm. The algorithm goes over all possible thresholds. For each threshold, it finds the optimal centers and calculates the cost accordingly. Then, it outputs the threshold cut that minimizes the $2$-medians cost.\n \n  \\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}{.6\\linewidth}\n        \\begin{algorithm}[H]\n    \t\\SetKwFunction{$2$-medians Optimal Threshold}{$2$-means medians Threshold}\n    \t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n    \t\\LinesNumbered\n    \t\\Input{%\n    \t\t{$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$} -- vectors in $\\mathbb{R}^d$\n    \t}\n    \t\\Output{%\n    \t\t\\xvbox{2mm}{$i$} -- Coordinate\\\\\n    \t\t\\xvbox{2mm}{$\\theta$} -- Threshold\n    \t}\n    \t\\BlankLine\n    \t\\setcounter{AlgoLine}{0}\n    \t\n    \t$\\xvar{best\\_cost} \\leftarrow \\infty$\\;\n    \t\n    \t$\\xvar{best\\_coordinate} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$\\xvar{best\\_threshold} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t\\ForEach {$i \\in [1, \\ldots, d]$}\n    \t{\n    \t\n    \t    $\\boldsymbol{\\mu}^2(0) \\leftarrow \\xfunc{median}(\\mathbf{x}^1, \\ldots \\mathbf{x}^n)$\\;\n    \t\n    \t    $\\xvar{cost} \\leftarrow \\sum_{j=1}^n \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^2(0)}_1$\\;\n    \t\n    \t    $\\mathcal{X} \\leftarrow \\xfunc{sorted}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n \\text{ by coordinate }i)$\\;\n    \t\n    \t\t\\ForEach {$j \\in [1, \\ldots, n - 1]$} \n    \t\t{ \n\t    \t\t$\\boldsymbol{\\mu}^1(j) \\leftarrow \\xfunc{median}(\\mathbf{x}^1, \\ldots \\mathbf{x}^j)$\\;\n    \t\t\n\t    \t\t$\\boldsymbol{\\mu}^2(j) \\leftarrow \\xfunc{median}(\\mathbf{x}^{j+1}, \\ldots \\mathbf{x}^n)$\\;\n\t    \t\t\n    \t\t\t$\\xvar{cost} \\leftarrow \\xvar{cost} + \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^1(j)}_1 - \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^2(j - 1)}_1$\\;\n    \t\t\t\n    \t\t\t\\If {$\\xvar{cost} < \\xvar{best\\_cost}$ and $x^j_i \\neq x^{j+1}_i$}\n    \t\t\t{\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_cost} \\leftarrow \\xvar{cost}$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_coordinate} \\leftarrow i$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_threshold} \\leftarrow x^j_i$\\;\n    \t\t\t}\n    \t\t}\n    \t}\n    \t\\Return $\\xvar{best\\_coordinate}, \\xvar{best\\_threshold}$\\;\n    \t\\caption{\\textsc{Optimal Threshold for $2$-medians}}\n    \t\\label{algo:2-medians-cut}\n    \\end{algorithm}\n    \\end{minipage}\n\\end{figure}\n \n \\paragraph{Updating cost.} \n To update the cost we need to show how to express $\\mathrm{cost}(p+1)$ in terms of $\\mathrm{cost}(p).$ We know that $\\mathrm{cost}(p+1)$ is equal to \n $$\\mathrm{cost}(p+1)=\\sum_{\\mathbf{x}\\in C_1}\\norm{x-\\boldsymbol{\\mu}^1(p+1)}_1+\\sum_{\\mathbf{x}\\in C_2}\\norm{x-\\boldsymbol{\\mu}^2(p+1)}_1.$$ \n For every feature $i \\in[d]$, there are $n-1$ thresholds to consider. After sorting by this feature, we can consider all splits into $C_1$ and $C_2$, where $C_1$ contains the $p$ smallest points,  and $C_2$ contains the $n-p$ largest points. We increase $p$ from $p=1$ to $p=n-1$, computing the clusters and cost at each step. If $p$ is odd then the median of $C_1$ (i.e., the optimal center of $C_1$) does not change compared to $p-1$. The only contribution to the cost is the point~$\\mathbf{x}$ that moved from $C_2$ to $C_1$. If $p$ is even, then at each coordinate there are two cases, depending on whether the median changes or not. If it changes, then let $\\Delta$ denote the change in cost of the points in $C_1$ that are smaller than the median. By symmetry, the change in the cost of the points that are larger is $-\\Delta$. Thus, the change of the cost is balanced by the points that are larger and smaller than the median. Similar reasoning holds for the other cluster $C_2.$ Therefore, we conclude that moving $\\mathbf{x}$ from $C_2$ to $C_1$ changes the cost by exactly $\\norm{\\mathbf{x}-\\boldsymbol{\\mu}^1(p+1)}_1 - \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^2(p)}_1$. Thus, we have the following connection between $\\mathrm{cost}(p+1)$ and $\\mathrm{cost}(p)$:\n $$\\mathrm{cost}(p+1)=\\mathrm{cost}(p) + \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^1(p+1)}_1 - \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^2(p)}_1.$$\n \n  \\paragraph{Updating centers.} For each $p$, the cost update relies on efficient calculations of the centers $\\boldsymbol{\\mu}^1(p)$ and  $\\boldsymbol{\\mu}^2(p + 1)$. The centers $\\boldsymbol{\\mu}^1(p), \\boldsymbol{\\mu}^2(p)$ are the medians of the clusters at the $p$th threshold. Note that moving from the $p$th thresold to the $(p+1)$th will only change the clusters by moving one vector from one cluster to the other. \n  We can determine the changes efficiently by using $d$ arrays, one for each coordinate. Each array will contain (pointers to) the input vectors $\\mathcal{X}$ sorted by their $i$th feature value. As we move the threshold along a single coordinate, we can read off the partition into two clusters, and we can compute the median of each cluster by considering the midpoint in the sorted list. \n  \n  Overall, this procedure computes the cost of each threshold, while also determining the partition into two clusters and their centers (medians). The time is $O(nd \\log n)$ to sort by each feature, and $O(nd^2)$ to compute $\\mathrm{cost}(p)$ for each $p \\in [n]$ and each feature. Therefore, the total time for the $2$-medians algorithm is \n  $O(nd^2 + nd \\log n).$\n \n \n \n \n\n\\subsection{Related work}\n\\label{sec:related}\n\nThe majority of work on explainable methods considers supervised learning, and in particular, explaining predictions of neural networks and other trained models~\\cite{\nalvarez2019weight, deutch2019constraints, garreau2020explaining, kauffmann2019clustering,\nlipton2018mythos,lundberg2017unified, molnar2019, murdoch2019interpretable, ribeiro2016should, ribeiro2018anchors, rudin2019stop, sokol2020limetree}. In contrast, there is much less work on explainable unsupervised learning. Standard algorithms for $k$-medians\/means use iterative algorithms to produce a good approximate clustering, but this leads to complicated clusters that depend on subtle properties of the data set~\\cite{aggarwal09,arthur2007k, kanungo02, ostrovsky2013effectiveness}. Several papers consider the use of decision trees for explainable clustering~\\cite{bertsimas2018interpretable, fraiman2013interpretable,geurts2007inferring,ghattas2017clustering, liu2005clustering}. However, all prior work on this topic is empirical, without any theoretical analysis of quality compared to the optimal clustering. We also remark that the previous results on tree-based clustering have not considered the $k$-medians\/means objectives for evaluating the quality of the clustering, which is the focus of our work. It is NP-hard to find the optimal $k$-means  clustering~\\cite{aloise2009np, dasgupta2008hardness} or even a very close approximation~\\cite{awasthi2015hardness}. In other words, we expect tree-based clustering algorithms to incur an approximation factor bounded away from one compared to the optimal clustering.\n\nOne way to cluster based on few features is to use dimensionality reduction.\nTwo main types of dimensionality reduction methods have been investigated for $k$-medians\/means. Work on {\\em feature selection} shows that it is possible to cluster based on $\\Theta(k)$ features and obtain a constant factor approximation for $k$-means\/medians~\\cite{boutsidis2009unsupervised, cohen2015dimensionality}. However, after selecting the features, these methods employ existing approximation algorithms to find a good clustering, and hence, the cluster assignments are not explainable. Work on {\\em feature extraction} shows that it is possible to use the Johnson-Lindenstrauss transform to $\\Theta(\\log k)$ dimensions, while preserving the clustering cost~\\cite{becchetti2019oblivious, makarychev2019performance}. Again, this relies on running a $k$-means\/medians algorithm after projecting to the low dimensional subspace. The resulting clusters are not explainable, and moreover, the features are arbitrary linear combinations of the original features.\n\nBesides explainability, many other clustering variants have received recent attention, such as fair clustering~\\cite{ahmadian2020fair, backurs2019scalable, bera2019fair,chiplunkar2020solve,  huang2019coresets,kleindessner2019fair, mahabadi2020individual, schmidt2019fair}, online clustering~\\cite{bhaskara20a, cohen2019online,hess2019sequential, liberty2016algorithm, moshkovitz2019unexpected}, and the use of same-cluster queries~\\cite{ailon2018approximate, ashtiani2016clustering, huleihel2019same, mazumdar2017clustering}. An interesting avenue for future work would be to further develop tree-based clustering methods by additionally incorporating some of these other constraints or objectives.\n\n\\section{Threshold trees with $k > 2$ leaves}\n\\label{sec:k-means}\n\nWe provide an efficient algorithm to produce a threshold tree with $k$ leaves that constitutes an approximate $k$-medians or $k$-means clustering of a data set $\\mathcal{X}$. Our algorithm, Iterative Mistake Minimization (IMM), starts with a reference set of cluster centers, for instance from a polynomial-time constant-factor approximation algorithm for $k$-medians or $k$-means~\\cite{aggarwal09}, or from a domain-specific clustering heuristic. \n\nWe then begin the process of finding an explainable approximation to this reference clustering, in the form of a threshold tree with $k$ leaves, whose internal splits are based on single features. The way we do this is almost identical for $k$-medians and $k$-means, and the analysis is also nearly the same. Our algorithm is deterministic and its run time is only $O(kdn \\log n)$, after finding the initial centers.\n \n \nAs discussed in Section~\\ref{sec:motivating-examples}, existing decision tree algorithms use greedy criteria that are not suitable for our tree-building process. However, we show that an alternative greedy criterion---minimizing the number of {\\em mistakes} at each split (the number of points separated from their corresponding cluster center)---leads to a favorable approximation ratio to the optimal $k$-medians or $k$-means cost.\n\n\n\\subsection{Our algorithm}\n\\label{sec:k-means-alg}\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\\centering\n\\begin{minipage}{.5\\textwidth}\n\\begin{algorithm}[H]\n\\SetKwFunction{Iterative Mistake Minimization}{Iterative Mistakes Minimization}\n\\SetKwFunction{BuildTree}{BuildTree}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\\SetKwInOut{Preprocess}{Preprocess}\n\\Input{%\n\t$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$ -- vectors in $\\mathbb{R}^d$\\\\\n\t$k$ -- number of clusters\\\\\n  }\n\\Output{%\n    root of the threshold tree\n}\n\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\n$\\boldsymbol{\\mu}^1, \\ldots \\boldsymbol{\\mu}^k \\leftarrow \\xfunc{k-Means}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n, k)$\\;\n\n\\ForEach {$j \\in [1, \\ldots, n]$}\n{\n    $y^j \\leftarrow \\argmin_{1 \\leq \\ell \\leq k} \\lVert \\mathbf{x}^j - \\boldsymbol{\\mu}^\\ell \\rVert$\\;\n}\n\n\\Return $\\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j=1}^n, \\{y^j\\}_{j=1}^n, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n\\SetKwProg{buildtree}{$\\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j=1}^m, \\{y^j\\}_{j=1}^m, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$:}{}{}\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\\buildtree{}{\n\\If{$\\{y^j\\}_{j=1}^m$ \\text{is homogeneous}}\n{\n    $\\xvar{leaf}.cluster \\leftarrow y^1$\\;\n    \n    \\Return $\\xvar{leaf}$\\;\n}\n\\ForEach {$i \\in [1, \\ldots, d]$}\n{\n    $\\ell_i \\leftarrow \\min_{1 \\leq j \\leq m} \\mu^{y^j}_i$\\;\n    \n    $r_i \\leftarrow \\max_{1 \\leq j \\leq m} \\mu^{y^j}_i$\\;\n}\n$i, \\theta \\leftarrow \\argmin_{i,\\ell_i \\leq \\theta < r_i} \\sum_{j=1}^m \\xfunc{mistake}(\\mathbf{x}^j, \\boldsymbol{\\mu}^{y^j}, i, \\theta)$\\;\\label{ln:k_dynamic}\n\n$\\xvar{M} \\leftarrow \\{j \\mid \\xfunc{mistake}(\\mathbf{x}^j, \\boldsymbol{\\mu}^{y^j}, i, \\theta) = 1\\}_{j=1}^m$\\; \n\n$\\xvar{L} \\leftarrow \\{j \\mid (x^j_i \\leq \\theta) \\wedge (j \\not \\in \\xvar{M})\\}_{j=1}^m$\\;\n\n$\\xvar{R} \\leftarrow \\{j \\mid (x^j_i > \\theta) \\wedge (j \\not \\in \\xvar{M})\\}_{j=1}^m$\\;\n\n$\\xvar{node}.condition \\leftarrow ``x_i \\leq \\theta\"$\\;\n\n$\\xvar{node}.lt \\leftarrow \\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j \\in \\xvar{L}}, \\{y^j\\}_{j \\in \\xvar{L}}, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n$\\xvar{node}.rt \\leftarrow \\xfunc{build\\_tree}(\\{\\mathbf{x}^j\\}_{j \\in \\xvar{R}}, \\{y^j\\}_{j \\in \\xvar{R}}, \\{\\boldsymbol{\\mu}^j\\}_{j=1}^k)$\\;\n\n\n\\Return $\\xvar{node}$\\;\n}\n\n\n\\SetKwProg{mistake}{$\\xfunc{mistake}(\\mathbf{x}, \\boldsymbol{\\mu}, i, \\theta)$:}{}{}\n\\LinesNumbered\n\\setcounter{AlgoLine}{0}\n\\BlankLine\n\\mistake{}{\n\\Return $(x_i \\leq \\theta) \\neq (\\mu_i \\leq \\theta)$  ? $1$ : $0$\\;\n}\n\\caption{\\textsc{\\newline Iterative Mistake Minimization}}\n\\label{algo:imm}\n\\end{algorithm}\n\\end{minipage} \\vspace{-2ex}\n\\end{wrapfigure}\n\nAlgorithm~\\ref{algo:imm} takes as input a data set $\\mathcal{X} \\subseteq \\mathbb{R}^d$. The first step is to obtain a reference set of $k$ centers $\\{\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k\\}$, for instance from a standard clustering algorithm. We assign each data point $\\mathbf{x}^j$ the label $y^j$ of its closest center. Then, the {\\tt{build\\_tree}} procedure looks for a tree-induced clustering that fits these labels. The tree is built top-down, using binary splits. Each node $u$ can be associated with the portion of the input space that passes through that node, a hyper-rectangular region $\\mbox{cell}(u) \\subseteq \\mathbb{R}^d$. If this cell contains two or more of the centers $\\boldsymbol{\\mu}^j$, then it needs to be split. We do so by picking the feature $i \\in [d]$ and threshold value $\\theta \\in \\mathbb{R}$ such that the resulting split $x_i \\leq \\theta$ sends at least one center to each side and moreover produces the fewest {\\em mistakes}: that is, separates the fewest points in $\\mathcal{X} \\cap \\mbox{cell}(u)$ from their corresponding centers in $\\{\\boldsymbol{\\mu}^j: 1 \\leq j \\leq k\\} \\cap \\mbox{cell}(u)$. We do not count points whose centers lie outside $\\mbox{cell}(u)$, since they are associated with mistakes in earlier splits. We find the optimal split $(i, \\theta)$ by searching over all pairs efficiently using dynamic programming. We then add this node to the tree, and discard the mistakes (the points that got split from their centers) before recursing on the left and right children. We terminate at a leaf node whenever all points have the same label (i.e., a {\\em homogeneous} subset). As there are $k$ different labels, the resulting tree has exactly $k$ leaves.\nFigure \\ref{fig:imm_example} depicts the operation of Algorithm~\\ref{algo:imm}. \n\nWe first discuss the running time, and we analyze the approximation guarantees of IMM in Section~\\ref{sec:imm-approx-main}.\n\n\n\n\n\\smallskip \\noindent {\\bf Time analysis of tree building.}\nWe sketch how to execute the algorithm in time $O(kdn\\log n)$ for an $n$-point data set. At each step of the top-down procedure, we find a coordinate and threshold pair that minimizes the mistakes at this node (line \\ref{ln:k_dynamic} in \\texttt{build\\_tree} procedure). We use dynamic programming to avoid recomputing the cost from scratch for each potential threshold. For each coordinate $i \\in [d]$, we sort the data and centers. Then, we iterate over possible thresholds. We claim that we can process each node in time $O(dn\\log n)$ because each point will affect the number of mistakes at most twice. Indeed, when the threshold moves, either a data point or a center moves to the other side of the threshold. Since we know the number of mistakes from the previous threshold, we count the new mistakes efficiently as follows. If a single point switches sides, then the number of mistakes changes by at most one. If a center switches sides, which happens at most once, then we update the mistakes for this center. Overall, each point affects the mistakes at most twice (once when changing sides, and once when its center switches sides). Thus, the running time for each internal node is $O(dn\\log n)$. As the tree has $k-1$ internal nodes, the total time is $O(kdn \\log n)$.\n\n\n\\input{imm-pic-steps}\n\n\\newpage\n\\subsection{Approximation guarantee for the IMM algorithm}\n\\label{sec:imm-approx-main}\n\nOur main theoretical contribution is the following result.\n\n\\begin{theorem}\\label{thm:main-k}\\label{thm:main-k-appendix}\n\tSuppose that IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$. Then, \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost is at most $$\\mathrm{cost}(T)\\leq (2H+1) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$$\n\t\t\\item The $k$-means cost is at most $$\\mathrm{cost}(T)\\leq (8Hk+2) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$$\n\t\\end{enumerate}\n\tIn particular, IMM achieves worst case approximation factors of $O(k)$ and $O(k^2)$ by using any $O(1)$ approximation algorithm (compared to the optimal $k$-medians\/means) to generate the initial centers.\n\\end{theorem}\n\nWe state the theorem in terms of the depth of the tree to highlight that the approximation guarantee may depend on the structure of the input data. If the optimal clusters can be easily identified by a small number of salient features, then the tree may have depth $O(\\log k)$. \nWe later provide a lower bound showing that an $\\Omega(\\log k)$ approximation factor is necessary for $k$-medians and $k$-means (Theorem~\\ref{thm:lb-k}). For this data set, our algorithm produces a threshold tree with depth $O(\\log k)$, and therefore, the analysis is tight for $k$-medians. We leave it as an intriguing open question whether the bound can be improved for $k$-means.\n\n\\subsubsection{Proof Overview for Theorem~\\ref{thm:main-k}}\n\nThe proof proceeds in three main steps. First, we rewrite the cost of IMM in terms of the minimum number of mistakes made between the output clustering and the clustering based on the given centers. Second, we provide a lemma that relates the cost of any clustering to the number of mistakes required by a threshold clustering. Finally, we put these two together to show that the output cost is at most an $O(H)$ factor larger than the $k$-medians cost and at most an $O(Hk)$ factor larger than the $k$-means cost, respectively, where $H$ is the depth of the IMM tree, and the cost is relative to $\\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k)$. \n\nThe approximation bound rests upon a  characterization of the excess clustering cost induced by the tree. For any internal node $u$ of the final tree $T$, let $\\mbox{cell}(u) \\subseteq \\mathbb{R}^d$ denote the region of the input space that ends up in that node, and let $B(u)$ be the bounding box of the centers that lie in this node, or more precisely,  $B(u) = \\{\\boldsymbol{\\mu}^j: 1 \\leq j \\leq k\\} \\cap \\mbox{cell}(u)$. We will be interested in the diameter of this bounding box, measured either by $\\ell_1$ or squared $\\ell_2$ norm, and denoted by $\\mathrm{diam}_1(B(u))$ and $\\mathrm{diam}_2^2(B(u))$, respectively.\n\n\n\\paragraph{Upper bounding the cost of the tree.} The first technical claim (Lemma~\\ref{lemma:tree-cost}) will show that if IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\\begin{itemize}\n\\item The $k$-medians cost of $T$ satisfies   $\\displaystyle \\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\mathrm{diam}_1(B(u)) $\n\\item The $k$-means cost of $T$ satisfies  $\\displaystyle  \\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\mathrm{diam}_2^2(B(u))$\n\\end{itemize}\n\nBriefly, any point $\\mathbf{x}$ that ends up in a different leaf from its correct center $\\boldsymbol{\\mu}^j$ incurs some extra cost. To bound this, consider the internal node $u$ at which $\\mathbf{x}$ is separated from $\\boldsymbol{\\mu}^j$. Node $u$ also contains the center $\\boldsymbol{\\mu}^i$ that ultimately ends up in the same leaf as $\\mathbf{x}$. For $k$-medians, the excess cost for $\\mathbf{x}$ can then be bounded by $\\|\\boldsymbol{\\mu}^i - \\boldsymbol{\\mu}^j\\|_1 \\leq \\mathrm{diam}_1(B(u))$. The argument for $k$-means is similar.\n\nThese $\\sum_u t_u \\mathrm{diam}(B(u))$ terms can in turn be bounded in terms of the cost of the reference clustering. \n\n\\paragraph{Lower bounding the reference cost.} \nWe next need to relate the cost of the centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ to the number of mistakes and the diameter of the cells in the tree. Lemma~\\ref{aux-claim:general_k_upper_bound} will show that if IMM makes $t_u$ mistakes at node $u \\in T$, then\n\\begin{itemize}\n\\item The $k$-medians cost satisfies \n$\\displaystyle \\sum_{u \\in T} t_u \\cdot \\mathrm{diam}_1(B(u)) \\leq 2H \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$\n\\item The $k$-means cost satisfies \n$\\displaystyle \\sum_{u \\in T} t_u \\cdot  \\mathrm{diam}_2^2(B(u)) \\leq 4Hk \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$\n\\end{itemize}\n\nThe proof for this is significantly more complicated than the upper bound mentioned above. Moreover, it contains the main new techniques in our analysis of tree-based clusterings. \n\nThe core challenge is that we aim to lower bound the cost of the given centers using only information about the number of mistakes at each internal node. Moreover, the IMM algorithm only minimizes the {\\em number} of mistakes, and not the {\\em cost} of each mistake. Therefore, we must show that if every axis-aligned cut in $B(u)$ separates at least $t_u$ points~$\\mathbf{x}$ from their centers, then there must be a considerable distance between the points in $\\mbox{cell}(u)$ and their centers.\n\nTo prove this, we analyze the structure of points in each cell. Specifically, we consider the single-coordinate projection of points in the box $B(u)$, and we order the centers in $B(u)$ from smallest to largest for the analysis. If there are $k'$ centers in node $u$, we consider the partition of $B(u)$ into $2(k'-1)$ disjoint segments, splitting at the centers and at the midpoints between consecutive centers. Since $t_u$ is the minimum number of mistakes, we must in particular have at least $t_u$ mistakes from the threshold cut at each midpoint. We argue that each of these segments is covered at least $t_u$ times by a certain set of intervals. Specifically, we consider the intervals between mistake points and their true centers, and we say that an interval \\textit{covers} a segment if the segment is contained in the interval. This allows us to capture the cost of mistakes at different distance scales. For example, if a point is very far from its true center, then it covers many disjoint segments, and we show that it also implies a large contribution to the cost. \nClaim~\\ref{claim:covering} in  Section~\\ref{sec:general_k_upper_bound} provides our main covering result, and we use this to argue that the cost of the given centers can be lower bounded in terms of the distance between consecutive centers in $B(u)$. For $k$-medians, we can directly derive a lower bound on the cost in terms of the $\\ell_1$ diameter $\\mathrm{diam}_1(B(u))$. For $k$-means, however, we employ Cauchy-Schwarz, which incurs an extra factor of $k$ in the bound with $\\mathrm{diam}_2^2(B(u))$. Overall, we sum these bounds over the height $H$ of the tree, leading to the claimed upper bounds in the above lemma. \n\n\\subsubsection{Preliminaries and Notation for Theorem~\\ref{thm:main-k}}\n\nLet  $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ be the reference centers, and let $T$ be the resulting IMM tree. Each internal node $u$ corresponds to a value $\\theta_u \\in \\mathbb{R}$ and a coordinate $i \\in [d]$. The tree partitions $\\mathcal{X}$ into $k$ clusters $\\widehat C_1, \\ldots, \\widehat C_k$ based on the points that reach the $k$ leaves in $T$, where we index the clusters so that leaf $j$ contains the centers $\\boldsymbol{\\mu}^j$ and $\\widehat \\boldsymbol{\\mu}^j$, where $\\widehat \\boldsymbol{\\mu}^j$ is the mean of $\\widehat C_j$ for $k$-means and the median of $\\widehat C_j$ for $k$-medians. This provides a bijection between old and new centers (and clusters).\nRecall that the map $c:\\mathcal{X} \\to \\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$ associates each point to its nearest center (i.e., $c(\\mathbf{x})$ corresponds to the cluster assignment given by the centers $\\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$).\n\n\nFor a node $u \\in T$, we let $\\mathcal{X}_u$  denote the surviving data set vectors at node $u \\in T$ based on the thresholds from the root to $u$.\nWe also define $J_u \\subseteq [k]$ be the set of surviving centers at node~$u$ from the set $\\{\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k\\}$, where these centers satisfy the thresholds from the root to~$u$. \nDefine $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$ to be the maximal (smallest and largest) coordinate-wise values of the centers in $J_u$, that is, for $i \\in [d]$, we set\n$$\n\\mu^{L,u}_i = \\min_{j \\in J_u} \\mu^j_i,\n\\qquad\\mathrm{and}\\qquad \n\\mu^{R,u}_i = \\max_{j \\in J_u} \\mu^j_i.\n$$\nIn other words, using the previous notation and recalling that $B(u) = \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\} \\cap \\mathrm{cell}(u)$, we have that\n$$\n\\mathrm{diam}_1(B(u)) = \\|\\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u}\\|_1\n\\qquad \n\\mbox{\\ and\\ }\n\\qquad\n\\mathrm{diam}_2^2(B(u)) = \\|\\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u}\\|_2^2.\n$$\n\nRecall that $t_u$ for node $u \\in T$ denotes the number of {\\em mistakes} incurred during the threshold cut defined by $u$, where a point $\\mathbf{x}$ is a mistake at node $u$ if $x$ reaches $u$, it was not a mistake before, and exactly one of the following two events occurs:\n$$\n\\{c(\\mathbf{x})_i \\leq \\theta_u \\ \\ \\mathrm{and}\\ \\  x_i > \\theta_u\\}\n\\qquad \\mathrm{or} \\qquad\n\\{c(\\mathbf{x})_i > \\theta_u \\ \\ \\mathrm{and}\\ \\   x_i \\leq \\theta_u\\}.\n$$\nLet $\\mathcal{X} = \\cX^{\\mathsf{cor}} \\cup \\cX^{\\mathsf{mis}}$ be a partition of the input data set into two parts, where $\\mathbf{x}$ is in $\\cX^{\\mathsf{cor}}$ if it reaches the same leaf node in $T$ as its center $c(\\mathbf{x})$, and otherwise, $\\mathbf{x}$ is in $\\cX^{\\mathsf{mis}}$. In other words, $\\cX^{\\mathsf{mis}}$ contains all points $\\mathbf{x} \\in \\mathcal{X}$ that are a mistake at any node $u$ in $T$, and the rest of the points are in $\\cX^{\\mathsf{cor}}$. We note that the notion of ``mistakes'' used here is different than the definition of ``changes'' used for the analysis of $2$-means\/medians, even though we reuse some of the same notation.\n\nWe need a standard consequence of the Cauchy\u2013Schwarz inequality to analyze the $k$-means cost.\n\\begin{claim}\\label{clm:cauchy_schwarz_k_means}\n\tFor any $a_1,\\ldots,a_{m}\\in\\mathbb{R},$ it holds that $\\sum_{i=1}^ka_i^2 \\geq \\frac{1}{k}\\left(\\sum_{i=1}^ka_i\\right)^2.$\n\\end{claim}\n\\begin{proof}\n\tDenote by $a$ the vector $(a_1,\\ldots,a_{m})$ and by $b$ the vector $(\\nicefrac{1}{\\sqrt{k}},\\ldots,\\nicefrac{1}{\\sqrt{k}}).$ \n\tBy the Cauchy\u2013Schwarz inequality\n\t$\\frac{1}{k}\\left(\\sum_{i=1}^ka_i\\right)^2=\\inner{a}{b}^2\\leq\\sum_{i=1}^ka_i^2$\n\\end{proof}\n\nWe also need two facts, which state the optimal center for a cluster corresponds to mean or median of the points in the cluster, respectively. The proofs of these facts can be found in standard texts~\\cite{schutze2008introduction}.\n\n\\begin{fact}\\label{fact:1-means-optimal center}\nFor any set $S=\\{\\mathbf{x}^1,\\ldots, \\mathbf{x}^n\\}\\subseteq \\mathbb{R}^d$, the optimal center under the $\\ell_2^2$ cost  is the mean $\\boldsymbol{\\mu} = \\frac1n\\sum_{\\mathbf{x}\\in S}\\mathbf{x}.$\n\\end{fact}\n\n\\begin{fact}\\label{fact:1-median-optimal center}\nFor any set $S=\\{\\mathbf{x}^1,\\ldots, \\mathbf{x}^n\\}\\subseteq \\mathbb{R}^d$, the optimal center $\\boldsymbol{\\mu}$ under the $\\ell_1$ cost is the coordinate-wise median, defined for $i\\in[d]$ as $\\mu_i = \\mathsf{median}(x^1_i,\\ldots, x^n_i).$\n\\end{fact}\n\n\\subsubsection{The Two Main Lemmas and the Proof of Theorem~\\ref{thm:main-k}}\n\nTo prove the theorem, we state two lemmas that aid in analyzing the cost of the given clustering versus the IMM clustering. \nThe theorem will follow from these lemmas, and we will prove the lemmas in the proceeding subsections. We start with the lemma relating the number of mistakes~$t_u$ at each node $u$ and the distance between $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$ to the cost incurred by the given centers.\n\n\\begin{lemma}\\label{lemma:tree-cost}\n\tIf IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost of the IMM tree satisfies  $$\\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\n\t\t\\item The $k$-means cost of the IMM tree satisfies  $$\\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.$$\n\t\\end{enumerate}\n\\end{lemma}\n\nWe next bound the cost of the given centers in the terms of the number of mistakes in the tree. The key idea is that if there must be many mistakes at each node, then the cost of the given centers must actually be fairly large.\n\n\\begin{lemma}\\label{aux-claim:general_k_upper_bound}\n\tIf IMM takes centers $\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k$ and returns a tree $T$ of depth $H$ that incurs $t_u$ mistakes at node $u \\in T$, then \n\t\\begin{enumerate}\n\t\t\\item The $k$-medians cost satisfies \n\t\t$$\\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1 \\leq 2H \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\n\t\t\\item The $k$-means cost \n\t\tsatisfies \n\t\t$$\\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2 \\leq 4Hk \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\n\t\\end{enumerate}\n\\end{lemma}\n\nCombining these two lemmas immediately implies Theorem~\\ref{thm:main-k}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main-k}.]\nFor $k$-medians, Lemmas~\\ref{lemma:tree-cost} and~\\ref{aux-claim:general_k_upper_bound}\ntogether imply that \n$$\\mathrm{cost}(T)\\leq \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1\n\\leq \n(2H+1) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).$$\nFor $k$-means, we have that \n\\begin{eqnarray*}\n\\mathrm{cost}(T)\\leq 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2\n\\leq (8Hk+2) \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k).\n\\end{eqnarray*}\n\n\\end{proof}\n\n\n\n\\subsubsection{Proof of Lemma~\\ref{lemma:tree-cost}}\n\nWe begin with the $k$-medians proof (the $k$-means proof will be similar). Notice that the cost can only increase when measuring the distance to the (suboptimal) center $\\boldsymbol{\\mu}^j$ instead of the (optimal) center $\\widehat \\boldsymbol{\\mu}^j$ for cluster $\\widehat C_j$, and hence,\n$$\n\\mathrm{cost}(T) = \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\widehat \\boldsymbol{\\mu}^j\\|_1\n\\leq \n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\boldsymbol{\\mu}^j\\|_1.$$\nWe can rewrite this sum using the partition  $\\cX^{\\mathsf{cor}}$ and $\\cX^{\\mathsf{mis}}$ of $\\mathcal{X}$, using the fact that\nwhenever $\\mathbf{x}\\in \\cX^{\\mathsf{cor}}$, then the distance is computed with respect to the true center $c(\\mathbf{x})$,\n\\begin{eqnarray*}\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\widehat C_j} \\|\\mathbf{x} - \\boldsymbol{\\mu}^j\\|_1 &=& \n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\ &=& \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\end{eqnarray*}\nStarting with the above cost bound, and using the triangle inequality, we see\n\\begin{eqnarray*}\n\\mathrm{cost}(T) \n&\\leq&\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\ &\\leq&  \n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_1 +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \n(\\|\\mathbf{x}- c(\\mathbf{x})\\|_1 + \\|c(\\mathbf{x})- \\boldsymbol{\\mu}^j\\|_1) \n\\\\ &=&  \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) +\n\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|c(\\mathbf{x})- \\boldsymbol{\\mu}^j\\|_1\n\\end{eqnarray*}\n\nTo control the second term in the final line, we must bound the cost of the mistakes. We decompose $\\cX^{\\mathsf{mis}}$ based on the node $u$ where $\\mathbf{x} \\in \\cX^{\\mathsf{mis}}$ is first separated from its true center $c(\\mathbf{x})$ due to the threshold at node~$u$.\nTo this end, consider some point $\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j$, where its distance is measured to the incorrect center $\\boldsymbol{\\mu}^{j} \\neq c(\\mathbf{x})$.  Both centers $c(\\mathbf{x})$ and $\\boldsymbol{\\mu}^j$ have survived until node $u$ in the threshold tree $T$, and hence, both vectors are part of the definitions of $\\boldsymbol{\\mu}^{L,u}$ and  $\\boldsymbol{\\mu}^{R,u}$. In particular, we can use the upper bound\n$$\\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_1 \\leq \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\nThere are $t_u$ points in $\\cX^{\\mathsf{mis}}$ caused by the threshold at node $u$, and we have that\n$$\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_1 \\leq \n\\sum_{u \\in T} t_u \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.$$\nTherefore, we have, as desired\n\\begin{eqnarray*}\n\\mathrm{cost}(T) &\\leq& \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|_1\n\\\\&\\leq& \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n\\end{eqnarray*}\n\\noindent\nAnalyzing $k$-means is similar; we incur a factor of two by using Claim~\\ref{clm:cauchy_schwarz_k_means} instead of the triangle inequality:\n\\begin{eqnarray*}\n\\mathrm{cost}(T) &\\leq& \n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|_2^2 + \n2\\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j}  (\\|\\mathbf{x}- c(\\mathbf{x})\\|_2^2 +\\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_2^2)\n\\\\&\\leq& 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{j=1}^k \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j}  \\|c(\\mathbf{x}) - \\boldsymbol{\\mu}^j\\|_2^2\n\\\\ &\\leq& 2\\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) + 2\\cdot \\sum_{u \\in T} t_u \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2\n\\end{eqnarray*}\n\n\n\\subsubsection{Proof of Lemma~\\ref{aux-claim:general_k_upper_bound}}\n\\label{sec:general_k_upper_bound}\n\t\nTo prove this lemma, we bound the cost at each node $u$ of tree in terms of the mistakes made at this node. For this lemma, we define $\\cX^{\\mathsf{cor}}_u$ to be the set of points in $\\mathcal{X}$ that reach node $u$ in $T$ along with their center $c(\\mathbf{x})$. We note that $\\cX^{\\mathsf{cor}}_u$ differs from $\\cX^{\\mathsf{cor}} \\cap \\mathcal{X}_u$ because a point $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$ may not make it to $\\cX^{\\mathsf{cor}}$ if there is a mistake later on (i.e., $\\cX^{\\mathsf{cor}}$ is the union of $\\cX^{\\mathsf{cor}}_u$ only over leaf nodes).\n\n\\begin{lemma}\\label{lemma:mistake-bound}\nFor any node $u \\in T$, we have that\n\t$$\n\t\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n\t\\geq \n\t\\frac{t_u}{2} \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n\t$$\nand\n\t$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n\\geq \n\\frac{t_u}{4k}   \\cdot  \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{lemma}\n\\begin{proof}\nFix a coordinate $i \\in [d]$ and a node $u \\in T$.\nTo simplify notation, we let $z_1 \\leq \\cdots \\leq z_{k'}$ denote the {\\em sorted} values of $i$th coordinate of the $k' \\leq k$ centers that survive until node $u$ (so that $z_1 = \\mu^{L,u}_i$ and $z_{k'} = \\mu^{R,u}_i$). Observe that for each $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$, the center $c(\\mathbf{x})$ must have survived until node $u$, and hence, $c(\\mathbf{x})_i$ equals one of the values $z_j$ for $j\\in[k']$. \n\nWe need a definition that allows us to relate the cost in coordinate $i$ to the distances between $z_1$ and $z_{k'}$. For consecutive values $(j,j+1)$, we say that the pair $(j,j+1)$ is {\\em covered} by $\\mathbf{x}$ if either \n\\begin{itemize}\n\t\\item The segment $[z_j, \\frac{z_j + z_{j+1}}{2})$ is contained in the segment $[x_i,c(\\mathbf{x})_i]$, or\n\t\\item The segment $[\\frac{z_j + z_{j+1}}{2}, z_{j+1})$ is contained in the segment $[x_i,c(\\mathbf{x})_i]$.\n\\end{itemize} \n\nWe prove the following claim, which enables us to relate the cost in the $i$th coordinate to the value $z_{k'} - z_1$ by decomposing this value into the distance between consecutive centers.\n\\begin{claim}\\label{claim:covering}\nFor each $j = 1,2,\\ldots, k'-1$, the pair $(j,j+1)$ is {covered} by at least~$t_u$ points $\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u$.\n\\end{claim}\n\\begin{proof}\nSuppose for contradiction that this does not hold. We argue that we can find a threshold value for coordinate $i$ that makes fewer than $t_u$ mistakes. To see this, assume that $(j,j+1)$ is covered by fewer than $t_u$ points $\\mathbf{x} \\in \\mathcal{X}_u$. In particular, setting the threshold to be $\\frac{z_j + z_{j+1}}{2}$ separates fewer than $t_u$ points $\\mathbf{x}$ from their centers $c(\\mathbf{x})$. This implies that there are fewer than $t_u$ mistakes at node $u$, which is a contradiction because the IMM algorithm chooses the coordinate and threshold pair that minimizes the number of mistakes.\n\\end{proof}\n\nNow this claim suffices to prove Lemma~\\ref{lemma:mistake-bound}. The only challenge is that we must string together the covering points $\\mathbf{x}$ to get a bound on $z_{k'}-z_1$.\n\nWe start with the $k$-medians proof. Using the above claim, we can lower bound the contribution of coordinate $i$ to the cost of the given centers. Notice that the values $z_1 \\leq \\cdots \\leq z_{k'}$ partition the interval between $z_1 = \\mu^{L,u}_i$ and $z_{k'} = \\mu^{R,u}_i$. Thus, each time $\\mathbf{x}$ covers a pair $(j,j+1)$, there must be a contribution of $\\frac{z_{j+1} - z_j}{2}$ to the cost $|x_i - c(\\mathbf{x})_i|$. Because each pair is covered at least $t_u$ times by Claim~\\ref{claim:covering}, we conclude that \n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} |x_i - c(\\mathbf{x})_i|\n\\geq t_u \\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)\n= \\frac{t_u}{2} (z_{k'} - z_1).\n$$\nTo relate the bound to $\\boldsymbol{\\mu}^{L,u}$ and $\\boldsymbol{\\mu}^{R,u}$, we note that the above argument holds for each coordinate $i \\in [d]$, and\nwe have that\n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n=  \\sum_{i \\in [d]} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}   |\\mathbf{x}_i - c(\\mathbf{x})_i|\n\\geq \n\\frac{t_u}{2} \\cdot  \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n$$\n\nFor the $k$-means proof, we apply the same argument as above, this time using Claim~\\ref{clm:cauchy_schwarz_k_means} to bound the sum of squared values as \n$$\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} |x_i - c(\\mathbf{x})_i|^2\n\\geq t_u \\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)^2\n\\geq \\frac{t_u}{k} \\left(\\sum_{j = 1}^{k'-1}\\left(\\frac{z_{j+1} - z_j}{2}\\right)\\right)^2\n= \\frac{t_u}{4k} (z_{k'} - z_1)^2,$$\nand therefore, summing over coordinates $i\\in[d]$, we have\n$$\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u} \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n=  \\sum_{i \\in [d]} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}   |x_i - c(\\mathbf{x})_i|^2\n\\geq \n\\frac{t_u}{4k} \\cdot \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{aux-claim:general_k_upper_bound}.]\n\tWe start with the $k$-medians proof.\nThe factor of $H$ arises because the same points $\\mathbf{x} \\in \\mathcal{X}$ can appear in at most $H$ sets $\\cX^{\\mathsf{cor}}_u$ because $H$ is the depth of the tree. More precisely, using Lemma~\\ref{lemma:mistake-bound} for each node $u$, we have that \n$$\nH \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) \\geq \\sum_{u \\in T} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}  \\|\\mathbf{x} - c(\\mathbf{x})\\|_1\n\\geq \\sum_{u \\in T} \\frac{t_u}{2} \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_1.\n$$\t\nApplying the same steps for the $k$-means cost, we have that\n$$\nH \\cdot \\mathrm{cost}(\\boldsymbol{\\mu}^1,\\ldots, \\boldsymbol{\\mu}^k) \\geq \\sum_{u \\in T} \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}_u}  \\|\\mathbf{x} - c(\\mathbf{x})\\|_2^2\n\\geq \\sum_{u \\in T} \\frac{t_u}{4k} \\| \\boldsymbol{\\mu}^{L,u} - \\boldsymbol{\\mu}^{R,u} \\|_2^2.\n$$\n\\end{proof}\n\n\n\n\n\n\\subsection{Approximation lower bound}\n\nTo complement our upper bounds, we show that a threshold tree with $k$ leaves cannot, in general, yield better than an $\\Omega(\\log k)$ approximation to the optimal $k$-medians or $k$-means clustering.\n\n\\begin{theorem}\\label{thm:lb-k}\nFor any $k \\geq 2$, there exists a data set with $k$ clusters such that any  threshold tree $T$ with $k$ leaves\nmust have $k$-medians and $k$-means cost at least\n$$\n\\mathrm{cost}(T) \\geq \\Omega(\\log k) \\cdot \\mathrm{cost}(opt).\n$$\n\\end{theorem}\n\nThe data set is produced by first picking $k$ random centers from the hypercube $\\{-1,1\\}^d$, for large enough~$d$, and then using each of these to produce a cluster consisting of the $d$ points that can be obtained by replacing one coordinate of the center by zero.  Thus the clusters have size $d$ and radius $O(1)$. To prove the lower bound, we use ideas from the study of pseudo-random binary vectors, showing that projecting the centers to any subset of $m\\lesssim\\log_2 k$ coordinates take on all $2^m$ possible values, with each occurring roughly equally often.\n Then, we show that (i) the threshold tree must be essentially a complete binary tree with depth $\\Omega(\\log_2 k)$ to achieve a clustering with low cost, and (ii) any such tree incurs a cost of $\\Omega(\\log k)$ times more than the optimal for this data set (for both $k$-medians and $k$-means).\nThe proof of Theorem~\\ref{thm:lb-k} appears in Appendix~\\ref{sec:general_k_lower_bound}.\n\n\n\\section{Motivating Examples}\n\\label{sec:motivating-examples}\n\n{\\bf Using $k-1$ features may be necessary.}\nWe start with a simple but important bound showing that trees with depth less than~$k$ (or fewer than $k-1$ features) can be arbitrarily worse than the optimal clustering. Consider the data set consisting of the $k-1$ standard basis vectors $\\mathbf{e}^1,\\ldots,\\mathbf{e}^{k-1} \\in \\mathbb{R}^{k-1}$ along with the all zeros vector. As this data set has $k$ points,  \nthe optimal $k$-median\/means cost is zero, putting each point in its own cluster. \nUnfortunately,  it is easy to see that for this data, depth $k-1$ is necessary for clustering with a threshold tree.  Figure~\\ref{fig:simplex} depicts an optimal tree for this data set. Shorter trees do not work because projecting onto any $k-2$ coordinates does not separate the data, as at least two points will have all zeros in these coordinates. Therefore, any tree with depth at most $k-2$ will put two points in the same cluster, leading to non-zero cost, whereas the optimal cost is zero. In other words, for this data set, caterpillar trees such as Figure~\\ref{fig:simplex} are necessary and sufficient for an optimal clustering. This example also shows that $\\Theta(k)$ features are tight for feature selection~\\cite{cohen2015dimensionality} and provides a separation with feature extraction methods that use a linear map to only a logarithmic number of dimensions~\\cite{becchetti2019oblivious, makarychev2019performance}.\n\n\\begin{figure}\n    \\centering\n    \\subfloat[Optimal threshold tree for the data set in $\\mathbb{R}^{k-1}$ consisting of the $k-1$ standard basis vectors and the all zeros vector. Any optimal tree must use all $k-1$ features and have depth~$k-1$.]{\n    \\begin{minipage}[t]{0.45\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n\t[inner\/.style={shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=gray!40, scale=.85},\n\tdots\/.style={shape=rectangle, align=center, top color=white, scale=.85},\n\tleaf\/.style={shape=rectangle, rounded corners, draw, align=center, scale=.85},\n\tlevel 1\/.style={sibling distance=2mm},\n\tlevel 2\/.style={sibling distance=2mm},\n\tlevel 3\/.style={sibling distance=2mm},\n\tlevel 4\/.style={sibling distance=2mm},\n\tlevel distance=8mm]\n\t\\Tree\n\t[.\\node[inner]{$x_{i_1} \\leq 0.5$};\n\t[.\\node[inner]{$x_{i_2} \\leq 0.5$};\n\t[.\\node[inner] {$\\ldots$};\n\t[.\\node[inner] {$x_{i_d} \\leq 0.5$};\n\t\\node[leaf]{\\textbf{$\\mathbf{0}$}};\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_d}$}};\n\t]    \n\t\\node[leaf]{\\textbf{$\\ldots$}};\n\t]\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_2}$}};\n\t]\n\t\\node[leaf]{\\textbf{$\\mathbf{e}^{i_1}$}};\n\t]\n\t\\node[top color=white, bottom color=white, scale=.75] at (-0.8,-4.1) {};\n\t\\end{tikzpicture}\n\t\\end{minipage}\n\t\\label{fig:simplex}}\n\t\\hfill\n    \\subfloat[The ID3 split results in a $3$-means\/medians clustering with arbitrarily worse cost than the optimal because it places the top two points in separate clusters. Our algorithm (Section~\\ref{sec:k-means}) instead starts with the optimal first split.\n    ]{\n    \\includegraphics[width=.4\\textwidth, trim=-2cm 0 -2cm 0, clip]{our_vs_id3.png}\n    \\label{fig:imm_vs_id3}}\n\\caption{Motivating examples showing that (a) threshold trees may need depth $k-1$ to determine $k$ clusters, and (b) standard decision tree algorithms such as ID3 or CART perform very badly on some data sets.}\n\\end{figure}\n\t\n\n\\paragraph{Standard top-down decision trees do not work.}\n\\label{sec:standard-dt-bad}\nA natural approach to building a threshold tree is to (1) find a good $k$-medians or $k$-means clustering using a standard algorithm, then (2) use it to label all the points, and finally (3) apply a supervised decision tree learning procedure, such as ID3~\\cite{quinlan1986induction,quinlan2014c4} to find a threshold tree that agrees with these cluster labels as much as possible. ID3, like other common decision tree algorithms, operates in a greedy manner, where at each step it finds the best split in terms of \\emph{entropy} or \\emph{information gain}. We will show that this is not a suitable strategy for clustering and that the resulting tree can have cost that is arbitrarily bad. \nIn what follows, denote by $\\mathrm{cost}({ID3}_{\\ell})$  the cost of the decision tree with~$\\ell$ leaves returned by ID3 algorithm.\n\n\n\nFigure \\ref{fig:imm_vs_id3} depicts a data set $\\mathcal{X} \\subseteq \\mathbb{R}^2$ partitioned into three clusters $\\mathcal{X} = \\mathcal{X}_0 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_1 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_2$.\nWe define two centers $\\boldsymbol{\\mu}^0=(-2,0)$ and $\\boldsymbol{\\mu}^1=(2,0)$ and for each  $i\\in \\{0, 1\\}$, we define~$\\mathcal{X}_i$ as $500$ i.i.d. points $\\mathbf{x} \\sim \\mathcal{N}(\\boldsymbol{\\mu}^i, \\epsilon)$ for some small $\\epsilon > 0$. \nThen, $\\mathcal{X}_2 = \\{(-2, v), (2, v)\\}$ where $v \\to \\infty$.  \nWith high probability, we have that the optimal $3$-means clustering is $(\\mathcal{X}_0, \\mathcal{X}_1, \\mathcal{X}_2)$, i.e. $\\mathbf{x} \\in \\mathcal{X}$ gets label $y\\in\\{0,1,2\\}$ such that $\\mathbf{x} \\in \\mathcal{X}_y$.\nThe ID3 algorithm minimizes the entropy at each step. In the first iteration, it splits between the two large clusters. As a result $(-2, v)$ and $(2, v)$ will also be separated from one another. Since $ID3_3$ outputs a tree with exactly three leaves, one of the leaves must contain a point from $\\mathcal{X}_2$ together with points from either $\\mathcal{X}_0$ or $\\mathcal{X}_1$, this means that $\\mathrm{cost}(ID3_3)= \\Omega(v) \\to \\infty$.\nNote that $\\mathrm{cost}((\\mathcal{X}_1, \\mathcal{X}_2, \\mathcal{X}_3))$ does not depend on $v$, and hence, it is substantially smaller than $\\mathrm{cost}(ID3_3)$.\nUnlike ID3, the optimal threshold tree first separates $\\mathcal{X}_2$ from $\\mathcal{X}_0 \\mathbin{\\mathaccent\\cdot\\cup} \\mathcal{X}_1$, and in the second split it separates $\\mathcal{X}_0$ and $\\mathcal{X}_1$. Putting the outliers in a separate cluster is necessary for an optimal clustering. It is easy to extend this example to more clusters or to when ID3 uses more leaves.\n\n\n\\section{Introduction}\nA central direction in machine learning is understanding the reasoning behind decisions made by learned models~\\cite{lipton2018mythos, molnar2019, murdoch2019interpretable}. Prior work on AI explainability focuses on the interpretation of a black-box model, known as {\\em post-modeling} explainability~\\cite{baehrens2010explain, ribeiro2018anchors}. While methods such as LIME~\\cite{ribeiro2016should} or Shapley explanations~\\cite{lundberg2017unified} have made progress in this direction, they do not provide direct insight into the underlying data set, and the explanations depend heavily on the given model. This has raised concerns about the applicability of current solutions, leading researchers to consider more principled approaches to interpretable methods~\\cite{rudin2019stop}.\n\nWe address the challenge of developing machine learning systems that are explainable by design, starting from an {\\em unlabeled} data set. Specifically, we consider {\\em pre-modeling} explainability in the context of {clustering}. \nA common use of clustering is to identify patterns or discover structural properties in a data set by quantizing the unlabeled points. For instance, $k$-means clustering may be used to discover coherent groups among a supermarket's customers. While there are many good clustering algorithms, the resulting cluster assignments can be hard to understand because the clusters may be determined using all the features of the data, and there may be no concise way to explain the inclusion of a particular point in a cluster. This limits the ability of users to discern the commonalities between points within a cluster or understand why points ended up in different clusters. \n\n\n\\input{cluster-pic-intro.tex}\n\n\n\nOur goal is to develop accurate, efficient clustering algorithms with concise explanations of the cluster assignments. There should be a simple procedure using a few features to explain why any point belongs to its cluster. Small decision trees have been identified as a canonical example of an easily explainable model~\\cite{molnar2019, murdoch2019interpretable}, and\nprevious work on explainable clustering uses an unsupervised decision tree~\\cite{bertsimas2018interpretable, fraiman2013interpretable,geurts2007inferring,ghattas2017clustering, liu2005clustering}. Each node of the binary tree iteratively partitions the data by thresholding on a single feature. We focus on finding $k$ clusters, and hence,  we use trees with $k$ leaves. Each leaf corresponds to a cluster, and the tree is as small as possible. \nWe refer to such a tree as a {\\em threshold tree}.\n\nThere are many benefits of using a small threshold tree to produce a clustering. Any cluster assignment is explained by computing the thresholds along the root-to-leaf path. By restricting to $k$ leaves, we ensure that each such path accesses at most $k-1$ features, independent of the data dimension. \nIn general, a threshold tree provides an initial quantization of the data set, which can be combined with other methods for future learning tasks. While we consider static data sets, new data points can be easily clustered by using the tree, leading to explainable assignments.\nTo analyze clustering quality, we consider the $k$-means and $k$-medians objectives~\\cite{macqueen, steinhaus}. The goal is to efficiently determine a set of $k$ centers that minimize either the squared $\\ell_2$ or the $\\ell_1$ distance, respectively, of the input vectors to their closest center. \n\nFigure~\\ref{fig:optimal_vs_tree} provides an example of standard and explainable $k$-means clustering on the same data set. Figure~\\ref{fig:optimal_clusters} on the left shows an optimal $5$-means clustering. Figure~\\ref{fig:tree_clusters} in the middle shows an explainable, tree-based $5$-means clustering, determined by the tree in Figure~\\ref{fig:decision_tree} on the right. The tree has five leaf nodes, and vectors are assigned to clusters based on the thresholds. Geometrically, the tree defines a set of axis-aligned cuts that determine the clusters. While the two clusterings are very similar, using the threshold tree leads to easy explanations, whereas using a standard $k$-means clustering algorithm leads to more complicated clusters. The difference between the two approaches becomes more evident in higher dimensions, because standard algorithms will likely determine clusters based on all of the feature values.\n\nTo reap the benefits of explainable clusters, we must ensure that the data partition is a good approximation of the optimal clustering. While many efficient algorithms have been developed for $k$-means\/medians clustering, the resulting clusters are often hard to interpret~\\cite{arthur2007k,kanungo02, ostrovsky2013effectiveness, shalev2014understanding}. For example, Lloyd's algorithm alternates between determining the best center for the clusters and reassigning points to the closest center~\\cite{lloyd1982least}. The resulting set of centers depends in a complex way to the other points in the data set. Therefore, the relationship between a point and its nearest center may be the result of an opaque combination of many feature values. This issue persists even after dimension reduction or feature selection, because a non-explainable clustering algorithm is often invoked on the modified data set. As our focus is on pre-modeling explanability, we aim for simple explanations that use the original feature vectors.\n\nEven though Figure~\\ref{fig:optimal_vs_tree} depicts a situation in which the optimal clustering is very well approximated by one that is induced by a tree, it is not clear whether this would be possible in general. Our first technical challenge is to understand the {\\em price of explainability} in the context of clustering: that is, the multiplicative blowup in $k$-means (or $k$-medians) cost that is inevitable if we force our final clustering to have a highly constrained, interpretable, form. The second challenge is to actually find such a tree {\\em efficiently}. This is non-trivial because it requires a careful, rather than random, choice of a subset of features. As we will see, the kind of analysis that is ultimately needed is quite novel even given the vast existing literature on clustering.\n\n\n\\subsection{Our contributions}\n\n\nWe provide several new theoretical results on explainable $k$-means and $k$-medians clustering. Our new algorithms and lower bounds are summarized in Table~\\ref{tab:results_summary}.  \n\n\\medskip \\noindent {\\bf Basic limitations.} \nA partition into $k$ clusters can be realized by a binary threshold tree with $k-1$ internal splits. This uses at most $k-1$ features, but is it possible to use even fewer, say $\\log k$ features? In Section~\\ref{sec:motivating-examples}, we demonstrate a simple data set that requires $\\Omega(k)$ features to achieve a explainable clustering with bounded approximation ratio compared to the optimal $k$-means\/medians clustering. In particular, the depth of the tree might need to be $k-1$ in the worst case.\n\nOne idea for building a tree is to begin with a good $k$-means (or $k$-medians) clustering, use it to label all the points, and then apply a supervised decision tree algorithm that attempts to capture this labeling. In Section~\\ref{sec:standard-dt-bad}, we show that standard decision tree algorithms, such as ID3, may produce clusterings with arbitrarily high cost. Thus, existing splitting criteria are not suitable for finding a low-cost clustering, and other algorithms are needed.\n\n\\medskip \\noindent  {\\bf New algorithms.} \nOn the positive side, we provide efficient algorithms to find a small threshold tree that comes with provable guarantees on the cost. We note that using a small number of clusters is preferable for easy interpretations, and therefore $k$ is often relatively small.\nFor the special case of two clusters ($k=2$), we show (Theorem~\\ref{thm:optimal_2_median_means}) that a single threshold cut provides a constant-factor approximation to the optimal $2$-medians\/means clustering, with a closely-matching lower bound (Theorem~\\ref{clm:2_median_lower_bound}), and we provide an efficient algorithm for finding the best cut. For general $k$, we show how to approximate any clustering by using a threshold tree with $k$ leaves (Algorithm~\\ref{algo:imm}). The main idea is to minimize the number of mistakes made at each node in the tree, where a mistake occurs when a threshold separates a point from its original center. Overall, the cost of the explainable clustering will be close to the original cost up to a factor that depends on the tree depth (Theorem~\\ref{thm:main-k}). In the worst-case, we achieve an approximation factor of $O(k^2)$ for $k$-means and $O(k)$ for $k$-medians compared to the cost of any clustering (e.g., the optimal cost). These results do not depend on the dimension or input size; hence, we get a constant factor approximation when $k$ is constant. \n\n\\paragraph{Approximation lower bounds.}\nSince our upper bounds depend on $k$, it is natural to wonder whether it is possible to achieve a constant-factor approximation, or whether the cost of explainability grows with~$k$. On the negative side, we identify a data set such that any threshold tree with $k$ leaves must incur an $\\Omega(\\log k)$-approximation for both $k$-medians and $k$-means (Theorem~\\ref{thm:lb-k}). \nFor this data set, our algorithm achieves a nearly matching bound for $k$-medians.\\vspace{2ex}\n\n\n\\begin{table}[!htb]\n\\renewcommand{\\arraystretch}{1.6}\n\\centering\n\\begin{minipage}{.8\\textwidth}\n\\centering\n    \\begin{tabular}{|c|cc|cc|}\n        \\hline\n        %\n        \\rowcolor[HTML]{F1F1F1}\n        & \\multicolumn{2}{c|}{\\textbf{$k$-medians}} &\n        \\multicolumn{2}{c|}{\\textbf{$k$-means}} \\\\\n       \n       \n        \\rowcolor[HTML]{F1F1F1}\n        & $k = 2$ & $k > 2$ & $k = 2$ & $k > 2$  \\\\\n         \\hline\n       \n        \\cellcolor[HTML]{F1F1F1}\n        \\textbf{Upper Bound} & {$2$} & {$O(k)$} & {$4$} & {$O(k^2)$} \\\\\n       \n        \\cellcolor[HTML]{F1F1F1} \\textbf{Lower Bound} & {$2 - \\frac{1}{d}$} & {$\\Omega(\\log k)$} &  {$3\\left(1 - \\frac{1}{d} \\right)^2$} & {$\\Omega(\\log k)$} \\\\\n       \n        \\hline\n    \\end{tabular}\n    \\caption{Summary of our upper and lower bounds on approximating the optimal $k$-medians\/means clustering with explainable, tree-based clusters. The values express the factor increase compared to the optimal solution in the worst case.\n    }\n    \\label{tab:results_summary}\n\\end{minipage}\n\\end{table}\n\n\n\\input{related.tex}\n\n\\section{Preliminaries}\n\n\\input{prelim.tex}\n\\input{motivation-examples.tex}\n\n\n\\input{2-means-short.tex}\n\\input{k-means-short-new.tex}\n\n\n\\section{Conclusion}\nIn this paper we discuss the capabilities and limitations of explainable clusters. For the special case of two clusters ($k=2$), we provide nearly matching upper and lower bounds for a single threshold cut. For general $k >2$, we present the IMM algorithm that achieves an $O(H)$ approximation for $k$-medians and an $O(Hk)$ approximation for $k$-means when the threshold tree has depth $H$ and $k$ leaves. \nWe complement our upper bounds with a lower bound showing that any threshold tree with $k$ leaves must have cost at least $\\Omega(\\log k)$ more than the optimal for certain data sets. \nOur theoretical results provide the first approximation guarantees on the quality of explainable unsupervised learning in the context of clustering. Our work makes progress toward the larger goal of explainable AI methods with precise objectives and provable guarantees. \n\n\n\nAn immediate open direction is to improve our results for $k$ clusters, either on the upper or lower bound side. One option is to use larger threshold trees with more than $k$ leaves (or allowing more than $k$ clusters). It is also an important goal to identify natural properties of the data that enable explainable, accurate clusters. For example, it would be interesting to improve our upper bounds on explainable clustering for well-separated data. Our lower bound of $\\Omega(\\log k)$ utilizes clusters with diameter $O(1)$ and separation $\\Omega(d)$, where the hardness stems from the randomness of the centers. In this case, the approximation factor $\\Theta(\\log k)$ is tight because our upper bound proof actually provides a bound in terms of the tree depth (which is about $\\log k$, see Appendix~\\ref{apx:lower_bound_log_k}). Therefore, an open question is whether a $\\Theta(\\log k)$ approximation is possible for any well-separated clusters (e.g., mixture of Gaussians with separated means and small variance). Beyond $k$-medians\/means, it would be worthwhile to develop other clustering methods using a small number of features (e.g., hierarchical clustering).\n\n\\paragraph{Acknowledgements.}\nSanjoy Dasgupta has been supported by NSF CCF-1813160. Nave Frost has been funded by the  European Research  Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant agreement No. 804302). The contribution of Nave Frost is part of a Ph.D. thesis research conducted at Tel Aviv University. \n\n\n\\section{Preliminaries}\nThroughout we use bold variables for vectors, and we use non-bold for scalars such as feature values.\nGiven a set of points $\\mathcal{X}=\\{\\mathbf{x}^1,\\ldots,\\mathbf{x}^n\\}\\subseteq\\mathbb{R}^d$ and an integer $k$ the goal of $k$-medians and $k$-means clustering is to partition $\\mathcal{X}$ into $k$ subsets and minimize the distances of the points to the centers of the clusters. It is known that the optimal centers correspond to means or medians of the clusters, respectively. Denoting the centers as  $\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k$, the aim of $k$-means is to find a clustering that minimizes the following objective\n$$\\mathrm{cost}_2(\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k)=\\sum_{\\mathbf{x}\\in \\mathcal{X}} \\norm{\\mathbf{x}-c_2(\\mathbf{x})}^2_2,$$ where $c_2(\\mathbf{x})=\\argmin_{\\boldsymbol{\\mu} \\in \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\}}{\\norm{\\boldsymbol{\\mu} - \\mathbf{x}}_2}$.\nSimilarly, the goal of $k$-medians is to minimize \n$$\\mathrm{cost}_1(\\boldsymbol{\\mu}^1, \\ldots, \\boldsymbol{\\mu}^k)=\\sum_{\\mathbf{x}\\in \\mathcal{X}} \\norm{\\mathbf{x}-c_1(\\mathbf{x})}_1,$$ where $c_1(\\mathbf{x})=\\argmin_{\\boldsymbol{\\mu} \\in \\{\\boldsymbol{\\mu}^1,\\ldots,\\boldsymbol{\\mu}^k\\}}{\\norm{\\boldsymbol{\\mu} - \\mathbf{x}}_1}$. \nAs it will be clear from context whether we are talking about $k$-medians or $k$-means, we abuse notation and write $\\mathrm{cost}$ and $c(\\mathbf{x})$ for brevity. We also fix the data set and use $opt$ to denote the optimal $k$-medians\/means clustering, where the optimal centers are the medians or means of the clusters, respectively; hence, $\\mathrm{cost}(opt)$ refers to the cost of the optimal $k$-medians\/means clustering. \n\n\n\\subsection{Clustering using threshold trees}\n\n\nPerhaps the simplest way to define two clusters is to use a \\emph{threshold cut}, which partitions the data based on a threshold for a single feature. More formally, the two clusters can be written as $\\widehat C^{\\theta,i}=(\\widehat C^1, \\widehat C^2)$, which is defined using a coordinate $i$ and a threshold $\\theta\\in\\mathbb{R}$ in the following way. For each input point $\\mathbf{x}\\in \\mathcal{X}$, we place $\\mathbf{x}=[x_1, \\ldots, x_d]$ in the first cluster $\\widehat C^1$ if $x_i\\leq\\theta$, and otherwise $\\mathbf{x} \\in \\widehat C^2$.\nA threshold cut can be used to explain $2$-means or $2$-medians clustering because a single feature and threshold determines the division of the data set into exactly two clusters. \n\n\nFor $k > 2$ clusters, we consider iteratively using threshold cuts as the basis for the cluster explanations. More precisely, \nwe construct a binary \\emph{threshold tree}. This tree is an unsupervised variant of a decision tree. Each internal node contains a single feature and threshold, which iteratively partitions the data, leading to clusters determined by the vectors that reach the leaves. We focus on trees with exactly $k$ leaves, one for each cluster $\\{1,2,\\ldots, k\\}$, which also limits the depth and total number of features to at most $k-1$. \n\nWhen clustering using such a tree, it is easy to understand why $\\mathbf{x}$ was assigned to its cluster: we may simply inspect the threshold conditions on the root-to-leaf path for $\\mathbf{x}$. This also ensures the number of conditions for the cluster assignment is rather small, which is crucial for interpretability. These tree-based explanations are especially useful in high-dimensional space, when the number of clusters is much smaller than the input dimension ($k \\ll d$). \nMore formally, a threshold tree $T$ with $k$ leaves induces a $k$-clustering of the data. Denoting these clusters as $\\widehat{C}^j \\subseteq \\mathcal{X}$, we define the $k$-medians\/means cost of the tree as\n$$\n\\mathrm{cost}_1(T) = \\sum_{j=1}^k \\sum_{x \\in \\widehat{C}^j} \\|x - \\mbox{median}(\\widehat{C}^j) \\|_1\n\\qquad \\mbox{} \\qquad \n\\mathrm{cost}_2(T) = \\sum_{j=1}^k \\sum_{x \\in \\widehat{C}^j} \\|x - \\mbox{mean}(\\widehat{C}^j) \\|_2^2 \n$$\nOur goal is to understand when it is possible to efficiently produce a tree $T$ such that $\\mathrm{cost}(T)$ is not too large compared to the optimal $k$-medians\/means cost. Specifically, we say that an algorithm is an {\\em $a$-approximation}, if the cost is at most $a$ times the optimal cost, i.e., if the algorithm returns threshold tree $T$ then we have\n$\\mathrm{cost}(T) \\leq a\\cdot \\mathrm{cost}(opt),$ where $opt$ denotes the optimal $k$-medians\/means clustering.\n\n \n\n\n\\section{Lower bounds for two clusters}\\label{sec:k2_lower_bound}\n\n\nWithout loss of generality we can assume that $d\\geq 2.$ We use the following dataset for both $2$-medians and $2$-means. It consists of $2d$ points, partitioned into two clusters of size $d$, which are the points with Hamming distance exactly one from the vector with all 1 entries and the vector with all $-1$ entries: \n\\begin{center}\n\\begin{tabular}{c c}\n    \\textbf{Optimal Cluster 1} & \\textbf{Optimal Cluster 2} \\\\\n    $(0,-1,-1,-1\\ldots,-1)$ & $(0,1,1,1\\ldots,1)$ \\\\\n    $(-1,0,-1,-1\\ldots,-1)$ & $(1,0,1,1\\ldots,1)$\\\\\n    $(-1,-1,0,-1\\ldots,-1)$ & $(1,1,0,1\\ldots,1)$\\\\\n    $\\vdots$ & $\\vdots$ \\\\\n    $(-1,-1,-1,-1\\ldots,0)$ & $(1,1,1,1\\ldots,0)$\\\\\n\\end{tabular}\n\\end{center}\n\n\nLet $\\widehat C = (\\widehat C^1, \\widehat C^2)$ be the best threshold cut. \n\n\\paragraph{2-medians lower bound.}\nThe cost of the cluster with centers $(1,\\ldots,1)$ and $(-1,\\ldots,-1)$ is $2d$, as each point is responsible for a cost of $1.$ Thus, $\\mathrm{cost}(opt)\\leq 2d.$\n\nThere is a coordinate $i$ and a threshold $\\theta$ that defines the cut $\\widehat C$. For any coordinate $i$, there are only three possible values: $-1,0,1$. Thus $\\theta$ is either in $(-1,0)$ or in $(0,1)$. Without loss of generality, assume that $\\theta\\in(-1,0)$ and $i=1$. Thus, the cut is composed of two clusters: one of size $d-1$ and the other of size $d+1$, in the following way:\n\n\\begin{center}\n\\begin{tabular}{c c}\n    $\\mathbf{Cluster\\ }\\widehat C^1$ & $\\mathbf{Cluster\\ } \\widehat C^2$ \\\\\n    $(-1,0,-1,-1\\ldots,-1)$ & $(1,0,1,1\\ldots,1)$\\\\\n    $(-1,-1,0,-1\\ldots,-1)$ & $(1,1,0,1\\ldots,1)$\\\\\n    $\\vdots$ & $\\vdots$ \\\\\n    $(-1,-1,-1,-1\\ldots,0)$ & $(1,1,1,1\\ldots,0)$\\\\\n     & $(0,1,1,1\\ldots,1)$\\\\\n     & $(0,-1,-1,-1\\ldots,-1)$\\\\\n\\end{tabular}\n\\end{center}\n\n\nUsing Fact~\\ref{fact:1-median-optimal center}, an optimal center of the first cluster is all $-1$, and the optimal center for the second cluster is all $1$. The cost of the first cluster is $d-1$, as each point costs $1$. The cost of the second cluster is composed of two terms $d$ for all points that include 1 in at least one coordinate and the cost of point $(0,-1,\\ldots,-1)$ is $2(d-1)+1$. So the total cost is $4d-2$. Thus $\\mathrm{cost}(\\widehat C)\\geq (2-1\/d) \\mathrm{cost}(opt).$\n\n\\paragraph{2-means lower bound.}\nFocus on the clustering with centers $$(\\nicefrac{(d-1)}{d},\\ldots,\\nicefrac{(d-1)}{d}) \\qquad \\mbox{and} \\qquad (-\\nicefrac{(d-1)}{d},\\ldots,-\\nicefrac{(d-1)}{d}).$$ \nThe cost of each point in the data is composed of (1) one coordinate with value zero, and the cost of this coordinate is $\\left(\\nicefrac{(d-1)}{d}\\right)^2$ (2) $d-1$ coordinates each with cost $\\nicefrac1d^2.$ Thus, each point has a cost of $\\nicefrac{(d-1)^2}{d^2}+\\nicefrac{d-1}{d^2}.$ Thus, the total cost is $\\frac{2(d-1)^2+2(d-1)}{d}=2(d-1)$. This implies that $\\mathrm{cost}(opt)\\leq 2(d-1).$\n\nAssume without loss of generality that $\\widehat C$ is defined using coordinate $i=1$ and threshold $-0.5$. The resulting clusters $\\widehat C^1$ and $\\widehat C^2$ are as in the case of $2$-medians. The optimal centers are (see Fact~\\ref{fact:1-means-optimal center}): \n$$\\left(-1,-\\frac{d-2}{d-1},\\ldots,-\\frac{d-2}{d-1}\\right) \\qquad \\mbox{and} \\qquad \\left(\\frac{d-1}{d+1},\\frac{d-2}{d+1},\\ldots,\\frac{d-2}{d+1}\\right).$$\nWe want to lower bound $\\mathrm{cost}(\\widehat C).$ We start with the cost of the first cluster, i.e. $\\widehat C^1$. To do so for each point in $\\widehat C^1$, we will evaluate the contribution of each coordinate to the cost (1) the first coordinate adds $0$ to the cost (2) the coordinate with value $0$, adds $\\left(\\frac{d-2}{d-1}\\right)^2$ to the cost (3) the rest of the $d-2$ coordinates adds $\\nicefrac{1}{(d-1)^2}.$ Thus, each point in $\\widehat C^1$ adds to the cost $\\left(\\frac{d-2}{d-1}\\right)^2 + \\frac{d-2}{(d-1)^2}=\\frac{d-2}{d-1}$. Since $\\widehat C^1$ contains $d-1$ points, its total cost is $d-2$.\n\nMoving on to evaluating the cost of $\\widehat C^2$, the cost of the point $(0,-1,\\ldots,-1)$ is composed of two terms (1) the first coordinate adds $\\left(\\frac{d-1}{d+1}\\right)^2$ to the cost (2) each of the other $d-1$ coordinates adds $\\left(1+\\frac{d-2}{d+1}\\right)^2\nto the cost. Thus, this point adds $$\\left(\\frac{d-1}{d+1}\\right)^2+(d-1)\\left(1+\\frac{d-2}{d+1}\\right)^2=\\frac{(d-1)d(4d-3)}{(d+1)^2}.$$\nSimilarly, the point $(0,1,\\ldots,1)$ adds to the cost $$\\left(\\frac{d-1}{d+1}\\right)^2+(d-1)\\left(1-\\frac{d-2}{d+1}\\right)^2=\\frac{(d-1)(d+8)}{(d+1)^2}.$$\nFinally, each of the $d-1$ remaining points in $\\widehat C^2$ adds to the cost $$\\left(1-\\frac{d-1}{d+1}\\right)^2+\\left(\\frac{d-2}{d+1}\\right)^2+(d-1)\\left(1-\\frac{d-2}{d-1}\\right)^2=\\frac{d^2+5d-1}{(d+1)^2}$$\nThus, the cost of $\\widehat C^2$ is $$\\frac{(d-1)(5d^2+3d+7)}{(d+1)^2}$$\nSumming up the costs of $\\widehat C^1$ and $\\widehat C^2$, for $d\\geq 2$\n$$\\mathrm{cost}(\\widehat C)\\geq (d-2)+\\frac{(d-1)(5d^2+3d+7)}{(d+1)^2}\\geq6(d-1)\\left(1-\\frac1d\\right)^2\\geq 3\\left(1-\\frac{1}{d}\\right)^2\\cdot \\mathrm{cost}(opt)$$\n\n\n\n\n\n\\section{Upper Bound Proof for 2-Means}\\label{sec:k2_upper_bound}\n\nWe show that there is a threshold cut $\\widehat C$ with $2$-means cost satisfying\n$\\mathrm{cost}(\\widehat C)\\leq 4 \\cdot \\mathrm{cost}(opt).$ We could just use the same proof idea as in the 2-medians case that first applies Lemma~\\ref{lemma:tree-cost} and then uses the matching result, Lemma~\\ref{lemma:matching}. This leads to a $6$-approximation, instead of $4$. The reason is that we apply twice Claim~\\ref{clm:cauchy_schwarz_k_means}, which is not tight. Improving the approximation to $4$ requires us to apply   Claim~\\ref{clm:cauchy_schwarz_k_means} only once. \n\nSuppose $\\boldsymbol{\\mu}^1,\\boldsymbol{\\mu}^2$ are optimal $2$-means centers for the clusters $C^1$ and $C^2$. \nLet $t = \\min(|C^1 \\Delta \\widehat C^1|, |C^1 \\Delta \\widehat C^2|)$ be the minimum number of changes for any threshold cut $\\widehat C^1, \\widehat C^2$, and define $\\cX^{\\mathsf{mis}}$ to the set of $t$ points in the symmetric difference, where $\\mathcal{X} = \\cX^{\\mathsf{cor}} \\cup \\cX^{\\mathsf{mis}}$ and $\\cX^{\\mathsf{cor}} \\cap \\cX^{\\mathsf{mis}} = \\emptyset$.\n\nUsing the same argument as in the proof of Lemma~\\ref{lemma:tree-cost}, we have\n\\begin{eqnarray}\\label{eq:2_means_upper_bound}\n\\mathrm{cost}(\\widehat{C}) &\\leq& \\sum_{j=1}^2 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|^2_2 +\n\\sum_{j=1}^2 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_j} \\|\\mathbf{x}- \\boldsymbol{\\mu}^j\\|^2_2 \\nonumber\n\\\\ &=& \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{cor}}} \\|\\mathbf{x}- c(\\mathbf{x})\\|^2_2 +\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2 \\nonumber\n\\\\ &\\leq& \\mathrm{cost}(opt) +\n\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray}\nThe goal now is to bound the latter two terms using $\\mathrm{cost}(opt).$\nThis term measures the distance of each $\\mathbf{x} \\in \\cX^{\\mathsf{mis}}$ from the ``other'' center, i.e., not $c(\\mathbf{x})$. \n\n\\begin{claim}\\label{claim:aux-2-means}\n\\begin{eqnarray*}\n\t\\mathrm{cost}(opt)\n\t\\geq \\frac13 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\frac13\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray*}\n\\end{claim}\n\nUsing Claim~\\ref{claim:aux-2-means},\ntogether with Inequality~(\\ref{eq:2_means_upper_bound}) we have\n$$\\mathrm{cost}(\\widehat{C})\\leq \\mathrm{cost}(opt) + 3\\cdot \\mathrm{cost}(opt)=4\\cdot \\mathrm{cost}(opt),$$ \nand this completes the proof. \n\n\n\\begin{proof}[Proof of Claim~\\ref{claim:aux-2-means}.]\nDenote the $t$ points in $\\cX^{\\mathsf{mis}}$  by $\\cX^{\\mathsf{mis}}=\\{\\mathbf{r}^1,\\ldots,\\mathbf{r}^t\\}.$ Assume that the first $\\ell$ points are in the first optimal cluster, $\\mathbf{r}^1,\\ldots,\\mathbf{r}^\\ell\\in C^1$, and the rest are in the second cluster, $\\mathbf{r}^{\\ell+1},\\ldots,\\mathbf{r}^t\\in C^2.$\n\nApplying Lemma~\\ref{lemma:matching} for each coordinate $i\\in[d]$ guarantees $t$ pairs of vectors $(\\mathbf{p}^1,\\mathbf{q}^1), \\ldots, (\\mathbf{p}^t,\\mathbf{q}^t)$ with the following properties. Each $p^j_i$ corresponds to the $i$th coordinate of some  point in $C^1$ and $q^j_i$ corresponds to the $i$th coordinate of some point in $C^2$. Furthermore, for each coordinate, the $t$ pairs correspond to $2t$ distinct points in $\\mathcal{X}$.\nFinally, we can assume without loss of generality that\n$\\mu^1_i \\leq \\mu^2_i$ and $q^j_i \\leq p^j_i$. \n\nFor each point $\\mathbf{r}^j$ in the first $\\ell$ points in $\\cX^{\\mathsf{mis}}$, if $r^j_i\\geq p^j_i$ then we can replace $\\mathbf{p}^j$ with $\\mathbf{r}^j$, thus we can assume without loss of generality that $p_i^j\\geq r^j_i.$ We next show that $\\mathrm{cost}(opt)$ is lower bounded by  a function of $t$. There will be two cases depending on whether $p^j_i\\leq \\mu^2_i$ or not. The harder case is the first where the improvement of the approximation from $6$ to $4$ arises. Instead of first bounding the distance between $\\mathbf{r}^j$ and its new center using the distance to its original center and then accounting for $\\norm{\\boldsymbol{\\mu}^1-\\boldsymbol{\\mu}^2}^2_2,$ we directly account for the distance between $\\mathbf{r}^j$ and its new center. \n\n\n\\paragraph{\\textbf{Case 1:}} if  $p^j_i\\leq \\mu^2_i$, then \nClaim~\\ref{clm:cauchy_schwarz_k_means} implies that\n\\begin{eqnarray*} \n(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n&\\geq& \\frac{1}{3}(\\mu^2_i-q^j_i+p^j_i-\\mu_i^1+\\mu_i^1-r^j_i)^2\\\\\n&=& \\frac{1}{3}((\\mu^2_i-q^j_i)+(p^j_i-r^j_i))^2\n\\geq \\frac{1}{3}(\\mu^2_i-r^j_i)^2.\n\\end{eqnarray*}  The last inequality follows from $q^j_i \\leq p^j_i$ and $r_i^j\\leq p_i^j,$ which imply that $(\\mu^2_i-q^j_i)+(p^j_i-r^j_i)\n\\geq \\mu^2_i-r^j_i\\geq 0,$ which means $((\\mu^2_i-q^j_i)+(p^j_i-r^j_i))^2\n\\geq (\\mu^2_i-r^j_i)^2.$ \n\\paragraph{\\textbf{Case 2:}} if $\\mu_i^2\\leq p_i^j$, then again Claim~\\ref{clm:cauchy_schwarz_k_means} implies that\n\\begin{eqnarray*} \n(p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2&\\geq& (\\mu_i^2-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2 \\geq \\frac12 (\\mu^2_i-\\mu^1_i+\\mu^1_i-r^j_i)^2=\\frac12(\\mu^2_i-r^j_i)^2,\n\\end{eqnarray*}\nwhere in the first inequality we use $(p^j_i-\\mu^1_i)^2\\geq(\\mu^2_i-\\mu^1_i)^2.$\n\nThe two cases imply that for $1\\leq j\\leq \\ell$  $$(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n\\geq  \\frac{1}{3}(\\mu^2_i-r^j_i)^2\n.$$\nSimilarly for each point $\\mathbf{r}^j$ in the last $t-\\ell$ points in $\\cX^{\\mathsf{mis}}$, we have $$(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^2-r^j_i)^2\n\\geq  \\frac{1}{3}(\\mu^1_i-r^j_i)^2\n.$$\nPutting these together we have \n\\begin{eqnarray*}\n\t\\mathrm{cost}(opt)&\\geq& \\sum_{i=1}^d\\sum_{j=1}^\\ell(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^1-r^j_i)^2\n\t+\n\\sum_{i=1}^d\\sum_{j=\\ell+1}^t(\\mu^2_i-q^j_i)^2 + (p^j_i-\\mu_i^1)^2 + (\\mu_i^2-r^j_i)^2\\\\\n\t&\\geq& \\frac{1}{3}\\sum_{j=1}^\\ell\\sum_{i=1}^d  (\\mu^2_i-r^j_i)^2\n\t+\\frac{1}{3}\\sum_{j=\\ell+1}^t\\sum_{i=1}^d  (\\mu^1_i-r^j_i)^2 \n\t\\\\ \t&=&\\frac{1}{3}\\sum_{j=1}^\\ell\\norm{\\mathbf{r}^j-\\boldsymbol{\\mu}^2}_2^2+\\frac{1}{3}\\sum_{j=\\ell+1}^t\\norm{\\mathbf{r}^j-\\boldsymbol{\\mu}^1}_2^2\\\\\n&=&\t\\frac13 \\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_1} \\|\\mathbf{x}- \\boldsymbol{\\mu}^1\\|^2_2+\\frac13\\sum_{\\mathbf{x} \\in \\cX^{\\mathsf{mis}} \\cap \\widehat C_2} \\|\\mathbf{x}- \\boldsymbol{\\mu}^2\\|^2_2\n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Efficient Implementation via Dynamic Programming for $k=2$}\\label{sec:DP_Implementation}\n\n\\subsection{The 2-means case}\nThe psudo-code for finding the best threshold for $k=2$ depicted in Algorithm~\\ref{algo:cut}.\n\n\\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}{.6\\linewidth}\n        \\begin{algorithm}[H]\n    \t\\SetKwFunction{$2$-means Optimal Threshold}{$2$-means Optimal Threshold}\n    \t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n    \t\\LinesNumbered\n    \t\\Input{%\n    \t\t{$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$} -- vectors in $\\mathbb{R}^d$\n    \t}\n    \t\\Output{%\n    \t\t\\xvbox{2mm}{$i$} -- Coordinate\\\\\n    \t\t\\xvbox{2mm}{$\\theta$} -- Threshold\n    \t}\n    \t\\BlankLine\n    \t\\setcounter{AlgoLine}{0}\n    \t\n        $\\xvar{best\\_cost} \\leftarrow \\infty$\\;\n    \t\n    \t$\\xvar{best\\_coordinate} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$\\xvar{best\\_threshold} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$u \\leftarrow \\sum_{j=1}^n \\norm{\\mathbf{x}^j}^2_2$\\;\n    \t\n    \t\\ForEach {$i \\in [1, \\ldots, d]$}\n    \t{\n    \t\t$\\mathbf{s} \\leftarrow \\xfunc{zeros}(d)$\\;\n    \t\t\n    \t\t$\\mathbf{r} \\leftarrow \\sum_{j=1}^n \\mathbf{x}^j$\\;\n    \t\t\n    \t\t$\\mathcal{X} \\leftarrow \\xfunc{sorted}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n \\text{ by coordinate }i)$\\;\n    \t\t\n    \t\t\\ForEach {$\\mathbf{x}^j \\in \\mathcal{X}$} \n    \t\t{ \n    \t\t\t$\\mathbf{s} \\leftarrow \\mathbf{s} + \\mathbf{x}^j$\\;\n    \t\t\t\n    \t\t\t$\\mathbf{r} \\leftarrow \\mathbf{r} - \\mathbf{x}^j$\\;\n    \t\t\t\n    \t\t\t$\\xvar{cost} \\leftarrow u - \\frac{1}{j}\\norm{\\mathbf{s}}^2_2- \\frac{1}{n - j}\\norm{\\mathbf{r}}^2_2$\\;\n    \t\t\t\n    \t\t\t\\If {$\\xvar{cost} < \\xvar{best\\_cost}$ and $x^j_i \\neq x^{j+1}_i$}\n    \t\t\t{\n\n    \t\t\t\t$\\xvar{best\\_cost} \\leftarrow \\xvar{cost}$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_coordinate} \\leftarrow i$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_threshold} \\leftarrow x^j_i$\\;\n    \t\t\t}\n    \t\t}\n    \t}\n    \t\\Return $\\xvar{best\\_coordinate}, \\xvar{best\\_threshold}$\\;\n    \t\\caption{\\textsc{Optimal Threshold for $2$-means}}\n    \t\\label{algo:cut}\n    \\end{algorithm}\n    \\end{minipage}\n\\end{figure}\n\n\nIn time $O(d)$ we can calculate $\\mathrm{cost}(p+1)$ and the new centers by using the value $\\mathrm{cost}(p)$ and the previous centers. Throughout the computation we save in memory \n\\begin{enumerate}\n\t\\item Two vectors $\\mathbf{s}^{p}=\\sum_{j=1}^p \\mathbf{x}^j$ and $\\mathbf{r}^{p}=\\sum_{j=p + 1}^n \\mathbf{x}^j$.\n\t\\item Scalar $u=\\sum_{j=1}^n \\norm{\\mathbf{x}^j}_2^2$\n\\end{enumerate}\nWe also make use of the identity:\n\\begin{eqnarray*}\n\t\\mathrm{cost}(p)\n\t&=& u - \\frac{1}{p}\\norm{\\mathbf{s}^{p}}_2^2 - \\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}_2^2.\n\\end{eqnarray*}\nThis identity is correct because \n\\begin{eqnarray*}\n    \\mathrm{cost}(p) &=& \\sum_{j=1}^{p}\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^1(p)}^2_2 + \\sum_{j=p+1}^{n}\\norm{\\mathbf{x}^j-\\boldsymbol{\\mu}^2(p)}^2_2\\\\\n    &=& \\sum_{j=1}^{p} \\norm{\\mathbf{x}^j}^2_2 - 2\\sum_{j=1}^{p}\\inner{\\mathbf{x}^j}{\\boldsymbol{\\mu}^1(p)} + \\sum_{j=1}^{p}\\norm{\\boldsymbol{\\mu}^1(p)}^2_2 + \\\\\n    && \\sum_{j=p+1}^{n} \\norm{\\mathbf{x}^j}^2_2 - 2\\sum_{j=p+1}^{n}\\inner{\\mathbf{x}^j}{\\boldsymbol{\\mu}^2(p)} + \\sum_{j=p+1}^{n}\\norm{\\boldsymbol{\\mu}^2(p)}^2_2\\\\\n    &=& \\sum_{j=1}^{n} \\norm{\\mathbf{x}^j}^2_2 - 2\\inner{\\sum_{j=1}^{p}\\mathbf{x}^j}{\\boldsymbol{\\mu}^1(p)} + \\frac{1}{p}\\norm{\\sum_{j=1}^{p}\\mathbf{x}^j}^2_2 - \\\\\n    && 2\\inner{\\sum_{j=p+1}^{n}\\mathbf{x}^j}{\\boldsymbol{\\mu}^2(p)} + \\frac{1}{n-p}\\norm{\\sum_{j=p+1}^{n}\\mathbf{x}^j}^2_2\\\\\n    &=& \\sum_{j=1}^{n} \\norm{\\mathbf{x}^j}^2_2 - \\frac{2}{p}\\inner{\\mathbf{s}^{p}}{\\mathbf{s}^{p}}+\\frac{1}{p}\\norm{\\mathbf{s}^{p}}^2_2 - \\frac{2}{n-p}\\inner{\\mathbf{r}^{p}}{\\mathbf{r}^{p}}+\\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}^2_2\\\\\n    &=& u - \\frac{1}{p}\\norm{\\mathbf{s}^{p}}^2_2 - \\frac{1}{n-p}\\norm{\\mathbf{r}^{p}}^2_2\n\\end{eqnarray*}\n\n\n By invoking this identity, we can quickly compute the cost of placing the first $p$ points in cluster one and the last $n-p$ points in cluster two. Each such partition can be achieved by using a threshold $\\theta$ between $x_i^p$ and $x_i^{p+1}$. Our algorithm computes these costs for each feature $i \\in[d]$. Then, we output the feature $i$ and threshold $\\theta$ that minimizes the cost. This guarantees that we find the best possible threshold cut.\n\n Overall, Algorithm \\ref{algo:cut} iterates over the $d$ features, and for each feature it sorts the $n$ vectors according to their values in the current feature. Next, the algorithm iterates over the $n$ vectors and for each potential threshold, it calculates the cost by evaluating the inner product of two $d$-dimensional vectors.\n Overall its runtime complexity is $O\\left(nd^2 + nd\\log n\\right)$.\n \n \\subsection{The 2-medians case}\n The high level idea of a finding an optimal 2-medians cut is similar to the 2-means algorithm. The algorithm goes over all possible thresholds. For each threshold, it finds the optimal centers and calculates the cost accordingly. Then, it outputs the threshold cut that minimizes the $2$-medians cost.\n \n  \\begin{figure}[!htb]\n  \\centering\n  \\begin{minipage}{.6\\linewidth}\n        \\begin{algorithm}[H]\n    \t\\SetKwFunction{$2$-medians Optimal Threshold}{$2$-means medians Threshold}\n    \t\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n    \t\\LinesNumbered\n    \t\\Input{%\n    \t\t{$\\mathbf{x}^1, \\ldots, \\mathbf{x}^n$} -- vectors in $\\mathbb{R}^d$\n    \t}\n    \t\\Output{%\n    \t\t\\xvbox{2mm}{$i$} -- Coordinate\\\\\n    \t\t\\xvbox{2mm}{$\\theta$} -- Threshold\n    \t}\n    \t\\BlankLine\n    \t\\setcounter{AlgoLine}{0}\n    \t\n    \t$\\xvar{best\\_cost} \\leftarrow \\infty$\\;\n    \t\n    \t$\\xvar{best\\_coordinate} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t$\\xvar{best\\_threshold} \\leftarrow \\textsc{null}$\\;\n    \t\n    \t\\ForEach {$i \\in [1, \\ldots, d]$}\n    \t{\n    \t\n    \t    $\\boldsymbol{\\mu}^2(0) \\leftarrow \\xfunc{median}(\\mathbf{x}^1, \\ldots \\mathbf{x}^n)$\\;\n    \t\n    \t    $\\xvar{cost} \\leftarrow \\sum_{j=1}^n \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^2(0)}_1$\\;\n    \t\n    \t    $\\mathcal{X} \\leftarrow \\xfunc{sorted}(\\mathbf{x}^1, \\ldots, \\mathbf{x}^n \\text{ by coordinate }i)$\\;\n    \t\n    \t\t\\ForEach {$j \\in [1, \\ldots, n - 1]$} \n    \t\t{ \n\t    \t\t$\\boldsymbol{\\mu}^1(j) \\leftarrow \\xfunc{median}(\\mathbf{x}^1, \\ldots \\mathbf{x}^j)$\\;\n    \t\t\n\t    \t\t$\\boldsymbol{\\mu}^2(j) \\leftarrow \\xfunc{median}(\\mathbf{x}^{j+1}, \\ldots \\mathbf{x}^n)$\\;\n\t    \t\t\n    \t\t\t$\\xvar{cost} \\leftarrow \\xvar{cost} + \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^1(j)}_1 - \\norm{\\mathbf{x}^j - \\boldsymbol{\\mu}^2(j - 1)}_1$\\;\n    \t\t\t\n    \t\t\t\\If {$\\xvar{cost} < \\xvar{best\\_cost}$ and $x^j_i \\neq x^{j+1}_i$}\n    \t\t\t{\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_cost} \\leftarrow \\xvar{cost}$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_coordinate} \\leftarrow i$\\;\n    \t\t\t\t\n    \t\t\t\t$\\xvar{best\\_threshold} \\leftarrow x^j_i$\\;\n    \t\t\t}\n    \t\t}\n    \t}\n    \t\\Return $\\xvar{best\\_coordinate}, \\xvar{best\\_threshold}$\\;\n    \t\\caption{\\textsc{Optimal Threshold for $2$-medians}}\n    \t\\label{algo:2-medians-cut}\n    \\end{algorithm}\n    \\end{minipage}\n\\end{figure}\n \n \\paragraph{Updating cost.} \n To update the cost we need to show how to express $\\mathrm{cost}(p+1)$ in terms of $\\mathrm{cost}(p).$ We know that $\\mathrm{cost}(p+1)$ is equal to \n $$\\mathrm{cost}(p+1)=\\sum_{\\mathbf{x}\\in C_1}\\norm{x-\\boldsymbol{\\mu}^1(p+1)}_1+\\sum_{\\mathbf{x}\\in C_2}\\norm{x-\\boldsymbol{\\mu}^2(p+1)}_1.$$ \n For every feature $i \\in[d]$, there are $n-1$ thresholds to consider. After sorting by this feature, we can consider all splits into $C_1$ and $C_2$, where $C_1$ contains the $p$ smallest points,  and $C_2$ contains the $n-p$ largest points. We increase $p$ from $p=1$ to $p=n-1$, computing the clusters and cost at each step. If $p$ is odd then the median of $C_1$ (i.e., the optimal center of $C_1$) does not change compared to $p-1$. The only contribution to the cost is the point~$\\mathbf{x}$ that moved from $C_2$ to $C_1$. If $p$ is even, then at each coordinate there are two cases, depending on whether the median changes or not. If it changes, then let $\\Delta$ denote the change in cost of the points in $C_1$ that are smaller than the median. By symmetry, the change in the cost of the points that are larger is $-\\Delta$. Thus, the change of the cost is balanced by the points that are larger and smaller than the median. Similar reasoning holds for the other cluster $C_2.$ Therefore, we conclude that moving $\\mathbf{x}$ from $C_2$ to $C_1$ changes the cost by exactly $\\norm{\\mathbf{x}-\\boldsymbol{\\mu}^1(p+1)}_1 - \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^2(p)}_1$. Thus, we have the following connection between $\\mathrm{cost}(p+1)$ and $\\mathrm{cost}(p)$:\n $$\\mathrm{cost}(p+1)=\\mathrm{cost}(p) + \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^1(p+1)}_1 - \\norm{\\mathbf{x}-\\boldsymbol{\\mu}^2(p)}_1.$$\n \n  \\paragraph{Updating centers.} For each $p$, the cost update relies on efficient calculations of the centers $\\boldsymbol{\\mu}^1(p)$ and  $\\boldsymbol{\\mu}^2(p + 1)$. The centers $\\boldsymbol{\\mu}^1(p), \\boldsymbol{\\mu}^2(p)$ are the medians of the clusters at the $p$th threshold. Note that moving from the $p$th thresold to the $(p+1)$th will only change the clusters by moving one vector from one cluster to the other. \n  We can determine the changes efficiently by using $d$ arrays, one for each coordinate. Each array will contain (pointers to) the input vectors $\\mathcal{X}$ sorted by their $i$th feature value. As we move the threshold along a single coordinate, we can read off the partition into two clusters, and we can compute the median of each cluster by considering the midpoint in the sorted list. \n  \n  Overall, this procedure computes the cost of each threshold, while also determining the partition into two clusters and their centers (medians). The time is $O(nd \\log n)$ to sort by each feature, and $O(nd^2)$ to compute $\\mathrm{cost}(p)$ for each $p \\in [n]$ and each feature. Therefore, the total time for the $2$-medians algorithm is \n  $O(nd^2 + nd \\log n).$\n \n \n \n \n\n\\subsection{Related work}\n\\label{sec:related}\n\nThe majority of work on explainable methods considers supervised learning, and in particular, explaining predictions of neural networks and other trained models~\\cite{\nalvarez2019weight, deutch2019constraints, garreau2020explaining, kauffmann2019clustering,\nlipton2018mythos,lundberg2017unified, molnar2019, murdoch2019interpretable, ribeiro2016should, ribeiro2018anchors, rudin2019stop, sokol2020limetree}. In contrast, there is much less work on explainable unsupervised learning. Standard algorithms for $k$-medians\/means use iterative algorithms to produce a good approximate clustering, but this leads to complicated clusters that depend on subtle properties of the data set~\\cite{aggarwal09,arthur2007k, kanungo02, ostrovsky2013effectiveness}. Several papers consider the use of decision trees for explainable clustering~\\cite{bertsimas2018interpretable, fraiman2013interpretable,geurts2007inferring,ghattas2017clustering, liu2005clustering}. However, all prior work on this topic is empirical, without any theoretical analysis of quality compared to the optimal clustering. We also remark that the previous results on tree-based clustering have not considered the $k$-medians\/means objectives for evaluating the quality of the clustering, which is the focus of our work. It is NP-hard to find the optimal $k$-means  clustering~\\cite{aloise2009np, dasgupta2008hardness} or even a very close approximation~\\cite{awasthi2015hardness}. In other words, we expect tree-based clustering algorithms to incur an approximation factor bounded away from one compared to the optimal clustering.\n\nOne way to cluster based on few features is to use dimensionality reduction.\nTwo main types of dimensionality reduction methods have been investigated for $k$-medians\/means. Work on {\\em feature selection} shows that it is possible to cluster based on $\\Theta(k)$ features and obtain a constant factor approximation for $k$-means\/medians~\\cite{boutsidis2009unsupervised, cohen2015dimensionality}. However, after selecting the features, these methods employ existing approximation algorithms to find a good clustering, and hence, the cluster assignments are not explainable. Work on {\\em feature extraction} shows that it is possible to use the Johnson-Lindenstrauss transform to $\\Theta(\\log k)$ dimensions, while preserving the clustering cost~\\cite{becchetti2019oblivious, makarychev2019performance}. Again, this relies on running a $k$-means\/medians algorithm after projecting to the low dimensional subspace. The resulting clusters are not explainable, and moreover, the features are arbitrary linear combinations of the original features.\n\nBesides explainability, many other clustering variants have received recent attention, such as fair clustering~\\cite{ahmadian2020fair, backurs2019scalable, bera2019fair,chiplunkar2020solve,  huang2019coresets,kleindessner2019fair, mahabadi2020individual, schmidt2019fair}, online clustering~\\cite{bhaskara20a, cohen2019online,hess2019sequential, liberty2016algorithm, moshkovitz2019unexpected}, and the use of same-cluster queries~\\cite{ailon2018approximate, ashtiani2016clustering, huleihel2019same, mazumdar2017clustering}. An interesting avenue for future work would be to further develop tree-based clustering methods by additionally incorporating some of these other constraints or objectives.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe dynamics of a quantum system composed by $N$ particles in $\\bR^d$, $d=1,2,3$, interacting via a zero-range, two-body interaction is described by the formal Hamiltonian\n\\beq\\label{hamnd}\n\\mathcal H =- \\sum_{i=1}^N \\f{1}{2m_i} \\Delta_{\\xv_i} + \\sum_{\\underset{i < j}{i,j=1}}^N \\mu_{ij} \\,\\delta(\\xv_i - \\xv_j),\n\\eeq\nwhere $\\xv_i \\in \\bR^d$, $i=1,\\ldots,N$,  is the coordinate of the $i$-th particle,   $m_i$ is the corresponding  mass, $\\Delta_{\\xv_i}$ is the Laplacian relative to  $\\xv_i $, and $\\mu_{ij} \\in \\bR$ is the strength  of the interaction between particles $i$ and $j$. To simplify the notation we set $\\hbar =1$. \nFormal Hamiltonians of the type \\eqref{hamnd} are widely used in physical applications. In particular they are relevant in the study of ultra-cold quantum gases, both in the bosonic and in the fermionic case, in the so-called unitary limit, i.e., for infinite two-body scattering length (see \\cite{bh},\\cite{wc1},\\cite{wc2},\\cite{cmp} and references therein).\n\nThe first step towards a rigorous approach to the analysis of the model is to give \nthe  mathematical definition  of such a Hamiltonian as a self-adjoint operator on the appropriate $L^2$-space. One first notices that  the interaction term in \\eqref{hamnd} is effective  only on the hyperplanes $\\cup_{i<j} \\{\\xv_i = \\xv_j\\}$. This suggests to consider the operator $\\dot{\\mathcal H}_0$ defined as the free Hamiltonian restricted to a domain of smooth functions vanishing in the neighbourhood of each hyperplane $\\{\\xv_i=\\xv_j\\}$. It is easily seen that $\\dot{\\mathcal H}_0$ is symmetric but not self-adjoint and a trivial self-adjoint extension is the free Hamiltonian on its natural domain. Then, by definition,    any non trivial self-adjoint extension of the operator $\\dot{\\mathcal H}_0$ is  a  Hamiltonian for a system of $N$ quantum particles in $\\bR^d$ with a two-body, zero-range interaction. \n \nThe second and more relevant problem  is the concrete construction of such self-adjoint extensions. It turns out that each extension is characterized by a specific generalized boundary condition satisfied by the wave function at the hyperplanes. \nThe two most frequently used techniques for the construction are Krein's theory of self-adjoint extensions and the approximation by regularized Hamiltonians, in the sense of the limit of the resolvent or of the quadratic form. The difficulty of the analysis depends on the dimension $d$. For $d=1$ the problem is greatly simplified by the fact that the interaction term is a small perturbation of the free Hamiltonian in the sense of quadratic forms. For $d=2$ a natural class of Hamiltonians with local zero-range interactions was constructed in \\cite{DFT} exploiting renormalised quadratic forms and it was also shown that such Hamiltonians are all bounded from below. For $d=3$ the problem is more delicate. In order to illustrate the main point, let us consider the special  case $n=2$  where, in the center of mass  reference frame,  one is reduced to study a one-body problem in the relative coordinate $\\xv$  with a  fixed $\\delta$-interaction placed at the origin. In this case the problem is completely understood (see, e.g., \\cite{al}) and the entire class of  Hamiltonians \ncan be explicitly constructed. It turns out that the domain of each Hamiltonian  consists of functions  $ \\psi \\in L^2(\\bR^3) \\cap H^2(\\bR^3 \\setminus \\{0\\})$ which exhibit the following singular behaviour for $  |\\xv| \\rightarrow 0$\n\\beq\\label{bc0}\n\\psi(\\xv) = \\f{q}{|\\xv |} + r  + o(1)\\, ,\t\\hspace{1cm}\t\\text{with}\\;\\; \\;r=\\alpha\\, q\\, ,\n\\eeq\nwhere $q \\in \\ci$ and $\\alpha \\in \\bR$ is a parameter proportional to the inverse of the scattering length. We underline that the relation $r=\\alpha q$ in \\eqref{bc0} should be understood as the generalized boundary condition satisfied at the origin by all the elements of the domain. In the general case $N>2$ the  characterization of all possible self-adjoint extensions of $\\dot{\\mathcal H}_0$ is more involved. However,  a  class of extensions  based on the analogy with the case $N=2$ can be explicitly constructed. More precisely, one considers the so-called Skornyakov-Ter-Martirosyan (STM) extension of $\\dot{\\mathcal H}_0$ which, roughly speaking, is a symmetric operator acting on functions $\\psi \\in L^2(\\bR^{3N}) \\cap H^2( \\bR^{3N} \\setminus \\cup_{i<j} \\{ \\xv_i=\\xv_j\\})$\nsatisfying the following condition for $|\\xv_i - \\xv_j| \\rightarrow 0$:\n\\beq\\label{bcn}\n\\psi(\\xv_1, \\ldots , \\xv_n)= \\f{q_{ij}}{|\\xv_i - \\xv_j|} + r_{ij} + o(1)\\,,\t\\hspace{1cm} \\text{with} \\;\\;\\; r_{ij}= \\alpha_{ij} q_{ij}\\, ,\n\\eeq\nwhere $q_{ij}$ is a  suitable function defined on the hyperplane $\\{\\xv_i = \\xv_j\\}$ and $\\{\\alpha_{ij}\\} $ is  a collection of real parameters labelling the extension.  Noticeably, the boundary condition \\eqref{bcn} defining the STM extension of $\\dot{\\mathcal H}_0$ is a  natural  generalization to the case $N>2$ of the condition \\eqref{bc0} that characterizes  the two-body case.   Unfortunately, unlike \\eqref{bc0}, \\eqref{bcn} does not necessarily define a self-adjoint operator. Indeed, for a system of three identical bosons it was shown in \\cite{fm} that the STM extension is not self-adjoint and all  its self-adjoint extensions are unbounded from below owing to the presence of an infinite sequence of energy levels $E_k$ going to $-\\infty$ for $k \\rightarrow \\infty$. In \\cite{MM} this result was generalized  to the case of three distinguishable  particles with different masses. This kind of instability  is known in the literature as the Thomas effect.  \nIt should be stressed that the Thomas effect is strongly related to the well-known Efimov effect (see, e.g., \\cite{bh}, \\cite{adfgl}, \\cite{ahkw}) even if, to our knowledge, a rigorous mathematical investigation of this  connection is still lacking. We also mention that if, instead of \\eqref{bcn},   one introduces a ``non-local'' boundary condition on the hyperplanes then it is possible to construct a positive Hamiltonian and to study its stability properties for $N$ large (see, e.g., \\cite{fs}). In this paper we do not consider this kind of Hamiltonians.\n \nIt is reasonable to expect that the Thomas effect does not occur if the Hilbert space of states is suitably restricted, e.g., introducing  symmetry constraints on the wave function. A remarkably important constraint is antisymmetry. In fact, a wave function that is antisymmetric under exchange of coordinates of two particles necessarily vanishes at the coincidence points of such two particles, thus making their mutual zero-range interaction ineffective. Analogously, in a mixture of fermions of different species subject to pairwise zero-range interaction, fermions of the same species cannot ``feel'' mutual zero-range interaction and therefore the interaction term in the Hamiltonian is less singular.\n\n\nIn this paper we consider the simplified model consisting of  $N$ identical fermions, with unit mass, and a different particle with mass $m$, interacting with the fermions through a zero-range potential. For such a model only partial results are available and it is remarkable that they strongly depend on the parameters $N$ and $m$. \n\nConcerning the physical literature, we mention that for $N=2$ it is known (see, e.g., \\cite{bh} and references therein)  that for $m < 0.0735= (13.607)^{-1}$ the Thomas effect is present while for $m> 0.0735$ the STM extensions are expected to be bounded from below. More recently (see \\cite{cmp}), it was shown by means of analytical and numerical arguments that in the case $N=3$ the Thomas effect occurs for $m< 0.0747= (13.384)^{-1}$. This, in particular, indicates that for $0.0735<m< 0.0747$ there is a sequence of  genuine four-body bound states with energy going to $-\\infty$.\n\nFrom the rigorous point of view, the case $N=2$ was investigated first in \\cite{m1} and \\cite{MS}, where it was proved that the STM extension is self-adjoint if $m=1$. In \\cite{Sh}, the existence of a critical mass  $m^*(2) \\simeq 0.0735$ was shown such that for $m<m^*(2)$ any STM extension -- more precisely its restriction to the subspace of angular momentum $l=1$ -- is not self-adjoint and its self-adjoint extensions are unbounded from below. Therefore, the system is unstable and the Thomas effect occurs. This result was extended in \\cite{FT}, where  it was shown that the quadratic form associated with the STM extension, restricted to the subspace of angular momentum $l$, is unbounded from below if an explicit condition on $m$ and $l$ is satisfied. Finally, in the case $N\\leq 4$ and $m$ sufficiently large it was shown in \\cite{m3} that the STM extension is self-adjoint.\n\nIn the present work we study the problem for generic $N$ and $m$ and following the line of \\cite{DFT} we construct a renormalised quadratic form $\\mathcal F_{\\alpha}$, $\\alpha \\in \\bR$, which is naturally associated with the STM symmetric extension  $H_{\\alpha}$. Here $\\alpha$ is the value that all the parameters $\\alpha_{ij}$ labelling the STM extension must be equal to as a consequence of the fermionic symmetry. \nNote that in the bosonic case the quadratic form would differ in a sign in front the non-diagonal term (see the remark after \\eqref{oqform}).\n\nThe first question we address is a non-trivial sufficient condition for the \\emph{stability} of the model. As a first main result (Theorem \\ref{clbou}) we prove that for any $N$ there is  a value of the mass $m^*(N)>0$  such that $\\mathcal F_{\\alpha}$ is closed and bounded from below if $m>m^*(N)$. This implies that $\\mathcal F_{\\alpha}$ is the quadratic form of a \\emph{unique} self-adjoint extension of the STM operator $H_{\\alpha}$, and this extension is bounded from below. It therefore describes a stable system, where the Thomas effect does not occur.\n\nSuch a critical mass  was first conjectured in \\cite{m2}  and is precisely the unique root of an explicit equation (see \\eqref{Lambda} and \\eqref{La=1} in Section \\ref{main results: sec}). It turns out that $m^*(N)$ is increasing with $N$ and that the condition $m>m^*(N)$ guarantees the stability also in the limit of infinitely many fermions, provided that the mass of the extra particle scales as $m \\propto N$. \n\n\n\nThe second question we address is a sufficient condition for the \\emph{instability} of the model. This can be seen by plugging suitable trial functions into $\\mathcal F_{\\alpha}$. An attempt in this direction is in \\cite[Section 7]{DFT}, but with trial functions that do not satisfy the fermionic symmetry: thus, the result stated there on the unboundedness from below of the quadratic form for $m=1$ and $N$ sufficiently large cannot be considered valid. Our second main result (Theorem \\ref{ub1}) fills in this gap and we prove that for any $N\\geq 2$ the quadratic form $\\mathcal F_{\\alpha}$ is unbounded from below for $m<m^*(2)$. In analogy with the bosonic case, we expect that in such a case $H_{\\alpha}$ is not self-adjoint and all its self-adjoint extensions are unbounded from below\n\nLet us make a few remarks on the above-mentioned results. First, we emphasize that in the case $N=2$ we fully characterize stability: the model is stable for $m>m^*(2)$ and unstable $m<m^*(2)$. Whereas the latter was already found in \\cite{Sh} by means of the theory of self-adjoint extensions, the former is proved here for the first time.\n\nSecond, if $N>2$ we expect the condition $m>m^*(N)$  to be far from optimal for stability. This is due to the crucial role played by the restriction to antisymmetric wave functions (see the discussion in Section \\ref{instability: sec}) so that the system might  be stable also if our condition  is violated.  \n\nThird, the fact that the $(N\\!+\\!1)$-particle system is unstable at least when $m$ is below the \\emph{same} threshold $m^*(2)$ for the instability of the $(2\\!+\\!1)$-particle system has a rather natural interpretation: the instability of a subsystem made of two fermions plus the different particle is responsible for the instability of the whole system. \n\n\nWe want to mention that in the final stage of the preparation of this work we became aware of a recent paper  \\cite{m4} where the case of two fermions plus a different particle is studied using the theory of self-adjoint extensions. We believe that a  comparison with our methods and  results would be of great interest for the further developments of the subject.\n\nThe paper is organized as follows. In Section \\ref{main results: sec} we introduce the renormalised quadratic form $\\mathcal F_{\\alpha}$ and the STM extension $H_{\\alpha}$ and we formulate our main results. In Section \\ref{stability: sec} we give the proof  of Theorem \\ref{clbou}. In Section \\ref{instability: sec} we give the proof of Theorem \\ref{ub1}. In the Appendix we  briefly outline the formal renormalisation  procedure to derive $\\form$.\n\n\\vs\n\nFor the convenience of the reader, we collect here  some useful notation that will be used throughout the paper. We use the notation $ \\ldf(\\R^{d}) $ (resp. $ \\hf(\\R^{d}) $, $ \\hcf(\\R^{d}) $, etc.)  for the space containing totally antisymmetric functions belonging to $ L^2(\\R^{d}) $ (resp. $ H^1(\\R^{d}) $, $ H^{-1\/2}(\\R^{d}) $, etc.). We often use the short-hand notation $\\|\\cdot\\|_{L^2}$, $\\|\\cdot\\|_{\\ldf}$, etc. for the associated norms $\\|\\cdot\\|_{L^2(\\bR^d)}$, $\\|\\cdot\\|_{\\ldf(\\R^{d})}$, etc.\n\\newline\nFor a vector $\\xv \\in \\bR^3$ we set $x = |\\xv|$. Moreover we define $ \\Kv := \\lf( \\kv_2,\\ldots , \\kv_{N-1} \\ri) $ and, for $ i = 1, \\ldots, N $,\n\t\\bdm\n\t\t\\breve{\\kkv}_i := \\lf( \\kv_1, \\ldots, \\kv_{i-1}, \\kv_{i+1}, \\ldots, \\kv_N \\ri).\n\t\\edm\nFor any $f \\in L^2(\\bR^d)$ the Fourier transform is defined by $ \\hat{f}(\\kv)=(2 \\pi)^{-d\/2} \\int_{\\bR^d} \\! \\diff \\xv \\, e^{-i \\kv \\cdot \\xv} f(\\xv)\\,.$\n\\newline\nThe functions $G_{\\lambda}$,  $\\pot$, $L_{\\lambda}$, with $\\lambda >0$, and $D(\\Kv)$ are defined in \\eqref{Green function}, \\eqref{pot xi}, \\eqref{Lla} and \\eqref{Dk} respectively.\n\n\n\n\n\n\n\n\n\n\\vs\n\n\\section{Main results}\n\\label{main results: sec}\n\nIn this section we introduce the quadratic form $\\form$,  the STM extension $H_{\\alpha}$ and we formulate our main results.\n\n\n\\subsection{The quadratic form $\\form$}\n\n\n\\n\nFor a three dimensional quantum system  composed by $N$ identical fermions, with mass  one, plus a different particles, with mass $m$, with a two-body zero-range interaction    the formal many-body Hamiltonian is \n\\beq\n\t\\label{formal Hamiltonian}\n\t\\tilde{H} : = - \\frac{1}{2 m}\\Delta_{\\xv_0} - \\f{1}{2} \\sum_{i =1}^N \\Delta_{\\xv_i} + \\mu   \\sum_{i = 1}^N  \\delta(\\xv_0 - \\xvi)\\, ,\n\\eeq\nwhere $ \\xv_i \\in \\RT $, $ i = 0, \\ldots, N $, and $\\mu \\in \\bR$. Introducing the centre of mass and relative coordinates \n\\beq\n\t\\lf\\{\n \t\t\\begin{array}{ll}\n\t\t\t\\mathbf{X} : =\\displaystyle\\frac{1}{m+N} \\Big( m\\xv_0 + \\sum_{i=1}^N \\xv_i \\Big), \t&\t\\mbox{}\t\\\\\n\t\t\t\\yv_i  : = \\xv_0 - \\xv_i ,\t\t&\t\\mbox{for} \\:\\:\\: i = 1,\\ldots N\\,,\n\t\t\\eay\n\t\\ri.\n\\eeq\none obtains\n\\beq\n\t\\tilde{H} = \\hcm + \\tx\\frac{m+1}{2m}  H \n\\eeq\nhere $ \\hcm : =- [2(m+N)]^{-1} \\Delta_{\\mathbf{X}} $ and\n\\beq\n\t\\label{cm Hamiltonian}\n\tH : = - \\sum_{i=1}^N \\Delta_{\\yv_i} -  \\frac{2}{m+1} \\sum_{i < j} \\nabla_{\\yv_i}  \\cdot \\nabla_{\\yv_j} +  \\mu \\sum_{i = 1}^N \\delta(\\yvi)\\,,\n\\eeq\n$\\nabla_{\\yv_i} $ denoting the gradient with respect to $\\yv_i$.  \nWe also introduce the free Hamiltonian\n\\beq\n\t\\label{free Hamiltonian}\n\t\t\\dom(H_0) := H^2_{\\mathrm{f}}(\\bR^{3N}),\t\\hspace{1cm}\tH_0 : = - \\sum_{i=1}^N \\Delta_{\\yv_i} -  \\frac{2}{m+1} \\sum_{i < j} \\nabla_{\\yv_i}  \\cdot \\nabla_{\\yv_j}\\,,\n\\eeq\nand its restriction to functions vanishing in a neighbourhood of the hyperplanes \n$ \\{\\yv_i = 0 \\}$\n\\beq\n\t\\label{free2 Hamiltonian}\n\\dom(\\dot{H}_0)  : = \\!\\bigg\\{ \\psi \\in \\hdf(\\R^{3N}) \\: \\bigg| \\! \\int_{\\R^3} \\!\\!\\diff \\kv_i \\: \\hat\\psi(\\kv_1, \\ldots, \\kv_N) = 0 \\,, \\;\\;\\;\\;i=1,\\ldots ,N \\bigg\\},\t\\hspace{1cm}\t\\dot{H}_0 : = H_0\\big|_{\\dom(\\dot{H}_0)}\\,.\n\\eeq\n\nIn order to give a rigorous meaning to the formal expression \\eqref{cm Hamiltonian} as a self-adjoint operator in  $ \\ldf(\\R^{3N}) $, one can use Krein's theory to construct self-adjoint extensions of the operator  \\eqref{free2 Hamiltonian} (see \\cite{m3}). Here, instead, we follow a different approach and we investigate the quadratic form associated with the expectation value $ \\bra{\\psi} H \\ket{\\psi} $. However, because of the singularity of the ``potential'' $ \\delta(\\yv_i) $ the quadratic form has to be defined via a renormalisation procedure in the Fourier space. The idea  of the construction is given in the Appendix (see also  \\cite{DFT},\\cite{FT}) and here we only give   the final result.\n\n\\n\nLet us denote for any $\\lambda>0$\n\\beq\n\t\\label{Green function}\n\t\\green(\\kv_1, \\ldots, \\kv_N) : = \\bigg[ \\sum_{i=1}^N k_i^2 + \\frac{2}{m+1}\\sum_{i < j} \\kv_i \\cdot \\kv_j + \\la \\bigg]^{-1}\\,,\n\\eeq\n\\beq\\label{Lla}\nL_{\\lambda}(\\kv_1,\\ldots,\\kv_{N-1}) := 2 \\pi^2 \\bigg( \\frac{m(m+2)}{(m+1)^2} \\sum_{i = 1}^{N-1} k_i^2 + \\frac{2m}{(m+1)^2} \\sum_{i <j} \\kv_i \\cdot \\kv_j + \\la \\bigg)^{1\/2}\\,,\n\\eeq\nand, for any ``charge'' $\\xi \\in   \\hcf(\\R^{3N-3})  $,  let us  denote the ``potential'' produced by $\\xi$ in the Fourier space by\n\\beq\n\t\\label{pot xi}\n\t\\lf( \\widehat{\\pot \\xi} \\ri)(\\kv_1, \\ldots, \\kv_N) : = \\sum_{i=1}^N (-1)^{i+1} \\green(\\kv_1, \\ldots, \\kv_N) \\, \\hat\\xi( \\breve{\\kkv}_i)\\,\n\\eeq\nwhere  $\\breve{\\kkv}_i := (\\kv_1, \\ldots, \\kv_{i-1}, \\kv_{i+1}, \\ldots, \\kv_N) $.  Then the quadratic form $\\form$  in the Hilbert space $\\ldf(\\R^{3N})$ is defined as follows\n\\beq\n\t\\label{form domain}\n\t\\dom(\\form) : = \\lf\\{ \\psi \\in \\ldf(\\R^{3N}) \\: \\Big| \\: \\exists \\xi \\in \\dom(\\qform) \\: \\;\\; \\mathrm{s.t.} \\;\\; \\: \\phi^{\\la} : = \\psi - \\pot \\xi \\in \\hf(\\R^{3N}) \\ri\\},\n\\eeq\n\\beq\n\t\\label{form}\n\t\\form[\\psi] : = \\F_0[\\phi^{\\la}] + \\la \\| \\phi^{\\la} \\|_{L^2(\\R^{3N})}^2 - \\la \\lf\\| \\psi \\ri\\|_{L^2(\\R^{3N})}^2 + N \\qform\\lf[\\xi\\ri],\n\\eeq\nwhere $\\lambda >0$, $ \\F_0[\\phi] : = \\bra{\\phi} H_0 \\ket{\\phi} $ and  $\\Phi_{\\alpha}^{\\lambda}$ is the following form on the charge $ \\xi $  \n\\beq\\label{domqform}\n \\dom(\\qform) := \\hcfpiu   (\\R^{3N - 3})\\,,\n\\eeq\n\\beq\n\t\\label{qform}\n\t\\qform[\\xi] : = \\dqform[\\xi] + \\oqform[\\xi]\\,,\n\\eeq\n\\beq\t\\label{dqform}\n\t\\dqform[\\xi] : = \\int_{\\R^{3N-3}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\lf[ \\al +\t\t\n\tL_{\\lambda}(\\kv_1,\\ldots,\\kv_{N-1}) \\ri],\n\\eeq\n\\bml{\n\t\\label{oqform}\n\t\\oqform[\\xi] : = (N-1) \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kv_2 \\cdots \\diff \\kv_{N-1} \\:\t \\hat\\xi^*(\\sv, \\kv_2, \\ldots, \\kv_{N-1}) \\hat\\xi (\\tv, \\kv_2, \\ldots, \\kv_{N-1}) \\cdot \t\\\\\n\t\\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) .\n}\n\n\\vspace{0.1cm}\n\n\\n\nWe notice that for $\\xi \\in  \\dom(\\qform)$ we have $\\pot \\xi \\in  \\ldf(\\R^{3N})$ and $\\pot \\xi \\notin \\hf(\\R^{3N})$. Therefore the decomposition in \\eqref{form domain} is meaningful. Moreover, as we shall see in Section \\ref{stability: sec}, the form $\\qform$ is well-defined on $\\dom (\\qform)$.\n\nIt is worth mentioning that the fermionic constraint implies not only that the wave function is totally antisymmetric but also that the form on the charges \\eqref{qform} differs from the bosonic case by a sign in front of the off-diagonal part \\eqref{oqform}. This fact results in a weaker effective interaction among the fermions and the stability problem is qualitatively different.\n\nIn order to formulate our main results on the form $\\form$ we first  introduce our definition of stability parameter. Let us consider the following function\n\\beq\\label{Lambda}\n\\Lambda (m,N) : =   2\\pi^{-1} (N-1)    (m+1)^2 \\bigg[ \\f{1}{ \\sqrt{m(m+2)}} - \\arcsin\\bigg(\\f{1}{m+1}\\bigg) \\bigg]\\,.\n\\eeq\nIt is easy to check that, for $N$ fixed, the function $\\Lambda(m,N)$ is decreasing in $m$ and \n\\beq\n\t\\lim_{m\\rightarrow 0} \\Lambda(m,N)=\\infty,\t\\hspace{1cm}\t\\lim_{m\\rightarrow \\infty} \\Lambda(m,N)=0\\,.\n\\eeq  \nThen we have\n\n\\begin{definition}[Stability parameter $m^*(N)$]\n\t\\mbox{}\t\\\\ \nFor $N$ fixed, we define $m^*(N)$ as the unique solution to the equation \n\\beq\\label{La=1}\n\\Lambda(m,N)=1\\,.\n\\eeq\n\\end{definition}\n\\vspace{0.2cm}\n\\n\nIf we define $\\theta := \\arctan \\sqrt{1+\\f{2}{m}}$,  where $\\theta \\in (\\f{\\pi}{4}, \\f{\\pi}{2})$, a direct computation shows that equation \\eqref{La=1} can be equivalently written as\n\\beq\\label{teta}\n \\cot 2 \\theta + 2 \\theta  -\\f{\\pi}{2}\\bigg( 1 - \\f{1}{N-1} \\cos^2 2 \\theta \\bigg)=0\\,.\n\\eeq\nWe remark that   \\eqref{teta}  reduces for $N=2$ to the equation found for the critical mass in  \\cite[p. 12871, note 10]{pc}. \nWe also notice that $m^*(N)$ is positive and increasing with $N$. In particular,  the condition $\\Lambda(m,N) <1$ is equivalent to  $m>m^*(N)$. This condition is crucial to guarantee closure and boundedness from below of the form $\\form$.\n\n\\begin{theorem}[Stability for $ m > m^*(N)$]\n\\label{clbou}\n\\mbox{}\t\\\\\nLet $N\\geq 2$ and  $m>m^*(N)$.  Then  the quadratic form $\\form$ is closed and bounded from below. In particular,  it is positive for $\\alpha \\geq 0$ and \n\\beq\\label{inff}\n\\form[\\psi] \\geq -  \\f{\\alpha^2}{4 \\pi^4 \\big(1\\!-\\!\\Lambda(m,N) \\big)} \\, \\|\\psi\\|^2_{L^2},\t\\hspace{1cm} \\psi \\in \\dom(\\form)\\,,\n\\eeq\nfor $\\alpha <0$.\n\\end{theorem}\n\n\\n\nThe proof   will be given in Section \\ref{stability: sec}. \n\nConcerning the instability problem, our  result is the following.\n\n\\begin{theorem}[Instability for $ m < m^*(2) $]\n\\label{ub1}\n\\mbox{}\t\\\\\nLet $N\\geq 2$ and $m<m^*(2)$. Then the quadratic form $\\form$ is unbounded from below for any $\\alpha \\in \\bR$.\n\\end{theorem}\n\n\\n\nThis theorem generalizes the result obtained for $N=2$ in \\cite{FT} and it is proved in Section \\ref{instability: sec}. In the case $N=2$, Theorems \\ref{clbou} and \\ref{ub1} show that the system is stable for $m>m^* (2)$ and unstable for $m<m^*(2)$, therefore our analysis is complete.  When  $N>2$ the problem remains open: no rigorous result is available for $m^*(2)<m<m^*(N)$. \n\n\n\\n\n\n\\vs\n\n\\subsection{The STM extension $H_{\\alpha}$}\n\nHere we recall the standard definition of the STM extension $H_{\\alpha}$ and its connection with the quadratic form $\\form$. Moreover, as a direct consequence of Theorem \\ref{clbou},  we prove our main result on $H_{\\alpha}$. \n\nAs we mentioned in the introduction, $H_{\\alpha}$  is    a distinguished  symmetric  extension of the operator  \\eqref{free2 Hamiltonian}. For the details of the construction we refer to \\cite{m3} and here we only give  the definition\n\\bml{\n\t\\label{domain Ha 2}\n\t\\dom(H_{\\al}) : = \\bigg\\{ \\psi \\in \\ldf(\\R^{3N}) \\: \\bigg| \\: \\exists \\, \\xi \\in H^{3\/2}_{\\mathrm{f}} (\\bR^{3N-3}) \\: \\;\\mathrm{s.t.} \\;\\: \\phi_{\\la} := \\psi - \\pot \\xi \\in \\hdf(\\R^{3N}),\t\\\\\n\t\\int_{\\R^3} \\diff \\kv_i \\: \\hat\\phi_{\\la}(\\kv_1, \\ldots, \\kv_N) =  (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i), \\, i = 1, \\ldots, N \\bigg\\}\\,,\n}  \n\\beq\n\t\\label{action Ha}\n\t(H_{\\al} + \\la) \\psi = (H_0 + \\la) \\phi_{\\la}\\,,\n\\eeq\nwhere $\\lambda>0$\n and\n\\beq\t\\label{operator A}\n(\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i) \n : = (-1)^{i+1} \\bigg[ \\alpha + L_{\\lambda}(\\breve{\\kkv}_i) \\bigg] \\hat{\\xi}(\\breve{\\kkv}_i) - \\!\\!  \\sum_{j=1, j\\neq i}^N \\!\\!  (-1)^{j+1}  \\!\\!  \\int_{\\R^3} \\diff \\kv_i \\:  G_{\\lambda} (\\kv_1,\\ldots, \\kv_N) \\, \\hat{\\xi}(\\breve{\\kkv}_j) \\,.\n\\eeq\nThe last equality in \\eqref{domain Ha 2} should be understood as the boundary condition satisfied by any $\\psi \\in \\dom(H_{\\alpha})$. In fact, \nby a straightforward computation, one verifies that such equality implies  the following asymptotic condition for $R\\rightarrow \\infty$ \n\\beq\n \\int_{k_i <R} \\diff \\kv_i \\: \\hat{ \\psi}(\\kv_1,\\ldots, \\kv_N)= 4 \\pi R \\, (-1)^{i+1} \\hat{\\xi}(\\breve{\\kkv}_i) + \\alpha \\,  (-1)^{i+1} \\hat{\\xi}(\\breve{\\kkv}_i) +o(1)\n \\eeq\nand this is exactly the standard boundary condition, formulated in the Fourier space,  characterizing the STM extension (see \\cite{m3}).\n\nThe next step is to establish the connection with the form $\\form$. For any $\\psi \\in \\dom (H_{\\alpha})$ one has\n\\begin{eqnarray}\n&& \\bra{\\psi} H_{\\alpha} +\\lambda  \\ket{\\psi}  = \\bra{\\psi} H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} = \\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} + \n \\bra{\\pot \\xi } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}}\\nonumber\\\\\n &&=  \\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} +\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) \\int_{\\bR^3} \\! \\diff \\kv_i \\, \\hat{\\phi}_{\\lambda} (\\kv_1,\\ldots,\\kv_N)\\nonumber\\\\\n &&=\\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} +\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i)\\,.\n\\end{eqnarray}\n\n\n\\n\nExploiting \\eqref{operator A} and the antisymmetry of $\\hat{\\xi}$  we have\n\\bml{ \\label{afi}\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i) \\\\ \n = \\sum_{i=1}^N \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; |\\hat{\\xi}(\\breve{\\kkv}_i)|^2 \\big[ \\alpha + L_{\\lambda}(\\breve{\\kkv}_i) \\big] \n -\\sum_{i,j=1, i\\neq j}^N \\!\\!\\!\\! (-1)^{i+j} \n \\int_{\\bR^{3N}}\\!\\! \\diff \\kv_1\\cdots \\diff \\kv_N \\,  \\hat\\xi^* (\\breve{\\kkv}_i) \\hat\\xi (\\breve{\\kkv}_j) \\, G_{\\lambda} (\\kv_1, \\ldots , \\kv_N) \\\\\n = N \\dqform [\\xi] + N(N-1) \\oqform [\\xi]\\,.\n}\nTherefore we conclude\n\\beq\\label{hfa}\n    \\bra{\\psi} H_{\\alpha} \\ket{\\psi}    =     \\form \\lf[\\psi \\ri], \t\\hspace{1cm}\t\\psi \\in \\dom(H_{\\al})\\,.\n\\eeq\n\n\\n\nFrom the above relation and Theorems \\ref{clbou}, \\ref{ub1},  we obtain the following result.\n\n\\begin{theorem}[Self-adjointness and boundedness from below]\n\\label{thhal}\n\\mbox{}\t\\\\\nOne has the  following two possible alternatives:\n\\ben[(i)]\n\t\\item If $N\\geq 2$ and $m>m^*(N)$, then $\\form$ defines a unique self-adjoint and bounded from below extension $\\hat{H}_{\\alpha}$ of the operator $H_{\\alpha}$. In particular, $\\hat{H}_{\\alpha}$ is positive for $\\alpha \\geq 0$ and \n\\beq\\label{infsp}\n\\inf \\sigma(\\hat{H}_{\\alpha}) \\geq - \\f{\\alpha^2}{4 \\pi^4 \\big(1 - \\Lambda(m,N) \\big)}, \t\\hspace{1cm}\t\\text{for} \\;\\;\\;\\alpha <0.\n\\eeq \n\t\\item  If $N \\geq 2$ and $m<m^*(2)$, then, for any $\\alpha \\in \\bR$, no self-adjoint extension $ \\hat{H}_{\\alpha} $ of $ H_{\\alpha} $ such that $ \\dom(\\hat{H}_{\\alpha}) \\subset  \\dom(\\form) $ can be both  self-adjoint and bounded from below.\n\\een\n\n\\end{theorem}\n\n\\begin{proof}\nThe assertion (i) immediately follows from Theorem \\ref{clbou} and the variational characterization of the infimum of the spectrum of a self-adjoint operator.\n\nThe converse (ii) can be obtained by exploiting the Birman-Krein-Vishik theory of positive self-adjoint extensions (see, e.g., \\cite{simon}, \\cite{P1},\\cite{P2}). We omit the details and refer to \\cite[Proposition 4.1]{FT} where an analogous results was proven.\n\\end{proof}\n\n\\vs\n\n\\n\nWe remark that, in the case $N\\geq 2$ and $m<m^*(2)$, an analogy with the bosonic case suggests that any self-adjoint extension of $H_{\\alpha}$ is unbounded from below and the Thomas effect occurs, although such a complete characterization is beyond our purposes. The most natural tool to approach the analysis of this case  is indeed the theory of self-adjoint extensions (see, e.g., \\cite{Sh},\\cite{m4}). \n\n\n\\n\nWe conclude this section with a brief comment relative to the special case $N=2$, $m>m^*(2)$. In particular we want to give the explicit characterization of our Hamiltonian $\\hat{H}_{\\alpha}$ in order to make more transparent a possible comparison of our result with the result in \\cite{m4}.\n\n\\n\nIn Section 3 we shall prove the estimate \\eqref{cC}, which in particular implies  that the form  $\\Phi^{\\lambda}_{\\alpha}$, defined on $H^{1\/2}(\\R^3)$, is closed and positive for $\\lambda$ sufficiently large. Therefore it defines a positive, self-adjoint operator $\\Gamma^{\\lambda}_{\\alpha}$, $\\dom (\\Gamma^{\\lambda}_{\\alpha})$ in $L^2(\\R^3)$ explicitly given by\n\\bdn\\label{opGa}\n\\dom(\\Gamma^{\\lambda}_{\\alpha})= \\!\\lf\\{ \\xi \\in H^{1\/2}(\\R^3)\\; \\Big| \\; \\Gamma^{\\lambda}_{\\alpha} \\hat\\xi \\in L^2(\\R^3) \\ri\\},\t\\nonumber\t\\\\\n\\lf(\\Gamma^{\\lambda}_{\\alpha} \\hat\\xi \\ri)(\\qv)\\!= [\\alpha + L_{\\lambda}(\\qv) ] \\hat\\xi(\\qv) + \\!\\!\\int_{\\R^3}\\!\\!\\! \\diff \\pv \\, G_{\\lambda}(\\pv, \\qv) \\hat\\xi(\\pv).\n\\edn\nExploiting this fact and following the same line of \\cite[Section5]{DFT}, we obtain\n\\bml{\n\t\\label{domain Ha3}\n\t\\dom(\\hat{H}_{\\al})  = \\bigg\\{ \\psi \\in \\ldf(\\R^{6}) \\: \\bigg| \\: \\exists \\, \\xi \\in \\dom(\\Gamma^{\\lambda}_{\\alpha})  \\: \\;\\mathrm{s.t.} \\;\\: \\phi_{\\la} := \\psi - \\pot \\xi \\in \\hdf(\\R^{6}),\t\\\\\n\t\\int_{\\R^3} \\diff \\pv \\: \\hat\\phi_{\\la}(\\pv, \\qv)  =  (\\Gamma^{\\lambda}_{\\alpha}  \\hat\\xi ) (\\qv) \\bigg\\}\\,,\n}  \n\\beq\n\t\\label{action Ha3}\n\t(\\hat{H}_{\\al} + \\la) \\psi = (H_0 + \\la) \\phi_{\\la}.\n\\eeq\nWe notice that the above operator differs from the operator \\eqref{domain Ha 2}, \\eqref{action Ha} only in a larger  class of admissible charges $ \\xi $, i.e., the domain $ \\dom(\\Gamma^{\\lambda}_{\\alpha}) $ strictly contains $ H^{3\/2}(\\R^3) $. We also underline that the boundary condition satisfied on the hyperplanes by an element of \\eqref{domain Ha3} is the standard STM boundary condition.\n\n\n\n\n\\section{Closure and boundedness from below of $ \\form $}\n\\label{stability: sec}\n\nThe proof of Theorem \\ref{clbou} is based on a careful estimate from below and from above of the form $\\qform$ on the charge $\\xi$.  \nIf $ N > 2 $ (with an obvious modification in the case $ N = 2 $) we rewrite \nboth the diagonal and the off-diagonal terms  of $\\Phi_{\\alpha}^{\\lambda}$, defined in  \\eqref{dqform}, \\eqref{oqform},  in a more manageable form (see \\cite{m3}), by introducing the change of coordinates\n\\bdm\n\t\\sv \\longrightarrow \\siv : = \\sv + \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i, \t\\hspace{1cm} \t\\tv \\longrightarrow \\tav : = \\tv + \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i.\t\t\n\\edm\nThen\n\\beq\n\t\\label{dqform 1}\n\t\\dqform[\\xi] = \\alpha \\, \\| \\xi\\|_{L^2(\\R^{3N-3})\n\t}^2 +\t\n\t2 \\pi^2 \\int_{\\R^{3N-3}} \\diff \\sigma \\diff \\Kv \\:\n\t\\big| \\xit(\\siv, \\Kv\n\t) \\big|^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} \\sigma^2 + \\dk+ \\la}\\,,\n\\eeq\n\\beq\n\t\\label{oqform 1}\n\t\\oqform[\\xi] = (N-1) \\int_{\\R^{3N}} \\diff \\siv \\diff \\tav \\diff \\Kv \\:\n\t\\xit^*(\\siv, \\Kv\n\t) \\: \\xit (\\tav, \\Kv\n\t)\n\t\\lf( \\sigma^2 + \\tau^2 + \\tx\\frac{2}{m+1} \\siv \\cdot \\tav + \\dk + \\la \\ri)^{-1}\\,,\n\\eeq\nwhere\n\\beq\n\\Kv := \\kv_2, \\ldots , \\kv_{N-1}\\,,\n\\eeq\n\\beq\n\t\\label{xit}\n\t\\xit \\lf(\\siv, \\Kv\n\t\\ri) := \\hat\\xi\\bigg(\\siv - \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i,  \\kv_2, \\ldots, \\kv_{N-1}\n\t\\bigg)\\,,\n\\eeq\n\\beq\n\t\\label{Dk}\n\t\\dk : =\n\t\\frac{m}{(m+1)(m+2)} \\bigg( (m+3) \\sum_{i=2}^{N-1} k_i^2 + 2 \\sum_{i < j} \\kv_i \\cdot \\kv_j \\bigg)\\,.\n\\eeq\nNotice that $\\dk$ satisfies the bound (see \\eqref{el12} in the following)\n\\beq\n\t\\label{estimate Dk}\n\t\\frac{m}{m+1} \\sum_{i=2}^{N-1} k_i^2 \\leq \\dk \\leq \\frac{m(m+N+1)}{(m+1)(m+2)} \\sum_{i=2}^{N-1} k_i^2\\,.\n\\eeq\nLast, setting \n\\beq\n\t\\siv := \\sqrt{\\dk + \\la} \\: \\pv,\t\\hspace{1cm}\t\\tav : = \\sqrt{\\dk + \\la} \\: \\qv\\,,\n\\eeq\nand\n\\beq\n\t\\label{charge Qk}\n\tQ_{\\kkv}(\\pv) : = (\\dk + \\la)^{3\/4} \\: \\xit \\lf(\\sqrt{\\dk + \\la} \\: \\pv, \\Kv\n\t\\ri)\\,,\n\\eeq\nwe obtain\n\\beq\n\t\\label{qform 1}\n\t\\Phi_0^{\\lambda}[\\xi]  = \\qform[\\xi] -  \\alpha \\| \\xi \\|_{L^2(\\R^{3N-3})}^2 =  \\int_{\\R^{3N-6}} \\diff \\K\n\t\\sqrt{\\dk \\!+\\! \\la} \\;\\,  F_1 \\! \\lf[ Q_{\\kkv} \\ri]\\,,\n\\eeq\nwhere for any $\\zeta \\geq 0$ we introduced the quadratic form in $L^2(\\R^3)$\n\\beq\n\t\\label{fform}\n\t\\dom(F_{\\zeta})\\!:=\\!\\dom(F_1)\\!= \\! \\bigg\\{ \\! f \\! \\in \\! L^2(\\R^3)\\,\\bigg|\\! \\int_{\\R^3} \\!\\! \\diff \\pv \\sqrt{p^2 \\!+\\!1} \\, |f(\\pv)|^2  <\\infty \\bigg\\},\t\\quad\tF_{\\zeta} \\lf[f\\ri\n\t: = \\dform_{\\zeta} \\lf[f\\ri]\n\t+ \\oform_{\\zeta}\\lf[f\\ri]\n\\eeq\nand \n\\beq\n\t\\label{dform}\n\t\\dform_{\\zeta} \\lf[f \\ri]\n\t= 2 \\pi^2 \\int_{\\R^{3}} \\diff \\pv  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} p^2 + \\zeta} \\: \\lf| f\n\t(\\pv) \\ri|^2 \\,,\n\\eeq\n\\beq\n\t\\label{oform}\n\t\\oform_{\\zeta} \\lf[f \\ri]\n\t=  (N-1) \\int_{\\R^{6}} \\diff \\pv \\diff \\qv \\,\t\\frac{f\n\t^*(\\pv) f\n\t(\\qv)}{  p^2 + q^2 + \\tx\\frac{2}{m+1} \\pv \\cdot \\qv + \\zeta}\\,   .\n\\eeq\n\n\\vspace{0.3cm}\n\\n\nUsing the representation \\eqref{qform 1}, \\eqref{fform}, \\eqref{dform}, \\eqref{oform} we obtain the following estimate for $\\Phi_0^{\\lambda}$, which is the crucial ingredient for the proof of Theorem \\ref{clbou}.   \n\n\n\\begin{proposition}[Upper and lower bounds for $ \\Phi_0^{\\lambda} $]\n\\label{stiqform}\n\\mbox{}\t\\\\\nFor any $\\xi \\in \\dom(\\qform)$ we have\n\\beq\\label{abqform}\n\\Big(1-\\Lambda(m,N) \\Big)\\; \\Phi_{0,\\lambda}^{\\mathrm{diag}}  [\\xi] \\;  \\leq \\;  \\Phi_0^{\\lambda} [\\xi]  \\; \\leq \\; \\Big( 1 + \\Gamma(m,N)\\Big) \\;\\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\\,, \n\\eeq\nwhere\n\\beq\\label{Gamma}\n\\Gamma(m,N) : = \\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri)\\,.\n\\eeq\n\\end{proposition}\n\n\n\\vspace{0.3cm}\n\nThe proof  is based on a careful analysis of the form $F_{\\zeta}$, $\\zeta \\in [0,1]$,  reduced to   each subspace with fixed  angular momentum $l$ and it is postponed to the end of this section. First we introduce some useful  notation and we prove some preliminary lemmas.\n\nFor any $f\\in L^2(\\R^3)$ we consider the  expansion\n\\beq\\label{exsh}\nf(\\pv) = \\sum_{l=0}^{\\infty}\\sum_{m=-l}^l f_{lm}(p) Y_{l}^m(\\theta_p, \\phi_p)\\,,\n\\eeq\nwhere $\\pv=(p, \\theta_p,\\phi_p)$ in spherical coordinates and $Y_{l}^m$ denotes the spherical harmonics of order $l,m$. We notice that $f\\in \\dom(F_1)$ is equivalent to\n\\beq\n\\sum_{l=0}^{\\infty}\\sum_{m=-l}^l \\int_0^{\\infty}\\!\\!dp\\, p^2 \\sqrt{p^2 +1}\\,  \\lf|f_{lm}(p)\\ri|^2 <\\infty\\,.\n\\eeq\nMoreover, we denote by $P_l$ the Legendre polynomial of order $l=0,1,\\ldots$ explicitly given by\n\\beq\\label{leg}\nP_l(y) = \\f{1}{2^l l!} \\f{d^l}{dy^l} (y^2 -1)^l \\,,\t\\hspace{1cm}\t y \\in [-1,1]\\,.\n\\eeq\n In the first lemma we decompose $F_{\\zeta}$ in each subspace of fixed angular momentum $l$.\n\n\\begin{lemma}[Decomposition of $ F_{\\zeta} $]\n\\label{lemma0}\n\\mbox{}\t\\\\\nFor $f\\in \\dom(F_{1})$ we have\n\\beq\t\\label{Fzl}\nF_{\\zeta}\\lf[f \\ri] \\; =\\;  \t\\sum_{l=0}^{\\infty}\\sum_{m=-l}^l  G_{\\zeta,l} \\lf[ f_{lm} \\ri] \\;  =:  \\; \\sum_{l=0}^{\\infty}\\sum_{m=-l}^l  \\lf( G^{\\mathrm{diag}}_{\\zeta} \\lf[f_{lm} \\ri] + G^{\\mathrm{off}}_{\\zeta,l} \\lf[f_{lm} \\ri]  \\ri)\\,,\n\\eeq  \nwhere for $g \\in L^2((0,\\infty), \\, p^2 \\sqrt{p^2+1}\\,dp)$\n\\begin{eqnarray}\n\t\\label{G diag off}\n&&G^{\\mathrm{diag}}_{\\zeta} \\lf[ g \\ri] : = 2 \\pi^2 \\!\\! \\int_0^{\\infty} \\!\\!\\! dp \\, p^2 \\sqrt{\\frac{m(m+2)}{(m+1)^2} p^2 +\\zeta}\\, \\, |g(p)|^2\\,, \\\\\n&& G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] : = 2\\pi (N-1) \\!\\int_0^{\\infty} \\!\\!\\!\\! \\!dp\\!\\!\\int_0^{\\infty}\\!\\!\\!\\!\\! dq\\, p^2 g^*(p) \\, q^2 g(q) \\!\\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}\\,.\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nFor a given $f\\in \\dom(F_{1})$ we consider the expansion \\eqref{exsh}. From \\eqref{dform} we see that  $\\dform_{\\zeta}[f]= \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l G^{\\text{diag}}_{\\zeta}[f_{lm}]$.  Concerning the off-diagonal term \\eqref{oform}, \nwe denote by $\\theta_{pq}$ the angle between the vectors $\\pv$ and $\\qv$ and we consider the following expansion in Legendre polynomials:\n\\begin{eqnarray}\\label{exle}\n&&\\f{1}{p^2 +q^2 + \\f{2}{m+1} pq \\cos \\theta_{pq} +\\zeta} = \\sum_{l=0}^{\\infty}\\f{2l+1}{2} \\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}  \\,\\, P_l(\\cos \\theta_{pq})\\nonumber\\\\\n&&= \\sum_{l=0}^{\\infty} 2\\pi  \\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}  \\sum_{m=-l}^{l} Y_{l}^{m *} (\\theta_p,\\phi_p) Y_{l}^m(\\theta_q,\\phi_q)\\,.\n\\end{eqnarray}\nIn the last line we used the addition formula for spherical harmonics (see, e.g., \\cite[Eq. (8.814)]{GR}). From (\\ref{exle}) we obtain $\\oform_{\\zeta}[f]=   \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l G^{\\text{off}}_{\\zeta,l}[f_{lm}]$.\n\\end{proof}\n\n\nIn the next lemma we give a new representation of $G^{\\text{off}}_{\\zeta,l}$ which is particularly useful to  control $G^{\\text{off}}_{\\zeta,l}$ in terms of $G^{\\text{off}}_{0,l}$ for any $\\zeta>0$. \n\n\\begin{lemma}[Estimates for $ G^{\\mathrm{off}}_{\\zeta,l}$]\n\\label{lemma01}\n\\mbox{}\t\\\\\nThe form $G^{\\mathrm{off}}_{\\zeta,l}$ can be written as\n\\begin{eqnarray}\\label{rapof}\n&&G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] = \\sum_{k=0}^{\\infty} B_{l,k} \\int_0^{\\infty}\\!\\!\\! d\\nu \\, \\nu^k e^{-\\zeta \\nu} \\lf| \\int_0^{\\infty}\\!\\!\\! dp \\, g(p) \\,p^{2+k} e^{-\\nu p^2} \\ri|^{2}\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{Blk}\n&&B_{l,k}=\n\t\t\\begin{cases}\n\t\t\t \\f{2\\pi (N-1)}{2^l l! \\,k!}  \\lf(\\! \\frac{-2}{m\\!+ \\!1} \\!\\ri)^{\\!k}  \\disp\\int_{-1}^1\\!\\!\\! dy \\, (1-y^2)^l \\frac{d^l}{dy^l} y^k \t& \\mbox{if} \\:\\:\\:\\: l \\leq k\\,,\t\\\\\n\t\t\t0\t&\t\\mbox{otherwise}\\,.\n\t\t\\end{cases}\n\\end{eqnarray}\n\n\\n \nMoreover for any $\\zeta > 0$ we have\n\\begin{eqnarray}\\label{stle}\n&&0\\leq  G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] \\leq G^{\\mathrm{off}}_{0,l} \\lf[ g\\ri] \\hspace{1cm} \\text{for $l$ even}\\,,\\\\\n&&\\nonumber\\\\\n&&G^{\\mathrm{off}}_{0,l} \\lf[ g\\ri] \\leq G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] \\leq 0 \\hspace{1cm} \\text{for $l$ odd}\\,.\\label{stlo}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nUsing the expansion\n\\beq\n\\f{1}{p^2+q^2+ \\f{2}{m+1} pqy + \\zeta}=\n \\f{1}{p^2+q^2+\\zeta}  \n\\sum_{k=0}^{\\infty} \\lf(\\!  \\f{-2}{m+\\!1}\\, \\f{pqy}{p^2+q^2 +\\zeta} \\!\\ri)^{\\!\\!k}\n\\eeq\nand formula (\\ref{leg}), we obtain\n\\beq\\label{exse1}\nG^{\\mathrm{off}}_{\\zeta,l}\\lf[ g \\ri]= \\f{2\\pi (N-1)}{ 2^l l! } \\!\\sum_{k=0}^{\\infty}  \\lf(\\! \\f{-2}{m+1} \\!\\ri)^{\\!\\!k} \\!\\! \\int_0^{\\infty}\\!\\!\\! \\!dp \\!\\int_0^{\\infty}\\!\\!\\!\\! dq\\, \\f{p^{2+k} g^*(p) q^{2+k} g(q)  }{(p^2+q^2+\\zeta)^{k+1}}\\!\n\\int_{-1}^1\\!\\!\\!\\!dy\\, y^k \\f{d^l}{dy^l} (y^2-1)^l \\,.\n\\eeq\nIntegrating  by parts $l$ times we find\n\\begin{eqnarray}\\label{exse2}\n&&G^{\\mathrm{off}}_{\\zeta,l}\\lf[ g \\ri]=\\sum_{k=0}^{\\infty} B_{l,k} \\, k! \\!\\int_0^{\\infty}\\!\\!\\! \\!dp \\!\\int_0^{\\infty}\\!\\!\\!\\! dq\\, \\f{p^{2+k} g^*(p) q^{2+k} g(q)  }{(p^2+q^2+\\zeta)^{k+1}}\\,.\n\\end{eqnarray}\nFinally we use the identity\n\\beq\\label{iden}\n\\f{k!}{(p^2+q^2+\\zeta)^{k+1}} = \\int_0^{\\infty}\\!\\!\\!\\! d\\nu \\, \\nu^{k} \\,e^{-(p^2+q^2+\\zeta)\\nu}\n\\eeq\nin (\\ref{exse2}) and we obtain (\\ref{rapof}). Let us fix $l$ even. Then the integral in (\\ref{Blk}) is different from zero only if $k$ is even and this implies that $G^{\\mathrm{off}}_{\\zeta,l}$ is positive and the estimate (\\ref{stle}) holds. Analogously, when $l$ is  odd  the integral in (\\ref{Blk}) is different from zero only if $k$ is odd and therefore $G^{\\mathrm{off}}_{\\zeta,l}$ is negative and (\\ref{stlo}) holds.\n\\end{proof}\n\n\n\n\n\n\n\n\\noindent\nNow we study the form $G_{0,l} = G^{\\text{diag}}_{0} + G^{\\text{off}}_{0,l}$ and we show that it can be diagonalized for each $l$. \n\\begin{lemma}[Diagonalization of $G_{0,l}$]\n\\label{lemma1}\n\t\\mbox{}\t\\\\\n\t For any $g \\in L^2((0,\\infty), p^2\\sqrt{p^2+1}\\, dp)$ we have\n\\begin{eqnarray} \\label{formula0}\n&&G^{\\mathrm{diag}}_0 [g] =   2 \\pi^2 \\f{\\sqrt{m(m+2)}}{m+1}  \\int_{\\R} \\diff k \\,  \\big|  g^{\\sharp} (k) \\big|^2\\,, \\\\\n&&G^{\\mathrm{off}}_{0,l}[g] = \\int_{\\R} \\diff k \\, S_l (k) \\big|  g^{\\sharp} (k) \\big|^2\\,,\n\\label{formula1} \n\\end{eqnarray}\nwhere\n\\beq \\label{formula2}\n g^{\\sharp} (k) : = \\frac{1}{\\sqrt{2\\pi}} \\int_{\\R} \\diff x \\, e^{-ikx}\\, e^{2x} \\, g(e^x)\n\\eeq\nand \n\\beq \\label{formula3}\nS_l (k) = 2\\pi^2 (N-1) \\int_{-1}^1 \\diff y \\, P_l (y) \n\\frac{\\sinh \\lf( k \\arccos \\frac{y}{m+1} \\ri) }{ \\sin  \\lf(  \\arccos \\frac{y}{m+1} \\ri) \\, \\sinh(\\pi k)}\\,.\n\\eeq\n\\end{lemma}\n\\begin{proof}\nThe proof of (\\ref{formula0}) is straightforward. To prove (\\ref{formula1}) we first introduce the new integration variables\n $p=e^{x_1}$ and $q=e^{x_2}$, so that  the form reads\n\\begin{equation}\\label{for2}\n\\begin{split}\nG^{\\mathrm{off}}_{0,l} \\lf[g \\ri]\\;&=\\;2 \\pi(N-1) \n\\int_{\\R}  dx_1 dx_2\\,\\,e^{3 x_1}{g}^* (e^{x_1})e^{3 x_2} g(e^{x_2})\n\\int_{-1}^1 dy  \\f{  P_l(y)   }{ e^{2 x_1} + e^{2x_2} +\\f{2y}{m+1} e^{x_1 +x_2}  } \\\\\n& = \\; \\pi(N-1) \n\\int_{\\R}  dx_1 dx_2\\,\\,e^{2 x_1}{g}^* (e^{x_1})e^{2 x_2} g(e^{x_2})\n\\int_{-1}^1 dy  \\f{  P_l(y)   }{ \\cosh (x_1-x_2) +\\f{y}{m+1}  }\\,.\n\\end{split}\n\\end{equation}\nThe kernel in (\\ref{for2}) is a convolution kernel and therefore it can be diagonalized by means of the Fourier transform. Using the explicit Fourier transform of the kernel (see, e.g., \\cite{erdely}) we finally arrive at  \\eqref{formula1}.\n\\end{proof}\n\n\nOwing to the previous lemma, the problem of finding  bounds for  the form $G^{\\text{off}}_{0,l} $ is reduced to finding \nbounds for the function $S_l(k)$.\nTaking into account the identity $\\arccos z = \\f{\\pi}{2} - \\arcsin z$ and the parity of $P_l$,  we represent $S_l(k)$ as\n\\begin{equation}\\label{reps}\nS_l(k) =\n\\begin{cases}\n   \\displaystyle  - \\pi^2 (N-1) \n\\int_{-1}^1 dy \\, P_l(y) \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)\\sinh \\lf(\\f{\\pi}{2} k\\ri) \\, } & \\text{for} \\;  l \\text{ odd}, \\\\\n  \\displaystyle  \\pi^2(N-1) \n\\int_{-1}^1 dy  \\, P_l(y) \\f{\\cosh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)\\cosh \\lf(\\f{\\pi}{2} k\\ri) \\, } & \\text{for} \\;  l \\text{ even}\\,.\n\\end{cases}\n\\end{equation}\nIt is evident from this representation that $S_l (k)$ is for any $l \\geq 0 $ an even $C^{\\infty}$-function of $k$ with $\\lim_{k \\rightarrow \\infty} S_l (k)=0$.   \nBefore discussing upper and lower bound of $S_l (k) $, we shall prove the following elementary lemma.\n\\begin{lemma}[Taylor expansions of $ \\wt S^{\\mathrm{o}}_k  $ and $ \\wt S^{\\mathrm{e}}_k $]\n\\label{forever}\n\t\\mbox{}\t\\\\\nFor any fixed $k\\geq 0$ the following functions \n\\beq\\label{s0y}\n\\wt S^{\\mathrm{o}}_k (y) = \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) \\, } \\,,\t\\hspace{1cm}\t \\wt S^{\\mathrm{e}}_k (y) = \\f{\\cosh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) \\, }\n\\eeq\nhave a Taylor expansion in the variable $y \\in [-1,1]$ with positive coefficients.\n\\end{lemma}\n\\begin{proof}\nLet $\\cP$ be the set of functions whose Taylor expansion has positive coefficients.\nFirst note that $\\arcsin y$, $\\sinh y$ and $\\cosh y$ belong to $\\cP$. \nThe derivative is a linear automorphism of $\\cP$ and therefore \n\\[\n\\frac{d}{dy} \\arcsin y = \\frac{1}{\\sqrt{1-y^2} } = \\frac{1}{\\cos \\arcsin y} \\in \\cP\\,.\n\\]\nMoreover, $\\cP$ is invariant under dilations, multiplications and compositions of functions in $\\cP$.\nThus, $\\wt S^{\\mathrm{o}}_k$ and  $\\wt S^{\\mathrm{e}}_k $ belong to $ \\cP$.\n\\end{proof}\n\n\\n\nIn the next lemma we compute  lower and upper bounds for $S_l (k) $.\n\\begin{lemma}[Bounds for $S_l (k) $]\n \\label{lem4}\n\t\\mbox{}\t\\\\\nFor  any $k \\in \\R$ we have\n\\begin{eqnarray}\\label{boe}\n0  \\; \\leq  \\;           S_l(k)  \\;  \\leq     \\;   2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf(\\f{1}{m\\!+\\!1}\\ri) &\t\\hspace{0,7cm}\t  & \\text{for $l$ even}\t\\\\\n\\label{boo}\n-4\\pi (N\\!-\\!1) (m\\!+\\!1) \\bigg[ 1- \\sqrt{m(m\\!+\\!2)} \\, \\arcsin  \\lf(\\f{1}{m\\!+\\!1}\\ri) \\bigg]\n\\; \\leq S_l(k)  \\;  \\leq 0 &\t\\hspace{0,7cm} & \\text{for $l$ odd}\\,.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nLet us prove \\eqref{boo} first. The upper bound follows from \\eqref{stlo} and \\eqref{formula1}. \nAs for the lower bound,  by means of \\eqref{s0y}  we write\n\\beq \\label{tonno}\nS_l (k) = - \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    }\\int_{-1}^{+1} \\diff y \\, P_l (y) \\, \\wt S^{\\mathrm{o}}_k (y) \\,.\n\\eeq\nThe first step is to  prove that $S_l (k)$ is an increasing function of $l$ for any  fixed $k$. \nFrom \\eqref{tonno},  using \\eqref{leg} and  integrating by parts, we obtain\n\\begin{align} \\label{tonno2}\nS_{l+2}(k) &= - \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\frac{d^{l+2}}{dy^{l+2} }(y^2-1)^{l+2}  \\wt S^{\\mathrm{o}}_k (y) \\no \\\\\n& = \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\frac{d^{2}}{dy^{2} }(y^2-1)^{l+2} \\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }(y) \\no \\\\\n& = \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\lf[(l+2)(l+1)(y^2-1)^{l}4y^2 + 2(l+2)(y^2-1)^{l+1}    \\ri] \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y) \\no \\\\\n&=  \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l} l!}   \n\\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l} \n\\frac{d^{l} \\wt S_k}{dy^{l} }  (y) + \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\f{1\\!+2(l\\!+\\!1)}{2^{l\\!+\\!1} (l\\!+\\!1)!} \\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l+1} \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y)\n\\no\\\\\n&= S_l(k)  + \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\f{1\\!+2(l\\!+\\!1)}{2^{l\\!+\\!1} (l\\!+\\!1)!} \\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l+1} \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y) \\,. \n\\end{align}\nBy Lemma \\ref{forever}, and taking into account that $l+1$ is even, we deduce that the last integral in \\eqref{tonno2} is positive and therefore  we conclude  $S_{l+2} (k) \\geq S_l(k)$. \nThis means that it is sufficient to find the minimum of $S_1 (k)$.\nFrom  \\eqref{reps} we have \n\\beq \\label{shotgun}\n S_1 (k) = \n-2\\pi^2 (N-1) \n\\int_{0}^1 \\diff y \\, \\f{y}{\\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)}   \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{\\sinh \\lf(\\f{\\pi}{2} k\\ri) \\, }\\,.\n\\eeq\nWe know that $S_1(0)<0$ and $\\lim_{k \\rightarrow \\infty} S_1(k)=0$. Moreover,  the derivative of $S_1(k)$ does not vanish for $k>0$, which follows from the fact that the derivative of the function\n\\[\n\\f{\\sinh ak}{\\sinh bk}\\,,\t\\hspace{1cm} 0<a<b,\n\\]\ndoes not vanish for $k>0$. Therefore, $S_1(k)$ is monotone increasing when $k>0$ and attains    its minimum at $k=0$. Thus,\n\\begin{equation}\\label{S10}\n\\begin{split}\nS_l (k) \\;\\geq\\; S_1 (k) \\;\\geq\\; S_1 (0)\\;&=\\;-4\\pi (N-1) \\int_{0}^1 \\diff y \\, y\n\\f{ \\arcsin  \\f{ y }{m+1} }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) } \\\\\n&=\\;-4\\pi (N-1) (m+1)^2 \\int_0^{\\arcsin \\f{ 1 }{m+1} } \\diff z \\, z \\sin z  \\\\\n&=\\;-4\\pi (N-1) (m+1) \\bigg[1- \\sqrt{m(m+2)} \\, \\arcsin \\lf(\\f{1}{m+1}\\ri) \\bigg]\n\\end{split}\n\\end{equation}\nand \\eqref{boo} is proved. The proof of \\eqref{boe}  is completely analogous. In this case the lower bound follows from \\eqref{stle} and \\eqref{formula1}.  Using the representation\n\\beq\nS_l (k) =  \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    }\\int_{-1}^{1} \\diff y \\, P_l (y) \\, \\wt S^e_k (y)\n\\eeq\nobtained from \\eqref{s0y}, one sees that $S_{l+2}(k) \\leqslant S_l(k)$ and therefore  it is enough to consider $S_0(k)$. Since\n\\beq\nS_0(0)= \\pi^2 (N\\!-\\!1) \\!\\int_{-1}^{1}\\!\\!\\! \\diff y \\f{1}{\\cos \\left(\\! \\arcsin \\f{y}{m+1} \\! \\right)} = 2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) >0\\,,\n\\eeq\n$\\lim_{k \\rightarrow \\infty} S_0(k)=0$, and the derivative of $S_0(k)$ does not vanish for $k>0$, we deduce that $S_0(0)$ is the maximum of $S_0(k)$.\n\\end{proof}\n\n\nUsing the results of the previous lemmas we can finally \nprove Proposition \\ref{stiqform}. \n \n\\begin{proof}[Proof of Proposition \\ref{stiqform}]  We prove first the estimate from below in \\eqref{abqform}. From Lemmas \\ref{lemma0},  \\ref{lemma01}, \\ref{lemma1}, \\ref{lem4} we have\n\\begin{equation}\\label{sotto}\n\\begin{split}\n\\oform_1[f]\\;&=\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ even}}}\n G^{\\mathrm{off}}_{1,l} \\lf[f_{lm} \\ri]   + \\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}}\n G^{\\mathrm{off}}_{1,l} \\lf[f_{lm} \\ri] \\;\\geq\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}}\n G^{\\mathrm{off}}_{0,l} \\lf[f_{lm} \\ri] \\;=\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}} \\int_{\\R} dk\\, S_l(k) \\, \\big| f^{\\sharp}_{lm} (k) \\big|^2 \\\\\n&\\geq\\; -4\\pi (N\\!-\\!1) (m\\!+\\!1)  \\bigg[ 1 - \\sqrt{m(m+2)} \\, \\arcsin \\lf( \\f{1}{m+1} \\ri) \\bigg]\n  \\;  \\sum_{l,m} \\int_{\\R} dk\\,  \\big| f^{\\sharp}_{lm} (k) \\big|^2 \\\\\n&=\\;- \\Lambda(m,\\!N) \\, \\f{2 \\pi^2 \\sqrt{m(m+2)}}{m+1}  \\sum_{l,m} \\int_{\\R} dk\\,  \\big| f^{\\sharp}_{lm} (k) \\big|^2\n\\end{split}\n\\end{equation}\nwhere $\\Lambda(m,\\!N)$ is defined in \\eqref{Lambda}. Then, by \\eqref{formula0},\n\\beq\\label{sotto1}\n\\oform_1[f] \\geq -\\Lambda(m,\\!N) \\! \\sum_{l,m} G^{\\mathrm{diag}}_{0} [f_{lm}]  \\geq -\\Lambda(m,\\!N) \\! \\sum_{l,m} G^{\\mathrm{diag}}_{1} [f_{lm} ] =  -\\Lambda(m,\\!N) \\, \\dform_1[f]\\,.\n\\eeq\nFrom the representation \\eqref{qform 1} and from \\eqref{sotto1} we have\n\\beq\n\\Phi_0^{\\lambda} [\\xi] \\geq \\big(1-\\Lambda(m,N)\\big) \\int_{\\bR^{3N-6}} \\!\\!\\! \\diff \\Kv \\, \\sqrt{D(\\Kv) +\\lambda} \\, F_1^{\\mathrm{diag}} [Q_{\\Kv}] = \\big(1-\\Lambda(m,N)\\big)\\, \\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\n\\eeq\nand the lower bound is proved. An analogous proof yields the upper bound in \\eqref{abqform}. We have\n\\begin{equation}\\label{sopra}\n\\begin{split}\n\\oform_1[f]\\;&\\leq\\;2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} \\int\\!\\! dk\\,  | f^{\\sharp}_{lm} (k)|^2 \\\\\n&=\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf(\\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} G^{\\mathrm{diag}}_{0} [f_{lm}] \\\\\n&\\leq\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} G^{\\mathrm{diag}}_{1} [f_{lm}] \\\\\n&=\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\dform_1[f]\n\\end{split}\n\\end{equation}\nwhich, together with \\eqref{qform 1}, yields the upper bound for $\\Phi_0^{\\lambda}$. \n\\end{proof}\n\n\nLet us briefly comment on the result of Proposition \\ref{stiqform}. By means of the elementary estimate\n\\beq\\label{el12}\n-\\f{1}{2} \\sum_{i=1}^{N-1} k_i^2 \\leq \\sum_{i<j} \\kv_i \\cdot \\kv_j \\leq \\f{N-2}{2} \\sum_{i=1}^{N-1} k_i^2\\,,\n\\eeq\nwe find\n\\bml{\\label{fiup}\n\t\\qform [\\xi] = \\alpha \\|\\xi\\|^2_{L^2} +\\Phi_0^{\\lambda} [\\xi]  \\leq \\alpha \\|\\xi\\|_{L^2} + \\Big( 1 + \\Gamma(m,N)\\Big) \\;\\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\t\\\\\n\t\\leq \\int_{\\bR^{3N-3}} \\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\! \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\! \\bigg[ \\al \\!+\\! 2 \\pi^2  \\Big(\\! 1\\! +\\! \\Gamma(m,N)\\Big)  \\bigg( \\frac{m(m\\!+\\!N)}{(m+1)^2} \\sum_{i = 1}^{N-1} k_i^2 + \\la \\bigg)^{\\!\\!1\/2}  \\bigg]\n}\nand\n\\beq\\label{fido}\n \\qform [\\xi] \\geq \\int_{\\bR^{3N-3}} \\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\! \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\! \\bigg[ \\al + 2 \\pi^2  \\Big(\\! 1 - \\Lambda(m,N)\\Big)  \\bigg( \\frac{m}{m\\!+\\!1} \\sum_{i = 1}^{N-1} k_i^2 + \\la \\bigg)^{\\!\\!1\/2}  \\bigg]\\,.\n\\eeq \nFrom \\eqref{fiup}, \\eqref{fido}, choosing $\\lambda$ sufficiently large if $\\alpha <0$, we conclude   \n\\beq\\label{cC}\nc\\, \\Big( 1- \\Lambda(m,N) \\Big)  \\|\\xi \\|_{\\hcfpiu} \\leq \\; \\qform[\\xi] \\; \\leq C \\;  \\|\\xi\\|_{\\hcfpiu}\\,,\n\\eeq\nwhere $c, C$ are two positive constants. Estimate  \\eqref{cC}  implies that  $\\qform[\\xi]$ is finite  for any  $\\xi \\in \\dom(\\qform)$ and therefore our definition of the quadratic forms \\eqref{form} and \\eqref{qform} is well-posed. \nOn the other hand, if in addition we assume $\\Lambda(m,N) <1$, from  \\eqref{cC}  we conclude that $\\qform$ defines a norm equivalent to the $H^{1\/2}$-norm. This is the crucial ingredient for the proof of Theorem \\ref{clbou}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{clbou}]  From \\eqref{form} and \\eqref{fido} we have\n\\bml{\n\\form [\\psi] \\geq -\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}} + N \\qform [\\xi] \\geq -\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}}\t\\\\\n + N \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\bigg[ \\al + 2 \\pi^2  \\Big(\\! 1 - \\Lambda(m,N) \\!\\Big)  \\bigg(\\! \\frac{m}{m\\!+\\!1} \\sum_{i = 1}^{N-1}\\! k_i^2 + \\la \\! \\bigg)^{\\!\\!1\/2}  \\bigg] \n}\nfor any $\\psi \\in \\dom(\\form)$ and $\\lambda >0$. Therefore, if $\\alpha \\geq 0$ the form $\\form$ is positive and if $\\alpha<0$ the lower bound \\eqref{inff} holds. Let us now prove that $\\form$ is closed. We choose \n\\beq\n\\lambda >\n\t\\begin{cases} \n\t\t0\t& \\text{if} \\;\\; \\alpha \\geq 0, \t\\\\\n\t\t\\f{\\alpha^2}{4 \\pi^4 \\big(1-\\Lambda(m,N)\\big) }\t&\t\\text{if} \\;\\; \\alpha <0\\,,\n\t\\end{cases}\n\\eeq\nand consider the form $\\form^{\\lambda}[\\psi] :=\\form [\\psi] +\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}}$ defined on $\\dom(\\form)$. Let $\\{\\psi_n \\}$ be a sequence in $\\dom(\\form)$ such that\n\\beq\n\\lim_{n \\rightarrow \\infty} \\|\\psi_n - \\psi \\|_{L^2_{\\mathrm{f}}} =0\\,, \\hspace{1cm}\t\\lim_{n,m \\rightarrow \\infty} \\form^{\\lambda}[\\psi_n - \\psi_m] =0\\,,\n\\eeq\nwhere $\\psi \\in L^2_{\\mathrm{f}}(\\bR^{3N})$. From the definition of $\\dom(\\form)$ and $\\form^{\\lambda}$  (see \\eqref{form}, \\eqref{form domain}) we have $\\psi_n = \\phi_n^{\\lambda} + \\pot \\xi_n$, with $\\phi_n^{\\lambda} \\in H^1_{\\mathrm{f}}(\\bR^{3N})$, $\\xi_n \\in H^{1\/2}_{\\mathrm{f}}(\\bR^{3N-3})$, and \n\\beq\n\\form^{\\lambda}[\\psi_n - \\psi_m]= \\mathcal{F}_0[\\phi_n^{\\lambda} - \\phi_m^{\\lambda}] + N \\qform[\\xi_n -\\xi_m]\\,.\n\\eeq\nThis, together with \\eqref{cC}, implies that $\\{\\phi_n^{\\lambda} \\}$ is a Cauchy sequence in $ H^1_{\\mathrm{f}}(\\bR^{3N})$ and $\\{ \\xi_n\\}$ is a Cauchy sequence in $H^{1\/2}_{\\mathrm{f}}(\\bR^{3N-3})$. Let us denote by $\\phi^{\\lambda}$ and $\\xi$ the corresponding limits. From the explicit expression of the potential \\eqref{pot xi} we notice that \n\\beq\n\\| \\pot \\xi \\|_{L^2_{\\mathrm{f}}} \\leq c \\, \\|\\xi\\|_{L^2_{\\mathrm{f}}}\\,,\n\\eeq\nwhere $c>0$. Hence,\n\\beq\n\\lim_{n \\rightarrow \\infty} \\|\\psi_n - (\\phi^{\\lambda} + \\pot \\xi) \\|_{L^2_{\\mathrm{f}}}= \\lim_{n \\rightarrow \\infty}  \\| (\\phi_n^{\\lambda} - \\phi^{\\lambda}) + \\pot ( \\xi_n - \\xi) \\|_{L^2_{\\mathrm{f}}}       =0\\,.\n\\eeq\nSince the limit of the $\\psi_n$'s is unique, $\\psi = \\phi^{\\lambda} + \\pot \\xi$. Therefore $\\psi \\in \\dom(\\form)$ and $\\lim_{n \\rightarrow \\infty} \\form^{\\lambda}[\\psi-\\psi_n]=0$. This shows that the form $\\form^{\\lambda}$ is closed and a fortiori $\\form$ is.\n\\end{proof}\n\n\n\n\n \n\n\n\n\n\\section{Unboundedness from Below of $ \\form $} \n\\label{instability: sec}\n\t\t\nThis section is devoted to the proof of Theorem \\ref{ub1}.\nAs we shall see, what makes an instability condition hard to prove is the restriction to antisymmetric wave functions. \n\n\nIn fact, the proof relies on the explicit evaluation of the charge form $ \\qform $ on a trial function, i.e., a convenient sequence of charges with energy going to $ -\\infty $. Identifying one such sequence is easy when $ N = 2 $ because $ \\qform $ is in practice the same as the reduced form $ F_{1} $ -- see \\eqref{michelebuliccio} below -- \nand the analysis performed in Section \\ref{stability: sec} suggests that a convenient $ Q_n(\\pv) $ has to be chosen\nin the subspace with angular momentum $ \\ell=1 $ and such that in the position representation it becomes peaked at the origin as $ n \\to \\infty $ (i.e., two identical fermions coming arbitrarily close).\n\nWhen $ N>2 $, on the other hand, a natural trial function satisfying the antisymmetry constraint would be the Slater determinant of $N$ one-particle charges, one of which is $ Q_n $ itself. A convenient choice is driven by the physical idea of a $N$-body configuration that contains precisely the $(2\\!+\\!1)$-body structure minimizing the energy with $ N = 2 $, whereas all remaining particles are placed far away in space so that there is no or negligible interference with the two-body state. This results in a $N$-particle Slater determinant  between $ Q_n $ and $N-1$ copy of a different component (see \\eqref{min sequence} below). The fermionic character of the trial function is thus fulfilled by construction and optimising the choice of the second component produces only higher order symmetry correlations.\n\n\tThroughout this section we assume that $ \\la $ is a positive number such that $ C_1 \\leq \\la^{-1} \\leq C_2 $ for two finite constants $ C_1, C_2 < \\infty $, which in particular will allow us to incorporate error factors proportional to $ \\la^{-1} $ into a constant $ C $.\n \t\n\t\\begin{proof}[Proof of Theorem \\ref{ub1}]\n\t\tIn order to prove instability of the form $ \\form $, it is enough to produce a sequence of normalised charges $ \\xi_n \\in H_{\\mathrm{f}}^{1\/2}(\\R^{3(N-1)}) $ such that\n\t\t\\beq\n\t\t\t\\lim_{n \\to \\infty} \\qform[\\xi_n] = - \\infty,\n\t\t\\eeq\n\t\tsince the sequence of states $ \\pot \\xi_n $ then satisfies\n\t\t\\beq\n\t\t\t\\form[\\pot \\xi_n] = - \\la \\lf\\| \\pot \\xi_n \\ri\\|_{L^2(\\R^{3N})}^2 + N \\qform[\\xi_n] \\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty.\n\t\t\\eeq\n\t\t\\underline{Case $ N = 2 $}. The result was already proved in \\cite{FT}, but we repeat here the argument for we use a slightly different trial function that turns out to be useful in the general case $ N > 2 $. Owing to  \\eqref{qform 1}, \n\t\t\\beq\\label{michelebuliccio}\n\t\t\t\\qform[\\xi] -  \\alpha \\| \\xi \\|_{L^2(\\R^{3})}^2 = \\sqrt{\\la} F_1 \\lf[ Q \\ri]\n\t\t\\eeq\n\t\twhere $ Q(\\pv) = \\la^{3\/4} \\xi(\\sqrt{\\la}\\pv) $ (recall \\eqref{charge Qk}). Then we only need to produce $ Q_n \\in L^2(\\RT) $ such that $ \\lim_{n \\to \\infty} F_1[Q_n] = - \\infty $. Note that no constraint is imposed on the symmetry properties of $ Q_n $. In fact, according to the discussion of Section \\ref{stability: sec} (see  \\eqref{formula0}, \\eqref{formula1} and \\eqref{S10}), we can take each $ Q_n $ in the subspace with angular momentum $ l = 1 $ and such that the support of its $\\sharp-$transform defined in \\eqref{formula2} gets concentrated at the origin. Explicitly, we choose\n\t\t\\beq\n\t\t\t\\label{qnga}\n\t\t\t\\qnga(\\kv) : = n^{-3\/2} \\qga(n^{-1} k) Y_1^{0}(\\vartheta_k),\n\t\t\\eeq\n\t\t\t\t\\beq\n\t\t\t\\label{Qgamma}\n\t\t\t\\qga(p) : =  \\pi^{-1\/4} c_{\\gamma} \\gamma^{1\/2} p^{-1} \\exp \\lf\\{ - \\hbox{$\\frac{1}{8\\gamma^{2} }$} \\ri\\} \\exp \\lf\\{ - \\half \\gamma^2 \\lf( \\log p \\ri)^2 \\ri\\} \\Theta(p-1),\n\t\t\\eeq\n\t\twhere $ \\gamma \\in (0, 1) $ is a variational parameter, $ \\Theta $ is the Heaviside function, i.e., $ \\Theta(p) = 1 $ if $ p \\geq0 $ and $ 0 $ otherwise, and $ c_{\\gamma} $ is a normalisation constant.\n\t\tBy direct computation,\n\t\t\\beq\n\t\t\t\\lf\\| \\qnga \\ri\\|^2_{L^2} = \\int_1^{\\infty} \\diff p \\: p^2 \\lf| \\qga(p) \\ri|^2 = \\tx\\frac{c^2_{\\gamma}}{\\sqrt{\\pi}} \\disp\\int_{0}^{\\infty} \\diff t \\: \\exp\\lf\\{ - t^2 + \\tx\\frac{t}{\\gamma} - \\tx\\frac{1}{4\\gamma^2} \\ri\\} = \\half c^2_{\\gamma} \\: \\lf( 1 + \\mathrm{erf}\\lf\\{ \\tx\\frac{1}{2\\gamma} \\ri\\} \\ri).\n\t\t\\eeq\n\t\tImposing $\\qnga$ to be normalised yields\n\t\t\\beq\n\t\t\t\\label{cgamma}\n\t\t\t1 \\leq c^2_{\\gamma} = 2 \\lf[ 1 + \\mathrm{erf}\\lf\\{ \\tx\\frac{1}{2\\gamma} \\ri\\} \\ri]^{-1} \\leq 1 + C\\gamma \\exp\\lf\\{-\\tx\\frac{1}{4\\gamma^2} \\ri\\}.\n\t\t\\eeq \n(see, e.g., \\cite[Eq. (7.1.13)]{AS}). It is also useful to compute the following integral for  $ a \\geq -1 $:\n\t\t\\bml{\n\t\t\t\\label{qga est}\n\t\t\t\\int_0^{\\infty} \\diff p \\: p^{2+a} \\lf| \\qga(p) \\ri|^2 = \\tx{\\frac{1}{\\sqrt{\\pi}}} \\gamma c_{\\gamma}^2 \\exp \\lf\\{ -\\tx{\\frac{1}{4\\gamma^2}} \\ri\\} \\disp\\int_{0}^{\\infty} \\diff t \\: \\exp \\lf\\{ - \\gamma^2 t^2 + (1+a) t \\ri\\}\t\\\\\n\t\t\t =\\half c_{\\gamma}^2 \\lf( 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\} \\ri) \\exp \\lf\\{ \\tx{\\frac{1}{4\\gamma^2}} (2a+a^2) \\ri\\} \\leq  \\lf( 1 + C \\gamma \\ri) \\exp \\lf\\{ \\tx{\\frac{1}{4\\gamma^2}} (2a+a^2) \\ri\\},\t\n\t\t}\n\twhere we used\n\t\t\\bdm\n\t\t\t\\half c_{\\gamma}^2 \\lf( 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\} \\ri) = \\frac{ 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\}}{ 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1}{2\\gamma} \\ri\\}} \\leq 1 + \\frac{2}{\\sqrt{\\pi}} \\int_{\\frac{1}{2\\gamma}}^{\\frac{1+a}{2\\gamma}} \\diff t \\: e^{-t^2} \\leq 1 + C \\gamma^{-1} \\exp\\lf\\{- \\tx\\frac{1}{4 \\gamma^2} \\ri\\} \\leq 1 + \\OO(\\gamma).\n\t\t\\edm\n\t\tUsing the decomposition \\eqref{Fzl}, as well as the scaling law of $ \\qnga $ with $n$, we have\n\t\t\\beq\n\t\t\t\\label{est F1 1}\n\t\t\tF_1[\\qnga] = G_{1}^{\\mathrm{diag}}\\lf[n^{-3\/2} \\qga(n^{-1} p)\\ri] + G_{1,1}^{\\mathrm{off}}\\lf[n^{-3\/2} \\qga(n^{-1} p)\\ri] = n \\lf[ G_{n^{-2}}^{\\mathrm{diag}}\\lf[\\qga\\ri] + G_{n^{-2},1}^{\\mathrm{off}}\\lf[\\qga\\ri] \\ri].\n\t\t\\eeq\nWe estimate the diagonal term in \\eqref{est F1 1} as\n\\begin{equation}\\label{approx Goff}\n\\begin{split}\nG_{n^{-2}}^{\\mathrm{diag}}\\lf[\\qga\\ri]\\;&\\leq\\;2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\lf(1 + \\OO(n^{-1}) \\ri) \\big\\| Q_{\\gamma}^{\\sharp} \\big\\|_{L^2}^2\t\\\\\n&\\leq\\;2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf[1 + \\OO(n^{-1}) + \\OO(\\gamma) \\ri],\n\\end{split}\n\\end{equation}\nwhere we used\n\t\t\\beq\n\t\t\t\\label{tilde Q norm}\n\t\t\t\\big\\| Q^{\\sharp}_{\\gamma} \\big\\|_{L^2}^2 = \\int_{\\R} \\diff k \\: e^{4k} \\: \\big| \\hat{Q}_{\\gamma}\\big(e^k\\big) \\big|^2 = \\int_{0}^{\\infty} \\diff p \\: p^3 \\: | Q_{\\gamma}(p) |^2 \\leq  (1 + C \\gamma ) \\exp \\lf\\{ \\tx{\\frac{3}{4\\gamma^2}} \\ri\\}.\n\t\t\\eeq\n\t\tAs for the off-diagonal term in \\eqref{est F1 1},\n\\beq\\label{Goff1}\nG^{\\mathrm{off}}_{n^{-2},1} \\lf[ \\qga \\ri] =   G^{\\mathrm{off}}_{0,1} \\lf[ \\qga \\ri] + \\mathcal{R},\n\\eeq\nwhere\n\\bml{\n \t\t\t|\\mathcal{R}| \\leq C \\int_{\\R^3} \\diff \\sv \\int_{\\R^3} \\diff \\tv \\: \\bigg| \\lf(s^2 + t^2 + \\tx\\frac{2}{m+1} \\sv \\cdot \\tv + n^{-2}\\ri)^{-1} - \\lf(s^2 + t^2 + \\tx\\frac{2}{m+1} \\sv \\cdot \\tv\\ri)^{-1} \\bigg| \\lf| \\qga(s) \\ri| \\lf| \\qga(t) \\ri|\t\\\\\n\t\t\t\\leq C n^{-2} \\int_{\\R^3} \\diff \\sv \\int_{\\R^3} \\diff \\tv \\: \\frac{\\lf| \\qga(s) \\ri| \\lf| \\qga(t) \\ri|}{(s^2 + t^2)^{2}}  \\leq C n^{-2} \\int_{\\R^3} \\diff \\sv \\: \\lf| \\qga(s) \\ri|^2 \\int_{1}^{\\infty} \\diff t \\: t^{-2} \\leq \\OO(n^{-2}). \\label{R1}\n\t\t}\nMoreover,\n\\begin{equation}\\label{Goff est}\n\\begin{split}\nG^{\\mathrm{off}}_{0,1} \\lf[ \\qga \\ri]\\;&=\\;\\int_{\\R} \\diff k \\: S_1(k) \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2  \\\\\n& \\leq\\; S_1(0) \\big\\| Q^{\\sharp}_{\\gamma} \\big\\|_{L^2}^2 + \\int_{\\R} \\diff k \\: \\lf( S_1(k) - S_1(0) \\ri) \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2 \\\\\n&\\leq\\; S_1(0)  \\exp \\lf\\{ \\tx{\\frac{3}{4\\gamma^2}} \\ri\\} \\lf(1 + \\OO(\\gamma)  \\ri) + C \\int_{\\R} \\diff k \\: \\sqrt{|k|} \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2 \\\\\n&=\\; - 2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\Lambda(m,2) \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf( 1 + \\OO(\\gamma)  \\ri) + C \\int_{\\R} \\diff k \\: \\sqrt{|k|} \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{S10} and the elementary estimate $S_1(k)-S_1(0) \\leq C \\sqrt{|k|}$. To estimate the last integral in \\eqref{Goff est} we observe that\n\\begin{equation}\\label{Qshes}\n\\begin{split}\nQ^{\\sharp}_{\\gamma}(k)\\;&=\\;\\pi^{-1\/4} c_{\\gamma} \\gamma^{1\/2}  \\exp\\lf\\{\\tx-\\f{1}{8 \\gamma^2} \\ri\\}    \\f{1}{\\sqrt{2\\pi}} \\int_0^{\\infty} \\!\\!\\!dx\\, \n\\exp\\lf\\{\\tx-ikx -\\f{\\gamma^2}{2} x^2 +x\\ri\\}  \\\\\n&=\\;\\pi^{-1\/4} c_{\\gamma} \\gamma^{-1\/2} \n\\exp\\lf\\{\\tx-\\f{1}{8 \\gamma^2}\\ri\\} \\bigg(\n\\exp\\lf\\{\\tx-\\f{k^2}{2\\gamma^2} -i\\f{k}{\\gamma^2} +\\f{1}{2\\gamma^2}\\ri\\} - \\f{1}{2 \\sqrt{\\pi} z}(1+r(z)) \\bigg),\n\\end{split}\n\\end{equation}\nwhere\n\\beq\nz= \\f{1-ik}{\\sqrt{2}\\, \\gamma}\\,, \\hspace{1cm} |r(z)|\\leq \\f{\\gamma^2}{\\sqrt{1+k^2}}\\,.\n\\eeq\nTherefore,\n\\begin{equation}\\label{sqrQ}\n\\begin{split}\n\\!\\!\\!\\!\\!\\!\\int_{\\bR}\\!\\!\\!dk\\, \\sqrt{|k|} |Q^{\\sharp}_{\\gamma} (k)|^2 \\;&\\leq\\;\\f{2 \\, c_{\\gamma}^2}{\\sqrt{\\pi} \\gamma}  \n\\exp\\lf\\{\\tx\\f{3}{4\\gamma^2}\\ri\\} \\int_{\\bR}\\!\\!\\! dk\\, \\sqrt{|k|} \\, \n\\exp\\lf\\{\\tx-\\f{k^2}{\\gamma^2}\\ri\\}  \n+ \\f{4 \\, c_{\\gamma}^2 \\, \\gamma}{\\pi^{3\/2}} \n\\exp\\lf\\{\\tx-\\f{1}{4\\gamma^2} \\ri\\}\n\\int_{\\bR} \\!\\!\\!dk \\, \\f{\\sqrt{|k|}}{1+\\, k^2} \\\\\n& \\leq\\;C \\, \\sqrt{\\gamma} \\; \\exp\\lf\\{\\tx\\f{3}{4 \\gamma^2} \\ri\\} \\Big( 1 + \\sqrt{\\gamma} \\, \n\\exp\\lf\\{\\tx-\\f{1}{\\gamma^2}\\ri\\} \\Big).\n\\end{split}\n\\end{equation}\nUsing \\eqref{approx Goff}, \\eqref{Goff1}, \\eqref{R1}, \\eqref{Goff est}, \\eqref{sqrQ} in  \\eqref{est F1 1} we finally obtain\n\t\t\\beq\n\t\t\tF_1[\\qnga] \\leq 2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} n \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf[ 1 - \\Lambda(m,2) + \\OO(\\sqrt{\\gamma}) + \\OO(n^{-1}) \\ri] \\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty,\n\t\t\\eeq\n\t\tif $ \\Lambda(m,2) > 1 $ and $ \\gamma $ is taken small enough (independent of $ n $).\n\n\\noindent \\underline{Case $N>2$}. \n\t\tAs mentioned at the beginning of this section, this case is more complicated, for the trial sequence $ \\xi_n $ must be antisymmetric under the exchange of any variable, i.e., $\\xi_n \\in \\ldf(\\R^{3N-3}) $, and at the same time we want $ \\hat\\xi_n(\\kv_1,\\ldots,\\kv_{N-1}) $ to behave like $ \\qnga(\\kv_1) $ once the other degrees of freedom are traced out. Looking for $\\xi_n$ matching these two requirements is an example of the well-known representability problem (see, e.g., \\cite{LS}), i.e., the search for sufficient conditions to impose on a one-particle density matrix so that it can be obtained as the reduced  density matrix of a fermionic many-body state. We remark that the solution is known only in some special cases and is non-trivial. Our choice here is a trial state that is as close as possible to an uncorrelated state, which is given by an antisymmetric wave function containing $ \\qnga $. Explicitly,\n\t\t\\beq\n \t\t\t\\label{min sequence}\n\t\t\t\\hat\\xi_n(\\kv_1, \\ldots, \\kv_{N-1}) : = \\frac{1}{\\sqrt{(N-1)!}}\n\t\t\t\\lf|\n\t\t\t\\begin{array}{ccccc}\n\t\t\t\t\\qnga(\\kv_1)\t&\t\\Xi_{\\beta,2}(\\kv_1)\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{1})\t\t\\\\\n\t\t\t\t\\qnga(\\kv_2)\t&\t\\Xi_{\\beta,2}(\\kv_2)\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{2})\t\t\\\\\n\t\t\t\t\\vdots \t&\t\\vdots \t&\t\\mbox{} \t&\t\\vdots\t\t\\\\\n\t\t\t\t\\qnga(\\kv_{N-1})\t&\t\\Xi_{\\beta,2}(\\kv_{N-1})\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{N-1})\t\n\t\t\t\\end{array}\n\t\t\t\\ri|\\,,\n\t\t\\eeq\n\t\twhere $ \\qnga $ is defined in \\eqref{qnga}, $ 0 < \\beta \\ll 1 $ is another variational parameter,\n\t\t\\beq\n\t\t\t\\label{chibm}\n\t\t\t\\chibm(\\kv) := (4\\pi)^{-1\/2} \\beta^{-3\/2} \\: \\Xi(\\beta^{-1} k) \\: \\exp\\lf\\{ i l \\varphi_k \\ri\\},\n\t\t\\eeq\n\t\t$ l \\in \\N $, $ \\Xi \\in C^{\\infty}_0(\\R^+) $ is real-valued, with support in $ (0,1) $, and such that\n\t\t\\beq\n\t\t\t\\label{G normalized}\n\t\t\t\\int_{0}^1 \\diff k \\: k^2 \\: \\Xi^2(k) = 1.\n\t\t\\eeq\n\t\tNote that, since the two functions $ \\qnga $ and $ \\chibm $, $ l > 0 $, are orthonormal by construction, the function \\eqref{min sequence} belongs to $ \\ldf(\\R^{3(N-1)}) $ and  is normalised. Moreover, the supports of $ \\qga $ and $ \\Xi $ do not intersect, which implies that the supports of $ \\qnga $ and $ \\chibm $ are disjoint as well, provided $ \\beta \\leq n $, which follows from the assumptions on $ \\beta $. \n\t\t\n\t\tWe can now evaluate $\\dqform[\\xi_n]$. We start by estimating the diagonal part. Using the exchange symmetry and the definition of $ L_{\\la} $ in \\eqref{Lla}),\n\t\t\\bml{\n \t\t\t\\label{energy est 0}\n\t\t\t\\dqform[\\xi_n] = \\al +  \\frac{1}{(N-2)!} \\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\diff \\kkv \\: L_{\\la}(\\kv_1,\\ldots,\\kv_{N-1}) \\lf| \\qnga(\\kv_1) \\ri|^2 \t\\cdot \\\\\n\t\t\t\\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-1}} \\prod_{l,j =2}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j}(\\kv_{\\tau(j)}),\n\t\t\t}\n\t\twhere $ \\mathcal{P}_{N-1} $ is the group of permutations of $ N-2 $ elements $2, \\ldots, N - 1 $ and $ \\sgn(\\sigma) $ denotes the sign of any $ \\sigma \\in \\mathcal{P}_N $. All the other terms vanish because of the integral of the product $ \\qnga(\\kv_i) \\chib(\\kv_i) $, which is pointwise zero thanks to the disjoint supports of the functions. Extracting the main factor\n\t\t\\bdm\n\t\t\t\\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la}\\,,\n\t\t\\edm\n\t\tand bounding the rest by means of the inequality\n\t\t\\beq\n\t\t\t\\label{useful ineq 1}\n\t\t\t\\sqrt{a + b} \\leq \\sqrt{|a|} + \\sqrt{|b|},\t\\hspace{1cm}\t\\mbox{for} \\:\\: a + b \\geq 0\\,,\t\n\t\t\\eeq\n\t\twe obtain \n\\begin{equation}\\label{Lla estimate}\n\\begin{split}\nL_{\\la}(\\kv_1, \\ldots, \\kv_{N-1}) \\;&\\leq\\; 2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\,\\bigg\\{ 1 + \\lf(\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la\\ri)^{-1\/2}  \\times \\\\\n& \\qquad\\qquad\\times\\bigg[ \\tx\\frac{m(m+2)}{(m+1)^2} \\disp\\sum_{i = 2}^{N-1} k_i^2 + \\tx\\frac{2m}{(m+1)^2} \\bigg| \\disp\\sum_{j > 1} \\kv_1 \\cdot \\kv_j +  \\disp\\sum_{1 < i <j} \\kv_i \\cdot \\kv_j \\bigg| \\bigg]^{1\/2} \\bigg\\}\t\\\\\n&\\leq\\;2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\bigg\\{ 1 + C_{N} \\disp\\sum_{i = 2}^{N-1} \\lf( k_i + \\sqrt{k_1k_i} \\ri) \\bigg\\}\\,.\n\\end{split}\n\\end{equation}\nThe diagonal term can be estimated as\t\n\t\t\\bml{\n \t\t\t\\label{energy est 1}\n\t\t\t\\dqform[\\xi_n] - \\al \\leq  2 \\pi^2\\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\:  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\lf| \\qnga(\\kv_1) \\ri|^2 \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2\t\\\\\n\t\t\t+ C_N \\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\lf|L_{\\la}(\\kv_1, \\kkv) - 2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\ri|  \\lf| \\qnga(\\kv_1) \\ri|^2 \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2,\n\t\t}\n\t\tthanks to the orthogonality of functions $ \\Xi_{\\beta,l} $  and  $ \\Xi_{\\beta,l^{\\prime}} $ for $ l \\neq l^{\\prime} $. Thus, by \\eqref{Lla estimate}, $ \\dqform[\\xi_n] - \\al $ is bounded from above by\n\\begin{equation}\n\\begin{split}\n\\!\\!\\!\\!\\!\\!\\!2 \\pi^2\\!\\!\\!& \\int_{\\R^3} \\diff \\kv_1 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\lf| \\qnga(\\kv_1) \\ri|^2 \\int_{\\R^{3(N-2)}} \\diff \\kkv \\bigg[ 1 + C_N \\bigg( k_2 + \\sqrt{k_1 k_2} \\bigg) \\bigg] \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2 \\\\\n&\\leq\\;2 \\pi^2 n \\int_{\\R^3} \\diff \\kv_1  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\frac{\\la}{n^2}}  \\lf| \\qga(\\kv_1) \\ri|^2 \\int_{0}^{1} \\diff k_2 \\: k_2^2\\lf[ 1 + C_N \\lf( \\beta k_2 + \\sqrt{n \\beta k_1 k_2} \\ri) \\ri] \\lf| \\Xi(k_2) \\ri|^2 \\\\\n& \\leq 2 \\pi^2 n \\tx\\frac{\\sqrt{m(m+2)}}{m+1} \\lf(1 + \\OO(n^{-1}) \\ri) \\disp\\int_{1}^{\\infty} \\diff k_1 \\: k_1^3 \\lf[ 1 + C_N \\lf( \\beta + \\sqrt{n \\beta k_1} \\ri) \\ri] \\lf| \\qga(k_1) \\ri|^2 \\\\\n& \\leq\\;2 \\pi^2 n \\tx\\frac{\\sqrt{m(m+2)}}{m+1}  \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\}  \\lf[1 + C _N \\lf( \\sqrt{n \\beta} \\exp \\lf\\{ \\tx{\\frac{9}{16 \\gamma^2}} \\ri\\} + \\gamma +  \\beta + n^{-1} \\ri) \\ri]\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{qga est}. We now compute the off-diagonal term (recall \\eqref{oqform}).  Owing to the exchange symmetry, the pre-factor $ N-1 $ cancels with the normalisation factor of $ \\xi_n $ and\n\t\t\\bml{\n \t\t\t\\label{energy est 3}\n\t\t\t\\oqform[\\xi_n] = \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kkv \\:\t\\green(\\sv, \\tv, \\kkv) \\bigg\\{ \\qnga^*(\\sv) \\qnga(\\tv) \\times \t\\\\\n\t\t\t\\times\\frac{1}{(N-2)!}\\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-1}} \\prod_{l,j =2}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j}(\\kv_{\\tau(j)}) \\bigg\\}  + \\mathcal{R}\n\t\t\n\t\t\n\t\t \n\t\t}\n\t\twhere $ \\mathcal{R} $ contains some remainder terms. We estimate the leading term (first term on the r.h.s.~of \\eqref{energy est 3}) from above by\n\t\t\\bml{\n \t\t\t\\label{energy est 4}\n\t\t\tn \\int_{\\R^{6}} \\diff \\sv \\diff \\tv \\:  G_{\\la\/n^2}(\\sv, \\tv) \\qga(s) \\qga(t) Y_1^0(\\vartheta_s) Y_1^0(\\vartheta_t)\t\\\\\n\t\t\t + C_N \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kkv \\:\t\\green(\\sv, \\tv, \\kkv) \\green(\\sv, \\tv) \\lf[ k_2^2 + (s+t) k_2 \\ri] \t\\lf| \\qnga(\\sv) \\ri| \\lf| \\qnga(\\tv) \\ri| \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2,\t\t\t}\n\t\twhere in the first term we replaced $ \\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) $ with $ \\green(\\sv, \\tv, \\zerov, \\ldots, \\zerov) = :  \\green(\\sv, \\tv) $ and we exploited the orthogonality of functions $ \\Xi_{\\beta,l} $  and  $ \\Xi_{\\beta, l^{\\prime}} $ for $ l \\neq l^{\\prime} $.\n\t\tThe first term in the expression above was bounded  in \\eqref{Goff est}.  Using\n\t\t\\beq\n\t\t\t\\label{kernel est 1}\n\t\t\t\\green(\\sv,\\tv,\\kv_2, \\ldots, \\kv_{N-1}) \\leq \\bigg[ \\frac{m}{m+1} \\big(s^2 + t^2\n\t\t\t\\big) + \\la \\bigg]^{-1}\n\t\t\\eeq \n\tand the elementary inequality $x+y \\leq \\sqrt{x^2 +1} \\sqrt{y^2 +1}$, $x,y \\geq 0$, \tthe second term in \\eqref{energy est 4} can be estimated as\n\\begin{equation}\\label{off2}\n\\begin{split}\nC_N \\int_{\\R^{3N}}& \\diff \\sv \\diff \\tv \\diff \\kkv   \\f{k_2^2 +(s+t) k_2}{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} \\big]^2}  \\lf| \\qnga(\\sv) \\ri| \\lf| \\qnga(\\tv) \\ri| \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2 \\\\\n&\\leq\\;C_N \\, n^{-1}  \\int_{\\R^{6}} \\!\\! \\diff \\sv \\diff \\tv  \\f{\\beta^2 +n \\beta (s+t) }{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} n^{-2} \\big]^2}  \\lf| \\qga(\\sv) \\ri| \\lf| \\qga(\\tv) \\ri|\\\\\n&\\leq\\; C_N n^{-1} (\\beta^2 + n \\beta)  \\bigg[ \\sup_{\\tv} \\int_{\\R^{3}} \\!\\! \\diff \\sv   \\f{1 }{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} n^{-2} \\big]^2} \\bigg] \\int_{\\R^3} \\!\\! \\diff \\pv \\, (p^2 +1) |Q_{\\gamma}(\\pv)|^2 \\\\\n&\\leq C_N (\\beta^2 + n \\beta) (1+\\gamma) \\exp \\lf\\{ \\tx\\frac{2}{\\gamma^2} \\ri\\}\\,.\n\\end{split}\n\\end{equation}\nThe rest $ \\mathcal{R} $ in \\eqref{energy est 3} contains several terms but it is not difficult to see that most of them vanish because of the disjoint supports of $ \\qnga $ and $ \\chib $. What remains is\n\\begin{equation}\\label{remainder 1}\n\\begin{split}\n[(N-3)!]^{-1} &\\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kv_2 \\cdots \\diff \\kv_{N-1} \\:\t\\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) \\: \\Xi_{\\beta,1}^*(\\sv) \\Xi_{\\beta,1}(\\tv)  \\lf| \\qnga(\\kv_2) \\ri|^2 \\times \\\\\n& \\qquad\\qquad\\times \\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-2}} \\prod_{l,j = 3}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j} (\\kv_{\\tau(j)}) \\\\ \n&\\leq\\;\tC_N \\int_{\\R^{6}} \\diff \\sv \\diff \\tv \\:  \\lf| \\Xi_{\\beta,1}(\\sv)  \\ri| \\lf| \\Xi_{\\beta,1}(\\tv) \\ri| \\leq C_N \\beta^3,\n\\end{split}\n\\end{equation}\nwhere we exploited the exchange symmetry again, as well as the properties of $ \\Xi $ (in particular $ \\supp(\\Xi) \\subset (0,1) $) and \\eqref{kernel est 1}. It is understood that the sum over permutations as well as the following factor in \\eqref{remainder 1} is not present when $N = 3 $. Putting together \\eqref{energy est 3}, \\eqref{energy est 4}, \\eqref{off2} and \\eqref{remainder 1}, we obtain \n\t\t\\beq\n \t\t\t\\label{energy est 8}\n\t\t\t\\oqform[\\xi_n] \\leq n G^{\\mathrm{off}}_{\\la\/n^{2},1} \\lf[ \\qga \\ri] + C_N \\lf(\\beta^2 + n \\beta \\ri) \\exp \\lf\\{ \\tx\\frac{2}{\\gamma^2} \\ri\\},\n\t\t\\eeq\n\t\tand finally\n\t\t\\bml{\n \t\t\t\\label{energy est final}\n\t\t\t\\qform[\\xi_n] \\leq 2 \\pi^2 n \\frac{\\sqrt{m(m+2)}}{m+1} \\exp\\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\Big\\{ 1 -  \\Lambda(m,2) \t\\\\\n\t\t\t   + C_N \\lf[ \\al \\,  n^{-1} + \\sqrt{\\gamma} + \\sqrt{n \\beta} \\exp \\lf\\{ \\tx{\\frac{9}{16 \\gamma^2}} \\ri\\} + (n^{-1} \\beta^2 +\\beta) \\exp\\lf\\{ \\tx\\frac{5}{4\\gamma^2} \\ri\\}  \\ri] \\Big\\}\\,.\n\t\t}\n\t\tBy assumption, $  1 -  \\Lambda(m,2) < 0 $ and we choose $ \\beta \\ll n^{-1} $ as, say, $ \\beta = n^{-2} $. Thus, we can always find some small $ \\gamma = \\OO(1) > 0 $, such that \n\t\t\\beq\n\t\t\t\\qform[\\xi_n]\n\t\t\t\\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty\n\t\t\\eeq\nwhich concludes the proof.\n\\end{proof}\n\n\n\n\\vspace{1cm}\n\\n\t\t\n{\\bf Acknowledgments}. M.C. acknowledges the support of the European Research Council under the European Community Seventh Framework Program (FP7\/2007-2013 Grant Agreement CoMBos No. 239694).\n\n\n\n\\vs\n\n\\section*{Appendix}\n\n\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{pro}{0}\n\\renewcommand{\\thesection}{A}\n\n\n\\label{Appendix A}\n\nHere we describe the formal procedure for the construction of the quadratic form $\\form$.   \nWe start  from the  Hamiltonian \\eqref{cm Hamiltonian} written in the Fourier space\n\\bml{\n(\\widehat{H \\psi})(\\kv_1,\\ldots, \\kv_N) = \\bigg(\\sum_{i=1}^N k_i^2 +\\f{2}{m\\!+\\!1}\\sum_{i<j} \\kv_i \\cdot \\kv_j \\bigg) \\hat{\\psi}(\\kv_1,\\ldots, \\kv_N) \\\\\n  +\\f{\\mu}{(2 \\pi)^3} \\int_{\\bR^3} \\!\\! \\diff \\sv \\,\\hat{\\psi} (\\kv_1,\\ldots, \\kv_{i-1},\\sv,\\kv_{i+1},\\ldots,\\kv_N)\n}\nand we consider the corresponding quadratic form, regularized by means of an ultra-violet cut-off\n\\bml{\n\t\\label{ren Form}\n\t\\renform[\\psi] : = \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\bigg\\{ \\sum_{i =1}^N k_i^2 + \\f{2}{(m+1)} \\sum_{i < j} \\kv_i \\cdot \\kv_j \\bigg\\} \\lf| \\hat{\\psi}(\\kv_1, \\ldots, \\kv_N) \\ri|^2\t\\\\\n\t + \\frac{\\mu(\\alpha,R)}{(2\\pi)^{3}} \\sum_{i = 1}^N   \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i)  \\hat{\\psi}^*(\\kv_1,\\ldots, \\kv_N) \\times \t\\\\\n\t\\times\\int_{\\RT} \\diff \\sv \\: \\chi_R(s) \\hat{\\psi}(\\kv_1,\\ldots, \\kv_{i-1},\\sv,\\kv_{i+1},\\ldots,\\kv_N)\\,.\n}\nHere $ \\psi \\in H^1(\\R^{3N}) $, $ \\chi_R(s) $ is the characteristic function of the three-dimensional ball $ s \\leq R $, and $ \\alpha $ is a  parameter that has the role of a renormalised coupling constant.  Note also that we introduced in $ \\mu$ a dependence on $ R $: the choice of such an explicit dependence will be the main content of the renormalisation procedure.\n\nWe now define the ``surface charges'' $ \\xi_{i}^R \\in L^2(\\R^{3N-3}) $ associated with $ \\psi \\in H^1(\\R^{3N}) $ as\n\\beq\n\t\\label{ren charges}\n\t\\hat{\\xi}_i^R(\\kv_1,\\ldots,\\kv_{N-1}) : = \\frac{\\mu(\\alpha,R)}{(2\\pi)^3} \\int_{\\RT} \\diff \\sv \\: \\chi_R(s) \\hat{\\psi}(\\kv_1,\\ldots,\\kv_{i-1},\\sv,\\kv_{i}, \\ldots, \\kv_{N-1}),\n\\eeq\nand the corresponding ``volume charges'' $\\hat{\\rho}_i^R (\\kv_1,\\ldots,\\kv_N):= \\chi_R(k_i)  \\: \\hat{\\xi}_i^R(\\breve \\kkv_i)$. Further, we introduce    the  ``potential'' \n\\beq\n\t\\label{ren potential}\n\t\\widehat{\\pot  \\rho^R}(\\kv_1, \\ldots, \\kv_N) : = \\sum_{i=1}^N \\green(\\kv_1,\\ldots,\\kv_N) \\: \\chi_R(k_i)  \\: \\hat{\\xi}_i^R(\\breve \\kkv_i),\n\\eeq\nwhere $G_{\\la}$ is defined in \\eqref{Green function} for any $ \\la > 0 $.\nSetting\n\\beq\n\t\\label{ren regular part}\n\t\\hat{\\phi}_{\\la}^R : = \\hat{\\psi} - \\widehat{\\pot \\rho^R},\n\\eeq\nwe have\n\\beq\n\t\\label{ren form decomposition}\n \t\\renform[\\psi] = \\F_0\\lf[\\phi_{\\la}^R\\ri] + \\la \\lf\\| \\phi_{\\la}^R \\ri\\|_{L^2(\\R^{3N})}^2 - \\la \\lf\\| \\psi \\ri\\|_{L^2(\\R^{3N})}^2 + \\renqform\\lf[\\xi^R\\ri],\n\\eeq\nwith $ \\F_0[\\phi] : = \\bra{\\phi} H_0 \\ket{\\phi} $, and\n\\bml{\n\t\\label{ren qform}\n\t\\renqform\\lf[\\xi\\ri] : = - \\sum_{i=1}^N \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i) \\: \\hat\\xi_i^*(\\breve \\kkv_i) \\lf[ \\hat{\\psi}(\\kv_1, \\ldots, \\kv_N) +  \\green(\\kv_1, \\ldots, \\kv_N) \\: \\hat\\xi_i(\\breve \\kkv_i) \\ri]\t\\\\\n\t- \\sum_{i < j} \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i) \\:\t\\hat\\xi_i^*(\\breve \\kkv_i) \\green(\\kv_1, \\ldots, \\kv_N) \\: \\chi_R(k_j) \\: \\hat\\xi_j(\\breve \\kkv_j)\\,.\n}\nIn the limit $ R \\to \\infty $ we assume that $\\rho^R_i , \\, \\xi^R_i \\rightarrow \\xi_i$. Moreover, we extract from the diagonal part of \\eqref{ren qform} only the terms not vanishing in that limit\n\\bmln{\n\t\\sum_{i=1}^N \\int_{\\R^{3N-3}} \\diff \\breve \\kkv_i \\: \\lf| \\hat\\xi_i (\\breve \\kkv_i) \\ri|^2 \\lf[ - \\frac{(2\\pi)^3}{\\mu(\\al,R)} - \\int_{\\RT} \\diff \\kv_i \\: \\chi_R(k_i) \\green(\\kv_1, \\ldots, \\kv_N) \\ri] \t\\\\\n\t= \\sum_{i=1}^N \\int_{\\R^{3N}}  \\diff \\breve \\kkv_i \\: \\lf| \\hat\\xi_i (\\breve \\kkv_i) \\ri|^2 \\bigg[ - \\frac{(2\\pi)^3}{\\mu(\\al,R)} - 4 \\pi R \\\\\t\n\t+ 2 \\pi^2 \\bigg[ \\frac{m(m+2)}{(m+1)^2} \\sum_{j \\neq i} k_j^2 + \\frac{2m}{(m+1)^2} \\sum_{i \\neq j} \\kv_i \\cdot \\kv_j + \\la \\bigg]^{1\/2}  + o(1) \\bigg].\n}\nIn order to remove the cut-off one is thus forced to set $ \\mu \\to 0 $ as $ R \\to \\infty $ and, although several choices are allowed, we set\n\\beq\n\t\\mu(\\al,R) : = - \\frac{(2\\pi)^3}{4 \\pi R + \\al},\n\\eeq\nthis way canceling the singular term proportional to $- 4\\pi R $ contained in the expression above. \n\nWe can now remove the cut-off taking the limit $ R \\to \\infty $ and so recovering the expression \\eqref{form}. Note that we exploit at this stage the fermionic symmetry, which in particular implies that all charges can be expressed in terms of a single function $ \\xi $, i.e., \n\\beq\n\t\\xi_i(\\xv_1,\\ldots,\\xv_{N-1}) = (-1)^{i+1} \\xi(\\xv_1,\\ldots,\\xv_{N-1}),\n\\eeq\nand $ \\xi $ itself is totally antisymmetric under exchange of coordinates. This in turns implies that the sign in front of the off-diagonal term is the opposite than in the bosonic case, implying a completely different behavior of the ground state. \n\n\n\n\n\\vs\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith the fast-paced advancement of the Artificial Intelligence (AI) and Robotics fields, there is an increasing potential to resort to\\textit{ robot assistants} (or \\textit{service robots}) to help with daily tasks. Service robots can take on many roles. They can operate as patient carers \\cite{bajones_hobbit_2018}, door-to-door garbage collectors \\cite{ferri_dustcart_2011}, Health and Safety monitors \\cite{dong_design_2018}, museum or tour guides \\cite{waldhart_reasoning_2019}, to name a few. Yet, even decades after the first robot vacuum cleaner was deployed (\\url{https:\/\/en.wikipedia.org\/wiki\/Robotic_vacuum_cleaner}), robot assistants still perform unreliably on more complex tasks. Succeeding in the real world is indeed a difficult challenge because it requires robots to make sense of the high-volume and diverse data coming through their perceptual sensors. Although different sensory modalities contribute to the robot's \\textit{sensemaking} abilities (e.g., touch, sound, temperature), in this work, we focus on the modality of vision. From this entry point, the problem then becomes one of enabling robots to correctly interpret the stimuli of their vision system, with the support of background knowledge sources, a capability also known as \\textit{Visual Intelligence} \\cite{chiatti_towards_2020}. The first prerequisite to Visual Intelligence is the ability to robustly recognise the different objects occupying the robot's environment (\\textit{object recognition}). Let us consider the case of HanS, the Health and Safety robot inspector currently under development at the Knowledge Media Institute (KMi). HanS is expected to monitor the Lab space in search of potentially dangerous situations, such as fire hazards. For instance, imagine that Hans was observing a flammable object (e.g., a paper cup) left on top of a portable heater. To conclude that it is in the presence of a potential fire hazard, the robot would first need to recognise that the cup and the portable heater are there. \n\nCurrently, the most common approach to tackling object recognition tasks is applying methods which are based on Machine Learning (ML). In particular, the state-of-the-art performance is defined by the latest approaches based on Deep Learning (DL) \\cite{liu_deep_2020,lecun_deep_2015}. Despite their popularity, these methods have received many critiques due to their brittleness and lack of transparency \\cite{marcus2018deep,parisi_continual_2019,pearl_theoretical_2018}. To compensate for these limitations, a more recent trend among AI researchers has been to combine ML with knowledge-based reasoning, thus adopting a \\textit{hybrid approach} \\cite{aditya_integrating_2019,gouidis_review_2020}. A question remains, however, on what type of knowledge resources and reasoning capabilities should be leveraged within this new class of hybrid methods \\cite{daruna2018sirok}.\n\nIn \\cite{chiatti_towards_2020}, we identified a set of \\textit{epistemic requirements}, i.e., a set of capabilities and knowledge properties, required for service robots to exhibit Visual Intelligence. We then mapped the identified requirements to the types of classification errors emerging from one of HanS' scouting rounds, where we relied solely on Machine Learning to recognise the objects. This error analysis highlighted that, in 74\\% of the cases, a more accurate object classification could in principle have been achieved if the relative size of objects was considered for their categorisation. For instance, back to HanS' case, the paper cup could be mistaken for a rubbish bin, due to its shape. However, rubbish bins are typically larger than cups. With this awareness, HanS would be able to rule out object categories, which, albeit being visual similarity to the correct class, are \\textit{implausible} from the standpoint of size.\n\nThese elements of \\textit{typicality} and \\textit{plausible reasoning} \\cite{davis_commonsense_2015} link size reasoning to the broader AI objective of developing systems which exhibit \\textit{common sense} \\cite{levesque_common_nodate}, especially with respect to understanding a set of intuitive physics rules governing the environment \\cite{hayes_second_1988,lake_building_2017}. This view is also supported by studies of human visual cognition, which suggest that our priors about the canonical size of objects play a role in how we categorise, draw and imagine objects \\cite{rosch_principles_1999,hoffman_visual_2000,konkle_canonical_2011}.\n\nOn a more practical level, knowledge representations which encode object sizes have already been applied effectively to Natural Language Processing (NLP) tasks. These include answering questions, such as, \"is object A larger than object B?\" \\cite{bagherinezhad_are_2016,elazar_how_2019}. However, despite this body of theoretical and empirical evidence, the role of size in object recognition has received little attention in the field of Computer Vision. To address this issue, in this paper we investigate the performance effects of augmenting a ML-based object recognition system both with background knowledge about the typical size of objects, as well as with a method to reason about the size of the observed objects. Namely, we propose: \n\n\\begin{itemize}\n\\item A hybrid method to validate ML-based predictions based on the typical size of objects. \n\\item A novel representation for size, which categorises objects differently along different dimensions contributing to their size (i.e., their front surface area, depth and aspect ratio). This representation also allows us to model object categories that include instances of varying size.\n\\end{itemize}\n\n\\section{Related work}\n\nState-of-the-art object recognition methods rely heavily on Machine Learning, as further discussed in Section 2.1. Because of the limitations of ML-based methods, hybrid methods, which combine ML with background knowledge and knowledge-based reasoning, have been recently proposed (Section 2.2). In particular, our hypothesis is that awareness of object size has the potential to drastically improve the performance of hybrid object recognition methods \\cite{chiatti_towards_2020}. Therefore, we will conclude our review of the literature by discussing existing approaches to representing the size of objects (Section 2.3). \n\n\\subsection{Machine Learning for Object Recognition}\nThe impressive performance exhibited by object recognition methods based on DL has led to significant advances on several Computer Vision benchmarks \\cite{liu_deep_2020,krizhevsky_imagenet_2017,he_deep_2016}. Deep Neural Networks (NNs), however, come with their own limitations. These models (i) are notoriously data-hungry, i.e., require thousands of annotated training examples to learn from, (ii) learn classification tasks offline, i.e., assume to operate in a closed world, on a fixed and pre-determined set of objects \\cite{mancini_knowledge_2019}, and (iii) learn representational patterns automatically, by iterating over a raw input set \\cite{lecun_deep_2015}. The latter trait can drastically reduce the start-up costs of feature engineering. However, it also complicates tasks such as explaining the obtained features and augmenting them with explicit knowledge statements \\cite{marcus2018deep,pearl_theoretical_2018}.\n\nThe issue of learning robust object representations to adequately reflect changes in the environment, even from minimal training examples, has inspired the development of few-shot metric learning methods. \\textit{Metric learning} is the task of learning an embedding (or feature vector) space, where similar objects are mapped closer to one another than dissimilar objects. In this setup, even objects unseen at training time can be categorised, by matching the learned representations against a support (reference) image set. In particular, in a \\textit{few-shot scenario}, the number of training examples and support images is kept to a minimum. Deep metric learning has been applied successfully to object recognition tasks \\cite{koch_siamese_2015,hoffer_deep_2015,schroff_facenet_2015}, even in real-world, robotic scenarios \\cite{zeng2018robotic}. Koch and colleagues \\cite{koch_siamese_2015} originally proposed to train two identical Convolutional Neural Networks (CNN) fed with images to be matched by similarity. This twin architecture is also known as Siamese Network. An extension of the Siamese architecture is the Triplet Network \\cite{hoffer_deep_2015,schroff_facenet_2015}, where the input data are fed as triplets including: (i) one image depicting a certain object class (i.e., \\textit{anchor}), (ii) a \\textit{positive example} of the same object, and (ii) a \\textit{negative example}, depicting a different object. The winning team for the object stowing task at the latest Amazon Robotic Challenge further tested the effects of learning weights independently on each CNN branch \\cite{zeng2018robotic}. Relaxing the weight-coupling constraint of Siamese and Triplet Networks was shown to benefit the matching of images across different visual domains, e.g., robot-collected images with product catalogue images. Hence, in what follows, we will use the two top-performing solutions in \\cite{zeng2018robotic} as a baseline to evaluate the object recognition performance of solutions which are purely based on Machine Learning. \n\n\\subsection{Hybrid Methods for Object Recognition}\nBroadly speaking, \\textit{hybrid reasoning methods} combine knowledge-based reasoning with Machine Learning. A detailed survey of hybrid methods developed to interpret the content of images can be found in \\cite{aditya_integrating_2019,gouidis_review_2020}. Many of these hybrid methods are specifically tailored on Deep NNs, which define the predominant approach to tackling object recognition problems. In this setup, background knowledge and knowledge-based reasoning can be integrated at four different levels of the NN \\cite{aditya_integrating_2019}: (i) in \\textbf{pre-processing}, to augment the training examples, (ii) within the \\textbf{intermediate layers}, (iii) as part of the \\textbf{architectural topology} or \\textbf{optimisation function}, and (iv) in the \\textbf{post-processing} stages, to validate the NN predictions.\n\nMethods in the first group rely on external knowledge to compensate for the lack of training examples. In \\cite{mancini_knowledge_2019}, auxiliary images depicting newly-encountered objects were first retrieved from the Web and then manually validated. As a result, significant supervision costs were introduced to compensate for the noisiness of data mined automatically.\n\nOther approaches have encoded the background knowledge directly in the inner layer representations of a Deep NN. In the RoboCSE framework, a set of knowledge properties of objects, (i.e., their typical location, fabrication material and affordances) were represented through multi-relational embeddings \\cite{daruna_robocse_2019}. This method was proven effective to infer the object's material, location and affordances from its class, but performed poorly on object categorisation tasks (i.e., when asked to infer the object's class from its properties).\n\nMore transparent and explainable than multi-relational embeddings, methods in the third group are either inspired by the topology of external knowledge graphs \\cite{marino_more_2017} or introduce reasoning components which are trainable end-to-end \\cite{serafini_logic_2016,manhaeve_deepproblog_2018,santoro_simple_2017,van_krieken_analyzing_2020}. Graph Search Neural Networks (GSNN) \\cite{marino_more_2017} resemble an input knowledge graph, where search seeds are selected based on a separate object detection module. In Logic Tensor Networks (LTN) \\cite{serafini_logic_2016}, entities are represented as distinct points in a vector space, based on a set of soft-logic assertions linking these entities. In this framework, symbolic rules (which may adhere to probabilistic logic - \\cite{manhaeve_deepproblog_2018}) are added as constraints to the NN's optimisation function. Authors in \\cite{santoro_simple_2017} proposed to incorporate reasoning in the form of a trainable Relational Reasoning Layer. Similarly, in \\cite{van_krieken_analyzing_2020}, differentiable knowledge statements (expressed in fuzzy logic) contribute to the training loss function, to aid digit classification on the MNIST dataset.\n\nFinally, the fourth family of hybrid methods uses knowledge-based reasoning to validate the object predictions generated through ML. In \\cite{young_towards_2016,young_semantic_2017} the results produced by the ML-based Vision module are first associated with the Qualitative Spatial Relationships extracted from the input image, and then also matched against the top-most related DBpedia concepts, if an unknown object is observed. As in the case of \\cite{santoro_simple_2017,van_krieken_analyzing_2020}, methods in this group can modularly interface with different NN architectures. Moreover, they make it possible to reason on objects unseen at training time, by querying external knowledge sources. For these reasons, in the approach proposed in this paper, knowledge-based reasoning is applied after generating the ML predictions. Because our focus is on reasoning about size, we cannot directly compare our approach against methods in \\cite{young_towards_2016,young_semantic_2017}, which focus on spatial reasoning and taxonomic reasoning (i.e., reasoning about the semantic relatedness of different object categories). \n\n\\subsection{Representing the Size of Objects}\nStudies on human visual cognition have suggested that there seems to be a set of preferred (or canonical) views which we use to mentally represent objects \\cite{rosch_principles_1999,hoffer_deep_2015}. These views have recognisable colour and shape features. Similarly, our perception of the object's sizes is influenced by a set of prototypical priors. Specifically, the \\textit{canonical size} we use to imagine and draw a certain object appears to be proportionally related to the logarithm of the object's assumed size, i.e., our \"prior knowledge about the size of objects in the world\" \\cite{konkle_canonical_2011}. Inspired by these findings, Bagherinezad et al. \\cite{bagherinezhad_are_2016} have modelled the object sizes through a log-normal distribution. The resulting distributions were then used to populate nodes in a graph, where objects which co-occurred frequently across the YFCC100M dataset \\cite{thomee_yfcc100m_2016} were linked together.\n\nThe size representation proposed in \\cite{bagherinezhad_are_2016} is both \\textit{quantitative} (i.e., expressed through statistical descriptors) and \\textit{qualitative }(i.e., smaller than or larger than, as symbolised by the graph's edges). Another quantitative representation was proposed in \\cite{elazar_how_2019}, where sizes (namely the object's length or volume) are represented as Distributions over Quantities (DoQ). In \\cite{zhu_reasoning_2014}, descriptors of the object's length are instead quantised with respect to three qualitative bins (i.e., <10in, 10\u2013100in and >100in). Compared to \\cite{zhu_reasoning_2014}, the statistical distributions in \\cite{bagherinezhad_are_2016,elazar_how_2019} can more expressively model the size variety within the same object class. Indeed, the same object class can comprise of many \\textit{instantiations} or \\textit{models} (a short novel and a dictionary are both books, although dictionaries are usually thicker). Moreover, the same object can be observed under different \\textit{appearances} (e.g., the book could be open or closed). Therefore, relying on a knowledge representation which can capture this within-class variability is crucial to ensure reuse across different real-world scenarios.\n\nAll the three approaches rely on data retrieved from the Web; this approach significantly reduces the cost of hardcoding a Knowledge Base about object sizes, but it is also more sensitive to noise. In addition, a limitation of the reviewed representations is that size is represented in one-dimensional terms, e.g., either through the object's volume or through its length. However, different physical dimensions (height, width and depth) contribute differently to characterising an object. For instance, recycling bins and coat stands may occupy a comparable volume, but bins are usually thicker than coat stands.\n\nThe size representations adopted in \\cite{bagherinezhad_are_2016,elazar_how_2019} have enabled to answer questions posed in natural language, such as \"are elephants bigger than butterflies?\". Furthermore, in \\cite{zhu_reasoning_2014}, the size features were used, among others, as an intermediate representation to predict the objects' \\textit{affordances}, or typical uses. Nonetheless, the role of size in object recognition is yet to be evaluated. In this paper, we propose a novel qualitative representation for the object's typical size, which requires minimal manual annotations and controls for the presence of noisy measurements.\n\n\\section{Methodology}\n\n\\subsection{Representing qualitative sizes in a Knowledge Base}\nWe identified 60 object categories which are commonly found in KMi, the setting in which we aim to deploy our robotic Health and Safety monitor, HanS. These include not only objects which are common to most office spaces (e.g., chairs, desktop computers, keyboards), but also Health and Safety equipment (e.g., fire extinguishers, emergency exit signs, fire assembly point signs), and objects which are, to some extent, specific to KMi (e.g., a foosball table, colorful hats from previous gigs of the KMi rock band, a KMi-branded welcome pod at the main entrance). The objective was then to associate each category in this catalogue to a series of typical size features, represented in qualitative terms. To this aim, we isolated three features contributing to the size of objects, namely their (i) \\textbf{front surface area} (i.e., the product of their width by their height), (ii) \\textbf{depth} dimension, and (iii) \\textbf{Aspect Ratio (AR)}, i.e., the ratio of their width to their height. With respect to the first dimension, we can characterise objects as \\textit{extra-small}, \\textit{small}, \\textit{medium}, \\textit{large} or \\textit{extra-large} respectively. Secondly, objects can be categorised as \\textit{flat}, \\textit{thin}, \\textit{thick}, or \\textit{bulky}, based on their depth. Thirdly, we can further discriminate objects based on whether they are \\textit{taller than wide} (\\textit{ttw}), \\textit{wider than tall} (\\textit{wtt}), or \\textit{equivalent} (\\textit{eq}), i.e., of AR close to 1. If the first two qualitative dimensions were plotted on a cartesian plane, a series of quadrants would emerge, as illustrated in Figure 1. Then, the AR can help to further separate the clusters of objects belonging to the same quadrant. For instance, doors and desks both belong to the extra-large and bulky group but doors, contrarily to desks, are usually taller than wide.\n\nHaving defined the cartesian plane of Figure 1, we can manually allocate the KMi objects to each quadrant and further sort the objects lying in the same quadrant. Sorting the objects manually ensures more reliable results than if the same information was retrieved automatically, especially given the paucity of resources encoding the relative size of objects \\cite{chiatti_towards_2020,bagherinezhad_are_2016}. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.2]{size_repr.pdf}\n  \\caption{Examples of object sorting across two dimensions: (i) the area of the front surface (the product of the width w by the height h), on the x axis; and (ii) the depth value d, on the y axis.}\n\\end{figure}\n\nMoreover, in the proposed representation, membership of each bin is mutually non-exclusive. Thus, with this representation, even classes which are extremely variable with respect to size, such as carton boxes and power cords, can be modelled. Indeed, boxes come in all sizes and power cords come in different lengths. Moreover, a box might lay completely flat, or appear bulkier, once assembled. Similarly, power cords, which are typically thinner than other pieces of IT equipment, might appear rolled up or tangled. \n\n\n\n\\subsection{Hybrid Reasoning Architecture}\n\nWe propose a modular approach to combining knowledge-based reasoning with Machine Learning for object recognition. In the proposed workflow, the knowledge of the qualitative size of objects (Section 3.1) is integrated in post-processing, after generating the ML-based object predictions. The proposed hybrid architecture is outlined in Figure 2 and consists of a series of sub-components, organised as follows. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth,trim=0 250 0 0, clip]{framework.pdf}\n  \\caption{The proposed architecture for hybrid object recognition. The knowledge-based reasoning module, which is aware of the typical size features of objects, can be modularly queried to validate the ML-based predictions.}\n\\end{figure}\n\n\\textbf{ML-based object recognition.} We rely on the state-of-the-art, ML-based object recognition methods of \\cite{zeng2018robotic}, to classify a set of pre-segmented object regions. Specifically, we classify objects by similarity to a reference image set, through a multi-branch Network. In this deep metric learning setting, predictions are ranked by increasing Euclidean (or L2) distance between each target embedding and the reference embedding set. Nevertheless, this configuration can be easily replaced by any other algorithm that provides, for each detected object, (i) a set of class predictions with an associated measure of confidence (whether similarity-based or probability-based) and (ii) the segmented image region enclosing the object. \n\n\\textbf{Prediction selection.} This checkpoint is conceived for assessing whether a ML-based prediction needs to be corrected or not. At the time of writing, we achieved good results simply by retaining those predictions which the ML algorithm is most confident about, and by running the remaining predictions through the size-based reasoner. Specifically, we avoid the knowledge-based reasoning steps if the top-1 class in the ML ranking: (i) has a ranking score smaller than $\\epsilon$ (i.e., the test image embedding lies near one of the reference embeddings, in terms of L2 distance); and also (ii) appears at least $i$ times in the top-K ranking. However, in Section 5, we also evaluate performance: (i) in the best case scenario where the ground truth labels are known and we know exactly which predictions to select for correction; and (ii) in the case where all ML-based predictions are passed to the size-based reasoner, without a pre-selection. \n\n\\textbf{Size estimation.} At this stage, the input depth image corresponding to each RGB scene is first converted to a 3D PointCloud representation. Then, statistical outliers are filtered out to reduce the impact of noisy points and extract the dense 3D region which best approximates the volume of the observed object. Specifically, all points which lie farther away than two standard deviations ($2\\sigma$) from their $n$ nearest neighbours are discarded. Because this outlier removal step is computationally expensive, especially for large 3D regions, we first sample each input PointCloud down so that 1 every $\\chi$ points is retained. Then, the Convex Hull algorithm is used to approximate the 3D box bounding the object region. Lastly, the x,y,z dimensions of the 3D bounding box are computed. Since the orientation of the object is not known a priori, we cannot unequivocally map any of the estimated dimensions to the object's real width and height. However, we can assume the object's depth to be the minimum of the returned x,y,z dimensions, due to the way depth measurements are collected through the sensor. Indeed, since we do not apply any 3D reconstruction mechanisms, we can expect that the measured depth underestimates the real depth occupied by the object.\n\n\\textbf{Size quantization.} The three dimensions obtained at the previous step are here expressed in qualitative terms. First, the two dimensions which were not marked as depth are multiplied together, giving a proxy of the object's surface area. The object is then categorised as extra-small, small, medium, large or extra-large, based on a set of cutoff thresholds $T$. Second, with respect to the estimated depth dimension, the object is categorised both as flat \/non-flat (based on a threshold $\\lambda_0$ ), and as flat, thin, thick, or bulky (based on a second set of thresholds $\\Lambda$, where $\\lambda_0 \\in \\Lambda$). Third, hypotheses are made about whether the object is taller than wide (ttw), wider than tall (wtt), or equivalent (eq), based on a cutoff $\\omega_0 \\in \\Omega$. It would be unfeasible to predict the object's Aspect Ratio from the estimated 3D dimensions, without knowing its current orientation. Therefore, we estimate the object's AR based on the width (w) and height (h) of the 2D box bounding the object region, as follows:\n\\begin{equation}\nAR = \\begin{cases}\nttw &\\text{if $h \\geq w \\land \\frac{h}{w} \\geq \\omega_0$}\\\\\nwwt &\\text{if $h<w \\land \\frac{w}{h} \\geq \\omega_0$}\\\\\neq &\\text{otherwise}\n\\end{cases}\n\\end{equation}\n\n\\textbf{Ranking validation.} The qualitative features returned by the size quantization module are then matched against the background KB introduced in Section 3.1, to identify the set of object categories which are plausible candidates for the observed object, based on their size. In Section 4, we test the effects of combining different size features (i.e., the area surface, thinness, and AR) to generate this candidate set. Ultimately, only those object classes in the original ML ranking which were validated as plausible with respect to the size are retained. \n\n\\section{Experiments}\n\\subsection{Data Preparation}\n\\subsubsection{KMi RGB Reference Set}\nFor training purposes, RGB images depicting 60 object classes commonly found in KMi were collected through a Turtlebot mounting an Orbbec Astra Pro RGB-Depth (RGB-D) monocular camera. Each object was captured against a neutral background and opportunely cropped to control for the presence of clutter and occluded parts. Coherently to our prior experimental setup \\cite{chiatti2020task}, we collected 10 images per class, with an 80\\%-20\\% training-validation split. Specifically, 5 images per class were used as anchor examples and paired up with their most similar image among the remaining 5 examples in that class (i.e., the multi-anchor switch strategy in  \\cite{zeng2018robotic}). Image similarity was assessed by: (i) extracting features from the pre-trained ResNet50 module, (ii) L2-normalising the resulting image embeddings, and (iii) computing their cosine similarity. Then, we also matched each anchor with the most similar image belonging to a different class. In this way, instead of picking negative examples randomly, i.e., the protocol followed in \\cite{zeng2018robotic}, we focused on triplets which are relatively harder to disambiguate.\n\n\\subsubsection{KMi RGB-D Test Set}\nFor testing purposes, additional RGB images and depth measurements of the KMi office environment were collected, during one of HanS' monitoring routines. As a result, the class cardinalities in this set reflect the natural, relative occurrence of objects in the observed domain. For instance, HanS is more likely to spot fire extinguishers and radiators than printers, on its scouting route. These data were initially recorded as Robot Operating System (ROS) bag files, with both RGB and depth messages logged at a 640x480 resolution (i.e., the maximum resolution allowed by the depth sensor). 622 clear RGB frames, where the robot camera was still, were manually selected from the original recording.\n\nBecause the recording of RGB and depth messages is asynchronous, each RGB frame had to be matched to its nearest depth image, within a time window of +\/- $\\mu$. Choosing a higher value for $\\mu$ increases the number of scenes for which a depth match is available, but also increases the chances that a certain scene is matched with the wrong depth measurements (e.g., because the robot has moved in the meantime). In our trials, we found that setting $\\mu$ to 0.2 seconds offered a good compromise. With this setup, a depth match was found for 509 (~82\\%) of the 622 original frames. After a visual inspection, only 2 (~0.4\\%) of the 509 depth matches were identified as inaccurate and discarded. This RGB-D set was further pruned to exclude identical images, where neither the robot's viewpoint nor the object arrangement had changed.\n\nWe annotated the remaining 213 images with respect to 60 reference object categories, through the Make Sense open-source annotation tool (https:\/\/www.makesense.ai\/). Besides labelling each object, we also annotated its bounding rectangular or polygonal region, depending on its 2D shape. Resorting to polygonal masks to segment non-rectangular or partially occluded objects allowed us to produce higher quality annotations and to mitigate the effects of marginal noise. These annotations were used to crop the original RGB images and store each object region as a separate test example. The same annotated regions were then reused to crop the depth image linked to each RGB frame. In this case, the output crop was stored as a 16bit grayscale PNG, where non-zero pixels encode the depth values in millimeters. Upon analysing the generated depth crop, we identified 19 object regions which did not enclose any depth measurement. This can happen, for instance, when the object falls outside the range of the depth sensor (i.e., 60 cm \u2013 8 m). To fairly compare the performance of the size-based reasoner (which relies on depth data) against the ML baseline, we discarded the empty depth crops from our test set, leaving us with a total 1414 object regions. \n\n\\subsection{Ablation Study}\nIn what follows, we list the different methods under evaluation and illustrate the changes introduced before each performance assessment.\n\n\\textbf{Baseline Nearest Neighbour (NN).} In this pipeline, feature vectors are extracted from a ResNet50 module pre-trained on ImageNet \\cite{he_deep_2016}, without re-training on the KMi RGB reference set. Namely, a 2048-dimensional embedding is extracted for each object region in the KMi RGB-D test set and matched to its nearest embedding in the KMi RGB reference set, in terms of L2 distance. This baseline provides us with a lower bound for the ML performance, before fine-tuning on the domain of interest.\n\n\\textbf{N-net} is the multi-branch Network which ensured the top performance on novel object classes, i.e., classes unseen at training time, in \\cite{zeng2018robotic}. In this configuration, each CNN branch is a ResNet50 module pre-trained on ImageNet. Image triplets are formed following the methodology discussed in Section 4.1.1. At training time, hyperparameters are updated so that the Triplet Loss is minimised. In this way, the Network optimisation is directed towards minimising the L2 distance between matching pairs, while also maximising the L2 distance between dissimilar pairs. At inference time, object regions in the KMi RGB-D test set are classified based on their nearest object in the KMi RGB reference set, as in the case of the baseline NN pipeline.\n\n\\textbf{K-net} is the multi-branch Network which led to the top performance on known object classes, i.e., classes seen at training time, in \\cite{zeng2018robotic}. K-net is a variation of N-net where a second loss component is added to the Triplet Loss. This auxiliary component of the loss derives from applying a SoftMax function to the last fully connected layer of the Network, to classify the input positive example over M classes. \n\n\\textbf{Hybrid (area).} This configuration follows the hybrid architecture introduced in Section 3.2. However, only the object's surface area is used as size feature to validate the ML predictions.\n\n\\textbf{Hybrid (area + flat).} With this ablation, we evaluate the effects of introducing a second feature to represent the size of objects. Specifically, here we consider not only the qualitative surface area of each object, but also whether they are flat or non-flat, based on their estimated depth. Then, the ML-based predictions are validated based on the set of object classes which both lie within the same area range and are also equally flat (or non-flat).\n\n\\textbf{Hybrid (area + thin)} is equivalent to the previous configuration, except the depth of objects is represented on a four-class scale: i.e., as flat, thin, thick, or bulky. The purpose of this ablation is testing the utility of introducing more granular bins in the y axis of Figure 1.\n\n\\textbf{Hybrid (area + flat + AR).} In this pipeline, we also integrate the qualitative Aspect Ratio (i.e., taller than wide, wider than tall, or equivalent) as a third knowledge property representing size. This setup is used for testing whether the qualitative AR can help further separating the object clusters exemplified in Figure 1.\n\n\\textbf{Hybrid (area + thin + AR) }follows the same configuration as the latter pipeline, except the object's depth is categorised as as flat, thin, thick, or bulky. \n\n\\subsection{Implementation Details}\n\n\\textbf{ML setup.} All the ML models illustrated in the previous Section were implemented in PyTorch \\cite{paszke_pytorch_2019}. Images in the KMi RGB reference set were resized to 224 \u00d7 224 frames and normalised to the same distribution as the ImageNet-1000 dataset, which was used for pre-training the ResNet50 modules in each CNN branch. The tested ablations were fine-tuned through an Adabound optimizer \\cite{luo_adaptive_2018}, over minibatches of 16 image triplets, with a learning rate set to start at 0.0001 and to trigger switching to Stochastic Gradient Descend optimization at 0.01. Parameters were updated for up to 1500 epochs, with an early stopping whenever the validation loss had not decreased for more than 100 epochs.\n\n\\noindent \\textbf{Knowledge-based reasoning parameters. }We relied on the Python Open3D library \\cite{zhou2018open3d} to process the PointCloud data and estimate the bounding 3D rectangles. During our trials we achieved the best results with threshold values set as follows. With respect to prediction selection, $\\epsilon$ was set to a distance of 0.04 and $i$ to 3 predictions, within a top-5 ranking, i.e., with $K=5$. The three cutoff sets in the size quantization module were defined so that $T=$\\{0.007,0.05,0.35,0.79\\} (with thresholds expressed in squared meters), $\\Lambda=$\\{0.1,0.2,0.4\\} (in meters) and $\\Omega$=\\{1.4 times\\}. \n\nAs an additional output of this work, we have also publicly released the RGB-D image set, annotated knowledge properties, and models used in these experiments: \\url{https:\/\/github.com\/kmi-robots\/object_reasoner}. \n\n\\section{Results and Discussion}\nPerformance on the KMi RGB-D test set was measured (i) in terms of the cross-class Accuracy, Precision (P), Recall (R) and F1 score of predictions in the top-1 result of the ranking; as well as (ii) based on the top-5 predictions in the ranking, in terms of mean Precision (P@5), mean normalised Discounted Cumulatve Gain (nDCG@5) and hit ratio (i.e., the ratio of the number of times the correct prediction appeared in the top-5 ranking to the total number of predictions). Specifically, the P, R and F1 metrics were aggregated across classes before and after weighing the averages by class support (i.e., based on the number of ground truth instances in each class). Measures@5 are unweighted. When comparing the different methods under evaluation, we prioritise improvements on the weighted F1 score, which accounts for the naturally imbalanced occurrence of classes in the test set. Moreover, at comparable top-1 results, we favour methods which provide higher quality top-5 rankings. Indeed, if the correct class was not ranked first but still appeared in the top-5 ranking, it would be easier for an additional reasoner (or human oracle) to correct the prediction.\n\n\\begin{table}\n  \\caption{ML baseline results on the KMi RGB-D test set.}\n  \\label{tab:ML}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Baseline NN & .33 &.36&.33&.25&.63&.33&.36&.23&.25&.54\\\\\n    N-net & .45 &.34&\\textbf{.40}&.31&.62&.45&.47&.33&.36&.63\\\\\n    K-net & \\textbf{.48} &\\textbf{.39}&\\textbf{.40}&\\textbf{.34}&\\textbf{.68}&\\textbf{.48}&\\textbf{.50}&\\textbf{.38}&\\textbf{.41}&\\textbf{.65}\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting only the wrong predictions.}\n  \\label{tab:best}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area) & .60&.52&.50&.47&.71&.60&.61&.43&.47&.72\\\\\n    Hybrid (area+flat) & .60&.55&.50&.48&.72&.60&.62&.44&.47&.72\\\\\n    Hybrid (area+thin) & .61&.55&.50&.48&.71&.61&.63&.44&.47&.71\\\\\n    Hybrid (area+flat+AR)& \\textbf{.62}&.59&.50&.49&\\textbf{.76}&\\textbf{.62}&\\textbf{.65}&\\textbf{.45}&\\textbf{.49}&\\textbf{.75}\\\\\n    Hybrid (area+thin+AR) & \\textbf{.62}&\\textbf{.62}&\\textbf{.51}&\\textbf{.52}&.74&\\textbf{.62}&\\textbf{.65}&\\textbf{.45}&.48&.74\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\nFirst, we evaluated all methods which are purely based on Machine Learning, i.e., before any background knowledge about the typical size of objects is integrated. The results of this first assessment are reported in Table 1. K-net is the ML baseline which led to the top performance, across all evaluation metrics. Therefore, we considered the K-net predictions as a baseline, for testing all the hybrid configurations.\n\nBecause the knowledge-based reasoner relies on different sub-modules (i.e., prediction selection, size estimation and ranking validation), and each one of these modules is likely to propagate its own errors, we initially tested performance assuming that the ground truth predictions are known and that we can accurately discern which ML predictions need to be corrected. Although unrealistic, this best-case scenario provides us with an upper bound for the reasoner's performance and aids the analysis of errors. As shown in Table 2, simply integrating knowledge about the qualitative surface area of objects already ensured a significant performance improvement, with a \\textbf{13\\%} increase of the unweighted F1 score and a \\textbf{11\\%} increase of the weighted F1 score. Overall, the best performance, both in terms of top-1 predictions as well as in terms of top-5 rankings, was achieved through the two hybrid configurations which included all the qualitative size features (i.e., surface area, thinness and AR). In particular, the unweighted F1 score increased up to \\textbf{18\\%} and the weighted F1 score up to \\textbf{15\\%}. Hence, the margin for improvement when complementing ML with size-based reasoning is significant. These results confirm the hypothesis laid in \\cite{chiatti_towards_2020}: the capability to compare objects by size and the access to background knowledge representing size play a crucial role in object categorisation. Notably, there is no significant difference between the results obtained when representing depth in binomial terms (i.e., as either flat or non-flat), as opposed to when more fine-grained categories are used (i.e., flat, thin, thick or bulky). Thus, we can hypothesise that the costs (and potential inaccuracies) associated with formalising additional priors for the objects' depth are not justified by a sufficient performance gain. \n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting all predictions.}\n  \\label{tab:worst}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area)& \\textbf{.49} &\\textbf{.34}&\\textbf{.33}&\\textbf{.29}&\\textbf{.59}&\\textbf{.49}&\\textbf{.50}&\\textbf{.37}&\\textbf{.40}&\\textbf{.62}\\\\\n    Hybrid (area+flat)& .47 &.33&.30&.27&.58&.47&.48&.37&.40&.60\\\\\n    Hybrid (area+thin)& .44 &.30&.25&.23&.53&.44&.44&.34&.37&.54\\\\\n    Hybrid (area+flat+AR)&.42 &.29&.22&.21&.58&.42&.44&.32&.35&.56 \\\\\n    Hybrid (area+thin+AR)& .39 &.28&.20&.19&.53&.39&.40&.30&.32&.50\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting only an automatically selected subset of predictions.}\n  \\label{tab:final}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area) & \\textbf{.55}&.42&\\textbf{.41}&\\textbf{.38}&.67&\\textbf{.55}&\\textbf{.57}&\\textbf{.43}&\\textbf{.46}&\\textbf{.69}\\\\\n    Hybrid (area+flat) & .54&\\textbf{.43}&.39&.37&.67&.54&.56&\\textbf{.43}&\\textbf{.46}&.68\\\\\n    Hybrid (area+thin) & .52&.39&.37&.35&.62&.52&.54&.42&.44&.64\\\\\n    Hybrid (area+flat+AR) & .53&\\textbf{.43}&.37&.36&\\textbf{.69}&.53&.56&\\textbf{.43}&.45&.68\\\\\n    Hybrid (area+thin+AR) & .51&\\textbf{.43}&.36&.36&.64&.51&.54&.41&.43&.63\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nThen, we ran all the predictions returned by the K-net method through the hybrid reasoning pipelines, without discriminating between correct and incorrect ML predictions (Table 3). This experiment is designed to evaluate the performance of the size-based reasoner in the worst-case scenario, i.e., when no assumption can be made about the ML algorithm confidence. As expected, the obtained results are comparable to or inferior to those achieved through a purely ML-based pipeline. These results indicate that applying the size priors indiscriminately led to inaccurate corrections of objects that the ML algorithm had correctly recognised. This issue is further emphasized the more size features (or knowledge priors) are introduced. As shown in Table 3, including other characteristics besides the object's surface area causes a performance drop.\n\nTherefore, to capitalise on the latent performance gains highlighted in Table 2, the ML and knowledge based outcomes need to be opportunely leveraged. To this aim, we introduced a meta-reasoning checkpoint (i.e., the prediction selection module described in Section 3.2) and automatically selected a subset of ML predictions to feed to the knowledge-based reasoner. The results of this last evaluation setup are summarised in Table 4. The tested hybrid configurations allowed to reach a compromise between the lower performance bound of Table 3 and the upper performance bound of Table 2. Specifically, the ML baseline was outperformed by up to \\textbf{4\\% }in terms of unweighted F1 and by up to \\textbf{7\\% } in terms of weighted F1. Moreover, introducing knowledge about the object's surface positively impacted the quality of the top-5 ranking: the mean P@5 and nDCG@5 both increased by \\textbf{5\\%}, and the hit ratio by \\textbf{4\\%}. Similarly to the results in Table 3, the qualitative surface area is the feature which led to the most consistent results across the different evaluation metrics. In other words, integrating additional knowledge beyond that first feature only led to comparable results, or even degraded the performance (i.e., in the case where a four-class scale instead of a binary one is used for the object's depth). Hence, in the experimental scenario of this paper, a size representation as minimalistic as indicating whether the object exposes an extra-small, small, medium, large, or extra-large front surface area is sufficient to ensure a significant boost in performance. \n\n\\section{Conclusion and Future Work}\nIn this paper we demonstrated that augmenting an object recognition pipeline based on Machine Learning with an effective representation of object sizes and a size reasoner can significantly improve the quality of predictions, as hypothesised in our prior work \\cite{chiatti_towards_2020}. These results are particularly promising, because they were achieved on image regions collected by a robot in its natural environment, i.e., in a more challenging experimental setup than the standard classification of benchmark image collections. In the proposed approach, we relied on a novel representation of the size of objects. Differently from prior knowledge representations, here we modelled size across three dimensions (the object's front surface area, depth and aspect ratio), to further separate the object clusters. Moreover, we allowed for annotating each object class with different size attributes, to adequately capture the size variability within each class.\n\nThe experiments presented in this paper also highlighted a series of directions of improvement, informing our future work. First, when estimating the object size from depth data we had to deal with hardware constraints: (i) objects falling outside the range of the depth sensors were excluded; (ii) highly reflective, absorptive or transparent materials (e.g., shiny metals, glass) altered the depth measurements. As such, access to a more advanced depth sensor would further improve the performance. Second, if the object was only partially visible in the original image, the estimated measurements (albeit accurate) would fail to represent the real object's size. Thus, incorporating the capability of moving towards the target object to refine the prediction through repeated measurements (i.e., Active Vision) is likely to benefit performance. Third, in the proposed method, we finetuned parameters through repeated trials. To increase the efficiency and scalability of our solution across different application scenarios, we will explore alternative solutions to (partially) automate this parameter tuning routine.\n\nNaturally, the relevance of background knowledge and knowledge-based reasoning for enabling Visual Intelligence spans way beyond the capability to reason about the typical size of objects. In \\cite{chiatti_towards_2020}, we have identified several other reasoners (e.g., spatial, compositional, motion-aware) which may enhance the robustness of state-of-the-art Machine Learning methods. Thus, in our future work, we will evaluate the performance impacts of integrating: (i) additional knowledge-based components, (ii) multiple sources of background knowledge, as well as (iii) effective meta-reasoning strategies, to reconcile the outcomes of different reasoners. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Statement of the problem and summary of results}\n\nLet $H$ be a matrix from the Gaussian unitary ensemble (GUE) and let each $G_i$ $(i=1,\\dots, M)$ denote a complex Ginibre matrix, i.e. a matrix with i.i.d. standard complex Gaussian entries. In this paper, we investigate the eigenvalues of the Hermitised product matrix\n\\begin{equation}\\label{W1}\nW_M = G_M^\\dagger \\cdots G_1^\\dagger H G_1 \\cdots G_M\n\\end{equation}\nunder the assumption that all matrices, $H$ and $G_i$ ($i=1,\\ldots,M$), are independent. We will see that the eigenvalues form a bi-orthogonal ensemble~\\cite{Bo98}. Furthermore, this ensemble is closely related (in a sense that will be specified in the next subsection) to the so-called Hermite Muttalib--Borodin ensemble~\\cite{Mu95,Bo98}. The latter is defined by the joint eigenvalue probability density function (PDF)\n\\begin{equation}\\label{MB-Hermite}\n\\tilde P(x_1,\\ldots,x_N)=\\frac1{\\tilde Z_N^{(M)}}\n\\prod_{1\\leq j<k\\leq N}(x_k-x_j)(x_k^{2M+1}-x_j^{2M+1})\n\\prod_{\\ell=1}^N|x_\\ell|^\\alpha e^{-x_\\ell^2},\n\\end{equation}\nwhere $\\tilde Z_N^{(M)}$ is a normalisation constant and $\\alpha$ is a non-negative constant.\n\nIt will transpire that the bi-orthogonal ensemble structure associated with the eigenvalue PDF of the product matrix~\\eqref{W1} is a corollary of the following more basic result.\n\n\\begin{thm}\\label{T1}\nLet $G$ be an $n \\times N$ ($n \\le N$) standard complex Gaussian matrix and let $A$ be an $n \\times n$\nHermitian matrix with eigenvalues $a_1,\\ldots,a_N$.\nIf the eigenvalues of $A$ are pairwise distinct and ordered as\n\\begin{equation}\\label{as}\na_1<a_2<\\cdots<a_{n_0}<0<a_{n_0+1}<\\cdots<a_n,\n\\end{equation}\nthen the PDF of the non-zero eigenvalues of matrix $X=G^\\dagger AG$ is given by\n\\begin{multline}\\label{as1}\nP^{n_0}_{n}(\\{a_j\\}_{j=1}^n;\\{x_j\\}_{j=1}^n)=\\\\\n\\prod_{l=1}^n\\frac1{|a_l|}\\frac{(x_l\/a_l)^{N-n}}{(N-l)!}\n\\prod_{1\\leq j<k\\leq n}\\frac{x_k-x_j}{a_k-a_j}\\det\\big[e^{-x_i\/a_j}\\big]_{i,j=1}^{n_0}\n\\det\\big[e^{-x_{i}\/a_{j}}\\big]_{i,j=n_0+1}^{n},\n\\end{multline}\nwhere\n\\begin{equation}\\label{as2}\nx_1< \\cdots < x_{n_0} < 0 < x_{n_0+1} < \\cdots < x_n.\n\\end{equation}\nIn particular, we see that $X$ has $n_0$ ($n-n_0$) negative (positive) eigenvalues, i.e. the same number as $A$.\nThe remaining $N-n$ eigenvalues are all identically zero.\n\\end{thm}\n\n\nWe remark that in Theorem \\ref{T1} the case of $n<N$ remains unanswered although  this  is certainly  of high  interest; see \\cite{DF06,ACLS} and references therein for a relevant  question.   \n\n\n\n\nThe rest of this paper is organised as follows:\nIn Section~\\ref{sec:product}, we use Theorem~\\ref{T1} to find the PDF for the eigenvalues of the product~\\eqref{W1} as a bi-orthogonal ensemble. Moreover, the explicit expression for the PDF is seen to reduce asymptotically to the functional form~\\eqref{MB-Hermite} specifying the Hermite Muttalib--Borodin ensemble.\nExplicit expressions for the bi-orthogonal functions are derived in Section~\\ref{sec:biortho}. Analogous to the theory of Hermite polynomials (see e.g.~(\\ref{HL}) below), we will see that it is convenient to consider bi-orthogonal functions of even and odd degree separately.\nSection~\\ref{sec:int} provides reformulations of the bi-orthogonal functions and the correlation kernel as integral representations, which are more suited for asymptotic analysis. These integral representations can also be expressed in terms of Meijer $G$-functions, and we will see that they are closely related to known formulae stemming from the product ensemble of Laguerre type.\nThe local scaling limit at the origin is derived and seen to be related to the Meijer $G$-kernel in Section~\\ref{sec:hard}. This result is also compared with the local scaling limit of the Hermite Muttalib--Borodin ensemble.\nSection~\\ref{sec:global} includes derivations of the global spectrum of the product~\\eqref{W1} as well as the Hermite Muttalib--Borodin ensembles. Since our product ensemble reduces asymptotically to the Hermite Muttalib--Borodin ensemble, they are as expected seen to have the same global spectrum, which in turn is given in terms of the Fuss--Catalan density.\nFinally, Theorem~\\ref{T1} is proven in the appendix. This theorem is an important result by itself. For this reason, we provide three separate proofs each with their own merits.\n\n\n\n\n\\subsection{First motivation: Muttalib--Borodin ensembles}\n\\label{sec:motivation}\n\nOrthogonal polynomial ensembles are point processes on (a subset of) the real line with a joint distribution given by\n\\begin{equation}\\label{jpdf-classical}\nP(dx_1,\\ldots,dx_n)=\\frac1{Z_n}\\Delta_n(\\{x\\})^2\\prod_{k=1}^nw(x_k)dx_k,\n\\end{equation}\nwhere $Z_N$ is a normalisation constant, $w(x)$ is a non-negative weight function, and $\\Delta_n(\\{x\\})$ denotes the Vandermonde determinant,\n\\begin{equation}\n\\Delta_{n}(\\{x\\})=\\det_{1\\leq i,j\\leq n}\\big[x_i^{\\,j-1}\\big]=\\prod_{1\\leq i<j\\leq n}(x_j-x_i).\n\\end{equation}\nLike the corresponding moment problem, it is often useful to distinguish between models with support on a finite, semi-infinite, and double-infinite interval. The canonical examples are the Jacobi-, Laguerre-, and Hermite-ensembles summarised in Table~\\ref{table:weights}. These ensembles are named according to the corresponding classical orthogonal polynomials. In fact, for $a\\neq0$ the latter would more appropriately be called the generalised Hermite ensemble.\n\\begin{table}[htbp]\n\\centering\n\\caption{Summary of weight functions and support for the three canonical orthogonal ensembles in random matrix theory given by the joint distribution~\\eqref{jpdf-classical}.}\n\\vspace*{.5em}\n\\label{table:weights}\n\\begin{tabular}{l@{\\qquad\\qquad}l@{\\qquad\\qquad}l}\n\\hline\\hline\nensemble  & \\hspace*{1.5em}weight & support\\\\ \\hline \\\\[-.9em]\nJacobi & $w(\\lambda)=\\lambda^a(1-\\lambda)^b$  & $\\lambda\\in(0,1)$ \\\\[.2em]\nLaguerre & $w(\\lambda)=\\lambda^ae^{-\\lambda}$ & $\\lambda\\in(0,\\infty)$ \\\\[.2em]\nHermite & $w(\\lambda)=|\\lambda|^ae^{-\\lambda^2}$ & $\\lambda\\in(-\\infty,\\infty)$\\\\[.1em] \\hline\\hline\n\\end{tabular}\n\\end{table}\n\nIn random matrix theory these three ensembles play a fundamental role as they appear as the distribution of the eigenvalues (or singular values) for the transfer (or truncated unitary) ensemble, the complex Wishart (or chiral) ensemble, and the Gaussian unitary ensemble, respectively; see e.g.~\\cite{Fo10}.\n\nA fundamental insight, which can be traced back to Wigner~\\cite{Wi57}, is that the joint distribution~\\eqref{jpdf-classical} allows an interpretation as the equilibrium measure for a one-dimensional gas of pairwise repulsive point particles in a confining potential.\nMore precisely, consider the Gibbs measure for a classical gas of $n$ point particles which are pairwise repulsive according to a two-point potential $U(x,y)$ and confined by a common one-point potential $V(x)$, i.e.\n\\begin{equation}\\label{gibbs}\nP(dx_1,\\ldots,dx_n)=\\frac{1}{Z_n}e^{-\\beta E(x_1,\\ldots,x_n)}\\prod_{k=1}^n dx_k\n\\end{equation}\nwith $\\beta$ denoting the inverse temperature and $E$ the energy functional\n\\begin{equation} \\label{H1}\nE(x_1,\\ldots,x_n)=\\frac12\\sum_{k=1}^n V(x_k)-\\sum_{1\\leq i<j\\leq n}U(x_j,x_i).\n\\end{equation}\nWe see that at $\\beta=2$ (sometimes referred to as the free fermion point) the Gibbs measure~\\eqref{gibbs} is identical to~\\eqref{jpdf-classical} provided we set $V(\\lambda)=-\\log w(\\lambda)$ and $U(x_i,x_j)=\\log|x_j-x_i|$. In this way, the eigenvalues of random matrices relate to the Boltzmann factor of a simple statistical mechanical system with one- and two-body interactions only.\n\nA recent development in random matrix theory is the study of exactly solvable product ensembles; see~\\cite{AI15} for a review.\nAs an example, let $G_i$ ($i=1,\\ldots,M$) be independent complex Ginibre matrices (matrices whose entries are i.i.d. standard complex Gaussians)\nand consider the Hermitian product\n\\begin{equation}\\label{old-product}\nG_{M}^\\dagger\\cdots G_1^\\dagger G_1\\cdots G_M.\n\\end{equation}\nFrom the work of Akemann et al.~\\cite{AKW13,AIK13}, we know that the explicit\nPDF for the eigenvalues of the matrix~\\eqref{old-product} is\n\\begin{equation}\\label{2}\nP^{(M)}(x_1,\\ldots,x_n)=\\frac1{Z_n^{(M)}}\\Delta_n(\\{x\\})\\det_{1\\leq i,j\\leq n}\\big[g_j^{(M)}(x_i)\\big],\n\\qquad x_i > 0 \\: (i=1,\\dots,n)\n\\end{equation}\nwhere $Z_n^{(M)}$ is a known normalisation constant and $g_j^{(M)}(x)$ ($j=1,\\ldots,n$) are given by certain Meijer $G$-functions.\nGenerally, such PDFs are known as polynomial ensembles~\\cite{KS14}.\n\nIt seems natural to ask whether these product ensembles also have (at least approximately) an interpretation as a Gibbs measure of the form~\\eqref{gibbs} and~\\eqref{H1}.\nHowever, unlike the Vandermonde determinant, the determinant in~\\eqref{2} cannot be evaluated as a product (for $M \\ge 2$).\nThis prohibits a literal interpretation of the eigenvalues of~\\eqref{old-product} as a statistical mechanical system\nwith only one- and two-body interactions. One could fear that this meant that there was no simple physical interpretation related to~\\eqref{2}.\nHowever, if we consider (\\ref{2}) with each $x_j$ large, the Meijer $G$-functions can be replaced by their asymptotic approximation~\\cite{Fi72}.\nAfter a change of variables, the joint density~\\eqref{2} to leading order in the asymptotic expansion becomes~\\cite{FLZ15}\n\\begin{equation}\\label{MB-laguerre}\n\\tilde P^{(M)}(x_1,\\ldots,x_n)=\n\\frac1{\\tilde Z_n^{(M)}}\\Delta_n(\\{x\\})\\Delta_n(\\{x^M\\})\\prod_{k=1}^n x_k^a\\,e^{-x_k},\n\\qquad x_k > 0 \\: (k=1,\\dots,n)\n\\end{equation}\nwhere $a$ is a known non-negative constant.\nThis does correspond to the Boltzmann factor of a statistical mechanical system with one- and two-body interactions only.\n\nA comparison between~\\eqref{2} and~\\eqref{MB-laguerre} can be done a posteriori.\nA connection between the two ensembles was first noted by Kuijlaars and Stivigny~\\cite{KS14}, who observed that the hard edge scaling limit of~\\eqref{MB-laguerre} found in~\\cite{Bo98} took the same functional form as the Meijer $G$-kernel found in the product ensemble~\\cite{KZ14}, albeit with a different choice of parameters. Due to recent progress, even more is known about the scaling limits of both models, and their similarities. Thus it has been established that the two ensembles also share the same global spectral distribution~\\cite{Mu02,BJLNS10,BBCC11,PZ11,FW15}. Furthermore, in both cases the local correlations in the bulk and near the soft edge are given by the familiar sine and Airy process, respectively~\\cite{LWZ14,Zh15}.\n\nThe ensemble~\\eqref{MB-laguerre} had, in fact, appeared in earlier random matrix literature.\nIt was first isolated by Muttalib~\\cite{Mu95}, who suggested it as a naive approximation to the transmission eigenvalues in a problem about quantum transport.\nA feature of the new interaction is that bi-orthogonal polynomials (rather than orthogonal polynomials) are needed in the study of correlation functions. Such bi-orthogonal ensembles were considered in greater generality by Borodin~\\cite{Bo98}, who devoted special attention to PDFs\n\\begin{equation}\\label{MB}\nP(x_1,\\ldots,x_n)=\\frac1{Z_n}\\prod_{j=1}^n w(x_l)\\prod_{1\\leq j<k\\leq n} \\big|x_k-x_j\\big|\\,\n\\big|\\sgn(x_k)|x_k|^{\\theta}-\\sgn(x_j)|x_j|^{\\theta}\\big|,\n\\end{equation}\nwith $\\theta>0$ and $w(x)$ representing one of the three classical weight functions from Table~\\ref{table:weights}.\nFollowing \\cite{FW15}, we will henceforth refer to these ensembles as the (Jacobi, Laguerre, Hermite) Muttalib--Borodin ensembles.\nWe note that the awkward dependence of signs in the last factor in~\\eqref{MB} disappears when the eigenvalues are non-negative (e.g. for Laguerre- and Jacobi-ensembles) and when $\\theta$ is an odd integer as in~\\eqref{MB-Hermite}.\n\nAt the time of their introduction,\nthe Muttalib--Borodin ensembles had no obvious relation to any random matrix models defined in terms of PDFs on their entries (except for the trivial case $\\theta=1$), and could merely be interpreted as a simple one-parameter generalisation of the classical ensembles.\nHowever, we now see that the Laguerre Muttalib--Borodin ensemble has a close connection\nto products of complex Gaussian random matrices~\\eqref{old-product} through the approximation~\\eqref{MB-laguerre}.\n\nKnowing that the Laguerre Muttalib--Borodin ensemble appears as an asymptotic approximation to the Gaussian product~\\eqref{old-product}, it seems natural to ask the reverse question: \\emph{Can we find product ensembles which reduce asymptotically to the Jacobi and Hermite Muttalib--Borodin ensembles?} If this is possible, it would be reasonable to say we have completed a link between the Muttalib--Borodin ensembles with classical weights and the new family of product ensembles.\n\nFor the Jacobi Muttalib--Borodin ensemble a link to products of random matrices is provided by looking at the squared singular values of a product of truncated unitary matrices~\\cite{KKS15,FW15}. In this paper, it is our aim to isolate a random matrix product structure for which the eigenvalue PDF reduces asymptotically to the functional form of the Hermite Muttalib--Borodin ensemble. This construction therefore completes the correspondence between product ensembles and the three Muttalib--Borodin ensembles with classical weights, i.e. Laguerre, Jacobi, Hermite. Furthermore, the relevant product ensemble provides by itself a new interesting class of integrable models, which unlike all previous product ensembles (see~the review \\cite{AI15}) allows for negative eigenvalues.\n\nAs the product ensemble in question must allow for negative eigenvalues, it is no longer sufficient to investigate Wishart-type matrices like~\\eqref{old-product} which are positive-definite by construction.\nIt turns out that the correct structure is the Hermitised product of a GUE matrix and $M$ complex Ginibre matrices given by~\\eqref{W1}.\nThe case $M = 1$ of~\\eqref{W1} has previously been isolated in the recent paper of Kumar~\\cite{Ku15} as an example\nof a matrix ensemble which permits an explicit eigenvalue PDF.\n\n\\subsection{Second motivation: hyperbolic Harish-Chandra--Itzykson--Zuber integrals}\n\nAnother reason that the Hermitised random matrix product~\\eqref{W1} is of particular interest is its relation to the so-called hyperbolic Harish-Chandra--Itzykson--Zuber (HCIZ) integral. By way of introduction on this point, we note that it is by now evident that the family of exactly solvable product ensembles is intimately linked to a family exactly solvable group integrals sometimes referred to as integrals of HCIZ type. For the study of products of Ginibre matrices~\\eqref{old-product} it was sufficient to know the familiar (and celebrated) HCIZ integral~\\cite{HC57,IZ80}:\n\\begin{equation}\\label{HCIZ}\n \\int_{U(N)\/U(1)^N}e^{-\\tr AVBV^{-1}}\\,(V^{-1}dV)=\\pi^{N(N-1)\/2}\n \\frac{\\det[e^{-a_ib_j}]_{i,j=1}^{N}}\n {\\prod_{1\\leq i<j\\leq N} (a_j-a_i)(b_j-b_i)},\n\\end{equation}\nwhere $(V^{-1}dV)$ denotes the Haar measure on the unitary quotient group $U(N)\/U(1)^N$, while $A$ and $B$ are Hermitian $N\\times N$ matrices with eigenvalues $a_1<\\cdots<a_N$ and $b_1<\\cdots<b_N$, respectively. However, for studies of products of spherical, truncated unitary, or coupled random matrices generalisations of the HCIZ integral are needed, see~\\cite{KKS15,AS16,Liu17} for the two latter cases. We emphasise that the product of truncated unitary matrices considered by Kieburg et al.~\\cite{KKS15} required a previously unknown generalisation of the HCIZ integral. Likewise, our study of the Hermitised random matrix product~\\eqref{W1} requires knowledge about the so-called hyperbolic HCIZ integral in which the integration on the left-hand side of~\\eqref{HCIZ} should be replaced with an integration over the pseudo-unitary group (see Section~\\ref{sec:fyodorov} for details). The study of such hyperbolic group integrals was initiated by Fyodorov~\\cite{Fy02,FS02}.\nAn interesting feature of the hyperbolic HCIZ integral is that the integration over the pseudo-unitary is non-compact, which forces us to introduce some additional constraints on the Hermitian matrices $A$ and $B$ to ensure convergence; this is a difficulty which does not arise in other HCIZ-type integrals. Finally, we mention that HCIZ-type integrals have other applications in theoretical and mathematical physics beyond products of random matrices, e.g.~the hyperbolic HCIZ integral was used to find the spectral properties of the Wilson--Dirac operator in lattice quantum chromodynamics~\\cite{KVZ13}. Moreover, HCIZ-type integrals represent a rich area of mathematical research, for example within the study of Lie groups, harmonic analysis, combinatorics, and probability (e.g. matrix-valued Brownian motion); see\ne.g.~the text \\cite{Te88a}.\n\n\n\\section{Products of random matrices and Hermite Muttalib--Borodin ensembles}\n\\label{sec:product}\n\nIn this section, we establish that the eigenvalue PDF of the matrix product~\\eqref{W1}\nis a polynomial ensemble and show that it reduces asymptotically to the Hermite Muttalib--Borodin ensemble~\\eqref{MB-Hermite}.\n\nAs stated in the introduction, the eigenvalue PDF of~\\eqref{W1} follows as a consequence of Theorem~\\ref{T1}. The idea is simple: let $A$ be a random matrix from a polynomial ensemble, i.e. it has an eigenvalue PDF of the form\n\\begin{equation}\\label{af}\nP_A(\\{a_k\\}_{k=1}^n)=\\frac1{Z_n}\\prod_{1\\leq i<j\\leq n}(a_j-a_i)\\det[w_j(a_i)]_{i,j=1}^n,\n\\end{equation}\nwhere $a_1\\leq a_{2}\\leq \\cdots\\leq  a_{n}$ are the (ordered) eigenvalues of $A$, $w_j$ ($j=1,\\ldots,n$) is a family of weight functions, and $Z_n$ is a normalisation constant. Now, let $G$ be an $n \\times N$ $(n \\le N)$ standard complex Gaussian matrix. Then Theorem~\\ref{T1} gives the eigenvalue PDF of $G^\\dagger AG$. Moreover, it is seen that this new eigenvalue PDF is also a polynomial ensemble. In other words, Theorem~\\ref{T1} provides a map from the class of polynomial ensembles into itself. Thus, we may apply Theorem~\\ref{T1} recursively to construct hierarchies of polynomial ensembles.\nLet us make this statement more precise.\n\n\\begin{lemma}\\label{C1}\nLet $G$ be an $n \\times N$ $(n \\le N)$ standard complex Gaussian matrix, and let $A$ be a random matrix from a\npolynomial ensemble with eigenvalue PDF (\\ref{af}), independent of $G$.\nThen the PDF for the non-zero eigenvalues of the random matrix product $G^\\dagger AG$ is equal to\n\\begin{equation}\\label{af2}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det_{1\\leq i,j\\leq n}\\bigg[\\int_0^\\infty \\frac{da\\,e^{-a}}{a^{n-N+1}}\\,w_{j}\\big(\\frac{x_i}{a}\\big)\\bigg]\n\\end{equation}\nwith the eigenvalues ordered  $x_1\\leq x_{2}\\leq \\cdots\\leq  x_{n}$.\n\\end{lemma}\n\n\n\\begin{proof}\nIn order to use Theorem~\\ref{T1}, we fix an $n_0\\in\\{0,1,\\ldots,n\\}$ and assume that the eigenvalues of $A$ are ordered as~\\eqref{as}.\nConsequently, the non-zero eigenvalues of $G^\\dagger AG$  can be ordered as~\\eqref{as2} almost surely.\nIt follows from the conditional eigenvalue PDF~\\eqref{as1} and~\\eqref{af} that the eigenvalue PDF of $G^\\dagger AG$ (up to $N-n$ eigenvalues which are identically zero) is given by\n\\begin{multline}\\label{recurrence-step}\n\\int_D P_A(\\{a_k\\}_{k=1}^n)P^{n_0}_{n}(\\{a_j\\}_{j=1}^n;\\{x_j\\}_{j=1}^n)\\, da_1\\cdots da_n\n=\\frac1{Z_n}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\\\\\n\\times\\int_D\\prod_{l=1}^n\\frac1{|a_l|}\\frac{(x_l\/a_l)^{N-n}}{(N-l)!}\n\\det\n\\begin{bmatrix}\n \\{e^{-x_i\/a_j}\\}_{i,j=1}^{n_0} & 0 \\\\\n 0 & \\{e^{-x_{i}\/a_{j}}\\}_{i,j=n_0+1}^{n}\n\\end{bmatrix}\n\\det[w_j(a_i)]_{i,j=1}^n\n\\, da_1\\cdots da_n,\n\\end{multline}\nwhere the domain of integration $D$ is given according to~\\eqref{as} and the eigenvalues $x_i$ ($i=1,\\ldots,n$) are ordered according to~\\eqref{as2}.\nWe note that the integral on the second line in~\\eqref{recurrence-step} is a close cousin to the well-known Andreief integral~\\cite{An83,Br55}.\nUpon expansion of the determinants, it is readily seen that~\\eqref{recurrence-step} may rewritten as\n\\begin{equation}\\label{recurrence-step2}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det\n\\begin{bmatrix}\n \\displaystyle\\bigg\\{-\\int_{-\\infty}^0\\frac{da}a (x_i\/a)^{N-n} e^{-x_i\/a}\n w_j(a)\\bigg\\}_{\\substack{i=1,\\ldots,n_0\\\\j=1,\\ldots,n}} \\\\\n \\displaystyle\\bigg\\{\\int^{\\infty}_0\\frac{da}a (x_i\/a)^{N-n} e^{-x_i\/a}\n w_j(a)\\bigg\\}_{\\substack{i=n_0+1,\\ldots,n\\\\j=1,\\ldots,n}}\n\\end{bmatrix}.\n\\end{equation}\nIf we make a change of variables $a\\mapsto -a$ in the first $n_0$ rows in the determinant in~\\eqref{recurrence-step2} then we get\n\\begin{equation}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det_{1\\leq i,j\\leq n}\\bigg[\\int_0^\\infty \\frac{da}a\\,\\frac{e^{-|x_i|\/a}}{(|{x_i}|\/a)^{n-N}}\\,w_j((\\sgn x_i)a)\\bigg];\n\\end{equation}\nrecall that $x_i$ ($i=1,\\ldots,n$) is ordered according~\\eqref{as2}. Finally, if we make another change of variables $a\\mapsto |x_i|\/a$ in the $i$-th row, then~\\eqref{af2} follows for any fixed $n_0$. Note that this result is independent of the choice of $n_0$, so we have the final result.\n\\end{proof}\n\n\\begin{remark}\nThe study of maps from the space of polynomial ensembles onto itself is an interesting endeavour, since such maps give rise to new random matrix ensembles without destroying integrability. In fact, the study of such maps is presently an active area of research in random matrix theory~\\cite{CKW15,Ku16,KK16,KR16}. Lemma~\\ref{C1} provides a new transformation to this class of maps, which cannot be obtained directly from any of the previously established transformations. We note that Lemma~\\ref{C1} includes the transformation~\\cite[Theorem 2.1]{KS14} as a special case arising when the matrix $A$ is positive definite. A restriction of Lemma~\\ref{C1} is that $n\\leq N$. Thus it is seen that the PDF~\\eqref{af2} develops a singularity for $n>N$ indicating that the formula is no longer generally valid in this case, depending on the properties of $w_j$. It would be interesting to extend the above results to include the case $n>N$ more generally.\n\\end{remark}\n\nWith Lemma~\\ref{C1} at hand, we are ready to write down the eigenvalue PDF for the product~\\eqref{W1}.\n\n\\begin{thm}\\label{cor-matrix}  Let $\\nu_0=0, \\nu_1,  \\ldots, \\nu_M$ be non-negative integers.\nSuppose that\n$H$ is an $n\\times n$ GUE matrix and  $G_1, \\ldots, G_M$  are independent  standard complex Gaussian matrices where  $G_m$ is of size $(\\nu_{m-1}+n) \\times (\\nu_{m} +n)$.\nThen the joint PDF for the non-zero eigenvalues of the matrix~\\eqref{W1} is given by\n\\begin{equation}\\label{PDF-matrix}\nP^{(M)}(x_1,\\ldots,x_n)=\\frac{1}{Z^{(M)}_n}\\prod_{1\\leq i<j\\leq n}(x_j-x_i)\\det_{1\\leq i,j\\leq n}\\big[g_{j-1}^{(M)}(x_i)\\big],\n\\end{equation}\nwhere the weight functions $g_j^{(M)}$ are defined recursively by\n\\begin{equation}\\label{weight-recursive}\ng_j^{(0)}(x)=x^{j}e^{-x^2}\n\\quad\\text{and}\\quad\ng_j^{(m)}(x)=\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_m}e^{-y}\\,g_j^{(m-1)}(x\/y),\\quad m=1,\\ldots,M\n\\end{equation}\nand the normalisation constant is\n\\begin{equation}\nZ^{(M)}_n={2^{-n(n-1)\/2}\\pi^{n\/2}}\\prod_{m=0}^M\\prod_{j=1}^n\\Gamma(\\nu_m+j).\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nFirst, let us consider the simplest situation, that is a product of square matrices, i.e. $\\nu_1=\\cdots=\\nu_M=0$. As the eigenvalue PDF of an $n\\times n$ GUE matrix is given by~\\eqref{PDF-matrix} with $M=0$, the theorem follows immediately by applying Lemma~\\ref{C1} $M$ times.\n\nWe need to be a little more careful when the case of rectangular matrices is considered.\nThe $M=1$ case of the theorem is again an immediate consequence of Lemma~\\ref{C1}, which gives us the eigenvalues of $W_1=G_1^\\dagger HG_1$. However, in order to apply Lemma~\\ref{C1} a second time and find the non-zero eigenvalues of $W_2=G_2^\\dagger W_1G_2$, we have to take into account that $W_1$ has a zero eigenvalue with multiplicity $\\nu_1$. To proceed, we can use the same idea as in~\\cite{IK14}. The unitary invariance of Gaussian matrices tells us that $W_1\\stackrel{d}=U^\\dagger W_1 U$ and $G_2\\stackrel{d}{=}VG_2$ for any $U,V\\in U(n+\\nu_1)$. It thus follows\n\\begin{equation}\\label{reduction}\nW_2=G_2^\\dagger W_1 G_2\\stackrel{d}{=}\n\\begin{bmatrix} \\tilde G_2^\\dagger & g_2^\\dagger \\end{bmatrix}\n\\begin{bmatrix} X_1 & 0 \\\\ 0 & 0 \\end{bmatrix}\n\\begin{bmatrix} \\tilde G_2 \\\\ g_2 \\end{bmatrix}\n=\\tilde G_2^\\dagger X_1 \\tilde G_2,\n\\end{equation}\nwhere $X_1=\\diag(x_1,\\ldots,x_n)$ is an $n\\times n$ diagonal matrix distributed according to~\\eqref{PDF-matrix} with $M=1$, while $\\tilde G_2$ and $g_2$ are standard Gaussian matrices of size $n\\times (n+\\nu_2)$ and $\\nu_1\\times (n+\\nu_2)$, respectively. Now, Lemma~\\ref{C1} can be applied to the right-hand side in~\\eqref{reduction}, which gives us the PDF of the non-zero eigenvalues of $W_2$. Repeating this procedure completes the proof.\n\\end{proof}\n\n\\begin{remark}\nWe note that the case $M=1$ of Theorem~\\ref{cor-matrix} is in agreement with the result stated by Kumar~\\cite[Eq.~(46) and~(47)]{Ku15}. However, the derivation therein is incomplete due to the reliance on the HCIZ integral\n(1.14), rather than its hyperbolic variant (A.38) below.\n\\end{remark}\n\nThere are many other representations for the weight functions in Theorem~\\ref{cor-matrix}\nbeyond the recursive definition~\\eqref{weight-recursive}.\nAs usual, it is particularly useful for analytic purposes to write the weight functions in their contour integral representation.\n\n\\begin{lemma}\nWe have\n\\begin{equation}\\label{contour-matrix}\ng_j^{(M)}(x)=\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^M\\Gamma(\\nu_m-s),\n\\end{equation}\nwhere $c<0$ is a negative constant.\n\\end{lemma}\n\n\\begin{proof}\nBy means of the residue theorem, it is seen that\n\\begin{equation}\ng_j^{(0)}(x)=x^{j}e^{-x^2}=\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big).\n\\end{equation}\nNow, assume that $g_j^{(M-1)}(x)$ is given by~\\eqref{contour-matrix}. From the recursive formula~\\eqref{weight-recursive}, we have\n\\begin{equation}\ng_j^{(M)}(x)=\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_M}e^{-y}\\,\\frac{(\\sgn x)^{j}}{4\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Big(\\frac{|x|}y\\Big)^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^{M-1}\\Gamma(\\nu_m-s).\n\\end{equation}\nIt is a straightforward exercise, considering the asymptotic decay, to show that\nwith $c<0$ the order of the integrals may be interchanged.   Thus we have\n\\begin{equation}\ng_j^{(M)}(x)=\n\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^{M-1}\\Gamma(\\nu_m-s)\\,\n\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_M-s}e^{-y}\n\\end{equation}\nand the lemma follows by induction.\n\\end{proof}\n\nAs already mentioned, functions with a contour integral representation like~\\eqref{contour-matrix} have certain properties which are useful for analytical purposes.\nIn fact, many of these properties may be found in the literature if we first recognise the contour integral as a Fox $H$-function\n\\begin{equation}\\label{weight-fox}\n g_j^{(M)}(x)=\\frac{(\\sgn x)^{j}}2\n\\FoxH{M+1}{0}{0}{M+1}{-}{(\\nu_1,1),\\ldots,(\\nu_M,1),(\\frac{j}2,\\frac12)}{|x|}\n\\end{equation}\nor as a Meijer $G$-function\n\\begin{equation}\\label{weight}\n g_j^{(M)}(x)=(\\sgn x)^{j}\\prod_{m=1}^M\\frac{2^{\\nu_m-1}}{\\sqrt{\\pi}}\n \\MeijerG{2M+1}{0}{0}{2M+1}{-}{\\frac{\\nu_1}{2},\\frac{\\nu_1+1}{2},\\ldots,\\frac{\\nu_M}{2},\\frac{\\nu_M+1}{2},\\frac{j}2}{\\frac{x^2}{4^M}}.\n\\end{equation}\nWe refer to the book~\\cite{MSH09} for an extensive review of these functions; the Fox $H$- and Meijer $G$-functions are defined by~\\cite[Def.~1.1]{MSH09} and~\\cite[Def.~1.5]{MSH09}, respectively.\n\nAs discussed in Section~\\ref{sec:motivation}, one of our goals is to find a `classical gas' approximation for~\\eqref{PDF-matrix}. For this purpose, we can use the asymptotic result~\\cite{Fi72}\n\\begin{equation}\\label{asymp}\n\\MeijerG{q}{0}{0}{q}{-}{b_1,\\ldots,b_q}{x}\\sim\\frac1{q^{1\/2}}\\Big(\\frac{2\\pi}{x^{1\/q}}\\Big)^{(q-1)\/2}\nx^{(b_1+\\cdots+b_q)\/{q}}e^{-qx^{1\/q}}\\big(1+O(x^{1\/q})\\big),\n\\end{equation}\nfor $x\\to\\infty$ (recall that a typical eigenvalue grows with $n$). This immediately allows us to find a Muttalib--Borodin ensemble~\\eqref{MB},\nwhich approximates the product ensemble~\\eqref{PDF-matrix}.\nHowever, for notational simplicity, it is convenient to first make a change of variables\n\\begin{equation}\\label{change}\nx_i\\mapsto x_i'=2^M\\Big(\\frac{y_i}{\\sqrt{2M+1}}\\Big)^{2M+1}\n\\end{equation}\nfor $i=1,\\ldots,n$. Using the asymptotic formula~\\eqref{asymp} and making a change of variable~\\eqref{change}, we find the approximate PDF\n\\begin{equation}\\label{hermite-MB}\n\\tilde P^{(M)}(y_1,\\ldots,y_n)=\\frac{1}{\\tilde Z^{(M)}}\\prod_{1\\leq i<j\\leq n} (y_j-y_i)(y_j^{2M+1}-y_i^{2M+1})\n\\prod_{k=1}^n|y_k|^\\alpha e^{-y_k^2},\n\\end{equation}\nwhere $\\tilde Z^{(M)}$ is a new normalisation constant and $\\alpha=\\sum_{m=1}^M(2\\nu_m+1)$.\nWe recognise~\\eqref{hermite-MB} as the Hermite Muttalib--Borodin ensemble~\\eqref{MB-Hermite}.\nNote that the approximation~\\eqref{asymp} is valid for large $x$ and that the absolute value of a typical eigenvalue grows with the matrix dimension $n$. Thus, one might suspect agreement between the two models in the large-$n$ limit except for local correlations near the origin. We will return to a comparison between the two models in Section~\\ref{sec:hard} and Section~\\ref{sec:global}.\n\n\nIt is worth noting that the following exact relation holds\n\\begin{equation}\\label{asymp-exact}\n\\MeijerG{q}{0}{0}{q}{-}{0,\\frac1q,\\ldots,\\frac{q-1}q}{x}=\\frac{(2\\pi)^{(q-1)\/2}}{q^{1\/2}}e^{-qx^{1\/q}}\n\\end{equation}\nfor integer $q$.\nThis may be proven by writing the Meijer $G$-function on the left-hand side as its integral representation and then using Gauss' multiplication formula for the gamma functions. The exact relation~\\eqref{asymp-exact} tells us that we can choose the parameters $b_k$ $(k=1,\\ldots,q)$ in~\\eqref{asymp} such that all subleading terms in the expansion vanish, and (\\ref{hermite-MB}) is exact.\n\n\n\\section{Biorthogonality and correlations}\n\\label{sec:biortho}\n\nGenerally polynomial ensembles  describe determinantal point processes. The correlation kernel may be written as\n\\begin{equation}\\label{kernel-sum}\nK_n(x,y)=\\sum_{k=0}^{n-1}\\frac{p_k(x)\\phi_k(x)}{h_k},\n\\end{equation}\nwhere $p_k(x)$ and $\\phi_k(x)$ are bi-orthogonal functions with normalisation $h_k$, i.e.\n\\begin{equation}\n\\int_{-\\infty}^\\infty dx\\, p_k(x)\\phi_k(x)=h_k\\delta_{k\\ell}.\n\\end{equation}\nFor both \\eqref{PDF-matrix} and \\eqref{hermite-MB} the $p_{k}(x)$ will be a monic polynomial of degree $k$, while\n\\begin{equation}\n\\phi_k(x)-g_k^{(M)}(x)\\in\\Span\\{g_{k-1}^{(M)}(x),\\ldots,g_{0}^{(M)}(x)\\}\n\\end{equation}\nfor the product ensemble~\\eqref{PDF-matrix} and\n\\begin{equation}\n\\phi_k(x)-x^{(2M+1)k} |x|^{\\alpha}e^{-x^2}\\in\\Span\\{x^{(2M+1)(k-1)} |x|^{\\alpha}e^{-x^2},\\ldots, |x|^{\\alpha}e^{-x^2}\\}\n\\end{equation}\nfor the Hermite Muttalib--Borodin ensemble~\\eqref{hermite-MB}. In the latter case, the bi-orthogonal structure is already known~\\cite{Ko67,Ca68,Bo98,FI16}. The main purpose of this section is to determine the bi-orthogonal functions --- and consequently the kernel~\\eqref{kernel-sum} --- for the product ensemble~\\eqref{PDF-matrix}.\n\n\\subsection{The oddness of being even}\n\nFor $M=0$, the bi-orthogonal functions are $p_n(x)=\\tilde H_n(x)$ and $\\phi_n(x)=\\tilde H_n(x)e^{-x^2}$ with $\\tilde H_n(x)$ denoting the Hermite polynomials in monic normalisation. We recall that the $n$-th Hermite polynomials is an even (odd) function when $n$ is even (odd);\nthis is due to the reflection symmetry of the Gaussian weight about the origin. A similar phenomenon is present for our product generalisation. In order to see this, we use an alternative form of the kernel~\\eqref{kernel-sum}. We have~\\cite{Co39,Bo98}\n\\begin{equation}\\label{kernel}\nK_n(x,y)=\\sum_{k,\\ell=0}^{n-1}(B_{n}^{(M)})^{-1}_{\\ell,k}\\,x^{k}g_{\\ell}^{(M)}(y),\n\\end{equation}\nwhere $B_n^{(M)}=(b_{i,j}^{(M)})_{i,j=0}^{n-1}$ is the $n$-th bi-moment matrix constructed from the bi-moments\n\\begin{equation}\\label{bimoment}\nb_{k,\\ell}^{(M)}=\\int_{-\\infty}^\\infty x^kg_\\ell^{(M)}(x)dx.\n\\end{equation}\nSimon~\\cite{Si08} refers  the inverse moment matrix representation~\\eqref{kernel} as the ABC (Aitken--Berg--Collar) theorem.\n\nIn the following, it will be useful to extend the concept of odd and even moments to bi-moments.\nWe say that the bi-moments are odd (even) when $k+\\ell$ is odd (even).\nNow, using that $g_\\ell^{(M)}(-x)=(-1)^\\ell g_\\ell^{(M)}(x)$,\nwe see that the bi-moments satisfy\n\\begin{equation}\n b_{k,\\ell}^{(M)}=(-1)^{k+\\ell}b_{k,\\ell}^{(M)},\\qquad k,\\ell=0,1,\\ldots,\n\\end{equation}\nwhich implies that all odd moments are equal to zero. In other words, the entries in the bi-moment matrix vanishes in a chequerboard pattern. Thus, by reordering rows and columns we may write the bi-moment matrix in a block diagonal form\n$B_n^{(M)}\\mapsto \\diag(B_n^{(M,\\text{even})},B_n^{(M,\\text{odd})})$ with $B_n^{(M,\\text{even})}=(b_{2k,2\\ell}^{(M)})_{k,\\ell}$ and\n$B_n^{(M,\\text{odd})}=(b_{2k+1,2\\ell+1}^{(M)})_{k,\\ell}$.\nUsing this reordering in the sum~\\eqref{kernel}, we see that the kernel splits into two parts\n\\begin{equation}\nK_n(x,y)=K_n^\\text{even}(x,y)+K_n^\\text{odd}(x,y),\n\\end{equation}\nwhere\n\\begin{align}\nK_n^\\text{even}(x,y)&=\\sum_{k,\\ell=0}^{\\lfloor \\frac{n-1}2\\rfloor}(B_n^{(M,\\text{even})})^{-1}_{\\ell,k}\\,x^{2k}g_{2\\ell}^{(M)}(y)\n=\\sum_{k=0}^{\\lfloor \\frac{n-1}2\\rfloor} \\frac{p_{2k}(x)\\phi_{2k}(x)}{h_{2k}}, \\label{kernel-even} \\\\\nK_n^\\text{odd}(x,y)&=\\sum_{k,\\ell=0}^{\\lfloor \\frac n2\\rfloor-1}(B_n^{(M,\\text{odd})})^{-1}_{\\ell,k}\\,x^{2k+1}g_{2\\ell+1}^{(M)}(y)\n=\\sum_{k=0}^{\\lfloor \\frac n2\\rfloor-1} \\frac{p_{2k+1}(x)\\phi_{2k+1}(x)}{h_{2k+1}}. \\label{kernel-odd}\n\\end{align}\nHere, the latter equality in both~\\eqref{kernel-even} and~\\eqref{kernel-odd} follow from comparison with~\\eqref{kernel-sum}.\nFinally, we note that\n\\begin{equation}\nK_{2n}^\\text{even}(x,y)=K_{2n-1}^\\text{even}(x,y)\n\\qquad\\text{and}\\qquad\nK_{2n}^\\text{odd}(x,y)=K_{2n+1}^\\text{odd}(x,y).\n\\end{equation}\nThus, in the following we can restrict our attention to kernels with an even subscript.\n\n\n\\subsection{Bi-orthogonal functions}\n\nWe are now ready to write down the bi-orthogonal functions for our product ensemble. As explained in the previous subsection, it is convenient to consider functions of odd and even degree separately.\n\\begin{prop}\\label{thm:bi-func-sum}\nThe ensemble defined by Theorem~\\ref{cor-matrix} is bi-orthogonalised by\n\\begin{align}\np_{2n}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\n\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+1)}{\\Gamma(\\nu_m+2\\ell+1)}x^{2\\ell}, &\n\\phi_{2n}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n)!}{(2\\ell)!}g_{2\\ell}^{(M)}(x), \\nonumber \\\\\np_{2n+1}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\n\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+2)}{\\Gamma(\\nu_m+2\\ell+2)}x^{2\\ell+1}, &\n\\phi_{2n+1}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n+1)!}{(2\\ell+1)!}g_{2\\ell+1}^{(M)}(x), \\nonumber \\\\\nh_n&=2^{-n}\\pi^{1\/2}\\prod_{m=0}^M\\Gamma(\\nu_m+n+1)\\label{h_n},\n\\end{align}\nwith notation as above (recall that $\\nu_0=0$).\n\\end{prop}\n\nThere are several different approaches to prove Proposition~\\ref{thm:bi-func-sum}. Here, we will present a method which emphasizes the relation to the Hermite polynomials (see~\\cite[Prop.~3.5]{Ip15} for the same method applied to the product ensemble of Laguerre type). In order to use this approach, it is convenient to first calculate the bi-moments.\n\n\\begin{lemma}\nThe bi-moments are given by\n\\begin{equation}\\label{bimoment-gamma}\nb_{k,\\ell}^{(M)}=\\Gamma\\Big(\\frac{k+\\ell+1}{2}\\Big)\\prod_{m=1}^M\\Gamma(\\nu_m+k+1),\n\\end{equation}\nfor $k+\\ell$ even and zero otherwise.\nThe bi-moment determinant is\n\\begin{equation}\\label{bi-det}\nD_n^{(M)}:=\\det_{0\\leq k,\\ell\\leq n}[b_{k\\ell}^{(M)}]=2^{-n(n+1)\/2}\\pi^{(n+1)\/2}\\prod_{m=0}^M\\prod_{j=0}^n\\Gamma(\\nu_m+j+1).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe insert the contour integral representation of the weight functions~\\eqref{contour-matrix} into the expression for the bi-moments~\\eqref{bimoment}, then we see that the even moments are\n\\begin{equation}\nb_{k\\ell}=\\frac{1}{2\\pi i}\\int_0^\\infty dx\\int_{c-i\\infty}^{c+i\\infty} ds \\,x^s\\,\\Gamma\\Big(\\frac{k+\\ell-s}{2}\\Big)\n\\prod_{m=1}^M\\Gamma(\\nu_m+k-s).\n\\end{equation}\nThe integrals in this expression can be recognised as a combination of a Mellin and an inverse Mellin transform, which yields~\\eqref{bimoment-gamma}.\n\nIn order to evaluate the bi-moment determinant~\\eqref{bi-det}, we note that\n\\begin{equation}\\label{bimoment-det}\nD_n^{(M)}=\\prod_{m=1}^M\\prod_{j=0}^n\\Gamma(\\nu_m+j+1)\\det_{0\\leq k,\\ell\\leq n}[b_{k\\ell}^{(M=0)}].\n\\end{equation}\nThis completes the proof, since the $M=0$ case is the well-known Hermite (or GUE) case.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{thm:bi-func-sum}]\nThe bi-orthogonal functions may be expressed by means of their bi-moments exactly as orthogonal polynomials through their moments.\nThus, we have\n\\begin{equation}\np_n(x)=\\frac{1}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M)} \\bigg\\vert\\, x^i\n\\end{bmatrix}\n\\qquad\\text{and}\\qquad\n\\phi_n(x)=\\frac{1}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n-1\\\\j=0,\\ldots,n}}\n\\bigg[\\begin{array}{@{}cc @{}}\nb_{i,j}^{(M)}  \\\\ \\hline\ng_j^{(M)}(x)\n\\end{array}\\bigg].\n\\end{equation}\nFurthermore, we have $h_n=D_n^{(M)}\/D^{(M)}_{n-1}$.\n\nThe constants $h_n$ are immediate from the above and~\\eqref{bi-det}. Thus, it remains only to find the bi-orthogonal functions. To do so, we first note that\n\\begin{align}\np_n(x)&=\\frac{\\prod_{k=0}^n\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M=0)} \\bigg\\vert\\, \\displaystyle\\frac{x^i}{\\prod_{m=1}^M\\Gamma(\\nu_m+i+1)}\n\\end{bmatrix},\\\\\n\\phi_n(x)&=\\frac{\\prod_{k=0}^{n-1}\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n-1\\\\j=0,\\ldots,n}}\n\\bigg[\\begin{array}{@{}cc @{}}\nb_{i,j}^{(M=0)}  \\\\ \\hline\ng_j^{(M)}(x)\n\\end{array}\\bigg].\n\\end{align}\nThis observation is important, since we know that the monic Hermite polynomials (with respect to the weight $e^{-x^2}$) are given by\n\\begin{equation}\n\\tilde H_n(x)=2^{-n}H_n(x)=\n\\frac{1}{D_{n-1}^{(M=0)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M=0)} \\bigg\\vert\\, x^i\n\\end{bmatrix}.\n\\end{equation}\nIt follows that the expressions for the bi-orthogonal function $p_n(x)$ and $\\phi_n(x)$ can be found using the known expressions for the Hermite polynomials and then making substitutions\n\\begin{equation}\nx^k\\mapsto \\frac{x^k}{\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}\n\\qquad\\text{and}\\qquad\nx^k\\mapsto g_k^{(M)}(x),\n\\end{equation}\nrespectively. We recall that\n\\[\n\\tilde H_{2n}(x)=\\sum_{\\ell=0}^n\\frac{(-1)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n)!}{(2\\ell)!}\\frac{(2x)^{2\\ell}}{2^{2n}}\n\\quad\\text{and}\\quad\n\\tilde H_{2n+1}(x)=\\sum_{\\ell=0}^n\\frac{(-1)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n+1)!}{(2\\ell+1)!}\\frac{(2x)^{2\\ell+1}}{2^{2n+1}},\n\\]\nwhich makes it a straightforward exercise to verify the proposition.\n\\end{proof}\n\n\\begin{remark}\nWe recall that the bi-orthogonal functions also can be obtained from the characteristic polynomial using that\n\\begin{equation}\np_N(x)=\\big\\langle\\det[x\\mathbb I_N-W_M]\\,\\big\\rangle\n\\qquad\\text{and}\\qquad\n\\int_{\\mathbb R} dx\\,\\frac{\\phi_{N-1}(x)}{z-x}=\\Big\\langle\\,\\frac1{\\det[z\\mathbb I_N-W_M]}\\,\\Big\\rangle\n\\end{equation}\nwith $\\langle\\cdots\\rangle$ denoting the matrix average and $z\\in\\mathbb C\\setminus\\mathbb R$. The first relation allows for an alternative method to calculate $p_N(x)$; see e.g.~\\cite{FL15}.\n\\end{remark}\n\n\n\\section{Integral representations and correlation kernels}\n\\label{sec:int}\n\nThe explicit expressions for the bi-orthogonal functions given by Proposition~\\ref{thm:bi-func-sum} allow us to write down an explicit form for the correlation kernel by insertion in~\\eqref{kernel-sum}. However, this formulation of the kernel is not optimal for asymptotic analysis. For this reason, in this section we will provide integral representations of the bi-orthogonal functions as well as the kernel.\n\n\\begin{prop}\\label{prop:bi-func-int}\nThe bi-orthogonal functions given by Proposition~\\ref{thm:bi-func-sum} have integral representations\n\\begin{align}\np_{2n}(|x|)&=\\frac{\\sqrt\\pi(2n)!}{(-1)^n2^{2n}}\\frac{1}{2\\pi i}\\oint_\\Sigma ds\\,|x|^{2s}\\,\n\\frac{\\Gamma(-s)}{\\Gamma(n+1-s)\\Gamma(s+\\frac12)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2n+1)}{\\Gamma(\\nu_m+2s+1)}, \\label{p2n-int} \\\\\np_{2n+1}(|x|)&=\\frac{\\sqrt\\pi(2n+1)!}{(-1)^n2^{2n+1}}\\frac{1}{2\\pi i}\\oint_\\Sigma ds\\,|x|^{2s+1}\\,\n\\frac{\\Gamma(-s)}{\\Gamma(n+1-s)\\Gamma(s+\\frac32)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2n+2)}{\\Gamma(\\nu_m+2s+2)}, \\label{p2n1-int} \\\\\n\\frac{\\phi_{2n}(|x|)}{h_{2n}}&=\\frac{(-1)^n\\,2^{2n}}{\\sqrt\\pi(2n)!}\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} dt\\,\n|x|^{-2t-1}\\frac{\\Gamma(n-t)\\Gamma(t+\\frac12)}{\\Gamma(-t)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2n+1)}, \\\\\n\\frac{\\phi_{2n+1}(|x|)}{h_{2n+1}}&=\\frac{(-1)^n\\,2^{2n+1}}{\\sqrt\\pi(2n+1)!}\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} dt\\,\n|x|^{-2t-2}\\frac{\\Gamma(n-t)\\Gamma(t+\\frac32)}{\\Gamma(-t)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+2)}{\\Gamma(\\nu_m+2n+2)},\n\\end{align}\nwhere the contour $\\Sigma$ encloses the integers $0,1,\\ldots,n$ in the negative direction and $-\\frac12<c<0$.\nWe recall that $p_{2n}(x)$ and $\\phi_{2n}(x)$ are even functions, while $p_{2n+1}(x)$ and $\\phi_{2n+1}(x)$ are odd functions.\n\\end{prop}\n\n\\begin{proof}\nThe integrands in~\\eqref{p2n-int} and~\\eqref{p2n1-int} has $n+1$ simple poles located at $0,1,\\ldots,n$, thus the series representations in Proposition~\\ref{thm:bi-func-sum} follow upon a straightforward application of the residue theorem.\n\nIn order to find the integral representation of the bi-orthogonal functions $\\phi_n(x)$, we first note that the weight functions can be written as\n\\begin{equation}\\label{weight-int-proof}\ng_\\ell^{(M)}(x)=(\\sgn x)^\\ell\\int_0^\\infty \\frac{dy}{y}\\,y^\\ell\\,e^{-y^2}\n\\MeijerG{M}{0}{0}{M}{-}{\\nu_1,\\ldots,\\nu_M}{\\frac{|x|}{y}};\n\\end{equation}\nthis is easily seen starting from the recursive definition~\\eqref{weight-recursive}. Now, using~\\eqref{weight-int-proof} in the expression for $\\phi_n(x)$ (cf. Proposition~\\ref{thm:bi-func-sum}), we see that\n\\begin{equation}\n\\phi_n(x)=\\int_0^\\infty \\frac{dy}{y}e^{-y^2}\\tilde H_n(y)\n\\MeijerG{M}{0}{0}{M}{-}{\\nu_1,\\ldots,\\nu_M}{\\frac{|x|}{y}}.\n\\end{equation}\nIn other words, the bi-orthogonal functions $\\phi_n(x)$ are an integral transform of the Hermite polynomials with respect to a Meijer $G$-function as integral kernel. The Hermite polynomial can itself be expressed as a Meijer $G$- or Fox $H$-function (see~\\cite[Sec.~1.8.1.]{MSH09}) and the remaining integral is well-known from the literature~\\cite[Sec.~2.3.]{MSH09}.\n\\end{proof}\n\n\n\\begin{prop}\\label{prop:kernel-finite}\nIntegral representations of the even and odd kernels are\n\\begin{align}\n K_{2n}^\\textup{even}(x,y)&=\\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t}\n \\frac{\\Gamma(-\\frac s2)\\Gamma(\\frac{t+1}2)}{\\Gamma(-\\frac t2)\\Gamma(\\frac{s+1}2)}\\frac{\\Gamma(\\frac {2n-t}2)}{\\Gamma(\\frac{2n-s}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)}, \\nonumber \\\\\n K_{2n}^\\textup{odd}(x,y)&=\\frac{\\sgn(xy)}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{s}\\,|y|^{-t-1}}{s-t}\\,\n \\frac{\\Gamma(\\frac{1-s}2)\\Gamma(\\frac{t+2}2)}{\\Gamma(\\frac{1-t}2)\\Gamma(\\frac{s+2}2)}\n \\frac{\\Gamma(\\frac{2n-t+1}2)}{\\Gamma(\\frac{2n-s+1}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)},\n\\end{align}\nwith $-1<c<0$ and the contour $\\Sigma$ is chosen such that it encircles $0,1,\\ldots,2n-1$ in the negative direction with $\\textup{Re}\\{s\\}>c$ for all $s\\in\\Sigma$.\n\\end{prop}\n\n\\begin{proof}\nAs the proofs for the odd and even kernels are almost identical, we provide only the proof for the even case. The odd case is easily verified by the reader.\n\nIt follows the definition of the even kernel~\\eqref{kernel-even} together with contour integral representation of the bi-orthogonal functions from Proposition~\\ref{prop:bi-func-int}, that\n\\begin{equation}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int dt\\oint_\\Sigma ds\\,|x|^{2s}|y|^{-2t-1}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}\n \\sum_{k=0}^{n-1}\\frac{\\Gamma(k-t)}{\\Gamma(k+1-s)}.\n\\end{equation}\nFollowing similar steps as in~\\cite{KZ14}, we note the sum allows a telescopic evaluation. This gives\n\\begin{multline}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\frac{\\Gamma(n-t)}{\\Gamma(n-s)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}\\\\\n -\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(t+\\frac12)}{\\Gamma(s+\\frac12)}\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}.\n\\end{multline}\nHere, the integrand on the second line is zero as it has no poles encircled by the contour $\\Sigma$ and, thus\n\\begin{equation}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\frac{\\Gamma(n-t)}{\\Gamma(n-s)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}.\n\\end{equation}\nFinally, the proposition follows by a change of variables $s\\mapsto s\/2$ and $t\\mapsto t\/2$.\n\\end{proof}\n\n\nThe above integral representations for the bi-orthogonal functions and the kernel are probably the most convenient form for further asymptotic analysis, as we will see in Section~\\ref{sec:hard}. However, it is also often helpful to express these formulae in terms of special functions as (for example) it allows for use of pre-defined mathematical software. Furthermore, such reformulations often guide us to recognise patterns which otherwise would have been left unseen.\n\nThe integral representations for the bi-orthogonal functions given by Proposition~\\ref{prop:bi-func-int} can also be recognised as several different types of special function; this includes generalised hypergeometric, Meijer $G$-, and Fox $H$-functions. Here, we will restrict ourselves to their Meijer $G$-function formulation.\n\nLet us first consider the bi-orthogonal polynomials which may be written as\n\\begin{align}\np_{2n}(x)&=\\frac{(-1)^n}{2^{2n}}\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+1)}{2^{\\,\\nu_m}\\pi^{-1\/2}}  \\nonumber\n\\MeijerG{1}{0}{1}{2M+2}{n+1}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2+\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2+\\frac12}{\\frac{x^2}{2^{2M}}}, \\\\\n\\frac{p_{2n+1}(x)}x&=\\frac{(-1)^n}{2^{2n}}\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+2)}{2^{\\,\\nu_m+1}\\pi^{-1\/2}}\n\\MeijerG{1}{0}{1}{2M+2}{n+1}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2-\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2-\\frac12}{\\frac{x^2}{2^{2M}}}.\n\\label{p_2n-meijer}\n\\end{align}\nIt is worth comparing these polynomials with the polynomial found in the study of the Laguerre-like matrix product~\\eqref{old-product}.\nAkemann et al.~\\cite{AIK13} found that in this case the bi-orthogonal polynomial is given by\n\\begin{equation}\nP_n^{(M)}(x)=(-1)^n\\prod_{m=0}^M\\Gamma(\\nu_m+n+1)\n\\MeijerG{1}{0}{0}{M+1}{n+1}{-\\nu_0,-\\nu_1,\\ldots,-\\nu_M}{x}.\n\\end{equation}\nIt is clear that the two families of polynomials are related as\n\\begin{equation}\np_{2n}(x)\\propto P_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\qquad\\text{and}\\qquad\np_{2n+1}(x)\\propto xP_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\end{equation}\nwith\n\\begin{equation}\\label{nu-map}\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}-1)\/2\\}_{m=0}^M\n\\qquad\\text{and}\\qquad\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}+1)\/2\\}_{m=0}^M,\n\\end{equation}\nrespectively.\nThis is a generalisation of the relation between Hermite and Laguerre polynomials. Recall that\n\\begin{equation}\\label{HL}\n\\tilde H_{2n}(x)=\\tilde L^{(-\\frac12)}_n(x^2)\n\\qquad\\text{and}\\qquad\n\\tilde H_{2n+1}(x)=x\\tilde L^{(+\\frac12)}_n(x^2),\n\\end{equation}\nwhere $\\tilde H_n(x)$ and $\\tilde L_n^{(\\alpha)}(x)$ denote the Hermite and Laguerre polynomials in monic normalisation.\n\nLikewise, the (non-polynomial) bi-orthogonal functions may be written as\n\\begin{align}\n\\frac{\\phi_{2n}(|x|)}{|x|}&=(-1)^n\\prod_{m=1}^M\\frac{2^{\\nu_m-2}}{\\pi^{1\/2}}\n\\MeijerG{2M+1}{1}{1}{2M+2}{-n}{\\frac{\\nu_M}2-\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2-\\frac12,\\frac{\\nu_0}2}{\\frac{x^2}{2^{2M}}}, \\\\\n\\phi_{2n+1}(|x|)&=(-1)^n\\prod_{m=1}^M\\frac{2^{\\nu_m-1}}{\\pi^{1\/2}}\n\\MeijerG{2M+1}{1}{1}{2M+2}{-n}{\\frac{\\nu_M}2+\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2+\\frac12,\\frac{\\nu_0}2}{\\frac{x^2}{2^{2M}}}.\n\\end{align}\nAgain, we want to compare to the formula in~\\cite{AIK13} which this time reads\n\\begin{equation}\n\\Phi_n^{(M)}(x)=(-1)^n\\MeijerG{M}{1}{1}{M+1}{-n}{\\nu_M,\\ldots,\\nu_1,\\nu_0}{x}.\n\\end{equation}\nEvidently, we have the following relations\n\\begin{equation}\\label{phi-relations}\n\\phi_{2n}(|x|)\\propto |x|\\Phi_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\qquad\\text{and}\\qquad\n\\phi_{2n+1}(|x|)\\propto \\Phi_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big),\n\\end{equation}\nwith~\\eqref{nu-map} as before.\nYet again, this is a generalisation of the relation between Hermite and Laguerre polynomials. In the simplest case the relations~\\eqref{phi-relations} reduces to\n\\begin{equation}\n\\tilde H_{2n}(x)w_\\text{H}(x)=|x|\\tilde L^{(-\\frac12)}_n(x^2)w_\\text{L}^{(-\\frac12)}(x^2)\n\\qquad\\text{and}\\qquad\n\\tilde H_{2n+1}(|x|)w_\\text{H}(x)=\\tilde L^{(+\\frac12)}_n(x^2)w_\\text{L}^{(\\frac12)}(x^2), \\nonumber\n\\end{equation}\nwhere $w_\\text{H}(x)=e^{-x^2}$ and $w_\\text{L}^{(\\alpha)}(x)=x^\\alpha e^{-x}$ are the Hermite and Laguerre weight functions.\n\nIt is, of course, well-known that there are relations between ensembles with reflection symmetry about the origin and ensembles on the half-line (albeit explicit formulae may be elusive). A general description of such relations in the Muttalib--Borodin ensembles can be found in~\\cite{FI16}.\n\n\n\n\n\\section{Scaling limits at the origin in product and Muttalib--Borodin ensembles}\n\\label{sec:hard}\n\n\nWith the integral representations of the correlation kernels established by Proposition~\\ref{prop:kernel-finite}, we can turn to a study of asymptotic properties. Perhaps the most interesting scaling regime is that of the local correlations near the origin, referred to as the hard edge when the eigenvalues are strictly positive. For other product ensembles~\\cite{KZ14,Fo14,KKS15,FL16}, it has been observed that correlations at the hard edge is determined by the so-called Meijer $G$-kernel, which generalises the more familiar Bessel kernel. Below, we will see that the Meijer $G$-kernel  appears once again, but this time\ninvolving a sum.\n\n\\begin{thm}\\label{thm:hard} Let $K_n(x,y)=K_n^\\textup{even}(x,y)+K_n^\\textup{odd}(x,y)$ with the even and odd kernels given by Proposition~\\ref{prop:kernel-finite}. For $x,y\\in\\mathbb R\\setminus\\{0\\}$ and $\\nu_1\\ldots,\\nu_M$ fixed, the microscopic limit near the origin is\n\\begin{equation}\\label{hard-limit}\n\\lim_{n\\to\\infty}\\frac{1}{\\sqrt n}K_{2n}\\Big(\\frac x{\\sqrt n},\\frac y{\\sqrt n}\\Big)\n=K^\\textup{even}(x,y)+K^\\textup{odd}(x,y)\n\\end{equation}\nwith\n\\begin{align}\nK^\\textup{even}(x,y)\n&=\\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t}\n \\frac{\\Gamma(-\\frac s2)\\Gamma(\\frac{t+1}2)}{\\Gamma(-\\frac t2)\\Gamma(\\frac{s+1}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)} \\label{hard-even} \\\\\nK^\\textup{odd}(x,y)&=\\frac{\\sgn(xy)}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}\\,|y|^{-t-1}}{s-t}\\,\n \\frac{\\Gamma(\\frac{1-s}2)\\Gamma(\\frac{t+2}2)}{\\Gamma(\\frac{1-t}2)\\Gamma(\\frac{s+2}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)} \\label{hard-odd},\n\\end{align}\nwhere $-1<c<-1\/2$ and $\\Sigma$ encloses the non-negative half-line in the negative direction starting and ending at $+\\infty$ such that $\\textup{Re}\\{s\\}>c$ for all $s\\in\\Sigma$.\n\\end{thm}\n\n\\begin{proof}\nWe only consider the even kernel in Proposition~\\ref{prop:kernel-finite} since the odd case is very similar.  After rescaling we rewrite  the integral representation of the even kernel in Proposition~\\ref{prop:kernel-finite} as\n\\begin{align}\\label{5.4}\n\\frac 1{\\sqrt n} K_{2n}^\\textup{even}(\\frac x{\\sqrt n},\\frac y{\\sqrt n})= \\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t} \\frac{f_{n}(s)}{f_{n}(t)} \\frac{g(t)}{g(s)},\n\\end{align}\n with \\begin{equation}\n f_{n}(s)= \\frac{\\Gamma(n)\\Gamma(-\\frac s2)}{n^{\\frac s2} \\Gamma(n-\\frac s2)}, \\qquad \n g(s)=\\Gamma\\big(\\frac{s+1}{2}\\big) \\prod_{m=1}^M \\Gamma(\\nu_m+s+1). \n \\end{equation}\n \n For any fixed $t\\in c+i \\mathbb{R}$  and $s \\in \\Sigma$, using \\cite[eq. 5.11.13]{NIST} we see \n\\begin{equation}\\label{5.6}\nf_{n}(s)= \\Gamma(-\\frac s2) \\big(1+O(\\frac{1}{n})\\big),  \\qquad f_{n}(t)= \\Gamma(-\\frac t2) \\big(1+O(\\frac{1}{n})\\big). \\end{equation}\nFormally, substituting (\\ref{5.6}) in (\\ref{5.4}) gives (\\ref{hard-even}). To proceed rigorously, we need to verify   a condition for the  exchange of limit and integration.  For this purpose, we  will proceed to \nfind two dominated functions respectively  corresponding to  $1\/|f_{n}(t)|$  and  $|f_{n}(s)|$.\n\nFirst,     using \\cite[eq. 5.11.13]{NIST} we have for sufficiently  large $n$\n\\begin{equation}\n \\frac{1}{|f_{n}(t)|} \\leq  \\frac{n^{\\frac c2} \\Gamma(n-\\frac c2) }{ \\Gamma(n)|\\Gamma(-\\frac t2)|} \\leq   \\frac{2}{|\\Gamma(-\\frac t2)|},   \\quad \\forall  t\\in c+i \\mathbb{R}. \\label{t-bound}\n\\end{equation}\n\nSecond, we require an upper bound  for $ | f_{n}(s)  |$. Noting the asymptotic expansion, that as $z\\rightarrow \\infty$ in the sector $|\\mathrm{arg}(z)|\\leq \\pi-\\delta$ (with $0<\\delta<\\pi$)\n\\begin{equation}\n\\Gamma(z)=e^{-z}z^{z-\\frac{1}{2}}\\sqrt{2\\pi} \\big(1+O(\\frac{1}{z})\\big), \\label{Agamma}\n\\end{equation} \nit is easy to see that for a given $y_0>0$ we can choose the contour $\\Sigma=\\Sigma_{l}\\cup \\Sigma_{r}$ with \\begin{equation} \\Sigma_{l}=\\big\\{\\frac{c}{2}+iy:|y|\\leq y_{0}\\big\\}\\cup \\big\\{x\\pm iy_0: \\frac{c}{2} \\leq x\\leq 1\\big\\}, \\quad \\Sigma_{r}=  \\big\\{x\\pm iy_0:   x> 1\\big\\}.\\end{equation}\nThus, we get from \\eqref{Agamma} and the boundedness of $\\Gamma(-s\/2)$ over $\\Sigma_l$ that  for large $n$ there exists a constant $C_1=C_1(y_0) > 0$ such that \n\\begin{equation}\n |f_{n}(s)| \\leq C_1, \\qquad \\forall s\\in  \\Sigma_{l}. \\label{upl}\n\\end{equation}\nIn order to estimate $f_{n}(s)$ with $s\\in  \\Sigma_{r}$, we use the integral representation  \n \\begin{equation}\n  f_{n}(s)=   \\frac{n^{-\\frac{s}{2}}}{ 2i \\sin \\frac{\\pi s}{2}}    \\int_{\\mathcal{C}_0} (1-u)^{n-1}(-u)^{-\\frac{s}{2}-1}du,\n\\end{equation}\nwhere  $\\mathcal{C}_0$ is a counter-clockwise path which begins and ends at $1$ and encircles the origin once; see e.g.  \\cite[eq. 5.12.10]{NIST}. Note that we choose $(-u)^{-1-s\/2}=e^{-(1+s\/2)\\log(-u)}$ with $-\\pi<\\mathrm{arg}(-u)<\\pi$. Change $u$ by $u\/n$ and deform the resulting contour into the path which starts from $n$, proceeds along the (upper) real axis to 1, describes a circle of radius one counter-clock round the origin and returns to $n$ along the (lower) real axis. That is, \n \\begin{equation}\n  f_{n}(s)=   \\frac{1}{ 2i \\sin \\frac{\\pi s}{2}}    \\int_{\\mathcal{C}} (1-\\frac{u}{n})^{n-1}(-u)^{-\\frac{s}{2}-1}du.\n\\end{equation}\nLet $s=v\\pm iy_0, v>1$. On the unit circle of the $u$-integral above  write $-u=e^{i\\theta}$. Then we easily obtain  for $n\\geq1$\n \\begin{equation}\n  |f_{n}(s)|\\leq    \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{-\\pi}^{\\pi} \\big(1+\\frac{1}{n}\\big)^{n-1} |e^{-(\\frac{s}{2}+1)i\\theta}|d\\theta\\leq \n  \\frac{\\pi e^{ 1+\\frac{\\pi y_{0}}{2}}}{   |\\sin \\frac{\\pi s}{2}|}. \\label{ub1}\n\\end{equation}\nOn the upper and lower  real axis,  we have \n \\begin{align}\n  |f_{n}(s)|&\\leq    \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{1}^{n} \\big(1-\\frac{u}{n}\\big)^{n-1}|u^{-\\frac{s}{2}-1} e^{-(\\frac{s}{2}+1)(\\mp i\\pi)}|du \\nonumber \\\\\n  &\\leq \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{1}^{n}   u^{-\\frac{v}{2}-1} e^{\\frac{1}{2}\\pi y_{0}} du \\nonumber\\\\\n & =\\frac{1}{  |\\sin \\frac{\\pi s}{2}|}  e^{\\frac{1}{2}\\pi y_{0}}    \\frac{1- n^{-\\frac{v}{2}} }{v}\\leq \\frac{1}{  |\\sin \\frac{\\pi s}{2}|}  e^{\\frac{1}{2}\\pi y_{0}}.  \\label{ub2}\n\\end{align} \nUsing the simple fact $|\\sin \\frac{\\pi s}{2}| \\geq |\\sinh \\frac{\\pi }{2} \\mathrm{Im}(s)|$, combination of \\eqref{ub1} and \\eqref{ub2} shows that \nthere exists a constant $C_2=C_2(y_0)>0$ such that \n\\begin{equation}\n |f_{n}(s)| \\leq C_2, \\qquad \\forall s\\in  \\Sigma_{r}.\n\\end{equation}\nTogether with \\eqref{upl} this gives us a bound $C>0$, that is, for large $n$  \n\\begin{equation}\n |f_{n}(s)| \\leq C, \\qquad \\forall s\\in  \\Sigma. \\label{s-bound}\n\\end{equation}\n\n\nFinally, use \\eqref{Agamma} and  the  asymptotic formula  that as $y\\rightarrow \\pm \\infty$ \n\\begin{equation}\n| \\Gamma(x + iy) |\\sim  \\sqrt{2\\pi} |y|^{x-\\frac{1}{2}}  e^{-\\frac{1}{2}\\pi |y|}\n\\end{equation}\nwith bounded real value of $x$ (see \\cite[eq. 5.11.9]{NIST}),  it is easy to conclude  that the  function  of  variables $s$ and $t$\n \\begin{equation}\n\n \\,\\frac{||x|^{s}|y|^{-t-1}|}{|s-t|}   \\frac{2}{|\\Gamma(-\\frac t2)|}   \\frac{|g(t)|}{|g(s)|},  \n \\end{equation}\n  is integrable along the chosen contours, \n whenever $-1<c<-1\/2$. Here  we emphasize that the assumption $-1\/2\\leq c<0$  does  not ensure the convergence in the special case $M=0$ while for  $M\\geq 1$ it can be relaxed to $-1<c<0$ as in Proposition~\\ref{prop:kernel-finite}.\n \nWith this, combing  \\eqref{t-bound} and \\eqref{s-bound}, we have indeed justified  the interchange of limit and integrals for every $M$ by the dominated convergence theorem, which completes the proof.\n\\end{proof}\n\n\n\nFor a comparison with other known results, it is useful to rewrite the hard edge correlation function of Theorem~\\ref{thm:hard} in terms of Meijer $G$-functions.\nUsing that\n\\begin{equation}\n\\int_0^1du\\,u^{s-t-1}=\\frac1{s-t},\n\\end{equation}\nwe see that the even and odd kernel can be written as\n\\begin{align}\nK^\\textup{even}(|x|,|y|)=\\frac{|y|}{2^{2M}}\\int_0^1du\\, \\nonumber\n&\\MeijerG{1}{0}{0}{2M+2}{-}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2+\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2+\\frac12}{\\frac{x^2}{2^{2M}}u} \\\\\n&\\times\\MeijerG{2M+1}{0}{0}{2M+2}{-}{\\frac{\\nu_M}2-\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2-\\frac12,\\frac{\\nu_0}2}{\\frac{y^2}{2^{2M}}u}, \\\\\nK^\\textup{odd}(|x|,|y|)=\\frac{|x|}{2^{2M}}\\int_0^1du\\, \\nonumber\n&\\MeijerG{1}{0}{0}{2M+2}{-}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2-\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2-\\frac12}{\\frac{x^2}{2^{2M}}u} \\\\\n&\\times\\MeijerG{2M+1}{0}{0}{2M+2}{-}{\\frac{\\nu_M}2+\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2+\\frac12,\\frac{\\nu_0}2}{\\frac{y^2}{2^{2M}}u},\n\\end{align}\nrespectively. We recall that the so-called Meijer $G$-kernel is given by~\\cite{KZ14}\n\\begin{equation}\nK_\\text{Meijer}^M(x,y)=\\int_0^1du\\,\\MeijerG{1}{0}{0}{M+1}{-}{-\\nu_0,\\ldots,-\\nu_M}{xu}\\MeijerG{M}{0}{0}{M+1}{-}{\\nu_M,\\ldots,\\nu_0}{yu} \\label{G-kernel}\n\\end{equation}\nwith $x,y>0$. We note that this kernel is single-sided ($x,y\\in\\mathbb R_+$) while the kernel from Theorem~\\ref{thm:hard} is double-sided ($x,y\\in\\mathbb R\\setminus\\{0\\}$). However, it is also evident that our new kernel may be re-expressed in terms of the Meijer $G$-kernel. We have\n\\begin{equation}\nK^\\textup{even}(|x|,|y|)=\\frac{|y|}{2^{2M}}K_\\text{Meijer}^{2M+1}\\Big(\\frac{x^2}{2^{2M}},\\frac{y^2}{2^{2M}}\\Big)\n\\quad\\text{and}\\quad\nK^\\textup{odd}(|x|,|y|)=\\frac{|x|}{2^{2M}}K_\\text{Meijer}^{2M+1}\\Big(\\frac{x^2}{2^{2M}},\\frac{y^2}{2^{2M}}\\Big)\n\\end{equation}\nwith\n\\begin{equation}\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}-1)\/2\\}_{m=0}^M\n\\qquad\\text{and}\\qquad\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}+1)\/2\\}_{m=0}^M,\n\\end{equation}\nrespectively. Thus, the random product matrix~\\eqref{W1} provides yet another appearance of the Meijer $G$-kernel; albeit this time in a double-sided version. For graphical representation of the Meijer $G$-kernel we refer to~\\cite[Fig.~3.2]{Ip15}, which shows plots of the local density (i.e. the kernel with $x=y$) for different values of $M$.\n\nA double-side hard edge scaling limit near the origin is also present in the Hermite Muttalib--Borodin ensemble.\nIn this case the kernel is found to be~\\cite{Bo98}\n\\begin{equation}\\label{kernel-borodin}\nK^\\text{even}(x,y)=K^{(\\frac{\\alpha-1}2,\\theta)}(x^2,y^2)\n\\qquad\\text{and}\\qquad\nK^\\text{odd}(x,y)=\\sgn(xy)|x|^\\theta|y|\\,K^{(\\frac{\\alpha+\\theta}2,\\theta)}(x^2,y^2)\n\\end{equation}\nwhere\n\\begin{equation}\\label{kernel-wright-bessel}\nK^{(\\alpha,\\theta)}(x,y)=\\theta\\int_0^1du(xu)^\\alpha J_{\\frac{\\alpha+1}\\theta,\\frac1\\theta}(xu)J_{\\alpha+1,\\theta}((yu)^\\theta)\n\\end{equation}\nwith $J_{a,b}(x)$ denoting Wright's Bessel function. In the case relevant to us~\\eqref{MB-Hermite}, we also have $\\theta=2M+1$. Furthermore, it is known from~\\cite{KS14} that the kernel~\\eqref{kernel-wright-bessel} is a Meijer $G$-kernel whenever $\\theta$ is a positive integer. In particular, we have\n\\begin{equation}\n\\Big(\\frac{x^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}-1}\nK^{(\\alpha,2M+1)}\\Big((2M+1)\\Big(\\frac{x^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}},(2M+1)\\Big(\\frac{y^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}}\\Big)\n=K_\\text{Meijer}^{2M+1}\\Big(\\frac{y^2}{2^{2M}},\\frac{x^2}{2^{2M}}\\Big),\n\\end{equation}\nwhere the Meijer $G$-kernel on the right-hand side has indices\n\\begin{equation}\n\\nu_m=\\frac{\\alpha+m-1}{2M+1},\\qquad m=1,\\ldots,2M+1,\n\\end{equation}\nand as always $\\nu_0=0$. It follows from~\\eqref{kernel-borodin} and~\\eqref{kernel-wright-bessel} that the hard edge correlations for the Hermite Muttalib--Borodin ensemble with appropriately chosen parameters may be expressed in terms of the Meijer $G$-kernel in a similar fashion as done for the product ensemble above. We note that the choice of variables in~\\eqref{kernel-wright-bessel} should be compared to the change of variables~\\eqref{change} performed in the derivation of the asymptotic reduction~\\eqref{hermite-MB}.\n\n\nIt is worth verifying consistency of the simplest scenario of $M=0$.\nWhen  $M=0$ our matrix ensemble~\\eqref{W1} reduces to the GUE, hence the kernel given by Theorem~\\ref{thm:hard} must reduce to the sine kernel for $M=0$. To see this, we use \n\\begin{equation}\n\\MeijerG{1}{0}{0}{2}{-}{0,\\frac12}{\\frac{x^2}{4}}=\\frac{\\cos x}{\\sqrt\\pi}\n\\qquad\\text{and}\\qquad\n\\MeijerG{1}{0}{0}{2}{-}{\\frac12,0}{\\frac{x^2}{4}}=\\frac{\\sin |x|}{\\sqrt\\pi}.\n\\end{equation}\nIt follows that\n\\begin{align}\nK^\\textup{even}(x,y)&=\\frac{1}{\\pi}\\int_0^1\\frac{du}{\\sqrt u}\\cos(2x\\sqrt u)\\cos(2y\\sqrt u)\n=\\frac1{\\pi}\\Big(\\frac{\\sin 2(x-y)}{2(x-y)}+\\frac{\\sin 2(x+y)}{2(x+y)}\\Big), \\\\\nK^\\textup{odd}(x,y)&=\\frac{1}{\\pi}\\int_0^1\\frac{du}{\\sqrt u}\\sin(2x\\sqrt u)\\,\\sin(2y\\sqrt u)\n=\\frac1{\\pi}\\Big(\\frac{\\sin 2(x-y)}{2(x-y)}-\\frac{\\sin 2(x+y)}{2(x+y)}\\Big),\n\\end{align}\nwhich upon insertion into~\\eqref{hard-limit} indeed reproduces the sine kernel.\n\nIn the end of this section, let us emphasize that there  also exists a contour integral  representation of the limiting kernel in Theorem~\\ref{thm:hard} which combines the odd and even into a single formula.\n\n\\begin{prop}\\label{prop:kernelrep2} With the same  notation as  in Theorem \\ref{thm:hard}, the limiting kernel at the origin can be rewritten as\n\\begin{align}\n K^\\textup{even}(x,y)+ K^\\textup{odd}(x,y)=2\\, \\mathcal{K}_{\\nu_{1},\\ldots,\\nu_{M}}(2x,2y), \\label{equivalence}\n \\end{align}\nwhere the kernel on the right-hand side is defined as\n\\begin{align}\n\\mathcal{K}_{\\nu_{1},\\ldots,\\nu_{M}}(x,y)&\n=\\int_{C_{R}} \\frac{dv}{2\\pi i}\n\\,\\MeijerG{1}{0}{0}{M+1}{-}{0,-\\nu_1, \\ldots,-\\nu_M}{-\\sgn(y)xv}\\MeijerG{M+1}{0}{0}{M+1}{-}{0, \\nu_1,\\ldots,\\nu_M}{|y|v},\n\\label{doubleG-kernel}\n\\end{align}\nwith $C_{R}$ denoting a path in the right-half plane from $-i$ to $i$.\n\\end{prop}\n\n\\begin{proof}\nUsing  Euler's reflection formula and duplication formula for the gamma function, we see that\n\\begin{equation*}\nK^\\textup{even}(x,y)+ K^\\textup{odd}(x,y)=\\frac{1}{(2\\pi i)^2}\\int dt\\int ds\\, (2|x|)^{s}(2|y|)^{-t-1}\n\\frac{ g(s,t)}{s-t} \\frac{\\Gamma(t+1)}{\\Gamma(s+1)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)},\n\\end{equation*}\nwhere\n\\begin{equation}\ng(s,t)=\\frac{\\sin\\frac{\\pi}{2}t}{\\sin\\frac{\\pi}{2}s}+\\sgn(xy) \\frac{\\cos\\frac{\\pi}{2}t}{\\cos\\frac{\\pi}{2}s}.\n\\end{equation}\nIn order to proceed, we will consider the cases $xy<0$ and $xy>0$ separately.\nFor $xy<0$,  it is seen that\n\\begin{equation}\ng(s,t)= \\frac{2}{\\sin\\pi s}\\sin\\frac{\\pi}{2}(t-s)=-\\frac{2}{\\pi} \\Gamma(-s)\\Gamma(1+s) \\, \\sin\\frac{\\pi}{2}(t-s).\n\\end{equation}\nNow~\\eqref{equivalence} can be obtained using the integral representation\n\\begin{equation}\n \\frac{1}{\\pi i} \\int_{C_{R}}dv \\,v^{s-t-1}= \\frac{1}{t-s}\\sin\\frac{\\pi}{2}(t-s),\n\\end{equation}\nwith the contour $C_R$ as above,\ntogether with the definition of Meijer $G$-function. For $xy>0$, we note that\n\\begin{equation}\ne^{i\\pi s}g(s,t)=\n\\left(\\frac{\\sin\\frac{\\pi}{2}t}{\\sin\\frac{\\pi}{2}s}-\\frac{\\cos\\frac{\\pi}{2}t}{\\cos\\frac{\\pi}{2}s} \\right)+2e^{i\\frac{\\pi}{2}(t+s)}.\n\\end{equation}\nThe $s$-variable integrand in the second part has no pole within the contour $\\Sigma$. Thus, the problem reduces to the proven situation.\n\\end{proof}\n\nThe simplest non-trivial case is $M=1$. Here, we get\n \\begin{equation}\n\\mathcal{K}_{\\nu}(x,y)\n=\\left(\\frac{y}{x}\\right)^{\\nu\/2}  \\frac{1}{\\pi i} \\int_{C_{R}}dv\n\\, I_{\\nu}\\big(2\\sqrt{\\sgn(y)xv}\\big)\\,  K_{\\nu}\\big(2\\sqrt{|y|v}\\big),  \\label{doubleM1-kernel}\n\\end{equation}\nwith the modified Bessel functions $I_{\\nu}$ and $K_{\\nu}$, which follows immediately from the fact that\n \\begin{align}\n\\MeijerG{1}{0}{0}{2}{-}{0,-\\nu}{-z}=z^{-\\nu\/2} I_{\\nu}(2\\sqrt{z}), \\qquad  \\MeijerG{2}{0}{0}{2}{-}{\\nu,0}{z}=2 z^{\\nu\/2} K_{\\nu}(2\\sqrt{z}).\n\\end{align}\n\n\n\n\\section{Global spectra in product and Muttalib--Borodin ensembles}\n\\label{sec:global}\n\nThe study of the scaling limit  at  the origin  in the previous section introduces a scale in which the average spacing between eigenvalues is of order unity. A very different, but still well-defined, limiting process is the so-called global scaling regime. In this regime the average spacing between eigenvalues tends to zero in such way that the spectral density tends to a quantity $\\rho(x)$ with compact support $I\\subset\\mathbb R$ and $\\int_I \\rho(x)dx=1$. Here $\\rho(x)$ is referred to as the global density.\nThroughout this section, the indices $\\nu_1\\ldots,\\nu_M$ are kept fixed.\n\nFor the Laguerre Muttalib--Borodin ensemble specified by the density~\\eqref{MB-laguerre} the global scaling limit corresponds to a change of variables $x_j\\mapsto nx_j$. Introducing the further change of variables $x_j\\mapsto Mx_j^M$, the global density is known to be the so-called Fuss--Catalan density with parameter $M$ \\cite{FW15}. It can be specified by the moment sequence\n\\begin{equation}\n\\text{FC}_M(k)=\\frac1{Mk+1}\\binom{(M+1)k}k,\\qquad k=0,1,\\ldots\\,.\n\\end{equation}\nThese are the Fuss--Catalan numbers (the Catalan numbers are the case $M=1$).\n\nNow, consider the product of $M$ standard complex Gaussian random matrices. Consistent with the discussion in Section~\\ref{sec:motivation}, the corresponding global density is again the Fuss--Catalan density with parameter $M$~\\cite{Mu02,AGT10,BBCC11,NS06}.\n\nIt is known that the Fuss--Catalan density, $\\rho^{(M)}_\\text{FC}(x)$ say, can also be characterised as the minimiser of the energy functional\n\\begin{equation}\\label{energy-laguerre}\nE[\\rho]=M\\int_0^Ldx\\,\\rho(x)x^{\\frac1M}-\\frac{1}{2}\\int_0^Ldx\\int_0^Ldy\\,\\rho(x)\\rho(y)\\log\\big(|x-y||x^{\\frac1M}-y^{\\frac1M}|\\big)\n\\end{equation}\nwith $L=(M+1)^{M+1}\/M^M$; see~\\cite{CR14,FL15,FLZ15}. Note that the energy functional~\\eqref{energy-laguerre} relates to~\\eqref{MB-laguerre} through the aforementioned change of variables. Similarly, the energy functional corresponding to~\\eqref{MB-Hermite} is\n\\begin{align}\n\\tilde E[\\tilde\\rho]&=\\theta\\int_{-\\tilde L}^{\\tilde L}dx\\,\\rho(x)x^{\\frac2\\theta}\n-\\frac{1}{2}\\int_{-\\tilde L}^{\\tilde L}dx\\int_{-\\tilde L}^{\\tilde L}dy\\,\n\\tilde\\rho(x)\\tilde\\rho(y)\\log\\big(|x-y||\\sgn x|x|^{\\frac1\\theta}-\\sgn y|y|^{\\frac1\\theta}|\\big) \\nonumber \\\\\n&=2\\theta\\int_{0}^{\\tilde L}dx\\,\\rho(x)x^{\\frac2\\theta}-\\int_{0}^{\\tilde L}dx\\int_{0}^{\\tilde L}dy\\,\n\\tilde\\rho(x)\\tilde\\rho(y)\\log\\big(|x^2-y^2||(x^2)^{\\frac1\\theta}-(y^2)^{\\frac1\\theta}|\\big)\n\\label{energy-hermite}\n\\end{align}\nwith $\\theta=2M+1$.\nWe note that changing variables $x^2\\mapsto x$ and $y^2\\mapsto y$, then setting $\\tilde\\rho(x)=x\\rho(x^2)$ reduces~\\eqref{energy-hermite} to~\\eqref{energy-laguerre} with $L=\\tilde L^2$. Thus, the minimiser in~\\eqref{energy-hermite} is given in terms of the Fuss-Catalan density\n\\begin{equation}\\label{double-sided-FC}\n\\tilde\\rho(x)=|x|\\rho_\\text{FC}^{(M)}(x^2)\n\\end{equation}\nand is symmetric about the origin.\n\nAs an illustration, let us consider the simplest case, $M=1$. The Fuss--Catalan density becomes the celebrated Mar\\v cenko--Pastur density,\n\\begin{equation}\n\\rho_\\text{FC}^{(M=1)}(x)=\\frac{1}{2\\pi}\\sqrt{\\frac{4-x}{x}},\\qquad 0<x<4.\n\\end{equation}\nThe formula~\\eqref{double-sided-FC} then gives the standard result (see e.g.~\\cite{PSbook}) that the energy functional\n\\begin{equation}\n \\tilde E[\\tilde\\rho]=\\int_{-2}^{2}dx\\,\\tilde \\rho(x)x^{2}\n-\\int_{-2}^{2}dx\\int_{-2}^{2}dy\\,\\tilde\\rho(x)\\tilde\\rho(y)\\log|x-y|\n\\end{equation}\nis minimised by\n\\begin{equation}\n\\rho_\\text{Wigner}(x)=\\frac{\\sqrt{4-x^2}}{2\\pi},\\qquad -2<x<2,\n\\end{equation}\nwhich is Wigner's semi-circle law.\n\nIt has been demonstrated in Section~\\ref{sec:product}, that the energy function implicit in~\\eqref{energy-hermite} underlies the eigenvalue distribution of the random matrix product~\\eqref{W1}. Thus, we can anticipate that after appropriate scaling the global density for the product ensembles is given by~\\eqref{double-sided-FC}. A direct proof of this can obtained through a number of different strategies. We consider first a method based on the characteristic polynomial.\n\nIn terms of the global scaled variables, the key equation relating the averaged characteristic polynomial to  the global is the asymptotic formula~\\cite{FW15}\n\\begin{equation}\\label{asymp-stieltjes}\n\\frac1n\\frac d{dz}\\log\\big\\langle\\det(z n^{M+\\frac12}\\mathbb I_n-W_M)\\big\\rangle=\\tilde G_M(z)+O(n^{-1}),\n\\end{equation}\nwhere $\\tilde G_M(z)$ is the Stieltjes transform of the global spectral density,\n\\begin{equation}\\label{stieltjes}\n\\tilde G_M(z)=\\int_{-\\tilde L}^{\\tilde L}dx\\,\\frac{\\tilde\\rho(x)}{z-x}.\n\\end{equation}\nFollowing the strategy first used in~\\cite{FL15}, the formula~\\eqref{asymp-stieltjes} leads to a characterisation of the Stieltjes transform~\\eqref{stieltjes}, upon realising that the characteristic polynomial satisfy a linear differential equation.\n\n\\begin{prop}\nConsider a matrix product~\\eqref{W1} with even matrix dimension, $n=2N$. Let\n\\begin{equation}\\label{char-poly}\nf(z)=\\big\\langle\\det(z\\mathbb I_{n}-W_M)\\big\\rangle\n\\end{equation}\ndenote the characteristic polynomial. Then~\\eqref{char-poly} is a solution to the $(2M+2)$-th differential equation,\n\\begin{equation}\\label{diff}\n2z^2\\Big(z\\frac d{dz}-n\\Big)f(z)\n=\\prod_{m=0}^M\\Big(z\\frac d{dz}+\\nu_m\\Big)\\Big(z\\frac d{dz}+\\nu_m-1\\Big)f(z),\n\\end{equation}\nwith asymptotic boundary condition $f(z)\\sim z^n$ for $|z|\\to \\infty$.\n\\end{prop}\n\n\\begin{proof}\nThe characteristic polynomial~\\eqref{char-poly} is identical to the bi-orthogonal polynomial $p_{2N}(z)$. As shown earlier, this polynomial is proportional to a Meijer $G$-function~\\eqref{p_2n-meijer}. It is well-known that such Meijer $G$-functions satisfy the differential equation~\\eqref{diff}. The asymptotic boundary condition follows trivially, since $f(z)$ is a monic polynomial.\n\\end{proof}\n\nChanging variables $z\\mapsto n^{M+\\frac12}\\hat z\/\\sqrt{2}$ in~\\eqref{diff} and using that~\\cite{FL15}\n\\begin{equation}\n\\frac{f^{(k)}(\\hat z)}{f(\\hat z)}\\sim\\Big(\\frac{f'(\\hat z)}{f(\\hat z)}\\Big)^k\n\\end{equation}\nto leading order in $n$, we see that for large $n$ the differential equation~\\eqref{diff} reduces to the algebraic equation (see e.g. \\cite{bai07} for $M=1$)\n\\begin{equation}\\label{alg-eq1}\nz^2(z\\tilde G_M(z)-1)=(z\\tilde G_M(z))^{2M+2}\n\\end{equation}\nwith asymptotic condition $\\tilde G_M(z)\\sim 1\/z$ as $|z|\\to\\infty$.\nThis equation is to be compared to the algebraic equation satisfied by the Stieltjes transform of the Fuss--Catalan density,\n\\begin{equation}\\label{alg-eq2}\nz(zG_M(z)-1)=(zG_M(z))^{M+1},\n\\end{equation}\nsee e.g.~\\cite{FL15}. With $z\\mapsto z^2$ and $M\\mapsto 2M+1$ and setting $\\tilde G_{M}(z)=zG_{2M+1}(z^2)$, we see that~\\eqref{alg-eq2} reduces to~\\eqref{alg-eq1}. This prescription is equivalent to ~\\eqref{double-sided-FC}, thus verifying this formula as the evaluation of the global density.\n\n\nThe same result can also be obtained using free probability techniques. To see this, we need some additional notation. Let $a$ be a non-commutative random variable with distribution $d\\mu(x)=\\rho(x)dx$. The Stieltjes transform $G_a(z)$ of the variable $a$ is defined analogous to~\\eqref{stieltjes}. The $S$-transform is defined as\n\\begin{equation}\nS_a(z)=\\frac{1+z}{z}\\gamma^{-1}(z)\n\\qquad\\text{with}\\qquad\n\\gamma(z)=-1+z^{-1}G_a(z^{-1}).\n\\end{equation}\nNow assume that $a$ and $b$ are two freely independent non-commutative random variables and that the Stieltjes transform $G_b(z)$ satisfies a functional equation $P(z,G_b(z))=0$.\nIt is known~\\cite{NS06} that under these conditions the Stieltjes transform $G_{ab}(z)$ of the product $ab$ satisfies\n\\begin{equation}\\label{functional-recursion}\nP\\Big(zS_a(zG_{ab}(z)-1),\\frac{zG_{ab}(z)}{S_a(zG_{ab}(z)-1)}\\Big)=0.\n\\end{equation}\nMoreover, we know that if $a$ is given by the free normal distribution (i.e. Wigner's semi-circle) and $b$ is given by the free Poisson distribution (i.e. Mar\\v cenko--Pastur), then\n\\begin{equation}\nS_a(z)=\\frac1{1+z} \\qquad\\text{and}\\qquad G_{b}(z)^2-zG_b(z)+1=0.\n\\end{equation}\nWe can now use that the limiting distributions for the GUE and the Wishart ensemble are the free normal and the free Poisson, respectively. Thus, using~\\eqref{functional-recursion} $M$ times, we see that our product~\\eqref{W1} indeed gives rise to the the functional equation~\\eqref{alg-eq1}.\n\nIt is also possible to construct a parametrisation of the global density in terms of elementary functions based on the polynomial equation \\eqref{alg-eq1}. With\n\\begin{equation}\nx_{0}^{2}=\\frac{\\big(\\sin((2M+2)\\varphi)\\big)^{2M+2}}{\\sin\\varphi\\,\\big(\\sin((2M+1)\\varphi)\\big)^{2M+1}},\n\\qquad  0\\leq\\varphi\\leq\\frac{\\pi}{2M+2},\n\\end{equation}\nwe have\n\\begin{equation}\n\\tilde\\rho(x_0\n=\\frac{1}{\\pi}\\sqrt{\\frac{\\sin\\varphi}{\\sin(2M+ 1)\\varphi}}\\left(\\frac{\\sin(2r+1)\\varphi}{\\sin(2M+ 2)\\varphi}\\right)^{ M}\\sin\\varphi,\n\\qquad  0\\leq\\varphi\\leq\\frac{\\pi}{2M+2};\n\\end{equation}\nsee e.g.~\\cite{FL15}.  We remark that it follows that the singularity at the origin blows up like\n\\begin{equation}\n\\tilde\\rho(x_0)\\sim  \\frac{1}{\\pi } \\sin\\frac{\\pi}{2M+2} \\, |x_0|^{-\\frac{M}{M+1}}\n\\end{equation}\nas $x_0\\to0$.\n\n\n\\section{Conclusion and outlook}\n\nIn this paper, we have shown that it is possible to construct a Hermitised random matrix product for which the eigenvalues form a determinantal point process  on the entire real line with an explicit kernel. This is a fundamental new contribution to the study of random matrix product ensembles, since all previous exactly solvable models of this type have had eigenvalues restricted to the positive half-line. Furthermore, we have argued that this Hermitised product ensemble can be considered a natural generalisation of the classical Hermite ensemble (i.e. GUE) in similar way as the squared singular values of matrix products with  Gaussian matrices~\\cite{AKW13,AIK13} and truncated unitary matrices~\\cite{KKS15} can be considered generalisations of the Laguerre and Jacobi ensembles, respectively.\nTo this point, we have shown that the joint eigenvalue PDF reduces asymptotically to the Muttalib--Borodin ensemble of Hermite type.\n\nOn another front, we have shown that the local scaling limit near the origin is described by a two-sided generalisation of the so-called Meijer $G$-kernel~\\cite{KZ14}. This two-sided kernel reduces to the sine kernel in the simplest case. We have also seen that the global density can be found explicitly and that it is expressed in terms of the so-called Fuss--Catalan distribution in a simple manner. Our result relies on an explicit double contour integral formulation of the correlation kernel (Proposition \\ref{prop:kernel-finite}).\nIt is worth stressing that we could make full use of this double contour integral formulation and give an analytical proof of the global density.  In fact, following almost exactly the same steps introduced in  \\cite{LWZ14}, it can be proven that the sine-kernel arises in the bulk\nand that Airy-kernel arises at the soft edge; cf. the proof of Theorems 1.1 and Theorem 1.3 as well as Remark 2 in~\\cite{LWZ14}. The full details are beyond the scope of this paper, so let us only mention that a basic starting  point is to approximate the integrand by elementary functions and rewrite the kernel,  say the even part, as\n\\begin{multline}\n\\frac{1}{n \\tilde\\rho(x_0)}\\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1}K_{2n}^\\text{even}\n\\bigg(\n\\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1} \\Big(\\frac{x_0}{\\sqrt{2}}+\\frac{x}{ \\tilde\\rho(x_0)n}\\Big), \\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1}(\\frac{x_0}{\\sqrt{2}}+\\frac{x}{ \\tilde\\rho(x_0)n}\\Big)\\bigg)\n\\\\ \\sim \\frac{\\sqrt{2}}{|x_0|\\tilde\\rho(x_0)}\\frac{1}{(2\\pi i)^2}\\int dt\\int ds\\,\\frac{e^{n(g(s)-g(t))}}{s-t}  \\Big|1+\\frac{\\sqrt{2}x}{ x_0\\tilde\\rho(x_0)n}\\Big|^{2ns} \\Big|1+\\frac{\\sqrt{2}y}{ x_0\\tilde\\rho(x_0)n}\\Big|^{-2nt-1}\n \\frac{h_{n}(s)}{h_{n}(t)},\n\\end{multline}\nwhere the phase function is given by\n\\begin{equation}\ng(z)=(2M+1)z-2(M+1)z\\log z+z\\log(z-1)-\\log(1-z)+z\\log x_{0}^2.\n\\end{equation}\nHence, the saddle point equation $g'(z)=0$ is exactly  expressed through the equation  \\eqref{alg-eq1}.   We stress that the above  parametrisation representation  plays a key role in the proof of the sine-kernel  via the steepest decent method.\n\nFinally, we emphasize that our construction of a Hermitised random product ensemble is based on a matrix transformation which maps  the space of polynomial ensembles onto itself (Theorem~\\ref{T1}). This type of matrix transformation are important since they preserve exact solvability. Our  proof of Theorem~\\ref{T1} is applicable  to the Hermitised product ensemble multiplied by a Gaussian matrix, crucially with the help of the   hyperbolic HCIZ integral over the pseudo-unitary group. However, it would be interesting to see whether this could be extended to the product ensemble multiplied by   other types of random matrices, say, truncated unitary matrices. For this, a possible way is to first extend the matrix integral formula stated  in \\cite[Theorem 2.3]{KKS15} from  the unitary group to  the  pseudo-unitary case, and then perform  the same steps as in Section \\ref{sec:fyodorov}. This will be an interesting and  challenging problem for us.\n\n\\paragraph{Acknowledgements}\nWe thank S. Kumar  for  useful discussions on his paper and M. Kieburg for comments on a first draft.\nWe acknowledge support by the Australian Research Council through grant DP170102028 (PJF),\nthe ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF,JRI), and  partially by ERC Advanced Grant \\#338804,  the National Natural Science Foundation of China  \\#11771417,  the Youth Innovation Promotion Association CAS  \\#2017491, the Fundamental Research Funds for the Central Universities \\#WK0010450002,   Anhui Provincial Natural Science Foundation \\#1708085QA03 (DZL).   D.-Z. Liu is particularly grateful to L\\'{a}szl\\'{o} Erd\\H{o}s  for funding   his one-year stay at  IST Austria.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec_introduction}\n\nNonlinear responses have been giving rise to a lot of research interest in condensed matter physics. For instance, the nonlinear optical response provides a powerful tool for spectrometry. It has been used to obtain a real-space imaging of the parity-violating magnetic order in insulators~\\cite{Fiebig2005SHG_review,VanAken2007ToroidalDomain} and to explore exotic order in spin-orbit coupled metals and superconductors~\\cite{Petersen2006SHG,Zhao2015SHG_Sr2IrO4,zhao2017global,Harter2017SHG_Cd2Re2O7}. In the optical nonlinear responses, energy of irradiating light  is usually larger than that of electron bands, and the observed signals are attributed to interband transitions~\\cite{Sturman1992Book}. On the other hand, intraband transitions are also important in conductivity measurements in which the frequency of the probe is usually lower than that of optical probes and comparable to the electronic energy scale. It is therefore expected that nonlinear responses are informative for investigating metallic compounds where the intraband transitions are relevant. \n\nRegarding the nonlinear response in metals, the second-order nonlinear conductivity (NLC) measurement has attracted much attention. Previous studies are mainly divided into two streams; field-induced NLC and field-free NLC. The former can be traced back to Rikken's seminal works~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy}. They realized the longitudinal NLC under an external magnetic field, and significant enhancement has recently been discovered in strongly spin-orbit coupled semiconductors and superconductors~\\cite{Tokura2018nonreciprocal_review}. The microscopic origin of the longitudinal NLC is attributed to a semiclassical contribution which we call Drude term~\\cite{Ideue2017,Wakatsuki2017,Itahashi2020STOnonreciprocal}. On the other hand, a lot of theoretical and experimental efforts have recently been devoted to the transverse NLC, that is, nonlinear Hall effect~\\cite{Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2}. The nonlinear Hall effect realized without the magnetic field is rooted in a geometric quantity named Berry curvature dipole (BCD)~\\cite{Sodemann2015}.\n\nAccording to the symmetry argument, the second-order NLC requires violation of parity symmetry \\Pa{}. The condition is satisfied by the acentric property of crystals which were previously studied~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Tokura2018nonreciprocal_review,Ideue2017,Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2}. In contrast, the \\Pa{}-symmetry breaking can also be accompanied by the magnetic order, that is called odd-parity magnetic multipole order~\\cite{Spaldin2008a,Watanabe2018grouptheoretical,Hayami2018Classification}. It is expected that counterparts of the NLC exist in magnetic metals.\n\nA key to the odd-parity magnetic multipole order is locally-noncentrosymmetric property of crystals. With such structure of crystals, the local site-symmetry of atoms does not have \\Pa{}-symmetry although the global \\Pa{}-symmetry is preserved owing to the sublattice degree of freedom~\\cite{Maruyama2012,Yanase2014zigzag}. Then, the anti-symmetric spin-orbit coupling  (ASOC) emerges in a sublattice dependent way and gives rise to exotic responses such as the antiferromagnetic Edelstein effect~\\cite{Yanase2014zigzag,Zelezny2014NeelorbitTorque,Hayami2014h,hikaruwatanabe2017}. Supposing the antiferromagnetic order preserving the translational symmetry, both of the \\Pa{} and \\T{}-symmetries may be violated while the combined symmetry, namely \\PT{}-symmetry, is preserved. Such parity-violating but \\PT{}-symmetric magnetic order is called odd-parity magnetic multipole order and has been discussed in the context of multipole physics~\\cite{Watanabe2018grouptheoretical,Hayami2018Classification} and antiferromagnetic spintronics~\\cite{Jungwirth2016,Manchon2019spin-orbit-torque_review,Watanabe2018}. More than 100 candidate materials such as BaMn$_2$As$_2$ and EuMnBi$_2$ have been identified~\\cite{Watanabe2018grouptheoretical}.\n\nIn this work, the NLC in odd-parity magnetic multipole metals are investigated. We present a general symmetry classification of NLC based on a quantum mechanical calculation. Supported by the microscopic analysis, we clarify NLC characteristic of magnetic metals with and without external magnetic field. We find that the NLC at $H=0$ is a measure of the ASOC. Furthermore, we reveal two types of field-induced NLC; the nematicity-assisted dichroism, and Berry curvature dipole effect induced by what we call magnetic ASOC. These phenomena originate from locally-noncentrosymmetric crystal structures and magnetic order, and hence have striking difference from the NLC in noncentrosymmetric (non-magnetic) crystals. We show the correspondence between \\T{}-symmetric and \\PT{}-symmetric systems in Table~\\ref{Table_NLC_MagneticField}.  \n\n\t\t\\begin{table}[htbp]\n\t\t\\caption{Second-order NLC in \\T{}\/\\PT{}-symmetric systems with\/without magnetic field $\\bm{H}$. Dominant contributions such as the BCD term are shown. The boldfaced terms are studied in this work.}\n\t\t\\label{Table_NLC_MagneticField}\n\t\t\\centering\n\t\t\\begin{tabular}{l|cc}\n\t\t\t\t\t&\\T{}\t&\\PT{}\t \\\\ \\hline\n\t\t$\\bm{H}=0$\t&BCD\t\t\t&\\textbf{Drude}\t \\\\\n\t\t$\\bm{H}\\ne 0$\t&magnetic Drude\t&\\textbf{magnetic BCD}\t \n\t\t\\end{tabular}\n\t\t\\end{table}\n\n\\section{Quantum theory and symmetry analysis}\\label{Sec_quantumtheory_symmetryanalysis}\n\nA theoretical treatment of the second-order NLC has been established in Sipe and his coworkers' works~\\cite{Sipe1993,Aversa1995,Sipe2000secondorder} where the nonlinear response functions are derived from straightforward extension of the linear response theory~\\cite{Kubo1957}. A detailed calculation of the NLC is shown in Appendix~\\ref{App_Sec_derivation_NLC}. Here we only describe the outline of derivation.\n\nA spatially-uniform electric field $\\bm{E}(t)$ is introduced so as to be compatible with calculations based on the Bloch states in the length gauge\nframework~\\cite{Sipe1993,Aversa1995,Sipe2000secondorder,Ventura2017,Passos2018,Parker2019}. Using the density matrix formalism, we derive the second-order electric current in the frequency domain as \n\t\t\\begin{align}\n\t\tJ^{\\mu (2)} (\\omega) = \\sum_{\\nu\\lambda} &\\int \\frac{d\\omega_1d\\omega_2}{2\\pi}  \\sigma^{\\mu;\\nu\\lambda} (\\omega; \\omega_1,\\omega_2 ) \\notag \\\\\n\t\t\t&\\times E^\\nu \\left( \\omega_1  \\right)E^\\lambda \\left( \\omega_2  \\right) \\delta (\\omega-\\omega_1-\\omega_2).\n\t\t\\end{align}\nAssuming the clean limit where the relaxation time $\\tau \\rightarrow \\infty$ within the transport regime $\\omega\\tau \\ll 1$, the NLC is classified by the dependence on the phenomenological relaxation time $\\tau = \\gamma^{-1}$ as\n\t\t\\begin{equation}\n\t\t\\sigma^{\\mu;\\nu \\lambda} = \\sigma^{\\mu;\\nu \\lambda}_\\text{D} + \\sigma^{\\mu;\\nu \\lambda}_\\text{BCD} + \\sigma^{\\mu;\\nu \\lambda}_\\text{int},\n\t\t\\label{nonlinear_conductivity}\n\t\t\\end{equation}\nin which the indices $\\nu,~\\lambda$ of applied electric fields are symmetric. The first two components are obtained as\n\t\t\\begin{align}\n\t\t\\sigma^{\\mu;\\nu \\lambda}_\\text{D} \n\t\t\t\t&= - \\frac{e^3}{\\gamma^2}\\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_a \\partial_\\mu \\partial_\\nu \\partial_\\lambda  \\epsilon_{\\bm{k}a}   f(\\epsilon_{\\bm{k}a}), \\label{drude}\\\\\n\t\t\\sigma^{\\mu;\\nu \\lambda}_\\text{BCD} \n\t\t\t\t&= \\frac{e^3}{2\\gamma}\\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_{a} \\epsilon_{\\mu\\nu\\kappa}   f(\\epsilon_{\\bm{k}a}) \\partial_\\lambda \\Omega^\\kappa_a + \\left[ \\nu \\leftrightarrow \\lambda \\right],\\\\\n\t\t\t\t&= \\frac{e^3}{2\\gamma} \\epsilon_{\\mu\\nu\\kappa}   \\mathcal{D}^{\\,\\lambda\\kappa} + \\left[ \\nu \\leftrightarrow \\lambda \\right], \\label{BCD_term}\n\t\t\\end{align}\nwhich are the Drude~\\cite{Ideue2017} and BCD~\\cite{Sodemann2015,deyo2009semiclassical,Moore2010} terms, respectively. The index $a$ represents the band index. We introduced the electron charge $e>0$ and the BCD defined as\n\t\t\\begin{equation}\n\t\t\\mathcal{D}^{\\,\\mu\\nu} = \\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_{a} f(\\epsilon_{\\bm{k}a}) \\partial_\\mu \\Omega^\\nu_a.\n\t\t\\end{equation}\nThese two terms are finite only in the metal state and divergent in the clean limit since both are Fermi surface terms. The remaining term $\\sigma^{\\mu;\\nu \\lambda}_\\text{int}$ comes from interband transitions, and it is not divergent in the clean limit~\\cite{Gao2014}. This term, therefore, gives a negligible contribution to the NLC in good metals.\n\nIn the group-theoretical classification of quantum phases, the parity-violating phases are classified into odd-parity electric\/magnetic multipole phases where \\T{}\/\\PT{}-symmetry is preserved~\\cite{Watanabe2018grouptheoretical,Hayami2018Classification}. It is known that these preserved symmetries impose strong constraints on the response functions~\\cite{hikaruwatanabe2017,Zelezny2017} in addition to equilibrium properties of the systems~\\cite{cracknell2016magnetism}. Thus, the symmetry analysis enables us to classify the NLC allowed in either \\T{}-symmetric or \\PT{}-symmetric systems based on the relaxation time dependence. The result is shown in Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity}.\n\nIn \\T{}-symmetric systems, all the terms in NLC are scaled by odd-order $O (\\tau^{2n+1})$. Recalling the linear response theory, the scattering rate $\\gamma$ can be replaced by the adiabaticity parameter whose sign represents irreversibility due to external fields~\\cite{Kubo1957}. Thus, the NLC should be accompanied by a dissipative response. This is consistent with previous theories~\\cite{Morimoto2018NonreciprocalElectronCorrelation,Hamamoto2019RachetScaling}. In contrast to the familiar linear conductivity, the Drude term is prohibited because it is even-order with respect to $\\tau$. The leading order term is the BCD term for the transverse NLC.\n\n On the other hand, the \\T{}-symmetry is broken by the magnetic order in the parity-violating \\PT{}-symmetric systems which we focus on. Therefore, the relaxation time dependence is even-order $O(\\tau^{2n})$, and intrinsic contributions $O(\\tau^{0})$ are allowed. The leading order term is the Drude term $O(\\tau^{2})$. We will show that the Drude term is a measure of the hidden ASOC characteristic of locally-noncentrosymmetric systems. The BCD term is prohibited to be consistent with the fact that the Berry curvature itself disappears due to the \\PT{}-symmetry. Although the effect of the \\T{}-symmetry breaking in acentric systems has been discussed in previous works~\\cite{Gao2014,Sodemann2015,Nandy2019,Du2019}, our classification has clarified the contrasting role of \\T{} and \\PT{}-symmetries in NLC. Below we will see that the \\PT{}-symmetry gives a clear insight into the NLC.\n\nIn our classification, extrinsic contributions such as the side jump and skew scattering are not taken into account~\\cite{Nandy2019,Du2019,CXiao2019ModifiedSemiclassics,Du2020quantumTheoryofNHE}. We, however, note that the extrinsic contributions may be similarly classified by the symmetries. Indeed, for nonmagnetic impurities with $\\delta$-function potential, we show that while extrinsic terms are allowed in the \\T{}-symmetric systems~\\cite{Du2019}, they are strongly suppressed by the \\PT{}-symmetry (See Appendix~\\ref{App_Sec_extrinsic}). This suppression is highly contrasting to the fact that the impurities play an important role in the NLC in \\T{}-symmetric materials such as WTe$_2$~\\cite{Kang2019}. When we focus on the \\PT{}-symmetric magnetic systems, the classification in Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity} is meaningful beyond the relaxation time approximation for impurity scattering.\n \n\t\t\\begin{table}[htbp]\n\t\t\\caption{Relaxation time dependence of the second-order NLC in \\T{}\/\\PT{}-symmetric systems. `N\/A' denotes that the component is forbidden by symmetry.}\n\t\t\\label{Table_relaxation_time_dependence_2nd_conductivity}\n\t\t\\centering\n\t\t$\n\t\t\t\t\\begin{array}{c|ccc}\n\t\t\t\t&\\sigma_\\text{D}\t&\\sigma_\\text{BCD}\t&\\sigma_\\text{int}\t\\\\ \\hline\n\t\t\t\t\\text{\\T{}}\t&\\text{N\/A}&O(\\tau)&O(\\tau^{-1})\\\\\n\t\t\t\t\\text{\\PT{}}&O(\\tau^2) &\\text{N\/A} &O(\\tau^{0})\n\t\t\t\t\\end{array}\n\t\t$\n\t\t\\end{table}\n\n\nAll the terms in NLC are allowed in the absence of both \\T{} and \\PT{}-symmetry. For instance, the Drude term becomes finite when we apply magnetic fields to originally \\T{}-symmetric systems~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Tokura2018nonreciprocal_review,Ideue2017}, that is described as `magnetic Drude' in Table~\\ref{Table_NLC_MagneticField}. Similarly, we expect a magnetic-field-induced NLC in originally \\PT{}-symmetric systems; the BCD term indeed arises from the \\PT{}-symmetry breaking (called `magnetic BCD' in Table~\\ref{Table_NLC_MagneticField}). This term is clarified in this work below. In the following, we consider the \\PT{}-preserving antiferromagnetic metal with or without the magnetic field, and discuss the Drude and BCD terms which are dominant in clean metals.\n\n\\section{NLC in odd-parity magnetic multipole systems}\\label{Sec_NLC_odd-parity_magnetic_multipole}\n\nWe introduce a minimal model of \\bma{} which undergoes odd-parity magnetic multipole order~\\cite{hikaruwatanabe2017}. Many magnetic compounds in the list of Ref.~\\cite{Watanabe2018grouptheoretical} belong to the same class. The Hamiltonian reads\n\t\t\\begin{equation}\n\t\t\tH(\\bm{k}) =\n\t\t\t\t\t\\epsilon(\\bm{k}) \\, \\tau_0+ \\bm{g} \\left( \\bm{k} \\right) \\cdot \\bm{\\sigma} \\, \\tau_z +\\bm{h} \\cdot \\bm{\\sigma} \\, \\tau_0 +  V_{\\rm AB} (\\bm{k}) \\, \\tau_x, \\label{BMA_model_Hamiltonian}\n\t\t\\end{equation}\nwhere $\\bm{\\sigma}$ and $\\bm{\\tau}$ are Pauli matrices representing the spin and sublattice degrees of freedom, respectively. In addition to the intra-sublattice and inter-sublattice hoppings, $\\epsilon (\\bm{k})$ and $V_\\text{AB}  (\\bm{k})$, we introduce the staggered $g$-vector $\\bm{g}  (\\bm{k})= \\bm{g}_0 (\\bm{k}) + \\bm{h}_\\text{AF}$ consisting of the sublattice-dependent ASOC $\\bm{g}_0  (\\bm{k})$~\\cite{Yanase2014zigzag,Zelezny2014NeelorbitTorque,Hayami2014h} and the molecular field $\\bm{h}_\\text{AF}= h_\\text{AF} \\hat{z}$ due to antiferromagnetic order in \\bma{} ~\\cite{Singh2009BaMn2As2_1,Singh2009BaMn2As2_2,Ramsal2013BMA_MagneticStructure}. The detailed material property of \\bma{} and expressions of $\\epsilon (\\bm{k})$, $V_\\text{AB}  (\\bm{k})$, and $\\bm{g}_0 (\\bm{k})$ are available in Appendix~\\ref{App_Sec_model_hamiltonian}. We also consider an external magnetic field $\\bm{h}$ to discuss field-induced NLC. \n\n\\subsection{Field-free nonlinear Hall effect}\\label{Sec_nonlinear_Hall_no_field}\n\nFirst, we show the NLC at zero magnetic field ($\\bm{h} =\\bm{0}$). Then, the NLC is mainly given by the Drude term (see Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity}), and it is determined by the anti-symmetric and anharmonic property of the energy dispersion [see Eq.~\\eqref{drude}]. Such dispersion is known to be a pronounced property of the odd-parity magnetic multipole systems~\\cite{Yanase2014zigzag,Hayami2014h,Sumita2016,hikaruwatanabe2017,Watanabe2018grouptheoretical}. In the case of \\bma{}, the anti-symmetric component was identified to be a cubic term $k_xk_yk_z$~\\cite{hikaruwatanabe2017}. Indeed, the energy spectrum of the model Eq.~\\eqref{BMA_model_Hamiltonian} is obtained as\n\t\t\\begin{equation}\n\t\tE^\\pm_{\\bm{k}}= \\epsilon (\\bm{k}) \\pm \\sqrt{V_{\\rm AB}(\\bm{k})^2 + \\bm{g}(\\bm{k})^2 }. \\label{energyspectrum_no_magnetic_field}\n\t\t\\end{equation}\nThe anti-symmetric distortion in the band structure arises from the coupling term $\\bm{g}_0 (\\bm{k}) \\cdot \\bm{h}_\\text{AF}$ which is approximated by $\\sim k_xk_yk_z$ near time-reversal-invariant momentum. Thus, $\\sigma^{z;xy}$ and its cyclic components of NLC tensor are allowed. This indicates the nonlinear Hall effect, namely, the second-order electric current $J^z$ generated from the electric field $\\bm{E} \\parallel [110]$. For the strong antiferromagnet, $|\\bm{h}_\\text{AF}| \\gg |\\epsilon (\\bm{k})|$, $|V_\\text{AB}  (\\bm{k})|$, $|\\bm{g}_0 (\\bm{k})|$, the Drude component is analytically obtained as \n \t\t\\begin{equation}\n\t\t\\sigma^{z;xy}_\\text{D} =\\sigma^{x;yz}_\\text{D} =\\sigma^{y;zx}_\\text{D} = \\frac{e^3\\alpha_{\\parallel}n }{4\\gamma^2} \\,\\text{sgn\\,} (h_\\text{AF}), \\label{Drude_no_external_field}\n\t\t\\end{equation}\nin the lightly-hole-doped region. Here $n$ denotes the carrier density of holes and $\\alpha_{\\parallel}$ represents the strength of ASOC parallel to the staggered magnetization $\\bm{h}_\\text{AF}$. It is noteworthy that Eq.~\\eqref{Drude_no_external_field} does not depend on the antiferromagnetic molecular field and therefore it is useful to evaluate the \\textit{sublattice-dependent} ASOC. Thus, the NLC provides a way to experimentally deduce the sublattice-dependent ASOC, although it was called \"hidden spin polarization\"~\\cite{Zhang2014HiddenSpin,Gotlieb2018HiddenSpinInCuprate} because it is hard to be measured. Equivalence of $\\sigma^{z;xy}_\\text{D} =\\sigma^{x;yz}_\\text{D} =\\sigma^{y;zx}_\\text{D}$ holds independent of parameters and it can be tested by experiments. Numerical calculations of Eq.~\\eqref{drude} are consistent with the above-mentioned symmetry argument and analytic formula as shown in Appendix~\\ref{App_Sec_NLC_no_magnetic_field}. A typical value of the nonlinear Hall response is obtained as $\\sigma^{z;xy}_\\text{D}\/[(\\sigma^{xx})^2 \\sigma^{zz}] \\sim \\mr{10^{-17}}{[A^{-2} \\cdot V \\cdot m^3]}$ and it is much larger than the experimental value of bilayer WTe$_2$, $\\sigma^{y;xx}\/(\\sigma^{xx})^3 \\sim \\mr{10^{-19}}{[A^{-2} \\cdot V \\cdot m^3]}$~\\cite{Ma2019BCD_experiment_WTe2}. Because the Drude term is more divergent with respect to $\\tau$ than the BCD term, we may see a giant nonlinear Hall response in the \\PT{}-symmetric antiferromagnet.\n\nThe NLC is a useful quantity not only to evaluate the sublattice-dependent ASOC but also to detect domain states in antiferromagnetic metals~\\cite{Watanabe2018grouptheoretical}. Indeed, the sign of the NLC depends on the antiferromagnetic domain and hence it may promote developments in the antiferromagnetic spintronics~\\cite{Jungwirth2016,Manchon2019spin-orbit-torque_review}. In fact, the read-out of antiferromagnetic domains has been successfully demonstrated by making use of the NLC~\\cite{Godinho2018AFM_reading}. For \\bma{} and related materials listed in Ref.~\\cite{Watanabe2018grouptheoretical}, the nonlinear Hall effect can be used to identify antiferromagnetic domain states. So far we considered intrinsic contributions. We have shown that the extrinsic contributions from impurity scattering are suppressed due to the  preserved \\PT{}-symmetry, and therefore, they are not relevant to the above discussions.\n\n\\subsection{Nematicity-assisted dichroism}\\label{Sec_nematicity_assisted}\n\nIn the absence of the external field, \\bma{}-type magnetic materials do not show the longitudinal NLC along the high symmetry axes, namely, $\\sigma^{\\mu;\\mu\\mu}=0$. Below, we show that the longitudinal NLC can be induced by magnetic fields. Since the BCD term contributes to only the transverse response, we have only to consider the Drude term. Generally speaking, to obtain a finite longitudinal electronic dichroism, the system is required to possess an anti-symmetric dispersion such as $k_\\mu^3$ or higher-order one. According to the group-theoretical classification, the `polarization' in the momentum-space denoted by $k_\\mu$ may share the same symmetry as $k_\\mu^3$~\\cite{hikaruwatanabe2017,Watanabe2018grouptheoretical}. Thus, the momentum-space polarization is a key to realize the longitudinal dichroism.\n\nIn \\bma{} and related materials, the momentum-space polarization can be induced by the nematicity. We can understand this by the discussion of the magnetopiezoelectric effect~\\cite{Varjas2016,hikaruwatanabe2017,Shiomi2019EuMnBi2_MPE,Shiomi2019CaMn2Bi2_MPE,Shiomi2020}. A magnetopiezoelectric effect means that the planer (electronic) nematicity is induced by the out-of-plane electric current. That is written as\n\t\t\\begin{equation}\n\t\t\\varepsilon^{xy} = e^{xy;z} J^z,\n\t\t\\end{equation}\nwhere $\\varepsilon^{\\mu\\nu}$ represents the strain tensor.  It was experimentally discovered in EuMnBi$_2$~\\cite{Shiomi2019EuMnBi2_MPE,Shiomi2020} and CaMn$_2$Bi$_2$~\\cite{Shiomi2019CaMn2Bi2_MPE} in accordance with theoretical prediction. The response is derived from the anti-symmetrically distorted Fermi surface and hence realizable in the odd-parity magnetic multipole systems. Similar to the conventional piezoelectric effect, we may expect an inverse effect. Given the in-plane nematic order or strain, the system should obtain the momentum-space polarization $P^{\\,k_z}$ whose symmetry is the same as the electric current $J^z$,\n\t\t\\begin{equation}\n\t\tP^{\\,k_z} = \\tilde{e}^{z;xy} \\varepsilon^{xy}.\n\t\t\\end{equation}\n Accordingly, the longitudinal dichroism $\\sigma^{z;zz}$ is allowed. Thus, nematicity-assisted dichroism which is unique to the odd-parity magnetic multipole systems is implied. \n\nThe nematicity can be induced by the magnetic field through the spin-orbit coupling. In the model for \\bma{} the sublattice-dependent ASOC plays an essential role. By $\\bm{h} \\ne 0$, the energy spectrum of the lower bands $E_{\\bm{k}}^-$ in Eq.~\\eqref{energyspectrum_no_magnetic_field} is modified as\n\t\t\\begin{equation}\n\t\tE_{\\bm{k}}^- = \\epsilon (\\bm{k}) - \\sqrt{V_{\\rm AB}(\\bm{k})^2  + \\bm{g}(\\bm{k})^2+ \\bm{h}^2 \\pm 2 |\\lambda|  }, \\label{energyspectrum_with_magnetic_field}\n\t\t\\end{equation}\nwhere $\\lambda^2 = V_{\\rm AB}(\\bm{k})^2\\,\\bm{h}^2+ \\left[ \\bm{g} (\\bm{k})\\cdot \\bm{h}\\right]^2 $. The magnetic field not only lifts the Kramers degeneracy but also causes the nematicity through the coupling $\\left[ \\bm{g}_0 (\\bm{k}) \\cdot \\bm{h}\\right]^2$ in $\\lambda$, although linear terms in $\\bm{h}$ are canceled out between sublattices in sharp contrast to acentric systems studied before~\\cite{Ideue2017}. For \\bma{} with Dresselhaus-type staggered ASOC~\\cite{Manchon2019spin-orbit-torque_review}, the nematicity denoted by $\\varepsilon^{xy}$ is maximally induced by the magnetic field $\\bm{h}$ parallel to $[110]$ or $[1\\bar{1}0]$.\n\nWe expect nematicity-assisted dichroism in \\bma{} under the magnetic field $\\bm{h} \\parallel [110]$ from the above discussions. In numerically calculated NLC $\\sigma_\\text{D}^{z;zz}$ with rotating the magnetic field in the azimuthal plane, the dichroism with {\\it two-fold field-angle dependence} is clearly seen (Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}). In this case the magnetic field is a \\textit{bipolar field} rather than a vector field, in sharp contrast to the magnetic Drude term for which the observed field-angle dependence is one-fold~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Ideue2017}. Although the field-induced NLC is tiny as evaluated in Appendix~\\ref{App_Sec_nematic_assisted_NLC}, it was actually detected in a recent experiment for \\bma{}~\\cite{KimataPrivate}.\n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{nonlinear_response_chemipot_-500_mev_allindex_hargEle_90_azimuthsweep_BaMnAs_unit_135.pdf}\n\t\t\\caption{Drude term of a longitudinal NLC $\\sigma^{z;zz}_\\text{D}$ as a function of the azimuthal angle of external magnetic fields $\\bm{h}=h(\\cos \\phi, \\sin \\phi, 0)$. Strength of the magnetic field $h=0.01$, temperature $T=0.01$, chemical potential $\\mu = -0.5$, relaxation time $\\gamma^{-1}=10^3$, and Brillouin zone mesh $N = 135^3$ are adopted. The other parameters and adopted energy scale are described in Appendix~\\ref{App_Sec_calc_NLC_Mn_magnet}.}\n\t\t\\label{Fig_drude_nematicity_assisted_azimuth_dependence} \n\t\t\\end{figure}\n\n\n\\subsection{Magnetic ASOC and Berry curvature dipole}\\label{Sec_magneticASOC_BCD}\n\nNow we consider the counterpart of the magnetic Drude term~\\cite{Ideue2017}, that is the magnetic BCD term. The \\PT{}-symmetry ensures Kramers doublet at each momentum $\\bm{k}$, and Berry curvature is completely canceled in the odd-parity magnetic multipole systems. The doublet, however, should be split when the \\PT{}-symmetry is broken by the external magnetic field. Let us consider \\bma{}-type magnet under the magnetic field $\\bm{h}=h_z \\hat{z}$ for an example. Then, while the total Berry curvature $\\int d\\bm{k} \\,\\Omega^z$ is trivially induced, the BCD also emerges. Using the allowed symmetry operations, the induced BCD is identified as\n\t\t\\begin{equation}\n\t\t\\mathcal{D}^{\\,xy} = \\mathcal{D}^{\\,yx}. \\label{induced_BCD_in_BMA_with_zField}\n\t\t\\end{equation}\n\nBecause the BCD has the same symmetry as the ASOC~\\cite{Manchon2019spin-orbit-torque_review}, emergence of one indicates the presence of the other.\nTherefore, the field-induced BCD is understood by discussing magnetically-induced ASOC in the following way. Although the sublattice-dependent ASOC is compensated with $\\bm{h}=0$, combination of the staggered exchange spitting $\\bm{h}_\\text{AF}\\cdot \\bm{\\sigma}~\\tau_z$ and uniform Zeeman field $\\bm{h} \\cdot \\bm{\\sigma}~\\tau_0$ leads to imbalance between the sublattices without Brillouin zone folding (Fig.~\\ref{Fig_magnetic_asoc}). One of sublattices obtains an increased carrier density, and consequently the sublattice-dependent ASOC is not compensated. The emergent ASOC has distinct properties compared to the conventional crystal ASOC since the former originates solely from the magnetic effects. We therefore name this field-induced ASOC `magnetic ASOC'. Interestingly, the magnetic ASOC is tunable by external magnetic fields. Thus, the concept of magnetic ASOC may be useful to design spin-momentum locking in more controllable way than the crystal ASOC which is determined by the crystal structure~\\cite{magASOC}. In the model for \\bma{} the magnetic ASOC and BCD with the same symmetry as Eq.~\\eqref{induced_BCD_in_BMA_with_zField} are actually obtained. \n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{magnetic_dresselhaus.pdf}\n\t\t\\caption{Mechanism of the magnetic ASOC and field-induced BCD. The blue-colored arrows denote the spin-polarization or Berry curvature at each $\\bm{k}$. (Left panel) A magnetic field along the $z$-axis splits the Fermi surface depending on the antiferromagnetic molecular field $\\bm{h}_\\text{AF}$. (Right panel) The split Fermi surface is viewed in the $xy$-plane which indicates the Dresselhaus-type ASOC and BCD.}\n\t\t\\label{Fig_magnetic_asoc} \n\t\t\\end{figure}\n\nThe field-induced BCD allows nonlinear Hall conductivity in accordance with Eq.~\\eqref{BCD_term}, which satisfies the relation \n\t\t\\begin{equation}\n\t\t\\sigma_\\text{BCD}^{z;xx} = -\\sigma_\\text{BCD}^{z;yy}= -2\\sigma_\\text{BCD}^{x;xz}= 2\\sigma_\\text{BCD}^{y;yz}.\n\t\t\\label{nonlinear_Hall_by_BCD}\n\t\t\\end{equation}\nFor example, we show the numerical result for $\\sigma_\\mathrm{BCD}^{z;xx}$ in Fig.~\\ref{Fig_BCD_with_z_field_elevation_dependence}, which reveals the dependence on the elevation angle of $\\bm{h}$. The induced BCD is inverted when the external field is flipped. Therefore, the field-angle dependence is one-fold in contrast to the nematicity-assisted dichroism (Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}).\n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{nonlinear_response_chemipot_-500_mev_allindex_elevatesweep_Azu_0_BaMnAs_unit_135.pdf}\n\t\t\\caption{BCD term of a nonlinear Hall conductivity $\\sigma^{z;xx}_\\text{BCD}$ as a function of the elevation angle of external magnetic fields $\\bm{h}=h(\\sin \\theta, 0, \\cos \\theta)$. Parameters and unit are the same as Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}.}\n\t\t\\label{Fig_BCD_with_z_field_elevation_dependence} \n\t\t\\end{figure}\n\nFinally, we comment on a linear Hall response. Because the systems under the external magnetic field possess neither the \\T{}- nor \\PT{}-symmetry, a linear Hall response is also allowed. This is in contrast to the previously studied acentric systems~\\cite{Moore2010,Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2} where the linear Hall response is forbidden because of the \\T{}-symmetry. However, the nonlinear Hall response can be distinguished from the linear one by symmetry. For example, the NLC, $\\sigma_\\text{BCD}^{z;xx}$ and $\\sigma_\\text{BCD}^{z;yy}$, in Eq.~\\eqref{nonlinear_Hall_by_BCD} represents the Hall response for which the linear response is forbidden.\n\n\\section{Conclusion and Discussions}\n\nThis work presents symmetry classification of the second-order NLC, and explores the NLC of odd-parity magnetic multipole systems. The Drude term gives rise to a giant nonlinear Hall conductivity at zero magnetic field, and provides an experimental tool for a probe of the sublattice-dependent ASOC. Thus, the hidden spin polarization in centrosymmetric crystals can be clarified. It also enables us to elucidate domain states in antiferromagnetic metals, and hence the NLC will be useful in the field of antiferromagnetic spintronics. Interestingly, the NLC induced by magnetic fields is significantly different from those studied in previous works. \nWe clarified the nematicity-assisted dichroism and the BCD-induced NLC due to the magnetic ASOC. \n\nIn accordance with our theoretical result, a recent experimental study actually detected nematicity-assisted electric dichroism under the magnetic field~\\cite{KimataPrivate}. \nWe believe that further studies of the nonlinear response in parity-violated magnetic systems will be motivated by our work.\n\n\n\\textbf{Acknowledgments}---\n\nThe authors are grateful to  A.~Shitade, A.~Daido,~Y. Michishita, M.~Kimata, and R.~Toshio for valuable comments and discussions. Especially, the authors thank M.~Kimata for providing experimental data and motivating this work. This work is supported by a Grant-in-Aid for Scientific Research on Innovative Areas ``J-Physics'' (Grant No.~JP15H05884) and ``Topological Materials Science'' (Grant No.~JP16H00991,~No,~JP18H04225) from the Japan Society for the Promotion of Science (JSPS), and by JSPS KAKENHI (Grant No.~JP15K05164, No.~JP15H05745, and No.~JP18H01178). H.W. is a JSPS research fellow and supported by JSPS KAKENHI (Grant No.~18J23115).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec::introduction}\n\nThis article develops a novel functional Bayesian network for modeling directed conditional independence and causal relationships of multivariate functional data, which arise in a wide range of applications. For example, learning brain effective connectivity networks from electroencephalogram (EEG) records is crucial for understanding brain activities and neuron responses. Another example is longitudinal medical studies where multiple clinical variables are recorded at possibly distinct time points across variables and\/or patients. Knowing causal dependence of these clinical variables may help physicians decide the right interventions. Functional data can also go beyond those defined on time domain e.g., spatial domain (environmental data, spatially-resolved genomics, etc).\n\nJoint analysis of multiple functional objects has attracted great attention in recent years with focuses mainly on reducing dimensionality and capturing functional dependence. For instance, \\cite{kowal2017bayesian} and \\cite{kowal2019integer} proposed to model time-ordered functional data through a time-varying parameterization for functional time series. Using basis transformation strategies, \\cite{zhang2016functional} built an autoregressive model for spatially correlated functional data, while \\cite{lee2018bayesian} modeled functional data in serial correlation semiparametrically. \\cite{chiou2014linear} developed a linear manifold model characterizing the functional dependence between multiple random processes.\n\n\\paragraph{Functional Graphical Models} In a similar but conceptually different manner, functional graphical models have been recently proposed to model conditional independence of multivariate functional data. Graphical models gives rise to compact probabilistic representation of high-dimensional data through the graph-encoded conditional independence constraints. One key challenge is that the graph is typically unknown and must be inferred from data. While graphical models have been extensively studied for vector- and matrix-variate data \\citep{yuan2007model, wang2009bayesian, leng2012sparse, ni2017sparse}, only recently have there been several developments for the functional data. \\cite{zhu2016bayesian} extended Markov and hyper Markov laws of decomposable undirected graphs for random vectors to those for random functions. \\cite{qiao2019functional} adopted the group lasso penalty on the precision matrix of coefficients extracted from the basis expansion of functions. \\cite{zapata2022partial} introduced the idea of partial separability to reduce the computational cost of \\cite{qiao2019functional}. \\cite{qiao2020doubly} further extended \\cite{qiao2019functional} and proposed to characterize the time-varying conditional independence of random functions through smoothing techniques. To relax the Gaussian process assumption of the aforementioned methods, \\cite{li2018nonparametric}, \\cite{solea2022copula}, and \\cite{lee2022nonparametric} proposed models based on additive conditional independence and copula Gaussian models.\n\nDespite these exciting developments of functional undirected graphical models, the work on functional \\textit{directed} graphical models is sparse. Generally, undirected graphs admit a different set of conditional independence constraints from directed graphs. For example, the directed graph in Figure \\ref{ex1} implies $X_2 \\perp X_3$ but $X_2 \\not\\perp X_3 | X_1$, yet there exists no undirected counterpart that admits the same set of conditional (in)dependence assertions. More importantly, causal discovery (i.e., generation of plausible causal hypotheses) is only possible with directed graphs given additional causal assumptions \\citep{pearl2000causality}. To the best of our knowledge, the functional structural equation model recently proposed by \\cite{lee2022functional} is the only work that infers directional relationships from multivariate functional data. However, as will become evident in Section \\ref{sec::fbn} and \\ref{sec::inference}, our model differs from theirs in several significant aspects.\n\n\\paragraph{Causal Discovery} As hinted earlier, one of the two important problems we intend to address in this work is discovering causality from functional observations. Causal discovery is one of the first steps to investigate the physical mechanism that governs the operation and dynamics of an unknown system. Given the learned causal knowledge, subsequent causal inference (e.g., deriving the interventional and counterfactual distributions) can be conducted under the celebrated do-calculus framework \\citep{pearl2000causality}. Therefore, inferring causal relationships potentially has more significant scientific impacts than learning associations since it may help answer fundamental questions about the nature. Bayesian networks paired with causal assumptions are among the most popular approaches in identifying unknown causal structure represented by a directed acyclic graph (DAG). One pressing obstacle of using Bayesian networks to discover causality from purely observational data is that in general, only Markov equivalence classes (MEC) can be learned based on conditional independence constraints alone. Causal interpretations of members in the same MEC can be drastically different, and, generally, only bounds on causal effects can be calculated \\citep{maathuis2009estimating}. For example, the three DAGs in Figure \\ref{ex2} constitute an MEC with the only conditional independence $X_2 \\perp X_3 | X_1$, but the causal directions are completely reversed in the last graph compared to the first one.\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{0.25\\textwidth}\n\\includegraphics[width=\\textwidth]{EX.pdf}\n\\caption{}\n\\label{ex1}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.72\\textwidth}\n\\includegraphics[width=\\textwidth]{MEC}\n\\caption{}\n\\label{ex2}\n\\end{subfigure}\n\\caption{Two Markov equivalence classes. (a) $X_2 \\perp X_3$. (b) $X_2 \\perp X_3 | X_1$.}\n\\end{figure}\n\nSince 2006, numerous researchers, however, have found that causal discovery (unique causal structure identification) is indeed possible with additional distributional assumptions on the data generating process, at least for finite-dimensional data. Examples include but are not limited to linear non-Gaussian models (LiNGAM, \\citealt{shimizu2006linear}), non-linear additive noise models \\citep{hoyer2008nonlinear}, and linear Gaussian models with equal error variances \\citep{peters2014identifiability}. See more related methods in a recent book of \\cite{peters2017elements}. Although remarkable progresses have been made in the causal discovery area for traditional finite-dimensional data, what remains lacking is method capable of discovering causality from general, purely observational, multivariate functional data. We remark that given a known causal graph, there are existing approaches that can be used to infer causal effects. For example, \\cite{lindquist2012functional} developed a causal mediation analysis framework where the treatment and outcome are scalars and the mediator is a univariate random function. Our scope is substantially different from this line of works in that we do not assume the causal graph to be known; in fact, learning the causal graph structure is precisely the focus of this paper.\n\n\\paragraph{Proposed Functional Bayesian Networks} We propose a novel functional Bayesian network model for multivariate functional data for which the conditional independence and causal relationships are represented by a DAG. As one would expect, the proposed functional Bayesian network factorizes over the DAG and respects all directed Markov properties (i.e., conditional independence constraints) encoded in the DAG via the notion of d-separation. Then for ease of exposition, we reformulate the proposed Bayesian network constructed in the functional space to an equivalent Bayesian network defined on the space of basis coefficients via basis expansion. Because in practice, functional data are almost always observed with noises, two essential ingredients are built in the proposed Bayesian networks to capture the functional dependence and to learn the causal structure. First, we capture the within-function dependence through a set of orthonormal basis functions chosen in a data-driven way. The resulting basis functions are interpretable and computationally efficient. Second, we encode the unknown causal structure by a structural equation model on the basis coefficients. Due to the equivalence of probability measures on the functional space and the space of basis coefficients, the conditional independence and causal relationships naturally transform back to the original random functions. To allow for unique DAG identification, we move away from the Gaussian process assumption often adopted by the existing functional graphical models and instead assume our random functions are generated from a discrete scale mixture of Gaussian distributions. We theoretically prove and empirically verify that the unique DAG identification is indeed possible even when the functions are observed with noises.\n\nTo conduct inference and uncertainty quantification from a finite amount of data, the proposed model is based on a Bayesian hierarchical formulation with carefully chosen prior distributions. Posterior inference is carried out through Markov chain Monte Carlo (MCMC). We perform simulation studies to demonstrate the capability of the proposed  model in recovering causal structure and key parameters of interest. A real data analysis with brain EEG records illustrates the applicability of the proposed framework in real world. We also apply the proposed model to a COVID-19 multivariate longitudinal dataset (shown in Section D of the Supplementary Material).\n\nThe rest of the paper is structured as follows. We provide an overview of Bayesian networks in Section \\ref{sec::overview}. The proposed functional Bayesian network is introduced in Section \\ref{sec::fbn}, which includes elaborations of the functional linear non-Gaussian model (Section \\ref{sec::FLiNG}) and the causal identifiability theory (Section \\ref{sec:ci}). Section \\ref{sec::inference} is devoted to Bayesian inference of the proposed model. We provide simulation studies and applications in Sections \\ref{sec::experiment} and \\ref{sec::eeg}, respectively. The main contributions of this paper are summarized in Section \\ref{sec::discussion} with some concluding remarks.\n \n\\section{Overview of Bayesian Networks} \\label{sec::overview}\n\nThroughout the paper, vectors and matrices are boldfaced whereas scalars and sets are not.\n\n\\paragraph{DAGs and Bayesian Networks} Let $\\bm{X} = (X_1, \\ldots, X_p)^T \\in \\mathcal{X}_1 \\times \\cdots \\times \\mathcal{X}_p$ denote a $p$-dimensional random vector. Denote $[m]: = \\{1, \\ldots, m\\}$ for any integer $m \\geq 1$. Let $\\bm{X}_S = (X_j)_{j\\in S}$ be a subvector of $\\bm{X}$ with $S\\subseteq [p]$. A DAG $G = (V, E)$ consists of a set of nodes $V = [p]$ and a set of directed edges represented by a binary adjacency matrix $\\bm{E} = (E_{j\\ell})$ where $E_{j\\ell} = 1$ if and only if $\\ell\\rightarrow j$ for $\\ell \\neq j\\in V$. DAGs do not allow directed cycles $j_0 \\to j_1 \\to \\cdots \\to j_k = j_0$. Each node $j \\in V$ represents a random variable $X_j \\in \\mathcal{X}_j$; we may use $j$ and $X_j$ interchangeably when no ambiguity arises. Each directed edge $\\ell \\to j$ and the lack thereof represent conditional dependence and independence of $X_\\ell$ and $X_j$, respectively. Note that although $X_j$ is often a scalar but it does not need to be. In fact, $X_j$ is a random function or an infinite dimensional random vector in this article. Denote $pa_G(j) = \\{\\ell\\in V: \\ell \\to j\\}$ the set of parents of $j$ in graph $G$. A Bayesian network (BN) $\\mathcal{B} = (G, P)$ on $\\bm{X}$ is a probability model where the joint probability distribution $P$ of $\\bm{X}$ factorizes with respect to $G$ in the following manner,\n\\begin{equation}\\label{eq:bnf}\nP(\\bm{X}) = \\prod_{j = 1}^p P_j(X_j | \\bm{X}_{pa_G(j)}),\n\\end{equation}\nwhere $P_j$ is the conditional distribution of $X_j$ given $\\bm{X}_{pa_G(j)}$ under $P$. Let $de_G(j) = \\{\\ell \\in V: j \\to \\cdots \\to \\ell\\}$ denote the descendants of $j$ in $G$ and let $nd_G(j) = V \\backslash de_G(j) \\backslash \\{j\\}$ denote the non-descendants of $j$. The BN factorization \\eqref{eq:bnf} directly implies the local directed Markov property -- any variable is conditionally independent of its non-descendants given its parents, $X_j \\perp \\bm{X}_{nd_G(j)\/pa_G(j)} | \\bm{X}_{pa_G(j)}, \\forall j \\in [p]$. In fact, the reverse is also true: if a distribution $P$ respects the local Markov property according to a DAG $G$, then $P$ must factorize over $G$ as in \\eqref{eq:bnf}. In summary, BN factorization and local Markov property are equivalent. We may omit the subscript $G$ of $pa_G(j)$ and $nd_G(j)$ and simply write $pa(j)$ and $nd(j)$ instead when $G$ is clear from the context.\n\n\\paragraph{Causal DAGs and Causal Bayesian Networks} A causal DAG $G$ is a DAG except that the directed edges are now interpreted causally, i.e., we say $X_\\ell$ is a direct cause (with respect to $V$) of $X_j$ and $X_j$ is a direct effect of $X_\\ell$ if $\\ell \\to j$. For simplicity, we will overload $nd(j)$ and $pa(j)$ to denote the noneffects and directed causes of $j$ in a causal DAG. To define a causal BN, we begin by asserting the local causal Markov assumption \\citep{spirtes2000causation,pearl2000causality} -- given a causal DAG $G$, a variable is conditionally independent of its noneffects given its direct causes. By noting the correspondence between noneffects and non-descendants, and between direct causes and parents in DAGs and causal DAGs, the local causal Markov assumption simply states that the distribution $P$ of $\\bm{X}$ respects the local Markov property of the causal DAG $G$, which in turn implies that $P$ must also factorize over $G$ (recall the equivalence between BN factorization and local Markov property). Therefore, a causal BN $\\mathcal{B} = (G, P)$ is a probability model where $P$ factorizes with respect to a causal DAG $G$ in the same way as in \\eqref{eq:bnf}.\n\n\\paragraph{Structural Equation Representation of Bayesian Networks} A BN is often represented by a structural equation model (SEM),\n\\begin{equation*}\nX_j = f_j(\\bm{X}, \\epsilon_j), ~ \\forall j \\in [p],\n\\end{equation*}\nwhere the transformation $f_j$ depends on $\\bm{X}$ only through its parents\/direct causes $\\bm{X}_{pa(j)}$, and the exogenous variables $\\bm{\\epsilon}=(\\epsilon_1,\\dots,\\epsilon_p)^T\\sim P_\\epsilon$ are assumed to be mutually independent. Denote the set of transformation functions as $F = \\{f_1, \\ldots, f_p\\}$. Since $F$ and $P_\\epsilon$ induce the joint distribution $P$ of $\\bm{X}$ and it is not difficult to show that the induced distribution $P$ factorizes over $G$, with a slight abuse of notation, we can rewrite the BN as $\\mathcal{B} = (G, F, P_\\epsilon)$.\n \n\\section{Functional Bayesian Networks} \\label{sec::fbn}\n\n\\subsection{General Framework} \\label{sec:gf}\n\nNow we introduce the construction of BNs for multivariate functional data. Denote the space of square integrable functions on domain $\\mathcal{D}$ with respect to measure $\\mu$ as $L^2(\\mathcal{D}) = \\{h: \\int_\\mathcal{D} h^2(\\omega) d\\mu(\\omega) < \\infty\\}$. We focus on a compact $\\mathcal{D} \\subset \\mathbb{R}$ (in fact, without loss of generality, $\\mathcal{D} = [0, 1]$) and the Lebesgue measure $\\mu$ for simplicity. Let $\\bm{Y} = (Y_1, \\ldots, Y_p)^T \\in L^2(\\mathcal{D}_1)\\times\\dots\\times L^2(\\mathcal{D}_p)$ be a collection of $p$ random functions. Denote $\\mathcal{H} = \\bigcup_{j = 1}^p \\{(\\omega, j): \\omega \\in \\mathcal{D}_j\\}$ the joint domain of $\\bm{Y}$ and $(L^2(\\mathcal{H}), \\mathcal{B}(L^2(\\mathcal{H})), P)$ its probability space. Similarly, for any subset $A \\subset [p]$, denote the joint domain $\\mathcal{H}_A = \\bigcup_{j \\in A} \\{(\\omega, j): \\omega \\in \\mathcal{D}_j\\}$ and $\\mathcal{B}(L^2(\\mathcal{H}_A))$ the Borel $\\sigma$-algebra on $L^2(\\mathcal{H}_A)$. Let $A, B, C$ be disjoint subsets of $[p]$. Following \\cite{zhu2016bayesian}, we say $\\bm{Y}_A$ is conditionally independent of $\\bm{Y}_B$ given $\\bm{Y}_C$ under $P$, if for any measurable set $D_A \\subset L^2(\\mathcal{H}_A)$, $P(\\bm{Y}_A \\in D_A | \\bm{Y}_B, \\bm{Y}_C)$ is $\\mathcal{B}(L^2(\\mathcal{H}_C))$ measurable and $P(\\bm{Y}_A \\in D_A | \\bm{Y}_B, \\bm{Y}_C) = P(\\bm{Y}_A \\in D_A | \\bm{Y}_C)$. We introduce a DAG $G = (V, E)$ where each node $j \\in V$ represents a random function $Y_j$. To begin with, we give the formal definition of a functional Bayesian network.\n\n\\begin{definition}[Functional Bayesian Networks]\nWe say $\\mathcal{B} = (G, P)$ is a functional Bayesian network for a set of random functions $\\bm{Y}$ if $P$ factorizes with respect to DAG $G$, \n\\begin{align*}\nP(Y_1 \\in D_1,\\dots,Y_p\\in D_p) = \\prod_{j = 1}^p P_j(Y_j \\in D_j | \\bm{Y}_{pa(j)} \\in D_{pa(j)}),\n\\end{align*}\nfor any measurable sets $D_j \\subset L^2(\\mathcal{D}_j), \\forall j \\in [p]$, where $P_j$ is the conditional probability measure of $Y_j$ given $\\bm{Y}_{pa(j)}$ under $P$.\n\\end{definition}\n\nJust like the ordinary finite-dimensional BN, the functional BN factorization implies the local Markov property and vice versa. \n\n\\begin{definition}[Functional Local Directed Markov Property]\nA probability measure $P$ of $\\bm{Y}$ satisfies the local directed Markov property with respect to $G$ if $Y_j \\perp \\bm{Y}_{nd(j)\/pa(j)} | \\bm{Y}_{pa(j)}$, i.e., $P(Y_j \\in D_j | \\bm{Y}_{nd(j)\/pa(j)}, \\bm{Y}_{pa(j)})$ is $\\mathcal{B}(L^2(\\mathcal{H}_{pa(j)}))$ measurable and $P(Y_j \\in D_j | \\bm{Y}_{nd(j)\/pa(j)}, \\bm{Y}_{pa(j)}) = P(Y_j \\in D_j | \\bm{Y}_{pa(j)})$ for any $D_j\\subset L^2(\\mathcal{D}_j)$.\n\\end{definition}\n\n\\begin{proposition}\nFunctional Bayesian network factorization is equivalent to functional local directed Markov property.\n\\end{proposition}\n\nProof is trivial. For modeling convenience, we use orthonormal basis expansion of random functions to (equivalently) redefine the functional BN in the space of basis coefficients. Let $\\{\\phi_{jk}\\}_{k = 1}^\\infty$ be a sequence of orthonormal basis functions of $L^2(\\mathcal{D}_j)$ and expand $Y_j = \\sum_{k = 1}^\\infty Z_{jk} \\phi_{jk}$, where $Z_{jk} = \\int_{\\mathcal{D}_j} Y_j(\\omega) \\phi_{jk}(\\omega) d\\omega$. The resulting coefficient sequence $\\bm{Z}_j = (Z_{jk})_{k = 1, \\ldots, \\infty}$ lies in the space of square summable sequences $\\ell_j^2 = \\{h_j: \\sum_{k = 1}^\\infty h_{jk}^2 < \\infty\\}$. The within-function and the between-function covariance can then be expressed in terms of the covariance of the coefficient sequences,\n\\begin{align*}\n\\text{cov}(Y_j(\\omega_j), Y_\\ell(\\omega_\\ell)) = \\sum_{k = 1}^\\infty \\sum_{h = 1}^\\infty \\phi_{jk}(\\omega_j) \\phi_{\\ell h}(\\omega_\\ell) \\text{cov}(Z_{jk}, Z_{\\ell h}), ~ \\forall \\omega_j \\in \\mathcal{D}_j, \\omega_\\ell \\in \\mathcal{D}_\\ell, ~ \\forall j, \\ell \\in [p].\n\\end{align*}\nBecause $L^2(\\mathcal{D}_j)$ and $\\ell_j^2$ are isometrically isomorphic for each $j$, for any disjoint subsets $A, B, C \\subset [p]$, $\\bm{Y}_A \\perp \\bm{Y}_B | \\bm{Y}_C$ if and only if $\\bm{Z}_A \\perp \\bm{Z}_B | \\bm{Z}_C$ where $\\bm{Z} = (\\bm{Z}_1, \\ldots, \\bm{Z}_p)^T$. Hence, if $\\bm{Y}$ follows the proposed BN model $\\mathcal{B} = (G, P)$, then the coefficient sequences $\\bm{Z}$ follows $\\mathcal{B}_Z = (G, P_Z)$ for some probability measure $P_Z$ of $\\bm{Z}$, and vice versa. Each node of the DAG $G$ either represents a random function $Y_j$ or, equivalently, its corresponding coefficient sequence $\\bm{Z}_j$. Moreover, the joint probability measure $P$ of $\\bm{Y}$ factorizes with respect to $G$ if and only if the joint probability measure $P_Z$ of $\\bm{Z}$ factorizes with respect to $G$. \n\n\\begin{proposition}\nSuppose $\\bm{Y}\\sim P$ and let $\\bm{Z}$ be the corresponding coefficient sequences from orthonormal basis expansion.\nThen \n\\begin{align*}\nP(Y_1 \\in D_1,\\dots,Y_p\\in D_p) = \\prod_{j = 1}^p P_j(Y_j \\in D_j | \\bm{Y}_{pa(j)} \\in D_{pa(j)}),\n\\end{align*}\nfor any measurable sets $D_j \\subset L^2(\\mathcal{D}_j), \\forall j \\in [p]$ if and only if\n\\begin{align*}\nP_Z(\\bm{Z}_1 \\in D_1',\\dots,\\bm{Z}_p\\in D_p') = \\prod_{j = 1}^p P_{Z j}(\\bm{Z}_j \\in D_j' | \\bm{Z}_{pa(j)} \\in D_{pa(j)}'),\n\\end{align*}\nfor any measurable sets $D_j' \\subset \\ell_j^2, \\forall j \\in [p]$.\n\\end{proposition}\n\nThe proof directly follows the preceding paragraph. Again, just like the ordinary finite-dimensional BN, if one makes the causal Markov assumption, the DAG $G$ in the proposed functional BN can be interpreted causally. Hereafter, by default, we always make the causal Markov assumption (hence $G$ is a causal DAG, the edge strength is interpreted as direct causal effect, etc) but all the results are simply reduced to those of a directed conditional independence model when the causal Markov assumption is dropped.\n \n\\subsection{Functional Linear Non-Gaussian Bayesian Networks} \\label{sec::FLiNG}\n\nSection \\ref{sec:gf} introduces a general framework for modeling directed conditional independence and causal relationships for multivariate functional data. In this subsection, we discuss in detail one specific case of the proposed general framework, namely the \\underline{F}unctional \\underline{Li}near \\underline{N}on-\\underline{G}aussian (FLiNG) BNs. Specifically, the FLiNG-BN assumes $\\bm{Z}$ follows a linear SEM,\n\\begin{align} \\label{eq1}\n\\bm{Z}_j = \\sum_{\\ell = 1}^p \\bm{B}_{j\\ell} \\bm{Z}_\\ell + \\bm{\\epsilon}_j, ~ \\forall j \\in [p],\n\\end{align}\nwhere $\\bm{\\epsilon}_j$ is an infinite-dimensional exogenous vector, $\\bm{B}_{j\\ell} = (B_{j\\ell}(k_j, k_\\ell))_{k_j=1,k_\\ell=1}^{\\infty,\\infty}$ is an infinite-dimensional direct causal effect matrix from $\\bm{Z}_\\ell$ to $\\bm{Z}_j$, and $\\ell\\to j$ is present in $G$ (i.e., $\\bm{Z}_\\ell$ is a direct cause of $\\bm{Z}_j$) if there exist $k_\\ell$ and $k_j$ such that $B_{j\\ell}(k_j, k_\\ell) \\neq 0$. Neither causal effects nor the causal graph is assumed to be known; therefore the main goal of this article is precisely to infer them from observational data. Because $L^2(\\mathcal{D}_j)$ and $\\ell_j^2$ are isometrically isomorphic for all $j \\in [p]$, the casual relationships of $\\bm{Z}$ encoded in DAG $G$ directly transfer to the the casual relationships of $\\bm{Y}$, i.e., $\\bm{Z}_\\ell$ is a direct cause of $\\bm{Z}_j$ if and if only if $Y_\\ell$ is a direct cause of $Y_j$.\n\nIn practice, the random functions $\\bm{Y}$ can only be measured on finite grids with random noises. In other words, we do not observe realizations of $\\bm{Y}$ but instead we observe realizations of $\\bm{W}=(\\bm{W}_1,\\dots,\\bm{W}_p)^T$ where $\\bm{W}_{j} = (W_{j}(1), \\ldots, W_{j}(m_j))$, which is the set of measurements of $Y_{j}$ on a finite grid $D_{j} = \\{\\omega_{j}(1), \\ldots, \\omega_{j}(m_j)\\} \\subset \\mathcal{D}_j$ with independent white noises $e_{j}(m) \\sim N(0, \\sigma_j)$, $\\forall m \\in [m_j]$, \n\\begin{align} \\label{eq2}\nW_j(m) = Y_{j}(\\omega_{j}(m)) + e_{j}(m).\n\\end{align}\nNote that $D_{j}$ can be different across $j$ (also across realizations).\n\nOne seemingly inconsequential element of the FLiNG-BN but turning out to be crucial for discovering causality is the specification of the probability distribution of the exogenous variables $\\bm{\\epsilon}_j= (\\epsilon_{jk})_{k=1}^\\infty$ in \\eqref{eq1}.\nA tempting choice may be Gaussian but it is the non-Gaussianity of $\\bm{\\epsilon}_j$'s that allows causal identification as we will show in Section \\ref{sec:ci}. Specifically, we assume $\\epsilon_{jk}$ to follow a finite scale mixture of Gaussian distributions, $\\epsilon_{jk} \\sim \\sum_{m = 1}^{M_{jk}} \\pi_{jkm} N(0, \\tau_{jkm})$,\nwhere $M_{jk}$ is the number of mixture components. The non-Gaussian exogenous variables lead to non-Gaussian coefficient sequences $\\bm{Z}$, which in turn lead to non-Gaussian-process distributed random functions $\\bm{Y}$. In addition to enabling causal identification, non-Gaussian-processes are robust against outlying curves \\citep{zhu2011robust}. For finite sample inference, we truncate the orthonormal basis at level $K$ such that $\\bm{\\phi}_j = (\\phi_{j1}, \\ldots, \\phi_{jK})^T$, as commonly done in existing functional data analysis literature. Consequently, \\eqref{eq2} is turned into\n\\begin{align} \\label{eq3}\nW_{j}(m) = \\sum_{k = 1}^K Z_{jk} \\phi_{jk}(\\omega_{j}(m)) + e_{j}(m).\n\\end{align}\n \n\\subsection{Causal Identifiability} \\label{sec:ci}\n\nThe proposed functional BNs are useful representations of directed conditional independence and causal relationships for multivariate functional data. The big remaining question is the learning of the underlying (causal) DAGs from observational data. Constraint-based methods, which are often model-free, have been popular for DAG learning. For the proposed functional BNs, we, in principle, can also use constraint-based methods, which test for conditional independence of pairs of functions. However, conditional independence tests are notoriously difficult and inefficient even for scalar random variables. Furthermore, even if we have access to oracle conditional independence tests for random functions, we can only hope for identifying the best MEC by definition (recall that an MEC contains DAGs with exactly the same set of conditional independence relationships). This may be acceptable if one is only interested in learning conditional independence relationships. But as mentioned in Section \\ref{sec::introduction}, for causal discovery, this is clearly unsatisfactory because the directionality of a potentially large number of edges of Markov equivalent DAGs may be left undetermined and hence the causal interpretations of these edges are unclear. Because the proposed FLiNG-BN is a proper probability model, we can exploit certain feature of the model, namely the non-Gaussianity, to uniquely identify the underlying causal DAG.\n \n\\begin{definition}[Causal Identifiability]\nSuppose $\\bm{Y}$ follows the FLiNG-BN $\\mathcal{B} = (G, P)$, and suppose $\\bm{W}$ is a noisy version of $\\bm{Y}$ with noise variances $\\bm{\\sigma}= (\\sigma_1, \\ldots, \\sigma_p)$ as defined in \\eqref{eq2}. Let $P_W$ denote the distribution of $\\bm{W}$ induced from FLiNG-BN and the noises. We say that the causal DAG of FLiNG-BN is identifiable from $\\bm{W}$ if there does not exist another BN $\\mathcal{B}' = (G', P')$ where $G'\\neq G$ and noise variances $\\bm{\\sigma}' = (\\sigma'_1, \\ldots, \\sigma'_p)$ such that the induced distributions on $\\bm{W}$, $P'_W$, is equivalent to $P_W$, i.e., $P_W(\\bm{W}) \\equiv P'_W(\\bm{W})$.\n\\end{definition}\n\n\\begin{theorem}[Causal Identifiability] \\label{them1}\nThe causal DAG of FLiNG-BN is identifiable if the number of Gaussian mixture components $M_{jk} > 1, \\forall j, k$.\n\\end{theorem}\n\nTheorem \\ref{them1} signifies that by examining the probability distribution $P_W$, to which we have access through the observational data alone, one can gauge the likelihood that a given causal DAG is the data generating DAG. With a finite dataset, we shall focus on weighing different candidate causal DAGs by their posterior probabilities. Here, we provide the outline of the proof; the complete proof is given in Section A of the Supplementary Material. Given a chosen set of basis functions, we show the result in the space of basis coefficients. The problem then transforms to prove that, given $\\bm{Z} = \\bm{B} \\bm{Z} + \\bm{\\epsilon}$ and observe $\\bm{W} = \\bm{Z} + \\bm{e}$, there does not exist another equivalent parameterization $\\bm{Z}' = \\bm{B}' \\bm{Z}' + \\bm{\\epsilon}'$ and $\\bm{W} = \\bm{Z}' + \\bm{e}'$. Since we assume each component of $\\bm{\\epsilon}$ follows a Gaussian scale mixture, the induced distribution on $\\bm{W}$ is a multivariate Gaussian mixture (with different precision matrices). We then prove the causal effect matrix $\\bm{B}$ is uniquely identifiable from such mixture model by combining the identification of Gaussian mixture components, uniqueness of LDL decomposition, and proof of causal ordering identification. We demonstrate the identifiability result with a toy example.\n\n\\begin{example} \\label{exp1}\nConsider a true functional causal graph $1\\to 2$ and the corresponding data generating model $Z_1 = \\epsilon_1$ with $\\epsilon_1 \\sim 0.5 N(0, 0.5) + 0.5 N(0, 1)$ and $Z_2 = Z_1 + \\epsilon_2$ with $\\epsilon_2 \\sim 0.5 N(0, 0.5) + 0.5 N(0, 1)$; note for simplicity, we assume in this example that the number of basis functions is $K=1$. Assume we observe with noises $W_1 = Z_1 + e_1$ and $W_2 = Z_2 + e_2$ with $e_1, e_2 \\sim N(0, 0.1)$. We sample $n = 1000$ observations from this model and index them by the subscript $i=1,\\dots,n$. For the purpose of illustration, suppose we know the mixture component assignment of each observation and define four groups of observations based on the combination of variances of $\\epsilon_1$ and $\\epsilon_2$,\n\\begin{equation*}\n\\begin{aligned}\n& C_1 = \\{i: \\text{Var}(\\epsilon_{i1}) = 0.5 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 0.5\\}, ~ C_2 = \\{i: \\text{Var}(\\epsilon_{i1}) = 0.5 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 1\\}, \\\\\n& C_3 = \\{i: \\text{Var}(\\epsilon_{i1}) = 1 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 0.5\\}, ~ C_4 = \\{i: \\text{Var}(\\epsilon_{i1}) = 1 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 1\\}.\n\\end{aligned}\n\\end{equation*}\nWe fit linear regression separately to all observations and to observations in each of the four groups with the true causal direction $1 \\to 2$ (regressing $W_2$ on $W_1$) and the  anti-causal direction $2 \\to 1$ (regressing $W_1$ on $W_2$), which are shown in Figure \\ref{demo}. We observe that the fitted lines are almost identical in the causal direction for all groups whereas they can be quite different across groups in the anti-causal direction. Therefore, only the true causal graph gives a unique regression coefficient among all groups. Notice that if there is only one mixture component (i.e., degeneration to the Gaussian case), no comparison can be made between the causal and anti-causal directions since there will be only one regression line.\n\\end{example}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1\\linewidth]{DEMO}\n\\caption{A toy example for demonstration of causal identification. The left (right) panel shows the linear regression of $W_2$ ($W_1$) on $W_1$ ($W_2$). Data are simulated from the causal graph $1\\to 2$. Colored lines are the fitted linear regressions for all observations and for the observations in groups $C_1$--$C_4$.}\n\\label{demo}\n\\end{figure}\n\nThe next counter example illustrates the necessity of the non-Gaussian assumption for causal identification.\n\n\\begin{example}\nConsider a similar bivariate case to Example \\ref{exp1} but now the exogenous variables are Gaussian instead of mixture of Gaussian. Suppose the true functional causal graph $1\\to 2$ and the corresponding data generating model $Z_1 = \\epsilon_1$ with $\\epsilon_1 \\sim N(0, \\tau_1)$ and $Z_2 = b Z_1 + \\epsilon_2$ with $\\epsilon_2 \\sim N(0, \\tau_2)$; note again for simplicity, we assume in this example that the number of basis functions is $K = 1$. Assume we observe with noises $W_1 = Z_1 + e_1$ and $W_2 = Z_2 + e_2$ with $e_1 \\sim N(0, \\sigma_1)$ and $e_2 \\sim N(0, \\sigma_2)$. The induced joint distribution on $\\bm{W} = (W_1, W_2)$ is then bivariate Gaussian with mean $0$ and covariance matrix\n$$\\begin{pmatrix}\n\\tau_1 + \\sigma_1 & b \\tau_1\\\\\nb \\tau_1 & b^2 \\tau_1 + \\tau_2 + \\sigma_2\n\\end{pmatrix}. $$\n\nFurther consider the anti-causal model $2 \\to 1$ where $Z_2' = \\epsilon_2'$ with $\\epsilon_2' \\sim N(0, \\tau_2')$ and $Z_1' = b' Z_2' + \\epsilon_1'$ with $\\epsilon_1' \\sim N(0, \\tau_1')$. Suppose $W_1 = Z_1' + e_1'$ and $W_2 = Z_2' + e_2'$ with $e_1' \\sim N(0, \\sigma_1')$ and $e_2' \\sim N(0, \\sigma_2')$. The induced joint distribution on $\\bm{W} = (W_1, W_2)$ is still bivariate Gaussian with mean 0 and covariance matrix\n$$\\begin{pmatrix}\nb^{'2} \\tau_2' + \\tau_1' + \\sigma_1' & b' \\tau_2'\\\\\nb' \\tau_2' & \\tau_2' + \\sigma_2'\n\\end{pmatrix}. $$ \nFor any chosen $\\tau_1', \\sigma_1' > 0$ such that $\\tau_1' + \\sigma_1' < \\tau_1 + \\sigma_1 - b^2 \\tau_1^2 \/ (b^2 \\tau_1 + \\tau_2 + \\sigma_2)$, if we set\n\\begin{equation*}\n\\begin{aligned}\n& b' = (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1') \/ b \\tau_1, \\\\\n&\\tau_2' = b^2 \\tau_1^2 \/ (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1'), \\\\\n& \\sigma_2' = b^2 \\tau_1 + \\tau_2 + \\sigma_2 - b^2 \\tau_1^2 \/ (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1'),\n\\end{aligned}\n\\end{equation*}\nthen the induced distribution coincides with that under the true causal model (i.e., Gaussian with mean 0 and the same covariance). Therefore, causal identification fails in this case.\n\\end{example}\n\n\\section{Bayesian Inference} \\label{sec::inference}\n\nThe inference of the proposed FLiNG-BN framework can be carried out in either a frequentist (e.g., maximizing penalized likelihood) or a Bayesian (e.g., sampling from posterior distribution) fashion. Existing frequentist functional graphical models \\citep{qiao2019functional,qiao2020doubly,zapata2022partial,solea2022copula,lee2022nonparametric,lee2022functional} often estimate graphs in two separate steps -- (i) estimate the basis coefficient sequence of each function marginally via functional principle component analysis, and (ii) learn an directed\/undirected graph based on the estimated coefficient sequences. However, the eigenfunctions that marginally explain the most variation of each individual function do not necessarily explain well the conditional\/causal relationships among a set of functions. Moreover, the estimation uncertainty is not propagated from the first step to the second, which may result in overly confident inference. To mitigate these potential drawbacks of the two-step approaches, we propose a fully Bayesian inference procedure that jointly infers basis coefficient sequences and the DAG structure. This joint inference approach constructs orthonormal basis functions adaptive to their conditional\/causal relationships and allows for finite-sample inference and uncertainty quantification. \n\n\\subsection{Adaptive Orthonormal Basis Functions}\n\nWe assume the basis functions to be shared across all random functions \\citep{kowal2017bayesian,zapata2022partial}, $\\phi_{jk}(\\omega) := \\phi_k(\\omega),\\forall j\\in [p]$, which are more parsimonious than models based on function-specific basis functions. Moreover, the common basis functions put the basis coefficient sequences $\\bm{Z}_j,\\forall j\\in[p]$ on an equal footing (e.g., the magnitudes of basis coefficients are directly comparable) so that they are directly comparable and the BN on $\\bm{Z}$ has a more coherent interpretation. In this case, the non-zero matrix block $\\bm{B}_{j\\ell}$ corresponds to a causal connection from $\\bm{Y}_\\ell$ to $\\bm{Y}_j$. Loosely speaking, if we regard the basis functions as signal channels, then a significant non-zero $B_{j\\ell}(k_j, k_\\ell)$ indicates that $\\bm{Y}_\\ell$ directly affects $\\bm{Y}_j$ through its signal transmission from the $k_\\ell$-th channel to the $k_j$-th channel. \n\nAs mentioned above, we do not pre-specify a fixed set of orthonormal basis functions but instead they are learned adaptively from data by further expanding them with spline basis functions \\citep{kowal2017bayesian}, $\\phi_k(\\omega) = \\sum_{\\ell = 1}^L A_{k\\ell} b_\\ell(\\omega)$, where $\\bm{b} = (b_1, \\ldots, b_L)^T$ is a set of cubic B-spline basis functions with equally spaced knots and $\\bm{A}_k = (A_{k1}, \\ldots, A_{kL})^T, \\forall k \\in [K]$ are spline coefficients. Because $\\bm{A}_k$'s are not fixed \\emph{a priori}, so are $\\phi_k$'s.\n\n\\subsection{Prior Model}\n\nWe summarize our model and its entailed parameters using a DAG shown in Figure \\ref{hmodel}. The prior distributions of the model parameters are introduced in this section. We simulate posterior samples through Markov chain Monte Carlo (MCMC). Details are given in Section B of the Supplementary Material. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1\\linewidth]{HMODEL}\n\\caption{A DAG illustrating the model hierarchy. Single-line arrows are stochastic relationships and double-line arrows are deterministic relationships. The observed node $\\bm{W}_j$ is shown in rectangle and other nodes are shown in circles.}\n\\label{hmodel}\n\\end{figure}\n\n\\paragraph{Prior on B-spline Coefficients $\\bm{A}_k$} The prior on $\\bm{A}_k$ serves three purposes. First, it forces $\\phi_k$'s to be orthonormal, i.e., $\\int \\phi_k(\\omega) \\phi_h(\\omega) d\\omega = I(k = h), \\forall k, h \\in[K]$. Second, it regularizes the roughness of $\\phi_k$'s to prevent overfitting and sorts the orthonormal basis functions by increasing roughness. Third, it enables posterior inference on the orthonormal basis functions simultaneously with the graph estimation without having to fix them \\emph{a priori}.\n\nWe summarize the main steps of prior specification and refer the details to \\cite{kowal2017bayesian}. First, to regularize the roughness of $\\phi_k$ in a frequentist framework, one would consider a penalized likelihood with the roughness penalty,\n$$\\lambda_k \\mathcal{P}(\\bm{A}_k) = \\lambda_k \\int [\\phi_k^{''}(\\omega)]^2 \\ d\\omega = \\lambda_k \\bm{A}_k^T \\bm{\\Omega} \\bm{A}_k,$$\nwhere $\\lambda_k > 0$ is the regularization parameter and $\\bm{\\Omega} = \\int \\bm{b}^{''}(\\omega)[\\bm{b}^{''}(\\omega)]^T \\ d\\omega$. As a Bayesian counterpart, the regularization term is equivalent to a prior on the B-spline coefficients $\\bm{A}_k \\sim N(\\bm{0}, \\lambda_k^{-1} \\bm{\\Omega}^{-})$, where $\\bm{\\Omega}^{-}$ is a pseudo-inverse (since $\\bm{\\Omega}$ is rank-deficient by 2). Let $\\bm{\\Omega} = \\bm{U} \\bm{D} \\bm{U}^T$ be the singular value decomposition of $\\bm{\\Omega}$. To facilitate efficient computation, we follow \\cite{wand2008semiparametric} and reparameterize $\\phi_k = \\sum_{\\ell=1}^L A_{k\\ell}b_\\ell = \\sum_{\\ell=1}^L \\tilde{A}_{k\\ell} \\tilde{b}_\\ell$ with $\\tilde{\\bm{b}}(\\omega) = (1, \\omega, \\bm{b}^T(\\omega) \\bm{U}_P \\bm{D}_P^{-1\/2})^T$ where $\\bm{D}_P$ is the $(L - 2) \\times (L - 2)$ submatrix of $\\bm{D}$ corresponding to non-zero singular values and $\\bm{U}_P$ is the corresponding $L \\times (L - 2)$ submatrix of $\\bm{U}$. The reparameterization induces a prior on $\\tilde{\\bm{A}}_k = (\\tilde{A}_{k1}, \\ldots, \\tilde{A}_{kL})^T\\sim N(0, S_k)$ where $S_k = \\mathrm{diag}(\\infty, \\infty, \\lambda_k^{-1}, \\ldots, \\lambda_k^{-1})$ with the first two dimensions corresponding to the unpenalized constant and linear terms. In practice, one can replace $\\infty$ by a large number, say $10^8$. \n\nSecond, we constrain the regularization parameters $\\lambda_1 > \\cdots > \\lambda_K > 0$ to identify the ordering of basis functions, which sorts the basis functions by decreasing smoothness. Unlike the functional principal component analysis (PCA) where the principal components are ordered by the proportion of variance explained, the adopted Bayesian approach is less prone to rough functions. Given the ordering constraint, a uniform prior is imposed such that $\\lambda_k \\sim U(L_k, U_k)$, where $U_1 = 10^8$, $L_k = \\lambda_{k + 1}$ for $k = 1, \\ldots, K - 1$, $U_k = \\lambda_{k - 1}$ for $k = 2, \\ldots, K$, and $L_K = 10^{-8}$.\n\nFinally, consider the orthonormal constraint\n\\begin{align} \\label{penmt}\n\\int \\phi_k(\\omega) \\phi_h(\\omega) = \\int \\tilde{\\bm{A}}_k^T \\tilde{\\bm{b}}(\\omega) \\tilde{\\bm{b}}^T(\\omega)\\tilde{\\bm{A}}_h \\ d\\omega = \\tilde{\\bm{A}}_k^T \\bm{J} \\tilde{\\bm{A}}_h = I(k = h),\n\\end{align}\nwith $\\bm{J} = \\int \\tilde{\\bm{b}}(\\omega) \\tilde{\\bm{b}}^T(\\omega) \\ d\\omega$. This constraint can be easily enforced by projection and normalization during the course of MCMC; see Section B of the Supplementary Material for details.\n\n\\paragraph{Priors on the DAG Adjacency Matrix $\\bm{E}$ and Direct Causal Effects $\\bm{B}$} The key problem we aim to address in this article is causal structure learning, i.e., inferring the adjacency matrix, $\\bm{E} = (E_{j\\ell})$ (recall $E_{j \\ell} = 1$ if and only if $\\ell \\to j$). We propose to use a beta-Bernoulli-like prior $E_{j \\ell} \\sim \\mathrm{Bernoulli}(r)$ with $r \\sim \\mathrm{Beta}(a_r, b_r)$, subject to the acyclicity constraint,\n$$P(\\bm{E} | r) \\propto \\prod_{j\\neq\\ell} r^{E_{j\\ell}} (1-r)^{1-E_{j\\ell}} I(G \\text{ is a DAG}).$$\nWe set $a_r = b_r = 1$. \\cite{scott2010bayes} showed that the beta-Bernoulli prior allows automatic multiplicity adjustment in sparse regression problem. In our context, the marginal distribution of $\\bm{E}$ with $r$ integrated out equals\n\\begin{align} \\label{mar}\nP(\\bm{E}) \\propto \\mathrm{Beta}\\left(\\sum_{j\\neq\\ell}E_{j\\ell}+1,\\sum_{j\\neq\\ell}(1-E_{j\\ell})+1\\right)I(G \\text{ is a DAG}).\n\\end{align}\nThe marginal distribution strongly prevents false discoveries by increasing the penalty against additional edges as the dimension $p$ grows. For example, the marginal \\eqref{mar} favors an empty graph over a graph with one edge by a factor of $p^2 - p$, which increases with $p$.\n\nConditional on $\\bm{E}$, we assume independent matrix-variate spike-and-slab priors on the direct causal effects,\n\\begin{align*}\n\\bm{B}_{j \\ell}|E_{j \\ell} \\sim (1 - E_{j \\ell}) \\delta_{\\bm{O}}(\\bm{B}_{j \\ell}) + E_{j \\ell} N(\\bm{B}_{j\\ell}|\\bm{O}, \\gamma \\bm{I}, \\bm{I}),\n\\end{align*}\nwhere $\\delta_{\\bm{O}}(\\cdot)$ is a point mass at a $K\\times K$ zero matrix $\\bm{O}$ and $N(\\cdot|\\bm{O}, \\gamma \\bm{I}, \\bm{I})$ is a centered matrix-variate normal distribution with row and column covariance matrices $\\gamma \\bm{I}$ and $\\bm{I}$ where $\\bm{I}$ is a $K\\times K$ identity matrix. The hyperparameter $\\gamma$ indicates the overall causal effect size and is assumed to follow a conjugate inverse-gamma prior, $\\gamma \\sim IG(a_\\gamma, b_\\gamma)$ with $a_\\gamma = b_\\gamma = 1$.\n\n\\paragraph{Prior on the Gaussian Scale Mixture} We choose conjugate priors,\n\\begin{align*}\n\\bm{\\pi}_{jk} = (\\pi_{jk1}, \\ldots, \\pi_{jkM}) \\sim \\text{Dirichlet}(\\alpha,\\dots,\\alpha), ~ \\tau_{jkm} \\sim IG(a_\\tau, b_\\tau), ~ \\forall j \\in [p], k \\in [K], m \\in [M],\n\\end{align*}\nwhich allows for straightforward Gibbs sampling. As default, we set $\\alpha = 1$ and $a_\\tau = b_\\tau = 1$. \n\n\\paragraph{Prior on Observation Noises} We complete the prior specification with a conjugate inverse-gamma prior on the variance of observation noises, $\\sigma_j \\sim IG(a_\\sigma, b_\\sigma), \\forall j \\in [p]$ with $a_\\sigma = b_\\sigma = 0.01$.\n\nFinally, we summarize the differences between the proposed FLiNG-BN and the work from \\cite{lee2022functional}. First, \\cite{lee2022functional} assume  their functions to be noiseless whereas we consider the scenario where functions are observed with noises. The causal identifiability theory is significantly more complicated when functions are noisy. Second, they assume their functions to be Gaussian whereas our functions are non-Gaussian; this difference leads to different learning algorithms and identifiability theory. Third, their inference is a two-step procedure based on causal ordering identification and sparse function-on-function regression, while the proposed Bayesian hierarchical model admits one-step inference procedure, which learns the graph structure by directly searching in the graph space without having to learn the causal ordering first.\n\n\\section{Simulation Studies} \\label{sec::experiment}\n\nWe conducted simulation studies to evaluate the proposed FLiNG-BN model. We considered two scenarios. In the first scenario, the functions were observed on an evenly spaced grid; this is the scenario commonly studied in the existing functional undirected graphical models \\citep{qiao2019functional} and is also similar to our later EEG application. In the second scenario, the functions were observed on an unevenly spaced grid, similar to the COVID-19 longitudinal application (the details are shown in Section C of the Supplementary Material). We compared the proposed FLiNG-BN with a functional undirected graphical model (FGLASSO; \\citealt{qiao2019functional}). We did not make comparison with \\cite{lee2022functional} due to lack of publicly available code at the time of submission. In addition, we compared FLiNG-BN with approaches based on two-step estimation procedures. In the first step, we extracted basis coefficients obtained from functional PCA using the package \\texttt{fdapace} \\citep{carroll2021}. In the second step, given the estimated basis coefficients, we constructed causal graphs using either the LiNGAM \\citep{shimizu2006linear} algorithm (termed FPCA-LiNGAM) or the PC \\citep{spirtes1991algorithm} algorithm (termed FPCA-PC). LiNGAM estimates a causal DAG based on the linear non-Gaussian assumption whereas PC generally returns only an equivalence class of DAGs based on conditional independence tests. Their implementations are available from R package \\texttt{pcalg} \\citep{kalisch2020overview}.\n\nTo mimic the EEG data application, we simulated data from FLiNG-BN with all the combinations of sample size $n \\in \\{50, 100, 200\\}$, number of functions $p \\in \\{30, 60, 90\\}$, and grid size $d \\in \\{125, 250\\}$. The grid spanned the unit interval $[0,1]$. We set the true number of basis functions to be $K = 5$. To generate basis functions, we first simulated the non-orthnormal functions $\\phi_k^U, \\forall k \\in [K]$ from a set of $L = 6$ cubic B-spline basis functions with evenly spaced knots, $\\phi_k^U = \\sum_{\\ell=1}^L A_{k\\ell}b_\\ell$, where $A_{k\\ell}$'s were generated from a standard normal distribution. We then empirically orthonormalized $(\\phi_1^U, \\ldots, \\phi_K^U)$ to get the orthonormal basis functions $(\\phi_1, \\ldots, \\phi_K)$. The simulation true causal graph $G$ was generated from the Erd\\H{o}s-R\\'{e}nyi model with connection probability $2\/p$, subject to the acyclicity constraint. Given the true graph $G$, each block of non-zero direct causal effects $\\bm{B}_{j\\ell}$ was generated independently from a standard matrix-variate normal distribution. Then the basis coefficient sequences $\\bm{Z}$ were generated from \\eqref{eq1} where the exogenous variables $\\bm{\\epsilon}_j$'s were generated from a centered Laplace distribution with scale $b = 0.5$. Note that when we fit FLiNG-BN to the simulated data, we still assumed the exogenous variables to be discrete scale Gaussian mixture although the simulation true exogenous variables were Laplace (i.e., continuous scale Gaussian mixture). Finally, noisy observations were simulated following \\eqref{eq3} with the signal-to-noise ratio, i.e., the mean value of $|y_j^{(i)}(\\omega_j^{(i)}(m))| \/ \\sigma_j$ across all samples $i \\in [n]$ and grid points $m \\in [m_{j}^{(i)}]$, set to 5.\n\nFor implementing the proposed FLiNG-BN, we set the number of mixture components to $M = 5$ and the number B-spline basis functions to $L=20$ (note that the simulation truth was $L=6$), and ran MCMC for 5,000 iterations (discarding the first half as burn-in and retaining every 5th iteration after burn-in). The causal graph $G$ was estimated by thresholding the posterior probability of inclusion at 0.5 (i.e., the median probability model). Parameters of competing methods were set to their default values. To assess the graph recovery performance, we calculated true positive rate (TPR), false discovery rate (FDR), and Matthews correlation coefficient (MCC),\n\\begin{align*}\n& \\text{TPR} = \\text{TP}(\\text{TP} + \\text{FN})^{-1}, ~~~~~ \n\\text{FDR} = \\text{FP}(\\text{TP} + \\text{FP})^{-1}, \\\\\n& \\text{MCC} = (\\text{TP} \\times \\text{TN} - \\text{FP} \\times \\text{FN})\\left[(\\text{TP} + \\text{FP}) \\times (\\text{TP} + \\text{FN}) \\times (\\text{TN} + \\text{FP}) \\times (\\text{TN} + \\text{FN})\\right]^{-1\/2},\n\\end{align*}\nwhere TP, TN, FP, and FN stand for the numbers of true positives, true negatives, false positives, and false negatives, respectively. MCC ranges from $-$1 to 1 with $0$ indicating a random guess and $1$ a perfect recovery. Since FGLASSO learns an undirected graph, we compared it with a moralization of the true graph\\footnote{Graph moralization converts a DAG to an undirected graph by first marrying all the unmarried parents and then removing all the directions. A probability distribution that respects the Markov property of a DAG must respect the Markov property of its moral graph.}. Similarly, since PC algorithm returns the MEC representation\\footnote{The MEC representation is shown in an essential graph, where any edge presented between two nodes is directed if and only if it follows the same direction in all members of this MEC. Otherwise, it is undirected.}, we compared it with the MEC of the true causal graph. \n\nThe results based on 50 repeat simulations are summarized in Table \\ref{s1}, from which we conclude that the proposed FLiNG-BN significantly outperformed all the competitors FGLASSO, FPCA-LiNGAM, and FPCA-PC across all combinations of $n$, $p$, and $d$. This is not surprising because (i) FGLASSO is not designed for learning directed graphs; they were compared with the proposed FLiNG-BN because of lack of alternative functional BN implementation. (ii) Although FPCA-LiNGAM and FPCA-PC are capable of learning directed graphs, they still performed poorly because they are implemented in a two-step procedure where there is little reason to believe that the basis coefficients extracted by the functional PCA in the first step are useful to capture the functional dependence in the second step. (iii) Unlike the proposed approach, none of the competing methods controls for false discovery and some impose the stringent Gaussian assumption, resulting in high FDR and\/or low TPR. Our suggested method to determine $K$ also worked well.\n\n\\begin{table}[ht]\n\\caption{Functions observed on evenly spaced grid. Average operating characteristics based on 50 repetitions are reported; standard deviations are given within the parentheses. Since LiNGAM is not applicable to cases where $q > n$ with $q = pK$ being the total number of extracted basis coefficients across all functions, the results from those cases are not available and indicated by -.}\n\\resizebox{\\textwidth}{!}{\n\\centering\n\\begin{tabular}{ccc|ccc|ccc|ccc|ccc}\n\\toprule\n\\multirow{2}{*}{$p$} & \\multirow{2}{*}{$d$} & \\multirow{2}{*}{$n$} & \\multicolumn{3}{c|}{FLiNG-BN} & \\multicolumn{3}{c|}{FGLASSO} & \\multicolumn{3}{c|}{FPCA-LiNGAM} & \\multicolumn{3}{c}{FPCA-PC} \\\\\n\\cmidrule(lr){4-6} \\cmidrule(lr){7-9} \\cmidrule(lr){10-12} \\cmidrule(lr){13-15}\n& & & TPR & FDR & MCC & TPR & FDR & MCC & TPR & FDR & MCC & TPR & FDR & MCC \\\\\n\\midrule\n30 & 125 & 50 & 0.62 (0.07) & 0.14 (0.07) & 0.72 (0.07) & 0.58 (0.02) & 0.88 (0.02) & 0.16 (0.02) & - & - & - & 0.22 (0.03) & 0.89 (0.02) & 0.12 (0.02) \\\\\n30 & 125 & 100 & 0.71 (0.08) & 0.19 (0.05) & 0.75 (0.06) & 0.63 (0.03) & 0.85 (0.03) & 0.20 (0.03) & 0.84 (0.02) & 0.85 (0.01) & 0.31 (0.02) & 0.30 (0.01) & 0.89 (0.01) & 0.13 (0.01) \\\\\n30 & 125 & 200 & 0.73 (0.05) & 0.13 (0.08) & 0.79 (0.06) & 0.69 (0.03) & 0.84 (0.05) & 0.19 (0.03) & 0.92 (0.04) & 0.87 (0.01) & 0.30 (0.01) & 0.13 (0.02) & 0.96 (0.01) & 0.02 (0.01) \\\\\n\\midrule\n30 & 250 & 50 & 0.68 (0.05) & 0.25 (0.08) & 0.73 (0.06) & 0.57 (0.02) & 0.88 (0.04) & 0.16 (0.04) & - & - & - & 0.30 (0.02) & 0.87 (0.01) & 0.15 (0.01) \\\\\n30 & 250 & 100 & 0.75 (0.04) & 0.26 (0.03) & 0.74 (0.03) & 0.64 (0.03) & 0.85 (0.04) & 0.18 (0.03) & 0.88 (0.04) & 0.86 (0.02) & 0.34 (0.02) & 0.18 (0.02) &0.92 (0.02) & 0.08 (0.01) \\\\\n30 & 250 & 200 & 0.85 (0.01) & 0.30 (0.06) & 0.79 (0.04) & 0.69 (0.02) & 0.83 (0.04) & 0.21 (0.02) & 0.97 (0.05) & 0.85 (0.03) & 0.35 (0.03) & 0.22 (0.02) & 0.94 (0.01) & 0.08 (0.02) \\\\\n\\midrule\n60 & 125 & 50 & 0.68 (0.03) & 0.05 (0.03) & 0.80 (0.02) & 0.57 (0.04) & 0.89 (0.06) & 0.11 (0.05) & - & - & - & 0.28 (0.02) & 0.87 (0.01) & 0.16 (0.01) \\\\\n60 & 125 & 100 & 0.68 (0.04) & 0.12 (0.04) & 0.75 (0.04) & 0.60 (0.03) & 0.85 (0.05) & 0.15 (0.04) & - & - & - & 0.28 (0.01) & 0.89 (0.01) & 0.15 (0.01) \\\\\n60 & 125 & 200 & 0.74 (0.02) & 0.11 (0.02) & 0.82 (0.02) & 0.61 (0.03) & 0.82 (0.04) & 0.17 (0.03) & 0.86 (0.03) & 0.89 (0.02) & 0.25 (0.02) & 0.22 (0.01) & 0.95 (0.01) & 0.11 (0.01) \\\\\n\\midrule\n60 & 250 & 50 & 0.70 (0.02) & 0.15 (0.02) & 0.77 (0.01) & 0.59 (0.04) & 0.82 (0.04) & 0.16 (0.03) & - & - & - & 0.35 (0.02) & 0.85 (0.02) & 0.21 (0.01) \\\\\n60 & 250 & 100 & 0.70 (0.01) & 0.13 (0.10) & 0.79 (0.05) & 0.62 (0.04) & 0.80 (0.04) & 0.17 (0.03) & - & - & - & 0.26 (0.01) & 0.89 (0.02) & 0.13 (0.01) \\\\\n60 & 250 & 200 & 0.76 (0.02) & 0.11 (0.01) & 0.85 (0.01) & 0.69 (0.05) & 0.80 (0.03) & 0.19 (0.04) & 0.91 (0.02) & 0.85 (0.02) & 0.33 (0.01) & 0.17 (0.01) & 0.84 (0.05) & 0.15 (0.03) \\\\\n\\midrule\n90 & 125 & 50 & 0.63 (0.04) & 0.10 (0.04) & 0.75 (0.03) & 0.52 (0.02) & 0.89 (0.03) & 0.10 (0.03) & - & - & - & 0.23 (0.01) & 0.88 (0.00) & 0.15 (0.01) \\\\\n90 & 125 & 100 & 0.66 (0.03) & 0.12 (0.03) & 0.74 (0.02) & 0.55 (0.04) & 0.87 (0.03) & 0.15 (0.02) & - & - & - & 0.18 (0.01) & 0.92 (0.01) & 0.10 (0.01) \\\\\n90 & 125 & 200 & 0.67 (0.02) & 0.13 (0.02) & 0.76 (0.01) & 0.57 (0.03) & 0.85 (0.04) & 0.17 (0.03) & - & - & - & 0.17 (0.01) & 0.94 (0.01) & 0.08 (0.01) \\\\\n\\midrule\n90 & 250 & 50 & 0.58 (0.03) & 0.09 (0.02) & 0.68 (0.03) & 0.54 (0.05) & 0.87 (0.04) & 0.11 (0.04) & - & - & - & 0.32 (0.01) & 0.84 (0.00) & 0.21 (0.01) \\\\\n90 & 250 & 100 & 0.65 (0.05) & 0.13 (0.04) & 0.73 (0.04) & 0.58 (0.04) & 0.82 (0.03) & 0.15 (0.03) & - & - & - & 0.18 (0.02) & 0.93 (0.01) & 0.11 (0.01) \\\\\n90 & 250 & 200 & 0.70 (0.02) & 0.12 (0.02) & 0.78 (0.01) & 0.61 (0.06) & 0.80 (0.04) & 0.18 (0.05) & - & - & - & 0.22 (0.01) & 0.89 (0.01) & 0.16 (0.01) \\\\\n\\bottomrule\t\t\t\t\t\n\\end{tabular}}\n\\label{s1}\n\\end{table}\n\nThe proposed FLiNG-BN has a few hyperparameters $L$, $M$, $\\alpha$, $(a_r, b_r)$, $(a_\\gamma, b_\\gamma)$, $(a_\\tau, b_\\tau)$, and $(a_\\sigma, b_\\sigma)$. We performed sensitivity analyses of these parameters at four different values with $(n, p, d) = (100, 30, 250)$. Results are summarized in Section C of the Supplementary Material. Our model appeared to be relatively robust within the tested ranges of hyperparameters.\n\n\\section{Applications} \\label{sec::eeg}\n\nWe applied the proposed FLiNG-BN to the brain EEG dataset downloaded from \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/eeg+database} \\citep{zhang1995event}. The dataset consists of 122 subjects with 77 in the alcoholic group and 45 in the control group, and was previously used to demonstrate functional undirected graphical models by \\cite{zhu2016bayesian} and \\cite{qiao2019functional}. The 64 electrodes placed on subjects' scalps (standard positions) measuring voltage values were sampled at 256 Hz for one second. Each subject completed 120 trials under one stimulus or two stimuli. See \\cite{zhang1995event} for details of the data collection procedure. We averaged all trials for each subject under the one stimulus condition. We separately analyzed these two groups to find their commonalities and differences of brain activity. Hence, we had $n = 77$ or $n = 45$ subjects and $p = 64$ functions representing the brain EEG signals at different scalp positions recorded at $d = 256$ time points. We focused on EEG signals filtered at $\\alpha$ frequency bands between 8 and 12.5 Hz using the \\texttt{eegfilt} function in the EEGLAB toolbox from MATLAB \\citep{delorme2004eeglab}.\n\nTo check the Gaussianity of the observed functions, we performed Shapiro--Wilk normality test \\citep{shapiro1965analysis} to each of $p = 64$ scalp positions at each of $d = 256$ time points. The null hypothesis (i.e., the observations are marginally Gaussian) was rejected for many combinations of scalp position and time point and therefore, the non-Gaussianity of the proposed model is deemed appropriate.\n\nFive orthonormal basis functions were selected for both the alcoholic and control group according to the procedure described in Section B of the Supplementary Material. We ran MCMC for 10,000 iterations, discarded the first half as burn-in, and retained every 10th iteration after burn-in. The estimated basis functions are shown in Figure \\ref{pfunction}. As evident from the plot, they are very similar across the two groups. The causal networks estimated by thresholding the posterior probability of inclusion at 0.9 are shown in Figure \\ref{fnetwork}. The sparsity level is approximately 3.0\\% for the alcoholic group and 2.5\\% for the\ncontrol group. \n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[ht]{1 \\textwidth}\n\\centering\n\\includegraphics[width = 1 \\textwidth]{FALCOHOL}\n\\caption{Alcoholic group.}\n\\end{subfigure}\n\\begin{subfigure}[ht]{1 \\textwidth}\n\\centering\n\\includegraphics[width = 1 \\textwidth]{FCONTROL}\n\\caption{Control group.}\n\\end{subfigure}\n\\caption{Estimated basis functions from brain EEG records that explained 90\\% of the variation.}\n\\label{pfunction}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width = 1 \\textwidth]{COMBINEAC}\n\\caption{Estimated causal brain networks from EEG records by FLiNG-BN with posterior probability of inclusion $\\geq 0.9$, separately for the alcoholic (left) and control (right) group.}\n\\label{fnetwork}\n\\end{figure}\n\nOur results reveal several interesting patterns. First, the connection is relatively dense in the frontal region for both groups. Second, the alcoholic group has more directed connections detected in the left temporal and occipital regions. Third, most brain locations tend to connect to adjacent positions, while distant locations are much less connected. Figure \\ref{dnetwork} shows the common and differential networks for the two groups, where a substantial connectivity difference is observed between the two groups.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width = 1 \\textwidth]{COMBINECD}\n\\caption{Common (left panel) and differential (right panel) connections for the two groups. Black arrows indicate common connections, red arrows indicate connections detected by the alcoholic group only, and green arrows indicate connections detected by the control group only. }\n\\label{dnetwork}\n\\end{figure}\n\nIn addition, we demonstrated the proposed FLiNG-BN model with an additional application to COVID-19 multivariate longitudinal data, which have unevenly spaced measurements, in Section D of the Supplementary Material.\n\n\\section{Discussion} \\label{sec::discussion}\n\nIn this paper, we have proposed a functional Bayesian network model for causal discovery from multivariate functional data. We have discussed in detail a specific case of functional Bayesian network, namely the functional linear non-Gaussian model, and proved the underlying causal structure is identifiable even if the functions are purely observational and observed with noises. A fully Bayesian inference procedure has been proposed to implement our framework. Through simulation studies and real data applications, we have demonstrated the ability of our model in causal discovery.\n\nWe briefly discuss several possible directions to extend our current work. First, we may replace the underlying DAG with cyclic graphs, chain graphs, or ancestral graphs for more general causal and conditional independence structures. We have chosen a linear non-Gaussian SEM on the basis coefficients but this model can be replaced with a nonlinear SEM. Second, instead of fixing the number of basis functions, one could resort to increasing shrinkage priors \\citep{bhattacharya2011sparse, legramanti2020bayesian} to adaptively truncate redundant components. Finally, since we have two groups of observations in the EEG application, it would be interesting to jointly estimate the brain networks or directly estimate the differential network.\n\n\\bibliographystyle{jasa} \n\\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Model architcture}\n\n\\lipsum[1-3]\n\n\\section{Introduction}\n\nThe unit commitment (UC) problem is an important optimization problem in the energy industry to compute the optimal operating schedules of power plants given demand over a fixed time period.\nThis problem is solved by electricity generating companies on a daily basis to determine which generators are to be used.\nGiven a demand forecast over a planning horizon, the timing of switching and the amount of power dispatch are to be optimized simultaneously.\nThis gives rise to a large-scale combinatorial problem and due to its practical importance it has been extensively studied over the last few decades.\nFor a recent survey, see \\citet{VanAckooijetal2018a}.\n\nThis work focuses on UC problems which are to be solved repeatedly with different data but with the same problem structures.\nIn practice, when a UC problem is solved as a day-ahead planning problem, the characterizations of generators such as generation costs and ramping rates remain the same and problems are solved with different demand forecasts.\n\nOne popular solution method for UC problems is to use Dantzig-Wolfe decomposition to decompose the problem by generators.\nThe reformulated problem is then solved with a column generation procedure.\nThis procedure can be seen as the dual of a cutting plane approach to Lagrangian relaxation, as discussed by \\citet{Briantetal2008}.\nThe DW reformulation is a relaxation of the original problem, however \\citet{Bertsekasetal1983} and \\citet{Bard1988} reported that the optimality gap introduced by the reformulation is typically small especially if the problem size is large.\nIn such cases, Dantzig-Wolfe decomposition with a suitable primal heuristic is likely to solve the UC problem to high precision.\n\nIn the aforementioned decomposition-based approaches, the dual variable plays a key role.\nIt is expected that using an appropriate dual value as an initial point may speed up the solution method.\nOne approach to generate such a dual value is to solve an approximation of the original problem.\n\\citet{Schulzeetal2017} solved the linear programming relaxation (LPR) of the original problem and obtained its optimal dual value.\nIt was then used to initialize Dantzig-Wolfe decomposition.\n\\citet{Takritietal1996} constructed an approximation by dropping some constraints, such as integrality constraints, minimum up\/down time constraints and minimum power output constraints.\nThe optimal dual value of the approximation was used to warmstart Lagrangian relaxation with the subgradient algorithm.\nHowever, as \\citet{Schulzeetal2017} reported, solving such an approximation may require a non-trivial amount of time.\n\nTo warmstart the column generation procedure, we propose to use a pre-trained model to generate initial dual values.\nOur work is inspired by the method for generating dual policies proposed by \\citet{Nairetal2018}.\nIn their work, neural networks were trained to compute dual values that yielded strong lower bounds on two-stage integer stochastic programming.\nIn this work, we propose to combine the dual policy with the aforementioned column generation procedure to solve UC.\nNamely, a model is used with problem data as input to generate dual values that yield tight lower bounds, as demonstrated by \\citet{Nairetal2018}.\nIn our work, the generated dual values are then used to warmstart the column generation procedure.\nWith this approach, we can exploit the strength of the dual policy while maintaining the desirable property of the solution method, such as exactness.\nThe above approach were tested with numerical experiments on large-scale UC problems, where our method outperformed baselines.\n\nThe rest of the paper is structured as follows.\nSection \\ref{sec_dantzig_wolfe} briefly introduces Dantzig-Wolfe decomposition and the column generation procedure.\nSection \\ref{sec_warmstarting} presents approaches and issues to warmstart the column generation procedure.\nA model based on a neural network is considered to generate an initial value for the algorithm.\nThe proposed approach is tested on large scale UC problems in Section \\ref{sec_neumerical_experiments}.\nFinally, in Section \\ref{sec_conclusion} conclusions and further extensions of this work are discussed.\n\n\\section{Dantzig-Wolfe decomposition and column generation}\n\\label{sec_dantzig_wolfe}\n\nIn this section, we briefly review Dantzig-Wolfe decomposition and the column generation procedure.\nFor further background, see \\citet{VanderbeckAndSavelsbergh2006}.\n\nConsider the following family of mixed-integer programmes parametrized by $\\omega$:\n\\begin{align}\nz(\\omega) = \\min_{x} \\ & \\sum_{s \\in S} c_s^T x_s\n\\label{eq_mip} \\\\\n\\text{s.t.} \\ &\n\\sum_{s \\in S} A_s x_s = a(\\omega), \\notag \\\\\n& x_s \\in X_s(\\omega)\n\\quad \\forall s \\in S, \\notag\n\\end{align}\nwhere\n$x = (x_s)_{s \\in S}$ is a vector of decision variables and\n$X_s(\\omega)$ is the set of feasible points given by\n\\[\nX_s(\\omega) = \\{x_s \\in \\mathbb{Z}^{n} \\times \\mathbb{R}^{m} \\mid D x_s \\le d(\\omega)\\} \\quad s \\in S.\n\\]\nFor simplicity, we assume that $X_s(\\omega)$ is non-empty and bounded for every $\\omega$ and $s \\in S$, and that the problem \\eqref{eq_mip} has an optimal solution for every $\\omega$.\nTo reduce clutter in what follows we drop the dependence on $\\omega$ unless it causes confusion.\n\nIn Dantzig-Wolfe decomposition, we consider a relaxation of \\eqref{eq_mip}, referred to as the master problem (MP), by replacing $X_s$ with $\\mathrm{conv}({X_s})$ for every $s \\in S$.\nLet $\\{x_{si} : i \\in I_s\\}$ denote the extreme points of $X_s$.\nGiven the boundedness assumption on $X_s$, the MP is written as\n\\begin{align}\n\\min_{p} \\ & \\sum_{s \\in S, i \\in I_s} c_{s}^T x_{si} p_{si}\n\\label{eq_mp} \\\\\n\\text{s.t.} \\ &\n\\sum_{s \\in S, i \\in I_s} A_s x_{si} p_{si} = a, \\notag \\\\\n& \\sum_{i \\in I_s} p_{si} = 1 \\quad \\forall s \\in S, \\notag \\\\\n& p_{si} \\ge 0 \\quad \\forall s \\in S, i \\in I_s. \\notag\n\\end{align}\n\nSince the MP is too large to formulate and solve explicitly, a column generation procedure is used.\nThe restricted master problem (RMP) is defined by replacing $I_s$ in the MP with a subset $\\hat{I}_s \\subset I_s$ for every $s \\in S$.\nSuppose that the RMP is feasible ($\\hat{I}_s$'s must have enough columns to satisfy the first constraint in \\eqref{eq_mp}) and let $y, \\sigma_s$ for $s$ in $S$ be the optimal dual solution to the RMP corresponding to the first and second constraint respectively.\nTo find columns to be added to the RMP, the following subproblems, known as the pricing problems, are solved:\n\\begin{equation}\nr_s(y) = \\min_{x_s} \\{(c_s - y^T A_s) x_s : x_s \\in X_s\\}\n\\quad s \\in S.\n\\label{eq_pricing_problem}\n\\end{equation}\nIf $r_s(y) \\ge \\sigma_s$ for all $s \\in S$, the RMP has found the optimal solution to the MP.\nOtherwise, the solutions to the pricing problems are added to the RMP and the above process is repeated.\nIt follows from LP duality that given a dual value $y$\n\\begin{equation}\nl(y) = a^T y + \\sum_{s \\in S} r_s(y)\n\\label{eq_lower_bound}\n\\end{equation}\nis a lower bound to the MP, which also bounds \\eqref{eq_mip}.\n\nIt is known that the naive column generation approach described above suffers from instability.\nTo mitigate the issue, quadratic regularization on the dual variable may be added as described by \\citet{Briantetal2008}.\nDualizing the RMP and adding quadratic regularization on the dual variable we get\n\\begin{align}\n\\max_{y, \\sigma} \\ & a^T y + \\sum_{s \\in S} \\sigma_s - \\frac{\\mu}{2} \\| y - \\bar{y} \\|_2^2\n\\label{eq_regularized_rmp} \\\\\n\\text{s.t.} \\ &\n\\sigma_s \\le (c_s - y^T A_s) x_{si} \\quad \\forall s \\in S, i \\in \\hat{I}_s, \\notag \\\\\n& y, \\sigma: \\mathrm{free}, \\notag\n\\end{align}\nwhere $\\bar{y}$ is a regularization centre and $\\mu$ is a parameter to adjust the strength of the regularization.\nIn the regularized column generation procedure, the regularized RMP is used in place of the RMP.\nThe optimal solution to \\eqref{eq_regularized_rmp} is computed and the pricing problems are solved based on this solution.\nThe regularization centre $\\bar{y}$ may be updated to the current dual value $y$ when the lower bound \\eqref{eq_lower_bound} gets improved.\n\n\n\\section{Warmstarting}\n\\label{sec_warmstarting}\n\nIn the above algorithm, the dual variable $y$ is updated iteratively to provide tighter lower bounds.\nConsider the case where a near-optimal dual value is used as an initial point.\nIt can be shown that the lower bound $l(y)$ is continuous (in fact concave) in the dual variable $y$.\nThus the algorithm would provide a near-optimal lower bound in the first iteration.\nAssuming the lower bound is sufficiently tight, the algorithm would terminate as soon as a primal feasible solution with sufficiently small suboptimality was found by primal heuristics.\nIt is expected that feeding a near-optimal dual value as an initial dual value helps the algorithm to terminate in a shorter execution time.\n\nIt is worth noting that regularization on the dual variable is crucial in this context.\nWithout regularization, even if a near-optimal dual value is used as an initial point, the algorithm is likely to be quite unstable, yielding dual values with poor lower bounds in the following iterations.\nSee \\citet{Briantetal2008} for a further discussion.\n\n\\subsection{Approximation}\n\nOne option to find a good dual value is to solve some approximation of \\eqref{eq_mip}.\nFor example, \\citet{Schulzeetal2017} used the LPR, which is a relaxation of \\eqref{eq_mip} where the binary variables are replaced with continuous variables.\nFirst the LPR was solved and the corresponding optimal dual value $y$ was obtained.\nThen, this was used as an initial dual value to warmstart the column generation procedure.\n\\citet{Takritietal1996} constructed an approximation by dropping some constraints, such as integrality constraints, minimum up\/down time constraints and minimum power output constraints.\nThe optimal dual value to this problem was used as the initial point for a subgradient method approach to Lagrangian relaxation.\nThese approaches do not require any offline computation or pre-trained models.\nOn the other hand solving the approximations requires a certain amount of time.\n\n\\subsection{Initialization methods}\n\nAn alternative approach, which we explore below, is to train a model which takes $\\omega$ as input and outputs a dual value $y$ that provides a tight lower bound.\nThe training approach is based on the work by \\citet{Nairetal2018} where dual decomposition was\napplied to solve parametrized stochastic two-stage programming.\n\nConsider a neural network model $f(\\omega, \\theta)$ which maps the pair of problem parameter $\\omega$ and model parameter $\\theta$ to $y$.\nWe shall learn a value of the parameter $\\theta$ so that given $\\omega$ the model outputs a dual value $y = f(\\omega, \\theta)$ that provides a tight lower bound $\\eqref{eq_lower_bound}$.\nIn this section, we explicitly state the dependency of the lower bound on $\\omega$ as $l(\\omega, y)$, which was suppressed in \\eqref{eq_lower_bound}.\nAssume that the distribution of $\\omega$ is given (e.g.\\ the empirical distribution based on historical data).\nOur goal is to maximize the expected lower bound\n\\begin{align*}\np(\\theta) = \\mathrm{E}_\\omega[l(\\omega, f(\\omega, \\theta))].\n\\end{align*}\n\nFrom \\eqref{eq_lower_bound} it follows that for each $\\omega$\n\\begin{align*}\nl(\\omega, y)\n= a^T y + \\sum_{s \\in S} r_s(y) = \\frac{1}{|S|} \\sum_{s \\in S} (a^T y + |S| r_s(y))\n= \\mathrm{E}_{t}[\\widetilde{l}(\\omega, y, t)],\n\\end{align*}\nwhere\n\\begin{align*}\n\\widetilde{l}(\\omega, y, t) = a^T y + |S| r_t(y)\n\\end{align*}\nand subproblem $t$ is sampled uniformly from $S$.\nTherefore, we get\n\\begin{align*}\np(\\theta) = \\mathrm{E}_{\\omega, t}[\\widetilde{l}(\\omega, f(\\omega, \\theta), t)].\n\\end{align*}\nWe follow the standard approach used when a neural network is trained with the stochastic gradient descent method.\nThat is, we sample $\\omega$ and $t$, compute the gradient of $\\widetilde{l}$ with respect to $\\theta$ and make a single gradient ascent step.\nWe then resample $\\omega$ and $t$ and repeat the process again.\n\nThe gradient of $\\widetilde{l}$ with respect to $\\theta$ can be computed as follows.\nFix problem parameter $\\omega$ and subproblem index $t$ and compute  the dual value $y = f(\\omega, \\theta)$ and the component $\\widetilde{l}(\\omega, y, t)$ of the lower bound corresponding to subproblem $t$.\nSuppose that the model output $f(\\omega, \\theta)$ is differentiable with respect to $\\theta$ and that the component $\\widetilde{l}(\\omega, y, t)$ of the lower bound and the optimal value $r_t(y)$ of the pricing problem \\eqref{eq_pricing_problem} are differentiable with respect to $y$.\nThen the gradient of $\\widetilde{l}(\\omega, y, t)$ with respect to $y$ is given by\n\\begin{equation*}\n\\frac{\\partial \\widetilde{l}}{\\partial y} = a - |S| A_t x_t^*\n\\end{equation*}\nwhere $x_t^*$ is the solution to the pricing problem $t$ \\eqref{eq_pricing_problem}.\nUsing the chain rule, we obtain\n\\begin{equation}\n\\frac{\\partial \\widetilde{l}}{\\partial \\theta} = J \\frac{\\partial \\widetilde{l}}{\\partial y},\n\\label{eq_sampled_gradient_dl_dtheta}\n\\end{equation}\nwhere $J$ is the Jacobian matrix of the neural network model output $y = f(\\omega, \\theta)$ with respect to $\\theta$, which is typically given by automatic differentiation.\n\nIt is worth noting that evaluating the above quantity \\eqref{eq_sampled_gradient_dl_dtheta} is much more faster than solving the original problem or the MP to optimality.\nThis is because it requires the solution of only a single pricing problem, which is typically substantially smaller than the original problem.\n\nOnce a model is trained, it can be used to compute an initial dual value in the regularized column generation procedure.\nIt is expected that unlike solving an approximation such as the LPR this model will run quickly.\nNevertheless it produces a near-optimal initial dual value with which to warmstart the column generation procedure.\n\n\\section{Numerical experiments}\n\\label{sec_neumerical_experiments}\n\nIn this section, the proposed method is compared with standard approaches.\nAll methods are implemented with Python.\nIBM ILOG CPLEX\\footnote{https:\/\/www.ibm.com\/products\/ilog-cplex-optimization-studio} is used as the optimization solver (to solve the pricing problems and the RMP) and PyTorch to implement the neural networks.\n\n\\subsection{Problem}\n\nIn the experiments, we consider a setup in which UC problems are solved repeatedly with a fixed set of generators but different demand forecasts.\nTo assess the scalability, we consider 3 different problem sizes; problems with 200, 600 and 1,000 generators.\nIn all cases, the length of the planning horizon is 48 hours with a time resolution of 1 hour.\nThe generator data is based on \\citet{Borghettietal2002}.\nSince their sets of generators contain 200 generators at most, we combine multiple sets to create larger ones.\nEach generator is distinct and combining the sets of generators does not introduce symmetry.\nThe demand data is based on the historical demand data in the UK published by National Grid ESO.\\footnote{https:\/\/www.nationalgrideso.com\/}\nFor each set of generators, the demand is scaled so that the median of the daily peaks becomes 50\\% of the total generation capacity.\nA detailed description of the problem formulation is found in the appendix.\n\nDantzig-Wolfe decomposition with the regularized column generation procedure is used to solve the UC problems.\nVarious initialization methods are compared, and these are described in the next section.\nIn each iteration of the column generation procedure, after the pricing problems are solved, a primal heuristic is run to find primal feasible solutions.\nImplementation details including the primal heuristics are given in the appendix.\n\n\\subsection{Initialization methods}\n\nWe consider the following 5 methods to initialize the dual variable.\n\n\\paragraph{Neural Network}\nAn initialization method based on a neural network is implemented as described in the previous section.\nThe model consists of 4 hidden layers of 1000 units per layer, with skip connections between hidden layers.\nFor each set of generators, a single neural network is trained for 24 hours with 8 CPU cores.\nThe parameters of the neural network are kept in shared memory and updated asynchronously in each worker.\nThis follows the procedure used by \\citet{Nairetal2018}, except that only a single neural network is trained in each setup to reduce the computational load.\nThe hyperparameters on the model architecture and the training are found in the appendix.\n\n\\paragraph{Random forest}\nAs a baseline of a trainable model, a method based on a random forest (\\citet{Briantetal2008}) is also evaluated.\nOn the same training budget (24 hours on 8 CPU cores), as many training instances as possible are solved to 0.25\\% optimality.\nDantzig-Wolfe decomposition with the LPR is used as a solver.\nThe parameter values and the corresponding optimal dual values of the training instances are saved.\nAfter the timelimit, a random forest is trained by supervised learning to predict the optimal dual value given a parameter value.\n\n\\paragraph{Nearest Neighbour}\nNearest neighbour is tested to provide an alternative trainable baseline.\nWe use the dataset created to train the random forest model.\nAt runtime, given a test instance, its parameter is compared against those of the training instances.\nThe training instance with the closest parameter value in terms of the Euclidean distance is selected, and the corresponding optimal dual value is used as an initial dual value.\n\n\\paragraph{LPR}\nThis is the method based on the LPR described in Section \\ref{sec_warmstarting}.\nGiven a test instance, we first solve the LPR by CPLEX.\nThen, the optimal dual value is used as an initial value of the column generation procedure.\nThis method does not require any training.\nThis is the method used by \\citet{Schulzeetal2017}.\n\n\\paragraph{Coldstart}\nAs a naive base line, column generation is run from $y=0$.\nThis method does not require any training.\n\n\\subsection{Evaluation}\n\nTo test the performances of the above methods, 40 test instances of each size are created and solved by each method.\nThe demand data to construct the test instances are sampled from a different year than those of the training instances.\nEach method is run sequentially on a single CPU core until a solution within 1\\%, 0.5\\% and 0.25\\% suboptimality is found or the time limit of 5 minutes is reached.\n\nTable \\ref{tab_lower_bound} shows the qualities of the lower and upper bounds obtained after one iteration of the column generation procedure and the required time (all times in this section are wall clock times) to run the initialization methods.\nOut of 40 test instances, 8 instances are sampled and the lower bounds of the costs are computed by running CPLEX for 4 hours per instance.\nThen the lower and upper bounds obtained from the column generation procedure are scaled so that the lower bounds from CPLEX becomes 1.0 (the upper bound obtained from CPLEX was very close to the lower bound from CPLEX, typically the gap was 0.04\\% in 1000-generator cases).\nThe table reports the average of the scaled bounds on the 8 instances.\n\nClearly, coldstart outputs dual values with poor lower bounds.\nAlthough the lower bounds of the remaining 4 methods are comparable, the lower bounds obtained from the method based on a neural network seems to be tighter than the lower bounds from the others.\nIn all cases, the upper bounds are looser than the lower bounds.\nAlso, methods with tighter lower bounds tend to have tighter upper bounds.\nHowever, the upper bounds are likely to vary if we use a different primal heuristic in the column generation procedure.\nThe solution time of the LPR grows rapidly and it takes significant time especially on larger problems.\nOn the other hand, the computational time of the other methods remains small.\n\n\\begin{table}\n\\TABLE\n{Average gap and computational time (seconds).\\label{tab_lower_bound}}\n{\\begin{tabular*}{15cm}{@{\\extracolsep{\\fill}} r rrr rrr rrr}\n    \\hline\n              & \\multicolumn{3}{c}{size: 200} & \\multicolumn{3}{c}{size: 600} & \\multicolumn{3}{c}{size: 1000} \\\\\n  \\cline{2-4}\n  \\cline{5-7}\n  \\cline{8-10}\n    method           &\n    \\multicolumn{1}{c}{lb} &\n    \\multicolumn{1}{c}{ub} &\n    \\multicolumn{1}{c}{time} &\n    \\multicolumn{1}{c}{lb} &\n    \\multicolumn{1}{c}{ub} &\n    \\multicolumn{1}{c}{time} &\n    \\multicolumn{1}{c}{lb} &\n    \\multicolumn{1}{c}{ub} &\n    \\multicolumn{1}{c}{time} \\\\\n  \\cline{1-1}\n  \\cline{2-10}\nnetwork     &  0.99945 &  1.00497 &  0.0 &  0.99948 &  1.00394 &   0.0 &  0.99959 &  1.00336 &   0.0 \\\\\nforest      &  0.99911 &  1.00605 &  0.0 &  0.99911 &  1.00519 &   0.0 &  0.99922 &  1.00521 &   0.0 \\\\\nnearest     &  0.99897 &  1.00646 &  0.0 &  0.99915 &  1.00507 &   0.0 &  0.99905 &  1.00512 &   0.0 \\\\\nLPR         &  0.99874 &  1.00512 &  4.6 &  0.99904 &  1.00387 &  17.8 &  0.99918 &  1.00316 &  29.6 \\\\\ncoldstart   &  0.00153 &  1.02122 &  0.0 &  0.00105 &  1.01981 &   0.0 &  0.00141 &  1.01961 &   0.0 \\\\\n    \\hline\n\\end{tabular*}}{}\n\\end{table}\n\nTable \\ref{tab_performance} shows the number of problems solved by the column generation procedure within the time limit of 5 minutes, the average computational time and the average number of iterations to close the optimality gap or to reach the time limit.\nThe computational time reported in this table includes the time to run the initialization methods such as solving LPR (this does not contain time spent to train the models).\nFor the instances that are not solved within the time limit, no penalty is added on the computation of the average and the time is set to be the 5 minutes, and the number of iterations reached by the 5 minute timelimit is used.\nFor comparison, we also solve the extensive form \\eqref{eq_mip} to the same tolerances without decomposition (i.e.\\ by branch and bound) using CPLEX.\n\n\\begin{table}\n\\TABLE\n{Number of problems solved, average computational time (seconds) and iterations.\\label{tab_performance}}\n{\\begin{tabular*}{15cm}{@{\\extracolsep{\\fill}} rr rrr rrr rrr}\n  \\hline\n   &\n   &\n   \\multicolumn{3}{c}{1\\% optimality} &\n   \\multicolumn{3}{c}{0.5\\% optimality} &\n   \\multicolumn{3}{c}{0.25\\% optimality} \\\\\n  \\cline{3-5}\n  \\cline{6-8}\n  \\cline{9-11}\n  size &\n  method &\n  solved & \\multicolumn{1}{c}{time} & \\multicolumn{1}{c}{iters} &\n  solved & \\multicolumn{1}{c}{time} & \\multicolumn{1}{c}{iters} &\n  solved & \\multicolumn{1}{c}{time} & \\multicolumn{1}{c}{iters} \\\\\n  \\hline\n200 & network   & 40\/40 &           4.6  &  1.0 & 40\/40 & \\textbf{  8.9} &  1.8 & 40\/40 & \\textbf{   34.0}&  6.6 \\\\\n    & forest    & 40\/40 & \\textbf{  4.5} &  1.0 & 40\/40 &          11.0  &  2.4 & 40\/40 &            39.0 &  7.9 \\\\\n    & nearest   & 40\/40 &           6.5  &  1.3 & 40\/40 &          20.0  &  3.8 & 40\/40 &            52.2 &  9.4 \\\\\n    & LPR       & 40\/40 &           7.9  &  1.0 & 40\/40 &          16.0  &  3.0 & 40\/40 &            45.4 &  8.8 \\\\\n    & coldstart & 37\/40 &         105.3  & 12.1 & 34\/40 &         173.4  & 14.0 & 27\/40 &           229.6 & 16.8 \\\\\n    & cplex     & 36\/40 &         195.9  &  0.0 & 35\/40 &         210.0  &  0.0 & 24\/40 &           264.8 &  0.0 \\\\\n  \\hline\n600 & network   & 40\/40 & \\textbf{ 13.4} &  1.0 & 40\/40 & \\textbf{ 20.5} &  1.5 & 40\/40 & \\textbf{   47.7}&  3.4 \\\\\n    & forest    & 40\/40 &          14.4  &  1.2 & 40\/40 &          30.1  &  2.3 & 40\/40 &            63.2 &  4.8 \\\\\n    & nearest   & 40\/40 &          17.6  &  1.2 & 40\/40 &          40.5  &  2.8 & 40\/40 &            81.9 &  5.7 \\\\\n    & LPR       & 40\/40 &          28.6  &  1.0 & 40\/40 &          43.3  &  2.1 & 40\/40 &            87.7 &  5.4 \\\\\n    & coldstart & 38\/40 &         221.5  & 11.4 & 28\/40 &         258.5  & 12.6 & 13\/40 &           290.2 & 13.8 \\\\\n    & cplex     &  0\/40 &         300.0  &  0.0 &  0\/40 &         300.0  &  0.0 &  0\/40 &           300.0 &  0.0 \\\\\n  \\hline\n1000& network   & 40\/40 &          21.9  &  1.0 & 40\/40 & \\textbf{ 34.4} &  1.5 & 40\/40 & \\textbf{   72.0}&  3.1 \\\\\n    & forest    & 40\/40 & \\textbf{ 21.6} &  1.1 & 40\/40 &          54.9  &  2.6 & 40\/40 &           106.9 &  4.9 \\\\\n    & nearest   & 40\/40 &          24.2  &  1.2 & 40\/40 &          66.3  &  3.0 & 40\/40 &           124.5 &  5.7 \\\\\n    & LPR       & 40\/40 &          45.9  &  1.0 & 40\/40 &          67.4  &  1.8 & 40\/40 &           119.4 &  4.2 \\\\\n    & coldstart & 23\/40 &         280.3  & 10.9 & 16\/40 &         295.5  & 11.5 &  6\/40 &           298.7 & 11.9 \\\\\n    & cplex     &  0\/40 &         300.0  &  0.0 &  0\/40 &         300.0  &  0.0 &  0\/40 &           300.0 &  0.0 \\\\\n    \\hline\n  \\end{tabular*}\n}{}\n\\end{table}\n\nThese results show that with appropriate regularization and a fast heuristic to generate suitable dual values, a column generation procedure can be warmstarted to find solutions in fewer iterations and in shorter time.\nSolving the problems without decomposition is the slowest.\nAmong the remaining 5 methods, which are based on the column generation procedure, solving the problem with coldstart is clearly the slowest in every case.\n\nThe method to generate initial dual values must balance its quality of the dual values and the overall computational time.\nOne extreme example is the LPR, which produces dual values with a tight lower bound.\nHowever the solution time grows rapidly as the problem size increases.\nThe method based on a neural network needs off-line training, but it runs very quickly when solving a new instance, resulting in significant speed up of the overall algorithm.\n\nA typical issue of learning-based heuristics is scalability.\nFor example, creating a dataset of optimal solutions can be prohibitive when large-scale problems are to be solved.\nIn such cases, methods based on supervised learning that uses optimal solutions become impractical.\nThe method used in our neural network approach, on the other hand, can be trained efficiently based on the decomposition, which allows the method to scale.\nThis may account for the performance difference between the neural network approach and the other learning-based approaches, which use supervised learning.\n\nThese observations indicate that the warmstarting method with a neural network tends to be competitive or outperform the other methods in terms of the computational time in almost many cases.\nThis trend is clearer with tighter tolerances.\n\nTable \\ref{tab_solution_time_breakdown} shows a breakdown of the average computational time of the column generation procedures based on the neural network and on the LPR for the 1000-generator instances for the 40 test cases.\nInitialization shows the time required to run the initialization methods.\nIn the LPR-based method, this is the time to solve the LPR, while the method based on a neural network this is the time to evaluate the neural network model (this does not include the time to train the neural network).\nIn cases of $1\\%$ tolerance, the time required to solve the LPR is longer than the sum of the time spent on the other routines.\nHowever as the tolerance becomes tighter, the number of iterations and the time spent in the other routines increases, and by $0.25\\%$ tolerance the LPR initialization time is only 25\\% of the total time.\n\n\\begin{table}\n\\TABLE\n{Breakdown of the average computational time (seconds).\\label{tab_solution_time_breakdown}}\n{\n  \\begin{tabular}{lr rrrr}\n    \\hline\ntolerance      &  method  & initialization & RMP &  subproblem & primal heuristic \\\\\n  \\cline{1-2}\n  \\cline{3-6}\n1\\%      & network  &   0.0  &  0.0  &   17.2  &   4.7 \\\\\n         & LPR      &  29.3  &  0.0  &   11.8  &   4.8 \\\\\n  \\cline{1-2}\n  \\cline{3-6}\n0.5\\%    & network  &   0.0  &  0.1  &   25.2  &   9.1 \\\\\n         & LPR      &  29.3  &  0.1  &   25.4  &  12.6 \\\\\n  \\cline{1-2}\n  \\cline{3-6}\n0.25\\%   & network  &   0.0  &  0.3  &   52.6  &  19.0 \\\\\n         & LPR      &  29.3  &  0.7  &   62.8  &  26.7 \\\\\n    \\hline\n  \\end{tabular}\n}{}\n\\end{table}\n\n\\section{Conclusion}\n\\label{sec_conclusion}\n\nWe considered methods to accelerate Dantzig-Wolfe decomposition with a column generation procedure to solve a parametrized UC problem.\nWe trained a neural network to generate values for the dual variables which yield tight lower bounds.\nThe training was done efficiently using dual decomposition.\nThe model was then used to warmstart the column generation procedure.\nWhen solving a new instance, the neural network provided a dual value quickly, which was used as the initial dual value of the column generation procedure.\nThe initial dual value gave a very tight lower bound and it successfully reduced the overall computational time to find a solution of prescribed suboptimality.\nThe numerical experiments show that this approach scales well and can be effective for solving large-scale unit commitment problems.\n\nA potential extension of this work is a UC problem with a stochastic demand forecast.\nIn many solution methods, UC problems with deterministic demand forecasts appear as subproblems and are solved repeatedly.\nSee \\citet{Takritietal1996} and \\citet{Schulzeetal2017} for instance.\nTypically the subproblems share the same problem structure and only differ in the cost coefficients and the demand forecasts.\nIn such cases, it would be of interest to use data gathered from previous instances and warmstart the algorithm when solving a new instance.\n\n\\begin{APPENDICES}\n\n\\section{Problem formulation}\n\nWe follow one of the standard formulations in literature and formulate the following constraints:\n\\begin{itemize}\n\\item \\textbf{Load balance}: Generators have to meet the all demand in each time period.\n\\item \\textbf{Reserve}: To deal with contingencies, it is required to keep a sufficient amount of back up in each time period, which can be activated quickly.\n\\item \\textbf{Power output bounds}: Each generator's power output has to be within its limit.\n\\item \\textbf{Ramp rate}: Generators can only change their outputs within the ramp rates.\n\\item \\textbf{Minimum up\/downtime}: If switched on (off), each generator has to stay on (off) for a given minimum period.\nThis is to avoid thermal stress in the generators which may cause wear and tear of the turbines.\n\\end{itemize}\n\nTo formulate the model, we use the following notation.\n\n\\begin{itemize}\n\\item{Sets}\n\\begin{itemize}\n\\item $G = \\{1, 2, \\ldots, n_G\\}$: set of generators\n\\item $T = \\{1, 2, \\ldots, n_T\\}$: set of time indices where decisions are taken\n\\end{itemize}\n\n\\item{Parameters}\n\\begin{itemize}\n\\item $C^{\\mathrm{nl}}_{g}$: no-load cost of generator $g$\n\\item $C^{\\mathrm{mr}}_{g}$: marginal cost of generator $g$\n\\item $C^{\\mathrm{up}}_{g}$: startup cost of generator $g$\n\\item $P^{\\max\/\\min}_{g}$: maximum\/minimum generation limit of generator $g$\n\\item $P^{\\mathrm{ru}\/\\mathrm{rd}}_{g}$: operating ramp up\/down limits of generator $g$\n\\item $P^{\\mathrm{su}\/\\mathrm{sd}}_{g}$: startup\/shutdown ramp limits of generator $g$\n\\item $T^{\\mathrm{u}\/\\mathrm{d}}_{g}$: minimum uptime\/downtime of generator $g$\n\\item $P^{\\mathrm{d}}_{t}$: power demand at time $t$\n\\item $P^{\\mathrm{r}}_{t}$: reserve requirement at time $t$\n\\end{itemize}\n\n\\item{Variables}\n\\begin{itemize}\n\\item $\\alpha_{gt} \\in \\{0, 1\\}$: 1 if generator $g$ is on in period $t$, and 0 otherwise\n\\item $\\gamma_{gt} \\in \\{0, 1\\}$: 1 if generator $g$ starts up in period $t$, and 0 otherwise\n\\item $\\eta_{gt} \\in \\{0, 1\\}$: 1 if generator $g$ shut down in period $t$, and 0 otherwise\n\\item $p_{gt} \\ge 0$: power output of generator $g$ in period $t$\n\\end{itemize}\n\\end{itemize}\n\nThe objective is the total cost\n\\[\n\\min \\sum_{t \\in T} \\sum_{g \\in G}\n\\left( C^{\\mathrm{nl}}_g \\alpha_{gt} + C^{\\mathrm{mr}}_g p_{gt} +\nC^{\\mathrm{up}}_g \\gamma_{gt}\n\\right).\n\\]\nThis is to be minimized subject to the following constraints.\n\\begin{itemize}\n\\item Load balance\n\\begin{equation*}\n\\sum_{g \\in G} p_{gt} \\ge P^{\\mathrm{d}}_{t}\n\\qquad t \\in T\n\\label{eq:uc_first_constraint}\n\\end{equation*}\n\n\\item Reserve\n\\begin{equation*}\n\\sum_{g \\in G} (P^{\\max}_{g} \\alpha_{gt} - p_{gt})\n\\ge P^{\\mathrm{r}}_t\n\\qquad t \\in T\n\\end{equation*}\n\n\\item Power output bounds\n\\begin{equation*}\nP^{\\min}_{g} \\alpha_{gt} \\le p_{gt} \\le P^{\\max}_{g} \\alpha_{gt}\n\\qquad g \\in G, t \\in T\n\\end{equation*}\n\n\\item Ramp rate\n\\begin{equation*}\np_{gt} - p_{g \\, t-1} \\le P^{\\mathrm{ru}}_g \\alpha_{g \\, t-1}\n+ P^{\\mathrm{su}}_g \\gamma_{gt}\n\\qquad g \\in G, t \\in T \\backslash \\{1\\}\n\\end{equation*}\n\\begin{equation*}\np_{g \\, t-1} - p_{gt} \\le P^{\\mathrm{rd}}_g \\alpha_{gt}\n+ P^{\\mathrm{sd}}_g \\eta_{gt}\n\\qquad g \\in G, t \\in T \\backslash \\{1\\}\n\\end{equation*}\n\n\\item Minimum up\/downtime\n\\begin{equation*}\n\\sum_{i=\\max\\{t-T^\\mathrm{u}_g+1, 1\\}}^t \\gamma_{gi} \\le \\alpha_{gt}\n\\qquad g \\in G, t \\in T\n\\end{equation*}\n\\begin{equation*}\n\\sum_{i=\\max\\{t-T^\\mathrm{u}_g+1, 1\\}}^t \\eta_{gi} \\le 1 - \\alpha_{gt}\n\\qquad g \\in G, t \\in T\n\\end{equation*}\n\n\\item Polyhedral\/Switching constraints (to enforce binaries to work as we expect)\n\\begin{equation*}\n\\alpha_{gt} - \\alpha_{g \\, t-1} = \\gamma_{gt} - \\eta_{gt}\n\\qquad g \\in G, t \\in T\n\\end{equation*}\n\\begin{equation*}\n1 \\ge \\gamma_{gt} + \\eta_{gt}\n\\qquad g \\in G, t \\in T\n\\label{eq:uc_last_constraint}\n\\end{equation*}\n\\end{itemize}\n\n\\section{Implementation details}\n\nIn the numerical experiment, a regularized column generation procedure is used.\nInitially, the regularization centre is set to the dual value given by the initialization method.\nIn each iteration, a lower bound is evaluated using the current dual value and the regularization centre is updated to the current dual value if the lower bound gets improved.\nFurthermore, if the lower bound gets improved the regularization parameter is divided by two, and otherwise multiplied by two.\nThis closely follows the implementation of a column generation procedure described by \\citet{Schulzeetal2017}.\n\nIn every iteration, after the pricing problems are solved, a primal heuristic based on the local search is run.\nGiven the solutions to the pricing problems, this primal heuristic checks the feasibility and switches the cheapest available generators on to make the solution feasible if necessary.\nNote that the solutions to the pricing problems are feasible generator schedules and infeasibility only arises from an insufficient generation capacity to meet the demand or insufficient spinning reserve.\nThis is similar to the methods used by \\citet{ZhuangAndGaliana1988} and \\citet{MerlinAndSnadrin1983} in the sense that a feasible solution is constructed from solutions to the pricing problems by committing additional generators.\nIn our experiments, we observe that this primal heuristic typically finds near-optimal primal solutions if this is used with a warmstarted column generation procedure.\nIf the column generation procedure using the local search primal heuristic fails to find a primal feasible solution within a given tolerance within 30 iterations, we use in addition a column combination based primal heuristic.\nThe column combination heuristic is a popular primal heuristic in the column generation procedure in general.\nThe idea of this heuristics is to solve (\\ref{eq_mip}) with restricted patterns of solutions, referred to as restricted master IP by \\citet{Vanderbeck2005}.\nIn the $k$th iteration, we replace $X_s$ in (\\ref{eq_mip}) with $\\{x_s^{(k - l)}\\}_{l = 0, 1, 2}$\nwhere $x_s^{(i)}$ is a solution to the $s$th pricing problem found in the $i$th iteration.\nThe resulting problem is still mixed-integer programming but the solution space is much smaller than the original problem.\nIn our numerical experiments, we observed that the problem is typically feasible and there is no need to add artificial variables (columns).\n\n\\section{Hyperparameters on neural networks}\n\nThe model consists of 4 hidden layers of 1000 units per layer, with skip connections between hidden layers.\nTanh is used as an activation function.\nThis is because in earlier experiments we observed that Tanh performed better than ReLU, which is a more standard activation function.\nAs \\citet{DeAndSmith2020} proposed, the weights on the residual connections are initialized to zero.\nThe other weights are initialized based on the methods by \\citet{GlorotAndBengio2010}.\nAll biases are initialized to zero.\n\nAdam (\\citet{KingmaAndJimmy2014}) is used to learn the parameters of the neural network.\nThe learning rate is initialized to $10^{-4}$.\nThe model performance is evaluated every 5 minutes, and if it fails to improve for successive 15 minutes, the learning rate is divided by 1.5.\n\n\\end{APPENDICES}\n\n\\ACKNOWLEDGMENT{%\n\n\n\n\n\\section{Introduction}\n\\label{sec_intro}\n\n\\lipsum[2-5]\n\n\n\n\\section{The problem}\n\\label{sec_problem}\n\n\\lipsum[6][1-2]\n\\begin{equation}\nf(x) = \\mathrm{E}[f(x; \\xi)],\n\\end{equation}\n\\lipsum[6][3-9]\n\n\\begin{theorem}\n\\lipsum[7][1-6]\n\\end{theorem}\n\n\\lipsum[8][1-3]\nInteger vitae justo \\cite{Someone1997,Another1999}.\n\\lipsum[8][5-9]\n\n\\lipsum[1-3]\n\n\\begin{figure}\n  \\centering\n  \\fbox{\\rule[-.5cm]{0cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n  \\caption{Sample figure caption.}\n\\end{figure}\n\n\\lipsum[4-6]\n\n\\begin{table}\n  \\caption{Sample table title}\n  \\label{sample-table}\n  \\centering\n  \\begin{tabular}{lll}\n    \\hline\n    & \\multicolumn{2}{c}{Something} \\\\\n    \\cline{2-3}\n    Name     & Description     & Size \\\\\n    \\hline\n    Apple    & Input terminal  & $\\sim$10  \\\\\n    Banana   & Output terminal & $\\sim$14  \\\\\n    Lemon    & Cell body       & up to 40  \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAmong the most precisely measured electroweak observables\nthe electromagnetic coupling constant, $\\alpha$,\nthe muon decay constant, $G_\\mu$, the anomalous magnetic moment $g-2$,\nand the ratio of charged to neutral current cross-sections\nin semi-leptonic neutrino scattering, feature prominently. Because of their\nhigh experimental precision $\\alpha$ and $G_\\mu$ are commonly used as\ninput in calculations of electroweak radiative corrections. Until relatively\nrecently the experimental errors on $\\alpha$ and $G_\\mu$ were entirely\nnegligible compared to those on the third input quantity, $M_Z$. Now,\nhowever, both $G_\\mu$ and $M_Z$ are both measured to 2 parts in $10^5$.\nAny further improvement on the accuracy of $M_Z$ would mean that the\nmeasurement of the muon lifetime in principle limits the accuracy with\nwhich electroweak theory can be tested.\n\nIn this arena, theoretical calculations currently lag experiment.\nCalculations are not available that could exploit the\n$2\\times10^{-5}$ accuracy available on $G_\\mu$ and $M_Z$. It is therefore\nof technical interest and practical necessity to develop electroweak\ncalculations that approach the $2\\times 10^{-5}$ accuracy of experiment.\nWhile a full 2-loop calculation would be ideal, it is likely that the analytic\nform of such results would be too complicated to usefully write down.\nOne would therefore probably wish to rely on computer algebra and\nintegration to obtain numerical results \\cite{Mainz}. However it\nis still true that certain dominant subclasses of 2-loop diagrams can be\ncalculated and written down in compact analytic form\n\\cite{BijHoogeveen,Barbieri,DennHollLamp,Fleischer}.\nSuch results are, of course, always preferable to numerical ones.\n\nThe set of 2-loop Feynman diagrams containing one fermion loop are such\na class. These we will refer to as ${\\cal O}(N_f\\alpha^2)$ corrections where\n$N_f$ is the number of fermions and $N_g=N_f\/8$ is the number of complete\ngenerations. In these corrections we will assume that fermions are massless\nwhich is an excellent approximation for all but the third generation. The\n${\\cal O}(N_f\\alpha^2)$ results can be used either with $N_g=3$ and mass\ncorrections computed for the third generation or $N_g=2$ and corrections for\nthe third generation computed separately.\n\nBecause $N_f$ is quite large --- up to 24 --- the ${\\cal O}(N_f\\alpha^2)$\ncorrections are expected to be a dominant subset of 2-loop graphs and\nsince $N_f$ provides a unique tag, the complete set of ${\\cal O}(N_f\\alpha^2)$\ncorrections contributing to a particular physical process will form a\ngauge-invariant set. Diagrams of this type have been considered in connection\nwith the muon anomalous magnetic moment \\cite{CKM1}.\n\n{\\it A priori\\\/} the ${\\cal O}(N_f\\alpha^2)$ corrections can be expected to\ncontribute at the level of $1.5\\times10^{-4}$. Without their calculation and\ninclusion, theoretical predictions are uncertain at this level.\n\nTo test the Standard Model, one needs at least four well-measured electroweak\nobservables. Three of these are used as input to the model and the fourth\nis then predicted and can be compared with experimental determinations. In\naddition, relating electroweak parameters, such as $M_Z$, extracted at high\nenergies with those, such as $\\alpha$ and $G_\\mu$, extracted at low energies\ninevitably involves an uncertainty arising from the hadronic contribution to\nthe photon vacuum polarization, $\\Pi_{\\gamma\\gamma}^\\prime(0)$. This hadronic\nuncertainty can be reduced by improved measurements of the cross-section\n$\\sigma(e^+e^-\\rightarrow{\\rm hadrons})$ and the use of dispersion relations\n\\cite{Zeppenfeld,Jegerlehner,Swartz} or eliminated completely by\nsacrificing one high precision observable \\cite{StuartHadron}.\n\nAt present the fourth well-measured electroweak observable, needed to test the\nStandard Model at very high precision, is missing but it is not unlikely that\nquantities, such as the polarization asymmetry, $A_{LR}$, or $g-2$ of the\nmuon, could fulfill this r\\^ole in the foreseeable future. In any event the\ncalculation of the ${\\cal O}(N_f\\alpha^2)$ electroweak radiative corrections\nare a necessary prerequisite to be able exploit the extremely accurate\nexperimental measurement that are the legacy of LEP.\n\n\\section{The Master Integral}\n\nIn what follows the general coupling of a fermion to a vector boson will\nbe denoted\n\\[\n\\raisebox{-0.7cm}{\\epsfig{file=loweng1.eps,width=1.5cm}}\n\\equiv i\\gamma_\\mu(\\beta_L\\gamma_L+\\beta_R\\gamma_R)\n\\]\nwhere $\\gamma_L$ and $\\gamma_R$ are the usual left- and right-handed\nhelicity projection operators and $\\beta_L$ and $\\beta_R$ are the\ncorresponding coupling constants. We assume throughout an anticommuting\n$\\gamma_5$. A Euclidean metric with the square of time-like momenta being\nnegative will be used and all calculations will be done in $R_{\\xi=1}$ gauge.\nThe sine and cosine of the weak mixing angle\n$\\theta_W$ will be denoted $s_\\theta$ and $c_\\theta$ respectively.\n\nWe will be interested in the ${\\cal O}(N_f\\alpha^2)$ corrections for massless\nfermions. In corresponding Feynman diagrams the fermions couple only to\nvector bosons and never to Goldstones or the physical Higgs particle.\nMany, but by no means all, such diagrams are obtained simply by inserting\na fermion loop into the boson propagator of a one-loop diagram,\nsee Fig.1a. Because the original one-loop diagram is logarithmically\ndivergent, the fermion loop insertion will need to be calculated to\n${\\cal O}(n-4)$ in dimensional regularization, where $n$ is the dimension\nof space-time. It turns out, however, that for all ${\\cal O}(N_f\\alpha^2)$\ndiagrams for low-energy processes, it is possible to obtain expressions that\nare exact in $n$.\n\nFor the massless fermion loop insertion, it may be shown that\n\\begin{eqnarray}\n\\raisebox{-0.3cm}{\\epsfig{file=loweng2.eps,width=2.5cm}}\n&=&-\\left(\\delta_{\\mu\\nu}-\\frac{p_\\mu p_\\nu}{p^2}\\right)\n   (\\beta_L\\beta_L^\\prime+\\beta_R\\beta_R^\\prime)\n   \\frac{\\displaystyle (n-2)}{\\displaystyle (n-1)}\n   \\int\\frac{d^n q}{\\displaystyle i\\pi^2}\\frac{p^2}{q^2(q+p)^2}\\\\\n&=&-\\left(p^2\\delta_{\\mu\\nu}-p_\\mu p_\\nu\\right)\n   (\\beta_L\\beta_L^\\prime+\\beta_R\\beta_R^\\prime)\n   4(\\pi p^2)^{\\frac{n}{2}-2}\n             \\frac{\\Gamma^2\\!\\left(\\frac{\\displaystyle n}\n                                        {\\displaystyle 2}\\right)}\n                  {\\Gamma(n)}\n             \\Gamma\\left(2-\\frac{\\displaystyle n}\n                                {\\displaystyle 2}\\right)\n\\end{eqnarray}\nwhere $\\beta_L$, $\\beta_R$ and $\\beta_L^\\prime$, $\\beta_R^\\prime$ of the\ncouplings of the attached vector bosons.\n\nA number of special cases of low energy 2-loop integrals have appeared in\nthe literature \\cite{BijVeltman,FleischerTarasov,DavyTausk}.\nBy using the projection operator techniques described in the following\nsections, most ${\\cal O}(N_f\\alpha^2)$ diagrams occurring in\nlow-energy electroweak processes, including box diagrams,\ncan be reduced to expressions involving a general\nmaster integral of the form\n\\begin{equation}\nI(j,k,l,m,n,M^2)=\n\\int\\frac{d^np}{i\\pi^2}\\frac{1}{[p^2]^j[p^2+M^2]^k}\n\\int\\frac{d^nq}{i\\pi^2}\\frac{1}{[q^2]^l[(q+p)^2]^m}\n\\label{eq:MasterIntDef}\n\\end{equation}\nprovided the diagram contains at least one massive boson.\nThe only exceptions to this are certain contributions to the photon\nvacuum polarization. In that case some diagrams must be calculated retaining\nthe fermion mass, $m_f$, and then applying the asymptotic expansion of\nref.\\cite{DavyTausk} before taking the limit $m_f\\rightarrow0$.\nOther ${\\cal O}(N_f\\alpha^2)$ diagrams that cannot be\nexpressed in terms of the integral (\\ref{eq:MasterIntDef}) are of the\npure QED type containing only internal photons and\nfermions. The relevant master integrals for this case can be found in\nref.\\cite{FleischerTarasov}.\n\nIn eq.(\\ref{eq:MasterIntDef}), the integration over $q$ can be performed\nusing standard Feynman parameter techniques and yields\n\\begin{equation}\n\\int\\frac{d^nq}{i\\pi^2}\\frac{1}{[q^2]^l[(q+p)^2]^m}=\n\\frac{\\pi^{\\frac{n}{2}-2}}{[p^2]^{l+m-\\frac{n}{2}}}\n\\frac{\\Gamma\\left(l+m-\\frac{\\textstyle n}{\\textstyle 2}\\right)\n      \\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}-l\\right)\n      \\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}-m\\right)}\n     {\\Gamma(l)\\Gamma(m)\\Gamma(n-l-m)}.\n\\end{equation}\nThe resulting integral with respect to $p$ in eq.(\\ref{eq:MasterIntDef})\nis independent of angle and hence\n\\begin{align}\n\\int\\frac{d^np}{i\\pi^2}\\ \\frac{1}{[p^2]^{j+l+m-\\frac{n}{2}}[p^2+M^2]^k}\n=\\frac{2\\pi^{\\frac{n}{2}-2}}\n        {\\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}\\right)}\n  &\\int_0^\\infty dp\\ \\frac{p^{2n-2j-2l-2m-1}}{[p^2+M^2]^k}\\nonumber\\\\\n=\\frac{\\pi^{\\frac{n}{2}-2}}{(M^2)^{k+j+l+m-n}}\n  &\\frac{\\Gamma(n-j-l-m)\\Gamma(k+j+l+m-n)}\n        {\\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}\\right)\\Gamma(k)}\n\\nonumber\\\\\n\\end{align}\nfrom which it follows\n\\begin{eqnarray}\n & &I(j,k,l,m,n,M^2)=\n    \\frac{\\pi^{n-4}}{(M^2)^{k+j+l+m-n}}\\nonumber\\\\\n & &\\ \\ \\ \\times\n    \\frac{\\Gamma(n-j-l-m)\\Gamma(k+j+l+m-n)\n          \\Gamma\\left(l+m-\\frac{\\textstyle n}{\\textstyle 2}\\right)\n          \\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}-l\\right)\n          \\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}-m\\right)}\n         {\\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}\\right)\n          \\Gamma(k)\\Gamma(l)\\Gamma(m)\\Gamma(n-l-m)}\n\\nonumber\\\\\n\\label{eq:MasterIntexpr}\n\\end{eqnarray}\n\n\\section{Self-energy Diagrams}\n\nFor electroweak processes, such as muon decay, in which the external\nfermions can be considered massless, only the self-energy diagrams of vector\nbosons need to be considered. General techniques are available to reduce the\ntensor integrals that occur to standard form factors\n\\cite{PassarinoVeltman}\nfor which scalar integral representations are known. For the present case\nof zero external momentum, $q=0$, the vector boson self-energies,\n$\\Pi_{\\mu\\nu}(q^2)$, can only take the form\n\\[\n\\Pi_{\\mu\\nu}(0)=\\delta_{\\mu\\nu}F\n\\]\nwhere $F$ is a function of the internal masses only and may be obtained\nfrom the tensor integral representation of $\\Pi_{\\mu\\nu}(0)$ by means of\nthe projection operator, $\\delta_{\\mu\\nu}\/n$. Thus\n\\begin{equation}\nF=\\left(\\frac{\\delta_{\\mu\\nu}}{n}\\right)\\Pi_{\\mu\\nu}(0).\n\\end{equation}\nThe resulting scalar integral can always written in terms the master integral,\n$I(j,k,l,m,n,M^2)$, of eq.(\\ref{eq:MasterIntexpr}) and results obtained that\nare exact for all $n$.\n\n\\section{Vertex Diagrams}\n\nAgain for processes with massless external fermions the only relevant\nvertex corrections are those involving vector bosons and these will\nnecessarily be purely vector and axial-vector in character. A general vertex\ncorrection can then be represented as\n\\[\n\\raisebox{-0.7cm}{\\epsfig{file=loweng3.eps,width=1.5cm}}\n\\equiv V_\\mu=i\\gamma_\\mu(V_L\\gamma_L+V_R\\gamma_R)\n\\]\nwhere $V_L$ and $V_R$ are functions only of the internal masses. The tensor\nintegral representation of $V_\\mu$ can easily be obtained by standard\ntechniques and from it the scalar integral representations of $V_L$ and\n$V_R$ follow by means of projection operators. Thus\n\\begin{equation}\nV_{L,R}=-\\frac{i}{2n}\\mathop{\\rm Tr}\\nolimits\\{V_\\mu\\gamma_\\mu\\gamma_{R,L}\\}.\n\\label{eq:vertexproj}\n\\end{equation}\nwhere $\\mathop{\\rm Tr}\\nolimits\\{\\gamma_\\mu\\gamma_\\mu\\}=4n$ is assumed.\nThis method for directly obtaining the scalar integral representation\nof the vertex form factors is particularly convenient when computer\nalgebra is being employed. Once again the resulting scalar integrals can\nbe written in terms of the master integral, $I(j,k,l,m,n,M^2)$.\n\n\\subsection{Example}\n\nThe diagram shown in Fig.\\ref{fig:example}(a) occurs in the calculation of the\n${\\cal O}(N_f\\alpha^2)$ electromagnetic charge renormalization. It will be\nassumed that the external fermion has $t_3=+1\/2$ in which case the fermion\nto which the photon couples has $t_3=-1\/2$. We will denote the charge of\nthe external fermion by $Q$. After obtaining the tensor integral for the\nFeynman diagram by usual methods, applying the projection operator as in\neq.(\\ref{eq:vertexproj}) gives the scalar integral representation\n\\begin{eqnarray*}\n & &\ni\\gamma_\\mu\\gamma_L g\\left(\\frac{g^2}{16\\pi^2}\\right)^2 Qs_\\theta\n    \\frac{(n-2)}{2n}\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\\\n    \\frac{[3p^2q^2-2(p\\cdot q)^2+q^2(p\\cdot q)]}\n         {p^2[p^2+M_W^2]^2 q^4(q+p)^2}\\\\\n & & \\ \\ \\ \\ \\\n=i\\gamma_\\mu\\gamma_L g\\left(\\frac{g^2}{16\\pi^2}\\right)^2 Q s_\\theta\n     \\frac{(n-2)}{2n}\\\\\n& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\times\n\\left\\{-\\frac{1}{2}I(1,2,2,-1,n,M_W^2)\n-\\frac{1}{2}I(-1,2,2,1,n,M_W^2)\n+\\frac{3}{2}I(0,2,1,1,n,M_W^2)\\right\\}\\\\\n & & \\ \\ \\ \\ \\\n=i\\gamma_\\mu\\gamma_L g\\left(\\frac{g^2}{16\\pi^2}\\right)^2\n   \\frac{Q}{2} s_\\theta(\\pi M_W^2)^{n-4}\n   \\Gamma(4-n)\\Gamma\\left(2-\\frac{n}{2}\\right)\\Gamma\\left(\\frac{n}{2}-1\\right)\n\\end{eqnarray*}\nexactly. These diagrams and their related topologies do not vanish when summed\nover a complete fermion generation.\n\n\\section{Box Diagrams}\n\n\\begin{figure}[t]\n\\epsfig{file=loweng4.eps,height=3cm}\n\\caption{Feynman diagrams treated in the examples.}\n\\label{fig:example}\n\\end{figure}\n\nFor 4-fermion processes one-loop box diagrams are finite but at\n${\\cal O}(N_f\\alpha^2)$ they develop logarithmic divergences.\nA useful set of identities for calculating one-loop box diagrams appears\nin ref.\\cite{SirlinBox}. They are, however, valid only for $n=4$\nbecause of their intended use at one-loop. For general $n$ it may be shown\nthat these relations become\n\\begin{eqnarray}\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\rho\\gamma_\\nu\\gamma_\\sigma\\gamma_{L,R}{]}_2\n&=&4\\delta_{\\mu\\nu}{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\alpha\\gamma_{L,R}{]}_2\n                 +(n-4){[}\\gamma_\\mu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\nu\\gamma_{L,R}{]}_2\\ \\ \\\n\\label{eq:firstBoxid}\\\\\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\rho\\gamma_\\nu\\gamma_\\sigma\\gamma_{R,L}{]}_2\n                   &=&4{[}\\gamma_\\nu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\mu\\gamma_{R,L}{]}_2\n                 +(n-4){[}\\gamma_\\mu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\nu\\gamma_{R,L}{]}_2\\ \\ \\ \\\\\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\sigma\\gamma_\\nu\\gamma_\\rho\\gamma_{L,R}{]}_2\n                   &=&4{[}\\gamma_\\nu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\mu\\gamma_{L,R}{]}_2\n                 +(n-4){[}\\gamma_\\mu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\nu\\gamma_{L,R}{]}_2\\ \\ \\ \\\\\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\sigma\\gamma_\\nu\\gamma_\\rho\\gamma_{R,L}{]}_2\n&=&4\\delta_{\\mu\\nu}{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\alpha\\gamma_{R,L}{]}_2\n                 +(n-4){[}\\gamma_\\mu\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\nu\\gamma_{R,L}{]}_2\\ \\ \\ \\ \\\n\\label{eq:fourthBoxid}\n\\end{eqnarray}\nwhere the square brackets $[\\ ]_1$ and $[\\ ]_2$ indicate that the enclosed\n$\\gamma$-matrices are associated with the external fermion currents\n$J_1$ and $J_2$ respectively.\n\nThe general ${\\cal O}(N_f\\alpha^2)$ box diagram for massless external\nfermions at $q=0$ therefore takes the form\n\\begin{equation}\nB_{\\mu\\nu}J_{1\\mu}J_{2\\nu}=B\\cdot J_{1\\alpha}J_{2\\alpha}.\n\\end{equation}\nHere $B_{\\mu\\nu}$ is a tensor integral and the product of\n$J_{1\\alpha}J_{2\\alpha}$ can be constructed from one or a combination of\nthe $\\gamma$-matrices appearing in\neq.(\\ref{eq:firstBoxid})--(\\ref{eq:fourthBoxid}). As for the self-energy\ncontributions, the tensor integral $B_{\\mu\\nu}$, can only be proportional\nto $\\delta_{\\mu\\nu}$ and hence, using the projection operator method of\nsection~2, the scalar integral representation for the form factor $B$ is\nseen to be $B=(\\delta_{\\mu\\nu}\/n)B_{\\mu\\nu}=B_{\\mu\\mu}\/n$.\nIn this case the identities\n(\\ref{eq:firstBoxid})--(\\ref{eq:fourthBoxid}) simplify to become\n\\begin{eqnarray}\n\\frac{\\delta_{\\mu\\nu}}{n}\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\rho\\gamma_\\nu\\gamma_\\sigma\\gamma_{L,R}{]}_2\n&=&\\frac{(5n-4)}{n}{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                   {[}\\gamma_\\alpha\\gamma_{L,R}{]}_2\\\\\n\\frac{\\delta_{\\mu\\nu}}{n}\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\rho\\gamma_\\nu\\gamma_\\sigma\\gamma_{R,L}{]}_2\n                   &=&{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\alpha\\gamma_{R,L}{]}_2\n\\label{eq:secondBoxid}\\\\\n\\frac{\\delta_{\\mu\\nu}}{n}\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\sigma\\gamma_\\nu\\gamma_\\rho\\gamma_{L,R}{]}_2\n                   &=&{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                       {[}\\gamma_\\alpha\\gamma_{L,R}{]}_2\\\\\n\\frac{\\delta_{\\mu\\nu}}{n}\n{[}\\gamma_\\rho\\gamma_\\mu\\gamma_\\sigma\\gamma_{L,R}{]}_1\n{[}\\gamma_\\sigma\\gamma_\\nu\\gamma_\\rho\\gamma_{R,L}{]}_2\n&=&\\frac{(5n-4)}{n}{[}\\gamma_\\alpha\\gamma_{L,R}{]}_1\n                   {[}\\gamma_\\alpha\\gamma_{R,L}{]}_2.\n\\end{eqnarray}\n\n\\subsection{Example}\n\nThe diagram shown in Fig.\\ref{fig:example}(b) occurs in the ${\\cal O}(N_f\\alpha^2)$\ncorrections to muon decay. Using (1) and\n(\\ref{eq:secondBoxid}) the resulting the Feynman diagram can be written as\n\\begin{eqnarray*}\n& &\n\\frac{g^2}{16c_\\theta^2}\\left(\\frac{g^2}{16\\pi^2}\\right)^2\n{[}\\gamma_\\alpha\\gamma_L{]}_\\mu{[}\\gamma_\\alpha\\gamma_L{]}_e\n\\frac{(n-2)}{(n-1)}\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\n\\frac{1}{[p^2+M_Z^2][p^2+M_W^2]^2 q^2(q+p)^2}\\\\\n & & \\ \\ \\ \\ \\ \\ \\ \\ \\\n=\\frac{g^2}{16c_\\theta^2}\\left(\\frac{g^2}{16\\pi^2}\\right)^2\n{[}\\gamma_\\alpha\\gamma_L{]}_\\mu{[}\\gamma_\\alpha\\gamma_L{]}_e\n\\frac{(n-2)}{(n-1)}\\\\\n & & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\times\n\\frac{1}{(M_Z^2-M_W^2)^2}\\Big\\{I(0,1,1,1,n,M_Z^2)-I(0,1,1,1,n,M_W^2)\\\\\n & & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n                +(M_Z^2-M_W^2)I(0,2,1,1,n,M_W^2)\n\\Big\\}\\\\\n & & \\ \\ \\ \\ \\ \\ \\ \\ \\\n=-\\frac{g^2}{24 M_W^2s_\\theta^4}\n\\left(\\frac{g^2}{16\\pi^2}\\right)\n{[}\\gamma_\\alpha\\gamma_L{]}_\\mu{[}\\gamma_\\alpha\\gamma_L{]}_e\n(\\pi M_W^2)^{-2\\epsilon}\\Gamma(\\epsilon)\\\\\n & & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\times\n\\left\\{(s_\\theta^2+\\ln c_\\theta^2)\n     \\left(1+\\epsilon\\left(\\frac{8}{3}-\\gamma\\right)\\right)\n            +\\epsilon\\ln^2 c_\\theta^2\\right\\}\n\\end{eqnarray*}\nup to ${\\cal O}(1)$ in $\\epsilon=2-n\/2$.\n\n\\section{Derivatives of Self-energy Diagrams}\n\nThe derivatives of self-energies with respect to the square of the\nexternal momentum, $k^2$, are required for the calculation physical\nprocesses. They are often thought of as wavefunction renormalization\nfactors although they occur even when wavefunction renormalization\nhas not been explicitly performed \\cite{ZMass2}.\nAs we are concerned with low-energy processes and massless fermions,\nthese derivatives will only be required at $k^2=0$.\nIn that case the scalar integral representation may be obtained by the\nfollowing method.\n\nFor a vector boson the self-energy may be written in the form\n\\begin{equation}\n\\Pi_{\\mu\\nu}(k^2)=\\left(\\delta_{\\mu\\nu}-\\frac{k_\\mu k_\\nu}{k^2}\\right)\\Pi_T(k^2)\n                 +\\left(\\frac{k_\\mu k_\\nu}{k^2}\\right)\\Pi_L(k^2)\n\\end{equation}\nwith $\\Pi_T$ and $\\Pi_L$ being the transverse and longitudinal form-factors\nrespectively.\n\\begin{eqnarray}\n\\Pi_T(k^2)&=&\\frac{1}{n-1}\\left(\\delta_{\\mu\\nu}-\\frac{k_\\mu k_\\nu}{k^2}\\right)\n             \\Pi_{\\mu\\nu}(k^2)\\\\\n\\Pi_L(k^2)&=&\\left(\\frac{k_\\mu k_\\nu}{k^2}\\right)\\Pi_{\\mu\\nu}(k^2).\n\\end{eqnarray}\n\nFor massless fermions self-energy diagrams take the form\n\\begin{equation}\n\\Sigma(k)=i\\slash{k}\\left(F_L(k^2)\\gamma_L+F_R(k^2)\\gamma_R\\right)\n\\end{equation}\nand\n\\begin{equation}\nF_{L,R}(k^2)=-\\frac{i}{2k^2}\\mathop{\\rm Tr}\\nolimits\\left\\{\\Sigma(k)\\slash{k}\\gamma_{R,L}\\right\\}\n\\end{equation}\n\nIn either case given expressions for the self-energies in the form of tensor\nintegrals, the scalar integral representations for the corresponding\nform factors $\\Pi_L$, $\\Pi_T$ or $F_L$, $F_R$ are easily obtained.\nIn general, then, these scalar integral representations may be written\n\\begin{equation}\nF(k^2)=\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\n      \\frac{F(k^2,p^2,q^2,k\\cdot p,k\\cdot q,p\\cdot q)}\n           {[(k+p)^2+M_1^2]^{N_1}[(k+q)^2+M_2^2]^{N_2}[(k+p+q)^2+M_3^2]^{N_3}}\n\\ \\ \\ \\ \\ \\ \\ {}\n\\label{eq:ScalarIntRep}\n\\end{equation}\nwhere $N_1$, $N_2$ and $N_3$ are integer powers.\n\nIn the region of small $k^2$ we may expand the propagators as a power series\nin $(k\\cdot p)$ and $(k\\cdot q)$. Thus, for example,\n\\begin{eqnarray*}\n\\frac{1}{[(k+p)^2+M_1^2]^{N_2}}&=&\\frac{1}{[k^2+p^2+M_1^2]^{N_2}}\n                          \\left(1-2N_2\\frac{(k\\cdot p)}{k^2+p^2+M_1^2}\n                                 +...\\right)\n\\label{eq:expansion}\\\\\n\\frac{1}{[(k+p+q)^2+M_3^2]^{N_3}}&=&\\frac{1}{[k^2+(p+q)^2+M_3^2]^{N_3}}\n                      \\left(1-2N_3\\frac{(k\\cdot p)+(k\\cdot q)}\n                                   {k^2+(p+q)^2+M_3^2}\n                             +...\\right)\\\\\n\\end{eqnarray*}\nwith the result that the integral in eq.(\\ref{eq:ScalarIntRep}) becomes\n\\begin{equation}\nF(k^2)=\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\n       \\sum_{i,j}(k\\cdot p)^i (k\\cdot q)^j F_{ij}(k^2,q^2,p^2).\n\\end{equation}\nUpon integration terms in which $i+j$ is odd vanish. Terms in the integrand\nwith $i+j=2$ are $(k\\cdot v_1)(k\\cdot v_2)F(k^2,q^2,p^2)$ in which\n$v_1$ and $v_2$ are either $p$ or $q$. Because $F$ is a function only\nof squared momenta the integration of $p_\\mu q_\\nu F(k^2,q^2,p^2)$ can only\nyield\na result proportional to $\\delta_{\\mu\\nu}$. Its coefficient may be extracted\nusing the projection operator $\\delta_{\\mu\\nu}\/n$ as in section 2. It follows\nthen that the terms $(k\\cdot v_1)(k\\cdot v_2)F(k^2,q^2,p^2)$ in the\nintegrand may be replaced by $k^2(v_1\\cdot v_2)\/n\\,F(k^2,q^2,p^2)$ without\nchanging the integral.\n\nIn a similar way for $i+j=4$ one can make the replacement\n\\begin{align*}\n(k\\cdot v_1)(k\\cdot v_2)&(k\\cdot v_3)(k\\cdot v_4)F(k^2,q^2,p^2)\n\\longrightarrow\\\\\n&k^4\\frac{(v_1\\cdot v_2)(v_3\\cdot v_4)\n         +(v_1\\cdot v_3)(v_2\\cdot v_4)\n         +(v_1\\cdot v_4)(v_2\\cdot v_3)}{n(n+2)}F(k^2,q^2,p^2)\n\\end{align*}\nwhere again the $v_i$ are $p$ or $q$.\n\nThe integral is thus transformed into a form whose only $k$ dependence\nis enters only through $k^2$.\n\nDifferentiation of the integrand with respect to $k^2$ can now be\nperformed and setting $k^2=0$ yields a result expressible in terms of the\nmaster integral (\\ref{eq:MasterIntexpr}).\n\nFor most practical purposes $i+j=4$ is as high as one needs to go in the\nseries expansion. The general term in this series can be found in\n\\cite{DavyTausk} and requires repeated application of the momentum-space\nd'Alembertian operator.\n\n\\subsection{Example}\n\nA self-energy diagram for a massless fermion is shown in Fig.1c and\nmay be written\n\\[\n\\Sigma(k)=i\\slash{k}A_L(k^2)\\gamma_L.\n\\]\nAfter obtaining the scalar integral representation for $A_L(k^2)$ and\nexpanding the propagator as in (\\ref{eq:expansion}) up to $(k\\cdot p)^2$\none obtains\n\\begin{eqnarray*}\nA_L(k^2)&=&-\\frac{1}{4k^2}\\left(\\frac{g^2}{16\\pi^2}\\right)^2\n\\frac{(n-2)}{(n-1)}\n\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\n\\frac{1}{[p^2+M_W^2]^2 q^2(q+p)^2}\\\\\n& &\\times\n\\left\\{\\frac{(3-n)k^2}{k^2+p^2}\n      +\\frac{2(p^2-k^2)(k^2+(n-2)p^2)}{p^2(k^2+p^2)^3}(k\\cdot p)^2\n\\right\\}\n\\end{eqnarray*}\nAfter the replacement $(k\\cdot p)^2\\rightarrow k^2p^2\/n$ the derivative\nwith respect to $k^2$ at $k^2=0$ is found to be\n\\begin{eqnarray*}\n\\frac{\\partial A_L(k^2)}{\\partial k^2}\\Bigg\\vert_{k^2=0}\n&=&\\left(\\frac{g^2}{16\\pi^2}\\right)^2\\frac{(n-2)(n-4)}{4n}\n\\int\\frac{d^np}{i\\pi^2}\\int\\frac{d^nq}{i\\pi^2}\n\\frac{1}{[p^2+M_W^2]^2 q^2 (q+p)^2}\\ \\ \\ \\ \\\\\n&=&\\left(\\frac{g^2}{16\\pi^2}\\right)^2\\frac{(n-2)(n-4)}{4n} I(0,2,1,1,n,M_W^2)\\\\\n&=&-\\left(\\frac{g^2}{16\\pi^2}\\right)^2(\\pi M_W^2)^{n-4}\\frac{1}{2n}\\Gamma(5-n)\n                \\Gamma\\left(2-\\frac{\\textstyle n}{\\textstyle 2}\\right)\n                \\Gamma\\left(\\frac{\\textstyle n}{\\textstyle 2}-1\\right)\n\\end{eqnarray*}\nexactly. This result is related to vertex diagrams by Ward identities that\nserve as a useful check.\n\n\\section{Acknowledgments}\nThis work was supported in part by the U.S. Department of Energy.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nA matroid is a structure which generalizes the notion of independence that rises in linear algebra, graph theory and other areas. From a matroid, we can construct a simplicial complex (\\newword{matroid complex}) and study its \\newword{h-vector} which provides the topological information of the complex. The following conjecture by Stanley is still wide open:\n\n\\begin{conjecture}[\\cite{Stanley1977}]\n\\label{con:stanley}\nThe h-vector of  a matroid is a pure O-sequence.\n\\end{conjecture}\n\nAn O-sequence is a sequence that can be obtained from counting monomials of each degree in an \\newword{order ideal}: a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. If all maximal monomials of $X$ have the same degree, we say it is \\newword{pure} and the degree sequence coming from it a pure O-sequence.\n\nThere are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in \\cite{Merino2001}, lattice-path matroids by Schweig in \\cite{Schweig2010}, cotransversal matroids by Oh in \\cite{Oh2013}, paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores  \\cite{MNRV2012}, internally perfect matroids by Dall in \\cite{Dal}, rank $3$ matroids by H\\'a, Stokes, and Zanello in \\cite{HaStokesZanello2013}, rank $3$ and corank $2$ matroids by DeLoera, Kemper, and Klee in \\cite{DelKemKle}, rank $4$ matroids by Klee and Samper in \\cite{KleeSamper2015}, rank $d$ matroids with $h_d \\leq 5$  by Constantinescu, Kahle, and Varbaro in \\cite{ConKahVar} and some small special classes of graphic matroids by Kook \\cite{Kook2011}, Preston et al \\cite{biconed} and David et al \\cite{triconed}.\n\nIn this paper the focus will solely be on the class of positroids, which is a special class of realizable matroids, for which the conjecture is still wide open. Positroids come from the study of the nonnegative part of the Grassmannian \\cite{Postnikov}, and have seen increased applications in physics, with use in the study of scattering amplitudes \\cite{arkani} and the study of shallow water waves \\cite{kodama}.\n\n\n\\section{Preliminary}\n\nThroughout the paper, we will use the following notation. When $A$ is a set and $a$ is a single element, instead of writing $A \\setminus \\{a\\}$ we will instead write $A \\setminus a$. Similarly, instead of writing $A \\cup \\{a\\}$ we will instead use $A \\cup a$.\n\n\\subsection{Positroids} \n\n\\begin{figure}\n\\begin{multicols}{2}\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lediagram}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{A $\\LE$-diagram.}\n  \\label{fig:lediagram}\n\n \\end{center}\n\n\\vfill\\null\n\\columnbreak\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lenetwork}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{The corresponding $\\LE$-network.}\n  \\label{fig:lenetwork}\n\n \\end{center}\n\\end{multicols}\n\\end{figure}\n\nWe will start with the definition of $\\LE$-diagrams and $\\LE$-graphs and use that to define positroids.\n\n\\begin{definition}[\\cite{Postnikov}, Definition 6.1] For a partition $\\lambda$, let us define a $\\LE$-diagram $L$ of shape $\\lambda$ as a filling of boxes of the Young diagram of shape $\\lambda$ such that, for any three boxes indexed $(i,j),(i',j),(i,j')$, where $i'<i$ and $j'>j$, if boxes on position $(i',j)$ and $(i,j')$ are filled, then the box on $(i,j)$ is also filled. This property is called the $\\LE$-property.\n\\end{definition}\n\nFrom each $\\LE$-diagram $L$, we can get a planar network called the $\\LE$-graph in Definition 6.3 of \\cite{Postnikov}, via the following way: Place a dot in the middle of each step in the boundary lattice path of the diagram and mark these dots by $1,\\ldots,n$ from top to bottom. We will call these vertices the \\newword{boundary vertices}. Now for each dot inside the $\\LE$-diagram, draw a horizontal line to its right, and vertical line to its bottom until it reaches another dot. Next orient all vertical edges downward horizontal edges to the right.  So the vertical labels will be the source vertices and the horizontal labels will be the sink vertices in the $\\LE$-graph. Since each $\\LE$-diagram encodes a unique $\\LE$-graph, we will abuse notation and go between a $\\LE$-diagram and its corresponding $\\LE$-graph freely in this paper.\n\nThe set $[n]$ (where $n$ is the length of the boundary lattice path) is called the \\newword{ground set} (being consistent with the notation for the positroid that will be indexed by the $\\LE$-diagram). Let $B_0$ denote the set of source vertices in the $\\LE$-graph of $L$. For example, in the $\\LE$-diagram $L$ drawn in Figure~\\ref{fig:lediagram}, we have $B_0 = 124$ as we can see from the corresponding $\\LE$-graph in Figure~\\ref{fig:lenetwork}. A \\newword{path} in a $\\LE$-graph is a directed path that starts at some boundary vertex and ends at some boundary vertex. A \\newword{family of non-touching paths} is a family of paths where no two paths intersects within the graph. We say that a family of non-touching paths consisting of paths $p_1,\\ldots,p_k$ \\newword{represents} a set $B_0 \\setminus \\{p_1^s,\\ldots,p_k^s\\} \\cup \\{p_1^e,\\ldots,p_k^e\\}$ where $p_i^s$ stands for the starting point and $p_i^e$ stands for the endpoint of the path $p_i$. For example, the bottom picture in Figure~\\ref{fig:allthebases} shows an empty family, which represents the set $124$. The leftmost picture at the top row in Figure~\\ref{fig:allthebases} shows a family consisting of two paths, one starting at $1$ and ending at $3$, the other one starting at $4$ and ending at $5$: this represents the set $124 \\setminus 14 \\cup 35 = 235$. Notice that any set represented by a family of non-touching paths in a $\\LE$-graph will have the same cardinality as $B_0$.\n\n\nFrom Theorem~6.5 of \\cite{Postnikov}, there is a bijection between positroids and $\\LE$-diagrams. Moreover, the same theorem gives the following result, which we will use as the definition of a positroid in this paper:\n\n\\begin{definition}\nFix a positroid $\\mathcal{M}_L$ that corresponds to a $\\LE$-diagram $L$. Then the set of bases of $\\mathcal{M}_L$ is exactly the collection of sets represented by a family of non-touching paths in the $\\LE$-graph of $L$.\n\\end{definition}\n\nFrom the above definition, the set $B_0$ is naturally the lexicographic minimal basis of $\\mathcal{M}_L$. We call this the \\newword{canonical basis} of the positroid. For example, take a look at the $\\LE$-diagram in Figure~\\ref{fig:lediagram}. All the bases we can get are:\n$$124,125,134,135,145,234,235,245,345,$$ \nas one can check from various families of disjoint paths drawn on the $\\LE$-graph as in Figure~\\ref{fig:allthebases}. \n\nAn element $e$ of the ground set is called a \\newword{loop} if it is not contained in any basis and is called a \\newword{coloop} if it is contained in every basis.\n\n\n\\subsection{Activity and the h-vector} \nFrom any ordering on the ground set $E$, we get an induced lexicographic ordering on the bases. An element $i$ of a base $B$ is \\newword{(internally) active} if $B \\setminus i \\cup j$ is not a base for any $j < i$. Otherwise, it is called \\newword{(internally) passive}. There is also the notion of externally active and externally passive elements, but in this paper we will only stick to interally active and passive elements, and hence will skip the usage of the word internally throughout. Given a base $B$ of a matroid, we call its \\newword{passivity} as the number of (internally) passive elements it contains.\n\nWe will use this activity property of h-vectors as the definition of h-vectors of matroids in this paper.\n\n\\begin{theorem}\nLet $(h_0,\\dots,h_r)$ be the $h$-vector of a matroid $\\mathcal{M}$. For $0 \\leq i \\leq r$, the entry $h_i$ is the number of bases of $\\mathcal{M}$ with $i$ passive elements.\n\\end{theorem}\n\nLet us again look at the running example of Figure~\\ref{fig:allthebases}. Take a look at $135$. Here $1$ is active since it cannot be replaced with a smaller element to get a basis of $\\mathcal{M}$. On the other hand, $3$ and $5$ are passive since we can replace either of them with $2$ to get a different basis of $\\mathcal{M}$. The $h$-vector of the corresponding positroid is $(1,2,3,3)$.\n\n\n\\begin{remark}\n\\label{rem:noloopcoloop}\nDeleting a loop (simply remove that element from the ground set) or contracting by a coloop (remove that element from the ground set and all the bases) does not change the $h$-vector of the matroid. For sake of proving Stanley's conjecture, we may assume that our matroid does not have any loops or coloops. Hence from now on, throughout the entire paper, we are only going to deal with positroids with no loops nor coloops.\n\\end{remark}\n\n\n\n\\subsection{Order ideal and pure O-sequences}\nAn \\newword{order ideal} is a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. Given an order ideal its \\newword{O-sequence} is a sequence that can be obtained from counting monomials of each degree. We say that the order ideal is \\newword{pure} if all the maximal monomials are of the same degree, and also call the O-sequence pure if it comes from a pure order ideal.\n\nFor example, take a look at Figure~\\ref{fig:allthebases} and focus only on the monomials. If we take any monomial among this collection, say $x_3x_5^2$, we can see that all divisors of it is again present in the collection. Hence this is an order ideal. Moreover, since the maximal monomials under division (which are usually not guaranteed to all have the same degree) here are $x_3^2x_5,x_3x_5^2,x_5^3$ which have the same degree, and hence this is a pure order ideal. Notice that the degree sequence of this pure order ideal is $(1,2,3,3)$. Hence we can see that the $h$-vector of the positroid coming from Figure~\\ref{fig:lediagram}, which was $(1,2,3,3)$, is indeed a pure $O$-sequence. We have just verified that Stanley's conjecture is true for this case. What we want to do in this paper is to similarly construct a pure order ideal for the class of positroids, hence proving the conjecture for this class.\n\n\n\\section{The main result}\n\nGiven a path from a source $s$ to sink $k$, we write it as $(s \\rightarrow k)$. Since each basis $B \\in \\mathcal{M}$ is coming from a family of non-crossing paths from set of sources to set of sinks, we write $B$ as a collection of $(s \\rightarrow k)$'s and call this the \\newword{non-crossing matching representation} (or simply a matching representation) of $B \\in \\mathcal{M}$. Note that all sources $s$ are contained in the canonical basis $B_0$, and all sinks $k$ are contained in $[n] \\setminus B_0$.\n\n\\begin{lemma}\nGiven a positroid $\\mathcal{M}$, for all $B \\in \\mathcal{M}$, its matching representation is unique.\n\\end{lemma}\n\n\\begin{proof}\n(I think this is theorem 3.2?)\n\\end{proof}\n\n\n\\begin{theorem}\n\\label{lem:one-to-one-matching}\nGiven a basis $B$, there is a one-to-one correspondence between the elements in $B_0$ but not in $B$ and the elements not in $B_0$ but in $B$ which creates noncrossing paths. \n\\end{theorem}\n\n\\begin{proof}\n\nIf there is a basis with $n$ elements of $B\\setminus B_0$, then we will also have exactly $n$ elements of $B_0$ in consideration.\n\nTo create a Dyck path of this basis, we will express the elements in $B_0$ as arrows pointing in the ``up\" direction and elements of $B\\setminus B_0$ as arrows pointing in the ``down\" direction. There will be an equal number of ``up\" arrows and ``down\" arrows.\n\nTo pair an ``up\" arrow with a ``down\" arrow, which is also pairing an element in $B_0$ and an element of $B\\setminus B_0$, there must be an equal number of ``up\" arrows and ``down\" arrows between the arrows that we are pairing. By [chrome-extension:\/\/oemmndcbldboiebfnladdacbdfmadadm\/https:\/\/math.mit.edu\/~apost\/courses\/18.204-2016\/18.204\\_Gabriella\\_Baracchini\\_final\\_paper.pdf], each set of ``up\" and ``down\" arrows produces a unique pairing; this means that each basis will have a unique correspondence between elements in $B_0$ and elements $B\\setminus B_0$.\n\nNote that the use od Dyck diagrams preserves the non-crossing nature of the partitions because any ``up\" and ``down\" arrows that are not adjacent will have other arrows within them that are strictly above the pair. \n\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:containment}\nFor each passive $b$ in $B\\cap B_0$, there is a path $(s \\rightarrow k)$ in the matching representation of $B$ containing $b$ (i.e. $s<b<k$).\n\\end{lemma}\n\n\\begin{proof}\nSuppose there is an element b in $B_0$ which is passive. We wish to claim that there is some interval $s\\to k$ represented by a path in the $LE$-diagram that contains b.\n    \nSince $b$ is passive in $B$, there is a $B_2$ which is the same as $B$ except $b$ has been replaced with something smaller. Since $b$ is in $B_0$, it must be on the right.\n    \nSince $b$ is not present in $B_2$, $b$ has to point to some element $k$ in the matching representation of $B_2$. (Note that this implies $b<k$.) Then, since the only change between $B$ and $B_2$ is that $b$ is replaced by some smaller element, this means that $k$ is also present in the $B$, which means that there is some element $s$ in $B_0$ which points to $k$, and hence $(s\\to k)$ is in the matching representation of $B$.\n\nNow, either $s$ is larger than $b$, or $s$ is smaller than $b$. If the former, this means that $s$ is not present in $B$, so $s$ is also not present in $B_2$. Then, $s$ has to point to some element $k_2$, so $k_2$ is present in $B_2$. Since $k_2>s>b$, $k_2$ thus is also present in $B$. Then, some element $s_2$ must point to $k_2$ in $B$, and thus $s_2$ must point to some $k_3$ in $B_2$, and so on. We get infinite recursion, and since we are only dealing with finite $LE$-diagrams, this cannot happen. Therefore, $s$ is smaller than $b$, and $b$ is contained in the path $(s\\to k)$ in the matching representation of $B$.\n    \n\\end{proof}\n\nThis leads us into our algorithm. We match every passive $b$ in a basis $B$ to a variable $x_i$. If $b$ is in $B\\setminus B_0$, then it will match to $x_b$. If $b$ is in $B\\cap B_0$, then by $\\ref{lem:containment}$ it is contained in at least one path $(s\\to k)$. Pick the smallest one, and match $b$ with the variable $x_k$. Using this algorithm, we convert every $B$ into a monomial.\n\n\\begin{proposition}\n\\label{lem:one-to-one-algom}\nWe claim that different matching representations will result in different monomials (in other words, that our algorithm is one-to-one).\n\\end{proposition}\n\n\\begin{proof}\nSuppose we are given a specific monomial $m$, which is obtained from a given matching representation. We wish to show that this matching representation is unique - in other words, that no other matching representation will give the same monomial. By \\ref{lem:one-to-one-matching} we know as long as we have the sets $B\\setminus B_0$ and $B_0\\setminus B$ (which must be equal in size) the pairings will be automatically determined. Hence, we will focus on proving that these sets are unique for a given monomial.\n\nAssume for the sake of contradiction that we have two bases $B_1\\neq B_2$ such that our algorithm gives the same monomial for both. Given $B_1$, there is one unique mapping (i.e matching representation) between $B_1$ $\\setminus$ $B_0$ and $B_0 \\setminus B_1$. The same thing is true for $B_2$. We write this mapping as $\\{(s_1 \\to k_1), (s_2 \\to k_1), \\dots\\}$ for $B_1$, where $k_i$ is an element in $B_1 \\setminus B_0$ and $s_i$ is the element in $B_0$ which maps to it, and $(t_i\\to k_i)$ for $B_2$. (Since they have the same monomial and the set of sinks $k_i$ define the variables in the monomial, we know $B_1$ and $B_2$ have the same $k_i$s.)\n\nWe know that (since $B_1 \\neq B_2$) there are some $k_i$ for which $s_i \\neq t_i$. Consider the set of all such $k_i$, and their corresponding $t_i$s. Then, there exists a specific $k_j$ for which the interval $(t_j \\to k_j)$ does not contain any other $t_i$ in the set. (i.e. there is a path that has changed from $B_1$ to $B_2$ which does not contain any other changed paths in $B_2$.) Proof: consider the $k_j$ for which $k_j - t_j$ is smallest and $k_j$ is in the set. Then, if $t_j \\to k_j$ contains an $t_i$, then the path $t_i\\to k_i$ would be contained inside $t_j \\to k_j$, and so $k_i - t_i$ would be smaller than $k_j - t_j$, a contradiction.\n\nSo, we know that the degree of the variable associated with $k_i$ in B1 is equal to the number of elements in  $B_1\\cap B_0$ within the interval $(s_i \\to k_i)$ but not contained in any other interval that can replace the path from $s_i$ to $k_i$ (plus 1 for $k_i$ itself).\n\nNow, we consider $k_j - s_j$ and $k_j - t_j$. We have two cases: either $k_j - t_j$ is larger than $k_j - s_j$, or $k_j - t_j$ is smaller than $k_j - s_j$. For the first case, this means that $k_j - t_j$ not only contains exactly all of the empowering elements of $k_j - s_j$ in $B_1$, but also at least 1 more. Therefore, the degree of $k_j$ in the monomial of $B_2$ will be higher than the degree of $k_j$ in $B_1$. Else, if $k_j - t_j$ is smaller, then it must contain at least 1 less. Hence, the monomial still changes. Therefore, distinct pairings result in distinct monomials.\n\\end{proof}\n\n\\begin{proposition}\n\\label{lem:multicomplex}\nFor any $\\LE$-diagram, our algorithm generates a set of monomials that is a multicomplex.\n\\end{proposition}\n\n\\begin{proof}\nSuppose you have a monomial $m$. Pick any variable $x_i$ in $m$ and consider its degree. If this degree is greater than 1, then this means that the variable corresponds to a matching $(s\\to i)$ which contains some number of passive elements of $B_0$. Then, find the first passive element of $B_0$ $t$ inside of this line. Replace the connection $(s\\to i)$ with $(t\\to i)$ This reduces the degree of $x_i$ by 1, since now there is one fewer passive element of $B_0$ within the connection. However, if $(s\\to i)$ was originally contained inside another matching, it is possible that this results in incrementing the degree of a different variable. In that case, keep applying this process to the outside connections until one is not incremented (which must occur since there is an outermost matching).\n\nIf this degree is equal to 1, then the only thing which contributes to the degree is just the element itself, so we simply delete the connection and replace $x_i$ with its $B_0$ counterpart $x_j$. If this ends up incrementing the degree of an outside degree, apply the same processes as before.\n\\end{proof}\n\n\\begin{proposition}\n\\label{lem:purity}\nFor any $LE$-diagram, our algorithm generates a set of monomials that is pure.\n\\end{proposition}\n\n\\begin{proof}\nWe look at the elements in $B_0$. For every basis, for each element $b$ in $B_0$, either $b$ is in the basis or $b$ is replaced by another element that is not in $B_0$ through a matching $(b\\to k)$. Consider an element $b$. If for every basis, $b$ is never passive and it is never replaced by an element not in $B_0$, then we can disregard $b.$\n\nAfter we have removed all elements corresponding to the above criteria, all maximal monomials will have degree equal to the size of $B_0$. (In other words, if a basis has an element which is not passive, it is possible to create another basis by replacing that not passive element with a passive element to get a ``larger\" monomial.) \n\nSuppose there is a basis $B$, and there is an element $s$ which is not passive. This element must be in $B_0$. Since we have deleted all non-passive elements that are never replaced by an element not in $B_0$, $b$ can be connected to an element $k$ not in $B_0$. Pick the largest such $k$. Then, we claim that if we connect $s$ to $k$, we get a larger monomial. There are a few cases here: if $k$ is not already connected to something, then we try $(b\\to k)$. If $k$ is already connected to an element in $B$, then replacing that matching with $(b\\to k)$ will make it larger (since if $b$ were already inside the existing matching, $b$ would be passive). If attempting to connect $b$ with $k$ would result in a collision, let the colliding line be $(a\\to j)$. Then, if $(a\\to j)$ starts before $(b \\to k)$, then this means that $b$ was passive because it could go to $j$, thus a contradiction. If $(a\\to j)$ starts after $(b\\to k)$, then we could take the path $(b\\to j)$, a contradiction since we assumed $k$ was the largest possible connection for $b$ and $j > k$. Hence we can always replace a nonpassive element with a passive element to get a ``larger\" monomial.\n\\end{proof}\n\n\n\n\\section{Example}\nBelow, we have the visual representation of all the bases of the matroid of this specific $LE$-Diagram. \n\nIn each basis, the white dots represent the elements present in the basis of the $LE$-diagram. (So for example, the $LE$-diagram in the top left represents the basis $\\{2,3,5\\}$.) The smallest basis, $B_0$, is $\\{1,2,4\\}$. The elements not in $B_0$, which in this case are $x_3$ and $x_5$, become the terms in the monomial. Hence, the empowerments are found by looking at the elements that are not in $B_0$ (hence, elements $3,5$) and elements that are in $B_0$ which are contained within a path. As a specific example, if we look at the top left $LE$-diagram, both elements $3$ and $5$ are present, which contributes $x_3x_5$, and the element $2$ is contained in the path ending with $3$, empowering $x_3$. Hence, we get $x_3^2x_5$.\n\nUsing the algorithm, we get the monomial for each basis and arrange them as shown in figure $\\ref{fig:allthebases}$. As we can see, the $LE$-Diagrams represent a multicomplex, as we have the maximal monomials at the top and the monomials that divide the maximal monomials underneath. Note that this is pure, for the maximal monomials all have the same degree of 3. The monomial $1$ represents the case in which the elements of the basis are exactly the same as the elements of $B_0.$ \n\n\n\n\n\n\\section{Introduction}\nA matroid is a structure which generalizes the notion of independence that rises in linear algebra, graph theory and other areas. From a matroid, we can construct a simplicial complex (\\newword{matroid complex}) and study its \\newword{h-vector} which provides the topological information of the complex. The following conjecture by Stanley is still wide open:\n\n\\begin{conjecture}[\\cite{Stanley1977}]\n\\label{con:stanley}\nThe h-vector of  a matroid is a pure O-sequence.\n\\end{conjecture}\n\n\nThere are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in \\cite{Merino2001}, lattice-path matroids by Schweig in \\cite{Schweig2010}, cotransversal matroids by Oh in \\cite{Oh2013}, paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores  \\cite{MNRV2012}, internally perfect matroids by Dall in \\cite{Dal}, rank $3$ matroids by H\\'a, Stokes, and Zanello in \\cite{HaStokesZanello2013}, rank $3$ and corank $2$ matroids by DeLoera, Kemper, and Klee in \\cite{DelKemKle}, rank $4$ matroids by Klee and Samper in \\cite{KleeSamper2015}, rank $d$ matroids with $h_d \\leq 5$  by Constantinescu, Kahle, and Varbaro in \\cite{ConKahVar} and some small special classes of graphic matroids by Kook \\cite{Kook2011}, Preston et al \\cite{biconed} and David et al \\cite{triconed}.\n\nIn this paper the focus will solely be on the class of positroids, which is a special class of realizable matroids, for which the conjecture is still wide open. Positroids come from the study of the nonnegative part of the Grassmannian \\cite{Postnikov}, and have seen increased applications in physics, with use in the study of scattering amplitudes \\cite{arkani} and the study of shallow water waves \\cite{kodama}.\n\n\n\\section{Preliminary}\n\nThroughout the paper, we will use the following notation. When $A$ is a set and $a$ is a single element, instead of writing $A \\setminus \\{a\\}$ we will instead write $A \\setminus a$. Similarly, instead of writing $A \\cup \\{a\\}$ we will instead use $A \\cup a$.\n\n\\subsection{Positroids} \n\n\\begin{figure}\n\\begin{multicols}{2}\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lediagram}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{A $\\LE$-diagram.}\n  \\label{fig:lediagram}\n\n \\end{center}\n\n\\vfill\\null\n\\columnbreak\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lenetwork}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{The corresponding $\\LE$-network.}\n  \\label{fig:lenetwork}\n\n \\end{center}\n\\end{multicols}\n\\end{figure}\n\nWe will start with the definition of $\\LE$-diagrams and $\\LE$-graphs and use that to define positroids. A $\\LE$-diagram will be a Young diagram with dots filled in some boxes. Consider the southeastern boundary path of the diagram. We call this the \\newword{boundary path} of the diagram. Starting from the northeast corner and heading towards the southwest corner, label each edge of the boundary path with integers $1,\\ldots,n$. We call these the \\newword{boundary labels}. We index each column and row of a diagram with the label of the unique boundary label it contains. A box at $(i,j)$ stands for the box of $L$ at row indexed with $i$ and column indexed with $j$. For example, a box of $L$ in Figure~\\ref{fig:lediagram} at $(2,5)$ is the box at the second row from top (indexed with $2$) and first column from left (indexed with $5$). We write $L_{i,j} = 1$ if there the box at $(i,j)$ contains a dot, and $L_{i,j} = 0$ otherwise.\n\n\\begin{definition}[\\cite{Postnikov}, Definition 6.1] For a partition $\\lambda$, let us define a $\\LE$-diagram $L$ of shape $\\lambda$ as a filling of boxes of the Young diagram of shape $\\lambda$ such that, for any three boxes at $(i,j),(i',j),(i,j')$, where $i'<i$ and $j'>j$, if $L_{i',j} = L_{i,j'} = 1$ then $L_{i,j} = 1$. This property is called the $\\LE$-property.\n\n\\end{definition}\n\nFrom each $\\LE$-diagram $L$, we can get a planar network called the $\\LE$-graph in Definition 6.3 of \\cite{Postnikov}, via the following way: place a dot in the middle of each step in the boundary lattice path of the diagram and mark these dots with the corresponding boundary labels (so $1,\\ldots,n$ from top to bottom). We will call these vertices the \\newword{boundary vertices}. For each dot inside the $\\LE$-diagram, draw a horizontal line to its right, and vertical line to its bottom until it reaches another dot. Next orient all vertical edges downward horizontal edges to the right.  So the vertical labels will be the source vertices and the horizontal labels will be the sink vertices in the $\\LE$-graph. Since each $\\LE$-diagram encodes a unique $\\LE$-graph, we will abuse notation and go between a $\\LE$-diagram and its corresponding $\\LE$-graph freely in this paper. An example of a $\\LE$-graph obtained from the $\\LE$-diagram in Figure~\\ref{fig:lediagram} is drawn in Figure~\\ref{fig:lenetwork}.\n\nThe set $[n]$ (where $n$ is the length of the boundary path) is called the \\newword{ground set} (being consistent with the notation for the positroid that will be indexed by the $\\LE$-diagram). Let $B_0$ denote the set of source vertices in the $\\LE$-graph of $L$. For example, in the $\\LE$-diagram $L$ drawn in Figure~\\ref{fig:lediagram}, we have $B_0 = 124$ as we can see from the corresponding $\\LE$-graph in Figure~\\ref{fig:lenetwork}. A \\newword{path} in a $\\LE$-graph is a directed path that starts at some boundary vertex and ends at some boundary vertex. A \\newword{family of non-touching paths} is a family of paths where no two paths intersects within the graph. We say that a family of non-touching paths consisting of paths $p_1,\\ldots,p_k$ \\newword{represents} a set $B_0 \\setminus \\{p_1^s,\\ldots,p_k^s\\} \\cup \\{p_1^e,\\ldots,p_k^e\\}$ where $p_i^s$ stands for the starting point and $p_i^e$ stands for the endpoint of the path $p_i$. For example, the bottom picture in Figure~\\ref{fig:allthebases} shows an empty family, which represents the set $124$. The leftmost picture at the top row in Figure~\\ref{fig:allthebases} shows a family consisting of two paths, one starting at $1$ and ending at $3$, the other one starting at $4$ and ending at $5$: this represents the set $124 \\setminus 14 \\cup 35 = 235$. Notice that any set represented by a family of non-touching paths in a $\\LE$-graph will have the same cardinality as $B_0$.\n\n\nFrom Theorem~6.5 of \\cite{Postnikov}, there is a bijection between positroids and $\\LE$-diagrams. Moreover, the same theorem gives the following result, which we will use as the definition of a positroid in this paper:\n\n\\begin{definition}\n\\label{def:pos}\nFix a positroid $\\mathcal{M}_L$ that corresponds to a $\\LE$-diagram $L$. Then the set of bases of $\\mathcal{M}_L$ is exactly the collection of sets represented by a family of non-touching paths in the $\\LE$-graph of $L$.\n\\end{definition}\n\nFrom the above definition, the set $B_0$ is naturally the lexicographic minimal basis of $\\mathcal{M}_L$. We call this the \\newword{canonical basis} of the positroid. For example, take a look at the $\\LE$-diagram in Figure~\\ref{fig:lediagram}. All the bases we can get are:\n$$124,125,134,135,145,234,235,245,345,$$ \nas one can check from various families of disjoint paths drawn on the $\\LE$-graph as in Figure~\\ref{fig:allthebases}. \n\nAn element $e$ of the ground set is called a \\newword{loop} if it is not contained in any basis and is called a \\newword{coloop} if it is contained in every basis.\n\n\n\n\\subsection{Activity and the h-vector} \nFrom any ordering on the ground set $E$, we get an induced lexicographic ordering on the bases. An element $i$ of a basis $B$ is \\newword{(internally) active} if $B \\setminus i \\cup j$ is not a base for any $j < i$. Otherwise, it is called \\newword{(internally) passive}. There is also the notion of externally active and externally passive elements, but in this paper we will only stick to interally active and passive elements, and hence will skip the usage of the word internally throughout. Given a basis $B$ of a matroid, we call its \\newword{passivity} as the number of (internally) passive elements it contains.\n\nWe will use this activity property of h-vectors as the definition of h-vectors of matroids in this paper.\n\n\\begin{theorem}\nLet $(h_0,\\dots,h_r)$ be the $h$-vector of a matroid $\\mathcal{M}$. For $0 \\leq i \\leq r$, the entry $h_i$ is the number of bases of $\\mathcal{M}$ with $i$ passive elements.\n\\end{theorem}\n\nLet us again look at the running example of Figure~\\ref{fig:allthebases}. Take a look at $135$. Here $1$ is active since it cannot be replaced with a smaller element to get a basis of $\\mathcal{M}$. On the other hand, $3$ and $5$ are passive since we can replace either of them with $2$ to get a different basis of $\\mathcal{M}$. The $h$-vector of the corresponding positroid is $(1,2,3,3)$.\n\n\\begin{remark}\n\\label{rem:noloopcoloop}\nDeleting a loop (simply remove that element from the ground set) or contracting by a coloop (remove that element from the ground set and all the bases) does not change the $h$-vector of the matroid. For sake of proving Stanley's conjecture, we may assume that our matroid does not have any loops or coloops. Hence from now on, throughout the entire paper, we are only going to deal with positroids with no loops nor coloops. In terms of the $\\LE$-diagram, it means there is no empty row nor column. \n\\end{remark}\n\n\n\n\\subsection{Order ideal and pure O-sequences}\nAn \\newword{order ideal} is a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. Given an order ideal its \\newword{O-sequence} is a sequence that can be obtained from counting monomials of each degree. We say that the order ideal is \\newword{pure} if all the maximal monomials are of the same degree, and also call the O-sequence pure if it comes from a pure order ideal.\n\nFor example, take a look at Figure~\\ref{fig:allthebases} and focus only on the monomials. If we take any monomial among this collection, say $x_3x_5^2$, we can see that all divisors of it is again present in the collection. Hence this is an order ideal. Moreover, since the maximal monomials under division (which are usually not guaranteed to all have the same degree) here are $x_3^2x_5,x_3x_5^2,x_5^3$ which have the same degree, and hence this is a pure order ideal. Notice that the degree sequence of this pure order ideal is $(1,2,3,3)$. Hence we can see that the $h$-vector of the positroid coming from Figure~\\ref{fig:lediagram}, which was $(1,2,3,3)$, is indeed a pure $O$-sequence. We have just verified that Stanley's conjecture is true for this case. What we want to do in this paper is to similarly construct a pure order ideal for each and every positroid, hence proving the conjecture for this class.\n\n\n\\section{The main result}\n\nOur goal this section is to come up with an algorithm that maps a family of non-crossing paths to a monomial, using the elements of $[n] \\setminus B_0$ as the variables. Then we will show that the image under this map is a pure order ideal to answer Stanley's conjecture for positroids.\n\nGiven a path from a source $s$ to sink $k$, we write it as $(s \\rightarrow k)$. Since each basis $B \\in \\mathcal{M}$ is coming from a family of non-crossing paths from set of sources to set of sinks, we write $B$ as a collection of $(s \\rightarrow k)$'s and call this the \\newword{non-crossing matching representation} (or simply a matching representation) of $B \\in \\mathcal{M}$. Note that all sources $s$ are elements of $B_0 \\setminus B$, and all sinks $k$ are elements of $B_0 \\setminus B$. \n\nWe will slightly abuse notation and go between the path $(s \\rightarrow k)$ and the corresponding interval $(s,k)$ freely. So for example, the path $(2 \\rightarrow 5)$ \\newword{contains} $3$ since $3 \\in (2,5)$. Given a matching representation $R$, we put a partial ordering $\\prec_R$ on elements of $B_0 \\setminus B$ (and also separately on elements of $B \\setminus B_0)$ by setting $s'  \\prec  s$ ($k' \\prec k$ respectively) if $(s \\rightarrow k),(s' \\rightarrow k') \\in R_1$ and $s < s' < k' < k$: i.e. if the path coming out of $s'$ is nested inside (as intervals on $[n]$) the path coming out of $s$. For example, in the middle picture of the top row of Figure~\\ref{fig:allthebases}, we have $2 \\prec_R 1$ (also $3 \\prec 5)$ since comparing the paths $2 \\rightarrow 3$ and $1 \\rightarrow 5$, the interval $(2,3)$ is nested inside the interval $(1,5)$. We put an ordering on the paths so it is compatible with the ordering we just discussed on the starting point: for example we have $(2,3) \\prec (1,5)$. \n\n\\begin{lemma}\n\\label{lem:uniquerep}\nFix a positroid $\\mathcal{M}_L$. For each basis $B \\in \\mathcal{M}_L$, there exists a unique matching representation of $B$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Definition~\\ref{def:pos}, there exists at least one matching representation of each $B \\in M_L$. For sake of contradication, assume that for some $B \\in M_L$ we have two different matching representations $R_1$ and $R_2$. Consider the partial order $\\prec_{R_1}$ on $B_0 \\setminus B$.  Under this ordering, pick any minimal element $s$ of $B_0 \\setminus B$ such that $(s \\rightarrow k) \\in R_1$ and $(s \\rightarrow k') \\in R_2$ with $k \\not = k'$. If $k < k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k) \\in R_2$. Since $R_2$ is non-crossing we must have $s < s'$. But then from the way $s$ was chosen, we need to have $(s' \\rightarrow k) \\in R_1$ as well and we get a contradiction. If $k > k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k') \\in R_1$. Since $R_1$ is non-crossing we must have $s < s' < k' < k$. But then from the way $s$ was chosen, we must have $(s' \\rightarrow k') \\in R_2$ and we again get a contradiction.\n\\end{proof}\n\nThanks to Lemma~\\ref{lem:uniquerep}, when considering the partial order $\\prec_R$ coming from a matching representation of some basis $B \\in \\mathcal{M}_L$, we may instead write it as $\\prec_B$. Using this, we slightly abuse notation and say $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is a path inside the unique matching representation of $B$. Given some fixed basis $B \\in \\mathcal{M}_L$, we say that an element $x \\in [n]$ is \\newword{associated under $B$} to a path $(s \\rightarrow k) \\in B$ if $(s \\rightarrow q)$ is the unique minimal path containing $x$ under the ordering $\\prec_B$. Next we show that each passive element $b$ in a basis of $B$ is closely related to the matching representation of $B$.\n\n\\begin{lemma}\n\\label{lem:containment}\nFix a positroid $\\mathcal{M}_L$ and a basis $B \\in \\mathcal{M}_L$. For each passive $b \\in B\\cap B_0$, there is a path $(s \\rightarrow k) \\in B$ containing $b$ (i.e. $s < b < k$).\n\\end{lemma}\n\n\\begin{proof}\nPick $b \\in B \\cap B_0$ that is passive. Assume for sake of contradiction there is no path $(s \\rightarrow k) \\in B$ such that $s < b < k$. From $b$ being passive, there is some $B' \\in \\mathcal{M}_L$ such that $B' = B \\setminus b \\cup b'$ with $b' < b$ and $b' \\not \\in B$. We have $(b \\rightarrow k) \\in B'$ for some $k \\in B' \\setminus B_0$. Then there is some $c$ such that $(c \\rightarrow k) \\in B$. From the assumption we have $c > b$. \n\nUnder the ordering $\\prec_{B'}$, pick any minimal element $x$ such that $(x \\rightarrow y) \\in B'$ and $(x' \\rightarrow y) \\in B$ with $x' > x \\geq b$. There exists such element thanks to the argument in the previous paragraph. Since $(b,n] \\cap B = (b,n] \\cap B'$, we have $x' \\not \\in B'$. So there is some $y'$ such that $(x' \\rightarrow y') \\in B'$, with $x <x'<y'<y$. Then, this implies $y'\\in B$, so there exists some $x''$ such that $(x'' \\rightarrow y')\\in B'$. Since $x'' \\prec_{B'} x$, we get a contradiction with $x$ being chosen as a minimal element.\n\n\n\\end{proof}\n\n\nWe are now ready to present the map $\\phi$ which will be mapping each basis of a positroid into a monomial supported on variables indexed by $[n] \\setminus B_0$. Given some basis $B \\in M_L$, for each element $b \\in B \\setminus B_0$, we simply map it to the variable $x_b$. For a passive element $b \\in B \\cap B_0$, we look at the unique minimum (under $\\prec_B$) path $s \\rightarrow k$ that contains $b$ (recall that we called this $b$ being associated to $s \\rightarrow k$), then map $b$ to $x_k$. Then take the product of all such variables to obtain a monomial, which is going to be supported on variables indexed by $[n] \\setminus B_0$ as we wanted. \n\nFor example, take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. This family of paths encodes the basis $235$. The passive elements of this basis is actually all of $2,3,5$. Since $3$ and $5$ are sinks of paths, $3$ gives $x_3$ and $5$ gives $x_5$. Now the minimum (under $\\prec_{235}$) path containing $2$ is $(1 \\rightarrow 5)$ so $2$ gives $x_3$. The resulting monomial is $x_3^2 x_5$.\n\n\\begin{proposition}\n\\label{prop:phi}\nLet $\\mathcal{M}_L$ be a positroid. Given a basis $B$, we have $\\phi(B) = x_{j_1}^{|T_1^B|+1} \\dots x_{j_k}^{|T_k^B|+1}$ where $B$ has the matching representation $(i_1 \\rightarrow j_1),\\ldots,(i_k \\rightarrow j_k)$ with $i_1 < \\dots < i_k$. The set $T_q^B$ consists of elements $x$ associated under $B$ to $(i_q \\rightarrow j_q)$ such that we can replace $(i_q \\rightarrow j_q)$ with $(x \\rightarrow j_q)$ in the matching representation of $B$ and still get a matching representation of some other basis in $\\mathcal{M}_L$.\n\\end{proposition}\n\\begin{proof}\nFrom the way $\\phi$ is defined, it is enough to show that $T_q^B$ consists exactly of the passive elements of $B \\cap B_0$ associated to the path $(i_q \\rightarrow j_q)$ of the matching representation of $B$. It is clear that if we can replace $(i_q \\rightarrow j_q)$ with $(x \\rightarrow j_q)$ then $x$ is passive. Hence we only need to show the other direction: if $x$ is passive then we can replace $(i_q \\rightarrow j_q)$ with $(x \\rightarrow j_q)$. \n\n\nSince $x$ is passive, there is some $x' < x$ such that $B' = B \\setminus x \\cup x' \\in \\mathcal{M}_L$. Consider the matching representation of $B$. Since we have a path $(i_q \\rightarrow j_q)$, we have equal number of source points and sink points within $(i_q,j_q)$. Moreover, since $x$ has $(i_q \\rightarrow j_q)$ as the minimal (under $\\prec$) path containing it, we have equal number of source points and sink points within $(i_q,x)$ and $(x,j_q)$. Now in the matching representation of $B'$, we have $x$ as a source, $j_q$ as a sink and equal number of sources and sinks in $(x,j_q)$. This leads to us having $(x \\rightarrow j_q)$ in $B'$.\n\n\n\n\\end{proof}\n\nThe Proposition~\\ref{prop:phi} tells us that it passes the passive degree to the degree of the monomial:\n\n\\begin{corollary}\nLet $\\mathcal{M}_L$ be positroid. Then the internally passive degree of $B$ is equal to the degree of $\\phi(B)$.\n\\end{corollary}\n\nFor example, take a look at the leftmost picture of second row from the top within Figure~\\ref{fig:allthebases}. We have the path $1 \\rightarrow 3$ so the corresponding basis is $234$. Here $2$ and $3$ are passive, but $4$ is active. So $234$ has passive degree to be $2$. The corresponding monomial is $x_3^2$, which again has degree of $2$.\n\n\n\\begin{lemma}\nThe map $\\phi$ is a one-to-one map.\n\\end{lemma}\n\n\n\n\\begin{proof}\nAssume for sake of contradiction that we can find some $B, B'$, two different bases of the positroid $\\mathcal{M}_L$ such that $\\phi(B) = \\phi(B')$. In $B$, under the ordering $\\prec_B$, pick the minimal path $(s \\rightarrow q)$ that isn't contained in $B'$. Since $\\phi(B) = \\phi(B')$, there has to be some path $(s' \\rightarrow q)$ with $s \\not = s'$. Without loss of generality, assume $s < s'$. Then since $s \\rightarrow q$ is replaceable with $s' \\rightarrow q$, we have $T_q^{B'} \\subsetneq T_q^{B}$ from Proposition~\\ref{prop:phi} and we get a contradiction.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:poi}\nThe map $\\phi$ gives a pure order ideal.\n\\end{proposition}\n\\begin{proof}\nWe first start with the order ideal property. We need to show that given any basis $B$, for each variable $x_q$ appearing in $\\phi(B)$, we can find a basis $B'$ such that $\\phi(B') = \\phi(B)\/x_q$. To use induction, we introduce a new statistic on $x_q$ within the poset $\\prec_B$ on $B \\setminus B_0$: we set $\\sigma(x)$ to stand for the length of the longest possible chain having $x$ as the lower end within the poset. \nWe are going to use induction on $\\sigma(x_q)$. We start with the base case, when $\\sigma(x_q) = 1$. The path $(s \\rightarrow q)$ in $B$ is a path that is not contained in any other path. If $|T_q^B| = 1$, we simply remove the path to get $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_q^B|$ to get our desired $B'$.\n\nNow assume for sake of induction we have proven the result for $\\sigma(x_q) < k$ and are now considering the case $\\sigma(x_q) = k$. Same as above, if $|T_q^B| = 1$, we simply remove the path to get $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_q^B|$ to get some other basis $B''$. Then $\\phi(B'')$ is obtained from $\\phi(B)$ by dividing by $x_q$, but the powers of other variables $x_{q'}$ might have gone up when the associated path $(s \\rightarrow q) \\prec_B (s' \\rightarrow q')$. Regardless, in such cases we have $\\sigma(x_{q'}) < k$, so we can find a new basis by shedding them off. This gives us the $B'$ we were looking for.\n\nNow for the pure part, fix a basis $B$. We know that every element of $B \\setminus B_0$ is automatically passive since we can simply undo the corresponding path to get a smaller basis. So potential active elements only occur within $B \\cap B_0$. We want to show that if there is some active element, we can always find some basis $B'$ such that $\\phi(B')$ is divisible by $\\phi(B)$. Let $a$ be the largest element of $B$ that is active. We do a case by case analysis based on whether $a+1$ is in $B_0$.\n\nWhen $a+1 \\not \\in B_0$: since there is no coloop in the positroid, we have a dot at the corner box with boundary vertices $a$ and $a+1$ in the $\\LE$-diagram $L$. If $a+1 \\not \\in B$, We simply add the path $a \\rightarrow a+1$ to get $B'$ we want. In particular, we have $\\phi(B') = \\phi(B) x_{a+1}$. If $a+1 \\in B$, then we have $s \\rightarrow a+1$ in $B$ which is replaceable with $a \\rightarrow a+1$ to get $B'$. This means $a$ is passive so we won't get this case.\n\n\n    \nWhen $a+1 \\in B_0$: first assume for sake of contradiction that $a+1 \\in B$. Then it is passive thanks to the way $a$ was chosen. This means we have $s \\rightarrow q$ in the matching representation $B$ that $a+1$ is associated with, such that $s \\rightarrow q$ is replaceable with $a+1 \\rightarrow q$. From the $\\LE$-property, the path $s \\rightarrow q$ is replaceable with $a \\rightarrow q$ as well, and we get a contradiction from Proposition~\\ref{prop:phi}. \n    \nHence we have that $a+1 \\in B_0 \\setminus B$. In the matching representation of $B$ we have some $a+1 \\rightarrow q$. Let $q'$ be the minimal element such that $a \\rightarrow q'$ is possible inside the $\\LE$-graph. If $q' \\leq q$, from the $\\LE$-property we can switch $a+1 \\rightarrow q$ to $a \\rightarrow q$ to get the $B'$ we desire (we have $\\phi(B') = \\phi(B) x_{a+1}$). In the case $q' > q$, from the way $q'$ was chosen and the $\\LE$-property, there are no dots in the quadrant weakly above the row indexed by $a$ and weakly to the right of column indexed by $q'$ in $L$. If the dot at row indexed by $a$ and column indexed by $q'$ is already occupied in the family of matchings representing $B$, we can replace the starting point of this path with $a$. This would contradict $a$ being active, so we may assume that the dot mentioned before is not occupied. If there is some path $s \\rightarrow q'$ with $s' > a$, then replace $s$ with $a$ to get the desired $B'$. Otherwise we can simply add $a \\rightarrow q'$ to get the $B'$ we desire (we have $\\phi(B') = \\phi(B) x_{q'}$).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n\n    \n\n\n\n\\end{proof}\n\n\nThanks to the above proposition, we have established Stanley's conjecture for the positroid case:\n\n\\begin{corollary}\nThe $h$-vector of any positroid is a pure $O$-sequence.\n\\end{corollary}\n\nAn example is shown in Figure~\\ref{fig:allthebases}. Using the $\\LE$-diagram from Figure~\\ref{fig:lediagram}, all the bases and the corresponding family of non-crossing paths are drawn in Figure~\\ref{fig:allthebases}. The monomials from the mapping $\\phi$ are written below each picture. Proposition~\\ref{prop:poi} guarantees that the collection of monomials we have obtained is indeed a pure order ideal.\n\n\n\\section*{Acknowledgements}\nThe research started under the $2021$-High school math camp hosted at Texas State University. The authors would like to thank the camp organizers for providing support and a great working environment.\n\n\n\n\n\\begin{figure}[h]\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/basis\/allthebases}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{Hasse diagram of the $\\LE$-Diagrams which form the bases of the matroid. Each $\\LE$-Diagram is labelled with its corresponding monomial.}\n  \\label{fig:allthebases}\n\n\\end{center}\n\\end{figure}\n\n\n\\section{Introduction}\nA matroid is a structure which generalizes the notion of independence that rises in linear algebra, graph theory and other areas. From a matroid, we can construct a simplicial complex (\\newword{matroid complex}) and study its \\newword{h-vector} which provides the topological information of the complex. The following conjecture by Stanley is still wide open:\n\n\\begin{conjecture}[\\cite{Stanley1977}]\n\\label{con:stanley}\nThe h-vector of  a matroid is a pure O-sequence.\n\\end{conjecture}\n\n\nThere are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in \\cite{Merino2001}, lattice-path matroids by Schweig in \\cite{Schweig2010}, cotransversal matroids by Oh in \\cite{Oh2013}, paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores  \\cite{MNRV2012}, internally perfect matroids by Dall in \\cite{Dal}, rank $3$ matroids by H\\'a, Stokes, and Zanello in \\cite{HaStokesZanello2013}, rank $3$ and corank $2$ matroids by DeLoera, Kemper, and Klee in \\cite{DelKemKle}, rank $4$ matroids by Klee and Samper in \\cite{KleeSamper2015}, rank $d$ matroids with $h_d \\leq 5$  by Constantinescu, Kahle, and Varbaro in \\cite{ConKahVar} and some small special classes of graphic matroids by Kook \\cite{Kook2011}, Preston et al \\cite{biconed} and David et al \\cite{triconed}.\n\nIn this paper the focus will solely be on the class of positroids, which is a special class of realizable matroids, for which the conjecture is still wide open. Positroids come from the study of the nonnegative part of the Grassmannian \\cite{Postnikov}, and have seen increased applications in physics, with use in the study of scattering amplitudes \\cite{arkani} and the study of shallow water waves \\cite{kodama}.\n\n\n\\section{Preliminary}\n\nThroughout the paper, we will use the following notation. When $A$ is a set and $a$ is a single element, instead of writing $A \\setminus \\{a\\}$ we will instead write $A \\setminus a$. Similarly, instead of writing $A \\cup \\{a\\}$ we will instead use $A \\cup a$.\n\n\\subsection{Positroids} \n\n\\begin{figure}\n\\begin{multicols}{2}\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lediagram}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{A $\\LE$-diagram.}\n  \\label{fig:lediagram}\n\n \\end{center}\n\n\\vfill\\null\n\\columnbreak\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lenetwork}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{The corresponding $\\LE$-network.}\n  \\label{fig:lenetwork}\n\n \\end{center}\n\\end{multicols}\n\\end{figure}\n\nWe will start with the definition of $\\LE$-diagrams and $\\LE$-graphs and use that to define positroids. A $\\LE$-diagram will be a Young diagram with dots filled in some boxes. Consider the southeastern boundary path of the diagram. We call this the \\newword{boundary path} of the diagram. Starting from the northeast corner and heading towards the southwest corner, label each edge of the boundary path with integers $1,\\ldots,n$. We call these the \\newword{boundary labels}. We index each column and row of a diagram with the label of the unique boundary label it contains. A box at $(i,j)$ stands for the box of $L$ at row indexed with $i$ and column indexed with $j$. For example, a box of $L$ in Figure~\\ref{fig:lediagram} at $(2,5)$ is the box at the second row from top (indexed with $2$) and first column from left (indexed with $5$). We write $L_{i,j} = 1$ if there the box at $(i,j)$ contains a dot, and $L_{i,j} = 0$ otherwise.\n\n\\begin{definition}[\\cite{Postnikov}, Definition 6.1] For a partition $\\lambda$, let us define a $\\LE$-diagram $L$ of shape $\\lambda$ as a filling of boxes of the Young diagram of shape $\\lambda$ such that, for any three boxes at $(i,j),(i',j),(i,j')$, where $i'<i$ and $j'>j$, if $L_{i',j} = L_{i,j'} = 1$ then $L_{i,j} = 1$. This property is called the $\\LE$-property.\n\n\\end{definition}\n\nFrom each $\\LE$-diagram $L$, we can get a planar network called the $\\LE$-graph in Definition 6.3 of \\cite{Postnikov}, via the following way: place a dot in the middle of each step in the boundary lattice path of the diagram and mark these dots with the corresponding boundary labels (so $1,\\ldots,n$ from top to bottom). We will call these vertices the \\newword{boundary vertices}. For each dot inside the $\\LE$-diagram, draw a horizontal line to its right, and vertical line to its bottom until it reaches another dot. Next orient all vertical edges downward horizontal edges to the right.  So the vertical labels will be the source vertices and the horizontal labels will be the sink vertices in the $\\LE$-graph. Since each $\\LE$-diagram encodes a unique $\\LE$-graph, we will abuse notation and go between a $\\LE$-diagram and its corresponding $\\LE$-graph freely in this paper. An example of a $\\LE$-graph obtained from the $\\LE$-diagram in Figure~\\ref{fig:lediagram} is drawn in Figure~\\ref{fig:lenetwork}.\n\nThe set $[n]$ (where $n$ is the length of the boundary path) is called the \\newword{ground set} (being consistent with the notation for the positroid that will be indexed by the $\\LE$-diagram). Let $B_0$ denote the set of source vertices in the $\\LE$-graph of $L$. For example, in the $\\LE$-diagram $L$ drawn in Figure~\\ref{fig:lediagram}, we have $B_0 = 124$ as we can see from the corresponding $\\LE$-graph in Figure~\\ref{fig:lenetwork}. A \\newword{path} in a $\\LE$-graph is a directed path that starts at some boundary vertex and ends at some boundary vertex. A \\newword{family of non-touching paths} is a family of paths where no two paths intersects within the graph. We say that a family of non-touching paths consisting of paths $p_1,\\ldots,p_k$ \\newword{represents} a set $B_0 \\setminus \\{p_1^s,\\ldots,p_k^s\\} \\cup \\{p_1^e,\\ldots,p_k^e\\}$ where $p_i^s$ stands for the starting point and $p_i^e$ stands for the endpoint of the path $p_i$. For example, the bottom picture in Figure~\\ref{fig:allthebases} shows an empty family, which represents the set $124$. The leftmost picture at the top row in Figure~\\ref{fig:allthebases} shows a family consisting of two paths, one starting at $1$ and ending at $3$, the other one starting at $4$ and ending at $5$: this represents the set $124 \\setminus 14 \\cup 35 = 235$. Notice that any set represented by a family of non-touching paths in a $\\LE$-graph will have the same cardinality as $B_0$.\n\n\nFrom Theorem~6.5 of \\cite{Postnikov}, there is a bijection between positroids and $\\LE$-diagrams. Moreover, the same theorem gives the following result, which we will use as the definition of a positroid in this paper:\n\n\\begin{definition}\n\\label{def:pos}\nFix a positroid $\\mathcal{M}_L$ that corresponds to a $\\LE$-diagram $L$. Then the set of bases of $\\mathcal{M}_L$ is exactly the collection of sets represented by a family of non-touching paths in the $\\LE$-graph of $L$.\n\\end{definition}\n\nFrom the above definition, the set $B_0$ is naturally the lexicographic minimal basis of $\\mathcal{M}_L$. We call this the \\newword{canonical basis} of the positroid. For example, take a look at the $\\LE$-diagram in Figure~\\ref{fig:lediagram}. All the bases we can get are:\n$$124,125,134,135,145,234,235,245,345,$$ \nas one can check from various families of disjoint paths drawn on the $\\LE$-graph as in Figure~\\ref{fig:allthebases}. \n\nAn element $e$ of the ground set is called a \\newword{loop} if it is not contained in any basis and is called a \\newword{coloop} if it is contained in every basis.\n\n\n\n\\subsection{Activity and the h-vector} \nFrom any ordering on the ground set $E$, we get an induced lexicographic ordering on the bases. An element $i$ of a basis $B$ is \\newword{(internally) active} if $B \\setminus i \\cup j$ is not a base for any $j < i$. Otherwise, it is called \\newword{(internally) passive}. There is also the notion of externally active and externally passive elements, but in this paper we will only stick to interally active and passive elements, and hence will skip the usage of the word internally throughout. Given a basis $B$ of a matroid, we call its \\newword{passivity} as the number of (internally) passive elements it contains.\n\nWe will use this activity property of h-vectors as the definition of h-vectors of matroids in this paper.\n\n\\begin{theorem}\nLet $(h_0,\\dots,h_r)$ be the $h$-vector of a matroid $\\mathcal{M}$. For $0 \\leq i \\leq r$, the entry $h_i$ is the number of bases of $\\mathcal{M}$ with $i$ passive elements.\n\\end{theorem}\n\nLet us again look at the running example of Figure~\\ref{fig:allthebases}. Take a look at $135$. Here $1$ is active since it cannot be replaced with a smaller element to get a basis of $\\mathcal{M}$. On the other hand, $3$ and $5$ are passive since we can replace either of them with $2$ to get a different basis of $\\mathcal{M}$. The $h$-vector of the corresponding positroid is $(1,2,3,3)$.\n\n\\begin{remark}\n\\label{rem:noloopcoloop}\nDeleting a loop (simply remove that element from the ground set) or contracting by a coloop (remove that element from the ground set and all the bases) does not change the $h$-vector of the matroid. For sake of proving Stanley's conjecture, we may assume that our matroid does not have any loops or coloops. Hence from now on, throughout the entire paper, we are only going to deal with positroids with no loops nor coloops. In terms of the $\\LE$-diagram, it means there is no empty row nor column. \n\\end{remark}\n\n\n\n\\subsection{Order ideal and pure O-sequences}\nAn \\newword{order ideal} is a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. Given an order ideal its \\newword{O-sequence} is a sequence that can be obtained from counting monomials of each degree. We say that the order ideal is \\newword{pure} if all the maximal monomials are of the same degree, and also call the O-sequence pure if it comes from a pure order ideal.\n\nFor example, take a look at Figure~\\ref{fig:allthebases} and focus only on the monomials. If we take any monomial among this collection, say $x_3x_5^2$, we can see that all divisors of it is again present in the collection. Hence this is an order ideal. Moreover, since the maximal monomials under division (which are usually not guaranteed to all have the same degree) here are $x_3^2x_5,x_3x_5^2,x_5^3$ which have the same degree, and hence this is a pure order ideal. Notice that the degree sequence of this pure order ideal is $(1,2,3,3)$. Hence we can see that the $h$-vector of the positroid coming from Figure~\\ref{fig:lediagram}, which was $(1,2,3,3)$, is indeed a pure $O$-sequence. We have just verified that Stanley's conjecture is true for this case. What we want to do in this paper is to similarly construct a pure order ideal for each and every positroid, hence proving the conjecture for this class.\n\n\n\\section{The main result}\n\nOur goal this section is to come up with an algorithm that maps a family of non-crossing paths to a monomial, using the elements of $[n] \\setminus B_0$ as the variables. Then we will show that the image under this map is a pure order ideal to answer Stanley's conjecture for positroids.\n\nGiven a path from a source $s$ to sink $k$, we write it as $(s \\rightarrow k)$. Since each basis $B \\in \\mathcal{M}$ is coming from a family of non-crossing paths from set of sources to set of sinks, we write $B$ as a collection of $(s \\rightarrow k)$'s and call this the \\newword{non-crossing matching representation} (or simply a matching representation) of $B \\in \\mathcal{M}$. Note that all sources $s$ are elements of $B_0 \\setminus B$, and all sinks $k$ are elements of $B_0 \\setminus B$. \n\nWe will slightly abuse notation and go between the path $(s \\rightarrow k)$ and the corresponding interval $(s,k)$ freely. So for example, the path $(2 \\rightarrow 5)$ \\newword{contains} $3$ since $3 \\in (2,5)$. Given a matching representation $R$, we put a partial ordering $\\prec_R$ on elements of $B_0 \\setminus B$ (and also separately on elements of $B \\setminus B_0)$ by setting $s'  \\prec  s$ ($k' \\prec k$ respectively) if $(s \\rightarrow k),(s' \\rightarrow k') \\in R_1$ and $s < s' < k' < k$: i.e. if the path coming out of $s'$ is nested inside (as intervals on $[n]$) the path coming out of $s$. For example, in the middle picture of the top row of Figure~\\ref{fig:allthebases}, we have $2 \\prec_R 1$ (also $3 \\prec 5)$ since comparing the paths $2 \\rightarrow 3$ and $1 \\rightarrow 5$, the interval $(2,3)$ is nested inside the interval $(1,5)$. We put an ordering on the paths so it is compatible with the ordering we just discussed on the starting point: for example we have $(2,3) \\prec (1,5)$. \n\n\\begin{lemma}\n\\label{lem:uniquerep}\nFix a positroid $\\mathcal{M}_L$. For each basis $B \\in \\mathcal{M}_L$, there exists a unique matching representation of $B$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Definition~\\ref{def:pos}, there exists at least one matching representation of each $B \\in M_L$. For sake of contradication, assume that for some $B \\in M_L$ we have two different matching representations $R_1$ and $R_2$. Consider the partial order $\\prec_{R_1}$ on $B_0 \\setminus B$.  Under this ordering, pick any minimal element $s$ of $B_0 \\setminus B$ such that $(s \\rightarrow k) \\in R_1$ and $(s \\rightarrow k') \\in R_2$ with $k \\not = k'$. If $k < k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k) \\in R_2$. Since $R_2$ is non-crossing we must have $s < s'$. But then from the way $s$ was chosen, we need to have $(s' \\rightarrow k) \\in R_1$ as well and we get a contradiction. If $k > k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k') \\in R_1$. Since $R_1$ is non-crossing we must have $s < s' < k' < k$. But then from the way $s$ was chosen, we must have $(s' \\rightarrow k') \\in R_2$ and we again get a contradiction.\n\\end{proof}\n\nThanks to Lemma~\\ref{lem:uniquerep}, when considering the partial order $\\prec_R$ coming from a matching representation of some basis $B \\in \\mathcal{M}_L$, we may instead write it as $\\prec_B$. Using this, we slightly abuse notation and say $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is a path inside the unique matching representation of $B$. Given some fixed basis $B \\in \\mathcal{M}_L$, we say that an element $x \\in [n]$ is \\newword{associated under $B$} to a path $(s \\rightarrow k) \\in B$ if $(s \\rightarrow q)$ is the unique minimal path containing $x$ under the ordering $\\prec_B$. Next we show that each passive element $b$ in a basis of $B$ is closely related to the matching representation of $B$.\n\n\\begin{lemma}\n\\label{lem:containment}\nFix a positroid $\\mathcal{M}_L$ and a basis $B \\in \\mathcal{M}_L$. For each passive $b \\in B\\cap B_0$, there is a path $(s \\rightarrow k) \\in B$ containing $b$ (i.e. $s < b < k$).\n\\end{lemma}\n\n\\begin{proof}\nPick $b \\in B \\cap B_0$ that is passive. Assume for sake of contradiction there is no path $(s \\rightarrow k) \\in B$ such that $s < b < k$. From $b$ being passive, there is some $B' \\in \\mathcal{M}_L$ such that $B' = B \\setminus b \\cup b'$ with $b' < b$ and $b' \\not \\in B$. We have $(b \\rightarrow k) \\in B'$ for some $k \\in B' \\setminus B_0$. Then there is some $c$ such that $(c \\rightarrow k) \\in B$. From the assumption we have $c > b$. \n\nUnder the ordering $\\prec_{B'}$, pick any minimal element $x$ such that $(x \\rightarrow y) \\in B'$ and $(x' \\rightarrow y) \\in B$ with $x' > x \\geq b$. There exists such element thanks to the argument in the previous paragraph. Since $(b,n] \\cap B = (b,n] \\cap B'$, we have $x' \\not \\in B'$. So there is some $y'$ such that $(x' \\rightarrow y') \\in B'$, with $x <x'<y'<y$. Then, this implies $y'\\in B$, so there exists some $x''$ such that $(x'' \\rightarrow y')\\in B'$. Since $x'' \\prec_{B'} x$, we get a contradiction with $x$ being chosen as a minimal element.\n\n\n\\end{proof}\n\n\nWe are now ready to present the map $\\phi$ which will be mapping each basis of a positroid into a monomial supported on variables indexed by $[n] \\setminus B_0$. Given some basis $B \\in M_L$, for each element $b \\in B \\setminus B_0$, we simply map it to the variable $x_b$. For a passive element $b \\in B \\cap B_0$, we look at the unique minimum (under $\\prec_B$) path $s \\rightarrow k$ that contains $b$ (recall that we called this as $b$ being associated to $s \\rightarrow k$ under $B$), then map $b$ to $x_k$. Then take the product of all such variables to obtain a monomial, which is going to be supported on variables indexed by $[n] \\setminus B_0$ as we wanted. \n\nFor example, take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. This family of paths encodes the basis $235$. The passive elements of this basis is actually all of $2,3,5$. Since $3$ and $5$ are sinks of paths, $3$ gives $x_3$ and $5$ gives $x_5$. Now the minimum (under $\\prec_{235}$) path containing $2$ is $(1 \\rightarrow 5)$ so $2$ gives $x_3$. The resulting monomial is $x_3^2 x_5$.\n\n\\begin{proposition}\n\\label{prop:phi}\nLet $\\mathcal{M}_L$ be a positroid. Given a basis $B$, we have $\\phi(B) = \\prod_{k \\in B \\setminus B_0} x_k^{|T_k^B|+1}$ where $T_k^B$ consists of elements $x$ associated to $(s \\rightarrow k) \\in B$ such that replacing $s \\rightarrow k$ with $x \\rightarrow k$ gives some other basis in $\\mathcal{M}_L$.\n\\end{proposition}\n\\begin{proof}\nFrom the way $\\phi$ is defined, it is enough to show that $T_k^B$ consists exactly of the passive elements of $B \\cap B_0$ associated to  $(s \\rightarrow k) \\in B$. It is clear that if we can replace $s \\rightarrow k$ with $x \\rightarrow k$ for $x \\in (s,k)$ then $x$ is passive. Hence we only need to show the other direction: if $x$ is passive then we can replace $s \\rightarrow k \\in B$ with $x \\rightarrow k$ to get another basis. \n\n\nSince $x$ is passive, there is some $x' < x$ such that $B' = B \\setminus x \\cup x' \\in \\mathcal{M}_L$. Consider the matching representation of $B$. Since we have a path $(s \\rightarrow k) \\in B$, from the non-crossing property we have equal number of source points and sink points within $(s,k)$. Moreover, since $x$ is associated to $s \\rightarrow k$, we have equal number of source points and sink points within $(s,x)$ and $(x,k)$. This property for $(x,k)$ carries over to the matching representation of $B'$ as well. From this property, we have $(x \\rightarrow k) \\in B'$.\n\\end{proof}\n\nFor example, again take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. The set $T_5^{235}$ is empty. The set $T_3^{235}$ consists of $2$, since we have $2 \\in (1,3)$ and we can replace $1 \\rightarrow 3$ with $2 \\rightarrow 3$ to get the set $135$, which is again a basis of the positroid.\n\nThe Proposition~\\ref{prop:phi} tells us that it passes the passive degree to the degree of the monomial:\n\n\\begin{corollary}\nLet $\\mathcal{M}_L$ be positroid. Then the internally passive degree of $B$ is equal to the degree of $\\phi(B)$.\n\\end{corollary}\n\nFor example, take a look at the leftmost picture of second row from the top within Figure~\\ref{fig:allthebases}. We have the path $1 \\rightarrow 3$ so the corresponding basis is $234$. Here $2$ and $3$ are passive, but $4$ is active. So $234$ has passive degree to be $2$. The corresponding monomial is $x_3^2$, which again has degree of $2$.\n\n\n\\begin{lemma}\nThe map $\\phi$ is a one-to-one map.\n\\end{lemma}\n\n\n\n\\begin{proof}\nAssume for sake of contradiction that we can find two different bases $B, B'$ of the positroid $\\mathcal{M}_L$ such that $\\phi(B) = \\phi(B')$. In $B$, under the ordering $\\prec_B$, pick the minimal path $(s \\rightarrow k)$ that isn't contained in $B'$. Since $\\phi(B) = \\phi(B')$, there has to be some path $(s' \\rightarrow k)$ with $s \\not = s'$. Without loss of generality, assume $s < s'$. From Proposition~\\ref{prop:phi}, we have either $T_k^{B'} \\subsetneq T_k^{B}$ (if $s < s'$) or $T_k^{B'} \\supsetneq T_k^{B}$ (if $s > s'$). This leads to $|T_k^B| \\neq |T_k^{B'}|$ and we get a contradiction.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:poi}\nThe map $\\phi$ gives a pure order ideal.\n\\end{proposition}\n\\begin{proof}\nWe first start with the order ideal property. We need to show that given any basis $B$, for each variable $x_k$ appearing in $\\phi(B)$, we can find a basis $B'$ such that $\\phi(B') = \\phi(B)\/x_k$. To use induction, we introduce a new statistic on the poset with elements $B \\setminus B_0$ and ordering $\\prec_B$: we set $\\sigma(k)$ to stand for the length of the longest possible chain having $k$ as the lower end within the poset. \n\nWe are going to use induction on $\\sigma(k)$. We start with the base case, when $\\sigma(k) = 1$, that is when the path $(s \\rightarrow k) \\in B$ is not contained in any other path. If $|T_k^B| = 1$, we simply remove the path to get $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get our desired $B'$.\n\nNow assume for sake of induction we have proven the result for $\\sigma(k) < t$ and are now considering the case $\\sigma(k) = t$. Same as above, if $|T_k^B| = 1$, we simply remove the path to get $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get some other basis $B''$. Then $\\phi(B'')$ is obtained from $\\phi(B)$ by dividing by $x_k$, but the powers of other variables $x_{k'}$ might have gone up for $k \\prec_B k'$. Regardless, in such cases we have $\\sigma(k') < t$, so we can find a new basis by shedding them off thanks to the induction assumption. This gives us the $B'$ we were looking for.\n\nNow for the pure part, fix a basis $B$. We know that every element of $B \\setminus B_0$ is automatically passive since we can simply undo the corresponding path to get a smaller basis. So potential active elements only occur within $B \\cap B_0$. We want to show that if there is some active element, we can always find some basis $B'$ such that $\\phi(B')$ is divisible by $\\phi(B)$. Let $a$ be the largest element of $B$ that is active. We do a case by case analysis based on whether $a+1$ is in $B_0$ or not.\n\nWhen $a+1 \\not \\in B_0$: from Remark~\\ref{rem:noloopcoloop} we have $L_{a,a+1} = 1$. If $a+1 \\in B$, then we have $s \\rightarrow a+1$ in $B$ which is replaceable with $a \\rightarrow a+1$ to get $B'$. This means $a$ is passive so we can ignore this case and say $a+1 \\not \\in B$. In this case, we may add the path $a \\rightarrow a+1$ to get $B'$ we want ($\\phi(B') = \\phi(B) x_{a+1})$. \n\nWhen $a+1 \\in B_0$: first assume for sake of contradiction that $a+1 \\in B$. Then it is passive thanks to the way $a$ was chosen. This means we have $s \\rightarrow k \\in B$ such that $a+1 \\in T_k^B$. From the $\\LE$-property, the path $s \\rightarrow k$ is replaceable with $a \\rightarrow k$ as well, and we get a contradiction from Proposition~\\ref{prop:phi}. \n    \nHence we have that $a+1 \\in B_0 \\setminus B$. This means we have some $(a+1 \\rightarrow k) \\in B$. Let $k'$ be the minimal element such that $a \\rightarrow k'$ is possible inside the $\\LE$-graph. If $k' \\leq k$, from the $\\LE$-property we can switch $a+1 \\rightarrow k$ to $a \\rightarrow k$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{a+1}$). In the case $k' > k$, from the way $k'$ was chosen and the $\\LE$-property, we have $L_{a,k'} = 1$ and $L_{i,j} = 0$ for $i \\leq a$ and $j \\leq k'$ except when $(i,j) = (a,k')$. If the dot at $(a,k')$ is already occupied in any family of non-crossing paths representing $B$, we can replace the starting point of this path with $a$. This would contradict $a$ being active, so we may assume that the dot at $(a,k')$ is never occupied. If there is some path $s \\rightarrow k' \\in B$ with $s' > a$, then replace $s \\rightarrow k'$ with $a \\rightarrow k'$ to get the desired $B'$ ($\\phi(B') = \\phi(B)x_{k'}^t$ with $t \\geq 1$). Otherwise we can simply add $a \\rightarrow k'$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{k'}$).\n\n\\end{proof}\n\n\nThanks to the above proposition, we have established Stanley's conjecture for the class of positroids:\n\n\\begin{corollary}\nThe $h$-vector of any positroid is a pure $O$-sequence.\n\\end{corollary}\n\nAn example is shown in Figure~\\ref{fig:allthebases}. Using the $\\LE$-diagram from Figure~\\ref{fig:lediagram}, all the bases and the corresponding family of non-crossing paths are drawn in Figure~\\ref{fig:allthebases}. The monomials from the mapping $\\phi$ are written below each picture. Proposition~\\ref{prop:poi} guarantees that the collection of monomials we have obtained is indeed a pure order ideal.\n\n\n\\section*{Acknowledgements}\nThe research started under the $2021$-High school math camp hosted at Texas State University. The authors would like to thank the camp organizers for providing support and a great working environment.\n\n\n\n\n\\begin{figure}[h]\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/basis\/allthebases}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{Hasse diagram of the $\\LE$-Diagrams which form the bases of the matroid. Each $\\LE$-Diagram is labelled with its corresponding monomial.}\n  \\label{fig:allthebases}\n\n\\end{center}\n\\end{figure}\n\n\n\\section{Introduction}\nA matroid is a structure which generalizes the notion of independence that rises in linear algebra, graph theory and other areas. From a matroid, we can construct a simplicial complex (\\newword{matroid complex}) and study its \\newword{h-vector} which provides the topological information of the complex. The following conjecture by Stanley is still wide open:\n\n\\begin{conjecture}[\\cite{Stanley1977}]\n\\label{con:stanley}\nThe h-vector of  a matroid is a pure O-sequence.\n\\end{conjecture}\n\n\nThere are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in \\cite{Merino2001}, lattice-path matroids by Schweig in \\cite{Schweig2010}, cotransversal matroids by Oh in \\cite{Oh2013}, paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores  \\cite{MNRV2012}, internally perfect matroids by Dall in \\cite{Dal}, rank $3$ matroids by H\\'a, Stokes, and Zanello in \\cite{HaStokesZanello2013}, rank $3$ and corank $2$ matroids by DeLoera, Kemper, and Klee in \\cite{DelKemKle}, rank $4$ matroids by Klee and Samper in \\cite{KleeSamper2015}, rank $d$ matroids with $h_d \\leq 5$  by Constantinescu, Kahle, and Varbaro in \\cite{ConKahVar} and some small special classes of graphic matroids by Kook \\cite{Kook2011}, Preston et al \\cite{biconed} and David et al \\cite{triconed}.\n\nIn this paper the focus will solely be on the class of positroids, which is a special class of realizable matroids, for which the conjecture is still wide open. Positroids come from the study of the nonnegative part of the Grassmannian \\cite{Postnikov} and have seen increased applications in physics, with use in the study of scattering amplitudes \\cite{arkani} and the study of shallow water waves \\cite{kodama}. Positroids contain the class of lattice-path matroids, but is incomparable with all other classes mentioned above. The closest class is perhaps that of transversal matroids since positroids and transversal matroids are both contained in the class called gammoids. Regardless, the class of positroids and the class of transversal matroids (and also the class of cotransversal matroids since dual of a positroid is again a positroid) are independent \\cite{marcott2019basis}.\n\n\n\n\\section{Preliminary}\n\nThroughout the paper, we will use the following notation. When $A$ is a set and $a$ is a single element, instead of writing $A \\setminus \\{a\\}$, we will  write $A \\setminus a$. Similarly, instead of writing $A \\cup \\{a\\}$, we will  use $A \\cup a$.\n\n\\subsection{Positroids} \n\n\\begin{figure}\n\\begin{multicols}{2}\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lediagram}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{A $\\LE$-diagram.}\n  \\label{fig:lediagram}\n\n \\end{center}\n\n\\vfill\\null\n\\columnbreak\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lenetwork}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{The corresponding $\\LE$-network.}\n  \\label{fig:lenetwork}\n\n \\end{center}\n\\end{multicols}\n\\end{figure}\n\nWe will start with the definition of $\\LE$-diagrams and $\\LE$-graphs and use that to define positroids. A $\\LE$-diagram will be a Young diagram with filled dots in some boxes. Consider the southeastern boundary path of the diagram. We call this the \\newword{boundary path} of the diagram. Starting from the northeast corner and heading towards the southwest corner, label each edge of the boundary path with integers $1,\\ldots,n$. We call these the \\newword{boundary labels}. We index each column and row of a diagram with the label of the unique boundary label it contains. A box at $(i,j)$ stands for the box of $L$ at row indexed with $i$ and column indexed with $j$. For example, a box of $L$ in Figure~\\ref{fig:lediagram} at $(2,5)$ is the box at the second row from top (indexed with $2$) and first column from left (indexed with $5$). We write $L_{i,j} = 1$ if there the box at $(i,j)$ contains a dot, and $L_{i,j} = 0$ otherwise.\n\n\\begin{definition}[\\cite{Postnikov}, Definition 6.1] For a partition $\\lambda$, let us define a $\\LE$-diagram $L$ of shape $\\lambda$ as a filling of boxes of the Young diagram of shape $\\lambda$ such that, for any three boxes at $(i,j),(i',j),(i,j')$, where $i'<i$ and $j'>j$, if $L_{i',j} = L_{i,j'} = 1$ then $L_{i,j} = 1$. This property is called the $\\LE$-property.\n\n\\end{definition}\n\nFrom each $\\LE$-diagram $L$, we can get a planar network called the $\\LE$-graph in Definition 6.3 of \\cite{Postnikov}, via the following way: place a dot in the middle of each step in the boundary lattice path of the diagram and mark these dots with the corresponding boundary labels (so $1,\\ldots,n$ from top to bottom). We will call these vertices the \\newword{boundary vertices}. For each dot inside the $\\LE$-diagram, draw a horizontal line to its right, and vertical line to its bottom until it reaches another dot. Next orient all vertical edges downward and horizontal edges to the right.  The vertical labels will be the source vertices and the horizontal labels will be the sink vertices in the $\\LE$-graph. Since each $\\LE$-diagram encodes a unique $\\LE$-graph, we will abuse notation and go between a $\\LE$-diagram and its corresponding $\\LE$-graph freely in this paper. An example of a $\\LE$-graph obtained from the $\\LE$-diagram in Figure~\\ref{fig:lediagram} is drawn in Figure~\\ref{fig:lenetwork}.\n\nThe set $[n]$ (where $n$ is the length of the boundary path) is called the \\newword{ground set} (being consistent with the notation for the positroid that will be indexed by the $\\LE$-diagram). Let $B_0$ denote the set of source vertices in the $\\LE$-graph of $L$. For example, in the $\\LE$-diagram $L$ drawn in Figure~\\ref{fig:lediagram}, we have $B_0 = 124$ as we can see from the corresponding $\\LE$-graph in Figure~\\ref{fig:lenetwork}. A \\newword{path} in a $\\LE$-graph is a directed path that starts at some boundary vertex and ends at some boundary vertex. A \\newword{family of non-touching paths} is a family of paths where no two paths intersect within the graph. We say that a family of non-touching paths consisting of paths $p_1,\\ldots,p_k$ \\newword{represents} a set $B_0 \\setminus \\{p_1^s,\\ldots,p_k^s\\} \\cup \\{p_1^e,\\ldots,p_k^e\\}$ where $p_i^s$ stands for the starting point and $p_i^e$ stands for the endpoint of the path $p_i$. For example, the bottom picture in Figure~\\ref{fig:allthebases} shows an empty family, which represents the set $124$. The leftmost picture at the top row in Figure~\\ref{fig:allthebases} shows a family consisting of two paths, one starting at $1$ and ending at $3$, the other one starting at $4$ and ending at $5$: this represents the set $124 \\setminus 14 \\cup 35 = 235$. Notice that any set represented by a family of non-touching paths in a $\\LE$-graph will have the same cardinality as $B_0$.\n\n\nFrom Theorem~6.5 of \\cite{Postnikov}, there is a bijection between positroids and $\\LE$-diagrams. Moreover, the same theorem gives the following result, which we will use as the definition of a positroid in this paper:\n\n\\begin{definition}\n\\label{def:pos}\nFix a positroid $\\mathcal{M}_L$ that corresponds to a $\\LE$-diagram $L$. Then the set of bases of $\\mathcal{M}_L$ is exactly the collection of sets represented by a family of non-touching paths in the $\\LE$-graph of $L$.\n\\end{definition}\n\nFrom the above definition, the set $B_0$ is naturally the lexicographic minimal basis of $\\mathcal{M}_L$. We call this the \\newword{canonical basis} of the positroid. For example, take a look at the $\\LE$-diagram in Figure~\\ref{fig:lediagram}. All the bases we can get are:\n$$124,125,134,135,145,234,235,245,345,$$ \nas one can check from various families of disjoint paths drawn on the $\\LE$-graph as in Figure~\\ref{fig:allthebases}. \n\nAn element $e$ of the ground set is called a \\newword{loop} if it is not contained in any basis and is called a \\newword{coloop} if it is contained in every basis.\n\n\n\n\\subsection{Activity and the h-vector} \nFrom any ordering on the ground set $E$, we get an induced lexicographic ordering on the bases. An element $i$ of a basis $B$ is \\newword{(internally) active} if $B \\setminus i \\cup j$ is not a base for any $j < i$. Otherwise, it is called \\newword{(internally) passive}. There is also the notion of externally active and externally passive elements, but in this paper we will only stick to interally active and passive elements, and hence will skip the usage of the word internally throughout. Given a basis $B$ of a matroid, we call its \\newword{passivity} as the number of (internally) passive elements it contains.\n\nWe will use this activity property of h-vectors as the definition of h-vectors of matroids in this paper.\n\n\\begin{theorem}\nLet $(h_0,\\dots,h_r)$ be the $h$-vector of a matroid $\\mathcal{M}$. For $0 \\leq i \\leq r$, the entry $h_i$ is the number of bases of $\\mathcal{M}$ with $i$ passive elements.\n\\end{theorem}\n\nLet us again look at the running example of Figure~\\ref{fig:allthebases}. Take a look at $135$. Here $1$ is active since it cannot be replaced with a smaller element to get a basis of $\\mathcal{M}$. On the other hand, $3$ and $5$ are passive since we can replace either of them with $2$ to get a different basis of $\\mathcal{M}$. The $h$-vector of the corresponding positroid is $(1,2,3,3)$.\n\n\\begin{remark}\n\\label{rem:noloopcoloop}\nThe class of positroids is closed under deletion and contraction \\cite{ARW}. Deleting a loop (simply remove that element from the ground set) or contracting by a coloop (remove that element from the ground set and all the bases) does not change the $h$-vector of the matroid. For sake of proving Stanley's conjecture, we may assume that our matroid does not have any loops or coloops. Hence, from now on, throughout the entire paper, we are only going to deal with positroids with no loops nor coloops. In terms of the $\\LE$-diagram, it means there is no empty row nor column. \n\\end{remark}\n\n\n\n\\subsection{Order ideal and pure O-sequences}\nAn \\newword{order ideal} is a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. We define a maximal monomial as the monomial such that all of other monomials in the order ideal are divisors of the maximal monomial. Given an order ideal its \\newword{O-sequence} is a sequence that can be obtained from counting monomials of each degree. We say that the order ideal is \\newword{pure} if all the maximal monomials are of the same degree, and also call the O-sequence pure if it comes from a pure order ideal.\n\nFor example, take a look at Figure~\\ref{fig:allthebases} and focus only on the monomials. If we take any monomial among this collection, say $x_3x_5^2$, we can see that all divisors of it is again present in the collection. Hence this is an order ideal. Moreover, since the maximal monomials under division (which are usually not guaranteed to all have the same degree) here are $x_3^2x_5,x_3x_5^2,x_5^3$ which have the same degree, and hence this is a pure order ideal. Notice that the degree sequence of this pure order ideal is $(1,2,3,3)$. Hence we can see that the $h$-vector of the positroid coming from Figure~\\ref{fig:lediagram}, which was $(1,2,3,3)$, is indeed a pure $O$-sequence. We have just verified that Stanley's conjecture is true for this case. What we want to do in this paper is to similarly construct a pure order ideal for each and every positroid, hence proving the conjecture for this class.\n\n\n\\section{The main result}\n\nOur goal in this section is to come up with an algorithm that maps a family of non-crossing paths to a monomial, using the elements of $[n] \\setminus B_0$ as the variables. Then we will show that the image under this map is a pure order ideal to answer Stanley's conjecture for positroids.\n\nGiven a path from a source $s$ to sink $k$, we write it as $(s \\rightarrow k)$. Since each basis $B \\in \\mathcal{M}$ is coming from a family of non-crossing paths from set of sources to set of sinks, we write $B$ as a collection of $(s \\rightarrow k)$'s and call this the \\newword{non-crossing matching representation} (or simply a matching representation) of $B \\in \\mathcal{M}$. Note that all sources $s$ are elements of $B_0 \\setminus B$, and all sinks $k$ are elements of $B \\setminus B_0$. \n\nWe will slightly abuse notation and go between the path $(s \\rightarrow k)$ and the corresponding interval $(s,k)$ freely. So for example, the path $(2 \\rightarrow 5)$ \\newword{contains} $3$ since $3 \\in (2,5)$. Given a matching representation $R$, we put a partial ordering $\\prec_R$ on elements of $B_0 \\setminus B$ (and also separately on elements of $B \\setminus B_0)$ by setting $s'  \\prec  s$ ($k' \\prec k$ respectively) if $(s \\rightarrow k),(s' \\rightarrow k') \\in R$ and $s < s' < k' < k$: i.e. if the path coming out of $s'$ is nested inside (as intervals on $[n]$) the path coming out of $s$. For example, in the middle picture of the top row of Figure~\\ref{fig:allthebases}, we have $2 \\prec_R 1$ (also $3 \\prec_R 5)$ since comparing the paths $2 \\rightarrow 3$ and $1 \\rightarrow 5$, the interval $(2,3)$ is nested inside the interval $(1,5)$. We put an ordering on the paths so it is compatible with the ordering we just discussed on the starting point: for example we have $(2,3) \\prec_R (1,5)$. \n\n\\begin{lemma}\n\\label{lem:uniquerep}\nFix a positroid $\\mathcal{M}_L$. For each basis $B \\in \\mathcal{M}_L$, there exists a unique matching representation of $B$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Definition~\\ref{def:pos}, there exists at least one matching representation of each $B \\in M_L$. For sake of contradication, assume that for some $B \\in M_L$ we have two different matching representations $R_1$ and $R_2$. Consider the partial order $\\prec_{R_1}$ on $B_0 \\setminus B$.  Under this ordering, pick any minimal element $s$ of $B_0 \\setminus B$ such that $(s \\rightarrow k) \\in R_1$ and $(s \\rightarrow k') \\in R_2$ with $k \\not = k'$. If $k < k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k) \\in R_2$. Since $R_2$ is non-crossing we must have $s < s'$. But then from the way $s$ was chosen, we need to have $(s' \\rightarrow k) \\in R_1$ as well and we get a contradiction. If $k > k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k') \\in R_1$. Since $R_1$ is non-crossing we must have $s < s' < k' < k$. But then from the way $s$ was chosen, we must have $(s' \\rightarrow k') \\in R_2$ and we again get a contradiction.\n\\end{proof}\n\nIt is worth noting that although we are calling $s \\rightarrow k$ a path, there might be multiple different paths in the $\\LE$-graph that are encoded by the same $s \\rightarrow k$. For example in the $\\LE$-graph of Figure~\\ref{fig:lediagram}, consider the path that starts from $1$, goes left, left, down, down, down to $5$. This path is encoded as $1 \\rightarrow 5$ in the matching representation. If we look at the path that starts from $1$, goes left, down, left, down, down to $5$, this is a different path from above but is still encoded as $1 \\rightarrow 5$. So even though Lemma~\\ref{lem:uniquerep} tells us that each $B$ has a unique matching representation, there may be a number of different families of non-crossing paths that correspond to that same matching representation.\n\nThanks to Lemma~\\ref{lem:uniquerep}, when considering the partial order $\\prec_R$ coming from a matching representation of some basis $B \\in \\mathcal{M}_L$, we may instead write it as $\\prec_B$. Using this, we slightly abuse notation and say $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is a path inside the unique matching representation of $B$. Given some fixed basis $B \\in \\mathcal{M}_L$, we say that an element $x \\in [n]$ is \\newword{associated under $B$} (or simply associated, when it is clear we are dealing with the basis $B$) to a path $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is the unique minimal path containing $x$ under the ordering $\\prec_B$. Next we show that each passive element $b$ in a basis of $B$ is closely related to the matching representation of $B$.\n\n\\begin{lemma}\n\\label{lem:containment}\nFix a positroid $\\mathcal{M}_L$ and a basis $B \\in \\mathcal{M}_L$. For each passive $b \\in B\\cap B_0$, there is a path $(s \\rightarrow k) \\in B$ containing $b$ (i.e. $s < b < k$).\n\\end{lemma}\n\n\\begin{proof}\nPick $b \\in B \\cap B_0$ that is passive. Assume for sake of contradiction there is no path $(s \\rightarrow k) \\in B$ such that $s < b < k$. From $b$ being passive, there is some $B' \\in \\mathcal{M}_L$ such that $B' = B \\setminus b \\cup b'$ with $b' < b$ and $b' \\not \\in B$. We have $(b \\rightarrow k) \\in B'$ for some $k \\in B' \\setminus B_0$. Then there is some $c$ such that $(c \\rightarrow k) \\in B$. From the assumption we have $c > b$. \n\nUnder the ordering $\\prec_{B'}$, pick any minimal element $x$ such that $(x \\rightarrow y) \\in B'$ and $(x' \\rightarrow y) \\in B$ with $x' > x \\geq b$. Such $x$ exists thanks to the argument in the previous paragraph. Since $(b,n] \\cap B = (b,n] \\cap B'$, we have $x' \\not \\in B'$. So there is some $y'$ such that $(x' \\rightarrow y') \\in B'$, with $x <x'<y'<y$. Then, this implies $y'\\in B$, so there exists some $x'' > x'$ such that $(x'' \\rightarrow y')\\in B$. Since $x' \\prec_{B'} x$, we get a contradiction with $x$ being chosen as a minimal element within $\\prec_B'$.\n\n\n\\end{proof}\n\n\nWe are now ready to present the map $\\phi$ which will be mapping each basis of a positroid into a monomial supported on variables indexed by $[n] \\setminus B_0$. Given some basis $B \\in M_L$, for each element $b \\in B \\setminus B_0$, we simply map it to the variable $x_b$. For a passive element $b \\in B \\cap B_0$, we look at the unique minimum (under $\\prec_B$) path $s \\rightarrow k$ that contains $b$ (recall that we called this as $b$ being associated to $s \\rightarrow k$ under $B$), then map $b$ to $x_k$. Now take the product of all such variables to obtain a monomial, which is going to be supported on variables indexed by $[n] \\setminus B_0$ as we wanted. \n\nFor example, take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. This family of paths encodes the basis $235$. The passive elements of this basis is actually all of $2,3,5$. Since $3$ and $5$ are sinks of paths, $3$ gives $x_3$ and $5$ gives $x_5$. Now the minimum (under $\\prec_{235}$) path containing $2$ is $(1 \\rightarrow 5)$ so $2$ gives $x_3$. The resulting monomial is $x_3^2 x_5$.\n\n\\begin{proposition}\n\\label{prop:phi}\nLet $\\mathcal{M}_L$ be a positroid. Given a basis $B$, we have $\\phi(B) = \\prod_{k \\in B \\setminus B_0} x_k^{|T_k^B|+1}$ where $T_k^B$ consists of elements $x$ associated to $(s \\rightarrow k) \\in B$ such that replacing $s \\rightarrow k$ with $x \\rightarrow k$ gives some other basis in $\\mathcal{M}_L$.\n\\end{proposition}\n\\begin{proof}\nFrom the way $\\phi$ is defined, it is enough to show that $T_k^B$ consists exactly of the passive elements of $B \\cap B_0$ associated to  $(s \\rightarrow k) \\in B$. It is clear that if we can replace $s \\rightarrow k$ with $x \\rightarrow k$ for $x \\in (s,k)$ then $x$ is passive. Hence we only need to show the other direction: if $x$ is passive then we can replace $s \\rightarrow k \\in B$ with $x \\rightarrow k$ to get another basis. \n\n\nSince $x$ is passive, there is some $x' < x$ such that $B' = B \\setminus x \\cup x' \\in \\mathcal{M}_L$. Consider the matching representation of $B$. Since we have a path $(s \\rightarrow k) \\in B$, from the non-crossing property we have equal number of source points and sink points within $(s,k)$. Moreover, since $x$ is associated to $s \\rightarrow k$, we have equal number of source points and sink points within $(s,x)$ and also within $(x,k)$. This property for $(x,k)$ carries over to the matching representation of $B'$ as well. From this property, we have $(x \\rightarrow k) \\in B'$.\n\\end{proof}\n\nFor example, again take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. The set $T_5^{235}$ is empty. The set $T_3^{235}$ consists of $2$, since we have $2 \\in (1,3)$ and we can replace $1 \\rightarrow 3$ with $2 \\rightarrow 3$ to get the set $135$, which is again a basis of the positroid.\n\nThe Proposition~\\ref{prop:phi} together with Lemma~\\ref{lem:containment} tells us that $\\phi$ maps the passivity of the basis to the degree of the monomial:\n\n\\begin{corollary}\nLet $\\mathcal{M}_L$ be positroid. Then the passivity of $B$ is equal to the degree of $\\phi(B)$.\n\\end{corollary}\n\nFor example, take a look at the leftmost picture of second row from the top within Figure~\\ref{fig:allthebases}. We have the path $1 \\rightarrow 3$ so the corresponding basis is $234$. Here $2$ and $3$ are passive, but $4$ is active. So $234$ has passive degree to be $2$. The corresponding monomial is $x_3^2$, which again has degree of $2$.\n\n\n\\begin{lemma}\nThe map $\\phi$ is a one-to-one map.\n\\end{lemma}\n\n\n\n\\begin{proof}\nAssume for sake of contradiction that we can find two different bases $B, B'$ of the positroid $\\mathcal{M}_L$ such that $\\phi(B) = \\phi(B')$. In $B$, under the ordering $\\prec_B$, pick the minimal path $(s \\rightarrow k)$ that isn't contained in $B'$. Since $\\phi(B) = \\phi(B')$, there has to be some path $(s' \\rightarrow k) \\in B'$ with $s \\not = s'$. If $s < s'$, from the way $s \\rightarrow k$ was chosen, we can replace $s \\rightarrow k$ with $s' \\rightarrow k$ within $B$ to get a basis so we have $T_k^{B'} \\subsetneq T_k^{B}$. If $s > s'$, we cannot have a path in $B'$ that starts within $(s',s)$ and ends within $(s,k)$ due to the way $s \\rightarrow k$ was chosen. This means we can replace $s' \\rightarrow k$ with $s \\rightarrow k$ within $B'$ to get a basis so we have $T_k^{B'} \\supsetneq T_k^{B}$.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:poi}\nThe map $\\phi$ gives a pure order ideal.\n\\end{proposition}\n\\begin{proof}\nWe first start with the order ideal property. We need to show that given any basis $B$, for each variable $x_k$ appearing in $\\phi(B)$, we can find a basis $B'$ such that $\\phi(B') = \\phi(B)\/x_k$. To use induction, we introduce a new statistic on the poset with elements $B \\setminus B_0$ and ordering $\\prec_B$: we set $\\sigma(k)$ to stand for the length of the longest possible chain having $k$ as the lower end within the poset. \n\nWe are going to use induction on $\\sigma(k)$. We start with the base case, when $\\sigma(k) = 1$, that is when the path $(s \\rightarrow k) \\in B$ is not contained in any other path. If $|T_k^B| = 1$, we simply remove the path to get the $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get our desired $B'$.\n\nNow assume for sake of induction we have proven the result for $\\sigma(k) < t$ and are now considering the case $\\sigma(k) = t$. Same as above, if $|T_k^B| = 1$, we simply remove the path to get the $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get some other basis $B''$. Then $\\phi(B'')$ is obtained from $\\phi(B)$ by dividing by $x_k$, but the powers of other variables $x_{k'}$ might have gone up for $k \\prec_B k'$. Regardless, in such cases we have $\\sigma(k') < t$, so we can find a new basis by shedding them off thanks to the induction assumption. This gives us the $B'$ we were looking for.\n\nNow for the pure part, fix a basis $B$. We know that every element of $B \\setminus B_0$ is automatically passive since we can simply undo the corresponding path to get a smaller basis. So potential active elements only occur within $B \\cap B_0$. We want to show that if there is some active element, we can always find some basis $B'$ such that $\\phi(B')$ is divisible by $\\phi(B)$. Let $a$ be the largest element of $B$ that is active. We do a case by case analysis based on whether $a+1$ is in $B_0$ or not.\n\nWhen $a+1 \\not \\in B_0$: from Remark~\\ref{rem:noloopcoloop} we have $L_{a,a+1} = 1$. If $a+1 \\in B$, then we have $s \\rightarrow a+1$ in $B$ which is replaceable with $a \\rightarrow a+1$ to get $B'$. This means $a$ is passive so we can ignore this case and say $a+1 \\not \\in B$. In this case, we may add the path $a \\rightarrow a+1$ to get $B'$ we want ($\\phi(B') = \\phi(B) x_{a+1})$. \n\nWhen $a+1 \\in B_0$: first assume for sake of contradiction that $a+1 \\in B$. Then it is passive thanks to the way $a$ was chosen. This means we have $s \\rightarrow k \\in B$ such that $a+1 \\in T_k^B$. From the $\\LE$-property, the path $s \\rightarrow k$ is replaceable with $a \\rightarrow k$ as well, and we get a contradiction from Proposition~\\ref{prop:phi}. \n    \nHence we have that $a+1 \\in B_0 \\setminus B$. This means we have some $(a+1 \\rightarrow k) \\in B$. Let $k'$ be the minimal element such that $a \\rightarrow k'$ is possible inside the $\\LE$-graph. If $k' \\leq k$, from the $\\LE$-property we can switch $a+1 \\rightarrow k$ to $a \\rightarrow k$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{a+1}$). In the case $k' > k$, from the way $k'$ was chosen and the $\\LE$-property, we have $L_{a,k'} = 1$ and $L_{i,j} = 0$ for $i \\leq a$ and $j \\leq k'$ except when $(i,j) = (a,k')$. If the dot at $(a,k')$ is already occupied in any family of non-crossing paths representing $B$, we can replace the starting point of this path with $a$. This would contradict $a$ being active, so we may assume that the dot at $(a,k')$ is never occupied. If there is some path $s \\rightarrow k' \\in B$ with $s' > a$, then replace $s \\rightarrow k'$ with $a \\rightarrow k'$ to get the desired $B'$ ($\\phi(B') = \\phi(B)x_{k'}^t$ with $t \\geq 1$). Otherwise we can simply add $a \\rightarrow k'$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{k'}$).\n\n\\end{proof}\n\n\nThanks to the above proposition, we have established Stanley's conjecture for the class of positroids:\n\n\\begin{corollary}\nThe $h$-vector of any positroid is a pure $O$-sequence.\n\\end{corollary}\n\nAn example is shown in Figure~\\ref{fig:allthebases}. Using the $\\LE$-diagram from Figure~\\ref{fig:lediagram}, all the bases and the corresponding family of non-crossing paths are drawn in Figure~\\ref{fig:allthebases}. The monomials from the mapping $\\phi$ are written below each picture. Proposition~\\ref{prop:poi} guarantees that the collection of monomials we have obtained is indeed a pure order ideal.\n\n\n\\section*{Acknowledgements}\nThe research started under the $2021$-High school math camp hosted at Texas State University. The authors would like to thank the camp organizers for providing support and a great working environment.\n\n\n\n\n\\begin{figure}[h]\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/basis\/allthebases}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{Hasse diagram of the $\\LE$-Diagrams which form the bases of the matroid. Each $\\LE$-Diagram is labelled with its corresponding monomial.}\n  \\label{fig:allthebases}\n\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Introduction}\nA matroid is a structure which generalizes the notion of independence that rises in linear algebra, graph theory and other areas. From a matroid, we can construct a simplicial complex (\\newword{matroid complex}) and study its \\newword{h-vector} which provides the topological information of the complex. The following conjecture by Stanley is still wide open:\n\n\\begin{conjecture}[\\cite{Stanley1977}]\n\\label{con:stanley}\nThe h-vector of  a matroid is a pure O-sequence.\n\\end{conjecture}\n\n\nThere are some specific classes of matroids for which the above conjecture has been established to be true. In particular, these include cographic matroids by Merino in \\cite{Merino2001}, lattice-path matroids by Schweig in \\cite{Schweig2010}, cotransversal matroids by Oh in \\cite{Oh2013}, paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores  \\cite{MNRV2012}, internally perfect matroids by Dall in \\cite{Dal}, rank $3$ matroids by H\\'a, Stokes, and Zanello in \\cite{HaStokesZanello2013}, rank $3$ and corank $2$ matroids by DeLoera, Kemper, and Klee in \\cite{DelKemKle}, rank $4$ matroids by Klee and Samper in \\cite{KleeSamper2015}, rank $d$ matroids with $h_d \\leq 5$  by Constantinescu, Kahle, and Varbaro in \\cite{ConKahVar} and some small special classes of graphic matroids by Kook \\cite{Kook2011}, Preston et al \\cite{biconed} and David et al \\cite{triconed}.\n\nIn this paper the focus will solely be on the class of positroids, which is a special class of realizable matroids, for which the conjecture is still wide open. Positroids come from the study of the nonnegative part of the Grassmannian \\cite{Postnikov} and have seen increased applications in physics, with use in the study of scattering amplitudes \\cite{arkani} and the study of shallow water waves \\cite{kodama}. Positroids contain the class of lattice-path matroids, but is incomparable with all other classes mentioned above. The closest class is perhaps that of transversal matroids since positroids and transversal matroids are both contained in the class called gammoids. Regardless, the class of positroids and the class of transversal matroids (and also the class of cotransversal matroids since dual of a positroid is again a positroid) are independent \\cite{marcott2019basis}.\n\n\n\n\\section{Preliminary}\n\nThroughout the paper, we will use the following notation. When $A$ is a set and $a$ is a single element, instead of writing $A \\setminus \\{a\\}$, we will  write $A \\setminus a$. Similarly, instead of writing $A \\cup \\{a\\}$, we will  use $A \\cup a$.\n\n\\subsection{Positroids} \n\n\\begin{figure}\n\\begin{multicols}{2}\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lediagram}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{A $\\LE$-diagram.}\n  \\label{fig:lediagram}\n\n \\end{center}\n\n\\vfill\\null\n\\columnbreak\n\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/lenetwork}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{The corresponding $\\LE$-network.}\n  \\label{fig:lenetwork}\n\n \\end{center}\n\\end{multicols}\n\\end{figure}\n\nWe will start with the definition of $\\LE$-diagrams and $\\LE$-graphs and use that to define positroids. A $\\LE$-diagram will be a Young diagram with filled dots in some boxes. Consider the southeastern boundary path of the diagram. We call this the \\newword{boundary path} of the diagram. Starting from the northeast corner and heading towards the southwest corner, label each edge of the boundary path with integers $1,\\ldots,n$. We call these the \\newword{boundary labels}. We index each column and row of a diagram with the label of the unique boundary label it contains. A box at $(i,j)$ stands for the box of $L$ at row indexed with $i$ and column indexed with $j$. For example, a box of $L$ in Figure~\\ref{fig:lediagram} at $(2,5)$ is the box at the second row from top (indexed with $2$) and first column from left (indexed with $5$). We write $L_{i,j} = 1$ if there the box at $(i,j)$ contains a dot, and $L_{i,j} = 0$ otherwise.\n\n\\begin{definition}[\\cite{Postnikov}, Definition 6.1] For a partition $\\lambda$, let us define a $\\LE$-diagram $L$ of shape $\\lambda$ as a filling of boxes of the Young diagram of shape $\\lambda$ such that, for any three boxes at $(i,j),(i',j),(i,j')$, where $i'<i$ and $j'>j$, if $L_{i',j} = L_{i,j'} = 1$ then $L_{i,j} = 1$. This property is called the $\\LE$-property.\n\n\\end{definition}\n\nFrom each $\\LE$-diagram $L$, we can get a planar network called the $\\LE$-graph in Definition 6.3 of \\cite{Postnikov}, via the following way: place a dot in the middle of each step in the boundary lattice path of the diagram and mark these dots with the corresponding boundary labels (so $1,\\ldots,n$ from top to bottom). We will call these vertices the \\newword{boundary vertices}. For each dot inside the $\\LE$-diagram, draw a horizontal line to its right, and vertical line to its bottom until it reaches another dot. Next orient all vertical edges downward and horizontal edges to the right.  The vertical labels will be the source vertices and the horizontal labels will be the sink vertices in the $\\LE$-graph. Since each $\\LE$-diagram encodes a unique $\\LE$-graph, we will abuse notation and go between a $\\LE$-diagram and its corresponding $\\LE$-graph freely in this paper. An example of a $\\LE$-graph obtained from the $\\LE$-diagram in Figure~\\ref{fig:lediagram} is drawn in Figure~\\ref{fig:lenetwork}.\n\nThe set $[n]$ (where $n$ is the length of the boundary path) is called the \\newword{ground set} (being consistent with the notation for the positroid that will be indexed by the $\\LE$-diagram). Let $B_0$ denote the set of source vertices in the $\\LE$-graph of $L$. For example, in the $\\LE$-diagram $L$ drawn in Figure~\\ref{fig:lediagram}, we have $B_0 = 124$ as we can see from the corresponding $\\LE$-graph in Figure~\\ref{fig:lenetwork}. A \\newword{path} in a $\\LE$-graph is a directed path that starts at some boundary vertex and ends at some boundary vertex. A \\newword{family of non-touching paths} is a family of paths where no two paths intersect within the graph. We say that a family of non-touching paths consisting of paths $p_1,\\ldots,p_k$ \\newword{represents} a set $B_0 \\setminus \\{p_1^s,\\ldots,p_k^s\\} \\cup \\{p_1^e,\\ldots,p_k^e\\}$ where $p_i^s$ stands for the starting point and $p_i^e$ stands for the endpoint of the path $p_i$. For example, the bottom picture in Figure~\\ref{fig:allthebases} shows an empty family, which represents the set $124$. The leftmost picture at the top row in Figure~\\ref{fig:allthebases} shows a family consisting of two paths, one starting at $1$ and ending at $3$, the other one starting at $4$ and ending at $5$: this represents the set $124 \\setminus 14 \\cup 35 = 235$. Notice that any set represented by a family of non-touching paths in a $\\LE$-graph will have the same cardinality as $B_0$.\n\n\nFrom Theorem~6.5 of \\cite{Postnikov}, there is a bijection between positroids and $\\LE$-diagrams. Moreover, the same theorem gives the following result, which we will use as the definition of a positroid in this paper:\n\n\\begin{definition}\n\\label{def:pos}\nFix a positroid $\\mathcal{M}_L$ that corresponds to a $\\LE$-diagram $L$. Then the set of bases of $\\mathcal{M}_L$ is exactly the collection of sets represented by a family of non-touching paths in the $\\LE$-graph of $L$.\n\\end{definition}\n\nFrom the above definition, the set $B_0$ is naturally the lexicographic minimal basis of $\\mathcal{M}_L$. We call this the \\newword{canonical basis} of the positroid. For example, take a look at the $\\LE$-diagram in Figure~\\ref{fig:lediagram}. All the bases we can get are:\n$$124,125,134,135,145,234,235,245,345,$$ \nas one can check from various families of disjoint paths drawn on the $\\LE$-graph as in Figure~\\ref{fig:allthebases}. \n\nAn element $e$ of the ground set is called a \\newword{loop} if it is not contained in any basis and is called a \\newword{coloop} if it is contained in every basis.\n\n\n\n\\subsection{Activity and the h-vector} \nFrom any ordering on the ground set $E$, we get an induced lexicographic ordering on the bases. An element $i$ of a basis $B$ is \\newword{(internally) active} if $B \\setminus i \\cup j$ is not a base for any $j < i$. Otherwise, it is called \\newword{(internally) passive}. There is also the notion of externally active and externally passive elements, but in this paper we will only stick to interally active and passive elements, and hence will skip the usage of the word internally throughout. Given a basis $B$ of a matroid, we call its \\newword{passivity} as the number of (internally) passive elements it contains.\n\nWe will use this activity property of h-vectors as the definition of h-vectors of matroids in this paper.\n\n\\begin{theorem}\nLet $(h_0,\\dots,h_r)$ be the $h$-vector of a matroid $\\mathcal{M}$. For $0 \\leq i \\leq r$, the entry $h_i$ is the number of bases of $\\mathcal{M}$ with $i$ passive elements.\n\\end{theorem}\n\nLet us again look at the running example of Figure~\\ref{fig:allthebases}. Take a look at $135$. Here $1$ is active since it cannot be replaced with a smaller element to get a basis of $\\mathcal{M}$. On the other hand, $3$ and $5$ are passive since we can replace either of them with $2$ to get a different basis of $\\mathcal{M}$. The $h$-vector of the corresponding positroid is $(1,2,3,3)$.\n\n\\begin{remark}\n\\label{rem:noloopcoloop}\nThe class of positroids is closed under deletion and contraction \\cite{ARW}. Deleting a loop (simply remove that element from the ground set) or contracting by a coloop (remove that element from the ground set and all the bases) does not change the $h$-vector of the matroid. For sake of proving Stanley's conjecture, we may assume that our matroid does not have any loops or coloops. Hence, from now on, throughout the entire paper, we are only going to deal with positroids with no loops nor coloops. In terms of the $\\LE$-diagram, it means there is no empty row nor column. \n\\end{remark}\n\n\n\n\\subsection{Order ideal and pure O-sequences}\nAn \\newword{order ideal} is a finite collection $X$ of monomials such that, whenever $M \\in X$ and $N$ divides $M$, we have $N \\in X$ as well. We define a maximal monomial as the monomial such that all of other monomials in the order ideal are divisors of the maximal monomial. Given an order ideal its \\newword{O-sequence} is a sequence that can be obtained from counting monomials of each degree. We say that the order ideal is \\newword{pure} if all the maximal monomials are of the same degree, and also call the O-sequence pure if it comes from a pure order ideal.\n\nFor example, take a look at Figure~\\ref{fig:allthebases} and focus only on the monomials. If we take any monomial among this collection, say $x_3x_5^2$, we can see that all divisors of it is again present in the collection. Hence this is an order ideal. Moreover, since the maximal monomials under division (which are usually not guaranteed to all have the same degree) here are $x_3^2x_5,x_3x_5^2,x_5^3$ which have the same degree, and hence this is a pure order ideal. Notice that the degree sequence of this pure order ideal is $(1,2,3,3)$. Hence we can see that the $h$-vector of the positroid coming from Figure~\\ref{fig:lediagram}, which was $(1,2,3,3)$, is indeed a pure $O$-sequence. We have just verified that Stanley's conjecture is true for this case. What we want to do in this paper is to similarly construct a pure order ideal for each and every positroid, hence proving the conjecture for this class.\n\n\n\\section{The main result}\n\nOur goal in this section is to come up with an algorithm that maps a family of non-crossing paths to a monomial, using the elements of $[n] \\setminus B_0$ as the variables. Then we will show that the image under this map is a pure order ideal to answer Stanley's conjecture for positroids.\n\nGiven a path from a source $s$ to sink $k$, we write it as $(s \\rightarrow k)$. Since each basis $B \\in \\mathcal{M}$ is coming from a family of non-crossing paths from set of sources to set of sinks, we write $B$ as a collection of $(s \\rightarrow k)$'s and call this the \\newword{non-crossing matching representation} (or simply a matching representation) of $B \\in \\mathcal{M}$. Note that all sources $s$ are elements of $B_0 \\setminus B$, and all sinks $k$ are elements of $B \\setminus B_0$. \n\nWe will slightly abuse notation and go between the path $(s \\rightarrow k)$ and the corresponding interval $(s,k)$ freely. So for example, the path $(2 \\rightarrow 5)$ \\newword{contains} $3$ since $3 \\in (2,5)$. Given a matching representation $R$, we put a partial ordering $\\prec_R$ on elements of $B_0 \\setminus B$ (and also separately on elements of $B \\setminus B_0)$ by setting $s'  \\prec  s$ ($k' \\prec k$ respectively) if $(s \\rightarrow k),(s' \\rightarrow k') \\in R$ and $s < s' < k' < k$: i.e. if the path coming out of $s'$ is nested inside (as intervals on $[n]$) the path coming out of $s$. For example, in the middle picture of the top row of Figure~\\ref{fig:allthebases}, we have $2 \\prec_R 1$ (also $3 \\prec_R 5)$ since comparing the paths $2 \\rightarrow 3$ and $1 \\rightarrow 5$, the interval $(2,3)$ is nested inside the interval $(1,5)$. We put an ordering on the paths so it is compatible with the ordering we just discussed on the starting point: for example we have $(2,3) \\prec_R (1,5)$. \n\n\\begin{lemma}\n\\label{lem:uniquerep}\nFix a positroid $\\mathcal{M}_L$. For each basis $B \\in \\mathcal{M}_L$, there exists a unique matching representation of $B$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Definition~\\ref{def:pos}, there exists at least one matching representation of each $B \\in M_L$. For sake of contradication, assume that for some $B \\in M_L$ we have two different matching representations $R_1$ and $R_2$. Consider the partial order $\\prec_{R_1}$ on $B_0 \\setminus B$.  Under this ordering, pick any minimal element $s$ of $B_0 \\setminus B$ such that $(s \\rightarrow k) \\in R_1$ and $(s \\rightarrow k') \\in R_2$ with $k \\not = k'$. If $k < k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k) \\in R_2$. Since $R_2$ is non-crossing we must have $s < s'$. But then from the way $s$ was chosen, we need to have $(s' \\rightarrow k) \\in R_1$ as well and we get a contradiction. If $k > k'$, then pick $s' \\in B_0 \\setminus B$ so that $(s' \\rightarrow k') \\in R_1$. Since $R_1$ is non-crossing we must have $s < s' < k' < k$. But then from the way $s$ was chosen, we must have $(s' \\rightarrow k') \\in R_2$ and we again get a contradiction.\n\\end{proof}\n\nIt is worth noting that although we are calling $s \\rightarrow k$ a path, there might be multiple different paths in the $\\LE$-graph that are encoded by the same $s \\rightarrow k$. For example in the $\\LE$-graph of Figure~\\ref{fig:lediagram}, consider the path that starts from $1$, goes left, left, down, down, down to $5$. This path is encoded as $1 \\rightarrow 5$ in the matching representation. If we look at the path that starts from $1$, goes left, down, left, down, down to $5$, this is a different path from above but is still encoded as $1 \\rightarrow 5$. So even though Lemma~\\ref{lem:uniquerep} tells us that each $B$ has a unique matching representation, there may be a number of different families of non-crossing paths that correspond to that same matching representation.\n\nThanks to Lemma~\\ref{lem:uniquerep}, when considering the partial order $\\prec_R$ coming from a matching representation of some basis $B \\in \\mathcal{M}_L$, we may instead write it as $\\prec_B$. Using this, we slightly abuse notation and say $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is a path inside the unique matching representation of $B$. Given some fixed basis $B \\in \\mathcal{M}_L$, we say that an element $x \\in [n]$ is \\newword{associated under $B$} (or simply associated, when it is clear we are dealing with the basis $B$) to a path $(s \\rightarrow k) \\in B$ if $(s \\rightarrow k)$ is the unique minimal path containing $x$ under the ordering $\\prec_B$. Next we show that each passive element $b$ in a basis of $B$ is closely related to the matching representation of $B$.\n\n\\begin{lemma}\n\\label{lem:containment}\nFix a positroid $\\mathcal{M}_L$ and a basis $B \\in \\mathcal{M}_L$. For each passive $b \\in B\\cap B_0$, there is a path $(s \\rightarrow k) \\in B$ containing $b$ (i.e. $s < b < k$).\n\\end{lemma}\n\n\\begin{proof}\nPick $b \\in B \\cap B_0$ that is passive. Assume for sake of contradiction there is no path $(s \\rightarrow k) \\in B$ such that $s < b < k$. From $b$ being passive, there is some $B' \\in \\mathcal{M}_L$ such that $B' = B \\setminus b \\cup b'$ with $b' < b$ and $b' \\not \\in B$. We have $(b \\rightarrow k) \\in B'$ for some $k \\in B' \\setminus B_0$. Then there is some $c$ such that $(c \\rightarrow k) \\in B$. From the assumption we have $c > b$. \n\nUnder the ordering $\\prec_{B'}$, pick any minimal element $x$ such that $(x \\rightarrow y) \\in B'$ and $(x' \\rightarrow y) \\in B$ with $x' > x \\geq b$. Such $x$ exists thanks to the argument in the previous paragraph. Since $(b,n] \\cap B = (b,n] \\cap B'$, we have $x' \\not \\in B'$. So there is some $y'$ such that $(x' \\rightarrow y') \\in B'$, with $x <x'<y'<y$. Then, this implies $y'\\in B$, so there exists some $x'' > x'$ such that $(x'' \\rightarrow y')\\in B$. Since $x' \\prec_{B'} x$, we get a contradiction with $x$ being chosen as a minimal element within $\\prec_B'$.\n\n\n\\end{proof}\n\n\nWe are now ready to present the map $\\phi$ which will be mapping each basis of a positroid into a monomial supported on variables indexed by $[n] \\setminus B_0$. Given some basis $B \\in M_L$, for each element $b \\in B \\setminus B_0$, we simply map it to the variable $x_b$. For a passive element $b \\in B \\cap B_0$, we look at the unique minimum (under $\\prec_B$) path $s \\rightarrow k$ that contains $b$ (recall that we called this as $b$ being associated to $s \\rightarrow k$ under $B$), then map $b$ to $x_k$. Now take the product of all such variables to obtain a monomial, which is going to be supported on variables indexed by $[n] \\setminus B_0$ as we wanted. \n\nFor example, take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. This family of paths encodes the basis $235$. The passive elements of this basis is actually all of $2,3,5$. Since $3$ and $5$ are sinks of paths, $3$ gives $x_3$ and $5$ gives $x_5$. Now the minimum (under $\\prec_{235}$) path containing $2$ is $(1 \\rightarrow 5)$ so $2$ gives $x_3$. The resulting monomial is $x_3^2 x_5$.\n\n\\begin{proposition}\n\\label{prop:phi}\nLet $\\mathcal{M}_L$ be a positroid. Given a basis $B$, we have $\\phi(B) = \\prod_{k \\in B \\setminus B_0} x_k^{|T_k^B|+1}$ where $T_k^B$ consists of elements $x$ associated to $(s \\rightarrow k) \\in B$ such that replacing $s \\rightarrow k$ with $x \\rightarrow k$ gives some other basis in $\\mathcal{M}_L$.\n\\end{proposition}\n\\begin{proof}\nFrom the way $\\phi$ is defined, it is enough to show that $T_k^B$ consists exactly of the passive elements of $B \\cap B_0$ associated to  $(s \\rightarrow k) \\in B$. It is clear that if we can replace $s \\rightarrow k$ with $x \\rightarrow k$ for $x \\in (s,k)$ then $x$ is passive. Hence we only need to show the other direction: if $x$ is passive then we can replace $s \\rightarrow k \\in B$ with $x \\rightarrow k$ to get another basis. \n\n\nSince $x$ is passive, there is some $x' < x$ such that $B' = B \\setminus x \\cup x' \\in \\mathcal{M}_L$. Consider the matching representation of $B$. Since we have a path $(s \\rightarrow k) \\in B$, from the non-crossing property we have equal number of source points and sink points within $(s,k)$. Moreover, since $x$ is associated to $s \\rightarrow k$, we have equal number of source points and sink points within $(s,x)$ and also within $(x,k)$. This property for $(x,k)$ carries over to the matching representation of $B'$ as well. From this property, we have $(x \\rightarrow k) \\in B'$.\n\\end{proof}\n\nFor example, again take a look at the leftmost picture in the top row of Figure~\\ref{fig:allthebases}. The set $T_5^{235}$ is empty. The set $T_3^{235}$ consists of $2$, since we have $2 \\in (1,3)$ and we can replace $1 \\rightarrow 3$ with $2 \\rightarrow 3$ to get the set $135$, which is again a basis of the positroid.\n\nThe Proposition~\\ref{prop:phi} together with Lemma~\\ref{lem:containment} tells us that $\\phi$ maps the passivity of the basis to the degree of the monomial:\n\n\\begin{corollary}\nLet $\\mathcal{M}_L$ be positroid. Then the passivity of $B$ is equal to the degree of $\\phi(B)$.\n\\end{corollary}\n\nFor example, take a look at the leftmost picture of second row from the top within Figure~\\ref{fig:allthebases}. We have the path $1 \\rightarrow 3$ so the corresponding basis is $234$. Here $2$ and $3$ are passive, but $4$ is active. So $234$ has passive degree to be $2$. The corresponding monomial is $x_3^2$, which again has degree of $2$.\n\n\n\\begin{lemma}\nThe map $\\phi$ is a one-to-one map.\n\\end{lemma}\n\n\n\n\\begin{proof}\nAssume for sake of contradiction that we can find two different bases $B, B'$ of the positroid $\\mathcal{M}_L$ such that $\\phi(B) = \\phi(B')$. In $B$, under the ordering $\\prec_B$, pick the minimal path $(s \\rightarrow k)$ that isn't contained in $B'$. Since $\\phi(B) = \\phi(B')$, there has to be some path $(s' \\rightarrow k) \\in B'$ with $s \\not = s'$. If $s < s'$, from the way $s \\rightarrow k$ was chosen, we can replace $s \\rightarrow k$ with $s' \\rightarrow k$ within $B$ to get a basis so we have $T_k^{B'} \\subsetneq T_k^{B}$. If $s > s'$, we cannot have a path in $B'$ that starts within $(s',s)$ and ends within $(s,k)$ due to the way $s \\rightarrow k$ was chosen. This means we can replace $s' \\rightarrow k$ with $s \\rightarrow k$ within $B'$ to get a basis so we have $T_k^{B'} \\supsetneq T_k^{B}$.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:poi}\nThe map $\\phi$ gives a pure order ideal.\n\\end{proposition}\n\\begin{proof}\nWe first start with the order ideal property. We need to show that given any basis $B$, for each variable $x_k$ appearing in $\\phi(B)$, we can find a basis $B'$ such that $\\phi(B') = \\phi(B)\/x_k$. To use induction, we introduce a new statistic on the poset with elements $B \\setminus B_0$ and ordering $\\prec_B$: we set $\\sigma(k)$ to stand for the length of the longest possible chain having $k$ as the lower end within the poset. \n\nWe are going to use induction on $\\sigma(k)$. We start with the base case, when $\\sigma(k) = 1$, that is when the path $(s \\rightarrow k) \\in B$ is not contained in any other path. If $|T_k^B| = 1$, we simply remove the path to get the $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get our desired $B'$.\n\nNow assume for sake of induction we have proven the result for $\\sigma(k) < t$ and are now considering the case $\\sigma(k) = t$. Same as above, if $|T_k^B| = 1$, we simply remove the path to get the $B'$ we want. Otherwise, we replace $s$ with the smallest element of $|T_k^B|$ to get some other basis $B''$. Then $\\phi(B'')$ is obtained from $\\phi(B)$ by dividing by $x_k$, but the powers of other variables $x_{k'}$ might have gone up for $k \\prec_B k'$. Regardless, in such cases we have $\\sigma(k') < t$, so we can find a new basis by shedding them off thanks to the induction assumption. This gives us the $B'$ we were looking for.\n\nNow for the pure part, fix a basis $B$. We know that every element of $B \\setminus B_0$ is automatically passive since we can simply undo the corresponding path to get a smaller basis. So potential active elements only occur within $B \\cap B_0$. We want to show that if there is some active element, we can always find some basis $B'$ such that $\\phi(B')$ is divisible by $\\phi(B)$. Let $a$ be the largest element of $B$ that is active. We do a case by case analysis based on whether $a+1$ is in $B_0$ or not.\n\nWhen $a+1 \\not \\in B_0$: from Remark~\\ref{rem:noloopcoloop} we have $L_{a,a+1} = 1$. If $a+1 \\in B$, then we have $s \\rightarrow a+1$ in $B$ which is replaceable with $a \\rightarrow a+1$ to get $B'$. This means $a$ is passive so we can ignore this case and say $a+1 \\not \\in B$. In this case, we may add the path $a \\rightarrow a+1$ to get $B'$ we want ($\\phi(B') = \\phi(B) x_{a+1})$. \n\nWhen $a+1 \\in B_0$: first assume for sake of contradiction that $a+1 \\in B$. Then it is passive thanks to the way $a$ was chosen. This means we have $s \\rightarrow k \\in B$ such that $a+1 \\in T_k^B$. From the $\\LE$-property, the path $s \\rightarrow k$ is replaceable with $a \\rightarrow k$ as well, and we get a contradiction from Proposition~\\ref{prop:phi}. \n    \nHence we have that $a+1 \\in B_0 \\setminus B$. This means we have some $(a+1 \\rightarrow k) \\in B$. Let $k'$ be the minimal element such that $a \\rightarrow k'$ is possible inside the $\\LE$-graph. If $k' \\leq k$, from the $\\LE$-property we can switch $a+1 \\rightarrow k$ to $a \\rightarrow k$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{a+1}$). In the case $k' > k$, from the way $k'$ was chosen and the $\\LE$-property, we have $L_{a,k'} = 1$ and $L_{i,j} = 0$ for $i \\leq a$ and $j \\leq k'$ except when $(i,j) = (a,k')$. If the dot at $(a,k')$ is already occupied in any family of non-crossing paths representing $B$, we can replace the starting point of this path with $a$. This would contradict $a$ being active, so we may assume that the dot at $(a,k')$ is never occupied. If there is some path $s \\rightarrow k' \\in B$ with $s' > a$, then replace $s \\rightarrow k'$ with $a \\rightarrow k'$ to get the desired $B'$ ($\\phi(B') = \\phi(B)x_{k'}^t$ with $t \\geq 1$). Otherwise we can simply add $a \\rightarrow k'$ to get the $B'$ we desire ($\\phi(B') = \\phi(B) x_{k'}$).\n\n\\end{proof}\n\n\nThanks to the above proposition, we have established Stanley's conjecture for the class of positroids:\n\n\\begin{corollary}\nThe $h$-vector of any positroid is a pure $O$-sequence.\n\\end{corollary}\n\nAn example is shown in Figure~\\ref{fig:allthebases}. Using the $\\LE$-diagram from Figure~\\ref{fig:lediagram}, all the bases and the corresponding family of non-crossing paths are drawn in Figure~\\ref{fig:allthebases}. The monomials from the mapping $\\phi$ are written below each picture. Proposition~\\ref{prop:poi} guarantees that the collection of monomials we have obtained is indeed a pure order ideal.\n\n\n\\section*{Acknowledgements}\nThe research started under the $2021$-High school math camp hosted at Texas State University. The authors would like to thank the camp organizers for providing support and a great working environment.\n\n\n\n\n\\begin{figure}[h]\n \\begin{center}\n \n  \\vspace{3.7mm}\n    \\input{tikzpics\/basis\/allthebases}\n    \\captionsetup{width=1.0\\linewidth}\n  \\captionof{figure}{Hasse diagram of the $\\LE$-Diagrams which form the bases of the matroid. Each $\\LE$-Diagram is labelled with its corresponding monomial.}\n  \\label{fig:allthebases}\n\n\\end{center}\n\\end{figure}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe field of quantum computing has witnessed astounding developments in the last decade.\nWhile the technology is improving quickly, so-called fault-tolerant quantum computing still seems a distant milestone.\nCurrent devices are limited to relatively few qubits and cannot reliably execute the deep circuits required by many paradigmatic quantum computing algorithms \\cite{preskill2018quantum}.\nYet, near-term computers with a few hundred qubits and low levels of noise can prepare entangled states that cannot be efficiently simulated classically, which can be a computational resource in itself if combined with classical compute appropriately \\cite{arute2019quantum,zhong2020quantum,wu2021strong,madsen2022quantum}, for instance, to mitigate the detrimental effects of noise \\cite{endo2018practical,endo2021hybrid,cai2022quantum,berg2022probabilistic}.\nThis is the reasoning behind most hybrid quantum-classical computing approaches \\cite{bharti2022noisy}.\nWithin this hybrid framework, it was recognised early on that one application stands out from the rest in terms of suitability: the simulation of many-body fermionic quantum systems.\n\n\nSolving electronic structure problems with near-term devices is of paramount importance in fields such as computational chemistry \\cite{kandala2017hardware,barkoutsos2018quantum,ollitrault2020quantum}, which itself has a crucial impact on many industries ranging from material science \\cite{lordi2021advances} to drug discovery \\cite{cao2018potential,blunt2022perspective}, amongst many others.\nIn many cases, the limitations of classical methods for chemistry stem from the inability to account for the complexity of electronic wave functions, which can easily involve a superposition of a combinatorially large number of electronic configurations.\nGiven that quantum processors can physically exhibit complex superpositions, this problem seems particularly appropriate for them to tackle.\nIndeed, in most near-term approaches, the physical state of the device is taken to represent the state of the many-body fermionic system of interest, and the physical properties of the latter are inferred by appropriately measuring physical properties of the former \\cite{mcardle2020quantum,tilly2022variational}.\nHowever, since the fermionic and the many-qubit wavefunctions live in fundamentally different Hilbert spaces, equipped with different algebraic structures, realising this prospect calls for a concrete way to establish connections between both, the so-called fermion-to-qubit mappings.\n\nThe mapping between fermions and qubits used has a direct impact on the quantum simulation.\nA fermionic wavefunction $\\ket{\\Psi_\\mathrm{f}}$ will be encoded into different many-qubit states $\\ket{\\Psi_\\mathrm{q}}$ by different mappings, and these different states will generally not be equally easy to prepare on a given quantum processor.\nUltimately, the latter point depends on the specifics of the hardware used, e.g., its connectivity, coherence time, etc.\nMoreover, one is not generally interested in merely simulating state $\\ket{\\Psi_\\mathrm{f}}$, but in determining physical properties, $\\bra{\\Psi_\\mathrm{f}} \\mathcal{O}_\\mathrm{f} \\ket{\\Psi_\\mathrm{f}}$, where $\\mathcal{O}_\\mathrm{f}$ is some fermionic quantity of interest.\nThe fermion-to-qubit mapping of choice will map said operator into its qubit counterpart, $\\mathcal{O}_\\mathrm{q}$, and the evaluation of $\\bra{\\Psi_\\mathrm{q}} \\mathcal{O}_\\mathrm{q} \\ket{\\Psi_\\mathrm{q}}$ will be carried out via physical measurements on the device.\nTherefore, in general, the measurement cost incurred will also depend on the choice of mapping.\nThe importance of the encoding has thus motivated considerable research towards designing convenient fermion-to-qubit mappings beyond the paradigmatic ones, such as Jordan-Wigner \\cite{jordan_uber_1928}, Bravyi-Kitaev (BK) \\cite{bravyi2002fermionic}, and Parity \\cite{Bravyi2017}.\nMuch work in relation to the design of fermion-to-qubit mappings is directed toward reduction of qubit and Pauli weight (the number of qubits that mapped fermionic operations involve) requirements on lattice models \\cite{Chiew2022, Setia2018, Steudtner2018, Whitfield, Chien2020, Steudtner2019}.\nOther works have addressed the measurement cost in certain fermionic simulation tasks, which directly depends on the Pauli weight of the mapping when using informationally complete measurements, and thus introduced a mapping with provable optimal Pauli weight \\cite{jiang2020optimal}.\nThe latter mapping is generated using regular ternary trees, and largely inspired the current work.\nHowever, the connection between ternary trees and mappings was established earlier.\nIn Ref.~\\cite{vlasov2019clifford}, it was used to find representations of the Clifford algebras and spin groups, and it was discussed that JW in BK can be generated from TT and binary subgraphs of them with the appropriate pairing of the resulting strings.\nMore recently, a framework to design mappings based on BK was put forward in Ref.~\\cite{Li2022}, but it cannot achieve optimal Pauli weight scaling.\n\nIn this paper, we consider the family of mappings generated by identifying Majorana operators with linearly and algebraically independent Pauli strings obeying equivalent anti-commutation relations.\nIn Section~\\ref{sec:theory}, we give an overview to fermion-to-qubit mappings and present a classification in terms of the number of qubits that the non-trivial overlap (NTO) between strings involves.\nWe then focus on one-qubit-wise anti-commuting (1-NTO) encodings, and introduce a ternary-tree-based framework to design mappings within this sub-class.\nMore precisely, we prove that any $n$-node connected ternary tree yields a valid mapping.\n\nImportantly, this framework presents a number of desirable properties.\nOn the one hand, it is clear and intuitive.\nIt enables understanding, and consequentially the control of many important properties of the mappings, such as their locality (that is, on how many qubits fermionic mode occupation is stored, which has been suggested to impact the resilience of quantum simulations to noise \\cite{sawaya2016error}) and their Pauli weight, in terms of simple properties of the underlying ternary trees generating them.\nPart of the discussion in Section \\ref{sec:theory} is aimed at explaining these aspects in a pedagogical manner.\nOn the other hand, it contains the aforementioned paradigmatic mappings, as well as the recently discovered optimal-weight fermion-to-qubit mapping \\cite{jiang2020optimal}, as specific instances.\nHence, we can regard our framework as being able to \\textit{interpolate between} and \\textit{combine} well-known mappings, as well as to generate completely novel ones that bear little resemblance with these.\nIn addition, these two properties together provide, as a byproduct, a clean and transparent perspective on these widely used encodings and their properties.\n\nWe also show that with the right assignment between Majorana operators and Pauli strings, for which we provide a recipe, the tree-based mapping design framework introduced here guarantees a crucial property of the sampled encodings: uncorrelated fermionic states, i.e., Fock basis states (including the vacuum state), are mapped to computational basis states in qubit space.\nThis property, which may be easily overlooked, is of utmost importance, as it ensures no additional quantum resources (more precisely, entanglement) are needed to prepare reference wavefunctions, such as the Hartree-Fock Slater determinant often used as a starting point in many quantum simulation algorithms.\n\nIn Section \\ref{sec:hardware_efficient_mappings}, we put our framework into use.\nWe exploit its versatility to devise an algorithm, which we name \\textit{the Bonsai algorithm}, that takes a hardware connectivity layout as an input and returns a tailored mapping.\nThe resulting encoding is designed to minimise SWAP overhead required in implementation of one- and two-electron excitation operations, which are the building blocks of many adaptive ansatz construction algorithms of the ADAPT-VQE family \\cite{tang2021qubit,grimsley2019adaptive,anastasiou2022tetris}.\nAt the same time, the algorithm aims at minimising the spread of fermionic occupancy over qubits.\nWhen applied to heavy-hexagon qubit connectivity graphs, the standard layout in current IBM devices, the Bonsai algorithm returns a mapping with no SWAP overhead for single-excitation operations and a quadratically lower Pauli weight than JW.\nMoreover, the mapping also drastically reduces circuit complexity of the worst-case scenario implementation of single- and double-excitation generated unitaries with respect to the latter mapping.\n\n\\section{Fermion-to-qubit mappings}\n\\label{sec:theory}\n\nThis section is devoted to discussion of the features of general fermion-to-qubit mappings. \n\n\\subsection{Fermionic systems}\nConsider an $N$-mode fermionic system in second quantisation described in terms of $N$ creation and annihilation operators, $\\{ a^\\dagger_i \\}_{i = 1, \\ldots, N}$ and $\\{ a_i \\}_{i = 1, \\ldots, N}$, which fulfil the usual fermionic canonical anti-commutation relations \n\\begin{align}\n   \\{ a_i, a_j \\} = \\{ a^\\dagger_i, a^\\dagger_j \\} = 0, \\qquad \\{ a^\\dagger_i, a_j\\} = \\delta_{ij} \\mathbbm{1}\n\\end{align}\nThe creation and annihilation operators act on the Fock space belonging to an $N$-dimensional one-particle space $\\mathcal{F}(\\mathbb{C}^N)$.\nThe Fock space $\\mathcal{F}(\\mathbb{C}^N)$ is a $2^N$ dimensional Hilbert space spanned by the fermionic vacuum $\\ket{\\text{vac}_\\text{f}}$ and the vectors obtained by applying  subsets of fermionic creation operators; this orthonormal basis, also called Fock basis, can be denoted as\n\\begin{align}\n    \\ket{n_1, n_2, \\ldots n_N}:= (a^\\dagger_1)^{n_1} (a^\\dagger_2)^{n_2} \\ldots (a^\\dagger_N)^{n_N} \\ket{\\text{vac}_\\text{f}}, \n\\end{align}\nwhere $n_j \\in \\{0,1 \\}$ are the so-called occupation numbers of mode $j$ and the notation $(a^\\dagger_j)^{0}=\\mathbbm{1}$ is used.\nThe fermion-number operator for mode $j$ is given as $\\hat{n}_j=a^\\dagger_j a_j$.\nIt is easy to show that the Fock basis states are eigenstates of these local fermion-number operators with eigenvalue given by the occupation numbers, e.g., $\\hat{n}_1 \\ket{n_1, n_2, \\ldots n_N}= n_1  \\ket{n_1, n_2, \\ldots n_N} $.\n\n\nBesides the fermionic creation and annihilation operators, another useful set of generators for the fermion observables are the $2N$ Majorana operators $\\{ m_k \\}_{k = 1, \\ldots, 2n}$ defined as\n\\begin{equation}\n    \\label{eq:fermion_to_majorana}\n    m_{2 j}=a_{j}^{\\dagger}+a_{j} \\quad \\text{and} \\quad m_{2 j-1}=i\\left(a_{j}^{\\dagger}-a_{j}\\right),\n\\end{equation}\nwhich are unitary, self-adjoint, and obey the majorana anticommutation relations\n\\begin{equation}\n    \\label{eq:majorana_antcomm}\n    \\{m_{i},m_{j}\\}=2\\delta_{ij}\\mathbbm{1}.\n\\end{equation}\nAny fermionic observable can be uniquely expressed as a linear combination of majorana monomials $m_{x_1}...m_{x_j}$. In particular, the number operator for mode $j$ is $\\hat{n}_j=\\frac{1}{2}(\\mathbbm{1}+im_{2j}m_{2j - 1})$. \n\n\\subsection{Jordan-Wigner transformation and general fermion-to-qubit mappings}\n\nThe fermionic Fock space $\\mathcal{F}(\\mathbb{C}^N)$ and the Hilbert space of $N$ qubits $(\\mathbb{C}^{2})^{\\otimes N}$ are both $2^{N}$ dimensional Hilbert spaces, thus one can map one into the other unitarily.\nA very natural unitary mapping is to map the Fock basis states of $\\mathcal{F}(\\mathbb{C}^N)$ to the computational basis states of the qubits such that the occupation number of the $j$-th fermionic mode matches with the state of the $j$-th qubit \\cite{jordan_uber_1928}:\n\\begin{equation}\n  \\mathcal{F}(\\mathbb{C}^N)  {\\ni} \\ket{n_1, n_2, ... n_N} \\mapsto \\ket{n_1} \\otimes \\ket{ n_2}\\otimes \\cdots \\otimes \\ket{n_N} {\\in} (\\mathbb{C}^{2})^{\\otimes N}.\n\\end{equation}\nOn the operator level this correspondence induces a linear mapping between the corresponding observable algebras given by:\n\\begin{align}\n m_{2j-1} \\mapsto \\prod_{k=1}^{j-1} Z_k Y_j, \\\\\n m_{2j} \\mapsto \\prod_{k=1}^{j-1} Z_k X_j,\n\\end{align}\nfor $j=1,2,\\ldots N$. Here and in the rest of the paper, we use the notation $P_j$ with $P \\in \\{X,Y,Z\\}$ for an operator that acts as the Pauli operator $P$ on the the $j$-th qubit and acts as identity on the other qubits. \n\nIn general, a unitary mapping between the fermionic and qubit Hilbert spaces induces a linear mapping on the corresponding observable algebras such that\n\\begin{align}\n    & m_{k} \\mapsto R_k, \\quad k=1, \\ldots 2N, \\label{eq:GenF2Q}\\\\\n    &\\text{(i) $R_k$'s are algebraically independent} ,\\nonumber \\\\\n    &\\text{(ii)}\\;\\{R_k, R_\\ell \\}= 2\\delta_{k\\ell}\\mathbbm{1}. \\nonumber\n\\end{align}\n\nConversely, any linear mapping between the fermionic and qubit observable algebras satisfying the properties (i) and (ii) in Eq.\\eqref{eq:GenF2Q} defines uniquely (up to a global phase factor) a unitary mapping between $\\mathcal{F}(\\mathbb{C}^N)$ and the Hilbert space of $N$ qubits $(\\mathbb{C}^{2})^{\\otimes N}$. \nThis unitary mapping between the two Hilbert spaces can be constructed as follows: Since  the fermionic vacuum state  $\\ket{\\text{vac}_\\text{f}}$ is the unique vector (up to a scalar factor) satisfying the relations $a_j \\ket{\\text{vac}_\\text{f}} =\\tfrac{1}{2}(m_{2j} + i m_{2j -  1}) \\ket{\\text{vac}_\\text{f}}= 0 $ for all $j=1, \\ldots, N$, the vacuum state is mapped to the state  $\\ket{\\psi}$ which satisfies that $\\tfrac{1}{2} (R_{2j} + i R_{2j -  1}) \\ket{\\psi} = 0$ for all $j=1, \\ldots, N$ (note that such a $\\ket{\\psi}$ is unique up to a phase factor). Any other Fock basis vector  $ a^\\dagger_{j_1} a^\\dagger_{j_2} \\ldots a^\\dagger_{j_\\ell} \\ket{\\text{vac}_\\text{f}}$ is mapped to $ \\tfrac{1}{2} (R_{2j_1} {-} i R_{2j_1 -  1}) \\tfrac{1}{2} (R_{2j_2} {-} i R_{2j_2 -  1})  ... \\tfrac{1}{2} (R_{2j_2} {-} i R_{2j_2 -  1}) \\ket{\\psi}$.\n\n\n\\subsection{Majorana string mappings}\\label{subsec:MString}\n\nWhen it comes to useful mappings of fermionic systems to qubit systems, the Pauli basis suggests a path: finding a suitable set $\\mathcal{S}$ of $2n$ Pauli strings (i.e., products of Pauli operators) $S_k$ ($k=1 \\ldots 2n$) fulfilling the anti-commutation $\\{ S_{i}, S_{j} \\} = 2 \\delta_{ij} \\mathbbm{1}$.\nFor this approach to result in a proper fermion-to-qubit mapping, the Pauli strings in $\\mathcal{S}$ must also be linearly and algebraically independent.\nLinear independence is trivially satisfied if all the strings are different, but algebraic independence is more subtle.\nThis means that it must not be possible to find two different subsets $A \\subseteq \\mathcal{S}$ and $B \\subseteq \\mathcal{S}$, $A \\neq B$, such that $\\prod_{S_i \\in A} S_i \\propto \\prod_{S_j \\in B} S_j$, given that the corresponding products of Majorana operators in fermion space result in distinct operators.\nThroughout this work, we will use the term \\textit{Majorana strings} to refer to the Pauli strings within a set $\\mathcal{S}$ satisfying these conditions.  \n\nSummarising the above, in this paper we will consider so called {\\it Majorana string fermion-to-qubit mappings}, which are linear mappings between the fermionic and qubit system observable algebras that satisfy the following criteria:\n\\begin{itemize}\n    \\item {\\bf Criterion~(A)}: Each Majorana operator is mapped to a  Pauli string,  \n    $m_j \\to  S_j \\in \\mathcal{S}$ for $j=1, \\ldots, N$. \n    \\item {\\bf Criterion~(B)}:  The above Pauli strings satisfy $ \\{ S_{i}, S_{j} \\} = 2 \\delta_{ij}\\mathbbm{1}$. \n    \\item {\\bf Criterion~(C)}: For any unequal subsets $A \\subseteq \\mathcal{S}$ and $B \\subseteq \\mathcal{S}$, $A \\neq B$, $\\prod_{S_i \\in A} S_i \\propto \\prod_{S_j \\in B} S_j$ is not fulfilled. \n\\end{itemize}\n\nFurthermore, one often considers Majorana string fermion-to-qubit mappings that satisfy an additional criterion:\n\\begin{itemize}\n    \\item {\\bf Criterion~(D)}: Vacuum preservation, the fermionic vacuum is mapped to the all-zero computational basis state, i.e.,  $\\ket{\\text{vac}_\\text{f}} \\mapsto \\ket{0}^{\\otimes N}$.\n\\end{itemize}\nMappings  satisfying also  Criterion (D) besides Criteria (A)-(C), will be called {\\it vacuum preserving} Majorana string fermion-to-qubit mappings.\n\n\nConsider a Majorana string $S_j$ which is (up to a phase factor) a product of (non-identity) Pauli operators over a subset of sites $A_j \\subset \\{1, \\ldots, N \\}$.\nWe call $A_j$ the {\\it support of  $S_j$}. Let $A_j$ and $A_k$ be the supports of $S_j$ and  $S_k$, respectively.\nWe call  $A_j \\cap A_k$ the {\\it overlapping sites} of $S_j$ and  $S_k$.\nThe subset of  $N_{j,k} \\subseteq A_j \\cap A_k$ where the local Paulis corresponding of the Majorana strings $S_j$ and  $S_k$ are different is called the {\\it non-trivial overlapping sites of $S_j$ and  $S_k$}.\nAs any two Majorana strings anticommute, the number of non-trivial overlapping sites of any pair of Majorana strings must be odd.\nGiven a Majorana string fermion-to-qubit mapping, let $k$ be the maximum of this odd number considering all the pairs of Majorana strings.\nWe call such a mapping a $k$-non-trivial-overlap ($k$-NTO) Majorana string fermion-to-qubit mapping.\nMost of the known fermion-to-qubit mappings (e.g., JW, BK, and Parity) are 1-NTO, but non-1-NTO mappings exist too (see Appendix \\ref{app:exotic}).\n\n\\section{Mappings originating from general ternary trees}\n\\label{sec:mappings_from_trees}\nIn this subsection, we explore the correspondence between a certain class of graphs, called ternary trees (TT), and vacuum preserving fermion-to-qubit mappings.\nAs mentioned previously, the connection between the two has been established before \\cite{vlasov2019clifford, jiang2020optimal}.\nIn Ref.~\\cite{jiang2020optimal}, the minimum-depth TT is used to find a fermion-to-qubit mapping with optimal Pauli weight.\nInspired by that work, we now extend the TT formalism and show that \\textit{any} TT can result in a valid mapping, the properties of which can be directly connected to the graph-theoretical properties of the tree.\nThis connection will be explored more carefully in the next subsection.\nMoreover, the mapping-generating method introduced here guarantees that, for any sampled mapping, the fermionic vacuum is mapped to the all-zeros qubit state, and Fock basis states are mapped to computational basis states; we refer to this property as \\textit{product preservation}.\n\n\\subsection{Ternary trees}\nIt will be useful for this section and the next one to start by reviewing some graph-theoretic concepts.\nA graph is a pair $\\mathcal{G} = (V,E)$ where $V$ is a set of vertices, or nodes, and $E \\subseteq\\left\\{(x, y) \\mid (x,y) \\in V^2, x \\neq y\\right\\}$ are the edges, or links, which are sets of paired vertices.\nWe consider undirected graphs, meaning that $(x, y) \\in E \\Rightarrow (y, x) \\in E$.\nA length-$\\ell$ path $\\mathbf{p}$ in a graph is a length-$\\ell$ ordered sequence of vertices $\\mathbf{p} = \\{ p_1, p_2, \\ldots, p_\\ell \\}$ such that any pair of consecutive vertices in the sequence are connected in the graph, that is, for every $l < \\ell$, $(p_l, p_{l+1}) \\in E$.\nWe call a graph $\\mathcal{G}$ connected if, for any pair of vertices $u, v$, there exists at least one path $\\mathbf{p}$ with $u$ and $v$ as endpoints.\nIn any such graph $\\mathcal{G}$, the path structure induces a well-defined metric distance $d(u, v)$ between all pairs of vertices, defined as the length $\\ell$ of the shortest path (or paths, as they may not be unique) with the two vertices as endpoints.\nIt can be easily shown that the set of distances $d(u, v)$ define a proper metric space, given that they are positive, symmetric, zero iff $u = v$, and they fulfil the triangle inequality, $d(u, v) \\leq d(u, w) + d(w, v), \\, \\forall u, v, w \\in V$.\n\nThe path structure also allows us to define a special kind of graph, the tree.\nA tree $\\mathcal{T}$ is a graph that contains no loops, that is, for which there are no paths $\\mathbf{p}$ containing any node more than once.\nIt is a fact that any connected $N$-node tree contains exactly $N - 1$ edges, and that any connected undirected graph with $N - 1$ edges is a tree.\nIt is also useful to define the degree of a node $u$ as the number of edges reaching $u$,\n\\begin{equation}\n    \\label{eq:degree}\n    \\kappa(u) = \\vert \\{v \\in V \\mid (u, v) \\in E\\} \\vert.\n\\end{equation}\n\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\columnwidth]{general_tree_example.pdf}\n   \\caption{Example of a mapping derived from a ternary tree.\n   The black lines represent the edges, which always connect two qubits, while the legs are depicted in red.\n   Throughout this paper, we always represent the Pauli labelling on the links (both edges and legs) by their position: the left-most link below a node is labelled with an $X$, while the right-most with a $Z$.\n   Every leg in the tree can be associated with a Pauli string by following the path from the root node (0, in this case) to the leg.\n   Every time a link with label $P$ stemming downwards from a qubit $u$ is crossed, the operator $P$ acting on qubit $u$ is added to the string.\n   The resulting string acts trivially on all qubits not visited along the path.\n   For example, with the tree in the figure, the leg labelled with $S_{0}$ generates the string $S_0 = X_0Z_1$, while $S_1 = Y_0 Z_2 Z_8 Z_{10}$.\n   The green lines depict the pairing between Majorana strings that guarantees product preservation.\n   Notice that, following the upward paths starting from the two legs of any such pair, the two paths meet at a qubit $u$ once they both cross their first link not labelled with a $Z$.\n   In the previous example with strings $S_0$ and $S_1$, this corresponds to the root node 0.}\n   \\label{fig:example_tree}\n\\end{figure}\n\nWith these definitions at hand, we can introduce the TT.\nEssentially, a TT is a tree in which the branching rate is at most three, that is, each node has at most three \\textit{descendants}.\nTo explain this concept more precisely, let us describe a process in which a TT is built by adding nodes sequentially.\nFirst, we start with a single node, the \\textit{root} $r$.\nNext, we add $k_r \\leq 3$ nodes to the graph and connect each of them to the root $r$.\nThe root now has degree $\\kappa(r) = k_r$.\nNext, for each of these new nodes (\\textit{descendants} of $r$) $u$, we add $0 \\leq k_u \\leq 3$ nodes, which connect to $u$.\nThe process can be iterated until the graph contains $N$ nodes.\nNotice that, since each time we add a node we add one link along with it, the total number of edges will be $N-1$ (this observation also shows why the resulting graph is a tree: it is not possible to close any loops in the graph by connecting one new node to a single existing one only).\nAny node without descendants will be denoted \\textit{leaf} throughout this work.\n\n\\subsection{From ternary trees to fermion-to-qubit mappings}\nThe starting point of this work is the observation that, following a simple procedure introduced in Refs.~\\cite{vlasov2019clifford,jiang2020optimal} for specific TT, any $N$-node TT can be used to generate valid fermion-to-qubit mappings.\nWe now explain this procedure and show its generality.\nThe scheme is illustrated in Fig.~\\ref{fig:example_tree} where we present an 11-qubit example.\n\nSuppose we have an $N$-qubit system that we want to use to simulate an $N$-mode fermionic system.\nWithout loss of generality, we may label each qubit  with an integer number $u = 0, \\ldots, N - 1$ in what follows.\nTo generate a fermion-to-qubit mapping, we first generate an $N$-node TT, by for instance following the iterative procedure introduced previously.\nNext, we assign a qubit label $u$ to each node.\nDifferent assignments will lead to different mappings (a degree of freedom that can be exploited), but any such labelling is admissible.\nIn the example (Fig.~\\ref{fig:example_tree}), this corresponds to the tree with blue nodes and black edges.\n\nThe next step in the process is to add $3 - k'_u$ \\textit{legs}, that is, edges without a node at the other end of the link, to each node $u$, where $k'_u$ is the number of descendants of $u$.\nNote that $k'_r = k_r$ for the root $r$, while $k'_u = k_u - 1$ for all other nodes.\nThese are the red links in Fig.~\\ref{fig:example_tree}.\nImportantly, doing this will result in $2N + 1$ legs for any tree.\nTo see this, let us denote the number of legs by $L$. \nEvery node except the root is now reached by four \\textit{links} (a term that we will use in this work to refer to both edges in the original tree and legs), while the root is reached by three.\nThus, if we sum all the new degrees (including legs) for all the nodes, we obtain $F = 4N - 1$.\nIn this sum, we have counted each of the edges in the original tree twice (once per each node at its endpoints), while the legs have been counted only once.\nHence, we have $F = |E| + L$, with $|E| = 2(N - 1)$ (since each edge is contained twice in $E$), so $L = 2N + 1$.\n\nOnce the legs have been added, and every node has exactly three descending links (either edges or legs), we distribute the labels $X$, $Y$, and $Z$ among said three links of each node.\nIn the figures in this work, in order to ease the graphical depictions of the trees, the left-most link implicitly carries label $X$, while the right-most one, $Z$, as in Ref.~\\cite{jiang2020optimal}.\n\nWith this labelling of nodes, edges, and legs, the tree can be used to generate Pauli strings in the following manner.\nFor every leg, there exists a unique path leading from it to the root $r$.\nThe path only includes one leg, the starting one, and it may cross some edges as well.\nTo every link in the path, we can associate a unique Pauli matrix, $P_u$, where $u$ is the parent node reaching the link, and $P$ is the label, $X$, $Y$, or $Z$, corresponding to the link.\nThe Pauli string is therefore formed by taking the tensor product these Pauli operators, along with identity on all nodes not along the path.\nSince every path results in a different Pauli string, this procedure generates $2N + 1$ strings for $N$ qubits.\n\nImportantly, all these Pauli strings anticommute with one another. \nThis can be seen by considering two Pauli strings $S_i$ and $S_j$ stemming from two different legs in the tree.\nThe two paths corresponding to the legs must meet, that is, when traversing them upwards towards the root, they must have a first node in common (which may be the root itself).\nIf the first common ancestor to both legs is not the root, both paths from that node upwards are equal, and hence both $S_i$ and $S_j$ contain the same Pauli operators for those qubits.\nThe first common ancestor, on the other hand, must be reached following two different descendants of said node, so the Pauli matrices for that qubit in the strings are distinct, and both different from identity (what we refer to as \\textit{non-trivial overlap}).\nIf the legs are not directly connected to their common ancestor, their paths include other nodes laying below the latter in the tree.\nHowever, notice that, by definition of first common ancestor, those nodes cannot be present in both paths, and hence one of the strings must act trivially on each such qubit.\nIn short, $S_i$ and $S_j$ have a single-qubit non-trivial overlap, that is, for every qubit different from the aforementioned first common ancestor, the corresponding Pauli matrices are either both equal or at least one of them is equal to identity.\n\nSince all the $2N + 1$ resulting Pauli strings are different, they are obviously linearly independent, but they are not algebraically independent.\nIn particular, any two disjoint subsets of strings $A$ and $B$, $A \\cap B = \\emptyset$ such that $A \\cup B$ is the whole set of strings fulfil $\\prod_{S_i \\in A} S_i \\propto \\prod_{S_j \\in B} S_j$.\nHowever, as we prove in in Appendix \\ref{app:alg_ind_proof}, any subset missing at least any one Pauli string is algebraically independent.\nTherefore, by dropping any of the strings, the remaining $2N$ ones can be readily identified with the $2N$ Majorana operators associated with the $N$-mode fermionic system, thus defining a valid fermion-to-qubit mapping.\n\n\n\\subsection{Majorana string pairing for product preservation}\\label{sec:pairing}\nThe previous discussion illustrates how any $N$-node TT can be used to obtain $N$-mode mappings.\nWhile any association between the generated Pauli strings and Majorana operators results in a legitimate mapping, not all are equally useful in practice.\nMany applications of fermion-to-qubit mappings require that at least some reference state (e.g., the vacuum) be known in qubit space.\nIn near-term quantum computing, for instance, it is also desirable that Fock basis states be mapped to computational basis states.\nWe now provide a simple recipe to enforce this product preservation feature of the map, including to guarantee that the fermionic vacuum is mapped to the state $\\ket{0}^{\\otimes N}$.\n\nFirst, let us introduce the concept of \\textit{pairing}.\nAccording to Eq.~\\eqref{eq:fermion_to_majorana}, there are two Majorana operators, $m_{2j}$ and $m_{2j - 1}$, associated to every fermionic mode $j$.\nTherefore, after identifying Majorana strings with Majorana operators, every creation and annihilation operators $a^{(\\dagger)}_j$ will be associated with two Pauli strings.\nThe key to product preservation lies in how the Pauli strings are paired into fermionic modes.\n\nLet the set of Majorana strings be the set obtained by removing the Pauli string corresponding to the path that only involves links with label $Z$, and consider the following pairing algorithm.\nFor every node $u$ in the TT, follow its downwards link labelled with $X$.\nIf the link is not a leg, keep travelling downwards taking always the $Z$-links until a leg is reached.\nDenote the leg by $s_x^{(u)}$.\nThe same procedure, starting from the link with label $Y$ will lead to a different final leg $s_y^{(u)}$.\nThe two Pauli strings $S_{s_x^{(u)}}$ and $S_{s_y^{(u)}}$ should then be paired together into some fermionic mode $j$, that is, one of them should be identified with $m_{2j}$ and the other one with $m_{2j - 1}$.\nThese pairings are illustrated with green lines in Fig.~\\ref{fig:example_tree}.\nThe simplest identification corresponds to the mapping\n\\begin{equation}\n    \\label{eq:pairing_identification}\n    m_{2j} \\leftrightarrow S_{s_x^{(u)}} \\quad \\text{and} \\quad m_{2j - 1} \\leftrightarrow S_{s_y^{(u)}},\n\\end{equation}\nbut it is worth mentioning that it is also possible to ensure that the mapped creation and annihilation operators are real in qubit space by associating with $m_{2j}$ the Pauli string that contains an even number of $Y$ operators and with $m_{2j - 1}$ the one containing an odd number of them.\nHowever, for the sake of simplicity, we will only consider the first type of identification explicitly throughout this work.\nImportantly, notice that the identification between modes $j$ and qubits $u$ in Eq.~\\eqref{eq:pairing_identification} implicitly establishes a bijection between the two sets.\n\n\\LinesNumbered\n\\begin{algorithm}[t]\n    \\label{alg:pairing_scheme}\n\n    \\caption{Pairing scheme}\n    \n        Choose a bijection $f$ between modes $j$ and qubits $u$, $j = f(u)$.\n\n        Let $V$ be the set of qubits in the tree.\n        \n        \\For{$u \\in V$}{\n        \n            Define $s$ as the $X$-labelled downward link stemming from $u$.\n            \n            \\While{$s$ not a leg}{\n            \n                Define $v$ as the qubit reached following $s$ downwards.\n                \n                Define $s$ as the $Z$-labelled downward link stemming from $v$.\n                \n            }\n            \n            Set $s \\rightarrow s_x^{(u)}$.\n            \n            Define $s$ as the $Y$-labelled downward link stemming from $u$.\n            \n            \\While{$s$ not a leg}{\n            \n                Define $v$ as the qubit reached following $s$ downwards.\n                \n                Define $s$ as the $Z$-labelled downward link stemming from $v$.\n                \n            }\n            \n            Set $s \\rightarrow s_y^{(u)}$.\n            \n        }\n        \n        Remove the unpaired right-most $Z$-leg from the tree.   \n        \n        Create mode operators $a_j$ and $a^{\\dagger}_j$ using Eq.~\\eqref{eq:mapped_creation_annihilation} with $j = f(u)$.\n        \n        Return mapped mode operators.\n        \n\\end{algorithm}\n\nBy inverting Eq.~\\eqref{eq:fermion_to_majorana}, we see that the fermionic creation and annihilation operators are mapped into qubit space according to\n\\begin{equation}\n    \\label{eq:mapped_creation_annihilation}\n    \\begin{aligned}\n    a^{\\dagger}_j &= \\frac{1}{2} (m_{2j} - i m_{2j - 1}) \\leftrightarrow \\frac{1}{2} \\left(S_{s_x^{(u)}} - i S_{s_y^{(u)}}\\right), \\\\\n    a_j &= \\frac{1}{2} (m_{2j} + i m_{2j - 1}) \\leftrightarrow \\frac{1}{2} \\left(S_{s_x^{(u)}} + i S_{s_y^{(u)}}\\right).\n    \\end{aligned}\n\\end{equation}\nFrom these expressions, it is possible to show that the fermionic vacuum is represented by the qubit state $\\ket{0}^{\\otimes N}$.\nTo do so, consider the two Pauli strings $S'_{s_x^{(u)}}$ and $S'_{s_y^{(u)}}$ obtained by substituting the $Z$ operators on the qubits that lie below $u$ on the tree (if any) with identity.\nClearly, $S'_{s_x^{(u)}}$ and $S'_{s_y^{(u)}}$ are equal except for the Pauli operator acting on qubit $u$, which implies that $S'_{s_x^{(u)}} + i S'_{s_y^{(u)}}$ contains Pauli operators on all qubits except for $u$, where it contains a $2 P^{-} = X + i Y$ operator.\nMoreover, since $S'_{s_x^{(u)}} \\ket{0}^{\\otimes N} = S_{s_x^{(u)}} \\ket{0}^{\\otimes N}$, and similarly for $S'_{s_y^{(u)}}$,\n\\begin{equation}\n    \\label{eq:vacuum_all_zeros}\n    \\left(S_{s_x^{(u)}} + i S_{s_y^{(u)}}\\right) \\ket{0}^{\\otimes N} = 0.\n\\end{equation}\nA similar argument for the creation operator can be used to show that $(S_{s_x^{(u)}} - i S_{s_y^{(u)}}) \\ket{0}^{\\otimes N}$ is proportional to a computational basis state: since $(S_{s_x^{(u)}} - i S_{s_y^{(u)}}) \\ket{0}^{\\otimes N} = (S'_{s_x^{(u)}} - i S'_{s_y^{(u)}}) \\ket{0}^{\\otimes N}$, its action is to flip qubit $u$'s state from $\\ket{0}$ to $\\ket{1}$, as well as possibly to flip any other qubits above $u$ in the tree for which the traversed links are labelled with an $X$ or a $Y$.\nIn Appendix \\ref{app:prod_pres_proof}, we extend this argument to prove that any Fock basis state is mapped into a computational basis state in qubit space.\n\nAfter applying the pairing scheme to a qubit-tree, we may write the general form of the $j$-th mode operator as\n\\begin{equation}\n    \\label{eq:general_form_tree_op}\n    a_j^{(\\dagger)} \\mapsto \\frac{1}{2} \\left( X_{u} \\prod_{k \\in \\mathcal{Z}^{(u)}_x} Z_k \\pm i Y_u \\prod_{k \\in \\mathcal{Z}^{(u)}_y} Z_k \\right) G_u,\n\\end{equation}\nwhere $\\mathcal{Z}^{(u)}_x$ and $\\mathcal{Z}^{(u)}_y$ are the sets of qubits on which $S_{s_x^{(u)}}$ and $S_{s_y^{(u)}}$ act non-trivially below qubit $u$ in the tree, and the string $G_u$ is a common Pauli operator to both Majorana strings, that is, involving all the qubits in the path from $u$ to the root $r$.\nThe sets $\\mathcal{Z}_{x\/y}$ may be empty, in which case the operator reduces to a form similar to that of the Jordan-Wigner mapping, $a_i^{(\\dagger)} \\rightarrow P_u^\\pm G_u$, where $G_u$ is a Pauli string which enforces anti-symmetry with other modes, exactly like the Jordan-Wigner $Z$-chain.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.95\\columnwidth]{delocalisation_explanation.pdf}\n    \\caption{Figures depicting the effect of edge labelling below a node $i$.\n    Its two descendants, $j$ and $k$, both have descendants, and the all-$Z$ paths underneath them have length $L_j$ and $L_k$, respectively.\n    We assume that all links in the tree except for the ones right below $i$ have the same label in both figures (a) and (b).\n    In the left figure, both paths contribute to occupation delocalisation of the mode associated with $i$, while in the right figure, only the path below $j$ does.\n    In the latter case, the path of length $L_k$ may contribute to the delocalisation of some other mode unless the path from $i$ to the root only crosses $Z$-labelled edges.}\n    \\label{fig:labelling_choice}\n\\end{figure}\n\n\\subsection{Properties of the mappings and the effect of labelling}\n\\label{sec:mapping_properties}\nIn the construction of a TT mapping, there are two main degrees of freedom: the tree and the labelling.\nIn this subsection we briefly discuss how the properties of these elements impact the resulting mappings.\n\nAn important feature of a fermion-to-qubit mapping is its \\textit{Pauli weight}.\nThe Pauli weight of a Pauli string is defined as the number of qubits on which the string acts non-trivially (that is, with a Pauli operator different from identity).\nIn the case of a mapping, this refers to the Pauli weight of the strings of the mapped fermionic operators.\nIn the case of TT mappings, this can be easily analysed by considering their Majorana strings.\nMore concretely, notice that the Pauli weight of a Majorana string generated from a TT is precisely the length of the corresponding path from root to leg.\nThus, there is a straightforward connection between the Pauli weight of the mapping and the average shortest path length to the root in the tree.\nMore generally, the topology of the tree impacts the sets of qubits involved when applying creation or annihilation operators, but not how (i.e., by which Pauli operators).\n\nThe labelling, instead, has a more subtle impact on the resulting map.\nWhile it cannot affect the average Pauli weight of the resulting Majorana strings, it impacts how the occupation of the fermionic modes is \\textit{delocalised} over qubits.\nMore precisely, consider the number operator $n_j = a^{\\dagger}_j a_j$ for a fermionic mode $j$ in qubit space.\nUsing Eq.~\\eqref{eq:mapped_creation_annihilation}, we see that\n\\begin{equation}\n    \\label{eq:number_op_majorana}\n    n_j = \\frac{1}{2} \\left( \\mathbbm{1} + i m_{2j} m_{2j - 1} \\right) \\leftrightarrow \\frac{1}{2} \\left( \\mathbbm{1} + i S_{s_x^{(u)}} S_{s_y^{(u)}} \\right).\n\\end{equation}\nSince the two Pauli strings $S_{s_x^{(u)}}$ and $S_{s_y^{(u)}}$ are equal on all qubits above $u$ in the tree, the product on the right-hand side results in the identity operator for those.\nOn $u$, on the other hand, the product yields $XY = iZ$.\nFor the qubits below $u$ in the strings, however, one string acts with a $Z$ operator while the other one with identity.\nTherefore, if we define the set $\\mathcal{Z}_u$ of qubits below $u$ on which $S_{s_x^{(u)}}$ and $S_{s_y^{(u)}}$ act non-trivially, and include $u$ itself too, we have\n\\begin{equation}\n    \\label{eq:number_op_z}\n    n_j = a^{\\dagger}_j a_j \\leftrightarrow \\frac{1}{2} \\left( \\mathbbm{1} - \\prod_{q \\in \\mathcal{Z}_u} Z_q\\right),\n\\end{equation}\nthat is, the occupation of a mode $j$ is encoded in the parity of the state of the qubits in $\\mathcal{Z}_u$.\nNow, given a qubit $u$ with edges directly below itself, the choice of label for each of these edges will generally affect the structure of sets $\\{ \\mathcal{Z}_u \\}_u$ in the resulting mapping and, with it, its delocalisation structure.\n\nTo analyse this in an illustrative manner, let us introduce a convenient definition of mode-specific delocalisation $D_u$ in terms of the qubit $u$ the mode is associated with in a mapping,\n\\begin{equation}\n    \\label{eq:def_delocalisation}\n    D_u = | \\mathcal{Z}_u | - 1.\n\\end{equation}\nNow, consider a generic example in which a node $i$, which is not the root, has two descendants $j$ and $k$ (see Fig.~\\ref{fig:labelling_choice}).\nNodes $j$ and $k$ themselves may have descendants.\nSuppose that we add legs to all qubits and we label all links in the resulting tree except for the three links stemming downwards from $i$.\nThe question is then how those three links should be labelled.\nOf course, there are three possibilities (three labels to be distributed among three links), but since swapping $X \\leftrightarrow Y$ labels between two links stemming from the same node has a simple impact on the mapping (switching the roles of $S_{s_x^{(u)}}$ and $S_{s_y^{(u)}}$), the only two situations to be discussed are whether the $Z$ label should be assigned to an edge or the leg.\n\nIn the left figure, we depict the case in which the leg is assigned the $Z$ label, and the two edges, $X$ and $Y$.\nIn this case, $\\mathcal{Z}_i$ contains $i$ and all the nodes in the $Z$-strings lying below $j$ and $k$, which in the example have length $L_j$ and $L_k$, respectively.\nTherefore, the occupation of the mode associated with qubit $i$ is delocalised among $D_i = L_j + L_k$ qubits.\nIn the opposite case, in which the $Z$ label is assigned to one of the edges, on the other hand, one of the two $Z$-strings no longer contributes to the delocalisation of node $i$.\nIn the illustration, we have $D_i =L_j$, that is, the occupation of the mode is less spread in the second case.\n\n\\begin{figure*}[t]\n   \\centering\n   \\includegraphics[width=0.8\\linewidth]{canonical_trees.pdf}\n   \\caption{Ternary trees generating four paradigmatic mappings.\n   a) Jordan-Wigner, b) Parity, c) Bravyi-Kitaev and d) JKMN mapping.\n   From the analysis presented throughout Sect.~\\ref{sec:mapping_properties}, we can see that both Jordan-Wigner and Parity have linear-scaling depth, and hence Pauli weight, while Bravyi-Kitaev and JKMN have $O(\\log_2 N)$ and $O(\\log_3 N)$ scaling, respectively.\n   Regarding the delocalisation, it can be seen that Jordan-Wigner minimises it, while Bravyi-Kitaev and Parity have no nodes other than the root in the all-$Z$ branch.\n   JKMN has a non-minimal all-$Z$ branch including $O(\\log_3 N)$ nodes, and hence is nearly maximally delocalised for large $N$.}\n   \\label{fig:paradigmatic_mappings}\n\\end{figure*}\n\nHowever, notice one important fact: assume we follow the path from $i$ upwards towards the root $r$ until we reach an edge labelled with an $X$ or a $Y$, and let us call $u$ the node reached by traversing that edge.\nIn the latter case (right-hand side of the figure), the $Z$-string along qubit $k$ is now part of one of the $Z$ strings directly below qubit $u$.\nIn other words, while the delocalisation of the mode associated with $i$ has decreased by an amount $L_k$, the delocalisation of the mode associated with $u$ has increased by the same amount.\nTherefore, under this assumption, the labelling cannot affect the average delocalisation of the mapping, but only its distribution among the qubits.\nCrucially, if the path from $i$ to $r$ only crosses $Z$-labelled edges, this is no longer true, and the second labelling does not increase the delocalisation of any other mode.\n\nFrom the above discussion, we can draw a very useful overall conclusion regarding the delocalisation structure of a mapping: the average delocalisation among nodes is given by\n\\begin{equation}\n    \\label{eq:avg_delocalisation}\n    \\left\\langle D_u \\right\\rangle = 1 - \\frac{h_Z}{N},\n\\end{equation}\nwhere $h_Z$ is simply the number of nodes that can reach the root node $r$ by traversing only $Z$-labelled edges, including the root itself.\nThis can be seen as follows.\nIf a node $i$ cannot be traced back to the root following $Z$-labelled edges, the path towards $r$ must cross an $X$- or $Y$-labelled edge attached to some node $u$, and thus $i$ contributes one unit to the delocalisation of node $u$.\nTherefore, the sum of all delocalisations must be equal to the number of nodes not in the $Z$-labelled path, that is, $\\sum_u D_u = N - h_Z$.\nThis observation implies that, in order to minimise the delocalisation of the modes, which may be a desirable property of a fermion-to-qubit mapping \\cite{sawaya2016error}, we must maximise the number of nodes along the $Z$-only path.\nInterestingly, since $h_Z \\in \\{ 1, \\ldots, N \\}$, the average delocalisation is bounded $\\left\\langle D_u \\right\\rangle \\in [0, 1 - 1\/N]$, that is, on average, the occupation of the fermionic modes is stored in less than two qubits, $\\left\\langle | \\mathcal{Z}_u | \\right\\rangle \\in [1, 2 - 1\/N]$.\n\n\\subsection{The ternary trees of paradigmatic mappings}\nIt is illustrative to analyse paradigmatic fermion-to-qubit mappings in this context.\nIn particular, Jordan-Wigner (JW), Bravyi-Kitaev (BK), Parity (P), and obviously the optimal mapping from Ref.~\\cite{jiang2020optimal} (JKMN) are all 1-NTO and can be generated from TT.\nIn Fig.~\\ref{fig:paradigmatic_mappings} we depict their corresponding trees.\nBy analysing their graph topologies, and following the insights from the previous discussion, we can easily understand their main properties.\n\nBoth JW and P are given by linear graphs.\nSince these are depth-$N$ trees, the Pauli weight of the resulting Majorana strings is $O(N)$.\nHowever, their occupation delocalisation is different.\nJW is an extreme case, given that all nodes are in the $Z$-labelled path and thus has average delocalisation $\\left\\langle D_u \\right\\rangle = 0$; this is the only possible TT mapping with no delocalisation.\nIn the case of P, the occupation is maximally delocalised, with each occupation encoded between two consecutive nodes in the chain, except for the last qubit, in which it is fully localised.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{JW_TT_4qubit_map.pdf}\n    \\caption{Qubit operators resulting from the product of two even Majorana operators, $m_{2i} m_{2j}$, implemented on a linear and a clustered topology.\n    The operators are mapped to qubit space using JW and a four-qubit TT mapping in which the root has degree three.\n    Each product of Majorana operators yields a Pauli string, which is written explicitly in the left-most column of each mapping.\n    The red (blue) highlight corresponds to qubits on which the Pauli strings act non-trivially.\n    A highlight line skipping over qubits denotes ones that are not present in the strings but are involved in the cascade of SWAP gates.\n    }\n    \\label{fig:swap_overhead_example}\n\\end{figure*}\n\nBK and JKMN, instead, are generated by trees with constant branching rate 2 and 3, respectively.\nTherefore, their depths, and hence the resulting Pauli weights, scale as $O(\\log N)$, with JKMN having a smaller depth owing to its higher branching rate (in fact, the authors prove the optimality of the Pauli weight of their mapping in Ref.~\\cite{jiang2020optimal}).\nIn both cases, however, the price to pay is the delocalisation of the occupation.\nMore precisely, notice that the higher up the tree the common ancestor of a given pairing is, the more $Z$-links are involved in the resulting Majorana strings.\nThus, only the lowest-lying nodes lead to completely localised modes.\n\n\\section{Growing hardware-efficient mappings with the bonsai algorithm}\n\\label{sec:hardware_efficient_mappings}\nTernary tree mappings can be used as a tool for the design of custom fermion to qubit mappings.\nThe framework introduced in Sect.~\\ref{sec:mappings_from_trees} is general and can help find mappings with specific desired properties by tailoring the trees according to different cost functions.\nIn what follows, we introduce an algorithm to produce mappings aimed at reducing the complexity of fermionic simulations on quantum computers by minimising the impact of limited qubit connectivity in the quantum processor.\nWe start this section by briefly introducing the problem, and we then present the Bonsai algorithm along with an illustrative and important use-case: heavy-hexagon qubit lattices, the topology of choice for current IBM quantum computers.\n\n\\subsection{Fermionic simulation under limited connectivity constraints}\n\\label{sec:limited_connectivity}\nThe simulation of fermionic many-body systems is one of the most promising applications of quantum computing, both in the near-term and in the fault-tolerant era.\nMany of the existing algorithms work in second quantisation, and thus typically require mapping the fermionic operators into qubit space.\nFermionic operations are then mapped to unitary gates among the qubits in the device.\nHowever, many platforms (such as superconducting qubits) have limited connectivity, meaning that many pairs of qubits in the processor cannot physically interact directly.\nThus, when a quantum gate involves qubits that are not physically connected, SWAP gates are iteratively applied so that the state of distant qubits are transported to neighbouring ones, and the gate then applied.\nWhile this is always possible in theory, in practice, the additional SWAP gates increase the circuit complexity, which results in longer runtimes and, consequently, in increased detrimental effect of noise.\n\nIn order to illustrate how limited connectivity impacts the circuit complexity, let us consider a minimal example with four fermionic modes simulated with four qubits on two different platforms, one with linear connectivity (the physical connectivity graph being a one-dimensional chain), and a second one with a star-like connectivity (three of the four qubits connected to the fourth, and no other connections).\nBoth are depicted in Fig.~\\ref{fig:swap_overhead_example}.\nWe now map four fermionic modes to these qubits using JW and a TT mapping.\nFor the latter, we consider the specific situation in which the ternary tree is congruent with the connectivity graph of the qubits: qubit 0 is the root, and the other three qubits are its descendants.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{HH_bonsai.pdf}\n    \\caption{a) Connectivity graph of a 37-qubit heavy-hexagon processor.\n    b) Unfolded tree, illuminating how the paths of the tree follow the hardware topology in order to create a hardware efficient mapping.\n    Colouring of vertices indicates where the three initial branches from the root-qubit unfold to.\n    c) The same tree as in b), folded in the layout used in the rest of this work.}\n    \\label{fig:heavy_hexagon_st}\n\\end{figure*}\n\nWe now examine the simulation of the even-even Majorana terms, $m_{2i} m_{2j}$ for $i\\neq j$, arising from single excitation terms, $a_i^{\\dagger} a_j$.\nThe resulting six Pauli strings are tabulated in the figure.\nIn many applications, these terms must be exponentiated and implemented as rotations, i.e., $\\exp{-i\\theta m_{2i} m_{2j}}$.\nThis requires entangling gates between the qubits not acted upon by an identity in the corresponding Pauli string.\nAs explained above, if two such qubits are not neighbours in the physical connectivity graph of the device, SWAPs must be applied.\nFigure \\ref{fig:swap_overhead_example} highlights the qubits involved in implementing the rotations.\nThe SWAP overhead is indicated by thin lines skipping over the qubits.\n\nIn Fig.~\\ref{fig:swap_overhead_example}, we observe that with the TT and linear connectivity, three operators, $m_{0}m_{6}, m_{2}m_{6}, m_{4}m_{6}$, involve all four qubits even though the actual Pauli strings only act on three qubits each.\nThe JW mapping, on the other hand, is more congruent with the underlying connectivity.\nIn the large $N$ limit, the regular TT mapping presents an advantage in terms of Pauli weight with respect to JW (the former scales as $O(\\log_3 N)$, while the latter as $O(N)$) so, in principle, each such rotation would involve much fewer qubits.\nHowever, the SWAP overhead with limited connectivity reduces the Pauli weight advantage for the regular TT (and similarly for BK) and an amount of CNOTs equivalent to JW may typically be required.\nIn the case of the star-graph connectivity, on the other hand, the TT mapping never requires SWAPs, as opposed to JW, and moreover no operation involves more than three qubits.\n\nThis simple example illustrates why limited connectivity can be an issue for the implementation of fermionic operations, and also that the right choice of mapping, in particular one that is congruent with the underlying connectivity, can help mitigate the overhead.\n\n\\subsection{The Bonsai algorithm}\n\\label{sec:bonsai_algorithm}\nIn this section, we introduce an algorithm to generate custom fermion-to-qubit mappings tailored to device-specific connectivity graphs.\nMore precisely, the problem is, given a quantum processor, to find a mapping such that: 1) it is product-preserving, 2) the resulting Pauli weight is low, and 3) mode occupancy is local in qubit space.\nThe first condition is satisfied by appropriate pairing as described in Sect.~\\ref{sec:pairing}.\nThe second and third points are suitably satisfied by finding a ternary tree that is a subgraph of the physical connectivity graph (or close to one), and then exploiting the labelling freedom to define how the mode occupancy is distributed over qubits in a rational manner.\nIn the following, we present this heuristic strategy in detail and illustrate it with an important application: designing mappings for heavy-hexagon quantum computers.\nThe steps of the algorithm are summarised in Algorithm \\ref{alg:bonsai}, while the specific subroutines are described in detail in Appendix \\ref{app:algorithms}.\n\n\\textit{Finding the ternary tree.}\nThe input of the Bonsai algorithm is a physical connectivity graph $\\mathcal{P} = (V_\\mathcal{P}, E_\\mathcal{P})$, in which the nodes are the qubits in the processor and the edges represent the pairs of qubits onto which it is possibly to physically apply entangling gates.\nIn Fig.~\\ref{fig:heavy_hexagon_st} a), we depict the physical connectivity graph $\\mathcal{P}$ of a 37-qubit heavy-hexagon computer.\nNow, the strategy to minimise the SWAP overhead is to find a TT, $\\mathcal{T} = (V_\\mathcal{T}, E_\\mathcal{T})$, that is congruent with the topology of $\\mathcal{P}$.\nMore precisely, suppose that $\\mathcal{T}$ is a subgraph of $\\mathcal{P}$ (that is, $V_\\mathcal{T} = V_\\mathcal{P}$ and $E_\\mathcal{T} \\subseteq E_\\mathcal{P}$).\nThen, any path from root to leaf in $\\mathcal{T}$ is a path in $\\mathcal{P}$ and, consequently, no SWAPs are required to apply a gate generated by a Majorana string.\nA similar argument can be used for gates generated by single-excitation operators.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=.95\\linewidth]{HH_delocalisation.pdf}\n    \n    \\caption{Mappings resulting from two different labelling strategies applied to the tree in Fig.~\\ref{fig:heavy_hexagon_st}.\n    a) By applying the homogeneous localisation labelling, modes associated with qubits with descendants are delocalised in a rather even fashion: nodes with one and two descendants have delocalisation $D_u = 1$ and $D_u = 2$, respectively.\n    The zoomed in area involving qubits 4, 7, 13, 19, and 20 further illustrate the occupancy distribution.\n    The modes associated with 4 and 13 involve three qubits, mode in qubit 7 involves 2, and the ones in 19 and 20, only one.\n    b) The application of the heterogeneous localisation strategy yields a very different delocalisation structure.\n    Nodes with one descendant are completely localised in this case ($D_u = 0$), but nodes with two descendants can be fairly delocalised.\n    For instance, looking again at the four nodes in the shaded area, we see that the modes in qubits 7 and 13 are more localised than in the previous case, while the one in qubit 4 is delocalised among more qubits, with $D_4 = 3$.\n    The particular case of the root node is a clear example, as it now exhibits delocalisation $D_0 = 14$.}\n    \\label{fig:mode_op_tree_folded}\n\\end{figure*}\n\nA tree subgraph $\\mathcal{T}$ that spans all the nodes in a graph $\\mathcal{P}$ is called a \\textit{spanning tree} (ST).\nIf $\\mathcal{P}$ is a tree itself, then the choice of ST is unique.\nA general graph, however, may have several STs.\nA degree-$\\kappa$ constrained ST is one such tree that has no vertices with degree greater than $\\kappa$.\nIn our case, since we need the subgraph $\\mathcal{T}$ to be a ternary tree in order to be useful in defining a mapping, we require it to be degree-4 constrained.\nThis implies that it is not always possible to find such tree (for instance, if $\\mathcal{P}$ is a tree but not degree-$\\kappa$ constrained with $\\kappa \\leq 4$).\nMoreover, even if a degree-4 constrained tree subgraph exists, finding it is generally hard (in fact, simply determining whether there is one is an NP-complete problem \\cite{garey1979computers}).\nFor our purposes, if $\\mathcal{T}$ is not a ST of $\\mathcal{P}$, it can nevertheless define a proper fermion-to-qubit mapping, although in such case some SWAPs may be needed to implement Majorana-generated unitary gates.\nTherefore, we propose using a greedy heuristic to find a TT that is close to a spanning tree, and is in fact guaranteed to find a ST for some specific topologies.\nThe routine is explained precisely in Appendix \\ref{app:algorithms} (Algorithm \\ref{alg:spanning_tree}).\n\n\\LinesNotNumbered\n\\begin{algorithm}[t]\n    \\label{alg:bonsai}\n    \\caption{Bonsai algorithm}\n    \n    Find a ternary tree $\\mathcal{T}$ congruent with the physical connectivity graph $\\mathcal{P}$ using Algorithm \\ref{alg:spanning_tree}, which consists of the routines:\n    \n        \\begin{itemize}\n        \n            \\item[a.] Find a degree-constrained tree subgraph $\\mathcal{T}$ using greedy search throughout $\\mathcal{P}$.\n            \n            \\item[b.] If the resulting tree does not span all nodes, add the remaining ones connecting them as to minimise the physical distance to nodes already in $\\mathcal{T}$.\n            \n        \\end{itemize}\n        \n    Add legs and introduce labels to $\\mathcal{T}$ using Algorithm \\ref{alg:branching_sub}, choosing among\n    \n        \\begin{itemize}\n        \n            \\item[a.] \\textit{Homogeneous localisation}: occupancy is spread evenly over qubits in the tree.\n            \n            \\item[b.] \\textit{Heterogeneous localisation}: a subset of mode-operators will act on many qubits, while reducing the amount that others act on.\n            \n        \\end{itemize}\n        \n    Pair the generated strings using Algorithm \\ref{alg:pairing_scheme}.\n    \n\\end{algorithm}\n\nThe idea is to start by defining $\\mathcal{T} = (V_\\mathcal{T}, E_\\mathcal{T})$, with empty $V_\\mathcal{T}$ and $E_\\mathcal{T}$, and grow the tree iteratively.\nFirst, choose a node to be the root $r$ of the TT, and define $\\mathcal{L}_0 = {r}$.\nThe choice of root has an impact on the resulting Pauli weight and average delocalisation of the mapping, as will be discussed later on; we now choose it to be central in $\\mathcal{P}$ (that is, such that it minimises the distance to its furthest node, $r = \\text{argmin}_u \\max_v d(u, v)$, where $d(u, v)$ is the topological distance between nodes $u$ and $v$ in $\\mathcal{P}$).\nIn Fig.~\\ref{fig:heavy_hexagon_st} b), this is the pale blue central node.\nNext, define an empty set $\\mathcal{L}_1$, and add to it $\\min(\\kappa(r), 3)$ neighbours of $r$ in $\\mathcal{P}$.\nFor every node $u$ that is added, add the link between $r$ and $u$ to $E_\\mathcal{T}$.\nNotice that $r$ may have degree $\\kappa(r) > 3$.\nIn that case, the choice is not unique.\nFor simplicity, we suggest choosing three of them randomly.\nThen, the process is repeated for each node in $\\mathcal{L}_1$: define $\\mathcal{L}_2 = \\emptyset$ and add to it up to three neighbours of each node in $\\mathcal{L}_1$ that have not yet been added, that is, not in $\\mathcal{L}_0 \\cup \\mathcal{L}_1 \\cup \\mathcal{L}_2$, and the corresponding links to $E_\\mathcal{T}$.\nBy iterating this process, at some point, all neighbours of all nodes in $\\mathcal{L}_D$ for some $D$ have been added to some $\\mathcal{L}_i$, so the procedure must stop.\nNow, let $V_\\mathcal{T} = \\bigcup_{i=1, \\ldots, D} \\mathcal{L}_i$.\nIf $V_\\mathcal{T} = V_\\mathcal{P}$, we have found a degree-4 constrained ST of $\\mathcal{P}$.\nNotice that this procedure succeeds with heavy-hexagon lattices, as shown in Fig.~\\ref{fig:heavy_hexagon_st} b) and c).\n\nIf the above procedure does not span all the qubits in $\\mathcal{P}$, we need to add the remaining nodes in $V_\\mathcal{P} \\setminus V_\\mathcal{T}$ to $\\mathcal{T}$ according to some criterion.\nNotice that it is always possible to include these nodes in $\\mathcal{T}$ through ``virtual edges'' that connect physically detached nodes at the expense of SWAPs in the compilation.\nIn order to minimise the resulting SWAP overhead, a good strategy is trying to minimise the physical topological distance between qubits connected in $\\mathcal{T}$.\nThis can be achieved following a greedy criterion: for every node $u$ in $V_\\mathcal{P} \\setminus V_\\mathcal{T}$, find amongst the nodes $v$ in $V_\\mathcal{T}$ with downwards degree less than three in $\\mathcal{T}$ the ones that minimise the distance $d(u, v)$ in $\\mathcal{P}$, and connect $u$ to one of them.\n\nIt is worth noting a few aspects of this method.\nOn the one hand, since at each step in the first part of the algorithm we add as many neighbours of each node as the topology allows, we are implicitly minimising the depth of the resulting tree.\nIndeed, notice that if the physical device is all-to-all connected, then the resulting graph is the TT from Ref.~\\cite{jiang2020optimal} with optimal depth $\\mathcal{O}(\\log _3 N)$.\nIn the case of the heavy-hexagon topology, the greedy algorithm succeeds in finding degree-constrained spanning trees with depth scaling as $O(\\sqrt{N})$, i.e., with quadratically lower Pauli weight than Jordan-Wigner.\nThis latter point can be seen through geometric arguments: the number of qubits at a given topological distance smaller than $R$ from a chosen root node $r$ scales as $R^2$.\nOn the other hand, if the connectivity graph is a chain, the algorithm, as presented above, would choose as root $r$ a node in the centre of the chain.\nWhile this would lead to a mapping with Pauli weight lower than JW, the occupation would be more delocalised than in the latter case.\nInstead, if one is interested in minimising delocalisation, a better choice of root is a node that lies on an extreme of a \\textit{diameter} of $\\mathcal{P}$ (that is, one of its longest shortest paths), so that the edges along the longest shortest path can be later labelled with $Z$, hence maximising $h_Z$.\nIn such case, JW would be obtained for a chain.\nIn general, this trade-off between Pauli weight and delocalisation can be easily controlled with the choice of root node.\n\n\\textit{Labelling the tree.}\nOnce the TT has been identified, the next step is to introduce terminating legs and Pauli labels to the links to create a qubit tree.\nAs discussed in the previous sections, in order to minimise the average delocalisation, we must label with $Z$ all edges from the root to the most distal node from it.\nIn the case of the heavy-hexagon in Fig.~\\ref{fig:heavy_hexagon_st}, we may do so with all the edges between nodes 0 (the root) and 36.\nUsing the different labelling techniques, we can decide to a certain degree how mode occupancy localisation is spread.\nWhile these cannot affect the average delocalisation, $\\langle D_u \\rangle$, they can determine how heterogeneously distributed among the qubits the occupancy can be.\nTo that end, we introduce two different labelling strategies, which we coin \\textit{homogeneous} and \\textit{heterogeneous} localisation, based on the discussion in Sect.~\\ref{sec:mapping_properties}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.99\\columnwidth]{HH_excitations.pdf}\n    \n    \\caption{The red and blue highlights indicate the qubits involved in the application of worst-case excitation terms, both for one- and two-particle terms.\n    The qubits to which modes are associated are circled in red (blue) for the JW (custom map).\n    For simplicity, we assume that the identification between modes $j$ and qubits $u$ in Eq.~\\eqref{eq:pairing_identification} is such that $j = u$.\n    In the case of JW (left), both $a_{0}^{\\dagger} a_{36}$ and $a_{0}^{\\dagger}a_{1}^{\\dagger}a_{2}a_{36}$ result in gates acting on all qubits.\n    The custom mapping generated by the Bonsai algorithm, on the other hand, presents much simpler worst-case scenarios.\n    The single-excitation term $a_{34}^{\\dagger}a_{35}$ and the two-particle one $a_{27}^{\\dagger}a_{34}^{\\dagger}a_{35}a_{36}$ involve much fewer qubits, and only two SWAPs are required to connect the separate highlighted regions in the latter case.}\n    \\label{fig:heavy_hexagon_worst_cases}\n\\end{figure}\n\nHomogeneous localisation proceeds by maximising the number of $XY$ branches amongst edges stemming from the same nodes, in a similar fashion as in Fig.~\\ref{fig:labelling_choice} (left).\nThus, pairs of edges below a node are assigned $X$ and $Y$ labels, while single edges, an $X$ label.\nHeterogeneous localisation, instead, maximises the number of $Z$ labels amongst edges: if a node has two descendants, one of the edges is assigned a $Z$ and the other one an $X$.\nIf the node has only one edge, it is assigned a $Z$.\nIn this way, the heterogeneous localisation assignment tends to localise the occupation of single-edge nodes at the expense of the delocalisation of nodes above them, hence resulting in typically more heterogeneous distributions of localisation.\n\nIn Fig.~\\ref{fig:mode_op_tree_folded}, we depict the two labelling outcomes for the heavy-hexagon topology.\nHomogeneous localisation produces many operators with occupancy depending on at worst case three qubits, giving a typical delocalisation $D_u = 2$ for those.\nThis is reflected also in the fact that the green lines representing the pairings do not span distant qubits.\nThe number of completely local operators, with specific delocalisation $D_u = 0$ is 16 in this case, whereas in the case of heterogeneous localisation, 27 operators are fully localised.\nThis has the side effect of creating fewer operators with higher delocalisation, such as the number operator for the mode associated with the root (qubit 0), with delocalisation $D_0 = 14$.\nWhether one should use one mapping or the other is application-specific.\nFor instance, in some applications, a few modes may be of little importance and can thus be delocalised without significant detrimental consequences.\nIn that case, it may be beneficial to use the heterogeneous localisation strategy in order to localise the relevant modes.\n\nWith these mappings, circuit cost is reduced in two ways.\nGiven that Majorana products follow paths along the paths in the tree structure, the number of SWAP gates needed for single excitations is zero, and the number is diminished for double excitations due to Majorana products following paths along the tree structure.\nThe number of entangling gates is further mitigated by lessening Pauli weight.\nThis is illustrated in Fig.~\\ref{fig:heavy_hexagon_worst_cases}, where we highlight the interaction maps of worst case single and double excitations for the mapping obtained through homogenous localisation, compared to the JW mapping.\nIn both cases, JW acts extensively on the whole system.\nThis is not the case for the custom encoding, where the reduction is approximately two thirds of the system for single and one third for double excitations.\nFor the latter case, two SWAP gates are required to bridge across the disconnected interaction regions.\n\nIt is worth mentioning that another interesting aspect of these custom mappings is that it simplifies the transpilation of the circuits.\nSince the mapping is designed to be congruent with the hardware, it is not necessary to search for the optimal qubit assignment, but only to solve the Steiner graph problem to determine where SWAP gates are required.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this work, we have considered fermion-to-qubit mappings relying on the identification of sets of Pauli strings obeying the anti-commutation relations of Majorana operators.\nWithin this context, we have focused on a specific class, arguably the simplest one to work with, in which the Pauli strings have a non-trivial overlap involving just one qubit.\nWe have then presented a framework that enables sampling such mappings while designing many of their resulting properties.\nAn important element of the methodology is the pairing algorithm that ensures the preservation of separability, that is, that uncorrelated fermionic states are mapped to uncorrelated qubit states. \nInterestingly, the framework contains paradigmatic mappings as particular instances, which allows us to interpolate between them at will.\n\nWith this framework at hand, we have devised an algorithm to design hardware-specific mappings with lower SWAP overhead than other paradigmatic mappings while retaining a fair localisation of the fermionic occupation in qubit space.\nWhen applied to the heavy-hexagon architecture, we obtain a mapping with a quadratically lower Pauli weight than JW.\nImportantly, the mapping enables applying single excitation operations with no SWAP overhead, and double excitations with a minor one.\nThis can result in a significant improvement in circuit complexity with respect to hardware-agnostic mappings.\n\nThe versatility of the approach here presented enables many other possibilities.\nIn terms of designing mappings, the choice of cost function to be optimised for is not unique, so the Bonsai algorithm can be naturally extended to produce encodings with different desirable properties.\nIn particular, an important application is extracting relevant physical quantities of the system using local informationally complete POVMs \\cite{jiang2020optimal, garcia2021learning, glos2022adaptive}.\nIn this case, the Pauli weight of the observable is the dominant figure of merit, which is why the authors proposed the logarithmic-depth regular ternary tree in  Ref.~\\cite{jiang2020optimal}.\nWhile that is the optimal mapping in terms of measurement cost for arbitrary fermionic reduced density matrix elements, in practice, one is typically interested in specific observables like the energy.\nIn that case, the mapping may be further optimised to reduce the measurement cost of e.g.~the Hamiltonian of the system.\nMoreover, it would be interesting to do so while limiting the incurred SWAP overhead on specific hardware.\n\nIn broader, more theoretical terms, we emphasise that the bulk of the work here presented is devoted to a specific subset of all the possible mappings, the 1-NTO class, which includes all the widely used encodings.\nAs we have proved, with the pairing we introduced, any root-containing connected ternary tree yields a valid, product-preserving fermion-to-qubit mapping.\nHowever, we have also shown with a counter-example that not all 1-NTO maps can be generated in this fashion, so the question of how to characterise and represent the space of 1-NTO encodings remains open.\nIn addition, as noted in Section \\ref{sec:theory}, $k$-NTO maps with $k > 1$ do exist.\nThis opens the interesting prospect of studying these somewhat exotic mappings.\n\n\\noindent {\\bf Additional information.} The Bonsai algorithm is part of \\emph{Aurora}'s suite of algorithms for chemistry simulation.\n\n\\bibliographystyle{apsrev4-2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nCarleson measures were first introduced by L.\\\nCarleson~\\cite{Carleson1,Carleson2}  in order to study the corona\nproblem in the classical Hardy spaces on the unit disc. The study of\nthese measures have flourished since then, and has been generalized to\nseveral different settings.\nForm a general perspective, they can be effectively defined as follows.\nGiven a locally compact space $\\Tc$   and a (quasi-)Banach space\n$X$ of (say) continuous functions on $\\Tc$, a positive Radon measure\n$\\mi$ on $\\Tc$ is said to be a $p$-Carleson measure for $X$ ($p\\in\n]0,\\infty[$) if there is a constant $C>0$ such that   \n\\[\n\\norm{f}_{L^p(\\mi)}\\meg C \\norm{f}_X\n\\]\nfor every $f\\in X$. Switching the roles of $X$ and $L^p(\\mi)$ in the\nabove inequality, one then gets the definition of a reverse\n$p$-Carleson measure for $X$. This kind of measures was essentially\nintroduced in a series of papers by D.\\ H.\\\nLuecking~\\cite{Luecking,Luecking3,Luecking4,Luecking2} for weighted\nBergman spaces on the unit disc or more general domains, and has been\nlater generalized to different settings by several authors\n(cf.~\\cite{FricainHR} and the references therein). \n\nIn this paper we study Carleson and reverse Carleson measures on\nweighted Bergman spaces on homogeneous Siegel domain, which we are now\nabout to introduce. \n\nLet $E$  be a complex hilbertian space of dimension $n$, $F$ a real\nhilbertian space of dimension $m>0$ and $\\Omega$ a  convex cone\nin $F$ not containing any straight lines.\\footnote{We shall generally\ndenote by $\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle$ bilinear pairings and\nreal scalar products, and by $\\langle\\,\\cdot\\,\\vert\\,\\cdot\\,\\rangle$\nsesquilinear pairings and complex scalar products, without\nspecifying the involved spaces.} \nWe also assume that $\\Omega$ is (affine-)homogeneous, that is the group $G(\\Omega)$ of linear transformation of $F$ that preserve $\\Omega$ acts transitively\non $\\Omega$. \n\nLet $\\Phi\\colon E\\times E\\to F_\\C $   be a non-degenerate hermitian\nmapping such that $\\Phi(\\zeta)\\coloneqq \\Phi(\\zeta,\\zeta)\\in\n\\overline\\Omega$ for all $\\zeta\\in E$.  Define the  Siegel domain of\ntype II associated with the cone $\\Omega$ and the mapping $\\Phi$ as  \n\\[\nD \\coloneqq \\Set{ (\\zeta, z) \\in E \\times F_\\C \\colon\n\\rho(\\zeta,z)\\coloneqq \\Ima z -  \\Phi(\\zeta) \\in \\Omega } .\n\\]\nIn the case $n=0$, i.e. $E=\\Set{0}$, $D$ is said to be of Type I, and\nis\noften called a {\\em  tubular domain} over the cone $\\Omega$.  \nThe domain $D$ is said to be homogeneous if for every\n$(\\zeta,z),(\\zeta',z')\\in D$ there is a biholomorphism $\\phi$ of $D$\nsuch that $\\phi(\\zeta,z)=(\\zeta',z')$. It turns out that one may then\nassume $\\phi$ to be affine, cf., e.g.,~\\cite[Theorem 2.3]{Xu}. More\nprecisely, $D$ is homogeneous if and only if for every $h,h'\\in\n\\Omega$ there are $t\\in GL(F)$ and $g\\in GL(E)$ such that $th=h'$ and\nsuch that $t\\Phi=\\Phi(g\\times g)$ (so that $g\\times t$ preserves\n$D$), cf., e.g.,~\\cite[Propositions 2.1 and 2.2]{Murakami}. In\nparticular, if $D$ is homogeneous, then $\\Omega$ is homogeneous.  \n\nThe domain $D$ is said to be symmetric if it is homogeneous and admits\nan involutive biholomorphism with an isolated fixed point. If $D$ is\nsymmetric, then $\\Omega$ is symmetric, that is, homogeneous and\nself-dual. Conversely, if $\\Omega$ is symmetric \\emph{and $D$ is a\ntubular domain}, then  $D$ is symmetric (cf.,\ne.g.,~\\cite[Theorem]{Satake} for more details on various\ncharacterizations of symmetric Siegel domains).   \n\n\n\n\nThe \\v{S}ilov boundary  of $D$ is the set \n\\[\nbD \\coloneqq \\Set{ (\\zeta, z) \\in E \\times F_\\C \\colon\\rho(\\zeta,z) =0 } ,\n\\]\nand it can be identified with $E\\times F$ via the mapping  $ (\\zeta,\nx+i\\Phi(\\zeta,\\zeta))\\mapsto(\\zeta,x)$. In addition, $bD$ admits a\nstep-2 nilpotent Lie group \nstructure,  whose product can be described as follows: \n\\[\n(\\zeta,x)(\\zeta',x')=(\\zeta+\\zeta',x+x'+2\\Ima\n\\Phi(\\zeta,\\zeta')),\n\\]\nfor $(\\zeta,x),(\\zeta',x')\\in E\\times F$, \nsee e.g.~\\cite[Section 1.1]{CalziPeloso}. We denote by\n$\\Nc$ the set $E\\times F$ endowed with this group structure. \n\nObserve that, by definition, $\\rho$ maps $ D$ into $\\Omega$, and that\nthe fibres of $\\rho$, namely the sets $b D+(0, i h)$ for $h\\in\n\\Omega$, give rise to a foliation of $D$. Given a function $f$\ndefined on $D$, we shall often denote by $f_h$ the restriction of $f$\nto $b D+(0,i h)$, interpreted as a function on $\\Nc$ for the sake of\nconvenience, so that \n\\[\nf_h (\\zeta,x)=f(\\zeta,x+i\\Phi(\\zeta)+i h)\n\\]\nfor every $h\\in \\Omega$ and for every $(\\zeta,x)\\in \\Nc$. Observe\nthat, identifying  $b D+(0,i h)$ with $\\Nc$ as above for every $h\\in\n\\Omega$, we get a left action of $b D$ on $D$ by affine\nbiholomorphisms.   \n\n\nFor $p,q\\in ]0,\\infty]$ and $\\vs\\in\\R^r$, the\nweighted Bergman spaces are defined as\\footnote{The definitions of the\nrank $r$ of $\\Omega$,  of the `generalized power functions'\n$\\Delta_\\Omega^{\\vect s}$ and of $\\vd\\in\\R^r$   are deferred to Section~\\ref{sec:2}.} \n\\begin{equation}\\label{eq:1}\nA^{p,q}_\\vs (D)\\coloneqq \\Set{  f\\in\\Hol(D)\\colon\\int_\\Omega \\Big(\n\\int_\\Nc\\abs{f_h(\\zeta,x)}^p \\,  \\dd (\\zeta,x) \\Big)^{q\/p}\n\\Delta^{q \\vs}_\\Omega (h)  \\, \\dd \\nu_\\Omega(h)<\\infty\\,  } \n\\end{equation}\n(modification if $\\max(p,q)=\\infty$), where $\\dd (\\zeta,x)$ denotes a\nHaar measure on $\\Nc$ and $\\nu_\\Omega$ denotes a positive\n$G(\\Omega)$-invariant measure   on $\\Omega$, both  fixed and  unique\nup to a \nmultiplicative constant. We shall often simply write $A^{p,q}_\\vs$ instead of $A^{p,q}_\\vs (D)$. \n\nWe remark that the spaces $A^{p,p}_\\vs$ are the weighted Bergman spaces,  \nthe unweighted case corresponding to the value $\\vs=-\\vd\/p$, while the\nspaces $A^{p,\\infty}_{\\vect 0}$ are the classical Hardy spaces.  \n\n\\medskip\n\nWe denote by  $\\cM_+(D)$ the space of positive Radon measures.\n\n\\begin{deff}\\label{def:dom-set}\nGiven  a Banach space $X$ of functions on $D$ and $p\\in ]0,\\infty[$, a\nmeasure $\\mi\\in\\cM_+(D)$  is said to be $p$-Carleson for $X$ if there\nexists a constant $C>0$ such that  \n\\[\n\\norm{f}_{L^p(\\mi)}\\le C \\norm{f}_X\n\\]\nfor every $f\\in X$.\nIf, in addition, the canonical mapping $X\\to L^p(\\mi)$ is compact,\nthen $\\mi$ is said to be a  compact or   vanishing $p$-Carleson for $X$. \n\nA measure $\\mu\\in\\cM_+(D)$ is said to be a reverse $p$-Carleson\nmeasure for $X$ if there exists $C>0$ such that  \n\\[\n\\norm{f}_X\\le C \\norm{f}_{L^p(\\mi)}.\n\\]\nfor every $f\\in X$.\n\nFinally, a measure $\\mu\\in\\cM_+(D)$ is said to be a sampling, or dominant, measure for $X$ \nif it is both Carleson and reverse Carleson for $X$, in other words,\nif $X$ embeds as a closed subspace of $L^p(\\mi)$.\n\\end{deff}\n\nNotice that, even though the definition of a $p$-Carleson and of a\nreverse $p$-Carleson measure is quite standard, the definition of a\nvanishing $p$-Carleson measure is slightly more\nproblematic. Originally, Carleson measures were characterized\nrequiring a suitable function  $F$ to be bounded, and vanishing\nCarleson measures were characterized requiring $F$ to vanish `at\ninfinity'. As noted in~\\cite{Power}, this corresponds to the\ncompactness of the canonical mapping $X\\to L^p(\\mi)$. In the\nliterature also appear apparently different, but still equivalent,\ndefinitions of a vanishing Carleson measure\n(cf.,e.g.,~\\cite{HuLvZhu}).  \n\n\nIn this paper we consider the  problem of obtaining necessary and\nsufficients conditions for the measure $\\mu$ to be a $p$-Carleson,  a vanishing\n$p$-Carleson, or a reverse $p$-Carleson measure for the spaces $A^{p,q}_\\vs$.\n\\medskip\n\nThere  exists a vast literature on Carleson measures on holomorphic\nfunction  spaces, in one and several variables, and it is impossible\nto give a proper and complete account of it.  Hence, we limit\nourselves to the papers most relevant for this work, and extend our\napologies to all other\nauthors. In~\\cite{Carleson1,Carleson2,Duren,Luecking8}, Carleson\nmeasures for the Hardy or Hardy--Sobolev spaces on the unit disc are\nstudied. In in~\\cite{Stegenga,Wu} Carleson measures for the Dirichlet\nand weighted Direchlet spaces still on the unit disc are\ncharacterized. In higher dimensions, we mention~\\cite{CimaWogen},\nwhich deals with Hardy spaces in the unit ball,~\\cite{Luecking7},\nwhich deals with Bergman spaces on general\ndomains,~\\cite{ArcozziRochbergSawyer3,ArcozziRochbergSawyer1,ArcozziRochbergSawyer5,ArcozziRochbergSawyer2,ArcozziRochbergSawyer4},\nwhich deal with holomorphic Sobolev--Besov spaces on the unit\nball,~\\cite{AbateSaracco,HuLvZhu,Abate,AbateRaissy,AbateMongodiRaissy},\nwhich deals with Bergman spaces on strongly \npseudoconvex domains,\nand~\\cite{NanaSehba,BekolleSehbaTchoundja,BekolleSehba}, which deal\nwith various function spaces  on Siegel domains. \n\n\nCarleson measures still draw a lot of attention, in connection with\nmany problems in analysis such \nas operator theory, interpolation, and boundary behavior problems,\njust to mention a few.  The results in this paper are applied in the\nstudy of Toeplitz and Ces\\`aro operators on weighted Bergman  spaces on\nSiegel domains of Type II, see~\\cite{T&C}.\n\nOn the other hand, reverse Carleson measures appeared at a later time,\nand much less is known about them.  They have been studied by D.\\ H.\\\nLuecking in the case of Bergman spaces on the unit disc\n\\cite{Luecking,Luecking2,Luecking3,Luecking4,Luecking5}. See also the recent\nsurvey~\\cite{FricainHR} and references therein.  \n\n\nAmong reverse Carleson measures, those which are also Carleson\nmeasures form a particularly tractable subclass, for which better\nresults are generally available. Such measures are often called\n\\emph{sampling measures}, since they can be considered as\ngeneralizations of sampling sequences.  \nA complete characterization of sampling measures is available on the\nunit disc (cf.~\\cite{Luecking5}), but not on more general\ndomains. Nonetheless, the techniques developed\nin~\\cite{Luecking,Luecking3,Luecking4} to characterize dominant (or\nsampling) sets can be effectively extended to general weighted Bergman\nspaces on homogeneous Siegel domains, as already noted\nin~\\cite{Luecking4}. A subset $G$ of $D$ is said to be \\emph{dominant} for\n$A^{p,p}_\\vs$ if the measure\n$\\chi_G(\\Delta_\\Omega^{p\\vs+\\vd}\\circ \\rho)\\cdot \\Hc^{2n+2m}$. As it turns out, this notion is\nindependent of both $\\vs$ and $p$. \n\n\nIn this paper we consider weighted Bergman spaces on a homogeneous\nSiegel domain $D$ of Type II. \nWe  provide necessary and sufficient conditions   for a measure\n$\\mi\\in\\cM_+(D)$ to be a $p$-Carleson measure for $A^{p_1,q_1}_\\vs(D)$ \nwhen $p_1,q_1\\le p$  (Theorems~\\ref{main:1} and~\\ref{teo:4}), as well\nas in the general case,  under suitable additional assumptions\n(Theorems~\\ref{main:2} and~\\ref{teo:4}).  \nWe also provide a sufficient condition for a measure to be a  reverse\nCarleson measure (Theorem~\\ref{prop:29}), as well as necessary\nconditions and sufficient conditions for a measure $\\mi\\in\\cM_+(D)$ to\nbe a sampling measure for $A^{p,p}_\\vs(D)$\n(Theorems~\\ref{prop:31},~\\ref{prop:30}, and~\\ref{prop:27}).   \nFinally, we remark that analysis of the theory of holomorphic\nfunction spaces, in particular on homogeneous Siegel domains of Type\nII, is a quite active area of research, see e.g. \\cite{ACMPS,AMPS,MPS} and\nreferences therein for some recent results.  \n\n\nThe paper is organized as follows. \nIn Section~\\ref{sec:C+} we review our main results in the particular\ncase $D=\\C_+$ for the ease of the reader.   \nIn Section~\\ref{sec:2} we give the\ndefinition of homogeneous Siegel domains of Type  II, while in\nSection~\\ref{sec:fnct-sp} we give the definitions of several function\nspaces on $D$ and recall some of their most relevant properties which\nare involved in our analysis.    \nIn Section~\\ref{sec:Carleson}, we prove necessary conditions and sufficient\nconditions for a measure $\\mi\\in\\cM_+(D)$ to be a Carleson measure for\nthe weighted Bergman space $A^{p,q}_\\vs$.  Section~\\ref{sec:5} is\ndevoted to an extension of a result by Hardy and Littlewood about\nembedding the Hardy space into a weighted Bergman space. Finally, in\nSection~\\ref{sec:6}, we prove our results concerning reverse Carleson and\nsampling measures. \n\n\n\\section{Carleson and Reverse Carleson Measures on $\\C_+$}\\label{sec:C+}\n\nIn this brief section we present our main results for weighted Bergman spaces on $\\C_+$, where they become particularly simple.\n\nIn this case, for $p,q\\in ]0,\\infty]$ and $s\\in\\R$,\n\\[\nA^{p,q}_s=\\Set{f\\in \\Hol(\\C_+)\\colon  \\int_0^\\infty y^{qs} \\left(\\int_\\R \\abs{f(x+iy)}^p\\,\\dd y\\right)^{q\/p}\\,\\frac{\\dd y}{y}<\\infty}\n\\] \n(modification when $\\max(p,q)=\\infty$). Then, $A^{p,q}_s=0$ if $s<0$ or $s=0$ and $q<\\infty$. In addition, $A^{p,\\infty}_0$ is the Hardy space $H^p$.\n\n\\begin{teo}\nTake $p_1,q_1,p\\in ]0,\\infty]$ with $p<\\infty$, and $s>0$. Take a compact neighbourhood $K$ of $i$ in $\\C_+$, $\\mi\\in \\cM_+(D)$, and define\n\\[\nM_K(\\mi)\\colon \\C_+ \\ni z\\mapsto \\mi(\\Rea z+(\\Ima z)K)\\in \\R_+. \n\\]\nThen, the following conditions are equivalent:\n\\begin{enumerate}\n\\item[(1)] $A^{p_1,q_1}_s$ embeds continuously into $L^p(\\mi)$;\n\n\\item[(2)] setting $p^*\\coloneqq \\max(1,p_1\/p)'$, $q^*\\coloneqq \\max(1,q_1\/p)'$, and $s^*\\coloneqq -\\max(1,p\/p_1)-p s$,\n\\[\n\\int_0^\\infty y^{q^*s^*} \\left(\\int_\\R \\abs{M_K(\\mi)(x+iy)}^{p^*}\\,\\dd y\\right)^{q^*\/p^*}\\,\\frac{\\dd y}{y}<\\infty\n\\]\n(modification when $\\max(p^*,q^*)=\\infty$).\n\\end{enumerate}\n\\end{teo}\n\n(Cf.~Theorem~\\ref{main:2} and Propositions~\\ref{prop:23bis} and~\\ref{prop:25} below.)\n\nEven though Carleson measures for the Hardy spaces\n$A^{p,\\infty}_0(\\C_+)$ have been characterized in the literature, it\nis known that the problem of determining the Carleson measures for\nHardy spaces on higher rank irreducible Siegel domains  is highly non-trivial. \n\n\nConcerning sampling measures, Theorem~\\ref{prop:27} becomes:\n\n\\begin{teo}\nTake $p,s>0$, and a $p$-Carleson measure $\\mi$ for $A^{p,p}_s$. Assume that the support of every vague cluster point of every sequence of measures of the form $ y^{p s+1}\\, (x+y\\,\\cdot)_*(\\mi) $  ($x\\in \\R, y>0$) is a set of uniqueness for $A^{q,q}_{(p\/q)s}$ for some (fixed) $q\\in ]0,p[$.\n\nThen, $\\mi$ is a $p$-sampling measure for $A^{p,p}_s$.\n\\end{teo}\n\nObserve that the above sufficient condition is likely to be necessary\nas well, \\emph{in this case}, since the corresponding assertion on the\nunit disc is true (cf.~\\cite{Luecking5}).  \n\nNonetheless, since the above sufficient condition need not be easily managed, other sufficient conditions can be provided.\n\n\\begin{prop}\\label{prop:20}\nTake $p,s,\\eps,C'>0$, and a compact neighbourhood $K$ of $i$ in\n$\\C_+$. Then, there is a compact neighbourhood $K'$ of $i$ in $\\C_+$\nsuch that, for every $\\mi\\in \\cM_+(\\C_+)$ such that \n\\[\nN(\\mi)\\coloneqq \\sup_{x\\in \\R,y>0} y^{-1-p s}\\mi(x+ y K')<\\infty\n\\]\nand such that\n\\[\n\\Hc^2\\left(\\Set{z\\in x+y K\\colon \\mi(\\Rea z+ (\\Ima z) K')\\Meg \\eps (\\Ima z)^{1+p s} N(\\mi)  }\\right)\\Meg C' y^2\n\\]\nfor every $x\\in \\R$ and for every $y>0$, is a $p$-sampling measure for $A^{p,p}_s$.\n\\end{prop}\n\nThe above result is a consequence of a characterization of dominant\nsubsets of $\\C_+$ (cf.~Theorem~\\ref{prop:31} below) which is very\nclose to what one obtains applying Proposition~\\ref{prop:20} to the\nmeasure $\\chi_G (\\Ima\\,\\cdot\\,)^{p s-1}\\cdot \\Hc^2$ and to some\n$\\eps\\in ]0,1[$, where $G$ is a measurable subset of $\\C_+$. \nHere and in what follows, we denote by $\\Hc^m$ the $m$-dimensional Hausdorff\nmeasure.     \n\n\\begin{prop}\\label{prop:21}\nA subset $G$ of $\\C_+$ is dominant if and only if there are a constant $C>0$ and a compact neighbourhood $K$ of $i$ such that\n\\[\n\\Hc^2(G\\cap (x+y K))\\Meg C y^2\n\\]\nfor every $x\\in \\R$ and for every $y>0$.\n\\end{prop}\n\n\nFinally, the following result provides a sufficient condition for a\n(not necessarily Carleson) measure to be reverse Carleson  (see\nTheorem~\\ref{prop:29}). \n\n\\begin{teo}  \nTake $p_1,q_1 ,p\\in ]0,\\infty]$ with $p<\\infty$, and $s>0$ or $s\\Meg0$ if $q_1=\\infty$. Define $p^*\\coloneqq\np_1\/(p-p_1)_+$ and $ q^*\\coloneqq q_1\/(p-q_1)_+$. \n\nThen, there are $\\delta>0$ and a constant $C>0$ such that every  $\\mi\\in\\cM_+(D)$ such that\n\\[\n\\norm*{  \\bigg( \\frac{2^{k p[s+1\/p_1]}}{\\mi([\\delta 2^k j,\\delta 2^k (j+1) ]+i [\\delta 2^k,\\delta2^{k+1}]  )} \\bigg)_{j,k}\n}_{\\ell^{p^*, q^*}(\\Z,\\Z)}  <\\infty,\n\\]\nis  a $p$-reverse Carleson for $A^{p_1,q_1}_s$.    \n\\end{teo}\n\n\n\n\n\\section{Homogeneous Siegel domains of Type  II}\n\\label{sec:2}\n\nFor the general theory of homogeneous Siegel domains of Type II and\nfurther details we refer the reader  to~\\cite{Vinberg,Gindikin,CalziPeloso}.  \nThroughout the paper we adopt the notation of~\\cite{CalziPeloso}.\n\n\\subsection{The structure of homogeneous cones and a description of $D$}\n\n\n\\begin{deff}\\label{def:A}\nTake $r\\in\\N$ and let $A=\\bigoplus_{j,k1}^r A_{j,k}$ be a graded algebra\nendowed with a linear involution~$^*$ such that the following hold:  \n\\begin{itemize}\n\\itb $A_{j,k}A_{k',\\ell}\\subseteq A_{j,\\ell}$ if $k=k'$ and\n$A_{j,k}A_{k',\\ell}=\\Set{0}$ if $k\\neq k'$;\n\\itb $A_0=\\Set{0}$ and $\\dim A_{j,j}=1$ for every $j=1,\\dots,r$;\n\\itb $A_0=\\Set{0}$ and $\\dim A_{j,j}=1$ for every $j=1,\\dots,r$;\n\\itb for every $j=1,\\dots, r$ there exists $e_j\\in A_{j,j}$ such\nthat the left multiplication by $e_j$ is the identity on\n$A_{j,k}$ and the  right multiplication by $e_j$ is the\nidentity on $A_{k,j}$, for every $k=1,\\dots, r$; \n\\itb if we define $\\tr a=\\sum_{j=1}^r \\langle e_j', a\\rangle$ for\nevery $a\\in A$, where $e'_j$ is the unique graded linear functional on\n$A$ which takes the value $1$ at $e_j$ ($j=1,\\dots,r$), then the\nmapping $(a,b)\\mapsto \\tr(a b)$ is symmetric; \n\\itb for every $a,b,c\\in A$, $\\tr(a(bc))=\\tr((ab)c)$;\n\\itb  $A^*_{j,k}=A_{k,j}$ for every $j,k=1,\\dots,r$ and $(ab)^*=b^* a^*$ for every $a,b\\in A$;\n\\itb the symmetric bilinear mapping $(a,b)\\mapsto \\tr(a^*b)$ is positive and non-degenerate;\n\\itb setting $T\\coloneqq\\bigoplus_{j\\meg k} A_{j,k}$, one has $t(u w)=(t u)w$ and $t( u u^*)=(t u) u^*$ for every $t,u,w\\in T$.\n\\end{itemize}\nThen, we say that $A$ is a $T$-algebra. \n\\end{deff}\n\n\\begin{deff}\nGiven a $T$-algebra $A$, define \n\\begin{equation}\\label{def:T,H,C(A)}\n\\begin{aligned}\nT_+&\\coloneqq \\Set{a\\in T\\colon \\langle e_j',a\\rangle>0 \\quad \\forall j=1,\\dots, r }, &\nH(A) &\\coloneqq\\Set{a\\in A\\colon a=a^*},\\\\\nC(A)&\\coloneqq \\Set{t t^*\\colon t\\in T_+}, &\nC'(A)&\\coloneqq \\Set{t^* t\\colon t\\in T_+}.\n\\end{aligned}\n\\end{equation}\nThe cones $C(A)$ and $C'(A)$ are said to have rank $r$.\\index{Rank!of a homogeneous cone}\n\\end{deff}\n\nThe following theorem was proved in~\\cite{Vinberg}.\n\n\\begin{teo}\\label{teo:Vinberg}\nLet $A$ be a $T$-algebra. Then, the following hold: \n\\begin{itemize}\n\\item[(i)] $T_+$, endowed with the product induced by $A$, is a Lie group;\n\n\\item[(ii)] $C(A)$ and $C'(A)$ are   homogeneous   cones, and are dual to\none another with respect to the scalar product on  $H(A)$ ;\\footnote{  In other words, $C(A)=\\Set{x\\in H(A)\\colon \\forall \\lambda\\in \\overline{C'(A)}\\setminus \\Set{0}\\quad \\tr(\\lambda x) >0 }$.}\n\\item[(iii)] $(t x) t^*= t(x t^*)$ and $(t^* x) t = t^* (x t)$ for every $t\\in T_+$ and for every $x\\in   H(A)$ ;\n\\item[(iv)] the mappings \n\\[\n(t,x)\\mapsto t x t^* \\qquad \\text{and}\\qquad (t,x)\\mapsto t^* x t\n\\]\nare simply transitive left and right actions of $T_+$ on $C(A)$ and\n$C'(A)$, respectively, which are dual to one another with respect to\nthe scalar product on   $H(A)$ . \n\\end{itemize}\n\nIn addition, if $\\Omega$ is a homogeneous cone in a finite-dimensional  vector space  $F$ over $\\R$, then, there exist a $T$-algebra $A'$  and an isomorphism $\\Psi\\colon H(A')\\to F'$ such that $\\Psi(C(A'))=\\Omega$.\n\\end{teo} \n\nBy the previous results, given a homogeneous cone $\\Omega$, we may select a  subgroup $T_+$ of $G(\\Omega)$ which acts simply transitively on $\\Omega$ and,  by transposition, on the dual con $\\Omega'$.  \nIn order to avoid some notational inconveniences, we shall write $t\\cdot h$\ninstead of $t(h)$ or $t h$, and   $\\lambda\\cdot t$ instead of $\\lambda\n\\circ t$ or $\\lambda t$, for every $t\\in T_+$, for every $h\\in\n\\Omega$, and for every $\\lambda\\in \\Omega'$. \nThen, the following hold (cf., e.g.,~\\cite[Lemma~2.9]{CalziPeloso}):\n\\begin{itemize}\n\\itb for every $t\\in T_+$ there exists $g\\in GL(E)$ such that\n$t\\cdot\\Phi = \\Phi (g\\times g)$;\\footnote{This follows from the\nrequirement that $D$ be homogeneous (for a suitable choice of\n$T_+$).} \n\\itb for every $\\vs\\in\\C^r$ the mapping $\\Delta^\\vs\\colon T_+\\to \\C^*$ defined  by\n$\\Delta^\\vs (t) \\coloneqq\\prod_{j=1}^r \\langle e_j', t\\rangle^{2 s_j}$  is a group\nhomomorphism;\n\\itb there is $\\vb\\in \\R^r$ such that\n$\\Delta^{-\\vb}(t)=\\det_{\\R}(g)=\\abs{\\det_\\C(g)}^2$ for every $t\\in T_+$ and for every $g\\in GL(E)$ such that $t\\cdot \\Phi=\\Phi(g\\times g)$.\n\\end{itemize}\nWe recall that $D$ is a tube domain, that is, $E=\\Set{0}$, if and only\nif $\\vb=\\vect 0$, cf.~\\cite[Remark 2.13]{CalziPeloso}.  \n\n\\subsection{The generalized  power and Gamma functions}\nUsing the characters $\\Delta^\\vs$ we define the {\\em\ngeneralized power functions} $\\Delta_\\Omega^\\vs$ and\n$\\Delta_{\\Omega'}^\\vs$ on $\\Omega$ and $\\Omega'$,  respectively.\n\n\\begin{deff}\nFix base points $e_\\Omega \\in \\Omega$ and  $e_{\\Omega'} \\in \\Omega'$, \nand  define the {\\em  generalized power functions} $\\Delta_\\Omega^\\vs$\nand $\\Delta_{\\Omega'}^\\vs$, for $\\vs\\in\\C^r$, by \n\\[\n\\Delta_\\Omega^\\vs (h) = \\Delta^\\vs(t) \\qquad \\text{and} \\qquad \\Delta_\\Omega^\\vs (\\lambda) = \\Delta^\\vs(t')\n\\]\nwhere $t,t'$ are the \\emph{unique} elements of $T_+$ such that\n$h=t\\cdot e_\\Omega$ and $\\lambda=e_\\Omega\\cdot t'$, respectively.  \nWe also set $m_{j,k}\\coloneqq\\dim A_{j,k}$ and define  \n\\begin{equation}\\label{def:m,m'}\n\\vm\\coloneqq\\Big(\\sum_{k>j} m_{j,k}\\Big)_{j=1,\\dots,r},\\qquad \\vm' \\coloneqq\\Big(\\sum_{k<j} m_{j,k}\\Big)_{j=1,\\dots,r} .\n\\end{equation}\nThe generalized Gamma functions on $\\Omega$ and $\\Omega'$ are defined for\n$\\Rea\\vs\\in \\frac{1}{2}{\\vect{m}}+(\\R_+^*)^r$ as\n\\[\n\\Gamma_\\Omega(\\vs)  \\coloneqq\\int_{\\Omega} \\ee^{-\\langle\ne_{\\Omega'},h\\rangle}\\Delta^{\\vs}_\\Omega(h)\\,\\dd \\nu_\\Omega(h)=\nc  \\prod_{j=1}^r \\Gamma\\left( s_j-\\frac{m_j}{2} \\right) \n\\]\nand, for $\\Rea\\vs\\in \\frac{1}{2}\\vect{m}'+(\\R_+^*)^r$,\n\\[\n\\Gamma_{\\Omega'}(\\vs)\\coloneqq \\int_{\\Omega'} \\ee^{-\\langle \\lambda,e_{\\Omega}\\rangle}\\Delta^{\\vs}_{\\Omega'}(\\lambda)\\,\\dd\\nu_{\\Omega'}(\\lambda) = c  \\prod_{j=1}^r \\Gamma\\left( s_j-\\frac{m'_j}{2} \\right),\n\\]\nwhere $c>0$ is a suitable constant.\n\nFurthermore, we set $  \\vd\\coloneqq -\\left(\\vect{1}_r+\\half\n\\vect{m}+\\half \\vect{m}'\\right)$  and define  \n\\begin{equation}\\label{def:d}\n\\nu_\\Omega\\coloneqq  \\Delta_\\Omega^{\\vect{d}}\\cdot \\Hc^m \\qquad\\text{and} \\qquad \\nu_{\\Omega'} \\coloneqq \\Delta^{\\vect{d}}_{\\Omega'}\\cdot \\Hc^m ,\n\\end{equation}\nwhere $\\Hc^m$ denotes the $m$-dimensional Hausdorff measure on\n$\\Omega$ and $\\Omega'$, respectively.  \n\\end{deff}\n\n\nWe point out  that  $\\nu_\\Omega$ and $\\nu_{\\Omega'}$ are two\n$G(\\Omega)$-invariant measures on $\\Omega$ and $\\Omega'$,\nrespectively:   cf.~\\cite[Lemma~2.18]{CalziPeloso} for\n$T_+$-invariance, and argue as in the proof of~\\cite[Proposition\nI.3.1]{FarautKoranyi} for $G(\\Omega)$-invariance.   \n\n\n\\begin{deff}\nWe denote by $(I^{\\vect s}_\\Omega)_{\\vs\\in \\C^r}$ the unique holomorphic family of tempered distributions on $F$ such that $I^{\\vect s}_\\Omega= \\frac{1}{\\Gamma_\\Omega(\\vs)}\\Delta^{\\vect s}_\\Omega \\cdot \\nu_\\Omega$ for $\\Rea\\vs\\in \\frac 1 2 \\vect m+(\\R_+^*)^r$ (cf.~\\cite[Lemma 2.26, Definition 2.27, and Proposition 2.28]{CalziPeloso}). We call `Riemann--Liouville operators' the operators of convolution by the $I^{\\vs}_\\Omega$.\n\\end{deff}\n\n\\subsection{Lattices on $D$}\n\nWe shall make extensive use of the notion of  a  lattice on the domain\n$D$.\nWe first introduce some Riemannian metrics on $\\Omega, \\Omega'$, and $D$.\n\n\\begin{deff}\nWe endow $D$ with the Bergman metric, that is, with the (complete K\\\"ahler) metric $\\vect k$ defined by\n\\[\n\\vect k_{(\\zeta,z)} v w\\coloneqq \\partial_v \\overline{\\partial_w} \\log (\\Delta^{\\vect b+2 \\vect d}\\circ \\rho)(\\zeta,z)\n\\]\nfor every $(\\zeta,z)\\in D$ and for every $v,w\\in E\\times F_\\C$ (cf.~\\cite[\\S 2.5]{CalziPeloso}). We denote by $B((\\zeta,z),R)$ the corresponding open ball of centre $(\\zeta,z)$ and radius $R$. \nWe denote by $\\nu_D$ the corresponding invariant measure $(\\Delta^{\\vect b+2\\vect d}\\circ \\rho) \\cdot \\Hc^{2 n+2m}$ on $D$ (cf., e.g.,~\\cite[Proposition 2.44]{CalziPeloso}).\n\nWe endow $\\Omega$ with the quotient metric induced by the submersion $\\rho\\colon D\\to \\Omega$, and $\\Omega'$ with the Riemannian metric induced by the correspondence $\\Omega\\ni t\\cdot e_\\Omega \\mapsto e_{\\Omega'}\\cdot t\\in \\Omega'$ (cf.~\\cite[Definition 2.45 and Lemma 2.46]{CalziPeloso}). We denote by $B_\\Omega(h,R)$ and $B_{\\Omega'}(\\lambda,R)$ the corresponding open balls of centre $h$ and $\\lambda$, respectively, and radius $R$. \n\\end{deff}\n\n\n\\begin{deff}\nGiven $\\delta>0$ and   $R>1$,\nwe say that a family $(h_{k})_{k\\in K}$ of elements of $\\Omega$  is a\n$(\\delta,R)$-lattice if the following hold:\n\\begin{itemize}\n\\itb the balls $B_\\Omega(h_k,\\delta)$ are pairwise disjoint;\n\\itb the balls  $B_\\Omega(h_k,R\\delta)$ cover $D$.\n\\end{itemize}\nWe define $(\\delta,R)$-lattices on $\\Omega'$ in an analogous fashion.  \n\nFurthermore, we say that a family\n$(\\zeta_{j,k},z_{j,k})_{j\\in J, k\\in  K}$ of elements of $D$ is a $(\\delta,R)$-lattice if the following\nhold: \n\\begin{itemize}\n\\itb  there is a $(\\delta,R)$-lattice $(h_k)_{k\\in K}$   on $\\Omega$ such that\n$h_k \\coloneqq \\rho(\\zeta_{j,k},z_{j,k})$  for every  $j\\in J$ and every $k\\in K$; \n\\itb the balls $B((\\zeta_{j,k},z_{j,k}),\\delta)$ are pairwise disjoint;\n\\itb the balls  $B((\\zeta_{j,k},z_{j,k}),R\\delta)$ cover $D$.\n\\end{itemize}\n\\end{deff}\n\nThe following  result  is~\\cite[Lemma~2.55]{CalziPeloso}, and  guarantees the\nexistence of lattices as above on $D$.\n\\begin{lem}\\label{lem:32}\nTake $\\delta>0$. Then, there is a $(\\delta,4)$-lattice on $D$. \n\\end{lem}\n\n\n\n\\subsection{Fourier transform on $\\Nc$}\n\nWe now recall the definition and a few properties of the Fourier\ntransform on $\\Nc$, cf.~\\cite[Section 1.2]{CalziPeloso} for  further  details.\n\n\nDefine $W\\coloneqq\\Set{\\lambda\\in F'\\colon \\exists \\zeta\\neq 0 \\ \\text{such\nthat}\\ \n\\langle \\lambda, \\Ima \\Phi(\\zeta, \\,\\cdot\\,)\\rangle=0 }$.\nObserve that, for every $\\lambda\\in F'\\setminus W$, the quotient\n$\\Nc\/\\ker \\lambda$ is a $(2 n+1)$-dimensional Heisenberg group, so\nthat the Stone--von Neumann theorem (cf., e.g.,~\\cite[Theorem\n1.50]{Folland}) implies that there is up to unitary equivalence)  a unique  irreducible unitary\nrepresentation $\\pi_\\lambda$ of $\\Nc$ in some hilbertian space\n$H_\\lambda$ such that $\\pi_\\lambda(0,i\nx)=\\ee^{-i\\langle \\lambda,x\\rangle}$ for every $x\\in F$. Using the\nPlancherel formula on the quotients $\\Nc\/\\ker \\lambda$ and integrating\nin $\\lambda$, one may then find the Plancherel formula for $\\Nc$\n(cf.,e.g.,~\\cite[Section 2]{AstengoCowlingDiBlasioSundari}). Namely,\nthere is a constant $c>0$ such that \n\\[\n\\norm{f}^2_{L^2(\\Nc)}=c \\int_{F'\\setminus W} \\norm{\\pi_\\lambda(f)}_{\\Lin^2(H_\\lambda)}^2 \\Delta_{\\Omega'}^{-\\vect b}(\\lambda)\\,\\dd \\lambda\n\\] \nfor every $f\\in L^1(\\Nc)\\cap L^2(\\Nc)$ (cf.~\\cite[Corollary 1.17 and Proposition 2.30]{CalziPeloso}), where $\\Lin^2(H_\\lambda)$ denotes the space of Hilbert--Schmidt endomorphisms of $H_\\lambda$. \n\nNotice, though, that $\\pi_\\lambda(f_h)=0$ for almost every $\\lambda\\in F'\\setminus (W\\cup \\Omega')$, for every $h\\in \\Omega$, and for every $f$ in the space $ A^{p,q}_{\\vs}$ to be defined below, $p\\in ]0,2]$ (cf.~\\cite[Corollares 1.37 and 3.3, and Proposition 3.2]{CalziPeloso}). For this reason, we shall only describe $\\pi_\\lambda$ for $\\lambda\\in \\Omega'$ (`Bargmann representation'). We define $H_\\lambda\\coloneqq \\Hol(E)\\cap L^2(\\nu_\\lambda)$, where $\\nu_\\lambda= \\ee^{-2\\langle\\lambda, \\Phi(\\,\\cdot\\,)\\rangle}\\cdot \\Hc^{2 n}$, and  \n\\begin{equation}\\label{eq:pi-lambda}\n\\pi_\\lambda(\\zeta,x)  \\psi  (\\omega)\n\\coloneqq \\ee^{\\langle \\lambda_\\C, -i x +2 \\Phi(\\omega, \\zeta)-\\Phi(\\zeta)\\rangle}  \\psi (\\omega- \\zeta),\n\\end{equation} \nfor every $\\psi \\in H_\\lambda$, for every $\\omega\\in E$, and for every $(\\zeta,x)\\in \\Nc$.\nDenote by $P_{\\lambda,0}$ the self-adjoint projector of $H_\\lambda$\nonto the space of constant functions, for every $\\lambda\\in \\Omega'$,\nand define $P_{\\lambda,0}=0$ for $\\lambda\\in F'\\setminus (W\\cup\n\\Omega')$. The introduction of $P_{\\lambda,0}$ is justified by the\nobservation that $\\pi_\\lambda(f_h)=\\pi_\\lambda(f_h)P_{\\lambda,0}$ for\nalmost every $\\lambda\\in F'\\setminus W$, for every $h\\in \\Omega$, and\nfor every $f$ in the space $ A^{p,q}_{\\vs}$ to be defined below, $p\\in\n]0,2]$ (cf.~\\cite[Corollares 1.37 and 3.3, and Proposition\n3.2]{CalziPeloso}). \n\n\n\n\n\n\n\n\\section{Function spaces}\\label{sec:fnct-sp}\nIn this section we introduce the functions spaces on $D$ and $\\Nc$ that we  shall consider  in this paper.  We refer\nthe reader to~\\cite{CalziPeloso} for  further  details and references.\n\n\\begin{deff}\\label{def:Lpqs}\nTake $\\vs\\in \\R^r$ and $p,q\\in ]0,\\infty]$. \nWe define \n\\[\nL^{p,q}_\\vs(D)\\coloneqq\\Set{ f\\colon D\\to \\C\\colon  f \\text{ is measurable}, \\int_\\Omega  \\Big(    \\Delta_\\Omega^{\\vs}(h) \\norm{f_h}_{L^p(\\Nc)}  \\Big) ^q\\,\\dd  \\nu_\\Omega(h)<\\infty  }  \n\\]\n(modification when $q=\\infty$), and  $L^{p,q}_{\\vs,0}(D)$ as the\nclosure of $C_c( D)$ in $L^{p,q}_{\\vs}(D)$.   Constistently with~\\eqref{eq:1}, we define\n\\[\nA^{p,q}_\\vs(D)= L^{p,q}_\\vs(D)\\cap \\Hol(D), \\qquad\\text{and}\\qquad\nA^{p,q}_{\\vs,0}(D)\n= L^{p,q}_{\\vs,0}(D) \\cap \\Hol(D).\n\\]\n\\end{deff}\nNotice that $L^{p,q}_{\\vs,0}(D)=L^{p,q}_{\\vs}(D)$  and $A^{p,q}_{\\vs,0}(D)=A^{p,q}_{\\vs}(D)$\nif (and only\nif) $p,q<\\infty$.\n\nIt is possible to characterize the values of $p,q, \\vs$ for which\n$A^{p,q}_{\\vs}(D)$ is non-trivial. The following is~\\cite[Proposition~3.5]{CalziPeloso}.\n\\begin{prop}\nTake $\\vs\\in \\R^r$ and $p,q\\in ]0,\\infty]$. Then,   $A^{p,q}_{\\vs,0}(D)\\neq \\Set{0}$ (resp.\\   $A^{p,q}_{\\vs}(D)\\neq \\Set{0}$) \nif and only if  $\\vs\\in \\frac{1}{2 q}\\vect m+(\\R_+^*)^r$ (resp.  $\\vs\\in \\R_+^r$ if $q=\\infty$). \n\\end{prop}\n\n\\begin{deff}\nFor $\\vs\\in\\C^r$ we define the kernel function $B^{\\vs}$ as\n\\[\nB_{(\\zeta',z')}^{\\vs} (\\zeta,z)\\coloneqq \\Delta_\\Omega^{\\vs}\\left( \\frac{z-\\overline{z'}}{2i}\n-\\Phi(\\zeta,\\zeta') \\right)\n\\]\nfor every $((\\zeta,z),(\\zeta',z'))\\in (D\\times \\overline D)\\cup (\\overline D \\times D)$. \n\\end{deff}\n\n\\begin{oss}\nIt turns out that, when $\\vs \\in \\frac14 {\\vect m}+ (\\R^*_+)^r$, \nthe\nreproducing kernel of $A^{2,2}_\\vs$ is  given by \n\\[\nK_\\vs ((\\zeta,z),(\\zeta',z'))\n\\coloneqq c\\frac{\\Gamma_{\\Omega'}(2 \\vs-\\vb-\\vd)}{\\Gamma_\\Omega(2\\vs)}\nB_{(\\zeta',z')}^{\\vd+\\vb-2\\vs} (\\zeta,z)\n\\]\nfor a suitable constant $c>0$, cf.~\\cite[Remark 3.12]{CalziPeloso}.\n\nIn addition, we can  also  describe the {\\em Cauchy--Szeg\\H o} kernel,\nthat is, the reproducing kernel of the Hardy space\n$A^{2,\\infty}_{\\vect 0}$. Indeed, for every $(\\zeta,z)\\in D$ and for every\n$(\\zeta',x')\\in \\Nc$, set \n\\[\nS_{(\\zeta,z)}(\\zeta',x') \\coloneqq\nc' \\big( B^{\\vb+\\vd}_{(\\zeta,z)}\\big)_0(\\zeta',x')\n\\]\nfor a suitable constant $c'>0$, cf.~\\cite[Lemma 5.1]{CalziPeloso}.  Then,  for every $f\\in A^{2,\\infty}_{\\vect 0}$, \n\\[\nf(\\zeta,z)=\\langle f_0| S_{(\\zeta,z)}\\rangle\n\\]\nfor every $(\\zeta,z)\\in D$, where $f_0\\coloneqq\\lim\\limits_{h\\to 0} f_h$ in $L^2(\\Nc)$. \n\\end{oss}\n\nFor $p,q\\in ]0,\\infty]$, and two sets $J$ and $K$, define\n\\[\n\\ell^{p,q}(J,K)\\coloneqq \\Set{\\lambda\\in \\C^{J\\times K}\\colon ((\\lambda_{j,k})_{j\\in J})_{k\\in K}\\in \\ell^q(K;\\ell^p(J))},\n\\]\nendowed with the corresponding quasi-norm, and define\n$\\ell^{p,q}_0(J,K)$ as the closure of $\\C^{(J\\times K)}$ in\n$\\ell^{p,q}(J,K)$.\n\n\n\\begin{deff}\\label{def:L-property}\nWe say that property $\\atomic^{p,q}_{\\vs,\\vect{s'},0}$ (resp.\\\n$\\atomic^{p,q}_{\\vs,\\vect{s'}}$) holds if for every $\\delta_0>0$\nthere is a $(\\delta,4)$-lattice $(\\zeta_{j,k},z_{j,k})_{j\\in J, k\\in K}$, with $\\delta\\in ]0,\\delta_0]$,  such that, defining\n$h_k\\coloneqq \\rho(\\zeta_{j,k},z_{j,k})$ for every $k\\in K$ and\nfor some (hence every) $j\\in J$, the mapping \n\\[\n\\Psi\\colon\\lambda \\mapsto \\sum_{j,k} \\lambda_{j,k}\nB_{(\\zeta_{j,k},z_{j,k})}^{\\vect{s'}} \\Delta_\\Omega^{(\\vb+\\vect\nd)\/p-\\vs-\\vect{s'}}(h_k) \n\\]\nis well defined (with locally uniform convergence of the sum) and maps\n$\\ell^{p,q}_0(J,K) $ into $A^{p,q}_{\\vs,0}(D)$ continuously\n(resp.\\ maps $\\ell^{p,q}(J,K) $ into $A^{p,q}_{\\vs}(D)$\ncontinuously). \n\nIf we may take $(\\zeta_{j,k},z_{j,k})_{j\\in J, k\\in K}$, for every\n$\\delta_0>0$ as above, in such a way that the corresponding mapping\n$\\Psi$ is onto, then we say that property $\\atomics^{p,q}_{\\vect\ns,\\vect{s'},0}$ (resp.\\ $\\atomics^{p,q}_{\\vs,\\vect{s'}}$) holds.  \n\\end{deff}\n\nWe shall present later (cf.~Proposition~\\ref{prop:25})  some sufficient conditions for property\n$\\atomics$ to hold. See also~\\cite{CalziPeloso} for a more thorough\ndiscussion and some necessary conditions.\n\n\n\n\nIn this paper we  are  also interested in a family of spaces of holomorphic\nfunctions on $D$,  denoted by $\\widetilde A^{p,q}_{\\vs}(D)$,\nwhich is  defined in connection with  the boundary values  of  the elements of \n$A^{p,q}_\\vs(D)$. We begin by introducing some \nBesov-type spaces defined on the \\v{S}ilov boundary  $bD$, that we\nidentify with $\\Nc$.\n\nWe define a space of test functions on $\\Nc$ by setting\n\\[\n\\Sc_\\Omega(\\Nc)\\coloneqq\\Set{ \\psi  \\in \\Sc(\\Nc)\\colon \\exists \\varphi\\in C^\\infty_c(\\Omega')\\;\\forall\\lambda\\in F'\\setminus W\\quad \\pi_\\lambda(\\psi)= \\varphi(\\lambda)P_{\\lambda,0} } .\n\\]\nThen, define $\\Sc_{\\Omega,L}(\\Nc)\\coloneqq\n\\Sc(\\Nc)*\\Sc_{\\Omega}(\\Nc)$, endowed with the inductive limit of the\ntopologies induced by $\\Sc(\\Nc)$ on its subspaces $\\Sc(\\Nc)*\\psi$,\n$\\psi\\in \\Sc_{\\Omega}(\\Nc)$.\\footnote{It is not hard to see that this  definition is equivalent to~\\cite[Definition 4.4]{CalziPeloso}.}\nDenote by  $\\Sc_{\\Omega,L}'(\\Nc)$ the dual of $\\Sc_{\\Omega,L}(\\Nc)$.\nSee~\\cite[Propositions 4.2 and 4.5, and Lemma~4.14]{CalziPeloso} for a proof of the following result. \n\\begin{prop}\\label{prop:4.2}\nThe following hold:\n\\begin{enumerate}\n\\item[\\em(1)] the mapping $\\Fc_\\Nc\\colon \\varphi\\mapsto [\\lambda\n  \\mapsto \\tr(\\pi_\\lambda(\\varphi) ) ] $ induces an isomorphism of\n  $\\Sc_\\Omega(\\Nc)$ onto $C^\\infty_c(\\Omega')$; \n\n\\item[\\em(2)]  given two $(\\delta,R)$-lattices $(\\lambda_k)_{k\\in K}$ and $(\\lambda'_{k'})_{k'\\in K'}$ on $\\Omega'$, and two families $(\\psi_k)_{k\\in K}, (\\psi'_{k'})_{k'\\in K'}$ of elements of $\\Sc_\\Omega(\\Nc)$ such that $((\\Fc_\\Nc \\psi_k)(\\,\\cdot\\, t_k))$ and $((\\Fc_\\Nc \\psi'_{k'})(\\,\\cdot\\, t'_{k'}))$ are bounded families of positive elements of $C^\\infty_c(\\Omega')$, where $t_k,t'_{k'}\\in T_+$ are such that $\\lambda_k=e_{\\Omega'}\\cdot t_k$ and $\\lambda'_{k'}=e_{\\Omega'}\\cdot t'_{k'}$, and\n\\[\n\\sum_k \\Fc_\\Nc\\psi_k, \\sum_{k'} \\Fc_\\Nc \\psi_{k'}\\Meg 1\n\\]\non $\\Omega'$,  one has  \n\\[\n\\Big\\| \\Delta_{\\Omega'}^{\\vs}(\\lambda_{k'}) \\big\\|u*\\psi'_{k'}\n\\big\\|_{L^p(\\Nc)}  \\Big\\|_{\\ell^q(K')}\n\\approx\\Big\\| \\Delta_{\\Omega'}^{\\vs}(\\lambda_k) \\big\\| u*\\psi_k\n\\big\\|_{L^p(\\Nc)}  \\Big\\|_{\\ell^q(K)},\n\\]\nfor every $u\\in \\Sc_{\\Omega,L}'(\\Nc)$.  \n\\end{enumerate} \n\\end{prop}\n\n\\begin{deff}\nLet $\\vs\\in \\R^r$, $p,q\\in ]0,\\infty]$. Take\n$(\\lambda_k)_{k\\in K}$ and $(\\psi_k)$ as in\nProposition~\\ref{prop:4.2}.\nThen, we define $B^{\\vs}_{p,q}(\\Nc,\\Omega)$ as the space of $u\\in \\Sc_{\\Omega,L}'(\\Nc)$ such that\n\\[\n(\\Delta^{\\vs}_{\\Omega'}(\\lambda_k) (u*\\psi_k))_k\\in  \\ell^q (K;L^p(\\Nc)),\n\\]\nendowed with the corresponding topology. We denote by  $\\mathring{B}^{\\vs}_{p,q}(\\Nc,\\Omega)$ the closure of $\\Sc_{\\Omega,L}(\\Nc)$ in $ B^{\\vs}_{p,q}(\\Nc,\\Omega)$, which is  the space of $u\\in\n\\Sc_{\\Omega,L}'(\\Nc)$ such that  \n\\[\n(\\Delta^{\\vs}_{\\Omega'}(\\lambda_k) (u*\\psi_k))_k\\in \\ell^q_0(K;L^p_0(\\Nc)).\n\\]\n\\end{deff}\n\nWe are now able  to define  an extension operator  from the Besov-type spaces $B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ on the \\v{S}ilov\nboundary $\\Nc$ to $D$ to the function  spaces $\\widetilde\nA^{p,q}_\\vs(D)$ and $\\widetilde A^{p,q}_{\\vs,0}(D)$ on $D$. \n\n\\begin{deff}\\label{def:tilde-spaces}\nTake $p,q\\in]0,\\infty]$ and  $\\vs\\in\n\\frac{1}{p}(\\vb+\\vd)+\\frac{1}{2 q'}\\vect{m'}+(\\R_+^*)^r$, so that, \nfor every $(\\zeta,z)\\in D$, $S_{(\\zeta,z)}\\in \\mathring\nB^{\\vs-(1\/p-1)_+(\\vect b+\\vect d)}_{p',q'}(\\Nc,\\Omega)$\n(cf.~\\cite[Lemma 5.1]{CalziPeloso}).  \nThen, define,  for every  $u\\in B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ and for every $(\\zeta,z)\\in D$,\n\\[\n\\Ec u  (\\zeta,z) \\coloneqq\\langle u| S_{(\\zeta,z)}\\rangle,\n\\]\nso that $\\Ec$ maps $B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ into $A^{\\infty,\\infty}_{\\vs-(\\vect b+\\vect d)\/p}(D)$ continuously (cf.~\\cite[Theorem 5.2]{CalziPeloso}).\nWe define \n\\[\n\\widetilde A^{p,q}_\\vs(D) \\coloneqq\\Ec(B^{-\\vs}_{p,q}(\\Nc,\\Omega))\\qquad\n\\text{and} \\qquad\\widetilde A^{p,q}_{\\vs,0}(D)\\coloneqq \\Ec(\\mathring B^{-\\vs}_{p,q}(\\Nc,\\Omega)),\n\\]\nand endow both spaces with the corresponding (direct image) topology.\n\nIn addition, we define the {\\em boundary value operator}\n$\\Bc \\colon A^{2,\\infty}_{\\vect 0}(D)\\to B^{\\vect 0}_{2,2} (\\Nc,\\Omega) $ \n\\[\n\\Bc\\colon A^{2,\\infty}_{\\vect 0}(D)\\ni f \\mapsto \\lim_{h\\to 0} f_h\\in L^2(\\Nc)\n\\]\n\\end{deff}\nWe remark that,  by~\\cite[Corollary 1.37]{CalziPeloso}, $\\Bc$  is\nwell defined and continuous, and that $\\Ec \\Bc=I$. \n\nIt is possible to describe the boundary values of functions in\n$A^{p,q}_\\vs(D)$, that is, the the limits of $f_h$, as $h\\to 0,h\\in \\Omega$, for $f\\in\nA^{p,q}_\\vs(D)$, as elements  of  $B^{-\\vs}_{p,q}(\\Nc,\\Omega)$.\nConversely, under  suitable  assumptions, every element of\n$B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ is the boundary value of a unique element\nof $A^{p,q}_{\\vs}(D)$.  Precisely, the following holds,\ncf.~\\cite[Theorem 5.2, Proposition 5.4 and its proof, and Corollary 5.11]{CalziPeloso}.\n\n\\begin{prop}\\label{prop:23bis}\nTake $p,q\\in ]0,\\infty]$, and $\\vs\\in \\sup\\Big( \\frac{1}{2 q}\\vect m,\n\\frac{1}{p}(\\vb+\\vd)+\\frac{1}{2 q'}\\vect{m'}\\Big)+(\\R_+^*)^r $. \nThen, the following hold:\n\\begin{enumerate}\n\\item[\\em(1)] $(\\Ec u)_h$ converges to $u$ in\n$B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ (resp.\\ in $\\Sc'_{\\Omega,L}(\\Nc)$)\nfor everu $u\\in \\mathring B^{-\\vs}_{p,q}(\\Nc,\\Omega)$ (resp.\\ for\nevery $u\\in B^{-\\vs}_{p,q}(\\Nc,\\Omega)$);  \n\n\\item[\\em(2)] the operator $\\Bc$ induces a continuous linear mapping\n\\[\n\\Bc\\colon A^{p,q}_{\\vs}(D)\\to B^{-\\vs}_{p,q}(\\Nc,\\Omega)\n\\qquad \\text{(resp.\\ } \\Bc\\colon A^{p,q}_{\\vs,0}(D)\\to \\mathring{B}^{-\\vs}_{p,q}(\\Nc,\\Omega))\n\\]\nsuch that $\\Ec \\Bc=I$;\n\n\\item[\\em(3)] there are continuous inclusions\n\\[\n\\Ec(\\Sc_{\\Omega,L}(\\Nc))\\subseteq A^{p,q}_{\\vs}(D) \\subseteq\n\\widetilde A^{p,q}_{\\vs}(D)\n\\qquad\n(\\text{resp.\\ } \\Ec(\\Sc_{\\Omega,L}(\\Nc))\\subseteq A^{p,q}_{\\vs,0}(D) \\subseteq \\widetilde A^{p,q}_{\\vs,0}(D))  ;\n\\]\n\n\\item[\\em(4)] if, further,\n$\\vs\\in\\sup\\Big(\\frac{1}{2 q}\\vect m+\\big( \\frac{1}{2 \\min(p,p')}-\\frac{1}{2q} \\big)_+\\vect{m'} , \\frac{1}{p}(\\vb+\\vd)+\\frac{1}{2 q'}\\vect{m'}\\Big) +(\\R_+^*)^r,\n$\nthen,\n\\[\nA^{p,q}_{\\vs}(D)=\\widetilde A^{p,q}_{\\vs}(D)  \\qquad\\text{and} \\qquad\nA^{p,q}_{\\vs,0}(D)=\\widetilde A^{p,q}_{\\vs,0}(D).\n\\]\n\\end{enumerate}\n\\end{prop}\n\nWe now present some sufficient conditions for property $\\atomics$. See~\\cite[Corollary 5.14]{CalziPeloso} for a proof of the following result.\n\n\\begin{prop}\\label{prop:25}\nTake $p,q\\in ]0,\\infty]$, $\\vs\\in \\sup\\Big( \\frac{1}{2 q}\\vect m,\n\\frac{1}{p}(\\vb+\\vd)+\\frac{1}{2 q'}\\vect{m'}\\Big)+(\\R_+^*)^r $, and\n$\\vs'\\in \\frac{1}{\\min(1,p)}(\\vect b+\\vect d)-\\frac{1}{2 q}\\vect{m'} -\n\\left(\\frac{1}{2 \\min(1,p)}-\\frac{1}{2q}  \\right)_+\\vect m -\n\\vs-(\\R_+^*)^r$. If $A^{p,q}_{\\vect s,0}(D)=\\widetilde A^{p,q}_{\\vs,0}(D)$\n(resp.\\ $A^{p,q}_{\\vect s}(D)=\\widetilde A^{p,q}_{\\vs}(D)$), then property\n$\\atomics^{p,q}_{\\vs, \\vs',0}$ (resp.\\ $\\atomics^{p,q}_{\\vs,\\vs'}$)\nholds. \n\\end{prop}\n\n\n\\medskip\n\nWe conclude this section with a  technical lemma that will be very useful later on. \n\\begin{lem}\\label{lem:5}\nTake $\\vs\\in \\R^r$, $p,q\\in ]0,\\infty]$, $t\\in T_+$, and $g\\in\nGL(E)$ such that $t\\cdot \\Phi=\\Phi\\circ (g\\times g)$. Then, \n\\[\n\\norm{f\\circ (g \\times t)}_{L^{p,q}_{\\vs}(D)}=\\Delta_\\Omega^{(\\vb+\\vd)\/p-\\vs}(t)\\norm{f}_{L^{p,q}_{\\vs}(D)}\n\\]\nfor every $\\nu_D$-measurable function $f$ on $D$.\n\\end{lem}\n\n\\begin{proof}\nObserve that \n\\[\n(f\\circ (g\\times t))_h(\\zeta,x)= f_{t\\cdot h}(g\\zeta,t\\cdot x)\n\\]\nfor every $(\\zeta,x)\\in D$ and for every $h\\in\\Omega$,  so that\n\\[\n\\norm{(f\\circ (g\\times t))_h}_{L^p(\\Nc)}= \\Delta^{(\\vb+\\vd)\/p}(t) \\norm{f_{t\\cdot h}}_{L^p(\\Nc)}\n\\]\nfor every $h\\in \\Omega$, thanks to~\\cite[Lemmas 2.9 and 2.18]{CalziPeloso}.\nThen,\n\\[\n\\norm{f\\circ (g\\times t)}_{L^{p,q}_{\\vs}(D)}=  \\Delta^{(\\vb+\\vd)\/p-\\vs}(t)\\norm{f}_{L^{p,q}_{\\vs}(D)},\n\\]\nwhence the result.\n\\end{proof}\n\n\\section{Carleson Measures}\\label{sec:Carleson}\n\nIn this section we prove our main results concerning Carleson measures\nfor $A^{p,q}_\\vs$.\nFor every  $\\mi\\in\\cM_+(D)$  and $R>0$, we set\n\\begin{equation}\\label{eq:MR-mi}\nM_R(\\mi)\\colon D\\ni (\\zeta,z)\\mapsto \\mi(B((\\zeta,z),R))\\in [0,\\infty[.\n\\end{equation}\n\n\n\n\n\n\\begin{lem}\\label{lem:1}\nTake $\\vs\\in \\R^r$, $p,q\\in ]0,\\infty]$, and $\\mi\\in\\cM_+(D)$. Then, the following conditions are equivalent:\n\\begin{enumerate}\n\\item[{\\em(1)}]  there is $R>0$ such that $M_R(\\mi)\\in L^{p,q}_{\\vs}(D)$;\n\n\\item[{\\em(2)}]  $M_R(\\mi)\\in L^{p,q}_{\\vs}(D)$ for every $R>0$;\n\n\\item[{\\em(3)}] there is a $(\\delta,R)$-lattice $(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ on $D$, with $\\delta>0$ and $R>1$, such that \n\\[\n\\Big(\\Delta_\\Omega^{\\vs-(\\vb+\\vect\nd)\/p}(\\rho(\\zeta_{j,k},z_{j,k}))M_{R\\delta}(\\mi)(\\zeta_{j,k},z_{j,k})\n\\Big)\\in\\ell^{p,q}(J,K);\n\\]\n\n\\item[{\\em(4)}] for every $(\\delta,R)$-lattice $(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ on $D$, with $\\delta>0$ and $R>1$, \n\\[\n\\big(\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(\\rho(\\zeta_{j,k},z_{j,k}))M_{R\\delta}(\\mi)(\\zeta_{j,k},z_{j,k}) \\big)\\in\\ell^{p,q}(J,K).\n\\]\n\\end{enumerate}\nThe same holds if one replaces $L^{p,q}_{\\vs}(D)$ and\n$\\ell^{p,q}(J,K)$ with $L^{p,q}_{\\vs,0}(D)$ and\n$\\ell^{p,q}_0(J,K)$, respectively. \n\\end{lem}\n\nThis extends~\\cite[Lemmas 2.9 and 2.12]{NanaSehba}, where the case in\nwhich $p=q\\in [1,\\infty]$, $D$ is an irreducible symmetric tube\ndomain, and $\\vect{s}\\in \\R\\vect 1_r$, is considered.\\footnote{Notice,\nthough, that in~\\cite[Lemmas 2.9 and 2.12]{NanaSehba} the use of the\n`pure-norm' spaces $L^p$ and $\\ell^p$ allows for the use of more\ngeneral lattices. } \n\n\\begin{proof}\nWe shall only prove the first assertion. The second assertion is clear\nif $\\mi$ has compact support, and then follows by approximation, since\nthe equivalence of conditions (1)--(4) is quantitative by the closed\ngraph theorem, or by the proof below. \n\n(1) $\\implies$ (2). Take $R'>0$. Observe that, since $\\overline B((0,i\ne_\\Omega),R')$ is compact by~\\cite[Proposition 2.44]{CalziPeloso},\nthere is a finite family $(\\zeta_j,z_j)_{j\\in J}$ of elements of $D$\nsuch that $B((0,i e_\\Omega),R')\\subseteq \\bigcup_{j\\in\nJ}B((\\zeta_j,z_j),R)$.  \nFor every $j\\in J$, choose $t_j\\in T_+$ and $g_j\\in GL(E)$ in such a\nway that $t_j\\cdot \\Phi=\\Phi\\circ (g_j\\times g_j)$ and $t_j\\cdot\ne_\\Omega= \\rho(\\zeta_j,z_j)$. If we define  \n\\[\n\\phi_j\\colon (\\zeta,z)\\mapsto (\\zeta_j,\\Rea z_j+i\\Phi(\\zeta_j))\\cdot (g_j \\zeta,t_j\\cdot z),\n\\]\nthen $\\phi_j$ is an affine automorphism of $D$ and\n$B((\\zeta,z),R')\\subseteq \\bigcup_{j\\in J} B(\\phi_j(\\zeta,z),R)$ for\nevery $(\\zeta,z)\\in D$. Therefore, in order to prove that\n$M_{R'}(\\mi)\\in L^{p,q}_{\\vs}(D)$, it will suffice to prove that\n$M_R(\\mi)\\circ \\phi_j\\in L^{p,q}_{\\vs}(D)$ for every $j\\in J$.\nSince this follows easily from Lemma~\\ref{lem:5}, this proves (2). \n\n(2) $\\implies$ (4). Let $(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ be a\n$(\\delta,R)$-lattice on $D$ for some $\\delta>0$ and $R>1$. Observe\nthat \n\\[\n\\sum_{j,k} M_{R\\delta}(\\zeta_{j,k},z_{j,k}) \\chi_{B((\\zeta_{j,k},z_{j,k}),\\delta)}\\meg M_{(R+1)\\delta}(\\mi)(\\zeta_{j,k},z_{j,k})\n\\]\non $D$, so that \n\\[\n\\sum_{j,k}  M_{R\\delta}(\\zeta_{j,k},z_{j,k})\\chi_{B((\\zeta_{j,k},z_{j,k}),\\delta)}\\in L^{p,q}_{\\vs}(D).\n\\]\nNow, observe that there is a constant $C_1>0$ such that\n\\[\n\\norm{(\\chi_{B((0,i e_\\Omega),\\delta)})_h}_{L^p(\\Nc)}\\Meg C_1\n\\]\nfor every $h\\in B_\\Omega(e_\\Omega,\\delta\/2)$,\\footnote{Observe that,\nsince $B((0,i e_\\Omega),\\delta)$ is an open set, the mapping\n$h\\mapsto \\norm{(\\chi_{B((0,i e_\\Omega),\\delta)})_h}_{L^p(\\Nc)}$ is\nlower semi-continuous, and that it vanishes nowhere on the compact\nset $\\overline B_\\Omega(e_\\Omega,\\delta\/2)$, thanks to~\\cite[Lemma\n2.46]{CalziPeloso}.} so that, by homogeneity, \n\\[\n\\norm{(\\chi_{B((\\zeta_{j,k},z_{j,k}),\\delta)})_h}_{L^p(\\Nc)}\\Meg C_1\\Delta_\\Omega^{-(\\vb+\\vd)\/p}(h_k)\n\\]\nfor every $h\\in B_\\Omega(h_k,\\delta\/2)$ and for every $(j,k)\\in J\\times K$, where $h_k\\coloneqq\\rho(\\zeta_{j,k},z_{j,k})$ for every $(j,k)\\in J\\times K$.\nIt then follows that\n\\[\n\\begin{split}\n&\\norm*{\\sum_{j,k} \\left( \\chi_{B((\\zeta_{j,k},z_{j,k}),\\delta)}\\right)_h M_{R\\delta}(\\zeta_{j,k},z_{j,k})}_{L^p(\\Nc)}\\\\\n&\\qquad\\Meg C_1 \\norm*{ \\big(\\Delta_\\Omega^{-(\\vb+\\vd)\/p}(h_k)M_{R\\delta}(\\zeta_{j,k},z_{j,k}) \\chi_{B_\\Omega(h_k,\\delta\/2)}(h)\\big)_{j,k}}_{\\ell^p(J\\times K)}\n\\end{split}\n\\]\nfor every $h\\in\\Omega$, whence\n\\[\n\\left(\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)M_{R\\delta}(\\zeta_{j,k},z_{j,k})\\right)\\in \\ell^{p,q}(J,K)\n\\]\nthanks to~\\cite[Corollary 2.49]{CalziPeloso}.\n\n(4) $\\implies$ (3). Obvious by~\\cite[Lemma 2.55]{CalziPeloso}.\n\n(3) $\\implies$ (1). Take a $(\\delta,R)$-lattice\n$(\\zeta_{j,k},z_{j,k})_{j\\in J, k\\in K}$ as in the statement, define\n$h_k\\coloneqq \\rho(\\zeta_{j,k},z_{j,k})$ for every $(j,k)\\in J\\times\nK$, and let us prove that \n\\[\n\\left(\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)M_{R' \\delta}(\\zeta_{j,k},z_{j,k}) \\right)\\in\\ell^{p,q}(J,K)\n\\] \nfor every $R'\\Meg R$. Indeed, for every $(j,k)\\in J\\times K$, define\n$V_{j,k}$ as the set of $(j',k')\\in J\\times K$ such that\n$d((\\zeta_{j,k},z_{j,k}),(\\zeta_{j',k'},z_{j',k'}))<(R+R')\\delta $,\nand observe that $N\\coloneqq \\sup\\limits_{j,k} \\card(V_{j,k})$ is finite\nby~\\cite[Proposition 2.56]{CalziPeloso}. In addition,\nby~\\cite[Corollary 2.49]{CalziPeloso} we may find a constant $C_2>0$\nsuch that \n\\[\n\\frac{1}{C_2}\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_{k'}) \\meg\n\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)\\meg\nC_2\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_{k'}) \n\\] \nfor every $(j,k)\\in J\\times K$ and for every $(j',k')\\in V_{j,k}$. Then,\n\\[\n\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)M_{R'\\delta}(\\zeta_{j,k},z_{j,k})\\meg\nC_2 \\sum_{(j',k')\\in V_{j,k}}\n\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_{k'})M_{R\\delta}(\\mi)(\\zeta_{j',k'},z_{j',k'}) \n\\]\nfor every $(j,k)\\in J\\times K$. Define $V'_k$ as the set of $k'\\in K$\nsuch that $d(h_k,h_{k'})\\meg (R+R')\\delta$, so that $k'\\in V'_k$ for\nevery $(j',k')\\in V_{j,k}$ and for every $j\\in J$ by~\\cite[Lemma\n2.46]{CalziPeloso}. In addition, $N'\\coloneqq  \\sup\\limits_{k} \\card(V'_{k})$\nis finite by~\\cite[Proposition 2.56]{CalziPeloso}. \nTherefore,\n\\[\n\\begin{split}\n&\\norm*{\\big(\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)M_{R'\\delta}(\\zeta_{j,k},z_{j,k})\\big)_j}_{\\ell^p(J)}\\\\\n&\\qquad\\qquad\\qquad\\qquad\n\\meg C_2 N^{\\max(1,1\/p)} \\norm*{\\big(\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_{k'})M_{R\\delta}(\\zeta_{j',k'},z_{j',k'})\\big)_{j',k'}}_{\\ell^p(J\\times V'_k)}\n\\end{split}\n\\]\nfor every $k\\in K$, so that\n\\[\n\\begin{split}\n&\\norm*{\\left( \\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p}(h_k)M_{R'\\delta}(\\zeta_{j,k},z_{j,k})\\right) _{j,k}}_{\\ell^{p,q}(J,K)}\\\\\n&\\qquad\\qquad\\qquad\\meg C_2 N^{\\max(1,1\/p)} N'^{\\max(1\/p,1\/q)}\n\\norm*{\\left( \\Delta_\\Omega^{\\vs-(\\vb+\\vect\nd)\/p}(h_{k'})M_{R\\delta}(\\mi)(\\zeta_{j',k'},z_{j',k'})\\right)\n_{j',k'}}_{\\ell^{p,q}(J,K)}. \n\\end{split}\n\\]\nwhence our claim.\n\nNow, let $(B_{j,k})_{j,k}$ be a Borel partition of $D$ such that $B_{j,k}\\subseteq B((\\zeta_{j,k},z_{j,k}),R\\delta)$ for every $(j,k)\\in J\\times K$, and observe that\n\\[\nM_\\delta(\\mi)\\meg \\sum_{j,k} \\chi_{B_{j,k}} M_{(R+1)\\delta}(\\zeta_{j,k},z_{j,k})\n\\] \non $D$. In addition, arguing as in the proof of~\\cite[Theorem\n3.23]{CalziPeloso}, we see that there is a constant $C_3>0$ such that \n\\[\n\\norm{(\\chi_{B((\\zeta,z),R\\delta)})_h}_{L^p(\\Nc)}\\meg C_3\n\\Delta^{-(\\vb+\\vd)\/p}_\\Omega(\\rho(\\zeta,z))\n\\chi_{B_\\Omega(\\rho(\\zeta,z),R\\delta)}(h) \n\\]\nfor every $(\\zeta,z)\\in D$ and for every $h\\in \\Omega$.\nHence,\n\\[\n\\norm*{M_\\delta(\\mi)_h}_{L^p(\\Nc)}\\meg C_3\\norm*{\\big(\n\\chi_{B(h_k,R\\delta)}(h) \\Delta^{-(\\vb+\\vect\nd)\/p}_\\Omega(h_k)M_{(R+1)\\delta}(\\zeta_{j,k},z_{j,k})\n\\big)_{j,k}}_{\\ell^p(J\\times K)}. \n\\]\nNow, by~\\cite[Corollary 2.49 and Proposition 2.56]{CalziPeloso} there\nare two constants $C_4>0$ and $N''\\in \\N$ such that \n\\[\n\\frac{1}{C_4}\\Delta^{\\vs}_\\Omega(h)\\meg  \\Delta^{\\vs}_\\Omega(h_k)\\meg C_4 \\Delta^{\\vs}_\\Omega(h)\n\\]\nfor every $h\\in B_\\Omega(h_k, R\\delta)$ and for every $k\\in K$, and\nsuch that $\\sum_k \\chi_{B_\\Omega(h_k,R\\delta)}\\meg N''\n\\chi_\\Omega$. Then, \n\\[\n\\norm*{M_\\delta(\\mi)}_{L^{p,q}_{\\vs}(D)}\\meg C_5\n\\norm*{\\big(\\Delta^{\\vs-(\\vb+\\vd)\/p}_\\Omega(h_k)\nM_{(R+1)\\delta}(\\zeta_{j,k},z_{j,k}) \\big)_{j,k}\n}_{\\ell^{p,q}(J,K)}, \n\\]\nwhere $C_5\\coloneqq C_3 C_4\n\\nu_\\Omega(B_\\Omega(e_\\Omega,R\\delta))^{1\/q}\nN''^{\\max(1\/p,1\/q)}$. Thus,  (1) follows.  \n\\end{proof}\n\n\nThe next proposition provides a first sufficient condition for a\nmeasure $\\mi\\in\\cM_+(D)$ to  be a $p$-Carleson measure or a  compact\n$p$-Carleson measure. \nIt extends~\\cite[Proposition 3.5 and Theorem\n3.8 (i)]{NanaSehba}, where the case in which $p_1=q_1,p\\in\n[1,\\infty[$, $D$ is an irreducible symmetric tube domain, and\n$\\vect{s}\\in \\R\\vect 1_r$, is considered. \n\nWe shall often use  the following notation. Given $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$ and $\\vs\\in\n\\R^r$, we set\\footnote{Recall that $t'\\coloneqq\\max(1,t)'$ for every $t\\in ]0,\\infty]$.}\n\\begin{equation}\\label{p*q*s'}\np^*\\coloneqq (p_1\/p)',\\quad\nq^*\\coloneqq(q_1\/p)',\\quad \\text{and}\\quad\n\\vs^*\\coloneqq \\max(1,p\/p_1)(\\vb+\\vd)-p\\vs.\n\\end{equation}\n\n\\begin{prop}\\label{prop:23}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and take $\\vs\\in\n\\R^r$ such that $\\vs\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ or\n$\\vs\\in \\R_+^r$ if $q_1=\\infty$. Let $p^*,q^*$, and $\\vs^*$ be as in~\\eqref{p*q*s'}. \nLet $\\mi\\in\\cM_+(D)$, and take $R>0$. \nAssume that \n\\[\nM_R(\\mi)\\in L^{p^*,q^*}_{\\vs^*}(D).\n\\]\nThen, $A^{p_1,q_1}_{\\vs}(D)$  embeds continuously into $L^{p}(\\mi)$.\n\nIf, in addition, \n\\[\nM_R(\\mi)\\in L^{p^*, q^*}_{\\vs^*,0}(D),\n\\]\nthen $A^{p_1,q_1}_{\\vs}(D)$  embeds compactly into $L^{p}(\\mi)$.\n\\end{prop}\n\n\n\n\n\\begin{proof}\nLet $(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ be an $(R\/4,4)$-lattice on\n$D$ (cf.~\\cite[Lemma 2.55]{CalziPeloso}), and define\n$S_+\\colon \\Hol(D)\\to \\C^{J\\times K}$ so that \n\\[\n(S_+ f)_{j,k}\\coloneqq \\Delta^{\\vs-(\\vb+\\vect\nd)\/p_1}_\\Omega(h_k) \\max_{\\overline B((\\zeta_{j,k},z_{j,k}), R)}\n\\abs{f} \n\\]\nfor every $(j,k)\\in J\\times K$, where $h_k\\coloneqq\n\\rho(\\zeta_{j,k},z_{j,k})$, so that there is a constant $C_1>0$ such\nthat \n\\[\n\\frac{1}{C_1} \\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\\meg \\norm{S_+\nf}_{\\ell^{p_1,q_1}(J,K)}\\meg C_1\\norm{f}_{A^{p_1,q_1}_{\\vs}(D)} \n\\] \nfor every $f\\in \\Hol(D)$, thanks to~\\cite[Theorem 3.23]{CalziPeloso}.\nIn addition, choose a Borel partition $(B_{j,k})$ of $D$ such that\n$B_{j,k}\\subseteq B((\\zeta_{j,k},z_{j,k}), R)$ for every $(j,k)\\in\nJ\\times K$. Then, for every $f\\in A^{p_1,q_1}_{\\vs}(D)$, \n\\[\n\\begin{split}\n\\norm{f}_{L^{p}(\\mi)}&\\meg \\norm*{\\sum_{(j,k)\\in J\\times K} (S_+\nf)_{j,k}  \\Delta^{(\\vb+\\vd)\/p_1-\\vect\ns}_\\Omega(h_k)\\chi_{B_{j,k}}}_{L^{p}(\\mi)}\\\\ \n&\\meg \\norm*{\\big((S_+ f)_{j,k}  \\Delta^{(\\vb+\\vd)\/p_1-\\vect\ns}_\\Omega(h_k)\nM_R(\\mi)(\\zeta_{j,k},z_{j,k})^{1\/p}\\big)_{j,k}}_{\\ell^{p}(J\\times\nK)}\\\\ \n&\\meg C_2\\norm*{S_+ f}_{\\ell^{p_1,q_1}(J,K)}\\\\ \n&\\meg C_1 C_2 \\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\n\\end{split}\n\\]\nwhere $C_2\\coloneqq \\norm*{\\Delta^{p[(\\vb+\\vd)\/p_1-\\vs]}_\\Omega(h_k) M_R(\\mi)(\\zeta_{j,k},z_{j,k}) }_{\\ell^{p^*,q^*}(J,K)}^{1\/p}$. \nThen, the first assertion follows from Lemma~\\ref{lem:1}.\n\nIn order to prove the second assertion, it will suffice to show that,\nif $\\mi$ has compact support, then the canonical mapping\n$A^{p_1,q_1}_{\\vs}(D)\\to L^{p}(\\mi)$ is compact. Since\n$A^{p_1,q_1}_{\\vs}(D)$ embeds continuously, hence compactly, into\nthe Fr\\'echet--Montel space $\\Hol(D)$,  which in turn embeds\ncontinuously into $L^p(\\mi)$, the assertion follows easily. \n\\end{proof}\n\n\\begin{prop}\\label{prop:26}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and take $\\vs\\in \\R^r$ such that $\\vs\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ (resp.\\ $\\vs\\in \\R_+^r$ if $q_1=\\infty$). \nLet $\\mi\\in\\cM_+(D)$ and  $R>0$. Assume that $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\ $A^{p_1,q_1}_{\\vs}(D)$) embeds continuously into $L^{p}(\\mi)$.\nThen,\n\\[\nM_R(\\mi)\\in L^{\\infty,\\infty}_{p(\\vb+\\vd)\/p_1-p\\vs}(D).\n\\]\nIf, in addition, $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\ $A^{p_1,q_1}_{\\vs}(D)$) embeds compactly into $L^{p}(\\mi)$ and $\\vect{s_1}\\in (\\R_+^*)^r$ if $p_1=q_1=\\infty$, then\n\\[\nM_R(\\mi)\\in L^{\\infty,\\infty}_{p(\\vb+\\vd)\/p_1-p\\vs,0}(D).\n\\]\n\\end{prop}\n\n\n\\begin{proof}\nTake $\\vect{s'}\\in \\R^r$ such that \n\\[\n\\vect{s'}\\in \\frac{1}{p_1} (\\vb+\\vd)- \\frac{1}{2 p_1}\\vect{m'}-(\\R_+^*)^r \\qquad \\text{and} \\qquad\n\\vs+\\vect{s'}\\in \\frac{1}{p_1} (\\vb+\\vd)- \\frac{1}{2 q_1}\\vect{m'}-(\\R_+^*)^r.\n\\]\nThen, $B^{\\vect{s'}}_{(\\zeta',z')}\\in A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\ $B^{\\vect{s'}}_{(\\zeta',z')}\\in A^{p_1,q_1}_{\\vs}(D)$) for every $(\\zeta',z')\\in D$, and\n\\[\n\\norm{B^{\\vect{s'}}_{(\\zeta',z')}}_{A^{p_1,q_1}_{\\vs}(D)}= C_1 \\Delta_\\Omega^{\\vs+\\vs^*-(\\vb+\\vd)\/p_1}(\\rho(\\zeta',z'))\n\\]\nfor a suitable constant $C_1>0$, thanks to~\\cite[Proposition 2.41]{CalziPeloso}. Hence, there is a constant $C_2>0$ such that\n\\[\n\\norm{B^{\\vect{s'}}_{(\\zeta',z')}}_{L^{p}(\\mi)}\\meg C_2 \\Delta_\\Omega^{\\vs+\\vs^*-(\\vb+\\vd)\/p_1}(\\rho(\\zeta',z'))\n\\]\nfor every $(\\zeta',z')\\in D$.\nIn addition, there is a constant $C_3>0$  such that\n\\[\n\\frac{1}{C_3}\\abs{B^{\\vect{s'}}_{(\\zeta',z')}}\\meg \\abs{B^{\\vect{s'}}_{(\\zeta,z)}}\\meg C_3\\abs{B^{\\vect{s'}}_{(\\zeta',z')}}\n\\]\non $D$, for every $(\\zeta',z')\\in D$ and for every $(\\zeta,z)\\in B((\\zeta',z'),r)$, thanks to~\\cite[Theorem 2.47]{CalziPeloso}. Therefore,\n\\[\n\\begin{split}\n\\norm{B^{\\vect{s'}}_{(\\zeta',z')}}_{L^{p}(\\mi)}&\\Meg \\frac{1}{C_3} \\Delta^{\\vect{s'}}_{\\Omega}(\\rho(\\zeta',z'))\\norm{ \\chi_{B((\\zeta',z'),R)} }_{L^{p}(\\mi)}\\\\\n&= \\frac{1}{C_3} \\Delta^{\\vect{s'}}_{\\Omega}(\\rho(\\zeta',z')) M_R(\\mi)(\\zeta',z')^{1\/p}\n\\end{split}\n\\]\nfor every $(\\zeta',z')\\in D$. It then follows that\n\\[\nM_R(\\mi)(\\zeta',z')\\meg C_2^{p} C_3^{ p} \\Delta_\\Omega^{p[\\vs-(\\vb+\\vd)\/p_1]  }(\\rho(\\zeta',z'))\n\\]\nfor every $(\\zeta',z')\\in D$.\n\nNow, assume that $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\ $A^{p_1,q_1}_{\\vs}(D)$) embeds compactly in $L^{p}(\\mi)$, and that $\\vs\\in (\\R_+^*)^r$ if $p_1=q_1=\\infty$.\nDefine $b_{(\\zeta,z)}\\coloneqq B^{\\vect{s'}}_{(\\zeta,z)}\\Delta_\\Omega^{(\\vb+\\vd)\/p_1-\\vs-\\vect{s'}}(\\rho(\\zeta,z))$ for every $(\\zeta,z)\\in D$, so that the family $(b_{(\\zeta,z)})_{(\\zeta,z)\\in D}$ is uniformly bounded in $A^{p_1,q_1}_{\\vs}(D)$. In addition, since $\\vs\\in \\frac{\\vb+\\vd}{p_1}+(\\R_+^*)^r$,~\\cite[Proposition 2.41]{CalziPeloso} implies that $B_{(\\zeta,z)}^{\\vect{s'}}\\in A^{\\infty,\\infty}_{(\\vb+\\vd)\/p_1-\\vs-\\vect{s'},0}(D)$, so that\n\\[\n\\lim_{(\\zeta',z')\\to \\infty} \\abs{b_{(\\zeta',z')}(\\zeta,z)}=\\lim_{(\\zeta',z')\\to \\infty} \\Delta_\\Omega^{(\\vb+\\vd)\/p_1-\\vs-\\vect{s'}}(\\rho(\\zeta',z')) \\abs{B^{\\vect{s'}}_{(\\zeta,z)}(\\zeta',z')}=0\n\\]\nfor every $(\\zeta,z)\\in D$.  Let $b$ be a cluster point of $(b_{(\\zeta,z)})$ in $L^p(\\mi)$ for $(\\zeta,z)\\to \\infty$, and observe that $b=0$ since $b_{(\\zeta,z)}$ converges pointwise (and also locally uniformly, thanks to~\\cite[Theorem 2.47]{CalziPeloso}) to $0$ as $(\\zeta,z)\\to 0$. Since $A^{p_1,q_1}_{\\vs}(D)$ embeds compactly into $L^{p}(\\mi)$, this suffices to prove that  $b_{(\\zeta,z)}$  converges to $0$ in $L^p(\\mi)$ as $(\\zeta,z)\\to \\infty$.\nTherefore, the computations of the proof of the implication (1) $\\implies$ (2) show that\n\\[\n\\lim_{(\\zeta,z)\\to \\infty}\\Delta^{p[(\\vb+\\vd)\/p_1-\\vs]}_\\Omega(\\rho(\\zeta,z))  M_R(\\mi)(\\zeta,z)=0,\n\\]\nwhence the result.\n\\end{proof}\n\n\nWe are now ready to prove our first main result.  It provides a\nnecessary and sufficient condition for a measure\n$\\mi\\in\\cM_+(D)$ to be a $p$-Carleson measure for $A^{p_1,q_1}_{\\vs}$.\nThis  result  is analogous to the classical characterization in the\ncase  of the unit ball, see e.g.~\\cite[Theorem 2.25]{Zhu2}, and it \nextends~\\cite[Theorem 3.5]{NanaSehba}, where the case in which \n$p_1=q_1,p\\in [1,\\infty[$, $D$ is an irreducible symmetric tube\ndomain, and $\\vs\\in \\R\\vect 1_r$, is considered.\n\n\n\n\\begin{teo}\\label{main:1}\nTake $p_1,q_1,p\\in ]0,\\infty[$ with $p_1,q_1\\meg p$, and take\n$\\vs\\in \\R^r$ such that $\\vs\\in \\frac{1}{2 q_1}\\vect\nm+(\\R_+^*)^r$. Let $\\mi\\in\\cM_+(D)$ and  $R>0$. Then, the following\nconditions are equivalent: \n\\begin{enumerate}\n\\item[{\\em (1)}] $A^{p_1,q_1}_{\\vs}(D)$  embeds continuously (resp.\\ compactly) into $L^{p}(\\mi)$;\n\\item[{\\em (2)}]  $M_R(\\mi)\\in L^{\\infty,\\infty}_{p[(\\vb+\\vect\nd)\/p_1-\\vs]}(D)$ (resp.\\  $M_R(\\mi)\\in\nL^{\\infty,\\infty}_{p[(\\vb+\\vd)\/p_1-\\vs],0}(D)$). \n\\end{enumerate}\n\\end{teo}\n\n\n\n\n\\begin{proof}\n(2) $\\implies$ (1). This follows from Proposition~\\ref{prop:23}.\n\n(1) $\\implies$ (2). This follows from Proposition~\\ref{prop:26}.\n\\end{proof}\n\n\nWe now deal with Carleson measures {\\em admiting a loss}, that is, with the case $p_1,q_1\\not \\meg p$.  Note that when dealing with the mixed norm case, this complementary condition is not as simple as in the pure norm one. \nWe provide a characterization of $p$-Carleson measures on\n$A^{p_1,q_1}_\\vs$ in two situations: when $A^{p_1,q_1}_\\vs$ admits a\nsuitable atomic decomposition (Theorem~\\ref{main:2}); when\n$A^{p_1,q_1}_\\vs$ embeds into $A^{p,p}_{\\vs+(1\/p-1\/p_1)(\\vect b+\\vect\nd)}$ (Theorem~\\ref{teo:4}). \n\nThe  next theorem extends~\\cite[Theorem 3.8 (ii)]{NanaSehba},   where the case in which $p_1=q_1,p\\in [1,\\infty[$, $D$ is an irreducible symmetric tube\ndomain, and $\\vect{s}\\in \\R\\vect 1_r$, is considered. \nRecall that property $\\atomic^{p_1,q_1}_{\\vs, \\vect{s'},0}$ is\ndefined   in Definition~\\ref{def:L-property}.\n\n\n\\begin{teo}\\label{main:2}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and $\\vs\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$.\nAssume that property $\\atomic^{p_1,q_1}_{\\vs,\\vect{s'},0}$ holds \nfor some $\\vect{s'}\\in \\R^r$, and let $\\mi\\in\\cM_+(D)$.  \nThen, the following conditions are equivalent: \n\\begin{enumerate}\n\\item[{\\em(1)}] $A^{p_1,q_1}_{\\vs}(D)$  embeds continuously  into $L^{p}(\\mi)$;\n\\item[{\\em(2)}] $A^{p_1,q_1}_{\\vs,0}(D)$  embeds continuously  into $L^{p}(\\mi)$;\n\\item[{\\em(3)}] for some (or, equivalently, every) $R>0$,\n$M_R(\\mi)\\in L^{p^*,q^*}_{\\vs^*}(D)$, where\n$p^*,q^*,\\vs^*$ are as in~\\eqref{p*q*s'}.\n\\end{enumerate}\nIf, in addition, $p< q_1$, then the following conditions are equivalent:\n\\begin{enumerate}\n\\item[{\\em(1$'$)}] $A^{p_1,q_1}_{\\vs}(D)$  embeds compactly  into $L^{p}(\\mi)$;\n\\item[{\\em(2$'$)}] $A^{p_1,q_1}_{\\vs,0}(D)$  embeds compactly  into $L^{p}(\\mi)$;\n\\item[{\\em(3$'$)}]for some (or, equivalently, every) $R>0$,\n$M_R(\\mi)\\in L^{p^*,q^*}_{\\vs^*,0}(D)$, where $p^*,q^*,\\vs^*$ are\nas in~\\eqref{p*q*s'}.\n\\end{enumerate}\n\\end{teo}\n\nBefore we pass to the proof, we need a lemma.\n\n\\begin{lem}\\label{lem:9}\nTake $p_1,q_1,p_2,q_2\\in ]0,\\infty]$, $\\vs\\in\\R^r$, $R>0$, and\n$\\mi\\in\\cM_+(D)$. Assume that $p_1\\meg p_2$ and that $q_1\\meg q_2$. Then, the following hold: \n\\begin{enumerate}\n\\item[{\\em(1)}] if $M_R(\\mi)\\in L^{p_1,q_1}_{\\vs}(D)$, then $M_R(\\mi)\\in L^{p_2,q_2}_{\\vs+(1\/p_2-1\/p_1)(\\vb+\\vd)}(D)$;\n\\item[{\\em(2)}] if $M_R(\\mi)\\in L^{p_1,q_1}_{\\vs,0}(D)$, then $M_R(\\mi)\\in L^{p_2,q_2}_{\\vs+(1\/p_2-1\/p_1)(\\vb+\\vd),0}(D)$;\n\\item[{\\em(3)}] if $q_1<\\infty$, $M_R(\\mi)\\in L^{p_1,q_1}_{\\vs}(D)\\cap\nL^{\\infty,\\infty}_{\\vs-(\\vb+\\vd)\/p_1,0}(D)$, then $M_R(\\mi)\\in\nL^{p_1,q_1}_{\\vs,0}(D)$. \n\\end{enumerate}\n\\end{lem}\n\n\n\n\n\\begin{proof}\nAll the assertions follow easily from Lemma~\\ref{lem:1} and the\nelementary inclusions $\\ell^{p_1,q_1}(J,K)\\subseteq \\ell^{p_2,q_2}(J,K)$,\n$\\ell^{p_1,q_1}_0(J,K)\\subseteq \\ell^{p_2,q_2}_0(J,K)$, and\n$\\ell^{p_1,q_1}(J,K)\\cap \\ell^{\\infty,\\infty}_0(J,K)\\subseteq\n\\ell^{p_1,q_1}_0(J,K)$ (when $q_1<\\infty$) for every two sets $J$ and\n$K$.  \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{main:2}.] \nClearly, (1) $\\implies$ (2), while  the implication (3) $\\implies$  (1) follows from  Proposition~\\ref{prop:23}. \nThus, it only remains to show that \n(2) $\\implies$ (3). \n\nBy assumption, there is a $(\\delta,4)$-lattice\n$(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ for some $\\delta\\meg R\/4$ such\nthat the mapping \n\\[\n\\Psi\\colon \\ell^{p_1,q_1}_0(J,K)\\ni \\lambda \\mapsto \\sum_{j,k}\n\\lambda_{j,k} B^{ \\vect{s'}}_{(\\zeta_{j,k},z_{j,k})} \\Delta_\\Omega^{(\\vect\nb+\\vd)\/p_1-\\vs- \\vect{s'}}(h_k)\\in A^{p_1,q_1}_{\\vect\ns,0}(D) \n\\]\nis well defined and continuous, where $h_k\\coloneqq\n\\rho(\\zeta_{j,k},z_{j,k})$ for every $j\\in J$ and for every $k\\in K$. \n\nNow, take a probability space $(X,\\nu)$ and a sequence\n$(r_\\beta)_{\\beta\\in \\N}$ of $\\nu$-measurable functions on $X$ such\nthat \n\\[\n\\Big(\\bigotimes_{\\beta\\in B} r_\\beta   \\Big)(\\nu)=\\frac{1}{2^{\\card(B)}}\\sum_{\\eps\\in \\Set{-1,1}^B} \\delta_\\eps\n\\]\nfor every finite subset $B$ of $\\N$ (cf.~\\cite[C.1]{Grafakos}). By\nKhintchine's inequality, there is a constant $C_1>0$ such that \n\\[\n\\frac{1}{C_1}\\Big( \\sum_{\\beta\\in\\N} \\abs{a_\\beta}^2\n\\Big)^{1\/2}\\meg \\bigg\\| \\sum_{\\beta\\in B} a_\\beta r_\\beta\n\\bigg\\|_{L^{p}(\\nu)}\\meg C_1 \\Big( \\sum_{\\beta\\in\\N} \\abs{a_\\beta}^2\n\\Big)^{1\/2} \n\\]\nfor every $(a_\\beta)\\in \\C^{(\\N)}$ (cf.~\\cite[C.2]{Grafakos}). Let\n$\\iota\\colon J\\times K\\to \\N$ be a bijection and define $\\vect{r}\\cdot\n\\lambda\\coloneqq ( r_{\\gamma(j,k)} \\lambda_{j,k})_{j,k}$ for every\n$\\lambda\\in \\C^{(J\\times K)}$. By the assumptions and the continuity\nof $\\Psi$, there is a constant $C_2>0$ such that, for every\n$\\lambda\\in \\C^{(J\\times K)}$, \n\\[\n\\norm{\\Psi(\\vect r\\cdot \\lambda)}_{L^{p}(\\mi)}\\meg C_2  \\norm{\\vect\nr\\cdot \\lambda}_{\\ell^{p_1,q_1}(J,K)}= C_2\n\\norm{\\lambda}_{\\ell^{p_1,q_1}(J,K)} \n\\]\n$\\nu$-almost everywhere.\nTherefore, by means of Tonelli's theorem we see that\n\\[\n\\norm*{\\left( \\sum_{j,k} \\abs*{\\lambda_{j,k}  B^{ \\vect{s'}}_{(\\zeta_{j,k},z_{j,k})}}^2    \\Delta_\\Omega^{2[(\\vb+\\vd)\/p_1-\\vs-\\vect{s'}]}(h_k)\n\\right)^{1\/2}}_{L^{p}(\\mi)}\\meg C_1 C_2\\norm{\\lambda}_{\\ell^{p_1,q_1}(J,K)} \n\\]\nfor every $\\lambda\\in \\C^{(J\\times K)}$. Now, observe that there is\n$N\\in\\N$ such that $\\sum_{j,k} \\chi_{B((\\zeta_{j,k},z_{j,k}),R)}\\meg\nN$ on $D$, thanks to~\\cite[Proposition 2.56]{CalziPeloso}. In\naddition, there is a constant $C_3>0$ such that  \n\\[\n\\frac{1}{C_3}\\Delta_\\Omega^{ \\vect{s'}}(h_k)\\meg\n\\abs*{B^{ \\vect{s'}}_{(\\zeta_{j,k},z_{j,k})}}\\meg C_3\n\\Delta_\\Omega^{ \\vect{s'}}(h_k) \n\\]\non $B((\\zeta_{j,k},z_{j,k}),R)$ for every $(j,k)\\in J\\times K$, thanks\nto~\\cite[Theorem 2.47]{CalziPeloso}. \nThen,\n\\[\n\\begin{split}\n&\\norm*{\\left( \\sum_{j,k} \\abs*{\\lambda_{j,k}\nB^{ \\vect{s'}}_{(\\zeta_{j,k},z_{j,k})}}^2\n\\Delta_\\Omega^{2[(\\vb+\\vd)\/p_1-\\vs- \\vect{s'}]}(h_k)\n\\right)^{1\/2}}_{L^{p}(\\mi)}\\\\ \n&\\qquad\\Meg \\frac{1}{C_3}\n\\norm*{\\left(\n\\sum_{j,k}\\chi_{B((\\zeta_{j,k},z_{j,k}),R)}\\abs*{\\lambda_{j,k}}^2 \\Delta_\\Omega^{2[(\\vb+\\vd)\/p_1-\\vs]}(h_k) \\right)^{1\/2}}_{L^{p}(\\mi)}\\\\ \n&\\qquad\\Meg \\frac{N^{-(1-p\/2)_+}}{C_3} \\left(  \\sum_{j,k}   \\abs*{\\lambda_{j,k}}^{p} \\Delta_\\Omega^{p[(\\vb+\\vd)\/p_1-\\vs]}(h_k) M_R(\\mi)(\\zeta_{j,k},z_{j,k}) \\right)^{1\/p}\n\\end{split}\n\\]\nfor every $\\lambda\\in \\C^{(J\\times K)}$. Using the natural duality between $\\ell^{p_1\/p,q_1\/p}_0(J,K)$ and $\\ell^{p^*,q^*}(J,K)$, we then see that\n\\[\n\\big(\\Delta_\\Omega^{p[(\\vb+\\vd)\/p_1-\\vs]}(h_k) M_R(\\mi)(\\zeta_{j,k},z_{j,k})\\big)\\in \\ell^{p^*,q^*}(J,K)  .\n\\]\nThen,  (3)   follows from Lemma~\\ref{lem:1}.\n\nNext, we turn to the second part of the statement.\nThe fact that (3$'$) $\\implies$ (1$'$) follows from\nProposition~\\ref{prop:23},   while it is   obvious that\n(1$'$) $\\implies$ (2$'$).  Finally, Proposition~\\ref{prop:26},\nLemma~\\ref{lem:9},   and the implication (2) $\\implies$ (3) prove that\n(2$'$) $\\implies$ (3$'$). \n\\end{proof}\n\n\nWith the same techniques, but using~\\cite[Corollary 5.14]{CalziPeloso}\ninstead of property $\\atomic^{p_1,q_1}_{\\vs,\\vect{s'},0}$, one\nalso proves the following result.  We recall that the spaces\n$\\widetilde A^{p,q}_{\\vs}(D)$ and $\\widetilde A^{p,q}_{\\vs,0}(D)$ are\nthe holomorphic extensions of  the analytic-type Besov spaces $B^{-\\vs}_{p,q}(\\Nc,\\Omega)$\nand $\\mathring B^{-\\vs}_{p,q}(\\Nc,\\Omega)$, respectively,   see Definition~\\ref{def:tilde-spaces}.\n\n\\begin{prop}\\label{prop:1}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and take  $\\vs\\in\n\\frac{1}{p_1}(\\vb+\\vd)+\\frac{1}{2\nq_1'}\\vect{m'}+(\\R_+^*)^r$.   Take $\\mi\\in\\cM_+(D)$ so   that  $\\widetilde A^{p_1,q_1}_{\\vs,0}(D)$  embeds continuously into $L^{p}(\\mi)$, and   take $p^*,q^*,\\vs^*$    as in~\\eqref{p*q*s'}. Then, \n\\[\nM_R(\\mi)\\in L^{p^*,q^*}_{\\vs^*}(D)\n\\]\nfor every $R>0$.\n\\end{prop}\n\nNotice, though, that an analogue of Proposition~\\ref{prop:23} for the\nspaces $ \\widetilde A^{p_1,q_1}_{\\vs,0}(D)$ is false, in general,\nas the following Proposition shows.   We denote by $\\Lc \\nu$  the Laplace\ntransform of $\\nu\\in\\cM_+(\\Omega)$, that is,\n\\[\n(\\Lc \\nu)(\\lambda)= \\int_\\Omega \\ee^{-\\langle \\lambda, h\\rangle}\\,\\dd \\nu(h)\n\\] \nfor every $\\lambda\\in F'$.\n\n\\begin{prop}\\label{prop:24}\nTake $\\vs\\in \\frac{\\vb+\\vd}{2}+\\frac{1}{4}\\vect{m'}+(\\R_+^*)^r$   and a $bD$-invariant   $\\mi\\in\\cM_+(D)$, and let $\\nu$ be the unique Radon measure $\\nu$ on\n$\\Omega$ such that $\\int_D f\\,\\dd \\mi= \\int_\\Omega \\int_{\\Nc}\nf_h(\\zeta,x)\\,\\dd (\\zeta,x)\\,\\dd \\nu(h)$ for every $f\\in C_c(D)$. Take\n$\\vect{s'}\\in -\\frac{1}{4}\\vect{m'}-\\vs-(\\R_+^*)^r$. Then, the\nfollowing conditions are equivalent: \n\\begin{enumerate}\n\\item[{\\em(1)}] $\\widetilde A^{2,2}_{\\vs}(D)$ embeds continuously (resp.\\ compactly) into $L^2(\\mi)$;\n\n\\item[{\\em(2)}] the function $\\Delta^{2\\vs}_{\\Omega'} \\Lc\\nu$ is\nbounded (resp.\\ is bounded and vanishes at $\\infty$) on $\\Omega'$; \n\n\\item[{\\em(3)}] the function\n$h\\mapsto\\Delta_\\Omega^{-\\vs-\\vect{s'}}(h)\n\\norm{\\Delta^{\\vect{s'}}(\\,\\cdot\\,+h)}_{L^2(\\nu)}$ is bounded\n(resp.\\ is bounded and vanishes at $\\infty$) on $\\Omega$. \n\\end{enumerate}\n\\end{prop}\n\nThis extends~\\cite[Theorem 3.1]{BekolleSehba}, where the case in which\n$\\vs=0$, $D$ is an irreducible symmetric tube domain, and $\\nu$ is\nabsolutely continuous with respect to Lebesgue measure, is\nconsidered. \n\n\nNotice that this shows that the necessary conditions of\nProposition~\\ref{prop:1} are not sufficient, in general. For example,\nwhen $\\vs=0$, then a $bD$-invariant positive Radon measure $\\mi$ on\n$D$ induces a continuous embedding of the Hardy space $\\widetilde\nA^{2,2}_{\\vect 0}(D)  = A^{2,\\infty}_{\\vect 0}(D)$ into $L^2(\\mi)$ if and only if the measure $\\nu$\ndefined as in the statement is integrable on $\\Omega$, while\nProposition~\\ref{prop:1} only requires the mapping $h\\mapsto\n\\nu(B_\\Omega(h,R))$ to be bounded for some (every) $R>0$. \n\n\\begin{proof}\n(2) $\\implies$ (1).  Take $f\\in \\widetilde A^{2,2}_{\\vs}(D)\\cap\nA^{2,2}_{-\\vd\/2}(D)$,  and observe that   there is a constant $c>0$ such that , defining\n$\\tau_f(\\lambda)\\coloneqq \\pi_\\lambda(f_{e_\\Omega}) \\ee^{\\langle\n\\lambda, e_\\Omega\\rangle}$ for almost every $\\lambda\\in \\Omega'$, \n\\[\n\\norm{f}_{L^2(\\mi)}^2=c\n\\int_{\\Omega'}  \\norm{\\tau_f(\\lambda)}_{\\Lin^2(H_\\lambda)}^2 (\\Lc\\nu)(2 \\lambda)\\Delta^{-\\vb}_{\\Omega'}(\\lambda)\\,\\dd \\lambda,\n\\]\narguing as in the proof of~\\cite[Corollary 1.38]{CalziPeloso}. In addition, we may choose a norm on $\\widetilde A^{2,2}_{\\vs}(D)$ so that\n\\[\n\\norm{f}_{\\widetilde A^{2,2}_{\\vs}(D)}^2=\\int_{\\Omega'}  \\norm{\\tau_f(\\lambda)}_{\\Lin^2(H_\\lambda)}^2 \\Delta^{-2\\vs-\\vb}_{\\Omega'}(\\lambda)\\,\\dd \\lambda\n\\]\nthanks to~\\cite[Lemma 2.26, Proposition 3.11 and Proposition 5.13]{CalziPeloso}. \nNow, define \n\\[\n\\Lc^2_{\\vs}(\\Omega')\\coloneqq \\Set{\\tau\\in \\int_{\\Omega'}^\\oplus \\Lin^2(H_\\lambda) \\Delta_{\\Omega'}^{-\\vect b-2\\vect s}(\\lambda)\\,\\dd \\lambda\\colon \\tau=\\tau P_{\\,\\cdot\\,,0} }\n\\]\n(cf.~\\cite[Definition 3.10]{CalziPeloso}), and observe that the image of the mapping $\\tau_f$, extended to $\\widetilde A^{2,2}_{\\vs}(D)$ by continuity, is the whole of $\\Lc^2_{\\vs}(\\Omega')$ (cf.~\\cite[Propositions 3.15, 3.17, 5.4, 5.13, and Corollary 5.11]{CalziPeloso}). Therefore,   $\\widetilde A^{2,2}_{\\vs}(D)$ embeds into $L^2(\\mi)$ if and only if there is a constant $C_1>0$ such that\n\\[\n\\Lc\\nu(2 \\lambda)\\meg C_1\\Delta^{-2\\vs}_{\\Omega'}(\\lambda)\n\\]\nfor every $\\lambda\\in \\Omega'$.\n\nNow, if $\\Delta^{2\\vs}_{\\Omega'} \\Lc\\nu$ is bounded and vanishes at infinity, then it is clear that the mapping $\\tau \\mapsto \\Delta^{2\\vs}_{\\Omega'} \\Lc\\nu \\,\\tau$ is a compact endomorphism of $\\Lc^2_{\\vs}(\\Omega')$, so that the preceding remarks show that  $\\widetilde A^{2,2}_{\\vs}(D)$ embeds compactly into $L^2(\\mi)$.\n\n(1) $\\implies$ (3). Observe first that $\\norm{B^{\\vect{s'}+(\\vb+\\vd)\/2}_{(\\zeta,z)}}_{\\widetilde A^{2,2}_{\\vs}(D)}= \\Delta_\\Omega^{\\vs+\\vect{s'}}(\\rho(\\zeta,z))$ for every $(\\zeta,z)\\in D$, for a suitable choice of a norm on $\\widetilde A^{2,2}_{\\vs}(D)$ (cf.~\\cite[Proposition 2.41 and Lemma 5.15]{CalziPeloso}). In addition, there is a constant $C_2>0$ such that \n\\[\n\\norm*{\\left(B^{\\vect{s'}+(\\vb+\\vd)\/2}_{(0, i h)}\\right)_{h'}}_{L^2(\\Nc)}=C_2 \\Delta_\\Omega^{\\vect{s'}}(h+h')\n\\]\nfor every $h,h'\\in \\Omega$, thanks to~\\cite[Lemma 2.39]{CalziPeloso}. Therefore, (3) follows arguing as in the proof of Proposition~\\ref{prop:26}.\n\n(3) $\\implies$ (2). Observe that there is a constant $C_3>0$ such that\n\\[\n\\norm{\\Delta^{\\vect{s'}}(\\,\\cdot\\,+h)  }_{L^2(\\nu_t)}\\meg C_3\\Delta_\\Omega^{\\vs+\\vect{s'}}(h) \\quad \\forall t\\in T_+ \\qquad\\text{(resp.\\ and   $\\lim\\limits_{t\\to \\infty }  \\norm{\\Delta^{\\vect{s'}}(\\,\\cdot\\,+h)  }_{L^2(\\nu_t)}=0$)}\n\\]\nfor  every $h\\in \\Omega$, where $\\nu_t= \\Delta^{2\\vs}(t)\n(t\\,\\cdot\\,)_* \\nu$, and that (2) is equivalent to saying that \n\\[\n\\sup_{t\\in t_+}\\Lc \\nu_t(e_{\\Omega}')<\\infty \\qquad \\text{(resp.\\ $\\lim\\limits_{t\\to\\infty}\\Lc \\nu_t(e_{\\Omega}')=0  $).}\n\\]\n\nNow,~\\cite[Corollary 2.36]{CalziPeloso} implies that\n\\[\n\\Delta^{2\\vect{s'}}(h+h')\\Meg \\Delta^{2 \\vect{s'}}(2 h)=2^{2\\vect{s'}} \\Delta^{2 \\vect{s'}}(h)\n\\]\nfor every $h\\in \\Omega$ and for every $h'\\in \\Omega \\cap (h-\\Omega)$,\nso that\n\\[\n\\nu_t(\\Omega\\cap (h-\\Omega))\\meg 2^{-2\\vect{s'}}\\Delta^{-2\n\\vect{s'}}(h)\\int_{\\Omega\\cap (h-\\Omega)}\n\\Delta^{2\\vect{s'}}(h+h')\\,\\dd \\nu_t(h') \\meg 2^{-2\\vect{s'}} C_3^2\n\\Delta^{2\\vs}(h) \n\\]\nfor every $h\\in\\Omega$ and for every $t\\in T_+$.\nIn addition, fix $R>0$ so that $\\langle e_{\\Omega'},h\\rangle\\Meg 1$\nfor every $h\\in \\Omega \\setminus (Re_\\Omega-\\Omega)$.  Then,  \n\\[\n\\begin{split}\n\\Lc\\nu_t(e_{\\Omega'})&=\\int_\\Omega \\ee^{-\\langle e_{\\Omega'}, h\\rangle}\\,\\dd \\nu_t(h)\\\\\n&\\meg \\nu_t(\\Omega\\cap (R e_\\Omega-\\Omega))+\\sum_{k\\in\\N}  \\ee^{-2^k R\n} \\nu_t(\\Omega\\cap (2^{k+1} R e_\\Omega-\\Omega)\\setminus (2^{k} R\ne_\\Omega-\\Omega))\\\\ \n&\\meg 2^{-2\\vect{s'}} C_3^2\\left( R^{2\\vs}+ \\sum_{k\\in\\N} (2^{k+1}\nR )^{2\\vs} \\ee^{- 2^k R}\\right) \n\\end{split}\n\\]\nfor every $t\\in T_+$. If, in addition, $\\lim\\limits_{t\\to \\infty }\n\\norm{\\Delta^{\\vect{s'}}(\\,\\cdot\\,+h)  }_{L^2(\\nu_t)}=0$ for every\n$h\\in\\Omega$, then the preceding computations show that $\n\\nu_t(\\Omega\\cap (2^{k+1} R e_\\Omega-\\Omega)\\setminus (2^{k} R\ne_\\Omega-\\Omega))\\to 0$ for $t\\to+\\infty$ for every $k\\in \\N$, so that\nit is readily seen that $\\Lc \\nu_t(e_{\\Omega'})\\to 0$ for\n$t\\to\\infty$. \n\\end{proof}\n\nWe now come to the last  main result of this section.  It provides a\nnecessary and sufficient condition for a measure $\\mi\\in\\cM_+(D)$  to be a\n$p$-Carleson  measure for $A^{p_1,q_1}_\\vs$ also in terms of\nintegrability properties of Bergman kernels.\n\n\\begin{teo} \\label{teo:4}\nTake    $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, $\\mi\\in\\cM_+(D)$,\n$\\vs\\in \\R^r$, and $\\vect{s'}\\in \\C^r$ such \nthat the following hold: \n\\begin{itemize}\n\\item[{\\em(i)}] $\\vs\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ (resp.\\\n$\\vs \\in \\R_+^r$ if $q_1=\\infty$); \n\\item[{\\em(ii)}]  $\\Rea\\vect{s'}\\in \\frac{1}{p_1}(\\vb+\\vect\nd)-\\frac{1}{2 p_1}\\vect{m'}-(\\R_+^*)^r $ (resp.\\ $\\Rea\\vect{s'}\\in\n-\\R_+^r $ if $p_1=\\infty$); \n\n\\item[{\\em(iii)}]  $\\vs+\\Rea\\vect{s'}\\in \\frac{1}{p_1}(\\vect\nb+\\vd)-\\frac{1}{2 q_1}\\vect{m'}-(\\R_+^*)^r $  (resp.\\ $\\vect\ns+\\Rea\\vect{s'}\\in  \\frac{1}{p_1}(\\vb+\\vd)-\\R_+^r $ if\n$q_1=\\infty$); \n\\item[{\\em(iv)}]  $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\\n$A^{p_1,q_1}_{\\vs}(D)$) embeds continuously into\n$A^{p,p}_{\\vect{s''}}(D)$, where $\\vect{s''}  \\coloneqq \\vect\ns+\\left(\\frac{1}{p}-\\frac{1}{p_1} \\right)(\\vb+\\vd)$. \n\\end{itemize}\nThen, the following conditions are equivalent for every $R>0$:\n\\begin{enumerate}\n\\item[{\\em(1)}] $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\\n$A^{p_1,q_1}_{\\vs}(D)$) embeds continuously into $L^{p}(\\mi)$;\n\n\\item[{\\em(2)}]  there is a constant $C>0$ such that\n\\[\n\\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p}(\\mi)}\\meg C\n\\Delta^{\\vect{s}+\\Rea\\vect{s'}-(\\vb+\\vd)\/p_1}(\\rho(\\zeta,z)) \n\\]\nfor every $(\\zeta,z)\\in D$;\n\\item[{\\em(3)}]  $M_R(\\mi)\\in L^{\\infty,\\infty}_{p[(\\vb+\\vd)\/p_1-\\vs]}(D)$.\n\\end{enumerate}\n\nIf, in addition, $\\vs,-\\vs-\\Rea\\vect{s'}\\in (\\R_+^*)^r$ when $p_1=q_1=\\infty$, then the following conditions are equivalent for every $R>0$:\n\\begin{enumerate}\n\\item[{\\em(1$'$)}] $A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\  $A^{p_1,q_1}_{\\vs}(D)$) embeds compactly into $L^{p}(\\mi)$;\n\\item[{\\em(2$'$)}]  the function $(\\zeta,z)\\mapsto\\Delta^{(\\vb+\\vd)\/p_1-\\vect{s}-\\Rea\\vect{s'}}(\\rho(\\zeta,z)) \\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p}(\\mi)}$ is bounded and vanishes at $\\infty$ on $D$;\n\\item[{\\em(3$'$)}]  $M_R(\\mi)\\in L^{\\infty,\\infty}_{p[(\\vb+\\vd)\/p_1-\\vs],0}(D)$.\n\\end{enumerate}\n\\end{teo}\n\n\nThis extends~\\cite[Theorem 1.1]{BekolleSehbaTchoundja}, where the case\nin which $\\vs=0$,  $\\vect{s'}\\in \\R\\vect 1_r$, $q=\\infty$, and $D$\nis an irreducible symmetric tube domain, is considered. \n\n\nNotice that, if (i) to (iv) hold, then (i), (ii), and (iii) hold with\n$\\vs$, $p_1$, and $q_1$ replaced by $\\vect{s''}$, $p$, and $p$,\nrespectively, thanks to~\\cite[Proposition 2.41]{CalziPeloso}.  \n\nConversely, if (2) implies (1) and (i), (ii), and (iii) hold with\n$\\vs$, $p_1$, and $q_1$ replaced by $\\vect{s''}$, $p$, and $p$,\nrespectively, then (iv) holds thanks to~\\cite[Proposition 2.41]{CalziPeloso} again. \n\nObserve that $A^{2,\\infty}_{\\vect 0}(D)=\\widetilde A^{2,2}_{\\vect 0}(D)\\subseteq \\widetilde A^{p,p}_{(1\/p-1\/2)(\\vb+\\vd)}(D)$ for every $p\\Meg 2$, thanks to~\\cite[Proposition 4.19]{CalziPeloso}, so that the assumptions are satisfied whenever $\\vect{s''}=(1\/p-1\/2)(\\vb+\\vd)$ and $\\widetilde A^{p,p}_{\\vect{s''}}(D)= A^{p,p}_{\\vect{s''}}(D)$. This is the case, for example, when (cf.~\\cite[Corollary 5.11]{CalziPeloso})\n\\[\n\\left( \\frac{1}{p}-\\frac 1 2 \\right)(\\vb+\\vd)\\in \\sup \\left(\\frac{1}{2 p}\\vect m+\\left(\\frac{1}{2}-\\frac{1}{p}\\right)\\vect{m'}, \\frac{1}{p}(\\vb+\\vd)+\\frac{1}{2 p'}\\vect{m'}    \\right)+(\\R_+^*)^r.\n\\]\n\\begin{comment}\n\\end{comment}\n\\begin{proof}\nWe prove only the equivalence of conditions (1)--(3). The equivalence of conditions (1$'$)--(3$'$) is proved similarly, using the techniques employed in the proof of Proposition~\\ref{prop:26}.\n\n(1) $\\implies$ (2). This follows from~\\cite[Proposition 2.41]{CalziPeloso}.\n\n(2) $\\implies$ (3). This follows easily from~\\cite[Theorem 2.47]{CalziPeloso}.\n\n(3) $\\implies$ (1). By Theorem~\\ref{main:1}, $A^{p,p}_{\\vs}(D)$ embeds continuously into $L^{p}(\\mi)$, so that the assertion\nfollows. \n\\end{proof}\n\nWith a similar proof, but using~\\cite[Lemma 5.15]{CalziPeloso} instead\nof~\\cite[Proposition 2.41]{CalziPeloso}, one may prove the following\nresult, which  deals with $p$-Carleson measures for the spaces\n$\\widetilde A^{p_1,q_1}_{\\vs}(D)$.  \n\n\\begin{teo}\nTake $p_1,q_1,p\\in ]0,\\infty[$, $\\mi\\in\\cM_+(D)$, $\\vs\\in \\R^r$, and $\\vect{s'}\\in \\C^r$ such that the following hold:\n\\begin{itemize}\n\\item[{\\em(i)}] $\\vs\\in \\frac{1}{ p_1}(\\vb+\\vd)+\\frac{1}{2 q_1'}\\vect{m'}+(\\R_+^*)^r$; \n\\item[{\\em(ii)}]  $\\vs+\\Rea\\vect{s'}\\in \\frac{1}{p_1}(\\vb+\\vd)-\\frac{1}{2 q_1}\\vect{m'}-(\\R_+^*)^r $  (resp.\\ $\\vs+\\Rea\\vect{s'}\\in  \\frac{1}{p_1}(\\vb+\\vd)-\\R_+^r $ if $q_1=\\infty$);\n\\item[{\\em(iii)}]  $\\widetilde A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\ $\\widetilde A^{p_1,q_1}_{\\vs}(D)$) embeds continuously into\n$A^{p,p}_{\\vect{s''}}(D)$, where $\\vect{s''}\\coloneqq\\vs+\\left(\\frac{1}{p}-\\frac{1}{p_1} \\right)(\\vb+\\vd)$. \n\\end{itemize}\nThen, the following conditions are equivalent for every $R>0$:\n\\begin{enumerate}\n\\item[{\\em(1)}] $\\widetilde A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\  $\\widetilde A^{p_1,q_1}_{\\vs}(D)$) embeds continuously into $L^{p}(\\mi)$;\n\n\\item[{\\em(2)}]  there is a constant $C>0$ such that\n\\[\n\\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p}(\\mi)}\\meg C \\Delta^{\\vect{s}+\\Rea\\vect{s'}-(\\vb+\\vd)\/p_1}(\\rho(\\zeta,z))\n\\]\nfor every $(\\zeta,z)\\in D$;\n\n\\item[{\\em(3)}]  $M_R(\\mi)\\in L^{\\infty,\\infty}_{p[(\\vb+\\vd)\/p_1-\\vs]}(D)$.\n\\end{enumerate}\nIf, in addition, $\\vs+\\Rea\\vect{s'}\\in -(\\R_+^*)^r$ when $p_1=q_1=\\infty$, then the following conditions are equivalent for every $R>0$:\n\\begin{enumerate}\n\\item[{\\em(1$'$)}] $\\widetilde A^{p_1,q_1}_{\\vs,0}(D)$ (resp.\\  $\\widetilde A^{p_1,q_1}_{\\vs}(D)$) embeds compactly into $L^{p}(\\mi)$;\n\n\\item[{\\em(2$'$)}]  the function\n$(\\zeta,z)\\mapsto\\Delta^{(\\vb+\\vd)\/p_1-\\vect{s}-\\Rea\\vect{s'}}(\\rho(\\zeta,z))\n\\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p}(\\mi)}$ is bounded and\nvanishes at $\\infty$ on $D$; \n\n\\item[{\\em(3$'$)}]  $M_R(\\mi)\\in L^{\\infty,\\infty}_{p[(\\vb+\\vd)\/p_1-\\vs],0}(D)$.\n\\end{enumerate}\n\\end{teo}\n\nNotice that, by Corollary~\\ref{cor:1} below, if (iii) holds, then $p_1,q_1\\meg p $.\n\n\n\n\n\n\\section{Inclusion Between the Spaces $\\widetilde A^{p,q}_{\\vs}(D)$\nand  $A^{p,q}_{\\vs}(D)$}\\label{sec:5}\n\nA classical result by  Hardy and Littlewood in dimension 1  shows that  \nthe Hardy space $H^2(\\C_+)$  embeds continuously into\n$A^{p,p}_{\\frac12-\\frac1p}(\\C_+)$ when $p\\ge4$.  In this section we\naddress a similar question. The main result is Theorem~\\ref{prop:2},\nin which we restrict ourselves to the setting of Siegel domains of Type I.\nIn addition to that, we also prove some necessary conditions for the spaces $A$ and $\\widetilde A$ to embed into one another. For sufficient condition, see~\\cite[Propositions 3.2, 3.7, and 4.19]{CalziPeloso}. \n\n\\begin{lem}\\label{lem:2}\nTake $p_1,p_2,q_1,q_2\\in ]0,\\infty]$ and $\\vect{s_1},\\vect{s_2}\\in \\R^r$, and assume that $\\vect{s_1}\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ (resp.\\ $\\vect{s_1}\\in \\R_+^r$ if $q_1=\\infty$) and that $A^{p_1,q_1}_{\\vect{s_1},0}(D)\\subseteq A^{p_2,q_2}_{\\vect{s_2}}(D)$ (resp.\\ $A^{p_1,q_1}_{\\vect{s_1}}(D)\\subseteq A^{p_2,q_2}_{\\vect{s_2}}(D)$).\nThen, \n\\[\np_1\\meg p_2\\qquad \\text{and} \\qquad \\vect{s_2}=\\vect{s_1}+\\left(\\frac{1}{p_2}-\\frac{1}{p_1}\\right)(\\vb+\\vd).\n\\]\n\\end{lem}\n\nAs Theorem~\\ref{prop:2} below shows, it is not possible to deduce $q_1\\meg q_2$ \\emph{in this generality}. Cf.~Corollary~\\ref{cor:1} below, though.\n\n\\begin{proof}\nBy assumption, there is a constant $C>0$ such that\n\\[\n\\norm{f}_{A^{p_2,q_2}_{\\vect{s_2}}(D)}\\meg C\\norm{f}_{A^{p_1,q_1}_{\\vect{s_1}}(D)}\n\\]\nfor every $f\\in A^{p_1,q_1}_{\\vect{s_1},0}(D)$ (resp.\\ for every $f\\in A^{p_1,q_1}_{\\vect{s_1}}(D)$). We assume that $C$ is the least possible constant for which such an inequality holds. \nNow, for every $t\\in T_+$ choose $g_t\\in GL(E)$ so that $t\\cdot \\Phi=\\Phi\\circ (g_t\\times g_t)$. Then,\n\\[\n\\norm{f\\circ (g_t\\times t)}_{A^{p_1,q_1}_{\\vect{s_1}}(D)}=\\Delta^{-\\vect{s_1}+(\\vb+\\vd)\/p_1}(t)\\norm{f}_{A^{p_1,q_1}_{\\vect{s_1}}(D)}\n\\]\nand\n\\[\n\\norm{f\\circ (g_t\\times t)}_{A^{p_2,q_2}_{\\vect{s_2}}(D)}=\\Delta^{-\\vect{s_2}+(\\vb+\\vd)\/p_2}(t)\\norm{f}_{A^{p_2,q_2}_{\\vect{s_2}}(D)}\n\\]\nfor every $f\\in \\Hol(D)$, thanks to Lemma~\\ref{lem:5}. Therefore, the arbitrariness of $t\\in T_+$ and the minimality of $C$ imply that\n\\[\n\\vect{s_2}=\\vect{s_1}+(1\/p_2-1\/p_1)(\\vb+\\vd).\n\\]\nNext, assume that $A^{p_1,q_1}_{\\vect{s_1},0}(D)\\neq 0$, that is, that $\\vect{s_1}\\in \\frac{1}{2 q_1}\\vect{m}+(\\R_+^*)^r$. \nThen,\n\\[\n\\begin{split}\n2^{1\/p_2}\\norm{f}_{A^{p_2,q_2}_{\\vect{s_2}}(D)}&=\\lim_{(\\zeta,x)\\to \\infty}\\norm{f-f((\\zeta,x+i\\Phi(\\zeta))\\,\\cdot\\,)}_{A^{p_2,q_2}_{\\vect{s_2}}(D)}\\\\\n&\\meg C\\lim_{(\\zeta,x)\\to \\infty}\\norm{f-f((\\zeta,x+i\\Phi(\\zeta))\\,\\cdot\\,)}_{A^{p_1,q_1}_{\\vect{s_1}}(D)}\\\\\n&=2^{1\/p_1} C\\norm{f}_{A^{p_1,q_1}_{\\vect{s_1}}(D)}\n\\end{split}\n\\]\nfor every $f\\in A^{p_1,q_1}_{\\vect{s_1},0}(D)$, so that $1\/p_1\\Meg 1\/p_2$, that is, $p_1\\meg p_2$, by the minimality of $C$.\n\nFinally, assume that $A^{p_1,q_1}_{\\vect{s_1},0}(D)=0$, so that\n$q_1=\\infty$ and  $(s_1)_j=0$ for some $j\\in \\Set{1,\\dots,r}$. Then,\nfor every $\\eps>0$ we may find $\\vect{s'}\\in \\R^r$ such that\n$s'_j=\\frac{1}{p_1}(b_j+d_j)-\\frac{1}{2 p_1}m'_j-\\eps$ and such that\n$B^{\\vect{s'}}_{(\\zeta,z)}\\in A^{p_1,q_1}_{\\vect{s_1}}(D)$ for every\n$(\\zeta,z)\\in D$ (cf.~\\cite[Proposition 2.41]{CalziPeloso}), so that\n$B^{\\vect{s'}}_{(\\zeta,z)}\\in\nA^{p_2,q_2}_{\\vect{s_2}}(D)$. Then,~\\cite[Proposition\n2.41]{CalziPeloso} implies that $s'_j\\meg\n\\frac{1}{p_2}(b_j+d_j)-\\frac{1}{2 p_2} m'_j$. Since $-b_j,m'_j\\Meg 0$\nand $-d_j>0$, the arbitrariness of $\\eps$ implies that $1\/p_1\\Meg\n1\/p_2$, that is, $p_1\\meg p_2$. \n\\end{proof}\n\n\\begin{lem}\\label{lem:3}\nTake $p_1,p_2,q_1,q_2\\in ]0,\\infty]$ and $\\vect{s_1},\\vect{s_2}\\in\n\\R^r$, and assume that the canonical mapping $\\Sc_{\\Omega,L}(\\Nc)\\to\nB^{\\vect{s_2}}_{p_2,q_2}(\\Nc,\\Omega)$ induces a continuous linear\nmapping $\\mathring B^{\\vect{s_1}}_{p_1,q_1}(\\Nc,\\Omega)\\to\nB^{\\vect{s_2}}_{p_2,q_2}(\\Nc,\\Omega)$. Then, $p_1\\meg p_2$, $q_1\\meg\nq_2$, and\n$\\vect{s_2}=\\vect{s_1}+\\left(\\frac{1}{p_1}-\\frac{1}{p_2}\\right)(\\vb+\\vd)$. \n\\end{lem}\n\n\n\n\\begin{proof}\nApplying the operator $u\\mapsto u*I^{\\vect{s'}}_\\Omega$ for a suitable $\\vect{s'}\\in \\R^r$, we may assume that $\\vect{s_1},\\vect{s_2} $ are sufficiently small so as to ensure that the mapping $\\Ec$ is defined and induces isomorphisms of $\\mathring B^{\\vect{s_1}}_{p_1,q_1}(\\Nc,\\Omega)$ and $B^{\\vect{s_2}}_{p_2,q_2}(\\Nc,\\Omega)$ onto $A^{p_1,q_1}_{-\\vect{s_1},0}(D)$ and $A^{p_2,q_2}_{-\\vect{s_2}}(D)$, respectively (cf.~\\cite[Theorem 4.26 and Corollary 5.11]{CalziPeloso}). Then, Lemma~\\ref{lem:2} implies that $p_1\\meg p_2$ and that $\\vect{s_2}=\\vect{s_1}+\\left(\\frac{1}{p_1}-\\frac{1}{p_2}\\right)(\\vb+\\vd)$.\nTherefore, it only remains to prove that $q_1\\meg q_2$. Then, take\n$\\vect{s'}\\in \\R^r$ and observe that~\\cite[Lemma 5.15]{CalziPeloso}\nimplies that $B^{\\vect{s'}}_{(\\zeta,z)}\\in\nA^{p_1,q_1}_{\\vect{s_1},0}(D)$ for some (or, equivalently, every)\n$(\\zeta,z)\\in D$ if and only if $\\vect{s_1}+\\vect{s'}\\in\n\\frac{1}{p_1}(\\vb+\\vd)-\\frac{1}{2 q_1}\\vect{m'}-(\\R_+^*)^r$, and that\n$B^{\\vect{s'}}_{(\\zeta,z)}\\in A^{p_2,q_2}_{\\vect{s_2}}(D)$ for some\n(or, equivalently, every) $(\\zeta,z)\\in D$ if and only if\n$\\vect{s_2}+\\vect{s'}\\in \\frac{1}{p_2}(\\vb+\\vd)-\\frac{1}{2\nq_2}\\vect{m'}-(\\R_+^*)^r$ and $q_2<\\infty$ or $\\vect{s_2}+\\vect{s'}\\in\n\\frac{1}{p_2}(\\vb+\\vd)-\\R_+^r$ and $q_2=\\infty$. Therefore, $q_1\\meg\nq_2$ \\emph{provided that $\\vect{m'}\\neq \\vect 0$}.   \n\nThus, we only need to consider the case $\\vect{m'}=\\vect 0$, that is,\n$r=m$. In this case, we may assume that $\\Omega=\\Omega'=(\\R_+^*)^r$,\nidentifying $\\R^r$ with its dual. \nDefine $K=\\Z^r$ and $t_k\\coloneqq\\lambda_k\\coloneqq\n2^k=(2^{k_1},\\dots, 2^{k_r})$ for every $k\\in K$. Notice that the\ndistance $(\\lambda, \\lambda')\\mapsto \\sum_{j=1}^r\n\\abs{\\log(\\lambda_j\/\\lambda'_j)}$ on $\\Omega'$ is\n$G(\\Omega')$-invariant and clearly locally bi-Lipschitz equivalent to\n$d_{\\Omega'}$, hence bi-Lipschitz equivalent to $d_{\\Omega'}$ near every point, with\nuniform constants. Therefore, it is readily verified that\n$(\\lambda_k)$ is a $(\\delta,R)$-lattice for some $\\delta>0$ and some\n$R>1$. \nNotice that $\\Omega$ is a group under pointwise multiplication, so  \nthat we may choose $T_+=\\Omega$ with the natural action. In this case, \n$\\Delta^{\\vs}(t)=t^{\\vs}=\\prod_{j=1}^r t_j^{s_j}$ for every $\\vs\\in\n\\C^r$. \nNow, choose a positive $\\phi\\in C_c^\\infty(\\Omega')$ so that\n$\\sum_{k\\in K} \\phi(\\,\\cdot\\, t_k)=1$ on $\\Omega'$, and define\n$\\psi_k\\coloneqq \\Fc_\\Nc^{-1}(\\phi(\\,\\cdot\\, t_k))$ for every $k\\in\nK$. \nThen,\n\\[\nu\\mapsto \\norm{u}_{B^{\\vs}_{p,q}(\\Nc,\\Omega)}\\coloneqq\\norm*{\n\\Delta^{\\vs}_{\\Omega'}(\\lambda_k) \\norm{u*\\psi_k}_{L^p(\\Nc)}}_{\\ell^q(K)} \n\\]\nis a quasi-norm which defines the topology of\n$B^{\\vs}_{p,q}(\\Nc,\\Omega)$ for every $p,q\\in ]0,\\infty]$ and for\nevery $\\vs\\in \\R^r$. Define $\\Psi_t(u)\\coloneqq\n\\Delta^{\\vect{s_1}-(\\vb+\\vd)(1-1\/p_1)}(t\\,\\cdot\\,)_*u$ for every $u\\in\n\\Sc_{\\Omega,L}'(\\Nc)$ and for every $t\\in T_+$. Then, clearly \n\\[\n\\norm{ \\Psi_{t_k}(u)}_{B^{\\vect {s_j}}_{p_j,q_j}(\\Nc,\\Omega)}=\\norm{ u}_{B^{\\vect {s_j}}_{p_j,q_j}(\\Nc,\\Omega)}\n\\]\nfor every $j=1,2$, for every $k\\in K$, and for every $u\\in B^{\\vect {s_j}}_{p_j,q_j}(\\Nc,\\Omega)$.\nIn addition, it is readily seen that\n\\[\n\\lim_{k\\to \\infty} \\norm{ f-\\Psi_{t_k}(f)}_{B^{\\vect {s_j}}_{p_j,q_j}(\\Nc,\\Omega)}=2^{1\/q_j}\\norm{ f}_{B^{\\vect {s_j}}_{p_j,q_j}(\\Nc,\\Omega)}\n\\]\nfor every $j=1,2$ and for every $f\\in \\Sc_{\\Omega,L}(\\Nc)$. Now, by assumption there is a constant $C>0$ such that\n\\[\n\\norm{f}_{B^{\\vect {s_2}}_{p_2,q_2}(\\Nc,\\Omega)}\\meg C \\norm{f}_{B^{\\vect {s_1}}_{p_1,q_1}(\\Nc,\\Omega)}\n\\]\nfor every $f\\in \\Sc_{\\Omega,L}(\\Nc)$. Then, applying the same inequality to $f-\\Psi_{t_k}(f)$ and passing to the limit for $k\\to \\infty$,\n\\[\n\\norm{f}_{B^{\\vect {s_2}}_{p_2,q_2}(\\Nc,\\Omega)}\\meg 2^{1\/q_1-1\/q_2}C\\norm{f}_{B^{\\vect {s_1}}_{p_1,q_1}(\\Nc,\\Omega)}\n\\]\nfor every $f\\in \\Sc_{\\Omega,L}(\\Nc)$. Since we may have assumed $C$ to be minimal, it is readily seen that $q_1\\meg q_2$.\n\\end{proof}\n\n\\begin{cor}\\label{cor:1}\nTake $p_1,p_2,q_1,q_2\\in ]0,\\infty]$ and $\\vect{s_1},\\vect{s_2}\\in\n\\R^r$ such that $\\vect{s_1}\\in \\frac{1}{p_1}(\\vb+\\vd)+\\frac{1}{2\nq_1'}\\vect{m'}+(\\R_+^*)^r$ and such that $\\widetilde\nA^{p_1,q_1}_{\\vect{s_1},0}(D)\\subseteq A^{p_2,q_2}_{\\vect{s_2}}(D)$. \nThen, $p_1\\meg p_2$, $q_1\\meg q_2$, and\n$\\vect{s_2}=\\vect{s_1}+\\left(\\frac{1}{p_1}-\\frac{1}{p_2}\\right)(\\vb+\\vd)$. \n\\end{cor}\n\nNotice that, if $p_1\\meg p_2$, $q_1\\meg q_2$, and\n$\\vect{s_2}=\\vect{s_1}+\\left(\\frac{1}{p_1}-\\frac{1}{p_2}\\right)(\\vb+\\vd)$, then $A^{p_1,q_1}_{\\vect{s_1}}\\subseteq A^{p_2,q_2}_{\\vect{s_2}}$ and $\\widetilde A^{p_1,q_1}_{\\vect{s_1}}\\subseteq \\widetilde A^{p_2,q_2}_{\\vect{s_2}}$ (cf.~\\cite[Propositions 3.2 and 4.19]{CalziPeloso}). Nonetheless, $\\widetilde\nA^{p_1,q_1}_{\\vect{s_1},0}$ need not embed into $ A^{p_2,q_2}_{\\vect{s_2}}$.  \n\n\\begin{proof}\nTake $\\vect{s'}\\in \\N_{\\Omega'}$ so that\n$A^{p_2,q_2}_{\\vect{s_2}+\\vect{s'}}(D)=\\widetilde\nA^{p_2,q_2}_{\\vect{s_2}+\\vect{s'}}(D)$ (cf.~\\cite[Corollary\n5.11]{CalziPeloso}). Observe that the mapping $f\\mapsto f*\nI^{-\\vect{s'}}_\\Omega$ induces an isomorphism of $\\widetilde\nA^{p_1,q_1}_{\\vect{s_1},0}(D)$ onto $\\widetilde\nA^{p_1,q_1}_{\\vect{s_1}+\\vect{s'},0}(D)$ by~\\cite[Proposition\n5.13]{CalziPeloso}, and a continuous mapping of\n$A^{p_2,q_2}_{\\vect{s_2}}(D)$ into\n$A^{p_2,q_2}_{\\vect{s_2}+\\vect{s'}}(D)$ by~\\cite[Corollary\n3.27]{CalziPeloso}. Thus, $\\widetilde\nA^{p_1,q_1}_{\\vect{s_1}+\\vect{s'},0}(D)\\subseteq \\widetilde\nA^{p_2,q_2}_{\\vect{s_2}+\\vect{s'}}(D)$, so that the conclusion follows\nfrom Lemma~\\ref{lem:3}. \n\\end{proof}\n\n\n\n\\begin{teo}\\label{prop:2}  \nAssume that $n=0$, and take $k\\in\\N^*$ and\n\\[\n\\vs\\in \\sup\\left(\\frac 1 2 \\vd+\\frac 1 4 \\vect{m'},\\frac{k-1}{2k}\n\\vd+\\frac{1}{4k} \\vect{m'}, \\frac{k-1}{2 k}\\vd+\\frac{1}{4 k}\\vect m\\right)+(\\R_+^*)^r.\n\\]\nThen, $\\widetilde A^{2,2}_{\\vs}(D)$ embeds continuously into\n$A^{q,q}_{\\vs-(1\/2-1\/q)\\vd}(D)$ for every $q\\in [2k,\\infty]$. \n\\end{teo}\n\n\nThis extends~\\cite[Theorem 1.3]{BekolleSehbaTchoundja}, where the case\nin which $\\vs=\\vect 0$ and $D$ is an irreducible symmetric tube domain is\nconsidered. Notice that~\\cite[Theorem 1.4]{BekolleSehbaTchoundja}\nprovides better results when $\\vs=\\vect 0$, $r=2$, and $m=3,4,5,6$. \n\nNotice, in addition, (cf.~\\cite[Remark 1.5]{BekolleSehbaTchoundja})\nthat if $A^{p_1,q_1}_{\\vect{s_1}}(D)$ embeds continuously into\n$A^{p_2,q_2}_{\\vect{s_2}}(D)$, then also $ A^{kp_1,kq_1}_{\\vect{s_1}\/k}(D)$\nembeds continuously into $A^{ k p_2,kq_2}_{\\vect{s_2}\/k}(D) $ forvevery $k\\in \\N^*$.\nIndeed, $f\\in A^{kp_1,k q_1}_{\\vect{s_1}\/k}(D)$ if and only if\n$f^k\\in  A^{p_1,q_1}_{\\vect{s_1}}(D)$, so that $f^k\\in\nA^{p_2,q_2}_{\\vect{s_2}}(D)$, which is equivalent to $f\\in A^{ k\np_2,kq_2}_{\\vect{s_2}\/k}(D)$.   \nUnfortunately, it is not known if the weighted Bergman spaces\ninterpolate in full generality, so that this fact cannot be used to\ndeduce similar embedding for every $k\\in [1,\\infty[$. \n\n\\begin{proof}\nDefine $\\tau\\colon \\widetilde A^{2,2}_{\\vs}(D)\\cap A^{2,\\infty}_{\\vect\n0}(D)\\to L^2(\\Omega', \\Delta^{-2 \\vs}\\cdot\\Hc^m)$ so that\n$\\tau(f)(\\lambda)=\\Fc(f_{e_\\Omega})(\\lambda) \\ee^{\\langle \\lambda,\ne_\\Omega\\rangle}$ for almost every $\\lambda\\in \\Omega'$. Observe\nthat we may choose a norm on $\\widetilde A^{2,2}_{\\vs}(D)$ such that \n\\[\n\\norm{f}_{\\widetilde A^{2,2}_{\\vs}(D)}^2=  \\int_{\\Omega'} \\abs{\\tau(f)(\\lambda)}^2 \\Delta_{\\Omega'}^{-2 \\vs}(\\lambda)\\,\\dd \\lambda\n\\]\nfor every $f\\in \\widetilde A^{2,2}_{\\vs}(D)\\cap A^{2,\\infty}_{\\vect 0}(D)$, so that we may extend $\\tau$ to the whole of $\\widetilde A^{2,2}_{\\vs}(D)$. In addition,\n\\[\nf(z)= \\frac{1}{(2 \\pi)^m} \\int_{\\Omega'} \\tau(f)(\\lambda) \\ee^{\\langle \\lambda_\\C, i z \\rangle}\\,\\dd \\lambda\n\\]\nfor every $z\\in D$ and for every $f\\in \\widetilde\nA^{2,2}_{\\vs}(D)$.\\footnote{This formula is clear when $f\\in\n\\widetilde A^{2,2}_{\\vs}(D)\\cap A^{2,\\infty}_{\\vect 0}(D)$ thanks\nto~\\cite[Proposition 1.39]{CalziPeloso}, and then follows by\ncontinuity in the general case, thanks to~\\cite[Propositions 2.19 and 5.4]{CalziPeloso}.} It then follows that $\\tau(f^k)=\n(2\\pi)^{-(k-1)m} \\tau(f)^{*k}$ for every $f\\in \\widetilde\nA^{2,2}_{\\vs}(D)\\cap \\Ec(\\Sc_{\\Omega,L}(\\Nc))$, where $\\tau(f)^{*k}$\ndenotes the convolution of $k$ functions all equal to $\\tau(f)$.  Now, \n\\[\n\\begin{split}\n\\abs{[\\tau(f)^{*k}](\\lambda)}&\\meg \\int_{\\Omega'^{k-1}} \\abs{\\tau(f)(\\lambda-\\lambda_1)\\cdots \\tau(f)(\\lambda_{k-2}-\\lambda_{k-1})}\\,\\dd (\\lambda_1,\\dots,\\lambda_{k-1})\\\\\n&\\meg\\left( \\int_{\\Omega'^{k-1}} (\\abs{\\tau(f)}^2 \\Delta_{\\Omega'}^{-2\\vs})(\\lambda-\\lambda_1)\\cdots (\\abs{\\tau(f)}^2\\Delta_{\\Omega'}^{-2\\vs})(\\lambda_{k-2}-\\lambda_{k-1})\\,\\dd (\\lambda_1,\\dots,\\lambda_{k-1})\\right) ^{1\/2}\\\\\n&\\qquad \\times \\left( \\int_{\\Omega'^{k-1}} \\Delta_{\\Omega'}^{2\\vs}(\\lambda-\\lambda_1)\\cdots \\Delta_{\\Omega'}^{2\\vs}(\\lambda_{k-2}-\\lambda_{k-1})\\,\\dd (\\lambda_1,\\dots,\\lambda_{k-1})\\right) ^{1\/2}\\\\\n&= \\abs{\\tau(f)^2 \\Delta_{\\Omega'}^{-2\\vs}}^{*k}(\\lambda)^{1\/2} (\\Delta_{\\Omega'}^{2 \\vs})^{*k}(\\lambda)^{1\/2}\n\\end{split}\n\\]\nfor every $\\lambda\\in \\Omega'$. In addition,~\\cite[Corollary 2.21]{CalziPeloso} shows that\n\\[\n(\\Delta_{\\Omega'}^{2 \\vs})^{*k}=\\frac{\\Gamma_{\\Omega'}(2\\vs-\\vd)^{k}}{\\Gamma_{\\Omega'}(2 k \\vs-(k-1)\\vd)} \\Delta^{2 k\\vs-(k-1)\\vd}_{\\Omega'}\n\\]\non $\\Omega'$. Therefore, $\\tau(f^k)\\in L^2(\\Omega', \\Delta_{\\Omega'}^{(k-1)\\vd-2k \\vs}\\cdot \\Hc^m)$ and\n\\[\n\\norm{\\tau(f^k)}_{L^2(\\Omega', \\Delta_{\\Omega'}^{(k-1)\\vd-2k \\vs}\\cdot \\Hc^m)}\\meg \\frac{\\Gamma_{\\Omega'}(2\\vs-\\vd)^{k\/2}}{(2 \\pi)^{(k-1)m}\\Gamma_{\\Omega'}(2k \\vs-(k-1)\\vd)^{1\/2}} \\norm{\\tau(f)}_{L^2(\\Omega', \\Delta_{\\Omega'}^{-2\\vs}\\cdot \\Hc^m)}^k.\n\\]\nHence,\n\\[\nf^k\\in \\widetilde A^{2,2}_{k \\vs-(k-1)\\vd\/2}(D)=A^{2,2}_{k \\vs-(k-1)\\vd\/2}(D)\n\\]\nsince $k \\vs-(k-1)\\vd\/2\\in \\frac 1 4 \\vect m+(\\R_+^*)^r$, so that $f\\in A^{2k,2k}_{\\vs-(k-1)\\vd\/(2k)}(D)$. Hence, there is a continuous embedding $\\widetilde A^{2,2}_{\\vs}(D)\\subseteq A^{2k, 2k}_{\\vs-(k-1)\\vd\/(2k)}(D)$. Since there is a continuous embedding $\\widetilde A^{2,2}_{\\vs}(D)\\subseteq A^{\\infty,\\infty}_{\\vs-\\vd\/2}(D)$ by~\\cite[Definition 5.3]{CalziPeloso}, the assertion follows.\n\\end{proof}\n\n\n\n\\section{Reverse Carleson and Sampling measures for\n$A^{p,p}_\\vs(D)$}\\label{sec:6} \n\nRecall that a measure $\\mi\\in \\cM_+(D)$ is $p$-sampling for\n$A^{p_1,q_1}_\\vs$ if this latter space embeds  as a closed subspace of\n$L^p(\\mi)$. As observe in~\\cite{Luecking6}, replacing $\\mi$ with a\nsuitable integral of positive measures (for instance, but not\nnecessarily, a disintegration of $\\mi$ along $\\rho$ with respect to\nsome of its image measures), it is also possible to extend this\ndefinition to mixed-norm Lebesgue spaces. Even though we prefer to\navoid such technicalities, we present here a result in this spirit,\nwhich shows how one may construct sampling measures out of\nlattices. See~\\cite[Theorem 3.22]{CalziPeloso} for a proof of a\nstronger version of the following result. \n\n\\begin{prop}\nTake $p,q\\in ]0,\\infty]$, $\\vs\\in \\R^r$, and $R_0>0$. Then, there is $\\delta_0>0$ such that, for every $(\\delta,R)$-lattice $(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ on $D$, with $\\delta\\in ]0,\\delta_0]$ and $R\\in ]1,R_0]$, the mapping\n\\[\nS\\colon \\Hol(D)\\ni f\\mapsto \\big(\\Delta_\\Omega^{\\vect s-(\\vect b+\\vect d)\/p}(\\rho(\\zeta_{j,k},z_{j,k})) f(\\zeta_{j,k},z_{j,k})  \\big) \\in \\C^{J\\times K}\n\\]\ninduces isomorphisms of $A^{p,q}_\\vs$ and $A^{p,q}_{\\vs,0}$ onto closed subspaces of $\\ell^{p,q}(J,K)$ and $\\ell^{p,q}_0(J,K)$, respectively. In addition, $A^{\\infty,\\infty}_{\\vs-(\\vect b+\\vect d)\/p}\\cap S^{-1}(\\ell^{p,q}(J,K))=A^{p,q}_\\vs$ and $A^{\\infty,\\infty}_{\\vs-(\\vect b+\\vect d)\/p}\\cap S^{-1}(\\ell^{p,q}_0(J,K))=A^{p,q}_{\\vs,0}$. \n\\end{prop}\n\n\nThe next result provides a necessary condition for a measure $\\mi$ for\nwhich the function $M_R (\\mi)$ is in some mixed-normed weighted\nLebesgue space on $D$ to be a sampling measure. \nIt extends~\\cite[Theorem 4.3]{Luecking2}, which deals with the case\nin which $D$ is the unit disc in $\\C$. \n\n\\begin{prop}\\label{prop:32}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and $\\vs\\in\n\\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ if $q_1<\\infty$, while $\\vs\\in\n\\R_+^r$ if $q_1=\\infty$. Let $p^*,q^*$, and $\\vs^*$ be as in~\\eqref{p*q*s'}.\nThen, for every $R,C,C'>0$, there are $R',C''>0$ such that for every\n$\\mi\\in\\cM_+(D)$ such that  \n\\[\n\\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\\meg C \\norm{f}_{L^{p}(\\mi)}\n\\]\nfor every $f\\in A^{p_1,q_1}_{\\vs}(D)$, and such that\n\\[\n\\norm{M_R (\\mi)}_{L^{p^*,q^*}_{\\vs^*}(D)}\\meg C',\n\\]\none has $p_1,q_1\\meg p$ and\n\\[\nM_{R'}(\\mi)(\\zeta,z)\\Meg C'' \\Delta_\\Omega^{p[\\vs-(\\vb+\\vd)\/p_1]}(\\rho(\\zeta,z))\n\\]\nfor every $(\\zeta,z)\\in D$.\n\\end{prop}\n\nNotice that, if $\\norm{M_R (\\mi)}_{L^{p^*,q^*}_{\\vs^*}(D)}$ is finite, then $\\mi$ is a $p$-Carleson measure for $A^{p_1,q_1}_\\vs$, thanks to Proposition~\\ref{prop:23}. Hence, a measure $\\mi$ as above is indeed a $p$-sampling measure for $A^{p_1,q_1}_\\vs$.  \n\n\\begin{proof}\nObserve that, by inspection of the proof of Proposition~\\ref{prop:23},\nit is readily verified that for every $R'>0$ there is a constant\n$C_{R'}>0$ such that  \n\\[\n\\norm{f}_{L^{p}(\\mi)}\\meg C_{R'}\\norm{M_R(\\mi)}_{L^{p^*,q^*}_{\\vs^*}(D)}^{1\/p}\\norm{\\chi_{B(\\Supp{\\mi},R')}f}_{L^{p_1,q_1}_{\\vs}(D)}\n\\]\nfor every $f\\in \\Hol(D)$ and for every positive Radon measure $\\mi$ on $D$, where \n\\[\nB(\\Supp{\\mi},R')\\coloneqq \\bigcup_{(\\zeta,z)\\in \\Supp{\\mi}} B((\\zeta,z),R').\n\\]\nTherefore, for every $(\\zeta,z)\\in D$ and for every $R'>0$,\n\\[\n\\norm{\\chi_{D\\setminus B((\\zeta,z), 2 R')}f}_{L^{p}(\\mi)}\\meg C_{R'}\\norm{M_R(\\mi)}_{L^{p^*,q^*}_{\\vs^*}(D)}^{1\/p}\\norm{\\chi_{D\\setminus B((\\zeta,z),R')}f}_{L^{p_1,q_1}_{\\vs}(D)}\n\\]\nfor every $f\\in \\Hol(D)$ and for every positive Radon measure $\\mi$ on $D$. Now, take $\\vect{s'}\\in \\R^r$ so that $B^{\\vect{s'}}_{(\\zeta,z)}\\in A^{p_1,q_1}_{\\vs}(D)$ for every $(\\zeta,z)\\in D$, and observe that there is a constant $C'''>0$ such that\n\\[\n\\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{A^{p_1,q_1}_{\\vs}(D)}=C''' \\Delta_\\Omega^{\\vs+\\vect {s'}-(\\vb+\\vd)\/p_1}(\\rho(\\zeta,z))\n\\]\nfor every $(\\zeta,z)\\in D$ (cf.~\\cite[Proposition 2.41]{CalziPeloso}). In addition, for every $R'>0$ there is a constant $C^{(4)}_{R'}>0$ such that\n\\[\n\\frac{1}{C^{(4)}_{R'}}\\Delta_\\Omega^{\\vect{s'}}(\\rho(\\zeta,z))\\meg \\abs{B^{\\vect{s'}}_{(\\zeta,z)}(\\zeta',z')}\\meg C^{(4)}_{R'}\\Delta_\\Omega^{\\vect{s'}}(\\rho(\\zeta,z)) \n\\]\nfor every $(\\zeta,z),(\\zeta',z')\\in D$ such that $d((\\zeta,z),(\\zeta',z'))\\meg R'$ (cf.~\\cite[Theorem 2.47]{CalziPeloso}).\nTherefore,\n\\[\n\\begin{split}\n&  (C^{(4)}_{2R'})^{p}\\Delta_\\Omega^{p\\vect{s'}}(\\rho(\\zeta,z))M_{2R'}(\\mi)(\\zeta,z) \\\\\n& \\quad \\Meg \n\\int_{B((\\zeta,z),2R')} \\abs{ B^{\\vect{s'}}_{(\\zeta,z)} }^{p}\\,\\dd \\mi\\\\\n&\\quad  =   \\norm{B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p}(\\mi)}^{p}- \\norm{\\chi_{D\\setminus B((\\zeta,z),2 R')}  B^{\\vect{s'}}_{(\\zeta,z)} }^{p}_{L^{p}(\\mi)}\\\\\n&  \\quad \n\\Meg \\left(\\frac{C'''}{C}\\Delta_\\Omega^{\\vs+\\vect{s'}-(\\vb+\\vd)\/p_1}(\\rho(\\zeta,z))\\right)^{p}-C' C_{R'}^{p} \\norm{\\chi_{D\\setminus B((\\zeta,z),R')} B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p_1,q_1}_{\\vs}(D)}^{p}\n\\end{split}\n\\]\nfor every $(\\zeta,z)\\in D$, for every $R'>0$, and for every $\\mi$ as in the statement. Now, observe that, by homogeneity, setting \n\\[\nC^{(5)}_{R'}\\coloneqq\\norm{\\chi_{D\\setminus B((0,ie_\\Omega),R')} B^{\\vect{s'}}_{(0, ie_\\Omega)}}_{L^{p_1,q_1}_{\\vs}(D)},\n\\]\none has\n\\[\n\\norm{\\chi_{D\\setminus B((\\zeta,z),R')} B^{\\vect{s'}}_{(\\zeta,z)}}_{L^{p_1,q_1}_{\\vs}(D)}= C^{(5)}_{R'} \\Delta_\\Omega^{\\vs+\\vs^*-(\\vb+\\vd)\/p_1}(\\rho(\\zeta,z))\n\\]\nfor every $(\\zeta,z)\\in D$ (cf.~Lemma~\\ref{lem:5}). Therefore,\n\\[\nM_{2 R'}(\\mi)(\\zeta,z)\\Meg (C^{(4)}_{2R'})^{-p} \\left( \\left(\\frac{C'''}{C}\\right)^{p}- C' (C_{R'} C^{(5)}_{R'})^{p}  \\right) \\Delta_\\Omega^{p[\\vs-(\\vb+\\vd)\/p_1]}(\\rho(\\zeta,z)) \n\\]\nfor every $(\\zeta,z)\\in D$ and for every $R'>0$. \n\nNow, observe that the function $(\\Delta_\\Omega^{p[(\\vect b+\\vect d)\/p_1-\\vs]}\\circ \\rho) M_{R'}$ is both bounded from below and in $L^{p^*,q^*}_{\\vs^*+p[\\vs-(\\vect b+\\vect d)\/p_1]  }$, thanks to Lemma~\\ref{lem:1}. It then follows easily that $p^*=q^*=\\infty$, that is, $p_1,q_1\\meg p$, in which case $\\vs^*=p[(\\vect b+\\vect d)\/p_1-\\vs]$. The proof is complete.  \n\\end{proof}\n\n\nIn the next result we establish a necessary and sufficient condition\nin order for a $\\nu_D$-measurable set $G$ to be a dominant (or sampling) set (recall\nDefinition~\\ref{def:dom-set}), that is, for the measure $\\chi_G(\\Delta_\\Omega^{p\\vs-(\\vect b+\\vect d)}\\circ \\rho)\\cdot\\nu_D$\nto be a $p$-sampling measure for $A^{p,p}_\\vs$.    It extends~\\cite[Main\nTheorem]{Luecking}, which deals with the case in which $D$ is the unit\ndisc in $\\C$. See also~\\cite[Theorem 1]{Luecking4}, which deals with\nweighted Bergman spaces on general homogeneous domains.  \n\n\n\\begin{teo}\\label{prop:31} \nTake $p\\in ]0,\\infty[$ and $\\vs\\in \\frac{1}{2 p}\\vect\nm+(\\R_+^*)^r$. Then,  for every $\\nu_D$-measurable subset $G$ of $D$\nthe following conditions are equivalent: \n\\begin{enumerate}\n\\item[{\\em(1)}] there are $R,C>0$ such that, for every  $(\\zeta,z)\\in D$,\n\\[\n\\norm*{\\chi_{B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)}\\meg C\n\\norm*{\\chi_{G\\cap B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)};\n\\]\n\\item[{\\em(2)}] there exists $C'>0$ such that, for every $f\\in A^{p,p}_{\\vs}(D)$,\n\\[\n\\norm*{f}_{A^{p,p}_{\\vs}(D)}\\meg C' \\norm*{\\chi_{G}f}_{L^{p,p}_{\\vs}(D)} ;\n\\]  \n\\item[{\\em(3)}]  there are $  R' ,C''>0$ such that $\\nu_D(G\\cap\nB((\\zeta,z),  R' ))\\Meg C'' $ for every $(\\zeta,z)\\in D$. \n\\end{enumerate}\n\\end{teo}\n\nNotice that condition (3) depends neither on $\\vs$, nor on $p$. \nBefore we pass to the proof, we need to establish some lemmas.\n\n\n\\begin{lem}\\label{lem:12}\nFix $R>0$, and define, for every $\\eps>0$  and for every $f\\in A^{p,p}_{\\vs}(D)$,\n\\[\nA_{f,\\eps}\\coloneqq \\Set{(\\zeta,z)\\in D\\colon \\abs{f(\\zeta,z)}\\meg \\eps \\left(\\dashint_{B((\\zeta,z),R)} \\abs{f}^p\\,\\dd \\nu_D\\right)^{1\/p}  }.\n\\]\nThen, there is a constant $C>0$ such that \n\\[\n\\norm{\\chi_{A_{f,\\eps}} f  }_{L^{p,p}_{\\vs}(D)}\\meg C\\eps \\norm{f  }_{L^{p,p}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$ and for every $\\eps>0$.\n\\end{lem}\n\nThis extends~\\cite[Lemma 2]{Luecking2}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\\begin{proof}\nObserve that, by Fubini's theorem,\n\\[\n\\begin{split}\n\\norm{\\chi_{A_{f,\\eps}} f  }_{L^{p,p}_{\\vs}(D)}^p&\\meg \\eps^p \\int_{A_{f,\\eps}} \\dashint_{B((\\zeta,z),R)} \\abs{f}^p\\,\\dd  \\nu_D\\, \\Delta_\\Omega^{p\\vs-(\\vb+\\vd)}(\\rho(\\zeta,z))\\,\\dd \\nu_D(\\zeta,z)\\\\\n&\\meg \\eps^p\\int_D \\abs{f(\\zeta',z')}^p \\dashint_{B((\\zeta',z'),R)}  \\Delta_\\Omega^{p\\vs-(\\vb+\\vd)}(\\rho(\\zeta,z)) \\,\\dd \\nu_D(\\zeta,z)\\,\\dd  \\nu_D(\\zeta',z').\n\\end{split}\n\\]\nNow, observe that, by homogeneity,\n\\[\n\\begin{split}\n\\dashint_{B((\\zeta',z'),R)}  \\Delta_\\Omega^{p\\vs-(\\vb+\\vd)}(\\rho(\\zeta,z)) \\,\\dd \\nu_D(\\zeta,z)&= C'\\Delta_\\Omega^{p\\vs-(\\vb+\\vd)}(\\rho(\\zeta',z'))  \n\\end{split}\n\\]\nfor a suitable constant $C'>0$. The assertion follows.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{lem:13}\nTake $R,\\eps,p\\in ]0,\\infty[$. Then, for every $\\delta>0$ there is $\\lambda>0$ such that, if we define\n\\[\nE_{f,\\lambda}(\\zeta,z)\\coloneqq \\Set{(\\zeta',z')\\in B((\\zeta,z),R)\\colon \\abs{f(\\zeta',z')}>\\lambda \\abs{f(\\zeta,z)} },\n\\]\nthen\n\\[\n\\nu_D( E_{f,\\lambda}(\\zeta,z)  )\\Meg (1-\\delta) \\nu_D(B((\\zeta,z),R))\n\\]\nfor every $(\\zeta,z)\\in D$ and for every $f\\in \\Hol(B((\\zeta,z),R))$ such that\n\\[\n\\abs{f(\\zeta,z)}\\Meg \\eps \\left( \\dashint_{B((\\zeta,z),R)} \\abs{f}^p\\,\\dd \\nu_D \\right)^{1\/p}.\n\\]\n\\end{lem}\n\nThis extends~\\cite[Lemma 2]{Luecking3}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\\begin{proof}\nObserve that, by homogeneity, we may reduce to proving the assertion for $(\\zeta,z)=(0,i e_\\Omega)$. Then, assume by contradiction that there are $\\delta>0$ and a sequence $(f_j)_{j\\in\\N}$ of elements of $\\Hol(D)$ such that\n\\[\n\\abs{f_j(0,i e_\\Omega)}\\Meg \\eps \\left( \\dashint_{B((\\zeta,z),R)} \\abs{f_j}^p\\,\\dd \\nu_D \\right)^{1\/p}\n\\]\nand such that\n\\[\n\\nu_D(E_{f_j,2^{-j}})<\\delta\\nu_D(B((0,i e_\\Omega),R))\n\\]\nfor every $j\\in \\N$. Observe that, up to multiplying each $f_j$ by a suitable constant, we may assume that  $\\dashint_{B((\\zeta,z),R)} \\abs{f_j}^p\\,\\dd \\nu_D=1$ for every $j\\in \\N$, so that $\\abs{f_j(0,i e_\\Omega)}\\Meg \\eps$ for every $j\\in\\N$. It is then readily verified that the sequence $(f_j)$ is bounded in $\\Hol(B((0,i e_\\Omega),R))$, so that we may assume that it converges locally uniformly to some $f$. Then, take $R'\\in ]0,R[$ so that $\\nu_D(B((0,i e_\\Omega),R'))\\Meg \\frac \\delta 2\\nu_D(B((0,ie_\\Omega),R))$, and observe that\n\\[\n\\nu_D(\\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon \\abs{f_j(\\zeta,z)}\\meg 2^{-j}  \\abs{f_j(0,i e_\\Omega)} })\\Meg \\frac \\delta 2\\nu_D(B((0,ie_\\Omega),R))\n\\]\nfor every $j\\in \\N$.\nObserve that $(f_j)$ converges uniformly to $f$ on $B((0,i e_\\Omega),R')$ (cf.~\\cite[Proposition 2.44]{CalziPeloso}), so that for every $j_0\\in\\N$ there is $j_1\\in \\N$ such that\n\\[\n\\abs{f_j-f}\\meg 2^{-j_0}\n\\]\non $B((0,i e_\\Omega),R')$for every $j\\Meg j_1$. Hence,\n\\[\n\\begin{split}\n&\\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon \\abs{f(\\zeta,z)}\\meg 2^{-j_0}+2^{-j}  \\abs{f_j(0,i e_\\Omega)} }\\\\\n&\\qquad \\qquad\\qquad \\qquad\\qquad \\qquad\\qquad \\qquad\\supseteq \\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon \\abs{f_j(\\zeta,z)}\\meg 2^{-j}  \\abs{f_j(0,i e_\\Omega)} }\n\\end{split}\n\\]\nfor every $j\\Meg j_1$, so that\n\\[\n\\nu_D(\\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon \\abs{f(\\zeta,z)}\\meg 2^{-j_0}+2^{-j}  \\abs{f_j(0,i e_\\Omega)} })\\Meg \\frac\\delta 2\\nu_D(B((0,ie_\\Omega),R)).\n\\]\nBy the arbitrariness of $j_0$, we then see that\n\\[\n\\begin{split}\n\\nu_D(\\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon f(\\zeta,z)=0})&=\\lim_{\\eta\\to 0^+} \\nu_D(\\Set{(\\zeta,z)\\in B((0,i e_\\Omega),R')\\colon \\abs{f(\\zeta,z)}\\meg \\eta })\\\\\n&\\Meg  \\frac\\delta 2\\nu_D(B((0,ie_\\Omega),R)).\n\\end{split}\n\\]\nThen, $f$ vanishes on a non-$\\nu_D$-negligible set, so that it vanishes identically on the connected set $B((0,i e_\\Omega),R)$ by holomorphy.\\footnote{Observe that the Bergman distance on $D$ is induced by a complete Riemannian metric by~\\cite[Proposition 2.44]{CalziPeloso}, so that $B((0,i e_\\Omega),R)=\\exp_{(0,i e_\\Omega)}(B(0,R))$ by the Hopf--Rinow theorem (cf., e.g.,~\\cite[the proof of Theorem 6.6]{Lang}).   Alternatively, every element of $B((0, i e_\\Omega),R)$ may be connected to $(0,i e_\\Omega)$ by a minimizing geodesic, whose image therefore lies entirely in $B((0,i e_\\Omega),R)$.} Nonetheless, $\\abs{f(0,i e_\\Omega)}\\Meg \\eps$: contradiction.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{prop:31}.]\nBy means of~\\cite[Corollary 2.49]{CalziPeloso}, condition\n(1) is readily seen to be equivalent to condition (3).  \n\n(2) $\\implies$ (1). This follows from Proposition~\\ref{prop:32}.\n\n\n(1) $\\implies$ (2). With the notation of Lemma~\\ref{lem:12}, choose $\\eps>0$  so small that\n\\[\n\\norm{f}_{A^{p,p}_{\\vs}(D)}\\meg 2\\norm{\\chi_{D\\setminus A_{f,\\eps}}f}_{L^{p,p}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$. In addition, with the notation of Lemma~\\ref{lem:13}, choose $\\lambda>0$ so that\n\\[\n\\norm{\\chi_{B((\\zeta,z),R)\\setminus E_{f,\\lambda}(\\zeta,z)}}_{L^{p,p}_{\\vs}(D)}\\meg \\frac{1}{2^{1\/p} C'}\\norm{\\chi_{B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$, and for every $(\\zeta,z)\\in D\\setminus A_{f,\\eps}$ (cf.~also~\\cite[Corollary 2.49]{CalziPeloso}), so that\n\\[\n\\norm{\\chi_{G\\cap E_{f,\\lambda}(\\zeta,z)}}_{L^{p,p}_{\\vs}(D)}\\Meg \\frac{1}{2^{1\/p} C'} \\norm{\\chi_{B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)}.\n\\]\nNow, the definition of $E_{f,\\lambda}(\\zeta,z)$ implies that\n\\[\n\\norm{\\chi_{G\\cap B((\\zeta,z),R)} f}_{L^{p,p}_{\\vs}(D)}\\Meg \\norm{\\chi_{G\\cap E_{f,\\lambda}(\\zeta,z)} f}_{L^{p,p}_{\\vs}(D)}\\Meg \\frac{\\lambda^p}{2^{1\/p} C'} \\abs{f(\\zeta,z)} \\norm{\\chi_{B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$, and for every $(\\zeta,z)\\in D\\setminus A_{f,\\eps}$.\nTherefore, by means of Fubini's theorem we see that\n\\[\n\\begin{split}\n\\norm{f}_{A^{p,p}_{\\vs}(D)}^p&\\meg 2^p\\norm{\\chi_{D\\setminus A_{f,\\eps}}f}_{L^{p,p}_{\\vs}(D)}^p\\\\\n&\\meg \\frac{2^{p+1} C'^p}{\\lambda^p} \\int_G \\abs{f(\\zeta',z')}^p \\int_{B((\\zeta',z'),R)} \\frac{\\Delta_\\Omega^{p\\vs+\\vd}(\\rho(\\zeta,z))}{\\norm{\\chi_{B((\\zeta,z),R)}}_{L^{p,p}_{\\vs}(D)}^p}\\,\\dd (\\zeta,z) \\,\\Delta_\\Omega^{p \\vs+\\vd}(\\rho(\\zeta',z'))\\,\\dd(\\zeta',z') \n\\end{split}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$. To conclude, it suffices to observe that the function\n\\[\n(\\zeta',z')\\mapsto \\int_{B((\\zeta',z'),R)} \\frac{\\Delta_\\Omega^{p\\vs+\\vd}(\\rho(\\zeta,z))}{\\norm{\\chi_{B((\\zeta,z),R)}}^p_{L^{p,p}_{\\vs}(D)}}\\,\\dd (\\zeta,z)=\\frac{\\nu_D(B((\\zeta',z'),R))  }{\\norm{\\chi_{B((0,i e_\\Omega),R)}}^p_{L^{p,p}_{\\vs}(D)}}\n\\]\nis constant, by homogeneity (cf.~Lemma~\\ref{lem:5}).\n\\end{proof}\n\n\n\n\\begin{teo}\\label{prop:30}\nTake $p\\in ]0,\\infty[$, $\\vs\\in \\frac{1}{2 p}\\vect m+(\\R_+^*)^r$,\n$\\eps,C',R_0>0$. Then, there are $R_1,C>0$ such that, for every $R\\in\n]0,R_1]$, for every $\\mi\\in\\cM_+(D)$ such that \n\\[\nN(\\mi)\\coloneqq \\norm{M_R(\\mi)}_{L^{\\infty,\\infty}_{\\vb+\\vd-p \\vs}(D)}<\\infty,\n\\]\nand such that\n\\[\n\\nu_D(G_\\mi\\cap B((\\zeta,z),R_0))\\Meg C'\n\\]\nfor every $(\\zeta,z)\\in D$,\nwhere\n\\[\nG_\\mi\\coloneqq \\Set{(\\zeta,z)\\in D\\colon \\Delta_\\Omega^{\\vb+\\vd-p\\vs}\n(\\rho(\\zeta,z))M_R(\\mi)(\\zeta,z)\\Meg \\eps N(\\mi)}, \n\\]\none has\n\\[\n\\norm{f}_{A^{p,p}_{\\vs}(D)}\\meg \\frac{C R^{(2 n+2m)\/p}}{N(\\mi)} \\norm{ f}_{L^{p}(\\mi)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$\n\\end{teo}\n\nThis extends~\\cite[Theorem 4.2]{Luecking2}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\nBefore we pass to the proof, we need another result.\n\n\\begin{prop}\\label{prop:28}\nTake $p\\in ]0,\\infty]$. Then, there are two constants $C,R_0>0$ such\nthat the following hold. For every\n$\\vect{s_1},\\vect{s_2},\\vect{s_3}\\in \\R^r$, for every $R,R'\\in\n]0,R_0]$, for every $\\mi_1,\\mi_2\\in\\cM_+(D)$ such that\n\\[\nC_1\\coloneqq \\sup_{(\\zeta,z)\\in D}\n\\Delta_\\Omega^{-\\vect{s_1}}(\\rho(\\zeta,z)) M_R(\\mi_1)(\\zeta,z)\\qquad\n\\text{and}\\qquad C_2\\coloneqq \\sup_{(\\zeta,z)\\in D}\n\\Delta_\\Omega^{-\\vect{s_2}}(\\rho(\\zeta,z)) M_{R+R'}(\\mi_2)(\\zeta,z) \n\\]\nare finite, and for every $f\\in \\Hol(D)$,\n\\[\n\\int_{d((\\zeta,z),(\\zeta',z'))< R}\\frac{\\abs{f(\\zeta,z)-f(\\zeta',z')}^p}{\\Delta_\\Omega^{\\vect{s_1}-\\vect{s_3}}(\\rho(\\zeta',z'))} \\,\\dd (\\mi_1\\otimes \\mi_2)((\\zeta,z),(\\zeta',z'))\\meg \\frac{C C_1 C_2 R^p }{R'^{p+2 n+2m}}   \\int_D   \\abs{f}^p (\\Delta_\\Omega^{\\vect{s_2}+\\vect{s_3}}\\circ \\rho)\\,\\dd \\nu_D,\n\\]\n\\end{prop}\n\nThis extends~\\cite[Theorem 2.3]{Luecking2}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\\begin{proof}\nBy~\\cite[Lemmas 3.24 and 3.25]{CalziPeloso}, there are two constants $C_1,R_0>0$ such that for every $R,R'\\in ]0,R_0]$, for every $f\\in \\Hol(D)$, and for every $(\\zeta,z),(\\zeta',z')\\in D$ such that $d((\\zeta,z),(\\zeta',z'))\\meg R$,\n\\[\n\\abs{f(\\zeta,z)-f(\\zeta',z')}^p\\meg C'\\frac{R^p}{R'^{p+2n+2m}} \\int_{B((\\zeta',z'),R+R')} \\abs{f}^p\\,\\dd \\nu_D.\n\\] \nTherefore, by means of~\\cite[Corollary 2.49]{CalziPeloso}, we see that there is a constant $C''>0$ such that\n\\[\n\\abs{f(\\zeta,z)-f(\\zeta',z')}^p\\Delta_\\Omega^{\\vect{s_3}}(\\rho(\\zeta',z'))\\meg C''\\frac{R^p}{R'^{p+2n+2m}} \\int_{B((\\zeta',z'),R+R')} \\abs{f}^p(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho)\\,\\dd \\nu_D.\n\\]\nfor every $R,R'\\in ]0,R_0]$, for every $f\\in \\Hol(D)$, and for every $(\\zeta,z),(\\zeta',z')\\in D$ such that $d((\\zeta,z),(\\zeta',z'))\\meg R$.\nThen, integrating in $(\\zeta,z)$ with respect to $\\mi_1$,\n\\[\n\\begin{split}\n&\\Delta_\\Omega^{\\vect{s_3}}(\\rho(\\zeta',z'))\\int_{ B((\\zeta',z'),R)} \\abs{f(\\zeta,z)-f(\\zeta',z')}^p\\,\\dd \\mi_1(\\zeta,z)\\\\\n&\\qquad\\meg C'' \\frac{R^p}{R'^{p+2n+2m}} M_R(\\mi_1)(\\zeta',z') \\int_{B((\\zeta',z'),R+R')} \\abs{f}^p(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho)\\,\\dd \\nu_D\\\\\n&\\qquad\\meg C'' \\frac{R^p}{R'^{p+2n+2m}}C_1\n\\Delta_\\Omega^{\\vect{s_1}}(\\rho(\\zeta',z')) \\int_{B((\\zeta',z'),R+R')}\n\\abs{f}^p(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho)\\,\\dd \\nu_D \n\\end{split}\n\\]\nso that, integrating in $(\\zeta',z')$ with respect to the measure $\\mi_2$,\n\\[\n\\begin{split}\n&\\int_{d((\\zeta,z),(\\zeta',z'))< R} \\abs{f(\\zeta,z)-f(\\zeta',z')}^p\\Delta_\\Omega^{\\vect{s_3}-\\vect{s_1}}(\\rho(\\zeta',z')) \\,\\dd (\\mi_1\\otimes \\mi_2)((\\zeta,z),(\\zeta',z'))\\\\\n&\\qquad \\qquad\\meg \\frac{C_1 C_2 C'' R^p}{ R'^{p+2n+2m}}  \\int_D   \\abs{f}^p (\\Delta_\\Omega^{\\vect{s_2}+\\vect{s_3}}\\circ \\rho)\\,\\dd \\nu_D,\n\\end{split}\n\\]\nwhence the result.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{prop:30}.]\nApply Proposition~\\ref{prop:28} with $\\mi_1=\\mi $, $\\mi_2=\\nu_D$,\n$\\vect{s_1}=\\vect{s_3}=p \\vs-(\\vb+\\vd)$,  and $\\vect{s_2}=\\vect 0$.\nThen, we find $R_0>0$ and $C_1>0$ such that\n\\[\n\\int_{d((\\zeta,z),(\\zeta',z'))< R} \\abs{f(\\zeta,z)-f(\\zeta',z')}^p\n\\,\\dd (\\nu_D\\otimes \\mi)((\\zeta,z),(\\zeta',z'))\\meg\n\\frac{C_1R^{p}}{R_0^{p+2 n+2 m}}\nN(\\mi) \\norm{f}_{A^{p,p}_{\\vs}(D)}^p\n\\]\nfor every $R\\in ]0,R_0]$ and for every $f\\in \\Hol(D)$.\nTherefore,\n\\[\n\\begin{split}\n&\\left( \\int_{d((\\zeta,z),(\\zeta',z'))< R} \\abs{f(\\zeta,z)}^p\\,\\dd (\\nu_D\\otimes \\mi)((\\zeta,z),(\\zeta',z'))\\right)^{1\/\\max(1,p)}\\\\\n&\\qquad\\meg \\left( \\int_{d((\\zeta,z),(\\zeta',z'))< R}\\abs{f(\\zeta',z')}^p \\,\\dd (\\nu_D\\otimes \\mi)((\\zeta,z),(\\zeta',z'))\\right) ^{1\/\\max(1,p)}\\\\\n&\\qquad\\qquad+ \\left( \\frac{C_1 R^{p}}{ R_0^{p+2 n+2 m}} N(\\mi) \\norm{f}_{A^{p,p}_{\\vs}(D)}^p\\right) ^{1\/\\max(1,p)}\n\\end{split}\n\\]\nfor every $f\\in \\Hol(D)$. Now, observe that Theorem~\\ref{prop:31} implies that there is a constant $C_2>0$ such that\n\\[\n\\begin{split}\n\\int_{d((\\zeta,z),(\\zeta',z'))< R} \\abs{f(\\zeta,z)}^p\\,\\dd (\\nu_D\\otimes \\mi)((\\zeta,z),(\\zeta',z'))\n&= \\int_D \\abs{f(\\zeta,z)}^p M_R(\\mi)(\\zeta,z)\\,\\dd \\nu_D(\\zeta,z) \\\\\n&\\Meg \\eps N(\\mi)\\int_{G_\\mi} \\abs{f}^p( \\Delta_\\Omega^{p\\vs-(\\vb+\\vd)}\\circ \\rho)\\,\\dd \\nu_D\\\\\n&\\Meg C_2 \\eps N(\\mi) \\norm{f}_{A^{p,p}_{\\vs}(D)}^p,\n\\end{split}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$,\nwhile clearly\n\\[\n\\begin{split}\n\\int_{d((\\zeta,z),(\\zeta',z'))< R}\\abs{f(\\zeta',z')}^p \\,\\dd (\\nu_D\\otimes \\mi)((\\zeta,z),(\\zeta',z'))&= \\nu_D(B((0,i e_\\Omega),R)  )\\int_D \\abs{f}^p\\,\\dd \\mi,\n\\end{split}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$.\nTherefore, \n\\[\n\\begin{split}\n\\left( C_2\\eps N(\\mi) \\norm{f}_{A^{p,p}_{\\vs}(D)}^p\\right)^{1\/\\max(1,p)}\n&\\meg\\left( \\nu_D(B((0,i e_\\Omega),R)  )\\int_D \\abs{f}^p\\,\\dd \\mi\\right) ^{1\/\\max(1,p)}\\\\\n&\\qquad+ \\left( \\frac{C_1 R^{p}}{R_0^{p+2 n+2 m}} N(\\mi)\\norm{f}_{A^{p,p}_{\\vs}(D)}^p\\right) ^{1\/\\max(1,p)}\n\\end{split}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$. It then follows that\n\\[\nC_3N(\\mi)\\norm{f}_{A^{p,p}_{\\vs}(D)}^p\\meg \\nu_D(B((0,i e_\\Omega),R)  )\\int_D \\abs{f}^p\\,\\dd \\mi\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$, where\n\\[\nC_3\\coloneqq \\left( \\left( C_2\\eps \\right)^{1\/\\max(1,p)}-    \\left( \\frac{C_1 R^{p}}{R_0^{p+2 n+2 m}}  \\right) ^{1\/\\max(1,p)}  \\right)^{\\max(1,p)}.\n\\]\nSince there is $R_1\\in ]0, R_0]$ such that $C_3$ is well defined and $>0$ whenever $R\\in ]0,R_1]$, the assertion follows.\n\\end{proof}\n\n\n\n\\begin{deff}\\label{def:1}\nGiven $\\vs\\in \\R^r$ and $\\mi\\in\\cM_+(D)$, we shall denote with\n$W_{\\vs}(\\mi)$ the vague closure of the set of measures of the\nform \n\\[\n\\Delta_\\Omega^{\\vs}(\\rho(\\zeta,z))(\\phi_{(\\zeta,z)})_*(\\mi),\n\\]\nfor every $(\\zeta,z)\\in D$, where $\\phi_{(\\zeta,z)}$ is an affine\nautomorphism of $D$ of the form $(\\zeta',z')\\mapsto (\\zeta,\\Rea\nz+i\\Phi(\\zeta))\\cdot (g\\zeta', t\\cdot z)$, with $t\\in T_+$, $g\\in\nGL(E)$, and $t\\cdot \\Phi=\\Phi\\circ (g\\times g)$.\\footnote{Notice that\n$\\phi_{(\\zeta,z)}$ is not uniquely determined by $(\\zeta,z)$ unless\n$n=0$, so that this definition may depend on the choice of the\nautomorphisms $\\phi_{(\\zeta,z)}$.} \n\\end{deff}\n\n\n\\begin{teo}\\label{prop:27}\nTake $p,q\\in ]0,\\infty[$, with $q<p$, $\\vs\\in\\R^r$, and\n$\\mi\\in\\cM_+(D)$ such that the following hold: \n\\begin{enumerate}\n\\item[{\\em(1)}] $\\vs\\in \\frac{1}{2 p}\\vect m+\\frac{q}{p(p-q)}\\vect{m'}+(\\R_+^*)^r$;\n\n\\item[{\\em(2)}]  $M_1(\\mi) \\in L^{\\infty,\\infty}_{\\vb+\\vd-p\\vs}(D)$;\n\n\\item[{\\em(3)}]   the support of   every element of $W_{p\\vs-(\\vb+\\vd)}(\\mi)$  is a set\nof uniqueness for $A^{q,q}_{(p\/q)\\vs}(D)$.  \n\\end{enumerate}\nThen, the canonical mapping $A^{p,p}_{\\vs}(D)\\to L^{p}(\\mi)$ is an\nisomorphism onto its image. \n\\end{teo}\n\nThis extends the implication (b)$\\implies$(a) in~\\cite[Theorem 5]{Luecking5}, which deals with the case in which $D$ is the unit disc in $\\C$. Notice that, as shown in~\\cite[Theorem 5]{Luecking5}, condition~(3) holds for some $q<p$ if $A^{p,p}_{\\vs}(D)$ embeds as a closed subspace of $L^p(\\mi)$ and  $D$ is the unit disc in $\\C$. \n\n\nBefore we pass to the proof, we need some lemmas.\n\n\n\\begin{lem}\\label{lem:10}\nTake $p_1,q_1,p\\in ]0,\\infty]$, with $p<\\infty$, and $\\vs\\in\n\\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ and let $p^*,q^*$ and $\\vs^*$ be as\nin~\\eqref{p*q*s'}.  Let $\\Ms$ be a set of Radon measures\non $D$ such that  \n\\[\n\\sup_{\\mi\\in M}\\norm{M_R(\\mi)}_{L^{p^*, q^*}_{\\vs^*}(D)}<\\infty.\n\\]\nThen, \n\\[\n\\lim_{\\mi,\\Ff}\\,  \\norm{f}_{L^{p}(\\mi)}= \\norm{f}_{L^{p}(\\mi_0)}\n\\]\nfor every filter $\\Ff$ on $\\Ms$ which converges vaguely to some  Radon\nmeasure $\\mi_0$ on $D$, and for every $f\\in A^{p_1,q_1}_{\\vs,0}(D)$.  \n\\end{lem}\n\nThis extends~\\cite[Theorem 1]{Luecking5}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\\begin{proof}\nBy inspection of the proof of Proposition~\\ref{prop:23}, it is readily verified that there is a constant $C'>0$ such that\n\\[\n\\norm{\\chi_{B((0,i e_\\Omega), R')}f}_{L^{q_2}(\\mi)}\\meg C' \\norm{\\chi_{B((0,i e_\\Omega), R'+1)} f}_{L^{q_1,q_2}_{\\vs}(D)}\n\\]\nfor every $\\mi\\in M$, for every $f\\in A^{p_1,q_1}_{\\vs,0}(D)$, and for every $R'>0$. The   assertion follows easily. \n\\end{proof}\n\n\n\n\\begin{lem}\\label{lem:14}\nTake $p_1,q_1,p\\in ]0,\\infty]$, $\\vs\\in \\frac{1}{2 q_1}\\vect\nm+(\\R_+^*)^r$ if $q_1<\\infty$ and $\\vs\\in \\R_+^r$ if $q_1=\\infty$, and\n$\\vect{s_2},\\vect{s_3}\\in \\R^r$. Let $p^*,q^*$ and $\\vs^*$  be as\nin~\\eqref{p*q*s'}.\nTake $\\eps>0$, and define, for every $(\\zeta,z)\\in D$,\n\\[\nU_\\eps(\\zeta,z)\\coloneqq \\Set{f\\in \\Hol(D)\\colon \\abs{f(\\zeta,z)}\\Meg\\eps \\Delta_\\Omega^{(\\vb+\\vd)\/p-\\vs-\\vect{s_2}}(\\rho(\\zeta,z))\\norm{f B^{\\vect{s_2}}_{(\\zeta,z)}}_{A^{p_1,q_1}_{\\vs}(D)}}.\n\\]\nTake $\\mi\\in\\cM_+(D)$ such that\n\\[\n\\norm{M_R(\\mi)}_{L^{p^*,q^*}_{\\vect {s'}}(D)}<\\infty,\n\\]\nand such that the support of every\nelement of $W_{p[\\vs-(\\vb+\\vd)\/p_1]}(\\mi)$  is a set of uniqueness for $A^{p_1,q_1}_{\\vect {s_1}}(D)$. \n\nThen, there is  a constant $C>0$ such that\n\\[\n\\norm{f B^{\\vect{s_2}}_{(\\zeta,z)}(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho) \n}_{L^{p}(\\mi')}\\Meg C\\Delta_\\Omega^{\\vect{s_3}}(\\rho(\\zeta,z))\\norm{f B^{\\vect{s_2}}_{(\\zeta,z)}}_{A^{p_1,q_1}_{\\vs}(D)}\n\\]\nfor every $(\\zeta,z)\\in D$, for every $f\\in U_\\eps(\\zeta,z)$ and for every $\\mi'\\in W_{p[\\vs-(\\vb+\\vd)\/p_1]}(\\mi)$.\n\\end{lem}\n\nThis extends~\\cite[Lemma 4]{Luecking5}, which deals with the case in which $D$ is the unit disc in $\\C$.\n\n\\begin{proof}\n\\textsc{Step I.} We prove the assertion for $(\\zeta,z)=(0,i e_\\Omega)$.\nDefine\n\\[\n\\mi_{(\\zeta,z)}\\coloneqq \\Delta_\\Omega^{p[\\vs-(\\vb+\\vd)\/p_1]}(\\rho(\\zeta,z))(\\phi_{(\\zeta,z)})_*(\\mi)\n\\]\nfor every $(\\zeta,z)\\in D$, where $\\phi_{(\\zeta,z)}$ is as in Definition~\\ref{def:1}.\nObserve that it will suffice to prove the assertion with $W_{p[\\vect{s_1}-(\\vb+\\vd)\/p_1]}(\\mi)$\nreplaced by $\\Set{\\mi_{(\\zeta,z)}\\colon (\\zeta,z)\\in D}$, thanks to Lemma~\\ref{lem:10}.\nThen assume, by contradiction, that there are a sequence $(f_j)$ of\nelements of $U_\\eps(0,i e_\\Omega)$, and a sequence $((\\zeta_j,z_j))$\nof elements of $D$ such that \n\\[\n\\norm{f_j B^{\\vect{s_2}}_{(0,i e_\\Omega)}}_{A^{p_1,q_1}_{\\vs}(D)}=1\n\\]\nfor every $j\\in \\N$, while\n\\[\n\\lim_{j\\to \\infty }\\norm{f_j B^{\\vect{s_2}}_{(0,i e_\\Omega)} (\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho) }_{L^{q_2}(\\mi_{(\\zeta_j,z_j)})}=0.\n\\]\nObserve that \n\\[\n\\norm{M_R(\\mi_{(\\zeta',z')})}_{L^{p^*,q^*}_{\\vs^*}(D)}= \\norm{M_R(\\mi)}_{L^{p^*,q^*}_{\\vs^*}(D)}\n\\]\nfor every $(\\zeta',z')\\in D$ (cf.~Lemma~\\ref{lem:5}), so that\n$W_{p[\\vect {s_1}-(\\vb+\\vd)\/p_1]}(\\mi)$ is bounded, hence compact and\nmetrizable, in the vague topology. Therefore, we may assume that\n$(\\mi_{(\\zeta_j,z_j)})  $ converges vaguely to some (positive Radon)\nmeasure $\\mi'$ on $D$. Analogously, we may assume that $(f_j)$\nconverges locally uniformly to some $f\\in A^{p_1,q_1}_{\\vs}(D)$, so\nthat $\\abs{f(0,i e_\\Omega)}\\Meg \\eps$. Let $(\\psi_k)_{k\\in K}$ be a\npartition of the unity on $D$ whose elements belong to $C_c(D)$, and\nobserve that \n\\[\n\\lim_{j\\to \\infty }\\norm{\\psi_k^{1\/p} f_j  }_{L^{p}(\\mi_{(\\zeta_j,z_j)})}= \\norm{\\psi_k^{1\/p} f }_{L^{p}(\\mi')}\n\\]\nby the previous remarks. Therefore, by Fatou's lemma,\n\\[\n\\begin{split}\n0&= \\lim_{j\\to \\infty }\\norm{f_j B^{\\vect{s_2}}_{(\\zeta,z)}(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho)  }_{L^{p}(\\mi_{(\\zeta_j,z_j)})}^{p}\\\\\n&= \\lim_{j\\to \\infty }\\sum_{k\\in K}\\norm{\\psi_k^{1\/p} f_j B^{\\vect{s_2}}_{(\\zeta,z)}(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho)  }_{L^{p}(\\mi_{(\\zeta_j,z_j)})}^{p}\\\\\n&\\Meg \\sum_{k\\in K}\\norm{\\psi_k^{1\/p} f B^{\\vect{s_2}}_{(\\zeta,z)}(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho) }_{L^{p}(\\mi')}^{p}\\\\\n&= \\norm{f B^{\\vect{s_2}}_{(\\zeta,z)}(\\Delta_\\Omega^{\\vect{s_3}}\\circ \\rho) }_{L^{p}(\\mi')}^{p}.\n\\end{split}\n\\]\nSince the support of $\\mi'\\in W_{p[\\vs-(\\vb+\\vd)\/p_1]}(\\mi)$ is a set\nof uniqueness for $A^{p_1,q_1}_{\\vs}(D)$, this implies that $f=0$,\nwhich is absurd, since $\\abs{f(0,i e_\\Omega)}\\Meg \\eps$. \n\n\\textsc{Step II.} We now prove the assertion for general $(\\zeta,z)\\in D$ and for $\\mi'=\\mi_{(\\zeta',z')}$, $(\\zeta',z')\\in D$. Define $\\psi_{(\\zeta,z),(\\zeta',z')}\\coloneqq  \\phi_{(\\zeta',z')}\\circ \\phi_{\\phi^{-1}_{(\\zeta',z')}(\\zeta,z)}$.\nThen, take $f\\in U_\\eps(\\zeta,z)$, and observe that $f\\circ \\psi_{(\\zeta,z),(\\zeta',z')}\\in U_\\eps(0,i e_\\Omega)$, since $B^{\\vect{s_2}}_{(\\zeta,z)}\\circ \\psi_{(\\zeta,z),(\\zeta',z')}= \\Delta_\\Omega^{\\vect{s_2}}(\\rho(\\zeta,z)) B^{\\vect{s_2}}_{(0,i e_\\Omega)}$ (use Lemma~\\ref{lem:5} again).\nApplying \\textsc{step I} to  $f\\circ \\psi_{(\\zeta,z),(\\zeta',z')}$ then yields the result, by means of another application of Lemma~\\ref{lem:5}.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{lem:15}\nTake $p\\in ]1,\\infty[$, $\\vect{s_1},\\vect{s_2},\\vect{s_3}\\in \\R^r$,\nand  $\\mi_1,\\mi_2\\in\\cM_+(D)$ such that the\nfollowing hold: \n\\begin{enumerate}\n\\item[{\\em(1)}] the mapping $M_1(\\mi_j)\\in L^{\\infty,\\infty}_{\\vb+\\vd-\\vect{s_j}}(D)$ for $j=1,2$;\n\n\\item[{\\em(2)}] $\\frac{1}{p'}\\vect{s_1}+\\frac 1 p \\vect{s_2}=\\vb+\\vd-\\vect{s_3}$;\n\n\\item[{\\em(3)}] $\\frac{1}{p'}\\vect{s_1}\\in \\frac{1}{2 p'}\\vect m+\\frac{1}{2 p}\\vect{m'}+(\\R_+^*)^r$;\n\n\\item[{\\em(4)}] $\\frac{1}{p}\\vect{s_2}\\in \\frac{1}{2 p}\\vect m+\\frac{1}{2 p'}\\vect{m'}+(\\R_+^*)^r$;\n\\end{enumerate}\nThen, the mapping\n\\[\nT\\colon C_c(D) \\ni f \\mapsto  \\int_D f(\\zeta,z) \\abs{B_{(\\zeta,z)}^{\\vect{s_3}}}\\,\\dd \\mi_1(\\zeta,z) \n\\]\ninduces a continuous linear mapping of $L^p(\\mi_1)$ into $L^p(\\mi_2)$.\n\\end{lem}\n\n\\begin{proof}\nDefine\n\\[\nT'\\colon C_c(D) \\ni f \\mapsto  \\int_D f(\\zeta,z) \\abs{B_{(\\zeta,z)}^{\\vect{s_3}}}\\,\\dd \\mi_2(\\zeta,z).\n\\]\nObserve that our assumptions and~\\cite[Theorem 2.47 and Corollary\n2.49]{CalziPeloso} imply that for every $\\vs',\\vect{s_5}\\in \\R^r$\nthere is a constant $C_1>0$ such that \n\\[\nT(\\Delta_\\Omega^{p'\\vs'}\\circ \\rho)(\\zeta,z)\\meg C_1 \\norm{B^{\\vect{s_3}}_{(\\zeta,z)}}_{A^{1,1}_{\\vect{s_1}+p'\\vs'}(D)}\n\\]\nand\n\\[\nT(\\Delta_\\Omega^{p\\vect{s_5}}\\circ \\rho)(\\zeta,z)\\meg C_1 \\norm{B^{\\vect{s_3}}_{(\\zeta,z)}}_{A^{1,1}_{\\vect{s_2}+p\\vect{s_5}}(D)}\n\\]\nfor every $ (\\zeta,z)\\in D$. In addition, by~\\cite[Proposition 2.41]{CalziPeloso}, there are two constants $C_2,C_3>0$ such that\n\\[\n\\norm{B^{\\vect{s_3}}_{(\\zeta,z)}}_{A^{1,1}_{\\vect{s_1}+p'\\vs'}(D)}=C_2 \\Delta_\\Omega^{\\vect{s_1}+\\vect{s_3}+p'\\vs'-(\\vb+\\vd)}(\\rho(\\zeta,z))\n\\]\nand\n\\[\n\\norm{B^{\\vect{s_3}}_{(\\zeta,z)}}_{A^{1,1}_{\\vect{s_2}+p\\vect{s_5}}(D)}=C_3 \\Delta_\\Omega^{\\vect{s_2}+\\vect{s_3}+p\\vect{s_5}-(\\vb+\\vd)}(\\rho(\\zeta,z))\n\\]\nfor every $(\\zeta,z)\\in D$ if and only if the following conditions are satisfied:\n\\begin{itemize}\n\\item[(i)] $p'\\vs'\\in \\frac 1 2 \\vect m-\\vect{s_1}+(\\R_+^*)^r$;\n\n\\item[(ii)] $\\vect{s_3}\\in \\vb+\\vd-\\frac1 2 \\vect{m'}-(\\R_+^*)^r$;\n\n\\item[(iii)] $ p' \\vs'\\in \\vb+\\vd-\\frac 1 2 \\vect{m'}-\\vect{s_1}-\\vect{s_3}-(\\R_+^*)^r$;\n\n\\item[(iv)] $p\\vect{s_5}\\in \\frac 1 2 \\vect m-\\vect{s_2}+(\\R_+^*)^r$;\n\n\\item[(v)] $ p \\vect{s_5}\\in \\vb+\\vd-\\frac 1 2 \\vect{m'}-\\vect{s_2}-\\vect{s_3}-(\\R_+^*)^r$.\n\\end{itemize}\nIn addition, \n\\[\n\\vect{s_1}+\\vect{s_3}+p'\\vs'-(\\vb+\\vd)= p'\\vect{s_5} \\qquad \\text{and} \\qquad \\vect{s_2}+\\vect{s_3}+p\\vect{s_5}-(\\vb+\\vd)=p \\vect{s_4}\n\\]\nif and only if\n\\[\n\\vect{s_5}=\\vect{s_4}+\\frac 1 p (\\vb+\\vd-\\vect{s_2}-\\vect{s_3}) \\qquad \\text{and} \\qquad \\frac{1}{p'}\\vect{s_1}+\\frac 1 p \\vect{s_2}=\\vb+\\vd-\\vect{s_3}.\n\\]\nIf these conditions are satisfied, then conditions (iv) and (v) become:\n\\begin{itemize}\n\\item[(iv$'$)] $p \\vect{s_4}\\in \\vect{s_3}-(\\vb+\\vd)+\\frac 1 2 \\vect m+(\\R_+^*)^r$;\n\n\\item[(v$'$)] $ p\\vect{s_4}\\in -\\frac{1}{2}\\vect{m'}-(\\R_+^*)^r$.\n\\end{itemize}\nBy our assumptions, we may take $\\vect{s_4}, \\vect{s_5}$ such that all the preceding conditions hold, so that the assertion follows by means of Schur's lemma (cf., e.g.,~\\cite[Lemma of I.2]{Grafakos}).\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:11}\nTake $p,q\\in ]0,\\infty[$, with $q<p$, and take $\\vect{s_1}, \\vect{s'_2},\\vect{s_3}\\in \\R^r$ such that the following hold:\n\\begin{enumerate}\n\\item $\\vect{s_1}-\\vect{s_3}\\in \\left( \\frac{1}{2 q}-\\frac{1}{2 p}  \\right)\\vect m+\\frac{1}{2 p}\\vect{m'}+(\\R_+^*)^r$;\n\n\\item $\\vect{s_1}+\\vect{s_2}-\\vect{s_3}\\in \\frac{1}{q}(\\vb+\\vd)-\\frac{1}{2 p}\\vect m-\\left( \\frac{1}{2 q}-\\frac{1}{2p}  \\right)\\vect{m'}-(\\R_+^*)^r$.\n\\end{enumerate}\nFor every $\\eps>0$ and for every $f\\in \\Hol(D)$, define\n\\[\nB_{\\eps,f}^{q,\\vect{s_1},\\vect{s_2}}\\coloneqq \\Set{ (\\zeta,z)\\in D\\colon \\abs{f(\\zeta,z)}\\meg \\eps  \\Delta_\\Omega^{(\\vb+\\vd)\/q-\\vect{s_1}-\\vect{s_2}}(\\rho(\\zeta,z))\\norm{f B^{\\vect{s_2}}_{(\\zeta,z)}}_{A^{q,q}_{\\vect{s_1}}(D)}}.\n\\] \nThen, there is a constant $C>0$ such that\n\\[\n\\norm{\\chi_{B_{\\eps,f}^{q,\\vect{s_1},\\vect{s_2}}} f}_{L^{p,p}_{\\vect{s_3}}(D)}\\meg C\\eps \\norm{f}_{A^{p,p}_{\\vect{s_3}}(D)}\n\\]\nfor every $\\eps>0$ and for every $f\\in A^{p,p}_{\\vect{s_3}}(D)$.\n\\end{lem}\n\nThis extends~\\cite[Lemma 2]{Luecking5}, which deals with the case in which $D$ is the unit disc in $\\C$. \n\n\n\\begin{proof}\nIt will suffice to prove that the operator\n\\[\nT\\colon f \\mapsto \\left[(\\zeta,z)\\mapsto\n\\Delta_\\Omega^{\\vb+\\vd-q_2(\\vect{s_1}+\\vect{s_2})}(\\rho(\\zeta,z))\n\\int_D f(\\zeta',z')\n\\abs{B^{q_2\\vect{s_2}}_{(\\zeta,z)}(\\zeta',z')}\\Delta_\\Omega^{q_2\\vect{s_1}-(\\vb+\\vd)}(\\rho(\\zeta',z'))\\,\\dd\n\\nu_D(\\zeta',z')   \\right] \n\\]\ninduces a continuous linear mapping  of\n$L^{p\/q,p\/q}_{q_2\\vect{s_3}}(D)$ into itself. This follows from\nLemma~\\ref{lem:15}. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{prop:27}.]\n\nSet $\\ell\\coloneqq p\/q$ and $\\vect{s''}\\coloneqq \\ell \\vs$.\nObserve that Lemma~\\ref{lem:14} implies that there is a constant $C'>0$ such that\n\\[\n\\norm{f B^{\\vect{s'}}_{(\\zeta,z)}\n}_{L^{q}(\\mi)}\\Meg C'\\norm{f B^{\\vect{s'}}_{(\\zeta,z)}}_{A^{q,q}_{\\vect {s''}}(D)}\n\\]\nfor every $(\\zeta,z)\\in D$ and for every $f\\in \\Hol(D)$ such that\n\\[\n\\abs{f(\\zeta,z)}>\\eps \\Delta_\\Omega^{(\\vb+\\vd)\/q-\\vect{s''}-\\vect{s'}}(\\rho(\\zeta,z))\\norm{f B^{\\vect{s'}}_{(\\zeta,z)}}_{A^{q,q}_{\\vect {s''}}(D)},\n\\]\nthat is, for every $f\\in \\Hol(D)$ and for every $(\\zeta,z)\\in D\\setminus B_{f,\\eps}^{q,\\vect {s''},\\vect{s'}}$, with the notation of Lemma~\\ref{lem:11}.\nIn addition, by~\\cite[Proposition 3.2]{CalziPeloso}, there is a constant $C''>0$ such that\n\\[\n\\norm{f}_{A^{\\infty,\\infty}_{\\vect{s''}-(\\vb+\\vd)\/q}(D)}\\meg C''\\norm{f}_{A^{q,q}_{\\vect {s''}}(D)}\n\\]\nfor every $f\\in A^{q,q}_{\\vect {s''}}(D)$, so that\n\\[\n\\norm{f B^{\\vect{s'}}_{(\\zeta,z)}\n}_{L^{q}(\\mi)}\\Meg \\frac{C'}{C''} \\Delta_\\Omega^{\\vect{s'}+\\vect {s''}-(\\vb+\\vd)\/q}(\\rho(\\zeta,z))\\abs{f(\\zeta,z)}\n\\]\nfor every $f\\in \\Hol(D)$ and for every $(\\zeta,z)\\in D\\setminus B_{f,\\eps}^{q,\\vect {s''},\\vect{s'}}$. By Lemma~\\ref{lem:11}, we may choose $\\eps$ so small that\n\\[\n\\norm{f}_{A^{p,p}_{\\vs}(D)}\\meg 2 \\norm{\\chi_{D\\setminus B_{f,\\eps}^{q,\\vect {s''},\\vect{s'}}}f}_{L^{p,p}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$, provided that\n\\[\n\\vect{s'}+ \\vect{s''}-\\vs\\in \\frac{1}{q}(\\vb+\\vd) -\\frac{1}{2p} \\vect{m}-\\left( \\frac{1}{2 q}-\\frac{1}{2 p}  \\right)\\vect{m'}-(\\R_+^*)^r.\n\\]\nTherefore,\n\\[\n\\norm{f }_{A^{p,p}_{\\vs}(D)}\\meg \\frac{2}{C' C''} \\norm*{(\\zeta,z)\\mapsto \\norm{f B^{\\vect{s'}}_{(\\zeta,z)} \n}_{L^{q}(\\mi)}  }_{L^{p,p}_{\\vs-\\vect{s'}-\\vect{s''}+(\\vb+\\vd)\/q}(D)}\n\\]\nfor every $f\\in A^{p,p}_{\\vs}(D)$.  Hence, it will suffice to show that the linear mapping\n\\[\nT\\colon f\\mapsto \\left[(\\zeta,z)\\mapsto  \\int_D f \\abs{ B^{q\\vect{s'}}_{(\\zeta,z)} }\\,\\dd \\mi  \\right]\n\\]\ninduces a continuous linear mapping from $L^{p \/q}(\\mi)$ into $L^{p\/q}((\\Delta_\\Omega^{p[\\vs-\\vect{s'}-\\vect{s''}+(1\/q-1\/p)(\\vb+\\vd) ]  }\\circ \\rho)\\cdot \\nu_D )$. \nBy Lemma~\\ref{lem:15}, this is the case if the following conditions are satisfied:\n\\begin{itemize}\n\\item[(i)] $\\frac{p}{\\ell'}\\vs\\in \\frac{1}{2 \\ell'}\\vect m+\\frac{1}{2 \\ell}\\vect{m'}+(\\R_+^*)^r $;\n\n\\item[(ii)] $q \\vect{s'}+\\frac{p}{\\ell'}\\vs\\in \\vb+\\vd- \\frac{1}{2 \\ell}\\vect m+\\frac{1}{2 \\ell'}\\vect{m'}-(\\R_+^*)^r $.\n\\end{itemize}\nThis is the case if $\\vect{s'}$ is sufficiently small.\n\\end{proof}\n\nFinally, we provide a sufficient condition for   a measure\n$\\mi\\in\\cM_+(D)$ to be a reverse Carleson, but not necessarily\nCarleson, measure for the weighted Bergman space $A^{p,q}_\\vs(D)$. A\nnecessary condition  (for sampling measures only)   has already been provided \nin Proposition~\\ref{prop:32}. \n\n\\begin{teo}\\label{prop:29}\nTake $p_1,q_1,  p \\in ]0,\\infty]$ with $  p <\\infty$, $R_0>1$, and\n$\\vs\\in \\frac{1}{2 q_1}\\vect m+(\\R_+^*)^r$ if $q_1<\\infty$, while\n$\\vs\\in \\R_+^r$ if $q_1=\\infty$. Define $  p^* \\coloneqq\np_1\/(  p -p_1)_+$ and $  q^* \\coloneqq q_1\/(  p -q_1)_+$. \n\nThen, there are $\\delta_0>0$ and a constant $C>0$ such that the\nfollowing hold. For every $(\\delta,R)$-lattice\n$(\\zeta_{j,k},z_{j,k})_{j\\in J,k\\in K}$ on $D$, with $\\delta\\in\n]0,\\delta_0]$ and $R\\in ]1,R_0]$, for every Borel partition\n$(B_{j,k})_{(j,k)\\in J\\times K}$ of $D$ such that \n\\[\nB((\\zeta_{j,k},z_{j,k}),\\delta)\\subseteq B_{j,k}\\subseteq B((\\zeta_{j,k},z_{j,k}),R\\delta)\n\\]\nfor every $(j,k)\\in J\\times K$, and for every  $\\mi\\in\\cM_+(D)$ such that\n\\[\nC'\\coloneqq \\norm*{  \\bigg( \\frac{\\Delta_\\Omega^{  p [\\vs-(\\vect\nb+\\vd)\/p_1]}(h_k)}{\\mi(B_{j,k})} \\bigg)_{j,k}\n}_{\\ell^{  p^*, q^* }(J,K)}  <\\infty,\n\\]\none has\n\\[\n\\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\\meg  C C'^{1\/  p} \\delta^{(2n+m)\/p_1+m\/q_1}  \\norm{f}_{L^{  p}(\\mi)}\n\\]\nfor every $f\\in A^{\\infty,\\infty}_{\\vs-(\\vb+\\vd)\/p_1}(D)$, where $h_k\\coloneqq \\rho(\\zeta_{j,k},z_{j,k})$ for every $(j,k)\\in J\\times K$.\n\\end{teo}\n\n\\begin{oss}\nWe point out  that the condition $C'<\\infty$ is sufficient, but far from\nbeing necessary.   As a matter of fact, using Theorem~\\ref{prop:31} one may easily construct reverse Carleson measures which vanish on any given compact subset of $D$. Moreover, the condition\n$f\\in A^{\\infty,\\infty}_{\\vs-(\\vb+\\vd)\/p_1}(D)$ can be relaxed (in the spirit of~\\cite[Theorem 3.23]{CalziPeloso})  but not\neliminated, as observed  in~\\cite{Luecking}.\n\\end{oss}\n\n\n\\begin{proof}\nBy~\\cite[Theorem 3.23]{CalziPeloso}, we may take $\\delta_0>0$ and $C_1>0$ so that, if we define \n\\[\nS_-\\colon \\Hol(D) \\ni f \\mapsto \\left( \\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p_1}(h_k) \\min_{\\overline\nB((\\zeta_{j,k},z_{j,k}),R\\delta)}\\abs{f} \\right)_{j,k}\\in\n\\R_+^{J\\times K}, \n\\]\nthen\n\\[\n\\frac{1}{C_1}\\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\\meg\\delta^{(2n+m)\/p_1+m\/q_1}\n\\norm{S_- f}_{\\ell^{p_1,q_1}(J,K)}\n\\meg C_1\\norm{f}_{A^{p_1,q_1}_{\\vs}(D)}\n\\]\nfor every $f\\in A^{\\infty,\\infty}_{\\vs-(\\vb+\\vd)\/p_1}(D)$ and for every lattice as in the statement.\nThen, take $f\\in A^{\\infty,\\infty}_{\\vs-(\\vb+\\vd)\/p_1}(D)$, and observe that, by H\\\"older's inequality,\n\\[\n\\begin{split}\n&\\norm*{  \\bigg( \\frac{\\Delta_\\Omega^{  p  [\\vs-(\\vb+\\vd)\/p_1]}(h_k)}{\\mi(B_{j,k})} \\bigg)_{j,k}  }^{1\/  p}_{\\ell^{  p^*,q^*}(J,K)} \\norm{f}_{L^{q_2}(\\mi)}\\\\\n&\\qquad\\Meg \\norm*{  \\bigg( \\frac{\\Delta_\\Omega^{\\vs-(\\vb+\\vd)\/p_1}(h_k)}{\\mi(B_{j,k})^{1\/  p}} \\bigg)_{j,k}  }_{\\ell^{  p p^*, p q^*}(J,K)} \\norm*{\\big(\\Delta_\\Omega^{(\\vb+\\vd)\/p_1-\\vs}(h_k) \\mi(B_{j,k})^{1\/  p}  (S_- f)_{j,k} \\big)_{j,k}  }_{\\ell^{  p}(J\\times K)}\\\\\n&\\qquad\\Meg \\norm{S_- f}_{\\ell^{p_1,q_1}(J,K)} \\\\\n&\\qquad\\Meg \\frac{1}{C_1\\delta^{(2n+m)\/p_1+m\/q_1} }  \\norm{f}_{A^{p_1,q_1}_{\\vs}(D)},\n\\end{split}\n\\]\nwhence the result.\n\\end{proof}\n\n\nWe conclude this section with some rather `pathological' examples of non-Carleson reverse Carleson measures for the spaces $A^{p,q}_s(\\C_+)$ which fail to satisfy the preceding sufficient condition.\n\n\\begin{oss}\nTake $q\\in ]0,\\infty]$ and $p\\in ]0,\\infty[$. In addition, fix $a<b$ in $\\R$ and $\\nu\\in \\cM_+(\\R_+^*)$ such that $\\nu(]0,1])=\\infty$. Define\n\\[\n\\mi\\coloneqq (\\chi_{[a,b]}\\cdot \\Hc^1)\\otimes \\nu,\n\\]\nand observe that $\\mi$ is a positive Radon measure on $\\C_+\\cong\\R\\times \\R_+^*$. Take a non-zero $f\\in H^q$, and let us prove that $\\norm{f}_{L^p(\\mi)}=\\infty$. Indeed, the function \n\\[\nf^*\\colon \\R\\ni x\\mapsto \\lim_{y\\to 0^+} f(x+i y)\\in \\C\n\\]\nis well-defined and non-zero almost everywhere (cf.~\\cite[Corollary to Theorem 11.1]{Duren2}). Hence, by Fatou's lemma,\n\\[\n\\liminf_{y\\to 0^+} \\int_a^b \\abs{f(x+i y)}^p\\,\\dd x\\Meg\\int_a^b \\abs{f^*(x)}^p\\,\\dd x>0,\n\\]\nso that clearly\n\\[\n\\int_{\\C_+}\\abs{f}^p\\,\\dd \\mi=+\\infty.\n\\]\nBy the arbitrariness of $f$, it is clear that $\\mi$ is (trivially) reverse $p$-Carleson for $H^q$.\n\\end{oss}\n\nIn order to extend the preceding example to the spaces $A^{p,q}_s(\\C_+)$ for $s>0$, we need the following lemma.\n\n\\begin{lem}\\label{lem:4}\nTake a non-zero $u\\in \\Sc'(\\R)$. If the Fourier transform of $u$ is supported in a half-line, then $\\Supp{u}=\\R$.\n\\end{lem}\n\n\\begin{proof}\nUp to replacing $u$ with $\\ee^{i \\xi\\,\\cdot\\,} u(\\eps\\,\\cdot\\,)$ for\nsome $\\xi\\in \\R$ and some $\\eps\\in \\Set{-1,1}$, we may assume that the\nFourier transform of $u$ is supported in $\\R_+$. Now, take a non-zero\n$g\\in \\Sc(\\R)$ whose Fourier transform is supported in $\\R_+$, and\nobserve that $g u$  is non-zero, and has a Fourier transform\nsupported in $\\R_+$, thanks to~\\cite[Theorems XIII and XIV of Chapter\nVI]{Schwartz}.    \nNow, fix $\\phi\\in C^\\infty_c(\\R)$ such that $\\int_\\R \\phi(x)\\,\\dd x=1$ and\n$\\Supp{\\phi}\\subseteq [-1,1]$, and define $\\phi_j\\coloneqq\n2^{j}\\phi(2^{j}\\,\\cdot\\,)$ for every $j\\in \\N$. Then, it is easily verified that $(g u)*\\phi_j$belongs to $\\Sc(\\R)$ and that its Fourier transform is supported in $\\R_+$ for every $j\\in \\N$. In addition, $(g u)*\\phi_j\\to g u$ in $\\Sc'(\\R)$,\nso that  $(g u)*\\phi_j\\neq 0$ if $j$ is sufficiently\nlarge. Therefore,~\\cite[Corollary to Theorem 11.1, and Theorem\n11.9]{Duren2} imply that $(g u)*\\phi_j$ is non-zero almost everywhere\nif $j$ is sufficiently large. \nNow, assume by contradiction that $\\Supp{u}\\neq \\R$, and take $a,b\\in\n\\R$ such that $a<b$ and $u$ vanishes on $]a,b[$. Then, clearly $(g\nu)*\\phi_j$ vanishes on $]a+2^{-j},b-2^{-j} [$, which is not empty if\n$b-a>2^{1-j}$, that is, if $j$ is sufficiently large: contradiction. \n\\end{proof}\n\n\\begin{oss}\nTake  $p_1,q_1\\in ]0,\\infty]$ and $s,p\\in ]0,\\infty[$. In addition, fix $a<b$ in $\\R$ and $\\nu\\in \\cM_+(\\R_+^*)$ such that $\\nu(]0,1])=\\infty$. Define\n\\[\n\\mi\\coloneqq (\\chi_{[a,b]}\\cdot \\Hc^1)\\otimes \\nu,\n\\]\nand observe that $\\mi$ is a positive Radon measure on $\\C_+\\cong\\R\\times \\R_+^*$. Take a non-zero $f\\in A^{p_1,q_1}_s$, and let us prove that $\\norm{f}_{L^q(\\mi)}=\\infty$. \nTake $(g^{(\\eps)})$ as in~\\cite[Lemma 1.22]{CalziPeloso}, and observe that $g^{(\\eps)}f\\in A^{1,1}_{s+(1\/p_1-1)_+}$ for every $\\eps>0$ (cf.~\\cite[Proposition 3.2]{CalziPeloso}). Hence,~\\cite[Theorem 1.7]{BekolleBonamiGarrigosRicci} implies that $(g^{(\\eps)}f)_y$ converges to some non-zero $f_0$ in $\\Sc'(\\R)$, for $y\\to 0^+$, and that the support of the Fourier transform of $f_0$ is contained in $\\R_+$. Therefore, Lemma~\\ref{lem:4} implies that $\\Supp{f_0}=\\R$, so that clearly \n\\[\n0<\\liminf_{y\\to 0^+} \\int_a^b \\abs{(g^{(\\eps)}f)(x+i y)}^q\\,\\dd x\\meg \\liminf_{y\\to 0^+} \\int_a^b \\abs{f(x+i y)}^q\\,\\dd x.\n\\]\nHence,\n\\[\n\\int_{\\C_+}\\abs{f}^q\\,\\dd \\mi=+\\infty.\n\\]\nBy the arbitrariness of $f$, it is clear that $\\mi$ is (trivially) reverse $p$-Carleson for $A^{p_1,q_1}_s$.\n\\end{oss}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn this paper, we study the optimal control for the following stochastic\nevolution equation with jumps\n\\begin{eqnarray} \\label{eq:1.1}\n  \\left\\{\n  \\begin{aligned}\n   d X (t) = & \\ [ A (t) X (t) + b ( t, X (t), u(t)) ] d t\n+ [B(t)X(t)+g( t, X (t), u(t)) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t, e,X(t-),u(t))\\tilde \\mu(de,dt),  \\\\\nX (0) = & \\  x , \\quad t \\in [ 0, T ]{\\color{blue},}\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nwith the cost functional\n\\begin{equation}\\label{eq:1.2}\nJ(u(\\cdot))= {\\mathbb E} \\bigg [ \\int_0^T l ( t, X (t), u (t) ) d t\n+ \\Phi ( X (T) ) \\bigg ],\n\\end{equation}\nand  state constraint\n\\begin{equation} \\label{eq:3.3}\n{\\mathbb E} [ \\phi(X(T)) ] =0 ,\n\\end{equation}\nin the framework of a Gelfand triple $V \\subset H= H^*\\subset V^{*},$ where $ H$ and  $V$ are\ntwo given Hilbert spaces. Here on  a given  filtrated probability space  $(\\Omega, \\mathscr{F},\\{\n{\\mathscr F}_t\\}_{0\\leq t\\leq T}, P),$ $W$  is  a  one-dimensional  Brownian motion  and  $\\tilde \\mu$\nis a Poisson random martingale measure on a fixed nonempty Borel measurable\nsubset ${E}$ of $\\mathbb R^1,$\n$A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$, $B\n  : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H ),$ $b:[0,T]\\times  \\Omega\\times H\n   \\times  U_{ad}\\longrightarrow H$, $g:[0,T]\\times\\Omega\\times H\n   \\times  U_{ad} \\longrightarrow H$ and $\\sigma:[0,T]\\times  \\Omega \\times\n   E\\times H  \\times  U_{ad}\\longrightarrow H$  are given random mappings, where the control variable $u$ takes value in a nonempty convex subset $U_{ad}$ of a real Hilbert space $U$. Here we denote by  $\\mathscr{L}(V,V^*)$ the space of bounded\nlinear transformations of V into $V^*$, by ${\\mathscr L} (V, H)$   the space of bounded\nlinear transformations of  $H$ into $V.$\n    An adapted solution of\n   \\eqref{eq:1.1} is  a $V$-valued, $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$-adapted process $X(\\cdot)$  which satisfies \\eqref{eq:1.1} under some appropriate sense.\n   The optimal control problem is to find an admissible control to minimize the cost functional  \\eqref{eq:1.2}\n over  the set of admissible controls.\n\n\nOne of the basic method to solve  stochastic optimal control\nproblems is the stochastic maximum principle whose objective is to\nestablish necessary (as well as sufficient) optimality conditions of\ncontrols. For  optimal control problems of infinite dimensional\nstochastic systems, many works are concerned with the\nstochastic systems and the corresponding stochastic maximum\nprinciples, see e.g.( \\cite{HuPe901, Ben830, Zhou93,  Alhu10,\nAlhu101, Alhu111, Alhu112, LuZh12,Guat11, DuMeng}.\n\n In contrast, there have not been\n  a\n number of   results on  the optimal control for stochastic partial\ndifferential equations driven by jump processes. In 2005,  {\\O}ksendal, Proske,  Zhang \\cite{Ok} studied the optimal control problem  of quasilinear semielliptic SPDEs driven by Poisson random measure   and gave sufficient maximum principle results, not necessary ones.  In 2017,  Tang and Meng \\cite{Tangmeng2016}  studied the optimal control problem\nfor a controlled  stochastic evolution equation \\eqref{eq:1.1} with the\ncost functional \\eqref{eq:1.2}, where  the  control domain is assumed to be  convex.  \\cite{Tangmeng2016} adopt the  convex variation method and the first adjoint duality analysis to show a necessary maximum principle.  And  Under the convexity assumption of the Hamiltonian and the terminal cost, a sufficient maximum principle for this optimal problem which is the so-called verification theorem is  obtained\n\n     The purpose of this paper is to  establish the\n maximum principle\n for the optimal control problem where the state process is driven by a  controlled stochastic evolution equation \\eqref{eq:1.1} with the cost\n functional \\eqref{eq:1.2} and the\n state constraint \\eqref{eq:3.3} by Ekland variational\nprinciple,  combining  the convex variation method\nand the duality technique.\n\n\nThe paper is organized as follows. In section 2 we formulate the problem and give\nvarious assumptions used throughout the paper. In section 3, we present a penalized optimal control problem. Section 4 is  devoted to derive necessary  optimality\nconditions in the form of stochastic maximum principles in a unified\nway. Some basic results on the SEE and the BSEE with jump are given in the Appendix which will been\nused in this paper.\n\\section{Problem formulation}\n\nIn this section, we introduce basic notation and standing assumptions, and state an optimal control\nproblem with state constraint under a  stochastic evolution equation with jumps\nin Hilbert space, which was considered by Tang and Meng\\cite{Tangmeng2016}.\n\n\nLet $(\\Omega, \\mathscr{F}, \\mathbb P)$ be a complete probability space\nequipped with a  one-dimensional standard Brownian motion $\\{W(t),\n0\\leq t\\leq T\\}$  and  a stationary Poisson point process\n$\\{\\eta_t\\}_{t\\geq 0}$ defined  on a fixed nonempty Borel measurable\nsubset ${E}$ of $\\mathbb R^1$.\n Denote by $\\mathbb E[\\cdot]$ the expectation\nunder the probability $\\mathbb P.$\n We denote by $\\mu(de,dt)$\n the counting measure induced by $\\{\\eta_t\\}_{t\\geq 0}$ and\n  by  $\\nu(de)$ the corresponding\n characteristic measure.  Then the compensatory\n random martingale measure is denoted by\n  $\\tilde{\\mu}(de, dt):={\\mu}(de,\ndt)-\\nu(de)dt$ which is assumed to be independent of the Brownian\nmotion  $\\{W(t),\n0\\leq t\\leq T\\}$.\n  Furthermore, we assume that\n$\\nu({E})<\\infty$. Let $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$ be the\nP-augmentation of the natural filtration generated by\n$\\{{W_t}\\}_{t\\geq 0}$ and $\\{\\eta_t\\}_{t\\geq 0}$.\n By  $\\mathscr{P}$  we denote the\npredictable $\\sigma$ field on $\\Omega\\times [0, T]$ and\nby $\\mathscr B(\\Lambda)$\n  the Borel $\\sigma$-algebra of any topological space $\\Lambda.$\n  Let $X$ be  a  separable Hilbert space with norm $\\|\\cdot\\|_X$.\n  Denote by  $M^{\\nu,2}( E; X)$ the set of all $X$-valued measurable\n  functions $r=\\{r(e), e\\in E\\}$ defined on the measure\n  space $(E, \\mathscr B(E); v)$\n  such that\n$\\|r\\|_{M^{\\nu,2}( E; X)}\\triangleq\n\\sqrt{{\\int_E\\|r(e)\\|_X^2v(de)}}<~\\infty,$ by\n${M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; X)}$ the  set of all\n$\\mathscr{P}\\times {\\mathscr B}(E)$-measurable $X$-valued processes\n$r=\\{r(t,\\omega,e),\\\n(t,\\omega,e)\\in[0,T]\\times\\Omega\\times E\\}$ such that\n$\\|r\\|_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; X)}}\\triangleq\n\\sqrt{{\\mathbb E\\bigg[\\displaystyle\\int_0^T\\displaystyle\\int_E\\displaystyle\\|r(t,e)\\|_X^2\n\\nu(de)dt\\bigg]}}<~\\infty,$\n by\n$M_{\\mathscr{F}}^2(0,T;X)$ the set of all ${\\mathscr{F}}_t$-adapted\n$X$-valued  processes $f=\\{f(t,\\omega),\\\n(t,\\omega)\\in[0,T]\\times\\Omega\\}$ such that\n$\\|f\\|_{M_{\\mathscr{F}}^2(0,T;X)}\n\\triangleq\\sqrt{\\mathbb E\\bigg[\\displaystyle\\int_0^T\\|f(t)\\|_X^2dt\\bigg]}<\\infty,$\nby  $S_{\\mathscr{F}}^2(0,T;X)$  the set of all\n${\\mathscr{F}}_t$-adapted  $X$-valued c\\`{a}dl\\`{a}g processes\n$f=\\{f(t,\\omega),\\ (t,\\omega)\\in[0,T]\\times\\Omega\\}$ such that\n$\\|f\\|_{S_{\\mathscr{F}}^2(0,T;X)}\\triangleq\\sqrt{\n\\mathbb E\\bigg[\\displaystyle\\sup_{0\n\\leq t \\leq T}\\|f(t)\\|_X^2}\\bigg]<+\\infty,$  by\n$L^2(\\Omega,{\\mathscr{F}},\\mathbb P;X)$ the set of all $X$-valued random\nvariables $\\xi$ on $(\\Omega,{\\mathscr{F}},\n\\mathbb P)$ such that\n$\\|\\xi\\|_{L^2(\\Omega,{\\mathscr{F}},\n\\mathbb P};X)\\triangleq\n\\sqrt{\\mathbb E[\\|\\xi\\|_X^2]}<\\infty.$\nThroughout this paper, we let  $C$ and $K$  be two generic positive constants, which may be different from line to line.\n\n\n\n\n\nIn what follows, we set up a Gelfand triple $(V, H, V^*)$, based on which the state process and the adjoint process is defined.\nIndeed, the state process is governed by a SEE with jumps, while the adjoint process is governed by a BSEE with jumps. We provide the existence, uniqueness\nand continuous dependence theorems for SEEs with jumps  and BSEEs with jumps in the appendix.\n\nLet $V$ and $H$ be two separable (real) Hilbert spaces such that $V$\nis densely embedded in $H$. We identify $H$ with its dual space by the Riesz mapping. Then we can take $H$ as a pivot space and get a Gelfand triple $(V, H,\nV^*)$ such that $V \\subset H = H^*\\subset V^{*}$. Let $(\\cdot,\\cdot)_H$ denote the inner product in $H$, and $\\la\\cdot,\\cdot\\ra$ denote the duality product between\n$V$ and $V^{*}$. Moreover, we write $\\mathscr{L}(V,V^*)$ for the space of bounded linear transformations of V into $V^*$.\n\n\n\nThe state process is governed by the following controlled SEE with jumps in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray} \\label{eq:4.1}\n  \\left\\{\n  \\begin{aligned}\n   d X (t) = & \\ [ A (t) X (t) + b ( t, X (t), u(t)) ] d t\n+ [B(t)X(t)+g( t, X (t), u(t)) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t, e,X(t-),u(t))\\tilde \\mu(de,dt),  \\\\\nX (0) = & \\  x , \\quad t \\in [ 0, T ]{\\color{blue},}\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nwhere the space of controls $U_{ad}$ is given by a nonempty closed convex subset of a separable real Hilbert space $U$.\n\n\\begin{definition}\nA stochastic process  $u(\\cdot)$ is an admissible control, if $u(t)\\in U_{ad}$ for almost $t\\in [0, T]$ and $ u(\\cdot)\\in\nM_{\\mathbb F}^2(0, T;U)$. The set of all admissible controls is denoted by ${\\cal A}$.\n\\end{definition}\n\nThe cost functional is given by\n\\begin{equation}\\label{eq:2.2}\nJ(u(\\cdot))= {\\mathbb E} \\bigg [ \\int_0^T l ( t, x (t), u (t) ) d t\n+ \\Phi ( x (T) ) \\bigg ].\n\\end{equation}\nWe assume that the control system (\\ref{eq:4.1})-(\\ref{eq:2.2}) is subject to the following state constraint\n\\begin{equation} \\label{eq:2.555}\n{\\mathbb E} [ \\phi(X(T)) ] =0 .\n\\end{equation}\nHere the coefficients $(A,B, b,g,\\sigma, l,\\Phi, \\phi)$ of the control system (\\ref{eq:4.1})-(\\ref{eq:3.3})\n\\begin{assumption}\\label{ass:2.5}\n\\begin{enumerate}\n\\item[]\n\\item[(i)]\n\nThe operator processes $A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B\n  : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n  are weakly predictable; i.e.,\n  $ \\langle A(\\cdot)x, y \\rangle$ and $(B(\\cdot)x, y)_H$\n  are both predictable process for every $x, y\\in V, $\n  and satisfy the coercive condition, i.e.,  there exist\n  some constants  $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and  each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n    \\begin{eqnarray}\n    \\begin{split}\n     - \\langle A(t)x, x \\rangle +\\lambda ||x||_H^2\\geq \\alpha\n      ||x||_V^2+||Bx||_H^2,\n    \\end{split}\n  \\end{eqnarray}\n  and  \\begin{eqnarray} \\label{eq:3.3}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\n\\item[(ii)]$b, g: [ 0, T ] \\times \\Omega \\times H \\times {\\mathscr U} \\rightarrow H$  are $\\mathscr P\\times\n   \\mathscr B(H)\\times \\mathscr B(\\mathscr U)\/\\mathscr B(H) $ measurable\n   mappings and $\\sigma:[0,T]\\times  \\Omega \\times\n   E\\times H \\times \\mathscr U \\longrightarrow H$ is a $\\mathscr P\\times \\mathscr B(E)\\times\n   \\mathscr B(H)\\times \\mathscr B(U)\/\\mathscr B(H)$-measurable mapping  such that $b ( \\cdot, 0, 0 ), g ( \\cdot, 0, 0 ) \\in {\nM}^2_{\\mathscr F} ( 0, T; H ), \\sigma(\\cdot, \\cdot, 0,0)\\in {M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; H)}.$ Moreover, for almost all $( t, \\omega, e ) \\in [ 0, T ] \\times \\Omega \\times E$,  $b$, $g$ and $\\sigma$ are  G\\^ateaux differentiable in $(x,u)$ with  continuous bounded G\\^ateaux  derivatives\n$b_x, g_x,\\sigma_x,  b_u, g_u$ and  $\\sigma_u$;\n\\item[(iii)]\n$l:[ 0, T ] \\times \\Omega \\times H \\times {\\mathscr U} \\rightarrow \\mathbb R $ is a\n ${\\mathscr P} \\otimes {\\mathscr B} (H) \\otimes {\\mathscr B} ({\\mathscr U})\/ {\\mathscr B} ({\\mathbb R})$-measurable mapping\n  and  $\\Phi,\\phi: \\Omega \\times H \\rightarrow {\\mathbb R}$\nis  a ${\\mathscr F}_T\\otimes {\\mathscr B} (H) \/ {\\mathscr B} ({\\mathbb R})$-measurable\nmapping.\nFor almost all $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$, $l$ is continuous G\\^ateaux\ndifferentiable in $(x,u)$\nwith continuous  G\\^ateaux derivatives $l_x$ and $l_u$, and $\\Phi$ and $\\phi$\n are G\\^ateaux differentiable in $x$\nwith  continuous\n G\\^ateaux derivative $\\Phi_x$ and\n $\\phi_x$.\nMoreover, for almost all $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$,   there exists a   constant $C > 0$ such that  for all $( x, u ) \\in H  \\times {\\mathscr U}$\n\\begin{eqnarray*}\n| l ( t, x, u  ) |\n\\leq  C ( 1 + \\| x \\|^2_H + + \\| u \\|_U^2 ) ,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n&& \\| l_x ( t, x,u) \\|_H +\n+ \\| l_u ( t, x, u ) \\|_U \\leq C ( 1 + \\| x \\|_H  + \\| u \\|_U  ) ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n& | \\Phi (x) | \\leq C ( 1 + \\| x \\|^2_H) , | \\phi (x) | \\leq C ( 1 + \\| x \\|^2_H)  \\\\\n& \\| \\Phi_x (x) \\|_H \\leq C ( 1 + \\| x \\|_H),\\| \\phi_x (x) \\|_H \\leq C ( 1 + \\| x \\|_H).\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{assumption}\n\n\n\n\n\nUnder Assumption \\ref{ass:2.5}, it can be shown from Lemma \\ref{thm:3.1} that for any $u(\\cdot)\\in {\\cal A},$\nthe state equation \\eqref{eq:4.1} admits a unique solution $X(\\cdot)\\in M_{\\mathscr{F}}^2(0,T;V)\\bigcap S_{\\mathscr{F}}^2(0,T;H)$. We also denote this solution as\n$X^u(\\cdot)$ whenever we want to emphasis its dependence on the control $u(\\cdot)$. Then we call\n$X(\\cdot)$ the state process corresponding to the control process $u(\\cdot)$ and $(u(\\cdot); X(\\cdot))$\nthe admissible pair. Furthermore, from  Assumption \\ref{ass:2.5} and the a priori estimate \\eqref{eq:3.4}, we can easily validate that\n\\begin{eqnarray*}\n|J(u(\\cdot))|<\\infty.\n\\end{eqnarray*}\n\nNow we state formally the optimal control problem\n\\begin{problem}\\label{pro:2.1}\nFind an admissible control $\\bar{u}(\\cdot)$ such that\n\\begin{eqnarray*}\\label{eq:b7}\nJ(\\bar{u}(\\cdot))=\\inf_{u(\\cdot)\\in {\\cal A}}J(u(\\cdot)),\n\\end{eqnarray*}\nsubject to \\eqref{eq:4.1} and \\eqref{eq:2.555}, where the cost functional is given by \\eqref{eq:2.2}.\n\\end{problem}\n\nAny $\\bar{u}(\\cdot)\\in {\\cal A}$ satisfying the above is called an optimal control process of Problem \\ref{pro:2.1};\nthe corresponding state process $\\bar{X}(\\cdot)$ is called an optimal state process; correspondingly,\n$(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$ is called an optimal pair of Problem \\ref{pro:2.1}.\n\n\\section{Penalized optimal control problem}\n\nIn this section, we relate the original constrained control problem with one without state constraint.\n\nThe results relies on the following Ekeland's principle.\n\n\\begin{lemma}[Ekeland's principle, \\cite{ekeland1974variational}]\nLet $(S,d)$ be a complete metric space and $\\rho (\\cdot ):S\n\\rightarrow {\\mathbb R}$ be lower-semicontinuous and bounded from\nbelow. For $\\varepsilon \\geq 0$, suppose $u^{\\varepsilon }\\in S$\nsatisfies\n\\begin{equation*}\n\\rho ( u^\\varepsilon ) \\leq \\inf_{u\\in S} \\rho (u) + \\varepsilon .\n\\end{equation*}\nThen for any $\\lambda >0$, there exists $u^\\lambda \\in S$ such that\n\\begin{eqnarray*}\n\\rho ( u^\\lambda ) \\leq \\rho ( u^\\varepsilon ) , \\quad d (\nu^\\lambda, u^\\varepsilon  ) \\leq \\lambda ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\rho ( u^\\lambda ) \\leq \\rho (u) + \\frac{\\varepsilon}{\\lambda} d\n(u^\\lambda, u) , {\\mbox { for all }} u \\in S.\n\\end{eqnarray*}\n\\end{lemma}\n\n Define\na metric $d$ on  the admissible\n controls set $\\cal A$ as\n\\begin{eqnarray}\\label{eq:3.13}\nd (u_1 (\\cdot), u_2 (\\cdot)) \\triangleq \\bigg \\{ {\\mathbb E} \\bigg [ \\int_0^T||u_1 (t) - u_2 (t)||^2_U dt\\bigg] \\bigg \\}^{\\frac{1}{2}} ,\n\\quad \\forall u_1(\\cdot), u_2 (\\cdot) \\in \\cal A.\n\\end{eqnarray}\n\n\nWe can assume   that\n$\\cal A$ is a bounded closed convex set in the sense of \\eqref{eq:3.13},\nthe unbounded case can be reduced to the\nbounded case.\n\n\n\n\n\n\nUnder this assumption of boundedness and closedness\nof $\\cal A$,\nwe have the following basic\nlemma which will be used in the sequence.\n\n\n\n\n\\begin{lemma}\\label{lem:4.2}\n$(\\Lambda, d)$ is a complete metric space.\n\\end{lemma}\n\n\\begin{proof}\nSince the control space $U$ is a Hilbert space $M_{\\mathbb{F}}^2(0,T;U)$ is also a Hilbert space under \\eqref{eq:3.13}. Therefore, $\\cal A$ is complete under the distance defined by \\eqref{eq:3.13}.\nsince $\\cal A$ is a  closed subset of $M_{\\mathbb{F}}^2(0,T;U).$\nThe proof is complete.\n\\end{proof}\n\n\nThe next lemma shows that a mapping from the control process in $\\cal A$ to the state process in ${\\cal M}^2_{\\mathbb F} (0, T)$, to be defined below, is bounded and continuous.\nTo simplify our notation, we write\n\\begin{eqnarray}\n{\\cal M}^2_{\\mathscr F} (0, T) \\triangleq S_ {\\mathscr{F}}^2(0,T;H) \\cap M_{\\mathscr{F}}^2(0,T;V)\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n|| X(\\cdot)||_{{\\cal M}^2_{\\mathscr F} (0, T)} \\triangleq \\sqrt{||X(\\cdot)||^2_{S^2_{\\mathscr F}(0,T;H)}\n+ ||X(\\cdot)||^2_{M^2_{\\mathscr F}(0,T;V)} }.\n\\end{eqnarray}\nThe next lemma shows that a mapping from the control process in ${\\cal A}$ to the state process in $ M_{\\mathscr{F}}^2(0,T;V)$ is bounded and continuous.\n\n\n\n\\begin{lemma}\\label{lem:4.7}\nLet Assumption \\ref{ass:2.5} be satisfied. Then the mapping ${\\cal I}: ({\\cal A}, d) \\rightarrow ({\\cal M}_{\\mathscr{F}}^2(0,T), || \\cdot ||_{{\\cal M}_{\\mathscr{F}}^2(0,T)})$\ndefined by\n\\begin{eqnarray*}\n{\\cal I} (u (\\cdot)) = X^u(\\cdot)\n\\end{eqnarray*}\nis bounded and continuous.\n\\end{lemma}\n\n\\begin{proof}\nBy the a priori estimate of SEE (Lemma \\ref{thm:3.2}), it can be shown that for any $u(\\cdot)\\in \\Lambda$,\n\\begin{eqnarray}\\label{eq:5.7}\n|| X^u(\\cdot) ||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)}\n&\\leq& K \\bigg \\{ {\\mathbb E} [ ||x||^2_H ] + {\\mathbb E} \\bigg [ \\int_0^T|| u(t)|| _{U}^{2}dt \\bigg ] + 1 \\bigg \\} \\nonumber \\\\\n&\\leq& K.\n\\end{eqnarray}\nHere $K$ is a positive constant independent of $u(\\cdot)$ and may change from line to line.\n\nOn the other hand, let $\\{v_n(\\cdot)\\}_{n \\geq 1}$ be a sequence in $\\cal A$ such that it  converges an admissible  $v(\\cdot)\\in \\cal A $ under the metric $d$. Suppose that $X_n(\\cdot)$, for each $n = 1, 2, \\cdots$,\nand $X(\\cdot)$ are the state processes corresponding to $v_n(\\cdot)$ and $v(\\cdot)$, respectively. By making use of the a priori estimate of SEE (Lemma \\ref{thm:3.2}), we can deduce that\n\\begin{eqnarray}\\label{eq5.9}\n&&|| X^{v_n}(\\cdot) - X^v(\\cdot)||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)}\n\\\\&\\leq& K \\bigg\\{{\\mathbb E} \\bigg [\\int_{0}^{T}||b(t,X^{v}(t), v_n(t))-b(t,X^{ v}(t), v(t)  )||^2_H dt\n \\bigg]+\n {\\mathbb E} \\bigg [\\int_{0}^{T}||g(t,X^{v}(t), v_n(t))-g(t,X^{ v}(t), v(t)  )||^2_H dt\n \\bigg]\\nonumber\n \\\\&&\\quad\\quad+\n {\\mathbb E} \\bigg [\\int_{0}^{T}||\\sigma(t,X^{v}(t), v_n(t))-\\sigma (t,X^{ v}(t), v(t)  )||^2_{M^{\\nu,2}( E; H)} dt\n \\bigg]\\bigg\\}\n  \\nonumber \\\\\n&\\leq& K {\\mathbb E} \\bigg [\\int_{0}^{T}||v_n(t)- v(t)||^2_U dt\\bigg] \\nonumber \\\\\n&=& K d^2(v_n(\\cdot), v(\\cdot)).\n\\end{eqnarray}\nSending $n\\rightarrow\\infty$ in (\\ref{eq5.9}) yields\n\\begin{eqnarray}\\label{eq:5.11}\n|| X^{v_n}(\\cdot) - X^v(\\cdot)||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)} \\rightarrow 0 .\n\\end{eqnarray}\nThis validates the continuity of ${\\cal I}$.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:4.4}\nLet Assumption \\ref{ass:2.5} be satisfied. Then the cost functional $J(u(\\cdot))$ is bounded and continuous on $\\cal A$ under the metric \\eqref{eq:3.13}.\n\\end{lemma}\n\n\\begin{proof}\nFor any $u(\\cdot)\\in {\\cal A}$, under Assumption \\ref{ass:2.5} and from Lemma \\ref{lem:4.7} we have\n\\begin{eqnarray}\n|J(u(\\cdot))| &\\leq& {\\mathbb E} \\bigg [ \\int_0^T |l(t,X^u(t),u(t))|dt + |\\Phi(X^u(T))| \\bigg ] \\nonumber \\\\\n&\\leq& K \\bigg [ 1 + ||X^u(\\cdot)||^2_{M_{\\mathscr{F}}^2(0,T;V)}\n+ ||u(\\cdot)||_{M_{\\mathscr{F}}^2(0,T;U)}^2 + ||X(T)||^2_{L^2(\\Omega,{\\mathscr{F}_T},\n\\mathbb P;H)} \\bigg] \\nonumber \\\\\n&\\leq& K \\bigg [ 1 + ||X^u(\\cdot)||^2_{{\\cal M}^2_{\\mathscr F}(0,T)} + ||{ u}(\\cdot)||_{M^2_{\\mathscr F}(0,T;U)}^2 \\bigg] \\nonumber \\\\\n&\\leq& K .\n\\end{eqnarray}\nHere $K$ is a positive constant independent of $u(\\cdot)$ and  may change from line to line. This implies the cost\nfunctional  $J(u(\\cdot))$ is bounded on ${\\cal A}$.\n\nTo show the continuity of the cost functional, as in the proof of Lemma \\ref{lem:4.7} we pick up the sequence $\\{v_n(\\cdot)\\}_{n \\geq 1}$\nand its converging point $v(\\cdot)$ in $\\cal A$ as well as the corresponding state processes $X_n(\\cdot)$ and $X(\\cdot)$.\nThus using Lemma \\ref{lem:4.7} and the Lebesgue dominated convergence theorem, we obtain\n\\begin{eqnarray}\\label{eq:3.4}\nJ(v^n(\\cdot))\\rightarrow J(v(\\cdot)) , \\quad \\mbox {as} \\ n \\rightarrow \\infty .\n\\end{eqnarray}\nThe completes the proof.\n\\end{proof}\n\nDefine a penalized cost functional associated with Problem \\eqref{pro:2.1} as\n\\begin{equation}\\label{eq:3.2000}\nJ^\\varepsilon(v(\\cdot)) \\triangleq \\bigg\\{ \\big [ J(v(\\cdot)) -J(\\bar u(\\cdot)) + \\varepsilon \\big ]^2+ \\big | {\\mathbb E} [ \\phi( X^v(T))] \\big |^2 \\bigg\\}^{\\frac{1}{2}} ,\n\\quad \\forall \\varepsilon>0 .\n\\end{equation}\nIt is worthwhile to point out that we will study this functional over $\\cal A$.\n\n\\begin{lemma}\\label{lem:4.5}\n$J^\\varepsilon(v(\\cdot))$ is bounded and continuous on ${\\cal A}$ under the metric \\eqref{eq:3.13}.\n\\end{lemma}\n\n\\begin{proof}\nThe proof can be obtained by Lemma \\ref{lem:4.4} and Lemma \\ref{lem:4.7} immediately.\n\\end{proof}\n\nNow we introduce an auxiliary optimal control problem without state constraint:\n\n\\begin{problem}[$(SC)^\\varepsilon$]\nFind an admissible control such that\n\\begin{eqnarray}\n\\inf_{v(\\cdot)\\in {\\cal A}} J^\\varepsilon(v(\\cdot)),\n\\end{eqnarray}\nwhere the state process is given by (\\ref{eq:4.1}) and the cost functional $J^\\varepsilon(v(\\cdot))$ is given by (\\ref{eq:3.4}).\n\\end{problem}\n\nFrom the definition of the penalized cost functional (\\ref{eq:3.2000}), we see that\n\\begin{eqnarray}\nJ^\\varepsilon(\\bar u(\\cdot)) = \\varepsilon\n \\leq  \\inf_{v(\\cdot)\\in {\\cal A}} J^\\varepsilon(v(\\cdot))+\\varepsilon.\n\\end{eqnarray}\nAn application of Ekeland's variational principle shows that there is a $u^\\varepsilon(\\cdot)\\in {\\cal A}$ such that\n\\begin{eqnarray}\\label{eq:5.1}\n\\left\\{\n\\begin{aligned}\n& J^\\varepsilon(u^\\varepsilon(\\cdot))\\leq J^\\varepsilon(\\bar u(\\cdot))=\\varepsilon  , \\\\\n& d(u^\\varepsilon(\\cdot),\\bar u(\\cdot))\\leq \\varepsilon^{\\frac{1}{2}}, \\\\\n& J^\\varepsilon(v(\\cdot))- J^\\varepsilon(u^\\varepsilon(\\cdot))\\geq -\\varepsilon^{\\frac{1}{2}}d(u^\\varepsilon(\\cdot),v(\\cdot)), \\quad \\forall v(\\cdot)\\in {\\cal A}.\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nDefine a convex perturbed control of ${u}^{\\varepsilon }\\left( \\cdot \\right)$ as\n\\begin{eqnarray}\\label{eq:5.2}\nu^{\\varepsilon ,\\rho}\\left( \\cdot \\right) \\triangleq {u}^{\\varepsilon }(\\cdot )+\\rho\nu\\left( \\cdot \\right)-u^{\\varepsilon }(\\cdot ) ),\n\\end{eqnarray}\nwhere $u\\left( \\cdot \\right)$ is an arbitrary admissible control in ${\\cal A}$ and $0\\leq \\rho\\leq 1$. It is easy to verify that\n$u^{\\varepsilon,\\rho }\\left( \\cdot \\right)$ is also in ${\\cal A}$. Suppose that $X^{\\epss,\\rho}(\\cdot)$ and\n$X^{\\epss}(\\cdot)$ are the state processes corresponding to $u^{\\epss, \\rho}(\\cdot)$ an $ u^\\epss(\\cdot)$, respectively.\nBy \\eqref{eq:5.1} and the fact\n\\begin{eqnarray}\\label{eq:5.3}\nd\\left( u^{\\varepsilon ,\\rho}\\left( \\cdot \\right) ,{u}^{\\varepsilon\n}\\left( \\cdot \\right) \\right) \\leq C\\rho,\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\\label{eq:3.9}\nJ^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))-J^\\varepsilon(u^\\varepsilon(\\cdot))\n\\geq {-\\varepsilon ^{\\frac{1}{2}}d\\left( u^{\\varepsilon ,\\rho}\\left(t\\right),{u}^{\\varepsilon}\\left( t\\right) \\right) }\n\\geq -\\varepsilon^{\\frac{1}{2}}C\\rho.\n\\end{eqnarray}\n\nOn the other hand,  from the definition of $J^\\varepsilon(\\bar u(\\cdot))$, we have\n\\begin{eqnarray}\\label{eq:3.17}\nJ^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))-J^\\varepsilon(u^\\varepsilon(\\cdot))\n&=& \\frac{[J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))]^2-[J^\\varepsilon(u^\\varepsilon(\\cdot))]^2}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))} \\nonumber \\\\\n&=& \\frac {J(u^{\\varepsilon,\\rho}(\\cdot))+J(u^\\varepsilon(\\cdot))-2J(\\bar u(\\cdot))+2\\varepsilon}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\\times[J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))] \\nonumber \\\\\n&& +\\frac {{\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]+{\\mathbb E}[\\phi(X^\\varepsilon(T))]}{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\times \\big \\{ {\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]-{\\mathbb E}[\\phi(X^\\varepsilon(T))] \\big \\} \\nonumber \\\\\n&=& {\\lambda}^{\\varepsilon,\\rho}[{J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))}]\n+\\mu^{\\varepsilon,\\rho} \\big \\{ {\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]-{\\mathbb E}[\\phi(X^\\varepsilon(T))] \\big \\} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\lambda}^{\\varepsilon,\\rho}\\triangleq\\frac {J(u^{\\varepsilon,\\rho}(\\cdot))+J(u^\\varepsilon(\\cdot))-2J(\\bar\nu(\\cdot))+2\\varepsilon}{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mu^{\\varepsilon,\\rho}\\triangleq\\frac{{\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]+{\\mathbb E}[\\phi(X^\\varepsilon(T))]}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))} .\n\\end{eqnarray}\nFrom \\eqref{eq:5.3}, we have\n\\begin{eqnarray} \\label{eq:5.8}\n\\lim_{\\rho\\rightarrow 0} d\\left( u^{\\varepsilon ,\\rho}\\left( \\cdot \\right) ,{u}^{\\varepsilon}\\left( \\cdot \\right) \\right) =0\n\\end{eqnarray}\nThen it follows from Lemma \\ref{lem:4.4} and Lemma \\ref{lem:4.5} that\n\\begin{eqnarray} \\label{eq:5.9}\n\\lim_{\\rho\\rightarrow 0} ||X^{\\epss,\\rho}(\\cdot)-X^{ \\epss}(\\cdot)||_{{\\cal M}^2_{\\mathscr F}(0,T)}^2=0\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\lim_{\\rho\\rightarrow 0} J^{\\epss}(u^{\\epss,\\rho}(\\cdot))= J^{\\epss}(u^{\\epss}(\\cdot)) .\n\\end{eqnarray}\nConsequently,\n\\begin{eqnarray}\\label{eq:5.10}\n\\lim_{\\rho\\rightarrow 0} { \\lambda}^{\\varepsilon,\\rho}= {\\lambda}^{\\varepsilon}, \\quad \\lim_{\\rho \\rightarrow 0} \\mu^{\\varepsilon,\\rho}= \\mu^{\\varepsilon} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\lambda}^{\\varepsilon}\\triangleq\\frac {J(u^{\\varepsilon}(\\cdot))-J(\\bar u(\\cdot))+\\varepsilon}{J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mu^{\\varepsilon}\\triangleq\\frac{{\\mathbb E}[\\phi(X^{\\varepsilon}(0))]}{J^\\varepsilon(u^{\\varepsilon}(\\cdot))} .\n\\end{eqnarray}\nNote that\n\\begin{eqnarray}\n|{\\lambda}^\\epss|^2+|\\mu^\\epss|^2=1.\n\\end{eqnarray}\n\nTherefore, there exists a subsequence $\\{({\\lambda}^\\epss, \\mu^\\epss)\\}_{\\epss > 0}$   ( still denoted also by $\\{({\\lambda}^\\epss, \\mu^\\epss)\\}_{\\epss > 0}$, such that\n\\begin{eqnarray} \\label{eq:5.13}\n\\lim_{\\epss\\rightarrow 0} {\\lambda}^{\\varepsilon}= {\\lambda}, \\quad \\lim_{\\epss \\rightarrow0} \\mu^{\\varepsilon}= \\mu ,\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:5.14}\n|{\\lambda}|^2+|\\mu|^2=1.\n\\end{eqnarray}\n\n\n\\section{Stochastic Maximum Principle}\n\nIn this section, we first drive a variational formula for the penalized cost functional $J^\\epss(u(\\cdot))$.\n\nTo simplify our notation, we write partial derivatives of $b,g \\sigma$ and $l$ as\n\\begin{eqnarray*}\n& \\varphi_x^{\\epss,\\rho}(t) \\triangleq \\varphi_x (t,{X}^{\\varepsilon,\\rho}(t),{u}^{\\varepsilon,\\rho}(t)), \\\\\n& \\varphi_x^{\\epss}(t) \\triangleq \\varphi_a (t,{X}^{\\varepsilon}(t),{u}^{\\varepsilon}(t)), \\\\\n& \\bar\\varphi_x(t) \\triangleq \\varphi_a (t,\\bar{X}(t),\\bar{u}(t)),\n\\end{eqnarray*}\nwhere $\\varphi = b, g,\\sigma$ and $l$.\n\nDefine the Hamiltonian ${\\cal H}: [ 0, T ] \\times \\Omega \\times H  \\times {\\mathscr U} \\times H\\times H \\times  M^{\\nu,2}( E; H)\\times \\mathbb R\n\\rightarrow {\\mathbb R}$ by\n\\begin{eqnarray}\\label{eq:5.3}\n{\\cal H} ( t, x, u, p, q, r(\\cdot),\\lambda ) := \\left ( b ( t, x, u ), p \\right )_H\n+\\left( g ( t, x,  u), q \\right)_H\n+\\int_{E}\\left( \\sigma ( t,e, x,  u), r(t,e) \\right)_H\\nu(de)\n+ \\lambda l ( t, x, u ) .\n\\end{eqnarray}\nUsing Hamiltonian ${\\cal H}$,\nthe adjoint equation \\eqref{eq:4.4}\ncan be written in the following form:\n\n\\begin{eqnarray}\\label{eq:5.4}\n\\begin{split}\n   \\left\\{\\begin{array}{ll}\nd\\bar p(t)=&-\\bigg[A^*(t)\\bar p(t)+B(t)^*\\bar q(t)+\\bar {\\cal H}_{x} (t)\\bigg]dt+\\bar q(t)dW(t)+\\displaystyle\n\\int_{{E}}\\bar r(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ \\bar p(T)=&\\Phi_x( \\bar X(T)),\n  \\end{array}\n \\right.\n \\end{split}\n  \\end{eqnarray}\nwhere we denote\n\\begin{eqnarray}\\label{eq:5.6}\n\\bar {\\cal H} (t) \\triangleq {\\cal H} ( t,\n\\bar x (t), \\bar u (t), \\bar p (t),\n\\bar q (t),\n\\bar r(t,\\cdot) ).\n\\end{eqnarray}\n\nSimilarly, for notational simplify, we write partial derivatives of $H$ as\n\\begin{eqnarray*}\n&{\\cal H}_a^{\\epss, \\rho}(t) \\triangleq\n{\\cal H}_a (t,{X}^{\\epss, \\rho}(t),{u}^{\\epss, \\rho}(t),{p}^{\\epss, \\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}), \\\\\n&{\\cal H}_a^{\\epss}(t) \\triangleq H_a (t,{X}^{\\epss}(t),{u}^{\\epss}(t),\n{p}^{\\epss}(t),q^{\\epss}(t),r^{\\epss}(t,\\cdot),\n{\\lambda}^{\\epss}), \\\\\n&{\\bar{\\cal H}}_a (t) \\triangleq H_a (t,\\bar{X}(t),\\bar{u}(t),\\bar{p}(t),{\\bar q}(t), \\bar r(t,\\cdot),{\\lambda}) .\n\\end{eqnarray*}\nwhere $a = x$ or $u$.\n\nFor the admissible pair $({u}^{\\varepsilon, \\rho}(\\cdot);{X}^{\\varepsilon,\\rho}(\\cdot))$ and $({u}^{\\varepsilon}(\\cdot);\n{X}^{\\varepsilon}(\\cdot))$ and the optimal pair $(\\bar{u}(\\cdot);\\bar{X}(\\cdot))$, the corresponding adjoint processes are denoted by\n$\\{ (p^{\\epss,\\rho}(t), q^{\\epss,\\rho}(t),\nr^{\\epss,\\rho}(t,\\cdot)), 0 \\leq t \\leq T \\}$, $\\{ (p^{\\epss}(t), q^{\\epss}(t),\nr^{\\epss,\\rho}(t)), 0 \\leq t \\leq T\\}$ and $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$.\nWe now define the adjoint equations for $\\{ (p^{\\epss,\\rho}(t), q^{\\epss,\\rho}(t),\nr^{\\epss,\\rho}(t,\\cdot)), 0 \\leq t \\leq T \\}$, $\\{ (p^{\\epss}(t), q^{\\epss}(t),\nr^{\\epss,\\rho}(t)), 0 \\leq t \\leq T\\}$ and $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$ as\n\\begin{eqnarray}\\label{eq:4.4}\n\\left\\{\n\\begin{aligned}\nd p^{\\epss,\\rho}(t)=&-\\bigg[A^*(t) p^{\\epss,\\rho}(t)+B(t)^* q^{\\epss,\\rho}(t)+ {\\cal H}_{x}^{\\epss,\\rho} (t)\\bigg]dt+ q^{\\epss,\\rho}(t)dW(t)+\\displaystyle\n\\int_{{E}}r^{\\epss,\\rho}(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\  p^{\\epss,\\rho}(T)=&\\lambda^{\\varepsilon, \\rho}\\Phi_x(  X^{\\varepsilon,\\rho}(T))+\\mu\n^{\\varepsilon, \\rho}\\phi_y(X^{\\varepsilon,\\rho}(T)),\n\\end{aligned}\n\\right.\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:4.5}\n\\left\\{\n\\begin{aligned}\nd p^{\\epss}(t)=&-\\bigg[A^*(t) p^{\\epss}(t)+B(t)^* q^{\\epss}(t)+ {\\cal H}_{x}^{\\epss} (t)\\bigg]dt+ q^{\\epss}(t)dW(t)+\\displaystyle\n\\int_{{E}}r^{\\epss}(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\  p^{\\epss}(T)=\n&\\lambda^{\\varepsilon}\\Phi_x(  X^{\\varepsilon}(T))+\\mu\n^{\\varepsilon}\\phi_y(X^{\\varepsilon}(T)),\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{eq:4.6}\n\\begin{split}\n   \\left\\{\\begin{array}{ll}\nd\\bar p(t)=&-\\bigg[A^*(t)\\bar p(t)+B(t)^*\\bar q(t)+\\bar {\\cal H}_{x} (t)\\bigg]dt+\\bar q(t)dW(t)+\\displaystyle\n\\int_{{E}}\\bar r(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ \\bar p(T)=&\\lambda\\Phi_x( \\bar X(T))\n+\\mu\\phi_x(\\bar X(T)),\n  \\end{array}\n \\right.\n \\end{split}\n  \\end{eqnarray}\nrespectively. In fact, the adjoint equations \\eqref{eq:4.4}, \\eqref{eq:4.5} and \\eqref{eq:4.6} are three linear BSEEs satisfying Assumptions \\ref{ass:3.1} and\n\\ref{ass:3.2}.\nHence by Lemma \\ref{lem:1.3}, it is easy to check that these three adjoint equations have unique solutions, respectively.\n\n\\begin{lemma}\\label{lem:4.1}\nUnder Assumptions \\ref{ass:2.5}, the following convergence results hold\n\\begin{eqnarray}\\label{eq:6.16}\n\\begin{split}\n&\\lim _{\\rho\\rightarrow 0} {\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon,\\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon, \\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon, \\rho}(t)-{q}^{\\varepsilon}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon, \\rho}(t,\\cdot)\n-{r}^{\\varepsilon}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; H)}} d t\\bigg]= 0 ,\n\\end{split}\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:6.13}\n\\begin{split}\n&\\lim _{\\epss\\rightarrow 0} {\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon}(t)\n-{\\bar p}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon}(t)-{\\bar p}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon}(t)-{\\bar q}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon}(t,\\cdot)\n-{\\bar r}^{}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; H)}} d t\\bigg]=0\n\\end{split}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nBy the continuous dependence theorem of BSEE (i.e., Lemma \\ref{lem:1.4}), we derive\n\\begin{eqnarray}\n\\begin{split}\n&{\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon,\\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon, \\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon, \\rho}(t)-{q}^{\\varepsilon}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon, \\rho}(t,\\cdot)\n-{r}^{\\varepsilon}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; H)}} d t\\bigg] \\nonumber \\\\\n& \\leq K \\bigg \\{ {\\mathbb E} \\bigg[ \\int_0^T ||(b_x^{\\varepsilon,\\rho}(t)\n-b_x^{\\epss}(t))\\cdot p^{\\epss}(t)\n+(g_x^{\\varepsilon,\\rho}(t)\n-g_x^{\\epss}(t))\\cdot q^{\\epss}(t)+\n\\int_E(\\sigma_x^{\\varepsilon,\\rho}(t,e)\n-\\sigma_x^{\\epss}(t,e))\\cdot r^{\\epss}(t,e)\n\\nu (de)\n\\\\&\\quad\\quad+\n {\\lambda}^{\\varepsilon,\\rho}l_x^{\\varepsilon,\\rho}(t)-{\\lambda} ^{\\varepsilon}l_x^{\\varepsilon}(t)||^2_H d t \\bigg ]+\\mathbb E\\bigg[||\\lambda^{\\varepsilon, \\rho}\\Phi_x(  X^{\\varepsilon,\\rho}(T))+\\mu\n^{\\varepsilon, \\rho}\\phi_y(X^{\\varepsilon}(T))-\n\\lambda^{\\varepsilon}\\Phi_x(  X^{\\varepsilon}(T))-\\mu\n^{\\varepsilon}\\phi_y(X^{\\varepsilon}(T))||_H.\\bigg]\n\\bigg\\}\n \\end{split}\n\\end{eqnarray}\nThen using \\eqref{eq:5.9} and \\eqref{eq:5.10} gives the desired result \\eqref{eq:6.16}.\nThe proof of \\eqref{eq:6.13} is similar and omitted here.\n\\end{proof}\n\n\nIn the next lemma, we give a representation of the difference $J^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss( u^\\epss(\\cdot))$\nin terms of the Hamiltonian $H$, the adjoint process $(p^{\\epss,\\rho}(\\cdot),\nq^{\\epss,\\rho}(\\cdot), r^{\\epss,\\rho}(\\cdot,\\cdot))$ and other relevant expressions associated with the admissible\npair $({u}^{\\epss, \\rho}(\\cdot);{X}^{\\epss,\\rho}(\\cdot))$.\n\n\\begin{lemma}\\label{lem:4.3}\nUnder Assumptions \\ref{ass:2.5}, it holds\n\\begin{eqnarray}\\label{eq:6.15}\nJ^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss(u^\\epss(\\cdot))\n&=& {\\mathbb E} \\bigg [ \\int_0^T \\big\\{ {\\cal H}^{\\epss, \\rho} (t)\n- {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& -{\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\\rho}(t)-X^{\\epss}(t)) \\big\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\phi^{\\epss}(X^{\\epss}(T))\n- \\phi_x(X^{\\epss,\\rho}(T))\n\\cdot(X^{\\epss,\\rho}(T)-X^{\\epss}(T)) \\big ] \\nonumber \\\\\n&& + {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\Phi^{\\epss}(X^{\\epss}(T))\n- \\Phi_x(X^{\\epss,\\rho}(T))\\cdot(X^{\\epss, \\rho}(T)-y^{\\epss}(T)) \\big ].\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of the Hamiltonian $\\cal H$ and $J^\\epss( u(\\cdot))$ (see \\eqref{eq:3.17}), we deduce\n\\begin{eqnarray}\\label{eq:4.10}\nJ^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss( u^\\epss(\\cdot))\n&=& {\\lambda}^{\\varepsilon,\\rho}[{J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))}]\n+\\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi(X^{\\varepsilon,\\rho}(T)) - \\phi(X^\\varepsilon(T)) \\big ] \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T \\bigg\\{ {\\cal H}^{\\epss, \\rho} (t) - {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& -  ( p^{\\epss,\\rho}(t), b^{\\epss,\\rho}(t)-b^{\\epss}(t)))_H - (q^{\\epss,\\rho}(t), g^{\\epss, \\rho} (t)-g^{\\epss} (t))_H \\nonumber\n\\\\&&-\\int_{E}\\bigg[(r^{\\epss,\\rho}(t,e)), \\sigma^{\\epss,\\rho}(t,e)-\\sigma^{\\epss} (t,e))_H\\nu(de)\\bigg]\\bigg\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi(X^{\\epss, \\rho}(T))-\\phi(X^{\\epss}(T)) \\big ]\n+ {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi(X^{\\epss, \\rho}(T))-\\Phi(X^{\\epss}(T)) \\big ] .\n\\end{eqnarray}\nOn the other hand,\n\\begin{eqnarray}\n  \\left\\{\n  \\begin{aligned}\n   d (X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n   = & \\ [ A (t)(X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n   + (b ( t, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-b ( t, X^{\\epss} (t), u^{\\epss}(t))) ] d t\n\\\\&+ [ B(t)(X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n   + (g( t, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-g ( t, X^{\\epss} (t), u^{\\epss}(t))) ]d W(t)\n \\\\&+\\int_E [\\sigma( t,e, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-\\sigma ( t,e, X^{\\epss} (t), u^{\\epss}(t))) ]\\tilde \\mu(de,dt),  \\\\\nX ^{\\epss,\\rho}(0)-X^\\epss(0) = & \\  0, \\quad t \\in [ 0, T ]\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nThen applying It\\^{o} formula to $( p^{\\epss,\\rho}(t),   X^{\\epss, \\rho}(t)-X^{\\epss}(t) )_H$ gives\n\\begin{eqnarray}\\label{eq:4.12}\n&& {\\mathbb E} \\bigg [ \\int_0^T  \\bigg\\{ (p^{\\epss,\\rho}(t), b^{\\epss, \\rho} (t)-b^{\\epss} (t))_H+(q^{\\epss,\\rho}(t), g^{\\epss, \\rho} (t)-g^{\\epss} (t))_H\n+\\int_{E}(r^{\\epss,\\rho}(t,e), \\sigma^{\\epss, \\rho} (t,e)-\\sigma^{\\epss} (t,e))_H\\nu(de) \\bigg\\} dt \\bigg ] \\nonumber \\\\\n&& = {\\mathbb E} \\bigg [ \\int_0^T   {\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\n\\rho}(t)-X^{\\epss}(t)) dt \\bigg ] \\nonumber +\\mu^{\\epss,\\rho} {\\mathbb E} \\big [ \\phi_x( X^{\\epss}(T))\\cdot(X^{\\epss, \\rho}(T)-X^{\\epss}(T)) \\big ]\n\\\\&&~~~~+{\\lambda}^{\\epss,\\rho} {\\mathbb E} \\big [ \\Phi_x( X^\\epss(T))\\cdot(X^{\\epss, \\rho}(T)-X^{\\epss}(T)) \\big ].\n\\end{eqnarray}\nPutting \\eqref{eq:4.12} into \\eqref{eq:4.10} leads to the desired representation \\eqref{eq:6.15}.\n\\end{proof}\n\nWe have the following  basic Lemma.\n\n\n\\begin{lemma}\\label{lem:3.2}\nUnder Assumptions \\ref{ass:2.5}, it follows that\n\\begin{eqnarray}\n\\| X^{\\varepsilon,\\rho}(\\cdot)\n-{X}^\\varepsilon(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2 = O (\\rho^2),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\| X^\\varepsilon(\\cdot)- {\\bar X}(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2 = O (\\varepsilon^2).\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nBy the continuous dependence theorem of BSEE (Lemma \\ref{lem:1.4}) and the uniform boundedness of the G\\^{a}teaux derivative $b_u$, we have\n\\begin{eqnarray*}\n&&\\| X^{\\varepsilon,\\rho}(\\cdot)\n-{X}^\\varepsilon(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2\n\\\\&\\leq& K {\\mathbb E} \\bigg [ \\int_0^T\n\\bigg\\{\\|b (t, {X}^\\varepsilon(t),  u^{\\varepsilon,\\rho}(t)) - b^\\varepsilon (t)\\big\\|^2_Hdt\n+\\| g(t, {y}^\\varepsilon(t), z^\\varepsilon(t), u^{\\varepsilon,\\rho}(t)) - g^\\varepsilon (t)\\big\\|^2_H\n\\\\&&\\quad\\quad+\\int_E\\bigg[\\|\\sigma (t, e,{X}^\\varepsilon(t), u^{\\varepsilon,\\rho}(t)) - \\sigma^\\varepsilon (t)\\big\\|^2_H\\bigg]\\nu(de)\\bigg\\}dt\\bigg] \\\\\n&\\leq& K {\\mathbb E} \\bigg [ \\int_0^T \\|u^{\\varepsilon,\\rho}(t)-{u^\\epss}(t)\\|^2_Udt\\bigg] \\\\\n&=& K \\rho^2 {\\mathbb E} \\bigg[ \\int_0^T \\|v(t)-{u^\\varepsilon}(t)\\|^2_Udt\\bigg] \\\\\n&\\leq& K \\rho^2  \\\\\n&=& O(\\rho^2).\n\\end{eqnarray*}\nHere $K$ is a generic positive constant and might change from line to line.\n\nIn the same vein, we deduce\n\\begin{eqnarray*}\n\\| X^\\varepsilon(\\cdot)-  {\\bar X}(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2\n&\\leq& K {\\mathbb E} \\bigg[ \\int_0^T \\|u^{\\varepsilon}(t)-{\\bar u}(t)\\|^2_Udt\\bigg] \\\\\n&=& K d^2(u^{\\varepsilon}(t),{\\bar u}(t)).\\\\\n&\\leq& K \\varepsilon^2\\\\\n&=& O(\\epss).\n\\end{eqnarray*}\nThe proof is complete.\n\\end{proof}\n\nNow we state the variational formula for the cost functional $J^\\epss(\\cdot)$.\n\n\\begin{theorem}\\label{them:3.1}\nUnder Assumptions \\ref{ass:2.5}, it follows that\nfor any admissible control $v(\\cdot),$ the cost functional $J(u(\\cdot))$ is\nG\\^{a}teaux differentiable at $ u^\\epss(\\cdot)$  in the direction $v(\\cdot)- u^\\epss(\\cdot)$\nand the corresponding G\\^{a}teaux derivative $J'$ is given by\n\\begin{eqnarray}\\label{eq:4.16}\n\\frac{d}{d\\rho}J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))|_{\\rho=0}\n&=& \\lim_{\\rho\\rightarrow 0} \\frac{J^\\epss(u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot))}{\\rho} \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u^\\epss(t), v(t)-{u}^\\epss(t))_Udt \\bigg ]\\nonumber\n\\\\\n&\\geq& -C\\varepsilon ^{\\frac{1}{2}}.\n\\end{eqnarray}\nHere $\\rho >0$ is a sufficiently small positive constant.\n\\end{theorem}\n\n\\begin{proof}\nBy \\eqref{eq:6.15}, we have\n\\begin{eqnarray}\\label{eq:6.23}\n&& J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot)) = I + II ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\nI &\\triangleq& {\\mathbb E} \\bigg [ \\int_0^T \\big\\{ {\\cal H}^{\\epss, \\rho} (t)\n- {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& -{\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\\rho}(t)-X^{\\epss}(t))\n -{\\cal H}_u^{\\epss, \\rho}(t) \\cdot(u^{\\epss,\\rho}(t)-u^{\\epss}(t))\\big\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\phi^{\\epss}(X^{\\epss}(T))\n- \\phi_x(X^{\\epss,\\rho}(T))\n\\cdot(X^{\\epss,\\rho}(T)-X^{\\epss}(T)) \\big ] \\nonumber \\\\\n&& + {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\Phi^{\\epss}(X^{\\epss}(T))\n- \\Phi_x(X^{\\epss,\\rho}(T))\\cdot(X^{\\epss, \\rho}(T)-y^{\\epss}(T)) \\big ].\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nII &\\triangleq& {\\mathbb E} \\bigg [ \\int_0^T {\\cal H}_u^{\\epss, \\rho}(t) \\cdot(u^{\\epss, \\rho}(t)-u^{\\epss}(t)) d t \\bigg ]\n\\end{eqnarray*}\nRecalling Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5} and using the Taylor Expansion for $H$ and the\ndominated convergence theorem, we obtain\n\\begin{eqnarray}\\label{eq:4.14}\nI = o(\\rho).\n\\end{eqnarray}\nOn the other hand, similarly, using Lemma \\ref{lem:4.1}, Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5} and using the Taylor Expansion for $H$ and the\ndominated convergence theorem, we deduce\n\\begin{eqnarray}\\label{eq:4.19}\nII =\\rho{\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u^\\epss(t), v(t)-{u}^\\epss(t))_Udt \\bigg ]+o(\\rho)\n\\end{eqnarray}\nHence, putting \\eqref{eq:4.14} and\n \\eqref{eq:4.19}into \\eqref{eq:6.23} and combing\n\\eqref{eq:3.9}, by  the\ndominated convergence theorem we conclude that\n\\begin{eqnarray}\n\\frac{d}{d\\rho}J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))|_{\\rho=0}\n&=& \\lim_{\\rho\\rightarrow 0} \\frac{J^\\epss(u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot))}{\\rho} \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}^\\epss_u(t), v(t)-{u}^\\epss(t))_U d t \\bigg ]\n\\geq - C \\varepsilon^{\\frac{1}{2}} .\n\\end{eqnarray}\n\\end{proof}\n\nNow we are ready to give the necessary condition of optimality for the existence of the optimal control of Problem \\ref{pro:2.1}.\n\n\\begin{theorem}\nLet Assumptions \\ref{ass:2.5} be satisfied. Let $(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$ be an optimal pair of Problem \\ref{pro:2.1}.\nThen there exist a $({\\lambda},\\mu)$ satisfying $|{\\lambda}|^2+|\\mu|^2=1$ such that\n\\begin{eqnarray}\\label{eq:4.20}\n( {\\cal H}_u(t,{\\bar X}(t),{\\bar u}(t),{\\bar p}(t),\\bar q(t),\\bar r(t,\\cdot), {\\lambda}), u-{\\bar u}(t))_U\\geq 0, \\quad \\forall u\\in U_{ad}, \\quad \\mbox{a.e.} \\ \\mbox{a.s.}.\n\\end{eqnarray}\nHere $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$ be  the solution of the corresponding adjoint equation \\eqref{eq:6.10} associated\nwith $(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$.\n\\end{theorem}\n\n\\begin{proof}\nFrom \\eqref{eq:5.14}, there exists a pair $({\\lambda},\\mu)$ satisfying $|{\\lambda}|^2+|\\mu|^2=1$. Note that\n\\begin{eqnarray}\\label{eq:4.22}\n\\lim_{\\epss \\rightarrow 0} d\\left( u^{\\varepsilon }\\left( \\cdot \\right) ,{\\bar u}\\left( \\cdot \\right) \\right) =0\n\\end{eqnarray}\nFrom \\ref{lem:4.1}, Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5}  and \\eqref{eq:5.13}, sending $\\varepsilon$ to $0$ on the both sides of  \\eqref{eq:4.16} and using\nthe dominated convergence theorem, we conclude that\n\\begin{eqnarray}\n{\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u(t,{\\bar X}(t),{\\bar u}(t),{\\bar p}(t),\\bar q(t),\\bar r(t,\\cdot), {\\lambda}), v(t)-{\\bar u}(t))_Udt \\bigg ] \\geq 0, \\quad \\forall v (\\cdot) \\in \\cal A,\n\\end{eqnarray}\nwhich implies that\n\\eqref{eq:4.20} holds.\nThis completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{1mm}\n\n\n\n\n\\section*{Appendix}\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\renewcommand{\\thedefinition}{A.\\arabic{definition}}\n\\renewcommand{\\thetheorem}{A.\\arabic{theorem}}\n\\renewcommand{\\thelemma}{A.\\arabic{lemma}}\n\\renewcommand{\\theassumption}{A.\\arabic{assumption}}\n\nIn this appendix, we introduce some preliminary results of SEEs and BSEEs, including\nexistence, uniqueness and continuous dependence theorems.\n\nConsider a SEE in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray} \\label{eq:3.1}\n  \\left\\{\n  \\begin{aligned}\n   d X (t) = & \\ [ A (t) X (t) + b ( t, X (t)) ] d t\n+ [B(t)X(t)+g( t, { X (t)}) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t,e, X(t-))\\tilde \\mu(de,dt),  \\\\\nX (0) = & \\  x \\in H , \\quad t \\in [ 0, T ],\n  \\end{aligned}\n  \\right.\n\\end{eqnarray}\nwhere $A,B,b,g$ and  $\\sigma $ are given random mappings\nwhich satisfy the following\nstandard  assumptions.\n\n\\begin{assumption} \\label{ass:3.1}\n  The operator processes $A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B\n  : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n  are weakly predictable; i.e.,\n  $ \\langle A(\\cdot)x, y \\rangle$ and $(B(\\cdot)x, y)_H$\n  are both predictable process for every $x, y\\in V, $\n  and satisfy the coercive condition, i.e., there exist\n  some constants  $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and  each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n    \\begin{eqnarray}\n    \\begin{split}\n     - \\langle A(t)x, x \\rangle +\\lambda ||x||_H^2 \\geq \\alpha\n      ||x||_V^2+||Bx||_H^2{\\color{blue},}\n    \\end{split}\n  \\end{eqnarray}\n  and  \\begin{eqnarray}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\\end{assumption}\n\n\\begin{assumption} \\label{ass:3.2}\n   The mappings $b:[0,T]\\times  \\Omega\\times H  \\longrightarrow H$ and $g:[0,T]\\times\\Omega\\times H  \\longrightarrow H$  are both $\\mathscr P\\times\n   \\mathscr B(H)\/\\mathscr B(H) $-measurable\n   such that $b(\\cdot,0), g(\\cdot,0)\\in M_{\\mathscr{F}}^2(0,T;H)\n   $; the mapping $\\sigma:[0,T]\\times  \\Omega \\times\n   E\\times H  \\longrightarrow H$ is $\\mathscr P\\times\\mathscr B(E)  \\times\n   \\mathscr B(H)\/\\mathscr B(H) $-measurable\n   such that $\\sigma(\\cdot,\\cdot, 0)\\in {M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times  E; H)}$.\n   And there exists a constant $C$  such that\n   for all $x, \\bar x\\in V$ and a.s.$(t,\\omega)\\in [0,T]\\times \\Omega,$\n   \\begin{eqnarray}\n     \\begin{split}\n       ||b(t,x)-b(t,x)||_H+ ||g(t,x)-g(t,x)||_H\n       +||\\sigma(t,\\cdot, x)-\\sigma(t,\\cdot,x)||\n       _{M^{\\nu,2}( E; H)} \\leq  C||x-\\bar x||_H.\n     \\end{split}\n   \\end{eqnarray}\n\n\\end{assumption}\n\n\\begin{definition}\n\\label{defn:c1}\nA $V$-valued, $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$-adapted process $X(\\cdot)$ is said to be a solution to the\nSEE \\eqref{eq:3.1}, if $X (\\cdot) \\in { M}_{\n\\mathscr F}^2 ( 0, T; V )$ such that for every $\\phi \\in V$\nand a.e. $( t, \\omega ) \\in [0, T ] \\times \\Omega$, it holds that\n\\begin{eqnarray}\n\\left\\{\n\\begin{aligned}\n( X (t), \\phi )_H =& \\ ( x, \\phi )_H + \\int_0^t \\left < A (s) X (s), \\phi \\right > d s\n+\\int_0^t ( b ( s, X (s){\\color{blue})}, \\phi )_H d s \\\\\n& + \\int_0^t ( B(s)X(s)+ g ( s, X (s) ), \\phi )_H d W (s)\n\\\\& + \\int_0^t \\int_{E} (\\sigma ( s,e, X (s-) ), \\phi )_H d \\tilde \\mu (de,ds), \\quad t \\in [ 0, T ] , \\\\\nX(0) =& \\ x\\in H  ,\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nor alternatively, $X (\\cdot)$ satisfies the following It\\^o's equation in $V^*$:\n\\begin{eqnarray}\n\\left\\{\n\\begin{aligned}\nX (t)=& \\  x+ \\int_0^t  A (s) X (s)d s\n+\\int_0^t  b ( s, X (s))d s + \\int_0^t\n[B(s)X(s)+ g ( s, X (s) )] d W (s)\n\\\\& + \\int_0^t \\int_{E} \\sigma ( s,e, X (s-) ) d \\tilde \\mu (de,ds), \\quad t \\in [ 0, T ] , \\\\\nX(t) =& \\ x\\in H .\n\\end{aligned}\n\\right.\n\\end{eqnarray}\n\\end{definition}\n\n\n\nNow we state our main result.\n\n\\begin{lemma} \\label{thm:3.1}\n  Let Assumptions \\ref{ass:3.1}-\\ref{ass:3.2} be\n  satisfied by any given coefficients\n  $(A,B,b,g,\\sigma)$ of the SEE \\eqref{eq:3.1}. Then for any initial\n    value $X(0)=x,$ the\n    SEE \\eqref{eq:3.1} has a unique\n  solution $X(\\cdot)\\in M_{\\mathscr{F}}^2(0,T;V) \\bigcap  S_{\\mathscr{F}}^2(0,T;H).$\n\\end{lemma}\nTo prove this  theorem, we first show the following\nresult on  the continuous dependence of  the solution to  the  SEE \\eqref{eq:3.1}.\n\n \\begin{lemma} \\label{thm:3.2}\n  Let  $ X(\\cdot)$  be a solution to\n  the  SEE   \\eqref{eq:3.1}\n   with the initial value $X(0)=x$ and\n    the coefficients $(A,B, b,g,\\sigma)$\n   which satisfy Assumptions \\ref{ass:3.1}-\\ref{ass:3.2}. Then\n  the following estimate holds:\n\\begin{eqnarray}\\label{eq:3.4}\n\\begin{split}\n&{\\mathbb E} \\bigg [ \\sup_{0 \\leq t \\leq T} \\| X (t) \\|_H^2 \\bigg\n]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| X (t) \\|_V^2 d t \\bigg ] \\\\\n& \\leq K \\bigg \\{ ||x||_H^2 + {\\mathbb E}\n\\bigg [ \\int_0^T \\| b ( t, 0) \\|_H^2 d t \\bigg ] + {\\mathbb E}\n\\bigg [ \\int_0^T \\| g ( t, 0) \\|_H^2 d t \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^T\\int_E \\| \\sigma (t,e,0) \\|^2_H  \\nu(de)d t \\bigg ] \\bigg \\}.\n\\end{split}\n\\end{eqnarray}\nFurthermore, suppose that   $ \\bar X(\\cdot)$  is a solution to\n  the  SEE   \\eqref{eq:3.1}\n   with the initial value $\\bar X(0)=\\bar x \\in H$ and the coefficients $(A,B,  \\bar b, \\bar g,\\bar\\sigma)$\n   satisfying Assumptions \\ref{ass:3.1}-\\ref{ass:3.2},\n   then we have\n\n   \\begin{eqnarray}\\label{eq:3.5}\n&& {\\mathbb E} \\bigg [ \\sup_{0 \\leq t \\leq T} \\| X (t) - {\\bar X} (t) \\|_H^2 \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| X (t) - {\\bar X} (t) \\|_V^2 d t \\bigg ] \\nonumber \\\\\n&& \\leq K \\bigg \\{  \\|x-\\bar x\\|^2_H\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| b ( t, {\\bar X} (t) )\n- {\\bar b} ( t, {\\bar X} (t) ) \\|_H^2 d t \\bigg ] \\\\\n&&+ {\\mathbb E} \\bigg [ \\int_0^T \\| g ( t, {\\bar X} (t) ) - {\\bar g}( t, {\\bar X} (t) ) \\|_H^2 d t \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\int_{E}\\| \\sigma ( t, e, {\\bar X} (t)) - {\\bar \\sigma}( t, e, {\\bar X} (t) ) \\|_H^2 \\nu(de)d t \\bigg ]   \\bigg \\} .\\nonumber\n\\end{eqnarray}\n \\end{lemma}\n\n\n\nNext we consider a BSEE in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray}\\label{eq:2.7}\n\\left\\{\n\\begin{aligned}\ndY(t)=& \\ [ A^*(t)Y(t)+B^*(t)Z(t)\n+f(t, Y(t), Z(t), R(t,\\cdot)) ] dt +Z(t)dW(t)+\\int_{E}R(t,e)\\tilde\\mu(dt,de),\\\\\nY(T)=& \\ \\xi,\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nwhere  $(A^*,B^*,f, \\xi)$  are given random mappings. Here $A^*$ and $B^*$\nare the adjoint operators of $A$ and\n$B$, respectively. Furthermore, we assume that the coefficients $(A^*, B^*, f, \\xi)$\nsatisfy the following conditions:\n\n\n\\begin{assumption} \\label{ass:3.1}\n  The operator processes $A^*:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B^*\n  : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n  are weakly predictable; i.e.,\n  $ \\langle A^*(\\cdot)x, y \\rangle$ and $(B^*(\\cdot)x, y)_H$\n  are both predictable process for every $x, y\\in V, $\n  and satisfy the coercive condition, i.e.,  there exist\n  some constants  $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and  each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n    \\begin{eqnarray}\n    \\begin{split}\n     - \\langle A^*(t)x, x \\rangle +\\lambda ||x||_H \\geq \\alpha\n      ||x||_V+||B^*x||_H{\\color{blue},}\n    \\end{split}\n  \\end{eqnarray}\n  and  \\begin{eqnarray}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A^* ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B^* ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\\end{assumption}\n\n\\begin{assumption} \\label{ass:3.2}\n   The mapping $\\xi: \\Omega \\rightarrow H$ is ${\\cal F}_T$-measurable such that\n   $\\xi\\in L^2(\\Omega,{\\mathscr{F}_T},\n   \\mathbb P;H).$\n    The mappings $f:[0,T]\\times  \\Omega\\times H \\times H\\times\n    M^{\\nu,2}( E; H) \\longrightarrow $   are both $\\mathscr P\\times\n   \\mathscr B(H)\\times \\mathscr B(H)\n   \\times\n   \\mathscr B(M^{\\nu,2}( E; H))\/\\mathscr B(H) $-measurable\n   such that $f(\\cdot,0,0,0)\\in M_{\\mathscr{F}}^2(0,T;H)\n   $.\n   And there exists a constant $C$  such that\n   for all $$(t,y,z,r,\\bar{y},\\bar{z},\n   \\bar r)\n\\in [0, T]\\times H\\times H\\times M^{\\nu,2}( E; H)\\times H\\times H \\times\nM^{\\nu,2}( E; H)$$ and a.s.$(t,\\omega)\\in [0,T]\\times \\Omega,$\n   \\begin{eqnarray}\n     \\begin{split}\n       ||f(t,y,z,r)-b(t,y,z,r)||_H \\leq  C\\bigg\\{||y-\\bar y||_H+||z-\\bar z||_H+\n       ||r-\\bar r||_{M^{\\nu,2}( E; H)}\\bigg\\}.\n     \\end{split}\n   \\end{eqnarray}\n\\end{assumption}\n\n\n\n\nIf the coefficients $(A^*, B^*,f,\\xi)$ satisfy Assumptions \\ref{ass:3.1} and\n\\ref{ass:3.2}, they are said to be a generator of BSEE \\eqref{eq:2.7}.\n\n\\begin{definition}\\label{defn:c1}\nA  $(V \\times H \\times M^{\\nu,2}( E; H) )$-valued, ${\\mathbb F}$-adapted process $( Y (\\cdot), Z (\\cdot), R (\\cdot, \\cdot) )$\nis called a solution to the BSEE \\eqref{eq:2.7}, if $Y (\\cdot) \\in { M}_{\\mathscr F}^2 ( 0, T; V )$,\n$Z (\\cdot) \\in { M}_{\\mathscr F}^2 ( 0, T; H )$ and $R (\\cdot, \\cdot) \\in { M}_{\\mathscr F}^{\\nu, 2}(0,T; H)$ such that for every $\\phi \\in V$ and\na.e. $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$, it holds that\n\\begin{eqnarray}\\label{eq:c5}\n( Y (t),  \\phi )_H  &=& (\\xi, \\phi)_H\n- \\int_t^T \\Big\\langle  A^* (s) Y (s) +B^*(s)Z(s)+f ( s, Y (s), Z (s), Y ( s  ), R(s, \\cdot) ), \\phi \\Big\\rangle d t \\nonumber \\\\\n&& - \\int_t^T ( Z (s), \\phi )_H d W (s) -\\int_t^T\\int_E (R(s,e), \\phi)_H\\tilde \\mu(ds,de), \\quad t \\in [0, T] ,\n\\end{eqnarray}\nor alternatively, $( Y (\\cdot), Z (\\cdot), R (\\cdot, \\cdot) )$ satisfies the following It\\^{o}'s equation in $V^*$:\n\\begin{eqnarray}\nY(t)&=& \\xi -\\int_t^T \\big[A^* (s) Y (s) d s+ B^* (s) Z (s)+f ( t, {Y} (s), {Z} (s),  R(s,\\cdot) ) \\big] d s \\nonumber \\\\\n&&- \\int_t^T Z (s) d W (s)-\\int_t^T \\int_E R (s,e) d \\tilde \\mu(ds,de) , \\quad t \\in [0, T] .\n\\end{eqnarray}\n\\end{definition}\n\n\\begin{lemma}[{\\bf Existence and Uniqueness of BSEE \\cite{Me}}]\\label{lem:1.3}\nFor any generator $(A^*,B^*, f,\\xi)$, BSEE \\eqref{eq:2.7} has a unique solution $(Y(\\cdot), Z(\\cdot), R(\\cdot,\\cdot)).$ Moreover, $Y(\\cdot)\\in S_{\\mathbb{ F}}^2(0,T;H)$.\n\\end{lemma}\n\n\\begin{lemma}[{\\bf  Continuous Dependence Theorem of BSEE}]\\label{lem:1.4}\nLet $(A^*, B^*, f,\\xi)$ and $(A^*,\nB^*,  \\bar{f},\\bar{\\xi})$ be two generators of BSEE \\eqref{eq:2.7}. Suppose that $(Y(\\cdot),Z(\\cdot), R(\\cdot,\\cdot))$ and $(\\bar{Y}(\\cdot),\\bar{Z}(\\cdot),\n\\bar R(\\cdot,\\cdot))$ are the\nsolutions of BSEE \\eqref{eq:2.7} corresponding to $(A^*,B^*, f,\\xi)$ and $(A^*,B^*, \\bar f,\\bar{\\xi})$, respectively. Then\n\\begin{eqnarray}\\label{eq:2.14}\n&& {\\mathbb E} \\bigg [ \\sup_{t \\in [0, T]}\\|Y(t) -\\bar{Y}(t)\\|_H^2 \\bigg ] + {\\mathbb E} \\bigg [ \\int_{0}^T\\|Y(t)-\\bar{Y}(t)\\|_V^2dt \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T\\|Z(t)-\\bar{Z}(t)\\|^2_Hdt \\bigg ]\n\\nonumber\n\\\\&&~~~+{\\mathbb E} \\bigg [ \\int_0^T\\int_E\\|R(t,e)-\\bar{R}(t,e)\\|^2_H\n\\nu(de)dt \\bigg ]  \\nonumber \\\\\n&& \\leq K \\bigg \\{ {\\mathbb E} [ \\|\\xi-\\bar{\\xi}\\|_H^2 ] + {\\mathbb E} \\bigg [ \\int_0^T\\|f(t,\\bar{Y}{(t),\\bar{Z}(t),\n\\bar R(t,\\cdot)})-\\bar{f}(t, \\bar{Y}{(t),\\bar{Z}(t)},\n\\bar R(t,\\cdot))\\|^2_Hdt \\bigg ] \\bigg \\} ,\n\\end{eqnarray}\nwhere $K$ is a positive constant depending only on $T$ and the constants $C,\\alpha, {\\lambda}$ in Assumption \\ref{ass:3.1}.\n\nIn particular, if $(A^*, B^*,\\bar{f},\\bar{\\xi})=(A^*, B^*, 0,0)$, the following a priori estimate holds\n\\begin{eqnarray}\\label{eq:2.15}\n&&{\\mathbb E} \\bigg [ \\sup_{t \\in [0, T]}\\|Y(t)\\|^2_H \\bigg ] + {\\mathbb E} \\bigg [ \\int_{0}^T\\|Y(t)\\|_V^2 d t \\bigg ] + {\\mathbb E} \\bigg [ \\int_0^T\\|Z(t)\\|^2_Hdt\n + {\\mathbb E} \\bigg [ \\int_0^T\\int_{E}\\|R(t,e)\\|^2_H\\nu(de)dt\\bigg ]\n\\\\&&\\leq K \\bigg\\{ {\\mathbb E} [ \\|\\xi\\|^2_H ] + {\\mathbb E} \\bigg [ \\int_0^T\\|f(t,0,0,0) \\|^2_Hdt \\bigg ] \\bigg \\}.\n\\end{eqnarray}\n\\end{lemma}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe theory of Discrete Exterior Calculus (DEC) is a relatively recent discretization  \\cite{HiraniThesis} of the classical theory of \nExterior Differential Calculus, a theory developed by \nE. Cartan \\cite{Cartan} which has been a fundamental tool in Differential Geometry and Topology for over a century. \nThe aim of DEC is to solve partial differential equations preserving their geometrical and physical\nfeatures as much as possible.\nThere are only a few papers about implementions of DEC to solve certain PDEs, such as \nthe Darcy flow and Poisson's equation \\cite{Hirani_K_N}, \nthe Navier-Stokes equations \\cite{Mohamedetal},\nthe simulation of  elasticity, plasticity and failure of isotropic materials \\cite{Dassiosetal}, some comparisons with \nthe finite differences and finite volume methods on regular flat meshes \\cite{Griebel_R_S}, as well as applications in digital geometry processing \\cite{Craneetal}.\n\nIn this paper, we describe a local formulation of DEC which is reminiscent of that of the Finite Element Method  (FEM) since,\nonce the local systems of equations have been established, they can be assembled into a global linear system.\nThis local formulation is also efficient and helpful in understanding various features of DEC that can otherwise remain unclear while dealing \nwith an entire mesh. We will, therefore, take a local approach when recalling all the objects required by DEC \\cite{Esqueda1}.\nOur main results are the following: \n\\begin{itemize}\n\\item We develop a local formulation of DEC analogous to that of FEM, which\nallows a natural treatment of heterogeneous material properties assigned to subdomains (element by element) and \neliminates the need of dealing with it through ad hoc modifications of the global discrete Hodge star operator. \n\n\\item Guided by the local formulation,\nwe also deduce a natural way to approximate the flux\/gradient-vector of a discretized function, as well as the anisotropic flux vector.\nWe carry out a comparison of the formulas defining the flux in both DEC and Finite Element Method with linear interpolation functions (FEML).\n\n\\item From the local formulation, we deduce the local DEC-discretization of the anisotropic Poisson equation. More precisely, in Exterior Differential Calculus the anisotropy tensor acts by {\\em pullback} on the differential of the unknown function. Here, we deduce how the anisotropy tensor acts on primal 1-forms.\nWe also carry out an algebraic comparison of the DEC and FEML local formulations of the anisotropic Poisson equation.\n\n\\item We present three numerical examples of the approximate solutions to the stationary anisotropic Poisson equation on different domains using DEC and FEML. The numerical examples show numerical convergence and a competitive performance \nof DEC, as well as a computational cost similar to that of FEML. In fact, the numerical solutions with both methods on fine meshes are identical, and\nDEC shows a slightly better performance than FEML on coarse meshes.\n\\end{itemize}\n\nThe paper is organized as follows.\nIn Section \\ref{sec: preliminaries}, we describe the local versions of the discrete derivative operator,\nthe dual mesh and the discrete Hodge star operator. \nIn Section \\ref{sec: anisotropy}, we deduce the natural way of computing flux vectors in DEC (which turns out to be equivalent to the FEML procedure), as well as the anisotropic flux vectors. \nIn Section \\ref{sec: comparison}, \nwe present the local DEC formulation of the 2D anisotropic Poisson equation\nand compare it with the local system of FEML, proving that the diffusion terms are identical while the source terms are discretized differently due to a different area-weight assignment for the nodes. \nIn Section \\ref{sec: extension}, we re-examine some of the local DEC quantities.\nIn Section \\ref{sec: examples}, we present and compare numerical examples of DEC and FEML approximate solutions to the 2D anisotropic \nPoisson equation on different domains with meshes of various resolutions. \nIn Section \\ref{sec: conclusions} we summarize the contributions of this paper.\n\n\n{\\em Acknowledgements}. The second named author was partially supported by a CONACyT grant,\nand would like to thank \nthe International Centre for Numerical Methods in Engineering (CIMNE) and the University of Swansea\nfor their hospitality. We gratefully acknowledge the support of\nNVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research.\n\n\\section{Preliminaries on DEC from a local viewpoint}\\label{sec: preliminaries}\n\nLet us consider a primal mesh made up of a single (positively oriented) triangle.\n    \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{Triangle01.png}%\n    \\captionof{figure}{Triangle $[v_1,v_2,v_3]$.}\n    \\end{center}\n\\subsection{Boundary operator}\nThere is a well known boundary operator \n\\begin{equation}\n\\partial [v_1,v_2,v_3]=[v_2,v_3]-[v_1,v_3]+[v_1,v_2],\\label{eq: boundary 2-1}\n\\end{equation}\nwhich describes the boundary of the triangle as an alternated sum of its oriented edges $[v_1,v_2]$, $[v_2,v_3]$ and $[v_3,v_1]$. \nSimilarly, one can compute the \nboundary of each edge\n\\begin{eqnarray}\n\\partial [v_1,v_2] &=& [v_2]-[v_1], \\nonumber\\\\ \n\\partial [v_2,v_3] &=& [v_3]-[v_2], \\label{eq: boundary 1-0}\\\\ \n\\partial [v_3,v_1] &=& [v_1]-[v_3]. \\nonumber\n\\end{eqnarray}\nIf we consider \n\\begin{itemize} \n\\item the symbol $[v_1,v_2,v_3]$ as a basis vector of a 1-dimensional vector space,\n\\item the symbols $[v_1,v_2]$, $[v_2,v_3]$, $[v_3,v_1]$ as an ordered basis of a 3-dimensional vector space,  \n\\item the symbols $[v_1]$, $[v_2]$, $[v_3]$ as an ordered basis of a 3-dimensional vector space,\n\\end{itemize}\nthen the map (\\ref{eq: boundary 2-1}), which sends the oriented triangle to a sum of its oriented edges, is represented by the matrix\n\\[\\left(\\begin{array}{r}\n1 \\\\ \n1 \\\\ \n1\n\\end{array}\\right) ,\\]\nwhile the map (\\ref{eq: boundary 1-0}), which sends the oriented edges to sums of their oriented vertices, is represented by the matrix\n\\[\\left(\\begin{array}{rrr}\n-1 & 0 & 1 \\\\ \n1 & -1 & 0 \\\\ \n0 & 1 & -1\n\\end{array} \\right).\\]\n\n\\subsection{Discrete derivative}\\label{subsec: discrete derivative}\nIt has been argued that the DEC discretization of the differential of a function is given by the transpose of the matrix of the \nboundary operator on edges (see \\cite{HiraniThesis, Esqueda1}). More precisely, suppose we have a function discretized by its values at the vertices\n\\[f\\sim \\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n              \\end{array}\n\\right).\\]\nIts discrete derivative, according to DEC, is \n\\begin{eqnarray*}\n\\left(\\begin{array}{rrr}\n-1 & 0 & 1 \\\\ \n1 & -1 & 0 \\\\ \n0 & 1 & -1\n\\end{array} \\right)^T\n\\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n              \\end{array}\n\\right)\n&=&\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n\\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n              \\end{array}\n\\right)\\\\\n&=&\n\\left(\\begin{array}{l}\nf_2-f_1\\\\\nf_3-f_2\\\\\nf_1-f_3\n              \\end{array}\n\\right) .\n\\end{eqnarray*}\nIndeed, such differences are rough approximations of the directional derivatives of $f$. For instance, $f_2-f_1$ is a rough approximation of the directional derivative of $f$ at $v_1$ in the direction of the vector $v_2-v_1$, i.e.\n\\[f_2-f_1 \\approx df_{v_1}(v_2-v_1) .\\]\nIt is precisely in this sense that, according to DEC, \n\\begin{itemize}\n\\item the value $f_2-f_1$ is assigned to the edge $[v_1,v_2]$, \n\\item the value $f_3-f_2$ is assigned to the edge $[v_2,v_3]$, \n\\item and the value $f_1-f_3$ is assigned to the edge $[v_3,v_1]$.\n\\end{itemize}\nLet\n\\[D_0:=\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right).\n\\]\n\n\\subsection{Dual mesh}\n\nThe dual mesh of the primal mesh consisting of a single triangle is constructured as follows:\n\\begin{itemize}\n \\item To the 2-dimensional triangular face $[v_1,v_2,v_3]$ will correspond the 0-dimensional point given by the circumcenter $c$ of the triangle.\n     \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{Triangle02.png}%\n    \\captionof{figure}{Circumcenter $c$ of the triangle $[v_1,v_2,v_3]$.}\n    \\end{center}\n\n \\item To the 1-dimensional edge $[v_1,v_2]$ will correspond the 1-dimensional straight line segment $[p_1,c]$ joining the midpoint $p_1$ of the edge $[v_1,v_2]$ to the circumcenter $c$. \n Similarly for the other edges.\n     \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{Triangle03.png}%\n    \\captionof{figure}{Dual segment $[p_1,c]$ of the edge $[v_1,v_2]$.}\n    \\end{center}\n\n \\item To the 0-dimensional vertex\/node $[v_1]$ will correspond the 2-dimensional quadrilateral $[v_1,p_1,c,p_3]$.\n    \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{Triangle04.png}%\n    \\captionof{figure}{Dual quadrilateral  $[v_1,p_1,c,p_3]$ of the vertex $[v_1]$.}\n    \\end{center}\n\n\\end{itemize}\n\n\\subsection{Discrete Hodge star}\n\nFor the Poisson equation in 2D, we need two matrices: one relating original edges to dual edges, and another relating vertices to dual cells.\n\\begin{itemize}\n\\item The discrete Hodge star map $M_1$ applied to the discrete differential of a discretized function $f\\sim (f_1,f_2,f_3)$ \nis given as follows:\n\\begin{itemize}\n \\item the value $f_2-f_1$ assigned to the edge $[v_1,v_2]$ is changed to the new value \n \\[{{\\rm length}[p_1,c]\\over {\\rm length}[v_1,v_2]}(f_2-f_1)\\]\n assigned to the segment $[p_1,c]$;\n \\item the value $f_3-f_2$ assigned to the edge $[v_2,v_3]$ is changed to the new value \n \\[{{\\rm length}[p_2,c]\\over {\\rm length}[v_2,v_3]}(f_3-f_2)\\]\n assigned to the segment $[p_2,c]$;\n \\item the value $f_1-f_3$ assigned to the edge $[v_3,v_1]$ is changed to the new value \n \\[{{\\rm length}[p_3,c]\\over {\\rm length}[v_3,v_1]}(f_1-f_3)\\]\n assigned to the edge $[p_3,c]$.\n\\end{itemize}\nIn other words, \n\\[M_1=\\left(\\begin{array}{ccc}\n{{\\rm length}[p_1,c]\\over {\\rm length}[v_1,v_2]} & 0 & 0 \\\\ \n0 & {{\\rm length}[p_2,c]\\over {\\rm length}[v_2,v_3]} & 0 \\\\ \n0 & 0 & {{\\rm length}[p_3,c]\\over {\\rm length}[v_3,v_1]}\n\\end{array} \\right).\\]\n\n\n\\item Similarly, the discrete Hodge star map $M_0$ on values on vertices is given as follows\n\\begin{itemize}\n \\item the value $f_1$ assigned to the vertex $[v_1]$ is changed to the new value \n \\[{\\rm Area}[v_1,p_1,c,p_3]f_1\\]\n assigned to the quadrilateral $[v_1,p_1,c,p_3]$;\n \\item the value $f_2$ assigned to the vertex $[v_2]$ is changed to the new value\n \\[{\\rm Area}[v_2,p_2,c,p_1]f_2\\]\n assigned to the quadrilateral $[v_2,p_2,c,p_1]$;\n \\item the value $f_3$ assigned to the vertex $[v_3]$ is changed to the new value\n \\[{\\rm Area}[v_3,p_3,c,p_2]f_2\\]\n assigned to the quadrilateral $[v_3,p_3,c,p_2]$.\n\\end{itemize}\nIn other words,\n\\[M_0=\\left(\\begin{array}{ccc}\n{\\rm Area}[v_1,p_1,c,p_3] & 0 & 0 \\\\ \n0 & {\\rm Area}[v_2,p_2,c,p_1] & 0 \\\\ \n0 & 0 & {\\rm Area}[v_3,p_3,c,p_2]\n\\end{array} \\right).\\]\n\\end{itemize}\n\n\n\n\\section{Flux and anisotropy}\\label{sec: anisotropy}\nIn this section, we deduce the DEC formulae for the local flux, the local anisotropic flux and the local anisotropy operator \nfor primal 1-forms.\n\n\\subsection{The flux in local DEC}\\label{subsec: local DEC flux}\nWe wish to find a natural construction for the discrete flux (discrete gradient vector) of a discrete function.\nRecall from Vector Calculus that the directional derivative of a differentiable function $f:\\mathbb{R}^2\\longrightarrow \\mathbb{R}$ at a point $p\\in \\mathbb{R}^2$ in the direction of $w\\in\\mathbb{R}^2$ is defined by\n\\begin{eqnarray*}\ndf_p(w)&:=&\\lim_{t\\rightarrow 0}{f(p+tw)-f(p)\\over t}\\\\\n&=&\\nabla f(p)\\cdot w.\n\\end{eqnarray*}\nThus, we have three Vector Calculus identities\n\\begin{eqnarray*}\n df_{v_1}(v_2-v_1) &=& \\nabla f(v_1)\\cdot (v_2-v_1)  ,\\\\\n df_{v_2}(v_3-v_2) &=& \\nabla f(v_2)\\cdot (v_3-v_2),\\\\\n df_{v_3}(v_1-v_3) &=& \\nabla f(v_3)\\cdot (v_1-v_3) .\n\\end{eqnarray*}\nAs in subsection \\ref{subsec: discrete derivative}, the rough approximations to directional derivatives of a function $f$ in the directions of the (oriented) edges are given as follows\n\\begin{eqnarray*}\n df_{v_1}(v_2-v_1) &\\approx& f_2-f_1  ,\\\\\n df_{v_2}(v_3-v_2) &\\approx& f_3-f_2  ,\\\\\n df_{v_3}(v_1-v_3) &\\approx& f_1-f_3  .\n\\end{eqnarray*}\nThus, if we want to find a discrete gradient vector $W_1$ of $f$\nat the point $v_1$,  we need to solve the equations of approximations\n\\begin{eqnarray}\nW_1\\cdot (v_2-v_1) &=& f_2-f_1 \\label{eq: W1-1}\\\\\nW_1\\cdot (v_3-v_1) &=& f_3-f_1.\\label{eq: W1-2}\n\\end{eqnarray} \nIf\n\\begin{eqnarray*}\nv_1&=&(x_1,y_1),\\\\\nv_2&=&(x_2,y_2),\\\\\nv_3&=&(x_3,y_3),\n\\end{eqnarray*}\nthen\n\\begin{eqnarray*}\nW_1\n&=&\n\\left({f_1y_2-f_1y_3-f_2y_1+f_2y_3+f_3y_1-f_3y_2\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\n,  -{f_1x_2-f_1x_3-f_2x_1+f_2x_3+f_3x_1-f_3x_2\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\\right)^T\n\\end{eqnarray*}\n\n\nNow, if we were to find \na discrete gradient vector $W_2$ of $f$\nat the point $v_2$,  we need to solve the equations\n\\begin{eqnarray}\nW_2\\cdot (v_1-v_2) &=& f_1-f_2\\label{eq: W2-1}\\\\\nW_2\\cdot (v_3-v_2) &=& f_3-f_2.\\nonumber\n\\end{eqnarray} \nThe vectors $W_2$ solving these equations is actually equal to $W_1$. \nIndeed, consider\n\\begin{eqnarray*}\nf_3-f_1 &=& W_1\\cdot (v_3-v_1) \\\\\n&=& W_1\\cdot (v_3-v_2+v_2-v_1)\\\\\n&=& W_1\\cdot (v_3-v_2)+W_1\\cdot (v_2-v_1)\\\\\n&=& W_1\\cdot (v_3-v_2)+f_2-f_1,\n\\end{eqnarray*} \nso that\n\\begin{equation}\n W_1\\cdot (v_3-v_2) = f_3-f_2  .\\label{eq: W1-3}\n\\end{equation} \nThus, adding up (\\ref{eq: W1-1}) and (\\ref{eq: W2-1}) we get\n\\begin{equation}\n(W_1-W_2)\\cdot (v_2-v_1)=0.\\label{eq: W1-W2-1}\n\\end{equation}\nSubtracting  (\\ref{eq: W1-2}) from (\\ref{eq: W1-3}) we get\n\\begin{equation}\n(W_1-W_2)\\cdot (v_3-v_2)=0.\\label{eq: W1-W2-2}\n\\end{equation}\nSince $v_2-v_1$ and $v_3-v_2$ are linearly independent \nand the two inner products in (\\ref{eq: W1-W2-1}) and (\\ref{eq: W1-W2-2}) vanish,\n\\[W_1-W_2=0.\\]\n\nAnalogously, the corresponding gradient vector $W_3$ of $f$ at the vertex $v_3$ is equal to $W_1$.\nThis means that the three approximate gradient vectors at the three vertices coincide. \nLet us call this unique vector $W$.\nNote that discrete flux $W$ satisfies \n\\begin{eqnarray*}\nW\\cdot (v_2-v_1) &=& f_2-f_1,\\\\\nW\\cdot (v_3-v_1) &=& f_3-f_1,\\\\\nW\\cdot (v_3-v_2) &=& f_3-f_2  .\n\\end{eqnarray*}\nThis means that the primal 1-form discretizing $df$ can be obtained by the dot products of the discrete flux $W$ with the \nvectors of the triangle's edges.\n\n\\vspace{.1in}\n\n{\\bf Remark}.\nMore generally, we can see that {\\em any vector which is constant on the triangle, naturally gives a primal 1-form on the edges of the triangle \nby means of its dot products with the triangle's edge-vectors}.\n\n\n\n\\subsubsection{Comparison of DEC and FEML local fluxes}\n\nThe local flux (gradient) of $f$ in FEML is given by\n\\[\n\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n\\left(\n\\begin{array}{c}\nf_1 \\\\ \nf_2 \\\\ \nf_3\n\\end{array} \n\\right),\n\\]\nwhere\n\\begin{eqnarray*}\nN_1 &=& {1\\over 2A}[(y_2-y_3)x+(x_3-x_2)y+x_2y_3-x_3y_2] ,\\\\\nN_2 &=& {1\\over 2A}[(y_3-y_1)x+(x_1-x_3)y+x_3y_1-x_1y_3] ,\\\\\nN_3 &=& {1\\over 2A}[(y_1-y_2)x+(x_2-x_1)y+x_1y_2-x_2y_1] ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nA &=& {1\\over 2} [(x_2y_3-x_3y_2) -(x_1y_3-x_3y_1)+(x_1y_2-x_2y_1)]\n\\end{eqnarray*}\nis the area of the triangle.\nExplicitly\n\\begin{eqnarray*}\n\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n&=&\n{1\\over 2A}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right)  \n\\end{eqnarray*}\nso that the FEML flux is given by\n\\[\n\\left(\n{[(y_2-y_3)f_1+(y_3-y_1)f_2+(y_1-y_2)f_3]\\over 2A} ,\n{[(x_3-x_2)f_1+(x_1-x_3)f_2+(x_2-x_1)f_3]\\over 2A}\n \\right)^T,\n\\]\nand we can see that its formula coincides with that of the DEC flux.\n\n\n\\subsection{The anisotropic flux vector in local DEC} \\label{subsec: discrete anisotropic flux}\n\nWe will now discuss how to discretize anisotropy in 2D DEC.\nLet $K$ denote the symmetric anisotropy tensor \n\\[K=\\left(\\begin{array}{cc}\nk_{11} & k_{12} \\\\ \nk_{12} & k_{22}\n\\end{array} \\right)\\]\nand recall the anisotropic Poisson equation\n\\[-\\nabla\\cdot (K\\, \\nabla f) = q.\\]\nAs in Subsection \\ref{subsec: local DEC flux}, we wish to find a vector $W'$ which will play the role of a discrete version of the {\\em anisotropic flux vector} $K \\nabla f$.\n\nFirst observe that, since $K$ is symmetric, for any $w\\in\\mathbb{R}^2$\n\\begin{eqnarray*}\n(K\\nabla f(p))\\cdot w \n   &=& \n  \\nabla f(p)\\cdot (K^Tw)\\\\\n   &=& \n  \\nabla f(p)\\cdot (Kw)\\\\\n   &=&\n  df_p(Kw)\\\\\n   &=&\n  (df_p\\circ K)(w)\\\\\n   &=:&\n  (K^*df_p)(w),\n\\end{eqnarray*}\nwhere $K^*df_p$ is called the {\\em pullback} of $df_p$ by $K$. \nThese identities mean that in order to discretize the anisotropic flux we need to understand the discretization\nof the linear functional $df_p\\circ K$. \nLet us suppose that {\\em $K$ is constant on our triangle}.\nAs before, we have three natural vectors on the triangle, \n\\begin{eqnarray*}\nw_1 &=& v_2-v_1,\\\\\nw_2 &=& v_3-v_2,\\\\\nw_3 &=& v_1-v_3.\n\\end{eqnarray*}\nGiven the vector $Kw_1$, we have the option to write it down as a linear combination of two of the three aforementioned vectors.\nSince $w_1$ is being used already, we use the other two vectors, i.e.\n\\begin{eqnarray*}\nKw_1 &=& \\lambda_1 w_2 + \\mu_1 w_3,\n\\end{eqnarray*}\nfor some $\\lambda_1,\\mu_1\\in\\mathbb{R}$.\nSimilarly,\n\\begin{eqnarray*}\nKw_2 &=& \\lambda_2 w_3 + \\mu_2 w_1,\\\\\nKw_3 &=& \\lambda_3 w_1 + \\mu_3 w_2,\n\\end{eqnarray*}\nfor some $\\lambda_2,\\mu_2,\\lambda_3,\\mu_3\\in\\mathbb{R}$.\nThese equations can be solved for $\\lambda_1,\\lambda_2,\\lambda_3,\\mu_1,\\mu_2,\\mu_3$.\nNow\n\\begin{eqnarray*}\n\\left(K\\nabla f(v_3)\\right)\\cdot w_1 \n   &=&\n  df_{v_3}(Kw_1)\\\\\n   &=&\n  df_{v_3}(\\lambda_1 w_2 + \\mu_1 w_3) \\\\\n   &=&\n  \\lambda_1 df_{v_3}(w_2) + \\mu_1 df_{v_1}(w_3) \\\\\n   &=&\n  \\lambda_1 df_{v_3}(v_3-v_2) + \\mu_1 df_{v_3}(v_1-v_3) \\\\\n   &=&\n  \\lambda_1 df_{v_3}(-(v_2-v_3)) + \\mu_1 df_{v_3}(v_1-v_3) \\\\\n   &=&\n  -\\lambda_1 df_{v_3}(v_2-v_3) + \\mu_1 df_{v_3}(v_1-v_3) .\n\\end{eqnarray*}\nSimilarly, for the vectors $w_2$ and $w_3$ we have the identities \n\\begin{eqnarray*}\n\\left(K\\nabla f(v_1)\\right)\\cdot w_2 &=& \\lambda_2 df_{v_1}(w_3) + \\mu_2 df_{v_1}(w_1),\\\\\n\\left(K\\nabla f(v_2)\\right)\\cdot w_3  &=& \\lambda_3 df_{v_3}(w_1) +\\mu_3 df_{v_3}(w_2).\n\\end{eqnarray*}\nThese equations lead to the three equations of approximations\n\\begin{eqnarray}\nW'\\cdot w_1\n   & = &\n  \\lambda_1 (f_3-f_2) + \\mu_1 (f_1-f_3), \\nonumber \\\\\nW'\\cdot w_2\n   & = &\n  \\lambda_2 (f_1-f_3) + \\mu_2 (f_2-f_1), \\label{eq: system equations 1 for anisotropic flux vector}\\\\\nW'\\cdot w_3\n   & = &\n  \\lambda_3 (f_2-f_1) + \\mu_3 (f_3-f_2). \\nonumber\n\\end{eqnarray}\nwhere $W'$ is the vector that should approximate $K\\nabla f(v_3)$, $K\\nabla f(v_3)$ and $K\\nabla f(v_3)$.\nThus, in order to find the discrete version $W'$ of the anisotropic flux vector $K\\nabla f$ on the triangle, we need to solve \nthe system (\\ref{eq: system equations 1 for anisotropic flux vector})\n\nThe system (\\ref{eq: system equations 1 for anisotropic flux vector}) has a unique solution. Indeed, since\n\\[w_1+w_2+w_3=0,\\] \nthen\n\\[Kw_1+Kw_2+Kw_3=0,\\] \ni.e.\n\\begin{eqnarray*}\n(\\lambda_3+\\mu_2-\\lambda_2-\\mu_1) w_1 + (\\lambda_1+\\mu_3-\\lambda_2-\\mu_1) w_2\n    =0.\n\\end{eqnarray*}\nSince $w_1$ and $w_2$ are linearly independent\n\\begin{eqnarray*}\n\\lambda_3+\\mu_2-\\lambda_2-\\mu_1&=&0\\\\\n\\lambda_1+\\mu_3-\\lambda_2-\\mu_1&=&0.\n\\end{eqnarray*}\ni.e.\n\\begin{eqnarray*}\n\\lambda_3&=&\\lambda_2+\\mu_1-\\mu_2\\\\\n\\mu_3&=&-\\lambda_1+\\lambda_2+\\mu_1.\n\\end{eqnarray*}\nThus, making the appropriate substitutions,  we see that\nthe third equation in (\\ref{eq: system equations 1 for anisotropic flux vector}) is dependent on the first two independent equations,\nand there is a unique vector $W'$ that solves the system.\n\nFor the sake of completeness, the values of the parameters are:\n\\begin{eqnarray*}\n\\lambda_1\n   &=&\n  {[k_{11}(x_2-x_1)+k_{12}(y_2-y_1)](y_1-y_3) -[k_{12}(x_2-x_1)+k_{22}(y_2-y_1)](x_1-x_3)\n  \\over (x_3-x_2)(y_1-y_3)-(x_1-x_3)(y_3-y_2)}\\\\ \n   &=&\n   -{J(w_3)\\cdot K(w_1) \\over 2A},\n   \\\\\n\\mu_1\n   &=&\n  -{[[k_{11}(x_2-x_1)+k_{12}(y_2-y_1)](y_3-y_2)-[k_{12}(x_2-x_1)+k_{22}(y_2-y_1)](x_3-x_2) \n  \\over (x_3-x_2)(y_1-y_3)-(x_1-x_3)(y_3-y_2)}\\\\ \n   &=&\n   {J(w_2)\\cdot K(w_1) \\over 2A},\n   \\\\\n\\lambda_2\n   &=&\n  {[k_{11}(x_3-x_2)+k_{12}(y_3-y_2)](y_2-y_1) -[k_{12}(x_3-x_2)+k_{22}(y_3-y_2)](x_2-x_1)\n  \\over (x_1-x_3)(y_2-y_1)-(x_2-x_1)(y_1-y_3)}\\\\ \n   &=&\n   -{J(w_1)\\cdot K(w_2) \\over 2A},\n   \\\\\n\\mu_2\n   &=&\n  -{[[k_{11}(x_3-x_2)+k_{12}(y_3-y_2)](y_1-y_3)-[k_{12}(x_3-x_2)+k_{22}(y_3-y_2)](x_1-x_3) \n  \\over (x_1-x_3)(y_2-y_1)-(x_2-x_1)(y_1-y_3)}\\\\ \n   &=&\n   {J(w_3)\\cdot K(w_2) \\over 2A},\n   \\\\\n\\lambda_3\n   &=&\n  {[k_{11}(x_1-x_3)+k_{12}(y_1-y_3)](y_3-y_2) -[k_{12}(x_1-x_3)+k_{22}(y_1-y_3)](x_3-x_2)\n  \\over (x_2-x_1)(y_3-y_2)-(x_3-x_2)(y_2-y_1)}\\\\ \n   &=&\n   -{J(w_2)\\cdot K(w_3) \\over 2A},\n   \\\\\n\\mu_3\n   &=&\n  -{[[k_{11}(x_1-x_3)+k_{12}(y_1-y_3)](y_2-y_1)-[k_{12}(x_1-x_3)+k_{22}(y_1-y_3)](x_2-x_1) \n  \\over (x_2-x_1)(y_3-y_2)-(x_3-x_2)(y_2-y_1)}\\\\ \n   &=&\n   {J(w_1)\\cdot K(w_3) \\over 2A},   \n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\nJ\n   &=&\n  \\left(\\begin{array}{cc}\n  0 & -1 \\\\ \n  1 & 0\n  \\end{array} \\right)\n\\end{eqnarray*}\nis the $90^\\circ$ counter-clockwise rotation,\nand\n\\begin{eqnarray*}\nW'\n   &=&\n\\left(\n\\begin{array}{c}\n{k_{11}[f_1(y_2-y_3)+f_2(y_3-y_1)+f_3(y_1-y_2)]\n+k_{12}[f_1(x_3-x_2)+f_2(x_1-x_3)+f_3(x_2-x_1)]\n\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2} \\\\\n{k_{12}[f_1(y_2-y_3)+f_2(y_3-y_1)+f_3(y_1-y_2)]\n+k_{22}[f_1(x_3-x_2)+f_2(x_1-x_3)+f_3(x_2-x_1)]\n\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\n\\end{array}\n\\right)\\\\\n&=&\n\\left(\\begin{array}{cc}\nk_{11} & k_{12} \\\\ \nk_{12} & k_{22}\n\\end{array} \\right)\n\\left(\n\\begin{array}{c}\n{[(y_2-y_3)f_1+(y_3-y_1)f_2+(y_1-y_2)f_3]\\over 2A} \\\\ \n{[(x_3-x_2)f_1+(x_1-x_3)f_2+(x_2-x_1)f_3]\\over 2A}\n\\end{array} \n\\right)\n\\end{eqnarray*}\nwhich is, in fact, {\\em the image under $K$ of the discrete isotropic flux} and shows the consistency of our {\\em local} reasoning.  \nAlso observe that this formula is the same as that of the FEML anisotropic flux.\n\n\\subsection{Anisotropy on primal 1-forms}\nThe system (\\ref{eq: system equations 1 for anisotropic flux vector}) can be rewritten in matrix form as follows\n\\begin{eqnarray}\n\\left(\\begin{array}{c}\nW'\\cdot w_1 \\\\ \nW'\\cdot w_2 \\\\ \nW'\\cdot w_3\n\\end{array} \\right)\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n \\left(\\begin{array}{l}\nf_2-f_1\\\\\nf_3-f_2\\\\\nf_1-f_3\n              \\end{array}\n\\right) \\nonumber\\\\\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n \\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n              \\end{array}\n\\right) \\nonumber\\\\\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\nD_0[f] .\\label{eq: dual of W}\n\\end{eqnarray}\nRecalling the Remark at the end of Subsection \\ref{subsec: local DEC flux}, the matrix identity (\\ref{eq: dual of W})\nstates that the primal 1-form dual to the anisotropic flux vector $W'$ is given by the local DEC discretization \nof the anisotropy tensor \n\\begin{eqnarray*}\nK^{DEC}\n   &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\\\\\n   &=&\n{1 \\over 2A}\n\\left(\n\\begin{array}{ccc}\n0 &    -J(w_3)\\cdot K(w_1) & J(w_2)\\cdot K(w_1) \\\\ \nJ(w_3)\\cdot K(w_2) & 0 & -J(w_1)\\cdot K(w_2) \\\\ \n-J(w_2)\\cdot K(w_3) & J(w_1)\\cdot K(w_3) & 0\n\\end{array} \n\\right).\n\\end{eqnarray*}\nacting on the primal 1-form $D_0[f]$.\n\n{\\bf Remark}. The matrix $K^{DEC}$ is the local DEC discretization on primal 1-forms of the pullback operator \n$K^*$ on 1-forms. In this case, the discretization of\n$K^*df:=df\\circ K$.\n\n\n\\subsubsection{Geometric interpretation of the entries of $K^{DEC}$}\nLet us examine $\\lambda_1$ in the anisotropic case.\nConsider the following figure\n    \\begin{center}\n    \\includegraphics[width=0.6\\textwidth]{DiscretizedAnisotropyOperator03.png}%\n    \\captionof{figure}{Geometric interpretation of the entries of the anisotropy tensor discretization $K^{ DEC}$}\\label{fig: anisotropy triangle 3}\n    \\end{center}\nWe have\n\\begin{eqnarray*}\n\\lambda_1\n   &= &   \n -{J(w_3)\\cdot K(w_1) \\over 2A}\\\\\n   &= &   \n {1\\over 2A} |-J(w_3)| |K(w_1)| \\cos(\\beta) \\\\\n   &= &   \n {1\\over 2A} |w_3| |K(w_1)| \\cos(\\alpha+\\pi\/2) \\\\\n   &= &   \n -{1\\over 2A} |w_3| |K(w_1)| \\sin(\\alpha) \\\\\n   &=&\n -{A'\\over A},\n\\end{eqnarray*}\nwhere $A'$ is the area of the red triangle and\nwe have used a well known formula for the area of a triangle in terms of an inner angle. Thus\n$\\lambda_1$ is the negative of the quotient of the area $A'$ of the red triangle and the area $A$ of the original triangle.\nThe calculations for the other entries are similar.\n\n\n\\subsubsection{Isotropic case}\nNow, let us assume $K=k\\,{\\rm Id}_{2\\times 2}$ on the triangle.\nThe previous calculations show that\n\\[K^{DEC}=\nk\\left(\\begin{array}{ccc}\n0 & -1 & -1 \\\\ \n-1 & 0 & -1 \\\\ \n-1 & -1 & 0\n\\end{array} \\right).\\]\nNote that, in this case,\n\\[K^{DEC} D_0= k\\, D_0.\\]\n\n\n\n\n\n\\section{2D anisotropic Poisson equation}\\label{sec: comparison}\nIn this section, we describe the local DEC discretization of the 2D anisotropic Poisson equation and compare it to that of FEML.\n\n\\subsection{Local DEC discretization of the 2D anisotropic Poisson equation}\nThe anisotropic Poisson equation reads as follows\n\\[-\\nabla\\cdot (K\\, \\nabla f) = q,\\]\nwhere $f$ and $q$ are two functions on a certain domain in $\\mathbb{R}^2$.\nIn terms of the exterior derivative $d$ and the Hodge star operator $\\star$ it reads as follows\n\\[-\\star d \\star (K^*df) = q\\]\nwhere $K^*df:=df\\circ K$\nand $K=K^T$.\nFollowing the discretization of the discretized divergence operator \\cite{Esqueda1},\nthe corresponding local DEC discretization of the anisotropic Poisson equation is\n\\[-M_0^{-1} \\, \\left(-D_0^T\\right)  \\, M_1\\, K^{DEC} \\, D_0 \\, [f] = [q], \\]\nor equivalently \n\\begin{equation}\nD_0^T  \\, M_1\\, K^{DEC}\\, D_0 \\, [f] = M_0 \\, [q]. \\label{eq: local DEC discretized anisotropic Poisson equation}\n\\end{equation} \nIn order to simplify the notation,\nconsider the lengths and areas defined in the Figure \\ref{fig: triangulo}.\n    \\begin{center}\n    \\includegraphics[width=0.35\\textwidth]{WellCenteredTriangle02-eps-converted-to.pdf}%\n    \\captionof{figure}{Triangle }\\label{fig: triangulo}\n    \\end{center}\nNow, the discretized equation (\\ref{eq: local DEC discretized anisotropic Poisson equation}) looks as follows:\n\\[\n\\left(\\begin{array}{ccc}\n-1 & 0 & 1\\\\\n1 & -1 & 0\\\\\n0 & 1 & -1\n      \\end{array}\n\\right)\n\\left(\\begin{array}{ccc}\n{l_1\\over L_1} & 0 & 0\\\\\n0 & {l_2\\over L_2}  & 0\\\\\n0 & 0 & {l_3\\over L_3} \n      \\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n\\left(\\begin{array}{c}\nf_1 \\\\ \nf_2 \\\\ \nf_3\n\\end{array} \\right)\n   =\n\\left(\\begin{array}{c}\nA_1q_1 \\\\ \nA_2q_2 \\\\ \nA_3q_3\n\\end{array} \\right).\n\\]\nThe diffusive term matrix \n\\begin{eqnarray*}\n&&\n\\left(\\begin{array}{ccc}\n-1 & 0 & 1\\\\\n1 & -1 & 0\\\\\n0 & 1 & -1\n      \\end{array}\n\\right)\n\\left(\\begin{array}{ccc}\n{l_1\\over L_1} & 0 & 0\\\\\n0 & {l_2\\over L_2}  & 0\\\\\n0 & 0 & {l_3\\over L_3} \n      \\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\\\\\n&=&\n\\left(\\begin{array}{ccc}\n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1} & {\\lambda_1l_1\\over L_1}+{(\\lambda_3-\\mu_3)l_3\\over L_3} \n          & -{(\\lambda_1-\\mu_1)l1\\over L_1}+{\\mu_3l_3\\over L3} \\\\ \n{\\mu_1l_1\\over L_1}-{(\\lambda_2-\\mu_2)l_2\\over L_2} & -{\\lambda_1l1\\over L_1} -{\\mu_2l_2\\over L_2}\n          &  {(\\lambda_1-\\mu_1)l_1\\over L_1}+{\\lambda_2l_2\\over L_2}\\\\ \n{(\\lambda_2-\\mu_2)l_2\\over L_2} + {\\lambda_3l_3\\over L_3} & {\\mu_2l_2\\over L_2}-{(\\lambda_3-\\mu_3)l_3\\over L_3} \n          & -{\\lambda_2l_2\\over L_2}-{\\mu_3l_3\\over L_3}\n\\end{array} \n\\right)\n\\end{eqnarray*}\nis actually symmetric (see Subsection \\ref{subsubsec: diffusive term DEC FEML}).\n\n\n\n\\subsection{Local FEML-Discretized 2D anisotropic Poisson equation}\n\n\n\nThe diffusive elemental matrix in FEM (frequently called \"stiffness matrix\") on an element $e$ is given by\n\\[K_e=\\int B^tDBdA,\\]\nwhere $D$ is the matrix representing the anisotropic diffusion tensor $K$ in this paper, \nand the matrix $B$ is given explicitly by\n\\[\nB=\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n=\n{1\\over 2A}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right) . \n\\]\nSince the matrix $B$ is constant on an element of the mesh, the integral is easy to compute. Thus, the difussive matrix $K_e$ for a linear triangular element (FEML) is given by\n\\begin{eqnarray*}\nK_e\n   &=&\n\\int B^T DB dA \\\\\n   &=&\n  B^T D B A_e \\\\\n   &=&\n  {1\\over 4A_e}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & x_3-x_2  \\\\ \ny_3-y_1 & x_1-x_3  \\\\\ny_1-y_2 & x_2-x_1\n\\end{array} \n\\right)\n  \\left(\\begin{array}{cc}\n  k_{11} & k_{12} \\\\ \n  k_{12} & k_{22}\n  \\end{array} \\right)\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right)\n\\end{eqnarray*}\nNow, let us consider the first diagonal entry of the local FEML anisotropic Poisson diffusive matrix $K_e$, \n\\begin{eqnarray*}\n  (K_e)_{11} \n    &=&\n{1\\over 4A}(k_{11}(y_2-y_3)^2 +(k_{12}+k_{12})(y_2-y_3)(x_3-x_2) + k_{22}(x_3-x_2)^2)\\\\\n   &=& \n{1\\over 4A}  (\\begin{array}{cc}\n  -(y_3-y_2), & x_3-x_2\n  \\end{array} )\n  \\left(\\begin{array}{cc}\n  k_{11} & k_{12} \\\\ \n  k_{12} & k_{22}\n  \\end{array} \\right)\n  \\left(\\begin{array}{c}\n  -(y_3-y_2) \\\\ \n  x_3-x_2\n  \\end{array} \\right)\\\\\n   &=& \n{1\\over 4A}  (\\begin{array}{cc}\n  x_3-x_2, & y_3-y_2\n  \\end{array} )\n  \\left(\\begin{array}{cc}\n  0 & 1 \\\\ \n  -1 & 0\n  \\end{array} \\right)\n  \\left(\\begin{array}{cc}\n  k_{11} & k_{12} \\\\ \n  k_{12} & k_{22}\n  \\end{array} \\right)\n  \\left(\\begin{array}{cc}\n  0 & -1 \\\\ \n  1 & 0\n  \\end{array} \\right)\n  \\left(\\begin{array}{c}\n  x_3-x_2 \\\\ \n  y_3-y_2\n  \\end{array} \\right)\\\\\n   &=&\n{1\\over 4A}  (J(v_3-v_2))^T K J(v_3-v_2)\\\\\n   &=&\n{1\\over 4A}  J(v_3-v_2)\\cdot K (J(v_3-v_2)),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\nJ\n   &=&\n  \\left(\\begin{array}{cc}\n  0 & -1 \\\\ \n  1 & 0\n  \\end{array} \\right)\n\\end{eqnarray*}\nis the $90^\\circ$ counter-clockwise rotation.\nIn this notation, the diffusive term in local FEML is given as follows\n{\\small\n\\[\n{1\\over 4A}\n\\left(\\begin{array}{ccc}\nJ(v_3-v_2)\\cdot K(J(v_3-v_2)) & J(v_3-v_2)\\cdot K(J(v_1-v_3)) & J(v_3-v_2)\\cdot K(J(v_2-v_1)) \\\\ \nJ(v_1-v_3)\\cdot K(J(v_3-v_2)) & J(v_1-v_3)\\cdot K(J(v_1-v_3)) & J(v_1-v_3)\\cdot K(J(v_2-v_1)) \\\\ \nJ(v_2-v_1)\\cdot K(J(v_3-v_2)) & J(v_2-v_1)\\cdot K(J(v_1-v_3)) & J(v_2-v_1)\\cdot K(J(v_2-v_1))\n\\end{array}\\right) . \\]\n}\n\n\n\\subsection{Comparison between local DEC and FEML discretizations}\nFor the sake of brevity, \nwe are only going to compare the entries of the first row and first column of each formulation.\nConsider the various lengths, areas and angles given in the triangle of Figure \\ref{fig: circumscribed triangle 01}.\n    \\begin{center}\n    \\includegraphics[width=0.33\\textwidth]{WellCenteredTriangle03-eps-converted-to.pdf}%\n    \\hspace{12mm}\n    \\includegraphics[width=0.33\\textwidth]{WellCenteredTriangle02-eps-converted-to.pdf}%\n    \\captionof{figure}{Circumscribed triangle.}\\label{fig: circumscribed triangle 01}\n    \\end{center}\nWe have the following:\n\\begin{eqnarray*}\n \\pi &=& 2(\\alpha_1+\\alpha_2+\\alpha_3),\\\\\n {2l_i\\over L_i} &=& \\tan(\\alpha_i),\\\\\n {l_i\\over R} &=& \\sin(\\alpha_i),\\\\\n {L_i\\over 2R} &=& \\cos(\\alpha_i),\\\\\n A_1&=& {L_1l_1\\over 4} + {L_3l_3\\over 4},\\\\\n A_2&=& {L_1l_1\\over 4} + {L_2l_2\\over 4},\\\\\n A_3&=& {L_2l_2\\over 4} + {L_3l_3\\over 4}.\n\\end{eqnarray*}\n\n\\subsubsection{The diffusive term}\\label{subsubsec: diffusive term DEC FEML}\nWe claim that\n\\[{J(v_3-v_2)\\cdot K(J(v_3-v_2))\\over 4A}\n= \n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1}.\n\\]\nIndeed,\n\\begin{eqnarray*}\n\\lambda_3\n   &=&\n  -{J(v_3-v_2)\\cdot K( v_1-v_3)\\over 2A},\\\\\n\\mu_1\n   &=&\n  {J(v_3-v_2)\\cdot K( v_2-v_1)\\over 2A}.\n\\end{eqnarray*}\nThus\n\\begin{eqnarray*}\n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1}\n   &=& \n   {J(v_3-v_2)\\cdot K(v_1-v_3)\\over 2A}{\\tan(\\alpha_3)\\over 2}\n   -{J(v_3-v_2)\\cdot K( v_2-v_1)\\over 2A}{\\tan(\\alpha_1)\\over 2}\\\\\n   &=& \n   {1\\over 4A}J(v_3-v_2)\\cdot K((v_1-v_3)\\tan(\\alpha_3)- (v_2-v_1)\\tan(\\alpha_1)).\n\\end{eqnarray*}\nAll we have to do is show that\n\\[(v_1-v_3)\\tan(\\alpha_3)- (v_2-v_1)\\tan(\\alpha_1) = J(v_3-v_2).\\]\nNote that, since $J(v_3-v_2)$ is orthogonal to $v_3-v_2$, $J(v_3-v_2)$ must be parallel to $c -  {v_2+v_3\\over 2}$. Thus, \n\\begin{equation}\nJ(v_3-v_2) = {L_2\\over l_2}\\left(c -  {v_2+v_3\\over 2}\\right). \\label{eq: J(w2)}\n\\end{equation}\nNow we are going to express $c$ in terms of $v_1,v_2,v_3$.\nLet us consider \n\\[c-v_1 = a (v_2-v_1) + b(v_3-v_1)\\]\nwhere $a,b$ are coefficients to be determined.\nTaking inner products with $(v_2-v_1)$ and $(v_3-v_1)$ we get the two equations\n\\begin{eqnarray*}\nR\\cos(\\alpha_1)\n   &=&\n  aL_1+bL_3\\cos(\\alpha_1+\\alpha_3),\\\\ \nR\\cos(\\alpha_3)\n   &=&\n  aL_1\\cos(\\alpha_1+\\alpha_3)+bL_3. \n\\end{eqnarray*}\nSolving for $a$ and $b$\n\\begin{eqnarray*}\na\n   &=&\n  {\\sin(\\alpha_3)\\over 2\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)},\\\\\nb\n   &=&\n  {\\sin(\\alpha_1)\\over 2\\cos(\\alpha_3)\\sin(\\alpha_1+\\alpha_3)}.\n\\end{eqnarray*}\nSubstituting all the relevant quantities in (\\ref{eq: J(w2)}) we have, for instance, that the coefficient of $(v_2-v_1)$ is\n\\begin{eqnarray*}\n2{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)\\over 2\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)}-{1\\over 2}\\right)\n   &=&\n{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)-\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)\\over \\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)}\\right)\\\\  \n   &=&\n{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)-\\cos(\\alpha_1)(\\sin(\\alpha_1)\\cos(\\alpha_3)+\\sin(\\alpha_3)\\cos(\\alpha_1))\\over \\cos(\\alpha_1)\\sin(\\pi\/2-\\alpha_2)}\\right)\\\\  \n   &=&\n{\\sin(\\alpha_1)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)\\sin(\\alpha_1)-\\cos(\\alpha_1)\\cos(\\alpha_3)\\over \\cos(\\alpha_1)}\\right)\\\\  \n   &=&\n  \\tan(\\alpha_1){-\\cos(\\alpha_1+\\alpha_3)\\over\\sin(\\alpha_2)}\\\\ \n   &=&\n  \\tan(\\alpha_1){-\\cos(\\pi\/2-\\alpha_2)\\over\\sin(\\alpha_2)}\\\\ \n   &=&\n  \\tan(\\alpha_1){-\\sin(\\alpha_2)\\over\\sin(\\alpha_2)}\\\\ \n   &=&\n  -\\tan(\\alpha_1), \n\\end{eqnarray*}\nand similarly for the coefficient of $(v_1-v_3)$.\nThe calculations for the remaining entries are similar.\n\nThus, the local DEC and FEML diffusive terms of the 2D anisotropic Poisson equation coincide.\n\n\n\\subsubsection{The source term}\nAs already observed in \\cite{Esqueda1}, the right hand sides of the local DEC and FEML systems are different\n\\[\\left(\\begin{array}{c}\nA_1q_1\\\\\nA_2q_2\\\\\nA_3q_3\n      \\end{array}\n\\right)\n\\not =\n{A\\over 3}\\left(\\begin{array}{l}\nq_1\\\\\nq_2\\\\\nq_3\n        \\end{array}\n\\right).\\]\nWhile FEML uses a barycentric subdivision to calculate the areas associated to each node\/vertex,\nDEC uses a circumcentric subdivision. Eventually, this leads the DEC discretization to a better approximation of the solution\n(on coarse meshes). \n\n\n\n\\section{Some remarks about DEC quantities}\\label{sec: extension}\n\n\n\\subsection{The discrete Hodge star quantities revisited}   \\label{subsec: determinants} \nThe numbers appearing in the local DEC matrices can be expressed both in terms of determinants and in terms of trigonometric functions.\nMore precisely, \n\\begin{eqnarray*}\n A_1\n&=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n                            \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n           \\end{array}\n\\right)\n\\right]\n \\quad =\\quad {R^2\\over 4}(\\sin(2\\alpha_1)+\\sin(2\\alpha_3)),\\\\\n A_2\n &=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n                            \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n           \\end{array}\n\\right)\n\\right]\n\\quad = \\quad {R^2\\over 4}(\\sin(2\\alpha_1)+\\sin(2\\alpha_2)),\\\\\n A_3\n &=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n                            \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n           \\end{array}\n\\right)\n\\right]\n\\quad = \\quad {R^2\\over 4}(\\sin(2\\alpha_2)+\\sin(2\\alpha_3)),\\\\\n{l_1\\over L_1}\n &=&\n{1\\over L_1^2}\n\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n                            \\end{array}\n\\right)\n\\quad = \\quad   {\\tan(\\alpha_1)\\over 2},\\\\\n{l_2\\over L_2}\n &=&\n{1\\over L_2^2}\n\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n                            \\end{array}\n\\right)\n \\quad = \\quad  {\\tan(\\alpha_2)\\over 2},\\\\\n{l_3\\over L_3}\n &=&\n{1\\over L_3^2}\n\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n                            \\end{array}\n\\right)\n  \\quad = \\quad {\\tan(\\alpha_3)\\over 2}.\n\\end{eqnarray*}\nThese expressions are valid regardless of the location of the circumcenter and can, indeed, take negative values.\nhe angles that are measured in the scheme can be negative as in the obtuse triangle of Figure \\ref{fig: negative angles}\n    \\begin{center}\n    \\includegraphics[width=0.3\\textwidth]{NonWellCenteredTriangle01a-eps-converted-to.pdf}%\n   \\hspace{8mm}\n    \\includegraphics[width=0.3\\textwidth]{NegativeAngles01-eps-converted-to.pdf}%\n    \\captionof{figure}{Negative (exterior) angles measured in an obtuse triangle.}\\label{fig: negative angles}\n    \\end{center}\nand some quantities can be zero or negative.\nFor instance, if \n\\[\\alpha_2={\\pi\\over 2}-2\\alpha_1,\\]\nhen\n\\[A_1=0.\\]\n\n\n\\subsection{Area weights assigned to vertices}\nIn order to understand how local DEC assigns area weights to vertices differently from FEML, let us\nconsider the obtuse triangle shown in Figure \\ref{fig: negative angles}\nLet $p_1,p_2,p_3$ be the middle points of the segments\n$[v_2,v_3],[v_2,v_3],[v_3,v_1]$ respectively.    \nAs shown in Figure \\ref{fig: obtuse triangle 01}, the triangle $[v_1,p_3,c]$ lies completely outside of the triangle $[v_1,v_2,v_3]$. Geometrically, \nthis implies that its area must be assigned a negative sign, which is confirmed \nby the determinant formulas of Subsection \\ref{subsec: determinants}. On the other hand, the triangle $[v_1,p_1,c]$ will have positive area. \nThus, their sum gives us the area $A_1$ in Figure \\ref{fig: obtuse triangle 01}.\n    \\begin{center}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle03a-eps-converted-to.pdf}%\n    \\hspace{3mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle04a-eps-converted-to.pdf}%\n    \\hspace{3mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle05a-eps-converted-to.pdf}%\n    \\captionof{figure}{Area weight assigned to $v_1$.}\\label{fig: obtuse triangle 01}\n    \\end{center}\nThe area $A_3$ is computed similarly, where the triangle $[p_3,v_3,c]$ is assigned negative area (see Figure \\ref{fig: obtuse triangle 02}).\n    \\begin{center}    \n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle06a-eps-converted-to.pdf}%\n    \\hspace{3mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle07a-eps-converted-to.pdf}%\n    \\hspace{3mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle08a-eps-converted-to.pdf}%\n    \\captionof{figure}{Area weight assigned to $v_3$.}\\label{fig: obtuse triangle 02}\n    \\end{center}\nNote that for $A_2$, the two triangles $[p_1,v_3,c]$ and $[v_2,p_2,c]$ both have positive areas (see Figure \\ref{fig: obtuse triangle 03}). \n    \\begin{center}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle09a-eps-converted-to.pdf}%\n    \\hspace{2mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle10a-eps-converted-to.pdf}%\n    \\hspace{2mm}\n    \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle11a-eps-converted-to.pdf}%\n    \\captionof{figure}{Area weight assigned to $v_2$.}\\label{fig: obtuse triangle 03}\n    \\end{center}\n\n \n\\section{Numerical Examples}\\label{sec: examples}\n\n\nIn this section, we present three examples in order to illustrate the performance of DEC resulting from the local formulation and its implementation. \nIn all cases, we solve the anisotropic Poisson equation.\nThe FEML methodology that we have used in the comparison can be consulted \\cite{Onate,Zienkiewicz1,Botello}.\n\n\n\\subsection{First example: Heterogeneity}\nThis example is intended to highlight how Local DEC deals effectively with heterogeneous materials. \nConsider the region in the plane given in Figure \\ref{fig: square example geometry}.\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.38\\textwidth]{Fig_12_SquareWithConditions.png}%\n\t\\captionof{figure}{Square and inner circle with different conditions.}\\label{fig: square example geometry}\n\\end{figure}\n\\begin{itemize}\n\t\\item The difussion constant for the region labelled mat1 is $k=12$ and its source term is $q=20$.\n\t\\item The difussion constant for the region labelled mat2 is $k=6$ and its source term is $q=5$.\n\\end{itemize}\nThe meshes used in this example are shown in Figure \\ref{fig: first example meshes} and vary from coarse to very fine.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m1.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m2.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m3.png}}\n\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m4.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m5.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m6.png}}\n\n\t\\captionof{figure}{Six of the meshes used in the first example.}\\label{fig: first example meshes}\n\\end{figure}\n\nThe numerical results for the maximum temperature value are exemplified in Table \\ref{table: square example numerical results}.\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Max. Temp. Value}}&\t\\multicolumn{2}{c|}{\\mbox{Max. Flux Magnitude}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\tFigure \\ref{fig: first example meshes}(a) &\n\t\t49 &\n\t\t80 &\n\t\t5.51836 &\n\t\t5.53345 &\n\t\t13.837 &\n\t\t13.453 \\\\\n\t\tFigure \\ref{fig: first example meshes}(b) &\n\t\t98 &\n\t\t162 &\n\t\t5.65826 &\n\t\t5.66648 &\n\t\t14.137 &\n\t\t14.024 \\\\\n\t\tFigure \\ref{fig: first example meshes}(c) &\n\t\t258 &\n\t\t466 &\n\t\t5.70585 &\n\t\t5.71709 &\n\t\t14.858 &\n\t\t14.770 \\\\\n\t\tFigure \\ref{fig: first example meshes}(d) &\n\t\t1,010 &\n\t\t1,914 &\n\t\t5.72103 &\n\t\t5.72280 &\n\t\t15.008 &\n\t\t15.006 \\\\\n\t\tFigure \\ref{fig: first example meshes}(e) &\n\t\t3,813 &\n\t\t7,424 &\n\t\t5.72725 &\n\t\t5.72725 &\n\t\t15.229 &\n\t\t15.228 \\\\\n\t\tFigure \\ref{fig: first example meshes}(f) &\n\t\t13,911 &\n\t\t27,420 &\n\t\t5.72821 &\n\t\t5.72826 &\n\t\t15.342 &\n\t\t15.337 \\\\\n\t\t&\n\t\t50,950 &\n\t\t101,098 &\n\t\t5.72841 &\n\t\t5.72842 &\n\t\t15.395 &\n\t\t15.396 \\\\\n\t\t&\n\t\t135,519 &\n\t\t269,700 &\n\t\t5.72845 &\n\t\t5.72845 &\n\t\t15.420 &\n\t\t15.417 \\\\\n\t\t&\n\t\t298,299 &\n\t\t594,596 &\n\t\t5.72848 &\n\t\t5.72848 &\n\t\t15.430 &\n\t\t15.429 \\\\\n\t\t&\n\t\t600,594 &\n\t\t1,198,330 &\n\t\t5.72848 &\n\t\t5.72848 &\n\t\t15.433 &\n\t\t15.433 \\\\\n\t\t&\n\t\t1,175,238 &\n\t\t2,346,474 &\n\t\t5.72849 &\n\t\t5.72849 &\n\t\t15.43724 &\n\t\t15.43724 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\captionof{table}{Numerical simulation results of the first example.}\\label{table: square example numerical results}  \n\\end{center}\nThe temperature and flux-magnitude distribution fields are shown in Figure \\ref{fig: square example temperature field}.    \n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.4\\textwidth]{Fig_14_Square_contour_temp.png}}\\hspace{3mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.4\\textwidth]{Fig_14_Square_contour_flux.png}}\n\t\\captionof{figure}{Temperature and flux-magnitude distribution fields of the first example.}\\label{fig: square example temperature field}\n\\end{figure}\n\nFigure \\ref{fig: first example diametral graphs} shows the graphs of the temperature and the flux-magnitude along a horizontal line crossing the inner circle for the first two meshes.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m1_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m1_diametral_flux.png}}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m2_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m2_diametral_flux.png}}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m3_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m3_diametral_flux.png}}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the first example along a cross-section of the domain for different meshes.}\\label{fig: first example diametral graphs}\n\\end{figure}\n\n\n\n\\newpage\n\n\\subsection{Second example: Anisotropy}\nLet us solve the Poisson equation in a circle of radius one centered at the origin $(0,0)$ under the following conditions (see Figure \\ref{fig: second example geometry}): \n\\begin{itemize}\n\t\\item heat anisotropic diffusion constants $K_x = 1.5, K_y=1.0$;\n\t\\item material angle $30^\\circ$;\n\t\\item source term $q= 1$;\n\t\\item Dirichlet boundary condition $u=10$.\n\\end{itemize}\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{Fig_16_CircleWithConditions.png}\n\t\\captionof{figure}{Disk of radius one.}\\label{fig: second example geometry}\n\\end{figure}\nThe meshes used in this example are shown in Figure \\ref{fig: second example meshes} and vary from very coarse to very fine.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh1.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh2.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh3.png}}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh4.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh5.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh6.png}}\n\n\t\\captionof{figure}{Six firsts meshes used for unit disk.}\\label{fig: second example meshes}\n\\end{figure}\nThe numerical results for the maximum temperature value ($u(0,0)=10.2$) are exemplified in Table \\ref{table: numerical results} where \na comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown. \n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Temp. Value at $(0,0)$}}&\t\\multicolumn{2}{c|}{\\mbox{Flux Magnitude at $(-1,0)$}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(a)}&\t17\t&\t20\t\t&\t10.20014&\t\t\t10.19002\t\t& 0.42133 & 0.43865\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(b)}&\t41\t&\t56\t\t&\t10.20007&\t\t\t10.19678\t\t& 0.48544 & 0.49387\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(c)}&\t201\t&\t344\t\t&\t10.20012&\t\t\t10.20158\t\t& 0.52470 & 0.52428\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(d)}&\t713\t&\t1304\t&\t10.20000&\t\t\t10.19969\t\t& 0.54143 & 0.54224\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(e)}&\t2455&\t4660\t&\t10.20000&\t\t\t10.19990\t\t& 0.54971 & 0.55138\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(f)}& \t8180&\t15862\t&\t10.20000&\t\t\t10.20002\t\t& 0.55326 & 0.55409\\\\\n\t\t& 20016\t&\t39198\t&\t10.20000&\t\t\t10.19999\t\t& 0.55470 & 0.55520\\\\\n\t\t& 42306\t&\t83362\t&\t10.20000&\t\t\t10.20000\t\t& 0.55540 & 0.55572\\\\\\hline\n\t\\end{tabular}    \n\t\\captionof{table}{Temperature value at the point $(0,0)$ and Flux magnitude value at the point $(-1,0)$ of the numerical simulations for the second example.}\\label{table: numerical results} \n\\end{center}\nThe temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure \\ref{fig: second example temperature field}.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.48\\textwidth]{Fig_18_CircleContourTemp.png}}\\hspace{2mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.48\\textwidth]{Fig_18_CircleContourFlux.png}}\n\t\\captionof{figure}{Temperature distribution and Flux magnitude fields for the finest mesh of the second example.}\\label{fig: second example temperature field}\n\\end{figure}\n\nFigures \\ref{fig: second example diametral graphs}(a), \\ref{fig: second example diametral graphs}(b) and \\ref{fig: second example diametral graphs}(c) \nshow the graphs of the temperature and flux magnitude values along a diameter of the circle for the different meshes of Figures \\ref{fig: second example meshes}(a), \\ref{fig: second example meshes}(b)\nand \\ref{fig: second example meshes}(c) respectively.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection01.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection01.png}}\\vspace{1mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection02.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection02.png}}\\vspace{1mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection03.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection03.png}}\\vspace{1mm}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the second example along a diameter of the circle for different meshes: \n\t\tmesh in Figure \\ref{fig: second example meshes}(a), a-Temperature, b-Flux; \n\t\tmesh in Figure \\ref{fig: second example meshes}(b), c-Temperature, d-Flux; \n\t\tmesh in Figure \\ref{fig: second example meshes}(c), e-Temperature, f-Flux; \n\t}\\label{fig: second example diametral graphs}\n\\end{figure}\n\n\n\\newpage\n\n\\subsection{Third example: Heterogeneity and anisotropy}\nLet us solve the Poisson equation in a circle of radius on the following domain (see Figure \\ref{fig: third example geometry}) with various material properties.\nThe geometry of the domain is defined by segments of ellipses passing through the given points which have centers at the origin $(0,0)$. \n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{Fig_20_HuevoGeometry.png}\n\t\\captionof{figure}{Egg-like domain with different materials.}\\label{fig: third example geometry}\n\\end{figure}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tPoint &$x$ &$y$ &Point &$x$ &$y$ \\\\\\hline\n\t\ta &-5 &0 &A &0 &-4 \\\\\n\t\tb &-4 &0 &B &0 &-3 \\\\\n\t\tc &-3 &0 &C &0 &-2 \\\\\n\t\td &-1 &0 &D &0 &-1 \\\\\n\t\te &1 &0 &E &0 &1 \\\\\n\t\tf &6 &0 &F &0 &2 \\\\\n\t\tg &7 &0 &G &0 &3 \\\\\n\t\th &8 &0 &H &0 &4 \\\\\\hline\n\t\\end{tabular}\n\\end{center}\n\\begin{itemize}\n\t\\item The Dirichlet boundary condition is $u=10$ and material properties (anisotropic heat diffusion constants, material angles and source terms) are given according to Figure \\ref{fig: third example dirichlet condition} and the table below.\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig_21_HuevoWithConditions.png}%\n\t\t\\captionof{figure}{Dirichlet condition.}\\label{fig: third example dirichlet condition}\n\t\\end{figure}\n\t\n\t\\begin{center} \n\t\t\\begin{tabular}{|l|c|c|c|c|}\\hline\n\t\t\t&    $K_x$\t&$K_y$&\tangle&\t$q$\\\\\\hline\n\t\t\tDomain mat1&\t5\t&25\t&30\t&15\\\\\\hline\n\t\t\tDomain mat2&\t25\t&5\t&0\t&5\\\\\\hline\n\t\t\tDomain mat3&\t50\t&12&\t45&\t5\\\\\\hline\n\t\t\tDomain mat4&\t10\t&35\t&0\t&5\\\\\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\n\\end{itemize}\nThe meshes used in this example are shown in Figure \\ref{fig: third example meshes}.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh1.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh2.png}}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh3.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh4.png}}\n\t\\captionof{figure}{Meshes for layered egg-like figure.}\\label{fig: third example meshes}\n\\end{figure}\nThe numerical results for the maximum temperature value ($u(0,0)=10.2$) are exemplified in Table \\ref{table: numerical results huevo} where \na comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown. \n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Max. Temp. Value}}&\t\\multicolumn{2}{c|}{\\mbox{Max. Flux Magnitude}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\tFigure \\ref{fig: third example meshes}(a) &\n\t\t342 &\n\t\t616 &\n\t\t2.79221 &\n\t\t2.79854 &\n\t\t18.41066 &\n\t\t18.40573 \\\\\n\t\tFigure \\ref{fig: third example meshes}(b) &\n\t\t1,259 &\n\t\t2,384 &\n\t\t2.83929 &\n\t\t2.84727 &\n\t\t18.93838 &\n\t\t18.91532 \\\\\n\t\tFigure \\ref{fig: third example meshes}(c) &\n\t\t4,467 &\n\t\t8,668 &\n\t\t2.85608 &\n\t\t2.85717 &\n\t\t19.13297 &\n\t\t19.13193 \\\\\n\t\tFigure \\ref{fig: third example meshes}(d)  &\n\t\t14,250  &\n\t\t28,506 &\n\t\t2.85994 &\n\t\t2.86056 &\n\t\t19.20982 &\n\t\t19.20909 \\\\\n\t\t&\n\t\t20,493 &\n\t\t40,316 &\n\t\t2.86120 &\n\t\t2.86177 &\n\t\t19.23120 &\n\t\t19.23457 \\\\\n\t\t&\n\t\t60,380 &\n\t\t119,418 &\n\t\t2.86219 &\n\t\t2.86231 &\n\t\t19.26655 &\n\t\t19.26628 \\\\\n\t\t&\n\t\t142,702 &\n\t\t283,162 &\n\t\t2.86249 &\n\t\t2.86256 &\n\t\t19.28045 &\n\t\t19.28028 \\\\\n\t\t&\n\t\t291,363 &\n\t\t579,360 &\n\t\t2.86263 &\n\t\t2.86267 &\n\t\t19.28727 &\n\t\t19.28755 \\\\\n\t\t&\n\t\t495,607 &\n\t\t986,724 &\n\t\t2.86275 &\n\t\t2.86269 &\n\t\t19.29057 &\n\t\t19.29081 \\\\\n\t\t&\n\t\t1,064,447 &\n\t\t2,122,160 &\n\t\t2.86272 &\n\t\t2.86273 &\n\t\t19.29385 &\n\t\t19.29389 \\\\\n\t\t&\n\t\t2,106,077 &\n\t\t4,202,536 &\n\t\t2.86274 &\n\t\t2.86274 &\n\t\t19.29618 &\n\t\t19.29615 \\\\\n\t\t&\n\t\t4,031,557 &\n\t\t8,049,644 &\n\t\t2.86275 &\n\t\t2.86275 &\n\t\t19.29763 &\n\t\t19.29765 \\\\\n\t\t\\hline\n\t\\end{tabular}    \n\t\\captionof{table}{Maximum temperature and Flux magnitude values in the numerical simulations of the third example.}\\label{table: numerical results huevo} \n\\end{center}\nThe temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure \\ref{fig: third example temperature field}.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.44\\textwidth]{Fig_23_HuevoContourTemp.png}}\\hspace{2mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.44\\textwidth]{Fig_23_HuevoContourFlux.png}}\n\t\\captionof{figure}{Temperature distribution and Flux magnitude fields for the finest mesh of the third example.}\\label{fig: third example temperature field}\n\\end{figure}\nFigure \\ref{fig: third example diametral graphs}\nshows the graphs of the temperature and flux magnitude values along a diameter of the circle for different meshes of Figure \n\\ref{fig: third example meshes}\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection01.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection01.png}}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection02.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection02.png}}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection03.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection03.png}}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the third example along a cross-section of the domain for different meshes: \n\t\tMesh in Figure \\ref{fig: third example meshes}(a), a-Temperature, b-Flux; \n\t\tMesh in Figure \\ref{fig: third example meshes}(b), c-Temperature, d-Flux; \n\t\tMesh in Figure \\ref{fig: third example meshes}(c), e-Temperature, f-Flux; \n\t}\\label{fig: third example diametral graphs}\n\\end{figure}\n\n\\newpage\n\n{\\bf Remark}. As can be seen from the previous examples, DEC behaves well\non coarse meshes. \nAs expected, the results of DEC and FEML are similar for fine meshes. \nWe would also like to point out the the computational costs of DEC and FEML are very similar.\n\n\n\n\\section{Conclusions}\\label{sec: conclusions}\n\nDEC is a relatively recent discretization scheme for PDE's which takes into account the geometric and analytic \nfeatures of the operators and the domains involved. \nThe main contributions of this paper are the following:\n\\begin{enumerate}\n \\item We have made explicit the local formulation of DEC, i.e. on each triangle of the mesh. As is customary, the local pieces can be assembled, which facilitates the implementation of DEC by the interested reader. \n Furthermore, the profiles of the assembled DEC matrices are equal to those of assembled FEML matrices.\n\n\\item Guided by the local formulation,\nwe have deduced a natural way to approximate the flux\/gradient vector of a discretized function as well as the anisotropic flux vector.\nWe have shown that the formulas defining the flux in DEC and FEML coincide. \n\n\\item We have deduced how the anisotropy tensor acts on primal 1-forms.\n\n\\item We have deduced the local DEC formulation of the 2D anisotropic Poisson equation, and\nhave proved that the DEC and FEML diffusion terms are identical, while the source terms are not \n-- due to the different area-weight allocation for the nodes.\n\n\n\\item Local DEC allows a simple treatment of heterogeneous material properties assigned to subdomains (element by element), \nwhich eliminates the need of dealing with it through ad hoc modifications of the global discrete Hodge star operator matrix. \n\n\\end{enumerate}\n\n\\vspace{.1in}\n\nOn the other hand we would like to point the following features:\n\\begin{itemize}\n\\item \n The area weights assigned to the nodes of the mesh when solving the 2D anisotropic Poisson equation can even be negative (when a triangle has an inner angle greater that $120^\\circ$), in stark contrast to the FEML formulation.\n\n\\item The computational cost of DEC is similar to that of FEML. While the numerical results of DEC and FEML on fine meshes are virtually identical, the DEC solutions are better than those of FEML on coarse meshes. Furthermore, DEC  solutions display numerical convergence.\n\n\\end{itemize}\n\n\\vspace{.1in}\n\nOur future work will include the DEC discretization of convective terms and DEC on 2-dimensional simplicial surfaces in 3D. \nPreliminary results on both problems are promising and competitive with FEML. \n\n\\bigskip\n\n{\\small\n\\renewcommand{\\baselinestretch}{0.5}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSequential Monte Carlo (SMC) is a class of algorithms that enable simulation from a target distribution of interest. These algorithms are based on defining a series of distributions, and generating samples from each distribution in turn. SMC was initially used in the analysis of state-space models. In this setting there is a time--evolving hidden state of interest, inference about which is based on a set of noisy observations \\citep{gordon1993,liuchen1998,SMCMiP,fearnhead2002}. The sequence of distributions are defined to be the posterior distributions of the state at consecutive time-points given the observations up to those time points. More recent work has looked at developing SMC methods that can analyse state-space models which have unknown fixed parameters. Such methods introduce steps into the algorithm to allow the support of the sample of parameter values to change over time, for example by using ideas from kernel density estimation \\citep{SMCMPliuwest}, or MCMC moves \\citep{gilks2001,storvik2002,fearnhead2002}.\n\nMost recently, SMC methods have been applied as an alternative to MCMC for standard Bayesian inference problems. \\citep{neal2001,chopin2002,delmoral2006,fearnhead2008}. In this paper the focus will be on methods for sampling from the posterior distribution of a set of parameters of interest. SMC methods for this class of targets introduce an artificial sequence of distributions that run from the prior to the posterior and sample recursively from these using a combination of Importance Sampling and MCMC moves. This approach to sampling has been demonstrated empirically to often be more effective than using a single MCMC chain \\citep{jasra2007,jasra2008}. There are heuristic reasons for why this may true in general: the annealing of the target and spread of samples over the support means that SMC is less likely to be become trapped in posterior modes.\n\nSimply invoking an untuned MCMC move within an SMC algorithm would likely lead to poor results because the move step would not be effective in combating sample depletion. The structure of SMC means that at the time of a move there is a sample from the target readily available, this can be used to compute posterior moments and inform the shape of the proposal kernel as in \\cite{jasra2008a}; however, further refinements can lead to even better performance. Such refinements include the scaling of estimated target moments by an optimal factor, see \\cite{roberts2001} for example. For general targets and proposals no theoretical results for the choice of scaling exist, and this has led to the recent popularity of adaptive MCMC \\citep{haario1998,andrieu2001,roberts2009,craiu2009,andrieu2008}. In this paper the idea of adapting the MCMC kernel within an SMC algorithm will be explored.\n\nTo date there has been little work at adapting SMC methods. Exceptions include the method of \\cite{jasra2008a}, whose method assumes a likelihood tempered sequence of target densities (see \\cite{neal2001}) and the adaptation procedure both chooses this sequence online, as well as computing the variance of a random walk proposal kernel used for particle dynamics. \\cite{cornebise2008} also considers adapting the proposal distribution within SMC for state-space models. Assuming that the proposal density belongs to a parametric family with parameter $\\theta$, their method proceeds by simulating a number of realisations for each of a range of values of $\\theta$ and selecting the value that minimises the empirical Shannon entropy of the importance weights; new samples are then re--proposed using this approximately optimal value. Further related work includes that of \\cite{douc2005a} and \\cite{cappe2008} on respectively population Monte Carlo and adaptive importance sampling.\n\nThe aims of this paper are to introduce a new adaptive SMC algorithm (ASMC) that automatically tunes MCMC move kernels and chooses between different proposal densities and to provide theoretical justification of the method. The algorithm is based on having a distribution of kernels and their tuning parameters at each iteration. Each current sample value, called a particle, is moved using an MCMC kernel drawn from this distribution. By observing the expected square jumping distance \\citep{craiu2009,sherlock2009} for each particle it is possible to learn which MCMC kernels are mixing better. The information thus obtained can then used to update the distribution of kernels. The key assumption of the new approach is that the optimal MCMC kernel for moving particles does not change much over the iterations of the SMC algorithm. As will be discussed, and shown empirically, in section \\ref{sect:resultsASMC} this can often be achieved by appropriate parameterisation of a family of MCMC kernels.\n\nThe structure of the paper is as follows. In the next section, the model of interest will be introduced and followed by a review of MCMC and SMC approaches. Then in Section \\ref{sect:newmethod}, the new adaptive SMC will be presented. Guidelines on implementing the algorithm as well as some theory on the convergence will be presented in Section \\ref{sect:theoryASMC}. In Section \\ref{sect:resultsASMC} the method will be evaluated using simulated data. The results show that the proposed method can successfully choose both an appropriate MCMC kernel and an appropriate scaling for the kernel. The paper ends with a discussion. \n\n\\section{Model}\n\nThe focus of this paper will be on Bayesian inference for parameters, $\\theta$, from a model where independent identically distributed data is available. Note that the ideas behind the proposed adaptive SMC algorithm can be applied more generally (see section \\ref{sect:discuss_extASMC}). Let $\\pi(\\theta)$ denote the prior for $\\theta$ and $\\pi(y|\\theta)$ the probability density for the observations. The aim will be to calculate the posterior density,\n\\begin{equation} \\label{eq:1}\n   \\pi(\\theta|y_{1:n})\\propto\\pi(\\theta)\\prod_{i=1}^n\\pi(y_i|\\theta),\n\\end{equation}\nwhere, here and throughout, $\\pi$ will be used to denote a probability density, and $y_{1:t}$ means $y_1,\\ldots,y_t$. \n\nIn general, $\\pi(\\theta|y_{1:n})$ is analytically intractable and so to compute posterior functionals of interest, for example expectations, Monte Carlo simulation methods are often employed. Sections \\ref{sect:MCMCintro} and \\ref{sect:SMCintro} provide a brief description of two such Monte Carlo approaches.\n\n\\subsection{MCMC\\label{sect:MCMCintro}}\n\nAn MCMC transition kernel, $K_h$, is an iterative rule for generating samples from a target probability density, for example a posterior. $K_h$ comprises a proposal kernel, here and throughout denoted $q_h$ (the subscript $h$ indicates dependence on a tuning parameter) and an acceptance ratio that depends on the target and, in general, the proposal densities (see \\cite{MCMCiP,gamermanlopes} for reviews of MCMC methodology). The most generally applicable MCMC method is Metropolis--Hastings, see Algorithm \\ref{alg:met_hast} \\citep{metropolis1953,hastings1970}.\n\n\\begin{algorithm}\n   \\caption{Metropolis--Hastings Algorithm \\citep{metropolis1953,hastings1970}}\n   \\label{alg:met_hast}\n   \\begin{algorithmic}[1]\n      \\STATE Start with an initial sample, $\\theta^{(0)}$, drawn from any density, $\\pi_0$.\n      \\FOR{$j=1,2,\\ldots$}\n         \\STATE Propose a move to a new location, $\\tilde\\theta$, by drawing a sample from $q_h(\\theta^{(i-1)},\\tilde\\theta)$.\n         \\STATE Accept the move (ie set $\\theta^{(i)} = \\tilde\\theta$) with probability,\n         \\begin{equation}\\label{eqn:MHacc}\n            \\min\\left\\{1,\\frac{\\pi(\\tilde\\theta|y_{1:n})}{\\pi(\\theta^{(i-1)}|y_{1:n})}\\frac{q_h(\\tilde\\theta,\\theta^{(i-1)})}{q_h(\\theta^{(i-1)},\\tilde\\theta)}\\right\\},\n         \\end{equation}\n         else set $\\theta^{(i)} = \\theta^{(i-1)}$.\n      \\ENDFOR\n   \\end{algorithmic}\n\\end{algorithm}\n\nProbably the simplest MH algorithm is the random walk Metropolis (RWM). The proposal kernel for RWM is a symmetric density centred on the current state, the most common example being a multivariate normal, $q_h(\\theta^{(i-1)},\\tilde\\theta)=\\mathcal{N}(\\tilde\\theta;\\theta^{(i-1)},h^2\\hat\\Sigma_\\pi)$, where $\\hat\\Sigma_\\pi$ is an estimate of the target covariance. Both the values of $\\hat\\Sigma_\\pi$ and $h$ are critical to the performance of the algorithm. If $\\hat\\Sigma_\\pi$ does not accurately estimate the posterior covariance matrix, then the likely directions of the random walk moves will likely be inappropriate. On the other hand, a value of $h$ that is too small will lead to high acceptance rates, but the samples will be highly correlated. If $h$ is too large then the algorithm will rarely move, which in the worst case scenario could lead to a degenerate sample. \n\nThese observations on the r\\^ole of $h$ point to the idea of an \\emph{optimal scaling}, a $h$ somewhere between the extremes that promotes the best mixing of the algorithm. In the case of elliptically symmetric unimodal targets, an optimal random walk scaling can sometimes be computed numerically; this class of targets includes the Multivariate Gaussian \\citep{sherlock2009}. Other theoretical results include optimal acceptance rates which are derived in the limit as the dimension of $\\theta$, $d\\rightarrow\\infty$ (see \\cite{roberts2001} for examples of targets and proposals). In general however, there are no such theoretical results. \n\nOne way of circumventing the need for analytical optimal scalings is to try to learn them online \\citep{andrieu2001,atchade2005}, this can include learning both a good scaling, $h$, and estimating the target covariance, $\\hat\\Sigma_\\pi$ \\citep{haario1998}. Recent research in adaptive MCMC has generated a number of new algorithms (see for example \\cite{andrieu2008,roberts2009,craiu2009}), though some care must be taken to ensure that the resulting chain has the correct ergodic distribution.\n\n\n\\subsection{Sequential Monte Carlo\\label{sect:SMCintro}}\n\nAn alternative approach to generating samples from a posterior is to use sequential Monte Carlo (SMC, see \\cite{delmoral2006} for a review). The main idea behind SMC is to introduce a sequence of densities leading from the prior to the target density of interest and to iteratively update an approximation to these densities. For the application considered here, it is natural to define these densities as $\\pi_t(\\theta)=\\pi(\\theta|y_{1:t})$ for $t=1,\\ldots,n$; this `data tempered' schedule will be used in the sequel. The approximations to each density are defined in terms of a set of weighted particles, $\\{\\theta_t^{(j)},w_t^{(j)}\\}_{j=1}^M$, produced so that as $M\\rightarrow\\infty$, Monte Carlo sums converge to their `correct' expectations:\n\\begin{equation*}\\label{eqn:properweight}\n   \\lim_{M\\rightarrow\\infty} \\left\\{\\frac{\\sum_{j=1}^Mw_t^{(j)}f(\\theta_t^{(j)})}{\\sum_{i=1}^Mw_t^{(i)}}\\right\\} = \\mathbb{E}_{\\pi_t(\\theta_t)}[f(\\theta_t)],\n\\end{equation*}\nfor all $\\pi_t$--integrable functions, $f$. One step of an SMC algorithm can involve importance reweighting, resampling and moving the particles via an MCMC kernel \\citep{gilks2001,chopin2002}. For concreteness, this paper will focus on the iterated batch importance sampling (IBIS) algorithm of \\cite{chopin2002}. \n\nThe simplest way to update the particle approximation in model (\\ref{eq:1}) is to let $\\theta_{t}^{(j)}=\\theta_{t-1}^{(j)}$ and\n$w_t^{(j)}=w_{t-1}^{(j)}\\pi(y_t|\\theta_{t}^{(j)})$. However such an algorithm will degenerate for large $t$, as eventually only one particle will have non-negligible weight. Within IBIS, resample--move steps (sometimes referred to here as simply `move steps') are introduced to alleviate this. In a move step, the particles are first resampled so that the expected number of copies of particle $\\theta_t^{(j)}$ is proportional to $w_t^{(j)}$. This process produces multiple copies of some particles. In order to create particle diversity, each resampled particle is moved by an MCMC kernel. The MCMC kernel is chosen to have stationary distribution $\\pi_t$. The resulting particles are then assigned a weight of $1\/M$.\n\nThe decision of whether to apply a resample-move step within IBIS is based on the effective sample size (ESS, see \\cite{kong1994,liuchen1998}). The ESS is a measure of variability of the particle weights; using this to decide whether to resample is justified by arguments within \\cite{liu1995} and \\cite{liu1998}. Full details of IBIS are given in Algorithm \\ref{alg:IBIS}.\n\n\\begin{algorithm}\n   \\caption{Chopin's IBIS algorithm}\n   \\label{alg:IBIS}\n   \\begin{algorithmic}[1]\n      \\STATE Initialise from the prior $\\{\\theta_0^{(j)},w_0^{(j)}\\}_{j=1}^M\\sim\\pi_0$.\n      \\FOR{$t=1,\\ldots,n$}\n         \\STATE Assume current $\\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M\\sim\\pi_{t-1}$\n         \\STATE Reweight $w_t^{(j)} = w_{t-1}^{(j)}\\pi_t(\\theta_{t-1}^{(j)})\/\\pi_{t-1}(\\theta_{t-1}^{(j)})$. Result: $\\{\\theta_{t-1}^{(j)},w_t^{(j)}\\}_{j=1}^M\\sim\\pi_t$.\n         \\IF{particle weights not degenerate (see text)}\n            \\STATE $\\{\\theta_{t}^{(j)},w_{t}^{(j)}\\}_{j=1}^M \\leftarrow \\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M$\n            \\STATE $t\\rightarrow t+1$.\n         \\ELSE\n            \\STATE \\label{alg:IBIS_state_resample} Resample: let $\\mathcal{K}=\\{k_1,\\ldots,k_M\\}\\subseteq\\{1,\\ldots,M\\}$ be the resampling indices, then $\\{\\theta_{t-1}^{(k)},1\/M\\}_{k\\in\\mathcal K}\\sim\\pi_t$. Relabel: $k_j \\leftarrow j$, the $j$th resampling index so that $\\{\\theta_{t-1}^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n            \\STATE Move via $\\pi_t$--invariant MCMC kernel. Result: $\\{\\theta_t^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n         \\ENDIF\n      \\ENDFOR\n   \\end{algorithmic}\n\\end{algorithm}\n\n\nChopin's IBIS algorithm is a special case of the resample--move (RM) algorithm of \\cite{gilks2001} and the general algorithm described by \\cite{delmoral2006} (note that the latter method applies beyond MCMC--within--SMC and provides a unifying framework for sampling from sequences of targets). The main difference between RM and IBIS is that, in their presentation of RM, \\cite{gilks2001} use resampling and move steps at each iteration of the sampler. Chopin noticed that at a particular iteration it may be better to just reweight the particles, rather than incur the computational cost and degeneracy induced by a resample--move step. Another related algorithm, a development of simulated annealing \\citep{kirkpatrick1983} due to \\cite{neal2001}, utilises an alternative `likelihood tempered' sequence of targets. The proposed target sequence is $\\pi_t(\\theta)=\\pi(\\theta)\\pi(y_{1:n}|\\theta)^{\\xi_t}$, where $\\{\\xi_t\\}$ is a sequence of real numbers starting at $0$ (the prior) and ending on $1$ (the posterior). Since each move step requires evaluation of the likelihood over all available observations, the main disadvantage of likelihood tempering is computational cost, although for models with sufficient statistics this is not an issue. Further disadvantages of Neal's proposed algorithm are the absence of resampling steps which eventually leads to sample degeneracy; and the lack of interpretability of intermediate target densities.\n\nThe efficiency of an SMC algorithm, such as IBIS, depends on the mixing properties of the associated MCMC kernel. Within SMC there is the advantage of being able to use the current set of particles to help tune an MCMC kernel. For example, the weighted particles can give an estimate of the posterior covariance matrix, which can be used within a random walk proposal. However even in this case, the proposal variance still needs to be appropriately scaled \\citep{roberts2001,sherlock2009}. In the next section the new adaptive SMC procedure will be introduced, the algorithm can learn an appropriate tuning for the MCMC kernel, and can also be used to choose between a set of possible kernels.\n\n\\section{The Adaptive SMC Sampler\\label{sect:newmethod}} \n\nFirst consider the case where the move step in the IBIS algorithm involves one type of MCMC kernel. Let $\\pi_t$ be an \\emph{arbitrary} continuous probability density (the target) and $K_{h,t}$ a $\\pi_t$--invariant MCMC kernel with tuning parameter, $h$. The parameter $h$ is to be chosen to maximise the following utility function,\n\\begin{eqnarray} \\label{eqn:optfun}\n   g^{(t)}(h) &=& \\int \\pi_t(\\theta_{t-1})K_{h,t}(\\theta_{t-1},\\theta_t)\\Lambda(\\theta_{t-1},\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\theta_t,\\\\\n   &=& \\mathbb{E}\\left[\\Lambda(\\theta_{t-1},\\theta_t)\\right], \\nonumber\n\\end{eqnarray}\nwhere $\\Lambda(\\theta_{t-1},\\theta_t)>0$ is a measure of mixing of the chain. Most MCMC adaptation criteria can be viewed in this way \\citep{andrieu2008}. Note that for simplicity of presentation, $\\Lambda$ only depends on the current and subsequent state, though the idea readily extends to more complex cost functionals, for example involving multiple transitions of the MCMC chain. The function $g^{(t)}$ is the average performance of the chain with respect to $\\Lambda$, which would normally be some measure of mixing. The ideal choice for $\\Lambda$ would be the integrated autocorrelation time (whence the goal would be to maximise $-g^{(t)}$), but a computationally simpler measure of mixing is the expected square jumping distance (ESJD). Maximising the ESJD is equivalent to minimising the lag-1 autocorrelation; this measure is often used within adaptive MCMC, see for example \\cite{sherlock2009,pasarica2010}. \n\nIn the following it will be assumed that the proposal distribution can depend on quantities calculated from the current set of particles (for example estimates of the posterior variance), but this will be suppressed in the notation.\nThe main idea of ASMC is to use the observed instances of $\\Lambda(\\theta_{t-1},\\theta_t)$ to help choose the best $h$. The tuning parameter will be treated as an auxiliary random variable. At time-step $t$ the aim is to derive a density for the tunings, $\\pi^{(t)}(h)$.  If a move step is invoked at this time, a sample of $M$ realisations from $\\pi^{(t)}(h)$, denoted $\\{h_t^{(j)}\\}_{j=1}^M$, will be drawn and `allocated' to particles at random. \n\nWhen moving the $j$th resampled particle, the tuning parameter $h_t^{(j)}$ will be used within the proposal distribution.\nFor notational simplicity this value will be denoted $h$ in the following.\nLet $\\theta_{t-1}^{(j)}$ be the $j$th resampled particle (see step \\ref{alg:IBIS_state_resample} of Algorithm \\ref{alg:IBIS}). In moving this particle, $\\tilde{\\theta}_t^{(j)}$ is drawn from $q_{h}(\\theta_{t-1}^{(j)},\\,\\cdot\\,)$, and accepted with probability \n$\\alpha_h(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$, given by (\\ref{eqn:MHacc}). If the proposed particle is accepted then $\\theta_t^{(j)}=\\tilde{\\theta}_t^{(j)}$ otherwise $\\theta_t^{(j)}={\\theta}_{t-1}^{(j)}$.\n\nThe utility function in (\\ref{eqn:optfun}) simplifies to,\n\\begin{equation*}\n   g^{(t)}(h) = \\int \\pi_t(\\theta_{t-1})q_h(\\theta_{t-1},\\tilde\\theta_t)\\tilde\\Lambda(\\theta_{t-1},\\tilde\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\tilde\\theta_t,\n\\end{equation*}\nwhere \n\\begin{equation*}\n   \\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})= \\alpha_h(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)}). \n\\end{equation*}\nSince by assumption the resampled particles are approximately drawn from $\\pi_t$ and proposed particles are drawn from $q_h(\\theta_{t-1},\\tilde\\theta_t)$, the quantity $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ can be viewed as an unbiased estimate of $g^{(t)}(h)$.\n\nThe approach in this paper is to use the observed $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ to update the distribution $\\pi^{(t)}(h)$ to a new distribution $\\pi^{(t+1)}(h)$. In particular each $h_t^{(j)}$ will be assigned a weight, $f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)}))$, for some function $f:\\mathbb{R}^+\\rightarrow\\mathbb{R}^+$. The new density of scalings will be defined,\n\\begin{equation} \\label{eq:pit}\n \\pi^{(t+1)}(h) \\propto \\sum_{i=1}^M f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})) R(h-h_t^{(j)}),\n\\end{equation}\nwhere $R(h-h_t^{(j)})$ is a density for $h$ which is centred on $h_t^{(j)}$. Simulating from $\\pi^{(t+1)}(h)$ is achieved by first resampling the $h_t^{(j)}$s with probabilities proportional to their weight and then adding noise to each resampled value; the distribution of this noise is given by $R(\\,\\cdot\\,)$. The motivation for adding noise to the resampled $h$--values is to avoid the distributions $\\pi^{(t)}(h)$ degenerating too quickly to a point-mass on a single value. Similar ideas are used in dynamic SMC methods for dealing with fixed parameters, for example \\cite{west1993,SMCMPliuwest}. In practice the variance of the noise can depend on the variance of $\\pi^{(t)}(h)$ and by analogy to Kernel density estimation should tend to 0 as the number of particles gets large.\n\nIf there is no resampling at step $t$ then set $\\pi^{(t+1)}(h)=\\pi^{(t)}(h)$. The scheme is initiated with an arbitrary distribution $\\pi(h)$.\nThe specific choice of $f$ considered in this paper is a simple linear weighting scheme,\n\\begin{equation*}\n   f(\\tilde\\Lambda) = a + \\tilde\\Lambda,\\qquad a\\geq 0.\n\\end{equation*} \nTheoretical justification for this choice is given in the next section.\n\nOne assumption of the proposed approach is that a good choice of $h$ at one time-step will be a good choice at nearby time-steps. Note that this is based on an implicit assumption within SMC that successive targets are similar (see \\cite{chopin2002,delmoral2006} for example). Furthermore, using estimates of posterior variances within the proposal distribution can also help ensure that good values of $h$ at one time-step will be a good choice at nearby time-steps. Some theoretical results concerning this matter will be presented in Section \\ref{sect:theoryASMC}.\n\nTo choose between different types of MCMC kernel is now a relatively straightforward extension of the above. Assume there are $I$ different MCMC kernels, each defined by a proposal distribution $q_{h,i}$, where $i \\in \\{1,\\ldots,I\\}$. The algorithm now learns a set of distributions, $\\pi^{(t)}(h,i)$, for the pair of kernel type and associated tuning parameter. Each particle is assigned a random kernel type and tuning drawn form this distribution, with the pair, $(h_{t-1}^{(j)},i_{t-1}^{(j)})$,  associated with $\\theta_{t-1}^{(j)}$. The algorithm proceeds by weighting this pair based on the observed $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ values as before, and updating the distribution, \n\\begin{equation}\\label{eqn:newkernelmixture}\n    \\pi^{(t)}(h,i) \\propto \\sum_{j=1}^M f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})) R(h-h_{t-1}^{(j)})\\delta_{i_{t-1}^{(j)}}(i).\n\\end{equation}\nwhere $\\delta_{i_{t-1}^{(j)}}(i)$ is a point mass on $i=i_{t-1}^{(j)}$. \nThe method is described in detail below, see Algorithm \\ref{alg:ASMC}. Within the specific implementation described, the sample of pairs, $(h,i)$, from $\\pi^{(t)}(h,i)$ are allocated to particles randomly immediately after the resample--move step at iteration $t$. These pairs are then kept until the next iteration a resample--move step is called.\n\n\\begin{algorithm}\n   \\caption{The Adaptive SMC algorithm. Here, $\\pi_0(\\,\\cdot\\,),\\ldots,\\pi_n(\\,\\cdot\\,)$ are an arbitrary sequence of targets; an MCMC kernel is assumed for particle dynamics.}\n   \\label{alg:ASMC}\n   \\begin{algorithmic}[1]\n      \\STATE Initialise from the prior $\\{\\theta_0^{(j)},w_0^{(j)}\\}_{j=1}^M\\sim\\pi_0$.\n      \\STATE Draw a selection of pairs of MCMC kernels with associated tuning parameters, $\\{(h_0^{(j)},K_{h,0}^{(j)})\\}_{j=1}^M\\equiv\\{(h_0^{(j)},i_0^{(j)})\\}_{j=1}^M\\sim\\pi(h,i)$, and attach one to each particle arbitrarily.\n      \\FOR{$t=1,\\ldots,n$}\n         \\STATE Assume current $\\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M\\sim\\pi_{t-1}$\n         \\STATE Reweight $w_t^{(j)} = w_{t-1}^{(j)}\\pi_t(\\theta_{t-1}^{(j)})\/\\pi_{t-1}(\\theta_{t-1}^{(j)})$. Result: $\\{\\theta_{t-1}^{(j)},w_t^{(j)}\\}_{j=1}^M\\sim\\pi_t$.\n         \\IF{particle weights not degenerate (see text)}\n            \\STATE $\\{\\theta_{t}^{(j)},w_{t}^{(j)}\\}_{j=1}^M \\leftarrow \\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M$\n            \\STATE $\\{(h_{t}^{(j)},K_{h,t}^{(j)})\\}_{j=1}^M \\leftarrow \\{(h_{t-1}^{(j)},K_{h,t-1}^{(j)})\\}_{j=1}^M$\n            \\STATE $t\\rightarrow t+1$.\n         \\ELSE\n            \\STATE Resample: let $\\mathcal{K}=\\{k_1,\\ldots,k_M\\}\\subseteq\\{1,\\ldots,M\\}$ be the resampling indices, then $\\{\\theta_{t-1}^{(k)},1\/M\\}_{k\\in\\mathcal K}\\sim\\pi_t$. Relabel: $k_j \\leftarrow j$, the $j$th resampling index so that $\\{\\theta_{t-1}^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$. DO NOT resample kernels or tuning parameters at this stage.\n            \\STATE Move $\\theta_{t-1}^{(j)}$ via the $\\pi_t$--invariant MCMC kernel, $K_{h,t}^{(j)}$, and tuning parameter $h_{t-1}^{(j)}$, denote the proposed new particle as $\\tilde\\theta_t^{(j)}$ and accepted\/rejected particle as $\\theta_t^{(j)}$. Result: $\\{\\theta_t^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n            \\STATE To obtain $\\{(h_{t}^{(j)},K_{h,t}^{(j)})\\}_{k=1}^M\\equiv\\{(h_t^{(j)},i_t^{(j)})\\}$, sample $M$ times from (\\ref{eqn:newkernelmixture}). Allocate the new selection to particles at random.\n         \\ENDIF\n      \\ENDFOR\n   \\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Theoretical Results \\label{sect:theoryASMC}}\n\nIn this section the proposed algorithm will be justified by a series of theoretical results; guidance as to how it should best be implemented will also be given. The results presented here apply in the limit as the number of particles, $M\\rightarrow\\infty$. As discussed above, in this limit, the variance of the kernel $R(\\,\\cdot\\,)$ in (\\ref{eq:pit}) tends to 0.\n\nTo simplify the discussion, it will be assumed that tunings are one dimensional (the arguments presented extend readily to the multivariate case). For a slight notational simplification, the criterion $\\Lambda$ will be used, rather than $\\tilde\\Lambda$ (as suggested in algorithm \\ref{alg:ASMC}); this does not affect the validity of any of the arguments, which also hold for $\\tilde\\Lambda$. The section is split into two parts.\n\nFirstly, in section \\ref{sect:onestep}, it is of interest to examine what happens to the distribution of the $h$s after one step of reweighting and resampling; this result will lead to a criterion for the choice of weight function that guarantees MCMC mixing improvement with respect to $\\Lambda$. In section \\ref{sect:manystep}, the sequential improvement of $h$s will be considered over many steps of the ASMC algorithm and with a changing target. General conditions for convergence of ASMC to the optimal kernel and tuning parameter will be provided.\n\n\\subsection{One Step Improvement and Weighting Function\\label{sect:onestep}}\n\nIn this section and in the relevant proofs, it is appropriate to temporarily drop the $t$ superscript, eg $g^{(t)}\\equiv g$, $\\theta_{t-1}\\equiv\\theta$ and $\\theta_t\\equiv\\theta'$. To study the effect of  reweighting and resampling on the distribution of the $h$s, suppose that currently $\\{h^{(j)}\\}_{j=1}^M\\stackrel{\\text{iid}}{\\sim}\\pi(h)$, the pdf of a random variable, $H$. The dependence on current and proposed particles means the weight attached to $h^{(j)}$ is random, but also, due to the independence of $h$ with the particles, is an unbiased estimator of the `true' weight, $\\mathbb{E}_{\\Theta,\\Theta'|H}[f(\\Lambda)|H=h^{(j)}]$, where $\\mathbb{E}_{\\Theta,\\Theta'|H}$ denotes the expectation with respect to the joint density of the random variables $\\Theta$ and $\\Theta'$ conditional on $H$. The true \\emph{weighting function} will be denoted,\n\\begin{equation}\\label{eqn:weightfun}\n   w(h) = \\mathbb{E}_{\\Theta,\\Theta'|H}[f(\\Lambda)|H=h].\n\\end{equation}\nThe following proposition, which is used repeatedly in subsequent results, shows how reweighting and resampling affects $\\pi(h)$.\n\n\\begin{Propn}\\label{propn:wboot}\n   Suppose that currently $\\{h^{(j)}\\}_{j=1}^M\\stackrel{\\text{iid}}{\\sim}\\pi(h)$, the pdf of a random variable, $H$, independent of $\\theta$. Let $w(h)$ be the weighting function defined as in (\\ref{eqn:weightfun}). Then in the limit as $M\\rightarrow\\infty$, the distribution of the reweighted and subsequently resampled $h$s is,\n   \\\n      \\pi^\\star(h) = \\frac{w(h)\\pi(h)}{\\int w(h)\\pi(h)\\mathrm{d} h}.\n   \\\n\\end{Propn}\n\n \\textbf{Proof:} See Appendix \\ref{sect:proofwboot}.\\qed\n\nSince ASMC uses a selection of $h$s, it is appropriate as a starting point to look for conditions under which their \\emph{distribution} is improved. It would be desirable if, over $\\pi^\\star(h)$, the objective function would on average take a higher value, for then the new distribution would on average perform better with respect to $\\Lambda$ than the old. This criterion can be stated in mathematical form: conditions on $f$ are sought for which,\n\\begin{equation*}\n   \\int\\pi^\\star(h)g(h)\\mathrm{d} h \\geq \\int\\pi(h)g(h)\\mathrm{d} h.\n\\end{equation*}\n\n\n\\begin{Lem}\\label{lem:improve_criterion}\nAssuming $g$ is $\\pi(h)$--integrable, in the limit as $M\\rightarrow\\infty$,\n\\begin{equation}\\label{eqn:cov_improve}\n   \\mathbb{E}_{\\pi^\\star(h)}[g(h)] \\geq \\mathbb{E}_{\\pi(h)}[g(h)]\\iff\\mathrm{cov}_{\\pi(h)}[g(h),w(h)] \\geq 0.\n\\end{equation}\nThat is, provided there is positive correlation between the objective function $g(h)$ and the weighting function, $w(h)$, the new distribution of $h$s will on average perform better (on $g(h)$) with respect to $\\Lambda$ than the old.\n\\end{Lem}\n\\textbf{Proof:} The result is obtained by expanding definitions in $(\\ref{eqn:cov_improve})$:\n\\begin{eqnarray*}\n   \\mathbb{E}_{\\pi^\\star(h)}[g(h)] &\\geq& \\mathbb{E}_{\\pi(h)}[g(h)],\\\\\n   \\iff\\mathbb{E}_{\\pi(h)}[w(h)g(h)] &\\geq& \\mathbb{E}_{\\pi(h)}[w(h)]\\mathbb{E}_{\\pi(h)}[g(h)],\\\\\n   \\iff\\mathrm{cov}_{\\pi(h)}[g(h),w(h)] &\\geq& 0.\n\\end{eqnarray*}\\qed\n\nAlthough this result does not directly yield a general form for $f$, it does give a simple criterion that must be fulfilled by any candidate function. An immediate corollary gives more concrete guidance:\n\n\\begin{Cor}\nA simple linear weighting scheme, $f(\\Lambda) = a + \\Lambda$, where $a\\geq0$, satisfies (\\ref{eqn:cov_improve}). \n\\end{Cor}\n\\textbf{Proof:} This is trivially verified using the linearity property of the covariance.\\qed\n\nA consequence of this lemma is that  the ASMC algorithm with linear weights will lead to sequential improvement with respect to $\\Lambda$ under very weak assumptions on the target and initial density for $h$. \nA linear weighting scheme may at first glance seem sub--optimal, and that it should be possible to learn $h$ more quickly using a function $f(\\Lambda)$ that increases at a super--linear rate. The present authors conjecture that such functions will not always guarantee an improvement in the distribution of $h$. For example consider $f(\\Lambda) = \\Lambda^2$, where the weighting function takes the form, $w(h) = g(h)^2 + \\mathbb{V}[\\Lambda|H=h]$. Because of the $\\mathbb{V}[\\Lambda|H=h]$ term, which may be large for values of $h$ where $g(h)$ is small, it is no longer true that $\\mathrm{cov}_{\\pi(h)}[g(h),w(h)]\\geq0$ in general.\n\n\\subsection{Convergence Over a Number of Iterations\\label{sect:manystep}}\n\nThe goal of this section is to provide a theoretical result concerning the ability of ASMC to update the distribution of $h$s with respect to a sequence of targets, $\\pi_1(\\theta_1),\\ldots,\\pi_n(\\theta_n)$. To simplify notation, it will be assumed that a move occurs at each iteration of the algorithm. The result can be extended to the case where moves occur intermittently, providing they incur infinitely often in the limit as the number of data points goes to infinity. \n\nDefine a set of functions, $\\{g^{(t)}(h)\\}_{t=1}^n$,\n\\begin{equation*}\n    g^{(t)}(h) = \\int \\pi_t(\\theta_{t-1})K_{h,t}(\\theta_{t-1},\\theta_t)\\Lambda(\\theta_{t-1},\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\theta_t\\geq0,\n\\end{equation*}\nwhere, for each $t$, $K_{h,t}$ is a $\\pi_t$--invariant MCMC kernel. \n\nFor a linear weighting scheme,\n\\begin{equation*}\n    \\pi^{(t)}(h) \\propto \\pi(h)\\prod\\limits_{s=1}^t(a+g^{(s)}(h)).\n\\end{equation*}\nBelow it will be shown that as $t\\rightarrow \\infty$ if the sequence of functions, $\\{g^{(t)}(h)\\}$, converge to a fixed function, $g(h)$, and if $g$ has a unique global maximum, $h_{\\text{opt}}$, then $\\pi^{(t)}(h)$ will converge to a point mass on $h_{\\text{opt}}$.\n\nThe key assumption of this theorem regards the convergence of the functions $\\{g^{(t)}(h)\\}$. This assumption is linked to the idea that a good value of $h$ for the target at time $t$ is required to be a good value at times later on. As mentioned above, the motivation behind SMC is that successive targets should be similar. Moreover, standard Bayesian asymptotic theory shows that as the number of observations, $n$, tends to infinity, the posterior tends in distribution to that of a Gaussian random variable. Thus, providing information from the current parameters about the posterior variance is used appropriately, it should be expected that the sequence of functions, $\\{g^{(t)}(h)\\}$, would also converge. This issue will be explored empirically in the next section.\n\n\\begin{Thm} \\label{thm:convvari}\n   Let $\\pi(h)$ be the initial density for the tuning parameter with support $\\mathcal{H}\\subseteq\\mathbb{R}$ and $a>0$. Define, as above, \n   \\begin{equation*}\n      \\pi^{(t)}(h) \\propto \\pi(h)\\prod\\limits_{s=1}^t(a+g^{(s)}(h)).\n   \\end{equation*}\n   Suppose there exists a function $g:\\mathcal{H}\\rightarrow\\mathbb{R}_{\\geq0}$ such that\n   \\begin{equation*}\n      \\sup_{h\\in\\mathcal H}|g^{(t)} - g| \\leq k_gt^{-\\alpha},\\qquad \\alpha\\in(0,1),\\ k_g>0.\n   \\end{equation*}\n   Furthermore, suppose $g$ has a unique global maximum, $h_{\\text{opt}}$, contained in the interior of $\\mathcal H$ and that $g$ is twice differentiable in an interval containing $h_{\\text{opt}}$. Then as $t\\rightarrow\\infty$, $\\pi^{(t)}(h)$ tends to a Dirac mass centred on the optimal scaling, $h_{\\text{opt}}$.\n\\end{Thm}\n\n\\textbf{Proof:} See Appendix \\ref{sect:proofvari}.\\qed\n\n\n\n\n\n\n\n\\section{Results \\label{sect:resultsASMC}}\n\nThis section is organised as follows. In Section \\ref{sect:convergeh}, the convergence of $h$ to an optimal scaling will be demonstrated empirically using a linear Gaussian model. Then in Section \\ref{sect:bayesian_mixtures} the problem of Bayesian mixture analysis will be introduced. In Sections \\ref{sect:detailsofSMCsims} and \\ref{sect:resultsofsims} the proposed method will be evaluated in simulation studies using the example of Bayesian mixture posteriors as defining the sequence of targets of interest.\n\nFollowing \\cite{sherlock2009}, the expected (Mahalanobis) square jumping distance will be considered as an MCMC performance criterion:\n\\begin{equation*}\n   \\Lambda(\\theta_{t-1},\\theta_t) =  (\\theta_{t-1}-\\theta_t)^T\\hat\\Sigma_{\\pi_t}^{-1}(\\theta_{t-1}-\\theta_t),\n\\end{equation*}\nwhere $\\theta_{t-1}$ and $\\theta_t$ are two points in the parameter space and $\\hat\\Sigma_{\\pi_t}$ is an empirical estimate of the target covariance obtained from the current set of particles. \n\nTwo different MCMC kernels will be considered within the SMC algorithm, these are defined by the following two proposals:\n\\begin{eqnarray*}\n   q_{\\text{rw}}(\\theta_{t-1},\\tilde\\theta_t) &=& \\mathcal{N}(\\theta_{t-1},h^2\\hat\\Sigma_{\\pi_t}),\\\\\n   q_{\\text{lw}}(\\theta_{t-1},\\tilde\\theta_t) &=& \\mathcal{N}(\\alpha\\theta_{t-1} + (1-\\alpha)\\bar{\\theta}_t,h^2\\hat\\Sigma_{\\pi_t}), \\qquad h\\in(0,1],\\ \\alpha=\\sqrt{1-h^2},\n\\end{eqnarray*}\nwhere $\\bar{\\theta}_t$ and $\\hat\\Sigma_{\\pi_t}$ are respectively estimates of the target and covariance. The first of these is a \\emph{random--walk} proposal. The second is based upon a method for updating parameter values in \\cite{SMCMPliuwest}, here named the `\\emph{Liu\/West}' proposal. \nThe Liu\/West proposal has mean shrunk towards the mean of the target and the imposed choice of $\\alpha=\\sqrt{1-h^2}$ sets the mean and variance of proposed particles to be the same as that of the current particles. Note that if the target is Gaussian, then this proposal can be shown to be equivalent to a Langevin proposal \\citep{roberts1996}.\n\n\\subsection{Convergence of $h$\\label{sect:convergeh}}\n\nIt is of interest to see an example $g(h)$ and demonstrate convergence of one of the proposed algorithms to the optimal scaling. This will be achieved using a Gaussian target, for which a useable analytic expression for the optimal scaling for the random walk kernel is available. The results in this section are based on 100 observations simulated from a 5--dimensional standard Gaussian density, $y_{1:100}\\stackrel{\\text{iid}}{\\sim}\\mathcal{N}(0,\\mathbb{I}_5)$, where $\\mathbb{I}_5$ is the $5\\times5$ identity matrix. The observation variance was assumed to be known and therefore the probability model, or likelihood, was specified as,\n\\begin{equation*}\n   \\pi(y|\\theta) = \\mathcal{N}(y;\\theta,\\mathbb{I}_5).\n\\end{equation*}\nThe prior on the unknown parameter, $\\theta$, the vector of means, was set to $\\mathcal{N}(0,5\\mathbb{I}_5)$. ASMC with a random walk proposal was used to generate $M=2000$ particles from the posterior. Resampling was invoked when the ESS dropped below $M\/2$ and no noise was added to the $h$s after resampling. The initial distribution for $h$ was chosen to be uniform on $(0,10)$. Note that this model admits exact inference via the Kalman Filter.\n\nThe left hand plot in Figure \\ref{fig:gaussiansimresults} shows $g(h)$ for this target. Note in this case that the sequence of functions, $\\{g^{(t)}\\}$, does not change much since each intermediate target is exactly Gaussian and the proposal is scaled by the approximate variance of the target. The optimum scaling, $h_{\\text{opt}}$, was computed using 1-dimensional numerical integration and Theorem 1 of \\cite{sherlock2009}. The right hand plot illustrates several features of the adaptive RWM; the resampling frequency, that the algorithm does indeed converge to the true optimal scaling and the approximate rate of this convergence.\n\n\n\\begin{figure}[htbp]\n \\begin{minipage}{0.5\\textwidth}\n      \\includegraphics[width=0.9\\textwidth,height=0.9\\textwidth]{5dimgaussiang_of_h.pdf}\n   \\end{minipage}\n   \\begin{minipage}{0.5\\textwidth}\n      \\includegraphics[width=0.9\\textwidth,height=0.9\\textwidth]{5dimgaussianconvergence.pdf}\n   \\end{minipage}\n   \\caption{\\label{fig:gaussiansimresults}Left plot: $g(h)$ for a 5--dimensional Gaussian Target, explored with RWM and with ESJD as the optimization criterion. Right hand plot: convergence of $h$ for the same density based on 100 simulated observations; the horizontal line is the approximately optimal scaling, $1.06$.}\n\\end{figure}\n\n\n\\subsection{Bayesian Mixture Analysis \\label{sect:bayesian_mixtures}}\n\nThe ability of the ASMC algorithm to learn MCMC tuning parameters in more complicated scenarios was evaluated using simulated data from mixture likelihoods (for a complete review of this topic, see \\cite{fruhwirth-schnatter}). Let $p_1,\\ldots,p_r>0$ be such that $\\sum_{i=1}^rp_i=1$. Let $\\mathcal{N}(\\,\\cdot\\,;\\mu,v)$ denote the normal density\nfunction with mean $\\mu$ and variance $v$. Let $\\theta=\\{p_{1:r-1},v_{1:r},\\mu_{1:r}\\}$.\n\nThe likelihood function for a single observation, $y_i$,is\n\\begin{equation}\\label{eqn:mixlik}\n   \\pi(y_i|\\theta) = \\sum_{j=1}^rp_j\\mathcal{N}(y_i;\\mu_j,v_j).\n\\end{equation}\nThe prior $\\theta$ was multivariate normal, on a transformed space using the generalised logit scale for the weights, log scale for variances, and leaving the means untransformed. The components of $\\theta$ were assumed independent \\emph{a priori}; the priors were $\\log(p_j\/p_r)\\sim\\mathcal{N}(0,1^2)$, $\\log(v_j)\\sim\\mathcal{N}(-1.5,1.3^2)$ and $\\mu_j\\sim\\mathcal{N}(0,0.75^2)$, where $j=1,\\ldots,r-1$ in the case of the weights and $j=1,\\ldots,r$ for the means and variances. The MCMC moves within the SMC algorithm were performed in the transformed space, using the appropriate inverse transformed values to compute the likelihood in (\\ref{eqn:mixlik}).\n\nAn issue with mixture models is that for the above choice of prior, the likelihood and posterior are invariant to permutation of the component labels \\citep{stephens2000}. As a result the posterior distribution has a multiple of $r!$ modes, corresponding to each possible permutation. One way of overcoming this problem is by introducing a constraint on the parameters, such as labelling the components so that $\\mu_1<\\mu_2<\\cdots<\\mu_r$, or so that $v_1<v_2<\\cdots<v_r$. This choice will affect the empirical moments of the resulting posterior and hence the proposal distribution of the MCMC kernel -- both the random walk and Liu\/West proposals depend on the posterior covariance, the latter also depending on the mean. In particular if there is a choice of ordering whereby the posterior is closer to Gaussian, then this is likely to lead to better mixing of the MCMC kernels. This phenomenon motivates the idea that it is also possible to choose between orderings on the parameter vector, which will be investigated in the sequel. \n\n\\subsection{Details of Implementation of ASMC \\label{sect:detailsofSMCsims}}\n\nIn analysing the simulated data, a number of SMC and ASMC algorithms were compared. These correspond to using the following MCMC kernels:\n\\begin{description}\n   \\item[RWfixed] Random walk ordered by means, with $h$ chosen based on the theoretical results for Gaussian targets \\citep{roberts2001,sherlock2009}.\n   \\item[RWadaptive] Adaptive random walk ordered by means with uniform prior on $h$\n   \\item[LWmean] Adaptive Liu\/West proposal ordered by means.\n   \\item[LWvariance] Adaptive Liu\/West proposal ordered by variances.\n   \\item[Kmix] Adaptive choice between random walk ordered by means, Liu\/West ordered on means and Liu\/West ordered on variances.\n\\end{description}\nIn each case the reference to ordering relates to how the component labels were defined, and thus affect the estimate of the posterior mean and covariance used. \n\nThe above methods were also compared with the adaptive MCMC algorithm of \\cite{haario1998}, denoted \\textbf{AMCMC}. The specific implementation used here is as follows. The prior densities were identical to those for ASMC, the parameter vector was ordered by means and the random walk tuning was computed using the approximately optimal Gaussian scaling of $h=2.4\/\\sqrt{3r-1}$. The AMCMC algorithm was run for 12000 iterations for the 5 dimensional datasets and for 30000 iterations for the 8 dimensional datasets: these values were chosen so as to approximately match the number of likelihood computations involved between the ASMC and AMCMC methods. The burn--in period was set to half of the number of iterations and the method was initialised by a draw from the prior. There was an initial non--adaptive phase, lasting 1000 iterations, where the proposal kernel was scaled by the prior covariance and after which scaling was via estimates of the posterior covariance computed from the chain to--date, this was updated every 100 iterations.\n\nFor the ASMC algorithms, the initial distribution of $h$s was chosen to be uniform on $(0,2)$ for the random walk and on $(0,1)$ for the Liu\/West proposal. In the case of the random walk, this range of $h$s can be justified by considering the optimal scaling for a random walk Metropolis on a multivariate Gaussian target in $5$ dimensions namely $2.38\/\\sqrt{5}=1.06$ (and decays with increasing dimension as $O(d^{-1\/2})$). For the Liu\/West, $h$ must be in $(0,1]$. \n\nIn each case a Gaussian kernel with variance $0.015^2$ was used in (\\ref{eq:pit}). A sensitivity analysis showed the effect of changing the variance of the noise slightly did not affect the conclusions of this research. The parameter for the linear $h$-weighting scheme was $a=0$. If any $h$ was perturbed below zero, a small value, $1\\times10^{-6}$, was imputed and similarly for the Liu\/West approach, any $h$ perturbed above 1 was replaced by 1.\n\nThe number of particles was set to $M=2000$ for the 2--mixture datasets and $M=5000$ for the 3--mixture datasets.  Each algorithm was run 100 times on each dataset with the order of observations randomised each time. For the MCMC based methods an ESS tolerance of $M\/2$ was used, as in \\cite{jasra2007}. Resampling of the particles was via residual sampling \\citep{whitley1994,liu1998}, but multinomial sampling was used in selecting $h$s. For ease of computing posterior quantities of interest, each of the above algorithms was forced to resample and move on the last iteration.\n\nTo compare the performance of different methods, a measure of the accuracy of the estimated predictive density was used. This is advantageous because it is invariant to re--labelling of the mixture components. The chosen accuracy measure was the variability of the predictive density (VPD) and was calculated as follows. Each run of the algorithm produces a weighted particle set, from which an estimate of $\\mathbb{E}[\\pi(y^{(i)}|y_{1:n})]$ can be obtained at 100 points, $\\{y^{(i)}\\}_{i=1}^{100}$, equi-spaced between -2.5 and 2.5. For each $i$, the 100 simulation runs produce 100 realisations of $\\mathbb{E}[\\pi(y^{(i)}|y_{1:n})]$; let $\\hat y^{(i,j)}$ be the estimate of $y^{(i)}$ obtained from run $j$.  The VPD measure used in this paper is\n\\begin{equation*}\n   \\mathrm{mean}_i[\\mathrm{var}_j(\\hat y^{(i,j)})],\n\\end{equation*}\nwhere $\\mathrm{mean}_i$ is the mean over the $i$s and $\\mathrm{var}_j$ is the variance of the estimates of $y^{(i)}$ obtained from the 100 simulations. The VPD gives an indication of the global variability of the predictive density across the simulations. In the tables, the relative VPD is used, which gives a scale--free comparison between methods. The SMC\/ASMC algorithm with a relative VPD of 1 is the reference algorithm and has the smallest VPD of the SMC\/ASMC methods; larger values indicate higher VPDs. For the AMCMC methods, the predictive densities were computed using all available samples ie with 6000 for the 2--mixture datasets and 15000 for the 3--mixture datasets. For the SMC\/ASMC methods a Rao--Blackwellised version of the predictive density was computed using all current and proposed particles available from the last iteration (that is, using 4000\/10000 sample points respectively for the 2\/3--mixture datasets).\n\n\n\\subsection{Results \\label{sect:resultsofsims}}\n\n100 realisations from were simulated from the following likelihoods:\n\\begin{eqnarray*}\n   \\text{Dataset 1:} & & \\pi(y|\\boldsymbol\\theta) = 0.5\\mathcal{N}(y;-0.25,0.5^2) + 0.5\\mathcal{N}(y;0.25,0.5^2),\\\\\n   \\text{Dataset 2:} & & \\pi(y|\\boldsymbol\\theta) = 0.5\\mathcal{N}(y;0,1^2) + 0.5\\mathcal{N}(y;0,0.1^2),\\\\\n   \\text{Dataset 3:} & & \\pi(y|\\boldsymbol\\theta) = 0.3\\mathcal{N}(y;-1,0.5^2) + 0.7\\mathcal{N}(y;1,0.5^2),\\\\\n   \\text{Dataset 4:} & & \\pi(y|\\boldsymbol\\theta) = 0.5\\mathcal{N}(y;-0.75,0.1^2) + 0.5\\mathcal{N}(y;0.75,0.1^2),\\\\\n   \\text{Dataset 5:} & & \\pi(y|\\boldsymbol\\theta) = \\T0.35\\mathcal{N}(y;-0.1,0.1^2) + 0.3\\mathcal{N}(y;0,0.5^2) + 0.35\\mathcal{N}(y;0.1,1^2),\\\\\n   \\text{Dataset 6:} & & \\pi(y|\\boldsymbol\\theta) = \\T0.25\\mathcal{N}(y;-0.5,0.1^2) + 0.5\\mathcal{N}(y;0,0.2^2) + 0.25\\mathcal{N}(y;0.5,0.1^2),\\\\\n\\end{eqnarray*}\n\nThis choice of datasets in combination with the selection of MCMC kernels allows several hypotheses to be tested empirically. Firstly, by comparing the performance of RWfixed with RWadaptive in these cases, it is possible to see whether anything is lost or gained by adapting the proposal kernel. Secondly, the impact of the different kernel orderings on MCMC mixing will become apparent by considering the performance of LWmean and LWvariance in these settings. Datasets 1, 3, 4 and 6 have well `separated' means and similar variances, so one might expect algorithms ordering by means to perform better; whereas datasets 2 and 5 have well separated variances and similar means, so perhaps the algorithms ordering by variances might do well here. Thirdly, the Kmix algorithm should be able to choose the best ordering and it is of interest to compare the results from this algorithm with an adaptive version of the individual kernels. \n\nThe simulation results from these datasets are presented in Table \\ref{tab:adaptive_1-4}. These give both the relative VPD for each method, but also an estimated mean ESJD for each method.\n\n\n\\begin{table}\n    \\centering\n\n   \\footnotesize\n\n\\vspace{0.5em}\n\\textbf{Dataset 1}\n\\vspace{0.5em}\n\n\n        \n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n         Method & Rel. VPD & JD & Acc. & $h$ & Propn. \\\\ \\hline\n        LWvariance & 1 & 1.869 & 0.3 & 0.941 &  \\\\\n        LWmean & 1.189 & 1.818 & 0.32 & 0.956 &  \\\\\n        Kmix & 1.258 & 1.845 & 0.317 & LWm 0.963 LWv 0.958 & LWm 0.785 LWv 0.215 \\\\\n        RWadaptive & 2.391 & 0.708 & 0.21 & 0.946 &  \\\\\n        AMCMC & 2.396 & 0.575 & 0.13 & 1.073 & $\\cdot$ \\\\\n        RWfixed & 3.414 & 0.641 & 0.18 & 1.064 &  \\\\ \\hline\n    \\end{tabular}\n\n\\vspace{0.5em}\n\\textbf{Dataset 2}\n\\vspace{0.5em}\n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n       \n        LWvariance & 1 & 9.139 & 0.873 & 0.978 &  \\\\\n        Kmix & 2.843 & 9.023 & 0.854 & LWm 0.984 LWv 0.978 & LWm 0.005 LWv 0.995 \\\\\n        AMCMC & 28.333 & 0.197 & 0.019 & 1.073 & $\\cdot$ \\\\\n        LWmean & 112.23 & 1.869 & 0.129 & 0.969 &  \\\\\n        RWadaptive & 188.094 & 0.77 & 0.134 & 0.584 &  \\\\\n        RWfixed & 219.907 & 0.596 & 0.041 & 1.064 &  \\\\ \\hline\n    \\end{tabular}\n\n\\vspace{0.5em}\n\\textbf{Dataset 3}\n\\vspace{0.5em}\n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n       \n        LWmean & 1 & 6.38 & 0.792 & 0.98 &  \\\\\n        Kmix & 1.54 & 6.378 & 0.806 & LWm 0.979 & LWm 1 \\\\\n        AMCMC & 7.465 & 0.847 & 0.146 & 1.073 & $\\cdot$ \\\\\n        RWfixed & 40.538 & 1.124 & 0.277 & 1.064 &  \\\\\n        RWadaptive & 45.739 & 1.057 & 0.369 & 1.045 &  \\\\\n        LWvariance & 148.827 & 0.737 & 0.064 & 0.966 &  \\\\ \\hline\n    \\end{tabular}\n\n\\vspace{0.5em}\n\\textbf{Dataset 4}\n\\vspace{0.5em}\n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n       \n         LWmean & 1 & 7.132 & 0.875 & 0.98 &  \\\\\n        Kmix & 1.099 & 7.127 & 0.877 & LWm 0.979 & LWm 1 \\\\\n        AMCMC & 24.024 & 0.462 & 0.057 & 1.073 & $\\cdot$ \\\\\n        RWadaptive & 48.606 & 1.143 & 0.274 & 1.086 &  \\\\\n        RWfixed & 51.919 & 1.167 & 0.298 & 1.064 &  \\\\\n        LWvariance & 1096.167 & 0.632 & 0.027 & 0.961 &  \\\\ \\hline\n    \\end{tabular}\n\n\\vspace{0.5em}\n\\textbf{Dataset 5}\n\\vspace{0.5em}\n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n       \n        AMCMC & 0.883 & 0.356 & 0.04 & 0.849 & $\\cdot$ \\\\\n        Kmix & 1 & 2.258 & 0.234 & LWm 0.964 LWv 0.971 & LWm 0.044 LWv 0.956 \\\\\n        LWvariance & 1.151 & 2.284 & 0.183 & 0.971 &  \\\\\n        LWmean & 2.792 & 1.007 & 0.092 & 0.961 &  \\\\\n        RWadaptive & 4.923 & 0.847 & 0.205 & 0.435 &  \\\\\n        RWfixed & 5.187 & 0.56 & 0.055 & 0.84 &  \\\\ \\hline\n    \\end{tabular}\n\n\\vspace{0.5em}\n\\textbf{Dataset 6}\n\\vspace{0.5em}\n\n    \\begin{tabular}{p{2.5cm}p{1.5cm}p{2cm}p{2cm}p{1.8cm}p{1.8cm}}\n       \n        LWmean & 1 & 4.099 & 0.277 & 0.972 &  \\\\\n        Kmix & 1.018 & 3.994 & 0.363 & LWm 0.973 & LWm 1 \\\\\n        AMCMC & 1.556 & 0.211 & 0.04 & 0.849 & $\\cdot$ \\\\\n        RWfixed & 3.244 & 0.996 & 0.429 & 0.84 &  \\\\\n        RWadaptive & 3.259 & 0.93 & 0.192 & 0.693 &  \\\\\n        LWvariance & 3.951 & 1.951 & 0.13 & 0.944 &  \\\\ \\hline\n    \\end{tabular}\n   \n\\caption{\\label{tab:adaptive_1-4} Rel. VPD is relative VPD, JD is the mean square jumping distance, Acc is the mean final acceptance probability, $h$ is the mean final scaling by kernel and Propn is the mean final kernel proportions. The kernels `LWm' and `LWv' indicate respectively a Liu\/West proposal ordering on means or variances.}\n\n\\end{table}\n\n\nAs would be hoped, a very strong correlation between lower VPD and higher ESJD is evident for the SMC\/ASMC algorithms, this empirically supports the use of ESJD as the chosen criteria for adapting the MCMC kernels.\n\nThere is relatively little difference across scenarios between the fixed and adaptive random walk methods. Furthermore, the adaptive random walk settles on a similar scaling as the fixed scaled version in datasets 3 and 4, whereas in datasets 1, 2, 5 and 6, RWadaptive settles to values below RWfixed. In datasets 1, 2, 4 and 5, the adaptive RW outperformed the fixed equivalent (though the difference was negligible in datasets 4 and 5); this is likely due to the fact that the covariance was not a good estimate and the adaptive version of the algorithm was able to rescale to compensate for this. In datasets 3 and 6, the fixed random walk marginally outperformed the adaptive.\n\nThe `correctly ordered' sequential Liu\/West algorithms considerably outperform the sequential RW--based methods in all six datasets and the incorrectly ordered versions perform worse or as poorly as the RW. For the Liu\/West proposals, the $h$ selected in each dataset was very close to 1: this special value corresponds to an independence kernel in the form of a moment--matched Gaussian approximation of the target. This is of interest as, in combination with the high acceptance rates of between $80$--$87$\\% in datasets 2--4, suggests that the `correct' ordering makes the target, ostensibly a very \\emph{complex} density function, approximately Gaussian in these cases.\n\nThe Kmix algorithm is able to choose between orderings; the advantages of this are clearly evidenced in the results, as it selects the best ordering in each case, with the exception of dataset 1 (where the means and variances are both similar). The Kmix sampler settles almost unanimously on one ordering above the others. These results show empirically that there is not much difference in using a single (correctly chosen) kernel compared with using a selection of kernels.\n\nThe performance of AMCMC was surpassed in all cases by the Kmix algorithm with the exception of dataset 5, where AMCMC was the best performing algorithm. In this latter case and in dataset 6, neither AMCMC nor the SMC\/ASMC algorithms performed well. AMCMC outperformed RWadaptive in each case apart from in dataset 1, where the difference was small. However, the results show the average jumping distance of the kernel used in the ASMC algorithm was greater than that of AMCMC in all cases, suggesting ASMC is able to adapt better to well-mixing kernels. To make this comparison more clear, two MCMC algorithms were run on each data-set, one using the final kernel found by AMCMC and one using a kernel based on the ASMC run, with the final estimated covariance matrix and the final mean value of the tuning parameter. The resulting MCMC algorithms performed very similarly in 3 cases (VPD of the two MCMC algorithms within 10\\% of each other) and the kernel found by ASMC performed better in the other 3 (VPD reduced by 30\\%, 40\\% and 80\\%). \n\n\n\n\\section{Discussion \\label{sect:discuss_extASMC}}\n\nThis paper introduces a new method for automatically tuning and choosing between different MCMC kernels. Where MCMC based SMC code already exists, adapting the $h$s would be a relatively straightforward means of enhancing performance, the main effort being in calculating the ratio of the proposed particles in the accept\/reject step. Probably the most important conclusion from the simulation studies presented is that there is not much lost in terms of performance in the adaption process -- the Kmix algorithm performed comparably to the respective best performing individual component and the adaptive random walk Metropolis performed similarly to the fixed, approximately optimally scaled version.\n\nAlthough the method as presented has assumed that i.i.d. observations are available from the likelihood, ASMC readily extends to the case of a dependent sequence. Furthermore, the extension to general sequences of target densities is immediate, and implied by the choice of notation in Algorithm \\ref{alg:ASMC}. The theoretical results presented in section \\ref{sect:theoryASMC} only apply to a one--dimensional $h$, in the case that the tuning parameter is a vector, the proposed algorithm and theoretical results still apply (with slight modifications), but convergence is likely to be at a slower rate.\n\nThe main assumption of ASMC is that a good $h$ at time $t$ is likely also to perform well at time $t+1$. One piece of evidence that supports this assumption is that the resampling frequency decreases with an increasing number of observations \\citep{chopin2002}. This implies that, although $\\pi_1$ and $\\pi_2$ may be quite different, $\\pi_{1001}$ and $\\pi_{1002}$ are likely to be less so, provided that the data provides sufficient information on the parameters. As mentioned earlier in the text, the assumption of similar successive target densities is also required for the non--adaptive version \\citep{chopin2002,delmoral2006}.\n\n\nASMC can be easily extended by considering other proposal densities. For example it is possible to formulate a $T$--distributed version of the Liu\/West proposal, this allows for heavier tailed proposals, the heaviness of which can be selected automatically by adaptively choosing the number of degrees of freedom; this $t$--based proposal includes the Liu\/West as a special case. Other interesting algorithms can be formulated using DE proposals \\citep{terbraak2006} (which generalises the snooker algorithm of \\cite{gilks1994}) or regional MCMC proposals \\citep{roberts2009,craiu2009} -- both of which appeal strongly to the particle structure of the new method.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe Casimir force \\cite{Casimir48} is the tiny long-range force between (neutral) macroscopic polarizable bodies that originates from the modification of the spectrum of quantum and thermal vacuum fluctuations of the electromagnetic  (em) field,  caused by the presence of the bodies. This phenomenon represents one of the rare manifestations of the quantum properties of the em field at the macroscopic scale. For  reviews   see Refs.\\cite{book1,parse,book2,woods,mehran}.\n\nA characteristic feature of the Casimir force is its {\\it non-additivity}, reflecting the   many-body character of fluctuation forces. This property  enormously complicates the  computation the Casimir force in  non-planar geometries. We recall that in his original paper Casimir worked out the force in the highly idealized geometry of two perfectly conducting plane-parallel surfaces at zero temperature.   Planar systems were the exclusive object of consideration also in the famous paper by Lifshitz \\cite{lifs},   who derived a general formula for the Casimir interaction between two plane-parallel slabs, taking into full account  realistic material properties of the plates, i.e. their frequency-dependent dielectric permittivity and finite temperature.\n\nUnfortunately, the theoretically simple planar geometry studied by Casimir and Lifshitz has been rarely used in experiments \\cite{sparnaay, gianni}, because of severe difficulties connected with controlling the parallelism of two macroscopic surfaces separated by a submicron gap.   To avoid these problems, the vast majority of Casimir experiments adopt the sphere-plate geometry (see for example  \\cite{lamor1,umar,semic,decca6,liq,ito,RSDPalasantzas2010,lamorth,chang2,bani2,deccamag} and Refs. therein), which is obviously immune from parallelism issues.  \n\nVery recently, a new experiment   \\cite{garrett}   has measured for the first time the (gradient of the) Casimir force   between two gold-coated spheres.  Following the practice of previous experiments, also in this new experiment the Casimir force has been computed using  the simple Proximity Force Approximation (PFA) \\cite{Derjaguin}, which expresses the Casimir force between two curved surfaces as the average of the plane-parallel force (as given by Lifshitz formula)  over the local separation between the surfaces.  The results of the experiment have been  found to be in good agreement with theoretical predictions based on Lifshitz formula.  By using  the measurements made with nine sphere-sphere and three sphere-plate systems of different radii,  the authors of  \\cite{garrett} could also place a bound  on the magnitude of  deviations from PFA,  using the same parametrization of the force that was used in the sphere-plate experiment of the IUPUI group  \\cite{ricardo}. \n\nMotivated by the new experiment, in this paper we compute beyond-PFA curvature corrections to the Casimir force gradient for   two gold spheres at room temperature, and we describe the correct parametrization of the force that should be used in the data analysis to measure corrections to PFA in this geometry.  We remind the reader that the computation of the Casimir force for non-planar  geometries has   been an untractable problem until recently.  Only in the early 2000's an exact scattering formula for the Casimir interaction between  dielectric objects of any shape, generalizing   early results of Balian and Duplantier \\cite{Balian}  and Langbein \\cite{langbein}, has been worked out  \\cite{sca1,sca2, kenneth}. At finite temperature $T$, the scattering formula has the form of a sum over so-called Matsubara (imaginary) frequencies $\\xi_n= 2 \\pi n k_B T\/\\hbar, \\;n=0,1,\\dots$ (with $k_B$ Boltzmann constant, and $\\hbar$ Planck constant) of functional determinants involving the multipole expansions of the  T-operators of the two bodies. Unfortunately, the scattering formula converges very slowly in the characteristic regime of Casimir experiments, in which the (minimum) separation $a$ between the  two surfaces  is very small compared to their characteristic radius of curvature $R$.  In order to obtain a precise estimate of the Casimir force, as it is needed  for a proper interpretation of current precision experiments,  it is necessary to push the computation  to very high multipole orders, which represents a very challenging task even for present day computers.  In \\cite{antoine1} simulations of the sphere-plate problem for Drude conductors were done up to  multipole order $l_{\\rm max}=24$, which were later \\cite{antoine2}  pushed to $l_{\\rm max}=45$, allowing to calculate the force for $a\/R \\ge 0.1$. Very recently \\cite{gert}, a  large-scale numerical simulation of the sphere-plate system going up to mutipole order $l_{\\rm max}=2 \\times 10^4$   reached for the first time the experimentally important region $a\/R \\sim 10^{-3}$. \n\n \nLarge numerical simulations of the scattering formula like that of \\cite{gert} require sophisticated algorithms for the computation of determinants of hierarchical matrices, that non every researcher may master or be willing to spend time on. This led us to investigate if the new  analytical tools that have been developed recently in the Casimir field,  could be exploited to construct easy-to-use formulae for the Casimir force having the high degree of precision demanded by current experiments.\nIn Ref. \\cite{bimonteprecise}   a formula with these features was constructed for the sphere-plate system. \n\nThe  approach followed in \\cite{bimonteprecise} can be described as follows.  As we said earlier, the scattering formula has the form of a sum over  Matsubara modes $\\xi_n$. The first term of this series, corresponding to $n=0$, represents a classical contribution to the Casimir interaction, which becomes dominant in the high temperature limit $a\/\\lambda_T \\gg 1$, where $\\lambda_T=\\hbar c\/2 \\pi k_B T$ ($\\lambda_T=1.2 \\;\\mu$m for room temperature)  is the thermal length. In Ref. \\cite{bimonteex1},  it was shown that this classical term can be evaluated {\\it exactly}  in the geometry of two metallic spheres of any radii, including the sphere-plate case as a special limit.  Of course, knowledge of the $n=0$ term, is not sufficient to compute the full Casimir energy. Unfortunately, the   $n>0$ terms of the scattering formula cannot be  computed exactly, but in \\cite{bimonteprecise}  it was shown that they can be computed very precisely using an  asymptotic small-distance formula,  which includes corrections to  PFA,    based on a recently proposed Derivative Expansion (DE) of the Casimir interaction \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3}. \n In  \\cite{bimonteprecise} it was proved that the semi-analytic approximate formula  for the sphere-plate Casimir force, resulting from the combination of  the exact $n=0$ term with the approximate expression of the $n>0$ terms,  is indeed  extremely precise  {\\it for all separations}. The   formula derived in \\cite{bimonteprecise} is in excellent agreement with the results of the large numerical simulation of \\cite{gert}.  In this paper, we extend the construction  of \\cite{bimonteprecise} to the sphere-sphere system.\n\nThe paper is organized as follows: in Sec. II we review the PFA for two spheres,  the scattering formula\nand display the exact  solution   for the classical Casimir energy of two Drude spheres  discovered in \\cite{bimonteex1}. In Sec. III we compute the contribution of the positive  Matsubara modes using the DE.   In Sec. IV we display our complete formula for the Casimir energy of two  spheres at finite temperature, and use it to compute deviations from PFA. In Sec. IV we also review the recent two-sphere experiment \\cite{garrett} and describe the correct  parametrization of corrections to PFA that should be used  in the data analysis of  experiments with two spheres. In Sec. V we present our conclusions. Finally, in Appendix A we review the DE and in Appendix B we use the DE to compute the leading curvature correction to the force gradient between two spheres. \n\n\n\n\\section{Casimir interaction of two spheres}\n\nWe consider a system composed by two spheres of respective radii $R_1$ and $R_2$ placed in a vacuum and separated by a gap of width $a$ (see Fig. 1). \n\nAs it was explained in the introduction,  until recently there were no tools to exactly compute the Casimir force in non-planar geometries, and so one had to resort to the old-fashioned PFA.  In the case of two spheres\n\\cite{parse} the PFA formula for the Casimir force is:\n\\begin{equation}\nF^{(\\rm PFA)}(a,R_1,R_2)= 2 \\pi {\\tilde R} \\;{\\cal F}^{(\\rm pp)}(a)\\;,\\label{PFA1}\n\\end{equation}     \nwhere ${\\tilde R}$ is the {\\it effective} radius of the two spheres\n\\begin{equation}\n{\\tilde R}=\\frac{R_1 R_2}{R_1+R_2}\\;,\n\\end{equation}\nand  ${\\cal F}^{(\\rm pp)}(a)$ is the unit-area Casimir free energy for two plane-parallel slabs respectively made of the same materials as the sphere and plate, whose expression was derived long ago by Lifshitz \\cite{lifs}:\n\\begin{eqnarray}\n&&\n{\\cal F}^{(\\rm pp)}(a,T)=\\frac{k_BT}{2\\pi} \\sum_{n \\ge 0}\\;\\!\\!'\n \\int_{0}^{\\infty}\nk_{\\bot}dk_{\\bot} \\nonumber\n\\\\\n&&\\times\n\\sum\\limits_{\\alpha={\\rm\nTE,TM}}\\ln\\left[1-\n{r_{\\alpha}^2({\\rm i}\\,\\xi_n,k_{\\bot})}e^{-2aq_n}\\right],\\label{PBeq3}\n\\end{eqnarray}\nwhere the prime in the sum over $n$ indicates that the $n=0$ term is taken with a weight 1\/2,  $T$ is the temperature of the plates, $k_{\\bot}$ is the in-plane momentum, $r_{\\alpha}({\\rm i}\\,\\xi_n,k_{\\bot})$ denotes the Fresnel reflection coefficient for polarization $\\alpha={\\rm TE, TM}$ of a thick slab, evaluated for the imaginary frequency $\\omega={\\rm i}\\, \\xi_n$ and \n$q_n=\\sqrt{\\xi_n^2\/c^2+k_{\\bot}^2}$. The PFA for the force gradient $F' \\equiv \\partial F\/\\partial a$  (here and in what follows a  prime shall denote a derivative with respect to the separation) easily follows from Eq. (\\ref{PFA1}):\n\\begin{equation}\n{F'}^{(\\rm PFA)}(a,R_1,R_2)=- 2 \\pi {\\tilde R}\\; {F}^{(\\rm pp)}(a)\\;,\\label{PFA2}\n\\end{equation}\nwhere ${F}^{(\\rm pp)}(a)=-\\partial  {\\cal F}^{(\\rm pp)}\/\\partial a$ is the unit-area Casimir force for two parallel slabs. The  PFA  force and force-gradient  for a  sphere of radius $R$ opposite a plane, is recovered from Eqs. (\\ref{PFA1}) and (\\ref{PFA2}), respectively, by taking the radius of one of the two spheres to infinity, i.e. substituting  ${\\tilde R}$ by $R$. It is important to remark that within the PFA, both $F$ and $F'$ depend on the radii of the two spheres only via the effective radius ${\\tilde R}$.\n \nFor a proper interpretation of current precision Casimir experiments it has become important to estimate curvature corrections beyond PFA. This has been impossible until recently, when \nan exact scattering formula providing the Casimir energy  of two compact dielectric bodies has been worked out \\cite{sca1,sca2,kenneth}.  The general structure of the scattering formula is:\n\\begin{equation}\n{\\cal F}=k_B T \\sum_{n \\ge 0}\\;\\!\\!' \\;{\\rm Tr} \\ln[1-\\hat{M}(\\rm {i} \\xi_n)]\\;,\\label{freeen}\n\\end{equation}  \nwhere the prime sign in the sum indicates again that the $n=0$ term is taken with weight 1\/2.  The trace ${\\rm Tr}$ in this equation is  over both spherical multipoles indices $(l,m)$ and   polarization indices $\\alpha={\\rm TE,TM}$:\n\\begin{equation}\n{\\rm Tr}= \\sum_{m=-\\infty}^{\\infty} \\sum_{l=  |m|}^{\\infty} {\\rm tr}\\;,\n\\end{equation}  \nwhere ${\\rm tr}$ denotes the trace over $\\alpha$. The matrix elements  $M_{lm\\alpha,m'l'\\alpha'}$ of ${\\hat M}$ shall not be reported here for brevity. Their explicit expressions can be found for example in Refs.\\cite{rahi,teo2}. We just recall that the matrix $M_{lm\\alpha,m'l'\\alpha'}$ involves a product of the T-matrices for the two bodies (i.e. the Mie scattering coefficients in the case of  two spheres), both evaluated for the imaginary Matsubara frequencies ${\\rm i} \\xi_n$,  intertwined  with  translation matrices that serve to convert the mutipole basis relative to either body into the multipole basis relative to the other body (see Refs.\\cite{rahi,teo2} for details). The expressions for the Casimir force $F=-  {\\cal F}'$ and its gradient $F'$  are obtained by taking  derivatives of Eq. (\\ref{freeen}) with respect to the separation $a$. Using the scattering formula  it has been possible to   prove eventually that the PFA formula is indeed asymptotically exact for small separation in the sphere-plate and cylinder-plate geometries  \\cite{bordag}. \n\nHaving at our disposal the exact representation  Eq. (\\ref{freeen}), it is natural to ask whether it can be used efficiently to accurately compute the Casimir force in concrete experimental situations. Unfortunately, this is not easy at all. Consider as an example the geometry of a sphere of radius $R$ at a minimum distance $a$ from a plate. The problem is that to obtain a precise estimate of the Casimir force for experimentally relevant sphere-plate separations (typical aspect ratios $a\/R \\sim 10^{-3}$) it is necessary to include a huge number of multipoles in the computation. Previous\nworks  \\cite{sca2,antoine1,antoine2,bimonteprecise,gert} found that the multipole order $l_{\\rm max}$ for which convergence  is achieved  scales as $l_{\\rm max} \\sim R\/a$. To date, the largest numerical simulation  of the sphere-plate scattering formula reached up to $l_{\\rm max}=2 \\times 10^4$   \\cite{gert},  which allowed the authors of \\cite{gert} to probe the Casimir force in  the experimentally relevant region $a\/R \\sim   10^{-3}$. Managing such a large number of multipoles on a computer is not easy at all, and sophisticated algorithms are needed to handle the problem. \n\n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{setup\n\\caption{\\label{setup}  The sphere-sphere Casimir setup. The sphere-sphere geometry is characterized by the {\\it effective} radius ${\\tilde R}=R_1 R_2\/(R_1+R_2)$ and the dimensionless parameter $u={\\tilde R}^2\/R_1 R_2$}\n\\end{figure}\n\nAt this point we turn to our objective of deriving a simple and very accurate formula for the Casimir force between two spheres. To do this, we go back to the general scattering formula Eq. (\\ref{freeen}). As we see, it has the form of a sum of terms ${\\cal F}=\\sum'_{n \\ge 0}{\\cal F}_{n}$  over the Matsubara frequencies $\\xi_n,\\;n=0,1,\\dots$. It is convenient for our purposes to separate the first term ${\\cal F}_{n=0}$ of the series from the the remaining terms with $n>0$. We accordingly decompose Eq. (\\ref{freeen}) as:\n\\begin{equation}\n{\\cal F}={\\cal F}_{n=0}+{\\cal F}_{n>0}\\;,\n\\end{equation}\nwhere we set ${\\cal F}_{n>0}=\\sum_{n > 0}{\\cal F}_{n}$.\nConsider first ${\\cal F}_{n=0}$. This term represents a {\\it classical} contribution to the Casimir energy, which provides the dominant contribution to the full Casimir energy ${\\cal F}$ of the system in the limit of large separations $a \\gg \\lambda_T$.  \n\nIn Ref. \\cite{bimonteex1}, it was shown that this classical term can be evaluated {\\it exactly}  in  the following two cases. The first one is that  of a scalar field obeying Dirichlet (D) boundary conditions (bc) on the surfaces of two spheres of arbitrary radii, including the sphere-plate geometry as a special case. The second case is that of a scalar field obeying so-called Drude bc on the surfaces of a sphere opposite a plate \\footnote{The sphere-plate solution with Drude bc presented in \\cite{bimonteex1} does not extend to the sphere-sphere geometry.}. The latter bc is identical to D bc, apart from the  fact that in the Drude case   charge monopoles (corresponding to index $l=0$) are excluded from  the scattering formula. Both sets of bc can be used to describe ohmic conductors, depending on the electric configuration of the system. Drude bc describe isolated conductors, whose total charge is fixed, while D bc  describe conductors whose voltages are fixed. The latter is the experimentally important situation,  since in all Casimir experiments (including the experiment \\cite{garrett}) one plate is  grounded, while the other is connected with a voltage generator  which serves to apply a bias potential, to compensate for unavoidable potential differences between the plates, resulting from differences in the respective work functions. The classical sphere-sphere Casimir energies  implied by D and Drude bc  are undistinuishable for separations much smaller than the spheres radii, while  the two models lead to distinct asymptotic behaviors in the limit of large separations $a \\gg (R_1,R_2)$, since   ${\\cal F}^{\\rm (D)}_{\\rm n=0} \\sim -k_B T R_1 R_2\/a^2$, while   ${\\cal F}_{\\rm n=0}^{(\\rm Dr)} \\sim -k_B T R_1^3 R_2^3\/a^6$. \n\nFor two  metallic spheres obeying D bc the exact solution worked out in \\cite{bimonteex1} is:  \n\\begin{equation}\n{\\cal F}_{n=0}^{(\\rm ex)}=\\frac{k_B T}{2} \\sum_{l=0}^{\\infty} (2 l +1) \\ln[1-Z^{2 l+1}] \\; ,\\label{enerDr}\n\\end{equation}\nwhere $Z$ is:\n\\begin{equation}\nZ=[1+x+x^2 u\/2+\\sqrt{(x+x^2 u\/2)\\,(2+x+x^2 u\/2)}]^{-1}\\;.\\label{Zdef}\n\\end{equation}\nIn the above Equation, $x=a\/\\tilde{R}$,  and $u=\\tilde{R}^2\/R_1 R_2$. The parameter $u$ depends only on the ratio between $R_1$ and $R_2$ and  takes values in the interval $[0,1\/4]$,  the upper bound $u=1\/4$ corresponding to two equal spheres $R_1=R_2$, while the lower bound $u=0$ is approached as either of the two radii becomes infinite. The  special case of a sphere of radius $R$ opposite a plate is thus recovered by taking $u=0$ into Eq. (\\ref{Zdef}), and setting $\\tilde{R}=R$.  The  range of the variable $Z$ in Eq. (\\ref{Zdef}) is the interval $[0,1]$,  the upper (lower) bound $Z=1$ ($Z=0$) corresponding to the limit of vanishing (infinite) separation $x\\rightarrow 0$ ($x\\rightarrow \\infty$). \nThe corresponding expression for the Casimir force $F_{n=0}^{(\\rm ex)}= - {{\\cal F}'}_{n=0}^{(\\rm ex)}$ and its derivative ${F'}_{n=0}^{(\\rm ex)}$ are easily obtained by deriving Eq. (\\ref{enerDr}) with respect to $a$.  As we see from the expression of $Z$ in Eq. (\\ref{Zdef}),  the exact classical energy ${\\cal F}_{n=0}^{(\\rm ex)}$ depends not only on the effective radius ${\\tilde R}$, but also on the ratio among the radii via the variable $u$. This feature marks an important difference with respect to the PFA formula.\n\nIn \\cite{bimonteex1} the small distance expansion of  ${\\cal F}_{n=0}^{(\\rm ex)}$ for the sphere-plate system was  worked out, by setting $Z=\\exp(-\\mu)$ and then taking the small-$\\mu$ asymptotic expansion of the series on the r.h.s. of Eq. (\\ref{enerDr}). Using the formulae of Ref. \\cite{bimonteex1}, it is easy to verify that for small separations  the sphere-sphere  force gradient has the expansion:\n\\begin{equation}\n {F'}_{n=0}^{(\\rm ex)}=k_B T\\frac{ \\zeta(3) {\\tilde R}}{4\\, a^3} \\left(1+\\frac{1}{12 \\zeta(3)} \\frac{a}{\\tilde R} +o(a\/{\\tilde R}) \\right)\\;,\\label{DEnzero}\n\\end{equation}\nwhere   $\\zeta(x)$ is Riemann zeta function.\nThe leading term coincides with the PFA Eq. (\\ref{PFA2}), since from Lifshitz formula we find  ${F}^{(\\rm pp)}_{n=0}=-k_B T \\zeta(3)\/(8 \\pi a^3)$ for two Drude-metal plates, while the next term provides the leading correction to PFA. Interestingly, like the PFA, also the latter correction is independent of the parameter $u$. The correction to PFA in Eq. (\\ref{DEnzero}) is consistent with the DE (see Eq. (\\ref{DEformula})).\n\nAt this point we need consider the contribution ${\\cal F}_{n>0}$ of the positive Matsubara modes $n>0$ to the free energy. Unfortunately, differently from the classical term ${\\cal F}_{n=0}$, the quantity ${\\cal F}_{n>0}$ cannot be computed exactly. Of course ${\\cal F}_{n>0}$ be computed numerically,  using the scattering formula truncated to a finite multipole order $l_{\\rm max}$. As we explained earlier, such a computation is however very challenging, because the multipole order $l_{\\rm max}$ that is necessary\nis very large for  experimentally relevant values of the radii and separation. Below we obtain a very precise and simple analytical formula for ${\\cal F}_{n>0}$,  by using the so-called Derivative Expansion  \n \\cite{fosco1,bimonte1,fosco2,bimonte2}.\n\n\\section{Derivative Expansion of ${\\cal F}_{n>0}$}\\\n\nThe DE \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3} is an analytical technique to compute curvature corrections to proximity forces beween two surfaces of small slope. For the benefit of the reader, we provide in Appendix A a short review to the DE and a guide to the relevant References.\n\nIn this Section we use the DE to estimate the contribution ${\\cal F}_{n>0}$ of the non-zero Matsubara modes to the Casimir energy. The DE is particularly well suited to this task, as we now explain. By it very nature, the DE is expected to be very precise in situations in which the slope of the surfaces  is small within the interaction region. A little reasoning shows that this condition is met in the problem at hand for all separations $a$ between the spheres, just provided (as it always is the case in current experiments) that the radii   of the spheres are both large compared to the thermal length $\\lambda_T$ ($\\lambda_T=1.2\\;\\mu$m at room temperature).   \n\nIt is a well known fact \\cite{parse,book2} that the Casimir interaction between two surfaces, with a characteristic radius of curvature $R$,  is localized inside a disk of radius $\\rho \\sim \\sqrt{a R}$ around the point of either surface  which is closest to the other. \nThus, one is led to expect  that the DE is applicable in general only  for  separations $a$ such that $\\rho\/R=\\sqrt{a\/R} \\ll1$. This condition is usually well satisfied in Casimir experiments, for which typically ${a\/R} < 0.01$. A  closer inspection reveals however that the DE of ${\\cal F}_{n>0}$   is in fact valid for {\\it all} separations, provided that $R_1$ and $R_2$ are both larger than $\\lambda_T$. The point to consider is that {\\it positive} Matsubara modes of (imaginary) frequency $\\xi_n$ can only propagate across a distance\nof order $\\ell_n=c\/\\xi_n =\\lambda_T\/n \\le \\lambda_T$.  Because of this constraint, the true size of the interaction region is actually not  $\\sqrt{a R}$, but instead $\\rho=\\min (\\sqrt{a R},\\sqrt{\\lambda_T R})$. This implies that  the DE for ${\\cal F}_{n>0}$ is actually valid for separations such that $\\rho\/R=\\min (\\sqrt{a\/R},\\sqrt{\\lambda_T\/R}) \\ll 1$. The latter condition is clearly satisfied for all separations, provided that $R_1$ and $R_2$ are both much larger than $\\lambda_T$.  \n\nWe have thus established that the DE is a valid method for ${\\cal F}_{n>0}$. In Appendix B it is shown that the DE leads to a simple general formula Eq. (\\ref{DEformula}) for the force gradient between two spheres. The  formula for the DE expansion of  ${\\cal F}_{n>0}$ can obtained by making into  Eq. (\\ref{DEformula}) the appropriate substitutions:     \n\\begin{equation}\n{ F}'_{n>0}= -2 \\pi {\\tilde R} {F}^{(\\rm pp)}_{n>0}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u \\,\\kappa(a) \\right) \\frac{a}{\\tilde R} \\right]\\;, \\label{DEformula1}\n\\end{equation}\nwhere the coefficients $\\tilde \\theta(a)$ and $\\kappa(a)$ are (see Eqs. (\\ref{thetacoe}) and (\\ref{kappacoe}))\n\\begin{eqnarray}\n{\\tilde \\theta}&=&  \\frac{{{\\cal F}}^{(\\rm pp)}_{n>0}(a) -2 \\alpha_{n>0}(a)}{a {F}^{(\\rm pp)}_{n>0}(a)}\\;,\\label{thetacoe1}\\\\\n\\kappa(a) &=&  1-2 \\frac{{ {\\cal F}}^{(\\rm pp)}_{n>0}(a)}{ a { F}^{(\\rm pp)}_{n>0}(a)} \\;.\\label{kappacoe1}\n\\end{eqnarray}\nIn the above Equations, $ {\\cal F}^{(\\rm pp)}_{n>0}$ denotes the contribution of the $n>0$ modes to Lifshitz formula Eq. (\\ref{PBeq3}), and ${F}^{(\\rm pp)}_{n>0}=-\\partial {\\cal F}^{(\\rm pp)}_{n>0}\/\\partial a$ is the corresponding (unit area) force.  \n\nAn important ingredient of Eqs. (\\ref{DEformula1}-\\ref{kappacoe1}) is the coefficient $\\alpha_{n>0}(a)$ which enters into the expression of $\\tilde{\\theta}$ in Eq. (\\ref{thetacoe1}). As it is explained in Appendix A, this coefficient can be extracted from the Green function  \n${\\tilde G}^{(2)}(k;a)$ of the second-order perturbative expansion of ${\\cal F}_{n>0}$ for a small amplitude deformation of one of the plates around the plane-parallel  geometry. The computation of this coefficient for gold plates at room temperature  can be carried out following the procedure described in \\cite{bimonte2}, and we address the interested reader to that Reference for details. The coefficients  $\\tilde \\theta$ and $\\kappa$ for perfect conductors (PC) in the limit of zero temperature are both independent of the separation.  Their values can be determined using the formulae listed in \\cite{bimonte1}:  \n\\begin{equation}\n\\tilde \\theta^{(\\rm PC)}_{T=0}=\\frac{20}{3 \\pi^2}-\\frac{1}{9}=0.564\\;,\\;\\;\\;\\;\\;\\;\\kappa^{(\\rm PC)}_{T=0}=\\frac{1}{3}\\;.\n\\end{equation}  For gold surfaces at room temperature, both coefficients depend on the separation (as well as on the temperature and on the material lengths characterizing the optical properties of gold, in particular the plasma length). In Table \\ref{tab.1} we list the values of $\\tilde \\theta(a)$ and $\\kappa(a)$ for gold at room temperature, that we computed using tabulated \\cite{palik} optical data \\footnote{The weighted Kramers-Kronig dispersion relations \\cite{bimonteKK} was used to compute precisely $\\epsilon({\\rm i} \\xi_n)$ starting from the real-frequency optical data given by Palik. }, for several values of the separation in the range from 100 nm to 2 micron.  \nUsing these values of $\\tilde \\theta(a)$ and $\\kappa(a)$ together with Eq. (\\ref{DEformula}), it is easily possible to compute  ${ F}'_{n>0}$ for any combinations of sphere radii. \n\nIn view of later applications, it is important to note that while in the Proximity Approximation  ${ F}'_{n>0}$ depends only on the effective radius ${\\tilde R}$ (see Eq. (\\ref{PFA2})),  the  more accurate expression of ${ F}'_{n>0}$ in Eq. (\\ref{DEformula1}), which includes curvature corrections to PFA, depends  also on the parameter $u$, i.e. on the ratio of the radii of the two spheres.\n\\begin{widetext}\n\\label{tab.1}\n\\begin{center}\n\\begin{table*}\n\\begin{tabular}{ccccccccccccc} \\hline\n$a (\\mu m)$\\;\\; &0.10& 0.15 & 0.2  \\;\\;& 0.25\\;\\;\\;& 0.3\\;\\; &0.35 \\;\\;& 0.4 \\;\\; & 0.45 & 0.5\\;\\;& 0.55 & 0.6\\;\\; & 0.65  \\\\ \\hline \\hline\n${\\tilde \\theta}$\\;\\;& 0.456  &0.4715 & 0.470\\;\\; &0.463 & 0.454 \\;\\;&0.4445 & 0.435 \\;&0.425 & 0.415 &0.4055 & 0.396& 0.387    \\\\  \\hline\n${\\kappa}$\\;\\;&0.245 &0.270 &\\; 0.289\\;\\; & 0.305 & 0.319 \\;\\;&0.331 & 0.342\\;\\;&0.353  & 0.362\\;\\;&0.371 & 0.380\\;\\; &0.389 \\;\\;    \\\\ \\hline \\\\\n \\hline\n$a (\\mu m)$\\; &0.70&0.75 & 0.8 & 0.85 & 0.9& 0.95&  1 &1.2&1.4 &1.6 & 1.8 & 2  \\\\ \\hline \\hline\n${\\tilde \\theta}$ \\;\\; &0.379 &0.370  &0.362 &0.3545 &0.347&0.3395&0.332 &0.306&0.282&0.261& 0.242 &0.225   \\\\  \\hline\n${\\kappa}\\;\\;$&0.397&\\;0.405 \\;&0.413 &0.421&0.429&0.437& 0.444\\;\\;&0.474&0.502&0.529&0.554&0.578 \\\\ \\hline  \\hline\n\\end{tabular}\n\\caption{Values of the coefficients ${\\tilde \\theta}$ and $\\kappa$ for Au at room temperature.}\n\\end{table*}\n\\end{center}\n\\end{widetext}\n\n\n\n\\section{Beyond-PFA corrections.}\n\nCombining Eq. (\\ref{enerDr}) with Eq. (\\ref{DEformula1}) we obtain the following formula for the force gradient between two gold spheres:\n\\begin{equation}\nF'=  {F'}_{n=0}^{(\\rm ex)}  -2 \\pi {\\tilde R} {F}^{(\\rm pp)}_{n>0}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u\\, \\kappa(a)\\right) \\frac{a}{\\tilde R} \\right]\\;,\\label{approx}\n\\end{equation}\nwhich constitutes the main result of the present work. A nice feature of the formula above is that, by construction,  it is {\\it exact in both limits} $a\/{\\tilde R}\\rightarrow 0$ and $a\/\\lambda_T\\rightarrow \\infty$. This is so because, on one hand, Eq. (\\ref{approx}) is exact  for $a\/{\\tilde R}\\rightarrow 0$, since in this limit the DE, which we used to compute the contribution of the positive Matsubara modes,  is asymptotically exact. On the other hand, Eq. (\\ref{approx}) is exact  also for $a\/\\lambda_T \\rightarrow \\infty$, because for separations larger than the thermal length $a \\gg \\lambda_T$ the relative contribution of the positive  Matsubara modes vanishes exponentially fast,  and then Eq. (\\ref{approx}) reduces to the exact  $n=0$ mode. Comparison with high precision numerical computations of the sphere-plate scattering formula in \\cite{bimonteprecise} revealed that Eq. (\\ref{approx}) is in fact very accurate also for {\\it all intermediate} separations.  For a gold sphere as small as 8 micron, the maximum error made by Eq. (\\ref{approx})  was only of 0.1$\\%$, and this was for the large aspect ratio $a\/R=0.12$. The error  is expected to be far smaller in the conditions of the experimet \\cite{garrett}, which used spheres with radii larger than 29.8 micron, and probed distances corresponding to aspect ratios smaller than 0.017.\n\nFor later use, it is useful to work out the small distance limit of our formula for the sphere-sphere force gradient. This can be easily done subsituting  $ {F'}_{n=0}^{(\\rm ex)} $ on the r.h.s. of Eq. (\\ref{approx}) by its small-distance expansion Eq. (\\ref{DEnzero}). After simple algebraic manipulations, one finds:\n\\begin{equation}\nF'=-2 \\pi {\\tilde R} {F}^{(\\rm pp)}(a) \\left[1- \\left(\\hat{\\theta}(a)+u\\, \\hat{\\kappa}(a)\\right) \\frac{a}{\\tilde R}+ o(a\/{\\tilde R}) \\right]\\;.\\label{DE2s}\n\\end{equation}\n The coefficients $\\hat{\\theta}(a)$ and $\\hat{\\kappa}(a)$ are \n\n\\begin{eqnarray}\n\\hat{\\theta}(a)&=&\\frac{ {F}^{(\\rm pp)}_{n>0}}{{F}^{(\\rm pp)}} \\; \\tilde{\\theta}-\\frac{{F}^{(\\rm pp)}_{n=0}}{{F}^{(\\rm pp)}} \\,\\frac{1}{12\\, \\zeta(3)}\\;,\\nonumber \\\\\n\\hat{\\kappa}(a)&=&\\frac{{F}^{(\\rm pp)}_{n>0}}{{F}^{(\\rm pp)}}\\;{\\tilde \\kappa}\\;.\\\\\n\\end{eqnarray}\n\nIn Table \\ref{tab.2}, we provide the values of the coefficients ${\\hat \\theta}$ and ${\\hat \\kappa}$ for gold at room temperature. We note that the coefficient ${\\hat \\theta}$ introduced here coincides with the opposite of the coefficient ${\\hat \\theta}_1$ of \\cite{bimonte2}.  \n\\begin{widetext}\n\\label{tab.2}\n\\begin{center}\n\\begin{table*}\n\\begin{tabular}{ccccccccccccccc} \\hline\n$a (\\mu m)$\\;\\;&0.05 &\\;0.10 & 0.15 & 0.2  \\;\\;& 0.25\\;\\;\\;& 0.3\\;\\; &0.35 \\;\\;& 0.4 & 0.45 & 0.5 & 0.55 & 0.6 \\\\ \\hline \\hline\n${\\hat \\theta}$\\;\\;& 0.378  &\\;\\;0.439 &\\; 0.449\\;\\; &\\;0.443\\;\\; &\\; 0.432 \\;\\;&0.419 & \\;0.405 \\;&\\;0.392&\\;\\; 0.378 &\\;\\;0.365 & \\;\\;0.352 &\\;\\;0.340      \\\\  \\hline\n${\\hat \\kappa}$\\;\\;&0.209 &\\;\\;0.237 &\\; 0.259\\;\\; & 0.275 &\\; 0.288 \\;\\;&0.298 &\\; 0.306\\;\\;&\\;0.313 &\\;\\;\\;0.320 &\\;\\;0.325 &\\;\\;0.330 &\\;\\; 0.334  \\\\\n  \\hline  \\hline\n\\end{tabular}\n\\caption{Values of the coefficients ${\\hat \\theta}$ and ${\\hat \\kappa}$ for Au at room temperature.}\n\\end{table*}\n\\end{center}\n\\end{widetext}\nIn order to show   the deviations of the force gradient  from PFA predicted by our formula, in Fig. \\ref{deviations} we  plot the quantity  $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$  for a system of two identical spheres of radius $R_1=R_2=30\\;\\mu$m (lower solid line) and $R_1=R_2=100\\;\\mu$m (lower dashed line).\n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{deviations\n\\caption{\\label{deviations}   Beyond-PFA corrections for the gradient of the Casimir force between two gold spheres at room temperature are shown as a function of the separation for two identical spheres of radius $R=30\\;\\mu$m (lower solid line) and $R=100\\;\\mu$m (lower dashed line). The upper pair of lines is for a sphere-plate system with sphere radius   $R=30\\;\\mu$m (upper solid line) and $R=100\\;\\mu$m (upper dashed line). }\n\\end{figure}\nThe upper solid and dashed  lines in Fig. \\ref{deviations} refer to a sphere-plate system, for  a sfere of radius $R=30\\;\\mu$m (solid line) and $R=100\\;\\mu$m (dashed line).  Fig. \\ref{deviations}  demonstrates that    deviations from PFA of the force gradient  are practically independent of the effective radius $\\tilde R$, for  fixed value of $u$. However deviations from PFA  depend significantly on the ratio among the radii of the spheres via the parameter $u$.  \n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{deviations2\n\\caption{\\label{deviations2}   Beyond-PFA corrections for the gradient of the Casimir force between two gold spheres at room temperature are shown as a function of the parameter $u={\\tilde R}^2\/ (R_1 R_2)$ (for constant ${\\tilde R}$ and $a$).  Solid lines are for   ${\\tilde R}=30\\;\\mu$m, dashed lines for ${\\tilde R}=100\\;\\mu$m. The four pairs of solid and dashed lines from top to bottom correspond to the four separations $a=1\\mu$m, 800 nm, 400 nm and 100 nm respectively. The extreme values $u=0$ and $u=1\/4$ correspond, respectively, to a sphere-plate configuration and to two spheres of equal radii.}\n\\end{figure}\nThis is clearly seen from Fig. \\ref{deviations2}, where the quantity $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$ is displayed versus the parameter $u$ (for constant ${\\tilde R}$ and $a$). In Fig. \\ref{deviations2} solid lines are for ${\\tilde R}=30\\;\\mu$m, while dashed lines are for  ${\\tilde R}=100\\;\\mu$m. The four pair of solid and dashed lines from top to bottom correspond to the four separations $a=1\\mu$m, 800 nm, 400 nm and 100 nm respectively. We recall that the extreme values $u=0$ and $u=1\/4$ correspond, respectively, to a sphere-plate and to two spheres of equal radii.  Fig. \\ref{deviations2} shows  that the $u$-dependence of the deviations from PFA  is linear for the considered separations.\n\n\nFor small values of $a\/{\\tilde R}$, the quantity displayed in Fig. \\ref{deviations} and in Fig. \\ref{deviations2} can be basically identified with the parameter $\\beta'$ that was introduced  in the sphere-plate experiment \\cite{ricardo}  as a measure of the deviation of the data from PFA. In \\cite{ricardo} starting from the force gradient $F'$,  an effective pressure $P^{(\\rm eff)}(a,R)$ was defined  as:\n\\begin{equation}\nP^{(\\rm eff)}(a,R)\\equiv-\\frac{F'}{2 \\pi R} \\;.\n\\end{equation}\nIf the PFA Eq. (\\ref{PFA2}) were exact, $P^{(\\rm eff)}(a,R)=F^{(\\rm pp)}(a)$.  However, the PFA is not exact, and so the authors of \\cite{ricardo}  parametrized  deviations  from PFA by a  coefficient $\\beta'(a)$ such that:\n\\begin{equation}\nP^{(\\rm eff)}=F^{(\\rm pp)}(a)\\left(1+\\beta' \\frac{a}{R}+o(a\/R)\\right)\\;.\\label{ricardo}\n\\end{equation} \n  It is clear from Eq. (\\ref{ricardo}) that, up to  higher order corrections, $\\beta'(a)$ coincides with  the quantity $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$.\nThe parameter $\\beta'$ was determined in \\cite{ricardo} by measuring the effective pressure  $P^{(\\rm eff)}$ for certain fixed sphere plate-separations using spheres of different radii, and then fitting  $P^{(\\rm eff)}$ versus $1\/R$ with a straight line.   \nThe experiment  \\cite{ricardo} placed a bound $|\\beta'|<0.4$ at 95 \\% CL in the separation range from 150 to 300 nm.  This bound is  in substantial agreement with the theoretical prediction (see the upper curves of Fig. \\ref{deviations}).  \n\nThe authors of \\cite{garrett} used the same procedure to  study  deviations of their data from PFA. In particular, they assumed that  the measured forces can be parametrized as:\n\\begin{equation}\n\\frac{F'}{\\tilde R}=-2 \\pi F^{(\\rm pp)}(a)\\left(1+\\beta' \\frac{a}{\\tilde R}+o(a\/{\\tilde R})\\right)\\;,\\label{garrett}\n\\end{equation} \ni.e. by a function of the same form as  that of the sphere-plate system, apart from the substitution of $R$ by the effective radius ${\\tilde R}$ of the two-sphere system. Importantly,  they assumed that $\\beta'$ is {\\it  independent of the radii of the spheres}.  Based on this assumption, the authors of \\cite{garrett} tried to determine $\\beta'(a)$ by doing a linear fit of $F'\/{\\tilde R}$ versus $1\/{\\tilde R}$  using for that purpose   12 measurements of $F'$. Three sets of data  were taken in sphere-plate setups (as in the experiment \\cite{ricardo}), using three different spheres of radii  $R=$ 40.7 $\\mu$m, 36.1 $\\mu$m and 34.2 $\\mu$m, while the remaining  nine data sets were taken with  nine different sphere-sphere setups,  corresponding to the different combinations of  each of three spheres with radii $R_1=$ 34.2  $\\mu$m, 36.1 $\\mu$m and 40.7 $\\mu$m,  with each of the three spheres of  radii  $R_2=$ 29.8  $\\mu$m, 38.0  $\\mu$m and 46.9  $\\mu$m. The measurements and the corresponding fits  were repeated for  26 values of the separation,  in the interval from 40 nm to $300$ nm.  It was found that  $\\beta'=-6 \\pm 27$ was within the 2$\\sigma$ confidence interval of their calculated $\\beta'$ for all  considered separations.\n\nThe procedure used in \\cite{garrett} to determine $\\beta'$ is not entirely correct, however, because the parametrization in Eq. (\\ref{garrett}) misses the dependence of $\\beta'$ on the parameter $u$. Indeed, by comparing Eq.(\\ref{garrett}) with the small-distance expansion of the sphere-sphere force Eq. (\\ref{DE2s}) one finds that $\\beta'$ has the expression\n\\begin{equation}\n\\beta'=- \\left(\\hat{\\theta}(a)+u\\, \\hat{\\kappa}(a)\\right)\\;.\\label{betass}\n\\end{equation}\nThis formula shows that in the two-sphere case, contrary to the assumption made in \\cite{garrett}, $\\beta'$ does depend on the radii of the spheres via the parameter $u$. Of course, this dependence disappears in the sphere-plate case, for which $u=0$.  The  linear dependence on $u$ of deviations from PFA is clearly visible from Fig. \\ref{deviations2}. Since in \\cite{garrett} the 12 combinations of radii  used  to determine $\\beta'$  correspond to values of $u$ that vary from zero (for the three sphere-plate setups) to 0.2498 (corresponding to  the sphere-sphere setup with  $R_1=36.1\\;\\mu$m and $R_2= 38\\;\\mu$m), the dependence of $\\beta'$ on $u$ should be considered in the data analysis.  Substituing Eq. (\\ref{betass}) into Eq. (\\ref{garrett}), we find that in a two-sphere system  $F'\/{\\tilde R}$ has the expression:\n\\begin{equation}\n\\frac{F'}{\\tilde R}=-2 \\pi F^{(\\rm pp)}(a)\\left(1- \\frac{a\\, \\hat{\\theta}}{\\tilde R}-\\frac{a\\,\\hat{\\kappa}}{R_1+R_2}+o(a\/{\\tilde R})\\right)\\;,\\label{garrett2}\n\\end{equation}\nwhere in the second term between the brackets we used the relation $u\/{\\tilde R}=1\/(R_1+R_2)$. This formula shows that the correct procedure to determine the coefficients  $\\hat \\theta$ and $\\hat \\kappa$,  is to make a joint 2-dimensional linear fit of $F'\/{\\tilde R}$ versus $1\/{\\tilde R}$ and $1\/(R_1+R_2)$. \n\nThe present sensitivity of the experiment \\cite{garrett} is not yet sufficient   to detect the small deviations from PFA predicted by Eq. (\\ref{garrett2}).  We estimate that an increase in the sensitivity by  over one  order of magnitude would be necessary for that purpose. It is hoped that future improvements of the apparatus  will achieve this goal.\n\n\\section{Conclusions}\n\nMotivated by the recent experiment in \\cite{garrett}, we have performed a precise computation of the gradient of the Casimir force between two gold spheres at room temperature.  Our computation  provides an accurate estimate of beyond PFA corrections for this system. The  semi-analytic formula  for the Casimir force  that we construct is valid for all separations and can be easily used to interpret future experiments in both the sphere-plate and sphere-sphere configurations. We have also described the correct parametrization of the corrections to PFA that should be used to carry out the data analysis in experiments using the sphere-sphere geometry. \n\nIn our computations we modelled the god plates as  ohmic conductors (connected to charge reservoirs). In recent years it has been argued by some researchers \\cite{book2} that a  better agreement with Casimir experiments is obtained if  metallic bodies are modelled as dissipationless plasmas. The main change introduced by this model is in the classical $n=0$ Matsubara term for TE polarization, which is zero within the Drude prescription, but different from zero in the plasma model. A detailed comparison between the Drude and plasma models for the sphere-plate configuration, based on a large scale numerical simulation of the scattering formula, has been reported in \\cite{gert}, where it was shown that the deviations from PFA engendered by the plasma prescription  have the same qualitative behavior as the Drude model, but are slightly larger in magnitude and show a more pronounced dependence on the aspect ration $a\/R$.  We plan to study the plasma prescription for the sphere-sphere case in a forthcoming work  \\cite{bimontenext}.\n\n\\acknowledgments\n\nThe author thanks T. Emig, N. Graham, M. Kruger, R. L. Jaffe and M. Kardar for valuable discussions.  \n\n\\section*{APPENDIX}\n\n\\subsection{The DE expansion}\n\nFor the convenience of the reader, in this Appendix we briefly review  general properties of the DE that are useful for the present work. The DE \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3}  is an asymptotic expansion that allows to compute  curvature corrections of any sufficiently {\\it local} functional ${\\hat {\\cal F}}$ that describes the interaction between two (non-intersecting) surfaces $\\Sigma_1$ and $\\Sigma_2$. The idea behind the DE is intuitive. One considers that the two surfaces can be described by {\\it smooth} height profiles $z=H_1(x,y)$ and $z=H_2(x,y)$, where $(x,y)$ are cartesian coordinates spanning some reference plane $\\Sigma$  and $z$ is a coordinate perpendicular to  $\\Sigma$ (see Fig. 1).  Since the two surfaces are non-intersecting, it can always be assumed that $H_2(x,y) < H_1(x,y)$.  At this point one considers that for surfaces of small slopes $| \\nabla H_i| \\ll 1,\\;i=1,2$ \\footnote{In fact it is sufficient that the small slope condition is satisfied only in the relevant interaction area around the point of closest approach between the two surfaces}  it should be possible to expand ${\\hat {\\cal F}}[H_1,H_2]$ in powers of {\\it derivatives} of  increasing order of the height profiles, at least up to some order.  It is rather easy to convince oneself that for a functional ${\\hat {\\cal F}}[H_1,H_2]$ that is invariant under simultaneous rotations and translations of $H_1$ and $H_2$  in the reference plane $\\Sigma$ (like the Casimir force between two plates made of a homogeneous and isotropic material) the most general expression of the DE valid to second order in the slopes of the surfaces is of the form:     \n\\begin{eqnarray}\n{\\hat {\\cal F}}[H_1,H_2]=\\int_{\\Sigma}  d^2 x \\left[{\\hat {\\cal F}}^{(\\rm pp)}(H)\\right. &+&\\left.  \\;\\alpha_1(H)  (\\nabla H_1)^2 \\right.  \\nonumber \\\\\n+ \\alpha_2(H)  (\\nabla H_2)^2 \\!\\!&+& \\!\\!\\alpha_{\\times}(H) \\nabla H_1 \\cdot \\nabla H_2 \\nonumber\\\\\n+  \\alpha_{-}(H) \\nabla H_1 \\!\\!& \\times &\\! \\!\\nabla H_2 \\left. \\right]+  \\rho^{(2)}\\;,\\label{derexp}\n\\end{eqnarray}\nwhere we set $H=H_1-H_2$ and $\\rho^{(2)}$ is a remainder that becomes negligible as the local radii of curvature of the surfaces go to infinity  for fixed minimum surface-surface distance $a$. \nNote that invariance of ${\\hat {\\cal F}}$ under translations of $\\Sigma$ in the $z$ direction implies that ${\\hat {\\cal F}}^{(\\rm pp)}$ and the $\\alpha$'s can depend only on the height difference $H$ an not on the individual heights $H_1$ and $H_2$.\nIt is evident that the quantity ${\\hat {\\cal F}}^{(\\rm pp)}(a)$ in Eq. (\\ref{derexp}) provides the (unit-area) interaction of two plane-parallel surfaces at distance $a$, and thus the first term on the r.h.s of Eq. (\\ref{derexp}) reproduces the Derjaguin Approximation (DA)  \\footnote{The DA approximation is sometimes referred to as the \"exact\" PFA.} for the functional ${\\hat {\\cal F}}$:  \n\\begin{equation}\n {\\hat {\\cal F}}^{(\\rm DA)} = \\int_{\\Sigma}  d^2 x  \\;{\\hat {\\cal F}}^{(\\rm pp)}(H)\\;,\n\\end{equation}\nThe integrals on the r.h.s. of Eq. (\\ref{derexp}) that are proportional to $\\alpha$'s represent curvature corrections beyond the DA, and thus we see that the DE  provides a systematic way to improve the old-fashioned DA. \nArbitrariness in the choice of the reference plane $\\Sigma$ further constraints the three coefficients $\\alpha$ in Eq. (\\ref{derexp}) \\cite{bimonte1}. In particular,   invariance of  ${\\hat {\\cal F}}$ with respect to tilting of $\\Sigma$ (for details, see \\cite{bimonte1}) implies:\n\\begin{eqnarray}\n&&2(\\alpha_1(H)+\\alpha_2(H)+\\alpha_{\\times}(H))+H \\frac{d  {\\hat {\\cal F}}^{(\\rm pp)} }{d H}-{\\hat {\\cal F}}^{(\\rm pp)}=0\\;,\\nonumber\\\\\n&&\\alpha_{-}(H)=0\\;.\\label{alpharel}\n\\end{eqnarray}\nThe above relations show that, to second order in the gradient expansion, the two-surface problem actually reduces to the simpler problem of a single curved surface opposite a plane, since $\\alpha_1$ and $\\alpha_2$ can be determined in that case, and then $\\alpha_{\\times}$ follows from the first of Eqs. (\\ref{alpharel}).  We now make the simplifying assumption that the  field(s) that mediate the interaction obeys the same boundary conditions on $\\Sigma_1$ and $\\Sigma_2$. Then \n\\begin{equation}\\alpha_1(H)=\\alpha_2(H)\\equiv\\alpha(H)\\;.\\label{onemat}\n\\end{equation}\nTaking advantage of Eqs. (\\ref{alpharel}) and (\\ref{onemat}) the DE  can then be recast in the form:\n$$\n{\\hat {\\cal F}}[H_1,H_2]= {\\hat {\\cal F}}^{(\\rm DA)} + \\int_{\\Sigma}  d^2 x \\; \\frac{}{} \\alpha(H)  (\\nabla H)^2  \n$$\n\\begin{equation}\n+ \\frac{1}{2} \\int_{\\Sigma}  d^2 x \\left( {\\hat {\\cal F}}^{(\\rm pp)}-H \\frac{d  {\\hat {\\cal F}}^{(\\rm pp)}  }{d H} \\right)\\nabla H_1 \\cdot \\nabla H_2 +  \\rho^{(2)}\\;.\\label{derexp2}\n\\end{equation}\nWe thus see  that to second order in the slope  the interaction  ${\\hat {\\cal F}}$ is fully determined by knowledge of the (unit area) interaction $  {\\hat {\\cal F}}^{(\\rm pp)}(a)$ of two parallel plates and by the single coefficient $\\alpha(H)$. \n\nThe latter coefficient can be determined by comparing the DE Eq. (\\ref{derexp2}) to a {\\it perturbative} expansion of  the functional ${\\hat {\\cal F}}[H,0]$ around flat plates $H=a + h(x,y)$ to second order in the deformation $h(x,y)$. Note that the latter perturbation requires a deformation of small amplitude  $h(x,y)\/a \\ll 1$,  while the DE relies on the condition that the slope of the surface be small. To second order in $h$ the perturbative expansion of ${\\hat {\\cal F}}$ reads:\n$$\n{\\hat {\\cal F}}[a+h({\\bf x})]=A {\\hat {\\cal F}}^{(\\rm pp)}(a)+\\mu(a) {\\tilde h}({\\bf 0})\n$$\n\\begin{equation}\n+\\int \\frac{d^2{\\bf k}}{(2 \\pi)^2}\\;{\\tilde G}^{(2)}(k;a)|{\\tilde h}({\\bf k})|^2+{\\tilde \\rho}^{(2)}[h]\\;,\n\\end{equation}\nwhere $A$ is the surface area, ${\\bf k}$ is the in-plane wave-vector,  ${\\tilde h}({\\bf k})$ is the Fourier transform of $h(x)$, and  ${\\tilde \\rho}^{(2)}[h]$  refers to higher order corrections. The function  $\\alpha(H)$ can now be determined if the kernel ${\\tilde G}^{(2)}(k;a)$ can be expanded to second order in $k$. Indeed, matching the expansion\n\\begin{equation}\n{\\tilde G}^{(2)}(k;a)= \\gamma(a)+\\delta(a) k^2+o(k^2)\\;,\\label{green1}\n\\end{equation}\nto Eq. (\\ref{derexp2}) one finds:\n\\begin{equation}\n {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'(a)=\\mu(a)\\;,\\;\\;{\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}''(a)=2 \\gamma(a)\\;,\\;\\;\\;\\alpha(a)=\\delta(a)\\;,\\label{green2}\n\\end{equation}\nwhere a prime denotes a derivative with respect to $a$. The above Equation shows that a {\\it necessary} condition for existence of the second order DE is existence of the Taylor expansion of the perturbative kernel ${\\tilde G}^{(2)}(k;a)$ to second order in the in-plane momentum. Indeed, it can be shown that the DE can be formally recovered by an (infinite) resummation of the perturbative series for small in-plane momenta \\cite{fosco3}.\n\nWhenever applicable, the DE has been successfully used to compute curvature corrections beyond the PA in various problems involving interactions among gently curved surfaces. In the context of Casimir physics, it was used in \\cite{fosco1} to  compute curvature corrections to the zero temperature Casimir energy for a scalar field obeying Dirichlet (D) boundary conditions (bc) in the  sphere-plate and cylinder-plate  geometries. The zero temperature Casimir problem for the em field with perfect conductor (PC) bc, as well as a scalar field obeying Neumann bc, or  mixed DN bc (i.e. D bc on one surface and N on the other), was studied in \\cite{bimonte1} for two spheres and for two inclined  cylinders. The curvature corrections obtained in the latter work  for the em field with PC bc in the sphere-plate and sphere-sphere geometries  were subsequently  confirmed in \\cite{bordagteo,teo2} by working out a rigorous small-distance expansion of the scattering formula. The experimentally important case of the Casimir interaction between gold sphere and plate at finite temperature was instead studied in \\cite{bimonte2}. Even in this case, the results obtained by the DE were later shown to be in agreement with the small-distance expansion of the scattering formula \\cite{teogold}. Curvature corrections obtained by the DE have also been found to be in agreement \nwith the small distance expansion of the rare exact Casimir energies in non planar geometries that have been discovered so far,  i.e. in the cases of two Drude or D spheres in the classical limit \\cite{bimonteex1}, and for two three-spheres with D or PC bc in four euclidean dimensions \\cite{euclidean}.    \nThe DE has been also used to study curvature effects in the Casimir-Polder interaction of a particle with a gently curved surface \\cite{CPbimonte1,CPbimonte2}, \nand to estimate the shifts of the rotational levels of a diatomic molecule due to its van der Waals interaction with a curved dielectric surface \\cite{CPbimonte3}. In a non Casimir context, the DE hase been also used to compute curvature corrections to the scattering amplitude for an em wave impinging on a curved surface \\cite{bimonteref} and to the electrostatic interaction among two curved plates \\cite{foscoel}.\n   \n\n\\subsection{Computing the leading curvature correction to the force gradient. }\n\n\nThe  small-slope approximation of the interaction energy ${\\hat {\\cal F}}$ provided by Eq. (\\ref{derexp2}) still involves a surface integral over $\\Sigma$ of functions depending on the height profiles of the surfaces. As such, \nEq.  (\\ref{derexp2}) is not very convenient for a practical use. A better route is to expand Eq. (\\ref{derexp2}) in powers of the small parameter $a\/R$, where $R$ is the characteristic radius of curvature of the surfaces. The leading order of this expansion will reproduce the standard PFA, while in the next order it shall provide us with the desired curvature correction beyond the PFA.  \nWe shall carry out this expansion not directly for the energy ${\\hat {\\cal F}}$, but rather for the gradient of the force ${\\hat F}'=-{\\hat {\\cal F}}''$, which is the quantity that was measured in the experiment \\cite{garrett}. Moreover, we shall restrict attention to the sphere-sphere system, which is again the geometry used in \\cite{garrett}.\n\nAccording to Eq. (\\ref{derexp2}), the formula for the force gradient ${\\hat F}'$ can be split as\n\\begin{equation}\n{\\hat F}'={\\hat { F'}}^{(\\rm DA)}+I_2+I_3\\;,\n\\end{equation}\nwhere\n\\begin{equation}\n{\\hat { F'}}^{(\\rm DA)}=\\int_{\\Sigma}  d^2 x  \\;{\\hat { F'}}^{(\\rm pp)}(H)\n\\end{equation}\nand we set\n\\begin{eqnarray}\n&I_2& =\\frac{1}{2} \\int_{\\Sigma}  d^2 x\\; (H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')'\\;\\nabla H_1 \\cdot \\nabla H_2\\;,\n\\nonumber \\\\\n&I_3&=- \\int_{\\Sigma}  d^2 x \\;  \\alpha''(H) \\;  (\\nabla H)^2\\;.\n\\end{eqnarray}\nWe consider proximity forces that decay rapidly wih the distance, like the Casimir force. For forces of this nature, the  interaction among the surfaces is localized within a small area, typically of radius $\\rho \\sim \\sqrt{a {\\tilde R}}$,  around the point of closest approach.  Under such circumstances,  it is legitimate to take the Taylor expansion of the  height profiles $H_1(x,y)$ and $H_2(x,y)$ of the two spheres around their tips, that we imagine placed at $x=y=0$. Since the position of the reference plane $\\Sigma$ in Fig. 1 is immaterial, we are free to take for $\\Sigma$ the tangent plane to the sphere of radius $R_2$, passing through the sphere tip. Then: \n\\begin{eqnarray}\nH_1(x,y) &=& a+\\frac{r^2}{2R_1}+\\frac{r^4}{8 R_1^3}+\\dots\\;,\\nonumber \\\\\nH_2(x,y) &=&-\\frac{r^2}{2R_2}-\\frac{r^4}{8 R_2^3}+\\dots\\;,\n\\end{eqnarray}\nwhere $r^2=x^2+y^2$.  To evaluate the integrals $I_j$ it is convenient to introduce polar coordinates $(r, \\theta)$ in the $(x,y)$ plane, and then substitute $r$ by the dimensionless quantity $\\xi=r^2\/a {\\tilde R}$.  An essential property of the integrals $I_j$ is that they involve derivatives of certain functions (i.e. ${\\hat {\\cal F}}^{(\\rm pp)}$ and $\\alpha$) of the height difference $H$ with respect to the separation $a$. These derivatives  can be converted into derivatives with respect to $\\xi$, using the identity\n\\begin{eqnarray}\nU'&=&U_{ ,\\xi} \\left(H_{,\\xi}\\right)^{-1} \\nonumber \\\\\n&=&2\\frac{U_{ ,\\xi} }{a}  \\left[1-\\frac{\\xi}{2} \\left(\\frac {a{\\tilde R}^2}{R_1^3}+\\frac{a{\\tilde R}^2}{R_2^3} \\right) +o(a\/{\\tilde R})\\right]\\;,\\label{dera}\n\\end{eqnarray}\n(comas denote derivatives) which holds for any function $U$ of $H$. We are ready now to take the  small-distance expansion of ${\\hat F}'$. We start from ${\\hat { F'}}^{(\\rm DA)}$. \nUsing Eq. (\\ref{dera}), and omitting corrections of order $o(a\/{\\tilde R})$ we find \n$$\n{\\hat { F'}}^{(\\rm DA)}\\!\\!\\!=2\\pi {\\tilde R} \\!\\!\\int_0^{\\infty} \\!\\!\\!\\!d \\xi \\,{\\hat {F}}^{(\\rm DA)}_{,\\xi}\\!\\left[1-\\frac{\\xi}{2} \\left(\\!\\frac{a{\\tilde R}^2}{R_1^3}+\\frac{a{\\tilde R}^2}{R_2^3}\\! \\right)\\right] \n$$\n$$\n=-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)-a\\, \\pi  \\left(\\frac{{\\tilde R}^3}{R_1^3}+\\frac{{\\tilde R}^3}{R_2^3} \\right)\n \\int_0^{\\infty} d \\xi \\;{\\hat {\\cal F}}^{(\\rm pp)}\\! \\frac{}{}'\n$$\n\\begin{equation}\n=-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)+ 2\\pi  \\left(\\frac{{\\tilde R}^3}{R_1^3}+\\frac{{\\tilde R}^3}{R_2^3} \\right) {\\hat {\\cal F}}^{(\\rm pp)}(a)\\;.\\label{first}\n\\end{equation}\nThe first term on the last line of Eq. (\\ref{first}) coincides with the standard PFA for the force gradient (see Eq. (\\ref{PFA2})), while its second term represents a curvature correction. \n\nBy following an analogous procedure for $I_2$, and again omitting higher order terms, we obtain:\n\\begin{eqnarray}\n&I_2 & = - \\pi\\frac{ a {\\tilde R}^2}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,(H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')_{,\\xi}\\,\\xi\n\\nonumber \\\\\n&=&\n \\pi \\frac{a {\\tilde R}^2}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,(H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')=\\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,H ({\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}')_{,\\xi}\n\\nonumber \\\\\n&=& \\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} a {\\hat F}^{(\\rm pp)}(a)-\\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\int_0^{\\infty} d \\xi\\, {\\hat {\\cal F}}^{(\\rm pp)}_{,\\xi} \\nonumber \\\\\n&=& \\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\left[a {\\hat F}^{(\\rm pp)}(a) +{\\hat {\\cal F}}^{(\\rm pp)}(a)\\right]\\;.\\label{I2}\n\\end{eqnarray}\nFinally, for $I_3$ we obtain:\n\\begin{equation}\nI_3 = - 4 \\pi \\int_0^{\\infty} d \\xi\\, \\alpha_{,\\xi \\xi} \\,\\xi =-4 \\pi \\alpha(a)\\;.\\label{I3}\n\\end{equation}\nUpon combining Eqs. (\\ref{first}-\\ref{I3}), after simple algebraic transformations, we obtain the following small-distance expansion of the force gradient, correct up to terms of order $o(a\/{\\tilde R})$:\n\\begin{eqnarray}\n{\\hat F}' &=&-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)+ 2 \\pi \\left[{\\hat {\\cal F}}^{(\\rm pp)}(a) -2 \\alpha(a)\\right]\n\\nonumber \\\\\n&+& 2 \\pi u \\left[ a {\\hat F}^{(\\rm pp)}(a)-2 {\\hat {\\cal F}}^{(\\rm pp)}(a) \\right]\\;\n\\nonumber \\\\\n&\\equiv& -2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u \\kappa(a) \\right) \\frac{a}{\\tilde R} \\right]\\;, \\label{DEformula}\n\\end{eqnarray}\nwhere the coefficients $\\tilde \\theta(a)$ and $\\kappa(a)$ are\n\\begin{eqnarray}\n{\\tilde \\theta} &=&  \\frac{{\\hat {\\cal F}}^{(\\rm pp)}(a) -2 \\alpha(a)}{a {\\hat F}^{(\\rm pp)}(a)}\\;,\\label{thetacoe}\n\\\\\n\\kappa(a) &=&  1-2 \\frac{{\\hat {\\cal F}}^{(\\rm pp)}(a)}{ a {\\hat F}^{(\\rm pp)}(a)} \\;.\\label{kappacoe}\n\\end{eqnarray}\n \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLike symplectic manifolds, contact manifolds have no local invariants.  Darboux's theorem tells us that locally all contact structures are the same. Moreover Gray stability tells us that there is no deformation theory.  Nonetheless there are contact manifolds which are not contactomorphic.  \nSometimes one can distinguish these different contact structure via the Chern classes of the underlying symplectic vector bundle defined by the contact distribution.  However this is insufficient.  In ~\\cite{Gi1} Giroux shows that the contact structures $$\\xi_{n} = ker(\\alpha_{n} =cos(n\\theta )dx + sin(n\\theta)dy)$$ are pairwise noncontactomorphic.  Moreover $c_{1}(\\xi_{n}) = 0$ for all $n$.\n\n\nCalculations of this type have been entirely dependent on the specific geometric conditions of the example.  However, we now have contact homology, a small piece of the symplectic field theory ~\\cite{EGH} of Elisahberg, Givental, and Hofer.  Using this powerful tool Ustilovsky ~\\cite{Ust} was able to find infinitely many exotic contact structures on odd dimensional spheres all in the same homotopy class of almost complex structures.  Similarly Otto Van Koert, in his thesis ~\\cite{OVK}, made a similar calculation for a larger class of Brieskorn manifolds using the Morse-Bott form of the theory.  It should be mentioned that it came to the author's  attention upon completion of this work that Miguel Abreu, using different methods, has, indepentently, computed a general formula for contact homology for toric contact manifolds with $c_{1}(\\xi)=0.$    \n\nThe ideas in this paper were originally motivated by examples related to a question of Lerman about contact structures on various $S^{1}$-bundles over $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ ~\\cite{Ler1}.  We are now able to distinguish these structures essentially using an extension of a theorem of Bourgeois for $S^1$-bundles over symplectic manifolds which admit perfect Morse functions ~\\cite{Bourg1}.  \n\\begin{thm}[Bourgeois]\nLet $(M, \\omega)$ be a symplectic manifold with $[\\omega] \\in H_2 (M, \\mathbb{Z})$  satisfying $c_1(TM) =\\tau[\\omega]$ for some $\\tau \\in \\mathbb{R}.$ Assume that $M$ admits a perfect Morse function. Let $V$ be a Boothby-Wang fibration over $M$ with its natural contact structure.  Then contact homology $HC^{cylindrical}_{\u2217} (V, \\xi)$ is the homology of the chain complex generated by infinitely many copies of $H_{*}(M, \\mathbb{R})$, with degree shifts $2 c k - 2$, $k \\in \\mathbb{N},$ where $c$ is the first Chern class of $T(M)$ evaluated on a particular homology class. The differential is then given in terms of the Gromov-Witten potential of $M.$\n\\end{thm}\n\nThis theorem exploits the fact that in the case of $S^{1}$-bundles the differential in contact homology is especially simple since there is essentially one type of orbit for each multiplicity (ie., simple orbits can be paramatrized so that their periods are all $1$).  Perfection of the Morse function kills the Morse-Smale-Witten part of the differential in Morse-Bott contact homology, so the chain complex reduces to the homology of $M.$  The grading in contact homology comes from the fact that the index calculations may be made via integration of $c_{1}(TM)$ over certain spherical two dimensional homology classes.  In the case of simply connected reduced spaces the cohomology ring of the base has a particularly nice form in terms of $2$ dimensional cohomology classes, obtained from the moment map as a Morse function.  Moreover all the two dimensional homology of the base in these cases is generated by spheres.  This fact makes Maslov\/Conley-Zehnder indices easier to compute without the need to find a stable trivialization of the contact distribution.  In this way we are able to extend the theorem of Bourgeois.  For homogeneous spaces the above theorem works ``out of the box'', but we must allow for non-monotone bases, ie., $c_{1}(\\xi) \\neq 0$.  To generalize the idea to general toric contact manifolds it is necessary to consider symplectic bases (of the circle bundle) which are orbifolds.\\footnote{It would be interesting to determine conditions under which the Reeb vector field could be perturbed so that the quotient of the Reeb action becomes an honest manifold when the contact manifold is simply connected.  There are known counter-examples, even in the toric case.  It would be interesting to find SFT obstructions to this.} The main new idea of this paper is to formalize this process for index computation from ~\\cite{Bourg1} and extend the result to an orbifold base, ie., the case where Reeb action is only locally free.     Moreover we can extend this to orbifolds bases which also admit a Hamiltonian action, since over $\\mathbb{C}$ their cohomology ring is still a polynomial ring in $H^{2}$ with spherical representatives of the ``diagonal'' homology classes.  \n\\begin{thm} \\label{thm:main}\n Let $M$ be a contact manifold, which is an $S^1$ bundle over the symplectic orbifold $\\mathcal{Z},$ where $\\mathcal{Z}$ admits a strongly Hamiltonian action of a compact Lie group.  Suppose that the curvature form, $d \\alpha$ of $M$ as a circle bundle over $\\mathcal{Z}$ is given by $$\\sum w_{j}\\pi^{*}c_{j},$$ where $c_{j}$ are the Chern classes associated to the Hamiltonian action.  Assume that the $c_{j}$ generate $H^{*}(\\mathcal{Z};\\mathbb{C})$ as variables in a truncated polynomial ring.  Let $\\tilde{w_{j}}$ be the coeffecient of $c_{j}$ in $c_{1}(T(\\mathcal{Z})).$  Assume further regularity of the $\\bar{\\partial}_{J}$-operator, and that $\\sum_{j} \\tilde{w_{j}} >1.$  Then contact homology is generated by copies of the homology of the critical submanifolds for any of the Morse-Bott functions given by the components of the moment map, with degree shifts given by twice-integer multiples of the sums of the $\\tilde{w_{j}}$ plus the dimension of the stratum, $S$, of $\\mathcal{Z}$ in which the given Reeb orbit is projected under $\\pi.$ \n\\end{thm}\nAs corollaries we obtain contact homology for both toric and homogeneous contact manifolds.  The reader should also beware that such calculations are only formal without some sort of transversality of the $\\bar{\\partial}_{J}$ operator.  Though several such statements and proofs have been published there are still certain persistent gaps.  There are many current developments around this issue and the underlying analysis.  Without such a result there is no way to know if the counts that we make are actually correct.  In section $3$ we give an alternate argument for transversality for homogeneous contact manifolds and for toric contact manifolds using only elementary tools from algebraic geometry, this is possible only because the almost complex structures involved are integrable in these cases.\n\n\\begin{ack}\n I would like to thank Charles Boyer for his patience, interest, and assistance in my work.  I would also like to thank Alexandru Buium for many useful conversations about the dark arts of algebraic geometry.\n\\end{ack}\n\n\\section{Preliminary Notions}\nIn this section we give some basic definitions, theorems, and ideas in order to fix notation and perspective.  \nFirst of all we define a contact structure:\n\\begin{defn}\n A \\textbf{contact structure} on a manifold $M$ of dimension $2n-1$ is a \\textbf{maximally non-integrable} $2n-2$-plane distribution, $\\xi \\subset T(M).$\nIn other words $\\xi$ is a field of $2n-2$-planes which is the kernel of some $1$-form $\\alpha$ which satisfies $$\\alpha \\wedge d\\alpha^{n-1} \\neq 0.$$  Such an $\\alpha$ is called a \\textbf{contact 1-form}.    \n\\end{defn}\nNotice that given such a $1$-form $\\alpha$, $f\\alpha$ is also a contact $1$-form for $\\xi$ whenever $f$ is smooth and non-vanishing.  In the following, a contact manifold has dimension $2n-1$, hence the symplectization has dimension $2n$ and our symplectic bases all have dimension $2n-2.$  Unless otherwise stated we assume $M$ compact without boundary. \nGiven a choice of contact $1$-form, $\\alpha$ we define the \\emph{Reeb vector field} of\n $\\alpha$ as the unique vector field $R_{\\alpha}$ satisfying $$i_{R_{\\alpha}} d \\alpha = 0$$ and $$i_{R_{\\alpha}} \\alpha = 1.$$   \n A contact form $\\alpha$ is called \\emph{quasi-regular} if, in flow-box coordinates with respect to the flow of the Reeb vector field, orbits intersect each flow box a finite number of times.  If that number can be taken to be $1$ we call the $\\alpha$ \\emph{regular}.  Note that if the manifold is compact this makes all Reeb orbits periodic\\footnote{By Poincar\\'{e} recurrence, for example.} of the same period.  In contrast, the standard contact form on $\\mathbb{R}^{2n-1}$ is regular.  We call a contact manifold (quasi)regular if there is a contact form $\\alpha$ for $\\xi$ such that $\\alpha$ is (quasi)regular.  Given $(M, \\alpha)$, we denote by $V$ the \\emph{symplectization} of $M$: $$V := (M \\times \\mathbb{R}, \\omega = d(e^t \\alpha)).$$  A contact structure defines a symplectic vector bundle with transverse symplectic form $d\\alpha.$  We can then choose an almost complex structure $J_{0}$ on $\\xi.$  We extend this to a complex structure $J$ on $V$ by $$J|_{\\xi}= J_{0}$$ and $$J\\frac{\\partial}{\\partial t} = R_{\\alpha}$$ where $t$ is the variable in the $\\mathbb{R}$-direction.  Note that we also get a metric, as usual,  on $V$ compatible with $J$ given by $$g(v, w) = \\omega(v, Jw).$$\n\nWe have the important result from ~\\cite{BG00b}, originally proved in the regular case in ~\\cite{BW58}:\n\\begin{thm}[Orbifold Boothby-Wang]\n Let $(M, \\xi)$ be a quasiregular contact manifold.  Then $M$ is a principal  $S^1$-orbibundle over a symplectic orbifold $(\\mathcal{Z}, \\omega)$ with connection $1$-form $\\alpha$ whose curvature satisfies $$d\\alpha = \\pi^*\\omega.$$  If $\\alpha$ is \\textbf{regular} then $\\mathcal{Z}$ is a manifold, and $M$ is the total space of a principal $S^1$-bundle. \n\\end{thm}\n\nThis enables us to study the nature of $M$ via the cohomology of $\\mathcal{Z}.$  As we will see, in nice enough cases, the cohomology of $\\mathcal{Z}$ along with the bundle data of the Boothby-Wang fibration will determine the contact homology as well.  Notice that if we really want to study the quasi-regular case via the base we are forced to consider symplectic orbifolds, a complete study of contact geometry with symmetries necessarily must include orbifolds.\n\nThe first important examples here are the homogeneous contact manifolds.  These are contact manifolds which have a transitive action of a Lie group via contactomorphisms.  In all of these examples we can apply the theorem of Bourgeois \\emph{almost} directly\\footnote{We say almost, since these manifolds are not always monotone. ie., that $c_{1}$ is a multiple of the symplectic form.}.  Next we have the toric contact manifolds.  The toric situation is not quite as nice as with the homogeneous manifolds, but we will still be able to make a similar statement and read off contact homology from the Boothby-Wang data.\nBefore we describe the contact situation in detail, we describe the corresponding symplectic objects over which we will construct our $S^1$ bundles.  We will have to work with symplectic orbifolds as in some of our examples the action of the Reeb vector field is only locally free.\n\\subsection{Toric Symplectic and Toric Contact Manifolds}\n\n\\begin{defn}\n Let $M$ be a symplectic orbifold.  Suppose that the Lie group $G$ acts on $M$ by (orbifold) symplectomorphisms.  The action is called \\textbf{Hamiltonian} is for each $\\tau \\in \\mathfrak{g}$ the associated vector field $X_{\\tau}$ on $M$ satisfies $$i_{X_{\\tau}}\\omega = dH$$ for some $H \\in C^{\\infty}(M , \\mathbb{R}).$\n\\end{defn} \nWe will actually be interested in something stronger.  We will assume that our actions are what some authors call \\emph{strongly} Hamiltonian, ie., that there exists a \\emph{moment map} $$\\mu : M \\rightarrow \\mathfrak{g}^*$$ such that \n $$d\\langle \\mu(p),\\tau \\rangle=-i_{X_{\\tau}} \\omega.$$\n\nThis result ~\\cite{LerTol} will be extremely important for us when we determine how to compute contact homology.\n\\begin{thm}(Lerman-Tolman)\n Let $(M, \\omega)$ be a symplectic orbifold.  Let $G$ be a compact Lie group acting effectively, and such that the action is strongly Hamiltonian with moment map $\\mu$.  Then each component of the moment map is a Morse-Bott function whose critical submanifolds are all even dimensional with even Morse-Bott index.  \n\\end{thm}\n\n\nWhen the group $G$ is a torus of maximal possible dimension we have a name for these manifolds:\n\\begin{defn} Suppose that a symplectic orbifold $(M, \\omega)$ of dimension $2n$ admits an effective Hamiltonian action of a torus $T^n$ of dimension $n.$ Then we call $(M, \\omega)$ a \\textbf{toric} symplectic orbifold.\n \\end{defn}\n\nNow we consider a compact Lie group $G$ acting via contactomorphisms on the contact manifold $(M^{2n-1}, \\xi).$   By averaging over $G$ we can always obtain a $G$-invariant contact form for $\\xi.$  We find it convenient to limit all of our actions.  First given $\\zeta \\in \\mathfrak{g}$, let $X_{\\zeta}$ denote the fundamental vector field associated to $\\zeta$ defined via the exponential map.  The following definition was introduced in ~\\cite{BG00}.  \n\n\\begin{defn}  Let $G$ be a Lie group which acts on the contact manifold $(M, \\xi).$  The action is said to be of \\textbf{Reeb type} if there is a contact $1$-form $\\alpha$ for $\\xi$ and an element $\\zeta \\in \\mathfrak{g}$, such that $X^{\\zeta}=R_{\\alpha}.$  \n\\end{defn}\n\n This is important since as long as such an action is proper, we know that our contact manifold is an $S^1$-bundle.  The proof of the following can be found in ~\\cite{BG} \n\n\\begin{prop}\n If an action of a torus $T^k$ is of Reeb type then there is a quasi-regular contact structure whose $1$-form satisfies, $ker(\\alpha)=\\xi.$ Moreover, then $M$ is the total space of a $S^1$ bundle over a symplectic orbifold which admits a Hamiltonian action of a torus $T^{k-1}.$  \n\\end{prop}\n\n\nWe now need the definition:\n\\begin{defn}\n A \\textbf{toric contact manifold} is a co-oriented contact manifold $(M^{2n-1}, \\xi)$ with an effective action of a torus, $T^{n}$ of maximal dimension $n$ and a moment map\\footnote{The contact moment map can be defined in terms of the symplectization, $V$, or intrinsically in terms of the annhilator of $\\xi$ in $TM^*$.  For more information about this see ~\\cite{Ler2} or ~\\cite{BG}}  into the dual of the Lie algebra of the torus.  \n\\end{defn}\n\nNotice now that by constraining to actions of Reeb type we get that all of our toric contact manifolds are circle bundles over symplectic orbifolds.  Even more is true, the bases are toric.  So we limit our study strictly to those tori of Reeb type.\\footnote{Actually, our homology calculations are potentially false otherwise, since in dimension $3$, there are toric manifolds with over-twisted contact structures. These always have vanishing contact homology (when contact homology is well-defined).}  An action being of Reeb type is quite a strong condition, but by the classification theorem of Lerman ~\\cite{Ler2} we see that it includes many cases of interest.  We only include his classification for $2n-1 >3.$\n\\begin{thm}\n Let $(M, \\xi)$ be a toric contact manifold of dimension greater than $3.$  \n\\begin{enumerate}\n \\item[i.] If the torus action is free, then $M$ is a principal torus bundle over a sphere.\n\\item[ii.] If the action is not free then the \\textbf{moment cone} is a \\textbf{good rational polyhedral cone}. \n\\end{enumerate}\n\n\\end{thm}\nWe are interested mostly in the second case and since we have the following proposition in~\\cite{BG} it is completely reasonable for us to restrict ourselves to actions of Reeb type.\n\\begin{prop}\n Let $M$ be a contact toric manifold of dimension greater than $3$.  Then the action of the torus is of Reeb type if and only if it is not free and the moment cone contains no non-zero linear subspace. \n\\end{prop}\n\n\\subsection{Cohomology Rings of Reduced Spaces}\nIn this section we follow ~\\cite{GlSt}.  The thing that really makes our calculation possible in its simple form is the wonderful structure of the cohomology rings of symplectically reduced spaces, ie, they are all truncated polynomial rings in the Chern classes.  Moreover in the simply connected case, we know that all of $H_{2}$ can be represented by spheres.  Even better, we can always relate all of these homology and cohomology classes to the moment map.       \n\nFirst let's work out what we get in general. Let $(M, \\omega)$ be a symplectic manifold of dimension $2n.$ Let $G$ be a compact connected Lie group of dimension d which acts via (strongly) Hamiltonian symplectomorphisms, $\\mathfrak{g} = Lie(G).$  Let $$\\mu: M \\rightarrow \\mathfrak{g^{*}}$$ denote the corresponding moment map.  Let $\\tau$ be a regular value of $\\mu$, let $$X_{\\tau}=\\mu^{-1}(\\tau).$$  Then $X_{\\tau}\/G$ is a symplectic orbifold of dimension $2(n-d)$.  Set $\\mathcal{Z} = X_{\\tau}\/G$.  Let $c_{1}, \\ldots, c_{n}$ be the Chern classes of the fibration $M \\rightarrow \\mathcal{Z}.$\n\n\\begin{thm}\n If the $c_{1}, \\ldots, c_{n}$ generate $H^{*}(\\mathcal{Z} ; \\mathbb{C})$ then\n$$H^{*}(\\mathcal{Z}; \\mathbb{C}) \\simeq \\mathbb{C}[x_{1}, \\ldots, x_{n}]\/ann(v_{top})$$\nwhere $ann(v_{top})$ is the annhilator of the highest order homogeneous part of the symplectic volume of $\\mathcal{Z}.$\n\\end{thm}\n\nAgain the following results are in ~\\cite{GlSt}\n\\begin{prop}\n Let $\\mathcal{Z}$ be a toric orbifold.  Then the Chern classes as above generate $H^{*}(\\mathcal{Z};\\mathbb{C}).$ \n\\end{prop}\nTo apply this to all homogeneous contact manifolds we need not only the case of flag manifolds but also of \\emph{generalized} flag manifolds.  These are quotients of a complex semi-simple Lie group $G$ by a \\emph{parabolic} subgroup $P.$  These include the flag manifolds.  We extend the result from ~\\cite{GlSt} about flag manifolds to $G\/P.$  For more about generalized flag manifolds see ~\\cite{BE} and ~\\cite{BGG}\n\\begin{prop}\n Let $G\/P$ be a generalized flag manifold.  Then the cohomology is generated by the Chern classes as above. \n\\end{prop}\n\\begin{proof}\n Since $P$ is parabolic, it contains a Borel subgroup.  Each Schubert cell in $G\/P$ lifts to one in $G\/B.$  This gives an injective map $$H^{*}(G\/P ; \\mathbb{C}) \\rightarrow H^{*}(G\/B;\\mathbb{C}).$$  Thus we need only to see that the Chern classes generate $H^{*}(G\/B; \\mathbb{C})$ which is known from ~\\cite{Bor1}.\n\\end{proof}\n\n\n\\subsection{The Maslov Index and the Conley-Zehnder Index}\nThe Maslov index associates to each path of symplectic matrices a rational number. This particular definition originally appeared in Salamon and Robbins.  This index determines the grading for the chain complex in contact homology.  The Maslov index should be thought of as analagous to the Morse index for a Morse function.  The analogy isn't perfect, since the actual Morse theory we consider should give information about the loop space of the contact manifold.  Also note that our action functional has an infinite dimensional kernel.   \nIt should be noted that we will describe three indices in the following.  Two of them will be called the Maslov index.  This is unfortunate, but it will always be clear which Maslov index we will use at any particular time.\n\\begin{remark}Historically, the Maslov index arose as an invariant of loops of Lagrangian subspaces in the Grassmanian of Lagrangian subspaces of a symplectic vector space V.  In this setting the Maslov index is the intersection number of a path of Lagrangian subspaces with a certain algebraic variety called the Maslov cycle.\nThis is of course related to our Maslov index of a path of symplectic matrices, since we can consider a path of Lagrangian subspaces given by the path of graphs of the desired path of symplectic matrices. For more information on this see ~\\cite{McDSal}, and ~\\cite{SalRob}.\n\\end{remark}\n\\begin{remark}For a symplectic vector bundle, $E$,  over a Riemann surface, $\\Sigma$ there is symplectic definition of the first Chern number $\\langle c_{1}(E),\\Sigma \\rangle$.  It turns out that this Chern number is the loop Maslov index of a certain loop of symplectic matrices, obtained from local trivializations of $\\Sigma$ decomposed along a curve $\\gamma \\subset \\Sigma$.  This Chern number agrees with the usual definition, considering $E$ as a complex vector bundle, and can be obtained via a curvature calculation. \n\\end{remark}\nLet $\\Phi(t)$, $t \\in[0, T]$ be a path of symplectic matrices starting at the identity such that $det(I - \\Phi(T)) \\neq 0$\\footnote{This is the \\emph{non-degeneracy} assumption.  In the context of the Reeb vector field, this condition implies that all closed orbits are isolated}.  We call a number $t \\in [0,T]$, a \\emph{crossing} if $det(\\Phi(t) - I)=0.$  A crossing is called \\emph{regular} if the \\emph{crossing form} (defined below) is non-degenerate.  One can always homotope a path of symplectic matrices to one with regular crossings, which, as we will see below, does not change the Maslov index. \n\nFor each crossing we define the \\emph{crossing form} \n$$\\Gamma(t)v = \\omega(v, D \\dot{\\Phi}(t)).$$ \nWhere $\\omega$ is the standard symplectic form on $\\mathbb{R}^{2n}.$\n\\begin{defn}\n The Conley-Zehnder of the path $\\Phi{t}$ under the above assumptions is given by:\n$$\\mu_{CZ}(\\Phi) = \\frac{1}{2}sign(\\Gamma(0)) + \\sum_{t\\neq 0\\,, \\,t\\, a\\,\\, crossing}sign(\\Gamma(t))$$\n\\end{defn}\nThe Conley-Zehnder index satisfies the following axioms: \n\\begin{itemize}\n\\item [Homotopy:] $\\mu_{CZ}$ is invariant under homotopies which fix endpoints.\n\\item [Naturality:] $\\mu_{CZ}$ is invariant under conjugation by paths in $Sp(n, \\mathbb{R}).$\n\\item [Loop:] For any path, $\\psi$ in $Sp(n, \\mathbb{R}),$ \nand a loop $\\phi$, $$\\mu_{CZ}(\\psi \\cdot \\phi) = \\mu_{CZ}(\\psi) + \\mu_{l}(\\phi).$$  Where $\\mu_{l}$ is the Maslov index for loops of symplectic matrices.\n\\item [Direct Sum:]  If $n = n' + n''$ and $\\psi_{1}$ is a path in $Sp(n', \\mathbb{R})$ \nand $\\psi_{2}$ is a path in $Sp(n'', \\mathbb{R})$ \nthen for the path $\\psi_{1} \\oplus \\psi_{2} \\in Sp(n', \\mathbb{R}) \\bigoplus Sp(n'', \\mathbb{R}),$ we have \n$$\\mu(\\psi_{1} \\oplus \\psi_{2}) = \\mu(\\psi_{1}) + \\mu(\\psi_{2}).$$\n\\item [Zero:]  If a path has no eigenvalues on $S^1$, \nthen its Conley-Zehnder index is 0.\n\\item [Signature:]  Let $S$ be symmetric and nondegenerate with $$||S|| < 2\\pi.$$  Let $\\psi(t) = exp(JSt)$, then $$\\mu_{CZ}(\\psi) = \\frac{1}{2}sign(S).$$  \n\\end{itemize}\nThe Conley-Zehnder index is still insufficient for our purposes since we need the assumption that at time $T=1$ the symplectic matrix has no eigenvalue equal to 1.\nWe introduce yet another index for arbitrary paths from ~\\cite{SalRob}.  We will call this index the Maslov index and denote it $\\mu.$  \n\nFor this new index we simply add half of the signature of the crossing form at the terminal time of the path to the formula for the Conley-Zehnder index.\n$$ \\mu(\\Phi(t)) = \\frac{1}{2}sign(\\Gamma(0)) + \\sum_{t\\neq 0\\,, \\,t\\, a\\,\\, crossing}sign(\\Gamma(t)) + \\frac{1}{2}sign(\\Gamma(T))$$\nThis Maslov index satisfies the same axioms as $\\mu_{CZ}$ as well as the new property of catenation.  This means that the index of the catenation of paths is the sum of the indices.\n\n\\subsection{Indices for homotopically trivial closed Reeb orbits}\nLet $\\gamma$ be a closed orbit of a Reeb vector field.  Choose a symplectic trivialization of this orbit in $M,$ ie., take a map $u : D \\rightarrow M$ from a disk into $M,$ with the property that the boundary of the image of $u$ is $\\gamma$ and a bundle isomorphism between $u^* \\xi$ and standard symplectic $\\mathbb{R}^{2n}$, $(\\mathbb{R}^{2n} , \\omega_{0}).$   Now we look at the Poincare time $T$ return map of the associated flow (with respect to this trivialization, choosing a framing), where $T$ is the period of $\\gamma.$  If the linearized flow  has no eigenvalue equal to $1$, we define the Conley-Zehnder index of $\\gamma$ to be the Conley-Zehnder index of the path of matrices given by the linearized Reeb flow.  If there are eigenvalues equal to $1$ we calculate the Maslov index of the path of matrices coming from the flow (in an appropriate symplectic trivialization.)  Note that when there is no eigenvalue equal to $1$, the two indices agree.  \n\nThe Conley-Zehnder and Maslov indices depends on the choice of spanning disk or Riemann surface used in the symplectic trivialization.  Different choices of disks will change the index by twice the first Chern class\\footnote{This is the reason that so often in the literature on contact homology authors insist that $c_{1}(\\xi) = 0$.  This index defines the grading of contact homology so if this Chern class is non-zero we must be careful to keep track of which disks we use to cap orbits.} of $\\xi$.  Intuitively, given a periodic orbit of the Reeb vector field, this index reveals how many times nearby orbits ``wrap around'' the given orbit.    \n\n\n\n\\section{Morse-Bott Contact Homology}\nOur invariants come from a sort of infinite dimensional Morse theory on the loop space of the symplectization $V$ of $M$.  In this context, the analogue of gradient trajectories is given by pseudo-holomorphic images of punctured Riemann surfaces.  For us a $J$-holomorphic curve is a $C^{\\infty}$ function from a Riemann surface into $V$ whose total derivative is almost-complex linear with respect to $J$ on $V$, and the standard complex structure on the Riemann surface.  \n\nWe want to count so-called \\emph{rigid} curves between critical points of a Morse functional.  To do this we study certain moduli spaces of \\emph{stable} maps of punctured Riemann surfaces into $V.$  As we will see these curves are \\emph{asymptotically cylindrical} over closed Reeb orbits.  The number of punctures corresponds to the number of orbits.  We denote such moduli spaces by $$\\mathcal{M}_{0,J}^A(\\gamma_{1}, \\ldots ,\\gamma_{s};\\gamma_{s+1}, \\ldots ,   \\gamma_{s+k},V).$$  This reads ``the set of genus $0$ ie., \\emph{rational} maps into $V$ with $s$ positive punctures and $k$ negative punctures representing the $2$ dimensional homology class $A$.  When no confusion can arise we will drop the ''$V$`` from the notation.  Also, when it is understood that there is only one positive puncture, we omit the semicolon.  \nThe trouble is that the relevant moduli spaces of curves have, at times, quite an elusive geometric structure.  We try to capture their structure as zero sets of a section of a certain infinite dimensional vector bundle over the space of maps from $\\Sigma$ into $V.$  The relevant section is a Fredholm operator with a nice index given in terms of the dynamics of the Reeb vector field of $\\alpha.$  The problem is that one cannot directly apply the implicit function theorem to determine the dimension of the moduli space since the linearized operator could have a nonzero cokernel.  Therefore, until we can prove that the operator is transverse to the zero section of the bundle we cannot conclude any kind of manifold structure or dimension formula from the Fredholm index of this operator. \n\nIn the cases at hand we have an integrable almost complex structure so we can try to use Dolbeault cohomology of an appropriate complex.  As it turns out, this works under suitable positivity conditions, so we can determine transversality from bounds on spherical chern numbers in $M.$  \n\nContact homology is the homology of the differential graded algebra generated by periodic orbits of the Reeb vector field and graded by the reduced Conley-Zehnder index equal to $\\mu_{CZ} +n-3.$  Whenever there are no Reeb orbits of reduced Conley-Zehnder index equal to $0$, $1$, or $-1$, we can actually use a simpler chain complex.  We then consider the graded \\emph{vector space} generated by the closed Reeb orbits, in this case we call the homology \\emph{cylindrical contact homology}.\\footnote{There is an alternative to this called \\emph{linearized contact homology} which ''chops off`` unwanted legs in pairs of pants via an augmentation.  This homology is well-defined even when cylindrical homology is not.  Whenever cylindrical homology \\emph{is} well-defined these two invariants are the same.}  The reader should be wary that when the first Chern class of the contact distribution\\footnote{We consider $\\xi$ as a symplectic vector bundle, defining a first Chern class.} is non-zero, then in order to define a reasonable grading we must keep track of all choices made, spanning surfaces for Reeb orbits, for example.    \nFor cylindrical contact homology the differential is given as follows:\n  $$ d\\gamma=\\sum_{\\gamma^{'}} n_{\\gamma , \\gamma^{'}} \\gamma^{'}.$$ Where $$n_{\\gamma, \\gamma^{'}} = 0$$\\\\*\nwhenever $$dim(\\mathcal{M}_{0,J}^{A}(\\gamma, \\gamma^{'}, V))\/\\mathbb{R} \\neq 0$$\notherwise it is equal to the signed count of these curves.\\footnote{Recall that we have an action of $\\mathbb{R}$ on the compact moduli space $\\mathcal{M}_{0}^{A}(\\gamma, \\gamma^{'}, V, J)$ by translation in $t$, thus we make a count when the \\emph{quotient} is $0$-dimensional.}\nThe following is from ~\\cite{EGH}.\n\\begin{thm}\n Suppose that there are no Reeb orbits of Conley-Zehnder index $0, -1, 1.$  Then $d\\circ d =0$, and cylindrical contact homology, $CH_{*}(M, \\xi)$ is defined to be the homology of the above chain complex.  Moreover it is independent of all choices made, including $J$ and $\\alpha$, in particular it is an invariant of the \\textbf{contact structure} $\\xi.$ \n\\end{thm}\n\nThis differential counts \\emph{rigid} \\footnote{Rigid, in this context means that these curves live in $0$-dimensional moduli spaces} $J$-holomporphic curves in the symplectization of $M$ asymptotically over $\\gamma$ at $\\infty$ and over $\\gamma^{'}$ at $- \\infty.$ \n\n\\begin{remark}\n Contact homology and in particular, the Floer-like cylindrical contact homology, are just the tip of the iceburg of the larger \\textbf{symplectic field theory} which contains many more subtle invariants.\n\\end{remark}\n\nTo use this complex, however, we really need the condition that the periodic Reeb orbits are isolated.  Since we are working exclusively with $S^1$-bundles, all Reeb orbits are periodic so are never isolated.  It turns out there there is a Morse-Bott version of all of this theory.  \n\nTo make this work we actually work with Morse theory on the base.  If we work in an orbifold we have different orbifold strata given by the various orbit types of the original action.  Recall that we assume all actions are strongly Hamiltonian.\nWe now introduce some terminology.\n\\begin{defn}\n The \\textbf{action functional} is the map $$\\mathcal{A} : C^{\\infty}(S^1, M) \\rightarrow \\mathbb{R}$$ given by $$\\gamma \\mapsto \\int_{\\gamma} \\alpha,$$ where $\\alpha$ is a contact 1-form for $(M, \\xi).$\n \\end{defn}\nNote that critical points of $\\mathcal{A}$ are periodic orbits of the Reeb vector field.  \n\\begin{defn} Let $(M, \\xi)$ be a contact manifold with contact form $\\alpha.$  The \\textbf{action spectrum}, $$\\sigma(\\alpha) = \\{r \\in \\mathbb{R} | r = \\mathcal{A}(\\gamma)\\}$$ for $\\gamma$ a periodic orbit of the Reeb vector field.\n \n\\end{defn}\n\\begin{defn} Let $T \\in \\sigma(\\alpha).$  Let $$N_{T} = \\{ p \\in M | \\phi_{p}^{T} = p\\},$$ $$S_{T} = N_{T}\/S^{1},$$ where $S^1$ acts on $M$ via the Reeb flow.  Then $S_{T}$ is called the \\textbf{orbit space} for period $T$.\n \\end{defn}\nHere the orbit spaces are precisely the orbifold strata when $M$ is considered as an $S^1$ orbibundle.\n\nFor our Morse-Bott set-up we assume that our contact form is of Morse-Bott type, ie.  \n\\begin{defn}\nA contact form is said to be of Morse-Bott type if\n\\begin{enumerate}\n \\item[i.] The action spectrum:\n$$\\sigma(\\alpha) := \\{r \\in \\mathbb{R} : \\mathcal{A}(\\gamma) = r,\\, for\\, some\\, periodic\\, Reeb\\, orbit\\, \\gamma.\\}$$ is discrete.\n\\item[ii.] The sets $N_{T}$ are closed submanifolds of M, such that the rank of $d\\alpha|_{N_{T}}$ is locally constant and $$T_{p}(N_{T}) = ker(d\\phi_{T} - I).$$\n\\end{enumerate}\n\\end{defn}\n\\begin{remark}These conditions are the Morse-Bott analogues for the functional on the loop space of $M.$\n\\end{remark}\n\n \nNotice that in the case of $S^1$ bundles this is always satisfied.  The key observation, as we soon shall see, is that we can relate $J$-holomorphic curves to Morse theory on the symplectic base.  Since we consider only quasi-regular contact manifolds here, we can always approximate the contact structure by one with a dense open set of periodic orbits of period 1, say, and a finite collection of strata of orbits of smaller period, each such stratum has even dimension and has codimension at least 2 (other than the dense set of regular points, of course).  \n\n\n\\subsection{Moduli}\nBefore we describe the Morse-Bott chain complex we need to describe the moduli spaces of pseudoholomorphic curves with which we will be working.  So, as before let $(M, \\xi)$ be a contact manifold, $V$ its symplectization, and $J$ an almost complex structure on $V$ compatible with the transverse symplectic structure on $M.$   The curves that we're interested in are $J$-holomorphic maps from punctured $S^2$'s into the the symplectization of our contact manifold.  Such curves are \\emph{asymptotically cylindrical} over closed Reeb orbits.\n\\begin{defn}\nLet $\\Sigma$ be a Riemann surface with a set of punctures $Z = \\{z_{1}, \\ldots, z_{k}\\}$  A map $$u = (a(s,t), f(s,t)): \\Sigma \\setminus Z \\rightarrow V$$ is called \\textbf{asymptotically cylindrical} over the set of Reeb orbits $\\gamma_{1}, \\ldots, \\gamma_{k}$ if for each $z_{j} \\in Z$ there are cylindrical coordinates centered at $z_{j}$ such that \n$$lim_{s \\rightarrow \\infty} f(s, t) = \\gamma (T t)$$ \nand\n$$lim_{s \\rightarrow \\infty} \\frac{a(s, t)}{s} = T$$\n\\end{defn}    \n\nHere are some precise statements from ~\\cite{BEHWZ}:\n\\begin{prop}Suppose that $\\alpha$ is of Morse,\\footnote{Here we mean that for each periodic Reeb orbit, the time 1 return map has no eigenvalue equal to $1$.} or Morse-Bott type. Let $$u = (a, f ) : \\mathbb{R}^{+} \\times \\mathbb{R}\/\\mathbb{Z} \\rightarrow (\\mathbb{R} \\times M, J)$$ be a $J$-holomorphic curve of finite energy.\\footnote{For the most general definitions of energy see ~\\cite{BEHWZ}}  Suppose that the image of $u$ is unbounded in $\\mathbb{R} \\times M$. Then there exist a number $T \\neq 0$ and a periodic orbit $\\gamma$ of $R_{\\alpha}$ of period $|T|$ such that\n\n$$lim_{s \\rightarrow \\infty} f(s, t) = \\gamma (T t)$$ \nand\n$$lim_{s \\rightarrow \\infty} \\frac{a(s, t)}{s} = T.$$\n\n\\end{prop}\n\nThis immediately implies \n\\begin{prop}\nLet $(S,j)$ be a closed Riemann surface and let $$Z=\\{z_1,\\ldots , z_k \\}\\subset S$$ be a set of punctures.  Every $J$-holomorphic curve $$F = (a,f):(S \\setminus Z)\\rightarrow \\mathbb{R} \\times M$$ of finite energy and without removable singularities is asymptotically cylindrical near each puncture $z_i$ over a periodic orbit $\\gamma_i$ of $R_{\\alpha}$.\n\\end{prop}\n\nThese propositions are extremely important to us because they show that it is reasonable to define gradient trajectories between Reeb orbits to be $J$-holomorphic curves in the symplectization.   \nWe have even more, ie., a Gromov compactness theorem, which says that although the moduli spaces are not necessarily compact, we can compactify them by adding stable curves of height $2$.  Even better, via a gluing construction ~\\cite{Bourg1} we show that the boundary of the moduli space is equal to set of height $2$ curves.  \n\nAs above we consider the moduli space of generalized $J$-holomorphic curves from an $s$-times punctured Riemann surface into $V$ asymptotically cylindrical over the orbit spaces $S_1, \\ldots, S_{s}$ representing the class $A$:\n$$\\mathcal{M}^{A}_{0,J}(S_{1}; \\ldots, S_{s}, V).$$\n\nIn this notation the first orbit space is from the \\emph{positive} puncture, all others are negative.  These moduli spaces are the analogues of the gradient trajectories of Morse theory.  We only count them when they come in zero dimensional families.  Thus we need a dimension formula for these spaces.  \n\\begin{prop}\n The virtual dimension for the moduli space of generalized genus $0$ $J$-holomorphic asymptotic over the orbit spaces $S_{T_{0}}, \\ldots, S_{T_{1}},\\ldots, S_{T_{s}}$ (with 1 positive, and $s$ negative punctures) representing $A$ is equal to\n$$(n-3)(1-s)  + \\mu(S_{T_{0}}) + \\frac{1}{2}dim(S_{T_{0}}) - \\sum_{i=0}^s (\\mu(S_{T_{i}}) + \\frac{1}{2} dim (S_{T_{i}})) + 2c_{1}(\\xi, \\Sigma),$$ where $\\Sigma$ is a Riemann surface used to define the symplectic trivialization and homology class $A.$ \n\\end{prop}\n\nIn cylindrical contact homology, since we only are keeping track of cylinders, we take $s=1$  and this formula reduces to $$ \\mu(S_{T_{+}}) + \\frac{1}{2}dim(S_{T_{+}}) + \\mu(S_{T_{-}}) + \\frac{1}{2} dim (S_{T^{-}}) + 2c_{1}(\\xi, \\Sigma).$$\nOf course if $\\xi$ has a regular structure this boils down to $$ \\mu(S_{T^{+}}) - \\mu(S_{T^{-}}) + 2n-2 + 2c_{1}(\\xi, \\Sigma).$$\nFor a proof of this formula see ~\\cite{Bourg1}.  Bourgeois' proof is of interest as traditionally these kinds of results come from a spectral flow analysis.  Bourgeois, however, makes interesting use of the Riemann-Roch theorem.\n \nWe want to understand the structure of the moduli space since our Morse-like chain complex uses these curves to construct the differential.  The reader should be aware that the formula for the dimension of the moduli space above is really a \\emph{virtual} dimension until some sort of transversality is achieved for some $\\bar{\\partial}_{J}$-operator. This formula is obtained via Fredholm analysis on the space  of $C^{\\infty}$ maps from $S^2 \\setminus \\{z_1, z_2, \\ldots, z_{j}\\}$ into $V.$ The $\\bar{\\partial}_{J}$ turns out to be a Fredholm section of a certain infinite dimensional bundle over this space whose kernel is precisely the set of $J$-holomorphic curves.  The Fredholm index $\\bar{\\partial}_{J}$ is the dimension formula above.  The trouble is that a priori, we cannot rule out a non-zero cokernal, hence our dimension formula could be \\emph{under counting} the relevant curves.  There have been many attempts at transversality proofs, and it seems as though the new \\emph{polyfold} theory of Hofer, Zehnder and Wysocki should solve the problem.  There are also proofs using virtual cycle techniques, (cf. ~\\cite{Bourg1}) however even here it seems that there may be potential gaps.  Therefore we show how, in some cases, we can justify the validity of our curve counts through more elementary geometric considerations.\n\nIn the cases that we are considering in this paper, the almost complex structure will be integrable, thus we can use algebro-geometric techniques to find conditions for regularity of $J$.\\footnote{The word \\emph{regularity} is over used.  Here we mean that for this $J$, the $\\bar{\\partial}_{J}$ operator is surjective, as a section in a suitable infinite dimensional vector bundle.}  \n\nNow let us describe the relationship between moduli spaces of stable curves in a symplectic orbifold and the moduli space of curves into the symplectization of its Boothby-Wang manifold.  Notice that the symplectization $W$ is just the associated line (orbi)bundle to the principle $S^1$-(orbi)bundle\\footnote{Of course, in the situations we are dealing with in this paper, $M$ is a manifold even if it is the total space of an orbibundle.}, $M$, with the zero section removed.   Given as many marked points as punctures we actually get a fibration, here curves upstairs are sections of $L$ with zeroes of order $k$ and poles of order $l$ once we fix the phase of a section we actually get unique curves.  This is described for the case of \\emph{regular} contact structures in ~\\cite{EGH}.  For $S^{1}$-bundles over $\\mathbb{CP}^1$ with isolated cyclic singularities Rossi extended this result in ~\\cite{PR}, we'll actually need an extension of this to higher dimension.  The point here is that we want to coordinate our curve counts upstairs with the ``Gromov-Witten'' curve count downstairs.  In the case where the base is an orbifold we must make sure that we can get an appropriate curve in the sense of Gromov-Witten theory on orbifolds.  It should be noted that in symplectizations all moduli spaces come with an $\\mathbb{R}$-action by translation.  Whenever we talk about $0$-dimensional moduli spaces, we really mean that we are considering $1$-dimensional moduli spaces quotiented out by the $\\mathbb{R}$-action giving $0$-dimensional manifolds\\footnote{Actually these moduli spaces in general are branched orbifolds with corners, however in the cases that we consider in this paper they really are manifolds.}.      \n\n\nThe following lemma comes from ~\\cite{CR}.\n\\begin{lemma}\n Suppose that $u$ is a $J$-holomorphic curve into the symplectic orbifold $\\mathcal{Z}$, then either $u$ is completely contained in the orbifold singular locus or it intersects it in only finitely many points.\n\\end{lemma}\nWe use this to prove:\n\\begin{lemma}\nLet $u : \\Sigma  \\rightarrow \\mathcal{Z}$ be a non-constant $J$-holomorphic map between a Riemann surface  and a symplectic orbifold. Then there is a unique orbifold structure on $\\Sigma$ and a unique germ of a $C^{\\infty}$-lift $\\tilde{u}$ of $u$ such that $u$ is an orbicurve.\n\\end{lemma}\n\\begin{proof}\nFirst let us assume that the marked points are all mapped into the singular locus, since otherwise the curve only intersects the singular locus in a finite number of points.  Now $u_{z_{i}}$ corresponds to a closed Reeb orbit of non-generic period, ie., a curve in $S_{T_{k}}$, say. Take an element from the moduli space of curves into $W$ asymptotically cylindrical over $S_{T_{k}}$ in some slot.  We need only to take a local uniformizing chart equivariant with respect to $\\mathbb{Z}_{T_{k}}.$\n\\end{proof}\n\nFrom this we actually get a fibration.\n\\begin{prop}\n There is a fibration $$pr :\\mathcal{M}_{0, J}(S_{T_{1}}; S_{T_{2}}, \\ldots , S_{T_{k}}) \\rightarrow \\mathcal{M}_{0, k}(a_{1}, \\ldots, a_{k}).$$ \n\\end{prop}\n\n\nWith this understanding of the moduli space, assuming now that $J$ is integrable we see that that the linearized Cauchy-Riemann operator is the $\\bar{\\partial}$ operator of the right Dolbeault complex on $\\mathcal{Z}$. This leads to the following criterion for regularity of $\\bar{\\partial}_{J}$ at each $u$ adapted to the $S^1$ bundle case from ~\\cite{McDSal2}.  First we note that $u: \\mathbb{CP}^1 \\rightarrow V$, we can look at $u^*T(V),$ whose characteristic classes look like those of $u^{*}\\xi.$  Over $\\mathbb{CP}^{1}$, $u^*\\xi$ splits as a sum of line bundles:\n$$u^*\\xi = \\bigoplus_{j} L_{j}.$$   \nThis splitting doesn't necessarily work for general orbifolds as ambient spaces (unless the sphere doesn't intersect the orbifold singular locus at all, but for toric orbifolds this splitting principle always works ~\\cite{Gmn97}.  Of course, in the case of a regular contact form, this is just the classical Groethendieck splitting principle. \n\\begin{thm}[Regularity criterion for genus $0$ moduli spaces when $J$ is integrable]\nSuppose that $J$ is integrable. Suppose that $$\\langle c_{1}(L_{j}), A \\rangle \\geq -2 + s -t$$ for every $A \\in H_{2}(\\mathcal{Z})$ which is represented by a 2-sphere.  Then the linearized Cauchy-Riemann operator is surjective and the genus $0$ moduli space of curves with $s$ positive punctures and $t$ negative punctures is a smooth manifold of dimension given by the Fredholm index.  In the case that $\\mathcal{Z}$ is an orbifold, we require that $$c_{1}(L_{j}) \\geq  \\sum_{\\alpha}(1-\\frac{1}{m_{\\alpha}})c_{1}(\\mathcal{O}(D_{\\alpha}))  -2-s-t,$$ where $D_{\\alpha}$ are branch divisors.\n\\end{thm}\n\\begin{proof}\nLet $z_{1}, \\ldots, z_{s}, \\ldots z_{s +t}$ be distinct points on $S^2.$  Consider the divisor $$D= k_{1}z_{1} + \\ldots + k_{s}z_{s} - k_{s+1}z_{s+1} - \\ldots - k_{s+t}z_{s+t}.$$  Then the Cauchy-Riemann operator is just the $\\bar{\\partial}$-operator of the Dolbeault complex for the line orbibundle $L_{j} \\otimes \\mathcal{O}(D).$\n$$ \\bar{\\partial} : \\Omega^{0}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)) \\rightarrow \\Omega^{0,1}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)).$$\nThe cokernel of $\\bar{\\partial}$ is just the $(0,1)$ cohomology of that complex.  But we have the following isomorphisms: \n$$H_{\\bar{\\partial}}^{0,1}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)) \\simeq H_{\\bar{\\partial}}^{1,0}(\\mathbb{CP}^1, (L_{j} \\otimes \\mathcal{O}(D))^*)^* \\simeq H_{\\bar{\\partial}}^{0,1}(\\mathbb{CP}^1, (L_{j} \\otimes \\mathcal{O}(D))^* \\otimes K).$$\nFor the last group to be $0$, we must have $$c_{1}(L_{j} \\otimes \\mathcal{O}(D))^* \\otimes K) <0.$$  This happens whenever $$c_{1}(L_{j}) > -2 - deg(D).$$  \nIn the quasi-regular case we check this in the orbifold sense. So we consider the orbifold first Chern class :\n$$c_{1}^{orb}= c_{1}(L_{j}) - \\sum_{\\alpha}(1-\\frac{1}{m_{\\alpha}})c_{1}(\\mathcal{O}(D_{\\alpha}))$$ where the sum at the end is non-zero only in the presence of branch divisors, $D_{\\alpha}.$  Here we must bound this below by $-2 -deg(D).$  \n\\end{proof}\n\\begin{thm}\n$$\\langle c_{1}(L), A \\rangle \\geq -2 + s -t$$ whenever $L$ is a line (orbi)bundle obtained by the Boothby-Wang fibration whose total space is either of the following:\n\\begin{enumerate}\n\\item[i.] a homogeneous contact manifold\n\\item[ii.] a toric Fano contact manifold.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\nThe proof of (i) is nearly the same Proposition 7.4.3 in ~\\cite{McDSal2} with $u^{*}TM$ replaced with $u^*(\\xi).$ Note that in case (i) we are always dealing with manifolds at each level rather than with orbifolds.\nFor (ii), for a splitting of $$u^*(\\xi) = \\bigoplus_j L_{j}$$ we get sections and positivity of Chern classes via the Fano condition.  Here the splitting takes place in the toric orbifold, then all the lines are pulled back to the sphere.  \n\\end{proof}   \n\n\\begin{cor}\n For homogeneous and toric contact manifolds the dimensions of all genus $0$ moduli spaces are given by the Fredholm index as predicted.\n\\end{cor}\nWe would like now to set up the Morse-Bott chain complex.  This was originally done in ~\\cite{Bourg1} and discussed for circle bundles in ~\\cite{EGH}.  We have already discussed some of the basic setup, now, much like the case in Morse theory we would like to relate the Morse-Bott case to the generic case.  The idea is to perturb our contact structure so it's periodic orbits are in $1$-$1$ correspondence with the critical points of some Morse-function.  In our case we would like to use our moment maps to get a perfect Morse or Morse-Bott function $f.$\nSo first for appropriate $\\epsilon$ and our Morse or Morse-Bott function$f$ we take the new contact form\n$$ \\alpha_{f} = (1 + \\epsilon f) \\alpha.$$\nThen critical points of $f$ correspond to periodic orbits of $\\alpha_{f}.$\nIf $f$ is a perfect Morse function as in the toric case we take this as our contact form.  Else, suppose $f$ is Morse-Bott, then we choose Morse functions on each critical submanifold.  Note that in the case of a Hamiltonian action of a compact Lie group all such submanifolds have even index and even dimension.  By $G$-invariance of the moment map this restricts to all orbit spaces.  \nIn this case the periodic orbits are in $1$-$1$ correspondence with critical points of Morse functions on the critical submanifolds of $f.$\nWe want to relate the Conley-Zehnder indices of the generic form with those of the original.  Since we have a $1$-$1$ correspondence between critical points and orbits, we will think of the chain complex associated to $\\alpha_{f}$ as critical points of $f.$  The index is the grading so we'll write the Conley-Zehnder index of the orbit corresponding to $p$, $\\mu_{CZ}(\\gamma_{p})$ as $|p|.$  In the Morse case we have\n$$ |p| = \\mu(S_{T_{k}}) -\\frac{1}{2}dim(S_{T_{k}}) + ind_{p} f.$$\\\\*  \nIn the case of a Morse-Bott function, we just proceed as in ordinary Morse theory to get a new Morse function on each critical submanifold, but now in this formula we use the Morse-Bott indices. \nNow we can define the differential for Morse-Bott contact homology.\n$$dp = \\partial p + \\sum_{q} n_{pq} q.$$\nWhere $\\partial p$ is just the Morse-Smale-Witten boundary operator, $n_{pq}$ is similar to the coeffecient in the generic case, and $ind_{p}$ is the Morse index for critical points.  Bourgeois proves, in his thesis ~\\cite{Bourg1}, that this homology computes contact homology.  For us, the particular form of the differential does not matter much since it will vanish for index reasons.  \n\n\\begin{thm}(Bourgeois)\n When the homology defined above exists it is isomorphic to the standard contact homology for non-degenerate contact forms. \n\\end{thm}\n\\section{Morse-Bott Contact Homology in the Homogeneous and Toric Cases}\nNow we apply the results from the previous section.  Let us first set some notation.  Suppose first that $(M, \\xi)$ is compact, simply connected, and admits a strongly Hamiltonian action of a Lie group as discussed in the introduction which is of Reeb type.  Then we know that there is a quasi-regular contact form $\\alpha$ for $(M, \\xi)$ equivariant with respect to the action.\nAs above, let $S_{T_{k}}$ denote the stratum in $\\mathcal{Z} = M\/(S^1)$ corresponding to Reeb orbits of period $T_{k}.$  Let $\\Gamma_j$ denote the local uniformizing group for the stratum $S_{T_{k}}.$  Recall that each stratum is a K\\\"{a}hler sub-orbifold of $\\mathcal{Z}.$ \nIn what follows assume that $H^{*}(\\mathcal{Z}; \\mathbb{C})$ is a truncated polynomial ring generated by elements in $H^{2}(\\mathcal{Z}; \\mathbb{C})$, ie., the Chern classes coming from the symplectic reduction defining $\\mathcal{Z}$ as a symplectically reduced orbifold.  Let us write such a basis of $H^2(\\mathcal{Z};\\mathbb{C})$ as $\\{c_{1}, \\ldots, c_{k} \\}.$  Now choose $1$ forms $\\tilde{c_{j}}$ representing the $c_{j}$'s   Now we just consider circle bundles over $\\mathcal{Z}$ by choosing connection $1$-forms $\\alpha$ with curvature $$ d \\alpha = \\sum_{j}\\pi^{*} w_{j} \\tilde{c_{j}}.$$  Notice that for $\\mathcal{Z}$ a toric orbifold, this construction yields all possible toric contact structures of Reeb type.   Note that we implicitly choose a symplectic form $\\omega = \\sum w_{i} \\tilde{c_{i}}$ on $\\mathcal{Z}$ during this process.  Then $$c_{1}(T(\\mathcal{Z})) = \\sum \\tilde{w_{i}} \\tilde{c_i},$$  where $\\tilde{w_i}$ is obtained via the spectral sequence for the Boothby-Wang fibration.\n\n\\begin{remark}\nIn the case of contact reduction in $\\mathbb{C}^n$ by a circle (where the action is of Reeb type) the coeffecients of $|z_{j}|^2$ in the (circle) moment map can be chosen to be the $\\tilde{w_{j}}$'s. \n\\end{remark}   \n\nNow we choose elements of $H_{2}(\\mathcal{Z}; \\mathbb{Z})$, $A_{1}, \\ldots, A_{n}$, with $$\\langle \\tilde{c_{i}},A_{i} \\rangle =1.$$  This is possible because the cohomology is a truncated polynomial ring generated by the $c_{j},$ all elements having even degree.  Now let $$A= \\sum_{j} A_{j}.$$ Then for any K\\\"{a}hler suborbifold $ i :S \\hookrightarrow \\mathcal{Z},$ $$\\sum_{i}\\langle i^{*}\\tilde{c_{i}},A \\rangle$$ is nonzero.  Thus we can also do this for each $S_{T_j}$ by pulling the Chern classes back along the inclusion maps, then choosing homology classes in each stratum as above in terms of $i_{j}^{*} \\tilde{c_{i}},$ where $i_{j}: S_{T_{j}} \\rightarrow \\mathcal{Z}$ is the inclusion, and $\\{c_{i} \\}$ are the Chern classes generating $H^{*}(\\mathcal(Z); \\mathbb{C}).$  Call the corresponding homology class $A_{S_{T_j}}.$ The purpose here is to find a nice diagonally embedded sphere with which to make our calculations.  Now let's use this set-up to do some index calculations. \nFirst we must find suitable trivializations and capping disks for Reeb orbits.  The idea here is to find two trivializations for each Reeb orbit, then use the loop property of the Maslov index to calculate the index via integration of $c_{1}(T(\\mathcal{Z}))$ over the sphere obtained by gluing the two disks (from the symplectic trivializations) along their boundaries.  The author first encountered this idea in ~\\cite{Bourg1} and ~\\cite{EGH}, however this was only for the regular\\footnote{Regular in the sense of foliation theory.} case.  So let $\\gamma_{S_{T_j}}$ be a Reeb orbit of period $T_{j}$, living, of course, in the stratum $S_{T_j}.$  We now pull back $\\xi$ via the inclusion map over $S_{T_j},$ $i_{j}.$  For the first disk we just cap off a tubular neighborhood of the Reeb orbit given by the product framing for $M$.  In this framing the Maslov index is $0$, since the return map is always the constant path in $Sp(2n-2, \\mathbb{R})$ given by the identity.  Now we need another disk to glue along the Reeb orbit to get a sphere.  In order to do this consider a holomorphic sphere, ie., a map $u:S^2 \\rightarrow S_{T_j}$ passing through $p \\in S_{T_{j}}$ such that $[u] =A_{S_{T_j}}.$  This is always possible since the moment map is invariant and since we assume $\\mathcal{Z}$ is simply connected, the Hurewicz homomorphism is surjective.  Now consider a holomorphic (orbi)section of $L$ over our sphere with a zero of order equal to the multiplicity of $\\gamma$ and no pole.  Such a section exists since we are talking about line (orbi)bundles over $\\mathbb{CP}^1.$  With this set-up we prove:\n\\begin{lemma}\n Let $M$ be an $S^1$-bundle over a symplectic orbifold admitting a Hamiltonian action of a compact Lie group, such that its cohomology is generated by the Chern classes associated to the action. Then the Maslov index of a Reeb orbit in the stratum $S_{T_{j}}$ of multiplicity $m$ is equal to $$\\frac{2m}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}}} i^{*}c_{1}(T(S_{T_{j}})),$$  moreover this number is an integer.\n\\end{lemma}\n\\begin{proof}\nBy the loop property of the Maslov index,  the Maslov index of the Reeb orbit is twice the Maslov index of the path of change of coordinate maps between the two disks glued along $\\gamma.$  Since the disk was obtained via an (orbi)section over a sphere representing $A_{S_{T_j}}$, we get $$\\mu(\\gamma) = 2 \\langle c_{1}(\\xi) , \\sigma(u) \\rangle =2 \\langle c_{1}(T(\\mathcal{Z}) , A_{S_{T_{j}}} \\rangle.$$  This is exactly $c_{1}^{orb}(T(S_{T_{j}}))$ evaluated on $A$.  Therefore the index of an orbit of multiplicity $m$ is $$2m \\int_{A_{S_{T_j}}} c_{1}^{orb}(T(S_{T_{j}})).$$  Now going back to the work of Satake ~\\cite{Sat57} to compute the integral of an orbifold characteristic class over a homology class, we take intersections with all orbifold strata and divide out by the orders of the local uniformizing groups and sum:$$2k \\int_{A_{S_{T_j}}} c_{1}^{orb}(T(S_{T_{j}}))=2m \\sum_{j}\\frac{1}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}} \\cap \\Sigma_{j}} c_{1}(T(S_{T_{j}}))$$ where $\\Gamma_{j}$ is a local uniformizing group in the orbifold stratum $\\Sigma_{j}=S_{T_j}.$  Now, since each such spherical class is completely contained in $S_{T_{j}},$ we can just compute the integral $$\\frac{2}{|\\Gamma_{j}|}\\int_{A_{S_{t_j}}} c_{1}(T(S_{T_{j}}))|_{S_{T_{j}}} = \\frac{2}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}}} i^{*}c_{1}(T(S_{T_{j}}))$$ for simple orbits, multiplying by $m$ for $m$-multiple orbits.  Note however that, although we may compute the integral on $\\mathcal{Z}$, this integral is equal to one which takes place as the evaluation of an integral form on the contact manifold, hence we always get an integer.  \n\\end{proof}\n\\begin{remark}\nThe idea above is that $A_{S_{T_j}}$ is a ``sufficiently diagonal'' sphere in $S_{T_k}.$  This ensures that we pick up as much information as possible about the line bundle as possible during the integration.  One should also note that in general $c_{1}(\\xi) \\neq 0$ so this grading scheme for contact homology is computed with respect to a \\emph{particular} choice of capping surface for each Reeb orbit.  When comparing contact manifolds which are $S^1$-orbibundles over the same base, care must be taken to make the same choices each time, so that the weights are realized via the Chern classes of each \\emph{specific toric structure}.   \n\\end{remark}\n\\begin{remark}\nThe reader may wonder what role branch divisors play in the index calculation above.  This is encoded in summing over the strata and dividing by the orders of local uniformizers.\n\\end{remark}\nWe want to use these calculations to compute cylindrical contact homology, however this is not well defined unless we can exclude Reeb orbits of degree $0$, $1$, $-1$.  To ensure this we must assume that for all $k$ $$2 (\\sum_i i^*c_{i}\\tilde{w_{i}}) -\\frac{1}{2} dim(S_{T_{k}})>0.$$ For this it is sufficient to assume that $$\\sum_{i}\\tilde{w_{i}} >1.$$  We take this as a standing assumption in the following.  \n\nNow we notice that there are no rigid $J$-holomorpic cylinders other than the trivial ones.  This follows from a simple index computation and comparison with the dimension formula for the relevant moduli spaces.  This means that the contact homology is given ompletely by the Morse-Smale-Witten complex of the moment map with degree shifts given by the Maslov indices.  The discussion above yields theorem ~\\ref{thm:main}].  We obtain the following corollaries.\n \n\n\n\\begin{cor}\n Let $(M, \\xi)$ be a simply connected compact homogeneous contact manifold.  Then $CH_{*}(M)$ is generated by copies of $H_{*}(\\mathcal{Z})$ with degree shifts given by $$2m \\int_{A} c_{1}(T(\\mathcal{Z})) = 2m \\sum_{i} \\tilde{w_{i}} -2$$.\n\\end{cor}\n\\begin{proof}In this case $M$ is an $S^{1}$-bundle over a generalized flag manifold, (recall that in this case there is a \\emph{regular} contact $1$-form, $\\alpha$ for $\\xi$).  The cohomology of the base is a polynomial ring as per the discussion earlier, and all the relevant homology classes are spherical.  By the regularity theorem for integrable $J$ the dimension of the moduli space is the one predicted by the Fredholm index.  Moreover by the index calculation above, and the dimension formula for the moduli spaces, there are no rigid $J$-holomorphic curves connecting orbit spaces.  This contact homology is given compeltely in terms of the Morse-Smale-Witten differential, which vanishes since the moment map deterines a perfect Morse function, thus we get a generator for each critical point of the norm squared of the moment map in degree given by the Maslov indices as calculated in the previous discussion.\n\\end{proof}\n\\begin{cor} Let $(M, \\xi)$ be a simply connected compact toric Fano contact manifold with a quasiregular contact form $\\alpha$.  Then $CH_{*}(M)$ is generated by copies of $H_{*}(\\mathcal{Z})$ with degree shifts given by the Maslov indices plus the dimension of the stratum containing the particular Reeb orbit as a point.  \nIf $\\xi$ has a regular contact form $\\alpha$ then the degree shifts are given by $$2m\\sum_{j}\\tilde{w_{j}}-2 ,$$ where the $\\tilde{w_{j}}$ are defined as above.  \n\\end{cor}\n\\begin{proof}The Fano condition gives transversality of the $\\bar{\\partial}_{J}$-operator via the Dolbeault complex.  If we assume transversality we can drop the Fano assumption.  Again, our cohomology ring is a truncated polynomial ring generated by all possible Chern classes, with spherical second homology because of simple connectivity.  The indices are given by the even multiples of the sum of the weights.  Again there are no non-trivial $J$-curves.  So the homology is that given by the Morse-Smale-Witten complex (whose differential again vanishes by perfection of the Morse function) with the degree shifts given by the Maslov indices as calculated above. \n\\end{proof}\n\n\\section{Examples} \n\\subsection{Wang-Ziller Manifolds}\nNow let's specialize to Wang-Ziller manifolds.  \nThese are toric manifolds either obtained from reduction in $\\mathbb{C}^2 \\times \\mathbb{C}^2$ via the moment map $\\mu(z,w) = k|w|^2 - l|z|^2.$  This manifold is also a homogeneous contact manifold.  This is how we get transversality.  Note that as a toric manifold, this manifold is non-Fano, so we really need to use the homogeneity to get transversaltiy via the Dolbeault complex.    \nWe can also see this manifold as a Boothby-Wang manifold.  Consider $\\mathcal{Z}= \\mathbb{C}\\mathbb{P}^1 \\times \\mathbb{C}\\mathbb{P}^1,$ and we take the standard symplectic form on each summand and multiply each piece by relatively prime integers $k$ and $l$.  \nWe take $P$ to be the circle bundle with a connection form $\\alpha$ satisfying $d \\alpha = \\pi^{*}(k c_{1} + l c_{2}),$ where the $c_{j}$ are actually the generators of the second cohomology of each sphere.  Then $$c_{1}(\\xi)=(2k - 2l)\\beta,$$ for $\\beta$ a generator of $H_2(S^2 \\times S^3 ; \\mathbb{Z}),$ and $$c_{1}(T\\mathcal{Z}) = (2kc_{1} + 2lc_{2}),$$ here $\\mathcal{Z}$ is topologically $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ with the toric structure obtained by with symplectic form determined by $k,l.$        \n$\\mathcal{Z}$ admits a perfect Morse function, and the Maslov indices in this case for orbits of multiplicity $m$ are given by $ 4m(k+l) $.  Thus the grading of contact homology is given by\n $$|p| = 4m(k+l) - 2 + d$$\nwhere $m\\in \\mathbb{Z} \\setminus\\{0\\}$, and $d$ ranges over all possible degrees of homology classes in $\\mathcal{Z},$ in this case $d=0,2,4.$b\nThis gives infinitely many distinct contact structure on $S^2 \\times S^3$ since for each choice of relatively prime $k$ and $l$, we get generators of contact homology in minimal dimension $4(k+l) -2.$  Of course, for all pairs such that $k-l = c$ we get a single first Chern class for the contact bundle ~\\cite{WZ}.  Choosing now all pairs with $k-l =c,$ we get infinitely many distinct contact structures in the same first Chern class.  In ~\\cite{Ler1} Lerman showed that these contact structures are all pairwise non-equivalent as toric contact structures, but he asked whether or not they were pairwise contactomorphic.  This answers that question in the negative. Via the above construction we get contact structures $\\xi_{k,l}$ on $S^{2} \\times S^{3}.$\n\\begin{cor}\n Fix $c \\in \\mathbb{Z},$  choose $k,l$ such that $gcd(k,l) =1,$ and $k-l =c$ then the contact structures $\\xi_{k,l}$ are pairwise non-contactomorphic all within the same first Chern class of $4$-plane distribution.  \n\\end{cor}\n\n\\begin{remark}\n This example suggests a Kunneth-type formula for the \\textbf{join} ~\\cite{BG07} construction for quasiregular contact manifolds provided each summand has suitable contact homology.  Suppose that $(\\mathcal{Z}_1, \\omega_{1})$ and $(\\mathcal{Z}_2 , \\omega_{2})$ are both simply connected symplectic orbifolds which are reduced spaces so that their cohomology rings are polynomials in the Chern classes.  Then we can build circle bundles over their product with curvature forms given as integer linear combinations of the $\\omega_{j}.$  By choosing appropriate spheres ``diagonally'' embedded into the product we can evaluate the first Chern class of this bundle in order to get the Maslov indices as above.  Assuming transversality of the $\\bar{\\partial}_{J}$-operator this always computes contact homology.   \n\\end{remark}\n\\begin{remark}\n One could do a similar computation with Boothby-Wang spaces over $\\mathbb{CP}^2 \\# \\overline{\\mathbb{CP}^2}.$\n\\end{remark}\n\n\\subsection{Circle bundles over weighted projective spaces.}      \nIn these examples first off, note that weighted projective spaces are toric Fano, (even in the orbifold sense.)\\footnote{These orbifolds generally have branch divisors.}  Note that the cohomology ring is then just the standard one, and we just need to find the right spherical classes.  Of course we just pullback $k$-multiples of the standard symplectic form to define our line bundles.  Now to compute contact homology of the bundle we integrate $c_{1}(T(\\mathcal{Z}))$ over the class of a line.  The base admits a perfect Morse function, so all we need to do is keep track of the strata.  Integrating $i^{*}_{j} c_{1}(T(\\mathcal{Z}))$ over spheres representing the K\\\"{a}hler class for each stratum.  So the grading of contact homology for an orbit of multiplicity $m$ will be $$2m \\sum_{j}( \\langle i_{j}^{*}c_{1}^{orb}(T\\mathbb{CP}^n), [S^2_{j}]\\rangle  + \\frac{1}{2} dim S_{T_{j}})+ d+n-3=(2km\\sum_{j}\\frac{1}{|\\Gamma_{j}|})+ \\frac{1}{2}dimS_{T_{j}} + d+n-3,$$ where the class $[S^{2}_{j}]$ is the class of a line in each stratum and $d$ corresponds to the possible degree of a homology class on $\\mathbb{CP}^{n}$, hence is an even number between $0$ and $2n$ and $S \\in \\mathbb{Z}^{+}.$  The dimension of the moduli space for genus $0$ and $1$ positive and $1$ negative puncture is then never $1.$  Notice that $c_{1}(\\xi) = 0$ in this case.  So again we see that these contact manifolds are distinguished by the bundle and orbifold data. \n\\begin{remark}\n One should be able to simplify the above formula when working with branch divisors.  We choose to stick with our earlier notation, in which any information about such branch divisors is encoded in the calculation.\n\\end{remark}\n\n\n \n\\newcommand{\\etalchar}[1]{$^{#1}$}\n\\def$'${$'$}\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"}}
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+{"text":"\n\n\\section{INTRODUCTION}\n    \\label{sec:intro}\n    \\input{sections\/1-intro}\n\n\\section{PROBLEM FORMULATION}\n    \\label{sec:problem}\n    \\input{sections\/2-problem}\n\n\\section{PRELIMINARIES}\n    \\label{sec:prelim}\n    \\input{sections\/3-prelim}\n    \n\\section{OFFLINE-ONLINE SYNTHESIS}\n    \\label{sec:synthesis}\n    \\input{sections\/4-synthesis}\n\n\\section{EVALUATIONS}\n    \\label{sec:evaluation}\n    \\input{sections\/5-evaluation}\n\n\\section{CONCLUSION}\n    \\label{sec:conclusion}\n    \\input{sections\/6-conclusion}\n\n\\bibliographystyle{ieeetr}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Offline Module}\n\\label{sec:offline}\n\nThe offline module first performs GP regression to estimate the unknown function $f(\\cdot,u)$ for each $u \\in U$ from the given initial dataset $\\mathrm{D}$.  This is achieved by using \\eqref{eq:post-mean} and \\eqref{eq:post-kernel} on each component of $f$.  Note that the corresponding regression error $|\\hat f^{(i)}(x,u) - f^{(i)}(x,u)|$, where $\\hat f^{(i)}$ is the estimate (GP mean) of the $i$-th component of $f$ obtained from \\eqref{eq:post-mean}, \nis given by \\eqref{eq:GPerror}.  \n\nGiven $\\hat f$, the method constructs the IMDP abstraction $\\mathcal{I} = (Q, A ,\\check{P}, \\hat{P}, AP, L)$ by partitioning of the compact set $X$.  Each resulting discrete region is associated with an IMDP state  $q \\in Q$. \nOne state of the IMDP is also used to represent the rest of the continuous space $\\mathbb{R}^n \\setminus X$.\nThe labeling function $L$ of the IMDP states is defined according to the labels of continuous states within each region.  We remark that, to ensure correct labeling, the space discretization performed above must respect the regions of interest in $R$.  This guarantees that the label of the discrete regions correctly hold for every continuous state in that region, i.e., for discrete region $q \\subset \\mathbb{R}^n$, $L(q) = L(x)$ for every $x \\in q$.  \n\nNext, the set of IMDP actions is defined to be the set of controls of Process~\\eqref{eq:sys}, i.e., $A = U$.  The transition probability bounds are then given by the following proposition, which uses the discrete regions, the regressed GP, and its corresponding error.\n\n\\begin{proposition}[\\hspace{-1.2mm} \\cite{jackson2021synthesis}, Theorem 1]\\label{prop:transition}\n    For a region $q \\subset \\mathbb{R}^n$ and control $u \\in A$, let $\\|h\\|^{q,u}_{\\infty}\\equiv \\sup_{x\\inq} |h(x,u)|$ .\n    Given a control $u \\inA$, regions $q,q' \\subset \\mathbb{R}^n$, dataset $\\mathrm{D}$, regression $\\hat f$, and \n    positive real vectors $\\epsilon\\in\\mathbb{R}^n$ and $\\eta\\in\\mathbb{R}^n$, then\n    \\begin{align}\n        \\hat{P}&(q,u,q') = \\max_{x\\in q} T(q' \\mid x, u) \\nonumber \\\\ \n            &\\leq \\big (\\interindicator{\\overline {q}' (\\epsilon+\\eta)}{ Im(q,u)} \n            \\prod_{i=1}^n P[|w^{(i)}|\\leq \\eta_i] + \\prod_{i=1}^n P[|w^{(i)}|>\\eta_i]\\big)~\\cdot  \\nonumber \\\\ \n            & \\prod_{i=1}^n P[\\|\\hat f^{(i)}-f^{(i)}\\|^{q,u}_\\infty \\leq \\epsilon_i] + \\prod_{i=1}^nP[\\|\\hat f^{(i)} - f^{(i)}\\|^{q,u}_\\infty >\\epsilon_i],\n            \\label{eq:upper-bound}\n    \\end{align}\n    \\begin{align}\n        \\check{P}&(q,u,q') = \\min_{x\\in q} T(q'\\mid x,u)  \\nonumber \\\\\n            & \\geq \\big(1-\\interindicator{X \\setminus \\underline {q}'(\\epsilon+\\eta)}{Im(q,u)}\\big)  \\prod_{i=1}^n P[\\|\\hat f^{(i)}-f^{(i)}\\|^{q,u}_\\infty \\leq \\epsilon_i]~\\cdot \\hspace{25mm} \\nonumber \\\\\n            & \\hspace{48mm} \\prod_{i=1}^n P[|w^{(i)}|\\leq \\eta_i].\n        \\label{eq:lower-bound}\n    \\end{align}\n    where $\\interindicator{A}{B}$ returns one if $A\\cap B\\neq\\emptyset$ otherwise zero, $\\overline{q}'(c)$ \n    and $\\underline {q}'(c)$ are the regions obtained by expanding and shrinking (each dimension of) $q'$ by the scalars in vector $c$, respectively, and $Im(q,u)$ is the image of $q$ under $\\hat f$ defined as $Im(q,u) = \\{\\hat f(x,u) \\mid x\\inq\\}$. \n\\end{proposition}\nFurther details of Proposition~\\ref{prop:transition} can be found in \\cite{jackson2021synthesis}. \nNote that the discretization (volume of $q$ and $q'$) plays a significant role in the transition probability bounds as the image of $q$ under the learned dynamics can be very large (especially when it is over-approximated).\nIn addition, the regression errors dictate both the $P[\\|\\hat f^{(i)}- f^{(i)}\\|^{q,u}_\\infty \\leq \\epsilon_i]$ terms along with affecting the magnitude of the expanded and reduced sets.  Hence, the larger these errors, the looser the bounds $\\check{P}$ and $\\hat{P}$, leading to larger uncertainty in the abstraction. \n\nGiven abstraction $\\mathcal{I}$ and specification $\\varphi$, a control strategy can be generated that is robust against these errors and maximizes the lower bound probability of satisfying $\\varphi$.  This is achieved via a product construction between abstraction $\\mathcal{I}$ and \\textsc{DFA}\\xspace $\\mathcal{A}_\\spec$, which represents $\\varphi$.\nThe result is the product \\textsc{IMDP}\\xspace (\\textsc{PIMDP}\\xspace) which is defined below.\n\n\n\\begin{definition}[Product \\textsc{IMDP}\\xspace]\\label{def:pimdp}\n    Given an \\textsc{IMDP}\\xspace $\\mathcal{I} = (Q, A, \\check{P},\\hat{P}, AP, L)$ and \\textsc{DFA}\\xspace $\\mathcal{A}_\\spec = (Z, 2^{AP}, \\delta, z_0,\\DFAStates_F)$, the \\textit{product \\textsc{IMDP}\\xspace} (\\textsc{PIMDP}\\xspace) $\\mathcal{P} \n   \n   \n    =(S, A, \\check{P}^{\\mathcal{P}}, \\hat{P}^{\\mathcal{P}}, S_0, S_F)$, where \n    $S = Q\\timesZ$, $S_F = Q\\timesZ_F,$\n    $$S_0 = \\{(q,z_{init}) \\mid  q \\in Q , \\; z_{init}=\\delta(z_0, L(q))\\},$$\n    and\n   \n   \n    \\begin{align*}\n        \\check{P}^{\\mathcal{P}}((q, z), u, (q', z')) &= \\begin{cases}\n            \\check{P}(q,u,q') & \\text { if } z'=\\delta(z,L(q))\\\\\n            0 & \\text{ otherwise}\n        \\end{cases} \\\\\n        \\hat{P}^{\\mathcal{P}}((q, z), u, (q', z')) &= \\begin{cases}            \\hat{P}(q,u,q') & \\text { if } z'=\\delta(z,L(q))\\\\\n            0 & \\text{ otherwise}.\n        \\end{cases} \n    \\end{align*}\n\\end{definition}\nThe $\\textsc{PIMDP}\\xspace$ accepting states $S_F$ encapsulate the satisfaction of the specification $\\spec$, i.e., a path that reaches $S_F$ satisfies $\\spec$.\nHence, we formulate the following optimization problem to compute a robust strategy $\\policy_{\\text{off}}:S\\toA$ that maximizes the lower bound of the probability of reaching a state in $S_F$:\n\\begin{equation}\n    \\label{eq:strategy-offline}\n    \\policy_{\\text{off}}(s) = \\arg\\max_{\\pi \\in \\Pi} \\;\\, \\min_{\\xi \\in \\Xi} P[\\path_\\PIMDPS \\models \\varphi \\mid \\pi, \\path_\\PIMDPS(0) = s].\n\\end{equation}\nWe solve this optimization problem using an interval-value iteration as detailed in \\cite{Lahijanian:TAC:2015}, which results in a memoryless strategy on $\\mathcal{P}$.  The interval-value iteration procedure also returns the lower- and upper-bounds of the probability of satisfaction of $\\varphi$ under $\\policy_{\\text{off}}$, i.e., \n\\begin{align}\n    \\label{eq:plow}\n    \\underline{p}_{\\policy_{\\text{off}}}(s) &= \\min_{\\xi \\in \\Xi} P[\\path_\\PIMDPS \\models \\varphi \\mid \\policy_{\\text{off}}, \\path_\\PIMDPS(0) = s], \\\\\n    \\overline{p}_{\\policy_{\\text{off}}}(s) &= \\max_{\\xi \\in \\Xi} P[\\path_\\PIMDPS \\models \\varphi \\mid \\policy_{\\text{off}}, \\path_\\PIMDPS(0) = s],\n    \\label{eq:pup}\n\\end{align}\nrespectively.  \nThese satisfaction intervals let us identify regions where success is more likely (a lower-bound near one), regions where violation is more likely (an upper-bound near zero), and regions with \nhigh uncertainty (large intervals).  By deploying the system with $\\policy_{\\text{off}}$ at $q$, the system is guaranteed to satisfy $\\spec$ with a probability at least  $\\underline{p}_{\\policy_{\\text{off}}}((q,z_{init}))$.  The goal of the online module is to synthesize controllers at runtime to improve this guarantee towards the optimal probability, which is evidently most helpful for the regions with high uncertainty. \n\n\n\n\\subsection{Online Module}\\label{sec:online}\n\nIn constructing the \\textsc{PIMDP}\\xspace $\\mathcal{P}$ and strategy $\\policy_{\\text{off}}$ offline, we are limited by the uncertainty introduced by the finite amount of data in $\\mathrm{D}$ and by the discretization of $X$. Our goal is to use the information collected at runtime to improve $\\policy_{\\text{off}}$ by reducing both sources of uncertainty.\nIn Alg. \\ref{alg:onlineloop}, we present a high-level description of our synergistic control framework that obtains both of these goals. \n\n\\begin{algorithm}[b!]\n\\caption{Online control loop}\\label{alg:onlineloop}\n\\begin{algorithmic}[1]\n\\Procedure{ControlLoop}{$\\mathcal{P}, \\GPFullSet, x_0, \\mathrm{D}$}\n  \\State $k\\gets 0$\n  \\State $s_0\\gets$\\textsc{InitialPIMDPState}$(\\mathcal{P}, x_0)$\n \\While{$s_k\\notinS_F$} \n  \\State $\\GPLocalSet\\gets$\\textsc{CreateLocGP}($\\GPFullSet, x_k, \\mathrm{D}$)  \\label{alg:line:localgp}\n  \\State $\\mathcal{P}\\gets$\\textsc{UpdatePIMDP}$(\\mathcal{P}, \\GPLocalSet)$  \\label{alg:line:pimdpupdate}\n  \\State $u^*\\gets$\n  \\textsc{GetOptimalAction}$(\\mathcal{P}, x_k)$ \\label{alg:line:optimalaction}\n  \\State $x_{k+1}\\gets f(x_k, u^*) + w_k$\n  \\State $s_{k+1}\\gets $\\textsc{UpdatePIMPDState}$(\\mathcal{P}, x_{k+1})$ \n  \\State $\\mathrm{D}\\gets \\mathrm{D}\\cup \\{(x_k,u^*,x_{k+1}) \\}$\n  \\State $k\\gets k+1$\n \\EndWhile\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\nOnline, at each time step, we observe the current state of the system $x_k$ and build a set of (local) GPs\nto make predictions for the transitions of Process \\eqref{eq:sys} to the states of the abstraction in one-time step starting from $x_k$ (Line 5).  Note that this set of GPs ($\\GPLocalSet$) are employed exclusively to make local predictions starting from $x_k$. Hence, to train them, we can simply use only data local to $x_k,$ thus making the training process efficient enough to be employed at runtime.\nThe resulting GPs are then employed to refine the PIMDP $\\mathcal{P}$ and get the optimal action $u^*$ using the values computed offline as explained below. Until we reach an accepting state of $\\mathcal{P}$, we continue the process and keep enlarging $\\mathrm{D}$ with the runtime observations.\nBelow, we give further details of how we update $\\mathcal{P}$ and we select the optimal action as well as how we train local GPs $\\GPLocalSet$.\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Optimal Actions}\nTo choose the optimal action, we first augment the \\textsc{PIMDP}\\xspace with the current state and then choose the optimal next action (Lines~\\ref{alg:line:localgp} and \\ref{alg:line:optimalaction} in Alg. \\ref{alg:onlineloop}).\nGiven the current state $x_k$, we begin \nby defining a new $\\mathcal{P}$ state\n$s' = (x_k,z)$, where $z$ is the current state of the \\textsc{DFA}\\xspace. By reasoning over a singleton ($x_k$ instead of region $q$ that contains $x_k$), we eliminate the discretization error over the next step. In particular, for $z'=\\delta(z,L(x_k))$ we can compute \n\\begin{align}\n    \\label{Eqn:improvedBound}\n    \\check{P}^{\\mathcal{P}}((x_k, z), u, (q', z'))=P[\\x_{k+1}\\in q' \\mid \\omega_{\\x}(k-1)=x_k,u],\n\\end{align} and similarly for the upper bound.\nThe bound in \\eqref{Eqn:improvedBound} can be computed directly using Proposition~\\ref{prop:transition} and is guaranteed to improve compared to $\\check{P}^{\\mathcal{P}}((q, z), u, (q', z'))$ as we can compute the image $Im(\\{x_k\\})$ under the learned dynamics exactly and mitigate this discretization error.\n\nWe now focus on how to choose the best action in order to reach a state in $S_F$. In particular, similar to the goal of the optimization problem in \\eqref{eq:strategy-offline}, we want to find the action $u^*$ \nthat maximizes (with an abuse of notation)\n\\begin{equation}\\label{eq:lbsat}\n    \\underline{p}(s',u) = \\min_{\\xi\\in\\Xi} \\sum_{s\\in \\Qpimdp^a(\\qpimdp')} \\underline{p}_{\\policy_{\\text{off}}}(s)\n\\xi(\\pathpimdp^{\\mathrm{fin}}, u)(s),\n\\end{equation}\nwhere $\\Qpimdp^a(\\qpimdp')=\\{s\\inS\\mid\\hat{P}^{\\mathcal{P}}(s',u,s) > 0\\}$.\nNote that $\\underline{p}(s',u)$ is the \\emph{worst-possible} expected lower-bound probability under action $u$.\nWe pick the best control as \n\\begin{equation}\\label{eq:opt-action}\n    u^* = \\arg \\max_{u \\in U} \\; \\underline{p}(s',u).\n\\end{equation}\n\n\nIn practice, it is common to have multiple actions that maximize\n$\\underline{p}(s',u)$ e.g. when it is $0$ for all actions.\nWe introduce secondary and tertiary metrics grounded encouraging progression, though maximizing $\\underline{p}(s',u)$ is always preferred. \n\n\n\n\n\\paragraph{Satisfaction Upper Bound}\nAs a secondary criteria, we consider the upper-bound of the probability of satisfaction from $s'$, which is defined as\n\\begin{equation}\\label{eq:ubsat}\n\\overline{p}(s',u) = \\min_{\\xi\\in\\Xi} \\sum_{s\\in \\Qpimdp^a(\\qpimdp')} \\overline{p}_{\\policy_{\\text{off}}}(s)\\xi(\\pathimdp^{\\mathrm{fin}}, u)(s).\n\\end{equation}\nIn particular, we select the action $u^*=\\argmax_{u \\in U^*} \\overline{p}(s',u),$ where $U^*$ is the set of actions returned by \\eqref{eq:opt-action}. Hence, $u^*$ is the one with the greatest lower- and upper-bound probability of satisfying  $\\spec.$\nNote that we are again selecting actions against the worst adversary, which is required in order to be robust against uncertainty.\n\nEven with these primary metrics defined, in practice they are not always sufficient to realize good online behavior, and there may be multiple $u^*$ optimizing both criteria.\nAs we are concerned with choosing $u^*$ that best improves the performance for the next step, we introduce two tertiary metrics that encourage one-step progression.\n\n\\paragraph{Sink-State Metric}\nThis metric prefers actions that do not violate $\\varphi$ by using the inherent \\textsc{DFA}\\xspace sink state, from which reaching the accepting state is impossible.\nFor each $s\\in\\Qpimdp^a(\\qpimdp')\n, we check if the \\textsc{DFA}\\xspace transitions to the sink state and prefer\nactions with a low chance of doing so.\n\n\\paragraph{Progression Metric}\nWe consider a progression metric that defines progress towards reaching $S_F$. \nGiven $s'$, we can measure its distance from an accepting state in $S_F$ by using the minimum distance to $S_F$ from each $s\\in\\Qpimdp^a(\\qpimdp')$, which is easily computed offline.\nOnce online, we choose the action that gives the shortest expected distance to $S_F$.\n\n\\subsubsection{Local GP and PIMDP Updates}\nAt runtime, we collect data that were not available offline. \nBy using this new data, we can refine the regression error and in turn obtain a less uncertain IMDP abstraction.\nRather than updating the full GP corresponding to action $u$, $\\GPFull^{u}$, we construct a local GP $\\GPLocal^{u}$ using the $\\ell$-nearest datapoints in $\\mathrm{D}$ to  $x_k$.  \nTraining then proceeds as for standard GP regression using the same hyperparameters as $\\GPFull^{u}$.\nThe hyperparameter $\\ell$ (number of data employed to train our local GP) induces a trade-off between regression accuracy and computation overhead. For more insight on picking $\\ell$ we refer to~\\cite{Nguyen-Tuong2010,das2018fast}.\n\n\nOur motivation for using local GP regression stems from computational tractability in the online setting. \nPerforming regression with local data is $O(\\ell^3)$ in the worst case.\nThus, by choosing $\\ell$ much smaller than the total number of available data, we can have a polynomial speed up that allows for runtime computations.\nNote that once we have $\\GPFull_{u^*}^{x_k}$, we can update the abstraction and product \\textsc{PIMDP}\\xspace by recalculating $\\check{P}(q,u^*,q')$ and $\\hat{P}(q,u^*,q')$ for all $q$ in a neighbourhood of $x_k$. \nThe transitions are then updated in $\\mathcal{I}$ and consequently in $\\mathcal{P}$ only if the resulting interval is tighter.\nOnce enough transitions are updated in the abstraction, the offline strategy $\\policy_{\\text{off}}$ can be updated using the refined \\textsc{PIMDP}\\xspace, denoted $\\pi_k$.  \nThis is done again using interval-value iteration, which is polynomial time (cubic) in the size of $\\mathcal{P}$.\n\n\n\n\\subsection{Correctness}\n\\noindent\nThe following theorem guarantees that Alg. \\ref{alg:onlineloop} monotonically reduces the uncertainty in the system predictions.\n\\begin{theorem}\n\\label{th:MonotonicImprovement}\n    Let $x_k$ be the state of Process \\eqref{eq:sys} at the $k$-th step of Alg. \\ref{alg:onlineloop}. Similarly, let $\\pi_k$ be the refinement of $\\policy_{\\text{off}}$ with $k$ additional datapoints computed according to \\eqref{eq:strategy-offline}. Then, for any $s \\in S$ it holds that \n    \\begin{equation}\n        [\\underline{p}_{\\pi_{k+1}}(s),\\overline{p}_{\\pi_{k+1}}(s)]\\subseteq [\\underline{p}_{\\pi_k}(s),\\overline{p}_{\\pi_k}(s)].    \n    \\end{equation}\n\\end{theorem}\nThe proof of Theorem \\ref{th:MonotonicImprovement} relies on the fact that, at runtime, we modify only the transition probabilities of $\\mathcal{P}$ that are improved by the new data. Hence, the set of adversaries of the updated \\textsc{PIMDP}\\xspace are a subset of those of the \\textsc{PIMDP}\\xspace at the previous time step.\n\nFinally, we remark that in the limit as $D$ becomes dense in $X$, \nthe regression error in \\eqref{eq:GPerror} goes to zero with probability one. \nAs a consequence, it is possible to show that as the size of the discretization goes to $0$ and as the coverage of $\\mathrm{D}$ approaches $X$, Alg. \\ref{alg:onlineloop} converges to the optimal strategy for Process \\eqref{eq:sys}.\n\n\n\n\n\n\n\\subsection{Gaussian Process Regression}\n{Gaussian Process} (GP) regression is a non-parametric Bayesian machine learning method \\cite{rasmussen2006gps4ml} that aims to infer an unknown function $f:\\mathbb{R}^n\\to \\mathbb{R}$ from noisy data.\nA standard assumption of GP regression is that $f$ is a sample from a GP with zero mean and covariance $k: \\mathbb{R}^n \\times \\mathbb{R}^n \\to \\mathbb{R}$. \nLet $\\mathrm{D}=\\{(\\mathrm{x}_i,\\mathrm{y}_i),i\\in \\{1,\\dots,m \\}\\}$ be a dataset, where \n$\\mathrm{y}_i$ is a sample of an observation of $f(\\mathrm{x}_i)$ with independent zero-mean noise ${v}$, which is assumed to be normally distributed with variance $\\sigma^2$ and  $\\mathrm{X}$ and $\\mathrm{Y}$ be ordered vectors with all points in $\\mathrm{D}$ such that $\\mathrm{X}_i = \\mathrm{x}_i$ and $\\mathrm{Y}_i = \\mathrm{y}_i$.  Further, call $K(\\mathrm{X},\\mathrm{X})$ the matrix with $K_{i,j}(\\mathrm{X}_i,\\mathrm{X}_j)=k(\\mathrm{x}_i,\\mathrm{x}_j)$, $K(\\mathrm{x},\\mathrm{X})$ the vector such that $K_{i}(\\mathrm{x},\\mathrm{X})=k(\\mathrm{x},\\mathrm{X}_i)$, and $K(\\mathrm{X},\\mathrm{x})$ defined accordingly. Then, the predictive distribution of $f$ at a test point $\\mathrm{x}$ is given by the conditional distribution of $f$ \ngiven $\\mathrm{D}$, which is Gaussian with mean $\\mu_\\mathrm{D}$ and variance $\\sigma_\\mathrm{D}^2$ given by\n\\begin{align}\n    \\begin{split}\\label{eq:post-mean}\n      & \\mu_{\\mathrm{D}}(\\mathrm{x}) = K(\\mathrm{x},\\mathrm{X}) \\big( K(\\mathrm{X},\\mathrm{X})+ \\sigma^2I_{m} \\big)^{-1} Y   \\\\\n    \\end{split}\\\\\n    \\begin{split}\\label{eq:post-kernel}\n      & \\sigma_{\\mathrm{D}}^2(\\mathrm{x}) = k(\\mathrm{x},\\mathrm{x})- K(\\mathrm{x},\\mathrm{X})\\big( K(\\mathrm{X},\\mathrm{X})+ \\sigma^2 I_{m} \\big)^{-1}K(\\mathrm{X},\\mathrm{x}),\n     \\end{split}\n\\end{align}\nwhere $I_{m}$ is the identity matrix of size $m \\times m$. \nFor stationary kernels, such as the squared-exponential kernel, $\\sigma_\\mathrm{D}(\\cdot)$ sees monotonic decay as $m$ increases.  \nAs $m\\to\\infty$, the posterior covariance $\\sigma_\\mathrm{D}(\\cdot)\\to 0$ and $\\mu_{\\mathrm{D}}\\to g$~\\cite{rasmussen2006gps4ml}.\n\nIn this work, contrary to the standard GP regression setting described above, we do not assume that $f$ is sampled from a given GP or that the observation noise is Gaussian. To quantify the estimation error in this more agnostic setting, under Assumption \\ref{assump:smoothnes}, we can rely on Theorem 2 from \\cite{chowdhury2017kernelized}, which bounds the regression error with high-probability in the form:\n\\begin{equation}\n    \\label{eq:GPerror}\n    P\\big[\\forall \\mathrm{x} \\in X, \\; |\\mu_\\mathrm{D}(\\mathrm{x}) - f(\\mathrm{x})| < \\beta(\\delta)\\sigma_{\\mathrm{D}}(\\mathrm{x}) \\big]\\geq 1-\\delta,\n\\end{equation}\nwhere $\\delta\\in (0,1)$ specifies the upper bound, and $\\beta$ is a compensator that depends on the choice of $k$, the dataset $\\mathrm{D}$, and $\\delta$. We refer to \\cite{chowdhury2017kernelized} for further information on $\\beta$ and \\eqref{eq:GPerror}.  \n\n\\subsection{Interval Markov Decision Processes (IMDPs)}\n We use \\textit{interval Markov decision process} (IMDPs) as the abstraction model for Process \\eqref{eq:sys}. IMDPs generalize Markov decision processes (MDPs) by allowing an interval of values for transition probabilities \\cite{givan2000bounded}.  \n\n\\begin{definition}[{\\textsc{IMDP}\\xspace}] \\label{def:imdp}\n    An interval Markov decision process ({\\textsc{IMDP}\\xspace}) is a tuple $\\mathcal{I} = (Q,A,\\check{P},\\hat{P},AP,L)$, where\n    \\begin{itemize}\n    \t\\setlength\\itemsep{1mm}\n    \t\\item $Q$ is a finite set of states,\n    \t\\item $A$ is a finite set of actions, and $A(q)$ denotes the set of available actions at state $q \\in Q$,\n        \\item $\\check{P}: Q \\times A \\times Q \\rightarrow [0,1]$ is a function, where $\\check{P}(q,a,\\qimdp^\\prime)$ defines the lower bound of the transition probability from state $q \\in Q$ to state $\\qimdp^\\prime \\in Q$ under action $a \\in A(q)$,\n        \\item $\\hat{P}: Q \\times A \\times Q \\rightarrow [0,1]$ is a function, where $\\hat{P}(q,a,\\qimdp^\\prime)$ defines the upper bound of the transition probability from state $q \\in Q$ to state $\\qimdp^\\prime \\in Q$ under action $a \\in A(q)$,\n        \\item $AP$ is a finite set of atomic propositions,\n        \\item $L: Q \\rightarrow 2^{AP}$ is a labeling function that assigns to each state $q \\in Q$ a subset of $AP$.\n    \\end{itemize}\n\\end{definition}\nFor all $q,\\qimdp^\\prime \\in Q$ and $a \\in A(q)$, it holds that $\\check{P}(q,a,\\qimdp^\\prime) \\leq \\hat{P}(q,a,\\qimdp^\\prime)$ and $\n    \\sum_{\\qimdp^\\prime \\in Q} \\check{P}(q,a,\\qimdp^\\prime) \\leq 1 \\leq \\sum_{\\qimdp^\\prime \\in Q} \\hat{P}(q,a,\\qimdp^\\prime)$.\n \nA path of an \\textsc{IMDP}\\xspace is a sequence of states $\\omega_{\\I} = q_0 \\xrightarrow{a_0} q_1 \\xrightarrow{a_1} q_2 \\xrightarrow{a_2}  \\ldots$ such that $a_k \\in A(q_k)$ and $\\hat{P}(q_k, a_k, \\allowbreak{q_{k+1}}) > 0$ for all $k \\in \\mathbb{N}$. We denote \nthe last state of a finite path $\\pathimdp^{\\mathrm{fin}}$ by $\\mathit{last}(\\pathimdp^{\\mathrm{fin}})$ and the set of all finite and infinite paths by $\\mathit{Paths}^{\\mathrm{fin}}$ and $\\mathit{Paths}$, respectively. \n\\begin{definition}[Strategy]\n\\label{def:strategy}\n    A strategy $\\pi$ of an \\textsc{IMDP}\\xspace model $\\mathcal{I}$ is\n    a function $\\pi: \\mathit{Paths}^{\\mathrm{fin}} \\rightarrow A$ that maps a finite path $\\pathimdp^{\\mathrm{fin}}$ of $\\mathcal{I}$ onto an action in $A(\\mathit{last}(\\mathit{Paths}^{\\mathrm{fin}}))$. If a strategy depends only on $\\mathit{last}(\\pathimdp^{\\mathrm{fin}})$, it is called a memoryless (stationary) strategy.  \n    The set of all strategies is denoted by $\\Pi$.\n\\end{definition}\n\nOnce an action is chosen by strategy $\\pi$, we evolve from the current state to the next state according to a probability distribution that respects the transition probability bounds of the \\textsc{IMDP}\\xspace.  There exist possibly infinitely many such distributions, and an \\textit{adversary} chooses this distribution.\n\\begin{definition}[Adversary]\n\\label{def:adversary}\n    For an \\textsc{IMDP}\\xspace $\\mathcal{I}$, an adversary is a function $\\xi: \\mathit{Paths}^{\\mathrm{fin}} \\times A \\rightarrow \\mathcal{D}(Q)$ that, for each finite path $\\pathimdp^{\\mathrm{fin}} \\in \\mathit{Paths}^{\\mathrm{fin}}$, state $q=\\mathit{last}(\\pathimdp^{\\mathrm{fin}})$, and action $a \\in A(\\mathit{last}(\\pathimdp^{\\mathrm{fin}}))$, assigns a feasible distribution $\\feasibleDist{q}{a}$ which satisfies\n   \n    $\\check{P}(q,a,\\qimdp^\\prime) \\leq \\feasibleDist{q}{a}(\\qimdp^\\prime) \\leq \\hat{P}(q,a,\\qimdp^\\prime).$\n   \n    The set of all adversaries is denoted by $\\Xi$.\n\\end{definition}\n\n\n\n\\noindent\nGiven a strategy $\\pi$ and an adversary $\\xi$, a probability measure over the paths of \\textsc{IMDP}\\xspace $\\mathcal{I}$ can be defined using the induced Markov chain \\cite{Lahijanian:TAC:2015}.\n\n\\subsection{Deterministic Finite Automaton (DFA)}\nGiven LTLf\\xspace formula $\\varphi$, a DFA can be constructed that precisely accepts the language of $\\varphi$ \\cite{de2013linear}.\n\\begin{definition}[DFA]\n    A \\textit{deterministic finite automaton} (DFA) constructed from an LTLf\\xspace formula $\\varphi$ defined over atomic propositions $AP$ is a tuple $\\mathcal{A}_\\spec=(Z, 2^{AP}, \\delta, z_0,\\DFAStates_F)$, where \n   \n   \n   \n   \n   \n   \n   \n    $Z$ is a finite set of states,\n    $2^{AP}$ is a finite set of input symbols,\n    $\\delta:Z\\times 2^{AP} \\toZ$ is the transition function, \n    $z_0 \\in Z$ is the initial state, and\n    $\\DFAStates_F \\subseteq Z$ is the set of accepting (final) states. \n\\end{definition} \nA finite \\emph{run} on $\\mathcal{A}_\\spec$ is a sequence of states $z_0z_1\\dotsz_{n+1}$ induced by trace $\\rho = \\rho_0\\rho_1\\dots\\rho_n$, where $\\rho_i \\in 2^{AP}$ and $z_{i+1} = \\delta(z_i,\\rho_i)$. A finite run on $\\rho$ is accepting if $z_{n}\\in\\DFAStates_F$.  If the run is accepting, then trace $\\rho$ is accepted by $\\mathcal{A}_\\spec$.  The set of all traces that are accepted by $\\mathcal{A}_\\spec$ is call the language of $\\mathcal{A}_\\spec$, denoted by $\\mathcal{L}(\\mathcal{A}_\\spec)$.  This language is equal to the language of $\\varphi$, i.e., $\\mathcal{L}(\\varphi) = \\mathcal{L}(\\mathcal{A}_\\spec)$. \n\n\\subsection{Specification Language}\n\\label{sec:ltlf}\nWe are interested in the properties of Process~\\eqref{eq:sys} in a compact set $X \\subset \\mathbb{R}^n$ with respect to a finite set of closed regions of interest $R = \\{r_1,\\ldots,r_{|R|}\\}$, where $r_i \\subseteq X$.  To this end, we associate to each region $r_i$ the atomic proposition $\\mathfrak{p}_i$ such that $\\mathfrak{p}_i = \\top$ (i.e., $\\mathfrak{p}_i$ is \\textit{true}) if $x \\in r_i$; otherwise $\\mathfrak{p}_i = \\bot$ (i.e., $\\mathfrak{p}_i$ is \\textit{false}).  \nLet $AP = \\{\\mathfrak{p}_1, \\ldots,\\mathfrak{p}_{|R|}\\}$ denote the set of all atomic propositions and $L: X \\rightarrow 2^{AP}$ be the labeling function that assigns to state $x$ the set of atomic propositions that are true at $x$.\nThen, we define the \\textit{trace} of path \n$\\path_{\\x}^{k}= x_0 \\xrightarrow{u_0} x_1 \\xrightarrow{u_1}  \\ldots \\xrightarrow{u_{k-1}} x_k$\nto be\n\\begin{equation*}\n\t\\rho = \\rho_0 \\rho_1 \\ldots \\rho_k,\n\\end{equation*}\nwhere $\\rho_i = L(x_i) \\in 2^{AP}$ for all $i \\leq k$. \nWith an abuse of notation, we use $L(\\path_{\\x}^k)$ to denote the trace of $\\path_{\\x}^k$.\n\nWe use a temporal logic to express desired properties of Process~\\eqref{eq:sys}.  Classical temporal logics such as LTL\\xspace  specifications \\cite{baier2008principles}, however, are interpreted over infinite behaviors (traces), and given the high levels of uncertainty of Process~\\eqref{eq:sys} (unknown dynamics as well as noise), its infinite behaviors have trivial probabilities.  Therefore, we instead employ recently developed \\textit{LTL\\xspace over finite traces} (LTLf\\xspace) \\cite{de2013linear}, which has the same syntax as LTL\\xspace  but its semantics is defined over finite traces. \n\n\\begin{definition}[LTLf\\xspace Syntax]\n    An LTLf\\xspace formula $\\varphi$ over a set of atomic propositions $AP$ is recursively defined as\n\t\\begin{equation*}\n\t\t\\varphi := \\top \\mid \\mathfrak{p}   \\mid   \\neg \\varphi   \\mid   \\varphi \\wedge \\varphi   \\mid  \\mathcal{X} \\varphi  \\mid  \\varphi\\, \\mathcal{U} \\varphi\n\t\\end{equation*}\n\twhere $\\mathfrak{p} \\in AP$, $\\neg$ (negation) and $\\wedge$ (conjunction) are Boolean operators, and $\\mathcal{X}$ (next) and $\\mathcal{U}$ (until) are temporal operators.\n\\end{definition}\n\n\\begin{definition}[LTLf\\xspace Semantics]\n\tThe semantics of an LTLf\\xspace formula $\\varphi$ are defined over finite traces. \n\tLet trace $\\rho \\in (2^{AP})^*$, $|\\rho|$ denote its length, and $\\rho_i$ be the $i$-th symbol of $\\rho$.  \n\tThen, the satisfaction of $\\varphi$ by the $i$-th step of $\\rho$, denoted by $\\rho, i \\models \\varphi$, is recursively defined as\n\t\\begin{itemize}\n\t\t\\item $\\rho , i \\models \\top,$\n\t\t\\item $\\rho , i \\models \\mathfrak{p} \\quad  \\Leftrightarrow  \\quad \\mathfrak{p} \\in \\rho_i,$\n\t\t\\item $\\rho , i \\models \\neg \\varphi \\quad \\Leftrightarrow \\quad \\rho, i \\not \\models \\varphi,$\n\t\t\\item $\\rho , i \\models \\varphi_1 \\wedge \\varphi_2 \\quad \\Leftrightarrow \\quad \\rho, i \\models \\varphi_1$ and $\\rho, i \\models \\varphi_2,$\n\t\t\\item $\\rho , i \\models \\mathcal{X} \\varphi \\quad \\Leftrightarrow \\quad |\\rho| > i+1$ and $\\rho, i+1 \\models \\varphi,$\n\t\t\\item $\\rho , i \\models \\varphi_1 \\mathcal{U} \\varphi_2 \\quad \\Leftrightarrow \\quad \\exists j$ s.t. $ i \\leq j < |\\rho|$ and $\\rho, j \\models \\varphi_2$ and $\\forall k$, $i \\leq k < j$, \\; $\\rho, k \\models \\varphi_1.$\n\t\\end{itemize}\n\tFinite trace $\\rho$ satisfies $\\varphi$, denoted by $\\rho \\models \\varphi$, if $\\rho, 0 \\models \\varphi$.\n\\end{definition}\n\\noindent\nSimilar to LTL\\xspace, the temporal operators $\\mathcal{F}$ (eventually) and $\\mathcal{G}$ (globally) are defined as: \n$\\mathcal{F} \\, \\varphi = \\top \\, \\mathcal{U} \\, \\varphi$ and $\\mathcal{G} \\, \\varphi = \\neg \\mathcal{F} \\, \\neg \\varphi.$\n\nAn LTLf\\xspace formula $\\varphi$ defines a language $\\mathcal{L}(\\varphi) = \\{\\rho \\in (2^{AP})^* \\mid \\rho \\models \\varphi\\}$, which is in fact a regular language.  \nSimilar to \\cite{wells2020ltlf}, we say that a path $\\path_{\\x}$ of Process~\\eqref{eq:sys} satisfies LTLf\\xspace formula $\\varphi$ if there exists a prefix of $\\path_{\\x}$ that lies entirely in $X$ and its trace is in  $\\mathcal{L}(\\varphi)$, i.e., \n\\begin{multline}\n    \\label{eq:satisfying-path}\n    \\path_{\\x} \\models \\varphi  \\quad \\Leftrightarrow \\quad \\exists k \\in \\mathbb{N}  \\;\\; s.t. \\;\\; L(\\path_{\\x}^k) \\in \\mathcal{L}(\\varphi) \\text{ and } \\\\ \\path_{\\x}^k(k') \\in X \\;\\; \\forall k' \\leq k,\n\\end{multline}\n \n\n\\subsection{Problem Statement}\n\nGiven an LTLf\\xspace specification $\\varphi$ and dataset $\\mathrm{D}$, our goal is to synthesize a control strategy $\\policy^*_\\x$ under which Process \\eqref{eq:sys} attempts to satisfy $\\varphi$ with the maximum probability.\n\n\n\\begin{problem}\\label{prob:main}{[Offline-Online Synthesis]}\nGiven an initial dataset $\\mathrm{D}$ and the capability of collecting data of Process \\eqref{eq:sys} at runtime, a compact set $X$, \nand\nan LTLf\\xspace property $\\varphi$ defined over the regions of interest in $X$, find a control strategy  $\\policy^*_{\\x}$ \nthat maximizes the probability of satisfying $\\varphi$, i.e.,\n$$ \\policy^*_{\\x} = \\arg \\max_{\\pi_\\x} P[\\path_{\\x} \\models \\varphi \\mid \\pi_\\x, \\path_{\\x}(0)=x_0].$$\n\\end{problem}\n\n\n\nThe fact that  Process \\eqref{eq:sys} is unknown makes solving Problem \\ref{prob:main} particularly challenging.  In this work, we develop a data-driven method that achieves the optimality objective of the problem at the limit.\nIn particular, we develop an offline-online synergistic framework, where the offline module uses GP regression on $\\mathrm{D}$ to build an initial abstraction of Process \\eqref{eq:sys} in terms of a finite uncertain Markov model and synthesizes a robust control strategy on the resulting abstraction.  \nThis abstraction and control strategy suffer from two types of errors: discretization and regression errors.  \nVia the online module, we iteratively reduce these errors as explained below.  \n\nThe offline strategy serves as a baseline controller for the online module and provides an initial performance guarantee in terms of the lower bound on the probability of satisfying $\\varphi$.\nOnline, while executing the system, we use the observed data to refine the abstraction, potentially improving the control strategy and its probability of satisfaction of $\\varphi$ in two ways: 1) we reduce the space discretization error in the abstraction by reducing the uncertainty in the transition probabilities of the abstraction, which can be done by leveraging our continuous state knowledge at every time step, and 2) using the data collected at each time step, we improve our learned model of Process~\\eqref{eq:sys}, hence, reducing the regression error. \nWe show that updates to the abstraction and baseline controller at runtime is computationally feasible by learning a series of local GPs instead of a single global GP.  Furthermore, we show that the framework formally accounts for the uncertainty in the learning process, and the online guarantees see monotonic improvement over those generated offline.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum chromodynamics (QCD) is generally considered as the fundamental theory of strong interactions. While it has been well established in the perturbative regime at high energies,\nQCD still lacks a comprehensive solution at low and intermediate energies, even 40 years\nafter its invention. In order to deal with the wealth of non-perturbative phenomena, various\napproaches are followed with limited validity and applicability. This is especially also true\nfor lattice QCD, various functional methods, or chiral perturbation theory, to name only\na few. In neither one of these approaches the full dynamical content of QCD can yet be included.\nBasically, the difficulties are associated with a relativistically covariant treatment\nof confinement and the spontaneous breaking of chiral symmetry (SB$\\chi$S), the latter\nbeing a well-established property of QCD at low and intermediate energies.\nAs a result, most hadron reactions, like\nresonance excitations, strong and electroweak decays etc., are nowadays only amenable\nto models of QCD. Most famous is the constituent-quark model (CQM), which essentially relies\non a limited number of effective degrees of freedom with the aim of encoding the essential\nfeatures of low- and intermediate-energy QCD.\n\nThe CQM has a long history, and it has made important contributions to the understanding of\nmany hadron properties, think only of the fact that the systematization of hadrons in the\nstandard particle-data base~\\cite{Nakamura:2010zzi} follows the valence-quark picture.\nOver the decades the CQM has ripened into a stage where its formulation and solution are\nwell based on a relativistic (or more generally Poincar\\'e-invariant) quantum theory. Relativistic\nconstituent-quark models (RCQM) have been developed by several groups, however, with\nlimited domains of validity. Of course, it is desirable to have a framework as universal\nas possible for the description of all kinds of hadron processes in the low- and\nintermediate-energy regions. This is especially true in view of the advent of ever\nmore data on heavy-baryon spectroscopy from present and future experimental facilities.\n\nHere, we present a RCQM that comprises all known baryons with flavors $u$, $d$, $s$,\n$c$, and $b$ within a single framework. There have been only a few efforts so far\nto extend a CQM from light- to heavy-flavor baryons. We may mention, for example, the\napproach by the Bonn group who have developed a RCQM, based on the 't Hooft instanton\ninteraction, along a microscopic theory solving the Salpeter equation~\\cite{Loring:2001kx}\nand extended their model to charmed baryons~\\cite{Migura:2006ep}, still not yet\ncovering bottom baryons. A further quark-model\nattempt has been undertaken by the Mons-Li\\`ege group relying on the large-$N_c$\nexpansion~\\cite{Semay:2007cv,Semay:2007ff}, partially extended to heavy-flavor\nbaryons~\\cite{Semay:2008wn}. Similarly, efforts are invested to expand other approaches\nto heavy baryons, such as the employment of Dyson-Schwinger equations together with either\nquark-diquark or three-quark calculations~\\cite{SanchisAlepuz:2011aa,SanchisAlepuz:2012}.\nAlso an increased amount of more refined lattice-QCD results has by now become available\non heavy-baryon spectra (see, e.g., the recent work by Liu et al.~\\cite{Liu:2009jc} and\nreferences cited therein).\n\n\nOur RCQM is based on the invariant mass operator\n\\begin{equation}\n\\hat{M}=\\hat{M}_{\\text{free}} + \\hat{M}_{\\text{int}} \\;,\n\\label{massop}\n\\end{equation}\nwhere the free part corresponds to the total kinetic energy of the three-quark system and\nthe interaction part contains the dynamics of the constituent quarks $Q$. In the rest frame\nof the baryon, where its three-momentum\n${\\vec P}=\\sum^3_i \\vec k^2_i=0$, we may express the terms as\n\\begin{equation}\n\\hat M_{\\text{free}}=\\sum^3_{i=1}\\sqrt{\\hat m^2_i+\\hat{\\vec k}_i^2} \\,,\n\\label{massopparts1}\n\\end{equation}\n\\begin{equation}\n\\hat M_{\\text{int}}=\\sum^3_{i<j}\\hat V_{ij}=\n\\sum^3_{i<j}(\\hat V^{\\text{conf}}_{ij}+\\hat V^{\\text{hf}}_{ij}) \\;.\n\\label{massopparts2}\n\\end{equation}\nHere, the $\\hat{\\vec k}_i$ correspond to the three-momentum operators of the\nindividual quarks with\nrest masses $m_i$ and the $Q$-$Q$ potentials $\\hat V_{ij}$ are composed of confinement and\nhyperfine interactions. By employing such a mass operator $\\hat M^2= \\hat P^\\mu \\hat P_\\mu$\\,, with baryon four-momentum $\\hat P_\\mu=(\\hat H, \\hat P_1, \\hat P_2, \\hat P_3)$, the Poincar\\'{e} algebra involving all ten generators $\\{\\hat H, \\hat P_i, \\hat J_i, \\hat K_i\\}$,\n ($i=1,2,3$), or equivalently $\\{\\hat P_\\mu, \\hat J_{\\mu\\nu}\\}$, ($\\mu,\\nu=0,1,2,3$),\nof time and space translations, spatial rotations as well as Lorentz boosts, can be guaranteed. The solution of the eigenvalue problem of the mass operator $\\hat M$ yields the\nrelativistically invariant mass spectra as well as the baryon eigenstates (the latter,\nof course, initially in the standard rest frame).\n\nWe adopt the confinement depending linearly on the $Q$-$Q$ distance $r_{ij}$\n\\begin{equation}\nV^{\\text{conf}}_{ij}(\\vec r_{ij})=V_0 +Cr_{ij}\n\\end{equation}\nwith the strength $C=2.33$ fm$^{-2}$, corresponding to the string tension of QCD.\nThe parameter $V_0=-402$ MeV is only necessary to set the ground state of the whole\nbaryon spectrum, i.e., the proton mass; it is irrelevant, if one considers only level\nspacings.\n\nThe hyperfine interaction is most essential to describe all of the baryon excitation spectra.\nIn a unified model the hyperfine potential must be explicitly flavor-dependent. Otherwise,\ne.g., the $N$ and $\\Lambda$ spectra with their distinct level orderings could not be\nreproduced simultaneously. At least for baryons with flavors $u$, $d$, and $s$ the type\nof hyperfine interaction taking into account SB$\\chi$S has been most successful over the\npast years. Obviously, it grabs the essential degrees of freedom governing the behavior of\nlow-energy baryons~\\cite{Manohar:1983md,Glozman:1995fu,Weinberg:2010bq}.\nThe RCQM constructed along this dynamical concept, i.e., on Goldstone-boson exchange (GBE),\nhas provided a comprehensive description\nof all light and strange baryons~\\cite{Glozman:1997ag,Glozman:1997fs}. This is not only true\nwith regard to the spectroscopy but to a large extent also for other baryon properties, like electromagnetic and axial form factors~\\cite{Boffi:2001zb} and a number of other\nreaction observables (for a concise summary see ref.~\\cite{Plessas:2010pk}).\nIt has been tempting\nto extend this successful concept even to the heavier flavors $c$ and $b$. By such studies\none should in addition learn about the proper light-heavy and heavy-heavy hyperfine $Q$-$Q$\ninteractions. Some exploratory work in this direction had already been undertaken some time\nago in ref.~\\cite{Glozman:1995xy}, hinting to promising results also for charm and bottom\nbaryons.\n\nTherefore we have advocated for the hyperfine interaction of our universal RCQM \nthe $SU(5)_F$ GBE potential\n\\begin{equation}\nV_{\\text{hf}}(\\vec{r}_{ij})=\\bigg[ V_{24}(\\vec{r}_{ij}) \\sum_{a=1}^{24} \\lambda_i^a \\lambda_j^a\n+ V_{0}(\\vec{r}_{ij}) \\lambda_i^0 \\lambda_j^0  \\bigg]\\vec{\\sigma}_i \\cdot \\vec{\\sigma}_j~.\n\\end{equation}\nHere, we take into account only its spin-spin component, which produces the most important\nhyperfine forces for the baryon spectra. While $\\vec{\\sigma}_i$ represent the Pauli spin\nmatrices of $SU(2)_S$, the $\\lambda_i^a$ are the generalized Gell-Mann flavor matrices of\n$SU(5)_F$ for quark $i$. In addition\nto the exchange of the pseudoscalar 24-plet also the flavor-singlet is included because of\nthe $U(1)$ anomaly. The radial form of the GBE potential resembles the one of the pseudoscalar\nmeson exchange\n\\begin{equation}\nV_{\\beta}(\\vec{r}_{ij})=\\frac{g_{\\beta}^2}{4\\pi}\\frac{1}{12 m_i m_j}\\bigg[ \\mu_{\\beta}^2 \\frac{e^{-\\mu_{\\beta}r_{ij}}}{r_{ij}}-4\\pi \\delta(\\vec{r}_{ij})\\bigg]\n\\label{delta}\n\\end{equation}\nfor $\\beta = 24$ and $\\beta = 0$. Herein the $\\delta$-function must be smeared out leading\nto~\\cite{Glozman:1997fs,Glozman:2000fu}\n\\begin{equation}\nV_{\\beta}(\\vec{r}_{ij})=\\frac{g_{\\beta}^2}{4\\pi}\\frac{1}{12 m_i m_j}\\bigg[ \\mu_{\\beta}^2 \\frac{e^{-\\mu_{\\beta}r_{ij}}}{r_{ij}}- \\Lambda_{\\beta}^2 \\frac{e^{-\\Lambda_{\\beta}r_{ij}}}{r_{ij}}\\bigg] \\,.\n\\end{equation}\nContrary to the earlier GBE RCQM~\\cite{Glozman:1997ag}, which uses several different\nexchange masses $\\mu_\\gamma$ and different cut-offs $\\Lambda_\\gamma$,\ncorresponding to $\\gamma=\\pi$, $K$, and $\\eta$=$\\eta_8$ mesons, we here managed to get\nalong with a universal GBE mass $\\mu_{24}$ and a single cut-off $\\Lambda_{24}$ for the 24-plet\nof $SU(5)_F$. Only the singlet exchange comes with another mass $\\mu_0$ and another cut-off\n$\\Lambda_0$ with a separate coupling constant $g_0$.\nConsequently the number of open parameters in the hyperfine interaction could be kept as\nlow as only three (see Tab.~\\ref{Free parameters}). \n\n\\begin{table}[h]\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\n\\multicolumn{3}{c}{Free Parameters} \\\\\n\\hline\n\\\\\n$(g_0 \/ g_{24})^2$ &$\\Lambda_{24}$ [fm$^{-1}$] &\\vspace{1mm} $\\Lambda_0$  [fm$^{-1}$] \\\\\n\\hline \\\\[-3mm]\n1.5 & 3.55 & 7.52 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption {Free parameters of the present GBE RCQM determined by a best fit to the baryon spectra.}\n\\label{Free parameters}\n\\end{table}\n\nAll other parameters entering the model have judiciously been predetermined by\nexisting phenomenological insights. In this way the constituent quark masses have been\nset to the values as given in Tab.~\\ref{Fixed parameters}. The 24-plet Goldstone-boson (GB)\nmass has been assumed as the value of the $\\pi$ mass and similarly the singlet mass as the\none of the $\\eta'$. The universal coupling constant of the 24-plet has been chosen according\nto the value derived from the $\\pi$-$N$ coupling constant via the Goldberger-Treiman relation.\n\n\\begin{table}[h]\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\n\\multicolumn{7}{c}{Fixed Parameters}\\\\\n\\hline\n\\\\\n\\multicolumn{4}{c}{Quark masses [MeV]} & \\multicolumn{2}{c}{Exchange masses [MeV]}& Coupling \\\\\n$m_u=m_d$ & $m_s$ & $m_c$ & $m_b$ & \\vspace{1mm} \\hspace{3mm} $\\mu_{24}$ & $\\mu_{0}$ & $g^2_{24}\/4\\pi$ \\\\\n\\hline \\\\[-3mm]\n340& 480 & 1675 & 5055 &  \\hspace{3mm} 139 & 958 & 0.7 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption {Fixed parameters of the present GBE RCQM predetermined from phenomenology and\nnot varied in the fitting procedure.}\n\\label{Fixed parameters}\n\\end{table}\n\n\nWe have calculated the baryon spectra of the relativistically invariant mass operator\n$\\hat M$ to a high accuracy both by the stochastic variational method~\\cite{Suzuki:1998bn}\nas well as the modified Faddeev integral equations~\\cite{Papp:2000kp,McEwen:2010sv}.\nThe present universal GBE\nRCQM produces the spectra in the light and strange sectors with\nsimilar or even better quality than the previous GBE RCQM~\\cite{Glozman:1997ag}\n(see Figs.~\\ref{fig:light} and~\\ref{fig:strange}). Most importantly, the right level\norderings specifically in the $N$, $\\Delta$ and $\\Lambda$ spectra as well as all other\n$SU(3)_F$ ground and excited states are reproduced in accordance with phenomenology.\nThe reasons are exactly the same as for the previous GBE RCQM, which has already been\nextensively discussed in the literature~\\cite{Glozman:1995fu,Glozman:1997ag,Glozman:1997fs}.\nUnfortunately, the case of the $\\Lambda$(1405) excitation could still not be resolved.\nIt remains as an intriguing problem for all three-quark CQMs.\n\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{nucleon_delta.pdf}\\\\\n\\caption{\\label{fig:light} Nucleon and $\\Delta$ excitation spectra (solid\/red levels)\nas produced by the universal GBE RCQM in comparison to phenomenological\ndata~\\cite{Nakamura:2010zzi}\n(the gray\/blue lines and shadowed\/blue boxes show the masses and their uncertainties).}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=8.6cm]{lambda_sigma.pdf}\\\\\n\\vspace{5mm}\n\\includegraphics[width=4.5cm]{xi_omega.pdf}\\\\\n\\caption{\\label{fig:strange} Same as Fig.~\\ref{fig:light} but for the strange baryons.}\n\\end{figure}\n\nWhat is most interesting in the context of the present work are the very properties of the\nlight-heavy and heavy-heavy $Q$-$Q$ hyperfine interactions. Can the GBE dynamics reasonably\naccount for them? In Figs.~\\ref{fig:charm} and~\\ref{fig:bottom} we show the spectra\nof all charm and bottom baryons that experimental data with at least three- or four-star status\nby the PDG~\\cite{Nakamura:2010zzi} are available for~\\footnote{Only $\\Xi_c$ and $\\Xi_b$ \nare missing, as we are presently not in the position to calculate baryons\nwith three different constituent-quark masses.}.\nAs is clearly seen, our universal GBE RCQM can reproduce all levels with respectable accuracy.\nIn the $\\Lambda_c$ and $\\Sigma_c$ spectra some experimental levels are not known with\nregard to their spin and parity $J^P$. They are shown in the right-most columns of\nFig.~\\ref{fig:charm}. Obviously they could easily be accommodated in accordance with the\ntheoretical spectra, once their $J^P$'s are determined. Furthermore the model predicts some\nadditional excited states for charm and bottom baryons that are presently missing\nin the phenomenological data base.\n\nOf course, the presently available data base on charm and bottom baryon states is not yet\nvery rich and thus not particularly selective for tests of effective $Q$-$Q$ hyperfine\nforces. The situation will certainly improve with the advent of further data from\nongoing and planned experiments. Beyond the comparison to experimental data, we note\nthat the theoretical spectra produced by our present GBE RCQM are also in good agreement\nwith existing lattice-QCD results for heavy-flavor baryons. This is especially\ntrue for the charm baryons vis-\\`a-vis the recent work by Liu et al.~\\cite{Liu:2009jc}.\nFurther comparisons with results from lattice QCD and alternative methods will\nbe given in a forthcoming more detailed article~\\cite{Day:2012}, where also a number\nof additional theoretical spectra up to $\\Omega_{bbb}$ will be presented (for which,\nhowever, no phenomenological data exist so far).\n\n\\begin{figure}[t]\n\\includegraphics[width=8.6cm]{one_charm.pdf}\\\\\n\\caption{\\label{fig:charm}Same as Fig.~\\ref{fig:light} but for charm baryons.}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=4cm,height=5cm]{one_bottom.pdf}\\\\\n\\caption{\\label{fig:bottom}Same as Fig.~\\ref{fig:light} but for bottom baryons.}\n\\end{figure}\n\n\nWe emphasize that the most important ingredients into the present RCQM are relativity,\nor more generally Poincar\\'e invariance, and a hyperfine interaction that is derived\nfrom an interaction Lagrangian built from effective fermion (constituent quark) and boson\n(Goldstone boson) fields connected\nby a pseudoscalar coupling~\\cite{Glozman:1995fu}. It appears that such kind of dynamics is\nquite appropriate for constituent quarks of any flavor. The effects of the hyperfine\nforces do not at all become tentatively small for baryons with charm and bottom flavors.\nIn some cases at least the heavy-light interactions are of the same importance for the level\nspacings as the light-light interactions. This has already been seen for charm baryons\nin the work by the Bonn group~\\cite{Migura:2006ep} and is also true for our universal GBE\nRCQM (as will be detailed in ref.~\\cite{Day:2012} too). It is furthermore in\nline with findings from earlier lattice-QCD calculations~\\cite{Woloshyn:2000fe}.\n\nAs a result we have demonstrated by the proposed GBE RCQM that a universal description of\nall known baryons is possible in a single model. Here, we have considered only\nthe baryon masses (eigenvalues of the invariant mass operator $\\hat M$). Beyond spectroscopy\nthe present model will be subject to further tests with regard to the baryon eigenstates,\nwhich are simultaneously obtained from the solution of the eigenvalue problem of $\\hat M$.\nThey must prove reasonable in order to make the model a useful tool for the treatment of\nall kinds of baryons reactions within a universal framework.\n\n\\section{Acknowledgement}\nThis work was supported by the Austrian Science Fund, FWF, through the Doctoral\nProgram on {\\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent\nThe quantum field theory in curved space-time has been a matter of\ngreat interest in recent years because of its applications to \ncosmology and astrophysics. The evidence of existence of strong\ngravitational fields in our Universe led to the study of the quantum \neffects of material fields in external classical gravitational field.\nAfter the appearance of Parker's paper on scalar fields \\cite{Par1} and \nspin-$\\frac{1}{2}$ fields \\cite{Par2}, several authors have studied this \nsubject. Although the Universe seems homogenous and isotropic at present,\nthere are no observational data guarantying the isotropy in the era prior\nto the recombination. In fact, there are theoretical arguments that sustain \nthe existence of an anisotropic phase that approaches an isotropic one \n\\cite{Mis}. Interest in studying Klein-Gordon and Dirac equations in\nanisotropic models has increased since Hu and Parker \\cite{Hu0} have shown\nthat the creation of scalar particles in anisotropic backgrounds can\ndissipate the anisotropy as the Universe expands. \n\n\\noindent\nA Bianchi type-I (B-I) Universe, being the straightforward generalization \nof the flat Robertson-Walker (RW) Universe, is one of the simplest models  \nof an anisotropic Universe that describes a homogenous and spatially flat\nUniverse. Unlike the RW Universe which has the same \nscale factor for each of the three spatial directions, a B-I Universe\nhas a different scale factor in each direction, thereby introducing an\nanisotropy to the system. It moreover has the agreeable property that\nnear the singularity it behaves like a Kasner Universe even in the \npresence of matter and consequently falls within the general analysis\nof the singularity given by Belinskii et al \\cite{Bel}. \nAnd in a Universe filled with matter for $p\\,=\\,\\gamma\\,\\varepsilon, \\quad \n\\gamma < 1$, it has been shown that any initial anisotropy in a B-I\nUniverse quickly dies away and a B-I Universe eventually evolve\ninto a RW Universe \\cite{Jac}. Since the present-day Universe is \nsurprisingly isotropic, this feature of the B-I Universe makes it a prime \ncandidate for studying the possible effects of an anisotropy in the early \nUniverse on present-day observations. In light of the importance of\nmentioned above, several authors have studied linear spinor field equations       \n\\cite{Chim}, \\cite{Cas} and the behavior of \ngravitational waves (GW's) \\cite{Hu}, \\cite{Mied}, \\cite{Cho}    \nin B-I Universe. Nonlinear spinor field (NLSF) in external cosmological\ngravitation field was first studied by G. N. Shikin in 1991  \n\\cite{Shik}. This study was extended by us for more general case where \nwe consider nonlinear term as an arbitrary function of all possible \ninvariants generated from spinor bilinear forms. In that paper we also\nstudied the possibility of elimination of initial singularity specially for\nKasner Universe \\cite{Ryb1}. \nIn a recent paper \\cite{Ryb} we studied \nthe behavior of self-consistent NLSF in B-I Universe that was \nfollowed by the papers \\cite{Alv}, \\cite{AlvIz} where we studied\nthe self-consistent system of interacting spinor and scalar fields.\nThe purpose of the paper is to extend our study for more general NLSF\nin presence of perfect fluid. In the section 2 we derive fundamental\nequations corresponding to the Lagrangian for the self-consistent system\nof spinor and gravitational fields in presence of perfect fluid and seek \ntheir general solutions. In section 3 we give a detail analysis of\nthe solutions obtained for different kinds of nonlinearity. In section 4\nwe study the role of perfect fluid and in section 5 we sum up the\nresults obtained.\n\\vskip 5mm\n\\section{Fundamental equations and general solutions}\n\\setcounter{equation}{0}\n\\noindent\nThe Lagrangian for the self-consistent system of  spinor  and gravitation \nfields in presence of perfect fluid is \n\\begin{equation} L=\\frac{R}{2\\kappa}+\\frac{i}{2} \n\\biggl[ \\bar \\psi \\gamma^{\\mu} \\nabla_{\\mu} \\psi- \\nabla_{\\mu} \\bar \n\\psi \\gamma^{\\mu} \\psi \\biggr] - m\\bar \\psi \\psi + L_N +L_m,\n\\end{equation} \nwith $R$ being the scalar curvature and $\\kappa$ being  the  \nEinstein's gravitational constant. The nonlinear term\n$L_N$ describes the self-interaction of spinor field and can be presented\nas some arbitrary functions of invariants generated from the real bilinear \nforms of spinor field having the form: \n$$S\\,=\\, \\bar \\psi \\psi, \\quad                    \nP\\,=\\,i \\bar \\psi \\gamma^5 \\psi, \\quad\nv^\\mu\\,=\\,(\\bar \\psi \\gamma^\\mu \\psi), \\quad\nA^\\mu\\,=\\,(\\bar \\psi \\gamma^5 \\gamma^\\mu \\psi), \\quad\nT^{\\mu\\nu}\\,=\\,(\\bar \\psi \\sigma^{\\mu\\nu} \\psi),$$\nwhere $\\sigma^{\\mu\\nu}\\,=\\,(i\/2)[\\gamma^\\mu\\gamma^\\nu\\,-\\,\n\\gamma^\\nu\\gamma^\\mu]$. Invariants, corresponding to the bilnear forms, look \n$$ I = S^2, \\quad J = P^2, \\quad \nI_v = v_\\mu\\,v^\\mu\\,=\\,(\\bar \\psi \\gamma^\\mu \\psi)\\,g_{\\mu\\nu}\n(\\bar \\psi \\gamma^\\nu \\psi),$$ \n$$I_A = A_\\mu\\,A^\\mu\\,=\\,(\\bar \\psi \\gamma^5 \\gamma^\\mu \\psi)\\,g_{\\mu\\nu}\n(\\bar \\psi \\gamma^5 \\gamma^\\nu \\psi), \\quad\nI_T = T_{\\mu\\nu}\\,T^{\\mu\\nu}\\,=\\,(\\bar \\psi \\sigma^{\\mu\\nu} \\psi)\\,\ng_{\\mu\\alpha}g_{\\nu\\beta}(\\bar \\psi \\sigma^{\\alpha\\beta} \\psi).$$ \nAccording to the Pauli-Fierz theorem \\cite{Ber} among the five invariants\nonly $I$ and $J$ are independent as all other can be expressed by them:\n$I_v = - I_A = I + J$ and $I_T = I - J.$ Therefore we choose the nonlinear\nterm $L_N = F(I, J)$, thus claiming that it describes the nonlinearity\nin the most general of its form. $L_m$ is the Lagrangian of \nperfect fluid.\\\\[2mm] \n\\noindent\nWe choose B-I space-time metric in the form \n\\begin{equation}\nds^2\\,=\\,dt^2 - \\gamma_{ij}(t)\\,dx^i\\,dx^j.\n\\end{equation}\nAs it admits no rotational matter, the spatial metric $\\gamma_{ij}(t)$\ncan be put into diagonal form. Now we can rewrite the B-I space-time metric \nin the form \\cite{Zel}:\n\\begin{equation} ds^2 = dt^2 - a^{2}(t)\\,dx^{2} - b^{2}(t)\\,dy^{2} -\nc^{2}(t)\\,dz^2,\n\\end{equation}\nwhere the velocity of light is taken to be unity.\nEinstein equations for $a(t), b(t)$ and $c(t)$ \ncorresponding to the metric (2.3) and Lagrangian (2.1) read~\\cite{Zel}:  \n\\begin{eqnarray}\n\\frac{\\ddot a}{a} +\\frac{\\dot a}{a} \\biggl(\\frac{\\dot b}{b}+\\frac{\\dot \nc}{c}\\biggr)= -\\kappa \\biggl(T_{1}^{1}- \\frac{1}{2}T\\biggr), \\\\\n\\frac{\\ddot b}{b} +\\frac{\\dot b}{b} \\biggl(\\frac{\\dot a}{a}+\\frac{\\dot \nc}{c}\\biggr)= -\\kappa \\biggl(T_{2}^{2}- \\frac{1}{2}T\\biggr),  \\\\\n\\frac{\\ddot c}{c} +\\frac{\\dot c}{c} \\biggl(\\frac{\\dot a}{a}+\\frac{\\dot \nb}{b}\\biggr)= -\\kappa \\biggl(T_{3}^{3}- \\frac{1}{2}T\\biggr),   \\\\\n\\frac{\\ddot a}{a} +\\frac{\\ddot b}{b} +\\frac{\\ddot \nc}{c}= -\\kappa \\biggl(T_{0}^{0}- \\frac{1}{2}T\\biggr),\n\\end{eqnarray}\nwhere points denote differentiation with respect to t, and \n$T=T_{\\mu}^{\\mu}.$ \n\n\\noindent\nNLSF equations  and  components  of  \nenergy-momentum tensor for the spinor field and perfect fluid \ncorresponding to (2.1) are\n\\begin{eqnarray}\ni\\gamma^\\mu \\nabla_\\mu \\psi -m\\psi + F_{I} 2 S \\psi + F_{J} 2P i \n\\gamma^5 \\psi\\,&=&\\,0, \\nonumber \\\\\ni \\nabla_\\mu \\bar \\psi \\gamma^\\mu + m \\bar\\psi - F_{I} 2 S \\bar \\psi \n- F_{J} 2P i \\bar \\psi \\gamma^5 \\,&=&\\,0, \n\\end{eqnarray}\nwhere $F_{I}:= \\partial F\/\\partial I$ and \n$F_{J}:= \\partial F\/\\partial J.$\n\\begin{equation}\nT_{\\mu}^{\\rho}=\\frac{i}{4} g^{\\rho\\nu} \\biggl(\\bar \\psi \\gamma_\\mu \n\\nabla_\\nu \\psi + \\bar \\psi \\gamma_\\nu \\nabla_\\mu \\psi - \\nabla_\\mu \\bar \n\\psi \\gamma_\\nu \\psi - \\nabla_\\nu \\bar \\psi \\gamma_\\mu \\psi \\biggr) \\,-\n\\delta_{\\mu}^{\\rho}L_{sp}+ T_{\\mu\\,(m)}^{\\rho},\n\\end{equation}\nwhile $L_{sp}$ on account of spinor field equations takes the form:\n$$ L_{sp}\\,=\\,-\\biggl[\\frac{1}{2}\\biggl(\\bar \\psi \\frac{\\partial \nL_N}{\\partial \\bar \\psi}+ \\frac{\\partial L_N}{\\partial \\psi} \\psi \n\\biggr)-L_N\\biggr]\\,= -\\bigl[2 I\\,F_{I}\\, + 2 J\\,\nF_{J} - L_N\\bigr].$$\nHere $T_{\\mu\\,(m)}^{\\rho}$ is the \nenergy-momentum tensor of perfect fluid. For a Universe filled with\nperfect fluid, in the concomitant system of reference \n$(u^0=1, \\, u^i=0, i=1,2,3)$ we have\n\\begin{equation}\nT_{\\mu (m)}^{\\nu}\\,=\\, (p + \\varepsilon) u_\\mu u^\\nu - \n\\delta_{\\mu}^{\\nu} p \\,=\\,(\\varepsilon,\\,- p,\\,- p,\\,- p),\n\\end{equation} \nwhere energy $\\varepsilon$ is related to the pressure $p$ by the\nequation of state $p\\,=\\,\\gamma\\,\\varepsilon$, the general solution\nhas been derived by Jacobs \\cite{Jac}. $\\gamma$ varies between the\ninterval $0\\,\\le\\, \\gamma\\,\\le\\,1$, whereas $\\gamma\\,=\\,0$ describes\nthe dust Universe, $\\gamma\\,=\\,\\frac{1}{3}$ presents radiation Universe,\n$\\frac{1}{3}\\,<\\,\\gamma\\,<\\,1$ ascribes hard Universe and $\\gamma\\,=\\,1$\ncorresponds to the stiff matter.   \nIn (2.8) and (2.9) $\\nabla_\\mu$ denotes the covariant \nderivative of spinor, having the form \\cite{Zelnor}:  \n\\begin{equation} \\nabla_\\mu \n\\psi=\\frac{\\partial \\psi}{\\partial x^\\mu} -\\Gamma_\\mu \\psi, \\end{equation} \nwhere $\\Gamma_\\mu(x)$ are spinor affine connection matrices.  \n$\\gamma^\\mu(x)$ matrices are defined for the metric (2.3) as follows. \nUsing the equalities  \\cite{Brill}, \\cite{Wein}\n$$ g_{\\mu \\nu} (x)= e_{\\mu}^{a}(x) e_{\\nu}^{b}(x) \\eta_{ab},\n\\qquad \\gamma_\\mu(x)\\,=\\,e_{\\mu}^{a}(x)\\bar\\gamma^a,$$ \nwhere $\\eta_{ab}= \\mbox{diag}(1,-1,-1,-1)$,\n$\\bar \\gamma_\\alpha$ are the Dirac matrices of Minkowski space and\n$e_{\\mu}^{a}(x)$ are the set of tetradic 4-vectors, we obtain \nthe Dirac matrices $\\gamma^\\mu(x)$ of curved space-time\n$$ \\gamma^0=\\bar \\gamma^0,\\quad \\gamma^1 =\\bar \\gamma^1 \/a(t),\\quad \n\\gamma^2= \\bar \\gamma^2 \/b(t),\\quad \\gamma^3 = \\bar \\gamma^3 \/c(t), $$\n$$ \\gamma_0=\\bar \\gamma_0,\\quad \\gamma_1 =\\bar \\gamma_1 a(t),\\quad \n\\gamma_2= \\bar \\gamma_2 b(t),\\quad \\gamma_3 = \\bar \\gamma_3 c(t). $$\n$\\Gamma_\\mu(x)$ matrices are defined by the equality $$\\Gamma_\\mu (x)= \n\\frac{1}{4}g_{\\rho\\sigma}(x)\\biggl(\\partial_\\mu e_{\\delta}^{b}e_{b}^{\\rho} \n- \\Gamma_{\\mu\\delta}^{\\rho}\\biggr)\\gamma^\\sigma\\gamma^\\delta, $$ \nwhich gives\n\\begin{equation} \\Gamma_0=0, \\quad \\Gamma_1=\\frac{1}{2}\\dot a(t) \n\\bar \\gamma^1 \\bar \\gamma^0, \\quad \\Gamma_2=\\frac{1}{2}\\dot b(t) \\bar \n\\gamma^2 \\bar \\gamma^0, \\quad \\Gamma_3=\\frac{1}{2}\\dot c(t) \\bar \\gamma^3 \n\\bar \\gamma^0.\\end{equation}\nFlat space-time matrices we choose in the form, given in \n\\cite{Bog}:\n\\begin{eqnarray}\n\\bar \\gamma^0&=&\\left(\\begin{array}{cccc}1&0&0&0\\\\0&1&0&0\\\\\n0&0&-1&0\\\\0&0&0&-1\\end{array}\\right), \\quad\n\\bar \\gamma^1\\,=\\,\\left(\\begin{array}{cccc}0&0&0&1\\\\0&0&1&0\\\\\n0&-1&0&0\\\\-1&0&0&0\\end{array}\\right), \\nonumber\\\\\n\\bar \\gamma^2&=&\\left(\\begin{array}{cccc}0&0&0&-i\\\\0&0&i&0\\\\\n0&i&0&0\\\\-i&0&0&0\\end{array}\\right), \\quad\n\\bar \\gamma^3\\,=\\,\\left(\\begin{array}{cccc}0&0&1&0\\\\0&0&0&-1\\\\\n-1&0&0&0\\\\0&1&0&0\\end{array}\\right).  \\nonumber\n\\end{eqnarray}\nDefining $\\gamma^5$ as follows\n\\begin{eqnarray}\n\\gamma^5&=&-\\frac{i}{4} E_{\\mu\\nu\\sigma\\rho}\\gamma^\\mu\\gamma^\\nu\n\\gamma^\\sigma\\gamma^\\rho, \\quad E_{\\mu\\nu\\sigma\\rho}= \\sqrt{-g}\n\\varepsilon_{\\mu\\nu\\sigma\\rho}, \\quad \\varepsilon_{0123}=1,\\nonumber \\\\\n\\gamma^5&=&-i\\sqrt{-g} \\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3 \n\\,=\\,-i\\bar \\gamma^0\\bar \\gamma^1\\bar \\gamma^2\\bar \\gamma^3 =\n\\bar \\gamma^5, \\nonumber\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\n\\bar \\gamma^5&=&\\left(\\begin{array}{cccc}0&0&-1&0\\\\0&0&0&-1\\\\\n-1&0&0&0\\\\0&-1&0&0\\end{array}\\right).\\nonumber\n\\end{eqnarray}\nWe study the space-independent solutions to NLSF equation (2.8).\nIn this case the first equation of the system (2.8) together  with (2.11) \nand (2.12) is\n\\begin{equation} i\\bar \\gamma^0 \n\\biggl(\\frac{\\partial}{\\partial t} +\\frac{\\dot \\tau}{2 \\tau} \\biggr) \\psi \n-m \\psi +{\\cal D} \\psi + i {\\cal G} \\gamma^5\\psi=0, \\quad  \n\\tau(t)=a(t)b(t)c(t),\n\\end{equation}\nwhere we denote ${\\cal D} := \\, 2 S\\, F_{I}$ and\n${\\cal G}:=\\,  2 P\\,F_{J}.$ For the components \n$\\psi_\\rho= V_\\rho(t)$, where $\\rho=1,2,3,4,$ from (2.13) one deduces\nthe following system of equations:  \n\\begin{eqnarray} \n{\\dot V}_1 +\\frac{\\dot \\tau}{2 \\tau} V_1 \n+i(m- {\\cal D}) V_1 - {\\cal G}V_3 &=& 0,  \\nonumber\\\\\n{\\dot V}_2 +\\frac{\\dot \\tau}{2 \\tau} V_2 \n+i(m- {\\cal D}) V_2 - {\\cal G}V_4 &=& 0,  \\nonumber\\\\\n{\\dot V}_3 +\\frac{\\dot \\tau}{2 \\tau} V_3 \n-i(m- {\\cal D}) V_3 + {\\cal G}V_1 &=& 0,  \\nonumber \\\\\n{\\dot V}_4 +\\frac{\\dot \\tau}{2 \\tau} V_4 \n-i(m- {\\cal D}) V_4 + {\\cal G}V_2 &=& 0.\n\\end{eqnarray}\n\n\\noindent\nLet us now define the equations for \n\\begin{eqnarray}\nP\\,=\\,i(V_1 V_{3}^{*}-V_{1}^{*}V_3 +V_2V_{4}^{*}-V_{2}^{*}V_4), \\nonumber \\\\\nR\\,=\\,(V_1 V_{3}^{*}+V_{1}^{*}V_3 +V_2V_{4}^{*}+V_{2}^{*}V_4), \\nonumber \\\\\nS\\,=\\,(V_{1}^{*} V_{1}+V_{2}^{*}V_2 -V_{3}^{*}V_{3}-V_{4}^{*}V_4). \n\\end{eqnarray}\nAfter a little manipulation one finds\n\\begin{eqnarray}\n\\frac{d S_0}{d t} -2 {\\cal G}\\, R_0\\,=\\,0, \\nonumber\\\\\n\\frac{d R_0}{d t}+2 (m- {\\cal D})\\, P_0 + 2 {\\cal G} S_0\\,=\\,0, \\nonumber\\\\ \n\\frac{d P_0}{d t}-2 (m- {\\cal D})\\, R_0\\,=\\,0,\n\\end{eqnarray}\nwhere $S_0 = \\tau S, \\quad P_0 = \\tau P, \\quad R_0 = \\tau R$.\nFrom this system we obtain\n\\begin{eqnarray}\nS_0 {\\dot S}_0 + R_0 {\\dot R}_0 +P_0 {\\dot P}_0\\,=\\,0, \\nonumber\n\\end{eqnarray}\nthat gives\n\\begin{equation}\nS^2 + R^2 + P^2 \\,=\\, C^2\/ \\tau^2, \\qquad C^2 = \\mbox{const.}\n\\end{equation}\n \n\\noindent\nLet us go back to the system of equations (2.14). It can be written as\nfollows if one defines $W_\\alpha\\,=\\,\\sqrt{\\tau}\\,V_\\alpha$:\n\\begin{eqnarray} \n{\\dot W}_1 +i\\Phi W_1 - {\\cal G}W_3 &=& 0, \\quad\n{\\dot W}_2 +i\\Phi W_2 - {\\cal G}W_4 = 0, \\nonumber\\\\\n{\\dot W}_3 -i\\Phi W_3 + {\\cal G}W_1 &=& 0, \\quad\n{\\dot W}_4 -i\\Phi W_4 + {\\cal G}W_2 = 0,\n\\end{eqnarray} \nwhere $\\Phi\\,=\\,m- {\\cal D}$. Defining  $U(\\sigma) = W (t)$, where \n$\\sigma = \\int\\,{\\cal G} dt$, we rewrite the foregoing system as:\n\\begin{eqnarray} \nU_{1}^{\\prime} + i (\\Phi\/{\\cal G}) U_{1} - U_{3} &=& 0, \\qquad\nU_{2}^{\\prime} + i (\\Phi\/{\\cal G}) U_{2} - U_{4} = 0, \\nonumber\\\\\nU_{3}^{\\prime} - i (\\Phi\/{\\cal G}) U_{3} + U_{1} &=& 0, \\qquad\nU_{4}^{\\prime} - i (\\Phi\/{\\cal G}) U_{4} + U_{2} = 0,\n\\end{eqnarray} \nwhere  prime ($^\\prime$) denotes differentiation with respect to $\\sigma$.\n \n\\noindent\nLet us now solve the Einstein equations. To do it we first write\nthe expressions for the components of the energy-momentum tensor \nexplicitly. Using the property of flat space-time Dirac matrices\nand the explicit form of covariant derivative $\\nabla_\\mu$ one can  \neasily find\n\\begin{equation}\nT_{0}^{0}= m\\,S \\,-\\,F(I,\\,J) + \\varepsilon, \\quad \nT_{1}^{1}=T_{2}^{2}=T_{3}^{3}= 2 I\\,F_{I}\\,+\\,2 J\\,F_{J} \n- F(I,\\,J) -\\,p. \n\\end{equation}\nSummation of Einstein equations (2.4), (2.5) and (2.6) leads to the equation \n\\begin{equation}\n\\frac{\\ddot \n\\tau}{\\tau}=-\\kappa(T_{1}^{1}+T_{2}^{2}+T_{3}^{3}-\\frac{3}{2}T)=\n\\frac{3\\kappa}{2}\\,\\bigl(mS +2 I\\,F_{I}\\,+\\,2 J\\,F_{J} \\,-\\,2\\,\nF(I,\\,J) \\,+\\varepsilon - \\,p\\bigr).\n\\end{equation} \nIn case if the right hand side of (2.21) be the \nfunction of $\\tau(t)\\,=\\,a(t)b(t)c(t)$, this equation takes the form \n\\begin{equation}\n\\ddot \\tau+\\Phi(\\tau)=0.            \n\\end{equation}\nAs is known this equation possesses exact solutions  for  \narbitrary  function $\\Phi(\\tau)$.  Giving the explicit \nform of $L_N \\,=\\,F(I,\\,J)$, from (2.21)  one  can find concrete \nfunction $\\tau(t)=abc$. Once the value of $\\tau$ is obtained, one can \nget expressions for components $V_\\alpha(t), \\quad \\alpha =1,2,3,4.$\nLet us express $a, b, c$ through $\\tau$. For this we notice that\nsubtraction of Einstein equations  (2.4) - (2.5)  leads  to  \nthe equation \n\\begin{equation}\n\\frac{\\ddot a}{a}-\\frac{\\ddot b}{b}+\\frac{\\dot a \\dot c}{ac}- \n\\frac{\\dot b \\dot c}{bc}= \\frac{d}{dt}\\biggl(\\frac{\\dot a}{a}- \n\\frac{\\dot b}{b}\\biggr)+\\biggl(\\frac{\\dot a}{a}- \\frac{\\dot b}{b} \\biggr) \n\\biggl (\\frac{\\dot a}{a}+\\frac{\\dot b}{b}+ \\frac{\\dot c}{c}\\biggr)= 0. \n\\end{equation} \nEquation (2.23) possesses the solution\n\\begin{equation}\n\\frac{a}{b}= D_1 \\mbox{exp} \\biggl(X_1 \\int \\frac{dt}{\\tau}\\biggr), \\quad \nD_1=\\mbox{const.}, \\quad X_1= \\mbox{const.} \\end{equation}\nSubtracting equations (2.4) - (2.6) and (2.5) - (2.6) one finds   \nthe equations similar to (2.23), having solutions \\begin{equation} \n\\frac{a}{c}= D_2 \\mbox{exp} \\biggl(X_2 \\int \\frac{dt}{\\tau}\\biggr), \\quad \n\\frac{b}{c}= D_3 \\mbox{exp} \\biggl(X_3 \\int \\frac{dt}{\\tau}\\biggr),  \n\\end{equation}\nwhere $D_2, D_3, X_2, X_3 $ are integration constants. There is a \nfunctional dependence between the constants \n$D_1,\\, D_2,\\, D_3,\\, X_1,\\, X_2,\\, X_3 $:  \n$$ D_2=D_1\\, D_3, \\qquad X_2= X_1\\,+\\,X_3.$$\nUsing the equations (2.24) and (2.25), we rewrite $a(t), b(t), c(t)$ in \nthe explicit form:  \n\\begin{eqnarray} a(t) &=& \n(D_{1}^{2}D_{3})^{\\frac{1}{3}}\\tau^{\\frac{1}{3}}\\mbox{exp}\\biggl[\\frac{2X_1 \n+X_3}{3} \\int\\,\\frac{dt}{\\tau (t)} \\biggr], \\nonumber \\\\\nb(t) &=& \n(D_{1}^{-1}D_{3})^{\\frac{1}{3}}\\tau^{\\frac{1}{3}}\\mbox{exp}\\biggl[-\\frac{X_1 \n-X_3}{3} \\int\\,\\frac{dt}{\\tau (t)} \\biggr], \\nonumber \\\\\nc(t) &=& \n(D_{1}D_{3}^{2})^{-\\frac{1}{3}}\\tau^{\\frac{1}{3}}\\mbox{exp}\\biggl[-\\frac{X_1 \n+2X_3}{3} \\int\\,\\frac{dt}{\\tau (t)} \\biggr]. \n\\end{eqnarray}\nThus the previous system of Einstein equations is completely integrated.  \nIn this process of integration only first three of the complete system of \nEinstein equations have been used. General solutions to these three second \norder equations have been obtained. The  solutions  contain  six arbitrary \nconstants: $D_1, D_3, X_1, X_3 $  and two others, \nthat  were obtained while solving  equation  (2.22).  \nEquation  (2.7)  is  the consequence of first three of Einstein equations.  \nTo  verify  the correctness of obtained solutions, it is necessary to put \n$a, b, c$ into (2.7). It should lead either to identity or to  some  \nadditional constraint between the constants. Putting $a, b, c$  from  (2.26) \ninto (2.7) one can get the following equality:  \n\\begin{equation}\n\\frac{1}{3 \\tau}\\biggl[3 \\ddot \\tau - 2\\frac{\\dot \\tau^2}{\\tau}+ \n\\frac{2}{3 \\tau}\\biggl(X_{1}^{2}+X_1 X_3 +X_{3}^{2}\\biggr)\\biggr] = \n-\\kappa \\biggl(T_{0}^{0}-\\frac{1}{2}T\\biggr), \n\\end{equation}\nthat guaranties the correctness of the solutions obtained.\nIn fact we can rewrite  (2.21) and (2.27) as\n\\begin{equation}\n\\frac{\\ddot \\tau}{\\tau} \\, = \\, \\frac{3 \\kappa}{2} \\,(T_{0}^{0} + T_{1}^{1}),\n\\end{equation}\nand \\begin{equation}\n\\frac{\\ddot \\tau}{\\tau} - \\frac{2}{3}\\,\\frac{\\dot{\\tau}^2}{\\tau^2} +\n\\frac{2}{9 \\tau^2} {\\cal X} \\, = \\, - \\frac{\\kappa}{2}\\, (T_{0}^{0} -\n3 T_{1}^{1}),\n\\end{equation}\nwhere ${\\cal X} := X_{1}^{2} + X_1 X_3 + X_{3}^{2}$. Combining (2.28) and \n(2.29) together one gets the solution for $\\tau$ in quadrature:\n\\begin{equation}\n\\int\\,\\frac{d\\tau}{\\sqrt{3\\kappa \\tau^2 T_{0}^{0} + {\\cal X}\/3}}\\,=\\,t.\n\\end{equation}  \nLet us note that in our further study we exploit the equations (2.21)\nto obtain $\\tau$ and (2.27) to estimate integration constants.\n\n\\noindent\nIt should be emphasized that we are dealing with cosmological problem\nand our main goal is to investigate the initial and the asymptotic behavior\nof the field functions and the metric ones. As one sees, all these functions\nare in some functional dependence with $\\tau$: $\\psi \\sim 1\/\\sqrt{\\tau}$\nand $a_i \\sim \\tau^{1\/3} e^{\\pm \\int dt\/\\tau}$. Therefore in our further\ninvestigation we mainly look for $\\tau$, though in some particular cases\nwe write down field and metric functions explicitly.\n\\vskip 5mm\n\\section{Analysis of the solutions obtained for some special choice\nof nonlinearity}\n\\setcounter{equation}{0}\n\\noindent\nLet us now study the system for some special choice of $L_N$. First we \nanalyze the system only for the NLSF which will be followed by the study \nwhen the Universe is filled with perfect fluid. But first of all  \nwe study the linear case. The reason to get the solution to \nthe self-consistent system of equations for the linear spinor and \ngravitational fields is the necessity of comparing this solution with that \nfor the  system  of equations   for the nonlinear spinor   and \ngravitational  fields  that permits  to clarify the  role of nonlinear \nspinor terms in the evolution of the cosmological model in question. \nUsing the equation (2.21) one gets\n\\begin{equation}\n\\tau(t)\\,=\\,(1\/2)\\,Mt^2\\,+\\,y_1t\\,+\\,y_0     \n\\end{equation}       \nwhere $M=\\frac{3}{2}\\kappa mC_0,$\\,\\,$C_0 = C_{1}^{2}+C_{2}^{2}-C_{3}^{2}\n-C_{4}^{2}$ and $y_1, y_0$ are the constants.  \nIn this case we get explicit expressions for the components of spinor\nfield functions and metric functions:\n\\begin{equation}\nV_r(t)=(C_r\/\\sqrt{\\tau})\\,e^{-imt},\\quad r = 1,2; \\quad\nV_l(t)=(C_l\/\\sqrt{\\tau})\\,e^{imt}, \\quad l = 3,4.\n\\end{equation}\n\\begin{eqnarray} a(t) &=& \n(D_{1}^{2}D_{3})^{\\frac{1}{3}}(\\frac{1}{2}Mt^2+y_1t+y_0)^{\\frac{1}{3}}\nZ^{2(2X_1+X_3)\/3B}, \\nonumber \\\\ \nb(t) &=& \n(D_{1}^{-1}D_{3})^{\\frac{1}{3}}(\\frac{1}{2}Mt^2+y_1t+y_0)^{\\frac{1}{3}}\nZ^{-2(X_1-X_3)\/3B}, \\nonumber \\\\ \nc(t) &=& \n(D_{1}D_{3}^{2})^{-\\frac{1}{3}}(\\frac{1}{2}Mt^2+y_1t+y_0)^{\\frac{1}{3}}\nZ^{-2(X_1+2X_3)\/3B}, \n\\end{eqnarray}\nwhere $Z = \\frac{(t-t_1)}{(t-t_2)},\\,\\, B = M(t_1 -t_2),$ \nand $t_{1,2} = -y_1\/M \\pm \\sqrt{(y_1\/M)^2 - 2y_0\/M}$\\, are the \nroots  of the quadratic equation \\, $Mt^2+2y_1t+2y_0 = 0.$ \nSubstituting $\\tau(t)$ into (2.27), one gets  \n\\begin{equation}\ny_{1}^{2}- 2My_0\\,=\\,(X_{1}^{2}+X_1X_3+X_{3}^{2})\/3\\,=\\,{\\cal X}\/3 \\,>\\,0.\n\\end{equation} \nThis means that the quadratic polynomial in (3.1) possesses real roots, \ni.e. $\\tau(t)$ in (3.1) turns into zero at $t=t_{1,2}$ and the solution \nobtained is the singular one. Let us now study the solutions (3.1) - (3.3) \nat $t \\to \\infty$. In this case we have \n$$\\tau(t) \\approx \\frac{3}{4}\\kappa mC_0 t^2, \\qquad\na(t) \\approx b(t) \\approx c(t) \\approx t^{2\/3}, $$\nthat leads to the conclusion about the asymptotical \nisotropization of the expansion process for the initially anisotropic \nB-I space.  Thus the solution to the self-consistent system of  \nequations for the linear spinor and gravitational fields  is  the singular \none at the initial time. In the initial state of evolution of the field \nsystem the expansion process of space is  anisotropic,  but at $t \\to \n\\infty$ there happens isotropization of the expansion process. \n\n\\noindent\nOnce the solutions to the linear spinor field equations and corresponding\nto them metric functions are obtained, let us now study the nonlinear case.  \n\\vskip 3mm\n\\noindent\n{\\bf I.} Let us consider the case when $L_N\\,=\\,F(I)$.\nIt is clear that in this case ${\\cal G}\\,=\\,0$.\nFrom (2.16) we find  \n\\begin{equation}\nS = C_0\/\\tau, \\quad C_0= \\mbox{const.}\n\\end{equation}\nAs in the considered case $L_N\\,=\\,F$ depends only on $S$, from (3.5) it\nfollows that $F(I)$ and $F_I(I)$ are functions of $\\tau= \nabc$.  Taking this fact into account, integration  of the system of \nequations (2.14) leads to the expressions \n\\begin{eqnarray} \nV_{r}(t) = (C_r\/\\sqrt{\\tau})\\,e^{-i\\Omega}, \\quad r=1,2, \\quad \nV_{l}(t) = (C_l\/\\sqrt{\\tau})\\,e^{i\\Omega}, \\quad l=3,4.  \n\\end{eqnarray} where $ C_r$ and \n$C_l$ are integration constants. Putting (3.6) into (2.15) one gets\n\\begin{equation}\nS = (C_{1}^{2}+C_{2}^{2}-C_{3}^{2}-C_{4}^{2})\/\\tau.\n\\end{equation}\nComparison of (3.5) with (3.7) gives $C_0 = \nC_{1}^{2}+C_{2}^{2}-C_{3}^{2}-C_{4}^{2}.$\n\n\\noindent\nLet us consider the concrete type of NLSF  \nequation with $F(I) =\\lambda I^{(n\/2)}=\\lambda S^n$ where $\\lambda$  \nis the coupling constant, $n>1$. \nIn this case for $\\tau$ one gets: \n\\begin{equation}\n\\ddot \\tau = (3\/2)\\kappa C_0 \\bigl[m+ \\lambda (n-2) \nC_{0}^{n-1}\/\\tau^{n-1}\\bigr].  \n\\end{equation}\nThe first integral of the foregoing equation takes form:\n\\begin{equation}\n\\dot \\tau^2 = 3\\kappa C_0 \\bigl[m\\tau - \\lambda \n\\, C_{0}^{n-1}\/\\tau^{n-2} + g^2\\bigr], \n\\end{equation} \nwhere from (2.27) one determines $g^2\\,=\\, {\\cal X}\/9\\kappa\\,C_0$.\nThe sign $C_0$  is determined by the positivity \nof the energy-density $T_{0}^{0}$  of linear spinor field:\n\\begin{equation}\nT_{0}^{0} = m C_0\/\\tau > 0.\n\\end{equation}   \nIt is obvious from (3.10) that $C_0 >0.$ Now one can write the\nsolution to the equation  (3.9) in quadratures:\n\\begin{equation}\n\\int \\frac{\\tau^{(n-2)\/2}d\\tau}{\\sqrt{m \\tau^{n-1} +g^2 \n\\tau^{n-2}-\\lambda C_{0}^{n-1}}}= \\sqrt{3\\kappa C_0}\\,t \n\\end{equation} \nThe constant of integration in (3.11) has been taken zero, as it only gives \nthe shift of the initial time.  Let us study the properties of solution to \nequation (3.8) for $n>2$. From (3.11) one gets \n\\begin{equation}\n \\tau(t)\\mid_{t \\to \\infty} \\approx (3\/4) \\kappa mC_0t^2,\n\\end{equation}\nwhich coincides with the asymptotic solution to the equation (3.3). It leads  \nto the conclusion about isotropization of the expansion process of the \nB-I space. It should be remarked that the isotropization takes place if and\nonly if the spinor field equation contains the massive term  [cf.  \nthe parameter m in (3.12)].  If m=0 the isotropization does not take place. \nIn this case from (3.11) we get \n\\begin{equation}\n\\tau(t)\\mid_{t \\to \\infty} \\approx \\sqrt{3\\kappa C_0 g^2}\\,t.\n\\end{equation} \nSubstituting (3.13) into (2.26) one comes  to  the conclusion  that the \nfunctions $a(t), b(t)$ and $c(t)$ are different.  Let us consider the \nproperties of solutions to equation  (3.8) when $t \\to 0.$ For $\\lambda<0$ \nfrom (3.11) we get \n\\begin{equation}\n\\tau(t)= \\bigl[(3\/4) n^2 \\kappa |\\lambda| \nC_{0}^{n}\\bigr]^{1\/n}t^{2\/n} \\to 0, \\end{equation} \ni.e. solutions are singular. For $\\lambda>0,$ from (3.11) it follows that \n$\\tau=0$ cannot be reached for any value of $t$ as in this case the \ndenominator of the integrand in (3.11) becomes imaginary. It means that \nfor $\\lambda>0$ there exist regular solutions to the previous system of \nequations \\cite{Ryb}. The absence of the initial singularity in the \nconsidered cosmological solution appears to be consistent with the \nviolation for $\\lambda>0$, of the dominant energy condition in the \nHawking-Penrose theorem \\cite{Zel}.  \n\n\\noindent\nLet us consider the Heisenberg-Ivanenko equation when in (3.8) n=2 \n\\cite{Ivan}. In this case  the equation  for $\\tau(t)$ does not contain \nthe nonlinear term and its solution coincides with that of the linear \nequation (3.3). With such $n$ chosen the metric functions $a, b, c$ are given by \nthe equality (3.2), and the spinor field functions are written as follows:  \n\\begin{equation}\nV_r = (C_r\/\\sqrt{\\tau})\\,e^{-imt}Z^{4i\\lambda C_0\/B}, \\quad\nV_l = (C_l\/\\sqrt{\\tau})\\,e^{imt}Z^{-4i\\lambda C_0\/B} \n\\end{equation}\nAs in the linear case, the obtained solution is singular at initial time \nand asymptotically isotropic as $t \\to \\infty$.\n\n\\noindent  \nWe now study the properties of solutions to equation (3.8) for \n$1<n<2.$ In this case it is convenient to present the solution (3.11) in the \nform:\n\\begin{equation}\n\\int \\frac{d \\tau}{\\sqrt{m\\tau -\\lambda \\tau^{2-n} \nC_{0}^{n-1}+g^2}}=\\sqrt{3\\kappa C_0}\\,t \n\\end{equation} \nAs $t \\to \\infty$, from (3.16) we get the equality (3.12), \nleading to the isotropization of the expansion process.  If  $m=0$ and \n$\\lambda>0,$ \\quad $\\tau(t)$ lies on the interval \n$$0 \\le \\tau(t) \\le \\bigl(g^2\/\\lambda C_{0}^{n-1}\\bigr)^{1\/(2-n)}.$$\nIf m=0 and $\\lambda<0,$ the relation (3.16) at $t \\to \\infty$ leads to the \nequality:  \n\\begin{equation}\n\\tau(t) \\approx \\bigl[(3\/4)n^2 \\kappa |\\lambda| C_{0}^{n} \n\\bigr]^{1\/n}t^{2\/n}.\n\\end{equation}  \nSubstituting (3.17) into (2.26) and taking into account that at $t \\to \n   \\infty$\n$$ \\int \\frac{dt}{\\tau} \\approx \\frac{n(3\\kappa |\\lambda| \nn^2C_{0}^{n})^{1\/n}}{(n-2)2^{2\/n}}t^{-2\/n+1} \\to 0 $$\ndue to $-2\/n+1 < 0,$ we obtain \n\\begin{equation}\na(t) \\sim b(t) \\sim c(t) \\sim \n[\\tau(t)]^{1\/3} \\sim t^{2\/3n} \\to \\infty.  \n\\end{equation} \nIt means that the solution obtained tends to the isotropic one.  In \nthis case the isotropization is provided not by the massive parameter, \nbut by the degree $n$ in the term $L_N = \\lambda S^n.$  \n(3.16) implies \n\\begin{equation}\n\\tau(t)\\mid_{t \\to 0} \\approx \\sqrt{3\\kappa C_0 g^2}\\,t \\to 0,\n\\end{equation} \nwhich means the solution obtained is initially singular. Thus, for $1<n<2$ \nthere exist only singular solutions at initial time. At $t \\to \\infty$ \nthe isotropization of  the expansion process of B-I space \ntakes place both for $m\\not= 0$ and for $m=0.$  \n\n\\noindent\nLet us finally study the \nproperties of   the solution to the equation (3.8) for $0<n<1.$ In this \ncase  we use the solution in the form (3.16). As now $2-n>1,$ then \nwith the increasing of $\\tau(t)$ in the denominator of the integrand in \n(3.16)  the second term $\\lambda \\tau^{2-n} C_{0}^{n-1}$ increases faster \nthan the first one.  Therefore the solution describing the space expansion \ncan be possible only for $\\lambda<0.$  In this case at $t\\to \\infty$, \nfor $m=0$ as well as for $m\\not= 0,$ one can get the asymptotic \nrepresentation (3.17) of the solution.  This solution,  as for the choice \n$1<n<2,$ provides asymptotically isotropic expansion of the B-I \nspace.  For $t \\to 0$ in this case we shall get only singular \nsolution of the form (3.19). \n\\vskip 3mm\n\\noindent\n{\\bf II.} We study the system when $L_N\\,=\\,F(J)$,\nwhich means in the case considered ${\\cal D}\\,=\\,0$. Let us \nnote that, in the unified nonlinear spinor theory of Heisenberg the \nmassive term remains absent, as according to Heisenberg, the particle \nmass should be obtained as a result of quantization of spinor prematter \n\\cite{Hei}. In nonlinear generalization of classical field equations, \nthe massive term does not possess the significance that it possesses \nin linear one, as by no means it defines total energy (or mass) of \nnonlinear field system. Thus without losing the generality we can \nconsider massless spinor field putting $m\\,=\\,0$ that leads to \n$\\Phi\\,=\\,0.$ This assumption metamorphoses (2.16) to get\n\\begin{equation}\nP(t)\\,=\\,D_0\/\\tau, \\,\\, D_0=\\,\\mbox{const.}\n\\end{equation}\nThe system of equations (2.19) in this case reads\n\\begin{eqnarray} \nU_{1}^{\\prime} -  U_{3} &=& 0, \\qquad\nU_{2}^{\\prime} -  U_{4} = 0, \\nonumber\\\\\nU_{3}^{\\prime} +  U_{1} &=& 0, \\qquad\nU_{4}^{\\prime} +  U_{2} = 0.\n\\end{eqnarray} \nDifferentiating the first equation of system (3.21) and taking into \naccount the third one we get \n\\begin{equation}\nU_{1}^{\\prime \\prime} +U_{1} =\\,0,\n\\end{equation}\nwhich leads to the solution\n\\begin{equation}\nU_1 = D_1 e^{i \\sigma} + iD_3 e^{-i \\sigma},\\quad\nU_3 =  i D_1 e^{i \\sigma} + D_3 e^{-i \\sigma}.\n\\end{equation}\nAnalogically for $U_2$ and $U_4$ one gets\n\\begin{equation}\nU_2 = D_2 e^{i \\sigma} + iD_4 e^{-i \\sigma},\\quad\nU_4 = i D_2 e^{i \\sigma} + D_4 e^{-i \\sigma}, \n\\end{equation}\nwhere $D_i$ are the constants of integration.\nFinally, we can write\n\\begin{eqnarray}\nV_1&=&(1\/\\sqrt{\\tau}) (D_1 e^{i \\sigma} + iD_3 \ne^{-i\\sigma}), \\quad\nV_2 = (1\/\\sqrt{\\tau}) (D_2 e^{i \\sigma} + iD_4\ne^{-i\\sigma}),\n\\nonumber \\\\\nV_3&=&(1\/\\sqrt{\\tau}) (iD_1 e^{i \\sigma} + D_3\ne^{-i \\sigma}), \\quad\nV_4 = (1\/\\sqrt{\\tau}) (iD_2 e^{i \\sigma} + D_4\ne^{-i\\sigma}).\n\\end{eqnarray} \nPutting (3.25) into the expressions (2.15) one finds\n\\begin{equation}\nP=2\\,(D_{1}^{2} + D_{2}^{2} - D_{3}^{2} -D_{4}^{2})\/\\tau.\n\\end{equation}\nComparison of (3.20) with (3.26) gives\n$D_0=2\\,(D_{1}^{2} + D_{2}^{2} - D_{3}^{2} -D_{4}^{2}).$\n\nLet us now estimate $\\tau$ using the equation\n\\begin{equation}\n\\ddot{\\tau}\/\\tau\\,=\\,3 \\kappa \\,\\lambda (n - 1) P^{2n},\n\\end{equation}\nwhere we chose $L_N\\,=\\,\\lambda P^{2n}$. Putting the value of $P$ into\n(3.20) and integrating one gets\n\\begin{equation}\n\\dot{\\tau}^2 \\, = \\,- 3\\kappa\\,\\lambda D_{0}^{2n} \\tau^{2 - 2n} + y^2,\n\\end{equation}\nwhere $y^2$ is the integration constant and can be defined from (2.27):\n$y^2\\,=\\,{\\cal X}\/3 > 0$. The solution to the equation (3.28) in quadrature \nreads\n\\begin{equation}\n\\int\\,\\frac{d\\tau}{\\sqrt{- 3 \\kappa\\lambda D_{0}^{2n}\\tau^{2 - 2n} + y^2}} \n\\, = \\, t.\n\\end{equation}\nLet us now analyze the solution obtained here. As one can see\nthe case $n = 1$ is the linear one.  In case of $\\lambda < 0$ for\n$n > 1$ i.e. $2 - 2n < 0$, we get\n$$ \\tau(t)\\mid_{t \\to 0} \\approx [(\\sqrt{3 \\kappa|\\lambda|}  \nD_{0}^{n}n) t]^{1\/n},$$\nand\n$$ \\tau\\mid_{t \\to \\infty} \\approx \\sqrt{3\\kappa y^2} \\,t, $$\nit means that for the term $L_N$ considered with $\\lambda < 0$ and\n$n > 1$ the solution is initially singular and the space-time is\nanisotropic at $t \\to \\infty.$ \nLet us now study it for $n < 1$. In \nthis case we obtain\n$$ \\tau\\mid_{t \\to 0} \\approx \\sqrt{3\\kappa y^2}\\, t, $$\nand\n$$ \\tau\\mid_{t \\to \\infty} \\approx \n[(\\sqrt{3\\kappa|\\lambda|} D_{0}^{n} n) t]^{1\/n}.$$\nThe solution is initially singular as in previous case, but as far as\n$ 1\/n > 1$, it provides asymptotically isotropic expansion of B-I space-time.\n\\vskip 3mm\n\\noindent\n{\\bf III.} In this case we study $L_N\\,=\\,F(I,\\,J)$. Choosing \n\\begin{equation}\nL_N\\,=\\,F(K_{\\pm}), \\quad K_{+} = I + J = I_v = -I_A, \\quad\nK_{-} = I - J = I_T,\n\\end{equation}\nin case of massless NLSF we find\n$$\n{\\cal D}\\,=\\,2 S F_{K_{\\pm}}, \\quad\n{\\cal G}\\,=\\, \\pm 2 P F_{K_{\\pm}}, \\quad F_{K_{\\pm}} = dF\/dK_{\\pm}.\n$$\nPutting them into (2.16) we find\n\\begin{equation}\nS_{0}^{2} \\pm P_{0}^{2} = D_{\\pm}.\n\\end{equation}\nChoosing $F = \\lambda K_{\\pm}^{n}$ from (2.21) we get\n\\begin{equation}\n\\ddot{\\tau}\\,=\\,3 \\kappa \\lambda (n - 1)\\,D_{\\pm}^{n}\\,\\tau^{1-2n},\n\\end{equation}\nwith the solution\n\\begin{equation}\n\\int\\,\\frac{\\tau^{n-1} d \\tau}{\\sqrt{g^2\\tau^{2n - 2} - 3\\kappa\\lambda \nD_{\\pm}^{n}}}\n\\,=\\, t,\n\\end{equation}\nwhere $g^2\\,=\\,{\\cal X}\/3.$ Let us study the case with $\\lambda < 0$.\nFor $n < 1$ from (3.33) one gets\n\\begin{equation}\n\\tau (t)\\mid_{t \\to 0} \\approx g t \\to 0,\n\\end{equation}\ni.e. the solutions are initially singular, and  \n\\begin{equation}\n\\tau (t)\\mid_{t \\to \\infty} \\approx \n[\\sqrt{(3\\kappa |\\lambda| D_{\\pm}^{n})}t]^{1\/n},\n\\end{equation}\nwhich means that the anisotropy disappears as the Universe expands.\nIn case of $n > 1$ we get \n$$ \\tau (t)\\mid_{t \\to 0} \\approx t^{1\/n} \\to 0, $$\nand\n$$ \\tau (t)\\mid_{t \\to \\infty} \\approx gt, $$\ni.e. the solutions are initially singular and the metric functions \n$a(t), b(t), c(t)$ are different at $ t \\to \\infty$, i.e.\nthe isotropization process remains absent. For $\\lambda > 0$ we \nget the solutions those are initially regular, but it violates the\ndominant energy condition in Hawking-Penrose theorem \\cite{Zel}.\nNote that one comes to the analogical conclusion choosing \n$L_N\\,=\\,\\lambda S^{2n}P^{2n}.$\n\\vskip 5mm\n\\section{Analysis of the results obtained when the B-I \nUniverse is filled with perfect fluid}\n\\setcounter{equation}{0}\n\n\\noindent\nLet us now analyze the system filled with perfect fluid.\nLet us recall that the energy-momentum tensor of perfect fluid is \n\\begin{equation}\nT_{\\mu (m)}^{\\nu}\\,=\\, (p + \\varepsilon) u_\\mu u^\\nu - \n\\delta_{\\mu}^{\\nu} p\\,=\\,(\\varepsilon, - p, - p, - p). \n\\end{equation}\nAs we saw earlier the introduction of perfect fluid does not change\nthe field equations, thus leaving the solutions to the NLSF equations\nexternally unchanged. Changes in the solutions performed by perfect fluid\ncarried out through Einstein equations, namely through $\\tau$. So, let us \nfirst see how the quantities $\\varepsilon$ and $p$ connected with $\\tau$.\nIn doing this we use the well-known equality $T_{\\mu;\\nu}^{\\nu}\\,=\\,0$,\nthat leads to\n\\begin{equation}\n\\frac{d}{dt}(\\tau \\varepsilon) + {\\dot \\tau} p\\,=\\,0,\n\\end{equation}\nwith the solution\n\\begin{equation}\n\\mbox{ln} \\tau\\,=\\,-\\int\\,\\frac{d \\varepsilon}{(\\varepsilon + p)}.\n\\end{equation}\nRecalling the equation of state \n$p\\,=\\,\\xi \\varepsilon,\\,\\, 0 \\le \\xi \\le 1$ finally we get\n\\begin{equation}\nT_{0 (m)}^{0}\\,=\\,\\varepsilon\\,=\\,\\frac{\\varepsilon_0}{\\tau^{1+\\xi}}, \\,\\,\nT_{1 (m)}^{1}\\,=\\,T_{2 (m)}^{2}\\,=\\,T_{3 (m)}^{3}\\,=\\,- p\\,=\\,-\n\\frac{\\varepsilon_0 \\xi}{\\tau^{1+\\xi}},\n\\end{equation}\nwhere $\\varepsilon_0$ is the integration constant. Putting them into (2.21)\nwe get \n\\begin{equation}\n\\frac{\\ddot \\tau}{\\tau}\\,=\\,\\frac{3 \\kappa}{2}\\frac{(\\xi - 1)\n\\varepsilon_0}{\\tau^{(\\xi + 1)}}\n\\end{equation}\nwhich shows that for stiff matter $(\\xi = 1)$ the contribution \nof fluid to the solution is missing. \nLet us now study the system with nonlinearity type {\\bf I}. In this case\nwe get\n\\begin{equation}\n\\int\\,\\frac{d \\tau}{\\sqrt{m C_0 \\tau - \\lambda C_{0}^{n}\/\\tau^{(n-2)} +\n\\varepsilon_0 \\tau^{(1 - \\xi)} + g^2}}\\,=\\,\\pm \\sqrt{3 \\kappa} t.\n\\end{equation}\nAs one can see in case of dust $(\\xi = 0)$  the fluid term can be combined \nwith the massive one, whereas in case of stiff matter $(\\xi = 1)$\nit mixes up with the constant. Analyzing the equation (4.6) one comes to the \nconclusion that the presence of perfect fluid does not influence the \nresult obtained earlier for the nonlinear term type {\\bf I}. One comes \nto the same conclusion analyzing the system with perfect fluid for the \nother types of nonlinear terms considered here. At least\nboth at $t \\to 0$ and at $t \\to \\infty$ the key role is \nplayed by the other terms rather than the term\npresenting fluid. \n\\vskip 5mm\n\\section{Conclusions}\n\n\\noindent\nExact solutions to the NLSF equations have been obtained for the \nnonlinear terms being arbitrary functions of the invariant \n$I = S^2$ and $J = P^2$, where $S=\\bar \\psi \\psi$ and \n$P= i \\bar \\psi \\gamma^5 \\psi$ are the real bilinear forms of spinor \nfield, for B-I space-time. Equations with power nonlinearity in  \nspinor field Lagrangian $L_N = \\lambda S^n$, where $\\lambda$ is the \ncoupling constant, have been thoroughly studied. In this case it is shown \nthat equations mentioned possess solutions both regular and singular at \nthe initial moment of time for $n>2$ . Singularity remains  absent  for  \nthe case of field system with broken dominant energy condition. It is also \nshown that if in the NLSF equation the massive parameter \n$m \\ne 0$ and $n\\ge 2$ then at $t \\to \\infty$ isotropization of  \nB-I space-time expansion takes place, while for $m=0$ the \nexpansion is anisotropic. Properties of solutions to the spinor field \nequation for $1<n<2$ and $0<n<1$ we also studied. It was found that in \nthese cases there does not exist solution that is initially regular. \nAt $t \\to \\infty$ the isotropization process of B-I\nspace-time takes place both for $m \\ne 0$ and for $m = 0$. In case of\nnonlinear term $L_N = \\lambda P^{2n}$, we found the solutions those are\ninitially singular and the isotropization process of B-I space-time\ndepends on the choice of $n$. For $L_N = \\lambda (I \\pm J)^{n}$ \nwe obtained the solutions those may be initially singular or\nregular depends on the sign of coupling constant $\\lambda$, but \nisotropization process depends on the value of power $n$. It is also \nshown that the results remain unchanged even in the case when the B-I \nspace-time is filled with perfect fluid.\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRelating the diameter of a polyhedron to its dimension and its number of facets is a classical topic in optimization. The {\\em combinatorial diameter} of a polyhedron is the maximum length of a shortest path between any two vertices in the graph (or $1$-skeleton) of the polyhedron. The famous Hirsch conjecture \\cite{d-63} claimed a bound of $f-d$ on the combinatorial diameter of any $d$-dimensional polyhedron with $f$ facets. It was first disproved for unbounded polyhedra \\cite{kw-67} and in a stronger setting requiring monotone walks \\cite{ToddExample} and only disproven much later for polytopes \\cite{msw-15,s-11}, i.e., bounded polyhedra. \nToday, the arguably most important open question in the field is the {\\em polynomial Hirsch conjecture}, which asks whether there is a polynomial bound on the diameter in terms of $f$ and $d$. Note that the existence of a strongly polynomial pivot rule for the Simplex method would require this conjecture to be true. The same holds for a polynomial version of the {\\em monotone Hirsch conjecture}, which concerns edge walks that are strictly decreasing and lead to a minimal vertex for some linear objective function.\n\nTo approach these long-standing questions, a number of abstractions and generalizations of edge walks on the $1$-skeleton have been introduced (see, e.g., \\cite{cs-22,ContinuousHirsch,ehrr-10,KimAbstraction,ks-10,s-13} and references therein). We focus on the so-called {\\em circuit diameter} of polyhedra introduced in \\cite{bfh-14}. {\\em Circuits} are a classical topic in oriented matroid theory  \\cite{oxley-06}, and correspond to the elementary vectors as introduced in \\cite{r-69}. In this work, we study whether the original counterexamples to the Hirsch conjecture and the monotone Hirsch conjecture can be transferred to the circuit setting. We begin with some necessary background in Section \\ref{sec:conjecture} and explain our contributions in Section \\ref{sec:contributions}. \n\n\\subsection{Circuit Diameters and the Circuit Diameter Conjecture}\\label{sec:conjecture}\n\nWe follow \\cite{bv-17,bv-22,dknv-22,dhl-15} for formal definitions and the important concepts. \n\n\\begin{definition}[Circuits]\nFor a rational polyhedron $P = \\{ \\mathbf{x} \\in \\mathbb{R}^n \\colon A \\mathbf{x} = \\mathbf{b}, B \\mathbf{x} \\leq \\mathbf{d} \\}$, the set of circuits of $P$, denoted $\\mathcal{C}(A,B)$, consists of all vectors $\\mathbf{g} \\in \\ker(A) \\setminus \\{ \\mathbf{0} \\}$, normalized to have coprime integer components, for which $B \\mathbf{g}$ is support-minimal in the set $\\{ B \\mathbf{x} \\colon \\mathbf{x} \\in \\ker(A) \\setminus \\{ \\mathbf{0} \\}\\}$.\n\\end{definition}\n\nThe set of circuits consists of all potential edge directions of the polyhedron as the right hand sides $\\mathbf{b}$ and $\\mathbf{d}$ vary \\cite{g-75}. In particular, it contains the set of all actual edge directions. Thus, a {\\em circuit walk}, referring to a sequence of maximal steps along circuits, is a direct generalization of an edge walk.\n\n \\begin{definition}[Circuit Walk]\\label{def:circuitwalk}\n Let $P = \\{ \\mathbf{x} \\in \\mathbb{R}^n \\colon A \\mathbf{x} = \\mathbf{b}, B \\mathbf{x} \\leq \\mathbf{d} \\}$ be a polyhedron. For two vertices $\\mathbf{v}_1$ and $\\mathbf{v}_2$ of $P$, we call a sequence $\\mathbf{v}_1 = \\mathbf{y}_0,\\dots,\\mathbf{y}_k = \\mathbf{v}_2$ a circuit walk of length $k$ from $\\mathbf{v}_1$ to $\\mathbf{v}_2$ in $P$ if, for all $i = 1,\\dots,k$, \n \n \\begin{myenumerate}\n     \\item $\\mathbf{y}_i \\in P$,\n     \\item $\\mathbf{y}_{i} = \\mathbf{y}_{i-1} +\\alpha_i \\mathbf{g}_i$ for some $\\mathbf{g}_i \\in \\mathcal{C}(A,B)$ and $\\alpha_i > 0$, and\n     \\item $\\mathbf{y}_{i-1} + \\alpha \\mathbf{g}_i \\notin P$ for all $\\alpha > \\alpha_i$.\n        \\label{def:circuitwalk-3}\n \\end{myenumerate}\n \\end{definition}\n \n We define the {\\em circuit diameter} of a polyhedron $P$ as the maximum length of a shortest circuit walk between any pair of vertices of $P$. Clearly, the circuit diameter is a lower bound on the combinatorial diameter of $P$. Note that, unlike edge walks, circuit walks are not necessarily \\emph{reversible}; the number of steps required to walk from $\\mathbf{v}_1$ to $\\mathbf{v}_2$ may not be the same as from $\\mathbf{v}_2$ to $\\mathbf{v}_1$. We use $\\Delta(f,d)$ to denote the maximum circuit diameter of any $d$-dimensional polyhedron with $f$ facets.  The circuit analogue of the Hirsch conjecture, the {\\em circuit diameter conjecture} \\cite{bfh-14}, asks whether $\\Delta(f,d)\\leq f-d$ and is open. So is its monotone variant: \nwe call a circuit walk $\\mathbf{y}_0,\\dots,\\mathbf{y}_k$ {\\em monotone} with respect to a linear objective function $\\mathbf{c}$ if the sequence $(\\mathbf{c}^\\top \\mathbf{y}_i)_{i=0,\\dots,k}$ is strictly decreasing. \nThe {\\em monotone circuit diameter} of a polyhedron $P$ is defined as the maximum length of a shortest monotone circuit walk from any vertex of $P$ to a vertex minimizer of $\\mathbf{c}$ across all possible choices of objective function $\\mathbf{c}$. \n\nCircuit diameters providing lower bounds on combinatorial diameters is just one of several reasons for interest in the study of the circuit analogues of the Hirsch conjecture and its variants. A resolution of the circuit diameter conjecture would reveal some information as to {\\em why} the Hirsch bound of $f-d$ is violated in the combinatorial setting \\cite{bdf-16,bsy-18}. More specifically, an affirmative answer to the circuit diameter conjecture implies that it is the restriction from circuit to edge steps that causes the violation. On the other hand, if not even the circuit diameter satisfies the Hirsch bound, the reason for this would be the maximality of steps in \\cref{def:circuitwalk}\\cref{def:circuitwalk-3}: if the step lengths are not required to be maximal, the so-called {\\em conformal sum property} \\cite{bk-84,r-69,z-95} guarantees the existence of a walk of at most $f-d$ circuit steps between any pair of vertices  (see also \\cref{sec:prelim}). \nFinally, circuit diameters are intimately related to the efficiency of {\\em circuit augmentation schemes} to solve linear programs \\cite{bv-19b,bv-22,dhl-15,env-21,gdl-14,gdl-15}. \n\nDespite results on circuit diameters for some polyhedra in combinatorial optimization (see, e.g., \\cite{bfh-16,kps-17}), and general upper bounds involving the so-called circuit imbalance \\cite{dknv-22,env-21} or the input bit-size \\cite{dks-22}, not much is known about the potential validity of the circuit diameter conjecture. It was shown in \\cite{bsy-18} that $\\Delta (8, 4) = 4$. In particular, this holds for the Klee-Walkup polyhedron in the original counterexample to the unbounded Hirsch conjecture \\cite{kw-67}: an unbounded $4$-dimensional polyhedron with $8$ facets and combinatorial diameter $5 >8-4$, but circuit diameter at most $4$ \\cite{sy-15}. The question that motivates our work is whether any of the other known counterexamples to variants of the Hirsch conjecture may be counterexamples to the respective circuit analogues. \n\n\n\\subsection{Contributions}\\label{sec:contributions}\n\nWe study four polytopes -- more precisely, so-called {\\em spindles} -- that appear in the construction of (monotone) Hirsch counterexamples: the $5$-dimensional spindles $S^{48}_5$ from \\cite{s-11} and $S^{25}_5$, $S^{28}_5$ from \\cite{msw-15} (named for their number of facets $f=48$ or $f=25,28$, respectively) that are the basis for a construction of counterexamples to the bounded Hirsch conjecture, as well as the $4$-dimensional spindle $M_4$ from \\cite{ToddExample} underlying the original counterexample for the monotone Hirsch conjecture. As our main contribution, we show that neither of these polytopes can lead to counterexamples for the (monotone) circuit diameter conjecture. We do so through a combination of theoretical arguments and computations. \n\nRecall that a spindle is a polytope with two distinguished vertices $\\mathbf{u}$ and $\\mathbf{v}$ such that each facet is incident to either $\\mathbf{u}$ or $\\mathbf{v}$. We call $\\mathbf{u}$ and $\\mathbf{v}$ the \\emph{apices} of the spindle. The {\\em length} of a spindle denotes the combinatorial distance of $\\mathbf{u}$ and $\\mathbf{v}$. A spindle can be defined as the intersection of two pointed cones emanating from $\\mathbf{u}$ and $\\mathbf{v}$ such that each apex is in the strict interior of the opposite cone.\n\nWe are interested in bounding the {\\em circuit length}, referring to the maximum length of a shortest circuit walk from one apex of the spindle to the other one. In \\cref{sec:prelim}, we define sign-compatible circuit walks, bound their length, and prove sufficient conditions for their existence on spindles in \\cref{thm:spindle-containment}. We then apply these results to the spindles $S^{25}_5,S^{28}_5,S^{48}_5$ and $M_4$ in \\cref{sec:counterexamples}.\n\nThe bounded Hirsch counterexamples in \\cite{msw-15,s-11} are constructed from highly degenerate $5$-dimensional spindles $S^{25}_5,S^{28}_5,S^{48}_5$ of length $6$. At the core of Santos' construction is the following observation: from a $d$-dimensional spindle with $f$ facets and length greater than $d$, one can obtain a $(d{+}1)$-dimensional spindle with $f+1$ facets which has length greater than $d+1$. By doing this $f-2d$ times, one obtains an $(f{-}d)$-dimensional spindle with $2f-2d$ facets whose length exceeds $f-d$. Santos applied this construction to $S^{48}_5$, leading to a $43$-dimensional spindle with $86$ facets and length at least $44$. The fact that $S^{28}_5$ and $S^{25}_5$ have fewer facets leads to counterexamples in lower dimensions $23$ and $20$, respectively.\n\nIn \\cref{sec:Santos}, we prove that $S^{25}_5,S^{28}_5$, and $S^{48}_5$ have circuit length at most $5$. In doing so, we show that the above construction does not lead to a counterexample for polytopes in the circuit setting (\\cref{cor:santos-width,thm:weibel-length}). \n   \nIn \\cref{sec:Todd}, we show that Todd's counterexample to the monotone Hirsch conjecture \\cite{ToddExample} is not a counterexample to the monotone circuit diameter conjecture. The Todd polytope $M_4$ is a $4$-dimensional spindle with $8$ facets. Todd found a linear objective function such that any monotone path from one apex to the other is of length at least $5$. In \\cref{sec:Todd}, we show in \\cref{thm:todd} that $M_4$ has monotone circuit diameter $4$. To do this, we study all orientations of the graph of $M_{4}$ induced by linear objective functions finding that only five contradict the monotone Hirsch bound for the combinatorial diameter.\n\n\nOne of the main challenges in studying circuit diameters is the fact that the circuit diameter of a polyhedron, unlike the combinatorial diameter, depends on its geometry. In particular, there can be two realizations of the same polyhedron with different circuit diameters (see, for example, \\cite{sy-15b}). We stress that our results in \\cref{sec:counterexamples} only apply to the specific realizations of the spindles in the literature (and some careful, mild perturbations). It will remain open whether {\\em all} realizations of these polytopes satisfy the circuit diameter bound. \n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nOur main tool for analyzing the spindles in \\cref{sec:counterexamples} is the concept of \\emph{sign-compatible} circuit walks.\nTwo vectors $\\mathbf{x},\\mathbf{y} \\in \\mathbb{R}^d$ are \\emph{sign-compatible} if $\\mathbf{x}_i \\mathbf{y}_i \\ge 0$ for all $i=1,\\dots,d$. Let $P = \\{ \\mathbf{x} \\in \\mathbb{R}^d \\colon B \\mathbf{x} \\le \\mathbf{d} \\}$ be a polyhedron for a matrix $B \\in \\mathbb{R}^{m \\times d}$ with rows $\\mathbf{b}_i \\in \\mathbb{R}^d$ for $i=1,\\dots,m$.\nWe call a circuit walk on $P$ with steps $\\mathbf{g}_j$ and step lengths $\\alpha_j$ for $j=1,\\dots,k$ a \\emph{sign-compatible circuit walk} if all $B \\mathbf{g}_j$ for $j=1,\\dots,k$ are pairwise sign-compatible.\nSuch walks are special cases of conformal sums of circuits, which correspond to sign-compatible circuit walks without the maximal step requirement in \\cref{def:circuitwalk}\\cref{def:circuitwalk-3}. In this weaker setting, the well-known \\emph{conformal sum property} \\cite{bk-84,r-69,z-95} guarantees that for any given pair of vertices $\\mathbf{u},\\mathbf{v}$ of a $d$-dimensional polyhedron, their difference $\\mathbf{v}-\\mathbf{u}$ can be written as a conformal sum of at most $d$ circuits. This contrasts with the situation for sign-compatible circuit walks (with maximal steps), which may not exist at all \\cite{bdf-16}. When they do, however, their length is at most $d$: note that once a sign-compatible walk enters a new facet, it may not leave it again. Since each step of the walk is maximal and must therefore enter a new facet, the number of steps is at most $d$, and thus satisfies the bound $f-d\\geq d$ for a spindle. \n\nMoreover, sign-compatible circuit walks are monotone for any linear objective function that is uniquely minimized at the ending vertex of the walk. To see this, let $P$ be a polyhedron given by $P = \\{ \\mathbf{x} \\in \\mathbb{R}^d \\colon B \\mathbf{x} \\le \\mathbf{d} \\}$. For the sake of simplicity, we only consider full-dimensional polyhedra here. We define the hyperplane arrangement $\\mathcal{H}(B) = \\bigcup_{i=1}^{m} \\{\\mathbf{x} \\in \\mathbb{R}^d \\colon \\mathbf{b}_i^\\top \\mathbf{x} = 0\\}$. Following \\cite{bv-17}, we call this arrangement the {\\em elementary arrangement} of $P$. \nSince the set of circuits of $P$ consists precisely of the (normalized) directions of the extreme rays of $\\mathcal{H}(B)$ (see \\cite{bv-17}), we obtain the following equivalent characterization of sign-compatibility:\n\n\\begin{lemma} \\label{prop:sign-comp-char}\nLet $P = \\{ \\mathbf{x} \\in \\mathbb{R}^d \\colon B \\mathbf{x} \\le \\mathbf{d} \\}$ be a polyhedron in $\\mathbb{R}^d$ and let $\\mathbf{u}$ and $\\mathbf{v}$ be two vertices of $P$. Denote by $C$ the minimal face of the elementary arrangement $\\mathcal{H}(B)$ containing $\\mathbf{v}-\\mathbf{u}$.\nFor a circuit walk $\\mathbf{u} = \\mathbf{y}_0,\\dots,\\mathbf{y}_k = \\mathbf{v}$ from $\\mathbf{u}$ to $\\mathbf{v}$ in $P$ with steps $\\mathbf{g}_j \\in \\mathcal{C}(B)$ for $j=1,\\dots,k$, the following statements are equivalent:\n\\begin{myenumerate}\n    \\item The walk $\\mathbf{y}_0,\\dots,\\mathbf{y}_k$ is sign-compatible. \\label{prop:sign-comp-char-1}\n    \\item $\\mathbf{g}_j \\in C$ for all $j=1,\\dots,k$. \\label{prop:sign-comp-char-2}\n    \\item The walk $\\mathbf{y}_0,\\dots,\\mathbf{y}_k$ is monotone for any linear objective function $\\mathbf{c} \\in \\mathbb{R}^d$ uniquely minimized over $-C$ at the origin $\\mathbf{0}$.  \\label{prop:sign-comp-char-3}\n\\end{myenumerate}\n\\end{lemma}\n\\begin{proof}\nTo see that \\cref{prop:sign-comp-char-1} and \\cref{prop:sign-comp-char-2} are equivalent, note that the set of all vectors in $\\mathbb{R}^d$ whose products with $B$ are pairwise sign-compatible and sign-compatible with $B(\\mathbf{v}-\\mathbf{u})$ is a polyhedral cone (see \\cite{bdf-16}) and coincides with the minimal face of $\\mathcal{H}(B)$ containing $\\mathbf{v}-\\mathbf{u}$.\n\nLet us now prove the equivalence of \\cref{prop:sign-comp-char-2} and \\cref{prop:sign-comp-char-3}. \nFrom standard polyhedral theory, $\\mathbf{c}$ is uniquely minimized over $-C$ at $\\mathbf{0}$ if and only if $-\\mathbf{c}$ is in the relative interior of the polar cone of $-C$, meaning that $\\mathbf{c}^\\top\\mathbf{x} < 0$ for all $\\mathbf{x} \\in C \\setminus \\{\\mathbf{0}\\}$.  Hence, if $\\mathbf{g}_{j} \\in C$ for all $j = 1, \\dots, k$ then $\\mathbf{c}^{\\top} \\mathbf{g}_{j} < 0$ for all such $j$ meaning that the path is monotone for all choices of $\\mathbf{c}$. Thus, \\cref{prop:sign-comp-char-2} implies \\cref{prop:sign-comp-char-3}. For the other direction, recall that polar duality is an involution, so $\\mathbf{x} \\in C$ if and only if, for all $\\mathbf{c}$ minimized at $\\mathbf{0}$ on $-C$, $\\mathbf{c}^\\top x < 0$. Thus, if the walk is monotone for all linear objective functions uniquely minimized over $-C$ at $\\mathbf{0}$, each step $\\mathbf{g}_{j}$ must be contained in $C$ meaning that \\cref{prop:sign-comp-char-3} implies \\cref{prop:sign-comp-char-2}.  \n\\qed\n\\end{proof}\n\nA key ingredient to our proofs is that we are able to guarantee the existence of sign-compatible circuit walks on spindles satisfying some restrictions. Recall that a spindle is the intersection of two pointed cones, one emanating from each apex.\nWe make the following key observation: one can always find sign-compatible walks between the two apices of a spindle formed by repeating the same cone twice.\n\n\\begin{lemma} \\label{lem:spindle-same-cone} \nLet $P \\subset \\mathbb{R}^d$ be a spindle with apices $\\mathbf{u}$ and $\\mathbf{v}$, given by $(C + \\mathbf{u}) \\cap (-C + \\mathbf{v})$ for a pointed cone $C$ with a unique vertex at the origin $\\mathbf{0}$. Then there is a sign-compatible circuit walk of length at most $d$ from $\\mathbf{u}$ to $\\mathbf{v}$ in $P$.\n\\end{lemma}\n\n\\begin{proof}\nWe proceed by induction on $d$. If $d=1$, then $P$ is an edge with endpoints $\\mathbf{u}$ and $\\mathbf{v}$ and the statement is clear. \nNow suppose that $d \\ge 2$. \nSince $C$ is a region of the elementary arrangement of $P$ and $\\mathbf{v}-\\mathbf{u}$ is in the interior of $C$, any circuit walk from $\\mathbf{u}$ to $\\mathbf{v}$ that only walks along directions of extreme rays of $C$ will be sign-compatible by \\cref{prop:sign-comp-char}. We construct such a walk as follows.\n\nStarting at $\\mathbf{u}$, we walk along any edge of $P$ incident with $\\mathbf{u}$ to some adjacent vertex $\\mathbf{u}'$. Then $\\mathbf{u}'$ and $\\mathbf{v}$ are contained in a common facet $F$ of $P$, which has to be of the form $F = P \\cap (-C' + \\mathbf{v})$  for some facet $C'$ of $C$. Now consider the spindle $P' = (C' + \\mathbf{u}') \\cap (-C' + \\mathbf{v})$. Since $C' + \\mathbf{u}' \\subset C + \\mathbf{u}$ and $F = (C + \\mathbf{u}) \\cap (-C' + \\mathbf{v})$, we have that $P' \\subset F$. By the induction hypothesis and \\cref{prop:sign-comp-char}, there is a circuit walk on $P'$ of length at most $d-1$ from $\\mathbf{u}'$ to $\\mathbf{v}$ with each step being in the direction of one of the extreme rays of $C'$ (and, thus, of $C$). Since each direction followed is in $C'$ and each step is maximal in $P'$, each step length must therefore be bounded by a facet-defining inequality for the cone $-C' + \\mathbf{v}$, which is a facet of $-C + \\mathbf{v}$ by hypothesis. So each step must also be maximal in $P$. Thus, appending this walk to the first edge step from $\\mathbf{u}$ to $\\mathbf{u'}$ yields a sign-compatible circuit walk from $\\mathbf{u}$ to $\\mathbf{v}$. \\qed\n\\end{proof}\n\nWe are now ready to prove the main result of this section.\n\n\\begin{theorem} \\label{thm:spindle-containment}\nLet $P \\subset \\mathbb{R}^d$ be a spindle with apices $\\mathbf{u}$ and $\\mathbf{v}$, given by $(C + \\mathbf{u}) \\cap (-D + \\mathbf{v})$ for two pointed cones $C$ and $D$  with a unique vertex at the origin $\\mathbf{0}$. Suppose that $D \\subseteq C$. Then there is a sign-compatible circuit walk of length at most $d$ from $\\mathbf{u}$ to $\\mathbf{v}$ in $P$.\nFor any linear objective function $\\mathbf{c} \\in \\mathbb{R}^d$ uniquely minimized over $P$ at $\\mathbf{v}$, this walk is monotone.\n\\end{theorem}\n\\begin{proof}\nDefine $P' = (D + \\mathbf{u}) \\cap (-D + \\mathbf{v})$. Since $P$ is a spindle, $\\mathbf{v}-\\mathbf{u}$ is in the interior of $C \\cap D = D$ by hypothesis. It follows then that $P'$ is a spindle with apices $\\mathbf{u}$ and $\\mathbf{v}$. Since $D \\subseteq C$, we further have that $P' \\subseteq P$. By \\cref{lem:spindle-same-cone}, there is a sign-compatible circuit walk of length at most $d$ steps from $\\mathbf{u}$ to $\\mathbf{v}$ in $P'$. Each step walks along the direction of some extreme ray of $D$ by \\cref{prop:sign-comp-char}. Therefore, the only facet-defining inequalities of $P'$ that can become tight at each step are facet-defining inequalities for the cone $-D + \\mathbf{v}$, which are facet-defining for $P$ as well. So each step is maximal in $P$ and the walk is therefore a circuit walk in $P$.\n\nSign-compatibility for $P$ follows from \\cref{prop:sign-comp-char} by noting that, since $D \\subseteq C$, $D$ is a region of the elementary arrangement which contains $\\mathbf{v}-\\mathbf{u}$ in its interior. For monotonicity, note that any linear objective function $\\mathbf{c}$ for which $\\mathbf{v}$ is the unique minimizer over $P$ is uniquely minimized at $\\mathbf{0}$ over $P - \\mathbf{v} \\subset -D$ and, hence, also over $-D$. By \\cref{prop:sign-comp-char}, the walk must therefore be monotone for any such $\\mathbf{c}$. \\qed\n\\end{proof}\n\n\n\\section{Circuit Walks on Hirsch Counterexamples}\\label{sec:counterexamples}\n\nWe are now ready to discuss the circuit lengths of spindles $S^{48}_5$, $S^{28}_5$, and $S^{25}_5$ and the monotone circuit diameter of $M_4$. In Section \\ref{sec:Santos}, we turn to the spindles used in the disproof of the bounded Hirsch conjecture. In Section \\ref{sec:Todd}, we turn to the spindle $M_4$ used in the disproof of the monotone Hirsch conjecture. \n\n\\subsection{Bounded Hirsch Counterexamples}\\label{sec:Santos}\n\nSantos' original counterexample in \\cite{s-11} is constructed from a $5$-dimensional spindle with $48$ facets and length $6$. We denote this spindle, as defined in \\cite[Theorem 3.1]{s-11}, by $S_5^{48}$. It is given by the description $S_5^{48} = \\{ \\mathbf{x} \\in \\mathbb{R}^5 \\colon A^+ \\mathbf{x} \\le \\mathbf{1}, A^- \\mathbf{x} \\le \\mathbf{1} \\}$ where $\\mathbf{1}$ denotes the all-one vector and $A^+$ and $A^-$ are the matrices\n\\[\n    A^+ = \\begin{pmatrix}\n    1 & \\pm 18 & 0 & 0 & 0 \\\\\n    1 & 0 & \\pm 18 & 0 & 0 \\\\\n    1 & 0 & 0 & \\pm 45 & 0 \\\\\n    1 & 0 & 0 & 0 & \\pm 45 \\\\\n    1 & \\pm 15 & \\pm 15 & 0 & 0 \\\\\n    1 & 0 & 0 & \\pm 30 & \\pm 30 \\\\\n    1 & 0 & \\pm 10 & \\pm 40 & 0 \\\\\n    1 & \\pm 10 & 0 & 0 & \\pm 40\n    \\end{pmatrix} \n    %\n    \\text{ and }\n    %\n    A^- = \\begin{pmatrix}\n    -1 & 0 & 0 & 0 & \\pm 18 \\\\\n    -1 & 0 & 0 & \\pm 18 & 0 \\\\\n    -1 & \\pm 45 & 0 & 0 & 0 \\\\\n    -1 & 0 & \\pm 45 & 0 & 0 \\\\\n    -1 & 0 & 0 & \\pm 15 & \\pm 15 \\\\\n    -1 & \\pm 30 & \\pm 30 & 0 & 0 \\\\\n    -1 & \\pm 40 & 0 & \\pm 10 & 0 \\\\\n    -1 & 0 & \\pm 40 & 0 & \\pm 10\n    \\end{pmatrix}\n\\]\nwith $24$ rows each. The two apices of $S_5^{48}$ are $\\mathbf{v}^+ = (1,0,0,0,0)$ and $\\mathbf{v}^- = (-1,0,0,0,0)$.\nThe spindle $S_5^{48}$ can be equivalently written as $S_5^{48} = (C^+ + \\mathbf{v}^+) \\cap (-C^- + \\mathbf{v}^-)$ for the two cones $C^+ = \\{ \\mathbf{x} \\in \\mathbb{R}^5 \\colon A^+ \\mathbf{x} \\le \\mathbf{0} \\}$ and $C^- = \\{ \\mathbf{x} \\in \\mathbb{R}^5 \\colon A^- \\mathbf{x} \\ge \\mathbf{0} \\}$. \n\n\\begin{theorem} \\label{cor:santos-width}\nThe circuit length of $S^{48}_5$ is $2$.\n\\end{theorem}\n\\begin{proof}\nFirst observe that, by the symmetry in the constraint matrices $A^{+}$ and $A^{-}$, for all $\\mathbf{g} \\in C^+$, the vector $\\mathbf{g}'$ obtained from $\\mathbf{g}$ by flipping the signs of all but the first component is contained in $C^+$, too (and similarly for $C^-$). \n\nLet $\\mathbf{g} = (-360,8,4,4,7)$ and $\\mathbf{g}' = (-360,-8,-4,-4,-7)$. By a direct computation, $\\mathbf{g}$ generates an extreme ray of the cone $C^+ \\cap C^-$. Since $C^+ \\cap C^-$ is a region of the elementary arrangement of $S_5^{48}$, the vector $\\mathbf{g}$ is a circuit. It is further contained in a facet of $C^+$ and a facet of $C^-$. By symmetry, all of the above also holds for $\\mathbf{g}'$.\n\nNow let $K$ be the cone spanned by $\\mathbf{g}$ and $\\mathbf{g}'$ and define $P = (K + \\mathbf{v}^+) \\cap (-K + \\mathbf{v}^-)$. Since $\\mathbf{g} + \\mathbf{g}' \\in K$ is a multiple of $\\mathbf{v}^+ - \\mathbf{v}^-$, $P$ is a nonempty spindle with apices $\\mathbf{v}^+$ and $\\mathbf{v}^-$. It further follows from $K \\subset C^+ \\cap C^-$ that $P \\subset S_5^{48}$.\nNote that $P$ is a parallelogram with two edge walks of length $2$ between $\\mathbf{v}^+$ and $\\mathbf{v}^-$. Since the edges of $P$ are generated by circuits of $S_5^{48}$ and each edge is contained in a facet of $S_5^{48}$, both edge walks are circuit walks in $S_5^{48}$.\nFurther, the vector $(\\mathbf{v}^+ - \\mathbf{v}^-)\/2 = (1,0,0,0,0)$ is not contained in any facet of $C^+$ or $C^-$ and thus cannot be a circuit of $S_5^{48}$. Hence, the circuit length of $S_{5}^{48}$ is $2$. \\qed\n\\end{proof}\n\nWe stress that the above bound heavily relies on the peculiar symmetry in the description of $S_5^{48}$. However, if $S_5^{48}$ is perturbed carefully, we may apply \\cref{lem:spindle-same-cone} to still obtain a bound of $5$ on the circuit length of the perturbed spindle. More precisely, suppose that the two steps $\\mathbf{g}$ and $\\mathbf{g}'$ from the proof of \\cref{cor:santos-width} are still circuits after a perturbation, and suppose that they can be used to walk from one of the apices to some point $\\mathbf{y}$ on the boundary of the perturbed spindle. Provided that the perturbation is sufficiently small, $\\mathbf{y}$ will be in a face of dimension at most $3$, close to the other apex of the perturbed spindle. We can then embed into this face a spindle that satisfies the requirements of \\cref{lem:spindle-same-cone} and has $\\mathbf{y}$ as one apex. By \\cref{lem:spindle-same-cone}, we can reach the other apex from $\\mathbf{y}$ within three more steps.\n\nSantos' original example constructed from $S_5^{48}$ is not the lowest-dimensional bounded Hirsch counterexample known to date. In \\cite{msw-15}, Matschke, Santos, and Weibel gave two smaller counterexamples, both of which are constructed from 5-dimensional spindles of length 6 with $28$ and $25$ facets, respectively. The first one, $S^{28}_5$, from \\cite[Corollary 2.9]{msw-15} has the same symmetries as $S_5^{48}$ and can be shown to have circuit length at most 2 by the same argument as in \\cref{cor:santos-width}. \nThe second spindle with 25 facets from \\cite[Theorem 2.14]{msw-15} is given by $S_5^{25} = \\{ \\mathbf{x} \\in \\mathbb{R}^5 \\colon A^+ \\mathbf{x} \\le \\mathbf{1}, A^- \\mathbf{x} \\le \\mathbf{1} \\}$ where\n\\begingroup\n\\renewcommand*{\\arraystretch}{1.1}\n\\[\n    A^+ = \\begin{pmatrix}\n    1 & 0 & 0 & -20 & -4\\\\\n    1 & 0 & 0 & 20 & -4\\\\\n    1 & 0 & 0 & 21 & -7\\\\\n    1 & 0 & 0 & -21 & -7\\\\\n    1 & 0 & 0 & 16 & -15\\\\\n    1 & 0 & 0 & -16 & -15\\\\\n    1 & 0 & 0 & 0 & 32\\\\\n    1 & 0 & 0 & 0 & -32\\\\\n    1 & \\frac{3}{50} & -\\frac{1}{25} & 0 & -30\\\\\n    1 & -\\frac{3}{50} & -\\frac{1}{25} & 0 & 30\\\\\n    1 & \\frac{3}{1000} & \\frac{7}{1000} & 0 & -\\frac{159}{5}\\\\\n    1 & -\\frac{3}{1000} & \\frac{7}{1000} & 0 & \\frac{159}{5}\n    \\end{pmatrix} \n    %\n    \\text{ and }\n    %\n    A^- = \\begin{pmatrix}\n    -1 & 60 & 0 & 0 & 0\\\\\n    -1 & 8 & -30 & 0 & 0\\\\\n    -1 & 0 & -33 & 0 & 0\\\\\n    -1 & -2 & -32 & 0 & 0\\\\\n    -1 & -55 & 0 & 0 & 0\\\\\n    -1 & -34 & 36 & 0 & 0\\\\\n    -1 & 0 & 76 & 0 & 0\\\\\n    -1 & 44 & 34 & 0 & 0\\\\\n    -1 & -20 & 0 & \\frac{1}{5} & -\\frac{1}{5}\\\\\n    -1 & \\frac{2999}{50} & 0 & -\\frac{3}{25} & -\\frac{1}{5}\\\\\n    -1 & \\frac{299999}{5000} & 0 & 0 & \\frac{1}{100}\\\\\n    -1 & -\\frac{549}{10} & 0 & \\frac{1}{5000} & \\frac{1}{800}\\\\\n    -1 & -54 & 0 & \\frac{1}{500} & -\\frac{1}{80}\\\\\n    \\end{pmatrix}.\n\\]\n\\endgroup\n\n\nUsing Polymake \\cite{polymake} and Python, we computed circuit walks of length $5$ between the apices $\\mathbf{v}^+ = (1,0,0,0,0)$ and $\\mathbf{v}^- = (-1,0,0,0,0)$ of $S_5^{25}$. An initial computation yielded that $S^{25}_5$ has $17454$ circuits, which is far too large for a brute force search across all length $5$ circuit walks. To find our length $5$ walks, we restricted to a suitably chosen subset of circuits, for which an enumerative approach was computationally feasible.\n\n\\begin{theorem} \\label{thm:weibel-length} \nThe circuit length of $S^{25}_5$ is at most $5$.\n\\end{theorem}\n\\begin{proof}\nWe exhibit explicit circuit walks between the apices of the $S^{25}_{5}$. We restricted to circuit walks using the rays of $C^{+} \\cap C^{-}$, $C^{+}$, and $C^{-}$. This restriction yielded $132$ pairs of circuits $\\pm \\mathbf{g}$. From this restricted set, we found the following circuit walks. \nFrom $\\mathbf{v}^{+}$ to $\\mathbf{v}^{-}$, the circuits used are: \n\\begin{align*} \n    &(-53592, -609, 1624, 2112, -1320), (-262276, 4060, -3451, -10336, -6460), \\\\\n    &(-339416, -4263, -4466, 13376, -8360), (-26752, -336, -352, -83979, 83381), \\\\\n    &\\text{and } (-83600, -1050, -1100, 3605210, -2060021).\n\\end{align*} \nFrom $\\mathbf{v}^{-}$ to $\\mathbf{v}^{+}$, the circuits used are: \n\\begin{align*}\n   &(53592, 609, -1624, -2112, 1320), (7197600, 118440, -208336, -404865, -224925),\\\\\n    &(5280, -96, 56, 297, -165), (160, -2500, 4000, 9, -5),\\\\\n    & \\text{and } (4320, 5500, -1500, 243, -135).\n\\end{align*}\n\nWe note that for both walks, the first three steps are circuits from $C^+ \\cap C^-$ and the final two are edge directions of the cones $C^-$ and $C^+$, respectively.\n\\qed\n\\end{proof}\n\n\n\\subsection{Todd's Monotone Hirsch Counterexample}\\label{sec:Todd}\n\nThe Todd polytope $M_{4}$ is given by $M_{4} = \\{ \\mathbf{x} \\in \\mathbb{R}^4 \\colon A \\mathbf{x} \\leq \\mathbf{b}, \\mathbf{x} \\geq \\mathbf{0} \\}$ where\n\\[ A = \\begin{pmatrix} 7 & 4 & 1 & 0\\\\ 4 & 7 & 0 & 1 \\\\ 43 & 53 & 2 & 5 \\\\ 53 & 43 & 5 & 2 \\end{pmatrix} \\text{ and } \\mathbf{b} = \\begin{pmatrix} 1 \\\\ 1 \\\\ 8 \\\\ 8 \\end{pmatrix}. \\]\n\nThe polytope has $8$ facets in dimension $4$. Consider the linear program $\\min \\{ (1,1,1,1)^\\top \\mathbf{x} \\colon \\mathbf{x} \\in M_{4} \\}$. Todd showed that the shortest monotone path from the vertex $(1,1,8,8)\/19$ to the optimum $\\mathbf{0}$ of this LP is of length at least $5$, a contradiction to the monotone Hirsch conjecture \\cite{ToddExample}.\n\nHowever, in the forty years since there has not been a detailed analysis of how many different orientations of $M_{4}$ have large monotone diameters. We will first address how rigid the selection of orientation is to obtain a counter-example to the monotone Hirsch conjecture. To do this computation, we first enumerate all orientations of the graph induced by a linear objective function and then we compute the monotone diameter using a breadth first search. In \\cite{edgotope}, the authors show that this set of orientations corresponds to the set of vertices of a zonotope they call the \\emph{edgotope}. To compute this zonotope for a polytope $P$ with vertices $V(P)$ and edges $E(P) = \\{(\\mathbf{u},\\mathbf{v}): \\mathbf{u}, \\mathbf{v} \\in V(P), \\mathbf{u} \\text{ is adjacent to } \\mathbf{v}\\}$, one computes the following:\n\\[EZ(P) = \\sum_{(\\mathbf{u},\\mathbf{v}) \\in E(P)} \\text{conv}(\\{\\mathbf{u},\\mathbf{v}\\}).\\]\nZonotopes are dual to hyperplane arrangements, so this statement is equivalent to the observation that the set of orientations are in bijection with the set of regions of the hyperplane arrangement \n\\[\\mathcal{H} = \\bigcup_{(\\mathbf{u},\\mathbf{v}) \\in E(P)} \\{ \\mathbf{x} \\colon (\\mathbf{u}-\\mathbf{v})^\\top \\mathbf{x} = 0\\}.\\] \nA region $R$ of $\\mathcal{H}$ is uniquely determined by its sign vector $\\mathbf{z} \\in \\{+,-\\}^{E(P)}$ with entry $\\mathbf{z}_{(\\mathbf{u},\\mathbf{v})}$ denoting whether $\\mathbf{c}^\\top(\\mathbf{u} - \\mathbf{v}) > 0$ or $\\mathbf{c}^\\top(\\mathbf{u} - \\mathbf{v}) < 0$ for each $(\\mathbf{u},\\mathbf{v}) \\in E(P)$ and all $\\mathbf{c} \\in R$. Equivalently, the sign vector determines whether $\\mathbf{c}^\\top \\mathbf{u} < \\mathbf{c}^\\top \\mathbf{v}$ or $\\mathbf{c}^\\top \\mathbf{v} > \\mathbf{c}^\\top \\mathbf{u}$ for all $(\\mathbf{u},\\mathbf{v}) \\in E(P)$ and $\\mathbf{c} \\in R$, which uniquely determines the orientation of the polytope.\n\nTo enumerate all the regions of the edgotope arrangement for $M_{4}$, we first found the graph $G(M_{4})$. This graph has precisely $40$ edges, which leads to an arrangement of $40$ hyperplanes. We used Sage \\cite{sage} to compute the set of regions of this arrangement and found that there are exactly $7112$ regions and therefore $7112$ orientations. We enumerated the possible oriented graphs of $M_{4}$ for those orientations, and only five have diameters that contradict the monotone Hirsch conjecture. Five representatives of choices of $\\mathbf{c}$ for which there is a bad orientation are $(1, 1, 1, 1),$ $(10716, 13680, 3477, 4465),$ $(13680, 10716, 4465, 3477),$ $(912, 1824, 513, 817),$  and $(1824, 912, 817, 513)$. These four orientations other than $(1,1,1,1)$ of the graph of $M_{4}$ are new and only differ from the Todd orientation on one edge in the first two cases and on two edges in the final two cases. \n\nEach of those orientations have $\\mathbf{0}$ as the optimum and the only vertex of distance $5$ away is $(1,1,8,8)\/19$. In all of the bad orientations, $(1,1,8,8)\/19$ is not a maximizer of $\\mathbf{c}$. It follows then that $M_{4}$ is not a counterexample to Ziegler's strict monotone Hirsch conjecture, which asks whether the Hirsch bound is satisfied for paths from maxima to minima across all orientations (see Chapter $3$ of \\cite{z-95} for more details). Furthermore, there are $1832$ orientations for which $\\mathbf{0}$ is the unique sink, so even among those oriented graphs, a large diameter is rare. \n\nWhile these observations are interesting on their own, in our context it allows us to reduce the set of orientations for which we need to prove there is always a monotone circuit walk down to those with $\\mathbf{0}$ as a unique sink and coming from $(1,1,8,8)\/19$. Note that the cones for this spindle are given by $\\{ \\mathbf{x} \\in \\mathbb{R}^{4} \\colon \\mathbf{x} \\geq \\mathbf{0} \\}$ and $\\{ \\mathbf{x} \\in \\mathbb{R}^{4} \\colon A \\mathbf{x} \\leq \\mathbf{b} \\}$. With these observations, we may prove our main result:\n\n\\begin{theorem} \\label{thm:todd}\nThe monotone circuit diameter of $M_{4}$ is $4$. \n\\end{theorem}\n\\begin{proof}\nFrom our computations, for all orientations that do not have $\\mathbf{0}$ as the unique optimum, there is always a monotone edge walk from any starting vertex of length at most $4$ and therefore always a monotone circuit walk of length at most $4$. Furthermore, the only case for which the shortest monotone edge walk is of length $5$ is when the starting vertex is $(1,1,8,8)\/19$. Note that $\\mathbf{0}$ is the apex of the cone $\\{ \\mathbf{x} \\in \\mathbb{R}^4 \\colon \\mathbf{x} \\geq \\mathbf{0} \\}$ and $(1,1,8,8)\/19$ is the apex of the cone $\\{ \\mathbf{x} \\in \\mathbb{R}^{4} \\colon A \\mathbf{x} \\leq \\mathbf{b} \\}$.\nObserve also that, since the entries of $-A$ are all non-positive, we have that $ \\{ \\mathbf{x} \\in \\mathbb{R}^{4} \\colon \\mathbf{x} \\geq \\mathbf{0}\\} \\subseteq \\left\\{ \\mathbf{x} \\in \\mathbb{R}^{4} \\colon -A  \\mathbf{x} \\leq \\mathbf{0} \\right\\}.$ Hence, by \\cref{thm:spindle-containment}, there must always exist a monotone circuit walk of length at most $4$ from $(1,1,8,8)\/19$ to $\\mathbf{0}$ for any orientation for which $\\mathbf{0}$ is minimal. Therefore, the monotone circuit diameter of the Todd example $M_4$ is at most $4$. We may show by a direct computation that it is exactly $4$ by noting that $(1,1,8,8)\/19$ is not a linear combination of any $3$ circuits meaning that there is no circuit walk of length $3$ from $\\mathbf{0}$ to $(1,1,8,8)\/19$. \\qed\n\\end{proof}\n\nA natural concern with our argument is whether it breaks after perturbation. Since $M_{4}$ is simple, perturbation does not change the graph of the polytope. However, any sufficiently generic perturbation will increase the set of possible orientations, since for such a perturbation, the set of edge directions will not satisfy any nontrivial linear relations. However, we suspect that the set of orientations is always the same for any sufficiently small perturbation even if it is distinct from the set of orientations we started with. If that is this case, then one can extend our argument to allow for perturbation by evaluating the circuit diameter for those new orientations.\n\n\n\\subsection*{Acknowledgements}\n We would like to thank Nicholas Crawford, Jes\\'{u}s A. De Loera, Michael Wigal, and Youngho Yoo for insightful discussions.\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCausality is the process from which an event influences the outcome of another process. In real-world data, detecting causal links may be hard. An intuitive way to infer causality consists of measuring lagged correlation between the states of two variables. Although this approach is in line with what is humanly perceived as a cause-effect relationship, these lagged correlations may, in many cases, come from the influences of hidden variables. Furthermore, observing significant lagged correlations is not a necessary condition for the inference of a cause-effect relationship because such a measure can seem negligible in a highly non-linear and noisy system.\n\nOne of the early approaches to cause-effect detection is Granger causality~\\cite{granger1969investigating}. The idea of Granger causality is to assume that \\textit{B causes A} is equivalent to \\textit{B forecasts A}. Forecasting at that time was typically addressed with auto-regressive models. Therefore, Granger causality is able to detect causality only when a linear model is a good approximation to the dynamics of $A$ and $B$. Needless to say, most real-world systems are not linear.\n\nMore generally, $A$ can only be predicted from $B$ if a significant amount of information is transferred from $B$ to $A$. \nTransfer Entropy (TE)~\\cite{schreiber2000measuring} measures the information gain from the past of $B$ in predicting the future of $A$ after discounting of self-information from the past of $A$. TE is more general than Granger causality. Both are equivalent for auto-regressive models with Gaussian variables~\\cite{barnett2009granger}. While TE can be defined for continuous non-linear models, it becomes model-free when the states of  $A$ and $B$ take or are approximated by discrete values. If the original models are not genuinely discrete, the longer the time series, the more refined the discrete description of the original time series can be. This is known as symbolic TE~\\cite{staniek2008symbolic}, which provides an unbiased estimator of the model-free TE. However, when only finite time series are available,  the error of sample probabilities and thus of the sample TE increases with the chosen number of distinct discrete values. Such an irreducible error makes it difficult to distinguish a real signal from random noise fluctuations (for more spurious sources of TE, see \\cite{smirnov2013spurious}).\n\nTesting the significance of TE can be done with data shuffling or resampling data, which is slow. Direct testing was proposed in Ref.~\\cite{barnett2012transfer}, which introduced a null model for TE and shows that TE asymptotically converges to a chi-square distribution with 2 degrees of freedom for binary time series. A generalization of this test to more generic discrete time series for Markovian systems is found in Ref.~\\cite{kontoyiannis2016estimating}. Finally, TE between asynchronous time series was recently defined in Ref.~\\cite{harre2015entropy}.\n\nOur main contributions are as follows: first, we derive an asymptotic statistical test to compare the amount of TE from $B$ to $A$ and from $C$ to $A$, which allows us to significantly rank predictors pairs based on their respective information content. Secondly, we quantify the finite-size effects of the asymptotic tests by comparing their outcomes with those of bootstrap tests. Thirdly,  we use time-shifting of the target time series in an asynchronous setting so as to infer significant time delays and the delay after which no statistically significant TE or TE difference takes place. Third, we demonstrate the effectiveness of our methods by inferring a statistically validated lead-lag network of tick-by-tick price changes from a large database of financial data and by characterising how TE changes as a function of time delays.\n\n\n\\section{Significance test for Transfer Entropy}\\label{sec:tepv}\n\nLet us fix some notations. For any time $t$, information about the past of time series $A$ is encoded by $n_A$ different states. The simplest case is binary information (two symbols) about some previous time step $t'< t$; binary information about $M$ different time steps amounts to $n_A=2^M$ possible states. Note that the number of possible symbols per time step may change for each $t'$ in a fixed way. The number of possible states that describe the future of $A$ is denoted by $n_A^+$. Similarly, the number of states that describe time series $B$ at time $t$ is $n_B$.\n\nThe transfer entropy from $B$ to $A$ measures the additional information provided by $B$ with respect to the information about the past of $A$  when one tries to predict $A$. It is defined by the difference between the conditional entropy $H(A^+|A)$ of the future of $A$, denoted by $A^+$, on its past, and the entropy $H(A^+|A,B)$ of $A^+$ conditional  on the past of $A$ and $B$, as follows\n\\begin{equation}\n\\mbox{TE}^{B \\to A} = H(A^+|A)- H(A^+|A,B).\n\\end{equation}\n \n\nRefs.\\ \\cite{barnett2012transfer,kontoyiannis2016estimating} show that the $\\mbox{TE}^{B\\to A}$ converges to a chi-square distribution for binary states and Markovian systems with constant numbers of symbols per time step. As a warm-up for the next section, let us derive this result from the comparison between two decision tree models (see Fig.\\ \\ref{fig:DTMs}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\columnwidth]{FIG\/tree_AB.pdf}\n\\includegraphics[width=0.45\\columnwidth]{FIG\/tree_A.pdf}\n\\caption{Example of the maximum likelihood binary tree (on the left) that includes $B$, and the alternative model (on the right) that does not include $B$. \\label{fig:DTMs}}\n\\end{figure}\n\n\n\n\nWhen testing for the presence of significant transfer entropy from $B$  to $A$, the null hypothesis $\\mathcal{H}_0$ and the alternative hypothesis $\\mathcal{H}_1$ are respectively\n\\begin{eqnarray}\n  \\mathcal{H}_0: \\mbox{TE}^{B \\to A}=0\\\\\n  \\mathcal{H}_1: \\mbox{TE}^{B \\to A}>0.\n\\end{eqnarray}\n\nThe likelihood of the decision tree model that includes the series $B$ equals\n\\begin{equation}\n    \\log \\mathcal{L}_{B A A^+} = T \\sum_{b=1}^{n^B} \\sum_{a=1}^{n^A} \\sum_{a^+=1}^{n^{A^+}}  \\mathcal{P}(a^+,a,b)  \\log \\mathcal{P}(a^+|a,b) = - T \\cdot  H(A^+|A,B),\n\\end{equation}\nwhere $\\mathcal{P}(a^+|a,b)$ and $\\mathcal{P}(a^+,a,b)$ are the sample probabilities.\n\nThe likelihood of the decision tree model that does not include the series $B$ is given by\n\\begin{equation}\n    \\log \\mathcal{L}_{A A^+} = T  \\sum_{a=1}^{n^A} \\sum_{a^+=1}^{n^{A^+}}  \\mathcal{P}(a^+,a)  \\log \\mathcal{P}(a^+|a) = - T \\cdot  H(A^+|A).\n\\end{equation}\n\n\n\\begin{figure}\n\\centering\n\n\\includegraphics[width=0.45\\columnwidth]{FIG\/TEnull_dist.pdf}\n\\includegraphics[width=0.45\\columnwidth]{FIG\/Vuong_dist.pdf}\n\n\\caption{The left panel shows the distribution of $2T\\cdot \\mbox{TE}^{B \\to A}$ obtained with reshuffling of the data-series $B$ for different values of the number of symbols. The lines are the theoretical expectation from the $\\mathcal{X}^2(\\nu)$ distribution. The right panel Distribution of the Vuong metric obtained with independent 100,000 realizations of a time series of T=10,000 records for $S=0.2$.\\label{fig:TEnull} }\n\\end{figure}\n\n\n\nIf the null hypothesis is true, the knowledge of $B$ does not improves significantly  predictability and thus the difference between both likelihoods will be due to over-fitting of the sampling error. In such a case, Wilks' theorem \\cite{wilks1938large} states that\n\\begin{eqnarray}\n   2 (\\log \\mathcal{L}_{B A A^+} - \\log \\mathcal{L}_{A A^+}) = -2 T \\left( H(A^+|A,B) - H(A^+|A) \\right) = \\nonumber \\\\\n   = 2 T \\cdot \\mbox{TE}^{B \\to A} \\xrightarrow[]{d} \\mathcal{X}^2\\left( \\nu \\right)\n\\end{eqnarray}\\label{eq:like_TE}\n  \nconverges to a chi-square distribution in the limit of large sample size with $\\nu$ degrees of freedom given by\n\\begin{equation}\\label{eq:df1}\n    \\nu = n^A \\left( n^{A^+} -1 \\right) \\left(n^B-1\\right).\n\\end{equation}\nThis number can be obtained by counting the difference in the branches of the two decision trees. Specifically the number of degrees of freedom of the model $\\mathcal{P}(a^+| a,b)$ is $\\nu^{AB} = n^A n^B (n^{A^+} -1)$ where $\\sum_{a^+=1}^{n^{A^+}} \\mathcal{P}(a^+| a,b) = 1$, whereas from the model $\\mathcal{P}(a^+| a)$ one finds $\\nu^{A} = n^A (n^{A^+} -1)$. Given that  $\\nu = \\nu^{AB} - \\nu^{A}$,  \\eqref{eq:df1} follows. If the number of symbols per time step is constant and if the maximum number of relevant time steps in the past is also constant, one recovers the result of Ref.\\ \\cite{hadash2018estimate}. Figure \\ref{fig:TEnull} (left) compares the distributions given by Eq.\\ \\eqref{eq:like_TE} and numerical simulations for various parameter values.\n\n\n\n\\section{Comparing Two Transfer Entropy measures}\\label{sec:over}\n\nThe same approach can be used to test if time series $B$ is more informative than $C$ to predict $A$. In other words, we aim to test if the real underlying TEs are such that\n\\begin{eqnarray}\n   \\mathcal{H}_0 :\\mbox{TE}^{B \\to A}=\\mbox{TE}^{C \\to A} \\\\ \n   \\mathcal{H}_1 :\\mbox{TE}^{B \\to A}> \\mbox{TE}^{C \\to A}. \n\\end{eqnarray}\n\nLet us introduce the difference between the estimated TEs\n\\begin{equation}\n    Q = \\mbox{TE}^{B \\to A} - \\mbox{TE}^{C \\to A} = H(A^+|A,C) - H(A^+|A,B).\n\\end{equation}\nThe likelihood of the model with $B$ is \n\\begin{equation}\n       \\log \\mathcal{L}_{B A A^+} = T \\sum_{x=1}^{n^B} \\sum_{a=1}^{n^A} \\sum_{a^+=1}^{n^{A^+}}  \\mathcal{P}(a^+,a,b)  \\log \\mathcal{P}(a^+|a,b) = - T \\cdot  H(A^+|A,B) \n\\end{equation}\nwhile the likelihood of the model with $C$ is \n\\begin{equation}\n       \\log \\mathcal{L}_{C A A^+} = T \\sum_{c=1}^{n^C} \\sum_{a=1}^{n^A} \\sum_{a^+=1}^{n^{A^+}}  \\mathcal{P}(a^+,a,c)  \\log \\mathcal{P}(a^+|a,c) = - T \\cdot  H(A^+|A,C).\n\\end{equation}\nThe difference in likelihood is therefore\n\\begin{equation}\n      \\Delta \\mathcal{L} = \\log \\mathcal{L}_{BAA^+} -  \\log  \\mathcal{L}_{CAA^+} = T \\cdot Q.\n\\end{equation}\n\nTo obtain the distribution of the difference in likelihood of the maximum likelihood models, let us call $\\omega$ the standard deviation of the point-wise likelihood difference\n\\begin{equation}\n    \\omega^2 = \\mbox{Var}\\left[ \\log \\mathcal{P}(a^+| a, b) -\\log \\mathcal{P}(a^+| a, c)  \\right]\n\\end{equation}\ncomputed over the data-series of length T of elements $(b,c,a,a^+)$.\n\nThen for Vuong's closeness test\\ \\cite{vuong1989likelihood} states that, if the null hypothesis is true,\n\\begin{equation}\n    \\mathcal{V}:=\\frac{\\Delta \\mathcal{L} -\\nu}{\\sqrt{T} \\omega} \\xrightarrow[]{d} \\mathcal{N}(0,1),\n\\end{equation}\nwhere $\\nu=E(\\Delta \\mathcal{L})$, converges to a standardized normal distribution for $T\\to\\infty$, see Fig.~\\ref{fig:TEnull} (right).\n\nIn fact, $\\nu$ is the difference between the number of degrees of freedom of the two trees associated with $B\\to A$ and $C\\to A$. While the estimation of $\\nu$ from the trees is hard for overlapping models, but given that the hypothesis of Vuong's test applies, it is possible to estimate it as follows: if the null hypothesis is true, $\\mbox{TE}^{B \\to A}=\\mbox{TE}^{C \\to A} = S > 0$ and  $\\Delta \\mathcal{L}$ is the difference between two non-central chi distribution~\\cite{barnett2012transfer}, divided by $2$, with expected value\n\\begin{eqnarray}\n  E[T\\, \\mbox{TE}^{C \\to A}] = \\frac{1}{2}\\left( \\nu^C + S\\right) =\\frac{1}{2}\\left[ n^A \\left( n^{A^+} -1 \\right) \\left(n^C-1\\right)+ S\\right]  \\\\\n  E[T\\, \\mbox{TE}^{B \\to A}] = \\frac{1}{2}\\left( \\nu^B + S\\right) =\\frac{1}{2}\\left[ n^A \\left( n^{A^+} -1 \\right) \\left(n^B-1\\right)+ S\\right].  \n\\end{eqnarray}\nThe degrees of freedom $\\nu^C$ and $\\nu^B$ are given by\\ \\eqref{eq:df1}. Therefore $E[\\Delta \\mathcal{L}- \\nu]=0$ if\n\\begin{equation}\n    \\nu = (n^B - n^C) \\frac{n^A}{2} (n^{A^+}-1).\n\\end{equation}\n\n\\begin{comment}\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.5\\columnwidth]{FIG\/Vuong_dist.pdf}\n    \\caption{Distribution of the Vuong metric obtained with independent 100,000 realizations of a time series of T=10,000 records for $S=0.2$. }\n    \\label{fig:my_label}\n\\end{figure}\n\\end{comment}\n\n\\subsection{Bootstrap p-value}\nLet us briefly describe two generic approaches to associate a p-value to TE from bootstrap resampling, which can of course be computationally intense. For example,    significant p-values corrected by  multiple comparison techniques  must be orders of magnitude smaller than the intended significance threshold, which requires a very large number of bootstrap samples. \nFor both approaches we must identify an event event matrix $E \\in  \\mathbb{R}^{T \\times 3}$ where $e_t = (a'_t, a_t, b_t)$ is a generic event. This  matrix can be  generalized to more than two variables. \n\n\\subsubsection{Significant TE via shuffling}\nTo estimate the p-value of sample $\\mbox{TE}^{B\\to A}=S$, it is enough to perform a permutation test of $B$, so as to break any link between the past of $B$ and the future of $A$. This kind of approach requires many  reshuffling of column $B$ of matrix $E$. Let us denote by  $E_{(r)}$ the data matrix $E$ whose $B$ column has been reshuffled, $r$ being the index of a particular permutation of column $B$. This new matrix preserves both $H(A^+|A)$ and $H(B)$ but breaks any information flow between $B$ and $A$. From $E_{(r)}$, one computes $\\mbox{TE}_{(r)}^{B \\to A}=S_{(r)}$. By performing $q$ independent re-shuffling and TE estimations, the p-value can be defined as\n\\begin{equation}\n    p\\mbox{-value} = \\frac{\\sum_{i=1}^q\\delta(S>S_{(i)})}{q},\n\\end{equation}\nwhere $\\delta(x)=1$ if the assertion is true and zero otherwise.\n\\subsubsection{Comparing TEs via bootstraps}\nAssume that both $\\mbox{TE}^{B \\to A}=S^{B}$ and $\\mbox{TE}^{C \\to A}=S^{B}$ are significantly larger than zero. A different bootstrap procedure must be adopted. In this case, indeed, the bootstrap is used to associate a confidence interval to $S^B$ and $S^C$. Let us start by defining the event matrix $E$ with elements $e_t = (a'_t, a_t, b_t, c_t)$, then a bootstrap realization of $E_{(r)}$ is obtained by re-sampling with replacement the events such that the bootstrap event matrix $E_{(r)}$ has the same sample size of $E$, i.e., $T$. This bootstrap procedure does not break down the information flow from $B$ and $C$ to $A$. For each bootstrap realization we estimate $S^{B}_{(r)}$ and $S^{C}_{(r)}$. Finally the p-value of $\\mbox{TE}^{B \\to A}>\\mbox{TE}^{C \\to A}=S^{C}$ is defined as\n\\begin{equation}\n    p\\mbox{-value} = \\frac{\\sum_{i=1}^q\\delta(S^{B}_{(i)}>S^{B}_{(i)})}{q} \n\\end{equation}\nwhere $q$ is the number of independent bootstrap re-sampling and $\\delta(x)=1$ if the assertion is true, zero otherwise.\n\n\n\\section{Finite size effects of the statistical tests}\n\\subsection{Testing for the significant TE}\\label{sec:null}\nIn this section, we show that the analytical p-value converges to the bootstrap one for various numbers of the number of states, hence of the degrees of freedom. In particular, we test first the convergence when the null hypothesis is true, then we test the power in detecting when the null hypothesis is false.\n\nThe true null hypothesis benchmark is obtained by sampling $\\mathcal{P}(a',a)$ and $P(b)$ from two Dirichlet distributions with parameters set to $\\alpha$. Then a synthetic dataset of sample size $T$ is obtained sampling a set of that from $\\mathcal{P}(a',a,b)=\\mathcal{P}(a',a)\\mathcal{P}(b)$. In the upper panel left of Fig.~\\ref{fig:bech} we show that both the analytical and bootstrap p-values converge to the same value irrespective of $n^{A^+}$, $n^A$, and $n^B$. However, we must be aware that this is not totally true. In fact, when $n^{A^+},n^A,n^B$ are large and when the sample size is small,  the likelihood ratio test should not be applied due to slow convergence to the chi-squared distribution (see the upper right panel of Fig.~\\ref{fig:bech}). In particular, a small $T$ leads to an under-estimation of the analytical p-values. In the lower-left panel of Fig.~\\ref{fig:bech}, we show that small  $n^{A^+},n^A,n^B$ are much more robust to finite sample size effects than the larger ones.  \n\nTo test the false null hypothesis, the probabilities $\\mathcal{P}(a',a,b)$ are sampled from a Dirichlet distribution with parameters set to $\\alpha$. In this set of simulation, the effect of $\\alpha$ is explored in order to test the power of the test: small values of $\\alpha$ produce very heterogeneous probabilities, while when $\\alpha>>1$,  all the probabilities become homogeneous, i.e.,  $\\mathcal{P}(a',a,b) \\to 1\/(n^{A^+}n^A n^B)$. While the former is the ideal condition for every test, the latter is a difficult case. In the lower-right panel of Fig.~\\ref{fig:bech}, we show the fraction of times the analytical and bootstrap p-values reach values smaller than $0.01$ over $1,000$ independent simulations. Note that both the asymptotic analytical and the bootstraps results agree.\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/Fig1a.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/Fig1b.pdf}\n    \n    \\includegraphics[width=0.45\\columnwidth]{FIG\/Fig1c.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/Fig1d.pdf}\n    \\caption{Tests for $\\mbox{TE}^{B \\to A}>0$: $a)$ comparison of bootstrap and analytical p-values for true null hypothesis with $T=10,000$ and all possible combinations of $(n^{A^+}n^A n^B) \\in [2,4]$ repeated $100$ times; $b)$ cumulative distributions computed with $10,000$ independent realizations of the bootstrap and analytical p-values for $(n^{A^+}n^A n^B)=(3,3,3)$ and $T=100$, the unbiased model should lie on the bisector; $c)$ Area under the CDF of analytic p-value for different sample size $T$, the area of the unbiased model is $0.5$;  $d)$ comparison of the bootstrap an analytic true positive rate obtained with a p-value$<0.01$ for false null hypothesis benchmark, the simulations refers to $1,000$ simulation with $(n^{A^+}n^A n^B)=2,2,2$ and $T=10,000$, the parameter $\\alpha$ controls the homogeneity of $\\mathcal{P}(a',a,b)$.  In all the plots the bootstrap p-value is computed with $10,000$ bootstrap re-sampling.  }\n    \\label{fig:bech}\n\\end{figure}\n\n\n\n\\subsection{Comparing TEs}\\label{sec:over}\nBenchmarking the true null hypothesis, in this case, is much harder. To do that, we should solve a non-linear minimization problem. Specifically, by using a Semidefinite Quadratic Linear Programming (SQLP), we build a set of probability $\\mathcal{P}(a',a,b,c)$ such that $\\left( \\mbox{TE}^{c \\to a}-S^{c})^2+(\\mbox{TE}^{b \\to a}-S^{b}\\right)^2$ computed on the real benchmark probabilities is minimal. Since SQLP depends on the initial conditions, we randomize them by starting the optimization procedure from a set of probability obtained from a Dirichlet distribution with parameter $\\alpha=0.5$.\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/pv_comp_vuong.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/Vuong_bias_dist.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/like_vuong_T.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/TPR_vuong.pdf}\n    \\caption{Benchmarks performance of the Vuong test. Upper left plot: scatter plot of the bootstrap p-values and the analytical ones for all possible combinations of $n^{A^+},n^{A},n^{B},n^{C} \\in [2,4]$ for $T=10,000$ for $S^B=S^C=0.1$, the dotted line is a linear regression. Upper right plot: distribution of $10,000$ p-values obtained for $n^{A^+}=n^{A}=n^{B}=n^{C}=4$ and $T=100$ and  $S^B=S^C=0.1$, the expected distribution should be homogeneous. Lower left plot: average element-wise likelihood of the Vuong metric being distributed as a $\\mathcal{N}(0,1)$ and $S^B=S^C=0.1$ while the dotted line is the likelihood corresponding to  $\\mathcal{N}(0,1)$. Right lower plot: true positive rate computed over $1,000$ simulations for $n^{A^+}=n^{A}=n^{B}=n^{C}=4$ as a function of $S^B$ for different values of $T$.}\n    \\label{fig:vuongbench}\n\\end{figure}\n\nIn the upper panel of Fig.~\\ref{fig:vuongbench} we show a scatter plot of the p-values obtained when the null hypothesis is true for a large sample size. Differently from Fig.~\\ref{fig:bech}, this time bootstrap and analytical p-values show a lower agreement, although on average unbiased, as depicted in the linear regression reported in the plot. \n\nIn the low sample size and large number of symbols regime (the upper right panel of Fig.~\\ref{fig:vuongbench}), both p-values are biased but in a different way. The bootstrap p-values are too restrictive, while the analytical ones are too permissive. In fact, the expected p-value distribution when the null hypothesis is true should be a homogeneous distribution. In this case, since the Vuong test is two-sided, looking at the area of the CDF of p-values may be misleading. Therefore, to inspect the speed of convergence of the analytical p-values, we measure the average element-wise likelihood that the Vuong metric $\\mathcal{V}$ is distributed as a $\\mathcal{N}(0,1)$ (see the lower-left plot of Fig.~\\ref{fig:vuongbench}). As expected, the bias is larger when the number of symbols is large and all the inspected cases converge to normality for $T>1,000$.\n\nFinally, in the lower right panel of Fig.~\\ref{fig:vuongbench}, we compare the true positive rate of the bootstrap and analytical p-values. The analytical p-values, as expected, show a higher recall than the bootstrap ones. Although this advantage is larger in the small sample size regime, it is worth stressing that it might be compensated by a lower control of the first type errors.\n\n\\section{Lagged Information transfer between asynchronous time series: limit order books}\n\nWe apply the analytical asymptotic p-values described above to measure the information flow between changes in the order books of different stocks. Specifically, for each stock, we consider the time series of mid-prices, defined as the instantaneous average between the best buy and sell quotes, and encode their changes with their signs. The inter-event time between two consecutive price variations is not constant, and two time series are not synchronous. Ref.~\\cite{harre2015entropy} generalized TE to asynchronous time series by matching a given event of time series $A$  with the last known event of time series $B$. This type of matching has two consequences: i) one event of $B$ may be matched to multiple events of $A$; ii) some event of $B$ might be not matched to an event of $A$, especially if the typical event frequency is much larger for $B$ than for $A$.\n\n\n\\begin{figure}[tbh]\n    \\centering\n    \\includegraphics[width=0.95\\columnwidth]{FIG\/Arrow_art-crop.pdf}\n    \\includegraphics[width=0.80\\columnwidth]{FIG\/Arrow_shift_art-crop.pdf}\n    \\caption{The upper plot shows an example of the backward matching among two asynchronous binary series. The lower plot shows the resulting match after a forward shift is applied to X.   }\n    \\label{fig:sync}\n\\end{figure}\n\nIt is interesting to observe that the reaction time is not immediate in real applications.\n Therefore, we expect that an event on $B$ might take some time to process and affect the future of $A$. To measure the amount of this transient, we apply a forward shift to the time series $A$ of $\\delta t$ and apply the matching again. (Fig.~\\ref{fig:sync}).\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/NEM_ABX.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/diff_OOB_RF_N.pdf}\n    \\caption{Left plot: TE from {\\tt NEM} to {\\tt ABX} as a function of the forward shift of {\\tt ABX}; when the shift is larger than $1$ second,   TE is no more significant. The right plot: TE from {\\tt STI} and {\\tt BBT} to {\\tt RF} as a function of the time shift of {\\tt RF} computed over 3 days of data. The difference in TE is significant according a two-sided Vuong test only in the red band}\n    \\label{fig:oob}\n\\end{figure}\n\nIn the left plot of Fig.~\\ref{fig:oob}, we show the TE from stock {\\tt NEM} to stock {\\tt ABX} computed on 2017-01-03, selecting only events between $09:35$ and $16:25$ (we removed 5 minutes at both ends of the trading day to reduce the effect of the auction phase). As expected, the TE drops down as the shift increases. By using the statistical test of sec.~\\ref{sec:tepv} we can detect the maximum $\\delta t\\simeq 1s$ after which the binary price changes of  {\\tt NEM} do not add any information to the predictability of those of {\\tt ABX}. Note that this delay does not allow one to make useful price predictions in a speculative setting, as prices move too little in 1 second to beat transaction costs.\nWe emphasize here that the ability to perform this test on long time series without bootstrapping opens the way to large-scale studies of the timescales of TE and delays. \n\nIn the right plot of Fig.~\\ref{fig:oob} we apply the TE comparison test with time shifts to similar data: we compare the TE from \\texttt{STI} and \\texttt{BBT} to {\\tt RF} as a function of the time shift $\\delta t$. In this case, it seems that when $\\delta t<0.01$s, {\\tt BBT}  transfers more information to {\\tt RF} than {\\tt BBT}, whereas on a larger time scales ($\\delta t>0.01$s) we observe the opposite. This, however, may simply be due to estimation noise. Indeed, the TE comparison test is only significant in the area highlighted in red at a $0.01$ threshold, whereas the comparison for $\\delta t<0.01$ is not.  \n\n\n\n\n\\subsection{Network inference}\n\nThe analytic p-values of sec.~\\ref{sec:tepv} can be used to extract the lead-lag network between different stocks. Specifically, we considered all the order books of NYSE and five German exchanges on a single day. Given that the information flow between the European and American exchange must be penalized due to physical distance, we expect to observe higher link homophily within a continent.\n\nWe restricted our analysis to a single day (2017-01-03) and selected only the price variations during the daytime when all the exchanges were simultaneously opened (14:30-16:30 UTC). In order to avoid small sample size biases of the analytic p-values, reported in sec.~\\ref{sec:null}, we removed all the stocks which have less than $1,000$ mid-price variations and we considered only binary variations, i.e, $P_{t+1}>P_{t}$ or $P_{t+1}<P_{t}$. After this data filtering, only $n=1,177$ stocks remain. Given that we are performing a multiple comparison test to control the number of false-positive selections, we adopted the Bonferroni correction~\\cite{millersimultaneous} which prescribes reducing the null hypothesis threshold for each test to $q\/n_t$ where $1-q$ is the confidence level at which we can guarantee that no false link is included in the inferred network ($q=0.01$) and $n_t=n(n+1)\/2=693,253$ are the number of tests performed. We want to stress that for such a problem using a bootstrap p-value will be computation unfeasible due to the large number of tests performed and the low Bonferroni threshold.\n\nOn the resulting statistically validated network \\cite{tumminello2011statistically}, shown in Fig.~\\ref{fig:net} left, only $0.3\\%$ of links were retained, and $767$ nodes were not isolated. As we expected\\cite{bongiorno2017core}, we observed a high assortativity coefficient~\\cite{newman2010networks} by exchange, i.e., $0.56$, which means that the number of links inside the same exchange is larger than the one expected from a configuration model.\n\nIn the right panel of Fig.~\\ref{fig:net}, we display a coarse-grained network, where all the nodes that belong to the same exchanges are grouped into a single node with a size proportional to the log of the number of nodes, and all the links are grouped with a width proportional to the log the number of links. From this analysis, one observes that the largest interconnection is between NYSE and Frankfurt stock exchanges, which are the most important stock exchanges in our dataset. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/network.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{FIG\/clust_network.pdf}\n    \\caption{Left plot: Bonferroni network obtained retaining only the significant TE among each pair of companies of NYSE and 4 German stock exchanges; the color code indicates the stock exchanges. Right plot:  network representation of the information flow among the stock exchanges. The size of the nodes is proportional to the log of the exchange's size, and the links' width is proportional to the log of the number of links among the considered groups. }\n    \\label{fig:net}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nHaving asymptotic tests for the presence of Transfer Entropy and a comparison of TE makes it possible to infer causality networks of a large number of nodes with long time series, especially since the minimum number of points required is relatively small. Using a time shift is especially relevant in systems in which latency is important, which deserves further study. Finally, being able to compare the TE from two sets of predictors of a given time series opens many possibilities, such as pair-wise ranking of feature importance.\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{S:intro}\n\nThe purpose of this note is to relate steady waves at the free surface of a two-dimensional, infinitely-deep and irrotational flow of an incompressible inviscid fluid, acted upon by gravity, to the nonlinear pseudodifferential equation\n\\begin{equation}\\label{E:deep}\n\\H w'=\\mu(w+w\\H w'+\\H(ww')). \\end{equation}\nIt will then be used to demonstrate the non-existence of solitary waves, for which the fluid surface asymptotically approaches a constant level over a nearly uniform flow. Here $w(x)$, $x\\in\\mathbb{R}$, measures the wave profile and $\\mu=g\/c^2$, where $g$ describes gravitational acceleration and $c$ denotes the speed of wave propagation; $\\mu>0$ is physically realistic. The Hilbert transform is given as\n\\begin{equation}\\label{D:H}\n\\H v(x)=\\frac{1}{\\pi} PV \\int^\\infty_{-\\infty} \\frac{v(t)}{x-t}~dt \\qquad (x \\in \\mathbb{R}), \\end{equation}\nwhere $PV$ stands for the Cauchy principal value; alternatively $\\widehat{\\H v}(\\xi)=-i\\text{sgn}(\\xi)\\hat{v}(\\xi)$ defines the operator in the Fourier space. Throughout $'=d\/dx$.\n\n\\medskip\n\nSolitary waves at the surface of water hold a central position in the theory of wave motion and they have historically stimulated a considerable part of its development, from Russell's famous horseback observations to the elucidation of the Korteweg-de Vries solitons. In the case where the flow depth is finite, the mathematical theory of solitary water waves dates back to constructions in \\cite{FrHy} and \\cite{Beale} of small-amplitude waves and it includes the global bifurcation result in \\cite{AmTo81} and extensions in \\cite{Hur08a} and \\cite{GrWa08} to waves with vorticity. The symmetry and monotonicity properties were discussed in \\cite{CrSt} and \\cite{Hur08b} while the regularity properties were in \\cite{Lewy} and \\cite{Hur10}. Moreover non-uniqueness and linear instability were addressed in \\cite{Plo} and in \\cite{Lin-IMRN} for waves near the ``extremal\" form. \n\nThe present result, in stark contrast, states that no solitary water waves exist in the infinite-depth case, which, in case gravity acts downwards, asymptotically tend to the quiescent state faster than linearly.\n\n\\subsection*{Notation} \nThe weighted H\\\"older space, written $C^{k+\\alpha}_\\rho(\\mathbb{R})$, for $k\\geq 0$ an integer, $\\alpha \\in (0,1)$ and for $\\rho\\in \\mathbb{R}$, is characterized via the norm\n\\begin{equation}\\label{D:wHolder} \n\\|v\\|_{C^{k+\\alpha}_\\rho(\\mathbb{R})} =\\sum_{j=0}^k\\sup_{x\\in\\mathbb{R}}\\left\\langle x\\right\\rangle^{\\rho}|v^{(j)}(x)|\n+ \\sup_{x\\in\\mathbb{R}}\\sup_{|x-t|\\leq 1}\\left\\langle x\\right\\rangle^{\\rho} \\frac{|v^{(k)}(x)-v^{(k)}(t)|}{|x-t|^\\alpha},\n\\end{equation}\nwhere $\\left\\langle x\\right\\rangle=(1+x^2)^{1\/2}$. If $v \\in C^{k+\\alpha}_\\rho(\\mathbb{R})$ then its derivatives of order up to $k$ vanish at least like $|x|^{-\\rho}$ as $|x| \\to \\infty$.\n\n\\begin{theorem}\\label{T:non-existence}\nIn the case of $\\mu>0$, if $w \\in H^1(\\mathbb{R})\\cap C^{1+\\alpha}_{1+\\epsilon}(\\mathbb{R})$  satisfies \\eqref{E:deep} for some $\\alpha \\in (0,1)$ and for some $\\epsilon \\in (0,1)$ then $w\\equiv 0$. \n\nIn the case of $\\mu\\leq0$, if $w \\in H^1(\\mathbb{R})\\cap C^{1+\\alpha}(\\mathbb{R})$ satisfies \\eqref{E:deep} for some $\\alpha \\in(0,1)$ and if in addition $\\inf_{x\\in\\mathbb{R}}(1+\\H w'(x))>0$ then $w\\equiv 0$. \\end{theorem}\n\nEstablishing an integral identity for the interface displacement, Sun in \\cite{Sun97} argued the non-existence of solitary waves between two fluids flows of infinite extent, provided that the profile vanishes at infinity faster than linearly and that it is either purely elevated or purely depressed. Craig in \\cite{Cra02} made an alternative proof based upon the maximum principle for the water wave problem in two and higher dimensions, without assuming decay of the fluid surface but requiring the positivity or negativity. Therefore a solitary wave on deep water cannot be everywhere positive (or negative). Do solitary water waves then exist, which oscillate about the mean fluid level? Theorem~\\ref{T:non-existence} answers it negatively. If the effects of surface tension are factored into, on the other hand, solitary water waves were shown to arise in \\cite{IK}, for instance, which necessarily change sign. \n\n\\medskip\n\nThe solitary wave problem on deep water is introduced in Section~\\ref{S:formulation} as a free boundary problem in potential theory. A ``regular\" solution of the system, for which the velocity vanishes nowhere at the fluid surface (see Section \\ref{SS:free-boundary}), then corresponds to a solution of \\eqref{E:deep}, which serves as the basis of the present account, and vice versa so long as the fluid surface lacks self-intersections and cusps. Exploiting a conformal mapping of the fluid domain, the proof resembles that for Babenko's equation in the Stokes (periodic) wave setting, which already led to much progress in \\cite{BDT00a} and \\cite{BDT00b} among others. \n\nObserve that \\eqref{E:deep} enjoys the scaling symmetry under \n\\[w(x)\\mapsto\\lambda^{-1}w(\\lambda x)\\quad\\text{and}\\quad\\mu\\mapsto\\lambda\\mu\\] \nfor any $\\lambda>0$ and that the vector field $xd\/dx$ generating the symmetry commutes with the Hilbert transform (see \\eqref{E:Hcomm}). They offers in Lemma~\\ref{L:non-existence} an extremely simple non-ex\\-is\\-tence proof for \\eqref{E:deep} whenever $xw'\\in L^2(\\mathbb{R})$, which is reminiscent of Pohozaev identities techniques. For  differential equations one may further employ a truncation argument to promote $L^2$-based spaces with weight to locally $L^\\infty$-based spaces; see \\cite{dBS}, for instance, for the Kadomtsev-Petviashivilli equation. But the Hilbert transform, being pseudodifferential, does not commute with functions as a rule. We instead make an effort to understand the asymptotic behavior of the solution to refine the conclusion.\n\nIn the case of $\\mu<0$, i.e. gravity acts oppositely to what is physically realistic, the linearized operator of \\eqref{E:deep} becomes $\\H d\/dx-\\mu$ plus a function after the Plotnikov transformation (see \\eqref{E:plotnikov}). Accordingly $\\mu$ may be regarded as an eigenvalue of $\\H d\/dx=(-d^2\/d x^2)^{1\/2}$ with potential. Spectral properties in \\cite{CaMaSi}, for instance, of a relativistic Schr\\\"odinger operator then reveals in Lemma~\\ref{L:decay-} that the derivative of a solution to \\eqref{E:deep} decays quadratically. In the Stokes wave setting, incidentally, a non-existence proof based upon duality is found in \\cite{Tol02}.\n\nIn case gravity acts downwards, on the other hand, $\\mu>0$ is contained in the essential spectrum of $\\H d\/dx$ and hence the proof of Lemma~\\ref{L:decay-} is not applicable. To attain decay nevertheless, we shall impose a solvability condition. Specifically, if a solution of \\eqref{E:deep} is assumed to decay faster than $x^{-1}$ as $|x|\\to \\infty$ then it does like $x^{-2}$ upon a bootstrapping argument in \\cite{CrSt} or \\cite{Sun97}, for instance. The author has not yet succeeded in removing the extra condition although it is desirable. For a broad class of interfacial fluids problems in the infinite-depth case, including the Benjamin-Ono equation and the water wave problem with surface tension (see \\cite{IK}, for instance), to compare, solitary waves vanish quadratically, suggesting that the decay rate of $x^{-1-}$ is not outrageous. \n\n\\medskip\n\nConcerning the non-stationary water wave problem in like setting, the associated linear operator $\\partial^2\/\\partial t^2+\\H\\partial\/\\partial x$ remains invariant under the vector field $\\frac12t\\partial\/\\partial t+x\\partial\/\\partial x$, where $t\\in\\mathbb{R}$ denotes the temporal variable, analogously to that $\\H d\/dx$ is invariant under $xd\/dx$ in the steady wave problem \\eqref{E:deep}. As a matter of fact dispersive estimates plus a profound understanding of the nonlinearity shed light in \\cite{Wu3} to the almost-global existence for small data. To interpret, solitary waves of small amplitude are unlikely to arise at the surface of a two-dimensional infinitely-deep flow of water, if the profile is sufficiently smooth and if in addition it vanishes sufficiently fast at infinity. Theorem \\ref{T:non-existence} bears it out. \n\nThe development in Section \\ref{S:formulation} of \\eqref{E:deep} relies upon conformal mapping techniques. It seems not to work in higher dimensions, leaving open the tantalizing possibility of steady water waves in three dimensions, which oscillate yet become flat at infinity.  \n\n\\section{Formulation}\\label{S:formulation}\nThe solitary wave problem on deep water is formulated as the nonlinear pseudodifferential equation \\eqref{E:deep}.\n\n\\subsection{The free boundary problem}\\label{SS:free-boundary}\nThe steady water wave problem in the simplest form concerns a two-dimensional, infinitely-deep and irrotational flow of an incompressible inviscid fluid and wave motions at the surface layer, under gravity.  By steady we mean that the flow as well as the surface wave move at a constant speed from right to left without changing their configuration. The effects of surface tension are neglected. The solitary wave problem then seeks for solutions, for which the fluid surface asymptotically tends to a constant level and the flow in the far field is nearly uniform.\n\nIn the dimensionless coordinates moving at the speed of wave propagation, let the parametric curve\n\\[\\Gamma=\\{ (u(x), v(x)): x \\in \\mathbb{R} \\}\\]\nrepresent the fluid surface. Assume that\n\\begin{subequations}\\label{E:stream}\n\\begin{align}\n& \\text{$x \\mapsto (u(x),v(x))$ is continuously differentiable}; \\label{E:gamma-injective}\\\\\n& \\text{$u'(x)>0$ and $u'(x)^2+v'(x)^2<\\infty$ for every $x \\in \\mathbb{R}$}; \\label{E:gamma-regular}\\\\\n& \\text{$(u(x)-x,v(x)) \\to 0$ as $|x| \\to \\infty$.}\\label{E:gamma-infty}\n\\end{align}\nLet $\\Omega$ denote the open region in the $(X,Y)$-plane below $\\Gamma$, occupied by the fluid. {\\em The solitary wave problem on deep water} is to find a curve $\\Gamma$ satisfying \\eqref{E:gamma-injective}-\\eqref{E:gamma-infty}, a parameter $\\mu \\in \\mathbb{R}$ and a function $\\psi$ defined over $\\overline{\\Omega}$ such that \n\\begin{align}\n& \\text{$\\psi \\in C^1(\\overline{\\Omega})\\cap C^2(\\Omega)$ and \n$\\psi$ is harmonic in $\\Omega$;} \\label{E:laplace} \\\\\n& \\text{$\\psi=0$ on $\\Gamma$;} \\label{E:kinematics-surface}\\\\\n& \\text{$\\psi(X,Y) -Y \\to 0$ as $|(X,Y)| \\to \\infty$;} \\label{E:psi-infty}\\\\\n& \\text{$\\lim_{(X,Y)\\to(u(x),v(x))} |\\nabla \\psi(X,Y)|^2\n+2\\mu v(x) = 1$ for every $x \\in \\mathbb{R}$.} \\label{E:bernoulli-weak}\n\\end{align}\\end{subequations}\nDetails are discussed in \\cite{Benjamin}, for instance. Here we merely hit the main points. \n\n\\medskip\n\nThe kinematic boundary condition \\eqref{E:kinematics-surface} states that the fluid surface itself makes a streamline while the dynamic boundary condition \\eqref{E:bernoulli-weak} manifests Bernoulli's law of constant pressure at the surface. They determine the free boundary $\\Gamma$, which by \\eqref{E:gamma-injective} and \\eqref{E:gamma-regular} is the graph of a $C^1$ function that is nowhere vertical. The (non-di\\-men\\-sion\\-alized) stream function $\\psi$ is to describe the fluid motion. Specifically $(-\\psi_Y, \\psi_X)$ denotes the steady velocity field. Boundary conditions \\eqref{E:gamma-infty} and \\eqref{E:psi-infty} mean, respectively, that the fluid surface becomes asymptotically horizontal and that the flow in the far field is nearly uniform. \n\n\\medskip\n\nA solution triple $\\Gamma$, $\\mu$ and $\\psi$ of \\eqref{E:stream} is said {\\em regular} if $|\\nabla \\psi|> 0$ on $\\Gamma$. A theorem of Lewy in \\cite{Lewy} results in that the fluid surface of a regular solution to \\eqref{E:stream} is real analytic. In the finite-depth case, on the other hand, $\\nabla \\psi=0$ at the crest of a wave of the maximum height, namely an ``extremal\" wave; see \\cite{AmTo81}, for instance.\n\n\\subsection{Complex Hardy spaces in the half plane}\\label{SS:preliminaries}\nThroughout \\[\\mathbb{P}=\\{x+iy\\in\\mathbb{C}:y <0\\}\\] denotes the open lower half plane in the complex field. \n\nThe following definitions and discussion are taken from \\cite[Chapter I and Chapter II]{Gar} and \\cite[Chapter 11]{Duren} among others.  \n\n\\medskip\n\nThe complex Hardy space in the lower half plane, written $\\mathbf{H}^p(\\mathbb{P})$, for $p \\in(0,\\infty]$, consists of holomorphic functions $f:\\mathbb{P} \\to \\mathbb{C}$ such that \n\\[ \\|f\\|_{\\mathbf{H}^p(\\mathbb{P})}=\\sup_{y \\in (-\\infty,0)}  \\| f(\\cdot+iy)\\|_{L^p(\\mathbb{R})} <\\infty. \\]\nIf $f \\in \\mathbf{H}^p(\\P)$, $p\\in (0,\\infty)$, then its non-tangential limit \n\\begin{equation}\\label{E:f*} \nf^*(x):=\\lim_{y \\to 0-} f(x+iy) \\end{equation} \nexists in $L^p(\\mathbb{R})$ and for almost all $x \\in\\mathbb{R}$. In the case of $f \\in \\mathbf{H}^\\infty(\\P)$ correspondingly $f^*$ exists almost everywhere on $\\mathbb{R}$; see \\cite[Theorem 11.1]{Duren} or \\cite[Chapter II, Theorem 3.1]{Gar}, for instance. If $f \\in \\mathbf{H}^p(\\P)$, $p \\in (0,\\infty]$, then either $f\\equiv 0$ or ${\\displaystyle \\int^\\infty_{-\\infty} \\frac{|\\log|f^*(x)||}{1+x^2}~dx <\\infty}$. If in addition $f^*=0$ on a set of positive measure then $f\\equiv 0$.  \n\nIf $f \\in \\mathbf{H}^p(\\mathbb{P})$, $p\\in [1,\\infty]$, then \\cite[Theorem 11.2]{Duren} or \\cite[Chapter II, Corollary 3.2]{Gar}, for instance, state that\n\\[ f(x+iy)=\\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\frac{y}{(x-t)^2+y^2}f^*(t)~dt.\\]\nThat is to say, $f$ agrees with the Poisson integral of its boundary function. Conversely, if $v \\in L^p(\\mathbb{R})$ and if ${\\displaystyle f(x+iy)=\\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\frac{y}{(x-t)^2+y^2}v(t)~dt}$ is holomorphic in $\\mathbb{P}$ then $f \\in\\mathbf{H}^p(\\mathbb{P})$. Moreover $f^*=v$ almost everywhere on $\\mathbb{R}$. \n\n\\medskip\n\nIf $u$ is harmonic in $\\P$ and if $\\sup_{y \\in (-\\infty,0)}\\|u(\\cdot+iy)\\|_{L^p(\\mathbb{R})} <\\infty$, $p \\in (1,\\infty]$, then the classical Fatou lemma (see \\cite[Chapter~I, Theorem~5.3]{Gar}, for instance) implies that its non-tangential limit $u^*$ exists almost everywhere on $\\mathbb{R}$ and $u$ is the Poisson integral of $u^*$. If in addition $u^*$ is uniformly continuous on $\\mathbb{R}$ then $u \\to u^*$ uniformly as $y \\to 0-$; see \\cite[Chapter~I, Theorem~3.1]{Gar}, for instance.\n\n\\medskip\n\nIn the case of $p=2$, in particular, the Paley-Wiener theorem (see \\cite[The\\-o\\-rem 11.9]{Duren}, for instance) implies that if $v \\in L^2(\\mathbb{R})$ then there exists a unique function $\\mathcal{R}v \\in \\mathbf{H}^2(\\mathbb{P})$ such that \n\\begin{equation}\\label{E:R} (\\mathcal{R}v)^*=\\H v+iv;\\end{equation} \nrecall that $\\H$ is the Hilbert transform (see \\eqref{D:H}). Conversely, if $f \\in \\mathbf{H}^2(\\mathbb{P})$ then $f^*=\\H v+iv$ for some $v \\in L^2(\\mathbb{R})$. \n\nElementary properties of the Hilbert transform include that $\\H: L^p(\\mathbb{R}) \\to L^p(\\mathbb{R})$, $p\\in (1,\\infty)$, is bounded and that the adjoint of $\\H$ is $-\\H$. Moreover the Privalov theorem states that $\\H:C^\\alpha(\\mathbb{R}) \\to C^\\alpha(\\mathbb{R})$, $\\alpha\\in (0,1)$, is bounded.\n\n\\medskip\n\nLastly, if $v \\in L^\\infty(\\mathbb{R})$, $v \\neq 0$ almost everywhere on $\\mathbb{R}$ and if $1\/v \\in L^\\infty(\\mathbb{R})$ then there exists $f \\in \\mathbf{H}^\\infty(\\P)$, called an {\\em outer function} of $v$, such that $|f^*|=v$ almost everywhere on $\\mathbb{R}$ if and only if ${\\displaystyle \\int^\\infty_{-\\infty}\\frac{|\\log|v(x)||}{1+x^2}~dx <\\infty}$. Moreover $1\/f \\in \\mathbf{H}^\\infty(\\P)$. We refer the reader to \\cite[Chapter~II, Theorem~4.4]{Gar}, for instance.\n\n\\subsection{Reduction to a single equation}\\label{SS:equation}\nThis subsection concerns the equivalence between the system \\eqref{E:stream} and the single nonlinear pseudodifferential equation \\eqref{E:deep}. \n\nIn what follows the two-dimensional space is identified with the complex plane whenever it is convenient to do so. \n\n\\begin{lemma}\\label{P:stream->bernoulli}\nSuppose that a curve $\\Gamma$ and a function $\\psi$ defined in the plane below $\\Gamma$ form a regular solution of \\eqref{E:stream} for some $\\mu \\in \\mathbb{R}$. Let  $-\\phi$ be a harmonic conjugate of $\\psi$. \nThen $\\phi+i\\psi$ is a conformal bijection from $\\overline{\\Omega}$ onto $\\overline{\\mathbb{P}}$ which maps $\\infty$ to $\\infty$.\n\nLet $w=\\emph{Im}\\,(\\phi+i\\psi)^{-1}(\\cdot+i0)$. If $w \\in H^1(\\mathbb{R})$ then\n\\begin{equation}\\label{E:bernoulli}\n(1-2\\mu w)((1+\\H w')^2+(w')^2)=1\\end{equation} \nand \n\\begin{equation}\\label{E:injective}\n\\inf_{x\\in\\mathbb{R}}(1+\\H w'(x))>0.\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $|\\nabla \\psi|>0$ on $\\Gamma$ by hypothesis, a theorem of Lewy in \\cite{Lewy} ensures that $\\Gamma$ is a real analytic curve. By \\eqref{E:gamma-regular}, moreover, $\\Gamma$ is the graph of a function that is nowhere vertical. \n\nNote from \\eqref{E:kinematics-surface}, \\eqref{E:gamma-infty} and \\eqref{E:psi-infty} that the harmonic function $\\psi$ is negative in the smooth domain $\\Omega$ and attains the maximum over $\\overline{\\Omega}$ at every point of $\\Gamma$. By the Hopf boundary point lemma, therefore, $\\phi_X=\\psi_Y>0$ at $\\Gamma$. Moreover $\\psi_Y$ is harmonic in $\\Omega$ and $\\psi_Y(X,Y) \\to 1$ as $|(X,Y)| \\to \\infty$ (see \\eqref{E:psi-infty}). The maximum principle then implies that $\\phi_X=\\psi_Y>0$ throughout $\\overline{\\Omega}$. Consequently $\\phi+i\\psi$ makes a conformal bijection from $\\overline{\\Omega}$ onto $  \\overline{\\P}$. Furthermore a streamline $\\{(X,Y) \\in \\Omega: \\psi(X,Y)=-\\psi_0\\}$ for each $\\psi_0 \\in [0,\\infty)$ is the graph of a smooth function on $\\mathbb{R}$. \n\n\\medskip\n\nLet $F:\\mathbb{P}\\to \\Omega$ be a conformal bijection such that $F(\\infty)=\\infty$. Then $F$ extends homeomorphically from $\\overline{\\P}$ onto $\\overline{\\Omega}$ by a famous theorem of Carath\\'eodory (see \\cite[pp. 89]{Gar}, for instance). Since $\\psi \\circ F$ is harmonic in $\\mathbb{P}$, continuous on $\\overline{\\mathbb{P}}$ and nonpositive, moreover, \\cite[Chapter I, Theorem 3.5]{Gar}, for instance, dictates that there exist a constant $\\alpha\\geq 0$ and a Borel measure $\\sigma \\geq 0$ on $\\mathbb{R}$ such that \n\\begin{equation}\\label{E:sigma} \n\\int^\\infty_{-\\infty} \\frac{d\\sigma(t)}{1+t^2}<\\infty \\quad \\text{and}\\quad\n\\psi\\circ F(x+iy)=\\alpha y+\\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\frac{y}{(x-t)^2+y^2}~d\\sigma(t)\\end{equation}\nfor $x+iy\\in\\mathbb{P}$. We claim that $\\sigma \\equiv 0$. \n\nLet $v$ be a continuous function on $\\mathbb{R}$ with bounded support, say $B \\subset \\mathbb{R}$. Clearly \n\\[ \\int^\\infty_{-\\infty} \\left| \\frac{v(x)y}{(x-t)^2+y^2}\\right|dx \\leq \\frac{C}{1+t^2}\\] \nfor some constant $C>0$ uniformly for $y$ near $0$. Since $\\psi \\circ F(x+iy) \\to 0$ as $y \\to 0-$ uniformly for $x$ in any bounded subset of $\\mathbb{R}$ (see \\eqref{E:kinematics-surface}), moreover,\n\\[ \\int^\\infty_{-\\infty} v(x) \\psi \\circ F(x+iy)~dx \\to 0 \\qquad \\text{as $y \\to 0-$}.\\] \nIn view of \\eqref{E:sigma}, therefore, Fubini's theorem and the dominated convergence theorem yield that\n\\begin{align*}\n0=&\\lim_{y\\to 0-} \\int^\\infty_{-\\infty} v(x) \\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\frac{y}{(x-t)^2+y^2}~d\\sigma(t)dx \\\\\n=&\\lim_{y \\to 0-} \\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\int^\\infty_{-\\infty} \\frac{v(x)y}{(x-t)^2+y^2}~dxd\\sigma(t)\n=\\int^\\infty_{-\\infty} v(t)~d\\sigma(t). \\end{align*}\nSince $v$ is arbitrary, this proves the claim. \n\nTo recapitulate, $(\\psi \\circ F)(x+iy)=\\alpha y$ for some $\\alpha\\geq 0$. Since $\\psi \\not\\equiv 0$ and since $(\\phi+i\\psi)\\circ F$ is holomorphic in $\\P$, furthermore, $\\alpha> 0$ and \n\\[((\\phi+i\\psi) \\circ F)(x+iy)=\\alpha(x+iy) +\\beta\\quad\\text{for some $\\beta \\in \\mathbb{R}$.}\\]  \nTo conclude, $F^{-1}=\\alpha^{-1}(\\phi+i\\psi-\\beta)$ and, in turn, $\\phi+i\\psi$ are conformal bijections from $\\overline{\\Omega}$ onto $\\overline{\\P}$ which map $\\infty$ to $\\infty$. Below let $Z=X+iY$ denote the complex coordinate in $\\overline{\\Omega}$ and we use $z=x+iy$ for $\\overline{\\P}$ such that $z=(\\phi+i\\psi)(Z)$. \n\n\\medskip\n\nIt remains to show \\eqref{E:bernoulli} and \\eqref{E:injective}. Let \n\\[\\zeta:=(\\phi+i\\psi)^{-1}:\\overline{\\P} \\to \\overline{\\Omega}.\\] \nThen it is a conformal bijection such that $\\zeta(\\infty)=\\infty$. Furthermore $0<|\\zeta'|<\\infty$ throughout $\\overline{\\P}$. Indeed, since $0<|\\nabla \\psi|<\\infty$ throughout $\\overline{\\Omega}$ by the maximum principle and by hypothesis and \\eqref{E:psi-infty},\n\\begin{equation}\\label{E:zeta'}\\zeta'(z)=(\\phi+i\\psi)' (Z)^{-1}=(\\psi_Y-i\\psi_X)(Z)^{-1}\\end{equation}\nis well-defined for each $z \\in \\overline{\\P}$, where $Z=\\zeta(z) \\in \\overline{\\Omega}$. Consequently $\\overline{\\Omega}$ may be parametrized by $x+iy \\in \\overline{\\P}$ and $\\Gamma$ may be parametrized as $x \\mapsto \\zeta(x+i0)$. \n\nLet $w(x)=\\text{Im}\\,\\zeta(x+i0)$, $x\\in\\mathbb{R}$. We claim that\n\\begin{equation}\\label{E:zeta}\n\\zeta(z)=z+\\mathcal{R}w(z)+\\alpha\\quad \\text{for every $z \\in \\overline{\\P}$}\n\\end{equation} for some $\\alpha \\in \\mathbb{R}$, \nwhere $\\mathcal{R}w$ is given in \\eqref{E:R} and well-defined thanks to the regularity hypothesis. \n\nNote that $\\text{Im}\\,\\zeta'$ is harmonic in $\\P$ and continuous on $\\overline{\\P}$. Since it is bounded throughout $\\overline{\\P}$ (see \\eqref{E:zeta} and \\eqref{E:psi-infty}), moreover, $\\text{Im}\\, \\zeta'(\\cdot +i0)=w'$ by \\cite[Chapter~I, Theorem~3.1]{Gar}, for instance; see Section~\\ref{SS:preliminaries}. Since $w' \\in L^2(\\mathbb{R})$ by hypothesis, on the other hand, the Paley-Wiener theorem (see \\cite[Theorem~11.9]{Duren}, for instance) ensures that $\\mathcal{R}w' \\in \\mathbf{H}^2({\\P})$ exists such that $(\\mathcal{R}w')^*=\\H w'+iw'$ in $L^2(\\mathbb{R})$; recall from \\eqref{E:f*} that $(\\mathcal{R}w')^*$ is the boundary function of $\\mathcal{R}w'$. In addition \\cite[The\\-o\\-rem~11.2]{Duren}, for instance, dictates that $\\mathcal{R}w'$ agrees with the Poisson integral of its boundary function; see Section~\\ref{SS:preliminaries}. Therefore $\\text{Im}\\,\\mathcal{R}w'$ is the Poisson integral of $w'$. Since $w'$ is smooth on $\\mathbb{R}$ in light of a Lewy theorem in \\cite{Lewy}, furthermore, the Fatou lemma (see \\cite[Chapter~I, Theorem~5.3]{Gar}, for instance) asserts that $\\text{Im}\\,\\mathcal{R}w' \\to w'$ everywhere on $\\mathbb{R}$, whence $\\text{Im}\\,\\zeta'\\equiv \\text{Im}\\,\\mathcal{R}w'$ in $\\overline{\\P}$. The Cauchy-Riemann equations then yield that\n\\[ \\zeta'=\\mathcal{R}w'+\\beta \\quad \\text{in $\\overline{\\P}$ for some $\\beta\\in\\mathbb{R}$.}\\] \nSince $\\zeta'(z)\\to 1$ by \\eqref{E:psi-infty}, moreover, $\\beta=1$. An integration therefore reveals \\eqref{E:zeta} for some $\\alpha \\in \\mathbb{C}$. But $\\alpha \\in \\mathbb{R}$ since $\\text{Im}\\,\\zeta(z)-y\\to 0$ as $|z| \\to \\infty$ by \\eqref{E:psi-infty}. This proves the claim. \n\nTo summarize, \\[ \\Gamma=\\{(x+\\H w(x) +\\alpha,w(x)):x \\in \\mathbb{R}\\}\\] \nand the former condition in \\eqref{E:gamma-regular} is written as \\eqref{E:injective}. Upon evaluating \\eqref{E:zeta'} at $z$ on the real axis, correspondingly, \\eqref{E:bernoulli-weak} becomes\n\\[ (1+\\H w'(x))^2+w'(x)^2=|\\zeta'(x)|^2=|\\nabla \\psi(Z(x+i0))|^{-2}=(1-2\\mu w(x))^{-1}\\] \nfor every $x \\in \\mathbb{R}$. Incidentally $1-2\\mu w>0$ everywhere on $\\mathbb{R}$.\\end{proof}\n\nNotice that \\eqref{E:bernoulli} recasts Beroulli's law of constant pressure at the fluid surface (see \\eqref{E:bernoulli-weak}) while \\eqref{E:injective} states that the surface is globally injective. \n\n\\begin{lemma}\\label{P:main<->bernoulli}\n{\\rm (a)} If $w \\in H^1(\\mathbb{R})$ satisfies \\eqref{E:deep} and if $w' \\in L^\\infty(\\mathbb{R})$ then $w$ satisfies \\eqref{E:bernoulli}. \n\n{\\rm (b)} If $w\\in H^1(\\mathbb{R})$ satisfies \\eqref{E:bernoulli} and \\eqref{E:injective} and if $w'$, $\\H w' \\in C(\\mathbb{R})$ then $w$ satisfies \\eqref{E:deep}.\n\\end{lemma}\n\n\\begin{proof} \n(a) The proof is similar to that in the first part of \\cite[Theorem~2.3]{BDT00a}. Hence we merely sketch the detail. \n\nSuppose that $w$ satisfies \\eqref{E:deep} with the stated regularity. Let $u=w-\\mu w^2$ and we rewrite \\eqref{E:deep} as  \n\\begin{equation}\\label{E:gradJ} \n(1-2\\mu w)(1+\\H w')+\\H u'=1. \\end{equation} \nSince $u' \\in L^2(\\mathbb{R})$ by a Sobolev inequality, the Paley-Wiener theorem (see \\cite[The\\-o\\-rem 11.9]{Duren}, for instance) ensures that $\\mathcal{R}u' \\in\\mathbf{H}^2(\\mathbb{P})$ exists such that $(\\mathcal{R}u')^*=\\H u'+iu'$ in $L^2(\\mathbb{R})$; recall from \\eqref{E:f*} that $(\\mathcal{R}u')^*$ is the non-tangential limit of $\\mathcal{R}u'$. Correspondingly $\\mathcal{R}w' \\in \\mathbf{H}^2(\\mathbb{P})$ exists such that $(\\mathcal{R}w')^*=\\H w'+iw'$ in $L^2(\\mathbb{R})$. \n\nLet \\[ V=\\text{Im}\\left(\\left(1-\\mathcal{R}u'\\right) (1+\\mathcal{R}w')\\right)\n=\\text{Im}(-\\mathcal{R}u'+\\mathcal{R}w'-\\mathcal{R}u'\\mathcal{R}w').\\] \nThen $V$ is harmonic in $\\mathbb{P}$ and $V(\\cdot+iy) \\to V(\\cdot+i0)$ in $L^2(\\mathbb{R})$ as $y \\to 0-$. Indeed, since $(\\H u'+iu')(\\H w'+iw')=(\\H u'\\H w'-u'w')+i(u'\\H w'+ w'\\H u')$ and since $u'\\H w'+ w'\\H u' \\in L^2(\\mathbb{R})$, the Cauchy integral formula furnishes that\n\\[\\H(u'\\H w'+w'\\H u')=\\H u'\\H w'-u'w' \\in L^2(\\mathbb{R})\\] \n(alternatively an explicit calculation in the Fourier space reveals the identity);\naccordingly $\\mathcal{R}u'\\mathcal{R}w'=\\mathcal{R}(u'\\H w'+w'\\H u') \\in \\mathbf{H}^2(\\mathbb{P})$ by \\cite[Chapter~II, Corollary~4.2]{Gar}, for instance. Evaluating $V$ on the real axis, we use \\eqref{E:gradJ} to obtain that\n\\begin{align*} \nV(\\cdot+0i)=&\\text{Im}\\left( (1-\\H u'-iu')(1+\\H w'+iw')\\right) \\\\\n=&\\text{Im}\\left((1-2\\mu w)((1+\\H w')^2+(w')^2)\\right)=0\n\\end{align*}\nfor almost all $x \\in \\mathbb{R}$. The Poisson integral technique then enforces that $V\\equiv 0$ in $\\mathbb{P}$.\nIn view of the Cauchy-Riemann equations, consequently, $\\text{Re}((1-\\mathcal{R}u') (1+\\mathcal{R}w'))$\nis constant in $\\mathbb{P}$. Since $(\\mathcal{R}u)^*(x), (\\mathcal{R}w')^*(x) \\to 0$ as $|x| \\to \\infty$, \nmoreover, the constant must be $1$ and\n\\begin{align*} \n1=\\text{Re}\\left(\\left(1-\\mathcal{R}u\\right) (1+\\mathcal{R}w')\\right)= (1-2\\mu w)((1+\\H w')^2+(w')^2)\n\\end{align*}\nalmost everywhere on $\\mathbb{R}$. \n\n\\medskip\n\n(b) Suppose that $w\\in H^1(\\mathbb{R})$ satisfies \\eqref{E:bernoulli}, \\eqref{E:injective} and enjoys the stated properties. Let $W=1+\\mathcal{R}w'$. Then \n\\begin{equation}\\label{E:W*}\nW^*=1+\\H w'+iw'\\end{equation} \neverywhere on $\\mathbb{R}$. Since $0<(1+\\H w')^2+(w')^2<\\infty$ everywhere on $\\mathbb{R}$ by hypothesis, $W \\in C^1(\\overline{\\P}) \\cap \\mathbf{H}^\\infty(\\P)$. Moreover, since $\\text{Re}\\,W^*=1+\\H w'$ is positive everywhere on $\\mathbb{R}$ by hypothesis, the maximum principle implies that $W$ is nowhere zero on $\\overline{\\P}$. Since ${\\displaystyle \\int^\\infty_{-\\infty} \\frac{|\\log |W^*(x)||}{1+x^2}~dx<\\infty}$, furthermore, \\cite[Chapter~II, Theorem~4.6]{Gar}, for instance, asserts that $W$ is an outer function; see at the end of Section~\\ref{SS:preliminaries}. Therefore $1\/W \\in \\mathbf{H}^\\infty(\\mathbb{P})$.\n\nRewriting \\eqref{E:bernoulli} as \\[|W^*|^2=1-2\\mu w,\\] \nsince $1\/W^*-1=(1-2\\mu w)\\overline{W^*}-1 \\in L^2(\\mathbb{R})$ by \\eqref{E:W*} and by a Sobolev inequality, $1\/W-1 \\in \\mathbf{H}^2(\\mathbb{P})$ in light of \\cite[Theorem~11.2]{Duren}, for instance; see Section~\\ref{SS:preliminaries}. The Paley-Wiener theorem (see \\cite[Theorem~11.9]{Duren}, for instance) then ensures that $1\/W^*-1=\\H u+iu$ for some $u \\in L^2(\\mathbb{R})$. At last a straightforward calculation reveals \\eqref{E:deep} since\n\\[ 1+\\H u=(1-2\\mu w)(1+\\H w') \\quad\\text{and}\\quad u=-(1-2\\mu w) w'.\\]\n\\end{proof}\n\n\\begin{corollary}\\label{C:main<->bernoulli}\nSuppose that $w \\in H^1(\\mathbb{R})\\cap C^{1+\\alpha}(\\mathbb{R})$, $\\alpha \\in (0,1)$, satisfies \\eqref{E:injective}. Then, $w$ satisfies \\eqref{E:deep} if and only if it satisfies \\eqref{E:bernoulli}.\n\\end{corollary}\n\n\\begin{proof}Since $w', \\H w' \\in C^\\alpha(\\mathbb{R})$ by Privalov's theorem for the Hilbert transform (see Section~\\ref{SS:preliminaries}), \\eqref{E:bernoulli} ensures that $0<(1+\\H w')^2+(w')^2<\\infty$  everywhere on $\\mathbb{R}$.\\end{proof}\n\nWe state the main result on equivalence.\n\n\\begin{proposition}\\label{T:equivalence}\nSuppose that a curve $\\Gamma$ and a function $\\psi$ defined in the plane below $\\Gamma$ form a regular solution of \\eqref{E:stream} for some $\\mu \\in \\mathbb{R}$. If $w \\in H^1(\\mathbb{R}) \\cap C^{1+\\alpha}(\\mathbb{R})$, $\\alpha \\in (0,1)$, where $w$ is in Proposition~\\ref{P:stream->bernoulli}, then $w$ satisfies \\eqref{E:deep} in addition to the conclusions of Lemma~\\ref{P:stream->bernoulli}. \n\nConversely, suppose that $w\\in H^1(\\mathbb{R}) \\cap C^{1+\\alpha}(\\mathbb{R})$, $\\alpha \\in (0,1)$, satisfies \\eqref{E:deep} \nfor some $\\mu \\in \\mathbb{R}$ as well as \\eqref{E:injective}. Let $\\Gamma=\\{(x+\\H w(x), w(x)): x\\in \\mathbb{R}\\}$ and let $\\Omega$ be the open domain below $\\Gamma$. Then, there exists a conformal bijection $\\Phi: \\Omega \\to \\mathbb{P}$ such that $\\Gamma$, $\\emph{Im}\\,\\Phi$ and $\\mu$ form a regular solution of \\eqref{E:stream}.\n\\end{proposition}\n\n\\begin{proof}\nIf $\\Gamma$ and $\\psi$, $\\mu$ form a regular solution of \\eqref{E:stream} with the stated properties then the former assertion follows from Lemma~\\ref{P:stream->bernoulli} and  Corollary~\\ref{C:main<->bernoulli}. \n\nThe proof of the converse is nearly identical to that in the second part of \\cite[Theorem 2.3]{BDT00a}. Hence we omit the detail. \\end{proof}\n\n\\section{Non-existence}\\label{S:proof}\nAttention is turned to the proof of Theorem \\ref{T:non-existence}. \n\n\\begin{lemma}\\label{L:non-existence}\nIf $w \\in H^1(\\mathbb{R})$ satisfies \\eqref{E:deep} for some $\\mu \\in\\mathbb{R}$ and if $xw' \\in L^2(\\mathbb{R})$ then $w\\equiv 0$. \\end{lemma}\n\n\\begin{proof}\nMultiplying \\eqref{E:deep} by $xw'$ and integrating over $\\mathbb{R}$ we use integration by parts and that the adjoint of $\\H$ is $-\\H$ to obtain that\n\\begin{align}\n\\int^\\infty_{-\\infty} xw' \\H w'~dx =&\\mu\\int^\\infty_{-\\infty} xww'~dx \n+ \\mu\\int^\\infty_{-\\infty}  xww'\\H w'~dx +\\mu\\int^\\infty_{-\\infty} xw' \\H(ww')~dx  \\notag \\\\\n=&-\\frac{\\mu}{2}\\int^\\infty_{-\\infty} w^2~dx\n-\\frac{\\mu}{2}\\int^\\infty_{-\\infty} (w^2)'[\\H,x]w'~dx, \\label{E:pohozaev}\n\\end{align}\nwhere $[\\,,]$ means the commutator. Note that all integrals make sense by hypotheses. In particular $w\\H w', \\H(ww') \\in L^2(\\mathbb{R})$.  \n\nA straightforward calculation (see \\eqref{D:H}) on the other hand reveals that\n\\begin{equation}\\label{E:Hcomm} \n[\\H, xd\/dx]w(x)=[\\H,x]w'(x)=\\frac{1}{\\pi} \\int^\\infty_{-\\infty} w'(x)~dx\\equiv 0.\\end{equation}\nThat is to say, $xd\/dx$ commutes with the Hilbert transform. Accordingly the second term on the right side of \\eqref{E:pohozaev} vanishes. Since\n\\[ \\int^\\infty_{-\\infty} x \\, w'\\H w' dx \n=-\\int^\\infty_{-\\infty} x(\\H w') w' dx -\\int^\\infty_{-\\infty} ([\\H,x]w')w'~dx\\equiv 0,\\]\nmoreover, \\eqref{E:pohozaev} reduces to that ${\\displaystyle \\int^\\infty_{-\\infty} w^2~dx =0}$. \\end{proof}\n\nThe proof is reminiscent of Pohozaev identities techniques. For differential equations (see \\cite{dBS}, for instance) one may further combine it with a truncation argument to promote the conclusion to the $L^\\infty_{loc}$-space setting; to compare, Lemma~\\ref{L:non-existence} is in the $L^2$-space setting with weight. But the Hilbert transform unfortunately does not commute with functions as a rule. \n\nWe instead make an effort to understand the asymptotic behavior of the solution. Note that if $xw'\\in L^2(\\mathbb{R})$ and if $w$ vanishes algebraically at infinity then, necessarily, $w'(x) \\to 0$ faster than $|x|^{-3\/2}$ as $|x| \\to \\infty$.\n\n\\begin{lemma}\\label{L:decay-}\nIf $w\\in H^1(\\mathbb{R})\\cap C^{1+\\alpha}(\\mathbb{R})$, $\\alpha\\in (0,1)$, satisfies \\eqref{E:deep} for some $\\mu<0$ as well as \\eqref{E:injective} then $w'(x)\\leq C(1+x^2)^{-1}$ for every $x \\in \\mathbb{R}$ for some constant $C>0$. \n\\end{lemma}\n\nIn light of Lemma~\\ref{L:non-existence} it proves the second part of Theorem~\\ref{T:non-existence}. \n\nColloquially speaking, in case gravity acts oppositely to what is physically realistic, localized steady waves cannot arise at the surface of a two-dimensional infinitely-deep flow of water so long as the profile lacks self-intersections and cusps. (\\eqref{E:injective} prevents the curve from developing self-intersections while $w \\in C^{1+\\alpha}(\\mathbb{R})$ excludes cusps.) In the Stokes wave setting, incidentally, a non-existence proof based upon duality is found in \\cite{Tol02}. \n\n\\begin{proof}\nSince $1-2\\mu w>0$ everywhere on $\\mathbb{R}$ by Corollary~\\ref{C:main<->bernoulli}, a bootstrapping argument and a Lewy theorem in \\cite{Lewy} lead to that $w$ is real analytic. The proof is nearly identical to that of \\cite[Theorem~3.7]{BDT00a} and hence we omit the detail. Accordingly \\eqref{E:deep}, or equivalently \\eqref{E:bernoulli} by Lemma \\ref{P:main<->bernoulli} (a), is written as\n\\begin{equation}\\label{E:bernoulli'} \n|W^*|^2(1-2\\mu w)\\equiv 1,\\end{equation}\nwhere $W=1+\\mathcal{R}w' $; recall from \\eqref{E:f*} and \\eqref{E:R}, respectively, that $W^*$ is the non-tangential limit of $W$ and that $\\mathcal{R}w'$ holomorphically extends to $\\mathbb{P}$ the boundary function $\\H w'+iw'$. Moreover $\\text{Re}\\,W^*=1+\\H w'>0$ everywhere on $\\mathbb{R}$ by hypothesis.\n\n\\medskip\n\nLet $\\mathcal{L}=\\mathcal{L}(\\mu,w):H^1(\\mathbb{R}) \\to L^2(\\mathbb{R})$ be the linearization of \\eqref{E:deep} at the solution pair $\\mu$ and $w$, defined as \n\\begin{equation}\\label{D:L}\n\\mathcal{L}v=\\H v'-\\mu(v+ w\\H v'+ v\\H w'+\\H(vw)').\\end{equation}\nClearly $\\mathcal{L}w'=0$. Following \\cite[Section~4]{Plo} in the finite-depth case or \\cite[Section~5]{BDT00a} in the Stokes wave setting we introduce the Plotnikov transformation \n\\begin{equation}\\label{E:plotnikov}\nPv:=\\text{Im}\\,(W^*(\\mathcal{R}v)^*)=(1+\\H w')v+w'\\H v.\\end{equation}\nA straightforward calculation then reveals that $P:H^1(\\mathbb{R}) \\to H^1(\\mathbb{R})$ is a ho\\-meo\\-mor\\-phism with the inverse\n\\begin{equation}\\label{E:A-1}v=\\text{Im}\\, \\frac{(\\mathcal{R}Pv)^*}{W^*}.\n\\end{equation}\nMoreover (see \\cite[Theorem~4.2]{Plo} and \\cite[Theorem~5.1]{BDT00a}, for instance)\n\\begin{equation*}\\label{E:linear} \n\\int^\\infty_{-\\infty} \\mathcal{L}PvPu~dx=\\int^\\infty_{-\\infty} (\\H v'-G(x)v)u~dx\\end{equation*}\nfor any $u,v\\in H^1(\\mathbb{R})$, where ${\\displaystyle G=\\text{Im}\\,\\frac{W^{*\\prime}}{W^*}+\\mu |W^*|^2(1+\\H w')}$. Note that $G(x) \\to \\mu$ as $|x| \\to \\infty$ since $w(x),w'(x) \\to 0$ and $W^*(x) \\to 1$ as $|x| \\to \\infty$.\n\n\\medskip\n\nIn view of \\eqref{E:A-1} and \\eqref{E:bernoulli'} we revamp $\\mathcal{L}w'=0$ as\n\\begin{equation}\\label{E:linear'} \n\\H v'+(\\mu-G(x))v=\\mu v,\\qquad v=(1-2\\mu w)w'\\end{equation}\nin the class of distributions. In other words, $v$ is an eigenfunction corresponding to the eigenvalue $\\mu<0$ of  $\\H d\/dx+\\mu-G(x)$. (The essential spectrum of $\\H d\/dx+\\mu-G(x)$ lies in $(0,\\infty)$.) Incidentally $\\H d\/dx=(-d^2\/dx^2)^{1\/2}$ is viewed as a relativistic Schr\\\"odinger operator in one dimension.\nSince $G(x)$ is continuous on $\\mathbb{R}$ ($G$ is real analytic) and since $\\mu -G(x) \\to 0$ as $|x| \\to \\infty$, a result in \\cite[Proposition~4.1]{CaMaSi}, for instance, results in that there exists a constant $C>0$ such that $|v(x)| \\leq C(1+x^2)^{-1}$ for every $x \\in \\mathbb{R}$ for some $C>0$. The proof then completes since $1-2\\mu w$ is positive everywhere on $\\mathbb{R}$ (see the latter in \\eqref{E:linear'}).\n\\end{proof}\n\n\nIn case gravity acts downwards, on the other hand, $\\mu>0$ is contained in the essential spectrum of $\\H d\/dx+\\mu -G(x)$ and hence the preceding proof is not applicable. To attain decay nevertheless, we shall impose a decay condition that guarantees the invertibility of $\\H d\/dx-G(x)$ but which is milder than the quadratic one, and bootstrap the decay rate. \n\n\\begin{lemma}\\label{L:decay+}\nIf $w\\in H^1(\\mathbb{R}) \\cap C^{1+\\alpha}_{1+\\epsilon}(\\mathbb{R})$ for some $\\alpha \\in (0,1)$ and for some $\\epsilon \\in(0,1]$ satisfies \\eqref{E:deep} then $w \\in C^{1+\\alpha}_2(\\mathbb{R})$, where the weighted H\\\"older spaces are in \\eqref{D:wHolder}. In particular $w'(x)\\leq C(1+x^2)^{-1}$ for every $x \\in \\mathbb{R}$ for some $C>0$.\n\\end{lemma}\n\nIn light of Lemma~\\ref{L:non-existence} it proves the first part of Theorem~\\ref{T:non-existence}.\n\n\\begin{proof}The proof closely resembles that in \\cite[Section 4]{Sun97} and hence we merely sketch the detail.\n\nIf $w \\in C^{1+\\alpha}(\\mathbb{R})$ then $\\H w'=\\partial V\/\\partial y(\\cdot,0) \\in C^\\alpha (\\mathbb{R})$, where $V$ is a (weak) solution of the boundary value problem\n\\[ \\Delta V=0 \\quad \\text{in $\\P$}, \\qquad V=w \\quad \\text{on $y=0$}\\]\nand $V(x,y) \\to 0$ as $|(x,y)| \\to \\infty$. Indeed,  \n\\[ \\frac{\\partial V}{\\partial y}(x,y)=\\frac{1}{\\pi}\\int^\\infty_{-\\infty} \\frac{(x-t)w'(t)}{(x-t)^2+y^2}~dt\n\\qquad ((x,y)\\in \\overline{\\P})\\]by the Poisson integral formula while Privalov's theorem for the Hilbert transform (see Section~\\ref{SS:preliminaries}) confirms the regularity. To interpret, $\\H d\/dx:C^{1+\\alpha}(\\mathbb{R}) \\to C^\\alpha(\\mathbb{R})$ is the Dirichlet-to-Neumann operator. Consequently one may associate \\eqref{E:deep}, or equivalently \\eqref{E:bernoulli} by Lemma \\ref{P:main<->bernoulli} (a), with the boundary value problem\n\\begin{subequations}\\label{E:BVP}\n\\begin{alignat}{2}\n\\Delta V=&\\,0\\qquad & &\\text{in } \\mathbb{P}, \\label{E:BVP-laplace}\\\\\nV=\\,&w  \\qquad & &\\text{on }y=0,\\label{E:BVP-dirichlet}\\\\\n\\frac{1}{V_x^2+(1+V_y)^2}&+2\\mu V=1\\qquad & &\\text{on } y=0, \\label{E:BVP-nonlinear}\n\\end{alignat} \n\\end{subequations}\nsubject to that $V(x,y), \\nabla V(x,y) \\to 0$ as $|(x,y)| \\to \\infty$. A straightforward calculation moreover manifests that \\eqref{E:BVP-nonlinear} is written as\n\\[ V_y-\\mu V=(\\mu V-1\/2)(V_x^2+V_y^2)+2\\mu VV_y=:G(V, \\nabla V) \\quad \\text{on $y=0$}.\\]\nTo recapitulate, \\eqref{E:deep} is reformulated as \n\\begin{equation}\\label{E:BVP'} \n\\H V'(\\cdot,0)-\\mu V(\\cdot,0)=G(V(\\cdot,0), \\nabla V(\\cdot,0)).\\end{equation}\n\n\\medskip\n\nWe claim that if $V(\\cdot,0) \\in C^{1+\\alpha}_{1+\\epsilon}(\\mathbb{R})$ satisfies \\eqref{E:BVP'} for some $\\alpha \\in (0,1)$ and for some $\\epsilon \\in (0,1]$ then $V(\\cdot,0) \\in C^{1+\\alpha}_{1+2\\epsilon}(\\mathbb{R})$. That is to say, if a solution of \\eqref{E:BVP'} decays like $x^{-1-\\epsilon}$ as $|x| \\to \\infty$ then it gains the additional decay of $x^{-\\epsilon}$.\n\nSince $G(V(\\cdot,0), \\nabla V(\\cdot,0))$ behaves quadratically as $|V(\\cdot,0)| \\to 0$, \n\\begin{equation}\\label{E:G}\n\\|G(V(\\cdot,0), \\nabla V(\\cdot,0))\\|_{C^{\\alpha}_{2(1+\\epsilon)}(\\mathbb{R})} \\leq \nC\\|V(\\cdot ,0)\\|^2_{C^{1+\\alpha}_{1+\\epsilon}(\\mathbb{R})}\\end{equation}\nfor some constant $C>0$, suggesting us to study the inhomogeneous  linear equation \n\\begin{equation}\\label{E:BVP''}\n\\H V'(\\cdot,0)-\\mu V(\\cdot,0)=G(\\cdot), \\qquad G \\in C^\\alpha_{2(1+\\epsilon)}(\\mathbb{R}). \\end{equation}\nUpon observing that \\eqref{E:BVP''} is the first equation in \\cite[(4.19)]{Sun97} we then run the argument in \\cite[Section~4, Case~II]{Sun97}, for instance, to show that $V(\\cdot,0)\\in C^{1+\\alpha}_{1+2\\epsilon}(\\mathbb{R})$. The idea is that the solution of \\eqref{E:BVP''} is given by the formula \\[ V(x,0)=\n\\frac{1}{\\mu}\\H \\left( \\int_x^\\infty \\sin\\mu (x-t)G(t)~dt\\right)'+\\int_x^\\infty\\sin\\mu(x-t)G(t)~dt,\\] where \nthe kernel associated with $\\H d\/dx$ decays like $x^{-2}$ as $|x|\\to \\infty$. This proves the claim. We refer the reader to \\cite[Section~4, Case~II]{Sun97} for the detail. \n\nTo summarize, if $w \\in C^{1+\\alpha}_{1+\\epsilon}(\\mathbb{R})$ satisfies \\eqref{E:deep} for some $\\alpha \\in (0,1)$ and for some $\\epsilon \\in (0,1]$ then $w=V(\\cdot,0) \\in C^{1+\\alpha}_{1+2\\epsilon}(\\mathbb{R})$, where $V$ solves \\eqref{E:BVP}. We then run a bootstrapping argument in \\cite{CrSt}, for instance, to conclude that $w=V(\\cdot,0) \\in C^{1+\\alpha}_2(\\mathbb{R})$. \n\\end{proof}\n\n\\subsection*{Acknowledgment} The author is supported by the National Science Foundation under grant No. DMS-1008885 and by the University of Illinois at Urbana-Champaign under the Campus Research Board grant No. 11162. She is grateful to anonymous referees for helpful comments.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{PSD simulation of MAXI\/GSC observation}\n\\label{gap}\n\n\n\n{\\bf A--1. How to simulate a MAXI\/GSC light curve}\n\nTo estimate effects of the data gaps and the sampling windows in our Cyg X-1 observation with the MAXI\/GSC,  \nwe make a simulated light curve in units of (counts s$^{-1}$ cm$^{-2}$), \nwhose PSD has a form of $Af^{-1}$.\nThe normalization $A$ is chosen to be close to what has actually been observed with the MAXI\/GSC.\nLet us assume first that the observation is continuous with each data bin having a length $\\Delta t$, \nand set it equal to a typical source transit time, $t_{\\rm scan} = 54$ s.\nAccording to the method described in \\citet{Isobe} and \\citet{Timmer}, the count rate $a_i$ at the $i$-th bin is generated as\n\\begin{eqnarray}\na_{i} = \\frac{2}{\\sqrt{N\\Delta t}}\\sum^{N_{F}-1}_{j=0}\\{c_{i}(f_{j}) + s_{i}(f_{j})\\} & \\rm{(counts~~s^{-1}~~cm^{-2})} \\nonumber \\\\\n                                                       & \\label{eq:A} \\\\\nc_{i}(f_{j}) = R_{{\\rm cos},j}\\cos(2\\pi f_{j}t_{i})   & \\nonumber \\\\\ns_{i}(f_{j}) = R_{{\\rm sin},j}\\sin(2\\pi f_{j}t_{i})   & \\nonumber \\\\\nt_{i} = i\\Delta t ~~~ (i = 0, ..., N-1)              & \\nonumber \\\\\nf_{j} = j\\Delta f ~~~ (j = 0, ..., N_{\\rm F}-1)      & \\nonumber \n\\end{eqnarray}\nHere $N$ is the total time-bin number, \nof the order of $5\\times 10^{5}$ (300d \/ 54s), \n$N_{\\rm F} = N\/2$ is the number of PSD bins, \n$\\Delta f$ is the width of each PSD bin [$\\Delta f = 1\/(N\\Delta t)$], $R_{{\\rm cos},j}$ and $R_{{\\rm sin},j}$ \nare independent gaussian-distributed random numbers with their mean values 0 and their standard deviations $\\sqrt{Af_{i}^{-1}}$, \nwhile $c_i(f_j)$ and $s_i(f_j)$ are the corresponding Fourier components.\nThe DC component, $c_0$, is chosen so as to reproduce the average source count rate.\nThe factor $\\frac{2}{\\sqrt{N\\Delta t}}$ is the coefficient arising from the inverse Fourier transformation.\nThus, \\{$a_i;\\ i = 1,2,...N$\\} provides a simulated light curve, but without including the Poisson error.\n\n\nTo simulate the Poisson noise, \na Poisson distributed random number $b_i$ is generated at the $i$-th bin as\n\\begin{eqnarray} \n  b_i = p(a_{i}\\cdot(S\\cdot t_{\\rm scan}) + \\overline{B})~~~ \\rm{(counts)}.\n\\label{eq:C}\n\\end{eqnarray}\nHere $p(x)$ means a Poisson distributed random number with the mean value of $x$, \n$S$ (cm$^2$) is the average effective area per scan, \n$a_{i}\\cdot(S\\cdot t_{\\rm scan})$ represents the simulated signal counts, \nand $\\overline{B}$ denotes the average background counts per transit of a source.\nWe ignore scan-by-scan scatter in $S$, $t_{\\rm scan}$, and $\\overline{B}$.\nFrom this $b_i$, we subtract $\\overline{B}$, and divide the result by $S\\cdot t_{\\rm scan}$, \nto obtain a simulated data point $Y_i$ as\n\\begin{equation} \n Y_i = (b_i - \\overline{B})\/(S\\cdot t_{\\rm scan})~~~ \\rm{(counts~~s^{-1}~~cm^{-2})}.\n  \\label{eq:Yi}\n\\end{equation}\nAn example of the light curve obtained in this way is shown in figure \\ref{fg:simlc}.\n\\\\\n\\begin{figure*}\n\\begin{center}\n   \\includegraphics[width=16cm]{lc5400.eps}\n \\end{center}\n \\caption{The left figure is a simulated light curve calculated with equation (\\ref{eq:Yi}).\n The right one is after applying the data gaps and filling them by linear interpolations.\n The sampling time is $\\Delta t =$ 5400 s.}\n\\label{fg:simlc}\n\\end{figure*}\n\nWe simulate one hundred light curves composed of continuous 54-s bins, with a total real observation time of 318 d  \nwhich is the longest observation segment in the LHS (data No.1-all in table \\ref{tb:hardem}).\nThese 100 light curves with 54-s bin were Fourier transformed into PSDs, \nwhich were averaged to give the PSD shown in black in figure \\ref{fg:alias+gap}.\nIt recovers the assumed $\\propto f^{-1}$ shape.\n\n{\\bf A--2. Sampling effect}\n\nThe MAXI exposure for a celestial object is far from being continuous.\nA MAXI light curve consists of discrete snap-shot scans, \nwhich are separated by $\\Delta t = 92 {\\rm min}\\simeq 5400 {\\rm s}$ and lasting for $\\sim t_{\\rm scan}$ each.\nIn such sparse-sampling observations with $t_{\\rm scan} \\ll \\Delta t$, significant alias appears in the high frequency end (subsection 3.2.2; \\cite{Kirchner}).\n\n\nTo estimate the effects of aliasing (sampling effects), \nwe retained one every 100th bin, \nand discarded the remaining 99 data points.\nAnalysis of these sparse light curves yielded the red PSD in figure \\ref{fg:alias+gap}.\nThe red points have more power than the black one towards the Nyquist frequency.\nThis is because the power from $10^{-4}$ to $10^{-2}$ Hz, \nwhich was smeared out in the black PSD by averaging over the original 100 data points, \nnow appears in frequencies below $10^{-4}$ Hz.\nThis excess represents the alias power.\n\n\n\n\n{\\bf A--3. Gap effect}\n\nNext, to simulate the observational gaps, \nthe sparse (5400s bin) light curves were multiplied with exactly the same sampling windows as in the actual data (data No.1-all).\nAn example is given in figure \\ref{fg:simlc} (right).\nAfter interpolating these gaps in the same manner as for the actual data, \nthe fake light curves were again Fourier transformed.\nThe obtained PSDs were averaged, and are shown in figure \\ref{fg:alias+gap} in blue.\nThe alias effect is still present.\n\\begin{figure*}\n\\begin{center}\n   \\includegraphics[width=12cm]{hkk_long-54-54gap.eps}\n \\end{center}\n \\caption{Simulations of the alias effects and sampling effects on the PSD.\n The black data are the PSD calculated from 100 fake light curves, \n each continuous with a bin width of 54s.\n The red one is the PSD calculated after sparse sampling \n (once every 5400s), to the original light curves, and blue one further considering the data gaps that are present in the read data.\n The bottom panel shows the transfer-function, \n obtained by the dividing the blue PSD by the black one.}\n\\label{fg:alias+gap}\n\\end{figure*}\n\n\n{\\bf A--4. Correction by a Transfer function}\n\nThe bottom panel of figure \\ref{fg:alias+gap} shows the ratio of the PSD calculated from the sparse and gap-applied light curves, \nmimicking the actual observation, to the original PSD.\nThis is to be called the transfer function.\nWe corrected the sampling and gap effects in the MAXI\/GSC observation, \nby dividing the PSD calculated from the real data by this transfer function.\n\n  \nTo evaluate systematic errors associated with the transfer function, \nwe also performed similar simulations by changing the PSD index to $-2$ and $-0.5$.\nAs expected, the transfer function became flatter ($\\sim 5$ at 10$^{-5}$) for the PSD index $-2$ ($\\sim 1$ at 10$^{-5}$ Hz), \nwhile more deviated from unity ($\\sim 5$ at $10^{-5}$ Hz) when the PSD index is set at $-0.5$.\nHowever, these effects are limited to frequencies above $\\sim 10^{-5}$ Hz.\n\n\n\n\n\n\n\\chapter{Estimate of the Poisson noise}\n\\label{poisson}\n\nThe observed data are subject to statistical fluctuations, or Poisson noise.\nSince this component is independent of the intrinsic source variability,\nthe derived raw PSDs are considered to be a direct sum of\nthe intrinsic and Poissionian contributions.\nAlthough the Poisson noise would usually be a constant,\nthis is not true in the present case,\nbecause of the data gaps, background variations, and other practical effects.\nWe hence estimate the Poisson-noise contributions\nto the PSDs by a numerical simulation;\nthe results were already used in subtraction in section \\ref{sec:psd}\nwhen calculating the PSDs of figure \\ref{fig:cygnpsd} and figure \\ref{fig:avepsd}.\n\nThe aim of the above simulation is to construct light curves\nthat would be observed if Cyg X-1 had a constant intensity throughout.\nThis has been carried out in the following steps.\n\\begin{enumerate}\n\\item[1]\nWe start from calculating the mean intensity $\\bar{y}$ of Cyg X-1\nin a specified energy band,\nas an average over the 5 years of the MAXI data.\n\\item[2]\nThe signal counts to be observed at the $i$-th scan\nof ``non-varying Cyg X-1\"\ncan be expressed as $\\bar{y} (S \\cdot  t_{\\rm scan})_i$,\nwhere the effective area $S$ and the source transit time $t_{\\rm scan}$\nare the same as in Appendix \\ref{gap}.\n\\item[3]\nThe total counts to be detected in the $i$-th scan can be expressed as\n$\\bar{y} (S \\cdot t_{\\rm scan})_i + \\alpha_i B_i$,\nwhere $B_i$ is the background which was estimated scan-by-scan\nand was already utilized in section \\ref{sec:obs} to derive the light curves in figure \\ref{fig:cygLC},\nwhile $\\alpha_i$ is a correction factor for the effective area in the $i$-th scan.\n\\item[4]\nThe above quantity is randomized into\n$b_i \\equiv p\\left( \\bar{y} (S \\cdot t_{\\rm scan} \\right)_i + \\alpha_i B_i)$\nin the same way as in Appendix \\ref{gap}.\nThen, $\\{b_i \\}$ represent the simulated raw counts data,\nbefore the background subtraction,\nto be detected in this fake observation.\n\\item[5]\nFrom this randomized counts $b_i$,\nwe subtract $\\alpha_i B_i$  in the same way as in section \\ref{sec:obs},\nand divide the result by $(S \\cdot t_{\\rm scan})_i$,\nto obtain the simulated light curve $\\{Y_i \\}$\nwhich should be observed from ``non-varying Cyg X-1\".\nThe procedure is summarized as\n\\begin{equation}\n       Y_i = \\left( b_i - \\alpha_i B_i \\right)\/ \\left( S \\cdot t_{\\rm scan} \\right)_i\n       ~~~ (\\rm{counts~s}^{-1}~\\rm{cm}^{-2})~.\n\\end{equation}\n\\end{enumerate}\n\nWe produced the simulated light curves $\\{ Y_i \\}$ 100 times,\nand calculated their PSDs in the same manner as for the actual data.\nIn comparison with the actual 4--10 keV PSD of Cyg X-1 in the HSS (black),\nthe 4--10 keV PSD thus simulated and averaged over the 100 runs\nis shown in figure \\ref{fig:psd+pl} in red.\nSince the source intensity has been assumed to be constant,\nthe red PSD is considered to represent pure Poisson fluctuations.\nWhile this PSD is approximately flat\non long time scales ($< 10^{-7}$ Hz)\nas expected for white noise,\nit decreases towards higher frequencies.\nThis is because the data gaps were filled with linear lines\nwhich do not contain statistical fluctuations:\nthe Poisson noises  are diluted on time scales shorter than those of the data gaps.\nIn any event, the estimated Poisson-noise contribution is\n$<15\\%$ of the signal PSD at all frequencies analyzed here,\nand particularly below a few percent at $<10^{-6}$ Hz.\n\\begin{figure}\n \\begin{center}\n  \\includegraphics[width=8cm]{hkkhosei_psd-pl.eps}\n\t\\hspace{30pt}\n \\end{center}\n \\caption{\nThe simulated 4--10 keV Poisson-noise contribution (red),\ncompared with the 4--10 keV PSD actually observed from Cyg X-1 in the HSS (black).\n}\n\\label{fig:psd+pl}\n\\end{figure}\n\n\n\n\n\n\\chapter{Mass accretion rates through optically thick accretion disk and corona}\n\\label{mdot}\n\nUtilizing schematic drawings in figure \\ref{fig:cartoon}, \nwe consider an accretion model of Cyg X-1 that can explain our results.\nThe optically-thick accretion disk (thick black line) is assumed to extend from outer regions inwards, but is truncated at a certain radius in the LHS.\nMeanwhile, in the HSS it is thought to extend down to the ISCO.\nIn both states, the optically-thin flow is assumed to extend from an outer region ($\\sim 6 \\times 10^{5} R_{\\rm s}$) to the ISCO.\nThe radii, observed low frequency variations of the mass flow are assumed to be \ngenerated in the optically-thin flow at large radii, \nand propagates through the optically-thin flow to the vicinity of the BH to make the hot corona vary on these long time scales.\n\nAfter \\citet{chura}, let us consider mass accretion rate through the optically-thick disk; $\\dot{M}_{d}$ \nand through the optically-thin flow; $\\dot{M}_{c}$.\nThe total accretion rate is given as $\\dot{M}_t = \\dot{M}_d + \\dot{M}_c$.\nWe further assume that \nthe Comptonized component in the energy spectrum is mainly powered by $\\dot{M}_{c}$, \nwhile the disk emission by $\\dot{M}_{d}$.\nLet us describe fractional variations in $\\dot{M}_{c}$ and $\\dot{M}_{d}$ as $f_c$ and $f_d$, respectively, \nand assume that these quantities depend neither  on the frequency nor on the spectral state.\nThen, the observed normalized power should be proportional to $[(f_{c}\\dot{M}_{c})^2+(f_{d}\\dot{M}_{d})^2]\/(\\dot{M}_{t})^2$.\nWhen the optically-thick flow is not variable, $f_{d}$ is equaled to zero.\n\nIn the LHS, the optically-thick accretion disk is truncated at a certain radius, \nand within that radius, only the optically-thin flow exists ($\\dot{M}_{t} = \\dot{M}_{c}$; \\cite{chura}). \nThen, since the time variation above 0.1~Hz originates only from the optically-thin flow,  \nthe normalized power of this range is $f_{c}^2$ itself; NPSD $= f_{c}^2$.\nOn the other hand, in the frequency less than 10$^{-3}$~Hz the NPSD is roughly $\\frac{1}{25}$  of the extrapolation of the NPSD above 0.1~Hz.\nTherefore $\\bigl[\\frac{(f_{c}\\dot{M}_{c})^2}{(\\dot{M}_{t})^2}\\bigr]_{\\rm LHS} = \\frac{1}{25} f_{c}^{2}$, \nwhere $f_{d}=0$ is assumed.\nThe suffix LHS specifies the states.\nSince we assume that $f_{c}$ is independent of frequency, \nwe obtain  $\\bigl[\\frac{\\dot{M}_{c}}{\\dot{M}_{t}}\\bigr]_{\\rm LHS} \\sim\\frac{1}{5}$, \nand hence $\\bigl[\\frac{\\dot{M}_{d}}{\\dot{M}_{t}}\\bigr]_{\\rm LHS} \\sim\\frac{4}{5}$.\nIn the hypothetical model, \nthe PSD flattening at $10^{-3} - 10^{-1}$ Hz (figure \\ref{fig:hkkchura}) is explained that \nthe higher-frequency variations ($> 10^{-1}$ Hz) are produced inside the disk-truncation radius, \nwhile the lower-frequency ones ($< 10^{-3}$ Hz) mainly originate in the region where the two flows co-exist.\n\nSimilar arguments can be performed on the HSS.\nIn the low frequency region below $10^{-4}$~Hz, the normalized power of the HSS, \n$\\bigl[ \\frac{(f_{c}\\dot{M}_{c})^2}{(\\dot{M}_{t})^2} \\bigr]_{\\rm HSS}$, \nis about six times larger than \nthat in the LHS, \n$\\bigl[ \\frac{(f_{c}\\dot{M}_{c})^2}{(\\dot{M}_{t})^2} \\bigr]_{\\rm LHS} = \\frac{1}{25} f_{c}^{2}$ (see figure \\ref{fig:hkkchura}).\nSince we assume that $f_{c}$ does not depend on the states, \n$\\bigl[\\frac{(f_{c}\\dot{M}_{c})^2}{(\\dot{M}_{t})^2}\\bigr]_{\\rm HSS} = \\frac{6}{25} f_{c}^{2}$. \nThen, $\\bigl[\\frac{\\dot{M}_{c}}{\\dot{M}_{t}}\\bigr]_{\\rm HSS} = \\sqrt{\\frac{6}{25}} \\sim \\frac{1}{2}$, \nand hence $\\bigl[\\frac{\\dot{M}_{d}}{\\dot{M}_{t}}\\bigr]_{\\rm HSS} \\sim\\frac{1}{2}$.\nThese values are supported by the fact that  \nthe luminosity of the disk component is roughly the same as that of the Comptonized component in the HSS (see table \\ref{tb:especcomp}).\nThe inferred fractions of the mass accretion rate through the assumed two flows are summarized in figure \\ref{fig:cartoon}.\n\n\\begin{figure*}[bhtp]\n  \\begin{center}\n   \\includegraphics[width=18cm]{CygX1_accmodel_hs.eps}\n  \\end{center}\n  \\caption{Ilustrations of accretion models in the LHS (left) and in the HSS (right).\n  The optically-thick accretion disk is shown with a thick black line, which is truncated at a certain radius in the LHS, \n  but extends down to the ISCO in the HSS.\n  In both states, the optically-thin flow extends up to $\\sim$ 10$^{5}R_{\\rm s}$, \n  while it turns into the Comptonizing corona at small radii. \n  The percentages mean the fractional mass accretion rate in each state.}\n\\label{fig:cartoon}\n\\end{figure*}\n\n\\section{Introduction}\nBlack-hole (BH) X-ray binaries emit X-rays through the mass accretion from their companion stars.\nThe matter accretes onto the BH forming an accretion disk, \nwhich is considered to work as an efficient energy-release engine.  \nThe properties of accretion disks have been studied by both theoretical and observational approaches.  \nFrom the theoretical studies, a disk is considered to evolve from the RIAF (Radiative Inefficient Accretion Flow ; \\cite{Ichimaru77, Narayan94}) state, \nthrough the standard-disk state \\citep{shakura}, up to the slim-disk state\n\\citep{abramowicz88}, as the accretion rate increases.  \nObservationally, galactic BH binaries show two spectral states, \nthe low\/hard state (hereafter LHS) which is dominated  by a power-law spectrum, possibly corresponding to the RIAF,  \nand the high\/soft state (hereafter HSS) which is dominated by an optically-thick thermal emission from an accretion disk, i.e. the standard-disk \\citep{tanaka,remillard2006,done}.\nIn the HSS, a hard tail component is seen in the energy spectrum, which often extends from 10 keV to several MeV by a power-law \\citep{gierlinski}.\nIn the LHS, there is also hard tail extending $\\sim 100$ keV with a clear cut-off,\nwhose origin is considered that thermal Comptonization of disk photons by hot plasma.\nThe origin of hard tail in the HSS is different from that in the LHS, and is still unknown.\n\n\nTime variations of the X-ray intensity are another important aspect of BH X-ray binaries.\nEspecially Cygnus X-1 (hereafter Cyg X-1), the leading galactic BH binary with an orbital period of 5.6 d,\nhas provided rich information on the short-term variability in both states (e.g. \\cite{chura, grinberg}).\nThe power spectrum density (PSD) in the LHS has a power-law shape with an index of about $-1$ in the frequency range from 10$^{-4}$ to 10$^{-2}$ Hz.\nBetween 10$^{-2}$ Hz and 10$^{-1}$ Hz, the PSD is constant as a white noise. \nAbove 10$^{-1}$ Hz, the index returns to about $-1$.\nThere is a break again at around 1 Hz, above which the index becomes steeper (e.g., \\cite{belloni, nowak2000, negoro, pot}).\nThese time scales are thought to reflect the dynamics of accretion flows, and the process of \nenergy dissipation or particle acceleration in a vicinity of the BH.\nThe break at 1 Hz in the LHS can be interpreted as a decay time scale of individual shots, which appear in a light curve \\citep{negoro}.\nIn the HSS, the PSD is approximated by a power-law also with an index of  $-1$,\nbut this form continues, without flattening, from $10^{-4}$ Hz up to 20 Hz, beyond which it steepens.\nThe PSD extending with a constant index $-1$ suggests that the accretion flows in the HSS have no characteristic time scales over the $10^{-4}$ -- 20 Hz range.\n\\citet{chura} explained the overall shape of the PSD in the LHS and the HSS of Cyg X-1 with \na phenomenological model that the optically-thick disk is sandwiched by optically-thin accretion flows extending up to a large distance from the BH. \nIn the LHS the optically-thick disk is truncated at some distances from the BH, \nturning into optically-thin and geometrically thick accretion flows \\citep{makishima}. \n\n\nOn frequencies lower than $\\sim 10^{-4}$ Hz (time scales of 0.1 d), \nthe variability of Cyg X-1 has been studied mainly by applying PSD analyses to the data obtained with RXTE observations. \n\\citet{reig} calculated PSDs of Cyg X-1 using more than eleven years of data obtained with the RXTE\/ASM.\nThey showed that PSDs in the $10^{-7}$ to $10^{-5}$ Hz range can be expressed by a power-law with an index of $-1$, \nand that the PSDs depend on the energy (over 1.3 -- 12.2 keV) \nby a factor of 3 $\\sim$ 5.\nThe long-term variability of hard X-rays from Cyg X-1 was studied by \\citet{Vikhlinin94}, down to 4$\\times$10$^{-4}$ Hz, using  the GRANAT\/SIGMA data.\nThey detected a very low frequency QPO (Quasi-Periodic Oscillation) around  0.04--0.07 Hz.\nHowever, on frequencies of $\\ltsim 10^{-4}$ Hz, our understanding of Cyg X-1 variability has remained much poorer, \nespecially in the HSS, \nthan the rich information accumulated on the short-term variability. \n\n\nThe Monitor of All-sky X-ray Image (MAXI) obtained a long-term light curve of Cyg X-1 for more than five years.\nThe object had been in the LHS until 2010 June, \nand after that it stayed mainly in the HSS \\citep{Negoro10}. \nBy analyzing PSDs and energy spectra from this unique data set, we investigated the characteristics of long-term variations of Cyg X-1 in both states.\n\n\n\n\n\\section{Observation}\n\\label{sec:obs}\nThe data of Cyg X-1 to be analyzed here were obtained with MAXI (\\cite{matsuoka}), \nwhich is attached to the International Space Station.\nAs the International Space Station orbits the earth in every 92 min, MAXI scans over nearly the entire sky with two kinds of X-ray cameras: \nthe Gas Slit Camera (GSC: \\cite{mihara}) covering the energy band of 2--20 keV,\nand the Solid-state Slit Camera (SSC: \\cite{tomida}) covering 0.7--7 keV.\n\nFrom the MAXI home page \\footnote{\\url{http:\/\/maxi.riken.jp\/top\/index.php?cid=1&jname=J1958+352}}, we downloaded one-day bin and 90-minute bin archival data of Cyg X-1.\nThese archival data were selected on the condition that X-ray incident angle to a camera is smaller than 38 $^\\circ$ and scan times of the source and the background are both longer than 15 s.\nEnergy spectra of the GSC and SSC were processed by the MAXI on-demand data web page\\footnote{\\url{http:\/\/maxi.riken.jp\/mxondem\/}} (\\cite{nakahira}).\nWe selected $2^{\\circ}.0$ for a source region and $3^{\\circ}.0$ for a background region.\n\nFigure \\ref{fig:cygLC} shows one-day bin light curves of Cyg X-1 obtained with the GSC from 2009 August 15 (MJD = 55058) to 2014 November 9 (MJD = 56970),\nin three energy bands (2--4 keV, 4--10 keV and 10--20 keV).\nTime histories of two kinds of hardness ratios (HR), $I$(4--10 keV)\/$I$(2--4 keV) and $I$(10--20 keV)\/$I$(4--10 keV), are also plotted.\nThe state of Cyg X-1 can be recognized by the values of the HR.\n\n\\begin{figure*}\n  \\begin{center}\n   \\includegraphics[width=16cm]{3bandlc_clean.eps}\\\\\n   \\vspace{1cm}\n   \\includegraphics[width=16cm]{hkklc4_h-s_2dan_log_lin_3d.eps}\n  \\end{center}\n  \\caption{(a) One-day bin light curves and HR histories of Cyg X-1 obtained with the MAXI\/GSC.\n  From the top to bottom panels, the 2--4 keV, 4--10 keV and 10--20 keV intensities,\n  and the $I$(4--10 keV)\/$I$(2--4 keV) and $I$(10--20 keV)\/$I$(4--10 keV) ratios are plotted.\n  The black and gray regions at the top indicate the LHS and the HSS periods, respectively.\n  Data points with large error bars, due to high background counts, were omitted.\n  (b) Expanded 10--20 keV light curves (three-day bin) of Data No.1-all and No.8-all,\n  representing the LHS and the HSS, respectively.\n  Ordinates are logarithmic.\n  (c) The same as panel (b), but with linear scales.\n  Data No.1-all and data No.8-all are shown LHS and HSS, respectively.\n  The data No. is indicated in top panel, \n  and the corresponding MJDs are listed in table \\ref{tb:hardem} and table \\ref{tb:softem}.\n  }\n\\label{fig:cygLC}\n\\end{figure*}\n\nSince the start of the MAXI observation in 2009 August, Cyg X-1 was in the LHS for about ten months.\nAt around MJD = 55378, the source made a transition to the HSS, where it stayed for another ten months.\nAfter several times of state transitions, it has been mainly in the HSS since MJD = 56078.\n\n\n\\section{Analysis and results}\n\\subsection{Definition of spectral states}\nAs shown in the HR histories in figure \\ref{fig:cygLC}, Cyg X-1 exhibited distinct LHS and HSS.\nSince the two HR histories have the same trend, we hereafter use the $I$(4--10 keV)\/$I$(2--4 keV) ratio, which has better statistics. \nThe top panel of figure \\ref{fig:cygHDI} shows a hardness-intensity diagram.\nThus the two states are clearly separated.\nThe bottom panel in figure \\ref{fig:cygHDI} shows a histogram of the HR, \nwhich exhibits two clear peaks corresponding to the HSS and the LHS.\nWe fit the histogram with two gaussian functions and determined their mean values and standard deviations.\nThen we defined the boundaries of each state as 3$\\sigma$ from the gaussian center, i.e.\nHR$<$0.43 for the HSS and 0.48$<$HR for the LHS.\n\nFrom figure \\ref{fig:cygLC} and figure \\ref{fig:cygHDI}, the periods of the two states have been obtained as listed in table \\ref{tb:trans}.\n\\begin{figure}[h]\n  \\begin{center}\n   \\includegraphics[width=8cm]{int-col_hst.eps}\n  \\end{center}\n  \\caption{\n  The hardness-intensity diagram (top panel) and the histogram of the 4--10 keV vs. 2--4 keV HR (bottom panel).\n  Blue and red data points specify the LHS and the HSS, respectively, while black points were taken during transitions.\n  }\n\\label{fig:cygHDI}\n\\end{figure}\n\n\\begin{table}\n\\small\n\\caption{Spectral states of Cyg X-1 during the MAXI observation.\nData numbers are indicated in figure \\ref{fig:cygLC}.\n}\n\\begin{center}\n\\begin{tabular}[b]{c c c c c c}\n\\hline\n\\hline\ndata         & spectrum & start & end & duration  \\\\ \nNo.          & state    & MJD   & MJD & day       \\\\\n\\hline\n1 & hard & 55058 & 55376 & 318 \\\\ \n2 & soft & 55378 & 55673 & 295 \\\\ \n3 & hard & 55680 & 55788 & 108 \\\\\n4 & soft & 55789 & 55887 & 98  \\\\ \n5 & hard & 55912 & 55941 & 29  \\\\\n6 & soft & 55943 & 56068 & 125 \\\\\n7 & hard & 56069 & 56076 & 7   \\\\\n8 & soft & 56078 & 56733 & 655 \\\\\n9 & hard & 56735 & 56741 & 6   \\\\\n10 & soft & 56742 & 56757 & 15 \\\\ \n11 & hard & 56781 & 56824 & 43 \\\\\n12 & soft & 56854 & 56970$\\sim$ & 116$\\sim$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tb:trans}\n\\end{table}\n\n\n\n\\subsection{Power spectra}\n\\label{sec:psd}\n\n\\subsubsection{Definition of PSDs}\n\nThe X-ray light curve with 90-minute bin was converted to the PSD by the discrete Fourier transformation as\n\\begin{eqnarray}\nF_{\\rm c}(f) = \\frac{2}{N}\\sum_{j=1}^{N}y_{j}\\cos(2\\pi f\\Delta t\\times j) \\nonumber \\\\\nF_{\\rm s}(f) = \\frac{2}{N}\\sum_{j=1}^{N}y_{j}\\sin(2\\pi f\\Delta t\\times j)\n\\label{eq:sincos}\n\\end{eqnarray}\nwhere $y_{i}$ is the intensity (photons s$^{-1}$ cm$^{-2}$) at the $i$-th bin, $N$ is the number of data after filling the gaps as described later, \nand $\\Delta t$ is a time interval of 5400 s, which is the regular sampling time of the MAXI public data, \n$f$ is the frequency which is an integer multiple of $\\Delta f$=$\\frac{1}{T}$, and  \n$T$ is the total time span of the observation.\nThe factor of $2\/N$ is employed so that $F_{c}(f)$ and $F_{s}(f)$ represent the amplitude of the cosine and sine components, respectively, regardless of $N$. \nThe PSD is the sum of squares of $F_{\\rm c}$ and $F_{\\rm s}$ as, \n\\begin{equation}\n  P(f) = \\frac{T}{2}\\bigl\\{F_{\\rm c}^2(f) + F_{\\rm s}^2(f)\\bigr\\}.\n\\label{eq:PSD}\n\\end{equation}\nBy multiplying with a factor of $T\/2$, the derived PSD becomes independent of $T$, and has a unit of (RMS$^2$\/Hz).\nThe PSD is normalized by the square of the average intensity following \\citet{miyamoto1994}.\nIt is called normalized PSD (NPSD).\n\n\n\\subsubsection{Corrections for sampling and gap effects}\n\\label{psd_effect}\n\nThe MAXI data are not completely regular-sampled.\nSometimes data gaps are caused by sun avoidance, high particle background regions,\n small dead regions at the scan poles which move with the precession period of the orbit of the International Space Station, and other effects.\nTherefore, we interpolated each gap with a linear line, \nwhich connects the pre-gap intensity (averaged over 5 data points) and the post-gap value (also averaged over 5 data points) \\citep{sugimoto}.\nFurthermore, the derived PSDs are strongly affected by aliasing effects, because MAXI measures the intensity of an X-ray source only for $40\\sim 70$ s, every 5400 s period.\nThat is, each data point is a very short snapshot, with a long interval to the next (or from the preceding) sampling.\nAs a result, the MAXI light curves are generally sensitive to source variations not only on time scale longer than 5400 s, \nbut also to these in between $40\\sim 70$ s and $\\sim 5400$ s.\n\n\nIn order to correct the PSDs for the effects of gaps (hereafter gap effects), and those of the short exposure with long-interval sampling (hereafter sampling effects), \nwe simulated MAXI data of a variable source with specified variability.\nSince the PSD of Cyg X-1 can be approximated by a power-law with an index of $-1$ (\\cite{reig}; \\cite{chura}),\nthis form of PSD was employed as the input variability.\nFurther assuming that the phases of the Fourier components are random, \nwe produced a hundred fake light curves, each of which has 54 s time bin and covers the same time span as figure \\ref{fig:cygLC}.\nDetails of the PSD simulation are given in Appendix \\ref{gap}.\nThen, for each simulation run, we picked up one data point from every 100 bins (= 5400 s = one orbit) and discarded the 99 points \nto simulate the sampling effects, and applied the same observing window as for Cyg X-1 in order to reproduce the gap effects.\nInterpolating the gaps in the same way as for the actual data, a PSD was calculated from a fake light curve.\nFinally, we took an average of the 100 simulated PSDs, and normalized it to the assumed input PSD, to obtain the transfer function \n(in Fourier space; figure \\ref{fg:alias+gap} in Appendix \\ref{gap}) \nof the present MAXI observation of Cyg X-1.\nBelow, PSDs obtained from the observed data are divided by the transfer function thus obtained, \nin order to correct for the gap effects and the sampling effects.\nPrior to this division, we subtract the Poisson noise, \nwhich is estimated by a numerical simulation (Appendix \\ref{poisson}). \n\n\n\n\\subsubsection{Results (1): using the longest data length}\n\nFor both the LHS and HSS, we calculated NPSDs from the light curves in the three energy bands, 2--4 keV, 4--10 keV and 10--20 keV.\nIn calculation, the longest span of data was used for each state, \ni.e. 318 days for the LHS \nand 477 days for the HSS.\nThese data segments are shown in table \\ref{tb:hardem} (as No. 1-all) and table \\ref{tb:softem} (as No. 8-all), respectively, together with other shorter segments available. \nOnly the GSC data were used, because the SSC data are much more sparse than those of the GSC.\nIn the HSS, the selected span is shorter than Data No. 8 in table \\ref{tb:trans} because short data bunches at the beginning and the end of the span were removed.\nAfter subtracting the Poisson noise and correcting for \tthe gap and the sampling effects (section \\ref{psd_effect}), \nwe obtained NPSDs down to 3$\\times$10$^{-8}$ Hz as shown in figure \\ref{fig:cygnpsd}.\nThe error of each data point was obtained by propagating errors of the observed light curve.\nThe NPSDs extend roughly with a power-law shape down to 3$\\times$10$^{-8}$ Hz, \nbut they scatter largely in the low frequency ($\\leq 3\\times10^{-7}$ Hz) region.\n\n\n\n\\subsubsection{Results (2): using shorter data lengths}\n\\label{psdresult2}\n\nBy shortening the data length to be used in a Fourier transform by equation (\\ref{eq:sincos}), \nwe can produce a larger number of NPSDs, and take their average. \nThe NPSD obtained in this way is expected to suffer smaller statistical errors, \nalthough it lacks the information in the lowest frequencies.\nThus, the data span was restricted to 40 and 43 days for the LHS and the HSS, respectively, to cover a frequency range down to $\\sim$ 3$\\times$10$^{-7}$ Hz \nat the sacrifice of the $3\\times$10$^{-8} - 3\\times$10$^{-7}$ Hz range.\nBy doing so, the fraction of data gaps was reduced from $\\sim$ 50 \\% to $\\sim$ 10\\%. \nAs listed in table \\ref{tb:hardem}, six data segments were extracted for the LHS, and eight data segments in table \\ref{tb:softem} for the HSS.\nThen, we converted these data segments individually into NPSDs, and took their averages, separately over the LHS and HSS.\nThe obtained ensemble-averaged NPSDs are shown in figure \\ref{fig:avepsd}, where the vertical error bars represent the NPSD scatter (standard deviation) within each ensemble.\nIn figure \\ref{fig:avepsd}, ordinate employs power times frequency, \ninstead of power itself.\n\n\nAs expected, the NPSDs in figure \\ref{fig:avepsd} are less scattered even in the low frequency range\nbelow $10^{-6}$ Hz, and show similar structures in all the energy bands, \nalthough they could still be subject to some structures including shallow dips around $3\\times 10^{-6}$ Hz, \nand the weak tendency of flattening below $3\\times 10^{-7}$ Hz.\nAs shown by a simulation in Appendix \\ref{gap}, the possible flattening is unlikely to be caused by the data gaps.\nInstead, it could be within the scatter of NPSDs, \nconsidering that the effect is seen in only the lowest-frequency data points in both states.\t\nWe also note that the energy dependence in figure \\ref{fig:cygnpsd} and figure \\ref{fig:avepsd} is slightly different.\nEspecially, the normalization of the 4--10 keV NPSD in the HSS in figure \\ref{fig:cygnpsd} is higher\nthan that in figure \\ref{fig:avepsd}.\nConsidering the ensemble-averaging procedure involved in the NPSDs in figure \\ref{fig:avepsd}, \nwe regard them as better representing the Cyg X-1 variability than those in figure \\ref{fig:cygnpsd}.\n\n\n\n\\begin{table}\n\\small\n\\caption{Data segments used in calculating NPSD in the LHSs.\n}\n\\begin{center}\n\\begin{tabular}[b]{c c c}\n\\hline\ndata         & start & end \\\\ \nNo.$^{*}$           & MJD   & MJD \\\\\n\\hline\n1-all & 55058 &  55376  \\\\\n\\hline\n1-1  &  55060  &  55100  \\\\ \n1-2  &  55138  &  55178  \\\\ \n1-3  &  55209  &  55249  \\\\\n1-4  &  55277  &  55317  \\\\\n3-1  &  55700  &  55740  \\\\\n11-1  &  56781  &  56821  \\\\\n\\hline \\\\\n\\end{tabular}\n\\end{center}\n* : ``$I$-$J$\" denotes the $J$-th segment in the $I$-th data.\n``$I$-all\" means as the segments in the $I$-th data number.\n\\label{tb:hardem}\n\\end{table}\n\n\\begin{table}\n\\small\n\\caption{The same as table \\ref{tb:hardem} but for the HSS.}\n\\begin{center}\n\\begin{tabular}[b]{c c c}\n\\hline\ndata         & start & end \\\\ \nNo.       & MJD   & MJD \\\\\n\\hline\n8-all & 56130 & 56607 \\\\\n\\hline\n2-1  &  55487 & 55530  \\\\ \n2-2  &  55623 & 55666  \\\\ \n8-1  &  56201 & 56244  \\\\\n8-2  &  56273 & 56316  \\\\\n8-3  &  56418 & 56461  \\\\\n8-4  &  56564 & 56607  \\\\\n8-5  &  56637 & 56680  \\\\\n12-1 &  56926 & 56969  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tb:softem}\n\\end{table}\n\n\n\\begin{figure*}\n  \\begin{center}\n   \\includegraphics[width=8cm]{hkkNPSDh_gaphosei_lmh.eps}\n   \\hspace{5pt}\n   \\includegraphics[width=8cm]{hkkNPSDs_gaphosei_lmh.eps}\n  \\end{center}\n  \\caption{NPSDs of Cyg X-1 in the LHS (left) and in the HSS (right), each calculated using the longest data span (No. 1-all in table \\ref{tb:hardem} and No. 8-all in table \\ref{tb:softem}).\n  The blue, black and red points represent NPSDs in the 2--4 keV, 4--10 keV and 10--20 keV band, respectively.\n  The vertical line at $2\\times10^{-6}$ Hz indicates the 5.6-day orbital period.}\n\\label{fig:cygnpsd}\n\\end{figure*}\n\n\n\\begin{figure*}\n  \\begin{center}\n   \\includegraphics[width=8cm]{hkkfNPSDh_gaphosei_lmh.eps}\n   \\hspace{5pt}\n   \\includegraphics[width=8cm]{hkkfNPSDs8_gaphosei_lmh.eps}   \n     \\end{center}\n  \\caption{\n  The NPSDs calculated over shorter data segments (40 d $\\times$ 6 for the LHS and 43 d $\\times$ 8 for the HSS), and averaged. \n  Here, the NPSDs are shown after multiplying with the frequency. \t\n  }\n\\label{fig:avepsd}\n\\end{figure*}\n\n\nAs seen in figure \\ref{fig:avepsd},\nthe NPSD slope does not appear to depend significantly either on the spectral state or the energy band.\nTo confirm this suggestion, we fitted the individual NPDSs with a power-law, over a frequency range of $10^{-6}$ and $9.2\\times10^{-5}$ Hz.\nThen, the 18 NPSDs before taking ensemble averages (6 segments times 3 energy bands) in the LHS gave an average slope and the associated RMS scatter as $-1.35\\pm 0.29$, \nwhereas the 24 HSS ones (8 segments times 3 energy bands) gave $-1.29 \\pm 0.23$.\nTherefore, the NPSD slope is consistent with being independent of the state.\nIn addition, slope differences among the 3 energy bands are at most within the scatter of $\\sim 0.3$ (in the LHS) and $\\sim 0.2$ (in the HSS).\n\n\nFigure \\ref{fig:avepsd} reveals two more important properties. \nOne is that the NPSD normalizations in the HSS are significantly energy dependent, increasing towards higher energies, \nwhereas those in the LHS show a much weaker and opposite trend.\nThe other is that Cyg X-1 is much more variable, at least above 4 keV, on the relevant time scales while it is in the HSS than in the LHS.\nTo quantify these properties, \nwe integrated the 6 (3 energy bands times two states) ensemble-averaged NPSDs from $10^{-6}$ to $9.2\\times10^{-5}$ Hz,\nand derived individually the fractional RMS variation $\\eta$ as \n\\begin{equation}\n  \\eta = \\sqrt{2\\Delta f \\sum_{i=1}^{N_{F}}NPSD(f_i)}\n\\label{eq:rms}\n\\end{equation}\nwhere $N_{F}$ is the number of data points in each NPSD.\nThe results, given in table \\ref{tb:rms}, indeed confirm the above inferences\n: the fractional RMS variation in 10--20 keV in the HSS ($\\eta = 0.70$) is larger than that in the LHS ($\\eta = 0.21$).\nThis property can be read directly from the light curve in a logarithmic scale presented in figure \\ref{fig:cygLC}b. \nMeanwhile, when plotted in a linear scale (figure \\ref{fig:cygLC}c), the absolute amplitude of variability is similar between the LHS and HSS.\nThese two properties can be explained as follows.\nThe 10--20 keV value of $\\eta$ is $0.70\/0.21\\sim 3.3$ times higher in the HSS than in the LHS, \nwhile the average 10--20 keV intensities (as read from figure \\ref{fig:cygLC}c) in the HSS is $0.089 \/ 0.19\\sim 0.47$ times that in the LHS.\nTherefore the absolute amplitude of variations in the HSS should be $3.3\\times 0.47 = 1.55$ times that in the LHS.\nThis is close to unity, though not exactly the same.\n\n\\begin{table*}\n\\small\n\\caption{The fractional RMS $\\eta$ of NPSDs from $10^{-6}$ to $9.2\\times10^{-5}$ Hz.}\n\\begin{center}\n\\begin{tabular}[b]{c c c c c c c}\n\\hline\nSpectral state        &                 &    LHS          &                  &                &    HSS          &               \\\\ \n\\hline\nEnergy band (keV)     &       2-4       &      4-10       &      10-20      &        2-4      &       4-10      &     10-20      \\\\\n\\hline\n$\\eta$                &  0.31$\\pm$0.09  &  0.24$\\pm$0.05  &   0.21$\\pm$0.04 &  0.31$\\pm$0.09  & 0.53$\\pm$0.12   &  0.70$\\pm$0.29\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tb:rms}\n\\end{table*}\n\n\n\nIn the LHS, a peak corresponding to the orbital period of 5.60 d is \nclearly seen in the low energy band (2--4 keV) at $2.06\\times$10$^{-6}$ Hz.\nHowever, it is not clear in the other two energy bands in the LHS, \nin agreement with the Ginga\/ASM results by \\citet{kitamoto2000}.\nIn the HSS, the orbital period is not clearly seen in any energy band.\n\n\nFigure \\ref{fig:hkkchura} shows our 4--10 keV NPSDs in both states, \nin comparison with previous works by \\citet{cui1997}, \\citet{pot}, and \\citet{reig}.\nIn both states, our results at 10$^{-4}$ Hz locate on simple extrapolations of the NPSDs \nobtained previously by the RXTE\/PCA in the frequency region above 10$^{-3}$ Hz.\nThe figure reconfirms and visualizes the higher variability in the HSS, \nalready presented in table \\ref{tb:rms}.\nThe NPSD in \\citet{reig} is located between the two states of the present work.\nThis difference is presumably because their data include some of the HSS and the transition periods, \nalthough the main part is in the LHS.\n\n\n\n\\begin{figure*}[htbp]\n  \\begin{center}\n   \\includegraphics[width=12cm]{hkk_hard_soft8_pot_cui_reig.eps}\n  \\end{center}\n  \\caption{A comparison of the MAXI 4--10 keV NPSDs in the LHS (blue) and the HSS (red), with the previous works.\n  The dashed line is the 2--13 keV NPSD in the LHS by \\citet{pot}, the dotted line is that in the HSS in 2--13 keV by \\citet{cui1997}, \n  and the dot-dashed line is that by \\citet{reig} where the most data are in the LHS.\n  }\n\\label{fig:hkkchura}\n\\end{figure*}\n\n\n\\subsection{Energy Spectra}\n\\label{energy}\n\nIn order to investigate the spectral components that are responsible for the observed long-term variations,\nwe analyzed energy spectra of Cyg X-1 obtained with the MAXI\/GSC and the SSC.\nWe used the same periods as in the PSD analysis,\ni.e. from MJD = 55058 to MJD = 55376 for the LHS,\nand from MJD = 56130 to MJD = 56607 for the HSS.\nThe SSC data, available only for about one third of the time, were also incorporated.\n\n\nIn section \\ref{sec:psd}, we found that the long-term variability of Cyg X-1,\nparticularly in the HSS, is significantly energy dependent.\nTo reconfirm the implied spectral variation in this frequency range,\nwe divided the observed data into bright and faint periods\nby comparing the individual one-day GSC intensities in 2--20 keV with their 15-day running averages.\nSince the 15-day time scale corresponds to a frequency of $8\\times 10^{-7}$ Hz, \nthis procedure means an extraction of time variations longer than $8\\times 10^{-7}$ Hz.\nThen, eight spectra in total were produced;\nthe GSC and SSC data in the bright and faint periods, from the LHS and HSS.\nThe background-subtracted eight MAXI spectra obtained in this way are shown in figure \\ref{fig:cygespeccomp}.\nThus, as expected, the HSS spectra are much softer than those of the LHS,\nin good agreement with the general understanding of the spectral states.\nFor reference, these spectra include periods of increased low-energy absorption (seen as dips in the 2--4 keV light curves), \nwhich often appear for $\\sim 15\\%$ of the orbital phase.\n\n\\begin{figure*}[hptb]\n  \\begin{center}\n   \\includegraphics[width=8cm]{hard_unfolded.eps}   \n   \\includegraphics[width=8cm]{soft_unfolded.eps}\n  \\end{center}\n  \\caption{Background-subtracted unfolded MAXI spectra in the LHS (left), and the HSS (right).\n  Red and black indicate the bright-period data of the SSC and GSC respectively, \n  while blue and green indicate the faint periods of the SSC and GSC, respectively.\n  The model is {\\em phabs*(diskbb+nthComp+gaussian)}, \n  with the parameters listed in table \\ref{tb:especcomp}.\n  The 1.5--2 keV energy range is ignored in the fitting.\n  The dotted lines represent contributions of the three components.\n  The bottom panels are residuals from the model.}\n\\label{fig:cygespeccomp}\n\\end{figure*}\n\nTo better visualize the intensity-correlated spectral changes in figure \\ref{fig:cygespeccomp},\nwe present, in figure \\ref{fig:pha},\nthe ratios of the GSC spectrum in the bright period to that in the faint period.\nThe SSC data are not shown here, because the errors are large.\nIn the LHS, the ratio decreases slightly with energy,\nwhereas the HSS ratio increases significantly towards higher energies.\nTo quantify these spectral results and examine their consistency with table \\ref{tb:rms},\nwe  calculated the ratio $R$ of the photon flux in the bright period $F_{\\rm b}$ to that in the faint period $F_{\\rm f}$, and show the results in table \\ref{tb:specrms}. \nWe may express $R$ as\n\\begin{equation}\nR = F_{\\rm b}\/F_{\\rm f} = (\\overline{F} +\\Delta F)\/(\\overline{F}-\\Delta F) ,\n\\label{eq:ratio}\n\\end{equation}\nwhere $\\overline{F} \\equiv \\frac{F_{\\rm b}+F_{\\rm f}}{2}$ is the average photon flux in the specified energy band\nand $\\Delta F \\equiv \\frac{F_{\\rm b}-F_{\\rm f}}{2}$ denotes the variable part.\nIn this formalism, the fractional RMS variation can be given as\n\\begin{equation}\n \\eta' \\equiv  a\\Delta F\/\\overline{F} = a (R-1)\/(R+1)\n  \\label{eq:eta}\n\\end{equation}\nwhere $a$ is an appropriate numerical factor of order unity,\nwhich can be approximated as energy independent.\nThis $a$ was introduced because the variation $\\eta$ of equation (\\ref{eq:rms}) calculated from NPSD can be different, by a constant factor, from that derived by equation (\\ref{eq:eta}).\nThe derived values of $\\eta'$ are also shown in table \\ref{tb:specrms},\nwere $a=2.9$ has been adopted. \nThus, the values of $\\eta'$ agree, within errors,\nwith $\\eta$ in table \\ref{tb:rms} which is based on the PSD  analysis.\n\n\\begin{figure*}[htbp]\n  \\begin{center}\n   \\includegraphics[width=16cm]{hkkphav2.eps}\n  \\end{center}\n  \\caption{Ratios of the GSC spectra in the bright period to that in the faint period,\n  in the LHS (left panel), and in the HSS (right panel).\n  }\n\\label{fig:pha}\n\\end{figure*}\n\n\n\nTo understand the physical origin of the behavior seen in figure \\ref{fig:pha},\nas well as in table \\ref{tb:rms} and table \\ref{tb:specrms}, we proceeded to simultaneous model fitting to the 2--20 keV GSC spectrum\nand the 0.7--7 keV SSC spectrum.\nThe 1.5--2 keV range of the SSC was excluded\nto avoid the known systematic uncertainty in the effective area (\\cite{tomida}).\nFirst, we employed a model composed of a multi-color disk ({\\em diskbb}) model (\\cite{shakura, mitsuda}) for the low-energy part,\nand a power-law ({\\em powerlaw}) as a hard tail.\nThis model has often been used in the previous studies of BH X-ray binaries including Cyg X-1 (e.g. \\cite{dotani}).\nThe photo-electric absorption model ({\\em phabs}) with abundances by \\citet{anders}\nand the Fe-K emission line model ({\\em gaussian}) were incorporated.\nThe model then becomes {\\em phabs$\\times$(diskbb+powerlaw+gaussian)} in the Xspec (\\cite{arnaud96}) terminology; \nhereafter, we call it {\\em diskbb+powerlaw} model.\nThe innermost radius of {\\em diskbb} is derived as $r_{\\rm{in}} = D\\sqrt{N_{\\rm{diskbb}}\/\\cos{i}}$,\nwhere $N_{\\rm{diskbb}}$ is the normalization of the {\\em diskbb} model.\nThe distance $D$ from the earth and the inclination angle $i$ from our line of sight\nwere adopted to be 1.86 kpc and 27$^{\\circ}$, respectively \\citep{orosz}.\nThe hydrogen column density $N_{\\rm H}$ and the gaussian parameters\nwere constrained to be common between the faint and the bright spectra  in each state,\nbut were allowed to differ between the two states.\n\n\nThe fit has been approximately successful,\nand yielded the best-fit parameters as summarized in table \\ref{tb:especpow}.\nThus, in both states, the  {\\em diskbb} parameters are the same, within errors,\nbetween the bright and faint spectra.\nIn the LHS, the {\\em powerlaw} index is steeper in the bright spectrum than in the faint one,\nwhile the opposite trend is found in  the HSS.\nThese are consistent with the spectral ratio shown in figure \\ref{fig:pha}.\nThe $r_{\\rm in}$ value for the LHS is consistent with that from a previous work \\citep{yamada}.\nOn the other hand,  that for the HSS is smaller than that reported, e.g., in \\citet{dotani} (even after corrections for the distance and the mass),\npresumably because the luminosity in the present result\nis roughly half  those in typical HSS observations made previously \\citep{zhang}.\n\nReferring to equation (\\ref{eq:eta}), we calculated $\\eta'$ also for the {\\em powerlaw} component only, and show the results in table \\ref{tb:specrms}. \nA comparison of the two values of $\\eta'$, one from the entire photon counts while the other from the {\\em powerlaw} contribution, \nreveals that the overall source variability in $> 4$ keV is determined solely by the power law variation, \nin agreement with the spectral decomposition in figure \\ref{fig:cygespeccomp} (right).\nIn the 2--4 keV band, in contrast, the overall  $\\eta'$ is reduced to $\\sim 60\\%$  of the {\\em powerlaw} variability; \nthis effect is readily attributed to the presence of the stable disk component.\nIn addition, table \\ref{tb:specrms} reveals slight increase of  $\\eta'$ (both total and {\\em powerlaw}) from the 10-\u201320 keV to 4--10 keV bands.\nThe behavior of $\\eta'$ agrees with that of $\\eta$ in table \\ref{tb:rms} and means that the {\\em powerlaw} slope slightly hardens as it gets stronger, \nas visualized in figure \\ref{fig:pha} (right) and quantified in the fitting results in table \\ref{tb:especpow}.\nThis finding is not necessarily consistent, however,  with previous reports on Cyg X-1 in the HSS (e.g., \\cite{jourdain}) and other HSS sources (e.g., \\cite{koyama}),\nthat the hard-tail slope is relatively independent of the source intensity.\nIt is possible that the present HSS was somewhat atypical, \nbecause the values of $\\Gamma \\sim 2.9$ we measured is steeper than those ( $\\Gamma \\sim 2.4$; \\cite{dotani}) observed from these BHBs in their typical HSS.\n\n\n\nFor a further analysis, we replaced the {\\em powerlaw} with a more realistic model {\\em nthComp} (\\cite{zdziarski}), which represents Compton \nscattering of some soft photons by hot electrons ({\\em diskbb+nthComp} model).\nThe seed photon source was set to be the {\\em diskbb} component representing the disk emission.\nSince the electron temperature $T_{\\rm e}$ of the {\\em nthComp} model cannot be determined with our data, we fixed it to a \ntypical value of 100 keV \\citet{Sunyaev79, makishima}. \nThe best-fit parameters are summarized in table \\ref{tb:especcomp} and the best-fit models are shown in figure \\ref{fig:cygespeccomp}.\nThe fits are not yet formally acceptable, but the model, as shown in figure \\ref{fig:cygespeccomp}, reasonably reproduces the spectra from 0.7 keV to 20 keV. \t\t\nIn this modeling, the disk photons are partially fed into {\\em nthComp}, \nso that the parameter $r_{\\rm in}$ in the {\\em diskbb} model is smaller than the true inner radius which is denoted $R_{\\rm in}$ hereafter. \nThis $R_{\\rm in}$ was calculated from the sum of 0.01--100 keV photon flux of the {\\em diskbb} component, $F_{\\rm{ph}}^{\\rm{disk}}$, \nand that of nthComp component, $F_{\\rm{ph}}^{\\rm{nth}}$, utilizing the innermost temperature $T_{\\rm in}$ of {\\em diskbb} and the equation by \\citet{kubota} as \n\\begin{eqnarray}\n\\lefteqn{F_{\\rm{ph}}^{\\rm{disk}}+F_{\\rm{ph}}^{\\rm{nth}}} \\nonumber \\\\\n\\lefteqn{= 0.0165\\Biggl[\\frac{R_{\\rm{in}}^2\\cos{i}}{(D\/10\\ \\rm{kpc})^2}\\Biggr]\\Biggl(\\frac{T_{\\rm{in}}}{1\\ \\rm{keV}}\\Biggr)^3\\ \\ \\rm{ph}\\ \\rm{s}^{-1}\\ \\rm{cm}^{-2}.} \\label{eq:r}\n\\end{eqnarray}\nThe derived values of $R_{\\rm in}$, shown in table \\ref{tb:especcomp}, are indeed larger than $r_{\\rm in}$ in table \\ref{tb:especpow} \n(although we do not discuss absolute values of $R_{\\rm in}$).\nThe LHS is characterized by about twice larger $R_{\\rm in}$ than the HSS, \nimplying a disk truncation in the LHS as noted before \\citep{makishima, yamada}.\nFinally, in either state, the disk parameters ($T_{\\rm in}$ and $R_{\\rm in}$) are found to be apparently constant as the source varies.\n\n\n\n\\begin{table*}\n\\small\n\\caption{The flux ratio $R$ of equation (\\ref{eq:ratio}), and the RMS variation $\\eta'$ of equation (\\ref{eq:eta}).}\n\\begin{center}\n\\begin{tabular}[b]{c c c c c c c}\n\\hline\nSpectral state        &                 &    LHS          &                  &                &    HSS          &               \\\\ \n                      &           \\multicolumn{3}{c}{$\\hrulefill$}           &          \\multicolumn{3}{c}{$\\hrulefill$}         \\\\\n\\hline\nEnergy band (keV)     &       2-4       &       4-10      &       10-20      &       2-4      &       4-10      &       10-20      \\\\\n\\hline\nTotal                 &                 &                 &                  &                &                 &                 \\\\\n~~~~~~~~~~~$R$        & 1.25$\\pm$ 0.11  & 1.19$\\pm$ 0.07  &  1.16$\\pm$ 0.08  & 1.17$\\pm$ 0.20 & 1.43$\\pm$ 0.24  &  1.64$\\pm$ 0.31  \\\\\n~~~~~~~~~~~$\\eta'$    & 0.32$\\pm$ 0.10  & 0.25$\\pm$ 0.06  &  0.21$\\pm$ 0.08  & 0.23$\\pm$ 0.19 & 0.52$\\pm$ 0.20  &  0.70$\\pm$ 0.23  \\\\\nPowerlaw              &                 &                 &                  &                &                 &                 \\\\\n~~~~~~~~~~~$R$        & 1.29$\\pm$ 0.12  & 1.19$\\pm$ 0.07  &  1.16$\\pm$ 0.08  & 1.33$\\pm$ 0.16 & 1.47$\\pm$ 0.22  &  1.64$\\pm$ 0.31 \\\\\n~~~~~~~~~~~$\\eta'$    & 0.37$\\pm$ 0.11  & 0.25$\\pm$ 0.06  &  0.21$\\pm$ 0.08  & 0.41$\\pm$ 0.14 & 0.55$\\pm$ 0.18  &  0.70$\\pm$ 0.23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tb:specrms}\n\\end{table*}\n\n\n\\begin{table*}[htpb]\n\\small\n\\caption{The best-fit parameters of the absorbed {\\em diskbb} plus {\\em powerlaw} model.}\n\\begin{center}\n\\begin{tabular}[b]{cccccc}\n\\hline \\hline\n\\multicolumn{6}{c}{Model = phabs*(diskbb+powerlaw+gaussian)}\\\\ \\hline\nComponent     & Parameter                    & LHS\/bright               & LHS\/faint                    & HSS\/bright              & HSS\/faint     \\\\ \\hline\nphabs         & $N_{\\rm H}^{*}$              & \\multicolumn{2}{c}{$5.8\\pm 0.6$}                      & \\multicolumn{2}{c}{$4.92\\pm 0.12$}     \\\\ \\hline\ndiskbb        & $T_{\\rm{in}}$ (keV)          & $0.24\\pm 0.02$           & $0.21\\pm 0.02$             & $0.50\\pm 0.01$          & $0.49\\pm 0.01$ \\\\ \n              & $r_{\\rm{in}}^{\\dagger}$ (km) & $84^{+34}_{-22}$         & $99^{+49}_{-26}$           & $33^{+1}_{-5}$          & $36^{+1}_{-5}$   \\\\ \\hline\npowerlaw      & Index                        & $1.66\\pm 0.01$           & $1.62\\pm 0.01$             & $2.85\\pm 0.03$          & $2.98\\pm 0.03$ \\\\\n              & Norm$^{\\ddagger}$                         & $2.09\\pm 0.04$           & $1.65^{+0.04}_{-0.03}$     & $19.5^{+1.0}_{-1.0}$  & $16.7\\pm 1.0$    \\\\ \\hline\ngaussian      & Line E (keV)                 & \\multicolumn{2}{c}{$6.66^{+0.17}_{-0.16}$}            & \\multicolumn{2}{c}{$6.62\\pm 0.07$}  \\\\ \n              & Sigma (keV)                  & \\multicolumn{2}{c}{$0.7$\\ (fixed)}                    & \\multicolumn{2}{c}{$0.70^{+0.09}_{-0.08}$}  \\\\ \n              & Norm$^{\\S}$            & \\multicolumn{2}{c}{$7.2\\pm 1.6$}                      & \\multicolumn{2}{c}{$28.3^{+3.7}_{-3.2}$}  \\\\ \\hline        \nfit\\ goodness & $\\chi ^{2}_{\\nu}(\\nu)$       & \\multicolumn{2}{c}{1.38 (420)}                       & \\multicolumn{2}{c}{1.31 (356)}    \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n* : In a unit of $10^{21}$ cm$^{-2}$.\\\\\n$\\dagger$ : The distance is assumed to be 1.86 kpc and the inclination angle is assumed to be 27$^{\\circ}$ \\citep{orosz}.\\\\\n$\\ddagger$ : In a unit of photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$ at 1 keV.\\\\\n$\\S$ : In a unit of 10$^{-3}$ photons cm$^{-2}$ s$^{-1}$.\\\\\n\\label{tb:especpow}\n\\end{table*}\n\\begin{table*}[htpb]\n\\small\n\\caption{The best-fit parameters of the model consisting of a {\\em diskbb} and {\\em nthComp} taking account of the photo-electric absorption.}\n\\begin{center}\n\\begin{tabular}[b]{cccccc}\n\\hline \\hline\n\\multicolumn{6}{c}{Model = phabs*(diskbb+nthComp+gaussian)}\\\\ \\hline\nComponent        & Parameter                                     & LHS\/bright              & LHS\/faint                     & HSS\/bright              & HSS\/faint       \\\\ \\hline\nphabs            & $N_{\\rm H}^{*}$                               & \\multicolumn{2}{c}{$6.1\\pm 0.6$}                      & \\multicolumn{2}{c}{$3.6\\pm 0.1$}  \\\\ \\hline\ndiskbb           & $T_{\\rm{in}}$ (keV)                           & $0.23\\pm 0.02$           & $0.20\\pm 0.02$             & $0.48\\pm 0.01$          & $0.47\\pm 0.01$    \\\\ \n                 & $F_{\\rm{ph}}^{\\rm{disk}\\dagger}$       &  47.0                    & 43.2                       & 69.6                    & 72.5   \\\\ \\hline\nnthComp$^{\\ddagger}$   & Gamma                                         & $1.68\\pm 0.01$           & $1.65\\pm 0.01$             & $2.81\\pm 0.03$          & $2.91\\pm 0.04$       \\\\\n                 & $F_{\\rm{ph}}^{\\rm{nth}\\dagger}$        & 12.9                     & 10.9                       & 36.3                    & 27.4    \\\\ \\hline\ngaussian         & LineE (keV)                  & \\multicolumn{2}{c}{$6.65\\pm 0.16$}                    & \\multicolumn{2}{c}{$6.64\\pm 0.07$}   \\\\ \n                 & Sigma (keV)                  & \\multicolumn{2}{c}{$0.7$\\ (fixed)}                    & \\multicolumn{2}{c}{$0.64^{+0.09}_{-0.08}$}  \\\\ \n                 & Norm$^{\\S}$            & \\multicolumn{2}{c}{$7.6^{+1.6}_{-1.5}$}               & \\multicolumn{2}{c}{$24.5^{+3.1}_{-2.8}$}   \\\\ \\hline\ninner radius     & $R_{\\rm{in}}^{l}$ (km)         & $116\\pm15$  & $127\\pm21$                    & $54\\pm 2$    & $53\\pm 2$ \\\\ \\hline\nLuminosity$^{\\#}$&                       & $1.76\\pm 0.04$           & $1.50\\pm 0.03$             & $2.71\\pm 0.02$          &$2.35\\pm 0.02$ \\\\ \\hline          \nCompton fraction$^{**}$ &                       & 0.87                     & 0.89                       & 0.51                    & 0.41 \\\\ \\hline   \nfit\\ goodness    & $\\chi ^{2}_{\\nu}(\\nu)$       & \\multicolumn{2}{c}{1.37 (420)}                        & \\multicolumn{2}{c}{1.38 (370)} \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n$*$ : In a unit of $10^{21}$ cm$^{-2}$.\\\\\n$\\dagger$: Photon flux in a unit of photons cm$^{-2}$ s$^{-1}$ in the energy range of 0.01-100 keV.\\\\\n$\\ddagger$ : The electron temperature $T_{\\rm{e}}$ is fixed at 100 keV. The spectrum of the seed photon is diskbb and the temperature $T_{\\rm{bb}}$ is fixed at $T_{\\rm{in}}$.\\\\\n$\\S$ : In a unit of $10^{-3}$ photons cm$^{-2}$ s$^{-1}$.\\\\\n$l$: The distance is assumed to be 1.86 kpc and the inclination angle is assumed to be 27$^{\\circ}$ \\citep{orosz}.\\\\\n$\\#$ : In a unit of 10$^{37}$ erg s$^{-1}$ and in the energy range of 0.5-100 keV.\\\\\n$**$ : The fractional luminosity in {\\em nthComp}, which is calculated from the disk luminosity $L_{\\rm{disk}}$ and the Compton luminosity $L_{\\rm{nth}}$ as \n$\\frac{L_{\\rm{nth}}}{L_{\\rm{disk}}+L_{\\rm{nth}}}$.\\\\\n\\label{tb:especcomp}\n\\end{table*}\n\n\n\n\\section{Discussion}\n\n\\subsection{Summary of data analysis}\n\nUsing 5 years archival data obtained with MAXI, we derived the NPSDs of Cyg X-1 in its LHS and HSS from $10^{-7}$ Hz to 10$^{-4}$ Hz in the three energy bands.\nIt is of particular importance that the long-term variations in the two states were studied in a unified way \nusing the same instrument, with the same analysis method, \nand under similar data statistics.\nIn addition, the long-term NPSD in the HSS was obtained for the first time thanks to a fortunate opportunity that Cyg X-1 stayed in the HSS in most of the time since 2010 June.\nThese results on the HSS are expected to provide some clues to the still unknown origin of the hard-tail component, \nwhich is nearly always observed in this spectral state.\n\nIn the LHS, the NPSD down to $10^{-7}$ Hz obeys a power-law, and  is approximately expressed as an extrapolation of the NPSD above 0.01 Hz \\citep{pot, nowak}.\nWe found that the NPSD in the low (2--4 keV) energy band is slightly larger than those in the 4--10 and 10--20 keV bands \n(figure \\ref{fig:cygnpsd} left, figure \\ref{fig:avepsd} left, and table \\ref{tb:rms}).\n\nThe newly obtained $10^{-7}-10^{-4}$ Hz NPSD in the HSS is also approximated by a power-law, \nand on an extrapolation of the NPSD previously obtained in frequencies above 10$^{-4}$ Hz \\citep{chura, cui1997}.\nThe NPSD (RMS$^2$\/mean$^2$ Hz$^{-1}$) in the HSS is about one order of magnitude larger than that in the LHS in 10--20 keV.  \nThis differs from the case of other BH binaries reported by \\citet{miyamoto1993}, although their data were limited to higher frequency range above 0.01 Hz.\nIn the HSS, the NPSD from $10^{-7}$ Hz to 10$^{-4}$ Hz has an energy dependence in such a way that it is 5$\\sim$6 times higher in the 10--20 keV band than that in the low energy band (2--4 keV).  \nThis finding extends the results of energy dependence in 0.2--200 Hz reported by \\citet{grinberg}. \nThese energy dependences of the long-term variation in the HSS were reconfirmed via the spectral analysis in section \\ref{energy}.\n\n\nIn the present paper,  \nwe used only the MAXI data \nbecause the energy band is suitable for simultaneous analyzing the variations of the disk component and the power-law component.\nIn order to further study the hard tail variability, \nthe Swift\/BAT data with good statistic above 20 keV should be utilized: \nwe consider that this is our future task.\n\n\n\\subsection{Comparison with short-term variability}\nSince the first discovery by \\citet{oda}, the aperiodic fast X-ray variation of Cyg X-1 has been\ninvestigated by a number of authors (e.g., \\cite{negoro1994}, \\cite{chura}),\nmainly in the LHS and typically over $10^{-3}-10^2$ Hz frequency range.\nSimilar studies have been performed on other BHBs as well, including in particular GX 339-4 (\\cite{maejima}; \\cite{miyamoto1994}).\nGiven these, let us briefly compare the present studies with the previous results on the short-term variability of Cyg X-1.\n\nOur results on the LHS has two similarities to the short-term variability in the same state.\nOne is the PSD slope, which is $\\sim 1.4$ over $10^{-7}-10^{-2}$ Hz and $\\sim 1$ in the $>10^{-1}$ Hz range (figure \\ref{fig:hkkchura}).\nThe other is the energy dependence; on both time scales, $\\eta$ decreases slowly towards higher energies, \nimplying ``softer when brighter\" characteristic, \nwhich Cyg X-1 shows when it is above $\\sim 1\\%$ of the Eddington luminosity.\nFor example, \\citet{grinberg} reported that the 0.125--256 Hz variation of Cyg X-1 in the LHS is about twice higher in 2.1--4.5 keV than in 5.7--9.4 keV,\nin approximate agreement with our table \\ref{tb:rms}.\nAlso, the Suzaku result obtained in the LHS,\nnamely, figure 8b of \\citet{makishima},\nreveals a similar energy dependence.\nThese two resemblances altogether suggest a common mechanism working on these very wide range of time scales.\nFor example, the higher short-term variability in softer energy bands may be a result of spectral softening as individual ``shots\" develop \n\\citep{negoro1994, yamadashot}.\nHowever, the phenomenon cannot be completely self-similar, since the PSD shows a considerable flattening over the $10^{-2}-10^{-1}$ range.\n\nThe fast variability in the HSS has been considerably less studied than that in the LHS.\nNevertheless, \nwe can point out again two similarities between the long-term vs. short-term variations.\nOne is that our NPSD index in the HSS ($\\sim 1.3$) is similar to the short-term HSS index of $\\sim 1$ by \\citet{cui1997}, \nas clearly seen figure \\ref{fig:hkkchura}.\nThe other is the energy dependence; the slight increase of variability from the 4--10 keV to 10--20 keV ranges \n(subsection \\ref{psdresult2} and section \\ref{energy}) \nis also observed in short-term results by \\citet{grinberg}.\t\nAs already described, this ``harder when brighter\" property is opposite to the behavior, ``softer when brighter\", in the LHS.\n\n\n\\subsection{Spectral variation}\n\nThe energy spectrum in the LHS is known to be approximated by a power-law with a high-energy cutoff around 100 keV (e.g., \\cite{makishima}).  \nThis is interpreted by a scenario that an optically-thick standard disk \\citep{shakura} is truncated at some distance \nfrom the BH and an optically-thin accretion flow (or a corona) is formed around the BH \\citep{esin}.  \nHigh temperature electrons, in the optically-thin corona, up-scatter seed photons presumably from the outer standard disk.\nIn this frame work, the fast hard X-ray variability in the LHS may be explained by considering \nthat the corona covers a variable fraction of the disk \\citep{makishima}; \nwhen this fraction increases, \nthe source gets brighter, \nand softer due to the enhanced Compton cooling of the corona.\nIn fact, the MAXI spectrum accumulated over the LHS (figure \\ref{fig:cygespeccomp} left) was described successfully with the two component model, \nconsisting of a low-temperature {\\em diskbb} \\citep{mitsuda} representing the emission from such a truncated standard disk, \nand the {\\em powerlaw}  or the {\\em nthComp} representing the Compton up-scattered component.\n\nLike in many other reports on BHBs in the HSS, the MAXI spectrum in the HSS has been expressed by a dominant optically-thick thermal spectrum, \naccompanied by a {\\em powerlaw} component extending into higher energies.\nThe former is again interpreted by an emission from a standard-disk \\citep{shakura}, which is now considered to extend down to \nthe innermost stable circular orbit (ISCO) around the BH.  \nThis is supported by the fact that the obtained inner disk radii in the HSS do not differ significantly between the faint and bright periods.\nLike the overall hard X-ray emission in the LHS, \nthe hard {\\em powerlaw} component in the HSS could also be a Compton up-scatters component produced by some hot electrons \n\\citep{cui1998, gierlinski}, but details are still unclear.\n\n\nIn order to better understand the NPSD results, we fitted the spectra by a model composed of a {\\em diskbb}, and the {\\em powerlaw} or the {\\em nthComp}, \nand studied how the spectrum changes as the source varies on frequencies range below $8\\times 10^{-7}$ Hz.\nThe obtained results, fully consistent with those from the NPSD studies, can be summarized into the following four points.\n\\begin{enumerate}\n\\item[1] In  both states, the {\\em powerlaw} component is responsible for the observed long-term intensity variations, while the disk emission is essentially constant.\n\\item[2] The power-law component in the HSS is concluded to be more variable, on this frequency range, than that in the LHS.\n\\item[3] The decreasing variability towards higher energies, observed in the LHS, can be attributed mostly to the ``softer when brighter\" property of the {\\em powerlaw} component.\n\\item[4] The variability increase with energies in the HSS results from a combination of the two effects,\nnamely, the ``harder when brighter\" property of the {\\em powerlaw} component in this state, and the presence of the stable disk emission at lower energies.\n\\end{enumerate}\n\n\nSimilarly high long-term variations were observed also from some BH transients, \nincluding GS 1124$-$684 and GS 2000+25 \\citep{tanaka, terada}.\nWhen their outburst decline was followed by sparse snap-shot observation with Ginga, the intensity of the disk component changed smoothly, \nbut the power-law component varied largely from one observation to the next.\nThus, high long-term variability of the hard component in the HSS may be common to all BH binaries, \nincluding persistent and transient sources.\n\n\n\n\\subsection{Origin of the time variation}\nLet us discuss the origin of the long-term variations which we detected down to $10^{-7}$ Hz.\nThe simplest possibility would be that  fluctuations in\nthe overall mass transfer rate from the companion star\nproduce the observed long-term X-ray variations.\nGenerally, the mass transfer in Cyg X-1 is thought to\noccur via capture of the stellar winds from the super giant companion star, HDE226868.\nThe winds are considered inhomogeneous as reported by several authors \\citep{Conti89,GiesBolton86,Hanke09},\nand as evidenced by variable X-ray absorption episodes\naround the superior conjunctions of the BH \\citep{kitamoto, remillard1984, church1997}.\nTherefore, the mass accretion rate $\\dot{M}$ must fluctuate,\nand could produce the long-term variability.\nHowever, besides the issue of how these fluctuations can propagate\nto the X-ray emission regions (see below),\nit is not trivial to explain with this scenario\nthe difference in the variation ($\\eta$ and $\\eta'$)\nobserved between the LHS and HSS.\n\n\n\nThe observed long-term X-ray variations may alternatively arise\nwhen some  fluctuations in $\\dot{M}$,\nproduced spontaneously in the accretion flow at various radii,\npropagate to the BH vicinity.\nAt a radius $r$ from the BH,\nsuch spontaneous fluctuations\nmay generally take place\non a time scale comparable to the thermal time scale\n$t_{\\rm th}=(\\alpha \\Omega_{\\rm K})^{-1}$,\nor the dynamical time scale $t_{\\rm d} = (\\Omega_{\\rm K})^{-1}$,\nwhere $\\Omega_{\\rm K}$ is the Keplerian angular velocity at $r$\nand $\\alpha$ is the viscosity parameter.\nSuch variations can propagate toward the center\nonly if their time scales are longer than the viscous time scale,\ngiven as $t_{\\rm v}=(\\alpha \\Omega_{\\rm K})^{-1}(r\/H)^{2}$,\nwhere $H$ is the disk half thickness.\nIn an optically thick and geometrically thin  disk,\nthis $t_{\\rm v}$ is much longer than $t_{\\rm th}$ and $t_{\\rm d}$ because $H \\ll r$.\nTherefore, variations in $\\dot{M}$ produced at any outer radii,\n being strongly dissipated, would not propagate down to the X-ray emitting region.\nHowever, if some fraction of $\\dot{M}$ streams through\na geometrically-thick ($H\\sim r$) and optically-thin flow,\nin which  we expect $t_{\\rm v} \\sim t_{\\rm th}$,\nthe fluctuations produced at various larger radii can propagate inwards,\nand cause the X-ray intensity to vary on a wide range of time scales.\nThis idea was invoked by \\citet{chura} to explain\nthe X-ray variations on  timescales down to $10^{-4}$ Hz.\nWe may extend this two-flow scenario from the time scale of $10^{-4}$ Hz to $10^{-7}$ Hz,\nidentifying the optically-thick part with the standard accretion disk,\nand assuming that the optically-thin part\ndevelops into the Comptonizing corona in a vicinity of the BH.\nAssuming a $15~M_{\\odot}$ BH and $\\alpha = 0.01$,\nthe observed time scale of $t_{\\rm th} \\sim 10^{7}$ s\ncan be explained if the optically-thin flow starts\nat $\\sim 3 \\times 10^{12}$ cm ($\\sim 6 \\times 10^{5} R_{\\rm s}$).\nFurthermore, as detailed in Appendix \\ref{mdot},\nthe larger long-term variability in the HSS can be explained\nif  the fraction of $\\dot{M}$ through the optically-thin part\ncarries $\\sim 0.5~\\dot{M}$ in the HSS,\nwhile $\\sim 0.2~\\dot{M}$ in the LHS.\n\n\nAlthough the two-flow picture, as constructed above after \\citet{chura},\ncan account for essential features of the observed long-term variability of Cyg X-1,\nit is still subject to a few issues to be solved.\nOne is that the required size of the optically-thin flow is comparable\nto the binary size of Cyg X-1 (0.2 AU).\nIt is not obvious whether such a large-scale optically-thin flow\ncan be actually created by possible candidate mechanisms,\nsuch as X-ray irradiation, vertical magnetic pressure,\nand initial scatter in the specific angular momentum of accepting blobs.\nEven if such a flow is produced,\nit is not obvious whether its low density can be compensated by its higher radial velocity,\nto  carry a considerable fraction of the total $\\dot{M}$.\nFurthermore, we need to assume that the state difference,\nwhich is usually thought to be triggered at close vicinities of the BH,\nis already present at such large radii:\nit is not clear, either,\nwhether some feedback mechanisms (e.g., X-ray irradiation)\ncan control large-scale accretion flows in the required manner.\n\nAt present, we cannot conclude for sure \nwhether the observed slow variations in the HSS can be explained by either the stellar wind fluctuation or the two-flow picture.\nWe might need to consider other origins of the slow variability,\nincluding their production at regions much closer to the BH.\nFurther discussion is beyond the scope of the present paper:\nalthough our ultimate goal is to identify the origin of the long-term variation, \nthe currently available data information is still insufficient for that purpose.\n\n\n\n\n\\section{Conclusion}\nBy  analyzing the three-band (2--4, 4--10 and 10--20 keV) MAXI data accumulated over 5 years,\nwe studied characteristics of the long term X-ray variation of Cyg X-1\nover the frequency range of $10^{-7}-10^{-4}$ Hz.\nThe long-term NPSD in the HSS was obtained for the first time.\nBy treating the LHS and the HSS data separately \nbut in the same manner, \nthe following results have been obtained.\n\\begin{enumerate}\n\\item \nIn the LHS and HSS, the index of the NPSD was obtained as $-1.35\\pm 0.29$ and $-1.29\\pm 0.23$, respectively.\nThey are consistent with those previously measured in the frequency range above 10$^{-3}$ Hz.\n\\item \nIn the 4--10 keV, and 10--20 keV bands, the fractional RMS variation $\\eta$ observed in the HSS was\n$\\sim 2$ and $\\sim 3$ times higher, respectively, than those measured in the LHS in the corresponding energies.\nIn the 2--4 keV band, $\\eta$ in the HSS was comparable to that in the LHS.\n\\item \nIn the LHS, $\\eta$ weakly decreased towards higher energies.\nThis property is consistent with that seen in the short-term variability during the LHS.\n\\item \nIn the HSS, $\\eta$ was found to increase significantly with energy,\nas a combination of  the following two effects.\nOne is that the disk component is stable while the hard tail varies.\nThe other is that the hard tail slope flattens as it becomes brighter.\n\n\\end{enumerate}\n\nWe have studied long-term variability of the hard tail component in the HSS.\nThis will provide important clues to the yet unidentified origin of this emission component.\n\n\n\nThis work was supported by RIKEN Junior Research Associate Program.\nThis work was  also partially supported by \nthe Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for Science Research 24340041,\nand the MEXT Supported Program for the Strategic Research Foundation at Private Universities, 2014-2018.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTremendous works have been made for more accurate and more efficient object detection\nin recent years. Data augmentation\\cite{2019AutoAugment,kisantal2019augmentation,zoph2020learning},\ndeeper layers of neural networks\\cite{fu2017dssd,liu2016ssd,2020EfficientDet},\nmore complex structured FPN modules\\cite{li2017fssd,lin2017feature,2020EfficientDet},\nand even more number of iterations make the model for object detection state-of-the-art. Undoubtedly these models\nhave achieved remarkable success, however, at the same time these models have huge parameters and unsatisfactory FLOPs,\nsuch as Detectron2 Mask R-CNN X101-FPN (parameters: 107M, FLOPs: 277B)\\cite{wu2019detectron2}, ResNet-50 + NAS-FPN (1280@384)\n(parameters: 104M, FLOPs: 1043B)\\cite{2019NAS}, AmoebaNet+ NAS-FPN +AA(1280) (parameters: 185M, FLOPs: 1317B)\\cite{zoph2020learning} and\nAmoebaNet+ NAS-FPN + AA(1536) (parameters: 209M, FLOPs: 3045B)\\cite{zoph2020learning}.\n\nThis paper focuses the performance improvement of object detection on the \\textbf{evaluation system}\nand \\textbf{feedback mechanism}\\cite{wen2016discriminative} (namely IoU modules and loss functions, combined called \\textbf{evaluation-feedback module}) of region proposals\nwithout increasing the number of parameters or FLOPs.\n\nThe evaluation-feedback modules have 3 main roles:\n(1) Evaluating region proposals, using ground truth as a criterion.\n(2) Ranking a set of region proposals (with the same ground truth criterion).\n(3) Feeding the gap between region proposals (RP) and ground truths (GT) to the neural network,\nwhich is used to correct the next evaluation module. Considering evaluation-feedback module is fundamental,\nthis module should be efficient and contain few parameters.\nA good evaluation-feedback module should meet the following 3 conditions:\n\\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n  \\item A measure overlapping area.\n  \\item The good ability to differentiate and a measure of the degree of difference\n(sometimes understood as centroid distance and aspect ratio).\n  \\item The IoUs calculation can be correlated with loss functions calculation.\n\\end{itemize}\n\nNumerous previous studies have tended to focus on the study of feedback mechanism\nat the expense of evaluation system. In this paper, the Control Distance IoU(\\textbf{CDIoU} for short)\nand the Control Distance IoU loss function (\\textbf{CDIoU loss} for short) are proposed and given the same importance.\nCDIoU has good continuity and derivability,\nand simplifies the calculation by measuring the distance\nbetween RP and GT in a unified way,\noptimizing the calculation of DIoU and CIoU\\cite{2020Distance} for centroid distance and aspect ratio,\nand completing the evaluation quickly.\nThe CDIoU loss function can be correlated with CDIoU calculation,\nwhich enables the feedback mechanism to characterize more accurately and\nfeed back the difference between RP and GT,\nthus making the objective function of the deep learning network converge faster\nand improving the overall efficiency. The CDIoU and CDIoU loss functions\nare highly adaptive and show significant improvements on several different models,\ncompared to traditional IoU modules and loss functions.\n The main contributions of this work can be summarized as:\n \\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n  \\item The \\textbf{evaluation-feedback module} is proposed to consist of \\textbf{evaluation system} and \\textbf{feedback mechanism}.\n  \\item \\textbf{CDIoU} is proposed as a new evaluation system and \\textbf{CDIoU loss} as a new feedback mechanism.\n  \\item Improving the results, while the number of parameters and running time are not increased.\n  \\item With wide applicability, make significant improvements on several models.\n\\end{itemize}\n\\section{Related Work}\n\nThe first culmination of deep learning for object detection was the proposal of R-CNN\\cite{2013Rich},\nFast R-CNN\\cite{2015Fast} and Faster R-CNN\\cite{2017Faster} models, which laid down the basic framework and data processing for deep learning applied to object detection. YOLO\\cite{redmon2016you} provides a more straightforward way by directly regressing the location of the bounding box and the class to which the\nbounding exploitation belongs in the output layer, thus transforming the object detection\nproblem into a regression problem. After this, YOLOv2\\cite{2017YOLO9000}, YOLOv3\\cite{farhadi2018yolov3},\nYOLOv4\\cite{bochkovskiy2020yolov4} and YOLOv5\\cite{glenn_jocher_2021_4418161} were proposed,\nwhich made the deep learning network not only improve in accuracy but also in computing speed.\nR-CNN series and YOLO series are the classical representatives of two-stage model\\cite{dai2016r,duan2020corner,he2017mask,he2015spatial,he2019multi} and\none-stage model\\cite{2020FCOS,wu2017squeezedet} in object detection.\n\n\\textbf{Neural network backbone and conv kernel}\n\nThe backbone networks of deep learning are also evolving. LeNet (1998)\\cite{1998Gradient},\nAlexNet (2012)\\cite{krizhevsky2017imagenet}, VGGNet (2014)\\cite{simonyan2014very},\nGoogLeNet (2014)\\cite{2014Going}, ResNet (2015)\\cite{2016Deep}, and MobileNet (2017)\\cite{howard2017mobilenets}\nare preserved in the path of deep learning development.\nEfficientNet (2019)\\cite{tan2019efficientnet} proposes a more generalized idea on the optimization of current\nclassification networks, arguing that the three common ways of enhancing network metrics,\nnamely widening the network, deepening the network and increasing the resolution,\nshould not be independent of each other.\n\nAlong with the backbone, the convolutional kernel\\cite{fu2017dssd,noh2015learning,radford2015unsupervised} is also evolving and changing.\nDeformable conv\\cite{dai2017deformable,2019Deformable} adds an offset variable to the position of each sampled point in the convolution kernel,\nenabling random sampling around the current position without being restricted to the previous regular grid points.\nDilated conv\\cite{chen2017rethinking,2016Multi} can effectively focus on the semantic information of the local pixel blocks,\ninstead of letting each pixel rub together with the surrounding blocks, which affects the detail of segmentation.\n\n\\textbf{Evaluation-feedback module}\n\nBased on IoU, GIoU\\cite{2019Generalized} focuses not only on overlapping regions but also on other non-overlapping regions,\nwhich can better reflect the overlap of RP and GT.\nDIoU\\cite{2020Distance} takes the distance between object and anchor, overlap rate and scale into consideration,\nwhich makes the object box regression more stable and does not have problems such as scattering during training like IoU and GIoU.\n\nGIoU loss\\cite{2016Multi} still has the problems of slow convergence and inaccurate regression.\nIt is found that GIoU first tries to overlap the object box by increasing the size of the detection box,\nand then uses the IoU loss term to maximize the overlap area with the object box.\n\nDIoU loss and CIoU loss\\cite{2020Distance} greatly enriched the connotation of IoU calculation results,\nadding the measurement of difference, including ``centroid distance\"\nand ``aspect ratio\" separately. DIoU loss cannot distinguish which RPs is more similar to\nGT when the center points of RPs are at the same position. We can know that the calculation process of CIoU loss is more time-consuming,\nwhich will eventually drag down the overall training and test time.\n\n\\section{Analysis of traditional IoUs and loss functions}\nIn object detection, the function of IoUs is to\nevaluate the similarity between RP and GT. Through the IoU method,\nthe evaluation between RP and GT is given,\nwhich plays a fundamental role in the selection of positive and negative samples. In the\nevaluation-feedback module, the most representative methods are IoU, GIoU, DIoU loss and CIoU loss.\nThey have played a fundamental role in the great progress of object detection,\nbut there still is much room for optimization.\n\n\\subsection{Analysis of traditional IoUs}\nIoU is a basic evaluation method, and as shown in Figure 1,\nthe relative position relationship between RP and GT is obviously\ndifferent. The human brain can clearly distinguish the disadvantages,\nbut the evaluation results of IoU are the same.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.3]{ioushortcoming3.pdf}\n\\caption{Different relative position relations, the same IoU results}\n\\label{fig:label}\n\\end{figure}\n\nBased on original IoU, many evaluation systems are derived, which enrich the evaluation\ndimensions of previous IoU from those different aspects.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.5]{giouloss.pdf}\n\\caption{Comparison of IoU GIoU CDIoU and Loss of GIoU \\& CDIoU}\n\\label{fig:label}\n\\end{figure}\n\nIoU only considers the calculation of overlapping area. Meanwhile, GIoU pays attention to\noverlapping area and non overlapping area, and strengthens the discussion of evaluation system.\nHowever, GIoU obviously ignored the ``measurement of difference\" between RP and GT.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.3]{ious2.pdf}\n\\caption{Comparison of multiple IoU losses}\n\\label{fig:label}\n\\end{figure}\n\nThe ``measurement of difference\" between RP and GT\ninclude the distance between the center points (centroid) and the ratio of length-width(aspect ratio).\nDIoU takes centroid into account in the calculation\nof the evaluation system, but omits aspect ratio. As shown in Figure 2, 3,\nDIoU can't recognize the difference between the high-thin region proposal and the short-fat region proposal,\nand gives them the same value. But in fact, the human brain can easily differentiate which one is better and which position is better.\n\n\\subsection{IoU:Smooth L1 Loss and IoU Loss}\n\nThe method of smooth loss is proposed from Fast RCNN\\cite{2015Fast},\nwhich initially solves the problem of characterizing the boundary box loss.\nAssuming that \\begin{math}x\\end{math} is the numerical difference between RP and GT,\n\\begin{math}L_{1}\\end{math} and \\begin{math}L_{2}\\end{math}\nloss are commonly defined as:\\\\\n\\begin{equation}\n\\mathcal{L}_{1}=|x| \\frac{d \\mathcal{L}_{2}(x)}{x}=2 x,\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{L}_{2}=x^{2}.\n\\end{equation}\nThe corresponding derivative\u00a0:\u00a0\n\\begin{equation}\n\\frac{d \\mathcal{L}_{1}(x)}{x}=\\left\\{\\begin{array}{ll}\n 1 & \\text{, if } x \\geq 0 \\\\\n-1 & \\text {, otherswise ,}\n\\end{array}\\right.\n\\end{equation}\n\n\\begin{equation}\n\\frac{d \\mathcal{L}_{2}(x)}{x}=2 x.\n\\end{equation}\n\nFrom the derivative of loss function to \\begin{math}x\\end{math}, we can know that the derivative of\nloss function \\begin{math}\\mathcal{L}_{1}\\end{math} to \\begin{math}x\\end{math} is constant. In the late training period,\nwhen \\begin{math}x\\end{math} is very small, if the learning rate is constant, the loss function will fluctuate\naround the stable value, and it is difficult to converge to higher accuracy. When the\nderivative of loss function \\begin{math}\\mathcal{L}_{2}\\end{math} to \\begin{math}x\\end{math} is large, its derivative\nis also very large and unstable at the beginning of training. \\begin{math}{smooth}_{\\mathcal{L}_{1}}(x)\\end{math} perfectly\navoids the shortcomings of \\begin{math}\\mathcal{L}_{1}\\end{math} and \\begin{math}\\mathcal{L}_{2}\\end{math} loss.\n\n\\begin{equation}\n\\text {smooth}_{\\mathcal{L}_{1}}(x)=\\left\\{\\begin{array}{ll}\n0.5 x^{2} & \\text {, if }|x|<1 \\\\\n|x|-0.5 & \\text {, otherswise   ,}\n\\end{array}\\right.\n\\end{equation}\n\n\n\\begin{equation}\n\\frac{d {smooth}_{\\mathcal{L}_{1}(x)}}{x}=\\left\\{\\begin{array}{ll}\nx & \\text{, if }|x|<1 \\\\\n\\pm 1 & \\text {, otherswise   ,}\n\\end{array}\\right.\n\\end{equation}\n\nHowever, in the actual object detection, the loss in box regression task is\n\n\\begin{equation}\n\\mathcal{L}_{l o c}\\left(t^{u}, v\\right)=\\sum_{i \\in\\{x, y, w, h\\}} \\text {smooth}_{\\mathcal{L}_{1}}\\left(t_{i}^{u}-v_{i}\\right),\n\\end{equation}\n\nWhere \\begin{math}\n   v=\\left(v_{x}, v_{y}, v_{w}, v_{h}\\right)\n\\end{math}represents the box coordinates of GT, and \\begin{math}\n   t^{u}=\\left(t_{x}^{u}, t_{y}^{u}, t_{w}^{u}, t_{h}^{u}\\right)\n   \\end{math} represents the predicted box coordinates,\nthat is to calculate the loss of four points respectively,\nand then add them as the bounding box regression loss.\n\n\\textbf{shortcomings:}\n\n1.When the above losses are used to calculate the bounding box loss of object detection,\nthe loss of four points is calculated independently, and then the final bounding box loss is obtained by adding.\nThe assumption of this method is that the four points are independent of each other,\nand there is a certain correlation in fact. The calculation of smooth can not be unified with IoU, which leads to errors of the feedback mechanism and evaluation system.\n\n2.The actual indicator of evaluation is to use IoU, which is not equivalent.\nIoU loss cannot avoid this scenario,``different RPs, same feedback results'' in Figure 2,3.\n\n\\begin{equation}\n\\mathcal{L}_{IoU}=-\\ln(IoU),\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{L}_{IoU}=1-IoU.\n\\end{equation}\n\n\\subsection{GIoU and GIoU Loss}\nOn the premise of not increasing the calculation time,\nGIoU\\cite{2019Generalized} initially optimized the calculation of IoU for overlapping area,\nand reduced the calculation error, but GIoU still did not take the measurement\nof the difference into account in the calculation results.\n\\begin{equation}\nG I o U=I o U-\\frac{|C \\backslash(A \\cup B)|}{|C|},\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{L}_{G I o U}=1-G I o U .\n\\end{equation}\n\nGIoU loss still has the problems of slow convergence and inaccurate regression.\nIt is found that GIoU first tries to overlap the object box (GT) by increasing size of the detection box (RP),\nand then uses IoU loss term to maximize the overlap area with the object box. At the same time,\nwhen the two boxes contain each other, GIoU loss will degenerate\ninto IoU loss. At this time, the alignment of the bounding box becomes more difficult and the convergence is slow.\n\n\\subsection{DIoU loss and CIoU Loss}\n\nDIoU loss and CIoU loss\\cite{2020Distance} greatly enriched the connotation of IoU calculation results,\nadding the measurement of difference, including ``centroid distance\"\nand ``aspect ratio\" separately.\n\n\\begin{equation}\n\\mathcal{L}_{D I o U}=1-I o U+\\frac{\\rho^{2}\\left(\\mathbf{b}, \\mathbf{b}^{g t}\\right)}{c^{2}}\n\\end{equation}\n\nFirst of all, DIoU loss cannot distinguish which region proposals are more similar to\n ground truth when the center points of region\nproposals are at the same position. Then when two boxes are completely coincident,\n\\begin{math}\n\\mathcal{L}_{I o U}=\\mathcal{L}_{G I o U}=\\mathcal{L}_{D I o U}=0\n\\end{math}; when two boxes do not intersect, GIoU loss can't distinguish region proposals exactly and \\begin{math}\n\\mathcal{L}_{G I o U}=\\mathcal{L}_{D I o U} \\rightarrow 2\n\\end{math}.\n\nAs shown in the Figure 4, in fact, as long as the center point of the region proposal is on arc \\begin{math}\nC\\end{math} of circle \\begin{math}O\\end{math},the penalty terms of DIoU loss are consistent.\nThis is DIoU, which loses the accuracy of the evaluation system.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.34]{dioulossshortcoming2.pdf}\n\\caption{DIoU loss: unuseful relative position relationship between RP and GT}\n\\label{fig:label}\n\\end{figure}\n\nThe penalty term of CIoU loss is composed of a factor \\begin{math}\n\\alpha v\n\\end{math} and DIoU loss penalty term,\nwhich takes that into account the aspect ratio of RP and GT.\n\\begin{equation}\n\\mathcal{L}_{C I o U}=1-I o U+\\frac{\\rho^{2}\\left(\\mathbf{b}, \\mathbf{b}^{g t}\\right)}{c^{2}}+\\alpha v\n\\end{equation}\n\nThe penalty is\n\n\\begin{equation}\nR_{C I o U}=\\frac{\\rho^{2}\\left(b, b^{g t}\\right)}{c^{2}}+\\alpha v,\n\\end{equation}\n\n\\begin{equation}\n\\alpha=\\frac{v}{(1-I o U)+v},\n\\end{equation}\n\n\\begin{equation}\nv=\\frac{4}{\\pi^{2}}\\left(\\arctan \\frac{w^{g t}}{h^{g t}}-\\arctan \\frac{w}{h}\\right)^{2}.\n\\end{equation}\n\nBecause the calculation of CIoU loss involves the inverse trigonometric function,\nand through comparative experiments, we can know that the calculation process of CIoU loss is more time-consuming,\nwhich will eventually drag down the overall training time. For detailed comparison tests, see ``\\textbf{Ablation studies}\".\n\n\\section{CDIoU and CDIoU loss functions}\n\nBased on  traditional IoUs and loss functions, CDIoU and CDIoU loss functions are proposed in this paper. Without increasing the operation time,\nthe running efficiency and AP are significantly improved. The CDIoU loss\nfunction converges faster and reduces the complexity of the operation significantly.\n\nControl Distance Intersection over Union (CDIoU) is an evaluation method that directly examines the similarity of RP and GT,\nand it does not directly measure the distance between their centroids and the similarity of their shapes. For detailed information, see Figure 5.\n\n\\begin{equation}\n\\begin{aligned}\n\\operatorname{diou} &=\\frac{\\|R P-G T\\|_{2}}{4\\mathrm{MBR}^{\\prime} s \\text { diagonal }} \\\\\n&=\\frac{A E+B F+C G+D H}{4 W Y}\n\\end{aligned}\n\\end{equation}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.22]{cdiou.pdf}\n\\caption{Calculation of CDIoU. Minimum bounding rectangle (MBR) is the smallest rectangle that can contain RP and GT.}\n\\label{fig:label}\n\\end{figure}\n\n\\begin{equation}\nC D I o U=I o U+\\lambda(1-\\operatorname{diou})\n\\end{equation}\n\nAlthough the formula for CDIoU does not mention ``centroid distance\" and ``aspect ratio\",\nthe final calculation results reflect a measure of the degree of difference between\nRP and GT. The higher the value of CDIoU,\nthe lower the degree of difference; the higher the value of CDIoU, the higher the similarity.\n\n\\begin{equation}\n   \\mathcal{L}_{C D I o U}=\\mathcal{L}_{I o U_{s}}+\\operatorname{diou}\n\\end{equation}\n\nIn order to cooperate with the calculation of CDIoU, this paper also proposes the CDIoU loss function.\nBy observing this formula, we can intuitively feel that after backpropagation,\nthe deep learning model tends to pull the four vertices of the region proposal toward the four vertices\nof the ground truth until they overlap. For detailed information, see \\textbf{Algorithm 1} and Figure 6.\n\nCDIoU and CDIoU loss as a new metric have the following properties:\n(1) \\begin{math}0 \\leq \\text { diou }<1 \\end{math}, \\begin{math}\\mathcal{L}_{I o U_{s}}\\end{math} is the lower limit of \\begin{math}\\mathcal{L}_{C D I o U} \\end{math}.\n(2) \\begin{math} \\text { diou } \\end{math} broadens the range of \\begin{math}\\mathcal{L}_{C D I o U} \\end{math}. If \\begin{math} \\mathcal{L}_{I o U_{s}} = \\mathcal{L}_{I o U} \\end{math}, then \\begin{math}0 \\leq  \\mathcal{L}_{C D I o U}<2 \\end{math}, if \\begin{math} \\mathcal{L}_{I o U_{s}} = \\mathcal{L}_{GI o U} \\end{math}, then \\begin{math}-1 \\leq  \\mathcal{L}_{C D I o U}<2 \\end{math}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.18]{turning.pdf}\n\\caption{Region proposal turning and variations by CDIoU loss}\n\\label{fig:label}\n\\end{figure}\n\n\\begin{algorithm}[htb]\n  \\caption{ CDIoU and CDIoU loss function }\n  \\label{alg:Framwork}\n  \\begin{algorithmic}[1]\n    \\Require\n      RP for region proposal;\n      GT for ground truth;\n    \\Ensure\n      CDIoU and CDIoU loss;\n    \\State For RP and GT, find MBR;\n    \\label{code:fram:extract}\n    \\State compute $I o U=\\frac{|RP \\cap GT|}{|RP \\cup GT|}$ ,\n    $\\operatorname{diou} =\\frac{\\|R P-G T\\|_{2}}{4\\mathrm{MBR}^{\\prime} s \\text { diagonal }}$;\n    \\label{code:fram:trainbase}\n    \\State compute $C D I o U=I o U+\\lambda(1-\\operatorname{diou})$;\n    \\label{code:fram:add}\n    \\State compute $\\mathcal{L}_{C D I o U}=\\mathcal{L}_{I o U_{s}}+\\operatorname{diou}$, $\\mathcal{L}_{I o U_{s}}$ could be $\\mathcal{L}_{IoU}=-\\ln(IoU)$, $\\mathcal{L}_{IoU}=1-IoU$, $\\mathcal{L}_{IoU}=1-IoU$ or $\\mathcal{L}_{DIoU}$, $\\mathcal{L}_{CIoU}$;\n    \\label{code:fram:classify}\n  \\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\nTo ensure the rigidity and richness of the experiments,\nwe did a lot of training and testing on representative models,\nsuch as Faster R-CNN, Cascade R-CNN\\cite{cai2018cascade}, YOLOs and ATSS\\cite{zhang2020bridging}.\nAlso, we try not to use\ntricks and allow individual models to compare differences purely due to changes in IoUs or loss functions.\n\n\\subsection{Working environment and preparation}\n\nThe following experiments were conducted on MS coco 2017 dataset using two GeForce RTX 2080 Ti GPUs or two\nTesla V100 PCIe 32GB GPUs. All models under pytorch or tenserflow framework are standard models without using any tricks.\n\n\\subsection{CDIoU and CDIoU loss for object detection}\n\nIn order to verify the effectiveness of CDIoU and CDIoU loss in object detection,\nexperiments are designed and applied to numerous models in this paper.\nThese models encompass existing classical models and emerging models,\nreflecting certain robustness and wide adaptability. We conduct several experiments to\nstudy the robustness of the hyperparameter $\\lambda$ in Table 1. Overall,\nthe only hyperparameter $\\lambda=0.001$ is quite robust and the proposed\nCDIoU can be nearly regarded as hyperparameter-free\\footnote {Code is available in https:\/\/github.com\/Alan-D-Chen\/CDIoU-CDIoUloss}.\n\n\\begin{table}[]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$\\lambda$                        & 1.0 & 0.1 & 0.01 & 0.001 & 0.0001 \\\\ \\hline\\hline\nFaster R-CNN                   &35.6 &36.5 &37.0  &38.5   &  38.0  \\\\\nATSS\\_R\\_50\\_FPN            &38.1 &39.0 &39.3  &39.5   &  38.9  \\\\ \\hline\n\\end{tabular}\n\\caption{Analysis(\\%) of different values of hyperparameter $\\lambda$ on the \\textbf{MS COCO} \\texttt{val} set.}\n\\end{table}\n\n\n\\begin{table*}[h]\n   \\centering\n   \\begin{threeparttable}\n   \\begin{tabular}{|l|c|c|c|c|c|c|c|c|}\n   \\hline\n   \\multicolumn{1}{|c|}{\\multirow{2}{*}{Models}}                            & \\multicolumn{4}{c|}{test-dev(\\%)} & \\multicolumn{4}{c|}{val(\\%)}                                                                                 \\\\ \\cline{2-9}\n   \\multicolumn{1}{|c|}{}                                                               & AP     & APs    & APm    & APl    & AP    & APs   & APm   & APl                                                                              \\\\ \\hline \\hline\n   Faster R-CNN\\cite{2015Fast}                                                    & 36.0   & -      & -      & -      & 36.0  & -     & -     & -                                                                                                    \\\\\n   YOLOv4\\cite{bochkovskiy2020yolov4}                                      & 41.2   & 20.4   & 44.4   & 56.0   & -     & -     & -     & -                                                                                             \\\\\n   RetinaNet-R101 (1024)\\cite{2017Focal}                                    & 40.8   & 24.1   & 44.2   & 51.2   & -     & -     & -     & -                                                                                              \\\\\n   ResNet-50 + NAS-FPN (1280@384)\\cite{2019NAS}                  & 45.4   & -      & -      & -      & -     & -     & -     & -                                                                                              \\\\\n   Detectron2 Mask R-CNN R101-FPN\\cite{wu2019detectron2}    & -      & -      & -      & -      & 44.3  & -     & -     & -                                                                                              \\\\\n   Cascade R-CNN\\cite{cai2018cascade}                                       & 42.8   & 23.7   & 45.5   & 55.2   & -     & -     & -     & -                                                                                              \\\\\n   FCOS\\cite{2020FCOS}                                                              & 43.2    & 26.5   & 46.2   & 53.3   & -     & -     & -     & -                                                                                              \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x\\cite{zhang2020bridging}                      & 39.2   & -      & -      & -      & 39.2  & -     & -     & -                                                                                              \\\\\n   ATSS\\_dcnv2\\_R\\_50\\_FPN\\_1x\\cite{zhang2020bridging}          & 43.0   & -      & -      & -      & 43.0  & -     & -     & -                                                                                              \\\\\n   ATSS\\_R\\_101\\_FPN\\_2x\\cite{zhang2020bridging}                    & 43.6   & -      & -      & -      & 43.5  & -     & -     & -                                                                                              \\\\\n   ATSS\\_dcnv2\\_R\\_101\\_FPN\\_2x\\cite{zhang2020bridging}        & 46.3   & -      & -      & -      & 46.1  & -     & -     & -                                                                                              \\\\\n   ATSS\\_X\\_101\\_32x8d\\_FPN\\_2x\\cite{zhang2020bridging}        & 45.1   & -      & -      & -      & 44.8  & -     & -     & -                                                                                             \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_32x8d\\_FPN\\_2x\\cite{zhang2020bridging}& 47.7   & -      & -      & -      & 47.7  & -     & -     & -                                                                                              \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_64x4d\\_FPN\\_2x\\cite{zhang2020bridging}& 47.7   & -      & -      & -      & 47.7  & -     & -     & -                                                                                             \\\\ \\hline\\hline\n   \\multicolumn{9}{|c|}{Comparison test 1}                                                                                                                                                                                                                                     \\\\ \\hline\\hline\n   ATSS\\_R\\_50\\_FPN\\_1x + IoU \\& loss                                       & 38.6  &  20.7    &  37.4    &  45.7    & 38.5  & 21.0  & 33.7  & 50.9                                                                                        \\\\\n   ATSS\\_dcnv2\\_R\\_50\\_FPN\\_1x + IoU \\& loss                           & 41.9  & 24.0    &  45.8     &  53.8    & 42.5  & 23.4  & 44.9  & 56.9                                                                                         \\\\\n   ATSS\\_dcnv2\\_R\\_101\\_FPN\\_2x + IoU \\& loss                         & 45.8  &  25.9   &  48.6     &  56.9    & 45.7  & 28.5  & 49.7  & 60.0                                                                                         \\\\\n   ATSS\\_X\\_101\\_32x8d\\_FPN\\_2x + IoU \\& loss                         & 44.5  &  26.8    &  47.2     &  53.2    & 44.7  & 27.7  & 46.1  & 55.9                                                                                         \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_32x8d\\_FPN\\_2x + IoU \\& loss             & 46.8  &  27.9    &  49.7     &  58.7    & 47.0  & 29.7  & 50.1  & 60.2                                                                                         \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_32x8d\\_FPN\\_2x(MS) + IoU \\& loss      & 49.6  &  31.2   &  50.5     &   60.3   & 49.9  & 32.7  & 50.9  &  61.4                                                                                         \\\\ \\hline\\hline\n   \\multicolumn{9}{|c|}{Comparison test 2}                                                                                                                                                                                                                                     \\\\ \\hline\\hline\n   Faster R-CNN + CDIoU \\& loss                                                   & \\textbf{38.3}   & 17.3  & 38.0  & 54.4  & \\textbf{38.5}  & 17.5  & 37.9  & 55.7                                                                                         \\\\\n   YOLOv4 + CDIoU \\& loss                                                            & \\textbf{41.4}   & 20.4  & 46.1  & 55.8  & \\textbf{41.9}  & 21.0  & 46.7  & 57.1                                                                                         \\\\\n   RetinaNet-R101 (1024) + CDIoU \\& loss                                      & \\textbf{41.2}   & 22.5  & 43.1  & 50.9  & \\textbf{41.5}  & 24.5  & 45.3  & 52.7                                                                                         \\\\\n   ResNet-50 + NAS-FPN (1280@384) + CDIoU \\& loss                   & \\textbf{45.8}   & 22.1  & 48.0  & 65.1  & \\textbf{46.0}  & 22.1  & 48.9  & 67.4                                                                                         \\\\\n   Detectron2 Mask R-CNN R101-FPN + CDIoU \\& loss                  & \\textbf{45.0}   & 21.3  & 46.0  & 64.9  & \\textbf{45.2}  & 21.9  & 45.7  & 66.5                                                                                         \\\\\n   Cascade R-CNN + CDIoU \\& loss                                                & \\textbf{43.0} & 25.0    & 45.7  & 66.1  & \\textbf{43.3}  & 25.3  & 46.1  & 67.2                                                                                         \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x + CDIoU \\& loss                                     & \\textbf{39.4}&  22.5    &  42.2     &  49.8    & \\textbf{39.5}  & 23.0  & 34.4  & 52.1                                                                                         \\\\\n   ATSS\\_dcnv2\\_R\\_50\\_FPN\\_1x + CDIoU \\& loss                         & \\textbf{43.1} & 24.4    &  46.0     &  55.8    & \\textbf{43.1}  & 25.4  & 46.9  & 57.8                                                                                         \\\\\n   ATSS\\_dcnv2\\_R\\_101\\_FPN\\_2x + CDIoU \\& loss                       & \\textbf{46.4} &  27.8   &  49.7     &  58.6    & \\textbf{46.3}  & 29.3  & 50.4  & 61.2                                                                                         \\\\\n   ATSS\\_X\\_101\\_32x8d\\_FPN\\_2x + CDIoU \\& loss                       & \\textbf{45.2} &  27.8    &  48.3     &  55.2    & \\textbf{45.1}  & 28.7  & 48.5  & 57.9                                                                                         \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_32x8d\\_FPN\\_2x + CDIoU \\& loss           & \\textbf{47.9} &  29.6    &  50.8     &  60.5    & \\textbf{48.1}  & 30.6  & 51.7  & 62.7                                                                                        \\\\\n   ATSS\\_dcnv2\\_X\\_101\\_32x8d\\_FPN\\_2x(MS) + CDIoU \\& loss    & \\textbf{50.7} &  33.2   &  52.6     &   62.7   & \\textbf{50.9}  & 35.8  & 53.4  &  65.3                                                                                         \\\\ \\hline\n   \\end{tabular}\n   \\caption{Detection models results with and without IoUs and IoU loss function on the \\textbf{MS COCO} \\texttt{val} and \\texttt{test-dev} set.}\n    \\begin{tablenotes}\n        \\footnotesize\n        \\item[1] \\textbf{MS} means multi-scale testing.\n        \\item[2] \\textbf{+ CDIoU \\& loss} means that this model uses CDIoU and CDIoU loss as evaluation-feedback module.\n        \\item[3] \\textbf{+ IoU \\& loss} means that this model uses IoU and IoU loss as evaluation-feedback module.\n        \\item[4] The original ATSS models use IoU and GIoU loss as evaluation-feedback module, and the original Faster R-CNN,YOLOv4,RetinaNet-R101,ResNet-50 + NAS-FPN,Detectron2 Mask R-CNN,Cascade R-CNN models use IoU and IoU loss or \\textit{L1-smooth} as evaluation-feedback module.\n        \\item[5] FCOS model uses the SSC method, so there is no comparison test.\n        \\item[6] Bold fonts indicate the best performance.\n        \\item[7] \\textit{1x} and \\textit{2x} mean the model is trained for 90K and 180K iterations, respectively.\n        \\item[8] All results are obtained with a single model and without any test time data augmentation such as \\textit{multi-scale, flipping} and etc..\n        \\item[9] \\textit{dcnv2} denotes deformable convolutional networks v2.\n      \\end{tablenotes}\n    \\end{threeparttable}\n   \\end{table*}\n\nCDIoU and CDIoU loss are universally adaptable, exhibiting differential performance gains on different models.\nAs shown in Table 2, we can see that the more complex the backbone structure is,\nthe less the CDIoU and CDIoU loss improvement is, while for the basic model,\nthe CDIoU and CDIoU loss improvement is more obvious. By using CDIoU and CDIoU loss,\nthe models in Table 2 improved AP by an average of 0.8\\%.\n\n\\subsection{Ablation studies}\n\nFaster R-CNN is a classical model, while ATSS is a new model recently. In Table 3, the evaluation system of all models is IoU, the most basic original one.\nBut in the loss function, we selects different calculation functions as feedback mechanism. \\textit{FPS} represents the number of images that can be processed per second. CDIoU loss does not increase the amount of computation and barely improves \\textit{FPS}.\n\n\\renewcommand\\arraystretch{1.1}\n\\begin{table}[h]\n   \\centering\n  \n   \\begin{threeparttable}\n   \\begin{tabular}{|c|c|c|c|c|c|}\n  \n   \\hline\n   IoUs loss                    & Model                                                & AP                     & FPS                   \\\\ \\hline\\hline\n   L1-smooth                 & Faster R-CNN\\cite{2017Faster}            & 36.0                   & 7.5                     \\\\\n   IoU loss                     & Faster R-CNN                                      & 36.8                   & 7.7                     \\\\\n   GIoU loss                   & Faster R-CNN                                     & 36.9                   & 8.5                     \\\\\n   DIoU loss                   & Faster R-CNN                                     & 38.0                   & 7.9                     \\\\\n   CIoU loss                   & Faster R-CNN                                      & 38.2                   & 6.3                    \\\\\n   CDIoU loss                  & Faster R-CNN                                     & 38.5                   & 7.7                    \\\\ \\hline\\hline\n   L1-smooth                 & ATSS\\_R\\_50\\_FPN\\_1x\\cite{zhang2020bridging} & 37.5    & 11.1                    \\\\\n   IoU loss                     & ATSS\\_R\\_50\\_FPN\\_1x                         & 38.0                   & 10.8                    \\\\\n   GIoU loss                   & ATSS\\_R\\_50\\_FPN\\_1x                        & 39.2                   & 11.0                   \\\\\n   DIoU loss                   & ATSS\\_R\\_50\\_FPN\\_1x                        & 39.0                   & 11.3                   \\\\\n   CIoU loss                   & ATSS\\_R\\_50\\_FPN\\_1x                         & 39.2                   & 8.8                    \\\\\n   CDIoU loss                 & ATSS\\_R\\_50\\_FPN\\_1x                        & 39.4                   & 11.2                   \\\\ \\hline\n   \\end{tabular}\n  \n   \\caption{Comparison of effects and running results(\\%) of various IoU losses on the \\textbf{MS COCO} \\texttt{val} set.}\n       \\begin{tablenotes}\n        \\footnotesize\n        \\item[1] The default evaluation module IoUs is IoU in this table.\n      \\end{tablenotes}\n    \\end{threeparttable}\n   \\end{table}\n\nAs shown in Table 3, we can accurately see that CDIoU loss function can significantly improve the AP results by 0.2$\\sim$1.9 $\\%$\ncompared to other loss functions, and this effect is more obvious in traditional and basic models.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.38]{fasterrcnnloss.png}\n\\caption{Loss Function value of Faster R-CNN}\n\\label{fig:label}\n\\end{figure}\n\n\\begin{figure}[h]\n\n\\centering\n\\includegraphics[scale=0.36]{atssloss.png}\n\\caption{Loss Function value of ATSS-R-50-FPN-1x}\n\\label{fig:label}\n\\end{figure}\n\n\\begin{figure}[h]\n\n\\centering\n\\includegraphics[scale=0.47]{fap.png}\n\\caption{The final AP of Faster R-CNN and ATSS-R-50-FPN-1x}\n\\label{fig:label}\n\\end{figure}\n\nFrom Figure 7,8 , it is clear that the loss function values of both Faster R-CNN and\nATSS-R-50-FPN-1x models drop and reach convergence faster after using CDIoU and CDIoU loss function\ncompared to the other loss functions. The Figure 9 shows that the CDIoU and CDIoU loss functions\ncan help the models achieve higher AP values with fewer iterations. These results show that the CDIoU\nand CDIoU loss functions have strong convergence and highlight their more accurate evaluation of region proposals.\n\n\nIn order to rigorously verify the effectiveness of CDIoU and CDIoU loss function\nproposed in this paper, a large number of comparative experiments are designed to suggest supporting evidence.\nWe can learn from Table 4 that CDIoU loss can indeed achieve better results than IoU loss,\nGIoU loss, DIoU loss, etc., while using the same IoU section.\n\n\\renewcommand\\arraystretch{1.1}\n\\begin{table*}[h]\n   \\centering\n  \n   \\begin{threeparttable}\n   \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n   \\hline\n   \\multirow{2}{*}{Model}     & \\multicolumn{2}{c|}{IoUs}             & \\multicolumn{5}{c|}{IoU loss functions}                 & \\multirow{2}{*}{AP}      \\\\ \\cline{2-8}\n                                          & GIoU      & CDIoU      & IoU loss & GIoU loss & DIoU loss & CIoU loss & CDIoU loss &                                   \\\\ \\hline \\hline\n   Faster R-CNN                               & \\checkmark&            &\\checkmark&           &           &           &            &   36.9                                 \\\\\n   Faster R-CNN                               & \\checkmark&            &          &\\checkmark &           &           &            &   37.3                                 \\\\\n   Faster R-CNN                               & \\checkmark&            &          &           & \\checkmark&           &            &   38.0                                 \\\\\n   Faster R-CNN                               & \\checkmark&            &          &           &           &\\checkmark &            &   38.2                                 \\\\\n   Faster R-CNN                               & \\checkmark&            &          &           &           &           & \\checkmark &   38.3                                 \\\\ \\hline\n   Faster R-CNN                               &           &\\checkmark  &\\checkmark&           &           &           &            &   37.1                                 \\\\\n   Faster R-CNN                               &           &\\checkmark  &          &\\checkmark &           &           &            &   37.3                                 \\\\\n   Faster R-CNN                               &           &\\checkmark  &          &           &\\checkmark &           &            &   38.1                                 \\\\\n   Faster R-CNN                               &           &\\checkmark  &          &           &           & \\checkmark&            &   38.3                                \\\\\n   Faster R-CNN                               &           &\\checkmark  &          &           &           &           &\\checkmark  &   \\textbf{38.5}                  \\\\ \\hline\\hline\n   ATSS\\_R\\_50\\_FPN\\_1x                       &\\checkmark &            &\\checkmark&           &           &           &            &   38.0                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &\\checkmark &            &          &\\checkmark &           &           &            &   39.3                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &\\checkmark &            &          &           & \\checkmark&           &            &   39.0                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &\\checkmark &            &          &           &           &\\checkmark &            &   39.2                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &\\checkmark &            &          &           &           &           & \\checkmark &   39.4                           \\\\ \\hline\n   ATSS\\_R\\_50\\_FPN\\_1x                       &           &\\checkmark  &\\checkmark&           &           &           &            &   38.1                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &           &\\checkmark  &          &\\checkmark &           &           &            &   39.3                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &           &\\checkmark  &          &           &\\checkmark &           &            &   39.2                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &           &\\checkmark  &          &           &           &\\checkmark &            &   39.2                           \\\\\n   ATSS\\_R\\_50\\_FPN\\_1x                       &           &\\checkmark  &          &           &           &           &\\checkmark  &  \\textbf{39.5}                   \\\\ \\hline\n   \\end{tabular}\n   \\caption{Comparison of effects and running results(\\%) of various IoUs and IoU loss functions on the \\textbf{MS COCO} \\texttt{val} set.}\n       \\begin{tablenotes}\n        \\footnotesize\n        \\item[1] For more information about results of evaluation module IoU, please refer to table 3.\n      \\end{tablenotes}\n    \\end{threeparttable}\n   \\end{table*}\n\nBy comparing the above experiments, we can observe that using different IoU modules with different IoU loss\nfunctions yields different and thought-provoking results. Combining Table 3,4 , we can analyze that under the same IoU loss function condition,\nusing CDIoU module alone can improve the final result by 0.3 $\\sim$ 1.8$\\%$; under the same IoU module,\nusing CDIoU loss function alone can improve the final result by 0.2 $\\sim$ 1.7$\\%$.\n\nAt the same time, if IoU module and IoU loss function could be unified in the computational form (eg.GIoU + GIoU loss function and CDIoU + CDIoU loss function),\nthe final result would seem to be better than the sum of the results of using the two optimization schemes independently.\n\nIn order to verify the independence and validity of CDIoU and CDIoU loss function, we designs the following comparison test in Table 5,\nusing original IoU + IoU loss(or DIoU loss) function and CDIoU + CDIoU loss function in training and validation stages of the program respectively,\nand finally comparing their AP results. We use CDIoU and CDIoU loss in the training and validation stages respectively,\nwhich can improve the final AP. If we used CDIoU and CDIoU loss in both training and validation stages, we could get more significant improvement results.\n\n\\subsection{Analysis of experiments}\n\nThe improvement effect of CDIoU + CDIoU loss tends to decrease as the model is updated. First, as the backbone of the model deepens,\nthe model itself enhances the strength of feature extraction. Second,\nthe continuous improvement of FPN modules also optimizes the function of traditional evaluation systems.\nThe above two points offset the advantages of CDIoU and CDIoU loss compared with the traditional evaluation-feedback modules.\n\nATSS bridges the gap between anchor-based and anchor-free detection via adaptive training sample selection.\nComparison tests on ATSS exclude the essential interference between anchor-based and anchor-free detection.\nIn these tests, the interference of positive and negative sample generation is eliminated, which give tests based on ATSS more representativeness.\n\n\\renewcommand\\arraystretch{1.1}\n\\begin{table*}[h]\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{Model}       & \\multicolumn{3}{c|}{Training}                                       & \\multicolumn{2}{c|}{validation}                & \\multirow{2}{*}{AP}       \\\\ \\cline{2-6}\n                                         & IoU+IoU loss & IoU+GIoU loss & CDIoU+CDIoU loss & Originals & CDIoU+CDIoU loss       &                                    \\\\ \\hline \\hline\nFaster R-CNN                    & \\checkmark   &               &                                         &\\checkmark &                        &         36.8                             \\\\\nFaster R-CNN                    & \\checkmark   &               &                                         &           &  \\checkmark            &         37.2                             \\\\ \\hline\nFaster R-CNN                    &                      &               &  \\checkmark                     &\\checkmark &                        &         37.7                             \\\\\nFaster R-CNN                    &                      &               &  \\checkmark                     &           &  \\checkmark            &\\textbf{38.5}                         \\\\ \\hline\\hline\nATSS\\_R\\_50\\_FPN\\_1x       &                      &  \\checkmark   &                                & \\checkmark&                        &         39.2                              \\\\\nATSS\\_R\\_50\\_FPN\\_1x       &                      &  \\checkmark   &                                 &           &  \\checkmark            &         39.3                             \\\\ \\hline\nATSS\\_R\\_50\\_FPN\\_1x       &                      &               &    \\checkmark                  & \\checkmark&                        &         39.3                              \\\\\nATSS\\_R\\_50\\_FPN\\_1x       &                      &               &    \\checkmark                   &           &  \\checkmark            &\\textbf{39.5}                         \\\\ \\hline\n\\end{tabular}\n\\caption{Comparison of effects and running results(\\%) in training and validation on the \\textbf{MS COCO} \\texttt{train} and \\texttt{val} set.}\n\\begin{tablenotes}\n        \\footnotesize\n        \\item[1] \\textit{Originals} means that original Faster R-CNN uses IoU+IoU loss as evaluation-feedback module and original ATSS\\_R\\_50\\_FPN\\_1x  uses IoU+GIoU loss as evaluation-feedback module.\n      \\end{tablenotes}\n    \\end{threeparttable}\n\\end{table*}\n\nFrom Table 5 , we can clearly observe that replacing IoU module and loss function separately can improve the results of the original model,\nand replacing IoU module and loss function at the same time also has a certain additive effect, achieving the synergy of ``one plus one is greater than two\".\nIt lies on the calculation form consistency between evaluation system and feedback mechanism. The numerical fluctuations of the feedback mechanism reflect the differences of the evaluation system, which makes the evaluation-feedback module more targeted.\n\n\\subsection{Tips to improve performances}\n\nIn this experiment, we also found some tips to improve AP. These tips are particularly useful for some basic models in this paper.\n\n\\textbf{Floating learning rate}\n\nIt is a consensus that the learning rate decreases as the iterative process in the experiment. Further,\nthis paper proposes to check the loss every $\\mathcal{K}$ iterations and increase the learning rate slightly, if the loss function does not decrease continuously.\nIn this way, the learning rate will decrease and float appropriately at regular intervals to promote the decrease of the loss function.\n\n\\begin{equation}\n\\begin{array}{ll}\nl r=\\left\\{\\begin{aligned}\n1.05lr \\textit{, Ls}<0 \\\\\nlr       \\textit{, Ls}>0\n\\end{aligned}\\right. \\\\\n\\textit{Ls}=\\operatorname{loss}_{i}-\\operatorname{loss}_{i-k}\n\\end{array}\n\\end{equation}\n\n\\textbf{Automatic GT clustering analysis}\n\nIt is well known that AP can be effectively improved by performing cluster analysis on GT in the original dataset.\nWe adjust anchor sizes and aspect ratios parameters based on the results of this cluster analysis.\nHowever, we do not know the number of clusters through the current approach.\nThe main solution is to keep trying the number of clusters $\\mathcal{N}$,\nand then judge by the final result AP. Obviously, this exhaustive method takes a lot of time.\n\nIn this paper, automatic GT clustering analysis is proposed, using \\textit{K-means\/PAM},\n\\textit{Hierarchical Clustering}, \\textit{Spectral Clustering}, \\textit{DBSCAN} and \\textit{Mean-shift} methods respectively,\nwhere \\textit{DBSCAN} and \\textit{Mean-shift} methods are able to obtain the number of clusters autonomously.\nThe above methods were evaluated using \\textit{SSE} (sum of the squared errors), \\textit{Silhouette Coefficient} and \\textit{Calinski-Harabaz},\nand then two recommended schemes are obtained. These recommended schemes include the number of clusters\nand the central GT of each cluster. We obtained anchor information from the central GT\nbefore executing the complex deep learning network, so that the experiments in this paper\nare much more efficient.\n\n\\section{Conclusion}\n\nIn this paper, we propose that a good evaluation-feedback module should focus on\nboth evaluation system and feedback mechanism. An evaluation-feedback module\nshould meet 3 conditions (overlapping area,\nthe degree of difference and correlated with the loss function)\nand 3 main roles (evaluation,ranking and feeding the gap).\nFinally this paper proposes CDIoU and CDIoU loss,\na unified interrelated evaluation-feedback module without increasing the model running time and system memory.\nThrough a large number of experiments,\nwe can demonstrate that CDIoU and CDIoU loss can effectively improve\nthe performance of multiple object detection models without  increasing parameters or FLOPs in models.\nBy using CDIoU and CDIoU loss, the models improved their AP by an average of 0.8\\%.\nAnd there are a maximum AP performance improvement of 1.9\\% and an average AP performance of 0.8\\% improvement on MS COCO\ndataset, comparing to traditional evaluation-feedback modules.\nIt lies on the calculation form consistency between evaluation system and feedback mechanism,\nwhich makes the evaluation-feedback module more targeted.\n\n\\section*{Acknowledgements}\nThis research was supported in part by the National Natural Science\nFoundation of China 61976158 and Grant Nos.~62006172.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and motivation}\n\nIf there is a quantum-mechanical theory of gravity, the big challenges in its formulation include understanding the fundamental description of its quantum states and observables, as well as the nature of the unitary evolution on its Hilbert space.  Approaches to this problem based on quantizing general relativity (GR) or related classical theories have run into vexing problems, initially nonrenormalizability\\footnote{For a review with further references, see \\cite{Burg}.} but likely more profoundly that of nonunitarity in the high-energy regime of black holes.  An alternative approach is to begin with the hypothesis that one is working with a quantum-mechanical theory, and investigate what mathematical structure of such a theory is necessary to describe gravity and consistently match the known and tested physics of local quantum fields propagating on a weakly-curved background, in the appropriate limits.  This might be referred to as a ``quantum-first\" approach \\cite{UQM,SGAlg,QFG,QGQF}.\\footnote{For related discussion, see \\cite{CCM,CaCa,CaSi}.}  One would like to understand the nature of the Hilbert space for gravity, and of its algebras of observables, symmetries, and unitary evolution law.\n\nSuch an approach does not argue for completely abandoning a perturbative quantization of GR.  The match to the known and tested physics of local quantum field theory (LQFT) on a weakly curved background, the confirmed existence of gravitational waves, and the apparent approximate validity of strong-field classical solutions suggests that such a perturbative treatment gives at least {\\it approximately} correct physics, in the weak-gravity regime, though one that is missing important effects in other regimes.  One can view this as a ``correspondence principle\" for quantum gravity.  What is interesting is that already in this limit, one encounters non-trivial new properties of quantum gravity that signal its departure from LQFT.  This also raises the hope that, by better understanding this structure in the perturbative limit, one may infer key properties of the more basic structure of a fundamental theory of quantum gravity.\n\nIn particular, a significant part of the difficulty of gravity seems to stem from the form of its gauge symmetries.  And, significant aspects of this non-trivial gauge structure appear to already be present at leading perturbative orders.  This suggests that a useful starting point is simply to better understand this structure at these leading orders.\n\nOne aspect of this structure is the lack of local gauge invariant observables\\cite{Torr}.  In short, any local observable clearly carries nontrivial Poincar\\'e charge (in the example of flat asymptotics), since it doesn't commute with translation generators, and this must source an associated gravitational field that extends to infinity\\cite{DoGi2}.  \n\nThere are different approaches to constructing {\\it non}local observables that respect gauge invariance, typically in a ``relational\" fashion.  One approach is to specify the position of a quantum operator relative to other quantum fields that vary in spacetime; we refer to the resulting operator as ``field-relational\" (or, observer-relational), and an example is provided by calculation of primordial perturbations in cosmology by referring to the time of reheating set by the inflaton in inflationary models.  Another alternative is to construct relational observables by using position information from the gravitational sector; perturbatively, one can begin with a local observable, and ``gravitationally dress\" it to construct gauge-invariant operators that no longer commute at spacelike separation\\cite{SGAlg}\\cite{DoGi1}\\cite{DoGi2}\\cite{DoGi3}\\cite{QGQF}\\cite{DoGi4,GiKi}.\\footnote{To clarify a difference in terminology, the recent work\n\\cite{CLPW} considers observables referred to an observer but calls them dressed, despite not having a gravitational component; here those would be referred to as field-relational.  Earlier work related to dressed observables includes \\cite{Heem} and \\cite{KaLigrav}; the first derived nontrivial commutators as arising from constraints, but didn't give the dressed operators, and the second focussed on deriving {\\it commuting} operators.}\nIn quantization of GR, the diffeomorphism symmetry is generated by the gravitational constraints, so in either case, a test for gauge invariance of such operators is that they commute with the constraints.  \n\n\nAnother key question for a quantum-mechanical theory is the structure of its states and their evolution.  The observables both characterize states, and furnish a means of constructing gauge-invariant states: one can act with a quantum observable on a ``vacuum\" state to create a nontrivial state.  The important question of evolution of these states, if it is unitary, can then be addressed by providing a hamiltonian.\\footnote{Even in LQFT, it is argued that the hamiltonian provides a more  fundamental description of evolution than an action; see {\\it e.g.} \\cite{Ramo}, sec.~2.2.}  In LQFT coupled to quantized GR, as we will review and further clarify, the hamiltonian and evolution are of course closely related to the constraints.  In a closed universe, the hamiltonian is given by the constraints, and so formally vanishes on their solution; in a universe with asymptotic spatial infinity, there is an additional term in the hamiltonian, that is important for evolution.   In particular, solving the quantum version of the hamiltonian constraint is commonly referred to as solving the Wheeler-de Witt (WdW) equation, and is accomplished by gravitationally dressing undressed operators or states.\\footnote{Ref.~\\cite{Rajuetal} also discusses perturbative solution of the WdW equation, but seems not to have realized that this is achieved by constructing gravitational dressing, nor recognized the relevance of preceding works on this subject.}\n An additional subtlety (see below) is that physical states may only be annihilated by ``half\" of these constraints.  The form of the evolution in the perturbative regime is expected to furnish clues about its nonperturbative completion.\n\n\n\nConstruction of the gravitational dressing, which can be explicitly treated at leading order in the gravitational coupling, is also relevant to the question of holography of gravity.  A leading proposed explanation of holography in anti de Sitter space (AdS) is that it follows from the hamiltonian being a boundary term\\cite{Maroholo,Maroholonost} when the constraints are satisfied; for additional discussion, see \\cite{Jacoholo,SGholo}. A closely similar argument is that momentum generators are also boundary terms, and so one can act both to translate a state to infinity, and to measure it, purely with operators at infinity\\cite{DoGi3}.  There are related arguments that even perturbative observables at infinity can determine bulk states\\cite{Rajuetal}, but their sensitivity is exponentially suppressed\\cite{SGsub}. These statements do rely on solution of the constraints, so implicitly on solution of the bulk dynamics\\cite{SGholo}.\\footnote{Entanglement wedge reconstruction appears to likewise assume solution of the constraints\\cite{JLMS}\\cite{SGholo}.}   \n\nHowever these arguments are ultimately understood, it is clear that the dressing modifies the locality properties of the algebra of operators\\cite{SGAlg,DoGi1}; it also apparently connects\\cite{SGinpro} to recent discussion of modification of the structure of von Neumann algebras from type III to II\\cite{Wittcross}.\n\nIn short, at the perturbative level it appears that we can begin to learn important aspects of the structure of observables, states, and their evolution.  This description is of course expected to miss crucial effects, particularly when treating strong gravitational configurations such as black holes.\\footnote{For an approach to parameterizing such effects as departures from LQFT evolution, see \\cite{NVU,BHQU} and references therein.}  But, a clearer understanding of the perturbative structure is also expected to help provide a basis and background for understanding the role of modifications in the strong gravity context, if the complete theory respects the correspondence principle and is consistent with its weak gravity limit.  \n\nIn the interest of such a deeper understanding of the interplay of evolution, gauge symmetry, the constraints, and gravitational dressing, in general contexts, this paper will investigate the perturbative structure of the hamiltonian and constraints, working perturbatively about a general background.  The next section begins with a simplified derivation of the hamiltonian, exhibiting it either in a more conventional local form familiar from LQFT, or as a term proportional to the constraints plus a boundary term.  Section three then outlines different approaches to perturbative quantization, and sets up a perturbative treatment of  the constraints in what we refer to as ``gauge-invariant canonical quantization.\"  Section four gives a leading perturbative construction of operators commuting with the constraints, working about a general background, in terms of a construction of the gravitational dressing that generalizes \\cite{DoGi1}\\cite{QGQF}\\cite{DoGi4,GiKi,SGsplit}.  Section five discusses construction of corresponding states, briefly discusses the form and characterization of their evolution, and illustrates application to the important cases of black holes and\/or AdS spacetimes.  Section six finishes with conclusions and further directions.\n\n\n\\section{Action, hamiltonian, and boundary terms}\n\n\\subsection{Action and boundary terms}\n\nThis section will review formulation of the action in Arnowitt-Deser-Misner (ADM)\\cite{ADM} variables, and describe a  simple approach to deriving the appropriate boundary terms.  This  paper focusses on quantization of Einstein gravity plus matter, perturbing around a general background.  The usual starting point is the action\\footnote{We find it most convenient to work with expressions for lagrangians and hamiltonians that are  densities.}\n\\begin{equation}\\label{action}\nS=\\int d^Dx \\left( \\frac{1}{16\\pi G} \\sqrt{|g|}R + {\\cal L}_m\\right) + S_\\partial\n\\end{equation}\nin $D$ spacetime dimensions, where $G$ is Newton's constant, ${\\cal L}_m$ is a matter lagrangian, and $S_\\partial$ is a boundary term.  If a specific matter action is needed, the scalar theory with\n\\begin{equation}\n{\\cal L}_m= -\\sqrt{|g|}\\left[\\frac{1}{2} (\\nabla \\phi)^2+V(\\phi)\\right]\n\\end{equation}\nfurnishes a useful example.   \n\n\\begin{figure}[!hbtp] \\begin{center}\n\\includegraphics[width=15cm]{fig1.pdf}\n\\end{center}\n\\caption{Shown are two members of a family of slices labelled by $t$.  Points at the same spatial coordinate $x^i$ are connected by the vector $\\xi^\\mu$, which can be decomposed in terms of normal and tangential components to give the lapse and shift; vectors in the figure are scaled by an implicit $\\delta t$.}\n\\label{Figslices}\n\\end{figure} \n\n\nSince our focus will be on the evolving quantum state describing perturbations about a background, we would like to find a corresponding hamiltonian.  We begin by \nintroducing a foliation of the spacetime by slices labelled by time $t$, and with spatial coordinate $x^i$, with relation to general coordinates given by\n\\begin{equation}\nx^\\mu= {\\cal X}^\\mu(t,x^i)\\ .\n\\end{equation}\nThe displacement vector between points of equal $x^i$ on nearby slices is given by $\\xi^\\mu = (\\partial {\\cal X}^\\mu\/\\partial t)_{x^i}$, and can be decomposed into pieces normal and tangential to a slice, \n\\begin{equation}\\label{xiton}\n\\xi^\\mu=N n^\\mu + N^\\mu\\ ,\n\\end{equation}\nwhere $N$ is the lapse, $N^\\mu$ is the shift, and $n^\\mu$ is the unit normal; in $(t,x^i)$ coordinates $N^\\mu=(0,N^i)$, and\nthese quantities are illustrated in Fig.~\\ref{Figslices}.\nIn the coordinates $(t,x^i)$ of the foliation the metric takes the ADM form\n\\begin{equation}\nds^2=-N^2 dt^2 +q_{ij}(dx^i+N^i dt) (dx^j+N^j dt)\\ \n\\end{equation}\nand the  unit normal to the slices has components\n\\begin{equation}\nn^\\mu=\\frac{1}{N} (1,-N^i)\\ .\n\\end{equation}\n\n\n\n\nThe gravitational lagrangian can then be derived in these variables, after introducing the extrinsic curvature of the slices,\\footnote{Note that there are differing sign conventions in the literature; {\\it e.g.} \\cite{Waldbook} differs by a sign.}\n\\begin{equation}\\label{excurv}\nK_{ij}= \\frac{1}{2N}\\left(-\\dot q_{ij} + D_iN_j + D_jN_i\\right)\\ ,\n\\end{equation}\nwith dot denoting $\\partial\/\\partial t$, $D_i$ the covariant derivative constructed from $q$, and latin indices raised\/lowered with the spatial metric $q$.  This lagrangian is given by\\cite{KuchII,KRV}\n\\begin{equation}\n \\sqrt{|g|}R = N\\sqrt q \\left[\\left(K_{ij}K^{ij} - K^2\\right) + R_q\\right] - 2\\partial_i\\left(\\sqrt q q^{ij} \\partial_j N\\right) + 2\\partial_i\\left(\\sqrt q KN^i\\right) -2\\partial_t(\\sqrt q K)\\ ,\n \\end{equation}\nwith $K=q^{ij}K_{ij}$ and $R_q$ the scalar curvature of $q$.  \nThe total derivative terms become boundary terms in the action, which can be cancelled by $S_\\partial$,\n\\begin{equation}\nS_\\partial = \\oint \\frac{dt dA^i}{8\\pi G} (\\partial_i N - K N_i) + \\int d^Dx \\frac{\\partial_t(\\sqrt q K)}{8\\pi G} + S_\\partial'\\ .\n\\end{equation}\nAn additional term $S_\\partial'$ is required, as is argued by requiring a well-defined variational principle in \\cite{ReTebdy} or finiteness of the action in \\cite{HaHo}.  \nThis can be described by introducing a background metric $g_0$, {\\it e.g.} the Minkowski or anti de Sitter metrics, depending on boundary conditions, and writing the full metric as\n\\begin{equation}\\label{metexp}\ng_{\\mu\\nu} = g_{0\\mu\\nu} + \\Delta g_{\\mu\\nu}\\quad ,\\quad q_{ij}= q_{0ij} +\\Delta q_{ij}\\ ;\n\\end{equation}\n it then takes the form\n\\begin{equation}\\label{Spp}\nS_\\partial'= - \\oint \\frac{dt dA^i}{16\\pi G} N \\left( D_0^j \\Delta q_{ij} - \\partial_i \\Delta q\\right)\\ ,\n\\end{equation}\nwith $dA^i$ the area element and $\\Delta q = q_0^{ij} \\Delta q_{ij}$.  This can be checked (using  equation \\eqref{hlinvar} below) to eliminate the problematic boundary terms in the variation of the action.\n\nThe resulting gravitational action can be written in terms of a local lagrangian,\n\\begin{equation}\nS_g = \\int d^D x {\\cal L}_g = \\int d^D x \\frac{N\\sqrt q}{16\\pi G} \\left[(K_{ij} K^{ij} - K^2) + R_q\\right] + S_\\partial'\\ ,\n\\end{equation}\nwith $S'_\\partial$ rewritten as a volume integral.\nThe structure of ${\\cal L}_g$ can for example be investigated by using the \nrelation\\cite{Edd,Sch,PMB}\n\\begin{equation}\\label{indrel}\n\\sqrt q R_q =\\sqrt q q^{lm}\\left(\\gamma^i_{jl} \\gamma^j_{im} - \\gamma^i_{lm} \\gamma^j_{ij} \\right) + \\partial_i\\left[\\sqrt q \\left(q^{jk}\\gamma^i_{jk} - q^{ij} \\gamma^k_{jk}\\right)\\right]\\ ,\n\\end{equation}\nwith $\\gamma^i_{jk}$ denoting the Christoffel symbols computed from the metric $q$.  If the metric is expanded about a background solution as in \\eqref{metexp}, the linear terms vanish by the equations of motion of the background or cancellation with the boundary term, and quadratic and higher-order terms in the expansion of \\eqref{indrel} give a lagrangian with quadratic contributions of the form $(\\partial \\Delta q)^2$, plus interaction terms.  \n\n\n\n\n \n \\subsection{Hamiltonian and constraints}\n \n Momenta conjugate to the spatial metric $q$ are defined as\n \\begin{equation}\\label{conjmom}\n P^{ij} = \\frac{\\delta S_g}{\\delta \\dot q_{ij}} = -\\frac{\\sqrt q}{16\\pi G}\\left(K^{ij}- q^{ij} K\\right)\\ ;\n \\end{equation}\n we find it easiest to work with the form of these which are tensor {\\it densities}.  The momenta conjugate to $N, N_i$ of course vanish, corresponding to the fact that the lapse and shift are Lagrange multipliers enforcing constraints.  The gravitational  action can then be rewritten in the canonical form\n\\begin{equation}\nS_g = \\int d^D x \\left( P^{ij}  \\dot q_{ij} -  {\\cal H}_g\\right)\\ .\n\\end{equation}\nThe hamiltonian  is found by a straightforward calculation to be\n\\begin{eqnarray}\\label{hamdens}\n\\int dt H_g &=& \\int d^{D} x \n {\\cal H}_g\\ = \\int d^{D} x (P^{ij}  \\dot q_{ij} -{\\cal L}_g)\\cr &=& \\int d^{D} x\\left[ \\frac{16\\pi G  N}{\\sqrt q} \\left( P^{ij} P_{ij} -\\frac{P^2}{D-2}\\right) - \\frac{N\\sqrt q}{16\\pi G}R_q +  2 P^{ij} D_i N_j \\right] - S_\\partial'\\ .\n \\end{eqnarray}\n\nThe hamiltonian can be rewritten in  different ways.  First, as expected from the description of $S_g$ given above, the expression ${\\cal H}_g$ in  \\eqref{hamdens} is quadratic in momenta and first derivatives of the metric perturbation.  Alternately,  \\eqref{hamdens} can be rewritten in terms of the Einstein tensor as\n\\begin{equation}\\label{HamEin}\n \\int dt H_g=\\int d^{D}x\\left[  -\\frac{\\sqrt q G_{nt}}{8\\pi G} + 2D_i(P^{ij} N_j)\\right] -S_\\partial' ,\n\\end{equation}\nwhere $G_{nt} = n^\\mu \\xi^\\nu G_{\\mu\\nu}$.  The matter hamiltonian likewise is given in terms of the stress tensor $T_{\\mu\\nu}$ for $\\phi$ as\n\\begin{equation}\n {\\cal H}_m = \\sqrt q T_{nt}\\ .\n\\end{equation}\nThen, the full hamiltonian becomes\n\\begin{equation}\\label{htot}\nH = \\int d^{D-1} x \\sqrt q\\, C_\\xi + H_\\partial\\ ,\n\\end{equation}\nwith \n\\begin{equation}\nC_\\xi\\coloneqq  \\xi^\\mu C_\\mu \\coloneqq \\xi^\\mu\\left( -\\frac{G_{\\mu\\nu}}{8\\pi G} + T_{\\mu\\nu} \\right) n^\\nu\n\\end{equation}\ngiving the usual gravitational constraints.  The boundary contribution is, from \\eqref{Spp} and \\eqref{HamEin},\n\\begin{equation}\\label{boundham}\nH_\\partial= \\oint dA^i \\left[\\frac{N}{16\\pi G} \\left(D_0^j \\Delta q_{ij} -\\partial_i \\Delta q\\right) + 2 \\frac{P^{ij}}{ \\sqrt q} N_j \\right]\\ ,\n\\end{equation}\nwhich is the expected boundary expression\nfor gravity\\cite{ReTebdy}, $-N(\\infty)P_0^{ADM}-N^i(\\infty) P_i^{ADM}$. As is well known, if the constraints are satisfied,\n\\begin{equation}\\label{constraints}\nC_\\mu=0\\ ,\n\\end{equation}\nthen the hamiltonian becomes simply this boundary expression \\eqref{boundham}.\n\nThe interplay of the expressions \\eqref{hamdens} and \\eqref{htot} is worth noting, and can be summarized in\n\\begin{equation}\\label{hreln}\nH = \\int d^{D-1}x \\sqrt q \\left(N C_n + N^i C_i\\right) + H_\\partial = \\int d^{D-1}x \\, ({\\cal H}_g +{\\cal H}_m)\n\\end{equation}\nwhere we have used  \\eqref{xiton} relating $\\xi$ to $n$, and define $C_n=n^\\mu C_\\mu$.   \nOn the one hand, using the expansion of ${\\cal L}_g$ described in connection with \\eqref{indrel},\n the rightmost expression in \\eqref{hreln}\nis of the general expected form for a field theory, with quadratic terms in momenta and derivatives of fields, as well as potential terms.  We can think of this as generating time translations in the usual way.  On the other hand, one can work with a solution of the constraints \\eqref{constraints}, in which case the hamiltonian reduces to the surface term $H_\\partial$.  The later observation has been argued to be connected to the ``holographic\" property of gravity \\cite{Maroholo,Maroholonost,Jacoholo}, but does rely\\cite{SGholo} on first solving the constraints, which behave as equations of motion.  \n\nAn important question is thus the role of the constraints and different forms for the hamiltonian in the quantum theory, as well as their possible corrections from a more complete quantum theory.\n\n\\section{Quantization and perturbative expansion}\n\n\\subsection{Quantization, constraints, and gauge invariance}\n\nOur goal is to find a consistent quantum theory reproducing the preceding classical structure.\\footnote{For an analogous discussion for QED, see the appendix.}  The canonical approach tells us to introduce canonical commutators,\n\\begin{equation}\\label{cancom}\n[P^{ij}(x,t),q_{kl}(x',t)] = -i \\delta^i_{(k}\\delta^j_{l)} \\delta^{D-1}(x-x')\\ ,\n\\end{equation}\nwith normalization \n\\begin{equation}\n\\delta^i_{(k}\\delta^j_{l)} = \\frac{1}{2}\\left( \\delta^i_{k}\\delta^j_{l} + \\delta^i_l\\delta^j_{k}\\right)\\ .\n\\end{equation}\nSince $N$ and $N^i$ have vanishing conjugate momenta, they are taken to be c-numbers, and \\eqref{hreln} shows their role as Lagrange multipliers for the constraints.  This means that these variables are not determined by the equations of motion, and this is of course connected to the gauge symmetry.  The gauge transformations are generated by $C_n$ and $C_i$. \n\nThe Heisenberg equations of motion take the form \n\\begin{equation}\\label{qeqn}\n\\partial_t q_{ij} = i [H,q_{ij}]\\ ,\n\\end{equation}\nwhich reproduces the expression \\eqref{excurv} for the extrinsic curvature, using \\eqref{hamdens}, and \n\\begin{equation}\\label{Peqn}\n\\partial_t P^{ij} = i [H,P^{ij}]\n\\end{equation}\nwhich gives the $ij$ Einstein equations.  These equations are not gauge invariant, unlike the case of QED (see appendix), since the gauge transformations act non-trivially on $q_{ij}$ and $P^{ij}$.  \n\nThe next question is how to describe physical states.  It is tempting to require $C_\\mu|\\psi\\rangle=0$ for physical states, but this would then imply that for general operators $O$\n\\begin{equation}\n\\langle \\psi| [C_\\mu,O]|\\psi\\rangle =0\\,\n\\end{equation}\nconflicting with the preceding equations of motion.  Thus, in order to correctly describe nontrivial evolution, the constraints should not be taken to vanish identically on the physical Hilbert space.\n\nMultiple related ways to proceed have been studied in the literature, in each of which the question of locality becomes nontrivial.  A brief summary is:\n\n\\begin{enumerate}\n\n\\item Dirac quantization.  Here one introduces gauge-fixing conditions and solves these and the constraints, and also introduces a new Dirac bracket (or alternately redefines operators) such that $[C_\\mu,O]_D=0$.  This appears to simplify commutators, but it is also true that solving the gauge conditions and constraints is nonlocal.  This nonlocality is then ``hidden\" in the structure of the Dirac brackets; a simple example of this for QED is described in sec. IV.A.3 of \\cite{DoGi1}.  \n\n\\item Covariant gauge ``fixing\" (breaking).  In this approach a gauge-violating term is added to the action, and then canonical commutators postulated for all components of the metric.  This was used to study gauge-invariant operators in \\cite{DoGi1,GiKi}; the gauge breaking term decouples for these.  These operators are in general nonlocal, due to gravitational dressing, which is found by requiring vanishing commutators with the constraints\\cite{DoGi2}.\n\n\\item BRST\/BFV quantization\\cite{FrVi,BaVi}.  Here extra fields, including ghosts, are added; extra conditions are necessary as well.\n\n\\item Refined algebraic quantization, in which group averaging of states on an auxiliary kinematic space induces an inner product on the space of states satisfying the constraints\\cite{ALMMT,MaroRAQ,GiuMa,Marogroupav}.\n\n\\item ``Gauge invariant canonical quantization.\" \n\n\\end{enumerate}\n\nThe latter approach, which is briefly described for QED in the appendix, is closely similar to covariant gauge breaking, and will be followed here.  This involves defining a suitable decomposition of the constraints,\n\\begin{equation}\nC_\\mu(x)= C_\\mu^+(x) + C_\\mu^-(x)\n\\end{equation}\ninto what may be called positive and negative frequency parts, and then imposing the physical state condition\n\\begin{equation}\\label{physstate}\nC_\\mu^+(x)|\\psi\\rangle=0\n\\end{equation}\nfor a $|\\psi\\rangle$ taken to be the vacuum state $|0\\rangle$, on a given time slice.  Since the constraints generate gauge transformations, gauge-invariant operators are those satisfying\n\\begin{equation}\\label{conscomm}\n[C_\\mu(x),O]=0\\ .\n\\end{equation}\nSuch an operator creates a non-trivial state, $|\\psi\\rangle = O |0\\rangle$, which also satisfies \\eqref{physstate} by virtue of $[C_\\mu^+,O]=0$.  \n\nSuppose we consider evolution of states, in a Schr\\\"odinger picture, via the hamiltonian  \\eqref{hreln}.\nSince the full constraints don't annihilate physical states, the  time dependence of states will depend on gauge; the gauge-dependent part of the change in the state for evolution for time $\\delta t$ via the hamiltonian \\eqref{hreln} is\n\\begin{equation}\n\\delta_{\\bf N} |\\psi\\rangle = i \\delta t \\int d^{D-1}x \\sqrt q \\left(N C_n^- + N^i C_i^-\\right)|\\psi\\rangle\\ .\n\\end{equation}\nHowever, this will be orthogonal to another physical state $|\\psi'\\rangle$.   Likewise, if we consider evolution of a matrix element of a gauge invariant operator $O$,\n\\begin{equation}\n\\partial_t \\langle \\psi'|O |\\psi\\rangle = i\\langle \\psi'|[H,O]|\\psi\\rangle\\ ,\n\\end{equation}\nwe find that this is also independent of the gauge-variant ($\\bf N$-dependent) part of the hamiltonian \\eqref{hreln}.  In short, while \nthere is a gauge ambiguity in the states, that is not present in matrix elements of gauge-invariant operators.  Of course, we find from the Heisenberg equations \\eqref{qeqn} and \\eqref{Peqn} that evolution of matrix elements of $q_{ij}$ and $P^{ij}$ is gauge dependent.\n\n\\subsection{Perturbative expansion}\n\nThe remainder of this paper will primarily focus on a perturbative construction of states and operators like we have just described.  We begin with a classical metric $g_{\\mu\\nu}\\leftrightarrow(N,N_i,q_{ij})$ satisfying Einstein's equations, including the constraints \\eqref{constraints}, possibly also with a classical stress tensor.  The corresponding quantum variables are denoted $\\tilde g_{\\mu\\nu}\\leftrightarrow(\\tilde N,\\tilde N_i,\\tilde q_{ij})$.  Introducing the parameter $\\kappa^2=32 \\pi G$, these may be expanded as\n\\begin{equation}\n\\tilde g_{\\mu\\nu} = g_{\\mu\\nu} + \\kappa h_{\\mu\\nu}\\ ,\n\\end{equation}\nand likewise for $(\\tilde N,\\tilde N_i,\\tilde q_{ij})$, in particular with\n\\begin{equation}\n\\tilde q_{ij}= q_{ij} + \\kappa h_{ij}\\ .\n\\end{equation}\nWe will likewise expand $\\tilde P^{ij}$ as\n\\begin{equation}\n\\tilde P^{ij} = P^{ij} + \\frac{p^{ij}}{\\kappa}\\ ,\n\\end{equation}\nin which case the canonical commutators \\eqref{cancom} also take the simple form\n\\begin{equation}\\label{pertcom}\n[p^{ij}(x,t),h_{kl}(x',t)] = -i \\delta^i_{(k}\\delta^j_{l)} \\delta^{D-1}(x-x')\\ .\n\\end{equation}\nNotice, from \\eqref{conjmom}, that for a non-trivial classical background, $P^{ij}\\propto 1\/\\kappa^2$.\n\nThe explicit form of the constraints (temporarily written without tildes) is\n\\begin{equation}\\label{hamcons}\n0=C_n=T_{nn} - \\frac{2}{\\kappa^2}R_q + \\frac{\\kappa^2}{2q} \\left( P^{ij} P_{ij} -\\frac{P^2}{D-2}\\right)\\ ,\n\\end{equation}\nand \n\\begin{equation}\\label{momcons}\n0=\\sqrt q C^i = \\sqrt q T_n^i  -2 D_j P^{ij}\\ .\n\\end{equation}\nThese can then be expanded in the perturbations $h_{ij}$, $p^{ij}$. \n\n In $\\tilde C_n$, we use the expansion of the Ricci scalar\n\\begin{equation}\nR_{\\tilde q} = R_q + \\kappa \\delta_h R_q + \\delta^{\\geq 2}_{\\kappa h} R_q\\ ,\n\\end{equation}\nwhere $\\delta_h R_q$ denotes the first-order change, with the variation replaced by $h$, and the last term summarizes all higher-order terms in $\\kappa h$.  Explicitly,\n\\begin{equation}\\label{hlinvar}\n\\delta_h R_q = D^i D^j h_{ij} -R_q^{ij}h_{ij} -D_iD^i(q^{kl} h_{kl}) :=L^{ij} h_{ij}\\ ,\n\\end{equation}\ndefining the second-order differential operator $L^{ij}$.\nExpanding the $P$-dependent terms then gives\n\\begin{equation}\n\\tilde C_n = C_n  - \\frac{2}{\\kappa}L^{ij} h_{ij}  +\\frac{2}{\\kappa} {\\cal P}^{ij}h_{ij} - \\frac{2}{\\kappa\\sqrt q}  K_{ij} p^{ij} \n+ T_{nn} + \\breve t_{nn}\n\\end{equation}\nwhere $K_{ij}$ is the extrinsic curvature of the slices in the background solution, related to the background $P_{ij}$ by \\eqref{conjmom}, and \n\\begin{equation}\n {\\cal P}^{ij} = \\frac{\\kappa^4}{2q}\\left[P^{ik}P_k^j - \\frac{PP^{ij}}{D-2} - \\frac{q^{ij}}{2}\\left(P^{kl}P_{kl} -\\frac{P^2}{D-2}\\right)\\right]\\ ;\n \\end{equation}\n recall that the classical $P^{ij}$ is ${\\cal O}(\\kappa^{-2})$, so ${\\cal P}^{ij}$ is ${\\cal O}(1)$.\n  Now $T_{nn}$ denotes just the quantum part of the matter stress tensor, and  the expression $\\breve t_{nn}$ collects all the terms of quadratic and higher order in $h$ and $p$, and can be thought of as a piece of a gravitational stress tensor.  \n\nExpansion of the constraint \\eqref{momcons} is handled similarly, giving\n\\begin{equation}\n\\sqrt{\\tilde q} \\tilde C^j = \\sqrt q C^j - \\frac{2}{\\kappa} D_i p^{ij} - 2(\\Delta D_i)\\left(P^{ij}+ \\frac{p^{ij}}{\\kappa}\\right) + 2(\\Delta \\sqrt{q}) \\frac{D_i P^{ij}}{\\sqrt{q}}+ \\sqrt{\\tilde q} T_n^j\n\\end{equation}\nwhere $\\Delta D_i = \\tilde D_i - D_i$, $\\Delta \\sqrt q = \\sqrt{\\tilde q}-\\sqrt q$.  We then reorganize this as\n\\begin{equation}\\label{pmomcons}\n\\sqrt{\\tilde q} \\tilde C^j = \\sqrt q C^j - \\frac{2}{\\kappa} D_i p^{ij} -2\\left( \\delta_{\\kappa h} D_i \\right) P^{ij} + D_iP^{ij}\\kappa h  + \\sqrt{\\tilde q}\\left( T_n^j + \\breve t_n^j\\right)\\ ,\n\\end{equation}\nwhere the quadratic and higher-order terms in $h$, $p$ have now been collected in $\\breve t_n^j$.  We also define the linear differential operator $\\cal Q$ by \n\\begin{equation}\n{\\cal Q}^{ijk} h_{jk} =  \\kappa \\left(\\delta_{\\kappa h} D_j\\right) P^{ij} - \\frac{\\kappa^2}{2} D_iP^{ij} q^{kl} h_{kl} \\ .\n\\end{equation}\n\nWorking about a classical background with $C_n=C^i=0$, the $\\kappa\\rightarrow0$ limit of the constraints gives the equations\n\\begin{equation}\\label{hamzero}\nL^{ij} h_{ij}  - {\\cal P}^{ij}h_{ij}+ \\frac{K_{ij}}{\\sqrt q} p^{ij} =0\n\\end{equation}\nand\n\\begin{equation}\\label{momzero}\nD_i p^{ij} + {\\cal Q}^{jkl} h_{kl}=0\n\\end{equation}\nconstraining linearized perturbations about the solution, {\\it i.e.} linearized gravitational waves.  These will be governed by a quadratic hamiltonian, which may be found from the rightmost expression in \\eqref{hreln}, and which is expected to give evolution similar to that for other quantum fields, {\\it e.g.} as treated in \\cite{GiPe}, such as Hawking production in a black hole background, {\\it etc.}  \n\nAt nonzero $\\kappa$, the constraints become\n\\begin{equation}\\label{hamone}\nL^{ij} h_{ij}  - {\\cal P}^{ij}h_{ij}+ \\frac{K_{ij}}{\\sqrt q} p^{ij} =\\frac{\\kappa}{2}\\left(T_{nn} + \\breve t_{nn}\\right)\n\\end{equation}\nand\n\\begin{equation}\\label{momone}\nD_i p^{ij} + {\\cal Q}^{jkl} h_{kl}=\\frac{\\kappa}{2}\\sqrt{\\tilde q}\\left(T_n^j + \\breve t_n^j\\right)\\ .\n\\end{equation}\nIn the classical theory the corresponding equations determine the perturbative ``Coulomb fields\" induced by matter, and at higher orders also incorporate the nonlinearities resulting from gravitational energy.  At leading order in $\\kappa$, the solutions to \\eqref{hamone}, \\eqref{momone} are of course highly nonunique, since a solution of the homogeneous equations \\eqref{hamzero}, \\eqref{momzero} may be added to any given solution.\n\n\n\\section{Leading perturbative dressing}\n\nIn the quantum theory, finding gauge-invariant operators $O$ that commute with the constraints, \\eqref{conscomm}, can be approached by perturbatively solving for operators that commute with \\eqref{hamone}, \\eqref{momone}.  Solutions can be found, beginning with an operator of the quantum field theory to which we couple gravity.  This is done by gravitationally dressing that operator, as has been described to leading order in perturbation theory about flat space in \\cite{SGAlg}\\cite{DoGi1}\\cite{DoGi2,DoGi3}\\cite{QGQF}\\cite{DoGi4,SGsplit} and about anti de Sitter space in \\cite{GiKi}.  Here we will extend those constructions to a more general background.\n\nThis gravitational dressing is most easily studied in the situation where the background satisfies $P^{ij}=0$, corresponding to vanishing extrinsic curvature of the time slices of the background metric.  This includes the case of flat and AdS backgrounds with standard time slicings.  However, we would also like to consider evolution that for example perturbs about black hole solutions, either with flat or AdS asymptotics.  For example in the case of the Schwarzschild solution, one may consider a general stationary slicing described by a spherically symmetric slicing\\cite{NVU}\\cite{SEHS}\\cite{GiPe}, \n\\begin{equation}\\label{sliceeq}\nx^+=t+S(r)\\ ,\n\\end{equation}\nspecified by a slicing function $S(r)$,\n where $x^+$ is the ingoing Eddington-Finkelstein coordinate.  The nontrivial components of the extrinsic curvature of the slices are then given by\n\\begin{equation}\nD_rN_r = \\partial_r N_r -\\gamma^r_{rr} N_r\\quad, \\quad D_\\theta N_\\theta = -\\gamma^r_{\\theta\\theta} N_r\\ ,\n\\end{equation}\nand so vanish if and only if $N_r=0$.  The expression\\cite{NVU} $N_r=1-fS'$, with $-f$ the coefficient of $dx^{+2}$ (see below), then implies this is true only for $S'=1\/f$, which is the case of Schwarzschild time slices.  These lead to a singular basis for perturbations at the horizon, and as explained in \\cite{GiPe} this can be avoided with a more general choice of slices.  But, this therefore requires considering $P^{ij}\\neq0$; similar statements hold for the case of black holes in AdS.\n\nThe construction of \\cite{SGAlg}\\cite{DoGi1}\\cite{DoGi2,DoGi3}\\cite{QGQF}\\cite{DoGi4,SGsplit} writes the linear order dressing of an underlying QFT operator $O_0$ as\n\\begin{equation}\\label{dressop}\nO = e^{i\\int d^{D-1}x \\sqrt{q} V^\\mu(x)(T_{n\\mu} + \\breve t_{n\\mu})} O_0 e^{-i\\int d^{D-1}x \\sqrt{q} V^\\mu(x)(T_{n\\mu} + \\breve t_{n\\mu})}\\ ;\n\\end{equation}\nas long as we work to linear order in $\\kappa$ the exponential is not strictly necessary, but is convenient and suggestive.  Here the dressing functions $V^\\mu(x)$ are functionals of the metric perturbation which are fixed by the condition that the operator $O$ commute with the constraints.\n\n\\subsection{Vanishing background extrinsic curvature: $P^{ij}=0$}\n\nSpecifically, generalizing the flat space construction\\cite{DoGi1,SGsplit}, we anticipate that $V^n(x)$ takes the form\n\\begin{equation}\nV^n(x) = -\\frac{\\kappa}{2} L^{-1}_{ij} p^{ij} = \\frac{\\kappa}{2} \\int d^{D-1} x'  \\check h_{ij}(x',x) p^{ij}(x')\\ ,\n\\end{equation}\nwhere the inverse $L^{-1}$ is given by a solution to\n\\begin{equation}\nL_{x'}^{ij}  \\check h_{ij}(x',x) = -\\frac{\\delta^{D-1}(x'-x)}{\\sqrt q}\\ .\n\\end{equation}\nOnce again, such solutions are highly non-unique, and for example in a flat background include explicit expressions either corresponding to line-like or Coulomb-like gravitational fields\\cite{DoGi1}, or more general instantaneous configurations of the field; these differ by homogeneous solutions, corresponding to source-free gravitational waves.\\footnote{This also means, as also described in \\cite{DoGi3,DoGi4,SGsplit} that soft charges are largely decorrelated with the quantum state of matter in a region.  For the most part they depend on the arbitrary choice of gravitational dressing (which may be specified {\\it e.g.} through imposition of boundary conditions), with the only necessary correlation through the total Poincar\\'e charges of the matter state.}\nAs a result, the commutator with the constraint $\\tilde C_n$ is easily seen to give\n\\begin{equation}\\label{Vncom}\n[\\tilde C_n(x),V^n(x')] = i \\frac{\\delta^{D-1}(x'-x)}{\\sqrt q} + {\\cal O}(\\kappa)\\ .\n\\end{equation}\nAs a consequence, using the expression \\eqref{dressop} and assuming the $V_i$ term doesn't contribute,\n\\begin{equation} \n[\\tilde C_n(x),O] = {\\cal O}(\\kappa)\\ ,\n\\end{equation}\nwith the commutator of the term proportional to $(T+t)$ in $\\tilde C^n$ being cancelled by the term arising from \\eqref{Vncom}.  \n\nTo solve the momentum constraint \\eqref{pmomcons}, we consider dressing functions of the general form\n\\begin{equation}\nV_i(x)=\\kappa\\int d^{D-1} x' G_i^{jk}(x',x) h_{jk}(x')\\ ,\n\\end{equation}\nand seek a solution of the equation\n\\begin{equation}\\label{Vicom}\n[\\sqrt{\\tilde q} \\tilde C^j(x), V_i(x')] = i \\delta^j_i \\delta^{D-1}(x'-x) +{\\cal O}(\\kappa)\\ .\n\\end{equation}\nFrom the canonical commutators \\eqref{pertcom}, we find this holds if\n\\begin{equation}\n2 D_k G_i^{jk}(x,x') = \\delta _i^j\\delta^{D-1}(x-x')\\ .\n\\end{equation}\nIn a flat background, solutions are given by\\cite{SGsplit}\n\\begin{equation}\nV_i(x)=\\int d^{D-1} x' \\check h^{jk}(x',x) \\gamma_{i,jk}\\ ,\n\\end{equation}\nand should exist more generally by the correspondence with solutions of the linearized equations for classical perturbations.  \nOne can also easily check that \n\\begin{equation}\\label{crosscom}\n[\\tilde C_n(x),V_i(x')]={\\cal O}(\\kappa)\\quad ,\\quad [\\sqrt{\\tilde q} \\tilde C^j(x), V^n(x')] ={\\cal O}(\\kappa)\\ .\n\\end{equation}\nThen, given the commutators \\eqref{Vncom}, \\eqref{Vicom}, \\eqref{crosscom}, we find that the constraints have been solved to leading nontrivial order in $\\kappa$ by the dressed operators \\eqref{dressop},\n\\begin{equation}\n[\\tilde C_\\mu(x),O]= {\\cal O}(\\kappa)\\ .\n\\end{equation}\n\n\\subsection{$P^{ij}\\neq 0$}\n\nOnce one sees this structure, it is apparent how one can generalize to the case of background $P^{ij}\\neq 0$.  We now define\n\\begin{equation}\\label{genvn}\nV^n(x)=\\frac{\\kappa}{2} \\int d^{D-1}x' \\left[ \\check h_{ij}(x',x) p^{ij}(x') - \\check p^{ij}(x',x) h_{ij}(x')\\right]\n\\end{equation}\nand\n\\begin{equation}\\label{genvi}\nV_i(x)= \\kappa \\int d^{D-1}x' \\left[ G_i^{jk}(x',x)h_{jk}(x') + H_{ijk}(x',x) p^{jk}(x')\\right]\\ \n\\end{equation}\nwhere $\\check h_{ij}$, $\\check p^{ij}$, $G_i^{jk}$, and $H_{ijk}$ are c-number functions.\nThen, the hamiltonian constraint gives\n\\begin{equation}\n[\\tilde C_n(x), V^n(x')] = -i\\left(L^{ij}-{\\cal P}^{ij}\\right)\\check h_{ij}(x,x') - i \\frac{K_{ij}}{\\sqrt q} \\check p^{ij}(x,x') + {\\cal O}(\\kappa)\\ ;\n\\end{equation}\nrequiring this commutator to be given by \\eqref{Vncom} then gives\n\\begin{equation}\n \\left(L^{ij}-{\\cal P}^{ij}\\right)\\check h_{ij}(x,x') +\\frac{K_{ij}}{\\sqrt q} \\check p^{ij}(x,x') = -\\frac{\\delta(x-x')}{\\sqrt q} \\ .\n \\end{equation}\n Leading order vanishing of the commutator of $V^n$ with the momentum constraint likewise gives\n \\begin{equation}\n D_i\\check p^{ij}(x,x') + {\\cal Q}^{jkl} \\check h_{kl}(x,x') =0\\ .\n \\end{equation}\n This corresponds to a generalized Green function problem, and we expect solutions for $\\check h_{ij}$ and $\\check p^{ij}$ to exist by the relation to the classical problem of finding linearized solutions on the background.\n \n Requiring the correct leading order commutators of the constraints with $V_i(x)$, \\eqref{Vicom} and \\eqref{crosscom}, likewise gives the equations\n\\begin{equation}\n2 D_k G_i^{jk}(x,x') - 2{\\cal Q}^{jkl}H_{ikl}(x,x') = \\delta _i^j\\delta^{D-1}(x-x')\n\\end{equation}\nand\n\\begin{equation}\n \\left(L^{ij}-{\\cal P}^{ij}\\right)H_{ijk}(x,x') -\\frac{K_{ij}}{\\sqrt q} G_i^{jk}(x,x')=0\\ ,\n \\end{equation}\n which we likewise expect to have solutions.  Once $\\check h_{ij}$, $\\check p^{ij}$, $G_i^{jk}$, and $H_{ijk}$ have been determined by solving these equations, together with specification of the homogenous part of the solution {\\it e.g.} through boundary conditions, then \n\\eqref{genvn} and \\eqref{genvi}, together with \\eqref{dressop}, give the dressed operator $O$ to leading nontrivial order in $\\kappa$.\n\nSignificant features of the role of gauge invariance can be seen from the leading order in $\\kappa$ construction of the dressed operators  \\eqref{dressop} given here.  For example, the dressing modifies the commutators from those of the underlying LQFT operators $O_0$, such that operators associated with spacelike-separated regions generically no longer commute; examples can be given extending the discussion of \\cite{DoGi1}.  Of course, to further understand the role and structure of the constraints and dressing, \none would like to go beyond to higher orders.  One does expect further difficulties here, in particular associated to infinities and the need to regulate operators.  We will leave further discussion of higher orders for future work, but will discuss some general features that already become apparent with these leading-order results.\n\n\\section{Description of evolution}\n\n\\subsection{General structure}\n\nIt is important to understand the general structure of evolution in quantum gravity, given the constraints of gauge invariance.  The leading-order construction of gauge invariant operators, and the more general structure of the hamiltonian and constraints, already appear to provide significant guidance to this structure.\n\nIn particular, we have given a leading-order construction of gauge invariant observables $O$, commuting with the constraints \\eqref{conscomm}.  These then lead to states that evolve via the hamiltonian \\eqref{hreln} of quantized general relativity, {\\it e.g.} of the form\n\\begin{equation}\\label{dressstate}\n|\\psi\\rangle = O|0\\rangle\\ .  \n\\end{equation}\nOne can alternately construct dressed states directly from undressed states,\n\\begin{equation}\n|\\psi\\rangle= e^{i\\int d^{D-1}x \\sqrt{q} V^\\mu(x)(T_{n\\mu} + \\breve t_{n\\mu})}|\\psi_0\\rangle\\ .\n\\end{equation}\nAs a simple example, one could begin with the basic scalar field operator, $O_0=\\phi(x)$, and then construct the corresponding dressed operator $O$, given to leading order by \\eqref{dressop}, or equivalently\\cite{DoGi1} by $\\Phi(x)=\\phi(x^\\mu + V^\\mu(x))$.  The resulting operator can be thought of as creating from the vacuum a quantum of the field $\\phi$, together with its corresponding gravitational field.  As we have emphasized, the gravitational part of the operator is non-unique, corresponding to the fact that there are different possible gravitational field configurations dressing the particle, differing at leading order by free gravitational excitations.\n\nEvolution can be thought of in two ways, related through the two expressions for the hamiltonian \\eqref{hreln}.  Since the rightmost expression is of the standard form of a LQFT hamiltonian, it determines evolution of the state by telling us how the $\\phi$ and gravitational parts of the state evolve like standard quantum fields.  This evolution is of course gauge dependent, through its dependence on $N$ and $N^i$.  We expect it to correspond to the quantum evolution of the matter state created by $O$, together with the quantum gravitational field created by the gravitational piece of the operator.\n\nOr, one can describe evolution in terms of the middle expression in \\eqref{hreln} which is written in terms of the constraints.  The boundary hamiltonian $H_\\partial$ contributes to the time dependence of the state, because the gravitational dressing generically extends into the asymptotic region\\cite{DoGi2,GiKi}.  One might have anticipated that the constraints $C^\\mu(x)$ annihilate the state, so that this is the only time dependence.  This would incorporate the statement that the state ``satisfies the Wheeler-DeWitt equation,\" since the constraint $C^n(x)$ corresponds to the Wheeler-DeWitt operator.  However, we have found that this would be inconsistent with the basic commutators and in particular with evolution such as described by the Heisenberg equations \\eqref{qeqn}, \\eqref{Peqn}.  Instead the state is annihilated by ``half\" of the constraints  (and of the Wheeler-DeWitt operator), \\eqref{physstate}.  This implies that the constraint terms in \\eqref{hreln} also contribute to gauge-dependent evolution of the state, though as we have argued above not to evolution of matrix elements of gauge-invariant operators.  Alternately, transition amplitudes of the form \n\\begin{equation}\n\\langle\\psi'| e^{-iHt}|\\psi\\rangle \n\\end{equation}\nwill also exhibit time dependence.  \n\n\\subsection{Bubble evolution, cosmology, and field-relational observables}\n\nThis raises the question of the description of evolution on slices that coincide at infinity, but not in a region in the interior of spacetime, so that the asymptotic lapse and shift vanish, implying $H_\\partial=0$.  Then, the full hamiltonian commutes with the gauge-invariant operators $O$.  This suggests that their evolution is trivial in such ``bubble\" evolution\\cite{Tomo1946,Kuch-bubble}.  This is also the case for closed cosmologies, with no boundary term.  In both of these cases the hamiltonian is typically explicitly time-dependent, with additional subtleties\\cite{ToVa,CMOV,CFMM,AgAs,MuOe}.  Once again, we can anticipate that the physical states are annihilated by ``half\" of the constraints.  An alternate way to then describe evolution is in terms of a different kind of relational observable that is not gravitationally dressed; an example of such a field-relational observable is\n\\begin{equation}\n\\int d^Dx \\sqrt{|g|} O_0(x) f(Z^I(x))\n\\end{equation}\nwhere $Z^I$ are $D$ dynamical ``locator\" fields and $f(Z^I)$ is chosen so that in a particular state for these fields its support is localized near a particular point.  An example is using the value of the inflaton field in inflation to localize in time to the reheating time; \nfor further discussion (including of limitations to localization) see  \\cite{GMH}.  We leave further exploration of such evolution for future work.  \n\n\\subsection{Other specific examples}\n\\label{exevol}\n\nBeyond a flat space background, it is of interest to better understand gravitational evolution in other backgrounds such as those of black holes, AdS, or black holes in AdS.  The static cases can be subsumed in the line element\n\\begin{equation}\nds^2=-f(r) dx^{+2} + 2 dx^+ dr + r^2 d\\Omega^2\n\\end{equation}\nwhere\n\\begin{equation}\nf(r)=1+\\frac{r^2}{R_\\Lambda^2} - \\left(\\frac{R_0}{r}\\right)^{D-3}\\ ,\n\\end{equation}\n$R_\\Lambda$ is the AdS radius, and \n\\begin{equation}\nR_0^{D-3} = \\frac{16\\pi G_D M}{(D-2) A_{D-2}}\n\\end{equation}\nwith $A_{D-2}$ the area of the unit sphere.  Then, introducing a stationary slicing \\eqref{sliceeq} given by a slice function $S(r)$ yields the ADM background solution\\cite{NVU}\n\\begin{equation}\nN^2=\\frac{1}{S'(2-fS')}\\quad,\\quad N_r= 1-fS' \\quad,\\quad q_{rr}= S'(2-fS')\\quad,\\quad\n\\end{equation} \nand with angular components the standard round metric of radius $r$.  \n\nEvolution may then be described perturbatively about this solution, using the preceding general construction.  Specifically, we may consider a dressed state \\eqref{dressstate} on a slice taken to be an initial slice.  The evolution of this state can be described via either of the forms of the hamiltonian \\eqref{hreln}.  The latter form in particular gives a standard description of field evolution, and so evolves the matter perturbation together with the perturbative gravitational field corresponding to its dressing in standard field theory fashion.\\footnote{While Lorentzian evolution for black holes has been considered previously, see {\\it e.g.} \\cite{CaTe}, the treatment outlined here extends to more general slicings than Schwarzschild, and including the black hole interior.}\n\nIn this way, one for example finds a perturbative expression for the bulk hamiltonian for an AdS black hole to leading order in $\\kappa$, or in the language of the AdS\/CFT correspondence, in $1\/N$, also making connection with the discussion in \\cite{GiPe} of this approach to defining such a hamiltonian.  \nAlternately, by virtue of the constraints, the hamiltonian is related to a boundary hamiltonian as in \\eqref{hreln}.\nWe  defer more detailed investigation of this evolution to future work.  \n\n\\section{Conclusion and directions}\n\nIn conclusion, we have shown that, starting from an ADM parameterization of the geometry and the corresponding construction of the hamiltonian, leading order perturbative gravitational states may be constructed and their evolution described.  The states and evolution have a gauge symmetry, generated by the constraints, and perturbative solution of the constraints to construct gauge-invariant operators and states can be accomplished by gravitational dressing operators of an underlying field theory.  Such a construction has been found to leading perturbative order about a general background, in terms of certain generalized Green functions of the given background, generalizing earlier constructions in a flat background\\cite{SGAlg}\\cite{DoGi1}\\cite{DoGi2,DoGi3}\\cite{QGQF}\\cite{DoGi4,SGsplit}.  The resulting gravitational part of the state is not uniquely determined, since it can be changed by addition of a piece corresponding to an arbitrary source-free propagating gravitational wave. The state gotten by acting with such a dressed operator on a vacuum state is then evolved by the hamiltonian, which may be described as a standard local QFT hamiltonian including the spin-two perturbative gravitational field, or alternately may be written in terms of a boundary hamiltonian, up to terms proportional to the constraints.\n\nThere are multiple directions for further work.  Within the framework of local QFT, as noted above, we would like to better understand bubble evolution, on slices that match at infinity, and the related description of cosmological evolution, and the connection to other field-relational observables that are more useful in that context. There are also related issues that occur when the background slicing has an explicit time dependence\\cite{ToVa,CMOV,CFMM,AgAs,MuOe}, which deserve to be more closely investigated.   It also seems useful to have more complete description of the evolving perturbative state of black holes, whether or not in AdS, {\\it e.g.} generalizing \\cite{GiPe} to include gravitational dynamics.  And, the problem of solving the constraints connects directly to a leading argument for the origin of holography \\cite{Maroholo,Maroholonost,Jacoholo,SGholo}, which is important to better understand.  This also connects to the question of in what sense information can be localized in a gravitational theory, either because of the argued existence\\cite{Maroholo,Maroholonost} of a holographic map, or a similar argument\\cite{DoGi3} that states internal to AdS may be observed by boundary observables, {\\it if} the constraints are solved.\\footnote{Such information has even been argued to be accessible perturbatively\\cite{Rajuetal}, although only if one can measure exponentially small quantities\\cite{SGsub}.} This connects directly to the question of the extent to which subsystems may be defined\\cite{QFG,QGQF,SGsub,SGsnow} in gravitational theories, whether exactly or approximately.  We note that a perturbative description of the evolution like we have outlined is consistent with the perturbative solution of the constraints, and appears to describe a black hole with a growing number of  internal states entangled with the exterior, and corresponding missing information if the black hole disappears at the end of evolution.  \nThus, while the perturbative solution of the constraints via the dressing does appear to provide some additional sensitivity to the black hole state, it does not appear to resolve the unitarity problem, in contrast to recent claims\\cite{Rajuetal}, and in particular does not obviously provide a mechanism for transfer of information from the black hole.\n\nIn clarifying these issues, understanding better the structure of higher-order solution to the constraints seems important (and it seems important to clarify  the challenges to finding higher-order solutions).  We would also like to better understand the structure of the algebras associated to dressed observables.  It can be observed that the leading-order perturbative dressed observables, \\eqref{dressop} and related expressions in \\cite{DoGi4}, have similar structure to the observables in the crossed-product construction, argued\\cite{Wittcross}\\cite{CLPW} to convert type III von Neuman algebras into type II.  Of course, the noncommutativity of perturbative observables associated with different regions\\cite{SGAlg,DoGi1}, due to the dressing, appears to be a likewise important modification of the underlying field theory structure; further investigation of these questions is in progress.\n\nThe modification of local algebras in gravity illustrate the general statement that locality is remarkably subtle in theories with structures like gauge symmetries.  In QED or standard nonabelian gauge theories, observables found by dressing underlying matter operators are generically nonlocal, as with the gravitationally dressed operators discussed above.  Put differently, the problem of solving the constraints is generically a nonlocal one, with a nonlocal solution.  However, in gauge theories based on an internal group, locality is still realized since there also exist gauge-invariant operators that are local, such as Wilson loops or other neutral observables confined to a neighborhood.  In gravity, the gauge symmetry is that of transformations including Poincar\\'e symmetries; any local observable thus carries nontrivial charge, and so is nonlocal when its gravitational dressing is included\\cite{DoGi2}.  \nIn short, gauge theories with an internal symmetry appear to be barely local, but it is less clear what locality properties quantum gravity has.  \n\nThere are also important directions that appear to go beyond the framework of local QFT.  For one thing, if we consider evolution of a black hole, like that described in \\ref{exevol}, that appears to lead to the breakdown of unitarity, also noted above, associated with the ``black hole information paradox\" or ``unitarity crisis.\"  This can be encapsulated in a ``black hole theorem\\cite{SGthm}\": unitary evolution, and the statements that black holes behave like subsystems, that field configurations outside them evolve independently of the black hole internal state, and that they disappear at the end of their evolution, are inconsistent.  \nWe expect modification of both the structure of the Hilbert space and the hamiltonian as compared to those given by quantization of GR like that we have described.  In a more complete description,  we expect that the fundamental quantum variables are likely {\\it not} those of fields moving on a background metric, with states labelled such as  $|q_{ij}(\\cdot),\\phi(\\cdot)\\rangle$, but that these variables only give an approximate description of the states.  An important question is what is the more accurate and complete description of the variables parameterizing the wavefunction.  This, then, closely relates to the question of what are the fundamental observables, and the ultimate form of the hamiltonian and its interactions, as well as the symmetries of the theory. \n\nIf in a more complete description of the Hilbert space black holes still effectively behave as subsystems, the ``black hole theorem\" tells us that unitarity apparently requires interactions that go beyond the quantized GR\/local QFT description, and specifically such that the evolution of the black hole exterior depends on the black hole internal state.  An approach to parameterizing such interactions has been developed in \\cite{SGmodels,BHQIUE,NVNL}\\cite{NVU,BHQU}.  The present work serves as an even firmer foundation on which to describe their parameterization, if they can be regarded as   corrections to the evolution governed by local QFT plus quantized GR.  \nIn short, one can describe corrections $\\Delta H$ to the hamiltonian of \\eqref{hreln}, constrain their properties, and investigate their possible observational effects for example in electromagnetic or gravitational wave observations of the near-horizon regime\\cite{BHQU}.  The contributions to $\\Delta H$ are plausibly small both far from the black hole and even in the near horizon region, but of course are expected not to be small in the deep black hole interior.  \nMore systematic analysis is planned for future work.  \n\n It is believed in a large segment of the quantum gravity community that a fundamental description like this may arise from a dual large-$N$ gauge theory; if this is true, it is important to understand and characterize the departures from the bulk LQFT description that this implies, for example as corrections to the hamiltonian \\eqref{hreln}.  But, it seems quite likely that the more fundamental description arises in connection with some other mathematical structure on Hilbert space\\cite{QFG,QGQF}, which it is our goal to infer and further describe.\n\n\n\n\\vskip.3in\n\\noindent{\\bf Acknowledgements} \n\nThis material is based upon work supported in part by the U.S. Department of Energy, Office of Science, under Award Number {DE-SC}0011702, and by Heising-Simons Foundation grant \\#2021-2819.   Work to complete this paper was carried out at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe computational capacity to store and manage an ever-changing social network is thought to depend roughly on neocortical size, which evolved driven by the need of managing increasingly large social groups \\cite{dunbar1992}. This is the statement of the far reaching \\emph{social brain hypothesis} \\cite{dunbar1998}, which links brain volume in humans, primates and other mammals with the size of their social groups. \nFor humans, the Dunbar's number constitutes an upper limit of $\\about150$ for the social group size. That is, the total number of active relationships that we can maintain at any given time; a cognitive constraint that seems to operate also in virtual environments, such as Twitter \\cite{GoncalvesVespignaniPONE2011}.\nCertainly, to monitor and handle social ties comes at a cost,  it takes time to cultivate these relationships \\cite{MiritelloMoroSR2013}, and the process is limited by cognitive constraints, such as memory and mentalising skills \\cite{StillerDunbarSN2007}.\nCurrent sociological studies rely on large digital datasets in order to build weighted networks of human relationships, where link weights encoding actor-to-actor interaction frequency are used as a proxy for emotional closeness \\cite{RobertsDunbarEHB2011, ArnaboldiCC2013}. \nIn this framework, an individual's social group is equivalent to its set of neighbors, which is often called its ego-network.\n\nEgo-networks are internally highly structured and their links can be sorted by their weights \\cite{SaramakiDunbarPNAS2014}. Moreover, links can be clustered into groups of increasing number of links and decreasing emotional closeness \\cite{ZhouDunbar2005}. \nThese layers form a nested hierarchy, where the cumulative sizes of consecutive groups follow a preferred scaling ratio of approximately $1\/3$, forming a sequence of typical group sizes of $\\about5$, $\\about15$, $\\about50$ and $\\about150$, which are sometimes called Dunbar's circles. \nA smaller inner layer of size $\\about1.5$ \\cite{DunbarPasarellaSN2015}, and two larger groups of sizes $\\about500$ and $\\about1500$ \\cite{DunbarBBS1993}, have also been reported. In each case, the scaling relationship between consecutive groups holds.\nThis hierarchical structure appears to be a fundamental organizational principle of human groups, and has been confirmed in online games \\cite{FuchsThurner2014}, online social networks \\cite{ArnaboldiFB2012, ArnaboldiTW2013} and telephone call detail records \\cite{MacCarronDunbarSN2016}.\n\nWhile several null models suitable for weighted social networks have been proposed recently \\cite{SagarraGuileraPRE2013, SquartiniGarlaschelliSR2013,SagarraGuileraPRE2015}, no such models have yet been proposed to analyze the observed ego-network hierarchical structure.\nIn this paper, we propose a grand-canonical ego-network statistical ensemble that reproduces qualitatively the hierarchical structure, and is able to fit experimental data successfully.\nWe simplify the problem by assigning the ties in each layer constant costs and postulate an abstract social capital \\cite{antonioni2014reds}, or resource, that is spent in placing the links into the different layers. \nThen, by fixing the actors average total degree and resource, the hierarchical structure emerges spontaneously.\nThis ensemble is an unbiased null model for ego-networks which offers a parsimonious explanation for the nested structure. It can also be used to generate synthetic data with the desired layer scaling, as well as for hypothesis testing against more complex models.\nThe ensemble probability distribution, as well as its thermodynamic limit, is presented in section \\ref{s:grand_canonical_ensemble}.\nIn section \\ref{s:hierarchical} we examine the hierarchical structure and prove that, in our setting, equispaced costs are a condition for a constant group size scaling in the outer layers. \nWe fit and compare the model to an empirical dataset in section \\ref{ss:RS_dunbar_structure}. Finally, section \\ref{s:discussion} is dedicated to the discussion.\n\n\n\n\n\n\n\n\n\n\n\\section{Grand-canonical ensemble for Dunbar's circles}\\label{s:grand_canonical_ensemble}\nLet us assume the existence of a social resource, $s$, that an individual can employ to establish ties or relationships of different emotional intensity with their $k$ acquaintances. \nWe consider $r=1,\\dots,K$ different relationship layers with respective costs, $s_r\\in{\\mathbb R}$, with $0 \\leq s_r \\leq s$ and sorted in order of  decreasing emotional intensity, such that $s_r > s_{r+1}$.  \nWe define the \\emph{layer-degree}, $k_r$, as the number of ties of cost $s_r$. \nIn sum, a given individual identified by index $j$ will have a total degree and social resource verifying\n\\begin{subequations}\\label{eq:micro-canonical_constraints}\n\t\\begin{alignat}{2}\n\t k(j) &= \\sum_r k_r(j), \\\\ \n\t s(j) &= \\sum_r s_r k_r(j) .\n\t\\end{alignat}\n\\end{subequations}\nThe Dunbar's circles are inclusive groupings of decreasing emotional closeness \\cite{ZhouDunbar2005}, hence the group at level $r$ includes all layers of cost higher or equal to $s_{r}$. In our setting, the variables corresponding to the size of Dunbar's circles are the cumulative \\emph{group-sizes}, defined as $n_{r}=\\sum_{l=1}^{r}k_l$. \n\n\nThus, an individual's ego-network in our setting is completely described by the configuration variables $k_r$. \nThis system can take any state, $\\{k_r\\}_{r=1,\\dots,K}$ that verifies the constraints Eqs. \\eqref{eq:micro-canonical_constraints}. \nThe problem is equivalent to that of distributing $k$ particles among $K$ energy levels $\\{s_r\\}$ in a quantum micro-canonical ensemble \\cite{BREY, KARDAR}. \nWithout making any assumption \\emph{a priori} about the probabilities of any given configuration, we could study the average layer structure in a micro-canonical ensemble. The micro-canonical ensemble assigns an homogeneous probability distribution on the configuration space, hence the problem would solved if we could compute the total number of allowed configurations, i.e. the partition function.\nHowever, as in classical statistical mechanics, it is simpler to formulate instead a generalized or grand-canonical ensemble, consisting on a large number of identical copies of the system, for which the constraints Eqs. \\eqref{eq:micro-canonical_constraints} are verified only on average. \nConsider a group of $N$ individuals or egos, each of which having different degree, $k(j)$ and resource $s(j)$; with given averages  \n\\begin{subequations}\\label{eq:grand_canonical_constraints}\n\t\\begin{alignat}{2}\n\t \\<k\\> &= \\sum_{j=1}^N k(j) P(\\{k_r(j)\\}), \\\\\n\t \\<s\\> &= \\sum_{j=1}^N s(j) P(\\{k_r(j)\\}).\n\t\\end{alignat}\n\\end{subequations}\n\n\nThe least biased distribution $P(\\{k_r\\})$ verifying the constraints Eqs. \\eqref{eq:grand_canonical_constraints} can be calculated following a maximum entropy principle \\cite{JAYNES}.  The distribution entropy is $S=-\\sum_{\\{k_r\\}}P(\\{k_r\\})\\ln P(\\{k_r\\})$, where the sum runs over all the allowed configurations.  Maximizing $S$ subject to the constraints Eqs. \\eqref{eq:grand_canonical_constraints} plus normalization, we obtain a Gibbs distribution\n\\begin{equation}\n\tP(\\{k_r\\}) = \\frac{1}{Z} D(\\{k_r\\}){\\mathrm e}^{H(\\{k_r\\})},\n\\end{equation}\nwhere $D(\\{k_r\\}) = \\binom{N}{k} k!\/\\prod_r k_r!$ is the degeneracy of the configuration $\\{k_r\\}$, which counts all possible ways of selecting $k=\\sum_r k_r$ links out of $N$ actors, and all ways of assigning $k$ distinguishable links into layers of degree $k_r$. Finally, $Z$ and $H$ are the partition function and cost function, respectively.\n\\begin{align}\n\n\t\\centering\n\tZ &= \\sum_{\\{k_r\\}} D(\\{k_r\\}) {\\mathrm e}^{H(\\{k_r\\})}, \\\\\n\tH(\\{k_r\\}) &= \\lambda k + \\mu s  + \\textstyle\\sum_r h_r k_r\\displaystyle,\n\\end{align}\nwhere  $\\lambda$ and $\\mu$ are the Lagrange multipliers and will act as fitting parameters in order to enforce the constraints Eqs. \\eqref{eq:grand_canonical_constraints}. We have also included the auxiliary fields $h_r$ for convenience.\nThe maximum entropy method has been applied to formulate a large number of complex network models with prescribed features, known generally as exponential random graphs \\cite{ParkNewmanPRE2004,GarlaschelliLoffredoPRE2008,BianconiPRE2009,SagarraGuileraPRE2013,SagarraGuileraPRE2015}. Our approach here seeks instead a probability distribution for ego-network configurations $\\{k_r\\}$. All the information of our system, including the cumulants of the layer degrees and group sizes, as well as their marginal distributions, are recovered from the partition function, which can be calculated analytically.\n\\begin{subequations}\\label{eq:_partition_function}\n\t\\begin{alignat}{3}\n\t &Z = \\sum_{\\{k_r\\}} \\binom{N}{k}\\frac{k!}{\\prod_r k_r!} {\\mathrm e}^{\\sum_r(\\lambda + \\mu s_r + h_r)k_r} \\\\\n\t &= \\sum_{k=0}^N \\binom{N}{k}  \\sum_{\\{k_r|k\\}} k! \\prod_r\\frac{{\\mathrm e}^{(\\lambda + \\mu s_r + h_r)k_r}}{k_r!} \\\\\n\t &= \\sum_{k=0}^N \\binom{N}{k}  \\( \\sum_r {\\mathrm e}^{\\lambda + \\mu s_r + h_r} \\)^k \n\t  = \\( 1 + \\sum_r {\\mathrm e}^{\\lambda + \\mu s_r + h_r} \\)^N. &\n\t\\end{alignat}\n\\end{subequations}\nwhere the symbol $\\sum_{\\{k_r|k\\}}$ on the second line denotes sums over configurations $\\{k_r\\}$ with total degree $k$, and we have used the multinomial and binomial sums on the second and third lines, respectively.\nThe cumulants of a single layer degree $k_r$ can be computed by taking derivatives with respect to the respective auxiliary field $h_r$. For instance, the average layer degree, variance, as well as the correlations between different layers are given by\n\\begin{eqnarray}\n\t\\<k_r\\> =& \\partial_{h_r} \\ln Z \\mid_{h_r=0} &= \\frac{N xy^{s_r}}{1+\\sum_s xy^{s_s}}, \\label{eq:canonical_mean_layer_degree}\\\\ \t\n\t\\sigma^2_{k_r} =& \\partial^2_{h_r} \\ln Z \\mid_{h_r=0} &= \\frac{1+\\sum_{s\\neq r}xy^{s_s}}{1+\\sum_s xy^{s_s}} \\<k_r\\>, \\label{eq:canonical_layer_degree_variance}\\\\\n\t\\sigma_{k_r, k_s} =& \\partial^2_{h_r, h_s} \\ln Z \\mid_{h_r=0} &= -\\frac{\\<k_r\\>\\<k_s\\>}{N}, \\label{eq:canonical_layer_degree_corr} \n\\end{eqnarray}\nwhere we have defined $x={\\mathrm e}^{\\lambda}$ and $y={\\mathrm e}^{\\mu}$.\nWe will prove later on that the layer degree marginal distributions become uncorrelated Poisson distributions in the thermodynamic limit $N\\to\\infty$. \nLet us first write, however, the saddle point equations, used to fix the $k$ and $s$ ensemble averages.\n\\begin{align}\n \t\\<k\\>  &= \\partial_{\\lambda} \\ln Z \\mid_{h_r=0}  = N\\frac{\\sum_r xy^{s_r}}{1+\\sum_r xy^{s_r}}, \\label{eq:saddle_k_canonical}\\\\\n\t\\<s\\>  &= \\partial_{\\mu}  \\ln Z \\mid_{h_r=0}   = N\\frac{\\sum_r s_r xy^{s_r}}{1+\\sum_r xy^{s_r}}, \\label{eq:saddle_s_canonical}\n\\end{align}\nNotice that the average degree and resource verify $\\<k\\>=\\sum_r \\<k_r\\>$ and $\\<s\\>=\\sum_r s_r \\<k_r\\>$, respectively. In our maximum entropy setting, Eqs. \\eqref{eq:saddle_k_canonical} and \\eqref{eq:saddle_s_canonical} are solved for \n$x$ and $y$ in order to obtain the parameter values that fix the desired $\\<k\\>$ and $\\<s\\>$.\n\nFurther derivatives recover the subsequent $k$ and $s$ cumulants. For instance, the variances are\n\\begin{align}\n \t\\sigma_k^2  &= \\partial_{\\lambda}^2 \\ln Z\\mid_{h_r=0}  = \\frac{\\<k\\>}{1+\\sum_r xy^{s_r}}, \\\\\n\t\\sigma_s^2  &= \\partial_{\\mu}^2  \\ln Z\\mid_{h_r=0}    =  \\sum_r s_r^2 \\<k_r\\> -\\frac{\\<s\\>^2}{N}.\n\\end{align}\nFinally, we can write the configuration probability function, which may be sampled with Monte Carlo methods.\n\\begin{equation}\n\n\tP(\\{k_r\\})= \\binom{N}{k}\\frac{k!}{\\prod_r k_r!}\\cdot\\frac{\\prod_r (xy^{s_r})^{k_r}}{\\(1+\\sum_r xy^{s_r}\\)^N} \\label{eq:canonical_prob}.\n\\end{equation}\n\n\n\n\n\n\n\n\\subsection*{Thermodynamic limit}\\label{s:ensemble_equiv}\n\nLet us study the thermodynamic limit for the grand-canonical ensemble, that is when the number of ego-networks $N\\to\\infty$ while keeping $\\<k\\>$ and $\\<s\\>$ constant. From Eq. \\eqref{eq:saddle_k_canonical} we can write $\\sum_r xy^{s_r} = \\<k\\>\/(N-\\<k\\>)$. This in turn, allows us to rewrite the partition function as\n\\begin{equation}\n\tZ=\\(1+\\frac{\\<k\\>\/N}{1-\\<k\\>\/N}\\)^N\\; \\xrightarrow[N\\to\\infty] \\; {\\mathrm e}^{\\<k\\>}=\\prod_r {\\mathrm e}^{\\<k_r\\>}.\n\\end{equation}\nThe expected layer degrees can be expressed as $\\<k_r\\>=xy^{s_r}(N-\\<k\\>) \\cong Nxy^{s_r} $, for which it is needed that $xy^{s_r}\\to0$ as $N\\to\\infty$. Then, the configuration probability distribution Eq. \\eqref{eq:canonical_prob}, reduces to\n\\begin{equation}\n\tP(\\{k_r\\}) = \\frac{N^{-k} N!}{(N-k)!} \\prod_r \\frac{\\<k_r\\>^{k_r}}{k_r!}{\\mathrm e}^{-\\<k_r\\>} \\; \\xrightarrow[N\\to\\infty] \\; \\prod_r p_r(k_r),\n\\end{equation}\nwhere $p_r(k_r)$ are the layer degree marginal distributions, and the prefactor tends to $1$ as $N\\to\\infty$, for finite $k$. Indeed, using Stirling's approximation and the exponential limit\n\\begin{equation}\n\t\\frac{N^{-k}N!}{(N-k)!} \\cong {\\mathrm e}^{-k}\\(\\frac{N}{N-k}\\)^{N-k} \\xrightarrow[N\\to\\infty]\\, 1.\n\\end{equation}\nThus, the resulting layer degree marginal distributions are Poisson distributions. \n\\begin{equation}\n\tp_r(k_r) =P(k_r|N,x,y)=\\frac{{\\mathrm e}^{N xy^{s_r}}}{k_r!}(N xy^{s_r})^{k_r}.\n\\end{equation}\nNoticing that $\\sum_r xy^{s_r}\\to0$, we can see that Eqs. \\eqref{eq:canonical_mean_layer_degree}, \\eqref{eq:canonical_layer_degree_variance} and \\eqref{eq:canonical_layer_degree_corr} reduce to \n\\begin{eqnarray}\n \t\\<k_r\\> &=& N x y^{s_r}, \\label{eq:micro-canonical_mean_layer_degree} \\\\\n \t\\sigma^2_{k_r} &=& \\<k_r\\>, \\label{eq:micro-canonical_layer_degree_variance}\\\\\n    \\sigma_{k_r, k_l} &=& 0. \\label{eq:micro-canonical_layer_degree_corr}\n\\end{eqnarray}\nMoreover, the saddle point equations, Eqs. \\eqref{eq:saddle_k_canonical} and \\eqref{eq:saddle_s_canonical}, become\n\\begin{align}\n\t k &= \\sum_r     Nx y^{s_r}, \\label{eq:saddle_k}\\\\\n\t s &= \\sum_r      s_r\\,Nx y^{s_r}. \\label{eq:saddle_s}\n\\end{align}\n\n\nThus, as we anticipated, the ego-network grand-canonical ensemble generates an uncorrelated layer structure in the thermodynamic limit. \nHence, sampling corresponds to drawing independent random Poisson variables from the layer degree marginal distributions.\n\n\n\n\n\n\n\n\n\\section{Hierarchical structure}\\label{s:hierarchical}\n\n\nBefore proceeding to study the constant group size scaling condition, let us first discuss the meanining of the ensemble parameters, $x$ and $y$.\nBoth $x$ and $y$ are positive, since they are defined as exponentials of the real Lagrange multipliers $\\lambda$ and $\\mu$. \nWe can identify $y^{s_r}$ as the relative weight of each layer when writing the average link weight\n\\begin{equation}\n\t\\bar s = \\ff{s}{k} = \\frac{\\sum_r s_r y^{s_r}}{\\sum_r y^{s_r}}.\t\n\\end{equation}\nMoreover, the parameter $y$ relates the expected layer-degree scaling with the difference of the link costs:\n\\begin{equation}\\label{eq:layer_degree_scaling}\n\\frac{\\<k_{r+1}\\>}{\\<k_r\\>}=y^{s_{r+1}-s_r}.\t\n\\end{equation}\nOn the other hand, $x$ acts a volumetric parameter which fixes the total degree through the constant product $Nx=k\/\\sum_r y^{s_r}$. \n\n\n\nFrom equation \\eqref{eq:layer_degree_scaling}, the hierarchical structure is made apparent, that is $\\<k_r\\> < \\<k_{r+1}\\>$, as long as $y<1$. \nLet us check that this is indeed the case. Notice that the partition function $Z$ measures the number of allowed configurations and that the layer costs $s_r$ are arbitrary positive numbers. \nConsider that we increase the cost of one of the layers, $s_c$, while keeping the imposed average values, $\\<k\\>$ and $\\<s\\>$ constant.\nFor large enough $s_c$ and fixed $\\<s\\>$, an overwhelming majority of ego-networks will not have any links placed in layer $s_c$, and the number of allowed configurations must decrease. \nIn the limit $s_c\\to\\infty$, a well behaved partition function demands $\\mu<0$. \nConsequently, $y\\in[0,1]$ and the layer degrees $\\<k_r\\>$ are monotonically decreasing with the layer cost $s_r$. \n\n\nIn sum, in a maximum entropy setting corresponding to the least unbiased guess about the ego-network configurations, a hierarchical structure arises naturally from the constraints Eqs. \\eqref{eq:grand_canonical_constraints}. Next, let us consider the condition of a constant group size scaling, as it is observed on empirical ego-networks.\n\n\n\n\n\n\\subsection*{Constant group size scaling condition}\\label{s:group_size_scaling}\n\nIn human social groups, a constant scaling is found on average between the cumulative sizes of consecutive layers. \nIn the grand-canonical ensemble, this is expressed by the expected group-size scaling, $\\<{n_r}\/{n_{r+1}}\\>$. This quantity is the quotient of two functions of the configuration variables and cannot be calculated analytically. \nInstead, we must approximate it in terms of $\\<k_r\\>$, $\\<k_r^2\\>$ and $\\<k_r k_s\\>$, as explained in the appendix \\ref{s:appendix_quotient_cumulants}. The ensemble average is given by\n\\begin{equation}\n\t\\left< \\ff{n_r}{n_{r+1}} \\right> = \\ff{\\<n_r\\>}{\\<n_{r+1}\\>}(1+\\epsilon_{r+1}), \n\\end{equation} \nwhere $\\epsilon_{r+1}$ is a second order correction term, which can be expressed as\n\\begin{equation}\n \t\\epsilon_{r+1}\n \t               =  \\frac{\\<n_r\\>\\<k_{r+1}^2\\>-\\<k_{r+1}\\>\\<n_r^2\\>}{\\<n_r\\>\\<n_{r+1}\\>^2}\n \t               =  \\frac{\\<k_{r+1}\\>^2-\\<k_{r+1}\\>\\<n_r\\>}{\\<n_{r+1}\\>^2}.\n \\end{equation} \nThe rightmost expression was obtained by using the identity for Poisson variables: $\\<k_r^2\\>=\\<k_r\\>+\\<k_r\\>^2$.\n\n\nLet us now consider the scaling of two consecutive group pairings $\\<n_{r+1}\/n_r\\>$ and $\\<n_r\/n_{r-1}\\>$. We will consider the implications of having a constant group-size scaling, such as observed in empirical relationship networks. Imposing $\\<n_{r+1}\/n_r\\>=\\<n_r\/n_{r-1}\\>$ we get\n\\begin{equation}\n\t\\<n_r^2\\>=\\<n_{r+1}\\>\\<n_{r-1}\\>R_r.\n\\end{equation}\nThe correction factor, $R_r=(1+\\epsilon_r)\/(1+\\epsilon_{r+1})$, tends to 1 provided that $\\<n_r\\> \\gg 1$. Indeed, this  is a good approximation for the outer layers, as shown in the inset of Fig. \\ref{fig:RS_scaling}. Considering $R_r\\approx 1$, simple manipulations lead to\n\\begin{equation}\\label{eq:group_size_scaling_approx}\n\n\t\\left<\\frac{n_{r-1}}{n_r}\\right> \\approx \\frac{\\<k_r\\>}{\\<k_{r+1}\\>} = y^{s_r - s_{r+1}}.\n\\end{equation}\nThis result states that a constant group-size scaling in the outer layers is possible only if the cost difference between them is constant.\n\\begin{equation}\\label{eq:equispaced_cost_condition}\n\ts_r-s_{r+1}=\\Delta,\\qquad (\\text{for  } \\,r\\, \\text{  s.t.  } \\,n_r \\gg 1).\t\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Fit to an empirical social network}\\label{ss:RS_dunbar_structure}\nThe grand-canonical ensemble presented above may function as a null model for ego-networks, providing a benchmark against which to test more complicated features. In order to accept the ensemble as a good model for social structure, the model should meet two demands: \\emph{(i)} It should generate ego-network instances with $k$ and $s$ values similar to the empirical ones (similar macrostate); and \\emph{(ii)} Those instances should present a nested layer structure. The assumption of constant layer costs makes the ensemble specially suited to model data from surveys, where the ties weights are chosen from predefined discrete scores or categories.\nWe have fitted the grand-canonical ensemble to the Reciprocity Survey (RS) dataset \\cite{AlmaatouqShmueliPONE2016}. In this experiment, a total of $N=84$ undergraduate students were asked to score their relationship with each of the other participants in a scale from $0$ to $5$, where $0$ meant no-relationship, and $1$ to $5$ represented increasing degree of friendship. \nWe have considered the zero weight as a no-link. Thus, the allowed layer costs are $\\{s_1,s_2,s_3,s_4,s_5\\}=\\{5,4,3,2,1\\}$, which verify the equispaced layer cost condition, Eq. \\eqref{eq:equispaced_cost_condition}.\nThe global and hierarchical structure of the empirical network is summarized in Table \\ref{table:RS_dunbar_structure}. The individual ego-networks show the expected Dunbar's structure, albeit with some remarks: \nBy design, the total active network is incomplete as the maximum possible degree is limited by the total number of experiment participants, $k_{\\max}=N-1$. Consequently, the outer layer degree, $\\<k_5\\>$, departs from the expected scaling. \nHowever, all the participants belong to the same course and live in the same campus, hence we can expect that a significant fraction of their actual social network is captured by the experiment. Indeed, the inner groups show an almost constant ratio of approximately $0.4$, which is consistent with the ratio of $\\sim 1\/3$ reported by larger scale studies \\cite{ArnaboldiFB2012, ArnaboldiTW2013, DunbarPasarellaSN2015, MacCarronDunbarSN2016}.\nThe degree distribution is peaked close to the number of participants, with low variance. The weights distribution, however, is more spread-out. \n\n\n\\begin{table}[t]\n\t\n\t\\begin{tabular}{|c|l||c|c|c|c|c|c|c|}\n\t\t\\hline\n\t\t \\multicolumn{2}{ |c ||}{global stats.} & layers & $s_r$ & $\\<k_r\\>$ & $\\sigma_{k_r}$ & $\\ff{\\<k_{r-1}\\>}{\\<k_{r}\\>}$ & $\\<n_r\\>$ & $\\left<\\ff{n_{r-1}}{n_{r}}\\right>$ \\\\\n\t\t\\hline\n\t\t$N$ & 84 & \t\t\t       $r=1$ & $5$ &  $2.93$  & $3.49$  &        & $2.93$  &     \\\\\n\t\t\\hline\n\t\t$\\<k\\>$\t& 73.63 &\t       $r=2$ & $4$ &  $3.68$  & $3.30$  & $0.80$ & $6.61 $ & $0.39$ \\\\\n\t\t\\hline\n\t\t$\\sigma_k$ & 17.17 &   $r=3$ & $3$ &  $9.50$  & $6.04$  & $0.39$ & $16.11$ & $0.40$ \\\\\t\t\n\t\t\\hline\n\t\t$\\<s\\>$ &  145.45 &        $r=4$ & $2$ &  $30.07$ & $16.78$ & $0.32$ & $46.18$ & $0.37$ \\\\\n\t\t\\hline\n\t\t$\\sigma_s$ & 42.71 &   $r=5$ & $1$ &  $27.45$ & $15.81$ & $1.10$ & $73.63$ & $0.61$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\caption{Global statistics and layer structure of the Reciprocity Survey dataset network \\cite{AlmaatouqShmueliPONE2016}.}\n\\label{table:RS_dunbar_structure}\n\\end{table}\n\nWe have fitted the grand-canonical ensemble to the observed data, by substituting the mean values of $\\<k\\>=73.63$ and $\\<s\\>=145.45$, along with $N=84$; into Eqs. \\eqref{eq:saddle_k_canonical} and \\eqref{eq:saddle_s_canonical}. \nSolving the saddle point equations numerically, we obtained the parameter values $x=0.74$ and $y=0.56$.  \nThe resulting distribution, along with the data are shown in Figure \\ref{fig:DOS}. We have employed a Wang-Landau algorithm \\cite{WangLandauPRL2001, FischerAltmannPRE2015} in order to compute the joint density of states in the $k$-$s$ space (the macrostate space). \nThis function is defined as $P(k,s) = \\sum_{\\{k_r\\}} P(\\{k_r\\})\\delta(k-\\sum_r k_r)\\delta(s-\\sum_r k_r s_r)$, where $P(\\{k_r\\})$ is the grand-canonical probability distribution from Eq. \\eqref{eq:canonical_prob}. We have made available online a python implementation of the algorithm \n\\footnote{It is straightforward to generalize a Wang-Landau algorithm to compute a joint bivariate density of states in an integer valued configuration space. The code in python is available at \\url{https:\/\/manu-jimenez.github.io\/2017\/04\/19\/Wang-Landau-for-joint-DOS.html}}. \nAs it can be seen on the figure, the presence of various outliers with very low $k$ and $s$ displaces the averages from the bulk of the distribution. Other than that, the probability distribution is a well behaved unimodal function, hence fulfilling our first demand, \\emph{(i)}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1.1\\linewidth]{.\/WangLandauDOS.pdf}\n\t\\caption{\n\tContour plot for the joint log-density of states in the $k$-$s$ space for the grand-canonical ensemble with parameter values $\\lambda=-0.3076$ and $\\mu= -0.5842$. \n    The red diamonds correspond to the dataset individual ego-networks and the red cross marks the empirical averages. The lines $s=k$ and $s=5k$ delimit the configuration minimum and maximum allowed total weight, respectively.}\n   \t\\label{fig:DOS}\n\\end{figure}\n\nNext, let us compare the RS layer structure with the layer structure generated by the grand-canonical ensemble. Figure \\ref{fig:RS_scaling} shows the empirical layer degree and layer group sizes distributions along with the ensemble averages. \nObserving the RS layer distributions we find that the empirical averages of the three first layers lie within one standard deviation of the ensemble expected values, both for degrees, $k_r$ , and group sizes, $n_r$. On the other hand, the outermost layers degrees suffer from finite size effects. The discrepancy between ensemble and data is however corrected for the accumulated variables, $n_r$, which follow closely the ensemble scaling trend. Thus, condition \\emph{(ii)} is verified as well.\nRemarkably, the obtained value of $y=0.56$ is comparable with the empirical group size scaling for the outer layers, $\\<n_4\/n_5\\>=0.61$. Indeed, the approximation from Eq. \\eqref{eq:group_size_scaling_approx} is better for the larger outer layers, where the correction factor, $R_r$, tends to $1$, as shown on the inset of Fig. \\ref{fig:RS_scaling}.\nWe have also fitted the ensemble thermodynamic limit to the empirical data. Interestingly, despite the small size of the RS network, the grand-canonical ensemble results are barely distinguishable from its thermodynamic limit. Both ensembles average layer degrees and group sizes are equivalent, and only the variances of the outer layer degrees are slightly larger in the thermodynamic limit.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{.\/RS_layer_group_scaling_vert.pdf}\n\t\\caption{\n\tLayer degrees (top) and layer group sizes (bottom) distributions. \n\tThe empirical distributions are represented by the black box-plots, where the box comprises the second and third quartiles, separated by the median line. The whiskers extend to the full distribution domain, and the averages are represented by the black diamonds.\n\tThe colored lines join the corresponding ensemble averages, $\\<k_r\\>$ (blue, left), and $\\<n_r\\>$ (red, right) from the grand-canonical ensemble with parameters $N=84$, $x=0.74$ and $y=0.56$. Shaded regions comprise one standard deviation.\n\n\tThe inset shows the numerical value of the micro-canonical correction factor, $R_r=(1+\\epsilon_r)\/(1+\\epsilon_{r+1})$, which tends to $1$ for the outer layers.}\n\t\\label{fig:RS_scaling}\n\\end{figure}\n\n\n\\section{Discussion}\\label{s:discussion}\n\n\nWe have proposed a grand-canonical ensemble as a null model for the hierarchical structure of social networks. \nThe ensemble generates the observed nested structure of increasingly large layers of decreasing link weight. Moreover we show that, at least for the outer layers, a constant group size scaling between consecutive group pairings is possible only if the difference between costs of consecutive layers is constant.\nIn the thermodynamic limit, that is, when the number of actors is large, the layer degrees are uncorrelated Poisson variables, which are related through the group size scaling, $y$. \nInterestingly, a recent paper providing more evidence on Dunbar's theory shows the good fit of Poisson distributions to the layer-specific degree distributions \\cite{kordsmeyer2017sizes}.\nIn the case of the dataset analyzed, after fitting the average values of the social bandwidth or resource $\\<s\\>$ and degree $\\<k\\>$, we find that typical samples of the ensemble are similar to the empirical ego-networks.\n\n\nThe proposed null model succeeds at modeling survey social data, where the available categories could be directly used as representations of layers. However, in larger scale studies of ego-network social structure, layers are not given \\emph{a priori}, but rather \\emph{inferred} from the interaction patterns. The link weights usually represent frequency of interaction which acts as a proxy for emotional closeness, and ties are then clustered into discrete groups according to these weights \\cite{ArnaboldiFB2012, ArnaboldiTW2013, DunbarPasarellaSN2015, MacCarronDunbarSN2016}. \nIt is important to assert that we do not identify our link weights, or costs, with interaction frequency. Rather, we introduce the abstract notion of social resource, which can be spent in assigning discrete weights to the social ties.\nIn order to justify this construction it can be argued that a limited social bandwidth may arise both from biological or temporal constraints, since maintaining a social relationship is costly \\cite{dunbar1992, GoncalvesVespignaniPONE2011}. \nNevertheless our model remains uninformative about the psychological or sociological nature of that cost.\nThe other main assumption of our model is the discrete nature of the layers. It is important to stress that we do not intend to neither justify nor provide any sort of explanation to their existence. We rather rely on the existing literature, where these layers have been consistently identified and even given specific names: the \\emph{support clique} of size $\\about5$, the \\emph{sympathy group} of size $\\about15$, the \\emph{affinity group} of size $\\about50$, and the total active network whose size equals the Dunbar's number, $\\about150$ \\cite{ZhouDunbar2005}.\nAssigning the layers a constant cost is not only a convenient simplification, but has also a rather natural interpretation. Indeed the very existence of the layers would implicitly define different \\emph{types} of relationships for the ego. We simply consider that relationships within a given layer are similar precisely because they have the same cost. Then, the problem we focused on was to compute in how many different ways a given number of actors can be distributed in an ego's network with the above mentioned restrictions.\nIn our setting, once the layers, and the total average degree and social resource are fixed, the hierarchical structure emerges in a natural way, as the number of possible configurations with few strong and many weak links is much higher than configurations made of only strong or weak links. This result suggests that the observed hierarchical structure could be a consequence of the existence of an internal discrete categorization in which individuals organize different types of relationships. However, the reason why those categories or layers would exist in the first place remains unknown and shall be further explored.\n\n\n\n\\begin{acknowledgements}\nM. J.-M. would like to thank Ignacio Morer, and  professors Conrad P\\'erez Vicente and Albert D\\'iaz-Guilera for their inspiring work and encouraging conversations.\nI. T. would like to thank Angel S\\'anchez and Robin Dunbar for inspiring conversations and acknowledges partial supporting from the EU through FET-Open Project IBSEN and from Fundaci\\'on BBVA through project DUNDIG.\nJ. R.-L. acknowledges partial supporting from MINECO through grant FIS2015-69167-C2-1.\n\\end{acknowledgements}\n\n\n\\vspace{1cm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe scope  of this work is the theoretical  study  of the welfare function\ndescribing the economics of shallow lakes. Pollution of shallow lakes is a quite often observed phenomenon \n because of  heavy use of fertilizers on\nsurrounding land and an increased inflow of waste water from human\nsettlements and industries.  The shallow lake system provides conflicting services as a resource, due   to the provision of ecological services of a clear lake, \nand as a waste sink, due  to agricultural activities. \n\\vskip.05in\n\nThe economic analysis of the problem requires the study of an optimal control problem or a differential game  in case of  common \nproperty resources  by various communities; see, for example,  \\cite{carpenter}, \\cite{brock-starrett} and \\cite{xepapadeas}.\n\\vskip.05in\n\nTypically the model is described in terms of the amount, $x(t)$, of phosphorus in algae which is assumed to evolve according to the stochastic differential equation (sde for short)\n\\begin{equation} \\label{sldyn}\n\\begin{cases} dx(t)=\\left( u(t)-b x(t)+\\dfrac{x^{2}(t)}{x^{2}(t)+1}\\right) dt+\\sigma x(t)dW(t), & \\\\\nx(0) = x. & \\end{cases}\n\\end{equation}\n\n The first   term, $u(t)$,  in the drift part of the dynamics represents the exterior load of phosphorus imposed by the agricultural community, which is assumed to be positive. The second term is\n  the rate of loss $bx(t)$, which consists   of sedimentation, outflow and\nsequestration in other biomass.  The third term  is the rate of recycling of  phosphorus due to  \nsediment resuspension resulting, for example, from waves, or oxygen depletion.  This \nrate is typically taken to be a sigmoid function; see  \\cite{carpenter}. The model also assumes an uncertainty in the  recycling rate  driven by a linear multiplicative white noise with  diffusion strenght $\\sigma$.   \n\\vskip.05in\n\nThe lake has value as a waste\nsink for agriculture modeled by $\\ln u$ and it provides ecological\nservices that decrease with the amount of phosphorus $-cx^{2}$. \nThe positive parameter $c$ reflects the relative weight of this welfare\ncomponent; large $c$ gives more weight to the ecological services\nof the lake.\n\\vskip.05in\n\nAssuming an infinite horizon at a discount rate $\\rho >0$, the total benefit is\n\\begin{equation}\\label{Jxu}\nJ(x;u)=\\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t}\\big(\\ln u(t)-cx^2(t)\\big)\\, dt\\right],\n\\end{equation}\nwhere $x(\\cdot)$ is the solution to (\\ref{sldyn}), for a given  $u(\\cdot)$, and $x(0)=x$.\nOptimal management  requires to maximize the total benefit, over all  exterior loads that act as controls by the social planner. Thus  the  welfare function is\n\\begin{equation}\\label{ihvf}\nV(x) = \\underset{u\\in \\mathfrak{U}_x} {\\sup}\\ J(x;u),\n\\end{equation}\nwhere   $\\mathfrak{U}_x$ is the set of admissible controls $u: [0,\\infty) \\rightarrow  \\mathbb{R}^+ $  which are specified in the next section.\n\\vskip.05in\n\nDynamic programming arguments lead, under the assumption that the welfare function is decreasing,  to the  Hamilton-Jacobi-Bellman equation\n$$\\rho V=\\left( \\dfrac{x^{2}}{x^{2}+1}-bx\\right)V_x-\\left( \\ln(-V_x)+x^{2}+1\\right)+\\dfrac{1}{2}\\sigma^{2}x^{2}V_{xx}, \\leqno\\mathrm{(OHJB)}$$\nand of the aims of this paper is to provide a rigorous justification of this fact.\n\\vskip.05in\n\nIn the deterministic case $\\sigma=0$, the optimal dynamics of the problem were fully investigated by analysing the possible equilibria of the  dynamics given by Pontryagin maximum principle; see, for example, \\cite{xepapadeas} and\n\\cite{Wagener}. The possibility to  steer the combined economic-ecological system\ntowards the trajectory of optimal management via optimal state-dependent taxes was also considered; see  \\cite{KPXAdeZ}. \n\\vskip.05in\n\nOn the other hand, there is not much in the literature about the stochastic problem. Formal  asymptotics expansions\nof the solution for small  $\\sigma $ for Hamilton-Jacobi-Bellman equations like  (OHJB) have been presented in \\cite{GKW}. In the same paper, the  authors also give a formal  phenomenological bifurcation analysis  based on a geometric invariant quantity, along with some  numerical computations of the stochastic bifurcations\nbased on  (formal) asymptotics  for small $\\sigma$.\n\\vskip.05in\n\nThe connection between  stochastic  control problems  and Hamilton-Jacobi-Bellman equation, which is based on the dynamic programing principle, has been studied extensively. \nThe correct mathematical framework \nis that of the  Crandall-Lions viscosity solutions introduced in \\cite{lions}; see  the review article \\cite{CIL}).   The deterministic case  leads to the study of a first order Hamilton-Jacobi equation; see, for example,   \n \\cite{BCD}, \\cite{barles} and references therein. For the general  stochastic \noptimal control problem we refer to \\cite{Kry}, \\cite{PLL1} - \\cite{PLL3}, \\cite{FS}, \\cite{JYXZ} and references therein.\n\\vskip.05in\n\nThe stochastic shallow lake problem has some nonstandard features and, hence, it requires some special analysis.  At first, the problem is formulated as a state constraint one on a\nsemi-infinite domain. Viscosity solutions with state\nconstraint boundary conditions were  introduced for first order equations by \\cite{So} and \\cite{CDL}.  For second order equations one should consult \\cite{Ka},\n\\cite{LL} and \\cite{ALL}. Apart from the left boundary condition, the correct asymptotic decay of the solution at infinity is necessary to establish a comparison result; see, for example,   \\cite{IL} and\n\\cite{DaLL}.\n\\vskip.05in\n\nThe unboundedness of the controls  along with  the logarithmic term in the cost functional lead to a logarithm of the gradient variable in (OHJB).  An \n a priori knowledge of the solution is required to guarantee that the Hamiltonian is well defined. Moreover, due the  presence of the logarithmic term it is necessary to construct an appropriate test function \n to establish a comparison proof. The commonly used  polynomial functions, see, for example,  \\cite{IL} and   \\cite{DaLL}, are not useful here since they  do not yield a supersolution \nof the equation. \n\\vskip.05in\n\nIn the present work we study the stochastic shallow lake problem   for a fixed $\\sigma$. We first prove the necessary stochastic analysis estimates for the welfare function (\\ref{ihvf}). We obtain directly various crucial\nproperties for the welfare function, that is,  boundary behaviour, local regularity, monotonicity and\nasymptotic estimates at infinity.\n\\vskip.05in\n\nWe prove, including the deterministic case, that (\\ref{ihvf}) is the unique decreasing constrained viscosity solution \nto (OHJB) with  quadratic growth  at infinity. The comparison theorem is  proved by considering a linearized equation and constructing a proper supersolution. Exploiting the well-posedness of the problem within the framework of constrained viscosity \nsolutions we investigate the exact asymptotic behavior of the solutions at infinity. The latter is used to construct a monotone convergent numerical scheme that\nalong with the optimal dynamics equation can be used  to  reconstruct numerically the stochastic optimal dynamics.\n\n\n\n\n\\section{The general setting and the main results}\n\nWe assume that there exists a filtered probability space $( \\Omega, \\mathcal{F},\\{ \\mathcal{F}_t\\}_{t\\ge 0},\\mathbb{P} )$\nsatisfying the usual conditions, and a Brownian motion $W(\\cdot)$ defined on that space. \nAn admissible control   $u(\\cdot)\\in\\mathfrak{U}_x$ is an $\\mathcal{F}_t$-adapted, $\\mathbb{P}$-a.s. locally integrable process with values in $U=(0,\\infty)$, satisfying \n\\begin{equation}\\label{ac_constraint}\n \\mathbb{E} \\left[ \\int_{0}^{\\infty}e^{-\\rho t}\\ln u(t)dt \\right] < \\infty,\n\\end{equation}\nsuch that  the problem (\\ref{sldyn}) has a unique strong solution $x(\\cdot)$.\n\\vskip.05in\n\n The shallow lake problem has an infinite horizon. Standard arguments  based on the dynamic programming principle   (see \\cite{FS}, Section III.7)  suggest that the welfare function  $V$  given by (\\ref{ihvf}) satisfies  the  HJB equation\n\\begin{equation}\\label{HJB}\n\\rho V=\\sup_{u\\in U } G(x,u,V_x,V_{xx}),\n\\end{equation}\nwith  $G $  defined by\n\\begin{equation}\\label{hG}\nG(x,u,p,P)=\\dfrac{1}{2}\\sigma^{2}x^{2}P+\\left( u-bx+\\dfrac{x^{2}}{x^{2}+1}\\right)p +\\ln u-x^{2}.\n\\end{equation}\n\nOne difficulty in  the study of this problem is related to the fact that the control functions $u$ take values in the unbounded set $ \\mathbb{R}^+$ so that supremum in (\\ref{HJB}) \nmight take infinite values. Indeed, when $U=\\mathbb{R}^+$, setting \n$$ H(x,p,P)= \\sup_{u\\in \\mathbb{R}^+ } G(x,u,p,P),$$\nwe find \n\\begin{equation}\\label{HJB-unbounded}\nH(x,p,P)=\n  \\left\\{\n  \\begin{array}{lr}  \\left( \\dfrac{x^{2}}{x^{2}+1}-bx\\right)p-\\left( \\ln(-p)+x^{2}+1\\right)+\\dfrac{1}{2}\\sigma^{2}x^{2}P & \\,\\, if \\,\\, p<0, \\\\\n                                                     + \\infty & \\,\\, if \\,\\, p \\geq 0. \\end{array} \\right.\n \\end{equation}\n \n \n It is natural to expect that  since shallow lake looses its value  with a higher  concentration  of phosphorus,  the welfare function is decreasing\n as the initial state of phosphorus increases. Assuming that $V_x<0$,  (\\ref{HJB}) becomes (OHJB).\n\\vskip.05in \n\nSince the problem is set in  $(0, \\infty)$, it is necessary to introduce boundary conditions   guaranteeing the well-posedness of  the corresponding boundary value problem. \n\\vskip.05in\n\nGiven the possible  degeneracies  of Hamilton-Jacobi-Bellman equations at $x=0$, the right framework \nis that of continuous viscosity solutions in which boundary conditions are considered in the weak sense.\nSince at the boundary point $x=0$\n\\begin{equation}\\label{control}\n\\underset{u\\in U} {\\inf} \\left\\{ - u+b x-\\dfrac{x^{2}}{x^{2}+1} \\right\\} < 0,\n\\end{equation}\nthat is,  there always exists a control that\ncan  drive the system inside $(0,\\infty)$, the problem should  be considered as a state constraint one on the interval $[0,\\infty)$, meaning that $V$ is a subsolution in $[0,\\infty)$ and a supersolution in $(0,\\infty)$. \n\\vskip.05in\n\nNext we present the main results of the paper. The proofs are given in Section 4. The first is about the relationship between the welfare function and (OHJB). \n\n\\begin{thm}\\label{constrained_vs}\nIf $\\sigma^2 < \\rho +2b$, the welfare function $V$  is a continuous in $[0,\\infty) $ constrained viscosity solution of the  equation {\\rm (OHJB)} in  $[0,\\infty)$. \n\\end{thm}\n\\noindent\nThe  second  result is the following  comparison principle for solutions of (OHJB).   \n\\begin{thm}\\label{comparison} Assume that  $u\\in C([0,\\infty))$ is a bounded from above  strictly decreasing subsolution of ${\\rm (OHJB)}$ in $[0, \\infty)$ and $v\\in C([0,\\infty))$ is a bounded from above strictly decreasing supersolution of\n${\\rm (OHJB)}$ in $(0, \\infty)$ such that  $v \\geq -c(1+x^2)$ and  $Du\\leq -\\frac{1}{c^*}$\nin the viscosity sense,\nfor $c$, $c^*$  positive constants. Then\n$u \\leq v$ in $[0, \\infty)$. \n\\end{thm}\n\\noindent\n\nThe next theorem describes the exact asymptotic behavior of (\\ref{ihvf}) at $+\\infty$. Let \n\\begin{equation*}\\label{K}\nA=\\frac{1}{\\rho+2b-\\sigma^2}\\,\\,\\, \\mbox{ and } \\,\\,\\, K=\\frac{1}{\\rho}\\left(\\frac{2b+\\sigma^2}{2\\rho}-\\frac{A(\\rho+2b)}{(b+\\rho)^2} -1\\right).\n\\end{equation*}\n\n\\begin{thm}\\label{asymptotic}\nAs $x\\rightarrow \\infty$,\n\\begin{equation}\\label{asympto_behav}\nV(x)= -A\\left(x+\\frac{1}{b+\\rho}\\right)^2-\\frac{1}{\\rho}\\ln \\left[2A(x+\\frac{1}{b+\\rho})\\right]+K+o(1). \n\\end{equation}\n\\end{thm}\n\\vskip.05in\n\nAn important ingredient of the analysis is the following proposition which collects some key properties of $V$ that  are used in \nthe proofs of Theorem \\ref{constrained_vs} and Theorem \\ref{asymptotic} and  show that $V$ satisfies the assumptions of Theorem \\ref{comparison}. The proof is presented in Section 3. \n\\begin{propos}\\label{vprop}\nSuppose $\\sigma^2<\\rho+2b$.\\\\[1mm]\n(i) There exist constants $K_1, K_2>0$, such that, for any $x\\ge 0$, we have\n\\begin{equation}\\label{Vbounds}\nK_1\\ \\le\\ V(x)+A\\left(x+\\frac{1}{b+\\rho}\\right)^2+\\frac{1}{\\rho}\\ln \\left(x+\\frac{1}{b+\\rho}\\right)\\ \\le\\ K_2.\n\\end{equation}\n\\noindent\n(ii) There exist  $C>0$ and  $\\Phi:[0,+\\infty)\\rightarrow \\mathbb{R} $ increasing such that, for any $x_1,x_2\\in [0,+\\infty)$ with $x_1<x_2$,\n\\begin{equation}\\label{DVbounds}\n-\\Phi(x_2)\\le \\frac{V(x_2)-V(x_1)}{x_2-x_1}\\le -C.\n\\end{equation}\n(iii) $V$ is differentiable at zero and \n\\begin{equation}\n\\ln\\big(-V'(0)\\big)+\\rho V(0)+1=0.\n\\label{bc}\n\\end{equation} \n\\end{propos}\n\n\n\n\nIt is shown in the next section that the assumption $\\sigma^2<\\rho+2b$ is necessary, otherwise $V(x)=-\\infty.$  \n\\vskip.05in\n\nRelation (\\ref{asympto_behav}) is important for the numerical approximation of (\\ref{ihvf}). Given that the  computational domain   is finite, the correct asymptotic behavior\nof (\\ref{ihvf}) at $+\\infty$ is necessary for an accurate computation of $V$.\nIn this connection, in Section 5 we present a monotone numerical scheme approximating (\\ref{ihvf}). \n\n \\section{The proof of Proposition \\ref{vprop}}\n\\noindent\n {\\bf Properties of the dynamics.} The first result states that, if $x\\geq0$,  the solution to (\\ref{sldyn}) stays nonnegative.\n\\begin{lemma}\\label{positive}\nIf $x\\ge 0$, $u\\in\\mathfrak{U}_{x}$, and $x(\\cdot)$ is the solution to (\\ref{sldyn}), then\n$\n\\mathbb{P}\\big[x(t)\\ge 0,\\ \\forall t\\ge 0\\big]=1.\n$\n\\end{lemma}\n\\noindent\n{\\em Proof:}\\ Elementary stochastic analysis calculations yield that \n\\begin{equation}\nx(t)=xZ_t+\\int_0^t \\frac{Z_t}{Z_s}\\left(u(s)+\\frac{x^2(s)}{1+x^2(s)}\\right) ds,\n\\label{implicit}\n\\end{equation}\nwhere \n\\begin{equation}\\label{fundsol}\nZ_t=e^{\\sigma W_t-(b+\\frac{\\sigma^2}{2})t},\n\\end{equation}\nand the claim is now obvious, since $u$ takes positive values.\\hfill$\\Box$\\\\[1mm]\n\nThe next assertion is  that the set of admissible controls is actually independent of the starting point $x$. \n\\begin{lemma}\\label{uxisu} \n  For all $x,y\\ge 0$,  $\\mathfrak{U}_x=\\mathfrak{U}_y.$\n\\end{lemma}\n\\noindent\n{\\em Proof:} Fix  $u\\in\\mathfrak{U}_x$ and $x\\in[0,\\infty)$ and let $x(\\cdot)$  the unique strong solution to (\\ref{sldyn}) with $x(0)=x$, and, for any $y\\ge 0$, consider the sde \n\\begin{equation}\\label{U_x-indep}\n\\begin{cases} dw(t)=\\Big\\{-bw(t)-\\frac{(x(t)-w(t))^2}{1+(x(t)-w(t))^2}+\\frac{x(t)^2}{1+x(t)^2}\\Big\\}dt+\\sigma w(t)dW(t),&\\\\\nw(0)=x-y,\\end{cases}\n\\end{equation}\nand note that the coefficients  are Lipschitz and grow at most linearly.\nIt follows that (\\ref{U_x-indep}) has a unique strong solution defined for all $t\\ge 0$.  It is easy to see now that the process $y(t)=x(t)-w(t)$ satisfies (\\ref{sldyn}) with $y(0)=y$. Moreover, the uniqueness of $y(\\cdot)$ follows from that of $x(\\cdot)$, so $u\\in\\mathfrak{U}_y$. \\hfill$\\Box$\\\\[2mm]\nIn view of Lemma \\ref{uxisu} we will denote the set of admissible controls by $\\mathfrak{U}$, regardless of the starting point $x$  in (\\ref{sldyn}).\n\n\n\\begin{lemma}\\label{monotone}\nSuppose $x(\\cdot),\\ y(\\cdot)$ satisfy (\\ref{sldyn}) with controls $u_1,\\ u_2\\in\\mathfrak{U}$, respectively, and $x(0)=x$, $y(0)=y$.\nIf $x\\le y$ and $\\mathbb{P}\\big[u_1(t)\\le u_2(t),\\ \\forall t\\ge 0\\big]=1$, then\n\\[\n\\mathbb{P}\\big[y(t)-x(t)\\ge (y-x)Z_t,\\ \\forall t\\ge 0\\big]=1.\n\\]\n\\end{lemma}\n\\noindent\n{\\em Proof:} The proof is an immediate consequence of (\\ref{implicit}) and Gronwall's inequality, since  $w(t)=y(t)-x(t)$ satisfies\n\\[\nw(t)=(y-x)Z_t+\\int_0^t \\frac{Z_t}{Z_s}\\big(u_2(s)-u_1(s)+w(s)\\frac{y(s)+x(s)}{\\big(1+y^2(s)\\big)\\big(1+x^2(s)\\big)}\\big)\\ ds.\n\\]\n $\\hfill\\Box$\\\\[5mm]\n\\noindent\n{\\bf Properties of the welfare function.} The results here follow from the properties of (\\ref{sldyn}).  In the rest of the paper, for the shake of convenience, we  \nassume that $c=1$. Throughout this section, we  refer to quantities depending only on $\\rho,b$ and $\\sigma^2$  as constants.\n\\vskip.05in\n\nWe remark that, if $\\sigma^2\\ge \\rho+2b$, then $V(x)\\equiv-\\infty$. Indeed, when $u\\in\\mathfrak{U}$ and $x(\\cdot)$ satisfies (\\ref{sldyn}), \n(\\ref{implicit}) implies that $\\mathbb{P}\\big[x(t)\\ge M_t(u),\\ \\forall t\\ge 0\\big]=1$, with \n\\begin{equation}\\label{Mf}\nM_t(u)=\\int_0^t\\frac{Z_t}{Z_s}u(s)\\ ds,\n\\end{equation}\nwhere $\\{Z_t\\}_{t\\ge 0}$ is as in \\eqref{fundsol}. \n\n\\vskip.05in \nIn view of this observation we will hereafter assume that \n\\[\n \\sigma^2<\\rho+2b.\n\\]\n\nWe will first prove three lemmata before we proceed with the proof of Proposition \\ref{vprop}.\n\\begin{lemma}\\label{Vdec}\n The function $x\\mapsto V(x)+Ax^2$ is decreasing on $[0,+\\infty)$. \n\\end{lemma}\n\n\\noindent\n{\\em Proof:} Fix $x_1,x_2\\ge 0$ with $x_1\\le x_2$. It suffices to show that, for any  control $u\\in\\mathfrak{U}$,\n\\[\nJ(x_2;u)+Ax_2^2\\le J(x_1;u)+Ax_1^2.\n\\]\n\nSince this holds trivially if $J(x_2;u)=-\\infty$, we may assume that $J(x_2;u)>-\\infty$. \n\\vskip.05in\n\nConsider now the solutions $x_1(\\cdot),x_2(\\cdot)$ to (\\ref{sldyn}) \nwith initial conditions $x_1, x_2$ and a common control $u\\in\\mathfrak{U}$. Lemma \\ref{monotone} implies that, $\\mathbb{P}$-a.s.  and for all $t\\geq0$ and $i=1,2,$\n\\[\nx_i(t)\\ge x_iZ_t, \\qquad\\text{and}\\qquad x_2(t)-x_1(t)\\ge (x_2-x_1)Z_t.\n\\]\n\nNote that since $u\\in\\mathfrak{U}$ and $J(x_2;u)>-\\infty$,\n\\[\n\\int_0^\\infty e^{-\\rho t}x_2^2(t)\\,dt<+\\infty\\Rightarrow\\int_0^\\infty e^{-\\rho t}x_1^2(t)\\,dt<+\\infty\\Rightarrow J(x_1;u)>-\\infty.\n\\]\n\nIn particular,\n\\begin{align*}\nJ(x_2;u)\\!-\\!J(x_1;u)&=\\mathbb{E}\\left[ \\int_{0}^{\\infty}\\!\\!\\!\\!e^{-\\rho t}\\big(\\ln u(t)-x_2^{2}(t)\\big)dt -\\!\\! \\int_{0}^{\\infty}\\!\\!\\!\\!e^{-\\rho t}\\big(\\ln u(t)-x_1^{2}(t)\\big)dt\\right]\\\\\n&=-\\mathbb{E}\\left[ \\int_{0}^{\\infty} e^{-\\rho t}\\big(x_2(t)-x_1(t)\\big)\\big(x_2(t)+x_1(t)\\big)dt \\right]\\\\\n&\\le -(x_2^2-x_1^2)\\int_0^\\infty e^{-\\rho t} \\mathbb{E}\\big[Z_t^2\\big]\\ dt= -A(x_2^2-x_1^2).\n\\end{align*}\n\\hfill$\\Box$\n\\begin{lemma}\\label{V0fin}\nThe welfare function at zero satisfies\n$\nV(0) \\le\\frac{1}{\\rho}\\ln\\left(\\frac{b+\\rho}{\\sqrt{2e}}\\right).\n$\n\\end{lemma}\n\\noindent\n{\\em Proof:} Recall that, for any $u\\in\\mathfrak{U}$,  $x(t)\\ge M_t(u)$.\nUsing Jensen's  inequality and part (i) of Lemma \\ref{Mest}, we find\n\\begin{align*}\n\\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t}\\ln u(t)\\ dt\\right] &\\le\\frac{1}{\\rho}\\ln\\mathbb{E}\\left[\\int_0^\\infty \\rho e^{-\\rho t}u(t)\\ dt\\right] \\\\\n&=\\frac{1}{\\rho}\\ln\\mathbb{E}\\left[\\int_0^\\infty \\rho(\\rho+b) e^{-\\rho t}M_t(u)\\ dt\\right] \\\\\n&\\le \\frac{\\ln(b+\\rho)}{\\rho}+\\frac{1}{\\rho}\\ln\\mathbb{E}\\left[\\int_0^\\infty \\rho e^{-\\rho t}x(t)\\ dt\\right] \\\\\n&\\le \\frac{\\ln(b+\\rho)}{\\rho}+\\frac{1}{2\\rho}\\ln\\mathbb{E}\\left[\\int_0^\\infty \\rho e^{-\\rho t}x^2(t)\\ dt\\right].\n\\end{align*}\n\nIn view of (\\ref{ac_constraint}), we need only consider  $u\\in\\mathfrak{U}$ such that $D=\\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t}x^2(t)\\ dt\\right]<\\infty$.\nThen\n\\[\n\\mathbb{E} \\left\\{ \\int_{0}^{\\infty}e^{-\\rho t}\\big[\\ln u(t)-x^{2}(t)\\big]dt \\right\\}\\le  \\frac{\\ln(b+\\rho)}{\\rho}+\\frac{\\ln(\\rho D)}{2\\rho}-D\\le\\frac{1}{\\rho}\\ln\\left(\\frac{b+\\rho}{\\sqrt{2e}}\\right),\n\\]\nand the assertion holds.\\hfill$\\Box$\\\\\n\nIt follows from Lemma \\ref{Vdec} and Lemma \\ref{V0fin} that $V<+\\infty$ in $[0,\\infty)$.\n\\noindent\nThe next result is a special case of the dynamic programming principle. \n\\begin{lemma}\\label{specialdpp}\nFix $x_1,x_2\\in[0,\\infty)$ with $x_1<x_2$, and, for  $u\\in\\mathfrak{U}$, let $x(\\cdot)$ be the solution to (\\ref{sldyn}) with control $u$ and $x(0)=x_1$.  If $\\tau_u$ is \nthe hitting time of $x(\\cdot)$ on $[x_2,+\\infty)$, that is,\n\\[\n\\tau_u=\\inf\\{t\\ge 0: x(t)\\ge x_2\\},\n\\]\nthen\n\n\\begin{equation}\\label{babydpp}\nV(x_1)=\\sup_{u\\in\\mathfrak{U}}\\mathbb{E}\\left[\\int_0^{\\tau_u} e^{-\\rho t}\\big(\\ln u(t)-x^2(t)\\big)\\, dt+e^{-\\rho\\tau_u}V(x_2)\\right].\n\\end{equation}\n\\end{lemma}\n\\noindent\n{\\em Proof:} We have\n\\begin{align*}\nJ(x_1;u)&=\\mathbb{E}\\left[\\int_0^{\\tau_u}\\hspace{-2mm}e^{-\\rho t}\\big(\\ln u(t)-x^2(t)\\big)\\, dt\\right]\n+\\mathbb{E}\\left[\\int_{\\tau_u}^\\infty \\hspace{-2mm}e^{-\\rho t}\\big(\\ln u(t) -x^{2}(t)\\big)\\,dt;\\, \\tau_u<+\\infty\\right].\n\\end{align*}\n\nConditioning on the $\\sigma$-field $\\mathcal{F}_{\\tau_u}$, and applying the strong Markov property, the rightmost term becomes\n\\begin{align*}\n\\mathbb{E}\\left[e^{-\\rho\\tau_u}\\mathbb{E}\\left[\\int_{\\tau_u}^\\infty \\hspace{-2mm}e^{-\\rho (t-\\tau_u)}\\big(\\ln u(t) -x^{2}(t)\\big)\\,dt\\ \\Big| \\mathcal{F}_{\\tau_u}\\right];\\, \\tau_u<+\\infty\\right]\\le \\mathbb{E}\\left[e^{-\\rho\\tau_u}\\right]V(x_2),\n\\end{align*}\nsince on the event $\\{\\tau_u<+\\infty\\}$, $x(\\tau_u+\\cdot)$ satisfies (\\ref{sldyn}) with initial condition $x(\\tau_u)=x_2$ and control $u(\\tau_u+\\cdot)$. Taking the supremum over $u\\in\\mathfrak{U}$ we\nsee that the left hand side of (\\ref{babydpp}) is less than or equal the right hand one.\n\\vskip.05in\nFor the reverse inequality, take any $u\\in\\mathfrak{U}$ and consider (\\ref{sldyn}) driven by the Brownian motion \n$B(t)=W(\\tau_u+t)-W(\\tau_u)$, and,  for $\\varepsilon>0$, choose  a control $u_\\varepsilon$ such that \n\\[\nV(x_2)<J(x_2;u_\\varepsilon)+\\varepsilon.\n\\]\n\nDefine now the new control $u_*\\in\\mathfrak{U}$ as \n\\[\nu_*(t)=\\begin{cases} u(t) & \\text{ for } t\\le\\tau_{u}\\\\ u_\\varepsilon(t-\\tau_u) & \\text{ for } t> \\tau_u.\\end{cases}\n\\]\n\nJust as in the proof of the upper bound we get\n\\begin{align*}\nV(x_1)&\\ge J(x_1;u_*)=\\mathbb{E}\\left[\\int_0^{\\tau_u} e^{-\\rho t}\\big(\\ln u(t)-x^2(t)\\big)\\, dt+e^{-\\rho\\tau_u}J(x_2;u_\\varepsilon)\\right]\\\\\n&>\\mathbb{E}\\left[\\int_0^{\\tau_u} e^{-\\rho t}\\big(\\ln u(t)-x^2(t)\\big)\\, dt+e^{-\\rho\\tau_u}V(x_2)\\right]-\\varepsilon,\n\\end{align*}\nwhich concludes the proof.\\hfill$\\Box$\\\\[2mm]\n\nWe next give the proof of Proposition \\ref{vprop}, which is subdivided in several parts. \\\\[1mm]\n\n\\noindent\n{\\bf Proof of Proposition \\ref{vprop}:} \n\\noindent\n{\\em Proof  lower bound, part (i):}\\quad The claim will follow by choosing the  control $u(t)=\\frac{1+x(t)}{1+x^2(t)}$, which is clearly admissible. \nThen, (\\ref{implicit}) gives\n\\begin{equation}\nx(t)=xZ_t+\\int_0^t \\frac{Z_t}{Z_s}\\ \\Big(1+\\frac{x(s)}{1+x^2(s)}\\Big)ds=xZ_t+M_t(1)+\\int_0^t \\frac{Z_t}{Z_s}\\ \\frac{x(s)}{1+x^2(s)}ds\n\\label{testX}\n\\end{equation}\nand, hence,\n\\begin{align*}\nx^2(t)&=x^2Z_t^2+2xZ_tM_t(1)+M_t^2(1)+\\Big(\\int_0^t \\frac{Z_t}{Z_s}\\ \\frac{x(s)}{1+x^2(s)}ds\\Big)^2\\\\\n&\\quad+M_t(1)\\int_0^t \\frac{Z_t}{Z_s}\\ \\frac{2x(s)}{1+x^2(s)}ds+\\int_0^t \\frac{Z_t^2}{Z_s}\\ \\frac{2x x(s)}{1+x^2(s)}ds.\n\\end{align*}\n\nTo estimate the rightmost term from above, note that, in view of (\\ref{testX}),  $x\\le x(s)Z_s^{-1}$,  while for the third and fourth terms of the sum we use that\n$\\frac{x(s)}{1+x^2(s)}\\le \\frac{1}{2}$. It follows that\n\\begin{equation}\nx^2(t)\\le x^2Z_t^2+2xZ_tM_t(1)+\\frac{9}{4}M_t^2(1)+2\\int_0^t \\frac{Z_t^2}{Z_s^2}\\ ds,\n\\label{testX2}\n\\end{equation}\n\nIt is easy to see that\n\\begin{equation}\n\\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t}x^2Z_t^2 dt\\right]=x^2\\int_0^\\infty e^{-(\\rho+2b-\\sigma^2)t}dt=Ax^2,\n\\label{Xsq}\n\\end{equation}\n\nLemma \\ref{Mest} (ii) gives\n\\begin{align}\\label{X1}\n\\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t} 2xZ_t M_t(1) dt\\right]&=2Ax\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[Z_t\\big]dt= \\frac{2Ax}{\\rho+b},\n\\end{align}\nwhile Lemma \\ref{Mest} (i) and Lemma \\ref{Mest} (iii)  yield\n\\begin{equation}\n\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[M_t^2(1)\\big]\\ dt = 2A\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[M_t(1)\\big]\\ dt =\\frac{2A}{\\rho(\\rho+b)}.\n\\label{Mtsq}\n\\end{equation}\n\nWe also have\n\\begin{align}\\label{lastterm}\n\\int_0^\\infty \\hspace{-2mm}e^{-\\rho t}\\int_0^t\\mathbb{E}\\left[\\frac{Z_t^2}{Z_s^2}\\right]ds\\ dt&=\\int_0^\\infty \\hspace{-2mm}e^{-\\rho t}\\int_0^t e^{(\\sigma^2-2b)(t-s)}ds\\ dt=\\frac{A}{\\rho}.\n\\end{align}\n\nUsing the last four observations in  (\\ref{testX2}), we find  for some constant $B$, \n\\begin{equation}\n\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[x^2(t)\\big]\\ dt \\le A\\left[\\big(x+\\frac{1}{\\rho+b}\\big)^2+B\\right].\n\\label{Xener}\n\\end{equation}\n\n On the other hand, using that, for all $x\\ge 0$, $(1+x)^2\\ge (1+x^2)$,  and Jensen's inequality,  we find\n\\begin{align*}\n\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[\\ln u(t)\\big]\\ dt &\\ge-\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[\\ln\\big(1+x(t)\\big)\\big]\\ dt \\\\\n&\\ge -\\frac{1}{\\rho}\\ln\\left(\\int_0^\\infty \\rho e^{-\\rho t} \\Big(1+\\mathbb{E}\\big[x(t)\\big]\\Big) dt\\right)\\\\\n&=-\\frac{1}{\\rho}\\ln\\left(1+\\rho\\int_0^\\infty  e^{-\\rho t} \\mathbb{E}\\big[x(t)\\big] dt\\right).\n\\end{align*}\n\nBy  (\\ref{testX}) it follows that\n$\\displaystyle\n\\mathbb{E}\\big[x(t)\\big]\\le x\\mathbb{E}\\big[Z_t\\big]+\\frac{3}{2}\\mathbb{E}\\big[M_t(1)\\big]=xe^{-bt}+\\frac{3}{2}\\mathbb{E}\\big[M_t(1)\\big].\n$\n\nHence, using Lemma \\ref{Mest} (i), we obtain\n\\[\n\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[\\ln u(t)\\big]\\ dt \\ge -\\frac{1}{\\rho}\\ln\\left(1+\\frac{\\rho x}{\\rho+b}+\\frac{3}{2(\\rho+b)}\\right).\n\\]\n\nThe preceding estimate and (\\ref{Xener}) together imply that, for some suitable constant $K_1$,\n\\begin{align*}\nV(x)&\\ge J(x;u) =\\mathbb{E}\\big[\\int_0^\\infty e^{-\\rho t}\\big(\\ln u(t)-x^2(t)\\big)\\ dt \\big]\\\\\n&\\ge -A\\left(x+\\frac{1}{b+\\rho}\\right)^2-\\frac{1}{\\rho}\\ln \\left(x+\\frac{1}{b+\\rho}\\right)+K_1.\n\\end{align*}\n\n\n\\noindent\n{\\em Proof of the upper bound, part (i) :}\\quad In view of Lemma \\ref{Vdec} and Lemma \\ref{V0fin}, it suffices to find  $K_2>0$, such that the asserted inequality holds for $x\\ge 1$.\n\\vskip.05in\n\nFix $u\\in\\mathfrak{U}$. Then \n\\begin{align*}\nx^2(t)&\\ge x^2Z_t^2+2xZ_t^2\\int_0^t\\frac{1}{Z_s}\\left(u(s)+\\frac{x^2(s)}{1+x^2(s)}\\right) ds\\\\\n&=x^2Z_t^2+2xZ_tM_t(1+u)-\\int_0^t\\frac{Z_t^2}{Z_s^2}\\frac{2xZ_s}{1+x^2(s)} ds.\n\\end{align*}\n\nSince\n\\[\n1+x^2(t)\\ge 1+x^2Z_t^2\\ge 2xZ_t,\n\\]\nso we can further estimate $x^2(t)$ from below by\n\\begin{align}\\label{mtuest}\nx^2(t)&\\ge x^2Z_t^2+2xZ_tM_t(1)+2xZ_tM_t(u)-\\int_0^t \\frac{Z_t^2}{Z_s^2} ds.\n\\end{align}\n\nUsing the elementary inequality that $\\ln a\\le ab-\\ln b-1$, which holds for all $a,b>0$, and Lemma \\ref{Mest} (ii), we obtain, for some B,\n\\begin{align*}\n&\\int_0^\\infty e^{-\\rho t}\\mathbb{E}\\big[\\ln u(t)\\big]dt \\le \\mathbb{E}\\left[\\int_0^\\infty e^{-\\rho t} \\Big\\{2AxZ_tu(t)-\\ln\\big(2AxZ_t\\big)\\Big\\}\\ dt\\right]\\nonumber\\\\\n&\\qquad=\\mathbb{E}\\left[\\int_0^\\infty\\hspace{-2mm} e^{-\\rho t} 2xZ_tM_t(u)\\, dt\\right]-\\frac{\\ln(2Ax) }{\\rho}+\\frac{2b+\\sigma^2}{2\\rho^2}\\nonumber\\\\\n&\\qquad\\le \\mathbb{E}\\left[\\int_0^\\infty \\hspace{-2mm}e^{-\\rho t} \\Big(x^2(t)-x^2Z_t^2-2xZ_tM_t(1)+\\int_0^t\\frac{Z_t^2}{Z_s^2}\\, ds\\Big)\\, dt\\right]-\\frac{\\ln x + B}{\\rho}\\nonumber,\n\\end{align*}\nwhere in the final step we have used (\\ref{mtuest}). \n\\vskip.05in\n\nIn view of (\\ref{Xsq}), (\\ref{X1}) and (\\ref{lastterm}), for every $u\\in\\mathfrak{U}$ there exists  $K_2>0$ such that\n\\[\nJ(x;u)\\le -A\\left(x+\\frac{1}{b+\\rho}\\right)^2-\\frac{1}{\\rho}\\ln \\big(x+\\frac{1}{\\rho+b}\\big)+K_2.\n\\]\n\nThe assertion now follows by taking the supremum over $u\\in\\mathfrak{U}$.\\\\[2mm]\n\\noindent\n{\\em Proof of the lower bound, part (ii):} Fix $x_1,x_2$ as in the statement. It follows from  Lemma \\ref{specialdpp} that for any $\\epsilon>0$ there exists a control $u_\\varepsilon\\in\\mathfrak{U}$ such that \n\\begin{equation}\\label{dppe}\nV(x_1)\\le \\mathbb{E}\\left[\\int_0^{\\tau_{\\varepsilon}}e^{-\\rho t}\\ln u_\\varepsilon(t)\\ dt\\right]+\\mathbb{E}\\big[e^{-\\rho\\tau_{\\varepsilon}}\\big]V(x_2)+\\varepsilon(x_2^2-x_1^2),\n\\end{equation}\nwhere $\\tau_\\varepsilon$ is the hitting time on $[x_2,+\\infty)$ of the solution $x_\\varepsilon(\\cdot)$ to (\\ref{sldyn}) with $x(0)=x_1$ and control $u_\\varepsilon$.\n\\vskip.05in\n\nUsing the elementary inequality\n\\[\n\\ln u_\\varepsilon(t)\\le \\ln c+\\frac{u_\\varepsilon(t)}{c}-1, \\quad\\text{with } c=e^{\\rho V(x_2)+1},\n\\]\nwe find\n\\begin{align}\\label{DVcont}\nV(x_1)-V(x_2)&\\le \\frac{1}{c}\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon} e^{-\\rho t}u_\\varepsilon(t)\\ dt\\right]+\\varepsilon(x_2^2-x_1^2).\n\\end{align}\n\nTo conclude it suffices to show that\n\\begin{equation}\n\\frac{1}{c}\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}u_\\varepsilon(t)\\ dt\\right]\\le \\Phi(x_2)(x_2-x_1).\n\\label{suffu}\n\\end{equation}\n\nTo this end, we apply It\\^{o}'s rule to the semimartingale $Y_t=e^{-\\rho t+\\gamma x_\\varepsilon(t)}$, where $\\gamma>0$ is a constant to be determined, and find\n\\begin{align*}\nY_t-e^{\\gamma x_1}&=\\int_0^t Y_s \\big(-\\rho\\, ds+ \\gamma\\, dx_\\varepsilon(s)+\\frac{\\gamma^2}{2}\\, d\\langle x_\\varepsilon\\rangle_s\\big)\\\\\n&=\\int_0^t Y_s\\Big(-\\rho+\\gamma\\big(u_\\varepsilon(s)-bx_\\varepsilon(s)+\\frac{x_\\varepsilon^2(s)}{1+x_\\varepsilon^2(s)}\\big)+\\frac{\\gamma^2\\sigma^2 x_\\varepsilon^2(s)}{2}\\Big)ds+M_t,\n\\end{align*}\nwhere $M_t$ stands for the martingale $\\gamma\\sigma \\int_0^t x_\\varepsilon(s)\\, dW(s)$. \n\\vskip.05in\nNext we  apply the optional stopping theorem for the bounded stopping time $\\tau_N=\\min\\{\\tau_\\varepsilon,N\\}$, with $N\\in\\mathbb{N}$, to find \n\\begin{align*}\n\\mathbb{E}\\big[Y_{\\tau_N}\\big]-e^{\\gamma x_1}&=\\mathbb{E}\\left[\\int_0^{\\tau_N} \\hspace{-2mm}Y_s\\Big(-\\rho+\\gamma\\big(u_\\varepsilon(s)-bx_\\varepsilon(s)+\\frac{x_\\varepsilon^2(s)}{1+x_\\varepsilon^2(s)}\\big)+\\frac{\\gamma^2\\sigma^2 x_\\varepsilon^2(s)}{2}\\Big)ds\\right].\n\\end{align*}\n\nSince  $0\\le x_\\varepsilon(s) \\le x_2$ in $[0,\\tau_\\varepsilon]$, \n\\begin{align*}\ne^{\\gamma x_2}\\mathbb{E}\\big[e^{-\\rho\\tau_N}\\big]-e^{\\gamma x_1}&\\ge \\mathbb{E}\\left[\\int_0^{\\tau_N} \\hspace{-2mm}Y_t\\Big(-\\rho+\\gamma\\big(u_\\varepsilon(t)-bx_\\varepsilon(t)\\big)+\\frac{\\gamma^2\\sigma^2 x_\\varepsilon^2(t)}{2}\\Big)dt\\right],\n\\end{align*}\nand\n\\begin{align*}\ne^{\\gamma x_2}-e^{\\gamma x_1}&\\ge \\gamma\\mathbb{E}\\left[\\int_0^{\\tau_N}\\hspace{-2mm}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right]+\\mathbb{E}\\left[\\int_0^{\\tau_N} \\hspace{-2mm}Y_t\\Big(-b\\gamma x_\\varepsilon(t)+\\frac{\\sigma^2\\gamma^2 x_\\varepsilon^2(t)}{2}\\Big)dt\\right].\n\\end{align*}\n\nNote that the term in the parenthesis above is nonnegative if $\\gamma x_\\varepsilon(t) \\ge 2b\/\\sigma^2$, and greater than or equal to $-b^2\/2\\sigma^2$ in any case. Hence, \n\\[\ne^{\\gamma x_2}-e^{\\gamma x_1}\\ge \\gamma\\mathbb{E}\\left[\\int_0^{\\tau_N}\\hspace{-2mm}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right]-\\frac{b^2e^{\\frac{2b}{\\sigma^2}}}{2\\sigma^2}\\mathbb{E}\\left[\\int_0^{\\tau_N} \\hspace{-2mm}e^{-\\rho t}dt\\right].\n\\]\n\nLetting $N\\to\\infty$  we get\n\\begin{equation}\\label{coerce}\ne^{\\gamma x_2}-e^{\\gamma x_1}\\ge \\gamma\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}\\hspace{-2mm}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right]-\\frac{b^2e^{\\frac{2b}{\\sigma^2}}}{2\\sigma^2}\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon} \\hspace{-2mm}e^{-\\rho t}dt\\right].\n\\end{equation}\n\nWith $\\gamma$ still at our disposal,  to show (\\ref{suffu}) it  suffices to control the term $\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon} e^{-\\rho t}dt\\right]$\nby $\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right]$. \n\\vskip.05in\n\nSince without loss of generality we may assume that $\\varepsilon<A$,  Lemma \\ref{Vdec} and (\\ref{dppe}) give\n\\[\n0\\le V(x_1)-V(x_2)-\\varepsilon(x_2^2-x_1^2)\\le \\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}\\ln u_\\varepsilon(t)\\ dt\\right]-\\rho V(x_2)\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}\\, dt\\right].\n\\]\n\nJensen's inequality then implies that\n\\[\n\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right] \\ge e^{\\rho V(x_2)}\\ \\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}\\, dt\\right],\n\\]\nand (\\ref{coerce}) gives, with $C(x_2)=\\frac{b^2}{2\\sigma^2}e^{\\frac{2b}{\\sigma^2}-\\rho V(x_2)}$,\n\\begin{equation}\\label{coerce2}\n\\gamma e^{\\gamma x_2}(x_2-x_1)\\ge e^{\\gamma x_2}-e^{\\gamma x_1}\\ge (\\gamma-C(x_2))\\ \\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right].\n\\end{equation}\n\nEven though (\\ref{suffu}), and hence the assertion of the Theorem, follows now with a suitable choice of $\\gamma$, we will optimize the preceding inequality for later use. \nChoosing $\\gamma=q(x_2)\/x_2$ in \\eqref{coerce2}, where\n\\begin{equation}\\label{qx2}\nq(x_2)=\\frac{x_2C(x_2)}{2}+\\sqrt{\\left(\\frac{x_2C(x_2)}{2}\\right)^2+x_2C(x_2)},\n\\end{equation}\nwe obtain\n\\[\n\\mathbb{E}\\left[\\int_0^{\\tau_\\varepsilon}e^{-\\rho t}u_\\varepsilon(t)\\, dt\\right] \\le (x_2-x_1)\\big(1+q(x_2)\\big)e^{q(x_2)}.\n\\]\n\nLetting $\\varepsilon \\rightarrow 0$, (\\ref{DVcont}) yields\n\\begin{equation}\\label{DVlower}\n\\frac{V(x_2)-V(x_1)}{x_2-x_1}\\ge -\\big(1+q(x_2)\\big)e^{q(x_2)-1-\\rho V(x_2)},\n\\end{equation}\nthe claim now follows.\\hfill$\\Box$ \\\\[2mm]\n\n\\noindent\n{\\em Proof of the  upper bound, part (ii):}  In view of Lemma \\ref{Vdec}, it suffices to assume that $x_2\\le b$, since otherwise we have\n\\[\nV(x_2)-V(x_1)\\le -A(x_2^2-x_1^2)< -Ab(x_2-x_1).\n\\]\n\nFor a positive constant $c$, choose  a $u_c\\in\\mathfrak{U}$ that is constant an equal to $c$ up to time $\\tau_c=\\tau_{u_c}$.\nThen, Lemma \\ref{specialdpp} yields\n\\[\nV(x_1)\\ge \\frac{\\ln c-x_2^2}{\\rho}(1-\\mathbb{E}\\big[e^{-\\rho\\tau_c}\\big])+\\mathbb{E}\\big[{e^{-\\rho\\tau_c}}\\big] V(x_2),\n\\]\nor equivalently,\n\\begin{equation}\n\\big(V(x_2)-V(x_1)\\big)\\mathbb{E}\\big[e^{-\\rho\\tau_c}\\big]\\le -\\big(\\ln c-\\rho V(x_1)-x_2^2)\\ \\mathbb{E}\\left[\\int_0^{\\tau_c}e^{-\\rho t}\\, dt\\right].\n\\label{ub}\n\\end{equation}\n\nConsider now the solution $x_c(\\cdot)$ to (\\ref{sldyn}) with $x(0)=x_1$ and control $u_c$. Applying It\\^o's formula to $e^{-\\rho t}x_c(t)$, \nfollowed by the optional stopping theorem for the bounded stopping time $\\tau_N=\\tau_c\\wedge N$, we get\n\\begin{align}\n\\mathbb{E}\\big[e^{-\\rho \\tau_N}x_c(\\tau_N)\\big]-x_1&=\\mathbb{E}\\left[\\int_0^{\\tau_N} e^{-\\rho t}\\big(c-(b+\\rho)x_c(t)+\\frac{x_c^2(t)}{1+x_c^2(t)}\\big)\\, dt\\right].\n\\label{lastone}\n\\end{align}\n\nThe leftmost term of \\eqref{lastone} is equal to \\(x_2\\mathbb{E}\\big[e^{-\\rho \\tau_c};\\tau_c\\le N\\big]+e^{-\\rho N}\\mathbb{E}\\big[x_c(\\tau_N); \\tau_c>N\\big]\\). \n\\vskip.05in\n\nOn the other hand, since we have assumed that $x_2\\le b$, we have $x_c(t)\\le b$ up to time $\\tau_c$. Thus, the right hand side \nof \\eqref{lastone} is bounded by \\( \\mathbb{E}\\left[\\int_0^{\\tau_N}e^{-\\rho t} c\\ dt\\right]\\). \n\\vskip.05in\n\nLetting $N\\to\\infty$ in $\\eqref{lastone}$, by the monotone convergence theorem,  we have\n\\[\nx_2 \\mathbb{E}\\big[e^{-\\rho \\tau_c}\\big]-x_1\\le c\\,\\mathbb{E}\\big[\\int_0^{\\tau_c} e^{-\\rho t}dt\\big] \\Longleftrightarrow    (x_2-x_1)\\mathbb{E}\\big[e^{-\\rho\\tau_c}\\big]\\le (c+\\rho x_1)\\ \\mathbb{E}\\big[\\int_0^{\\tau_c} e^{-\\rho t}dt\\big].\n\\]\n\nSubstituting this in (\\ref{ub}) and choosing $\\ln c=\\rho V(x_1)+1+x_2^2$, we find\n\\begin{equation}\\label{lowb}\nV(x_2)-V(x_1)\\le -(x_2-x_1)\\left(e^{\\rho V(x_1)+1+x_2^2}+\\rho x_1\\right)^{-1} \\!\\!\\!.\n\\end{equation}\n\nThe assertion now follows letting $C=Ab\\wedge \\left(e^{\\rho V(0)+1+b^2}+\\rho b\\right)^{-1}>0$. \\hfill$\\Box$ \\\\[2mm]\n\n\\noindent\n{\\em Proof of part (iii):} It follows from (\\ref{lowb}) that, for any $\\varepsilon\\in(0,b]$,\n\\[\n\\frac{V(\\varepsilon)-V(0)}{\\varepsilon}\\le -e^{-\\rho V(0)-1-\\varepsilon^2}.\n\\]\n\nLetting $\\varepsilon\\to 0$ we get\n\\[\n\\limsup_{\\varepsilon\\to 0}\\frac{V(\\varepsilon)-V(0)}{\\varepsilon}\\le -e^{-\\rho V(0)-1},\n\\]\nwhile (\\ref{DVlower}) gives\n\\[\n\\frac{V(\\varepsilon)-V(0)}{\\varepsilon}\\ge -\\big(1+q(\\varepsilon)\\big)e^{q(\\varepsilon)-1-\\rho V(\\varepsilon)}.\n\\]\n\nLetting $\\varepsilon\\to 0$ and noting that $q(\\varepsilon)\\to 0$, and $\\ V(\\varepsilon)\\to V(0)$, we have\n\\[\n\\liminf_{\\varepsilon\\to 0}\\frac{V(\\varepsilon)-V(0)}{\\varepsilon}\\le -e^{-\\rho V(0)-1},\n\\]\nwhich proves the claim. \\hfill$\\Box$ \\\\[2mm]\n\nWe conclude observing that since $V\\in C\\left([0,\\infty)\\right)$, the general dynamic programming principle is also satisfied. For a proof we refer to \\cite{Tou}.\n\n\n\n\n\\section{Viscosity solutions and the Hamilton--Jacobi--Bellman equation}\n\nSince the Hamiltonian (\\ref{HJB-unbounded}) can take infinite values  we have a singular stochastic control problem and the welfare  function (\\ref{ihvf}) should satisfy the proper variational inequality;  see \\cite{FS},\nSection VIII and \\cite{pham}, Section 4. The proof of the  next Lemma, except for the treatment of the boundary conditions, follows the lines of Proposition 4.3.2 of  \\cite{pham}. \n\\begin{lemma}\\label{singular_control_problem}\nIf $\\sigma^2 < \\rho +2b$, the welfare function $V$ defined by (\\ref{ihvf}) is a continuous constrained viscosity solution of \n\\begin{equation}\\label{HJB-var}\n  \\min \\Big[ \\rho V -\\sup_{u\\in \\mathbb{R}^+ } \\left( \\dfrac{1}{2}\\sigma^{2}x^{2}V_{xx}+( u-bx+\\dfrac{x^{2}}{x^{2}+1})V_x +\\ln u-x^{2}\\right), -V_x   \\Big] =0, \\quad \\text{ in }  [0,\\infty).\n \\end{equation}\n\\end{lemma}\n\\noindent\n{\\em Proof:} That $V$ is a viscosity solution in $(0, \\infty)$ follows as in \\cite{pham}, so we omit the details. \n\\vskip.05in\n\nHere we briefly discuss the subsolution property at $x=0$. Let $\\phi$ be a test function \nsuch that $V-\\phi$ has a maximum at $x=0$ with $V(0)-\\phi(0)=0$, and, proceeding by contradiction,  we assume that \n\\begin{align}\\label{subat0}\n  \\rho \\phi(0) -\\sup_{u\\in \\mathbb{R}^+ } G\\big(0,u,\\phi'(0),\\phi''(0)\\big) >0 \\,\\, \\text{ and }\\,\\, -\\phi'(0) >0. \n\\end{align}\n\nSince  $-\\phi'(0)>0$, the supremum in the above inequality is finite, the Hamiltonian takes the standard form, and the first inequality in \\eqref{subat0} becomes\n\\[\n\\rho \\phi(0)+1+\\ln\\big(-\\phi'(0)\\big)>0.\n\\]\n\nOn the other hand, in view of  Proposition \\ref{vprop} (iii),  we have \\(\\rho V(0)+1+\\ln\\big(-V'(0)\\big)=0\\), hence \\(V'(0)>\\phi'(0)\\), contradicting that \\(V-\\phi\\) has a maximum at $x=0$.\n\\vskip.05in\n\nWe have now obtained all the necessary material for the proof of Theorem \\ref{constrained_vs}.\n\\vskip.05in\n\n\\noindent\n{\\em  Proof:} \nThe fact that (\\ref{ihvf}) is a continuous constrained viscosity solution of the equation ${\\rm (OHJB)}$  is a direct consequence of the above Lemma  and the fact that inequality  (\\ref{DVbounds})  \nimplies that $p\\leq-C$ for any $p\\in D^\\pm V(x)$, with $x\\in (0, \\infty)$. The regularity of $V$ in $(0,\\infty)$ follows from the classical results for uniformly elliptic \noperators. \\hfill$\\Box$ \\\\\n\nDue to the extra regularity of   the welfare function,   the following verification equation holds in $(0, \\infty)$, for any optimal pair $(\\overline{u}(\\cdot)),\\overline{x}(\\cdot))$, \n\\begin{align*}\n\\rho V(\\overline{x}(t))=& \\sup_{u\\in U } G(\\overline{x}(t),u,V_x(\\overline{x}(t)),V_{xx}(\\overline{x}(t))) \\\\\n=&\\left( \\dfrac{\\overline{x}^{2}(t)}{\\overline{x}^{2}(t)+1}-b\\overline{x}(t)\\right)V_x(\\overline{x}(t))-\\Big( \\ln(-V_x(\\overline{x}(t)))+\\overline{x}^{2}(t)+1\\Big) \\\\\n& +\\dfrac{1}{2}\\sigma^{2}\\overline{x}^{2}(t)V_{xx}(\\overline{x}(t)), \n\\ \\ \\ \\ \\  \\ t\\in[0,\\infty)- a. e.\\ ,\\ \\ \\mathbb{P}- a.e.;\n\\end{align*}\n\\noindent\n see   \\cite{FS} and \\cite{JYXZ}.\n\\vskip.05in\n\nNext we prove the proper  comparison principle for ${\\rm (OHJB)}$. The proof is along the lines of  the strategy in \\cite{HI}, \nwhere given a subsolution $u$ and a supersolution $v$ \nof (OHJB),  $u-v$ is a subsolution of the corresponding linearized equation. Then, one concludes by comparing $u-v$ with the appropriate supersolution of the linearized equation; see also \n\\cite{DaLL} and \\cite{Za}. The difference with the existing results is that, \ndue to the presence of the logarithmic term,\nthe commonly used  functions of simple polynomials do not yield a supersolution of the equation.  \n\\vskip.05in\n\nHaving in mind that we are looking for a viscosity solution that is strictly decreasing \nand satisfies (\\ref{DVbounds}),  we prove the following lemma. \n\n\\noindent\n\\begin{lemma} \\label{intermed} Suppose  $u$, $v$ satisfy the assumptions of Theorem \\ref{comparison}.\nThen $\\psi=u-v$ is a subsolution of \n\\begin{equation}\\label{u-v}\n \\rho \\psi +bxD\\psi-\\left(1+c^*\\right)|D\\psi| -\\frac{1}{2}\\sigma^2x^2 D^2 \\psi = 0 \\,\\, \\mbox{ in } \\,\\, [0, \\infty).\n\\end{equation}\n\\end{lemma} \n\\noindent\n{\\em Proof:}\nLet  $\\bar{x}\\ge 0$ a maximum point of  $\\psi -\\phi$ for some smooth function $\\phi$ and set, following \\cite{So},\n$$\\theta(x,y)=\\phi(x)+\\frac{(x-y+\\varepsilon L)^2}{\\varepsilon}+\\delta(x-\\bar{x})^4,$$\nwhere $L,\\delta$ are positive constants.\n\nThe assumptions on $u,v$ imply that the function $(x,y) \\mapsto u(x)-v(y)-\\theta(x,y)$ is bounded from above and achieves its maximum at, say, $(x_{\\varepsilon},y_{\\varepsilon})$. It follows that  $x \\mapsto u(x)-v(y_{\\varepsilon})-\\theta(x,y_{\\varepsilon})$ has a local maximum at\n$x_{\\varepsilon}$ and  $y \\mapsto v(y)-u(x_{\\varepsilon})+\\theta(x_{\\varepsilon},y)$ has a local minimum at $y_{\\varepsilon}$. Moreover, (see Proposition 3.7 in \\cite{CIL}), as  $\\varepsilon \\rightarrow 0$, \n\\begin{equation}\\label{doubling}\n \\frac{|x_{\\varepsilon}-y_{\\varepsilon}|^2}{\\varepsilon} \\rightarrow 0 ,\\, x_\\varepsilon\\rightarrow \\bar{x}, \\, \\mbox{ and }\nu(x_{\\varepsilon})-v(y_{\\varepsilon}) \\rightarrow \\psi(\\bar{x}).\n\\end{equation}\n\nThe inequalities\n\\[\nu(x_\\varepsilon)-v(y_\\varepsilon)-\\theta(x_\\varepsilon,y_\\varepsilon)\\le \\psi(\\bar{x})-\\phi(\\bar{x})+v(x_\\varepsilon)-v(y_\\varepsilon)-\\frac{|x_{\\varepsilon}-y_{\\varepsilon}+\\varepsilon L|^2}{\\varepsilon}-\\delta(x_\\varepsilon-\\bar{x})^4\n\\]\nand \n\\[\nu(x_\\varepsilon)-v(y_\\varepsilon)-\\theta(x_\\varepsilon,y_\\varepsilon)\\ge u(\\bar{x})-v(\\bar{x}+\\varepsilon L)-\\theta(\\bar{x},\\bar{x}+\\varepsilon L)\\ge \\psi(\\bar{x})-\\phi(\\bar{x})\n\\]\ntogether imply that \n\\[\n\\frac{|x_{\\varepsilon}-y_{\\varepsilon}+\\varepsilon L|^2}{\\varepsilon}+\\delta(x_\\varepsilon-\\bar{x})^4\\le v(x_\\varepsilon)-v(y_\\varepsilon).\n\\]\n\nSince $v$ is decreasing we must have $y_\\varepsilon>x_\\varepsilon.$ In particular, $y_\\varepsilon\\in(0,\\infty)$.\n\nTherefore, setting $p_{\\varepsilon}=2\\frac{x_{\\varepsilon}-y_{\\varepsilon}+\\varepsilon L}{\\varepsilon}$  and $q_\\varepsilon= \\phi_x(x_{\\varepsilon})+4\\delta(x_\\varepsilon-\\bar{x})^3$, Theorem 3.2 in \\cite{CIL} implies that,\nfor every $\\alpha >0$, there exist $X, \\, Y \\in \\mathbb{R}$ such that\n\\begin{equation}\\label{sub-super}\n \\rho u(x_{\\varepsilon})-H(x_{\\varepsilon},p_\\varepsilon+q_{\\varepsilon},X) \\leq 0 \\mbox{   \\,\\,and\\,\\,  } \\rho v(y_{\\varepsilon})- H(y_{\\varepsilon},p_{\\varepsilon},Y) \\geq 0\n\\end{equation}\nand \n\\begin{equation}\\label{jet-inequality}\n  -(\\frac{1}{\\alpha}+\\|M\\|) I \\leq \\left(\n\\begin{array}{cc}\nX & 0 \\\\\n0 & -Y  \n\\end{array} \\right) \\leq M+\\alpha M^2\n\\end{equation}\nwith $M=D^2\\theta(x_{\\varepsilon},y_{\\varepsilon}).$\n\nBy subtracting the two inequalities in  (\\ref{sub-super}) we obtain \n\\begin{align}\\label{comp-inequ_1}\n \\rho & u(x_{\\varepsilon})-\\rho v(y_{\\varepsilon}) +b(x_{\\varepsilon}-y_{\\varepsilon})p_{\\varepsilon}+\\Big(\\frac{y_{\\varepsilon}^2}{y_{\\varepsilon}^2+1}-\\frac{x_{\\varepsilon}^2}{x_{\\varepsilon}^2+1}\\Big)p_{\\varepsilon}\n - \\Big(\\frac{x_{\\varepsilon}^2}{x_{\\varepsilon}^2+1}-bx_{\\varepsilon} \\Big) q_\\varepsilon  \\\\\n & +\\ln{(-p_{\\varepsilon}- q_\\varepsilon)}- \\ln{(-p_\\varepsilon)} \\nonumber + x_{\\varepsilon}^2- y_{\\varepsilon}^2  -\\frac{1}{2}\\sigma^2x_{\\varepsilon}^2 X + \\frac{1}{2}\\sigma^2y_{\\varepsilon}^2 Y \\leq 0. \\nonumber\n\\end{align}\n\nOur assumption on $u$ implies that $p_\\varepsilon+q_\\varepsilon\\le -1\/c^*$. Thus, the difference of the logarithmic terms in the above inequality can be estimated from below as\n\\[\n\\ln\\Big(\\frac{p_\\varepsilon+q_\\varepsilon}{p_\\varepsilon}\\Big)\\ge \\frac{q_\\varepsilon}{p_\\varepsilon+q_\\varepsilon}\\ge -c^*|q_\\varepsilon|,\n\\]\nand inequality (\\ref{comp-inequ_1}) gives\n\\begin{align}\\label{comp-inequ_2}\n\\frac{1}{2}\\sigma^2x_{\\varepsilon}^2 X + \\frac{1}{2}\\sigma^2y_{\\varepsilon}^2 Y&\\ge\\rho u(x_{\\varepsilon})-\\rho v(y_{\\varepsilon}) + bx_\\varepsilon q_\\varepsilon- (1+c^*)|q_\\varepsilon| +\\nonumber\\\\\n &\\qquad p_\\varepsilon(x_\\varepsilon-y_\\varepsilon)\\Big(b-\\frac{x_\\varepsilon+y_\\varepsilon}{(1+x_\\varepsilon^2)(1+y_\\varepsilon^2)}\\Big)+x_\\varepsilon^2-y_\\varepsilon^2\n\\end{align}\n\nOn the other hand, the right-hand-side in (\\ref{jet-inequality}) yields\n\\begin{equation}\\label{X-Y}\n \\frac{1}{2}\\sigma^2x_{\\varepsilon}^2 X - \\frac{1}{2}\\sigma^2y_{\\varepsilon}^2 Y \\leq \\frac{1}{2}\\sigma^2x_{\\varepsilon}^2 \\big(\\phi_{xx}(x_{\\varepsilon})+12\\delta(x_\\varepsilon-\\bar{x})^2\\big)\n +\\frac{\\sigma^2}{\\varepsilon}(x_{\\varepsilon}-y_{\\varepsilon})^2+m(\\frac{\\alpha}{\\varepsilon^2}), \n\\end{equation}\nwith $m$ a modulus of continuity independent of $\\alpha$, $\\varepsilon$. \n\\vskip.05in\n\nBy combining \\eqref{comp-inequ_2} with \\eqref{X-Y}, we conclude the proof taking first $\\alpha  \\rightarrow 0$, then $\\varepsilon \\rightarrow 0$ and using \\eqref{doubling}. \\hfill$\\Box$ \\\\\n\n\\noindent\n We continue with the \\\\[1mm]\n \\noindent\n{\\it Proof of Theorem \\ref{comparison}:} The main step is the construction of a solution of the linearized equation. For this, we  consider the ode \n\\begin{equation}\\label{ode1}\n\\rho w +\\big(bx-(1+c^*)\\big)w'-\\frac{1}{2}\\sigma^2x^2w''=0,\n\\end{equation}\nwhich has a solution of the form \n\\begin{equation}\\label{sol}\n w(x)=x^{-k}\\mathcal{J}(\\frac{2+2c^*}{\\sigma^2x}),\n\\end{equation}\nwhere $k$ is a root of  \n\\begin{equation}\\label{root}\n k^2+\\Big(1+\\frac{2b}{\\sigma^2}\\Big)k-\\frac{2\\rho}{\\sigma^2}=0\n\\end{equation}\nand   $\\mathcal{J}$ a solution of the degenerate hypergeometric equation \n\\begin{equation}\\label{confluent}\nxy''+(\\tilde{b}-x)y'-\\tilde{a}y=0\n\\end{equation}\nwith $\\tilde{a}=k$ and $\\tilde{b}=2(k+1+b\/\\sigma^2)$.\n\nSince we are looking for a solution of \\eqref{ode1} with superquadratic growth at $+\\infty$, we choose $k$ to be the negative root of \\eqref{root}. The assumption $\\sigma^2 < \\rho+2b$ implies $-k>2$.\n\nWe further choose $\\mathcal{J}$ to be the Tricomi solution of \\eqref{confluent} which satisfies\n\\[\n\\mathcal{J}(0)>0\\qquad\\text{and} \\qquad \\mathcal{J}(x)=x^{-k}\\big(1+\\frac{2\\rho}{\\sigma^2 x}+o(x^{-1})\\big)\\quad\\text{as }x\\to\\infty.\n\\]\n\nWith this choice, the function $w$ defined in \\eqref{sol} for $x>0$ and by continuity at $x=0$, satisfies $w(0), w'(0) >0$ and $w(x)\\sim \\mathcal{J}(0) x^{-k}$, as $x\\to\\infty$. \n\nNote that $w$ is increasing in $[0,\\infty)$ since it would otherwise have a positive local maximum and this is impossible by \\eqref{ode1}. In particular, $w$ satisfies \\eqref{u-v}.\n\nSet now $\\psi=u-v$ and consider $\\epsilon>0$. Since $\\psi-\\epsilon w<0$  in a neighborhood of infinity, there exists $x^\\epsilon \\in [0,\\infty)$ such that\n\\[\n\\max_{x\\ge 0} \\big(\\psi(x)-\\epsilon w(x)\\big)=\\psi(x^\\epsilon)-\\epsilon w(x^\\epsilon).\n\\]\n\nBy Lemma \\ref{intermed} $\\psi $ is a subsolution of \\eqref{u-v}. We now use $\\epsilon w$ as a test function to find tha\n\\begin{equation*}\n0\\ge \\rho \\psi({x}^\\epsilon) +\\epsilon b{x}^\\epsilon w({x}^\\epsilon)-\\epsilon\\left(1+c^*\\right)|w'({x}^\\epsilon))| -\\frac{1}{2}\\sigma^2{(x^\\epsilon)}^2 w''({x}^\\epsilon) = \\psi({x}^\\epsilon)-\\epsilon w(x^\\epsilon).\n\\end{equation*}\n\nHence, $\\psi(x)\\le \\epsilon w(x)$ for all $x\\in [0,\\infty)$. Since $\\epsilon$ is arbitrary, this proves the claim. \\hfill$\\Box$ \\\\[1mm]\n\nThe stability property of viscosity solutions yields the following theorem. \n\n\\begin{thm}\\label{constrained_vs_det}  As  $\\sigma \\rightarrow 0$, the welfare function $V$ defined by (\\ref{ihvf}) converges locally uniformly to the constrained viscosity solution $V^{(d)}$   of the deterministic \nshallow lake equation in $[0, \\infty)$,\n$$\\rho V^{(d)}=\\left( \\dfrac{x^{2}}{x^{2}+1}-bx\\right)V_{x}^{(d)}-\\Big( \\ln(-V_{x}^{(d)})+x^{2}+1\\Big). \\leqno\\mathrm{(OHJB_d)}$$\n\\end{thm}\n\\vskip.05in\nWe next prove Theorem  \\ref{asymptotic} that describes the asymptotic behaviour of $V$ as $x \\rightarrow \\infty$. The proof is based on a scaling argument and the stability properties of the viscosity solutions.\\\\\n\n\\noindent\n{\\it Proof of Theorem \\ref{asymptotic}: } We write  $V$ as \n\\begin{equation*}\n V(x)= -A\\left(x+\\frac{1}{b+\\rho}\\right)^2-\\frac{1}{\\rho}\\ln \\left(2A(x+\\frac{1}{b+\\rho})\\right)+K+v(x). \n\\end{equation*}\n\nStraightforwad calculations yield that   $v$ is a viscosity solution in $(0, \\infty)$ of the equation\n\\begin{multline}\n\\rho v+\\left(bx-\\dfrac{x^{2}}{x^{2}+1}\\right)v'+\\ln\\left(1+\\frac{1-\\rho\\big(x+\\frac{1}{b+\\rho}\\big)v'}{2A\\rho\\left(x+\\frac{1}{b+\\rho}\\right)^2} \\right)-\\dfrac{1}{2}\\sigma^{2}x^{2}v''+f=0,\n\\end{multline}\nwhere \n\\[\nf(x)=\\frac{b+\\frac{\\sigma^2}{2}+(b+\\rho)\\frac{x^2}{1+x^2}}{\\rho\\big(1+x(b+\\rho)\\big)}+\\frac{\\sigma^2x(b+\\rho)}{2\\rho\\big(1+x(b+\\rho)\\big)^2}-2A\\frac{\\big(1+x(b+\\rho)\\big)}{1+x^2}.\n\\]\n\nNote $f$ is smooth on $[0,\\infty)$ and vanishes as $x\\to\\infty$.\n\\vskip.05in\n\n Let $v_\\lambda(y)=v(\\frac{y}{\\lambda})$ and observe that, if $v_\\lambda(1) \\rightarrow 0$ as $\\lambda \\rightarrow 0$, then $v(x) \\rightarrow 0$ as $x \n \\rightarrow \\infty$. It turns out that  $v_\\lambda$ solves\n \\begin{multline*}\n\\rho v_\\lambda+ \\left(bx-\\dfrac{\\lambda x^{2}}{x^{2}+\\lambda^2}\\right)v'_\\lambda+\\ln\\left(1+\\frac{\\lambda^2\\big(1-\\rho\\big(x+\\frac{\\lambda}{b+\\rho}\\big)v'_\\lambda\\big)}{2A\\rho\\left(x+\\frac{\\lambda}{b+\\rho}\\right)^2} \\right)-\\dfrac{1}{2}\\sigma^{2}x^{2}v''_\\lambda+f\\big(\\frac{x}{\\lambda}\\big)=0.\n \\end{multline*}\n\nSince, by (\\ref{Vbounds}) $v_\\lambda $ is uniformly bounded, we consider the half-relaxed limits $v^*(y)=\\limsup_{x \\rightarrow y, \\lambda\\rightarrow 0}v_\\lambda(x)$ and $v_*(y)=\\liminf_{x \\rightarrow y, \\lambda\\rightarrow 0}v_\\lambda(x)$ \n in $(0, \\infty)$, which are (see \\cite{BP}) respectively sub- and super-solutions of \n\\begin{equation}\n \\rho w +byw'-\\frac{1}{2} \\sigma^2 y^2 w''=0.\n\\end{equation}\n\nIt is easy to check that for any $y>0$ we have $v^*(y)=\\limsup_{x\\to\\infty}v(x)$ and $v_*(y)=\\liminf_{x\\to\\infty}v(x)$. \n\\vskip.05in\nThe subsolution property of $v^*$ and the supersolution property of $v_*$ give \n\\[\n\\limsup_{x\\to\\infty}v(x)\\le 0\\le\\liminf_{x\\to\\infty}v(x)\\le \\limsup_{x\\to\\infty}v(x).\n\\]\\qed\n\n\n\n\\section{A numerical scheme and optimal dynamics}\n\n A general argument to prove the convergence of monotone schemes for viscosity solutions of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations has been introduced in \\cite{BS}. \n Their methodology  has been extensively\n used to approximate solutions to first-order equations, see for example  \\cite{RT},  \\cite{Se2}, \\cite{KMS}, \\cite{Qi}. \n \\vskip.05in\n \n On the other hand, it is not always possible to construct  monotone schemes for second-order equations in their full generality. However, various types of nonlinear second-order equations have been approximated\n via monotone schemes based on \\cite{BS}; see, for example,   \\cite{OF}, \\cite{BJ}, \\cite{BZ}, \\cite{FO}.\n \\vskip.05in\n \n Next, following \\cite{KosZoh} which considered the deterministic problem, we construct a monotone finite difference scheme to approximate numerically the welfare function and recover numerically the stochastic optimal dynamics. \n\\vskip.05in  \n\nLet\n$\\Delta x$ denote the step size of a uniform partition $0 = x_0 < x_1 < \\ldots < x_{N-1} < x_N = l$ of $[0, l]$ for \n$l > 0$ sufficiently large. Having in mind (\\ref{DVbounds}),  if $V_{i}$ is the approximation of $V$ at $x_i$,  we employ a backward finite difference discretization  to approximate the first derivative in the linear term of the (OHJB),  \na forward finite difference discretization for the derivative in the logarithmic term and a central finite difference scheme to approximate the second derivative. \n\n\\vskip.05in\nThese considerations yield, for $i = 1,\\ldots, N-1$,   the approximate equation\n\\begin{multline}\\label{DHJB}\n V_i - \\frac{1}{\\rho} \\Big(\\frac{x_i^2}{x_i^2+1} - bx_i \\Big) \\frac{ V_{i} - V_{i-1} }{ \\Delta x} + \\frac{1}{\\rho}\n\\left[x_i^2 + 1 +\\ln \\left( - \\frac{V_{i+1} - V_i}{\\Delta x}\\right)\\right] \\\\\n- \\frac{\\sigma^2}{2\\rho}\n\\frac{ V_{i+1} + V_{i-1} - 2 V_i}{(\\Delta x)^2}=0.\n\\end{multline}\n\nSetting \n\\begin{multline}\n g(x,w,c,d)=\\Big[(\\Delta x)^2 - \\frac{1}{\\rho}\\Big(\\frac{x^2}{x^2+1} - bx \\Big) \\Delta x +\\frac{\\sigma^2}{\\rho}\\Big]w+ \\\\ \n \\frac{1}{\\rho}\n(x^2 + 1 ) (\\Delta x)^2+\\frac{1}{\\rho}(\\Delta x)^2\\ln \\left( - \\frac{c - w}{\\Delta x}\\right)+\\frac{1}{\\rho}\\Delta x \\Big(\\frac{x^2}{x^2+1} - bx \\Big) d- \\frac{\\sigma^2}{2\\rho}\n( c + d ),\n\\end{multline}\n the numerical  approximation of $V$ satisfies \n\\begin{equation}\\label{fdscheme}\n g(x_i,V_i,V_{i+1},V_{i-1})=0, \\mbox{\\,\\,\\,for\\,\\, \\, i = 1,\\ldots, N-1 },\n\\end{equation}\nand the consistency is immediate.\n\\vskip.05in\n\nFor the monotonicity we observe that for  two different approximation grid vectors $(U_0, \\ldots, U_N)$ and $(V_0, \\ldots, V_N)$ with $U_i \\geq V_i$ and $U_i=V_i=w$, we have \n\\begin{equation}\n g(x_i,w,U_{i+1},U_{i-1})\\leq g(x_i,w,V_{i+1},V_{i-1}),\n\\end{equation}\nprovided $\\Delta x (\\frac{x^2}{x^2+1} - bx)\\leq \\sigma^2\/2$. This condition is satisfied if $b \\geq 0.5$ or if we take $\\Delta x\\leq\\sigma^2\/2$.\n\\vskip.05in\nSince the welfare function solves a state constraint problem the equation is satisfied on the left boundary point.\n\\vskip.05in\n\nIt follows that  the  numerical scheme is monotone in the sense of \\cite{BS} and converges to the unique constrained\nviscosity solution. \n\\vskip.05in\n\n Since the computational  domain of the problem is finite, a boundary condition has to be imposed at $x=l$, for $l$ sufficiently large, by exploiting  the asymptotic behaviour of the welfare function $V$ as $x\\to+\\infty$. The boundary condition at\nthe right endpoint $x_N$ is provided by  the asymptotic estimate \\eqref{asympto_behav}.  \n\\vskip.05in\n\nThe scheme above suggests a numerical algorithm for the computation of optimal dynamics governing the shallow lake problem. In this direction, the nondegeneracy of the shallow lake equation in $(0,\\infty)$ induces  extra regularity for the function $V$ in $(0,\\infty)$. Hence, the optimal dynamics for the shallow lake problem  are described by \n \\begin{equation}\\label{opt_dyn}\n \\left\\{ \\begin{array}{l} d\\bar{x}(t)=\\left(-\\dfrac{1}{V'(\\bar{x}(t))} -b \\bar{x}(t)+\\dfrac{\\bar{x}^{2}(t)}{\\bar{x}^{2}(t)+1}\\right)dt+\\sigma \\bar{x}(t)dW(t), \\\\\n   \\bar{x}(0)=x       \n  \\end{array} \\right.\n\\end{equation}\n \nUsing the numerical representation  of $V$ via \\eqref{fdscheme} and  properly discretizing the SDE (\\ref{opt_dyn}), we can  reconstruct numerically the optimal dynamics. \nThis is a direct approach to investigate numerically the stochastic properties of the optimal dynamics of the shallow lake problem for the various parameters $\\rho$, $b$, $c$, $\\sigma$ of the problem.\n\\vskip.05in\n\nThe exact numerical algorithm for the computation of the constrained viscosity solution along with the numerical study of the optimal dynamics and their stochastic properties \n for various $\\sigma$'s will be presented elsewhere. \n \n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\nWe know that in Hamiltonian systems a dynamic function $f(q,p)$ develops in time according to\n\\be\n\\label{1}\n\\dot{f} = [f,H]_{P.B} \\, .\t\t\t\t\t\t\t\n\\ee\nIn classical mechanics, we are used to studying the time development of a physical system by employing the Hamiltonian and the \nPoisson brackets $(P.B.)$. \nBut one can extend the latter, using the so-called Lie brackets $(L.B.)$ to great advantage.\nA Lie algebra is an algebraic structure in which the connections between its elements  \nare determined with the help of $L.B$. \nIn order to do this, we begin with the set $\\cD$\nof all dynamic functions $A(q,p), B(q,p), \\ldots \\epsilon \\cD$ and require that $\\alpha A, \\alpha A + \\beta B, AB$ and $A^{-1}\n(\\alpha, \\beta$ constants) be defined and likewise should belong to $\\cD$. \nWhen we consider elements like $A, B, \\ldots$ as building blocks, then one can construct a large class of dynamic functions, \ni.e., polynomials, analytic functions, meromorphic functions, Fourier series, etc. One only gets a really new characteristic of \n$\\cD$ if one imposes an algebraic structure by way of the Lie bracket $[A,B] \\epsilon \\cD$, where $L.B.$\nsatisfies the following relations:\n\\begin{align}\n \\label{2}\n{[A,B]} & = -[B,A] \\, , \\\\\n \\label{3}\n [\\alpha A,B] & = \\alpha [A,B] \t\\, , \\\\\n \\label{4}\n [\\alpha A + \\beta B,C] & =  \\alpha [A,C] + \\beta [B,C] \\, , \\\\\n\\label{5}\n\t\t\t\t\t[AB,C] & = A[B,C] + B[A,C] \\, .  \t\t\n \\end{align}\n\t\t\tThe operation $L.B.$ is not associative. Instead the Jacobi identity applies:\n\\be\n\\label{6}\n\t\t\t\t\t[[A,B],C] = [[B,C],A] + [[C,A],B]] = 0 \\, .\n\\ee\nA well-known example is given by a 3-dimensional vector space with cross product:\n\\begin{align}        \n\t\t\t\t \\vA \\times \\vB & = - \\vB \\times \\vA \\, , \\nonumber\\\\\n\t\t\t\t(\\alpha \\vA + \\beta \\vB) \\times \\vC & = \\alpha \\vA \\times  \\vC +  \\beta \\vB \\times \\vC \\, , \\nonumber\\\\\n\t\t\t\t\\vA \\times  (\\vB \\times \\vC) + \\vB \\times (\\vC \\times \\vA) + \\vC \\times  (\\vA \\times \\vB) & = 0 \\, .\\nonumber \n\\end{align}\nThe three rules (\\ref{3}, \\ref{4}, \\ref{5}) can be realized with the help of first-order differential operators:\n\\be\n\\label{7}\n[A (q, p), B] = \\sli^N_{a = 1} \\lk \\frac{\\partial A}{\\partial q_a} \\left[ q_a, B \\right] + \\frac{\\partial A}{\\partial p_a} [p_a, B] \\right) \\, .\n\\ee\nThe proof for (\\ref{3}) and (\\ref{4}) is trivial, as is the proof for (\\ref{5}):\n\\begin{align}\n [AB, C] &= \\frac{\\partial (AB)}{\\partial q} [q, C] + \\frac{\\partial (AB)}{\\partial p} [p, C] \\nonumber\\\\\n &= \\lk \\frac{\\partial A}{\\partial q} B + A \\frac{\\partial B}{\\partial q} \\right) [q, C] + \n \\lk \\frac{\\partial A}{\\partial p} B + A \\frac{\\partial B}{\\partial p} \\right) [p, C] \\nonumber\\\\\n &= A \\lk \\frac{\\partial B}{\\partial q} [q, C] + \\frac{\\partial B}{\\partial p} [p, C] \\right) + \n B \\lk \\frac{\\partial A}{\\partial q} [q, C] + \\frac{\\partial A}{\\partial p} [p, C] \\right) \\nonumber\\\\\n &= A [B, C] + B [A, C]  \\, .\\nonumber\n\n\\end{align}\nIf one iterates relation (\\ref{7}), one finds:\n \\begin{align}\n  [A (q, p), B (q, p)] &= \\sli^N_{a = 1} \n \\sli^N_{b = 1} \\Bigg( \\frac{\\partial A}{\\partial q_a} \\frac{\\partial B}{\\partial q_b} \n [q_a, q_b] + \\frac{\\partial A}{\\partial q_a} \\frac{\\partial B}{\\partial p_b} [q_a,  p_b] \n  \\nonumber \\\\\n  &  + \\frac{\\partial A}{\\partial p_a} \\frac{\\partial B}{\\partial q_b} [p_a, q_b] + \\frac{\\partial A}{\\partial p_a} \n  \\frac{\\partial B}{\\partial p_b} [p_a, p_b] \\Bigg) \\, . \n\\label{8}\n \\end{align}\nThe $q$'s and $p$'s here are arbitrary pairs of phase-space variables - not necessarily canonical conjugate variables as they appear \nin the Hamiltonian equations.\n\nThe quantities $[q_a, q_b], [q_a, p_b], [p_a, p_b]$\nare called fundamental $L.B$. If one knows these for all $a, b = 1, \\ldots N$, then one can calculate the Lie brackets in (\\ref{8}).\nIf, however, we define the fundamental L.B. according to\n\\be\n\\label{9}\n[q_i, q_j] = 0 \\, , \\quad [p_i, p_j] = 0 \\, , \\quad [q_i, p_j] = \\delta_{ij} \\, \\quad \\mbox{for all} \\, i, j \\, , \n\\ee\nthe $L.B.$ go over to the $P.B.$:\n\\be\n\\label{10}\n[A, B]_{q, p} = \\sli^N_{i, j = 1} \\lk \\frac{\\partial A}{\\partial q_i} \\frac{\\partial B}{\\partial p_j} - \\frac{\\partial A}{\\partial p_i}\n\\frac{\\partial B}{\\partial q_j} \\right) \\, .\n\\ee\nThe variables $q_i, p_i$ are then said to be canonically conjugate.\nFor a Hamiltonian system $H$, the time evolutionary equations then immediately follow  from (\\ref{8}) for the variables $q$ and $p$:\n\t\t\t\t\t\t\t\t\t\t\t\t\\begin{align}\n\t\t\t\t\t\t\t\t\t\t\t\t \\label{11}\nA = q, B = H &: \\dot{q}_i = [q_i, H] = [q_i, p_j] \\frac{\\partial H}{\\partial p_j} = \\delta_{ij} \\frac{\\partial H}{\\partial p_j}\n= \\frac{\\partial H}{\\partial p_i} \\, , \\\\\n\\label{12}\nA = p, B = H &: \\dot{p}_i = [p_i, H] = [p_i, q_j] \\frac{\\partial H}{\\partial q_j} = - \\delta_{ij} \\frac{\\partial H}{\\partial q_j}\n= \\frac{\\partial H}{\\partial q_i} \\, .\n\t\t\t\t\t\t\t\t\t\t\t\t\\end{align}\nWe now combine the canonical coordinates $q_i$ and the momenta $p_i$ into a new set of $N$ generalized coordinates:\n\\be\n\\label{13}\n                                 z     = (q_1, \\ldots, q_{N\/2}, p_1, \\ldots, p_{N\/2}) \\, .\n\t\t  \t\t\t\t\t\t\t\t       \\ee\n\tThen the canonical $P.B.$ from (\\ref{9}) can be very elegantly written with the aid of the Poisson tensor $\\omega$ as\n\t\\be\n\t\\label{14}\n\t\t\t[z_a, z_b] = \\omega_{ab} \\, , \\quad \\quad    det (\\omega_{ab}) \\neq 0 \\quad    (1 \\, \\mbox{for canonical coordinates})  \t            \t\t\t\t\t\t\t\n\\ee\nwith\n\\begin{align}\n\\label{15}\n\\omega_{ab} = \\,\\,\\,\n  \\begin{blockarray}{cccccc}\n           & \\scriptstyle{q_1}       & \\scriptstyle{p_1}       & \\scriptstyle{q_2}    & \\scriptstyle{p_2}    & \\cdots \\\\\n    \\begin{block}{c(ccccc)}\n    \\scriptstyle{q_1}    & [q_1,q_1] & [q_1,p_1] & \\cdots & \\cdots & \\cdots \\\\\n    \\scriptstyle{p_1}    & [p_1,q_1] & [p_1,p_1] & \\cdots & \\cdots & \\cdots \\\\\n    \\scriptstyle{q_2}    &           &           & \\ddots &        &        \\\\\n    \\scriptstyle{p_2}    &           &           &        & \\ddots &        \\\\\n    \\vdots &           &           &        &        & \\ddots \\\\\n    \\end{block} \n  \\end{blockarray} \n\\,\\,\\,=\\,\\, \n\\begin{pmatrix}\n \\dboxed{\\begin{matrix} 0 & 1 \\\\ -1 & 0 \\end{matrix}} &  & \\\\\n & \\dboxed{\\begin{matrix}0 & 1 \\\\ -1 & 0 \\end{matrix}} & \\\\\n & & \\ddots\n\\end{pmatrix}\n\\end{align}\n\nor\n\\be\n\\label{16}\n\\omega_{ab} =   \\mathrm{diag} \\left[ \\lk \\begin{array}{cc} \n                                 0 & 1 \\\\\n                                 - 1 & 0\n                                \\end{array} \\right) \\, , \\ldots\n                                \\lk\n                                \\begin{array}{cc} \n                                 0 & 1 \\\\\n                                 - 1 & 0\n                                \\end{array} \\right) \\right] \\, .\n\t\t\t\t\t\t\t\t\\ee\nThe inverse of $\\omega_{ab} , \\omega^{ab} = -\\omega_{ba}$, is defined according to\n\\be\n\\omega_{ac} \\omega^{cb} = \\delta_a^b \\nonumber\n\\ee\n\\be\n\\mbox{e.g., for} \\quad N = 2, \\quad  (q_1, p_1):\n\\lk  \\begin{array}{cc}\n      0 & 1\\\\\n      - 1 & 0\n     \\end{array} \\right) \n     \\lk  \\begin{array}{cc}\n      0 & - 1\\\\\n      1 & 0\n     \\end{array} \\right)\n     \\lk  \\begin{array}{cc}\n      1 & 0\\\\\n      0 & 1\n     \\end{array} \\right)\n\\, . \\nonumber\n\\ee\nThen our $P.B.$ (\\ref{8}) can now be written in abbreviated form as\n\\be\n\\label{17}\n\t[A (z), B (z)]_{P.B.} = \\frac{\\partial A}{\\partial z_a} [z_a, z_b] \\frac{\\partial B}{\\partial z_b} = \\omega_{ab}\n\t\\frac{\\partial A}{\\partial z_a} \\frac{\\partial B}{\\partial z_b} = \\partial^a A \\omega_{ab} \n\t\\partial^b B \\, .\n\t\\ee\n\tAccordingly, the Hamiltonian equations are given by\n\\begin{align}\n \\label{18}\n \\dot{z}_a &= [z_a, H] = \\omega_{ab}  \\partial^b H \\, , \\quad \\quad \\omega_{ab} \\, \\mbox{is independent of} \\, z \\nonumber\\\\\n \\lk {\\dot{q}  \\atop \\dot{p}} \\right) &= \\lk \\begin{array}{cc}\n                                             0 & 1 \\\\\n                                             - 1 & 0 \n                                            \\end{array} \\right) \n                                            \\lk \\begin{array}{c}\n                                                 \\frac{\\partial H}{\\partial q} \\\\\n                                                 \\frac{\\partial H}{\\partial p}\n                                                \\end{array} \\right) \\quad : \\quad \\dot{q} = \n                                                \\frac{\\partial H}{\\partial p} \\, , \\quad \\dot{p} = - \\frac{\\partial H}{\\partial q} \\, .\n\\end{align}\nWe call a transformation from the original variables $(q_i, p_i)$ to new variables $(Q_i, P_i)$\n\\begin{align}\n \\label{19}\n Q_i &= Q_i (q_i, p_i) \\, , \\nonumber\\\\\n P_i &= P_i (q_i, p_i) \\, ,\n\\end{align}\n\\underline{canonical}, if the fundamental $P.B.$ for the new $(Q_i, P_i)$ also are of form (\\ref{9}):\n\\be\n\\label{20}\n[Q_i, Q_j] = 0 = [P_i, P_j] \\, , \\quad \\quad [Q_i, P_j] = \\delta_{ij} \\, .\n\\ee\nIt can be easily shown that as a consequence of formulas (\\ref{20}), \nthe Jacobi determinant of a canonical transformation is equal to one (''the canonicity is preserved''): \n\\be\n\\label{21}\n\t\\mathrm{det}(\\omega_{qp}) = \\mathrm{det} (\\omega_{QP}) = 1 \\, .\n\t\\ee\nTo prove (\\ref{21}), we use $N = 2$. With the help of (\\ref{10}), the following simply applies:\n\\be\n\\label{22}\n[Q, P]_{q, p} = \\lk \\frac{\\partial Q}{\\partial q} \\frac{\\partial P}{\\partial p} - \n\\frac{\\partial Q}{\\partial p} \\frac{\\partial P}{\\partial q} \\right) \\, .\n\\ee\nIf $Q$ and $P$ are to be a pair of canonical conjugate variables, then according to equations (\\ref{20}), $[Q,P] = 1$ is valid; thus\n\\be\n[Q, P] = \\left| \\begin{array}{cc}\n                 \\frac{\\partial Q}{\\partial q} & \\frac{\\partial Q}{\\partial p} \\\\ \n\\frac{\\partial P}{\\partial q} &  \\frac{\\partial P}{\\partial p} \n                \\end{array}\\right|  = 1\\, . \\nonumber\n\\ee\nThe main characteristic of a canonical transformation is, however, that in the new variables, the canonical equations also hold:\nthe Hamiltonian canonical equations are form invariant (covariant) under (\\ref{19}), \nbut with a new Hamiltonian function $K$:\n\\be\n\\label{23}\nH (q (Q, P), p (Q, P)) = K    (Q, P) \\, .\n\\ee\nWe show this again for one degree of freedom:\n\\begin{align}\n \\dot{Q} = [Q, H]_{q, p} &= \\frac{\\partial Q}{\\partial q} \\frac{\\partial H}{\\partial p} - \n \\frac{\\partial Q}{\\partial p} \\frac{\\partial H}{\\partial q} \\nonumber\\\\\n &= \\frac{\\partial Q}{\\partial q} \\lk \\frac{\\partial H}{\\partial Q} \\frac{\\partial Q}{\\partial p} + \n \\frac{\\partial H}{\\partial P} \\frac{\\partial P}{\\partial p} \\right) - \n \\frac{\\partial Q}{\\partial p} \\lk \\frac{\\partial H}{\\partial Q}  \\frac{\\partial Q}{\\partial q}\n + \\frac{\\partial H}{\\partial P}  \\frac{\\partial P}{\\partial q} \\right) \\nonumber\\\\\n & = \\frac{\\partial H}{\\partial P} \\lk \\frac{\\partial Q}{\\partial q} \\frac{\\partial P}{\\partial p} - \n \\frac{\\partial Q}{\\partial p} \\frac{\\partial P}{\\partial q} \\right) = \n  \\frac{\\partial H}{\\partial P} [Q, P] \\equiv   \\frac{\\partial K}{\\partial P} (Q, P) \\, .\n\\nonumber\n\\end{align}\nSimilarly for $\\dot{P}$.\n\\bi\n\n\\no\nThe literature on Hamiltonian dynamics is dominated by canonical transformations. But this an unfortunate situation, \nsince it is often impossible to introduce convenient  variables which are also canonical. In a ``pseudocanonical''\ntransformation, the new variables must not be canonical, i.e., they need \\underline{not} satisfy (\\ref{20}). \nHowever, the transformation equations, as before, have to be invertible.\n\\bi\n\n\\no\nFor non-canonical phase-space variables, \ninstead of (\\ref{18}), we now write\n\\be\n\\dot{z}_a = [z_a, H] = \\sigma_{ab} (z) \\partial^b H \\, , \\quad \\quad a, b = 1, 2, \\ldots N \\,  \\nonumber\n\\ee\nwith $\\sigma_{ab} (z)$ instead of (\\ref{14})\n\\be\n\\label{24}\n[z_a, z_b] = \\sigma_{ab} (z) \\, .\n\\ee \nProperties of $\\sigma_{ab} (z)$\n\\be\n\\label{25}\n[z_a, z_b] = - [z_b, z_a] : \\quad \\quad \\sigma_{ab} (z) = - \\sigma_{ba} (z) \\, .\n\\ee\nInverse\n\\be\n\\label{26}\n\\sigma_{ac} (z) \\sigma^{cb} (z) = \\delta^b_a \\, .\n\\ee\nFrom the Jacobi identity\n\\be\n[z_a, [z_b, z_c]] + [z_b, [z_c, z_a]] + [z_c, [z_a, z_b]] = 0 \\nonumber\n\\ee\nfollows as a condition for the $\\sigma$'s:\n\\be\n\\label{27}\n\\partial_a \\sigma_{bc} (z)  + \\partial_b \\sigma_{ca} (z) + \\partial_c \\sigma_{ab} (z) = 0 \\, .\n\\ee\nIf we change the phase-space coordinates according to the transformation\n   \\be\n   z \\to         z'(z) \\nonumber\n   \\ee\nwe get as $L.B.$ for the new coordinates\n\\begin{align}\n \\sigma'_{ab} &= [z'_a, z'_b] = [z'_a (z), z'_b (z)] \\nonumber\\\\\n &=\\frac{\\partial z'_a}{\\partial z_c} [z_c, z_d] \\frac{\\partial z'_b}{\\partial z_d} = \n \\frac{\\partial z'_a}{\\partial z_c} \\sigma_{cd} (z) \\frac{\\partial z'_b}{\\partial z_d} \\, . \\nonumber\n\\end{align}\nTherefore\t\t\t\t\t\t\n\\be\n\\label{28}\n\\sigma'_{ab} = \\frac{\\partial z'_a}{\\partial z_c} \\frac{\\partial z'_b}{\\partial z_d} \\sigma_{cd} \\, .\n\\ee\nHere the $\\sigma_{ab}$ turn out to be components of a two-rank contravariant tensor.\n\\bi\n\n\\no\nMoreover, we already know that the $L.B.$ for two arbitrary phase-space functions $A(z)$, $B(z)$ can be written as\n\\begin{align}\n \\label{29}\n [A (z), B (z)] &= \\frac{\\partial A}{\\partial z_a} [z_a, z_b] \\frac{\\partial B}{\\partial z_b} \\, , \\nonumber\\\\\n &= \\frac{\\partial A}{\\partial z_a} \\sigma_{ab} \\frac{\\partial B}{\\partial z_b} \\, .\n\\end{align}\nIf $z_a =  (q_1, \\ldots, q_{N\/2}, p_1, \\ldots, p_{N\/2})$ are canonical coordinates, then we get \nfor the transformation $z \\to z'$   ($z'$ is not necessarily canonical)\n\\be\n\\sigma'_{ab} = \\frac{\\partial {z'}_a}{\\partial z_c} \\frac{\\partial {z'}_b}{\\partial z_d} \\omega_{cd} \\, . \\nonumber\n\\ee\nHere, $\\omega_{ab}$ is again the ``flat metric'' in the canonical $q-p$ basis (\\ref{16}).\n\\bi\n\n\\no\t\t\t\t\t\t\t\t\t\t\t\t\t\nWe now want to illustrate the practicality of the noncanonical coordinates using three very important examples from \nphysics. Let us begin with the mathematical pendulum. The energy (not Hamiltonian) of a particle in the Earth's \ngravitational field is given in right-angle Cartesian coordinates by\n\\be\n\\label{30}\nH' (x, p_x, p_y) = \\frac{1}{2m} (p^2_x + p^2_y) - mgx .\n\\includegraphics[width=.2\\textwidth]{cartesian.eps}\n\\ee\nbut subject to the constraint \n\\be\n\\label{31}\n\tx^2 + y^2 = l^2 \\, .            \t\t\t\t\t           \\ee\nIt should be observed that the variables in (\\ref{30}) \nare \nnoncanonical, so that they do not satisfy the conventional Hamiltonian canonical equations of motion, because of the additional\nconstraint (\\ref{31}). It is, however, well known that we can express $H'$ in canonical coordinates\n$(\\varphi, p_\\varphi)$ with the resultant Hamiltonian:\n\t\t\t\\be\n\t\t\t\\label{32}\n\t\t\tH (\\varphi, p_\\varphi) = \\frac{1}{2 m l^2} p^2_\\varphi - m g l \\cos \\varphi \\, .\n\\ee\nHence we can express the equations of motion with a single degree of freedom. The associated phase space \n$(\\varphi, p_\\varphi)$ is two dimensional and endowed with a Poisson bracket structure:\n\\be\n\\label{33}\n[\\varphi, \\varphi]_{P. B} = 0 = [p_\\varphi, p_\\varphi] \\, , \\quad \\quad [\\varphi, p_\\varphi]_{P. B} = 1 \\, .\n\t\\ee\nThis can also be formulated by saying that the symplectic structure of this two-dimensional phase space is given by the symplectic matrix\n\\be\n\\label{34}\n\\omega = \n\\bordermatrix{\n  & \\varphi\t& p_\\varphi   \\cr\n\\varphi & 0 & 1 \\cr\np_\\varphi & - 1 & 0 \\cr\n} \\, .\n\\ee\nThe two-dimensional matrix representation is typical for every canonical pair $(q,p)$.\nOur next goal is to enlarge the so far two-dimensional phase pace into four dimensions with pseudo-canonical variables \n$(r, p_r, \\varphi, p_\\varphi)$. We call them pseudo-canonical variables because, as will be shown, $r$\nand $p_r$ satisfy certain constraints and hence cannot be canonical variables. This means that there are no Poisson brackets for the \n$(r, p_r)$ variables. Instead, the Lie brackets take over and are the building elements of the symplectic matrix.\nHere are the details. We begin with the pseudo-Hamiltonian    \n\\be\n\\label{35}\n\tH' (r, p_r, \\varphi, p_\\varphi)\t= \\frac{1}{2 m} \\lk p^2_r + \\frac{1}{r^2} p^2_\\varphi \\right)\t- m g r \\cos \\varphi \\, .\n\t\\ee\nThe pseudo-canonical transformation is given by\n\\be\n(x, p_x, y, p_y ) \\to (r, p_r, \\varphi, p_\\varphi) \\nonumber\n\\ee\nwith\n\\begin{align}\n x &= r \\cos \\varphi \\, , \\quad \\quad p_x = p_r \\cos \\varphi - \\frac{p_\\varphi}{r} \\sin \\varphi \\, , \\nonumber\\\\\n y &= r \\sin \\varphi \\, , \\quad \\quad p_y = p_r \\sin \\varphi + \\frac{p_\\varphi}{r} \\cos \\varphi \\, . \\nonumber\n\\end{align}\nInverting these equation yields\n\\begin{align}\n\\cos \\varphi &= \\frac{x}{r} \\, , \\quad \\quad p_r = p_x \\cos \\varphi + p_\\varphi \\sin \\varphi \\, , \\nonumber\\\\\n \\sin \\varphi &= \\frac{y}{r} \\, , \\quad \\quad p_\\varphi = - p_x r \\sin \\varphi + p_y  r \\cos \\varphi \\, . \\nonumber\n\\end{align}\nHere are our two constants of motion (constraints):\n\\be\n\\label{36}\n(x^2 + y^2) = r^2 \\, , \\quad \\quad \\frac{x p_y + y p_y}{(x^2 + y^2)^{1\/2}} = p_r \\, .\n\\ee\nThese quantities are called Casimirs and are constants of motion because of the form of the bracket rather \nthan the form of the Hamiltonian. In $(\\varphi, p_\\varphi)$\nspace, of course, the brackets (as well as the bracket matrix $\\omega$) are canonical.\nBefore introducing the constraints (\\ref{36}), the four-dimensional matrix $\\omega'$ is given by\n\\begin{align}\n \\label{37}\n \\omega' = \\begin{array}{c}\n \\\\\n            r \\\\\n            p_r \\\\\n            \\varphi \\\\\n            p_\\varphi \n           \\end{array}          \n                      \\lk\n           \\begin{array}{cccc}\n            r & p_r & \\varphi & p_\\varphi \n                       \\\\\n            \\left[r, r\\right] & \\left[r, p_r\\right] & \\left[r, \\varphi \\right] & \\left[r, p_\\varphi \\right]                    \n            \\\\\n            \\left[p_r, r\\right] & \\left[p_r, p_r\\right] & \\ldots \\ldots &  \\ldots \\ldots \\\\\n            \\vdots && \\vdots & \\\\\n            \\left[p_\\varphi, r\\right] & \\ldots & \\left[p_\\varphi, \\varphi\\right] & \\left[p_\\varphi, p_\\varphi\\right] \n           \\end{array}\n           \\right) \n                                 \\begin{array}{l}\n            \\mbox{constraints} \\\\\n            \\longrightarrow \\\\\n            \\left[r, r \\right] = 0 \\\\\n            \\mbox{etc.}\n           \\end{array}\n           \\lk \\begin{array}{cc}\n                O_2 & O_2 \\\\\n                O_2 &  \\lk \\begin{array}{cc}\n                       0 & 1 \\\\\n                       - 1 & 0\n                      \\end{array}\n\\right) \n               \\end{array}\n\\right) \\, .\n\\end{align}\nAfter taking into account the two constraints in (\\ref{36}), \nthis four-dimensional $\\omega'$ structure is reduced to a two-dimensional phase subspace $(N = 2)$\nwith canonical coordinates $(\\varphi, p_\\varphi)$ and the well-known symplectic matrix $\\lk \\begin{array}{cc}\n                       0 & 1 \\\\\n                       - 1 & 0\n                      \\end{array}\n\\right)$, meaning one degree of freedom, so that our dynamical \nsystem ``mathematical pendulum'' is soluble.\n\\bi\n\n\\no\nWhen we use the one-dimensional harmonic oscillator in the $x$ representation, we have for the Schr\\\"odinger Hamiltonian\n\\be\nH (x, \\partial_x) =  - \\frac{\\hbar^2}{2 m} \\partial^2_x + \\frac{1}{2} m \\omega^2 x^2 \\quad \\mbox{with} \\quad\n[\\partial_x, x] = 1 \\, , \\quad [\\partial^2_x, x] =  2 \\partial_x \\, . \\nonumber\n\\ee\nTo set up the corresponding Lie algebra we define the following ``operators'':\n\\be\nL_+ = \\frac{1}{2} x^2 \\, , \\quad \\quad L_- = - \\frac{1}{2} \\partial^2_x \\, , \\qquad \\quad L_3 = \\frac{1}{2} x \\partial_x + \\frac{1}{4} \n\\nonumber\n\\ee\nso that the pseudo-Hamiltonian can be written as\n\\be\nH' = \\frac{\\hbar^2}{m} L_- + m \\omega^2 L_+ \\, . \\nonumber\n\\ee\nNow it is easy to   prove that\n\\be\n\\left[L_+, L_- \\right] = 2 L_3 \\, , \\quad \\quad \\left[L_3, L_\\pm\\right] = L_\\pm \\, . \\nonumber\n\\ee\nThis is the $SO(3)$ Lie algebra of angular momenta, which is locally isomorphic to the $SU(2)$ Lie algebra\nand thus our problem is naturally soluble. \n\\bi\n\n\\no\nOur next example is the description of a charged particle in an external constant magnetic field, i.e., with the Hamiltonian function\n\\be\n\\label{38}\nH (\\vq, \\vp) = \\frac{1}{2 m} \\lk \\vp - \\frac{e}{c} \\vA (\\vq) \\right)^2 \\, , \\quad \\quad\t\n\\vA (\\vq) = \\frac{1}{2} \\vB \\times \\vq \\, .\n\\ee\nThis choice of vector potential (gauge) guarantees that $\\vB$ indeed is constant, because\n\\begin{align}\n \\vB &= \\vec{\\nabla}_{\\vq} \\times \\vA (\\vq) = \\frac{1}{2} \\vec{\\nabla} \\times (\\vB \\times \\vq) \\nonumber\\\\\n &= \\frac{1}{2} \n \\Big[ (\\vq \\cdot \\vec{\\nabla}) \\vB - \n  (\\vB \\cdot \\vec{\\nabla}) \\vq + \\vB (\\vec{\\nabla} \\cdot \\vq) - \n \\vq (\\vec{\\nabla} \\cdot B) \\Big]  \n \\nonumber\\\\\n &= \\frac{1}{2} \\left[ 0 - \\vB + 3 \\vB - 0 \\right] = \\frac{1}{2} (- \\vB + 3 \\vB) = \\vB \\, . \\nonumber\n\\end{align}\nHerewith we get as Hamiltonian equations of motion\n\\begin{align}\n \\label{39}\n \\dot{p}_i &= - \\frac{\\partial H}{\\partial q_i} = \\frac{e}{mc} \\lk p_j  - \\frac{e}{c} A_j (\\vq) \\right) \n \\frac{\\partial}{\\partial q_i} A_j (\\vq) \\, , \\quad \\quad A_j (\\vq) = \\frac{1}{2} \\epsilon_{jkl} B_k q_l \\nonumber\\\\\n &= \\frac{e}{mc} \\lk p_j - \\frac{e}{c} A_j (\\vq) \\right) \\frac{1}{2} \\epsilon_{jki} B_k \\nonumber\\\\\n &= \\frac{e}{2 mc} \\left[ \\lk \\vp - \\frac{e}{c} \\vA (\\vq) \\right) \\times \\vB \\right]_i \\\\\n \\label{40}\n \\dot{q}_i &= \\frac{\\partial H}{\\partial p_i} = \\lk \\vq - \\frac{e}{c} \\vA (\\vq) \\right)_i \\, .\n \\end{align}\nNow we consider the noncanonical transformation\n\\begin{align}\n(\\vq, \\vp)   & \\longrightarrow (\\vx', \\vec{v}') \\equiv \\vz'_a \\nonumber\n\\\\\n               & \\mbox{noncanonical coordinates} \\nonumber\n\\end{align}\nso:  \n\\begin{align}\n \\label{41}\n \\vx' &= \\vq \\nonumber\\\\\n \\vv' &= \\frac{1}{m} \\lk \\vp - \\frac{e}{c} \\vA (\\vq) \\right) \\equiv\n \\frac{1}{m} \\vec{\\Pi} \\, \\mbox{(noncanonical, gauge invariant)} \\, .\n\\end{align}\n$\n(\\vx', \\vv')$ parametrize the phase-space just as well as $(\\vq, \\vp)$.\nNow it holds for the fundamental Lie brackets:\n\\begin{align}\n \\label{42}\n {\\vv'}_i &= q_i : \\quad \\quad \\left[{\\vx'}_i, {x'}_j \\right]_{q, p} =   \n \\frac{\\partial {x'}_i}{\\partial q_k} \\frac{\\partial x'_j}{\\underbrace{\\partial p_k}_{= 0}}  -\n \\frac{\\partial x'_i}{\\underbrace{\\partial p_k}_{= 0}} \n \\frac{\\partial {x'}_j}{\\partial q_k} = 0\n \\nonumber\\\\\n {v'}_i &= \\frac{1}{m} \\lk p_i - \\frac{e}{c} A_i (q_j) \\right): \\quad \\quad \\left[{x'}_i, {v'}_j \\right]_{q, p} = \n \\frac{1}{m} \\delta_{ij} = \n \\left[{v'}_i, {x'}_j \\right]_{q, p} \\nonumber\\\\\n A_i (\\vq) &= \\frac{1}{2} \\epsilon_{ijk} B_j q_k\n \\nonumber\\\\\n \\left[{v'}_i, {v'}_j \\right]_{q, p} &= \\frac{e}{m^2 c} \\lk \\frac{\\partial A_j}{\\partial q_i} - \\frac{\\partial A_i}{\\partial q_j} \\right) = \n \\frac{e}{m^2c} B_{ij} = \\frac{1}{m^2} \\left[ \\Pi_i, \\Pi_j \\right] \\nonumber\\\\\n & \\hspace{5cm} \\frac{e}{c} B_{ij} = F_{ij} \\nonumber\\\\\n &= \\frac{e}{m^2 c} \\epsilon_{ijk} B_k = \\frac{1}{m} \\Omega_{ij} \\nonumber\\\\\n \\overset{\\leftrightarrow}{\\Omega} &= \\Omega_{ij} = \\frac{e}{m c} \\epsilon_{ijk} B_k \\, .\n\\end{align}\nSo we have found:\n\\begin{align}\n \\label{43}\n \\sigma'_{a b} &= \\frac{1}{m} \\lk \\begin{array}{cc}\n                                   O_3 & 1_3\\\\\n                                   - 1_3 &  \\overset{\\leftrightarrow}{\\Omega}\n                                  \\end{array}\n                                  \\right) \\\\\n                                  \\label{44}\n                                  \\sigma'^{ab} &= m \n\\lk \\begin{array}{cc}\n                                   \\overset{\\leftrightarrow}{\\Omega} & - 1_3\\\\\n                                   1_3 & O\n                                  \\end{array}\n                                  \\right) \\, ,\n\\end{align}\t\nso that indeed\n\\begin{align}\n \\sigma'_{ac} \\sigma'^{cb} &= \\lk \\begin{array}{cc}\n                                   0 & 1\\\\\n                                   -1 &  \\overset{\\leftrightarrow}{\\Omega}\n                                  \\end{array}\n                                  \\right) \n                                  \\lk \\begin{array}{cc}\n                                    \\overset{\\leftrightarrow}{\\Omega} & - 1\\\\\n                                   1 & 0\n                                  \\end{array}\n                                  \\right) \\nonumber\\\\\n                                  &= \\lk \\begin{array}{cc}\n                                   1 & 0\\\\\n                                   0 & 1\n                                  \\end{array}\n                                  \\right) = 1 \\, . \\nonumber\n\\end{align}\nIn canonical coordinates $(\\vq, \\vp)$:\t\t\t\t\t\t\t\t\t\n\\be\n\\label{45}\nH (\\vq, \\vp) = \\frac{1}{2 m} \\lk \\vp - \\frac{e}{c} \\vA (\\vq) \\right)^2 \\, .\n\\ee\nIn non-canonical coordinates \n\\be\n\\label{46}\n(\\vx', \\vv'): H' (\\vx', \\vv') = \\frac{m}{2}  \\vv'^2  = \\frac{1}{2 m} \\vec{\\Pi}^2\t\t\t\t\\, .\n\\ee\nNon-canonical equations of motion:\n\\be\n\\frac{d z'_a}{d t} = [z'_a, H'] = \\sigma'_{ab} \\frac{\\partial H'}{\\partial z'_b}\\nonumber\n\\ee\nor\n\\begin{align}\n\\label{47}\n \\frac{d}{dt} \\lk {\\vx' \\atop \\vv'} \\right) &= \\frac{1}{m} \n\\lk \\begin{array}{cc} \n     0 & 1 \\\\\n     - 1 &  \\overset{\\leftrightarrow}{\\Omega}\n         \\end{array}\\right)\n    \\lk {\\frac{\\partial H'}{\\partial\\vx'} \\atop \\frac{\\partial H'}{\\partial \\vv'}} \\right) = \n    \\lk\n    \\begin{array}{c}\n     \\vv' \\\\\n     \\frac{e}{m c} \\epsilon_{ijk} B_k v'_j\n    \\end{array}\n    \\right) \\nonumber\\\\\n    &= \\lk\n    \\begin{array}{c}\n     \\vv' \\\\\n     \\frac{e}{mc} \\vv' \\times \\vB\n    \\end{array}\n    \\right) \n\\end{align}\n\\be\n\\label{48}\n\\mbox{Newton-Lorentz equation.} \\atop (\\mbox{No} \\, \\vA \\, , \\mbox{gauge invariant!}) \\, .\n\\ee\nOne should note that in $H'(\\vx', \\vv') = \\frac{m}{2} \\vv'^2 = \\frac{1}{2m} \\vec{\\Pi}^2$, \nthe nonphysical magnetic vector potential $\\vA$ has disappeared ! ? \nThe components of the $\\sigma'_{ab}$\ntensor can be used to calculate the $L.B.$ of two phase-space functions $f (\\vx', \\vv'), g (\\vx',\\vv')$:\n\\be\n[f, g] = \\frac{\\partial f}{\\partial {z'}_a} \\sigma'_{ab} \\frac{\\partial g}{\\partial {z'}_b} \\, . \\nonumber\n\\ee\nThe result follows directly from (\\ref{43})\n\\begin{align}\n \\label{49}\n [f, g] &= \\frac{\\partial f}{\\partial \\vx'} \n \\mathclap{\\raisebox{3.5ex}{\\scalebox{3}[1]{$\\frown$}}}\n\\mathclap{\\raisebox{-3.5ex}{\\scalebox{3}[1]{$\\smile$}}}\n \\frac{\\partial f}{\\partial \\vv'} \\frac{1}{m} \n \\lk \\begin{array}{cc}\n      0 & 1\\\\\n      - 1 & \\overset{\\leftrightarrow}{\\Omega}\n     \\end{array}\n     \\right)\n      \\lk \\begin{array}{c}        \n      \\frac{\\partial g}{\\partial \\vx'}\\\\\n      \\frac{\\partial g}{\\partial \\vv'}\n     \\end{array}\n     \\right) \\nonumber\\\\\n     &= \\frac{1}{m} \n\\frac{\\partial f}{\\partial \\vx'} \n \\mathclap{\\raisebox{3.5ex}{\\scalebox{3}[1]{$\\frown$}}}\n\\mathclap{\\raisebox{-3.5ex}{\\scalebox{3}[1]{$\\smile$}}}\n\\frac{\\partial f}{\\partial \\vv'}  \n\\begin{pmatrix}\n \\frac{\\partial g}{\\partial \\vv'} \\\\\n - \\frac{\\partial g}{\\partial \\vx'} \n +  \\overset{\\leftrightarrow}{\\Omega} \\cdot \\frac{\\partial g}{\\partial \\vv'}\n\\end{pmatrix} \\nonumber\\\\\n&= \\frac{1}{m} \\lk\n\\frac{\\partial f}{\\partial \\vx'} \\cdot \\frac{\\partial g}{\\partial \\vv'} - \\frac{\\partial f}{\\partial \\vv'} \\cdot \n\\frac{\\partial g}{\\partial \\vx'} \\right) + \\frac{e}{m^2 c} \\vB \\cdot \n\\lk \\frac{\\partial f}{\\partial \\vv'} \\times \\frac{\\partial g}{\\partial \\vv'} \\right) \n\\, ,\n\\end{align}\nwhich for\n\\be\nf = \\Pi_i (\\vv') = m v'_i \\, , \\quad \\quad g = \\Pi_j (\\vv') = m v'_j \\, \\mbox{reproduces} \\, (\\ref{42}). \\nonumber\n\\ee\nThe $\\vB$ or, respectively, the $\\vA$ field, which disappeared in $H' = \\frac{m}{2} \\vv'^2$, \ncan be rediscovered in the altered symplectic structure $\\sigma'$ (\\ref{43}) of the gauge invariant  pseudo-Hamiltonian system\n$H' = \\frac{1}{2 m} \\vec{\\Pi}^2$.\n\\bi\n\n\\no\nA further important example of the application of noncanonical coordinates can be found in force-free rigid bodies. \nHere $\\dot{\\vL}  = 0$ holds in the inertial system. Relative to the fixed-body reference system, the \\underline{same}\nangular-momentum vector has the components $\\{K_i\\}_{i = 1, 2, 3} = \\vL^{body}$.\n\\bi\n\n\\no\nNow we perform a non-canonical transformation\n\\begin{align}\n\\begin{array}{lll}\n(\\phi, \\theta, \\psi, p_\\phi, p_\\theta, p_\\psi)  &  \\longrightarrow  & (K_1, K_2, K_3) \\nonumber\\\\\n\\mbox{6-dimensional canonical phase space} & &   \\mbox{3-dimensional reduced noncanonical phase space} \n\\end{array} \\nonumber\n\\end{align}\nThe\nformation equations are\n\\begin{align}\n K_1 &= \\lk p_\\phi \\frac{1}{\\sin \\theta}  - p_\\psi \\cot \\theta \\right) \\sin \\psi + p_\\theta \\cos \\psi \\, , \\nonumber\\\\\n  K_2 &= \\lk p_\\phi \\frac{1}{\\sin \\theta}  - p_\\psi \\cot \\theta \\right) \\cos \\psi - p_\\theta \\sin \\psi \\, , \\nonumber\\\\\n  K_3 &= p_\\psi \\, . \\nonumber\n\\end{align}\nWith the help of fundamental canonical brackets $[\\phi, p_\\phi], \\ldots$ one can prove the following validity for the fundamental $L.B.$:\n\\begin{align}\n\\label{50}\n\t[K_1, K_2] &  = -K_3 \\nonumber\\\\\n\t[K_2, K_3] &  = -K_1 \\quad \\quad \\mbox{Note the minus signs!} \\nonumber\\\\\n\t[K_3, K_1] & = -K_2 \n\\end{align}\n                  or     \n                  \\be\n                  \\label{51}\n                  [K_\\alpha, K_\\beta] = - \\epsilon_{\\alpha \\beta \\gamma} K^\\gamma \\, ,\n                  \\ee\n                                 i.e.,\n                                 \\be\n                                 \\sigma^{\\alpha \\beta} = - \\epsilon^{\\alpha \\beta \\gamma} K_\\gamma \\nonumber\n\\ee\n                  or\t\t\t\t                                                                         \t\t\t\t\t\t\t\t\t  \n\\be\n\\label{52}\n\\sigma = \\lk \\begin{array}{ccc}\n              0 & - K_3 & K_2 \\\\\n              K_3 & 0 & - K_1 \\\\\n              - K_2 & K_1 & 0\n             \\end{array} \\right) \\, .\n\\ee\nHerewith one can form the (noncanonical) $L.B.$ of two arbitrary functions $A (\\vK), B (\\vK)$ according to\n\\begin{align}\n \\label{53}\n [A (\\vK), B (\\vK)] &= \\frac{\\partial A}{\\partial K^\\alpha} [K^\\alpha, K^\\beta] \\frac{\\partial B}{\\partial K^\\beta} \\nonumber\\\\\n &= - \\epsilon^{\\alpha \\beta \\gamma} K_\\gamma \\frac{\\partial A}{\\partial K^\\alpha} \\frac{\\partial B}{\\partial K^\\beta} \\nonumber\\\\\n &= - \\vK \\cdot \\lk \\frac{\\partial A}{\\partial \\vK} \\times \\frac{\\partial B}{\\partial \\vK} \\right) \\, .\n\\end{align}\nWith the pseudo-Hamiltonian function with $I_i$ the principal axis of inertia\n\\be\nH' = \\frac{K^2_1}{2 I_3} + \\frac{K^2_2}{2 I_2} + \\frac{K^2_3}{2 I_3}\n\\nonumber\n\\ee\nfollows for the equations of motion:\t\t\t\t\t\t\t\t\t\n\\be\n\\label{54}\n\\frac{d}{dt} K^\\alpha = [K^\\alpha, H'] = \\sigma^{\\alpha \\beta} \\frac{\\partial H'}{\\partial K^\\beta} \\, ,\n\\ee\ni.e.,\n\\be\n\\frac{d}{dt} \\lk \\begin{array}{c}\nK^1\\\\ K^2 \\\\ K^3\n                                 \\end{array}\n\\right) = \\lk\n\\begin{array}{ccc}\n 0 & - K_3 & K_2 \\\\\n K_3 & 0 & - K_1 \\\\\n - K_2 & K_1 & 0\n\\end{array} \\right) \n\\lk \n\\begin{array}{c}\n \\frac{\\partial H'}{\\partial K_1} \\\\\n \\frac{\\partial H'}{\\partial K_2} \\\\\n \\frac{\\partial H'}{\\partial K_3} \n\\end{array} \\right) \\, , \\quad \\quad\n\\frac{\\partial H'}{\\partial K^\\beta} = \\frac{K_\\beta}{I_\\beta} \\, . \\nonumber \n\\ee\nFor example:\n\\be\n\\label{55}\n\\frac{d}{dt} K_1 = K_3 \\frac{K_2}{I_2} + K_2 \\frac{K_3}{I_3} \\, ,\n\\ee\n  i.e.\n  \\begin{align}\n  \\dot{K}_1 = & \\lk \\frac{1}{I_3}  - \\frac{1}{I_2} \\right) K_2 K_3 \\nonumber\\\\\n\\mbox{and cyclic order} &. \\nonumber               \n   \\end{align}\nThese are Euler's equations of motion for the force-free rigid body.\nWith $K_i = I_i \\omega_i (i = 1,2,3)$ we obtain the well-known formula for the force-free rigid body\n\\be\n\\label{56}\n(I_i - I_j) \\omega_i \\omega_j - \\sli^3_{k = 1} \\epsilon_{ijk} I_k \\dot{\\omega}_k = 0 \\, .\n\\ee\nIt is remarkable that without further calculation our system proves to be soluble. This follows from the Casimir\n\\be\nC= K^2_1 + K^2_2 +  K^2_3 \\, , \\nonumber\n\\ee\nwhich is conserved by the form of the brackets. \nThus the motion of a free rigid body actually lives in a two-dimensional state space on the sphere $C = constant$, \nequivalent to a one-degree-of-freedom system, and therefore the motion is integrable.\n\\bi\n\n\\no\nTo show how useful our former results are, let us calculate the non-relativistic propagation function for a charged particle \nin three dimensions in presence of a constant $B$ field in the $z$ direction. \n\\bi\n\n\\no\nWe rewrite our findings (\\ref{47}) in terms of the pseudo-momentum $\\vec{\\Pi} = m \\vv$:\n\\begin{align}\n \\label{57}\n \\frac{d \\vx (t)}{dt} &= \\frac{\\vec{\\Pi}}{m} (t) \\, , \\nonumber\\\\\n \\frac{d \\vec{\\Pi} (t)}{dt} &= \\frac{e}{c}  \\overset{\\leftrightarrow}{\\Omega} \\cdot \\vec{\\Pi} (t) \\, , \\quad \\quad\n ( \\overset{\\leftrightarrow}{\\Omega})_{ij} = \n \\frac{e}{mc} \\epsilon_{ijk} B^k \\, .\n\\end{align}\nFrom now on we choose the axis of the magnetic field in the $z$ direction so that our pseudo-Hamiltonian $\\cH$ reads:\n\\be\n\\label{58}\n\\cH = \\frac{1}{2m} (\\Pi^2_1 + \\Pi^2_2) + \\frac{P^2_3}{2m} = \\cH_\\perp + \\frac{P^2_3}{2m} \\, .\n\\ee\nThe free-particle propagation in the third direction can be easily treated, so we drop it and introduce\n\\be\n\\vr (t) = \\lk {x_1 (t) \\atop x_2 (t)} \\right) \\, , \\quad \\quad \\vec{\\Pi} (t) = \n\\lk {\\Pi_1 (t) \\atop \\Pi_2 (t)} \\right) \\, . \\nonumber\n\\ee\nWe leave it as an exercise for the reader to show that in the now two-dimensional problem the matrix $\\sigma$ is given by\n\\begin{align}\n\\label{59}\n\\sigma_{ab} = \\frac{1}{m} \n\\lk \\begin{array}{cc}\n     0_2 & 1_2 \\\\\n     - 1_2 & \\frac{e B}{mc} \\overset{\\leftrightarrow}{\\epsilon}\n    \\end{array}\n    \\right) \\, , \\quad \\quad \n    (\\overset{\\leftrightarrow}{\\epsilon})_{ab} = \n  \\lk \\begin{array}{cc}\n     0 & 1 \\\\\n     - 1 & 0\n    \\end{array}\n    \\right) \\, , \\quad \\quad  \n    ( \\overset{\\leftrightarrow}{\\epsilon})^{ab} = \n  \\lk \\begin{array}{cc}\n     0 & -1 \\\\\n      1 & 0\n    \\end{array}\n    \\right)  & = - \\epsilon_{ab} \\, , \\nonumber\\\\\n    & \\epsilon_{ac} \\epsilon^{cb} = \\delta_a^b \\, .\n\\end{align}\nHence the non-canonical equations of motion take the form\n\\begin{align}\n \\label{60}\n \\lk \\dot{\\bar{x}} \\atop \\dot{\\vv} \\right) &= \\frac{1}{m} \n \\lk \\begin{array}{cc}\n     0 & 1 \\\\\n      -1 & \\frac{e B}{m c} \\overset{\\leftrightarrow}{\\epsilon}\n    \\end{array}\n    \\right) \n \\lk \\begin{array}{c}\n     0  \\\\\n      \\frac{\\partial \\cH_\\perp}{\\partial \\vv}\n    \\end{array}\n    \\right)    = \\frac{1}{m}\n    \\lk \\begin{array}{cc}\n     0_2 & 1_2 \\\\\n      -1 & \\frac{e B}{m c} \n       \\overset{\\leftrightarrow}{\\epsilon}\n          \\end{array}\n    \\right) \n    \\lk \\begin{array}{c}\n     0  \\\\\n      m \\vv\n    \\end{array}\n    \\right)     \n    \\\\\n    \\label{61}\n    & \\mbox{or} \\quad \\frac{d \\vr}{d t} = \\frac{\\vec{\\Pi}}{m}  \\\\\n    \\label{62}\n     \\frac{d \\vec{\\Pi}}{d t} & = \\frac{e B}{mc} \n      \\overset{\\leftrightarrow}{\\epsilon}\n     \\cdot \\vec{\\Pi} \\, , \\quad \\quad \n    \\Omega = \\frac{e B}{m c} \\quad \\mbox{cyclotron frequency} \\nonumber\\\\\n    & = \\Omega \n         \\overset{\\leftrightarrow}{\\epsilon}\n\\cdot \\vec{\\Pi} \\, .\n\\end{align}\nThe solution of (\\ref{62}) for $\\vec{\\Pi}$ is needed for determining the explicit expression for $\\cH_\\perp (t) = \\frac{\\vec{\\Pi}^2}{2m}$.                \nThis information enables us to calculate the three-dimensional non-relativistic Feynman propagation function. \nSolving equations (\\ref{61}) and (\\ref{62}) for $\\vr (t)$ and $\\vec{\\Pi} (t)$, is equivalent to solving the following Lie\nbracket relations\n\\begin{align}\n \\label{63}\n \\left[ r_i (t), \\Pi^2 (t) \\right] &= 2 \\Pi_i (t) \\, , \\nonumber\\\\\n \\left[ \\Pi_i (t), \\Pi^2 (t) \\right] &= 2 \\frac{e}{c} B_{ij} \\Pi^j (t) \\nonumber\\\\\n &= F_{ij} \\Pi^j = 2 \\frac{e}{c} \\epsilon_{ij3} B^3 \\Pi^j (t) \\, .\n\\end{align}\nOn the way to calculating the Feynman propagation function we could use Schwinger's proper time method. But instead of doing so, we prefer\nanother method, making use of the classical action and the Van Vleck determinant.\n\\bi\n\n\\no\nTo begin with, we remember from the harmonic oscillator the classical action \\cite{1}\n\\be\nS_{el} = \\frac{m \\omega}{2 \\sin (\\omega \\tau)} \\left[ (x^2 + {x'}^2) \\cos (\\omega \\tau) - 2 x x' \\right] \\, . \\nonumber\n\\ee\nUsing the Van Vleck determinant\n\\be\nD = \\mathrm{det} \\left[ - \\lk \\frac{\\partial^2 S_{el}}{\\partial x \\partial x'} \\right) \\right] = \\frac{m \\omega}{\\sin (\\omega \\tau)} : \\quad \\quad\n\\sqrt{D} = \\sqrt{\\frac{m \\omega}{\\sin (\\omega \\tau)}} \\nonumber\n\\ee\nwe obtain for the propagation function of the linear harmonic oscillator\n\\begin{align}\n K (x, x'; \\tau) &= \\sqrt{\\frac{1}{2 \\pi i \\hbar}} \\sqrt{D} e^{\\frac{i}{\\hbar} S_{el}} \\nonumber\\\\\n &= \\lk \\frac{m \\omega}{2 \\pi i \\hbar \\sin (\\omega \\tau)} \\right)^{1\/2} \\exp \n \\left\\{ \\frac{i m \\omega}{2 \\hbar \\sin (\\omega \\tau)} \\left[ (x^2 + x'^2) \\cos (\\omega \\tau) - 2 x x' \\right] \\right\\} \\, . \\nonumber\n\\end{align}\nIf we now follow the same path for a charged particle gauge fixed for a magnetic field in $z$ direction we obtain \n                                                                                                                  \\begin{align}\n \\label{64}\n \\lk \\vr \\right. & = \\left. (x_1, x_2) \\, , \\quad {\\vr}' = ({x'}_1, {x'}_2) \\, , \\quad  (\\overset{\\leftrightarrow}{\\epsilon})_{ij} =                                                                                                         \n \\lk \\begin{array}{cc}\n 0 & 1 \\\\\n - 1 & 0\n           \\end{array} \\right) \\right) \\nonumber\\\\\n           S_{el} &= \\frac{m}{2} \\left\\{ \\frac{\\omega}{2} \\cot (\\omega \\tau) (\\vr - \\vr')^2 + \\omega \\vr \\cdot \n           \\overset{\\leftrightarrow}{\\epsilon} \\cdot \n           \\vr' \\right\\} \\, , \\quad \\omega = \\frac{e B}{m c} \\, .\n\\end{align}\nThe results of the two space derivatives we need for the Van Vleck determinant are given by\n\\begin{align}\n & - \\frac{m}{2} \\omega \\cot \\lk \\frac{\\omega \\tau}{2} \\right) 1 + \\frac{m}{2} \\omega \\overset{\\leftrightarrow}{\\epsilon} = - \\frac{m}{2}\n \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)} \\lk \\cos \\lk \\frac{\\omega \\tau}{2} \\right) 1 - \n\\overset{\\leftrightarrow}{\\epsilon}\\sin \\lk \\frac{\\omega \\tau}{2} \\right) \\right) \\nonumber\\\\\n &= - \\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)} \\lk \\cos \\lk \\frac{\\omega \\tau}{2} \\right) 1  - \n i \\sigma_2 \\sin \\lk \\frac{\\omega \\tau}{2} \\right) \\right) = - \\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)}\n e^{- i \\sigma_2 \\lk \\frac{\\omega \\tau}{2} \\right)} \\, . \\nonumber\n\\end{align}\nFrom here we obtain the determinant\n\\be\nD = \\lk - \\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)} \\right)^2 : \\quad \\quad \\sqrt{D} = \n\\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)} \\, . \\nonumber\n\\ee\nThe full result for the propagation function in a gauge that fixes the magnetic field in the third direction is then given by \n($\\omega = \\frac{eB}{m c}$, cyclotron frequency)\n\\begin{align}\n \\label{65}\n K (\\vr, \\vr'; \\tau) K (x_3, x'_3; \\tau) &= \\frac{1}{2 \\pi  i \\hbar} \\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)}\n e^{\\frac{i}{\\hbar} \\frac{m}{2} \\left[ \\frac{\\omega}{2} \\cot \\lk \\frac{\\omega \\tau}{2} \\right) \n (\\vr - \\vr')^2 + \\omega \\vr \\cdot \\vec{\\epsilon} \\cdot \\vr' \\right]} \\nonumber\\\\\n & \\cdot \\sqrt{\\frac{m}{2 \\pi i \\hbar \\tau}} e^{\\frac{i}{\\hbar} \\frac{m}{2} \\frac{(x_3 - x'_3)^2}{\\tau}} \\, .\n\\end{align}\nIt might be interesting to write in $K (\\vr, \\vr'; \\tau)$: \n($L$ indicates a straight path connecting $\\vr'$ with $\\vr$):\n \\begin{align}\n \\label{66}\n e^{\\frac{i}{\\hbar} e \\il^{\\vr}_{\\vr'} d \\vec{\\zeta} \\cdot \\vA (\\vec{\\zeta})} \n &= \n e^{\\frac{i}{\\hbar} \\frac{m}{2} \\omega {\\vr} \\cdot \\vec{\\epsilon} \\cdot {\\vr'}} \n \\nonumber\\\\   \n &= e^{\\frac{i}{\\hbar} \n \\frac{m}{2} \\frac{e B}{m c} (x_1 {x'}_2 - x_2 {x'}_1)} \n \\nonumber\\\\\n &= e^{\\frac{i}{\\hbar}  \\frac{e}{2 c} \\phi} \\, , \\quad \\quad \\phi  = A r e a \\cdot B \\, .\n\\end{align}\nHence we can also write\n\\be\n\\label{67}\nK (\\vr, \\vr'; \\tau) = \\frac{1}{2 \\pi i \\hbar} \\frac{m}{2} \\frac{\\omega}{\\sin \\lk \\frac{\\omega \\tau}{2} \\right)} \ne^{\\frac{1}{\\hbar} e \\il^{\\vr}_{\\vr'_L} d \\vec{\\zeta} \\cdot \\vA (\\vec{\\zeta})}\ne^{\\frac{i}{\\hbar} \\frac{m}{2} \\frac{\\omega}{2} \\cot \\lk \\frac{\\omega \\tau}{2} \\right) (\\vr - \\vr')^2} \\, , \n\\ee\nan expression we will meet again in the full relativistic case of a charged particle travelling in a \nconstant electromagnetic field. \n\\bi\n\n\\no\nSo let us turn to the calculation of the propagation function using the relativistic coordinate representation\nof $K (x', s; x'', 0)$. One can think of $s$ as   of proper time introduced by V. Fock and J. Schwinger \\cite{2}. \nWe will not adopt this method but instead follow the former non-relativistic strategy, i.e., making use of the \nVan Vleck determinant in $d = 4$ dimensions. We also set $\\hbar = 1$.\n\\bi\n\n\\no\nAs before, we begin with a Lagrangian, in our case with a matrix-valued Lagrangian, set up\nthe equations of motion, and compute the classical action. So let us start with \\cite{3}\n\\be\n\\label{68}\nL   = \\frac{1}{4} \\dot{x}_\\mu \\dot{x}^\\mu + e A^\\mu \\dot{x}_\\mu + \\frac{e}{2} \\sigma^{\\mu \\nu} F_{\\mu \\nu} (x (s)) \\, .\n\\ee\nThe associated (``pseudo-'') Hamiltonian is clearly $\\lk \\Pi = \\frac{\\dot{x}}{2} = (p - eA) \\right)$\n\\begin{align}\n H &= p \\dot{x} - L = \\lk \\frac{\\dot{x}}{2} + e A \\right) \\dot{x} - \\frac{1}{4} \\dot{x}^2 - e A \\dot{x} - \\frac{e}{2} \\sigma F = \n \\frac{\\dot{x}^2}{4} - \\frac{e}{2} \\sigma F \\nonumber\\\\\n \\mbox{or} \\quad H &= \\Pi^2 - \\frac{e}{2} \\sigma F \\, . \\nonumber\n\\end{align}\nFrom (\\ref{68}) we obtain the equations of motion:\n\\begin{align}\n \\frac{d}{ds} \\lk \\frac{\\partial L}{\\partial \\dot{x}^\\mu} \\right) - \\frac{\\partial L}{\\partial {x}^\\mu} &= 0 : \\quad \\quad\n \\frac{\\partial L}{\\partial \\dot{x}^\\mu} = \\frac{1}{2} \\dot{x}_\\mu + e A_\\mu \\, , \\nonumber\\\\\n & \\hspace{2cm} \\frac{\\partial L}{\\partial x^\\mu} =  e \\frac{\\partial A_\\lambda}{\\partial x^\\mu} \\dot{x}^\\lambda + \\frac{e}{2}\n \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu} \\nonumber\\\\\n  \\frac{d}{ds} \\lk \\frac{1}{2} \\dot{x}_\\mu + e A_\\mu \\right) &= e \\frac{\\partial A_\\lambda}{\\partial x^\\mu} \\dot{x}^\\lambda + \\frac{e}{2} \n  \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu} \\, , \\nonumber\\\\\n  \\frac{1}{2} \\ddot{x}_\\mu + e \\frac{\\partial A_\\mu}{\\partial x^\\lambda} \\dot{x}^\\lambda : \\quad \\quad \\frac{1}{2} \\ddot{x}_\\mu &= e \\lk \n  \\frac{\\partial A_\\lambda}{\\partial x^\\mu } - \\frac{\\partial A_\\mu}{\\partial x^\\lambda} \\right) \\dot{x}^\\lambda + \n  \\frac{e}{2} \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu} \\, . \\nonumber\n\\end{align}\nThis provides us with the equation of motion\n\\be\n\\label{69}\n\\ddot{x}_\\mu = 2 e F_{\\mu \\nu} \\dot{x}^\\nu + e \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu} \\, .\n\\ee\nUsing\n\\be\n\\dot{x}_\\mu = 2 \\Pi_\\mu \\nonumber\n\\ee\nwe have\n\\be\n\\dot{\\Pi}_\\mu = \\frac{1}{2} \\ddot{x}_\\mu = e F_{\\mu \\nu} \\dot{x}^\\nu + \\frac{e}{2} \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu}\n\\nonumber\n\\ee\nor\n\\be\n\\label{70}\n\\dot{\\Pi}_\\mu = 2 e F_{\\mu \\nu} \\Pi^\\nu + \\frac{e}{2} \\sigma^{\\lambda \\nu} \\frac{\\partial F_{\\lambda \\nu}}{\\partial x^\\mu} \\, .\n\\ee\nFrom now on we consider constant fields only, i.e., $F_{\\mu \\nu} = const.$. Since we want to calculate the classical action, i.e.,\n\\be\n\\label{71}\nS_{el} = \\il^s_0 d \\lambda L (\\dot{x} (\\lambda), x (\\lambda))\n\\ee\nwe first have to integrate (in matrix notation)\n\\be\n\\ddot{x} = 2 e F \\dot{x} \\, , \\nonumber\n\\ee\nwhich can be solved with the Ansatz\n\\be\nx (\\lambda) = e^{2 e F \\lambda} \\dot{x}  (0) \\nonumber\n\\ee\nand a further integration yields\n\\be\nx (\\lambda) - x (0) = \\il^\\lambda_0 d \\lambda' e^{2 e F \\lambda'} \\dot{x} (0) = \\frac{1}{2 e F} \\left[ e^{2 e F \\lambda} - 1 \\right] \n\\dot{x} (0) \\nonumber\n\\ee\nor with the initial condition $x (\\lambda = 0) = x''$ and $x (\\lambda = s) = x'$ we obtain\n\\be\nx' - x'' = \\frac{1}{2 e F} \\left[ e^{2 e F s} - 1 \\right] \\dot{x} (0) \\nonumber\n\\ee\nor\n\\be\n\\dot{x} (0) = \\frac{1}{e^{2 e F s} - 1} 2 e F (x' - x'') \\, . \\nonumber\n\\ee\nSo far we have\n\\begin{align}\n \\label{72}\n x (\\lambda) - x (0) &= \\frac{1}{2 e F} \\left[ e^{2  e F \\lambda} - 1 \\right] \\frac{1}{e^{2 e F s} - 1} 2 e F (x' - x'') \\nonumber\\\\\n &= \\frac{e^{2 e F \\lambda} - 1}{e^{2 e Fs} - 1} (x' - x'') \\, .\n\\end{align}\nWe need\n\\be\n\\label{73}\nS = \\il^s_0 d \\lambda \\left\\{ \\frac{1}{4} \\dot{x}^T (\\lambda) \\dot{x} (\\lambda) + e \\dot{x}^T (\\lambda) A (\\lambda) \\right\\}\n+ \\frac{e}{2} s \\sigma F \\, .\n\\ee\nIn the present case ${g (F)^T} = g (F^T) = g (- F)$. \nTherefore with\n\\begin{align}\n \\dot{x} (\\lambda) & {=} \n e^{2 e F \\lambda} \\dot{x} (0)  {=} \\frac{e^{2 e F \\lambda}}{e^{2 e Fs} - 1} \n 2  e F (x' - x'')\n \\nonumber\\\\\n \\dot{x}^T (\\lambda) \\dot{x} (\\lambda) &= (x' - x'')^T \\frac{(- 2 e F)}{e^{- 2 e Fs} - 1} \\cancel{e^{- 2 e F \\lambda}} \n \\cancel{e^{2 e F \\lambda}}  \\frac{2  e F}{e^{2 e F s} - 1} 2 e F (x' - x'') \\nonumber\\\\\n&=  (x' - x'')^T \\frac{(- 2 e F)}{\\cancel{e^{- e Fs}} \\lk e^{- e Fs} - e^{e Fs} \\right)} \n\\frac{(2 e F)}{\\cancel{e^{e Fs}} \\lk e^{e Fs} - e^{- eFS} \\right)} (x' - x'') \\nonumber\\\\\n&= (x' - x'')^T \\frac{e F}{\\sin h (e Fs)} \\frac{e F}{\\sin h (e Fs)} (x' - x'') \\, , \\nonumber\\\\\n\\to \\frac{1}{4} \\dot{x}^T (\\lambda) \\dot{x} (\\lambda) &= \\frac{1}{4} (x' - x'') ^T \\frac{(e F)^2}{\\sin h^2 (e Fs)} (x' - x'') \\nonumber\n\\end{align}\nwe obtain for the first term in (\\ref{73})\n\\be\n\\label{74}\n\\frac{1}{4} \\il^s_0 d \\lambda \\dot{x} (\\lambda)^T \\dot{x} (\\lambda)  = \\frac{1}{4} (x' - x'')^T \\frac{(eF)^2 s}{\\sin h^2 (e Fs)} (x' - x'') \\, .\n\\ee\nNow we turn to the expression \n\\begin{align}\n \\label{75}\n \\il^s_0 d \\lambda \\{ e \\dot{x}^T A \\} &= \\il^{x'}_{x''} d x^T (e A) \\nonumber\\\\\n &= e  \\il^{x'}_{x''}  d x^T (e A + \\frac{1}{2} F \\cdot (x - x'')) - \n e \\frac{1}{2} \\il^{x'}_{x''} d x^T F \\cdot (x - x'') \\, .\n\\end{align}\nThe first integral on the right-hand-side of (\\ref{75}) is independent of the path of \nintegration, since the curl of the integrand \nvanishes: \n\\begin{align}\n & \\partial_\\mu \\lk A_\\nu + \\frac{1}{2} F_{\\nu \\lambda} (x - x'')^\\lambda \\right) - \n \\partial_\\nu \\lk A_\\mu + \\frac{1}{2} F_{\\mu \\lambda} (x - x'')^\\lambda \\right) \\nonumber\\\\\n &= \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu + \\frac{1}{2} F_{\\nu \\mu} - \\frac{1}{2} F_{\\mu \\nu} \\nonumber\\\\\n &= F_{\\mu \\nu} - F_{\\mu \\nu} = 0 \\, . \\nonumber\n\\end{align}\nHence instead of integrating along the classical path $x (\\lambda)$ we can integrate along the straight path\nconnecting $x'$ and $x''$:\n\\be\n\\label{76}\nq (\\lambda) = x'' + (x' - x'') \\frac{\\lambda}{s} \\, , \\quad \\quad q (0) = x'' \\, , \\quad \\quad\nq (s) = x'\n\\ee\n\\begin{align}\n & e \\il^{x'}_{x''} d x^T \\lk A + \\frac{1}{2} F \\cdot (x - x'') \\right) \\nonumber\\\\\n &= e \\il^{x'}_{x''} d  q^T A + \\frac{e}{2} \\il^{x'}_{x''} d  q^T F \\cdot (q - x'') \\nonumber\\\\\n &= e \\il^{x'}_{x''} d  q^\\mu A_\\mu (q) + \\frac{e}{2} \\il^s_0 \\frac{d \\lambda}{s} \n \\underbrace{(x' - x'') F (x' - x'')}_{(x' - x'')^\\mu F_{\\mu \\nu} \\cdot (x' - x'')^\\nu = 0 \\atop F_{\\mu \\nu} = \n - F_{\\nu \\mu}} \\frac{\\lambda}{s} \\nonumber\\\\\n &= e \\il^{x'}_{x''} d q^\\mu A_\\mu (q) \\, . \\nonumber\n\\end{align}\nFinally we need the second term in (\\ref{75})\n\\be\n\\label{77}\n- \\frac{e}{2} \\il^{x'}_{x''} d x^T F \\cdot (x - x'') = \\frac{e}{2} \\il^{x'}_{x''} d \\lambda \n\\cdot{x}^T F \\cdot (x - x'') \\, .\n\\ee\n\\begin{align}\n \\dot{x}^T F (x - x'') &= (x - x'')^T \\frac{(- 2 e F)}{e^{- 2 e Fs} - 1} \n e^{- 2 e F \\lambda} F \\cdot (x - x'') \\nonumber\\\\\n &= (x - x'')^T \\frac{(- 2 e F)}{e^{- 2 e Fs} - 1} e^{- 2 e F \\lambda} F \n \\frac{e^{2 e F \\lambda} - 1}{e^{2 e F s} - 1} (x' - x'') \\nonumber\\\\\n &= (x' - x'')^T \\frac{(- 2 e F)}{\\cancel{e^{- e Fs}} (e^{- e Fs} - e^{e Fs})} F \n \\frac{1 - e^{- 2 e F \\lambda}}{\\cancel{e^{e Fs}} (e^{e Fs} - e^{- Fs})} (x' - x'') \\nonumber\\\\\n .\/. &= (x' - x'')^T \\frac{e F^2}{\\sin h e Fs} \\cdot \\frac{1}{2} \n \\frac{(1 - e^{- 2 e F \\lambda})}{\\sin h e Fs} (x' - x'') \\nonumber\\\\\n \\il^s_0 d \\lambda .\/. &= \\frac{1}{2} (x' - x'')^T \\frac{e F^2}{\\sin h^2 e Fs} \n \\left[ s + \\frac{1}{2 e F} \\lk e^{- 2e Fs} - 1 \\right) \\right] (x' - x'') \\nonumber\\\\\n &= \\frac{1}{2} (x' - x'')^T \\left[ \\frac{e F^2 s}{\\sin h^2 e Fs} + \\frac{e F^2}{\\sin h^2 e Fs} \n \\frac{1}{2 e F} e^{- e Fs} \\lk e^{- e Fs} - e^{e Fs} \\right) \\right] (x' - x'') \\nonumber\\\\\n &=  \\frac{1}{2} (x' - x'')^T  \\left[ \\frac{e F^2 s}{\\sin h^2 e Fs}  - \\frac{e F^2}{\\sin h^2 e F s} \\frac{e^{- e FS}}{e F} \\sin h e Fs \n \\right] (x' - x'') \\nonumber\\\\\n &= \\frac{1}{2} (x' - x'')^T \\left[ \\frac{e F^2 s}{\\sin h^2 e Fs} - e^{- e Fs} \\frac{F}{\\sin h e Fs} \\right] (x' - x'') \\nonumber\n\\end{align}\nSo the contribution of the integral is given by\n\\be\n\\label{78}\n- \\frac{e}{2} \\il^{x'}_{x''} d x^T F \\cdot (x - x'') = - \\frac{1}{4} (x' - x'')^T \\left[ \\frac{(e F)^2 s}{\\sin h^2 e Fs} - \n\\frac{e^{- e Fs} (e F)}{\\sin h e Fs} \\right] (x' - x'') \\, .\n\\ee\nThe first term in (\\ref{78}) cancels the result (\\ref{74}). So we obtain for the classical action \n\\begin{align}\n S (x', x''; s) & \\underset{\\text{($\\ref{13}$)}}{=}  e \\il^{x'}_{x''} d q^\\mu A_\\mu (q) + \\frac{1}{4} (x' - x'')^T\n e^{- e Fs} \\frac{eF}{\\sin h e Fs} (x' - x'') + \\frac{e}{2} s \\sigma F \\nonumber\\\\\n &= e \\il^{x'}_{x''} d q_\\mu  A^\\mu (q) + \\frac{1}{4} (x' - x'')^T e F \n \\underbrace{\\frac{\\cos h e Fs - \\sin h e Fs}{\\sin h e Fs}}_{(\\cot h e F s - \\cancel{1})_{(x' - x'') F (x' - x'') = 0}} \n (x' - x'') + \\frac{e}{2} \\sigma F s \\nonumber\\\\\n &= e \\il^{x'}_{x''} d q_\\mu A^\\mu (q) + \\frac{1}{4} (x' - x'')^T e F \\cot h (e Fs) (x' - x'') + \\frac{e}{2} \\sigma_{\\mu \\nu}\n F^{\\mu \\nu} s \\, .\\nonumber\n\\end{align}\nThen the final result reads\n\\be\n\\label{79}\nS (x', x''; s) = e \\il^{x'}_{x''} d q_\\mu A^\\mu (q) + \\frac{1}{4} (x' - x'')^\\alpha e F_\\alpha^\\beta [\\cot h (e Fs)]_\\beta^\\gamma (x' - x'')_\\gamma\n+ \\frac{e}{2} \\sigma_{\\mu \\nu} F^{\\mu \\nu} s \\, .\n\\ee\nWith this information we are able to compute the propagation function \nby a classical WKB approximation which is exact for the constant-field case\n\\begin{align}\n \\label{80}\n K (x', s; x'', 0) &= \\il_{x (0) = x'' \\atop x (s) = x'} [d x (\\lambda)] e^{i S [x (\\lambda)]} \\nonumber\\\\\n S &= \\il^s_0 d \\lambda L (x, \\dot{x}) = (\\ref{79})\n\\end{align}\nor\n\\be\n\\label{81}\nK (x', s; x'', 0) = (2 \\pi i)^{- \\frac{n}{2}} \\sqrt{D} e^{i S}\n\\ee\nwith the Van Vleck determinant\n\\be\n\\label{82}\nD (x', x''; s)^{n = 4} = (- 1)^4 \\det \\lk \\frac{\\partial^2 S}{\\partial x'^\\mu \\partial x''^\\nu} \\right) \\, .\n\\ee\nWe begin with\n\\begin{align}\n \\label{83}\n & \\frac{\\partial^2}{\\partial {x'}^\\mu \\partial {x''}^\\nu} \\left[ \\frac{1}{4} (x' - x'')^\\alpha  e F_\\alpha^\\beta \n ( \\cot h e F s)_\\beta^\\gamma \n (x' - x'')_\\gamma \\right] \\nonumber\\\\\n &= \\frac{\\partial}{\\partial x''^\\nu} \\left[ \\frac{1}{4} e F_\\mu^\\beta (\\cot h e Fs)_\\beta^\\gamma (x' - x'')_\\gamma \n + \\frac{1}{4} (x' - x'')^\\alpha e F_\\alpha^\\beta (\\cot h e Fs)_{\\beta \\mu} \\right] \\nonumber\\\\\n &= \\frac{\\partial}{\\partial x''^\\nu} \\left[ \\frac{1}{2} (x' - x'')^\\alpha e F_\\alpha^\\beta (\\cot h e Fs)_{\\beta \\mu} \\right] \\nonumber\\\\\n &= - \\frac{1}{2} e F_\\nu^\\beta (\\cot h e Fs)_{\\beta \\mu} \\, .\n \\end{align}\nFurthermore we have to compute\n\\begin{align}\n\\label{84}\n\\frac{\\partial}{\\partial {x'}^\\mu} \\il^{x'}_{x''} d q^\\alpha A_\\alpha (q) &= \\frac{\\partial}{\\partial {x'}^\\mu} \\il^s_0 d \\lambda\n\\frac{\\partial q^\\alpha}{\\partial \\lambda} A_\\alpha (q) \\nonumber\\\\\n&= \\il^s_0 d \\lambda \\left[ \\lk \\frac{\\partial}{\\partial \\lambda} \\frac{\\partial q^\\alpha}{\\partial {x'}^\\mu} \\right) A_\\alpha (q) + \n\\frac{\\partial q^\\alpha}{\\partial \\lambda} \\frac{\\partial A_\\alpha (q)}{\\partial q^\\beta} \\frac{\\partial q^\\beta}{\\partial {x'}^\\mu} \\right]\n\\nonumber\\\\\n&= \\left. \\frac{\\partial q^\\alpha}{\\partial {x'}^\\mu} A_\\alpha (q) \\right]^{\\lambda = s}_{\\lambda = 0} + \\il^s_0 d \\lambda\n\\left[ - \\frac{\\partial q^\\alpha}{\\partial {x'}^\\mu} \\frac{\\partial A_\\alpha}{\\partial q^\\beta} \\frac{\\partial q^\\beta}{\\partial \\lambda}\n+ \\frac{\\partial q^\\alpha}{\\partial \\lambda} \\frac{\\partial A_\\alpha (q)}{\\partial q^\\beta} \n\\frac{\\partial q^\\beta}{\\partial {x'}^\\mu} \\right]\\nonumber\\\\\nq (s) = x' \\atop q (0) = x'' &= A_\\mu (x') + \\il^s_0 d \\lambda \\frac{\\partial q^\\alpha}{\\partial \\lambda} \n\\left[ \\frac{\\partial A_\\alpha}{\\partial q^\\beta} - \\frac{\\partial A_\\beta}{\\partial q^\\alpha} \\right] \n\\frac{\\partial q^\\beta}{\\partial {x'}^\\mu} \n\\nonumber\\\\\nq (x', x'', \\lambda) &= x'' + (x' - x'') \\frac{\\lambda}{s} : \\quad \\frac{\\partial q^\\alpha}{\\partial \\lambda} = \\frac{(x' - x'')^\\alpha}{s}\n\\nonumber\\\\\n& \\hspace{5cm} \\frac{\\partial q^\\beta}{\\partial {x'}^\\mu} = \\delta_\\mu^\\beta \\frac{\\lambda}{s} \\nonumber\\\\\n\\frac{\\partial}{\\partial {x'}^\\mu} \\il^{x'}_{x''} d q^\\alpha A_\\alpha (q) &= A_\\mu (x') + \\il^s_0 d \\lambda\n(x' - x'')^\\alpha \\frac{1}{s} F_{\\beta \\alpha} \\delta_\\mu^\\beta \\frac{\\lambda}{s} \\nonumber\\\\\n&= A_\\mu (x') + \\frac{1}{2} F_{\\mu \\alpha} (x' - x'')^\\alpha \\, .\n \\end{align}\n   Therefore\n \\be\n \\label{85}\n \\frac{\\partial^2}{\\partial x''^\\nu \\partial x'^\\mu} \\il^{x'}_{x''} d q^\\alpha A_\\alpha (q) = - \\frac{1}{2} F_{\\mu \\nu} \\, .\n \\ee\n Together with (\\ref{83}) we obtain\n \\begin{align}\n  \\label{86}\n  \\frac{\\partial' S}{\\partial x'^\\mu \\partial x''^\\nu} &= - \\frac{1}{2} e F_{\\mu \\nu} - \\frac{1}{2} e\n  F_\\nu^\\beta (\\cot h e Fs)_{\\beta \\mu} \\nonumber\\\\\n  &= - \\frac{1}{2s} \\left[ e Fs + e Fs (\\cot h e Fs) \\right]_{\\mu \\nu} \\nonumber\\\\\n  &= - \\frac{1}{2s} \\left[ e Fs \\frac{\\sin h e Fs + \\cos h e Fs}{\\sin h e Fs} \\right]_{\\mu \\nu} \\nonumber\\\\\n   &= - \\frac{1}{2s} \\left[ e Fs \\frac{e^{e Fs}}{\\sin h e Fs} \\right]_{\\mu \\nu} \\, .\n \\end{align}\nGiven (\\ref{86}), the Van Vleck determinant is given by\n\\begin{align}\n \\label{87}\n n = 4: \\quad D  &= \\det \\left\\{ (-1) \\frac{1}{2s} e Fs \\frac{e^{e Fs}}{\\sin h e Fs} \\right\\} \\nonumber\\\\\n &= - 1 \\frac{1}{16 s^4} \\det \\lk e F s \\frac{e{Fs}}{\\sin h e F s} \\right) \\nonumber\\\\\n &= -  1 \\frac{1}{16 s^4} e^{tr \\ln \\lk \\frac{e Fs}{\\sin h e Fs} e^{e Fs} \\right)} \\nonumber\\\\\n &= - \\frac{1}{16 s^4} e^{- tr \\ln \\frac{\\sin h e Fs}{e Fs} + es \\overbrace{tr F}^{= 0}} \\nonumber\\\\\n \\sqrt{D} &= \\frac{i}{4 s^2} e^{- \\frac{1}{2} tr \\ln \\frac{\\sin h e Fs}{e Fs}} \\, .\n\\end{align}\nWe finally end up with\n\\begin{align}\n  K (x', s; x'', 0)  &= - \\frac{1}{4 \\pi^2} \\frac{i}{4 s^2} e^{- \\frac{1}{2} tr \\ln \\frac{\\sin h e Fs}{e Fs}} e^{i S} \\nonumber\n\\end{align}\nor\n\\begin{align}\n \\label{88}\n   K (x', s; x'', 0)  &= - \\frac{i}{16 \\pi^2} \\frac{1}{s^2} e^{- \\frac{1}{2} tr \\ln \\lk \\frac{\\sin h e Fs}{e Fs} \\right)} \n  \\nonumber\\\\\n  & \\cdot e^{i e \\il^{x'}_{x''} d q_\\mu A^\\mu (q)} e^{i \\left[ \\frac{1}{4} (x' - x'') \\lk e F \\cot h e Fs \\right) \n  (x' - x'') \\right]} e^{i \\frac{e}{2} \\sigma F s} \\, \\\\\n  & \\mbox{holonomy factor} \\quad \\quad \\mbox J.S., P.R. {\\bf 82}, 664 (51) \\nonumber\\\\\n  & \\mbox{counter gauge factor} \\quad \\quad (3.20) \\nonumber\\, .\n\\end{align}\nThis result should be compared with Julian Schwinger's superb article, Physical Review {\\bf 82}, 664 (1951) [ref. 2], formula (3.20).\nA useful result is then given by (c.f. \\ref{80})\n\\begin{align}\n \\label{89}\n & \\il_{x (0) = x'' \\atop x (s) = x'} \\left[d x (\\lambda) \\right] e^{i \\il^s_0 d \\lambda \\left[ \\frac{1}{4} \n \\dot{x}^2 (\\lambda) + e A (x (\\lambda)) \\dot{x} (\\lambda) \\right]} \\nonumber\\\\\n &= \\underbrace{e^{i e \\il^{x'}_{x''} d q_\\mu A^\\mu (q)}}_{\\mbox{gauge dependence} \\atop \\mbox{isolated!}} \n \\frac{- i}{(4 \\pi)^2} \\frac{1}{s^2} e^{- \\frac{1}{2} tr \\ln\n \\lk \\frac{\\sin h e Fs}{e Fs} \\right)} \\nonumber\\\\\n & \\cdot e^{i \\frac{1}{4} (x' - x'') (e F \\cot h e Fs) (x' - x'')} \\, .\n \\end{align}\nIt takes a little more to show that (\\ref{88}) makes its appearance when calculating the Green's function of an \nelectron in an external constant electromagnetic field (in an arbitrary function gauge):\n\\begin{align}\n \\label{90}\n G (x, x' | A ) &= \\phi (x, x' |  A) \\frac{1}{(4 \\pi)^2} \\il^\\infty_0 \\frac{ds}{s^2} \n \\left[ m - \\frac{1}{2} \\gamma^\\mu [f (s) + e F]_{\\mu \\nu} (x - x')^\\nu \\right] \\nonumber\\\\\n& \\cdot e^{- i m^2 s - L (s)} \\exp \\left[ \\frac{1}{4} (x - x') f (s) (x - x') \\right]e^{i \\frac{e}{2} \\sigma Fs} \\, . \n\\end{align}\nHere we have used the abbreviations\n\\be\nf (s) = e F \\cot h (e Fs) \\quad \\mbox{and} \\quad L (s) = \\frac{1}{2} tr \\ln \\frac{\\sin h (e Fs)}{e Fs} \\, , \\nonumber\n\\ee\nand\n\\be\n\\phi (x, x' | A) = \\exp \\left\\{ i e \\il^x_{x'} d \\zeta_\\mu \\left[ A^\\mu (\\zeta) + \\frac{1}{2} F^{\\mu \\nu} (\\zeta - x')_\\nu \\right]\n\\right\\} \\nonumber\n\\ee\ncarries completely the gauge dependence of the propagation function. Needless to say, with (\\ref{90}) in our hands, we can compute several \nimportant processes in QED, as there are the effective Lagrangian, in particular the one-loop effective Lagrangian, the pair-production rate, the Heisenberg-Euler\nLagrangian, photon-photon scattering, the axial vector anomaly, dispersion effects for low-frequency photons and many more \\cite{3}.\n\\bi\n\n\\no\nAlso note that in our last relativistic problem we never used operator quantum field theory.\nWe always dealt with measured physical quantities like, for instance, the charge and mass of the electron,\nand nowhere were we confronted with any kind of renormalisation procedure. \n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and terminology}\n\n\nIn the non-Archimedean setting, at least two notions of differentiability have been defined: classical and strict derivative. Classical derivative has some unpleasant and strange behaviors, but it has been long known that the strict differentiability in the sense of Bourbaki is the hypothesis most useful to geometric applications, such as inverse theorem. Let us recall  definitions.\n\nLet $\\mathbb K$ be a valued field containing $\\mathbb{Q}_p$ such that $\\mathbb K$ is complete (as a metric space), and  $X$ be a nonempty subset of $\\mathbb K$ without isolated points.\nLet $f\\colon X \\to \\mathbb K$ and $r$ be a natural number. Set\n\t$$\n\t\\nabla^{r} X:=\\begin{cases}\n\tX & \\text{if } r=1,\\\\\n\t\\left\\{\\left(x_1, x_2,\\ldots, x_r\\right)\\in X^r \\colon x_i\\neq x_j \\text{ if } i\\neq j \\right\\} & \\text{if } r\\geq 2.\n\t\\end{cases}\n\t$$\nThe $r$-th difference quotient $\\Phi_rf: \\nabla^{r+1} X \\to \\mathbb K$ of $f$, with $r\\geq 0$, is recursively given by $\\Phi_0f=f$ and, for $r\\geq 1$, $(x_1, x_2,\\ldots, x_{r+1})\\in \\nabla^{r+1} X$ by \n\t$$\n\t\\Phi_r f(x_1, x_2,\\ldots, x_{r+1})=\\frac{\\Phi_{r-1}f(x_1, x_3,\\ldots, x_{r+1})-\\Phi_{r-1}f(x_2, x_3,\\ldots, x_{r+1})}{x_1-x_2}.\n\t$$\nThen a function  $f\\colon X \\to \\mathbb K$ at a point $a\\in X$  is said to be:\n\\begin{itemize}\n\t\\item \\textbf{differentiable}  if     $\\lim_{x\\to a}f(x)-f(a)\/(x-a)$ exists;\n\t\n\t\\item \\textbf{strictly differentiable of order $r$} if  $\\Phi_r f$ can be extended to a continuous function $\\overline{\\Phi}_r  f: X^{r+1} \\to \\mathbb K$. We then set $D_rf(a)=\\overline{\\Phi}_r  f(a, a,\\ldots, a)$.\n\\end{itemize}\n\n\n\nOur aim in this work is to study these notions through a recently coined approach--the lineability theory. Searching  for large algebraic structures in the sets of objects with a special  property,  we, in this approach, can get  deeper understanding of the behavior of  the objects under discussion.   In \\cites{jms1,jms2}  authors studied lineability notions in the $p$-adic analysis; see\nalso \\cites{preprint-June-2019,jms3, fmrs}. The study of lineability  can be traced back to   Levine\nand Milman \\cites{levinemilman1940} in 1940, and  V. I. Gurariy \\cite{gur1} in 1966.  These works, among\nothers, motivated the introduction of the notion of   lineability  in 2005 \\cite%\n{AGS2005} (notion coined by V. I. Gurariy). Since then it has been a rapidly developing trend in mathematical analysis. There are extensive works on the classical lineability theory, see e.g.  \\cites{ar,AGS2005,ar2,survey,bbf,bay}, whereas some recent topics can be found in \\cites{TAMS2020,abms,bns,cgp,nak,CS2019,bo,egs,bbls,ms}. It is interesting to note that Mahler in \\cite{Ma2} stated that:\n\\begin{quote}\n``On the other hand, the behavior of continuous functions of a $p$-adic variable is quite distinct from that of real continuous functions, and many basic theorems of real analysis have no $p$-adic analogues. \n\n$\\ldots$ there exist infinitely many linearly independent non-constant functions the derivative of which vanishes identically $\\ldots$''. \n\\end{quote}  \n\nBefore further going, let us recall three essential notions. \nLet $\\kappa$ be  a cardinal number. We say that a subset $A$ of a vector space $V$ over a field $\\mathbb{K}$ is \n\\begin{itemize}\n\n\t\\item \\textbf{$\\kappa$-lineable} in $V$   if there exists a vector space $M$ of dimension $\\kappa$ and $%\n\tM\\setminus \\{0\\} \\subseteq A$;\n\\end{itemize}\t \n\tand following \\cites{ar2, ba3}, if $V$ is\n\tcontained in a (not necessarily unital) linear algebra,  then $A$ is called\n\\begin{itemize}[resume*]\t\n\t\\item \\textbf{$\\kappa$-algebrable} in $V$ if there exists an algebra $M$ such that $%\n\tM\\setminus \\{0\\} \\subseteq A$ and $M$ is a $\\kappa$-dimensional vector\n\tspace;\n\t\n\t\\item \\textbf{strongly $\\kappa$-algebrable} in $V$ if there exists an $\\alpha$-generated free algebra $M$ such that $M\\setminus \\{0\\}\\subseteq A$.\n\\end{itemize}\nNote that if $V$ is also contained in a commutative algebra, then a set $B\\subset V$ is a generating set of a free algebra contained in $A$ if and only if for any $n\\in \\mathbb N$ with $n\\leq \\text{card}(B)$ (where $\\text{card}(B)$ denotes the cardinality of $B$), any nonzero polynomial $P$ in $n$ variables with coefficients in $\\mathbb K$ and without free term, and any distinct $b_1,\\ldots,b_n\\in B$, we have $P(b_1,\\ldots,b_n)\\neq 0$.\n\nNow we can give an outline of our work. In Section \\ref{bac} we  recall some standard concepts and notations concerning\nnon-Archimedean   analysis. Then, in the section of Main results, we \nshow, among other things, that: \n(i) the set of  functions $\\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ with continuous derivative that are not strictly differentiable   is $\\mathfrak{c}$-lineable ($\\mathfrak{c}$ denotes the cardinality of the continuum), \n(ii) the set of strictly differentiable functions $\\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ of order $r$ but not strictly differentiable of order $r+1$ is $\\mathfrak c$-lineable,\n(iii) the set of all strictly differentiable functions $\\mathbb{Z}_p\\rightarrow \\mathbb K$ with zero derivative  that are not Lipschitzian of any order $\\alpha>1$ is $\\mathfrak{c}$-lineable and $1$-algebrable, \n(iv) the set of differentiable functions $\\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ which derivative is unbounded is strongly $\\mathfrak{c}$-algebrable, \n(v) the set of continuous functions $\\mathbb{Z}_p\\rightarrow \\mathbb{Q}_p$ that are differentiable with bounded derivative on a full set for any positive real-valued Haar measure on $\\mathbb{Z}_p$ but not differentiable on its complement having cardinality $\\mathfrak{c}$ is $\\mathfrak c$-lineable.\n\n\n\\section{Preliminaries for $p$-adic analysis}\n\\label{bac}\n\nWe   summarize  some basic definitions of $p$-adic analysis (for a\nmore profound treatment of this topic we refer the interested reader to \\cites{go,kato,va,sc}).\n\nWe shall use standard set-theoretical notation. As usual, $\\mathbb{N}, \n\\mathbb{N}_0, \\mathbb{Z}, \\mathbb{Q}$, $\\mathbb{R}$ and $\\mathbb P$ denote\nthe sets of all natural, natural numbers including zero, integer, rational, real, and prime numbers, respectively.\nThe restriction of a function $f$ to a set $A$ and the characteristic function of a set $A$ will be denoted\nby $f\\restriction A$ and $1_A$, respectively.\n\nFrequently,  we use a theorem due to Fichtenholz-Kantorovich-Hausdorff \\cites{fich,h}: For any set $X$ of infinite cardinality there exists a family $\\mathcal{B}\\subseteq \n\\mathcal{P}(X)$ of cardinality $2^{\\text{card}(X)}$ such that for any finite\nsequences $B_1,\\ldots , B_n\\in \\mathcal{B}$ and $\\varepsilon_1, \\ldots,\n\\varepsilon_n \\in \\{0,1\\}$ we have $B_1^{\\varepsilon_1}\\cap \\ldots \\cap\nB_n^{\\varepsilon_n}\\neq \\emptyset$, where $B^1=B$ and $B^0=X\\setminus B$.  \nA family of subsets of $X$ that satisfy the latter condition is called a family of \\textit{independent subsets} of $X$.\nHere $\\mathcal{P}(X)$ denotes the power set of $X$. In what follows we fix  $\\mathcal{N}$, $\\mathcal N_0$ and $\\mathcal P$ for families of independent subsets of $\\mathbb{N}$, $\\mathbb N_0$ and $\\mathbb P$, respectively,\nsuch that $\\text{card}(\\mathcal N)=\\text{card}(\\mathcal N_0)=\\text{card}(\\mathcal P)=\\mathfrak c$.\n\n\nNow let us recall that given a field $\\mathbb K$, an absolute value on $\\mathbb K$ is a function \n\t$$\n\t|\\cdot|\\colon \\mathbb K\\to [0,+\\infty)\n\t$$ \nsuch that:\n\t\\begin{itemize}\n\t\t\\item $|x|=0$ if and only if $x=0$,\n\t\t\n\t\t\\item $|x y|=|x||y|$, and\n\t\t\n\t\t\\item $|x+y|\\leq |x|+|y|$,\n\t\\end{itemize}\nfor all $x,y\\in \\mathbb{K}$. \nThe last condition is the so-called \\textit{triangle inequality}.\nFurthermore, if $(\\mathbb K,|\\cdot|)$ satisfies the condition $|x+y|\\leq \\max\\{|x|, |y|\\}$ (the \\textit{strong triangle inequality}), then $(\\mathbb K,|\\cdot|)$ is called a non-Archimedean field.\nNote that $(\\mathbb K,|\\cdot|)$ is a normed space since $\\mathbb K$ is a vector space in itself.\nFor simplicity, we will denote for the rest of the paper $(\\mathbb K,|\\cdot|)$ by $\\mathbb K$.\nClearly, $|1|=|-1|=1$ and, if $\\mathbb K$ is a non-Archimedean field, then $\\underbrace{|1+\\cdots+1|}_{n \\text{ times}}\\leq 1$ \nfor all $n\\in \\mathbb{N}$. An immediate consequence of the strong triangle\ninequality is that $|x|\\neq |y|$ implies $|x+y|=\\max\\{|x|, |y|\\}$.\nNotice that if $\\mathbb K$ is a finite field, then the only possible absolute value that can be defined on $\\mathbb K$ is the trivial absolute value, that is, $|x|=0$ if $x=0$, and $|x|=1$ otherwise.\nFurthemore, given any field $\\mathbb K$, the topology endowed by the trivial absolute value on $\\mathbb K$ is the discrete topology.\n\n\n\nLet us fix a prime number $p$ throughout this work. For any non-zero integer \n$n\\neq 0$, let $\\text{ord}_p (n)$ be the highest power of $p$ which divides $%\nn$. Then we define $|n|_p=p^{-\\text{ord}_p (n)}$, $|0|_p=0$ and $|\\frac{n}{m}%\n|_p=p^{-\\text{ord}_p (n)+\\text{ord}_p (m)}$, the $p$-adic absolute value. The completion of the field of\nrationals, $\\mathbb{Q}$, with respect to the $p$-adic absolute value\nis called the field of $p$-adic numbers $\\mathbb{Q}_p$. \nAn important property of $p$-adic numbers is that each nonzero $x\\in \\mathbb{Q}_p$ can be represented as\n\t$$\n\tx=\\sum_{n=m}^\\infty a_n p^n,\n\t$$\nwhere $m\\in \\mathbb Z$, $a_n\\in \\mathbb F_p$ (the finite field of $p$ elements) and $a_m\\neq 0$.\nIf $x=0$, then $a_n=0$ for all $n\\in \\mathbb Z$.\nThe $p$-adic absolute value\nsatisfies the strong triangle inequality. Ostrowski's Theorem states that every nontrivial\nabsolute value on $\\mathbb{Q}$ is equivalent (i.e., defines the same\ntopology) to an absolute value $|\\cdot|_p$, where $p$ is a prime number, or\nthe usual absolute value (see \\cite{go}). \n\nLet $a\\in \\mathbb{Q}_{p}$ and $r$ be a positive number. The set $%\nB_r(a)=\\{x\\in \\mathbb{Q}_{p}\\colon |x-a|_{p}<r\\}$ is called the open ball of\nradius $r$ with center $a$, $\\overline{B}_{r}(a)=\\{x\\in \\mathbb{Q}%\n_{p}\\colon |x-a|_{p}\\leq r\\}$ the closed ball of radius $r$ with center $a$, and $%\nS_r(a)=\\{x\\in \\mathbb{Q}_{p}\\colon |x-a|_{p}=r\\}$ the sphere of radius $r$ and\ncenter $a$. \nIt is important to mention that $B_r(a)$, $\\overline{B}_{r}(a)$ and $S_r(a)$ are clopen sets in $\\mathbb{Q}_p$.\nThe closed unit ball\n $$\n \\mathbb{Z}_{p}=\\{x\\in \\mathbb{Q}_{p}\\colon |x|_{p}\\leq 1\\}=\\left\\{x\\in \\mathbb{Q}_{p}\\colon x=\\sum_{i=k}^\\infty a_ip^i,\\ a_i\\in \\{0,1,\\ldots ,p-1\\}, \n \\ k\\in \\mathbb N_0 \\right\\}\n $$\n is called the ring of $p$-adic integers in $\\mathbb{Q}_{p}$. We know that $\\mathbb{Z}_{p}$ is a compact set and $\\mathbb{N}$ is\ndense in $\\mathbb{Z}_{p}$ (\\cite{go}). \n\n\nThroughout this article we shall consider all vector spaces and algebras taken over the\nfield $\\mathbb{Q}_p$ (unless stated otherwise).\n\n\n\\section{Main results}\n\nWe are ready to present our results. For the rest of this work, $X$ will denote a nonempty subset of $\\mathbb K$ without isolated points.\nLet us fix two notations:\n\t$$\n\tC^1(X,\\mathbb K)=\\{f\\colon X \\to \\mathbb K \\colon f \\text{ has continuous (classical) derivative on }X\\},\n\t$$\n\t$$\n\tS^r(X,\\mathbb K)=\\{f\\colon X \\to \\mathbb K \\colon f \\text{ is strictly differentiable of order }r \\text{ on }X\\}.\n\t$$\n\nOur first result shows that, unlike the classical case, strict  differentiability is a stronger condition than simply having continuous derivative. \nAn analogue to part (ii) of the result  for the classical case can be found in \\cite{bbf}.\n\n\\begin{theorem}\n\\label{34} \n~\\begin{itemize}\n\\item[(i)] The set $C^1(\\mathbb{Q}_p,\\mathbb{Q}_p)\\setminus S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)$ is $\\mathfrak{c}$-lineable.\n\\item[(ii)] The set $S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)\\setminus S^{2}(\\mathbb{Q}_p,\\mathbb{Q}_p)$ is $\\mathfrak{c}$-lineable. In general, $S^{r}(\\mathbb{Q}_p,\\mathbb{Q}_p)\\setminus S^{r+1}(\\mathbb{Q}_p,\\mathbb{Q}_p)$ is $\\mathfrak{c}$-lineable for any $r\\geq 1$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\t\n\\textit{(i).} Notice that $B_{p^{-2n}}(p^n)\\subset S_{p^{-n}}(0)$ for every $n\\in\\mathbb N$, therefore $B_n\\cap B_m=\\emptyset$ if, and only if, $n\\neq m$.\nAlso, let us define $f_N\\colon \\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ for every $N\\in\\mathcal N$ as follows:\n$$\nf_N(x)=\\begin{cases}\np^{2n} & \\text{if }x\\in B_{p^{-2n}}(p^n) \\text{ with } n\\in N,\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n$$\n\tFirst, notice that $f_N$ is locally constant outside $0$ and, thus, $f_N$ is differentiable everywhere except maybe at  $0$ with $f_N'(x)=0$ for every $x\\in\\mathbb{Q}_p\\setminus\\{0\\}$.\n\tMoreover, we have \n\t$$\n\t\\left|\\frac{f_N(x)-f_N(0)}{x}\\right|_p=\\left|\\frac{f_N(x)}{x}\\right|_p=\\begin{cases}\n\tp^{-n} & \\text{if } x\\in B_{p^{-2n}}(p^n) \\text{ with } n\\in N,\\\\\n\t0 & \\text{otherwise},\n\t\\end{cases}\n\t$$\n\ti.e., $\\left|\\frac{f_N(x)-f_N(0)}{x}\\right|_p\\rightarrow 0$ as $x\\rightarrow 0$.\n\tHence, $f_N^\\prime(0)=0$.\n\tTherefore, $f_N^\\prime$ exists everywhere and is continuous since $f_N^\\prime\\equiv 0$, that is, $f_N\\in C^1(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\t\n\tIt is enough to show that the family of functions $V=\\{f_N\\colon N\\in\\mathcal N \\}$ is linearly independent over $\\mathbb{Q}_p$ and the vector space generated by $V$, denoted by $\\text{span}(V)$, satisfies $\\text{span}(V)\\setminus \\{0\\}\\subset C^1(\\mathbb{Q}_p,\\mathbb{Q}_p)\\setminus S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\tTake $f=\\sum_{i=1}^m \\alpha_i f_{N_i}$, where $\\alpha_1,\\ldots, \\alpha_m\\in\\mathbb{Q}_p$, $N_1,\\ldots, N_m\\in\\mathcal N$ are distinct and $m\\in \\mathbb N$.\n\tAssume that $f\\equiv 0$ then, by taking $x=p^n$ with $n\\in N_1^1\\cap N_2^0\\cap\\cdots \\cap N_m^0$, we have that $0=f(x)=\\alpha_1 f_{N_1}(x)=\\alpha_1 p^{2n}$, i.e., $\\alpha_1=0$.\n\tTherefore, applying similar arguments, we arrive at $\\alpha_i=0$ for every $i\\in\\{1,\\ldots, m \\}$.\n\tAssume now that $\\alpha_i\\neq 0$ for every $i\\in\\{1,\\ldots, m \\}$.\n\tSince $C^1(\\mathbb{Q}_p,\\mathbb{Q}_p)$ forms a vector space, we have that $f\\in C^1(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\tMoreover, by construction we have $f'\\equiv 0$.\n\tIt remains to prove that $f\\notin S^1(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\tTo do so, take the sequences $$(x_n)_{n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0}=(p^n)_{n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0}$$ and $$(y_n)_{n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0}=(p^n-p^{2n})_{n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0}.$$\n\tNotice that both sequences converge to $0$, and $f(x_n)=\\alpha_1 p^{2n}$ and $f(y_n)=0$ for each $n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0$.\n\tHence, \n\t\t$$\n\t\t\\frac{f(x_n)-f(y_n)}{x_n-y_n}=\\alpha_1,\n\t\t$$\n\tfor every $n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_m^0$.\n\tSince $\\alpha_1\\neq 0$, we have the desired result.\n\t\n\t\\textit{(ii).} We will prove the case when $r=1$ since the general case can be easily deduced.\n\tFor every nonempty subset $N$ of $\\mathbb N$, let us define $g_N \\colon \\mathbb{Q}_p\\rightarrow\\mathbb{Q}_p$ as follows: for every $x=\\sum_{n=m}^\\infty a_n p^n\\in\\mathbb{Q}_p$, take\n\t\t$$\n\t\tg_N(x)=\\sum_{n=0}^\\infty b_n p^{2n},\n\t\t$$\n\twhere\n\t\t$$\n\t\tb_n=\\begin{cases}\n\t\ta_n & \\text{if } n\\in N,\\\\\n\t\t0 & \\text{otherwise},\n\t\t\\end{cases}\n\t\t$$\n\tfor every $n\\in\\mathbb N_0$.\n\tFor any $x,y\\in \\mathbb{Q}_p$, notice that \n\t\t$$\n\t\t|g_N(x)-g_N(y)|_p\\leq |x-y|_p^2.\n\t\t$$\n\tHence, \n\t\t$$\n\t\t\\left|\\frac{g_N(x)-g_N(z)}{x-z} \\right|_p\\leq |x-z|_p\\rightarrow 0\n\t\t$$\n\tas $x\\rightarrow z$ for any $z\\in\\mathbb{Q}_p$, that is, $g_N$ is differentiable and $g_N^\\prime\\equiv 0$.\n\tMoreover,\n\t\t$$\n\t\t\\left|\\frac{g_N(x)-g_N(y)}{x-y} \\right|_p\\leq |x-y|_p\\rightarrow 0,\n\t\t$$\n\twhen $(x,y)\\rightarrow (z,z)$ for any $z\\in\\mathbb{Q}_p$, and where $(x,y)\\in \\nabla^2 \\mathbb{Q}_p$.\n\tThus, $g_N\\in S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\n\tWe will prove that the family of functions $W=\\{g_N \\colon N\\in\\mathcal N \\}$ is linearly independent over $\\mathbb{Q}_p$ and $\\text{span}(W)\\setminus\\{0\\}\\subset S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)\\setminus S^{2}(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\tIt is easy to see that any linear combination of the functions in $W$ belongs to $S^{1}(\\mathbb{Q}_p,\\mathbb{Q}_p)$.\n\tTake now $g=\\sum_{i=1}^k \\beta_i g_{N_i}$, where $\\beta_1,\\ldots,\\beta_k\\in \\mathbb{Q}_p$, $N_1,\\ldots,N_k\\in\\mathcal N$ are distinct and $k\\in\\mathbb N$.\n\tAssume that $g\\equiv 0$ then, by taking $x=p^{n}$, with $n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap  N_k^0$ fixed, we have that $0=g(x)=\\beta_1 p^{2n}$, i.e., $\\beta_1=0$.\n\tBy applying similar arguments we see that $\\beta_i=0$ for every $i\\in \\{1,\\ldots,k \\}$.\n\tTherefore, assume that $\\beta_i\\neq 0$ for every $i\\in \\{1,\\ldots,k \\}$.\n\tFor every $n\\in  N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0$, denote\n\t\t$$\n\t\tn_{+}=\\min\\{l\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0 \\colon l>n \\}.\n\t\t$$\n\tNow consider the sequences $\\overline x=(x_n)_{N_1^1\\cap N_2^0\\cap \\cdots \\cap  N_k^0}$, $\\overline y=(y_n)_{N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0}$ and $\\overline z=(z_n)_{n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0}$ defined as $x_n=p^n$, $y_n=0$ and $z_n=p^{n}+p^{n_+}$ for every $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0$.\n\t(Notice that the sequences $\\overline{x}$, $\\overline{y}$ and $\\overline{z}$ converge to $0$.)\n\tThen, \n\t\t\\begin{align*}\n\t\t\t& \\left|(y_n-z_n)^{-1}\\right|_p \\left|\\frac{g(x_n)-g(y_n)}{x_n-y_n}-\\frac{g(x_n)-g(z_n)}{x_n-z_n} \\right|_p\\\\\n\t\t\t& = \\left|(p^n+p^{n_+})^{-1}\\right|_p \\left|\\frac{\\beta_1 p^{2n}}{p^n}-\\frac{\\beta_1 p^{2n}-\\beta_1 p^{2n}-\\beta_1 p^{2n_{+}} }{p^n-p^n-p^{n_+}} \\right|_p\\\\\n\t\t\t& = |\\beta_1|_p \\left|\\frac{p^n-p^{n_+}}{p^n+p^{n_+}} \\right|_p=|\\beta_1|_p\\neq 0,\n\t\t\\end{align*}\n\tfor every $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap  N_k^0$.\n\tHowever, $g^{\\prime\\prime}\\equiv 0$.\n\tThis finishes the proof.\n\\end{proof}\n\nLet $\\mathbb K$ be a non-Archimedean field with absolute value $|\\cdot|$ that contains $\\mathbb{Q}_p$.\nFor any $\\alpha>0$, the space of Lipschitz functions from $X$ to $\\mathbb K$ of order $\\alpha$ is defined as\n\t$$\n\t\\text{Lip}_\\alpha (X,\\mathbb K)=\\{f\\colon X\\rightarrow \\mathbb K \\colon \\exists M>0(|f(x)-f(y)|\\leq M|x-y|^\\alpha),\\forall x,y\\in X \\}.\n\t$$\nLet\n\t$$\n\tN^1(X,\\mathbb K)=\\{f\\in S^1(X,\\mathbb K)\\colon f' \\equiv 0 \\}.\n\t$$\nIn view of \\cite{sc}*{Exercise~63.C} we have\n\t$$\n\tN^1(\\mathbb{Z}_p,\\mathbb K)\\setminus \\bigcup_{\\alpha>1} \\text{Lip}_\\alpha (\\mathbb{Z}_p, \\mathbb K)\\neq \\emptyset.\n\t$$\n\t\nTo prove the next theorem, we need a  characterization of the spaces $N^1(\\mathbb{Z}_p,\\mathbb K)$ and $\\text{Lip}_\\alpha (\\mathbb{Z}_p,\\mathbb K)$ from \\cite{sc}. \nFor this we will denote by $(e_n)_{n\\in \\mathbb N_0}$ the van der Put base of $C(\\mathbb{Z}_p,\\mathbb K)$, which is given by $e_0\\equiv 1$ and $e_n$ is  the characteristic function of $\\{x\\in\\mathbb{Z}_p \\colon |x-n|_p<1\/n \\}$ for every $n\\in \\mathbb N$.\n\n\\begin{proposition}\\label{thm:1}~\\begin{itemize}\n\t\\item[(i)] \tLet $f=\\sum_{n=0}^\\infty a_n e_n \\in C(\\mathbb{Z}_p,\\mathbb K)$.\n\tThen $f\\in N^1(\\mathbb{Z}_p,\\mathbb K)$ if and only if  $(|a_n| n)_{n\\in \\mathbb N_0}$ converges to $0$ (see \\cite{sc}*{Theorem~63.3}).\\label{thm:63.3}\n\t\n\t\\item[(ii)]\n\tLet $f=\\sum_{n=0}^\\infty a_n e_n \\in C(\\mathbb{Z}_p,\\mathbb K)$ and $\\alpha>0$.\n\tThen $f\\in \\text{Lip}_\\alpha (\\mathbb{Z}_p,\\mathbb K)$ if and only if \n\t$$\n\t\\sup \\{|a_n| n^\\alpha \\colon n\\in\\mathbb N_0 \\}<\\infty\n\t$$ \n\t(see \\cite{sc}*{Exercise~63.B}).\\label{ex:63.B}\n\\end{itemize}\n\\end{proposition} \n\nThe next result shows that there is a vector space of maximum dimension of  strictly differentiable functions with zero derivative that are not Lipschitzian.\nOur next three results can be compared with some results obtained in \\cites{jms, bbfg,bq} for the classical case.\n\n\\begin{theorem}\n\tThe set $N^1(\\mathbb{Z}_{p},\\mathbb K)\\setminus\\bigcup_{\\alpha>1}{\\rm{Lip}}_{\\alpha }(\\mathbb{Z}_{p},\\mathbb K)$ is $\\mathfrak{c}$-lineable (as a $\\mathbb K$-vector space).\n\\end{theorem}\n\n\\begin{proof}\n\tFix $n_1\\in\\mathbb N$ and take $B_1=\\{x\\in \\mathbb Z_p\\colon |x-n_1|_p<1\/n_1 \\}$.\n\tSince $\\mathbb Z_p$ and $B_1$ are clopen sets, we have that $\\mathbb Z_p\\setminus B_1$ is open and also nonempty.\n\tTherefore there exists an open ball $D_1\\subset \\mathbb Z_p\\setminus B_1$.\n\tFurthermore, as $\\mathbb N$ is dense in $\\mathbb Z_p$, there exists $n_2\\in \\mathbb N\\setminus \\{n_1\\}$ such that $B_2=\\{ x\\in \\mathbb Z_p\\colon |x-n_2|_p<1\/n_2 \\}\\subset D_1$.\n\tBy recurrence, we can construct a  set  $M=\\{n_k\\colon k\\in\\mathbb N \\}\\subset \\mathbb N$ such that the balls $B_k=\\{x\\in\\mathbb Z_p\\colon |x-n_k|_p<1\/n_k \\}$ are pairwise disjoint.\n\t\n\tLet $\\sigma\\colon \\mathbb N_0\\rightarrow M$ be the increasing bijective function and let $(m_n)_{n\\in\\mathbb N}\\subset \\mathbb N$ be an increasing sequence such that $p^{-m_n}n\\rightarrow 0$ and $p^{-m_n}n^\\alpha\\rightarrow\\infty$ for every $\\alpha>1$ when $n\\rightarrow\\infty$. (This can be done for instance by taking $m_n=\\left\\lfloor \\ln(n \\ln(n))\/\\ln(p) \\right\\rfloor$.)\n\tFor every $N\\in\\mathcal N_0$, define $f_N\\colon \\mathbb Z_p\\rightarrow \\mathbb K$ as\n\t\t$$\n\t\tf_N=\\sum_{n=0}^\\infty 1_N(n) p^{m_{\\sigma(n)}} e_{\\sigma(n)}.\n\t\t$$\n\tSince every $N\\in\\mathcal N_0$ is infinite, we have that $|1_N(n)| p^{-m_{\\sigma(n)}} \\sigma(n)\\rightarrow 0$ when $n\\rightarrow \\infty$ and \n\t\t$$\n\t\t\\{|1_N(n)| p^{-m_{\\sigma(n)}} \\sigma(n)^\\alpha \\colon n\\in\\mathbb N_0 \\}\n\t\t$$ \n\tis unbounded for every $\\alpha>1$.\n\tHence, by Theorem~\\ref{thm:1}, we have $f_N\\in N^1(\\mathbb{Z}_{p},\\mathbb K)\\setminus\\bigcup_{\\alpha>1}{\\rm{Lip}}_{\\alpha }(\\mathbb{Z}_{p},\\mathbb K)$ for every $N\\in\\mathcal N_0$.\n\t\n\tWe will prove now that the functions in $V=\\{f_N\\colon N\\in\\mathcal N_0 \\}$ are linearly independent over $\\mathbb K$ and such that any nonzero linear combination of $V$ over $\\mathbb K$ is contained in $N^1(\\mathbb{Z}_{p},\\mathbb K)\\setminus\\bigcup_{\\alpha>1}{\\rm{Lip}}_{\\alpha }(\\mathbb{Z}_{p},\\mathbb K)$.\n\tTake $f=\\sum_{i=1}^r a_i f_{N_i}$, where $a_1,\\ldots, a_r\\in\\mathbb K$, $N_1,\\ldots, N_r\\in\\mathcal N_0$ are distinct and $r\\in\\mathbb N$.\n\tAssume that $f\\equiv 0$. Fix $n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_r^0$ and take $x\\in\\mathbb Z_p$ such that $x\\in B_{\\sigma(n)}$, then $0=f(x)=a_1 p^{m_{\\sigma(n)}}$, i.e., $a_1=0$.\n\tBy applying similar arguments we have $a_i=0$ for every $i\\in\\{1,\\ldots,r \\}$.\n\tAssume for the rest of the proof that $a_i\\neq 0$ for every $i\\in\\{1,\\ldots,r \\}$.\n\tNotice that $f=\\sum_{n=0}^\\infty \\left(\\sum_{i=1}^r a_i 1_{N_i}\\right)(n) p^{m_{\\sigma(n)}} e_{\\sigma(n)}$, where $\\left|\\left(\\sum_{i=1}^r a_i 1_{N_i}\\right)(n) p^{m_{\\sigma(n)}}\\right|\\leq |p^{m_{\\sigma(n)}}| \\max\\{|a_i|\\colon i\\in\\{1,\\ldots,r \\} \\}=p^{-m_{\\sigma(n)}} \\max\\{|a_i|\\colon i\\in\\{1,\\ldots,r \\} \\}$.\n\tTherefore $\\left|\\left(\\sum_{i=1}^r a_i 1_{N_i}\\right)(n) p^{m_{\\sigma(n)}}\\right| \\sigma(n)\\rightarrow 0$ when $n\\rightarrow \\infty$.\n\tFinally, as $N_1^1\\cap N_2^0\\cap \\cdots\\cap N_r^0$ is infinite, we have that \n\t\t\\begin{align*}\n\t\t\t& \\left\\{\\left|\\left(\\sum_{i=1}^r a_i 1_{N_i}\\right)(n) p^{m_{\\sigma(n)}}\\right| \\sigma(n)^\\alpha \\colon n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_r^0 \\right\\}\\\\\n\t\t\t& = \\{|a_1| p^{-m_{\\sigma(n)}} \\sigma(n)^\\alpha \\colon n\\in N_1^1\\cap N_2^0\\cap \\cdots\\cap N_r^0 \\}\n\t\t\\end{align*}\n\tis unbounded for every $\\alpha>1$.\n\\end{proof}\n\n\n\nThe next lineability result can be considered as a non-Archimedean counterpart of  \\cite{g7}*{Theorem 6.1}.\nTo prove it we will make use of the following lemma.\n(For more information on the usage of the lemma see \\cite[Section~5]{fmrs}.)\nIn order to understand it, let us consider the functions $x\\mapsto (1+x)^\\alpha$ where $x\\in p\\mathbb Z_p$ and $\\alpha\\in \\mathbb Z_p$.\nIt is well known that $(1+x)^\\alpha$ is defined analytically by $(1+x)^\\alpha = \\sum_{i=0}^\\infty {\\alpha \\choose i} x^i$.\nMoreover $x\\mapsto (1+x)^\\alpha$ is well defined (see \\cite[Theorem~47.8]{sc}), differentiable with derivative $\\alpha(1+x)^{\\alpha-1}$, and $x\\mapsto (1+x)^\\alpha$ takes values in $\\mathbb Z_p$ (in particular $(1+x)^\\alpha=1+y$ for some $y\\in p\\mathbb Z_p$, see \\cite[Theorem~47.10]{sc}).\n\n\\begin{lemma}\\label{lem:1}\n\tIf $\\alpha_1,\\ldots,\\alpha_n\\in \\mathbb Z_p\\setminus \\{0\\}$ are distinct, with $n\\in \\mathbb N$, then every linear combination $\\sum_{i=1}^n \\gamma_i (1+x)^{\\alpha_i}$, with $\\gamma_i\\in \\mathbb Q_p\\setminus \\{0\\}$ for every $1\\leq i\\leq n$, is not constant.\n\\end{lemma}\n\n\n\\begin{theorem}\n\t\\label{16} The set of everywhere differentiable functions $\\mathbb{Q}_{p}\\rightarrow \\mathbb{Q}_{p}$\n\twhich derivative is unbounded is strongly $\\mathfrak{c}$-algebrable.\n\\end{theorem}\n\n\\begin{proof}\n\tLet $\\mathcal H$ be a Hamel basis of $\\mathbb Q_p$ over $\\mathbb Q$ contained in $p\\mathbb Z_p$, and for each $\\beta \\in \\mathbb Z_p\\setminus\\{0\\}$ define $f_\\beta : \\mathbb Q_p\\to \\mathbb Q_p$ by\n\t\t$$\n\t\tf_\\beta (x)=\\begin{cases}\n\t\tp^{-n} (1+y)^\\beta & \\text{if } x=\\sum_{k=-n}^0 a_k p^k +y, \\text{ where } a_n\\neq 0,\\ n\\in \\mathbb N_0 \\text{ and } y\\in p\\mathbb{Z}_p,\\\\\n\t\t0 & \\text{otherwise}.\n\t\t\\end{cases}\n\t\t$$\n\tThe function $f_\\beta$ is differentiable everywhere for any $\\beta\\in \\mathbb Z_p$.\n\tIndeed, firstly we have that $f_\\beta$ is locally constant on $p\\mathbb Z_p$ as $f_\\beta\\restriction p\\mathbb Z_p \\equiv 0$.\n\tLastly it remains to prove that $f_\\beta$ is differentiable at $x_0=\\sum_{k=-n}^0 a_k p^k +y_0$, i.e., the limit\n\t\t\\begin{equation}\\label{equ:2}\n\t\t\t\\lim_{x\\rightarrow x_0} \\frac{p^{-m}(1+y)^\\beta-p^{-n}(1+y_0)^\\beta}{x-x_0}\n\t\t\\end{equation}\n\texists, where the values $x$ are of the form $x=\\sum_{k=-m}^0 b_k p^k +y$.\n\tMoreover, as $x$ approaches $x_0$ in the limit \\eqref{equ:2}, we can assume that $x=\\sum_{k=-n}^0 a_k p^k +y$.\n\tHence, the limit in \\eqref{equ:2} can be simplified to\n\t\t\\begin{align*}\n\t\t\t\\lim_{x\\rightarrow x_0} \\frac{p^{-m}(1+y)^\\beta-p^{-n}(1+y_0)^\\beta}{x-x_0} & =\\lim_{x\\rightarrow x_0} \\frac{p^{-n}(1+y)^\\beta-p^{-n}(1+y_0)^\\beta}{x-x_0}\\\\\n\t\t\t& =p^{-n} \\lim_{y\\rightarrow y_0} \\frac{(1+y)^\\beta-(1+y_0)^\\beta}{y-y_0}=p^{-n}\\beta(1+y_0)^{\\beta-1}.\n\t\t\\end{align*}\n\tIn particular the derivative of $f_\\beta$ is given by\n\t\t$$\n\t\tf_\\beta^\\prime (x)=\\begin{cases}\n\t\tp^{-n} \\beta (1+y)^{\\beta-1} & \\text{if } x=\\sum_{k=-n}^0 a_k p^k +y, \\text{ where } a_n\\neq 0,\\ n\\in \\mathbb N_0, y\\in p\\mathbb{Z}_p,\\\\\n\t\t0 & \\text{otherwise},\n\t\t\\end{cases}\n\t\t$$\n\tand it is unbounded since $\\lim_{n\\rightarrow \\infty} |p^{-n}\\beta(1+y)^{\\beta-1}|_p=\\lim_{n\\rightarrow \\infty} p^{n}|\\beta(1+y)^{\\beta-1}|_p=|\\beta|_p\\lim_{n\\rightarrow \\infty} p^{n}=\\infty$, where we have used the fact that $\\beta\\neq 0$.\n\t\n\tLet $h_1,\\ldots,h_m \\in \\mathcal H$ be distinct and take $P$ a polynomial in $m$ variables with coefficients in $\\mathbb Q_p\\setminus \\{0\\}$ and without free term, that is, $P(x_1,\\ldots,x_m)=\\sum_{r=1}^d \\alpha_r x_1^{k_{r,1}}\\cdots x_m^{k_{r,m}}$, where $d\\in \\mathbb N$, $\\alpha_i\\in\\mathbb{Q}_p\\setminus \\{0\\}$ for every $1\\leq r\\leq d$, $k_{r,i}\\in\\mathbb N_0$ for every $1\\leq r\\leq d$ and $1\\leq i\\leq m$ with $\\overline{k}_r:=\\sum_{i=1}^m k_{r,i}\\geq 1$, and the $m$-tuples $(k_{r,1},\\ldots,k_{r,m})$ are pairwise distinct.\n\tAssume also without loss of generality that $\\overline{k}_1\\geq \\cdots\\geq \\overline{k}_d$.\n\tWe will prove first that $P(f_{h_1},\\ldots,f_{h_m})\\not\\equiv 0$, i.e., the functions in $\\{f_h \\colon h\\in \\mathcal  H \\}$ are algebraically independent.\n\tNotice that $P(f_{h_1},\\ldots,f_{h_m})$ is of the form\n\t\t$$\n\t\t\\begin{cases}\n\t\t\\sum_{r=1}^d p^{-n\\overline{k}_r} \\alpha_r (1+y)^{\\beta_r} & \\text{if } x=\\sum_{k=-n}^0 a_k p^k +y, \\text{ with } a_n\\neq 0,\\ n\\in \\mathbb N_0,  y\\in p\\mathbb{Z}_p,\\\\\n\t\t0 & \\text{otherwise},\n\t\t\\end{cases}\n\t\t$$\n\twhere the exponents $\\beta_r:=\\sum_{i=1}^m k_{r,i} h_i$ belong to $p\\mathbb{Z}_p\\setminus\\{0\\}$ and are pairwise distinct because $\\mathcal H$ is Hamel basis of $\\mathbb Q_p$ over $\\mathbb Q$ contained in $p\\mathbb{Z}_p$, $k_{r,i}\\in\\mathbb N_0$, $\\overline{k}_r\\neq 0$ and the numbers $h_1,\\ldots,h_m$ as well as the $m$-tuples $(k_{r,1},\\ldots,k_{r,m})$ are pairwise distinct.\t\n\tFix $n\\in\\mathbb N_0$.\n\tSince $p^{-n\\overline{k}_r} \\alpha_r \\neq 0$ for every $1\\leq r\\leq d$, by Lemma~\\ref{equ:2}, there is $y\\in p\\mathbb{Z}_p$ such that $\\sum_{r=1}^d p^{-n\\overline{k}_r} \\alpha_r (1+y)^{\\beta_r}\\neq 0$.\n\tHence, by taking $x=p^{-n}+y$, we have $P(f_{h_1},\\ldots,f_{h_m})(x)\\neq 0$.\n\t\n\tFinally, let us prove that $P(f_{h_1},\\ldots,f_{h_m})'$ exists and it is unbounded.\n\tClearly $P(f_{h_1},\\ldots,f_{h_m})$ is differentiable and the derivative is given by\n\t\t\\begin{equation}\\label{equ:6}\n\t\t\t\\begin{cases}\n\t\t\t\\displaystyle \\sum_{r=1}^d p^{-n\\overline{k}_r} \\alpha_r \\beta_r (1+y)^{\\beta_r-1} & \\displaystyle \\text{if } x=\\sum_{k=-n}^0 a_k p^k +y, \\text{ with } a_n\\neq 0,\\ n\\in \\mathbb N_0, y\\in p\\mathbb{Z}_p,\\\\\n\t\t\t0 & \\text{otherwise}.\n\t\t\t\\end{cases}\n\t\t\\end{equation}\n\tNotice that $\\beta_r\\neq 1$ for every $1\\leq r\\leq d$ since $\\beta_r\\in p\\mathbb{Z}_p$.\n\tWe will now rewrite formula \\eqref{equ:6} in order to simplify the proof.\n\tNotice that if some of the exponents $\\overline{k}_r$ are equal, i.e., for instance $\\overline{k_i}=\\cdots=\\overline{k}_j$ for some $1\\leq i<j\\leq d$, then $p^{-n\\overline{k}_i}$ is a common factor in each summand $p^{-n\\overline{k}_i} \\alpha_i \\beta_i (1+y)^{\\beta_i-1},\\ldots,p^{-n\\overline{k}_j} \\alpha_j \\beta_j (1+y)^{\\beta_j-1}$.\n\tTherefore, $P(f_{h_1},\\ldots,f_{h_m})^\\prime(x)$ can also be written as\n\t\t\\begin{equation}\\label{equ:7}\n\t\t\t\\begin{cases}\n\t\t\t\\displaystyle \\sum_{q=1}^{\\widetilde{d}} p^{-n\\widetilde{k}_q} \\sum_{s=1}^{m_q}\\alpha_{q,s}\\beta_{q,s} (1+y)^{\\beta_{q,s}-1} & \\displaystyle \\text{if } x=\\sum_{k=-n}^0 a_k p^k +y, \\text{ where } a_n\\neq 0,\\\\\n\t\t\t& \\displaystyle n\\in \\mathbb N_0 \\text{ and } y\\in p\\mathbb{Z}_p,\\\\\n\t\t\t0 & \\displaystyle \\text{otherwise},\n\t\t\t\\end{cases}\n\t\t\\end{equation}\n\twhere $\\tilde{d}\\in \\mathbb{N}$, the $\\widetilde{k}_q$'s represent the common exponents $p^{-n\\overline{k}_i}$ with $\\widetilde{k}_1>\\cdots>\\widetilde{k}_{\\widetilde{d}}$, and $\\alpha_{q,s}$ and $\\beta_{q,s}$ are the corresponding terms $\\alpha_r$ and $\\beta_r$, respectively.\n\tNow, since $\\alpha_{1,s} \\beta_{1,s}\\neq 0$ and the exponents $\\beta_{1,s}-1$ are pairwise distinct and not equal to $0$ for every $1\\leq s\\leq m_1$, by Lemma~\\ref{equ:2}, there exists $y_1\\in p\\mathbb{Z}_p$ such that $\\sum_{s=1}^{m_1}\\alpha_{1,s}\\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1}\\neq 0$.\n\tTake the sequence $(x_n)_{n=1}^\\infty$ defined by $x_n=p^{-n}+y_1$.\n\tSince $\\widetilde{k}_1>\\cdots>\\widetilde{k}_{\\widetilde{d}}$, there exists $n_0\\in \\mathbb N$ such that\n\t\t\\begin{align*}\n\t\t\t|P(f_{h_1},\\ldots,f_{h_m})^\\prime (x_n)|_p & =\\left|\\sum_{q=1}^{\\widetilde{d}} p^{-n\\widetilde{k}_q} \\sum_{s=1}^{m_q}\\alpha_{q,s}\\beta_{q,s} (1+y_1)^{\\beta_{q,s}-1} \\right|_p\\\\\n\t\t\t& =\\left|p^{-n\\widetilde{k}_1}\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1} + \\right.\\\\ \n\t\t\t&\\left. + \\sum_{q=2}^{\\widetilde{d}} p^{-n\\widetilde{k}_r} \\sum_{s=1}^{m_q}\\alpha_{q,s} \\beta_{q,s} (1+y_1)^{\\beta_{q,s}-1} \\right|_p\\\\\n\t\t\t& =\\left|p^{-n\\widetilde{k}_1}\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1}\\right|_p\\\\\n\t\t\t& =p^{n\\widetilde{k}_1}\\left|\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1}\\right|_p,\n\t\t\\end{align*}\n\tfor every $n\\geq n_0$. \n\tHence, $\\lim_{n\\to \\infty} |P(f_{h_1},\\ldots,f_{h_m})^\\prime (x_n)|_p=\\infty$.\n\\end{proof}\n\nThe reader may have noticed that the functions in the proof of Theorem~\\ref{16} have unbounded derivative but the derivative is bounded on each ball of $\\mathbb{Q}_p$.\nThe following result (which proof is a modification of the one in Theorem~\\ref{16}) shows that we can obtain a similar optimal result when the derivative is unbounded on each ball centered at a fixed point $a\\in \\mathbb{Q}_p$.\nThe functions will not be differentiable at $a$.\n\n\\begin{corollary}\n\\label{15} Let $a\\in \\mathbb{Q}_p$.\nThe set of continuous functions $\\mathbb{Q}_{p}\\rightarrow \\mathbb{Q}_{p}$ that are differentiable except at $a$ and which derivative is unbounded on $\\mathbb{Q}_p\\setminus (a+\\mathbb{Z}_p)$ and on $(a+\\mathbb{Z}_p)\\setminus\\{a\\}$ is strongly $\\mathfrak{c}$-algebrable.\n\\end{corollary}\n\n\\begin{proof}\n\tFix $a\\in \\mathbb{Q}_p$.\n\tLet $\\mathcal H$ be a Hamel basis of $\\mathbb Q_p$ over $\\mathbb Q$ contained in $p\\mathbb Z_p$.\n\tFor every $\\beta\\in \\mathbb{Z}_p\\setminus\\{0\\}$ take the function $f_\\beta$ defined in the proof of Theorem~\\ref{16} and also define $g_\\beta\\colon \\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ by\n\t\t$$\n\t\tg_\\beta(x)=\\begin{cases}\n\t\tp^{n}[p^{-n^2}(x-a)]^\\beta & \\text{if } x\\in \\overline{B}_{p^{-(n^2+1)}}(a+p^{n^2}) \\text{ for some } n\\in \\mathbb N,\\\\\n\t\t0 & \\text{otherwise}.\n\t\t\\end{cases}\n\t\t$$\n\tNotice that by applying the change of variable $y=x-a$ we can assume, without loss of generality, that $a=0$.\n\tSince for any $x\\in \\overline{B}_{p^{-(n^2+1)}}(p^{n^2})$ with $n\\in\\mathbb N$, $x$ is of the form $p^{n^2}+\\sum_{k=n^2+1}^\\infty a_k p^k$ with $a_{k}\\in \\{0,1,\\ldots,p-1\\}$ for every integer $k\\geq n^2+1$, we have that $p^{-n^2}x=1+\\sum_{k=n^2+1}^\\infty a_k p^{k-n^2}\\in 1+p\\mathbb{Z}_p$.\n\tThus $g_\\beta$ is well defined.\n\tNow, for every $\\beta\\in \\mathbb{Z}_p\\setminus\\{0\\}$, let $F_{\\beta}:=f_\\beta+g_\\beta$.\n\tIt is easy to see that $F_\\beta$ is differentiable at every $x\\in \\mathbb{Q}_p\\setminus\\{0\\}$ and, in particular, continuous on $\\mathbb{Q}_p\\setminus\\{0\\}$.\n\tLet us prove now that $F_\\beta$ is continuous at $0$.\n\tFix $\\varepsilon>0$ and take $n\\in\\mathbb N$ such that $p^{-n}<\\varepsilon$. \n\tIf $|x|_p<p^{1-n^2}$, then \n\t\t\\begin{align*}\n\t\t\t|F_\\beta(x)| & =\\begin{cases}\n\t\t\t|p^n (p^{-n^2}x)^\\beta|_p & \\text{if } x\\in \\overline{B}_{p^{-(n^2+1)}}(p^{n^2}),\\\\\n\t\t\t0 & \\text{otherwise},\n\t\t\t\\end{cases}\\\\\n\t\t\t& =\\begin{cases}\n\t\t\tp^{-n} & \\text{if } x\\in \\overline{B}_{p^{-(n^2+1)}}(p^{n^2}),\\\\\n\t\t\t0 & \\text{otherwise}.\n\t\t\t\\end{cases}\n\t\t\\end{align*}\n\tIn any case, $|F_\\beta(x)|<\\varepsilon$.\n\tHence $F_\\beta$ is continuous.\n\tMoreover, $F_\\beta$ is not differentiable at $0$.\n\tIndeed, by considering the sequence $(x_n)_{n=1}^\\infty=(p^{n^2}+p^{n^2+1})_{n=1}^\\infty$ which converges to $0$ when $n\\to \\infty$ we have\n\t\t\\begin{align*}\n\t\t\t\\lim_{n\\to \\infty} \\frac{|F_\\beta(x_n)-F_\\beta(0)|_p}{|x_n|_p} & =\\lim_{n\\to \\infty} \\frac{|p^{n} [p^{-n^2}(p^{n^2}+p^{n^2+1})]^\\beta|_p}{|p^{n^2}+p^{n^2+1}|_p}\\\\\n\t\t\t& =\\lim_{n\\to \\infty} \\frac{p^{-n} |(1+p)^\\beta|_p}{p^{-n^2}}=\\lim_{n\\to \\infty} p^{n^2-n}=\\infty.\n\t\t\\end{align*}\n\tIn particular, by the chain rule, the derivative of $F_\\beta$ on $\\mathbb{Q}_p\\setminus \\{0\\}$ is as follows\n\t\t$$\n\t\t\\hspace*{-1cm} F_\\beta^\\prime(x)=\\begin{cases}\n\t\tp^{-n} \\beta (1+y)^{\\beta-1} & \\displaystyle \\text{if } x=\\sum_{k=n}^0 a_k p^k +y, \\text{ with } a_n\\neq 0,\\ n\\in \\mathbb Z\\setminus \\mathbb N,  y\\in p\\mathbb{Z}_p,\\\\\n\t\tp^{n-n^2}\\beta(p^{-n^2}x)^{\\beta-1} & \\text{if } x\\in \\overline{B}_{p^{-(n^2+1)}}(p^{n^2}) \\text{ for some } n\\in \\mathbb N,\\\\\n\t\t0 & \\text{otherwise}.\n\t\t\\end{cases}\n\t\t$$\n\t\t\n\tBy the proof of Theorem~\\ref{16} the functions in the set $V=\\{F_h \\colon h\\in \\mathcal H \\}$ are algebraically independent, and every function in the algebra $\\mathcal A$ generated by $V$ over $\\mathbb{Q}_p$ that is not the $0$ function is continuous, differentiable on $\\mathbb{Q}_p\\setminus \\{0\\}$ and has unbounded derivative on $\\mathbb{Q}_p\\setminus \\mathbb{Z}_p$.\n\tIt remains to prove that any nonzero algebraic combination in $V$ has unbounded derivative on $\\mathbb{Z}_p\\setminus\\{0\\}$.\n\tTo do so, let $h_1,\\ldots,h_m \\in \\mathcal H$ be distinct and take $P$ a polynomial in $m$ variables with coefficients in $\\mathbb Q_p\\setminus \\{0\\}$ and without free term.\n\tThen, by the chain rule, $P(f_{h_1},\\ldots,f_{h_m})^\\prime$ on $\\overline{B}_{p^{-(n^2+1)}}(p^{n^2})$ is of the form\n\t\t$$\n\t\tp^{-n^2}\\sum_{q=1}^d p^{nk_q} \\sum_{s=1}^{m_q} \\alpha_{q,s} \\beta_{q,s} (p^{-n^2}x)^{\\beta_{q,s}-1},\n\t\t$$\n\tsee the proof of Theorem~\\ref{16}.\n\tAssume without loss of generality that $k_1<\\cdots<k_d$.\n\tSince $\\alpha_{1,s} \\beta_{1,s}\\neq 0$ and the exponents $\\beta_{1,s}- 1$ are pairwise distinct and not $0$ for every $1\\leq s\\leq m_1$, by Lemma~\\ref{lem:1}, there exists $y_1\\in p\\mathbb{Z}_p$ such that $\\sum_{s=1}^{m_1} \\alpha_{1,s} \\beta_{1,s}(1+y_1)^{\\beta_{1,s}-1}\\neq 0$.\n\tFor every $n\\in\\mathbb N$, take $x_n=p^{n^2}(1+y_1)$.\n\tThen, notice that there exists $n_0\\in\\mathbb N$ such that \n\t\t\\begin{align*}\n\t\t\t|P(F_{h_1},\\ldots,F_{h_m})^\\prime (x_n)|_p & =\\left|p^{-n^2}\\sum_{q=1}^{d} p^{nk_q} \\sum_{s=1}^{m_q}\\alpha_{q,s}\\beta_{q,s} (p^{-n^2}x_n)^{\\beta_{q,s}-1} \\right|_p\\\\\n\t\t\t& =p^{n^2}\\left|\\sum_{q=1}^{d} p^{nk_q} \\sum_{s=1}^{m_q}\\alpha_{q,s}\\beta_{q,s} (1+y_1)^{\\beta_{q,s}-1} \\right|_p\\\\\n\t\t\t& =p^{n^2}\\left|p^{nk_1}\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1} + \\right. \\\\\n\t\t\t& \\left. + \\sum_{q=2}^{d} p^{nk_r} \\sum_{s=1}^{m_q} \\alpha_{q,s} \\beta_{q,s} (1+y_1)^{\\beta_{q,s}-1} \\right|_p\\\\\n\t\t\t& =p^{n^2}\\left|p^{nk_1}\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1}\\right|_p\\\\\n\t\t\t& =p^{n^2-nk_1}\\left|\\sum_{s=1}^{m_1 }\\alpha_{1,s} \\beta_{1,s} (1+y_1)^{\\beta_{1,s}-1}\\right|_p,\n\t\t\t\\end{align*}\n\tfor every $n\\geq n_0$. \n\tTherefore $\\lim_{n\\to \\infty} |P(F_{h_1},\\ldots,F_{h_m})^\\prime (x_n)|_p=\\infty$.\n\\end{proof}\n\nIn Corollary~\\ref{15} we can replace unbounded derivative with being not Lipschitzian although the conclusion is weaker in terms of lineability as it is shown in the following proposition.\n\n\\begin{proposition}\n\t\\label{26} The set of continuous functions $\\mathbb{Q}_{p}\\rightarrow \\mathbb{Q}_{p}$ which are differentiable except at $0$, with bounded derivative on $\\mathbb{Q}_p\\setminus\\{0\\}$ and not Lipschitzian of order $\\alpha>0$ is $\\mathfrak{c}$-lineable and 1-algebrable.\n\\end{proposition}\n\n\\begin{proof}\n\tLet us prove first the lineability part.\n\tFor any $N\\in\\mathcal N$, let $f_N\\colon \\mathbb{Q}_p\\rightarrow \\mathbb{Q}_p$ be:\n\t$$\n\tf_N(x)=\\begin{cases}\n\t\tp^{n} & \\text{if } x\\in S_{p^{-n^2}}(0) \\text{ and } n\\in N,\\\\\n\t\t0 & \\text{otherwise}.\n\t\\end{cases}\n\t$$\n\tFor every $x\\in\\mathbb{Q}_p\\setminus\\{0\\}$, it is clear that there exists a neighborhood of $x$ such that $f_N$ is constant since the spheres are open sets.\n\tThus, $f_N$ is locally constant on $\\mathbb{Q}_p\\setminus\\{0\\}$ which implies that $f_N$ is continuous, differentiable on $\\mathbb{Q}_p\\setminus\\{0\\}$ and $f_N^\\prime(x)=0$ for every $x\\in \\mathbb{Q}_p\\setminus\\{0\\}$.\n\tMoreover, it is easy to see that $f_N$ is continuous at $0$.\n\tHowever, $f_N$ is not differentiable at $0$.\n\tIndeed, take $x_n=p^{n^2}$ for every $n\\in N$.\n\tIt is clear that the sequence $(x_n)_{n\\in N}$ converges to $0$ and also, for every $\\alpha>0$,\n\t$$\n\t\\frac{|f_N(x_n)|_p}{|x_n|_p^\\alpha}=p^{\\left(-1+\\alpha n\\right) n}\\rightarrow \\infty,\n\t$$\n\twhen $n\\in N$ tends to infinity.\n\tTherefore $f_N$ is not differentiable at $0$.\n\tFurthermore, notice that for any $M>0$ there are infinitely many $x\\in \\mathbb{Z}_p$ such that $|f_N(x)|_p> M|x|_p^\\alpha$.\n\tHence $f_N$ is not Lipschitzian of order $\\alpha>0$.\n\t\n\tIt remains to prove that the functions in $V=\\{f_N\\colon N\\in\\mathcal N \\}$ are linearly independent over $\\mathbb{Q}_p$ and such that the functions in $\\text{span}(V)\\setminus\\{0\\}$ are differentiable except at $0$, with bounded derivative on $\\mathbb{Q}_p\\setminus\\{0\\}$ and not Lipschitzian of order $\\alpha>0$.\n\tLet $f=\\sum_{i=1}^m \\alpha_i f_{N_i}$, where $\\alpha_1,\\ldots, \\alpha_m\\in \\mathbb{Q}_p$, $N_1,\\ldots, N_m\\in \\mathcal N$ are distinct and $m\\in \\mathbb N$.\n\tAssume that $f\\equiv 0$ and take $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_m^0$.\n\tThen $0=f(p^{n^2})=\\alpha_1 p^{n}$ which implies that $\\alpha_1=0$.\n\tApplying similar arguments we have that $\\alpha_i=0$ for every $i\\in \\{1,\\ldots, m \\}$.\n\tFinally, assume that $\\alpha_i\\neq 0$ for every $i\\in \\{1,\\ldots, m \\}$.\n\tIt is clear that $f$ is continuous on $\\mathbb{Q}_p$ and differentiable on $\\mathbb{Q}_p\\setminus\\{0\\}$ with $f_N^\\prime(x)=0$ for every $x\\in \\mathbb{Q}_p\\setminus \\{0\\}$.\n\tLet $x_n=p^{n^2}$ for every $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_m^0$ and notice that $(x_n)_{n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_m^0}$ converges to $0$.\n\tMoreover, for every $\\alpha>0$,\n\t$$\n\t\\frac{|f(x_n)|_p}{|x_n|_p^\\alpha}=|\\alpha_1|_p p^{\\left(-1+\\alpha n\\right) n}\\rightarrow \\infty,\n\t$$\n\twhen $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_m^0$ tends to infinity.\n\tHence, $f$ is not differentiable at $0$, and also for every $M>0$ there are infinitely many $x\\in \\mathbb{Z}_p$ such that $|f(x)|_p> M|x|_p^\\alpha$.\n\t\n\tFor the algebrability part, let $g\\colon \\mathbb{Q}_p\\rightarrow\\mathbb{Q}_p$ be defined as:\n\t$$\n\tg(x)=\\begin{cases}\n\t\tp^n & \\text{if } x\\in S_{p^{-n^2}}(0) \\text{ for some } n\\in\\mathbb N,\\\\\n\t\t0 & \\text{otherwise}.\n\t\\end{cases}\n\t$$\n\tBy applying similar arguments used in the first part of the proof, we have that $g$ is continuous, differentiable on $\\mathbb{Q}_p\\setminus\\{0\\}$ with $g^\\prime(x)=0$ for every $x\\in \\mathbb{Q}_p\\setminus\\{0\\}$ and not Lipschitzian of order $\\alpha>0$.\n\tTo finish the proof, let $G=\\beta g^k$ where $\\beta\\in\\mathbb{Q}_p\\setminus\\{0\\}$ and $k\\in\\mathbb N$.\n\tIt is obvious that $G$ is continuous, differentiable on $\\mathbb{Q}_p\\setminus\\{0\\}$ and $G^\\prime(x)=0$ for every $x\\in\\mathbb{Q}_p\\setminus\\{0\\}$.\n\tNow, let $x_n=p^{n^2}$ for every $n\\in\\mathbb N$.\n\tIt is easy to see that $(x_n)_{n\\in\\mathbb N}$ converges to $0$ and\n\t$$\n\t\\frac{|G(x_n)|_p}{|x_n|_p^\\alpha}=|\\beta|_p p^{\\left(-k+\\alpha n\\right) n}\\rightarrow \\infty,\n\t$$\n\twhen $n\\rightarrow\\infty$.\n\\end{proof}\n\nLet $\\mathcal B$ be the $\\sigma$-algebra of all Borel subsets of $\\mathbb{Z}_p$ and $\\mu$ be any non-negative real-valued Haar measure on the measurable space $(\\mathbb{Z}_p,\\mathcal B)$.\nIn particular, if $\\mu$ is normalized, then $\\mu\\left(x+p^n\\mathbb{Z}_p\\right)=p^{-n}$ for any $x\\in \\mathbb{Z}_p$ and $n\\in\\mathbb N$.\nFor the rest of the paper $\\mu$ will denote a non-negative real-valued Haar measure on $(\\mathbb{Z}_p,\\mathcal B)$.\nAs usual, a Borel subset $B$ of $\\mathbb{Z}_p$ is called a null set for $\\mu$ provided that $\\mu(B)=0$.\nWe also say that a Borel subset of $\\mathbb{Z}_p$ is a full set for $\\mu$ if $\\mathbb{Z}_p\\setminus B$ is a null set. (See \\cite[Section~2.2]{Fo} for more details on the Haar measure.)\n\n\nIt is easy to see that the singletons of $\\mathbb{Z}_p$ are null sets for any Haar measure $\\mu$ on $(\\mathbb{Z}_p,\\mathcal B)$.\nTherefore Proposition~\\ref{26} states in particular that, for any Haar measure $\\mu$ on $(\\mathbb{Z}_p,\\mathcal B)$, the set of continuous functions $\\mathbb{Z}_p\\rightarrow\\mathbb{Q}_p$ that are differentiable except on a null set for $\\mu$ of cardinality $1$, with bounded derivative elsewhere, is $\\mathfrak{c}$-lineable.\nThe following result shows that a similar version can be obtained when we consider null sets of cardinality $\\mathfrak{c}$ for any Haar measure $\\mu$ on $(\\mathbb{Z}_p,\\mathcal B)$.\nIn order to prove it, we recall the following definition and result from probability theory.\n\n\\begin{definition}\\label{def:1}\n\tLet $(\\Omega,\\mathcal F,P)$ be a probability space and $Y$ be a measurable real-valued function on $\\Omega$. We say that $Y$ is a random variable.\n\\end{definition}\n\n\\begin{theorem}\\normalfont{(Strong law of large numbers, see \\cite[Theorem~22.1]{Bi}).}\n\tLet $(Y_n)_{n\\in \\mathbb N_0}$ be a sequence of independent and identically distributed real-valued random variables on a probability space $(\\Omega,\\mathcal F,P)$ such that, for each $n\\in\\mathbb N_0$, $E[Y_n]=m$ for some $m\\in\\mathbb R$ (where $E$ denotes the expected value).\n\tThen\n\t$$\n\tP\\left(\\left\\{x\\in \\Omega\\colon \\exists\\lim_{n\\rightarrow \\infty} \\frac{\\sum_{k=0}^{n-1} Y_k(x)}{n}=m \\right\\} \\right)=1.\n\t$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:2}\n\tLet $\\mu$ be a Haar measure on $(\\mathbb{Z}_p,\\mathcal B)$.\n\tThe set of continuous functions $\\mathbb{Z}_p\\rightarrow\\mathbb{Q}_p$ which are differentiable on a full set for $\\mu$ with bounded derivative but not differentiable on the complement having cardinality $\\mathfrak c$ is $\\mathfrak c$-lineable.\n\\end{theorem}\n\n\\begin{proof}\n\tWe will prove the result for $\\mu$ the normalized Haar measure on $(\\mathbb{Z}_p,\\mathcal B)$ since any null set for the normalized Haar measure is also a null set for any other non-negative real-valued Haar measure on $(\\mathbb{Z}_p,\\mathcal B)$.\n\tThis is an immediate consequence of Haar's Theorem which states that Haar measures are unique up to a positive multiplicative constant (see \\cite[Theorem~2.20]{Fo}).\n\t\n\n\t\n\tLet $f\\colon \\mathbb{Z}_p\\rightarrow \\mathbb{Z}_p$ be defined as follows: for every $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$, we have\n\t\t\\begin{equation}\\label{equ:3}\n\t\t\tf(x)=\\begin{cases}\n\t\t\t\tx & \\text{if } (x_{2i},x_{2i+1})\\neq (0,0) \\text{ for all } i\\in \\mathbb N_0,\\\\\n\t\t\t\t\\displaystyle \\sum_{i=0}^{2n+1} x_i p^i & \\text{if } (x_{2i},x_{2i+1})\\neq (0,0) \\text{ for all } i\\leq n \\\\\n\t\t\t\t & \\text{with } n\\in \\mathbb N_0 \\text{ and } (x_{2n+2},x_{2n+3})= (0,0),\\\\\n\t\t\t\t0 & \\text{if } (x_0,x_1)=(0,0).\n\t\t\t\\end{cases}\n\t\t\\end{equation}\n\tThe function $f$ is continuous.\n\tIndeed, let $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$ and fix $\\varepsilon>0$.\n\tTake any $m\\in \\mathbb N_0$ such that $p^{-(2m+1)}<\\varepsilon$.\n\tThen for any $y\\in \\mathbb{Z}_p$ such that $|x-y|_p<p^{-(2m+1)}$ we have that $y$ is of the form $\\sum_{i=0}^{2m+1} x_i p^i + \\sum_{i=2m+2}^\\infty y_i p^i$.\n\tHence, notice that in any possible case of $x$ given in \\eqref{equ:3}, we have that $|f(x)-f(y)|_p< p^{-(2m+1)}<\\varepsilon$.\n\t\n\tLet us define, for every $i\\in \\mathbb N_0$, the random variables $Y_i\\colon \\mathbb{Z}_p\\rightarrow \\{0,1\\}$ in the following way: for any $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$,\n\t$$\n\tY_i(x)=\\begin{cases}\n\t\t1 & \\text{if } (x_{2i},x_{2i+1})=(0,0),\\\\\n\t\t0 & \\text{if } (x_{2i},x_{2i+1})\\neq (0,0).\n\t\\end{cases}\n\t$$\n\tNotice that the random variables $(Y_i)_{i\\in \\mathbb N_0}$ are independent and identically distributed with $E[Y_i]=\\frac{1}{p^2}$ for every $i\\in \\mathbb N_0$.\n\tThus, by the strong law of large numbers, the set\n\t$$\n\tD=\\left\\{x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p \\colon \\exists\\lim_{n\\rightarrow\\infty} \\frac{\\sum_{i=0}^{n-1} Y_i(x)}{n}=\\frac{1}{p^2} \\right\\}\n\t$$\n\thas measure $1$.\n\tNow, since for every $i\\in \\mathbb N_0$, $Y_i(x)=0$ for all $x=\\sum_{j=0}^\\infty x_j p^j$ that satisfy $(x_{2j},x_{2j+1})\\neq (0,0)$ for each $j\\in \\mathbb N_0$, we have that $\\lim_{n\\to \\infty} \\frac{\\sum_{i=0}^{n-1} Y_i(x)}{n}=0$ for all such $x$.\n\tHence, it is clear that $E:=\\left\\{\\sum_{i=0}^\\infty  x_i p^i\\in \\mathbb{Z}_p\\colon (x_{2i},x_{2i+1})\\neq (0,0)  \\text{ for all } i\\in \\mathbb N_0 \\right\\}\\subset \\mathbb{Z}_p\\setminus D$.\n\tMoreover, by construction $\\text{card}(E)=\\mathfrak{c}$.\n\tNotice that it is not obvious that $E$ is a Borel set since any Haar measure $\\mu$ on $(\\mathbb{Z}_p,\\mathcal B)$ is not complete.\n\tHowever, as $\\mathbb{Z}_p\\setminus D$ is a null set, we have that if $E$ were a Borel set, then $E$ would be a null set.\n\tLet us prove that $E$ is a Borel set.\n\tConsider the finite field of $p$ elements $\\mathbb F_p$ endowed with an absolute value $|\\cdot|_T$--the trivial absolute value.\n\tThen $\\mathbb F_p$ is a discrete topological space, which implies that the product space $\\mathbb F_p^2:=\\mathbb F_p\\times \\mathbb F_p$ has the discrete topology. (Recall that the finite product of discrete topological spaces has the discrete topology.)\n\tFor every $n\\in \\mathbb N_0$, let $\\pi_n\\colon \\mathbb{Z}_p\\to \\mathbb F_p^2$ be defined as $\\pi_n(x)=(x_{2n},x_{2n+1})$ for every $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$.\n\tLet $n\\in \\mathbb N_0$, $x\\in \\mathbb{Z}_p$ and fix $\\varepsilon>0$.\n\tTake an integer $m>n$.\n\tThen for every $y\\in \\mathbb{Z}_p$ such that $|x-y|_p<p^{-(2m+1)}$ we have that $\\pi_n(x)=\\pi_n(y)$, i.e., $|\\pi_n(x)-\\pi_n(y)|_T=0<\\varepsilon$.\n\tHence $\\pi_n$ is continuous.\n\tNote that $E=\\bigcap_{n=0}^\\infty \\pi_n^{-1}(\\{(x,y)\\in \\mathbb F_p^2 \\colon (x,y)\\neq (0,0) \\})$, where $\\pi_n^{-1}(\\{(x,y)\\in \\mathbb F_p^2 \\colon (x,y)\\neq (0,0) \\})$ is closed since $\\pi_n$ is continuous and $\\{(x,y)\\in \\mathbb F_p^2 \\colon (x,y)\\neq (0,0) \\}$ is closed in $\\mathbb F_p^2$.\n\tHence, $E$ is closed since it is the countable intersection of closed sets and, therefore, a Borel set.\n\t\n\tLet us analyze now the differentiability of $f$.\n\tOn the one hand, if for $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$ there exists $i_0\\in\\mathbb N_0$ such that $(x_{2i_0},x_{2i_0+1})=(0,0)$, then it is clear that $f$ is constant on some neighborhood of $x$, and hence differentiable at $x$.\n\tOn the other hand, if $f$ were differentiable at $x=\\sum_{i=0}^\\infty x_i p^i\\in \\mathbb{Z}_p$ satisfying $(x_{2i},x_{2i+1})\\neq (0,0)$ for all $i\\in \\mathbb N_0$, then we would have $f^\\prime (x)=1$.\n\tAssume, by means of contradiction, that $f$ is differentiable at $x$.\n\tFor every $n\\in \\mathbb N_0$, take $\\overline{x}_n:=\\sum_{i=0}^{2n+1} x_i p^i + \\sum_{i=2n+4}^\\infty y_i p^i$ with $y_{2n+4}\\neq 0$, then\n\t\\begin{align*}\n\t\t\\frac{f(x)-f(\\overline{x}_n)}{x-\\overline{x}_n} & = \\frac{\\sum_{i=0}^\\infty x_i p^i-\\sum_{i=0}^{2n+1} x_i p^i}{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i} }\\\\\n\t\t& = \\frac{\\sum_{i=2n+2}^\\infty x_{i} p^{i}}{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i}}\\\\\n\t\t& = \\frac{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i}+\\sum_{i=2n+4}^\\infty y_i p^{i} }{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i}}\\\\\n\t\t& = 1+\\frac{\\sum_{i=2n+4}^\\infty y_i p^{i}}{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i}}.\n\t\\end{align*}\n\tNow, as $y_{2n+4} \\neq 0$, we have\n\t\t$$\n\t\t\\left|\\frac{\\sum_{i=2n+4}^\\infty y_i p^{i}}{\\sum_{i=2n+2}^\\infty x_{i} p^{i}-\\sum_{i=2n+4}^\\infty y_i p^{i}} \\right|_p=\\begin{cases}\n\t\tp^{-2} & \\text{if } x_{2n+2}\\neq 0,\\\\\n\t\tp^{-1} & \\text{if } x_{2n+3}\\neq 0.\n\t\t\\end{cases}\n\t\t$$\n\tThus we have $\\lim_{n\\to \\infty} |x-\\overline{x}_n|_p=0$ and $\\lim_{n\\to \\infty} \\left|\\frac{f(x)-f(\\overline{x}_n)}{x-\\overline{x}_n}-1\\right|_p\\geq p^{-2}\\neq 0$, a contradiction.\n\t\n\tLet us define the function $g\\colon \\mathbb{Z}_p\\rightarrow \\mathbb{Q}_p$ by:\n\t$$\n\tg(x)=\\begin{cases}\n\t\tp^n f(x^\\prime) & \\text{if } x=p^n+p^{n+1} x^\\prime \\text{ with } n\\in\\mathbb N \\text{ and } x^\\prime \\in \\mathbb{Z}_p,\\\\\n\t\t0 & \\text{otherwise}.\n\t\\end{cases}\n\t$$\n\tNotice that $g$ is well defined since the sets $B_n:=p^n+p^{n+1}\\mathbb{Z}_p$ are pairwise disjoint.\n\t(The sets $B_n$ are the closed balls $\\overline{B}_{p^{-(n+1)}}(p^n)$.)\n\tIf $x\\in \\mathbb{Z}_p\\setminus \\left(\\{0\\} \\cup \\bigcup_{n=1}^\\infty B_n\\right)$, then there exists an open neighborhood $U^x$ of $x$ such that $g$ is identically zero on $U^x$, i.e., $g$ is differentiable at $x$.\n\tNow, as $g(p^n+p^{n+1}x)=p^n f(x)$ for every $n\\in\\mathbb N$ and $x\\in \\mathbb{Z}_p$, and since $f$ is continuous, it is obvious that $g$ is continuous on $\\bigcup_{n=1}^\\infty B_n$.\n\tMoreover, $g$ is also continuous at $0$.\n\tTo prove it fix $\\varepsilon>0$ and take $n\\in \\mathbb N$ such that $p^{-n}<\\varepsilon$.\n\tIf $x\\in \\mathbb{Z}_p$ is such that $|x|_p=p^{-n}$, then $x=x_n p^n+p^{n+1}x^\\prime$ with $x_n\\neq 0$.\n\tFurthermore,\n\t\t\\begin{align*}\n\t\t\t|g(0)-g(x)|_p & =\\begin{cases}\n\t\t\t|p^n f(x^\\prime)|_p & \\text{if } x_n=1,\\\\\n\t\t\t0 & \\text{otherwise},\n\t\t\t\\end{cases}\n\t\t\t=\\begin{cases}\n\t\t\tp^{-n}|f(x^\\prime)|_p & \\text{if } x_n=1,\\\\\n\t\t\t0 & \\text{otherwise}.\n\t\t\t\\end{cases}\n\t\t\\end{align*}\n\tHence, $|g(0)-g(x)|_p\\leq p^{-n}<\\varepsilon$.\n\tTherefore we have proven that $g$ is continuous on $\\mathbb{Z}_p$.\n\tMoreover, $g$ is differentiable also on $\\bigcup_{n=1}^\\infty (B_n\\setminus E_n)$ (with bounded derivative) as $f$ is differentiable on $\\bigcup_{n=1}^\\infty (B_n\\setminus E_n)$, where $E_n:=p^n+p^{n+1}E$; and $g$ is not differentiable on $\\bigcup_{n=1}^\\infty E_n$ since $f$ is not differentiable on $\\bigcup_{n=1}^\\infty E_n$.\n\tNotice that once again $\\text{card}(E_n)=\\mathfrak{c}$ for every $n\\in \\mathbb N_0$.\n\t\n\tLet us prove that $E_n$ is a Borel set with $\\mu(E_n)=0$ for every $n\\in \\mathbb N$.\n\tTo do so, let us consider the restricted measure $\\mu_n=p^{n+1}\\mu$ on the measurable space $(B_n,\\mathcal B_n)$, where $\\mathcal B_n$ is the $\\sigma$-algebra of all Borel subsets of $B_n$.\n\tNotice that $\\mathcal B_n=\\{B\\cap B_n\\colon B\\in \\mathcal B \\}$ and $(B_n,\\mathcal B_n,\\mu_n)$ is a probability space.\n\tDefine now for every $i\\in \\mathbb N_0$ the random variables $Y_{n,i}\\colon B_n\\rightarrow\\{0,1\\}$ as follows: for $x=p^n+p^{n+1}\\sum_{i=0}^\\infty x_i p^i\\in B_n$, we have\n\t$$\n\tY_{n,i}(x)=\\begin{cases}\n\t\t1 & \\text{if } (x_{2i},x_{2i+1})=(0,0),\\\\\n\t\t0 & \\text{if } (x_{2i},x_{2i+1}) \\neq (0,0).\n\t\\end{cases}\n\t$$\n\tOnce again the random variables $(Y_{n,i})_{i\\in \\mathbb N_0}$ are independent and identically distributed with $E[Y_{n,i}]=\\frac{1}{p^2}$ for every $i\\in\\mathbb N_0$.\n\tThus, the set \n\t$$\n\t\\left\\{x=p^n+p^{n+1}\\sum_{i=0}^\\infty x_i p^i\\in B_n \\colon \\exists\\lim_{m\\rightarrow\\infty} \\frac{\\sum_{i=0}^{m-1} Y_{n,i}(x)}{m}=\\frac{1}{p} \\right\\}=p^n+p^{n+1}D=:D_n\n\t$$\n\tis a full set for $\\mu_n$.\n\tBy considering for each $k\\in \\mathbb N_0$ the function $\\pi_{n,k}\\colon B_n\\to \\mathbb F_p^2$ given by $\\pi_{n,k}(x)=(x_{2k},x_{2k+1})$ for every $x=p^n+p^{n+1}\\sum_{i=0}^\\infty x_i p^i\\in B_n$ and applying similar arguments as above, we have that $\\pi_{n,k}$ is continuous.\n\tHence $E_n=\\bigcap_{k=0}^\\infty \\pi_{n,k}(\\{(x,y)\\in \\mathbb F_p^2\\colon (x,y) \\neq (0,0) \\})$ is once again a Borel set.\n\tFurthermore, since $E_n\\subset B_n\\setminus D_n$ we have that $E_n$ is a null set for $\\mu_n$.\n\tThus $\\mu(E_n)=p^{-(n+1)}\\mu_n(E_n)=0$ for every $n\\in \\mathbb N$.\n\t\n\tFinally let us prove that $g$ is not differentiable at $0$.\n\tSince every neighborhood containing $0$ on $\\mathbb{Z}_p$ contains points $x$ such that $g(x)=0$, if $g$ were differentiable at $0$ then $g^\\prime (0)=0$.\n\tAssume that $g$ is differentiable at $0$.\n\tAs $p^n+p^{n+1}\\sum_{i=0}^\\infty p^i=p^n \\sum_{i=0}^\\infty p^i\\in B_n$ for every $n\\in \\mathbb N$, we have that\n\t\t$$\n\t\t\\left|\\frac{g\\left(p^n+p^{n+1}\\sum_{i=0}^\\infty p^i \\right)-g(0)}{p^n+p^{n+1}\\sum_{i=0}^\\infty p^i} \\right|_p=\\left|\\frac{p^n f\\left(\\sum_{i=0}^\\infty p^i \\right)}{p^n \\sum_{i=0}^\\infty p^i} \\right|_p=\\left|\\frac{p^n \\sum_{i=0}^\\infty p^i}{p^n \\sum_{i=0}^\\infty p^i}  \\right|_p=1,\n\t\t$$\n\twhere $\\lim_{n\\to \\infty } \\left|p^n+p^{n+1}\\sum_{i=0}^\\infty p^i \\right|_p=\\lim_{n\\to \\infty} p^{-n}=0$, a contradiction.\n\t\n\n\tFor every $N\\in \\mathcal N$, let us define $f_N\\colon \\mathbb{Z}_p\\rightarrow \\mathbb{Q}_p$ by:\n\t$$\n\tf_N(x)=g(x)\\sum_{n\\in N} 1_{B_n}(x).\n\t$$\n\tSince $B_n\\cap B_m=\\emptyset$ for every distinct $n,m\\in \\mathbb N$, the function $f_N$ is well defined.\n\tFurthermore, since each $N\\in \\mathcal N$ is infinite, we can apply the above arguments to prove that $f_N$ is continuous and differentiable on a full set for $\\mu$ with bounded derivative but not differentiable on the complement having cardinality $\\mathfrak c$.\n\t\n\tIt remains to prove that the functions in $V=\\{f_N\\colon N\\in \\mathcal N \\}$ are linearly independent over $\\mathbb{Q}_p$ and any nonzero linear combination over $\\mathbb{Q}_p$ of the functions in $V$ satisfies the necessary properties.\n\tLet $F:=\\sum_{i=1}^k a_i f_{N_i}$, where $k\\in \\mathbb N$, $a_1,\\ldots,a_k\\in \\mathbb{Q}_p$, and $N_1,\\ldots,N_k\\in \\mathcal N$ are distinct.\n\tWe begin by showing the linear independence.\n\tAssume that $F\\equiv 0$.\n\tFix $n\\in N_1^1\\cap N_2^0\\cap \\cdots \\cap N_k^0$ and take $x=p^n+p^{n+1}\\sum_{i=0}^\\infty p^i\\in B_n$.\n\tThen, $0=F(x)=a_1 f_{N_1}(x)=a_1 \\sum_{i=0}^\\infty p^i$ if and only if $a_1=0$.\n\tBy repeating the same argument, it is easy to see that $a_i=0$ for every $i\\in \\{1,\\ldots,k \\}$.\n\tAssume now that $a_i\\neq 0$ for every $i\\in \\{1,\\ldots,k \\}$.\n\tThen $F$ is continuous but also differentiable on\n\t\t$$\n\t\t\\Delta:=\\mathbb{Z}_p\\setminus \\left(\\{0\\}\\cup \\left(\\bigcup_{n\\in \\bigcup_{i=1}^k N_i} E_n \\right) \\right)\n\t\t$$\n\t(with bounded derivative).\n\t\tApplying similar arguments as above, we have that $F$ is not differentiable at $0$.\n\tLet $x\\in E_n$ with $n\\in \\bigcup_{i=1}^k N_i$.\n\tWe will analyze the differentiability of $F$ at $x$ depending on the values that $F$ takes on $B_n$.\n\tWe have two possible cases.\n\t\n\t\\textit{Case 1:} If $F$ is identically $0$ on $B_n$, then $F$ is differentiable at $x$.\n\t\n\t\\textit{Case 2:} If $F$ is not identically $0$ on $B_n$, then there exists $a\\in \\mathbb{Q}_p\\setminus\\{0\\}$ such that $F=ag$.\n\tHence $F$ is not differentiable at $x$ since $g$ is not differentiable at $x$.\n\tNotice that Case 2 is always satisfied.\n\t\n\tTo finish the proof, it is enough to show that $\\Delta$ is a full set for $\\mu$, but this is an immediate consequence of the fact that $\\{0\\}\\cup \\left(\\bigcup_{n\\in \\bigcup_{i=1}^k N_i} E_n \\right)$ is the countable union of null sets for $\\mu$ since it implies that\n\t\t$$\n\t\t\\mu\\left(\\{0\\}\\cup \\left(\\bigcup_{n\\in \\bigcup_{i=1}^k N_i} E_n \\right) \\right)=0.\n\t\t$$\n\\end{proof}\n\n\\begin{bibdiv}\n\\begin{biblist}\n\t\n\\bib{abms}{article}{\n\tAUTHOR = {Ara\\'{u}jo, G.},\n\tAUTHOR = {Bernal-Gonz\\'{a}lez, L.},\n\tAUTHOR = {Mu\\~{n}oz-Fern\\'{a}ndez, G. A.},\n\tAUTHOR = {Seoane-Sep\\'{u}lveda, J. B.},\n\tTITLE = {Lineability in sequence and function spaces},\n\tJOURNAL = {Studia Math.},\n\tVOLUME = {237},\n\tYEAR = {2017},\n\tNUMBER = {2},\n\tPAGES = {119--136},\n}\n\t\n\t\n\n\\bib{ar}{book}{\n\tauthor={Aron, R.M.},\n\tauthor={Bernal Gonz\\'{a}lez, L.},\n\tauthor={Pellegrino, D.M.},\n\tauthor={Seoane Sep\\'{u}lveda, J.B.},\n\ttitle={Lineability: the search for linearity in mathematics},\n\tseries={Monographs and Research Notes in Mathematics},\n\tpublisher={CRC Press, Boca Raton, FL},\n\tdate={2016},\n\tpages={xix+308},\n\tisbn={978-1-4822-9909-0},\n}\n\n\n\n\\bib{AGS2005}{article}{\n\tauthor={Aron, R. M.},\n\tauthor={Gurariy, V. I.},\n\tauthor={Seoane-Sep\\'{u}lveda, J. B.},\n\ttitle={Lineability and spaceability of sets of functions on $\\mathbb R$},\n\tjournal={Proc. Amer. Math. Soc.},\n\tvolume={133},\n\tdate={2005},\n\tnumber={3},\n\tpages={795--803},\n}\n\n\\bib{ar2}{article}{\n\tauthor={Aron, R.M.},\n\tauthor={P\\'{e}rez-Garc\\'{\\i }a, D.},\n\tauthor={Seoane-Sep\\'{u}lveda, J.B.},\n\ttitle={Algebrability of the set of non-convergent Fourier series},\n\tjournal={Studia Math.},\n\tvolume={175},\n\tdate={2006},\n\tnumber={1},\n\tpages={83--90},\n}\n\n\\bib{bbf}{article}{\n\tAUTHOR = {Balcerzak, Marek},\n\t\tAUTHOR = {Bartoszewicz, Artur},\n\t\tAUTHOR = {Filipczak, Ma\\l gorzata},\n\tTITLE = {Nonseparable spaceability and strong algebrability of sets of\n\t\tcontinuous singular functions},\n\tJOURNAL = {J. Math. Anal. Appl.},\n\tVOLUME = {407},\n\tYEAR = {2013},\n\tNUMBER = {2},\n\tPAGES = {263--269},\n\t}\n\\bib{bbfg}{article}{\n\tAUTHOR = {Bartoszewicz, Artur},\n\t\tAUTHOR = {Bienias, Marek},\n\t\tAUTHOR = {Filipczak, Ma\\l gorzata},\n\t\tAUTHOR = {G\\l \\c{a}b, Szymon},\n\tTITLE = {Strong {$\\germ{c}$}-algebrability of strong\n\t\t{S}ierpi\\'{n}ski-{Z}ygmund, smooth nowhere analytic and other sets\n\t\tof functions},\n\tJOURNAL = {J. Math. Anal. Appl.},\n\tVOLUME = {412},\n\tYEAR = {2014},\n\tNUMBER = {2},\n\tPAGES = {620--630},\n}\n\n\n\\bib{barglab}{article}{\n\tauthor={Bartoszewicz, A.},\n\tAUTHOR = {Bienias, M.},\n\tauthor={G\\l \\c {a}b, S.},\n\tTITLE = {Independent {B}ernstein sets and algebraic constructions},\n\tJOURNAL = {J. Math. Anal. Appl.},\n\tVOLUME = {393},\n\tYEAR = {2012},\n\tNUMBER = {1},\n\tPAGES = {138--143},\n}\n\n\n\n\\bib{bar20}{article}{\n\tAUTHOR = {Bartoszewicz, Artur},\n\tAUTHOR ={ Filipczak, Ma\\l gorzata},\n\tAUTHOR = {Terepeta, Ma\\l gorzata},\n\tTITLE = {Lineability of {L}inearly {S}ensitive {F}unctions},\n\tJOURNAL = {Results Math.},\n\tVOLUME = {75},\n\tYEAR = {2020},\n\tNUMBER = {2},\n\tPAGES = {Paper No. 64},\n}\n\n\n\\bib{ba3}{article}{\n\tauthor={Bartoszewicz, A.},\n\tauthor={G\\l \\c {a}b, S.},\n\ttitle={Strong algebrability of sets of sequences and functions},\n\tjournal={Proc. Amer. Math. Soc.},\n\tvolume={141},\n\tdate={2013},\n\tnumber={3},\n\tpages={827--835},\n}\n\n\n\n\n\n\\bib{bay}{article}{\n\tAUTHOR = {Bayart, Fr\\'{e}d\\'{e}ric},\n\tTITLE = {Linearity of sets of strange functions},\n\tJOURNAL = {Michigan Math. J.},\n\tVOLUME = {53},\n\tYEAR = {2005},\n\tNUMBER = {2},\n\tPAGES = {291--303},\n\t}\n\n\n\\bib{bq}{article}{\n\tAUTHOR = {Bayart, Fr\\'{e}d\\'{e}ric},\n\tAUTHOR = {Quarta, Lucas},\n\tTITLE = {Algebras in sets of queer functions},\n\tJOURNAL = {Israel J. Math.},\n\tVOLUME = {158},\n\tYEAR = {2007},\n\tPAGES = {285--296},\n}\n\n\\bib{bbls}{article}{\n\tAUTHOR = {Bernal-Gonz\\'{a}lez, L.},\n\tAUTHOR = {Bonilla, A.},\n\tAUTHOR = {L\\'{o}pez-Salazar, J.},\n\tAUTHOR = {Seoane-Sep\\'{u}lveda, J. B.},\n\tTITLE = {Nowhere h\\\"{o}lderian functions and {P}ringsheim singular\n\t\tfunctions in the disc algebra},\n\tJOURNAL = {Monatsh. Math.},\n\tVOLUME = {188},\n\tYEAR = {2019},\n\tNUMBER = {4},\n\tPAGES = {591--609},\n}\n\n\n\\bib{TAMS2020}{article}{\n\tauthor={Bernal-Gonz\\'{a}lez, L.},\n\tauthor={Cabana-M\\'{e}ndez, H. J.},\n\tauthor={Mu\\~{n}oz-Fern\\'{a}ndez, G. A.},\n\tauthor={Seoane-Sep\\'{u}lveda, J. B.},\n\ttitle={On the dimension of subspaces of continuous functions attaining\n\t\ttheir maximum finitely many times},\n\tjournal={Trans. Amer. Math. Soc.},\n\tvolume={373},\n\tdate={2020},\n\tnumber={5},\n\tpages={3063--3083},\n}\n\n\\bib{b20}{article}{\n\tauthor={Bernal-Gonz\\'{a}lez, L.},\n\tauthor={Mu\\~{n}oz-Fern\\'{a}ndez, G. A.},\n\tauthor={Rodr\\'{\\i }guez-Vidanes, D. L.},\n\tauthor={Seoane-Sep\\'{u}lveda, J. B.},\n\ttitle={Algebraic genericity within the class of sup-measurable functions},\n\tjournal={J. Math. Anal. Appl.},\n\tvolume={483},\n\tdate={2020},\n\tnumber={1},\n\tpages={123--576},\n}\n\n\\bib{bo}{article}{\n\tAUTHOR = {Bernal-Gonz\\'{a}lez, Luis},\n\tAUTHOR = {Ord\\'{o}\\~{n}ez Cabrera, Manuel},\n\tTITLE = {Lineability criteria, with applications},\n\tJOURNAL = {J. Funct. 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Appl.},\n\tVOLUME = {489},\n\tYEAR = {2020},\n\tNUMBER = {1},\n\tPAGES = {124132, 11},\n}\n\n\\bib{Bi}{book}{\n\tauthor={Billingsley, Patrick},\n\ttitle={Probability and measure},\n\tseries={Wiley Series in Probability and Mathematical Statistics},\n\tedition={3},\n\tnote={A Wiley-Interscience Publication},\n\tpublisher={John Wiley \\& Sons, Inc., New York},\n\tdate={1995},\n\tpages={xiv+593},\n}\n\n\\bib{cgp}{article}{\n\tAUTHOR = {Calder\\'{o}n-Moreno, M. C.},\n\tAUTHOR = {Gerlach-Mena, P. J.},\n\tAUTHOR = {Prado-Bassas, J. A.},\n\tTITLE = {Lineability and modes of convergence},\n\tJOURNAL = {Rev. R. Acad. Cienc. Exactas F\\'{\\i}s. Nat. Ser. A Mat. RACSAM},\n\tVOLUME = {114},\n\tYEAR = {2020},\n\tNUMBER = {1},\n\tPAGES = {Paper No. 18, 12},\n}\n\n\n\n\\bib{CS2019}{article}{\n\tauthor={Ciesielski, Krzysztof C.},\n\tauthor={Seoane-Sep\\'{u}lveda, Juan B.},\n\ttitle={Differentiability versus continuity: restriction and extension theorems and monstrous examples},\n\tjournal={Bull. 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An approach via algebraic genericity},\n   status={Preprint (2021)},\n}\n\n\\bib{preprint-June-2019}{article}{\n\tauthor={Fern\\'andez-S\\'anchez, J.},\n\tauthor={Mart\\'inez-G\\'omez, M.E.},\n\tauthor={Seoane-Sep\\'ulveda, J.B.},\n\ttitle={Algebraic genericity and special properties within sequence spaces and series},\n\tjournal={Preprint (2019)},\n}\n\n\\bib{fich}{article}{\n\tauthor={Fichtenholz, G.},\n\tauthor={Kantorovich, L.},\n\ttitle={Sur les op\\'{e}rations dans l'espace des functions born\\'{e}es},\n\tjournal={Studia Math.},\n\tvolume={5},\n\tdate={1934},\n\tnumber={},\n\tpages={69-98},\n\t}\n\n\\bib{Fo}{book}{\n\tauthor={Folland, Gerald B.},\n\ttitle={A course in abstract harmonic analysis},\n\tseries={Textbooks in Mathematics},\n\tedition={2},\n\tpublisher={CRC Press, Boca Raton, FL},\n\tdate={2016},\n\tpages={xiii+305 pp.+loose errata},\n}\n\n\\bib{g7}{article}{\n\tAUTHOR = {Garc\\'{\\i}a-Pacheco, F. J.},\n\tAUTHOR = {Palmberg, N.},\n\tAUTHOR = {Seoane-Sep\\'{u}lveda, J. B.},\n\tTITLE = {Lineability and algebrability of pathological phenomena in analysis},\n\tJOURNAL = {J. Math. Anal. Appl.},\n\tVOLUME = {326},\n\tYEAR = {2007},\n\tNUMBER = {2},\n\tPAGES = {929--939},\n}\n\n\n\n\\bib{go}{book}{\n\tauthor={Gouv\\^{e}a, F.Q.},\n\ttitle={$p$-adic numbers, An introduction},\n\tseries={Universitext},\n\tedition={2},\n\tpublisher={Springer-Verlag, Berlin},\n\tdate={1997},\n\tpages={vi+298},\n}\n\n\n\n\\bib{gur1}{article}{\n\tauthor={Gurari\\u {\\i }, V.I.},\n\ttitle={Subspaces and bases in spaces of continuous functions},\n\tlanguage={Russian},\n\tjournal={Dokl. Akad. Nauk SSSR},\n\tvolume={167},\n\tdate={1966},\n\tpages={971--973},\n}\n\n\\bib{h}{article}{\n\tauthor={Hausdorff, F.},\n\ttitle={Uber zwei Satze von G. Fichtenholz und L. Kantorovich},\n\tlanguage={German},\n\tjournal={Studia Math.},\n\tvolume={6},\n\tdate={1936},\n\tpages={18--19},\n}\n\n\\bib{jms}{article}{\n\tAUTHOR = {Jim\\'{e}nez-Rodr\\'{\\i}guez, P.},\n\t\tAUTHOR = {Mu\\~{n}oz-Fern\\'{a}ndez, G. A.},\n\t\tAUTHOR = {Seoane-Sep\\'{u}lveda, J. B.},\n\t\t\tTITLE = {Non-{L}ipschitz functions with bounded gradient and related\n\t\tproblems},\n\tJOURNAL = {Linear Algebra Appl.},\n\tVOLUME = {437},\n\tYEAR = {2012},\n\tNUMBER = {4},\n\tPAGES = {1174--1181},\n\t}\n\n\n\n\\bib{kato}{book}{\n\tAUTHOR = {Katok, S.},\n\tTITLE = {{$p$}-adic analysis compared with real},\n\tSERIES = {Student Mathematical Library},\n\tVOLUME = {37},\n\tPUBLISHER = {American Mathematical Society, Providence, RI; Mathematics\n\t\tAdvanced Study Semesters, University Park, PA},\n\tYEAR = {2007},\n\tPAGES = {xiv+152},\n}\n\n\\bib{jms1}{article}{\n\tauthor={Khodabendehlou, J.},\n\tauthor={Maghsoudi, S.},\n\tauthor={Seoane-Sep{\\'u}lveda, J.B.},\n\ttitle={Algebraic genericity and summability within the non-Archimedean setting},\n\tjournal={Rev. R. Acad. Cienc. Exactas F\\'{\\i}s. Nat. Ser. A Mat. RACSAM},\n\tvolume={115},\n\tnumber={21},\n\tdate={2021},\n}\n\n\n\n\\bib{jms2}{article}{\n\tauthor={Khodabendehlou, J.},\n\tauthor={Maghsoudi, S.},\n\tauthor={Seoane-Sep{\\'u}lveda, J.B.},\n\ttitle={Lineability and algebrability within $p$-adic function spaces},\n\tjournal={Bull. Belg. Math. Soc. Simon Stevin},\n\tvolume={27},\n\tpages={711\u2013729},\n\tdate={2020},\n}\n\n\\bib{jms3}{article}{\n\tauthor={Khodabendehlou, J.},\n\tauthor={Maghsoudi, S.},\n\tauthor={Seoane-Sep{\\'u}lveda, J.B.},\n\ttitle={Lineability, continuity, and antiderivatives in the\n\t\tnon-Archimedean setting},\n\tjournal={Canad. Math. Bull.},\n\tvolume={64},\n\tdate={2021},\n\tnumber={3},\n\tpages={638--650},\n}\n\n\n\\bib{levinemilman1940}{article}{\n\tauthor={Levine, B.},\n\tauthor={Milman, D.},\n\ttitle={On linear sets in space $C$ consisting of functions of bounded variation},\n\tlanguage={Russian, with English summary},\n\tjournal={Comm. Inst. Sci. Math. M\\'{e}c. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4)},\n\tvolume={16},\n\tdate={1940},\n\tpages={102--105},\n}\n\n\n\n\n\\bib{mahler}{book}{\n\tAUTHOR = {Mahler, K.},\n\tTITLE = {$p$-adic numbers and their functions},\n\tSERIES = {Cambridge Tracts in Mathematics},\n\tVOLUME = {76},\n\tEDITION = {Second},\n\tPUBLISHER = {Cambridge University Press, Cambridge-New York},\n\tYEAR = {1981},\n\tPAGES = {xi+320},\n\t}\n\n\\bib{Ma2}{article}{\n\tauthor={Mahler, K.},\n\ttitle={An interpolation series for continuous functions of a $p$-adic\n\t\tvariable},\n\tjournal={J. Reine Angew. Math.},\n\tvolume={199},\n\tdate={1958},\n\tpages={23--34},\n}\n\n\\bib{ms}{article}{\n\tAUTHOR = {Moothathu, T. K. Subrahmonian},\n\tTITLE = {Lineability in the sets of {B}aire and continuous real\n\t\tfunctions},\n\tJOURNAL = {Topology Appl.},\n\tVOLUME = {235},\n\tYEAR = {2018},\n\tPAGES = {83--91},\n}\n\n\n\n\\bib{nak}{article}{\n\tAUTHOR = {Natkaniec, Tomasz},\n\tTITLE = {On lineability of families of non-measurable functions of two\n\t\tvariable},\n\tJOURNAL = {Rev. R. Acad. Cienc. Exactas F\\'{\\i}s. Nat. Ser. A Mat. RACSAM},\n\tVOLUME = {115},\n\tYEAR = {2021},\n\tNUMBER = {1},\n\tPAGES = {Paper No. 33, 10},\n}\n\n\n\n\\bib{ro}{book}{\n\tAUTHOR = {Robert, A. M.},\n\tTITLE = {A course in $p$-adic analysis},\n\tSERIES = {Graduate Texts in Mathematics},\n\tVOLUME = {198},\n\tPUBLISHER = {Springer-Verlag, New York},\n\tYEAR = {2000},\n\tPAGES = {xvi+437},\n}\n\n\\bib{sc}{book}{\n\tauthor={Schikhof, W.H.},\n\ttitle={Ultrametric calculus, An introduction to $p$-adic analysis},\n\tseries={Cambridge Studies in Advanced Mathematics},\n\tvolume={4},\n\tpublisher={Cambridge University Press, Cambridge},\n\tdate={1984},\n\tpages={viii+306},\n}\n\n\n\n\\bib{juanksu}{book}{\n\tauthor={Seoane-Sep\\'{u}lveda, J.B.},\n\ttitle={Chaos and lineability of pathological phenomena in analysis},\n\tnote={Thesis (Ph.D.)--Kent State University},\n\tpublisher={ProQuest LLC, Ann Arbor, MI},\n\tdate={2006},\n\tpages={139},\n}\n\n\n\n\n\\bib{va}{book}{\n\tauthor={van Rooij, A.C.M.},\n\ttitle={Non-Archimedean functional analysis},\n\tseries={Monographs and Textbooks in Pure and Applied Math.},\n\tvolume={51},\n\tpublisher={Marcel Dekker, Inc., New York},\n\tdate={1978},\n\tpages={x+404},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe microbiome, a dynamic ecosystem of microorganisms (bacteria, archaea, fungi, and viruses) that live in and on us, plays a vital role in host-immune responses resulting in significant effects on host health (see, for example, \\cite{metwally2018review}). Dysbiosis of the microbiome has been linked to diseases including asthma, obesity, diabetes, transplant rejection, and inflammatory bowel disease~\\citep{Vatanen2016VariationHumans,pflughoeft2012human,Cho2012TheDisease,rani2016diverse}. \nThese observations suggest that modulation of the microbiome could become an important therapeutic modality for some diseases~\\citep{ sehrawat2021probiotics, gupta2020therapies, Yatsunenko2012HumanGeography}. \n\nModeling microbiome data is very challenging due to the exceeding number of zeros in the data~\\citep{gloor2017microbiome, knights2011supervised}. Dealing with zeros is one of the biggest challenges in microbiome and transcriptomics studies \\citep{xia2018statistical}. It is challenging to model those features which are skewed and zero-inflated \\citep{chen2012statistical}. Table~\\ref{countTableExample} is a toy example of taxonomic profile with a dimension of $3\\times 6$, where $3$ denotes the number of microbial features and $6$ denotes the number of metagenomic samples. The table shows the sparsity of the mirobiome data. Therefore, zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), Poisson hurdle (PH), and negative binomial hurdle (NBH) models are commonly used to model microbiome data \\citep{metwally2018review}.\n\\begin{table}[]\n\\caption{A Toy Example of Taxonomic Profile Count Table}\n\\label{countTableExample}\n\\begin{center}\n    \\begin{tabular}{||c| c c c c c c || c||} \n\t\t\t\\hline\n\t\t\tSpecies\/Sample & $S_1$ & $S_2$ &$S_3$ &$S_4$ &$S_5$ &$S_6$ &Total \\\\ [0.5ex] \n\t\t\t\\hline\\hline\n\t\t\t$\\textit{Streptococcus pneumoniae}$ & $0$ & 0& $102$ &0 & $3$ & 0 & $105$ \\\\ \n\t\t\t\\hline\n\t\t\t$\\textit{Escherichia coli}$ & $13$&0 & 0 &75 & $0$ & $0$ & $88$ \\\\\n\t\t\t\\hline\n\t\t\t$\\textit{Staphylococcus aureus}$ &  0&$14$& 0 & $0$ &138 & $0$& $152$ \\\\ \n\t\t\t\\hline\\hline\n\t\t\tTotal& $13$ & $14$& 102 & $75$ & 141& $0$ &345 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\\vspace{1em}\n\\end{center}\n\\end{table}\n\n\nThe  selection  of  an  appropriate  probabilistic  model  is critical for microbiome studies. For example, in order to determine if  there  is  an  association  between  a microbiome  feature (such  as a bacteria), and the disease, we may need to detect the significance of the difference between two groups of records. With appropriate probabilistic models identified successfully, we can improve the power of the statistical test significantly.\nRecently, \\cite{aldirawi2019identifying} proposed a statistical procedure for identifying the most appropriate discrete probabilistic models for zero-inflated or hurdle models based on the p-value of the discrete Kolmogorov-Smirnov (KS) test. The same procedure could be used for more general zero-inflated or hurdle models, including the ones with continuous baseline distributions. More specifically, the goal is to test if the sample ${\\mathbf X} = \\{ X_{1},X_{2},..,X_{n}  \\}$ comes from a discrete or mixed distribution with cumulative distribution function (CDF) $F_{\\boldsymbol\\theta}(x)$ where the parameter(s) $\\boldsymbol\\theta$ is unknown. Algorithm~1, which is regenerated from \\cite{aldirawi2019identifying}, provides our procedures in details.\n\n\\begin{algorithm}\n\\caption{Estimating p-value of KS test}\n\\begin{algorithmic}[1]\n\\State \\emph{Given ${\\mathbf X}=(X_1, X_2, \\cdots X_n$)}\n\\State \\emph{For $b = 1, \\ldots, B$,\nresample $X$ with replacement to get a bootstrapped sample ${\\mathbf X}^{(b)} = \\{X^{(b)}_{1}, \\cdots, X_{n}^{(b)}\\}$.}\n\\State \\emph{For each b, calculate the MLE $\\hat{\\boldsymbol\\theta}^{(b)}$ of $\\boldsymbol\\theta$.}\n\\State \\emph{Simulate ${\\mathbf X}^{(c)} = \\{X^{(c)}_1, \\ldots, X^{(c)}_n\\}$ iid from $F_{\\hat{\\boldsymbol\\theta}^{(b)}}$, which is the CDF $F_{\\boldsymbol\\theta}(x)$ with parameter $\\boldsymbol\\theta = \\hat{\\boldsymbol\\theta}^{(b)}$. }\n\\State \\emph{Calculate the KS statistic $D_{n}^{(b)}={\\rm sup}_{x}\\lvert \\hat F^{(c)}_{n}(x)- F_{\\hat{\\boldsymbol\\theta}^{(b)}}(x)\\rvert$, where $\\hat F^{(c)}_{n}(x)$ is the empirical distribution function of ${\\mathbf X}^{(c)}$. }\n\\State \\emph{Estimate the p-value by \n$\\frac{\\#\\{b \\mid D_{n}^{(b)} > D_{n}\\}+1}{B+1}$\nwhere $D_{n} = {\\rm sup}_{x}\\lvert \\hat F_{n}(x)- F_{\\hat{\\boldsymbol\\theta}}(x)\\rvert$ is the KS statistic based on the original data and its MLE $\\hat{\\boldsymbol\\theta}$.}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\nAlthough the following procedure and algorithm were described, their theoretical justifications were not  provided in ~\\cite{aldirawi2019identifying}. One major step in the above algorithm is to estimate the distribution parameters (step 3) using the maximum likelihood estimate (MLE) method. In this paper, we develop a general MLE procedure for estimating the parameters for general zero-inflated and hurdle models. In addition, we discuss the asymptotic properties and Fisher information matrix for MLEs, which can be used for building up the confidence intervals of the distribution parameters. For modeling microbiome data, we recommend zero-inflated (or hurdle) beta binomial or beta negative binomial models.  \n\n \n\n\n\n\n\n\\section{Zero-altered or hurdle models and their MLEs}\\label{sec:hurdle}\n\nZero-altered models, also known as {\\it hurdle models}, have been used for modeling data with an excess or deficit of zeros (see, for example, ~\\cite{metwally2018review}, for a review). A general hurdle model consists of two components, one generating the zeros and the other generating non-zeros (or positive values for many applications). Given a baseline distribution $f_{\\boldsymbol\\theta}(y)$, which could be the probability mass function (pmf) of a discrete distribution or the probability density function (pdf) of a condinuous distribution, with parameter(s) $\\boldsymbol\\theta = (\\theta_1, \\ldots, \\theta_p)^T$,  the distribution function of the corresponding hurdle model can be written as follows:\n\\begin{equation}\\label{eq:hurdle}\nf_{\\rm ZA}(y\\mid \\phi, \\boldsymbol\\theta) = \\phi {\\mathbf 1}_{\\{y=0\\}} + (1-\\phi) f_{\\rm tr}(y\\mid \\boldsymbol\\theta) {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{equation}\nwhere $\\phi \\in [0, 1]$ is the weight parameter of zeros, $f_{\\rm tr}(y\\mid \\boldsymbol\\theta) = [1-p_0(\\boldsymbol\\theta)]^{-1} f_{\\boldsymbol\\theta} (y), y\\neq 0$ is the pmf or pdf of the zero-truncated baseline distribution, and $p_0(\\boldsymbol\\theta) = f_{\\boldsymbol\\theta}(0)$ for discrete baseline distributions or simply $0$ for continuous baseline distributions.\nExamples with discrete baseline distributions include zero-altered Poisson (ZAP) or Poisson hurdle (PH), zero-altered negative binomial (ZANB) or negative binomial hurdle (NBH) models and others, where model~\\eqref{eq:hurdle} provides a new pmf. Examples with continuous basedline distributions include zero-altered Gaussian (ZAG) or Gaussian hurdle (GH), zero-altered lognormal or lognormal hurdle models and others, where model~\\eqref{eq:hurdle} is indeed a mixture distribution with a discrete part with a probability mass $\\phi$ at $[Y=0]$ and a continuous component in $[Y \\neq 0]$ with density function $(1-\\phi) f_{\\boldsymbol\\theta}(y), y\\neq 0$.\n\nThe zero-altered models can actually be defined with a fairly general baseline distribution equiped with a cumulative distribution function (cdf) $F_{\\boldsymbol\\theta}(y) = P_{\\boldsymbol\\theta} (Y\\leq y)$. Its corresponding cdf is defined as follows:\n\\[\nF_{\\rm ZA}(y\\mid \\phi, \\boldsymbol\\theta) = P_{\\rm ZA}(Y \\leq y\\mid \\phi, \\boldsymbol\\theta) = \\phi {\\mathbf 1}_{\\{y \\geq 0\\}} + (1-\\phi) F_{\\rm tr}(y\\mid \\boldsymbol\\theta) \n\\]\nwhere $F_{\\rm tr}(y\\mid \\boldsymbol\\theta) = [F_{\\boldsymbol\\theta}(y) - P_{\\boldsymbol\\theta}(Y=0) {\\mathbf 1}_{\\{y \\geq 0\\}}]\/[1-P_{\\boldsymbol\\theta}(Y=0)]$ is a zero-truncated cdf. \nIn this paper, we assume that the baseline distribution is either discrete or continuous with pmf or pdf $f_{\\boldsymbol\\theta}(y)$.\n\n\\subsection{Maximum likelihood estimate for zero-altered or hurdle model}\n\nThe parameters of hurdle model~\\eqref{eq:hurdle} include both $\\phi$ and $\\boldsymbol\\theta$. Let $Y_1, \\ldots, Y_n$ be a random sample from model~\\eqref{eq:hurdle}. \nThe likelihood function of $(\\phi, \\boldsymbol\\theta)$ is\n\\begin{equation}\\label{eq:hurdelmle}\nL(\\phi, \\boldsymbol\\theta) = \\phi^{n-m} (1-\\phi)^m \\cdot \\prod_{i:Y_i\\neq 0} f_{\\rm tr}(Y_i\\mid \\boldsymbol\\theta)\n\\end{equation}\nwhere $m=\\#\\{i:Y_i\\neq 0\\}$ is the number of nonzero observations. Since $\\phi$ and $\\boldsymbol\\theta$ are separable in the likelihood function, we obtain the following theorem.\n\n\\begin{theorem}\\label{thm:hurdlemle}\nFor model~\\eqref{eq:hurdle} with zero-truncated pmf or pdf $f_{\\rm tr}(y\\mid \\boldsymbol\\theta)$, the maximum likelihood estimate (MLE) maximizing \\eqref{eq:hurdelmle} is\n\\[\n\\hat{\\phi}=1-\\frac{m}{n},\\>\\>\\> \n\\hat{\\boldsymbol\\theta}={\\rm argmax}_{\\boldsymbol\\theta} \\prod_{i:Y_i\\neq 0} f_{\\rm tr}(Y_i\\mid \\boldsymbol\\theta)\n\\]\n\\end{theorem}\nRecall that $f_{\\rm tr}(y\\mid \\boldsymbol\\theta) = [1-p_0(\\boldsymbol\\theta)]^{-1} f_{\\boldsymbol\\theta}(y), y\\neq 0$ for discrete baseline functions or simply $f_{\\boldsymbol\\theta}(y), y\\neq 0$ for continuous baseline functions.\n\n\\begin{example}\\label{ex:ZABB}{\nFor zero-altered beta binomial or beta binomial hurdle (BBH) distribution, the pmf of the baseline distribution is \n\\[\nf_{\\boldsymbol\\theta} (y) = \\dbinom{n}{y} \\frac{{ \\rm beta}(y + \\alpha, n - y + \\beta)}{{ \\rm beta}(\\alpha,\\beta)}\n\\]\nwhere $\\boldsymbol\\theta = (n, \\alpha, \\beta)$, $y=0, 1, \\ldots, n$ and\n\\begin{equation*}\np_0(\\boldsymbol\\theta) = \\frac{\\Gamma(n + \\beta) \\Gamma(\\alpha + \\beta)}{\\Gamma(n + \\alpha + \\beta)  \\Gamma(\\beta)}\n\\end{equation*}\n\tLet $L(\\boldsymbol\\theta)$ be the likelihood of zero-truncated beta binomial distribution, then \n\n\\begin{eqnarray*}\n\tL(n,\\alpha,\\beta)&=& {argmax}_{\\boldsymbol\\theta}  \\frac{\\prod_{i: y_i \\neq 0}f_{\\theta}(y_i)}{[1-p_0(\\boldsymbol\\theta)]^m}=\\left(\\frac{\\Gamma(\\alpha+n+\\beta)\\Gamma(\\beta)}{\\Gamma(\\alpha+n+\\beta)\\Gamma(\\beta)-\\Gamma(n+\\beta)\\Gamma(\\alpha+\\beta)}\\right)^{m}\\cdot\\\\\t\n\t& &\\prod_{i=1}^{m}   \\left(\\frac{\\Gamma(n+1)\\Gamma(y_i+\\alpha)\\Gamma(n-y_i+\\beta)\n\t\t\\Gamma(\\alpha+\\beta)}{\\Gamma(y_{i}+1)\\Gamma(n-y_i+1)\\Gamma(\\alpha+n+\\beta)\\Gamma(\\alpha) \\Gamma(\\beta)}\\right)  \n\\end{eqnarray*}\n\t\n The loglikelihood of zero-truncated beta binomial is given by:\n\\begin{eqnarray*}  \nl(n,\\alpha,\\beta)&=& m\\log\\Gamma(n+1)+m\\log\\Gamma(\\alpha+\\beta)-m\\log\\Gamma(\\alpha)+\\sum_{i=1}^{n}\\log \\Gamma(y_{i}+\\alpha) \\\\ \n&-& m\\log\\left(\\Gamma(\\alpha+n+\\beta)\\Gamma(\\beta)-\\Gamma(n+\\beta)\\Gamma(\\alpha+\\beta)\\right)+\\sum_{i=1}^{m}\\log \\Gamma(n-y_{i}+\\beta)\\\\ &-& \\sum_{i=1}^{m}\\log \\Gamma(y_{i}+1) -\\sum_{i=1}^{m}\\log \\Gamma(n-y_{i}+1)\n\\end{eqnarray*}\n\t\n\t\n\nLet $\\Psi(\\cdot) = \\Gamma'(\\cdot)\/\\Gamma(\\cdot)$, known as the {\\it digamma} function. In order to calculate the MLE, the following formulae are needed:\n\\begin{eqnarray*}\n\t \\frac{\\partial l(n,\\alpha,\\beta)}{\\partial n}& =& m\\left(\\frac{\\exp(\\log B-\\log A)(\\psi(n+\\beta)-\\psi(n+\\alpha+\\beta))}{1-\\exp(\\log B-\\log A)}\\right)+m\\psi(n+1)\\\\\n\t &&+\n\\sum_{i=1}^{m} \\psi(n-y_{i}+\\beta) -\\sum_{i=1}^{m} \\psi(n-y_{i}+1)-m\\psi(\\alpha+n+\\beta) \\\\\n\t \\frac{\\partial l(n,\\alpha,\\beta)}{\\partial \\alpha} &=& m\\left(\\frac{\\exp(\\log B-\\log A)(\\psi(n+\\beta)-\\psi(n+\\alpha+\\beta))}{1-\\exp(\\log B-\\log A)}\\right)\\\\\n\t &&+\\sum_{i=1}^{m} \\psi(y_{i}+\\alpha) +m\\psi(\\alpha+\\beta)-m\\psi(\\alpha+n+\\beta)-m\\psi(\\alpha) \\\\\n\t \\frac{\\partial l(n,\\alpha,\\beta)}{\\partial \\beta} &=& m\\left(\\frac{\\exp(\\log B-\\log A)(\\psi(n+\\beta)+\\psi(\\alpha+\\beta)-\\psi(\\alpha+n+\\beta)-\\psi(\\beta))}{1-\\exp(\\log B-\\log A)}\\right)\\\\\n\t &&-m\\psi(\\beta)+\\sum_{i=1}^{m} \\psi(n-y_{i}+\\beta) +m\\psi(\\alpha+\\beta)-m\\psi(\\alpha+n+\\beta) \n\\end{eqnarray*}\nwhere $A=\\Gamma(n+\\alpha+\\beta) \\Gamma(\\beta)$, and $B=\\Gamma(n+\\beta) \\Gamma(\\alpha+\\beta)$.\n}\\hfill{$\\Box$}\n\\end{example}\n\n\n\n\n\n\\subsection{Asymptotic properties and Fisher information matrix of hurdle MLEs}\n\nLet $Y_1, \\ldots, Y_n$ be a random sample from hurdle model~\\eqref{eq:hurdle}. In order to find the MLE numerically, we often consider the log-likelihood function\n\\begin{eqnarray*}\nl(\\phi, \\boldsymbol\\theta) = \\log L(\\phi, \\boldsymbol\\theta) &&= \\log\\phi \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i=0\\}} + \\log(1-\\phi) \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\\\\\n&&-\\ \\log [1-p_0(\\boldsymbol\\theta)] \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}} + \\sum_{i=1}^n \\log f_{\\boldsymbol\\theta}(Y_i) {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\n\\end{eqnarray*}\nwhose first derivatives are\n\\begin{eqnarray*}\n\\frac{\\partial l}{\\partial\\phi} &=& \\frac{1}{\\phi} \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i=0\\}} - \\frac{1}{1-\\phi} \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\\\\\n\\frac{\\partial l}{\\partial \\boldsymbol\\theta} &=& \\frac{p_0(\\boldsymbol\\theta)}{1-p_0(\\boldsymbol\\theta)}\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}} + \\sum_{i=1}^n \\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\n\\end{eqnarray*}\n\n\\begin{lemma}\\label{lem:hurdelmlefirst}\n\\[\nE\\left(\\frac{\\partial l}{\\partial \\phi}\\right) = 0 \\>\\>\\mbox{ and }\\>\\> E\\left(\\frac{\\partial l}{\\partial \\boldsymbol\\theta}\\right) = \\frac{n(1-\\phi)}{1-p_0(\\boldsymbol\\theta)} \\cdot E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta} \\right]\n\\]\nwhere $Y'$ follows the baseline distribution $f_{\\boldsymbol\\theta}(y)$.\n\\end{lemma}\n\n\n\\medskip\\noindent\n{\\bf Proof of Lemma~\\ref{lem:hurdelmlefirst}:} \nSince $P(Y_i=0) = \\phi$, then $E(\\partial l\/\\partial\\phi) = 0$. Let $Y_1', \\ldots, Y_n'$ be iid $\\sim f_{\\boldsymbol\\theta} (y)$. Then\n\\begin{eqnarray*}\nE\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\\right] \\ &&=\\ \\frac{1-\\phi}{1-p_0(\\boldsymbol\\theta)} \\cdot E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i'\\neq 0\\}}\\right]  \\\\\n&&= \\frac{1-\\phi}{1-p_0(\\boldsymbol\\theta)} \\cdot \\left\\{ E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} \\right] - p_0(\\boldsymbol\\theta)\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta}\\right\\}\n\\end{eqnarray*}\nThen\n\\[\nE\\left(\\frac{\\partial l}{\\partial \\boldsymbol\\theta}\\right) = \\frac{1-\\phi}{1-p_0(\\boldsymbol\\theta)} \\cdot \\sum_{i=1}^n E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} \\right] = \\frac{n(1-\\phi)}{1-p_0(\\boldsymbol\\theta)} \\cdot E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta} \\right]\n\\]\n\\hfill{$\\Box$}\n\nAs a direct corollary of Theorem~17 in \\cite{ferguson1996}, the MLEs of hurdle model have strong consistency under fairly general conditions.\n\n\n\\begin{theorem}\\label{thm:hurdlemleconsistency}\nLet $Y_1, \\ldots, Y_n$ be a random sample from hurdle model~\\eqref{eq:hurdle} with true parameter value $(\\phi_0, \\boldsymbol\\theta_0) \\in (0,1)\\times \\boldsymbol\\Theta$, where $\\boldsymbol\\Theta$ is compact. Let\n$\\hat{\\phi}=n^{-1} \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i =0\\}}$ and $\\hat{\\boldsymbol\\theta}={\\rm argmax}_{\\boldsymbol\\theta \\in \\boldsymbol\\Theta} \\prod_{i:Y_i\\neq 0} f_{\\rm tr}(Y_i\\mid \\boldsymbol\\theta)$ be the MLEs.\nSuppose (1) $f_{\\boldsymbol\\theta}(y)$ is continuous in $\\boldsymbol\\theta$ for all $y$; (2) $f_{\\boldsymbol\\theta}(y) = f_{\\boldsymbol\\theta_0}(y)$ for all $y$ always implies $\\boldsymbol\\theta = \\boldsymbol\\theta_0$; and (3) there exists a nonnegative function $K(y)$ such that $E[K(Y)] < \\infty$ for $Y\\sim f_{\\rm tr}(y; \\boldsymbol\\theta_0)$ and $\\log[f_{\\rm tr}(y\\mid \\boldsymbol\\theta)\/f_{\\rm tr}(y \\mid \\boldsymbol\\theta_0)] \\leq K(y)$ for all $y\\neq 0$ and $\\boldsymbol\\theta \\in \\boldsymbol\\Theta$. Then $\\hat\\phi \\stackrel{\\rm a.s.}{\\longrightarrow} \\phi_0$ and $\\hat{\\boldsymbol\\theta}_n \\stackrel{\\rm a.s.}{\\longrightarrow} \\boldsymbol\\theta_0$ as $n$ goes to infinity.\n\\end{theorem}\n\n\nUnder regularity conditions, see for example, Chapter 18 in \\cite{ferguson1996} or Section~5f in \\cite{rao1973linear}, $E[\\partial \\log f_{\\boldsymbol\\theta}(Y')\/\\partial \\boldsymbol\\theta]=0$ and thus $E\\left(\\partial l\/\\partial \\boldsymbol\\theta\\right) =0$ according to Lemma~\\ref{lem:hurdelmlefirst}. We can further calculate the Fisher information matrix of the random sample\n\\begin{equation}\\label{eq:fisher_information}\n{\\mathbf F}(\\phi, \\boldsymbol\\theta) = \n-\\left[\n\\begin{array}{cc}\nE\\left(\\frac{\\partial^2 l}{\\partial \\phi^2}\\right) & E\\left(\\frac{\\partial^2 l}{\\partial \\phi \\partial \\boldsymbol\\theta^T}\\right) \\\\\nE\\left(\\frac{\\partial^2 l}{\\partial\\boldsymbol\\theta \\partial \\phi}\\right) & E\\left(\\frac{\\partial^2 l}{\\partial \\boldsymbol\\theta \\partial\\boldsymbol\\theta^T}\\right)\n\\end{array}\\right]\n\\end{equation}\n\n\\begin{theorem}\\label{thm:hurdle_Fisher}\nLet $Y_1, \\ldots, Y_n$ be a random sample from hurdle model~\\eqref{eq:hurdle}. Under regularity conditions, the Fisher information matrix of the sample is\n\\[\n{\\mathbf F}_{\\rm ZA} = \nn \\left[\\begin{array}{cc}\n\\phi^{-1} (1-\\phi)^{-1} & {\\mathbf 0}^T\\\\\n{\\mathbf 0} & {\\mathbf F}_{\\rm ZA22}\n\\end{array}\\right]\n\\]\nwhere \n\\[\n{\\mathbf F}_{\\rm ZA22} = -\\frac{1-\\phi}{1-p_0(\\boldsymbol\\theta)} \\left(E\\left[\\frac{\\partial^2 \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} \\right] + \\frac{p_0(\\boldsymbol\\theta)}{1-p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta}\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T}\\right)\n\\]\nand $Y'$ follows the baseline distribution $f_{\\boldsymbol\\theta}(y)$.\n\\end{theorem}\n\n\n\\medskip\\noindent\n{\\bf Proof of Theorem~\\ref{thm:hurdle_Fisher}:}\nSince \n\\begin{eqnarray*}\n\\frac{\\partial\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi} &=& \\phi^{-1} {\\mathbf 1}_{\\{y=0\\}} - (1-\\phi)^{-1} {\\mathbf 1}_{\\{y\\neq 0\\}}\\\\\n\\frac{\\partial\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} &=& \\frac{p_0(\\boldsymbol\\theta)}{1-p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{y\\neq 0\\}} + \\frac{\\partial \\log f_{\\boldsymbol\\theta}(y)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{eqnarray*}\nThen \n\\begin{eqnarray*}\n\\frac{\\partial^2\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi^2} &=& -\\phi^{-2} {\\mathbf 1}_{\\{y=0\\}} - (1-\\phi)^{-2} {\\mathbf 1}_{\\{y\\neq 0\\}}\\\\\n\\frac{\\partial^2\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi \\partial \\boldsymbol\\theta^T} &=& {\\mathbf 0}^T\\\\\n\\frac{\\partial^2\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\phi} &=& {\\mathbf 0}\\\\\n\\frac{\\partial^2\\log f_{\\rm ZA} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} &=&  \\frac{p_0(\\boldsymbol\\theta)}{[1-p_0(\\boldsymbol\\theta)]^2} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T}{\\mathbf 1}_{\\{y\\neq 0\\}}\\\\\n&+&\\frac{p_0(\\boldsymbol\\theta)}{1-p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial^2\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} {\\mathbf 1}_{\\{y\\neq 0\\}} +\\frac{\\partial^2 \\log f_{\\boldsymbol\\theta}(y)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{eqnarray*}\nThe rest of the conclusions can be obtained via $l(\\phi, \\boldsymbol\\theta) = \\sum_{i=1}^n \\log f_{\\rm ZA} (Y_i \\mid \\phi, \\boldsymbol\\theta)$.\n\\hfill{$\\Box$}\n\nAs direct conclusions of Theorem~18 in \\cite{ferguson1996}: (i) $\\sqrt{n}(\\hat\\phi - \\phi_0)  \\stackrel{{\\cal L}}{\\rightarrow} N(0, \\phi_0 (1-\\phi_0))$; (ii) $\\sqrt{n}(\\hat{\\boldsymbol\\theta} - \\boldsymbol\\theta_0)  \\stackrel{{\\cal L}}{\\rightarrow} N\\left(0, {\\mathbf F}_{\\rm ZA22}^{-1} \\right)$; and (iii) $\\hat\\phi$ and $\\hat{\\boldsymbol\\theta}$ are asymptotically independent. These results can be used for building up confidence intervals of $\\phi$ and $\\boldsymbol\\theta$.\n\n\\begin{example}\\label{ex:ZAP}{\\rm\nFor zero-altered Poisson or Poisson hurdle  distribution, the pmf of the baseline distribution is $f_\\lambda(y)= e^{-\\lambda}\\lambda^{y}\/y!$ with $p_0(\\lambda) = e^{-\\lambda}$. It can be verified that \n\\[\nE\\left(\\frac{\\partial\\log f_\\lambda(Y')}{\\partial \\lambda}\\right) = 0\n\\]\nif $Y' \\sim f_\\lambda (y)$. The truncated pmf is\n\\[\n\\rm f_{\\rm tr}(y \\mid \\lambda) = \\frac{e^{-\\lambda}}{1-e^{-\\lambda}} \\cdot \\frac{\\lambda^y}{y!}, \\>\\>\\> y=1,2,\\ldots \n\\]\nThe loglikelihood for the zero-truncated Poisson is\n\\[\nl(\\lambda) = -m\\lambda-m\\log(1-e^{-\\lambda}) + \\sum_{i:Y_i>0} Y_i \\cdot \\log\\lambda - \\log(\\prod_{i:Y_i>0} Y_i!)\n\\] \nThe MLE $\\hat\\lambda$ of $\\lambda$ solves the likelihood equation $\\lambda = \\bar{Y} (1-e^{-\\lambda})$ with $\\bar{Y} = m^{-1} \\sum_{i:Y_i>0} Y_i$~, which can be solved numerically. If the true value $\\lambda_0 \\in [\\lambda_1, \\lambda_2]$ for some $0 < \\lambda_1 < \\lambda_2 < \\infty$, then $K(y)$ in Theorem~\\ref{thm:hurdlemle} can be chosen as\n\\[\nK(y) = \\log\\frac{\\lambda_2}{\\lambda_1} \\cdot y + \\log\\frac{1-e^{-\\lambda_2}}{1-e^{-\\lambda_1}} + \\lambda_2 - \\lambda_1\n\\]\nSince there is no difference in practice as long as $0 < \\lambda_1 < \\hat\\lambda < \\lambda_2 < \\infty$, we know $\\hat\\lambda \\stackrel{\\rm a.s.}{\\rightarrow} \\lambda_0$ as $n$ goes to infinity.\n\nAccording to Theorem~\\ref{thm:hurdle_Fisher}, the Fisher information matrix of the Poisson hurdle sample is\n\\[\n{\\mathbf F}_{\\rm PH} = \\left[\\begin{array}{cc}\n\\frac{1}{\\phi(1-\\phi)} & 0 \\\\\n0 & \\frac{1-\\phi}{1-e^{-\\lambda}} \\left(\\frac{1}{\\lambda} - \\frac{e^{-\\lambda}}{1-e^{-\\lambda}}\\right)\n\\end{array}\\right]\n\\]\nNote that $\\frac{1}{\\lambda} - \\frac{e^{-\\lambda}}{1-e^{-\\lambda}} > 0$ as long as $\\lambda > 0$.\n}\\hfill{$\\Box$}\n\\end{example}\n\n\n\n\\begin{example}\\label{ex:ZANB}{\\rm\nFor zero-altered negative binomial or negative binomial hurdle distribution, the pmf of the baseline distribution with parameters $\\boldsymbol\\theta = (r, p) \\in (0, \\infty)\\times [0,1]$ is given by\n$f_{\\boldsymbol\\theta}(y)= \\frac{\\Gamma(y+r)}{\\Gamma(y+1) \\Gamma(r)} p^{y}(1-p)^{r}$, $y \\in \\{0, 1, 2, \\ldots\\}$. Then $p_0(\\boldsymbol\\theta) = (1-p)^r$. In order to apply Lemma~\\ref{lem:hurdelmlefirst}, we obtain\n\\begin{eqnarray*}\n\\log f_{\\boldsymbol\\theta}(y) &=& \\log\\Gamma(y+r) - \\log\\Gamma(y+1) - \\log\\Gamma(r) + y\\log p + r\\log(1-p)\\\\\n\\frac{\\partial\\log f_{\\boldsymbol\\theta}(y)}{\\partial r} &=& \\Psi(y+r)-\\Psi(r)+\\log(1-p)\\\\\n\\frac{\\partial\\log f_{\\boldsymbol\\theta}(y)}{\\partial p} &=& \\frac{y}{p}-\\frac{r}{1-p}\n\\end{eqnarray*}\nwhere $\\Psi(\\cdot) = \\Gamma'(\\cdot)\/\\Gamma(\\cdot)$ is known as the {\\it digamma} function.\n\nIf $Y' \\sim f_{\\boldsymbol\\theta}(y)$, then $E(Y') = pr\/(1-p)$ and $E\\left(\\partial\\log f_{\\boldsymbol\\theta}(Y')\/\\partial p\\right) = 0$. On the other hand,\nsince $\\Gamma(y) = \\int_0^\\infty t^{y-1} e^{-t} dt$ and $\\Gamma'(y) = \\int_0^\\infty t^{y-1} e^{-t} \\log t dt$ for $y>0$, then \n\n\\begin{eqnarray*}\nE\\left(\\Psi(Y'+r)\\right) &=& \\sum_{y=0}^\\infty \\frac{\\Gamma'(y+r)}{\\Gamma(y+r)}\\cdot \\frac{\\Gamma(y+r)}{\\Gamma(y+1)\\Gamma(r)}p^y (1-p)^r\\\\\n&=& \\frac{(1-p)^r}{\\Gamma(r)} \\sum_{y=0}^\\infty \\frac{p^y}{y!} \\Gamma'(y+r)\\\\\n&=& \\frac{(1-p)^r}{\\Gamma(r)} \\sum_{y=0}^\\infty \\frac{p^y}{y!} \\int_0^\\infty t^{y+r-1} e^{-t}\\log tdt\\\\\n&=& \\frac{(1-p)^r}{\\Gamma(r)} \\int_0^\\infty \\left(\\sum_{y=0}^\\infty \\frac{(pt)^y}{y!} e^{-pt}\\right) \\cdot t^{r-1} e^{-t(1-p)} \\log t dt\\\\ \t\n&=& \\frac{(1-p)^r}{\\Gamma(r)} \\int_0^\\infty t^{r-1} e^{-t(1-p)} \\log t dt\\>\\>\\> (\\mbox{let }s=(1-p)t)\\\\ \t\n&=& \\frac{(1-p)^r}{\\Gamma(r)} \\int_0^\\infty s^{r-1} e^{-s}\\left[\\log s - \\log(1-p)\\right] ds \\cdot (1-p)^{-r}\\\\\t\n&=& \\frac{1}{\\Gamma(r)} \\left[\\int_0^\\infty s^{r-1} e^{-s}\\log s ds - \\int_0^\\infty s^{r-1} e^{-s} \\log(1-p) ds\\right]\\\\\n&=& \\frac{1}{\\Gamma(r)} \\left[\\Gamma'(r) - \\log(1-p) \\Gamma(r)\\right]\\\\\n&=& \\Psi(r) - \\log(1-p)\n\\end{eqnarray*}\nTherefore, $E\\left(\\partial\\log f_{\\boldsymbol\\theta}(Y')\/\\partial r\\right) =  E\\left(\\Psi(Y'+r)\\right) - \\Psi(r)+\\log(1-p) = 0$.\n\nAccording to Theorem~\\ref{thm:hurdle_Fisher}, the Fisher information matrix of a random sample from the negative binomial hurdle distribution is\n\\[\n{\\mathbf F}_{\\rm NBH} = \\left[\\begin{array}{cc}\n\\frac{1}{\\phi(1-\\phi)} & {\\mathbf 0}^T\\\\\n{\\mathbf 0} & {\\mathbf F}_{\\rm NBH22}\n\\end{array}\\right]\n\\]\nwith\n\\[\n{\\mathbf F}_{\\rm NBH22} = \n\\frac{1-\\phi}{1-(1-p)^r} \\left\\{\n\\left[\\begin{array}{cc}\nA(y,r) & \\frac{1}{1-p}\\\\\n\\frac{1}{1-p} & \\frac{r}{p(1-p)^2}\n\\end{array}\\right] -\n\\frac{(1-p)^r}{1-(1-p)^r}\n\\left[\\begin{array}{cc}\n\\log^2(1-p) & -\\frac{r\\log (1-p)}{1-p}\\\\\n-\\frac{r\\log(1-p)}{1-p} & \\frac{r^2}{(1-p)^2}\n\\end{array}\\right]\\right\\}\n\\]\n$A(y,r)=\\Psi_1(r) - E\\Psi_1(Y' + r)$\nwhere $\\Psi_1(\\cdot) = \\Psi'(\\cdot)$ is known as the {\\it trigamma} function. Since $E\\Psi_1(Y' + r)$ does not have a simple form for computation, a numerical solution for estimating it has been proposed by \\cite{guo2020}. \n}\\hfill{$\\Box$}\n\\end{example}\n\n\n\\section{Zero-inflated models and their MLEs}\\label{sec:zimodel}\n\nUnlike zero-altered models, a zero-inflated model always assumes an excess of zeros. Besides zeros coming from the baseline distribution, such as Poisson or negative binomial, there are additional zeros modeled by a weight parameter $\\phi \\in [0,1]$. \n\nWhen the baseline distribution is discrete with a pmf $f_{\\boldsymbol\\theta}(y)$, the corresponding zero-inflated model has a pmf $f_{\\rm ZI}(y\\mid \\phi, {\\boldsymbol\\theta}) = \\phi {\\mathbf 1}_{\\{y=0\\}} + (1-\\phi) f_{{\\boldsymbol\\theta}}(y)\n$ as well. In order to cover both pmf and pdf, we would rather write the distribution function of the zero-inflated distribution as\n\\begin{equation}\\label{eq:zimodel}\nf_{\\rm ZI}(y\\mid \\phi, {\\boldsymbol\\theta}) = [\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)] {\\mathbf 1}_{\\{y=0\\}} + (1-\\phi) f_{{\\boldsymbol\\theta}}(y) {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{equation}\nRecall that $p_0(\\boldsymbol\\theta) = f_{\\boldsymbol\\theta}(0)$ for pmf and $0$ for pdf. Similar to zero-altered models, when the baseline function $f_{\\boldsymbol\\theta}(y)$ is a pdf, the zero-inflated model is a mixture distribution with a probability mass on $[Y=0]$ and a density in $[Y\\neq 0]$. It should be noted that when the baseline distribution is either continuous with a pdf $f_{\\boldsymbol\\theta}(y)$ or discrete but with $p_0(\\boldsymbol\\theta)=0$, the zero-inflated model is essentially the same as the corresponding zero-altered model.\n\nCommonly used baseline distributions include Gaussian, half-normal, Poisson, negative binomial, beta binomial, etc. The corresponding zero-inflated models are known as zero-inflated Gaussian (ZIG), zero-inflated half-normal (ZIHN), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-inflated beta binomial (ZIBB), respectively.\n\n \n\\subsection{Maximum likelihood estimate for zero-inflated models}\\label{sec:zimle}\n\nGiven a random sample $Y_1, \\ldots, Y_n$ from the zero-inflated model~\\eqref{eq:zimodel}, we adopt the maximum likelihood estimate $\\hat\\phi$ for $\\phi$ and $\\hat{\\boldsymbol\\theta}$ for $\\boldsymbol\\theta$. Similar as in Section~\\ref{sec:hurdle}, we denote $m= \\#\\{i:Y_i \\neq 0\\}$.\n\nIf the baseline distribution satisfies $P_{\\boldsymbol\\theta}(Y=0)=0$, then the likelihood function $L(\\phi, \\boldsymbol\\theta) = \\phi^{n-m} (1-\\phi)^m \\cdot \\prod_{i: Y_i \\neq 0} f_{\\boldsymbol\\theta}(Y_i)$. Then $\\hat\\phi = 1-m\/n$ and $\\hat{\\boldsymbol\\theta} = {\\rm argmax}_{\\boldsymbol\\theta} \\prod_{i: Y_i \\neq 0} f_{\\boldsymbol\\theta}(Y_i)$, which are the same as the ones for hurdle models.\n\nFor general cases, the likelihood function for model~\\eqref{eq:zimodel} is\n\\begin{equation}\\label{eq:discreteL}\nL(\\phi, \\boldsymbol{\\theta}) \n=\\left[\\phi + (1-\\phi)p_0({\\boldsymbol\\theta}) \\right]^{n-m} (1-\\phi)^{m}\n \\left(1 - p_0(\\boldsymbol\\theta)\\right)^{m} \\prod_{i: y_i \\neq 0} \\rm f_{\\rm tr}(Y_{i} \\mid \\boldsymbol\\theta)\n\\end{equation}\nwhere $f_{\\rm tr}(y \\mid \\boldsymbol\\theta) = f_{\\boldsymbol\\theta}(y)\/[1-p_0(\\boldsymbol\\theta)], y\\neq 0$ is the pmf or pdf of the zero-truncated version of the baseline distribution. \nBy reparametrization, we let\n\\begin{equation}\n\t\\psi = 1- [\\phi + (1-\\phi) p_0({\\boldsymbol\\theta}) ] = (1-\\phi) [1-p_0({\\boldsymbol\\theta})]\n\\end{equation}\nThen $\\phi = 1 - \\psi\/[1 - p_0(\\boldsymbol\\theta)]$ and the likelihood of $\\psi$ and $\\boldsymbol\\theta$ is\n\\[\nL(\\psi, \\boldsymbol\\theta)\n=(1-\\psi)^{n-m}\\psi^{m}\\cdot \\prod_{i: Y_i \\neq 0} \\rm f_{\\rm tr}(Y_i \\mid \\boldsymbol\\theta)\n\\] \nwhich is separable for $\\psi$ and $\\boldsymbol\\theta$.\n\n\\begin{theorem}\\label{thm:zimle}\nLet $\\boldsymbol\\theta_* = {\\rm argmax}_{\\boldsymbol\\theta} \\prod_{i: Y_i \\neq 0} f_{\\rm tr}(Y_i \\mid \\boldsymbol\\theta)$. The maximum likelihood estimate $(\\hat\\phi, \\hat{\\boldsymbol\\theta})$ maximizing \\eqref{eq:discreteL} can be obtained as follows:\n\\begin{itemize}\n    \\item[(1)] If $m\/n \\leq 1-p_0({\\boldsymbol\\theta_*})$, then $\\hat{\\boldsymbol\\theta} = \\boldsymbol\\theta_*$ and $\\hat\\phi = 1 - m\/n\\cdot (1-p_0({\\boldsymbol\\theta_*}))^{-1}$.\n\\item[(2)] Otherwise, $\\hat{\\boldsymbol\\theta} = {\\rm argmax}_{\\boldsymbol\\theta}  (1-\\psi(\\boldsymbol\\theta))^{n-m} \\psi(\\boldsymbol\\theta)^m  \\prod_{i: Y_i \\neq 0} f_{\\rm tr}(Y_i \\mid \\boldsymbol\\theta)$ and $\\hat\\phi = 1 - \\psi(\\hat{\\boldsymbol\\theta}) \\cdot (1-p_0(\\hat{\\boldsymbol\\theta}))^{-1}$, where $\\psi(\\boldsymbol\\theta) = \\min\\{m\/n, 1-p_0({\\boldsymbol\\theta})\\}$.\n\\end{itemize}\n\\end{theorem}\n\n\\medskip\\noindent\n{\\bf Proof of Theorem~\\ref{thm:zimle}:}\nFirst of all, we denote \n$\\psi_* = {\\rm argmax}_{\\psi} (1-\\psi)^{n-m} \\psi^m$ and $\\boldsymbol\\theta_* = {\\rm argmax}_{\\boldsymbol\\theta} \\prod_{i: Y_i \\neq 0} f_{\\rm tr}(Y_i \\mid \\boldsymbol\\theta)$. It can be verified that $\\psi_* = m\/n$.\n\nOn the other hand, $\\psi = (1-\\phi) [1-p_0({\\boldsymbol\\theta})]$ with $\\phi \\in [0,1]$, which implies $\\psi \\in [0, 1-p_0({\\boldsymbol\\theta})]$. If $m\/n \\leq 1-p_0({\\boldsymbol\\theta_*})$, then $\\hat\\psi = m\/n$, $\\hat{\\boldsymbol\\theta} = \\boldsymbol\\theta_*$ is the mle. In this case, the mle of $\\phi$ is $\\hat\\phi = 1 - \\hat\\psi (1-p_0({\\boldsymbol\\theta_*}))^{-1}$.\n\nOtherwise, we have $m\/n > 1-p_0({\\boldsymbol\\theta_*})$. Then $\\hat\\psi = \\psi(\\boldsymbol\\theta) = \\min\\{m\/n, 1-p_0({\\boldsymbol\\theta})\\}$ is the mle of $\\phi$ given $\\boldsymbol\\theta$. In order to find the mle of $\\phi$ and $\\boldsymbol\\theta$, we first find $\\boldsymbol\\theta^* = {\\rm argmax}_{\\boldsymbol\\theta} L(\\psi(\\boldsymbol\\theta), \\boldsymbol\\theta)$. Then $\\hat{\\boldsymbol\\theta} = {\\boldsymbol\\theta}^*$ and $\\hat\\psi = \\psi(\\boldsymbol\\theta^*)$.\n\\hfill{$\\Box$}\n\nTheorem~\\ref{thm:zimle} actually makes the connection between MLEs for zero-inflated models and zero-altered models when $m\/n \\leq 1 - p_0(\\boldsymbol\\theta_*)$. It can also be used for finding MLEs numerically.\n\n\n\\begin{example} {\\rm\\quad\nLet $\\boldsymbol\\theta = (n, \\alpha, \\beta)$. The pmf of beta binomial distribution is\n\\[\nf_{\\boldsymbol\\theta} (y) = \\dbinom{n}{y} \\frac{{\\rm beta}(y + \\alpha, n - y + \\beta)}{{\\rm beta}(\\alpha,\\beta)}\n\\]\nwith $y=0, 1, \\ldots, n$ and\n\\begin{eqnarray*}\np_0(\\boldsymbol\\theta) &=& \\frac{\\Gamma(n + \\beta) \\Gamma(\\alpha + \\beta)}{\\Gamma(n + \\alpha + \\beta)  \\Gamma(\\beta)}\\\\\n\\frac{p_0(\\boldsymbol\\theta)}{1 - p_0(\\boldsymbol\\theta)} &=& \\frac{\\Gamma(n + \\beta) \\Gamma(\\alpha + \\beta)}{\\Gamma(n + \\alpha + \\beta) \\Gamma(\\beta) - \\Gamma(n + \\beta) \\Gamma(\\alpha + \\beta)}\n\\end{eqnarray*}\nLet $\\Psi(\\cdot) = \\Gamma'(\\cdot)\/\\Gamma(\\cdot)$, known as the {\\it digamma} function. In order to apply Theorem~\\ref{thm:zimle}, we need the following formulas:\n\\begin{eqnarray*}\n\\frac{\\partial\\log f_{\\boldsymbol\\theta}(y)}{\\partial n} &=& \\Psi(n+1) - \\Psi(n-y+1) + \\Psi(n-y+\\beta) - \\Psi(n+\\alpha+\\beta)\\\\\n\\frac{\\partial\\log f_{\\boldsymbol\\theta}(y)}{\\partial \\alpha} &=& \\Psi(y+\\alpha) - \\Psi(n+\\alpha+\\beta) + \\Psi(\\alpha+\\beta) - \\Psi(\\alpha)\\\\\n\\frac{\\partial\\log f_{\\boldsymbol\\theta}(y)}{\\partial \\beta} &=& \\Psi(n-y+\\beta) - \\Psi(n+\\alpha+\\beta) +  \\Psi(\\alpha+\\beta) - \\Psi(\\beta) \\\\\n\\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial n} &=& \\Psi(n+\\beta) - \\Psi(n+\\alpha+\\beta) \\\\\n\\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\alpha} &=& \\Psi(\\alpha+\\beta) - \\Psi(n+\\alpha+\\beta) \\\\\n\\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\beta} &=& \\Psi(n+\\beta) + \\Psi(\\alpha+\\beta) - \\Psi(n+\\alpha+\\beta) - \\Psi(\\beta)\n\\end{eqnarray*}\n}\\hfill{$\\Box$}\n\\end{example}\n\n\\subsection{Asymptotic properties and Fisher information matrix for zero-inflated MLEs}\n\nLet $Y_1, \\ldots, Y_n$ be a random sample from the zero-inflated model~\\eqref{eq:zimodel}. The log-likelihood function of $\\phi$ and $\\boldsymbol\\theta$ is\n\\begin{eqnarray*}\nl(\\phi, \\boldsymbol\\theta) = \\log L(\\phi, \\boldsymbol\\theta) &&= \\log [\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)] \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i=0\\}}\n+ \\log(1-\\phi) \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}} \\\\\n&&+ \\sum_{i=1}^n \\log f_{\\boldsymbol\\theta}(Y_i) {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\n\\end{eqnarray*}\nThen \n\\begin{eqnarray*}\n\\frac{\\partial l}{\\partial\\phi} &=& \\frac{1 - p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i=0\\}} - \\frac{1}{1-\\phi} \\cdot \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\\\\\n\\frac{\\partial l}{\\partial \\boldsymbol\\theta} &=& \\frac{(1-\\phi)p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)}\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} \\sum_{i=1}^n {\\mathbf 1}_{\\{Y_i = 0\\}} + \\sum_{i=1}^n \\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\n\\end{eqnarray*}\n\n\n\\begin{lemma}\\label{lem:zimodelmlefirst}\nSuppose $0\\leq \\phi <1$. Then \n\\[\nE\\left(\\frac{\\partial l}{\\partial \\phi}\\right) = 0 \\>\\>\\mbox{ and }\\>\\>\nE\\left(\\frac{\\partial l}{\\partial \\boldsymbol\\theta}\\right) = n(1-\\phi) E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta} \\right]\n\\]\nwhich is $0$ if and only if $E[\\partial \\log f_{\\boldsymbol\\theta}(Y')\/\\partial \\boldsymbol\\theta]=0$, where $Y'$ follows the baseline distribution $f_{\\boldsymbol\\theta}(y)$.\n\\end{lemma}\n\n\\medskip\\noindent\n{\\bf Proof of Lemma~\\ref{lem:zimodelmlefirst}:}\nSince $P(Y_i=0) = \\phi + (1-\\phi) p_0(\\boldsymbol\\theta)$ and $P(Y_i \\neq 0) = (1-\\phi) [1- p_0(\\boldsymbol\\theta)]$, then $E(\\partial l\/\\partial\\phi) = 0$. Let $Y_1', \\ldots, Y_n'$ be iid $\\sim f_{\\boldsymbol\\theta} (y)$. Then\n\\begin{eqnarray*}\nE\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i\\neq 0\\}}\\right] &&= (1-\\phi) \\cdot E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{Y_i'\\neq 0\\}}\\right] \\\\ \n&&= (1-\\phi) \\left\\{ E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} \\right] - p_0(\\boldsymbol\\theta)\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta}\\right\\}\n\\end{eqnarray*}\nand\n\\[\nE\\left(\\frac{\\partial l}{\\partial \\boldsymbol\\theta}\\right) = (1-\\phi) \\sum_{i=1}^n E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y_i')}{\\partial \\boldsymbol\\theta} \\right] = n(1-\\phi) E\\left[\\frac{\\partial \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta} \\right]\n\\]\n\\hfill{$\\Box$}\n\n\nUnder regularity conditions, $E[\\partial \\log f_{\\boldsymbol\\theta}(Y')\/\\partial \\boldsymbol\\theta]=0$ and thus $E\\left(\\partial l\/\\partial \\boldsymbol\\theta\\right) =0$ according to Lemma~\\ref{lem:zimodelmlefirst}. \n\n\\begin{theorem}\\label{thm:zi_Fisher}\nLet $Y_1, \\ldots, Y_n$ be a random sample from zero-inflated model~\\eqref{eq:zimodel}. Under regularity conditions, the Fisher information matrix of the sample is\n\\[\n{\\mathbf F}_{\\rm ZI} = \nn \\left[\\begin{array}{cc}\n\\frac{1-p_0(\\boldsymbol\\theta)}{[\\phi + (1-\\phi)p_0(\\boldsymbol\\theta)] (1-\\phi)} & \\frac{p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T} \\\\\n\\frac{p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta}  & {\\mathbf F}_{\\rm ZI22}\n\\end{array}\\right]\n\\]\nwhere \n\\[\n{\\mathbf F}_{\\rm ZI22} = -(1-\\phi) \\left(E\\left[\\frac{\\partial^2 \\log f_{\\boldsymbol\\theta}(Y')}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} \\right] + \\frac{\\phi p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta}\\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T}\\right)\n\\]\nand $Y'$ follows the baseline distribution $f_{\\boldsymbol\\theta}(y)$.\n\\end{theorem}\n\n\n\\medskip\\noindent\n{\\bf Proof of Theorem~\\ref{thm:zi_Fisher}:}\nSince \n\\begin{eqnarray*}\n\\frac{\\partial\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi} &=& \\frac{1-p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} {\\mathbf 1}_{\\{y=0\\}} - \\frac{1}{1-\\phi} {\\mathbf 1}_{\\{y\\neq 0\\}}\\\\\n\\frac{\\partial\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} &=& \\frac{(1-\\phi) p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{y = 0\\}} + \\frac{\\partial \\log f_{\\boldsymbol\\theta}(y)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{eqnarray*}\nThen \n\\begin{eqnarray*}\n\\frac{\\partial^2\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi^2} &=& -\\frac{[1-p_0(\\boldsymbol\\theta)]^2}{[\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)]^2} {\\mathbf 1}_{\\{y=0\\}} - \\frac{1}{(1-\\phi)^2} {\\mathbf 1}_{\\{y\\neq 0\\}}\\\\\n\\frac{\\partial^2\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\phi \\partial \\boldsymbol\\theta^T} &=& -\\frac{p_0(\\boldsymbol\\theta)}{[\\phi + (1-\\phi)p_0(\\boldsymbol\\theta)]^2} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T} {\\mathbf 1}_{\\{y=0\\}}\\\\\n\\frac{\\partial^2\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\phi} &=& -\\frac{p_0(\\boldsymbol\\theta)}{[\\phi + (1-\\phi)p_0(\\boldsymbol\\theta)]^2} \\cdot \\frac{\\partial \\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} {\\mathbf 1}_{\\{y=0\\}}\\\\\n\\frac{\\partial^2\\log f_{\\rm ZI} (y \\mid \\phi, \\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} &=& \n\\frac{\\phi (1-\\phi) p_0(\\boldsymbol\\theta)}{[\\phi + (1-\\phi)p_0(\\boldsymbol\\theta)]^2} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta} \\cdot \\frac{\\partial\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta^T}{\\mathbf 1}_{\\{y = 0\\}}\\\\\n&&+\\frac{(1-\\phi) p_0(\\boldsymbol\\theta)}{\\phi + (1-\\phi) p_0(\\boldsymbol\\theta)} \\cdot \\frac{\\partial^2\\log p_0(\\boldsymbol\\theta)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} {\\mathbf 1}_{\\{y = 0\\}} +\\frac{\\partial^2 \\log f_{\\boldsymbol\\theta}(y)}{\\partial \\boldsymbol\\theta \\partial \\boldsymbol\\theta^T} {\\mathbf 1}_{\\{y\\neq 0\\}}\n\\end{eqnarray*}\nThe rest of the conclusions can be obtained via $l(\\phi, \\boldsymbol\\theta) = \\sum_{i=1}^n \\log f_{\\rm ZI} (Y_i \\mid \\phi, \\boldsymbol\\theta)$, $P(Y_i = 0) = \\phi + (1-\\phi)p_0(\\boldsymbol\\theta)$ and $P(Y_i \\neq 0) = (1-\\phi) [1-p_0(\\boldsymbol\\theta)]$.\n\\hfill{$\\Box$}\n\nAs direct conclusions of Theorem~18 in \\cite{ferguson1996}, under regularity conditions, we have (i) \n\\[\n\\sqrt{n}(\\hat\\phi - \\phi_0)  \\stackrel{{\\cal L}}{\\longrightarrow} N\\left(0, \\frac{[\\phi + (1-\\phi)p_0(\\boldsymbol\\theta)] (1-\\phi)}{1-p_0(\\boldsymbol\\theta)}\\right)\n\\] \nand (ii) $\\sqrt{n}(\\hat{\\boldsymbol\\theta} - \\boldsymbol\\theta_0)  \\stackrel{{\\cal L}}{\\rightarrow} N\\left(0, {\\mathbf F}_{\\rm ZI22}^{-1} \\right)$. However, $\\hat\\phi$ and $\\hat{\\boldsymbol\\theta}$ are usually not asymptotically independent unless $p_0(\\boldsymbol\\theta) = 0$ or does not depend on $\\theta$. These results can be used for building up confidence intervals of $\\phi$ and $\\boldsymbol\\theta$ as well.\n\n\n\\begin{example}\\label{ex:zipFisher} {\\rm\nFor zero-inflated Poisson (ZIP) model, the pmf of the baseline distribution is $f_\\lambda(y)= e^{-\\lambda}\\lambda^{y}\/y!$ with $y=0, 1, \\ldots$, where $p_0(\\lambda)= e^{-\\lambda}$. Then $\\partial \\log f_\\lambda(y)\/\\partial \\lambda = y\/\\lambda -1$.\n\nAccording to Theorem~\\ref{thm:zi_Fisher}, the Fisher information matrix of the Poisson sample is\n\\[\n{\\mathbf F}_{\\rm ZIP} = \nn \\left[\\begin{array}{cc}\n\\frac{1-e^{-\\lambda}}{[\\phi + (1-\\phi)e^{-\\lambda}] (1-\\phi)} & -\\frac{e^{-\\lambda}}{\\phi + (1-\\phi) e^{-\\lambda}} \\\\\n-\\frac{e^{-\\lambda}}{\\phi + (1-\\phi) e^{-\\lambda}}  &  (1-\\phi) \\left(\\frac{1}{\\lambda}  - \\frac{\\phi e^{-\\lambda}}{\\phi + (1-\\phi) e^{-\\lambda}} \\right)\n\\end{array}\\right]\n\\]\nNote that $\\frac{1}{\\lambda}  - \\frac{\\phi e^{-\\lambda}}{\\phi + (1-\\phi) e^{-\\lambda}} > 0$ as long as $\\lambda > 0$ and $0\\leq \\phi < 1$.\n\\hfill{$\\Box$}\n}\\end{example}\n\n\n\n\n\n\n\n\n\\section{Microbiome Data Application}\nAs an application, the bootstrap KS test with unknown parameters (Algorithm 1) has been used to a list of 229 bacterial and fungal OTUs~\\citep{aldirawi2019identifying,tipton2018fungi}. We are interested in knowing how many of the 229 OTU follows each of the following distributions given that the distribution parameters are unknown: Poisson, negative binomial, beta binomial, beta negative binomials, and the corresponding zero-inflated and hurdle models. Table~\\ref{ks}, which was regenerated  from \\cite{aldirawi2019identifying}, summarizes the number of features that do\nnot show significant divergence (p-value $>$ 0.05). \n\n\n\n\nPoisson, negative binomial, ZIP, ZINB have been used commonly for modeling microbiome data. However, as shown in the above table, Poisson, zero-inflated Poisson, and Poisson hurdle are not appropriate distributions to model sparse microbial features as only 0.4\\%, 2\\%, 1\\% out of 229 the features were able to be appropriately fitted using these distributions, respectively. On the other hand, binomial and negative binomial families can be used to approximate sparse microbial data, with BNBH as the best distribution to model such dataset (being able to appropriately fit 53\\% of the 229 features) using the proposed conservative method. In addition, ZIBNB fits about half of the features followed by BBH and ZIBB which they fit about 40\\% of the features. \n\n\nBased on the above table, we conclude that zero-inflated and hurdle beta binomial or beta negative binomial are more appropriate than commonly used models.\n\n\n\n\n\\begin{table}[]\n\\caption{Number and percentage of species out of 229 species that don't show significant difference (KS p-value $>0.05$)}\n\\label{ks}\n\\begin{center}\n\\resizebox{0.85\\textwidth}{!}{\n\\begin{tabular}{|| c| c |c||}\n\\hline\n     \\textbf{Distribution} & \\textbf{Number} & \\textbf{Percentage}    \\\\\n     \\cline{1-3}\n     Poisson & $1 $ & 0.4\\%  \\\\\n     \\hline\n     negative binomial (NB) & 23 & 10\\%    \\\\\n     \\hline\n     beta binomial (BB) & 76 & 33\\%   \\\\\n     \\hline\n     beta negative binomial (BNB) &  60 & 26\\%  \\\\\n     \\hline\n     zero-inflated Poisson (ZIP)&  3 & 2\\%         \\\\\n     \\hline\n      zero-inflated negative binomial (ZINB)&  25 &  11\\%         \\\\\n     \\hline\n      zero-inflated beta binomial (ZIBB)&   89 & 39\\%        \\\\\n     \\hline\n      zero-inflated beta negative binomial (ZIBNB)&  110& 48\\%         \\\\\n     \\hline\n     Poisson hurdle (PH)&  2 & 1\\%       \\\\\n     \\hline\n     negative binomial hurdle (NBH)& 56 & 24\\%           \\\\\n     \\hline\n     beta binomial hurdle (BBH)&  92 & 40\\%        \\\\\n     \\hline\n     beta negative binomial hurdle (BNBH)& 121 & 53\\%          \\\\\n     \\hline\n \\end{tabular}\n\\label{tab1}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{conclusion}\n\nUnderstanding the role of the microbiome in human health and how it can be modeled is becoming increasingly relevant for preventive medicine and for the medical management of chronic diseases \\citep{calle2019statistical}. However, the microbiome data is highly sparse and skewed. It is very challenging to select an appropriate probabilistic model. \n\nIn this paper, we use the MLE approach to estimate the parameters of general zero-inflated and hurdle models. We also derive the corresponding Fisher information matrices for exploring the estimator's asymptotic properties to build up the confidence intervals of the parameters.\n\nIn the literature, Poisson and negative binomial models have been commonly used for modeling microbiome data. Based on a real dataset analysis, we show that zero-inflated (or hurdle) beta binomial or beta negative binomial models are more appropriate.\n\n\n\\section*{Acknowledgments}\n\nThis work was partly supported by NSF grant DMS-1924859.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKaluza-Klein (KK) compactification \\cite{KAL}\\footnote{For a recent\nreview see Ref. \\cite{D}.} in its original sense is\nthe procedure by which one unifies the pure 4-dimensional (4-d) gravity\nwith gauge theories through the dimensional reduction of\nthe higher dimensional pure gravity theories.  The basic idea is that\nthe isometry symmetry of the internal space manifests itself as the\ngauge symmetry in 4-d after the compactification of the extra space.\nIn addition to gauge fields, associated with the isometry group, the\nscalar fields, corresponding to the degrees of freedom of the\ninternal metric, arise upon compactification.\n\nA class of interesting solutions for such an effective 4-d KK theory\nconstitutes configurations with a non-trivial 4-d space-time\ndependence for the additional scalar fields, gauge fields,\nas well as for the 4-d space-time metric\n\\footnote{One can view \\cite{CY} this class of configurations as a subset of\nblack holes in the heterotic string theory \\cite{Sen} compactified on a six\ntorus. Such configurations, therefore, constitute a\nclass of non-trivial solutions in low-energy 4-d string theory.}.\nIn particular, spherically symmetric charged configurations correspond to\ncharged black hole (BH) solutions with additional scalar fields\nvarying with the spatial radial coordinate.  We shall refer to such\nconfigurations as KK BH's.\n\nExamples of KK BH's in 5-d \\cite{FIV,DM,GW} KK theories\nwith Abelian isometry have been studied in the\npast.\\footnote{See also Ref. \\cite{GM}.}  Dobiasch and Maison \\cite{DM}\ndeveloped a formalism by which one can generate the most general\nstationary, spherically symmetric solutions in Abelian KK theories.\nTheir formalism was applied primarily to obtain the explicit\nsolutions for the BH's in 5-d KK theory, only.\n\nAmong a class of non-trivial configurations, those which saturate\nthe Bogomol'nyi bound on their energy (ADM mass \\cite{ADM})\ncan be viewed as non-trivial vacuum configurations -- solitons --\nand are thus of special interest.\nWhen embedded in supersymmetric theories such configurations\ncorrespond to bosonic configurations which are invariant under\n(constrained) supersymmetry transformations, {\\it i.e.}, they\nsatisfy the corresponding Killing spinor equations.  One refers\nto such configurations as supersymmetric ones.\n\nSupersymmetric KK BH's in 5-d KK theory were first discussed by Gibbons\nand Perry \\cite{GIBB}.  Their result was recently generalized \\cite{EX}\nto supersymmetric KK BH's in $(4+n)$-d Abelian KK theory, {\\it i.e.},\nthose with the $U(1)^n$ internal isometry group.  With a diagonal internal\nmetric, such solutions exist if and only if the isometry group of\nthe internal space is broken down to the $U(1)_M \\times U(1)_E$ group.\n\nThe aim of this paper is to address static 4-d charged BH solutions,\ncompatible with the corresponding Bogomol'nyi bound,\nin $(4+n)$-d KK theory with Abelian isometry group $U(1)^n$.\nExplicit solutions with diagonal internal metric\nturn out to correspond to configurations with at most one electric\nand one magnetic charges, which can either come from the same\n$U(1)$-gauge field (corresponding to dyonic BH's in effective\n5-d KK theory) or from different ones (corresponding to BH's with\nthe $U(1)_M \\times U(1)_E$ isometry in an effective 6-d KK theory).\nTheir global space-time and thermal properties are explored.\nA class of Abelian KK BH solutions with constrained charge configurations\nis further obtained by performing global $SO(n)$ transformations on\nthe above solutions.\n\nThe paper is organized in the following way. In Section II we discuss\ndimensional reduction of the $(4+n)$-d gravity. In Section III we\nderive the constraints on charges for charged KK BH's with the diagonal\ninternal metric Ansatz.  In Section IV we discuss the explicit form of\nthe non-extreme $U(1)_M \\times U(1)_E$ solutions, their global\nspace-time structure and thermal properties.  In Section V we discuss\nthe electric-magnetic duality of a more general class of the solutions\ngenerated by global $SO(n)$ rotations on the internal metric.\nConclusions are given in Section VI.\n\\section{Dimensional Reduction of $(4+n)$-Dimensional Gravity}\nThe starting point is the pure Einstein-Poincar\\' e gravity in\n$(4+n)$-d:\n\\begin{equation}\n{\\cal L} = -{1 \\over {2\\kappa^2}} \\sqrt{-g^{(4+n)}}{\\cal R}^{(4+n)}\\ \\ ,\n\\label{highlag}\n\\end{equation}\nwhere the Ricci scalar ${\\cal R}^{(4+n)}$ and the determinant\n$g^{(4+n)}$ are defined in terms of a $(4+n)$-d metric\n$g^{(4+n)}_{\\Lambda\\Pi}$, and $\\kappa$ is the $(4+n)$-d\ngravitational constant.  For the notation of space-time indices, we\nshall follow the convention of Ref. \\cite{EX}, {\\it i.e.}, the upper-\n(or lower-) case letters are for those running over $(4+n)$-d (or 4-d)\nspace-time.  Those with tilde are reserved for the $n$ extra spatial\ndimensions.  Latin (or greek) letters denote flat (or curved) indices.\nWe shall use the mostly positive signature convention\n$(-++\\cdot\\cdot\\cdot +)$ for the $(4+n)$-d metric.\n\nThe dimensional reduction of the higher dimensional gravity is\nachieved by splitting the $(4+n)$-d metric $g^{(4+n)}_{\\Lambda\\Pi}$\ninto the following form:\n\\begin{equation}\ng^{(4+n)}_{\\Lambda \\Pi} =\n\\left [ \\matrix{{\\rm e}^{-{1 \\over \\alpha}\\varphi}g_{\\lambda \\pi} +\n{\\rm e}^{{2\\varphi} \\over {n\\alpha}}\\rho_{\\tilde{\\lambda}\n\\tilde{\\pi}}A^{\\tilde{\\lambda}}_{\\lambda} A^{\\tilde{\\pi}}_{\\pi} &\n{\\rm e}^{{2\\varphi} \\over {n\\alpha}}\\rho_{\\tilde{\\lambda}\n\\tilde{\\pi}}A^{\\tilde{\\lambda}}_{\\lambda}\n\\cr {\\rm e}^{{2 \\varphi} \\over {n\\alpha}}\\rho_{\\tilde{\\lambda}\n\\tilde{\\pi}}A^{\\tilde{\\pi}}_{\\pi} & {\\rm e}^{{2\\varphi} \\over\n{n\\alpha}}\\rho_{\\tilde{\\lambda} \\tilde{\\pi}}} \\right ]\\ \\ ,\n\\label{ansatz}\n\\end{equation}\nwhere $\\rho_{\\tilde{\\lambda} \\tilde{\\pi}}$ is the unimodular part,\n{\\it i.e.}, ${\\hbox{det}}\\rho_{\\tilde{\\lambda} \\tilde{\\pi}}=1$,\nof the internal metric, $\\varphi$ is the determinant of the\ninternal metric, which we will refer to as dilaton, and\n$\\alpha = \\sqrt{{n+2}\\over n}$.\n\nThe effective theory in 4-d is then obtained by imposing ``the right\ninvariance'' \\cite{CHB} of the $(4+n)$-d metric $g_{\\Lambda\\Pi}$\nunder the action of an isometry of the internal space.  This requirement\nfixes the dependence of the metric components on the internal coordinates.\nIt turns out that the internal coordinate dependence of the transformation\nlaws (under the general coordinate transformations) of the fields in\n(\\ref{ansatz}) factors out, and $(4+n)$-d Einstein Lagrangian density\n(\\ref{highlag}) becomes independent of the internal coordinates.  Then,\n4-d effective action after a trivial integration over the internal space\nassumes the form (see for example Eq.(8) of Ref. \\cite{CHB}):\n\\begin{eqnarray}\n{\\cal L}= -{1 \\over 2}\\sqrt{-g}[{\\cal R} +\n{\\rm e}^{-\\alpha\\varphi}{\\cal R}_K + {1 \\over 4}{\\rm e}^{\\alpha\\varphi}\n\\rho_{ij}F^i_{\\mu \\nu}F^{j \\mu \\nu}\n+ {1 \\over 2}\\partial_{\\mu}\\varphi \\partial^{\\mu}\\varphi \\nonumber \\\\\n+ {1 \\over 4}\\rho^{ij}\\rho^{k\\ell}\n(D_{\\mu}\\rho_{ik})(D^{\\mu}\\rho_{j\\ell})\n+ \\chi (\\det\\rho_{ij} - 1)]\\ \\ ,\n\\label{efflag}\n\\end{eqnarray}\nwhere the Ricci scalar ${\\cal R}_K$ is defined in terms of the unimodular\npart $\\rho_{ij}$ $(i,j=1,2,...,n)$\n\\footnote{From now on, for the simplicity of\nnotation, we shall denote the internal space\nindices as $i\\equiv \\tilde{\\alpha} - 3, etc.$ }\nof the internal metric, $F^i_{\\mu \\nu} \\equiv \\partial_{\\mu}\nA^i_{\\nu} - \\partial_{\\nu}A^i_{\\mu} - gf^i_{jk} A^j_{\\mu} A^k_{\\nu}$,\nwhere $f^i_{jk}$ is the structure constant for the internal isometry\ngroup and $g$ is the gauge coupling constant of the isometry group,\nis the field strength of the gauge field $A^i_{\\mu}$,\n$D_{\\mu} \\rho_{ij} = \\partial_{\\mu}\\rho_{ij} - f^\\ell_{kj}A^k_{\\mu}\n\\rho_{i \\ell}$ is the corresponding gauge covariant derivative\nand $\\chi$ is the Lagrangian multiplier.\nThe 4-d gravitational constant $\\kappa_4$ has been set equal to 1.\nWhen the isometry group is Abelian, {\\it i.e.}, one compactifies on\n$n$-torus $T^n$, the structure constant $f^i_{jk}$ vanishes.  In this\ncase, the gauge covariant derivatives in (\\ref{efflag}) become ordinary\nones and the Ricci scalar ${\\cal R}_K$, which in general describes\nthe self-interactions among scalar fields, vanishes.\n\\section{Diagonal Internal Metric Ansatz and Constraints on Solutions}\nWe shall first address the spherically symmetric configurations\nwith the diagonal internal metric Ansatz.  Then, the unimodular\npart of the internal metric is of the form:\n\\begin{equation}\n\\rho_{ij} = {\\rm diag}(\\rho_1 ,..., \\rho_{n-1},\n\\prod^{n-1}_{k=1} \\rho^{-1}_k )\\ \\ ,\n\\label{diag}\n\\end{equation}\nthe spherically symmetric {\\it Ansatz} for the 4-d space-time metric\nis of the form:\n\\begin{equation}\n{\\rm d} s^2 = g_{\\mu\\nu} {\\rm d}x^{\\mu} {\\rm d} x^{\\nu} =\n-\\lambda (r) {\\rm d} t^2 + \\lambda^{-1} (r) {\\rm d} r^2\n+ R(r) ( {\\rm d} \\theta^2 + \\sin^2 \\theta {\\rm d} \\phi^2 )\n\\label{sph}\n\\end{equation}\nand the scalar fields $\\varphi$ and $\\rho_i$ depend on the radial\ncoordinate $r$, only.  The electromagnetic vector potentials take the form:\n\\begin{equation}\nA^i_{\\phi} = P_i \\cos \\theta \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\nA^i_t = \\psi_i (r) \\ \\ ,\n\\label{vector}\n\\end{equation}\nwhere $E_i (r) = -\\partial_r \\psi_i (r) = {{\\tilde{Q}_i} \\over\n{R {\\rm e}^{\\alpha \\varphi} \\rho_i}}$\n\\footnote{The physical Electric charge $Q_i$ defined in terms of the\nasymptotic behavior $E_i \\sim {Q_i \\over r^2}$ ($r \\rightarrow \\infty$)\nof the electric field is related to $\\tilde{Q}_i$ through\n$\\tilde{Q}_i = {\\rm e}^{\\alpha\\varphi_{\\infty}}\\rho_{i \\infty} Q_i$. Here,\n$\\varphi_\\infty$ and $\\rho_\\infty$ are the constant values of the\ncorresponding scalar fields at $r\\rightarrow \\infty$.}.\n\nThe {\\it Ansatz} with all the off-diagonal components of the internal\nmetric turned off has to be consistent with the equations of motion.\nIn fact, such a consistency restricts the allowed charge\nconfigurations.  Namely, the Euler-Lagrange equations for the components\n$\\rho_{ij}$ of the unimodular part of the metric is given by:\n\\begin{equation}\n{1 \\over 2}{\\rm e}^{\\alpha\\varphi(r)}[R(r)E_i(r) E_j(r) -\nR^{-1}(r)P_i P_j ] + \\chi R(r)\\rho^{ij}(r) =\n{1 \\over 2}{{\\rm d} \\over {{\\rm d}r}}\n[\\lambda(r)R(r){{{\\rm d}\\rho^{ij}(r)}\\over {{\\rm d}r}}]\n\\ \\ ,\n\\label{rhoij}\n\\end{equation}\nwhich implies that for the diagonal metric {\\it Ansatz} (\\ref{diag}) the\nfollowing constraints have to be satisfied:\n\\begin{equation}\nQ_iQ_j-e^{2\\alpha\\varphi}\\rho_i\\rho_j\\, P_iP_j=0\\ \\ \\\ni\\neq j\\ \\ .\n\\label{ij}\n\\end{equation}\nEqs. (\\ref{ij}) can be satisfied if and only if\n\\footnote{A more general constraint $Q_iQ_j-c_ic_jP_iP_j=0$ with\n$c_i\\equiv\\rho_ie^{\\alpha\\varphi}$ being constant would imply the equation\nof motion for $\\rho_ie^{\\alpha\\varphi}$ with $Q_i=P_i=0$.  Thus, the\nconstraint  $Q_iQ_j-c_ic_jP_iP_j=0$ in turn reduces to the subset of\nconstraints (\\ref{chcon}).}\n\\begin{equation}\nQ_i Q_j = 0\\ \\ \\ \\ {\\rm and}\\ \\ \\ \\ P_i P_j = 0 ,\\ \\ \\ \\ i \\neq j \\ \\ .\n\\label{chcon}\n\\end{equation}\nConstraints (\\ref{chcon}) imply that the same type of charge,\n{\\it i.e.}, either electric or magnetic one, can appear in at most\none gauge field.  Consequently, the internal isometry group $U(1)^n$\nis broken down to at most $U(1)\\times U(1)$, with one electric and\none magnetic charges, only.\n\nWhen, say, the first ($n-2$) gauge fields are turned off the first\n($n-2$) components of the diagonal internal metric become constant\n(${\\rm e}^{{2\\varphi}\\over {n\\alpha}}\\rho_i = const.$\n($i=1,...n-2$)).  Thus, the KK BH's are those of effective 6-d KK:\n\\begin{eqnarray}\n{\\tilde {\\cal L}} = -{1\\over 2}\\sqrt{-g}\n[{\\cal R} + {1\\over 2} \\partial_{\\mu}\\Phi \\partial^{\\mu}\\Phi +\n{1\\over 2} \\partial_{\\mu}\\chi_{n-1}\\partial^{\\mu}\\chi_{n-1} +\n{1\\over 2} \\partial_{\\mu}\\chi_n \\partial^{\\mu}\\chi_n  \\nonumber \\\\\n+ {1\\over 4}{\\rm e}^{\\sqrt{2}(\\Phi + \\chi_{n-1})}F^{n-1}_{\\mu\\nu}\nF^{n-1 \\ \\mu\\nu} + {1\\over 4}{\\rm e}^{\\sqrt{2}(\\Phi +\\chi_n)}\nF^n_{\\mu\\nu} F^{n \\ \\mu\\nu}]\\ \\ ,\n\\label{effec}\n\\end{eqnarray}\nwhere $\\Phi \\equiv {\\sqrt{2} \\over \\alpha}\\varphi$ and\n$\\chi_i \\equiv {1 \\over \\sqrt{2}}[\\ln\\rho_i + {{2-n}\\over {n\\alpha}}\n\\varphi ]$ $(i = n-1, n)$ with the constraint $\\chi_{n-1} + \\chi_n =\nconst$.\n\nIn general, one can, therefore, have the following qualitatively\ndifferent classes of configurations:\n\\begin{itemize}\n\\item  $Q_{n-1}=P_{n-1}=Q_n=P_n=0$, which corresponds to the ordinary 4-d\nSchwarzschild BH's.\n\\item  Say, $Q_n = P_n =0$, which corresponds to KK BH's in effective\n5-d KK theory.\n\\item Say, $Q_{n-1}=P_n=0$, which corresponds to a class of\nKK BH's in effective 6-d KK theory, where electric and magnetic\ncharges arise from different $U(1)$ groups.\n\\end{itemize}\nSchwarzschild BH's are well understood, and BH's in 5-d KK theory have\nbeen extensively studied in Refs. \\cite{FIV,DM,GW}.\nWe, therefore, concentrate on the study of the last class of solutions,\nwhich correspond to the non-extremal generalization of $U(1)_M \\times\nU(1)_E$ supersymmetric solution, studied in Ref. \\cite{EX}.\n\\section{Non-extreme $U(1)_M \\times U(1)_E$ Solutions}\nIf only a pair of gauge fields are non-zero, the Lagrangian density\n(\\ref{efflag}) becomes that of effective 6-d KK theory \\cite{EX}.\nWithout loss of generality, one assumes that the first\n[second] gauge field to be magnetic [electric].  The Einstein's and\nEuler-Lagrange equations can be cast in following form:\n\\begin{equation}\n(\\lambda R)^{\\prime \\prime} = 2\n\\label{beqn}\n\\end{equation}\n\\begin{equation}\n2(\\lambda^{\\prime}R)^{\\prime} = R^{-1}{\\rm e}^{-\\sqrt{2}\\Phi}\n\\rho \\tilde{Q}^2 +R^{-1}{\\rm e}^{\\sqrt{2}\\Phi}\\rho P^2\n\\label{aeqn}\n\\end{equation}\n\\begin{equation}\n{1 \\over \\sqrt{2}}\n(\\lambda R \\Phi^{\\prime})^{\\prime} + (\\lambda R \\rho^{-1}\n\\rho^{\\prime})^{\\prime}=R^{-1}{\\rm e}^{\\sqrt{2}\\Phi}\\rho P^2\n\\label{ceqn}\n\\end{equation}\n\\begin{equation}\n-{1 \\over \\sqrt{2}}(\\lambda R \\Phi^{\\prime})^{\\prime} +\n(\\lambda R \\rho^{-1} \\rho^{\\prime})^{\\prime}=R^{-1}\n{\\rm e}^{-\\sqrt{2} \\Phi}\\rho \\tilde{Q}^2 \\ \\ ,\n\\label{deqn}\n\\end{equation}\nwhere $\\rho \\equiv{\\rm e}^{\\sqrt{2} \\chi_{n-1}}$ and the prime denotes\ndifferentiation with respect to $r$. Recall, ${\\tilde Q} =\n{\\rm e}^{\\sqrt{2}\\Phi_{\\infty}} \\rho_{\\infty}Q$.\nEqs.(\\ref{aeqn})-(\\ref{deqn}) exhibit manifest electric-magnetic\nduality symmetry:\n$P \\leftrightarrow \\tilde{Q}$ and $\\Phi \\rightarrow -\\Phi$.\n\nEquation (\\ref{beqn}) implies\n$\\lambda R = (r - r_+ )(r - r_+ + 2\\beta )$, while\nthe resultant equation obtained by subtracting Eqs. (\\ref{ceqn}) and\n(\\ref{deqn}) from Eq. (\\ref{aeqn}) yields\n$\\lambda = {\\rho \\over \\rho_{\\infty}}\\left({{r-r_+ }\n\\over {r-r_+ +2\\beta}} \\right )$.\nHere $r_+$ is defined to be the outermost horizon and $\\beta > 0$\nis the non-extremality parameter\n\\footnote{Note, the role of the non-extremality parameter $\\beta$\nis very similar to the one used in describing non-extreme\nsupergravity walls \\cite{CS,NON}.}.\nSubstitution of these relations into Eqs. (\\ref{ceqn}) and (\\ref{deqn}),\nyields the following ordinary differential equations\n\\footnote{These equations are reminiscent of equations for Toda molecule.}:\n\\begin{eqnarray}\n2{\\bf P}^2{{{\\rm e}^X} \\over {(r-r_{+})^2}} =\n[(r-r_{+})(r-r_{+}+2\\beta)X^{\\prime}]^{\\prime}\\ \\ \\  \\nonumber \\\\\n2{\\bf Q}^2{{{\\rm e}^Y} \\over {(r-r_{+})^2}} =\n[(r-r_{+})(r-r_{+}+2\\beta)Y^{\\prime}]^{\\prime} \\ \\ ,\n\\label{toda}\n\\end{eqnarray}\nwhere $X \\equiv \\sqrt{2}\\Phi + 2\\ln \\lambda - \\sqrt{2}\\Phi_{\\infty}$\nand $Y \\equiv -\\sqrt{2}\\Phi + 2\\ln \\lambda + \\sqrt{2} \\Phi_{\\infty}$.\nHere, ${\\bf P}$ and ${\\bf Q}$ are the ``screened'' \\cite{EX} magnetic\nand electric monopole charges, {\\it i.e.}, ${\\bf P} \\equiv\n{\\rm e}^{{1 \\over \\sqrt{2}}\\Phi_{\\infty}} \\rho^{1\\over 2}_{\\infty}P$ and\n${\\bf Q} \\equiv {\\rm e}^{{1 \\over \\sqrt{2}}\\Phi_{\\infty}}\n\\rho^{-{1\\over 2}}_{\\infty}Q$.  The two equations in (\\ref{toda})\ncan be solved explicity, yielding the following explicit form of\nthe solutions with regular horizons:\n\\begin{equation}\n\\lambda ={{r-r_+} \\over {(r-r_+ +\\hat{P})^{1\/2}(r-r_+ + \\hat{Q})^{1\/2}}}\n\\label{asol}\n\\end{equation}\n\\begin{equation}\nR = r^2 (1-{{r_+ - 2\\beta}\\over r})(1-{{r_+ - \\hat{P}}\\over r})^{1 \\over 2}\n(1-{{r_+ - \\hat{Q}} \\over r})^{1 \\over 2}\n\\label{bsol}\n\\end{equation}\n\\begin{equation}\n{\\rm e}^{\\sqrt{2}(\\Phi - \\Phi_{\\infty})} =\n{{r - r_+ + \\hat{Q}} \\over {r - r_+ + \\hat{P}}}\n\\label{csol}\n\\end{equation}\n\\begin{equation}\n\\rho = \\rho_\\infty {{r -r_+ +2\\beta} \\over {(r - r_+ + \\hat{P})^{1\/2}\n(r - r_+ + \\hat{Q})^{1\/2}}}  \\ \\ ,\n\\label{dsol}\n\\end{equation}\nwhere $r_+ = \\beta + {{|{\\bf P}| \\sqrt{{\\bf P}^2 + \\beta^2} -\n|{\\bf Q}| \\sqrt{{\\bf Q}^2 + \\beta^2}} \\over {|{\\bf P}| - |{\\bf Q}}|}$,\n$\\hat{P} = \\beta + \\sqrt{{\\bf P}^2 + \\beta^2}$ and $\\hat{Q} = \\beta +\n\\sqrt{{\\bf Q}^2 + \\beta^2}$, while the ADM mass of the configurations\nis of the form:\n\\begin{equation}\nM = 2\\beta + \\sqrt{{\\bf P}^2 + \\beta^2} + \\sqrt{{\\bf Q}^2 + \\beta^2}\n\\ \\ .\\label{mass}\n\\end{equation}\nIn the limit $\\beta \\rightarrow 0$, the above expressions\nreduce to those with $r_+=r_H=|{\\bf P}|+|{\\bf Q}|$, $\\hat{P}=|{\\bf P}|$,\n$\\hat{Q}=|{\\bf Q}|$ and the ADM mass saturates the Bogomol'nyi bound\n$M_{ext}=|{\\bf P}|+|{\\bf Q}|$. These are supersymmetric\nsolutions of Ref. \\cite{EX}.\n\nNow, we shall discuss the global space-time structure and\nthermal properties of the solution.  We first describe the\nsingularity structure of 4-d space-time defined by the metric\ncoefficients (\\ref{asol}) and (\\ref{bsol}).  For the non-extreme\nsolutions, {\\it i.e.}, $\\beta > 0$, there is a space-like singularity\nat $r = r_+ - 2\\beta$ which is hidden behind a horizon at\n$r =r_+ $.  The global space-time of the non-extreme solutions\nis that of the Schwarzschild BH (see Fig. 1).\nIn the supersymmetric (extreme) limit ($\\beta\\rightarrow 0$)\nthe singularity becomes null, {\\it i.e.}, it coincides with the horizon\n(see Fig. 2a).  In the extreme limit with either $Q$ or $P$ zero,\n{\\it i.e.}, supersymmetric 5-d KK BH \\cite{GIBB},\nthe singularity becomes naked (see Fig. 2b).\n\\begin{figure}[p]\n\\vfill\n\\iffiginc\n\\hfill\\psfig{figure=warsaw1.eps}\\hfill\n\\fi\n\\caption{The Penrose diagram (in the ($r,t$) plane) for non-extreme\n$U(1)_M \\times U(1)_E$ 4-d Kaluza-Klein black holes.\nThe space-like singularity (jagged line) at $r=r_+ - 2\\beta$\n($\\beta$ is the non-extremality parameter) is hidden behind the\nhorizon (dashed line) at $r=r_+$.}\n\\label{Fig. 1}\n\\end{figure}\n\\begin{figure}\n\\vfill\n\\iffiginc\n\\psfig{figure=warsaw2.eps}\\hfill\n\\fi\n\\caption{The Penrose diagram (in the ($r,t$) plane) for the\nsupersymmetric (extreme) $U(1)_M \\times U(1)_E$ 4-d Kaluza-Klein black\nholes, {\\it i.e.}, those with {\\it both} $Q$ and $P$ charges\nnon-zero, is given in fig. 2a.  The Penrose diagram for the\nsupersymmetric $U(1)_E$ (or $U(1)_M$) 4-d Kaluza-Klein black\nholes,{\\it i.e.}, those with $P$ or $Q$ charge nonzero,\nis given in fig. 2b.  Note a null singularity (jagged line) in the\nformer case, and a naked singularity in the latter one.}\n\\label{Figure 2}\n\\end{figure}\n\nThe Hawking temperature\\ cite{HAW}, which can be calculated by\nidentifying the inverse of the imaginary time period \\cite{GH} of a\nfunctional path integral, turns out to be\n$T_H = {{|\\lambda^{\\prime}(r_+ )|} \\over {4\\pi}}$.\nWith the explicit solution (\\ref{asol}) for $\\lambda$ one obtains:\n\\begin{equation}\nT_H ={1 \\over {4\\pi [\\beta + ({\\bf Q}^2 + \\beta^2 )^{1 \\over 2}]^{1\\over 2}\n[\\beta +({\\bf P}^2 +\\beta^2 )^{1\\over 2}]^{1\\over 2}}} \\ \\ .\n\\label{temp}\n\\end{equation}\nAs $\\beta$ decreases $T_H$ increases and reaches the upper bound\n$T_{H\\, ext}=1\/{4\\pi|{\\bf P Q}|}$ in the extreme limit\n$\\beta = 0$. In the extreme limit, when either\n${\\bf Q}$ or ${\\bf P}$ becomes zero $T_H$ is infinite.\n\nThe entropy $S$ of the system, determined as $S = {1 \\over 4}\\times$\n(the surface area of the event horizon) \\cite{BEK}, is of the\nfollowing form:\n\\begin{equation}\nS = 2\\pi \\beta [\\beta +({\\bf Q}^2 +\\beta^2)^{1\\over 2} ]^{1\\over 2}\n[\\beta +({\\bf P}^2 +\\beta^2 )^{1\\over 2}]^{1\\over 2}\\ \\ .\n\\label{ent}\n\\end{equation}\nwhere we have used the explicit solution (\\ref{bsol}) for $R(r)$.\nIn the extreme limit the entropy becomes zero.\n\\section{Electric-Magnetic Duality Transformations}\nIn addition to the 4-d general coordinate invariance and gauge symmetry,\nwhich arise from the $(4+n)$-d general coordinate invariance of the\n$(4+n)$-d Einstein action (\\ref{highlag}), the Lagrangian density\n(\\ref{efflag}) has a global $SO(n)$ invariance:\n\\begin{equation}\n\\rho_{ij} \\rightarrow U_{ik} \\rho_{k\\ell} (U^{T})_{\\ell j} \\ \\ \\ \\ \\ \\ \\\nA^i_{\\mu} \\rightarrow U_{ij} A^j_{\\mu}\\ \\ , \\label{son}\n\\end{equation}\nwhere $U$ is an $SO(n)$ rotation matrix.\nOne convenient parameterization of $U$ is in terms of\nsuccessive rotations in all the possible planes in $\\Re^n$:\n\\begin{equation}\nU = \\hat{U}_{12} \\cdot \\cdot \\cdot \\hat{U}_{1n}\\hat{U}_{23}\n\\cdot \\cdot \\cdot \\hat{U}_{2n} \\cdot \\cdot \\cdot \\hat{U}_{i, i+1}\n\\cdot \\cdot \\cdot \\hat{U}_{in} \\cdot \\cdot \\cdot \\hat{U}_{n-1, n} \\ \\ ,\n\\label{repre}\n\\end{equation}\nwhere $\\hat{U}_{k\\ell}$ is the rotation matrix in the $(k,\\ell)$-plane\nwith a rotational angle $\\theta_{k\\ell}$.\n\nThe two types of solutions are obtained by performing\nthe $SO(n)$ transformation on BH solutions of the effective 5-d and\n6-d KK theories, respectively.  Since the 4-d space-time metric and\nthe dilaton field are not affected by the $SO(n)$ transformations,\nthe global space-time and the thermal properties in each class of the\nsolutions remain the same.\nThe above transformations also generate non-diagonal\ninternal metric coefficients; the ${{n(n-1)}\/2}$ degrees of freedom\nassociated with the rotational angles of the $SO(n)$ matrix\ngenerate the $n(n-1)\/2$ off-diagonal internal metric components.\n\nOn the other hand, the two classes of the solutions correspond to\nKK BH's with constrained charge configurations, and therefore do not\nconstitute the most general set of static 4-d Abelian KK BH's solutions.\nNamely, the subset of rotational matrices, which\ntransform the charge configuration of the\n$U(1)_M\\times U(1)_E$ solution to a new type of charge configurations,\ncorresponds to the coset space $SO(n)\/SO(n-2)$. This subset of\ntransformations therefore provides $(2n-3)$ additional degrees of\nfreedom for the (electric and magnetic) charge configuration.\nThus, the resultant number of degrees of freedom for the charge\nconfiguration is $2n-1$.  In fact, $2n$ ($n$-electric and $n$-magnetic)\ncharges $(\\vec{Q},\\vec{P})$ satisfy the constraint\n$\\vec{Q} \\cdot \\vec{P} = 0$.  Similarly, for solutions generated by\nthe $SO(n)\/SO(n-1)$-transformations on the charge configuration of\nthe effective 5-d KK solution, the resultant number of charge degrees\nof freedom is $n+1$.\n\nIn the following we shall concentrate on the electric-magnetic duality of\nsolutions generated by $SO(n)$ transformations on BH's of the effective\n6-d KK theory, since they involve less trivial transformations than those\nacting on BH's of the effective 5-d KK theory.  The unimodular part of\nthe internal metric and the set of $n$ gauge fields transform in\nthe following way:\n\\begin{equation}\n\\rho_{ij}^\\prime = \\rho_k {U}_{ik} {U}_{jk} \\ \\ \\ \\\nA^{\\prime i}_\\phi= U_{i \\,(n-1)} P  \\cos \\theta \\ \\ \\ \\\nA^{\\prime i}_t  = U_{in}\\psi (r)  \\ \\ ,\n\\label{nondiag}\n\\end{equation}\nwhere the electric and magnetic charges of new gauge field\n$A^{\\prime i}_\\mu$ are, therefore, given by $\\tilde{Q}^{\\prime i} =\nU_{in}\\tilde{Q} $ and $P^{\\prime i} = U_{i\\,(n-1)}P $ ($i=1,...,n$).\nThe expression for 4-d metric components in the new configuration\ncan be obtained from (\\ref{asol}) - (\\ref{bsol}) by replacing\n$\\tilde{Q}$ and $P$ by $\\sqrt{\\sum^n_{i=1} (\\tilde{Q}^{\\prime i})^2}$ and\n$\\sqrt{\\sum^n_{i=1}( P^{\\prime i})^2}$, respectively.\n\nThe $SO(n)$ transformations generate the continuous\nelectric-magnetic duality transformations which rotate the\n$U(1)_M \\times U(1)_E$ configuration, {\\it i.e.}, $P^i = \\delta_{n-1}^i P$\nand $\\tilde{Q}^i = \\delta_n^i \\tilde{Q}$, to general charge\nconfigurations $P^{\\prime i} = U_{i\\,(n-1)}P $ and $\\tilde{Q}^{\\prime i} =\nU_{in}\\tilde{Q}$, with the constraint $\\sum^n_{i=1}P^{\\prime\ni}Q^{\\prime i}=0$. This constraint is a consequence of\n$\\sum^n_{i=1} U_{i n}U_{i\\, (n-1)} = 0$.\n\nA subset of $SO(n)$ transformations corresponding to the\nrotation in the $(n-1,n)$-plane, {\\it i.e.}, $U=\\hat U_{(n-1)\\, n}$\n(see Eq. (\\ref{repre})), mixes the monopole charges in the\n$(n-1)$-$th$ and $n$-$th$ gauge fields and induces the corresponding\noff-diagonal terms in the internal metric of the effective 6-d KK\ntheory.  In particular, the discrete change of the rotation angle\n$\\theta_{(n-1)\\, n}$ from 0 to $\\pi \\over 2$ corresponds to a discrete\nelectric-magnetic duality transformation, which interchanges the magnetic\nand electric monopole charges in the $(n-1)$-$th$ and $n$-$th$\ngauge fields \\cite{EX}.\n\n\n\\section{Conclusions}\nIn this paper, we studied a class of static, spherically\nsymmetric solutions in $(4+n)$-d KK theory with Abelian isometry\n($U(1)^n$).  In particular, for a diagonal internal metric {\\it Ansatz},\nthe consistency of the equations of motion imposes strong constraints\non the possible charge configurations of such solutions; BH's\nexist only for configurations with at most one non-zero electric\nand one non-zero magnetic charges.  The case of electric and magnetic\ncharges coming from the same gauge field corresponds to BH's\nin effective 5-d KK theory.  Configurations with electric and magnetic\ncharges arising from different $U(1)$ gauge factors, {\\it i.e.},\nthe isometry group is $U(1)_M \\times U(1)_E$, correspond to BH's\nin effective 6-d KK theory.\n\nNon-extreme BH's with $U(1)_M\\times U(1)_E$ symmetry, which are compatible\nwith the corresponding Bogomol'nyi bound, are parameterized in terms of\nthe non-extremality parameter $\\beta>0$.  They have a global\nspace-time structure of Schwarzschield BH's, with the temperature\n$T_H$ [entropy $S$] increasing [decreasing] as $\\beta$ decreases.\nIn the extreme limit $\\beta \\to 0$, the solutions correspond to\nthe supersymmetric BH's \\cite{EX}.  In this limit, the corresponding\nBogomol'nyi bound for the ADM mass is saturated, $T_H$ [$S$] reaches\nthe upper [lower] limit $T_{H\\, ext}=1\/(4\\pi\\sqrt{|PQ|})$ [$S=0$],\nand the space-like singularity becomes null.\nNotably, the extreme limit is reached {\\it smoothly}.\n\nA class of solutions with non-diagonal internal metrics and\ngeneral charge distributions among gauge fields are obtained by\napplying the global $SO(n)$ transformations on the solutions with\ndiagonal internal metric.  These solutions correspond to a subset of\nstatic 4-d Abelian KK BH's with constrained charge configurations.\nSince $SO(n)$ transformations act on the unimodular part of the\ninternal metric and the gauge fields, only, the 4-d global space-time\nand the thermal properties of the general class of the 4-d abelian KK\nBH's are the same as those of the corresponding 4-d abelian KK BH's\nwith the diagonal internal metric.\nThis class of solutions exhibits continuous electric-magnetic\nduality symmetry parameterized by rotational angles of $SO(n)$ rotations.\n\\acknowledgments\nThe work is supported by U.S. DOE Grant No. DOE-EY-76-02-3071,\nand the NATO collaborative research grant CGR\n940870.  We would like to acknowledge useful discussions with E. Kiritsis,\nC. Kounnas, J. Russo and D. Waldram.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nAt a distance of 3.8~Mpc \\citep{sak04}, \\object{NGC~5253}, in the Centaurus A \/ M~83 group \\citep{kar07}, is one of the closest Blue Compact Dwarf (BCD) galaxies. This galaxy is well known for presenting several peculiarities, whose detailed study are closely connected to its proximity and high surface brightness. For example, it contains a deeply embedded very dense compact H\\,\\textsc{ii}\\ region at its nucleus (hereafter, \"the supernebula\"), detected in the radio at 1.3~cm and 2~cm \\citep{tur00} that host two very massive Super Star Clusters \\citep[SSCs,][]{alo04} and is embedded in a larger (i.e. $\\sim$100~pc$\\times$80~pc) Giant H\\,\\textsc{ii}\\ Region (hereafter, the central GH\\,\\textsc{ii}R). Recently, mid-infrared observations showed how its kinematics is compatible with a model for the supernebula in which gas is outflowing from the molecular cloud \\citep{bec12}. Indeed, the whole central region of the galaxy is dominated by an intense burst of star formation in the form of a large collection of compact young ($\\sim1-12$~Myr) star clusters \\citep[e.g.][]{har04}. In contrast to this, the main body of \\object{NGC~5253} resembles that of a dwarf elliptical galaxy and recently, three potentially massive ($\\gsim10^5$~M$_\\odot$) and old ($1-2$ Gyr) star clusters have been found in the outskirts of the galaxy \\citep{har12}.\nFinally, \\object{NGC~5253} is best-known  for being one of the few examples (and the closest) of a galaxy presenting  a confirmed local excess in nitrogen \\citep[see e.g.][]{wal89}.\n\n\\defcitealias{mon10}{Paper~I}\n\\defcitealias{mon12}{Paper~II}\n\\defcitealias{wes13}{Paper~III}\n\nWe are carrying out  a detailed study of this galaxy using Integral Field Spectroscopy (IFS). The results obtained so far have further highlighted its peculiar nature.\nIn \\citet[][hereafter Paper I]{mon10}, we found that the emission line profiles were complex and consistent with an scenario where the two SSCs produce an outflow \\citep[see also][]{bec12}. Also, we delimited very precisely the area polluted with extra nitrogen. Moreover, we detected nebular He\\,\\textsc{ii}$\\lambda$4686 in several locations, some associated with WN-type Wolf-Rayet (WR) stars (as traced by the blue bump at around 4680\\AA) and some not, but not necessarily coincident with the area exhibiting extra nitrogen. In \\citet[][hereafter Paper II]{mon12}, we studied the 2D distribution of  the main physical (electron temperature and density, degree of ionization) and chemical properties (metallicity and relative abundances of several light elements) of the ionized gas. A new area of enhanced nitrogen abundance at $\\sim$130~pc from the main area of enhancement and not reported so far was found. In \\citet[][hereafter Paper III]{wes13} several locations showing emission characteristic of WC-type WR stars (via the red bump at around 5810\\AA) were identified. The fact that WR stars are spread over $\\sim$350~pc gives an idea of the area over which the recent starburst has occurred.\nThe chemical analysis was extended with the finding that, with the exception of the aforementioned localised N excess, the $O\/H$ and $N\/H$ distributions are flat within the whole central 250~pc.\n\n\nAn issue not addressed in detail so far in NGC~5253 is the 2D determination and distribution of the He$^+$ abundance. In a cosmological context, this is particularly relevant since the joint determination of metallicity (as traced by the $O\/H$ abundance) and $^4He$ abundance ($Y$\\footnote{Here, we use $Y$ for  the helium mass fraction and $y$ for the number density of helium relative to hydrogen. Assuming $Z = 20(O\/H)$, they are related as $Y = \\frac{4y(1-20(O\/H))}{1+4y}$. }) for extragalactic H\\,\\textsc{ii}\\ regions and star-forming galaxies at low-metallicity was proposed as a means to estimate the primordial helium abundance, $Y_P$ \\citep{pei76,pag86}\n serving as a test-bench for the standard hot big band model of nucleosynthesis. However, the  density of baryonic matter depends weakly on $Y_P$. Therefore, to put useful constrains on $Y_P$, $^4He$ abundance of  individual objects  has to be determined with accuracies $\\lsim$1\\%. Nowadays, emission\nline flux data of this quality can be achieved and, indeed, the astronomical community is actively working on getting and improving the estimation of $Y_P$ \\citep[see e.g. ][for recent estimations by the different groups]{ave10,izo10,fuk06,pei07,izo07,oli01,pei02}.\nHowever, He abundance determinations are influenced by several effects and systematic errors which are not, in principle, straightforward to quantify and correct  \\citep[see for example][]{oli01}. Specifically, the intensity of He\\,\\textsc{i}\\ emission lines may intrinsically deviate from the recombination values due to collisional and radiative transfer effects. Moreover, the emitted spectrum also depends on the physical conditions of the ionized gas (e.g. temperature, density and ionization structure). Also, on top of these specific properties of the H\\,\\textsc{ii}\\ region, extinction by dust and a possible underlying stellar absorption component can also affect the observed spectrum. \nAll these effects contribute to the uncertainties associated with the estimation of \\emph{ionized} helium abundance ($y^+=He^+\/H^+$).\nA final extra source of uncertainty is associated with the estimation of the amount of existing neutral helium (i.e. the estimation of the ionization correction factor, icf(He)) and the calculation of the total helium abundance, $y=y_{tot}=\\rm{icf(He)}\\times y^+$.\n  \nTentative values for  $y^+$ were presented in \\citetalias{mon10} based on the He\\,\\textsc{i}$\\lambda$6678 line, which is almost insensitive to collisional and self-absorption effects. However, 2D distributions of all the relevant physical properties for the ionized gas were not available at that time. Moreover, other He\\,\\textsc{i}\\ lines, in particular  He\\,\\textsc{i}$\\lambda$7065, can indeed be affected by collisional and self-absorption effects, specially in conditions of relatively high electron temperature ($T_e$) and density ($n_e$), as it is the case in the main GH\\,\\textsc{ii}R\\ of NGC~5253. Supporting this expectation, long-slit measurements predict too high He$^+$ abundances from the  He\\,\\textsc{i}$\\lambda$7065 when only recombination effects are taken into account \\citep{sid10} and a non-negligible optical depth $\\tau$(3889) when the He$^+$ abundance is derived using several emission lines in a consistent manner \\citep{lop07}.\n\nAt present, a complete 2D characterization of the physical properties of the ionized gas in the main GH\\,\\textsc{ii}R\\ of \\object{NGC~5253} is available. Therefore, we are in an optimal situation for both mapping the collisional and radiative transfer effects in 2D, and re-visiting the derivation of the He$^+$ abundance map taking into account many He\\,\\textsc{i}\\ lines. This will be the main purpose of this work. Given \nthat the metallicity of the object  \\citepalias[$12+\\log(O\/H)=8.26$,][]{mon12} is only moderately low, our focus will not to be in achieving the $\\lsim1\\%$ accuracy required in the determination of the primordial He abundance, but to explore the effects of a parameter not taken into account so far: namely spatial resolution.\nIn addition, we will explore the 2D relationship between the properties of the ionized gas derived so far and the He\\,\\textsc{i}\\ collisional and self-absorption effects. To our knowledge this work constitutes the first attempt to study in 2D the collisional and self-absorption effects  in He\\,\\textsc{i} in any extra-galactic object.\nMoreover, irrespective of the spatial resolution,  due to the characteristics inherent in IFS data, there is the guarantee that the set of $\\sim100$ spectra utilized in this work have been processed in a homogeneous manner all the way from the observations (i.e. a given observable was taken with the same observing conditions for the whole set of data) to the final $y^+$ and $y_{tot}$ derivations. \n\nThe paper is structured as follows:  Sec. 2 describes the characteristics of the data utilized in our analysis; Sec. 3 contains an evaluation of the He\\,\\textsc{i}\\ collisional and self-absorption effects in 2D as well as the derivation of the $y^+$ and $y$ maps. Sec. 4 discusses the relation between radiative transfer effects and other quantities (e.g. kinematics of the gas, relative $N\/O$ abundance). Our main conclusions are summarized in Sec. 5.\n\n\n\\section{The data}\n\nWe will focus our study on the area associated with the main GH\\,\\textsc{ii}R\\ in \\object{NGC~5253} (see Fig. \\ref{apuntado}).\nThis is a portion of the full area studied in \\citetalias{mon10}\nand the location where: i) the gas presents relatively high electron temperature and density, and therefore, important collisional and radiative transfer effects are expected; and ii) several  helium lines can be detected over a relatively large area with sufficient quality, and therefore a 2D analysis based on the information in individual spaxels is feasible. The utilized data were collected during several observing runs using the FLAMES-Argus and the GMOS Integral Field Units (IFUs) and together cover the whole optical spectral range. In the following, we briefly describe the basic instrumental characteristics of each set of data and compile the information that was extracted. \n\n\n \n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.48\\textwidth,clip=]{.\/fg1.eps}\n   \\caption[Area under study]{\n\\emph{Left:} False colour image in filters $F658N$ (H$\\alpha$, cyan channel),  $F550M$ ($V$, yellow channel), and $F814W$ ($I$, magenta channel) for the central part of \\object{NGC~5253} using the HST-ACS images (programme 10608, P.I.: Vacca). The area  studied here is marked with a black rectangle.\n\\emph{Right:}   \n   Ionized gas  distribution as traced by the extinction corrected H$\\beta$\\ map derived from a portion of the original FLAMES data.   The position of the two main peaks of continuum emission are marked with crosses.\nThe map is presented in logarithmic scale in order to emphasize  the relevant morphological features and cover  a range of 2.0~dex.\nFlux units are arbitrary. Note the existence of three dead spaxels at $\\sim[5\\farcs0,-1\\farcs0]$ as well as absence of signal in the spaxels at the two left corners of the field of view.  \\label{apuntado}}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2f.ps}\\\\\n   \\caption[Maps of the extinction corrected fluxes of the utilized lines]{Extinction corrected flux maps normalized to H$\\beta$\\ for the lines utilized in this work.  The position of the two main peaks of continuum emission are marked with crosses in this and all subsequent maps.\nWe marked in white the areas where a given line was not observed. Specifically, for the He\\,\\textsc{i}$\\lambda$6678,  He\\,\\textsc{i}$\\lambda$4922 and He\\,\\textsc{i}$\\lambda$7065, they correspond to dead fibers. For He\\,\\textsc{i}$\\lambda$5876, He\\,\\textsc{i}$\\lambda$4471 and He\\,\\textsc{i}$\\lambda$3889, these areas were not covered by the corresponding GMOS or FLAMES-Argus field of view.\n \\label{hetohb}}\n \\end{figure}\n\n\n\\subsection{FLAMES-Argus data}\n\nData were obtained with FLAMES \\citep{pas02} at VLT UT2 in Paranal. We used the Argus IFU with the sampling of 0.52$^{\\prime\\prime}$\/lens, covering a field of view (f.o.v.) of 11\\farcs5$\\times$7\\farcs3, and four low resolution gratings (LR1, LR2, LR3, and LR6).\nAll together, they  offer a spectral coverage of $3\\,610-5\\,070$~\\AA\\ plus  $6\\,440-7\\,180$~\\AA\\ at a dispersion of 0.2~\\AA~pix$^{-1}$.\n Details of the observations, data reduction, and cube processing,\nas well as maps for most of the physical and chemical properties utilized in this work, have already been presented in \\citetalias{mon10} and \\citetalias{mon12}. \nMaps for the helium lines were derived by independently fitting in each spaxel the  He\\,\\textsc{i}\\ line profiles with a single Gaussian function with MPFITEXPR \\citep{mar09}. \nFor the particular case of He\\,\\textsc{i}$\\lambda$3889,  which is blended with H8 at our spectral resolution, we created an extinction corrected map by subtracting from the H8+He\\,\\textsc{i}$\\lambda$3889 map, a map of 0.659$\\times$ the H7 map. This H8 line intensity is that predicted by \\citet{sto95}  for Case B, $T_e = 10^4$~K and $n_e = 100$~cm$^{-3}$.\nThe final set of FLAMES-Argus maps utilized here are:\n\\begin{enumerate}[i)]\n\\item an extinction map derived from the H$\\alpha$\/H$\\beta$\\ line ratio;\n\\item a map for the H$\\beta$\\ equivalent width  ($EW$(H$\\beta$));\n\\item a map of electron temperature $T_e$ as derived from the \\textsc{[O\\,iii]}$\\lambda\\lambda$4959,5007\/ \\textsc{[O\\,iii]}$\\lambda$4363 line ratio: $T_e$(\\textsc{[O\\,iii]}). In those spaxels where no determination of $T_e(\\textsc{[O\\,iii]})$ was available, we assumed $T_e(\\textsc{[O\\,iii]})=10\\,500$~K \\citepalias[see][for typical $T_e$(\\textsc{[O\\,iii]}) values outside the main GH\\,\\textsc{ii}R]{mon12};\n\\item a map for the electron density ($n_e$) as derived from the \\textsc{[S\\,ii]}$\\lambda$6717\/\\textsc{[S\\,ii]}$\\lambda$6731 line ratio; \n\\item maps for the $O^+\/H^+$, $O\/H$ and $S^+\/H^+$ abundances, as derived from collisional lines using the direct method, to estimate the icf(He);\n\\item maps for different tracers of the excitation degree (i.e. \\textsc{[S\\,ii]}$\\lambda\\lambda$6717,6731\/H$\\alpha$\\ and \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ line ratio) to be used in the estimation of the icf(He) at those locations where no measure of the $O^+\/H^+$, $O\/H$ and $S^+\/H^+$ abundances is available and to explore the dependence of $y^+$ and $y_{tot}$ on the excitation;\n\\item a map of the relative abundance of nitrogen, $N\/O$, as derived from collisional lines using the direct method;\n\\item maps for the $\\lambda$3889, $\\lambda$4471, $\\lambda$4922, $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\ equivalent widths and extinction corrected  line fluxes using our extinction map and  the extinction curve of \\citet{flu94}. The extinction corrected line flux maps normalized to H$\\beta$\\ are presented in Fig. \\ref{hetohb}. \nNote that in order to minimize uncertainties associated to aperture matching, absolute flux calibration and extinction, lines were measured relative to a bright Balmer line observed simultaneously with a given helium line. Then, we assumed the theoretical Balmer line intensities obtained from \\citet{sto95} for Case B, $T_e = 10^4$~K and $n_e = 100$~cm$^{-3}$. \nAdditional He\\,\\textsc{i}\\ lines were covered by the FLAMES set-up but not used here. Specifically, He\\,\\textsc{i}$\\lambda$5016 and He\\,\\textsc{i}$\\lambda$4713 emission lines were detected over most of the FLAMES f.o.v. He\\,\\textsc{i}$\\lambda$5016 is relatively close to the much brighter (i.e. $\\sim$150-400 times)  \\textsc{[O\\,iii]}$\\lambda$5007 line and, in the main GH\\,\\textsc{ii}R, the wings of the \\textsc{[O\\,iii]}\\ line profile prevented us from measuring a reliable line flux. Regarding the relatively weak $\\lambda$4713 line, at the spectral resolution of these data, this line is strongly blended with [Ar\\,\\textsc{iv}]$\\lambda$4711 and  the uncertainties associated with the deblending of these lines in the GH\\,\\textsc{ii}R\\ are relatively large due to the existence of several distinct kinematic components \\citepalias[see][]{mon10,wes13};\n\\item maps with the extinction corrected line fluxes in $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\  and H$\\alpha$, for the different kinematic components presented in \\citetalias{mon10}.\n\\end{enumerate}\n\n\\subsection{GMOS data}\n\nThe Gemini-South Multi-Object Spectrograph (GMOS) data were taken using the one-slit mode of its IFU \\citep{all02}. In this mode, the IFU covers a f.o.v. of $5\\farcs0\\times3\\farcs5$ sampled by 500 contiguous hexagonal lenslets of 0\\farcs2 diameter.\nThe utilized grating (R381) gives a spectral coverage of $4\\,750-6\\,850$~\\AA\\ at a dispersion of 0.34~\\AA~pix$^{-1}$,  thus complementing the FLAMES-Argus data. We refer to \\citetalias{wes13}  for further details on the observations and data reduction. The product of the reduction is a datacube per pointing with a uniformly sampled grid of 0\\farcs1. In this work, we utilized the two pointings (out of four) that mapped the central GH\\,\\textsc{ii}R. As we did with the FLAMES's data, in each spaxel all the lines of interest were independently fit with a single Gaussian function with MPFITEXPR. \nThe final set of GMOS maps utilized here are:\n\n\\begin{enumerate}[i)]\n\\item Maps for H$\\alpha$\\ and H$\\beta$\\ fluxes. These images were utilized to check the consistency between the FLAMES and GMOS data, both in terms of observed structure and derived extinction map, to estimate the offset and rotation that was necessary to be applied to the GMOS data, and to correct for extinction the  He\\,\\textsc{i}$\\lambda$5876 map;\n\\item A map for He\\,\\textsc{i}$\\lambda$5876 flux. This is the strongest He\\,\\textsc{i}\\ line and therefore, one of the key observables for the present study;\n\\item  Equivalent widths and extinction corrected line flux maps of He\\,\\textsc{i}$\\lambda$5876 flux normalized to H$\\beta$. These were derived from the maps previously mentioned and were rotated and reformatted to match the FLAMES data using the \\texttt{drizzle} task of the Space Telescope Science Data Analysis System (STSDAS) package of IRAF\\footnote{The Image Reduction and Analysis Facility \\emph{IRAF} is distributed by the National Optical Astronomy Observatories which is operated by the association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.}. They were the only GMOS maps utilized jointly with the FLAMES's maps. The flux map is shown in Fig. \\ref{hetohb} together with the other He\\,\\textsc{i}\\ line flux maps. \n\\end{enumerate}\n\n\n\\section{Results}\n\n\\subsection{Collisional effects as traced by the theoretical C\/R ratio \\label{seccoli}}\n\nCollisional excitation in hydrogen can be important in regions of very low metallicities ($Z\\lsim1\/6$~Z$_\\odot$)\n due to their relatively high temperatures \\citep{lur09}. However, for the typical temperatures and densities found in H\\,\\textsc{ii} regions in general, and in the main Giant H\\,\\textsc{ii} Region (GH\\,\\textsc{ii}R) of \\object{NGC~5253} in particular, the collisional excitation in hydrogen is negligible in comparison with recombination \\citep[see e.g. Fig. 9 of ][]{ave10}.  This is not the case for  helium.  The He\\,\\textsc{i}\\ $2^3S$ level \\citep[see Fig. 1 in ][for a representation of the Grotian diagram for He\\,\\textsc{i}\\ singlet and triplet ladders]{ben99} is highly metastable, and collisional transition from it can be important \\citep{ost06}. Specifically, at relatively high densities this level can be depopulated via collisional transitions to the $2^3P^0$, $2^1P^0$ and $2^1S$ and, to a lesser extent, to higher singlets and triplets (mainly $3^3P^0$). The effect on the observed He\\,\\textsc{i}\\ emission lines is always an increase in the observed flux. The relative importance of the collisional effects on a given emission line is characterized by the $C\/R$ factor, i.e. the ratio of the collisional component to that arising from recombination which is given by:\n\n\\begin{equation}\n\\frac{C}{R} = \\frac{n_{2^3S}k_{eff}}{n^+_{He}\\alpha_{eff}}\n\\end{equation}\n\nwhere $n_{2^3S}$, and $n^+_{He}$ are the densities of the $2^3S$ state and He$^+$ respectively, $\\alpha_{eff}$ is the effective recombination coefficient for the line, and $k_{eff}$ is the effective collisional rate coefficient.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3f.ps}\\\\\n   \\caption[C\/R ratios]{Collisional effects as traced by the $C\/R$ ratio for the He\\,\\textsc{i}\\ lines utilized in this work. Note that a common scale was used for all the lines in order to facilitate comparison of the relative effects between the different lines. Also, a logarithmic color stretch is used to emphasize the variations \\emph{within} the region for a given line.\n \\label{c2rratios}}\n \\end{figure}\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4f.ps}\\\\\n   \\caption[Differences on the $C\/R$ factors  for different electron temperature]{\n Maps for the ratio between the $C\/R$ factors derived for two assumptions of the $T_e(He\\,\\textsc{i})$: $Q(\\lambda)=\\frac{C\/R(\\lambda)_{Case 1}}{C\/R(\\lambda)_{Case 2}}$. Note that areas with constant values for which a $T_e$(\\textsc{[O\\,iii]})=10\\,500~K was assumed have not been included in the comparison.\n %\n A logarithmic color stretch is employed to emphasize the variations \\emph{within} the region for a given line.\n \\label{c2rratiosdif}}\n \\end{figure}\n\nHere, we will estimate the 2D contribution to the collisional component using the relations derived by  \\citet{por07}, where  theoretical $C\/R$ factors are calculated as functions of $n_e$ and $T_e$, and assuming that the density from \\textsc{[S\\,ii]}\\ traces well that from helium.\nPlasma temperatures as traced by helium lines can typically be $\\sim$50\\% of those traced by oxygen in planetary nebulae \\citep{zha05}. For H\\,\\textsc{ii}\\ the situation is not so clear. For the specific case of \\object{NGC~5253}, values between 82\\% and 96\\% have been reported in specific apertures \\citep{lop07}. We made two assumptions for the electron temperature:\n\\begin{itemize}\n\\item Case 1: The temperature from oxygen traces well that from helium: $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{[O\\,iii]});\n\\item Case 2: The temperature from helium is proportional to the temperature from oxygen, with the constant of proportionality estimated as the mean of the ratios between both temperatures provided by \\citet{lop07}:  $T_e$(He\\,\\textsc{i}) = 0.87\\,$T_e$(\\textsc{[O\\,iii]}).\n\\end{itemize}\n\nMaps for the $C\/R$ factors using the first assumption are presented in Fig. \\ref{c2rratios} while the ratio between the two estimations is presented in Fig. \\ref{c2rratiosdif}.\nThis is the first time that collisional effects for a set of helium lines have been mapped in an\nextra-galactic source. Several results can be extracted from these maps.\n\nFirstly, all the $C\/R$ maps display the same structure. That is: higher ratios at the peak of emission for the ionized gas and towards the northwest half of the GH\\,\\textsc{ii}R\\ and a decrease of the collisional contribution outwards. This reproduces the observed density structure \\citepalias[see e. g. Fig. 6 in ][]{mon10}.\n\nSecondly, \nlines corresponding to transitions in the singlet cascade have a negligible contribution from collisional effects (e.g. $C\/R$ factor for $\\lambda$4922 varies between  $\\sim0.001-0.006$) while for those lines in the triplet cascade ($\\lambda$7065, $\\lambda$5876, $\\lambda$4471,  and $\\lambda$3889), the contributions from  collisional effects can be important.  In particular,  the  $C\/R$ factor  for $\\lambda$7065 ranges between $\\sim0.02$ and $\\sim0.22$, meaning it reaches  $\\sim$20\\% in the nucleus of the galaxy. \n\nThirdly, the assumed temperature has some influence in the estimation of the collisional effects. In our particular case the assumed temperature in Case 2 was only $\\sim15\\%$ smaller than that in Case 1. However, this implies a smaller contribution of the collisional effects by $\\sim$25-30~\\% for $\\lambda$7065  and $\\sim$30-35~\\% for $\\lambda$3889 (the two lines most affected by collisional effects) and up to $\\sim50$\\% for the other lines under study.  Interestingly, areas of lower temperature are more sensitive to the assumption on $T_e$. \nIt is important to note that the uncertainties associated to the errors due to the measurement of the line fluxes involved in the determination of $T_e$(\\textsc{[O\\,iii]}) were typically $\\lsim$1\\,000~K \\citepalias{mon12}. This is smaller than the difference  in the assumed temperature between the two reasonable assumptions, Case 1 and 2, which range between $\\sim$1\\,500~K at the peak of emission to $\\sim$1\\,300~K in the areas of lowest surface brightness. \nThis implies that, at this level of data quality, more than errors associated to line fluxes, it is systematic errors associated with assumptions in density and temperature that dominate the uncertainties in the evaluation of the collisional effects.\n\n\\subsection{Radiative transfer effects as traced by $\\tau(3889)$ and derivation of He$^+$ abundance \\label{secymas}}\n\nSingle ionized helium abundance, $y^+$,   can be calculated as follows:\n\n\\begin{equation}\ny^+(\\lambda) = \\frac{F(\\lambda)}{F(H\\beta)}\n\\frac{E(H\\beta)}{E(\\lambda)}\n\\frac{\\frac{EW(\\lambda) + a_{He\\,I}(\\lambda)}{EW(\\lambda)}}{\\frac{EW(H\\beta) + a_{H}(H\\beta)}{EW(\\lambda)}} \\frac{1}{f_\\tau(\\lambda)}\n\\label{eqymas}\n\\end{equation}\n\nwhere $F(\\lambda)\/F$(H$\\beta$) is the extinction corrected flux for the He\\,\\textsc{i}\\ line scaled to H$\\beta$, $E$(H$\\beta$) and $E$(He\\,\\textsc{i}) are the theoretical emissivities; $f_{\\tau}(\\lambda)$ is a factor that takes into account  radiative transfer effects. The remaining term in Eq. \\ref{eqymas}, with $a(\\lambda)$ is the equivalent width in absorption, which takes into account the effect of the underlying stellar population.\nIn the following, we will describe how each of these terms were evaluated.\n\n\\subsubsection{Calculation of the emissivities}\n\nFor H$\\beta$\\  emissivities, we utilized the function from \\cite{sto95} which is, in units of 10$^{25}$ erg cm$^{-3}$ s$^{-1}$:\n\n\\begin{equation}\nE(H\\,\\beta) =  4\\pi j_{H\\,\\beta}\/n_e n_{H^+} = 1.37 t_e^{-0.982} \\exp(-0.104\/t_e)\n\\end{equation}\n\nwith $t_e = T_e\/10^4 $.\nFor the He\\,\\textsc{i}\\ lines, we utilized those emissivities originally provided by \\citet{por12} and recently corrected by \\citet{por13}. These are the most recent He\\,\\textsc{i}\\ emissivities and have the collisional effects included. Therefore, there was no need to include an extra term in Eq. \\ref{eqymas} to take into account collisional effects. They are tabulated for discrete values of $n_e$ and $T_e$. However, the $n_e$ in the main GH\\,\\textsc{ii}R\\  varies between $\\lsim100$~cm$^{-3}$ and $\\sim660$~cm$^{-3}$ and $T_e$ ranges between $\\sim9\\,000$ and $\\sim12\\,000$~K. Therefore, to evaluate the emissivities at each individual spaxel, we fitted the values provided for $n_e= 100$ and 1\\,000~cm$^{-3}$  and $T_e$ ranging from 5\\,000~K to 25\\,000~K to functions with the same parametrization as $E(H\\,\\beta)$, $a  t_e^{b} \\exp(c\/t_e)$, and then interpolated on a logarithmic scale as follows:\n\n\\begin{equation}\nE(\\lambda,\\log n_e) = E(\\lambda,2) + (E(\\lambda,3) - E(\\lambda,2)) (\\log n_e -2)\n\\end{equation}\n\nThe coefficients and standard deviations of the fits are compiled in Table \\ref{coeficientes} while Fig. \\ref{compaemi} shows a \ncomparison between the fitted function and the discrete values provided by \\citet{por12} for the range of densities and temperatures covered in the GH\\,\\textsc{ii}R. A comparison between this figure and Fig. \\ref{c2rratios} shows the correspondence between the degree of dependence on the density and the contribution of the collisional effects.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5f.ps}\\\\\n   \\caption[Fitted function to the Porter et al. 2012 emissivities]{Fitted functions to the \\citet{por12} emissivities. The black lines are the functions for $n_e=100$~cm$^{-3}$ (continuous line) and for $n_e=1000$~cm$^{-3}$ (long dashed line). The intemediate short dashed lines represent the interpolated emissivities for $\\log(n_e) = 2.2, 2.4, 2.6, 2.8$. \n \\label{compaemi}} \n \\end{figure}\n\n\n\\begin{table}\n\\small\n     \\centering\n     \\caption[]{Coefficients for the equations fitted to the \\citet{por12} Table 2. \\label{coeficientes}}\n            \\begin{tabular}{lcccccccccccc}\n            \\hline\n            \\noalign{\\smallskip}\n\n  Line                   & $a$  &  $b$  &  $c$  & Std Dev (\\%)  \\\\\n\\hline\n\\multicolumn{5}{c}{$n_e=100$~cm$^{-3}$}\\\\\n\\hline\n\\noalign{\\smallskip}  \n3889 & 1.453 & $-0.724$ & $-0.036$ & 0.010 \\\\\n4471 & 0.679 & $-1.078$ & $-0.105$ & 0.003\\\\\n4922 & 0.184 & $-1.091$ & $-0.105$ & 0.001\\\\\n5876 & 1.680 & $-1.061$ & $+0.004$ & 0.042\\\\\n6678 & 0.473 & $-1.065$ & $+0.013$ & 0.015\\\\\n7065 & 0.269 & $-0.367$ & $+0.100$ & 0.007\\\\ \n\\hline\n\\multicolumn{5}{c}{$n_e=1000$~cm$^{-3}$}\\\\\n\\hline\n3889 &  1.198 & $-0.336$ & $+0.207$ & 0.032\\\\\n4471 &  0.507 & $-0.694$ & $+0.202$ & 0.048\\\\\n4922 & 0.131 & $-0.642$ & $+0.252$ & 0.016\\\\\n5876 & 0.895 & $-0.195$ & $+0.671$ & 0.255\\\\\n6678 & 0.237 & $-0.138$ & $+0.738$ & 0.094\\\\\n7065 & 0.320 &  $+0.171$ & $+0.163$ & 0.096\\\\ \n\n\\noalign{\\smallskip}  \n\\hline\n\n\\end{tabular}\n\\end{table}\n\n\n\n\\subsubsection{Correction for underlying stellar population}\n\nTo take into account the effect of the underlying stellar population it is necessary to estimate the equivalent width both in emission and absorption for H$\\beta$\\ and the helium lines.\nThe equivalent width of H$\\beta$\\ in emission (not shown) ranges typically from $\\sim$240~\\AA\\ at the peak of emission for the ionized gas to $\\sim$65~\\AA\\ in the most outer regions of the area sampled here.\nWe assumed a component in absorption of 2~\\AA, which is adequate for very young starbursts, as the one at the nucleus of \\object{NGC~5253}  \\citep[e.g.][]{gon05,alo10}. \nThis implies a correction for H$\\beta$\\ in absorption from a negligible value (i.e. $\\lsim1\\%$) at the peak of emission to $\\sim$4\\% in the outer parts of the covered area. \n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6d.ps}\\\\\n   \\caption[Maps for the estimated correction factors for an underlying stellar population]{\n Maps of the correction factor due to a component in absorption in the He\\,\\textsc{i}\\ lines.\n %\nDifferent scales were utilized in the different maps to emphasize the structure.\n The corresponding map for the H$\\beta$\\ correction factor (not shown) is similar to those for $\\lambda$6678 and $\\lambda$7065.\n %\n \\label{corrabs}}\n \\end{figure}\n\nThe correction due to the underlying stellar population is not straightforward to estimate for the helium lines. \nAs explained in \\citetalias{mon12}, for the \ndata obtained with FLAMES Giraffe and the LR1 or LR2 gratings, the contribution of the stellar population was separated from that of the gas using the STARLIGHT code \\citep{cid05,cid09} to match the stellar continuum. Therefore, \nwe can assume $a_{HeI}(\\lambda4471)=a_{HeI}(\\lambda3889)=0$.\nFor the other lines,  a common approach assumes identical stellar equivalent widths for all the helium lines. However,  He\\,\\textsc{i}\\ lines are produced by the bluest stars. Therefore, in the context of a galaxy  suffering a burst of star formation on top of an older population, redder lines should have smaller equivalent widths in absorption, since older stars contribute more to the stellar continuum.\nSpecifically, typical estimations of the equivalent width in absorption for redder lines would be $\\sim40-80$\\% that of $a_{He\\,\\textsc{i}}(\\lambda4471)$ \\citep{ave10}. This is one of the lines observed with the LR2+FLAMES configuration for which nebular and stellar information have been disentangled. Therefore $a_{He\\,\\textsc{i}}(\\lambda4471)$ could be measured from our emission line free cube. Typical values were extremely low (i.e. $\\sim0.11\\pm0.03$~\\AA) and without any obvious variation following the structure of the GH\\,\\textsc{ii}R\\ or the location of the Super Star Clusters.  Taking this as reference, and the relative values between the different $a_{He\\,\\textsc{i}}(\\lambda)$'s reported by \\citet{ave10}, which were in turn derived using the models presented by \\citet{gon05} and \\citet{mar05b}, we assumed $a_{He\\,\\textsc{i}}(\\lambda)=0.09, 0.08, 0.06$ and 0.05~\\AA\\ for $\\lambda$4922, $\\lambda$5876, $\\lambda$6678, $\\lambda$7065, respectively. \nEquivalent widths for the  $\\lambda$4922, $\\lambda$5876, $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\ emission lines were $\\sim0.3-4$~\\AA, $\\sim20-60$~\\AA, $\\sim3-20$~\\AA, and $\\sim3-35$~\\AA, respectively. With these values, the largest correction for absorption was for He\\,\\textsc{i}$\\lambda$4922, with typical values between 3\\% and 6\\%. However, corrections in the most external spaxels could reach up to  $\\sim$25\\%. For the other lines the correction was more moderate, with values between $\\sim$1\\% and $\\lsim$8\\% for He\\,\\textsc{i}$\\lambda$5876 and comparable to the correction in H$\\beta$\\ for the  He\\,\\textsc{i}$\\lambda$6678 and  He\\,\\textsc{i}$\\lambda$7065 lines. This is illustrated in Fig. \\ref{corrabs}.\n\n\\subsubsection{Estimation of radiative transfer effects}\n\nRadiative transfer effects can be important in recombination radiation. Given the metastable character of the $2^3 S$ level, under certain conditions, the optical depths in lower $2^3S - n^3 P^0$ lines imply non-negligible effects on the emission line strengths \\citep{ost06}. Specifically $\\lambda$10\\,830 photons are only scattered, but absorbed photons, corresponding to transitions to higher levels, can be converted to several photons of lower energy. The most prominent example is\n$\\lambda$3889 photons that can be converted to $\\lambda4.3~\\mu$m $3^3 S-3^3 P^0$, plus $\\lambda 7065\\, 2^3 S-3^3 P^0$, plus $\\lambda10\\,830\\, 2^3 S-2^3 P^0$. The net effect  is that lines associated with transitions from the $2^3 P$ level upwards (e.g. $\\lambda3889$) are weakened by self-absorption, while lines associated with several transitions from higher levels (e.g. $\\lambda$7065) are strengthened by resonance fluorescence. Contrary to collisional effects, radiative transfer effects do not affect photons in the singlet cascade.\nThe relative importance of radiative transfer effects is quantified by a correction factor, $f_\\tau(\\lambda)$,  for each line which is a function of the optical depth at $\\lambda$3889, $\\tau(3889)$. Here, we  parametrized these factors by fitting the values provided by \\citet{rob68} to a non-expanding nebula to the functional form $f_\\tau(\\lambda) _{\\omega=0} = 1 + a \\tau^b$. The corresponding functions for the lines utilized in this work are:\n\n\\begin{equation}\nf_\\tau(7065)_{\\omega=0} = 1 + 0.741\\tau^{0.341} \\label{eqfac7065}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(6678)_{\\omega=0} = 1    \\label{eqfac6678}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(5876)_{\\omega=0} = 1 + 0.0126\\tau^{0.496}   \\label{eqfac5876}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(4922)_{\\omega=0} = 1    \\label{eqfac4922}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(4471)_{\\omega=0} = 1 + 0.0022\\tau^{0.728}   \\label{eqfac4471}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(3889)_{\\omega=0} = 1 - 0.261\\tau^{0.305}  \\label{eqfac3889}\n\\end{equation}\n\nNote that no values were provided for He\\,\\textsc{i}$\\lambda$4922 in the original work of \\citet{rob68}. However, as it happens for He\\,\\textsc{i}$\\lambda$6678, this is a singlet line, and therefore $f_\\tau(4922) = 1$ can be assumed. \n\nTypically, optical depth ($\\tau(3889)$) and He\\,\\textsc{i}\\ abundance ($y^+$)  (and other parameters) are determined simultaneously by minimizing $\\chi^2$, defined as the difference between each helium line's abundance (weighted according to a reasonable criterion like the line flux) and the average. In this methodology it is implicit that all the lines trace the same location in the nebula\/galaxy. However, the area under study in this work suffers from heavy extinction \\citepalias[see Fig. 3 in][]{mon10}.\nTherefore,  \\emph{a priori}, it is not possible to assume that all the lines equally penetrate the nebula interior and that bluer and redder lines trace zones with the same $y^+$. Because of that, we grouped the lines in two sets according to their wavelengths, called hereafter the \\emph{blue} ($\\lambda$3889, $\\lambda$4471, $\\lambda$4922) and \\emph{red} ($\\lambda$5876, $\\lambda$6678, $\\lambda$7065) \\emph{sets}, which will be analyzed independently. Each set is made out of: i) a line from the singlet cascade, and therefore not affected by radiative transfer effects; ii) a line from the triplet cascade, thus highly sensitive to radiative transfer effects; iii) a line from the triplet cascade mildly sensitive to radiative transfer effects. \n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg7a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg7b.ps}\\\\\n   \\caption[Abundance maps from singlets]{Maps for $y^+$ derived from lines of the singlet cascade for the blue (left) and red (right) sets.\nA common scale was used in both maps to facilitate comparison between them.\n  \\label{abunmapsinglet}}\n \\end{figure}\n\nA first estimation of the $y^+$ abundance structure as traced by the red and blue lines can be obtained from the singlet lines, since they are not affected by radiative transfer effects. This is shown in Fig. \\ref{abunmapsinglet}. The mean ($\\pm$ standard deviation) are \n80.7($\\pm$5.1) and 76.8($\\pm$1.8)\nfor the $\\lambda$4922 and $\\lambda$6678 lines averaged over the mapped area. These values indicate that even if the red and blue lines are not tracing exactly the same gas columns, at least they sample areas with a the same  $y^+$ within $\\sim$5\\%. \n\nA comparison of the initial $y^+$ maps  with those obtained from a line highly sensitive to radiative transfer effects in each set ($\\lambda$3889 and $\\lambda$7065), allowed us to determine the respective $f_\\tau(\\lambda)$ map for each set, which can, in turn, be converted into $\\tau(3889)$ maps. These are shown in Fig. \\ref{maptau}. The optical depth has the same structure in both maps, with the peak at the main super star cluster(s). The shape of the area presenting $\\tau(3889)>0$ is circular but resolved. With a FWHM=1\\farcs6$\\sim$30~pc, this is larger than the seeing ($\\sim$0\\farcs9). This is consistent with a picture where large optical depths are not restricted to the deepest core of the galaxy, associated with the supernebula, but extend over a larger region. Also, the fact that\nthe optical depths derived from the red set of lines are larger than those derived from the blue set is in harmony with a picture where deeper layers in the nebula, which we interpreted as denser and hotter \\citep[][see also Beck et al. 2012]{mon12}, suffer from larger radiative transfer effects and suggests a link between He\\,\\textsc{i}\\ optical depth and dust optical depth.\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg8a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg8b.ps}\\\\\n   \\caption[Maps for $\\tau(3889)$]{Maps for $\\tau(3889)$ as derived from the $\\lambda$4922 and $\\lambda$3889  (left) and the $\\lambda$6678 and $\\lambda$7065 (right) emission lines.\n \\label{maptau}}\n \\end{figure}\n\n\\subsubsection{Final derivation of He$^+$ abundance}\n\nFinally for both sets, the information of the  mildly sensitive line was added and abundances from each line were recalculated using their corresponding $\\tau(3889)$. The final abundance maps for each set  were  made using a weighted average.  We used the mean fluxes of the He\\,\\textsc{i}\\ in the utilized area to determine the respective weights. These were 3:1:1 for the $\\lambda$5878:$\\lambda$6678:$\\lambda$7065 and 25:5:1 for $\\lambda$3889:$\\lambda$4471:$\\lambda$4922.  The mean ($\\pm$ standard deviation) for the red and blue sets for $y^+$ are\n79.3($\\pm$2.5) and 82.0($\\pm$3.8),\ni.e., results from the red and the blue sets agree to within $\\sim$3\\%. \nSince differences between the  maps derived from the blue and red sets are consistent with tracing the same $y^+$ per spaxel, we derived a final map using the information of all the lines by averaging these two maps with a \\emph{blue:red} weight of 0.8:1.0, the mean ratio between the brightest lines in the blue and red sets (i.e. $\\lambda$3889 and $\\lambda$5876).  This is shown in Fig. \\ref{abundancefinal}. The mean ($\\pm$ standard deviation) is\n 80.3($\\pm$2.7).\n  This value compares well with other values of $y^+$ reported in the literature for this area \\citep{kob97,lop07,sid10}.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg9.ps}\n   \\caption[Final $y^+$ map]{Final map for $y^+$ derived from a weighted average of those derived for the lines in the blue and red sets.  Note we used a different scale than in Fig. \\ref{abunmapsinglet}  to increase the contrast and emphasize the structure following the excitation structure. Instead, the scale is common with that in the right column of Fig. \\ref{icfhe} to make easier the comparison between total and ionic helium abundances.\n  \\label{abundancefinal}}\n  \\end{figure}\n\nTo estimate the emissivities, the calculation of these abundances was made with assumed $n_e$(\\textsc{[S\\,ii]}) and $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{H\\,i}) = $T_e$(\\textsc{[O\\,iii]}). As for the mapping of the collisional effects, in order to evaluated the influence of the selection of $T_e$,  the derivation presented here was repeated assuming   $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{H\\,i}) = 0.87 $T_e$(\\textsc{[O\\,iii]}) and finding a mean  ($\\pm$ standard deviation) of\n79.4($\\pm$3.0).\nThis implies that the precise selection of $T_e$ has a small effect ($\\sim$1\\%) in the determination of $y^+$, in comparison to other factors like e.g.  the correction for absorption or radiative transfer effects, as long as one keeps this selection within reasonable values.\n\nFinally, it is worth mentioning that throughout all the derivation of $y^+$, we have assumed that He\\,\\textsc{i} singlets are formed under Case B conditions (i.e. lines are formed in the limit of infinite Lyman line  optical depth and there is no optical depth in transitions arising from excited states). This is the standard framework for the derivation of helium abundances.\nHowever, under certain conditions, helium Ly$\\alpha$ $\\lambda$584 (and higher Lyman transitions) may be attenuated due to two effects: absorption of the helium Ly$\\alpha$ line by dust and hydrogen absorption \\citep{fer80,shi93}.\nIf these effects were important, the Case B assumption would no longer be applicable. The fact that we have recovered similar helium\nabundances from the two sets of lines supports the Case B assumption.\n \n\\subsection{Determination of He abundance}\n\nThe favorite scenario to explain the extra nitrogen found within the area studied here is that this has been produced by Wolf-Rayet stars and presumedly those at the two Super Star Clusters. In this scenario, as it happens with the nitrogen, an overabundance of helium is also expected to be observed.\nSpecifically, typical stellar atmospheric $N\/He$ ratios  would be $\\sim3\\times10^{-3}$ and $\\sim5\\times10^{-4}$ for WN and WC stars, respectively \\citep{smi82}. This could be use as a reference for the expected $N\/He$ ratios of the material newly incorporated into the warm ISM. Are our data supporting an enrichment in helium abundance compatible with these ratios and, therefore, supporting the Wolf-Rayet stars hypothesis?\n\n\n  \\begin{figure}[ht!]\n   \\centering\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg10.ps}\n\\caption{\nHe$^+$ abundance vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. The first-degree polynomial fit is shown with a black line. The 1-$\\sigma$ and 3-$\\sigma$ levels are marked with thick and thin red lines respectively. Mean and standard deviation of each 0.05 dex bin in $\\log$(\\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$) are shown with blue diamonds and error bars respectively. The green horizontal dashed line shows the mean value. Data points corresponding to spaxels with $\\log(N\/O)>-1.3$ have been marked with orange circles.\n}\n \\label{ionidegvsymas}\n \\end{figure}\n\nAs can be clearly seen in Fig. \\ref{abundancefinal}, the $y^+$ map presents some structure that follows the excitation, even if the standard deviation for $y^+$ is small ($\\sim$4\\% of the mean value). This is better seen in Fig. \\ref{ionidegvsymas} which is an updated version of Fig. 14 in \\citetalias{mon10} but restricted to the area studied here. \nThe figure presents $y^+$ for each individual spaxel versus  a tracer of the excitation, in this case \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. \nFor each bin of 0.05~dex, we overplotted the mean and standard deviation of $y^+$ values with blue diamonds and error bars and fitted all our data points with a first-degree polynomial.\nConsidering that if we find $<y^+>$  in the bin of highest \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ to be larger than $<y^+>+\\sigma(y^+)$ in the bin of lowest \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ indicates the presence of a gradient, then this plot is consistent with a positive gradient in $y^+$.\nMoreover, spaxels with an extra amount of nitrogen have, on average higher $y^+$.\nIs this increase due \\emph{only} to variations in the ionization structure or are we witnessing an enrichment in the helium abundance?  \nTo answer this question, one needs to estimate and correct for the  unseen neutral helium, removing in this way, the dependence on the excitation.\nThis is not straight forward  as examplified by previous studies \\citep[e.g.][]{vie00,sau02,gru02}.\n\n  \\begin{figure}[ht!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg11a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg11b.ps}\n   \\caption[Ionic abundance maps]{Maps for the ionic abundances utilized in the derivation of the ionization correction factors. \\emph{Left:} $O^+\/H^+$. \\emph{Right:} $S^+\/H^+$ }\n \\label{ionicabundancemap}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12f.ps}\\\\\n   \\caption[icf(He)]{\\emph{Left:} Maps for the icf(He) as estimated from the expressions proposed by \\citet[][\\emph{top}]{kun83},  \\citet[][\\emph{center}]{pei77} and assuming no gradient in helium abundance \\emph{bottom}. \\emph{Right:} Corresponding helium abundance maps. }\n \\label{icfhe}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13a.ps}\\\\\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13b.ps}\\\\\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13c.ps}\\\\\n\\caption{\nHelium abundance as derived using the icf(He) proposed by  \\citet[][\\emph{top}]{kun83},  \\citet[][\\emph{center}]{pei77} and assuming no gradient in helium abundance (\\emph{bottom}) vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. Symbol and color codes are as in Fig. \\ref{ionidegvsymas}.\n}\n \\label{ionidegvsy}\n \\end{figure}\n\nWithin the spirit of keeping the analysis simple, here we compare the results using three approaches. The first two make use the icf's proposed by  \\citet{kun83} and  \\citet{pei77}:\n\n\\begin{equation}\n\\rm icf(He)_{K\\&S83} = (1 - 0. 25\\,O^+ \/ O)^{-1}                             \\label{eqks83}\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{P\\&TP77} = (1 - 0. 35\\,O^+ \/ O- 0.65\\,S^+ \/ S)^{-1}  \\label{eqptp77}\n\\end{equation}\n\nFor the third one, we assumed \\emph{a priori} a functional form similar to that of \\citet{kun83} and iteratively determined the required coefficients to obtain a relation consistent with no gradient of helium abundance. This would represent the case of largest reasonable correction for unseen helium:\n\n\\begin{equation}\n\\rm icf(He)_{M13} = (1 - 0. 46\\, O^+ \/ O)^{-1}                                  \\label{eqm13}\n\\end{equation}\n\nThese icf's depend on the ionic and total abundances of oxygen and sulfur. A map for the total abundance of oxygen was presented in \\citetalias{mon12}. We utilized a constant sulfur abundance determined as the mean of those presented in \\citetalias{wes13}. Regarding the ionic abundances of oxygen and sulfur, they were derived as part of the work presented in \\citetalias{mon12}. However, maps were not included there and are displayed in Fig. \\ref{ionicabundancemap} for completeness. An extra assumption is needed to calculate the icf's in those areas where no measurement for the ionic abundance is available. For that, we utilized the information in the other spaxels to fit  a  first-degree polynomial to the relation between  the \\textsc{[S\\,ii]}$\\lambda\\lambda$6717,6731\/H$\\alpha$\\ line ratio and icf(He):\n\n\n\\begin{equation}\n\\rm icf(He)_{K\\&S83} =  (0.354\\pm0.012) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.033\\pm0.002)\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{P\\&TP77} =  (0.447\\pm0.026) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.054\\pm0.003)\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{M13} =  (0.763\\pm0.025) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.057\\pm0.004)\n\\end{equation}\n\nThe  maps for the three icf(He) estimations are presented in the left column of Fig. \\ref{icfhe}. In all three cases, the icf is smallest at the peak of emission for the ionized gas and increases outwards, following the ionization structure. \n\nThe icf(He) values are close to one another but variable within a range $\\sim1.04-1.09$,  $\\sim1.06-1.12$ and $\\sim1.09-1.20$ when using Eqs. \\ref{eqks83}, \\ref{eqptp77}, and  \\ref{eqm13}, respectively.\nThe corresponding maps with the total helium abundance are presented in the right column of this figure, while the dependence on the excitation is presented in Fig. \\ref{ionidegvsymas}.\nNote that, when using the \\citet{kun83} and \\citet{pei77} icf's, the slope of the relation between $y$ and \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ is slightly positive within the errors of the fit. However, according to the criterion presented for the $y^+$ vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ relation, the derived helium abundance for these icf's would be effectively consistent with no positive gradient.\nThe mean total helium abundance ranges between 0.086 and 0.091, depending on the assumed icf.  This range is larger by a factor $\\sim$2 than  the uncertainties due to the measurements  (assumed to be traced by  the standard deviation).  This implies that the main source of uncertainty is still in the assumptions taken on the way to the derivation of the final abundance and highlight how,\neven with a level of data quality as high as those utilized here (i.e. data allowing to determine locally the physical conditions of the gas, the fluxes of several helium lines and estimations of the contribution of underlying stellar populations), achieving an uncertainty $\\lsim1$\\% is extremely difficult.\n\nThe main conclusion of Fig. \\ref{ionidegvsy} is that the relation between the total helium abundance, $y_{tot}$, and the excitation of the gas, as traced by \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\  is consistent with a lack of gradient in  helium abundance.\nHowever, if the extra helium were produced by the Wolf-Rayet stars, the required amount to be detected would be tiny in comparison to the pre-existing helium. This implies that  a positive slope in Fig. \\ref{ionidegvsy} consistent with this enrichment would be indistinguishable of the presented fits.\nExploiting these data to their limits, we can derive the average excess in $N\/He$, by comparing the mean abundances in the N-enriched and non N-enriched areas.\nA map for the $N\/H$ abundance can be derived using the $O\/H$ and $N\/O$ maps presented in \\citetalias{mon12}. Assuming a value of $\\log(N\/O=-1.3)$ as the limit above which the interstellar medium is enriched in nitrogen, we find a mean $N\/H$ of $8.4\\times10^{-6}$ and $17.4\\times10^{-6}$ in the spaxels without and with enrichment. This implies a flux weighted  excess in nitrogen of $N\/H_{exc}=8.9\\times10^{-6}$.\nProceeding in the same manner with the different estimations of the total helium abundance, we find mean $He\/H$ values ranging between $8.49\\times10^{-2}$ and $9.05\\times10^{-2}$ for the non-enriched spaxels and between $8.68\\times10^{-2}$ and $9.14\\times10^{-2}$, depending on the assumed icf(He). Differences in helium abundances between the enriched and non-enriched zones are $\\sim0.9-1.9\\times10^{-3}$. The lowest value was derived using our largest icf(He) (i.e. Eq. \\ref{eqm13}) which, by construction, was defined to minimize any helium abundance gradient. The largest value compares well with the stellar atmospheric $N\/He$ ratios for an N-type Wolf-Rayet star \\citep[e.g.][]{smi82}. However, it is also at the limit of the uncertainties ($\\sim2\\times10^{-3}$assumed to be traced by the standard deviation for the spaxels under consideration in each separate group). \nTherefore, these data appear to be marginally in accord with the hypothesis of a putative enrichment of helium due to the Wolf-Rayet population in the main H\\,\\textsc{ii}\\ region, but more importantly, they stress the difficulties in pushing this methodology further in order to  confirm this contamination without doubt.\n\n\\section{Discussion}\n\n\\subsection{Kinematics and radiative transfer effects}\n\nThe  standard approach in the literature to estimate radiative transfer effects assumes\na negligible influence of the movements of the gas in the H\\,\\textsc{ii}\\ region. i.e. a static nebula \\citep[$\\omega=v\/v_{th}=0$, in the formalism presented by ][]{rob68}. This is the approach utilized in Sec. \\ref{secymas}. \nThe general use of this assumption is motivated by the difficulty of obtaining data of sufficient  quality to allow, on top of measuring the flux of several helium lines with good signal to noise, tracing their profiles and clearly identify the different kinematic components in a consistent manner in all of them.\nHowever, it is not unusual for starburst galaxies or H\\,\\textsc{ii}\\ regions to present velocity gradients and\/or relatively high velocity dispersion that can be attributed to outflows (or expanding structures).\nIn particular, the kinematic study presented in \\citetalias{mon10} and  \\citetalias{wes13} showed that movements in this H\\,\\textsc{ii}\\ region are significant and can indeed be attributed to an outflow caused by the two embedded Super Star Clusters. Specifically, in  \\citetalias{mon10}, we detected a relatively static (i.e. small velocity gradient) narrow component on top of a much broader component ($\\sigma\\sim20-25$ km s$^{-1}$) with a velocity gradient of $\\Delta v\\sim70$~km~s$^{-1}$. An additional component was also detected with the higher resolution GMOS data  \\citepalias{wes13}. Therefore, these data provide a very good opportunity to explore what is the impact  of the kinematics in the derivation of the optical depths from an empirical point of view.  \n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg14a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg14b.ps}\\\\\n   \\caption[$\\tau(3889)$ for the two kinematic components]{ $\\tau(3889)$ for the two main kinematic components presented in Fig. 19 of \\citetalias{mon10} derived from the fits to relations presented by \\citet{rob68} for $T_e$=10\\,000~K. \\emph{Left:} Narrow kinematic component and $\\omega=0$.  \\emph{Right:} Broad kinematic component and $\\omega=3$. The $\\tau$(3889) map for this component using the relation for $\\omega=0$ (not shown) displays the same structure but with values $\\sim4-6$ times smaller.}\n \\label{taucompos}\n \\end{figure}\n\nSince this is only an exploratory analysis and a consistent approach based on the identification and modeling of multiple kinematic components in \\emph{several} emission lines would be more complex (and\nnot supported by the signal-to-noise of all the He\\,\\textsc{i}\\ data) than the one presented in Sec. \\ref{secymas}, we opt for a simpler analysis based on the  He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065 lines. These were the pair of lines utilized to estimate $\\tau(3889)$ in the \\emph{red set}. Also both were observed with the same instrumental set-up, which was the same as for H$\\alpha$\\ in the multicomponent analysis presented in  \\citetalias{mon10}. Therefore, we could link on a spaxel-by-spaxel basis the multicomponent analysis utilized in \\citetalias{mon10} to similar components in  He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065.\nThe contribution to the total flux from each component varies from spaxel to spaxel as well as with the line in consideration. Specifically, the broad component presents a larger contribution  (i.e. $\\sim35-60$\\%) to He\\,\\textsc{i}$\\lambda$7065 than to He\\,\\textsc{i}$\\lambda$6678 ($\\sim30-50$\\%). This implies different correction factors, $f_\\tau(7065)$, for the narrow and broad component that can be converted into optical depths as in Sec. \\ref{secymas}.\nGiven the velocity gradient and the velocity dispersion measured for the broad component, values for $\\omega=v\/v_{th}=3$ are more appropriate for this component. Fitting the same functional form as in Eqs. \\ref{eqfac7065}-\\ref{eqfac3889} to the values reported in \\citet{rob68} for  $\\omega=v\/v_{th}=3$ and $T_e=10\\,000$~K, we derived the following relation:\n\n\\begin{equation}\nf_\\tau(7065)_{\\omega=3} = 1 + 0.279\\tau^{0.521}\n\\end{equation}\n\nThe structure of the derived optical depth for both the narrow and the broad components is shown in Fig. \\ref{taucompos} and several conclusions can be drawn from this figure. Firstly, a comparison between the left panel of this figure with those presented in Fig. \\ref{maptau} highlights the similarity between the optical depth maps derived for the narrow component and for the line integrated analysis. To our knowledge, this is the first empirical evidence supporting the traditional assumption that movements in extragalactic H\\,\\textsc{ii}\\ regions with  a known outflow have a negligible effect in the estimation of the global optical depth. \n\nSecondly, a comparison between the two maps in Fig. \\ref{taucompos}  shows how the \\emph{structure} of  the two $\\tau(3889)$ is different: while the narrow component $\\tau(3889)$ displays a clear peak associated with the peak of emission for the GH\\,\\textsc{ii}R\\ and decreases outwards, the broad component  remains relatively high over a large area of $\\sim4^{\\prime\\prime}\\times2^{\\prime\\prime}$ ($\\sim$74~pc$\\times$37~pc) centered on the double cluster.\n\nThirdly, the comparison between the specific values displayed in these two graphics shows that the broad component suffers from radiative transfer effects to a larger degree than the narrow one. This is a consequence of the adopted relation between $f_\\tau(7065)$ and $\\tau(3889)$, which is motivated by our kinematic results. If we had used Eq. \\ref{eqfac7065} (i.e. assuming $\\omega=0$ for the \\emph{broad} component), we would have obtained comparable values for the $\\tau(3889)$ in both components (although, still, with a different structure since this is determined by the $f_\\tau(7065)$ itself, which in turns depends on the relative flux between He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065).\n\nThere is certainly room for improvement in this analysis. For example, although the multicomponent analysis treats in a consistent way the line profiles of He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065, the same physical conditions (i.e. $T_e$, $n_e$, extinction) had to be assumed for both components. Moreover, we used here only two helium lines, in contrast with the more canonical methodology for determining helium abundances based in a larger set of helium lines. These improvements suggest a new approach to the study of the helium content in galaxies with IFS. Specifically,  the 2D analysis presented in the previous sections together with the separate analysis of different kinematic components along the line of sight (i.e. in a given spaxel) constitute a 3D view of the radiative transfer effects in the nebula.\n\n\\subsection{Relation between collisional and radiative transfer effects and other properties of the ionized gas} \n\nWe have evaluated  locally both collisional and radiative transfer effects in the helium lines, as well as several important physical conditions ($n_e$, $T_e$, excitation) and chemical properties (i.e. relative abundance $N\/O$) of the ionized gas. Therefore, the central part of \\object{NGC~5253} constitutes a good case study to explore whether there is a (strong) spatial relation of collisional and\/or radiative transfer effects with other gas properties.\n\nAs discussed in Sec. \\ref{seccoli}, theory predicts a strong dependence of the collisional effects on the electron density and, to a lesser extent, on the electron temperature. This implies a strong spatial correlation with the these properties. Indeed, when all the spaxels in the area under study are considered, the relations between $C\/R(\\lambda7065)$ vs.  $n_e$ and $T_e$ have Pearson's correlation coefficients of 0.99 and 0.71, respectively. Regarding the other two properties,  Fig. \\ref{c2rvscosas} presents the relation between the collisional effects (as traced by $C\/R$(7065)) and the excitation (as traced by \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$) and the relative abundance of nitrogen: $N\/O$.  The excitation presents a degree of correlation similar to  $n_e$. The correlation for $N\/O$ is not as strong as that for $n_e$ but still comparable to that for $T_e$.\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg15a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg15b.ps}\\\\\n   \\caption[C\/R ratios]{Relation between collisional effects as traced by the C\/R ratio for the $\\lambda$7065 line and the excitation (\\emph{left}) and relative abundance of nitrogen, N\/O (\\emph{right}). The Pearson's correlation coefficients are indicated in the corners of the individual plots.\n \\label{c2rvscosas}}\n \\end{figure}\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16d.ps}\\\\\n   \\caption[$\\tau(3889)$ vs. physical and chemical properties]{Relation between the self-absorption effects as traced by $\\tau(3889)$  and different physical and chemical properties in \\object{NGC~5253}.  From left to right and from top to bottom, these are electron density and temperature, excitation and relative abundance of nitrogen, N\/O.  The correlation coefficients  are indicated in the corners of the individual plots.\n \\label{tauvscosas}}\n \\end{figure}\n\nLikewise, Fig. \\ref{tauvscosas} shows the relation between radiative effects, as traced by $\\tau$(3889) and different properties of the gas. As expected, there is a good correlation with electron density. For electron temperatures lower than $\\sim$10\\,700~K, He\\,\\textsc{i}\\ emission lines do not suffer from radiative transfer effects in a significant way. However, above this threshold there is a clear correlation between the temperature and the relevance of the radiative transfer effects. \n\nMore interestingly, as happened with the collisional effects, there is a good correlation between the excitation and the relevance of radiative transfer effects (lower left panel in Fig. \\ref{tauvscosas}).  It is not clear why the excitation should correlate with the contribution of  collisional effects (mainly) or the radiative transfer effects (to a lesser extent). \n\nSomething similar occurs with the relative abundance, $N\/O$. At the typical $\\log(N\/O)$ values for this galaxy ($\\lsim-1.3$), He\\,\\textsc{i}\\ radiative transfer effects appear to be negligible. However, those locations displaying extra nitrogen appear to show significant radiative transfer effects. Rather than a discernible correlation between individual spaxels, there appears a correlation between the upper limit  to the contribution of the radiative transfer effects and the amount of extra nitrogen. To our knowledge, there is no reason to link these two properties from the theoretical point of view. The fact that both the relations with $T_e$ and $N\/O$ present bimodal behavior (i.e. with \"turning points\"), and that there is a similar structure for the electron temperature and relative abundance \\citepalias{mon12}, implying a good correlation (i.e. $r=0.91$) between $T_e$ and $N\/O$, points towards the local electron temperature as a common cause.\n\n\\section{Conclusions}\n\nThis is the fourth in a series of articles that make use of IFS-based data to study in detail the 2D physical and chemical properties of the gas in the main GH\\,\\textsc{ii}R\\ of the nearby BCD \\object{NGC~5253}. The main goal of this article was to estimate the contribution of the collisional and radiative transfer effects on the helium emission lines and to map, in turn, the helium abundance.\nThe major conclusions can be summarized as follows:\n\n   \\begin{enumerate}\n      \\item The collisional effects on the different helium transitions have been mapped for the first time in an extragalactic object. As expected, they reproduce the electron density structure. They are negligible (i.e. $\\sim$0.1-0.6\\%) for transitions in the singlet cascade while relatively important  for those transitions in the triplet cascade. In particular, they can contribute up to 20\\% of the flux in the He\\,\\textsc{i}$\\lambda$7065 line.\n      \n      \\item The contribution of the collisional effects is sensitive to the assumed $T_e$ for helium. Specifically, we found  differences $\\sim25-35$\\% for the $\\lambda$3889 and $\\lambda$7065 lines two reasonable assumptions for the $T_e$ sensitivity. Relative differences for the other lines were larger. However, this does not have important consequences  since the contribution of the collisional effect to the observed spectrum for these lines is negligible.\n\n      \\item We present a map for the optical depth at $\\lambda$3889 in the main GH\\,\\textsc{ii}R\\ of \\object{NGC~5253}. $\\tau(3889)$ is elevated over an extended and circular area of $\\sim$30~pc in diameter, centered at the Super Star Cluster(s), where it reaches its maximum. \n\n      \\item The singly ionized helium abundance, $y^+$, has been mapped using extinction corrected fluxes of six He\\,\\textsc{i}\\ lines, realistic assumptions for $T_e$, $n_e$, and the stellar absorption equivalent width as well as the most recent emissivities. We found a mean($\\pm$ standard deviation of $10^3 y^+ \\sim80.3(\\pm2.7)$ over the mapped area.\n      \n      \\item We derived total helium abundance maps using three possible icf(He)'s. \nThe relation between the excitation and the total helium abundances is consistent with no abundance gradient.\nDifferences between the derived total abundances according to the three methods are \\emph{larger} than statistical errors associated with the data themselves, emphasizing how uncertainties in the derivation of helium abundances are dominated by the adopted assumptions.\n      \n      \\item We illustrated the difficulty of detecting a putative helium enrichment due to the presence of Wolf-Rayet stars in the main GH\\,\\textsc{ii}R. This is due to the comparatively large amount of preexisting helium. The data are marginally consistent with an excess in the $N\/He$ ratio in the nitrogen enriched area of the order of the atmospheric $N\/He$ ratios in W-R stars. However, this excess is also of the same order of the uncertainty estimated for the $N\/He$ ratios in the nitrogen enriched and non-enriched areas.\n      \n      \\item We explored the influence of the kinematics in the evaluation of the He\\,\\textsc{i}\\ radiative transfer effects. Our data empirically support  the use of the traditional assumption that motions in an extragalactic H\\,\\textsc{ii}\\ region have a negligible effect in the estimation of the global optical depths. However, individually, the broad kinematic component (associated with an outflow) is affected by radiative transfer effects in a much more significant way than the narrow one.\n      \n      \\item The local relationships between the contribution of collisional and radiative transfer effects to the helium lines and different physical and chemical properties of the gas have been explored. Interestingly, we found a relation between the amount of extra nitrogen and the upper limit of the contribution from radiative transfer effects that requires further investigation. We suggest the electron temperature as perhaps a common agent causing this relation.\n      \n \\end{enumerate}\n\n\\begin{acknowledgements}\n\nWe are very grateful to the referee for the careful and diligent reading of the manuscript as  well as for the useful comments that helped us to clarify and improve the first submitted version of this paper.\nAlso, we thank R. L. Porter for advice on use of his tabulated He\\,\\textsc{i}\\ emissivities and for so promptly informing us of the Corrigendum.\n\nBased on observations carried out at the European Southern\nObservatory, Paranal (Chile), programmes 078.B-0043(A) and\n383.B-0043(A). This paper uses \nthe plotting package \\texttt{jmaplot}, developed by Jes\\'us\nMa\\'{\\i}z-Apell\\'aniz,\n\\texttt{http:\/\/dae45.iaa.csic.es:8080\/$\\sim$jmaiz\/software}. This \nresearch made use of the NASA\/IPAC Extragalactic \nDatabase (NED), which is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under contract with the National Aeronautics and Space\nAdministration.\n\n\nA.~M.-I. is supported by the Spanish Research Council within the program JAE-Doc, Junta para la Ampliaci\\'on de Estudios, co-funded by the FSE.\nA.M-I is also grateful to ESO - Garching, where part of this work was carried out, for their hospitality and funding via their visitor program. \nThis work has been partially funded by the Spanish PNAYA, project AYA2010-21887 of the Spanish MINECO. \nThe research leading to these results has received funding from the European Community's Seventh Framework Programme (\/FP7\/2007-2013\/) under grant agreement No 229517.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\nTime series forecasting is playing a vital role in many application scenarios of a variety of domains, such as solar power generation forecasting in renewable energy \\citep{10,11,mtgnn}, traffic forecasting in transportation system \\citep{traffic1,traffic2},\nelectricity consumption forecasting in social life \\citep{10,11,mtgnn}, market trend prediction in financial investment~\\citep{finance1,finance2}, etc.\nIn many of them, there exist multiple forecasting entities, e.g. power stations in the solar system, stations in the traffic system, stocks in financial market, and commodities in retailing business. \n\n\nA straightforward forecasting solution is to mine temporal dependency for each individual entity by 1d-convolution neural network (1D-CNN) \\citep{tcn-2018}, recurrent neural network (RNN) \\citep{lstm,gru1}, transformer \\citep{attention-is-all-you-need}, and etc., while it will overlook important relations between these entities, such as explicit relation defined by human knowledge (e.g. competitive, cooperative, causal, and geospatial relation) and other implicit relation behind the data.\nIn reality, such relations could provide valuable signals towards accurate forecasting for each individual entity.\nFor example, in traffic system, the station can affect its geospatial adjacent stations;\nthe stock price of upstream companies in the supply-chain can substantially indicate that of downstream ones in the financial market scenario; the similarity relation between entities behind the data can also increase the robustness of model. \nSpatial-temporal graph neural networks \\cite{dcrnn,stmetanet,graphwavenet,mtgnn}, in which the entity relations are fully utilized, are good examples with better forecasting performance.\n\n\nIn many real-world scenarios, however, beside some explicit relations, there usually exist crucial yet implicit relations between entities.\nRecently, a growing number of research works pay attention to graph neural network (GNN) \\citep{gcn,graphsage,gat,protein1,physical-system1} to leverage the implicit relations between entities~\\citep{ddgf,graphwavenet,Gumbel-graph-neural-network-2019,mtgnn}. \nOne common idea among these works is to assume all entities composing a complete graph and let GNNs automatically learn the pairwise correlations between any two entities~\\citep{ddgf,graphwavenet}. \nTo avoid the computation complexity and over-smoothing issues of above-mentioned studies, the operation analogous to sparse encoding~\\citep{mtgnn} has been further proposed.\nNevertheless, an imprudent employment of sparse constraints at the earlier learning phase may limit the model capacity of discovering crucial relation, while the uni-directional relation is not suitable for every real scenario.\n\n\nMoreover, there also exist critical challenges regarding the utilization of multiple relations, one of which is the information aggregation through different types of relations, especially the co-existence of explicit relation as well as implicit relation.\nA straightforward method is to employ different graph convolution blocks for different relation types and then conduct a direct fusion \\citep{graphwavenet}. Some others conduct fusion based on correlation between nodes by leveraging the meta information of both nodes and edges \\citep{stmetanet}. In addition, Graph multi-attention network (GMAN) \\citep{gman} employs a gated fusion mechanism to fuse the spatial and temporal representations. \nHowever, all of these efforts rely on static fusion of multiple relations reflecting either pre-existed or latent connections. Indeed, given the distinct characteristics of multiple relations, accurate forecasting usually relies on dynamic reliance on them under various circumstances.\n\n\n\n\\begin{figure}[!htbp]\n\\centerline{\\includegraphics[width=1\\linewidth]{photo\/demo_show_case_of_a2gnn.png}}\n\\caption{Diagrammatic representation of the relationships between entities.}\n\\label{fig:Diagrammatic representation}\n\\end{figure}\n\nThus, the problem lies in two crucial aspects, i.e., implicit relation {\\em discover} and relations' effective {\\em utilization}.\nWe take the $6$ entities in Fig. \\ref{fig:Diagrammatic representation} to illustrate the importance of them.\nIn Fig. \\ref{fig:Diagrammatic representation}, entity-(a), entity-(b), and entity-(c) have a similar curve, thus, the similarity can be employed to enhance the robustness of the model.\nAlthough there does not exist a similar curve for entity-(e), it is not hard to see that when entity-(c) or entity-(d) is in lowest peak, entity-(e) is also in lowest peak. Thus, entity-(e) can use this relation for better forecasting.\nThe periodicity of entity-(f) is strong so that only employing its own information can also get an accurate forecasting result (i.e. the explicit and implicit relation may be not necessary for entity-(f)). \n\n\nIn this paper, in order to mine the implicit relation between entities as much as possible and effectively utilize the relations to improve the forecasting performance, we propose an attentional multi-graph neural network with automatic graph learning (A2GNN).\nIn particular, we propose a novel auto graph learner based on Gumbel-Softmax~\\citep{gumbelsoftmax} to sample all feasible entity pairs so that the relation between any entity pair has the chance to be reserved in the learned graph, while we leverage the sparse matrix to ensure the computing efficiency. Moreover, we propose an attentional relation learner so that every entity can dynamically pay attention to useful relations, resulting in more flexible utilization of multiple relations and consequently better forecasting performance.\n\n\n\n\nThe main contributions of this paper include:\n\\begin{itemize}\n\\item We propose a new graph neural network framework, A2GNN, with automatic discovery and dynamic utilization of relations for time series forecasting.\n\\item Within A2GNN, a novel auto graph learner based on Gumbel-Softmax~\\citep{gumbelsoftmax} can effectively discover the implicit relation between entities while significantly reducing complexity.\n\\item Within A2GNN, an attentional relation learner enables every entity dynamically pay more attention to their preferable relations, resulting in more flexible utilization of multiple relations.\n\\item \nOur proposed A2GNN is quite general such that it can be applied to both time series and spatial-temporal forecasting tasks.\nAnd, extensive experiments have shown that our method outperforms the state-of-the-art methods on a couple of well-known benchmark datasets.\n\\end{itemize}\n\n\n\n\\section{RELATED WORK}\n\\label{related work}\nOur work is related to three lines of research: time series forecasting methods, spatial-temporal graph neural networks, and graph neural networks sparsification methods.\n\n\\subsection{Time Series Forecasting Methods}\n\nThere are plenty of works on time series forecasting problem \\citep{5,7,8,gru1,gru2,lstm}. Recently, long and short-term time-series network (LSTNet) \\citep{10} utilizes the convolution neural network (CNN) \\citep{cnn} and recurrent neural network (RNN) to extract short-term local dependency and long-term patterns for time series trends. Shih proposes a temporal pattern attention (TPA-LSTM)\\citep{11} to select relevant time series, and leverages its frequency domain information for multivariate forecasting. A Lot of studies \\citep{unfoldingtemporaldynamics-2016-aaai,restful-wu-2018,dsanet-huang-2019} also focus on multi-scale temporal information extraction. Existing methods focus more on time series information utilization for forecasting, but they neglect the implicit relation among entities. \n\n\n\n\n\\subsection{Spatial-Temporal Graph Neural Networks}\n\nA spatial-temporal forecasting task has pre-defined relation (i.e. explicit relation) by human knowledge, and traffic forecasting is a typical and hot problem of spatial-temporal forecasting as the natural geographic relation. Similar like time series task, a lot of works focus on temporal information mining, such as deep spatio-temporal residual networks (ST-ResNet) \\citep{stresnet}, spatio-temporal graph convolutional network (STGCN) \\citep{stgcn}, Graph-WaveNet \\citep{graphwavenet,wavenet}. Some works focus on natural geographic relation utilization, such as diffusion convolutional recurrent neural network (DCRNN) \\citep{dcrnn}, spatial-temporal forecasting with meta knowledge (ST-MetaNet) \\citep{stmetanet}, graph multi-attention network (GMAN) \\citep{gman}, multi-range attentive bicomponent GCN (MRA-BGCN) \\citep{mrabgcn}, spatial temporal graph neural network (STGNN) \\citep{STGNN},  Spectral temporal graph neural network (StemGNN) \\citep{StemGNN}, and etc. \n\n\n\n\n\n\n\nImplicit relation discover aroused researcher's attention. Graph convolutional neural network with data driven graph filter (DDGF) \\citep{ddgf} breaks this limitation and discovers implicit relation to replace the pre-defined relation, by calculating all pairwise correlations between nodes. However, the risk of over smoothing and computation complexity increase when taking all nodes as neighbors. \nGumbel Graph Network (GGN) \\citep{Gumbel-graph-neural-network-2019}  proposes a model-free, data-driven deep learning framework to accomplish the reconstruction of network connections, while the one time sampler will curb the efficiency of connection reconstruction.\nMultivariate time graph neural network (MTGNN)~\\citep{mtgnn} accelerates the computation efficiency in traffic forecasting by employing a sparse uni-directional graph to learn hidden spatial dependencies among variables. Nevertheless, a imprudent employment of sparse constraints at the earlier learning phase may limit the model capacity of discovering crucial relation and the assumption of uni-direction limits its application in many real scenarios. \n\n\n\n\\subsection{Graph Neural Networks Sparsification Methods}\nGraph sparsification aims at finding small subgraphs from given implicit large graphs that\nbest preserve desired properties. \nFor instance, Fast learning with graph neural networks (FastGCN) \\citep{fastgcn-2018-ICLR}  interpret graph convolutions as integral transforms of embedding functions under probability measures and uses Monte Carlo approaches to consistently estimate the integrals.\nNeuralSparse \\citep{graphsparse-2020-ICML} considers node\/edge features as parts of input and optimizes graph sparsification by supervision signals from errors made in downstream tasks.\n\n\n\\begin{figure*}[ht!]\n\\centerline{\n\\includegraphics[width=0.75\\linewidth]{photo\/framework_of_A2GNN.png}\n}\n\\caption{The framework of A2GNN. AGL is the auto graph learner for implicit graph discover. ARL is the attentional relation learner that enables each entity dynamically pay attention to different relations. $\\mathbf{m}_{v_i}$ is the learnable node embedding of node $v_i$. $\\mathbf{x}_{v_i}$ is historical observed features. \n$\\mathbf{h}^1_{v_i}$, $\\mathbf{h}^2_{v_i}$, and $\\mathbf{h}^3_{v_i}$ mean own temporal information, neighbor information by implicit relation, and neighbor information by explicit relation, respectively. $p_{v_i,1}$, $p_{v_i,2}$, and $p_{v_i,3}$ are  dynamic weight given by ARL.}\n\\label{fig:framework of A2GNN}\n\\end{figure*}\n\n\\section{PROBLEM FORMULATION}\nIn this section, we first provide a detailed problem formulation. Suppose there are $N$ forecasting entities, we use $x_t\\in R^{N\\times d}$ to stand for the values of these $N$ entities at time step $t$, and $d$ is the feature dimension. $y_{t}\\in R^{N\\times 1}$ is the feature\/variable need to forecast. The historical observations of past $t_{in}$ steps before $t$ is defined as $X=\\{x_{t-t_{in}+1},x_{t-t_{in}+2},\\cdots,x_t \\}$.\nOur goal is to build a function to predict a sequence of values $Y=\\{y_{t+1},y_{t+2},... ,y_{t+t_{out}}\\}$ for future $t_{out}$ steps.\n\nWe give formal definitions of graph-related concepts:\n\n\\textbf{\\textit{Graph: }}\nA graph is formulated as $G=(V,E)$ where $V$ is the set of nodes, and $E$ is the set of edges. We use $N=|V|$ to denote the number of nodes in the graph. In particular, in forecasting scenario, we consider each forecasting entity as a node in the graph.\n\n\\textbf{\\textit{Node neighbors: }}\nFor any node $v_i,v_j\\in V$, $(v_i,v_j)\\in E$ denote an edge between $v_i$ and $v_j$. The neighbors of $v_i$ are defined as $N_{v_i}=\\{v_k ~|~ (v_k,v_i)\\in E \\land v_k\\in V \\}$.\n\n\\textbf{\\textit{Weighted adjacency matrix: }}\nThe weighted adjacency matrix is a mathematical representation of the relation. We denote the weighted adjacency matrix as $A\\in R^{N\\times N}$, where $A_{i,j}>0$ if there is an edge between the node $v_i$ and node $v_j$, otherwise, $A_{ij}=0$. The weight $A_{i,j}$ stands for some metrics of relation between these two nodes. \n\n\n\n\\section{Proposed Framework}\nIn this section, we introduce the proposed attentional multi-graph neural network with automatic graph learning (A2GNN). \nAs shown in Figure~\\ref{fig:framework of A2GNN}, the whole framework of A2GNN consists of 5 parts: temporal encoder, auto graph learner (AGL), graph neural network (GNN), and attentional relation learner (ARL). The key parts of this study are AGL and ARL. In the end, we employ the multi-layer perception (MLP) for efficient inference instead of RNN.\n\nThe algorithm is shown in Alg. \\ref{alg:algorithm}. \nIt can be easily applied both spatial-temporal and time series forecasting tasks, and the difference between them is whether exist pre-defined relation. For better understand A2GNN, we make a brief introduction about its temporal encoder and a detailed introduction about the key parts: AGL and ARL.\n\n\n\n\n\\newcommand{\\parag}[1]{\\vspace{1mm}\\textit{#1}:\\ }\n\\parag{Temporal encoder} 1D-CNN~\\citep{tcn-2018,cnn}, RNN~\\citep{gru1,gru2,lstm}, and transformer~\\citep{attention-is-all-you-need}, can be employed to extract the temporal information for each node.\nThe representation will be used as input for the further graph neural network. \nIn this paper, we use LSTM to extract temporal information.\n\n\n\\begin{algorithm}[ht]\n\\caption{A2GNN}\n\\label{alg:algorithm}\n\\textbf{Input}: Node temporal information $X\\in R^{N\\times T_{in} \\times D}$; Pre-defined graph $A^p \\in R^{N\\times N}$(Optional).\\\\\n\\textbf{Parameter}: Node embedding $M\\in R ^ {N \\times D_m}$; Random initialized adjacent matrix $A\\in R^{N \\times N}$; Parameter of graph neural network; Parameter of auto relation learner.\\\\\n\\textbf{Output}: $\\hat{Y}\\in R^{N\\times T_{out} \\times D}$\n\\begin{algorithmic}[1]\n\\STATE{\/* Discover implicit relation *\/}\n\\IF {training} \n\\STATE $A^* = \\textsc{AGL}_{training}(A)$\n\\ELSIF{inference}\n\\STATE $A^* = \\textsc{AGL}_{inference}(A)$\n\\ENDIF\n\\STATE Extract temporal information: $S=\\textsc{LSTM}(X)$\n\\STATE Extract node own information: $H^1=\\textsc{MLP}(S)$\n\\STATE Aggregate implicit neighbor: $H^2=\\textsc{GNN}(S,A^*)$\n\\STATE Aggregate pre-defined neighbor: $H^3=\\textsc{GNN}(S,A^p)$\n\\STATE Utilize multiple relations: $Z=\\textsc{ARL}(M,H^1,H^2,H^3)$\n\\STATE \/* efficient inference *\/\n\\STATE $Y=ZW$\n\\STATE \\textbf{return} $\\hat{Y}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Auto Graph Learner} \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{photo\/auto_graph_learner_process.png}\n    \\caption{The process of auto graph learner in training. }\n    \\label{fig:auto graph learner process}\n\\end{figure}\n\n\nAuto graph learner (AGL) kicks off the learning for implicit relation from a randomly initiated adjacency matrix. Towards minimizing the over smoothing risk, we use sample-based method to select potential edges. Moreover, multiple edges of each node are sampled for an efficient training.\nHowever, the vanilla sample operation is not differentiable, which hinders the back propagation for edges' weight optimization. To address this issue, we employ Gumbel-Softmax \\citep{gumbelsoftmax} method to make a differentiable sampling. There are some differences in training and inference phases and the detailed procedures are shown in below.\n\n\n\\parag{Training} Given the randomly initiated adjacency matrix $A\\in R^{N\\times N}$, the calculation steps include: \n\n1. $\\forall v_j\\in N_{v_i}$, we employ a softmax function to compute the sampling probability of each edge:\n\\begin{equation}\n    \\pi_{v_i,v_j}=\\frac{\\exp(a_{v_i,v_j})} {\\sum_{v_k\\in N_{v_i}} \\exp(a_{v_i,v_k})},\n    \\label{eqn:matrix softmax}\n\\end{equation}\nwhere $\\pi_{v_i,v_j}$ is the weight of edge $(v_i,v_j)$ after softmax, and $a_{v_i,v_j}$ is the weight of edge $(v_i,v_j)$ in adjacency matrix $A$. \n\n2. We then generate differentiable samples through Gumbel-Softmax:\n\\begin{equation}\n\t\t\t\\pi_{v_i,v_j}^{G}=\\frac{\\exp( (\\log(\\pi_{v_i,v_j})+\\epsilon_{v_j})\/\\tau )}   {\\sum_{v_k\\in N_{v_i}} \\exp( (\\log(\\pi_{v_i,v_k})+\\epsilon_{v_k})\/\\tau )},\n\\label{eqn:gumbel softmax}\n\\end{equation}\nwhere $\\pi_{v_i,v_j}^{G}$ is the weight of edge $(v_i,v_j)$ after Gumbel-Softmax.  $\\epsilon=-\\log(-\\log(s))$, with randomly generated $s$ from Uniform distribution U$(0,1)$, and $\\tau \\in {(0,+\\infty)}$ is a hyper-parameter called temperature. As the softmax temperature $\\tau$ approaches $0$, samples from the Gumbel-Softmax distribution approximate one-hot vector, which means discrete.\n\n\n3. Repeating above procedure $C$ times, we can obtain $C$ samples. Finally, we apply normalization over all samples to get the final correlations for all nodes:\n\\begin{equation}\n\t\t\ta^*_{v_i,v_j}= \\frac{\\sum_{c=1}^{C} \\pi ^{G,c}_{v_i,v_j} }   {\\sum_{c=1}^{C}\\sum_{v_k\\in N_{v_i}} \\pi^{G,c}_{v_i,v_k}},\n\\label{eqn:training rate}\n\\end{equation}\nwhere $a^*_{v_i,v_j}$ is the weight of edge $(v_i,v_j) $and will be further used in the calculation procedures in the graph neural network. $\\pi ^{G,c}$ is the $c$-th Gumbel-Softmax result.\n\n\n\\parag{Inference} Since the auto graph learner has already learned the nodes' relation by updating matrix $A$ in the training phase, we replace Gumbel-Softmax sample operation by selecting the top related $C$ neighbors $\\textsc{top}_C (N_{v_i})$ for each node $v_i$ (the edge weight is used as the metric for $\\textsc{top}$ operation). Then softmax is applied to normalize these edge weights:\n\\begin{equation}\n    a^*_{v_i,v_j}=\\frac{\\exp(a_{v_i,v_j})} {\\sum_{v_k\\in \\textsc{top}_C (N_{v_i})} \\exp(a_{v_i,v_k})}, ~~v_j\\in \\textsc{top}_C (N_{v_i})\n    \\label{eqn:evaluation matrix softmax}\n\\end{equation}\n\nThe $C$ times sampling operation ensures the rationality and efficacy of relation learning process.\nAt the begin of training, all the edges have the probability to be sampled, and the sampled edge's weight can be properly updated.\nAs time goes on, the useful edges will have larger weights through the updating with back propagation algorithm.\n\n\nThe output of AGL is $A^*\\in R^{N\\times N}$, and the element of which is $a^*_{v_i,v_j}$.\n\n\n\\subsection{Graph Neural Network}\nThe node's time series information is processed by LSTM before input into the graph neural network. Specifically, the hidden state from LSTM in each time step will be concatenated as the output. The output from LSTM block (i.e. temporal encoder) is defined as $S=\\textsc{LSTM}(X)$, and $S\\in R^{N\\times d_{enc}}$. \n\n\n\nWith the assistance of the output from LSTM, the node itself information is processed by multi-layer perceptron (MLP) like $H^1=\\textsc{MLP}(S)$.\nThe neighbor information by implicit relation is aggregated by graph neural network like \n$H^2=A^* \\cdots (A^* S W^{(1)}_2) \\cdots W^{(l)}_2$, and $l$ is the layer depth, $W$ is learnable parameter for feature transportation. For some scenarios, such as traffic, there always exists natural geographic relation. Thus, the pre-defined natural relation can also be employ by graph neural network like $H^3=A^p \\cdots (A^p S W^{(1)}_3) \\cdots W^{(l)}_3$, and $A^p$ us the pre-dedined weighted adjacency matrix.\n\nThus, after the graph neural network, for each node $v_i$, the own information $h^1_{v_i}$, the aggregated information based on implicit relation $h^2_{v_i}$, and the aggregated information based on explicit relation $h^3_{v_i}$ are available from $H^1$, $H^2$, and $H^3$, respectively.\n\n\n\n\\subsection{Attentional Relation Learner}\n\nThe representations $H^1$, $H^2$, and $H^3$ are fed into the attentional relation learner (ARL).  Unlike previous studies that always rely on static fusion of multiple relations (e.g. explicit or implicit relation), AGL enables each node dynamically rely on different relations under various circumstances. Furthermore, node itself information, which can be called own relation, is maintained through $H^1$ instead of skip connection. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{photo\/attentional_relation_learner_process.png}\n    \\caption{The process of attentional relation learner.}\n    \\label{fig:attentional relation learner process}\n\\end{figure}\n\n\nAs show in Fig. \\ref{fig:attentional relation learner process}, in ARL, each node has a random initialized embedding vector, that is $M\\in R^{N\\times d_m}$. \nEach node learns its attention coefficient on multiple representations, that is $H^1$, $H^2$, and $H^3$. In particular, in Eq. \\eqref{eqn:attentional relation coefficient}, the dot production and $\\textsc{Softmax}$ operation are employed to calculate the attention coefficient.\n\n\n\n\n\\begin{equation}\n    p_{v_i,k\\in \\{1,2,3\\}} =\\textsc{Softmax}_k\\left( \\frac{(m_{v_i}W_{query})\\cdot(h^k_{v_i}W_{key}) }{\\sqrt{d}}\\right)\n    \\label{eqn:attentional relation coefficient}\n\\end{equation}\nwhere $p_{v_i,k\\in \\{1,2,3\\}}$ is the attention coefficient scalar of node $v_i$ on the representation $h^k_{v_i}$. $m_{v_i}$ is node $v_i$'s embedding from $M$. $h^k_{v_i}$ is node $v_i$'s representation from corresponding $H^k$. $W_{query} \\in R^{d_m \\times d}$ and $W_k \\in R^{d_h \\times d}$ are learnable parameters for feature transportation. $d_m$ is the dimension of $m_{v_i}$, $d_h$ is the dimension of $h^k_{v_i}$, $d$ is the output dimension of $W_{query}$ and $W_{key}$. $\\cdot$ means dot production.\n\n\n\n\n\n\nAfter that, the attention coefficient scalar is employed to merge multiple representations for each node $v_i$ as:\n\\begin{equation}\n\t\t\tz_{v_i} = \\textsc{Concat}_{k\\in \\{1,2,3\\}}\\left(p_{v_i}(h^k_{v_i}W_{value}) \\right)\n\\label{eqn: attentional relation learner attention concat}\n\\end{equation}\nwhere $W_{value}$ is the learnable parameter. \n\n\n\\subsection{Optimizaiton}\nFor each node $v_i$, final forecasting module will use the $z_{v_i}$ to predict, that is $\\hat{y}_{v_i} = z_{v_i}W$. The Root Mean Squared Error (RMSE) is employed as the loss function.\n\n\n\n\n\\section{Experiments}\nWe evaluate the proposed A2GNN framework in $5$ datasets: Solar-energy, Traffic, Electricity, METR-LA, and PEMS-BAY.\nParticularly, the first $3$ datasets are used to prove that A2GNN can be employed to model time series forecasting problem, and they have no pre-defined relation. And then, we make further experiments on $2$ well known spatial-temporal traffic datasets compared with state-of-the-art graph neural network method, and these datasets have pre-defined relation (i.e. natural geographic relation).  Details of these datasets are introduced in appendix.\n\n\n\n\nTo  evaluate  the  model  performances,  we adopt five metrics, which are Mean Absolute Error (MAE), Root Mean  Squared  Error  (RMSE),  Mean  Absolute  Percentage  Error(MAPE), Relative Squared Error (RSE) and Empirical Correlation Coefficient (CORR). The mathematical formulas are shown in appendix.\nAll the experiments $5$ times and report the average score in order to remove the influence of randomness (e.g. instability of Gumbel-Softmax operation and randomness of model initialized parameters). More settings of our experiments are shown in appendix.\n\n\n\n\\subsection{Baseline Methods for Comparison}\nAs we mentioned above, the biggest difference between time series forecasting task and spatial-temporal forecasting task lies in whether there exists a pre-defined relation. \nAll the methods mentioned in Section Related Work are concluded. The details of these baselines are shown in the appendix.\n\n\n\n\n\n\n\\subsection{Result Comparison}\n\\subsubsection{Result Comparison on Time Series Dataset}\nWe compare the performances of the proposed A2GNN model with above-mentioned baseline methods on $3$  well-known time series forecasting datasets, and we want to prove that our method can find the implicit relation and make a better prediction compared with other previous time series forecasting methods.\n\n\n\\begin{table}[ht!]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\n\\caption{Experiments on time series forecasting datasets.}\n\\begin{center}\n\\scalebox{0.66}{\n\\begin{tabular}{cc|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{2}{c|}{Dataset} & \\multicolumn{3}{c|}{Solar-Energy} & \\multicolumn{3}{c|}{Traffic} & \\multicolumn{3}{c}{Electricity} \\\\ \\hline\n\n\\multicolumn{2}{c|}{ } & \\multicolumn{3}{c|}{ Horizon} & \\multicolumn{3}{c|}{ Horizon} & \\multicolumn{3}{c}{ Horizon} \\\\ \\hline\n\n\n\\hline\nMethods & Metrics & 6 & 12 & 24  & 6 & 12 & 24 & 6 & 12 & 24\\\\ \\hline\nAR & RSE$\\downarrow$  & 0.379 & 0.591 & 0.869  & 0.621 & 0.625 & 0.630  & 0.103 & 0.105 & 0.105\\\\\n & CORR$\\uparrow$  & 0.926 & 0.810 & 0.531  & 0.756 & 0.754 & 0.751  & 0.863 & 0.859 & 0.859\\\\ \\hline\nVAR-MLP & RSE$\\downarrow$  & 0.267 & 0.424 & 0.684  & 0.657 & 0.602 & 0.614  & 0.162 & 0.155 & 0.127\\\\\n & CORR$\\uparrow$  & 0.965 & 0.905 & 0.714  & 0.769 & 0.792 & 0.789 & 0.838 & 0.819 & 0.867\\\\ \\hline\nGP & RSE$\\downarrow$  & 0.328 & 0.520 & 0.797  & 0.677 & 0.640 & 0.599  & 0.190 & 0.162 & 0.127\\\\\n & CORR$\\uparrow$  & 0.944 & 0.851 & 0.597 & 0.740 & 0.767 & 0.790  & 0.833 & 0.839 & 0.881\\\\ \\hline\nRNN-GRU & RSE$\\downarrow$  & 0.262 & 0.416 & 0.485  & 0.552 & 0.556 & 0.563  & 0.114 & 0.118 & 0.129\\\\\n & CORR$\\uparrow$  & 0.967 & 0.915 & 0.882  & 0.840 & 0.834 & 0.830 & 0.862 & 0.847 & 0.865\\\\ \\hline\nLSTNet & RSE$\\downarrow$ & 0.255 & 0.325 & 0.464  & 0.489 & 0.495 & 0.497 & 0.093 & 0.100 & 0.100\\\\\n & CORR$\\uparrow$ & 0.969 & 0.946 & 0.887 & 0.869 & 0.861 & 0.858 & 0.913 & 0.907 & 0.911\\\\ \\hline\nTPA-LSTM  & RSE$\\downarrow$ & 0.234 & 0.323 & 0.438 &  0.465 & 0.464 & 0.476 & 0.091 & 0.096 & 0.100\\\\\n& CORR$\\uparrow$ &  0.974 & 0.948 & 0.908 & 0.871 & 0.871 & 0.862 & 0.933 & 0.925 & 0.913\\\\ \\hline\nMTGNN & RSE$\\downarrow$ & 0.234 & 0.310 & 0.427 & 0.475 & 0.446 & 0.453 & 0.087 & 0.091 & \\textbf{0.095}\\\\\n & CORR$\\uparrow$ &  0.972 & 0.950 & 0.903 & 0.866 & 0.879 & 0.881  & 0.931 & 0.927 & 0.923\\\\ \\hline \\hline\n A2GNN & RSE$\\downarrow$  &\\textbf{ 0.223} &\\textbf{ 0.288} & \\textbf{0.407} & \\textbf{0.427}  &\\textbf{ 0.437 }& \\textbf{0.448} &  \\textbf{0.0858} &\\textbf{ 0.0903} & 0.0970\\\\ \n(ours) & CORR$\\uparrow$ &\\textbf{ 0.976} &\\textbf{ 0.958} &\\textbf{ 0.910}  &\\textbf{ 0.890 }& \\textbf{0.885} & \\textbf{ 0.881}  & \\textbf{0.934 }& \\textbf{0.929} & \\textbf{0.930}\\\\\n\\bottomrule\n\\multicolumn{11}{l}{$\\downarrow$ means lower is better and $\\uparrow$ means higher is better} \\\\\n\\end{tabular}}\n\\vspace{-0.5cm}\n\\label{tab:one step forecasting}\n\\end{center}\n\\end{table}\n\n\nFrom the Table \\ref{tab:one step forecasting}, we can see that A2GNN achieves best performance over almost all time steps on Solar-Energy, Traffic, and Electricity data. In particular, compared to previously state-of-the-art methods, A2GNN can achieve significant improvements in terms of the RSE score on Solar-Energy dataset with $4.6\\%$, $7.3\\%$ and $4.5\\%$ error reduction when $t_{out}$ is set to $6$, $12$ and $24$, respectively. Furthermore, on Traffic dataset, the error reduction in terms of RSE is $10.1\\%$, $1.8\\%$ and $1.0\\%$ correspondingly. \n\n\n\n\n\\begin{table}[ht]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\\caption{Experiments on spatial-temporal forecasting datasets.}\n\\begin{center}\n\\scalebox{0.70}{\n\n\\begin{tabular}{c|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{1}{c|}{Dataset} & \\multicolumn{9}{c}{METR-LA} \\\\  \\hline\n\n\\multicolumn{1}{c}{} & \\multicolumn{3}{|c|}{Horizon 3 } & \\multicolumn{3}{|c|}{Horizon 6} & \\multicolumn{3}{|c}{Horizon 12}\\\\\n\n\\hline\nMethods\/Metrics & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE \\\\ \\hline\nDCRNN & 2.77 & 5.38 & 7.30\\% & 3.15 & 6.45 & 8.80\\% & 3.60 & 7.60 & 10.50\\%   \\\\\nSTGCN & 2.88 & 5.74 & 7.62\\% & 3.47 & 7.24 & 9.57\\% & 4.59 & 9.40 & 12.70\\% \\\\\nGraph-WaveNet & 2.69 & 5.15 & 6.90\\% & 3.07 & 6.22 & 8.37\\% & 3.53 & 7.37 & 10.01\\%\\\\\nST-MetaNet & 2.69 & 5.17 & 6.91\\% & 3.10 & 6.28 & 8.57\\% & 3.59 & 7.52 & 10.63\\%\\\\\nMRA-BGCN & 2.67 & 5.12 & 6.80\\% & 3.06 & 6.17 & 8.30\\% & 3.49 & 7.30 & 10.00\\%\\\\\nGMAN & 2.77 & 5.48 & 7.25\\% & 3.07 & 6.34 & 8.35\\% & 3.40 & 7.21 & 9.72\\%\\\\\nMTGNN & 2.69 & 5.18 & 6.86\\% & 3.05 & 6.17 & 8.19\\% & 3.49 & 7.23 & 9.87\\%\\\\\nA2GNN (ours)& \\textbf{2.63} & \\textbf{4.99} & \\textbf{6.74\\%} & \\textbf{2.95} & \\textbf{5.95} &\\textbf{ 8.02}\\% & \\textbf{3.34} & \\textbf{7.00} & \\textbf{9.65}\\% \\\\ \\hline\nDataset & \\multicolumn{9}{c}{PEMS-BAY} \\\\ \\hline\nDCRNN& 1.38 & 2.95 & 2.90\\% & 1.74 & 3.97 & 3.90\\% & 2.07 & 4.74 & 4.90\\% \\\\\nSTGCN& 1.36 & 2.96 & 2.90\\% & 1.81 & 4.27 & 4.17\\% & 2.49 & 5.69 & 5.79\\% \\\\\nGraph-WaveNet& 1.30 & 2.74 & 2.73\\% & 1.63 & 3.70 & 3.67\\% & 1.95 & 4.52 & 4.63\\% \\\\\nST-MetaNet& 1.36 & 2.90 & 2.82\\% & 1.76 & 4.02 & 4.00\\% & 2.20 & 5.06 & 5.45\\% \\\\\nMRA-BGCN& 1.29 & 2.72 & 2.90\\% & 1.61 & 3.67 & 3.80\\% & 1.91 & 4.46 & 4.60\\% \\\\\nGMAN& 1.34 & 2.82 & 2.81\\% & 1.62 & 3.72 & 3.63\\% & 1.86 & 4.32 &\\textbf{ 4.31}\\% \\\\\nMTGNN& 1.32 & 2.79 & 2.77\\% & 1.65 & 3.74 & 3.69\\% & 1.94 & 4.49 & 4.53\\% \\\\\nA2GNN (ours)& \\textbf{1.28} & \\textbf{2.70} & \\textbf{2.72}\\% &\\textbf{ 1.58} & \\textbf{3.62} & \\textbf{3.61}\\% & \\textbf{1.85} & \\textbf{4.29} & 4.39\\%\\\\\n\\bottomrule  \n\n\\multicolumn{10}{c}{%\n  \\begin{minipage}{12cm} %\n    \\small In these experiments, our goal is to forecast a sequence of values $Y=\\{x_{t+1},... ,x_{t+12}\\}$. We show the scores for $x_{t+3},\\,x_{t+6},\\text{ and } x_{t+12}$ for simplicity. %\n  \\end{minipage}%\n}\\\\\n\\end{tabular}\n\\label{tab:multi step forecasting}\n}\n\\vspace{-0.7cm}\n\\end{center}\n\n\\end{table}\n\n\n\n\n\n\\begin{table}[ht]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\\caption{Experiments on spatial-temporal forecasting datasets (mean scores).}\n\\center\n\\scalebox{0.7}{\n\\begin{tabular}{c|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{1}{c|}{Dataset} & \\multicolumn{9}{c}{METR-LA}  \\\\ \\hline\n\n\\multicolumn{1}{c}{} & \\multicolumn{3}{|c|}{Horizon 1$\\sim$3 } & \\multicolumn{3}{|c|}{Horizon 1$\\sim$6} & \\multicolumn{3}{|c}{Horizon 1$\\sim$12} \\\\\n\n\\hline\nMethods\/Metrics & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE\\\\ \\hline\nStemGNN & 2.56& 5.06 & 6.46\\% & 3.01 & 6.03 & 8.23\\% & 3.43 & 7.23 & 9.85\\% \\\\\nSTGNN & 2.62 &  4.99 &  6.55\\%  & 2.98 &  5.88  & 7.77\\% &  3.49  & 6.94 &  9.69\\% \\\\\nA2GNN & \\textbf{2.43} & \\textbf{4.44} & \\textbf{6.12\\%} & \\textbf{2.65} & \\textbf{5.05} & \\textbf{6.93\\%} & \\textbf{2.92} & \\textbf{5.83} & \\textbf{8.05\\%} \\\\ \\hline\n& \\multicolumn{9}{c}{PEMS-BAY}\\\\ \\hline\nStemGNN& 1.23 & 2.48 & 2.63 & - & - & - & - & - & - \\\\\nSTGNN&  1.17 &  2.43 &  2.34\\%  & 1.46 &  3.27 &  3.09\\%  & 1.83 &  4.20 &  4.15\\% \\\\\nA2GNN& \\textbf{1.08} & \\textbf{2.14} & \\textbf{2.23}\\% & \\textbf{1.29} & \\textbf{2.75} & \\textbf{2.81\\%} & \\textbf{1.52} & \\textbf{3.42} & \\textbf{3.42\\%} \\\\ \\bottomrule  \n\\multicolumn{10}{c}{%\n  \\begin{minipage}{12cm} %\n    \\small As STGNN and StemGNN published the mean score of all predicted horizons in their paper. Thus, we show the mean score (e.g. Horizon 1$\\sim$3 represents the mean score of horizon $1,2,3$). %\n  \\end{minipage}%\n}\\\\\n\\end{tabular}\n\\vspace{-2cm}\n\n}\n\\label{tab:multi step forecasting (mean score)}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Result Comparison on Spatial-temporal Dataset}\nIn order to further verify the performance of our model in the traditional spatial-temporal forecasting problem, we make further experiments on $2$ well known spatial-temporal datasets in traffic. \nWe compare the performance of A2GNN with above-mentioned baseline methods, and these methods can not be employed to model time series problem without pre-defined graph. \n\n\nWe present the results on spatial-temporal forecasting tasks in Table \\ref{tab:multi step forecasting} and compares corresponding performance of A2GNN with other spatial-temporal graph neural network methods. From this table, we can see that A2GNN achieves the best performance in terms of both RMSE and MAPE over almost all steps. In particular, A2GNN reduces the RMSE by $2.2\\%$, $3.3\\%$, $4.3\\%$ on the METR-LA dataset and $3.0\\%$, $4.4\\%$ and $4.6\\%$ on PEMS-BAY dataset when horizon is set to $3$, $6$ and $12$, respectively. \n\nOur method achieves the fastest training and inference time compared with other graph based models. The details are shown in appendix.\n\n\n\n\n\n\n\n\n\\subsection{Ablation Study}\n\\label{sec:ablation}\nTo gain a better understanding of the effectiveness of A2GNN's key components, we perform ablation studies through the time series forecasting task on Soloar-Energy dataset as well as the spatial-temporal forecasting task on META-LA dataset. The settings are summarized below:\n\\begin{itemize}\n\\item \\textbf{A2GNN} is A2GNN method.\n\\item \\textbf{w\/o AGL} is A2GNN without auto graph learner.\n\\item \\textbf{w\/o $A^p$} is A2GNN without pre-defined relation.\n\\item \\textbf{w\/o ARL} represents A2GNN  attentional relation learner, which is replaced by $\\rm{concatenate}$ operation.\n\\item \\textbf{w\/o A2} represents A2GNN without auto graph learner and attentional relation learner.\n\\end{itemize}\n\n\\begin{table}[t]\n\\caption{Ablation study.}\n\\renewcommand\\arraystretch{1.05}        \n\\centering\n\\renewcommand\\tabcolsep{3.0pt}\n\\scalebox{0.7}{\n\\begin{tabular}{c|cc|cc|cc|cc|cc}\n    \\toprule\n& \\multicolumn{4}{c}{Solar-Energy}  & \\multicolumn{6}{|c}{METR-LA} \\\\\n \\hline\n & \\multicolumn{2}{c}{RSE$\\downarrow$}  & \\multicolumn{2}{|c|}{CORR$\\uparrow$} & \\multicolumn{2}{c}{MAE$\\downarrow$}  & \\multicolumn{2}{|c|}{RMSE$\\downarrow$} &   \\multicolumn{2}{c}{MAPE$\\downarrow$} \\\\ \\hline\nMethods & Valid & Test & Valid & Test & Valid & Test & Valid & Test & Valid & Test\\\\ \\hline\nw\/o AGL & 0.37 & 0.35     & 0.94 & 0.93 & 2.73 & 2.97     & 5.33 & 5.99     & 7.57\\% & 8.27\\%      \\\\\nw\/o $A^p$ & - & - & - & - & 2.75  & 2.98     & 5.29  & 5.97    & 7.53\\%  & 8.17\\%  \\\\\nw\/o ARL & 0.33 & 0.30     & 0.95 & 0.95 & 2.73 & 2.98     & 5.26 & 5.92     & 7.60\\% & 8.28\\%      \\\\ \nw\/o A2  & 0.39 & 0.37     & 0.94 & 0.93 & 2.82 & 3.05     & 5.35 & 6.04     & 7.87\\% & 8.57\\%  \\\\ \\hline\nA2GNN & \\textbf{0.32} & \\textbf{0.28} & \\textbf{0.95} & \\textbf{0.95} & \\textbf{2.71} & \\textbf{2.93}     & \\textbf{5.23} & \\textbf{5.82}     & \\textbf{7.50\\%} & \\textbf{8.14\\%}      \\\\\n \\bottomrule\n\\end{tabular}}\n\n\\label{tab:ablation study}\n\n\\end{table}\n\n\n\n\n\nTo perform the studies for time series forecasting tasks, we run experiments on the Solar-Energy dataset with $t_{out} = 12$. To perform the ablation study on spatial-temporal forecasting tasks, we report the experiments on the METR-TA dataset on all $12$ steps. In each experiment, the model is trained for $50$ epochs, and $10$ repeated runs ensure the reliability.\nTable \\ref{tab:ablation study} show the performance in terms of evaluation scores on both validation and test sets. From the table, we can see that, without auto graph learner, the performance drops drastically, which indicates that our auto graph learner plays an indispensable role in A2GNN for achieving more accurate forecasting. \nSimilarly, the attentional relation learner is also responsible for a considerable performance gain by A2GNN. \nSpecially, in spatial-temporal forecasting task, pre-defined graph\/relation by human knowledge is a further information to be employed, which can also influence the model performance.\nIn the end, the experiment without auto graph learner and attentional relation learner prove the superiority of our overall approach.\n\n\\textit{The Effect of Neighbor Amount: }\nAuto graph Learner (AGL) is designed to discover the implicit relation. In AGL, the $C$ is the key factor to control the neighbor amount for each node. Therefore, we set multiple $C$ to study the influence. As shown in Table \\ref{tab:Effect of neighbor amount}, $C$ value that is too large or too small will make the model's performance worse.\nA reasonable $C$ value will have a big improvement compared with the w\/o AGL. Especially, the implicit relation discovered by AGL have a significant for time series forecasting task.\n\n\n\\begin{table}[]\n\\caption{Effect of neighbor amount. $C$ is the number of edges for each node in AGL.}\n\\renewcommand\\arraystretch{1.05}        \n\\centering\n\\renewcommand\\tabcolsep{3.0pt}\n\\scalebox{0.75}{\n\\begin{tabular}{c|cc|cc|cc|cc|cc}\n    \\toprule\n& \\multicolumn{4}{c}{Solar-Energy}  & \\multicolumn{6}{|c}{METR-LA} \\\\\n \\hline\n & \\multicolumn{2}{c}{RSE$\\downarrow$}  & \\multicolumn{2}{|c}{CORR$\\uparrow$} & \\multicolumn{2}{|c}{MAE$\\downarrow$}  & \\multicolumn{2}{|c|}{RMSE$\\downarrow$} &   \\multicolumn{2}{c}{MAPE$\\downarrow$}\\\\\n \\hline\n $C$ & Valid & Test & Valid & Test & Valid & Test & Valid & Test & Valid & Test\\\\ \\hline\n 1 & 0.349  & 0.320  & 0.951  & 0.946  & 2.759  & 2.960  & 5.328  & 5.915  & 7.54\\%  & 8.18\\% \\\\\n 3 & 0.337  & 0.302  & 0.954  & 0.952  & 2.725  & 2.945  & 5.233  & 5.848  & 7.50\\%  & 8.16\\% \\\\\n 10 & 0.327  & 0.296  & 0.956  & 0.954 & \\textbf{2.715} & \\textbf{2.935}     & 5.233 & \\textbf{5.825}     & \\textbf{7.50\\%} & \\textbf{8.14\\%} \\\\\n 15 & \\textbf{0.322}  & \\textbf{0.289}  & \\textbf{0.958}  & \\textbf{0.957}  & 2.715  & 2.974  & 5.202  & 5.906  & 7.35\\%  & 8.03\\%\\\\\n 30 & 0.333  & 0.302  & 0.954  & 0.952  & 2.720  & 2.953  & \\textbf{5.210}  & 5.860  & 7.58\\% & 8.22\\%\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:Effect of neighbor amount}\n\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Interpretability Analysis}\n\\subsubsection{Analysis of Attentional Relation Learner}\nTo further reveal the effectiveness of the attentional relation learner, we provide intensive interpretability analysis on it (the pre-defined relation branch is disabled). The corresponding experiments are conducted on the META-LA dataset. \n\n\n\n\nTo show the relation attentions learned by attentional relation learner, we first sample some stations in the dataset, each of which corresponds to a node in the graph, and plot their attention coefficient on different relations in Figure \\ref{fig:attentional relation learner attention matrix}. From this figure, we can see that various stations pay variant attention to different relations. For example, station $16$ and station $196$ pay more attention to identity adjacency matrix, i.e. they are more concerned about their own information. \nFurthermore, Figure \\ref{fig:attentional relation learner station curve} visualizes stations with greater attention on the automatically learned adjacency matrix, i.e., station $49$ and $129$, on the left compared with stations with more concentrated attention on the identity matrix, i.e., station $16$ and $196$, on the right. It is obvious that the curves of station $49$ and $129$ are not stable with a couple of randomly occurred sudden drops. The curves of station $16$ and $196$, on the other hand, have a very clear and stable change pattern, where both the peak and the valley appear alternately and periodically. Therefore, it is much easier for station $16$ and $196$ to make accurate enough forecasting mainly based on their own information.\n\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=0.32\\textwidth]{photo\/attentional_relation_learner_visualize.png}\n    \\caption{Visualization for the attention coefficients (over own information and neighbor information by AGL) learned by attentional relation learner for station 16, 49, 129, 196. }\n    \\label{fig:attentional relation learner attention matrix}\n\\end{figure}\n\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{photo\/station_original_curve.jpg}\n    \\caption{Station speed visualization for $49,129,16,196$. The speed curves of station $16$ and $196$ are similar to the sine waveform, while the speed curves of station $49$ and $129$ seems not stable.}\n    \\label{fig:attentional relation learner station curve}\n\\end{figure}\n\n\n\\subsubsection{Analysis of Auto Graph Learner}\nIn this section, we make a further analysis on whether the learning process of auto graph learner is reasonable or not and whether the extra useful information exists in the learned neighbors from auto graph learner. In the first place, we visualize the correlation between station $49,129$ and all other stations during the training process in figure \\ref{fig:auto graph learner learning process}. As we can see, as the training process goes on, the correlations between station $49$ and some stations, such as station $150,120$, become stronger and stronger (from light to dark in color). Moreover, we visualize station $49,129$'s neighbors with strong correlation learned from auto graph learner in figure \\ref{fig:top related node curve for station49}. As we can see, the neighbor of station $49,129$ look like a similar curve of station $49,129$. The information aggregation of neighbors will enhance the predicted results (i.e. improve the robustness) of current station.\nThus, the learned neighbors can significantly improve the forecasting of station itself.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{photo\/learning_process.jpg}\n    \\caption{The learning process of auto graph learner for station $49,129$. }\n    \\label{fig:auto graph learner learning process}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{photo\/station_and_its_neighbor_curve.jpg}\n    \\caption{Speed visualization of station and its related neighbors learnt by auto graph learner. As we can see, the station and its implicit\/learnt neighbors have a similar trend, which will improve the robustness. }\n    \\label{fig:top related node curve for station49}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions and Future Work}\nIn this paper, we propose attentional multi-graph neural network with automatic graph learning (A2GNN). Compared with previous studies, our framework can automatically learn sparse relation by using Gumbel-Softmax with facilitating each node to dynamically pay more attention to preferred relation graphs. Experiments on a couple of real-world datasets have demonstrated the effectiveness of A2GNN on a variety of time series forecasting tasks.\n\n\n\n\n\n\n\n\n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the year 2012, an important task of particle physics is to fully\nmeasure the properties of the Higgs boson with mass $M_\\text{H}\\approx\n125$\\,GeV discovered at the Large Hadron Collider\n({\\abbrev LHC}{})~\\cite{Aad:2012tfa,Chatrchyan:2012xdj}.  At the same time, the\nsearch for additional Higgs bosons, which are predicted in many extended\ntheories, is among the main missions of the {\\abbrev LHC}{} experiments. For this\npurpose, the knowledge of the corresponding production cross sections\nwith high precision is of great relevance. The latest efforts in this\ndirection are regularly summarized in the reports of the ``{\\abbrev LHC}{} Higgs\ncross section working\ngroup''\\cite{Dittmaier:2011ti,Dittmaier:2012vm,Heinemeyer:2013tqa,deFlorian:2016spz}.\n\nIn this paper, we describe the new features that have been implemented\nin version {\\tt 1.6.0{}} of the program\n\\sushi{}~\\cite{Harlander:2012pb,sushiwebpage}.  \\sushi{} is a Fortran\ncode which calculates Higgs-boson production cross sections through\ngluon fusion and bottom-quark annihilation in the Standard Model ({\\abbrev SM}),\ngeneral Two-Higgs-Doublet Models ({\\abbrev 2HDM}), the Minimal Supersymmetric\nStandard Model ({\\abbrev MSSM}) as well as its next-to-minimal extension\n({\\abbrev NMSSM}), see \\citere{Liebler:2015bka}.\\footnote{Other codes to obtain\ninclusive Higgs-boson cross sections through gluon fusion in the {\\abbrev SM}{} and beyond are described in\n\\citeres{Spira:1995mt,Bagnaschi:2011tu,Anastasiou:2011pi,Anastasiou:2009kn,\nCatani:2007vq,Catani:2008me,Ball:2013bra,Bonvini:2014jma,Bonvini:2016frm}.}\nSome of these additions to\n\\sushi{} directly improve the theoretical predictions of the cross\nsection; others are provided to allow for more sophisticated uncertainty\nestimates of these predictions. The new features are the following:\n\\begin{itemize}\n\\item \\sushi{} now includes the next-to-next-to-next-to-leading order\n  (\\nklo{3}) terms for the gluon-fusion cross section of a {\\abbrev CP}{}-even\n  Higgs boson in the heavy-top limit as described in\n  \\citeres{Anastasiou:2014lda,Anastasiou:2015ema,Anastasiou:2015yha,Anastasiou:2016cez}.\n\\item It provides the so-called soft expansion of the gluon-fusion cross\n  section around the threshold of Higgs-boson production at \n  $x\\equivM_\\phi^2\/\\hat s=1$, where $\\hat{s}$ denotes the partonic\n  center-of-mass energy and $M_\\phi$ the Higgs-boson mass. This expansion\n  is available for the cross sections in the heavy-top limit up to\n  \\nklo{3} for {\\abbrev CP}{}-even Higgs bosons. At next-to-leading order ({\\abbrev NLO}{}) and\n  next-to-{\\abbrev NLO}{} ({\\abbrev NNLO}{}), the exact $x$-dependence is still available,\n   of course, and remains the default.\n\\item In addition, \\sushi{\\_\\sushiversion}{} includes top-quark mass effects to the\n  gluon-fusion cross section of a {\\abbrev CP}{}-even Higgs boson in the\n  heavy-top limit up to {\\abbrev NNLO}{}, implemented through an expansion in\n  inverse powers of the top-quark mass as described in\n  \\citeres{Marzani:2008az,Harlander:2009my,Harlander:2009mq,%\n    Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs}.\n  The exact top-mass dependence at lowest order can be factored out.\n  We remark that this feature is most interesting at {\\abbrev NNLO}{}, of course,\n  since at leading order ({\\abbrev LO}{}) and {\\abbrev NLO}{}, \\sushi{} also provides the full quark-mass\n  dependence.\n\\item A matching of the soft expansion to the high-energy\n  limit~\\cite{Marzani:2008az,Harlander:2009my,Harlander:2009mq},\n  i.e.\\ $x\\to 0$, is available through \\nklo{3}.\n\\item The renormalization-scale dependence of the gluon-fusion cross\n  section within an arbitrary interval is calculated in a single\n  \\sushi{} run.\n\\item The effect of \\dimension{5} operators to the gluon-fusion cross\n  section can be taken into account through \\nklo{3} {\\abbrev QCD}{} for the inclusive\n  cross section, and at {\\abbrev LO}{} and {\\abbrev NLO}{} (i.e.\\ $\\alpha_s^3$) for the Higgs\n  transverse momentum ($\\pt{}$) distribution and (pseudo)rapidity\n  distribution, respectively.\n\\item Higgs-boson production cross sections through heavy-quark\n  annihilation are implemented along the lines of\n  \\citere{Harlander:2015xur}, both for the {\\abbrev NNLO}{} {\\abbrev QCD}{} inclusive\n  cross section, as well as for more exclusive\n  cross sections up to {\\abbrev NLO}{} {\\abbrev QCD}{}.\n\\end{itemize}\n\nAll of the described features are applicable to Higgs-boson production\nin the theoretical models currently implemented in \\sushi{}, even though\nsome only work for low Higgs masses below the top-quark threshold $M_\\phi\n< 2M_\\text{t}$ or for {\\abbrev CP}{}-even Higgs bosons.\n\nOur paper is organized as follows: We start with a brief general\noverview of the code \\sushi{} in \\sct{sec:sushi}, and subsequently\npresent the new features implemented for the prediction of the\ngluon-fusion cross section in \\sct{sec:gluonfusion}. This includes a\ntheoretical description of the soft expansion, the inclusion of\n\\nklo{3} terms and the top-quark mass effects in\n\\scts{sec:softexp}--\\ref{sec:mt}. We proceed with a description of the\n``{\\scalefont{.9} RGE}{} procedure'' to determine the renormalization-scale dependence\nof the gluon-fusion cross section in \\sct{sec:scaledep}, and finally\ndescribe the implementation of an effective Lagrangian including\n\\dimension{5} operators in \\sct{sec:dim5}. The implementation of\nheavy-quark annihilation cross sections is described in\n\\sct{sec:heavyquark}. Numerical results are presented in\n\\sct{sec:numerics}; they also include a comparison of our results with\nthe most recent literature. In \\ref{app:inputfile} we\npresent a collection of example input blocks of \\sushi{}, which contain\nexample settings for the various input entries introduced in previous and the newest release.\n\n\\section{The program \\sushi{}}\n\\label{sec:sushi}\n\n\\sushi{} is a program originally designed to describe Higgs production\nin gluon fusion and bottom-quark annihilation in the {\\abbrev MSSM}{}.  It\ncollects a number of results from the literature valid through \\nklo{3}\nin the strong coupling constant, and combines them in a consistent way.\nWe subsequently discuss the present theoretical knowledge of the\ncalculation of the gluon fusion and bottom-quark annihilation cross\nsections and their inclusion in \\sushi{}.\n\nIt is well-known that {\\abbrev QCD}{} corrections to the gluon-fusion process\n$gg\\rightarrow \\phi$~\\cite{Georgi:1977gs}, mediated through heavy quarks\nin the {\\abbrev SM}{}, are very large. {\\abbrev NLO}{} {\\abbrev QCD}{} corrections are known for\ngeneral quark\nmasses~\\cite{Djouadi:1991tka,Dawson:1990zj,Spira:1995rr,Harlander:2005rq,Anastasiou:2006hc,Aglietti:2006tp}.\nIn the heavy-top limit, an effective theory can be constructed by\nintegrating out the top quark. In this case, {\\abbrev NNLO}{} corrections have\nbeen calculated a long time\nago~\\cite{Harlander:2002wh,Anastasiou:2002yz, Ravindran:2003um}.  The\n\\nklo{3} contributions were only recently obtained in\n\\citeres{Anastasiou:2014vaa,Li:2014afw,Anastasiou:2014lda,Anastasiou:2015ema,\n  Anastasiou:2016cez}, while various parts of the \\nklo{3} calculation\nhave been calculated\nindependently\\,\\cite{Hoschele:2012xc,Baikov:2009bg,Lee:2010cga,Gehrmann:2010ue,\n  Gehrmann:2010tu,Gehrmann:2011aa,Kilgore:2013gba,Duhr:2014nda,\n  Anastasiou:2015yha,Duhr:2013msa,Anastasiou:2013srw,Hoschele:2014qsa,\n  Dulat:2014mda,Anzai:2015wma,Li:2014bfa}.  Approximate \\nklo{3} results\nwere presented in\n\\citeres{Ball:2013bra,Bonvini:2014jma,deFlorian:2014vta}.  Effects of a\nfinite top-quark mass at {\\abbrev NNLO}{} were approximately taken into account\nin~\\citeres{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs,Marzani:2008az}.\n\nMany of these effects can be taken into account in the latest\nversion of \\sushi{}; this will be discussed in detail in\n\\sct{sec:gluonfusion}. Electroweak\ncorrections~\\cite{Actis:2008ug,Aglietti:2004nj,Bonciani:2010ms} can be\nincluded as well, either in terms of the full {\\abbrev SM}{} electroweak\ncorrection factor, or restricted to the corrections mediated by light\nquarks, the latter being a more conservative estimate in certain {\\abbrev BSM}{}\nscenarios. For completeness, we note that effects beyond fixed order\nhave been addressed through soft-gluon resummation \\cite{Catani:2003zt,\n  Moch:2005ky,Idilbi:2005ni,Idilbi:2006dg,Ravindran:2006cg,Ahrens:2008nc,\n  Schmidt:2015cea,Bonvini:2016frm}, but those are not included in\n\\sushi{}.\n\nIf requested in the input file, \\sushi{} uses the {\\abbrev SM}{} results\ndescribed above also for the {\\abbrev 2HDM}{}, the {\\abbrev MSSM}{} or the {\\abbrev NMSSM}{}\nthrough the proper rescaling of the Yukawa couplings.  In supersymmetric\nmodels, also squarks induce an interaction of the Higgs boson to two\ngluons.  In the {\\abbrev MSSM}{}, the corresponding {\\abbrev NLO}{} virtual contributions,\ninvolving squarks, quarks and gluinos, are either known in an expansion\nof inverse powers of heavy {\\abbrev SUSY}{}\nmasses~\\cite{Degrassi:2010eu,Degrassi:2011vq,Degrassi:2012vt} or in the\nlimit of a vanishing Higgs mass, see\n\\citeres{Harlander:2003bb,Harlander:2004tp,Harlander:2005if,Degrassi:2008zj}.\nIn this limit, even {\\abbrev NNLO}{} corrections of stop-induced contributions\nare known, see \\citeres{Pak:2010cu,Pak:2012xr}; an approximation of\nthese effects\\,\\cite{Harlander:2003kf} is included in \\sushi{}, see\n\\citere{Bagnaschi:2014zla}.  Whereas for the {\\abbrev MSSM}{} \\sushi{} relies on\nboth expansions, for the {\\abbrev NMSSM}{} the {\\abbrev NLO}{} virtual corrections are\npurely based on an expansion in heavy {\\abbrev SUSY}{}\nmasses~\\cite{Liebler:2015bka}.\nWe note that numerical results for the exact {\\abbrev NLO}{} virtual\ncontributions involving squarks, quarks, and gluinos were presented in\n\\citeres{Anastasiou:2008rm,Muhlleitner:2010nm}, and analytic results for\nthe pure squark-induced contributions can be found in\n\\citeres{Anastasiou:2006hc,Aglietti:2006tp,Muhlleitner:2006wx}.\n\nThe associated production of a Higgs boson with bottom quarks,\n$pp\\rightarrow b\\bar b\\phi$, is of particular relevance for Higgs\nbosons, where the Yukawa coupling to bottom quarks is enhanced. This\nhappens in models with two Higgs doublets, for example, if $\\tan\\beta$,\nthe ratio of the vacuum expectation values of the two neutral Higgs\nfields, is large. \\sushi{} includes the cross section for this process\nin the so-called 5-flavor scheme, i.e.\\ for the annihilation process\n$b\\bar b\\to \\phi$. The inclusive cross section for this process is\nimplemented at {\\abbrev NNLO}{} {\\abbrev QCD}{} \\cite{Maltoni:2003pn,Harlander:2003ai}; it\nis reweighted by effective Yukawa couplings in the model under\nconsideration. \\sushi{\\_\\sushiversion}{} now also includes general heavy-quark\nannihilation cross sections\\,\\cite{Harlander:2015xur} at {\\abbrev NNLO}{} {\\abbrev QCD}{},\nwhich we will describe in \\sct{sec:heavyquark}.\n\nFor completeness we note that \\sushi{} can be linked to {\\tt\nFeynHiggs}~\\cite{Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh}\nand {\\tt 2HDMC}~\\cite{Eriksson:2009ws} to obtain consistent sets of\nparameters in the {\\abbrev MSSM}{} or the {\\abbrev 2HDM}{}, respectively.\n\n\\sushi{} is controlled via an {\\scalefont{.9} SLHA}-style \\cite{Skands:2003cj}\ninput file. In the following, we will refer to the entries of a {\\tt\n  Block \"NAME\"} and their possible values as\n\\blockentry{NAME}{ENTRY}{=VALUE}. If more than one value is required, we\nwrite \\blockentry{NAME}{ENTRY}{=\\{VALUE1,VALUE2,$\\dots$\\}} or, when\nreferring only to one specific value,\n\\blockentry{NAME}{ENTRY,1}{=VALUE1}, etc.\nIn \\ref{app:inputfile} we include input blocks with\nexample settings for the various new input entries.\n\n\\section{Higgs production through gluon fusion}\n\\label{sec:gluonfusion}\n\nThe hadronic cross section for Higgs production in gluon fusion can be\nwritten as\n\\begin{equation}\n\\begin{split}\n\\sigma(pp\\to H+X) = \\sum_{i,j\\in\\{q,\\bar q,g\\}}\\tilde{\\phi}_i\\otimes\n\\tilde{\\phi}_j\\otimes\\hat\\sigma_{ij}\\,,\n\\label{eq:convolution}\n\\end{split}\n\\end{equation}\nwhere $\\phi_i(x,\\mu_\\text{F})=\\tilde{\\phi}_i(x,\\mu_\\text{F})\/x$ are parton\ndensities, $q$ ($\\bar q$) denotes the set of all (anti-)quarks ($q=t$\nand $\\bar q=\\bar t$ can be neglected), and $\\otimes$ is the convolution\ndefined as\n\\begin{equation}\n\\begin{split}\n(f\\otimes g)(z) \\equiv \\int_0^1{\\rm d} x_1\\int_0^1{\\rm d} x_2\n  f(x_1)g(x_2)\\delta(z-x_1x_2)\\,.\n\\end{split}\n\\end{equation}\nThe perturbative expansion of the partonic cross section,\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma_{ij,\\text{\\nklo{n}}}&= \n\\sum_{l=0}^n\\hat\\sigma^{(l)}_{ij}\\,,\n\\label{eq:sigmanij}\n\\end{split}\n\\end{equation}\ncan be represented in terms of Feynman diagrams where the external\npartons couple to the Higgs bosons through a top-, bottom-, or\ncharm-quark loop (contributions from lighter quarks are\nnegligible).\n\nThe first two terms in the perturbative expansion of $\\hat\\sigma_{ij}$\n($l=0,1$ in \\eqn{eq:sigmanij}) are known for general quark mass and\nincluded in \\sushi{\\_\\sushiversion}.\\footnote{We focus on the {\\abbrev SM}{} contributions\n  here, but also {\\abbrev SUSY}{} contributions can be added in the first two\n  terms and partially even at {\\abbrev NNLO}{}, see the description in\n  \\sct{sec:sushi}.} For the top-quark contribution, the\n{\\abbrev NNLO}{} term $\\hat\\sigma^{(2)}_{ij}$ has been evaluated on the basis of\nan effective Higgs-gluon interaction vertex which results from\nintegrating out the top quark from the {\\abbrev SM}{} Lagrangian. At {\\abbrev NLO}{}, it\nhas been checked that this results in an excellent approximation of the\n{\\abbrev NLO}{} {\\abbrev QCD}{} correction factor to the {\\abbrev LO}{} cross section, even for\nrather large Higgs-boson masses.  At {\\abbrev NNLO}{}, the validity of the\nheavy-top limit for the {\\abbrev QCD}{} corrections factor was investigated\nthrough the calculation of a number of terms in an expansion around\n$M_\\text{t}^2\\gg \\hat s,M_\\phi^2$, and matching it to the high-energy limit of\n$\\hat\\sigma^{(2)}_{ij}$\\,\\cite{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs,Marzani:2008az}.\nIt was found that the mass effects to the {\\abbrev QCD}{} correction factor are\nat the sub-percent level.\n\nRecently, also the \\nklo{3}-term $\\hat\\sigma_{ij}^{(3)}$ has become\navailable in terms of a soft expansion and assuming the heavy-top\nlimit. We will comment on its implementation in the latest release of\n\\sushi{} in \\sct{sec:n3lo}.\n\nThe exact {\\abbrev NLO}{} and the approximate higher order results for the cross\nsection are combined in \\sushi{} through the formula\n\\begin{equation}\n\\begin{split}\n\\sigma_\\text{{\\scalefont{.9} X}} &= \\sigma_\\text{{\\abbrev NLO}} +\n\\Delta_\\text{{\\scalefont{.9} X}} \\sigma^t\\,,\\qquad\\Delta_\\text{{\\scalefont{.9} X}}\n\\sigma^t \\equiv (1+\\delta_\\text{EW})\\sigma^t_\\text{{\\scalefont{.9} X}} - \\sigma_\\text{{\\abbrev NLO}}^t\\,,\n\\label{eq:sigmasushi}\n\\end{split}\n\\end{equation}\nwhere $\\sigma_\\text{{\\abbrev NLO}}$ refers to the {\\abbrev NLO}{} cross section with exact\ntop-, bottom- and charm-mass dependence, while $\\sigma^t_\\text{{\\scalefont{.9} X}}$\n({\\scalefont{.9} X}$=$\\nklo{n}, $n\\geq 1$) is obtained in the limit of a large top-quark\nmass.  Electroweak effects\\,\\cite{Actis:2008ug}, encoded in\n$\\delta_\\text{EW}$, are included by assuming their full factorization\nfrom the {\\abbrev QCD}{} effects, as suggested by \\citere{Anastasiou:2008tj} for\na {\\abbrev SM}{} Higgs boson.  In {\\abbrev BSM}{} scenarios, this assumption may be no\nlonger justified. \\sushi{} therefore provides an alternative way to\ninclude electroweak effects which is based solely on the light-quark\ncontributions to the electroweak correction factor; for details, we\nrefer the reader to \\citeres{Bagnaschi:2014zla,Harlander:2012pb}. For\nour purpose, it suffices to assume \\eqn{eq:sigmasushi}. The new release\nof \\sushi{} provides various approximations to evaluate\n$\\sigma^t_\\text{{\\scalefont{.9} X}}$, in particular through expansions in $1\/M_\\text{t}$, and\nexpansions around $\\hat s=M_\\phi^2$.\n\nIn addition to $\\sigma_\\text{{\\scalefont{.9} X}}$, which can be found in {\\tt Block\n  SUSHIggh}, \\sushi{} also outputs the individual terms of\n\\eqn{eq:sigmasushi}. The exact {\\abbrev LO}{} and {\\abbrev NLO}{} cross sections are\ncollected in {\\tt Block XSGGH}, while the $\\sigma^t_\\text{{\\scalefont{.9} X}}$ are given\nin {\\tt Block XSGGHEFF}, which also contains the electroweak correction\nterm $\\delta_\\text{EW}$, if requested.\n\nIt is understood that the \\nklo{n} terms in \\eqn{eq:sigmasushi} are\nevaluated with \\nklo{n} {\\abbrev PDF}{}s.\\footnote{Since \\nklo{3} {\\abbrev PDF}{}s are not\nyet available, we use {\\abbrev NNLO}{} {\\abbrev PDF}{}s for the evaluation of the\n\\nklo{3} cross section in this paper. The user of \\sushi{} can specify\nthe {\\abbrev PDF}{} set at each order individually.} Note that this means that,\nfor example, $\\Delta_\\text{{\\abbrev NNLO}}\\sigma^t$ is not simply the convolution\nof $\\hat\\sigma^{t,(2)}$ with {\\abbrev NNLO}{} {\\abbrev PDF}{}s, but retains a sensitivity\nto $\\hat\\sigma^{t,(1)}$. Thus, the final result for the {\\abbrev NNLO}{}\ngluon-fusion cross section obtained from \\sushi{} through\n\\eqn{eq:sigmasushi} depends on the approximation applied to the\nevaluation of both $\\hat\\sigma^{t,(2)}$ and $\\hat\\sigma^{t,(1)}$. If\nelectroweak effects are included, this even holds for \\sushi{}'s final\nresult for $\\sigma_\\text{{\\abbrev NLO}}$ due to the definition of\n$\\Delta_\\text{X}\\sigma^t$ in \\eqn{eq:sigmasushi}.\n\nIn the remainder of this section, we first discuss the soft expansion\naround the threshold of Higgs production, $\\hat{s}=M_\\phi^2$, in\n\\sct{sec:softexp}. The implementation of \\nklo{3} contributions is\ndescribed in \\sct{sec:n3lo}, and top-quark mass effects through {\\abbrev NNLO}{}\nas well as the matching to the high-energy limit in \\sct{sec:mt}. While\nthese features are only available for {\\abbrev CP}{}-even Higgs bosons (partially\nin a certain range of Higgs-boson masses $M_\\phi$ only), the analytic\ncalculation of the $\\mu_\\text{R}$ dependence of the gluon-fusion cross section\ndescribed in \\sct{sec:scaledep} is available for all Higgs bosons. The\ninclusion of \\dimension{5} operators is discussed in \\sct{sec:dim5}.\n\n\\subsection{Soft expansion}\\label{sec:softexp}\n\nThe {\\abbrev NLO}{} and {\\abbrev NNLO}{} coefficients of $\\sigma^t$ are approximated very\nwell by the first few terms\\footnote{The first $16$ terms in\n this expansion lead to an accuracy of better than $1$\\% with respect to the heavy-top\n limit with exact $x$-dependence at {\\abbrev NNLO}{}, for example. For more details see\n below.} in an expansion around the ``soft limit'', $x\\to 1$. In fact,\nthe gain of the full $\\hat s$-dependence becomes doubtful anyway when\nworking in the heavy-top limit, since the latter formally breaks down\nfor $\\hat s>4M_\\text{t}^2$, meaning $x\\lesssim 0.13$ for $M_\\text{H}=125$\\,GeV.\nApart from the exact $\\hat s$-dependence at {\\abbrev LO}{}, {\\abbrev NLO}{},\nand {\\abbrev NNLO}{}, \\sushi{\\_\\sushiversion}{} provides the soft expansion of the cross\nsection for {\\scalefont{.9} CP}-even Higgs production through order\n$(1-x)^{16}$ at these perturbative orders, and also at \\nklo{3} (for\nmore details on the latter, see \\sct{sec:n3lo}).\n\nThe precise way in which the soft expansion is applied is governed by\nthe new {\\tt Block GGHSOFT}. Each line in this block contains four\nintegers:\n\\begin{lstlisting}\nBlock GGHSOFT\n  <entry> <n1> <n2> <n3>\n\\end{lstlisting}\nFollowing \\sct{sec:sushi}, we will refer to such a line as\n\\blockentry{GGHSOFT}{<entry>}{\\tt=\\{<n1>,<n2>,<n3>\\}} in the text, and\nto the individual entries as \\blockentry{GGHSOFT}{<entry>,1}{\\tt=<n1>},\netc. The integer \\blockentry{GGHSOFT}{$n$,1}{} determines\n  whether the soft expansion is applied ({\\tt =1}) or not ({\\tt =0}) at\norder \\nklo{n}.\n  Setting \\blockentry{GGHSOFT}{$n$}{=\\{1,$N,a$\\}} evaluates the soft\n  expansion of $\\hat\\sigma^{t,(n)}_{ij}$ in the following way:\n\\begin{equation}\n  \\begin{split}\n    \\hat \\sigma^t_{ij}\\to\n    \\hat\\sigma^t_{ij,N} \\equiv x^{a}{\\cal\n      T}^x_{N}\\left(\\frac{\\Delta\\hat\\sigma^t_{ij}}{x^{a}}\\right)\\,,\n    \\label{eq:softexp}\n  \\end{split}\n\\end{equation}\nwhere ${\\cal T}^x_N$ denotes the asymptotic expansion around $x=1$\nthrough order $(1-x)^N$, and $a$ is a non-negative integer. Setting\n\\blockentry{GGHSOFT}{$n$,2}{=-1} will keep only the soft and collinear\nterms, whose $x$ dependence is given by\n\\begin{equation}\n\\begin{split}\n  \\delta(1-x)\\quad\\text{or}\\quad\\left(\\frac{\\ln^k(1-x)}{1-x}\\right)_+\\,,\\quad\n  k\\geq 0\n\\end{split}\n\\end{equation}\nby definition.  Here $(\\cdot)_+$ denotes the usual plus distribution,\ndefined by\n\\begin{equation}\n  \\begin{split}\n    \\int_z^1{\\rm d} x\\left(f(1-x)\\right)_+ g(x) = \\int_z^1{\\rm d} x\n    f(x)\\left[g(x)-g(1)\\right] + g(1)\\int_0^z{\\rm d} xf(x)\\,.\n  \\end{split}\n\\end{equation}\nThe parameters \\blockentry{GGHSOFT}{$n$}{} apply to all partonic\nsubchannels at order \\nklo{n}, and to all terms in the $1\/M_\\text{t}$\nexpansion as requested by the input {\\tt Block GGHMT}, see \\sct{sec:mt}\nbelow.\n\nThe exact $x$-dependence is obtained by setting\n\\blockentry{GGHSOFT}{$n$,1}{=0} (only available for $n\\leq 2$). The\nother two entries in \\blockentry{GGHSOFT}{$n$}{} are then irrelevant.\nThe default values for the block {\\tt GGHSOFT} through \\nklo{3} are\n\\begin{equation}\n{{\\footnotesize\n\\begin{split}\n\\fbox{\\it default:}\\quad\n&\\text{\\blockentry{GGHSOFT}{1,1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHSOFT}{2,1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHSOFT}{3}{=\\{1,16,0\\}}}\\,.\n\\end{split}}}\n\\end{equation}\nAgain, all terms of the soft expansion are available including the full\n$\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependence.\n\nA sample input block reads\n\\begin{lstlisting}\nBlock GGHSOFT\n  1 0 0 0\n  2 1 16 1\n  3 1 16 1\n\\end{lstlisting}\nwhich provides the result including the exact $x$-dependence at {\\abbrev NLO}{},\nand the soft expansion through $(1-x)^{16}$ at {\\abbrev NNLO}{} and \\nklo{3}\nafter factoring out a factor of $x$ ($a=1$ in \\eqn{eq:softexp}).  We\nrecall that these settings only affect the heavy-top results\n$\\hat\\sigma^t_\\text{{\\scalefont{.9} X}}$ in \\eqn{eq:sigmasushi}; $\\sigma_\\text{{\\abbrev NLO}}$ is\nalways calculated by taking into account the full quark-mass and\n$x$-dependence. The soft expansion is available for all {\\abbrev CP}{}-even\nHiggs bosons of arbitrary masses.\n\n\\subsection{\\nklo{3} terms}\\label{sec:n3lo}\n\nRecently, the \\nklo{3} {\\abbrev QCD}{} corrections to the Higgs production cross\nsection through gluon fusion have become\navailable\\,\\cite{Anastasiou:2014lda,Anastasiou:2015ema,Anastasiou:2015yha,Anastasiou:2016cez}. More\nspecifically, the result was provided in terms of the soft expansion\nthrough order $(1-x)^{37}$ of the leading term in $1\/M_\\text{t}$ for\n$\\mu=\\mu_\\text{R}=\\mu_\\text{F}$.  We implemented this expansion through $(1-x)^{16}$;\nhigher order terms do not change the result within the associated\nuncertainty. In addition, we included the $\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependent\nterms at the same order.  Experience from {\\abbrev NNLO}{} lets one expect that\nthese terms are sufficient to obtain an excellent approximation of the\n{\\abbrev QCD}{} correction factor to the {\\abbrev LO}{} cross section, at least for\nHiggs masses in the validity range of the effective theory description.\n\nThe \\nklo{3} result is accessible in \\sushi{\\_\\sushiversion}{} by setting the input\nparameter \\blockentry{SUSHI}{5}{=3}. This will evolve $\\alpha_s(M_Z)$ to\n$\\alpha_s(\\mu_\\text{R})$ at 4-loop order when calculating the cross section,\nwhere $\\mu_\\text{R}\/M_\\phi$ is defined in \\blockentry{SCALES}{1}{}. Note that\nwith this setting, the hadronic cross section will formally still suffer\nfrom an inconsistency because \\nklo{3} {\\abbrev PDF}{} sets are not yet\navailable.\nAs described in \\sct{sec:softexp}, the depth of the soft\nexpansion at \\nklo{3}, as well as the power $a$ in \\eqn{eq:softexp} can\nbe controlled through the input variables\n\\blockentry{GGHSOFT}{3}{}.\\footnote{The setting\n  \\blockentry{GGHSOFT}{3,1}{=0} is not available, of course.}\n\nFinally, we remark that, also at \\nklo{3}, the full $\\mu_\\text{R}$- and\n$\\mu_\\text{F}$-dependence is available, again accessible through the variables\n{\\tt SCALES(1)} and {\\tt SCALES(2)}, respectively. It follows from\ninvariance of the hadronic result under these scales, and only requires\nthe {\\abbrev NNLO}{} result as input, as well as the {\\abbrev QCD}{} $\\beta$ function and\nthe {\\abbrev QCD}{} splitting functions through three loops. The required\nconvolutions can be evaluated with the help of the program {\\tt\n  MT.m}\\,\\cite{Hoeschele:2013gga}, for example.\n\n\\subsection{Top-quark mass effects}\\label{sec:mt}\n\nIn versions before \\sushi{\\_\\sushiversion}{}, only the formally leading terms in\n$1\/M_\\text{t}$ were available for $\\hat\\sigma^{t,(2)}_{ij}$. However, in\norder to allow for thorough studies of the theoretical uncertainty\nassociated with the gluon-fusion cross section, \\sushi{\\_\\sushiversion}{} includes\nalso subleading terms in $1\/M_\\text{t}$ for the production of a {\\abbrev CP}{}-even\nHiggs (\\blockentry{SUSHI}{2}{}$\\in\\{${\\tt\n11,12,13}$\\}$). There are a number of options provided by\n\\sushi{\\_\\sushiversion}{} associated with this; they are controlled by the new input\n{\\tt Block GGHMT}.\n\nFirst of all,\n\\blockentry{GGHMT}{$n$}{=$P\\in\\{$0,1,$\\ldots,P_n^\\text{max}\\}$} provides\nthe expansion of $\\hat\\sigma^{t,(n)}_{ij}$ through $1\/M_\\text{t}^{P}$ (note\nthat terms with odd $P$ vanish). In addition (or alternatively), one may\ndefine the depth of the expansion individually for each partonic channel\n$\\hat\\sigma^{t,(n)}_{ij}$ through the parameters\n\\blockentry{GGHMT}{$nm$}{=$P\\in\\{$0,1,$\\ldots,P_n^\\text{max}\\}$}, where\n$ij=(gg,qg,q\\bar q,qq,qq')$ corresponds to $m=(1,2,3,4,5)$,\nrespectively. Currently, the maximal available depths of expansion\nare\\footnote{For the $q\\bar q$-channel, the maximum reduces to\n  $P_2^\\text{max}=4$, if a soft expansion beyond $N=13$ is requested.}\n$P_0^\\text{max}=P_1^\\text{max}=10$ and $P_2^\\text{max}=6$.\n\nThe default settings are\n\\begin{equation}\n{\\footnotesize\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad \n&\\text{\\blockentry{GGHMT}{0}{=-1}}\\,;\\quad\n\\text{\\blockentry{GGHMT}{1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHMT}{2}{=0}}\\,;\\\\\n&\\text{\\blockentry{GGHMT}{1$i$}{=GGHMT(1)}}\\,,\\ i=1,\\ldots,3\\,;\\quad\n\\text{\\blockentry{GGHMT}{2$i$}{=GGHMT(2)}}\\,,\\ i=1,\\ldots,5\\,,\n\\end{split}}\n\\end{equation}%\nwhere \\blockentry{GGHMT}{0}{=-1} means to keep the full top mass\ndependence. Let us recall that these settings only affect the heavy-top\nresults $\\hat\\sigma^t_\\text{{\\scalefont{.9} X}}$ in \\eqn{eq:sigmasushi}; $\\sigma_\\text{{\\abbrev NLO}}$\nis always calculated by taking into account the full quark-mass dependence.\n\nAs an example, consider the input\n\n\\begin{lstlisting}\nBlock GGHMT\n  1    10\n  13   0\n  2    6\n  23   0\n  24   0\n  25   0\n\\end{lstlisting}\n\nwhich will cause \\sushi{\\_\\sushiversion}{} to\n\\begin{itemize}\n\\item keep the full top mass dependence at {\\abbrev LO}{}\n\\item expand the {\\abbrev NLO}{} terms $\\hat\\sigma^{t,(1)}_{gg}$ and\n$\\hat\\sigma^{t,(1)}_{qg}$ through $1\/M_\\text{t}^{10}$\n\\item expand the {\\abbrev NNLO}{} terms $\\hat\\sigma^{t,(2)}_{gg}$ and\n  $\\hat\\sigma^{t,(2)}_{qg}$ through $1\/M_\\text{t}^{6}$\n\\item keep only the terms of order $1\/M_\\text{t}^0$ for the pure quark\n  channels at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.\n\\end{itemize}\nThis also shows that the variables \\blockentry{GGHMT}{$nm$}{} overrule\nthe setting of \\blockentry{GGHMT}{$n$}{} for the individual\nchannels. This may be desirable as it is known that the pure quark\nchannels show a rather bad convergence\nbehavior~\\cite{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs},\nso one may want to include only a small number of terms for them in the\n$1\/M_\\text{t}$ expansion.  By convention, \\blockentry{GGHMT}{$n$}{} must\nalways be at least as large as the maximum of \\blockentry{GGHMT}{$nm$}{};\nif this is not the case in the input file, \\sushi{} will override the\nuser's definition of \\blockentry{GGHMT}{$n$}{} and set it to the maximum\nof all \\blockentry{GGHMT}{$nm$}{}.\n\nIn the strict heavy-top limit (i.e., $P=0$), the quality of the\napproximation improves considerably if one factors out the {\\abbrev LO}{} mass\ndependence $\\sigma^t_0$\\,\\cite{Graudenz:1992pv,Spira:1995rr} before the\nexpansion, given by\n\\vspace{-5mm}\n\\begin{equation}\n\\begin{split}\n\\sigma^t_0 &= \\frac{\\pi\\sqrt{2}G_{\\rm F}}{256}\\left(\\api{}\\right)^2\n\\tau^2\\left|1+(1-\\tau)\\arcsin^2\\frac{1}{\\sqrt{\\tau}}\\right|^2\\,,\\qquad\n\\tau = \\frac{4M_\\text{t}^2}{M_\\phi^2}\\,,\n\\label{eq:sigma0}\n\\end{split}\n\\end{equation}\nwhere $G_\\text{F}\\approx 1.16637\\cdot 10^{-5}\\,\\text{GeV}^{-2}$~\\cite{Agashe:2014kda} is\nFermi's constant. The generalization to higher orders in $1\/M_\\text{t}$\ncorresponds to\n\\vspace{-5mm}\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^{t,(n)}_{ij} = \\sigma^t_0 \\frac{{\\cal\n    T}_{P_{n,ij}}\\hat\\sigma^{t,(n)}_{ij}}{{\\cal T}_{P_n}\\sigma^t_0}\\,,\n\\label{eq:defexpmt}\n\\end{split}\n\\end{equation}\nwhere ${\\cal T}_{P}$ denotes an operator that performs an asymptotic\nexpansion through order $1\/M_\\text{t}^P$.  In a strict sense, it should be\n$P_n=P_{n,ij}$; however, \\sushi{} allows only for a global value of\n$P_n$ here, which applies to all sub-channels $ij$ and is set to\n\\blockentry{GGHMT}{$n$}{}.\n\nSetting \\blockentry{GGHMT}{-1}{=$n$} factors out the {\\abbrev LO}{} mass\ndependence through order $n$, i.e.\\\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^t_{ij} &= \\sigma^t_0\\, \\sum_{k=0}^n \\frac{{\\cal\n    T}_{P_{k,ij}}\\hat\\sigma^{t,(k)}_{ij}}{{\\cal T}_{P_k}\\sigma^t_0} +\n\\sum_{k\\geq n+1}{\\cal T}_{P_{k,ij}}\\hat\\sigma^{t,(k)}_{ij}\\,.\n\\label{eq:plainexp}\n\\end{split}\n\\end{equation}\nThis will affect all partonic channels. The default setting is\n\\begin{equation}\n{\\footnotesize\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad& \\text{\\tt GGHMT(-1)=3}\n\\end{split}}\n\\end{equation}\nwhich means that the {\\abbrev LO}{} $M_\\text{t}$ dependence is factored out from\nall available orders.\n\nIt was observed that higher orders in $1\/M_\\text{t}$ in general spoil the\nvalidity of the expansion, since its radius of convergence is formally\nrestricted to $\\hat s<4M_\\text{t}^2$. This manifests itself in the expansion\ncoefficients containing positive powers of $\\hat s\/M_\\phi^2$. In order to\ntame the corresponding divergence as $\\hat s\\to \\infty$, it was\nsuggested to match the result to the asymptotic behavior in this limit,\nwhich is known from \\citeres{Marzani:2008az,Harlander:2009my}. Whether\nor not such a matching is performed for $\\hat\\sigma^{t,(n)}_{ij}$ is\ngoverned by the parameter \\blockentry{GGHMT}{$n\\cdot$10}{}\n(i.e.\\ \\blockentry{GGHMT}{10}{}, \\blockentry{GGHMT}{20}{}, \\ldots).  By\ndefault,\\\\[-2.5em]\n\n\\begin{equation}\n{\\footnotesize%\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad& \\text{\\tt\n  GGHMT($n\\cdot$10)=0}\\,,\\ n=1,\\ldots,3\\,, \n\\end{split}}\n\\end{equation}\nmeaning that no matching is done; setting \\blockentry{GGHMT}{$n\\cdot$10}{=1}\nswitches the matching on for all partonic subchannels at order \\nklo{n}.\n\nAs we will find in \\sct{sec:numerics}, the matching to $x\\to 0$ is\nhelpful in approximating the full cross section even at $1\/M_\\text{t}^0$.\nThus, we provide the possibility to do this matching also at \\nklo{3},\neven though top-mass suppressed terms are not yet known at this order.\nThe form of the matching through {\\abbrev NNLO}{} has been introduced in\n\\citeres{Harlander:2009my,Harlander:2009mq}; here we adopt the same\nstrategy, generalized to \\nklo{3}:\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^{t,(n)}_{ij}(x) = \\hat\\sigma^{t,(n)}_{ij,N}(x)\n&+ \\,\\sigma^t_0 \\sum_{l=1}^{n-1}A^{(n,l)}_{ij}\\left[ \\ln \\frac{1}{x} -\n  \\sum_{k=1}^N\\frac{1}{k}(1-x)^k  \\right]^l\\\\&\n+ (1-x)^{N+1}\\,\\left[\\sigma^t_0 B_{ij}^{(n)} -\n  \\hat\\sigma^{t,(n)}_{ij,N}(0)\n  \\right]\\,,\n\\label{eq:match}\n\\end{split}\n\\end{equation}\nwhere $\\sigma^t_0 B^{(0)}_{ij} = \\hat\\sigma^{t,(0)}_{ij,N}(0)=0$, and\n$\\hat\\sigma^{t,(n)}_{ij,N}(x)$ denotes the soft expansion of the cross\nsection through order $(1-x)^N$, see \\eqn{eq:softexp}.  The coefficients\n$B^{(1)}_{ij}$ and $A^{(2,1)}_{ij}$ are given in numerical form in\n\\citeres{Marzani:2008az,Marzani:diss,Harlander:2009my},\\footnote{The\n  notation for $A^{(2,1)}_{ij}$ is $A^{(2)}_{ij}$ in that paper.}  while\n$A^{(3,2)}_{gg}$ can be found in \\citere{Marzani:diss} (where it is\ncalled $C_\\text{A}^3{\\cal C}^{(3)}$).  For the unknown coefficients\nthrough \\nklo{3}, {\\it we assume} \n\\begin{equation}\n  \\begin{split}\n\\sigma^t_0 B^{(n)}_{ij} &=\n\\hat\\sigma^{t,(n)}_{ij,N}(0)\\quad\\text{for}\\quad n\\geq 2\\,,\\qquad\nA^{(3,1)}_{ij}=0\\,.\n  \\end{split}\n\\end{equation}\nThe technical consequence of the matching procedure implemented in\n\\sushi{} is that it {\\it requires} the cross section to be expressed in\nterms of the soft expansion, i.e., one needs to set\n\\blockentry{GGHSOFT}{$n$,1}{=1} if the \\nklo{n} cross section is requested.\n\nThe effect of the matching at \\nklo{3} is shown in \\fig{fig:sgg3match}:\nthe soft expansion tends to a constant towards $x\\to 0$ by construction,\nand cannot reproduce the $\\ln^2x$-behavior of the exact result. The\nmerging of the two limits is very smooth and suggests that the matched\ncurve is not too far from the full result. Of course, the fact that some\ncoefficients in \\eqn{eq:match} are unknown introduces a theoretical\nuncertainty. However, we observe a change in the final cross section of\nonly about 0.5\\% when setting $A^{(3,1)}_{gg}=A^{(3,2)}_{gg}$ for a\n{\\abbrev SM}{} Higgs, for example.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{figures\/N3LOmatch.pdf}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{Partonic cross section $\\hat{\\sigma}_{gg}^{t,(3)}\/\\sigma^t_0$\n  in $10^4$ according to \\eqn{eq:match} as a function of\n  $x=M_\\phi^2\/\\hat{s}$ with and without matching to the high-energy\n  limit. The order of the soft expansion applied in both cases is\n  $(1-x)^{16}$.}\n\\label{fig:sgg3match}\n\\end{figure}\n\nIt remains to say that all terms in $1\/M_\\text{t}$ are available including\nthe full $\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependence. As in earlier versions of\n\\sushi{}, these parameters are accessible through the input parameters\n{\\tt SCALES(1)} and {\\tt SCALES(2)}.  Since top-quark mass effects\nare not known for the \\nklo{3} cross section, all settings\nof {\\tt GGHMT} involving $n=3$ except from\n\\blockentry{GGHMT}{30}{} have no effect in the current version\n\\sushi{\\_\\sushiversion}{}. The inclusion of $1\/M_\\text{t}$ terms is only\navailable for $M_\\phi<2M_\\text{t}$ and the matching to the\nhigh-energy limit only in a mass range $M_\\phi \\in\n\\left[100\\,\\text{GeV},300\\,\\text{GeV}\\right]$.\n\n\\subsection{Renormalization scale dependence}\\label{sec:scaledep}\n\nThe renormalization scale ($\\mu_\\text{R}$) dependence of the partonic cross\nsection can be written as\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma_{ij} = \\sum_{n\\geq\n  0}\\sum_{l=0}^n\\left(\\api{(\\mu_\\text{R})}\\right)^{n+2}\n\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)\\,l_\\text{R0}^l\\,,\n\\label{eq:murdepparton}\n\\end{split}\n\\end{equation}\nwhere $l_\\text{R0}=2\\,\\ln(\\mu_\\text{R}\/\\mu_0)$, and $\\mu_0$ is an arbitrary reference\nscale. The coefficients $\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)$ are explicitly\ncontained in \\sushi{} (for $\\mu_0=M_\\phi$).  The dependence of the\ncross section on $\\mu_\\text{R}$ can be studied with \\sushi{} by varying the\ninput parameter \\blockentry{SCALES}{1}{}, which contains the numerical\nvalue for $\\mu_\\text{R}\/M_\\phi$. \\sushi{} will then insert this value into\n\\eqn{eq:murdepparton} and convolve the resulting partonic cross section\nover the {\\abbrev PDF}{}s. A decent picture of the $\\mu_\\text{R}$ dependence may require\nto perform this ``standard procedure'' ten times or more.\n\n\\sushi{\\_\\sushiversion}{} provides a considerably faster way to obtain the $\\mu_\\text{R}$\ndependence of the cross section by convolving the\n$\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)$ with the {\\abbrev PDF}{}s {\\it before} varying\n$\\mu_\\text{R}$,\n\\begin{equation}\n  \\begin{split}\n    \\kappa^{(n,l)}(\\mu_0)=\n    \\hat\\kappa^{(n,l)}_{ij}(\\mu_0)\\otimes \\tilde\\phi_i\\otimes \\tilde\\phi_j\\,.\n    \\label{eq:kappahad}\n  \\end{split}\n\\end{equation}\nWe will refer to this as the ``{\\scalefont{.9} RGE} procedure''.  Due to the\nrenormalization group equation\\footnote{The power $n+3$\n  takes into account the fact that the {\\abbrev LO}{} cross section is of order\n  $\\alpha_s^2$.}\n\\begin{equation}\n\\begin{split}\n\\deriv{}{\\mu_\\text{R}^2}\\sigma_\\text{\\nklo{n}} = \\order{\\alpha_s^{n+3}}\n=\\deriv{}{\\mu_\\text{R}^2}\\hat\\sigma_{ij,\\text{\\nklo{n}}}\\,,\n\\label{eq:rginv}\n\\end{split}\n\\end{equation}\nwhich holds both at the partonic and the hadronic level, it suffices to\ncalculate the coefficients $\\kappa^{(n,l)}(\\mu_0)$ for $l=0$ and $n\\leq\n2$ if the \\nklo{3} result is requested. \\sushi{\\_\\sushiversion}{} does this by\ninitially assuming\n$\\mu_0\/M_\\phi=\\mu_\\text{R}\/M_\\phi=$~\\blockentry{SCALES}{1}{} in\n\\eqn{eq:murdepparton}.  All other coefficients are then determined via\nthe {\\abbrev QCD}{} $\\beta$ function, defined through\n\\begin{equation}\n\\begin{split}\n\\deriv{}{\\mu_\\text{R}^2}\\alpha_s(\\mu_\\text{R}) = \\alpha_s(\\mu_\\text{R})\\beta(\\alpha_s)\\,,\n\\qquad \\beta(\\alpha_s) = - \\api{}\\sum_{n\\geq 0}\n\\left(\\api{}\\right)^n\n\\beta_n\\,.\n\\label{eq:betafun}\n\\end{split}\n\\end{equation}\nExplicitly, one finds\n\\begin{equation}\n\\begin{split}\n  \\kappa^{(1,1)} &= 2\\,\\beta_0\\,\\kappa^{(0,0)}\\,,\\qquad \\kappa^{(2,2)} =\n  \\frac{3}{2}\\,\\beta_0\\,\\kappa^{(1,1)}\\,,\\qquad \\kappa^{(2,1)} =\n  2\\,\\beta_1\\,\\kappa^{(0,0)} +\n  3\\,\\beta_0\\,\\kappa^{(1,0)}\\,,\\\\ \\kappa^{(3,3)} &=\n  \\frac{4}{3}\\beta_0\\,\\kappa^{(2,2)}\\,,\\qquad \\kappa^{(3,2)} =\n  \\frac{3}{2}\\,\\beta_1\\,\\kappa^{(1,1)} +\n  2\\,\\beta_0\\,\\kappa^{(2,1)}\\,,\\\\ \\kappa^{(3,1)} &=\n  2\\,\\beta_2\\,\\kappa^{(0,0)} + 3\\,\\beta_1\\,\\kappa^{(1,0)} +\n  4\\,\\beta_0\\,\\kappa^{(2,0)}\\,.\n\\label{eq:murrec}\n\\end{split}\n\\end{equation}\nInserting these coefficients into the hadronic analog of\n\\eqn{eq:murdepparton}, it is possible to obtain the hadronic cross\nsection at any value of $\\mu_\\text{R}$ without any further numerical\nintegration. Since the $\\mu_\\text{R}$ dependence is typically much larger than\nthe $\\mu_\\text{F}$ dependence for gluon fusion, this feature of \\sushi{} saves a\nsignificant amount of computing time when aiming for an estimate of the\ntheoretical uncertainty of the cross section.\n\nThus, in addition to the usual output file {\\tt <outfile>}, running\n\\sushi{\\_\\sushiversion}{} with the standard command\\\\[-2em]\n\n{\\footnotesize\n\\begin{verbatim}\n.\/bin\/sushi <infile> <outfile>\n\\end{verbatim}}\n\n\\vspace*{-1em}\nwill produce an additional file {\\tt <outfile>\\_murdep} which contains\nthe gluon-fusion cross section for several values of $\\mu_\\text{R}$ in the form\n\\begin{equation}\n\\begin{array}{rrrrr}\n  \\mu_\\text{R}\\text{\/GeV}\\quad\n  & \\sigma_\\text{{\\abbrev LO}}\\text{\/pb}\\quad\n  & \\sigma_\\text{{\\abbrev NLO}}\\text{\/pb}\\quad\n  & \\sigma_\\text{{\\abbrev NNLO}}\\text{\/pb}\\quad\n & \\sigma_\\text{\\nklo{3}}\\text{\/pb}\n\\end{array}\n\\end{equation}\nwhere all cross sections are evaluated following\n\\eqn{eq:sigmasushi}, i.e.\\ they potentially contain quark-mass\neffects, {\\abbrev SUSY}{} corrections, and\/or electroweak effects.  The values\nof $\\mu_\\text{R}$ to be scanned over can be set in {\\tt <infile>}\nthrough\n\\begin{lstlisting}\nBlock SCALES\n    1  <mu0mh>\n  102  <min0>  <max0>  <N>\n\\end{lstlisting}\nwhich will cause \\sushi{\\_\\sushiversion}{} to evaluate the cross section at $N+1$\nequidistant points for $\\log\\mu_\\text{R}$ between $\\log\\mu_\\text{min}$ and\n$\\log\\mu_\\text{max}$, meaning\\footnote{{\\tt <N>}=$N$, {\\tt\n    <min0>}=$\\mu_\\text{min}\/\\mu_0$, {\\tt <max0>}=$\\mu_\\text{max}\/\\mu_0$,\n  {\\tt <mu0mh>}=$\\mu_0\/M_\\phi$.}\n\\begin{equation}\n  \\begin{split}\n    \\mu_\\text{R}\n    &=\n    \\mu_\\text{min}\\left(\\frac{\\mu_\\text{max}}{\\mu_\\text{min}}\\right)^{i\/N}\\,,\\qquad\n    i\\in\\{0,1,\\ldots,N\\}\\,.\n  \\end{split}\n  \\label{eq:mursampling}\n\\end{equation}\nIn addition, \\sushi{\\_\\sushiversion}{} includes a theoretical error estimate on the\ninclusive cross section into the standard output file {\\tt <outfile>},\ngiven as the maximal and minimal deviation (in pb) within the interval\n$\\mu_\\text{R}\\in [\\mu_1,\\mu_2]$ from the value at $\\mu_\\text{R}=\\mu_0$, using the\nsampled values of $\\mu_\\text{R}$ defined in \\eqn{eq:mursampling}, and the cross\nsections at the two boundaries $\\mu_\\text{R}=\\mu_1$ and $\\mu_2$. The interval is\nspecified as\n\\blockentry{SCALES}{101}{=\\{$\\mu_{1}\/\\mu_0$,$\\mu_2\/\\mu_0$\\}} (recall\nthat $\\mu_0\/M_\\phi=$\\blockentry{SCALES}{1}{}); it defaults to\n$[\\mu_1,\\mu_2]=[\\mu_0\/2,2\\mu_0]$.\n\nWe remark that this feature works at all perturbative orders through\n\\nklo{3}, for any settings in the blocks {\\tt GGHMT} or {\\tt GGHSOFT},\nand for any model under consideration. The only restriction is that all\nparameters except for the strong coupling constant need to be defined\non-shell. If this is not the case, \\sushi{\\_\\sushiversion}{} will not produce {\\tt\n  <outfile>\\_murdep}. Note that, due to \\eqn{eq:sigmasushi}, the\nprocedure implemented in \\sushi{\\_\\sushiversion}{} is a slightly refined version of\nthe one described above. In particular, this implies that the {\\abbrev NNLO}{}\n$\\mu_\\text{R}$ dependence is {\\it exact}, since it is fully determined by the\nexact {\\abbrev NLO}{} cross section $\\sigma_\\text{{\\abbrev NLO}}$. On the other hand, the\nrenormalization-scale dependence at \\nklo{3} derived from the {\\scalefont{.9}\n  RGE} procedure inherits whatever approximations were made (or not\nmade) at {\\abbrev NNLO}{}.  Thus, the results obtained through the standard and\nthe {\\scalefont{.9} RGE} procedure are usually not identical. For example, if\none keeps the full $x$-dependence at {\\abbrev NNLO}{}, one also obtains the full\n$x$-dependence of the $\\mu_\\text{R}$-terms at \\nklo{3} with the {\\scalefont{.9} RGE}\nprocedure, while the standard procedure would only provide them in the\nsoft expansion.\n\n\\subsection{Effective Lagrangian - \\dimension{5} operators}\\label{sec:dim5}\n\nLet us start from a particular well-defined theory {\\abbrev TH}{}; in the\ncurrent version of \\sushi{}, this could be the {\\abbrev SM}{}, a general {\\abbrev 2HDM}{},\nthe {\\abbrev MSSM}{}, or the {\\abbrev NMSSM}{}. We may now include additional gauge\ninvariant \\dimension{5} operators to {\\abbrev TH}{} which couple the neutral\nHiggs bosons of {\\abbrev TH}{} to gluons in the following\nway\\footnote{{\\abbrev CP}{}-even and -odd scalars, which couple\nthrough \\dimension{5} operators only, can also be studied, see the description after \\eqn{eq:yukfactors}.}:\n\\begin{equation}\n\\begin{split}\n{\\cal L} &= {\\cal L}_\\text{{\\abbrev TH}} +\n\\sum_{i=1}^{N_1}\\frac{\\alpha_s}{12\\pi v}c_{5,1i}\\,H_{1i}G^a_{\\mu\\nu}G^{a,\\mu\\nu} +\n\\sum_{i=1}^{N_2}\\frac{\\alpha_s}{8\\pi v}c_{5,2i}\\,H_{2i}G^a_{\\mu\\nu}\\tilde\nG^{a,\\mu\\nu}\\,.\n\\label{eq:leff}\n\\end{split}\n\\end{equation}\nHere, ${\\cal L}_\\text{{\\abbrev TH}}$ is the Lagrangian of the initial theory\n{\\abbrev TH}{}, $G^a_{\\mu\\nu}$ is the gluonic field strength tensor with\ncolor index $a$ and Lorentz indices $\\mu$ and $\\nu$, and $\\tilde\nG_{\\mu\\nu}^a\\equiv \\varepsilon_{\\mu\\nu\\rho\\sigma}G^{a,\\rho\\sigma}$ is\nits dual ($\\varepsilon^{0123}=+1$). As usual, $\\alpha_s$ is the strong\ncoupling constant and $v$ the {\\abbrev SM}{} Higgs-boson vacuum expectation\nvalue, which we express in terms of Fermi's constant\n$v=1\/\\sqrt{\\sqrt{2}G_F}$.  $N_1$ and $N_2$ are the numbers of {\\scalefont{.9}\n  CP}-even and {\\scalefont{.9} CP}-odd Higgs bosons of the theory,\nrespectively. The particles themselves are generically denoted by\n$H_{1i}$ and $H_{2i}$ (cf.\\ also Table\\,\\ref{tab:htype} below).\n\nThe $c_{5,ni}$ denote dimensionless Wilson coefficients which are\nunderstood as perturbative series in $\\alpha_s$:\n\\begin{equation}\n c_{5,ni} = \\sum_{k=0}^3 \\left(\\frac{\\alpha_s}{\\pi}\\right)^k c_{5,ni}^{(k)}\\,.\n\\end{equation}\nThe normalization is such that $c_{5,ni}^{(0)}=1$ corresponds to the\n{\\abbrev LO}{} contribution of an infinitely heavy up-type quark $u'$ with\n{\\abbrev SM}{}-like couplings.\\footnote{``{\\abbrev SM}-like'' refers to the interaction\n  Lagrangian ${\\cal L}_\\text{int} = -(m_{u'}\/v)H_{1i}\\bar{u'} u'$ for a\n  {\\scalefont{.9} CP}-even, and ${\\cal L}_\\text{int} =\n  -i(m_{u'}\/v)H_{2i}\\bar{u'}\\gamma_5 u'$ for a {\\scalefont{.9} CP}-odd Higgs\n  boson.} The {\\abbrev NLO}{} term for a {\\scalefont{.9} CP}-even Higgs in this case\nwould be $c_{5,11}^{(1)}=\\tfrac{11}{4}$, etc. In a theory that obeys\nnaturalness, on the other hand, the order of magnitude of the Wilson\ncoefficients would be $c_{5,ni} = \\order{v\/\\Lambda}$, where $\\Lambda$ is\na scale of physics beyond the {\\abbrev SM}{}.\n\nThe basic structures for the implementation of the effective Lagrangian\nin \\eqn{eq:leff} have already been present in earlier versions of\n\\sushi{}. The reason for this is that the very same operators result\nfrom integrating out the top quark or heavy squarks and gluinos from\n${\\cal L}_\\text{{\\abbrev TH}}$. In fact, the {\\abbrev NNLO}{} corrections due to top\nquarks, as well as the {\\abbrev NLO}{} corrections due to top, stop, and gluino\nare evaluated on the basis of these \\dimension{5}\noperators.\n\nThus, \\sushi{\\_\\sushiversion}{} does not implement any new results; it simply re-uses\npreviously available {\\tt functions} and {\\tt subroutines} in order\nto extend the gluon-fusion amplitudes to take into account the effect of\nthe additional terms in \\eqn{eq:leff}.  The numerical values for the\ncoefficients $c_{5,ni}$ in \\eqn{eq:leff} are specified through the newly\nintroduced {\\tt Block DIM5}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|ccc|}\n\\hline\n{\\tt <htype>} & {\\abbrev SM} & {\\abbrev 2HDM}\/{\\abbrev MSSM} & {\\abbrev NMSSM} \\\\\n\\hline\n11 & $H$ & $h$ & $H_1$\\\\\n12 & $-$ & $H$ & $H_2$\\\\\n13 & $-$ & $-$ & $H_3$\\\\\n21 & $A$ & $A$ & $A_1$\\\\\n22 & $-$ & $-$ & $A_2$\\\\\n\\hline\n\\end{tabular}\n\\caption[]{\\label{tab:htype} Assignment of the \\sushi{} input parameter\n\\blockentry{SUSHI}{2}{=<htype>} to the type of Higgs boson in the\nvarious models. A dash ($-$) means that the assignment is not\nmeaningful; it will lead to a fatal error in \\sushi{}. }\n\\end{center}\n\\end{table}\n\nFor example, within the {\\abbrev MSSM}{},\n\\begin{lstlisting}\nBlock DIM5\n   11   1.00000000E-04    # c5h0\n   12   4.00000000E-05    # c5H0\n   21  -3.00000000E-07    # c5A0\n\\end{lstlisting}\ncorresponds to $c_{5,11}^{(0)} \\equiv c_{5,h}^{(0)}=10^{-4}$,\n$c_{5,12}^{(0)} \\equiv c_{5,H}^{(0)}=4\\cdot 10^{-5}$, and\n$c_{5,21}^{(0)} \\equiv c_{5,A}^{(0)}=-3\\cdot 10^{-7}$.  Note that\n\\sushi{} calculates the cross section of only one particular type of\nHiggs boson per run (defined in \\blockentry{SUSHI}{2}{}),\nsee \\citere{Harlander:2012pb}. Correspondingly, only the pertinent entry in\n{\\tt Block DIM5} will have an effect on the result, the other\nentries will be ignored. The corrections at higher orders are\nspecified by setting \\blockentry{DIM5}{<k><ni>}{} for\ncoefficients $c_{5,ni}^{(k)}$ with $k\\geq 1$.  At {\\abbrev NLO}{} the\ncontribution of an infinitely heavy up-type quark $u'$ is thus\nreproduced by setting \\blockentry{DIM5}{11}{=1} and\n\\blockentry{DIM5}{111}{=2.75}.\n\nThe scale dependence of the \\dimension{5} Wilson coefficient can be\nderived from the non-renormalization of the trace anomaly\nterm\\,\\cite{Crewther:1972kn,Chanowitz:1972vd,Chanowitz:1972da,Collins:1976yq},\n\\begin{equation}\n  \\begin{split}\n    \\mu^2\\deriv{}{\\mu^2} \\beta(\\alpha_s) G_{\\mu\\nu}G^{\\mu\\nu} \\equiv 0\\,,\n  \\end{split}\n\\end{equation}\nwhere $\\beta(\\alpha_s)$ is given in \\eqn{eq:betafun}. Since also\n$\\alpha_sc_{5,1i}G_{\\mu\\nu}G^{\\mu\\nu}$ must be scale invariant, this immediately\nleads to~\\cite{Spira:1995mt,Brooijmans:2016vro}\n\\begin{equation}\n  \\begin{split}\n    c_{5,1i}(\\mu_\\text{R})=c_{5,1i}(\\mu_{\\phi})\n    \\frac{(\\beta\/\\alpha_s)|_{\\mu_\\text{R}}}{(\\beta\/\\alpha_s)|_{\\mu_{\\phi}}}\\,.\n    \\label{eq:c5evolve}\n  \\end{split}\n\\end{equation}\nPerturbatively, we can write this as\n\\begin{equation}\n  c_{5,1i}(\\mu_\\text{R})=\n  \\sum_{n\\geq 0}\\sum_{l=0}^n\\left(\\frac{\\alpha_s(\\mu_\\text{R})}{\\pi}\\right)^n\n  c_{5,1i}^{(n,l)}(\\mu_{\\phi})l_{R\\phi}^l\\,,\n\\label{eq:pertc5} \n\\end{equation}\nwith $l_{R\\phi}=2\\ln(\\mu_\\text{R}\/\\mu_{\\phi})$,\n\\begin{equation}\n  c_{5,1i}^{(n,0)} = c_{5,1i}^{(n)}(\\mu_{\\phi})\\,,\\quad\n  c_{5,1i}^{(n,n)} = 0\\qquad \\forall\\ n\n\\end{equation}\nand, through {\\abbrev NNLO}{},\n\\begin{equation}\n\\begin{split}\n  &c_{5,1i}^{(2,1)}=\\beta_0c_{5,1i}^{(1,0)}-\\beta_1c_{5,1i}^{(0,0)}\\,,\\\\&\nc_{5,1i}^{(3,2)}=\\beta_0(\\beta_0c_{5,1i}^{(1,0)}-\\beta_1c_{5,1i}^{(0,0)})\\,,\\quad\nc_{5,1i}^{(3,1)}=2(\\beta_0c_{5,1i}^{(2,0)}-\\beta_2c_{5,1i}^{(0,0)})\\,.\n\\end{split}\n\\end{equation}\nSetting \\blockentry{DIM5}{0}{=1} makes \\sushi{} evolve the Wilson\ncoefficient perturbatively, i.e.\\ according to \\eqn{eq:pertc5}; this\nis the default. On the other hand, one can also employ\n\\eqn{eq:c5evolve} for the evolution by setting \\blockentry{DIM5}{0}{=2},\nsimilar to the implementation in {\\tt HIGLU}~\\cite{Spira:1995mt}.\nThe evolution can also be switched off\n(i.e.\\ $c_{5,1i}(\\mu_\\text{R})=c_{5,1i}(\\mu_{\\phi})$) by setting \\blockentry{DIM5}{0}{=0}.\nThe {\\scalefont{.9} RGE} procedure described in the previous\nsection is only applicable for \\blockentry{DIM5}{0}{=1}.\n\\sushi{} will assume the Wilson coefficient provided in the input {\\tt\nBlock DIM5} to be renormalized at $\\mu_{\\phi}=M_\\phi$. The corresponding\nvalues at $\\mu_\\text{R}$ (where $\\mu_\\text{R}$ is given in\n\\blockentry{SCALES}{1}{}) are output in {\\tt Block DIM5OUT}.\n\nMoreover, the inclusion of \\dimension{5} operators is not compatible with\nthe inclusion of $1\/M_\\text{t}$ terms, i.e.\\ \\sushi{} stops if \\blockentry{GGHMT}{1}{$\\neq$0} or\n\\blockentry{GGHMT}{2}{$\\neq$0}. The {\\abbrev LO}{} dependence including quark-mass\neffects must not be factored out, i.e. \\sushi{} only accepts\nthe setting \\blockentry{GGHMT}{-1}{=-1}, in order not to reweight\nthe \\dimension{5} operator contributions with top-quark mass effects.\n\nWe note that through the {\\tt Block FACTORS}, which existed also in\nearlier versions, \\sushi{} allows to alter the couplings of the Higgs\nboson to quarks and squarks. Thus, for example additional factors\n$\\kappa_t$ and $\\kappa_b$ for the Higgs-boson coupling to top and bottom\nquarks can be chosen. In case of the {\\abbrev SM}{} the corresponding Lagrangian\nthen takes the following form for the {\\abbrev CP}{}-even Higgs boson $H_{11}=H$\n\\begin{equation}\n \\mathcal{L}_\\text{{\\abbrev TH}} \\ni -\\kappa_t\\sqrt{2}\\frac{M_\\text{t}}{v}t\\bar t H -\\kappa_b\\sqrt{2}\\frac{M_\\text{b}}{v}b\\bar b H\\,.\n \\label{eq:yukfactors}\n\\end{equation}\nIt is therefore easily possible to perform an analysis as presented in\n\\citere{Grojean:2013nya} in \\sushi{}, where the dependence of the\ngluon-fusion cross section on $\\kappa_t$ and $c_{5,1i}$ is discussed.\nWe will later also focus on this dependence for a very boosted Higgs\ntaking into account the bottom-quark induced contribution in addition.\nMoreover, by setting the couplings to quarks and gauge bosons to\nzero through the settings in {\\tt Block FACTORS} and \\blockentry{SUSHI}{7}{=0}, respectively,\nalso {\\abbrev CP}{}-even or -odd scalars beyond the implemented models can be studied.\nWe will demonstrate this option by providing inclusive cross sections for a scalar with\na mass of $750$\\,GeV at the $13$\\,TeV {\\abbrev LHC}{} in \\sct{sec:numdim5}.\n\n\\section{Heavy-quark annihilation}\\label{sec:heavyquark}\n\nIn this section we shortly comment on the implementation of the total\ninclusive {\\abbrev NNLO}{} Higgs-production cross sections through heavy-quark\nannihilation, ${\\scriptstyle Q}'\\bar{\\scriptstyle Q}\\to \\phi$, as described in\n\\citere{Harlander:2015xur}. Its activation is through the presence of\nthe {\\tt Block QQH} in the input file, which has the following form:\n\\begin{lstlisting}\nBlock QQH\n         1      <parton1>\n         2      <parton2>\n        11      <v*y>\n        12      <mu>\n\\end{lstlisting}\nHere, {\\tt <parton1>}$\\in\\{1,\\dots,5\\}$ denotes the initial-state quark\nflavor ${\\scriptstyle Q}'$, and {\\tt <parton2>}$\\in\\{-1,\\dots,-5\\}$ the initial-state\nanti-quark flavor $\\bar {\\scriptstyle Q}$. {\\tt <v*y>} is the $\\Q'\\bar \\Q\\phi$ coupling in the\n$\\overline{\\mbox{\\abbrev MS}}$ scheme at scale {\\tt <mu>}$=\\mu$\/GeV, normalized such that the\n{\\abbrev SM}{} value of the $q\\bar qH$ coupling is {\\tt <v*y>}$=m_q(\\mu)$\/GeV.\nFor further details regarding the implementation in \\sushi{\\_\\sushiversion}{} and\nresults we refer to \\citere{Harlander:2015xur}.\n\nIf the {\\tt Block QQH} is provided, \\sushi{} will not calculate the\ngluon-fusion cross section. The calculation of heavy-quark annihilation\ncross sections is also compatible with cuts on the (pseudo)rapidity or\ntransverse momentum of the Higgs boson up to $\\mathcal{O}(\\alpha_s^3)$, controlled\nthrough the settings in {\\tt Block DISTRIB}.\nAlso $\\pt{}$ distributions (\\blockentry{DISTRIB}{1}{=1}) can be requested.\nSince all quarks\nare assumed massless in this approach, the underlying theory is chirally\nsymmetric. Therefore the results for a scalar and a pseudo-scalar Higgs\nare identical and the setting of \\blockentry{SUSHI}{2}{} is irrelevant.\nNote also that the collision of an up-type quark with a down-type\nanti-quark (or vice versa) implies that $\\phi$ carries an electric\ncharge.  The only model dependence of the $\\Q'\\bar \\Q\\phi$ cross section as\ncalculated by \\sushi{} is through the setting of the Yukawa coupling in\n\\blockentry{QQH}{11}, such that a calculation in the {\\abbrev SM}{}-mode is\nsufficient (\\blockentry{SUSHI}{1}{=0}), unless the Higgs mass should be\nobtained from some external code like {\\tt FeynHiggs}.\n\nOther parameters of the $\\Q'\\bar \\Q\\phi$ calculation are determined by the same\ninput values as they are used for the $b\\bar b\\phi$ cross section when no input\n{\\tt Block QQH} is present. In particular, the perturbative order of\n$\\Q'\\bar \\Q\\phi$ is controlled through \\blockentry{SUSHI}{6}{=$n$}, where\n$n=1,2,3$ results in the {\\abbrev LO}{}, {\\abbrev NLO}{}, or {\\abbrev NNLO}{} prediction,\nrespectively, and the renormalization and factorization scales (relative\nto $M_\\phi$) are defined through \\blockentry{SCALES}{11}{} and\n\\blockentry{SCALES}{12}{}, respectively.\n\n\\section{Numerical results}\\label{sec:numerics}\n\nThis section demonstrates the newly implemented features of \\sushi{\\_\\sushiversion}{}\nwith the help of exemplary numerical results. We start with a discussion\nof the convergence of the soft expansion at individual perturbative\norders up to \\nklo{3}, proceed with top-quark mass effects in the\neffective field-theory approach, move to the {\\scalefont{.9} RGE} procedure to\ndetermine the renormalization-scale dependence, before we use these\nfeatures to provide a prediction for the cross section of the {\\abbrev SM}{}\nHiggs boson. Finally, we study the effect of higher dimensional\noperators to the transverse momentum~$\\pt{}$ of the {\\abbrev SM}{} Higgs boson\nand provide inclusive cross sections for a {\\abbrev CP}{}-even scalar with a\nmass of $750$\\,GeV. For numerical results concerning heavy-quark annihilation, we refer the\nreader to \\citere{Harlander:2015xur}.\n\nIf not stated otherwise, the setup for the numerical evaluations is as\nfollows: The {\\abbrev LHC}{} center-of-mass energy is set to $\\sqrt{s}=13$\\,TeV,\nand the {\\abbrev SM}{} Higgs mass to $M_\\text{H}=125$\\,GeV. We employ {\\tt\nPDF4LHC15}~\\cite{Butterworth:2015oua,Dulat:2015mca,Harland-Lang:2014zoa,\nBall:2014uwa,Gao:2013bia,Carrazza:2015aoa,Carrazza:2015hva}\nas parton distribution functions\n({\\abbrev PDF}{}), where the {\\tt (n)nlo\\_mc} Monte Carlo is used by default, and\nthe {\\tt (n)nlo\\_100} Hessian sets if noted.  Since \\nklo{3} {\\abbrev PDF}{} sets\nare not available, we use the {\\abbrev NNLO}{} set also for the evaluation of the\n\\nklo{3} terms. Nevertheless, in the \\nklo{3} calculation, we evolve\n$\\alpha_s$ at $4$-loop level; using 3-loop running of $\\alpha_s$\ninstead, the final prediction of the cross section for a {\\abbrev SM}{} Higgs\nboson changes at the level of $10^{-5}$. The remaining input follows the\nrecommendation of the {\\abbrev LHC}{} Higgs cross section working group,\nsee~\\citere{Denner:2047636}. The on-shell charm-quark mass is set to\n$m_c^{\\rm OS} = 1.64$\\,GeV, which is the upper edge of the range given\nin \\citere{Denner:2047636}.  The central scale choice for the\nrenormalization and factorization scale is $\\mu_\\text{R}=\\mu_\\text{F}=M_\\text{H}\/2$.\n\nNote that the results of\n\\scts{sec:numsoftexp}--\\ref{sec:bestandrenorm} are obtained for\na {\\abbrev SM}{} Higgs boson. However, \\sushi{\\_\\sushiversion}{} allows to take into account\nthe effects of \\nklo{3} contributions in the heavy-top limit and\n$1\/M_\\text{t}$ terms to the {\\abbrev NNLO}{} contributions for any {\\abbrev CP}{}-even Higgs\nboson in the implemented models, as long as the mass of the Higgs boson\nunder consideration is sufficiently light, i.e.\\ below $2M_\\text{t}{}$.\nEffects of \\dimension{5} operators (see \\sct{sec:dim5} and\n\\ref{sec:numdim5}), on the other hand, can be taken into account for any\nof the neutral Higgs bosons of the implemented models and\n{\\abbrev CP}{}-even and -odd scalars, which couple through \\dimension{5} operators\nonly.\n\n\\subsection{Soft expansion up to \\nklo{3}}\n\\label{sec:numsoftexp}\n\nIn this section, we study the behavior of the expansion around the\n``soft limit'', $x\\rightarrow 1$, for the gluon-fusion cross section,\nsee also \\sct{sec:softexp}. For the sake of clarity, top-quark\nmass effects beyond {\\abbrev LO}{} will be neglected in this section, although\nthe {\\abbrev LO}{} cross section including the full top-quark mass dependence is\nfactored out to all orders (i.e.\\ we set \\blockentry{GGHMT}{-1}{=3}, see\n\\sct{sec:mt}).\nIn order to discuss the convergence of the soft expansion, we define the\nquantity\n\\begin{equation}\n \\left(\\frac{\\delta \\sigma}{\\sigma}\\right)^{\\text{\\nklo{n}}} =\n \\frac{\\sigma^t_{\\text{\\nklo{n}},N,a}}{\\sigma^t_{\\text{\\nklo{n-1}}}}-1\\quad\\text{\n   with } n\\geq 1 \\,,\n\\label{eq:delsigbysig}\n\\end{equation}\nwhere $\\sigma^t_\\text{\\nklo{n}}$ has been introduced in\n\\eqn{eq:softexp}.  Through $\\order{\\alpha_s^{n+1}}$, the exact\n$x$-dependence is taken into account. In the highest-order terms,\ni.e.\\ the terms of order $\\order{\\alpha_s^{n+2}}$ in\n$\\sigma^t_{\\text{\\nklo{n}},N,a}$, the soft expansion is applied\naccording to \\eqn{eq:softexp} up to order $(1-x)^N$ with $N\\leq\n16$. All studies in this subsection were performed without matching the\ncross section to the result at $x\\to 0$, i.e., we set\n\\blockentry{GGHMT}{$n\\cdot$10}{=0} for $n=1,2,3$.\n\nAt infinite order of the soft expansion, the value of the parameter $a$\nin \\eqn{eq:softexp} is obviously irrelevant.  If only a finite number of\nterms in the expansion is available, the dependence of the result on the\nparameter $a$ has been studied in detail in\n\\citere{Anastasiou:2016cez}. It was shown that the soft expansion seems\nto converge particularly well for small, non-negative values of $a$. The\ndifferences among the final results for different values of $a$ are smaller\nat higher orders, as we demonstrate subsequently. One\nobserves that the $\\mu_\\text{F}$-dependent terms of $\\hat\\sigma$ at {\\abbrev NLO}{} are\npolynomial in $x$, which means that they are {\\it identical} to their\nsoft expansion for $a=0$ once it is taken to sufficiently high order ($N=3$, to be\nspecific). This is no longer true with the choice $a > 0$. Let us add\nthat, since the $\\mu_\\text{R}$-dependent terms at {\\abbrev NLO}{} are proportional to\n$\\delta(1-x)$, they are the same whether the soft expansion is applied\nor not.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_NLO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_NNLO.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the {\\abbrev NLO}{} cross section as a function of\n$N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence of the\n{\\abbrev NNLO}{} cross section as a function of $N$ for $a=0,1,2,3$ in\n\\eqn{eq:softexp}.  In both figures the colors depict $a=0$ (red),\n$a=1$ (blue), $a=2$ (green), $a=3$ (black). The black, dashed line\ncorresponds to the exact result in the heavy-top limit. \nThe results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:softexp}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_N3LO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_N3LO_gg.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the \\nklo{3} cross section as a\nfunction of $N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence\nof the gg channel of the \\nklo{3} as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp}.  A zoom for larger values of $N$ is\nprovided in the upper right corner of the figures.  In both figures\nthe colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green), $a=3$\n(black). The results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}. }\n\\label{fig:softexp2}\n\\end{figure}\n\n\\figs{fig:softexp}\\,(a) and (b) show the convergence of the soft\nexpansion at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, respectively.  Both figures also\ninclude the result without soft expansion as dashed black line,\ni.e.\\ where $\\sigma^t_{\\text{\\nklo{n}},N,a}$ is replaced by\n$\\sigma^t_{\\text{\\nklo{n}}}$ in \\eqn{eq:delsigbysig}. At {\\abbrev NLO}{}, the\ncase $a=0$ appears to be clearly preferable; for larger value of $a$,\nthe soft expansion is further away from the exact\n$x$-dependence. For $N\\geq 9$, the deviation for $a=0$ is less\nthan $2.5$\\%.\\footnote{Note that this refers to the absolute {\\abbrev NLO}{}\n{\\it correction} term in pb; with respect to the total cross\nsection, this translates into an approximation which is better than\n$1.6$\\%.} It decreases down to $1.3$\\% at $N=16$, while the result\nfor $a=1$ is still more than $7$\\% off.  \n\nAt {\\abbrev NNLO}, convergence of the soft expansion appears to be a bit faster,\nwith no significant impact of the terms higher than $(1-x)^6$ both for\n$a=0$ and $a=1$. For $N\\geq 9$, the result for $a=0$ ($a=1$)\napproximates the exact $x$-dependence of the correction term to better\nthan $5$\\% ($2$\\%) (translating into about $0.9$\\% ($0.3$\\%) for the total cross\nsection).\n\n\\fig{fig:softexp2}~(a) depicts the convergence of the soft expansion for\nthe cross section at \\nklo{3}. Above $N=11$, the spread among the\ncurves for $a=0,1,2,3$ is of the order of $3$\\% of\n$\\delta\\sigma\/\\sigma$, which means about $0.1$\\% of the total cross\nsection. For completeness, the same plot for the dominant $gg$ channel\nalone is shown in \\fig{fig:softexp2}~(b). Note that in this case, we\nonly include the $gg$ channel also in the denominator of\n\\eqn{eq:delsigbysig}. At lower orders of the soft expansion, the curve\nfor $a=0$ behaves less smoothly compared to $a\\geq 1$; at sufficiently\nhigh orders though, all results can be considered consistent with each\nother at the level of accuracy indicated above.\n\n\\subsection{Top-quark mass effects through {\\abbrev NNLO}{} and matching to the high-energy limit}\n\\label{sec:topquarkmass}\n\nIn this section we comment on top-quark mass effects beyond the\nheavy-top limit, which can be taken into account in \\sushi{} up to\n$1\/M_\\text{t}^{10}$ at {\\abbrev LO}{} and {\\abbrev NLO}{} and up to $1\/M_\\text{t}^6$ at {\\abbrev NNLO}{}. As\nalready pointed out in \\sct{sec:mt}, a naive expansion of the\npartonic cross section in $1\/M_\\text{t}$ breaks down. Thus, in this section,\nwe apply the matching to the high-energy limit as described in\n\\sct{sec:mt}, i.e.\\ we set\n\\blockentry{GGHMT}{$n\\cdot$10}{=1} for $n=1,2,3$.\n\nRecall that the matching procedure of\n\\citeres{Harlander:2009mq,Harlander:2009my} requires the soft expansion\nof the partonic cross section. Thus before discussing the relevance of\nthe top-quark mass effects, it is necessary to study the convergence of\nthe soft expansion also for these terms.  For the result at {\\abbrev NLO}{} we\ncan compare to the result in the heavy-top limit, but also to the exact\ntop-quark mass dependence; the difference between these two results is\nabout $1$\\%.  At {\\abbrev NNLO}{}, on the other hand, only a comparison to the\nheavy-top limit is possible.  The results are shown in\n\\fig{fig:softexpmt}, including terms through $1\/M_\\text{t}^8$ at {\\abbrev NLO}{}, and\nthrough $1\/M_\\text{t}^4$ at {\\abbrev NNLO}{} (for the $gg$ and the $qg$ channels\nalso $1\/M_\\text{t}^6$ terms are implemented in \\sushi{} but provide a\nnegligible contribution, see \\fig{fig:mtterms} below).\nFollowing \\eqn{eq:delsigbysig}, we keep the exact\n$x$-dependence one order below to allow for a better comparison with\nthe figures of \\sct{sec:numsoftexp}. At {\\abbrev NLO}{}, one observes a nice convergence of\nthe soft expansion to the exact result, provided $a=0$. Terms beyond\n$(1-x)^{10}$ have only negligible effects on the final result in this\ncase. At {\\abbrev NNLO}{}, convergence of the soft expansion is significantly\nslower, but the available number of terms in this expansion seems\nsufficient for a prediction of the mass effects with permille level\naccuracy, provided that $a=0$ is indeed the most reliable choice for the\nparameter defined in \\eqn{eq:softexp}.  \\fig{fig:softexpmt2} shows\nthe \\nklo{3} result with matching to the high-energy limit as\ndescribed in \\sct{sec:mt}. The convergence of the soft expansion as a\nfunction of $N$ is slightly worse compared to the result without\nmatching, but shows a similar behavior as the results at {\\abbrev NLO}{} and\n{\\abbrev NNLO}{} depicted in \\fig{fig:softexpmt}. The correction\nat $N=16$ is comparable to the result without matching, see\n\\fig{fig:softexp2}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_NLO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_NNLO.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Convergence of the {\\abbrev NLO}{} cross section as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp} with top-quark mass effects up to $1\/M_\\text{t}^8$;\n(b) Convergence of the {\\abbrev NNLO}{} cross section as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp} with top-quark mass effects up to $1\/M_\\text{t}^4$.\nIn both figures the colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green),\n$a=3$ (black). The black, dashed line corresponds to\nthe exact result in the heavy-top limit, the black, dot-dashed line\nto the exact result with full top-quark mass dependence (only known at {\\abbrev NLO}{}).\nThe results are obtained for a {\\abbrev SM}{} Higgs with $M_\\text{H}=125$\\,GeV\nat the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.\n}\n\\label{fig:softexpmt}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_N3LO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_N3LO_gg.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the \\nklo{3} cross section as a\nfunction of $N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence\nof the $gg$ channel of the \\nklo{3} as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp}.  A zoom for larger values of $N$ is\nprovided in the upper right corner of the figures.  In both figures\nthe colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green), $a=3$\n(black). In contrast to \\fig{fig:softexp2} the \\nklo{3} result\nis matched to the high-energy limit. \nThe results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:softexpmt2}\n\\end{figure}\n\nLet us now discuss the top-quark mass effects at different orders\n$1\/M_\\text{t}^P$ in more detail, while applying the soft expansion through\n$(1-x)^{16}$ with $a=0$ (see \\eqn{eq:softexp}).  The result is presented\nin \\fig{fig:mtterms}, where the relative difference\n\\begin{equation}\n  \\begin{split}\n    \\left(\\frac{\\delta\\sigma}{\\sigma}\\right)_{M_\\text{t}}\n    =\\frac{\\sigma_{P}}%\n    {\\sigma_\\text{htl}}\n    -1\n    \\label{eq:delsigmt}\n  \\end{split}\n\\end{equation}\nto the heavy-top limit at the corresponding perturbative order is shown.\nAt {\\abbrev NLO}{}, $\\sigma_{P}$ is obtained by including terms of order\n$1\/M_\\text{t}^P$ in the partonic cross section and matching it to the $x\\to0$\nlimit (i.e.\\ \\blockentry{GGHMT}{1}{=$P$}, \\blockentry{GGHMT}{10}{=1},\n\\blockentry{GGHSOFT}{1}{=\\{1,16,0\\}}), while $\\sigma_\\text{htl}$ is the\nheavy-top limit at {\\abbrev NLO}{}\n(i.e.\\ \\blockentry{GGHMT}{1}{=}\\blockentry{GGHMT}{10}{=0},\n\\blockentry{GGHSOFT}{1}{=\\{0,0,0\\}}). In both cases, the value for the\ncross section provided by \\sushi{} in \\blockentry{XSGGHEFF}{1}{} is used.\n\nAt {\\abbrev NNLO}{}, we use \\eqn{eq:sigmasushi} which corresponds to the \\sushi{}\noutput \\blockentry{SUSHIggh}{1}, neglecting bottom- and charm-quark, and\nelectroweak effects\n(\\blockentry{FACTORS}{1}{=}\\blockentry{FACTORS}{3}{=}\\blockentry{SUSHI}{7}{=0}).\nFurthermore, we make sure that only the genuine {\\abbrev NNLO}{} effects of the\n$1\/M_\\text{t}$ terms are shown, by fixing the approximation used at\n$\\order{\\alpha_s^3}$; specifically, we set \\blockentry{GGHMT}{1}{=6},\n\\blockentry{GGHMT}{10}{=1}, and \\blockentry{GGHSOFT}{1}{=\\{1,16,0\\}}, both for\n$\\sigma_P$ and $\\sigma_\\text{htl}$. For the $\\mathcal{O}(\\alpha_s^4)$-terms, we apply\nthe analogous settings of the {\\abbrev NLO}{} case described above. I.e., we\ninclude terms of order $1\/M_\\text{t}^P$ in $\\sigma_P$ (modulo the restriction\nto $1\/M_\\text{t}^4$ for the pure quark channels, see above), and match them\nto the $x\\to 0$ limit, while we apply the usual heavy-top limit\nfor $\\sigma_\\text{htl}$.\n\nThe results are shown in \\fig{fig:mtterms}, together with the relative\ndifference of the {\\it exact} {\\abbrev NLO}{} cross section to its heavy-top\nlimit (black dashed).\nThe points at $P=0$ illustrate the effect of using the soft expansion\ncombined with matching to the result at $x=0$, as opposed to keeping the\nfull $x$ dependence (without matching). Both at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, this\neffect is obviously larger than the genuine $1\/M_\\text{t}$-terms.  This\nunderlines that, as long as one works in a heavy-top approximation,\nwhich is strictly valid only for $x>M_\\phi^2\/(4M_\\text{t}^2)$, the full\n$x$-dependence is not necessarily an improvement w.r.t.\\ the soft\nexpansion, in particular if additional information like the $x\\to 0$\nlimit is available.\n\nBoth at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, the $1\/M_\\text{t}$ terms exhibit\na nice convergence behavior. However, the observation at {\\abbrev NLO}{} is that,\nwhile the $1\/M_\\text{t}^0$ result almost exactly reproduces the full mass\ndependence after matching to the high-energy limit and employing the soft expansion,\nincluding higher-order mass effects moves the approximation\n{\\it away} from the exact result. Thus, we cannot expect that their\ninclusion at {\\abbrev NNLO}{} leads to an improved result w.r.t.\\ the heavy-top\nlimit.  Nevertheless, we believe that their overall behavior allows to\nderive an upper bound on the top-mass effects to the heavy-top limit of\nthe order of\n$1$\\%\\,\\cite{Harlander:2009my,Harlander:2009mq,Pak:2009dg}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{figures\/step2_scale05_mt_rel.pdf}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ Relevance of $1\/M_\\text{t}^P$ terms to the cross section in the\nheavy-top limit in percent at {\\abbrev NLO}{} (black) up to $P=10$ and at {\\abbrev NNLO}{}\n(red) up to $P=6$ with respect to the exact heavy-top limit at the\ncorresponding order.  The black, dashed line corresponds to the exact\n{\\abbrev NLO}{} result. The results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.  }\n\\label{fig:mtterms}\n\\end{figure}\n\n\\subsection{Cross section prediction for the {\\abbrev SM}{} Higgs boson and scale dependence}\n\\label{sec:bestandrenorm}\n\nHaving discussed the top-quark mass terms to the {\\abbrev NLO}{} and {\\abbrev NNLO}{}\ncross section in the heavy-top limit and the convergence of the soft\nexpansion, we can finally provide a prediction for the cross section of\nthe {\\abbrev SM}{} Higgs boson including its scale uncertainty. In this section\nwe make use of the Hessian {\\abbrev PDF}{} sets {\\tt PDF4LHC15\\_(n)nlo\\_100}.\nFollowing the arguments of the preceding sections, the best prediction\nof \\sushi{} is obtained with the following settings: use the\nperturbative result through \\nklo{3}, i.e.\\ set\n\\blockentry{SUSHI}{5}{=3}; at each order of the effective-theory result,\napply the soft expansion through $(1-x)^{16}$ with $a=0$, i.e.\\ set\n\\blockentry{GGHSOFT}{$n$}{=\\{1,16,0\\}} for $n\\in\\{1,2,3\\}$; take into\naccount top-quark mass terms to the predictions of the {\\abbrev NLO}{} and\n{\\abbrev NNLO}{} cross sections in the heavy-top limit through the settings\n\\blockentry{GGHMT}{$n$}{=4} for $n\\in\\{1,2\\}$, i.e.\\ $1\/M_\\text{t}^4$ terms\nare taken into account at {\\abbrev NLO}{} and {\\abbrev NNLO}{};\nmatch to the high-energy limit $x\\rightarrow 0$ at {\\abbrev NLO}{}, {\\abbrev NNLO}{},\nand \\nklo{3}, i.e.\\ set \\blockentry{GGHMT}{$n\\cdot$10}{=1} for\n$n=1,2,3$. The choice of $a=0$ is motivated through the reproduction\nof the correct scale dependence at {\\abbrev NLO}{} and the observations in\n\\sct{sec:numerics}. Also note that for all predictions in the effective\nfield-theory approach, we factor out the full top-quark mass dependence,\ni.e.\\ \\blockentry{GGHMT}{-1}{=3}.  Finally, we include the electroweak\ncorrection factor according to \\eqn{eq:sigmasushi}, i.e.\\ we set\n\\blockentry{SUSHI}{7}{=2}.  The exact {\\abbrev NLO}{} cross section of\n\\eqn{eq:sigmasushi} contains the contributions from the three heaviest\nquarks: top, bottom, and charm.  The numbers can be reproduced with the\ninput file {\\tt SM-N3LO\\_best.in} in the {\\tt example}-folder of the\n\\sushi{\\_\\sushiversion}{} distribution.\n\n\\newpage\nWith this setup, we obtain\n\\vspace{-7mm}\n\\begin{equation}\n\\begin{split}\n \\text{{\\abbrev NNLO}{}}:                 &\\qquad \\sigma = 43.55\\,\\text{pb}\\pm 4.44\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{\\nklo{3}}:               &\\qquad \\sigma = 45.20\\,\\text{pb}\\pm 1.61\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{$1\/M_\\text{t}$ effects at {\\abbrev NLO}{} and {\\abbrev NNLO}{}}\\,\\, &\\\\\n +\\text{matching ($x\\to 0$) at {\\abbrev NLO}{}, {\\abbrev NNLO}{} and \\nklo{3}}:&\\qquad \\sigma = 45.80\\,\\text{pb}\\pm 1.87\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{electroweak corrections}:&\\qquad \\sigma = 48.28\\,\\text{pb}\\pm 1.97\\,\\text{pb}(\\mu_\\text{R})\\,, \n \\label{eq:res}\n\\end{split}\n\\end{equation}\nwhere the uncertainty $\\pm \\Delta(\\mu_\\text{R})$ only takes into account the\nrenormalization-scale dependence. Here, $\\Delta(\\mu_\\text{R})$ is the maximum\ndeviation of the cross section within the interval $\\mu_\\text{R}\/M_\\text{H}\n\\in[1\/4,1]$ from the value at $\\mu_\\text{R}=M_\\text{H}\/2$. Each line of\n\\eqn{eq:res}, including the uncertainty, has been obtained in a single\nrun of \\sushi{}, which takes a few seconds on a modern desktop\ncomputer. The final result is perfectly consistent within its\nuncertainties with the prediction $48.58\\,\\text{pb}\\pm 1$\\,pb($\\mu_\\text{R}$) given\nin \\citere{Anastasiou:2016cez} and the result $48.1\\,\\text{pb}\\pm 2.0$\\,pb\n(without resummation) employing the Cacciari-Houdeau Bayesian approach~\\cite{Cacciari:2011ze}\nto estimate higher unknown orders presented in \\citere{Bonvini:2016frm}.\nWe note that the result of\n\\citere{Anastasiou:2016cez} was computed with the {\\abbrev NNLO}{} {\\abbrev PDF}{} set at\nall orders, whereas we employ the {\\abbrev NLO}{} {\\abbrev PDF}{} set for the {\\abbrev NLO}{} terms\nin \\eqn{eq:sigmasushi}.  If we employ {\\tt PDF4LHC15\\_nnlo\\_100} instead\nat all orders, we obtain $48.37$\\,pb.  Other uncertainties need to be\nadded as described in \\citeres{deFlorian:2016spz,Anastasiou:2016cez}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/hessian_scale_analytic.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/hessian_scale_num.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) {\\abbrev LO}{} (red, dotted), {\\abbrev NLO}{} (green, dashed), {\\abbrev NNLO}{}\n(blue, dot-dashed) and \\nklo{3} (black, solid) gluon-fusion cross\nsection in pb (see \\eqn{eq:sigmasushi}) as a function of\n$\\mu_\\text{R}\/M_\\text{H}$ (obtained in a single run); (b) Best prediction cross\nsection in pb as a function of $\\mu_\\text{F}\/M_\\text{H}$ (together with\n$\\mu_\\text{R}=M_\\text{H}\/2$) (blue) and $\\mu_\\text{F}\/M_\\text{H}=\\mu_\\text{R}\/M_\\text{H}$ (red) and\n$\\mu_\\text{R}\/M_\\text{H}$ (together with $\\mu_\\text{F}=M_\\text{H}\/2$) (black). Each curve\nis shown twice, once for $a=0$ (solid) and $a=1$ (dashed) in the soft\nexpansion at \\nklo{3}. The dotted, thin black line depicts the \\nklo{3}\nresult from (a). Both figures are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}. }\n\\label{fig:scaledep}\n\\end{figure}\n\nRunning the input file {\\tt SM-N3LO\\_best.in} also generates a file\nincluding the renormalization-scale dependence. Its content is shown in\n\\fig{fig:scaledep}~(a). The dependence clearly\nreduces successively from {\\abbrev NLO}{} to \\nklo{3}.  Note that at\neach order we follow \\eqn{eq:sigmasushi} and thus include the\nelectroweak correction factor beyond {\\abbrev LO}{}.  The flat behavior around\n$\\mu_\\text{R}=M_\\text{H}\/2$ leads to a highly asymmetric scale variation around the\ncentral value, suggesting a symmetrization of the corresponding\nuncertainty band as done in \\eqn{eq:res}.  As explained in\n\\sct{sec:scaledep}, the $\\mu_\\text{R}$ dependence obtained through the {\\scalefont{.9}\n  RGE} procedure at \\nklo{n} is as precise as the calculation at\n\\nklo{n-1}, while in the standard procedure (by manually varying\n\\blockentry{SCALES}{1}{}), its precision is determined by the \\nklo{n}\ncalculation.  We show the result of the standard procedure in\n\\fig{fig:scaledep}~(b) (black lines). In addition, the $\\mu=\\mu_\\text{F}$\ndependence for $\\mu_\\text{R}=M_\\text{H}\/2$ (blue) and the combined $\\mu=\\mu_\\text{F}=\\mu_\\text{R}$\ndependence (red) are shown. In each case, the solid and dashed line\ncorresponds to setting $a=0$ and $a=1$ in \\eqn{eq:softexp},\nrespectively.  The differences between these two cases, as well as\nbetween the standard and the {\\scalefont{.9} RGE} procedure are small, except for small\nvalues of $\\mu$. We also observe that the behavior at low values of $\\mu$ in\n\\fig{fig:scaledep}~(b) is dependent on the soft expansion and the\nmatching performed at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.\nHowever, within the interval $\\mu\\in\\left[M_\\text{H}\/4,M_\\text{H} \\right]$\nwhich we use for the uncertainty determination, the agreement is good.\n\n\\subsection{Dimension~$5$ operators}\n\\label{sec:numdim5}\n\nIn order to study the effect of the \\dimension{5} operators, it is\nhelpful to consider the fraction of events where the {\\abbrev SM}{} Higgs boson is\nproduced at transverse momenta above a certain value~$\\pt{}^\\text{cut}$. We define\n\\begin{equation}\n\\begin{split}\nR(\\pt{}^\\text{cut})\n=\\frac{1}{\\sigma^\\text{tot}}\\sigma(\\pt{}^\\text{cut})\\quad\\text{with}\\quad\n\\sigma(\\pt{}^\\text{cut})\n=\\int_{\\pt{}>\\pt{}^\\text{cut}}\n{\\rm d}\\pt{}\\frac{{\\rm d}\\sigma}{{\\rm d}\\pt{}}\\,,\n\\label{eq:ptrat}\n\\end{split}\n\\end{equation}\nwhere $\\sigma\\equiv \\sigma_{ni}(c_{5,ni})$ denotes the cross section for\nthe production of a Higgs boson $H_{ni}$ within the theory defined by\n\\eqn{eq:leff}, and follow the numerical setup described at the beginning\nof \\sct{sec:numerics}.  However, we do not take into account charm-quark\nand electroweak contributions and choose a $\\pt{}$-dependent\nrenormalization and factorization scale for the result presented in\n\\fig{fig:corrfactors}.\nIf not stated otherwise, the relative Yukawa couplings to top- and\nbottom quarks are set to one, i.e.\\ we discuss the specific model\n{\\abbrev TH}{} with additional \\dimension{5} operator.\nIn the subsequent {\\abbrev NLO}{} analysis, we set $c_{5}^{(1)}=\\tfrac{11}{4}c_{5}^{(0)}$,\ni.e.\\ our \\dimension{5} operator assumes the same (rescaled) {\\abbrev NLO}{}\ncorrection as for the top-quark induced Wilson coefficient.\n\nThe ratio $R(\\pt{}^\\text{cut})$ of \\eqn{eq:ptrat} is shown in\n\\fig{fig:r-11} for the {\\abbrev SM}{} Higgs boson as a function of\n(a)~$\\pt{}^\\text{cut}$ for various values of $c_{5,H}^{(0)}$, and (b)~$c_{5,H}^{(0)}$\nfor various values of $\\pt{}^\\text{cut}$.  Similarly, \\fig{fig:r-21}\nshows the ratio for a {\\abbrev CP}{}-odd Higgs boson with mass $125$\\,GeV. For\n\\fig{fig:r-11}~(a) and \\fig{fig:r-21}~(a), $\\sigma^\\text{tot}$ is chosen\nsuch that each $R(\\pt{}^{\\text{cut}})$ is normalized to its {\\abbrev NLO}{}\ninclusive cross section.  For \\fig{fig:r-11}~(b) and \\fig{fig:r-21}~(b),\n$\\sigma^{\\text{tot}}=\\sigma(\\pt{}^{\\text{cut}})$ for $c_5=0$ to ensure\nthat all curves start at one.  The minima, which are clearly visible\naround $\\pt{}^{\\text{cut}}=50$\\,GeV, are induced by the negative\ninterference with the bottom-quark induced contributions to gluon\nfusion, which turns into a positive interference for higher values of\n$\\pt{}^{\\text{cut}}$. Accordingly, these minima affect also the\ndependence on $c_5$ in \\fig{fig:r-11}~(b) and\n\\fig{fig:r-21}~(b), i.e.\\ the lowest curve is obtained for a value of $\\pt{}^{\\text{cut}}$\naround $40$\\,GeV. Apart from the impact on the\ninclusive cross section, the point-like\ninteraction encoded in the coefficient $c_5$ thus distorts the shape of\nthe $\\pt{}$ distributions with respect to the loop-induced massive top-\nand bottom-quark contributions, as expected.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/c11.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/p11.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Ratio of $R(\\pt{}^\\text{cut})$ with different $c_{5,H}^{(0)}$ (see figure)\nand $R(\\pt{}^\\text{cut})$ with $c_{5,H}=0$ as a function of $\\pt{}^\\text{cut}$ in GeV;\n(b) Ratio of $R(\\pt{}^\\text{cut})$ and $R(\\pt{}^\\text{cut})$ with $c_{5,H}=0$\nas a function of $c_{5,H}^{(0)}$ for different $\\pt{}^\\text{cut}$ (see figure).\nIn both figures we set $c_{5,A}^{(1)}=\\tfrac{11}{4}c_{5,A}^{(0)}$.\nBoth figures are obtained for a {\\abbrev CP}{}-even {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:r-11}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/c21.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/p21.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Ratio of $R(\\pt{}^\\text{cut})$ with different $c_{5,A}^{(0)}$ (see figure)\nand $R(\\pt{}^\\text{cut})$ with $c_{5,A}=0$ as a function of $\\pt{}^\\text{cut}$ in GeV;\n(b) Ratio of $R(\\pt{}^\\text{cut})$ and $R(\\pt{}^\\text{cut})$ with $c_{5,A}=0$\nas a function of $c_{5,A}^{(0)}$ for different $\\pt{}^\\text{cut}$ (see figure).\nIn both figures we set $c_{5,A}^{(1)}=\\tfrac{11}{4}c_{5,A}^{(0)}$.\nBoth figures are obtained for a {\\abbrev CP}{}-odd Higgs\nwith $m_A=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.\n}\n\\label{fig:r-21}\n\\end{figure}\n\nFollowing the study performed in \\citere{Grojean:2013nya}, we now work\nout the dependence of the cross section with a minimal cut on $\\pt{}$ on\nthe factors $\\kappa_t$ and $c_{5,H}^{(0)}$ for the {\\abbrev SM}{} Higgs\nboson\\footnote{Our $c_{5,H}^{(0)}$ corresponds to $\\kappa_g$ in\n\\citere{Grojean:2013nya}.}.  In addition, we include the dependence on\nthe bottom-quark induced contribution through the factor $\\kappa_b$,\nsince the latter is non-negligible for $\\pt{}^\\text{cut}<200$\\,GeV.  For\nthis study we also choose $\\pt{}$-dependent renormalization and\nfactorization scales $\\mu_\\text{R}=\\mu_\\text{F}=\\sqrt{M_\\text{H}^2+\\pt{}^2}\/2$, which is\npossible through the setting \\blockentry{SCALES}{3}{=1}.  We define\n$\\tilde{\\sigma}(\\pt{}^\\text{cut})$, which just includes the top-quark\ninduced contribution, i.e.\\ we set $\\kappa_t=1$ and $c_{5,H}=\\kappa_b=0$,\nand then perform a fit of\n\\begin{equation}\n \\frac{\\sigma(\\pt{}^\\text{cut})}{\\tilde{\\sigma}(\\pt{}^\\text{cut})}\n =(\\kappa_t+c_{5,H}^{(0)})^2+\\delta\\kappa_tc_{5,H}^{(0)}+\\epsilon (c_{5,H}^{(0)})^2\n +\\delta_{bt}\\kappa_b\\kappa_t+\\delta_{bg}\\kappa_bc_{5,H}^{(0)}+\\epsilon_b\\kappa_b^2\\,, \n\\end{equation}\nwhere we set $c_{5,H}^{(1)}=\\tfrac{11}{4}c_{5,H}^{(0)}$ and\n$\\delta$ and $\\epsilon$ are defined identically to\n\\citere{Grojean:2013nya}.\nIn addition, however, we include the\nbottom-quark induced contribution, which is understood as pure\ncorrection entering through $\\delta_{bg}$, $\\delta_{bt}$, and\n$\\epsilon_{b}$.  The values for $\\delta$ and $\\epsilon$ coincide at the\npercent level with the values of Table~$1$ in \\citere{Grojean:2013nya},\nwhere for completeness we note that our calculation also includes the\n$qq$ induced contribution to gluon fusion. For our numerical setup we\nshow the dependence of the five correction factors on the lower cut\n$\\pt{}^\\text{cut}$ in \\fig{fig:corrfactors}.\n\n\\begin{figure}[htp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/deleps_ptmin.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/delepswithb_ptmin.pdf}  \\\\[-0.4cm]\n (a)  & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{(a) Correction factors $\\delta$, $\\epsilon$ as a function of the\nlower cut $\\pt{}^\\text{cut}$ in GeV and in addition (b) $\\delta_{bg}$, $\\delta_{bt}$ and $\\epsilon_{b}$\nas a function of $\\pt{}^\\text{cut}$. Both figures are obtained for a {\\abbrev SM}{}\nHiggs with $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:corrfactors} \n\\end{figure}\n\nAs can be seen in \\fig{fig:corrfactors}~(a), the larger the lower cut\n$\\pt{}^\\text{cut}$, the more the degeneracy between $\\kappa_t$ and\n$c_{5,H}$, which are indistinguishable in the inclusive cross section, is\nbroken.  On the other hand \\fig{fig:corrfactors}~(b) points out that for\nlow $\\pt{}^\\text{cut}<200$\\,GeV bottom-quark induced contributions\nshould also be taken into account.  The interferences of the latter with\nthe top-quark induced contributions on the one hand and with the\neffective coupling $c_{5,H}$ on the other hand, encoded in $\\delta_{bt}$\nand $\\delta_{bg}$, are identical only for low $\\pt{}^\\text{cut}$.  We\nnote that the cross section prediction for the {\\abbrev SM}{} Higgs boson of\ncourse should include the full correction by bottom quarks given by\n$\\delta_{bt}$ and $\\epsilon_b$.  For completeness we partially also\nreproduced Fig.~$2$ of \\citere{Grojean:2013nya}, which illustrates the\ndisentanglement of the degeneracy between $\\kappa_t$ and $c_{5,H}$.\n\nAs a last example we discuss the calculation of the gluon-fusion\ncross section for an arbitrary scalar, which couples to gluons\nthrough an effective operator $c_{5}^{(0)}=1$ only. Motivated\nby the background deviation in the diphoton channel at $750$\\,GeV\nin both {\\abbrev LHC}{} experiments~\\cite{CMS:2016owr,ATLAS750}, we choose\nthe mass of the scalar to be $m_X=750$\\,GeV. We pick an input\nfile for the {\\abbrev SM}{}, set the {\\abbrev SM}{} Higgs-boson mass to $M_\\text{H}{}=750$\\,GeV,\ninclude a \\dimension{5} operator through \\blockentry{DIM5}{11}{=1},\nbut set the {\\abbrev SM}{} Higgs-boson couplings\nto quarks and gauge bosons to zero in {\\tt Block FACTORS} and \nthrough \\blockentry{SUSHI}{7}{=0}. The results are shown\nin \\tab{tab:750}. We include the renormalization scale\nuncertainty $\\pm \\Delta(\\mu_\\text{R})$, which was obtained simultaneously. \nAgain $\\Delta(\\mu_\\text{R})$ is the maximum\ndeviation of the cross section within the interval $\\mu_\\text{R}\n\\in[1\/4,1]m_X$ and $\\mu_\\text{R} \\in[1\/2,2]m_X$ for\nthe central scale choices $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ and\n$\\mu_\\text{R}=\\mu_\\text{F}=m_X$, respectively. For this\npurpose the Wilson coefficient is evolved perturbatively,\ni.e. \\blockentry{DIM5}{0}{=1}. At \\nklo{3} the\nsoft expansion is performed up to $(1-x)^{16}$ with $a=0$.\nThe matching to the high-energy limit, $x\\to 0$, is not applied.\nSimilar to the {\\abbrev SM}{} Higgs boson we observe a good convergence of\nthe perturbative series with a renormalization scale\nuncertainty of less than $\\pm 1.3$ and $\\pm 2.9$\\% at \\nklo{3} {\\abbrev QCD}{}\nfor the central scale choices $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ and $\\mu_\\text{R}=\\mu_\\text{F}=m_X$,\nrespectively.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\n\\hline\n$\\sigma(gg\\to X)$ [fb] & $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ & $\\mu_\\text{R}=\\mu_\\text{F}=m_X$ \\\\\n\\hline\n{\\abbrev LO}{}      & $246.2  \\pm 52.8$ & $185.8  \\pm 36.0$\\\\\n{\\abbrev NLO}{}     & $368.7  \\pm 43.1$ & $316.3  \\pm 39.1$\\\\\n{\\abbrev NNLO}{}    & $410.0  \\pm 19.1$ & $384.9  \\pm 24.0$\\\\\n\\nklo{3}   & $414.6  \\pm 5.4$  & $407.2  \\pm 11.7$\\\\\n\\hline\n\\end{tabular}\n\\caption[]{\\label{tab:750} Inclusive gluon-fusion cross section in fb\nfor a {\\abbrev CP}{}-even scalar with mass $m_X=750$\\,GeV, which couples to gluons through $c_{5}^{(0)}=1$ only.\nThe results are given at different orders \\nklo{k}, $k=0,1,2,3$, in {\\abbrev QCD}{}\nfor the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{} for two renormalization\nand factorization scale choices.\nThe depicted uncertainty is the renormalization-scale uncertainty $\\pm \\Delta(\\mu_\\text{R})$.}\n\\end{center}\n\\end{table}\n\n\\section{Conclusions}\n\nWe presented the new features implemented in version {\\tt 1.6.0}\nof the code \\sushi{}.  Aside from the implementation of heavy-quark\nannihilation, many new features aim at the improvement of the\ngluon-fusion cross-section prediction and its associated uncertainty\nestimate. In particular, \\sushi{} now provides the soft expansion around\nthe threshold of Higgs production and the matching to the high-energy\nlimit for {\\abbrev CP}{}-even Higgs bosons, at {\\abbrev NLO}{}, {\\abbrev NNLO}{} and\n\\nklo{3} {\\abbrev QCD}{}. Top-quark mass effects beyond the usual infinite top-mass\nlimit can be taken into account at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.  We investigated\nthe relevance of these effects for a {\\abbrev SM}{}-like Higgs boson with a mass\nof $125$\\,GeV and provide a prediction of the corresponding gluon-fusion\ncross section at the {\\abbrev LHC}{} with a center-of-mass energy of $13$\\,TeV.\nBoth for {\\abbrev CP}{}-even and -odd Higgs bosons, \\sushi{} now calculates the\nrenormalization-scale uncertainty simultaneously to the calculation of\nthe gluon-fusion cross section at the central scale.  Moreover, the\neffects of \\dimension{5} operators can be studied in any model currently\nsupported by \\sushi{}.  We showed how the degeneracy between the\ntop-quark mass contribution and a point-like \\dimension{5} operator\ncontribution can be broken at large values of the transverse momentum of\na Higgs boson with mass $125$\\,GeV. The implementation of arbitrary\n\\dimension{5} operators is also particularly suited for the study of new\n{\\abbrev CP}{}-even and -odd scalars beyond the implemented models. We showed the\nconvergence of the perturbative series for the inclusive gluon-fusion\ncross section of a scalar with mass $750$\\,GeV at the $13$\\,TeV {\\abbrev LHC}{}.\n\nOur description and the subsequent appendix include examples how the user can control the new features through\nthe setting of blocks in the input file of \\sushi{}. Example input files are\ncontained in the {\\tt example}-folder of the current \\sushi{} release to be found\nat \\cite{sushiwebpage}.\n\n\\section*{Acknowledgments}\n\nRVH would like to thank {\\scalefont{.9} DFG} for financial support.  SL\nacknowledges support by the {\\scalefont{.9} SFB} 676 ``Particles, Strings and\nthe Early Universe''. Many of the calculations presented in this paper\nhave been performed on the {\\scalefont{.9} FUGG} cluster at Bergische\nUniversit\\\"at Wuppertal.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\n\\subsection{Background}\n\\label{sec:Bkgrnd}\n\n\\begin{enumerate}\n\\item \\label{Bkgrndi} In \\cite{Q}*{\\S1} Quillen proved his Theorem~B\n  which states that, for a functor $f\\colon \\cat X \\to \\cat Z$ and an\n  object $Z \\in \\cat Z$, the rather simple \\emph{over category}\n  $\\commacat{f}{Z}$ is a \\emph{homotopy fiber} of $f$ if $f$ has a\n  certain \\emph{property $B_{1}$}.\n\\item \\label{Bkgrndii} This was generalized in \\cite{DKS}*{\\S6} where\n  it was shown that increasingly weaker \\emph{properties} $B_{n}$\n  ($n>1$) allowed for increasingly less simple description of these\n  homotopy fibers as \\emph{$n$-arrow overcategories}\n  $\\subcommacat{f}{n}{Z}$.\n\\item \\label{Bkgrndiii} It was also noted that a sufficient condition\n  for a functor $f\\colon \\cat X \\to \\cat Z$ to have property $B_{n}$\n  was that the category $\\cat Z$ had a certain property $C_{n}$.\n\\end{enumerate}\n\n\\subsection{The current paper}\n\\label{sec:CurPap}\n\nOur main results in this paper are the following.\n\\begin{enumerate}\n\\item \\label{CurPapi} We show that for a zigzag $\\zigzag{f}{\\cat\n    X}{\\cat Z}{\\cat Y}{g}$ between categories in which $f$ has\n  property $B_{n}$ \\eqref{Bkgrndii} (and in particular if $\\cat Z$ has\n  property $C_{n}$) \\eqref{Bkgrndiii}, its \\emph{homotopy pullback}\n  admits a description similar to the one mentioned in\n  \\ref{sec:Bkgrnd} namely as a \\emph{$n$-arrow pullback category\n    $\\subcommacat{f\\cat X}{n}{g\\cat Y}$}.  Moreover its\n  \\emph{pullback} comes with a monomorphism into the homotopy pullback\n  and hence is itself a homotopy pullback if the monomorphism is a\n  weak equivalence.\n\\item \\label{CurPapii} We then deduce from this similar results for\n  zigzags in the categories of \\emph{relative categories} and\n  \\emph{$k$-relative categories} ($k>1$) and do this not only with\n  respect to their \\emph{Reedy structure}, but also with respect to\n  their \\emph{Rezk structure} which turns them into models for the\n  theories of \\emph{$(\\infty,1)$- and $(\\infty,k)$-categories}\n  respectively.\n\\item \\label{CurPapiii} We also note that a sufficient condition for a\n  category, a relative category or a $k$-relative category to have\n  \\emph{property $C_{3}$} is that it admits what we will call a\n  \\emph{strict $3$-arrow calculus}.\n\\item \\label{CurPapiv} Our main tool for proving all this consists of\n  the \\emph{sharp maps} of Hopkins and Rezk \\cite{R2} which, because\n  of their fibration-like properties we prefer to call\n  \\emph{fibrillations}.  They are the dual of Hopkins' \\emph{flat\n    maps} which have similar cofibration-like properties and which we\n  therefore call \\emph{cofibrillations}.  These cofibrillations do not\n  play any role in the current paper, except for a surprise appearance\n  in \\ref{def:strcalculus}(iii)$'$.\n\\end{enumerate}\n\n\\subsection{The genesis of the current paper}\n\\label{sec:genesis}\n\nThe original version of this paper consisted of only the results\nmentioned in \\ref{CurPapi} and the corresponding part of\n\\ref{CurPapiii}.  That was exactly what we needed in \\cite{BK3} where\nit enabled us to reduce the proof that certain pullbacks were\nhomotopy pullbacks to a rather straightforward calculation.  However\nour proof of these results was rather ad hoc and not very\nsatisfactory.\n\nFortunately two things happened.\n\\begin{enumerate}\n\\item \\label{genesisi} We discovered a manuscript of Charles Rezk\n  \\cite{R2} in which he studied the \\emph{sharp maps} of Mike Hopkins.\n  These maps seemed to be exactly what we needed.  Just like\n  fibrations, they could be used in a \\emph{right proper} model\n  category to obtain pullbacks which were homotopy pullbacks (which by\n  the way made us call them by the more suggestive name of\n  \\emph{fibrillations}).\n\\item \\label{genesisii} Moreover when subsequently we took a closer\n  look at the lemma of Quillen \\cite{Q}*{\\S1} which started it all and\n  which he used to prove his Theorem~B, we noted that this lemma was\n  essentially just an elegant way of constructing fibrillations, some\n  of which were exactly the ones we needed.\n\\end{enumerate}\n\nCombining all this with some simple properties of fibrillations we\nthen not only obtained a much better proof of the above mentioned\nresult \\ref{CurPapi}, but also realized that with relatively little\neffort this proof could be extended to a proof of similar results for\n\\emph{relative categories} and the \\emph{$k$-relative categories} of\n\\cite{BK2}, which as result the current manuscript.\n\n\\subsection{Acknowledgements}\n\\label{sec:Ack}\n\nWe would like to thank Phil Hirschhorn, Mike Hopkins and especially\nDan Dugger for several useful conversations.\n\n\n\\setcounter{tocdepth}{1}\n\\tableofcontents\n\n\\section{An overview}\n\\label{sec:Oview}\n\nThe paper consists of three parts, each of which consists of three\nsections.\n\n\\boldcentered{Part I: Categorical preliminaries}\n\nIn sections 2, 3 and 4 we discuss the various models and relative\ncategories involved and some of the relative functors between them.\n\\subsection{The categories involved}\n\\label{sec:CatInvlv}\n\n\\begin{enumerate}\n\\item \\label{CatInvlvi} The categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k \\ge 0$) of small\n  \\emph{$k$-relative categories} of \\cite{BK2}.\n\n  For $k=1$ this is just the category $\\mathbf{RelCat}$ of small\n  \\emph{relative categories} of \\cite{BK1}.\n\n  For $k=0$ this is the category of small \\emph{maximal} relative\n  categories (i.e.\\ in which all maps are weak equivalences).  As\n  $\\mathbf{Rel}^{0}\\mathbf{Cat}$ thus is isomorphic, although not equal, to the category\n  $\\mathbf{Cat}$ of small categories, we \\emph{will often denote $\\mathbf{Rel}^{0}\\mathbf{Cat}$ by\n  $\\widehat{\\mathbf{Cat}}$}.\n\\item \\label{CatInvoviii} The categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and\n  $\\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) which are respectively the categories\n  of \\emph{$k$-simplicial diagrams} in $\\widehat{\\mathbf{Cat}}$ and the categories of\n  small \\emph{$k$-simplicial spaces}, i.e.\\ $(k+1)$-simplicial sets.\n\\end{enumerate}\n\n\\subsection{The functors involved}\n\\label{sec:FuncInvov}\n\n\\begin{enumerate}\n\\item \\label{FuncInvovi} The \\emph{$k$-simplicial nerve functors}\n  $\\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) of\n  \\cite{BK2}, which for $k=0$ is just he \\emph{classical nerve\n    functor $\\N$}.\n\\item \\label{FuncInfovii} The functors $\\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}\n  \\to \\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) induced by $\\N$.\n\\item The (unique) functors $w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to\n  \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) such that $\\N_{*}w_{*} = \\mathrm{s}^{k}\\N$.\n\\end{enumerate}\n\n\\subsection{The model and relative structures}\n\\label{sec:ModRlSt}\n\n\\begin{enumerate}\n\\item \\label{ModRlSti} We endow $\\cat S = \\mathrm{s}^{0}\\cat S$ with the\n  classical model structure and $\\widehat{\\mathbf{Cat}}$ with the Quillen equivalent\n  Thomason structure \\cite{T2}.\n\\item \\label{ModRlStii} We endow $\\mathrm{s}^{k}\\cat S$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) with the resulting Reedy model structure.\n\\item \\label{ModRlStiii} We endow $\\mathbf{RelCat}$ and $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$)\n  respectively with the Quillen equivalent Reedy \\emph{model}\n  structure which \\cite{BK1} is lifted from the Reedy structure on\n  $\\mathrm{s}\\cat S$ and the homotopy equivalent \\emph{relative} Reedy\n  structure which \\cite{BK2} is lifted from the Reedy structure on\n  $\\mathrm{s}^{k}\\cat S$.\n\\item \\label{ModRlStiv} In the application of our main result we will\n  consider the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$)\n  as models for the theory of $(\\infty,k)$-categories by endowing them\n  with a \\emph{Rezk} structure which has more weak equivalences than\n  the Reedy structure and we will denote those categories so endowed\n  by $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\\end{enumerate}\n\n\\subsection{Properties of the functors involved}\n\\label{sec:FncPrp}\n\nThe functors considered in \\ref{sec:FuncInvov} above all have the same\nnice properties as the classical nerve functors (and we will therefore\nrefer to them as \\emph{abstract nerve functors}).\n\nIn particular they all are \\emph{relative functors} which are\n\\emph{homotopy equivalences}.\n\n\\boldcentered{Part II: Homotopy pullbacks and potential homotopy\n  pullbacks}\n\n\n\\subsection{}\n\\label{sec:empty}\n\nIn section \\ref{sec:HmtpyPlbk} we will define homotopy pullbacks of\nzigzags in a model category in a more general fashion than is usually\ndone, but that enables us to extend the definition to certain\nsaturated relative categories.\n\\begin{enumerate}\n\\item \\label{emptyi} In a model category we define a homotopy pullback\n  of a zigzag as \\emph{any} object which is weakly equivalent to its\n  image under a \\emph{``homotopically correct'' homotopy limit functor}.\n\\item \\label{emptyii} In a saturated relative category we then define\n  a homotopy pullback of a zigzag of a \\emph{any} object which is\n  weakly equivalent to its image under what we will call a \\emph{weak\n    homotopy limit functor} which is a functor which has only some of\n  the properties of the above (i) homotopy limit functors.\n\\item \\label{emptyiii} Our main result then is a \\emph{Global\n    equivalence lemma} which states that\n  \\begin{itemize}\n  \\item \\emph{if $f\\colon \\cat C \\to \\cat D$ is a homotopy equivalence\n    between saturated relative categories,} then \\emph{$\\cat C$ admits\n  weak homotopy limit functors iff $\\cat D$ does, and in that case $f$\n  preserves homotopy pullbacks.}\n  \\end{itemize}\n\\item \\label{emptyiv} This lemma not only\n  \\begin{itemize}\n  \\item implies that any two weak homotopy limit functors on the same\n    category yield the same notion of homotopy pullbacks of zigzags, and\n  \\item enables us, in view of the fact that model categories admit\n    such weak homotopy limit functors, to prove their existence\n    elsewhere.\n  \\end{itemize}\n  but also plays a role in the proof of our main result in section\n  \\ref{sec:PrfThm}.\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep6}\n\nIn section \\ref{sec:PotHmPb} we discuss, for a zigzag $\\zigzag{f}{\\cat\n  X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge\n0$) the \\emph{$n$-arrow pullback object} $\\subcommacat{f\\cat\n  X}{n}{g\\cat Y}$ which was mentioned in \\ref{sec:CurPap} above.\n\n\\subsection{}\n\\label{sec:onep7}\n\nIn section \\ref{sec:PrpBnCn} we recall the notions of \\emph{property\n  $B_{n}$} and \\emph{property $C_{n}$} for $\\widehat{\\mathbf{Cat}}$ which were\nmentioned in \\ref{sec:Bkgrnd} above and then extend them to the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for $k\\ge 1$.\n\n\\boldcentered{Part III: The main results and their proof}\n\n\n\\subsection{}\n\\label{sec:onep8}\n\nIn section \\ref{sec:MainRlt} we formulate our main results, Theorems\n\\ref{thm:MpPropBn}, \\ref{thm:SuffCn}, \\ref{thm:QFibQfib} and\n\\ref{thm:MpBn}.\n\nWe also prove Theorem \\ref{thm:SuffCn} and, assuming Theorem\n\\ref{thm:MpPropBn}, Theorems \\ref{thm:QFibQfib} and \\ref{thm:MpBn} as\nwell, but defer the proof of Theorem \\ref{thm:MpPropBn} itself until\nsection \\ref{sec:PrfThm}.\n\n\\begin{enumerate}\n\\item \\label{onep8i} Theorem \\ref{thm:MpPropBn} states that, for a\n  zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or\n  $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge 0$), for which the map $f\\colon \\cat X \\to\n  \\cat Z$ has property $B_{n}$ or the object $\\cat Z$ has property\n  $C_{n}$ \\eqref{sec:onep7}, the \\emph{$n$-arrow pullback object}\n  $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ \\eqref{sec:onep6} is actually a\n  \\emph{homotopy pullback} \\eqref{sec:empty} of that zigzag.\n\\item \\label{onep8ii} Theorem \\ref{thm:MpPropBn} states that a\n  sufficient condition on an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$)\n  in order that it has property $C_{3}$ is that $\\cat Z$ admits what\n  we will call a \\emph{strict $3$-arrow calculus}.\n\\item \\label{onep8iii} If $k=1$, then this (ii) is in particular the\n  case if $\\cat Z$ is a \\emph{partial model category}, i.e.\\ a\n  relative category which has the \\emph{two out of six} property and\n  admits a \\emph{$3$-arrow calculus}.\n\n  These partial model categories have the property that \\cite{BK3}, if\n  $\\mathbf{RelCat}$ is endowed with the \\emph{Rezk} structure\n  \\eqref{ModRlStiv}, then\n  \\begin{itemize}\n  \\item every partial model category is Reedy equivalent to a fibrant\n    object in $\\mathrm{L}\\mathbf{RelCat}$ \\eqref{ModRlStiv}, and\n  \\item every fibrant object in $\\mathrm{L}\\mathbf{RelCat}$ is Reedy equivalent to a\n    partial model category.\n  \\end{itemize}\n  Consequently one has the following result for\n  $(\\infty,1)$-categories.\n\\item \\label{onep8iv} Theorem \\ref{thm:QFibQfib} that states that, for\n  every zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{RelCat}$\n  between partial model categories, the \\emph{$3$-arrow pullback\n    object} $\\subcommacat{f\\cat X}{3}{g\\cat Y}$ is a \\emph{homotopy\n    pullback} of this zigzag not only in $\\mathbf{RelCat}$, but also in\n  $\\mathrm{L}\\mathbf{RelCat}$, i.e.\\ in $(\\infty,1)$-categories.\n\\item \\label{onep8v} Finally in Theorem \\ref{thm:MpBn} we state a\n  similar result for $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$), i.e.\\ for\n  $(\\infty,k)$-categories, which however is much weaker than that\n  Theorem \\ref{thm:QFibQfib}, as we have no model structure on\n  $\\mathbf{Rel}^{k}\\mathbf{Cat}$ for $k>1$ nor an analog for partial model categories.\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep9}\n\nIn section \\ref{sec:HopRzF} we review the \\emph{fibrillations} of\nHopkins and Rezk which can be defined in any \\emph{relative category\n  with pullbacks} and describe the following three lemmas which we\nneed in section \\ref{sec:PrfThm} to prove Theorem \\ref{thm:MpPropBn}.\n\n\\begin{enumerate}\n\\item \\label{onep9i} A \\emph{Quillen fibrillation lemma} which is a\n  reformulation and a slight strengthening of the lemma that Quillen\n  used to prove his Theorem~B, and which enables us to obtain the\n  needed fibrillations in $\\widehat{\\mathbf{Cat}}$.\n\n  If, for an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ and a not necessarily\n  relative functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$, $\\Gr F \\in \\widehat{\\mathbf{Cat}}$\n  denotes its Grothendieck construction and $\\pi\\colon \\Gr F \\to \\cat\n  D \\in \\widehat{\\mathbf{Cat}}$ denotes the associated projection functor, then this\n  lemma states that\n  \\begin{itemize}\n  \\item the projection functor $\\pi\\colon \\Gr F \\to \\cat D \\in\n    \\widehat{\\mathbf{Cat}}$ is a \\emph{fibrillation} iff the functor $F\\colon \\cat D\n    \\to \\widehat{\\mathbf{Cat}}$ is a \\emph{relative} functor, i.e.\\ sends every map\n    of $\\cat D$ to a weak equivalence in $\\widehat{\\mathbf{Cat}}$.\n  \\end{itemize}\n\\item \\label{onep9ii} A \\emph{fibrillation lifting lemma} which\n  enables us to obtain from the fibrillation in $\\widehat{\\mathbf{Cat}}$ obtained in\n  (i) above corresponding fibrillations in the categories\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for $k\\ge 1$.\n\\item \\label{onep9iii} A \\emph{Hopkins-Rezk fibrillation lemma} which\n  enables us to use the fibrillations of (ii) above to obtain the\n  desired homotopy pullback in the categories $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge\n  0$), i.e.\\ it states that\n  \\begin{itemize}\n  \\item for every pullback square in a right proper model category\n    \\begin{displaymath}\n      \\xymatrix{\n        {\\cat U} \\ar[r] \\ar[d]\n        & {\\cat Y} \\ar[d]\\\\\n        {\\cat V} \\ar[r]\n        & {\\cat Z}\n      }\n    \\end{displaymath}\n    in which the map $\\cat C \\to \\cat Z$ is a fibrillation, the object\n    $\\cat U$ is a homotopy pullback of the zigzag $\\cat V \\to \\cat Z\n    \\gets \\cat Y$.\n  \\end{itemize}\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep10}\n\nIn section \\ref{sec:PrfThm} we finally prove Theorem\n\\ref{thm:MpPropBn}.  We prove this first for the categories\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) by using the above \\eqref{sec:onep9}\n\\emph{three lemmas} and then lift these results to the categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) by means of the \\emph{Global equivalence lemma}\nwhich was mentioned in \\ref{emptyiii} above.\n\nActually for $\\mathbf{RelCat}$ we did not have to use this fourth lemma as we\ncould have obtained this also using only the first three.\n\n\\part{Categorical Preliminaries}\n\n\n\\section{Relative categories}\n\\label{sec:RelCat}\n\n\\subsection{Summary}\n\\label{sec:RlCtSum}\nWe start with a brief review of\n\\begin{enumerate}\n\\item \\emph{relative categories} and \\emph{relative functors} between them,\n\\item \\emph{saturated}, \\emph{maximal} and \\emph{minimal} relative\n  categories,\n\\item a \\emph{homotopy relation} between relative functors,\n\\item \\emph{strict $3$-arrow calculi} on relative categories,\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{5}\n\\item the associated \\emph{simplicial nerve functor} to simplicial\n  spaces, i.e.\\ bisimplicial sets.\n\\end{resumeenumerate}\n\n\\boldcentered{Relative Categories}\n\n\\subsection{Definition}\n\\label{def:relcat}\n\n\\begin{enumerate}\n\\item \\label{relcati} A \\textbf{relative category} is a pair $(\\cat C,\n  w\\cat C)$ (usually denoted by just $\\cat C$) consisting of an\n  \\textbf{underlying category} $\\cat C$ (sometimes denoted by\n  $\\und\\cat C$) and a subcategory $w\\cat C \\subset \\cat C$ which is\n  \\emph{only} required to contain \\emph{all} the objects of $\\cat C$\n  (and hence their identity maps).\n\\item \\label{relcatii} The maps or $w\\cat C$ are called \\textbf{weak\n    equivalences} and two objects of $\\cat C$ are called\n  \\textbf{weakly equivalent} if they can be connected by a finite\n  zigzag of weak equivalences.\n\\item \\label{relcatiii} By a \\textbf{functor} $f\\colon \\cat C \\to \\cat\n  D$ between two relative categories we will mean just a functor\n  $f\\colon \\und\\cat C \\to \\und\\cat D$ (i), and\n\\item \\label{relcativ} such a functor will be called a\n  \\textbf{relative functor} if it preserves weak equivalences, i.e.\\\n  if $f(w\\cat C) \\subset w\\cat D$.\n\\end{enumerate}\n\n\\boldcentered{Saturated, maximal and minimal relative categories}\n\n\n\n\\subsection{Definition}\n\\label{def:saturated}\n\n\\begin{enumerate}\n\\item \\label{saturi} A relative category $\\cat C$ will be called\n  \\textbf{saturated} if a map in $\\cat C$ is a weak equivalence iff\n  its image in $\\Ho\\cat C$ (i.e.\\ the category obtained from $\\cat C$\n  by formally inverting all the weak equivalences) is an isomorphism.\n\\end{enumerate}\nThis is in particular the case if\n\\begin{resumeenumerate}{2}\n\\item \\label{saturii} $\\cat C$ is a \\textbf{maximal} relative\n  category, i.e.\\ \\emph{all} maps of $\\cat C$ are weak equivalences\n\\end{resumeenumerate}\nor if\n\\begin{resumeenumerate}{3}\n  \\item \\label{saturiii} $\\cat C$ is a \\textbf{minimal} relative\n    category, i.e.\\ the identity maps re the \\emph{only} weak\n    equivalences.\n\\end{resumeenumerate}\n\n\n\\boldcentered{A homotopy relation between relative functors}\n\n\n\\subsection{Definition}\n\\label{def:HomRel}\n\nLet $\\boldsymbol 1^{w}$ denote the maximal relative category\n(Df.~\\ref{saturii}) which  consists of two objects $0$ and $1$ and\none map $0\\to 1$ between them.  Then $\\boldsymbol 1^{w}$ gives rise to the\nfollowing notions.\n\\begin{enumerate}\n\\item \\label{HomReli} Given two relative functors $f,g\\colon \\cat C\n  \\to \\cat D$ between relative categories, a \\textbf{strict homotopy}\n  $h\\colon f \\to g$ between them will be a natural weak equivalence,\n  i.e.\\ a map\n  \\begin{displaymath}\n    h\\colon \\cat C\\times\\boldsymbol 1^{w} \\longrightarrow \\cat D\n  \\end{displaymath}\n  such that $h(c,0) = fc$ and $h(c,1) = gc$ for every object or map $c\n  \\in \\cat C$.\n\\item \\label{HomRelii} Two relative functors $\\cat C \\to \\cat D$ then\n  are called \\textbf{homotopic} if they can be connected by a finite\n  zigzag of strict homotopies, and\n\\item \\label{HomReliii} a functor $f\\colon \\cat C \\to \\cat D$ is\n  called a \\textbf{homotopy equivalence} if $f$ is a relative functor\n  and if there exists a relative functor $f'\\colon \\cat D \\to \\cat C$\n  (called a \\textbf{homotopy inverse} of $f$) such that the\n  compositions $f'f$ and $ff'$ are homotopic to $1_{\\cat C}$ and\n  $1_{\\cat D}$ respectively.\n\\end{enumerate}\n\n\n\\boldcentered{$3$-arrow calculi}\n\n\\subsection{Definition}\n\\label{def:calculus}\n\nA relative category $\\cat C$ is said to admit a \\textbf{$3$-arrow\n  calculus} if there exist subcategories $\\cat U$ and $\\cat V \\subset\nw\\cat C$ which behave like the categories of the \\emph{trivial\n  cofibrations} and \\emph{trivial fibrations} in a model category in\nthe sense that\n\\begin{enumerate}\n\\item \\label{calculusi} for every map $u \\in \\cat U$, its pushouts in\n  $\\cat C$ exist and are again in $\\cat U$,\n\\item \\label{calculusii} for every map $v \\in \\cat V$, it's pullbacks\n  in $\\cat C$ exist and are again in $\\cat C$, and\n\\item \\label{calculusiii} the maps $w \\in w\\cat C$ admit a\n  \\emph{functorial} factorization $w = vu$ with $u \\in \\cat U$ and $v\n  \\in \\cat V$.\n\\end{enumerate}\n\n\\subsection{Remark}\n\\label{rem:easier}\n\nIt should be noted that the conditions \\ref{calculusi} and (ii) are\nstronger than the ones in \\cite{DK}*{8.2} and \\cite{DHKS}*{36.1}.\nHowever we prefer them as they are easier to use and are likely to be\nautomatically satisfied.\n\n\\subsection{Remark}\n\\label{rem:whystrict}\n\nIn \\cite{DK}*{8.2} and \\cite{DHKS}*{36.1} $3$-arrow calculi were\ndefined in the presence of the \\emph{two out of three} and the\n\\emph{two out of six} properties respectively with the result that\n\\emph{a $3$-arrow calculus on $(\\cat C,w\\cat C)$ automatically\n  restricted to a $3$-arrow calculus on $(w\\cat C, w\\cat C)$}.\n\nAs we do not want to assume the presence of the two out of\nthree or the two out of six property, however, we define a notion of\n\\emph{strict $3$-arrow calculi} as follows.\n\n\n\\boldcentered{Strict $3$-arrow calculi}\n\n\\subsection{Definition}\n\\label{def:strcalculus}\n\nA \\textbf{strict $3$-arrow calculus} on a relative category $(\\cat\nC,w\\cat C)$ will be a $3$-arrow calculus (Df.~\\ref{def:calculus})\nwhich restricts to a $3$-arrow calculus on $(w\\cat C, w\\cat C)$.\n\nAnother way of saying this by adding in Df.~\\ref{def:calculus} to the\nconditions (i) and (ii) the conditions that\n\\begin{enumerate}\n\\item [(i)$'$] every pushout of a map in $w\\cat C$ along a map in\n  $\\cat U$ is again in $w\\cat C$, and\n\\item [(ii)$'$] every pullback of a map in $w\\cat C$ along a map in\n  $\\cat V$ is again in $w\\cat C$.\n\\end{enumerate}\n\nWith other words the maps in $\\cat U$ and $\\cat V$ behave like trivial\ncofibrations and trivial fibrations in a \\emph{proper} model category.\n\nA more compact way of saying all this is that\n\\begin{enumerate}\n\\item [(iii)$'$] a strict $3$-arrow calculus on a relative category is\n  a functorial factorization of its weak equivalences into a\n  \\emph{trivial cofibrillation} followed by a \\emph{trivial fibrillation}\n\\end{enumerate}\nwhere fibrillations are as defined in Df.~\\ref{def:fibril} below and\ncofibrillations are defined dually.\n\n\n\\boldcentered{The simplicial nerve functor}\n\n\\subsection{Definition}\n\\label{def:smpnv}\n\nLet $\\mathbf{RelCat}$ denote the category of (small) \\emph{relative\n  categories} (Df.~\\ref{def:relcat}) and let $\\mathrm{s}\\cat S$ denote the\ncategory of \\textbf{simplicial spaces}, i.e.\\ \\emph{bisimplicial}\nsets.\n\nFurthermore, for every integer $p \\ge 0$, let $\\cat p$ denote the\n$p$-arrow category\n\\begin{displaymath}\n  0 \\longrightarrow \\cdots \\longrightarrow p\n\\end{displaymath}\nand let $\\cat p^{v}$ and $\\cat p^{w}$ denote respectively the\n\\emph{minimal} and \\emph{maximal} relative categories\n(Df.~\\ref{ModRlStiii} and (ii)) which have $\\cat p$ as their\nunderlying category.\n\nThe \\textbf{simplicial nerve functor} then is the functor\n\\begin{displaymath}\n  \\mathrm{s}\\N\\colon \\mathbf{RelCat} \\longrightarrow \\mathrm{s}\\cat S\n\\end{displaymath}\nwhich sends each object $\\cat C \\in \\mathbf{RelCat}$ to the bisimplicial set\nwhich in bidimension $(p,q)$ consists of the maps\n\\begin{displaymath}\n  \\cat p^{v}\\times \\cat q^{w} \\longrightarrow \\cat C \\in \\mathbf{RelCat}\n\\end{displaymath}\n\n\\section{The categories and functors involved}\n\\label{sec:CtFncInv}\n\n\n\\subsection{Summary}\n\\label{sec:CtFncSumm}\n\nWe now describe the categories and functors which we need in the paper\nand note that\n\\begin{enumerate}\n\\item \\label{CtFncSummi} these categories come with obvious notions of\n  \\emph{strict homotopies} and \\emph{homotopy equivalences} (as in\n  Df.~\\ref{rem:easier}),\n\\item \\label{CtFncSummii} these functors \\emph{preserve} these strict\n  homotopies and homotopy equivalences, and\n\\item \\label{CtFncSummiii} these functors all have \\emph{left\n    adjoints} which are \\emph{left inverses}.\n\\end{enumerate}\n\nIn section \\ref{sec:AbsNv} we then endow these categories with model\nor relative structures which\n\\begin{resumeenumerate}{4}\n\\item \\label{CtFncSummiv} are compatible with these homotopy\n  equivalences in the sense that \\emph{every homotopy equivalence is a\n  weak equivalence}, and\n\\item \\label{CtFncSummv} turn the functors between these categories\n  into relative functors which are \\emph{homotopy equivalences}.\n\\end{resumeenumerate}\n\n\\boldcentered{$k$-relative categories}\n\n\\subsection{Definition}\n\\label{def:krel}\n\nWe extend the definition of $k$-relative categories for $k \\ge 1$ of\n\\cite{BK2}*{2.3} to include also the case $k=0$ as follows.\n\nA \\textbf{$k$-relative category} ($k>0$) will be a $(k+2)$-tuple\n\\begin{displaymath}\n  \\cat C = (a\\cat C, v_{1}\\cat C, \\ldots, v_{k}\\cat C, w\\cat C)\n\\end{displaymath}\nconsisting of an \\textbf{ambient category} $a\\cat C$ and subcategories\n\\begin{displaymath}\n  v_{1}\\cat C, \\ldots, v_{k}\\cat C \\text{ and } w\\cat C\n  \\subset a\\cat C\n\\end{displaymath}\nthat each contain all the objects of $a\\cat C$ and form a\ncommutative diagram with $2k$ arrows of the form\n\\begin{displaymath}\n  \\xymatrix@C=0.5em{\n    & {w\\cat C} \\ar[dl] \\ar[dr]\\\\\n    {v_{1}\\cat C} \\ar[dr]\n    & {{\\cdot\\;\\cdot\\;\\cdot}}\n    & {v_{k}\\cat C} \\ar[dl]\\\\\n    & {a\\cat C}\n  }\n\\end{displaymath}\nand where $a\\cat C$ is subject to the conditions that\n\\begin{enumerate}\n\\item \\label{kreli} every map in $a\\cat C$ is a finite composition of\n  maps in the $v_{i}\\cat C$ ($1 \\le i \\le k$), and\n\\item \\label{krelii} every relation in $a\\cat C$ is the consequence of\n  the commutativity relations in $v_i\\cat C$ ($1\\leq i\\leq n$) and the commutativity in $a\\cat C$ of diagrams of the form\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cdot} \\ar[r]^{x_{1}} \\ar[d]_{y_{1}}\n      & {\\cdot} \\ar[d]^{y_{2}}\\\\\n      {\\cdot} \\ar[r]_{x_{2}}\n      & {\\cdot}\n    }\n  \\end{displaymath}\n  in which $x_{1}, x_{2} \\in v_{i}\\cat C$ and $y_{1}, y_{2} \\in\n  v_{j}\\cat C$ and $1\\leq i<j\\leq n$.\n\\end{enumerate}\n\nThe maps of $w\\cat C$ are called \\textbf{weak equivalences} and a\n\\textbf{relative functor} $f\\colon \\cat C \\to \\cat D$ between two\n$k$-relative categories will be a functor $f\\colon a\\cat C \\to a\\cat\nD$ such that\n\\begin{displaymath}\n  f(w\\cat C) \\subset w\\cat D\n  \\qquad\\text{and}\\qquad\n  f(v_{i}\\cat C) \\subset v_{i}\\cat D\n  \\qquad \\text{for all $i \\le i < k$.}\n\\end{displaymath}\n\n\\subsection{Remark}\n\\label{rem:onerel}\n\nIf $\\cat C$ is a $1$-relative category, then $a\\cat C = v_{1}\\cat C$\nand hence\n\\begin{itemize}\n\\item \\emph{$1$-relative categories are the same as relative\n    categories} (Df.~\\ref{def:relcat}).\n\\end{itemize}\n\n\\subsection{Remark}\n\\label{rem:zerorel}\n\nIf $\\cat C$ is a $0$-relative category, then $a\\cat C = w\\cat C$ and\nhence\n\\begin{itemize}\n\\item $0$-relative categories are the same as maximal relative\n  categories (Df.~\\ref{saturii}).\n\\end{itemize}\n\n\\boldcentered{Strict $3$-arrow calculi and $k$-relative categories}\n\n\\subsection{Definition}\n\\label{def:Str3ar}\n\nA \\textbf{strict $3$-arrow calculus} on a $k$-relative category $\\cat\nC$ is a $3$-arrow calculus (Df.~\\ref{def:calculus}) on $(a\\cat C,w\\cat\nC)$ which restricts to a $3$-arrow calculus on\n\\begin{displaymath}\n  (w\\cat C,w\\cat C)\\qquad\\text{and}\\qquad (v_{i}\\cat C,w\\cat C)\n  \\qquad \\text{for all $i \\le i \\le k$.}\n\\end{displaymath}\n\n\\boldcentered{The $k$-simplicial nerve functor}\n\n\\subsection{Definition}\n\\label{def:kSmpNv}\n\nLet $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$) denote the category of (small) $k$-relative\ncategories (Df.~\\ref{def:krel}) and let $\\mathrm{s}^{k}\\cat S$ denote the\ncategory of (small) \\textbf{$k$-simplicial spaces}, i.e.\\\n$(k+1)$-simplicial sets.\n\nFurthermore, for every integer $p \\ge 0$ let $\\abs{\\cat p} \\subset\n\\cat p$ (Df.~\\ref{def:smpnv}) denote the subcategory which consists of\nthe objects and their identity maps only and let\n\\begin{displaymath}\n  \\text{$\\cat p^{w}$ and\n  $\\cat p^{v_{i}}$ ($1 \\le i \\le k$) $\\in \\mathbf{Rel}^{k}\\mathbf{Cat}$}\n\\end{displaymath}\nbe such that\n\\begin{gather*}\n  a\\cat p^{w} = v_{i}\\cat p^{w} = w\\cat p^{w} = \\cat p\n  \\qquad \\text{($1 \\le i \\le k$), and}\\\\\n  a\\cat p^{v_{i}} = v_{i}\\cat p^{v_{i}} = \\cat p\n  \\quad\\text{and}\\quad\n  v_{j}\\cat p^{v_{i}} = w\\cat p^{v_{i}} = \\abs{\\cat p}\n  \\quad\\text{for $j \\ne i$.}\n\\end{gather*}\n\nThe \\textbf{$k$-simplicial nerve functor} then is the functor\n\\begin{displaymath}\n  \\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\longrightarrow \\mathrm{s}^{k}\\cat S\n\\end{displaymath}\nwhich each object $\\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ to the $(k+1)$-simplicial set\nwhich in multidimension $(p_{k}, \\ldots, p_{1}, q)$ consists of the\nmaps\n\\begin{displaymath}\n  \\cat p_{k}^{v_{k}} \\times \\cdots \\times \\cat p_{1}^{v_{1}}\\times\n  \\cat q^{w} \\longrightarrow \\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}\n\\end{displaymath}\n\n\\subsection{Remark}\n\\label{rem:agree}\n\nIf $k=1$, then (Rk.~\\ref{rem:onerel}) $\\mathbf{Rel}^{k}\\mathbf{Cat} = \\mathbf{RelCat}$\n(Df.~\\ref{def:smpnv}) and hence in that case Df.~\\ref{def:kSmpNv}\nagrees with Df.~\\ref{def:smpnv}.\n\n\\subsection{Notation}\n\\label{not:kzero}\n\nIf $k=0$, then\n\\begin{enumerate}\n\\item \\label{kzeroi} the category $\\mathbf{Rel}^{0}\\mathbf{Cat}$ is, in view of\n  Rk.~\\ref{rem:zerorel}, isomorphic, although not identical, with the\n  category $\\mathbf{Cat}$ of (small) categories and we will therefore\n  \\begin{itemize}\n  \\item \\emph{often denote $\\mathbf{Rel}^{0}\\mathbf{Cat}$ by $\\widehat{\\mathbf{Cat}}$.}\n  \\end{itemize}\n\\item \\label{kzeroii} $\\mathrm{s}^{0}\\cat S = \\cat S$, the category of\n  (small) simplicial sets, and\n\\item \\label{kzeroiii} the functor $\\mathrm{s}^{0}\\N\\colon \\mathbf{Rel}^{0}\\mathbf{Cat} \\to\n  \\mathrm{s}^{0}\\cat S$ is just the \\textbf{classical nerve functor} which\n  we will therefore denote by\n  \\begin{displaymath}\n    \\N\\colon \\widehat{\\mathbf{Cat}} \\longrightarrow \\cat S\n  \\end{displaymath}\n\\end{enumerate}\n\nClosely related to the $k$-simplicial nerve functor are the following\ntwo functors.\n\n\n\\boldcentered{The levelwise nerve functor}\n\n\\subsection{Definition}\n\\label{def:lvlnrv}\n\nLet $k\\ge 0$.  Then the \\textbf{levelwise nerve functor}\n\\begin{displaymath}\n  \\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\longrightarrow \\mathrm{s}^{k}\\cat S\n\\end{displaymath}\nwill be the functor in which $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ denotes the category\nof $k$-simplicial diagrams in $\\widehat{\\mathbf{Cat}}$ and $\\N_{*}$ is dimensionwise\napplication of the classical nerve functor $\\N$\n(Nt.~\\ref{rem:onerel}).\n\n\\boldcentered{The higher equivalence functor}\n\n\\subsection{Definition}\n\\label{def:HighEq}\n\nFor every integer $k \\ge 0$ we denote by\n\\begin{displaymath}\n  w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\longrightarrow \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}\n\\end{displaymath}\nthe \\textbf{higher equivalence functor} which, for every $k$-fold\ndimension $p_{*} = (p_{k},\\ldots,p_{1})$ sends each object $\\cat C \\in\n\\mathbf{Rel}^{k}\\mathbf{Cat}$ to the object $w_{p_{*}}\\cat C \\in \\widehat{\\mathbf{Cat}}$ which has as its\nobjects the maps\n\\begin{displaymath}\n  \\cat p_{k}^{v_{k}} \\times \\cdots \\times \\cat p_{1}^{v_{1}}\n  \\longrightarrow \\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}\n\\end{displaymath}\nand as maps the natural weak equivalences between them.\n\n\\intro The above functors have the following properties.\n\\subsection{Proposition}\n\\label{prop:RlCtsmpS}\n\n\\emph{The following diagram commutes}\n\\begin{displaymath}\n  \\vcenter{\n    \\xymatrix{\n      {\\mathbf{Rel}^{k}\\mathbf{Cat}} \\ar[rr]_-{\\mathrm{s}^{k}\\N} \\ar[dr]_{w_{*}}\n      && {\\mathrm{s}^{k}\\cat S}\\\\\n      & {\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}} \\ar[ur]_{\\N_{x}}\n    }\n  }\n  \\qquad\\qquad\n  k\\ge 0\n\\end{displaymath}\n\n\\begin{proof}\n  This follows directly from the definitions.\n\\end{proof}\n\n\\subsection{Proposition}\n\\label{prop:SmpCtAdj}\n\n\\begin{em}\n  For every integer $k \\ge 0$ the functors\n  \\begin{enumerate}\n  \\item $\\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\to \\mathrm{s}^{k}\\cat S$,\n  \\item $\\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\cat S$, and\n  \\item $w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$\n  \\end{enumerate}\n  have a left adjoint which is a left inverse.\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item It clearly suffices to prove this for the functor $\\N$.  But\n    this was already done long ago in \\cite{GZ}*{II,~4.7}.\n  \\item This was shown in \\cite{BK2}*{4.3}.\n  \\item This follows from Pr.~\\ref{prop:RlCtsmpS} above and the\n    following result.  \\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\\subsection{Proposition}\n\\label{prop:LfAdjInvComp}\n\nIf, for two functors $g_{1}\\colon \\cat A \\to \\cat B$ and $g_{2}\\colon\n\\cat B \\to \\cat C$, both $g_{2}$ and $g_{2}g_{1}$ have left adjoints\nwhich are left inverses, then so does $g_{1}$.\n\n\\begin{proof}\n  Consider the diagram\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cat A} \\ar@<-0.25ex>[r]_{g_{1}} \\ar@<-0.75ex>[r];[]_{f_{1}}\n      & {\\cat B} \\ar@<-0.25ex>[r]_{g_{2}} \\ar@<-0.75ex>[r];[]_{f_{2}}\n      & {\\cat C} \\ar `u[ll] `[ll]_{f_{12}} [ll]\n    }\n  \\end{displaymath}\n  in which $f_{2}$ and $f_{12}$ are left adjoints of $g_{2}$ and\n  $g_{2}g_{1}$ and\n  \\begin{displaymath}\n    f_{2}g_{2} = 1,\\qquad\n    f_{12}g_{2}g_{1} = 1\n    \\qquad\\text{and}\\qquad\n    f_{1} = f_{12}g_{2}.\n  \\end{displaymath}\n  Then $f_{1}g_{1} = f_{12}g_{2}g_{1} = 1$ and it thus remains to show\n  that $f_{1}$ is a left adjoint of $g_{1}$ and this one does by\n  noting that every pair of objects $A \\in \\cat A$ and $B \\in \\cat B$\n  gives rise to a natural sequence of identity maps and isomorphisms\n  \\begin{displaymath}\n    \\vcenter{\n    \\xymatrix@C=1em@R=1.5ex{\n      {\\cat A(f_{1}B,A)} \\ar@{=}[d]\n      && {\\cat B(B,g_{1}A)} \\ar@{=}[d]\\\\\n      {\\cat A(f_{12}g_{2}B,A)} \\ar@{}[r]|{\\approx}\n      & {\\cat C(g_{2}B, g_{2}g_{1}A)} \\ar@{}[r]|{\\approx}\n      & {\\cat B(f_{2}g_{2}B,g_{1}A)}\n    }\n    }  \\qedhere\n  \\end{displaymath}\n\\end{proof}\n\n\\intro\nIt remains to deal with the results that were promised in \n\\ref{CtFncSummi} and (ii).\n\\boldcentered{Homotopy relations on $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n    $\\mathrm{s}^{k}\\mathbf{Cat}$}\n\n\\subsection{Definition}\n\\label{def:StrHmtp}\n\nOne can define \\textbf{strict homotopies} and \\textbf{homotopy\n  equivalences} in the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ as in Df.~\\ref{def:HomRel} by choosing an ``obvious\nanalog'' of $\\boldsymbol 1^{w}$ as follows.\n\\begin{enumerate}\n\\item \\label{StrHmtpi} For $\\mathbf{Rel}^{k}\\mathbf{Cat}$ this will be the object $\\boldsymbol\n  1^{w} \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ (Df.~\\ref{def:kSmpNv}),\n\\item \\label{StrHmtpii} for $\\mathrm{s}^{k}\\cat S$ this will be the\n  standard multisimplex $\\Delta[0,\\ldots,0,1] \\in \\mathrm{s}^{k}\\cat S$,\n  and\n\\item \\label{StrHmtpiii} for $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ this will be the\n  object $\\boldsymbol 1^{w}_{x} \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ such that, for every\n  $k$-fold dimension $p_{*} = (p_{k}, \\ldots, p_{1})$, $\\boldsymbol\n  1^{w}_{p_{*}} = \\boldsymbol 1^{w} \\in \\widehat{\\mathbf{Cat}}$.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:PrsStHmt}\n\n\\begin{em}\n  The functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and $\\N_{*}$\n  \\begin{enumerate}\n  \\item send strictly homotopic maps to strictly homotopic maps\n  \\end{enumerate}\n  and hence\n  \\begin{resumeenumerate}{2}\n    \\item send homotopy equivalences to homotopy equivalences.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item The classical nerve functor $\\N\\colon \\widehat{\\mathbf{Cat}} \\to \\cat S$ has\n    the property that (\\cite{La}*{\\S2} and \\cite{Le}), for every two\n    maps $f,g\\colon \\cat A \\to \\cat B \\in \\widehat{\\mathbf{Cat}}$\n    \\begin{itemize}\n    \\item $\\N$ sends every strict homotopy between $f$ and $g$ to a\n      strict homotopy between $\\N f$ and $\\N g$, and\n    \\item conversely, every strict homotopy in $\\cat S$ between $\\N f$\n      and $\\N g$ is obtained in this fashion.\n    \\end{itemize}\n  \\item This readily implies that $\\N_{*}$ has the same properties.\n  \\item The proof for $\\mathrm{s}^{k}\\N$ is just a higher dimensional\n    version of the proof for $k=1$ in \\cite{BK1}*{7.5}.\n  \\item A proof for $w_{*}$ then is obtained by combining\n    Pr.~\\ref{prop:RlCtsmpS} with (ii) and (iii) above.\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\\section{Abstract nerve functors}\n\\label{sec:AbsNv}\n\n\\subsection{Summary}\n\\label{sec:AbsNvSm}\n\nWe now endow the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) with a \\emph{relative} or \\emph{model}\nstructure which will turn the functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and\n$\\N_{*}$ into what we will call \\emph{abstract nerve functors}, i.e.\\\nfunctors which, in addition to the properties that wee mentioned in\nPr.~\\ref{prop:SmpCtAdj} and Pr.~\\ref{prop:PrsStHmt}, have the property\nthat\n\\begin{enumerate}\n\\item \\label{AbsNvSmi} in the categories involved all \\emph{homotopy\n    equivalences} are \\emph{weak equivalences}, and\n\\item \\label{AbsNvSmii} the functors between them are \\emph{homotopy\n    equivalences}.\n\\end{enumerate}\n\nWe do this in two different ways.  In the first we endow the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge\n1$) with \\emph{Reedy} structures.  In the second we endow them with\n\\emph{Rezk} structures which have more weak equivalences and which\nturn these categories into models for the theory of\n$(\\infty,k)$-categories.\n\n\\boldcentered{Abstract nerve functors}\n\n\\subsection{Definition}\n\\label{def:AbsNvFn}\n\nA functor $f\\colon \\cat C \\to \\cat D$ between saturated relative\ncategories (Df.~\\ref{def:saturated}) will be called an\n\\textbf{abstract nerve functor} if it has the following four\nproperties:\n\\begin{enumerate}\n\\item \\label{AbsNvFni} The relative categories $\\cat C$ and $\\cat D$\n  come with \\emph{strict homotopies} for which the associated\n  \\emph{homotopy equivalences} are \\emph{weak equivalences}.\n\\item \\label{AbsNvFnii} The functor $f$ is a relative functor which is\n  a \\emph{homotopy equivalence}.\n\\item \\label{AbsNvFniii} The functor $f$ sends strictly homotopic maps\n  to strictly homotopic maps and hence sends \\emph{homotopy\n    equivalences} to \\emph{homotopy equivalences}.\n\\item \\label{AbsNvFniv} The functor $f$ has a \\emph{left adjoint} which\n  is a \\emph{left inverse}.\n\\end{enumerate}\n\n\n\\boldcentered{The Reedy structures}\n\n\\subsection{Definition}\n\\label{def:ReeStrct}\n\nWe endow\n\\begin{enumerate}\n\\item \\label{ReeStrcti} the category $\\cat S$ with the usual model\n  structure,\n\\item \\label{ReeStrctii} the category $\\widehat{\\mathbf{Cat}}$ with the \\emph{Quillen\n  equivalent} Thomason structure \\cite{T2},\n\\item \\label{ReeStrctiii} the categories $\\mathrm{s}^{k}\\cat S$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) with the resulting Reedy model\n  structures,\n\\item \\label{ReeStrctiv} the category $\\mathbf{RelCat}$ with the \\emph{Quillen\n  equivalent} Reedy model structure of \\cite{BK1}*{6.1} which was\nlifted from the Reedy structure on $\\mathrm{s}\\cat S$, and\n\\item \\label{ReeStrctv} the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$) with the\n  \\emph{homotopy equivalent} relative Reedy structures of\n  \\cite{BK2}*{3.3} which were lifted from the Reedy structures on\n  $\\mathrm{s}^{k}\\cat S$.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:AbsNvFncs}\n\n\\begin{em}\n  If the categories $s^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) are endowed with the relative\n  structures of \\ref{def:ReeStrct}, then the functors $\\N_{*}$,\n  $\\mathrm{s}^{k}\\N$ and $w_{*}$ are abstract nerve functors.\n\\end{em}\n\n\\begin{proof}\n  In view of Pr.~\\ref{prop:SmpCtAdj} and \\ref{prop:PrsStHmt} it\n  suffices to show that\n  \\begin{enumerate}\n  \\item the functors $N_{*}$, $\\mathrm{s}^{k}\\N$ ($k\\ge 1$) and $w_{*}$ are\n    homotopy equivalences, and\n  \\item the homotopy equivalences in $\\mathrm{s}^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n    $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge 0$) are weak equivalences.\n  \\end{enumerate}\n\n  To prove (i) we note:\n  \\begin{lettered}\n  \\item As Dana Latch \\cite{La} has shown that the functor $\\N$ is a\n    homotopy equivalence it readily follows that so are the functors\n    $\\N_{*}$.\n  \\item That the functors $\\mathrm{s}^{k}\\N$ are homotopy equivalences was\n    shown in \\cite{BK2}*{3.4}.\n  \\item That the functors $w_{*}$ are homotopy equivalences then\n    follows a), b) and Pr.~\\ref{prop:RlCtsmpS} and the observation\n    that homotopy equivalences have the two out of three property.\n  \\end{lettered}\n  To prove (ii) we note\n  \\begin{resumelettered}{4}\n  \\item As the homotopy equivalences in $\\cat S$ are weak\n    equivalences, it readily follows that so are the homotopy\n    equivalences in $\\mathrm{s}^{k}\\cat S$.\n  \\item That the homotopy equivalences in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n    $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ are also weak equivalences then follows from d)\n    and the fact that the functors $\\mathrm{s}^{k}\\N$ and $\\N_{*}$ preserve\n    homotopy equivalences and reflect weak equivalences.\\qedhere\n  \\end{resumelettered}\n\\end{proof}\n\n\n\\boldcentered{The Rezk structures}\n\n\\subsection{Definition}\n\\label{def:RzkStrct}\n\nIn \\cite{R1} Charles Rezk constructed a left Bousfield localization of\nthe Reedy structure on $\\mathrm{s}\\cat S$ and showed it to be a model for\nthe theory of $(\\infty,1)$-categories.\n\nFurthermore it was noted in \\cite{B} (and a proof thereof can be found\nin \\cite{Lu}*{\\S1}) that iteration of Rezk's construction yields for\nevery integer $k>1$, a left Bousfield localization of the Reedy\nstructure on $\\mathrm{s}^{k}\\cat S$ which is a model for the theory of\n$(\\infty,k)$-categories.\n\nWe therefore will denote\n\\begin{enumerate}\n\\item \\label{RzkStrcti} by $\\mathrm{L}\\mathrm{s}^{k}\\cat S$ ($k\\ge 1$) the\n  category $\\mathrm{s}^{k}\\cat S$ with this (iterated) Rezk structure,\n\\item \\label{RzkStrctii} by $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) the\n  induced \\cite{H}*{3.3.20} \\emph{Quillen equivalent} Rezk model\n  structure, and\n\\item \\label{RzkStrctiii} by $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ the \\emph{Quillen} or\n  \\emph{homotopy equivalent} Rezk structure lifted \\cite{BK2}*{4.2}\n  from the Rezk structure on $\\mathrm{s}^{k}\\cat S$ (or the Quillen\n  equivalent Rezk structure on $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$) which categories\n  therefore are all models for the theory of $(\\infty,k)$-categories.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:AllAbsnv}\n\n\\begin{em}\n  If the categories $\\mathrm{s}^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ are endowed with the Rezk structures of\n  \\ref{def:RzkStrct}, then the functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and\n  $\\N_{*}$ are abstract nerve functors.\n\\end{em}\n\n\\begin{proof}\n  This follows from Pr.~\\ref{prop:AbsNvFncs} and the fact that the\n  Rezk structures have more weak equivalences than the Reedy ones.\n\\end{proof}\n\n\\part{Homotopy pullback and potential homotopy pullback}\n\n\n\\section{Homotopy pullbacks}\n\\label{sec:HmtpyPlbk}\n\n\\subsection{Summary}\n\\label{sec:HmtpyPlbkSumm}\n\nAs we are concerned not only with homotopy pullbacks in the\n\\emph{model categories} $\\mathbf{RelCat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$),\nbut  also in the \\emph{saturated relative categories} $\\mathbf{Rel}^{k}\\mathbf{Cat}$\n($k>1$) on which we do \\emph{not} have a model structure, we will\ndefine homotopy pullback in a more general fashion than is usually\ndone.\n\\begin{enumerate}\n\\item \\label{HmtpyPlbkSummi} In a model category we define a homotopy\n  pullback of a zigzag as \\emph{any object} which is weakly equivalent\n  to its image under a \\emph{``homotopically correct'' homotopy limit functor}.\n\\item \\label{HmtpyPlbkSummii} In a saturated relative category we then\n  define a homotopy pullback of a zigzag as \\emph{any object} weakly\n  equivalent to its image under what we will call a \\emph{weak\n    homotopy limit functor} which is a functor which has only some of\n  the properties of the above (i) homotopy limit functors.\n\\item \\label{HmtpyPlbkSummiii} Our main result then is a \\emph{global\n    equivalence lemma} which states that, if $f\\colon \\cat C \\to \\cat\n  D$ is a homotopy equivalence between saturated relative categories,\n  then $\\cat C$ \\emph{admits weak homotopy limit functors iff $\\cat D$\n    does}, and in that case $f$ \\emph{preserves homotopy pullbacks}.\n\\end{enumerate}\n\nIn view of Df.~\\ref{AbsNvFnii} and Pr.~\\ref{prop:AbsNvFncs} this\nresult not only takes care of the notion of homotopy pullback in the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$), but it enables us, in the proof of our\nmain result in section \\ref{sec:PrfThm}, to lift our results from the\nmodel categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k>1$) to the relative categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$).\n\n\\subsection{Remark}\n\\label{rem:Arbtr}\n\nThe results of this section actually hold for homotopy limit functors\n(and dually homotopy colimit functors) on \\emph{arbitrary} diagram\ncategories.\n\n\\subsection{Remark}\n\\label{rem:MdlLim}\n\nAs our definition of homotopy pullbacks is much less rigid that the\nusual ones, it might be more correct to refer to them as \\emph{models\n  for homotopy limits}.\n\n\\boldcentered{Homotopy pullbacks in model categories\\\\\n  and their left Bousfield localizations}\n\n\\subsection{Definition}\n\\label{def:HomELim}\n\nLet $\\cat E$ denote the $2$-arrow category $\\cdot \\to \\cdot \\gets\n\\cdot$.\n\nGiven a model category $\\cat M$, we then mean by a \\textbf{homotopy\n  $\\cat E$-limit functor} on $\\cat M$ a ``homotopically correct''\nhomotopy limit functor\n\\begin{displaymath}\n  \\holim^{\\cat E}\\colon \\cat M^{\\cat E} \\longrightarrow \\cat M \\rlap{\\enspace ,}\n\\end{displaymath}\ni.e.\\ a functor which, as for instance in \\cite{DHKS}*{20.1}, sends\nevery object of $\\cat M^{\\cat E}$ to a \\emph{fibrant} object of $\\cat\nM$ and every (objectwise) weak equivalence in $\\cat M^{\\cat E}$ to a\n\\emph{weak equivalence} in $\\cat M$.\n\nIt has the following property.\n\\subsection{Proposition}\n\\label{prop:HolimRtAdj}\n\n\\begin{em}\n  The functor\n  \\begin{displaymath}\n    \\Ho\\holim^{\\cat E}\\colon \\Ho(\\cat M^{\\cat E})\n    \\longrightarrow \\Ho\\cat M\n  \\end{displaymath}\n  is a right adjoint of the constant diagram functor $\\Ho\\cat M \\to\n  \\Ho(\\cat M^{\\cat E})$.\n\\end{em}\n\n\\begin{proof}\n  This is a special case of \\cite{DHKS}*{20.2}.\n\\end{proof}\n\n\\subsection{Definition}\n\\label{def:HmtpyPlbk}\n\nGiven an object $\\cat B \\in \\cat M^{\\cat E}$, we will say that an\nobject $\\cat U \\in \\cat M$ is a \\textbf{homotopy pullback} of $\\cat\nB$, if $\\cat U$ is weakly equivalent to $\\holim^{\\cat E}\\cat B$.\n\n\\boldcentered{Quasi-fibrant objects in left Bousfield localizations\\\\ of\n    left proper model categories}\n\n\\subsection{Definition}\n\\label{def:QFib}\n\nLet $\\cat M$ be a model category and let $\\mathrm{L}\\cat M$ be a left\nBousfield localization of $\\cat M$, i.e.\\ \\cite{H}*{3.3.3} a model\ncategory with the \\emph{same} cofibrations but \\emph{more} weak\nequivalences.\n\nIf $\\cat M$ is left proper, then an object $\\cat D \\in \\mathrm{L}\\cat M$ will\nbe called \\textbf{quasi-fibrant} if\n\\begin{enumerate}\n\\item \\label{QFibi} $\\cat X$ is weakly equivalent in $\\cat M$ to a\n  fibrant object in $\\mathrm{L}\\cat M$\n\\end{enumerate}\nor equivalently \\cite{H}*{3.4.6(1)}\n\\begin{resumeenumerate}{2}\n\\item \\label{Qfibii} one (and hence every) fibrant approximation of\n  $\\cat X$ in $\\cat M$ is fibrant in $\\mathrm{L}\\cat M$.\n\\end{resumeenumerate}\n\n\\subsection{Proposition}\n\\label{prop:QFpb}\n\nLet $\\mathrm{L}\\cat M$ be a left Bousfield localization of a left proper\nmodel category $\\cat M$.  Then, for every zigzag between quasi-fibrant\nobjects, its homotopy pullbacks in $\\cat M$ are quasi-fibrant in\n$\\mathrm{L}\\cat M$ and also homotopy pullbacks of this zigzag in $\\mathrm{L}\\cat M$.\n\n\\begin{proof}\n  This follows readily from \\cite{H}*{3.4.6(1)} and\n  \\cite{H}*{19.6.5}.\n\\end{proof}\n\n\n\\boldcentered{Homotopy pullbacks in saturated relative categories}\n\n\\subsection{Definition}\n\\label{def:WHLF}\n\nGive a saturated relative category $\\cat R$ (Df.~\\ref{saturi}), a\n\\textbf{weak homotopy $\\cat E$-limit functor} will be a relative functor\n\\begin{displaymath}\n  \\wholim^{\\cat E}\\colon \\cat R^{\\cat E} \\longrightarrow \\cat R\n\\end{displaymath}\nfor which the induced functor\n\\begin{displaymath}\n  \\Ho\\wholim^{\\cat E}\\colon \\Ho(\\cat R^{\\cat E}) \\longrightarrow\n  \\Ho\\cat R\n\\end{displaymath}\nis a right adjoint of the constant diagram functor $\\Ho c\\colon\n\\Ho\\cat R \\to \\Ho(\\cat R^{\\cat E})$.\n\n\\subsection{Definition}\n\\label{def:HmptPlbk}\n\nGiven an object $\\cat B \\in \\cat R^{\\cat E}$, we will say that an\nobject $\\cat U \\in \\cat R$ is a \\textbf{homotopy pullback} of $\\cat B$\nif $\\cat U$ is weakly equivalent to $\\wholim^{\\cat E}\\cat B$.\n\n\\subsection{Remark}\n\\label{rem:LocEq}\n\nWhile for a model category any two homotopy limit functors\n(Df.~\\ref{def:HomELim}) are naturally weakly equivalent, this need not\nbe the case for these \\emph{weak} homotopy limit functors.  However\nthey still have the following \\emph{local equivalence} property which,\nbecause of our \\emph{non-functorial} definition of homotopy pullbacks,\nis all we will need.\n\n\\subsection{Proposition}\n\\label{prop:HPBInd}\n\n\\begin{em}\nThe notion of homotopy pullbacks does not depend on the choice of weak\nhomotopy $\\cat E$-limit functor as\n  \\begin{enumerate}\n  \\item \\label{HPBindi}\n    for any two such functors\n    \\begin{displaymath}\n      \\wholim^{\\cat E}_{1} \\quad\\text{and}\\quad\n      \\wholim^{\\cat E}_{2}\\colon \\cat R^{\\cat E}\\longrightarrow\n      \\cat R\n    \\end{displaymath}\n    the induced functors\n    \\begin{displaymath}\n      \\Ho\\wholim^{\\cat E}_{1} \\quad\\text{and}\\quad\n      \\Ho\\wholim^{\\cat E}_{2}\\colon \\Ho(\\cat R^{\\cat E})\n      \\longrightarrow \\Ho\\cat R\n    \\end{displaymath}\n    are naturally isomorphic\n  \\end{enumerate}\n  which implies that\n  \\begin{resumeenumerate}{2}\n  \\item \\label{HPBindii} for every object $\\cat B \\in \\cat R^{\\cat\n      E}$, the objects $\\wholim^{\\cat E}_{1}\\cat B$ and $\\wholim^{\\cat\n      E}_{2}\\cat B$ are weakly equivalent.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  This follows readily from the uniqueness of adjoints and the\n  saturation of $\\cat R$.\n\\end{proof}\n\n\\intro\nWe end with another useful property.\n\\boldcentered{A global equivalence lemma}\n\n\\subsection{Lemma}\n\\label{lem:WHLEx}\n\n\\begin{em}\n  Let $f\\colon \\cat R_{1} \\to \\cat R_{2}$ be a homotopy equivalence\n  (Df.~\\ref{HomReliii}) between saturated relative categories.  Then\n  \\begin{enumerate}\n  \\item \\label{WHLExi} there exist weak homotopy $\\cat E$-limit\n    functors on $\\cat R_{1}$ iff they exist on $\\cat R_{2}$\n  \\end{enumerate}\n  in which case\n  \\begin{resumeenumerate}{2}\n  \\item \\label{WHLExii} an object $\\cat U \\in \\cat R_{1}$ is a\n    homotopy pullback of an object $B \\in \\cat R_{1}^{\\cat E}$ iff the\n    object $f\\cat U \\in \\cat R_{2}$ is a homotopy pullback of the\n    object $f\\cat B \\in \\cat R_{2}^{\\cat E}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  If $\\wholim^{\\cat E}_{1}\\colon \\cat R_{1}^{\\cat E} \\to \\cat R_{1}$\n  is a weak homotopy limit functor and $g\\colon \\cat R_{2} \\to \\cat\n  R_{1}$ is a homotopy inverse (Df.~\\ref{HomReliii}) of $f$, then it\n  suffices to show that the following composition is also a weak\n  homotopy limit functor\n  \\begin{displaymath}\n    \\xymatrix@C=4em{\n      {\\cat R_{2}}\n      & {\\cat R_{1}} \\ar[l]_-{f}\n      & {\\cat R_{1}^{\\cat E}} \\ar[l]_-{\\wholim_{1}^{\\cat E}}\n      & {\\cat R_{2}^{\\cat E}} \\ar[l]_-{g^{\\cat E}}\n    }\n  \\end{displaymath}\n  To do this we successively note the following.\n  \\begin{enumerate}\n  \\item The maps\n    \\begin{displaymath}\n      \\Ho f\\colon \\Ho\\cat R_{1}\\longrightarrow \\Ho\\cat R_{2}\n      \\qquad\\text{and}\\qquad\n      \\Ho g\\colon \\Ho\\cat R_{2} \\longrightarrow \\Ho\\cat R_{1}\n    \\end{displaymath}\n    are inverse equivalences of categories and hence are both left and\n    right adjoint.\n  \\item This implies the existence of the sequence of adjunctions\n    \\begin{displaymath}\n      \\xymatrix@C=4.5em{\n        {\\Ho\\cat R_{2}} \\ar@<0.75ex>[r]^-{\\Ho g}\n        \\ar@<0.25ex>[r];[]^-{\\Ho f}\n        & {\\Ho\\cat R_{1}} \\ar@<0.75ex>[r]^-{\\Ho i_{1}}\n        \\ar@<0.25ex>[r];[]^-{\\Ho \\wholim^{\\cat E}_{1}}\n        & {\\Ho(\\cat R_{1}^{\\cat E})} \\ar@<0.75ex>[r]^-{\\Ho f^{\\cat E}}\n        \\ar@<0.25ex>[r];[]^-{\\Ho g^{\\cat E}}\n        & {\\Ho(\\cat R_{2}^{\\cat E})}\n      }\n    \\end{displaymath}\n  \\item The composition of the left adjoints equals the composition\n    \\begin{displaymath}\n      \\xymatrix@C=3em{\n        {\\Ho\\cat R_{2}} \\ar[r]^-{\\Ho g}\n        & {\\Ho \\cat R_{1}} \\ar[r]^-{\\Ho f}\n        & {\\Ho\\cat R_{2}} \\ar[r]^-{\\Ho i_{2}}\n        & {\\Ho(\\cat R_{2}^{\\cat E})}\n      }\n    \\end{displaymath}\n  \\item As $(\\Ho f)(\\Ho g)$ is naturally isomorphic to the identity of\n    $\\Ho\\cat R_{2}$, the composition in (iii) is naturally isomorphic to\n    \\begin{displaymath}\n      \\Ho i_{2}\\colon \\Ho\\cat R_{2} \\longrightarrow \\Ho(\\cat\n      R_{2}^{\\cat E})\n    \\end{displaymath}\n    which implies that this map is a left adjoint of the composition\n    of the right adjoints in (iii).\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\n\\section{Potential homotopy pullbacks}\n\\label{sec:PotHmPb}\n\n\\subsection{Summary}\n\\label{sec:PHPBsum}\n\nFor every integer $n \\ge 1$ we will in a functorial manner embed every\nzigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in the categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) (Df.~\\ref{def:kSmpNv} and\n\\ref{def:lvlnrv}) in a commutative diagram of the form\n\\begin{displaymath}\n  \\xymatrix{\n    {\\pullback{\\cat X}{\\cat Z}{\\cat Y}} \\ar[r]^-{k} \\ar[d]\n    & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^-{\\pi} \\ar[d]\n    & {\\cat Y} \\ar[d]^{g}\\\\\n    {\\cat X} \\ar[r]^-{h}\n    & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^-{\\pi}\n    & {\\cat Z}\n  }\n\\end{displaymath}\nin which\n\\begin{enumerate}\n\\item the squares are pullback squares,\n\\item $h$ is a weak equivalence, and\n\\item the object $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a\n  \\emph{potential} homotopy pullback of this zigzag in the sense that,\n  under suitable restrictions on the map $f\\colon \\cat X \\to \\cat Z$\n  (which will be discussed in section \\ref{sec:PrpBnCn}), this object\n  is indeed a homotopy pullback of this zigzag.\n\\end{enumerate}\n\nWe start with an auxiliary construction.\n\\boldcentered{$n$-arrow path objects in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:ArrPthOb}\n\nGiven an integer $n \\ge 1$ and an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge\n0$) we denote by $\\subcommacat{\\cat Z}{n}{\\cat Z} \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ the\n\\textbf{$n$-arrow path object} which has\n\\begin{enumerate}\n\\item as objects the $n$-arrow zigzags\n  \\begin{displaymath}\n    \\xymatrix{\n      {Z_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}}\n      & {Z_{2}}\n      & {Z_{1}} \\ar[l] \\ar[r]\n      & {Z_{0}}\n    }\\quad\\text{in $w\\cat Z$,}\n  \\end{displaymath}\n\\item as maps in $w\\subcommacat{\\cat Z}{n}{\\cat Z}$ and\n  $v_{i}\\subcommacat{\\cat Z}{n}{\\cat Z}$ ($1 \\le i \\le k$) the\n  commutative diagrams of the form\n  \\begin{displaymath}\n    \\vcenter{\n      \\xymatrix{\n        {Z_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}} \\ar[d]\n        & {Z_{2}} \\ar[d]\n        & {Z_{1}} \\ar[l] \\ar[r] \\ar[d]\n        & {Z_{0}} \\ar[d]\\\\\n        {Z'_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}}\n        & {Z'_{2}}\n        & {Z'_{1}} \\ar[l] \\ar[r]\n        & {Z'_{0}}\n      }\n    }\n    \\qquad \\text{in $a\\cat Z$}\n  \\end{displaymath}\n  in which the vertical maps are in $w\\cat Z$ and $v_{i}\\cat Z$\n  respectively, and\n\\item as maps in $a\\subcommacat{\\cat Z}{n}{\\cat Z}$ those commutative\n  diagrams as above which are finite compositions of maps in the\n  $v_{i}\\subcommacat{\\cat Z}{n}{\\cat Z}$ ($1 \\le i \\le k$).\n\\end{enumerate}\nFurthermore\n\\begin{resumeenumerate}{4}\n\\item we denote by\n  \\begin{displaymath}\n    \\cat Z \\xrightarrow{\\enskip \\pi_{n}\\enskip}\n    \\subcommacat{\\cat Z}{n}{\\cat Z}\n    \\xrightarrow{\\enskip\\pi_{0}\\enskip} \\cat Z\n    \\qquad\\text{and}\\qquad\n    \\cat Z \\xrightarrow{\\enskip j\\enskip} \\subcommacat{\\cat Z}{n}{Z}\n  \\end{displaymath}\n  the restrictions of $\\subcommacat{\\cat Z}{n}{\\cat Z}$ to the first\n  and last entries respectively and the map which sends each object of\n  $\\cat Z$ to the alternating zigzag of its identity maps.\n\\end{resumeenumerate}\n\nThese maps have the following nice properties.\n\\subsection{Proposition}\n\\label{prop:HEWeEq}\n\\begin{em}\n  \\begin{enumerate}\n  \\item \\label{HEWeEqi} $\\pi_{n}j = \\pi_{0} j = 1$,\n  \\item \\label{HEWeEqii} $j$ is a homotopy equivalence which has\n    $\\pi_{n}$ and $\\pi_{0}$ as homotopy inverses\n  \\end{enumerate}\n  and hence (Df.~\\ref{ReeStrctiv} and (v) and\n  Pr.~\\ref{prop:AbsNvFncs})\n  \\begin{resumeenumerate}{3}\n  \\item \\label{HEWeEqiii} all three maps are weak equivalences in\n    $\\mathbf{Rel}^{k}\\mathbf{Cat}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  This is a straightforward computation.\n\\end{proof}\n\n\n\\boldcentered{$n$-arrow fibers and pullback objects in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\n  \\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:NArrFibr}\n\nGiven an integer $n \\ge 1$ and a zigzag in $\\zigzag{f}{\\cat X}{\\cat\n  Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k \\ge 0$)\n\\begin{enumerate}\n\\item \\label{NArrFibri} we denote by $\\subcommacat{f\\cat X}{n}{\\cat\n    Z}$ the \\textbf{$n$-arrow fibers object} which is defined by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{\\cat Z}\n    = \\pullback{\\cat X}{\\cat Z}{\\subcommacat{\\cat Z}{n}{\\cat Z}}\n    = \\lim\\bigl(\\cat X \\xrightarrow{f}\\cat Z\n    \\xleftarrow{\\pi_{n}} \\subcommacat{\\cat Z}{n}{\\cat Z}\\bigr)\n  \\end{displaymath}\n  and note that it comes with a \\textbf{projection map}\n  \\begin{displaymath}\n    \\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\longrightarrow \\cat Z\n  \\end{displaymath}\n  induced by the map $\\pi_{0}\\colon \\subcommacat{\\cat X}{n}{\\cat Z}\n  \\to \\cat Z$, and\n\\item \\label{NArrFibrii} we denote by $\\subcommacat{f\\cat X}{n}{g\\cat\n    Y}$ the \\textbf{$n$-arrow pullback object} which is defined by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{g\\cat Y}\n    = \\pullback{\\subcommacat{f\\cat X}{n}{\\cat Z}}{\\cat Z}{\\cat Y}\n    = \\lim\\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z}\n    \\xrightarrow{\\pi} \\cat Z \\xleftarrow{g} \\cat Y\\bigr)\n  \\end{displaymath}\n  and note that it comes with a \\textbf{projection map}\n  \\begin{displaymath}\n    \\pi\\colon \\subcommacat{f\\cat X}{n}{g\\cat Y} \\longrightarrow \\cat Y\n  \\end{displaymath}\n  obtained by ``restriction to $\\cat Y$''.\n\\end{enumerate}\n\n\\boldcentered{$n$-arrow fibers and pullback objects in\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k \\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:NArrPB}\n\nAs (Nt.~\\ref{rem:zerorel}) $\\mathbf{Rel}^{0}\\mathbf{Cat} = \\widehat{\\mathbf{Cat}}$, the case\n$\\mathrm{s}^{0}\\widehat{\\mathbf{Cat}} = \\widehat{\\mathbf{Cat}}$ has already been taken care of in\nDf.~\\ref{def:NArrFibr}.\n\nGiven an integer $n \\ge 1$ and a zigzag $\\zigzag{f}{\\cat X}{\\cat\n  Z}{\\cat Y}{g}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\le 1$) we can therefore\ndefine the \\textbf{$n$-arrow fibers and pullback objects}\n$\\subcommacat{f\\cat Z}{n}{\\cat Z}$ and $\\subcommacat{f\\cat X}{n}{g\\cat\n  Y}$ and the associated \\textbf{projection maps}\n\\begin{displaymath}\n  \\pi\\colon\\subcommacat{f\\cat X}{n}{\\cat Z} \\longrightarrow \\cat Z\n  \\qquad\\text{and}\\qquad\n  \\pi\\colon\\subcommacat{f\\cat X}{n}{g\\cat Y}\\longrightarrow \\cat Y\n\\end{displaymath}\nby the requirement that, for every $k$-fold dimension $p_{*} =\n(p_{k},\\ldots,p_{1})$\n\\begin{align*}\n  \\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z}\n  \\xrightarrow{\\pi}\\cat Z\\bigr)_{p_{*}}\n  &= \\bigl(\\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n  \\xrightarrow{\\pi} \\cat Z_{p_{*}}\\bigr) \\in \\widehat{\\mathbf{Cat}},\n  \\quad\\text{and}\\\\\n  \\bigl(\\subcommacat{f\\cat X}{n}{g\\cat Y}\n  \\xrightarrow{\\pi}\\cat Y\\bigr)_{p_{*}}\n  &= \\bigl(\\subcommacat{f_{p_{*}}X_{p_{*}}}{n}{g_{p_{*}}\n    \\cat Y_{p_{*}}} \\xrightarrow{\\pi} \\cat Y_{p_{*}} \\bigr) \\in\n  \\widehat{\\mathbf{Cat}}\n\\end{align*}\n\n\\intro\nThe definitions \\ref{def:NArrFibr} and \\ref{def:NArrPB} are closely\nrelated as follows.\n\\subsection{Proposition}\n\\label{prop:wpicom}\n\n\\begin{em}\n  For every integer $n \\ge 1$ and zigzag $\\zigzag{f}{\\cat X}{\\cat\n    Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) (Df.~\\ref{def:HighEq})\n  \\begin{align*}\n    w_{*}\\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z} \\xrightarrow{\\pi}\\cat\n    Z\\bigr) &= \\bigl(\\commacat{w_{*}fw_{*}\\cat X}{w_{*}\\cat Z}\n    \\xrightarrow{\\pi} w_{*}\\cat Z\\bigr) \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}},\n    \\quad\\text{and}\\\\\n    w_{*}\\bigl(\\subcommacat{f\\cat X}{n}{g\\cat Y}\\xrightarrow{\\pi}\\cat\n    Y\\bigr) &= \\bigl(\\subcommacat{w_{*}fw_{*}\\cat X}{n}{w_{*}gw_{*}Y}\n    \\xrightarrow{\\pi} w_{*}Y\\bigr) \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}.\n  \\end{align*}\n\\end{em}\n\\begin{proof}\n  This follows readily from the observation that, for every $k$-fold\n  dimension $p_{*} = (p_{k},\\cdots,p_{1})$ (Df.~\\ref{def:ArrPthOb})\n  \\begin{displaymath}\n    w_{p_{*}}\\bigl(\\subcommacat{\\cat Z}{n}{\\cat Z}\n    \\xrightarrow{\\pi_{0}} \\cat Z\\bigr) =\n    \\bigl(\\subcommacat{w_{p_{*}}\\cat\n      Z}{n}{w_{p_{*}}Z}\\xrightarrow{\\pi_{0}} w_{p_{*}}Z\\bigr) \\in\n    \\widehat{\\mathbf{Cat}}. \\qedhere\n  \\end{displaymath}\n\\end{proof}\n\n\\intro\nIt remains the pull it all together as we promised in\n\\ref{sec:PHPBsum}.\n\\subsection{Proposition}\n\\label{prop:zzEmbed}\n\n\\begin{em}\n  For every integer $n \\ge 1$, every zigzag $\\zigzag{f}{\\cat X}{\\cat\n    Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) can\n  in a functorial manner be embedded in a commutative diagram of the\n  form (Df.~\\ref{def:NArrFibr} and \\ref{def:NArrPB})\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\pullback{\\cat X}{\\cat Z}{\\cat Y}} \\ar[r]^-{k} \\ar[d]\n      & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^-{\\pi} \\ar[d]\n      & {\\cat Y} \\ar[d]^{g}\\\\\n      {\\cat X} \\ar[r]^-{h}\n      & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^-{\\pi}\n      & {\\cat Z}\n    }\n  \\end{displaymath}\n  in which $h$ and $k$ send each object $X \\in \\cat X$ and $(X,Y) \\in\n  \\pullback{\\cat X}{\\cat Z}{\\cat Y}$ to the alternating zigzag of\n  identity maps of $fX$, and\n  \\begin{enumerate}\n  \\item the squares are pullback squares, and\n  \\item the map $h$ is a weak equivalence.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  That the square on the right is a pullback square follows from\n  Df.~\\ref{def:NArrFibr} and \\ref{def:NArrPB} and that the one on the\n  left is so is a simple calculation.\n\n  That $h$ is a weak equivalence follows readily from\n  Pr.~\\ref{prop:HEWeEq}.\n\\end{proof}\n\n\\section{Properties \\texorpdfstring{$B_{n}$}{Bn} and \\texorpdfstring{$C_{n}$}{Cn}}\n\\label{sec:PrpBnCn}\n\n\\subsection{Summary}\n\\label{sec:BnCnsum}\n\nIn final preparation for the formulation of our main results (in\nsection \\ref{sec:MainRlt}) we recall from \\cite{DKS}*{\\S6} the notions\nof \\emph{properties} $B_{n}$ and $C_{n}$ in $\\widehat{\\mathbf{Cat}}$ and then extend\nthese notions to the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for\n$k \\ge 1$.\n\nAs these notions in $\\widehat{\\mathbf{Cat}}$ are closely related to the\n\\emph{Grothendieck construction} we start with a discussion of the\nlatter.\n\\subsection{Definition}\n\\label{def:GrConst}\n\nGiven an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ and a \\emph{not necessarily\n  relative} functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$\n(Df.~\\ref{relcatiii}), the \\textbf{Grothendieck construction} on $F$\nis the object $\\Gr F \\in \\widehat{\\mathbf{Cat}}$ which has\n\\begin{enumerate}\n\\item \\label{GrConsti} as \\emph{objects} the pairs $(D,A)$ of objects\n  $D \\in \\cat D$ and $A \\in FD$, and\n\\item \\label{GrCoonstii} as \\emph{maps} $(D_{1},A_{1}) \\to\n  (D_{2},A_{2})$ the pairs $(d,a)$ of maps\n  \\begin{displaymath}\n    d\\colon D_{1} \\longrightarrow D_{2} \\in \\cat D\n    \\qquad\\text{and}\\qquad\n    a\\colon (FD_{1})A_{1} \\longrightarrow A_{2} \\in FD_{2}\n  \\end{displaymath}\n  of which the \\emph{compositions} are given by the formula\n  \\begin{displaymath}\n    (d',a')(d,a) = \\bigl(d'd, a'\\bigl((Fd)a\\bigr)\\bigr).\n  \\end{displaymath}\n\\end{enumerate}\n\nSuch a Grothendieck construction $\\Gr F$ comes with a\n\\textbf{projection functor}\n\\begin{displaymath}\n  \\pi\\colon \\Gr F \\longrightarrow \\cat D\\in \\widehat{\\mathbf{Cat}}\n\\end{displaymath}\nwhich sends an object $(D,A)$ (resp.\\ a map $(d,a)$) to the object $D$\n(resp.\\ the map $d$) in $\\cat D$.\n\n\\intro\nThe usefulness of Grothendieck constructions is due to the following\nproperty which was noted by Bob Thomason \\cite{T1}*{1.2}.\n\\subsection{Proposition}\n\\label{prop:GrCnHmCl}\n\\begin{em}\n  \\begin{enumerate}\n  \\item The Grothendieck construction is a homotopy colimit functor on\n    $\\widehat{\\mathbf{Cat}}$\n  \\end{enumerate}\n  and hence\n  \\begin{resumeenumerate}{2}\n  \\item it is homotopy invariant in the sense that every natural weak\n    equivalence between two functors $F_{1},F_{2}\\colon \\cat D \\to\n    \\widehat{\\mathbf{Cat}}$ induces a weak equivalence $\\Gr F_{1} \\to \\Gr F_{2}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\subsection{Example}\n\\label{ex:GrCn}\n\nGiven a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ and an integer $n\n\\ge 1$\n\\begin{enumerate}\n\\item \\label{GrCni} denote, for every object $Z \\in \\cat Z$ by\n  (Df.~\\ref{def:NArrFibr})\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{Z} \\subset\n    \\subcommacat{f\\cat X}{n}{\\cat Z} \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the category consisting of the objects and maps which end at $Z$ or\n  $1_{Z}$, and\n\\item \\label{GrCnii} denote by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{-}\\colon \\cat Z \\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the \\emph{not necessarily relative} functor (Df.~\\ref{relcatiii})\n  which sends each object $Z \\in \\cat Z$ to $\\subcommacat{f\\cat\n    X}{n}{Z}$ and each map $z\\colon Z \\to Z' \\in \\cat Z$ to the\n  functor $\\subcommacat{f\\cat X}{n}{Z} \\to \\subcommacat{f\\cat\n    X}{n}{Z'}$ obtained by ``composition with $z$''.\n\\end{enumerate}\n\nThen one readily verifies that\n\\begin{resumeenumerate}{3}\n\\item \\label{GrCniii} $\\subcommacat{f\\cat X}{n}{\\cat Z} =\n  \\Gr\\subcommacat{f\\cat X}{n}{-}$, and\n\\item \\label{GrCniv} $\\bigl(\\subcommacat{f\\cat X}{n}{\\cat\n    Z}\\xrightarrow{\\pi} \\cat Z\\bigr) = \\bigl(\\Gr\\subcommacat{f\\cat\n    X}{n}{-}\\xrightarrow{\\pi} \\cat Z\\bigr)$.\n\\end{resumeenumerate}\n\n\\subsection{Example}\n\\label{ex:GrCnExAn}\n\nGiven a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ and an integer $n\n\\ge 1$\n\\begin{enumerate}\n\\item \\label{GrCnExAni} denote, for every pair of objects $X \\in \\cat\n  X$ and $Z \\in \\cat Z$, by (Ex.~\\ref{ex:GrCn})\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z} = \\subcommacat{f\\cat X}{n}{Z}\n  \\end{displaymath}\n  the category consisting of the objects and maps which start at $fX$\n  or $1_{fX}$, and\n\\item \\label{GrCnExAnii} denote by\n  \\begin{displaymath}\n    \\subcommacat{f-}{n}{Z}\\colon \\cat X\\longrightarrow \\widehat{\\mathbf{Cat}}\n    \\qquad\\text{or}\\qquad\n    \\subcommacat{f-}{n}{Z}\\colon \\cat X^{\\mathrm{op}}\\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the functor which sends each object $X \\in \\cat X$ to\n  $\\subcommacat{fX}{n}{Z}$ and each map $x\\colon X \\to X' \\in \\cat X$\n  to the induced functor\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z}\\longrightarrow \\subcommacat{fX'}{n}{Z}\n    \\qquad\\text{or}\\qquad\n    \\subcommacat{fX'}{n}{Z}\\longrightarrow \\subcommacat{fX}{n}{Z}\n  \\end{displaymath}\n  depending on whether $n$ is even or odd.\n\\end{enumerate}\n\nThen one readily verifies that\n\\begin{resumeenumerate}{3}\n\\item \\label{GrCnExAniii}\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{Z} = \\left\\{\n    \\begin{aligned}\n      \\Gr\\bigl(\\subcommacat{f-}{n}{Z}&\\colon \\cat X\n      \\longrightarrow \\widehat{\\mathbf{Cat}}\\bigr)\\\\\n      \\text{or}&\\\\\n      \\Gr\\bigl(\\subcommacat{f-}{n}{Z}&\\colon \\cat X^{\\mathrm{op}}\n      \\longrightarrow \\widehat{\\mathbf{Cat}}\\bigr)\n    \\end{aligned}\\right.\n  \\end{displaymath}\n\\end{resumeenumerate}\n\n\\boldcentered{Properties $B_{n}$ and $C_{n}$ in $\\widehat{\\mathbf{Cat}}$}\n\n\\subsection{Definition}\n\\label{def:BnCn}\n\n\\begin{enumerate}\n\\item \\label{BnCni} A map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ is\n  said to have \\textbf{property $B_{n}$} ($n\\ge 1$) if the functor\n  $Ex.~\\ref{ex:GrCn}$\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{-}\\colon \\cat Z \\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  is a \\emph{relative functor} (Df.~\\ref{relcativ}), and\n\\item \\label{BnCnii} an object $\\cat Z \\in \\widehat{\\mathbf{Cat}}$ is said to have\n  \\textbf{property $C_{n}$} ($n\\ge 1$) if every map\n  (Df.~\\ref{def:smpnv})\n  \\begin{displaymath}\n    \\boldsymbol 0^{w} \\longrightarrow \\cat Z \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  has property $B_{n}$.\n\\end{enumerate}\n\n\\intro\nThe usefulness of property $C_{n}$ is due to the following result of\n\\cite{DKS}*{\\S6}.\n\\subsection{Proposition}\n\\label{prop:BnCn}\n\\begin{em}\n  If, given a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$, the object\n  $\\cat Z$ has property $C_{n}$ ($n \\ge 1$), then the map $f\\colon\n  \\cat X \\to \\cat Z$ has property $B_{n}$.\n\\end{em}\n\n\\begin{proof}\n  In view of Ex.~\\ref{GrCnii} one has to show that every map $z\\colon\n  Z \\to Z' \\in \\cat Z$ induces a weak equivalence $\\subcommacat{f\\cat\n    Z}{n}{Zj \\to \\subcommacat{f\\cat X}{n}{Z'}}$, or equivalently\n  (Ex.~\\ref{GrCnExAniii}) a weak equivalence\n  \\begin{displaymath}\n    \\Gr\\subcommacat{f-}{n}{Z}\\longrightarrow\n    \\Gr\\subcommacat{f-}{n}{Z'}\n  \\end{displaymath}\n  But this follows readily from Ex.~\\ref{GrCnExAniii} and\n  Df.~\\ref{GrCnExAnii} and the fact that, in view of property $C_{n}$,\n  for every object $X \\in \\cat X$ the map\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z} \\longrightarrow \\subcommacat{fX}{n}{Z'}\n  \\end{displaymath}\n  is a weak equivalence.\n\\end{proof}\n\n\\boldcentered{Properties $B_{n}$ and $C_{n}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$)}\n\n\\subsection{Definition}\n\\label{def:MpPrpB}\n\n\\begin{enumerate}\n\\item \\label{MpPrpBi} A map $f\\colon \\cat X \\to \\cat Z \\in\n  \\mathrm{s}^{k}\\mathbf{Cat}$ has \\textbf{property $B_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k},\\ldots, p_{1})$,\n    the map $f_{p_{*}}\\colon \\cat X_{p_{*}} \\to \\cat Z_{p_{*}} \\in\n    \\widehat{\\mathbf{Cat}}$ has property $B_{n}$ (Df.~\\ref{BnCni}),\n  \\end{itemize}\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{2}\n\\item \\label{MpPrpBii} An object $\\cat Z \\in \\mathrm{s}^{k}\\mathbf{Cat}$ has\n  \\textbf{property $C_{n}$} ($n \\ge 1$) if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*}= (p_{k},\\ldots, p_{1})$\n    the object $Z_{p_{*}} \\in \\widehat{\\mathbf{Cat}}$ has property $C_{n}$\n    (Df.~\\ref{BnCnii}).\n  \\end{itemize}\n\\end{resumeenumerate}\n\n\\subsection{Definition}\n\\label{def:MpPrBrel}\n\n\\begin{enumerate}\n\\item \\label{MpPrBreli} A map $f\\colon \\cat X \\to \\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$\n  has \\textbf{property $B_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item the map $w_{*}f\\colon w_{*}\\cat X \\to w_{*}\\cat Z \\in\n    \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ has property $B_{n}$ (Df.~\\ref{MpPrpBi}\n  \\end{itemize}\n  or equivalently if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k},\\ldots, p_{1})$\n    the map $w_{p_{*}}f\\colon w_{p_{*}}\\cat X \\to w_{p_{*}}\\cat Z\n    \\in \\widehat{\\mathbf{Cat}}$ has property $B_{n}$\n  \\end{itemize}\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{2}\n\\item \\label{MpPrBrelii} an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ has\n  \\textbf{property $C_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item the object $w_{*}Z \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ has property $C_{n}$\n    (Df.~\\ref{MpPrpBii})\n  \\end{itemize}\n  or equivalently if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k}, \\ldots, p_{1})$\n    the object $w_{p_{*}}\\cat Z \\in \\widehat{\\mathbf{Cat}}$ has property $C_{n}$.\n  \\end{itemize}\n\\end{resumeenumerate}\n\n\\part{The main results and their proofs}\n\n\n\\section{The main results}\n\\label{sec:MainRlt}\n\n\\subsection{Summary}\n\\label{sec:MnRsltSum}\n\nOur main results are\n\\begin{enumerate}\n\\item \\label{MpRsltSumi} \\emph{Theorem} \\ref{thm:MpPropBn} which\n  states that the presence of properties $B_{n}$ and $C_{n}$ ensure\n  that the potential homotopy pullbacks of section~\\ref{sec:PotHmPb},\n  i.e.\\ the \\emph{$n$-arrow pullback objects}, are indeed\n  \\emph{homotopy pullbacks}, and\n\\item \\label{MpRsltSumii} \\emph{Theorem} \\ref{thm:SuffCn} which states\n  that the presence of a \\emph{strict $3$-arrow calculus} implies\n  \\emph{property $C_{3}$}.\n\\end{enumerate}\n\nThese results then are applied to $(\\infty,1)$-categories and\n$(\\infty,k)$-categories for $k>1$ to prove\n\\begin{resumeenumerate}{3}\n\\item \\label{MpRsltSumiii} \\emph{Theorem} \\ref{thm:QFibQfib} which\n  combines Theorem \\ref{thm:MpPropBn} and \\ref{thm:SuffCn} with\n  results from \\cite{BK3} to show that \\emph{homotopy pullbacks in\n    $(\\infty,1)$-categories} can be described as \\emph{$3$-arrow\n    pullback objects} of zigzags between \\emph{partial model\n    categories}, i.e.\\ relative categories which have the \\emph{two\n    out of six} property and admit a \\emph{$3$-arrow calculus}, and\n\\item \\label{MpRsltSumiv} \\emph{Theorem} \\ref{thm:MpBn} which is an\n  application of Theorem \\ref{thm:MpPropBn} to $(\\infty,k)$-categories\n  for $k\\ge 1$, which is much weaker than Theorem \\ref{thm:QFibQfib},\n  because we have no model structure on $\\mathbf{Rel}^{k}\\mathbf{Cat}$ for $k\\ge 1$, nor\n  an analog for partial model categories.\n\\end{resumeenumerate}\n\nWe also give proofs of Theorems \\ref{thm:SuffCn}, \\ref{thm:QFibQfib}\nand \\ref{thm:MpBn}, but postpone the proof of Theorem\n\\ref{thm:MpPropBn} until section \\ref{sec:PrfThm}.\n\n\n\\boldcentered{The main results}\n\n\\subsection{Theorem}\n\\label{thm:MpPropBn}\n\n\\begin{em}\n  Given an integer $n \\ge 1$, let $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat\n    Y}{g}$ be a zigzag in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$)\n  (Df.~\\ref{def:kSmpNv} and \\ref{def:lvlnrv}) with the property that\n  \\begin{enumerate}\n  \\item \\label{MpPropBni} the map $f\\colon \\cat X \\to \\cat Z$ has\n    property $B_{n}$ (Df.~\\ref{BnCni}, \\ref{MpPrpBi} and\n    \\ref{MpPrBreli})\n  \\end{enumerate}\n  which (Pr.~\\ref{prop:BnCn}) is in particular the case if\n  \\begin{resumeenumerate}{2}\n  \\item \\label{MpPropBnii} the object $\\cat Z$ has property $C_{n}$\n    (Df.~\\ref{BnCnii}, \\ref{MpPrpBii} and \\ref{MpPrBrelii}).\n  \\end{resumeenumerate}\n  Then\n  \\begin{resumeenumerate}{3}\n  \\item \\label{MpPropBniii} the $n$-arrow pullback object\n    $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ (Df.~\\ref{def:NArrFibr} and\n    \\ref{def:NArrPB}) is a homotopy pullback (Df.~\\ref{def:HmtpyPlbk}\n    and \\ref{def:HmptPlbk}) of this zigzag.\n  \\end{resumeenumerate}\n\\end{em}\n\nThe main tools for proving this are the \\emph{fibrillations} of\nHopkins and Rezk which we will discuss in section \\ref{sec:HopRzF} and\nwe therefor postpone the proof of Theorem \\ref{thm:MpPropBn} until\nsection \\ref{sec:PrfThm}.\n\n\\subsection{Corollary}\n\\label{cor:SuffHPB}\n\n\\begin{em}\n  A sufficient condition in order that the pullback $\\pullback{\\cat\n    X}{\\cat Z}{\\cat Y}$ of the above zigzag is also a homotopy\n  pullback is that the obvious map\n  \\begin{displaymath}\n    \\pullback{\\cat X}{\\cat Z}{\\cat Y} \\xrightarrow{\\enspace k\\enspace}\n    \\subcommacat{f\\cat X}{n}{g\\cat Y}\n  \\end{displaymath}\n  of Pr.~\\ref{prop:zzEmbed} is a weak equivalence.\n\\end{em}\n\n\\subsection{Theorem}\n\\label{thm:SuffCn}\n\n\\begin{em}\n  A sufficient condition in order that an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$\n  ($k\\ge 0$) has property $C_{3}$ (Df.~\\ref{MpPrBrelii}) is that $\\cat\n  Z$ admits a strict $3$-arrow calculus (Df.~\\ref{def:strcalculus} and\n  \\ref{def:Str3ar}).\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item The case $k=0$.  This follows from Rk.~\\ref{rem:whystrict},\n    Df.~\\ref{def:strcalculus} and \\cite{DK}*{3.3, 6.1, 6.2 and 8.2}.\n  \\item The case $k\\ge 1$.  It follows readily from\n    Df.~\\ref{def:Str3ar} and \\ref{def:HighEq} that, for every $k$-fold\n    dimension $p_{*} = (p_{k},\\ldots, p_{1})$, the object\n    $w_{p_{*}}\\cat Z \\in \\widehat{\\mathbf{Cat}}$ admits a strict $3$-arrow calculus\n    and hence (i) has property $C_{3}$.  The desired result then\n    follows from Df.~\\ref{MpPrBrelii}.\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\n\\boldcentered{Applications to $(\\infty,1)$-categories and\n  $(\\infty,k)$-categories for $k>1$}\n\nBefore formulating the application mentioned in \\ref{MpRsltSumiii} we\nrecall some results from \\cite{BK3}.\n\n\\subsection{Remark}\n\\label{rem:CSSfib}\n\nRecall from \\cite{BK3} that a \\textbf{partial model category} is an\nobject $\\cat X \\in  \\mathbf{RelCat}$ which admits a $3$-arrow calculus\n(Df.~\\ref{def:calculus}) and has the \\textbf{two out of six property}\nthat, for every three maps $r$, $s$ and $t \\in \\cat X$ for which the\n\\emph{two} compositions $sr$ and $ts$ exist and are weak equivalences,\nthe other $four$ maps $r$, $s$, $t$ and $tsr$ are also weak\nequivalences.\n\nIt then was shown in \\cite{BK3} that\n\\begin{enumerate}\n\\item \\label{CSSfibi} \\emph{for every partial model category $\\cat X\n    \\in \\mathbf{RelCat}$, one (and hence every) Reedy fibrant approximation to\n  $\\mathrm{s}\\N\\cat X \\in \\mathrm{s}\\cat S$ (Df.~\\ref{def:smpnv}) is a complete\n  Segal space, i.e.\\ a fibrant object in $\\mathrm{L}\\mathrm{s}\\cat S$\n  (Df.~\\ref{RzkStrcti})} \n\\end{enumerate}\nwhich in view of \\cite{BK1}*{6.1} implies that\n\\begin{resumeenumerate}{2}\n\\item \\label{CSSfibii} \\emph{$\\cat X$ is a quasi-fibrant object\n    (Df.~\\ref{def:QFib}) of $\\mathrm{L}\\mathbf{RelCat}$ (Df.~\\ref{RzkStrctiii})}.\n\\end{resumeenumerate}\nMoreover\n\\begin{resumeenumerate}{3}\n\\item \\label{CSSfibiii}\\emph{every complete Segal space is Reedy\n    equivalent to the simplicial nerve of a partial model category.}\n\\end{resumeenumerate}\n\n\\subsection{Theorem}\n\\label{thm:QFibQfib}\n\nIf $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ is a zigzag in $\\mathbf{RelCat}$ in\nwhich\n\\begin{enumerate}\n\\item $\\cat X$ and $\\cat Y$ are quasi-fibrant in $\\mathrm{L}\\mathbf{RelCat}$ and\n  $\\cat Z$ is a partial model category,\n\\end{enumerate}\nwhich in particular is the case if\n\\begin{resumeenumerate}{2}\n\\item $\\cat X$, $\\cat Y$ and $\\cat Z$ are all three partial model\n  categories,\n\\end{resumeenumerate}\nthen\n\\begin{resumeenumerate}{3}\n\\item \\emph{$\\subcommacat{f\\cat X}{3}{g\\cat Y}$ is a quasi-fibrant\n    object of $\\mathrm{L}\\mathbf{RelCat}$, which is a homotopy pullback of this\n    zigzag not only in $\\mathbf{RelCat}$, but also in $\\mathrm{L}\\mathbf{RelCat}$.}\n\\end{resumeenumerate}\n\n\\begin{proof}\n  This follows readily from Rk.\\ref{rem:whystrict},\n  Pr.~\\ref{prop:QFpb} and Th.~\\ref{thm:MpPropBn} and \\ref{thm:SuffCn}.\n\\end{proof}\n\n\\intro It remains to deal with the result that was mentioned in\n\\ref{MpRsltSumiv}.\n\\subsection{Theorem}\n\\label{thm:MpBn}\n\n\\begin{em}\n  Let $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ be a zigzag in\n  $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) for which\n  \\begin{enumerate}\n  \\item $\\mathrm{s}^{k}\\N\\cat X$, $\\mathrm{s}^{k}\\N\\cat Z$ and $\\mathrm{s}^{k}\\N\\cat\n    Y$ are quasi-fibrant objects of $\\mathrm{L}\\mathrm{s}^{k}\\cat S$\n    (Df.~\\ref{RzkStrcti})\n  \\end{enumerate}\n  or equivalently (Pr.~\\ref{prop:AbsNvFncs})\n  \\begin{resumeenumerate}{2}\n  \\item $w_{*}\\cat X$, $w_{*}\\cat Z$ and $w_{*}\\cat Y$ are\n    quasi-fibrant objects of $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$\n  \\end{resumeenumerate}\n  and assume that for some integer $n \\ge 1$\n  \\begin{resumeenumerate}{3}\n  \\item the map $f\\colon \\cat X \\to \\cat Z$ has property $B_{n}$\n  \\end{resumeenumerate}\n  which (Pr.~\\ref{prop:BnCn}) in particular is the case if\n  \\begin{resumeenumerate}{4}\n  \\item the object $\\cat Z$ has property $C_{n}$,\n  \\end{resumeenumerate}\n  then\n  \\begin{resumeenumerate}{5}\n  \\item $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback of\n    this zigzag not only in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ but also in $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$\n    (Df.~\\ref{RzkStrctiii}).\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  In view of Df.~\\ref{def:NArrPB} and \\ref{def:MpPrBrel},\n  Pr.~\\ref{prop:wpicom} and Th.~\\ref{thm:MpPropBn}\n  $w_{*}\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback if\n  the zigzag\n  \\begin{displaymath}\n    \\zigzag{w_{*}f}{w_{*}\\cat X}{w_{*}\\cat Z}{w_{*}\\cat Y}{w_{*}g}\n  \\end{displaymath}\n  in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ as well as, in view of Pr.~\\ref{prop:QFpb}, in\n  $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\n  The desired result now follows readily from the \\emph{Global\n    equivalence lemma} \\ref{lem:WHLEx} and the fact that\n  (Df.~\\ref{def:AbsNvFn} and Pr.~\\ref{prop:AbsNvFncs}) $w_{*}$ is a\n  \\emph{homotopy equivalence}.\n\\end{proof}\n\n\n\\section{Hopkins-Rezk fibrillations}\n\\label{sec:HopRzF}\n\n\\subsection{Summary}\n\\label{sec:HopRzFSum}\n\nOur proof of Theorem~\\ref{thm:MpPropBn} (in section \\ref{sec:PrfThm})\nwill consist of two parts.  The first consists of a proof for the\n\\emph{model} categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and $\\mathbf{RelCat}$.  In the\nsecond we lift these results for the model categories\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k>1$) to the \\emph{relative} categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ by means of the \\emph{Global equivalence lemma}\n\\ref{lem:WHLEx}.\n\nOur aim in this section is to describe three lemmas which we will need\nfor the first part.  They involve, each in a different way, the\n\\emph{fibrillations} of Hopkins and Rezk as follows.\n\\begin{enumerate}\n\\item \\label{HopRzFSumi} The first lemma is a \\emph{Quillen\n    fibrillation lemma} which will produce the fibrillations to get us\n  started.\n\n  It is essentially a reformulation in terms of relative functors and\n  fibrillations as well as a slight strengthening of the lemma that\n  Quillen used to prove his Theorem~B and states that\n  \\begin{itemize}\n  \\item given an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$, a functor $f\\colon \\cat\n    D \\to \\widehat{\\mathbf{Cat}}$ is a \\emph{relative} functor iff the projection\n    functor from its Grothendieck construction to $\\cat D$\n    (Df.~\\ref{def:GrConst}) is a \\emph{fibrillation} in $\\widehat{\\mathbf{Cat}}$.\n  \\end{itemize}\n\\item \\label{HopRzFSumii} The second lemma is a \\emph{Fibrillation\n    lifting lemma} which enables us to obtain more fibrillation as it\n  provides\n  \\begin{itemize}\n  \\item a sufficient condition on a relative functor in order that it\n    \\emph{reflects fibrillations}.\n  \\end{itemize}\n\\item \\label{HopRzFSumiii} The third lemma is a \\emph{Hopkins-Rezk\n    fibrillation lemma} which shows how, in a right proper model\n  category, some of the fibrillations obtained in (i) and (ii) can be\n  used to construct homotopy pullbacks.\n\\end{enumerate}\n\n\n\\boldcentered{Fibrillations}\n\nWe start with recalling from \\cite{R2}*{\\S2} the notion of what\nCharles Rezk called \\emph{sharp maps} but which, because of their\nfibration-like properties (see \\ref{prop:PBfibril} below), we prefer\nto call \\emph{fibrillations}.\n\n\\subsection{Definition}\n\\label{def:fibril}\n\nGiven a relative category $\\cat R$ \\textbf{with pullbacks} (i.e.\\\nwhich is closed under pullbacks) a map $p \\in \\cat R$ is called a\n\\textbf{fibrillation} if every diagram in $\\cat R$ of the form\n\\begin{displaymath}\n  \\xymatrix{\n    {\\cdot} \\ar[r]_{j} \\ar[d]\n    & {\\cdot} \\ar[r] \\ar[d]\n    & {\\cdot} \\ar[d]^{p}\\\\\n    {\\cdot} \\ar[r]_{i}\n    & {\\cdot} \\ar[r]\n    & {\\cdot}\n  }\n\\end{displaymath}\nin which\n\\begin{itemize}\n\\item the squares are pullback squares, and\n\\item $i$ is a weak equivalence\n\\end{itemize}\nhas the property that\n\\begin{itemize}\n\\item $j$ is also a weak equivalence.\n\\end{itemize}\n\nThis definition readily implies that, just like the \\emph{fibrations}\nin a \\emph{right proper} model category, fibrillations have the\nfollowing properties.\n\n\\subsection{Proposition}\n\\label{prop:PBfibril}\n\n\\begin{em}\n  \\begin{enumerate}\n  \\item \\label{PBfibrili} Every pullback of a fibrillation is again a\n    fibrillation, and\n  \\item \\label{PBfibrilii} every pullback of a weak equivalence along\n    a fibrillation is again a weak equivalence.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  This is straightforward.\n\\end{proof}\n\n\\boldcentered{The fibrillation lifting lemma}\n\n\n\\subsection{Lemma}\n\\label{lem:FibLftLm}\n\n\\begin{em}\n  Let $f\\colon \\cat R_{1} \\to \\cat R_{2}$ be a relative functor\n  between relative categories with pullbacks which\n  \\begin{itemize}\n  \\item preserves pullbacks (e.g.\\ is a right adjoint), and\n  \\item reflects weak equivalences.\n  \\end{itemize}\n  Then\n  \\begin{itemize}\n  \\item it also reflects fibrillations.\n  \\end{itemize}\n\\end{em}\n\n\\begin{proof}\n  Given a pullback diagram in $\\cat R_{1}$ of the form\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cdot} \\ar[r]_{j} \\ar[d]\n      & {\\cdot} \\ar[r] \\ar[d]\n      & {\\cdot} \\ar[d]^{f}\\\\\n      {\\cdot} \\ar[r]_{i}\n      & {\\cdot} \\ar[r]\n      & {\\cdot}\n    }\n  \\end{displaymath}\n  in which $i$ is a weak equivalence and of which the image in $\\cat\n  R_{2}$ is a similar diagram in which $fj$ is a fibrillation.  Then\n  $fj$ is a weak equivalence and hence so is $j$.\n\\end{proof}\n\n\\subsection{Example}\n\\label{ex:FibLm}\n\nExamples of functors which satisfy the conditions of the fibrillation\nlifting lemma \\ref{lem:FibLftLm} are the abstract nerve functors of\nDf.~\\ref{def:AbsNvFn} and Pr.~\\ref{prop:AbsNvFncs}.\n\n\n\\boldcentered{The Hopkins-Rezk fibrillation lemma}\n\nWe recall from \\cite{R2}*{\\S2} the following\n\\subsection{Lemma}\n\\label{lem:HPBzz}\n\n\\begin{em}\n  Let\n  \\begin{displaymath}\n    \\xymatrix{\n      & {\\cat A} \\ar[r] \\ar[d]\n      & {\\cat Y} \\ar[d]^{g}\\\\\n      {\\cat X} \\ar[r]^{h}\n      & {\\cat B} \\ar[r]^{\\pi}\n      & {\\cat Z}\n    }\n  \\end{displaymath}\n  be a diagram in a right proper model category in which\n  \\begin{itemize}\n  \\item the square is a pullback square\n  \\item $h$ is a weak equivalence, and\n  \\item $\\pi$ is a fibrillation.\n  \\end{itemize}\n  Then $\\cat A$ is a homotopy pullback of the zigzag\n  \\begin{displaymath}\n    \\zigzag{\\pi}{\\cat B}{\\cat Z}{\\cat Y}{g}\n  \\end{displaymath}\n  and hence (Df.~\\ref{def:HmtpyPlbk}) also of the zigzag\n  \\begin{displaymath}\n    \\zigzag{\\pi h}{\\cat X}{\\cat Z}{\\cat Y}{g}\n  \\end{displaymath}\n\\end{em}\n\n\\boldcentered{The Quillen fibrillation lemma}\n\n\\subsection{Lemma}\n\\label{lem:QFLm}\n\n\\begin{em}\n  Given an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ (Df.~\\ref{kzeroi}) and a\n  functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$ (Df.~\\ref{relcatiii}), the\n  following three statements are equivalent.\n  \\begin{enumerate}\n  \\item \\label{QFLmi} $F$ is a relative functor (Df.~\\ref{relcativ}),\n  \\item \\label{QFLmii} the map $\\N\\pi\\colon \\N\\Gr F \\to \\N\\cat D \\in\n    \\cat S$ (Nt.~\\ref{kzeroiii} and Df.~\\ref{def:GrConst}) is a\n    fibrillation, and\n  \\item \\label{QFLmiii} the map $\\pi\\colon \\Gr F \\to \\cat D \\in\n    \\widehat{\\mathbf{Cat}}$ is a fibrillation.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  To prove (i)$\\Rightarrow$(ii) we note that\n  \\begin{itemize}\n  \\item Quillen's proof of this lemma \\cite{Q}*{\\S1} implies that, for\n    every integer $p \\ge 0$ and map $\\Delta[p]= \\N\\cat p \\to \\N\\cat\n    D$, the pullback\n    \\begin{displaymath}\n      \\pullback{\\N\\cat p}{\\N\\cat D}{\\N\\Gr F}\n    \\end{displaymath}\n    of the zigzag $\\N\\cat p \\to \\N\\cat D\\gets \\N\\Gr F$ is a homotopy\n    pullback\n  \\end{itemize}\n  and that\n  \\begin{itemize}\n  \\item in view of \\cite{R2}*{4.1(i and (ii))} this implies that the\n    map $\\N\\pi \\to \\N\\Gr F \\to \\N D \\in \\cat S$ is a fibrillation.\n  \\end{itemize}\n\n  That (ii)$\\Rightarrow$(iii) then follows from the \\emph{Fibrillation\n    lifting lemma} \\ref{lem:HPBzz} and Ex.~\\ref{ex:FibLm}.\n\n  Finally (iii)$\\Rightarrow$(i) follows by a simple calculation from\n  the fact that, in view of the \\emph{Hopkins-Rezk fibrillation lemma}\n  \\ref{lem:HPBzz} the pullbacks of the form\n  (Df.~\\ref{def:strcalculus})\n  \\begin{displaymath}\n    \\pullback{\\boldsymbol 0^{w}}{\\cat D}{\\Gr F}\n    \\qquad\\text{and}\\qquad\n    \\pullback{\\boldsymbol 1^{w}}{\\cat D}{\\Gr F}\n  \\end{displaymath}\n  are both homotopy pullbacks.\n\\end{proof}\n\n\n\\section{A proof of Theorem \\ref{thm:MpPropBn}}\n\\label{sec:PrfThm}\n\n\\subsection{Preliminaries}\n\\label{sec:Prelim}\n\nWe start with recalling from Pr.~\\ref{prop:zzEmbed} that\n\\begin{enumerate}\n\\item \\label{Prelimi} every zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat\n    Y}{g}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k \\ge 0$) or $\\mathbf{RelCat}$ can, for\n  every integer $n \\ge 1$, in a functorial manner be embedded in a\n  commutative diagram of the form (Df.~\\ref{def:NArrFibr} and\n  \\ref{def:NArrPB})\n  \\begin{displaymath}\n    \\xymatrix{\n      & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^{\\qquad\\pi} \\ar[d]\n      & {Y} \\ar[d]^{g}\\\\\n      {X} \\ar[r]^{h\\qquad}\n      & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^{\\qquad\\pi}\n      & {Z}\n    }\n  \\end{displaymath}\n  in which\n  \\begin{itemize}\n  \\item the square is a pullback square, and\n  \\item $h$ is a weak equivalence\n  \\end{itemize}\n\\end{enumerate}\nand note that, in view of the \\emph{Hopkins-Rezk fibrillation lemma}\n\\ref{lem:HPBzz}\n\\begin{resumeenumerate}{2}\n\\item \\label{Prelimii} if the map $\\pi\\colon \\commacat{f\\cat X}{\\cat\n    Z} \\to \\cat Z$ is a fibrillation, then the object\n  $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback of that\n  zigzag.\n\\end{resumeenumerate}\n\n\n\\subsection{A proof for the category $\\widehat{\\mathbf{Cat}}$}\n\\label{sec:PrfCtht}\n\n\\begin{enumerate}\n\\item \\label{PrfCthti} It follows from Df.~\\ref{def:GrConst} and\n  \\ref{BnCni}, Ex.~\\ref{GrCniii} and the \\emph{Quillen fibrillation\n    lemma} \\ref{lem:QFLm} that the map $\\pi\\colon \\subcommacat{f\\cat\n    X}{n}{\\cat Z} \\to \\cat Z$ is a fibrillation, and\n\\item \\label{PrfCthtii} the desired result now follows from\n  \\ref{sec:Prelim}.\n\\end{enumerate}\n\n\\subsection{A proof for the categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$)}\n\\label{sec:PrfSmpk}\n\n\\begin{enumerate}\n\\item In view of \\ref{PrfCthti} and Df.~\\ref{def:MpPrpB}, for every\n  $k$-fold dimension $p_{*}$, the map\n  \\begin{displaymath}\n    \\pi_{p_{*}}\\colon\n    \\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n    \\longrightarrow \\cat Z_{p_{*}} \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  is a fibrillation, and\n\\item if $\\prod_{p_{*}} p_{*}$ denotes the product of these maps for\n  all $k$-fold dimensions and $\\prod_{p_{*}} \\widehat{\\mathbf{Cat}}$ denotes the\n  corresponding product of copies of $\\widehat{\\mathbf{Cat}}$, then clearly the same\n  holds for the map\n  \\begin{displaymath}\n    \\prod_{p_{*}}\\bigl(\\pi_{p_{*}}\\colon\n        \\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n        \\to \\cat Z_{p_{*}}\\bigr) \\in \\prod_{p_{*}} \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n\\item Moreover one readily verifies that, in view of\n  Df.~\\ref{def:NArrPB}, the obvious map $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\to\n  \\prod_{p_{*}}\\widehat{\\mathbf{Cat}}$ satisfies the conditions of the\n  \\emph{Fibrillation lifting lemma} \\ref{lem:FibLftLm}, which implies\n  that the map $\\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\to \\cat Z\n  \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ is also a fibrillation.\n\\item The desired result now follows from \\ref{sec:Prelim}.\n\\end{enumerate}\n\n\\subsection{Two proofs for the category $\\mathbf{RelCat}$}\n\\label{sec:TwoPrf}\n\n\\begin{enumerate}\n\\item \\label{TwoPrfi} As the map $f\\colon \\cat X \\to \\cat Z \\in\n  \\mathbf{RelCat}$ has property $B_{n}$ (by assumption), so does, in view of\n  Df.~\\ref{def:MpPrBrel}, the map $w_{*}f\\colon w_{*}\\cat X \\to\n  w_{*}\\cat Z \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\\item \\label{TwoPrfii} It follows from \\ref{sec:PrfSmpk} that the map\n  $\\pi\\colon \\commacat{w_{*}fw_{*}\\cat X}{w_{*}\\cat Z} \\to w_{*}\\cat Z\n  \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ is a fibrillation.\n\\item \\label{TwoPrfiii} Moreover as (Ex.~\\ref{ex:FibLm}) $w_{*}$\n  satisfies the conditions of the \\emph{Fibrillation lifting lemma}\n  \\ref{lem:FibLftLm} it follows from \\ref{prop:wpicom} that the map\n  $\\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\to \\cat Z \\in \\mathbf{RelCat}$\n  is also a fibrillation and the desired result then follows from\n  \\ref{sec:Prelim}.\n\\end{enumerate}\n\nHowever if one does not want to use the model structure on $\\mathbf{RelCat}$\none can also, instead of using (ii) and (iii) proceed as follows:\n\\begin{itemize}\n\\item [(ii)$'$] It follows from \\ref{sec:PrfSmpk} that the object\n  $\\commacat{w_{*}fw_{*}\\cat X}{w_{*}gw_{*}\\cat Y}$ is a homotopy\n  pullback of the zigzag $\\zigzag{w_{*}f}{w_{*}\\cat X}{w_{*}\\cat\n    Z}{w_{*}\\cat Y}{w_{*}g}$, and\n\\item [(iii)$'$] as (Pr.~\\ref{prop:AbsNvFncs}) $w_{*}$ is a homotopy\n  equivalence the desired result now follows from Pr.~\\ref{prop:wpicom}\n  and the \\emph{Global equivalence Lemma} \\ref{lem:WHLEx}.\n\\end{itemize}\n\n\\subsection{A proof for the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$)}\n\\label{sec:PrfRlkCt}\n\nThis is essentially the same as the second proof for $\\mathbf{RelCat}$\n(\\ref{TwoPrfi}, (ii)$'$ and (iii)$'$).\n\n\n\n\n\n\n\n\n\\begin{bibdiv} \n  \\begin{biblist}\n\n\n    \\bib{B}{thesis}{\n      label={B},\n      author={Barwick, Clark},\n      title={$(\\infty,n)$-$\\mathbf{Cat}$ as a closed model category},\n      organization={University of Pennsylvania},\n      date={2005}\n    }\n\n    \\bib{BK1}{article}{\n      label={BK1},\n      author={Barwick, Clark},\n      author={Kan, Daniel M.},\n      title={Relative categories: another model for the homotopy theory 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G.},\n      author={Kan, D. M.},\n      title={Calculating simplicial localizations},\n      journal={J. Pure Appl. Algebra},\n      volume={18},\n      date={1980},\n      number={1},\n      pages={17--35},\n    }\n\n    \\bib{DKS}{article}{\n      label={DKS},\n      author={Dwyer, W. G.},\n      author={Kan, D. M.},\n      author={Smith, J. H.},\n      title={Homotopy commutative diagrams and their realizations},\n      journal={J. Pure Appl. Algebra},\n      volume={57},\n      date={1989},\n      number={1},\n      pages={5--24},\n    }\n\n    \\bib{GZ}{book}{\n      label={GZ},\n      author={Gabriel, P.},\n      author={Zisman, M.},\n      title={Calculus of fractions and homotopy theory},\n      series={Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35},\n      publisher={Springer-Verlag New York, Inc., New York},\n      date={1967},\n    }\n\n    \\bib{H}{book}{\n      label={H},\n      author={Hirschhorn, Philip S.},\n      title={Model categories and their localizations},\n      series={Mathematical Surveys and Monographs},\n      volume={99},\n      publisher={American Mathematical Society},\n      place={Providence, RI},\n      date={2003},\n    }\n\n    \\bib{La}{article}{\n      label={La},\n      author={Latch, Dana May},\n      title={The uniqueness of homology for the category of small categories},\n      journal={J. Pure Appl. Algebra},\n      volume={9},\n      date={1976\/77},\n      number={2},\n      pages={221--237},\n    }\n\n    \\bib{Le}{article}{\n      label={Le},\n      author={Lee, Ming Jung},\n      title={Homotopy for functors},\n      journal={Proc. Amer. Math. Soc.},\n      volume={36},\n      date={1972},\n      pages={571--577; erratum, ibid. 42 (1973), 648--650},\n    }\n\n    \\bib{Lu}{article}{\n      label={Lu},\n      author={Lurie, Jacob},\n      title={$(\\infty,2)$-categories and the Goodwillie calculus I},\n      note={Available at \\url{http:\/\/arxiv.org\/abs\/0905.0462}},\n      status={To appear}\n    }\n    \n    \\bib{Q}{article}{\n      label={Q},\n      author={Quillen, Daniel},\n      title={Higher algebraic $K$-theory. I},\n      conference={\n        title={Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf.,\n          Battelle Memorial Inst., Seattle, Wash., 1972)},\n      },\n      book={\n        publisher={Springer},\n        place={Berlin},\n      },\n      date={1973},\n      pages={85--147. Lecture Notes in Math., Vol. 341},\n    }\n\n    \\bib{R1}{article}{\n      label={R1},\n      author={Rezk, Charles},\n      title={A model for the homotopy theory of homotopy theory},\n      journal={Trans. Amer. Math. Soc.},\n      volume={353},\n      date={2001},\n      number={3},\n      pages={973--1007 (electronic)},\n    }\n\n    \\bib{R2}{misc}{\n      label={R2},\n      author={Rezk, Charles},\n      title={Fibrations and homotopy colimits of simplicial sheaves},\n    }\n\n    \\bib{T1}{article}{\n      label={T1},\n      author={Thomason, R. W.},\n      title={Homotopy colimits in the category of small categories},\n      journal={Math. Proc. Cambridge Philos. Soc.},\n      volume={85},\n      date={1979},\n      number={1},\n      pages={91--109},\n      issn={0305-0041},\n    }\n\n    \\bib{T2}{article}{\n      label={T2},\n      author={Thomason, R. W.},\n      title={Cat as a closed model category},\n      journal={Cahiers Topologie G\\'eom. Diff\\'erentielle},\n      volume={21},\n      date={1980},\n      number={3},\n      pages={305--324},\n      issn={0008-0004}\n    }\n\n\n  \\end{biblist}\n\\end{bibdiv}\n\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThanks to extremely sensitive near-infrared (NIR) imaging obtained\nusing Wide Field Camera 3 (WFC3) on the {\\em Hubble Space Telescope}\nit is now possible to routinely identify galaxies \\ at very high\nredshift ($z>6$: e.g., Oesch et al.\\ 2010; Bunker et al.\\ 2010;\nWilkins et al.\\ 2010; Wilkins et al.\\ 2011a; Bouwens et al.\\ 2011b;\nFinkelstein et al.\\ 2010, 2012; Oesch et al.\\ 2012; McLure et\nal.\\ 2013; Schmidt et al.\\ 2014) with the first samples now being\nidentified at $z\\sim 10$ and beyond, less than 500 Myr after the Big\nBang (Bouwens et al.\\ 2011a; Zheng et al.\\ 2012; Ellis et al.\\ 2013;\nOesch et al.\\ 2013, 2014, 2015b; Zitrin et al.\\ 2014; Ishigaki et\nal.\\ 2015; Bouwens et al.\\ 2015b).\n\nOne area of significant interest in the study of distant galaxies\nregards their spectral characteristics.  At very early times, we might\nexpect galaxies to potentially have very different SEDs than at latter\ntimes.  In particular, one would expect galaxies to be bluer in terms\nof their rest-frame UV colours and UV-optical breaks, both due to a\nyounger stellar population (Wilkins et al.\\ 2013a) and a lower dust\ncontent (e.g., Wilkins et al.\\ 2011b; Bouwens et al.\\ 2009, 2012;\nFinkelstein et al.\\ 2012).  In fact, there has been some evidence for\nbluer UV colours at high redshift (Lehnert \\& Bremer 2003; Papovich et\nal.\\ 2004; Stanway et al.\\ 2005; Bouwens et al.\\ 2006, 2009, 2012,\n2014a; Wilkins et al.\\ 2011b; Finkelstein et al.\\ 2012; Kurczynski et\nal.\\ 2014) though the strength of the evolution at the highest\nredshifts has been subject to some debate (Dunlop et al.\\ 2012;\nRobertson et al.\\ 2013).  Evolution in the size of the Balmer break is\nless easily inferred (Stark et al.\\ 2009; Gonzalez et al.\\ 2010;\nMcLure et al.\\ 2011) largely due to the presence of strong nebular\nemission lines (Shim et al.\\ 2011; Schaerer \\& de Barros 2010; Wilkins\net al. 2013b), but now seems clear from $z\\sim6$ to $z\\sim2$ (Stark et\nal.\\ 2013; Gonzalez et al.\\ 2014; Schaerer \\& de Barros 2013;\nLabb{\\'e} et al.\\ 2013; Smit et al.\\ 2014; Salmon et al.\\ 2015).\n\nThe spectral characteristics of galaxies at $z\\sim8$ and $z\\sim8.5$\nhave also been explored (Bouwens et al.\\ 2010, 2013; Finkelstein et\nal.\\ 2010, 2012; Dunlop et al.\\ 2013), but are more difficult to\nrobustly quantify due to the limited leverage in wavelength available\nfor constraining these slopes and corrections required to remove the\nimpact of the IGM absoprtion (Bouwens et al.\\ 2014a).  This is\nparticularly true for $UV$-continuum slope determinations at\n$z\\sim8.5$-9.5.  Given the challenges in deriving the spectral\ncharacteristics of galaxies at $z\\sim8.5$-9.5, it may seem that\nfurther advances may need to wait until the {\\em James Webb Space\n  Telescope} (JWST).\n\nFortunately, as we will show, we can make immediate progress on this\nissue by taking advantage of the deep IRAC observations available for\ngalaxy samples at $z\\sim10$ from the {\\em Spitzer Space Telescope}.\nOver the redshift interval $z\\sim9.5$-10.5, the {\\em Hubble} $H$-band\nband and the IRAC $3.6\\mu$m band fall in the $UV$-continuum.  The\nwavelength leverage is sufficient between these bands that one can\nplausibly quantify the UV-continuum slopes of galaxies more accurately\nat $z\\sim10$ than is possible at $z\\sim8$ and especially at\n$z\\sim8.5$-9, as we demonstrate in \\S\\ref{sec:observations.slope} of\nthis paper.\n\nOpportunities to use $z\\sim10$ samples to perform such studies now\nexist thanks to the deep near-IR data available from {\\em Hubble} in\nthe Cosmic Assembly Near-Infrared Deep Extragalactic Survey (CANDELS,\nGrogin et al. 2011; Koekemoer et al. 2011), {\\em Hubble} Ultra-Deep\nField 2009\/2012 programs (Bouwens et al.\\ 2011; Koekemoer et al. 2013;\nEllis et al. 2013)\\footnote{The HUDF12 programme \\ obtained\n  $Y_{f105w}$ and $J_{f160w}$. Both the HUDF09 and HUDF12\n  observations, along with imaging from various other sources have\n  been released as part of the eXtreme Deep Field (XDF) project\n  (Illingworth et al. 2013).}, and Cluster Lensing and Supernovae\nsurvey with {\\em Hubble} (CLASH: Postman et al.\\ 2012) programs.  The\nsamples range from particularly faint sources over the HUDF\/XDF (Ellis\net al. 2013; Oesch et al. 2013) and the Frontier Fields (Zitrin et\nal.\\ 2014) to brighter sources located over CANDELS (Oesch et\nal. 2014) and CLASH (e.g. Zheng et al. 2012, Coe et al. 2013, and\n\\ Zitrin et al. 2014).\\footnote{It is important to note that not all\n  such candidates necessarily lie at $z\\sim 10$; indeed UDFj-39546284\n  (Bouwens et al. 2011a) for instance, which was initially reported to\n  be a $z\\sim 10$ candidate, no longer has a favoured high-redshift\n  identification (Ellis et al.\\ 2013; Brammer et al. 2013, Bouwens et\n  al. 2013a).}\n\nParticularly valuable for probing the spectral characteristics of\ngalaxies in the early universe are those $z\\sim10$ candidates that are\nintrinsically bright or lensed, since those sources have sufficient\nS\/N with IRAC that one can use them to quantify the mean\n$UV$-continuum slope of galaxies at very early times.  To date, five\nsuch galaxies have been identified in the magnitude range 26-27 mag\nfrom the CANDELS fields (Oesch et al.\\ 2014) and behind lensing\nclusters (Zheng et al.\\ 2012).\n\nIn this paper, we make use of these 5 particularly bright $z\\sim10$\ncandidates to derive, for the first time, the mean $UV$-continuum\nslope $\\beta$ at $z\\sim10$ for a multi-object sample.  We compare the\nobserved properties with predictions from recent cosmological galaxy\nformation simulations to provide some context.  Such simulations are\nvaluable given that the $UV$-continuum slopes and $UV$-optical colours\ncan be affected by a complex mixture of different factors including\nthe joint distribution of stellar masses, ages, metallicities, and the\nescape fraction -- which makes it difficult to interpret the\nobservations in terms of a single variable (Bouwens et al.\\ 2010;\nWilkins et al.\\ 2011).\n\nThis paper is organised as follows: in Section \\ref{sec:observations}\nwe describe recent observations of candidate $z\\sim 10$ star forming\ngalaxies. In Section \\ref{sec:dust_interp} we interpret observations\nof these systems in the context of dust emission.  Finally, in Section\n\\ref{sec:conclusions} we present our conclusions. Throughout this work\nmagnitudes are calculated using the $AB$ system (Oke \\& Gunn 1983). In\ncalculating absolute magnitudes we assume a $\\Omega_{M}=0.3$,\n$\\Omega_{\\Lambda}=0.7$, $h=0.7$ cosmology.\n\n\\section{Observations of the UV-continuum slope}\\label{sec:observations}\n\n\\subsection{Data and Sample Selection}\\label{sec:data}\n\nThis paper is based on the bright $z\\sim10$ galaxy sample selected\nover the GOODS fields from Oesch et al. (2014) as well the $z\\sim9.6$\nCLASH source behind MACS1149 identified in Zheng et al. (2012).  For\ndetails on the datasets, we refer the reader to the discovery\npapers. We summarise the photometry of these sources in Table\n\\ref{tab:photometry}.\n\nIn brief, the candidates from Oesch et al. (2014) were identified\nusing the complete {\\em Hubble} dataset from the CANDELS survey\n(Grogin et al. 2011, Koekemoer et al. 2011) in addition to ancillary\nAdvanced Camera for Surveys (ACS) data mostly from the GOODS\nsurvey. The central area of the GOODS-South and North fields represent\nthe CANDELS Deep survey, which reach to H$_{f160w}=27.8$ mag\n($5\\sigma$), while the outer regions are covered with slightly\nshallower data H$_{f160w}=27.1$ mag.  Galaxy candidates at $z\\sim10$\nwere identified based on the spectral break shortward of\nLyman-$\\alpha$ resulting in a red color of\n$($J$_{f125w}-$H$_{f160w})>1.2$ and complete non-detection in all\nshorter wavelength filters. While Oesch et al. (2014) also selected\n$z\\sim9$ galaxies with a weaker continuum break, here we restrict our\nanalysis to the redder color selection resulting in galaxies with\n$z_{phot}\\gtrsim9.5$.\n\nHere we make use of a redetermination of the IRAC photometry for the\nfour sources in the Oesch et al.\\ (2014) sample.  For these\nmeasurements, we take advantage of new reductions of the Spitzer\/IRAC\n3.6\\,$\\mu$m and 4.5\\,$\\mu$m imaging data over the GOODS fields from\nLabbe et al.\\ (2015).  These reductions include all data from the\noriginal GOODS, the Spitzer Extended Deep Survey (SEDS: Ashby et\nal.\\ 2013), the IRAC Ultra Deep Field (IUDF: Labb\\'e et al.\\ 2015),\nand S-CANDELS (Ashby et al.\\ 2015) programs.  The average 5$\\sigma$\ndepths of the IRAC data within 1\\arcsec\\ radius apertures are 27.0 and\n26.7 mag in the two channels, respectively.\n\nOur measurements again make use of the \\textsc{mophongo} software\n(Labb{\\'e} et al. 2006, 2010, 2013) to model and subtract the flux\nfrom neighboring sources, but include significantly improved PSF\nmodeling.  Our derived PSFs account for the orientation of all\nexposures that contribute to our IRAC reductions.  Our rederived IRAC\nfluxes are completely consistent within the errors with those given in\nOesch et al. (2014).  Three of the four $z\\sim10$ galaxy candidates\nfrom Oesch et al. (2014) are significantly detected ($>4.5\\sigma$) in\nthese data in at least one filter. In particular, the brightest source\nGN-z10-1 with a photometric redshift of $z_{phot} = 10.2\\pm0.4$ is\nrobustly detected in the 3.6\\,$\\mu$m channel at $\\sim7\\sigma$,\nallowing for an individual estimate a UV-continuum slope for this\nsource.\n\nEqually important for robust measurements of the $H_{160}-[3.6]$\ncolors are accurate measurements of the total $H_{160}$-band flux.\nOur determination of the total $H_{160}$-band flux is as described in\nOesch et al.\\ (2014) and includes all the light inside an elliptical\naperture extending to 2.5 Kron (1980) radii.  The measured flux inside\nthe utilized Kron aperture is corrected to total (typically a\n$\\sim$0.2 mag correction) based on the expected light outside this\naperture (Dressel et al.\\ 2012).\n\nAs a correction to total is performed for both the HST $H_{160}$-band\nphotometry and for the Spitzer\/IRAC photometry, colors derived from\nHST to Spitzer\/IRAC should not suffer from significant biases.\nNevertheless, it was worthwhile to verify that this was the case by\napplying the HST-to-Spitzer PSF-correction kernel to the HST data\n(derived from \\textsc{mophongo}) and then measuring total magnitudes\nfrom the HST data in the same way as the Spitzer\/IRAC data.  Given the\nrelatively small number of $z\\sim10$ candidates in our sample and\ntheir limited S\/N after convolving with the IRAC PSF correction\nkernel, we perform this test on a sample of bright ($H_{160,AB}<25$)\n$z\\sim4$ galaxies from the Bouwens et al.\\ (2015a) catalogs.  We found\nthat the total magnitudes we recover by applying this procedure to the\nHST observations of these $z\\sim4$ sources were consistent to $<0.1$\nmag in the median with that derived using our primary method.\n\nTo probe intrinsically fainter galaxies while maintaining a sufficiently high signal-to-noise, we can take advantage of sources which are lensed by foreground galaxies (or clusters of galaxies). Several $z\\sim 10$ candidates have now been discovered in cluster searches (Zheng et al. 2012; Coe et al. 2013; Zitrin et al. 2014), with more likely to be identified in the near future as a result of the ongoing {\\em Hubble} Frontier Fields observations. \n\nOf the three currently known $z\\sim 10$ candidates we concentrate on MACS1149-JD (Zheng et al. 2012). The object presented by Zitrin et al. (2014) does not currently have sufficiently deep IRAC imaging to provide anything other than a weak upper limit on the rest-frame UV-continuum slope. The object presented by Coe et al. (2013) has a photometric redshift of $z\\sim 10.7$, at which redshift the $H_{160}$-band flux could be affected by the position of the Lyman-$\\alpha$ break within the $H_{160}$ band and\/or Lyman-$\\alpha$ line emission. \n\nMACS1149-JD was identified in a search of 12 CLASH clusters and has strong detections in both the JH$_{f140w}$ and H$_{f160w}$ bands with weaker detections in both Y$_{f105w}$ and J$_{f125w}$ and non-detections in several optical bands. Photometric redshift fitting of the sources photometry suggests it is $z\\sim 9.6$.\\footnote{Bouwens et al.\\ 2014b estimate a photometric redshift of 9.7$\\pm$0.1.}  Zheng et al. (2012) presented {\\em Spitzer}\/IRAC photometry of MACS1149-JD ([3.6]$<160\\,{\\rm nJy},\\,1\\sigma$) based on observations taken under Program ID 60034 (PI: Egami). We augment these observations with Frontier Fields observations and archival data taken as part of the Spitzer UltRa Faint SUrvey Program (Surfs'Up, Brada{\\v c} et al. 2014) and measure a [3.6] flux of $175\\pm44$ nJy (see Bouwens et al.\\ 2014b). This is consistent with that reported by both Zheng et al. (2012) ($<160\\,{\\rm nJy},\\,1\\sigma$) and Brada{\\v c} et al. (2014) ($190\\pm 87\\,{\\rm nJy}$).\n\n\n\\begin{table*}\n\\caption{Photometry and derived properties for the $z\\sim10$ sources, and the bright stack, considered in this work.\\label{tab:photometry}}\n\\begin{tabular}{lccccccccc}\n\\hline\n & $z$ & $M_{1500}$ & H$_{f160w}$\/nJy & [3.6]\/nJy & H$_{f160w}-$[3.6] & $\\beta_{\\rm obs}$ & $A_{1500,{\\rm C00}}$ & $A_{1500,{\\rm SMC}}$ &\\\\\n& & & & & & & \\multicolumn{2}{c}{(assumes $\\beta_{\\rm int} = -2.54$)$^7$} \\\\\n\\hline\nGN-z10-1$^{1,2}$ & $10.2$ & $-21.6$ & $152\\pm10$ & $136\\pm29$ & $-0.1\\pm0.2$ & $-2.1\\pm0.3$ & $0.9\\pm0.6$ & $0.5\\pm0.3$ \\\\\nGN-z10-2$^{1,2}$ & $9.8$ & $-20.7$ & $68\\pm9$ & $45\\pm26$ & $-0.5\\pm0.6$ & $-2.5\\pm0.7$ & $0.1\\pm1.6$ & $0.1\\pm0.9$ \\\\\nGN-z10-3$^{1,2}$ & $9.5$ & $-20.6$ & $73\\pm8$ & $59\\pm24$ & $-0.2\\pm0.5$ & $-2.3\\pm0.5$ & $0.6\\pm1.1$ & $0.3\\pm0.6$ \\\\\nGS-z10-1$^{1,2}$ & $9.9$ & $-20.6$ & $66\\pm9$ & $71\\pm27$ & $0.1\\pm0.4$ & $-1.9\\pm0.5$ & $1.4\\pm1.1$ & $0.8\\pm0.6$ \\\\\nMACS1149-JD1$^{4}$ & $9.6$ & $-19.4$$^4$ & $190\\pm13.3$$^5$ & $177\\pm44$ & $-0.1\\pm0.3$ & $-2.1\\pm0.3$ & $1.0\\pm0.7$ & $0.6\\pm0.4$ \\\\\n\\hline\n\\bf STACK & - & - &  - &  - & $-0.1\\pm0.2$$^6$ & $-2.1\\pm0.3$$^6$ & $0.9\\pm0.6$ & $0.5\\pm0.3$  \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\n$^1$ Oesch et al. (2014). $^{2}$ included in stack. $^3$ Zheng et al. (2012), Bouwens et al.\\ (2014b). $^{4}$ MACS1149-JD is gravitationally lensed by a foreground cluster, we determine the un-lensed absolute magnitude using the best-fit magnification of $14.5$. $^{5}$ We independently measure the [3.6] flux for MACS1149-JD using the Frontier Fields {\\em Spitzer}\/IRAC observations combined with observations taken as part of the Spitzer UltRa Faint SUrvey Program (SURFSUP, Brada{\\v c} et al. 2014) and Program ID 60034 (PI: Egami). Our flux measurements are consistent with those reported by both Zheng et al. (2012) ($<160\\,{\\rm nJy},\\,1\\sigma$) and Brada{\\v c} et al. (2014) ($190\\pm 87\\,{\\rm nJy}$).  $^6$ The uncertainty on the mean increases to $\\pm0.3$, if we assume there is significant intrinsic scatter in the $\\beta$ distribution as observed at $z\\sim4$-5 (Bouwens et al.\\ 2009, 2012; Castellano et al.\\ 2012; Rogers et al.\\ 2014: see \\S2.2).  $^7$ The intrinsic $UV$-continuum slope predicted in dynamical simulation at $z\\sim 10$ assuming $f_{\\rm esc}=0$ (\\S3).\n\\end{table*}\n\n\n\n\\subsubsection{Bright Stack}\\label{sec:bright_stack}\n\nThe uncertainties on the $UV$-continuum slopes of individual sources\nis large enough that it is useful to consider constraints for the\naverage $z\\sim10$ source from the Oesch et al.\\ (2014) bright sample.\nWe therefore stack the photometry of the 4 $z\\sim 10$ candidates\npresented in Oesch et al. (2014).  The H$_{f160w}-$[3.6] colour of the\nstack is $-0.1\\pm0.2$. If we stack only GN-z10-1 and GN-z10-2 (i.e. excluding sources without JH$_{f140w}$ detections) we find H$_{f160w}-$[3.6]$=-0.2\\pm0.2$.\n\n\\subsection{Measuring the UV continuum slope}\\label{sec:observations.slope}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/measuring_beta.eps}\n\\caption{Relative observed near-IR photometry of a model star-forming galaxy at $z\\in\\{7,8,9,10\\}$ highlighting the bands available to measure the rest-frame UV-continuum slope. At $z>9.6$, the {\\em Spitzer}\/IRAC [3.6] band can be combined with the H$_{f160w}$ band to measure the UV continuum slope over a large wavelength baseline, minimising its uncertainty. At $z\\sim 8$ only the JH$_{f140w}$ and H$_{f160w}$ bands are uncontaminated by the Lyman-$\\gamma$ break providing only a small wavelength baseline and leaving the uncertainty on the observed UV continuum slope very large.}\n\\label{fig:measuring_beta}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/wavelength_baseline.eps}\n\\caption{{\\em Top panel} - The number of {\\em Hubble}\/WFC3 (Y$_{f098m}$, J$_{f125w}$, JH$_{f140w}$, and H$_{f160w}$) and {\\em Spitzer}\/IRAC ([3.6]) bands probing the rest frame $1216<\\lambda_{\\rm rest}\/{\\rm\\AA}<3700$ UV continuum as a function of redshift. {\\em Second panel} - The rest-frame UV wavelength baseline accessible by {\\em Hubble}\/WFC3 and {\\em Spitzer}\/IRAC observations as a function of redshift. {\\em Third panel } - The ratio of the mean wavelength of the bluest and reddest filters to probe the rest-frame UV continuum. The arrows at the top denote the redshift range over which an individual band probes the rest-frame $1216<\\lambda_{\\rm rest}\/{\\rm\\AA}<3700$ UV continuum. {\\em Bottom panel } - The expected uncertainty on the measurement of $\\beta$ as a function of redshift. This assumes a $\\beta=-2$, $m=26$ source, for which $\\beta$ is measured from the colour providing the longest usable wavelength baseline.}\n\\label{fig:wavelength_baseline}\n\\end{figure}\n\nFor $z>7$ galaxies, observations of the rest-frame UV continuum are\ntypically limited to just 3 WFC3\/IR bands uncontaminated by the\nLyman-$\\alpha$ break (i.e., $J_{f125w}$, $JH_{f140w}$, and\n$H_{f160w}$), and therefore suitable to measure the UV continuum slope\n$\\beta$. This is demonstrated, in Figure \\ref{fig:measuring_beta}, for\nsources at $z\\in\\{7,8,9,10\\}$.\n\nWhen only two bands are utilised for $UV$-continuum slope estimates, the relationship\\footnote{Assuming the underlying spectral flux density is described by a power-law, i.e. $f_{\\nu}\\propto\\lambda^{\\beta+2}$} between the observed colour and UV continuum slope can simply be written as:\n\n\\begin{equation}\n\\beta = p\\times (m_{1}-m_{2})_{\\rm AB}-2,\n\\end{equation}\nwhere $(m_{1}-m_{2})_{\\rm AB}$ is the observed colour assuming the AB magnitude system and $p$ is a value sensitive to the choice of bands. The value $p$ is approximately related to the ratio of the effective wavelengths ($\\lambda_1$, $\\lambda_2$) of the filters used to probe the slope\\footnote{The value of $p$ is also sensitive to the shape of the filter and when calculating the final value appropriate for our observations we take this into account.}, i.e.,\n\n\\begin{equation}\np^{-1} = 2.5\\cdot\\log_{10}(\\lambda_1\/\\lambda_2).\n\\end{equation}\n\nThe value of $\\log_{10}(\\lambda_1\/\\lambda_2)$ and the rest-frame UV wavelength baseline ($\\lambda_2-\\lambda_1$) accessible by {\\em Hubble}\/WFC3 and {\\em Spitzer}\/IRAC observations as a function of redshift are shown in Figure \\ref{fig:wavelength_baseline}.\n\nThis accessible wavelength baseline is important as the uncertainty on $\\beta$ will also scale with $p$. For example, at $z\\sim 8$ where we only have observations using the JH$_{f140w}$ and H$_{f160w}$ bands, the value of $p$ using these bands is approximately $9.0$ (Bouwens et al.\\ 2014a). A typical error on $m_{1}$ and $m_{2}$ of $0.1$ then translates into a large uncertainty on $\\beta$ ($\\approx 9.0\\cdot\\sqrt{0.1^{2}+0.1^{2}}\\approx 1.3$). At $z>9.5$ the {\\em Spitzer}\/IRAC [3.6]-band probes the UV continuum, which, when combined with H$_{f160w}$-band observations, provides an especially extended wavelength baseline.  The end result is a particularly small value for $p$ ($\\approx 1.1$) and thus small uncertainty on $\\beta$. This long wavelength baseline compensates for the lower sensitivity of observations with {\\em Spitzer}\/IRAC [3.6], allowing the the UV continuum to be estimated much more robustly than at $z\\sim 8$ and on a par with $z\\sim 6-8$ for the same observed apparent magnitude. This is demonstrated in the bottom panel of Figure \\ref{fig:wavelength_baseline}. \n\nThe observed values of the UV-continuum slope $\\beta$ for the $z\\sim\n10$ candidates (and the stack) are listed in Table\n\\ref{tab:photometry} and shown in Figure \\ref{fig:M_beta}. For the\nbrightest candidate (GN-z10-1) we find $\\beta_{\\rm obs}=-2.1\\pm 0.2$\nwhile for the bright stack (see \\S\\ref{sec:bright_stack}) we find\n$\\beta_{\\rm obs}=-2.1\\pm 0.3$.  If we stack only those sources which\nhave $JH_{140}$-band observations (providing a second WFC3\/IR filter\nwhere the $z\\sim10$ candidates are detected), i.e., GN-z10-1 and\nGN-z10-2, we find $\\beta_{\\rm obs}=-2.2\\pm 0.3$.\n\nWhile the formal random error on the mean $\\beta$ is 0.2, at lower\nredshifts the $\\beta$ distribution for luminous galaxies appears to\nshow a significant intrinsic scatter of $\\sigma_{\\beta}\\sim0.35$\n(Bouwens et al.\\ 2009, 2012; Castellano et al.\\ 2012; Rogers et\nal.\\ 2014).  Assuming a similar scatter at $z\\sim10$ translates to a\nslightly larger random error on the mean $\\beta$ of 0.3.\n\n\n\\subsubsection{Selection Biases}\n\nIt is useful to consider briefly whether our mean $\\beta$ results\ncould be biased because of the selection criteria that were applied in\nsearching for $z\\sim10$ galaxies.  Such issues became an important\naspect of the debate regarding $UV$-continuum slopes at $z\\sim7$\n(e.g., Wilkins et al.\\ 2012; Dunlop et al.\\ 2012; Bouwens et\nal.\\ 2012, 2014a), and it is important that we ensure that such issues\ndo not become important again.\n\n\nIn identifying bright $z\\sim9$-10 candidates over CANDELS GOODS-S and\nGOODS-N, Oesch et al. (2014) did not consider sources with\nparticularly red H$_{f160w}-$[4.5]$>2$ colors redward of the Lyman\nbreak.  Although the H$_{f160w}-$[4.5] colour, unlike the\nH$_{f160w}-$[3.6] colour, does not directly probe the UV continuum\nslope, they are correlated and noise in the two colors is not\nindependent.\\footnote{This is particularly true at\n  H$_{f160w}-$[4.5]$>0$ where the colour is dominated by the effect of\n  dust reddening.}. A colour of H$_{f160w}-[4.5]\\sim2$ corresponds to\n$\\beta\\approx -0.1$.  Given that all of our candidates have measured\ncolors of $\\beta<-1.8$ with small uncertainties, it is unlikely that a\n$\\beta<-0.1$ selection will have an important impact on the mean\n$\\beta$ measured for the sample, since the limit is $>3\\sigma$ away\nfrom the mean $\\beta$ observed and $3\\sigma$ away from the mean\n$\\beta$ assuming no evolution from $z\\sim5$-7.\n\nIt is encouraging that Bouwens et al.\\ (2015a) selected exactly the\nsame $z\\sim10$ candidates as Oesch et al.\\ (2014), utilizing a\nslightly modified criteria (H$_{f160w}-$[3.6]$<1.6$: corresponding to\n$\\beta<-0.2$).  This further confirms that the Oesch et al. (2014)\nsample of bright sources is not affected by strong $\\beta$-dependent\nselection biases.\n\n\\subsection{Possible Evolution in the observed UV-continuum slope}\n\nIn the previous section, we presented the first determination of the\nmean $UV$-continuum slope $\\beta$ for a multi-object sample at\n$z\\sim10$.  Previously, UV continuum slopes for similarly luminous\ngalaxies could only be determined up to $z\\sim8$ (Finkelstein et\nal.\\ 2012).\n\nWith these new measurements in hand, it is interesting to look for\nevidence of evolution in the mean $\\beta$ of galaxies versus redshift.\nFigure~\\ref{fig:beta_z_M} presents the current measurements and\ncompares it against previous measurements at $z\\sim4$-7 from Bouwens\net al.\\ (2014a) and Rogers et al.\\ (2014).\n\nAs is evident from Figure~\\ref{fig:beta_z_M}, the observed UV\ncontinuum slopes of individual galaxies at $z\\sim 10$ are found to be\nmuch bluer ($\\Delta\\beta\\approx 0.4$) than those at $z\\sim 4$-7.\nThese new results are interesting, as they suggest either a gradual\nevolution in $\\beta$ towards bluer colors or no evolution over a wide\nredshift baseline.\n\nThe comparison of measurements at $z=4$-7 with $z\\sim 10$ is not\nnecessarily straightforward as the $UV$-continuum slope is measured\nover very different wavelength baselines which may introduce a\nsystematic bias.  This is explored in more detail in Appendix A where\nwe conclude that this is unlikely to be an important factor in this\ncase, but could be as large as $\\Delta\\beta\\sim 0.2$.  If we allow for\npotential $\\sim$10\\% systematics in our HST - IRAC color measurements\n(possible if the total magnitudes we estimate from the HST or\nSpitzer\/IRAC data are not quite identical), the total systematic error\nrelevant to the inferred evolution in $\\beta$ could be as large as\n$\\sim0.22$.\n\nAccounting for both random and systematic errors, we find the observed\n$\\beta$ for luminous galaxies at $z\\sim10$ is only $1\\sigma$ bluer\nthan at $z\\sim7$.  Therefore, our results are consistent with either a\nmild reddening of the $UV$-continuum slopes with cosmic time or no\nevolution at all.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_beta.eps}\n\\caption{Mean value of the UV-continuum slope as a function of absolute rest-frame UV magnitude for candidate high-redshift ($z=4-10$) star forming galaxies (Bouwens et al.\\ 2014; Rogers et al.\\ 2014). The thin horizontal line denotes the intrinsic slope implicit in the empirical Meurer et al. (1999) relation and the two horizontal bands show the range of intrinsic slopes expected from the {\\sc MassiveBlack-II} hydrodynamical simulation at $z\\sim 10$ assuming both $f_{\\rm esc}=1$ (pure stellar) and $f_{\\rm esc}=0$. It is important to note that at lower redshift the {\\sc MassiveBlack-II}  simulation predicts significantly bluer intrinsic slopes (see Wilkins et al. 2013a).}\n\\label{fig:M_beta}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/beta_z_M.eps}\n\\caption{The observed UV continuum slope as a function of redshift for candidate high-redshift ($z=4-10$) star forming galaxies. Results at $z=4-7$ are based on the bi-weight mean of galaxies with $M_{1500}\\approx -20.75$ from Bouwens et al. (2014a).}\n\\label{fig:beta_z_M}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Inferred Dust Attenuation}\\label{sec:dust_interp}\n\n\n\\subsection{Relating the observed UV continuum slope to dust attenuation}\n\nThe presence of dust causes the observed UV continuum slope to redden relative to the intrinsic slope. The relationship between between the observed slope $\\beta_{\\,\\rm obs}$ and the attenuation $A_{\\lambda}$ can be written as (e.g. Meurer et al. 1999, Wilkins et al. 2013a),\n\\begin{equation}\\label{eq:abeta}\nA_{\\lambda}=D_{\\lambda}\\times[\\beta_{\\,\\rm obs}-\\beta_{\\,\\rm int}],\n\\end{equation} \nwhere $\\beta_{\\rm int}$ is the intrinsic UV continuum slope, $\\beta_{\\rm obs}$ is the observed slope, and $D_{\\lambda}$ ($={\\rm d}A_{\\lambda}\/{\\rm d}\\beta$) describes the change in the attenuation as a function of the change in $\\beta$, and is sensitive to the choice of attenuation\/extinction curve. The intrinsic UV continuum slope is sensitive to a range of properties, including the star formation and metal enrichment histories, and the ionising photon escape fraction (see \\S\\ref{sec:dust_interp.intbeta}, and Wilkins et al. 2012, Wilkins et al. 2013a). \n\nBoth $\\beta_{\\rm int}$ and $D_{\\lambda}$ can be constrained empirically (e.g. Meurer et al. 1999, Heinis et al. 2013) using a combination of rest-frame UV and far-IR observations of a sample of galaxies. $D_{\\lambda}$ can be determined for any attenuation curve which extends over the rest-frame UV, and thus doesn't necessarily require far-IR observations to be constrained. \n\n\\subsection{The intrinsic UV continuum slope}\\label{sec:dust_interp.intbeta}\n\nEmpirical constraints on $\\beta_{\\rm int}$ are, at present, due to the lack of sufficiently deep far-IR\/sub-mm observations, limited to low-intermediate redshift (e.g. Heinis et al. 2013). The value of $\\beta_{\\rm int}$ at low\/intermediate redshift is unlikely to reflect that at very-high redshift as $\\beta_{\\rm int}$ is sensitive to both the age and metallicity of the stellar population, both of which are expected to decrease in typical star forming galaxies to high-redshift (e.g. Wilkins et al. 2013a).\n\nThe sensitivity of the intrinsic UV-continuum slope $\\beta_{\\rm int}$ to various properties, including the joint distribution of stellar masses, ages, and metallicities (themselves determined the recent star formation and metal enrichment histories, and initial mass function) and the presence of nebular continuum, and to a lesser extent, line emission is discussed in Wilkins et al. (2012) and Wilkins et al. (2013a). We demonstrate this sensitivity in Figure \\ref{fig:colours_ageZ_constant_z10} utilising the {\\sc Pegase.2} stellar population synthesis code (Fioc \\& Rocca-Volmerange 1997, 1999). We determine the H$_{f160w}-$[3.6] colour (our proxy for the UV continuum slope at $z\\sim 10$) as a function of the duration of previous constant star formation, for two stellar metallicities, and also assuming $f_{\\rm esc}=0$ and $f_{\\rm esc}=1$ (i.e. a pure stellar continuum). Increasing the metallicity, the strength of nebular emission (i.e. decreasing the escape fraction $f_{\\rm esc}$), and the duration of previous star formation all result in the intrinsic UV continuum slope becoming redder. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/colours_ageZ_constant_z10.eps}\n\\caption{The sensitivity of the H$_{f160w}$-[3.6] colour (our proxy for the UV continuum slope) to the duration of previous (constant) star formation. The blue and purple lines show the result for $Z=0.02$ and $Z=0.0004$ respectively, while the solid and dashed lines show the result assuming $f_{\\rm esc}=0$ (i.e. including nebular continuum and line emission) and $f_{\\rm esc}=1$ (i.e. pure stellar emission) respectively. The solid and hatched horizontal bands show the predictions from the {\\sc MassiveBlack-II} simulations assuming $f_{\\rm esc}=0$ and $f_{\\rm esc}=1$ respectively.}\n\\label{fig:colours_ageZ_constant_z10}\n\\end{figure}\n\n\\subsubsection{Sensitivity to the candidate redshift}\n\nIn addition to the physical properties outlined above the UV continuum slope inferred from observations will also be sensitive to the redshift of the source. This is demonstrated in Figure \\ref{fig:colours_z} where the predicted intrinsic H$_{f160w}-$[3.6] colour (our proxy for the observed UV-continuum slope) of a stellar population that has been forming stars constantly for 50 Myr is shown as a function of redshift. Three different nebular emission scenarios are shown: (i) $f_{\\rm esc}=1$ (i.e. pure stellar), (ii) $f_{\\rm esc}=0$, and (iii) $f_{\\rm esc}=1$ but with Lyman-$\\alpha$ suppressed. In all three cases, in the interval $z=9.6-10.4$, the observed colour exhibits virtually no variation. At $z<9.6$ various strong emission lines, beginning with [OII]$\\lambda 3727{\\rm\\AA}$, enter the {\\em Spitzer}\/IRAC [3.6]-band resulting in generally redder colours\\footnote{The {\\em Spitzer}\/IRAC [3.6]-band also no longer probes the rest-frame UV continuum}.\n\nAt $z>10.4$ the H$_{f160w}$-band filter overlaps with the wavelength\nof rest-frame Lyman-$\\alpha$. Assuming no Lyman-$\\alpha$ emerges\n(scenario {\\em iii}) this results in the observed H$_{f160w}$-band\nencompassing the Lyman-$\\alpha$ break resulting in a decreased flux,\nand consequently redder H$_{f160w}-$[3.6] colours. If strong\nLyman-$\\alpha$ emerges the H$_{f160w}-$[3.6] colour would rapidly\nbecome very blue before gradually becoming redder.  Spectroscopic\nfollow-up of $z\\gtrsim6.5$ galaxies strongly points towards very\nlittle Ly$\\alpha$ emission in $z\\gtrsim10.4$ galaxies (e.g., Stark et\nal.\\ 2010; Ono et al.\\ 2012; Schenker et al.\\ 2012; Pentericci et\nal.\\ 2011; Caruana et al.\\ 2012; Treu et al.\\ 2013; Finkelstein et\nal.\\ 2013; Oesch et al.\\ 2015a: but see also Zitrin et al.\\ 2015).\n\n\\subsubsection{Simulation Predictions for the Intrinsic Slope}\n\nWhile empirical constraints on the intrinsic slope $\\beta_{\\rm int}$ exist they are only available at low-redshift, and are therefore unlikely to be representative of the very-high redshift Universe. We then also employ predictions from galaxy formation models for the intrinsic slope and spectral energy distribution. Specifically, we utilise predictions from the {\\sc MassiveBlack} and {\\sc MassiveBlack-II} hydro-dynamical simulations (see Khandai et al. 2015 for a general description of the simulations, and Wilkins et al. 2013a for predictions of the intrinsic UV continuum slope). At $z=10$ these simulations predict a median intrinsic slope of $\\beta_{\\rm int}\\approx -2.54$ (assuming $f_{\\rm esc}=0$) and $\\beta_{\\rm int}\\approx -2.78$ for a pure stellar SED. In both cases the {\\sc Pegase.2} SPS model is assumed, however other commonly used models models produce similar results (Wilkins et al. 2013ab).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/beta_z.eps}\n\\caption{{\\em lower-panel} The sensitivity of the H$_{f160w}$-[3.6] colour (our proxy for the UV continuum slope) to the redshift for a stellar population which has been constantly forming stars for 50 Myr (with $Z_{}=0.004$, i.e. $\\approx 1\/5\\,Z_{\\odot}$). The three lines show the result of different nebular emission scenarios: ({\\em dashed}) $f_{\\rm esc}=1$ (i.e. pure stellar), ({\\em solid}) $f_{\\rm esc}=0$, and ({\\em dotted}) $f_{\\rm esc}=1$ but with Lyman-$\\alpha$ suppressed. The predictions from {\\sc MassiveBlack-II} are shown by the shaded horizontal bands. {\\em upper-panel} The redshift probability distributions of the 4 bright $z\\sim 10$ candidates.}\n\\label{fig:colours_z}\n\\end{figure}\n\n\n\\subsection{Inferred Dust Attenuation}\n\nBy combining the observed UV continuum slope $\\beta_{\\rm obs}$ with a choice of intrinsic slope $\\beta_{\\rm int}$ and dust curve we can infer the level of dust attenuation using Equation \\ref{eq:abeta}. We do this both utilising the empirical Meurer relation and by assuming the intrinsic slope predicted by the {\\sc MassiveBlack-II} simulation combined with various attenuation\/extinction curves. \n\nIn Figure \\ref{fig:M_A1500_M99} we first show the inferred UV\nattenuation assuming the empirical Meurer et al. (1999) relation. The\ncombination of very blue observed colours and the $\\beta\\approx -2.23$\nintrinsic slope implicit in the Meurer relation result in individual objects\nhaving best fit attenuations formally consistent with $A_{1500}=0$. The attenuation inferred from the bright stack is $A_{1500}=0.3\\pm 0.5$.\n\nFigure \\ref{fig:M_A1500} is similar but instead shows the attenuation when we assume the intrinsic slope predicted by the {\\sc MassiveBlack-II} simulation (assuming $f_{\\rm esc}=0.0$) along with both the Calzetti et al. (2000) and SMC (from Pei et al. 1992) dust curves. In this case all of the individual observations as well of the stack yield positive values of $A_{1500}$. If instead a higher escape fraction is assumed (which would make the intrinsic slope bluer) the inferred attenuation increases by between $0.25-0.5$ mags, depending on the choice of attenuation\/extinction curve.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_A1500_M99.eps}\n\\caption{The inferred rest-frame UV attenuation $A_{1500}$ (right-hand axes) assuming the Meurer et al. (1999) relation.}\n\\label{fig:M_A1500_M99}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_A1500.eps}\n\\caption{The inferred rest-frame UV attenuation $A_{1500}$ (right-hand axes) assuming the intrinsic slope predicted by {\\sc MassiveBlack} along with the Calzetti et al. (2000) {\\em starburst} and Pei et al. (1992) SMC dust curves.}\n\\label{fig:M_A1500}\n\\end{figure}\n\n\\subsubsection{Evolution of dust attenuation}\n\nWe now focus on the evolution of the rest-frame UV attenuation from $z\\sim 10$ to $z=4-7$. In Figure~\\ref{fig:A1500_zevo} we compare our predictions for the UV attenuation at $z\\sim 10$ to those at $z=4-7$ applying the same methodology - using both the Meurer et al. 1999 relation and by combining predictions of the intrinsic slope with the Calzetti et al. (2000) dust curve. Assuming a constant intrinsic slopes hints at a significant increase in the UV attenuation from $z\\sim 10$ to $z\\sim 7$ followed by little evolution between $z=7$ and $z=4$. Utilising the intrinsic slopes predicted by {\\sc MassiveBlack} (which vary with redshift) weakens this evolution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/A1500_zevo.eps}\n\\caption{The evolution of the rest-frame UV attenuation $A_{1500}$ inferred from observations of the UV continuum slope in bright $z\\in\\{4,5,6,7,10\\}$ star forming galaxies. Both the attenuation calculated using the Meurer et al. (1999) relation and combining the Calzetti et al. (2000) attenuation curve with the {\\sc MassiveBlack} predictions are shown.}\n\\label{fig:A1500_zevo}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nHere we make use of the deep {\\em Hubble} and Spitzer observations available\nover 5 particularly bright $z\\sim10$ candidates (Oesch et al.\\ 2014;\nZheng et al.\\ 2012) to provide a first characterization of the mean\nUV-continuum slope for a multi-object sample of galaxies at\n$z\\sim10$.\n\nWe find:\\\\\n\n\\begin{itemize}\n\n\\item Combining {\\em Hubble} and {\\em Spitzer} we have measured the\n  mean UV continuum slope of star formation galaxy candidates at\n  $z\\sim 10$.  We find a mean $\\beta$ of $-2.1\\pm0.3\\pm0.2$.  We allow\n  for up to a $0.2$ error in this measurement due to systematic errors\n  that may derive from the wavelength baseline used to derive $\\beta$\n  (Appendix A) or from small systematics in the photometry.  The\n  average observed UV continuum slope of a stack of bright $z\\sim 10$\n  sources is only bluer than those at $z<8$ ($\\beta_{\\rm obs}\\approx\n  -1.7$) by $1\\sigma$.  These measurements are more robust than those\n  at $z\\sim 8$ due to the wide wavelength baseline provided by the\n  combination of the {\\em Hubble}\/WFC3 H$_{f160w}$-band and {\\em\n    Spitzer}\/IRAC [3.6]-band.  The only previous measurement of\n  $\\beta$ at $z\\sim10$ was by Oesch et al.\\ (2014) for the most\n  luminous $z\\sim10$ galaxy in their selection.\\\\\n\n\\item These slopes are redder than the intrinsic slope predicted by galaxy formation models, suggesting the existence of some dust attenuation, even at $z\\approx 10$. For the brightest candidate, GN-z10-1 (Oesch et al. 2014), we infer a rest-frame UV attenuation of $\\approx 0.9$ assuming the Calzetti et al. (2000) attenuation curve ($\\approx 0.5$ assuming an SMC extinction law), while for the stack of bright candidates we find $0.9\\pm 0.4$ assuming the Calzetti et al. (2000) law ($0.6\\pm 0.2$ assuming an SMC extinction law). \\\\\n\n\\end{itemize}\n\n\n\n\\subsection{Acknowledgements}\n\nWe acknowledge useful conversations with Dan Coe on this topic, who\nhas been attempting similar measurements on his triply imaged\n$z\\sim11$ candidate.  SMW acknowledges support from the Science and\nTechnology Facilities Council.  IL acknowledges support from the\nEuropean Research Council grant HIGHZ no. 227749 and the Netherlands\nOrganisation for Scientific Research Spinoza grant. The Dark Cosmology\nCentre is funded by the Danish National Research Foundation.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nVariable radio emission is a hallmark of energetic objects such as\ncoronally active stars, supernovae, neutron stars, black holes, and\nactive galactic nuclei (AGN). Indeed, radio variability is often\nindicative of high-energy processes and, in principle, can be\nvaluable for finding examples of relatively rare objects. However,\nsurveys for variability are themselves quite rare --- blind sky\nsurveys are almost never repeated owing to the scarcity of telescope\ntime. The exceptions are mostly in the optical regime. Comparisons\nbetween POSS1, POSS2, and SDSS have been useful for studying\nvariability (de~Vries et al.\\ 2005). Gravitational microlensing\nstudies (e.g., Alcock et al.\\ 1997) and supernova searches (e.g.,\nAstier et al.\\ 2006, Miknaitis et al.\\ 2007) have produced a wealth\nof data on optical variability from targeted sky regions, and several\nupcoming experiments such as Pan-STARRS (Kaiser et al.\\ 2002), the\nPalomar Transit Factory (Rau et al.\\ 2009), and LSST (Tyson 2002)\nwill make the coming decade one in which time-domain astronomy plays\na prominent role.\n\nVariability studies in the radio band have typically targeted bright\nextragalactic sources (see de~Vries et al.\\ 2004 for a review of searches\nfor, and mechanisms of, radio variability). Comparisons between blind\nradio surveys are often hampered by differences in angular resolution and\nthe confusing presence of interferometric sidelobe patterns. For example,\nthere has been no systematic search for radio variability between the\ntwo largest radio sky surveys, FIRST (Becker et al.\\ 1995) and NVSS (Condon\net al.\\ 1998). The FIRST survey did observe one area twice at 1400 MHz, an\nequatorial strip $\\sim1.5$ degrees wide in the range\n$21^{\\mathrm h}20^{\\mathrm m} < RA < 03^{\\mathrm h} 20^{\\mathrm\nm}$. A search for variable sources in this area was reported in\nde~Vries et al.\\ (2004). The search covered $\\sim120$~deg$^2$ of\nextragalactic sky with a sensitivity similar to the Galactic plane\nsearch reported here; it thus serves as a useful control from which\nto estimate how many of the sources we find are background extragalactic\nradio sources.\n\nThe most systematic search for radio variablity in the Galactic plane\nused the NRAO 91-m telescope in Greenbank, WV, operating at a\nfrequency of 5~GHz (Gregory \\& Taylor 1986). Over a five-year period\nthe plane was observed 16 times, leading to the detection of 32\nvariable radio sources. The survey had a flux density threshold of\n$\\sim20$~mJy and an angular resolution of 3\\arcmin. Using the\nVery Large Array\\footnote{The\nVery Large Array is an instrument of the National Radio Astronomy\nObservatory, a facility of the National Science Foundation operated\nunder cooperative agreement by Associated Universities, Inc.}\n(VLA), the Galactic plane has been surveyed at this same frequency\nbut with much higher sensitivity and angular resolution, although\nwith minimal repetition (Becker et al.\\ 1994).\n\nRecently, a new Galactic plane survey at 6~cm (4.86~GHz) has begun\nat the VLA. The new survey (CORNISH\\footnote{The Co-Ordinated Radio\n'N' Infrared Survey for High-mass star formation; see\n\\url{http:\/\/www.ast.leeds.ac.uk\/Cornish}}; Purcell et al.\\ 2008)\nhas substantial overlap with our previous survey; both surveys have\na flux density threshold of $\\sim1$~mJy. Comparison between these\ntwo data sets allows for a search for Galactic radio sources that\nexhibit variability over the fifteen-year interval between the\nsurveys. The search is complicated because the two surveys use\ndifferent VLA configurations and hence have different angular\nresolutions (5\\arcsec\\ versus 1.5\\arcsec).\nNonetheless, it is possible to identify strongly varying sources.\nIn this paper we will compare results between the original survey\nand two epochs of data from the new survey. In section 2 we discuss\nthe parameters of the two surveys, while in section 3 we present\nthe results from a comparison of the two samples and adduce evidence\nfor variability. We describe the properties of the variable sources including\ntheir spatial distribution, spectral indices, and counterparts at other wavelengths\n(\\S4) and end with a discussion of our limited knowledge of the nature of\nthese objects (\\S5).\n\n\\begin{figure*}\n\\epsscale{1.0}\n\\plotone{f1.eps}\n\\caption{\nSky coverage for overlapping regions from the 1990${+}$ (red), 2005\n(blue), and 2006 (gray) 6~cm survey epochs. Darker regions have\nhigher rms noise values, while white areas are outside the survey.\nThe noise is higher at the edges of the coverage and in fields with\nbright or complex extended sources.  Typical rms values are $\\sim\n0.1$~mJy in both surveys, but the old 6~cm data were acquired with\na more widely spaced pointing grid and so display greater variation\nwith position. The areas of overlap are given in Table~1.\n}\n\\label{fig-coverage}\n\\end{figure*}\n\n\\section{The 6~cm Surveys}\n\nThe original VLA 6~cm Galactic plane survey was carried out between\n1989 and 1991 in the C and BnC configurations (Becker et al.\\ 1994).\nIt covered a longitude range $-10^{\\circ} < l < 42^{\\circ}$ within\n$\\pm 0.4^{\\circ}$ of the plane for a total of 43~deg$^2$. The data\nwere re-reduced in 2005 using much improved data processing algorithms\nand some additional data (White et al.\\ 2005).  The new catalog\nreaches a flux density limit of $\\sim1$~mJy and contains over 2700\nradio sources. Since the data were taken in C and BnC configurations,\nthe angular resolution is $\\sim5\\arcsec$.\n\nThe new CORNISH survey (Purcell et al.\\ 2008) is meant to complement\nthe {\\it Spitzer} GLIMPSE Legacy program (Benjamin et al.\\ 2003).\nWhen completed, it will cover a longitude range $10^{\\circ} < l <\n65^{\\circ}$ within $\\pm 1^{\\circ}$ of the plane. The data are being\ntaken in the B configuration and hence will have an angular resolution\nof $\\sim1.5\\arcsec$. The new survey will also achieve a flux\ndensity sensitivity of $\\sim1$~mJy. The ultimate areal coverage\nwill be 110~deg$^2$. A pilot study of 10~deg$^2$ near $l = 30^{\\circ}$\nwas carried out in the spring of 2005. The first 64~deg$^2$ of the\nfull survey (including repeated observations of the pilot area) was\nobserved in summer 2006.\nWe retrieved these data from the VLA archive and reduced them using\nthe AIPS procedures we developed for the FIRST survey (White et\nal.\\ 1997). Our source detection algorithm HAPPY was run on the\nfinal co-added images.\n\nWe henceforth refer to the three epochs as~I (1990${+}$), II~(2005)\nand III~(2006).  Note that while the epoch~II and~III data were\ntaken over short periods of time (spanning about 2 months in each\ncase), the epoch~I data were taken over a much greater time period (hence\nour choice of the label ``1990${+}$'').\nFor the overlapping area used for this paper, 70\\% of the epoch~I\nobservations were taken between June 1989 and December 1990, and\n30\\% were taken between February and April of 2004.  Consequently\nthe time span between epoch~I and the later two epochs varies by a\nlarge factor depending on the source location.  We will report the\nmean observational epoch of the flux density measurements for\nindividual objects in the following discussion.\n\nFor this paper we restrict our attention\nto sources in sky regions with coverage at two or three epochs.\nTable~1 describes the areas of overlap between the various epochs\nand the number of sources from each catalog included in those areas.\nTo ensure source reliability, we restrict our sample to sources\nthat are strongly detected ($>8.5\\sigma$) in one of the epochs or\nthat are confirmed by detections at multiple epochs.  We also check\nfor detections at 20~cm, either from our MAGPIS survey (Helfand et\nal.\\ 2006) or in the catalog of White et al.\\ (2005); a 20~cm detection is\nrequired as confirmation for sources detected in only one 6~cm epoch.\n\nIn\nFigure~\\ref{fig-coverage} we show the sky coverage for the three\nepochs in the vicinity of the overlap region.  Note that the 2005\npilot area is entirely covered by the 2006 data, so all of the sky\narea covered by 1990${+}$ and 2005 observations also has 2006\nobservations.\n\n\\begin{deluxetable}{ccccc}\n\\label{table-area}\n\\tablecolumns{5}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Sky Regions with Multiple Epochs of Observations}\n\\tablehead{\n\\colhead{Epochs} & \\colhead{Area Covered} & \\multicolumn{3}{c}{Number of Sources} \\\\\n& \\colhead{(deg$^2$)} & 1990${+}$ & 2005 & 2006 \\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{(5)}\n}\n\\startdata\n1990${+}$, 2006 & 13.4 & 541 & \\nodata & 347 \\\\\n2005, 2006 & \\phn5.9 & \\nodata & 144 & 142 \\\\\n1990${+}$, 2005, 2006 & \\phn3.9 & 161 & 168 & 133 \\\\\n\\enddata\n\\end{deluxetable}\n\nThe other significant difference between the two surveys is their\nangular resolution. For unresolved radio sources, the flux densities\nfrom the two surveys are directly comparable; the difficulty comes\nin knowing which sources are true point sources. For sources that\nare partially resolved by the new survey, the flux density will be\nlower than that measured a decade and a half ago, even in the absence\nof variability. Hence, partially resolved sources will give a\nfalse-positive variability signal. By the same token, however, any\nsource significantly brighter in the newer survey is almost certainly\nvariable. All images from both surveys can be found at the MAGPIS\nwebsite (\\url{http:\/\/third.ucllnl.org\/gps}).\n\n\\section{Search for Variability}\n\nA match among the three 6~cm data sets resulted in 503 distinct\nsources detected at two or more epochs.\nTo ensure reliability,\nwe restrict our sample to sources that are detected in at least two 6~cm\nepochs or that have confirming detections at 20~cm.  Sources detected only\nin a single 6~cm epoch and not at 20~cm are excluded.\nSources are considered a\nmatch if their positions agree to within 1.5\\arcsec\\ for the epoch~II\nand~III catalogs or to within 5\\arcsec\\ between the epoch~I and\nlater-epoch catalogs.  These relatively large match radii are chosen\nto include extended sources, which can have larger positional\noffsets.  For the higher resolution epoch~II and~III data, the\nmedian position difference is 0.2\\arcsec, and 80\\% of the sources\nhave positions that agree to within 0.4\\arcsec.  For comparisons\nbetween the low-resolution epoch~I data and the more recent\nobservations, the median separation is 0.7\\arcsec, and 80\\% of the\nsources have positions that differ by 1.6\\arcsec\\ or less. To avoid\npotential confusion, we have removed from the match list sources\nthat have ambiguous matches owing to multiple components within the\nmatching radius.\n\nA comparison of the flux densities determined from the old and new\ndata is plotted in Figure~\\ref{fig-flux}; sources that fall along\nthe diagonal have comparable flux densities from the two measurements.\nThere is a clear bias for sources to be weaker in the newer\nobservations, a direct consequence of the higher angular resolution,\nwhich results in slightly extended sources having some of their\nflux resolved out in the newer data.  Certainly some of these sources\ncould be variable, but it is difficult to distinguish between a\ndecrease due to variability and a decrease due to resolution effects.\nHappily, the reverse is not true; sources that brighten between the\ntwo epochs are likely to be truly variable.\n\n\\begin{figure*}\n\\epsscale{0.8}\n\\plotone{f2.eps}\n\\caption{\nComparison of 6~cm integrated flux densities for old (epoch I\/1990${+}$) and\nnew (epochs II\/2005 and III\/2006) catalogs. Source in all three epochs are plotted\ntwice to show both the 2005 and 2006 flux densities.\nRed symbols indicate the variable\nsources, with upper limits shown for variables detected at only one\nepoch. Most sources have similar flux densities in the two epochs,\nbut extended sources tend to have lower flux densities in the newer\nsurvey because those data were taken in a higher-resolution VLA\nconfiguration that resolves out some of the extended radio emission.\nConsequently, in our variable source search there is a bias toward\nobjects that are brighter in the 2005\/2006 epoch.\n}\n\\label{fig-flux}\n\\end{figure*}\n\nFigure~\\ref{fig-flux2} displays a comparison of the flux densities\nmeasured in the two high-resolution epochs (II and~III).  The area\nof overlap is smaller (5.9~deg$^2$ versus 17.3~deg$^2$ for\nFig.~\\ref{fig-flux}), but it is clear that the scatter is considerably\nreduced.  This is expected because the observations are taken in\nthe same VLA configuration and so have the same resolution.  (The\nshorter time baseline for variability also presumably contributes\nslightly to the reduced scatter.)\n\n\\begin{figure*}\n\\epsscale{0.8}\n\\plotone{f3.eps}\n\\caption{\nComparison of the integrated flux densities for the epoch~II and~III\ncatalogs.  These observations were taken in the same VLA configuration,\nwhich makes the flux densities directly comparable. The two epochs\nare also much closer in time, reducing the amplitude of the expected\nvariability signal.  The symbols are the same as in Fig.~2.\n}\n\\label{fig-flux2}\n\\end{figure*}\n\nThe sources falling in a region covered by at least two of the three\ncatalogs yielded a list of potential variable sources using a\n$5\\sigma$ variability threshold\\footnote{The difference in the peak\nflux densities was required to be greater than\n$5 \\times (\\sigma_{\\rm old}^2 + \\sigma_{\\rm new}^2)^{1\/2}$ if the source\nwas detected at both epochs; if the object was detected at only one\nepoch, the flux density at the undetected epoch was conservatively\ntaken to be twice the rms value at that epoch.}.  A visual inspection\nof the pairs of images led us to reject many as suspect owing either\nto source confusion or to clear angular extent in the higher\nresolution observations.  There remained 39 sources regarded as\nhaving a high likelihood of being true radio variables (Table~2).\nWe were cautious about including sources that were brigher in epoch~I\ndue to the resolution difference discussed above.  Only 5 of the\ncandidates rely on a bright epoch~I measurement to establish\nvariability; the other 34 either are brighter in the high-resolution\ndata or show variability between epochs~II and~III.  We have retained\nthe five sources that fade from their epoch~I flux measurements\nbecause they appear to be point-like in all epochs at both 6~cm and 20~cm,\nbut we cannot\ncategorically exclude a small source extent being responsible for\nthe lower flux density observed in the more recent data.  The\nvariability significance in column 16 of Table~2 is shown in bold\nitalic type for these less reliable sources.\n\nIn examining candidate variables, we were alert to the possibility\nof calibration errors or bad data causing systematic flux density\ndifferences.  Consequently we looked carefully at the close pair\nof sources, G37.7347${-}$0.1126 and G37.7596${-}$0.1001, both of\nwhich were undetected in epoch~I and were bright ($>10$~mJy) in\nepoch~III.  To confirm the reality of these sources, we examined\nthe individual grid images that contributed to the coadded images\nfor each source.  The grid images confirmed the variation: in epoch~I\neach source would have been detected at more than $5\\sigma$\nsignificance in two different grid images if the source was as\nbright as in epoch~III, but neither showed any evidence for emission.\nAnd in epoch~III the sources were detected in two or more independent\nobservations at flux densities consistent with the coadded image\ndetection.\n\nOf the five single-epoch detections in the list, one appears only\nin epoch~I (and so is one of the less reliable sources), one appears\nonly in epoch~II, and three appear only in epoch~III.  Note that\nsingle-epoch sources both must be relatively bright in the detected\nepoch for the variability to be considered and also must have\nconfirming detections at 20~cm.  In fact, all of the multi-epoch\nsources are also detected at 20~cm; we use the spectral indices\nderived from the 6 and 20~cm flux densities below, although caution\nis warranted since the 20~cm and 6~cm observations are not\ncontemporaneous, so the variability for which we are selecting will\nalso affect the spectral index estimates.  In fact, roughly half\nof the MAGPIS 20~cm flux densities are inconsistent with our original\ncompact source survey at this wavelength undertaken in the 1980s\n(see catalogs in White et al.\\ 2005), underscoring the case for\nvariability. The degree of variability at 6~cm ranges from 20\\% to\na factor of 18. The flux density distribution ranges from $<1$ to\n65~mJy with a median of $\\sim8$~mJy in the second-epoch data.\n\n\\section{Characteristics of the Variable Sources}\n\nOnly a few papers have reported variability results for\ncentimetric radio sources from extragalactic surveys\non time scales of years and with sensitivities\nin the mJy range. Bower et al.\\ (2007) examined archival VLA data\nfrom a frequently observed calibration field (944 observations over\n22 years) to search for transient radio sources at 6~cm to a flux\ndensity limit of 370~$\\mu$Jy; a transient source was defined as one\nthat only appeared at a single epoch (or over a short range of\ncontiguous epochs $<2$ months in length). They measured a two-epoch rate of\n$1.5\\pm0.4$ transients per square degree, and\nestimated that the number of transients scales with flux density\nas $S^{-1.5}$. Our faintest variable source has a 6~cm flux density\nof 2.8~mJy (at the brighter epoch; fainter objects would not have\npassed the $5\\sigma$ variability threshold). Thus, we should expect\n$\\sim1$ true extragalactic transient in our survey area of\n23.2~deg$^2$ (the total pair-wise area covered -- see Table~1). While\nthere are five sources detected at one epoch only in our sample,\nall four are detected independently at 20~cm (at one or more different\nepochs), and thus cannot be considered true transients. The detection\nof zero transients when one is expected is construed as consistent\nwith the results of Bower et al.\\ (2007) while setting a weak limit\non the number of Galactic transient sources.\n\nExamining data collected for a deep survey of the Lockman Hole\nspaced at intervals of 19 days and 17 months, Carilli et al.\\ (2003)\nconcluded that only 2\\% of sources between 50 and 100~$\\mu$Jy at\n1.4~GHz are highly variable ($>50$\\%). These observations did,\nhowever, yield nine variable objects in the flux density range 1\nto 25~mJy in a field with a FWHM 32\\arcmin; only one of these\nsources varied by more than 50\\%. \n\nThe survey by de~Vries et al.\\ (2004) mentioned in the introduction\n(\\S1) offers the best comparison sample against which to assess the\nfraction of our variable sources likely to be extragalactic. They\nfind 123 sources variable at $>4\\sigma$ significance in 120.2~deg$^2$\nof high-latitude sky, or roughly 1 variable source per square degree.\nThe median flux density of the extragalactic sample is 13.5~mJy at\n$\\lambda=20$~cm. As noted above, we have 20~cm flux densities (albeit\nnon-contemporaneous ones) for all members of our sample; the median\nflux density is 12.9~mJy, very similar to that of the de~Vries\nsample.\n\n\\begin{deluxetable}{ccccc}\n\\label{table-vardist}\n\\tablenum{3}\n\\tablecolumns{5}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Distribution of Variability}\n\\tablehead{\n\\colhead{Fractional} & \\colhead{de~Vries} & \\colhead{This Paper} & \\colhead{Predicted} & \\colhead{Galactic}\\\\\n\\colhead{Variability $f$} & \\colhead{Counts} & \\colhead{Counts} & \\colhead{Extragalactic} & \\colhead{Excess} \\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{(5)}\n}\n\\startdata\n  $<$ 1.25   &    51 &      1 & 7.4 &   \\nodata \\\\\n1.25 -- 1.50 &    39 &      5 & 5.7 &   0.0 \\\\\n1.50 -- 1.75 &    19 &      5 & 2.8 &   2.2 \\\\\n1.75 -- 2.00 & \\phn4 &      4 & 0.6 &   3.4 \\\\\n2.00 -- 2.25 & \\phn4 &      2 & 0.6 &   1.4 \\\\\n2.25 -- 2.50 & \\phn2 &      3 & 0.3 &   2.7 \\\\\n2.50 -- 2.75 & \\phn1 &      2 & 0.1 &   1.9 \\\\\n2.75 -- 3.00 & \\phn1 &      0 & 0.1 &   0.0 \\\\\n  $>$ 3.0    & \\phn2 & 17\\phn & 0.3 &  16.7\\phn \\\\[\\smallskipamount]\nTotal        & 123\\phn & 39\\phn & 17.9\\phn &  28\n\\enddata\n\\tablecomments{\nCol.~(1): Ratio of brightest to faintest flux density measurements.\nCol.~(2): Variable source counts from de~Vries et al.\\ (2004).\nCol.~(3): Variable source counts from this paper.\nCol.~(4): de~Vries counts scaled to match area covered in this paper.\nCol.~(5): Net excess of variables in this paper compared with de~Vries counts.\n}\n\\end{deluxetable}\n\nHowever, an examination of the fractional variability of the two\nsamples (defined as $f$, the highest flux density recorded over the\nlowest) reveals drastic differences. Table~3 displays the distribution\nof fractional flux density variation for the de~Vries\nextragalactic sample and for our Galactic plane catalog of\nvariables. A total of $73\\pm4$\\% of the extragalactic sample has a\nfractional variability of $f<1.5$ while only 6 of the 39 Galactic\nvariable (15\\%) vary this little. At the other end of the distribution,\nonly 2\/123 extragalactic objects vary by as much as a factor of 3,\nwhile fully 17\/39 (44\\%) of the Galactic plane sources are this\nvariable.\n\nWe can use the de~Vries sample to estimate the number of extragalactic\nvariables present in our survey area. The ratio of areas is 23.2\ndeg$^2$\/120.2 deg$^2$ or 0.193. We cannot simply scale by area,\nhowever, because of the resolution bias, discussed above, that\ndiscriminates against sources which faded between epoch~I and\nepochs~II and~III. Of the 30 sources whose variability was established\non the basis of a change between epoch~I and a later epoch, five\nsources faded and 25 sources brightened in the later epochs. Since\nthe distribution should be inherently symmetrical, we can assume\nwe eliminated roughly 20 fading sources to protect against resolution\neffects. Thus, the total number of true variables is reduced by\n20\/50 or 40\\%; note that this correction factor applies only to the\narea in which 1990${+}$ data is compared to later data (19.3 deg$^2$).\nThe effective sky area covered by our survey when this inefficiency\nis taken into account is $A_{\\rm eff} = 0.6 \\times 19.3 + 5.9 =\n17.5\\,\\hbox{deg}^2$, and the expected number of extragalactic\nvariables is therefore $123 \\times A_{\\rm eff}\/120.2 = 18$.\n\nIf we distribute these 18 sources with the fractional variability\nof the extragalactic sample (column 4 of Table~3), we expect\n$\\sim13$ at $f<1.5$, 3 with $1.5<f<2.0$, and $\\sim2$ to vary by\nmore than a factor of 2.  The distribution for Galactic variables\nis 6 at $f<1.5$, 9 with $1.5<f<2.0$, and 24 varying by more than a\nfactor of two.  Our Galactic sample includes few sources that vary\nby less than $25\\%$ because the complexity of radio emission in the\nGalactic plane combined with differences in the VLA configuration\nrequired a higher threshold for confident detection of variability;\nthat is the reason for our use of a $5\\sigma$ threshold rather than\nthe $4\\sigma$ threshold adopted by de~Vries et al.  Ignoring this\nlowest bin as below our survey sensitivity, the de~Vries catalog\nhas 72 variable sources, yielding an expectation of $\\sim10.5$\nsources in our area.  Subtracting these numbers from our catalog\nsuggests that while many of the sources at $f<1.5$ are extragalactic,\nonly $\\sim5$ of the 33 sources with greater variability are background\nobjects.  Scaling by the effective area $A_{\\rm eff}$ derived above,\nwe find a surface density of Galactic variable sources of 1.6~deg$^{-2}$,\nnearly six times the 0.3~deg$^{-2}$ density of extragalactic\nsources that vary by more $50\\%$ on time scales of years.\n\nA complicating factor in this comparison is that the de~Vries\nvariables were selected at 20~cm rather than 6~cm wavelength.  The\nmost variable AGN have beamed nuclear radio emission that usually\nhas a flatter spectral index than the non-variable extended radio\nemission. Consequently at a shorter wavelength, the beamed emission\nwill be more dominant, which will increase the apparent amplitude\nof variability.  Without a good extragalactic comparison sample at\n6~cm, it is difficult to determine exactly how large this effect\nwill be.  But we consider it very unlikely that the changes resulting\nfrom the wavelength difference could explain the large differences\nseen between the extragalactic and Galactic samples in the frequency\nof large-amplitude variables.\n\n\\begin{figure*}\n\\epsscale{0.9}\n\\plotone{f4.eps}\n\\caption{\nDistribution of variable sources (red) compared with\nthe population of sources that are point-like at\n6~cm and that either are detected at multiple\nepochs or are detected at greater than\n$8.5\\sigma$ significance in a single epoch.\n}\n\\label{fig-posdist}\n\\end{figure*}\n\n\\begin{figure*}\n\\epsscale{0.6}\n\\plotone{f5.eps}\n\\caption{\nLongitude distribution of variable sources (red) compared with\nthe same comparison sample as in Fig.~4.  The top panel (a) shows\nthe raw counts, while the bottom panel (b) shows the fraction of\nvariable sources in each bin, with error bars computed from the binomial\nprobability distribution (and a $2\\sigma$ upper limit).  The variability\nfraction increases dramatically toward the Galactic center, which\nindicates that the variable sources are mainly a Galactic\nrather than extragalactic population.\n}\n\\label{fig-longhist}\n\\end{figure*}\n\nThe spatial distribution of the sources in our catalog strongly\nsupports the view that they are dominated by a Galactic population.\nIn Figure~\\ref{fig-posdist} we compare the distribution of variable\nsources with a parent population consisting of point-like sources\n(defined by an integrated flux density less than 25\\% larger than\nthe peak flux density) that either are detected at multiple 6~cm\nepochs or that are strong single-epoch detections ($>8.5\\sigma$).\nSuch sources could have been detected as variables.  The latitude\ndistribution shows a bias toward negative latitudes, consistent\nwith earlier studies that show that $b=0.0$ lies above the Galactic\nplane in the first quadrant. \nIn Figure~\\ref{fig-longhist}, we display the\nlongitude distribution for all sources (which is distorted by\ndifferential coverage) and for the variable sources. The fraction\nof variables, displayed in the lower panel, shows a clear rise\ntoward the Galactic center.\n\n\\begin{figure*}\n\\epsscale{0.6}\n\\plotone{f6.eps}\n\\caption{\nVariability fraction as a function of the spectral index $\\alpha$ (\n$F_\\nu \\sim \\nu^\\alpha$) between 6~cm and 20~cm.\nThe comparison sample includes the sources from\nFigs.~4 and 5 that have 20~cm flux measurements.  The spectral\nindex was computed from the lowest 6~cm flux density at any epoch\nin order to reduce the selection bias toward flatter spectra in variable\nsources.\n}\n\\label{fig-spindhist}\n\\end{figure*}\n\nThe spectral index distribution (Figure~\\ref{fig-spindhist}) also\nsuggests that the variable sources represent a distinct population.\nThere is a trend toward greater variability as the radio spectrum\nbecomes flatter (increasing spectral index).  For any individual\nsource, the spectral index calculated between our 6~cm and 20~cm\ncatalogs is unreliable, as the measurements were 1) obtained with\ndifferent spatial resolutions, and 2) far from contemporaneous.\nSince the 20~cm observations have lower resolution, they detect\nmore flux in extended sources and so tend to produce spectral indices\nthat are too steep.  On the other hand, the observed variability\nin the 6~cm flux density tends to bias the index toward flatter\nvalues, since sources that brighten at 6~cm are more likely to be\nrecognized as variable.  To compensate partially for the latter\neffect, the spectral index in Figure~\\ref{fig-spindhist} is computed\nusing the smallest 6~cm flux measured at any of the three epochs.  That\nmay also be responsible for the variable sources with very steep\nspectral indices ($\\alpha < -2$), which may have been in a bright\nphase when measured at 20~cm.\n\nThe interpretation of the spectral index distribution is therefore\nnot straightforward.  The apparent increase in variability for\nflatter spectrum sources could result from either a Galactic\npopulation (optically thin or thick thermal emission) or \nan extragalactic population (beamed emission from AGN\/blazars).\n\nAnother line of evidence that many of the variable sources are\nGalactic derives from their counterparts at other wavelengths.  We\nexamined images from the {\\it Spitzer} GLIMPSE survey (3.6, 4.5,\n5.8 and 8.0$\\mu$m; Benjamin et al.\\ 2003), the {\\it Spitzer} MIPSGAL\n24$\\mu$m survey (Carey et al.\\ 2009), and the 1.1~mm Bolocam Galactic\nPlane Survey (BGPS; Aguirre et al.\\ 2009).  Of the 39 variable\nsources, 7 are MIPSGAL sources, with 6 of those also found to be\nGLIMPSE sources, and 4 are detected in the BGPS millimeter observations;\nall are described in greater detail below.  None of the counterparts\nare expected to be the result of chance coincidences, implying that\nall of these objects must be in the Galaxy.  Infrared\/mm counterparts\nare significantly more common among the variable sources than\namong the non-variable 6~cm sources.  We examined a sample of 40\nnon-variable radio sources, selected as unresolved sources detected\nin at least 2 epochs with 6~cm flux densities that are consistent\nwithin $2\\sigma$ at all epochs.  Only 2 of the non-variable objects\nwere found to have MIPSGAL counterparts, and none had BGPS matches.\nWe conclude that the existence of these counterparts is related to\nthe nature of the variable sources.\n\n\\section{Discussion}\n\n\\subsection{Source Identification -- What They Are Not}\n\nHaving established the existence of a population of highly variable\nGalactic sources, the obvious question is, What are they? Three\nclasses of Galactic variable radio emitters can be easily eliminated from\nconsideration: coronally active radio-emitting stars, pulsars, and\nmasers. We justify our exclusion of these source classes in turn.\n\nIn a survey of 122 RS CVn and related active binary systems which\nare among the most luminous stellar radio sources, Drake et al.\n(1989) found only 18 detected above a quiescent flux density of\n1~mJy at 6~cm; the faintest optical counterpart was $V=10.0$. Even\nassuming an extreme flare of a factor of 100 (Osten 2008), the\nfaintest possible counterpart would have $V=15$; none of our variables\nhas a counterpart this bright. As for dMe flare stars, the other\nmain class of variable stellar radio emitters, the most luminous\nquiescent emission is $\\sim10^{14.2}$ erg s$^{-1}$ Hz$^{-1}$ (Gudel\net al.\\ 1993) corresponding to a flux density of $\\sim1$~mJy at a\ndistance of 13~pc. Even an extreme flare with an increase of a\nfactor of 500 over the quiescent level (Osten 2008 and references\ntherein) would fall below our flux density threshold for a distance\n$>290$~pc. Stars with spectral types later than M6 would have\ncounterparts fainter than 20th magnitude and could be represented\nin our sample. However, statistically, M-stars cannot be a significant\ncontributor; Helfand et al.\\ (1999) found only $\\sim5$ M stars in\n5000 deg$^2$ of the {\\sl FIRST} survey to a flux density limit of\n0.7~mJy, whereas our variables have a surface density of 1.6~deg$^{-2}$.\n\nWhile nearby radio pulsars scintillate strongly in the ISM leading\nto large-amplitude variability, pulsars have very steep radio\nspectra, and most have not been detected at 6~cm (none of our objects\nare coincident with one of the 1827 known pulsars; Manchester et\nal. 2005). For a typical spectral index of $-1.5$, our weakest\nsource would be a $\\sim100$~mJy pulsar at 400~MHz, and most unlikely\nto have been missed in pulsar surveys.  The small duty cycle of the\nrecently discovered RRATs (Rotating RAdio Transients -- McLaughlin\net al.\\ 2006) makes them equally unlikely to explain our variable\nsources.\n\nFinally, radio masers are known to be highly variable, but no known\nmaser transitions fall within our bandpass. As noted below, however,\nthree of our variables are coincident with methanol masers.\n\nTwo classes of extragalactic radio transients --- supernovae and GRB\nafterglows --- are also highly improbable counterparts for our events.\nBoth have rise times of at most tens of days and cannot, in the\nabsence of a steady underlying source of radio emission, account for\nthe bulk of our sources which show a flux density increase over\nmany years. In addition, their rarity makes them statistically\nunlikely counterparts. The one extragalactic population that does\nshow variability on the time scales we probe, AGN, are shown above\nto have variability amplitudes which exclude them from explaining\nall but a handful of our events.\n\nThe remaining known classes of variable radio sources include\nmicroquasars (accreting, high-mass X-ray binaries that produce\nrelativistic jets: e.g., SS433, Cyg X-3 and GRS 1915${+}$105), radio\nmagnetars (Camilo et al.\\ 2006; Camilo et al.\\ 2007), and the recently\ndescribed Galactic Center Transient sources (Hyman et al.\\ 2009 and\nreferences therein).  The first two of these have signatures at\nother wavelengths; we explore below the fragmentary data outside\nthe radio band that is available for our variable objects.\n\n\\subsection{Source Identification -- Multiwavelength Data}\n\nCounterparts at other wavelengths can be useful in suggesting the\norigin of radio variability. At our MAGPIS website (Helfand et al.\n2006), we have collected the following Galactic plane data in\naddition to the three-epoch 6~cm data described herein: two epochs\nof 20~cm observations for these same fields including the principal\nMAGPIS survey, 90~cm observations of the same regions, the 3.6,\n4.5, 5.8, and 8.0 $\\mu$m data from the Spitzer {\\sl GLIMPSE} survey\n(Benjamin et al.\\ 2003), 24 $\\mu$m images from MIPSGAL (Carey et\nal.\\ 2009), 20 $\\mu$m data from the MSX survey (Price et al.\\ 2001)\nand the 1.1~mm Bolocam Galactic plane survey (Aguirre et al.\\ 2009).\nIn addition, we have queried the SIMBAD database for each of our\nsources and have examined the Digitized Sky Survey images; in one\ncase, we have obtained optical observations of a source. We report\nthe results of this multi-wavelength inquiry here.\n\n\\subsubsection{Mid-IR and mm observations}\n\nSeven of our variables are detected at 24 $\\mu$m in the MIPSGAL\nsurvey, and six of these are also detected in at least one GLIMPSE\nmid-IR band. Four of the objects are also detected at 1.1~mm in the\nBolocam survey. In all seven cases at least two bands are available,\nand in all seven cases the sources are red; i.e., they are faintest\nin the short-wavelength bands and brightest in the long-wavelength\nbands. In two cases (G31.1595${+}$0.0449 and G37.7347${-}$0.1126) multiple\ncomponents with different IR spectral shapes are present, with the\nradio source identified with the brighter component in the first\ncase, and the redder component in the second. Three of the IR-detected\nobjects have associated methanol masers; this, coupled with their\nIR spectra demonstrate they represent activity associated with\nstar formation in compact or ultracompact \\ion{H}{2} regions.\n\nFor one IR-detected source, G29.5779${-}$0.2685, we have obtained\nfollowup observations at the MDM Observatory (J.~Halpern, private\ncommunication).  R-band and H$\\alpha$ images were obtained on\n23~August 2009, and show a barely resolved ($\\sim1\\arcsec$)\nobject, brighter in H$\\alpha$ and coincident with\nthe radio source. A spectrum obtained the same\nnight shows no continuum, but very strong nebular emission\nlines. The object appears to be a very compact planetary nebula.\nIts radio flux history is thus perplexing: 6.9~mJy at 6~cm in $\\sim1990$,\nrising to 10.5~mJy in 2005 and falling again to 5.8~mJy in 2006.\nThe 20~cm flux density in the MAGPIS survey (epoch 2001--2004)\nis only 1.3~mJy, suggesting the source may be optically thick.\nFurther simultaneous multi-frequency observations are required to\nmeasure the radio spectrum and derive clues as to the nature of the\nsource's variability.\n\n\\begin{figure*}\n\\epsscale{1.0}\n\\plotone{f7.eps}\n\\caption{\nMAGPIS 20~cm image of supernova remnants W41 (G23.3${-}$0.3) and\nG22.7${-}$0.2.  The boxes mark the positions of three\nvariable 6~cm sources (G22.7194${-}$0.1939, \nG22.9116${-}$0.2878, and G22.9743${-}$0.3920).\n}\n\\label{fig-snrs}\n\\end{figure*}\n\n\\subsubsection{X-ray observations}\n\nThe brightest variable, G21.6552${-}$0.3611, is coincident with a\npoint-like X-ray source catalogued in the XMM Galactic Plane survey\n(Hands et al.\\ 2004). It has a hard band (2--6~keV) flux of 0.0051\nct s$^{-1}$ and is undetected in the soft (0.4--2.0~keV) band. For\nan intrinsic power-law spectrum with spectral index $\\Gamma = 1.9$,\nthe expected absorption column density through the Galactic plane\nof $\\sim10^{23}$ cm$^{-2}$ is consistent with the non-detection\nin the soft band; the inferred intrinsic flux in the 0.2--10~keV\nband would be $7 \\times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$. For an\nextragalactic AGN at 1 Gpc, this corresponds to a luminosity of $8\n\\times 10^{42}$ erg s$^{-1}$, while for a Galactic object at 5~kpc,\nthe X-ray luminosity would be a modest $2 \\times 10^{33}$ erg\ns$^{-1}$; for a column density of only $10^{22}$ cm$^{-2}$, the\nluminosity estimates are lower by a factor of 3. While this source\nis the brightest of our variables, it has one of the lowest modulation\nfactors (decreasing by just $\\sim50\\%$ over 16 years). The\n(non-contemporaneous) 20~cm flux density is lower than either of\nthe 6~cm values, suggesting a mildly inverted spectrum source. It\nis not detected at any other wavelength. The most likely explanation\nof this object is a flat-spectrum extragalactic radio source, one\nof a handful we expect in our sample.\n\nOne other source, G30.4460${-}$0.2148, lies 27\\arcsec\\ from\nthe position of an ASCA Galactic Plane Survey catalog entry (Sugizaki\net al.\\ 2001). The uncertainty in the X-ray position is 1\\arcmin;\none other (brighter) radio source lies within the X-ray error circle\nalthough at twice the distance from its centroid. The X-ray source\nis a marginal detection ($4.6 \\sigma$) with a 0.7--7.0~keV unabsorbed\nflux of $2.6 \\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ for an intrinsic\npower law index of $\\Gamma = 1.9$ and an absorption column density\nof $10^{23}$ cm$^{-2}$; the flux is roughly four times lower for\n$N_H = 10^{22}$ cm$^{-2}$. Assuming the identification is correct,\nthe X-ray to radio flux ratio is thus $\\sim20$ times greater than\nour other X-ray detection, although still within the X-ray to radio\nluminosity ratios characteristic of AGN. The primary distinguishing\nfeature, however, is that the radio source is coincident with a\nvery bright mid-IR source (saturated in all but the 3.6 $\\mu$m band)\nwhich is also detected at 1.1~mm.\n\nThe ASCA Galactic Plane Survey covered an area encompassing all but six of\nour variables to a flux density level of approximately $10^{-12.5}$\nerg cm$^{-2}$ s$^{-1}$; no other X-ray sources are coincident to\nwithin 1\\arcmin. The higher-resolution coverage of the Einstein,\nROSAT, XMM, and Chandra is much spottier; no further matches are\nfound within the 10\\arcsec\\ error circles of these other\ncatalogs.\n\n\\subsubsection{Low-frequency radio detections}\n\nThree of the variable sources are detected at 90~cm. G22.9116${-}$0.2878\n(Fig.~\\ref{fig-snrs}) has a 90~cm flux density of $\\sim180$~mJy; this is consistent with\na nonthermal spectral index of $\\sim-0.9$ if one takes the most\nrecent (but far from contemporaneous) 6, 20, and 90~cm measurements.\nThe 6~cm flux density increased by more than a factor of three since\n1990, making it one of the higher amplitude variables, but no other\ninformation is available on this source. G30.6724${+}$0.9637 (the highest\nlatitude source detected) has a 90~cm flux density of $\\sim70$\nmJy, below that of the 20~cm flux density (90~mJy), possible\nadditional evidence for variability, as the 20:6~cm flux density\nratio is 3:1 (again, all non-contemporaneous). This is the smallest\namplitude variable in our sample and, given its distance from the\nGalactic plane, an extragalactic counterpart is the most likely\nexplanation.\n\nThe third 90~cm detection is G22.7194${-}$0.1939, perhaps the most\nintriguing source in our sample. An image of the region surrounding\nthis source is given in Figure~\\ref{fig-snrs}. The source lies very near\nto the geometric center of a 30\\arcmin-diameter supernova remnant,\nG22.7${-}$0.2 (Green 2004 and references therein) and 4\\arcmin\\ from\na fairly bright \\ion{H}{2} region. There is no counterpart detected at mm,\nIR, or optical wavelengths. The source brightened by a factor of\nfour between 2003 and 2006 at 6~cm; its 20~cm flux density is 12~mJy,\nthree times higher than the higher of the two 6~cm measurements.\nThe distance to the remnant is unknown, although its large angular\ndiameter would suggest it is not very remote (its diameter would\nbe $\\sim45$~pc at 5~kpc). X-ray observations could reveal whether\nor not this source is likely to be a compact object associated with\nthe supernova remnant.\n\n\\subsection{Summary}\n\nWe have discovered a relatively high surface density (2 deg$^{-2}$)\nof variable radio sources in the Galactic plane and have argued\nthat the large majority of these ($\\sim80$\\%) are Galactic objects. While a few\nare associated with young star formation activity, the identity\nof the majority is unknown.\nFollow up radio observations are required to confirm the variability\nin these sources, establish the variability time scale(s), and\nobtain contemporaneous spectral indices. Observations at optical,\ninfrared, and X-ray wavelengths could help establish counterparts\nand identify the origin of the variable radio emission.\n\n\n\\acknowledgments\n\nRHB and DJH acknowledge the support of the National Science Foundation\nunder grants AST-05-07598 and AST-02-6-55.  RHB's work was supported\nin part under the auspices of the US Department of Energy by Lawrence\nLivermore National Laboratory under contract W-7405-ENG-48.  DJH\nwas also supported in this work by NASA grant NAG5-13062.  RLW\nacknowledges the support of the Space Telescope Science Institute,\nwhich is operated by the Association of Universities for Research\nin Astronomy under NASA contract NAS5-26555. The first three authors\nare grateful for the hospitality of Quest University Canada\n(\\url{http:\/\/www.questu.ca}), an innovative new undergraduate university in\nBritish Columbia, where this manuscript was completed.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\nFrom a purely theoretical point of view, given the underlying conditional probability distribution of a dependent variable $C$ and a set of features $\\mathbf{X}$, the Bayes decision rule can be applied to construct the optimum induction algorithm. However, in practice learning machines are not given access to this distribution, $Pr(C|\\mathbf{X})$. Therefore, given a feature vector or variables $\\mathbf{X}\\in R^N$, the aim of most machine learning algorithms is to approximate this underlying distribution or estimate some of its characteristics. Unfortunately, in most practically relevant data mining applications, the dimensionality of the feature vector is quite high making it prohibitive to learn the underlying distribution. For instance, gene expression data or images may easily have more than tens of thousands of features. While, at least in theory, having more features should result in a more discriminative classifier, it is not the case in practice because of the computational burden and curse of \ndimensionality. \n\nHigh-dimensional data poses different challenges on induction and prediction algorithms. Essentially, the amount of data to sustain the spatial density of the underlying distribution increases exponentially with the dimensionality of the feature vector, or alternatively, the sparsity increases exponentially given a constant amount of data. Normally in real-world applications, a limited amount of data is available and obtaining a sufficiently good estimate of the underlying high-dimensional probability distribution is almost impossible unless for some special data structures or under some assumptions (independent features, etc). \n\nThus, dimensionality reduction techniques, particularly feature extraction and feature selection methods, have to be employed to reconcile idealistic learning algorithms with real-world applications.  \n\nIn the context of feature selection, two main issues can be distinguished. The first one is to define an appropriate measure function to assign a score to a set of features. The second issue is to develop a search strategy that can find the optimal (in a sense of optimizing the value of the measure function) subset of features among all feasible subsets in a reasonable amount of time. \n\nDifferent approaches to address these two problems can roughly be categorized into three groups: Wrapper methods, embedded methods and filter methods.\n\nWrapper methods \\cite{kohavi:96} use the performance of an induction algorithm (for instance a classifier) as the measure function. Given an inducer $\\mathcal{I}$, wrapper approaches search through the space of all possible feature subsets and select the one that maximizes the induction accuracy. Most of the methods of this type require to check all the possible $2^N$ subsets of features and thus, may rapidly become prohibitive due to the so-called combinatorial explosion. Since the measure function is a machine learning (ML) algorithm, the selected feature subset is only optimal with respect to that particular algorithm, and may show poor generalization performance over other inducers. \n\nThe second group of feature selection methods are called embedded methods \\cite{neumann:04} and are based on some internal parameters of the ML algorithm. Embedded approaches rank features during the training process and thus simultaneously determine both the optimal features and the parameters of the ML algorithm. Since using (accessing) the internal parameters may not be applicable in all ML algorithms, this approach cannot be seen as a general solution to the  feature selection problem. In contrast to wrapper methods, embedded strategies do not require to run the exhaustive search over all subsets since they mostly evaluate each feature individually based on the score calculated from the internal parameters. However, similar to wrapper methods, embedded methods are dependent on the induction model and thus the selected subset is somehow tuned to a particular \ninduction algorithm.\n\nFilter methods, as the third group of selection algorithms, focus on filtering out irrelevant and redundant features in which irrelevancy is defined according to a predetermined measure function. Unlike the first two groups, filter methods do not incorporate the learning part and thus show better generalization power over a wider range of induction algorithms. They rely on finding an optimal feature subset through the optimization of a suitable measure function. Since the measure function is selected independently of the induction algorithm, this approach decouples the feature selection problem from the following ML algorithm. \n\nThe first contribution of this work is to analyze the popular mutual information measure in the context of the feature selection problem. We will expand the mutual information function in two different series and show that most of the previously suggested information-theoretic criteria are the first or second order truncation-approximations of these expansions. The first expansion is based on generalization of mutual information and has already appeared in literature while the second one is new, to the best of our knowledge. The well-known minimal Redundancy Maximal Relevance (mRMR) score function can be immediately concluded from the second expansion. We will discuss the consistency and accuracy of these approximations and experimentally investigate the conditions in which these truncation-approximations may lead to high estimation errors.  \n\nAlternatively, feature selection methods can be categorized based on the search strategies they employ. Popular search approaches can be divided into four categories: Exhaustive search, greedy search, projection and heuristic. A trivial approach is to exhaustively search in the subset space as it is done in wrapper methods. However, as the number of features increases, it can rapidly become infeasible. Hence, many popular search approaches use greedy hill climbing, as an approximation to this NP-hard combinatorial problem. Greedy algorithms iteratively evaluate a candidate subset of features, then modify the subset and evaluate if the new subset is an improvement over the old one. This can be done in a forward selection setup which starts with an empty set and adds one feature at a time or with a backward elimination process which starts with the full set of features and removes one feature at each step. The third group of the search algorithms are based on targeted projection pursuit which is a linear \nmapping \nalgorithm to pursue an optimum projection of data onto a low dimensional manifold that scores highly with respect to a measure function \\cite{friedman:74}. In heuristic methods, for instance genetic algorithms, the search is started with an initial subset of features which gradually evolves toward better solutions.\n\nRecently, two convex quadratic programing based methods, QPFS in \\cite{rod:10} and SOSS in \\cite{naghibi:13} have been suggested to address the search problem. QPFS is a deterministic algorithm and utilizes the Nystr\\\"{o}m method to approximate large matrices for efficiency purposes. SOSS on the other hand, has a randomized rounding step which injects a degree of randomness into the algorithm in order to generate more diverse feature sets.     \n\nDeveloping a new search strategy is another contribution of this paper. Here, we introduce a new class of search algorithms based on Semi-Definite Programming (SDP) relaxation. We reformulate the feature selection problem as a (0-1)-quadratic integer programming and will show that it can be relaxed to an SDP problem, which is convex and hence can be solved with efficient algorithms \\cite {boyd:04}. Moreover, there is a discussion about the approximation ratio of the proposed algorithm in subsection 3.2. We show that it usually gives better solutions than greedy algorithms in the sense that its approximate solution is more probable to be closer to the optimal point of the criterion.\n\\vspace{-3mm}\n\\section{Mutual Information Pros and Cons}\n\\label{mutualinformation}\nLet us consider an $N$ dimensional feature vector $\\mathbf{X}=[X_1,X_2,...,X_N]$ and a dependent variable $C$ which can be either a class label in case of classification or a target variable in case of regression. The mutual information  function is defined as a distance from independence between  $\\mathbf{X}$ and $C$ measured by the Kullback-Leibler divergence \\cite{cover:91}. Basically, mutual information measures the amount of information shared between $\\mathbf{X}$ and $C$ by measuring their dependency level. Denote the joint pdf of $\\mathbf{X}$ and $C$ and its marginal distributions by $Pr(\\mathbf{X},C)$, $Pr(\\mathbf{X})$ and  $Pr(C)$, respectively. The mutual information between the feature vector and the class label can be defined as follows:\n\\begin{align} \n\\label{eq_1} \nI(X_1,X_2,\\dots,& X_N;C)\\! = I(\\mathbf{X};C) = \\notag \\\\\n& \\! \\int \\!Pr(\\mathbf{X},C)\\! \\log{\\frac{Pr(\\mathbf{X},C)}{Pr(\\mathbf{X})Pr(C)}}\\,\\mathrm{d} \\mathbf{X}\\,\\mathrm{d}C\n\\end{align}\nIt reaches its maximum value when the dependent variable is perfectly described by the feature set. In this case mutual information is equal to $H(C)$, where $H(C)$ is the Shannon entropy of $C$.\n\nMutual information can also be considered a measure of set intersection \\cite{reza:61}. Namely, let $\\mathbb{A}$ and $\\mathbb{B}$ be event sets corresponding to random variables $A$ and $B$, respectively. It is not difficult to verify that a function $\\mu$ defined as:\n\\begin{equation} \n\\label{eq_2}\n\\mu({\\mathbb{A} \\cap \\mathbb{B}}) = I(A;B)\n\\end{equation}\nsatisfies all three properties of a formal measure over sets \\cite{yeung:91} \\cite{bog:07}, i.e., non-negativity, assigning zero to empty set and countable additivity. However, as we see later, the generalization of the mutual information measure to more than two sets will no longer satisfy the \\textit{non-negativity} property and thus can be seen as a signed measure which is the generalization of the concept of measure by allowing it to have negative values.   \n\nThere are at least three reasons for the popularity of the use of mutual information in feature selection algorithms. \n\n1. Most of the suggested non information-theoretic score functions are not formal set measures (for instance correlation function). Therefore, they cannot assign a score to a set of features but rather to individual features. However, mutual information as a formal set measure is able to evaluate all possible informative interactions and complex functional relations between features and as a result, fully extract the information contained in a set of features.\n\n2. The relevance of the mutual information measure to misclassification error is supported by the existence of bounds relating the probability of misclassification of the Bayes classifier, $P_e$, to the mutual information. More specifically, Fano's weak lower bound \\cite{fano:61} on $P_e$,\n\\begin{equation} \n\\label{eq_3}\n1+P_e\\text{log}_2(n_y{-}1)\\ge H(C)-I(\\mathbf{X};C)\n\\end{equation}\nwhere $n_y$ is the number of classes and the Hellman-Raviv \\cite{hellman:70} upper bound,\n\\begin{equation} \n\\label{eq_4}\nP_e\\le \\frac{1}{2}(H(C)-I(\\mathbf{X};C))\n\\end{equation}\non $P_e$, provide somewhat a performance guarantee. \n\nAs it can be seen in \\eref{eq_3} and \\eref{eq_4}, maximizing the mutual information between $\\mathbf{X}$ and $C$ decreases both upper and lower bounds on misclassification error and guarantees the goodness of the selected feature set. However, there is somewhat of a misunderstanding of this fact in the literature. It is sometimes wrongly claimed that maximizing the mutual information results in minimizing the $P_e$ of the optimal Bayes classifier. This is an unfounded claim since $P_e$ is not a monotonic function of the mutual information. Namely, it is possible that a feature vector $\\mathbf{A}$ with less relevant information-content about the class label $C$ than a feature vector $\\mathbf{B}$ yields a lower classification error rate than $\\mathbf{B}$. The following example may clarify this point.\n\n\\textbf{Example 1}: Consider a binary classification problem with equal number of positive and negative training samples and two binary features $X_1$ and $X_2$. The goal is to select the optimum feature for the classification task. Suppose the first feature $X_1$ is positive if the outcome is positive. However, when the outcome is negative, $X_1$ can take both positive and negative values with the equal probability. Namely, $Pr(X_1{=}1|C{=}1) = 1$ and $Pr(X_1{=} -1 | C {=} -1) = 0.5$. In the same manner, the likelihood of $X_2$ is defined as $Pr(X_2 {=} 1 | C {=}1) = 0.9$ and $Pr(X_2 {=} -1 | C{=} -1) = 0.7$. Then, the Bayes classifier  with feature $X_1$ yields the classification error:\n\\begin{align} \n\\label{ex_1}\nP_{e1}= & Pr(C{=}{-}1)Pr(X_1{=}1|C{=}{-}1) \\notag \\\\ \n        & +Pr(C{=}1)Pr(X{=}{-}1|C{=}1)=0.25 \n\\end{align}\nSimilarly, the Bayes classifier with $X_2$ yields $P_{e1}=0.2$ meaning that, $X_2$ is a better feature than $X_1$ in the sense of minimizing the probability of misclassification. However, unlike their error probabilities, $I(X_1;C) = 0.31$, is greater than \n$I(X_2;C) = 0.29$. That is, $X_1$ conveys more information about the class label in the sense of Shannon mutual information than $X_2$.\n\nA more detailed discussion can be found in \\cite{ben:12}. However, it is worthwhile to mention that although using mutual information may not necessarily result in the highest classification accuracy, it guarantees to reveal a salient feature subset by reducing the upper and lower bounds of $P_e$.\n\n3- By adapting classification error as a criterion, most standard classification algorithms fail to correctly classify the instances from minority classes in imbalanced datasets. Common approaches to address this issue are to either assign higher misclassification costs to minority classes or replace the classification accuracy criterion with the area under the ROC curve which is a more relevant criterion when dealing with imbalanced datasets. Either way, the features should also be selected by an algorithm which is insensitive (robust) with respect to class distributions (otherwise the selected features may not be informative about minority classes, in the first place). Interestingly, by internally applying unequal class dependent costs, mutual information provides some robustness with respect to class distributions. Thus, even in an imbalanced case, a mutual information based feature selection algorithm is likely (though not guaranteed) to not overlook the features that represent the minority classes. In \\\ncite{bao:11}, the concept of the mutual information classifier is investigated. Specifically, the internal cost matrix of the mutual information classifier is derived to show that it applies unequal misclassification \ncosts when dealing with imbalanced data and showed that the mutual information classifier is an optimal classifier in the sense of maximizing a weighted classification accuracy rate. The following example shows this robustness.\n\n\\textbf{Example 2}: Assume an imbalanced binary classification task where $Pr(C{=}1)=0.9$. As in Example 1, there are two binary features $X_1$ and $X_2$ and the goal is to select the optimum feature. Suppose $Pr(X_1{=}1|C{=}1) = 1$ and $Pr(X_1 {=} -1 | C {=} -1) = 0.5$. Unlike  the first feature, $X_2$ can much better classify the minority class $Pr(X_2 {=} {-}1 | C {=}{-}1) = 1$ and $Pr(X_2 {=}1 | C {=} 1) = 0.8$. It can be seen that the Bayes classifier with $X_1$ results in 100\\% classification rate for the majority class while only 50\\% correct classification for the minority. On the other hand, using $X_2$ leads to 100\\%  correct classification for the minority class and 80\\% for the majority. Based on the probability of error, $X_1$ should be preferred since its probability of error is  $P_{e1} = 0.05$ while  $P_{e2} = 0.18$. However, by using $X_1$ the classifier can not learn the rare event (50\\% classification rate) and thus randomly classifies the minority class which is the class of interest in \nmany applications. Interestingly, unlike the Bayesian error probabilities, mutual information prefers $X_2$ over $X_1$, since $I(X_2;C) = 0.20$ is greater than $I(X_1;C) = 0.18$. That is, mutual information is to some extent robust against imbalanced data.\n\nUnfortunately, despite the theoretical appeal of the mutual information measure, given a limited amount of data, an accurate estimate of the mutual information would be impossible. Because to calculate mutual information, estimating the high-dimensional joint probability $Pr(\\mathbf{X},C)$ is inevitable which is, in turn, known to be an NP hard problem \\cite{karger:01}.\n\nAs mutual information is hard to evaluate, several alternatives have been suggested \\cite{battiti:94}, \\cite{peng:05}, \\cite{kwak:02}. For instance, the Max-Relevance criterion approximates \\eref{eq_1} with the sum of the mutual information values between individual features $X_i$ and $C$:\n\\begin{equation} \n\\label{eq_5}\n\\text{Max-Relevance} = \\sum_{i=1}^{N} I(X_i;C)\n\\end{equation}\nSince it implicitly assumes that features are independent, it is likely that selected features are highly redundant. To overcome this problem, several heuristic corrective terms have been introduced to remove the redundant information and select mutually exclusive features. Here, it is shown that most of these heuristics are derived from the following expansions of mutual information with respect to $X_i$. \n\n\\subsection{First Expansion: Multi-way Mutual Information Expansion}\n\nThe first expansion of mutual information that is used here, relies on the natural extension of mutual information to more than two random variables proposed by McGill \\cite{mcgill:54} and Abramson \\cite{abramson:63}. According to their proposal, the three-way mutual information between random variables $Y_i$ is defined by:\n\\begin{align} \n\\label{eq_6}\nI(Y_1;Y_2;Y_3) = & I(Y_1;Y_3)+I(Y_2;Y_3)-I(Y_1,Y_2;Y_3) \\notag\\\\\n               = &  I(Y_1;Y_2) - I(Y_1;Y_2|Y_3)\n\\end{align}\nwhere ``,'' between variables denotes the joint variables. Note that, similar to two-way mutual information, it is symmetric with respect to $Y_i$ variables, i.e., $ I(Y_1;Y_2;Y_3) = I(Y_2;Y_3;Y_1)$. Generalizing over $N$ variables:\n\\begin{align} \n\\label{eq_7}\nI(Y_1;Y_2;\\dots;Y_N) =\\,  & I(Y_1;\\dots;Y_{N-1}) \\notag \\\\\n                        &- I(Y_1;\\dots;Y_{N-1}|Y_N)\n\\end{align}\nUnlike 2-way mutual information, the generalized mutual information is not necessarily nonnegative and hence, can be interpreted as a signed measure of set intersection \\cite{han:80}. Consider \\eref{eq_6} and assume $Y_3$ is class label $C$, then positive $I(Y_1;Y_2;C)$ implies that $Y_1$ and $Y_2$ are redundant with respect to $C$ since $I(Y_1,Y_2;C) \\le I(Y_1;C)+I(Y_2;C)$. However, the more interesting case is when $I(Y_1;Y_2;C)$ is negative, i.e., $I(Y_1,Y_2;C) \\ge I(Y_1;C)+I(Y_2;C)$. This means, the information contained in the interactions of the variables is greater than the sum of the information of the individual variables \\cite {gurban:09}.\n\nAn artificial example for this situation is the binary classification problem depicted in Figure \\ref{fig1}, where the classification task is to discriminate between the \nellipse class (class samples depicted by circles) and the line class (star samples) by using two features: values of $x$ axis and values of $y$ axis. As can be seen, since $I(x;C)\\! \\approx \\! 0$ and $I(y;C)\\! \\approx\\! 0$, there is no way to distinguish between these two classes by just using one of the features. However, it is obvious that employing both features results in almost perfect classification, i.e., $I(x,y;C)\\! \\approx\\! H(C)$.\n\\begin{figure}\n\\centering\n\\vspace{-0mm}\n\\includegraphics[scale=.45]{fig1.ps}\n\\vspace{0mm}\n\\caption{Synergy between $x$ and $y$ features. While information of each individual feature about the class label (ellipse or line) is almost zero, their joint information can almost completely remove the class label ambiguity.}\n\\label{fig1}\n\\vspace{-6mm}\n\\end{figure}\nThe mutual information in \\eref{eq_1} can be expanded out in terms of generalized mutual information between the features and the class label as:\n\\begin{align} \n\\label{eq_8}\nI(\\mathbf{X};C) =& \\sum_{i_1=1}^{N} I(X_{i_1};C) -  \\sum_{i_1=1}^{N-1} \\sum_{i_2=i_1+1}^{N} I(X_{i_1};X_{i_2};C)  \\notag  \\\\\n   &  +\\dots +   (-1)^{N-1} I(X_1;\\dots;X_N;C) \n\\end{align}\nFrom the definition in \\eref{eq_7} it is straightforward to infer this expansion. However, the more intuitive proof is to use the fact that mutual information is a measure of set intersection, i.e.,  $I(Y_1;Y_2;Y_3) = \\mu(\\mathbb{Y_1}\\cap \\mathbb{Y_2}\\cap\\mathbb{Y_3})$, where $\\mathbb{Y_i}$ is the corresponding event set of the $Y_i$ variable. Now, expanding the $N$-variable measure function results in:\n\\begin{align} \n\\label{eq_9}\nI(\\mathbf{X};C&) = \\mu((\\bigcup_{i=1}^{N} \\mathbb{X}_i) \\cap \\mathbb{C}) =  \\mu(\\bigcup_{i=1}^{N} (\\mathbb{X}_i \\cap \\mathbb{C})) \\\\\n      &=  \\sum_{i=1}^{N} \\mu(\\mathbb{X}_i \\cap \\mathbb{C}) -   \\sum_{i_1=1}^{N-1} \\sum_{i_2=i_1+1}^{N} \\mu(\\mathbb{X}_{i_1}\\cap \\mathbb{X}_{i_2} \\cap \\mathbb{C})  \\notag  \\\\  \n      & +\\dots+ (-1)^{N-1}\\mu(\\mathbb{X}_1\\cap \\mathbb{X}_2 \\dots \\cap \\mathbb{X}_N \\cap \\mathbb{C})   \\notag\n\\end{align}\nwhere the last equation follows directly from the addition law or sum rule in set theory. The proof is complete by recalling that all measure functions with the set intersection arguments in the last equation can be replaced by the mutual information functions according to the definition of mutual information in \\eref{eq_2}.\n\\subsection {Second Expansion: Chain Rule of Information}\n\nThe second expansion for mutual information is based on the \\textit{chain rule of information} \\cite{cover:91}:\n\\begin{align} \n\\label{eq_10}\nI(\\mathbf{X};C) = \\sum_{i=1}^{N} I(X_i;C|X_{i-1},\\dots,X_1)\n\\end{align}\nThe chain rule of information leaves the choice of ordering quite flexible. For example, the right side can be written in the order $(X_1,X_2,\\dots,X_N)$ or $(X_N,X_{N-1},\\dots,X_1)$. In general, it can be expanded over $N!$ different permutations of the feature set $\\{X_1,\\dots,X_N\\}$. Taking the sum over all possible expansions yields,\n\\begin{align} \n\\label{eq_10.5}\n(N!)&I(\\mathbf{X};C) =  (N{-}1)! \\sum_{i=1}^{N} I(X_i;C) \\\\ \n                    &   + (N{-}2)! \\sum_{i_1=1}^{N} \\sum_{i_2{\\in}\\{1,...,N\\}\/i_1} I(X_{i_2};C|X_{i_1}) \\notag \\\\\n                    & + \\cdots + (N{-}1)! \\sum_{i=1}^{N} I(X_{i};C|\\{X_1,\\dots,X_N\\}_{\\backslash{X_{i}}}) \\notag\n\\end{align}\nDividing both sides by $(N{-}1)!\/2$, and using the following equation $I(X_{i_1};C|X_{i_2})= I(X_{i_1};C)-I(X_{i_1};X_{i_2};C)$ to replace $I(X_{i_1};C|X_{i_2})$ terms, our second expansion can be expressed as\n\\begin{align} \n\\label{eq_11}\n\\frac{N}{2}& I( \\mathbf{X};C)  = \\sum_{i=1}^{N} I(X_{i};C)  \\\\\n& -\\frac{1}{N-1} \\sum_{i_1=1}^{N-1} \\sum_{i_2=i_1+1}^{N} I(X_{i_1};X_{i_2};C)  \\notag \\\\\n   &+ \\dots + \\frac{1}{2} \\sum_{i=1}^{N} I(X_{i};C|\\{X_1,\\dots,X_N\\}_{\\backslash{X_{i}}}) \\notag\n \\end{align}\n Ignoring the unimportant multiplicative constant $N\/2$ on the left side of equation \\eref{eq_11}, the right side can be seen as a series expansion form of mutual information (up to a known constant factor). \n \n \\subsection{Truncation of the Expansions}\n In the both proposed expansions \\eref{eq_8} and \\eref{eq_11}, mutual information terms with more than two features represent higher-order interaction properties. Neglecting the higher order terms yields the so-called truncated approximation of the mutual information function. If we ignore the constant coefficient in \\eref{eq_11}, the truncated forms of suggested expansions can be written as:\n\\begin{align} \n\\label{eq_12}\nD_1 = & \\sum_{i=1}^{N} I(X_{i};C) -  \\sum_{i=1}^{N-1} \\sum_{j=i+1}^{N} I(X_{i};X_{j};C)\\notag \\\\ \nD_2 = & \\sum_{i=1}^{N} I(X_{i};C) -\\frac{1}{N-1}  \\sum_{i=1}^{N-1} \\sum_{j=i+1}^{N} I(X_{i};X_{j};C)   \n\\end{align}\n where $D_1$ is the truncated approximation of \\eref{eq_8} and $D_2$ is for \\eref{eq_11}. Interestingly, despite the very similar structure of the expressions in \\eref{eq_12}, they have intrinsically different behaviors. This difference seems to be rooted in different functional forms they employ to approximate the underlying high-order pdf with lower order distributions ( i.e., how they combine these lower order terms). For instance, the functional form that MIFS employs to approximate $Pr(\\mathbf{x})$ is shown in \\eref{kirk}. While $D_1$ is not necessarily a positive value, $D_2$ is guaranteed to be a positive approximation since all terms  in \\eref{eq_10.5} are positive. However, $D_2$ may highly underestimate the mutual information values since it may violate the fact that \\eref{eq_1} is always greater than or equal to $\\max_{i} {I(X_i;C)}$.\n \\subsubsection{JMI, mRMR \\& MIFS Criteria}\n Several known criteria including Joint Mutual Information (JMI) \\cite{meyer:06}, minimal Redundancy Maximal Relevance (mRMR) \\cite{peng:05} and Mutual Information Feature Selection (MIFS) \\cite{battiti:94} can immediately be derived from $D_1$ and $D_2$. \n \n Using the identity: $I(X_i;X_j;C) = I(X_i;C)+I(X_j;C)-I(X_i,X_j;C)$ in $D_2$ reveals that $D_2$ is equivalent to JMI.\n \\begin{align} \n\\label{jmi}\n\\text{JMI} \\!=D_2=  \\sum_{i=1}^{N-1}\\! \\sum_{j=i+1}^{N} \\!I(X_{i},X_{j};C)    \n\\end{align}\n\n Using $I(X_{i};X_{j};C) =I(X_{i};X_{j}) -I(X_{i};X_{j}|C)$ and ignoring the terms containing more than two variables, i.e., $I(X_{i};X_{j}|C)$, in the second approximation $D_2$, one may immediately recognize the popular score function\n\\begin{align} \n\\label{eq_13}\n\\text{mRMR} \\!= \\!\\sum_{i=1}^{N} I(X_{i};C)\\! -\\!\\frac{1}{N-1}\\!  \\sum_{i=1}^{N-1}\\! \\sum_{j=i+1}^{N} \\!I(X_{i};X_{j})    \n\\end{align}\nintroduced by Peng et al. in \\cite{peng:05}. That is, mRMR is a truncated approximation of mutual information and not a heuristic approximation as suggested in \\cite{brown:12}. \n\nThe same line of reasoning as for mRMR can be applied to $D_1$ to achieve MIFS with $\\beta$ equal to 1. \n\\begin{align} \n\\label{eq_13.5}\n\\text{MIFS} \\!= \\!\\sum_{i=1}^{N} I(X_{i};C)\\! -  \\sum_{i=1}^{N-1}\\! \\sum_{j=i+1}^{N} \\!I(X_{i};X_{j})    \n\\end{align} \n\n\\textbf{Observation}:  A constant feature is a potential danger for the above measures. While adding an informative but correlated feature may reduce the score value (since $I(X_i;X_j|C)-I(X_i;X_j)$ can be negative), adding a non-informative constant feature $Z$ to a feature set does not reduce its score value since both $I(Z;C)$ and $I(Z;X_i;C)$ terms are zero, that is, constant features may be preferred over informative but correlated features. Therefore, it is essential to remove constant features by some preprocessing before using the above criteria for feature selection. \n\\subsubsection{Implicitly Assumed Distribution}\n A natural question arising in this context with respect to the proposed truncated approximations is: Under what probabilistic assumptions do the proposed approximations become valid mutual information functions? That is, which structure should a joint pdf admit, to yield mutual information in the forms of $D_1$ or $D_2$?  \n \n For instance, if we assume features are mutually and class conditionally independent, i.e., $Pr(\\mathbf{X}) = \\prod_{i=1}^{N}Pr(X_i)$ and $Pr(\\mathbf{X},C) = Pr(C)\\prod_{i=1}^{N}Pr(X_i|C)$, then it is easy to verify that mutual information has the form of Max-Relevance introduced in \\eref{eq_5}. These two assumptions, define the adapted \\textit{independence-map} of $Pr(\\mathbf{X},C)$ where the independence-map of a joint probability distribution is defined as follows.\n \n \\textbf{Definition 1}: \\textit{An independence-map (i-map) is a look up table or a set of rules that denote all the conditional and unconditional independence between random variables. Moreover, an i-map is consistent if it leads to a valid factorized probability distribution}.\n \n That is, given a consistent i-map, a high-order joint probability distribution is approximated with product of low-order pdfs and the obtained approximation is a valid pdf itself (e.g., $ \\prod_{i=1}^{N}Pr(X_i)$ is an approximation of the high-order pdf $Pr(\\mathbf{X})$ and it is also a valid probability distribution). \n \n The question regarding the implicit consistent i-map that MIFS adopts has been investigated in \\cite{balagani:10}. However, the assumption set (i-map) suggested in their work is inconsistent and leads to the incorrect conclusion that MIFS upper bounds the Bayesian classification error via the inequality \\eref{eq_4}. As we show in the following theorem, unlike the Max-Relevance case, there is no i-map that can produce mutual information in the forms of mRMR of MIFS (ignoring the trivial solution that reduces mRMR or MIFS to Max-Relevance). \n \n \\textbf{Theorem 1.} \\textit{Ignoring the trivial solution, i.e., the i-map indicating that random variables are mutually and class conditionally independent, there is no consistent i-map that can produce mutual information functions in the forms of mRMR \\eref{eq_13} or MIFS \\eref{eq_13.5} for arbitrary number of features}.\n \n \\textbf{Proof}: The proof is by contradiction. Suppose there is a consistent i-map, where its corresponding joint pdf $\\hat{P}r(\\mathbf{X},C)$ (which is the approximation of $Pr(\\mathbf{X},C)$) can generate mutual information in the forms of \\eref{eq_13} or \\eref{eq_13.5}. That is, if this i-map is adopted, by replacing $\\hat{P}r(\\mathbf{X},C)$ in \\eref{eq_1} we get mRMR or MIFS. This implies that mRMR and MIFS are \\textit{always} valid set measures for all datasets regardless of their true underlying joint probability distributions. Now, if we show (by any example) that they are not valid mutual information measures, i.e., they are not always positive and monotonic, then we have contradicted our assumption that $\\hat{P}r(\\mathbf{X},C)$ exists and is a valid pdf. It is not so difficult to construct an example in which mRMR or MIFS can get negative values. Consider the case where features are independent of class label, $I(X_i;C)=0$, while they have nonzero dependencies among themselves, $I(X_i;X_j)\\neq0$. \nIn this case, both mRMR and MIFS \ngenerate negative values which is not allowed by a valid set measure. This contradicts our assumption that they are generated by a valid distribution, so we are forced to conclude that there is no consistent i-map that results in mutual information in the mRMR or MIFS forms.$\\blacksquare$\n \n The same line of reasoning can be used to show that $D_1$ and $D_2$ are also not valid measures. \n  \n However, despite the fact that no valid pdf can produce mutual information of those forms, it is still valid to ask for which low-order approximations of the underlying high-order pdfs, mutual information reduces to a truncated approximation form. That is, we do not restrict an approximation to be a valid distribution anymore. Any functional form of low-order pdfs may be seen as an approximation of the high-order pdfs and may give rise to MIFS or mRMR. In the next subsection we reveal these assumptions for the MIFS criterion. \n \\subsubsection{MIFS Derivation from Kirkwood Approximation}\n \n It is shown in \\cite{killian:07} that truncation of the joint entropy $H(\\mathbf{X})$ at the $r$th-order is equivalent to approximating the full-dimensional pdf $Pr(\\mathbf{X})$ using joint pdfs with dimensionality of $r$ or smaller. This approximation is called $r$th order Kirkwood approximation. The truncation order that we choose, partially determines our belief about the structure of the function that we are going to estimate the exact $Pr(\\mathbf{X})$ with.\n \n The 2nd order Kirkwood approximation of $Pr(\\mathbf{X})$, can be denoted as follows \\cite{killian:07}: \n  \\begin{equation} \n \\label{kirk}\n\\hat{Pr}(\\mathbf{X})= \\frac{\\prod_{i=1}^{N-1}\\prod_{j=i+1}^{N}Pr(X_i,X_j)}{\\big[\\prod_{i=1}^{N}Pr(X_i)\\big]^{N-2}}\n \\end{equation} \n Now, assume the following two assumptions hold:\n \n \\textbf{Assumption 1}: Features are class conditionally independent, that is: $Pr(\\mathbf{X}|C)=\\prod_{i=1}^{N}Pr(X_i|C)$\n \n \\textbf{Assumption 2}: $Pr(\\mathbf{X})$ is well approximated by a 2nd order Kirkwood superposition approximation in \\eref{kirk}.\n \n Then, writing the definition of mutual information and applying the above assumptions yields the MIFS criterion\n   \\begin{align} \n \\label{identity}\nI(\\mathbf{X};C) &= H(\\mathbf{X})-H(\\mathbf{X}|C) \\\\\n\t\t&  \\stackrel{(a)}{\\approx} \\sum_{i=1}^{N} H(X_i) - \\sum_{i=1}^{N-1}\\sum_{j=i+1}^{N} I(X_i;X_j) -H(\\mathbf{X}|C) \\notag \\\\\n\t\t&  \\stackrel{(b)}{=} \\sum_{i=1}^{N} I(X_i;C) - \\sum_{i=1}^{N-1}\\sum_{j=i+1}^{N} I(X_i;X_j)  \\notag\n \\end{align} \n In the above equation, (a) follows the second assumption by substituting the 2nd order Kirkwood approximation \\eref{kirk} inside the logarithm of the entropy integral and (b) is an immediate consequence of the first assumption.\n \n The first assumption has already appeared in previous works \\cite{brown:12} \\cite{balagani:10}. However, the second assumption is novel and, to the best of our knowledge, the connection between the Kirkwood approximation and the MIFS criterion has not been explored before. \n\n It is worth to mention that, in reality, both assumptions can be violated. Specifically, the Kirkwood approximation may not precisely reproduce dependencies we might observe in real-world datasets. Moreover, it is important to remember that the Kirkwood approximation is not a valid probability distribution.\n \\vspace{-1mm}\n \\subsection{$\\mathbf{D}_2$ Approximation}\n From our experiments,  which we omit because of space constraints, $D_2$ tends to underestimate the mutual information while $D_1$ shows a large overestimation for independent features and a large underestimation (even becoming negative) in the presence of dependent features. In general, $D_2$ shows more robustness than $D_1$. The same results can be observed for mRMR which is derived from $D_2$ and MIFS derived from $D_1$. Previous work also arrived to the same results and reported that mRMR performs better and more robustly than MIFS especially when the feature set is large. Therefore, in the following sections we use $D_2$ as the truncated approximation. For simplicity, its subscript is dropped and it is rewritten as follows:   \n\\begin{align} \n\\label{eq_14}\nD(\\{X_1,\\dots,X_N\\}) = &\\sum_{i=1}^{N}  I(X_{i};C)  \\\\\n- & \\frac{1}{N-1} \\sum_{i=1}^{N-1} \\sum_{j=i+1}^{N} I(X_{i};X_{j};C)\\notag\n\\end{align}\nNote that although $D$ in \\eref{eq_14} is not a formal set measure any more, it still can be seen as a score function for sets. However, it is noteworthy that unlike formal measures, the suggested approximations are no longer monotonic where the monotonicity merely means that a subset of features should not be better than any larger set that contains the very same subset. Therefore, as explained in \\cite{narendra:77} the branch and bound based search strategies can not be applied to them. \n\nA very similar approach has been applied \\cite{brown:09} (by using $D_1$ approximation) to derive several known criteria like MIFS \\cite{battiti:94} and mRMR \\cite{peng:05}. However, in \\cite{brown:09} and most of other previous works, the set score function in \\eref{eq_14} is immediately reduced to an individual-feature score function by fixing $N{-}1$ features in the feature set. This will let them to run a greedy selection search method over the feature set which essentially is a one-feature-at-a-time selection strategy. It is clearly a naive approximation of the optimal NP-hard search algorithm and may perform poorly under some conditions. In the following, we investigate a convex approximation of the binary objective function appearing in feature selection inspired by the Goemans-Williamson maximum cut approximation approach \\cite{goemans:95}.\n\n\n\n\\section{Search Strategies}\n\\label{search}\nGiven a measure function\\footnote{By some abuse of terminology, we refer to any set function in this section as a measure, no matter whether they satisfy the formal measure properties.} $D$, the Subset Selection Problem (SSP) can be defined as follows:\n\n\\textbf{Definition 2}: Given $N$ features $X_i$ and a dependent variable $C$, select a subset of $P \\! \\ll \\! N$ features that maximizes the measure function. Here it is assumed that the cardinality $P$ of the optimal feature subset is known.\n\nIn practice, the exact value of $P$ can be obtained by evaluating subsets for different values of cardinality $P$ with the final induction algorithm. Note that it is intrinsically different than wrapper methods. While in wrapper methods $2^N$ subsets have to be tested, here at most $N$ runs of the learning algorithm are needed to evaluate all possible values of $P$.  \n\nA search strategy is an algorithm trying to find a feature subset in the feature subset space with $2^N$ members\\footnote{Given a $P$, the size of the feature subset space reduces to  ${N\\choose P}$.} that optimizes the measure function. The wide range of proposed search strategies in the literature can be divided into three categories: 1- Exponential complexity methods including exhaustive search \\cite{kohavi:96}, branch and bound based algorithms \\cite{narendra:77}. 2- Sequential selection strategies with two very popular members, forward selection and backward elimination methods. 3- Stochastic methods like simulated annealing and genetic algorithms \\cite{vafai:93}, \\cite{doak:92}.\n\nHere, we introduce a fourth class of search strategies which is based on the convex relaxation of the 0-1 integer programming and explore its approximation ratio by establishing a link between SSP and an instance of the maximum-cut problem in graph theory. In the following, we briefly discuss the two popular sequential search methods and continue with the proposed solution: a close to optimal polynomial-time complexity search algorithm and its evaluation on different datasets.   \n\\vspace{-1mm}\n\\subsection{Convex Based Search}\n\\label{sdp}\nThe forward selection (FS) algorithm selects a set $\\mathbb{S}$ of size $P$ iteratively as follows:\n\\begin{enumerate}\n\\item Initialize $\\mathbb{S}_0 = \\emptyset$. \n\\item In each iteration $i$, select the feature $X_m$ maximizing $D(\\mathbb{S}_{i{-}1}\\cup X_m)$, and set $\\mathbb{S}_{i} = \\mathbb{S}_{i{-}1} \\cup {X_m}$. \n\\item Output $\\mathbb{S}_P$. \n\\end{enumerate}\nSimilarly, backward elimination (BE) can be described as: \n\\begin{enumerate}\n\\item Start with the full set of feature $\\mathbb{S}_N$. \n\\item Iteratively remove a variable $X_m$ maximizing $D(\\mathbb{S}_i\\backslash{X_m})$, and set $\\mathbb{S}_{i-1} =\\mathbb{S}_i\\backslash{X_m}$, where removing $X$ from $\\mathbb{S}$ is denoted by $\\mathbb{S}\\backslash{X}$.\n\\item Output $\\mathbb{S}_P$.\n\\end{enumerate}\nAn experimentally comparative evaluation of several variants of these two algorithms has been conducted in \\cite{aha:94}. From an information theoretical standpoint, the main disadvantage of the forward selection method is that it only can evaluate the utility of a single feature in the limited context of the previously selected features. The artificial binary classifier in Figure \\ref{fig1} may illustrate this issue. Since the information content of each feature ($x$ and $y$) is almost zero, it is highly probable that the forward selection method fails to select them in the presence of some other more informative features. \n\nContrary to forward selection, backward elimination can evaluate the contribution of a given feature in the context of all other features. Perhaps this is why it has been frequently reported to show superior performance than forward selection. However, its overemphasis on feature interactions is a double-edged sword and may lead to a sub-optimal solution. \n\n\\textbf{Example 3}: Imagine a four dimensional feature selection problem where $X_1$ and $X_2$ are class conditionally and mutually independent of $X_3$ and $X_4$, i.e., $Pr(X_1,X_2,X_3,X_4)=Pr(X_1,X_2)Pr(X_3,X_4)$ and $Pr(X_1,X_2,X_3,X_4|C)=Pr(X_1,X_2|C)Pr(X_3,X_4|C)$. Consider $I(X_1;C)$ and $I(X_2;C)$ are equal to zero, while their interaction is informative. That is, $I(X_1,X_2;C) = 0.4$. Moreover, assume $I(X_3;C) = 0.2$, $I(X_4;C)=0.25$ and $I(X_3,X_4;C) = 0.45$. The goal is to select only two features out of four. Here, backward elimination will select $\\{X_1,X_2\\}$ rather than the optimal subset $\\{X_3,X_4\\}$ because, removing any of $X_1$ or $X_2$ features will result in $0.4$ reduction of the mutual information value $I(X_1,\\dots,X_4;C)$, while eliminating $X_3$ or $X_4$ deducts at most $0.25$. One may draw the conclusion that backward elimination tends to sacrifice the individually-informative features in favor of the merely cooperatively-informative features. As a remedy, several hybrid \nforward-backward sequential search methods have been proposed. However, they all fail in one way or another and more importantly cannot guarantee the goodness of the solution.\n\nAlternatively, a sequential search method can be seen as an approximation of the combinatorial subset selection problem. To propose a new approximation method, the underlying combinatorial problem has to be studied. To this end, we may formulate the SSP defined in the beginning of this section as:\n \\begin{align}\n\\label{eq_15}\n &\\max_{\\mathbf{x}} \\, {\\mathbf{x}^T\\mathbf{Q}\\mathbf{x}}  \\notag \\\\\n  & \\sum_{i=1}^{N} x_i = P \\\\ \n & x_i \\in \\{0,1\\} \\text {   for   } i=1,\\dots,N \\notag\n \\end{align}\n where $\\mathbf{Q}$ is a symmetric mutual information matrix constructed from the mutual information terms in \\eref{eq_14}:\n \\begin{equation} \n\\label{eq_16}\n  \\mathbf{Q} = \\begin{pmatrix}\n   I(X_1;C) &  \\cdots & -\\frac{\\lambda}{2}I(X_1;X_N;C)\\\\\n   -\\frac{\\lambda}{2}I(X_1;X_2;C) & \\cdots &  -\\frac{\\lambda}{2}I(X_2;X_N;C)\\\\\n  \\vdots   & \\ddots & \\vdots  \\\\\n  -\\frac{\\lambda}{2}I(X_1;X_N;C) & \\cdots & I(X_N;C)\\\\\n\\end{pmatrix}\n \\end{equation}\nwhere $\\lambda =\\frac{1}{P-1}$ and $\\mathbf{x} = [x_1,\\dots,x_N]$ is a binary vector where the variables $x_i$ are set-membership binary variables indicating the presence of the corresponding features $X_i$ in the feature subset.\nIt is straightforward to verify that for any binary vector $\\mathbf{x}$, the objective function in \\eref{eq_15} is equal to the score function $D(\\mathbb{X}_{nz})$ where $\\mathbb{X}_{nz} =\\{X_i | x_i =1;i=1,\\dots,N \\}$. \nNote that, for mRMR $I(X_i;X_j;C)$ terms have to be replaced with $I(X_i;X_j)$.\n\nThe (0,1)-quadratic programming problem \\eref{eq_15} has attracted a great deal of theoretical study because of its importance in combinatorial problems \\cite[and references therein]{poljak:95}. This problem can simply be transformed to a \\text{(-1,1)-quadratic} programming problem,\n\\begin{align} \n\\label{eq_17}\n &\\max_{\\mathbf{y}} {\\frac{1}{4}\\mathbf{y}^T\\mathbf{Q}\\mathbf{y} + \\frac{1}{2}\\mathbf{y}^T\\mathbf{Q}\\mathbf{e} + c}  \\notag \\\\\n & \\sum_{i=1}^{N} y_i = 2P-N \\\\ \n & y_i \\in \\{-1,1\\} \\text {   for   } i=1,\\dots,N \\notag\n \\end{align}\nvia $\\mathbf{y} = 2\\mathbf{x}-\\mathbf{e}$ transformation, where $\\mathbf{e}$ is an all ones vector. Additionally $c$ in the above formulation is a constant equal to $\\frac{1}{4}\\mathbf{e}^T\\mathbf{Qe}$ and it can be ignored because of its independence of $\\mathbf{y}$. In order to homogenize the objective function in \\eref{eq_17}, define an $(N{+}1)\\!\\times\\!(N{+}1)$ matrix $\\mathbf{Q}^u$ by adding a 0-th row and column to $\\mathbf{Q}$ so that:\n\\begin{align} \n\\label{eq_18}\n\\mathbf{Q}^u = \\begin{pmatrix}\n  0 & \\mathbf{e}^T\\mathbf{Q}\\\\\n  \\mathbf{Q}^T\\mathbf{e} & \\mathbf{Q}\n\\end{pmatrix} \n \\end{align}\n Ignoring the constant factor $\\frac{1}{4}$ in \\eref{eq_17}, the equivalent homogeneous form of \\eref{eq_15} can be written as:\n\\begin{align} \n\\label{eq_19}\n & S_{\\text{SSP}} = \\max_{\\mathbf{y}} {\\mathbf{y}^T\\mathbf{Q}^u\\mathbf{y}}  \\notag \\\\\n  \\text{$\\langle$SSP$\\rangle$}\\quad \\quad  & \\sum_{i=1}^{N} y_iy_0 = 2P-N\\\\ \n & y_i \\in \\{-1,1\\} \\text {   for   } i=0,\\dots,N \\notag\n\\end{align}\nNote that $\\mathbf{y}$ is now an $N+1$ dimensional vector with the first element $y_0 = \\pm 1$ as a reference variable. Given the solution of the problem above, i.e., $\\mathbf{y}$, the optimal feature subset is obtained by $\\mathbb{X}_{op} =\\{X_i|y_i=y_0\\}$. \n\nThe optimization problem in \\eref{eq_19} can be seen as an instance of the maximum-cut problem \\cite{goemans:95} with an additional cardinality constraint, also known as the k-heaviest subgraph or maximum partitioning graph problem. The two main approaches to solve this combinatorial problem are either to use the linear programming relaxation by linearizing the product of two binary variables \\cite{frieze:83}, or the semidefinite programming (SDP) relaxation suggested in \\cite {goemans:95}. The SDP relaxation has been proved to have\nexceptionally high performance and achieves the approximation ratio of 0.878 for the original maximum-cut problem. The SDP relaxation of \\eref{eq_19} is:\n\\begin{align} \n\\label{eq_20}\n &S_{\\text{SDP}} = \\max_{\\mathbf{Y}} {\\text{tr}\\{\\mathbf{Q}^u\\mathbf{Y}\\}}  \\notag \\\\\n & \\sum_{i,j=1}^{N} Y_{ij} = (2P-N)^2  \\notag \\\\\n\\text{$\\langle$SDP$\\rangle$}\\quad \\quad & \\sum_{i=1}^{N} Y_{i0} = (2P-N)  \\\\\n & \\text{diag}(\\mathbf{Y}) =\\mathbf{e}  \\notag \\\\\n & \\mathbf{Y}\\succeq 0   \\notag\n\\end{align}\nwhere $\\mathbf{Y}$ is an unknown $(N+1)\\times(N+1)$ positive semidefinite matrix and\n$\\text{tr}\\{\\mathbf{Y}\\}$ denotes its trace. Obviously, any feasible solution $\\mathbf{y}$ for\n$\\langle$SSP$\\rangle$ is also feasible for its SDP relaxation by $\\mathbf{Y} =\n\\mathbf{y}\\mathbf{y}^T$. Furthermore, it is not difficult to see that any rank one solution,\n$\\text{rank}(\\mathbf{Y})=1$, of $\\langle$SDP$\\rangle$ is a solution of $\\langle$SSP$\\rangle$. \n\n The $\\langle$SDP$\\rangle$ problem can be solved within an additive error $\\gamma$ of the optimum by for example interior point methods \\cite{boyd:04} whose computational complexity are polynomial in the size of the input and $log(\\frac{1}{\\gamma})$. However, since its solution is not necessarily a rank one matrix, we need some more steps to obtain a feasible solution for $\\langle$SSP$\\rangle$. The following three steps summarize the approximation algorithm for $\\langle$SSP$\\rangle$ which in the following will be referred to as convex based relaxation approximation (COBRA) algorithm.\n\n\\vspace{2mm}\n\\noindent\\textbf{COBRA Algorithm}:\n\\vspace{1.5mm}\n\\begin{enumerate}\n\\item SDP: Solve $\\langle$SDP$\\rangle$ and obtain $\\mathbf{Y}_{sdp}$. Repeat the following steps many times and output the best solution.\n\n\\item  Randomized rounding: Using the multivariate normal distribution with a zero mean and a covariance matrix $\\mathbf{R} = \\mathbf{Y}_{sdp}$ to sample $\\mathbf{u}$ from distribution $\\mathcal{N}(0,\\mathbf{R})$ and construct $\\hat{\\mathbf{x}} = \\text{sign}(\\mathbf{u})$. Select $\\mathbb{X} = \\{X_i | \\hat{x}_i = \\hat{x}_0\\}$.\n\n\\item Size adjusting: By using the greedy forward or backward algorithm, resize the cardinality of $\\mathbb{X}$ to $P$. \n\\end{enumerate}\n\nThe randomized rounding step is a standard procedure to produce a binary solution from the real-valued solution of $\\langle$SDP$\\rangle$ and is widely used for designing and analyzing approximation algorithms \\cite{ragh:88}. The third step is to construct a feasible solution that satisfies the cardinality constraint. Generally, it can be skipped since in feature selection problems the exact satisfaction of the cardinality constraint is not required.\n\nWe use the SDP-NAL solver \\cite{zhao:10} with the Yalmip interface \\cite{lof:04} to implement this algorithm in Matlab. SDP-NAL uses the Newton-CG augmented Lagrangian method to efficiently solve SDP problems. It can solve large scale problems ($N$ up to a few thousand) in an hour on a PC with an Intel Core i7 CPU. Even more efficient algorithms for low-rank SDP have been suggested claiming that they can solve problems with the size up to $N{=}30000$ in a reasonable amount of time \\cite{grippo:12}. Here we only use the SDP-NAL for our experiments.     \n\\vspace{-3mm}\n\\subsection{Approximation Analysis}\n\\begin{center}\n \\begin{table*}\n \\addtolength{\\tabcolsep}{5mm}\n \\centering\n   \\begin{tabular}{l|| c c c c c c c} \n\\hline\\rule[-0.0ex]{0ex}{2.5ex}\n\\bf{Values of $P$}     &  $N\/2$     &  $N\/3$ & $N\/4$ & $N\/6$ &$N\/8$& $N\/10$ &$N\/20$\n\\\\\n\\hline \n\\hline\n\\ \\bf{BE}            & 0.4         & 0.25   & 0.16    & 0.10    & 0.071    & 0.055    & 0.026 \\rule[-0.0ex]{0ex}{2.5ex}    \\\\\n\n\\ \\bf{COBRA}          & 0.48       & 0.33   & 0.24    &0.13     & 0.082    &  0.056   & 0.015     \\\\ \n\\hline\n\\end{tabular}\n\\caption{Approximation ratios of BE and COBRA for different $N\/P$ values.}\n\\vspace{-2mm}\n\\label{app-ratio}\n\\end{table*}  \n\\end{center}\n\\vspace{-3mm}\nIn order to gain more insight into the quality of a measure function, it is essential to be able to directly examine it. However, since estimating the exact mutual information value in real data is not feasible, it is not possible to directly evaluate the measure function. Its quality can only be indirectly examined through the final classification performance (or other measurable criteria). However, the quality of a measure function is not the only contributor to the classification rate. Since SSP is an NP-hard problem, the search strategy can only find a local optimal solution. That is, besides the quality of a measure function, the inaccuracy of the search strategy also contributes to the final classification error. Thus, in order to draw a conclusion concerning the quality of a measure function, it is essential to have an insight about the accuracy of the search strategy in use. In this section, we compare the accuracy of the proposed method with the traditional backward elimination approach.\n\nA standard approach to investigate the accuracy of an optimization algorithm is by analyzing how close it gets to the optimal solution. Unfortunately, feature selection is an NP-hard problem and thus achieving the optimal solution to use as reference is only feasible for small-sized problems. In such cases, one wants a provable solution's quality and certain properties about the algorithm, such as its approximation ratio. Given a maximization problem, an algorithm is called $\\rho$-approximation algorithm if the approximate solution is at least $\\rho$ times the optimal value. That is, in our case $\\rho S_{SSP}\\le S_{COBRA}$, where $S_{COBRA}=D(\\mathbb{X}_{COBRA})$. The factor $\\rho$ is usually referred to as the approximation ratio in the literature.  \n\nThe approximation ratios of BE and COBRA can be found by linking the SSP to the k-heaviest subgraph problem (k-HSP) in graph theory. k-HSP is an instance of the max-cut problem with a cardinality constraint on the selected subset, that is, to determine a subset $S$ of k vertices such that the weight of the subgraph induced by $S$ is maximized \\cite{sriva:98}. From the definition of k-HSP, it is clear that SSP with the criterion \\eref{eq_14} is equivalent to the $P$-heaviest subgraph problem since it selects the heaviest subset of features with the cardinality $P$, where heaviness of a set is the score assigned to it by $D$.\n\nAn SDP based algorithm for k-HSP has been suggested in \\cite{sriva:98} and its approximation ratio has been analyzed. Their results are directly applicable to COBRA since both algorithms use the same randomization method (step 2 of COBRA) and the randomization is the main ingredient of their approximation analysis. The approximation ratio of BE for k-HSP has been investigated in \\cite{asahiro:00}. It is a deterministic analysis and their results are also valid for our case, i.e., using BE for maximizing $D$. \n\nThe approximation ratios of both algorithms for different values of $P$, as a function of $N$ (total number of features), have been listed in Table \\ref{app-ratio} (values are calculated from the formulas in \\cite{asahiro:00}). As can be seen, as $P$ becomes smaller, the approximation ratio approaches zero yielding the trivial lower bound 0 on the approximate solution. However, for larger values of $P$, the approximation ratio is nontrivial since it is bounded away from zero. For all cases shown in the table except the last one, COBRA gives better guarantee bound than BE. Thus, we may conclude that COBRA is more likely to achieve better approximate solution than BE.\n\nIn the following section, we will focus on comparing our search algorithm with sequential search methods in conjunction with different measure functions and over different classifiers and datasets.\n\n\\section{Experiments}\n\nThe evaluation of a feature selection algorithm is an intrinsically difficult task since there is no direct way to evaluate the goodness of a \\textit{selection process} in general. Thus, usually a selection algorithm is scored based on the performance of its output, i.e., the selected feature subset, in some specific classification (regression) system. This kind of evaluation can be referred to as the goal-dependent evaluation. However, this method obviously cannot evaluate the generalization power of the selection process on different induction algorithms. To evaluate the generalization strength of a feature selection algorithm, we need a goal-independent evaluation. Thus, for evaluation of the feature selection algorithms, we propose to compare the algorithms over different datasets with multiple classifiers. This method leads to a more classifier-independent evaluation process. \n\nSome properties of the eight datasets used in the experiments are listed in Table \\ref{datasets}. All datasets are available on the UCI machine learning archive \\cite{frank:10}, except the NCI data which can be found in the website of Peng et al. \\cite{peng:05}. These datasets have been widely used in previous feature selection studies \\cite{peng:05}, \\cite{ciarelli:10}. The goodness of each feature set is evaluated with five classifiers including Support Vector Machine (SVM), Random Forest (RF), Classification and Regression Tree (CART), Neural Network (NN) and Linear Discriminant Analysis (LDA). To derive the classification accuracies, 10-fold cross-validation is performed except for the NCI, DBW and LNG datasets where leave-one-out cross-validation is used.\n  \nAs explained before, filter-based methods consist of two components: A measure function and a search strategy. The measure functions we use for our experiments are mRMR and JMI defined in \\eref{eq_13} and \\eref{jmi}, respectively. To unambiguously refer to an algorithm, it is denoted by measure function + search method used in that algorithm, eg., mRMR+FS. \n\\vspace{-20mm}\n\\begin{center}\n\\begin{table*}\n\\addtolength{\\tabcolsep}{1mm}\n\\centering\n\\begin{tabular}{l ||c  c  c  c  c  c  c  c } \n\\hline \\rule[-0.0ex]{0ex}{2.5ex}\n \\bf{Dataset Name}         & Arrhythmia & NCI             & DBWorld e-mails    & CNAE-9  & Internet Adv. & Madelon & Lung Cancer & Dexter          \\\\ \n  \\bf{Mnemonic}        & ARR        & NCI                  & DBW        & CNA     & IAD           & MAD     &LNG          &DEX                             \\\\ \\hline\\hline\n  \\# \\bf{Features}    & 278        & 9703                 &  4702      & 856     & 1558          & 500     &56           &20000                            \\\\   \n  \\# \\bf{Samples}     & 370        & 60                   & 64         & 1080    & 3279          & 2000      & 32          &300                         \\\\   \n  \\# \\bf{Classes}     & 2          & 9                    & 2          & 9       & 2             & 2       &3             &2                                \\\\  \n  \\hline\n\\end{tabular}\n\\caption{Datasets descriptions}\n\\label{datasets}\n\\end{table*}\n\\end{center}\n\n\\begin{table}\n\\def\\\\[1.6mm]{\\\\[1.6mm]}\n\\framebox{\n\\parbox{80mm}{\n{\\normalsize \nSet $P$: $\\mathbb{P}\\!=\\!\\{P_1,\\dots,P_L\\}$.\n \\begin{algorithmic}\n\\ForAll {$P$ in $\\mathbb{P}$}\n  \n     \\State Run the COBRA algorithm and output the solution $\\mathbb{X}$.\n     \\State Derive the classifier error-rate by applying K-fold \n     \\State cross-validation and save it in $CL(P)$. \n   \n\\EndFor \n\\end{algorithmic}\n Output: $ P_{opt} = \\underset{P}{\\operatorname{argmin}}\\, CL(P)$\n }}\n }\n \\vspace{-1mm}\n \\caption{ Estimating $P$ by searching over an admissible set that minimizes the classification error-rate.}\n \\label{algo}\n\\end{table} \n\n\\begin{center}\n  \\begin{table*}[!ht]\n  \\addtolength{\\tabcolsep}{3.5mm}\n   \\hfill{}\n    \\begin{tabular}{l||  c@{\\hspace{3pt}}c c@{\\hspace{3pt}}c c@{\\hspace{3pt}}c c@{\\hspace{3pt}}c c@{\\hspace{3pt}}c c  } \n      \\hline  \\rule[-0.0ex]{0ex}{2.5ex} \n   \\textbf{Classifiers}                    &  \\multicolumn{2}{|c|}{\\bf{SVM}}                      &  \\multicolumn{2}{|c|}{\\bf{LDA}}                &  \\multicolumn{2}{|c|}{\\bf{CART}}            &\\multicolumn{2}{|c|}{\\bf{RF}}                     & \\multicolumn{2}{|c|}{\\bf{NN}}        & \\bf{Average}            \\\\   \\hline  \\hline   \n                                                                &   \\multicolumn{11}{c}{\\bf{NCI Dataset}}  \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                     & (54) & 81.7               & (95) & 78.3         & (20) & 45.0          & (71)& 88.3           &  (60) & 75.0 & \\textbf{73.67}                          \\\\  \n   \\bf{mRMR+FS}                        &  (32) & 78.3  \t           &  (11) & 68.3        &  (2) & 45.0        &   (12) & 83.3        &  (99) & 70.0    & 69.00                \\\\ \n   \\bf{mRMR+BE}                        &  (26) & 76.6              &  (11) & 68.3        &  (2) & 45.0        &   (13) & 85.0        &  (31) & 71.7    & 69.33                                \\\\  \n    \n   \\bf{JMI+COBRA}                      &  (72)& 85.0              &  (70)& 75.0        &  (28)& 45.0       &   (45)& 90.0             &  (93)& 75.0      & 74.00                          \\\\  \n   \\bf{JMI+FS}                         &  (27)& 75.0              &  (17)& 68.3       &  (82)& 45.0       &   (17)& 86.6             &  (78)&  70.0     & 69.00                   \\\\  \n   \\bf{JMI+BE}                         &  (23)& 76.6              &  (20)&  76.6       &  (7)& 33.3      &   (19)& 86.6             &  (89)& 76.6     & 70.00                                \\\\  \\hline\\hline\n   \n                                                                  &   \\multicolumn{11}{c}{ \\bf{DBW Dataset}}  \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                    &  (38)&   96.9             &  (152)&   92.2        &  (38)&   86.0         &    (33)&   92.2       &    (33)&  98.4  & \\textbf{93.12}                    \\\\ \n   \\bf{mRMR+FS}                       &  (31)&   93.7             &  (4)&   89.0        &  (4)&   86.0          &   (7)&    90.6       &    (9)&  92.2  & 90.31         \\\\ \n   \\bf{mRMR+BE}                       &  (110) &   93.7             &  (6)     &   89.0        &  (4)   &   82.8          &   (29)   &   92.2       &    (9)   &  92.2  & 90.00          \n   \\\\           \n   \\bf{JMI+COBRA}                     & (35)  & 93.7                  &  (14) & 89.0            &  (8)  & 82.8          &  (24)  & 92.2      &  (108)  & 93.7  & 90.31                         \\\\  \n   \\bf{JMI+FS}                        & (23) &93.7                    &  (6)  & 89.0            &  (5)  & 82.8          &  (34)  & 92.2      &  (96)  & 92.2   & 90.00                \\\\  \n   \\bf{JMI+BE}                        & (24) & 93.7                   &  (6) &  89.0            &  (5)  & 82.8          &  (23)  & 92.2      &  (149)  & 92.2  & 90.00                               \\\\  \\hline\\hline\n    \n                                                                     &   \\multicolumn{11}{c}{ \\bf{CNA Dataset}} \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                    &   (200)  & 94.0 &  (183)  &  92.7      &  (63)  & 75.0         &  (183) & 90.8       & (187)  & 92.0     & 88.91     \\\\  \n   \\bf{mRMR+FS}                       &   (149)  & 90.6 &  (142)  &  90.4      &  (7)   & 70.2         &  (138) & 87.7       & (78)   & 85.5     & 84.88        \\\\ \n   \\bf{mRMR+BE}                       &   (199)  & 94.0 &  (165)  &  92.5      &  (47)  & 75.0         &  (176) & 90.8       & (84)   & 92.2     & 88.90    \\\\  \n   \n   \\bf{JMI+COBRA}                     &  (140)  & 92.6  &  (146)  & 92.2  &  (47)  & 75.0    &  (148)  & 90.4  &  (148)  & 91.4  & 88.30                        \\\\  \n   \\bf{JMI+FS}                        &  (150)  & 92.7  &  (142)  & 92.1  &  (48)  & 75.3    &  (148)  & 90.7  &  (145)  & 91.3  & 88.40               \\\\ \n   \\bf{JMI+BE}                        &  (150)  & 92.7  &  (142)  & 92.1  &  (48)  & 75.0    &  (144)  & 90.4  &  (134)  & 91.2  & 88.30                            \\\\  \\hline\\hline\n   \n                                                                       &   \\multicolumn{11}{c}{ \\bf{IAD Dataset}} \\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\\n   \\bf{mRMR+COBRA}                    & (165) & 96.5            &  (140) & 96.1      &  (28) & 96.4           &  (160) & 97.2      &   (68) & 97.1   \t  & 96.64         \\\\ \n   \\bf{mRMR+FS}                       & (109) & 96.2            &  (127) & 95.8      &  (127) & 96.7          &  (25)  & 97.0      &   (52) & 97.2\t  & 96.58               \\\\ \n   \\bf{mRMR+BE}                       & (22)  & 96.3            &  (163) & 95.9       &  (121) & 96.1          &  (109) & 97.2      &  (148) & 97.4  \t  & 96.58             \\\\  \n   \n   \\bf{JMI+COBRA}                     & (112)  & 96.3  &  (4)  & 96.3  &  (9)  & 96.3  &  (57)  & 97.3  &  (140)  & 100& 97.24                        \\\\ \n   \\bf{JMI+FS}                        & (9)  & 96.2    &  (4)  & 96.2 &  (52)  & 96.4  &   (7)  & 96.8  &  (7)  & 97.8  & 96.68             \\\\  \n   \\bf{JMI+BE}                        & (4)  & 96.6    &  (17)  & 95.8  &  (79) & 96.3  &  (13) & 96.5  &  (10) & 97.2  & 96.48                            \\\\  \\hline\\hline\n   \n                                                                     &   \\multicolumn{11}{c}{ \\bf{MAD Dataset}}  \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                    &   (12)  & 83.2  &  (13)  & 60.4  &  (26)  & 80.5  &  (12)  & 88.0 &  (11)  & 62.2    & \\textbf{74.81}        \\\\  \n   \\bf{mRMR+FS}                       &   (32)  & 55.3  &  (5)   & 55.5  &  (12)  & 58.2  &  (49)  & 57.3  &  (5)    & 52.7  & \\textbf{55.82}               \\\\  \n   \\bf{mRMR+BE}                       &   (14)  & 55.3  &  (11)  & 54.8  &  (31)  & 57.3  &  (26)  & 56.4  &  (115)  & 48.6  & 54.50             \\\\    \n   \n   \\bf{JMI+COBRA}                     &  (13)  & 82.5  &  (12)  & 60.7  &  (40)  & 80.7  &  (13)  & 87.6  &  (4)   & 61.1  & 74.54                         \\\\  \n   \\bf{JMI+FS}                        &  (13)  & 82.5  &  (12)  & 60.7  &  (58)  & 80.5  &  (13)  & 87.9  &  (19)  & 59.2  & 74.20               \\\\  \n   \\bf{JMI+BE}                        &  (13)  & 82.5  &  (12)  & 60.7  &  (58)  & 80.5  &  (13)  & 87.3  &  (20)  & 60.1  & 74.25                               \\\\  \\hline\\hline\n   \n\t\t\t\t\t\t\t\t    &   \\multicolumn{11}{c}{ \\bf{LNG Dataset}} \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\  \n   \\bf{mRMR+COBRA}                    &  (23) & 75.0  &  (28) & 96.9   &  (13) & 71.8  &  (28) & 68.7  &  (27) & 71.8  & \\textbf{76.87}         \\\\  \n   \\bf{mRMR+FS}                       &  (7)  & 81.2  &  (5)  & 68.7   &  (5)  & 71.8  &  (5)  & 75.0  &  (6)  & 71.8  & 73.75                 \\\\  \n   \\bf{mRMR+BE}                       &  (7)  & 81.2  &  (4)  & 68.7   &  (4)  & 71.8  &  (4)  & 75.0  &  (4)  & 75.0  & 74.37            \\\\ \n   \n   \\bf{JMI+COBRA}                     &  (7)  & 78.1       &   (6) & 71.8      &  (5)  & 71.8     &   (5)  & 75.0       &  (5) &  68.7 & 73.12                        \\\\  \n   \\bf{JMI+FS}                        &  (7)  & 78.1        &  (4) & 71.8       &  (4)  & 71.8     &  (8)  & 78.1       &  (5) &  68.7 & 73.75              \\\\ \n   \\bf{JMI+BE}                        &  (7)  & 78.1        &  (6) & 71.8       &  (5)  & 71.8     &  (6)  & 78.1       &  (6) &  71.8 & 74.37                              \\\\  \\hline\\hline\n   \n\t\t\t\t\t\t\t\t  &   \\multicolumn{11}{c}{ \\bf{ARR Dataset}} \t\\rule[-0.0ex]{0ex}{2.5ex}\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                    &   (45)  & 81.9  &  (48)  & 76.3  &  (30)  & 75.4  &  (43)  & 82.2  &  (57) & 72.9 & 77.75          \\\\\n   \\bf{mRMR+FS}                       &   (34)  & 81.3  &  (43)  & 76.1  &  (7)   & 78.3  &  (34)  & 81.3  &  (5)  & 75.7  & 78.56                \\\\\n   \\bf{mRMR+BE}                       &   (36)  & 81.6  &  (43)  & 76.3  &  (22)  & 78.0  &  (25)  & 82.9  &  (8)  & 76.1  & 79.02               \\\\ \n   \n   \\bf{JMI+COBRA}                     &   (26)  & 80.6  &  (51)  & 74.7  &  (15)  & 78.3  &  (51)  & 81.5  &  (13)  & 71.9       & 77.41                          \\\\  \n   \\bf{JMI+FS}                        &   (47)  & 74.3  &  (38)  & 73.5  &  (26)  & 76.9  &  (37)  & 79.2  &  (54)  & 70.0       & 74.80                \\\\ \n   \\bf{JMI+BE}                        &   (47)  & 74.3  &  (38)  & 73.5  &  (26)  & 76.9  &  (25)  & 80.0  &  (29)  & 68.6       & 74.66                               \\\\  \\hline\\hline\n   \n   \t\t\t\t\t\t\t\t    &   \\multicolumn{11}{c}{ \\bf{DEX Dataset}}  \\rule[-0.0ex]{0ex}{2.5ex}\t\t\t                                             \\\\ \n   \\bf{mRMR+COBRA}                    &    (3)  & 92.0    &  (131) & 86.3   &  (24)  & 80.7  &  (3)  & 93.0   &(3)   & 81.3            & 86.66         \\\\  \n   \\bf{mRMR+FS}                       &    (3)  & 90.3  &  (56)  & 87.0   &  (94)  & 80.3    &  (3)  & 92.0   &(3)   & 80.0            & 86.00               \\\\  \n   \\bf{mRMR+BE}                       &    (3)  & 90.0  &  (131)  & 87.3  &  (18)  & 80.3    &  (3)  & 91.6   &(99)  & 78.6            & 85.53              \\\\  \n   \n   \\bf{JMI+COBRA}                     &  (88)   & 91.6  &  (13)   & 83.0  &  (12)  & 80.3    &  (3)    & 94.0  &  (3)   & 81.0          &   86.00                      \\\\ \n   \\bf{JMI+FS}                        &  (149)  & 91.0  &  (129)  & 87.6  &  (95)  & 80.3    &  (119)  & 92.3  &  (94)  & 80.6          &   86.40              \\\\  \n   \\bf{JMI+BE}                        &  (149)  & 90.0  &  (128)  & 87.3  &  (22)  & 81.0    &  (146)  & 92.0  &  (138) &  78.0         &   85.60                             \\\\  \\hline\n   \n  \\end{tabular}\n      \\hfill{}\n       \\caption{Comparison of COBRA with the greedy search methods over different datasets. For each classifier and combination of search method and measure function, the values in parentheses is the number of selected features and the second value is the classification accuracy. The last column reports the average of the classification accuracies for each algorithm.}\n       \\label{ssp-be}\n  \\end{table*} \n \\end{center}\n\\vspace{0mm}\n\n  A simple algorithm listed in Table \\ref{algo} is employed to search for the optimal value of the subset cardinality $P$, where $P$ ranges over a set $\\mathbb{P}$ of admissible values. In the worst case, $\\mathbb{P} =\\{1,\\dots,N\\}$. \n\n  Table \\ref{ssp-be} shows the results obtained for the 8 datasets and 5 classifiers. Friedman test with the corresponding Wilcoxon-Nemenyi post-hoc analysis was used to compare the different algorithms. However, looking at the classification rates even before running the Friedman tests on them reveals a few interesting points which are marked in bold font.\n  \n  First, on the small size datasets (NCI, DBW and LNG), mRMR+COBRA consistently shows higher performance than other algorithms. The reason lies in the fact that the \\textit{similarity ratio} of the feature sets selected by COBRA is lower than BE or FS feature sets. The \\textit{similarity ratio} $S_i$ is defined as the number of features in the intersection of $i$th and $i{+}1$th feature sets divided by the cardinality of the $i$th feature set. From its definition it is clear that for BE and FS this ratio is always equal to 1. However, because of the randomization step this ratio may widely vary for COBRA. That is, COBRA generates \nquite diverse feature sets. Some of these feature sets have relatively low scores as compared with BE or FS sets. However, since for small datasets the estimated mutual information terms are highly inaccurate, features that rank low with our noisy measure function may in fact be better for classification. The average of the similarity ratios of 50 subsequent feature sets ($\\frac{1}{50}\\sum_{i=5}^{55} S_i$) have been reported for 4 datasets in Table \\ref{s-ratio}. As seen, for NCI the averaged similarity ratio is significantly smaller than 1 while for CNA which is a relatively larger dataset, it is almost constant and equal to 1.\n  \n\\begin{table}\n     \\addtolength{\\tabcolsep}{2mm}\n   \\hfill{}\n    \\begin{tabular}{l|| c c c c} \n      \\hline \\rule[-0.0ex]{0ex}{2.5ex}  \n   \\textbf{Datasets}                     &  \\bf{NCI}            &     \\bf{DBW}  &\\bf{IAD}              &  \\bf{CNA}                          \\\\ \\hline  \\hline  \\rule[-0.0ex]{0ex}{2.5ex} \n   \\bf{S-ratio}                          &   0.7717             &     0.8929    & 0.9266               &   0.9976 \\\\ \\hline \n     \\end{tabular}\n      \\hfill{}\n       \\caption{The average (over 50 similarity ratios) similarity ratio for 4 datasets.}\n       \\label{s-ratio}\n\\end{table} \n\n  \\begin{figure}\n\\resizebox{8cm}{!}{\n  \n\\begin{tikzpicture}[x=1pt,y=1pt]\n\\definecolor[named]{fillColor}{rgb}{1.00,1.00,1.00}\n\\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (505.89,505.89);\n\\begin{scope}\n\\path[clip] ( 49.20, 61.20) rectangle (480.69,456.69);\n\\definecolor[named]{fillColor}{rgb}{0.00,1.00,0.00}\n\n\\path[fill=fillColor] ( 78.50,267.80) --\n\t(185.04,267.80) --\n\t(185.04,423.22) --\n\t( 78.50,423.22) --\n\tcycle;\n\\definecolor[named]{drawColor}{rgb}{0.00,0.00,0.00}\n\n\\path[draw=drawColor,line width= 1.2pt,line join=round] ( 78.50,346.62) -- (185.04,346.62);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (131.77,210.02) -- (131.77,267.80);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (131.77,442.04) -- (131.77,423.22);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (105.13,210.02) -- (158.40,210.02);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (105.13,442.04) -- (158.40,442.04);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 78.50,267.80) --\n\t(185.04,267.80) --\n\t(185.04,423.22) --\n\t( 78.50,423.22) --\n\t( 78.50,267.80);\n\\definecolor[named]{fillColor}{rgb}{0.75,0.75,0.75}\n\n\\path[fill=fillColor] (211.67,242.34) --\n\t(318.22,242.34) --\n\t(318.22,283.52) --\n\t(211.67,283.52) --\n\tcycle;\n\n\\path[draw=drawColor,line width= 1.2pt,line join=round] (211.67,258.95) -- (318.22,258.95);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (264.94,238.80) -- (264.94,242.34);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (264.94,324.70) -- (264.94,283.52);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (238.31,238.80) -- (291.58,238.80);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (238.31,324.70) -- (291.58,324.70);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (211.67,242.34) --\n\t(318.22,242.34) --\n\t(318.22,283.52) --\n\t(211.67,283.52) --\n\t(211.67,242.34);\n\\definecolor[named]{fillColor}{rgb}{0.00,1.00,0.00}\n\n\\path[fill=fillColor] (344.85, 81.83) --\n\t(451.39, 81.83) --\n\t(451.39,250.31) --\n\t(344.85,250.31) --\n\tcycle;\n\n\\path[draw=drawColor,line width= 1.2pt,line join=round] (344.85,134.96) -- (451.39,134.96);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (398.12, 75.85) -- (398.12, 81.83);\n\n\\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] (398.12,302.12) -- (398.12,250.31);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (371.49, 75.85) -- (424.76, 75.85);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (371.49,302.12) -- (424.76,302.12);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (344.85, 81.83) --\n\t(451.39, 81.83) --\n\t(451.39,250.31) --\n\t(344.85,250.31) --\n\t(344.85, 81.83);\n\\end{scope}\n\\begin{scope}\n\\path[clip] (  0.00,  0.00) rectangle (505.89,505.89);\n\\definecolor[named]{drawColor}{rgb}{0.00,0.00,0.00}\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (131.77, 61.20) -- (398.12, 61.20);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (131.77, 61.20) -- (131.77, 55.20);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (264.94, 61.20) -- (264.94, 55.20);\n\n\\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (398.12, 61.20) -- (398.12, 55.20);\n\n\\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.80] at (131.77, 39.60) {CO - 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80.41);\n\\end{scope}\n\\end{tikzpicture}\n\n   }\n\\caption{Comparing the search strategies for mRMR. Results of the post-hoc tests for each classifier.}\n\\label{mrmr-cl}\n\\end{figure}\n  The second interesting point is with respect to the Madelon dataset. As can be seen, mRMR with greedy search algorithms perform poorly on this dataset. Several authors have already utilized this dataset to compare their proposed criterion with mRMR and arrived at the conclusion that mRMR cannot handle highly correlated features, as in Madelon dataset. However, surprisingly the performance of the mRMR+COBRA is as good as JMI on this dataset meaning that it is not the criterion but the search method that has difficulty to deal with highly correlated  features. Thus, any conclusion with respect to the quality of a measure has to be drawn carefully since, as in this case, the effect of the non optimum search method can be decisive.\n   \n  To discover the statistically meaningful differences between the algorithms, we applied the Friedman test following with Wilcoxon-Nemenyi post-hoc analysis, as suggested in \\cite{hollander:99}, on the average accuracies (the last column of Table \\ref{ssp-be}). Note that since we have 8 datasets, there are 8 independent measurements available for each algorithm. The results of this test for mRMR based algorithms have been depicted in Figure \\ref{mrmr-mean}. In all box plots, CO stands for COBRA algorithm. Each box plot compares a pair of the algorithms. The green box plots represent the existence of a significant difference between the corresponding algorithms. The adjusted p-values for each pair of algorithms have also been reported in Figure \\ref{mrmr-mean}. The smaller the p-value, the stronger the evidence against the null hypothesis. As can be seen, COBRA shows meaningful superiority over both greedy algorithms. However, if we set the significance level at $p=0.05$, only FS rejects the null \nhypothesis and shows a meaningful difference with COBRA.\n  \n  The same test was run for each classifier and its results can be found in Figure \\ref{mrmr-cl}. While three of the classifiers show some differences between FS and COBRA, neither of them reveal any meaningful difference between BE and COBRA. At this point, the least we can conclude is that independent of the classification algorithm we choose, it is a good chance that FS performs worse than COBRA. \n  \n  For JMI, however, the performances of all algorithms are comparable and with only 8 datasets it is difficult to draw any conclusion. Thus, the Wilcoxon-Nemenyi test results for JMI is not shown here because of the lack of space.\n \\vspace{-10mm}\n    \n\\begin{center}\n  \\begin{table*}\n    \\addtolength{\\tabcolsep}{4.4mm}\n    \\hfill{}\n    \\begin{tabular}{l|| c c c c c} \n      \\hline \\rule[-0.0ex]{0ex}{2.5ex}\n   \\textbf{Datasets}                     & \\bf{MAD}         & \\bf{NCI}              &     \\bf{IAD}               &  \\bf{ARR}             &  \\bf{CNA}                         \\\\ \\hline\\hline \\Ib \n   \\bf{mRMR+COBRA}                       &  74.81$\\pmS$0.65  & 73.67$\\pmS$2.41        &      96.64$\\pmS$0.16        &   77.75$\\pmS$1.03      & 88.91$\\pmS$0.31                      \\\\  \n   \\bf{mRMR+QPFS}                        &  71.44$\\pmS$0.57  & 71.00$\\pmS$1.84        &      95.02$\\pmS$0.21        &   78.73$\\pmS$0.84      & 86.93$\\pmS$0.45                       \\\\ \n   \\bf{mRMR+SOSS}                        &  71.36$\\pmS$0.53  & 72.65$\\pmS$2.13        &      96.64$\\pmS$0.28        &   79.86$\\pmS$1.18      & 85.43$\\pmS$0.49                        \\\\  \\hline \\hline \\rule[-0.0ex]{0ex}{2.5ex}\n                                       \n   \\bf{Time COBRA}                       &  175${\\,+\\,}$24  &368${\\,+\\,}$341          &     540${\\,+\\,}$121        &  6${\\,+\\,}$14          & 120${\\,+\\,}$50                      \\\\  \n   \\bf{Time QPFS}                        &    11            &  180                    &      202                   &    1                   & 25                       \\\\ \n   \\bf{Time SOSS}                        &  175${\\,+\\,}$5   &  368${\\,+\\,}$27         &     540${\\,+\\,}$12          &  6${\\,+\\,}$4          & 120${\\,+\\,}$7                        \\\\  \\hline\n                                             \n      \\end{tabular}\n      \\hfill{}\n       \\caption{Comparison of COBRA with QPFS and SOSS over 5 datasets. Average classification rates and their standard deviations are reported in the top three rows of the table. In the next three rows, the computational times in second are shown where the first value for COBRA and SOSS is for calculating the mutual information matrix and the second value is the time needed to solve the optimization problems.}\n       \\label{qpfs}\n  \\end{table*} \n \\end{center}\n  In the next experiment COBRA is compared with two other convex programming based feature selection algorithms, SOSS \\cite{naghibi:13} and QPFS \\cite{rod:10}. Both SOSS and QPFS employ quadratic programing techniques to maximize a score function. SOSS, however, uses an instance of randomized rounding to generate the set-membership binary values while QPFS ranks the features based on their scores (achieved from solving the convex problem) and therefore, sidesteps the difficulties of generating binary values. Note that both COBRA and SOSS first need to calculate the mutual information matrix $\\mathbf{Q}$. Once it is calculated, they can solve their corresponding convex optimization problems for different values of $P$. The first 3 rows of Table \\ref{qpfs} report the average (over 5 classifiers) classification accuracies of these three algorithms and the standard deviation of these mean accuracies (calculated over the cross-validation folds). In the next three rows of the table, the computational times of each \nalgorithm for a single run (in second) are shown, i.e., the amount of time needed to select a feature set with (given) $P$ features. The reported times for COBRA and SOSS consist of two values. The first value is the time needed to calculate the mutual information matrix $\\mathbf{Q}$ and the second value is the amount of time needed to solve the corresponding convex optimization problem. All the values were measured on a PC with an Intel Core i7 CPU. As seen, QPFS is significantly faster than COBRA and SOSS. This computational superiority, however, seems to come at the expense of lower classification accuracy. For large datasets such as IAD, CNA and MAD, the  Nystr\\\"{o}m approximation used in QPFS to cast the problem into a lower dimensional subspace does not yield a precise enough approximation and results in lower classification accuracies. An important remark to interpret these results is that, for NCI dataset (in all the experiments) we first filtered out the features with the low mutual information \nvalues with the class label and only kept 2000 informative features (similarly for DEX and DBW datasets). Thus, the dimension is 2000 and not 9703 as mentioned in Table \\ref{datasets}. \n\n  The generalization power of the COBRA algorithm over different classifiers is another important issue to test. As can be observed in Table \\ref{ssp-be}, the number of selected features varies quite markedly from one classifier to another. However, based on our experiments, the optimum feature set of any of the classifiers, usually (for large enough datasets) achieves a near-optimal accuracy in conjunction with other classifiers as well. This is shown in Table \\ref{general} for 4 classifiers and 3 datasets. The COBRA features of the LDA classifier in Table \\ref{ssp-be} is used here to train other classifiers. Table \\ref{general} lists the accuracies obtained by using the LDA features and the optimal features, repeated from Table \\ref{ssp-be}. Unlike the CNA and  IAD datasets, a significant accuracy reduction can be observed in the case of ARR data which has substantially less training data than CNA and IAD. It suggests that for small size datasets, a feature \nselection scheme should take the induction algorithm into account since the learning algorithm is sensitive to small changes of the feature set.       \n\\begin{table}[ht]\n  \\addtolength{\\tabcolsep}{1.3mm}\n\n\\begin{tabular}{l|c||c c c c } \n\\hline \n  \\multicolumn{2}{c||}{\\bf{Classifiers}}            & \\textbf{SVM}         & \\textbf{CART}                   & \\textbf{RF}          & \\textbf{NN}              \\rule[-0.0ex]{0ex}{2.5ex}          \\\\ \\hline\\hline\n                                 \n   \\multirow{2}{*}{\\textbf{ARR}}       &   LDA feat.     &78.4         & 73.7                   & 77.1        & 68.00                           \\\\  \\cline{2-2}\n                              &   Optimum       &81.9         & 75.4                   & 82.2        & 72.9                                        \\\\  \\hline\\hline\n  \\multirow{2}{*}{\\textbf{CNA}}        &  LDA feat.      & 92.6        & 75.0                   & 90.5        & 91.1                       \\\\  \\cline{2-2}\n                              & Optimum         & 94.0        & 75.0                   & 90.8        & 92.0                        \\\\  \\hline\\hline\n                              \n  \\multirow{2}{*}{\\textbf{IAD}}        &  LDA feat      & 95.8        & 96.0                   & 97.2        & 96.3                          \\\\  \\cline{2-2}\n                              &  Optimum       & 96.5        & 96.4                   & 97.2        & 97.1                           \\\\  \\hline\n\n   \\end{tabular}\n\\caption{The performance of the classification algorithms when trained with COBRA features optimized for the LDA classifier. This table shows the generalization power of the COBRA features on the classifiers.}\n\\label{general}\n\\end{table}\n\\vspace{-5mm}\n\\section{Conclusion}\n\\label{con}\nA convex based parallel search strategy for feature selection, COBRA, was suggested in this work. Its approximation ratio was derived and compared with the approximation ratio of the backward elimination method. It was experimentally shown that COBRA outperforms sequential search methods especially in the case of sparse data.  Moreover, we presented two series expansions for mutual information, and showed that most mutual information based score functions in the literature including mRMR and MIFS are truncated approximations of these expansions. Furthermore, the underlying connection between MIFS and the Kirwood approximation was explored, and it was shown that by adopting the class conditional independence assumption and the Kirkwood approximation for $Pr(\\mathbf{X})$, mutual information reduces to the MIFS criterion. \n\\section{Acknowledgments}\nThis work has partly been supported by Swiss National Science Foundation (SNSF).\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $f=(f_1,\\ldots,f_p)$ be a holomorphic mapping from the unit ball $\\mathbb{B} \\subset \\mathbb{C}^n$ to $\\mathbb{C}^p$.\nIf $p=1$ and $f$ is a monomial it is elementary to show, e.g., by integrations by parts or by a Taylor expansion,\nthat the principal value current\n$\\varphi \\mapsto \\lim_{\\epsilon \\to 0}\\int_{|f|^2>\\epsilon}\\varphi\/f$, $\\varphi \\in \\mathscr{D}_{n,n}(\\mathbb{B})$, exists and defines a \n$(0,0)$-current $1\/f$. From Hironaka's theorem it then follows that such limits exist \nin general for $p=1$ and also that $\\mathbb{B}$ may \nbe replaced by a complex space, \\cite{HL}. \nThe $\\bar{\\partial}$-image, $\\bar{\\partial} (1\/f)$, is the residue current of $f$. It has the useful property that its \nannihilator ideal is equal to the principal ideal $\\langle f \\rangle$ and by Stokes' theorem it is \ngiven by $\\varphi \\mapsto \\lim_{\\epsilon \\to 0}\\int_{|f|^2=\\epsilon}\\varphi\/f$, $\\varphi \\in \\mathscr{D}_{n,n-1}(\\mathbb{B})$. \nFor $p>1$, Coleff-Herrera, \\cite{CH}, proposed the following generalization. Define the residue integral\n\\begin{equation}\\label{CHresint}\nI_f^{\\varphi}(\\epsilon)=\\int_{T(\\epsilon)}\\varphi\/(f_1\\cdots f_p), \\quad \\varphi \\in \\mathscr{D}_{n,n-p},\n\\end{equation}   \nwhere $T(\\epsilon)=\\cap_1^p\\{|f_j|^2=\\epsilon_j\\}$ is oriented as the distinguished boundary of the corresponding \npolyhedron. Coleff-Herrera showed that if $\\epsilon \\to 0$ {\\em along an admissible path}, which means that\n$\\epsilon\\to 0$ inside $(0,\\infty)^p$\nin such a way that $\\epsilon_j\/\\epsilon_{j+1}^k\\to 0$\nfor all $k\\in \\mathbb{N}$, then the limit of $I_f^{\\varphi}(\\epsilon)$ exists and defines a $(0,p)$-current.\nWe call this current the Coleff-Herrera product associated to $f$. \n\nIf $f$ defines a complete intersection, Coleff-Herrera\nshowed that the Coleff-Herrera product associated to $f$ depends only in an alternating fashion on the \nordering of $f$, (see \\cite{JebHs} and \\cite{HasamArkiv} for stronger results implying this).\nMoreover, in the complete intersection case, \nit has turned out that the Coleff-Herrera product is a good notion of a multivariable residue\nof $f$. In particular, its annihilator ideal is equal to $\\langle f \\rangle$, (\\cite{DS}, \\cite{Pdr}).\nMoreover, the Coleff-Herrera product is the ``minimal'' extension to a current of Grothendieck's cohomological residue\n(see, e.g., \\cite{Pdr} for the definition)\nin the sense that it is annihilated by anti-holomorphic functions vanishing on $\\{f=0\\}$. \nThis is also related to the fact that the Coleff-Herrera product \nhas the so called {\\em Standard Extension Property}, SEP, which means that\nit has no mass concentrated on the singular part of $\\{f=0\\}$, (see, e.g., \\cite{JebAbel} and \\cite{CH}).\n\nThe Coleff-Herrera product in the complete intersection case has also found \napplications, e.g., to explicit division-interpolation formulas and Brian\\c con-Skoda type results \n(\\cite{MatsAM}, \\cite{BGVY}), explicit versions\nof the fundamental principle (\\cite{BePa}), the $\\bar{\\partial}$-equation on complex spaces (\\cite{AS}, \\cite{HePo}), \nexplicit Green currents in arithmetic intersection theory \\cite{BYJAM}, etc. \nHowever, if $f$ does not define a complete intersection, then the Coleff-Herrera product does not depend in\nany simple way on the ordering of $f$. For example, the Coleff-Herrera product associated to \n$(zw,z)$ is zero while the Coleff-Herrera product associated to \n$(z,zw)$ is equal to $\\bar{\\partial} (1\/z^2)\\wedge \\bar{\\partial} (1\/w)$, which is to be interpreted simply as a tensor product. \nNevertheless, it has turned out that the Coleff-Herrera product indeed describes interesting phenomena \nalso in the non-complete intersection case. \nFor instance, the St\\\"{u}ckrad-Vogel intersection algorithm in excess intersection theory can \nbe described by the Coleff-Herrera method of multiplying currents; \nthis is shown in a forthcoming paper by M.\\ Andersson, the second author, E.\\ Wulcan, and A.\\ Yger. \n\n\\smallskip\n\nIn this paper, we describe various approaches to Coleff-Herrera type products, both in general and in the \ncomplete intersection case. More precisely, we study\nto which extent (exterior) products of natural regularizations of the individual \ncurrents $1\/f_j$ and $\\bar{\\partial} (1\/f_j)$ \nyield regularizations of the corresponding Coleff-Herrera products. Moreover, we do this globally on a \ncomplex space and we also consider products of Cauchy-Fantappi\\`e-Leray type currents.\n\n\\smallskip\n\nLet $Z$ be a complex space of pure dimension $n$, let $E_1^*,\\ldots,E_p^*$ be hermitian holomorphic\nline bundles over $Z$, and let $f_j$ be a holomorphic section of $E_j^*$. \nThen $1\/f_j$ is a meromorphic section of the dual bundle $E_j$ and we define it as a current on $Z$ by\n\\begin{equation*}\n\\frac{1}{f_j}:= \\frac{|f_j|^{2\\lambda_j}}{f_j} \\, \\Big|_{\\lambda_j = 0}.\n\\end{equation*}\nThe right-hand side is a well-defined and analytic current-valued function\nfor $\\mathfrak{Re}\\, \\lambda_j \\gg 1$ and  \nwe will see in Section~\\ref{formulering}\nthat it has a current-valued analytic continuation to $\\lambda_j = 0$; it is well-known and easy to show\nthat this definition of the current $1\/f$ indeed coincides with the principal value definition of \nHerrera-Lieberman described above, (see, e.g.,\nLemma \\ref{pmmultlemma} below).\nThe residue current of $f_j$ is then defined as the $\\bar{\\partial}$-image of $1\/f_j$, i.e.,\n\\begin{equation*}\n\\bar{\\partial} \\frac{1}{f_j} = \\frac{\\bar{\\partial} |f_j|^{2\\lambda_j}}{f_j} \\, \\Big|_{\\lambda_j = 0}.\n\\end{equation*}\nIt follows that $\\bar{\\partial} (1\/f_j)$ coincides with \nthe limit of the residue integral associated to $f_j$. A conceptual reason for this equality is that \n$\\bar{\\partial} |f_j|^{2\\lambda_j}\/f_j$ in fact is the Mellin transform of the residue integral. \nThe technique of using analytic continuation in \nresidue current theory has its roots in the work of Atiyah, \\cite{At},\nand Bernstein-Gel'fand, \\cite{BG}, and has turned out to be very useful. In the context of residue currents \nit has been developed by several authors, e.g., Barlet-Maire, \\cite{BaMa}, Yger, \\cite{Y},\nBerenstein-Gay-Yger,\\cite{BGY}, Passare-Tsikh, \\cite{PTCanad},\nand recently by the second author in \\cite{HasamArkiv}.\n\nWe use this technique to define products of the residue currents \n$\\bar{\\partial} (1\/f_j)$ by defining recursively\n\\begin{equation} \\label{eqproddef}\n\\bar{\\partial} \\frac{1}{f_k} \\wedge \\dots \\wedge \\bar{\\partial} \\frac{1}{f_1} := \\frac{\\bar{\\partial} |f_k|^{2\\lambda_k}}{f_k} \n\\wedge \\bar{\\partial} \\frac{1}{f_{k-1}} \\wedge \\dots \\wedge \\bar{\\partial} \\frac{1}{f_1} \\, \\Big|_{\\lambda_k = 0}.\n\\end{equation}\nThe existence of the right-hand side of \\eqref{eqproddef} follows from the fact that this type of products\nof residue currents are pseudomeromorphic, see Section \\ref{formulering} for details.\n\n\\smallskip\n\nA natural way of regularizing the current $\\bar{\\partial} (1\/f_j)$ inspired by Passare, \\cite{PCrelle}, is as\n$\\bar{\\partial} \\chi(|f_j|^2\/\\epsilon)\/f_j$, where $\\chi$ is a smooth approximation of ${\\bf 1}_{[1,\\infty)}$,\n(the characteristic function of $[1,\\infty)$). This regularization corresponds to a mild average of the \nresidue integral $I_{f_j}^{\\varphi}(\\epsilon)$ and again, it is well-known and easy to show that \n$\\lim_{\\epsilon\\to 0}\\bar{\\partial} \\chi(|f_j|^2\/\\epsilon) \/f_j =\\bar{\\partial} (1\/f_j)$, (see, e.g., Lemma \\ref{pmmultlemma}). \nWe define the regularized residue integral associated to $f$ by\n\\begin{equation}\\label{resint}\n\\mathcal{I}_f^{\\varphi}(\\epsilon)=\n\\int_Z \\frac{\\bar{\\partial} \\chi_p^{\\epsilon}}{f_p}\\wedge \\cdots \\wedge \\frac{\\bar{\\partial} \\chi_1^{\\epsilon}}{f_1} \\wedge \\varphi,\n\\end{equation}\nwhere $\\chi_j^{\\epsilon}=\\chi(|f_j|^2\/\\epsilon_j)$ and $\\varphi$ is a test form with values in \n$\\Lambda (E_1^*\\oplus\\cdots \\oplus E_p^*)$. \nNotice that if $\\chi={\\bf 1}_{[1,\\infty)}$ (and the $E_j$ are trivial), then \\eqref{resint} becomes \\eqref{CHresint}.\n\n\\begin{theorem}\\label{sats1}\nWith the notation of Definition \\ref{limitdef}, we have\n\\begin{equation*}\n\\bar{\\partial} \\frac{1}{f_p}\\wedge \\cdots \\wedge \\bar{\\partial} \\frac{1}{f_1}.\\, \\varphi = \\lim_{\\epsilon_1 \\ll \\cdots \\ll \\epsilon_p\\to 0}\n\\mathcal{I}_f^{\\varphi}(\\epsilon).\n\\end{equation*}\nMoreover, if we allow $\\chi={\\bf 1}_{[1,\\infty)}$ in \\eqref{resint}, then the limit of \\eqref{resint}\nalong any admissible path also equals\n$\\bar{\\partial} (1\/f_p)\\wedge \\cdots \\wedge \\bar{\\partial} (1\/f_1).\\, \\varphi$.\n\\end{theorem}\n\n\\begin{remark}\nThe requirement that $\\epsilon \\to 0$ along an admissible path if $\\chi={\\bf 1}_{[1,\\infty)}$ is not \nreally necessary. However,\nsince it is not completely obvious what, e.g., $(\\bar{\\partial} \\chi(|f_2|^2\/\\epsilon_2))\/f_2 \\wedge \\bar{\\partial} (1\/f_1) $ means if \n$\\chi={\\bf 1}_{[1,\\infty)}$ we prefer to add the requirement.\n\\end{remark}\n\nTheorem~\\ref{sats1} thus says that the Coleff-Herrera product associated to $f$ equals the successively defined \ncurrent in \\eqref{eqproddef} and also that it can be smoothly regularized by \\eqref{resint}.\nIt also follows that \n$\\bar{\\partial} (1\/f_p)\\wedge \\cdots \\wedge \\bar{\\partial} (1\/f_1)=\\lim_{\\epsilon\\to 0}(\\bar{\\partial}\\chi(|f_p|^2\/\\epsilon)\/f_p)\\wedge \\bar{\\partial} (1\/f_{p-1})\\wedge \\cdots \\wedge \\bar{\\partial} (1\/f_1)$. \n\nTheorem~\\ref{sats1} is a special case of Theorem~\\ref{main} below, where we show a similar result for \nproducts of Cauchy-Fantappi\\`e-Leray type currents, which can be thought of as analogues of the currents\n$1\/f_j$ and $\\bar{\\partial} (1\/f_j)$ in the case when the bundles $E_j$ have ranks $>1$. Products of such currents \nwere first defined in \\cite{WArkiv}, but the definition of the products given there is in general not the same as our.\nThe proof of Theorem~\\ref{sats1} (and Theorem~\\ref{main})\nis very similar to the proof of Proposition 1 in \\cite{PCrelle} but it needs to be modified in our case\nsince extra technical difficulties arise when the metrics of the bundles $E_j$ are not supposed to be trivial.\n\n\\smallskip\n\nTo give some intuition for Theorem~\\ref{sats1}, we recall Bj\\\"{o}rk's realization of the Coleff-Herrera \nproduct, see, e.g., \\cite{JebAbel}, \\cite{MatsAAFS}, or \\cite{JebHs} for proofs.\nGiven a holomorphic function $f_1$ in $\\mathbb{B}\\subset \\mathbb{C}^n$, \nthere exists a holomorphic differential operator $Q$, a holomorphic function $h$, and a holomorphic $(n-1)$-form\n$dX$ such that\n\\begin{equation}\\label{DLrep}\n\\bar{\\partial} \\frac{1}{f_1}.\\, \\varphi\\wedge dz=\\lim_{\\epsilon\\to 0}\\int_{\\{f_1=0\\}} \\chi\\left(\\frac{|h|^2}{\\epsilon}\\right)\n\\frac{Q(\\varphi)\\wedge dX}{h}, \\quad \n\\varphi \\in \\mathscr{D}_{0,n-1}(\\mathbb{B}),\n\\end{equation} \nwhere $\\chi={\\bf 1}_{[1,\\infty)}$ or a smooth approximation thereof.\nThis representation makes it possible to define the principal value of $1\/f_2$ {\\em on} the current $\\bar{\\partial} (1\/f_1)$.\nIn fact, \n$\\lim_{\\epsilon \\to 0}\\int_{\\{f_1=0\\}}\\chi(|hf_2|^2\/\\epsilon)Q(\\varphi\/f_2)\\wedge dX\/h$ exists and defines a current\n$(1\/f_2)\\bar{\\partial} (1\/f_1)$.\nThe $\\bar{\\partial}$-image of this current is then well-defined and, (e.g., by Theorem~\\ref{sats1}), it\nequals $\\bar{\\partial}(1\/f_2)\\wedge \\bar{\\partial}(1\/f_1)$.\nBut $\\bar{\\partial}(1\/f_2)\\wedge \\bar{\\partial}(1\/f_1)$ has a representation similar to \\eqref{DLrep} and one can\nthus define the principal value of $1\/f_3$ on $\\bar{\\partial}(1\/f_2)\\wedge \\bar{\\partial}(1\/f_1)$, and so on. \nIntuitively, this procedure corresponds to first letting $\\epsilon_1\\to 0$ in \\eqref{CHresint} (or \\eqref{resint}),\nthen letting $\\epsilon_2\\to 0$ etc.\n\n\n\n\\smallskip\n\nWe now turn to the case that the sections $f_j$ define a complete intersection on $Z$. \nThen we know that the Coleff-Herrera product\nis anti-commutative but we have in fact the following result generalizing Theorem 1 in\n\\cite{JebHs}.\n\n\\begin{theorem}\\label{jebhs+}\nAssume that $f_1,\\ldots,f_p$ define a complete intersection. Then\n\\begin{equation*}\n\\big|\\mathcal{I}_f^{\\varphi}(\\epsilon)-\\bar{\\partial} \\frac{1}{f_p}\\wedge \\cdots \\wedge \\bar{\\partial} \\frac{1}{f_1}.\\ \\varphi \\big|\\leq\nC\\|\\varphi\\|_{C^M}(\\epsilon_1^{\\omega_1}+\\cdots +\\epsilon_p^{\\omega_p}),\n\\end{equation*}\nwhere the positive constants $M$ and $\\omega_j$ only depend on $f$, $Z$, and $\\supp \\varphi$\nwhile $C$ also depends on the $C^M$-norm of the $\\chi$-functions appearing in the regularized residue integral\n$\\mathcal{I}_f^{\\varphi}$, \\eqref{resint}.  \n\\end{theorem}\n\nWe also have a similar statement for products of Cauchy-Fantappi\\`e-Leray currents, Theorem \\ref{epsilon-main} below.\nNotice that it is necessary that the $\\chi$-functions are smooth; if $p\\geq 2$ and $\\chi={\\bf 1}_{[1,\\infty)}$\nin \\eqref{resint}, \nthen the corresponding statement is false in view of the examples \nby Passare-Tsikh, \\cite{PTex}, and Bj\\\"{o}rk, \\cite{JebAbel}.\n\nWe also have a generalization of Theorem 1 in \\cite{HasamArkiv} to products of Cauchy-Fantappi\\`e-Leray\ncurrents, namely our Theorem~\\ref{lambda-main} in Section~\\ref{formulering}. \nIn the special case of line bundles discussed here,\nTheorem \\ref{lambda-main} becomes the following Theorem~\\ref{lambda-budget}. \nHowever, Theorem \\ref{lambda-budget} also\nfollows from the results in \\cite{HasamArkiv}; the presence of non-trivial metrics does not cause any additional \nproblems.\n\\begin{theorem}\\label{lambda-budget}\nAssume that $f_1,\\dots,f_p$ define a complete intersection. If $\\varphi$ is a test form, then\n\\begin{equation*}\n    \\Gamma^\\varphi(\\lambda) := \\int \\frac{\\bar{\\partial} |f_p|^{2\\lambda_p}}{f_p} \\wedge \\dots \n\\wedge \\frac{\\bar{\\partial} |f_1|^{2\\lambda_1}}{f_1} \\wedge \\varphi\n\\end{equation*}\nhas an analytic continuation to a neighborhood of the half space $\\{ \\mathfrak{Re}\\, \\lambda_j \\geq 0 \\}$.\n\\end{theorem} \n\nIn the classical case, $\\Gamma^{\\varphi}(\\lambda)$ is the iterated Mellin transform of the residue integral \n\\eqref{CHresint} and it \nis well known that it has a meromorphic continuation to $\\mathbb{C}^p$ that is analytic in\n$\\cap_1^p\\{\\mathfrak{Re}\\, \\lambda_j>0\\}$; (this is also true in the non-complete intersection case). \nThe analyticity of $\\Gamma^{\\varphi}(\\lambda)$ in a neighborhood of \n$0$ when $p=2$ was proved by Berenstein-Yger (see, e.g., \\cite{BGVY}).  \n\n\\smallskip \n\nIn Section~\\ref{formulering}, we give the necessary background and the general formulations of our results.\nSection \\ref{bevis} contains the proof of Theorems \\ref{sats1} and \\ref{main}.\nThe proof of Theorems \\ref{jebhs+}, \\ref{epsilon-main}, and \\ref{lambda-main} is the content of \nSection~\\ref{bevis2}; the crucial part is Lemma~\\ref{divlemma} which enables us to effectively use the assumption \nabout complete intersection. \n\n\\section{Formulation of the general results}\\label{formulering}\nLet $E_1^*,\\ldots,E_q^*$ be holomorphic hermitian vector bundles \nover a reduced complex space $Z$ of pure dimension $n$. \nThe metrics are supposed to be smooth in the following sense.\nWe say that $\\varphi$ is a smooth $(p,q)$-form on $Z$ if $\\varphi$ is smooth on $Z_{reg}$, and\nfor a neighborhood of any $p\\in Z$, there is a smooth \n$(p,q)$-form $\\tilde{\\varphi}$ in an ambient complex manifold such that the pullback of $\\tilde{\\varphi}$\nto $Z_{reg}$ coincides with $\\varphi\\lvert_{Z_{reg}}$ close to $p$.\nThe $(p,q)$-test forms on $Z$, $\\mathscr{D}_{p,q}(Z)$, are defined as the smooth compactly supported \n$(p,q)$-forms (with a suitable topology) and the $(p,q)$-currents on $Z$, $\\mathscr{D}'_{p,q}(Z)$, is the\ndual of $\\mathscr{D}_{n-p,n-q}(Z)$; see, e.g., \\cite{larkang} for a more thorough discussion.\n\nWe recall from \\cite{AWCrelle} the definition of {\\em pseudomeromorphic} currents, $\\mathcal{PM}$.\nA current is pseudomeromorphic if it is a (locally finite) sum of push-forwards of elementary currents \nunder modifications of $Z$. A current, $T$, is elementary if it is a current on $\\mathbb{C}^n_x$\nof the form\n\\begin{equation}\\label{T}\nT=\\frac{1}{x^{\\alpha}}\\bigwedge_{\\beta_j\\neq 0}\\bar{\\partial} \\frac{1}{x_j^{\\beta_j}}\\wedge \\vartheta,\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are multiindices with disjoint supports and $\\vartheta$ is a smooth compactly \nsupported (possibly bundle valued) form. \n(We are abusing notation slightly; $\\Lambda_{\\beta_j\\neq 0}\\bar{\\partial} (1\/x_j^{\\beta_j})$\nis only defined up to a sign.) Elementary currents are thus merely tensor products of one-variable\nprincipal value currents $1\/x_i^{\\alpha_i}$ and $\\bar{\\partial}$-images of such (modulo smooth forms).\n\n\\begin{lemma} \\label{pmmultlemma}\n    Let $f$ be a holomorphic function, and let $T \\in \\mathcal{PM}(Z)$.\n    If $\\tilde{f}$ is a holomorphic function such that $\\{ \\tilde{f} = 0 \\} = \\{ f = 0 \\}$\n    and $v$ is a smooth non-zero function, then $(|\\tilde{f} v|^{2\\lambda}\/f) T$ and $(\\bar{\\partial} |\\tilde{f}v|^{2\\lambda}\/f)\\wedge T$ have\n    current-valued analytic continuations to $\\lambda = 0$ and the values at $\\lambda = 0$ are pseudomeromorphic\n    and independent of the choices of $\\tilde{f}$ and $v$.\n    Moreover, if $\\chi={\\bf 1}_{[1,\\infty)}$, or a smooth approximation thereof, then\n    \\begin{equation} \\label{eqepsilonlambda}\n        \\left.\\frac{|\\tilde{f} v|^{2\\lambda}}{f} T\\right|_{\\lambda = 0} = \\lim_{\\epsilon \\to 0^+} \\frac{\\chi^\\epsilon}{f} T\n        \\quad\\text{and}\\quad \n        \\left.\\frac{\\bar{\\partial} |\\tilde{f} v|^{2\\lambda}}{f} \\wedge T\\right|_{\\lambda = 0} = \\lim_{\\epsilon \\to 0^+} \\frac{\\bar{\\partial}\\chi^\\epsilon}{f}\\wedge T,\n    \\end{equation}\nwhere $\\chi^\\epsilon = \\chi(|\\tilde{f}v|^2\/\\epsilon)$.\n\\end{lemma}\n\n\\begin{proof}\n    The first part is essentially Proposition 2.1 in \\cite{AWCrelle}, except that there, $Z$ is a complex manifold, $\\tilde{f} = f$\n    and $v \\equiv 1$. However, with suitable resolutions of singularities, the proof in \\cite{AWCrelle} goes through in the\n    same way in our situation, as long as we observe that in $\\mathbb{C}$\n    \\begin{equation*}\n        \\frac{|x^{\\alpha'}v|^{2\\lambda}}{x^\\alpha} \\frac{1}{x^\\beta} \\quad \\text{and} \\quad\n        \\frac{|x^{\\alpha'}v|^{2\\lambda}}{x^\\alpha} \\bar{\\partial} \\frac{1}{x^\\beta}\n    \\end{equation*}\n    have analytic continuations to $\\lambda = 0$, and the values at $\\lambda = 0$ are $1\/x^{\\alpha + \\beta}$\n    and $0$ respectively, independently of $\\alpha'$ and $v$, as long as $\\alpha' > 0$ and $v \\neq 0$\n    (and similarly with $\\bar{\\partial} |x^{\\alpha'}v|^{2\\lambda}\/x^\\alpha$).\n\n    By Leibniz rule, it is enough to consider the first equality in \\eqref{eqepsilonlambda}, since if we have proved the first equality, then\n    \\begin{align*}\n        & \\lim_{\\epsilon \\to 0} \\frac{\\bar{\\partial} \\chi^\\epsilon}{f}\\wedge T = \\lim_{\\epsilon \\to 0} \\bar{\\partial} \\left( \\frac{\\chi^\\epsilon}{f} T  \\right)\n        - \\frac{\\chi^\\epsilon}{f} \\bar{\\partial} T \\\\\n        &= \\left.\\left(\\bar{\\partial}\\left(\\frac{|\\tilde{f}v|^{2\\lambda}}{f} T\\right) -\n        \\frac{|\\tilde{f}v|^{2\\lambda}}{f} \\bar{\\partial} T\\right)\\right|_{\\lambda = 0}\n        = \\left.\\frac{\\bar{\\partial} |\\tilde{f}v|^{2\\lambda}}{f} \\wedge T \\right|_{\\lambda = 0}.\n    \\end{align*}\n    To prove the first equality in \\eqref{eqepsilonlambda}, we observe first that in the same way as in the first part, we\n    can assume that $f = x^{\\gamma} u$ and $\\tilde{f} = x^{\\tilde{\\gamma}} \\tilde{u}$,\n    where $u$ and $\\tilde{u}$ are non-zero holomorphic functions.\n    Since $T$ is a sum of push-forwards of elementary currents,\n    we can assume that $T$ is of the form \\eqref{T}. \n    Note that if $\\supp \\gamma \\cap \\supp \\beta \\neq \\emptyset$, then\n    $(|\\tilde{f} v|^{2\\lambda}\/f) T = 0$ for $\\mathfrak{Re}\\, \\lambda \\gg 1$ and $(\\chi(|\\tilde{f}v|^2\/\\epsilon) \/f) T = 0$\n    for $\\epsilon > 0$, since $\\supp T \\subseteq \\{ x_i = 0, i \\in \\supp \\beta \\}$.\n    Thus, we can assume that $\\supp \\gamma \\cap \\supp \\beta = \\emptyset$. By a smooth (but non-holomorphic) change of variables,\n    as in Section~\\ref{bevis} (equations \\eqref{varbyte}), we can assume that $|\\tilde{u} v|^2 \\equiv 1$.\n    Thus, since $(|x^{\\tilde{\\gamma}}|^{2\\lambda} \/ x^\\gamma) (1\/x^\\alpha)$, $(\\chi(|x^{\\tilde{\\gamma}}|^2\/\\epsilon)\/x^\\gamma) (1\/x^\\alpha)$\n    depend on variables disjoint from the ones that $\\wedge_{\\beta_i \\neq 0} \\bar{\\partial} (1\/x_i^{\\beta_i})$ depends on,\n    it is enough to prove that\n    \\begin{equation*}\n        \\left.\\frac{|x^{\\tilde{\\gamma}}|^{2\\lambda}}{x^\\gamma} \\frac{1}{x^\\alpha}\\right|_{\\lambda = 0} =\n        \\lim_{\\epsilon \\to 0} \\frac{\\chi(|x^{\\tilde{\\gamma}}|^2\/\\epsilon)}{x^\\gamma} \\frac{1}{x^\\alpha},\n    \\end{equation*}\n    which is Lemma 2 in \\cite{JebHs}.\n\\end{proof}\n\nLet $f_j$ be a holomorphic section of $E_j^*$, $j=1,\\ldots,q$, and let $s_j$ be the section of $E_j$\nwith pointwise minimal norm such that $f_j \\cdot s_j=|f_j|^2$. Outside $\\{f_j=0\\}$, define\n\\begin{equation*}\n    u^j_k = \\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}}.\n\\end{equation*}\nIt is easily seen that if $f_j = f_j^0 f_j'$, where $f_j^0$ is a holomorphic function and $f_j'$ is a non-vanishing section,\nthen $u^j_k = (1\/f_j^0)^k (u')^j_k$, where $(u')^j_k$ is smooth across\n$\\{ f_j = 0 \\}$.\nWe let\n\\begin{equation}\\label{Udef}\nU^j=\\sum_{k=1}^{\\infty} \\left.|\\tilde{f}_j|^{2\\lambda} u^j_k \\right|_{\\lambda=0},\n\\end{equation}\nwhere $\\tilde{f}_j$ is any holomorphic section of $E_j^*$ such that $\\{\\tilde{f}_j=0\\}=\\{f_j=0\\}$.\nThe existence of the analytic continuation is a local statement, so we can assume that \n$f_j = \\sum f_{j,k} \\mathfrak{e}_{j,k}^*$,\nwhere $\\mathfrak{e}_{j,k}^*$ is a local holomorphic frame for $E_j^*$. After principalization\nwe can assume that the ideal $\\langle f_{j,1},\\dots,f_{j,k_j} \\rangle$ is generated by, e.g., $f_{j,0}$.\nBy the representation $u^j_k = (1\/f_{j,0})^k (u')^j_k$,\nthe existence of the analytic continuation of $U^j$ in \\eqref{Udef} then follows from Lemma \\ref{pmmultlemma}.\nLet $U^j_k$ denote the term of $U^j$ that takes values in $\\Lambda^kE_j$; $U^j_k$ is thus\na $(0,k-1)$-current with values in $\\Lambda^kE_j$. Let $\\delta_{f_j}$ denote interior multiplication\nwith $f_j$ and put $\\nabla_{f_j}=\\delta_{f_j}-\\bar{\\partial}$; it is not hard to verify that \n$\\nabla_{f_j}U=1$ outside $f_j=0$. \nWe define the Cauchy-Fantappi\\`e-Leray type residue current, $R^j$, of $f_j$ by $R^j=1-\\nabla_{f_j}U^j$.\nOne readily checks that \n\\begin{eqnarray}\\label{residydef}\nR^j &=& R^j_0+\\sum_{k=1}^{\\infty}R^j_k \\\\\n&=& (1-|\\tilde{f}_j|^{2\\lambda})|_{\\lambda=0}+\\sum_{k=1}^{\\infty}\n\\left.\\bar{\\partial}|\\tilde{f}_j|^{2\\lambda}\\wedge\\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}}\\right|_{\\lambda=0}, \\nonumber\n\\end{eqnarray} \nwhere, as above, $\\tilde{f}_j$ is a holomorphic section such that $\\{\\tilde{f}_j=0\\}=\\{f_j=0\\}$.\n\n\\begin{remark}\nNotice that if $E_j$ has rank $1$, then $U_j$ simply equals $1\/f_j$ and \n$R^j=1-\\nabla_{f_j} (1\/f_j)=1-f_j\\cdot (1\/f_j)+\\bar{\\partial} (1\/f_j)=\\bar{\\partial} (1\/f_j)$.\n\\end{remark}\n\nWe now define a non-commutative calculus for the currents $U^i_k$ and $R^j_{\\ell}$ recursively as follows.\n\\begin{definition}\\label{proddef}\nIf $T$ is a product of some $U^i_k$:s and $R^j_{\\ell}$:s, then we define\n\\begin{equation*}\n\\bullet \\,\\,\\, U^j_k\\wedge T=\n\\left.|\\tilde{f}_j|^{2\\lambda}\\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}}\\wedge T \\right|_{\\lambda=0} \\hspace{.6cm}\n\\end{equation*}\n\n\\begin{equation*}\n\\bullet \\,\\,\\, \\left.R^j_0\\wedge T=(1-|\\tilde{f}_j|^{2\\lambda})T \\right|_{\\lambda=0} \\hspace{2.5cm}\n\\end{equation*}\n\n\\begin{equation*}\n\\bullet \\,\\,\\, R^j_k\\wedge T=\n\\left.\\bar{\\partial}|\\tilde{f}_j|^{2\\lambda}\\wedge \\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}}\\wedge T \\right|_{\\lambda=0},\n\\end{equation*}\nwhere $\\tilde{f}_j$ is any holomorphic section of $E^*_j$ with $\\{\\tilde{f}_j=0\\}=\\{f_j=0\\}$.\n\n\\end{definition}\n\nNote first that $U^j$ and $R^j$ are pseudomeromorphic. Hence, in the same way as the analytic continuation\nin the definition of $U^j$ and $R^j$ exist, we see that the analytic continuations in the definition\nof the currents in Definition \\ref{proddef} exist and also are pseudomeromorphic.\n\n\\begin{remark}\nUnder assumptions about complete intersection, these products have the suggestive\ncommutation properties, e.g., if $\\textrm{codim}\\, \\{f_i=f_j=0\\}= \\rank E_i + \\rank E_j$,\nthen $R^i_k\\wedge R^j_{\\ell}=R^j_{\\ell}\\wedge R^i_k$, $R^i_k\\wedge U^j_{\\ell}=U^j_{\\ell}\\wedge R^i_k$, and\n$U^i_k\\wedge U^j_{\\ell}=-U^j_{\\ell}\\wedge U^i_k$, (see, e.g., \\cite{MatsAArk}). \nIn general, there are no simple relations.\nHowever, products involving only $U$:s are always anti-commutative.\n\\end{remark}\n\nNow, consider collections $U=\\{U^q_{k_q},\\ldots,U^{p+1}_{k_{p+1}}\\}$ and\n$R=\\{R^p_{k_p},\\ldots,R^1_{k_1}\\}$ and put\n$(P_q,\\ldots,P_1)=(U^q_{k_q},\\ldots,R^p_{k_p},\\ldots,R^1_{k_1})$. For a permutation $\\nu$ of \n$\\{1,\\ldots,q\\}$ we define\n\\begin{equation}\\label{URdef}\n(UR)^{\\nu}=P_{\\nu(q)}\\wedge \\cdots \\wedge P_{\\nu(1)}.\n\\end{equation}\nWe will describe various natural ways to regularize products of this kind. For $q=1$ we see \nfrom \\eqref{Udef} and \\eqref{residydef} that we have a natural $\\lambda$-regularization, $P^{\\lambda}_j$, \nof $P_j$ and from Definition \\ref{proddef} we have\n$(UR)^{\\nu}=P^{\\lambda_q}_{\\nu(q)}\\wedge \\cdots \\wedge P^{\\lambda_1}_{\\nu(1)}|_{\\lambda_1=0}\\cdots |_{\\lambda_q=0}$.\nWe have the following result that is proved in a forthcoming paper by M.\\ Andersson, the second author, E.\\ Wulcan, and \nA.\\ Yger.\n\n\\begin{theorem}\\label{aswy}\nLet $a_1>\\cdots >a_q>0$ be integers and $\\lambda$ a complex variable. Then we have\n\\begin{equation*}\n(UR)^{\\nu}=\\left.P^{\\lambda^{a_q}}_{\\nu(q)}\\wedge \\cdots \\wedge P^{\\lambda^{a_1}}_{\\nu(1)} \\right|_{\\lambda=0}.\n\\end{equation*}\n\\end{theorem}\n\nWe see that one does not need to put $\\lambda_1=0$ first, then $\\lambda_2=0$ etc., one just has to \nensure that $\\lambda_1$ tends to zero much faster than $\\lambda_2$ and so on.\nThe current $(UR)^{\\nu}$ can thus be obtained as the value at zero of a one-variable $\\zeta$-type\nfunction. From an algebraic point of view, this is desirable since one can derive functional equations\nand use Bernstein-Sato theory to study $(UR)^{\\nu}$. \n\n\\smallskip\n\nThere are also natural $\\epsilon$-regularizations of the currents\n$U^i_k$ and $R^j_{\\ell}$ inspired by \\cite{CH} and \\cite{PCrelle}. Let $\\chi={\\bf 1}_{[1,\\infty)}$, or \na smooth approximation thereof that is $0$ close to $0$ and $1$ close to $\\infty$. \nIt follows from \\cite{hasamJFA}, or after principalization from Lemma \\ref{pmmultlemma}, that \n\\begin{equation}\\label{Uepsilon}\nU^j_k=\\lim_{\\epsilon\\to 0^+}\\chi(|\\tilde{f}_j|^2\/\\epsilon) \\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}}.\n\\end{equation}\n\\begin{equation}\\label{Repsilon}\nR^j_k=\\lim_{\\epsilon\\to 0^+}\\bar{\\partial}\\chi(|\\tilde{f}_j|^2\/\\epsilon)\\wedge \n\\frac{s_j\\wedge (\\bar{\\partial} s_j)^{k-1}}{|f_j|^{2k}},\\,\\,\nk>0,\n\\end{equation}\nand similarly for $k=0$; as usual, $\\{\\tilde{f}_j=0\\}=\\{f_j=0\\}$. \nOf course, the limits are in the current sense and if $\\chi={\\bf 1}_{[1,\\infty)}$,\nthen $\\epsilon$ is supposed to be a regular value for $|f_j|^2$ and $\\bar{\\partial}\\chi(|f_j|^2\/\\epsilon)$ is to be \ninterpreted as integration over the manifold $|f_j|^2=\\epsilon$. We denote the regularizations \ngiven by \\eqref{Uepsilon} and \\eqref{Repsilon} by $P_j^{\\epsilon}$.\n\\begin{definition}\\label{limitdef}\nLet $\\vartheta$ be a function defined on $(0,\\infty)^q$. We let\n\\begin{equation*}\n\\lim_{\\epsilon_1 \\ll \\cdots \\ll \\epsilon_q\\to 0}\\vartheta(\\epsilon_1,\\ldots,\\epsilon_q)\n\\end{equation*}\ndenote the limit (if it exists and is well-defined) \nof $\\vartheta$ along any path $\\delta\\mapsto \\epsilon (\\delta)$ towards the origin\nsuch that for all $k\\in \\mathbb{N}$ and $j=2,\\ldots,q$ there are positive constants $C_{jk}$ such that\n$\\epsilon_{j-1}(\\delta) \\leq C_{jk}\\, \\epsilon_j^k(\\delta)$.\nHere, we extend the domain of definition of $\\vartheta$ to points $(0,\\dots,0,\\epsilon_{m+1},\\dots,\\epsilon_q)$,\nwhere $\\epsilon_{m+1},\\dots,\\epsilon_q > 0$, by defining recursively\n\\begin{equation*}\n    \\vartheta(0,\\dots,0,\\epsilon_{m+1},\\dots,\\epsilon_q) = \\lim_{\\epsilon_{m} \\to 0} \\vartheta(0,\\dots,0,\\epsilon_m,\\epsilon_{m+1},\\dots,\\epsilon_q),\n\\end{equation*}\nif the limits exist.\n\\end{definition}\n\n\\begin{remark}\\label{rem1}\nThe paths considered here are very similar to the admissible paths of Coleff-Herrera, but we also allow \npaths where, e.g., $\\epsilon_1$ attains the value $0$ before the other parameters tend to zero.\n\\end{remark}\n\n\nWe have the following analogue of Theorem \\ref{aswy}.\n\n\\begin{theorem}\\label{main}\nLet $U=\\{U^q_{k_q},\\ldots,U^{p+1}_{k_{p+1}}\\}$ and\n$R=\\{R^p_{k_p},\\ldots,R^1_{k_1}\\}$ be collections of currents defined in \\eqref{Udef} and \\eqref{residydef}.\nLet $\\nu$ be a permutation of $\\{1,\\ldots,q\\}$ and let $(UR)^{\\nu}$ be the product defined in \\eqref{URdef}.\nThen\n\\begin{equation*}\n(UR)^{\\nu}=\\lim_{\\epsilon_1 \\ll \\cdots \\ll \\epsilon_q \\to 0}P_{\\nu(q)}^{\\epsilon_q}\\wedge \\cdots \\wedge P_{\\nu(1)}^{\\epsilon_1},\n\\end{equation*}\nwhere, as above, $(P_q,\\ldots,P_1)=(U^q_{k_q},\\ldots,R^p_{k_p},\\ldots,R^1_{k_1})$ and \n$P_{\\nu(j)}^{\\epsilon_j}$ is an $\\epsilon$-regularization defined in \\eqref{Uepsilon} and \\eqref{Repsilon} of\n$P_{\\nu(j)}$. If $\\chi={\\bf 1}_{[1,\\infty)}$, we require that $\\epsilon\\to 0$ along an admissible path.\n\\end{theorem}\n\n\\smallskip\n\n\\subsection{The complete intersection case}\n\nNow assume that $f_1,\\ldots,f_q$ defines a complete intersection, i.e., that \n$\\textrm{codim}\\, \\{f_1=\\cdots =f_q=0\\}=e_1+\\cdots +e_q$, where $e_j=\\rank E_j$. Then we know that \nthe calculus defined in Definition \\ref{proddef} satisfies the suggestive commutation properties, but \nwe have in fact the following much stronger results.\n\n\\begin{theorem}\\label{epsilon-main}\nAssume that $f_1,\\ldots,f_q$ defines a complete intersection on $Z$, let\n$(P_1,\\ldots,P_q)=(R^1_{k_1},\\ldots,R^p_{k_p},U^{p+1}_{k_{p+1}},\\ldots,U^q_{k_q})$, and let\n$P^{\\epsilon_j}_{j}$ be an $\\epsilon$-regularization of $P_j$ defined by \\eqref{Uepsilon} and \\eqref{Repsilon}\nwith smooth $\\chi$-functions. Then we have\n\\begin{equation*}\n\\left| \\int_Z P^{\\epsilon_1}_1\\wedge \\cdots \\wedge P_q^{\\epsilon_q}\\wedge \\varphi -\nP_1\\wedge \\cdots \\wedge P_q .\\, \\varphi \\right| \\leq C \\|\\varphi\\|_M (\\epsilon_1^{\\omega} + \\dots + \\epsilon_q^\\omega),\n\\end{equation*}\nwhere $M$ and $\\omega$ only depend on $f_1,\\ldots, f_q$, $Z$, and $\\supp \\varphi$ while\n$C$ also depends on the $C^M$-norm of the $\\chi$-functions.\n\\end{theorem}\n\n\n\\begin{theorem}\\label{lambda-main}\nAssume that $f_1,\\ldots,f_q$ defines a complete intersection on $Z$, let\n$(P_1,\\ldots,P_q)=(R^1_{k_1},\\ldots,R^p_{k_p},U^{p+1}_{k_{p+1}},\\ldots,U^q_{k_q})$, and let\n$P^{\\lambda_j}_{j}$ be the $\\lambda$-regularization of $P_j$ given by \\eqref{Udef} and \\eqref{residydef}.\nThen the current valued function\n\\begin{equation*}\n\\lambda \\mapsto P_1^{\\lambda_1}\\wedge \\cdots \\wedge P_q^{\\lambda_q},\n\\end{equation*}\na priori defined for $\\mathfrak{Re}\\, \\lambda_j \\gg 1$, has an analytic continuation\nto a neighborhood of the half-space $\\cap_1^q \\{\\mathfrak{Re}\\, \\lambda_j \\geq 0\\}$.\n\\end{theorem}\n\n\\begin{remark}\nIn case the $E_j$:s are trivial with trivial metrics, Theorems \\ref{epsilon-main} and \\ref{lambda-main} follow\nquite easily from, respectively, Theorem 1 in \\cite{JebHs} and Theorem 1 in \\cite{HasamArkiv} by taking averages.\nAs an illustration, let \n$\\varepsilon_1,\\ldots,\\varepsilon_r$ be a nonsense basis and let $f_1,\\ldots,f_r$ be holomorphic functions. \nThen we can write $s=\\bar{f}\\cdot \\varepsilon$ and so \n$u_k=(\\bar{f}\\cdot \\varepsilon)\\wedge (d\\bar{f}\\cdot \\varepsilon)^{k-1}\/|f|^{2k}$.\nA standard computation shows that \n\\begin{equation*}\n\\int_{\\alpha \\in \\mathbb{CP}^{r-1}}\\frac{|\\alpha \\cdot f|^{2\\lambda}\\alpha \\cdot \\varepsilon}{\n(\\alpha\\cdot f)|\\alpha|^{2\\lambda}}dV\n=A(\\lambda)|f|^{2\\lambda}\\frac{\\bar{f}\\cdot \\varepsilon}{|f|^2},\n\\end{equation*}\nwhere $dV$ is the (normalized) Fubini-Study volume form and $A$ is holomorphic with $A(0)=1$. It follows that\n\\begin{equation*}\n\\int_{\\alpha_1,\\ldots,\\alpha_k\\in \\mathbb{CP}^{r-1}}\\bigwedge_1^k\\frac{\\bar{\\partial} |\\alpha_j\\cdot f|^{2\\lambda}}{\\alpha_j\\cdot f}\n\\wedge \\frac{\\alpha_j \\cdot \\varepsilon}{|\\alpha_j|^{2\\lambda}}dV(\\alpha_j)=\nA(\\lambda)^k\\bar{\\partial} (|f|^{2k\\lambda}u_k).\n\\end{equation*}\nElaborating this formula and using Theorem 1 in \\cite{HasamArkiv} one can show Theorem \\ref{lambda-main} in the case\nof trivial $E_j$:s with trivial metrics. The general case can probably also be handled in a similar manner but the \ncomputations become more involved and we prefer to give direct proofs.\n\\end{remark}\n\n\n\n\\section{Proof of Theorem \\ref{main}}\\label{bevis}\nWe start by making a Hironaka resolution of singularities, \\cite{Hiro}, of $Z$ such that the pre-image of\n$\\cup_j\\{f_j=0\\}$ has normal crossings. We then make further toric resolutions (e.g., as in \\cite{PTY}) \nsuch that, in local charts, the pullback of each $f_i$ is a monomial, $x^{\\alpha_i}$, times a \nnon-vanishing holomorphic tuple. One checks that the pullback of $P_j^{\\epsilon}$ is of one of the following\nforms: \n\\begin{equation*}\n\\frac{\\chi(|x^{\\tilde{\\alpha}}|^2\\xi\/\\epsilon)}{x^{\\alpha}}\\, \\vartheta, \\quad\n1-\\chi(|x^{\\tilde{\\alpha}}|^2\\xi\/\\epsilon),\\quad\n\\frac{\\bar{\\partial} \\chi(|x^{\\tilde{\\alpha}}|^2\\xi\/\\epsilon)}{x^{\\alpha}}\\wedge \\vartheta,\n\\end{equation*}\nwhere $\\xi$ is smooth and positive, $\\supp \\tilde{\\alpha}= \\supp \\alpha$,\nand $\\vartheta$ is a smooth bundle valued form; by localizing on the blow-up we may also suppose that \n$\\vartheta$ has as small support as we wish. If the $\\chi$-functions are smooth, the following special\ncase of Theorem \\ref{main} now immediately follows from Lemma \\ref{pmmultlemma}:\n\\begin{equation}\\label{eq1}\n(UR)^{\\nu}=\\lim_{\\epsilon_q\\to 0}\\cdots \\lim_{\\epsilon_1\\to 0}P^{\\epsilon_q}_{\\nu(q)}\\wedge \\cdots \\wedge \nP^{\\epsilon_1}_{\\nu(1)}.\n\\end{equation} \n\n\\bigskip\n\nFor smooth $\\chi$-functions we put\n\\begin{equation*}\n\\mathcal{I}(\\epsilon)=\n\\int \\frac{\\bar{\\partial} \\chi_1^{\\epsilon}\\wedge \\cdots \\wedge \\bar{\\partial} \\chi_p^{\\epsilon}\n\\chi_{p+1}^{\\epsilon}\\cdots \\chi_q^{\\epsilon}}{x^{\\alpha_1+\\cdots +\\alpha_p+\\cdots +\\alpha_{q'}}}\\wedge \\varphi,\n\\end{equation*}\nwhere $q'\\leq q$, $\\varphi$ is a smooth $(n,n-p)$-form with support close to the origin, and \n$\\chi_j^{\\epsilon}=\\chi (|x^{\\tilde{\\alpha_j}}|^2\\xi_j\/\\epsilon_j)$ for smooth positive $\\xi_j$.\nWe note that we may replace the $\\bar{\\partial}$ in $\\mathcal{I}(\\epsilon)$ by $d$ for bidegree reasons.\nIn case $\\chi={\\bf 1}_{[1,\\infty)}$ we denote the corresponding integral by $I(\\epsilon)$. \nWe also put $\\mathcal{I}^{\\nu}(\\epsilon_1,\\ldots,\\epsilon_q)=\\mathcal{I}(\\epsilon_{\\nu(1)},\\ldots,\\epsilon_{\\nu(q)})$\nand similarly for $I^{\\nu}$. In view of \\eqref{eq1}, the special case of Theorem \\ref{main}\nwhen the $\\chi$-functions are smooth will be proved if we can show that\n\\begin{equation}\\label{eq2}\n\\lim_{\\epsilon_1 \\ll \\cdots \\ll \\epsilon_q \\to 0}\\mathcal{I}^{\\nu}(\\epsilon)\n\\end{equation}\nexists. \nThe case with $\\chi={\\bf 1}_{[1,\\infty)}$ will then follow if we can show\n\\begin{equation}\\label{eq3}\n\\lim_{\\delta\\to 0} (\\mathcal{I}^{\\nu}(\\epsilon(\\delta))-I^{\\nu}(\\epsilon(\\delta)))=0,\n\\end{equation}\nwhere $\\delta \\mapsto \\epsilon(\\delta)$ is any admissible path.\n\nFor notational convenience, we will consider $\\mathcal{I}^\\nu(\\epsilon)$ (unless otherwise stated), but our \narguments apply just as well to $I^\\nu(\\epsilon)$ until we arrive at the integral \\eqref{Iepsilon}.\n\nDenote by $\\tilde{A}$ the $q\\times n$-matrix with rows $\\tilde{\\alpha}_i$. \nWe will first show that we can assume that $\\tilde{A}$ has full rank. The idea is the same as in \\cite{CH} and \\cite{PCrelle},\nhowever because of the paths along which our limits are taken, we have to modify the argument slightly.\nThe following lemma follows from the proof of Lemma III.12.1 in \\cite{TsikhBook}.\n\\begin{lemma}\\label{ranklemma}\nAssume that $\\alpha$ is a $q \\times n$-matrix with rows $\\alpha_i$ such that there exists $(v_1,\\dots,v_q) \\neq 0$ with $\\sum v_i\\alpha_i = 0$.\nLet $j = \\min \\{ i; v_i \\neq 0 \\}$. Then there exist constants $C,c > 0$ such that if $\\epsilon_{j} < C(\\epsilon_{j+1}\\dots\\epsilon_q)^c$,\nthen $\\chi(|x^{\\alpha_j}|^2\\xi_j\/\\epsilon_j) \\equiv 1$ and $\\bar{\\partial}\\chi(|x^{\\alpha_j}|^2\\xi_j\/\\epsilon_j) \\equiv 0$ for\nall $x \\in \\Delta \\cap \\{ |x^{\\alpha_i}|^2 \\geq C_i\\epsilon_i, i = j+1,\\dots,q \\}$, where $\\Delta$ is the unit polydisc.\n\\end{lemma}\nAssume that $\\tilde{A}$ does not have full rank, and let $v$ be a column vector such that $v^t \\tilde{A} = 0$. Since\n$(\\epsilon_1,\\dots,\\epsilon_q)$ is replaced by $(\\epsilon_{\\nu(1)},\\dots,\\epsilon_{\\nu(q)})$ in $\\mathcal{I}^\\nu(\\epsilon)$,\nwe choose instead $j_0$ such that $\\nu(j_0) \\leq \\nu(i)$ for all $i$ such that $v_i \\neq 0$.\nIf $j_0 \\leq p$, we let $\\widetilde{\\mathcal{I}}^\\nu(\\epsilon) = 0$, and if $j_0 \\geq p+1$, we let $\\widetilde{\\mathcal{I}}^\\nu(\\epsilon)$\nbe $\\mathcal{I}^\\nu(\\epsilon)$ but with $\\chi_{j_0}^\\epsilon$ replaced by $1$.\nIf $\\epsilon = \\epsilon(\\delta)$ is such that $\\epsilon_{\\nu(j_0)} > 0$, then $\\mathcal{I}^\\nu(\\epsilon)$ is a current\nacting on a test form with support on a set of the form\n\\begin{equation*}\n    \\Delta \\cap \\{ |x^{\\alpha_i}|^2 \\geq C_i\\epsilon_{\\nu(i)}; \\text{for all $i$ such that } \\nu(i) \\geq \\nu(j_0) \\}.\n\\end{equation*}\nIn particular, if $\\epsilon_{\\nu(j_0)}(\\delta)$ is sufficiently small compared to $(\\epsilon_{\\nu(j_0)+1}(\\delta),\\dots,$\n$\\epsilon_q(\\delta))$,\nthen by Lemma \\ref{ranklemma}, if $j_0 \\leq p$, the factor $\\bar{\\partial}\\chi_{j_0}^\\epsilon$ is identically $0$, and if $j_0 \\geq p+1$,\nthe factor $\\chi_{j_0}^\\epsilon$ is identically $1$\nand thus is equal to $\\widetilde{\\mathcal{I}}^\\nu(\\epsilon)$ for such $\\epsilon$.\nSimilarly, if $\\epsilon_{\\nu(j_0)} = 0$, we have that $\\mathcal{I}^\\nu(\\epsilon)$ is defined as a limit along $\\epsilon_{\\nu(j_0)} \\to 0$,\nwith $\\epsilon_{\\nu(j_0)+1},\\dots,\\epsilon_q$ fixed and in the limit we get again that for sufficiently small $\\epsilon_{\\nu(j_0)}$,\nwe can replace $\\mathcal{I}^\\nu(\\epsilon)$ by $\\widetilde{\\mathcal{I}}^\\nu(\\epsilon)$.\nThus we have\n\\begin{equation*}\n\\lim_{\\epsilon_1 \\ll \\dots \\ll \\epsilon_q \\to 0} \\mathcal{I}^\\nu(\\epsilon) = \n\\lim_{\\epsilon_1 \\ll \\dots \\ll \\epsilon_q \\to 0} \\widetilde{\\mathcal{I}}^\\nu(\\epsilon),\n\\end{equation*}\nand we have reduced to the case that $\\tilde{A}$ is a $(q-1)\\times n$-matrix of the same rank. We continue\nthis procedure until $\\tilde{A}$ has full rank.\n\n\\smallskip\n\nBy re-numbering the coordinates, we may suppose that the minor\n$A=(\\tilde{\\alpha}_{ij})_{1\\leq i,j\\leq q}$ of $\\tilde{A}$ is invertible and we put $A^{-1}=B=(b_{ij})$. \nWe now use complex notation to \nmake a non-holomorphic, but smooth change of variables:\n\\begin{equation}\\label{varbyte}\ny_1=x_1\\,\\xi^{b_1\/2},\\ldots, y_q=x_q\\,\\xi^{b_q\/2}, y_{q+1}=x_{q+1},\\ldots, y_n=x_n,\n\\end{equation}\n\\begin{equation*}\n\\hspace{.7cm} \\bar{y}_1=\\bar{x}_1\\,\\xi^{b_1\/2},\\ldots, \\bar{y}_q=\\bar{x}_q\\,\\xi^{b_q\/2}, \n\\bar{y}_{q+1}=\\bar{x}_{q+1},\\ldots, \\bar{y}_n=\\bar{x}_n,\n\\end{equation*}\nwhere $\\xi^{b_i\/2}=\\xi_1^{b_{i1}\/2}\\cdots \\xi_q^{b_{iq}\/2}$. One easily checks that \n$dy\\wedge d\\bar{y}=\\xi^{b_1}\\cdots \\xi^{b_q}\\,$ $dx\\wedge d\\bar{x}+ O(|x|)$, so \\eqref{varbyte} defines a\nsmooth change of variables between neighborhoods of the origin. \nA simple linear algebra computation then shows that\n$|x^{\\tilde{\\alpha_i}}|^2\\xi_i=|y^{\\tilde{\\alpha_i}}|^2$. Of course, this change of variables does not preserve \nbidegrees so $\\varphi(y)$ is merely a smooth compactly supported $(2n-p)$-form.\nWe thus have\n\\begin{equation}\\label{I1(y)}\n\\mathcal{I}^\\nu(\\epsilon)=\n\\int_{\\Delta} \\frac{d\\chi_1^{\\epsilon}\\wedge \\cdots \\wedge d \\chi_p^{\\epsilon}\n\\chi_{p+1}^{\\epsilon}\\cdots \\chi_q^{\\epsilon}}{y^{\\alpha_1+\\cdots +\\alpha_p+\\cdots +\\alpha_{q'}}}\\wedge \\varphi'(y),\n\\end{equation}\nwhere $\\chi_j^{\\epsilon}=\\chi(|y^{\\tilde{\\alpha}_j}|^2\/\\epsilon_{\\nu(j)})$ and \n$\\varphi'(y)=\\sum_{|I|+|J|=2n-p}\\psi_{IJ}dy_I\\wedge d\\bar{y}_J$. By linearity we may assume that the sum only consists\nof one term $\\varphi'(y)=\\psi dy_K\\wedge d\\bar{y}_L$, and by scaling, we may assume that \n$\\supp \\psi \\subseteq \\Delta$, $\\Delta$ being the unit polydisc.\nBy Lemma 2.4 in \\cite{CH}, we can write the function $\\psi$ as\n\\begin{equation}\\label{taylor}\n\\psi(y)=\\sum_{I+J<\\sum_1^{q'}\\alpha_j-{\\bf 1}}\\psi_{IJ}\\, y^I\\bar{y}^J +\n\\sum_{I+J=\\sum_1^{q'}\\alpha_j-{\\bf 1}}\\psi_{IJ}\\, y^I\\bar{y}^J,\n\\end{equation}\nwhere $a<b$ for tuples $a$ and $b$ means that $a_i<b_i$ for all $i$. In the decomposition \\eqref{taylor}\neach of the smooth functions $\\psi_{IJ}$ in the first sum on the left-hand side \nis independent of some variable. We now show that this implies that the first sum on the left-hand side of\n\\eqref{taylor} does not contribute to the integral \\eqref{I1(y)}. In case $\\varphi'(y)$ has bidegree $(n,n-p)$\nthis is a well-known fact but we must show it for an arbitrary $(2n-p)$-form.\n\nWe change to polar coordinates:\n\\begin{equation*}\ndy_K\\wedge d\\bar{y}_L=d(r_{K_1}e^{i\\theta_{K_1}})\\wedge \\cdots \\wedge d(r_{L_1}e^{-i\\theta_{L_1}})\\wedge \\cdots\n\\end{equation*}\nSince $\\chi_j^{\\epsilon}$ in \\eqref{I1(y)} is independent of $\\theta$, it follows that we must have full\ndegree $=n$ in $d\\theta$. The only terms in the expansion of $dy_K\\wedge d\\bar{y}_L$ above that will contribute\nto \\eqref{I1(y)} are therefore of the form\n\\begin{equation*}\nc\\, r_1\\cdots r_ne^{i\\theta \\cdot \\gamma}\\, dr_M\\wedge d\\theta,\n\\end{equation*}\nwhere $|M|=n-p$, $c$ is a constant, and $\\gamma$ is a multiindex with entries equal to $1$, $-1$, or $0$.\nSubstituting this and a term $\\psi_{IJ}y^I\\bar{y}^J=\\psi_{IJ}r^{I+J}e^{i\\theta \\cdot (I-J)}$ from \\eqref{taylor}\ninto \\eqref{I1(y)} gives rise to an ``inner'' $\\theta$-integral (by Fubini's theorem):\n\\begin{equation*}\n\\mathscr{J}_{IJ}(r)=\n\\int_{\\theta \\in [0,2\\pi)^n}\\psi_{IJ}(r,\\theta) \\ e^{i\\theta \\cdot (I-J-\\sum_1^{q'}\\alpha_j+\\gamma)}\\, d\\theta.\n\\end{equation*}\nIf $I+J<\\sum_1^{q'}\\alpha_j-{\\bf 1}$, then $I-J-\\sum_1^{q'}\\alpha_j+\\gamma <0$ and $\\psi_{IJ}$ is independent \nof some $y_j=r_je^{i\\theta_j}$. Integrating over $\\theta_j\\in [0,2\\pi)$ thus yields $\\mathscr{J}_{IJ}=0$\nif $I+J<\\sum_1^{q'}\\alpha_j-{\\bf 1}$. If instead $I+J=\\sum_1^{q'}\\alpha_j-{\\bf 1}$, then $\\mathscr{J}_{IJ}(r)$\nis smooth on $[0,\\infty)^n$.\n\nSumming up, we see that we can write \\eqref{I1(y)} as\n\\begin{equation} \\label{Iepsilon}\n\\mathcal{I}^\\nu(\\epsilon)=\n\\int_{r\\in (0,1)^n} d\\chi_1^{\\epsilon}\\wedge \\cdots \\wedge d \\chi_p^{\\epsilon}\n\\chi_{p+1}^{\\epsilon}\\cdots \\chi_q^{\\epsilon}\\, \\mathscr{J}(r)\\, dr_M,\n\\end{equation}\nwhere $\\chi_j^{\\epsilon}=\\chi(r^{2\\alpha_j}\/\\epsilon_{\\nu(j)})$, $\\mathscr{J}$ is smooth, and $|M|=n-p$.\n\nAfter these reductions, the integral \\eqref{Iepsilon} we arrive at is the same as equation (16) in \\cite{PCrelle},\nand we will use the fact proven there, that $\\lim_{\\delta \\to 0} \\mathcal{I}^\\nu(\\epsilon(\\delta))$ exists\nalong any admissible path $\\epsilon(\\delta)$, and is well-defined independently of the choice of\nadmissible path. (This is not exactly what is proven there, but the fact that if $b \\in \\mathbb{Q}^p$,\nthen $\\lim_{\\delta \\to 0} \\epsilon(\\delta)^b$ is either $0$ or $\\infty$ independently of the\nadmissible path chosen is the only addition we need to make for the argument to go through in our case.)\nUsing this, if we let $\\epsilon(\\delta)$ be any admissible path, we will show by induction over $q$ that\n\\begin{equation*}\n    \\lim_{\\epsilon_1 \\ll \\dots \\ll \\epsilon_q \\to 0 } \\mathcal{I}^\\nu(\\epsilon) = \\lim_{\\delta \\to 0} \\mathcal{I}^\\nu(\\epsilon(\\delta)).\n\\end{equation*}\nFor $q = 1$ this is trivially true, so we assume $q > 1$. Let $\\epsilon^k$ be any sequence satisfying the conditions\nin Definition~\\ref{limitdef}.\nConsider a fixed $k$, and let $m$ be such that $\\epsilon^k = (0,\\dots,0,\\epsilon^k_{m+1},\\dots,\\epsilon^k_{q})$ with $\\epsilon_{m+1}^k > 0$.\nLet $I_1 = \\nu^{-1}(\\{1,\\dots,m\\})\\cap\\{1,\\dots,p\\}$ and $I_2 = \\nu^{-1}(\\{1,\\dots,m\\})\\cap\\{p+1,\\dots,q\\}$.\nWe consider $\\epsilon^k_{m+1},\\dots,\\epsilon^k_{q}$ fixed in $\\mathcal{I}^\\nu(\\epsilon)$, and define\n\\begin{equation*}\n    \\mathcal{I}_k(\\epsilon_1,\\dots,\\epsilon_m) = \\int_{[0,1]^n} \\bigwedge_{i \\in I_1} d\\chi(r^{\\alpha_i}\/\\epsilon_{\\nu(i)})\n    \\prod_{i \\in I_2} \\chi(r^{\\alpha_i}\/\\epsilon_{\\nu(i)})\\mathscr{J}_k(r)dr_M,\n\\end{equation*}\noriginally defined on $(0,\\infty)^p$, but extended according to Definition \\ref{limitdef}, where\n\\begin{equation*}\n    \\mathscr{J}_k(r) = \\pm \\bigwedge_{i \\in \\{ 1,\\dots,p \\} \\setminus I_1} d\\chi(r^{\\alpha_i}\/\\epsilon^k_{\\nu(i)})\n    \\prod_{i \\in \\{ p+1,\\dots,q\\} \\setminus I_2} \\chi(r^{\\alpha_i}\/\\epsilon^k_{\\nu(i)})\\mathscr{J}(r)\n\\end{equation*}\n(where the sign is chosen such that $\\mathcal{I}_k(0) = \\mathcal{I}^\\nu(\\epsilon^k)$).\nSince $m < q$ and $\\mathscr{J}_k$ is smooth, we have by induction that\n\\begin{equation*}\n    \\mathcal{I}_k(0) = \\lim_{\\epsilon_m \\to 0} \\dots \\lim_{\\epsilon_1 \\to 0} \\mathcal{I}_k(\\epsilon_1,\\dots,\\epsilon_m) =\n    \\lim_{\\delta \\to 0} \\mathcal{I}_k(\\epsilon'(\\delta)),\n\\end{equation*}\nwhere $\\epsilon'(\\delta)$ is any admissible path, and the first equality follows by definition of $\\mathcal{I}_k(0)$.\nWe fix an admissible path $\\epsilon'(\\delta)$. For each $k$ we can choose $\\delta_k$ such that\nif $\\epsilon^{k'} = (\\epsilon'_1(\\delta_k),\\dots,\\epsilon'_m(\\delta_k))$, then\n$\\lim_{k \\to \\infty} (\\mathcal{I}_k(\\epsilon^{k'}) - \\mathcal{I}_k(0)) = 0$ and\nif $\\tilde{\\epsilon}^k = (\\epsilon^{k'},\\epsilon^k_{m+1},\\dots,\\epsilon^k_{q})$, then $\\tilde{\\epsilon}^k$ forms\na subsequence of an admissible path.\nSince $\\mathcal{I}_k(0) = \\mathcal{I}^\\nu(\\epsilon^k)$, and $\\mathcal{I}_k(\\epsilon^{k'}) = \\mathcal{I}^\\nu(\\tilde{\\epsilon}^k)$, we thus have\n\\begin{equation*}\n    \\lim_{k \\to \\infty} \\mathcal{I}^\\nu(\\epsilon^k) = \\lim_{k \\to \\infty} \\mathcal{I}^\\nu(\\tilde{\\epsilon}^k) =\n    \\lim_{\\delta \\to 0} \\mathcal{I}^\\nu(\\epsilon(\\delta))\n\\end{equation*}\nwhere the second equality follows from the existence and uniqueness of $\\mathcal{I}^\\nu(\\epsilon(\\delta))$\nalong any admissible path.\nHence we have shown that the limit in \\eqref{eq2} exists and is well-defined.\n\nFinally, if we start from \\eqref{Iepsilon}, as (23) in \\cite{PCrelle} shows, either\n\\begin{equation*}\n    \\lim_{\\epsilon_1 \\ll \\dots \\ll \\epsilon_q \\to 0} \\mathcal{I}^\\nu(\\epsilon) = \\pm \\int_{r_M \\in (0,1)^{n-p}} \n    \\mathscr{J}(0,r_M) dr_M,\n\\end{equation*}\nor the limit is $0$, depending only on $\\alpha$. If we consider $I^{\\nu}(\\epsilon)$ instead, we get the same limit,\nsee \\cite[p. 79--80]{TsikhBook}, and \\eqref{eq3} follows.\n\n\n\n\n\\section{Proof of Theorems \\ref{epsilon-main} and \\ref{lambda-main}}\\label{bevis2}\n\nRecall that $(P_1,\\ldots,P_q)=(R^1_{k_1},\\ldots,R^p_{k_p},U^{p+1}_{k_{p+1}},\\ldots,U^q_{k_q})$  and that \n$P_j^{\\epsilon_j}$ and $P_j^{\\lambda_j}$ are the $\\epsilon$-regularizations with smooth $\\chi$ \n(given by \\eqref{Uepsilon}, \\eqref{Repsilon}) \nand the $\\lambda$-regularizations (cf., \\eqref{Udef}, \\eqref{residydef}) respectively of $P_j$.  \nWe will consider the following two integrals:\n\\begin{equation*}\n\\mathcal{I}(\\epsilon)=\\int_Z\nP_1^{\\epsilon_1}\\wedge \\cdots \\wedge P_q^{\\epsilon_q}\\wedge \\varphi\n\\end{equation*}\n\\begin{equation*}\n\\Gamma(\\lambda) =\\int_Z P_1^{\\lambda_1}\\wedge \\cdots \\wedge P_q^{\\lambda_q}\\wedge \\varphi,  \n\\end{equation*}\nwhere $\\varphi$\nis a test form on $Z$, supported close to a point in $\\{f_1=\\cdots=f_q=0\\}$, \nof bidegree $(n,n-k_1-\\cdots- k_q +q-p)$ with values in $\\Lambda (E_1^*\\oplus \\cdots \\oplus E_q^*)$. \nIn the arguments below, we will assume for notational convenience that $\\tilde{f}_j=f_j$ \n(cf., e.g., \\eqref{Udef}); the modifications to the general case are straightforward.\n\nThe crucial parts of the proofs of Theorems \\ref{epsilon-main} and \\ref{lambda-main}\nare contained in the following propositions.\n\n\\begin{proposition}\\label{epsilonpropp}\nAssume that $f_1,\\ldots,f_q$ define a complete intersection. For $p<s\\leq q$ we have\n\\begin{equation*}\n\\big| \\mathcal{I}(\\epsilon)-\\mathcal{I}(\\epsilon_1,\\ldots,\\epsilon_{s-1},0,\\ldots,0)\\big|\\leq\nC\\|\\varphi\\|_M (\\epsilon_{s}^{\\omega}+\\cdots +\\epsilon_q^{\\omega}).\n\\end{equation*}\nNote that $\\mathcal{I}(\\epsilon_1,\\ldots,\\epsilon_{s-1},0,\\ldots,0)$ is well-defined; it is the action \nof $U^s_{k_s}\\wedge \\cdots \\wedge U^q_{k_q}$ on a smooth form. \n\\end{proposition}\n\n\\begin{proposition}\\label{lambdapropp}\nAssume that $f_1,\\ldots,f_q$ define a complete intersection.\nThen $\\Gamma(\\lambda)$ has a meromorphic continuation to all of $\\mathbb{C}^q$ and its only possible poles in a neighborhood \nof $\\cap_1^q\\{\\mathfrak{Re}\\, \\lambda_j\\geq 0\\}$ are along hyperplanes of the form\n$\\sum_{j=1}^p\\lambda_j \\alpha_j=0$,\nwhere $\\alpha_j\\in \\mathbb{N}$ and at least two $\\alpha_j$ are positive.\nIn particular, for $p=1$, $\\Gamma(\\lambda)$ is analytic in \na neighborhood of $\\cap_{1}^q\\{\\mathfrak{Re}\\, \\lambda_j\\geq 0\\}$.\n\\end{proposition} \n\nUsing that\n\\begin{equation}\\label{nablalikhet}\n\\bar{\\partial} |f_j|^{2\\lambda}\\wedge u^j_k=\\bar{\\partial} (|f_j|^{2\\lambda}u^j_k)-f_j\\cdot (|f_j|^{2\\lambda}u^j_{k+1}),\n\\end{equation}\nthe proof of Theorem \\ref{lambda-main} follows from Proposition \\ref{lambdapropp} in a similar way as\nTheorem 1 in \\cite{HasamArkiv} follows from Proposition 4 in \\cite{HasamArkiv}.\n\n\\bigskip\n\nWe indicate one way Proposition~\\ref{epsilonpropp} can be used to prove Theorem~\\ref{epsilon-main}.\nTo simplify notation somewhat, we let $R^j$ denote any $R^j_k$ and $R^j_{\\epsilon}$ denotes a smooth \n$\\epsilon$-regularization of $R^j$; $U^j$ and $U^j_{\\epsilon}$ are defined similarly.\n The uniformity in the estimate of Proposition \\ref{epsilonpropp} implies that\nwe have estimates of the form\n\\begin{equation}\\label{foljd}\n\\left|\\bigwedge_1^m R^j_{\\epsilon}\\wedge \\bigwedge_{m+1}^p R^j \\wedge \\bigwedge_{p+1}^q U^j_{\\epsilon}-\n\\bigwedge_1^m R^j_{\\epsilon}\\wedge \\bigwedge_{m+1}^p R^j \\wedge \\bigwedge_{p+1}^q U^j\\right|\\lesssim \n(\\epsilon_{p+1}^{\\omega}+\\cdots +\\epsilon_q^{\\omega}),\n\\end{equation}\nwhere, e.g., $R^{m+1}\\wedge \\cdots \\wedge R^p$ a priori is defined as a Coleff-Herrera product.\nWe prove (a slightly stronger result than) Theorem \\ref{epsilon-main} by induction over $p$.\nLet $R^*$ denote the Coleff-Herrera product of some $R^j$:s with $j>p$ and let $U^*$ and $U^*_{\\epsilon}$\ndenote the product of some $U^j$:s and $U^j_{\\epsilon}$:s respectively, also with $j>p$ but only $j$:s not\noccurring in $R^*$. We prove\n\\begin{equation*}\n\\big|R^1_{\\epsilon}\\wedge \\cdots \\wedge R^p_{\\epsilon}\\wedge R^*\\wedge U^*_{\\epsilon}-\nR^1\\wedge \\cdots \\wedge R^p\\wedge R^*\\wedge U^*\\big|\\lesssim \\epsilon^{\\omega},\n\\end{equation*} \ni.e., we prove Theorem \\ref{epsilon-main} {\\em on} the current $R^*$.\nThe induction start, $p=0$, follows immediately from \\eqref{foljd}. If we add and subtract\n$R^1_\\epsilon\\wedge \\dots \\wedge R^p_\\epsilon \\wedge R^*\\wedge U^*$, the induction step follows easily\nfrom \\eqref{nablalikhet} (construed in setting of $\\epsilon$-regularizations) and estimates like \\eqref{foljd}.\n \n\n\n\n\\begin{proof}[Proof of Propositions \\ref{epsilonpropp} and \\ref{lambdapropp}]\nWe may assume that $\\varphi$ has arbitrarily small support. Hence, we may assume that $Z$ is an analytic subset\nof a domain $\\Omega\\subseteq \\mathbb{C}^N$ and that all bundles are trivial,\nand thus make the identification $f_j=(f_{j1},\\ldots,f_{je_j})$, where $f_{ji}$ are holomorphic in $\\Omega$. \nWe choose a Hironaka resolution $\\hat{Z}\\rightarrow Z$ such that the pulled-back ideals $\\langle\\hat{f}_j\\rangle$\nare all principal, and moreover, so that in a fixed chart with coordinates $x$\non $\\hat{Z}$ (and after a possible re-numbering), \n$\\langle\\hat{f}_j\\rangle$ is generated by $\\hat{f}_{j1}$ and $\\hat{f}_{j1}=x^{\\alpha_j}h_j$, where $h_j$ is holomorphic \nand non-zero. We then have\n\\begin{equation*}\n|\\hat{f}_j|^2=|\\hat{f}_{j1}|^2\\xi_j, \\quad \\hat{u}^j_{k_j}=v^j\/\\hat{f}^{k_j}_{j1},\n\\end{equation*}\nwhere $\\xi_j$ is smooth and positive and $v^j$ is a smooth (bundle valued) form. We thus get\n\\begin{equation*}\n\\bar{\\partial} \\chi_j(|\\hat{f}_j|^2\/\\epsilon_j)=\n\\tilde{\\chi}_j(|\\hat{f}_j|^2\/\\epsilon_j)\\left(\\frac{d\\bar{\\hat{f}}_{j1}}{\\bar{\\hat{f}}_{j1}}+\n\\frac{\\bar{\\partial} \\xi_j}{\\xi_j}\\right),\n\\end{equation*}\nwhere $\\tilde{\\chi}_j(t)=t \\chi_j'(t)$, and \n\\begin{equation*}\n\\bar{\\partial} |\\hat{f}_j|^{2\\lambda_j}=\\lambda_j |\\hat{f}_j|^{2\\lambda_j} \\left(\\frac{d\\bar{\\hat{f}}_{j1}}{\\bar{\\hat{f}}_{j1}}+\n\\frac{\\bar{\\partial} \\xi_j}{\\xi_j}\\right).\n\\end{equation*}\nIt follows that $\\mathcal{I}(\\epsilon)$ and $\\Gamma(\\lambda)$ are finite sums of integrals which\nwe without loss of generality can assume to be of the form\n\\begin{equation}\\label{eq4}\n\\pm \\int_{\\mathbb{C}^n_x}\\prod_1^p \\tilde{\\chi}_j^{\\epsilon} \\prod_{p+1}^q\\chi_j^{\\epsilon}\n\\bigwedge_1^m \\frac{d\\bar{\\hat{f}}_{j1}}{\\bar{\\hat{f}}_{j1}}\\wedge \\bigwedge_{m+1}^p\n\\frac{\\bar{\\partial}\\xi_j}{\\xi_j}\\wedge \\bigwedge_1^q\\frac{v^j}{\\hat{f}^{k_j}_{j1}}\\wedge \\varphi \\rho,\n\\end{equation}\n\\begin{equation}\\label{eq4'}\n\\pm \\lambda_1\\cdots \\lambda_p \\int_{\\mathbb{C}^n_x}\n\\prod_1^q |\\hat{f}_j|^{2\\lambda_j}\n\\bigwedge_1^m \\frac{d\\bar{\\hat{f}}_{j1}}{\\bar{\\hat{f}}_{j1}}\\wedge \\bigwedge_{m+1}^p\n\\frac{\\bar{\\partial}\\xi_j}{\\xi_j}\\wedge \\bigwedge_1^q\\frac{v^j}{\\hat{f}^{k_j}_{j1}}\\wedge \\varphi \\rho,\n\\end{equation}\nwhere $\\rho$ is a cutoff function.\n\n\\smallskip\n\nRecall that $\\hat{f}_{j1}=x^{\\alpha_j}h_j$ and let $\\mu$ be the number of vectors in a maximal \nlinearly independent subset of $\\{\\alpha_1,\\ldots,\\alpha_m\\}$; say that \n$\\alpha_1,\\ldots,\\alpha_{\\mu}$ are linearly independent. We then can define new holomorphic coordinates\n(still denoted by $x$) so that $\\hat{f}_{j1}=x^{\\alpha_j}$, $j=1,\\ldots,\\mu$, see \\cite[p.~46]{PCrelle} for details.\nThen we get\n\\begin{eqnarray}\\label{hack}\n\\bigwedge_1^m d\\hat{f}_{j1} &=& \\bigwedge_1^{\\mu}dx^{\\alpha_j} \\wedge\n\\bigwedge_{\\mu+1}^m(x^{\\alpha_j}dh_j+h_jdx^{\\alpha_j}) \\\\\n&=& x^{\\sum_{\\mu+1}^m \\alpha_j}\\bigwedge_1^{\\mu}dx^{\\alpha_j}\\wedge \\bigwedge_{\\mu+1}^m dh_j, \\nonumber\n\\end{eqnarray}\nwhere the last equality follows because $dx^{\\alpha_1}\\wedge \\cdots \\wedge dx^{\\alpha_{\\mu}}\\wedge dx^{\\alpha_j}=0$,\n$\\mu+1\\leq j \\leq m$, since $\\alpha_1,\\ldots,\\alpha_{\\mu},\\alpha_j$ are linearly dependent.\nFrom the beginning we could also have assumed that $\\varphi=\\varphi_1\\wedge \\varphi_2$, where\n$\\varphi_1$ is an anti-holomorphic $(n-\\sum_1^q k_j +q-p)$-form and $\\varphi_2$ is a (bundle valued)\n$(n,0)$-test form on $Z$. We now define\n\\begin{equation*}\n\\Phi=\\bigwedge_{\\mu+1}^m \\frac{d\\bar{h}_j}{\\bar{h}_j}\\wedge \\bigwedge_{m+1}^p\\frac{\\bar{\\partial} \\xi_j}{\\xi_j}\\wedge\n\\bigwedge_1^q v^j \\wedge \\hat{\\varphi}_1.\n\\end{equation*}\nUsing \\eqref{hack} we can now write \\eqref{eq4} and \\eqref{eq4'} as\n\\begin{equation}\\label{eq5}\n\\pm \\int_{\\mathbb{C}^n_x}\\frac{\\prod_1^p \\tilde{\\chi}_j^{\\epsilon} \\prod_{p+1}^q\\chi_j^{\\epsilon}}{\\prod_1^q \\hat{f}^{k_j}_{j1}}\n\\frac{d\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{x}^{\\alpha_{\\mu}}}{\\bar{x}^{\\alpha_{\\mu}}}\n\\wedge \\Phi \\wedge \\hat{\\varphi}_2\\rho,\n\\end{equation}\n\\begin{equation}\\label{eq5'}\n\\pm \\lambda_1\\cdots \\lambda_p \\int_{\\mathbb{C}^n_x}\n\\frac{\\prod_1^q |\\hat{f}_j|^{2\\lambda_j}}{\\prod_1^q \\hat{f}^{k_j}_{j1}}\n\\frac{d\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{x}^{\\alpha_{\\mu}}}{\\bar{x}^{\\alpha_{\\mu}}}\n\\wedge \\Phi \\wedge \\hat{\\varphi}_2\\rho.\n\\end{equation}\n\n\\begin{lemma}\\label{divlemma}\nLet $\\mathcal{K}=\\{i;\\, x_i \\, \\big| \\, x^{\\alpha_j}, \\, \\textrm{some} \\,\\, p+1\\leq j \\leq q\\}$.\nFor any fixed $r\\in \\mathbb{N}$, one can replace $\\Phi$ in \\eqref{eq5} and \\eqref{eq5'} by\n\\begin{equation*}\n\\Phi':=\\Phi -\n\\sum_{J\\subseteq \\mathcal{K}}(-1)^{|J|}\\sum_{k_1,\\dots,k_{|J|} = 0}^{r+1}\n\\left.\\frac{\\partial^{|k|}\\Phi}{\\partial x_J^k}\\right|_{x_J=0}\n\\frac{x_J^k}{k!}\n\\end{equation*}\nwithout affecting the integrals. Moreover, for any $I\\subseteq \\mathcal{K}$, we have that \n$\\Phi'\\wedge \\Lambda_{i\\in I}(d\\bar{x}_i\/\\bar{x}_i)$ is $C^r$-smooth.\n\\end{lemma}\n\nWe replace $\\Phi$ by $\\Phi'$ in \\eqref{eq5} and \\eqref{eq5'} and we \nwrite $d=d_{\\mathcal{K}}+d_{\\mathcal{K}^c}$, where $d_{\\mathcal{K}}$ differentiates with respect to the \nvariables $x_i$, $\\bar{x}_i$ for $i\\in \\mathcal{K}$ and $d_{\\mathcal{K}^c}$ differentiates with respect to the rest.\nThen we can write \n$(d\\bar{x}^{\\alpha_1}\/\\bar{x}^{\\alpha_1})\\wedge \\cdots \\wedge (d\\bar{x}^{\\alpha_{\\mu}}\/\\bar{x}^{\\alpha_{\\mu}})\\wedge \\Phi'$ \nas a sum of terms, which we without loss of generality can assume to be of the form \n\\begin{equation*}\n\\frac{d_{\\mathcal{K}^c}\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}}\\wedge \\cdots \\wedge \n\\frac{d_{\\mathcal{K}^c}\\bar{x}^{\\alpha_{\\nu}}}{\\bar{x}^{\\alpha_{\\nu}}}\\wedge\n\\frac{d_{\\mathcal{K}}\\bar{x}^{\\alpha_{\\nu+1}}}{\\bar{x}^{\\alpha_{\\nu+1}}}\\wedge \\cdots \\wedge \n\\frac{d_{\\mathcal{K}}\\bar{x}^{\\alpha_{\\mu}}}{\\bar{x}^{\\alpha_{\\mu}}}\\wedge \\Phi'\n\\end{equation*}\n\\begin{equation*}\n=\\frac{d_{\\mathcal{K}^c}\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}}\\wedge \\cdots \\wedge \n\\frac{d_{\\mathcal{K}^c}\\bar{x}^{\\alpha_{\\nu}}}{\\bar{x}^{\\alpha_{\\nu}}}\\wedge\n\\Phi''\\wedge d\\bar{x}_{\\mathcal{K}},\n\\end{equation*}\nwhere $\\Phi''$ is $C^r$-smooth and of bidegree $(0,n-\\nu-|\\mathcal{K}|)$ (possibly, $\\Phi''=0$).\nThus, \\eqref{eq5} and \\eqref{eq5'} are finite sums of of integrals of the following type\n\\begin{equation}\\label{eq6}\n\\int_{\\mathbb{C}^n_x}\\frac{\\prod_1^p \\tilde{\\chi}_j^{\\epsilon} \\prod_{p+1}^q\\chi_j^{\\epsilon}}{\\prod_1^q \\hat{f}^{k_j}_{j1}}\n\\frac{d\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{x}^{\\alpha_{\\nu}}}{\\bar{x}^{\\alpha_{\\nu}}}\n\\wedge \\psi \\wedge d\\bar{x}_{\\mathcal{K}}\\wedge dx,\n\\end{equation}\n\\begin{equation}\\label{eq6'}\n\\lambda_1\\cdots \\lambda_p \\int_{\\mathbb{C}^n_x}\n\\frac{\\prod_1^q |\\hat{f}_j|^{2\\lambda_j}}{\\prod_1^q \\hat{f}^{k_j}_{j1}}\n\\frac{d\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{x}^{\\alpha_{\\nu}}}{\\bar{x}^{\\alpha_{\\nu}}}\n\\wedge \\psi \\wedge d\\bar{x}_{\\mathcal{K}}\\wedge dx,\n\\end{equation}\nwhere $\\psi$ is $C^r$-smooth and compactly supported.\n\n\\bigskip\n\nWe now first finish the proof of Proposition \\ref{lambdapropp}. First of all, it is well known that\n$\\Gamma(\\lambda)$ has a meromorphic continuation to $\\mathbb{C}^q$. We have\n\\begin{equation*}\n\\frac{d\\bar{x}^{\\alpha_1}}{\\bar{x}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{x}^{\\alpha_{\\nu}}}{\\bar{x}^{\\alpha_{\\nu}}}\n\\wedge d\\bar{x}_{\\mathcal{K}} =\n\\sum_{\\stackrel{|I|=\\nu}{I\\subseteq \\mathcal{K}^c}}C_I\\frac{d\\bar{x}_I}{\\bar{x}_I}\\wedge d\\bar{x}_{\\mathcal{K}}.\n\\end{equation*}\nLet us assume that $I=\\{1,\\ldots,\\nu\\}\\subseteq \\mathcal{K}^c$ and consider the contribution to \n\\eqref{eq6'} corresponding to this subset. This contribution equals\n\\begin{equation}\\label{eq7'}\nC_I\\lambda_1\\cdots \\lambda_p \\int_{\\mathbb{C}^n_x}\n\\frac{|x^{\\sum_1^q \\lambda_j\\alpha_j}|^2}{x^{\\sum_1^q k_j\\alpha_j}} \\bigwedge_1^{\\nu}\\frac{d\\bar{x}_j}{\\bar{x}_j}\\wedge\n\\Psi(\\lambda,x)\\wedge d\\bar{x}_{\\mathcal{K}}\\wedge dx\n\\end{equation}\n\\begin{eqnarray*}\n&=& \\frac{C_I\\prod_1^p\\lambda_j}{\\prod_{i=1}^{\\nu}(\\sum_1^q\\lambda_j\\alpha_{ji})}\n\\int_{\\mathbb{C}^n_x} \\frac{\\bigwedge_{i=1}^{\\nu}\\bar{\\partial} |x_i|^{2\\sum_1^q\\lambda_j\\alpha_{ji}}\n\\prod_{i=\\nu+1}^n|x_i|^{2\\sum_1^q\\lambda_j\\alpha_{ji}}}{x^{\\sum_1^q k_j\\alpha_j}}\\wedge \\\\\n& & \\hspace{7cm}\\wedge \\Psi(\\lambda,x)\\wedge d\\bar{x}_{\\mathcal{K}}\\wedge dx,\n\\end{eqnarray*}\nwhere $\\Psi(\\lambda,x)=\\psi(x)\\prod_1^q(\\xi_j^{\\lambda_j}\/h_j^{k_j})$.\nIt is well known (and not hard to prove, e.g., by integrations by parts as in \\cite{MatsAB}, Lemma 2.1) that the \n{\\em integral} on the right-hand side of \\eqref{eq7'} has an analytic continuation in $\\lambda$ to \na neighborhood of $\\cap_1^q\\{\\mathfrak{Re}\\, \\lambda_j \\geq 0\\}$.\n(We thus choose $r$ in Lemma \\ref{divlemma} large enough so that we can integrate by parts.)\nIf $p=0$, then the coefficient in front of \nthe integral is to be interpreted as $1$ and Proposition \\ref{lambdapropp} follows in this case. \nFor $p>0$, we see that the poles of \\eqref{eq7'}, and consequently\nof $\\Gamma(\\lambda)$, in a neighborhood of $\\cap_1^q\\{\\mathfrak{Re}\\, \\lambda_j \\geq 0\\}$ are along\nhyperplanes\nof the form $0=\\sum_1^q\\lambda_j\\alpha_{ji}$, $1\\leq i \\leq \\nu$. But if $j>p$ and $i\\leq \\nu$, then $\\alpha_{ji}=0$\nsince $\\{1,\\ldots,\\nu\\}\\subseteq \\mathcal{K}^c=\\{i;\\, x_i \\nmid x^{\\alpha_j},\\, \\forall j=p+1,\\ldots,q\\}$.\nThus, the hyperplanes are of the form $0=\\sum_1^p\\lambda_j\\alpha_{ji}$ and Proposition \\ref{lambdapropp} is proved\nexcept for the statement that at least for two $j$:s, the $\\alpha_{ji}$ are non-zero. However, we see from \n\\eqref{eq7'} that if for some $i$ we have $\\alpha_{ji}=0$ for all $j$ but one, then the appearing $\\lambda_j$ in\nthe denominator will be canceled by the numerator. Moreover, we may assume that \nthe constant $C_I=\\det (\\alpha_{ji})_{1\\leq i,j\\leq \\nu}$ is non-zero which implies that we cannot have any\n$\\lambda_j^2$ in the denominator. \n\n\\bigskip\n\nWe now prove Proposition \\ref{epsilonpropp}. Consider \\eqref{eq6}. We have that $\\alpha_1,\\ldots,\\alpha_{\\nu}$\nare linearly independent so we may assume that $A=(\\alpha_{ij})_{1\\leq i,j\\leq \\nu}$ is invertible\nwith inverse $B=(b_{ij})$. We make the non-holomorphic change of variables \\eqref{varbyte}, where the ``$q$'' of \n\\eqref{varbyte} now should be understood as $\\nu$. Then we get $x^{\\alpha_j}=y^{\\alpha_j}\\eta_j$, where \n$\\eta_j>0$ and smooth and $\\eta_j^2=1\/\\xi_j$, $j=1,\\ldots,\\nu$. \nHence, $|\\hat{f}_j|^2=|y^{\\alpha_j}|^2$, $j=1,\\ldots,\\nu$.\nExpressed in the $y$-coordinates we get that \n$\\Lambda_1^{\\nu}(d\\bar{x}^{\\alpha_j}\/\\bar{x}^{\\alpha_j})\\wedge \\psi \\wedge d\\bar{x}_{\\mathcal{K}}\\wedge dx$ \nis a finite sum of terms of the form \n\\begin{equation}\\label{hack2}\n\\frac{d\\bar{y}^{\\alpha_1}}{\\bar{y}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{y}^{\\alpha_{\\nu'}}}{\\bar{y}^{\\alpha_{\\nu'}}}\n\\wedge \\bar{y}_{\\mathcal{K}'}\\, d\\bar{y}_{\\mathcal{K}''} \\wedge \\psi_1,\n\\end{equation}\nwhere $\\nu'\\leq \\nu$, $\\psi_1$ is a $C^r$-smooth compactly supported form, and\n$\\mathcal{K}'$ and $\\mathcal{K}''$ are disjoint sets such that $\\mathcal{K}'\\cup \\mathcal{K}''=\\mathcal{K}$.\nIn order to give a contribution to \\eqref{eq6} we see that $\\psi_1$ must contain $dy$. \nIn \\eqref{hack2} we write $d=d_{\\mathcal{K}}+d_{\\mathcal{K}^c}$, and arguing as we did\nimmediately after Lemma \\ref{divlemma}, \\eqref{hack2} is a finite sum of terms of the form\n\\begin{equation*}\n\\frac{d\\bar{y}^{\\alpha_1}}{\\bar{y}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{y}^{\\alpha_{\\nu''}}}{\\bar{y}^{\\alpha_{\\nu''}}}\n\\wedge \\psi_2 \\wedge d\\bar{y}_{\\mathcal{K}}\\wedge dy,\n\\end{equation*}\nwhere $\\nu''\\leq \\nu$ and $\\psi_2$ is $C^r$-smooth and compactly supported.\nWith abuse of notation we thus have that \\eqref{eq6} is a finite sum of integrals of the form\n\\begin{equation}\\label{eq7}\n\\int_{\\mathbb{C}^n_x}\\frac{\\prod_1^p \\tilde{\\chi}_j^{\\epsilon} \\prod_{p+1}^q\\chi_j^{\\epsilon}}{\\prod_1^q \\hat{f}^{k_j}_{j1}}\n\\frac{d\\bar{y}^{\\alpha_1}}{\\bar{y}^{\\alpha_1}} \\wedge \\cdots \\wedge \\frac{d\\bar{y}^{\\alpha_{\\nu}}}{\\bar{y}^{\\alpha_{\\nu}}}\n\\wedge \\psi \\wedge d\\bar{y}_{\\mathcal{K}}\\wedge dy\n\\end{equation}\n\\begin{equation*}\n=\\int_{\\mathbb{C}^n_x}\\frac{\\bigwedge_1^{\\nu}d\\chi_j^{\\epsilon}\n\\prod_{\\nu+1}^p \\tilde{\\chi}_j^{\\epsilon} \\prod_{p+1}^q\\chi_j^{\\epsilon}}{y^{\\sum_1^q k_j\\alpha_j}}\n\\wedge \\Psi \\wedge d\\bar{y}_{\\mathcal{K}}\\wedge dy,\n\\end{equation*}\nwhere $\\Psi$ is a $C^r$-smooth compactly supported $(n-|\\mathcal{K}|-\\nu)$-form; the equality follows since\n$\\chi_j^{\\epsilon}=\\chi_j(|y^{\\alpha_j}|^2\/\\epsilon_j)$, $j=1,\\ldots,\\nu$. Now, \\eqref{eq7} is essentially equal\nto equation (24) of \\cite{JebHs} and the proof of Proposition \\ref{epsilonpropp} is concluded as in the \nproof of Proposition 8 in \\cite{JebHs}.\n\\end{proof}\n\n\\bigskip\n\n\\begin{proof}[Proof of Lemma \\ref{divlemma}]\nThe proof is similar to the proof of Lemma 9 in \\cite{JebHs} but some modifications have to be done.\nFirst, it is easy to check by induction over $|\\mathcal{K}|$ that \n$\\Phi'\\wedge \\Lambda_{i\\in I}(d\\bar{x}_i\/\\bar{x}_i)$ is $C^r$-smooth for any $I\\subseteq \\mathcal{K}$; for\n$|\\mathcal{K}|=1$ this is just Taylor's formula for forms. It thus suffices to show that\n\\begin{equation*}\nd\\bar{x}^{\\alpha_1}\\wedge \\cdots \\wedge d\\bar{x}^{\\alpha_{\\mu}}\\wedge \\left.\\frac{\\partial^{|k|} \\Phi}{\\partial x_I^k}\\right|_{x_I=0}=0,\\quad\n\\forall I\\subseteq \\mathcal{K}, \\, k=(k_{i_1},\\ldots,k_{i_{|I|}}).\n\\end{equation*}\nTo show this, fix an $I\\subseteq \\mathcal{K}$ and let $L=\\{j; \\, x_i \\nmid x^{\\alpha_j}\\,\\, \\forall i\\in I\\}$. \nSay for simplicity that \n\\begin{equation*}\nL=\\{1,\\ldots,\\mu',\\mu+1,\\ldots,m',m+1,\\ldots,p',p+1,\\ldots,q'\\},\n\\end{equation*}\nwhere $\\mu'\\leq \\mu$, $m'\\leq m$, $p'\\leq p$, and $q'<q$. The fact that $q'<q$ follows from the definitions\nof $\\mathcal{K}$, $I$, and $L$.\n\nConsider, on the base variety $Z$, the smooth form\n\\begin{equation*}\nF=\\bigwedge_1^{\\mu'}d\\bar{f}_{j1} \\bigwedge_{\\mu+1}^{m'}d\\bar{f}_{j1}\n\\bigwedge_{m+1}^{p'}(|f_{j1}|^2\\bar{\\partial} |f_j|^2-\\bar{\\partial} |f_{j1}|^2 |f_j|^2)\n\\bigwedge_{j\\in L}|f_j|^{2k_j}u^j_{k_j} \\wedge \\varphi_1.\n\\end{equation*}\nIt has bidegree $(0,n-\\sum_{j\\in L^c}k_j +q-q')$ so $F$ \nhas a vanishing pullback to $\\cap_{j\\in L^c}\\{f_j=0\\}$ since this set has dimension \n$n-\\sum_{j\\in L^c}e_j < n-\\sum_{j\\in L^c}k_j +q-q'$ by our assumption about complete intersection.\nThus, $\\hat{F}$ has a vanishing pullback to $\\{x_I=0\\}\\subseteq \\cap_{j\\in L^c}\\{\\hat{f}_j=0\\}$.\nIn fact, this argument shows that \n\\begin{equation}\\label{Feq}\n\\hat{F}=\\sum\\phi_j,\n\\end{equation}\nwhere the $\\phi_j$ are smooth linearly independent forms such that each $\\phi_j$ is divisible by \n$\\bar{x}_i$ or $d\\bar{x}_i$ for some $i\\in I$. (It is the pull-back to $\\{x_I=0\\}$ of the anti-holomorphic\ndifferentials of $\\hat{F}$ that vanishes.)\nFor the rest of the proof we let $\\sum \\phi_j$ \ndenote such expressions and we note that they are invariant under holomorphic differential operators.\nComputing $\\hat{F}$ we get\n\\begin{equation*}\n\\hat{F}=\\prod_{m+1}^{p'} |\\hat{f}_{j1}|^4\\prod_{j\\in L}\\frac{|\\hat{f}_j|^{2k_j}}{\\hat{f}_{j1}^{k_j}}\n\\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j} \\bigwedge_{\\mu+1}^{m'}d(\\bar{x}^{\\alpha_j}\\bar{h}_j)\n\\bigwedge_{m+1}^{p'}\\bar{\\partial} \\xi_j\n\\bigwedge_{j\\in L}v^j \\wedge \\hat{\\varphi}_1.\n\\end{equation*}\nThe ``coefficient'' $\\prod_{m+1}^{p'} |\\hat{f}_{j1}|^4\\prod_{j\\in L}(|\\hat{f}_j|^{2k_j}\/\\hat{f}_{j1}^{k_j})$\ndoes not contain any $\\bar{x}_i$ with $i\\in I$ so we may divide \\eqref{Feq} by it \n(recall that the $\\phi_j$ are linearly independent) and we obtain\n\\begin{eqnarray*}\n\\sum\\phi_j &=&\n\\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j} \\bigwedge_{\\mu+1}^{m'}d(\\bar{x}^{\\alpha_j}\\bar{h}_j)\n\\bigwedge_{m+1}^{p'}\\bar{\\partial} \\xi_j\n\\bigwedge_{j\\in L}v^j \\wedge \\hat{\\varphi}_1 \\\\\n&=& \\prod_{\\mu+1}^{m'}\\bar{x}^{\\alpha_j} \\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j} \\bigwedge_{\\mu+1}^{m'}d\\bar{h}_j\n\\bigwedge_{m+1}^{p'}\\bar{\\partial} \\xi_j\n\\bigwedge_{j\\in L}v^j \\wedge \\hat{\\varphi}_1 \\\\\n& & + \\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j} \\wedge \\sum_{\\mu+1}^{m'}d\\bar{x}^{\\alpha_j}\\wedge \\tau_j\n\\end{eqnarray*}\nfor some $\\tau_j$. We multiply this equality with \n\\begin{equation*}\n\\bigwedge_{m'+1}^md\\bar{h}_j \\bigwedge_{p'+1}^p \\bar{\\partial} \\xi_j \\bigwedge_{j\\in L^c}v^j\/\\left(\\prod_{\\mu+1}^m\\bar{h}_j\n\\prod_{m+1}^p\\xi_j\\right)\n\\end{equation*}\nand get\n\\begin{equation*}\n\\prod_{\\mu+1}^{m'}\\bar{x}^{\\alpha_j} \\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j}\\wedge \\Phi+\n\\bigwedge_1^{\\mu'}d\\bar{x}^{\\alpha_j} \\wedge \\sum_{\\mu+1}^{m'}d\\bar{x}^{\\alpha_j}\\wedge \\tau_j\n=\\sum\\phi_j\n\\end{equation*}\nfor some new $\\tau_j$. We apply the operator $\\partial^{|k|}\/\\partial x_I^k$ to this equality and then we\npull back to $\\{x_I=0\\}$, which makes the right-hand side vanish;\n(we construe however the result in $\\mathbb{C}^n_x$). Finally, taking the exterior product with\n$\\Lambda_{\\mu'+1}^{\\mu}d\\bar{x}^{\\alpha_j}$, which will make each term in under the summation sign on the left\nhand side vanish, we arrive at\n\\begin{equation*}\n\\prod_{\\mu+1}^{m'}\\bar{x}^{\\alpha_j} \\bigwedge_1^{\\mu}d\\bar{x}^{\\alpha_j}\\wedge \n\\frac{\\partial^{|k|}\\Phi}{\\partial x_I^k}\\big|_{x_I=0}=0\n\\end{equation*} \nand we are done.\n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUltracold atoms confined within optical lattice potentials are of great current interest,\nboth theoretically\\cite{key-1,key-2,key-3,key-4,key-5} and experimentally\\cite{key-6,key-7}. Of particular interest are the\nconnections to solid state regimes where optical lattices can be manufactured with\nextraordinary control so as to simulate more complicated, less perfect and, often, less\nmanipulable systems, initially of interest in a condensed matter sphere\n(eg. Heisenberg or Ising Hamiltonians )\\cite{key-8,key-9,key-10}.  Ultracold atoms in \noptical lattices have also been proposed as candidates for\nquantum information processing\\cite{key-11}, which has applications\nin quantum cryptography and quantum computing. Various methods of\nloading Bose-Einstein concensates into optical lattices have been\nproposed\\cite{key-12,key-13,key-14,key-15}, and this is now routinely performed\\cite{key-14},\nas is the manipulation and control of the atoms in such a lattice\\cite{key-16}.\n\nThe focus of this paper is on the microscopic treatment of Bose-Einstein\ncondensates in a one dimensional optical lattice. \nWe extend previous treatments at zero temperature\\cite{key-2,key-3} to finite \ntemperature using the \nHartree-Fock-Bogoliubov mean-field treatment as applied to a discrete\nBose-Hubbard model yielding modified Gross-Pitaevskii\nand Bogoliubov-de Gennes equations. This set of\nequations is solved for both the case of a translationally invariant\nlattice (no external trapping potential) and an inhomogeneous lattice\n(optical lattice in an external harmonic trapping potential). The use of the mean field\nbased treatment means we are only considering the superfluid phase for the atoms in the\noptical lattice. The model is not valid in the Mott Insulator regime. We use the model to\nestimate the superfluid to normal phase\ntransition temperature in each case. Thus one is able to obtain a phase diagram for\nthe superfluid and normal gas states in the low effective interaction strength limit.\n\n\n\n\\section{Formalism}\n\nWe consider a one-dimensional optical lattice with $I$ lattice\nsites. We begin from the Bose-Hubbard\nHamiltonian for atoms in a one-dimensional\noptical lattice \\cite{key-1,key-2,key-3}\n\\begin{equation}\n\\hat{H}=\\sum_{i=1}^{I}\\hat{n}_{i}\\epsilon_{i}-J\\sum_{i=1}^{I}\\left(\\hat{a}_{i+1}^{\\dag}\\hat{a}_{i}+\\hat{a}_{i}^{\\dag}\\hat{a}_{i+1}\\right)+\\frac{V}{2}\\sum_{i=1}^{I}\\hat{n}_{i}\\left(\\hat{n}_{i}-1\\right),\\label{eq1}\n\\end{equation}\n where $J$ represents the coupling strength between adjacent lattice\nsites, $V$ is the interaction potential acting between atoms on the\nsame site, and $\\hat{a}_{i}$ is the Bose field operator for the $i^{th}$\nlattice site, and $\\hat{n}_{i}=\\hat{a}_{i}^{\\dag}\\hat{a}_{i}$ . $\\epsilon_{i}$\nis the energy on each lattice site $i$ due to the trapping potential. The\nusual commutation relations apply for the Bose field operator $\\hat{a}_{i}$.\nAssuming a macroscopic occupation of the ground state, we express\nthe Bose annihilation operator for each lattice site $i$ in terms\nof a complex mean field part $z_{i}$ and a fluctuation operator part $\\hat{\\delta}_{i}$\n\\cite{key-17}, $\\hat{a}_{i}=\\left(z_{i}+\\hat{\\delta}_{i}\\right)\\exp\\left(\\frac{-i\\mu t}{\\hbar}\\right)$, where $\\mu$ is the eigenvalue for the generalised Gross-Pitaevskii equation to be discussed below, and take the self-consistent mean-field approximation such that \n$\\hat{\\delta}_{i}^{\\dag}\\hat{\\delta}_{i}\\hat{\\delta}_{i}=2\\left\\langle \\hat{\\delta}_{i}^{\\dag}\\hat{\\delta}_{i}\n\\right\\rangle \\hat{\\delta}_{i}+\\left\\langle \\hat{\\delta}_{i}\\hat{\\delta}_{i}\\right\\rangle \\hat{\\delta}_{i}^{\\dag}$,\netc. The condensate density is given by $n_{c_{i}}=\\left|z_{i}\\right|^{2}$\nand $\\tilde{n}_{i}=\\left\\langle \\hat{\\delta}_{i}^{\\dag}\\hat{\\delta}_{i}\\right\\rangle $ is\nthe excited population density. The anomalous density is neglected\\cite{key-3,key-17,key-18,key-19}.\n\nThe resulting Hamiltonian can now be diagonalised using the Bogoliubov (canonical) transformation\\begin{equation}\n\\hat{\\delta}_{i}=\\sum_{q}\\left[u_{i}^{q}\\hat{\\alpha}_{q}e^{-i\\omega_{q}t}-v_{i}^{q^{*}}\\hat{\\alpha}_{q}^{\\dag}e^{i\\omega_{q}t}\\right]\\label{eq10}\\end{equation}\nand\n\n\\begin{equation}\n\\hat{\\delta}_{i}^{\\dag}=\\sum_{q}\\left[u_{i}^{q^{*}}\\hat{\\alpha}_{q}^{\\dag}e^{i\\omega_{q}t}-v_{i}^{q}\\hat{\\alpha}_{q}e^{-i\\omega_{q}t}\\right],\\label{eq11}\\end{equation}\n where $u_{i}^{q}$ and $v_{i}^{q}$ are the quasiparticle amplitudes,\n$\\omega_{q}$ are the quasiparticle excitation frequencies, and $\\hat{\\alpha}_{q}^{\\dag}$\n ($\\hat{\\alpha}_{q}$) is the quasiparticle creation (annihilation) operator. This yields\n\\cite{key-3} a set\nof coupled equations comprising of a modified\nGross-Pitaevskii equation\n\\begin{equation}\n\\mu^{\\prime}z_{i}=\\epsilon_{i}^{\\prime}z_{i}-\\left(z_{i+1}+z_{i-1}\\right)+V_{\\rm{eff}}\\left(n_{c_{i}}z_{i}+2\\tilde{n}_{i}z_{i}+\\tilde{m}_{i}z_{i}\\right)\\label{eq12}\n\\end{equation}\nand the Bogoliubov-de Gennes equations\n\\begin{equation}\n\\begin{array}{c}\n\\hbar\\omega_{q}^{\\prime}u_{i}^{q}+c_{q}^{\\prime}z_{i}=\\left[2V_{\\rm{eff}}\\left(n_{c_{i}}+\\tilde{n}_{i}\\right)-\\mu^{\\prime}+\\omega_{i}^{\\prime}\\right]u_{i}^{q}-\\left[u_{i+1}^{q}+u_{i-1}^{q}\\right]-V_{\\rm{eff}}z_{i}^{2}v_{i}^{q}\\\\\n-\\hbar\\omega_{q}^{\\prime}v_{i}^{q}-c_{q}^{\\prime}z_{i}=\\left[2V_{\\rm{eff}}\\left(n_{c_{i}}+\\tilde{n}_{i}\\right)-\\mu^{\\prime}+\\epsilon_{i}^{\\prime}\\right]v_{i}^{q}-\\left[v_{i+1}^{q}+v_{i-1}^{q}\\right]-V_{\\rm{eff}}z_{i}^{2}u_{i}^{q}\\end{array},\\label{eq13}\n\\end{equation}\n with\n \\begin{equation}\n\\tilde{n}_{i}=\\left\\langle \\hat{\\delta}_{i}^{\\dag}\\hat{\\delta}_{i}\\right\\rangle =\\sum_{q}\\left[\\left|v_{i}^{q}\\right|^{2}+\\left(\\left|u_{i}^{q}\\right|^{2}+\\left|v_{i}^{q}\\right|^{2}\\right)N_{BE}\\left(\\hbar\\omega_{q}^{\\prime}\\right)\\right]\\label{eq15}\n\\end{equation}\n where\n \\begin{equation}\nN_{BE}\\left(\\hbar\\omega_{q}^{\\prime}\\right)=\\frac{1}{Z^{-1}e^{\\beta\\hbar\\omega_{q}^{\\prime}}-1}\\label{eq16}\n\\end{equation}\nis the usual Bose distribution, the primed quantities are measured in units of $J$ (which depends on the depth of the optical lattice), and where we have assumed without loss of\ngenerality the condensate amplitudes $z_{i}$ to be real. $\\mu$ is the energy eigenvalue\nfor the modified Gross-Pitaevskii equation (and approximates the chemical\npotential closely for values of temperature well below the transition\ntemperature) and $\\hbar\\omega_{q}$ is the energy eigenvalue of the Bogoliubov-de Gennes equations. The $c^{q}$'s\nare necessary in order to ensure the orthogonality of the condensate\nwith the excited states\\cite{key-3,key-18}. $Z$ is a fugacity term resulting from the difference between the\ntrue chemical potential $\\mu_{T}$, and the chemical potential as\nestimated using the eigenvalue $\\mu$ corresponding to the ground\nstate of the modified Gross-Pitaevskii equation.\nThus $Z=\\exp\\left(\\beta\\left(\\mu_{T}-\\mu\\right)\\right)$. \n\nIn the case of a homogeneous gas with a large\nnumber of particles in the ground state (ie. $n_{c}\\gg1$), the fugacity\nmay be approximated by\\cite{key-18}\\begin{equation}\nZ=1+1\/n_{c}\\label{eq17}\\end{equation}\n In the lattice, this is not always a good approximation, since the condition $n_{c}\\gg1$\ndoes not always hold, and we will take $Z=1$. This will lead to increased\nvalues for the excited atom population, and will therefore result\nin under-estimated values of the transition temperature.\n\nSuperfluid flow of the condensate occurs when there is a phase gradient.\nA phase gradient of the condensate modifies the hopping term of the\nHamiltonian by the introduction of Peierls phase factors. The resulting\nenergy shift may be estimated using second order perturbation theory.\nSince this energy shift is due entirely to the kinetic energy associated\nwith the superfluid flow, and hence due to the superfluid fraction,\nit follows that the superfluid fraction $f_{s}$ may be calculated\nin terms of this energy shift\\cite{key-3}\n\\begin{equation}\nf_{s}=\\frac{1}{N}\\frac{E_{\\phi}-E_{0}}{J\\left(\\Delta\\phi\\right)^{2}}=-\\frac{1}{2NJ}\\left\\langle \\psi_{0}\\right|\\hat{T}\\left|\\psi_{0}\\right\\rangle -\\frac{1}{NJ}\\sum_{\\nu\\neq0}\\frac{\\left|\\left\\langle \\psi_{\\nu}\\right|\\hat{J}\\left|\\psi_{\\nu}\\right\\rangle \\right|^{2}}{E_{\\nu}-E_{0}}\n\\end{equation}\nwhere $\\hat{J}=iJ\\sum_{i=1}^{I}\\left(\\hat{a}_{i+1}^{\\dag}\\hat{a}_{i}-\\hat{a}_{i}^{\\dag}\\hat{a}_{i+1}\\right)$\nand $\\hat{T}=J\\sum_{i=1}^{I}\\left(\\hat{a}_{i+1}^{\\dag}\\hat{a}_{i}+\\hat{a}_{i}^{\\dag}\\hat{a}_{i+1}\\right)$.\nThe superfluid fraction can then be expressed\nin terms of the condensate and quasiparticle amplitudes\n\\begin{equation}\nf_{s}=f_{s}^{(1)}-f_{s}^{(2)}\\label{eq20}\n\\end{equation}\nwhere\n\\begin{equation}\nf_{s}^{(1)}=\\frac{1}{2N}\\sum_{i=1}^{I}\\left[\\left(z_{i+1}z_{i}^{*}+z_{i+1}^{*}z_{i}\\right)+\\sum_{q}\\left(v_{i}^{q}v_{i+1}^{q^{*}}+v_{i}^{q^{*}}v_{i+1}^{q}\\right)\\right]\\label{eq21}\n\\end{equation}\nand\n\\begin{equation}\nf_{s}^{(2)}=\\frac{J}{N}\\sum_{q,q^{\\prime}}\\left[\\frac{\\left|\\sum_{i}\\left(u_{i+1}^{q}v_{i}^{q^{\\prime}}-u_{i}^{q}v_{i+1}^{q^{\\prime}}\\right)\\right|^{2}}{\\hbar\\omega_{q}+\\hbar\\omega_{q}^{\\prime}}+\\delta_{qq^{\\prime}}\\frac{\\left|\\sum_{i}\\left(u_{i+1}^{q}v_{i}^{q}-u_{i}^{q}v_{i+1}^{q}\\right)\\right|^{2}}{2\\hbar\\omega_{q}}\\right].\\label{eq22}\n\\end{equation}\n\n\n\n\\subsection{Translationally Invariant Lattice\\label{sub:3.1}}\n\nIn the case of a translationally invariant lattice, periodic boundary\nconditions apply, and the quasi-particle amplitudes are given by\n\\begin{equation}\n\\begin{array}{ccccc}\nu_{j}^{q}=\\frac{u^{q}e^{i(qja)}}{\\sqrt{I}} & , & v_{j}^{q}=\\frac{v^{q}e^{i(qja)}}{\\sqrt{I}} & , & 1\\leq j\\leq I-1\\end{array}\\label{eq32}\n\\end{equation}\n Furthermore the condensate amplitudes are equal for each site $j$\nand, since there is no trapping potential, $\\epsilon_{i}=\\epsilon=0$. Solving for the \nquasiparticle amplitudes we obtain \n\\begin{equation}\n\\left|u^{q}\\right|^{2}=\\frac{Vz^{2}+4J\\sin^{2}\\left(\\frac{qa}{2}\\right)+\\hbar\\omega_{q}}{2\\hbar\\omega_{q}}\\label{eq36}\\end{equation}\n\\begin{equation}\n\\left|v^{q}\\right|^{2}=\\frac{Vz^{2}+4J\\sin^{2}\\left(\\frac{qa}{2}\\right)-\\hbar\\omega_{q}}{2\\hbar\\omega_{q}},\\label{eq37}\n\\end{equation}\n and\n\\begin{equation}\n\\hbar\\omega_{q}=\\sqrt{4J\\sin^{2}\\left(\\frac{qa}{2}\\right)\\left[2n_{c}V+4J\\sin^{2}\\left(\\frac{qa}{2}\\right)\\right]}.\\label{eq38}\\end{equation}\n Thus from equations (\\ref{eq36}), (\\ref{eq37}) and (\\ref{eq38}),\nfor the translationally invariant lattice, the superfluid fraction\nis given by\\begin{equation}\nf_{s}=\\frac{1}{N}\\left(I\\left|z\\right|^{2}+\\sum_{j=1}^{I-1}\\left|v^{q}\\right|^{2}\\cos\\left(\\frac{2\\pi j}{I}\\right)\\right),\\label{eq43}\\end{equation}\n since $q=2\\pi j\/Ia$ for $1\\leq j\\leq I-1$.\nThe condensate fraction for a given lattice site is given by $f_{c}=n_{c}\/n_{0}$\nwhere $n_{0}=N\/I$ is the number of atoms per site. \n\nTo calculate the condensate and superfluid fraction, we first determine the condensate amplitude. Then\ndefining\\begin{equation}\ng_{j(n)}=\\left|z\\right|_{(n-1)}^{2}V_{\\rm{eff}}+4\\sin^{2}\\left(\\frac{\\pi j}{I}\\right)\\label{eq49}\\end{equation}\n\\begin{equation}\ne_{j(n)}=2\\sin\\left(\\frac{\\pi j}{I}\\right)\\sqrt{2\\left|z\\right|_{(n-1)}^{2}V_{\\rm{eff}}+4\\sin^{2}\\left(\\frac{\\pi j}{I}\\right)}\\label{eq50}\\end{equation}\n and\\begin{equation}\nN_{BE_{j(n)}}=\\frac{1}{\\exp\\left(\\beta^{\\prime}e_{j_{(n-1)}}\\right)-1},\\label{eq51}\\end{equation}\n where $\\beta^{\\prime}=J\/k_{B}T$,\n$k_{B}$ is Boltzman's constant and where the subscript\n$(n)$ refers to the variable in question at the $n^{\\mathrm{th}}$\niteration, one can solve for $\\left|z\\right|^{2}$ using an iterative\nscheme\\begin{equation}\n\\left|z\\right|_{(n)}^{2}=\\frac{1}{I}\\left(N-\\sum_{j=1}^{I-1}\\left[\\frac{g_{j(n)}-e_{j(n)}}{2e_{j(n)}}+\\frac{g_{j(n)}}{e_{j(n)}}N_{BE_{j(n)}}\\right]\\right).\\label{eq52}\\end{equation}\n As an initial guess, the value $\\left|z\\right|_{(n)}^{2}=\\frac{N}{I}$\nis used, and the calculations (\\ref{eq49}), (\\ref{eq50}), (\\ref{eq51})\nand (\\ref{eq52}) repeated until convergence is attained (ie. $\\left|\\left|z\\right|_{(n)}^{2}-\\left|z\\right|_{(n-1)}^{2}\\right|<$\nError Tolerance), or the maximum number of iterations is exceeded\n(divergent solution)). \n\nOne first calculates $g_{j(n)}$, $e_{j(n)}$ and $N_{BE_{j(n)}}$using\nequations (\\ref{eq49}), (\\ref{eq50}) and (\\ref{eq51}). The quasiparticle\namplitudes (given by equations (\\ref{eq36}) and (\\ref{eq37})) may\nbe calculated using the equations\\begin{equation}\n\\left|u^{(j)}\\right|_{(n)}^{2}=\\frac{g_{j(n)}+e_{j(n)}}{2e_{j(n)}}\\label{eq53}\\end{equation}\n and\\begin{equation}\n\\left|v^{(j)}\\right|_{(n)}^{2}=\\frac{g_{j(n)}-e_{j(n)}}{2e_{j(n)}}\\label{eq54}\\end{equation}\n The condensate and superfluid fractions are then readily determined.\n\n\n\\subsection{Inhomogeneous Lattice\\label{sub:3.2}}\n\nThe condensate amplitudes $z_{i}$ are found by solving equation (\\ref{eq12}),\nwhere\nthe trapping potential is given by $\\epsilon_{i}=\\Omega (i-(I+1)\/2)^{2}$ for\nsite $i$, with $\\Omega=\\frac{1}{2}m\\omega^{2}a^{2}$, and $a$ is\nthe inter-lattice spacing. The quasi-particle amplitudes for site\n$i$, $u_{i}^{q}$and $v_{i}^{q}$are found by solving equations (\\ref{eq13}).\n\nThis set of equations can again be solved iteratively. In performing\nthe calculation, we actually \nset the $c^{q}$'s to zero when solving the Bogoliubov-de\nGennes equations, but do so in the Hartree-Fock basis, thus ensuring orthogonality\nof the ground state and the excited states \\cite{key-18}. The Hartree-Fock basis\nis given by the normalised solutions to the eigenvalue problem given\nby equation (\\ref{eq12}), but where the ground state (zero energy\nsolution) is excluded. \n\nFirst, let us rewrite the Bogoliubov-de Gennes equations (\\ref{eq13})\nin matrix form\\begin{equation}\n\\hbar\\omega_{q}\\left[\\begin{array}{c}\n\\mathbf{u}^{q}\\\\\n\\mathbf{v}^{q}\\end{array}\\right]=\\left[\\begin{array}{cc}\n\\hat{\\mathcal{L}} & M\\\\\n-M & -\\hat{\\mathcal{L}}\\end{array}\\right]\\left[\\begin{array}{c}\n\\mathbf{u}^{q}\\\\\n\\mathbf{v}^{q}\\end{array}\\right],\\label{eq61}\\end{equation}\nwhere\\begin{equation}\n\\begin{array}{ccc}\n\\mathbf{u}^{q}=\\left[\\begin{array}{c}\nu_{1}^{q}\\\\\n\\vdots\\\\\nu_{i}^{q}\\\\\n\\vdots\\\\\nu_{I}^{q}\\end{array}\\right] & \\mathrm{,} & \\mathbf{v}^{q}=\\left[\\begin{array}{c}\nv_{1}^{q}\\\\\n\\vdots\\\\\nv_{i}^{q}\\\\\n\\vdots\\\\\nv_{I}^{q}\\end{array}\\right],\\end{array}\\label{eq60}\\end{equation}\n\\begin{equation}\n\\hat{\\mathcal{L}}=2V_{\\rm{eff}}\\left(n_{c_{i}}+\\tilde{n}_{i}\\right)-\\mu+\\Omega (i-(I+1)\/2)^{2}-\\hat{\\mathcal{J}}\\label{eq57}\\end{equation}\nand\\begin{equation}\nM=-V_{\\rm{eff}}z_{i}^{2}\\label{eq57a}\\end{equation}\nHere $\\hat{\\mathcal{J}}$ is defined as the operator acting on $z_{i}$, $u_{i}^{q}$,\nand $v_{i}^{q}$ as follows:\\begin{equation}\n\\hat{\\mathcal{J}}u_{i}^{q}=u_{i+1}^{q}+u_{i-1}^{q}.\\label{eq58}\\end{equation}\n\nNow, let $\\left\\{ z_{i}^{q}\\right\\} $constitute the eigenstates of\nequation (\\ref{eq12}) with eigenvalues $\\mu^{q}$. We order the\neigenvalues into ascending order, and order the normalised eigenstates accordingly,\ncall these $\\left\\{ \\xi_{i}^{q}\\right\\} $. The state $\\xi_{i}^{0}$\ncorresponds to the Goldstone mode, we must exclude this in order to\nobtain the Hartree-Fock-Bogoliubov basis. Let us define the matrix\\begin{equation}\n\\mathbf{U}=\\left[\\begin{array}{ccccc}\n\\xi^{1} & \\ldots & \\xi^{q} & \\ldots & \\xi^{I-1}\\end{array}\\right],\\label{eq55}\\end{equation}\nwhere\\begin{equation}\n\\xi^{q}=\\left[\\begin{array}{c}\n\\xi_{1}^{q}\\\\\n\\vdots\\\\\n\\xi_{i}^{q}\\\\\n\\vdots\\\\\n\\xi_{I}^{q}\\end{array}\\right].\\label{eq56}\\end{equation}\n Let $\\left\\{ \\mathbf{u}_{HFB}^{q},\\mathbf{v}_{HFB}^{q}\\right\\} $be\nthe solution to the Bogoliubov-de Gennes equations in the Hartree-Fock-Bogoliubov\nbasis, then the solution to the matrix equation\\begin{equation}\n\\hbar\\omega_{q}\\left[\\begin{array}{c}\n\\mathbf{u}_{HFB}^{q}\\\\\n\\mathbf{v}_{HFB}^{q}\\end{array}\\right]=\\left[\\begin{array}{cc}\nU^{T}\\hat{\\mathcal{L}}U & U^{T}MU\\\\\n-U^{T}MU & -U^{T}\\hat{\\mathcal{L}}U\\end{array}\\right]\\left[\\begin{array}{c}\n\\mathbf{u}_{HFB}^{q}\\\\\n\\mathbf{v}_{HFB}^{q}\\end{array}\\right]\\label{eq62}\\end{equation}\ngives the Bogoliubov quasiparticle amplitudes in the Hartree-Fock-Bogoliubov\nbasis. To obtain the Bogoliubov quasiparticle amplitudes$\\left\\{ u_{i}^{q},v_{i}^{q}\\right\\} $\none transforms back using\\begin{equation}\n\\begin{array}{c}\n\\mathbf{u}^{q}=U\\mathbf{u}_{HFB}^{q}\\\\\n\\mathbf{v}^{q}=U\\mathbf{v}_{HFB}^{q}\\end{array}.\\label{eq63}\\end{equation}\n\nThe iterative solution of these equations is then obtained numerically.\n\n\\section{Results}\n\nWe present results for a lattice of depth of approximately $16.8E_R$, where the recoil energy is defined as $E_R=h^{2}\/8ma^{2}$ with $a$ the inter-site spacing. This is defined for the optical lattice parameters defined in reference \\cite{key-20} and using the band structure calculations of \\cite {key-21}.\n\n\n\\subsection{Translationally Invariant Lattice}\n\n\n\n\\begin{figure}\n\\includegraphics[%\n  scale=0.70,\n  angle=270]{fig1.eps}\n\n\n\\caption{Translationally invariant lattice with ten atoms and ten sites. Overall condensate (a) and superfluid (b) fractions as a function of temperature. The corresponding phase diagram \nis shown in panel (c).\n\\label{cap:fig1}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[%\n  scale=0.7,\n  angle=270]{fig2.eps}\n\n\n\\caption{Translationally invariant lattice with ten atoms per site. Condensate Fraction\n(a), and\nexcitation spectrum as a function of temperature (b) with the corresponding phase diagram (c).\n\\label{cap:fig2}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[%\n  scale=0.7,\n  angle=270]{fig3.eps}\n\n\n\\caption{Inhomogeneous lattice with forty one lattice sites and ten atoms in total for\n$V_{\\rm{eff}}=1$, at (a) $T=0$ nK, (b) $T=1$ nK (c)\n$T=1.6$ nK. Circles represent the number of condensate atoms, squares the number of excited atoms and crosses the total number of atoms\\label{cap:fig3}}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[%\n  scale=0.7,\n  angle=270]{fig4.eps}\n\n\n\\caption{Overall condensate and superfluid fractions as a\nfunction of temperature, and the corresponding phase diagram for\nan optical lattice in a harmonic potential with forty lattice sites (even case) and \nten atoms. (a) corresponds to the condensate\nfraction , (b) to the superfluid\nfraction and (c) to the phase diagram.\\label{cap:fig4}}\n\\end{figure}\n\n\nIn figures \\ref{cap:fig1} and \\ref{cap:fig2} we present results for a \ntranslationally invariant lattice with one atom and ten atoms per site\nrespectively for various values\nof the effective interaction potential $V_{\\rm{eff}}$. In both cases ten lattice sites, with\nperiodic boundary conditions, were used. One observes, a decrease in both the condensate\nand superfluid fractions with temperature and we interpret the point at which these\ndensities approach zero as indicative of the critical temperature \n$T_{c}$ for the superfluid to normal gas phase transition. This interpretation\nis supported by an examination of the low lying excitation spectrum which, for\nthe case $V_{\\rm{eff}}=20$, is shown in panel (b) of figure \\ref{cap:fig2}. The\n``softening'' of the modes, indicative of the phase transition, is clearly seen and\ncoincides with\nthe transition temperature obtained by noting the temperature at which the condensate and\nsuperfluid densities approach zero.\n\nWe obtain such a transition temperature for each value of the effective\ninteraction potential $V_{\\rm{eff}}$, producing a phase diagram as shown in figures\n\\ref{cap:fig1}(c) and \\ref{cap:fig2}(c), where the superfluid phase and the normal phase\nare as indicated, the superfluid lying to the left of the curve. The\ntransition to the Mott insulator phase cannot be determined by this\nanalysis, but one would expect the transition from the superfluid\nphase to the Mott insulator phase to occur when the site coupling\nstrength $J$ is decreased below a certain point, depending on the on-site interaction strength $V_{\\rm{eff}}$. In practice the Mott insulator phase transition will occur when $V_{\\rm{eff}}=V\/J$ exceeds some critical value. Thus a transition to the Mott insulator phase is also possible by increasing the on-site interaction strength $V$ for a given coupling strength $J$.\nWe note here that, in all instances, the transition temperature increases\nwith the effective interaction potential, as is indeed the case with\na homogeneous Bose gas (ie. a Bose gas in the absence of an optical\nlattice and of a confining potential)\\cite{key-22,key-23}. This reentrant behaviour has also been\npredicted for the translationally invariant lattice by Kleinert {\\em et al.}\n\\cite{key-24}. Our results are consistent with their conclusions. In addition, they predict a\nreduction of the critical temperature as the effective interaction strength is increased\nfurther. We, however, are unable to explore this regime as it extends beyond the validity\nof our model.\n\n\n\n\n\\subsection{Inhomogeneous Lattice}\n\nIn this section we present results for the case of an optical lattice in\nthe presence of a harmonic trapping potential. The condensate fraction\n$f_{c}$ and the superfluid fraction $f_{s}$ are evaluated as a function\nof the effective potential $V_{\\rm{eff}}$ and the temperature. \n\nFigure \\ref{cap:fig3} shows plots of the number\nof condensate atoms, excited atoms and of the total number of atoms\nfor each lattice site for the case of an inhomogeneous optical lattice\nconsisting of forty one lattice sites (odd case) with ten atoms in\ntotal, for $V_{\\rm{eff}}=1$ at various temperatures ranging from $T=0$ nK\nto $T=1.6$ nK. At zero temperature, the condensate atom distribution\nis bell-shaped, peaked at the central lattice site.\nThere is a small quantum depletion even at zero temperature, and the\ndistribution of excited atoms is shaped as a bimodal distribution,\ncentred about the central lattice site. As the temperature increases, \nthe condensate population decreases,\nbut the distribution still remains bell-shaped, and the excited population\nincreases.  In figure \\ref{cap:fig4} we present the overall condensate\nand superfluid fractions and the corresponding phase diagram\nfor optical lattices in a harmonic potential consisting of forty lattice sites \n(even case). Panel (a) corresponds to the condensate fraction, (b) the superfluid fraction, \nand (c) the phase diagram. We note that for higher temperatures, calculations performed using forty one lattice sites begin to show a marked difference. This is indicative of the\nfact that we are pushing the bounds of validity \nof our model. In particular we are seeing significant finite size effects, which affect\nthe value of the chemical potential and, ultimately, a failure of the mean field\napproximation. The dotted continuation of the lines in figure \\ref{cap:fig4} indicate\nwhere our calculations become unreliable, but are included for completness. We are still\nable to use our model to obtain an estimate of the critical temperature and hence the\ntrend in its dependence upon the effective interaction strength.\nWe therefore conclude from figure \\ref{cap:fig4} that the superfluid to\nnormal phase transition temperature increases with increasing $V_{\\rm{eff}}$. \nIt is clear, then, that\nthe shift in critical temperature with effective interaction potential\nis positive definite for a Bose gas in a one-dimensional optical lattice,\nregardless of whether the system is confined in a (harmonic) trapping\npotential or not. This is in contrast to the case of a three-dimensional\nBose gas, where $\\Delta T_{c}$ changes sign for the trapped gas.\n\n\n\n\n\\section{Conclusions}\n\nWe have applied the descretized Hartree-Fock-Bogoliubov formulation to the \nBose-Hubbard model in order to calculate the dependence of the condensate and superfluid\nfractions on the temperature. We have used this to estimate the critical temperature for\nthe superfluid to normal phase transition for both a translationally\ninvariant optical lattice (no external trap present), and an inhomogeneous\noptical lattice (contained within an external harmonic trap).\nThis has enabled us to investigate the phase diagram for both cases\nand we observe that the transition temperature increases with increasing\neffective interaction potential $V_{\\rm{eff}}$. Unlike the\nhomogeneous case with no optical lattice, this positive shift in the\ncritical temperature with interaction strength is present in both\nthe translationally invariant case and when a parabolic confining\npotential is imposed. In the homogeneos gas the shift in the critical\ntemperature is only positive in\nthe absence of a confining potential. These conclusions are consistent with previous\nwork for the translationally invariant case\\cite{key-24}, extending this result to\ninclude parabolic confinement.\n\n\\section{Acknowledgements}\nWe would like to thank the Marsden Fund of the Royal Society of New\nZealand and the University of Otago for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRadio-frequency (RF) ion traps are used in many different contexts including quantum computing~\\cite{bruzewicz_trapped-ion_2019}, cold reactions~\\cite{PhysRevLett.97.243005}, cooling highly charged ions~\\cite{RN174}, mass spectrometry~\\cite{march_practical_1995} and quantum simulations~\\cite{bruzewicz_trapped-ion_2019}.\n\nIn the context of quantum computing and simulation, novel scalable ion trap architecture, in particular the surface ion trap approach \\cite{pino_demonstration_2021} has gained significant traction in recent years. Unlike a 3D linear ion trap geometry, the surface ion traps are relatively easy to micro-fabricate and hence the electrodes can be designed in an arbitrary shape and size~\\cite{hucul_transport_2008, amini_toward_2010, mokhberi_optimised_2017}. However, implementing a large scale quantum algorithms require more than just a scalable ion trap; it is necessary a develop a scalable computing architecture capable of controlling $100$-$1000$s of qubits. Among such architectures, a quantum charged coupled device (QCCD)~\\cite{RN134} has recently being realised in a rudimentary application~\\cite{pino_demonstration_2021}. This requires the ions to be shuttled around while keeping alive the quantum state. This invariably requires full temporal and spatial control over the ion chain, thus requiring a large number of DC electrodes. Separation of two ions in different trapping regions~\\cite{warschburger_experimental_2014}, transport of ions around the corners~\\cite{hucul_transport_2008} are some of the examples challenges that needs careful consideration. While this approach may be viable for $100$s of qubits, it is often challenged by heating of the motional degree of freedom, an essential for two qubit gate operations. An alternative approach is based on distributed computing architecture, where each computing node either implements a QCCD architecture or uses multiple addressing beams for multiple ions without the need to shuttle the ions around~\\cite{RN135}. The nodes are then connected via optical quantum communication channels. Considering the later option, the ions are not only addressed individually but also photons are collected from individual ions to create remote entanglement. Therefore, ion-ion separation is an important parameter to consider in distributed computing scheme. \n\nIn case of a 1D chain of ion trapped and laser cooled in a harmonic potential, the ions are distributed non-uniformly. The ion-ion distance increases from the trap centre~\\cite{kamsap_experimental_2017} to the edges. Technically, such a configuration is not convenient to address the ions using multiple laser beams. Furthermore, an uniform chain of trapped and laser cooled ions are test-beds for simulating a variety of quantum many body systems like the homogeneous Kibble-Zureck mechanism~\\cite{RN161}, adiabatic entanglement generation in a Bose-Hubbard model~\\cite{RN73} $\\textit{etc}$. Adiabatic entanglement generation using the Bose-Hubbard model is easier to implement when the hopping probability is uniform over the ion chain, thus requiring uniform distribution of the ions. On the contrary the homogeneous Kibble-Zureck mechanism demands an uniform density of the ions in the chain as it is a strong assumption in the model.   \n\nIt was first suggested by Lin et al~\\cite{lin_large-scale_2009} that the use of a quartic rather than a harmonic potential generates a near uniform ion distribution. More recently, Xie et al~\\cite{xie_creating_2017} developed a feedback method to achieve a high degree of homogeneity on a surface trap experiment. Despite these developments, designing electrodes to generate uniform chain of ion with uniformity below $10\\%$, remains a challenge. Often, the electric field generated by a non-regular shaped electrodes do not have an analytical solution, making it harder to calculate the required voltages and frequencies to operate such a surface ion trap. Most often, one relies on specialised software based on the Finite Element Method~(FEM) or the Boundary Element Methods~(BEM), to solve the Laplace\/Poisson equations. An alternative numerical approach assumes that there is no gap between the electrodes: the gap-less(GL) approximation~\\cite{house_analytic_2008}, which reflects the reality in case the electrodes are much longer than the real gaps.\n\nHere, we report an alternative approach for designing multi-electrode surface ion trap in order to achieve an homogeneous ion chain. The homogeneity of the chain is aimed at reducing the qubit addressability error from individual laser beams and more importantly for experiments where the uniform density of strongly coupled oscillators are required such as in the Kibble-Zureck mechanism. The later puts stringent condition on the required uniformity for which the currently available techniques fail. \n\nThe article is organised as follows: starting with a brief introduction to the surface ion trap architecture, the novel algorithm to calculate the voltages needed to generate a desired ion chain distribution is presented; It is then used to explore the possibility to obtain the desired topology in function of several experimental constraints and finally, molecular dynamics simulations are used to verify the degree of homogeneity of the ion-chain for various optimal trap parameters as obtained from the novel algorithm. \n\n\\section{Basics of Surface Ion Traps}\nA Radio-Frequency~(RF) linear Paul trap uses a time dependent harmonic potential in three dimensions to generate a time averaged pseudo-potential to confine charge particles in 3D. The stability of the ion on the trap depends on the charge-to-mass ratio of the particle as well as the voltage and frequency of the applied time dependent field. A surface ion trap generates a similar pseudo-potential by planarizing the 3D electrodes to a plane. In this case, see figure~\\ref{fig:designs}, the ions are trapped above the surface at a given height, $h$, corresponding to the location of the minimum of the pseudo-potential, $V_\\text{PS}(\\vec{r})$ at $\\vec{r}$. The $V_\\text{PS}(r)$ is given by:\n\\begin{eqnarray}\\label{eq:Vps}\n    V_{PS}(\\vec{r}) = \\frac{Q^{2}}{4 m \\Omega^{2}}|\\vec{E}_{RF}(\\vec{r})|^{2}\n\\end{eqnarray}\nwhere $Q$ and $m$ are the charge and the mass of the trapped particles, and $\\vec{E}_\\text{RF}(\\vec{r})$ is the electric field generated by the RF electrodes at $\\vec{r}$.\n\n\\begin{figure}\n\\center\n\\scalebox{0.15}{\\includegraphics{figures\/design.png}}\n\\caption{Example of a basic surface trap design with six pairs of control electrodes. For the ease of fabrication, it is usual to have the control electrodes of equal length except the outer ones which are larger. The surface trap electrodes are assumed to be in the $xz$ plane.}\n\\label{fig:designs}\n\\end{figure}\n\nThe methodology to calculate the electric field due to a planar infinite gap-less electrode has been discussed previously in~\\cite{house_analytic_2008,schmied_electrostatics_2010}. Essentially, it consists of performing the following surface integral for each surface:\n\\begin{eqnarray}\\label{eq:general_integral}\n\\Phi(x,y,z) = \\int \\frac{\\phi(x',y')|z|}{2\\pi ( (x-x')^2 +  (y-y')^2 +z^2  ) ^{3\/2}} dx' dy'\n\\end{eqnarray}\nwith the surface potential $\\phi(x,y) = \\Phi(x,y,0)$ known.\n\nFor the particular case of a square surface, equation~\\ref{eq:general_integral} can be solve analytically~\\cite{house_analytic_2008}.\n\nSuch analytical results are used to compute important parameters, like the height at which the ions will be trapped, $h$. The parameter $h$ plays in fact an important role due to the strong dependency of the heating effects on $h$: $\\propto h^4$ for the anomalous heating~\\cite{turchette_heating_2000} and $\\propto h^2$ for Johnson noise~\\cite{wineland_experimental_1998}.\n\nIn the context of the GL approximation and for the case of equal width of the two RF electrodes (which is already assumed in figure~\\ref{fig:designs}), the value of $h$ can be obtained analytically~\\cite{house_analytic_2008}: $h = \\sqrt{ab + a^2}$. Notice that the expression of $h$ depend only on geometric considerations and not on the applied voltages.\n\nThe trap depth can also be obtained analytically for this particular case~\\cite{house_analytic_2008}:\n\\begin{equation}\nV_\\text{depth} = \\frac{Q^2 V_{RF}^2}{\\pi^2 m \\Omega^2} \\left( \\frac{b}{(a+b)^2 + (a+b)\\sqrt{2ab+a^2}} \\right)^2\n\\end{equation}\nwhere $V_\\text{RF}$ and $\\Omega$ are the voltage amplitude and the frequency of the applied voltage on the RF electrode.\n\nRegarding the axial direction, the potential is defined by the control electrodes. Their total number, $n$, their width $c$, and the voltages applied on them, $V_{cc}$, will essentially determine the potential along the trap axis. In the following, we will present a methodology to obtain the $V_\\text{cc}$ that will best approximate the desired potential.\n\n\\section{Optimization problem}\nThe objective is, given a number of ions, a desired ion-ion distance, $d$ and ion height, $h$, to find a set of parameters, $n$, $c$ and $V_{cc}$ that best approximated the desired experimental situation: that an equally distributed ion chain represents the equilibrium topology.\n\nWe start by defining the position of the ions as a uniform chain at $y=0$, $z=h$ and ion-ion distance of $d$. By imposing that the total force along all directions are equal to zero, we arrive at the following system of equations:\n\n\\begin{align}\n\\sum_{k=1}^{n}{ V_k E^{CC,x}_k(r_i) } = - E^{RF,x}_k(r_i) - E^{DC,x}_k(r_i) \\\\\n\\sum_{k=1}^{n}{ V_k E^{CC,y}_k(r_i) } = C_i -E^{RF,y}_k(r_i) - E^{DC,y}_k(r_i) \\\\\n\\sum_{k=1}^{n}{ V_k E^{CC,z}_k(r_i) } = - E^{RF,z}_k(r_i) - E^{DC,z}_k(r_i)\n\\end{align}\n\nwhere $r_i = (x_i,y_i,z_i)$, $C_i = k_c Q \\sum_j{ \\frac{\\vec{r}_{ij}}{|r_{ij}|^3}}$, $k_c$ is the Coulomb's constant and $E^{CC,x}_k$ ($E^{CC,y}_k$, $E^{CC,z}_k$) denotes the electric field generated by the control electrode $k$ in the $x$ ($y$, $z$) direction when 1V is applied to it. Similar for the RF and DC electrodes.\n\nThe above equations can be re-written as a linear least-squares problem~\\cite{bevington_data_2003}.\n\\begin{eqnarray}\\label{eq:leasst_square}\n   \n    \\chi^2 = || A x - b ||^2\n\\end{eqnarray}\nwhere $A$ is the design matrix and $b$ is the target vector.\n\nIn order to find the $V_i$ values that minimise $\\chi^2 $, we have used the optimization solver found on the Scipy package~\\cite{virtanen_scipy_2020} which allows us to impose bounds on the solutions. While the unbounded linear least-square problems lead to a unique solution based on matrix inversion, the bounded algorithms are iterative. The Scipy solver implements the Trust Region Reflective algorithm~\\cite{branch_subspace_1999}, which iteratively solves trust-region sub-problems based on the on the different variables and the Bounded-Variable least-squares method~\\cite{stark_bounded-variable_1995} where the variables are categorised in free or active, then the algorithm solves the case with the active variables fixed and the free ones as unbounded, effectively solving successively the unconstrained least-square problem. The presence of bounds is a requirement as the voltages supplies or the dielectric strength of the material used, will impose limitations on the maximum allowed voltages.\n\nIn the following, we will assume that only a half of the electrodes are operating independently. The other half is paired up. Following the notation of figure\\ref{fig:designs}, it means that $V_i = V_{-i}$.\n\nFor the particular case of 10 independent control electrodes ($n=10$), $h=100um$, $a=100\\mu m$, $N = 64$, $d_{ion} = 10 \\mu m$, $V_{DC}=0$, $V_{RF} = 200V$, $\\Omega\/2\\pi = 20MHz$ and $V_{max} = 10V$, figure~\\ref{fig:Chi2_1Case}a) shows the evolution of $\\chi^2$ with the width of the control electrodes, $c$ (all taken to be equal, except the last outer ones, which are taken to be $c' = 4c$). The results of the two different algorithms are shown. The noise on the final residue is explained by the large values of the condition number of the $A$ matrix, shown in figure~\\ref{fig:Chi2_1Case}b). In the following, both algorithms will be used and the one with the lower value of $\\chi^2$ will be kept.\n\n\\begin{figure}\n\\center\n\\scalebox{0.35}{\\includegraphics{figures\/CondNumber_h100um_N64_d10um_Vmax10V_nc10}} \\caption{Top: Evolution of $\\chi^2$ vs the control electrode width $c$. The other parameters are given in the main text}\n\\label{fig:Chi2_1Case}\n\\end{figure}\n\nThe dependency of $\\chi^2$ with the electrode width, $c$ and the maximum allowed control voltage, $V_{max}$, is shown in figure~\\ref{fig:vmax_vs_c}. The other parameters remain the same. We observe that by increasing the maximum voltage allowed, a region can be found  with a significantly lower value of $\\chi^2$: from $\\chi^2(V_{max} = 60, c=45um ) = 1.5\\cdot10{-6}$ to $\\chi^2(V_{max} = 70, c=45um ) = 6\\cdot10{-14}$.\n\n\\begin{figure}\n\\center\n\\scalebox{0.5}{\\includegraphics{figures\/N64_d10.0_RFw150.0_VRF200_DCw50.0_Vdc0_nn10}} \\caption{Dependency of $\\chi^2$ (in color-scale) vs the maximum voltage allowed, $V_{max}$, and the control electrodes width,$c$. The color axis correspond to $\\log{\\chi^2}$}.\n\\label{fig:vmax_vs_c}\n\\end{figure}\n\nAnother variable that influences the uniformity, is the number of control electrodes for a fixed value of $V_{max}$. For the case of $V_{max} = 80$V,  we obtain figure~\\ref{fig:n_vs_c}.\n\n\\begin{figure}\n\\center\n\\scalebox{0.5}{\\includegraphics{figures\/N64_d10.0_RFw150.0_VRF200_DCw50.0_Vdc0_Vmax80}} \\caption{Evolution of $\\chi^2$ with the number of control electrodes $n$, and their width, $c$. The color axis correspond to $\\log{\\chi^2}$}\n\\label{fig:n_vs_c}\n\\end{figure}\n\nNotice that in figure~\\ref{fig:n_vs_c}, there is a region with few number of control electrodes that leads to a very low value of $\\chi^2$. However, the voltages provided by the optimization algorithm in such cases leads to axial potential depths (APD) that are also very low. For example, for $c = 45\\mu m$ and $n=2$ and $n=4$, the axial potential depth is $V_{APD}\\approx 3.5meV$, while for $n=10$ we obtain $V_{APD}\\approx 70meV$. A higher value of $V_{APD}$ can be obtained by allowing a higher value $V_{max}$: for $V_{max}=100V$, $c=74\\mu m$ and $n=10$, leads to $V_{APD} \\approx 280meV$.\n\n\n\nThe results regarding the value of $\\chi^2\/N$ for a given set of trap and ion chain parameters provide useful information for the decision making during the design process. However, it does not directly provide information about the degree of homogeneity of the ion chain. Such issue is the topic of the next section.\n\n\\section{Molecular dynamic simulations}\n\nIn the following we will use the values of $V_{cc}$ obtained using the methodology presented above to perform realistic molecular dynamic simulations of the trapped ions. The equations of motion to be solved for each ion $i$, are given by:\n\\begin{align}\nm \\partial_{tt} x_i = Q^2 k_C\\sum_{j=1,j\\neq i}^N{\\frac{x_i - x_{j}}{|\\vec{r}_{i}-\\vec{r}_{j}|^3}} + Q E^x(r_i)- \\Gamma\\partial_t x_i \\nonumber\\\\\nm \\partial_{tt} y_i = Q^2 k_C\\sum_{j=1,j\\neq i}^N{\\frac{y_i - y_{j}}{|\\vec{r}_{i}-\\vec{r}_{j}|^3}} + Q E^y(r_i)- \\Gamma\\partial_t y_i \\nonumber \\\\\nm \\partial_{tt} z_i = Q^2 k_C\\sum_{j=1,j\\neq i}^N{\\frac{z_i - z_{j}}{|\\vec{r}_{i}-\\vec{r}_{j}|^3}} + Q E^z(r_i)- \\Gamma\\partial_t z_i \n\\end{align}\nwhere $\\Gamma$ is the friction coefficient due to laser cooling, $r = (x,y,z)$ and the electric field of trap is given by:\n\n\\begin{align}\nE^j(r_i) & = V_{RF}\\cos(\\Omega t) E^{RF,j}_k(r_i) + V_{cc}E^{CC,j}_k(r_i)  \\nonumber \\\\\n         & + \\sum_{k=1}^{n_c}{ V_k E^{CC,j}_k(r_i) } \n\\end{align}\nwhere $j$ denotes the orthogonal spatial components $(x,y,z)$. The C code which implements the GL approximation analytical solutions is a modification of a code reported in~\\cite{pedregosa-gutierrez_direct_2021}.\n\nThe ions are initialised at the desired position, i.e., as an homogenous ion chain with inter-ion distance equal $d$ at a height $h$. The trap frequency used is $\\Omega \/ 2 \\pi = 20MHz$, $V_{RF} = 200V$, $V_{DC} = 0V$, the friction term used is $\\Gamma = 2.0\\cdot 10^{-19} kg\/s$ and the time-step of the Velocity-Verlet algorithm is $dt = \\Omega \/ 100 = 2ns$. The final positions after $1~$ms simulation are used to evaluate the final inhomogeneity of the ions corresponding to a particular set of $V_{cc}$. The inhomogeneity is defined as the ratio between the spacing standard deviation and the mean of such spacing~\\cite{pagano_cryogenic_2018}: $\\xi = \\sigma_{\\Delta x} \/ \\bar{\\Delta x}$. By using the same parameter space as figure~\\ref{fig:vmax_vs_c}, we obtain figure~\\ref{fig:example_dd_N64}, where the colour code represents $\\log_{10}(100\\xi)$.\n\n\\begin{figure}\n\\center\n\\scalebox{0.60}{\\includegraphics{figures\/example_dd_N64.png}} \\caption{Left: Dependency of inhomogeneity at the end of MD given by $\\xi$, see text for details, versus the maximum voltage allowed, $V_{max}$ and the width of the control electrodes, $c$. The colour code indicates $\\log{100\\xi}$. The Black contour represents $\\xi = 2\\%$, while the Red contour represents the $\\xi = 1.75\\%$. Right: Two examples of final configurations. Big Blue Dots: $V_{max} = 10V$ and $c = 89\\mu m$ leading to $\\xi = 3.2\\%$. Small Orange Dots: $V_{max} = 100V$ and $c = 80\\mu m$ leading to $\\xi = 1.6\\%$.}\n\\label{fig:example_dd_N64}\n\\end{figure}\n\nOn the right side of figure~\\ref{fig:example_dd_N64}, two examples of the ion-ion distances are shown. It is clear that the two extremes ions are responsible for a significant increase on the $\\xi$ parameter. If such two ions are excluded on the computation of $\\xi$, then the $\\xi$ parameter would decrease from $3.2\\% \\to 1.5\\%$ ($1.6\\% \\to 0.9\\%$) for the $V_{max} = 10V$ ($V_{max} = 100V$) case shown. \n\nIn fact, the two extreme ions can be removed from the optimization problem, i.e., we find the set of $V_{cc}$ that leads to the more homogeneous ion chain, with the exception of the two extreme ions. If we perform again molecular dynamic simulations with such new sets of $V_{cc}$, it leads to figure~\\ref{fig:N64_2D_Without_Extremes}. The range of parameters where the degree of homogeneity is lower than 2\\% has significantly increased and it is now possible to achieve values of $\\xi<1.5$ (green contour). The approach can be extended to exclude a greater number of ions from each side, which it will improve even further the degree of homogeneity. Such flexibility could be exploited to compensate for technical limitations on the possible maximum values or sizes of the electrodes.\n\n\\begin{figure}\n\\center\n\\scalebox{0.5}{\\includegraphics{figures\/Inh_N64_c_vs_Vmax_without_extremes.png}} \\caption{Left: Similar to figure~\\ref{fig:example_dd_N64} where the two extreme ions have been excluded of the optimization method to obtain the set of $V_{cc}$ voltages.}\n\\label{fig:N64_2D_Without_Extremes}\n\\end{figure}\n\n\\section{From Harmonic trap to homogeneous ion chain}\n\nThe set of voltages leading to an homogeneous ion chain are not necessarily the more appropriate choice to trap and cool the ion chain down to ground state. For this reason, we perform molecular dynamic simulations where the ions are initialised in the well known and studied~\\cite{RN133} harmonic potential, followed by a transition to the values corresponding to the homogeneous ion chain.\n\nThe half-length of an ion chain in an harmonic potential of secular frequency, $\\omega$, can be estimated to be~\\cite{dubin_minimum_1997}:\n\\begin{equation}\nL^3= \\frac{3 N Q^2 k_C}{m \\omega^2} (\\ln N + \\ln + \\gamma_e -7\/2 )\n\\end{equation}\n\nWe have use an approach similar as above to find the needed voltages to apply in order to obtain an harmonic potential with $\\omega \/ 2\\pi = 212kHz$, which should lead to a total ion chain length similar for both the harmonic potential and the homogeneous potential case.\n\nAfter initializing the ions in the harmonic potential, the simulation waits for $50\\mu$s before the transition starts to the new set of voltage values. A transition time of $t_g = 50\\mu$s has been used. After the transition has finished, the simulation continues during $50\\mu$s more. In order to change smoothly the voltages from the harmonic to the homogenous case, we have used an hyperbolic tangent transition, given by~\\cite{pedregosa-gutierrez_ion_2015,kamsap_fast_2015, kamsap_fast_2015-1}:\n\\begin{equation}\n    V(t) = V_{1} + \\frac{V_{2} - V_{1} }{2} \\left( \\frac{\\tanh(n_h(2t\/t_g - 1))}{\\tanh(n_h)}  + 1\\right)\n\\end{equation}\nwhere $V_1$ correspond to the set of voltages needed for the harmonic case, $V_2$ correspond to the set of voltages needed for the homogeneous case and $n_h$ parameter is used to go from a linear evolution ($n_h = 1$) to a step function ($n_h \\to \\infty$). The evolution of the axial position of the ions is shown in figure~\\ref{fig:From_Harmo_to_Homo}, where $n_h = 4$ has been used. It illustrates that the approach proposed should be feasible and applicable to existing surface traps.\n\n\\begin{figure}\n\\center\n\\scalebox{0.35}{\\includegraphics{figures\/From_Harmo_to_Homo.png}} \\caption{Evolution of the axial position when the trapping voltages are evolved using a \"tangent gate\" (see main text for details) from an harmonic potential to the homogeneous one.}\n\\label{fig:From_Harmo_to_Homo}\n\\end{figure}\n\n\\section{Conclusion}\nIn summary, we have developed a novel algorithm capable to obtain the control electrodes voltages of a surface ion trap needed to generate, with high fidelity, an uniform ion distribution and an harmonic ion distribution. The approach followed is general and therefore, it should be applicable to a large range of desired ion distributions. The approach is based on local minimization of the total force field at each individual ion position. While the present work has used the obtained the needed electric-fields using the GL approximation, the method can be use electric-fields obtained by FEM\/BEM methods to more accurately represent the final geometry.\n\nThe validity of the different voltages sets obtained by our method have been  cross-checked against a robust molecular dynamics simulation. The simulation results show that even for a relatively large ion number, uniformity below $2\\%$ is achievable with only $10$ pairs of electrodes and limited voltage applied to them. Therefore, it opens up the possibilities to implement an uniform chain of trapped and laser cooled ions for realizing homogeneous KZM, implementing large scale entanglement using the BHM and scaling up the nodes of a distributed quantum computing architecture.     \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOnline learning has been widely studied and efficiently applied to sequentially arriving data problems that cannot be solved by batch learning \\cite{cesa2006prediction, shalev2011online, hoi2018online}. In online binary classification, Perceptron \\cite{rosenblatt1958perceptron} is one of the most popular algorithms based on the first-order information. The  Perceptron adopts the following step loss function to minimize misclassification:\n\\begin{equation} \\label{eq.l_sign}\n\\ell_{step}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0 & \\mathbf{\\pmb{w} \\cdot x}^{}_{t} \\geq 0\\\\\n1 & \\mathbf{\\pmb{w} \\cdot x}^{}_{t} < 0\n\\end{array} \\right. ,\n\\end{equation}\nwhere $\\pmb{w} \\in \\mathbb{R}^{d}$ is a parameter vector, and $\\mathbf{x}_{t} \\in \\mathbb{R}^{d}$ is a sample vector at time $t$.\nIn view of margin based learning, online passive-aggressive learning  (PA)  \\cite{crammer2006online} has been successfully developed as a notable online learning method. The PA updates their learning parameters with a large margin obtained by the following hinge loss function:\n\\begin{equation} \\label{eq.hinge_loss}\n\\ell_{hinge}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0  \t\t\t\t\t\t\t\t\t\t\t \t\t& y_{t}\\left(\\mathbf{\\pmb{w}\\cdot x}_{t}\\right) \\geq 1\\\\\n1-y_{t}\\left(\\mathbf{\\pmb{w}\\cdot x}_{t}\\right) \t& y_{t}\\left(\\mathbf{\\pmb{w}\\cdot x}_{t}\\right) < 1\n\\end{array} \\right. ,\n\\end{equation}\nwhere $\\displaystyle y_{t} \\in \\{-1,1\\}$ is a target label.\nThe PA also accommodates non-separable data problems by not assuming data separability.\nIn the use of the second-order information, \nthe second-order perceptron \\cite{cesa2005second}, the confidence weighted learning \\cite{dredze2008confidence}, the adaptive regularization of weights learning \\cite{crammer2009adaptive} have received considerable attention.\nAlthough the second-order information ensures the faster optimization convergence, the use of the first-order information is still reasonably attractive due to its simplicity.\n\nDifferent from the margin based learning, Total-Error-Rate minimization (TER) based learning \\cite{toh2008deterministic} has been introduced  with a classification error approximation for binary classification (see Section \\ref{sec.TER}). The TER takes two different step loss functions for false negative and false positive involved in the confusion matrix of binary classification. In order to achieve a deterministic global solution and avoid local solutions, the TER objective function adopts a quadratic approximation to the step function for the desired convexity.  \nHowever, the above TER based solution has constructed a batch learning solution that has  poor scalability for large-scale and real-time applications.\n\nIn this paper, we introduce a new online learning algorithm\nwhich can put in use the properties of PA and TER and overcome their individual shortcomings by combining them in which we call PATER for online binary classification. \nIn addition, motivated by a weighted accuracy scheme \\cite{brodersen2010balanced}, we also propose a weighted PATER (wPATER) method for data imbalance problems.\nOur empirical results demonstrate that the proposed methods show promising performances on 31 real-world data sets in terms of efficiency and effectiveness.\n\n\n\n\n\\section{Preliminaries}\n\\subsection{Online Passive-Aggressive Learning (PA)}\nWe introduce online Passive-Aggressive learning (PA) which is one of the popular online learning methods of linear classifiers for binary classification \\cite{crammer2006online}. The optimization problem of the PA  is as follows:\n\\begin{equation} \\label{eq.PA_objective}\n{{\\pmb{w}}_{t + 1}} = \\arg \\min \\frac{1}{2}{\\left\\| {{\\pmb{w}} - {{\\pmb{w}}_t}} \\right\\|^2}   \\text{~~~} s.t.  \\text{~~~} \\ell_{hinge}(\\pmb{w}) = 0,\n\\end{equation}\nBy solving this optimization problem, the PA solution is given by:\n\\begin{equation}\n{{\\pmb{w}}_{t + 1}} = {{\\pmb{w}}_t} + \\tau_t y_t \\mathbf{x}_t   ,\n\\end{equation}\nwhere $\\tau_t = \\frac{\\ell_{hinge}(\\pmb{w})}{\\left\\| \\mathbf{x}_t \\right\\|^2}  $ is a Lagrange multiplier under the Karush-Khun-Tucker (KKT) condition \\cite{boyd2004convex}. The online PA solution is passively updated when the loss is close zero,  whereas it is aggressively updated when it suffered from a significant loss.\n\n\n\n\n\\subsection{Total-Error-Rate (TER) Minimization}\n\\label{sec.TER}\n\n\n\n\n\nIn \\cite{toh2008deterministic}, a Total-Error-Rate (TER) has been utilized in an optimization problem for binary classification. The TER is defined as the sum of the False Positive Rate (FPR) and the False Negative Rate (FNR):\n\\begin{equation} \\label{eq.l_step_TER}\n\\begin{aligned}\n\\ell _{step}^{TER}(\\pmb{w}) &= FPR + FNR \\\\\n&=\\frac{1}{n^{-}}\\sum ^{n^{-}}_{i=1} \\ell _{step}^{FP}(\\pmb{w}) +\\frac{1}{n^{+}}\\sum ^{n^{+}}_{j=1} \\ell _{step}^{FN}(\\pmb{w}) ,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq.l_step_FP_FN}\n\\begin{array}{c}\n\\ell_{step}^{FP}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0 & -\\mathbf{\\pmb{w} \\cdot x}^{-}_{t} \\geq 0\\\\\n1 & -\\mathbf{\\pmb{w} \\cdot x}^{-}_{t} < 0\n\\end{array} \\right. ,\\\\ \\\\\n\\ell_{step}^{FN}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0 & \\phantom{-}\\mathbf{\\pmb{w} \\cdot x}^{+}_{t} \\geq 0\\\\\n1 & \\phantom{-}\\mathbf{\\pmb{w} \\cdot x}^{+}_{t} < 0\n\\end{array} \\right. .\n\\end{array}\n\\end{equation}\n$n^-$ and $n^+$ is the numbers of negative and positive samples respectively $(n = n^- + n^+)$.\nThe two superscripts, $-$ and $+$, are for  indication of each class.\nThe TER objective function has been presented based on the quadratic approximation to the two step loss functions of FP and FN as follows: \n\\begin{equation} \\label{eq.TER_J}\n\\footnotesize\n\\begin{aligned} {{\\pmb{w}}} &= \\arg \\min \\frac{1}{n^{-}}\\sum ^{n^{-}}_{i=1} \\ell _{quad}^{FP}(\\pmb{w}) +\\frac{1}{n^{+}}\\sum ^{n^{+}}_{j=1} \\ell _{quad}^{FN}(\\pmb{w}) ,\n\\\\\n&= \\arg \\min \\frac{1}{{2n_{}^ - }}\\sum\\limits_{i = 1}^{n_{}^ - } {{{\\left( {y_i^ -  - ({\\pmb{w }}\\cdot{\\bf{x}}_i^ -) } \\right)}^2}}  + \\frac{1}{{2n_{}^ + }}\\sum\\limits_{j = 1}^{n_{}^ + } {{{\\left( {y_j^ +  - ({\\pmb{w }}\\cdot{\\bf{x}}_j^ +) } \\right)}^2}}, \n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation} \\label{eq.l_quad_FP_FN}\n\\begin{array}{c}\n\\ell_{quad}^{FP}(\\pmb{w}) = {{{\\left( {y^ -  - ({\\pmb{w }}\\cdot{\\bf{x}}^ -) } \\right)}^2}} , \\\\ \\\\\n\\ell_{quad}^{FN}(\\pmb{w}) = {{{\\left( {y^ +  - ({\\pmb{w }}\\cdot{\\bf{x}}^ +) } \\right)}^2}} ,\n\\end{array}\n\\end{equation}\nwhich yields a closed-form solution related to weighted least-squares as \n\\begin{equation} \\label{eq.TER_sol}\n\\pmb{w} = (\\mathbf{X}^T\\mathbf{WX} )^{-1}\\mathbf{X}^T\\mathbf{Wy},\n\\end{equation}\nwhere $\\mathbf{X} =\\left[\n\\mathbf{X}^{-},  \\mathbf{X}^{+}\n\\right]^{T}$ includes two sample matrices for negative and positive classes.\n$\\mathbf{W} =\ndiag\\begin{pmatrix}\\begin{bmatrix}\\frac{1}{n^-}\\,,\\,\\dotsc\\,,\\,\\frac{1}{n^-}\\,,\\,\\frac{1}{n^+}\\,,\\,\\dotsc\\,,\\,\\frac{1}{n^+}\n\\end{bmatrix}\\end{pmatrix}$ is a weight matrix that includes two different weights for the negative and positive classes.  \n$\\mathbf{y} = \\begin{bmatrix} -1,\\,\\dotsc,\\,-1,\\,1,\\,\\dotsc,\\,1\\end{bmatrix}$ is the target vector.\nThe batch mode TER solution handles binary classification problems by approximately counting the misclassified samples using the two different  weights, $1 \\over n^-$  and $1 \\over n^+$. \n\n\\section{Online Passive-Aggressive Total-Error-Rate Minimization (PATER)}\n\\label{sec.PROPOSED1}\n\n\nIn this section, based on the online PA learning, we present an online TER minimization algorithm  (PATER).\nWe start by addressing the two step loss functions $\\ell_{step}^{FP}(\\pmb{w})$ and $\\ell_{step}^{FN}(\\pmb{w})$ of TER.\nSimilar to the hinge loss function in \\eqref{eq.hinge_loss}, these $\\ell_{step}^{FP}(\\pmb{w})$ and $\\ell_{step}^{FN}(\\pmb{w})$ can be rewritten as:\n\\begin{equation} \\label{eq.l_hinge_FP_FN}\n\\begin{array}{c}\n\\ell_{hinge}^{FP}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0                                  \t& -\\mathbf{\\pmb{w} \\cdot x}^{-}_{t} \\geq 1\\\\\n1+\\mathbf{\\pmb{w} \\cdot x}^{-}_{t} \t& -\\mathbf{\\pmb{w} \\cdot x}^{-}_{t}    < 1\n\\end{array} \\right. ,\\\\ \\\\\n\\ell_{hinge}^{FN}(\\pmb{w}) =\\left\\{\\begin{array}{ c c }\n0  \t\t\t\t\t\t\t\t\t& \\phantom{-}\\mathbf{\\pmb{w} \\cdot x}^{+}_{t} \\geq 1\\\\\n1-\\mathbf{\\pmb{w} \\cdot x}^{+}_{t} \t& \\phantom{-}\\mathbf{\\pmb{w} \\cdot x}^{+}_{t}    < 1\n\\end{array} \\right. .\n\\end{array}\n\\end{equation}\nBased on $\\ell_{hinge}^{FP}(\\pmb{w})$ and $\\ell_{hinge}^{FN}(\\pmb{w})$ in \\eqref{eq.l_hinge_FP_FN}, a new TER loss function is given by:\n\\begin{equation} \\label{eq.l_hinge_TER}\n\\ell _{hinge}^{TER}(\\pmb{w}) =\\frac{1}{n^{-}}\\sum ^{n^{-}}_{i=1} \\ell _{hinge}^{FP}(\\pmb{w}) +\\frac{1}{n^{+}}\\sum ^{n^{+}}_{j=1} \\ell _{hinge}^{FN}(\\pmb{w})\n\\end{equation}\n\nSimilar to the PA objective function in \\eqref{eq.PA_objective}, a new TER objective function using \\eqref{eq.l_hinge_TER} can be written as:\n\\begin{equation} \\label{eq.TER_objective}\n{{\\pmb{w}}_{t + 1}} = \\arg \\min \\frac{1}{2}{\\left\\| {{\\pmb{w}} - {{\\pmb{w}}_t}} \\right\\|^2}   \\text{~~~} s.t.  \\text{~~~} \\ell _{hinge}^{TER}(\\pmb{w})  = 0.\n\\end{equation}\nThe Lagrangian of optimization problem for TER minimization can then be defined as follows:\n\\begin{equation}  \\label{eq.Lagrangian_TER}\n\\begin{aligned}\n\\mathcal{L}( \\pmb{w},\\tau ) &= \\frac{1}{2}\\Vert \\pmb{w}-\\pmb{w}_{t}\\Vert ^{2}\\\\\n&\\phantom{=} +\\tau _{t}\\left(\\frac{1}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1} 1+\\mathbf{\\pmb{w}\\cdot x}^{-}_{i} +\\frac{1}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1} 1-\\mathbf{\\pmb{w}\\cdot x}^{+}_{j}\\right) .\n\\end{aligned}\n\\end{equation}\nTaking the derivative of $\\mathcal{L}( \\pmb{w},\\tau )$ with respect to $\\pmb{w}$ and setting it to zero as:\n\\begin{equation}\n\\nabla _{\\pmb{w}}\\mathcal{L}( \\pmb{w},\\tau ) =\\pmb{w}-\\pmb{w}_{t} +\\tau _{t}\\left(\\frac{1}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1}\\mathbf{x}^{-}_{i} -\\frac{1}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1}\\mathbf{x}^{+}_{j}\\right) =0 ,\n\\end{equation}\nthe PATER solution $\\pmb{w}$ is given by:\n\\begin{equation} \\label{eq.PATER_solution}\n\\pmb{w}=\\pmb{w}_{t} -\\tau _{t}\\left(\\frac{1}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1}\\mathbf{x}^{-}_{i} -\\frac{1}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1}\\mathbf{x}^{+}_{j}\\right) .\n\\end{equation}\nIn order to transform the two summation terms into vectorized recursive forms in \\eqref{eq.PATER_solution}, \nthe PATER solution $\\pmb{w}$ can be rewritten as:\n\\begin{equation}\\label{eq.PATER_solution2}\n\\begin{aligned}\n\\pmb{w} &= \\pmb{w}_{t} -\\tau _{t}\\left(\\left(\\frac{n^{-}_{t-1}}{n^{-}_{t}}\\sum ^{n^{-}_{t-1}}_{i=1}\\frac{1}{n^{-}_{t-1}}\\mathbf{x}^{-}_{i} +\\lambda ^{-}_{t}\\frac{1}{n^{-}_{t}}\\mathbf{x}^{}_{t}\\right)  \\right.\\\\\n&\\phantom{=} \\left. -\\left(\\frac{n^{+}_{t-1}}{n^{+}_{t}}\\sum ^{n^{+}_{t-1}}_{j=1}\\frac{1}{n^{+}_{t-1}}\\mathbf{x}^{+}_{j} +\\lambda ^{+}_{t}\\frac{1}{n^{+}_{t}}\\mathbf{x}^{}_{t}\\right) \\right)\\\\\n&= \\pmb{w}_{t} -\\tau _{t}\\left(\\left(\\frac{n^{-}_{t-1}}{n^{-}_{t}}\\mathbf{z}^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\mathbf{x}^{}_{t}\\right) -\\left(\\frac{n^{+}_{t-1}}{n^{+}_{t}}\\mathbf{z}^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\mathbf{x}^{}_{t}\\right)\\right)\\\\\n&= \\pmb{w}_{t} -\\tau _{t}\\left(\\mathbf{z}^{-}_{t} -\\mathbf{z}^{+}_{t}\\right)\\\\\n&= \\pmb{w}_{t} +\\tau _{t}\\mathbf{z}_{t} ,\n\\end{aligned}\n\\end{equation}\nwhere  $ \\lambda ^{-}_{t} =\\frac{1-y_{t}}{2}$ and $ \\lambda ^{+}_{t} =\\frac{1+y_{t}}{2}$ are  indicators to select one class between the two classes at time $t$. \n$ \\mathbf{z}_{t} =\\mathbf{z}^{+}_{t} - \\mathbf{z}^{-}_{t}$, \n$ \\mathbf{z}^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}}\\mathbf{z}^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\mathbf{x}^{}_{t}$, and\n$ \\mathbf{z}^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}}\\mathbf{z}^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\mathbf{x}^{}_{t}$.\n\nSubstituting the parameter vector $\\pmb{w}$ with \\eqref{eq.PATER_solution2} in \\eqref{eq.Lagrangian_TER}, we get\n\\begin{equation} \\label{eq.Lagrangian2_TER}\n\\begin{aligned}\n\\mathcal{L}( \\tau ) & =-\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}\\\\\n&\\phantom{=} +\\tau \\left(\\frac{1}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1} 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{-}_{i} +\\frac{1}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1} 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{+}_{j}\\right) .\n\\end{aligned}\n\\end{equation}\nIn \\eqref{eq.Lagrangian2_TER}, the linearly increasing computational complexity $ \\mathcal{O}( dn_{t})$ of the summation terms is observed and caused by the dot product of $\\pmb{w}_{t}\\mathbf{\\cdot x}$ for each sample. \nTo avoid this computation, two different versions, namely PATER-I and PATER-II, of \\eqref{eq.Lagrangian2_TER} are given by:\n\\begin{equation} \\label{eq.PATER1}\n\\begin{aligned}\n\\text{PATER-I: }\\mathcal{L}( \\tau ) & =-\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\tau \\left( \\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right. \\\\\n&\\phantom{=} \\left. +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right) ,\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{eq.PATER2}\n\\begin{aligned}\n\\text{PATER-II: }\\mathcal{L}( \\tau ) & =-\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\tau \\left(\\frac{n^{-}_{t-1}}{n^{-}_{t}} k^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right. \\\\\n&\\phantom{=} \\left. +\\frac{n^{+}_{t-1}}{n^{+}_{t}} k^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right)\\\\\n&= -\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\tau \\left( k^{-}_{t} +k^{+}_{t}\\right) ,\n\\end{aligned}\n\\end{equation}\nwhere $ k^{-}_{0} =k^{+}_{0} =0$,\n$ k^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}} k^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)$, and\n$ k^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}} k^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)$.\n\n\nTaking the derivative of $\\mathcal{L}( \\tau )$ in \\eqref{eq.PATER1} and \\eqref{eq.PATER2} with respect to $\\tau$ and setting it zero as:\n\\begin{equation}  \n\\begin{aligned}\n\\text{PATER-I: }\\frac{\\partial \\mathcal{L}( \\tau )}{\\partial \\tau } & =-\\tau \\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\\\\n&\\phantom{=} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)=0 ,\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \n\\text{PATER-II: }\\frac{\\partial \\mathcal{L}( \\tau )}{\\partial \\tau } =-\\tau \\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +k^{-}_{t} +k^{+}_{t}=0 ,\n\\end{equation}\nwe get:\n\\begin{equation} \\label{eq.PATER1_tau}\n\\text{PATER-I: } \\tau _{t} =\\frac{\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right) +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}} ,\n\\end{equation}\n\\begin{equation} \\label{eq.PATER2_tau}\n\\text{PATER-II: } \\tau _{t} =\\frac{k^{-}_{t} +k^{+}_{t}}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}} .\n\\end{equation}\n\n\n\\begin{algorithm}[t]\n\t\\caption{Passive-Aggressive TER Minimization}\n\t\\label{alg.PATER1}\n\t\\begin{algorithmic}\n\t\t\\STATE {\\bfseries Input:} a sample $\\mathbf{x}_t \\in \\mathbb{R}^{d}$, \\\\\n\t\t\\hskip3.1em a label $y_t \\in \\{-1,+1\\}$\n\t\t\\STATE {\\bfseries Initialize:} $\\pmb{w}_0= \\mathbf{z}_0^-=\\mathbf{z}_0^+=\\mathbf{0}$, \\\\\n\t\t\\hskip4.5em $n_0^-=n_0^+=k_t^-=k_t^+=0$\n\t\n\t\t\n\t\t\n\t\t\n\t\t\\FOR{$t=1,\\dots$ }\n\t\t\\STATE Set $ \\lambda ^{-}_{t} =\\frac{1-y_{t}}{2}$ and $ \\lambda ^{+}_{t} =\\frac{1+y_{t}}{2}$\n\t\t\\STATE Update $n_t^-=n_{t-1}^- + \\lambda ^{-}_{t}$,  \\\\ \n\t\t\\hskip3.1em $n_t^+=n_{t-1}^+ + \\lambda ^{+}_{t}$\n\t\t\\STATE Update $ \\mathbf{z}^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}}\\mathbf{z}^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\mathbf{x}^{}_{t}$, \\\\\n\t\t\\hskip3.1em $ \\mathbf{z}^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}}\\mathbf{z}^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\mathbf{x}^{}_{t}$\n\t\t\\STATE Update $ k^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}} k^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)$, \\\\\n\t\t\\hskip3.1em $ k^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}} k^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)$\n\t\t\\STATE Set $\\mathbf{z}_{t} = \\mathbf{z}^{+}_{t} - \\mathbf{z}^{-}_{t}$\n\t\t\\STATE Set $\\tau _{t}$ as:  \\\\\n\t\t\\hskip2.2em PATER-I:  $\\tau _{t} = \\frac{\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right) +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}}$ \\\\  \n\t\t\\hskip2.2em PATER-II: $\\tau _{t} =\\frac{k^{-}_{t} +k^{+}_{t}}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}}$\n\t\t\n\t\n\t\n\t\n\t\n\t\t\\STATE Update $\\pmb{w}_{t} = \\pmb{w}_{t-1} +\\tau _{t}\\mathbf{z}_{t}$\n\t\t\\ENDFOR\n\t\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Summary}\nThe pseudo code of the proposed PATER algorithm is given in Algorithm \\ref{alg.PATER1}.\nThe main difference from the PA learning is that the PATER solution is given by the newly derived objective function \\eqref{eq.TER_objective} utilizing the PA and the TER learning together.\nThe main difference between PATER-I and PATER-II  is that PATER-I includes an instantaneous loss while PATER-II uses an approximately accumulated loss in $\\tau _{t}$.\n\n\n\n\n\\section{Weighted PATER Minimization (wPATER)}\n\\label{sec.PROPOSED2}\nIn this section, we present a weighted PATER algorithm (wPATER) inspired by the concept of a balanced accuracy for performance evaluation in \\cite{brodersen2010balanced}. \nIn order to address  imbalanced data sets, a balanced accuracy is defined as \n$\\displaystyle bAcc=\\frac{1}{2}\\left(\\frac{TN}{n^{-}} +\\frac{TP}{n^{+}}\\right)$, \nwhere TN is the True Negative, and TP is the True Positive. \nIn view of a generalized design, a weighted accuracy can be written as: \n\\begin{equation} \\label{eq.def_wTER}\nwAcc= \\alpha^- \\frac{TN}{n^{-}} + \\alpha^+ \\frac{TP}{n^{+}},\n\\end{equation}\nwhere $bAcc$ is given by $\\alpha^- = \\alpha^+ = 0.5$.\nIn a similar manner to this weighted accuracy, a weighted TER (wTER)  can be defined as \n\\begin{equation} \\label{eq.def_wTER}\nwTER = \\alpha^- \\frac{FP}{n^{-}} + \\alpha^+  \\frac{FN}{n^{+}}.\n\\end{equation}\nBased on the wTER definition, similar to \\eqref{eq.l_hinge_TER}, \na new wTER loss function is given by:\n\\begin{equation} \\label{eq.l_hinge_wTER}\n\\ell _{hinge}^{wTER}(\\pmb{w}) =\\frac{\\alpha^- }{n^{-}}\\sum ^{n^{-}}_{i=1} \\ell _{hinge}^{FP}(\\pmb{w}) +\\frac{\\alpha^+ }{n^{+}}\\sum ^{n^{+}}_{j=1} \\ell _{hinge}^{FN}(\\pmb{w}) .\n\\end{equation}\nAccordingly, the wTER objective function can be written as:\n\\begin{equation} \\label{eq.wTER_objective}\n{{\\pmb{w}}_{t + 1}} = \\arg \\min \\frac{1}{2}{\\left\\| {{\\pmb{w}} - {{\\pmb{w}}_t}} \\right\\|^2}   \\text{~~~} s.t.  \\text{~~~} \\ell_{hinge}^{wTER}(\\pmb{w}) = 0.\n\\end{equation}\nSimilar to \\eqref{eq.Lagrangian_TER}, the Lagrangian of the optimization problem can then be defined as:\n\\begin{equation}  \\label{eq.Lagrangian_wTER}\n\\begin{aligned}\n\\mathcal{L}( \\pmb{w},\\tau ) &= \\frac{1}{2}\\Vert \\pmb{w}-\\pmb{w}_{t}\\Vert ^{2}\\\\\n&\\phantom{=} + \\tau _{t}\\left(\\frac{\\alpha^-}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1} 1+\\mathbf{\\pmb{w}\\cdot x}^{-}_{i} +\\frac{\\alpha^+}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1} 1-\\mathbf{\\pmb{w}\\cdot x}^{+}_{j}\\right) .\n\\end{aligned}\n\\end{equation}\nSimilarly, the wPATER solution $\\pmb{w}$ is given by:\n\\begin{equation} \\label{eq.wPATER_solution}\n\\pmb{w}=\\pmb{w}_{t} + \\tau _{t}\\mathbf{z}_{t},\n\\end{equation}\nwhere $\\mathbf{z}_{t} = \\alpha^+ \\mathbf{z}^{+}_{t} - \\alpha^- \\mathbf{z}^{-}_{t}$.\nNext, similar to \\eqref{eq.Lagrangian2_TER}, we get:\n\\begin{equation} \\label{eq.Lagrangian2_wTER}\n\\begin{aligned}\n\\mathcal{L}( \\tau ) & =-\\frac{1}{2} \\tau^2\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}\\\\\n&\\phantom{=} + \\tau \\left(\\frac{\\alpha^-}{n^{-}_{t}}\\sum ^{n^{-}_{t}}_{i=1} 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{-}_{i} +\\frac{\\alpha^+}{n^{+}_{t}}\\sum ^{n^{+}_{t}}_{j=1} 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{+}_{j}\\right) .\n\\end{aligned}\n\\end{equation}\n\n\\begin{algorithm}[t]\n\t\\caption{Weighted PATER Minimization}\n\t\\label{alg.wPATER1}\n\t\\begin{algorithmic}\n\t\t\\STATE {\\bfseries Input:} a sample $\\mathbf{x}_t \\in \\mathbb{R}^{d}$, \\\\\n\t\t\\hskip3.1em a label $y_t \\in \\{-1,+1\\}$\n\t\t\\STATE {\\bfseries Initialize:} $\\pmb{w}_0= \\mathbf{z}_0^-=\\mathbf{z}_0^+=\\mathbf{0}$, \\\\\n\t\t\\hskip4.5em $n_0^-=n_0^+=k_t^-=k_t^+=0$, \\\\\n\t\t\\hskip4.5em $\\alpha^- > 0, \\alpha^+ > 0$\n\t\n\t\t\n\t\t\n\t\t\n\t\t\\FOR{$t=1,\\dots$ }\n\t\t\\STATE Set $ \\lambda ^{-}_{t} =\\frac{1-y_{t}}{2}$ and $ \\lambda ^{+}_{t} =\\frac{1+y_{t}}{2}$\n\t\t\\STATE Update $n_t^-=n_{t-1}^- + \\lambda ^{-}_{t}$, \\\\\n\t\t\\hskip3.1em  $n_t^+=n_{t-1}^+ + \\lambda ^{+}_{t}$\n\t\t\\STATE Update $ \\mathbf{z}^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}}\\mathbf{z}^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\mathbf{x}^{}_{t}$, \\\\\n\t\t\\hskip3.1em $ \\mathbf{z}^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}}\\mathbf{z}^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\mathbf{x}^{}_{t}$\n\t\t\\STATE Update $ k^{-}_{t} =\\frac{n^{-}_{t-1}}{n^{-}_{t}} k^{-}_{t-1} +\\frac{\\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)$, \\\\\n\t\t\\hskip3.1em $ k^{+}_{t} =\\frac{n^{+}_{t-1}}{n^{+}_{t}} k^{+}_{t-1} +\\frac{\\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)$\n\t\t\\STATE Set $\\mathbf{z}_{t} = \\alpha^+ \\mathbf{z}^{+}_{t} - \\alpha^- \\mathbf{z}^{-}_{t}$\n\t\t\\STATE Set $\\tau _{t}$ as:  \\\\\n\t\t\\hskip2.2em wPATER-I:  $\\tau _{t} = \\frac{\\frac{\\alpha^- \\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right) +\\frac{\\alpha^+ \\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t-1}\\mathbf{\\cdot x}^{}_{t}\\right)}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}}$ \\\\  \n\t\t\\hskip2.2em wPATER-II: $\\tau _{t} =\\frac{\\alpha^- k^{-}_{t} +\\alpha^+ k^{+}_{t}}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}}$\n\t\t\n\t\n\t\n\t\n\t\n\t\t\\STATE Update $\\pmb{w}_{t} = \\pmb{w}_{t-1} +\\tau _{t}\\mathbf{z}_{t}$\n\t\t\\ENDFOR\n\t\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{table*}[t!]\n\t\\caption{Summary of the 31 real-world data sets for binary classification.} \\label{tbl.dataSummary}\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\fontsize{6}{8}\\selectfont\n\t\t\t\t\\centering \n\t\t\t\t\\begin{tabular}{c@{\\hskip0.5pt}llllllc@{\\hskip3pt}llllll}\n\t\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\t\t& No. {\\rule{0pt}{3ex}} & Data sets          & \\#cases(original)    & \\#feat & Ratio & \\#miss & No. {\\rule{0pt}{3ex}} & Data sets          & \\#cases(original)    & \\#feat & Ratio  & \\#miss   \\\\ \\hline\n\t\t\t\t\t& 1 \n\t\t\t\t\t{\\rule{0pt}{3ex}}  \t\t   \n\t\t\t\t\t& Monks-3     \t   & 122(423)   & 6      & 0.98       & No      \n\t\t\t\t\t& 17  & Blood-transfusion  & 748        & 4      & 0.31       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 2   & Monks-1            & 124(423)   & 6      & 1.01       & No       \n\t\t\t\t\t& 18  & Pima-diabetes      & 768        & 8      & 0.54       & No\t\t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 3   & Monks-2            & 169(423)   & 6      & 0.62       & No       \n\t\t\t\t\t& 19  & Mammographic       & 830(961)   & 5      & 0.95       & 131\t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 4   & Wpbc               & 194(198)   & 33     & 0.31       & 4        \n\t\t\t\t\t& 20  & Tic-tac-toe        & 958        & 9      & 0.53       & No\t\t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 5   & Parkinsons         & 195        & 22     & 3.15       & No       \n\t\t\t\t\t& 21  & Statlog-german     & 1,000      & 24     & 2.34       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 6   & Sonar              & 208        & 60     & 1.16       & No       \n\t\t\t\t\t& 22  & Ozone-eight        & 1,847(2,534) & 72   & 0.07       & 687 \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 7   & SPECTF-heart       & 267        & 44     & 3.89       & No       \n\t\t\t\t\t& 23  & Ozone-one          & 1,848(2,536) & 72   & 0.03       & 688 \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 8   & StatLog-heart      & 270        & 13     & 0.80       & No       \n\t\t\t\t\t& 24  & 20News-talk        & 1,848 \t    & 3      & 1.04       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t& 9   & BUPA-liver         & 345        & 6      & 1.39       & No       \n\t\t\t\t\t& 25  & 20News-comp        & 1,937      & 3      & 0.98       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 10  & Ionosphere         & 351        & 34     & 0.56       & No       \n\t\t\t\t\t& 26  & 20News-sci         & 1,971      & 3      & 1.01       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 11  & Votes              & 435        & 16     & 1.60       & Yes      \n\t\t\t\t\t& 27  & Spambase           & 4,601      & 57     & 0.65       & No \t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 12  & Musk-clean-1       & 476        & 166    & 0.77       & No       \n\t\t\t\t\t& 28  & Mushroom           & 5,644(8,124) & 22   & 1.62       & Attr\\#11 \t\t \t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 13  & Wdbc               & 569        & 30     & 1.69       & No       \n\t\t\t\t\t& 29  & Cod-rna            & 59,535     & 8      & 0.50       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 14  & Credit-app         & 653(690)   & 15     & 1.21       & 37       \n\t\t\t\t\t& 30  & Ijcnn1             & 141,691    & 22     & 0.11       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 15  & Breast-cancer-W    & 683(699)   & 9      & 0.54       & 16 \t\t\t\n\t\t\t\t\t& 31  & Skin-nonskin       & 245,057    & 3      & 0.26       & No \t\t\t\\\\\n\t\t\t\t\t\n\t\t\t\t\t& 16  & Statlog-australian & 690        & 14     & 0.81       & Yes \t\t\t\n\t\t\t\t\t&     &      &     &      &      &   \t\t\t\\\\ \\hline\n\t\t\t\t\\end{tabular} \n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table*}\n\n\\begin{table*}[t!]\n\t\n\t\\caption{Comparison of average accuracies with its standard deviations and ranks.} \\label{tbl.avgacc}\n\t\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\fontsize{6.5}{8.5}\\selectfont\n\t\t\t\t\\centering \n\t\t\t\t\\begin{tabular}{l@{\\hskip0.5pt}l@{\\hskip2pt}c@{\\hskip2pt}c@{\\hskip2pt}c@{\\hskip2pt}c@{\\hskip2pt}c@{\\hskip2pt}c@{\\hskip2pt}}\n\t\t\t\t\t\\hline\n\t\t\t\t\tNo. {\\rule{0pt}{3ex}} & Data sets          & PE                        & PA                        & PATER-I                    & PATER-II                    & wPATER-I                   & wPATER-II                   \\\\ \\hline\n\t\t\t\t\t1   {\\rule{0pt}{3ex}} & Monks-3            & 70.902 $\\pm$ 6.446 (6)    & 71.066 $\\pm$ 4.795 (5)    & 75.574 $\\pm$ 4.218 (4)    & 75.738 $\\pm$ 3.821 (3)    & 75.984 $\\pm$ 4.297 (2)    & \\textbf{77.295 $\\pm$ 4.675 (1)}    \\\\\n\t\t\t\t\t2   & Monks-1            & 64.355 $\\pm$ 6.407 (5)    & 64.355 $\\pm$ 7.760 (6)    & 64.919 $\\pm$ 7.454 (4)    & 67.177 $\\pm$ 3.747 (2)    & 67.097 $\\pm$ 5.976 (3)    & \\textbf{69.919 $\\pm$ 3.891 (1)}    \\\\\n\t\t\t\t\t3   & Monks-2            & 54.195 $\\pm$ 6.741 (3)    & 52.426 $\\pm$ 6.664 (4)    & 49.235 $\\pm$ 4.245 (6)    & 50.049 $\\pm$ 5.094 (5)    & 61.720 $\\pm$ 4.682 (2)    & \\textbf{62.132 $\\pm$ 4.186 (1)}    \\\\\n\t\t\t\t\t4   & Wpbc               & 67.835 $\\pm$ 5.274 (3)    & 65.670 $\\pm$ 5.558 (4)    & 59.021 $\\pm$ 3.899 (5)    & 55.309 $\\pm$ 5.375 (6)    & \\textbf{72.113 $\\pm$ 4.443 (1)}    & 69.278 $\\pm$ 4.749 (2)    \\\\\n\t\t\t\t\t5   & Parkinsons         & 76.563 $\\pm$ 5.060 (3)    & 78.150 $\\pm$ 6.388 (2)    & 67.586 $\\pm$ 3.124 (5)    & 66.408 $\\pm$ 2.809 (6)    & \\textbf{82.618 $\\pm$ 4.467 (1)}    & 74.822 $\\pm$ 5.296 (4)    \\\\\n\t\t\t\t\t6   & Sonar              & 69.904 $\\pm$ 5.202 (5)    & 72.067 $\\pm$ 3.424 (4)    & 72.788 $\\pm$ 4.335 (2)    & 69.615 $\\pm$ 5.242 (6)    & \\textbf{74.471 $\\pm$ 3.618 (1)}    & 72.115 $\\pm$ 6.378 (3)    \\\\\n\t\t\t\t\t7   & SPECTF-heart       & 72.361 $\\pm$ 3.659 (3)    & 72.473 $\\pm$ 4.124 (2)    & 55.580 $\\pm$ 3.530 (5)    & 53.633 $\\pm$ 2.681 (6)    & \\textbf{76.405 $\\pm$ 3.015 (1)}    & 70.447 $\\pm$ 3.898 (4)    \\\\\n\t\t\t\t\t8   & Statlog-heart      & 78.667 $\\pm$ 4.303 (5)    & 77.667 $\\pm$ 5.370 (6)    & 81.000 $\\pm$ 2.867 (4)    & 82.815 $\\pm$ 2.704 (2)    & 82.111 $\\pm$ 2.956 (3)    & \\textbf{84.037 $\\pm$ 2.046 (1)}    \\\\\n\t\t\t\t\t9   & BUPA-liver         & 58.233 $\\pm$ 4.359 (4)    & 57.769 $\\pm$ 3.575 (5)    & 59.535 $\\pm$ 2.958 (2)    & 54.463 $\\pm$ 3.414 (6)    & \\textbf{61.161 $\\pm$ 3.785 (1)}    & 58.550 $\\pm$ 4.726 (3)    \\\\\n\t\t\t\t\t10  & Ionosphere         & 80.573 $\\pm$ 4.352 (5)    & 84.302 $\\pm$ 2.641 (2)    & 81.711 $\\pm$ 5.333 (3)    & 73.478 $\\pm$ 6.391 (6)    & \\textbf{85.128 $\\pm$ 3.707 (1)}    & 81.367 $\\pm$ 4.494 (4)    \\\\\n\t\t\t\t\t11  & Votes              & 90.481 $\\pm$ 3.674 (3)    & 91.605 $\\pm$ 3.796 (2)    & 90.369 $\\pm$ 1.994 (4)    & 87.999 $\\pm$ 1.633 (6)    & \\textbf{92.368 $\\pm$ 1.608 (1)}    & 88.644 $\\pm$ 1.396 (5)    \\\\\n\t\t\t\t\t12  & Musk-clearn-1      & 70.399 $\\pm$ 5.390 (4)    & 70.546 $\\pm$ 5.285 (3)    & 74.958 $\\pm$ 3.450 (2)    & 62.836 $\\pm$ 6.602 (6)    & \\textbf{75.546 $\\pm$ 3.032 (1)}    & 63.592 $\\pm$ 5.455 (5)    \\\\\n\t\t\t\t\t13  & Wdbc               & 94.728 $\\pm$ 2.591 (4)    & 95.518 $\\pm$ 1.985 (3)    & 96.010 $\\pm$ 0.963 (2)    & 93.145 $\\pm$ 1.369 (6)    & \\textbf{96.116 $\\pm$ 0.886 (1)}    & 93.479 $\\pm$ 1.225 (5)    \\\\\n\t\t\t\t\t14  & Credit-app         & 80.503 $\\pm$ 5.775 (6)    & 81.682 $\\pm$ 5.579 (5)    & 84.349 $\\pm$ 1.957 (4)    & 85.284 $\\pm$ 1.519 (2)    & 84.441 $\\pm$ 1.818 (3)    & \\textbf{85.376 $\\pm$ 1.581 (1)}    \\\\\n\t\t\t\t\t15  & Breast-cancer-W    & 95.623 $\\pm$ 2.029 (5)    & 95.491 $\\pm$ 2.126 (6)    & 96.969 $\\pm$ 0.858 (3)    & 96.881 $\\pm$ 0.581 (4)    & 97.115 $\\pm$ 0.792 (2)    & \\textbf{97.438 $\\pm$ 0.593 (1)}    \\\\\n\t\t\t\t\t16  & Statlog-australian & 78.841 $\\pm$ 5.582 (6)    & 79.493 $\\pm$ 5.908 (5)    & 84.217 $\\pm$ 2.009 (4)    & 85.507 $\\pm$ 1.128 (2)    & 84.290 $\\pm$ 1.702 (3)    & \\textbf{85.580 $\\pm$ 1.289 (1)}    \\\\\n\t\t\t\t\t17  & Blood-transfusion  & 68.436 $\\pm$ 8.363 (3)    & 66.444 $\\pm$ 11.36 (4)   & 59.024 $\\pm$ 9.134 (6)    & 62.794 $\\pm$ 3.059 (5)    & 76.377 $\\pm$ 1.439 (2)    & \\textbf{76.912 $\\pm$ 1.255 (1)}    \\\\\n\t\t\t\t\t18  & Pima-diabetes      & 68.099 $\\pm$ 3.994 (5)    & 69.505 $\\pm$ 3.482 (4)    & 66.797 $\\pm$ 5.272 (6)    & 72.057 $\\pm$ 1.466 (2)    & 70.208 $\\pm$ 3.635 (3)    & \\textbf{74.740 $\\pm$ 1.262 (1)}    \\\\\n\t\t\t\t\t19  & Mammographic       & 70.482 $\\pm$ 16.87 (6)   & 72.241 $\\pm$ 13.58 (5)   & 76.494 $\\pm$ 8.903 (4)    & 81.349 $\\pm$ 1.096 (2)    & 78.012 $\\pm$ 6.420 (3)    & \\textbf{81.602 $\\pm$ 1.023 (1)}    \\\\\n\t\t\t\t\t20  & Tic-tac-toe        & 55.929 $\\pm$ 5.301 (5)    & 56.013 $\\pm$ 5.805 (4)    & 52.453 $\\pm$ 4.740 (6)    & 56.983 $\\pm$ 2.743 (3)    & 65.376 $\\pm$ 1.731 (2)    & \\textbf{67.056 $\\pm$ 1.786 (1)}    \\\\\n\t\t\t\t\t21  & Statlog-german     & 68.470 $\\pm$ 3.223 (4)    & 66.410 $\\pm$ 4.967 (5)    & 63.120 $\\pm$ 2.671 (6)    & 68.630 $\\pm$ 1.454 (3)    & 71.910 $\\pm$ 1.475 (2)    & \\textbf{75.630 $\\pm$ 1.260 (1)}    \\\\\n\t\t\t\t\t22  & Ozone-eight        & 89.924 $\\pm$ 3.083 (3)    & 90.921 $\\pm$ 3.164 (2)    & 52.528 $\\pm$ 1.977 (6)    & 53.059 $\\pm$ 1.154 (5)    & \\textbf{92.534 $\\pm$ 0.891 (1)}    & 67.845 $\\pm$ 5.597 (4)    \\\\\n\t\t\t\t\t23  & Ozone-one          & 92.516 $\\pm$ 5.130 (3)    & 95.070 $\\pm$ 3.190 (2)    & 50.168 $\\pm$ 1.519 (5)    & 49.800 $\\pm$ 0.945 (6)    & \\textbf{96.374 $\\pm$ 0.478 (1)}    & 79.015 $\\pm$ 10.88 (4)   \\\\\n\t\t\t\t\t24  & 20News-talk        & 49.957 $\\pm$ 1.196 (4)    & 49.854 $\\pm$ 1.030 (5)    & 49.811 $\\pm$ 1.829 (6)    & 50.060 $\\pm$ 0.986 (3)    & \\textbf{51.292 $\\pm$ 2.147 (1)}    & 50.838 $\\pm$ 1.230 (2)    \\\\\n\t\t\t\t\t25  & 20News-comp        & 54.544 $\\pm$ 4.127 (5)    & 53.376 $\\pm$ 5.999 (6)    & 58.839 $\\pm$ 6.779 (3)    & 57.495 $\\pm$ 6.426 (4)    & \\textbf{63.676 $\\pm$ 6.350 (1)}    & 59.172 $\\pm$ 9.193 (2)    \\\\\n\t\t\t\t\t26  & 20News-sci         & 72.572 $\\pm$ 10.24 (4)   & 69.980 $\\pm$ 9.689 (5)    & 73.310 $\\pm$ 9.626 (3)    & 65.591 $\\pm$ 5.703 (6)    & 74.420 $\\pm$ 10.16 (2)   & \\textbf{74.425 $\\pm$ 6.073 (1)}    \\\\\n\t\t\t\t\t27  & Spambase           & 87.142 $\\pm$ 2.570 (5)    & 86.707 $\\pm$ 2.714 (6)    & 89.961 $\\pm$ 1.743 (2)    & 89.048 $\\pm$ 0.462 (4)    & \\textbf{90.915 $\\pm$ 0.894 (1)}    & 89.641 $\\pm$ 0.411 (3)    \\\\\n\t\t\t\t\t28  & Mushroom           & 91.357 $\\pm$ 5.435 (4)    & 93.698 $\\pm$ 3.097 (3)    & 93.820 $\\pm$ 1.721 (2)    & 87.509 $\\pm$ 0.593 (6)    & \\textbf{94.341 $\\pm$ 0.891 (1)}    & 88.521 $\\pm$ 0.761 (5)    \\\\\n\t\t\t\t\t29  & Cod-rna            & 90.590 $\\pm$ 1.627 (2)    & \\textbf{90.669 $\\pm$ 1.822 (1)}    & 87.965 $\\pm$ 1.843 (4)    & 69.606 $\\pm$ 0.246 (6)    & 89.420 $\\pm$ 2.441 (3)    & 77.571 $\\pm$ 0.110 (5)    \\\\\n\t\t\t\t\t30  & Ijcnn1             & 89.655 $\\pm$ 1.811 (4)    & 90.342 $\\pm$ 1.673 (3)    & 59.430 $\\pm$ 3.303 (6)    & 65.847 $\\pm$ 0.690 (5)    & \\textbf{92.029 $\\pm$ 0.266 (1)}    & 90.426 $\\pm$ 0.070 (2)    \\\\\n\t\t\t\t\t31  & Skin-nonskin       & 88.917 $\\pm$ 5.564 (3)    & \\textbf{91.104 $\\pm$ 4.185 (1)}    & 89.302 $\\pm$ 1.610 (2)    & 66.539 $\\pm$ 4.338 (6)    & 88.209 $\\pm$ 0.790 (4)    & 87.725 $\\pm$ 0.077 (5)    \\\\\n\t\t\t\t\t& Average {\\rule{0pt}{3ex}} & 75.573 $\\pm$ 5.012 (4.19) & 75.891 $\\pm$ 4.863 (3.87) & 71.511 $\\pm$ 3.737 (4.06) & 69.571 $\\pm$ 2.757 (4.52) & \\textbf{79.477 $\\pm$ 2.929 (1.77)} & 76.619 $\\pm$ 3.125 (2.58) \\\\ \\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table*}\n\nIn order to establish wPATER-I and wPATER-II, similar to \\eqref{eq.PATER1} and \\eqref{eq.PATER2}, we get:\n\\begin{equation} \\label{eq.wPATER1}\n\\begin{aligned}\n\\text{wPATER-I: }\\mathcal{L}( \\tau ) & =-\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\tau \\left( \\frac{\\alpha^- \\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right. \\\\\n&\\phantom{=} \\left. +\\frac{\\alpha^+ \\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\right) ,\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{eq.wPATER2}\n\\begin{aligned}\n\\text{wPATER-II: }\\mathcal{L}( \\tau ) &  = -\\frac{1}{2} \\tau ^{2}\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\tau \\left(\\alpha^- k^{-}_{t} +  \\alpha^+ k^{+}_{t}\\right) .\n\\end{aligned}\n\\end{equation}\nTaking the derivative of $\\mathcal{L}( \\tau )$ in \\eqref{eq.wPATER1} and \\eqref{eq.wPATER2} with respect to $\\tau$ and setting it zero as:\n\\begin{equation}  \n\\begin{aligned}\n\\text{wPATER-I: }\\frac{\\partial \\mathcal{L}( \\tau )}{\\partial \\tau } & =-\\tau \\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} +\\frac{\\alpha^- \\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)\\\\\n&\\phantom{=} +\\frac{\\alpha^+ \\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)=0 ,\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \n\\text{wPATER-II: }\\frac{\\partial \\mathcal{L}( \\tau )}{\\partial \\tau } =-\\tau \\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2} + \\alpha^- k^{-}_{t} +\\alpha^+ k^{+}_{t}=0 ,\n\\end{equation}\nwe get:\n\\begin{equation} \\label{eq.wPATER1_tau}\n\\text{wPATER-I: } \\tau _{t} =\\frac{\\frac{\\alpha^- \\lambda ^{-}_{t}}{n^{-}_{t}}\\left( 1+\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right) +\\frac{\\alpha^+ \\lambda ^{+}_{t}}{n^{+}_{t}}\\left( 1-\\pmb{w}_{t}\\mathbf{\\cdot x}^{}_{t}\\right)}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}} ,\n\\end{equation}\n\\begin{equation} \\label{eq.wPATER2_tau}\n\\text{wPATER-II: } \\tau _{t} =\\frac{\\alpha^- k^{-}_{t} + \\alpha^+ k^{+}_{t}}{\\left\\Vert \\mathbf{z}_{t}\\right\\Vert ^{2}} .\n\\end{equation}\n\n\n\\begin{figure*}[ht]\n\t\\vskip 0.2in\n\t\\begin{center}\n\t\t\\subfigure[20News-sci]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth] \t{20NewsSci_merged_cma.eps\n\t\t}  \n\t\t\\subfigure[Spambase]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth] \t{spam_merged_cma.eps\n\t\t} \n\t\t\\subfigure[Mushroom]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth]  {mushroom_merged_cma.eps\n\t\t} \n\t\t\\subfigure[Cod-rna]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth] \t{codrna_merged_cma.eps\n\t\t}  \n\t\t\\subfigure[Ijcnn1]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth] \t{ijcnn1_merged_cma.eps\n\t\t} \n\t\t\\subfigure[Skin-nonskin]{ \n\t\t\t\\includegraphics[width=0.31\\textwidth] \t{skin_merged_cma.eps\n\t\t} \n\t\n\t\t\\caption{The cumulative classification accuracies along the online learning process. The enlarged figure is also drawn for detailed comparison. The brackets include the average accuracies.}\n\t\t\\label{fig.cma}\n\t\\end{center}\n\t\\vskip -0.2in\n\\end{figure*}\n\n\\subsection{Summary}\nThe pseudo code of the proposed wPATER algorithm is given in Algorithm \\ref{alg.wPATER1}.\nThe main difference from PATER is that \nwPATER has two weights, $\\alpha^-$ and $\\alpha^+$, as hyperparameters in its loss function \\eqref{eq.l_hinge_wTER} to address the imbalance data issue.\n\n\n\n\n\n\n\\section{Experiments}\nIn this section, we evaluate the performance of the proposed PATER and wPATER for binary classification. \nIn terms of efficiency and effectiveness, the performance comparison is presented based on 31 real-world data sets. \n\n\n\n\\subsection{Data sets and experimental settings}\nIn our evaluation, 31 real-world data sets are obtained from \nUCI machine learning repository\\footnote{\\url{https:\/\/archive.ics.uci.edu\/ml\/index.php}} \\cite{Lichman2013}, \nLIBSVM\\footnote{\\url{https:\/\/www.csie.ntu.edu.tw\/~cjlin\/libsvm\/}} \\cite{chang2011libsvm}\nand 20 Newsgroups\\footnote{\\url{http:\/\/qwone.com\/~jason\/20Newsgroups\/}} which are popular in the NLP community.\nThe description of the 31 data sets is summarized in Table \\ref{tbl.dataSummary}.\nEach data set is conducted in a z-score normalization.\nThe data set size ranges from 122 to 245,057 samples.\nThe data imbalance ratio is calculated by $\\frac{n^+}{n^-}$.\n\n\n\n\n\nFor comparison with competing state-of-the-arts, the following algorithms are used as baselines: the perceptron (PE) \\cite{rosenblatt1958perceptron}  and \nthe online passive-aggressive algorithm (PA) \\cite{crammer2006online}.\nThe average accuracies and CPU times are recorded using 10 runs of 2-fold cross-validation for each algorithm.\n\nIn wPATER, the two weights are varied as $\\alpha^-, \\alpha^+ \\in \\{0.01,\\ 0.1,\\ 0.3,\\ 0.5,\\ 0.9,\\ 0.99\\}$. When one weight is changed, another weight is fixed at $1$ (i.e., $\\alpha^-=0.1$, $\\alpha^+ =1$) to only address one side population of the imbalance binary classification problems.\n\n\n\n\\begin{table*}[t]\n\t\n\t\\caption{Evaluation of the best weights between $\\alpha^-$ and $\\alpha^+ $ of the proposed wPATER  for the data imbalance problems. } \\label{tbl.eval_weights}\n\t\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\fontsize{7}{8.5}\\selectfont\n\t\t\t\t\\centering \n\t\t\t\t\n\t\t\t\t\\begin{tabular}{l c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip4.5pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}c@{\\hskip3pt}}\n\t\t\t\t\t\\hline\n\t\t\t\t\t{\\rule{0pt}{3ex}} & \\multicolumn{31}{l}{Evaluation}                                                                                                                 & Sum \\\\ \\hline\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\tData No.  {\\rule{0pt}{3ex}} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 &     \\\\\n\t\t\t\t\tNeeded Weight (NW)     & P & N & P & P & N & N & N & P & N & P  & N  & P  & N  & N  & P  & P  & P  & P  & P  & P  & N  & P  & P  & N  & P  & N  & P  & N  & P  & P  & P  &     \\\\\n\t\t\t\t\tBest Weight (BW)       & P & P & P & P & N & N & N & P & N & P  & N  & P  & N  & P  & N  & P  & P  & P  & P  & P  & N  & P  & P  & N  & P  & P  & P  & N  & -  & P  & -  &     \\\\\n\t\t\t\t\tResults (BW equals NW) & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1  & 1  & 1  & 1  & 0  & 0  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 0  & 1  & 1  & 0  & 1  & 0  & 25  \\\\ \\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\\begin{flushleft}\n\t\t\t\t\tNotes: `N' and `P' indicate the negative and posivie classes respectively. \n\t\t\t\t\tIn NW, `N' is given if $\\frac{n^+}{n^-} \\geq 1$; otherwise `P' is given. \n\t\t\t\t\t$\\frac{n^+}{n^-} $ is the data imbalance ratio shown in Table \\ref{tbl.dataSummary}. \n\t\t\t\t\tIn BW, `N' is given if the best performance is obtained from the $\\alpha^- $ variation; otherwise `P' is given.\n\t\t\t\t\n\t\t\t\t\\end{flushleft}\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table*}\n\n\\begin{table}[t]\n\t\n\t\\caption{Comparison of average CPU times in seconds.} \\label{tbl.avgtime}\n\t\n\t\\vskip 0.15in\n\t\\begin{center}\n\t\t\\begin{small}\n\t\t\t\\begin{sc}\n\t\t\t\t\\fontsize{6}{8.5}\\selectfont\n\t\t\t\t\\centering \n\t\t\t\t\n\t\t\t\t\\begin{tabular}{l@{\\hskip0.5pt}l@{\\hskip2pt}c@{\\hskip3.5pt}c@{\\hskip3.5pt}c@{\\hskip2.5pt}c@{\\hskip2.5pt}c@{\\hskip2.5pt}c@{\\hskip2.5pt}}\n\t\t\t\t\t\\hline\n\t\t\t\t\tNo. {\\rule{0pt}{3ex}}& Data sets          & PE     & PA     & PATER-I  & PATER-II & wPATER-I & wPATER-II \\\\ \\hline\n\t\t\t\t\t1   {\\rule{0pt}{3ex}}& Monks-3            & 0.0008 & 0.0013 & 0.0017 & 0.0019 & 0.0006  & 0.0009  \\\\\n\t\t\t\t\t2   & Monks-1            & 0.0006 & 0.0006 & 0.0008 & 0.0009 & 0.0006  & 0.0008  \\\\\n\t\t\t\t\t3   & Monks-2            & 0.0007 & 0.0008 & 0.0010 & 0.0012 & 0.0009  & 0.0012  \\\\\n\t\t\t\t\t4   & Wpbc               & 0.0006 & 0.0008 & 0.0011 & 0.0014 & 0.0010  & 0.0014  \\\\\n\t\t\t\t\t5   & Parkinsons         & 0.0006 & 0.0007 & 0.0010 & 0.0014 & 0.0009  & 0.0013  \\\\\n\t\t\t\t\t6   & Sonar              & 0.0007 & 0.0009 & 0.0013 & 0.0016 & 0.0011  & 0.0015  \\\\\n\t\t\t\t\t7   & SPECTF-heart       & 0.0008 & 0.0011 & 0.0015 & 0.0019 & 0.0014  & 0.0018  \\\\\n\t\t\t\t\t8   & Statlog-heart      & 0.0008 & 0.0010 & 0.0014 & 0.0019 & 0.0012  & 0.0018  \\\\\n\t\t\t\t\t9   & BUPA-liver         & 0.0011 & 0.0014 & 0.0019 & 0.0024 & 0.0016  & 0.0022  \\\\\n\t\t\t\t\t10  & Ionosphere         & 0.0010 & 0.0012 & 0.0019 & 0.0025 & 0.0017  & 0.0024  \\\\\n\t\t\t\t\t11  & Votes              & 0.0012 & 0.0013 & 0.0021 & 0.0031 & 0.0018  & 0.0029  \\\\\n\t\t\t\t\t12  & Musk-clearn-1      & 0.0016 & 0.0020 & 0.0028 & 0.0036 & 0.0025  & 0.0034  \\\\\n\t\t\t\t\t13  & Wdbc               & 0.0015 & 0.0015 & 0.0027 & 0.0041 & 0.0023  & 0.0038  \\\\\n\t\t\t\t\t14  & Credit-app         & 0.0019 & 0.0022 & 0.0031 & 0.0046 & 0.0028  & 0.0043  \\\\\n\t\t\t\t\t15  & Breast-cancer-W    & 0.0017 & 0.0017 & 0.0030 & 0.0048 & 0.0028  & 0.0045  \\\\\n\t\t\t\t\t16  & Statlog-australian & 0.0019 & 0.0023 & 0.0033 & 0.0048 & 0.0029  & 0.0046  \\\\\n\t\t\t\t\t17  & Blood-transfusion  & 0.0022 & 0.0026 & 0.0038 & 0.0051 & 0.0038  & 0.0049  \\\\\n\t\t\t\t\t18  & Pima-diabetes      & 0.0023 & 0.0028 & 0.0039 & 0.0054 & 0.0035  & 0.0051  \\\\\n\t\t\t\t\t19  & Mammographic       & 0.0024 & 0.0028 & 0.0039 & 0.0058 & 0.0037  & 0.0054  \\\\\n\t\t\t\t\t20  & Tic-tac-toe        & 0.0031 & 0.0038 & 0.0051 & 0.0065 & 0.0046  & 0.0063  \\\\\n\t\t\t\t\t21  & Statlog-german     & 0.0031 & 0.0038 & 0.0053 & 0.0070 & 0.0046  & 0.0066  \\\\\n\t\t\t\t\t22  & Ozone-eight        & 0.0056 & 0.0064 & 0.0110 & 0.0136 & 0.0086  & 0.0135  \\\\\n\t\t\t\t\t23  & Ozone-one          & 0.0054 & 0.0061 & 0.0109 & 0.0136 & 0.0084  & 0.0135  \\\\\n\t\t\t\t\t24  & 20News-talk        & 0.0061 & 0.0073 & 0.0096 & 0.0119 & 0.0088  & 0.0122  \\\\\n\t\t\t\t\t25  & 20News-comp        & 0.0062 & 0.0075 & 0.0097 & 0.0129 & 0.0090  & 0.0123  \\\\\n\t\t\t\t\t26  & 20News-sci         & 0.0058 & 0.0072 & 0.0093 & 0.0132 & 0.0085  & 0.0127  \\\\\n\t\t\t\t\t27  & Spambase           & 0.0152 & 0.0151 & 0.0238 & 0.0354 & 0.0200  & 0.0316  \\\\\n\t\t\t\t\t28  & Mushroom           & 0.0171 & 0.0169 & 0.0266 & 0.0415 & 0.0222  & 0.0376  \\\\\n\t\t\t\t\t29  & Cod-rna            & 0.4536 & 0.3357 & 0.4590 & 0.7083 & 0.2496  & 0.4397  \\\\\n\t\t\t\t\t30  & Ijcnn1             & 2.1150 & 1.1109 & 2.0940 & 2.2597 & 0.6436  & 1.9357  \\\\\n\t\t\t\t\t31  & Skin-nonskin       & 3.3478 & 1.9708 & 2.7245 & 3.9448 & 1.0431  & 1.9172  \\\\\n\t\t\t\t\t& Average   {\\rule{0pt}{3ex}}   & 0.1938 & 0.1136 & 0.1752 & 0.2299 & 0.0667  & 0.1449 \\\\ \\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\n\t\t\t\\end{sc}\n\t\t\\end{small}\n\t\\end{center}\n\t\\vskip -0.1in\n\\end{table}\n\n\n\\begin{figure}[ht]\n\t\\vskip 0.2in\n\t\\begin{center}\n\t\t\\subfigure[Accuracy rank]{ \n\t\t\t\\includegraphics[width=0.45\\textwidth]{acc_Nemenyi_PATER.eps\n\t\t} \\hspace{0.9cm}\n\t\t\\subfigure[Computational rank]{ \n\t\t\t\\includegraphics[width=0.45\\textwidth]{time_Nemenyi_PATER.eps\n\t\t} \n\t\n\t\t\\caption{The Nemenyi test to check statistical significance of the average (a) accuracies and (b) CPU times. The connected algorithms by the Critical Difference (CD) have no statistical significance.}\n\t\t\\label{fig.Nemenyi}\n\t\\end{center}\n\t\\vskip -0.2in\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Results and Discussion}\nTable \\ref{tbl.avgacc} shows the average accuracies and its standard deviations on the 31 real-world data sets.\nAdditionally, ranks for each algorithm are shown in brackets.\nIn terms of the average accuracy, the proposed wPATER-I performs better accuracy performance than that of the other algorithms. \nMoreover, the proposed wPATER-II is also shown as the second best.\nIn Table \\ref{tbl.avgacc}, both wPATER-I and wPATER-II achieve the best performances on the 29 data sets. \nIn order to investigate the validity  of the proposed wPATER for the data imbalance issue, \nin Table \\ref{tbl.eval_weights}, we evaluate a best weight (BW), which gives the best accuracy, between $\\alpha^-$ and $\\alpha^+ $  for each data set. \nWe then present matching results between the best weights and needed weights (NW) which are given by the data imbalance ratios from Table \\ref{tbl.dataSummary}.\nConsequently,  Table \\ref{tbl.eval_weights} shows that \nour assumption of the proposed wPATER on addressing the data imbalance problems properly works in the 25 data sets.\nFigure \\ref{fig.cma} presents cumulative classification accuracies along the online learning process on the last 6 data sets (the data sets 26-31) that have large training samples.\nExcept the `Ijcnn1' data set in Figure \\ref{fig.cma}(e), the PATER based algorithms maintain the best performers in terms of the cumulative accuracies.\nFor time complexity, Table \\ref{tbl.avgtime} shows the average CPU times on the aforementioned 31 data sets.\nIn terms of the average CPU time, the proposed wPATER-I shows better CPU performance than that of the other algorithms.\n\nTo evaluate the statistical significance on the reported accuracy and CPU time comparisons, \nFriedman tests (see \\cite{demvsar2006statistical}) and then Nemenyi plots are performed as a post-hoc test to statistically group connected algorithms at a confidence threshold (e.g., $p = 0.05$).\nIn terms of the accuracy rank, Fig. \\ref{fig.Nemenyi}(a) has three groups of algorithms, namely \n1) wPATER-I---wPATER-II, 2) wPATER-II---PA and 3) PA---PATER-I---PE---PATER-II.\nThe proposed wPATER-I and wPATER-II are shown as the best group.\nIn terms of the computational rank, Fig. \\ref{fig.Nemenyi}(b) shows four groups of algorithms, namely \n1) PE---PA---wPATER-I,\n2) wPATER-I---PATER-I,\n3) PATER-I---wPATER-II and\n4) wPATER-II---PATER-II.\nThe proposed wPATER-I is included in the best group, although this is shown as the third best individually. In summary, the proposed wPATER-I is seen as the best algorithm in terms of  efficiency and effectiveness.\n\n\n\n\n\\section{Conclusion}\nWe have presented a passive-aggressive total-error-rate (PATER) minimization for online learning. \nConsequently, the proposed PATER had the useful properties of PA and TER together.\nIn addition, a weighted PATER method (wPATER) has also presented to address the data imbalance problems.\nThe proposed wPATER effectively and efficiently outperformed the state-of-the-art algorithms, including the proposed PATER  on the real-world data sets.\nThis work was based on the first-order information. We will improve this work to extend it to the second-order methods.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Significance statement}\n\nThe description of the correlated activity of large populations of neurons\nis essential to understand how the brain performs computations and\nencodes sensory information.\nThese correlations can manifest themselves in the coupling of single\ncells to the total firing rate of the\nsurrounding population, as was recently demonstrated in the\nvisual cortex, but how to build this dependence into an explicit\nmodel of the population activity is an open question. \nHere we introduce a general and tractable model based on the principle\nof maximum entropy to describe this population coupling. By applying our approach to\nmulti-electrode recordings of retinal ganglion cells, we find complex forms\nof coupling, with the unexpected tuning of many neurons to a preferred\npopulation rate.\n\n\\vskip .3cm\n\\end{minipage}\n}\n\n\\bigskip\n\n\nAn important feature of neural population codes is the correlated firing of neurons. Manifestations of collective activity are observed in the correlated firing of individual pairs of neurons \\citep{Arnett1978}, and through the coupling of single neurons to the activity in its surrounding population \\citep{Arieli1996,Tsodyks1999}.\nThese correlations, whether they are evoked by common inputs or result from interactions between neurons, imply that the neural code must be studied through the collective patterns of activity rather than by individual neuron.     \n  \nAs the number of possible firing patterns in a population grows exponentially with its size, they cannot be sampled exhaustively for large populations. \nSeveral modeling approaches have been suggested to describe the collective activity patterns of neural population \\citep{Grun1995,Schneidman2003,Schneidman2006,Pillow2008,Cocco2009a,Tkacik2014a}. In these approaches, a small number of statistics (e.g. mean firing rate, pairwise correlations) is measured to constrain the parameters of the model. Models are then evaluated on their ability to predict statistics of the population activity that were not fitted to the data. These models are computationally hard to infer, and one must usually have recourse to approximate methods to fit them.\n\nRecently, \\cite{Okun2015} investigated how the activity of the whole population influenced the behavior of single neurons in the primary visual cortex of awake mice and monkeys. In particular, they studied the role of the correlation between neurons and the summed activity of the population, called the population rate. To assess whether these couplings between neurons and population activity were sufficient to describe the correlative structure of the code, synthetic spike trains preserving these couplings were generated and compared to data. However, the numerical method used to generate synthetic spike trains is computationally heavy, and is unable to predict the probability of particular patterns of spikes, as most of them are unlikely to ever occur. \n\nHere we introduce a new method, based on the principle of maximum entropy, to exactly account for the coupling between individual neurons and the population rate.\n\\CG{This model is tractable, meaning that predictions for the statistics of the activity can be derived analytically. The gradient and Hessian of the model's likelihood can thus also be computed efficiently, allowing for fast inference using Newton's method.}\n Compared to previous methods \\citep{Okun2015}, our method can fit hours of large-scale recordings of large populations in a few seconds on a standard laptop computer. \n We tested it on recordings of the salamander retina (160 neurons). We uncovered new ways for individual neurons to be coupled to the population, where a single neuron is tuned to a particular value of the population rate, rather than being monotonically coupled to the population.\n\n\n\n\n\\section{Materials and Methods}\n\\subsection{Recordings from retinal ganglion cells}\nWe analyzed previously published {\\em ex vivo} recordings from  retinal ganglion cells of the tiger salamander (Ambystoma tigrinum) \\citep{Tkacik2014a}. In brief, animals were euthanized according to institutional animal care standards. The retina was extracted from the animal, maintained in an oxygenated Ringer solution, and recorded on the ganglion cell side with a 252 electrode array. Spike sorting was done with a custom software \\citep{Marre2012}, and $N=160$ neurons were selected for the stability of their spike waveforms and firing rates, and the lack of refractory period violation.\n\n\\subsection{Maximum entropy models}\nWe are interested in modeling the probability distribution of population responses in the retina. The responses are first binned into 20-ms time intervals. The response of neuron $i$ in a given interval is represented by a binary variable $\\sigma_i$, which takes value 1 if the neuron spikes in this interval, and 0 if it is silent. The population response in this interval is represented by the vector ${\\bm \\sigma}=(\\sigma_1,\\ldots,\\sigma_N)$ of all neuron responses (Figure ~\\ref{fig:1}A). We define the population rate $K$ as the number of neurons spiking in the interval: $K({\\bm \\sigma}) = \\sum_{i=1}^N \\sigma_i$.\n\nWe build three models for the probability of responses, $P({\\bm \\sigma})$. These models reproduce some chosen statistics, meaning that these statistics have same value in the model and in empirical data. The first model reproduces the firing rate of each neuron and the distribution of the population rate. The second model also reproduces the correlation between each neuron and the population rate. The third model reproduces the whole joint probability of single neurons with the population rate. It is a hierarchy of models, because the statistics of each model are also captured by the next one. \\\\\n\n{\\em Minimal model.}\nWe build a first model that reproduces the firing rate of each neuron, $P(\\sigma_i\\!=\\!1)=\\<\\sigma_i\\>$, and the distribution of the population rate, $P(K)$. We also want the model to have no additional constraints, and thus be as random as possible. In statistical physics and information theory, the randomness of a distribution $P$ is measured by its entropy $S(P)$:\n\\begin{eqnarray}\nS(P) = - \\sum_{\\bm \\sigma} P({\\bm \\sigma}) \\; \\ln P({\\bm \\sigma}),\n\\end{eqnarray}\nwhere the sum runs over all possible states. The maximum entropy model is the distribution that maximizes this entropy while reproducing the constrained statistics. Using the technique of Lagrange multipliers (see Mathematical derivations), one shows that the model must take the form:\n\\begin{equation}\nP({\\bm \\sigma}) = \\dfrac{1}{Z} \\exp\\left( \\sum_{i=1}^N (\\alpha_i + \\beta_{K({\\bm \\sigma})}) \\; \\sigma_i  \\right),\n\\label{eqn:intermediate_model1}\n\\end{equation}\t\t\nwhere the parameters $\\alpha_i$, $i\\!=\\!1,\\ldots,N$ and $\\beta_K$, $K=0,\\ldots,N$ must be fitted so that the distribution of Eq.~\\ref{eqn:intermediate_model1} matches the statistics $\\<\\sigma_i\\>$ and $P(K)$ of the data. $Z$ is a normalization factor. \\CG{Note that $\\beta_{K({\\bm \\sigma})}$ depends on the state ${\\bm \\sigma}$ through $K({\\bm \\sigma})$.} We refer to this distribution as the minimal model, as no explicit dependency between the activity of individual neurons and the population rate is constrained. \\\\\n\n{\\em Linear-coupling model.}\nThe second  model reproduces $\\<\\sigma_i\\>$ and $P(K)$ as before, as well as the linear correlation $\\<K\\cdot \\sigma_i\\>$ between each neuron response $\\sigma_i$ and the population rate $K$, for $i=1,\\ldots,N$. It takes the form (see Mathematical derivations):\n\\begin{align}\nP({\\bm \\sigma}) &= \\dfrac{1}{Z} \\exp\\left( \\sum_{i=1}^N ( \\alpha_i + \\beta_{K({\\bm \\sigma})} + \\gamma_i K) \\; \\sigma_i  \\right).\n\\label{eqn:linear_model}\n\\end{align}\nAnalogously to the minimal model, the parameters $\\alpha_i$, $\\beta_K$ and $\\gamma_i$ are inferred so that the model agrees with the mean statistics $\\< \\sigma_i\\>$, $P(K)$ and $\\<K\\cdot \\sigma_i\\>$ of the data. Importantly, despite their common notation, the values of the fitted parameters $\\alpha_i$ and $\\beta_K$ are different from the ones fitted in the minimal model (see Mathematical derivations). \\\\\n\n{\\em Complete coupling model.}\nThe third model reproduces the joint probability distributions of the response of each neuron and the population rate, $P(\\sigma_i, K)$. It takes the form (see Mathematical derivations):\n\\begin{align}\\label{eq:ModelDef_methods}\n P({\\bm \\sigma}) &= \\dfrac{1}{Z} \\exp \\left(\\sum_{i=1}^N h_{iK({\\bm \\sigma})} \\: \\sigma_i \\right).\n\\end{align}\nThe parameters $(h_{iK})_{i=1,\\ldots,N;K=0,\\ldots,N}$ are inferred so that the model agrees with the data on $P(\\sigma_i, K)$ for each $(i,K)$ pair. \\CG{Note that $h_{i K({\\bm \\sigma})}$ depends on the state ${\\bm \\sigma}$.} We refer to this model as the complete coupling model since it reproduces exactly the joint probability between each neuron and the population rate.\n\n\\subsection{Model solution}\nThe minimal and linear-coupling models can be written in the same form as the complete coupling model (Eq.~\\ref{eq:ModelDef_methods}),\nbut with constraints on the form of $h_{iK}$. In the minimal model, the matrix $h_{iK}$ is constrained to have the form $\\alpha_i + \\beta_K $. In the linear-coupling model, it is constrained to have the form $\\alpha_i + \\beta_K + \\gamma_i K$.  In the complete coupling model, the matrix $h_{iK}$ has no imposed structure, and all its elements must be learned from the data.\n\nSince all the considered models can be viewed as sub-cases of the complete coupling model (Eq.~\\ref{eq:ModelDef_methods}), we only describe the mathematical solution to this general case. First we describe how to solve the {\\em direct} problem, {\\em i.e.} how to compute statistics of interest, such as $P(\\sigma_i,K)$, from the parameters $h_{iK}$. In the next section, we explain how to solve the {\\em inverse} problem -- the reverse task of inferring the model parameters from the statistics -- which relies on the solution to the direct problem.\n\nA model is considered tractable if there exists an analytical expression for the normalization factor,\n\\begin{eqnarray}\\label{Z}\nZ = \\sum_{{\\bm \\sigma}} \\exp\\left( \\sum_{i=1}^N h_{iK(\\sigma)} \\sigma_i \\right),\n\\end{eqnarray}\nallowing for its rapid computation, e.g. in polynomial time in $N$.\nAll statistics of the model, such as $P(\\sigma_i,K)$ or covariances $\\<\\sigma_i\\sigma_j\\>-\\<\\sigma_i\\>\\<\\sigma_j\\>$ between pairs of neurons, can then be calculated efficiently through derivatives of $Z$ (see Mathematical derivations). \nIn general maximum entropy models are not tractable, because sums of the kind in Eq.~\\ref{Z} involve a sum over an exponential number of terms ($2^N$).\nFortunately, in our case, the technique of probability-generating functions provides an expression for $Z$ which is amenable to fast computation using polynomial algebra (see Mathematical derivations):\n\\begin{eqnarray}\nZ=\\sum_{K=0}^N \\mathrm{Coeff}\\left[\\prod_{i=1}^N (1+Xe^{h_{iK(\\sigma)}}),X^K\\right],\n\\end{eqnarray}\nwhere $\\mathrm{Coeff}[Q,X^n]$ denotes the coefficient of polynomial $Q$ of order $X^n$.\n\n\\subsection{Model inference}\nWe now describe how to fit the models to experimental data. The inference of the model parameters  is equivalent to a problem of likelihood maximization \\citep{Ackley1988}. The model reproduces the empirical statistics exactly when the parameters maximize the likelihood of experimental data measured  by the model, $L=\\prod_{\\alpha=1}^n P({\\bm \\sigma}^{(\\alpha)})$, where $({\\bm \\sigma}^{(1)},\\ldots,{\\bm \\sigma}^{(n)})$ are the $n$ activity patterns recorded in the experiment, assumed to be independently drawn.\n\nIn practice, we maximized the normalized log-likelihood $\\mathcal{L} = (1\/n) \\log L$ instead of $L$, which is equivalent theoretically but more convenient for computation. We used Newton's method  to perform the maximization. This method requires to compute the first and second derivatives of the normalized log-likelihood. These derivatives can be expressed as functions of mean statistics of the model and can be calculated using the solution to the direct problem sketched in the previous section, and detailed in the Mathematical derivations. Because the model is tractable, these mean statistics can be computed quickly and the model can be inferred rapidly.\n\nFor the minimal model, the optimization was performed over the parameters $(\\alpha_i)_{i=1,\\ldots,N}$ and  $(\\beta_K)_{K=0,\\ldots,N}$. For the linear-coupling model, the optimization was done over these two sets of parameters, as well as $(\\gamma_i)_{i=1,\\ldots,N}$.\nFor the complete coupling model, the optimization was performed over all elements of the matrix $(h_{iK}) _{i=1,\\ldots,N;K=0,\\ldots,N}$. We stopped the algorithm when the fitting error was smaller than $10^{-6}$ (see Mathematical derivations).\n\n\\subsection{Regularization}\nPrior to learning the model, we regularized the empirical population rate distribution $P(K)$ and conditional firing rates $P(\\sigma_i|K)$ to mitigate the effects of low-sampling noise. This regularization allowed us to remove zeros from the mean statistics, avoiding issues with the fitting procedure.\nWe performed this regularization using pseudocounts (see Mathematical derivations).\n\n\\subsection{Tuning curves in the population rate}\nWe define the tuning curve of neuron $i$ in the population rate as the conditional probability of neuron $i$ to spike given the summed activity of all neurons but $i$, $K_{\\backslash i} \\!\\!=\\!\\! \\sum_{j\\neq i} \\sigma_j$. It is equal to:\n\\begin{eqnarray}\n P(\\sigma_i\\!\\!=\\!\\!1|K_{\\backslash i}) =\\frac{P(\\sigma_i\\!\\!=\\!\\!1,K_{\\backslash i})}{P(\\sigma_i\\!\\!=\\!\\!0,K_{\\backslash i})+P(\\sigma_i\\!\\!=\\!\\!1,K_{\\backslash i})}\n\\end{eqnarray}\nwhere we can use $P(\\sigma_i\\!\\!=\\!\\!1,K_{\\backslash i}) = P(\\sigma_i\\!\\!=\\!\\!1,K\\!\\!=\\!\\!K_{\\backslash i}\\!\\!+\\!\\!1)$ and $P(\\sigma_i\\!\\!=\\!\\!0,K_{\\backslash i}) = P(\\sigma_i\\!\\!=\\!\\!1,K\\!\\!=\\!\\!K_{\\backslash i})$. Each of these quantities can be computed using the solution to the direct problem (see Mathematical derivations).\n\nWe then tested for each neuron if its tuning curve had significant local maxima. We first identified the set of $K_{\\backslash i}$ for which $P(\\sigma_i=1|K_{\\backslash i})$ was significantly larger than points below and above $K_{\\backslash i}$. To assess significance, we measured the standard deviation of the difference across 100 training sets consisting in random halves of the dataset. The difference was said to be significant when it was 5 standard deviations above 0.\n\nFor the cells for which the presence of a maximum was determined, we then evaluated the location of the maximum, $K_{\\backslash i}^*$, by taking the median of the maxima determined for each training set. We inferred the presence and position of minima in a similar way.\n\nTo estimate the quality of the model prediction for the tuning curve, we quantified how the model differed from the data. We trained the model on 100 random training sets and computed  $D_{KL}({\\rm test}\\|{\\rm model})$, the difference between $P(\\sigma_i,K)$ in the testing data and predicted by the model, measured by the Kullback Leibler (KL) divergence. The KL divergence between two distributions $P$ and $Q$ of a random variable $x$ is: $ D_{KL}(P \\| Q) = \\sum_x P(x) \\text{log}[{P(x)}\/{Q(x)}]$. We regularized $P(\\sigma_i,K)$ in the testing set before computing the KL divergence.\nTo measure sampling noise we computed the difference between the testing and the training sets,  $D_{KL}($test$\\|$train$)$, where $P(\\sigma_i, K)$ was regularized in both sets. The normalized KL divergence, $z,$ is defined as the difference between $D_{KL}($test$\\|$model$)$ and $D_{KL}($test$\\|$train$)$, divided by the standard deviation:\n\\begin{equation}\n z = \\dfrac{\\text{mean}\\left(D_{KL}(\\text{test}\\|\\text{model}) - D_{KL}(\\text{test}\\|\\text{train}) \\right)}{\\text{std}\\left(D_{KL}(\\text{test}\\|\\text{model}) - D_{KL}(\\text{test}\\|\\text{train}) \\right)}.\n\\end{equation}\nIn other words, it measures by how many standard deviations the data differs from the model.\n\n\\subsection{Quality of the model}\n{\\em Pairwise correlations.}\nIn order to measure the quality of the predictions of \ncorrelations between pairs of neurons $\\sigma_i$ and $\\sigma_j$, we used cross-validation. We randomly divided the dataset into 100 training and testing sets half the size of data, and learned the model on the training sets. The correlations of each testing set, $c_{{\\rm test},\\, ij}$ were then predicted with the model $c_{{\\rm model},\\,ij }$. The quality of the model prediction was measured \\CG{by a goodness-of-fit index quantifying the amount of correlations predicted by the model. We define it as:}\n\\begin{equation} \\label{eq:explained_cc}\n C= \\dfrac{\\sum_{i<j}c_{{\\rm test},ij }^2 - \\sum_{i<j}\\left(c_{{\\rm test},ij } - c_{{\\rm model},\\,ij }\\right)^2}{\\sum_{i<j}c_{{\\rm test},ij }^2 - \\sum_{i<j}\\left(c_{{\\rm test},ij } - c_{{\\rm train},ij }\\right)^2},\n\\end{equation}\nwhere $c_{{\\rm train},\\, ij}$ is the correlation in the corresponding training set. The numerator of Eq.~\\eqref{eq:explained_cc} is the part of the correlations in the testing set that is predicted by the model, and the lower one is a normalization correcting for sampling noise. We have $C=1$ when the model perfectly accounts for the correlations of the training set. When the model completely ignores correlations, as in a model of independent neurons, $c_{ij}=0$, then $C=0$. \\\\\n\n{\\em Likelihood.}\nUsing the models learned on the same 100 training sets, we computed the likelihood of responses in the testing sets for the minimal, linear-coupling and complete coupling models. In this paper the log-likelihood is expressed in bits, using binary logarithms.\nWe then computed the improvement in mean log-likelihood compared to the minimal model as the ratio $\\< \\, \\< \\log P({\\bm \\sigma})\\>_{\\bm \\sigma} \\, \/ \\, \\< \\log P({\\bm \\sigma}) \\>_{\\bm \\sigma} \\, \\>_\\text{test}$, where $\\< \\cdot \\>_{\\text{test}}$ is the mean over training sets and $\\< \\cdot \\>_{{\\bm \\sigma}}$ is the mean over responses in each testing set.\\\\\n\n{\\em Multi-information.}\nThe multi-information \\citep{Cover,Schneidman2003} quantifies the amount of correlative structure captured by a model. It is defined as the difference between the entropy of a model of independent neurons reproducing firing rates, and empirical data: $I = S_{\\rm indep} - S_{\\rm data}$. Here $S_{\\rm data}=-\\sum_{\\bm \\sigma} P_{\\rm data}({\\bm \\sigma})\\log P_{\\rm data}({\\bm \\sigma})$ is the entropy of the spike patterns measured by their frequencies $P_{\\rm data}({\\bm \\sigma})$ in the data, and $S_{\\rm indep}$ is the entropy if all neurons were independent.\n\nThe entropy of a maximum entropy model is by construction higher than that of the real data, $S_{\\rm model}>S_{\\rm data}$, because the model has maximum entropy given the statistics it reproduces. Its entropy is also smaller than $S_{\\rm indep}$ provided that constraints include the spike rates, because the model has more structure, reproduces more statistics, than if neurons were independent, Thus the fraction of correlations that is accounted by the maximum entropy model, $0<I_{\\rm model}\/I<1$, where $I_{\\rm model}=S_{\\rm indep}-S_{\\rm model}$, can be viewed as a measure of how well the model captures the correlative structure of responses. The true multi-information $I$ can only be calculated for small groups of neurons ($N\\leq 20$), because $P_{\\rm data}$ requires to evaluate $2^N$ pattern frequencies, which is prohibitive for large networks.\n\n\n\n\n\\section{Results}\n\\subsection{Tractable maximum entropy model for coupling neuron firing to population activity}\n\nThe principle of maximum entropy \\citep{Jaynes1957a,Jaynes1957} provides a powerful tool to explicitly construct probability distributions that reproduce key statistics of the data, but are otherwise as random as possible.\nWe introduce a novel family of maximum entropy models of spike patterns that preserve\n the firing rate of each neuron, the distribution of the population rate, and the correlation between each neuron and the population rate, with no additional assumptions (Figure \\ref{fig:1}A). \n Under these constraints, the maximum entropy distribution over spike patterns in a fixed 20-ms time window is given by (see Materials and Methods):\n\\begin{equation}\n\\label{ModelDef_linear}\n P(\\sigma_1,\\ldots,\\sigma_N) = \\frac{1}{Z} \\exp\\left( \\sum_{i=1}^N ( \\alpha_i + \\beta_K + \\gamma_i K)\\,  \\sigma_i \\right)\n\\end{equation}\nwhere $\\sigma_i$ equals 1 when neuron $i$ spikes within the time window, and 0 otherwise, $K = \\sum_i \\sigma_i$ is the population rate, and $Z$ is a normalization constant. The parameters $(\\alpha_i)_{i=1,\\ldots,N}$, $(\\gamma_i)_{i=1,\\ldots,N}$ and $(\\beta_K)_{K=0,\\ldots,N}$ must be fitted to empirical data. We refer to this model as the linear-coupling model, because of the linear term $\\gamma_i K \\sigma_i$ in the exponential.\n\nUnlike maximum entropy models in general, this model is tractable, meaning that its prediction for the statistics of spike patterns has an analytical expression that can be computed efficiently using polynomial algebra. This allows us to infer the model parameters rapidly for large populations on a standard computer, using Newton's method (see Materials and Methods). We learned this model in the case of a population of $N=160$ salamander retinal ganglion cells, stimulated by a natural movie.\nIt took our algorithm 14 seconds to fit the $3N-2$ model parameters (see Mathematical derivations) so that the maximum discrepancy between the model and the data was smaller than $10^{-6}$ (Figure~\\ref{fig:1} B-D).\n\nThe linear-coupling model provides a rigorous mathematical formulation to the hypotheses underlying the modeling approach of \\cite{Okun2015} applied to cortical populations. In that work, synthetic spike trains were generated by shuffling spikes from the original data so as to match the three constraints listed above on the single-neuron spike rates, the distribution of population rates, and their linear correlation. Shuffling data, {\\em i.e.} increasing randomness and hence entropy, while constraining mean statistics, has previously been shown to be equivalent to the principle of maximum entropy in the context of pairwise correlations \\citep{Bialek2008a}. Our formulation provides a fast way to learn the model and to make predictions from it, as we shall see below. In addition, it allows us to calculate the probability of individual spike patterns, Eq.~\\ref{ModelDef_linear}, which a generative procedure such as the one in \\cite{Okun2015} cannot. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig1.pdf}\n\\caption{\\textbf{A maximum entropy model for population coupling.} \\textbf{A}, Spikes trains are recorded with a multieletrode array and binned in 20ms time windows. We study the dependence between each neuron's binned response, $\\sigma_i$, and the population rate $K$, defined as the summed activity of all neurons. \\textbf{B-D}, The linear-coupling model fits three observables with high accuracy: the population rate distribution (B), the cells firing rates (C), and population couplings (D). For each observable the model fit is plotted against empirical values.}\n\\label{fig:1}\n\\end{figure}\n\n\n\\subsection{Tuning curves of single neurons to the population activity}\nWe wondered whether the linear-coupling model could explain how the response of single neurons depended on the population rate.\nWe examined the firing probability of neuron~$i$ as a function of the summed activity of the other neurons $K_{\\backslash i} = \\sum_{j \\neq i}\\sigma_j$, denoted by $P(\\sigma_i\\!=\\!1 | K_{\\backslash i})$. \nThis quantity can be viewed as the tuning curve of neuron~$i$ in response to the rest of the population.\nIt can be calculated analytically from the parameters of the model (see Materials and Methods), and compared to empirical values. \nThe tuning curves of four representative cells are shown in Figure~\\ref{fig:Pi_condKo_selectcells}A-D.\n\nThe linear-coupling model predicts a variety of tuning curves  (in red), from sub-linear to super-linear.\nAlthough its prediction was qualitatively close to the empirical value for some cells (Figure ~\\ref{fig:Pi_condKo_selectcells}A), the model generally did not account well for the coupling between $\\sigma_i$ and $K_{\\backslash i}$. A majority of cells (85 out of 160) displayed a local maximum in their empirical tuning curves, at some prefered value $K_{\\backslash i}^*$ of the population activity to which the neuron is tuned. The model did not predict the existence of this maximum in 47 out of these 85 cells (Figure~\\ref{fig:Pi_condKo_selectcells}C). Even when it did, the location of the maximum,  $K_{\\backslash i}^*$, was poorly predicted, as can be seen by the distribution of the difference between model and data (Figure~\\ref{fig:Pi_condKo_selectcells}E). In six cases, the tuning curve had two local maxima, while \nthe model only predicted one. Another 27 cells had a minimum in their empirical tuning curve, which was never reproduced by the model (Figure~\\ref{fig:Pi_condKo_selectcells}D).\nInterestingly, no cells were tuned to fire when the rest of the population is silent; even cells whose spiking activity was anti-correlated to the rest of the population had a non-zero preferred population rate, $K_{\\backslash i}^*>0$. \n\nThe model performance can be quantified by computing the Kullback Leibler (KL) divergence between data and model for the joint probability $P(\\sigma_i,K)$ of the neuron and population activity (Figure~\\ref{fig:Pi_condKo_selectcells}F). The KL divergence is a measure of the dissimilarity between two distributions $P$ and $Q$ \\citep{Cover}, which quantifies the amount of information that is lost if we use $Q$ to approximate $P$.\nWe calculated a normalized KL divergence (see Materials and Methods) measuring by how many standard deviations the KL divergence \nbetween linear-coupling model and data\ndeviated from what would be expected from sampling noise (Figure~\\ref{fig:Pi_condKo_selectcells}F). A majority of cells (143 out of 160) deviated by more than 2 standard deviations, meaning that their tuning curve was not well accounted for by the linear-coupling model.\nThis observation is consistent with the model's failure to account for the qualitative properties of their tuning curves.\n\nTaken together, these results indicate that the full dependency between single cells and the population rate cannot be explained by their linear correlation only.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig2.pdf}\n\\caption{\\textbf{Tuning curves of single neurons as a function of the population rate.} \\textbf{A-D}, Spiking probability of neuron $i$ conditioned on the summed activity of all other neurons, $P(\\sigma_i =1 | K_{\\backslash i})$, as observed in the data (black curves, standard error shaded in grey), and predicted by linear-coupling model (red curves). \nEach subfigure corresponds to a different representative cell. \n\\textbf{E}, Histogram of the difference between the preferred population rate -- at which the tuning curve is maximal -- observed in the data, $K_{\\rm data}^*$, and predicted by the linear-coupling model, $K_{\\rm model}^*$. Data are shown for the 38 cells that had at least one local maximum both in the linear model and in the data. When the empirical tuning curve had 2 local maxima, the closest one to the model prediction was chosen.\n\\textbf{F}, Histogram of the normalized Kullback-Leibler divergence between the observed joint distributions $P(\\sigma_i, K)$ and its prediction by linear-coupling model.\nThe arrows indicate the value for the four example cells A-D. The vertical line shows a normalized divergence of 2, meaning that cells sitting on its right deviate from the linear-coupling model by more than 2 standard deviations.}\n\\label{fig:Pi_condKo_selectcells}\n\\end{figure}\n\n\n\\subsection{A refined maximum entropy model}\nTo overcome the limitations of the linear-coupling model, and to fully account for the variety of tuning curves found in data, we built a maximum entropy model constrained to match all joint probabilities of the population rate with each single neuron response, $P(\\sigma_i,K)$. This model takes the form (see Materials and Methods):\n\\begin{equation}\n\\label{ModelDef_general}\n P(\\sigma_1,\\ldots,\\sigma_N) = \\frac{1}{Z} \\exp\\left( \\sum_{i=1}^N h_{iK}  \\sigma_i \\right),\n\\end{equation}\nwhere the parameters $h_{iK}$ for $i=1,\\ldots,N$ and $K=0,\\ldots,N$ are fitted to empirical data, and $Z$ is a normalization constant.\nNote that the linear-coupling model can be viewed as a particular case of this model, with parameters $h_{iK}$ constrained to take the form $h_{iK} = \\alpha_i + \\beta_K + \\gamma_i K$.\nBy construction, this model exactly reproduces the tuning curves of Figs.~\\ref{fig:Pi_condKo_selectcells}A-D.\n\nAlthough this model has many more parameters than the simpler linear-coupling model, it is still tractable, and we could infer its $N(N-1)+1$ parameters (see Mathematical derivations) in 7 seconds for the whole population of 160 neurons. Hereafter, we refer to this model as the complete coupling model.\n\n\\subsection{Pairwise correlations}\nThe models introduced thus far are only constrained to reproduce the firing rate of each neuron, the distribution of population rates, and the coupling of each neuron with the population rate.\nWe asked whether these simple models\ncould account for correlations between individual pairs of cells, which were not fitted to the data.\nThe correlation between two neurons, $\\<\\sigma_i\\sigma_j\\>-\\<\\sigma_i\\>\\<\\sigma_j\\>$, can be calculated analytically from the model parameters (see Materials and Methods) and directly compared to the data (Figure ~\\ref{fig:correlations_hist}A and B). \n\nIn order to understand the importance of the population rate coupling for the prediction of pairwise correlations, we built a null model only constrained by the firing rates of each neuron and the distribution of the population rate. This simpler maximum entropy model reads:\n\\begin{equation}\n\\label{ModelDef_minimal}\n P(\\sigma_1,\\ldots,\\sigma_N) = \\frac{1}{Z} \\exp\\left( \\sum_{i=1}^N (\\alpha_i + \\beta_K) \\,  \\sigma_i \\right).\n\\end{equation}\nWe call it the minimal model. Interactions between neurons only derive from the fluctuations of the population activity, rather than from an explicit coupling.  This model has $2N-1$ parameters (see Mathematical derivations), which are inferred using the same techniques as before.\n\nTo quantify the performance of the different models, we calculated a goodness-of-fit index ranging from 0 when the correlations were not predicted at all to 1 when they were predicted perfectly (Materials and Methods). This index was 0.380 $\\pm$ 0.001 for the minimal model, 0.526 $\\pm$ 0.002 for the linear-coupling model, and 0.544 $\\pm$ 0.002 for the complete coupling model. Thus, a substantial amount of pairwise correlations could be explained from the coupling of neurons to the population. By this measure, the complete model performed slightly (but significantly) better than the linear-coupling model.\n\nFigure~\\ref{fig:correlations_hist}C shows the distribution of the pairwise correlations in the data and as predicted by the three models.\nThe minimal model fails to reproduce the long tail of large correlations, and predicts no negative correlations, while 28\\% of empirical correlations are negative. By contrast, the linear and complete coupling models predict 7.6\\% and 7.7\\% of negative interactions respectively, and a longer tail of large correlation coefficients.\nThus the coupling to the population rate is important to reproduce both large correlations and the strong asymetry of the distribution.\n \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{fig4.pdf}\n\\caption{\\textbf{Maximum entropy models of population coupling partly\n    account for pairwise correlations.} \\textbf{A-B}, The observed correlation\n  coefficient between all pairs of neurons is compared to its prediction according to\n  the linear (A) and complete (B) coupling models. \\textbf{C},\n  Distribution of pairwise correlation coefficients, as observed in the data and predicted by minimal, linear and complete coupling models.}\n\\label{fig:correlations_hist}\n\\end{figure}\n\n\n\n\\begin{table*}\n \\begin{center}\n\n\\begin{tabular}{cc|c|c|c|c|} \n\\cline{3-6}\n& & \\multicolumn{1}{ |c| }{Data} & Minimal & Linear & Complete \\\\ \\cline{1-6}\n\\multicolumn{1}{ |c  }{\\multirow{2}{*}{$N= 10$} } &\n\\multicolumn{1}{ |c| }{$I$} &0.0713  $\\pm$    0.0398  &    0.0343  $\\pm$    0.0268  &    0.0478  $\\pm$    0.0315  &    0.0498  $\\pm$    0.0326 \\\\ \\cline{2-6}\n\\multicolumn{1}{ |c  }{}                        &\n\\multicolumn{1}{ |c| }{${I}\/{I_{\\rm data}}$} & 1 & 0.444 $\\pm$ 0.135 & 0.649 $\\pm$ 0.108  & 0.677 $\\pm$ 0.108  \\\\ \\cline{1-6}\n\\multicolumn{1}{ |c  }{\\multirow{2}{*}{$N= 20$} } &\n\\multicolumn{1}{ |c| }{$I$} &0.3188  $\\pm$    0.0967  &    0.1291  $\\pm$    0.0558  &    0.1702  $\\pm$    0.0592  &    0.1789  $\\pm$    0.0606 \\\\ \\cline{2-6}\n\\multicolumn{1}{ |c  }{}                        &\n\\multicolumn{1}{ |c| }{${I}\/{I_{\\rm data}}$} & 1 & 0.393 $\\pm$ 0.071 & 0.531 $\\pm$ 0.055 & 0.557 $\\pm$ 0.055\\\\ \\cline{1-6}\n\\end{tabular}\n\\caption{Mean ($\\pm$ standard deviation) of the multi-information $I$ estimated either directly from the data, or from maximum entropy models, for random sub-populations of 10 and 20 neurons (100 sub-populations each), as well as the ratio of the multi-information between model and data. Results are in bits.}\n\\label{table}\n\\end{center}\n\\end{table*}\n\n\n\\subsection{Prediction of probabilities of spike patterns}\nWe quantified the capacity of models to describe population responses by computing the probability of responses predicted by each model. The mean log-likelihood of responses was -33.10 $\\pm$  0.06 bits for the minimal model, -30.12  $\\pm$  0.06 bits for the linear-coupling model and -29.49 $\\pm$  0.06 bits for the complete coupling model. \nThe improvement in mean log-likelihood compared to the minimal model was 51.3 $\\pm$ 0.5 \\% higher for the complete coupling model than for the linear-coupling model, meaning that nonlinear couplings to the population are important to model the probability of responses.\n\nThe multi-information $I$ \\citep{Cover} quantifies, in bits, the amount of correlations in the response, whether they are pairwise or of higher order (see Materials and Methods). To assess the performance of the models in capturing the collective behavior of the networks, we calculated the ratio of the multi-information explained by the model to that estimated directly from the data, $I_{\\rm model}\/I_{\\rm data}$. This ratio gives a measure of how well the probability of particular spike patterns are predicted by the model:\nit is 1 when the model is a perfect description of the data, and 0 when the model assumes independent neurons with no correlation between them. Because it requires to estimate the probability of all possible spike patterns of the populations, the multi-information can only be calculated for small populations of at most 20 cells.\n\nWith this measure, the linear coupling model could account for $65\\%$ of the multi-information for groups of $10$ neurons, and $53\\%$ for groups of 20 neurons. The complete model slightly improved these ratios to $68\\%$ and $56\\%$, respectively (see Table \\ref{table}). Thus, more than half of the correlative structure in the spike patterns could be explained by the coupling to the population rate alone.\n\n\n  \n\\section{Discussion}\n\nIn this work we have introduced a general computational model for coupling individual neurons to the population rate.\nThis model formalizes and simplifies the generative procedure proposed by \\cite{Okun2015} to study population coupling, and overcomes its computational difficulties. In addition, it allows for non-linear coupling to the population rate. \n\nWe have used our model to investigate population coupling in large recordings of $N=160$ retinal ganglion cells. We found that most cells had a non-linear coupling to the population rate. In particular, a large fraction of cells were tuned to a preferred value of the population rate. Even more strikingly, a few cells had a least preferred population rate, {\\em i.e.} they were more likely to spike at lower or higher populations rates. We found no cell that was maximally active when all other neurons were silent, even among cells that were anti-correlated with the population rate.\nThese results emphasize the need for the non-linear coupling afforded by our model\n, as they  uncover new dependencies that do not fit within the proposed division between soloists and choristers \\citep{Okun2015}, such as the tuning to a specific population rate. \nIt would be interesting to test if these non-linear couplings can also be found at the cortical level. \n\nOverall, our model reaches a similar predictive performance than what was found in the cortex. The coupling to the population rate accounted for more than half of\ncorrelations between pairs of neurons.\nIn \\cite{Okun2015}, a custom measure of the fraction of explained pairwise correlations (different from the one used in the present work) gave 0.34. Applying the same measure to our case yields a similar value, 0.33. However, this similarity in performance can be due to different underlying mechanisms. In the retina, most correlations are due to common input from previous layers \\citep{Trong2008}, while ganglion cells do not make synaptic connections to each other. In contrast, at the cortical level, a larger part of the variability in the activity should be due to internal dynamics generated by recurrent connections \\citep{vanVreeswijk1996,Arieli1996,Tsodyks1999}. It would be interesting to test our model on cortical data to see if these differences result in different types of non-linear population coupling. \n\nOur maximum entropy model of population coupling is complementary to maximum entropy models reproducing correlations between all pairs of neurons. Pairwise models have been shown to accurately describe the collective activity of retinal ganglion cells \\citep{Schneidman2006,Shlens2006,Shlens2009,Ganmor2011,Ganmor2011a,Tkacik2014a}, and in cortical networks {\\em in vitro} \\citep{Tang2008} and {\\em in vivo} \\citep{Yu2008}, but they are not tractable,\nrequiring to sum over $2^N$ all possible spiking states in order to implement Boltzmann-machine learning \\citep{Ackley1988}. Alternative methods based on mean-field approximations \\citep{Cocco2009a,Cocco2011} or Monte-Carlo simulations \\citep{Broderick2008} have been proposed. However, Monte-Carlo methods require hours of computations, although recent efforts have tried to lower these computation times for moderately large populations \\citep{Ferrari2015}.\n\nBy contrast, the models of population couplings introduced here are much easier to solve. \nThey are tractable, so their predictions can be computed analytically in time $N^3$, and their parameters can be inferred in a few seconds on a personal computer from large-scale, hour-long recordings of spike trains for a population of $N=160$ neurons. \nThese models can then be used to generate synthetic spike trains, to calculate analytically response statistics such as pairwise correlations, or to estimate the probability of particular spike trains. \nCompared to the shuffling method described in \\cite{Okun2015}, which is equivalent to the linear-coupling model, our method is simpler and computationally less intensive.\n\nThe procedure is general and can be applied to any multi-neuron recording of individual spikes.\nThe speed of model inference could prove an important advantage when studying very large populations, which can now reach a thousand cells \\citep{SchwarzD2014}. \nIn the case of the linear-coupling model, the number of\nparameters is also smaller, scaling with the population\nsize $N$ rather than $N^2$ for pairwise correlations model.\n\nNote that the population coupling models introduced here belong to a different class than pairwise models. Each class captures features of neural responses that the other cannot:\nmodels of population  coupling should be sufficient for studying the global properties of collective activity, while pairwise models are still needed to account for the detailed structure of the response statistics. Pairwise models have been reported to capture 90\\% of the correlations as measured by the multi-information for populations of size $N=10$ \\citep{Schneidman2006}, while our model captures at most $70\\%$ (Table I). Yet pairwise models can also miss important aspects of the collective activity such as the probability of large population rates \\citep{Tkacik2014a}, which is captured by our population model.\n\nBoth classes of models consider same-time spike patterns, with no regard for the dynamics of spike trains and their temporal correlations. Generalizations of pairwise maximum entropy models to temporal statistics are even \nharder to solve computationally \\citep{Vasquez2012,Nasser2013a}. By contrast, our models of \npopulation coupling are fully compatible with any model describing the dynamics of the population rate such as \\cite{Mora2015}.\n\n\n\\bibliographystyle{natbib}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Model}\\label{sec:model}\n\n\\subsection{Kac models, clustering and the Kirkwood instability}\n\nOne can introduce a mean-field treatment of particle systems in\ndifferent ways.  One may, for example, use explicit infinite range\ninteractions \\cite{dotsenko2004infinite,dotsenko2005mean}.\nGlassiness is obtained by choosing a potential imposing a strong\nfrustration, and there is in principle no need for quenched disorder.\nNext, one may consider long, but finite ranges, in the spirit of Kac\ninteractions.  Although quite intuitive, the choice of the potential\nis in practice difficult, as crystallization \\cite{gils2007absence} or\ninstabilities in the liquid phase (like the Kirkwood instability\n\\cite{kirkwood1941statistical,grewe1977kirkwood2,klein1994repulsive,mel1995long,likos2007ultrasoft,fragner2007interparticle})\neasily set in as soon as the interaction range is finite.\n\nConsider, for example, the model studied by Dotsenko\n\\cite{dotsenko2004infinite,dotsenko2005mean}. Particles are in\na confining potential $V_{conf}({\\bf x_a})$ (which may be harmonic)\nand interact with a long-range, oscillatory potential:\n\\begin{equation}\nH= \\frac{1}{\\sqrt{N}}\\sum_{a \\neq b} \\cos(|{\\bf x_a}-{\\bf x_b}|) + \\sum_a V_{conf}({\\bf x_a})\n\\label{dots}\n\\end{equation}\nDotsenko showed that the system indeed has a mean-field glass phase, induced by \nthe frustration due to the conflict of attraction and repulsion. \nThe next step, in an ordinary Kac program, would be to introduce the model\nwith a finite range $\\gamma$:\n\\begin{equation}\n H= \\frac{1}{\\sqrt{\\gamma}}\\sum_{a \\neq b} \\cos(|{\\bf x_a}-{\\bf x_b}|) e^{-|{\\bf x_a}-{\\bf x_b}|\/\\gamma}+ \\sum_a V_{conf}({\\bf x_a})\n\\label{dots1}\n\\end{equation}\nThe result is disappointing: instead of giving a glass, the system\narranges as follows: it forms clusters of many particles, themselves\ndisposed in a crystalline arrangement, at the optimal distance so that\nthe interaction between clusters is minimized.  The way to avoid the\ncrystal-of-clusters mesophase is, of course, to add a hard-core that\nhampers the clustering, but then the model is no longer mean field.\nAnother related difficulty with this strategy is that even the liquid\nphase may have a transition to a `liquid' with spatial modulation, the\nKirkwood instability~\\cite{kirkwood1941statistical,\n  grewe1977kirkwood2, klein1994repulsive, mel1995long}. This phase is\nnot without interest of its own (see Fig.~\\ref{fig:kirkwood} for a\nnumerical simulation), but it is not what we are wishing to study\nhere.\n\nAnother way to construct a mean-field model is to work with particles\non a Bethe lattice\n\\cite{PhysRevE.67.057105,rivoire2003glass,PhysRevLett.88.025501,tarzia2004glass,mari2009jamming}.  This introduces quenched disorder and the mean-field nature at the\nsame time.  By increasing the graph connectivity, one can, at least\nformally, recover the original finite dimensional model by setting the\ngraph connectivity to infinity.\n\n\\subsection{Kraichnan's proposal and beyond.}\n\nIn this paper, we will follow another route.\nWe study family of models which are defined through the Hamiltonian:\n\\begin{equation}\n\\label{eq:model}\nH(\\{\\mathbf{x}\\}, \\{\\mathbf{A}\\})=\\sum_{<i,j>} V(\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}})  \n\\end{equation}\nwhere $V$ is a short-ranged interaction potential.  The $\\mathbf{x}$'s\nare the positions of the particles and the $\\mathbf{A}$'s are quenched\nrandom variables with a probability distribution $P(|\\mathbf{A}|)$,\nwhich has a variance $\\lambda^2$. We also impose  that\n$\\mathbf{A}_{ij}=\\mathbf{A}_{ji}$. The model for  $\\lambda = \\infty$ was\nfirst introduced by Kraichnan some fifty years ago\n\\cite{kraichnan1962stochastic1}.\n\nWhen $\\lambda=0$,  $P(|\\mathbf{A}|)=\\delta(|\\mathbf{A}|)$, and the\nmodel reduces to an usual $d$-dimensional system. On the other hand,\nwhen $\\lambda \\to \\infty$, one particle $i$ can interact with\nparticles $j$ which are possibly anywhere in the system, as long as\n$|\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}|$ is of the order of the\nrange of the potential $V$. Therefore, the system tends to have a\nmean-field nature in this limit even though   one particle\neffectively interacts with a \\textit{finite} number of other particles, as it\ndoes in a conventional finite-dimensional hard sphere system \n(see  Fig.~\\ref{fig:shifts}). Thus by tuning $\\lambda$, the\nmodel (\\ref{eq:model}) goes from a finite dimensional system ($\\lambda =\n0$) to a mean-field realization of the same system ($\\lambda \\to\n\\infty$).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.4\\columnwidth]{shifts.pdf}  \n  \\caption{The black particle effectively interacts with a finite\n    number of particles (in red) that have appropriate random shifts\n    (blue arrows), but these particles may be anywhere in the sample.}\n  \\label{fig:shifts}\n\\end{figure}\n\n\nIn the liquid phase, the mean-field limit of the model has an entropy\nof an ideal gas plus only the first virial correction, just like a van\nder Waals gas. Physically, this comes from the fact that it is very\nunlikely that three (or more) spheres effectively interact\nsimultaneously with one another, as is sketched in\nFig.~\\ref{fig:three_body}.  For instance, in the hard-sphere case, one\nwould need to have, for particles $i$, $j$, and $k$ (having a diameter\n$D$):\n\\begin{equation}\n  \\begin{array}{c}\n  |\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}|\\sim D  \\\\\n  |\\mathbf{x_j}-\\mathbf{x_k}-\\mathbf{A_{jk}}|\\sim D  \\\\\n  |\\mathbf{x_k}-\\mathbf{x_i}-\\mathbf{A_{ki}}|\\sim D\n  \\end{array}\n\\end{equation}\nwhich requires that the random shifts satisfy\n$|\\mathbf{A_{ij}}+\\mathbf{A_{jk}}+\\mathbf{A_{ki}}|\\sim D$. This of\ncourse is very unlikely for shifts $|\\mathbf{A}|\\sim L$, where $L$ is\nthe linear size of the system.\n\n\\begin{figure}\n  \\centering\n  \\hspace{0cm}\n  \\includegraphics[width=0.3\\columnwidth]{three_body}\n  \\hspace{0.5cm}\n  \\includegraphics[width=0.3\\columnwidth]{three_body_RS}\n  \\hspace{0.5cm}\n  \\includegraphics[width=0.3\\columnwidth]{three_body_RS_2}\n  \\caption{\\textbf{Left :} Effective three-body correlation in a\n    finite-dimensional hard-sphere system (with no random shifts),\n    leading to corrections beyond the first virial\n    term. \\textbf{Center :} Same situation in the random shift model,\n    from the point of view of particle 1. Particle 1 sees particles 2\n    and 3 nearby. \\textbf{Right :} From the point of view of particle\n    2, there is no effective three-body interaction, as the random\n    shifts displaces particle 3 very far from particle 2.}\n  \\label{fig:three_body}\n\\end{figure}\n\nIn this work, we will concentrate mostly on the hard-sphere potential,\nbut the procedure is a very generic way to obtain a mean-field limit,\nand can even be generalized  to objects with rotational\ndegrees of freedom, where one can introduce a rotational disorder.\nWe studied both a monodisperse  system, for conceptual simplicity;\nand a bidisperse one, to be able to work with arbitrary small $\\lambda$ without \nhaving to deal with crystallisation.\n\nFrom now on, we set the diameter $D$ of a sphere (in the monodisperse\ncase) to $D=1$, which will be used as length scale. The temperature\n$T$ is set to $1$ almost everywhere, as it is irrelevant for hard\nspheres. The only part where we keep it explicit is in the section\ndealing with the dynamics of the model.  For simplicity we introduce\nthe following notations: a sphere with diameter $1$ in dimension $d$\nhas a volume $2^{-d}v_d$, and a surface $s_d$.\n\n\n\\section{Statics of the liquid phase}\\label{sec:eos_liq}\n\n\\subsection{Grand-canonical formalism and Mayer expansion}\n\nIn this section, we work with a monodisperse system.  We note $V(u)$\nthe interaction potential:\n\\begin{equation}\n  \\begin{array}{r|l}\n    \\multirow{2}{*}{V(u)=}& \\infty \\quad \\text{if} \\quad u>1 \\\\\n    &  0 \\quad \\text{otherwise}    \n  \\end{array}\n\\end{equation}\nThe canonical partition function of the system reads:\n\\begin{equation}\nZ_{\\{\\mathbf{A} \\}}= \\int \\prod_{k} d\\mathbf{x_k} \\exp \\(( -\\sum_{ij}V(\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}) \\))    \n\\end{equation}\nOne would like to study the entropy of the system averaged over disorder:\n\\begin{equation}\n  S=-\\overline{\\ln Z_{\\{\\mathbf{A} \\}}}\n\\end{equation}\nIn the liquid phase, we can treat this average as an {\\em annealed} average over the disorder:\n\\begin{equation}\n  S=S_{an}=-\\ln \\overline{ Z_{\\{\\mathbf{A} \\}}}\n\\end{equation}\nSo the problem reduces to the study of:\n\\begin{equation}\\label{eq:canonical_annealed_partition}\n  \\overline{Z_{\\{\\mathbf{A} \\}}}= \\frac{1}{N!}\\int \\prod_{lm}P(\\mathbf{A}_{lm}) d\\mathbf{A}_{lm} \\int \\prod_{k} d\\mathbf{x_k} \\exp \\(( -\\sum_{ij}V(\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}) \\))    \n\\end{equation}\nWe introduced a $1\/N!$ prefactor to be able to perform the next step, a\nMayer expansion. We just have to remember to compensate  this factor at\nthe end of the computation. To do the Mayer expansion, we have to\ntranslate our problem in a grand-canonical formalism. We introduce the\ngrand-canonical partition function:\n\\begin{equation}\n  \\overline{\\Theta}=\\sum_N z^N \\overline{Z_{\\{\\mathbf{A} \\}}}\n\\end{equation}\nwhere $z$ is the activity, related to the chemical potential $\\mu$ by\n$z=e^{\\mu}$, and we rewrite Eq.~(\\ref{eq:canonical_annealed_partition}) as:\n\\begin{equation}\\label{eq:part_func_mayer}\n  \\overline{Z_{\\{\\mathbf{A} \\}}} = \\int \\prod_{k} d \\mathbf{x_k} \\prod_{ij} \\[[ \\overline{f}(\\mathbf{x_i}-\\mathbf{x_j})+ 1 \\]]  \n\\end{equation}\nwhere $\\overline{f}$ is an annealed Mayer function:\n\\begin{equation}\n  \\label{eq:mayer_f}\n  \\overline{f}(\\mathbf{x}-\\mathbf{y}) = \\int d \\mathbf{A} P(\\mathbf{A}) \\[[ \\exp \\(( - V(\\mathbf{x}-\\mathbf{y}-\\mathbf{A}) \\)) -1 \\]] = -\\int d \\mathbf{A} P(\\mathbf{A}) \\chi ( |\\mathbf{x}-\\mathbf{y}-\\mathbf{A}|)\n\\end{equation}\nwith $\\chi(r)$ the step function such that $\\chi(r)=1$ if $r<1$ and $\\chi(r)=0$ otherwise.\nWe can then introduce a diagrammatic representation to express the\ngrand-canonical potential $G=\\ln \\overline{\\Theta}$ as\nusual\\cite{hansen2006theory}:\n\\begin{equation}\n\\begin{array}[b]{lllll}\n  \\label{eq:G_Mayer}\n  G=\\ln \\overline{\\Theta}=\\{ \\text{ connected diagrams } \\}= \n    \\xymatrix{\\bullet} + &\n    \\segment{\\bullet}{\\bullet} + &\n    \\dipod{\\bullet}{\\bullet}{\\bullet}+ &\n    \\triangle{\\bullet}{\\bullet}{\\bullet}+ &\n    \\dots\n\\end{array}\n\\end{equation}\nwhere the diagram vertices represent a factor $z$ while the bonds\nrepresent the Mayer function $\\overline{f}$.\n\n\\subsection{Mean-field equation of state}\n\nAs already noted by Kraichnan \\cite{kraichnan1962stochastic1}, only the first \nvirial correction survives in the mean-field limit. The entropy can then be expressed in term of the density field $\\rho( \\mathbf{x})$:\n\\begin{equation}\n  \\label{eq:F_mean_field}\nS_{an}[\\rho( \\mathbf{x})]  =  -\\int \\text{d} \\mathbf{x} \\rho( \\mathbf{x}) \\left[ \\ln \\rho( \\mathbf{x})-1 \\right] + \\frac{1}{2} \\int  \\text{d} \\mathbf{x} \\text{d} \\mathbf{y} \\rho( \\mathbf{x}) \\rho( \\mathbf{y}) \\overline{f}(\\mathbf{x}-\\mathbf{y})+N\\ln N\n\\end{equation}\n The $\\ln N$ contribution to the entropy density reflects the fact that\nparticles are not, in this model, truly indistinguishable -- just as\nthey are not in a system particles of polydisperse sizes.\n\nThe fact that the first terms contribute  can be understood by noticing that:\n\\begin{equation}\n  \\overline{f}(\\mathbf{x}-\\mathbf{y})=-v_d\/V  \n\\end{equation}\nThus, in the Mayer expansion Eq~(\\ref{eq:G_Mayer}), a diagram with $n$\nvertices and $m$ bonds reads $V^{n-m}z^n(-v_d)^m$. If $m \\ge  n$, the\ndiagram vanishes in the thermodynamic limit, because this requires that several random shifts add to zero. Only the diagrams having $m<n$ contribute, and they are the tree ones\n(having $m=n-1$):\n\\begin{equation}\n  \\label{eq:G_Mayer_tree}\n    \\ln \\overline{\\Theta}= \n    \\xymatrix{\\bullet } + \n    \\segment{\\bullet}{\\bullet} + \n    \\dipod{\\bullet}{\\bullet}{\\bullet} + \n    \\arch{\\bullet}{\\bullet}{\\bullet}{\\bullet} + \n    \\tripod{\\bullet}{\\bullet}{\\bullet}{\\bullet} + \n    \\dots\n\\end{equation}\n\nThe next step is to do the Legendre transform~\\cite{hansen2006theory} $z \\to \\rho$, which leads to:\n\\begin{equation}\\label{eq:saddle_rho_liquid}\n  \\ln \\rho( \\mathbf{x})=\\ln z + \\xymatrix{  *{\\circ} \\ar@{-}[r] *{\\otimes} & }\n\\end{equation}\nwhere now the $\\otimes$ vertex denotes a factor $\\rho( \\mathbf{x})$,\nand ${\\circ}$ is $1$.  Eq.~(\\ref{eq:saddle_rho_liquid}) is the saddle\npoint equation which minimizes the grand-canonical potential functional\n$G[\\rho( \\mathbf{x})]$, which can be translated back into the\ncanonical ensemble, leading to the entropy Eq.~(\\ref{eq:F_mean_field}).\n\nIn the liquid phase with density $\\rho$, the entropy is thus:\n\\begin{equation}\n  \\frac{S_{an}}{N}=1 - \\ln \\rho - \\frac{1}{2} \\rho v_d + \\ln N\n\\end{equation}\nand the system has a van der Waals equation of state:\n\\begin{equation}\n   P = \\rho+\\frac{1}{2} v_d\\rho^2  \n\\end{equation}\nIt is remarkable that this equation of state (ideal gas with first\nvirial correction) is the one of hard spheres when $d \\to \\infty$\n(see \\cite{frisch1985classical,frisch1999high,PhysRevE.62.6554}). This is a first indication that  the\nlimit $\\lambda \\to \\infty$ and the limit $d \\rightarrow \\infty$ are of the same nature, as we shall see below.\n\n\n\\subsection{Pair correlation function}\n\nAs in high-dimensional liquids or in systems with Kac interactions,\nthe very simple form of the Mayer expansion allows to compute exactly\nthe two-point correlation functions in the liquid phase.\nA `naive' pair correlation function is defined as~\\cite{hansen2006theory}:\n\\begin{equation}\\label{eq:pair_correl_def}\n  \\begin{array}{rl}\n  g(\\mathbf{x}-\\mathbf{y})&=\\overline{<\\delta(\\mathbf{x_1}-\\mathbf{x})\\delta(\\mathbf{x_2}-\\mathbf{y})>} \\\\\n  &=\\dfrac{N(N-1)}{\\rho^2 N!\\overline{Z_{\\{\\mathbf{A} \\}}}} \\int \\prod_{lm}P(\\mathbf{A}_{lm}) d\\mathbf{A}_{lm} \\\\\n  & \\qquad \\int \\prod_{k} d\\mathbf{x_k} \\delta(\\mathbf{x_1}-\\mathbf{x}) \\delta(\\mathbf{x_2}-\\mathbf{y}) \\exp \\(( -\\sum_{ij}V(\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}) \\)) \\\\\n  &=\\dfrac{2}{\\rho N} \\overline{e}(\\mathbf{x_1}-\\mathbf{x_2}) \\dfrac{\\delta \\ln \\overline{Z_{\\{\\mathbf{A} \\}}}}{\\delta \\overline{e}(\\mathbf{x_1}-\\mathbf{x_2})}\n  \\end{array}\n\\end{equation}\nFrom Eq.~(\\ref{eq:F_mean_field}) we then get:\n\\begin{equation}\\label{eq:pair_correl_non_shifted_result}\n  g(\\mathbf{x}-\\mathbf{y})=1+\\overline{f}(\\mathbf{x}-\\mathbf{y})\\simeq 1-v_dP(\\mathbf{x}-\\mathbf{y})\n\\end{equation}\n\n\\noindent When $\\lambda = \\infty$ the pair correlation function is simply:\n\\begin{equation}\n  g(\\mathbf{x}-\\mathbf{y})=1\n\\end{equation}\nThis expresses  the absence of structure in the system due to the\nquenched disorder, which totally blurs the hard core repulsion when\n$\\lambda \\to \\infty$. \nOf course, this does not mean that there are no\nreal pair correlations  in the system. \nIndeed, a  more interesting\nquantity is the pair correlation function `seen from one particle':\n\\begin{equation}\\label{eq:pair_correl_shifted_def}\n\\begin{array}{rl}\n  g_S(\\mathbf{x}-\\mathbf{y})&=\\overline{<\\delta(\\mathbf{x_1}-\\mathbf{x})\\delta(\\mathbf{x_2}+\\mathbf{A}_{12}-\\mathbf{y})>} \\\\\n  &=\\dfrac{N(N-1)}{\\rho^2 N!\\overline{Z_{\\{\\mathbf{A} \\}}}} \\int \\prod_{lm}P(\\mathbf{A}_{lm}) d\\mathbf{A}_{lm} \\\\  & \\qquad \\qquad \\int \\prod_{k} d\\mathbf{x_k} \\delta(\\mathbf{x_1}-\\mathbf{x}) \\delta(\\mathbf{x_2}+\\mathbf{A}_{12}-\\mathbf{y}) \\exp \\(( -\\sum_{ij}V(\\mathbf{x_i}-\\mathbf{x_j}-\\mathbf{A_{ij}}) \\)) \\\\\n  &=\\dfrac{2}{\\rho N} \\exp \\left(-V(\\mathbf{x}-\\mathbf{y})\\right) \\dfrac{\\delta \\ln \\overline{Z_{\\{\\mathbf{A} \\}}}}{\\delta \\overline{e}(\\mathbf{x}-\\mathbf{y})}\n  \\end{array}\n\\end{equation}\nInserting Eq.~(\\ref{eq:F_mean_field}), we obtain:\n\\begin{equation}\\label{eq:pair_correl_shifted_result}\n  g_S(\\mathbf{x}-\\mathbf{y})=\\exp \\left(-V(\\mathbf{x}-\\mathbf{y})\\right)\\end{equation}\n\n\\noindent This pair correlation function is\nidentical to the one obtained for a hard sphere system in infinite\ndimensions.\n\n\n\n\n\\subsection{Relation with a Kac potential in the statics of the liquid phase}\n\nThe connection between the two ways of approaching  a mean-field limit\n(sending $d \\to \\infty$ or introducing random shifts with range\n$\\lambda \\to \\infty$) is to be compared with a similar connection in\nsystems with Kac type potentials \\cite{klein1986instability}. We can\npush forward this analogy by an explicit mapping of our model onto a system with a\nKac potential, valid (only) for static quantities in the low density liquid phase, when $\\lambda$ is\nlarge but finite.\n\nIn this case, the Fourier transform of the Mayer function defined by\nEq.~(\\ref{eq:mayer_f}) is the product of a Bessel function having a\nrange $\\sim 1$ with the Fourier transform $\\tilde{P}$ of the shifts\ndistribution, having a range $\\sim 1\/\\lambda$:\n\\begin{equation}\n  \\tilde{f}(k)= -(2\\pi)^{d\/2} \\tilde{P}_{\\lambda}(k) \\frac{J_{\\frac{d}{2}}(k)}{k^{d\/2}} \n\\end{equation}\nThus, as long as $\\lambda\\gg1$, we have:\n\\begin{equation}\n  \\overline{f}(r) \\simeq -v_d P_{\\lambda}(r) = - \\gamma^{-d} K (\\gamma^{-1}r)  \n\\end{equation}\nwith $K$ a short ranged bounded positive function.\nThis function obviously has a range $\\gamma \\sim \\lambda$, as\nlong as $\\lambda\\gg1$. Thus, (\\ref{eq:F_mean_field}) reads:\n\\begin{equation}\n S_{an}[\\rho( \\mathbf{x})]  =  - \\int \\text{d} \\mathbf{x} \\rho( \\mathbf{x}) \\left[ \\ln \\rho( \\mathbf{x}) -1 \\right] - \\frac{ \\gamma^{-d}}{2} \\int  \\text{d} \\mathbf{x} \\text{d} \\mathbf{y} \\rho( \\mathbf{x}) \\rho( \\mathbf{y}) K(\\gamma^{-1}|\\mathbf{x}-\\mathbf{y}|)+N\\ln N\n\\end{equation}\n\nThis is exactly the functional obtained in the so-called mean-field\napproximation for a potential $K$ (with temperature $\\beta=1$), and it\nis exact in the Kac limit $\\gamma \\to \\infty$\n\\cite{grewe1977kirkwood1}. Indeed, the naive pair correlation function\nEq.~(\\ref{eq:pair_correl_non_shifted_result}) is equivalent to the\none obtained for a Kac potential~\\cite{zachary2008gaussian}.\n\nThis mapping concerns the {\\em statics} of the {\\em liquid}\nphase, and   does not mean that the {\\em dynamics} of our model has anything to\ndo with the one of a Kac model, even in the liquid phase. As a matter of fact, in the following\nsections we present  results showing that a dynamical glass\ntransition occurs in our model with a Gaussian distribution for the\nshifts, whereas the related Kac model, which is the Gaussian Core\nModel~\\cite{stillinger1976phase} becomes in the high density limit an\nideal gas~\\cite{stillinger1976phase,lang2000fluid}.\n\n\n\\subsection{Avoiding the Kirkwood instability}\n\nIf $\\lambda$ is finite but large, we can take\nEq.~(\\ref{eq:F_mean_field}) as a good approximation of the liquid\nphase, but we have to be careful in our choice of the random shift\ndistribution $P(\\mathbf{A})$. It is well known that the mean-field\nentropy functional for Kac models can be unstable above a given\ndensity (the so-called Kirkwood instability~\\cite{grewe1977kirkwood2})\ntowards a phase with spatial density modulations.  This can be seen\nvia a linear stability analysis of Eq.~(\\ref{eq:F_mean_field}). If we\nperturb the uniform liquid phase solution $\\rho( \\mathbf{x})=\\rho$\nwith a small oscillatory term $\\rho_{\\epsilon}(\n\\mathbf{x})=(1+\\epsilon \\exp(i\\mathbf{k} \\mathbf{x}))\\rho$, we get:\n\\begin{equation}\\label{eq:linear_perturb_entopy}\n  S_{an}[\\rho_{\\epsilon}(\\mathbf{x})]=S_{an}[\\rho] + \\frac{\\epsilon^2 N \\rho }{2} \\text{TF} \\left[ \\dfrac{\\delta^2 S_{an}}{\\delta \\rho(\\mathbf{x}) \\delta \\rho(\\mathbf{y})} \\right] (\\mathbf{k})\n\\end{equation}\nwith:\n\\begin{equation}\n  \\frac{\\delta^2 S_{an}}{\\delta \\rho(\\mathbf{x}) \\delta \\rho(\\mathbf{y})}=-\\frac{1}{\\rho} \\delta(\\mathbf{x}-\\mathbf{y})+ \\overline{f}(\\mathbf{x}-\\mathbf{y})\n\\end{equation}\nFrom Eq.~(\\ref{eq:linear_perturb_entopy}), it is clear that the uniform liquid phase is stable only if:\n\\begin{equation}\\label{eq:stability_cond1}\n  \\text{TF} \\left[ \\dfrac{\\delta^2 S_{an}}{\\delta \\rho(\\mathbf{x}) \\delta \\rho(\\mathbf{y})} \\right] (\\mathbf{k})=-\\frac{1}{\\rho} + \\tilde{f}(\\mathbf{k})  < 0 \\qquad \\forall \\mathbf{k}\n\\end{equation}\nIf $\\tilde{f}(\\mathbf{k})$ takes positive values for some wave vector\n$k$, there is a value of $\\rho$ above which the condition\nEq.~(\\ref{eq:stability_cond1}) is not fulfilled: this is the Kirkwood\ninstability. The only way to avoid this transition is to have a Mayer function\n with a negative Fourier transform:\n\\begin{equation}\n  \\label{eq:stability_cond2}\n \\tilde{f}(\\mathbf{k})\n< 0 \\qquad \\qquad \\forall \\mathbf{k}\n\\end{equation}\nSince:\n\\begin{equation}\n\\tilde{f}(\\mathbf{k})= -(2\\pi)^{d\/2} \\tilde{P}_{\\lambda}(k) \\frac{J_{\\frac{d}{2}}(k)}{k^{d\/2}}\n\\end{equation}\nwe have to tune the distribution of the random shifts to get the\ndesired property. If $\\lambda\\gg1$, the range of $\\tilde{P}_{\\lambda}$\n($\\sim \\lambda^{-1})$ is much smaller than the range of\n$J_{\\frac{d}{2}}$, therefore $-\\tilde{f}(\\mathbf{k})= v_d\\tilde{P}_{\\lambda}(\\mathbf{k})$ and taking\n$\\tilde{P}_{\\lambda}(\\mathbf{k})>0$ is enough to ensure condition\n(\\ref{eq:stability_cond2}). \nIn this work, we will take a Gaussian\ndistribution for the shifts:\n\\begin{equation}\n  P_{\\lambda}(\\mathbf{A})=\\frac{1}{(2\\pi \\lambda)^d}\\exp\\(( -\\frac{|\\mathbf{A}|^2}{2\\lambda^2}\\))  \n\\end{equation}\nAs an example of the Kirkwood instability, we show in Fig.~\\ref{fig:kirkwood} a dense configuration of the random shift model with a flat density of shifts.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{RS_kirkwood.pdf}\n  \\caption{Modulated phase due to a Kirkwood instability. Here the\n    random shift distribution is flat within  a sphere with radius\n    $\\lambda=2$.}\n  \\label{fig:kirkwood}\n\\end{figure}\n\n\\subsection{Corrections beyond Mean-field: the role of high dimensionality.}\n\nWe can consider corrections to the mean-field equation of \nstate. As we shall see, for finite $\\lambda$ they vanish with dimensionality as $\\propto \\lambda^{-d}$.\n In the Mayer expansion of the entropy, the dominant correction will \ncome from diagrams with $m=n$, which are \nthe ring diagrams:\n\\begin{equation}\\label{eq:ring_entropy}\n  S_{an,\\lambda}= N\\ln N -\\int \\text{d} \\mathbf{x} \\rho( \\mathbf{x}) \\left[ \\ln \\rho( \\mathbf{x})-1 \\right] + \\segment{\\otimes}{\\otimes} + \\triangle{\\otimes}{\\otimes}{\\otimes} + \\square{\\otimes}{\\otimes}{\\otimes}{\\otimes} + \\pentagon{\\otimes}{\\otimes}{\\otimes}{\\otimes}{\\otimes} + \\dots\n\\end{equation}\n\n\\noindent The resummation of these diagrams has been done by \nMontroll and Mayer \\cite{mayer1941statistical}:\n\\begin{equation}\\label{eq:S_rings}\n\\frac{S_{an,\\lambda}}{N} = 1 -  \\ln \\rho - \\frac{1}{2} \\rho v_d + \\frac{1}{2}\\frac{1}{(2\\pi)^d} \\rho^2 \\int \\Omega_d k^{d-1} \\text{d} k \\frac{ \\[[\\tilde{f}(k)\\]]^3 }{ 1-\\rho \\tilde{f}(k) }+\\ln N\n\\end{equation}\nand gives, for $\\lambda\\gg1$:\n\\begin{equation}\n  \\begin{array}{rl}\n    \\frac{S_{an,\\lambda}}{N} & =  1 -  \\ln \\rho - \\frac{1}{2} \\rho v_d \n    - \\frac{1}{2}\\frac{1}{(2\\pi)^d} \\rho^2 \\int \\Omega_d k^{d-1} \\text{d} k \\frac{ \\[[ v_d\\tilde{P}_{\\lambda}(k)\\]]^3 }{ 1+ \\rho v_d \\tilde{P}_{\\lambda}(k) } +\\ln N\\\\ \n    & = 1 -  \\ln \\rho -  \\frac{1}{2}\\rho v_d  - \\frac{1}{2}\\frac{1}{(2\\pi)^d \\lambda^d} \\rho^2 I_{31}(\\rho)+\\ln N\n  \\end{array}\n\\end{equation}\nwhere we have introduced the $\\lambda$-independent factor:\n\\begin{equation}\n  I_{ab}(\\rho)=\\int \\Omega_d k^{d-1} \\text{d} k \\frac{ \\[[ v_d\\tilde{P}_1(k)\\]]^a }{ \\left[1+ \\rho v_d \\tilde{P}_1(k) \\right]^b}\n\\end{equation}\nThe equation of state is then:\n\\begin{equation}\n  P=\\rho+\\frac{1}{2} \\rho^2 v_d  + \\frac{\\rho^3}{(2\\pi)^d \\lambda^d} \\left[ I_{31}(\\rho) -\\frac{\\rho}{2} I_{42}(\\rho) \\right]\n\\end{equation}\nThis expression gives a quantitative estimation of  how the approximation becomes better\nat higher dimensions, through the factor  $\\lambda^d$. Already in \n$d=3$, the ring corrections are less than 1\\% at $\\phi=1$ for a range as small f\nas $\\lambda=1$. When $d>3$, one needs to take a very small range of \nrandom shifts to feel any finite dimensional effect.\n\n\\subsection{Estimation of the  density as $P \\rightarrow \\infty$}\n\nThe model with $\\lambda \\rightarrow \\infty$ has a maximal packing density that diverges in the thermodynamic limit,\na rather awkward property from the thermodynamic point of view.\nWhen $\\lambda \\to \\infty$ we can derive an estimation of the maximum\ndensity of the random shift model.  To do this, we consider the following algorithm:\n\n\\noindent {\\em i)} Starting a configuration with $N$ spheres in a volume $V$, generate\n$N$ random shifts,\n\n\\noindent {\\em ii)} Locate positions where one can add, without any\noverlap, a new sphere\ninteracting with other spheres via the $N$ random shifts,\n\n\\noindent {\\em iii)} If such a position exists, add this\n$(N+1)$th sphere, and go back to step $1$.\n\nUp to which density such an algorithm can work? When $\\lambda \\to\n\\infty$, the $(N+1)$th sphere will see the other spheres as having\ncompletely random positions. Then the probability for a position\n$\\mathbf{x}$ to satisfy the hard sphere constraints imposed by the $N$\nother spheres is simply:\n\\begin{equation}\n  P_N(\\mathbf{x})=\\left(1-\\frac{v_d}{V}\\right)^N \\simeq e^{-\\frac{Nv_d}{V}}=e^{-\\phi}\n\\end{equation}\n\nThen the available volume to place the $(N+1)$th sphere will be\n$Ve^{-\\phi}$. The algorithm will not be able to find a solution once\nthe available volume is of the order of $v_d$.  If we note\n$\\phi_{max}$ the volume fraction for which this happens, we have:\n\\begin{equation}\n  e^{-\\phi_{max}} \\sim \\frac{v_d}{V}\n\\end{equation}\nwhich gives:\n\\begin{equation}\\label{eq:phi_max}\n  \\phi_{max} \\sim \\ln \\frac{V}{v_d} = \\ln N -\\ln (\\ln N) + \\ln (\\ln (\\ln N)) ...\n\\end{equation}\nWe implemented this algorithm for the random shift model in\n$d=1$. Results are presented in Fig.~\\ref{fig:sequential_addition},\nshowing the $\\ln N$ dependence of $\\phi_{max}$.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{sequential_addition.pdf}  \n  \\caption{Maximum density $\\phi_{max}$ obtained by the algorithm\n    described in the main text, showing the $\\ln N$ dependence\n   of Eq.~(\\ref{eq:phi_max}).}\n  \\label{fig:sequential_addition}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Dynamics in the liquid phase}\\label{sec:dynamics}\n\n\\subsection{A warming up exercise: computation of the canonical partition function}\n\nThe essential steps followed to compute the exact dynamic equation are\nsimilar to the ones performed in making an equilibrium computation in the {\\em  canonical} ensemble. It is\nthus instructive to understand them in this context first.\nStarting again from Eq.~(\\ref{eq:part_func_mayer}), and\nintroducing the density $\\rho(\\underline{\\mathbf{x}})$ by:\n\\begin{equation}\n  \\rho(\\mathbf{x})=\\sum_i \\delta(\\mathbf{x}-\\mathbf{x_i})  \n\\end{equation}\nwith ${\\bf x_i} $ the coordinate of the $i-th$ particle,\nwe can rewrite $\\overline{Z_{\\{\\mathbf{A} \\}}}$ as:\n\\begin{equation}\n  \\overline{Z_{\\{\\mathbf{A} \\}}} =  \\int \\text{d} \\mathbf{x_i} \\int \\mathscr{D}[\\rho( \\mathbf{x})]  \\delta \\left(\\rho( \\mathbf{x})-\\sum_i \\delta(\\mathbf{x}-\\mathbf{x}_i) \\right) \n\\exp \\left[\\frac{1}{2} \\int \\text{d} \\mathbf{x} \\text{d} \\mathbf{y}  \\rho(\\mathbf{x}) \\rho(\\mathbf{y}) \\ln\\left[ 1+\\overline{f}(\\mathbf{x_i}-\\mathbf{x_j})\\right] \\right]\n\\end{equation}\n Now we can exponentiate the $\\delta$ constraint using a\nsecond field $\\hat{\\rho}$, and integrate over the\n$\\underline{\\mathbf{x_i}}$'s:\n\\begin{equation}\n  \\begin{array}{rl}\n    \\overline{Z_{\\{\\mathbf{A} \\}}} =  \\int & \\mathscr{D}[\\rho( \\mathbf{x})] \\mathscr{D}[\\hat{\\rho}( \\mathbf{x})] \\\\\n& \\exp\\left\\{ i \\int \\text{d} \\mathbf{x} \\hat{\\rho}( \\mathbf{x}) \\rho( \\mathbf{x}) +  N \\ln \\int  \\text{d} \\mathbf{x} e^{-i \\hat{\\rho}( \\mathbf{x})}+ \\frac{1}{2} \\int \\text{d} \\mathbf{x} \\text{d} \\mathbf{y}  \\rho(\\mathbf{x}) \\rho(\\mathbf{y}) \\ln\\left[ 1+\\overline{f}(\\mathbf{x_i}-\\mathbf{x_j})\\right] \\right\\}\n  \\end{array}\n\\end{equation}\n Again noticing that:\n\\begin{equation}\n  \\overline{f}(\\mathbf{x}-\\mathbf{y}) = -\\frac{v_d}{V} \n\\end{equation}\n we can expand the logarithm to get:\n\\begin{equation}\n  \\begin{array}{rl}\n    \\overline{Z_{\\{\\mathbf{A} \\}}} =  \\int & \\mathscr{D}[\\rho( \\mathbf{x})] \\mathscr{D}[\\hat{\\rho}( \\mathbf{x})] \\\\\n& \\exp\\left\\{ i\\int \\text{d} \\mathbf{x} \\hat{\\rho}( \\mathbf{x}) \\rho( \\mathbf{x}) + N \\ln \\int  \\text{d} \\mathbf{x} e^{-i \\hat{\\rho}( \\mathbf{x})}+ \\frac{1}{2} \\int \\text{d} \\mathbf{x} \\text{d} \\mathbf{y}  \\rho(\\mathbf{x}) \\rho(\\mathbf{y}) \\overline{f}(\\mathbf{x}-\\mathbf{y})  \\right\\}\n  \\end{array}\n\\end{equation}\nThis integral may be evaluated by saddle point with respect to the \nfields $\\rho$ and $\\hat{\\rho}$ (note that each term in the exponential is of order $N$, including the last one, due to the value of $\\overline{f}$). This gives:\n\\begin{equation}\\label{eq:saddle_statics}\n  \\begin{array}{rl}\n    \\rho( \\mathbf{x}) & = N\\frac{e^{-i \\hat{\\rho}( \\mathbf{x})}}{\\int \\text{d} \\mathbf{x} e^{-i \\hat{\\rho}( \\mathbf{x})} } \\\\\n    \\hat{\\rho}( \\mathbf{x}) & = i \\int \\text{d} \\mathbf{y} \\rho(\\mathbf{y}) \\overline{f}(\\mathbf{x}-\\mathbf{y})\n  \\end{array}\n\\end{equation}\nThus, the logarithm of the partition function can be written in terms of $\\rho$ as:\n\\begin{equation}\\label{eq:replicated_FT_partition0}\n  \\ln \\overline{Z_{\\{\\mathbf{A} \\}}} = - \\int \\text{d} \\mathbf{x} \\rho( \\mathbf{x}) \\ln \\rho( \\mathbf{x}) + \\frac{1}{2} \\int  \\text{d} \\mathbf{x} \\text{d} \\mathbf{y} \\rho( \\mathbf{x}) \\rho( \\mathbf{y}) \\overline{f}(\\mathbf{x}-\\mathbf{y})+N\\ln N\n\\end{equation}\nBelow, we shall follow the same steps, but with trajectories playing the role of particles.\n\n\n\n\\subsection{Derivation of the exact dynamics as a partition function of trajectories}\\label{sec:RS_dynamics}\n\nWe wish to study the  dynamics of the model in the mean-field limit, starting from the Langevin equation:\n\\begin{equation}\\label{eq:langevin}\n  \\dot{\\mathbf{x}}_i(t)=-\\sum_{j\\neq i} \\nabla_i\nV(\\mathbf{x}_i(t)-\\mathbf{x}_j(t)-\\mathbf{A}_{ij})+\\boldsymbol{\\eta}_i(t)+\\mathbf{h}_i(t),\n\\end{equation}\nwhere we have introduced an external field $\\mathbf{h}_i(t)$, which\nacts as a source term, and will later be set to zero. The vector\n$\\boldsymbol{\\eta}_i(t)$ is a Gaussian white noise with variance $2T$,\nwhere $T$ is the temperature, that we will keep explicit for all the\nderivation of the dynamics equations:\n\\begin{equation}\n    <\\boldsymbol{\\eta}_i(t)\\boldsymbol{\\eta}_j(t')>=2Td\\; \\delta_{ij}\\; \\delta(t-t')\n\\end{equation}\n\nWe denote $P(\\{\\mathbf{x}^0\\},\\{\\mathbf{x}^\\tau\\}, \\tau)$ the\nprobability of having particle $i$ in $\\mathbf{x}^0_i$ at time $t=0$\nand in $\\mathbf{x}^\\tau_1$ at time $t=\\tau$, in the absence of\nexternal field $\\mathbf{h}$.  We may express this as a sum over paths\nusing the Martin-Siggia-Rose\/\nDeDominicis-Jensen~\\cite{zinn2002quantum} formalism. Averaging over\nthe noise, this gives:\n\\begin{eqnarray}\n  \\label{eq:P_MSR}\n    P(\\{\\mathbf{x}^0\\},\\{\\mathbf{x}^\\tau\\}, \\tau)&=&\\int \\prod_{ij}\n    \\text{d} \\mathbf{A}_{ij} P(\\mathbf{A}_{ij}) \\nonumber \\\\\n& & \\quad     \\int_{\\{\\mathbf{x}(0)\\}=\\{\\mathbf{x}^0\\}}^{\\{\\mathbf{x}(\\tau)\\}=\\{\\mathbf{x}^\\tau\\}}\n    \\prod_{i} \\text{D}[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t)] \\exp \\left[\n      -\\mathcal{S}[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}]\n      \\right],\n\\end{eqnarray}\n with\n\\begin{equation}\n  \\label{eq:action}\n  \\begin{array}{rl}\n    \\mathcal{S}[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}] &= \n     \\int_0^\\tau \\mbox{d} t \\left[\\sum_i \\left(\\dot{\\mathbf{x}}_i(t)\\hat{\\mathbf{x}}_i(t) +\n\\hat{\\mathbf{x}}_i(t)\\hat{\\mathbf{x}}_i(t)\\right)\\right] \\\\\n    & \\;\\;\\; +\\int_0^\\tau \\mbox{d} t \\left[\\frac{1}{2}\\sum_{<ij>}\\left(\n\\hat{\\mathbf{x}}_i(t) \\nabla_i V(\\mathbf{x}_i -\n\\mathbf{x}_j-\\mathbf{A}_{ij}))+\\hat{\\mathbf{x}}_j(t) \\nabla_j V(\\mathbf{x}_j -\n\\mathbf{x}_i - \\mathbf{A}_{ji}) \\right) \\right] \\\\\n    &= \\sum_i \\Phi[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t)] + \\sum_{ij} W_{ij}\n[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t),\\mathbf{x}_j(t),\\hat{\\mathbf{x}}_j(t)]\n\\end{array}\n\\end{equation}\nwhere the last two lines define $\\Phi$ and $W_{ij}$.\nIntegrations in Eq.~(\\ref{eq:P_MSR}) of the variables\n$\\hat{\\mathbf{x}}_i(t)$ are along the imaginary axis.  The quantity\n$P[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}]=\\exp \\left[\n  -\\mathcal{S}[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}] \\right]$\nmay seem mysterious, because of the variables $\\hat{\\mathbf{x}}_i(t)$,\nwhich do not have an immediate physical meaning.  In order to\nunderstand them, we consider the probability of paths \\textit{in the\n  presence } of external fields $\\{\\mathbf{h}_i(t)\\}$:\n\\begin{equation}\n P[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}, \\{\\mathbf{h}_i(t)\\}\n]=\\exp \\left\\{\n  -\\mathcal{S}[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}]-\\hat{\\mathbf{x}}_i(t)\\mathbf{h}_i(t)\\right\\},\n\\end{equation}\n Integrating over the `hat' variables, we find that:\n\\begin{equation}\n  \\begin{array}{rl}\n  P[\\{\\mathbf{x}_i(t)\\}, \\{\\mathbf{h}_i(t)\\}]& =\\int \\prod_{i}\n\\text{D}[\\hat{\\mathbf{x}}_i(t)] P[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\},\n\\{\\mathbf{h}_i(t)\\}]\\\\\n&=\n  \\int \\prod_{i} \\text{D}[\\hat{\\mathbf{x}}_i(t)]\ne^{\\hat{\\mathbf{x}}_i(t)\\mathbf{h}_i(t)}P[\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\}]\n  \\end{array}\n\\end{equation}\nIn other words, the `hat' variables $\\{\\hat{\\mathbf{x}}_i(t)\\}$ are\nthe Fourier-transform variables of the fields: the probability of a\ntrajectory $(\\{\\mathbf{x}_i(t)\\},\\{\\hat{\\mathbf{x}}_i(t)\\})$ in the\nabsence of field, is the Fourier transform of the corresponding\nphysical probability $\\{\\mathbf{x}_i(t)\\}$ in the presence of an\nexternal field $\\{\\mathbf{h}_i(t)\\}$.\n\nThe functional formalism casts the dynamical problem into a form that\nresembles a partition function, but with one-dimensional objects (the\ntrajectories) replacing the point particles.  We may exploit the\nanalogy to repeat the canonical computation in the preceding\nsubsection. In particular, we may define the `dynamical Mayer\nfunction' as:\n\\begin{equation}\\label{eq:mayer_f_dyn_def}\n1+\\overline{f}_d[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t),\\mathbf{x}_j(t),\\hat{\\mathbf{x}}_j(t)]\n=  \\int \n    \\text{d} \\mathbf{A}_{ij} P(\\mathbf{A}_{ij}) \\exp \\left\\{ - W_{ij}\n[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t),\\mathbf{x}_j(t),\\hat{\\mathbf{x}}_j(t)]\n\\right\\} \n\\end{equation}\n\n\\noindent  and introduce the dynamical `partition function':\n\n\\begin{equation}\\label{eq:dyn_annealed_partition}\n  \\begin{array}{rl}\n    \\overline{Z_{N,\\tau}}& = \\int \\prod_i \\text{d} \\mathbf{x}^0_i \\text{d} \\mathbf{x}^\\tau_i \nP_{\\{\\mathbf{A}\\}}(\\{\\mathbf{x}^0\\},\\{\\mathbf{x}^\\tau\\}, \\tau)\\\\\n& =  \\int \\prod_{i} \\left(\\text{D}[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t)]\ne^{-\\Phi[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t)]} \\right) \\prod_{ij}\n\\left(1+\\overline{f}_d[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t),\\mathbf{x}_j(t),\\hat{\\mathbf{x}}_j(t)]\\right)\n  \\end{array}\n\\end{equation}\nIn the above expression the path integral is now performed over paths\nwith free boundary conditions.  The initial conditions are weighted\nwith a flat distribution. With these small reinterpretations,\n$\\overline{Z_{N,\\tau}}$ looks like the (static) partition function of $N$\n'polymers' $(\\mathbf{x}(t),\\hat{\\mathbf{x}}(t))$, in an external\nfield $\\Phi$, interacting via a potential $\\ln \\left[1+\\overline{f}_d\\right]$.\n\nNext we introduce a density field:\n\n\\begin{equation}\\label{eq:traj_rho}\n \\rho[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]=\\sum_i\n\\delta[\\mathbf{x}(t)-\\mathbf{x}_i(t)]\n\\delta[\\hat{\\mathbf{x}}(t)-\\hat{\\mathbf{x}}_i(t)]\n\\end{equation}\nNote that $\\delta$ is here a Dirac function in the sense of\ntrajectories, i.e. a product of ordinary deltas, one for each time.\nInserting this field in the partition function we obtain:\n\\begin{equation}\n\\prod_{i} e^{-\\Phi[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t)]}=\\exp \\left[ -  \\int\n\\text{D}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)] \\rho[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]\n\\Phi[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)] \\right],\n\\end{equation}\nand:\n\\begin{equation}\\label{eq:field_introduction}\n  \\begin{array}{rl}\n    \\prod_{i\\neq\nj}\\left(1+\\overline{f}_d[\\mathbf{x}_i(t),\\hat{\\mathbf{x}}_i(t),\\mathbf{x}_j(t),\\hat{\\mathbf{x}}_j(t)]\\right)\n  &  =  \\exp \\bigg[ \\frac{1}{2} \\int\n\\text{D}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]\\text{D}[\\mathbf{y}(t),\\hat{\\mathbf{y}}(t)] \n      \\\\ \n &  \\hspace{-2.7cm} \\rho[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]\n\\rho[\\mathbf{y}(t),\\hat{\\mathbf{y}}(t)] \\ln\n\\left(1+\\overline{f}_d[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t),\\mathbf{y}(t),\\hat{\\mathbf{y}}(t)]\n\\right) \\bigg].\n    \\end{array}\n\\end{equation}\n\nNote that the right-hand side contains a self-interaction term that\nwas not present on the left-hand side, but this term is negligible\ncompared to the interparticle interactions.  \n\nAs in the static case, looking more closely to the function\n$\\overline{f}_d$ defined by Eq.~(\\ref{eq:mayer_f_dyn_def}), we see that\nin the integral over disorder, the integrand is $1$ if the two\ntrajectories do not interact. These trajectories do interact if the\nrandom shift is able to bring them close to one  another. If the\ntrajectories explore a finite volume during between times $0$ and\n$\\tau$ (\\textit{ie} $\\tau$ is not too large, at least much smaller\nthan the ergodic time), this is only possible for a finite volume of\nintegration on the random shift. As the distribution of the shifts is\n$P(\\mathbf{A})=1\/V$, this means that the function $\\overline{f}_d$ is\nof order $\\Gamma\/V$, where $\\Gamma$ is the typical volume covered by a\ntrajectory during an time interval $\\tau$. Then, just as in the static calculation, we may use that\n$\\ln(1+\\overline{f}_d)\\simeq \\overline{f}_d$.  Now, imposing the\ncondition Eq.~(\\ref{eq:traj_rho}) via:\n\\begin{equation}\n\\begin{array}{rl}\n  & \\delta \\bigg[ \\rho[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]-\\sum_i\n\\delta[\\mathbf{x}(t)-\\mathbf{x}_i(t)]\n\\delta[\\hat{\\mathbf{x}}(t)-\\hat{\\mathbf{x}}_i(t)] \\bigg]  = \\int \nD\\left[ \\hat{\\rho}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]\\right]  \\\\ & \\qquad  \\qquad e^{i\\int\n\\text{D}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]  \\rho[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)]\n\\hat{\\rho}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)] -\ni\\hat{\\rho}[\\mathbf{x}(t),\\hat{\\mathbf{x}}(t)] \\sum_i\n\\delta[\\mathbf{x}(t)-\\mathbf{x}_i(t)]\n\\delta[\\hat{\\mathbf{x}}(t)-\\hat{\\mathbf{x}}_i(t)] }\n\\end{array}\n\\end{equation}\nand integrating over $\\mathbf{x}_i(t)$'s and\n$\\hat{\\mathbf{x}}_i(t)$'s, we get, for a system at equilibrium at time\n$0$ (we dropped the time dependence of the paths to simplify the notation):\n\\begin{equation}\\label{eq:field_theory_dyn_part_func}\n  \\begin{array}{rl}\n    Z_{N,\\tau}  &= \\int D[\\rho] D[\\hat{\\rho}]\\exp \\bigg\\{  i\\int\n\\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}]\\rho[\\mathbf{x}, \\hat{\\mathbf{x}}]\n\\hat{\\rho}[\\mathbf{x}, \\hat{\\mathbf{x}}] + N\\ln \\int\n\\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}] e^{-i\\hat{\\rho}[\\mathbf{x}, \\hat{\\mathbf{x}}]}\n\\\\ \n& - \\int \\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}] \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}]\n\\Phi[\\mathbf{x}, \\hat{\\mathbf{x}}]\n + \\frac{1}{2} \\int \\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}]\n\\text{D}[\\mathbf{y},\\hat{\\mathbf{y}}]  \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}] \n\\rho[\\mathbf{y}, \\hat{\\mathbf{y}}]\n\\overline{f}_d[\\mathbf{x},\\hat{\\mathbf{x}},\\mathbf{y},\\hat{\\mathbf{x}}]\n\\bigg\\}\n  \\end{array}\n\\end{equation}\nFor the mean-field dynamics we can take the saddle-point with respect to $\\rho$ and $\\hat \\rho$ of the last\nequation. This reads:\n\\begin{equation}\n\\begin{array}{rl}  \n  \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}] & =N\\frac{e^{-i\\hat{\\rho}[\\mathbf{x},\n      \\hat{\\mathbf{x}}]}}{\\int \\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}]\n    e^{-i\\hat{\\rho}[\\mathbf{x}, \\hat{\\mathbf{x}}]}}, \\\\\n  \\hat{\\rho}[\\mathbf{x}, \\hat{\\mathbf{x}}] & =-i\\Phi[\\mathbf{x}, \\hat{\\mathbf{x}}] + \n  i\\int \\text{D}[\\mathbf{y},\\hat{\\mathbf{y}}]  \\rho[\\mathbf{y}, \\hat{\\mathbf{y}}]\n  \\overline{f}_d[\\mathbf{x},\\hat{\\mathbf{x}},\\mathbf{y},\\hat{\\mathbf{y}}].\n\\end{array}\n\\end{equation}\nwhich gives a closed equation on the density of paths $\\rho[\\mathbf{x},\n\\hat{\\mathbf{x}}]$:\n\\begin{equation}\\label{eq:saddle_dynamics}\n  \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}]  = \\frac{1}{\\mathcal{N}} e^{-\\Phi[\\mathbf{x}, \\hat{\\mathbf{x}}] + \\int \\text{D}[\\mathbf{y},\\hat{\\mathbf{y}}]  \\rho[\\mathbf{y}, \\hat{\\mathbf{y}}]\n    \\overline{f}_d[\\mathbf{x},\\hat{\\mathbf{x}},\\mathbf{y},\\hat{\\mathbf{y}}] }\n\\end{equation}\nwhere $\\mathcal{N}$ ensures that the density is normalized to $N$.\n\n\\noindent Reinserting this equation in the partition function, we get the functional $\\mathcal{S}=- \\ln Z$, which reads:\n\\begin{equation}\\label{eq:free_energy_dynamics}\n\\begin{array}{rl}\n\\mathcal{S}[\\rho] &=   \\int \\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}] \\rho[\\mathbf{x},\n\\hat{\\mathbf{x}}] \\ln \\left[\\rho[\\mathbf{x}, \\hat{\\mathbf{x}}] \\right]  + \\int\n\\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}] \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}] \\Phi[\\mathbf{x},\n\\hat{\\mathbf{x}}]  +\\mathcal{S}_{int}[\\rho] -N\\ln N \\\\\n & \\mathcal{S}_{int}[\\rho]=\n - \\frac{1}{2} \\int \\text{D}[\\mathbf{x},\\hat{\\mathbf{x}}]\n\\text{D}[\\mathbf{y},\\hat{\\mathbf{y}}]  \\rho[\\mathbf{x}, \\hat{\\mathbf{x}}] \n\\rho[\\mathbf{y}, \\hat{\\mathbf{y}}]\n\\overline{f}_d[\\mathbf{x},\\hat{\\mathbf{x}},\\mathbf{y},\\hat{\\mathbf{y}}]\n\\end{array}\n\\end{equation}\nEq.~(\\ref{eq:saddle_dynamics})  is the counterpart  of the saddle point\nequation~(\\ref{eq:saddle_statics}) obtained in the previous subsection for the average density.\nAs happens often in this kind of problems, even the mean-field equations are hard to solve, in this case\nbecause the complexity of an object like $\\rho[\\mathbf{x}, \\hat{\\mathbf{x}}]$\nmakes the problem  intractable. But, just as in equilibrium calculations, we can\ntry, as a further approximation, to find extrema of the free energy\nEq.~(\\ref{eq:saddle_dynamics}) in a well chosen restricted subspace of\nall possible $\\rho[\\mathbf{x}, \\hat{\\mathbf{x}}]$. The next subsection\nis devoted to the search of such an ansatz.\n\n\n\\subsection{An approximation in terms of two-point functions}\n\nIn terms of physical quantities, \nthe simplest trial form for the probabilities is to propose a Gaussian form:\n\\begin{eqnarray}\n & &\\ln  P[\\{\\mathbf{x}(t)\\}, \\{\\mathbf{h}(t)\\}] = \\nonumber \\\\\n  & & \\;\\;\\;\\;\\;\\; -\\frac{1}{2} \\int \\text{d} t_a \\; \\text{d} t_b \\; \\{ A_1 (t_a,t_b) {\\bf h}(t_a) {\\bf h}(t_b) +\n A_2 (t_a,t_b) {\\bf h}(t_a){\\bf  x}(t_b) +A_3 (t_a,t_b) {\\bf x}(t_a) {\\bf x}(t_b) \\}\n\\end{eqnarray}\nThis leads, equivalently, to proposing an ansatz that is Gaussian in the $\\{{\\bf x,{\\hat x}}\\}$ variables.\nThis ansatz has to be invariant with respect to a time-independent translation \nof the trajectory, which may be imposed, in terms of the ${\\bf x, {\\hat{x}}}$ variables, by the form:\n\\begin{equation}\n  \\begin{array}{rl}\n &  P[\\{\\mathbf{x}(t)\\},\n  \\{      {    \\bf  {\\hat{x}}(t) }    \\}] \n  = \\int \\text{d} {\\bf \\bar x} \\; \\; \\exp \\bigg[\n    -\\frac{1}{2} \\int \\text{d} t_a \\; \\text{d} t_b \\; \\{ {\\tilde B} (t_a,t_b)  {    \\bf  {\\hat{x}}}(t_a) {    \\bf  {\\hat{x}}}(t_b)  \\\\\n  &   \\;\\;\\;\\;\\;\\;\\; + {\\tilde R} (t_b,t_a)  {    \\bf  {\\hat{x}}}(t_b) ( {\\bf x}(t_a) -{\\bf \\bar x})+\n {\\tilde R} (t_a,t_b)  {    \\bf  {\\hat{x}}}(t_a) ( {\\bf x}(t_b) -{\\bf \\bar x}) + {\\tilde D} (t_a,t_b) ({\\bf x}(t_a)-{\\bf \\bar x})( {\\bf x}(t_b)-{\\bf \\bar x)} \\}\\bigg]\n \\end{array}\n\\end{equation}\nwhere the integration over ${\\bf \\bar x}$ is over the whole volume and implements the translational invariance of the\nansatz, i.e. that the quadratic form has a zero-mode in the translations. (A similar strategy has been previously used\nin a static replica ansatz, and it is the same idea as the Hill-Wheeler integral that imposes rotational invariance in nuclear theory).\nIt will turn out that, in equilibrium, the ansatz satisfies the causality and fluctuation-dissipation relations :\n\\begin{equation}\n \\begin{array}{rl}\n{\\tilde D}(t,t')&=0  \\\\\n{\\tilde R(t,t')} &= -\\frac{1}{T} \\frac{\\partial }{\\partial t} {\\tilde B}(t-t') \\Theta(t-t')  \n \\end{array}\n\\end{equation}\n\nThe calculation is cumbersome, and we leave it for Appendix B.\nThe result is expressed in terms of the two-time correlation function:\n\\begin{equation}\nB(t-t') = \\frac{1}{N} \\sum_a \\; \\langle ({\\bf x}_a(t) - {\\bf x}_a(t'))^2 \\rangle\n\\label{bbb}\n\\end{equation} \nwhere the average is over the Langevin noise and the initial conditions.\nIt may be written in the (superficially) Mode-Coupling-like format:\n\\begin{equation}\\label{eq:dynamic_equation}\n  \\frac{\\partial B(t)}{\\partial t_a}=\\int  \\text{d} t' \\Sigma_R[B](t-t') B(t') + 2T\n  \\end{equation}\n  with  $t>0$.\nThis form is quite general, what defines our equation is the kernel $\\Sigma_R$, which in Mode-Coupling equations is a simple function of $B$, and here is computed as follows. Define first the response function $R(t)$, which satisfies the Fluctuation-dissipation theorem:\n\\begin{equation}\nR(t) = -\\frac{1}{T} \\frac{\\partial B(t)} { \\partial t} \\Theta(-t)\n\\end{equation}\n\n\\noindent Then the kernel is:\n\\begin{equation}\n    \\Sigma_R(t_a-t_b)= \\frac{2}{T}\\int \\text{d} t_{a'} \\text{d} t_{b'} \\;  {R}^{-1} (t_a-t_{a'})\n\\;  \\frac{ \\partial }{ \\partial t_{a'}}\\langle    \\mathbf{x}(t_{a'}) \\mathbf{x}(t_{b'}) \\rangle_{int}  \\;   {R}^{-1}(t_{b'}-t_b)\n\\end{equation}\nwhere $R^{-1}$ is the inverse through convolution of $R$:\n\\begin{equation}\n\\int \\text{d} t' \\; R(t-t') R^{-1}(t') = \\delta(t)\n\\label{inverse}\n\\end{equation}\nand may be obtained from $R$ easily with Laplace transforms.\nSelf-consistency is imposed by the definition \n\\begin{equation}\n  \\begin{array}{rl}\n{\\tilde R}&= R^{-1}  \\\\\n{\\tilde B} (t)&=  \\int \\text{d} t\" \\text{d} t' \\; R^{-1} (t-t'' )B(t\"-t')  R^{-1} (t' )\n  \\end{array}\n\\end{equation}\nThe trajectories that contribute to the averages $\\langle \\bullet \\rangle_{int} $ are those in which two particles\n  enter {\\em at any intermediate time} within the interaction range, as depicted in figure \\ref{fig:dynamics_interaction}, otherwise their contribution vanishes, as in a static Mayer expansion.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7cm]{dynamics_interaction.pdf}\n  \\caption{The trajectories contributing to a Mayer diagram are those that come into interaction range at some time.}\n\\label{fig:dynamics_interaction}\n\n\\end{figure}\n\n\n\n\\section{Numerical results from $\\lambda=\\infty$ to $\\lambda=0$}\\label{sec:num_liq}\n\nIn order to investigate the system beyond  mean-field, we studied \nthe finite $\\lambda$ case by numerical simulations. We used two different\n Monte-Carlo algorithms, an isobaric-isothermal one for the study of the \nequation of state, and an isovolumic-isothermal one for the study of the \ndynamics close to the glass transition.\nWe worked with systems of $864$ particles. The system we looked at for the \nbidisperse case is a 50:50 mixture of particles \nof diameter $1$ and $1.4$, a common choice for a three dimensional \nbidisperse glass former \\cite{o2002random,PhysRevLett.102.085703}.\n\nWe equilibrate the system at increasing pressures by annealing simulations. We \ncheck carefully that equilibrium is reached at every studied pressure by \nperforming annealings with several compression rates, and ensuring that they \ngive the same values for a given observable, \\textit{e. g.} density. Close to \nthe glass transition, we also check that no aging is visible in the system.\n\n\\subsection{Simulations in the mean-field case}\n\nIn the $\\lambda \\to \\infty$ limit, equilibrating the system can be\nmuch easier, thanks to the `planting' technique\n\\cite{krzakala2009hiding,achlioptas2008algorithmic}: for\nmean-field problems with quenched random variables, if the annealed\nfree energy is exact ($\\overline{\\ln Z}=\\ln \\overline{Z}$), one can\ncreate an instance in thermal equilibrium  by\ntaking a random set of variables (here particle positions), and\nlooking for a disorder (here random shifts) compatible with this set,\n\\textit{ie} leading to $H(\\{\\mathbf{x}\\}, \\{\\mathbf{A}\\})=0$. One can show that, within mean-field, this \ncreates no bias in the measure.\nIn our model, this means that we can generate safely an equilibrium\ninstance as long as the annealed entropy Eq.~(\\ref{eq:F_mean_field}) is\nvalid, \\textit{ie} as long as the liquid is the equilibrium phase.\nFinding an equilibrium\ninstance in this ensemble requires at most a few tens of seconds for a\nsystem with $1000$ particles on a desktop computer. This of course, is\nvery useful as we can get easily equilibrated configurations with\ndensities close and even \\textit{above} the dynamic glass transition\ndensity, as long as we do not reach the Kauzmann transition point, if it\nexists.\n \n\n\\subsection{Equation of state}\n\nAlready at the level of the equation of state, we can notice that the\nsystem behaves in a mean-field way even for random shifts as small as\n$\\lambda=1$. As shown in Fig. \\ref{fig:EOS}, all the distance between\nthe mean-field and the $3d$ hard sphere equation of state is covered by systems\nwith $0<\\lambda<1$.  The  inclusion of ring diagrams does not show any\nnoticeable difference with the leading order  for $\\lambda>1$, but it takes the agreement down to\n $\\sim \\; \\lambda=0.5$, which is far beyond its expected domain of applicability ($\\lambda\\gg1$). \n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{eq_etat.pdf}\n  \\caption{The equation of state of the model for different values of the \nshift range $\\lambda$. Solid lines are the analytical equation of state for the mean-field case \n(for which it is exact), and for the monodisperse $3d$ hard-sphere system \n(the  Carnahan-Starling approximation). Points are simulation \nresults. The equation of state sticks to the mean-field value for shifts \nlarger than $\\lambda=1$, except at high pressures where a glass transition prevents \nequilibration.}\n\\label{fig:EOS}\n\\end{figure}\n\nThe  monodisperse hard-sphere system without disorder undergoes a first\norder transition towards a crystal, as does the system with  small\nvalues of $\\lambda$. We have checked that nothing occurs in\nthe equation of state for systems with shifts as small as\n$\\lambda=0.15$. For smaller shifts, a weak first order transition is\nvisible, which increases when we decrease $\\lambda$.\n\n\n\\subsection{Pair correlation function}\n\nWe compare the pair correlation function obtained by MC simulations\nwith analytical results of section \\ref{sec:eos_liq}. In\nFig.~\\ref{fig:pair_correlation} we show that for\n$g_S(\\mathbf{x}-\\mathbf{y})$ (Eq.~(\\ref{eq:pair_correl_shifted_def})),\nthe analytical form derived for large shifts is verified.\n\\begin{figure}\n  \\centering\n    \\includegraphics[width=0.7\\columnwidth]{pair_correlation.pdf}\n    \\caption{Pair correlation function $g_S(\\mathbf{x}-\\mathbf{y})$\n      (Eq.~(\\ref{eq:pair_correl_shifted_def})) of a system  of\n      $N=864$ particles with several values of $\\lambda$.}\n  \\label{fig:pair_correlation}\n\\end{figure}\nFor finite $\\lambda$, the usual structure of the $3d$ hard-sphere\nmodel appears gradually as we reduce $\\lambda$.\n\n\n\\section{Dynamic glass transition}\\label{sec:dyn_glass}\n\nIn this section, we show that in $d=3$, we observe numerically the\npresence of a dynamic glass transition at finite pressure, and we\nstudy some dynamical properties close to the transition. A numerical\nsimulation cannot exclude that the transition pressure scales with the\nsystem size (for example as $P_c \\sim \\ln N$), but we have given\narguments in the previous sections that favor the finite-$P_c$\nscenario.\nNote that, on the contrary, we expect that the Kauzmann pressure $P_K$ might well scale\nas $\\log N$, but we shall not try to prove this in this paper.\n\nOur model allows to study various interesting aspects of the dynamic\nglass transition. First, we can follow the qualitative evolution of\nthe glass transition from mean-field to finite-dimensions. In\nparticular we ask if we can find in the mean-field limit generic\nfeatures of the mode-coupling theory, and try to see where these\nfeatures break down as we move away from mean-field. Secondly, as we\ncan generate equilibrium mean-field configurations at pressures larger\nthan the dynamic glass transition, we can numerically access any\ndesired property in the region between dynamic and equilibrium\ntransitions $P_c<P<P_K$ (but only at infinite $\\lambda$).\n\n\n\\subsection{Dynamic glass transition}\n\nAs is well known, mean field approximations give a dynamic transition (e.g. the mode-coupling transition)\nwhich is in fact avoided, thanks to activated processes. Here the situation is conceptually more clear:\nfor $\\lambda=\\infty$ we expect activated processes to be absent, and for finite $\\lambda$ we expect them\nto destroy any pure dynamic transition. For sufficiently large $\\lambda$, one may expect that the trace of the \ndynamic transition is quite clear. The (avoided) dynamic transition pressure will still have a dependence on the range,\nsince even the non-activated dynamics depends on the $\\lambda$. \n\n\nTo specify this glass transition we look at three standard\nquantities: relaxation time, diffusion coefficient and dynamic\nsusceptibility $\\chi_4(t)$.  First we look at the relaxation time of\nthe system obtained from a two-time correlation function:\n\\begin{equation}\n  \\label{eq:def_correlation}\n  C(q,t)=\\frac{1}{3N} \\sum_{i=1}^{N} \\sum_{\\alpha=1}^{d} \\cos \\left(q(\\mathbf{x}_i^{\\alpha}(0)-\\mathbf{x}_i^{\\alpha}(t)) \\right)\n\\end{equation}\nThe curves shown is this article were obtained with $q=\\pi$.\nThe relaxation time is defined by the time rescaling that gives the best time-pressure superposition in the $\\alpha$ regime as shown in Fig. \\ref{correl_RS10}.\n\n\\begin{figure}\n  \\centering\n  \\hspace{-1.cm} \n  \\includegraphics[width=0.48\\columnwidth]{correl_RS10.pdf}\n   \\hspace{0.8cm}\n  \\includegraphics[width=0.49\\columnwidth]{correl_rescaled_RS10.pdf}\n  \\caption{\\textbf{Left} Two-time correlation Eq.~(\\ref{eq:def_correlation}) for the mean-field model for densities between $\\phi=0.531$ and $\\phi=1.7478$. There is a plateau appearing upon compression, and the time needed to escape this plateau increases strongly with pressure, signaling the existence of a dynamic glass transition. \\textbf{Right}: Same data rescaled by the relaxation time $\\tau_{\\alpha}$. For high densities, time-density superposition holds.}\n\\label{correl_RS10}\n\\end{figure}\n\nThe relaxation time shows a super-Arrhenius dependence with pressure\n(Fig. \\ref{fig:tau_vs_P_RS10}), and seems to diverge for a finite\npressure (or density) value, which is characteristic of a fragile\nglass former.  This is independent of the range of the\ndisorder. Varying $\\lambda$ from $\\infty$ to $0.2$ increases the glass\ntransition pressure $P_d$ by  at most 50\\%, while the transition\ndensity $\\phi_d$ drops by almost a factor 2. This is to be related to\nthe large shift in the equation of state as we go from a mean-field to a $3d$ system, as shown in the previous\nsection .\n\\begin{figure}\n  \\centering\n  \\hspace{-1.cm} \n  \\includegraphics[width=0.48\\columnwidth]{tau_vs_P_RS10.pdf}\n   \\hspace{0.8cm}\n  \\includegraphics[width=0.48\\columnwidth]{tau_vs_Phi_RS10.pdf}\n  \\caption{\\textbf{Left:} Relaxation time $\\tau_{\\alpha}$ as a function of the pressure for several values of $\\lambda$. The relaxation is clearly super-Arrhenius, and there is a  divergence at a finite pressure for $\\lambda=\\infty$. For finite $\\lambda$,\n  the (apparent) dynamic transition pressure  (the analogue of the Mode-coupling transition)  slowly decreases  when $\\lambda$ decreases. \\textbf{Right:} Relaxation time $\\tau_{\\alpha}$ as a function of the density. The glass transition density shows a much broader change with the range of the disorder.}\n\\label{fig:tau_vs_P_RS10}\n\\end{figure}\n\nThe same divergence is found for the diffusion coefficient $D$. In\nFig.~\\ref{fig:D_vs_tau}, we plot the relation between $D$ and\n$\\tau_{\\alpha}$, showing a weak violation of the Stokes-Einstein\nrelation, which is commonly observed for supercooled liquids. Note\nthat this curve shows no dependence on $\\lambda$, and extrapolates smoothly \nto the mean-field limit, which is quite\nremarkable, as the violation of Stokes-Einstein relation is often\nexplained by the existence of dynamic heterogeneities, which depend\non $\\lambda$.\n\\begin{figure}\n  \\centering\n  \\hspace{-0.5cm}\n  \\includegraphics[width=0.48\\columnwidth]{D_vs_tau.pdf}\n  \\hspace{0.5cm}\n  \\includegraphics[width=0.48\\columnwidth]{Dtau_vs_Phi.pdf}\n  \\caption{\\textbf{Left} Diffusion coefficient as a function of the\n    relaxation time for several values of $\\lambda$. The diffusion\n    coefficient indicates a glass transition at the same density than\n    the relaxation time. \\textbf{Right} Evolution of the product $D\n    \\tau_{\\alpha}$ with density. The Stokes-Einstein relation predicts a\n    constant $D\\tau_{\\alpha}$.}\n\\label{fig:D_vs_tau}\n\\end{figure}\n\n\nA natural quantity to study in a fragile glass former is the $\\chi_4(t)$ susceptibility, a measure\nof dynamic heterogeneities. It is defined as the variance of the correlation $C$:\n\\begin{equation}\\label{eq:def_X4}\n  \\chi_4(t)=N(<C(q,t)^2>-<C(q,t)>^2)\n\\end{equation}\nFor fragile glass former, this quantity exhibits a peak\non a time scale $\\sim \\tau_{\\alpha}$ with a  height that sharply\nincreases when approaching the glass transition from the liquid (low pressure)\nside. For pressures above the dynamic glass transition (if there is one),\nwe expect $\\chi_4(t)$ to  saturate to a plateau for a time $\\sim\n\\tau_{\\beta}$, the characteristic time of the $\\beta$-relaxation, with a \nheight that increases closer to the dynamic transition. When $\\lambda$\nis finite, we cannot reach the glass  region,  as it is impossible to\nequilibrate a system with an infinite relaxation time\n$\\tau_{\\alpha}$. However, when $\\lambda \\to \\infty$, we can use the\n'planted-configuration' ensemble mentioned above to generate an\nequilibrium configuration even at a pressure larger than the dynamic\nglass transition pressure.\n\nA typical result in the mean-field limit of our model for $\\chi_4(t)$\nis shown in Fig.~\\ref{fig:X4}. The two expected behaviors -- growth from above and from below -- are observed,\nindicating that we crossed a dynamic glass transition. The location of\nthis qualitative change coincides with the point where the relaxation\ntime seems to diverge, and also to the analytic estimate of the transition pressure we give below.\n\nThe dependence of $\\chi_4(t)$ on $\\lambda$ (at a given relaxation\ntime) again shows little difference between the $\\lambda=1$ and the\nmean-field cases (see Fig.~\\ref{fig:X4}). Further from mean-field,\nthe height of $\\chi_4(t)$ depends on $\\lambda$, indicating a clear\nenhancement of the heterogeneities when we get close to the $3d$\nsystem. This is consistent with the idea that a long ranged disorder \ncorrelates large regions and thus inhibits heterogeneities occurring on \nsmaller scales.\n\\begin{figure}\n  \\hspace{-2cm}\n  \\includegraphics[width=0.5\\columnwidth]{X4_RS10.pdf}\n  \\includegraphics[width=0.5\\columnwidth]{X4_compare.pdf}\n  \\caption{\\textbf{Left:} Dynamic susceptibility $\\chi_4(t)$ (\\ref{eq:def_X4}) for the mean-field model at several densities. The peak position and value grow  \nrapidly on approaching  to the glass transition.\n\\textbf{Right:} $\\chi_4(t)$ with different values of the range of random \nshifts $\\lambda$, for equal relaxation times. Going from $\\lambda=\\infty$ to \n$\\lambda=1$ does not affect the dynamic heterogeneities, while for smaller values of $\\lambda$ (closer to \nfinite a dimensional system), heterogeneities  increase strongly.}\n\\label{fig:X4}\n\\end{figure}\n\nWe can infer the location of the transition by considering the  divergence of the\npeak value $\\chi_4^*$ of the dynamic susceptibility. This leads to\nthe results compatible with those obtained from  the relaxation time. We plotted the density\ndependence of the peak value $\\chi_4^*$ in\nFig.~\\ref{fig:X4_vs_phi}. Note that in the mean-field case, the\ndivergence can be observed on both sides of the glass transition.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{X4_vs_Phi.pdf}\n  \\caption{Peak of the dynamic susceptibility $\\chi_4^*$ as a function of the density for several values of $\\lambda$. This quantity shows a slight difference between $\\lambda=1$ and the mean-field model.}\n\\label{fig:X4_vs_phi}\n\\end{figure}\n\n\n\n\nIt is striking that, for all these quantities, the mean-field\nbehavior seems to be quite close to the $3d$ hard-sphere model. There\nis no dramatic change between $\\lambda \\to \\infty$ and\n$\\lambda=1$. This is also probably true for other choices of the\npotential $V$, suggesting  that one can create very simply a mean-field\ncaricature  of any finite-dimensional glass former by adding random shifts\nwith a range of the order of the range of the potential.\n\n\n\\subsection{The approach to the dynamic transition}\n\nThe mode-coupling approximation predicts\nthat the timescale \n$\\tau_{\\alpha}$ diverges algebraically with the \ndistance to the glass transition density $\\phi_{c}$ with an exponent \n$-\\gamma$. \n\\begin{equation}\n  \\tau_{\\alpha} \\sim \\left(\\frac{\\phi - \\phi_{c}}{ \\phi_{c}} \\right)^{-\\gamma}\n  \\label{mctt}\n\\end{equation}\n\n\n\n\n\nIn real life, the mode-coupling transition becomes at best a crossover, and the question\nis to what extent should one believe, and in what temperature-pressure range, extrapolations\nwithin  mode-coupling functional forms.\nIn our case, we also expect a divergence   in the limit $\\lambda=\\infty$. What is interesting about this model, is that we\nmay make $\\lambda$ gradually smaller and follow the transition as it becomes a crossover, and keep track on\nthe interpolations as they become less and less obvious, right down to the original particle model.  \n\n\nIn order to test the behavior (\\ref{mctt}), we have to fit our simulation data\nwith two free parameters $\\phi_{c}$ and $\\gamma$. In practice, this\nis a delicate task as one needs to have a relaxation time running\nover many decades to be able to chose unambiguously the couple $\\{\n\\phi_{c},\\gamma \\}$ (see for example\n\\cite{PhysRevLett.105.199605}). A way to help this procedure is to\nlook at the four-point dynamic susceptibility $\\chi_4(t)$. Within MCT,\nthe maximum $\\chi_4^*$ should diverge~\\cite{berthier2007spontaneous2} like $\\chi_4^*(\\phi)\\sim\n\\left(\\frac{\\phi-\\phi_{c}}{\\phi_{c}}\\right)^{-1}$\n. This gives us  an independent\nmeasure of the $\\gamma$ exponent, using the relation $\\chi_4^*\\sim\n\\tau_{\\alpha}^{1\/\\gamma}$ if there is a region where mode-coupling\nscaling holds. We can then look for the pair of parameters  $\\{\\phi_{c},\\gamma\n\\}$ giving the best fit for both measures.\n\\begin{figure}\n  \\centering\n  \\hspace{-1.cm}\n  \\includegraphics[width=0.48\\columnwidth]{X4_MCT.pdf}\n  \\hspace{1.cm}\n  \\includegraphics[width=0.48\\columnwidth]{X4_vs_tau.pdf}\n  \\caption{\\textbf{Left:} Peak of the dynamic susceptibility $\\chi_4^*$ as a function of the distance to the dynamic glass transition for different values of $\\lambda$. The MCT-like  scaling $\\left(\\frac{\\phi-\\phi_{c}}{\\phi_{c}}\\right)^{-1}$ is clearly visible. \\textbf{Right:} $\\chi_4^*$ as a function of $\\tau_{\\alpha}$ in the same regime. The power law relation obtained in MCT  also seems to hold here for every values of $\\lambda$. }\n\\label{fig:X4_vs_tau}\n\\end{figure}\n\\begin{figure}\n  \\includegraphics[width=0.7\\columnwidth]{tau_MCT.pdf}\n  \\caption{$\\tau_{\\alpha}$ as a function of $\\frac{\\phi-\\phi_{c}}{\\phi_{c}}$ for $\\phi_{c}$ giving the broadest range of densities verifying MCT-like  behaviour. The agreement is perfect for values of $\\lambda$ down to $0.5$, \nwhereas for $\\lambda=0.2$, the dynamic transition seems to be missed.}\n\\label{fig:tau_MCT}\n\\end{figure}\n\nThe results are shown in Fig~\\ref{fig:X4_vs_tau} and Fig.~\\ref{fig:tau_MCT}.\nWe get an excellent agreement with power law divergence on 6 decades of relaxation time\nfor the mean-field model, as expected. For small values of $\\lambda$, however, we\nobserve a deviation to the power law scaling when we get close to the\ntransition, presumably due to activated processes, a fact that is well attested  for a $3d$ binary hard sphere\nsystem~\\cite{PhysRevLett.102.085703}. This seems to confirm the idea\nthat mean field theories give  correct qualitative features for the relaxation time,\nbut  only over a limited range of density not too close to  the glass transition. Quite interestingly,\nintermediate values of $\\lambda$ show a good  power-law scaling up to\n$\\lambda=1$, and for $\\lambda>1$  we were not able to observe any evidence of activation \n within the range\nof relaxation times that are reachable with our simulations. \n\nOne would like to take these simulations  all the way  down to\n$\\lambda=0$. Unfortunately, this is not possible with a monodisperse\nsystem, but one can use a bidisperse system and follow the same\nsteps. This is what is done in Fig.~\\ref{fig:tau_MCT_bi}. \nHere we see that the effect is more clear:\nin term of relaxation time, the algebraic divergence is followed on\ntwo decades when $\\lambda=0$, but extends to more than three decades\nwhen $\\lambda=1$.\n\\begin{figure}\n  \\centering\n  \\hspace{-1.cm}\n  \\includegraphics[width=0.48\\columnwidth]{tau_MCT_bi.pdf}\n  \\hspace{1.cm}\n  \\includegraphics[width=0.48\\columnwidth]{X4_vs_tau_bi}\n  \\caption{\\textbf{Left:} $\\tau_{\\alpha}$ as a function of\n    $\\frac{\\phi-\\phi_{c}}{\\phi_{c}}$ for $\\phi_{c}$ giving the\n    broadest range of densities verifying MCT-like  behaviour. The agreement\n    is perfect for values of $\\lambda$ down to $0.5$, whereas for\n    $\\lambda=0.2$, the dynamic transition seems to be missed.}\n\\label{fig:tau_MCT_bi}\n\\end{figure}\nIn figure\n\\ref{fig:gamma_vs_lambda} we show the behavior of the extrapolated dynamical  transition density and\nthe $\\gamma$ exponent as a function of  $\\lambda$. We observe that as \n$\\lambda$ is lowered,  these values continuously approach the known values for $3d$\nsystems, both in the monodiperse~\\cite{van1994glass} and in the\nbidisperse case~\\cite{PhysRevLett.102.085703}.\n\\begin{figure}\n  \\centering\n  \\hspace{-0.9cm}\n  \\includegraphics[width=0.48\\columnwidth]{gamma_mono.pdf}\n  \\hspace{1.cm}\n  \\includegraphics[width=0.48\\columnwidth]{gamma_bi.pdf}\n  \\caption{The values of the parameters of the MCT-like divergence,\n    for different values of $\\lambda$, in the monodisperse (left) and\n    bidisperse (right) cases. These quantities show a continuous\n    behavior from the $3d,\\; \\lambda=0$ system to the mean field\n    $\\lambda=\\infty$ one.}\n\\label{fig:gamma_vs_lambda}\n\\end{figure}\n\nThe relaxation curves for the correlations contain  information beyond that of the timescale $\\tau_\\alpha$.\nThe shape of the relaxation curves does depend on the range of the\ndisorder, as is presented on Fig. \\ref{fig:correl_compare}, with the\n$\\beta$-relaxation part becoming slower  when $\\lambda$ decreases, \nboth in the case of monodisperse and of bidisperse systems.  There is no visible\ndifference between $\\lambda=1$ and $\\lambda=\\infty$, yet  another\nindication that for this range of disorder, the system behaves in a\nmean-field manner. \nFor small $\\lambda$, we do not observe a complete\nseparation between $\\alpha$ and $\\beta$-relaxations, and the plateau is\nnot well defined. These features however should strongly depend on\nthe microscopic dynamics, and it is known that Monte-Carlo dynamics\nleads to a longer $\\beta$-relaxation than molecular dynamics (see for example\n\\cite{kob2002supercooled}).\n\n Remarkably, the $\\alpha$-relaxation \nshape is almost insensitive to the value of $\\lambda$ in the\nmonodisperse case. We can just barely  notice an increase of the slope when\n$\\lambda$ gets smaller, a feature that appears much more clearly in the\nbinary mixture case, as we can access lower $\\lambda$ values.\n\\begin{figure}\n  \\centering\n  \\hspace{-0.5cm}\n  \\includegraphics[width=0.48\\columnwidth]{correl_compare.pdf}\n  \\hspace{0.5cm}\n  \\includegraphics[width=0.48\\columnwidth]{correl_compare_bi_B.pdf}\n  \\caption{Relaxation curves $C(q,t)$ with different values of the\n    range of random shifts $\\lambda$, for equal relaxation\n    times. \\textbf{Left:} monodisperse case.  The main evolution while\n    tuning $\\lambda$ is a lengthening the\n    $\\beta$-relaxation. \\textbf{Right:} Bidisperse case. The evolution\n    is emphasized here. The shape of the $\\alpha$-relaxation depends\n    varies strongly between the mean-field case and small values of\n    $\\lambda$.}\n\\label{fig:correl_compare}\n\\end{figure}\n\n\\subsection{Small or large exponent?}\n\nWe wish to stress again  that our goal is this work is to\nperform a data analysis of the same type as performed with MCT, and in\nparticular we wanted to have an exponent $\\gamma$ that fits the\ndivergence of $\\tau_{\\alpha}$ on the broadest possible range\n\\textit{and} satisfies the relation between $\\chi_4^*$ and\n$\\tau_{\\alpha}$. It is clear that by relaxing the constraint given by $\\chi_4^*$ , we can fit the divergence of $\\tau_{\\alpha}$ on a\nbroader range, and possibly on the whole available range, just\nby taking an  exponent of the order of four, and a larger density for the\ndivergence point (see for instance~\\cite{PhysRevLett.105.199605}).\n\n Indeed, if one looks at the relation $\\chi_4^*$ versus $\\tau_{\\alpha}$\nvery close to the transition (something we have not done here but\ncan be found  in Fig.~3 of Brambilla \\textit{et\n  al.}~\\cite{PhysRevLett.102.085703}), one can fit the relation with a\nlarger $\\gamma$ on a restricted range.\n\nThen, one can legitimately think that the exponent $\\gamma \\simeq 4.5$\nwe find in the mean-field limit  is also the one of the $3d$ hard sphere system, as long\nas one accepts to relax the constraint on the (mode-coupling inspired) relation of  $\\chi_4^*$ versus\n$\\tau_{\\alpha}$. \nTherefore, what our work proves is that,\nprovided we restrict ourselves to orthodox MCT fitting, the\nglass transition is more and more mode-coupling-like when we approach\nmean-field and the part of mode-coupling divergence which is observed\nin finite dimension is somehow a shadow of the mean-field one. What is\nappealing in this point of view is that the analysis gives for the\n$3d$ bidisperse case an exponent $\\gamma \\simeq 2.5$, which is\ncompatible with what has already been found in experiments and\nsimulations of comparable\nsystems~\\cite{PhysRevE.58.6073,PhysRevLett.102.085703,el2009dynamic},\nand is reasonably close to the real MCT (not only its phenomenology)\nfor a monodisperse system~\\cite{van1993glass}.\n\n\n\n\\subsection{Analytic estimation  of the dynamic transition}\n\nOne quick way to determine dynamic (mode coupling-like) transition\npoints, is to make a static calculation of the equilibrium state,\nconsidered as an ensemble of metastable ergodic components; or,\nequivalently, to look for the lowest pressure at which the {\\em\n  effective potential} (free-energy at fixed distance between\nconfigurations) still has two minima.\n  \nIn Appendix A we do this, leading to the determination in the figure\nbelow:\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\columnwidth]{dynamic_transition_gaussian.pdf}  \n    \\includegraphics[width=0.48\\columnwidth]{cages.pdf}\n    \\caption{\\textbf{Left:} The function defined in Eq.(\\ref{bla}), in\n      terms of the \"cage size\" parameter $K$, for $m=0.96$.  The\n      solution for the transition pressure is obtained by searching\n      for the point with the largest gradient, which is $1\/\\rho_d$.\n      One also obtains the estimated cage size at this\n      point. \\textbf{Right:} Density profiles of a single particle in\n      its cage, defined by Eq.~(\\ref{eq:cage_density}), obtained from\n      simulations (in black), and from analytical approach with a\n      Gaussian ansatz (in red).}\n  \\label{fig:dynamic_transition_gaussian}\n\\end{figure}\nThe estimated transition density is $\\phi_d=2^{-d}\\rho_d= 1.65$ (see\nleft-hand side of figure~\\ref{fig:dynamic_transition_gaussian}) to be\ncompared with the value estimated numerically $\\phi_d=1.82$. Note that\nthe analytic computation is not exact because it relies on a Gaussian\napproximation.\n\nThe estimated cage size is also consistent with the simulated\nvalues. In the right-hand side figure~\\ref{fig:dynamic_transition_gaussian}, we plot the\nquantity:\n\\begin{equation}\\label{eq:cage_density}\n  g_0(\\mathbf{r})= \\frac{1}{N} <\\sum_i \\delta\\left[(\\mathbf{x}_i-<\\mathbf{x}_i>)-\\mathbf{r}\\right]>\n\\end{equation}\n\\noindent which is as the density profile of a single particle in its\ncage, and is the long time limit of the so-called van Hove\nself-correlation function.\n\n\\section{Onset Pressure}\\label{sec:onset}\n\nA decade ago, Sastry \\textit{et al.} introduced a  temperature\nscale $T_{on}>T_g$ in supercooled liquids, the so-called onset\ntemperature~\\cite{sastry1998signatures}. They noticed that in a\nLennard-Jones binary mixture, the energy of the inherent structures\n(the configurations reached after a quench at $T=0$) associated with\nequilibrium configurations at a temperature $T$ shows a crossover from\na roughly constant value above $T_{on}$ to a regime where it decreases\nwhen $T$ decreases. They argued that this temperature is the one at\nwhich the dynamics becomes landscape-influenced, with a\nsuper-Arrhenius dependence of the relaxation time.  It was a few years\nlater argued~\\cite{brumer2004mean} that there is a connection between\n$T_{on}$ and the computed mode-coupling density temperature $T_c$,\nopening the possibility that in mean-field systems $T_{on}$ and\n$T_g(=T_c)$ might coincide. The question is legitimate, since one expects that \nfor the spherical $p$-spin glass, the two temperatures do indeed coincide.\n\nIn this section, we show that this connection does not exist for our\nmodel.  Because we are dealing with hard spheres, an inherent\nstructure is the (infinite pressure) configuration reached after a\nrapid compression  process, like the one introduced\nin~\\cite{stillinger1964systematic}. In figure ~\\ref{fig:onset} we\nplot the inherent structure density reached after such a process, in\nterms of the initial density, {\\em for the model with\n  $\\lambda=\\infty$} . The crossover, corresponding to the `onset\ndensity' (which is the relevant quantity for hard spheres, instead of\ntemperature, see previous section) is clearly visible.  The dynamic\ntransition in this mean-field limit ($d=3$) occurs at density\n$\\phi_g=1.82$, while we find here\n$\\phi_{on}\\simeq 1.15$.  Because $\\phi_g$ is a well defined quantity\nhere, we see beyond doubt that both densities do not coincide.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.7\\columnwidth]{onset_density.pdf}\n  \\caption{Inherent structure density $\\phi_J$ as a function of the\n    equilibrium liquid density $\\phi_{eq}$. The crossover density is\n    clearly different from the dynamic glass transition point,\n    indicated by the vertical blue line.}\n\\label{fig:onset}\n\\end{figure}\nAs expected, the crossover density coincides with  the point where\nthe relaxation (\\ref{eq:def_correlation}) starts to show a\nshoulder. Thus, also in the mean-field limit, $\\phi_{on}$ marks the\nonset of a qualitative change in the dynamics (see\nFig.\\ref{correl_RS10}), which becomes landscape influenced.\n\n\\section{Conclusions}\n\nWe have introduced an approximation scheme for particle systems that\nis close in spirit to the Mode-Coupling approximation, but has the\nadvantage that one may construct a continuous range of models, with at\none end the original one, and at the other end one for which the\napproximation is exact.  The approximation becomes better at higher\ndimensionality of space, everything else remaining equal.\n\nAs it stands, the present scheme is derived from the microscopic\nmodel. This was originally the case also with the Mode-Coupling\nequations, although the standard practice has become to modify freely\nthe interactions in such a way as to obtain the observed static\nstructure factor and transition temperature or pressure.  In our case,\nthe analogous procedure would be to substitute the true potential by\none based on the pair correlation function\n(\\ref{eq:pair_correl_shifted_def})\n\\begin{equation}\nV_{eff} = - T \\ln[g(r)]\n\\end{equation}\nWe have not tried this strategy.\n\nIn this paper we have not discussed the possible static Kauzmann transition, and we suspect that  it might happen at divergent pressures, of the order of $\\ln \\lambda$, \\textit{i.e.} divergent in the mean-field limit.\n\nBy following the model from the original problem to its mean-field\nlimit, we have used the present construction to give new arguments on\nthe existence of a vestige of the genuine mean-field dynamic\ntransition, following a mode-coupling-like behavior. We also argued\nconclusively that the `onset' temperature (or pressure) should not be\nidentified with the dynamic transition.  More generally, one may\nfollow this strategy to decide whether features found in true system\nthat one `explains' within random first order theory, really\nextrapolate to the corresponding feature in the limit in which the\ntheory is exact.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Ludovic Berthier, Florent Krzakala, Marco\nTarzia and Francesco Zamponi for discussing with us various points of\nthis work.\n\n\\section*{Appendix A: An estimation of the dynamical transition pressure from replicas}\\label{sec:ideal_glass}\n\n\\subsection*{Physical discussion}\n\nThe dynamic glass transition is related to the existence of an\nexponential number of amorphous metastable states. Above a given\ndynamic glass transition density $\\phi_d$, the liquid phase can be\nseen as the sum of all these states. \n\nTo derive the properties of the model at high density in order to test\nthe above scenario, we will study the partition function of $m$ copies\nof the original system:\n\\begin{equation}\\label{eq:m_replicated_partition}\n  Z_m=\\int  \\prod_{\\alpha=1}^{m} \\prod_{k} \\text{d} \\mathbf{x_k}^{\\alpha} \\exp \\(( -\\sum_{\\alpha=1}^{m}\\sum_{ij}V(\\mathbf{x_i}^{\\alpha}-\\mathbf{x_j}^{\\alpha}-\\mathbf{A_{ij}}) \\))    \n\\end{equation}\n\n\nThe idea is the following~\\cite{monasson1995structural,\n  mezard1996tentative, mezard1999thermodynamics, mezard1999first}: if\nwe force the $m$ copies to be close one another  by adding a\nsmall coupling term between them, we can expect that if we study the\nsystem at a density higher than $\\phi_d$, and switch off the coupling\nafter taking the thermodynamic limit, the $m$ copies will be confined in\nthe same metastable state.\n\nThis is done in practice by looking at the entropy of the replicated system:\n\\begin{equation}\\label{eq:replicated_entropy_def}\n  S=\\overline{\\ln Z_m}=\\lim_{n\\to 0} \\frac{1}{n}\\left(\\overline{Z_m^n}-1 \\right)\n\\end{equation}\nwhere we use the replica trick to average over disorder the logarithm of $Z_m$.\nLooking more closely to $\\overline{Z_m^n}$, we see that it can be interpreted as\nthe partition function of $N$ `molecules'\n$\\underline{\\mathbf{x_i}}=\\{\\mathbf{x}_i^1,...,\\mathbf{x}_i^{nm}\\}$ made\nof $nm$ spheres:\n\\begin{equation}\\label{eq:mn_replicated_partition}\n\\begin{array}{rl}\n  \\overline{Z_m^n} & =\\int \\prod_{ij} \\text{d} \\mathbf{A_{ij}} P(\\mathbf{A_{ij}}) \\int  \\prod_{\\alpha=1}^{mn} \\prod_{k} \\text{d} \\mathbf{x_k}^{\\alpha} \\exp \\(( -\\sum_{\\alpha=1}^{mn}\\sum_{ij}V(\\mathbf{x_i}^{\\alpha}-\\mathbf{x_j}^{\\alpha}-\\mathbf{A_{ij}}) \\))\\\\\n& =\\int \\prod_{k} \\text{d} \\underline{\\mathbf{x_k}} \\prod_{ij}\\int \\text{d} \\mathbf{A_{ij}} P(\\mathbf{A_{ij}}) \\exp \\(( -\\sum_{\\alpha=1}^{mn}\\sum_{ij}V(\\mathbf{x_i}^{\\alpha}-\\mathbf{x_j}^{\\alpha}-\\mathbf{A_{ij}}) \\))\n  \\end{array}\n\\end{equation}\nThen,  using standard liquid theory techniques, we are able to write the\nentropy~Eq.~(\\ref{eq:replicated_entropy_def}) as a functional of the\ndensity of molecules $\\rho(\\underline{\\mathbf{x}})$. The metastable\nstates of the system are then the maxima of this entropy with respect\nto $\\rho(\\underline{\\mathbf{x}})$ when $m=1$, which we expect to\ncouple replicas.\n\n\n\\subsection*{Replicated canonical formalism}\n\nWe have to study the following partition function\nEq.~(\\ref{eq:mn_replicated_partition}).\nIntroducing $\\underline{\\mathbf{x_i}}=\\{\\mathbf{x_i}^1,...,\n\\mathbf{x_i}^{nm}\\}$, we can see\nEq.~(\\ref{eq:mn_replicated_partition}) as the partition function of a\n$N$ molecules interacting via a potential $\\tilde{V}$ such that:\n\\begin{equation}\\label{eq:replicated_mayer}\n \\left[ 1+ \\overline{f}_r(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}})\\right]=\\exp\\left(-\\tilde{V}(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}})\\right)=\\int P(\\mathbf{A}) \\text{d} \\mathbf{A} \\exp \\left( -\\sum_{\\alpha=1}^{mn}\\sum_{ij}V(\\mathbf{x}^{\\alpha}-\\mathbf{y}^{\\alpha}-\\mathbf{A}) \\right)\n\\end{equation}\nEach `molecule' $\\underline{\\mathbf{x_i}}$ is made of $n$ independent\nsets of $m$ coupled (in the same state) original particles.\nThen, Eq.~(\\ref{eq:mn_replicated_partition}) reads:\n\\begin{equation}\\label{eq:mn_replicated_partition2}\n  \\overline{Z_m^n} = \\int \\prod_{i} \\text{d} \\underline{\\mathbf{x_i}} \\prod_{ij}\\left[ 1+ \\overline{f}_r(\\underline{\\mathbf{x_i}}-\\underline{\\mathbf{x_j}})\\right]\n\\end{equation}\n\n\nWe wish  to do a Mayer expansion like in the non-replicated case,\nbut we cannot add a convenient combinatoric prefactor in\n(\\ref{eq:mn_replicated_partition2}), as we do not know what it should be\n\\textit{a priori}. However the specific properties of the\nrandom-shift model allows us  to do a canonical treatment of the\npartition function as we show in the following.\nIntroducing the density of molecules $\\rho(\\underline{\\mathbf{x}})$ by:\n\\begin{equation}\n  \\rho(\\underline{\\mathbf{x}})=\\sum_i \\delta(\\underline{\\mathbf{x}}-\\underline{\\mathbf{x_i}})  \n\\end{equation}\n we can rewrite $\\overline{Z_m^n}$ as:\n\\begin{equation}\n  \\overline{Z_m^n} =  \\int \\text{d} \\underline{\\mathbf{x_i}} \\int \\text{D}[\\rho( \\mathbf{x})]  \\delta \\left(\\rho( \\mathbf{\\underline{x}})-\\sum_i \\delta(\\mathbf{\\underline{x}}-\\mathbf{\\underline{x}}_i) \\right) \n\\exp \\left[\\frac{1}{2} \\int \\text{d} \\mathbf{\\underline{x}} \\text{d} \\mathbf{\\underline{y}}  \\rho(\\mathbf{\\underline{x}}) \\rho(\\mathbf{\\underline{y}}) \\ln\\left[ 1+ \\overline{f}_r(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}})\\right] \\right]\n\\end{equation}\nWe may  now exponentiate the $\\delta$ constraint at the cost of adding a second \nfield $\\hat{\\rho}$, and integrating over the $\\underline{\\mathbf{x_i}}$'s:\n\\begin{equation}\n  \\begin{array}{rl}\n    \\overline{Z_m^n} =  \\int & \\text{D}[\\rho( \\underline{\\mathbf{x}})] \\text{D}[\\hat{\\rho}( \\underline{\\mathbf{x}})] \\\\\n& \\exp\\left\\{ i \\int \\text{d} \\underline{\\mathbf{x}} \\hat{\\rho}( \\underline{\\mathbf{x}}) \\rho( \\underline{\\mathbf{x}}) + N \\ln \\int  \\text{d} \\underline{\\mathbf{x}} e^{-i \\hat{\\rho}( \\underline{\\mathbf{x}})}+ \\frac{1}{2} \\int \\text{d} \\underline{\\mathbf{x}} \\text{d} \\underline{\\mathbf{y}}  \\rho(\\underline{\\mathbf{x}}) \\rho(\\underline{\\mathbf{y}}) \\ln\\left[ 1+ \\overline{f}_r(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}})\\right] \\right\\}\n  \\end{array}\n\\end{equation}\n\n\n The next important step is to look more closely at the\nfunction\n$\\overline{f}_r(\\mathbf{\\underline{x}}-\\mathbf{\\underline{y}})$, as we did \nin the non-replicated case. In the integral of\nEq.~(\\ref{eq:replicated_mayer}), the exponential of  the potential\n$V$ vanishes $1$ whenever the two molecules in\n$\\mathbf{\\underline{x}}$ and $\\mathbf{\\underline{y}}+\\mathbf{A}$\noverlap, and gives $1$ otherwise. Then we have:\n\\begin{equation}\n  -nm\\frac{v_d}{V} \\leq  \\overline{f}_r(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}})\\leq -\\frac{v_d}{V} \n\\end{equation}\nThus, by expanding the logarithm, we get:\n\\begin{equation}\n  \\begin{array}{rl}\n    \\overline{Z_m^n} =  \\int & \\text{D}[\\rho( \\underline{\\mathbf{x}})] \\text{D}[\\hat{\\rho}( \\underline{\\mathbf{x}})] \\\\\n    & \\exp\\left\\{  i\\int \\text{d} \\underline{\\mathbf{x}} \\hat{\\rho}( \\underline{\\mathbf{x}}) \\rho( \\underline{\\mathbf{x}}) + N\\ln \\int  \\text{d} \\underline{\\mathbf{x}} e^{-i \\hat{\\rho}( \\underline{\\mathbf{x}})}+ \\frac{1}{2} \\int \\text{d} \\underline{\\mathbf{x}} \\text{d} \\underline{\\mathbf{y}}  \\rho(\\underline{\\mathbf{x}}) \\rho(\\underline{\\mathbf{y}}) \\overline{f}_r(\\underline{\\mathbf{x}}-\\underline{\\mathbf{y}}) \\right\\}\n\\end{array}\n\\end{equation}\nWe wish then to evaluate this integral by   saddle point  with\nrespect to the fields $\\rho$ and $\\hat{\\rho}$ (because each term in\nthe exponential is of order $N$, including the last one, due to the\nvalue of $f$). This gives:\n\\begin{equation}\\label{eq:saddle}\n  \\begin{array}{rl}\n    \\rho( \\mathbf{\\underline{x}}) & = N\\frac{e^{-i \\hat{\\rho}( \\mathbf{\\underline{x}})}}{\\int \\text{d} \\mathbf{\\underline{x}} e^{-i \\hat{\\rho}( \\mathbf{\\underline{x}})} } \\\\\n    \\hat{\\rho}( \\mathbf{\\underline{x}}) & = i  \\int \\text{d} \\mathbf{\\underline{y}} \\rho(\\mathbf{\\underline{y}})\\overline{f}_r(\\mathbf{\\underline{x}}-\\mathbf{\\underline{y}})\n  \\end{array}\n\\end{equation}\n Thus, the logarithm of the partition function is:\n\\begin{equation}\\label{eq:replicated_FT_partition}\n  \\ln \\overline{Z_m^n} = - \\int \\text{d} \\mathbf{\\underline{x}} \\rho( \\mathbf{\\underline{x}}) \\ln \\rho( \\mathbf{\\underline{x}}) + \\frac{1}{2} \\int  \\text{d} \\mathbf{\\underline{x}} \\text{d} \\mathbf{\\underline{y}} \\rho( \\mathbf{\\underline{x}}) \\rho( \\mathbf{\\underline{y}}) f(\\mathbf{\\underline{x}}-\\mathbf{\\underline{y}})+N\\ln N\n\\end{equation}\n We wish to stress the similarity of\nEq.~(\\ref{eq:replicated_FT_partition}) with\nEq.~(\\ref{eq:F_mean_field}). In fact the average over disorder does\nexactly the same job for the replicated liquid than for the bare\nnon-replicated liquid: it disallows `three-molecule effective\ninteractions', just like in a high-dimensional system.\n\n\\subsection*{Location of the dynamic transition, within the Gaussian  ansatz}\n\nWe may obtain an approximation for $\\rho$ by assuming it has a Gaussian\nform.  Of course, this ansatz will not  be a solution of the full saddle\npoint equation Eq.~(\\ref{eq:saddle}), but we can still extremize the\nentropy Eq.~(\\ref{eq:replicated_FT_partition}) within this ansatz. A\nnatural choice is the  ansatz\\cite{parisi2010mean}:\n\\begin{equation}\n  \\label{eq:ansatz}\n  \\rho( \\mathbf{\\underline{x}})=\\frac{N}{V^{n}}\\prod_{\\gamma=1}^n\n\\int \\text{d} \\mathbf{X_{\\gamma}} \\frac{1}{(2\\pi K)^{md\/2}}\\exp \\left[ \\sum_{\\alpha=\\gamma (m-1)+1}^{\\gamma m} \\frac{\\left(\\mathbf{x^{\\alpha}} - \\mathbf{X_{\\gamma}}\\right)^2}{2K} \\right]\n\\end{equation}\nNote the similarity with the dynamic ansatz.\n\nThis is a 1-step replica symmetry breaking (1-RSB) ansatz.  Injecting\nEq.~(\\ref{eq:ansatz}) in the partition function\nEq.~(\\ref{eq:replicated_FT_partition}) leads to integrals exactly\nsimilar to the ones one has compute in the hard sphere system (without\nrandom shifts).  This has been done in \\cite{parisi2010mean}.\nFollowing the computations along the lines of \\cite{parisi2010mean},\none finds:\n\\begin{equation}\\label{eq:action_gaussian}\n  \\begin{array}{rl}\n   \\frac{S[\\rho( \\mathbf{\\underline{x}})]}{N}= \\ln N+ 1-\\ln \\rho - \\frac{d}{2}(1-m) \\ln (2\\pi K) + \\frac{d}{2}\\ln m - \\frac{d}{2}(1-m) - \\frac{\\rho}{2} I(m,K)\n  \\end{array}\n\\end{equation}\nwhere $I(m,K)$ is the integral:\n\\begin{equation}\n  I(m,K)=\\int \\text{d} \\mathbf{X} \\left[\\int \\text{d} \\mathbf{x} \\text{d} \\mathbf{y} \\frac{1}{(2\\pi K)^{d}}\\exp \\left(\\frac{\\left(\\mathbf{x} - \\mathbf{X}\\right)^2}{2K} \\right)\\exp \\left(\\frac{\\mathbf{y}^2}{2K} \\right) \\chi(\\mathbf{x}-\\mathbf{y}) \\right]^m\n\\end{equation}\nThen, the saddle point equation on $K$ reads:\n\\begin{equation}\n  \\frac{d(m-1)}{\\rho}= \\frac{\\partial I(m,K)}{\\partial \\ln K}\n\\end{equation}\n\n Thus, there exist metastable states in the liquid phase\n(which is recovered in the limit $m\\to 1$) if there is a cage size $K$\nwhich verifies:\n\\begin{equation}\n  \\frac{1}{\\rho}=\\lim_{m \\to 1}\\frac{1}{d(m-1)} \\frac{\\partial I(m,K)}{\\partial \\ln K}\n  \\label{bla}\n\\end{equation}\nThe dynamic glass transition density is $\\rho_d$, beynd  which no\nsolution to this equation can be found.  It is possible to compute\n$I(m,K)$ numerically, and we find $\\phi_d=2^{-d} \\rho_d \\simeq 1.65$\nfor $d=3$ (see Fig.~\\ref{fig:dynamic_transition_gaussian}), not far\nfrom the value ($1.82$) found in numerical simulations. The value of\n$K$ at the transition is also comparable with what is found in the\nsimulations (see Fig.~\\ref{fig:dynamic_transition_gaussian}).\n\n\n\n\\section*{Appendix B : Gaussian ansatz for the mean-field dynamics}\n\n\\subsection{Notation}\n\nThe calculation we are going to follow is quite heavy. It may be made\nsomewhat more compact, and one may follow the analogy with the static\ntreatment better, by using the supersymmetric\nnotation~\\cite{zinn2002quantum} (see\n\\cite{kurchan1992supersymmetry,semerjian2004stochastic}).  One\nintroduces two extra Grassmann variables $\\theta$ et\n$\\overline{\\theta}$. Denoting $ a=(t,\\theta,\\overline{\\theta})$, the\ntrajectories $\\mathbf{x}(t)$ and $\\hat{\\mathbf{x}}(t)$ may be encoded\nin a superfield $\\psi(a)$:\n\\begin{equation}\n  \\psi(a)=\\mathbf{x}(t)+ \\theta \\overline{\\theta} \\hat{\\mathbf{x}}(t)\n\\end{equation}\nDefining the operator  $D_a$  as\n\\begin{equation}\n  D_a=T\\frac{\\partial^2}{\\partial \\theta  \\partial \\overline{\\theta}}+\\theta\\frac{\\partial^2}{\\partial \\theta  \\partial t}-\\frac{\\partial}{ \\partial t}\n\\end{equation}\nwe have:\n\\begin{equation}\n\\Phi[\\mathbf{x}, \\hat{\\mathbf{x}}] =  \\int \\text{d} a \\text{d} b \\; \\delta(b-a) D_a  \\left(\\psi(a)-\\psi(b)\\right)^2 \n\\end{equation}\nSimilarly:\n\\begin{equation}\n  \\begin{array}{rl}\n  1+\\overline{f}_d[\\mathbf{x},\\hat{\\mathbf{x}},\\mathbf{x}',\\hat{\\mathbf{x}}']&= \n  \\int \\text{d} \\mathbf{A} P(\\mathbf{A}) \\exp \\left\\{ - \\frac{1}{2} \\int_0^{\\tau} \\text{d} t\\; \\hat{\\mathbf{x}}(t) \\nabla_{\\mathbf{x}} V(\\mathbf{x} - \\mathbf{y}-\\mathbf{A}))+\\hat{\\mathbf{y}}(t) \\nabla_{\\mathbf{y}} V(\\mathbf{y} - \\mathbf{x} - \\mathbf{A})  \\right\\} \\\\\n&=\\int \n    \\text{d} \\mathbf{A} P(\\mathbf{A}) \\exp \\left\\{-\\frac{1}{2} \\int \\text{d} a\\;  V\\left(\\psi(a)-\\psi'(a)-\\mathbf{A} \\right) \\right\\}\\\\\n&=1+\\overline{f}_d[\\psi,\\psi']\n  \\end{array}\n\\end{equation}\n The action can be written in the following compact way:\n \\begin{equation}\\label{eq:susy_action}\n  \\begin{array}{rl}\n   \\mathcal{S}\\left[\\rho [\\psi ] \\right] =  &\\int \\text{D} \\psi \\rho[\\psi] \\ln \\left[\\rho[\\psi] \\right] +\\int \\text{D} \\psi \\rho[\\psi] \\int \\text{d} a\\; \\psi(a) D_a \\psi(a) \\\\\n& \\qquad - \\frac{1}{2} \\int \\text{D}[\\psi,\\psi'] \\rho[\\psi] \\rho[\\psi']\\overline{f}_d[\\psi-\\psi']-N\\ln N\n\\end{array}\n\\end{equation}\nBy analogy with the statics, one may make a `Gaussian' ansatz\n$\\rho$~\\cite{semerjian2004stochastic}:\n\\begin{equation}\n  \\rho[\\psi] = \\frac{\\sqrt{2}}{V\\; \\det^{\\frac 1 2}(\\mathcal{B})} \\int d{\\bf{\\bar x}} \\; \\exp \\left[ - \\int d a d b  \\; {\\mathcal B}^{-1}(a,b) \\;  (\\psi(a)-\\mathbf{\\bar{x}})(\\psi(b)-\\mathbf{\\bar{x}})  \\right]\n\\end{equation}\nwith\n\\begin{equation}\n\\mathcal{B} (a,a)=0  \\;\\;\\; \\forall \\;\\;\\; a\n\\end{equation}\nIn components, the super-correlators read:\n\\begin{eqnarray}\n \\mathcal{B}(a,b)&=&{ B}(t_a,t_b) - \\overline{\\theta}_a \\theta_a { R} (t_b,t_a)\n  -  {\\bar \\theta}_b  \\theta_b { R}(t_a,t_b) + { D} (t_a,t_b) {\\bar \\theta}_a \\theta_a {\\bar \\theta}_b \\theta_b\\nonumber \\\\\n  \\mathcal{B}^{-1}(a,b)&=&{\\tilde B}(t_a,t_b) - \\overline{\\theta}_a \\theta {\\tilde R} (t_b,t_a)\n  -  {\\bar \\theta}_b  \\theta_b {\\tilde R}(t_a,t_b) + {\\tilde D} (t_a,t_b) {\\bar \\theta}_a \\theta_a {\\bar \\theta}_b \\theta_b\n  \\label{inverse1}\n\\end{eqnarray}\nand they are related through:\n\\begin{equation}\n\\int \\text{d}{\\bar \\theta}_b \\text{d} \\theta_b \\text{d} t_b \\;  \\mathcal{B}(a,b) \\mathcal{B}^{-1}(b,c)= \\delta({\\bar \\theta_a}-{\\bar \\theta_c})\n\\delta( \\theta_a-{\\theta_c}) \\delta(t_a-t_c)\n\\end{equation}\nThis inversion formula may be developed, to obtain the \"tilde\" variables in terms of the ones without tilde.\n\n\\noindent In equilibrium, the fluctuation-dissipation theorem implies that ${\\cal B}$ and ${\\cal B}^{-1}$ take the form:\n\\begin{eqnarray}\n \\mathcal{B}(a,b)&=&{ B}(t_a-t_b) - \\overline{\\theta}_a \\theta_a { R} (t_b-t_a)\n  -  {\\bar \\theta}_b  \\theta_b { R}(t_a-t_b) \\nonumber \\\\\n  \\mathcal{B}^{-1}(a,b)&=&{\\tilde B}(t_a-t_b) - \\overline{\\theta}_a \\theta_a {\\tilde R} (t_b-t_a)\n  -  {\\bar \\theta}_b  \\theta_b {\\tilde R}(t_a-t_b) \n\\end{eqnarray}\nwhere \n\\begin{equation}\n  R(t) = -\\frac{ 1}{T} \\frac{\\partial B}{\\partial t} \\;\\;\\;  ; \\;\\;\\; {\\tilde R}(t)= -\\frac{ 1}{T} \\frac{\\partial{\\tilde  B}}{\\partial t}\n\\end{equation}\nwhere $R$ and ${\\tilde R}$ are zero for negative time-differences, due to causality.\n\n\\noindent It is easy to show that:\n\\begin{equation}\n \\Phi[\\mathbf{x}, \\hat{\\mathbf{x}}] =  \\frac{1}{V} \\int \\text{d} {\\bf{\\bar x}} \\;  \\int \\text{d} a \\text{d} b \\; \\delta(b-a) D_a  (\\psi(a)-\\mathbf{\\bar{x}})(\\psi(b)-\\mathbf{\\bar{x}}) \n\\end{equation}\nThe super-correlator  $\\mathcal{B}$ is given by:\n\\begin{equation}\n  \\mathcal{B}(a,b)=\\int \\text{D}\\psi  \\left(\\psi(a)-\\psi(b)\\right)^2\\rho[\\psi]\n\\end{equation}\n\n\\noindent Inserting this ansatz in Eq.~(\\ref{eq:susy_action}), we get:\n\\begin{equation}\n \\mathcal{S}\\left( \\mathcal{B} \\right)=-\\frac{1}{2} \\text{Tr}\\ln \\mathcal{B}+ \\int \\text{d} a \\text{d} b\\; \\delta(a-b)D_a\\mathcal{B}(a,b)-\\mathcal{S}_{int}(\\mathcal{B})\n\\end{equation}\nwith\n\\begin{equation}\n  \\begin{array}{rl}\n    \\mathcal{S}_{int}(\\mathcal{B})&=  \\frac{1}{V^2 \\det \\mathcal{B}} \\int \\text{D} [\\psi, \\psi'] \\int \\text{d} \\bar{\\mathbf{x}} \\text{d} \\bar{\\mathbf{y}} \\\\\n    & \\quad \\exp \\left[  - \\int \\text{d} a \\text{d} b \\;  \\mathcal{B}^{-1}(a,b) \\left[(\\psi(a)-\\bar{\\mathbf{x}})(\\psi(b)-\\bar{\\mathbf{x}})+(\\psi'(a)-\\bar{\\mathbf{y}})(\\psi'(b)-\\bar{\\mathbf{y}})\\right] \\right]\\;  \\overline{f}_d[\\psi,\\psi'] \n  \\end{array}\n\\end{equation}\nThe integrand is invariant with respect to independent translations of ${\\bf x} $ and $ {\\bf y}$, because they may be absorbed into the shift $\\mathbf{A}$, so we may write:\n\\begin{equation}\n  \\mathcal{S}_{int}(\\mathcal{B})=   \\frac{1}{\\det \\mathcal{B}} \\int \\text{D} [\\psi, \\psi'] \\; \\exp \\left[  - \\int \\text{d} a \\text{d} b \\; \\mathcal{B}^{-1}(a,b) \\left[\\psi(a)\\psi(b)+\\psi'(a)\\psi'(b)\\right] \\right]\\; \\overline{f}_d[\\psi,\\psi']\n\\end{equation}\nThe saddle point equation gives:\n\\begin{equation}\\label{eq:dynamics_saddle}\n  0=\\frac{\\delta \\mathcal{S}}{\\delta \\mathcal{B}(a,b)} = -\\frac{1}{2}\\mathcal{B}^{-1}(a,b)+ \\delta(a-b)D_{a}- \\frac{\\delta \\mathcal{S}_{int}}{\\delta \\mathcal{B}(a,b)}\n\\end{equation}\nwith:\n\\begin{equation}\n  \\frac{\\delta \\mathcal{S}_{int}}{\\delta \\mathcal{B}(a,b)}= \\frac{1}{2}\\mathcal{B}^{-1}(a,b) - \\left[ \\mathcal{B}^{-1}\\otimes\\langle \\psi(a')\\psi(b') \\rangle_{int}\\otimes  \\mathcal{B}^{-1} \\right](a,b)\n\\end{equation}\n and:\n\\begin{equation}\n   \\langle \\bullet \\rangle_{int}  = \\int \\text{D} \\psi \\int \\text{D} \\psi' \\bullet \\rho[\\psi]\n\\rho[\\psi']\\; \\overline{f}_d[\\psi,\\psi']\n\\end{equation}\nMaking the convolution product with $\\mathcal{B}(b,c)$, we obtain:\n\\begin{equation}\\label{eq:super_dynamics}\n  0=    D_a\\mathcal{B}(a,b) + \\int \\text{d} c \\; \\Sigma(a,c)\\mathcal{B}(c,b),\n\\end{equation}\nwhere $\\Sigma$ is given by \n\\begin{equation}\n  \\Sigma(a,b)= - 2\\left[ \\mathcal{B}^{-1}\\otimes \\langle \\psi(a') \\psi(b')\\rangle_{int} \\otimes  \\mathcal{B}^{-1} \\right](a,b),\n\\end{equation}\n also of the form (\\ref{inverse1}):\n\\begin{equation}\n {\\Sigma}(a,b)={ \\Sigma_B}(t_a-t_b) - \\overline{\\theta}_a \\theta_a { \\Sigma_R} (t_b-t_a)\n  -  {\\bar \\theta}_b  \\theta_b { \\Sigma_R}(t_a-t_b) \n\\end{equation}\n$\\Sigma_B$ and $\\Sigma_R$  satisfy also a fluctuation-dissipation relation:\n\\begin{equation}\n\\Sigma_R(t) = -\\frac{ 1}{T} \\frac{\\partial \\Sigma_B}{\\partial t} \n\\end{equation}\nBecause correlation and response satisfy a fluctuation-dissipation relation, we may write everything\nexclusively in terms of correlations:\n\\begin{equation}\n  \\frac{\\partial B(t_a-t_b)}{\\partial t_a}=-TR(t_b-t_a)+\\int \\text{d} t_c \\; \\Sigma_R(t_a-t_c) B(t_c-t_b)-\\left.\\Sigma_C(t_b-t_c) B(t_b-t_c)\\right|_{-\\infty}^{t_b},\n\\end{equation}\nwith:\n\\begin{equation}\n   \\Sigma_R(t_a-t_b)= \\frac{2}{T}\\int \\text{d} t_{a'} \\text{d} t_{b'} \\;  {R}^{-1} (t_a-t_{a'})\n\\;  \\frac{ \\partial }{ \\partial t_{a'}}\\langle    \\mathbf{x}(t_{a'}) \\mathbf{x}(t_{b'}) \\rangle_{int}  \\;   {R}^{-1}(t_{b'}-t_b)\n\\end{equation}\nGiven that, as $t_a \\rightarrow t_b$, $B(t_a-t_b) \\sim 2T  |t_a-t_b|$, we have that:\n\\begin{equation}\n  \\begin{array}{rl}\n  \\frac{\\partial B(t_a,t_b)}{\\partial t_a}&=-TR(t_b,t_a)+\\int  \\text{d} t_c\n\\; \\Sigma_R(t_a - t_c) B(t_c - t_b) +2T \\\\\n2T&=-\\left.\\Sigma_B(t_b- t_c) B(t_b- t_c)\\right|_{-\\infty}^{t_b},\n  \\end{array}\n\\end{equation}\nwhich is the result (\\ref{eq:dynamic_equation}).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\\subsection{The Systematic Normal Form : a \"smoothed analysis\" of lattices}\n\nWe introduce a certain canonical form of lattices called the systematic normal form (or SNF for short).\nAn SNF matrix is a matrix of this form:\n\\begin{eqnarray}\nB = \\left[\\begin{array}{ccccc} \nN & b_{2} & b_{3} & \\ldots & b_{n} \\\\\n& 1 & & &\\\\\n& & 1& &  \\\\\n& & & \\ddots & \\\\\n& & & & 1\n\\end{array}\n\\right]\n\\end{eqnarray}\nwhere all unspecified entries are $0$, and $N$ is a prime number.\nThe SNF form allows for a \"smoothed analysis\" of lattices in the following sense:\nGiven a matrix $B$ in Hermite normal form (HNF) one can efficiently generate a matrix $B_{SNF}$ in systematic normal form \nsuch that any solution of a problem of interest on $B_{SNF}$ can be translated easily to a solution for the original problem on $B$,\nalbeit with a slight degradation in accuracy.\nFurthermore, $B_{SNF}$ is in a specialized canonical form with several important properties which are used in the reduction.\nHence, the reduction maps each lattice to a \"nearby\" lattice with the \"right\" properties.\n\nThe defining property of SNF, is that it maps the quotient structure ${\\mathbb{Z}}^n \/ L$ as a quotient of infinite groups,\nto a quotient  of finite groups $\\mathbb{F}_N^n \/ L_N$, where $L_N = L \\cap\\mathbb{F}_N^n$ and $N = \\det(B)$.\nIn addition, the very sparse structure of SNF allows for several other important properties among which we name:\n\\begin{enumerate}\n\\item\nThe dual to $L_N$ in $\\mathbb{F}_N^n$, denoted by $(NL^*)_N$ also has a very structured matrix.\n\\begin{eqnarray}\nB = \\left[\\begin{array}{ccccc} \n1 & &&& \\\\\n-b_{2} & 1& & &\\\\\n-b_{3} & & 1 & &  \\\\\n\\vdots & & & \\ddots & \\\\\n-b_{n} & & & & 1 \n\\end{array}\n\\right]\n\\end{eqnarray}\nThe duality between these two matrix forms has intriguing connections to well-known problems in cryptography:\nOn one hand, the primal SNF form is shown here to be synonymous with the problem finding short-integer-solutions ${\\rm SIS}$ (see Definition \\ref{def:sis}, and connection in Proposition \\ref{prop:equiv}),\nwhereas solving the closest vector problem on the dual lattice is very similar to the problem of approximate common divisor problem (\\cite{HG01}). \n\\item\nIt is straightforward to sample uniformly from both $L_N$ and $(NL^*)_N$:\nto sample uniformly from $L_N$ one samples $n-1$ coordinates $x_n,\\hdots, x_2$ uniformly at random from $\\mathbb{F}_N$, and \nthen sets $x_1$ as $x_1 = \\sum_{i=2}^m b_i x_i (\\mod \\ N)$.\nFor the dual lattice: one samples $a$ uniformly at random from $\\mathbb{F}_N$, and then sets the dual vector as $(a,-b_2 \\cdot a(\\mod \\ N), \\hdots, -b_n \\cdot a (\\mod \\ N))$.\n\\item\nFor every $x\\in {\\cal P}(L)$, i.e. the basic parallelotope of $L$, there exists a unique element in $(NL^*)_N$ that corresponds to $x$.\nFurthermore, this element can be computed efficiently from $x$.\nThis is shown in Claim \\ref{cl:dual}.\n\\item\nThe SNF form can be naturally associated with a rank-$1$ instance of ${\\rm SIS}$ modulo a prime number field:\nGiven a vector $g\\in \\mathbb{F}_N^m$ we define:\n$$\nL(g)_1^{\\perp}:= \\{ h\\in \\mathbb{F}_N^{m+1}, \\langle (1,g), h \\rangle =0 \\in F_N\\},\n$$\nOne can then show that any SNF matrix $B$ has $L(B) = L(g)_1^{\\perp}$ where $g$ is the vector of non-trivial entries in $B$, i.e. $b_2,\\hdots, b_n$.\nThis is the subject of Proposition \\ref{prop:equiv}.\n\\end{enumerate}\n\nThe formal connection between a given lattice basis $B$ and its nearby SNF matrix is given in Lemma \\ref{lem:snf} which we sketch here:\n\\begin{lemma*}(sketch)\nLet $B$ denote some integer matrix.\nThere exists an efficient algorithm that computes an SNF matrix $B_{SNF}$ and a number $T \\leq 2^{{\\rm poly}(n)}$ such that\n$$\\left\\| B - \\frac{1}{T} B_{SNF}\\right\\| = 2^{-\\tilde{\\Omega}(n)},$$ \nand for any $v\\in L(B_{SNF})$, the vector $\\hat{v} = B B_{SNF}^{-1} v$ has $\\|v\/T - \\hat{v}\\| = {\\rm O}(n^{-k})$,\nfor some fixed $k$.  \n\\end{lemma*}\n\n\n\\subsection{Worst-to-average case lattice reductions}\n\n\nOne of the most attractive features of lattice-based crypto-systems is the ability to reduce \nworst-case instances of such problems to average instances with a small loss in approximation error,\nthus potentially paving the way to prove that they are one-way functions under standard complexity-theoretic assumptions,\nnamely ${\\rm NP} \\subsetneq {\\rm BPP}$.\n\nUsually, when speaking of worst-to-average case reductions on lattices, one thinks of a theorem\nof the following form, as stated in the seminal work of Ajtai \\cite{Ajt04}\n\\begin{theorem*}(Ajtai)\nLet $A \\in Z_q^{m \\times n}$ be some random matrix, where $q, m = {\\rm poly}(n)$, \nand consider $\\Lambda(A)^{\\perp} = \\{x\\in \\mathbb{F}_N^n, A x = 0 (\\mbox{mod} \\ q)\\} \\subseteq {\\mathbb{Z}}^n$.\nIf ${\\cal A}$ is an algorithm that finds a vector $v\\in \\Lambda(A)^{\\perp}$, $\\|v\\| \\leq \\beta<< q$ with probability at least, say $1\/n$, then \nthere exists an algorithm ${\\cal B}$ such that ${\\cal B}^{\\cal A}$ can solve ${\\rm gapSVP}_{\\gamma}$  \nfor $\\gamma = \\beta {\\rm poly}(n)$, for any instance,\nwhere ${\\rm SVP}_{\\gamma}$ is the problem of finding a lattice vector which is at most $\\gamma$ longer than the shortest one.\n\\end{theorem*}\nThe problem of finding a short vector in $\\Lambda(A)^{\\perp}$ is known as the Short-Integer-Solution or ${\\rm SIS}$,\nand the statement above means that solving ${\\rm SIS}$ on average is at least as hard as approximating ${\\rm SVP}$\nfor suitable choice of parameters.\n\nThis line of research, pioneered by Ajtai \\cite{Ajt04}, was later improved in the subsequent works of \nMicciancio and Regev \\cite{MR07}, Gentry, Peikert and Vaikuntanathan \\cite{GPV08}, \nand Micciancio and Peikert \\cite{MP14}.\nSpecifically, in \\cite{MR07}, the authors introduced the use of Gaussian measures, and harmonic (Fourier) analysis on the lattice\nas a means to simply the reduction, and achieve better parameters.\nMost prominently, they reduced the loss in accuracy: i.e. the ratio $\\gamma\/\\beta$, to ${\\rm O}(n \\sqrt{{\\rm log}(n)})$, using a sequence of adaptive reductions.\n \nEssentially, all prior reductions follow the same path, originally due to Ajtai.\nAt a very high level, they\namount to querying the ${\\rm SIS}$ oracle in order to iteratively improve (decrease the length) of a set of linearly independent\nlattice vectors.\nThe oracle is applied to the vector of coefficients of the lattice vectors from previous rounds, \nwhich are then shown to be random (modulo ${\\mathbb{Z}}_q^m$), so long as these vectors are of length which is comparable to the smoothing parameter of the lattice (see Definition \\ref{def:smooth}).\n\nUsing the SNF structure, we propose a new worst-to-average case reduction that diverges from the template reduction due to Ajtai.\nSpecifically, we show a simplified scheme that reduces known lattices problem such as Guaranteed-Distance-Decoding (${\\rm GDD}$), and Shortest-Independent-Vector-Problem (${\\rm SIVP}$) to\n${\\rm SIS}$.\n\nDenote the smoothing parameter of a lattice by $\\eta_{\\varepsilon}(L)$ (see Definition \\ref{def:smooth}).\nWe consider the problem of Guaranteed-distance-decoding (${\\rm GDD}$) where we are given a lattice $L = L(B)$, and a target vector $v$, and asked to find a vector $x\\in L$, such that ${\\rm dist}(v,x) \\leq \\eta_{\\varepsilon}(L)$.\nWe also consider rank-$1$ instances of the Short-Integer-Solution problem as follows:\nFix some prime number $N$.\nGiven $g = (g_1,\\hdots, g_n)$, \nwhere $g_i\\in \\mathbb{F}_N$\none can then define a lattice constrained by a \"parity\" check matrix,\nas follows:\n$$\n\\Lambda(g)^{\\perp} = \\{ h\\in \\mathbb{F}_N^n, \\ \\ \\langle h, (g_1,\\hdots, g_n) \\rangle = 0 \\in \\mathbb{F}_N^n \\}.\n$$\nThe ${\\rm SIS}(N)$ problem (see Definition \\ref{def:sis}), is then to find short vectors in $\\Lambda(g)^{\\perp}$, i.e. of length, say ${\\rm O}(n)$.\nWe note that while formally the ensemble above is a proper ${\\rm SIS}$ ensemble, \nfor cryptographic applications\none usually uses a completely different range of parameters: the parity check matrix has rank $m = {\\rm poly}(n)$ (and not $1$) and the field of interest is usually taken as $q = {\\rm poly}(n)$, which is related to the encryption complexity.\n\n\nWe show the following theorem:\n\\begin{theorem*}(sketch)\nLet $(B,v)$ be an input to ${\\rm GDD}$\nwhere $B$ is an $n\\times n$ integer matrix.\nSuppose that\n$\\eta_{\\varepsilon}(L) \\leq \\Phi$ for some $\\varepsilon = 2^{-n}$.\nSuppose that ${\\cal A}$ returns w.p. at least $1\/{\\rm poly}(n)$ a solution to\n${\\rm SIS}$.  \nThen for $x_{out} = {\\cal B}(B,v)$ we have that $x_{out}\\in L$ and w.p. $\\Omega(1)$\n$$\n\\|x_{out} - v\\| \\leq \\Phi \\cdot n^{1.5} \\cdot \\max \\{{\\rm log} \\det(B), n\\}. \n$$\n\\end{theorem*}\nA similar theorem (and algorithm) can be shown for reducing the problem of Shortest-Independent-Vector-Problem (${\\rm SIVP}$) to ${\\rm SIS}$.\nFollowing is a sketch of the algorithm:\n\\begin{algorithm*}(sketch)\n\\item\nInput: A lattice $L = L(B)$ for some $n\\times n$ HNF matrix $B$, and vector $v\\in \\mathbb{F}_N^n$.\n\\begin{enumerate}\n\\item \nReduce $L$ to SNF form.\nDenote matrix basis by $B_{SNF}$, matrix $M$, and constant $T$.\nDenote $L_{SNF} = L(B_{SNF})$,  $N = \\det(L_{SNF})$, $L_N = L_{SNF} \\cap [N]^n$,  $NL^* = L(N \\cdot B_{SNF}^{-T})$.\n\\item\nPut \n$m$ as the minimal positive integer for which $\\sqrt{m}^m \\geq N$.\nSet $s = T \\cdot \\eta_{\\varepsilon}(L)$ for some $\\varepsilon = {\\rm O}(1)$.\n\\item\nChoose $c$ uniformly at random from $\\mathbb{F}_N \\cap [- m\\sqrt{m}, m\\sqrt{m}]$.\nChoose $u_{rand}\\in L_N$ uniformly at random from $L_N$.\nPut $v_{target} = (c^{-1}\\cdot T \\cdot v + u_{rand}) (\\mod \\ N)$.\n\\item\nRepeat $m$ times:\n\\begin{enumerate}\n\\item\nSample $x_i = (x_{i,1},\\hdots, x_{i,n})\\sim \\rho_{\\mathbb{F}_N^n,s,v_{target}}$.\n\\item\nCompute $y_i$ as the point in $NL^*$ corresponding to $x_i$.\nDenote by $a_i$ the first coordinate of $y_i$.\n\\end{enumerate}\n\\item\nPut $\\{\\alpha_i\\}_{i=1}^{m} = {\\cal A}(a_1,\\hdots, a_m)$.\n\\item\nIf ${\\cal A}$ fails or $\\sum_{i=1}^m \\alpha_i \\neq c$ return FAIL.\n\\item\nCompute $x_0 = (\\sum_{i=1}^m \\alpha_i (y_i + x_i) - c u_{rand})(\\mod \\ N)$.\n\\item\nReturn $x_{out} := B (B_{SNF}^{-1}M^{-1} \\cdot x_0) $.\n\\end{enumerate}\n\\end{algorithm*}\n\nThe algorithm, given target vector $t$, tries to find a \"relatively\" close lattice vector to $t$.\nThis is, in a sense, the same problem as ${\\rm CVP}$ (the closest-vector problem), except here we are only interested in an approximation which\nis comparable to the \"smoothing parameter\" of the lattice (see Definitions \\ref{def:gdd} and \\ref{def:cvp} for a formal statement).\nUsing Lemma \\ref{lem:snf}, which shows how to reduce an arbitrary lattice to SNF form, and then translate\nback the solution to the original lattice, we will describe the algorithm assuming the input lattice is already in SNF form.\n\nThe reduction begins by sampling from a discrete Gaussian around the target vector $t$, with sufficiently large variance.\nDenote these samples by $x_1,\\hdots, x_n\\in \\mathbb{F}_N^n$.\nBy the SNF structure, the algorithm then computes for each sampled point $x_i$, a corresponding point $y_i\\in (NL^*)_N$.\nsuch that\n$$\n\\forall i \\ \\ x_i + y_i \\in L_N. \\\n$$\nWe do not know whether such a bijection exists for general lattices. \n\nThe structure of the dual lattice implies that the points $y_i$ are completely characterized by their first coordinate.\nWe then show that for sufficiently large Gaussian variance, the first coordinate of these points\nis uniformly distributed on $\\mathbb{F}_N^n$.\nThis implies that, given as input the first coordinate $a_i$ of each $y_i$, the random oracle for SIS will succeed with high probability.\nSuppose that the oracle to ${\\rm SIS}$ returns coefficients $\\{\\alpha_i\\}_{i=1}^m$\nsuch that $\\sum_i \\alpha_i a_i = 0 \\in \\mathbb{F}_N$, and consider the linear combination of the vectors $x_i$ using these coefficients:\n$$\nx_0 = \\sum_{i=1}^m \\alpha_i (x_i+y_i)\n$$\nWe note that the above is a lattice point in $L_N$ because it is an integer combination of vectors in $L_N$.\nBy the SNF structure this implies that $\\sum_i \\alpha_i y_i = 0$.\nHence, the distance of $x_0$ to the target $c$ is given by the vector:\n$$\n\\sum_{i=1}^m \\alpha_i x_i = \\sum_i \\alpha_i t + \\sum_{i=1}^m \\alpha_i {\\cal E}_i,\n$$\nwhere ${\\cal E}_i$ is a discrete Gaussian on $\\mathbb{F}_N^n$ centered around $0$ with the same variance as $x_i$.\nSince the $\\alpha_i$'s are small (say $\\sqrt{n}$) and the variance of ${\\cal E}_i$ is $s^2$, \nthen to upper bound the length of $x_0-t$ it is sufficient to make sure that the linear combination of $\\alpha_i$'s is in fact {\\it affine}, i.e.\n$\\sum_{i=1}^m \\alpha_i = 1$.\n\nIt is not immediately clear why a short-integer-solution to a random instance should be affine.\nWe do know however, that although it may not be affine, the sum of coefficients is quite small, say ${\\rm O}(n\\sqrt{n})$.\nThus, \na natural scheme would be to \"guess\"\nthe sum of coefficients $c = \\sum_{i=1}^m \\alpha_i$ in advance out of a small interval, and then sample $x_i$'s from \na Gaussian centered around a scaled target - $\\rho_{\\mathbb{F}_N^n, s, c^{-1} t}$.\nFor such a scheme to work, we need to rule out the possibility that since we change the original samples to be centered around $c^{-1} t$, the oracle to ${\\rm SIS}$, despite returning a good answer for almost all inputs, will \"adversarially\" pick\nan answer $(\\alpha_1,\\hdots, \\alpha_m)$ such that $\\sum_{i=1}^m \\alpha_i \\neq c$ {\\it always}.\n\nTo rule out this possibility, we introduce extra randomization to make sure that the random oracle has no knowledge of which target vector we are looking for.\nWe add a random lattice vector to $t$, for which we compensate at the end, and then argue that the random oracle can only gain information about the value of $t$ modulo the basic parallelotope ${\\cal P}(L)$.  But since the $x_i$'s are sampled with variance above the smoothing parameter, it implies that modulo ${\\cal P}(L)$ the values of the $x_i$'s are almost completely uniform.\n\n\\subsection{Discussion and prior art}\n\nThe above theorem improves on prior art, and specifically \\cite{MR07} in the following ways:\n\\begin{enumerate}\n\\item\nThe reduction is direct:  in \\cite{MR07} the reduction from ${\\rm GDD}$ to ${\\rm SIS}$ goes through the problem of finding a short linearly independent set called\n${\\rm SIVP}$, and through a variant of ${\\rm GDD}$ called ${\\rm INC-GDD}$.\nHence our reduction is somewhat simpler to follow.\n\\item\nIt does not rely on the Ajtai form of reductions.\nHence it allows to consider other problems as hard random instances: for example, \na linear congruence modulo a large prime number $N$, instead of simultaneous linear equations modulo a small field $q$\n\\cite{Ajt04}.\n\\item\nIt is inherently non-adaptive: the oracle calls to ${\\rm SIS}$ can be made once in parallel, following which the algorithm returns\nan answer to the original problem (${\\rm GDD}$ or ${\\rm SIVP}$).\n\\end{enumerate}\n\nBy \\cite{AR04} it is known that approximating ${\\rm SVP},{\\rm CVP}$ to a factor at most $\\sqrt{n}$\nis in ${\\rm NP} \\cap {\\rm coNP}$, and by \\cite{GG} and \\cite{GMR} a $\\sqrt{n\/{\\rm log}(n)}$ approximation is in ${\\rm NP} \\cap {\\rm coAM}$.\nHence, much larger improvement is needed in order to approach the domain of NP-hardness.\nIn addition, since the reduction is non-adaptive, then by \\cite{BT} it cannot be used to \nshow a reduction from an NP complete problem to an average case problem, unless the polynomial hierarchy collapses.\nThis suggests, that perhaps an adaptive variant of our algorithm can result a further improvement of the approximation factor.\n\nFinally, we point out that it is not known \\cite{RS} whether ${\\rm GDD}_p$ is as hard as ${\\rm SVP}_q$ for any polynomials $p,q$, hence\nit is possible that approximate ${\\rm GDD}$ (or ${\\rm SIVP}$) is in fact an {\\it easier} problem than approximate ${\\rm SVP},{\\rm CVP}$.\n\n\n\\section{Preliminaries}\n\n\\subsection{Notation}\n\nThe $n$-dimensional Euclidean space is denoted by $\\mathbb{R}^n$.\nThe Euclidean norm of a vector $x\\in \\mathbb{R}^n$ is $\\|x\\| = \\sqrt{\\sum_{i=1}^n |x_i|^2}$.\nA Euclidean lattice $L$ is written as $L = L(B)$ where $B$ is some basis of $L$.\n$N$ is used to denote a prime number, and $\\mathbb{F}_N$ the prime number field corresponding to $N$.\nWe define $\\Delta$ as the statistical distance between distributions $(p,\\Omega), (q,\\Omega)$,  i.e. $\\Delta(p,q) = \\int_{\\Omega} |p(x) - q(x)| dx$.\nGiven a set $S$, $U(S)$ is the uniform distribution on $S$.\nFor any $v\\in \\mathbf{R}^n$ define: \n$\n|v| = max_i |v_i|.\n$ \nFor real number $s>0$ and vector $c\\in R^n$, $\\bar{B}_s(c)$ is the closed Euclidean ball of radius $s$ around $c$.\nFor integer $n\\geq 1$, the notation $[n]$ stands for the set of indices $\\{1,\\hdots, n\\}$.\nGiven a set $S\\subseteq \\mathbb{R}^n$, and a vector $v\\in \\mathbb{R}^n$, we denote ${\\rm dist}(v,S): = \\min_{x\\in S} \\|v - x\\|$.\nFor a positive integer $M$, we denote by $[M]$ as the interval of integers $[0,\\hdots, M-1]$.\n\nWe say that a problem $P$ is efficiently computable if there exists an algorithm that runs in time ${\\rm poly}(n)$,\nwhere $n$ is the size of the description of an instance of $P$.\n\n\\subsection{Background on lattices}\nWe start by stating some standard facts about lattices.\n\n\\begin{definition}\n\n\\textbf{Euclidean Lattice}\n\n\\noindent\nA Euclidean lattice $L\\subseteq \\mathbb{R}^n$ is the set of all integer linear combinations of a set of linearly independent vectors\n$b_1,\\hdots, b_m$:\n$$\nL = \\left\\{ \\sum_{i=1}^m z_i b_i, \\ \\ z_i\\in {\\mathbb{Z}}, \\right\\} \\subseteq \\mathbb{R}^n\n$$\nThis set $\\{b_i\\}_{i=1}^n$ is called the {\\em basis} of the lattice.\nWe denote by $L = L(B)$, where $B$ is the matrix whose columns are $b_1,\\hdots, b_m$.\nIn this paper, we will always assume that $L$ is full-dimensional, i.e. $m=n$.\n\\end{definition}\n\\noindent\nFor lattice $L = L(B)$, ${\\cal P}(B)$ is the basic parallelotope of $L$ according to ${\\cal B}$:\n$$\n{\\cal P}(B) := \\left\\{ v= \\sum_{i\\in [n]} x_i b_i, \\ \\ x_i\\in [0,1) \\right\\}.\n$$\nSometimes, it will be more convenient to use ${\\cal P}(L)$ which is independent of the basis.\nThis then refers to ${\\cal P}(B)$ for some basis $B$ for $L$.\n\n\\begin{definition}\n\n\\textbf{The Dual Lattice}\n\n\\noindent\nThe dual of a lattice is the lattice generated by\nthe columns of $B^{-T}$. \n\\end{definition}\n\n\n\\begin{definition}\n\n\\textbf{Successive minima of a lattice}\n\n\\noindent\nGiven a lattice $L$ of rank $n$, its successive minima $\\lambda_i(L)$ for all $i\\in [n]$ are defined as follows:\n$$\n\\lambda_i(L) = \n\\inf \n\\left\\{ \nr | \\dim( {\\rm span} (L \\cap \\bar{B}_r(0)) ) \\geq i \n\\right\\}.\n$$\n\\end{definition}\n\n\\begin{definition}\n\n\\textbf{Unimodular matrix}\n\n\\noindent\nThe group of unimodular matrices $GL_n({\\mathbb{Z}})$ is the set of $n\\times n$\ninteger matrices with determinant $1$.\nUnimodular matrices preserve a lattice: $L(B) = L(B')$ if and only if $B = B' \\cdot A$, for some unimodular matrix $A$.\n\\end{definition}\n\n\n\n\\begin{definition}\n\n\\textbf{The determinant of a lattice}\n\n\\noindent\nFor a lattice $L = L(B)$ we define $\\det(L) = \\det(B)$, and denote by $N$.\n\\end{definition}\nThe determinant of a lattice is well-defined, since if $L(B') = L(B)$, then by the above $B = B' \\cdot A$\nfor some unimodular matrix $A$, in which case $\\det(B') = \\det(B) \\det(A) = \\det(B)$.\n\nThe lattice $L$ is periodic modulo $N$. In other words, if we add $N$ to \nany coordinate of a lattice point, we reach another\nlattice point. Thus, a cube of side length $N$ gives a subset of the lattice\nwhich generates the whole lattice when acted on by translations by $N$ in\nany direction. We let $L_N$ denote the lattice restricted to a cube of side\nlength $N$.  \n\nIn particular, if $L = L(B)$ is an integer lattice, with $\\det(L) = N$, for prime $N$, this implies that $L_N$ is a lattice of $\\mathbb{F}_N^n$:\n\\begin{proposition}\nLet $\\mathbb{F}_N^n$ denote the additive group of $n$-dimensional vectors of integers, where in each coordinate summation is carried out modulo $N$.\nThen $L_N$ is an additive sub-group of $\\mathbb{F}_N^n$, that contains the $0$ point.\nIn particular $L_N$ is a lattice of $\\mathbb{F}_N^n$.\n\\end{proposition}\n\n\\noindent\nA canonical representation of integer lattices is called the Hermite normal form (HNF):\n\\begin{definition}\n\n\\textbf{Hermite Normal Form}\n\n\\noindent\nAn integer matrix $A\\in {\\mathbb{Z}}^{n\\times n}$ is said to be in Hermite normal form (HNF) if $A$ is \nupper-triangular, and $a_{i,i}> a_{i,j}\\geq 0$ for all $j>i$, and all $i\\in [n]$.\n\\end{definition}\n\n\\noindent\nIt is well-known that every integer matrix can be efficiently transformed into HNF:\n\\begin{fact}\n\n\\textbf{Unique, efficiently-computable, Hermite normal form}\n\n\\noindent\nFor every full-rank integer matrix $A\\in Z^{n\\times n}$, there exists a unique unimodular\nmatrix $U\\in GL_n({\\mathbb{Z}})$, such that $H = U \\cdot A$, and $H$ is HNF.\n$U$ can be computed efficiently.\n\\end{fact}\n\n\n\n\\subsection{Gaussians on lattices}\n\nThe use of the Gaussian measure in the context of lattices is well-known in math.\nIn the context of lattices, the use of the Gaussian measure has been shown in recent years to be very useful to derive \nimportant geometric facts about lattices \\cite{Ban93}\ncomplexity-theoretic results \\cite{AR04},\nthe well-known Learning-with-Errors public-key crypto-system \\cite{Reg09},\nhard random lattices \\cite{MR07} and the fastest classical algorithms for\nthe shortest vector problem \\cite{ADR+15}.\n\n\\begin{definition}\n\n\\textbf{The discrete Gaussian measure over lattices}\n\n\\noindent\nFor any $s > 0$ define the Gaussian function on $\\mathbb{R}^n$ centered at $c$ with parameter $s$:\n$$\n\\forall x\\in R^n , \\ \\rho_{s,c}(x)  = \\exp( - \\pi \\|x - c\\|^2 \/ s^2 ).\n$$\nFor any $c\\in \\mathbb{R}^n$, real number $s>0$, and $n$-dimensional lattice $L$,\nthe discrete Gaussian distribution over $L$ is:\n$$\n\\forall x\\in L, \\ \\ D_{s,c}(x) = \\frac{\\rho_{s,c}(x)}{\\rho_{s,c}(L)},\n$$\nwhere\n$$\n\\rho_{s,c}(L) = \\sum_{x\\in L} \\rho_{s,c}(x).\n$$\n\\end{definition}\n\n\\noindent\nIn \\cite{MR07}\nMicciancio and Regev introduced a lattice quantity called the smoothing parameter:\n\\begin{definition}[MR07]\\label{def:smooth}\n\n\\textbf{The smoothing parameter of a lattice}\n\n\\noindent\nFor any $n$-dimensional lattice and positive real $\\varepsilon>0$, the smoothing parameter\n$\\eta_{\\varepsilon}(L)$ is the smallest real number $s>0$ for which\n$\\rho_{1\/s}(L^* - \\{0\\}) \\leq \\varepsilon$.\n\\end{definition}\n\\noindent\nWe mention several important facts by Micciancio, Regev \\cite{MR07}.\nThe first one can be regarded, in a sense\nas the defining property of the smoothing parameter:\n\\begin{fact}\\label{fact:mr}\nLet $L$ be some integer lattice $L\\subseteq \\mathbb{R}^n$, $\\varepsilon\\in (0,1)$.  If $s\\geq \\eta_{\\varepsilon}(L)$ then for all $c\\in \\mathbb{F}_N^n$\n$$\n\\Delta \\left(D_{\\mathbb{F}_N^n,s,c} (\\mbox{mod} {\\cal P}(L)), U({\\cal P}(L)) \\right)\\leq \\varepsilon.\n$$\n\\end{fact}\n\\noindent\nThe second one states that the smoothing parameter can be chosen arbitrarily close to the $n$-th minima of the lattice:\n\\begin{fact}\\label{fact:nmin}\nFor any lattice $L$, and $\\varepsilon = n^{-k}$ for some constant $k$, we have $\\eta_{\\varepsilon}(L) = {\\rm O}(\\lambda_n(L) \\cdot \\sqrt{{\\rm log}(n)})$.\n\\end{fact}\n\\noindent\nThe third one - is that if we sample from the discrete Gaussian on a lattice $L$ with variance $s^2$, then typically\na returned vector will have length $s \\sqrt{n}$.\n\\begin{fact}\\label{fact:ban}\nFor any $n$-dimensional lattice $L$, $\\varepsilon = o(1)$, real vector $c\\in \\mathbb{R}^n$, and real number $s \\geq \\eta_{\\varepsilon}(L)$, we have \n$$\nD_{L,s,c}(L - B_{s}(c)) = 2^{-\\Omega(n)}.\n$$\n\\end{fact}\n\n\n\n\n\n\\subsection{Lattice problems}\n\n\n\n\\begin{definition}\n\n\\textbf{Closest-vector problem \/ Shortest-vector problem}\n\n\\noindent\nThe closest-vector problem is defined as follows:\nGiven is a lattice $L = L(B)$, and a vector $v\\in \\mathbf{R}^n$.\nFind a lattice vector $w$ for which $\\|v - w\\| = {\\rm dist}(v,L)$.\nThe shortest-vector problem is defined as follows:\nGiven $L = L(B)$ find a non-zero lattice vector of minimal length.\n\\end{definition}\n\n\\begin{definition}\\label{def:cvp}\n\n\\textbf{Approximate closest-vector (search problem)}\n\n\\noindent\nThe approximate shortest-vector problem ${\\rm CVP}_{\\beta}$  is the following problem:\ngiven a lattice $L$, and a vector $v\\in \\mathbf{R}^n$\nreturn $w\\in L$ such that ${\\rm dist}(v,w) \\leq \\beta \\cdot {\\rm dist}(v,L)$.\n\\end{definition}\n\n\n\\begin{definition}\\label{def:gdd}\n\n\\textbf{Guaranteed-distance decoding - ${\\rm GDD}$}\n\n\\noindent\nWe are given an $n$-dimensional integer lattice $L = L(B)$, and $v\\in {\\mathbb{Z}}^n$ be some vector.\nFix $\\varepsilon = {\\rm poly}(n)$.\nThe problem ${\\rm GDD}$ is to find a vector $w\\in L$, such that ${\\rm dist}(w,v) \\leq \\eta_{\\varepsilon}(L)$.\n\\end{definition}\n\\noindent\nThe approximate version ${\\rm GDD}_{\\gamma}$ is to find such a vector $w\\in L$ with  ${\\rm dist}(w,v) \\leq \\eta_{\\varepsilon}(L) \\cdot \\gamma$.\n\n\\begin{definition}\\label{def:sivp}\n\n\\textbf{Shortest Independent Vectors Problem}\n\n\\noindent\nLet $L = L(B)$ be some $n$-dimensional lattice.\nThe problem ${\\rm SIVP}$ is to find a set of linearly independent vectors in $L$, of maximal length at most $\\lambda_n(L)$.\n\\end{definition}\n\\noindent\nThe approximate version ${\\rm SIVP}_{\\gamma}$ is to find such vectors whose length is at most $\\lambda_n(L) \\cdot \\gamma$.\n\n\n\n\n\n\\section{The Systematic Normal Form (SNF)}\n\n\\begin{definition}\\label{def:snf}\n\n\\textbf{Systematic Normal Form (SNF)}\n\n\\noindent\nA matrix $B$ is said to be SNF if $B_{i,i}=1$ for all $i>1$,\nand $B_{1,1}=N$ where $N$ is a prime number, and in addition, for all $i>1$ $B_{i,j}=0$ for all $i\\neq j$.\n\\end{definition} \nThis form is called suggestively \"systematic\" because for every $v\\in L(B)$, the last $n-1$ coordinates,\nare in fact the last $n-1$ coefficients of the vector under the basis $B$, which in error-correcting terminology can be considered as the \"message\" to be encoded by the matrix $B$.\n\n\\noindent\nThe following facts will be useful later on:\n\\begin{proposition}\nIf $B$ is in SNF form, then $NB^{-T}$, i.e. the matrix spanning the scaled dual of $L(B)$ assumes the following form:\n\\begin{eqnarray}\\label{eq:ndual}\nB = \\left[\\begin{array}{ccccc} \n1 & &&& \\\\\n-b_{2} & 1& & &\\\\\n-b_{3} & & 1 & &  \\\\\n\\vdots & & & \\ddots & \\\\\n-b_{n} & & & & 1 \n\\end{array}\n\\right]\n\\end{eqnarray}\n\\end{proposition}\n\\begin{proposition}\nThere are $N^{n-1}$ points of $L = L(B_{SNF})$ in $\\mathbb{F}_N^n$, and there are $N$ lattice points of $NL^*$ in that cube.\nHence there are $N$ points of $\\mathbb{F}_N^n$ inside ${\\cal P}(L)$, and $N^{n-1}$ points in ${\\cal P}(NL^*)$. \n\\end{proposition}\n\nIn fact, a somewhat stronger statement is true:\n\\begin{proposition}\\label{prop:bij}\nThere exist bijections $\\Phi_1: \\mathbb{F}_N^n \/ L_N \\mapsto {\\cal P}(L)$,\n$\\Phi_2: {\\cal P}(L) \\mapsto NL^*$ as follows:\nFor every coset of $L_N$ in $\\mathbb{F}_N^n$ $\\Phi_1$ returns a unique element of ${\\cal P}(L)$, \nand $\\Phi_2$ maps each element of ${\\cal P}(L)$ uniquely to an element of $(NL^*)_N = NL^* \\cap \\mathbb{F}_N^n$ in that coset.\nThus, in particular for every $x\\in {\\cal P}(L)$, there exists a unique $z\\in (NL^*)_N$, such that $x + z\\in L_N$.\n\\end{proposition}\n\nPerhaps more interestingly, though, for SNF one can compute efficiently, for each integer vector $x$, a corresponding\ndual-lattice vector $y$, such that their sum is in $L_N$:\n\\begin{claim}\\label{cl:dual}\n\n\\textbf{Compute dual vector for any vector}\n\n\\noindent\nFor $x\\in \\mathbb{F}_N^n$, the map $\\Phi_3:= \\Phi_2 \\circ \\Phi_1(x)$ \ncan be computed efficiently.\n\\end{claim}\n\\begin{proof}\nLet $x\\in \\mathbb{F}_N^n$.\nWe would like to find (the unique) $y = \\Phi_3(x) =  \\Phi_2 \\circ \\Phi_1(x) \\in NL^*$ for which $x + y\\in L_N$.\nEach point in $y\\in NL^*$ is characterized uniquely by an element $a\\in \\mathbb{F}_N$ as follows:\n\\begin{eqnarray}\\label{eq:nl}\ny = (a, -b_2 a  (\\mod \\ N),\\hdots, -b_n a(\\mod \\ N)).\n\\end{eqnarray}\nThus, to find $y$ we would like to solve the following vector equality over $a,z_2,\\hdots, z_n\\in \\mathbb{F}_N$:\n\\begin{eqnarray}\n(x_1,\\hdots, x_n)^T + (a, -b_2 a (\\mod \\ N),\\hdots, -b_n a (\\mod \\ N))\n=\n\\left(\\sum_{i=2}^n b_i z_i (\\mod \\ N), z_2,\\hdots, z_n\\right)^T\n\\end{eqnarray}\nConsider the first coordinate. We have:\n\\begin{eqnarray}\nx_1 + a = \\sum_{i=2}^n b_i z_i (\\mod \\ N).\n\\end{eqnarray}\nSubstituting in the above $z_i = x_i - a b_i (\\mod \\ N)$ for all $i\\geq 2$ implies:\n\\begin{eqnarray}\nx_1 - \\sum_{i=2}^n x_i b_i =  - a \\cdot \\left( \\sum_{i=2}^n b_i^2 + 1 \\right) (\\mod \\ N).\n\\end{eqnarray}\nSince $N$ is prime, then the number\n$\\sum_{i=2}^n b_i^2 + 1$ has an inverse.\nThus, the parameter $a$ can be\ncomputed uniquely from the equation above, which implies that $y$ can be determined uniquely and efficiently.\n\\end{proof}\n\n\n\\subsection{Reduction to SNF form}\n\nIn this section we provide an efficient reduction from an arbitrary lattice to a lattice in SNF form,\nthat preserves all important properties of the lattice.\nSpecifically - \nit allows the reduction of any computational problem on an arbitrary lattice $L$ to another problem on an SNF lattice $L_{SNF}$\nsuch that any solution to the reduced problems allows to find {\\it efficiently} a solution to the problem on $L$.\n\n\\begin{lemma}\\label{lem:snf}\n\n\\textbf{Efficient reduction to SNF}\n\n\\noindent\nThere exists an efficient algorithm that for any LLL-reduced upper-triangular matrix $B$, \nand numbers $a>0,b>0$, \ncomputes efficiently a tuple $\\langle B_{SNF},M,T \\rangle$, where\n$B_{SNF}$ is an SNF matrix, $T$ is a positive integer\n$T = \\max\\{ 2^{{\\rm O}(n)}\/ \\lambda_1(L(B)),\\det(B) \\}$, and $M\\in GL_n({\\mathbb{Z}})$, such that the following holds:\nFor any $v\\in L(B_{snf}\/T)$ put $\\hat{v} = B \\cdot (T B_{snf}^{-1} M v)$.\nIf $\\|v\\| \\leq \\det(B) \\cdot n^a$\nthen $\\hat{v}\\in L(B)$ and  $\\left\\|\\hat{v} - v\\right\\|  = {\\rm O}(n^{-b})$.\nAlso $\\det(B_{SNF}) = {\\rm O}(\\det(B) \\cdot T^n)$.\n\\end{lemma}\n\n\\noindent\nBefore presenting the proof,\nlet us bound the coefficients of any short vector in a lattice. \n\n\n\\begin{proposition}\\label{prop:lll1}\nLet $B$ be an $LLL$-reduced matrix, and $v\\in L$ be some lattice vector.\nThen $v$ can be represented in the basis $B$ using a vector coefficients of length at most $ \\|v\\|\\cdot 2^{{\\rm O}(n)}\/ \\lambda_1(L(B))$.\n\\end{proposition}\n\\begin{proof}\nWrite $v = Bx$, where $x\\in {\\mathbb{Z}}^n$.\nConsider the QR decomposition of $B$ as $B = Q \\cdot R$, where $Q$ is a unitary matrix,\nand $R$ is an upper-triangular matrix.\nThus $Q\\cdot R$ correspond to the Gram-Schmidt decomposition of $B$.\nWe express $x$ as:\n\\begin{eqnarray}\n\\|x\\| = \\|B^{-1} v\\| \\leq \\sqrt{n} R_{n,n}^{-1} \\|v\\|,\n\\end{eqnarray}\nwhere the second inequality follows from unitarity of $Q$.\nBy LLL we know that for some constant $\\beta>0$ we have:\n\\begin{eqnarray}\n\\lambda_1:= \\lambda_1(L) \\leq R_{1,1} \\leq 2^{\\beta n} \\cdot R_{n,n}.\n\\end{eqnarray}\nTherefore\n\\begin{eqnarray}\nR_{n,n} \\geq 2^{- \\beta n} \\lambda_1.\n\\end{eqnarray}\nThis implies that\n\\begin{eqnarray}\n\\|x\\| \\leq \\sqrt{n} R_{n,n}^{-1} \\|v\\| \\leq \\|v\\| \\sqrt{n} \\cdot 2^n\/\\lambda_1  = \\|v\\| \\cdot 2^{{\\rm O}(n)} \/ \\lambda_1.\n\\end{eqnarray}\n\\end{proof}\n\n\n\\noindent\nThe following are easy corollaries of the above:\n\n\n\n\\begin{proposition}\\label{prop:close}\n\\label{two-close-lattices}\nLet $B_1 = \\{v_i\\}_{i=1}^n$ be some LLL-reduced basis\nand another basis $B_2 = \\{w_i\\}_{i=1}^n$ for lattice $L_2 = L(B_2)$.\nSuppose that $\\|v_i - w_i\\| \\leq \\alpha$. \nLet $v = \\sum_{i=1}^n c_i v_i$ be a point in $L_1$ and $w = \\sum_{i=1}^n\nc_i w_i$ be the corresponding point in $L_2$. Then $\\left\\|v - w \\right\\| \\leq \\|v\\| \\alpha 2^{{\\rm O}(n)}\/\\lambda_1(L_1) $. \n\\end{proposition}\n\n\\begin{proof}\nBy the triangle inequality we have:\n\\begin{eqnarray}\n\\left\\|v-w \\right\\| &= &  \\left\\|\\sum_{i=1}^n c_i v_i  -  \\sum_{i=1}^n c_i w_i  \\right\\| \\\\\n&\\leq & \\sum_{i=1}^n \\| v_i -w_i \\| |c_i| \\\\\n&\\leq & n\\alpha \\|v\\| \\cdot 2^{{\\rm O}(n)} \/\\lambda_1(L_1)\\\\\n&=     & \\alpha \\|v\\| \\cdot 2^{{\\rm O}(n)}\/\\lambda_1(L_1),\n\\end{eqnarray}\nwhere the inequality before last follows from Proposition \\ref{prop:lll1}. \n\\end{proof}\n\n\n\\subsection{Proof of Lemma \\ref{lem:snf}}\n\n\\begin{proof}\nWe first use $T$ as a parameter and determine it later on in the proof.\nWe start from an upper-triangular LLL-reduced matrix $B_1$:\n\\begin{eqnarray}\nB_1 = \\left[\\begin{array}{ccccc} \nb_{1,1} & b_{1,2} & b_{1,3} & \\ldots & b_{1,n} \\\\\n& b_{2,2} & b_{2,3} & \\ldots & b_{2,n} \\\\\n& & b_{3,3} & \\ldots & b_{3,n} \\\\\n& & & \\ddots & \\vdots \\\\\n& & & & b_{n,n} \n\\end{array}\n\\right]\n\\end{eqnarray}\nadd $1\/{T}$ along the subdiagonal,\nand truncate each non-zero entry to its nearest integer multiple of $1\/T$:\n\\begin{eqnarray}\n\\label{add-subdiagonal}\nB_2 = \\left[\\begin{array}{ccccc} \nb_{1,1}' & b_{1,2}' & b_{1,3}' & \\ldots & b_{1,n}' \\\\\n\\frac{1}{T}& b_{2,2}' & b_{2,3}' & \\ldots & b_{2,n}' \\\\\n& \\frac{1}{T}& b_{3,3}' & \\ldots & b_{3,n}' \\\\\n& & \\ddots & \\ddots & \\vdots \\\\\n& & & \\frac{1}{T}& b_{n,n}'\n\\end{array}\n\\right],\n\\end{eqnarray}\nwhere $b_{i,j}' = \\round{b_{i,j} T} \/T$.\nWe note that\n\\begin{eqnarray}\\label{eq:entry}\n\\forall i,j \\ \\ |B_2(i,j) - B_1(i,j)| \\leq 1\/T.\n\\end{eqnarray}\n\n\nWe now use column operations to make rows $2$, $3$, $\\ldots$,\n$n$ of the lattice zero except for the subdiagonal. This involves subtracting integer multiples\nof the $i$th column from all later columns. We obtain a lattice of the form.\n\\begin{eqnarray}\\label{eq:b3}\n\\label{after-Gaussian}\nB_3 = \\left[\\begin{array}{ccccc} \nb_{1,1}' & b_{1,2}'' & b_{1,3}'' & \\ldots & b_{1,n}'' \\\\\n\\frac{1}{T}& 0 & 0 & \\ldots & 0 \\\\\n& \\frac{1}{T}& 0 & \\ldots & 0 \\\\\n& & \\ddots & \\ddots & \\vdots \\\\\n& & & \\frac{1}{T}& 0 \n\\end{array}\n\\right]\n\\end{eqnarray}\nObserve that if we move the $n$th column to the first column and multiply all entries by\n$T$, we now have a lattice which is a $1\/T$ multiple of an SNF lattice, \nexcept possibly from the entry $b_{1,n}''$ which may not be prime.\n\n\nWe now show how to make determinant of the new lattice prime,\nby rounding\n$b_{1,n}''$ to a prime.  \nBy standard number-theory conjecture\n\\footnote{Generalized Riemann Hypothesis (GRH)} we assume:\n\\begin{eqnarray}\\label{eq:delta}\n\\exists \\delta = {\\rm O}({\\rm log}(T b_{1,n}'')), \\ \\ T b_{1,n}'' + \\delta \\mbox{ is prime }.\n\\end{eqnarray}\n\nWe need to compute the matrix that transforms the basis given in equation\n(\\ref{add-subdiagonal}) to the basis given in equation (\\ref{after-Gaussian}).\nThat is, we want the matrix $M$ such that $B_3 =  B_2 M$. The diagonal and\nsuperdiagonal of the matrix can be easily calculated: \n\\begin{eqnarray}\nM &=& \\left( \n\\begin{array}{ccccccc}\n1 & T b_{2,2}' & T b_{2,3}' &  \\ldots & T b_{2,n-1}'&   T b_{2,n}' \\\\\n  & 1 & T b_{3,3}' & \\ldots & T b_{3,n-1}' & T b_{3,n}' \\\\\n  &   & 1 & \\ldots & T b_{4,n-1}' & T b_{4,n}' \\\\\n  &      &   & \\ddots &   \\vdots & \\vdots \\\\\n  &   & & & 1 & T b_{n,n}'  \\\\ \n  &   &  & &   & 1 \n\\end{array}\n\\right)^{-1} \\\\\n&=&\n\\left( \n\\begin{array}{ccccccc}\n1 & - T b_{2,2}' &   \\ldots &   \\\\\n  & 1 & - T b_{3,3}' & \\ldots &  \\\\\n  &   & 1 & - T b_{4,4}' & \\ldots\\\\\n  &      &   & \\ddots &   \\ddots  \\\\\n  &   & & & 1 & - T b_{n,n}'  \\\\ \n  &   &  & &   & 1 \n\\end{array}\n\\right)\n\\end{eqnarray}\nBy the above, $M$ is a unimodular matrix, hence $\\det(B_2) = \\det(B_3)$.\nNote that both $M$ and $M^{-1}$ are upper triangular matrices with $1$s along the diagonal.\nThe determinant of $TB_3$ is $T b_{1,n}''$. \nTo obtain a lattice with a prime determinant, we need to round $\\det(T B_3)$ to a \nnearby prime.  Let us assume that $\\det{T B_3} + \\delta$ is prime. This rounding \ncorresponds to adding $\\delta\/T$ to the entry $B_3(1,n)$. \n\nWhat effect does this change have on the basis of the lattice in $B_2$?\nLet $\\Delta$ be the matrix with\n$\\Delta(1,n) = \\delta\/T$ and all other entries $0$. Then our matrix in the SNF basis\nis $B_3 + \\Delta$. To see what the effect on $B_2$ is, we merely need to multiply by\n$M^{-1}$. That is,\n\\begin{eqnarray}\nB_2 + \\Delta M^{-1} = ( B_3 + \\Delta )M^{-1} .\n\\end{eqnarray}\nUsing the form we derived above for $M^{-1}$, we see that because there are 1s along the\ndiagonal of $M$ then \n$\\Delta M^{-1} = \\Delta$. Thus, we can make $B_2$ obtain a prime determinant by simply\nadding $\\delta\/T$ to $B_2(1,n) = b_{1,n}'$. This\nchanges the length of the $n$th basis vector by at most\n$\\delta\/T$.\n\nLet $B_4$ denote then the output SNF matrix.\n\\begin{eqnarray}\nB_4 = \\left[\\begin{array}{ccccc} \nT b_{1,n}'' + \\delta & T b_{1,1}' & T b_{1,2}'' & \\ldots & T b_{1,n-1}'' \\\\\n0 & 1 & 0 & \\ldots & 0 \\\\\n   & 0 & 1 & \\ldots & 0 \\\\\n& & \\ddots & \\ddots & \\vdots \\\\\n& & & 0& 1 \n\\end{array}\n\\right]\n\\end{eqnarray}\nBy Equations \\ref{eq:entry} and equation \\ref{eq:delta} :\n\\begin{eqnarray}\n\\forall i,j\\in [n] |(M^{-1} B_4(i,j))\/T - B_1| = {\\rm O}({\\rm log}(T b_{1,n}'')\/T).\n\\end{eqnarray}\nThat is, the basis $M^{-1} B_4\/T$ of $L(B_4)\/T$ is entry-wise close to $B_1$.\nBy assumption $B_1$ is in particular LLL-reduced.\nHence we can invoke Proposition \\ref{prop:close} w.r.t. these two bases:\nConsider some $v\\in L(B_4\/T)$.\nLet $\\beta$ denote the implicit constant in the bound of Proposition \\ref{prop:close}.\nApplying Proposition \\ref{prop:close} implies that \nthe \ncorresponding vector $\\hat{v} = B_1 \\cdot (TB_4^{-1}M) \\cdot v\\in L(B_1)$\nhas\n\\begin{eqnarray}\\label{eq:bound2}\n\\left\\|\\hat{v}- v\\right\\| \\leq \\frac{2^{\\beta n} \\|v\\| {\\rm log}(T b_{1,n}'')}{T\\lambda_1(L(B_1))}.\n\\end{eqnarray}\nBy assumption $\\|v\\| \\leq n^a \\det(B_1)$.\nThen together with Equation \\ref{eq:bound2} we conclude that there exists\n\\begin{eqnarray}\nT = {\\rm poly}(n) \\cdot \\max \\{ \\det(B_1)\/\\lambda_1(L(B_1)), 2^{\\beta n} \\}.\n\\end{eqnarray}\nsuch that\n\\begin{eqnarray}\n\\left\\|\\hat{v}- v\\right\\| \\leq  n^{-b}.\n\\end{eqnarray}\nFinally, the entry $B_{SNF}(1,1) = \\det(B_{SNF}) \\leq 2 b_{1,n}'' \\cdot T^n$, hence the determinant $\\det(B_{SNF})$ is upper-bounded by \n\\begin{eqnarray}\n\\det(B_{SNF})  = {\\rm O}(\\det(B) T^n).\n\\end{eqnarray}\n\n\\end{proof}\n\nThe lemma above implies that one can reduce the standard lattice problems, given for an arbitrary lattice, to the same problem on a lattice in SNF,\nand then translate the output solution efficiently to a solution for the original lattice:\n\\begin{corollary}\\label{cor:cvp}\n\n\\textbf{SNF reduction preserves approximate ${\\rm CVP}$}\n\n\\noindent\nLet $(B,v)$ be an input to ${\\rm CVP}_{\\gamma}$ for some $\\gamma$, \nwhere $B$ is an LLL-reduced upper-triangular matrix.\nLet $\\langle B_{SNF},T,M \\rangle$ denote the tuple returned by the SNF reduction, for parameters $a>1\/2,b$.\nSuppose that for $x_0\\in L(B_{SNF})$, \n$v\\in [\\det(B)]^n \\cap {\\mathbb{Z}}^n$ \nwe have\n$\\|x_0 - Tv\\| \\leq \\gamma \\cdot {\\rm dist}(Tv, L(B_{SNF}))$, for $\\gamma = n^{a - 1\/2}$.\nThen the vector $x_{out} := B(B_{SNF}^{-1} M) x_0\\in L(B)$ has\n$$\n\\| x_{out} - v\\| \\leq \\gamma \\cdot {\\rm dist}(v,L(B)).\n$$\n\\end{corollary}\n\n\\begin{proof}\nDenote $L = L(B), L_{SNF} = L(B_{SNF})$.\nBy the triangle inequality:\n\\begin{eqnarray}\n\\left\\|x_{out} - v\\right\\| =\n\\left\\| x_{out} - x_0\/T +x_0\/T - v\\right\\| \n\\leq\n\\left\\| x_0\/T - v \\right\\| + \\| x_0\/T - x_{out}\\|\n\\end{eqnarray}\nSince\n\\begin{eqnarray}\nx_0\\in L_{SNF} \\ \\mbox{ and }\n\\| x_0 - T v \\| \\leq \\gamma \\cdot {\\rm dist}(Tv, L_{SNF})\n\\end{eqnarray}\nthen together with the above we have that $x_{out}\\in L$ and\n\\begin{align}\n\\|x_{out} - v\\| \n&\\leq \n\\frac{1}{T} \\left\\| x_0 - Tv \\right\\| + \\| x_0\/T - x_{out}\\|\\\\\n&\\leq\n\\frac{1}{T} \\gamma \\cdot {\\rm dist}(Tv, L_{SNF})  + \\| x_0\/T - x_{out}\\|\\\\\n&\\leq\\label{eq:align3}\n\\gamma \\cdot {\\rm dist}(v,L) + \\| x_0\/T - x_{out}\\|,\n\\end{align}\nwhere in the last inequality we used again Lemma \\ref{lem:snf}.\nSince $v$ is an integer vector, we can assume that the length of $x_0$ is bounded by:\n\\begin{eqnarray}\n\\| x_0 \\| \\leq \\gamma \\|Tv\\| = \\gamma T \\|v\\|,   \n\\end{eqnarray}\nhence\n\\begin{eqnarray}\\label{eq:bound1}\n\\| x_0 \/ T \\| \\leq \\gamma \\|v \\| \\leq \\gamma \\det(B) \\sqrt{n} = n^a \\det(B).\n\\end{eqnarray}\nBy assumption $x_0\\in L_{SNF}$ so $x_0\/T \\in L(B_{SNF}\/T)$.\nHence,\nby Lemma \\ref{lem:snf} and together with the above equation \\ref{eq:bound1} then $x_{out}\\in L(B)$ and we have\n\\begin{eqnarray}\n\\left\\|x_{out} - x_0\/T\\right\\| =  {\\rm O}(n^{-b}).\n\\end{eqnarray}\nPlugging this inequality into Equation \\ref{eq:align3} implies the proof.\n\n\n\\end{proof}\n\n\\subsection{Uniform distribution on SNF-dual}\n\n\\begin{fact}\\label{fact:1}\nLet $L\\subseteq \\mathbb{F}_N^n$ be some SNF lattice, $\\det(L) = N$, and\nlet ${\\cal D}^*$ denote the distribution on $NL^*$ defined by sampling $x\\sim D_{\\mathbb{F}_N^n,s,c}$,\nand computing the corresponding element $y\\in (NL^*)_N = \\Phi_3(x)$.\nIf $s\\geq \\eta_{\\varepsilon}(L)$ then $\\Delta({\\cal D}^*, U[(NL^*)_N]) \\leq \\varepsilon$.\n\\end{fact} \n\n\\begin{proof}\nBy  Fact \\ref{fact:mr} we have $\\Delta((D_{\\mathbb{F}_N^n,s,c} \\mbox{mod} {\\cal P}(L), U({\\cal P}(L))) \\leq \\varepsilon$.\nBy Proposition \\ref{prop:bij} $\\Phi_3$ is a bijection between ${\\cal P}(L)$ and $(NL^*)_N$, so therefore:\nsampling $x\\sim \\rho_{\\mathbb{F}_N^n,s,c}$, and computing $y = y(x)\\in (NL^*)_N$ results in a distribution $D^*$\nwhich is $\\varepsilon$ close to $U[(NL^*)_N]$.\n\\end{proof}\n\n\\section{Rank-$1$ ${\\rm SIS}$ with Prime Modulus}\n\nIn this work we will use a somewhat different variant of the Ajtai ensemble \nthat arises naturally from the SNF reduction.\nWe define formally our variant of SIS as follows:\n\\begin{definition}\\label{def:sis}\n\n\\textbf{Short-Integer-Solution ${\\rm SIS}(N,\\delta)$ - homogeneous}\n\n\\noindent\nLet $N$ be some prime number, and $\\delta >0$ some constant.\nFix $n$ as the minimal positive integer\nfor which $N \\leq n^{\\delta n}$.\nGiven are $n$ numbers $g_1,\\hdots, g_n \\in \\mathbb{F}_N$.\nThe Short-Integer-Solution problem is to find $n$ numbers $h_1,\\hdots, h_n\\in \\mathbb{F}_N$, such that\n$$\n\\sum_{i=1}^n h_i g_i = 0 \\in \\mathbb{F}_N^n,\\mbox{ and } \\max_{i=1}^n |h_i| \\leq  2 n^{\\delta}.\n$$ \n\\end{definition}\n\\noindent\nAlternatively, ${\\rm SIS}(N)$ asks for a short vector in the lattice\n$$\nL = L(g)^{\\perp} := \\{ h\\in \\mathbb{F}_N^{n}, \\ \\ \\langle (g_1,\\hdots, g_n),h \\rangle = 0\\in \\mathbb{F}_N \\}.\n$$\n\nWe note \\cite{VV16} that one can formulate ${\\rm SIS}(N)$ also as a non-homogeneous congruence of the following\nform\n\\begin{definition}\\label{def:affine}\n\n\\textbf{Short-Integer-Solution ${\\rm SIS}(N,\\delta)$ - non-homogeneous}\n\n\\noindent\nLet $N$ be some prime number, and $\\delta>0$ some constant.\nFix $n$ as the minimal positive integer\nfor which $N \\leq n^{\\delta n}$.\nGiven are $n$ numbers $g_1,\\hdots, g_n \\in \\mathbb{F}_N$.\nThe Short-Integer-Solution problem is to find $n+1$ numbers $h_0,h_1,\\hdots, h_n\\in \\mathbb{F}_N$, such that\n$$\n\\sum_{i=1}^n h_i g_i = h_0 \\in \\mathbb{F}_N,\\mbox{ and } \\max_{i=0}^n |h_i| \\leq  2 n^{\\delta}.\n$$ \n\\end{definition}\nSimilarly to the homogeneous case we can define ${\\rm SIS}(N)$ as asking for short vectors in the following $n+1$-dimensional lattice:\n$$\nL = L(g)_1^{\\perp} := \\{ h\\in \\mathbb{F}^{n+1}, \\ \\ \\langle (1,g_1,\\hdots, g_n),h \\rangle = 0\\in \\mathbb{F}_N \\}.\n$$\n\nTo see why the two problems are equivalent, note that we can implement an oracle ${\\cal O}_h$ for the homogeneous\n${\\rm SIS}(N,\\delta)$ using oracle calls to ${\\cal O}_n$ for the non-homogeneous ${\\rm SIS}(N,\\delta\/2)$ as follows:\nGiven $g$, call ${\\cal O}_n(g) = \\{h_i\\}_{i=0}^n$, i.e. \n$$\n\\sum_i h_i g_i = h_0 (\\mod \\ N).\n$$\nThen call \n$$\n{\\cal O}_n(h_0^{-1} g) = \\{h_i'\\}_{i=0}^n.\n$$\nThen by definition\n$$\n\\sum_{i=1}^n h_i' h_0^{-1} g_i = h_0'\n$$\nso \n$$\n\\sum_{i=1}^n h_i' g_i = h_0 h_0' = h_0' \\sum_{i=1}^n h_i g_i,\n$$\nwhich implies that \n$$\n\\sum_{i=1}^n (h_i' - h_0' h_i) g_i = 0 (\\mod \\ N),\n$$ \nand since $|h_i|, |h_i'|$ are bounded by $n^{\\delta\/2}$ for all $i$ this implies that\n$\\{h_i' - h_0' h_i\\}_{i=1}^n$\nare bounded by $n^{\\delta\/2} + n^{\\delta\/2} n^{\\delta\/2} \\leq 2 n^{\\delta}$, hence it\n is a valid solution to ${\\cal O}_h(g)$.\n\nThe significance of the non-homogeneous version is evident in the following \nequivalence of definitions:\n\\begin{proposition}\\label{prop:equiv}\nLet $B$ be an SNF matrix, with $\\det(B) = N$.\nThen $L(B) = L(g)_1^{\\perp}$ where $g$ is the $n-1$-dimensional vector $g = (b_2,\\hdots, b_n)$.\n\\end{proposition}\n\nThe random - approximate SIS problem ($\\rm rSIS(N)$) is then defined by having $g_1,\\hdots, g_n$ be chosen independently and uniformly at random.\nWe also define an approximation variant called ${\\rm SIS}_{\\gamma}$ in which the solution\n$\\{h_i\\}$ must satisfy $max_i |h_i| \\leq n^{\\delta} \\gamma$.\nAs in previous works, one can establish the existence of a solution to the SIS problem for any input,\nusing the pigeonhole principle:\n\\begin{fact}\\label{fact:sol}\nFor any $N,\\delta$ the (homogeneous \/ non-homogeneous) ${\\rm SIS}(N,\\delta)$ problem has a solution.\n\\end{fact}\n\\begin{proof}\nFix some $h_0\\in \\mathbb{F}_N$.\nBy definition, there are at least $n^{\\delta n}$ vectors of coefficients \n$(h_1,\\hdots, h_{n})$\nwhose magnitude is at most $n^{\\delta}$. Since $N\\leq n^{\\delta n}$ there are at least two $n$-dimensional coefficient vectors $h \\neq h'$\nfor which $\\sum_{i=1}^n h_i g_i = \\sum_{i=1}^n h_i' g_i = h_0 (\\mod \\ N)$.\nHence the vector $(h_1-h_1', \\hdots, h_{n} - h_{n}')$ is \na non-zero $n$-dimensional vector which is a valid solution.\n\\end{proof}\n\nNext, we adapt an argument that appeared in \\cite{AP11} which discussed\nthe Ajtai ensemble of lattices, for our ensemble of random lattices - $\\rm rSIS(N)$:\n\\begin{lemma}\n\n\\textbf{Random ${\\rm SIS}$ lattices are dense}\n\n\\noindent\nLet $N$ be some prime number, $\\delta\\leq 1\/2$, and consider the lattice defined by ${\\rm SIS}(N,\\delta)$,\nfor some numbers $g_1,\\hdots, g_n\\in \\mathbb{F}_N$:\n$$\nL = L(g)^{\\perp} := \\{ z\\in {\\mathbb{Z}}^{n}, \\ \\ \\langle (g_1,\\hdots, g_n),z \\rangle = 0 \\}.\n$$\nWith high probability when choosing the numbers $\\{g_i\\}_{i=1}^n$ \nindependently and uniformly at random from $\\mathbb{F}_N$ we have that $\\lambda_i(L(g)^{\\perp})= \\Theta(\\sqrt{n})$ for all $i\\in [n]$.\n\\end{lemma}\n\n\\begin{proof}\nBy definition $n$ is the minimal positive integer for which $N\\leq n^{\\delta n}$.  \nThere exists such $n$ for which $N \\geq n^{\\delta (n-1)}$.\n\\footnote{assuming Bertrand-Chebyshev theorem, and sufficiently large $n$.}\nThe argument has two parts:\nFirst, is that $\\det(L) \\leq N$ - this is immediate because there are at most N congruence classes of ${\\mathbb{Z}}^{n} \/ L$ by definition.\nNext, for any integer $n$ and constant $\\alpha$ let: \n\\begin{eqnarray}\nN_{\\alpha,n} = \\left| \\{ z\\in \\mathbb{F}_N^n, \\ \\ \\|z\\| \\leq \\alpha\\sqrt{n} \\} \\right|.\n\\end{eqnarray}\nBy \\cite{MO90} for every $\\varepsilon$, there exists an $\\alpha$ such that $N_{\\alpha,n} \\leq \\varepsilon^n$.\nFix a vector $v\\in N_{\\alpha,n}$.  We have that \n${\\sf{P}}_{g\\sim U[\\mathbb{F}_N^{n}]} ( v\\in L(g)^{\\perp}) = 1\/N$.\nHence by the union bound \n\\begin{eqnarray}\n{\\sf{P}}\n\\left( \\{ z\\in \\mathbb{F}_N^n, \\ \\ \\|z\\| \\leq n^{\\delta} \\} \\cap L(g)^{\\perp} \\neq \\Phi \n\\right) \n\\leq \n\\varepsilon^{n} \/ N \\leq \\varepsilon^{n} \/ n^{\\delta(n-1)} <<1.\n\\end{eqnarray}\nTogether with the assumption $\\delta \\leq 1\/2$ this implies that w.p. $1-{\\rm O}(1)$ we get $\\lambda_1(L(g)^{\\perp})  \\geq \\alpha n^{\\delta}$.\nThis implies by definition that  $\\lambda_i(L) \\geq \\alpha n^{\\delta}$ for all $i\\in [n]$.\nTogether with the fact that $\\det(L) \\leq N \\leq n^{\\delta n}$,\nit implies that $\\lambda_i(L) = \\Theta(n^{\\delta})$ for all $i\\in [n]$.\n\\end{proof}\n\n\n\\section{Reduction from worst-case GDD to random SIS}\n\nIn this section we show\na reduction from a variant of ${\\rm CVP}$ called Guaranteed-Distance-Decoding, or ${\\rm GDD}$,\n(see Definition \\ref{def:gdd})\nto random SIS.\nLet ${\\cal A}$ denote an algorithm for ${\\rm SIS}(N)$.\nWe define the following algorithm:\n\\begin{algorithm*}${\\cal B}$\n\\item\nInput: An integer matrix $B$,  and vector $v\\in {\\mathbb{Z}}^n$, and parameter $\\delta>0$.\n\\begin{enumerate}\n\\item\nReduce $B$ via LLL, denote by $B_{LLL}$.  \n\\item\nDecompose $B_{LLL}$ by QR decomposition, $B_{LLL} = Q \\cdot R$,\nset $\\hat{v} = Q^{\\dag} v$.\n\\item \nReduce $R$ to SNF form: $\\langle B_{SNF},M,T \\rangle$.\nDenote $L_{SNF} = L(B_{SNF})$,  $N = \\det(L_{SNF})$, $L_N = L_{SNF} \\cap \\mathbb{F}_N^n$,  $NL^* = L(N \\cdot B_{SNF}^{-T})$.\n\\item\nPut \n$m$ as the minimal positive integer for which $m^{\\delta m} \\geq N$.\nLet $\\Phi$ denote the minimal number for which: $\\eta_{\\varepsilon}(L) \\leq \\Phi$ for $\\varepsilon = m^{-5}$.\nSet $s = T \\cdot \\Phi$.\n\\item\nChoose $c$ uniformly at random from $\\mathbb{F}_N \\cap [- m^{1 + \\delta}, m^{1 + \\delta}]$.\nChoose $u_{rand}\\in L_N$ uniformly at random from $L_N$.\nPut $v_{target} = (c^{-1}\\cdot T \\cdot \\hat{v} + u_{rand}) (\\mod \\ N)$.\n\\item\nRepeat $m$ times:\n\\begin{enumerate}\n\\item\nSample $x_i = (x_{i,1},\\hdots, x_{i,n})\\sim D_{\\mathbb{F}_N^n,s,v_{target}}$.\n\\item\nCompute $y_i = \\Phi_3(x_i)$.\nDenote by $a_i$ the first coordinate of $y_i$.\n\\end{enumerate}\n\\item\nPut $\\{\\alpha_i\\}_{i=1}^{m} = {\\cal A}(a_1,\\hdots, a_m)$.\n\\item\nIf ${\\cal A}$ fails or $\\sum_{i=1}^m \\alpha_i \\neq c$ return FAIL.\n\\item\nCompute $x_0 = \\sum_{i=1}^m \\alpha_i (y_i + x_i) - c u_{rand}(\\mod \\ N)$.\n\\item\nReturn $x_{out} := Q \\cdot R \\cdot (B_{SNF}^{-1}M^{-1} \\cdot x_0) $.\n\\end{enumerate}\n\\end{algorithm*}\n\nThe theorem below is stated so that for an oracle ${\\cal A}$ that returns a correct answer w.p. $1-o(1)$ the algorithm ${\\cal B}$\ncomputes the correct answer w.p. $1\/{\\rm poly}(n)$.\nHowever, by making multiple (parallel) calls to the oracle ${\\cal A}$, and algorithm ${\\cal B}$, the probability of success can be amplified to an arbitrary  constant, while using an oracle that returns a correct answer w.p. $1\/{\\rm poly}(n)$.\nWe note that by Lemma \\ref{lem:snf}, the minimal $m$ for which $m^{\\delta m} \\geq N$,\nwhere $N = \\det(B_{SNF})$ is polynomial in $n$, i.e. $m = {\\rm poly}(n)$.\nThus, for simplicity, in our main theorem we state the probability of success as a function of $m$\ninstead of $n$.\n\\begin{theorem}\nLet $(B,v)$ be an input to ${\\rm GDD}$\nwhere $B$ is an $n\\times n$ integer.\nSuppose that ${\\cal A}$ returns w.p. at least $1-m^{-3}$ a solution to\n${\\rm SIS}(N,\\delta)$.  Then for $x_{out} = {\\cal B}(B,v)$ we have $x_{out}\\in L$ and w.p. $\\Omega(m^{-2})$ we have:\n$$\n\\|x_{out} - v\\| \\leq \\Phi \\cdot {\\rm dist}(v,L) \\cdot {\\rm O}(n^{1.5+\\delta} \\cdot \\max\\{n,{\\rm log}\\det(B)\\}^{1+\\delta}).\n$$\n\\end{theorem}\n\n\\begin{proof}\n$B_{LLL}$ is LLL-reduced, hence the matrix $R$ as a basis for $L = L(R)$ is trivially also LLL-reduced.\n$R$ is thus LLL-reduced and upper-triangular which means we can invoke Lemma \\ref{lem:snf} w.r.t. $R$, and derive\nthe tuple $\\langle B_{SNF},M,T\\rangle$ which has the properties of the lemma.\nIn particular, by Corollary \\ref{cor:cvp} \nif \n\\begin{eqnarray}\n(\\star)\\ \\ x_0\\in L_{SNF} \\ \\mbox{ and }\n\\| x_0 - T \\hat{v} \\| \\leq \\gamma \\cdot T\\Phi\n\\end{eqnarray}\nthen\n\\begin{eqnarray}\\label{eq:star1}\n R \\cdot (B_{SNF}^{-1}M^{-1} \\cdot x_0)\\in L(R) \\mbox{ and }\n\\| R \\cdot (B_{SNF}^{-1}M^{-1} \\cdot x_0) - \\hat{v}\\| \n\\leq\n\\gamma \\Phi + {\\rm O}(n^{-k}),\n\\end{eqnarray}\nand since $Q$ is a unitary matrix then\n\\begin{eqnarray}\nx_{out} \\in L(B), \\ \\ \n\\| x_{out} - v \\|\n=\n\\| R \\cdot (B_{SNF}^{-1}M^{-1} \\cdot x_0) - \\hat{v}\\|\n\\leq\n\\gamma \\Phi + {\\rm O}(n^{-k}).\n\\end{eqnarray}\nWe will first show that when ${\\cal B}$ terminates then the output vector has property $(\\star)$.\nThen, we will show that ${\\cal B}$ succeeds with high probability.\n\n\\item\n\\textbf{Property $(\\star)$ - quality of estimate:}\n\n\\noindent\nBy Claim \\ref{cl:dual} we have:\n\\begin{eqnarray}\\label{eq:inlattice}\n\\forall i\\in [m], \\ \\ x_i + \\Phi_3(x_i) = x_i + y_i \\in L.\n\\end{eqnarray}\nSince $x_i\\sim D_{\\mathbb{F}_N^n,s,v_{target}}$, for $v_{target}\\in \\mathbb{F}_N^n$ we can write \n\\begin{eqnarray}\\label{eq:expand}\nx_i = c^{-1} T \\hat{v} + u_{rand} + {\\cal E}_i,\\ \\ \n{\\cal E}_i \\sim D_{\\mathbb{F}_N^n,s,0}.\n\\end{eqnarray}\nSince we choose $m$ such that $N\\leq \\sqrt{m}^{m}$ then by Fact \\ref{fact:sol} there exists a short solution, i.e. whenever ${\\cal A}$\nsucceeds it returns\na set of coefficients $\\alpha_1,\\hdots, \\alpha_m$, $|\\alpha_i| \\leq \\beta\\sqrt{m}$, \nfor some constant $\\beta$, and in addition \n$$\n\\sum_{i=1}^m \\alpha_i a_i = 0.\n$$\nAssume from now on that this is the case.\nUsing again Equation \\ref{eq:nl} implies that\n\\begin{eqnarray}\\label{eq:aout}\n\\sum_{i=1}^m \\alpha_i y_i = 0 \\in \\mathbb{F}_N, \\ \\ |\\alpha_i| \\leq \\beta m^{\\delta}, \\ \\forall \\ 1\\leq i\\leq n\n\\end{eqnarray}\nNow, assume that $\\sum_{i=1}^m \\alpha_i = c$, in which case the algorithm ${\\cal B}$ \ndeclares success and consider the vector: \n\\begin{eqnarray}\nx_0 := \n\\sum_{i=1}^m \\alpha_i (x_i + y_i)- c u_{rand} .\n\\end{eqnarray}\nOn one hand: by definition $x_0 \\in L_{SNF}$, because the $\\alpha_i$'s and $c$ are integers, $u_{rand}\\in L_N$, and by Equation \\ref{eq:inlattice} $x_i + y_i\\in L_N$ for all $i$.\nOn the other hand by Equation \\ref{eq:expand}\n\\begin{eqnarray}\\label{eq:1}\nT \\hat{v} - x_0\n=\nT \\hat{v} - \n\\sum_{i=1}^m \\alpha_i y_i - \n\\sum_{i=1}^m \\alpha_i (c^{-1} T \\hat{v}+u_{rand}) - \n\\sum_{i=1}^m \\alpha_i {\\cal E}_i + c u_{rand}\n\\end{eqnarray}\nIn addition, by the assumption that $\\sum_{i=1}^m \\alpha_i = c$ we get:\n\\begin{eqnarray}\n\\sum_{i=1}^m \\alpha_i c^{-1} T \\hat{v} = \nT \\hat{v} c^{-1} \\cdot \\sum_{i=1}^m \\alpha_i = T \\hat{v} \\cdot c^{-1} \\cdot c = T \\hat{v} ,\n\\end{eqnarray}\nand similarly \n\\begin{eqnarray}\n\\sum_{i=1}^m \\alpha_i u_{rand} = c u_{rand}.\n\\end{eqnarray}\nHence\n\\begin{eqnarray}\nT \\hat{v} - x_0 = \\sum_i \\alpha_i {\\cal E}_i\n\\end{eqnarray}\nBy Equation \\ref{eq:expand} ${\\cal E}_i\\sim D_{\\mathbb{F}_N^n, s,0}$ for all $i$, so by Fact \\ref{fact:ban}\n\\begin{eqnarray}\n\\forall i, \\ \\ \n{\\sf{P}}( \\|{\\cal E}_i\\| \\leq s \\sqrt{n}) = 1-2^{-\\Omega(n)},\n\\end{eqnarray}\nand so by the union bound:\n\\begin{eqnarray}\n{\\sf{P}}( \\forall i, \\ \\|{\\cal E}_i\\| \\leq s \\sqrt{n}) = 1- 2^{-\\Omega(n)} = 1 - o(1).\n\\end{eqnarray}\nNext, by Equation \\ref{eq:aout} the random variables $\\alpha_i$ are bounded by $m^{\\delta}$ hence\n\\begin{eqnarray}\n{\\sf{P}}( \\|T \\hat{v} - x_0 \\| \\leq s \\cdot m^{1+\\delta} \\cdot \\sqrt{n}) = 1 - o(1).\n\\end{eqnarray} \nWe note that in the above we assume a worst-case scenario where\nthe summation $\\sum_i \\alpha_i {\\cal E}_i$ is coherent, i.e. the $\\alpha_i$'s\nare completely correlated with the ${\\cal E}_i$'s.\nTogether with our choice $s = T \\Phi$ this implies that\n\\begin{eqnarray}\n{\\sf{P}}( \\| T \\hat{v} - x_0 \\| \\leq T \\phi m^{1+\\delta} \\sqrt{n}) = 1 - o(1).\n\\end{eqnarray}\nSince  $m$ is the minimal integer for which\n$m^{\\delta m} \\geq N$ \nthen by Bertrand-Chebyshev theorem for sufficiently large $m$ we have:\n\\begin{eqnarray}\nm^{\\delta (m-1)}\\leq N.\n\\end{eqnarray}\nIn addition, by Lemma \\ref{lem:snf} we have $T\\leq \\max\\{ \\det(B), 2^{{\\rm O}(n)} \\}$ and \n$N = {\\rm O}(\\det(B) T^n)$.\nHence $m = {\\rm O}(n \\max\\{n,{\\rm log}\\det(B)\\})$.\nThus, w.p. $1 - o(1)$ we have \n\\begin{eqnarray}\n\\| T \\hat{v} - x_0\\| = {\\rm O}( n^{1.5+\\delta} T \\Phi \\max\\{n,{\\rm log}\\det(B)\\}^{1+\\delta}),\n\\end{eqnarray}\nin which case by Equation \\ref{eq:star1}\n\\begin{eqnarray}\\label{eq:out}\n\\| v - x_{out} \\| = \\Phi \\cdot {\\rm O}( n^{1.5+\\delta} \\max\\{n,{\\rm log}\\det(B)\\}^{1+\\delta}) .\n\\end{eqnarray}\n\n\n\n\\item\n\\textbf{Probability of success:}\n\n\\noindent\nNow we would like to lower-bound the probability of the algorithm succeeding.\nBy our choice of $s = T \\cdot \\eta_{\\varepsilon}(L)$, \nand Lemma \\ref{lem:snf} we have: $s \\geq \\eta_{\\varepsilon}(L_{SNF})$.\nHence, by Fact \\ref{fact:1} we have that \n\\begin{eqnarray}\n\\Delta(y_i , U[(NL^*)_N]) \\leq \\varepsilon.\n\\end{eqnarray}\nBy equation \\ref{eq:nl} this implies that $\\Delta(a_i, U[N]) \\leq \\varepsilon$,\nand since the $a_i$'s are independent then\n\\begin{eqnarray}\n\\Delta((a_1,\\hdots,a_m), U[N^m]) \\leq 1 - (1-\\varepsilon)^m \\leq m^{-3},\n\\end{eqnarray}\nwhere the last inequality follows from our choice of $\\varepsilon = m^{-5}$.\nBy assumption, algorithm ${\\cal A}$ for ${\\rm SIS}(N)$ returns a correct answer w.p at least $1- m^{-3}$, for \na uniformly random vector $(a_1,\\hdots, a_m)$.\nTherefore\n\\begin{eqnarray}\n{\\sf{P}}({\\cal A} \\mbox{ succeeds } ) \\geq 1 - m^{-3} -  m^{-3} = 1-{\\rm O}(m^{-3}). \n\\end{eqnarray}\nNext, we would like to make sure that ${\\cal A}$ is not \"adversarial\", and given $c$, returns\n$\\alpha_1,\\hdots, \\alpha_m$ whose sum is different than $c$ almost all the time.\nUsing Fact \\ref{fact:2}, for any $c_1,c_2\\in [N]$ we have:\n\\begin{eqnarray}\\label{eq:ratio}\n\\Delta \\left( {\\sf{P}}(\\alpha_1,\\hdots,\\alpha_m | c_1) ,  {\\sf{P}}(\\alpha_1,\\hdots,\\alpha_m | c_2) \\right) \\leq m^{-3}.\n\\end{eqnarray}\nHence, given a choice of $c$ by the algorithm,\nthe probability that the same $c$ is going to be the value of $\\sum_i \\alpha_i$ is lower-bounded by:\n\\begin{eqnarray}\n{\\sf{P}}\\left( \\sum_{i=1}^m \\alpha_i = c | c\\right) \\geq \\max_{c'} {\\sf{P}}\\left(\\sum_i \\alpha_i = c'\\right) - m^{-3} = \\Omega\\left(\\frac{1}{m^{1 + \\delta}}\\right),\n\\end{eqnarray}\nwhere the inequality follows because\n$|\\alpha_i| \\leq 2m^{\\delta}$ for each $i$, and so $\\left|\\sum_i \\alpha_i \\right| \\leq 2m^{1 + \\delta} $.\nBy the union bound we thus have:\n\\begin{eqnarray}\n{\\sf{P}}({\\cal B} \\mbox{ succeeds }) =\n{\\sf{P}} \\left({\\cal A} \\mbox{ succeeds } \\wedge \\sum_{i=1}^m  \\alpha_i= c\\right)\n=\n\\Omega(m^{-1-\\delta}) - O(m^{-3})= \\Omega(m^{-2}).\n\\end{eqnarray}\n\\end{proof}\n\n\n\n\\begin{fact}\\label{fact:2}\nFor any $c_1,c_2\\in [N]$:\n$$\n\\Delta\n\\left(\n{\\sf{P}}(\\alpha_1,\\hdots,\\alpha_m | c_1), \n{\\sf{P}}(\\alpha_1,\\hdots,\\alpha_m | c_2)\n\\right) \n\\leq m^{-3}.\n$$\n\\end{fact}\n\\begin{proof}\nSince $y_i$ is uniquely determined from $x_i$ for each $i$, and $(\\alpha_1,\\hdots,\\alpha_m)$ is determined (possibly probabilistically) from the pairs \n$(y_i,x_i)$ for all $i\\in [n]$ then it is sufficient to show that for any $c_1\\neq c_2$, we have\n\\begin{eqnarray}\n\\Delta\n\\left(\n{\\sf{P}}(x_1,\\hdots,x_m | c_1), \n{\\sf{P}}(x_1,\\hdots,x_m | c_2)\n\\right) \n\\leq m^{-3}.\n\\end{eqnarray}\nBy definition, we have:\n\\begin{eqnarray}\n{\\sf{P}}(x_1,\\hdots, x_m | c) \\propto \\prod_{i=1}^m e^{-\\pi \\|x_i - v c - u_{rand}\\| \/ s^2}.\n\\end{eqnarray}\nSince $u_{rand}$ is uniform on $L_N$, \nand chosen independently of $c$, then it is sufficient to consider the a-posteriori probabilities\nof the vectors $x_i$ modulo ${\\cal P}(L)$ for each $i$:\n\\begin{eqnarray}\n{\\sf{P}}(x_1^L ,\\hdots,x_m^L  | c), \\ \\ x_i^L := x_i\\  \\mbox{mod} {\\cal P}(L).\n\\end{eqnarray}\nFinally, since the $x_i$'s are independent given $c$ then:\n\\begin{eqnarray}\n{\\sf{P}}(x_1^L,\\hdots,x_m^L | c)\n=\n\\prod_{i=1}^m {\\sf{P}}( x_i^L |c)\n\\end{eqnarray}\nHence, for $c_1\\neq c_2$ we have:\n\\begin{eqnarray}\\label{eq:allx}\n\\Delta\n\\left(\n{\\sf{P}}(x_1,\\hdots,x_m | c_1), \n{\\sf{P}}(x_1,\\hdots,x_m | c_2)\n\\right) \n=\n\\Delta \\left(\n\\prod_{i=1}^m {\\sf{P}}( x_i^L |c_1) ,\n\\prod_{i=1}^m {\\sf{P}}( x_i^L |c_2)\n\\right).\n\\end{eqnarray}\nSince $s \\geq \\eta_{\\varepsilon}(L_{SNF})$ then by Fact \\ref{fact:mr} and the triangle inequality we have:\n\\begin{eqnarray}\n\\forall i\\in [m] \\ \\ \n\\Delta \\left( {\\sf{P}}( x_i^L |c_1), {\\sf{P}}( x_i^L |c_2) \\right) \\leq 2\\varepsilon\n\\end{eqnarray}\nwhich implies by Equation \\ref{eq:allx}\n\\begin{eqnarray}\n\\Delta\n\\left(\n{\\sf{P}}(x_1,\\hdots,x_m | c_1), \n{\\sf{P}}(x_1,\\hdots,x_m | c_2)\n\\right) \n\\leq\n1 - (1-2\\varepsilon)^m\n= O(m^{-4}),\n\\end{eqnarray}\nwhere the last equality follows from our choice $\\varepsilon = m^{-5}$.\n\\end{proof}\n\n\\noindent\nWe note that by a simple modification of the algorithm above one can obtain a reduction from approximate ${\\rm SIVP}$ to $\\rm rSIS(N)$:\n\\begin{theorem}\nGiven is an integer matrix $B$.\nLet $\\Phi$ denote the number defined by: $\\eta_{\\varepsilon}(L) \\leq \\Phi$ for $\\varepsilon = m^{-5}$.\nSuppose that ${\\cal A}$ returns w.p. at least $1-m^{-3}$ a solution to\n$\\rm rSIS(N,\\delta)$.  \nThere exists an algorithm ${\\cal B}$ such that ${\\cal B}^{\\cal A}$ returns w.p. $1 - o(1)$ \na vector $x_{out}\\in L$, such that\n$$\n\\|x_{out} \\| \\leq \\Phi \\cdot {\\rm O}(n^{1.5+\\delta} \\max\\{n,{\\rm log}\\det(B)\\}^{1+\\delta}).\n$$\n\\end{theorem}\n\\begin{proof}\nWe slightly modify algorithm ${\\cal B}$ as follows: we set $v_{target} = u_{rand} = v= 0$ in step (3).\nIn Step (6) we terminate successfully whenever ${\\cal A}$ succeeds (i.e. regardless of the sum of coefficients).\nSimilar to Equation \\ref{eq:out} we have:\n\\begin{eqnarray}\nx_{out}\\in L, \n{\\sf{P}}( \\|x_{out}\\| \\leq \\Phi \\cdot {\\rm O}( n^{1.5+\\delta} \\max\\{n,{\\rm log}\\det(B)\\}^{1+\\delta})) = 1 -o(1).\n\\end{eqnarray}\nHence, sampling $n$ such vectors $x_{out}$ independently returns w.h.p. a set of $n$ linearly-independent vectors,\nwith the desired approximation ratio.\n\\end{proof}\n\n\n\n\\section{Acknowledgements}\nThe authors are thankful to \nOded Regev and Vinod Vaikuntanathan for their important insights regarding ${\\rm SIS}$ and to\nthe following people for useful discussion and insightful comments:  \nShafi Goldwasser, Sean Hallgren, David Jerison,\nDaniele Micciancio, Noah Stephens-Davidowitz.\nLE is thankful to the Templeton foundation for their support of this work.\nPWS was supported by the US Army Research Laboratory's Army Research Office through grant number W911NF-12-1-0486, the NSF through grant number CCF-121-8176, and by the NSF through the STC for Science of\nInformation under grant number CCF0-939370.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{rts54q43}\n\n\n\nThe classical combinatorics deals with finite structures, e.g.\\ cardinalities of\nfinite sets, which, in particular, may represent populations. This brings clear\nintuitive interpretations of numerous combinatorial objects, especially in their\nprobabilistic applications, see e.g.\\ {\\cite[Chapter~II]{Feller}}. The state\nspace is hence the set of nonnegative integers, $\\mathbb N_0=\\{0,1,2,\\dots\\}$. \nDefined initially for numbers from $\\mathbb N_0$, many combinatorial objects admit natural extensions to  a variable from $\\mathbb F=\\mathbb R$ or $\\mathbb C$. This leads to important relations between combinatorics and analysis, in particular,   difference calculus.\n \n When studying population dynamics in biology or ecology, one is often interested not only in the size of the population in a certain region but also how the population is spatially distributed, see e.g.\\  \\cite{DLM}. Let $X$ denote the space in which the population is located. For a sufficient generality, we assume that $X$ is a locally compact Polish space. The spatial distribution of a population in $X$ is modelled by a configuration $\\gamma$, a locally finite subset of $X$. We denote by $\\Gamma(X)$ the space of all configurations $\\gamma$ in $X$. \n \n \nWe note that configuration spaces are widely  used in different branches of mathematics and applications. We only mention the theory of point processes (e.g.\\ \\cite{DVJ}), statistical physics of continuous systems (e.g.\\ \\cite{Preston}), geometry and topology (e.g.\\ \\cite{AKR,FH}), spatial ecology (e.g.\\ \\cite{DLM}).\n Contrary  to $\\mathbb N_0$, the configuration space $\\Gamma(X)$ possesses both continuous topological properties (arising from the topology in $X$)\n  and  the discrete structure of each particular configuration.\\footnote{Although a configuration is a locally finite set, the total number of points of a configuration can be infinite. This feature is important for many branches of mathematics, in particular, for the theory of point processes and statistical physics of continuous systems.} \n  \n Having in mind the spatial distribution of a population, we may naturally think about extending certain notions and results of the classical combinatorics to the configuration space $\\Gamma(X)$. The aim of this paper is to show that  Stirling numbers and many related results have their natural counterparts in the  spatial combinatorics. In turn, our results within the spatial combinatorics cast new light on the classical combinatorics related to Stirling numbers.\n\nLet us now briefly describe what we mean under a spatial counterpart of  Stirling numbers. Let $n\\in\\mathbb N=\\{1,2,3,\\dots\\}$. For $k\\in\\mathbb N_0$, $(k)_n=\\frac1{n!}\\binom kn= k(k-1)\\dotsm(k-n+1)$ is called a {\\it falling factorial}.  The latter expression allows one to define  falling factorials as polynomials of a variable $z\\in\\mathbb F$: $(z)_n=z(z-1)\\dotsm(z-n+1)$. The {\\it Stirling numbers of the first kind}, $s(n,k)$, are defined through the expansion $(z)_n=\\sum_{k=1}^ns(n,k)z^k$, while the {\\it Stirling numbers of the second kind}, $S(n,k)$, are defined through the inverse expansion $z^n=\\sum_{k=1}^nS(n,k)(z)_k$. We refer e.g.\\ to the monographs \\cite{QG,MS} and the references therein for studies of  Stirling numbers in combinatorics and their applications in mathematical physics.\n \n It is standard to interpret a configuration $\\gamma=\\{x_i\\}\\in\\Gamma(X)$ as a discrete  measure on $X$: $\\gamma=\\sum_{i}\\delta_{x_i}$ where $\\delta_{x_i}$ is the Dirac measure with mass at $x_i$. Then, for $n\\in\\mathbb N$, one can naturally define a {\\it (spatial) falling factorial} $(\\gamma)_n$ as the discrete measure on $X^n$ given by\n \\begin{equation}\\label{rts5w53w}\n (\\gamma)_n:=\\sum_{x_1\\in\\gamma}\\sum_{x_2\\in\\gamma\\setminus\\{x_1\\}}\\dotsm\\sum_{x_n\\in\\gamma\\setminus\\{x_1,\\dots,x_{n-1}\\}}\\delta_{x_1}\\otimes\\delta_{x_2}\\otimes\\dots\\otimes\\delta_{x_n},\\end{equation}\n see \\cite{FKLO}. \n If $A\\subset X$ is compact and $\\gamma(A)=k\\in\\mathbb N_0$ (i.e., the configuration $\\gamma$ has $k$ points in $A$), then $(\\gamma)_n(A^n)=(k)_n$\\,.\nSimilarly to the classical case, one can now extend the definition of the falling factorials $(\\gamma)_n$ to a linear space.  More precisely, let $M(X)$ denote the space of $\\mathbb F$-valued Radon measures on $X$. For each $\\omega\\in M(X)$,  we define an $\\mathbb F$-valued Radon measure $(\\omega)_n$ on $X^n$ by\\footnote{In the case where $X$ is a smooth manifold, the definition of the falling factorials can be further extended to the case where $\\omega$ is a generalized function on $X$,  see \\cite{BKKL,FKLO,KunaInfusino}. This brings further connections with infinite dimensional analysis. However, in this paper, we will not discuss this wider extension.} \n\\begin{align}\n&(\\omega)_n(dx_1\\dotsm dx_n):=\\omega(dx_1)\\big(\\omega(dx_2)-\\delta_{x_1}(dx_2)\\big)\\notag\\\\\n&\\qquad\\times\\dotsm\\times\\big(\\omega(dx_n)-\\delta_{x_1}(dx_n)-\\delta_{x_2}(dx_n)-\\dots-\\delta_{x_{n-1}}(dx_n)\\big).\\label{xesa332a}\\end{align}\nEach $(\\omega)_n$ is a symmetric measure, i.e., it remains invariant under the natural action of the symmetric group on $X^n$. \n \n\n For $n\\in\\mathbb N$, let $\\mathcal F^{(n)}(X)$ denote the space of measurable, bounded, compactly supported, symmetric functions $f^{(n)}:X^n\\to\\mathbb F$. We define {\\it Stirling operators of the first kind} as the linear operators \n $\\mathbf s(n,k):\\mathcal F^{(n)}(X)\\to\\mathcal F^{(k)}(X)$ satisfying\n\\begin{equation}\\label{cxta5wq5y43w}\n\\int_{X^n}f^{(n)}\\,d(\\omega)_n=\\sum_{k=1}^n\\int_{X^k}\\mathbf s(n,k)f^{(n)}\\,d\\omega^{\\otimes k},\\quad\\text{for all }\\omega\\in M(X),\\end{equation}\n and {\\it Stirling operators of the second kind} as the linear operators \n $\\mathbf S(n,k):\\mathcal F^{(n)}(X)\\to\\mathcal F^{(k)}(X)$ satisfying\n\\begin{equation}\\label{cde5w35}\n\\int_{X^n}f^{(n)}\\,d\\omega^{\\otimes n}=\\sum_{k=1}^n\\int_{X^k}\\mathbf S(n,k)f^{(n)}\\,d(\\omega)_k, \\quad\\text{for all }\\omega\\in M(X).\\end{equation}\nIn the case where the underlying space $X$ has a single point (hence the whole population is located at this point), we may obviously identify the operators $\\mathbf s(n,k)$ and $\\mathbf S(n,k)$ with the Stirling numbers $s(n,k)$ and $S(n,k)$, respectively. In the case where $X=\\{x_1,\\dots,x_n\\}$ with $n\\in\\mathbb N$, $n\\ge2$,  the Stirling operators $\\mathbf s(n,k)$ and $\\mathbf S(n,k)$ act in finite-dimensional spaces, hence they are multivariate extensions of the corresponding Stirling numbers.\n\nLet us now describe the content of the paper. In Section \\ref{sa45wq43q}, we discuss the preliminaries. In particular, following \\cite{FKLO}, we discuss the binomial property of the falling factorials $(\\omega)_n$ and their lowering operators. \n\nIn Section~\\ref{sa4t2q}, we discuss basic results on the Stirling operators. These include the explicit formulas for the action of the Stirling operators, the recurrence relations satisfied by the Stirling operators, and the explicit form of their generating functions. We also introduce  {\\it Lah operators}, which connect the rising factorials, $(\\omega)^{(n)}=(-1)^n(-\\omega)_n$\\,, with the falling factorials, $(\\omega)_n$. We derive the explicit form of the Lah operators and their generating function. \n\nIn Section \\ref{few53q}, we discuss an infinite dimensional counterpart of Euler's formula for the Stirling operators of the second kind,  compare with \\cite[Section 9.1]{QG}. The proof of our result uses the binomial property of the falling factorials, $(\\omega)_n$.  We also discuss how the infinite dimensional Euler formula is related to the correlation measure of a point process in $X$. \n\nOlson's identity (see e.g.\\ \\cite[Section 12.2]{QG}) is a generalization of the orthogonality identity satisfied by the Stirling numbers. In Section~\\ref{tes5w}, by using  Euler's formula from Section \\ref{few53q}, we prove an infinite dimensional counterpart of Olson's identity.  Even in the case of a single-point space $X$, the obtained identity yields an extension of the classical Olson's identity for  Stirling numbers. We also discuss a couple of other identities satisfied by the Stirling operators. The proofs of these identities significantly use the binomial property of  $(\\omega)_n$. \n\nOur studies of the Stirling operators naturally lead us to a generalization of  Poisson point process. Recall that, if $\\sigma$ is a non-atomic positive Radon measure on $X$, one can define the Poisson point process, with intensity measure $\\sigma$ as a probability measure on the configuration space $\\Gamma(X)$. We denote by $\\mathbb E_\\sigma$ the expectation with respect to this probability measure. Let $p$ be a polynomial on $M(X)$, i.e., a function $p:M(X)\\to\\mathbb F$ which is of the form \n\\begin{equation}\\label{dftwfedt}\np(\\omega)=f^{(0)}+\\sum_{k=1}^n\\int_{X^k}f^{(k)}\\,d\\omega^{\\otimes k},\\end{equation}\nwhere $f^{(0)}\\in\\mathbb F$ and $f^{(k)}\\in\\mathcal F^{(k)}(X)$ for $k=1,\\dots,n$. \nThen $p$ is integrable  and we can evaluate its expectation, $\\mathbb E_\\sigma(p)$. If we denote by $\\mathcal P(M(X))$ the space of all polynomials on $M(X)$,  the expectation \n$\\mathbb E_\\sigma$ determines a linear functional $\\mathbb E_\\sigma:\\mathcal P(M(X))\\to\\mathbb F$. In Appendix, we prove that such a functional can be naturally defined for any $\\mathbb F$-valued Radon measure $\\sigma\\in M(X)$. We call $\\mathbb E_\\sigma$ the {\\it Poisson functional with intensity measure} $\\sigma$. We prove that several standard properties of the Poisson point process (for example, Mecke identity, see Proposition~\\ref{g6ew4}) still hold for the Poisson functional $\\mathbb E_\\sigma$. The main result of Section~\\ref{ufxdqyr} (Theorem~\\ref{e6ew63}) states a connection between the Poisson functional $\\mathbb E_\\sigma$ and the Stirling operators of the second kind. More precisely, for a monomial $p(\\omega)=\\int_{X^n}f^{(n)}\\,d\\omega^{\\otimes n}$, it holds that \n$\\mathbb E_\\sigma(p)=\\sum_{k=1}^n\\int_{X^k}\\mathbf S(n,k)f^{(n)}\\,d\\sigma^{\\otimes k}$.\n\nLet $a^+$, $a^-$ be a pair of (adjoint) operators satisfying the canonical commutation relation \n\\begin{equation}\\label{xzzaaaa}\na^-a^+=a^+a^-+1,\\end{equation}\nwhere  $a^+$ is called a {\\it creation operator} and $a^-$ an {\\it annihilation operator}. The operator $\\rho=a^+a^-$ is called a {\\it number operator}, or a {\\it particle density}. Due to the commutation relation, the operator $\\rho^n$ can be represented as a linear combination of the {\\it Wick (normally) ordered operators}, $(a^+)^k(a^-)^k$. Katriel \\cite{Katriel} in 1974 found the explicit formula, $\\rho^n=\\sum_{k=1}^nS(n,k)(a^+)^k(a^-)^k$. Katriel's result includes, as a special case,  the classical formula of Gr\\\"unert (1843) for the action of the Euler operator $t\\frac d{dt}$,\n$$\\bigg(t\\frac{d}{dt}\\bigg)^n=\\sum_{k=1}^nS(n,k)\\,t^k\\bigg(\\frac{d}{dt}\\bigg)^k.$$ \n The seminal paper \\cite{Katriel} led to numerous generalizations of  Stirling numbers; we refer to \\cite[Subsection~ 1.2.3]{MS} for a long list of references. The main result of Section~\\ref{gwfdytwyc} (Theorem~\\ref{taw4aq234q}) is an extension of Katriel's formula to the case of a family of operators $(a^+(x),a^-(x))_{x\\in X}$ satisfying the canonical commutation relations, see formula~\\eqref{r4wqa4y} below. In fact, it is only in the case where $X$ is a discrete set that the operators $a^+(x)$ and $a^-(x)$ are well-defined. In the general case, these are operator-valued distributions, hence one needs to choose a reference measure $\\sigma$ on $X$ and consider smeared operators $a^+(\\varphi) =\\int_X\\varphi(x)a^+(x)\\sigma(dx)$, $a^-(\\varphi)=\\int_X\\varphi(x)a^-(x)\\sigma(dx)$, and $\\rho(\\varphi)=\\int_X\\varphi(x)a^+(x)a^-(x)\\sigma(dx)$. \nTheorem~\\ref{taw4aq234q} states the explicit formula for the product $\\rho(\\varphi_1)\\dotsm\\rho(\\varphi_n)$ in terms of the Wick ordered terms. This formula uses the Stirling operators $\\mathbf S(n,k)$ at the place of the Stirling numbers $S(n,k)$ in Katriel's formula. We also consider an infinite dimensional generalization of Gr\\\"unert's formula (Theorem~\\ref{dwq3q32q}), which appears to be a special case of the formula from Theorem~\\ref{taw4aq234q}. \n\nIn Theorem~\\ref{y6weu43}, for an arbitrary measure $\\sigma\\in M(X)$,  we consider a certain  representation of the canonical commutation relations and a {\\it vacuum functional} $\\tau_\\sigma$ on the commutative unital algebra generated by the particle densities  $\\rho(\\varphi)$ and the identity operator. We show that the functional $\\tau_\\sigma$ can be naturally identified with the Poisson functional with intensity measure $\\sigma$. This extends the available results on the Fock space realization of the classical Poisson point process (e.g.\\ \\cite{GGPS,HP,Surgailis}) to the case of an $\\mathbb F$-valued intensity measure $\\sigma$.\n\nThe classical {\\it Touchard (or exponential) polynomials} are given by \n$$T_n(z)=\\sum_{k=1}^nS(n,k)z^k,$$\n while $B_n=T_n(1)$ are called the {\\it Bell numbers}. ($B_n$ counts the number of all unordered partitions of a set of $n$ elements.)\nIn Section~\\ref{ydrsdese3aqw}, for each $n\\in\\mathbb N$ and $\\omega\\in M(X)$,  we define an $\\mathbb F$-valued Radon measure $T_n(\\omega)$ on $X^n$ that satisfies \n$$\\int_{X^n}f^{(n)}\\,dT_n(\\omega)=\\sum_{k=1}^n\\int_{X^k}\\mathbf S(n,k)f^{(n)}\\,d\\omega^{\\otimes k}\\quad\\text{for all }f^{(n)}\\in \\mathcal F^{(n)}(X).$$\nWe call $T_n$ {\\it Touchard (or exponential)  polynomials on} $M(X)$.  Comparing the definition of $T_n(\\omega)$ with Theorem~\\ref{e6ew63} gives us an immediate connection between $T_n(\\omega)$ and the Poisson functional~$\\mathbb E_\\omega$ (Corollary~\\ref{sesawawqa2q}). We prove several further properties of $T_n(\\omega)$, including their explicit form, connections with the infinite dimensional Gr\\\"unert's formula, the binomial property, recurrence formulas, and the explicit form of their generating function.\nIf $\\nu$ is a probability measure on $X$, we call $B_n(\\nu)=T_n(\\nu)$ the {\\it $n$th Bell measure corresponding to~$\\nu$}. Proposition~\\ref{qwdwswxe} shows that $B_n(\\nu)$ is related to the expectation of a certain infinite sum involving an infinite sequence of independent random variables that have distribution~$\\nu$.\n\nFinally, in Section \\ref{xzrwaq4yq4q}, we discuss an open problem related to Theorem~\\ref{taw4aq234q}. \n\n\n\\section{Preliminaries}\\label{sa45wq43q}\n\n\\subsection{Polynomials on Radon measures}\n\nLet $X$ be a locally compact Polish space. Let $\\mathcal B(X)$ denote the Borel $\\sigma$-algebra on $X$ and let $\\mathcal B_0(X)$ denote the collection of all precompact sets from $\\mathcal B(X)$. Recall that a positive Radon measure $\\sigma$ on  $(X,\\mathcal B(X))$ is  a  measure satisfying $\\sigma(A)<\\infty$ for all $A\\in\\mathcal B_0(X)$; a real-valued (signed) Radon measure $\\omega$ on $X$ has the form $\\omega=\\sigma_1-\\sigma_2$, where $\\sigma_1$ and $\\sigma_2$ are positive Radon measures; and a complex-valued Radon  measure $\\omega$ on $X$ has the form $\\omega=\\omega_1+i\\omega_2$, where $\\omega_1$ and $\\omega_2$ are real-valued  Radon measures. Note that, generally speaking,  a real-valued or complex-valued Radon measure  is well-defined on $\\mathcal B_0(X)$ only.\nLet $\\mathbb F$ denote either $\\R$ or $\\mathbb C$, and we denote by  $M(X)$ the set of all $\\mathbb F$-valued Radon measures on $X$. \n\nA configuration $\\gamma$ in $X$ is a subset of $X$ that contains only a finite number of elements in each compact set $K\\subset X$. In particular, the set $\\gamma$ is either  finite (possibly empty) or countable. We denote by $\\Gamma(X)$ the collection of all configurations in $X$.  By identifying each configuration $\\gamma=\\{x_i\\}$ with the (positive) Radon measure $\\gamma=\\sum_i\\delta_{x_i}$, we get the inclusion\n$\\Gamma(X)\\subset M(X)$.\n\nWe denote by $M^{(n)}(X)$ the \nset of all symmetric Radon measures on $X^n$.  Obviously, for each $\\omega\\in M(X)$, the product measure $\\omega^{\\otimes n}$ belongs to $M^{(n)}(X)$. Let also $M^{(1)}(X):=M(X)$, $M^{(0)}(X):=\\mathbb F$ and $\\omega^{\\otimes 0}:=1$ for $\\omega\\in M(X)$.\n\nWe denote by $\\mathcal F(X)$ the space of all $\\mathbb F$-valued bounded measurable functions on $X$ with compact support, and by $\\mathcal{CF}(X)$ the space of all $\\mathbb F$-valued continuous functions on $X$ with compact support.\nFor each $n\\ge 2$, we similarly define the spaces $\\mathcal F(X^n)$ and $\\mathcal {CF}(X^n)$. We denote by $\\mathcal F^{(n)}(X)$ and $\\mathcal{CF}^{(n)}(X)$ the spaces of all symmetric functions from $\\mathcal F(X^n)$ and $\\mathcal {CF}(X^n)$, respectively. We also denote $\\mathcal F^{(1)}(X):=\\mathcal F(X)$, $\\mathcal{CF}^{(1)}(X):=\\mathcal{CF}(X)$ and $\\mathcal F^{(0)}(X):=\\mathbb F$.\n\n\nFor $\\mu^{(n)}\\in M^{(n)}(X)$ and $f^{(n)}\\in\\mathcal F^{(n)}(X)$, we denote\n\\begin{equation}\\label{dera4tq2y4}\n\\langle\\mu^{(n)},f^{(n)}\\rangle:=\\int_{X^n}f^{(n)}\\,d\\mu^{(n)},\\quad n\\in\\mathbb N,\\end{equation}\nand $\\langle\\mu^{(0)},f^{(0)}\\rangle:=\\mu^{(0)}f^{(0)}$ for $n=0$.  \n\n\nLet $\\mathfrak S_n$ denote the symmetric group of degree $n$.\nFor a function $f^{(n)}:X^n\\to\\mathbb F$, we denote by $P_nf^{(n)}$ the  symmetrization of $f^{(n)}$,\n$$(P_nf^{(n)})(x_1,\\dots,x_n):=\\frac1{n!}\\sum_{\\pi\\in \\mathfrak S_n}f(x_{\\pi(1)},\\dots,x_{\\pi(n)}).$$\n (For $n=1$, $P_1$ is just the identity.)\n\nFor functions $f^{(n)}:X^n\\to\\mathbb F$ and $g^{(m)}:X^m\\to\\mathbb F$, we define $f^{(n)}\\otimes g^{(m)}:X^{n+m}\\to\\mathbb F$ by $(f^{(n)}\\otimes g^{(m)})(x_1,\\dots,x_{n+m}):=f^{(n)}(x_1,\\dots,x_n)g^{(m)}(x_{n+1},\\dots,x_{n+m})$. \n For any $f^{(n)}\\in\\mathcal F^{(n)}(X)$ and $g^{(m)}\\in\\mathcal F^{(m)}(X)$, the symmetric tensor product of $f^{(n)}$ and $g^{(m)}$ is the function \n$$f^{(n)}\\odot g^{(m)}:=P_{n+m}(f^{(n)}\\otimes g^{(m)})\\in\\mathcal F^{(n+m)}(X).$$\n Obviously, if $f^{(n)}\\in\\mathcal {CF}^{(n)}(X)$ and $g^{(m)}\\in\\mathcal {CF}^{(m)}(X)$, then $f^{(n)}\\odot g^{(m)}\\in\\mathcal {CF}^{(n+m)}(X)$. \n \n Similarly, for $\\mu^{(n)}\\in M^{(n)}(X)$ and $\\nu^{(m)}\\in M^{(m)}(X)$, we define the symmetric product measure $\\mu^{(n)}\\odot\\nu^{(m)}\\in M^{(n+m)}(X)$. This measure can be characterized as the unique element of $M^{(n+m)}(X)$ that satisfies\n$$\\langle \\mu^{(n)}\\odot\\nu^{(m)} ,f^{(n+m)}\\rangle=\\langle \\mu^{(n)}\\otimes\\nu^{(m)} ,f^{(n+m)}\\rangle\\quad \\text{for all }f^{(n+m)}\\in\\mathcal F^{(n+m)}(X). $$\n\nBy the polarization identity, each $\\mu^{(n)}\\in M^{(n)}(X)$ is uniquely characterized by the values $\\langle \\mu^{(n)},\\xi^{\\otimes n}\\rangle$ with $\\xi\\in \\mathcal {F}(X)$.  Note that, for $\\mu^{(0)}\\in M^{(0)}(X)=\\mathbb F$ and $\\xi\\in\\mathcal F(X)$,  $\\langle \\mu^{(0)},\\xi^{\\otimes 0}\\rangle=\\mu^{(0)}$.\n\n\n \nA {\\it polynomial on $M(X)$} is a function $p:M(X)\\to\\mathbb F$ of form \\eqref{dftwfedt}. Using the notation~\\eqref{dera4tq2y4}, we may write formula \\eqref{dftwfedt} as follows:\n\\begin{equation}\\label{tew53w5}\np(\\omega)=\\sum_{k=0}^n\\langle \\omega^{\\otimes k},f^{(k)}\\rangle.\\end{equation}\nIf $f^{(n)}\\ne 0$, we say that $p$ is a polynomial of degree $n$. Recall that $\\mathcal P(M(X))$ denotes the space of all polynomials on $M(X)$.\n\n The space $M(X)$ can be naturally equipped with the vague topology.\\footnote{The vague topology on $M(X)$ is the minimal topology on $M(X)$ such that, for each $f\\in\\mathcal {CF}(X)$, the  mapping $M(X)\\ni\\omega\\mapsto\\langle\\omega,f\\rangle\\in\\mathbb F$ is continuous.} Thus, if in formula \\eqref{tew53w5}, $f^{(k)}$  belongs to $\\mathcal {CF}^{(k)}(X)$ for each $k=1,\\dots,n$, then $p$ is a continuous function on $M(X)$. This is why the subset of $\\mathcal P(M(X))$ consisting of all polynomials of form \\eqref{tew53w5} with $f^{(k)}\\in \\mathcal {CF}^{(k)}(X)$ will be called the {\\it space of continuous polynomials on $M(X)$} and denoted by $\\mathcal {CP}(M(X))$. \n\n\n\\subsection{Falling and rising factorials on Radon measures}\n\nFollowing \\cite{FKLO} (in particular, Subsections 5.1 and 5.2), we will now \nrecall some basic properties of the falling and rising factorials on $M(X)$.\n For each $n\\in\\mathbb N$, the {\\it falling factorial on $M(X)$ of degree $n$} is defined to be the mapping\n$M(X)\\ni\\omega\\mapsto (\\omega)_n\\in M^{(n)}(X)$\ngiven by \\eqref{xesa332a}. (It is not difficult to see that the  Radon measure on the right hand side of formula \\eqref{xesa332a} is indeed symmetric.) Similarly, for each $n\\in\\mathbb N$, the {\\it rising factorial on $M(X)$ of degree $n$} is defined to be the mapping\n$M(X)\\ni\\omega\\mapsto (\\omega)^{(n)}\\in M^{(n)}(X)$\ngiven by\n\\begin{align}\n&(\\omega)^{(n)}(dx_1\\dotsm dx_n):=\\omega(dx_1)\\big(\\omega(dx_2)+\\delta_{x_1}(dx_2)\\big)\\notag\\\\\n&\\qquad\\times\\dotsm\\times\\big(\\omega(dx_n)+\\delta_{x_1}(dx_n)+\\delta_{x_2}(dx_n)+\\dots+\\delta_{x_{n-1}}(dx_n)\\big).\\label{fdeseas}\\end{align}\nClearly, \n\\begin{equation}\\label{ctsrea4w}\n(\\omega)^{(n)}=(-1)^n(-\\omega)_n,\\quad n\\in\\mathbb N.\\end{equation}\nWe will also  use the notation $(\\omega)_0=(\\omega)^{(0)}:=1$.\n\n\nIt follows directly from formulas \\eqref{xesa332a} and \\eqref{fdeseas} that, for each  $\\omega\\in M(X)$ and any $A\\in\\mathcal B_0(X)$, \n\\begin{equation}\\label{s5w53w}\n(\\omega)_n(A^n)=(\\omega(A))_n,\\quad (\\omega)^{(n)}(A^n)=(\\omega(A))^{(n)}.\\end{equation}\nHere, for $z\\in\\mathbb F$, $(z)_n$ and $(z)^{(n)}=(-1)^n(-z)_n$ are the classical falling and rising factorials, respectively. \n\n\\begin{remark}\\label{fdryde65re4w}\nAssume that the underlying space $X$ has only a single point. In that case, we can identify each $M^{(n)}(X)$ with $\\mathbb F$ and formula \\eqref{s5w53w} means that $(\\omega)^{(n)}$ and $(\\omega)_n$ are the classical rising and falling factorials, respectively.\n\\end{remark}\n\nFor $\\omega\\in M(X)$ and $n\\in\\mathbb N$, we define the {\\it (spatial) binomial coefficient}\n$\\binom \\omega n:=\\frac1{n!}\\,(\\omega)_n$.\nFor each configuration $\\gamma=\\sum_{i\\ge1}\\delta_{x_i}\\in\\Gamma(X)$,\n\\begin{equation}\\label{reaq43q42}\n\\binom \\gamma n=\\sum_{\\{i_1,\\dots,i_n\\}} \\delta_{x_{i_1}}\\odot \\delta_{x_{i_2}}\\odot\\dots\\odot \\delta_{x_{i_n}},\\end{equation}\ncompare with \\eqref{rts5w53w}.\n\n\\begin{remark}\\label{c4q4q}\nIt follows from \\eqref{rts5w53w} or \\eqref{reaq43q42} that, if a configuration $\\gamma$ has strictly less than $n$ points, then $\\binom \\gamma n=(\\gamma)_n=0$.\n\\end{remark}\n\nBoth the falling factorials and the rising factorials have the binomial property:\n\\begin{align}\\label{5w53w2}\n(\\omega+\\sigma)_n&=\\sum_{k=0}^n\\binom nk(\\omega)_k\\odot(\\sigma)_{n-k},\\\\\n\\label{5w53w22} (\\omega+\\sigma)^{(n)}&=\\sum_{k=0}^n\\binom nk(\\omega)^{(k)}\\odot(\\sigma)^{(n-k)}, \\quad\\omega,\\sigma\\in M(X)\n\\end{align}\n Furthermore, the following lowering property holds: for each $n\\in\\mathbb N_0$, $\\omega\\in M(X)$, and $x\\in X$,\n\\begin{equation}\\label{s5wuwu}\n(\\omega+\\delta_x)_n-(\\omega)_n=n\\delta_x\\odot(\\omega)_{n-1},\\quad (\\omega)^{(n)}-(\\omega-\\delta_x)^{(n)}=n\\delta_x\\odot(\\omega)^{(n-1)}.\\end{equation}\n\nIt follows from \\eqref{xesa332a} and \\eqref{fdeseas}, that for each $f^{(n)}$ from $\\mathcal {F}^{(n)}(X)$ or $\\mathcal {CF}^{(n)}(X)$, the functions $\\langle(\\omega)_n,f^{(n)}\\rangle$ and  $\\langle(\\omega)^{(n)},f^{(n)}\\rangle$ belong to $\\mathcal P(M(X))$ or $\\mathcal {CP}(M(X))$, respectively.\nWith an abuse of terminology, we will sometimes call these polynomials {\\it falling and rising factorials}, respectively. The linear span of all falling (or all  rising) factorials forms the whole space $\\mathcal P(M(X))$.\n\n\nThe falling factorials satisfy the recurrence relation\n\\begin{gather}\n(\\omega)_0=1,\\notag\\\\\n\\langle(\\omega)_{n+1},\\xi^{\\otimes(n+1)}\\rangle=\\langle(\\omega)_{n},\\xi^{\\otimes n}\\rangle\\langle\\omega,\\xi\\rangle-n\\langle(\\omega)_n,\\xi^2\\odot\\xi^{\\otimes(n-1)}\\rangle,\\quad\\xi\\in\\mathcal F(X).\n\\label{fseraw4qa}\n\\end{gather}\nFrom here a similar recurrence relation for the rising factorials follows.\n\nThe following generating functions uniquely characterize the falling and rising factorials:\n\\begin{align}\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle(\\omega)_n,\\xi^{\\otimes n}\\rangle&=\\exp\\big[\\langle\\omega,\\log(1+\\xi)\\rangle\\big],\\label{dw5w}\\\\\n\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle(\\omega)^{(n)},\\xi^{\\otimes n}\\rangle&=\\exp\\big[\\langle\\omega,-\\log(1-\\xi)\\rangle\\big],\\quad\\xi\\in \\mathcal F(X).\\label{esa4q42}\\end{align}\nBoth formulas \\eqref{dw5w} and \\eqref{esa4q42} are understood as equalities of formal power series, see Subsection~2.2 and Appendix in \\cite{FKLO}. To obtain the  falling and rising factorials, one has to set in  formulas \\eqref{dw5w}, \\eqref{esa4q42} $\\xi=z\\psi$, where $\\psi\\in \\mathcal F(X)$ and $z\\in\\mathbb F$, formally expand  in powers of $z$, and equate the coefficients by each $z^n$. \n\nFor vector spaces $V$ and $W$, we denote by $\\mathcal L(V,W)$ the space of linear operators from $V$ into $W$. We also denote   $\\mathcal L(V):=\\mathcal L(V,V)$.\nFor each $x\\in X$, we define $\\partial_x,D_x\\in \\mathcal L(\\mathcal P(M(X)))$ by\n\\begin{align}\n\\partial_xp(\\omega)&=\\lim_{z\\to 0}\\frac{p(\\omega+z\\delta_x)-p(\\omega)}z,\\label{qdftyfe}\\\\\nD_xp(\\omega)&=p(\\omega+\\delta_x)-p(\\omega)\\label{r3q3q2sqg}\n\\end{align}\nfor each $p\\in \\mathcal P(M(X))$.  (The right-hand side of \\eqref{qdftyfe} is just the derivative of $p$ in direction $\\delta_x$.)\nIn particular, for each $f^{(n)}\\in \\mathcal F^{(n)}(X)$,\n\\begin{equation}\\label{ydsrqqsqwdx}\n\\partial_x\\langle\\omega^{\\otimes n},f^{(n)}\\rangle=n\\langle\\omega^{\\otimes(n-1)},f^{(n)}(x,\\cdot)\\rangle,\\end{equation}\nand by \\eqref{s5wuwu},\n\\begin{equation}\\label{rsa4q}\nD_x\\langle(\\omega)_n,f^{(n)}\\rangle=n\\langle(\\omega)_{n-1},f^{(n)}(x,\\cdot)\\rangle.\n\\end{equation}\nIt follows from \\eqref{dw5w} that\n\\begin{equation}\\label{cxrwq5w4}\n D_x=e^{\\partial_x}-1=\\sum_{n=1}^\\infty\\frac1{n!}\\,\\partial_x^n. \\end{equation}\nNote that for each $p\\in\\mathcal P(M(X))$ of degree $n$, we have $\\partial_x^kp=0$ for all $k\\ge n+1$, which justifies the use of the infinite series of powers of $\\partial_x$ in \\eqref{cxrwq5w4}.\n\n\\section{Stirling operators}\\label{sa4t2q}\n\nIn Section \\ref{rts54q43}, for $n,k\\in\\mathbb N$, $k\\le n$, we defined the Stirling operators of the first kind, $\\mathbf s(n,k)$, and the Stirling operators of the second kind, $\\mathbf S(n,k)$. Note that, using the notation~\\eqref{dera4tq2y4}, we may write formulas~\\eqref{cxta5wq5y43w} and~\\eqref{cde5w35} as follows:\n\\begin{align}\n\\langle (\\omega)_n,f^{(n)}\\rangle&=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},\\mathbf s(n,k)f^{(n)}\\rangle,\\label{yrdqs}\\\\\n\\langle \\omega^{\\otimes n},f^{(n)}\\rangle&=\\sum_{k=1}^n\\langle (\\omega)_k,\\mathbf S(n,k)f^{(n)}\\rangle.\\label{fdr6ew456u4w}\\end{align}\n Note that, for each polynomial $p$ on $M(X)$ of degree $n$, its representation as in formula~\\eqref{tew53w5} is unique, hence formula~\\eqref{yrdqs} indeed uniquely identifies the operators $\\mathbf s(n,k)$.\nSince $\\mathbf s(n,n)$ is the identity operator, we similarly conclude that formula~\\eqref{fdr6ew456u4w} indeed uniquely identifies the operators $\\mathbf S(n,k)$.\n\nWe further define the {\\it unsigned Stirling operators of the first kind}, $\\mathbf c(n,k)\\linebreak\\in\\mathcal L(\\mathcal F^{(n)}(X),\\mathcal F^{(k)}(X))$, through the formula  \n\\begin{equation}\n\\langle(\\omega)^{(n)},f^{(n)}\\rangle=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},\\mathbf c(n,k)f^{(n)}\\rangle.\\label{drtrehgy}\n\\end{equation}\n\n\n\\begin{remark} By \\eqref{ctsrea4w}, we obtain\n\\begin{equation}\\label{xfdyf}\n\\mathbf s(n,k)=(-1)^{n-k}\\mathbf c(n,k),\\end{equation}\nand\n$$\\langle\\omega^{\\otimes n},f^{(n)}\\rangle=\\sum_{k=1}^n\\langle(\\omega)^{(k)},(-1)^{n-k}\\,\\mathbf S(n,k)f^{(n)}\\rangle, \\quad f^{(n)}\\in\\mathcal F^{(n)}(X).$$\n\\end{remark}\n\n\n\n\n Each space $\\mathcal{CF}^{(n)}(X)$ ($n\\in\\mathbb N$) can be endowed with a natural locally convex topology, as a direct limit of the Banach spaces of $\\mathbb F$-valued continuous symmetric functions with  supports in  compact subsets of $X^n$, equipped with the supremum norm. Note that the operators $\\mathbf s(n,k)$, $\\mathbf c(n,k)$, and $\\mathbf S(n,k)$ belong to $\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$. Furthermore, it will follow from  Proposition~\\ref{a4w5qwgb} below that these operators are  continuous.\n\nThe linear topological space $\\mathcal{CF}^{(n)}(X)$ is Hausdorff and its dual space is  $M^{(n)}(X)$, see e.g.\\ \\cite[\\S 29]{Bauer} or \\cite[Chapter 7]{Simonnet}.\n  Let $A\\in\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$ be a linear  continuous operator. Then, by e.g.\\ \\cite[Theorem 8.11.3]{NB}, the operator $A$ has the adjoint operator $A^*\\in\\mathcal L(M^{(k)}(X),M^{(n)}(X))$, satisfying $\\langle \\mu^{(k)},Af^{(n)}\\rangle=\\langle A^*\\mu^{(k)},f^{(n)}\\rangle$ for all $\\mu^{(k)}\\in M^{(k)}(X)$ and $f^{(n)}\\in\\mathcal {CF}^{(n)}(X)$. Formulas \\eqref{yrdqs}--\\eqref{drtrehgy} then imply that, for $\\omega\\in M(X)$, \n$$(\\omega)_n=\\sum_{k=1}^n\\mathbf s(n,k)^*\\omega^{\\otimes k},\\quad(\\omega)^{(n)}=\\sum_{k=1}^n\\mathbf c(n,k)^*\\omega^{\\otimes k},\\quad \\omega^{\\otimes n}=\\sum_{k=1}^n\\mathbf S(n,k)^*(\\omega)_k.$$\n\n\n\n\n\\begin{remark}\\label{vyd6wq4a24}\n Let $A\\in\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$ be   a continuous linear operator, and let $A^*\\in\\mathcal L(M^{(k)}(X),M^{(n)}(X))$ be its adjoint. For each $\\mu^{(k)}\\in M^{(k)}(X)$, the measure $A^*\\mu^{(k)}\\in M^{(n)}(X)$ is completely identified by the integrals \n  $$\\langle A^*\\mu^{(k)},\\xi^{\\otimes n}\\rangle=\\langle \\mu^{(k)},A\\xi^{\\otimes n}\\rangle,$$\n   where $\\xi$ runs through $\\mathcal {CF}(X)$. This implies that the linear operator $A$ is completely identified by $A\\xi^{\\otimes n}$ where $\\xi$ runs through $\\mathcal {CF}(X)$. (Equivalently, the linear span of the set $\\{\\xi^{\\otimes n}\\mid\\xi\\in\\mathcal{CF}(X)\\}$ is dense in $\\mathcal{CF}^{(n)}(X)$.)\n\\end{remark}\n\n\n\\begin{remark}\\label{rtsdqwsuqwds6qe}\nFor technical reasons, it is useful to define operators $\\mathbf s(n,k)$, $\\mathbf c(n,k)$, and $\\mathbf S(n,k)$ from $\\mathcal L(\\mathcal F^{(n)}(X),\\mathcal F^{(k)}(X))$ to be zero \nif either $k>n\\ge0$ or $k=0<n$ and to be equal to 1 if $k=n=0$. \n\\end{remark} \n\n\n\n\n\nWe will now present explicit formulas for the action of the operators $\\mathbf c(n,k)$ and $\\mathbf S(n,k)$. Let $k,n\\in\\mathbb N$, $k\\le n$  and let $i_1,\\dots,i_k\\in\\mathbb N$ be such that $i_1+\\dots+i_k=n$. \nWe define $\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\in\\mathcal L(\\mathcal F^{(n)}(X),\\mathcal F^{(k)}(X))$ by\n$$\n(\\mathbb D^{(n)}_{i_1,\\dots,i_k}f^{(n)})(x_1,\\dots,x_k):=P_k\\big[f^{(n)}\\big(\\underbrace{x_1,\\dots,x_1}_{\\text{$i_1$ times}},\\dots,\\underbrace{x_k,\\dots,x_k}_{\\text{$i_k$ times}}\\big)\\big],\\quad f^{(n)}\\in\\mathcal F^{(n)}(X). $$\nNote that\n\\begin{equation}\\label{cre4a543w6}\n\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\xi^{\\otimes n}=\\xi^{i_1}\\odot\\dots\\odot\\xi^{i_k},\\quad \\xi\\in\\mathcal F(X).\n\\end{equation}\nObviously, we also have $\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\in\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$.  Note that these operators are continuous.\nChoosing $k=1$, gives the operator $\\mathbb D^{(n)}:=\\mathbb D^{(n)}_n$, \n\\begin{equation}\\label{cq5y42w}\n(\\mathbb D^{(n)}f^{(n)})(x)=f^{(n)}(x,\\dots,x).\n\\end{equation}\n\n\\begin{remark}\\label{se5wu563} If $X$ has only a single point, all the spaces $\\mathcal F^{(n)}(X)$ can be identified with $\\mathbb F$, in which case all operators $\\mathbb D^{(n)}_{i_1,\\dots,i_k}$ can be identified with number 1.\n\\end{remark}\n\n\\begin{remark}\n We will sometimes need an extension of $\\mathbb D^{(n)}$ to $\\mathcal F(X^n)$, which will act according to the same formula \\eqref{cq5y42w}.\n\\end{remark}\n\nNext, for $k,n\\in\\mathbb N$, $k\\le n$, we denote by $\\operatorname{UP}(n,k)$ the collection of all \nunordered  partitions of $\\{1,\\dots,n\\}$ into $k$ non-empty parts.  For any $\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in\\operatorname{UP}(n,k)$, we define $\\mathbb D^{(n)}_\\lambda\\in\\mathcal L(\\mathcal F(X^n),\\mathcal F^{(k)}(X))$ as follows: \nlet $f^{(n)}\\in \\mathcal F(X^n)$; in $f^{(n)}(x_1,\\dots,x_n)$ replace each $x_j$ with $y_i$ where $j\\in\\lambda_i$;  symmetrize the obtained function of the $y_1,\\dots,y_k$ variables; this function is $(\\mathbb D^{(n)}_\\lambda f^{(n)})(y_1,\\dots,y_k)$. Note that, although in our definition of $\\mathbb D^{(n)}_\\lambda$ we used an order of the elements $\\lambda_1,\\dots,\\lambda_k$ from \n$\\lambda$, the obtained function does not depend on this order. \n \n\\begin{remark} Let $\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in\\operatorname{UP}(n,k)$. We obviously have\n\\begin{equation}\\label{cs5w5a}\n\\mathbb D^{(n)}_\\lambda f^{(n)}=\\mathbb D^{(n)}_{|\\lambda_1|,\\dots,|\\lambda_k|}f^{(n)},\\quad f^{(n)}\\in\\mathcal F^{(n)}(X).\\end{equation} \nHere $|\\lambda_l|$ denotes the number of elements of the set $\\lambda_l$.  \nOn the other hand, the operator $\\mathbb D^{(n)}_\\lambda$ was defined for not necessarily symmetric functions, and we will use this fact below.\n\\end{remark}\n\n\\begin{proposition}\\label{a4w5qwgb} For any $n,k\\in\\mathbb N$, $k\\le n$, \n\\begin{align}\n\\mathbf c(n,k)&=\\frac{n!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\frac1{i_1\\dotsm i_k}\\,\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\label{cx5waq4}\\\\\n&=\\sum_{\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in \\operatorname{UP}(n,k)}(|\\lambda_1|-1)!\\dotsm (|\\lambda_k|-1)!\\,\\mathbb D^{(n)}_\\lambda,\\label{dts5ws53qq}\\\\\n\\mathbf S(n,k)&=\\frac{n!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\frac1{i_1!\\dotsm i_k!}\\,\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\label{cta4waq2wed}\\\\\n&=\\sum_{\\lambda\\in \\operatorname{UP}(n,k)}\\mathbb D^{(n)}_\\lambda.\\label{xsa43wqa4q}\n\\end{align}\nIn particular, the operators $\\mathbf s(n,k),\\mathbf c(n,k),\\mathbf S(n,k)\\in \\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$ are continuous.\n\\end{proposition}\n\n\\begin{proof}\nIt easily follows from  \\eqref{esa4q42} that\n\\begin{align*}\n\\langle(\\omega)^{(n)},\\xi^{\\otimes n}\\rangle&=\n\\sum_{k=1}^n \\frac{n!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\frac1{i_1\\dotsm i_k}\\,\\langle\\omega,\\xi^{i_1}\\rangle\\dotsm \\langle\\omega,\\xi^{i_k}\\rangle\n\\\\\n&=\\sum_{k=1}^n \\frac{n!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\frac1{i_1\\dotsm i_k}\\,\\langle\\omega^{\\otimes k},\\xi^{i_1}\\odot\\dots\\odot\\xi^{i_k}\\rangle.\\end{align*}\nThis and \\eqref{cre4a543w6} imply \\eqref{cx5waq4}.  Next, formula \\eqref{dw5w} implies\n$$\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle(\\omega)_{n},(e^{\\xi}-1)^{\\otimes n}\\rangle=\\exp\\big[\\langle\\omega,\\xi\\rangle\\big],\\quad\\xi\\in \\mathcal F(X).$$\nFrom here, analogously to the proof of \\eqref{cx5waq4},  we derive  \\eqref{cta4waq2wed}.\nBy  \\eqref{cs5w5a}, \\eqref{cx5waq4}, and \\eqref{cta4waq2wed}, formulas \\eqref{dts5ws53qq} and \\eqref{xsa43wqa4q} easily follow. Finally, the statement about the continuity of the Stirling operators follows  from the continuity of $\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\in\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$ and formulas  \\eqref{xfdyf}, \\eqref{cx5waq4}, and \\eqref{cta4waq2wed}.\n\\end{proof}\n\n\\begin{remark}\\label{tewu6534e} For $n,k\\in\\mathbb N$, $n\\le k$, we denote by $\\mathfrak S(n,k)$ the collection of all permutations $\\pi\\in\\mathfrak S_n$ that have $k$ cycles. Each $\\pi \\in \\mathfrak S(n,k)$ determines a partition $\\lambda\\in\\operatorname{UP}(n,k)$ through the cycles of $\\pi$. Furthermore, for each $\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in \\operatorname{UP}(n,k)$ there are $(|\\lambda_1|-1)!\\dotsm (|\\lambda_k|-1)!$ permutations $\\pi\\in\\mathfrak S(n,k)$ that give rise to $\\lambda$. Thus, denoting $\\mathbb D^{(n)}_\\pi:=\\mathbb D^{(n)}_\\lambda$, we can rewrite formula \\eqref{dts5ws53qq} as follows:\n\\begin{equation}\\label{fstsdys}\n\\mathbf c(n,k)=\\sum_{\\pi\\in\\mathfrak S(n,k)}\\mathbb D^{(n)}_\\pi.\\end{equation}\n\\end{remark}\n\n\\begin{remark}\nFor $A\\in\\mathcal B_0(X)$, let $\\chi_{A^n}$ denote the indicator function of the set $A^n$. Then, by \\eqref{xfdyf}, \\eqref{xsa43wqa4q}, and \\eqref{fstsdys},\n\\begin{equation}\\label{tsa4waAA}\n\\mathbf c(n,k)\\chi_{A^n}=c(n,k)\\chi_{A^k}\\,,\\quad \\mathbf s(n,k)\\chi_{A^n}=s(n,k)\\chi_{A^k}\\,,\\quad \\mathbf S(n,k)\\chi_{A^n}=S(n,k)\\chi_{A^k}\\,.\\end{equation}\nHere $c(n,k)$, $s(n,k)$, and $S(n,k)$ denote the classical unsigned and signed Stirling numbers of the first kind and the classical Stirling numbers of the second kind, respectively. (Recall that $c(n,k)$ is equal to the number of permutations in $\\mathfrak S(n,k)$, and $S(n,k)$ is equal to the number of partitions in $\\operatorname{UP}(n,k)$.) If $X$ has only a single point, then by \\eqref{tsa4waAA} and Remark~\\ref{se5wu563}, the linear operators $\\mathbf c(n,k)$, $\\mathbf s(n,k)$, $\\mathbf S(n,k)$ can be identified with the  numbers $c(n,k)$, $s(n,k)$, $S(n,k)$, respectively.\n\\end{remark}\n\n\\begin{remark} Let $\\mathbb P(n,k)$ denote the collection of all $(r_1,\\dots,r_{n-k+1})\\in\\mathbb N_0^{n-k+1}$ such that $r_1+r_2+\\dots+r_{n-k+1}=k$ and $r_1+2r_2+\\dots+(n-k+1)r_{n-k+1}=n$. For each $(r_1,r_2,\\dots,r_{n-k+1})\\in\\mathbb P(n,k)$, we denote\n$$\\alpha(r_1,\\dots,r_{n-k+1}):=\\frac{n!}{r_1!\\,r_2!\\dotsm r_{n-k+1}!\\,(1!)^{r_1}(2!)^{r_2}\\dotsm((n-k+1)!)^{r_{n-k+1}}},$$\nwhich is the number of partitions from $\\operatorname{UP}(n,k)$ that contains $r_1$ parts that have exactly  one element, $r_2$ parts that contain exactly two elements, and so on. By \\eqref{xsa43wqa4q},\n\\begin{multline*}\n\\mathbf S(n,k)=\\sum_{(r_1,\\dots,r_{n-k+1})\\in\\mathbb P(n,k)}\\alpha(r_1,\\dots,r_{n-k+1})\\\\\n\\times P_k\\big((\\mathbb D^{(1)})^{\\otimes r_1}\\otimes (\\mathbb D^{(2)})^{\\otimes r_2}\\otimes\\dots\\otimes (\\mathbb D^{(n-k+1)})^{\\otimes r_{n-k+1}}\\big).\\end{multline*}\nThis formula resembles the definition of the {\\it (partial) exponential Bell polynomials} \\cite{Bell}. The latter multivariate polynomials  are defined by\n\\begin{equation}\\label{rw5w2u53}\nB_{n,k}(z_1,\\dots,z_{n-k+1})=\\sum_{(r_1,\\dots,r_{n-k+1})\\in\\mathbb P(n,k)}\\alpha(r_1,\\dots,r_{n-k+1})\\,z_1^{r_1}z_2^{r_2}\\dotsm z_{n-k+1}^{r_{n-k+1}}\\end{equation}\nfor $(z_1,\\dots,z_{n-k+1})\\in\\mathbb F^{n-k+1}$. In \\cite{Schreiber}, these polynomials were called the {\\it multivariate Stirling polynomials of the second kind}. Furthermore, by \\eqref{dts5ws53qq}, we obtain\n\\begin{multline*}\n\\mathbf s(n,k)=\\sum_{(r_1,\\dots,r_{n-k+1})\\in\\mathbb P(n,k)}\\beta(r_1,\\dots,r_{n-k+1})\\\\\n\\times P_k\\big((\\mathbb D^{(1)})^{\\otimes r_1}\\otimes (\\mathbb D^{(2)})^{\\otimes r_2}\\otimes\\dots\\otimes (\\mathbb D^{(n-k+1)})^{\\otimes r_{n-k+1}}\\big),\\end{multline*}\nwhere\n$$\\beta(r_1,\\dots,r_{n-k+1}):=\\frac{n!}{r_1!\\,r_2!\\dotsm r_{n-k+1}!\\,1^{r_1}\\,2^{r_2}\\dotsm (n-k+1)^{r_{n-k+1}}}.$$\nNote, however, that if we defined a new sequence of multivariate polynomials by replacing in formula~\\eqref{rw5w2u53}, the $\\alpha$ coefficients with the $\\beta$ coefficients, then these would differ from the multivariate Stirling polynomials of the first kind as defined in \\cite{Schreiber}. \n\\end{remark}\n\n\nWe will now discuss recurrence relations satisfied by the Stirling operators. For linear operators $L_i\\in\\mathcal L(\\mathcal F(X^{n_i}),\\mathcal F(X^{k_i}))$,\n $i=1,2$, we can define their tensor product $L_1\\otimes L_2\\in\\mathcal L(\\mathcal F(X^{n_1+n_2}),\\mathcal F(X^{k_1+k_2}))$. \n\n\\begin{proposition}\\label{cdrttes} Let $\\mathbf 1^{(n)}$ denote the identity operator on $\\mathcal F(X^n)$. We have:\n\\begin{gather}\n\\mathbf S(n,n)=\\mathbf s(n,n)=\\mathbf 1^{(n)},\\label{rses3waw2}\\\\\n\\mathbf S(n+1,k)=P_k\\big(\\mathbf 1^{(1)}\\otimes\\mathbf S(n,k-1)\\big)+kP_k\\big((\\mathbb D^{(2)}\\otimes \\mathbf 1^{(k-1)})(\\mathbf 1^{(1)}\\otimes\\mathbf S(n,k))\\big), \\label{crtw5w}\\\\\n\\mathbf s(n+1,k)=P_k\\big(\\mathbf 1^{(1)}\\otimes\\mathbf s(n,k-1)\\big)-n\\mathbf s(n,k)P_n\\big(\\mathbb D^{(2)}\\otimes\\mathbf 1^{(n-1)}\\big).\\label{dw4w34}\n\\end{gather}\nfor $n\\in\\mathbb N$ and $k=1,\\dots, n$.\n\\end{proposition}\n\n\\begin{remark}\nFormulas \\eqref{xfdyf}, \\eqref{rses3waw2}, and \\eqref{dw4w34} immediately imply a recurrence relation for $\\mathbf c(n,k)$.\n\\end{remark}\n\n\\begin{remark} In the case of a single-point underlying space $X$, formulas  \\eqref{crtw5w}, \\eqref{dw4w34} become  the classical recurrence formulas $S(n+1,k)=S(n,k-1)+kS(n,k)$ and $s(n+1,k)=s(n,k-1)-ns(n,k)$.\n\\end{remark}\n\n\\begin{proof}[Proof of Proposition \\ref{cdrttes}] We denote by $\\operatorname{OP}(n,k)$ the colection of all ordered partitions  of $\\{1,\\dots,n\\}$ into $k$ non-empty parts. For each $\\lambda=(\\lambda_1,\\dots,\\lambda_k)\\in \\operatorname{OP}(n,k)$, we define the operator $\\mathbb D^{(n)}_\\lambda\\in\\mathcal L(\\mathcal F(X^n),\\mathcal F(X^k))$ analogously to the case when $\\lambda$ was an unordered partition but without the symmetrization of the obtained function in the end. By \\eqref{xsa43wqa4q},\n$\\mathbf S(n,k)=\\frac1{k!}\\sum_{\\lambda\\in \\operatorname{OP}(n,k)}\\mathbb D^{(n)}_\\lambda$. Note also that if a function $g^{(k)}:X^k\\to\\R$ is symmetric in its $x_2,\\dots,x_k$ variables, then\n$$(P_kg^{(k)})(x_1,\\dots,x_k)=\\frac1k\\sum_{i=1}^k g^{(k)}(x_2,\\dots,x_{i-1},x_1, x_i,\\dots,x_k).$$\nTherefore,\n\\begin{align*}\n&P_k\\big(\\mathbf 1^{(1)}\\otimes\\mathbf S(n,k-1)\\big)=\\frac1{k!}\\sum_{\\lambda=(\\lambda_1,\\dots,\\lambda_k)\\in\\operatorname{OP}(n+1,k):\\  \\{1\\}=\\lambda_i\\text{ for some }i}\\mathbb D^{(n)}_\\lambda,\\\\\n&kP_k\\big((\\mathbb D^{(2)}\\otimes \\mathbf 1^{(k-1)})(\\mathbf 1^{(1)}\\otimes\\mathbf S(n,k))\\big)=\\frac1{k!}\\sum_{\\lambda=(\\lambda_1,\\dots,\\lambda_k)\\in\\operatorname{OP}(n+1,k):\\  \\lambda_i\\ne\\{1\\}\\text{ for all }i}\\mathbb D^{(n)}_\\lambda,\n\\end{align*}\nwhich implies \\eqref{crtw5w}. While formula \\eqref{dw4w34} could be derived from \\eqref{xfdyf} and \\eqref{fstsdys}, it is in fact a direct consequence of the recurrence formula \\eqref{fseraw4qa}.\n\\end{proof}\n\n\n\\begin{remark}\\label{ts5as2}\n\n   Let $A\\in\\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$ be   a continuous linear operator, and let $A^*\\in\\mathcal L(M^{(k)}(X),M^{(n)}(X))$ be its adjoint. Each $f^{(n)}\\in\\mathcal F^{(n)}(X)$ determines a linear functional \n\\begin{equation}\\label{tew53wq}\nM^{(k)}(X)\\ni\\mu^{(k)}\\mapsto \\langle A^*\\mu^{(k)},f^{(n)}\\rangle.\\end{equation}\n Assume that, for each $f^{(n)}\\in\\mathcal F^{(n)}(X)$, there exists a function $g^{(k)}\\in\\mathcal F^{(k)}(X)$ that allows to represent the functional \n\\eqref{tew53wq} as $\\langle\\mu^{(k)},g^{(k)}\\rangle$. (The function $g^{(k)}$ is then obviously unique.)\nThen, we can extend $A$ to  the linear operator from  $\\mathcal L(\\mathcal F^{(n)}(X),\\mathcal F^{(k)}(X))$ defined by $Af^{(n)}:=g^{(k)}$. We may call it the {\\it canonical extension} of $A$. In particular, in view of Remark~\\ref{vyd6wq4a24}, the canonical extension is completely identified by $A\\xi^{\\otimes n}$ where $\\xi$ runs through $\\mathcal {CF}(X)$, or through the larger set $\\mathcal {F}(X)$. Finally, we note that canonical extensions of the Stirling operators $\\mathbf s(n,k),\\mathbf S(n,k)\\in \\mathcal L(\\mathcal {CF}^{(n)}(X),\\mathcal {CF}^{(k)}(X))$  exist and are equal to the Stirling operators  $\\mathbf s(n,k)$ and $\\mathbf S(n,k)$ from $\\mathcal L(\\mathcal {F}^{(n)}(X),\\mathcal {F}^{(k)}(X))$. \\end{remark}\n\n\nThe following proposition gives the explicit form of the  generating functions of the Stirling operators of the first and second kind. In view of Remark~\\ref{ts5as2}, these generating functions uniquely characterize the Stirling operators. \n\n\\begin{proposition}\\label{cesseas} We have, for each $k\\in\\mathbb N$,\n\\begin{align}\n\\sum_{n=k}^\\infty\\frac1{n!}\\,\\mathbf S(n,k)\\xi^{\\otimes n}&=\\frac1{k!}\\,(e^\\xi-1)^{\\otimes k},\\label{xtsaer5aw5}\\\\\n\\sum_{n=k}^\\infty\\frac1{n!}\\,\\mathbf s(n,k)\\xi^{\\otimes n}&=\\frac1{k!}\\,(\\log(1+\\xi))^{\\otimes k},\\quad\\xi\\in\\mathcal F(X).\\label{dre6ew6ww}\n\\end{align}\n\\end{proposition}\n\n\\begin{remark} In the case of a single-point underlying space $X$, formula \\eqref{xtsaer5aw5} becomes \nthe classical formula for the generating function of the Stirling numbers of the second kind: $\\sum_{n=k}^\\infty\\frac{\\xi^n}{n!}\\,S(n,k)=\\frac1{k!}(e^\\xi-1)^k$ ($\\xi\\in\\mathbb F$). The respective interpretation of  \\eqref{dre6ew6ww} is similar.\n\\end{remark}\n\n\\begin{remark}\nFormulas \\eqref{xfdyf} and \\eqref{dre6ew6ww} yield the generating function of the operators $\\mathbf c(n,k)$, which is equal to $\\frac1{k!}\\,(-\\log(1-\\xi))^{\\otimes k}$.\n\\end{remark}\n\n\\begin{proof}[Proof of Proposition \\ref{cesseas}] We have\n$$\\frac1{k!}\\,(e^\\xi-1)^{\\otimes k}=\\frac1{k!}\\,\\bigg(\\sum_{i=1}^\\infty\\frac{\\xi^i}{i!}\\bigg)^{\\otimes k}=\\frac1{k!}\\sum_{n=k}^\\infty \\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\frac1{i_1!\\dotsm i_k!}\\,\\xi^{i_1}\\odot\\dots\\odot\\xi^{i_k}.$$\nHence, \\eqref{xtsaer5aw5} follows from \\eqref{cta4waq2wed}. The proof of \\eqref{dre6ew6ww} is analogous.\n\\end{proof}\n\n\n\n\\begin{corollary} For $\\omega\\in M(X)$ and $\\xi\\in\\mathcal F(X)$,\n\\begin{align}\n\\langle\\omega^{\\otimes k},\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle&=\\frac1{k!}\\,\\frac{d^n}{dz^n}\\Big|_{z=0}\\langle\\omega, e^{z\\xi}-1\\rangle^n,\\label{vctrs5e}\\\\\\langle\\omega^{\\otimes k},\\mathbf s(n,k)\\xi^{\\otimes n}\\rangle&=\\frac1{k!}\\,\\frac{d^n}{dz^n}\\Big|_{z=0}\\langle\\omega, \\log(1+z\\xi)\\rangle^n. \\label{dxerts}\\end{align}\n\\end{corollary}\n\n\\begin{proof} By \\eqref{xtsaer5aw5}, for $\\omega\\in M(X)$, $\\xi\\in\\mathcal F(X)$, and $z\\in\\mathbb F$,\n$$\\sum_{n=k}^\\infty\\frac{z^n}{n!}\\,\\langle\\omega^{\\otimes k},\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle=\\langle\\omega^{\\otimes k},\\frac1{k!}\\,(e^{z\\xi}-1)^{\\otimes k}\\rangle=\\frac1{k!}\\langle\\omega, e^{z\\xi}-1\\rangle^k,$$\nwhich is an equality of formal power series in $z$. This immediately implies \\eqref{vctrs5e}. The proof of \\eqref{dxerts} is analogous. \n\\end{proof}\n\nThe following proposition shows that the classical orthogonality identities for the Stirling numbers (e.g.\\ \\cite[Section~12.2]{QG}) admit a generalization to our setting.\n\n\\begin{proposition}\\label{fye6e}\nFor any $i,n\\in\\mathbb N$,\n\\begin{equation}\\label{yr6e3}\n\\sum_{k=1}^n\\mathbf s(k,i)\\mathbf S(n,k)=\\sum_{k=1}^n\\mathbf S(k,i)\\mathbf s(n,k)=\\delta_{ni}\\mathbf 1^{(i)},\n\\end{equation}\nwhere $\\delta_{ni}$ is the Kronecker symbol.\n\\end{proposition}\n\n\\begin{proof} The proof is essentially the same as in the classical case. For the reader's convenience, we still present it. Formula \\eqref{yr6e3} obviously holds when $i>n$. Let $i\\le n$. \nFor each $f^{(n)}\\in\\mathcal F^{(n)}(X)$, we get\n\\begin{align*}\n\\langle\\omega^{\\otimes n},f^{(n)}\\rangle&=\\sum_{k=1}^n \\langle(\\omega)_k,\\mathbf S(n,k)f^{(n)}\\rangle\\\\\n&=\\sum_{k=1}^n\\sum_{i=1}^k\\langle\\omega^{\\otimes i},\\mathbf s(k,i)\\mathbf S(n,k)f^{(n)}\\rangle\\\\\n&=\\sum_{i=1}^n\\big\\langle\\omega^{\\otimes i},\\sum_{k=i}^n \\mathbf s(k,i)\\mathbf S(n,k)f^{(n)}\\big\\rangle\\\\&=\\sum_{i=1}^n\\big\\langle\\omega^{\\otimes i},\\sum_{k=1}^n \\mathbf s(k,i)\\mathbf S(n,k)f^{(n)}\\big\\rangle,\n\\end{align*}\nwhich proves the first equality in \\eqref{yr6e3}. The proof of the second equality  is similar.\n\\end{proof}\n\n\n\nWe finish this section with a short discussion of Lah operators. For $n,k\\in\\mathbb N$, $k\\le n$, we define the {\\it Lah operator} $\\mathbf L(n,k)\\in\\mathcal L(\\mathcal F^{(n)}(X),\\mathcal F^{(k)}(X))$ by\n\\begin{equation}\\label{xtesa5y}\n\\langle(\\omega)^{(n)},f^{(n)}\\rangle=\\sum_{k=1}^n\\langle(\\omega)_k\\,,\\mathbf L(n,k)f^{(n)}\\rangle, \\quad f^{(n)}\\in\\mathcal F^{(n)}(X).\\end{equation}\nIn view of \\eqref{ctsrea4w}, we equivalently have\n$$\\langle(\\omega)_n,f^{(n)}\\rangle=\\sum_{k=1}^n\\langle(\\omega)^{(k)},(-1)^{n-k}\\,\\mathbf L(n,k)f^{(n)}\\rangle, \\quad f^{(n)}\\in\\mathcal F^{(n)}(X).$$\n\n\\begin{proposition} (i) For $n,k\\in\\mathbb N$, $k\\le n$, we have\n\\begin{align}\n\\mathbf L(n,k)&=\\sum_{i=k}^n\\mathbf S(i,k)\\mathbf c(n,i)\\label{ra4q4}\\\\\n&=\\frac{n!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=n}}\\mathbb D^{(n)}_{i_1,\\dots,i_k}\\label{dcutwr}\\\\\n&=\\sum_{\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in \\operatorname{UP}(n,k)}|\\lambda_1|!\\,\\dotsm |\\lambda_k|!\\,\\,\\mathbb D^{(n)}_\\lambda.\\label{CWD}\n\\end{align}\n\n(ii) For each $k\\in\\mathbb N$,\n\\begin{equation}\\label{vtsw5w3}\n\\sum_{n=k}^\\infty\\frac1{n!}\\,\\mathbf L(n,k)\\xi^{\\otimes n}=\\frac{1}{k!}\\,\\bigg(\\frac{\\xi}{1-\\xi}\\bigg)^{\\otimes k},\\quad\\xi\\in\\mathcal F(X).\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nEquality \\eqref{ra4q4} follows immediately from \\eqref{fdr6ew456u4w}, \\eqref{drtrehgy},   and \\eqref{xtesa5y}. By \\eqref{cx5waq4}, \\eqref{cta4waq2wed}, and \\eqref{ra4q4}, we obtain\n\\begin{align}\n\\mathbf L(n,k)&=\\sum_{j=k}^n\\frac{j!}{k!}\\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=j}}\\frac1{i_1!\\dotsm i_k!}\\,\\mathbb D^{(j)}_{i_1,\\dots,i_k}\\,\\frac{n!}{j!}\\,\\sum_{\\substack{(l_1,\\dots,l_j)\\in\\mathbb N^j\\\\l_1+\\dots+l_j=n}}\\frac1{l_1\\dotsm l_j}\\,\\mathbb D^{(n)}_{l_1,\\dots,l_j}\\notag\\\\\n&=\\frac{n!}{k!}\\sum_{j=k}^n \\sum_{\\substack{(i_1,\\dots,i_k)\\in\\mathbb N^k\\\\i_1+\\dots+i_k=j}}\\sum_{\\substack{(l_1,\\dots,l_j)\\in\\mathbb N^j\\\\l_1+\\dots+l_j=n}}\\frac1{i_1!\\dotsm i_k!}\\,\\frac1{l_1\\dotsm l_j}\\,\\mathbb D^{(j)}_{i_1,\\dots,i_k}\\,\n\\mathbb D^{(n)}_{l_1,\\dots,l_j}\\notag\\\\\n&=\\frac{n!}{k!}\\,\\sum_{\\substack{(l_1,\\dots,l_k)\\in\\mathbb N^k\\\\l_1+\\dots+l_k=n}} r(l_1)\\dotsm r(l_k)\\mathbb D^{(n)}_{l_1,\\dots,l_k}.\\label{seqw54q327}\n\\end{align}\nHere, for $l\\in\\mathbb N$,\n\\begin{equation}\nr(l):=\\sum_{m=1}^l\\frac1{m!}\\,\\sum_{\\substack{(i_1,\\dots,i_m)\\in\\mathbb N^m\\\\i_1+\\dots+i_m=l}}\\frac1{i_1\\dotsm i_m}=\\frac1{l!}\\sum_{m=1}^lc(l,m)=\\frac{l!}{l!}=1,\\label{erq4332q}\n\\end{equation}\nwhere we used Remark \\ref{tewu6534e}. Formulas \\eqref{seqw54q327} and \\eqref{erq4332q} imply \\eqref{dcutwr}. Next, \\eqref{cs5w5a} and \\eqref{dcutwr} imply \\eqref{CWD}. Finally, formula \\eqref{vtsw5w3} easily follows from \\eqref{dcutwr}  by analogy with the proof of Proposition~\\ref{cesseas}. \n\\end{proof}\n\n\\begin{corollary}\nFor any $i,n\\in\\mathbb N$,\n$$\\sum_{k=1}^n(-1)^{n-k}\\,\\mathbf L(k,i)\\,\\mathbf L(n,k)=\\delta_{ni}\\mathbf 1^{(i)}.\n$$\n\\end{corollary}\n\n\\begin{proof}\nImmediate by \\eqref{xfdyf}, \\eqref{ra4q4}, and Proposition~\\ref{fye6e}.\n\\end{proof}\n\n\n\\section{Euler's formula for the Stirling operators of the~second kind}\\label{few53q}\n\n\\subsection{Euler's formula}\n\nWe denote by $\\Gamma_0(X)$ the subset of the configuration \nspace $\\Gamma(X)$ consisting of all finite (possibly empty) configurations. Thus, each element $\\eta\\in \\Gamma_0(X)$  is understood as both a finite subset $\\eta=\\{x_1,\\dots,x_n\\}$ of $X$ and a finite measure $\\delta_{x_1}+\\dots+\\delta_{x_n}$. (Note that the empty set is associated with the zero measure.)\nWe now need an extension of $\\Gamma_0(X)$, the set of multiple finite configurations in $X$, denoted by $\\ddot\\Gamma_0(X)$. Each element $\\eta\\in \\ddot\\Gamma_0(X)$  is understood as both a finite multiset $\\eta=[x_1,\\dots,x_n]$ and the corresponding finite measure $\\delta_{x_1}+\\dots+\\delta_{x_n}$.\nAs a subset of $M(X)$, $\\ddot\\Gamma_0(X)$ is the set of all positive finite integer-valued measures on $X$. \n\nThe following theorem gives an infinite dimensional counterpart of Euler's formula for $S(n,k)$, cf.\\ \\cite[Section 9.1]{QG}.\n\n\\begin{theorem}\\label{rsdtresw45w}\nLet $p\\in\\mathcal P(M(X))$ be a polynomial of degree $n$. Then\n\\begin{equation}\\label{x4tw5wq}\np(\\omega)=\\sum_{k=0}^n\\langle(\\omega)_k,g^{(k)}\\rangle,\n\\end{equation} where $g^{(0)}=p(0)$ and, for each $k=1,\\dots,n$,\n\\begin{align}\ng^{(k)}(x_1,\\dots,x_k)&=\\frac1{k!}\\,(D_{x_1}\\dotsm D_{x_k}\\,p)(0)\\label{cddssw4}\\\\\n&=\\frac{(-1)^k}{k!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_k]}(-1)^{|\\eta|}p(\\eta).\\label{rtdesse}\\end{align}\nHere $|\\eta|$ is the cardinality of the multiset $\\eta$, equivalently $|\\eta|=\\eta(X)$.\n\nIn particular, for each $n\\in\\mathbb N$, $k=1,\\dots,n$, and \n$f^{(n)}\\in\\mathcal F^{(n)}(X)$,\n\\begin{equation}\\label{fxqdqs}\n(\\mathbf S(n,k)f^{(n)})(x_1,\\dots,x_k)=\\frac{(-1)^k}{k!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_k]}(-1)^{|\\eta|}\\langle\\eta^{\\otimes n},f^{(n)}\\rangle.\\end{equation}\n\\end{theorem}\n\n\\begin{proof}  Formula \\eqref{cddssw4} follows from \\eqref{rsa4q} and the polynomial expansion theorem \\cite[Proposition 4.6]{FKLO}. Note that, in \\cite{FKLO}, this theorem was proved under slightly different assumptions. Nevertheless, an easy check shows that it remains true in our setting.\n\n By \\eqref{r3q3q2sqg} and the induction, we easily conclude that\n\\begin{equation}\\label{dt5w56uw}\n(D_{x_1}\\dotsm D_{x_k}\\,p)(\\omega)=(-1)^k\\sum_{\\eta\\subseteq[x_1,\\dots,x_k]}(-1)^{|\\eta|}p(\\omega+\\eta).\\end{equation}\nBy \\eqref{cddssw4} and \\eqref{dt5w56uw}, we get \\eqref{rtdesse}.\nSetting $p(\\omega)=\\langle\\omega^{\\otimes n},f^{(n)}\\rangle$ into formulas \\eqref{x4tw5wq}, \\eqref{rtdesse}, and using \\eqref{fdr6ew456u4w}, we obtain \n  \\eqref{fxqdqs}.\n\\end{proof}\n\n\\begin{remark}\\label{tdr652r6} In view of Remark \\ref{rtsdqwsuqwds6qe}, formula \\eqref{fxqdqs} holds, in fact, for all $n,k\\in\\mathbb N_0$.\n\\end{remark}\n\n\\begin{remark}\nFormulas \\eqref{x4tw5wq}, \\eqref{rtdesse}  imply that each polynomial $p\\in\\mathcal P(M(X))$ is uniquely determined by its values on $\\ddot\\Gamma_0(X)$. \n\\end{remark}\n\n\\begin{remark} In the case of a single-point space $X$, formula \\eqref{fxqdqs} becomes  the classical Euler's formula\n$S(n,k)=\\frac{(-1)^k}{k!}\\sum_{l=1}^k(-1)^l\\binom k l l^n$.\n\\end{remark}\n\n\\subsection{$\\mathcal K$-transform}\\label{fdsters5w5}\n\n\nLet us now briefly discuss how Theorem \\ref{rsdtresw45w} is related to the\ntheory of point processes. \n\nDenote by $\\mathcal F(\\ddot\\Gamma_0(X))$ the algebraic direct sum of the $\\mathcal F^{(n)}(X)$ spaces, $n\\in\\mathbb N_0$. \nThus, each $f\\in \\mathcal F(\\ddot\\Gamma_0(X))$ is of the form $f=(f^{(n)})_{n=0}^\\infty$ with $f^{(n)}\\in \\mathcal F^{(n)}(X)$ and, for some $N\\in\\mathbb N$, $f^{(n)}=0$ for all $n\\ge N$. We may identify $\\mathcal F(\\ddot\\Gamma_0(X))$ with a class of functions on $\\ddot\\Gamma_0(X)$. Indeed, for each $f=(f^{(n)})_{n=0}^\\infty$, define $f(\\varnothing):=f^{(0)}$ and $f([x_1,\\dots,x_n]):=f^{(n)}(x_1,\\dots,x_n)$.\nBelow, with an abuse of notation, we will use both interpretations of elements of $\\mathcal F(\\ddot\\Gamma_0(X))$.\n\nSimilarly to \\cite[Subsection~3.1]{KunaInfusino} (see also \\cite{Len75b}), we define a bijective map \n$\\mathcal K:\\mathcal F(\\ddot\\Gamma_0(X))\\to\\mathcal P(M(X))$ by \n$$ (\\mathcal Kf)(\\omega)=\\sum_{n=0}^\\infty\\left\\langle \\binom \\omega n,f^{(n)}\\right\\rangle,$$\nthe sum being in fact finite. In particular,  by formula \\eqref{reaq43q42}, for \n$\\omega=\\gamma\\in\\Gamma(X)$, we get\n\\begin{equation}\\label{drtw5wsd}\n(\\mathcal Kf)(\\gamma)=\\sum_{\\eta\\subseteq\\gamma,\\ \\eta\\in\\Gamma_0(X)}f(\\eta).\\end{equation}\nTheorem \\ref{rsdtresw45w} implies that the inverse map $\\mathcal K^{-1}:\\mathcal P(M(X))\\to\\mathcal F(\\ddot\\Gamma_0(X))$ \n is given by\n\\begin{equation}\\label{5w45q43q}\n(\\mathcal K^{-1}p)(\\eta)=\\sum_{\\sigma\\subseteq\\eta}(-1)^{|\\eta|-|\\sigma|}p(\\sigma).\\end{equation}\n\nIn the theory of point processes, one considers a slightly different map, denoted by $K$. Let $\\mathcal F(\\Gamma_0(X))$ denote the class of function on $\\Gamma_0(X)$ obtained as restrictions of functions from $\\mathcal F(\\ddot\\Gamma_0(X))$. (Note that, for each $f\\in \\mathcal F(\\Gamma_0(X))$, there are infinitely many functions from $\\mathcal F(\\ddot\\Gamma_0(X))$ whose restriction to $\\Gamma_0(X)$ coincides with $f$.) Similarly, let $\\mathcal P(\\Gamma(X))$ denote the class of functions on $\\Gamma(X)$ obtained as restrictions of polynomials from $\\mathcal P(M(X))$.  One defines a bijective map $K:\\mathcal F(\\Gamma_0(X))\\to\\mathcal P(\\Gamma(X))$ by formula \\eqref{drtw5wsd} in which $\\mathcal K$ is replaced by $K$ and $f\\in \\mathcal F(\\Gamma_0(X))$. The inverse operator, $K^{-1}$, is then given by formula \\eqref{5w45q43q} in which $\\mathcal K$ replaced by $K$ and $p\\in \\mathcal P(\\Gamma(X))$.  Note that, in this case, formula \\eqref{5w45q43q} is just a straightforward application of the inclusion-exclusion principle. \n\nThe main reason for introducing the $K$-transform in the theory of point processes is that, for a point process on $X$ (equivalently a probability measure on $\\Gamma(X)$), the measure $\\theta$ on $\\Gamma_0(X)$ defined by $\\mathbb E(Kf)=\\int_{\\Gamma_0(X)}f\\,d\\theta$ is called the {\\it correlation measure} of the point process, and (under certain weak assumptions) $\\theta$ uniquely determines the point process. See e.g.\\ \\cite{KK,Len1,Len75b} for details. \n\nOne defines a binary operation $\\star$ on $\\mathcal F(\\Gamma_0(X))$ so that, for any $f,g\\in\\mathcal F(\\Gamma_0(X))$, one has $\\big(K(f\\star g)\\big)(\\gamma)=(Kf)(\\gamma)(Kg)(\\gamma)$. An easy calculation show that \n\\begin{equation}\\label{sea453q}\n(f\\star g)(\\eta)=\\sum_{ \\substack{\\sigma_1,\\sigma_2,\\sigma_3\\in\\Gamma_0(X) \\\\ \\sigma_1+\\sigma_2+\\sigma_3=\\eta} }f(\\sigma_1+\\sigma_2)g(\\sigma_2+\\sigma_3),\\quad\\eta\\in\\Gamma_0(X),\\end{equation}\nsee \\cite{KK}. We now extend the binary operation $\\star$ to $\\mathcal F(\\ddot\\Gamma_0(X))$ by requiring that\n\\begin{equation}\\label{stw6us}\n\\big(\\mathcal K(f\\star g)\\big)(\\omega)=(\\mathcal Kf)(\\omega)(\\mathcal Kg)(\\omega)\\quad\\text{for all } \\omega\\in M(X).\n\\end{equation} \n\n\n\\begin{remark} In the theory of point processes, the $K$-transform is often thought of as a counterpart of the Fourier transform, see e.g.\\ \\cite{KK}.  Hence, in view of formula \\eqref{stw6us}, it is natural to interpret the binary operation $\\star$ as a convolution of functions from $\\mathcal F(\\ddot\\Gamma_0(X))$.\n\\end{remark}\n\n\n\n\\begin{proposition}\\label{sweawqa}\nFor any $f,g\\in \\mathcal F(\\ddot\\Gamma_0(X))$, the $\\star$ product of $f$ and $g$ is given by formula \\eqref{sea453q} in which $\\Gamma_0(X)$ is replaced by $\\ddot\\Gamma_0(X)$. \n\\end{proposition}\n\n\\begin{proof} The statement can be immediately concluded from (the proof of)\\cite[Proposition~3.4]{KunaInfusino}. For the reader's convenience, we will now outline an (alternative) proof of it. \nFormula \\eqref{fseraw4qa} implies that, for any $f^{(n)}\\in\\mathcal F^{(n)}(X)$ and $\\xi\\in \\mathcal F(X)$,\n\\begin{equation}\\label{fsre5w5}\nf^{(n)}\\star\\xi=(n+1)f^{(n)}\\odot\\xi+N(\\xi)f^{(n)}.\\end{equation}\nHere $N(\\xi)f^{(n)}\\in\\mathcal F^{(n)}(X)$ is defined by\n\\begin{equation}\\label{cresa5}\n(N(\\xi)f^{(n)})(x_1,\\dots,x_n):=f^{(n)}(x_1,\\dots,x_n)(\\xi(x_1)+\\dots+\\xi(x_n)).\\end{equation}\n\nIt suffices to prove formula \\eqref{sea453q} for $f=f^{(n)}\\in\\mathcal F^{(n)}(X)$ and $g=\\xi^{\\otimes m}\\in\\mathcal F^{(m)}(X)$.  For $m=1$, the result follows directly from formula~\\eqref{fsre5w5}. Assume that it holds for $m$. By \\eqref{fsre5w5},\n$$f^{(n)}\\star\\xi^{\\otimes (m+1)}=\\frac1{m+1}\\,\\big((f^{(n)}\\star\\xi^{\\otimes m})\\star\\xi-f^{(n)}\\star (N(\\xi)\\xi^{\\otimes m})\\big),$$\nthen one uses the induction assumption applied to $f^{(n)}\\star\\xi^{\\otimes m}$ and $f^{(n)}\\star (N(\\xi)\\xi^{\\otimes m})$, and then formula~\\eqref{sea453q} again. \\end{proof}\n\n\\begin{remark}\nLet us consider the one-dimensional counterpart of Proposition \\ref{sweawqa}, i.e., the case when $X$ has a single point. Consider $\\mathcal F(\\mathbb N_0)$, the space of  $\\mathbb F$-valued functions on $\\mathbb N_0$ (sequences) that vanish at all but finitely many points, and denote by $\\mathcal P(\\mathbb F)$ the space of polynomials on $\\mathbb F$.  Consider the bijective map $\\mathcal K:\\mathcal F(\\mathbb N_0)\\to\\mathcal P(\\mathbb F)$ given by $( Ka)(z):=\\sum_{n=0}^\\infty a(n)(z)_n$\\,, and define a binary operation $\\star$ on $\\mathcal F(\\mathbb N_0)$ by requiring $\\big(K(a\\star b)\\big)(z)=(Ka)(z)(Kb)(z)$. Then, by  Proposition \\ref{sweawqa}, \n$$(a\\star b)(n)=\\sum_{ \\substack{ i,j,k\\in\\mathbb{N}_0 \\\\ i+j+k=n} }\\binom n {i\\ j\\ k} a(i+j)b(j+k).$$\n\\end{remark}\n\n\n\\section{Identities for the Stirling operators}\\label{tes5w}\n\nWe will now discuss a few identities satisfied by the Stirling operators. All of them will yield classical identities for the Stirling numbers when the space $X$ has a single point. \n\n\n\\begin{proposition}[Infinite dimensional Olson's identity]\\label{dtrsews3a}\n Let $m,n,i\\in\\mathbb N$ and denote $l:=m+n$. Then, for each $f^{(l)}\\in\\mathcal F^{(l)}(X)$, we have\n\\begin{align}\n&\\bigg(\\sum_{k=1}^n \\mathbf S(k+m,i)P_{m+k}\\big(\\mathbf 1^{(m)}\\otimes \\mathbf s(n,k)\\big)f^{(l)}\\bigg)(x_1,\\dots,x_i)\\notag\\\\\n&\\quad =\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i],\\, |\\eta|\\ge n}(-1)^{|\\eta|}\\langle(\\eta)_n\\odot \\eta^{\\otimes m},f^{(l)}\\rangle.\\label{xtsw5w}\n\\end{align}\nIf either $i<n$ or $i>l$, the right hand side of \\eqref{xtsw5w} is equal to zero, and if $i=n$, it is equal to\n\\begin{equation}\\label{cxsesa53w}\n\\big\\langle(\\delta_{x_1}+\\dots+\\delta_{x_n})^{\\otimes m},f^{(l)}(x_1,\\dots,x_n,\\cdot)\\big\\rangle=\\sum_{j_1=1}^n\\dotsm\\sum_{j_m=1}^n f^{(l)}(x_1,\\dots,x_n,x_{j_1},\\dots,x_{j_m}).\\end{equation}\n\\end{proposition}\n\n\\begin{proof} We first note that formula \\eqref{reaq43q42} and Remark \\ref{c4q4q} remain true when $\\gamma\\in\\ddot\\Gamma_0(X)$.\n\n\nJust to simplify notations, we assume that $f^{(l)}=\\varphi^{\\otimes l}$ with $\\varphi\\in\\mathcal F(X)$.\nUsing Theorem~\\ref{rsdtresw45w} (see also Remark~\\ref{tdr652r6}), and Proposition \\ref{fye6e}, we get\n\\begin{align}\n&\\bigg(\\sum_{k=1}^n \\mathbf S(k+m,i)P_{m+k}\\big(\\mathbf 1^{(m)}\\otimes \\mathbf s(n,k)\\big)\\varphi^{\\otimes l}\\bigg)(x_1,\\dots,x_i)\\notag\\\\\n&\\quad=\\bigg(\\sum_{k=1}^n \\mathbf S(k+m,i)\\big(\\varphi^{\\otimes m}\\odot(\\mathbf s(n,k)\\varphi^{\\otimes n})\\big)\\bigg)(x_1,\\dots,x_i)\\notag\\\\\n&\\quad=\\sum_{k=1}^n\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i]}(-1)^{|\\eta|}\\langle\\eta^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\langle\\eta^{\\otimes k},\\mathbf s(n,k)\\varphi^{\\otimes n}\\rangle\\notag\\\\\n&\\quad=\\sum_{k=1}^n\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i]}(-1)^{|\\eta|}\\langle\\eta^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\sum_{j=1}^n\\langle (\\eta)_j,\\mathbf S(k,j)\\mathbf s(n,k)\\varphi^{\\otimes n}\\rangle\\notag\\\\\n&\\quad=\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i]}(-1)^{|\\eta|}\\langle\\eta^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\sum_{j=1}^n\\Big\\langle (\\eta)_j,\\sum_{k=1}^n\\mathbf S(k,j)\\mathbf s(n,k)\\varphi^{\\otimes n}\\Big\\rangle\\notag\\\\\n&\\quad=\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i]}(-1)^{|\\eta|}\\langle\\eta^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\langle(\\eta)_n,\\varphi^{\\otimes n}\\rangle\\notag\\\\\n&\\quad=\\frac{(-1)^i}{i!}\\sum_{\\eta\\subseteq[x_1,\\dots,x_i],\\, |\\eta|\\ge n}(-1)^{|\\eta|}\\langle\\eta^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\langle(\\eta)_n,\\varphi^{\\otimes n}\\rangle,\\notag\n\\end{align}\nwhich proves formula \\eqref{xtsw5w}. If $i<n$, the right hand side of \\eqref{xtsw5w} obviously vanishes. If $i>l$, the left hand side of \\eqref{xtsw5w} vanishes. If $i=n$, the only $\\eta\\subseteq[x_1,\\dots,x_i]$ that satisfies $|\\eta|\\ge i$ is $\\eta=[x_1,\\dots,x_i]$. Furthermore, by formula \\eqref{reaq43q42}, we get \n$$(\\delta_{x_1}+\\dots+\\delta_{x_i})_i=i!\\,(\\delta_{x_1}\\odot\\dots\\odot\\delta_{x_i}),$$\nwhich implies that the right hand side of \\eqref{xtsw5w} becomes \\eqref{cxsesa53w}.\n\\end{proof}\n\n\\begin{remark}\nIn the case of a single-point space $X$, Proposition~\\ref{dtrsews3a} implies \n$$\\sum_{k=1}^nS(k+m,i)s(n,k)=\\begin{cases}\n0,&\\text{if either $i<n$ or $i>n+m$},\\\\\ni^m,&\\text{if $i=n$},\\\\\n\\sum_{l=n}^i(-1)^{i+l}\\frac{l^m}{(i-l)!\\,(l-n)!},&\\text{if }i=n+1,\\dots,n+m.\n\\end{cases}$$\nThe case $i\\le n$ is attributed in \\cite[Section 12.2]{QG} to Frank Olson (1963). The case $i=n+1,\\dots,n+m$ does not seem to have been considered before.\n\\end{remark}\n\n\n\n\\begin{remark} For a fixed $f^{(l)}$ as in Proposition~\\ref{dtrsews3a}, denote\n$$g^{(i)}:=\\sum_{k=1}^n \\mathbf S(k+m,i)P_{m+k}\\big(\\mathbf 1^{(m)}\\otimes \\mathbf s(n,k)\\big)f^{(l)}.$$\nThen, by Theorem~\\ref{rsdtresw45w} and Proposition~\\ref{dtrsews3a}, we get\n$$\\langle (\\omega)_n\\odot\\omega^{\\otimes m},f^{(l)}\\rangle=\\sum_{i=n}^{n+m}\\langle(\\omega)_i,g^{(i)}\\rangle.$$\n\\end{remark}\n\n\n\n\\begin{proposition} For any $n\\in\\mathbb N$, $i=0,1,\\dots,n$, and $j=0,1,\\dots,n-i$,\n\\begin{equation}\\label{ceraq435q}\n\\binom{i+j}i\\mathbf s(n,i+j)=\\sum_{k=i}^{n-j}\\binom nk P_{i+j}\\big(\\mathbf s(k,i)\\otimes\\mathbf s(n-k,j)\\big).\\end{equation}\nIn formula \\eqref{ceraq435q}, the Stirling operators of the first kind can be replaced with the Stirling operators of the second kind. \n\\end{proposition}\n\n\\begin{proof} To simplify the notation, we assume that $f^{(n)}\\in\\mathcal F^{(n)}(X)$ is of the form $f^{(n)}=\\varphi^{\\otimes n}$ with $\\varphi\\in\\mathcal F(X)$. By \\eqref{5w53w2} and \\eqref{yrdqs}, we have,  for any $\\omega,\\sigma\\in M(X)$,\\begin{align}\n\\langle(\\omega+\\sigma)_n,f^{(n)}\\rangle&=\\sum_{k=0}^n\\binom nk\\langle(\\omega)_k,\\varphi^{\\otimes k}\\rangle\\langle(\\sigma)_{n-k},\\varphi^{\\otimes(n-k)}\\rangle\\notag\\\\\n&=\\sum_{k=0}^n\\binom nk\\sum_{i=0}^k\\langle\\omega^{\\otimes i},\\mathbf s(k,i)\\varphi^{\\otimes k}\\rangle\\sum_{j=0}^{n-k}\\langle\\sigma^{\\otimes j},\\mathbf s(n-k,j)\\varphi^{\\otimes(n-k)}\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\sum_{j=0}^{n-i}\\sum_{k=i}^{n-j}\\binom nk \\langle\\omega^{\\otimes i},\\mathbf s(k,i)\\varphi^{\\otimes k}\\rangle\\langle\\sigma^{\\otimes j},\\mathbf s(n-k,j)\\varphi^{\\otimes(n-k)}\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\sum_{j=0}^{n-i}\\Big\\langle\\omega^{\\otimes i}\\otimes\\sigma^{\\otimes j},\\sum_{k=i}^{n-j}\\binom nk \\big(\\mathbf s(k,i)\\otimes\\mathbf s(n-k,j)\\big)f^{(n)}\\Big\\rangle.\\label{cdtw5w}\n\\end{align}\nOn the other hand,\n\\begin{align}\n\\langle(\\omega+\\sigma)_n,f^{(n)}\\rangle&=\n\\sum_{k=0}^n\\langle(\\omega+\\sigma)^{\\otimes k},\\mathbf s(n,k)f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\binom ki\\langle\\omega^{\\otimes i}\\odot\\sigma^{\\otimes(k-i)},\\mathbf s(n,k)f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\sum_{j=0}^{n-i}\\Big\\langle\\omega^{\\otimes i}\\odot\\sigma^{\\otimes j},\\binom{i+j}i\\mathbf s(n,i+j)f^{(n)}\\Big\\rangle.\\label{sea3qa}\n\\end{align}\nLet $u,v\\in\\mathbb F$, $\\varkappa\\in M(X)$ and set in formulas \\eqref{cdtw5w}, \\eqref{sea3qa} $\\omega=u\\varkappa$ and $\\sigma=v\\varkappa$. This gives\n\\begin{align*}\n&\\sum_{i=0}^n\\sum_{j=0}^{n-i} u^iv^j \\Big\\langle\\varkappa^{\\otimes (i+j)},\\sum_{k=i}^{n-j}\\binom nk P_{i+j}\\big[\\mathbf s(k,i)\\otimes\\mathbf s(n-k,j)\\big]f^{(n)}\\Big\\rangle\\\\\n&\\quad=\\sum_{i=0}^n\\sum_{j=0}^{n-i} u^iv^j \\Big\\langle\\varkappa^{\\otimes (i+j)},\\binom{i+j}i\\mathbf s(n,i+j)f^{(n)}\\Big\\rangle.\n\\end{align*}\nThis implies formula \\eqref{ceraq435q}. The proof for the Stirling operators of the second kind is similar. One just uses formulas \\eqref{5w53w22} and \\eqref{fdr6ew456u4w} instead of \\eqref{5w53w2} and \\eqref{yrdqs}, respectively.  \n\\end{proof}\n\n\nRecall that, in Subsection \\ref{fdsters5w5}, we defined $\\mathcal F(\\ddot\\Gamma_0(X))$ as the algebraic direct sum of the $\\mathcal F^{(n)}(X)$ spaces. For each $x\\in X$, we define a linear operator $\\mathcal D_x\\in\\mathcal L(\\mathcal F(\\ddot\\Gamma_0(X)))$  that maps each $\\mathcal F^{(n)}(X)$ into $\\mathcal F^{(n-1)}(X)$ so that, for each $f^{(n)}\\in\\mathcal F^{(n)}(X)$, we have \n$(\\mathcal D_xf^{(n)})(\\cdot):=nf^{(n)}(x,\\cdot)$. Note that, by \\eqref{ydsrqqsqwdx}, $\\partial_x\\langle\\omega^{\\otimes n},f^{(n)}\\rangle=\\langle\\omega^{\\otimes(n-1)},\\mathcal D_xf^{(n)}\\rangle$.\n\n\\begin{proposition}\\label{drtes5w3}\nFor each $n,i\\in\\mathbb N$, $i<n$, and $f^{(n)}\\in\\mathcal F^{(n)}(X)$, \n\\begin{align}\\label{ts6uw65uxz}\n\\sum_{k=1}^{n-i}\\frac1{k!}\\, (-\\mathcal D_x)^k\\,\\mathbf s(n,i+k)f^{(n)}&=\\sum_{k=1}^{n-i}\\mathbf s(n-k,i)(-\\mathcal D_x)^kf^{(n)},\\\\\n\\sum_{k=1}^{n-i}(-\\mathcal D_x)^k\\,\\mathbf S(n,i+k)f^{(n)}&=\\sum_{k=1}^{n-i}\\frac1{k!}\\,\\mathbf S(n-k,i)(-\\mathcal D_x)^kf^{(n)}.\\label{utfrtrz}\n\\end{align}\n\\end{proposition}\n\n\\begin{proof} To simplify the notation, we assume that $f^{(n)}\\in\\mathcal F^{(n)}(X)$ is of the form $f^{(n)}=\\varphi^{\\otimes n}$ with $\\varphi\\in\\mathcal F(X)$. By \\eqref{xesa332a}, we get $(-\\delta_x)_n=(-1)^nn!\\,\\delta_x^{\\otimes n}$. Hence, by \\eqref{5w53w2}, for each $\\omega\\in M(X)$,\n$$(\\omega-\\delta_x)_n=\\sum_{k=0}^n (-1)^{n-k}\\,(n)_{n-k}\\, (\\omega)_k\\odot\\delta_x^{\\otimes(n-k)},$$\nwhich implies\n\\begin{align}\n\\langle (\\omega-\\delta_x)_n,f^{(n)}\\rangle&=\n\\sum_{k=0}^n(-1)^{n-k}(n)_{n-k}\\,\\langle(\\omega)_k,\\varphi^{\\otimes k}\\rangle\\,\\varphi^{n-k}(x)\\notag\\\\\n&=\\sum_{k=0}^n(-1)^{n-k}(n)_{n-k}\\,\\sum_{i=0}^k\\langle\\omega^{\\otimes i},\\mathbf s(k,i)\\varphi^{\\otimes k}\\rangle\\,\\varphi^{n-k}(x)\\notag\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\langle\\omega^{\\otimes i},\\mathbf s(k,i)(-\\mathcal D_x)^{n-k}f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\Big\\langle\\omega^{\\otimes i},\\sum_{k=i}^n \\mathbf s(k,i)(-\\mathcal D_x)^{n-k}f^{(n)}\\Big\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\Big\\langle\\omega^{\\otimes i},\\mathbf s(n,i)f^{(n)}+\\sum_{k=1}^{n-i}\\mathbf s(n-k,i)(-\\mathcal D_x)^{k}f^{(n)}\\Big\\rangle.\n\\label{w54qw5}\\end{align}\nOn the other hand,\n\\begin{align}\n\\langle (\\omega-\\delta_x)_n,f^{(n)}\\rangle&=\\sum_{k=0}^n\\langle(\\omega-\\delta_x)^{\\otimes k},\\mathbf s(n,k)f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\binom ki\\langle \\omega^{\\otimes i}\\odot (-\\delta_x)^{\\otimes(k-i)},\\mathbf s(n,k)f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\frac1{(k-i)!}\\,\\langle\\omega^{\\otimes i},(-\\mathcal D_x)^{k-i}\\mathbf s(n,k)f^{(n)}\\rangle\\notag\\\\\n&=\\sum_{i=0}^n\\Big\\langle\\omega^{\\otimes i},\\mathbf s(n,i)f^{(n)}+\\sum_{k=1}^{n-i} \\frac1{k!}\\, (-\\mathcal D_x)^{k}\\mathbf s(n,i+k)f^{(n)}\\Big\\rangle.\\label{rsw553q5}\n\\end{align}\nNow formula \\eqref{ts6uw65uxz} follows from \\eqref{w54qw5} and \\eqref{rsw553q5}. The proof of \\eqref{utfrtrz} is similar, one just starts with the polynomial $\\langle(\\omega-\\delta_x)^{\\otimes n},f^{(n)}\\rangle$.\n\\end{proof}\n\n\\begin{remark}\nIn the case of a single-point space $X$, Proposition~\\ref{drtes5w3} gives the following identities:\n\\begin{align}\n\\sum_{k=1}^{n-i}(-1)^k\\binom{i+k}{k}s(n,i+k)&=\\sum_{k=1}^{n-i}(-1)^k (n)_k\\, s(n-k,i),\\label{ds5w53wq}\\\\\n\\sum_{k=1}^{n-i}(-1)^k(i+k)_k\\,S(n,i+k)&=\\sum_{k=1}^{n-i}(-1)^k\\binom nk S(n-k,i).\\notag\n\\end{align}\nFormula \\eqref{ds5w53wq} is the equality (12.15) in \\cite{QG}. \n\\end{remark}\n\n\\begin{remark}\nIn equalities \\eqref{ceraq435q} and \\eqref{ts6uw65uxz},  the Stirling operators of the first kind can be replaced with the Lah operators.\n\\end{remark}\n\n\\section{Poisson functional and Stirling operators}\\label{ufxdqyr}\n\nLet $\\omega\\in M(X)$ be a positive Radon measure and assume that $\\omega$ is non-atomic, i.e., $\\omega(\\{x\\})=0$ for all $x\\in X$. Then one can define a Poisson point process on $X$ with intensity measure $\\omega$, which is a probability measure on the configuration space $\\Gamma(X)$ equipped with the cylinder $\\sigma$-algebra, see e.g.\\ \\cite{Kingman}. Denote by $\\mathbb E_\\omega$ the expectation with respect to this point process. The restriction of each polynomial $p\\in\\mathcal P(M(X))$ to $\\Gamma(X)$ is a random variable on $\\Gamma(X)$. Let $p\\in \\mathcal P(M(X))$ be of the form $p(\\omega)=\\sum_{k=0}^n\\langle\\omega^{\\otimes k},f^{(k)}\\rangle$, and choose a set $\\Lambda\\in\\mathcal B_0(X)$ such that, for each $k=1,\\dots,n$, the function $f^{(k)}$ vanishes outside $\\Lambda^k$. Then, it follows from the definition of the Poisson point process that\n\\begin{equation}\\label{ufyde}\n\\mathbb E_\\omega(p)=e^{-\\omega(\\Lambda)}\\sum_{n=0}^\\infty\\frac1{n!}\\int_{\\Lambda^n}p([x_1,\\dots,x_n])\\,\\omega^{\\otimes n}(dx_1\\dotsm dx_n).\\end{equation}\n(In the sum on the right-hand side of formula \\eqref{ufyde}, the term corresponding to $n=0$ is meant to be $p(0)$.) Note that the value of the right hand side of formula \\eqref{ufyde}  does not depend on the choice of the set $\\Lambda$. Furthermore, since the measure $\\omega$ is non-atomic, for $\\omega^{\\otimes n}$-a.a.\\ $(x_1,\\dots,x_n)\\in \\Lambda^n$, $[x_1,\\dots,x_n]\\in \\Gamma_0(X)$. \n\nFrom now on, we assume that $\\omega\\in M(X)$ is arbitrary. We define a  (linear) {\\it Poisson functional\\\/  $\\mathbb E_\\omega:\\mathcal P(M(X))\\to\\R$ with intensity measure $\\omega$} by formula \\eqref{ufyde}, provided the set $\\Lambda$ satisfies the above assumption that each $f^{(k)}$ vanishes outside $\\Lambda^k$. In Appendix,  we prove that the value of the right-hand side of formula~\\eqref{ufyde} still does not depend on the choice of such a set $\\Lambda$.  \nFurthermore, we discuss in Appendix several properties of the Poisson functional which generalize the corresponding facts about the Poisson point process.\n\n\n\n\n\\begin{theorem} \\label{e6ew63} Let $f^{(n)}\\in\\mathcal F^{(n)}(X)$ and let $p(\\omega)=\\langle\\omega^{\\otimes n},f^{(n)}\\rangle$,  $\\omega\\in M(X)$. Then\n$$\\mathbb E_\\omega(p)=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},\\mathbf S(n,k)f^{(n)}\\rangle.$$\n\\end{theorem}\n\n\\begin{proof}\nWe need equivalently to prove that, for each falling factorial $p\\in\\mathcal P(M(X))$ of the form $p(\\omega)=\\langle(\\omega)_k,g^{(k)}\\rangle$ with $g^{(k)}\\in\\mathcal F^{(k)}(X)$, we have $\\mathbb E_\\omega(p)=\\langle\\omega^{\\otimes k},g^{(k)}\\rangle$. We prove the statement by induction on $k$. For $k=1$, the result follows immediately from Proposition~\\ref{g6ew4}.\n\nAssume that the statement holds for $k$ and let us prove it for $k+1$. Choose $\\Lambda\\in\\mathcal B_0(X)$ such that $g^{(k+1)}$ vanishes outside of $\\Lambda^{k+1}$. Let $\\eta\\in\\ddot\\Gamma_0(\\Lambda)$, $\\eta=[x_1,\\dots,x_m]$, $m\\in\\mathbb N$. By \\eqref{xesa332a},\n\\begin{align}\n\\langle(\\eta)_{k+1},g^{(k+1)}\\rangle&=\\sum_{i_1\\in\\{1,\\dots,m\\}}\\,\\sum_{i_2\\in\\{1,\\dots,m\\}\\setminus\\{i_1\\}}\\dotsm \\sum_{i_{k+1}\\in\\{1,\\dots,m\\}\\setminus\\{i_1,\\dots,i_{k}\\}}g^{(k+1)}(x_{i_1},\\dots,x_{i_{k+1}})\\notag\\\\\n&=\\int_\\Lambda\\eta(dx)\\int_{\\Lambda^k}(\\eta-\\delta_x)_k(dx_1\\dotsm dx_k)g^{(k+1)}(x,x_1,\\dots,x_k).\\label{crse3aq3}\n\\end{align}\nBy \\eqref{crse3aq3}, Proposition \\ref{g6ew4}, and the induction assumption, we have, for $p(\\omega)=\\langle(\\omega)_{k+1},g^{(k+1)}\\rangle$,\n\\begin{align*}\n\\mathbb E_\\omega(p)&=\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\int_\\Lambda \\eta(dx)\\int_{\\Lambda^k}(\\eta-\\delta_x)_{k}(dx_1\\dotsm dx_k)g^{(k+1)}(x,x_1,\\dots,x_k)\\\\\n&=\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\int_\\Lambda \\omega(dx)\\int_{\\Lambda^k}(\\eta)_k(dx_1\\dotsm dx_k)g^{(k+1)}(x,x_1,\\dots,x_k)\\\\\n&=\\int_\\Lambda \\omega(dx)\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\int_{\\Lambda^k}(\\eta)_k(dx_1\\dotsm dx_k)g^{(k+1)}(x,x_1,\\dots,x_k)\\\\\n&=\\langle\\omega^{\\otimes(k+1)},g^{(k+1)}\\rangle.\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{remark} Consider a linear operator $U\\in\\mathcal L(\\mathcal P(M(X)))$ defined by $(Up)(\\omega):=\\mathbb E_\\omega(p)$ for $p\\in\\mathcal P(M(X))$. It follows from the proof of Theorem~\\ref{e6ew63} that $U$ maps each falling factorial $\\langle(\\omega)_k,g^{(k)}\\rangle$ into the monomial $\\langle\\omega^{\\otimes k},g^{(k)}\\rangle$. Note that  both the falling factorials and the monomials have the binomial property. Hence, by analogy with umbral calculus in dimension one (e.g.\\ \\cite{Roman}), we can think of $U$ as an {\\it umbral operator}.\n\\end{remark}\n\n\n\\begin{corollary}\\label{f6we5w}\nLet $p\\in \\mathcal P(M(X))$ be of the form $p(\\omega)=\\sum_{k=0}^n\\langle\\omega^{\\otimes k},f^{(k)}\\rangle$, and choose a set $\\Lambda\\in\\mathcal B_0(X)$ such that, for each $k=1,\\dots,n$, the function $f^{(k)}$ vanishes outside $\\Lambda^k$. Then \n$$\\mathbb E_\\omega(p)=\\sum_{i=0}^n\\frac1{i!}\\int_{\\Lambda^i}p([x_1,\\dots,x_{i}])\\,\\omega^{\\otimes i}(dx_1\\dotsm dx_{i}) \\sum_{k=0}^{n-i} \\frac{\\big(-\\omega(\\Lambda)\\big)^{k}}{k!}.$$\n\\end{corollary}\n\n\\begin{proof} By Theorem \\ref{rsdtresw45w} and (the proof of) Theorem~\\ref{e6ew63}, we have\n\\begin{align*}\n\\mathbb E_\\omega(p)&=\\sum_{k=0}^n\\frac{(-1)^k}{k!}\\int_{\\Lambda^k}\\sum_{i=0}^k(-1)^i\\sum_{\\{l_1,\\dots,l_i\\}\\subset\\{1,\\dots,k\\}}p([x_{l_1},\\dots,x_{l_i}])\\,\\omega^{\\otimes k}(dx_1\\dotsm dx_k)\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\frac{(-1)^{k-i}}{k!}\\binom ki\n\\int_{\\Lambda^k}p([x_1,\\dots,x_i])\\,\\omega^{\\otimes k}(dx_1\\dotsm dx_k)\\\\\n&=\\sum_{k=0}^n\\sum_{i=0}^k\\frac{(-1)^{k-i}}{i!(k-i)!}\\,\\omega(\\Lambda)^{k-i}\n\\int_{\\Lambda^i}p([x_1,\\dots,x_i])\\,\\omega^{\\otimes i}(dx_1\\dotsm dx_i),\\end{align*}\nwhich implies the statement.\n\\end{proof}\n\n\\begin{remark}\nIn the case of a single-point space $X$, Corollary \\ref{f6we5w} states that, if $p$ is a polynomial on $\\mathbb F$ of degree $n$, then for each $z\\in\\mathbb F$,\n$$\\sum_{k=0}^\\infty\\frac{z^k}{k!} \\, p(k)=e^z \\sum_{i=0}^n \\frac{p(i)}{i!}\\sum_{k=0}^{n-i}\\frac{(-1)^k z^{k+i}}{k!},\n$$\nwhich is Theorem~9.2 in \\cite{QG}.\n\\end{remark}\n\n\n\n\\section{Infinite dimensional Gr\\\"unert's and Katriel's formulas}\\label{gwfdytwyc}\n\n\\subsection{Infinite dimensional Gr\\\"unert's formula}\n\nFor each $\\xi\\in\\mathcal F(X)$, we consider the following linear operator on $\\mathcal P(M(X))$:\n$$\\langle\\omega,\\xi\\partial\\rangle=\\int_X\\omega(dx)\\xi(x)\\partial_x,$$\n where $\\partial_x$ is defined by \n\\eqref{qdftyfe} and $\\omega\\in M(X)$ is the variable of the polynomial this operator is applied to. More precisely, by \\eqref{ydsrqqsqwdx} and \\eqref{cresa5}, for each $f^{(n)}\\in\\mathcal F^{(n)}(X)$, we get\n\\begin{equation}\\label{fde5w44}\n\\langle\\omega,\\xi\\partial\\rangle\\langle\\omega^{\\otimes n},f^{(n)}\\rangle=n\\int_X\\omega(dx)\\xi(x)\\langle\\omega^{\\otimes(n-1)},f^{(n)}(x,\\cdot)\\rangle=\\langle\\omega^{\\otimes n},N(\\xi)f^{(n)}\\rangle.\n \\end{equation}\n We similarly define, for $f^{(n)}\\in\\mathcal F^{(n)}(X)$, the operator\n $$\\langle\\omega^{\\otimes n},f^{(n)}\\partial^{\\otimes n}\\rangle=\\int_{X^n}\\omega^{\\otimes n}(dx_1\\dotsm dx_n)f^{(n)}(x_1,\\dots,x_n)\\partial_{x_1}\\dotsm\\partial_{x_n}.$$\n\n\\begin{theorem}[Infinite dimensional Gr\\\"unert's formula]\\label{dwq3q32q}\n For any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$,\n\\begin{equation}\n\\langle\\omega,\\xi_1\\partial\\rangle\\dotsm \\langle\\omega,\\xi_n\\partial\\rangle=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},\\big(\\mathbf S(n,k)(\\xi_1\\odot\\dots\\odot\\xi_n)\\big)\\partial^{\\otimes k}\\rangle.\\label{cdrts6ew6}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof} We start with the following\n\n\\begin{lemma}\\label{yd6effd6} For any $\\xi\\in\\mathcal F(X)$ and $f^{(n)}\\in\\mathcal F^{(n)}(X)$,\n\\begin{equation}\\nota\n\\langle\\omega,\\xi\\partial\\rangle\n\\langle\\omega^{\\otimes n},f^{(n)}\\partial^{\\otimes n}\\rangle=\\langle\\omega^{\\otimes n},(N(\\xi)f^{(n)})\\partial^{\\otimes n}\\rangle+\\langle\\omega^{\\otimes(n+1)},(\\xi\\odot f^{(n)})\\partial^{\\otimes(n+1)}\\rangle.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} Let $g^{(m)}\\in\\mathcal F^{(m)}(X)$, and to simplify the notation we assume that $g^{(m)}=\\varphi^{\\otimes m}$ with $\\varphi\\in\\mathcal F(X)$. Then\n\\begin{align}\n&\\langle\\omega,\\xi\\partial\\rangle\n\\langle\\omega^{\\otimes n},f^{(n)}\\partial^{\\otimes n}\\rangle\\langle\\omega^{\\otimes m},g^{(m)}\\rangle\\notag\\\\\n&\\quad=\\langle\\omega,\\xi\\partial\\rangle(m)_n\\langle\\omega^{\\otimes n},f^{(n)}\\varphi^{\\otimes n}\\rangle\\langle\\omega^{\\otimes(m-n)},\\varphi^{\\otimes(m-n)}\\rangle\\notag\\\\\n&\\quad=(m)_n \\langle\\omega^{\\otimes n},N(\\xi)(f^{(n)}\\varphi^{\\otimes n})\\rangle\\langle\\omega^{\\otimes(m-n)},\\varphi^{\\otimes(m-n)}\\rangle\\notag\\\\\n&\\qquad+(m)_{n+1}\\langle\\omega^{\\otimes n},f^{(n)}\\varphi^{\\otimes n}\\rangle\\langle\\omega,\\xi\\varphi\\rangle\\langle\\omega^{\\otimes(m-n-1)},\\varphi^{\\otimes(m-n-1)}\\rangle\\notag\\\\\n&\\quad=\\langle\\omega^{\\otimes n},(N(\\xi)f^{(n)})\\partial^{\\otimes n}\\rangle\\langle\\omega^{\\otimes m},\\varphi^{\\otimes m}\\rangle\\notag\\\\\n&\\qquad+(m)_{n+1}\\langle\\omega^{\\otimes(n+1)},(f^{(n)}\\odot\\xi)\\varphi^{\\otimes(n+1)}\\rangle\\langle\\omega^{\\otimes(m-n-1)},\\varphi^{\\otimes(m-n-1)}\\rangle\\notag\\\\\n&\\quad=\\big[\\langle\\omega^{\\otimes n},(N(\\xi)f^{(n)})\\partial^{\\otimes n}\\rangle+\\langle\\omega^{\\otimes(n+1)},(\\xi\\odot f^{(n)})\\partial^{\\otimes(n+1)}\\rangle\\big]\\langle\\omega^{\\otimes m},\\varphi^{\\otimes m}\\rangle.\\notag\\qedhere\n\\end{align}\n\\end{proof}\n\n\n\nWe now prove \\eqref{cdrts6ew6} by induction on $n$. For $n=1$, \\eqref{cdrts6ew6}  is just trivial. Assume that \\eqref{cdrts6ew6} holds for $n$. Noting that $(\\langle\\omega,\\xi\\partial\\rangle)_{\\xi\\in\\mathcal F(X)}$ is a family of commuting operators, we get by the induction assumption, \\eqref{rses3waw2}, \\eqref{crtw5w}, and Lemma~\\ref{yd6effd6}, \n\\begin{align}\n&\\langle\\omega,\\xi_1\\partial\\rangle\\dotsm \\langle\\omega,\\xi_{n+1}\\partial\\rangle=\\sum_{k=1}^n\\langle\\omega,\\xi_{n+1}\\partial\\rangle\\langle\\omega^{\\otimes k},(\\mathbf S(n,k)(\\xi_1\\odot\\dots\\odot\\xi_n))\\partial^{\\otimes k}\\rangle\\notag\\\\\n&\\quad=\\langle\\omega^{\\otimes (n+1)},(\\mathbf S(n+1,n+1)(\\xi_1\\odot\\dots\\odot\\xi_{n+1}))\\partial^{\\otimes {n+1}}\\rangle\\notag\\\\\n&\\qquad+\\sum_{k=2}^n\\big[\\langle\\omega^{\\otimes k},(\\xi_{n+1}\\odot(\\mathbf S(n,k-1)(\\xi_1\\odot\\dots\\odot\\xi_n)))\\partial^{\\otimes k}\\rangle\\notag\\\\\n&\\qquad+\\langle\\omega^{\\otimes k},(N(\\xi_{n+1})\\mathbf S(n,k)(\\xi_1\\odot\\dots\\odot\\xi_n))\\partial^{\\otimes k}\\rangle\\big]+\\langle\\omega,\\xi_{n+1}(\\mathbf S(n,1)(\\xi_1\\odot\\dots\\odot\\xi_n))\\rangle\\notag\\\\\n&\\quad=\\sum_{k=1}^{n+1}\\langle\\omega^{\\otimes k},(\\mathbf S(n+1,k)(\\xi_1\\odot\\dots\\odot\\xi_{n+1}))\\partial^{\\otimes k}\\rangle.\\notag\\qedhere\n\\end{align}\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{vctrse45w}\nFor any $\\xi,\\varphi\\in\\mathcal F(X)$ and $n\\in\\mathbb N$, we have\n\\begin{align}\n&\\langle\\omega,\\xi\\partial\\rangle^n\\frac1{1-\\langle\\omega,\\varphi\\rangle}=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},(\\mathbf S(n,k)\\xi^{\\otimes n})\\varphi^{\\otimes k}\\rangle\\frac{k!}{(1-\\langle\\omega,\\varphi\\rangle)^{k+1}},\\label{f6ew5w}\\\\\n&\\langle\\omega,\\xi\\partial\\rangle^n e^{\\langle\\omega,\\varphi\\rangle}=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},(\\mathbf S(n,k)\\xi^{\\otimes n})\\varphi^{\\otimes k}\\rangle e^{\\langle\\omega,\\varphi\\rangle}.\\label{drtwe54w}\n\\end{align}\n\\end{corollary}\n\n\\begin{proof}\nBoth formulas \\eqref{f6ew5w} and \\eqref{drtwe54w} follow directly from Theorem \\ref{dwq3q32q} by differentiation of the functions $\\frac1{1-\\langle\\omega,\\varphi\\rangle}$ and $e^{\\langle\\omega,\\varphi\\rangle}$, respectively.\n\\end{proof}\n\n\\begin{remark}\nFormula \\eqref{f6ew5w} is an infinite dimensional extension of formula (9.47) in \\cite{QG}.\n\\end{remark}\n\n\\subsection{Wick ordering and infinite dimensional Katriel's formula}\n\n Let $V$ be a vector space. For linear operators $A,B\\in\\mathcal L(V)$,   we denote $[A,B]:=AB-BA$, called the {\\it commutator of $A$ and $B$}. \n\nLet us fix a reference measure $\\sigma\\in M(X)$.  Let us consider linear operators $a^+(\\xi),a^-(\\xi)\\in\\mathcal L(V)$ that linearly depend  on $\\xi\\in\\mathcal F(X)$ and satisfy the {\\it canonical commutation relations\\footnote{We drop the standard assumption that the measure $\\sigma$ appearing in the canonical commutation relations is positive. (Hence, we neither assume that $V$ is a Hilbert space, nor that  $a^-(\\xi)$ is the adjoint of $a^+(\\xi)$.)} (CCR)}:\n\\begin{gather}\n[a^+(\\varphi),a^+(\\xi)]=[a^-(\\varphi),a^-(\\xi)]=0,\\notag\\\\\n[a^-(\\varphi),a^+(\\xi)]=\\int_X\\varphi(x) \\xi(x)\\sigma(dx),\\quad \\varphi,\\xi\\in\\mathcal F(X).\\label{xra453}\n\\end{gather}\nThe operators $a^+(\\xi)$ and $a^-(\\xi)$ are called {\\it creation} and {\\it annihilation operators}, respectively. \n\n Our aim now is to introduce in $V$ the corresponding operators of particle density and Wick product of these operators. We will initially do this heuristically. \n\nWe define in $V$ {\\it creation operators $a^+(x)$} and {\\it annihilation operators \n$a^-(x)$ at points $x\\in X$} that satisfy\n$$a^+(\\xi)=\\int_X \\xi(x)a^+(x)\\sigma(dx),\\quad a^-(\\xi)=\\int_X\\xi(x)a^-(x)\\sigma(dx)\\quad\\text{for all }\\xi\\in \\mathcal F(X).$$\nIn terms of these operators, the CCR \\eqref{xra453} become\n\\begin{equation}\\label{r4wqa4y}\n[a^+(x),a^+(y)]=[a^-(x),a^-(y)]=0,\\quad [a^-(x),a^+(y)]=\\delta(x,y),\n\\end{equation}\nwhere the distribution $\\delta(x,y)$ satisfies \n$$\\int_{X^2}\\varphi(x)\\xi(y) \\delta(x,y)\\sigma(dx)\\sigma(dy)=\\int_X\\varphi(x)\\xi(x)\\sigma(dx),\\quad\\xi,\\varphi\\in\\mathcal F(X).$$\n\n \nWe define the {\\it particle density} $\\rho(x):=a^+(x)a^-(x)$ for $x\\in X$, and the corresponding {\\it operators of particle density} $\\rho(\\xi):=\\int_X\\xi(x)\\rho(x)\\sigma(dx)$ for $\\xi\\in\\mathcal F(X)$.\nThe CCR \\eqref{r4wqa4y} then imply the commutation relations\n\\begin{equation}\\label{esa4q4z}\n[\\rho(\\varphi),\\rho(\\xi)]=0,\\quad [\\rho(\\varphi),a^+(\\xi)]=a^+(\\varphi\\xi),\\quad [a^-(\\xi),\\rho(\\varphi)]=a^-(\\varphi\\xi).\\end{equation}\nfor all $\\varphi,\\xi\\in\\mathcal F(X)$.\n\nLet $\\sharp_1,\\dots,\\sharp_n\\in\\{+,-\\}$. We define the {\\it Wick product} ${:}\\,a^{\\sharp_1}(x_1)\\dotsm a^{\\sharp_n}(x_n){:}$ as the product of the operators $a^{\\sharp_1}(x_1),\\dots, a^{\\sharp_n}(x_n)$ that is Wick ordered, i.e.,  all the creation operators are to the left of all the annihilation operators.  In particular, \n\\begin{equation}\\label{cd55uw}\n{:}\\,\\rho(x_1)\\dotsm \\rho(x_n){:}=a^+(x_1)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_1).\\end{equation}\n\n\n\n\n\\begin{lemma}\\label{erws5w3}\n The CCR \\eqref{r4wqa4y}  imply the following   recurrence formulas:\n\\begin{gather}\n{:}\\rho(x){:}=\\rho(x),\\notag\\\\\n{:}\\rho(x_1)\\dotsm \\rho(x_n){:}=\\rho(x_1)\\,{:}\\rho(x_2)\\dotsm \\rho(x_n){:}\n-\\sum_{i=2}^n\\delta(x_1,x_i)\\,{:}\\rho(x_2)\\dotsm \\rho(x_n){:}\\, ,\\quad n\\ge2.\\notag\n\\end{gather}\n\\end{lemma}\n\n\\begin{proof} This formal result is known, see e.g.\\ \\cite[Section~2]{MeSh}. For the reader's convenience, we present the explicit calculations:\n\\begin{align*}\n&\\rho(x_1)\\,{:}\\rho(x_2)\\dotsm \\rho(x_n){:}=a^+(x_1)a^-(x_1)a^+(x_2)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&=\\delta(x_1,x_2)a^+(x_1)a^+(x_3)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&\\quad+a^+(x_1)a^+(x_2)a^-(x_1)a^+(x_3)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&=\\delta(x_1,x_2){:}\\rho(x_2)\\dotsm \\rho(x_n){:}+\\delta(x_1,x_3)a^+(x_1)a^+(x_2)a^+(x_4)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&\\quad+a^+(x_1)a^+(x_2)a^+(x_3)a^-(x_1)a^+(x_4)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&=\\sum_{i=2}^3\\delta(x_1,x_i){:}\\rho(x_2)\\dotsm \\rho(x_n){:}\\\\\n&\\quad+a^+(x_1)a^+(x_2)a^+(x_3)a^-(x_1)a^+(x_4)\\dotsm a^+(x_n)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&=\\sum_{i=2}^n\\delta(x_1,x_i){:}\\rho(x_2)\\dotsm \\rho(x_n){:}+a^+(x_1)\\dotsm a^+(x_n)a^-(x_1)a^-(x_n)\\dotsm a^-(x_2)\\\\\n&=\\sum_{i=2}^n\\delta(x_1,x_i){:}\\rho(x_2)\\dotsm \\rho(x_n){:}+\n{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,.\\qedhere\n\\end{align*}\n\\end{proof}\n\n  We are now in position to rigorously treat the  Wick product of operators of particle density. Assume that we are given a family of creation operators $a^+(\\xi)$ and annihilation operators $a^-(\\xi)$ acting in a vector space $V$ and satisfying the CCR \\eqref{xra453}. Further  assume that the corresponding particle density $\\rho(\\xi)$ ($\\xi\\in\\mathcal F(X)$) is well defined as a family of linear operators in $V$ that linearly depend on $\\xi$. Hence, the commutation relations \\eqref{esa4q4z} hold.\n\n\n Inspired by Lemma~\\ref{erws5w3},  we  define, for $\\xi\\in\\mathcal F(X)$, $\\int_X \\xi(x){:}\\rho(x){:}\\,\\sigma(dx):=\\rho(\\xi)$ and for $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$ ($n\\ge2$),\n\\begin{align}\n&\\int_{X^n}\\xi_1(x_1)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\notag\\\\\n&:=\\rho(\\xi_1)\\int_{X^{n-1}}\\xi_2(x_2)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_2)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes (n-1)}(dx_2\\dotsm dx_n)\n\\notag\\\\\n\n-\\sum_{i=2}^n\\int_{X^n}\\xi_2(x_2)\\dotsm (\\xi_1\\xi_i)(x_i)\\dotsm\n\\xi_n(x_n)\\,{:}\\rho(x_2)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes (n-1)}(dx_2\\dotsm dx_{n}).\n\\label{tesw5yw}\n\\end{align}\nExtending this by linearity, we obtain a linear operator\n$$\\int_{X^n}f^{(n)}(x_1,\\dots,x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\in\\mathcal L(V)$$\n for each $f^{(n)}:X^n\\to\\mathbb F$ that is a linear combination of functions $\\xi_1\\otimes\\dots\\otimes \\xi_n$ with  $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$.\n\n\\begin{theorem}[Infinite dimensional Katriel's formula] \\label{taw4aq234q}\nLet  $\\rho(\\xi)$ ($\\xi\\in\\mathcal F(X)$) be the operators of particle density for a family of creation  operators $a^+(\\xi)$ and annihilation operators $a^-(\\xi)$ satisfying the CCR  \\eqref{xra453}. Then, for each $n\\in\\mathbb N$ and $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$,\n\\begin{align}\n&\\rho(\\xi_1)\\dotsm\\rho(\\xi_n)\\notag\\\\\n&=\\sum_{k=1}^n\\int_{X^k}(\\mathbf S(n,k)(\\xi_1\\odot\\dotsm\\odot\\xi_n))(x_1,\\dots,x_k)\\,{:}\\rho(x_1)\\dotsm\\rho(x_k){:}\\,\\sigma^{\\otimes k}(dx_1\\dotsm dx_k).\\label{ds5w5u4} \n\\end{align}\n\\end{theorem}\n\n\\begin{remark}\nIn the case where $X=\\mathbb R^d$ and $\\sigma$ is the Lebesgue measure, a formula equivalent to \\eqref{ds5w5u4} was discussed, at a formal level,  by \nMenikoff and Sharp in \\cite[Section~2]{MeSh}.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{taw4aq234q}] As easily follows from \\eqref{tesw5yw}, for any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$ and a permutation $\\pi\\in \\mathfrak S_n$,\n\\begin{align*}\n&\\int_{X^n}\\xi_1(x_1)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\\\\n&\\quad=\\int_{X^n}\\xi_{\\pi(1)}(x_1)\\dotsm\\xi_{\\pi(n)}(x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n),\n\\end{align*}\nwhich implies\n\\begin{align}\n&\\int_{X^n}\\xi_1(x_1)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\notag\\\\\n&\\quad=\\int_{X^n}(\\xi_{1}\\odot\\dotsm\\odot\\xi_n)(x_1,\\dots,x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n).\\label{cdtrs5w}\n\\end{align}\nBy \\eqref{cresa5}, \\eqref{tesw5yw}, and \\eqref{cdtrs5w}, for any $\\varphi,\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$,\n\\begin{align}\n&\\rho(\\varphi)\\int_{X^n}(\\xi_{1}\\odot\\dotsm\\odot\\xi_n)(x_1,\\dots,x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\notag\\\\\n&\\quad=\\int_{X^{n+1}}(\\varphi\\odot\\xi_{1}\\odot\\dotsm\\odot\\xi_n)(x_1,\\dots,x_{n+1})\\,{:}\\rho(x_1)\\dotsm \\rho(x_{n+1}){:}\\,\\sigma^{\\otimes (n+1)}(dx_1\\dotsm dx_{n+1})\\notag\\\\\n&\\qquad+\\int_{X^n}(N(\\varphi)(\\xi_{1}\\odot\\dotsm\\odot\\xi_n))(x_1,\\dots,x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n).\\nota\n\\end{align}\nNow to prove formula \\eqref{ds5w5u4} we employ the same arguments as those used to derive formula \\eqref{cdrts6ew6} from Lemma \\ref{yd6effd6}.\n\\end{proof}\n\n\\begin{remark}\\label{tesw5}\nWe note that Theorem~\\ref{dwq3q32q} is actually a special case of Theorem~\\ref{taw4aq234q}. Indeed, let $V=\\mathcal P(M(X))$, and for each $\\xi\\in\\mathcal F(X)$, we define $a^+(\\xi)=\\langle\\omega,\\xi\\rangle$, the operator of multiplication by $\\langle\\omega,\\xi\\rangle$, and $a^-(\\xi)=\\langle\\sigma,\\xi\\partial\\rangle$. These operators satisfy the CCR \\eqref{xra453}. In this case, $a^-(x)=\\partial_x$ is a well-defined operator on $V$, while $a^+(x)$ is an operator-valued distribution---the operator of multiplication by the (generally speaking distribution) $\\frac{d\\omega}{d\\sigma}(x)$. The corresponding particle density is\n$\\rho(x)=\\frac{d\\omega}{d\\sigma}(x)\\partial_x$, which yields, for each $\\xi\\in\\mathcal F(X)$,\n$$\\rho(\\xi)=\\int_X\\xi(x)\\frac{d\\omega}{d\\sigma}(x)\\partial_x\\,\\sigma(dx)=\\int_X\\omega(dx)\\xi(x)\\partial_x=\\langle\\omega,\\xi\\partial\\rangle.$$\n As easily seen, the operators $a^+(\\xi)$, $a^-(\\xi)$, $\\rho(\\xi)$ indeed satisfy \\eqref{esa4q4z}. Furthermore, using \\eqref{tesw5yw}, we conclude that,  for any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$,\n\\begin{align}\n&\\int_{X^n}\\xi_1(x_1)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_1)\\dotsm \\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dotsm dx_n)\\notag\\\\\n&\\quad=\\langle\\omega^{\\otimes n},(\\xi_1\\otimes\\dots\\otimes \\xi_n)\\partial^{\\otimes n}\\rangle= \\langle\\omega^{\\otimes n},(\\xi_1\\odot\\dots\\odot \\xi_n)\\partial^{\\otimes n}\\rangle.\\notag\n\\end{align}\nHence, formula \\eqref{cdrts6ew6} is  a consequence of the commutation relations between the operators of multiplication by $\\langle\\omega,\\xi\\rangle$ ($\\xi\\in\\mathcal F(X)$) and the differentiation operators $\\partial_x$ ($x\\in X$). Note also that, in this case, the choice of the reference measure $\\sigma\\in M(X)$ was irrelevant. \n\\end{remark}\n\n\n\\begin{remark} Formula \\eqref{ds5w5u4} can be inverted:\n\\begin{align*}\n&\\int_{X^n}\\xi_1(x_1)\\dotsm\\xi_n(x_n)\\,{:}\\rho(x_1)\\dotsm\\rho(x_n){:}\\,\\sigma^{\\otimes n}(dx_1\\dots,dx_n)\\\\\n&\\quad=\\sum_{k=1}^n\\int_{X^k}(\\mathbf s(n,k)(\\xi_1\\odot\\dotsm\\odot\\xi_n))(x_1,\\dots,x_k)\\,\\rho(x_1)\\dotsm\\rho(x_k)\\,\\sigma^{\\otimes k}(dx_1\\dotsm dx_k).\n\\end{align*}\nThis follows immediately from  \\eqref{ds5w5u4} and Proposition \\ref{fye6e}.\n\\end{remark}\n\n\n\\subsection{Quantum Poisson process}\n\nLet the conditions of Theorem \\ref{taw4aq234q} be satisfied. \n Let $\\mathbf A$ denote the unital algebra generated by the operators $a^+(\\xi)$, $a^-(\\xi)$,  $\\rho(\\xi)$ ($\\xi\\in\\mathcal F(X)$). Due to the commutation relations \\eqref{xra453}, \\eqref{esa4q4z} and the polarization identity, each element of the algebra $\\mathbf A$ is a linear combination of the identity operator $\\mathbf 1$ and operators of the form $a^+(\\varphi)^i\\rho(\\psi)^ja^-(\\xi)^k$ with $\\varphi,\\psi,\\xi\\in\\mathcal F(X)$ and $i,j,k\\in\\mathbb N_0$, $i+j+k\\ge1$.\n \n \n \n  We define the {\\em vacuum functional} $\\tau_\\sigma$ on $\\mathbf A$ by setting \n\\begin{gather}\n\\tau_\\sigma(\\mathbf 1)=1,\\notag\\\\\n\\tau_\\sigma\\left(a^+(\\varphi)^i\\rho(\\psi)^ja^-(\\xi)^k\\right)=0,\\quad \\varphi,\\psi,\\xi\\in\\mathcal F(X),\\ i,j,k\\in\\mathbb N_0,\\ i+j+k\\ge1,\\label{vcyrs5a}\n\\end{gather}\n and extending it by linearity to the whole $\\mathbf A$. \n \n  For each $\\xi\\in\\mathcal F(X)$, we define $R(\\xi)\\in\\mathbf A$ by \n\\begin{equation}\\label{vcttfts5yw3}\nR(\\xi):=a^+(\\xi)+a^-(\\xi)+\\rho(\\xi)+\\langle\\sigma,\\xi\\rangle.\\end{equation} \nBy the commutation relations \\eqref{xra453}, \\eqref{esa4q4z}, for any $\\xi_1,\\xi_2\\in\\mathcal F(X)$, we have $[R(\\xi_1),R(\\xi_2)]=0$.\n\n\n\\begin{remark} Consider the representation of the CCR discussed in Remark~\\ref{tesw5}. Then each $A\\in\\mathbf A$ is a linear operator on $\\mathcal P(M(X))$. Recall also that $a^+(\\xi)$ is the multiplication by $\\langle\\omega,\\xi\\rangle$, $a^-(\\xi)=\\langle\\sigma,\\xi\\partial\\rangle$, and $\\rho(\\xi)=\\langle\\omega,\\xi\\partial\\rangle$. If $1$ denotes the  monomial on $M(X)$ that is identically equal to 1, then \n$$\\big(a^+(\\varphi)^i\\rho(\\psi)^ja^-(\\xi)^k 1\\big)(\\omega)=\\begin{cases}\n0,&\\text{if }\\max\\{j,k\\}\\ge1,\\\\\n\\langle\\omega^{\\otimes i},\\varphi^{\\otimes i}\\rangle,&\\text{if $i\\ge1$ and $j=k=0$}.\n\\end{cases}\n$$\nTherefore, by \\eqref{vcyrs5a}, the vacuum functional on $\\mathbf A$ is given by $\\tau_\\sigma(A)=(A1)(0)$, i.e., one has to apply the operator $A$ to 1 and then evaluate the obtained polynomial at zero.  Furthermore, it follows from \\eqref{vcttfts5yw3} that, in this case, \n$$R(\\xi)=\\langle\\omega+\\sigma,\\xi(\\partial+1)\\rangle,\\quad\\xi\\in\\mathcal F(X).$$\n\\end{remark}\n\n\n\n\\begin{theorem}\\label{y6weu43} Let  $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$, $n\\in\\mathbb N$ . Under the above assumptions, we then have\n$$\\tau_\\sigma\\big(R(\\xi_1)\\dotsm R(\\xi_n)\\big)=\\mathbb E_\\sigma(p). $$\n Here $\\mathbb E_\\sigma$ is the Poisson functional with intensity measure $\\sigma$ and $p(\\omega)=\\langle\\omega,\\xi_1\\rangle\\dotsm \\langle\\omega,\\xi_n\\rangle$.\n\n\\end{theorem}\n\nLet $\\mathbf A'$ denote the commutative unital subalgebra of $\\mathbf A$ that is generated by the operators $R(\\xi)$ ($\\xi\\in\\mathcal F(X)$). Let $\\mathcal P_{\\mathrm{alg}}(M(X))$ denote the subset of $\\mathcal P(M(X))$  consisting of all polynomials $p$ of the form \\eqref{tew53w5}, where each $f^{(k)}$ is a finite sum of functions of the form $\\xi_1\\odot\\dots\\odot\\xi_k$ with $\\xi_1,\\dots,\\xi_k\\in\\mathcal F(X)$. Theorem~\\ref{y6weu43} states an isomorphism between $(\\mathbf A',\\tau_\\sigma)$ and $(\\mathcal P_{\\mathrm{alg}}(M(X)),\\mathbb E_\\sigma)$. Hence, $(\\mathbf A',\\tau_\\sigma)$ can be thought of as the {\\it quantum Poisson process with intensity measure $\\sigma$}. This is a well-known result in the case where $\\sigma$ is a positive non-atomic measure, see e.g.\\ \\cite{GGPS,HP,Surgailis}.\n\nWe first prove the following\n\n\n\\begin{lemma}\\label{rqdwes6eqw}\nFor any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$, we have\n\\begin{align}\n&R(\\xi_1)\\dotsm R(\\xi_n)=\\sum_{k=1}^n\\int_{X^k}(\\mathbf S(n,k)(\\xi_1\\odot\\dotsm\\odot\\xi_n))(x_1,\\dots,x_k)\\notag\\\\\n&\\times\n(a^+(x_1)+1)\\dotsm(a^+(x_k)+1)(a^-(x_k)+1)\\dotsm(a^-(x_1)+1)\\,\n\\sigma^{\\otimes k}(dx_1\\dotsm dx_k).\\label{eraq42q}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nFirst, we note that the integrals on the right-hand side of formula \\eqref{eraq42q} well define  linear operators on the vector space $V$. Indeed, since all creation operators (respectively all annihilation operators) commute, each integral is a finite sum of terms of the form\n\\begin{align*}\n&\\int_{X^k}\\varphi_1(x_1)\\dotsm\\varphi_k(x_k)a^+(x_1)\\dotsm a^+(x_{i+j})a^-(x_{i+j})\\dotsm a^-(x_{i+1})\\\\\n&\\qquad\\times a^-(x_{i+j+1})\\dotsm a^-(x_k)\\sigma^{\\otimes k}(dx_1\\dotsm dx_k)\\\\\n&\\quad=a^+(\\varphi_1)\\dotsm a^+(\\varphi_i)\\,{:}\\rho(\\varphi_{i+1})\\dotsm\\rho(\\varphi_{i+j}){:}\\,a^-(\\varphi_{i+j+1})\\dotsm a^-(\\varphi_k),\n\\end{align*}\nwhere $\\varphi_1,\\dots\\varphi_k\\in\\mathcal F(X)$, $i,j\\in\\mathbb N_0$, $0\\le i+j\\le k$.\n\nThe formal proof of formula \\eqref{eraq42q} is straightforward. Indeed, for $\\xi\\in\\mathcal F(X)$, denote $A^+(\\xi):=a^+(\\xi)+\\langle\\sigma,\\xi\\rangle$, $A^-(\\xi):=a^-(\\xi)+\\langle\\sigma,\\xi\\rangle$. These operators obviously satisfy the CCR. Writing $A^+(\\xi)=\\int_X \\xi(x)A^+(x)\\sigma(dx)$ and $A^-(\\xi)=\\int_X\\xi(x)A^-(x)\\sigma(dx)$, we get $A^+(x)=a^+(x)+1$, $A^-(x)=a^-(x)+1$. Hence, the corresponding particle density is\n$$\\int_X\\xi(x)A^+(x)A^-(x)\\sigma(dx)=\\int_X\\xi(x)(a^+(x)+1)(a^-(x)+1)\\sigma(dx)=R(\\xi).$$\nTherefore, formula \\eqref{eraq42q} follows from Theorem \\ref{taw4aq234q}.\n\nTo make these arguments rigorous, we proceed as follows. For any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$, denote\n\\begin{align}\n{:}R(\\xi_1)\\dotsm R(\\xi_n){:}=&\\int_{X^n}\\xi_1(x_1)\\dotsm \\xi_n(x_n) A^+(x_1)\n\\dotsm A^+(x_n)\\notag\\\\\n&\\times A^-(x_n)\\dotsm A^-(x_1)\\sigma^{\\otimes n}(dx_1\\dotsm dx_n).\\label{y6se6qesq}\\end{align}\n(Recall that the integral on the right-hand side of formula \\eqref{y6se6qesq} indeed determines a linear operator on $V$.) Using the commutation relations \\eqref{xra453}, \\eqref{esa4q4z}, one easily checks that\n$${:}R(\\xi_1)\\dotsm R(\\xi_n){:}=R(\\xi_1){:}R(\\xi_2)\\dotsm R(\\xi_n){:}-\\sum_{i=2}^n {:}R(\\xi_2)\\dotsm R(\\xi_1\\xi_i)\\dotsm R(\\xi_n){:}\\,. \\qedhere$$\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{y6weu43}] Applying the vacuum functional $\\tau_\\sigma$ to equality \\eqref{eraq42q}, we easily obtain\n$$\\tau_\\sigma\\big(R(\\xi_1)\\dotsm R(\\xi_n)\\big)= \\sum_{k=1}^n\\langle\\sigma^{\\otimes k},\\mathbf S(n,k)(\\xi_1\\odot\\dots\\odot\\xi_n)\\rangle.$$\nNow Theorem \\ref{e6ew63} yields the statement. \n\\end{proof}\n\n\n\n\\section{Touchard polynomials}\\label{ydrsdese3aqw}\n\nAccording to Remark \\ref{ts5as2}, for each $n,k\\in\\mathbb N$, $k\\le n$, we  define the linear operator $\\mathbf S(n,k)^*\\in\\mathcal L(M^{(k)}(X),M^{(n)}(X))$. For each $n\\in\\mathbb N$, we define the {\\it Touchard (or exponential) polynomial of degree $n$} to be the mapping \n\\begin{equation}\\label{d5w5w4}\nT_n: M(X)\\to M^{(n)}(X),\\quad T_n(\\omega)=\\sum_{k=1}^n\\mathbf S(n,k)^*\\omega^{\\otimes k}.\\end{equation}\nWe also set $T_0(\\omega):=1$ for all $\\omega\\in M(X)$.\n\n\\begin{remark}\nIn the case of a single-point space $X$, the polynomials \\eqref{d5w5w4} become the classical Touchard polynomials.\n\\end{remark}\n\nLet $\\omega\\in M(X)$ and $i\\in\\mathbb N$. Using a notation from \\cite[Section~5]{RW}, we define a {\\it diagonal measure} $\\omega^{[i]}_{\\hat 1}\\in M^{(i)}(X)$ by\n$$\\omega^{[i]}_{\\hat 1}(dx_1\\dotsm dx_i):=\\omega(dx_1)\\delta_{x_1}(dx_2)\\dotsm\\delta_{x_1}(dx_i).$$\nBy \\eqref{cq5y42w}, we have, for each $f^{(i)}\\in\\mathcal F^{(i)}(X)$,\n$$\\langle \\omega^{[i]}_{\\hat 1},f^{(i)}\\rangle=\\langle\\omega,\\mathbb D^{(i)}f^{(i)}\\rangle.$$\nHence, in view of formulas \\eqref{cs5w5a} and \\eqref{xsa43wqa4q}, we have, for each $n\\in\\mathbb N$,\n\\begin{equation}\\label{dtw5ua}\nT_n(\\omega)=\\sum_{k=1}^n\\sum_{\\lambda=\\{\\lambda_1,\\dots,\\lambda_k\\}\\in \\operatorname{UP}(n,k)}\\omega^{[\\,|\\lambda_1|\\,]}_{\\hat 1}\\odot\\dots\\odot \\omega^{[\\,|\\lambda_k|\\,]}_{\\hat 1}.\\end{equation}\n\n\n\nFor each $f^{(n)}\\in\\mathcal F^{(n)}(X)$, \n\\begin{equation}\\label{ctresw53wq}\\langle T_n(\\omega),f^{(n)}\\rangle=\\sum_{k=1}^n\\langle\\omega^{\\otimes k},\\mathbf S(n,k)f^{(n)}\\rangle\\end{equation}\nis a polynomial from $\\mathcal P(M(X))$, and if $f^{(n)}\\in\\mathcal {CF}^{(n)}(X)$, then this polynomial belongs to $\\mathcal {CP}(M(X))$. \n\nThe following statement is an immediate consequence of  Theorem \\ref{e6ew63}.\n\n\\begin{corollary}\\label{sesawawqa2q}\nFor each $\\omega\\in M(X)$,  and $f^{(n)}\\in\\mathcal F^{(n)}(X)$, $n\\in\\mathbb N$, we have\n\\begin{equation*\n\\langle T_n(\\omega),f^{(n)}\\rangle=\\mathbb E_\\omega(p),\n \\end{equation*}\nwhere $p(\\omega)=\\langle\\omega^{\\otimes n},f^{(n)}\\rangle$. In particular, for each $\\xi\\in\\mathcal F(X)$,\n\\begin{equation}\\label{cydye6edd}\n\\langle T_n(\\omega),\\xi^{\\otimes n}\\rangle=\\mathbb E_\\omega(m^{(n)}_{\\xi}),\\end{equation}\nwhere\n\\begin{equation}\\label{vydye}\nm_\\xi^{(n)}(\\omega)=\\langle\\omega,\\xi\\rangle^n.\n\\end{equation}\n\\end{corollary}\n\n\n\n\n\nFor $\\omega\\in M(X)$ and $\\varphi\\in\\mathcal F (X)$, we define $\\omega\\varphi\\in M(X)$ by $(\\omega\\varphi)(dx):=\\omega(dx)\\varphi(x)$. \nThen, Corollary~\\ref{vctrse45w} can be written in the following equivalent form.\n\n\\begin{corollary}For any $\\xi,\\varphi\\in\\mathcal F(X)$ and $n\\in\\mathbb N$, we have\n\\begin{align}\n&\\langle\\omega,\\xi\\partial\\rangle^n\\frac1{1-\\langle\\omega,\\varphi\\rangle}=\n\\langle T_n(\\omega\\varphi),\\xi^{\\otimes n}\\rangle\\,\\frac{k!}{(1-\\langle\\omega,\\varphi\\rangle)^{k+1}},\\notag\\\\\n&\\langle\\omega,\\xi\\partial\\rangle^n e^{\\langle\\omega,\\varphi\\rangle}=\\langle T_n(\\omega\\varphi),\\xi^{\\otimes n}\\rangle\\, e^{\\langle\\omega,\\varphi\\rangle}.\\notag\n\\end{align}\n\n\\end{corollary}\n\n\nThe following proposition is a generalization of statement (7) of Theorem~3.29 in \\cite{MS}, which is formulated there for the number operator for a pair of operators satisfying CCR.\n\n\\begin{proposition}\nFor each $\\xi\\in\\mathcal F(X)$ and $n\\in\\mathbb N_0$,\n\\begin{equation*\n\\langle T_{n+1}(\\omega),\\xi^{\\otimes(n+1)}\\rangle=\n\\langle\\omega,\\xi(\\partial+1)\\rangle\\,\\langle T_{n}(\\omega),\\xi^{\\otimes n}\\rangle.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof} By \\eqref{crtw5w}, \\eqref{cresa5}, \\eqref{fde5w44} and \\eqref{ctresw53wq},\n\\begin{align*}\n&\\langle T_{n+1}(\\omega),\\xi^{\\otimes(n+1)}\\rangle=\\sum_{k=1}^{n+1}\\langle\\omega^{\\otimes k},\\mathbf S(n+1,k)\\xi^{\\otimes (n+1)}\\rangle\\\\\n&\\quad=\\sum_{k=1}^{n+1}\\big[\n\\langle\\omega,\\xi\\rangle\\langle\\omega^{\\otimes(k-1)},\\mathbf S(n,k-1)\\xi^{\\otimes n}\\rangle+\\langle\\omega^{\\otimes k},N(\\xi)\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle\\big]\\\\\n&\\quad=\\sum_{k=1}^{n+1}\\big[\n\\langle\\omega,\\xi\\rangle\\langle\\omega^{\\otimes(k-1)},\\mathbf S(n,k-1)\\xi^{\\otimes n}\\rangle+\\langle\\omega,\\xi\\partial\\rangle\\langle\\omega^{\\otimes k},\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle\\big]\\\\\n&\\quad=\\langle\\omega,\\xi\\rangle\\sum_{k=2}^{n+1}\\langle\\omega^{\\otimes(k-1)},\\mathbf S(n,k-1)\\xi^{\\otimes n}\\rangle+\\langle\\omega,\\xi\\partial\\rangle\\sum_{k=1}^n\n\\langle\\omega^{\\otimes k},\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle\\\\\n&\\quad=\\big(\\langle\\omega,\\xi\\partial\\rangle+\\langle\\omega,\\xi\\rangle\\big)\\langle T_{n}(\\omega),\\xi^{\\otimes n}\\rangle.\\qedhere\n\\end{align*}\n\\end{proof}\n\nThe following proposition gives the explicit form of the generating function of the Touchard polynomials.\n\n\\begin{proposition}\\label{cts65}\n We have\n$$\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle T_n(\\omega),\\xi^{\\otimes n}\\rangle=\\exp\\big[\\langle\\omega,e^{\\xi}-1\\rangle\\big],\\quad \\xi\\in\\mathcal F(X).$$\n\n\n\n\\begin{proof}\nWe will now present two proofs of this result, the first proof giving a connection with  the generating function of the Stirling operators of the second kind, the second proof giving a connection with the Laplace transform of the Poisson functional. \n\n\n{\\it Proof 1}. We have, by \\eqref{xtsaer5aw5},\n\\begin{align*}\n&\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle T_n(\\omega),\\xi^{\\otimes n}\\rangle=\n\\sum_{n=0}^\\infty\\frac1{n!}\\,\\sum_{k=0}^n\\langle\\omega^{\\otimes k},\\mathbf S(n,k)\\xi^{\\otimes n}\\rangle\\\\\n&\\quad=\\sum_{k=0}^\\infty\\Big\\langle\\omega^{\\otimes k},\\,\\sum_{n=k}^\\infty\\frac1{n!}\\,\\mathbf S(n,k)\\xi^{\\otimes n}\\Big\\rangle=\\sum_{k=0}^\\infty\\frac1{k!}\\,\\langle\\omega^{\\otimes k},(e^\\xi-1)^{\\otimes k}\\rangle\\\\\n&\\quad=\\sum_{k=0}^\\infty\\frac1{k!}\\,\\langle\\omega,e^\\xi-1\\rangle^k=\\exp\\big[\\langle\\omega,e^{\\xi}-1\\rangle\\big].\n\\end{align*}\n\n{\\it Proof 2}. Using formulas  \\eqref{cydye6edd}, \\eqref{vydye}, \\eqref{vuffr7}, and Lemma \\ref{xseaq43},\n  we have\n$$\n\\sum_{n=0}^\\infty\\frac1{n!}\\,\\langle T_n(\\omega),\\xi^{\\otimes n}\\rangle=\n\\sum_{n=0}^\\infty \\frac1{n!}\\,\\mathbb E_\\omega(m^{(n)}_\\xi)=\\exp\\big[\\langle\\omega,e^{\\xi}-1\\rangle\\big].\\qquad \\qedhere\n$$\n\\end{proof}\n\n\\end{proposition}\n\n\\begin{corollary}\nThe Touchard polynomials satisfy the binomial property:\n$$T_n(\\omega+\\sigma)=\\sum_{k=0}^n\\binom nk T_k(\\omega)\\odot T_{n-k}(\\sigma),\\quad\\omega,\\sigma\\in M(X),\\ n\\in\\mathbb N.$$\n\\end{corollary}\n\n\\begin{proof}\nThe statement follows immediately from Proposition \\ref{cts65} and the proof of the implication $\\mathrm{(BT4)}\\Rightarrow \\mathrm{(BT1)}$ in \n\\cite[Theorem 4.1]{FKLO}. \n\\end{proof}\n\n\\begin{proposition}\\label{rseaopti}\nFor each $n\\in\\mathbb N_0$ and $\\omega\\in M(X)$,\n$$T_{n+1}(\\omega)=\\sum_{k=0}^n\\binom nk T_k(\\omega)\\odot\\omega^{[n+1-k]}_{\\hat 1}. $$\n\\end{proposition}\n\n\\begin{proof}\nFor each $\\xi\\in\\mathcal F(X)$ with support in $\\Lambda\\in\\mathcal B_0(X)$ and $\\omega\\in M(X)$, we have, by Proposition~\\ref{g6ew4} and formulas  \\eqref{cydye6edd}, \\eqref{vydye},\nand \\eqref{tyfdqwd6rqwx},\n\\begin{align*}\n\\langle T_{n+1}(\\omega),\\xi^{\\otimes(n+1)}\\rangle&=\\mathbb E_\\omega(m^{(n+1)}_\\xi)\\\\\n&=\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\int_\\Lambda\\eta(dx)\\xi(x)\\langle\\eta,\\xi\\rangle^n\\\\\n&=\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\int_\\Lambda\\omega(dx)\\xi(x)\\langle\\eta+\\delta_x,\\xi\\rangle^n\\\\\n&=\\sum_{k=0}^n\\binom nk\\int_{\\ddot\\Gamma_0(\\Lambda)}\\mathbb P_\\omega^\\Lambda(d\\eta)\\langle\\eta,\\xi\\rangle^k\\int_\\Lambda\\omega(dx)\\xi^{n+1-k}(x)\\\\\n&=\\Big\\langle\\sum_{k=0}^n\\binom nk T_k(\\omega)\\odot\\omega^{[n+1-k]}_{\\hat 1},\\xi^{\\otimes (n+1)}\\Big\\rangle.\\qedhere\n\\end{align*}\n\\end{proof}\n\nLet $M_1(X)\\subset M(X)$ denote the set of all probability measures on $X$. \nFor $\\nu\\in M_1(X)$, we define  $B_n(\\nu):=T_n(\\nu)\\in M^{(n)}(X)$ and call $B_n(\\nu)$ the {\\it $n$th Bell measure corresponding to $\\nu$}. By \\eqref{dtw5ua}, $B_n(\\nu)$ is a finite positive measure on $X^n$ such that the  $B_n(\\nu)$ measure of the whole $X^n$ is equal to $B_n$, the classical Bell number.\n\n\\begin{proposition}\\label{qwdwswxe}\nLet $n\\in\\mathbb N$ and $f^{(n)}:X^n\\to\\mathbb F$ be measurable and bounded. Let $\\nu\\in M_1(X)$ and let $(Z_n)_{n=1}^\\infty$ be a sequence of independent, identically distributed $\\mathbb F$-valued random variables whose distribution is equal to $\\nu$. Let $\\mathbb E(\\cdot)$ denote the corresponding expectation. Then\n\\begin{equation}\\label{re64u4e}\n\\langle B_n(\\nu),f^{(n)}\\rangle=e^{-1}\\,\\mathbb E\\bigg(1+\\sum_{k=1}^\\infty\\frac1{k!}\\sum_{i_1=1}^k\\dotsm\\sum_{i_n=1}^k f^{(n)}(Z_{i_1},\\dots,Z_{i_n})\\bigg).\\end{equation}\n\\end{proposition}\n\n\\begin{proof}Without loss of generality we may assume that the function $f^{(n)}$ is symmetric. If additionally, $f^{(n)}$ has a compact support, i.e., $f^{(n)}\\in\\mathcal F^{(n)}(X)$, then formula \\eqref{re64u4e} easily follows from Corollary~\\ref{sesawawqa2q} and Lemma~\\ref{gdxtssd}.  For a general function $f^{(n)}$, formula~\\eqref{re64u4e} can then be shown by approximating $f^{(n)}$ with functions $f^{(n)}\\chi_{k}^{(n)}$, where $\\chi_{k}^{(n)}$ is the indicator function of the set $\\Lambda_k^n$, and \n$\\{\\Lambda_k\\}_{k=1}^\\infty$ is an increasing sequence of compact subsets of $X$ such that  $\\Lambda_k\\uparrow\\ X$. \\end{proof}\n\n\\begin{remark}\nBy setting $f^{(n)}\\equiv 1$ in formula \\eqref{re64u4e}, we obtain Dobi\\'nski's formula $B_n=e^{-1}(1+\\sum_{k=1}^\\infty\\frac{k^n}{k!})$.\n\\end{remark}\n\n\nFor $\\nu\\in M_1(X)$, we define $D_n(\\nu):=T_n(-\\nu)\\in M^{(n)}(X)$ and call $D_n(\\nu)$ the {\\it $n$th Rao--Uppuluri--Carpenter measure corresponding to $\\nu$}. The measures $D_n(\\nu)$ generalize the corresponding numbers $D_n$ in classical combinatorics, see e.g. \\cite[Section 9.4]{QG}.   The $D_n(\\nu)$ is a signed measure on $X^n$ of finite total variation, and the $D_n(\\nu)$ measure of the whole $X^n$ is equal to $D_n$. \nProposition \\ref{rseaopti} implies\n$$D_{n+1}(\\nu)=-\\sum_{k=0}^n\\binom nk D_k(\\nu)\\odot\\nu^{[n+1-k]}_{\\hat 1}. $$\nUnder the assumptions of Proposition \\ref{qwdwswxe}, we easily obtain\n$$\\langle D_n(\\nu),f^{(n)}\\rangle=e\\,\\mathbb E\\bigg(1+\\sum_{k=1}^\\infty\\frac{(-1)^k}{k!}\\sum_{i_1=1}^k\\dotsm\\sum_{i_n=1}^k f^{(n)}(X_{i_1},\\dots,X_{i_n})\\bigg).$$\n\n\n\n\n\\section{An open problem}\\label{xzrwaq4yq4q}\n\nAmong several open problems arising from the present paper, let us mention only a problem related to Theorem~\\ref{taw4aq234q}. \n\n\n\n\nThe idea of Katriel's theorem \\cite{Katriel} was used in order to define and study several deformations of Stirling numbers, see e.g.\\ \\cite{KK92,MS}.  \nOne may similarly use the idea of Theorem~\\ref{taw4aq234q} in order to define and study generalizations of  Stirling operators in the case of various deformed commutation relations, see e.g.\\ \\cite{BEH,bozejkolytvynovwysoczanskiCMP2012,BLW-Q,BS}. Indeed, assume that we are given a family of creation operators $a^+(\\xi)$ and annihilation operators $a^-(\\xi)$ ($\\xi\\in\\mathcal F(X)$) that satisfy certain deformed commutation relations. Then, similarly to the CCR case, we can define operators $\\rho(\\xi)=\\int_X\\xi(x)a^+(x)a^-(x)\\sigma(dx)$ and the Wick product ${:}\\,\\rho(x_1)\\dotsm \\rho(x_n){:}$ by formula \\eqref{cd55uw}. Now, the question is: Can we define deformed Stirling operators, $\\mathbf S(n,k)$, in such a way that formula \\eqref{ds5w5u4}  holds in this setting? In the following two cases, one can easily answer this question. \n\nFirst, consider the case of the anyon commutation relations \\cite{bozejkolytvynovwysoczanskiCMP2012},\n\\begin{gather*}\na^+(x)a^+(y)-Q(y,x)a^+(x)a^+(y)=0,\\quad a^-(x)a^-(y)-Q(y,x)a^-(x)a^-(y)=0,\\\\\na^-(x)a^+(y)-Q(x,y)a^+(y)a^-(x)=\\delta(x,y).\n\\end{gather*} \nHere $Q(x,y)$ is a complex-valued function on $X^2$ such that $|Q(x,y)|=1$ and $Q(y,x)=\\overline{Q(x,y)}$. An easy calculation shows that   Lemma~\\ref{erws5w3}  holds true in this case, hence the operators $\\mathbf S(n,k)$ remain the same as in the case of the CCR. \n\nSecond, consider the case of the free commutation relation, \n$$a^-(x)a^+(y)=\\delta(x,y). $$\n As easily seen, \n   $$\\rho(\\xi_1)\\dotsm\\rho(\\xi_n)=\\rho(\\xi_1\\dotsm\\xi_n)$$\n   for any $\\xi_1,\\dots,\\xi_n\\in\\mathcal F(X)$. \n Hence, in this case, $\\mathbf S(n,1)$ is the operator $\\mathbb D^{(n)}$ given by \\eqref{cq5y42w}, and $\\mathbf S(n,k)=0$ if $k\\ne1$. \n \n\n\nNow one can naturally ask which generalizations of the Stirling operators appear if the operator-valued distributions $(a^+(x),a^-(x))_{x\\in X}$ satisfy the $q$-commutation relation\nfor $q\\in(-1,1)$ with  $q\\ne0$:\n\\begin{equation}\\label{vcyei76vhvv}\na^-(x)a^+(y)-qa^+(y)a^-(x)=\\delta(x,y),\\end{equation}\nsee \\cite{BS}. If the set $X$ has a single point, the commutation relation \\eqref{vcyei76vhvv} becomes\n$$a^-a^+-qa^+a^-=1,$$\ncompare with \\eqref{xzzaaaa}. In this one-mode case, the corresponding Stirling  numbers were studied in \\cite{KK92}, see also  \\cite[Section 7.2]{MS} and the references therein. However, in the general case of  \\eqref{vcyei76vhvv}, an explicit representation of the product $\\rho(\\xi_1)\\dotsm\\rho(\\xi_n)$ through Wick ordered operators is an open, interesting problem. \n\n\n\nOne may also extend this problem to the case of the commutation relation discussed in \\cite{BLW-Q}, in which case the parameter $q$ is replaced by a complex-valued function $Q(x,y)$ on $X^2$ such that $|Q(x,y)|\\le 1$ and $Q(y,x)=\\overline{Q(x,y)}$. Another important generalization of \\eqref{vcyei76vhvv} was considered in \\cite{BEH} and is related to Coxeter groups of type B.\n\n\n\\subsection*{Acknowledgments}\n\nMJO was supported by the Portuguese \nnational funds through FCT--Funda{\\c c}\\~ao para a Ci\\^encia e a Tecnologia, \nwithin the project UIDB\/04561\/2020.  EL and MJO are grateful to the London Mathematical Society for partially supporting the visit of MJO to Swansea University.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\chapter*{Survival, absorption and escape of interacting diffusing particles}\\label{ra_ch1}\n\n\\author{Tal Agranov  and Baruch Meerson}\n\n\\address{Racah Institute of Physics, Hebrew University of\n\tJerusalem, Jerusalem 91904, Israel}\n\n\\begin{abstract}\nAt finite concentrations of reacting molecules,  kinetics of diffusion-controlled reactions is affected by intra-reactant interactions. As a result, multi-particle reaction statistics cannot be deduced from  single-particle results. Here we briefly review a recent progress in overcoming this fundamental difficulty. We show that the fluctuating hydrodynamics and macroscopic fluctuation theory provide a simple, general and versatile framework for studying a whole class of problems  of survival, absorption and escape of interacting diffusing particles.\n\\end{abstract}\n\\markright{Survival, absorption and escape}\n\\body\n\n\\tableofcontents\n\n\\section{Introduction}\\label{1}\nKinetics of many diffusion-controlled reactions is affected by intra-reactant interactions. This happens when the density of the reacting molecules is not too small.  Although the importance of interactions may have been recognized for a long time, there has been very little progress in their account in theory. Here we will briefly review, and slightly generalize, one promising approach toward solving this long-standing problem \\cite{MVK,surv,fullabsorb,multi,narrow}. We will consider several prototypical gas settings.  The first group of settings -- interior settings -- deals with interacting diffusing molecules inside a domain (think about a living cell).  The second group -- exterior settings -- deals with molecules surrounding a domain. In both cases the domain boundary, or part of it, absorbs the molecules upon impact, signaling that a reaction occurred.\n\nThe interior settings give simplified descriptions of inter-cellular transport in the living cell, where molecules search for a correct location within a cell  membrane. The efficacy of the inter-cellular transport is determined by the absorption rate of the molecules \\cite{bress}. The interior settings include the \\emph{narrow escape problem}, see the right panel of Fig.\\ref{hure}, which is well studied in the case of non-interacting diffusing molecules trying to escape from a closed domain via a small hole in its boundary \\cite{keller1,bress,beni,bookz,rev,russians}.\n\nThe exterior problems are different but closely related. The case when the boundary of the domain is fully absorbing is known as the target search, or target survival problem \\cite{rice,osh,red}. This describes the situation where the molecules of one reactant -- a minority -- can be viewed as big and immobile, whereas the molecules of another reactant  -- a majority -- are small and mobile. A different scenario happens when molecules get absorbed only through some absorbing patches -- receptors -- distributed on the otherwise reflecting domain boundary \\cite{berg,bergbook}, see the left panel of Fig. \\ref{hure}.\n\nIf the diffusing molecules are treated as noninteracting random walkers, the calculation of the effective reaction rates and its fluctuation  statistics boils down to calculating a single-particle probability.  Interactions invalidate the single-particle picture and make the problem very difficult.  Fortunately, a new simplification emerges if there are sufficiently many interacting diffusing particles in the relevant region of space. In this case one can use the fluctuating hydrodynamics, which goes back to Landau and Lifshitz \\cite{LL}, and a large deviation theory for it. As a convenient and well controlled first-principle model, Refs. \\cite{MVK,surv,fullabsorb,multi,narrow} adopted\ndiffusive lattice gases, where the fluctuating hydrodynamics is well established \\cite{Spohn}. The corresponding large-deviation theory has recently become available under the name of macroscopic fluctuation theory (MFT) \\cite{MFTreview}.\n\n\\begin{figure} [ht]\\label{hure}\n \t\\includegraphics[width=0.49\\textwidth,clip=]{fig1a.pdf}\n  \t\\includegraphics[width=0.49\\textwidth,clip=]{fig1b.pdf}\n \t\\caption{Left panel: An exterior problem with multiple absorbing patches shown in white. The rest of the boundary is reflecting. Right panel: Narrow escape of multiple particles through a small hole in the boundary, the rest of the boundary being reflecting. }\n \\end{figure}\n\n\\section{Fluctuating Hydrodynamics and the MFT}\nFluctuating hydrodynamics is a coarse-grained description of a gas of particles in terms of the particle number density $\\rho(\\mathbf{x},t)$ \\cite{Spohn,KL}. The average particle density of a lattice gas obeys a diffusion equation $\\partial_t \\rho = \\nabla \\cdot \\left[D(\\rho) \\nabla \\rho\\right]$, whereas macroscopic fluctuations are described by the conservative Langevin equation\n\\begin{equation}\n\\partial_t \\rho = -\\nabla \\cdot \\mathbf{J}\\,,\\quad\\mathbf{J}=-D(\\rho) \\nabla \\rho-\\sqrt{\\sigma(\\rho)}\\boldsymbol{\\eta}(\\mathbf{x},t),\\label{lang}\n\\end{equation}\t\nwhere $\\boldsymbol{\\eta}(\\mathbf{x},t)$ is a zero-mean Gaussian noise, delta-correlated in space and in time \\cite{Spohn,KL,Liggett}.  The diffusivity $D(\\rho)\\geq 0$ and the mobility $\\sigma(\\rho)\\geq 0$ are to be obtained, for each lattice gas, from the microscopic model. The simplest case, whose coarse-grained behavior coincides with that of non-interacting Brownian particles is the gas of Random Walkers (RWs)\\cite{Paulbook}, where one has $D(\\rho)=D_0$ and $\\sigma(\\rho)=2D_0\\rho$ \\cite{Spohn}. A model with interactions, which we will focus on, is the symmetric simple exclusion process (SSEP) \\cite{Spohn} which accounts, in a simple way, for excluded volume interactions. The SSEP's average behavior coincides with that of the RW's, as they share the same density-independent diffusivity $D_0$. Their fluctuations, however, are different as the SSEP's mobility  $\\sigma(\\rho)=2D_0\\rho(1-\\rho a^3)$ is a non-linear function of $\\rho$ \\cite{Spohn,KL}. Here $a$ is the lattice constant which we set to unity, so that  $0\\leq\\rho\\leq1$. Like many other lattice gases, the SSEP behaves in its dilute limit as non-interacting RWs.\n\nTo develop a large-deviation theory for Eq. (\\ref{lang}), one starts from a path integral for the probability of observing a joint density and flux histories $\\rho(\\mathbf{x},t), \\mathbf{J}(\\mathbf{x},t)$, constrained by the continuity equation (\\ref{lang}):\n\\begin{eqnarray}\\nonumber\n&&\\!\\!\\!\\!\\!\\!\\mathcal P =\\int\\mathcal{D}\\rho\\mathcal{D}\\mathbf{J}\n\\prod_{\\mathbf{x},t}\\delta(\\partial_{t}{\\rho}+\\nabla \\cdot \\mathbf{J})\\,\n\\exp\\left(-\\mathcal{S}\\right),\\label{path0}\\\\\n&&\\!\\!\\!\\!\\!\\!\\mathcal{S}\\left[\\rho(\\mathbf{x},t),\\mathbf{J}(\\mathbf{x},t)\\right]=\\int_0^Tdt\\int d^3\\mathbf{x}\\frac{\\left[\\mathbf{J}+D(\\rho)\\nabla \\rho\\right]^2}{2\\sigma(\\rho)}.\\label{path1}\n\\end{eqnarray}\nIn the interior and exterior settings, presented above,  we condition the process on a specified (zero or non-zero) particle absorption current by a given time $T$. Therefore, we need to evaluate the path integral over only those density and flux histories which led to the specified current. Assuming that all characteristic length scales involve large numbers of particles, the dominant contribution for $\\mathcal P$ comes from the \\emph{optimal fluctuation}: the most probable history $\\rho(\\mathbf{x},t), \\,\\mathbf{J}(\\mathbf{x},t)$ \\cite{MSR}.\nThe ensuing minimization procedure yields the Euler-Lagrange equation which can be cast into a Hamiltonian form, known as the MFT equations \\cite{MFTreview}. The minimization procedure also generates problem-specific boundary conditions. Evaluating the minimum action $S$ over the solutions to the minimization problem yields the desired probability ${\\mathcal P}$ up to a pre-exponential factor,\n\\begin{eqnarray}\n\\label{actionmain}\n-\\ln {\\mathcal P}\n\\simeq S \\equiv\\min_{\\rho,\\mathbf{J}}\\mathcal{ S}\\left[\\rho(\\mathbf{x},t),\\mathbf{J}(\\mathbf{x},t)\\right].\n\\end{eqnarray}\nIn general, the minimization problem is not solvable analytically. Considerable simplifications arise in the limits of very long and very short times compared to a characteristic diffusion time of the problem, see below. We will first address the long-time limit, where the optimal gas density and flux become \\emph{stationary}, and devote the last chapter \\ref{nonstat} to the non-stationary regime relevant for short times. Following Refs. \\cite{MVK,surv,fullabsorb,multi,narrow}, we will consider two types of initial conditions. The first is a random (or annealed) initial condition, where particles are randomly distributed in space with an average density $\\rho_0$. It describes the situation where the gas has enough time to equilibrate before the process starts (for example, before the receptor becomes available). The other is deterministic (or quenched) initial condition with a uniform density $\\rho_0$. When considering long times the details of the initial condition become irrelevant. In contrast, the short-time statistics strongly depends on the initial condition.\n\n\\section{The exterior problem}\nSuppose a gas of diffusing particles fills the whole space outside of a simply connected $3d$ domain of a linear size $L$.\nThe domain boundary $\\Omega$ is composed of a reflecting part $\\Omega_r$ and a complementary absorbing part $\\Omega_a$. Whenever a particle hits $\\Omega_a$, it is immediately absorbed, which sets\n\\begin{eqnarray}\\label{blang}\n&&\\rho(\\mathbf{x}\\in{\\Omega_a},t)=0.\n\\end{eqnarray}\nWhenever a particle hits $\\Omega_r$, it is reflected, which sets a zero-flux boundary condition\n\\begin{eqnarray}\\label{blang2}\n&&\\mathbf{J}(\\mathbf{x}\\in{\\Omega_r},t)\\cdot\\hat{n}=0,\n\\end{eqnarray}\nwhere $\\hat{n}$ denotes a local unit vector normal to the domain boundary and directed into the domain.\nFor a fully absorbing domain there is no reflecting part. The simpler latter setting is known as ``the target search problem\" \\cite{rice,osh,red}; it captures the essence of many diffusion-controlled chemical reactions.\nA more involved setting is a domain whose boundary has several disjoint absorbing patches (receptors) \\cite{berg,bergbook}.\nFor both random and deterministic initial conditions the boundary condition at infinity is\n \\begin{eqnarray}\\label{blang3}\n \\rho(|\\mathbf{x}|\\rightarrow\\infty,t)=\\rho_0.\n \\end{eqnarray}\nThe quantity of interest is the probability $\\mathcal{P}(n,T,\\rho_0)$ that $N$ gas particles were absorbed during the time interval $0<t<T$, where $n\\equiv N\/T$ is the absorption current. For multiple absorbing patches one is interested in the corresponding multivariate probability.\n\nAt times $T$ much longer than the diffusion time $L^2\/D(\\rho_0)$ the system reaches a non-equilibrium steady state, where the average gas density $\\bar{\\rho}(\\mathbf{x})$ is independent of time. In its turn, the average number of absorbed particles, $\\bar{N}$, is proportional to time, so that the\naverage absorption current $\\bar{n}=\\bar{N}\/T$ is independent of time. Similarly, for a whole class of lattice gases the \\emph{optimal} density and flux, conditioned on a specified current $n\\neq\\bar{n}$, also become stationary \\cite{layers}.\nAs a result, ${\\mathcal P}(n,T,\\rho_0)$  exponentially decays with time $T$. A similar situation occurs in the context of stationary fluctuations of current in diffusive lattice gases, driven by density reservoirs at the boundaries. There the stationarity of the optimal gas density and flux is known under the name of the ``additivity principle\"  \\cite{bd}, and we use this term here as well.\n\n\\subsection{Target survival}\\label{exsurv}\nThe authors of\\cite{MVK} considered a fully absorbing domain ($\\Omega_a=\\Omega$) and studied the probability $\\mathcal{P}(n=0,T,\\rho_0)$ that not a single particle  gets absorbed by time $T$. This probability is often called the survival probability; it is a key quantity in determining the distribution of absorption times of the first particle. The latter is given by $\\mathcal{P}_{\\text{first}}(T,\\rho_0)=-\\partial_T\\mathcal{P}(n=0,T,\\rho_0)$. As a result, the mean absorption time of the first particle, which determines the average reaction rate, is $\\langle T \\rangle=\\int_0^Tdt \\mathcal{P}(n=0,T,\\rho_0)$.\n\nPreviously, the target survival has been extensively studied, by exploiting single particle results, in the case when the particles are noninteracting RWs. The probability that the target survives until a long time $T$ decays exponentially in time,\n\\begin{equation}\\label{steady}\n-\\ln{\\mathcal P}(n=0,T,\\rho_0)\\simeq T s(\\rho_0),\n\\end{equation}\nwith the decay rate \\cite{9,10,11,12,13,14,15,16,17}\n\\begin{equation}\ns(\\rho_0)= 4\\pi CD_0 \\rho_0,\n\\label{soutrw}\n\\end{equation}\nwhere $C$ is the electrical capacitance of a conductor whose shape is $\\Omega$. For a sphere of  radius  $R$ one has $C=R$.\nAs shown in\\cite{MVK},  the long-time expression (\\ref{steady}) holds for interacting lattice gases as well, and the steady-state MFT calculations yield model-specific $s(\\rho_0)$. Here is a scheme of the calculations. As one can show,  the stationary particle flux, optimal for survival, vanishes everywhere \\cite{surv,MVK}. In other words, the\nfluctuating contribution to the optimal flux exactly counterbalances the deterministic contribution, thus preventing the particles from being absorbed. One is left with finding the optimal  density profile. Upon the ansatz $\\mathbf{J}=0$ and $\\rho=\\rho(\\mathbf{x})$ in Eq.~(\\ref{path1}), the action $\\mathcal{S}$ becomes proportional to $T$, and the problem reduces to minimizing the \\emph{action rate} functional\n\\begin{equation}\\label{lag}\n\\mathfrak{s}\\left[\\rho\\left(\\mathbf{x}\\right)\\right]=\\int d^3\\mathbf{x}\\frac{\\left[D(\\rho)\\nabla \\rho\\right]^2}{2\\sigma(\\rho)},\n\\end{equation}\nsubject to the boundary conditions (\\ref{blang}) and (\\ref{blang3}).\nIt is convenient to make the transformation  $u(\\mathbf{x})=f\\left[\\rho\\left(\\mathbf{x}\\right)\\right]$,\nwhere \\cite{conv,MVK}\n\\begin{equation}\nf(\\rho)=\\int_0^\\rho dz \\frac{D(z)}{\\sqrt{\\sigma (z)}}.\\label{transform1}\n\\end{equation}\nWe denote the inverse function, $f^{-1}$, by $F$.\nThe transformation (\\ref{transform1}) reduces the minimization problem to solving the Laplace's equation\n\\begin{equation}\n\\nabla^2 u=0. \\label{ustatnd}\n\\end{equation}\nReturning to the original variables, the solution is given in terms of the effective electrostatic potential around a conductor with boundary $\\Omega$ kept at unit voltage $\\phi\\left(\\mathbf{x}\\right)$. In simple cases (\\textit{e.g.}, when $\\Omega$ is a disk, a sphere or a spheroid), $\\phi(\\mathbf{x})$ can be found explicitly \\cite{jackson} . The stationary density profile, optimal for the particle survival, is a function of this potential alone:\n\\begin{equation}\\label{rho0}\n \\rho(\\mathbf{x})=F\\{f(\\rho_0)[1-\\phi(\\mathbf{x})]\\}.\n \\end{equation}\nThe action rate~(\\ref{lag}), evaluated over the solution (\\ref{rho0}), yields the decay rate $s(\\rho_0)$ entering Eq. (\\ref{steady}). It is given by the electrostatic energy created by a conductor $\\Omega$ held at voltage $f(\\rho_0)$:\n\\begin{equation}\n s\\left(\\rho_0\\right)= 2\\pi  C f^2(\\rho_0).\n\\label{sout}\n\\end{equation}\n\nRemarkably, the entire effect of interactions is encoded in\nthe density dependence $f(\\rho_0)$, coming from the nonlinear transformation~(\\ref{transform1}). The geometry dependence is universal for all gases of this class and is given by the capacitance $C$. When specialized to the RWs, Eq.~(\\ref{sout}) reduces to Eq.~(\\ref{soutrw}), as to be expected.\n\nFor the SSEP  Eqs.~(\\ref{transform1}) and (\\ref{sout}) yield\n\\begin{equation}\ns(\\rho_0)=4\\pi C D_0 \\arcsin^2(\\sqrt{\\rho_0}).\\label{soutssep}\n\\end{equation}\nThis decay rate is larger than that of the Rws~(\\ref{soutrw}), as to be expected because of the effective\nmutual repulsion of the particles, see the left panel of Fig.~\\ref{t}. Earlier works \\cite{kuz,cum1,cum2,seki1,seki2} on the target survival for the SSEP only established some bounds on $\\mathcal{P}(n=0,T,\\rho_0)$.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth,clip=]{fig2a.pdf}\n\\includegraphics[width=0.45\\textwidth,clip=]{fig2b.pdf}\n\t\\caption{Left panel: the function $\\arcsin^2\\sqrt{\\rho_0}$ which describes the density dependence of the decay rate, Eq.~(\\ref{soutssep}), of the target survival probability for the SSEP. The straight line is the same quantity for the RWs, Eq.~(\\ref{soutrw}). Right panel: the large-deviation function $\\tilde{s}\\left(N\/\\bar{N},\\rho_0\\right)$ from Eq.~(\\ref{fullssep}), which describes the full statistics of absorption in the exterior problem for the SSEP with $\\rho_0=0.75$. The dashed line is the same quantity for the RWs, see Eq.~(\\ref{pois}). The fat point at $N=0$ shows the survival probability~(\\ref{soutssep}).}\n\t\\label{t}\n\\end{figure}\n\n\\subsection{Full statistics of absorption}\\label{exfull}\nWhen conditioning on arbitrary $n>0$, we impose the constraint $\\oint_{\\mathbf{x}\\in\\Omega}\\mathbf{J}\\cdot\\hat n=n$.\nThe stationary version of  Eq.~(\\ref{lang}) is $\\nabla \\cdot \\mathbf{J} = 0$. This fact, alongside with the boundary conditions and the additional assumption that the field $\\mathbf{J}$ is irrotational, uniquely defines $\\mathbf{J}$  \\cite{irot,irot2}. It is given in terms of the average steady-state flux field\n\\begin{equation}\n\\bar{\\mathbf{J}}\\left(\\mathbf{x}\\right)=-D(\\bar{\\rho})\\nabla\\bar{\\rho},\\label{j1}\n\\end{equation}\nwhich is also solenoidal and irrotational but obeys  $\\oint_{\\mathbf{x}\\in\\Omega}\\bar{\\mathbf{J}}\\cdot\\hat n=\\bar{n}$. Then it follows that the optimal absorption flux is simply\n\\begin{equation}\n\\mathbf{J}=\\frac{n}{\\bar{n}}\\bar{\\mathbf{J}},\\label{j}\n\\end{equation}\nsee also\\cite{fullabsorb,shpiel}. The average flux (\\ref{j1}) is given by the effective electrostatic potential $\\phi\\left(\\mathbf{x}\\right)$, defined in the previous section \\cite{berg,multi}:\n\\begin{equation}\n\\bar{\\mathbf{J}}=V\\left(\\rho_0\\right)\\nabla\\phi\\left(\\mathbf{x}\\right),\\quad V(\\rho_0)\\equiv\\int_{0}^{\\rho_0}D(z)dz.\\label{j2}\n\\end{equation}\nThis potential plays the role of the natural ``spatial coordinate\" of the problem.  The optimal density field $\\rho$ is a function of the potential alone, and the problem is effectively one-dimensional with respect to this coordinate \\cite{shpiel,multi,fullabsorb}:\n\\begin{equation}\n\\rho\\left(\\mathbf{x}\\right)=\\rho_1\\left[\\phi\\left(\\mathbf{x}\\right)\\right].\n\\end{equation}\nThe function $\\rho_1\\left(u\\right)$ is defined on the segment $u\\in[0,1]$ and obeys the boundary conditions $\\rho_1\\left(0\\right)=\\rho_0$ and $\\rho_1\\left(1\\right)=0$.\nIt is to be determined by a one-dimensional variational problem which involves the minimization of the action~(\\ref{actionmain}). Upon the ansatz $\\mathbf{J}=(n\/\\bar{n})\\bar{\\mathbf{J}}$ and $\\rho\\left(\\mathbf{x}\\right)=\\rho_1\\left[\\phi\\left(\\mathbf{x}\\right)\\right]$ in Eq.~(\\ref{path1}), the action becomes proportional to $T$, so one needs to minimize an action rate functional. After some algebra the problem is reduced to minimizing a one-dimensional functional in terms of $\\rho_1\\left(u\\right)$\\cite{fullabsorb,shpiel,multi}:\n\\begin{equation}\n\\mathfrak{s}\\left[\\rho_1\\left(u\\right)\\right]=4\\pi C\\times\\int_0^1\\frac{\\left\\{\\frac{n}{4\\pi C}+D\\left[\\rho_1\\left(u\\right)\\right]\\rho_1^{\\prime}\\left(u\\right)\\right\\}^2}{2\\sigma\\left[\\rho_1\\left(u\\right)\\right]}du\\label{s1},\n\\end{equation}\nwhere, as in the previous section, $C$ is the capacitance of the absorbing domain. The same functional appears in the context of the long-time statistics of the current in a lattice gas on a segment, driven by two reservoirs with different gas densities at the segment's ends \\cite{bd}. In the latter setting, the probability of having the current $n$  decays exponentially in time, $-\\ln\\mathcal{P}_{\\text{1d}}\\left(n,T,\\rho_0\\right)\\simeq Ts_{\\text{1d}}\\left(n,\\rho_0\\right)$. The action rate $s_{\\text{1d}}\\left(n,\\rho_0\\right)$ is simply related to the action rate, obtained by minimization of Eq.~(\\ref{s1}) $s(n,\\rho_0)$:\n\\begin{equation}\ns\\left(n,\\rho_0\\right)=4\\pi C s_{\\text{1d}}\\left(\\frac{n}{4\\pi C},\\rho_0\\right) .\\label{fulls}\n\\end{equation}\nThis sets a universal relation between the different problems \\cite{fullabsorb,shpiel,multi}.\nThe geometry enters only through the capacitance of the domain.\n\nThe one-dimensional problem is exactly solvable in quadratures \\cite{bd,fullabsorb}. For the SSEP, the result can be written as\\cite{fullabsorb}\n\\begin{equation}\n-\\ln\\mathcal{P}\\left(N,T,\\rho_0\\right)\\simeq\\bar{N}\\tilde{s}\\left(\\frac{N}{\\bar{N}},\\rho_0\\right),\\label{fullssep}\n\\end{equation}\nwhere $\\bar{N} = 4 \\pi C D_0 \\rho_0 T$. The function $\\tilde{s}(N\/\\bar{N},\\rho_0)$ is shown in the right panel of Fig. \\ref{t}. In the limit of $\\rho_0\\to 0$ the function $\\tilde{s}(N\/\\bar{N},\\rho_0)$ describes the RWs and corresponds to the $N \\gg 1$ limit of the Poisson distribution with mean $\\bar{N}$:\n\\begin{equation}\\label{pois}\n\\tilde{s}\\left(\\frac{N}{\\bar{N}},\\rho_0\\to 0\\right)=\\frac{N}{\\bar{N}}\\ln\\frac{N}{\\bar{N}}-\\frac{N}{\\bar{N}}+1.\n\\end{equation}\n\n\n\\subsection{Multiple absorbing patches}\nRef.\\cite{multi} considered particle absorption by \\emph{multiple} patches $\\Omega_{i},\\,i=1,2,\\dots,s$, distributed on an otherwise reflecting boundary, see the left panel of Fig.~\\ref{hure}.\nThe results brought some surprises. To start with, the optimal particle flux field, conditioned on a specified joint absorption statistics $\\left\\{n_i\\right\\}_{i=1}^s$, exhibits a large-scale \\emph{vorticity} $\\vec{\\omega}=\\mathbf{\\nabla}\\times{\\mathbf{J}}\\neq 0$ \\cite{multi}. The vorticity emerges even when the particles are non-interacting RWs, and for any geometry, as long as there are more than one absorbing patch. This makes the problem more involved as one should consider a joint variational problem for the flux and the density given by Eq.~(\\ref{actionmain}). A simplification comes when considering the statistics of typical, small fluctuations, $\\delta n_i=n_i-\\bar{n}_i \\ll \\bar{n}_i$, of the absorption currents around their mean values. Here one can linearize the MFT equations around the mean values $\\bar{\\rho}$ and $\\bar{\\mathbf{J}}$. The resulting solution\\cite{multi} describes a multivariate Gaussian distribution\n\\begin{equation}\\label{mainresult}\n{\\mathcal P} \\simeq\n\\frac{T^{s\/2}}{(2\\pi)^{s\/2}\\,{[\\text{det}\\,\\boldsymbol C]^{1\/2}}}\\exp\\left(-\\frac{T}{2}\\sum_{i,j=1}^{s}\\delta n_iC_{ij}^{-1}\\delta n_j\\right).\n\\end{equation}\nHere  $\\boldsymbol C$ is an $s\\times s$ positive-definite symmetric matrix which depends on $\\rho_0$ and on the geometry of the problem, but is independent of time. Equation~(\\ref{mainresult}) suffices for the evaluation of the variance of the\njoint probability distribution \\cite{KrMe,multi}. Each diagonal element of $\\boldsymbol C$\ndescribes the variance of the current into the corresponding patch:\n\\begin{equation}\\label{vari}\n\\overline{\\delta n_i^2} = \\frac{C_{ii}}{T},\n\\end{equation}\nwhere the overline denotes averaging with respect to the Gaussian distribution (\\ref{mainresult}).\nThe off-diagonal elements of $\\boldsymbol C$ describe cross-correlations between the currents into different patches:\n\\begin{equation}\\label{corr}\n\\overline{ \\delta n_i\\delta n_j } = \\frac{C_{ij}}{T}.\n\\end{equation}\nThe optimal density field can again be presented via an electrostatic analogue which involves $s$ characteristic potentials $\\phi_i\\left(\\mathbf{x}\\right)$. Each of the potentials appears when the corresponding conducting patch $\\Omega _i$ is held at unit voltage, the rest of the conducting patches are grounded, and the Neumann boundary condition is specified at the reflecting part of the boundary. The potentials $\\phi_i$-s can be found explicitly in simple cases \\cite{jackson,multi}. The covariance matrix $\\boldsymbol C$ is given in terms of the volume integrals involving the characteristic potentials:\n\\begin{equation}\nC_{ij}= \\int d\\mathbf{x} \\,\\sigma (\\bar{\\rho})\\nabla\\phi_i\\cdot\\nabla\\phi_j.\\label{mat}\n\\end{equation}\nRemarkably, general properties of the cross-correlations turn out to be independent of the system's geometry, and are determined solely by the functions $D(\\rho)$ and $\\sigma(\\rho)$ \\cite{multi}. Of course, there are no cross-correlations if the particles do not interact.\nWhat is the sign of cross-correlations for an interacting gas? As Ref.\\cite{multi} showed, if\n\\begin{equation}\nD(\\bar{\\rho})\\sigma^{\\prime\\prime}(\\bar{\\rho})<D^{\\prime}(\\bar{\\rho})\\sigma^{\\prime}(\\bar{\\rho}), \\label{condition}\n\\end{equation}\nfor any value of $\\bar{\\rho}\\in[0,\\rho_0]$,\nthen the currents into different patches $i\\neq j$ are all anti-correlated, $\\overline{ \\delta n_i\\delta n_j }<0$, regardless of the system's geometry. In particular, this is always true for the SSEP.\n\nInterestingly, the same condition (\\ref{condition}) guarantees the validity of the additivity principle (that is, stationarity of the optimal density profile in the long-time limit) for an arbitrary value of current \\cite{ber,main}, and also determines the sign of the two-point \\emph{density} correlation function \\cite{bertinistat,tridib}, in single-current systems.\n\n\n\n\\section{The interior problem}\n\nFor many non-interacting particles the theory is based on the well-established single-particle results\\cite{keller1,bress,beni,bookz,russians,RoKim,many2}\nAs in the exterior problem, the long-time survival probability in this case\ndecays exponentially in time, $-\\ln\\mathcal P\\left(T,\\rho_0\\right)\\simeq Ts\\left(\\rho_0\\right)$. The geometry dependence of $s\\left(\\rho_0\\right)$ is, however,  different\\cite{keller1,bress,beni,bookz,russians,RoKim,many2}:\n\\begin{equation}\\label{suas}\ns(\\rho_0)=D_0 \\mu_0^2 \\rho_0V,\n\\end{equation}\nwhere  $V$ is the domain's volume, and $\\mu_0^2$ is the principal eigenvalue of the eigenvalue problem $\\nabla^2 \\Psi+\\mu^2 \\Psi=0$ inside the domain with the mixed boundary conditions $\\Psi(\\mathbf{x} \\in  \\Omega_a,t) = \\nabla\\Psi(\\mathbf{x} \\in \\Omega_r,t)\\cdot\\hat{n}=0$.\n\n\nWhat happens for interacting particles? As in section \\ref{exsurv}, the optimal flux field, conditioned on the survival of all particles, vanishes identically. As a result, we can determine the optimal gas profile for survival by minimizing the same action rate functional~(\\ref{lag}), but now the integration is carried over the space inside the domain. One distinct feature of the interior survival problem is conservation of the total number of particles, which enters the variational problem as a constraint,\n\\begin{equation}\\label{cons}\n\\int d^3\\mathbf{x}\\,\\rho(\\mathbf{x})=\\rho_0 V\n\\end{equation}\nand calls for a Lagrange multiplier $\\Lambda$.  The transformation of variables (\\ref{transform1}) proves useful in the interior case as well. The resulting Euler-Lagrange equation for $u$ has the form of\na non-linear Poisson equation \\cite{surv},\n\\begin{equation}\n\\nabla^2 u +  \\Lambda \\frac{d F(u)}{du}=0, \\label{ustatnd2}\n\\end{equation}\nwith the mixed boundary conditions \\cite{boundary},\n\\begin{equation}\nu(\\mathbf{x}\\in{\\Omega_a})=0 ,\\quad\\nabla u(\\mathbf{x}\\in{\\Omega_r})\\cdot\\hat{n}=0.\n\\label{bqu2}\n\\end{equation}\nFor the RWs Eq.~(\\ref{transform1}) yields\n$F\\left( u\\right)=u^2\/2D_0$, and Eq.~(\\ref{ustatnd2}) becomes the Helmholtz equation\n\\begin{equation}\\label{helm2}\n\\nabla^2 u+\\mu^2 \\,u=0 ,\n\\end{equation}\nwith $ \\mu^2\\equiv\\Lambda\/D_0$ playing the role of the eigenvalue. The minimum action is achieved for the fundamental mode,\nand the resulting expression for $\\mathfrak{s}[u(\\mathbf{x})]= s(\\rho_0)$ reproduces the result quoted in Eq.~(\\ref{suas})\\cite{surv,narrow}.\nFor the SSEP, upon rescaling $U= \\sqrt{2\/D_0}\\, u$ and $C= \\Lambda\/D_0$, Eq.~(\\ref{ustatnd2}) becomes the stationary sine-Gordon equation\n\\begin{equation}\n\t\\nabla^2 U +  C\\sin\\,U=0. \\label{ustatssepnd1}\n\\end{equation}\n\n\\subsection{Particle survival inside a fully absorbing domain}\n\nEquation~(\\ref{ustatssepnd1}) can be solved exactly in some simple geometries. Among them are a one-dimensional segment (where the problem is exactly solvable for any gas model) and a rectangle \\cite{surv}. For a sphere of radius $R$ one can solve Eq.~(\\ref{ustatssepnd1}) numerically, and also explore analytically the low- and high-density limits. In the dilute limit $\\rho_0 \\ll 1$ one reproduces the RWs result (\\ref{suas}) which becomes\n\\begin{equation}\\label{ssphere}\ns_{\\text{RWs}}(\\rho_0)=\\frac{4\\pi^3}{3}RD_0 \\rho_0,\n\\end{equation}\nAt the other extreme, $\\rho_0 \\to 1$, the stationary optimal density profile $\\rho$ stays very close to $1$ across most of the domain, and drops to $0$ in a narrow boundary layer of characteristic width $\\delta =1-\\rho_0$ along the domain boundary. As a result, the problem becomes effectively one-dimensional in the direction normal to the domain boundary.\nThe solution for this one-dimensional problem can be found exactly, and the action rate, Eq.~(\\ref{lag}), mostly comes from the boundary layer. The final result, for a general domain shape, is\n\\begin{equation}\\label{highs}\ns(\\rho_0) \\simeq \\frac{D_0 A^2 }{V\\left(1-\\rho_0\\right)},\n\\end{equation}\nwhere $A$ is the surface area of the boundary.\nFor a sphere of radius $R$ one obtains\n\\begin{equation}\ns(\\rho_0)\\simeq\n\\frac{12\\pi D_0R}{1-\\rho_0}. \\label{divergence3d}\n\\end{equation}\nFigure~\\ref{s3d} shows the numerically found $s(\\rho_0)\/D_0R$,  alongside with the asymptotics (\\ref{ssphere})  and (\\ref{divergence3d}) \\cite{surv}.\n\n\\begin{figure}\n\t\\includegraphics[width=0.50\\textwidth,clip=]{fig3.pdf}\n\t\\caption{Solid line: the rescaled action rate $s(\\rho_0)\/D_0R$ for a sphere, vs. $\\rho_0$, obtained by numerically solving Eq.~(\\ref{ustatssepnd1}) and using Eq.~(\\ref{lag}). Also shown are the low-density asymptotic~(\\ref{ssphere}) (dashed line) and the high-density asymptotics~(\\ref{divergence3d}) (dotted line).}\n\t\\label{s3d}\n\\end{figure}\n\n\n\n\n\\subsection{Narrow escape of interacting particles}\nIn the narrow escape problem, particles can escape only through a small escape hole $\\Omega_a$, of size $\\epsilon \\ll L$, see the right panel of Fig.~\\ref{hure}.\nThe mean escape time of the first particle (MET) in this setting determines the rates of important processes in molecular and cellular biology \\cite{diffapfere,rna,den,bress}.\n\nFor the non-interacting RWs, one can evaluate $\\mu_0^2$ in Eq.~(\\ref{suas}) perturbatively with respect to the small parameter $\\epsilon\/L$.\nIn the leading  order $\\mu_0^2$ can be expressed through the electrical capacitance $C_{\\epsilon}$ of the conducting patch $\\Omega_a$ in an otherwise empty space: $\\mu_0^2\\simeq 2\\pi C_{\\epsilon}\/V$ \\cite{rayleigh}. The capacitance  $C_{\\epsilon}$ scales as $\\epsilon$. When $\\Omega_a$ is a disk  of radius $\\epsilon$, one has $C_{\\epsilon}=2\\epsilon\/\\pi$ \\cite{jackson}.\nThe resulting survival probability decay rate (\\ref{suas}) is \\cite{RoKim,narrow}\n\\begin{equation}\\label{asuas}\ns(n_0,\\epsilon)\\simeq 2\\pi C_{\\epsilon}  D_0\\rho_0.\n\\end{equation}\nFor interacting particles one can exploit the small\nparameter $\\epsilon\/L$ in a similar way \\cite{narrow}. The leading-order contribution to the action rate (\\ref{lag}) comes from only a vicinity of the escape hole. That is, to leading order in $\\epsilon\/L$, the solution for a finite domain coincides with the one for\na gas of particles occupying the infinite half-space on one side of an infinite reflecting plane with the hole $\\Omega_a$ on it. This reduces the problem to the \\emph{unconstrained} minimization procedure of the exterior survival problem of section \\ref{exsurv}.\nThe solution is therefore given by Eq.~(\\ref{rho0}) where $\\phi(\\mathbf{x})$ is the electrostatic potential  of a conducting patch $\\Omega_a$ kept at unit voltage on an otherwise insulating infinite plane. If the escape hole is a circle, $\\phi(\\mathbf{x})$ can be found explicitly \\cite{jackson}. Then\nEq.~(\\ref{lag}) yields the decay rate of the non-escape probability to order $\\epsilon\/L$:\n\\begin{equation}\ns\\left(\\rho_0,\\epsilon\\right) \\simeq \\pi  C_{\\epsilon} f^2(\\rho_0).\n\\label{sout2}\n\\end{equation}\nAs in the exterior survival problem, the gas-specific interactions are encoded in\nthe density dependence $f(\\rho_0)$, whereas the geometry dependence $C_{\\epsilon}$ is universal. To leading order it only depends on the shape of the escape hole and is independent of the domain shape.\nFor the SSEP  inside a domain with a small circular escape hole\nof radius $\\epsilon$ one obtains\\cite{narrow}\n\\begin{equation}\ns(\\rho_0,\\epsilon)\\simeq 4D_0\\epsilon \\arcsin^2(\\sqrt{\\rho_0}).\\label{soutssep2}\n\\end{equation}\nThe density dependence of Eq.~(\\ref{soutssep2}) is the same as in Eq.~(\\ref{soutssep}), see Fig.~\\ref{t}.\nAs argued in \\cite{narrow} , for the SSEP with random initial condition, the exponential decay of ${\\mathcal P}$ with time $T$ holds as soon as $T$ is much longer than the diffusion time across the escape hole \\cite{russians,RoKim,narrow}. For sufficiently low gas densities, $\\rho_0\\epsilon^3 \\ll 1$, the MET of the first particle, $\\langle T \\rangle$ is also much longer than this diffusion time, and is thus given by $\\langle T \\rangle\\simeq1\/s(\\rho_0,\\epsilon)$ \\cite{RoKim,narrow}.\n\n\n\\section{Short-time statistics: non-stationary fluctuations}\\label{nonstat}\nAt short times, $T \\ll L^2\/D\\left(\\rho_0\\right)$, the particle absorption statistics, in both exterior and interior settings, strongly depend on the initial condition, whereas the optimal density profile explicitly depends on time \\cite{MVK}.\nHere we must return to the full time-dependent MFT formulation given by Eqs.~(\\ref{path1}) and (\\ref{actionmain}). A universal simplification comes from the fact that, for very short times, the domain size is irrelevant. As a result, the process is effectively one-dimensional in the direction normal to the absorbing part of the boundary \\cite{narrow,MVK,RoKim}, and\nthe absorption statistics can be expressed through that of a gas on the infinite half-line $x>0$, with absorbing boundary conditions at $x=0$ \\cite{1d1,1d2,1d3,1d4,1d5,1d6,1d7,1d8,1d9,MVK}.\nThe particle \\emph{survival} probability on the half-line is well studied \\cite{1d1,1d2,1d3,1d4,1d5,1d6,1d7,1d8,1d9,MVK}. For the RWs with random initial conditions one obtains $-\\ln {\\mathcal P_{1d}}^{\\text{(rand)}} \\simeq (2\/\\sqrt{\\pi})  \\rho_0 \\sqrt{D_0T}$. The corresponding result for the deterministic (or quenched) setting differs by a numerical factor \\cite{1d2,MVK} .\nRemarkably, for the SSEP one obtains the same stretched-exponential\ndecay with time as for the RWs: $-\\ln {\\mathcal P_{1d}} \\simeq s_{\\text{1d}}(\\rho_0)\\sqrt{D_0T}$, but the density dependence $s_{\\text{1d}}(\\rho_0)$ is now different for the different initial conditions.\nThe low-density expansion of $s_{\\text{1d}}(\\rho_0)$  was recently calculated: $s^{\\text{(rand)}}_{\\text{1d}}(\\rho_0) = (2\/\\sqrt{\\pi})[\\rho_0 + (\\sqrt{2}-1) \\rho_0^2 + \\dots]$ \\cite{MVK,Santos}. For larger $\\rho_0$, $s_{\\text{1d}}(\\rho_0)$ can be computed numerically \\cite{MVK}. To evaluate the survival probability ${\\mathcal P}(T,\\rho_0)$, one should multiply the action $s_{\\text{1d}}(\\rho_0)$ by\nthe surface area $A$ of the absorbing part $\\Omega_a$ \\cite{MVK,narrow,RoKim} :\n\\begin{equation}\n-\\ln {\\mathcal P}(T,\\rho_0)\\simeq A s_{\\text{1d}}(\\rho_0) \\sqrt{D(\\rho_0)T} .\n\\label{shorttimeint}\n\\end{equation}\nAt sufficiently high densities, $\\rho_0L^3\\gg 1$, the short-time expression (\\ref{shorttimeint}) suffices for the evaluation of the MET. Indeed, in this regime the MET is much shorter then the diffusion time across the domain, $L^2\/D_0$, and we obtain $\\langle T \\rangle \\simeq 2 [A^2 D(\\rho_0) s_{\\text{1d}}^2(\\rho_0)]^{-1}$\\cite{RoKim,narrow}. For the narrow escape problem the relevant diffusion time scale is $\\epsilon^2\/D_0$ \\cite{RoKim,narrow}.\nAs an example, consider a circular absorbing patch of radius $\\epsilon$. In this case we have for the RWs $\\langle T_{\\text{RWs}} \\rangle^{\\text{(rand)}}\\simeq (2\\pi  D_0 \\rho_0^2 \\epsilon^4)^{-1}$ \\cite{RoKim,narrow}. For the SSEP the MET is shorter because of the effective particle repulsion: $\\langle T \\rangle^{\\text{(rand)}}\\simeq \\langle T_{\\text{RWs}} \\rangle^{\\text{(rand)}}\\left[1-2 (\\sqrt{2}-1) \\rho_0 + \\dots\\right]$.\n\n\\section{Summary}\n\nThe fluctuating hydrodynamics and macroscopic fluctuation theory provide a simple, general and versatile framework for the study of kinetics of diffusion-controlled reactions in multi-particle systems where intra-reactant interactions are important. We demonstrated the versatility of these approaches in several exterior and interior settings of particle survival, absorption and escape. More complicated settings and geometries can be also considered. The approach can be extended in different directions. For example, it can accommodate simple reactions among, and a finite lifetime of, the particles \\cite{ElgartKamenev,reac1,reac2,hurtado2,reac3}.\n\n\n\n\\section{Acknowledgments}\n\nThis research was supported\nby the United States-Israel Binational Science\nFoundation (BSF) (Grant No. 2012145) and  by the Israel Science Foundation (Grant No. 807\/16).\n\n\n\\chapter{Kinetics of survival and escape of interacting diffusing particles}\\label{ra_ch1}\n\n\\author{Baruch Meerson \\footnote{meerson@mail.huji.ac.il} and Tal Agraonv\\footnote{tal.agranov@mail.huji.ac.il}}\n\n\\address{Racah Institute of Physics, Hebrew University of\n\tJerusalem, Jerusalem 91904, Israel}\n\n\\begin{abstract}\n\n\\end{abstract}\n\\body\n\n\n\\section{Using Other Packages}\\label{ra_sec1}\nThe WSPC class file has already loaded the packages \n\\section{Acknowledgments}\nAcknowledgments to funding bodies etc. may be placed in a separate\nsection at the end of the text, before the Appendices. This should not\nbe numbered so use \\verb|\\section*{Acknowledgements}|.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}  The usual equation for both motions of a single planet around the sun and electrons  in the deterministic  Rutherford-Bohr  atomic model is conservative with a singular potential at the origin. In the previous paper \\cite{Tired}, the author raised the hypothesis that some phenomena such as carboniferous gigantism of arthropods and vegetals (cf.\\cite {HKV, Parks} ), the appearent expansion of the universe  (cf.\\cite{Hubble})or slight deformation of electrical devices over decades might be explained by an atomic contraction phenomenon (with respect to the size of the proton) coming from a very weak environmental dissipation produced by what we consider to be vacuum.  When a dissipation is added to the basic equation $$ mu'' = - \\frac{q^2}{4\\pi \\varepsilon_0} \\frac {u} {|u|^3} $$  modelling Coulomb's central force (with $q$ the elementary charge, m the mass of the electron  and $\\varepsilon_0$  the vacuum permittivity) or its  equivalent  Newton's law for planets (where $G$ is the gravitational constant and $M_S$ the mass of the sun) $$ u'' = - GM_S \\frac {u} {|u|^3} $$ written in complex form in the orbital plane with a suitable set of axes and length unit,  new interesting phenomena appear. This leads us to consider,  as the simplest possible dissipative perturbation of the conservative problem \\begin{equation} \\label{cons} u'' + c_{0}  \\frac {u} {|u|^3} = 0 \\end{equation}  including both equations,  the modified ODE \\begin{equation}\\label{diss} u'' + \\delta u' + c_{0}  \\frac {u} {|u|^3} = 0 \\end{equation}In \\cite{Tired}, considering the difficulty of exhibiting decaying solutions for the equation \\eqref{diss}, the author investigated a slightly more complicated case where we allow the charge $q$ to decay exponentially in time. This led to the equation\n  \\begin{equation} \\label{tired} u'' + \\delta u' + c_{0} e^{-\\alpha t} \\frac {u} {|u|^3} = 0 \\end{equation}\n   for which explicit exponentially decaying solutions depending on two real parameters can be exhibited. More precisely for $\\alpha = \\delta$ we found the fast rotating spiraling solutions \n   $$ u_\\pm(t) = U e^{-\\delta t}  \\exp ( \\pm i \\frac{ \\sqrt{c_0}}{\\delta U^{3\/2}} e^{\\delta t} + i \\phi) $$ whereas for $\\alpha = \\frac{3}{2}\\delta$ , we found the uniformly rotating spiraling solutions  $$ v_\\pm(t) =V e^{-\\frac{\\delta}{2} t}  \\exp ( \\pm i t \\sqrt{V^{-3}c_0 +  \\frac{\\delta^2}{4} }  + i \\phi) $$\nAlthough it would probably be interesting to do, at the level of \\cite{Tired} we did not try to prove well-posedness (global existence) for general initial data not leading to collision with the center, nor did we try to elucidate global behavior of trajectories other than the previous special  solutions. Before deciding to do that, one must first try to see whether or not the simpler model \\eqref{diss} is sufficient to identify a contraction phenomenon, and possibly determine the extent of its stability. \\\\\n\nIn the present paper, we derive some partial results on \\eqref{diss} together with some heuristic discussion. More precisely, in Section 2, we prove that whenever the momentum is not zero, the moving particle does not  reach the center in finite time and its displacement does not blow-up in finite time either, even in the classical context where arbitrarily large velocities are allowed. In Section 3, we prove that  $u(t)$ remains bounded under a smallness condition involving both initial radius and velocity. In Section 4, we prove that all bounded non-vanishing solutions converge to $0$ at infinity in $t$. Finally, in Section 5, some formal calculations suggest the existence of special orbits with an asymptotically spiraling convergence to the center. We conclude by a few heuristic remarks.\n \\section{A global existence result}  For convenience we recall our basic equation \\eqref{diss}\n\\begin{equation}\\label{dis} u'' + \\delta u' + c  \\frac {u} {|u|^3} = 0 \\end{equation}  where  we simply wrote $c$ instead of $c_0$, since for mathematicians $c$ is not always the velocity of light. In order to solve this equation with $u = u(t) \\in \\mathbb{R}^2 = \\mathbb{C}$ we introduce the amplitude and the phase \n$$ u(t) = r(t) e^{i\\theta(t)} $$ and we shall drop the variable $t$ when it does not lead to confusion. Here all derivatives are with respect to $t$. From the formulas \n$$ u' = r' e^{i\\theta} + ir \\theta'  e^{i\\theta}= ( r' + ir \\theta' )e^{i\\theta} =  ( r' + ir \\theta' ) \\frac{u} {|u|} $$ \n$$ u''= [(r'' -r \\theta'^2) + i( 2r'  \\theta'  + r \\theta'' )] \\frac{u} {|u|} $$ we conclude that \\eqref{dis} is equivalent to the system of two real equations \n\\begin{equation}\\label{r} r'' -r \\theta'^2+ \\delta r' +  \\frac {c} {r^2} = 0 \\end{equation} \n\\begin{equation}\\label{theta} 2r'  \\theta'  + r \\theta'' + \\delta r\\theta' = 0 \\end{equation} \nWe observe first that when trying to solve \\eqref{dis} for $t\\ge 0$, the initial value $u(0) = 0$ is excluded by the singularity, while whenever $u(0)\\not= 0$ we shall obtain  at least a local solution for any initial velocity $u'(0)$. Moreover, multiplying \\eqref{theta} by $r$ we reduce it to  \n $$ ( r^2 \\theta') '   + \\delta r^2\\theta' = 0 $$ so that \\begin{equation}\\label{theta1}\\forall t\\ge 0, \\quad r^2(t) \\theta' (t) = M e^{-\\delta t}\\end{equation}  with $$ M =:  r^2(0) \\theta' (0) $$  This relation expresses the variation of the momentum of the solution, which was constant in the conservative case $\\delta =0$ and decays exponentially when $\\delta>0$. By plugging in \\eqref{r} the value of $\\theta'$ given by \\eqref{theta1}, we are left with an equation involving $r(t) $ only:\n  \\begin{equation}\\label{r1} r'' -\\frac{M^2e^{-2\\delta t}}{r^3} +  \\frac {c} {r^2} + \\delta r' = 0 \\end{equation} \nWe now state our first result \\begin{thm} Assuming $ M \\not=0 $, the unique local non-vanishing solution $r= r(t)$ of \\eqref{r1} with any initial data $r(0)= r_0\\not=0; r'(0) = r'_0 \\in \\mathbb{R} $   is global and we have for some constants $C>0, \\eta>0 $ \n$$  \\forall t\\ge 0, \\quad | r' (t)| \\le C e^{\\delta t}$$ \n$$  \\forall t\\ge 0, \\quad r (t)  \\ge \\eta  e^{-2\\delta t}$$  \\end{thm} \n\\begin{proof} The existence and uniqueness of local non-vanishing solutions is obvious. Let us introduce the scalar function $F$, defined as long as the solution $r$ exists and does not vanish  by the formula:\n \\begin{equation}\\label{F} F(t) : = \\frac{r'^2(t)}{2}  - \\frac {c} {r(t)} + \\frac{M^2}{2} \\frac{e^{-2\\delta t}}{ r^2(t)}  \\end{equation} We have immediately\n   $$ F'(t) = r' (t) ( r''(t) + \\frac {c} {r^2(t)} -  \\frac{M^2  e^{-2\\delta t}}{ r^3(t)}  ) -  \\frac{M^2 \\delta e^{-2\\delta t}}{ r^2(t)}  =  -\\delta r'^2(t) -  \\frac{M^2 \\delta e^{-2\\delta t}}{ r^2(t)} < 0 $$ \n    In particular for all $ t\\in [0, Tmax)$ we have \n    $$  \\frac{r'^2(t)}{2}  - \\frac {c} {r(t)} + \\frac{M^2}{2} \\frac{e^{-2\\delta t}}{ r^2(t)} \\le F_0: = F(0) $$ which can be rearranged in the more convenient form \n    \n    \\begin{equation} \\frac{r'^2(t)}{2}  + \\frac{M^2}{2} \\left(\\frac{e^{-\\delta t}}{ r(t)}- \\frac{c}{M^2} e^{\\delta t}\\right)^2 \\le F_0 + \\frac{c^2}{2M^2} e^{2\\delta t}  \\end{equation} providing  at once the two inequalities \n    $$ \\forall t\\in [0, Tmax), \\quad r'^2(t) \\le 2(F_0 + \\frac{c^2}{2M^2} e^{2\\delta t}) \\le (2 F_0 + \\frac{c^2}{M^2} ) e^{2\\delta t} $$ and \n    $$ \\forall t\\in [0, Tmax), \\quad \\frac{e^{-\\delta t}}{ r(t)} \\le \\frac{c}{M^2} e^{\\delta t} + \\sqrt{\\frac{2}{M^2}F_0 +\\frac{ c^2 }{M^4} e^{2\\delta t}  } \\le Ke^{\\delta t} $$ The conclusions follow immediately with $$ C = (2 F_0 + \\frac{c^2}{M^2} )^{1\/2}; \\quad \\eta = \\frac{1}{K}. $$\n \\end{proof}\n \n  \\begin{cor} Assume  \\begin{equation} Im ( \\overline {u_0 } u'_0 ) \\not = 0 \\end{equation} Then the local solution of \\eqref{dis} with initial conditions $ u(0) = u_0 ; u'(0) = u'_0$  is global and satisfies for some constants $D>0, \\eta>0 $ \n\\begin{equation} \\forall t\\ge 0, \\quad | u' (t)| \\le D e^{\\delta t}\\end{equation}\n\\begin{equation}  \\forall t\\ge 0, \\quad |u (t)| \\ge \\eta  e^{-2\\delta t}\\end{equation}  \n  \n \\end{cor} \n\\begin{proof} I suffices to prove that the two above inequalities are satisfied for all $ t\\in [0, Tmax)$, the maximal existence time for a non-vanishing solution $u$ of \\eqref{dis} with initial conditions $ u(0) = u_0 ; u'(0) = u'_0$.  Selecting for $\\theta$ a continuous determination of the argument of $u(t)$, the pair $(r, \\theta) $ satisfies the equivalent system of equations \\eqref{r}-\\eqref{theta}. In addition an immediate calculation shows that $\\theta' =  Im ( \\frac{\\overline{u}} {|u|} u' )$  and in particular $$ Im ( \\overline {u_0 } u'_0 ) = r(0) \\theta'(0)$$ By the previous theorem, the solution $r(t)$ of the scalar equation for the radius is global and satisfies the concluding inequalities. Then the pair $(r, \\theta) $ correspond to the non-vanishing solution $u$ of \\eqref{dis} with initial conditions $ u(0) = u_0 ; u'(0) = u'_0$ which is therefore global with $ |u (t)| \\ge \\eta  e^{-2\\delta t}$.  For the last inequality we observe that $$ |\\theta'(t) r(t) \\le Me^{-\\delta t} r^{-1}(t) \\le \\frac{M}{\\eta} e^{\\delta t}  $$ Then $$ |u'(t)| = | r' + ir \\theta'| \\le | r'(t)| + | r (t)\\theta'(t)| \\le (C+  \\frac{M}{\\eta} ) e^{\\delta t}  : = D  e^{\\delta t} .$$\\end{proof}\n\n \\section{Bounded trajectories}  The next result does not require the condition $M\\not=0 $. \\begin{thm}  Let  $u_0\\not=0 $ and assume  the initial smallness condition  \n \\begin{equation}\\label{small} |u_0||u'_0|^2 < 2c \\end{equation}  Then the local solution $u$  of \\eqref{dis} on $[0, T)$ with initial conditions $ u(0) = u_0 ; u'(0) = u'_0$   satisfies the inequality \n \\begin{equation}\\label{bound} \\forall t\\in [0, T) , \\quad |u(t)| \\le \\frac{2c|u_0|}{2c - |u_0||u'_0|^2}\\end{equation} In particular if $u$ does not vanish in finite time, $u$ is a global bounded non-vanishing solution .  \n  \\end{thm} \n  \n  \\begin{proof} We start from the inequality  $$  \\frac{r'^2(t)}{2}  - \\frac {c} {r(t)} + \\frac{M^2}{2} \\frac{e^{-2\\delta t}}{ r^2(t)} \\le F_0=   \\frac{r'^2_0}{2}  - \\frac {c} {r_0} + \\frac{M^2}{2 r^2_0}  $$  and we observe that if $F_0<0$, the inequality   $ \\displaystyle \\frac {c} {r(t)} \\ge - F_0 = |F_0| $  implies $ \\displaystyle r(t) \\le  \\frac {c} {|F_0|}. $ Now we compute $F_0$ in terms of the initial data.  In fact we have the formula $$ |u'|^2 = r'^2 + \\theta'^2 r^2 = r'^2+ \\frac{M^2}{ r^2} $$ so that for $t=0$ we find \n  \n  $$  F_0=   \\frac{|u'_0|^2}{2}  - \\frac {c} {r_0} = \\frac{|u'_0|^2}{2} - \\frac {c} {|u_0|} $$ therefore the condition $F_0<0$ is equivalent to $ |u_0||u'_0|^2 < 2c$.  The previous observation now gives that under condition \\eqref {small}, we have \n  $$ r(t) \\le  \\frac {c} {\\displaystyle  \\frac {c} {|u_0|}- \\frac {|u'_0|^2}{2} } =  \\frac{2c|u_0|}{2c - |u_0||u'_0|^2} $$  \\end{proof}\n  \n   \\section{Convergence to 0 of non-vanishing bounded solutions } As in the case of the conservative equation \\eqref{cons} which is well known to have elliptic and parabolic trajectories,  we suspect that \\eqref{dis} may have some unbounded solutions. But the next result shows that in sharp constrast with  \\eqref{cons}, \\eqref{dis} does not have any periodic trajectory at all. \\begin{thm} For any solution $u$ of \\eqref{dis}  such as $|u(t)|$ is global, positive and bounded on $\\mathbb{R}^+$, we have \n   \n \\begin{equation}\\label{conv0} \\lim_ {t\\to +\\infty} |u(t)| = 0\\end{equation}   \n  \\end{thm} \n  \n  \\begin{proof} We introduce the total energy  $$  E(t) : =   \\frac{1}{2} |u'|^2(t )- \\frac {c} {|u(t)|} $$  Since $u$ never vanishes, it is clear that $u\\in C^2(\\mathbb{R}^+)$ and we have\\begin{equation} \\label{E'} E'(t) = -\\delta |u'(t)|^2\\end{equation}   In particular, $E(t)$ is non-increasing. Then we have two possibilities \\medskip\n  \n { \\bf Case 1}  \\begin{equation}\\label{1} \\lim_ {t\\to +\\infty} E(t)  =  -\\infty \\end{equation} Then since $\\displaystyle \\frac {c} {|u(t)|}\\ge -E(t) $ we conclude that \n $$ \\lim_ {t\\to +\\infty} |u(t)| = 0$$.\\medskip\n\n  \n { \\bf Case 2}  \\begin{equation}\\label{2} \\lim_ {t\\to +\\infty} E(t)  = E^* > -\\infty \\end{equation} Then  $$ \\forall t\\in \\mathbb{R}^+, \\quad E(0)- E(t) = \\delta\\int _0^{t} |u'(s|^2ds  $$  In particular $u'\\in L^2(\\mathbb{R}^+, \\mathbb{C}).$  We have assumed that $u(t) $ is bounded, hence  precompact in $\\mathbb{R}^+$ with values in $\\mathbb{C}$. Therefore if \\eqref{conv0} is not satisfied we may assume that for some sequence $t_n$ tending to $+\\infty$  \\begin{equation}\\label{convw} \\lim_ {n\\to +\\infty} u(t_n) = w \\not= 0\\end{equation}  On the other hand we have \n  \\begin{equation}\\label{conv'} \\lim_ {n\\to +\\infty} u'(t_n+s) =  0\\end{equation} in the strong topology of $L^2(0, 1)$ and in particular, by Cauchy-Schwarz-inequality we find  \\begin{equation}\\label{convw+} \\lim_ {n\\to +\\infty} u(t_n+s) = w \\end{equation} uniformly on $[0, 1]$. Since $w \\not= 0$ this implies \n  \\begin{equation}\\label{convf} \\lim_ {n\\to +\\infty} c \\frac{u(t_n+s)}{|u(t_n+s)|^2} = c \\frac{w}{|w|^2 } \\end{equation}  uniformly on $ [0, 1]. $  But then by the equation  \\begin{equation}\\label{conv''} \\lim_ {n\\to +\\infty} u''(t_n+s) =  c \\frac{w}{|w|^2 } \\end{equation} in the strong topology of $L^2(0, 1)$. This is contradictory with \\eqref {conv'}. Indeed it implies for instance \n  $$  \\lim_ {n\\to +\\infty} \\int_0^1 s(1-s) u''(t_n+s) ds = c \\int_0^1 s(1-s)  ds \\frac{w}{|w|^2} = z\\not=0 $$ while on the other hand \n   $$   \\int_0^1 s(1-s) u''(t_n+s) ds =  -   \\int_0^1 (1-2s)  u'(t_n+s) ds \\longrightarrow  0 $$ This contradiction concludes the proof. \n  \\end{proof} \n  \n  As an immediate consequence of the previous theorem and the results of Sections 2 and 3, we obtain\n  \n   \\begin{cor} Assume  $ Im ( \\overline {u_0 } u'_0 ) \\not = 0  $ and  $|u_0||u'_0|^2 < 2c$. Then the local solution of \\eqref{dis} with initial conditions $ u(0) = u_0 ; u'(0) = u'_0$  tends to $0$ as $t$ tends to infinity.   \n \\end{cor} \n \n \n \n \\section{A formal calculation for spiraling solutions} We now  try  a formal calculation suggesting that  solutions with norm equivalent to a decreasing exponential for $t$ large might however  exist.  We would like to see which sort of solutions would replace the circular trajectories of the conservative problem \\eqref{cons} (cf. also section 2.) Observing that such trajectories correspond to a constant angular velocity and for them the nonlinear term of the radial equation vanishes, it is natural to try \n $$ r(t) = \\frac{M^2}{c} e^{-2\\delta t}. $$  By the argument above his cannot be an exact solution of the equation and indeed  $$ r''+ \\delta r' = \\frac{M^2}{c} 2\\delta^2  e^{-2\\delta t} \\not = 0 $$ but at least the result is non-singular and even becomes very small if $\\delta$ is small. This gives the idea to look for a solution of the form \\begin{equation}  r(t) = \\frac{M^2}{c} e^{-2\\delta t} + \\varepsilon (t)\\end{equation}  where $\\varepsilon (t)$ would be negligible with respect to $\\frac{M^2}{c} e^{-2\\delta t}$, either uniformly, or at least for $t$ large.  Plugging this in the equation we find $$ c\\varepsilon(t) = -r^3(r''+\\delta r')  $$ If now we replace both \n $$r^3\\sim  \\frac{M^6}{c^3} e^{-6\\delta t}; \\quad r''+\\delta r'\\sim \\frac{M^2}{c} (2\\delta^2) e^{-2t} $$ by their alleged ``main part\" we find \n \n $$ \\varepsilon(t) \\sim  - 2 \\delta^2 \\frac{M^8}{c^5}  e^{-8\\delta t}  $$ which is indeed quite negligible with respect to $\\frac{M^2}{c} e^{-2\\delta t}$ for $t$ large and even uniformly if $\\delta$ and  $\\frac{M^6}{c^4}$ are both small enough. To see the rule we main consider trying the next step. the calculations are slightly more involved but we get rather easily \n\\begin{equation} \\varepsilon(t) \\sim  - 2 \\delta^2 \\frac{M^8}{c^5}  e^{-8\\delta t} + 124 \\delta^4 \\frac{M^{14}}{c^9}e^{-14\\delta t} ...\\end{equation}\n From the algebraic point of view, it is not difficult to see that the process can be continued indefinitely with the appearance of a single new exponential term at each step. So that we can dream of a solution of the form \n \\begin{equation} r(t) \\sim \\frac{M^2}{c} \\sum _{n=0}^{\\infty} (-1)^n C_n \\delta^ {2n } \\left(\\frac{M^6}{c^4}\\right)^ne^{-(2+6n)\\delta t}\\end{equation}\n The only difficulty for convergence would be to control the growth of $C_n $.  Actually $C_n $ seems to grow like a factorial, precluding any sort of convergence for any fixed value of $t$. Then in which sense would the above calculations approach a solution? At each step  we have an approximate solution with a very close accuracy for $t$ large. For the time being the moral of the story remains a mystery for the author.  In any case  it is clear that a different method is needed to find the solutions we are looking for.\n  \n  \\section{Concluding remarks} The previous calculations suggest the possible existence of special orbits with an asymptotically (exponentially fast) spiraling convergence to the center. More precisely if we succeed in finding, by whichever method, a solution $r$ of \\eqref {r1}with  $$ r(t) \\sim Ce^{-2\\delta t} $$ for a certain $C= C(M)>0$, from this we can build a solution $u= re^{i\\theta}$ with $$ \\theta'^2\\sim K e^{6\\delta t }$$ so that $ |u'|^2 = r'^2 + \\theta'^2 r^2 \\ge k e^{2\\delta t } $ for some positive $k$ and $t$ large. The kinetic energy blows-up exponentially in such a case, which is not absurd since the potential energy also. \\begin{rmk} This construction seems interesting, but even in case it works  this is not entirely satisfactory for the following reason: as in the case of \\eqref{tired}, the family of spiraling orbits built in this way would depend on only two real parameters (the moment and the initial phase) whereas the phase space has four dimensions. \\end{rmk} \n \\begin{rmk} For the moment, despite the formal calculation of the previous section,  we know nothing about the rate of decay of any solution to $0$ . The only certain information that we have on asymptotics is the inequality  $ r (t)  \\ge \\eta  e^{-2\\delta t}$ for some $\\eta>0$. This inequality can be refined asymptotically, relying on  the inequality  \n  $$ \\forall t\\in [0, Tmax), \\quad \\frac{e^{-\\delta t}}{ r(t)} \\le \\frac{c}{M^2} e^{\\delta t} + \\sqrt{\\frac{2}{M^2}F_0 +\\frac{ c^2 }{M^4} e^{2\\delta t}  }  $$ \n to yield $$ \\mathop{\\underline{\\rm lim}}_{t\\to\\infty}  r (t)  e^{2\\delta t} \\ge \\frac{M^2}{2c}.$$ \\end{rmk} \n  \\begin{rmk} In the case  $M=0$, it should be possible to study if, as in the conservative case, solutions with  initial velocity directed towards the center vanish in finite time, if necessary under some additional conditions. We did not study this question here since it is not our main concern.  \\end{rmk} \n  \n  \\begin{rmk} After the simple model of Rutherford-Bohr explained in \\cite{Bohr, Rutherford} and after the introduction of undulatory mechanics by L. De Broglie, the purely probabilistic model of E. Schrodinger (cf. \\cite{Schrodinger}) has  been considered the best atomic model since 1926. It does not  seem obvious at all to introduce a damping mechanism in that model, and it might very well happen that in order to do that, an effective coupling between deterministic corpuscular and probabilistic wave conceptions of particles has to be devised, as was always advocated by L. De Broglie and even suggested by  E. Schrodinger himself in \\cite{Schrodinger}.  \\end{rmk}  \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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{}\n\\fi\n\\let\\DOTSI\\relax\n\\def\\RIfM@{\\relax\\ifmmode}%\n\\def\\FN@{\\futurelet\\next}%\n\\newcount\\intno@\n\\def\\iint{\\DOTSI\\intno@\\tw@\\FN@\\ints@}%\n\\def\\iiint{\\DOTSI\\intno@\\thr@@\\FN@\\ints@}%\n\\def\\iiiint{\\DOTSI\\intno@4 \\FN@\\ints@}%\n\\def\\idotsint{\\DOTSI\\intno@\\z@\\FN@\\ints@}%\n\\def\\ints@{\\findlimits@\\ints@@}%\n\\newif\\iflimtoken@\n\\newif\\iflimits@\n\\def\\findlimits@{\\limtoken@true\\ifx\\next\\limits\\limits@true\n \\else\\ifx\\next\\nolimits\\limits@false\\else\n \\limtoken@false\\ifx\\ilimits@\\nolimits\\limits@false\\else\n \\ifinner\\limits@false\\else\\limits@true\\fi\\fi\\fi\\fi}%\n\\def\\multint@{\\int\\ifnum\\intno@=\\z@\\intdots@                         \n \\else\\intkern@\\fi                                                   \n \\ifnum\\intno@>\\tw@\\int\\intkern@\\fi                                  \n \\ifnum\\intno@>\\thr@@\\int\\intkern@\\fi                                \n \\int\n\\def\\multintlimits@{\\intop\\ifnum\\intno@=\\z@\\intdots@\\else\\intkern@\\fi\n \\ifnum\\intno@>\\tw@\\intop\\intkern@\\fi\n \\ifnum\\intno@>\\thr@@\\intop\\intkern@\\fi\\intop}%\n\\def\\intic@{%\n    \\mathchoice{\\hskip.5em}{\\hskip.4em}{\\hskip.4em}{\\hskip.4em}}%\n\\def\\negintic@{\\mathchoice\n {\\hskip-.5em}{\\hskip-.4em}{\\hskip-.4em}{\\hskip-.4em}}%\n\\def\\ints@@{\\iflimtoken@                                             \n \\def\\ints@@@{\\iflimits@\\negintic@\n   \\mathop{\\intic@\\multintlimits@}\\limits                            \n  \\else\\multint@\\nolimits\\fi                                         \n  \\eat@\n \\else                                                               \n \\def\\ints@@@{\\iflimits@\\negintic@\n  \\mathop{\\intic@\\multintlimits@}\\limits\\else\n  \\multint@\\nolimits\\fi}\\fi\\ints@@@}%\n\\def\\intkern@{\\mathchoice{\\!\\!\\!}{\\!\\!}{\\!\\!}{\\!\\!}}%\n\\def\\plaincdots@{\\mathinner{\\cdotp\\cdotp\\cdotp}}%\n\\def\\intdots@{\\mathchoice{\\plaincdots@}%\n {{\\cdotp}\\mkern1.5mu{\\cdotp}\\mkern1.5mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}}%\n\\def\\RIfM@{\\relax\\protect\\ifmmode}\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\RIfM@\\expandafter\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@\\else\\expandafter\\mbox\\fi}\n\\let\\nfss@text\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@#1{\\mathchoice\n   {\\textdef@\\displaystyle\\f@size{#1}}%\n   {\\textdef@\\textstyle\\tf@size{\\firstchoice@false #1}}%\n   {\\textdef@\\textstyle\\sf@size{\\firstchoice@false #1}}%\n   {\\textdef@\\textstyle \\ssf@size{\\firstchoice@false #1}}%\n   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\\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\leftarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\\mkern-6mu\\mathord-$}%\n\\def\\leftrightarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\n\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\overrightarrow{\\mathpalette\\overrightarrow@}%\n\\def\\overrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\\rightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\let\\overarrow\\overrightarrow\n\\def\\overleftarrow{\\mathpalette\\overleftarrow@}%\n\\def\\overleftarrow@#1#2{\\vbox{\\ialign{##\\crcr\\leftarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\overleftrightarrow{\\mathpalette\\overleftrightarrow@}%\n\\def\\overleftrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\n   \\leftrightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\underrightarrow{\\mathpalette\\underrightarrow@}%\n\\def\\underrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n  $\\crcr\\noalign{\\nointerlineskip}\\rightarrowfill@#1\\crcr}}}%\n\\let\\underarrow\\underrightarrow\n\\def\\underleftarrow{\\mathpalette\\underleftarrow@}%\n\\def\\underleftarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n  $\\crcr\\noalign{\\nointerlineskip}\\leftarrowfill@#1\\crcr}}}%\n\\def\\underleftrightarrow{\\mathpalette\\underleftrightarrow@}%\n\\def\\underleftrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\n  \\hfil#1#2\\hfil$\\crcr\n \\noalign{\\nointerlineskip}\\leftrightarrowfill@#1\\crcr}}}%\n\n\\def\\qopnamewl@#1{\\mathop{\\operator@font#1}\\nlimits@}\n\\let\\nlimits@\\displaylimits\n\\def\\setboxz@h{\\setbox\\z@\\hbox}\n\n\n\\def\\varlim@#1#2{\\mathop{\\vtop{\\ialign{##\\crcr\n \\hfil$#1\\m@th\\operator@font lim$\\hfil\\crcr\n \\noalign{\\nointerlineskip}#2#1\\crcr\n \\noalign{\\nointerlineskip\\kern-\\ex@}\\crcr}}}}\n\n \\def\\rightarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n  $#1\\copy\\z@\\mkern-6mu\\cleaders\n  \\hbox{$#1\\mkern-2mu\\box\\z@\\mkern-2mu$}\\hfill\n  \\mkern-6mu\\mathord\\rightarrow$}\n\\def\\leftarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n  $#1\\mathord\\leftarrow\\mkern-6mu\\cleaders\n  \\hbox{$#1\\mkern-2mu\\copy\\z@\\mkern-2mu$}\\hfill\n  \\mkern-6mu\\box\\z@$}\n\n\n\\def\\qopnamewl@{proj\\,lim}{\\qopnamewl@{proj\\,lim}}\n\\def\\qopnamewl@{inj\\,lim}{\\qopnamewl@{inj\\,lim}}\n\\def\\mathpalette\\varlim@\\rightarrowfill@{\\mathpalette\\varlim@\\rightarrowfill@}\n\\def\\mathpalette\\varlim@\\leftarrowfill@{\\mathpalette\\varlim@\\leftarrowfill@}\n\\def\\mathpalette\\varliminf@{}{\\mathpalette\\mathpalette\\varliminf@{}@{}}\n\\def\\mathpalette\\varliminf@{}@#1{\\mathop{\\underline{\\vrule\\@depth.2\\ex@\\@width\\z@\n   \\hbox{$#1\\m@th\\operator@font lim$}}}}\n\\def\\mathpalette\\varlimsup@{}{\\mathpalette\\mathpalette\\varlimsup@{}@{}}\n\\def\\mathpalette\\varlimsup@{}@#1{\\mathop{\\overline\n  {\\hbox{$#1\\m@th\\operator@font lim$}}}}\n\n\\def\\stackunder#1#2{\\mathrel{\\mathop{#2}\\limits_{#1}}}%\n\\begingroup \\catcode `|=0 \\catcode `[= 1\n\\catcode`]=2 \\catcode `\\{=12 \\catcode `\\}=12\n\\catcode`\\\\=12 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\\frenchspacing\\@vobeyspaces \\@alignverbatim\nYou are using the \"align\" environment in a style in which it is not defined.}\n\\let\\endalign=\\endtrivlist\n \n\\@namedef{align*}{\\@verbatim\\@salignverbatim\nYou are using the \"align*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalign*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\alignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@alignatverbatim\nYou are using the \"alignat\" environment in a style in which it is not defined.}\n\\let\\endalignat=\\endtrivlist\n \n\\@namedef{alignat*}{\\@verbatim\\@salignatverbatim\nYou are using the \"alignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\xalignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@xalignatverbatim\nYou are using the \"xalignat\" environment in a style in which it is not defined.}\n\\let\\endxalignat=\\endtrivlist\n \n\\@namedef{xalignat*}{\\@verbatim\\@sxalignatverbatim\nYou are using the \"xalignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endxalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\gather{\\@verbatim \\frenchspacing\\@vobeyspaces \\@gatherverbatim\nYou are using the \"gather\" environment in a style in which it is not defined.}\n\\let\\endgather=\\endtrivlist\n \n\\@namedef{gather*}{\\@verbatim\\@sgatherverbatim\nYou are using the \"gather*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endgather*\\endcsname =\\endtrivlist\n\n\n\\def\\multiline{\\@verbatim \\frenchspacing\\@vobeyspaces \\@multilineverbatim\nYou are using the \"multiline\" environment in a style in which it is not defined.}\n\\let\\endmultiline=\\endtrivlist\n \n\\@namedef{multiline*}{\\@verbatim\\@smultilineverbatim\nYou are using the \"multiline*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endmultiline*\\endcsname =\\endtrivlist\n\n\n\\def\\arrax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@arraxverbatim\nYou are using a type of \"array\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endarrax=\\endtrivlist\n\n\\def\\tabulax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@tabulaxverbatim\nYou are using a type of \"tabular\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endtabulax=\\endtrivlist\n\n \n\\@namedef{arrax*}{\\@verbatim\\@sarraxverbatim\nYou are using a type of \"array*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endarrax*\\endcsname =\\endtrivlist\n\n\\@namedef{tabulax*}{\\@verbatim\\@stabulaxverbatim\nYou are using a type of \"tabular*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endtabulax*\\endcsname =\\endtrivlist\n\n\n\n \\def\\endequation{%\n     \\ifmmode\\ifinner\n      \\iftag@\n        \\addtocounter{equation}{-1}\n        $\\hfil\n           \\displaywidth\\linewidth\\@taggnum\\egroup \\endtrivlist\n        \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false\n        \\global\\@ignoretrue   \n      \\else\n        $\\hfil\n           \\displaywidth\\linewidth\\@eqnnum\\egroup \\endtrivlist\n        \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false\n        \\global\\@ignoretrue \n      \\fi\n     \\else   \n      \\iftag@\n        \\addtocounter{equation}{-1}\n        \\eqno \\hbox{\\@taggnum}\n        \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false%\n        $$\\global\\@ignoretrue\n      \\else\n        \\eqno \\hbox{\\@eqnnum\n        $$\\global\\@ignoretrue\n      \\fi\n     \\fi\\fi\n } \n\n \\newif\\iftag@ \\@ifnextchar*{\\@tagstar}{\\@tag}@false\n \n \\def\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}{\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}}\n \\def\\@TCItag#1{%\n     \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n     \\global\\def\\@taggnum{(#1)}}\n \\def\\@TCItagstar*#1{%\n     \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n     \\global\\def\\@taggnum{#1}}\n\n  \\@ifundefined{tag}{\n     \\def\\@ifnextchar*{\\@tagstar}{\\@tag}{\\@ifnextchar*{\\@tagstar}{\\@tag}}\n     \\def\\@tag#1{%\n         \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n         \\global\\def\\@taggnum{(#1)}}\n     \\def\\@tagstar*#1{%\n         \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n         \\global\\def\\@taggnum{#1}}\n  }{}\n\n\n\n\\makeatother\n\\endinput\n\n\\section{Introduction}\\label{sec:introduction}\n\nDynamical response analysis is an essential ingredient of control\nengineering, and is also the basis of controller design. For example,\nimpulse response, step response, and frequency response are standard\nmaterials in modern control textbooks, see, e.g., \\cite{KS72}, \\cite{AM79}, \n\\cite{ZDG96}, \\cite{QZ09}. Fluctuation analysis of a dynamical\nsystem driven by white noise underlies the celebrated Kalman filter and\nlinear quadratic Gaussian (LQG) control. Likewise, in the quantum regime,\nthe response of quantum linear systems to quantum Gaussian white noise is\nthe basis of quantum filtering and measurement-based feedback control, see,\ne.g., \\cite{VPB80}, \\cite{VPB83}, \\cite{HP84}, \\cite{VPB92}, \\cite{VPB93}, \n\\cite{vHSM05}, \\cite{WM10}, \\cite{DP10}, \\cite{AT12}, \\cite{ZWL12}, \\cite{ZLW+15}, \\cite{GY16} and references therein.\n\nIn addition to quantum Gaussian noise commonly dealt with in quantum optical\nlaboratories, in recent years, highly nonclassical signals such as\nsingle-photon states, multi-photon states, and Schr\\\"{o}dinger's cat states\nhave been attracting growing interest due to their promising applications\nin quantum information technology \\cite{NC00}. Roughly speaking, an $\\ell $-photon state of a light beam means that the light field contains \\textit{exactly} $\\ell $ photons. In this paper, we are concerned with\ncontinuous-mode $\\ell $-photon states, that is, these photons are specified\nby their frequency profiles centered at the carrier frequency of the light\nfield. Continuous-mode single- and multi-photon states have found important\napplications in quantum computing, quantum communication, and quantum\nmetrology, see, e.g., \\cite{GEP+98}, \\cite{NC00}, \\cite{RL00}, \\cite{Ou07}, \\cite{Milburn08}, \\cite{CMR09}, \\cite{GJN11}, \\cite{GJNC12}, \\cite{BCB+12}, \\cite{BDS+12}, \\cite{Yukawa:13}, \\cite{zhang13}, \\cite{zhang14}, \\cite{YJ14}, \n\\cite{NKM+15}.\n\nIn the quantum control community, the response of quantum systems to single-\nand multi-photon states has been studied in the past few years. The\nphenomenon of cross phase shift on a coherent signal induced by a single\nphoton pulse was investigated in \\cite{MNM10}. Gough \\textit{et al.} derived\nquantum filters for Markov quantum systems driven by single-photon states or\nSchr\\\"{o}dinger's cat states \\cite{GJN11}, \\cite{GJNC12}. The theory in \\cite{GJN11} and \\cite{GJNC12} was applied to the study of phase modulation in \n\\cite{CHJ12}. Quantum master equations for an arbitrary quantum system\ndriven by multi-photon states were derived in \\cite{BCB+12}. Quantum filters\nfor multi-photon states were derived in \\cite{SZX15}, for both homodyne\ndetection and photodetection measurements. Numerical simulations carried out\nin \\cite{SZX15} for a two-level system driven by a 2-photon state revealed\ninteresting and complicated nonlinear behavior in the photon-atom\ninteraction. When a two-level atom, initialized in the ground state, is\ndriven by a single photon, the exact form of the output field state was\nderived in \\cite{PZJ15}. More discussions can be found in, e.g., \\cite{FKS10}, \\cite{LL13}, \\cite{NKM+15} and references therein.\n\n\nIn \\cite{zhang13}, an analytic expression of the output field state of a\nquantum linear system driven by a single-photon state was derived.  The research initialized in \\cite{zhang13} was continued and extended in \\cite{zhang14}, where multi-photon states were considered. Unfortunately, the multi-photon states studied in  \\cite{zhang14} are either with very limited quantum correlation or mathematically formidable. More specifically, the multi-photon states defined in \\cite[Eqs. (23) and (41)]{zhang14} are separable states, i.e., there exists no entanglement among the channels.  A class of photon-Gaussian states was defined in \\cite[Eq. (34)]{zhang14}. On the one hand, this class of states appears mathematically complicated. On the other hand, because each pulse shape is indexed by three parameters only,  the feature of the multi-channel entanglement is unclear.  A class of multi-channel multi-photon states was defined in \\cite[Eq. (43)]{zhang14}, which, due to the presence of an $m$-fold product,  looks rather complicated mathematically.\n\n\n\nThe purpose of this paper is to provide a direct study of the dynamical\nresponse of quantum linear systems to initially entangled multi-channel\nmulti-photon states. Unlike those separable  states studied in \\cite{zhang13} and \\cite{zhang14}, the multi-channel multi-photon states proposed in this paper are able to capture the entanglement among channels.  Examples presented in this paper demonstrate that these types\nof multi-channel multi-photon states can be easily processed by quantum linear\nsystems. Furthermore, the proposed multi-channel multi-photon states are very general\nas they contain many types of multi-channel multi-photon states as special\ncases, see, e.g., \\cite[Chapter 6]{RL00}, \\cite{RMS07}, \\cite{BDS+15}.\nFinally, these states are mathematically more tractable than those in \\cite[%\nEqs. (34) and (43)]{zhang14}. Therefore, the study carried out in this paper is more\nrelevant to quantum linear feedback networks and control.\n\nThree types of multi-channel multi-photon states are studied in this paper.\nCase 1): $m$ photons are superposed among $m$ channels. Specifically, the $m$-channel $m$-photon states are defined in Subsection \\ref{subsec:m_photon}.\nWhen the underlying quantum linear system is passive, an analytic\nexpression of the output intensity is presented in Subsection \\ref{subsec:intensity}, see Theorem \\ref{thm:intensity}. Moreover, the\nsteady-state output field state is investigated in Subsections \\ref{sec:passive_mm} and \\ref{subsec:invariant}, see Theorems \\ref{thm:passive_mm} and \\ref{thm:passive_invariance}. When the underlying\nquantum linear system is non-passive, the steady-state output field state is no\nlonger an $m$-channel $m$-photon state, an explicit form of the output field\nstate is given in Subsection \\ref{subsec:nonpassive}, see Theorem \\ref{thm:nonpassive}. Case 2): $N$ photons are superposed among $m$ channels\nwhere $N\\geq m$. For this case, we assume the underlying quantum linear\nsystem is passive. The analytic expressions of the output field state are\nderived, see Theorems \\ref{thm:aug23} and \\ref{thm:passive_invariance_general}. Case 3): $N$ photons are superposed among $m$ channels, where $N$ is an arbitrary positive integer.  Specifically, a class of $m$-channel $N$-photon states are first presented in Subsection \\ref{subsec:mN_input}, then in\nSubsection \\ref{subsec:mN_output}, the steady-state output field state of a\nquantum linear passive system driven by an $m$-channel $N$-photon input\nstate is derived, see Theorem \\ref{thm:general}.\n\n\\emph{Notation}. The imaginary unit $\\sqrt{-1}$ is denoted by $\\mathrm{i}$.\nGiven a column vector of complex numbers or operators $x=[x_{1}~\\cdots\n~x_{k}]^{T}$, define a column vector $x^{\\#} \\triangleq [x_{1}^{\\ast }~\\cdots\n~x_{k}^{\\ast }]^{T}$, where the superscript ``$\\ast $'' stands for complex\nconjugation of a complex number or Hilbert space adjoint of an operator.\nDefine a row vector $x^{\\dag }\\triangleq (x^{\\#})^{T}= [x_{1}^{\\ast }~\\cdots\n~x_{k}^{\\ast }]$. Define a \ndoubled-up column vector $\\breve{x}\\triangleq \\lbrack x^{T}~x^{\\dag }]^{T}$.\nLet $I_{k}$ be an identity matrix and $0_{k}$ a zero square matrix, both of\ndimension $k$. Denote $J_{k}=\\mathrm{diag}(I_{k},-I_{k})$.  Given a matrix $X\\in \\mathbb{C}^{2j\\times 2k}$, define $X^{\\flat\n}\\triangleq J_{k}X^{\\dag }J_{j}$. Given a matrix $A$, let $A^{jk}$ denote\nthe entry on the $j$th row and $k$th column. Let $m$ be the number of input\nchannels. Let $n$ be the number of degrees of freedom of a given quantum\nlinear system, namely the number of quantum harmonic oscillators. The ket $|\\phi \\rangle $ denotes the initial state of the system of interest, and $|0\\rangle $\nstands for the vacuum state of free fields. The convolution of two functions \n$f$ and $g$ is denoted as $f\\circledast g$. Given two matrices $U$, $V\\in \n\\mathbb{C}^{r\\times k}$, define a doubled-up matrix $\\Delta (U,V)\\triangleq\n\\lbrack U~V;V^{\\#}~U^{\\#}]$. Given two operators $\\mathcal{A}$ and $\\mathcal{B}$, their commutator is defined to be $[\\mathcal{A},\\mathcal{B}]\\triangleq \n\\mathcal{AB}-\\mathcal{BA}$. The Kronecker delta function is denoted by $\\delta _{jk}$, whereas the Dirac delta function is denoted by $\\delta (t)$.\nThe $m$-fold integral $\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty\n}^{\\infty }dt_{1}\\cdots dt_{m}$ is sometimes denoted by $\\int d\\overrightarrow{t}$. Given a function $f(t)$ in the time domain, define its \n\\textit{two-sided} Laplace transform \\cite[Eq. (13)]{Sog10} to be $F[s]\\equiv \\mathscr{L}_{b}\\{f(t)\\}(s)\\triangleq \\int_{-\\infty }^{\\infty\n}e^{-st}f(t)dt$. The $m$ dimensional Fourier transform of an $m$-variable function $f(t_1,\\ldots,t_m)$ is, \\cite{Bracewell99}, \n\\begin{equation}\\label{eq:fourier}\nf(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\\omega _{m})\n\\triangleq\n \\frac{1}{(2\\pi )^{m\/2}}\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty\n}dt_{1}\\cdots dt_{m}\\ e^{-\\mathrm{i}(\\omega _{1}t_{1}+\\cdots +\\omega\n_{m}t_{m})}f(t_{1},\\ldots ,t_{m}). \n \\end{equation}\nWe set $\\hbar =1$ throughout this paper.\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nIn this section, quantum linear systems are briefly introduced; more\ndiscussions can be found in, e.g., \\cite{GZ00}, \\cite{YK03a}, \\cite{YK03b}, \n\\cite{WM08}, \\cite{GJ09}, \\cite{WM10}, \\cite{JG10}, \\cite{IRP11}, \\cite%\n{zhang11}, \\cite{zhang12}, \\cite{NY13}, \\cite{HIN14}, and \\cite{zhang16}.\nSome tensors and their associated operations are also discussed.\n\n\\subsection{Quantum linear systems}\\label{sec:systems}\n\n\\begin{figure}[ptb]\n\\centering\n\\includegraphics[width=2.0in]{system}\n\\caption{Quantum linear system $G$ with input $b_{\\mathrm{in}}$ and output $\nb_{\\mathrm{out}}$}\n\\label{system}\n\\end{figure}\n\nA quantum linear system $G$ is shown schematically in Fig.~\\ref{system}. In\nthis model, the quantum linear system $G$ consists of a collection of $n$\n(interacting) quantum harmonic oscillators represented by $a=[a_{1} \\ \\cdots\n\\ a_{n}]^{T}$. Here, $a_{j}$ ($j=1,\\ldots ,n$), defined on a Hilbert space $\n\\mathfrak{H}$, is the annihilation operator of the $j$th quantum harmonic\noscillator. The adjoint operator of $a_{j}$, denoted by $a_{j}^{\\ast }$, is\ncalled a creation operator. These operators satisfy the following canonical\ncommutation relations: $[a_{j},a_{k}^{\\ast }]=\\delta _{jk}$, and $\n[a_{j},a_{k}]=[a_{j}^{\\ast },a_{k}^{\\ast }]=0$, $(j,k=1,\\ldots ,n)$. The\ninput fields are represented by a vector of annihilation operators $b_{\n\\mathrm{in}}(t)=[b_{\\mathrm{in},1}(t)\\ \\cdots \\ b_{\\mathrm{in},m}(t)]^{T}$;\nthe entry $b_{\\mathrm{in},j}(t)$ ($j=1,\\ldots ,m$), defined on a Fock space $\n\\mathfrak{F}$, is the annihilation operator for the $j$th input channel. The\nadjoint operator of $b_{\\mathrm{in},j}(t)$, denoted by $b_{\\mathrm{in}\n,j}^{\\ast }(t)$, is also called a creation operator. However, unlike $a_{j}$\nand $a_{k}^{\\ast }$, the annihilation and creation operators for the input\nfields satisfy the following singular commutation relations, \\cite{GZ00}, \n\\cite[Eq. (20)]{GJN10}, \n\\begin{equation}\n\\lbrack b_{\\mathrm{in},j}(t),b_{\\mathrm{in},k}^{\\ast }(r)] =\\delta\n_{jk}\\delta (t-r), \\ \\ \\lbrack b_{\\mathrm{in},j}(t),b_{\\mathrm{in},k}(r)]\n=[b_{\\mathrm{in},j}^{\\ast }(t),b_{\\mathrm{in},k}^{\\ast }(r)]=0,\\\nj,k=1,\\ldots ,m,~\\forall t,r\\in \\mathbb{R}.  \\label{eq:ccr}\n\\end{equation}\nNotice the presence of the Dirac delta function $\\delta (t-r)$ in Eq. (\\ref\n{eq:ccr}). Mathematically, it is often more convenient to work with\nintegrated annihilation and creation operators, which are defined\nrespectively to be $B_{\\mathrm{in}}(t)\\triangleq \\int_{t_{0}}^{t}b_{\\mathrm{\nin}}(\\tau )d\\tau $ and $B_{\\mathrm{in}}^{\\#}(t)\\triangleq \\int_{t_{0}}^{t}b_{\n\\mathrm{in}}^{\\#}(\\tau )d\\tau $, where the lower limit $t_{0}$ of the\nintegrals is the initial time, namely the time when the system and the\nfields start to interact. The input gauge process (also called {\\it number\n} process) is defined by the following $m$-by-$m$ matrix of operators, \\cite[\nChapter 11]{GZ00}, \\cite[Section III.A]{GJN10}, \\cite[Eq. (11)]{zhang13},\n\\begin{equation}\n\\Lambda _{\\mathrm{in}}(t)\\triangleq \\int_{t_{0}}^{t}b_{\\mathrm{in}\n}^{\\#}(\\tau )b_{\\mathrm{in}}^{T}(\\tau )d\\tau .  \\label{july20_1}\n\\end{equation}\nIn this paper, we deal with \\emph{canonical }quantum input fields, that is,\nthe only non-zero It\\^{o} products for the input fields are, \\cite[Chapter 11]{GZ00}, \\cite{GJ09}\n, \\cite{GJN10}, \\cite[Eq. (12)]{zhang13}, \n\\begin{align}\n&\ndB_{\\mathrm{in},j}(t)dB_{\\mathrm{in},k}^{\\ast }(t) =\\delta\n_{jk}dt,~d\\Lambda _{\\mathrm{in}}^{jk}(t)dB_{\\mathrm{in},l}^{\\ast }(t)=\\delta\n_{kl}dB_{\\mathrm{in},j}^{\\ast }(t), \n \\nonumber\n  \\\\\n&\n dB_{\\mathrm{in},j}(t)d\\Lambda _{\\mathrm{in}}^{kl}(t) =\\delta _{jk}dB_{\n\\mathrm{in},l}(t),~d\\Lambda _{\\mathrm{in}}^{jk}(t)d\\Lambda _{\\mathrm{in}\n}^{lr}(t)=\\delta _{kl}d\\Lambda _{\\mathrm{in}}^{jr}(t),~j,k,l,r=1,\\ldots\n,m,~\\forall t\\in \\mathbb{R}, \n \\label{Eq:CCR2}\n\\end{align}\nwhere $\\Lambda _{\\mathrm{in}}^{jk}(t)$ is the entry of the matrix $\\Lambda _{\\mathrm{in}}(t)$ on the $j$th row and $k$th column, as introduced in the {\\it Notation} part.\n\nThe dynamics of the open quantum linear system $G$ can be described\nconveniently in the $(S_{-},L,H)$ formalism \\cite{GJ09}, \\cite{zhang12}.\nHere, $S_{-}$ is a constant unitary matrix of dimension $m$, which can be\nused to model static devices such as phase shifters and\nbeamsplitters. The operator $L$ describes how the system is coupled to the\nfields, and is of the form $L=C_{-}a+C_{+}a^{\\#}$ with $C_{-},C_{+}\\in \n\\mathbb{C}^{m\\times n}$. For example, when a single-mode (namely, $n=1$) optical cavity is driven by a\nlight field, $L$ can be of the form $L=\\sqrt{\\kappa }a$, where $a$ is the\nannihilation operator of the quantum harmonic operator for the cavity\n(also called the cavity mode) and $\\kappa >0$ is the coupling strength between\nthe cavity and the field. The operator $H$ stands for the initial system\nHamiltonian, which can be written as $H=\\frac{1}{2}\\breve{a}^{\\dag }\\Delta\n\\left( \\Omega _{-},\\Omega _{+}\\right) \\breve{a}$ with constant matrices $\n\\Omega _{-},\\Omega _{+}\\in \\mathbb{C}^{n\\times n}$ satisfying $\\Omega\n_{-}=\\Omega _{-}^{\\dag }$ and $\\Omega _{+}=\\Omega _{+}^{T}$. For example,\nfor the optical cavity above mentioned, $H=\\frac{1}{2 } [ a^\\ast \\ \\ a] \n\\left[\n\\begin{smallmatrix}\n\\omega _{d} & 0 \\\\ \n0 & \\omega _{d}\n\\end{smallmatrix}\n\\right] [a \\ \\ a^\\ast]^T = \\omega _{d}a^{\\ast }a+\\frac{1}{2}\\omega _{d} $,\nwhere $\\omega _{d} \\in \\mathbb{R}$ is the detuning frequency between the cavity mode and\nthe center frequency of the input light field. (The term $\\frac{1}{2}\\omega\n_{d}$ introduces a global phase shift and leads to no consequence.) With\nthese parameters, in It\\^{o} form, Schr\\\"{o}dinger's equation for the\ntemporal evolution of the open quantum linear system in Fig.~\\ref{system}\nis, \\cite{HP84}, \\cite[Eq. (30)]{GJ09}, \\cite[Eq. (22)]{GJN10}, \\cite[Eq.\n(13)]{zhang13}, \n\\begin{equation}  \\label{eq:U}\ndU(t, t_{0})=\\left\\{ \\mathrm{Tr}[(S_{-}-I_{m})d\\Lambda _{\\mathrm{in}\n}(t)^{T}]+dB_{\\mathrm{in}}^{\\dag }(t)L-L^{\\dag }S_{-}dB_{\\mathrm{in}}(t)-(\n\\frac{1}{2}L^{\\dag }L+\\mathrm{i}H)dt\\right\\} U(t, t_{0}),~t\\geq t_{0}\n\\end{equation}\nwith  $U(t, t_{0})=I$ (identity operator) for all $t\\leq t_{0}$.\n\nIn the Heisenberg picture, system operators evolve according to $\\breve{a}\n(t)=U(t, t_{0})^{\\ast }\\breve{a}(t_{0})U(t, t_{0})$ (component-wise for the\ncomponents of $\\breve{a}(t_{0})$). Moreover, the output field $b_{\\mathrm{out}}(t)$\ncarries away information of the system after interaction, and is defined by \n\\begin{equation}  \\label{eq:b_out_dec22}\n\\breve{b}_{\\mathrm{out}}(t)\\triangleq U(t, t_{0})^{\\ast }\\breve{b}_{\\mathrm{in\n}}(t)U(t, t_{0})\n\\end{equation}\n(component-wise for the components of $\\breve{b}_{\\mathrm{in}}(t)$).\nConsequently, by Eq. (\\ref{eq:U}) and quantum It\\^{o} calculus \\cite{HP84},\nHeisenberg's equation of motion for the system in Fig.~\\ref{system} is, \\cite\n[ Eq. (26)]{GJN10}, \\cite[Eqs. (14)-(15)]{zhang13}, \n\\begin{align}\n\\dot{\\breve{a}}(t)& =\\mathbf{A}\\breve{a}(t)+\\mathbf{B}\\breve{b}_{\\mathrm{in}\n}(t),  \n\\nonumber \\\\\n\\breve{b}_{\\mathrm{out}}(t)& =\\mathbf{C}\\breve{a}(t)+\\mathbf{S}\\breve{b}_{\n\\mathrm{in}}(t),\\ \\ \\breve{a}(t_{0})=\\breve{a},  \\label{system-out}\n\\end{align}\nin which the constant system matrices are \n\\begin{equation}\n\\mathbf{S}=\\Delta (S_{-},0),\\ \\mathbf{C}=\\Delta (C_{-},C_{+}),\\ \\mathbf{B}=-\n\\mathbf{C}^{\\flat }\\mathbf{S},\\ \\mathbf{A}=-\\frac{1}{2}\\mathbf{C}^{\\flat }\n\\mathbf{C}-\\mathrm{i}J_{n}\\Delta (\\Omega _{-},\\Omega _{+}) \n \\label{eq:ABCD}\n\\end{equation}\nwith the matrix $J_{n}=\\mathrm{diag}(I_{n},-I_{n})$ introduced in the \\textit{\nNotation} part. The gauge process $\\Lambda _{\\mathrm{out}}(t)$ of the output\nfields, \n\\begin{equation}\n\\Lambda _{\\mathrm{out}}(t)\\triangleq \\int_{t_{0}}^{t}b_{\\mathrm{out}\n}^{\\#}(\\tau )b_{\\mathrm{out}}^{T}(\\tau )d\\tau =U(t, t_{0})^{\\ast }\\Lambda _{\n\\mathrm{in}}(t)U(t, t_{0}),  \\label{output_gauge}\n\\end{equation}   \nsatisfies the following quantum stochastic differential equation (QSDE), \n\\cite{GJ09}, \\cite[Eq. (16)]{zhang12}, \n\\begin{equation}\nd\\Lambda _{\\mathrm{out}}(t)=S_{-}^{\\#}d\\Lambda _{\\mathrm{in}\n}(t)S_{-}^{T}+S_{-}^{\\#}dB_{\\mathrm{in}}^{\\#}(t)L^{T}(t)+L^{\\#}(t)dB_{\n\\mathrm{in}}^{T}(t)S_{-}^{T}+L^{\\#}(t)L^{T}(t)dt.  \\label{eq:output_gauge}\n\\end{equation}\nIn quantum optics, the diagonal elements of $\\Lambda _{\\mathrm{out}}(t)$ are operators for the\ntotal number of photons in each of the $m$ output channels, counted from\ntime $t_{0}$ to $t$. The intensity of the output field, namely the rate of\nchange of the number process $\\Lambda _{\\mathrm{out}}(t)$, is given by, \\cite\n[Eq. (45)]{zhang13}, \n\\begin{equation}\n\\bar{n}_{\\mathrm{out}}(t)\\triangleq \\langle \\phi \\Psi _{\\mathrm{in}}|b_{\n\\mathrm{out}}^{\\#}(t)b_{\\mathrm{out}}^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n.  \\label{n_out2}\n\\end{equation}\nIn Eq. (\\ref{n_out2}), $|\\phi \\rangle $ is the initial system state and $\n|\\Psi _{\\mathrm{in}}\\rangle $ is the initial input field state. Therefore, the ket vector $ |\\phi\\rangle \\otimes |\\Psi _{\\mathrm{in}}\\rangle \\equiv |\\phi \\Psi _{\\mathrm{in}}\\rangle $\nis the initial joint system-field state. The bra vector $\\langle \\phi \\Psi _{\n\\mathrm{in}}|$ is the Hilbert space conjugate of the ket vector $|\\phi \\Psi\n_{\\mathrm{in}}\\rangle $. In this paper, $|\\phi \\rangle $ is always assumed\nto be the vacuum state, while the specific form of $|\\Psi _{\\mathrm{in}\n}\\rangle $ will be given in due course.\n\nThe quantum linear system $G$ is said to be \\textit{asymptotically stable}\nif the matrix $\\mathbf{A}$ in Eq. (\\ref{eq:ABCD}) is Hurwitz stable, \\cite[Sec.~III-A]{zhang11}. In\nanalogy to classical (namely non-quantum) control theory, the impulse\nresponse function of the system $G$ is, \\cite[Eq. (18)]{zhang13}, \n\\begin{equation*}\ng_{G}(t)\\triangleq \\left\\{ \n\\begin{array}{ll}\n\\delta (t)\\mathbf{S}-\\mathbf{C}e^{\\mathbf{A}t}\\mathbf{C}^{\\flat }\\mathbf{S},\n& t\\geq 0, \\\\ \n0, & t<0,\n\\end{array}\n\\right.  \\label{eq:gg}\n\\end{equation*}\nwhich enjoys the following doubled-up form \n\\begin{equation}\ng_{G}(t)=\\Delta ( g_{G^{-}}(t),g_{G^{+}}(t)) ,  \\label{eq:impulse}\n\\end{equation}\nwith matrix functions \n\\begin{align}\ng_{G^{-}}(t)& \\triangleq \\left\\{ \n\\begin{array}{ll}\n\\delta (t)S_{-}-[\n\\begin{array}{cc}\nC_{-} & C_{+}\n\\end{array}\n]e^{\\mathbf{A}t}\\left[ \n\\begin{array}{c}\nC_{-}^{\\dag } \\\\ \n-C_{+}^{\\dag }\n\\end{array}\n\\right] S_{-}, & t\\geq 0, \\\\ \n0, & t<0,\n\\end{array}\n\\right.  \\nonumber \\\\\ng_{G^{+}}(t)& \\triangleq \\left\\{ \n\\begin{array}{ll}\n-[\n\\begin{array}{cc}\nC_{-} & C_{+}\n\\end{array}\n]e^{\\mathbf{A}t}\\left[ \n\\begin{array}{c}\n-C_{+}^{T} \\\\ \nC_{-}^{T}\n\\end{array}\n\\right] S_{-}^{\\#}, & t\\geq 0, \\\\ \n0, & t<0.\n\\end{array}\n\\right.  \\label{eq:io}\n\\end{align}\nNext, we express the output field in terms of the impulse function $g_G(t)$. In fact, solving Eq. (\\ref{system-out}) we have \n\\begin{equation}\n\\breve{b}_{\\mathrm{out}}(t)=\\mathbf{C}e^{\\mathbf{A}(t-t_{0})}\\breve{a}\n+\\int_{t_{0}}^{t}g_{G}(t-r)\\breve{b}_{\\mathrm{in}}(r)dr.  \n\\label{eq:out_tf}\n\\end{equation}\nFurthermore, if the system is asymptotically stable, then in the limit $t_{0}\\rightarrow\n-\\infty $, Eq. (\\ref{eq:out_tf}) reduces to\n\\begin{equation}\n\\breve{b}_{\\mathrm{out}}(t)\n=\n\\int_{-\\infty }^{t}g_{G}(t-r)\\breve{b}_{\\mathrm{\nin}}(r)dr = g_{G}\\circledast \\breve{b}_{\\mathrm{in}}(t). \n \\label{eq:out_tf_2}\n\\end{equation}\n\n\n\n\\begin{remark}\nIf the interaction starts in the remote past, namely $t_{0}\\rightarrow\n-\\infty $, and if the system is asymptotically stable, Eq. (\\ref{eq:out_tf_2}) indicates that the initial system information has no influence on the\noutput field. This is also true in classical control theory, see, e.g., \\cite{KS72}.\n\\end{remark}\n\nDefine a matrix function \n\\begin{equation}\ng_{G^{-1}}(t)\\triangleq \\Delta (g_{G^{-}}(-t)^{\\dagger\n},-g_{G^{+}}(-t)^{T}) .  \\label{eq:G_inv}\n\\end{equation}\nIt can be verified that the following convolution relations \n\\begin{equation}\ng_{G}\\circledast g_{G^{-1}}\\circledast f(t)=g_{G^{-1}}\\circledast\ng_{G}\\circledast f(t)=f(t)  \\label{eq:may9_1}\n\\end{equation}\nhold for any function $f(t)$ of suitable dimension provided that the involved integrals converge. Thus, $g_{G^{-1}}(t)$\nis the inverse function of the impulse response function $g_{G}(t)$.\nAccording to Eqs. (\\ref{eq:out_tf_2}) and (\\ref{eq:may9_1}), in the limit $\nt_{0}\\rightarrow -\\infty $ we have\n\\begin{equation}\n\\breve{b}_{\\mathrm{in}}(t)=g_{G^{-1}}\\circledast \\breve{b}_{\\mathrm{out}}(t).\n\\label{eq:out_tf_3}\n\\end{equation}\n\n\n\nA class of \\textit{passive} quantum linear systems is obtained when $C_{+}=0$\nand $\\Omega _{+}=0$ in Eq. (\\ref{eq:ABCD}). For this type of systems, it is\nsufficient to work in the annihilation-operator representation. To be\nspecific, it suffices to study \n\\begin{eqnarray}\n\\dot{a}(t) &=&Aa(t)+Bb_{\\mathrm{in}}(t),  \\nonumber \\\\\nb_{\\mathrm{out}}(t) &=&Ca(t)+S_{-}b_{\\mathrm{in}}(t),~~~a(t_{0})=a,\n\\label{sys_passive}\n\\end{eqnarray}\nwhere \n\\begin{equation*}\nA=-\\mathrm{i}\\Omega _{-}-\\frac{1}{2}C_{-}^{\\dag }C_{-},~~~B=-C_{-}^{\\dag\n}S_{-},~~~C=C_{-}.\n\\end{equation*}\nIn this case, Eq. (\\ref{eq:io}) reduces to\n\\begin{equation}\ng_{G^{-}}(t)=\\left\\{ \n\\begin{array}{ll}\n\\delta (t)S_{-}-C_{-}e^{At}C_{-}^{\\dag }S_{-}, & t\\geq 0, \\\\ \n0, & t<0,\n\\end{array}\n\\right. \\ \\ \\ \\ \\ \\  g_{G^{+}}(t)=0.  \\label{july17_1}\n\\end{equation}\nAccordingly, Eqs.  (\\ref{eq:impulse}) and (\\ref{eq:G_inv}) reduce to \n\\begin{equation}\ng_{G}(t)=\\Delta \\left( g_{G^{-}}(t),0\\right) ,   \\ \\ \ng_{G^{-1}}(t)\\triangleq \\Delta \\left( g_{G^{-}}(-t)^{\\dagger },0\\right) ,\n\\label{dec1_g_inv}\n\\end{equation}\nrespectively.\n\nIt is well-known that in linear classical control theory,  if an asymptotically stable finite-dimensional linear time-invariant (FDLTI) system is driven by Gaussian white noise, then the steady-state output is again a Gaussian stationary process, see, e.g., \\cite[Section 11, Chapter 1]{KS72}, \\cite{AM79}, and  \\cite{BH97}.  The following result is the quantum counterpart. \n\n\\begin{lemma}\n\\label{lem:gaussian}\\cite[Theorem 2]{zhang13} Let the asymptotically stable\nquantum linear system $G$ be initialized in the vacuum state $|\\phi \\rangle $\nand let the input field $|\\Psi _{\\mathrm{in}}\\rangle $ be in the vacuum\nstate $|0\\rangle $. Then the steady-state \noutput field state is a zero-mean Gaussian state, whose power spectral density\nmatrix $R_{\\mathrm{out}}[\\mathrm{i}\\omega ]$ is given by \n\\begin{equation}\nR_{\\mathrm{out}}[\\mathrm{i}\\omega ]=G[\\mathrm{i}\\omega ]\\left[ \n\\begin{array}{cc}\nI_{m} & 0 \\\\ \n0 & 0_{m}\n\\end{array}\n\\right] G[\\mathrm{i}\\omega ]^{\\dag },  \\label{eq:temp_gaussian}\n\\end{equation}\nwhere $G[\\mathrm{i}\\omega ]=\\int_{-\\infty }^{\\infty }e^{-\\mathrm{i}\\omega\nt}g_{G}(t)dt$ is the two-sided Laplace transform of $g_{G}(t)$ with $s=\\mathrm{i}\\omega $, as\nintroduced in the Notation part. In particular, if the system is passive,\nthen the output is in a vacuum state with power spectral density matrix \n\\begin{equation*}\nR_{\\mathrm{out}}[\\mathrm{i}\\omega ]=\\left[ \n\\begin{array}{cc}\nI_{m} & 0 \\\\ \n0 & 0_{m}\n\\end{array}\n\\right] .  \\label{R_out_vac}\n\\end{equation*}\n\\end{lemma}\n\n\n\\subsection{Tensors}\\label{subsec:tensor}\n\nTensors and their associated operations are essential\nmathematical machinery for the research carried out in this paper \\cite\n{QSW07}, \\cite{KB09}, \\cite{zhang14}. In this subsection, we discuss several\ntensors.\n\nGiven an $m$-variable function $\\psi (t_{1},\\ldots ,t_{m})$ and an $m$-dimensional column vector $x(t)=[x_{1}(t)\\ \\cdots \\ x_{m}(t)]^{T}$, denote\n\\begin{equation}\n\\psi \\circ ^{m}x\\equiv \\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty\n}^{\\infty }dt_{1}\\cdots dt_{m}\\ \\psi (t_{1},\\ldots ,t_{m})x_{1}(t_{1})\\cdots\nx_{m}(t_{m}).  \\label{dec3_circ}\n\\end{equation}\nGiven an $m$-way $m$-dimensional tensor function $\\varphi =\\left( \\varphi\n_{j_{1}\\ldots j_{m}}(t_{1},\\ldots ,t_{m})\\right) $, ($j_{1},\\ldots\n,j_{m}=1,\\ldots ,m$), and an $m$-dimensional column vector $\nx(t)=[x_{1}(t)\\ \\cdots \\ x_{m}(t)]^{T}$, denote \n\\begin{equation}\n\\varphi \\circledcirc ^{m}x\n\\equiv\n \\sum_{j_{1}=1}^{m}\\cdots \\sum_{j_{m}=1}^{m}\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty\n}dt_{1}\\cdots dt_{m}\\ \\varphi _{j_{1}\\ldots j_{m}}(t_{1},\\ldots\n,t_{m})x_{j_{1}}(t_{1})\\cdots x_{j_{m}}(t_{m}). \n \\label{dec3_circ^2}\n\\end{equation}\nWe may update an $m$-variable function $\\psi (t_{1},\\ldots ,t_{m})$ to an $m$\n-way $m$-dimensional tensor function $\\psi ^{\\uparrow }=( \\psi^{\\uparrow }\n_{j_{1}\\ldots j_{m}}(t_{1},\\ldots ,t_{m})) $ with entries \n\\begin{equation}\n\\psi _{j_{1}\\ldots j_{m}}^{\\uparrow }(t_{1},\\ldots ,t_{m})\\triangleq \\left\\{ \n\\begin{array}{ll}\n\\psi (t_{1},\\ldots ,t_{m}), & \\mathrm{if\\ }j_{1}=1,j_{2}=2,\\ldots ,j_{m}=m,\n\\\\ \n0, & \\mathrm{otherwise}.\n\\end{array}\n\\right.  \\label{dec_up1}\n\\end{equation}\nThen Eq. (\\ref{dec3_circ}) can be re-written as Eq. (\\ref{dec3_circ^2}),\nspecifically,\n\\begin{equation*}\n\\psi \\circ ^{m}x=\\psi ^{\\uparrow }\\circledcirc ^{m}x.\n\\end{equation*}\nLet\\ $\\psi (t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m})$ be an $N$-variable function, where the positive integers\n$k_{1},\\ldots ,k_{m}$ satisfy $\\sum_{i=1}^{m}k_{i}=N$. Let $\nx(t)=[x_{1}(t)\\ \\cdots \\ x_{m}(t)]^{T}$ be an $m$-dimensional column vector.  Denote \n\\begin{eqnarray}\n&&\n\\psi \\ \\ \\cdot _{k_{1}\\cdots k_{m}}^{N}\\ \\ x  \n\\label{dec3_dot} \\\\\n&\\equiv &\n\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty\n}dt_{1}^{1}\\cdots dt_{k_{1}}^{1}\\cdots dt_{1}^{m}\\cdots dt_{k_{m}}^{m}\\ \\psi\n(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m})x_{1}(t_{1}^{1})\\cdots x_{1}(t_{k_{1}}^{1})\\cdots\nx_{m}(t_{1}^{m})\\cdots x_{m}(t_{k_{m}}^{m}).  \n\\nonumber\n\\end{eqnarray}\n\nGiven an $N$-way $m$-dimensional tensor function $\\varphi =\\left( \\varphi\n_{j_{1}^{1}\\ldots j_{k_{1}}^{1}\\ldots j_{1}^{m}\\ldots\nj_{k_{m}}^{m}}(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m})\\right) $ and an $m$-dimensional vector $\nx(t)=[x_{1}(t)\\ \\cdots \\ x_{m}(t)]^{T}$, denote\n\\begin{eqnarray}\n&&\n\\varphi  \\odot _{k_{1}\\cdots k_{m}}^{N}x  \n\\label{dec3_dot^2} \n\\\\\n&\\equiv & \\sum_{j_{1}^{1}=1}^{m}\\cdots \\sum_{j_{k_{1}}^{1}=1}^{m}\\cdots\n\\sum_{j_{1}^{m}=1}^{m}\\cdots \\sum_{j_{k_{m}}^{m}=1}^{m}\\int_{-\\infty\n}^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }dt_{1}^{1}\\cdots\ndt_{k_{1}}^{1}\\cdots dt_{1}^{m}\\cdots dt_{k_{m}}^{m}\\   \\nonumber \\\\\n&& \\ \\ \\ \\times \\varphi _{j_{1}^{1}\\ldots j_{k_{1}}^{1}\\ldots\nj_{1}^{m}\\ldots j_{k_{m}}^{m}}(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots\n,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})x_{j_{1}^{1}}(t_{1}^{1})\\cdots\nx_{j_{k_{1}}^{1}}(t_{k_{1}}^{1})\\cdots x_{j_{1}^{m}}(t_{1}^{m})\\cdots\nx_{j_{k_{m}}^{m}}(t_{k_{m}}^{m}).  \\nonumber\n\\end{eqnarray}\nUpdate the $N$-variable function $\\psi (t_{1}^{1},\\ldots\n,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})$ in Eq. (\\ref\n{dec3_dot}) to an $N$-way $m$-dimensional tensor function $\\psi ^{\\uparrow }(t_{1}^{1},\\ldots\n,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})$, whose elements are\ndefined as \n\\begin{eqnarray}\n&&\\psi _{j_{1}^{1}\\ldots j_{k_{1}}^{1}\\ldots j_{1}^{m}\\ldots\nj_{k_{m}}^{m}}^{\\uparrow }(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots\n,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})  \\nonumber \\\\\n&\\triangleq &\\left\\{ \n\\begin{array}{ll}\n\\psi (t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m}), & \\mathrm{if\\ }j_{1}^{1}=1,\\cdots\n,j_{k_{1}}^{1}=k_{1},\\cdots ,j_{1}^{m}=\\sum\\limits_{j=1}^{m-1}k_{j}+1,\\cdots\n,j_{k_{m}}^{m}=N, \\\\ \n0, & \\mathrm{otherwise}.\n\\end{array}\n\\right.   \\label{dec_up2}\n\\end{eqnarray}\nThen\n\\begin{equation*}\n\\psi \\ \\ \\cdot _{k_{1}\\cdots k_{m}}^{N}\\ \\ x=\\psi ^{\\uparrow }\\odot\n_{k_{1}\\cdots k_{m}}^{N}x.\n\\end{equation*}\n\nIn the above, we have defined several operations between tensors \nand vectors.  In the following, we look at operations between tensors and matrices.\n\nGiven an $m\\times m$ matrix function $\\mathcal{A}(t)$ and an $m$-way $m$-dimensional tensor function $\\varphi =\\left( \\varphi _{j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})\\right) $, $(j_{1},\\ldots ,j_{m}=1,\\ldots ,m)$,\ndefine another $m$-way $m$-dimensional tensor function $\\tilde{\\varphi}=(\n\\tilde{\\varphi}_{i_{1}\\ldots i_{m}}(r_{1},\\ldots ,r_{m}))$ in such a way\nthat, for all $i_{1},\\ldots ,i_{m}=1,\\ldots ,m$, \n\\begin{equation}\n\\tilde{\\varphi}_{i_{1}\\ldots i_{m}}(r_{1},\\ldots ,r_{m})=\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty\n}dt_{1}\\cdots dt_{m}\\ \\mathcal{A}^{i_{1}j_{1}}(r_{1}-t_{1})\\cdots \\mathcal{A}\n^{i_{m}j_{m}}(r_{m}-t_{m})\\varphi _{j_{1}\\ldots j_{m}}(t_{1},\\ldots ,t_{m}).\n\\label{eq:product_m_a}\n\\end{equation}\nEq. (\\ref{eq:product_m_a}) may be re-written in a more compact form \n\\begin{equation}\n\\tilde{\\varphi}=\\varphi \\circledast _{t}^{m}\\mathcal{A},\n\\label{eq:product_m_b}\n\\end{equation}\nwhere the subscript ``$t$'' indicates the\ntime domain, while the superscript ``$m$''\nimplies the $m$-fold convolution. Applying the $m$-dimensional Fourier\ntransform (\\ref{eq:fourier}) to Eq. (\\ref{eq:product_m_a}), we get\n\\begin{equation}\n\\tilde{\\varphi}_{i_{1}\\ldots i_{m}}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\n\\omega _{m})=\\sum_{j_{1},\\ldots ,j_{m}=1}^{m}\\mathcal{A}^{i_{1}j_{1}}[\n\\mathrm{i}\\omega _{1}]\\cdots \\mathcal{A}^{i_{m}j_{m}}[\\mathrm{i}\\omega\n_{m}]\\varphi _{j_{1}\\ldots j_{m}}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\n\\omega _{m}),  \\label{eq:product_m_c}\n\\end{equation}\nwhere\n\\begin{equation*}\n\\mathcal{A}^{i_{k}j_{k}}[\\mathrm{i}\\omega _{k}]=\\int_{-\\infty }^{\\infty }e^{-\n\\mathrm{i}\\omega _{k}t}\\mathcal{A}^{i_{k}j_{k}}(t)dt\n\\end{equation*}\nis the two-sided Laplace transform of $\\mathcal{A}^{i_{k}j_{k}}(t)$. In\nanalogy to Eq. (\\ref{eq:product_m_b}), we may also write Eq. (\\ref\n{eq:product_m_c}) in the following compact form \n\\begin{equation*}\n\\tilde{\\varphi}=\\varphi \\circledast _{\\omega }^{m}\\mathcal{A},\n\\label{eq:product_m_d}\n\\end{equation*}\nwhere the subscript ``$\\omega $'' indicates\nthe frequency domain.\n\nGiven an $N$-way $m$-dimensional tensor function $\\varphi =\\left( \\varphi\n_{j_{1}^{1}\\ldots j_{k_{1}}^{1}\\ldots j_{1}^{m}\\ldots\nj_{k_{m}}^{m}}(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m})\\right) $, and an $m\\times m$ matrix function $\\mathcal{A}(t)$%\n, define a new $N$-way $m$-dimensional tensor function $\\tilde{\\varphi}$ by \n\\begin{eqnarray*}\n&&\\tilde{\\varphi}_{l_{1}^{1}\\ldots l_{k_{1}}^{1}\\ldots l_{1}^{m}\\ldots\nl_{k_{m}}^{m}}(r_{1}^{1},\\ldots ,r_{k_{1}}^{1},\\ldots ,r_{1}^{m},\\ldots\n,r_{k_{m}}^{m})  \n\\label{dec11_1} \\\\\n&\\triangleq &\\sum_{i_{1}^{1}=1}^{m}\\cdots \\sum_{i_{k_{1}}^{1}=1}^{m}\\cdots\n\\sum_{i_{1}^{m}=1}^{m}\\cdots \\sum_{i_{k_{m}}^{m}=1}^{m}\\int_{-\\infty\n}^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }dt_{1}^{1}\\cdots\ndt_{k_{1}}^{1}\\cdots dt_{1}^{m}\\cdots dt_{k_{m}}^{m}\\ \\mathcal{A}\n^{l_{1}^{1}i_{1}^{1}}(r_{1}^{1}-t_{1}^{1})\\cdots \\mathcal{A}\n^{l_{k_{1}}^{1}i_{k_{1}}^{1}}(r_{k_{1}}^{1}-t_{k_{1}}^{1})\\times \\cdots \n\\nonumber \\\\\n&&\\;\\times \\mathcal{A}^{l_{1}^{m}i_{1}^{m}}(r_{1}^{m}-t_{1}^{m})\\cdots \n\\mathcal{A}^{l_{k_{m}}^{m}i_{k_{m}}^{m}}(r_{k_{m}}^{m}-t_{k_{m}}^{m})\\varphi\n_{i_{1}^{1}\\ldots i_{k_{1}}^{1}\\ldots i_{1}^{m}\\ldots\ni_{k_{m}}^{m}}(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m}),  \\nonumber\n\\end{eqnarray*}\nwhich may be re-written in a more compact form \n\\begin{equation}\n\\tilde{\\varphi}=\\varphi \\circledast _{t, k_{1}\\cdots k_{m}}^{N}\\mathcal{A}.\n\\label{dec11_2}\n\\end{equation}\n\nGiven an $m$-way $m$-dimensional tensor function $\\varphi = \\varphi _{j_1\n\\dots j_m}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\\omega _{m})$ in the frequency domain, denote\n\\begin{equation*}\n\\left\\Vert \\varphi (\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\\omega\n_{m})\\right\\Vert \\equiv \\sqrt{\\sum_{j_{1},\\ldots ,j_{m}=1}^{m}\\left\\vert\n\\varphi _{j_{1}\\ldots j_{m}}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\\omega\n_{m})\\right\\vert ^{2}},~~\\forall \\omega _{1},\\ldots ,\\omega _{m}\\in \\mathbb{R}.  \\label{eq:norm2}\n\\end{equation*}\nWe end this subsection by citing the following result.\n\n\n\n\\begin{lemma}\n\\label{lem:norm} \\cite[Theorem 3.3]{QSW07} Let two tensors $\\varphi $ and $\n\\tilde{\\varphi}$ be related by Eq. (\\ref{eq:product_m_c}), or equivalently\nEq. (\\ref{eq:product_m_a}). If $\\mathcal{A}[\\mathrm{i}\\omega ]$ is unitary\nfor all $\\omega \\in \\mathbb{R}$, then\n\\begin{equation*}\n\\left\\Vert \\tilde{\\varphi}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\\omega\n_{m})\\right\\Vert =\\left\\Vert \\varphi (\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i\n}\\omega _{m})\\right\\Vert ,~~\\forall \\omega _{1},\\ldots ,\\omega _{m}\\in \n\\mathbb{R}.  \\label{eq:norm3}\n\\end{equation*}\n\\end{lemma}\n\n\n\n\\section{$m$ photons superposed among $m$ input channels}\\label{sec:mm}\n\nIn this section, we investigate how a quantum linear system responds to a class of $m$-photon input states. We first define $m$-photon input states in Subsection \\ref\n{subsec:m_photon}, then derive the output intensity in Subsection \\ref\n{subsec:intensity}, after that, we present an analytic form of the output\nfield state when the underlying quantum linear system is passive in\nSubsections \\ref{sec:passive_mm} and \\ref{subsec:invariant}, finally we turn\nto the non-passive case in Subsection \\ref{subsec:nonpassive}.\n\n\n\\subsection{$m$-photon input states}\n\n\\label{subsec:m_photon}\n\nIn this subsection, we introduce a class of $m$-photon input states. For\nease of presentation, we start with the \\textit{single-channel\nsingle-photon} state case. In this case, $m=1$. A single-channel\nsingle-photon input state can be defined by \n\\begin{equation*}\n\\left\\vert \\Psi _{\\mathrm{in}}\\right\\rangle \\triangleq \\int_{-\\infty\n}^{\\infty }dt\\ \\psi _{\\mathrm{in}}(t)b_{\\mathrm{in}}^{\\ast }(t)\\left\\vert\n0\\right\\rangle .  \\label{eq:1_photon_t}\n\\end{equation*}\nHere, the function $\\psi _{\\mathrm{in}}$ is square integrable, more\nspecifically, $\\psi _{\\mathrm{in}}\\in L_{2}(\\mathbb{R},\\mathbb{C})$. The  Euclidean norm of $\\psi _{\\mathrm{in}}$, $\\Vert \\psi _{\\mathrm{in}}\\Vert \\triangleq \n\\sqrt{\\int_{-\\infty }^{\\infty }|\\psi _{\\mathrm{in}}(t)|^{2}dt}$, is equal to\n1. Consequently, the inner product $\\langle \\Psi _{\\mathrm{in}}|\\Psi _{\n\\mathrm{in}}\\rangle =1$. That is, $\\left\\vert \\Psi _{\\mathrm{in}\n}\\right\\rangle $ is a normalized state. Moreover, it can be easily shown\nthat \n\\begin{equation}\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\langle \\Psi _{\\mathrm{\nin}}|\\Lambda _{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\rangle =1,  \\label{aug1_1}\n\\end{equation}\nwhere $\\Lambda _{\\mathrm{in}}(t)$ is the input gauge process defined in Eq.\n(\\ref{july20_1}). Eq. (\\ref{aug1_1}) indicates that there is one\nphoton in the field. On the other hand, it can be readily verified that \n\\begin{equation}\n\\langle \\Psi _{\\mathrm{in}}|b_{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\rangle\n=\\langle \\Psi _{\\mathrm{in}}|b_{\\mathrm{in}}^{\\ast }(t)|\\Psi _{\\mathrm{in}\n}\\rangle =0,~\\forall t\\in \\mathbb{R}.  \\label{aug1_2}\n\\end{equation}\nThat is, the average field amplitude is zero. Finally, it is worth noting\nthat $|\\Psi _{\\mathrm{in}}\\rangle $ is not a single-photon \\textit{coherent}\nstate which can be defined to be \n\\begin{equation*}\n|\\alpha _{\\psi _{\\mathrm{in}}}\\rangle \\triangleq \\exp \\left( \\int_{-\\infty\n}^{\\infty }dt\\ \\alpha \\psi _{\\mathrm{in}}(t)b_{\\mathrm{in}}^{\\ast\n}(t)-\\int_{-\\infty }^{\\infty }dt\\ (\\alpha \\psi _{\\mathrm{in}}(t))^{\\ast }b_{\n\\mathrm{in}}(t)\\right) |0\\rangle ,  \\label{coherent}\n\\end{equation*}\nwhere $\\alpha =e^{i\\theta }$ is a complex number. In fact, for the\nsingle-photon coherent state $|\\alpha _{\\psi _{\\mathrm{in}}}\\rangle $, Eq. (\\ref{aug1_1}) still holds, but Eq. (\\ref{aug1_2}) does not.\n\nNext, we look at \\textit{single-channel} \\textit{two-photon} states, which\ncan be defined as \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \\triangleq \\int_{-\\infty }^{\\infty\n}\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}\\ \\psi _{\\mathrm{in}}(t_{1},t_{2})b_{\n\\mathrm{in}}^{\\ast }(t_{1})b_{\\mathrm{in}}^{\\ast }(t_{2})|0\\rangle .\n\\label{aug1_3}\n\\end{equation}\nHere, the function $\\psi _{\\mathrm{in}}(t_{1},t_{2})$ is required to\nnormalize the state, namely $\\langle \\Psi _{\\mathrm{in}}|\\Psi _{\\mathrm{in}\n}\\rangle =1$, which is equivalent to \n\\begin{equation*}\n2\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}\\ \\left\\vert\n\\psi _{\\mathrm{in}}(t_{1},t_{2})\\right\\vert ^{2}=1.\n\\end{equation*}\nMoreover, swapping $t_{1}$ and $t_{2}$ in Eq. (\\ref{aug1_3}) yields \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\int_{-\\infty }^{\\infty }\\int_{-\\infty\n}^{\\infty }dt_{1}dt_{2}\\ \\psi _{\\mathrm{in}}(t_{2},t_{1})b_{\\mathrm{in}\n}^{\\ast }(t_{1})b_{\\mathrm{in}}^{\\ast }(t_{2})|0\\rangle . \n \\label{aug1_4}\n\\end{equation}\nComparing Eqs. (\\ref{aug1_3}) and (\\ref{aug1_4}) we see that $\\psi _{\\mathrm{\nin}}(t_{1},t_{2})=\\psi _{\\mathrm{in}}(t_{2},t_{1})$. Moreover, it can be\nverified that \n\\begin{equation*}\n\\displaystyle{\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }}\\langle\n\\Psi _{\\mathrm{in}}|\\Lambda _{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\rangle =2,\n\\end{equation*}\ni.e., there are two photons in the field. \n\nNext, let us look at \n\\textit{two-channel} \\textit{two-photon} states, which can be defined to be \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \\triangleq \\int_{-\\infty }^{\\infty\n}\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}\\ \\psi _{\\mathrm{in}}(t_{1},t_{2})b_{\n\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})|0_{1}\\rangle\n\\otimes |0_{2}\\rangle .  \\label{aug1_5}\n\\end{equation}\nAgain, the function $\\psi _{\\mathrm{in}}(t_{1},t_{2})$ is required to\nnormalize the state. This is guaranteed by \n\\begin{equation}\n\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}\\ \\left\\vert\n\\psi _{\\mathrm{in}}(t_{1},t_{2})\\right\\vert ^{2}=1.  \\label{aug1_7}\n\\end{equation}\n(Notice that in this case, the condition $\\psi _{\\mathrm{in}\n}(t_{1},t_{2})=\\psi _{\\mathrm{in}}(t_{2},t_{1})$ is not necessary.) It can\nbe easily shown that \n\\begin{equation}\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\langle \\Psi _{\\mathrm{\nin}}|\\Lambda _{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\rangle =\\left[ \n\\begin{array}{cc}\n1 & 0 \\\\ \n0 & 1\n\\end{array}\n\\right] .  \\label{aug1_18}\n\\end{equation}\nEq. (\\ref{aug1_18}) implies that each channel contains one photon. However, these two photons can form an entangled state.  Moreover,\nif we use the single-photon state $\\int_{-\\infty }^{\\infty }\\gamma (\\tau )b_{\n\\mathrm{in},2}^{\\ast }(\\tau )d\\tau \\ |0_{2}\\rangle $ to measure the second\nchannel, the resulting state for the first channel is\ngiven by \n\\begin{equation*}\n\\int_{-\\infty }^{\\infty }\\left[ \\int_{-\\infty }^{\\infty }\\gamma ^{\\ast\n}(\\tau )\\psi _{\\mathrm{in}}(t,\\tau )d\\tau \\right] b_{\\mathrm{in},1}^{\\ast\n} (t) dt\\ |0_{1}\\rangle .  \\label{aug1_19}\n\\end{equation*\nIn general, Eq. (\\ref{aug1_5}) defines a state for which the two photons are\nentangled. However, for the special case that $\\psi _{\\mathrm{in}\n}(t_{1},t_{2})=\\xi _{1}(t_{1})\\xi _{2}(t_{2})$, we end up with a product\nstate \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\int_{-\\infty }^{\\infty }\\xi _{1}(t_{1})b_{\n\\mathrm{in},1}^{\\ast }(t_{1})dt_{1}|0_{1}\\rangle \\otimes \\int_{-\\infty\n}^{\\infty }\\xi _{2}(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{2})dt_{2}|0_{2}\\rangle .  \\label{aug1_17}\n\\end{equation}\nFor the state defined in Eq. (\\ref{aug1_17}), there exists no entanglement between these two photons.\n\nWe are ready to introduce a class of $m$-channel $m$-photon input states.\nSuch states can be of the form \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \\triangleq \\int_{-\\infty }^{\\infty }\\cdots\n\\int_{-\\infty }^{\\infty }dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{in}\n}(t_{1},\\ldots ,t_{m})b_{\\mathrm{in},1}^{\\ast }(t_{1})\\cdots b_{\\mathrm{in}\n,m}^{\\ast }(t_{m})|0_{1}\\rangle \\otimes \\cdots \\otimes |0_{m}\\rangle .\n\\label{eq:aug23_10}\n\\end{equation}\nFor convenience, in the sequel we use the shorthand notation $|0^{\\otimes\nm}\\rangle $ for the tensor product of the vacuum input fields $|0_{1}\\rangle\n\\otimes \\cdots \\otimes |0_{m}\\rangle $. In the notation introduced in\nSubsection \\ref{subsec:tensor}, Eq. (\\ref{eq:aug23_10}) may be re-written as \n\\begin{equation*}  \\label{dec23_1}\n|\\Psi _{\\mathrm{in}}\\rangle =\\psi _{\\mathrm{in}}\\circ ^{m}b_{\\mathrm{in}\n}^{\\#}\\ |0^{\\otimes m}\\rangle .\n\\end{equation*}\nBy analogy with Eq. (\\ref{aug1_7}), it can be readily shown that the\nnormalization condition for $|\\Psi _{\\mathrm{in}}\\rangle $ is \n\\begin{equation}\n\\left\\Vert \\psi _{\\mathrm{in}}\\right\\Vert ^{2}\\equiv \\int_{-\\infty }^{\\infty\n}\\cdots \\int_{-\\infty }^{\\infty }dt_{1}\\cdots dt_{m}\\ \\left\\vert \\psi _{\n\\mathrm{in}}(t_{1},\\ldots ,t_{m})\\right\\vert ^{2}=1.  \\label{aug1_6}\n\\end{equation}\nThe bra vector $\\left\\langle \\Psi _{\\mathrm{in}}\\right\\vert $, namely the\nconjugate of the ket vector $|\\Psi _{\\mathrm{in}}\\rangle $, is \n\\begin{equation}\n\\left\\langle \\Psi _{\\mathrm{in}}\\right\\vert \n=\n\\langle 0^{\\otimes m}\\vert \\int_{-\\infty }^{\\infty }\\cdots\n\\int_{-\\infty }^{\\infty }dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{in}}^{\\ast\n}(t_{1},\\ldots ,t_{m})b_{\\mathrm{in},1}(t_{1})\\cdots b_{\\mathrm{in}\n,m}(t_{m}).  \n\\label{aug22_1}\n\\end{equation}\n\nFor the $m$-channel $m$-photon state $|\\Psi _{\\mathrm{in}}\\rangle $, it is\nclear that\n\\begin{equation}\n\\left\\langle \\Psi _{\\mathrm{in}}|\\breve{b}_{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in\n}}\\right\\rangle =0,~\\forall t\\in \\mathbb{R}.  \n\\label{eq:amplitude}\n\\end{equation}\nThat is, the average field amplitude of the input light field is $0$. Next,\nwe look at two-time correlations $\\langle \\Psi _{\\mathrm{in}}|\\breve{b}_{\\mathrm{in}}(t)\\breve{b}_{\\mathrm{in}}^{\\dag\n}(r)|\\Psi _{\\mathrm{in}}\\rangle$ with $t,r\\in \\mathbb{R}$. For each $k=1,\\ldots ,m$, introduce the\nnotation \n\\begin{equation}\n\\zeta _{k}(\\tau ,r) \\equiv \\psi _{\\mathrm{in}}(\\tau _{1},\\ldots ,\\tau\n_{k-1},r,\\tau _{k+1},\\ldots ,\\tau _{m}).  \n\\label{eq:zeta_k}\n\\end{equation}\nSpecifically,\n\\begin{eqnarray*}\n\\zeta _{1}(\\tau ,r) &=&\\psi _{\\mathrm{in}}(r,\\tau _{2},\\ldots ,\\tau _{m}), \\\\\n\\zeta _{2}(\\tau ,r) &=&\\psi _{\\mathrm{in}}(\\tau _{1},r,\\tau _{3},\\ldots\n,\\tau _{m}), \\\\\n&&\\vdots \\\\\n\\zeta _{m}(\\tau ,r) &=&\\psi _{\\mathrm{in}}(\\tau _{1},\\ldots ,\\tau _{m-1},r).\n\\end{eqnarray*}\nAlso, define a diagonal matrix function\n\\begin{equation}\n\\Lambda \\left( t, r\\right) \\triangleq \\left[ \n\\begin{array}{ccc}\n\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }d\\tau _{2}\\cdots\nd\\tau _{m}\\ \\zeta _{1}(\\tau ,t)^{\\ast }\\zeta _{1}(\\tau ,r) &  &  \\\\ \n& \\ddots &  \\\\ \n&  & \\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }d\\tau\n_{1}\\cdots d\\tau _{m-1}\\ \\zeta _{m}(\\tau ,t)^{\\ast }\\zeta _{m}(\\tau ,r)\n\\end{array}\n\\right] ,\\ \\forall t, r\\in \\mathbb{R}.  \n\\label{eq:input_Lambda}\n\\end{equation}\nClearly, $\\Lambda \\left( t, r\\right) ^\\dag =\\Lambda \\left( r, t\\right) $, and\nby Eq. (\\ref{aug1_6}), $\\int_{-\\infty }^{\\infty }dt\\ \\Lambda \\left(\nt, t\\right) =I_{m}$. Furthermore, it can be shown that the two-time\ncorrelation $\\langle \\Psi _{\\mathrm{in}}|\\breve{b}_{\\mathrm{in}}(t)\\breve{b}_{\\mathrm{in}}^{\\dag\n}(r)|\\Psi _{\\mathrm{in}}\\rangle$ has the form \n\\begin{equation}\n\\left\\langle \\Psi _{\\mathrm{in}}|\\breve{b}_{\\mathrm{in}}(t)\\breve{b}_{\\mathrm{in}}^{\\dag\n}(r)|\\Psi _{\\mathrm{in}}\\right\\rangle =\\delta (t-r)\\left[ \n\\begin{array}{cc}\nI_{m} & 0 \\\\ \n0 & 0_{m}\n\\end{array}\n\\right] +\\left[ \n\\begin{array}{cc}\n\\Lambda \\left( r,t\\right) & 0 \\\\ \n0 & \\Lambda \\left( t,r\\right)\n\\end{array}\n\\right].  \\label{eq:input_non_Markovian}\n\\end{equation}\n\n\n\\begin{remark}\nIf all the input fields are in the vacuum state, i.e., $|\\Psi_{\\rm in}\\rangle = |0^{\\otimes m}\\rangle$, it is well-known that \n\\begin{equation}\n\\left\\langle 0^{\\otimes m}|\\breve{b}_{\\mathrm{in}}(t)\\breve{b}_{\\mathrm{in}\n}^{\\dag }(r)|0^{\\otimes m}\\right\\rangle =\\delta (t-r)\\left[ \n\\begin{array}{cc}\nI_{m} & 0 \\\\ \n0 & 0_{m}\n\\end{array}\n\\right] .\n\\end{equation}\nIn this case, the field is Markovian. The second term on the right-hand side\nof Eq. (\\ref{eq:input_non_Markovian}) reveals the \\textit{non-Markovian}\nnature of the $m$-channel $m$-photon input fields. Moreover, due to the\npresence of the pulse shape $\\psi _{\\mathrm{in}}$ in all the diagonal\nentries of $\\Lambda \\left(t, r\\right) $, the inputs can be regarded as\ncorrelated non-Markovian noise inputs.\n\\end{remark}\n\n\n\n\\subsection{The passive case: output intensity}\\label{subsec:intensity}\n\nIn this subsection, for the \\textit{passive }quantum linear system (\\ref\n{sys_passive}) driven by an $m$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle $\ndefined in Eq. (\\ref{eq:aug23_10}), we derive a formula for the output\nintensity $\\bar{n}_{\\mathrm{out}}(t)$ defined in Eq. (\\ref{n_out2}).\n\nRecall that in the passive case the matrix $C_{+}=0$. Substitution of $\nL(t)=C_{-}a(t)$ into Eq. (\\ref{eq:output_gauge}) yields \n\\begin{equation}\nd\\Lambda _{\\mathrm{out}}(t)=S_{-}^{\\#}d\\Lambda _{\\mathrm{in}\n}(t)S_{-}^{T}+S_{-}^{\\#}dB_{\\mathrm{in}}^{\\#}(t)a^{T}(t)C_{-}^{T}+C_{-}^{\n\\#}a^{\\#}(t)dB_{\\mathrm{in}}^{T}(t)S_{-}^{T}+C_{-}^{\\#}a^{\n\\#}(t)a^{T}(t)C_{-}^{T}dt.  \\label{eq:output_gauge_2}\n\\end{equation}\nInspired by the second term on the right-hand side of Eq. (\\ref\n{eq:output_gauge_2}), we define an $n$-by-$m$ matrix function $f(t)$ as \n\\begin{equation}\nf(t)\\triangleq \\langle \\phi \\Psi _{\\mathrm{in}}|b_{\\mathrm{in}\n}^{\\#}(t)a^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle ^{T}.  \\label{eq:m+2}\n\\end{equation}\nMoreover, define an $n\\times n$ matrix function $\\Sigma (t)$ to be \n\\begin{equation}\n\\Sigma (t)\\triangleq \\langle \\phi \\Psi _{\\mathrm{in}}|a(t)a^{\\dag }(t)|\\phi\n\\Psi _{\\mathrm{in}}\\rangle ,\\ \\ \\ t\\geq t_{0}.  \\label{eq:sigma}\n\\end{equation} \nClearly, $\\Sigma(t) = \\Sigma(t)^\\dag$. \n\nThe following theorem is the main result of this subsection, which gives an\nexplicit procedure for computing the output intensity $\\bar{n}_{\\mathrm{out}}(t)$.\n\n\\begin{theorem}\\label{thm:intensity} \nFor the \\textit{passive }quantum linear system (\\ref\n{sys_passive}) initialized in the vacuum state $|\\phi\\rangle$ and  driven by the $m$-channel $m$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle $\ndefined in Eq. (\\ref{eq:aug23_10}),\n the matrix function $f(t)$ defined\nin Eq. (\\ref{eq:m+2}) has the following form \n\\begin{equation}\nf(t)=-\\int_{t_{0}}^{t}e^{A(t-r)}C_{-}^{\\dagger } S_- \\Lambda \\left(\nt, r\\right) dr, \n \\label{eq:m_+_dot}\n\\end{equation}\nwhere the matrix function $ \\Lambda (t, r)$ is given in Eq. (\\ref{eq:input_Lambda}). The output intensity $\\bar{n}_{\\mathrm{out}}(t)$ is given by\n\\begin{equation}  \\label{eq:intensity}\n\\bar{n}_{\\mathrm{out}}(t) = S_{-}^{\\#}\\Lambda \\left( t, t\\right) S_{-}^{T}\n+S_{-}^{\\#}f(t)^{T}C_{-}^{T}+C_{-}^{\\#}f(t)^{\\#}S_{-}^{T}-C_{-}^{\\#}C_{-}^{T}+C_{-}^{\\#}\\Sigma (t)^{T}C_{-}^{T}, \n\\end{equation}\nin which the covariance function $\\Sigma (t)$ solves the following matrix\nequation \n\\begin{equation}\n\\dot{\\Sigma}(t)=A\\Sigma (t)+\\Sigma (t)A^{\\dagger }+C_{-}^{\\dagger\n}C_{-}-C_{-}^{\\dagger }S_{-}f(t)^{\\dagger }-f(t)S_{-}^{\\dagger }C_{-}\n\\label{eq:Sigma_dot_2}\n\\end{equation}\nwith the initial condition $\\Sigma (t_{0})=I_{n}$.\n\\end{theorem}\n\n\n\\textbf{Proof.} ~ We prove this theorem in three steps.\n\n\\textit{Step 1.} We establish Eq. (\\ref{eq:m_+_dot}). Firstly, it can be\nreadily shown that\n\\begin{equation}\nb_{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\rangle =\\left[ \n\\begin{array}{c}\n|\\zeta _{1}(t)\\rangle \\\\ \n\\vdots \\\\ \n|\\zeta _{m}(t)\\rangle\n\\end{array}\n\\right] ,  \\label{eq:mar27_1}\n\\end{equation}\nwhere the following {\\it notation}\n\\begin{equation}\n|\\zeta _{j}(t)\\rangle \\equiv \\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty\n}^{\\infty }d\\tau _{1}\\cdots d\\tau _{j-1}d\\tau _{j+1}\\cdots d\\tau _{m}\\;\\psi\n_{\\mathrm{in}}(\\tau _{1},\\ldots ,\\tau _{j-1},t,\\tau _{j+1},\\ldots ,\\tau\n_{m})\\tprod\\limits_{k=1,k\\neq j}^{m}b_{\\mathrm{in},k}^{\\ast }(\\tau\n_{k})|0^{\\otimes m}\\rangle  \\label{eq:zeta_0}\n\\end{equation}\nhas been used, ($j=1,\\ldots ,m)$.  By the notation in Eq. (\\ref{eq:zeta_k}), Eq. (\\ref{eq:zeta_0}) can be re-written as\n\\begin{equation}\n|\\zeta _{j}(t)\\rangle = \\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty\n}^{\\infty }d\\tau _{1}\\cdots d\\tau _{j-1}d\\tau _{j+1}\\cdots d\\tau _{m}\\;\\zeta _{j}(\\tau, t)\\tprod\\limits_{k=1,k\\neq j}^{m}b_{\\mathrm{in},k}^{\\ast }(\\tau\n_{k})|0^{\\otimes m}\\rangle.\n  \\label{eq:zeta}\n\\end{equation}\n As a result, \n\\begin{equation}\n\\langle \\phi \\Psi _{\\mathrm{in}}|b_{\\mathrm{in}}^{\\#}(t)b_{\\mathrm{in}\n}^{T}(r)|\\phi \\Psi _{\\mathrm{in}}\\rangle \n=\n\\langle \\Psi _{\\mathrm{in}}|b_{\n\\mathrm{in}}^{\\#}(t)b_{\\mathrm{in}}^{T}(r)|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\left[\n\\begin{array}{ccc}\n\\langle \\zeta _{1}(t)|\\zeta _{1}(r)\\rangle &  &  \\\\ \n& \\ddots &  \\\\ \n&  & \\langle \\zeta _{m}(t)|\\zeta _{m}(r)\\rangle\n\\end{array}\n\\right] =\\Lambda \\left( t,r\\right) ,  \n\\label{eq:mar27_2}\n\\end{equation}\nwhere Eq.  (\\ref{eq:input_Lambda}) has been used in the last step. \nMoreover, \n\\begin{equation}\n\\langle \\phi \\Psi _{\\mathrm{in}}|b_{\\mathrm{in}}^{\\#}(t)a^{T}(t)|\\phi \\Psi _{\n\\mathrm{in}}\\rangle =\\left[ \n\\begin{array}{c}\n\\langle \\phi \\Psi _{\\mathrm{in}}|b_{\\mathrm{in},m}^{\\ast }(t)a^{T}(t)|\\phi\n\\Psi _{\\mathrm{in}}\\rangle \\\\ \n\\vdots \\\\ \n\\langle \\phi \\Psi _{\\mathrm{in}}|b_{\\mathrm{in},m}^{\\ast }(t)a^{T}(t)|\\phi\n\\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] =\\left[ \n\\begin{array}{c}\n\\langle \\phi \\zeta _{1}(t)|a^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle \\\\ \n\\vdots \\\\ \n\\langle \\phi \\zeta _{m}(t)|a^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] .  \\label{eq:mar28_temp2}\n\\end{equation}\nSubstituting Eq. (\\ref{eq:mar28_temp2}) into Eq. (\\ref{eq:m+2}) yields\n\\begin{equation}\nf(t)=\\left[ \n\\begin{array}{ccc}\n\\langle \\phi \\zeta _{1}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle & \\cdots & \n\\langle \\phi \\zeta _{m}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] .  \n\\label{eq:mar28_temp3}\n\\end{equation}\nSecondly, solving Eq. (\\ref{sys_passive}) we get \n\\begin{equation}\na(t)=e^{A(t-t_{0})}a-\\int_{t_{0}}^{t}e^{A(t-r)}C_{-}^{\\dagger }S_- b_{\n\\mathrm{in}}(r)dr,\\ \\ t\\geq t_{0}.  \n\\label{eq:a_t}\n\\end{equation}\nPartition the $n$-by-$m$ matrix function $e^{At}C_{-}^{\\dagger }S_-$ into $m$\ncolumns, specifically, \n\\begin{equation}\ne^{At}C_{-}^{\\dagger }S_-=[\n\\begin{array}{ccc}\nc_{1}(t) & \\cdots & c_{m}(t)\n\\end{array}\n].  \\label{eq:c}\n\\end{equation}\nBy Eqs. (\\ref{eq:a_t}), (\\ref{eq:c}), (\\ref{eq:mar27_1}), and (\\ref{eq:mar27_2}), we have%\n\\begin{eqnarray}\n&&\n\\left[ \n\\begin{array}{ccc}\n\\langle \\phi \\zeta _{1}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle & \\cdots & \n\\langle \\phi \\zeta _{m}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right]  \\nonumber \\\\\n&=&\n-\\left[ \n\\begin{array}{ccc}\n\\langle  \\zeta _{1}(t)|\\int_{t_{0}}^{t}dr\\ e^{A(t-r)}C_{-}^{\\dagger }S_-b_{\n\\mathrm{in}}(r)| \\Psi _{\\mathrm{in}}\\rangle & \\cdots & \\langle \n\\zeta _{m}(t)|\\int_{t_{0}}^{t}dr\\ e^{A(t-r)}C_{-}^{\\dagger }S_-b_{\\mathrm{in}\n}(r)| \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right]  \\nonumber \\\\\n&=&\n-\\int_{t_{0}}^{t}\\left[ \n\\begin{array}{ccc}\n\\sum_{j=1}^{m}\\langle  \\zeta _{1}(t)|c_{j}(t-r)b_{\\mathrm{in},j}(r)|\n\\Psi _{\\mathrm{in}}\\rangle & \\cdots & \\sum_{j=1}^{m}\\langle \\zeta\n_{m}(t)|c_{j}(t-r)b_{\\mathrm{in},j}(r)| \\Psi _{\\mathrm{in}}\\rangle%\n\\end{array}\n\\right] dr  \\nonumber \\\\\n&=&\n-\\int_{t_{0}}^{t}\\left[ \n\\begin{array}{ccc}\n\\sum_{j=1}^{m}\\langle \\zeta _{1}(t)|c_{j}(t-r))|\\zeta _{j}(r)\\rangle & \\cdots & \\sum_{j=1}^{m}\\langle \\zeta\n_{m}(t)|c_{j}(t-r)|\\zeta _{j}(r)\\rangle\n\\end{array}\n\\right] dr  \\nonumber \\\\\n&=&-\\int_{t_{0}}^{t}\\left[ \n\\begin{array}{ccc}\n\\langle \\zeta _{1}(t)|c_{1}(t-r)|\\zeta _{1}(r)\\rangle & \\cdots & \\langle\n\\zeta _{m}(t)|c_{m}(t-r)|\\zeta _{m}(r)\\rangle\n\\end{array}\n\\right] dr  \\nonumber \\\\\n&=&-\\int_{t_{0}}^{t}\\left[ \n\\begin{array}{ccc}\nc_{1}(t-r) & \\cdots & c_{m}(t-r)\n\\end{array}\n\\right] \\left[ \n\\begin{array}{ccc}\n\\langle \\zeta _{1}(t)|\\zeta _{1}(r)\\rangle &  &  \\\\ \n& \\ddots &  \\\\ \n&  & \\langle \\zeta _{m}(t)|\\zeta _{m}(r)\\rangle\n\\end{array}\n\\right] dr  \\nonumber \\\\\n&=&-\\int_{t_{0}}^{t}e^{A(t-r)}C_{-}^{\\dagger }S_-\\Lambda \\left( t,r\\right)\ndr.  \\label{eq:mar28_temp1}\n\\end{eqnarray}\nSubstituting Eq. (\\ref{eq:mar28_temp1}) into Eq. (\\ref{eq:mar28_temp3})\ngives Eq. (\\ref{eq:m_+_dot}).\n\n\\textit{Step 2.} We establish Eq. (\\ref{eq:Sigma_dot_2}). By  It\\^{o}\ncalculus and Eq. (\\ref{eq:m+2}), we have \n\\begin{eqnarray}\n&&\nd\\Sigma (t)  \\nonumber \\\\\n&=&\nd\\langle \\phi \\Psi _{\\mathrm{in}}|a(t)a^{\\dag }(t)|\\phi \\Psi _{\\mathrm{in}\n}\\rangle  \\nonumber \\\\\n&=&\n\\langle \\phi \\Psi _{\\mathrm{in}}|(da(t))a^{\\dag }(t)|\\phi \\Psi _{\\mathrm{\nin}}\\rangle +\\langle \\phi \\Psi _{\\mathrm{in}}|a(t)(da^{\\dag }(t))|\\phi \\Psi\n_{\\mathrm{in}}\\rangle +\\langle \\phi \\Psi _{\\mathrm{in}}|(da(t))(da^{\\dag\n}(t))|\\phi \\Psi _{\\mathrm{in}}\\rangle  \\nonumber \\\\\n&=&\nA\\Sigma (t)dt+\\Sigma (t)A^{\\dagger }dt\n-C_{-}^{\\dagger }S_{-}\\langle \\phi \\Psi _{\\mathrm{in}}|dB_{\\mathrm{in}\n}(t)a^{\\dag }(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle -\\langle \\phi \\Psi _{\n\\mathrm{in}}|a(t)dB_{\\mathrm{in}}^{\\dagger }(t)|\\phi \\Psi _{\\mathrm{in}\n}\\rangle S_{-}^{\\dagger }C_{-}+C_{-}^{\\dagger }C_{-}dt  \\nonumber \\\\\n&=&A\\Sigma (t)dt+\\Sigma (t)A^{\\dagger }dt+C_{-}^{\\dagger }C_{-}dt  \\nonumber \\\\\n&&-C_{-}^{\\dagger }S_{-}\\left[ \n\\begin{array}{c}\n\\langle \\phi \\Psi _{\\mathrm{in}}|a^{\\dag }(t)dB_{\\mathrm{in},1}(t)|\\phi \\Psi\n_{\\mathrm{in}}\\rangle \\\\ \n\\vdots \\\\ \n\\langle \\phi \\Psi _{\\mathrm{in}}|a^{\\dag }(t)dB_{\\mathrm{in},m}(t)|\\phi \\Psi\n_{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] -\\left[ \n\\begin{array}{ccc}\n\\langle \\phi \\Psi _{\\mathrm{in}}|dB_{\\mathrm{in},1}^{\\ast }(t)a(t)|\\phi \\Psi\n_{\\mathrm{in}}\\rangle & \\cdots & \\langle \\phi \\Psi _{\\mathrm{in}}|dB_{\n\\mathrm{in},m}^{\\ast }(t)a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] S_{-}^{\\dagger }C_{-}  \\nonumber \\\\\n&=&A\\Sigma (t)dt+\\Sigma (t)A^{\\dagger }dt+C_{-}^{\\dagger }C_{-}dt\n-C_{-}^{\\dagger }S_{-}\\left[ \n\\begin{array}{ccc}\n\\langle \\phi \\zeta _{1}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle & \\cdots & \n\\langle \\phi \\zeta _{m}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] ^{\\dagger }dt  \\nonumber \\\\\n&&-\\left[ \n\\begin{array}{ccc}\n\\langle \\phi \\zeta _{1}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle & \\cdots & \n\\langle \\phi \\zeta _{m}(t)|a(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\end{array}\n\\right] S_{-}^{\\dagger }C_{-}dt  \\nonumber \\\\\n&=&A\\Sigma (t)dt+\\Sigma (t)A^{\\dagger }dt+C_{-}^{\\dagger }C_{-}dt\n-C_{-}^{\\dagger }S_{-}f(t)^{\\dagger }dt-f(t)S_{-}^{\\dagger }C_{-}dt .\n\\label{eq:Sigma_dot}\n\\end{eqnarray}\nIn Eq. (\\ref{eq:Sigma_dot}), the commutation relations $[a_{j}(t),dB_{\\mathrm{in}\n,k}(t)]=[a_{j}^{\\ast }(t),dB_{\\mathrm{in},k}(t)]=[a_{j}(t),dB_{\\mathrm{in}\n,k}^{\\ast }(t)]=[a_{j}^{\\ast }(t),dB_{\\mathrm{in},k}^{\\ast }(t)]=0$ ($\nj=1\\ldots ,n$, $k=1,\\ldots ,m$) have been used to get the 4th step, and Eq. (\\ref{eq:mar28_temp3}) has been used to get the 6th step (which is the last\nstep). Dividing both sides of Eq. (\\ref{eq:Sigma_dot}) by $dt$ yields Eq. (\\ref{eq:Sigma_dot_2}).\n\n\\textit{Step 3.} We establish Eq. (\\ref{eq:intensity}). By the canonical\ncommutation relation $[a_{j},a_{k}^{\\ast }]=\\delta _{jk}$ ($j, k=1,\\ldots ,n$\n), we have \n\\begin{eqnarray}\n\\Sigma (t) &=&\\langle \\phi \\Psi _{\\mathrm{in}}|a(t)a^{\\dag }(t)|\\phi \\Psi _{\n\\mathrm{in}}\\rangle  \\nonumber \\\\\n&=&\\langle \\phi \\Psi _{\\mathrm{in}}|\\left( I+(a^{\\#}(t)a^{T}(t))^{T}\\right)\n|\\phi \\Psi _{\\mathrm{in}}\\rangle  \\nonumber \\\\\n&=&I+\\langle \\phi \\Psi _{\\mathrm{in}}|a^{\\#}(t)a^{T}(t)|\\phi \\Psi _{\\mathrm{\nin}}\\rangle ^{T}.  \\label{aug1_temp1}\n\\end{eqnarray}\nThis, together with Eqs. (\\ref{eq:output_gauge_2}) and (\\ref{eq:mar28_temp3}), yields\n\\begin{eqnarray}\n&&\\langle \\phi \\Psi _{\\mathrm{in}}|d\\Lambda _{\\mathrm{out}}(t)|\\phi \\Psi _{\n\\mathrm{in}}\\rangle  \\nonumber \\\\\n&=&S_{-}^{\\#}\\langle \\phi \\Psi _{\\mathrm{in}}|d\\Lambda _{\\mathrm{in}\n}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle S_{-}^{T}+S_{-}^{\\#}\\langle \\phi \\Psi _{\n\\mathrm{in}}|dB_{\\mathrm{in}}^{\\#}(t)a^{T}(t)|\\phi \\Psi _{\\mathrm{in}\n}\\rangle C_{-}^{T}  \\nonumber \\\\\n&&+C_{-}^{\\#}\\langle \\phi \\Psi _{\\mathrm{in}}|a^{\\#}(t)dB_{\\mathrm{in}\n}^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle S_{-}^{T}+C_{-}^{\\#}\\langle \\phi\n\\Psi _{\\mathrm{in}}|a^{\\#}(t)a^{T}(t)|\\phi \\Psi _{\\mathrm{in}}\\rangle\nC_{-}^{T}dt  \\nonumber \\\\\n&=&S_{-}^{\\#}\\langle \\Psi _{\\mathrm{in}}|d\\Lambda _{\\mathrm{in}}(t)|\\Psi _{\n\\mathrm{in}}\\rangle\nS_{-}^{T}+S_{-}^{\\#}f(t)^{T}C_{-}^{T}dt+C_{-}^{\\#}f(t)^{\n\\#}S_{-}^{T}dt-C_{-}^{\\#}C_{-}^{T}dt+C_{-}^{\\#}\\Sigma (t)^{T}C_{-}^{T}dt.\n\\label{eq:temp2bb}\n\\end{eqnarray}\nBy Eq. (\\ref{eq:mar27_2}),\n\\begin{equation}\n\\langle \\phi \\Psi _{\\mathrm{in}}|d\\Lambda _{\\mathrm{in}}(t)|\\phi \\Psi _{\n\\mathrm{in}}\\rangle =\\Lambda \\left( t, t\\right) dt.  \\label{eq:temp1bb}\n\\end{equation}\nSubstituting Eq. (\\ref{eq:temp1bb}) into Eq. (\\ref{eq:temp2bb}) and dividing\nboth sides of the resulting equation by $dt$ yield Eq. (\\ref{eq:intensity}). $\\blacksquare $\n\n\n\n\n\\subsection{The passive case: state transfer}\\label{sec:passive_mm}\n\nIn this subsection, we derive an analytical form of the output field state of the \n\\textit{passive} quantum linear system (\\ref\n{sys_passive}) driven by the $m$-photon input state $|\\Psi\n_{\\mathrm{in}}\\rangle $ defined in Eq. (\\ref{eq:aug23_10}).\n\nThe following is the main result of this subsection.\n\n\\begin{theorem} \\label{thm:passive_mm}\nIf the asymptotically stable \\textit{passive} quantum linear system (\\ref\n{sys_passive}) is initialized in the vacuum state and is driven by the $m$-channel $m$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle $\ndefined in Eq. (\\ref{eq:aug23_10}), then the steady-state output field state is an $m$-channel $m$-photon state of the form \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\psi _{\\mathrm{out}}\\circledcirc ^{m}b_{\n\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle , \n \\label{dec12_thm2}\n\\end{equation}\nwhere the operation $\\circledcirc ^{m}$ has been defined in Eq. (\\ref{dec3_circ^2}), and the output pulse $\\psi _{\\mathrm{out}}$ is given by the $m$-fold\nconvolution\n\\begin{equation}\n\\psi _{\\mathrm{out},j_{1}\\ldots j_{m}}(r_{1},\\ldots ,r_{m})=\\int_{-\\infty\n}^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }dt_{1}\\cdots\ndt_{m}\\;g_{G^{-}}^{j_{1}1}(r_{1}-t_{1})\\cdots\ng_{G^{-}}^{j_{m}m}(r_{m}-t_{m})\\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m}) \n\\label{dec23_2}\n\\end{equation}\n($j_{1}, \\ldots, j_{m}=1,\\ldots, m$) with the impulse response function $g_{G^{-}}(t)$ given in Eq. (\\ref{july17_1}). If\nwe update the $m$-variable function $\\psi _{\\mathrm{in}}$ in Eq. (\\ref{eq:aug23_10}) to a tensor $\\psi _{\\mathrm{in}}^{\\uparrow }$ with entries\n\\begin{equation}\n\\psi _{\\mathrm{in,}j_{1}\\ldots j_{m}}^{\\uparrow }(r_{1},\\ldots\n,r_{m})\\triangleq \\left\\{ \n\\begin{array}{ll}\n\\psi _{\\mathrm{in}}(r_{1},\\ldots ,r_{m}), & \\mathrm{if\\ }j_{1}=1,j_{2}=2,\n\\ldots ,j_{m}=m, \\\\ \n0, & \\mathrm{otherwise},\n\\end{array}\n\\right.  \\label{july31_5}\n\\end{equation}%\nas has been done in Eq. (\\ref{dec_up1}), then the output pulse $\\psi _{\\mathrm{\nout}}$ can be written in a compact form \n\\begin{equation}\n\\psi _{\\mathrm{out}}=\\psi _{\\mathrm{in}}^{\\uparrow }\\circledast\n_{t}^{m}g_{G^{-}},\n \\label{eq:aug26_10}\n\\end{equation}\nwhere the operation $\\circledast_{t}^{m}$ has been defined in Eq. (\\ref{eq:product_m_b}).\n\\end{theorem}\n\n\\textbf{Proof.}~ To prove this result, we use both the Schr\\\"{o}dinger picture and  Heisenberg picture.  We first work in  the Heisenberg picture. By Eqs. (\\ref{sys_passive}) and (\\ref{eq:a_t}),\n\\begin{equation*}\nb_{\\mathrm{out}}(t)\n=\nCe^{A(t-t_{0})}a-\\int_{t_{0}}^{t}Ce^{A(t-r)}C^{\\dagger\n}S_{-}b_{\\mathrm{in}}(r)dr+S_{-}b_{\\mathrm{in}}(t)\n\\end{equation*}\nIn terms of Eq. (\\ref{july17_1}), the above equation can be re-written as\n\\begin{equation*}\nb_{\\mathrm{out}}(t)=Ce^{A(t-t_{0})}a+\\int_{t_{0}}^{t}g_{G^{-}}(t-r)b_{\n\\mathrm{in}}(r)dr,\\ \\ \\ t\\geq t_{0},\n\\end{equation*}\nwhose adjoint operator $b_{\\mathrm{out}}^{\\#}(t)$ satisfies \n\\begin{equation}\nb_{\\mathrm{out}}^{\\#}(t)=C^{\\#}e^{A^{\\#}(t-t_{0})}a^{\\#}+\n\\int_{t_{0}}^{t}g_{G^{-}}(t-r)^{\\#}b_{\\mathrm{in}}^{\\#}(r)dr,\\ \\ \\ t\\geq\nt_{0}.  \\label{dec11_5}\n\\end{equation}\nOn the other hand, notice that in the Heisenberg picture, Eq. (\\ref{eq:b_out_dec22}) gives \n\\begin{equation}\nb_{\\mathrm{out}}^{\\#}(t)=U(t, t_{0})^{\\ast }b_{\\mathrm{in}}^{\n\\#}(t)U(t, t_{0}),\\ \\ \\ t\\geq t_{0}  \\label{dec11_6}\n\\end{equation}\n(component-wise for the components of $b^\\#_{\\rm in}(t)$).\nEqs. (\\ref{dec11_5})-(\\ref{dec11_6}) yield \n\\begin{eqnarray}\nb_{\\mathrm{in}}^{\\#}(t)\n&=&C^{\\#}e^{A^{\\#}(t-t_{0})}U(t, t_{0})a^{\\#}U(t, t_{0})^{\\ast\n}+\\int_{t_{0}}^{t}g_{G^{-}}(t-r)^{\\#}U(t, t_{0})b_{\\mathrm{in}\n}^{\\#}(r)U(t, t_{0})^{\\ast }dr  \n\\nonumber \n\\\\\n&=&C^{\\#}e^{A^{\\#}(t-t_{0})}U(t, t_{0})a^{\\#}U(t, t_{0})^{\\ast\n}+\\int_{t_{0}}^{t}g_{G^{-}}(t-r)^{\\#}U(r, t_{0})b_{\\mathrm{in}\n}^{\\#}(r)U(r, t_{0})^{\\ast }dr,  \n\\label{dec11_7}\n\\end{eqnarray}\nwhere the fact \n\\begin{equation*}\nU(t, t_{0})b_{\\mathrm{in}}^{\\#}(r)U(t, t_{0})^{\\ast }=U(r, t_{0})b_{\\mathrm{in}\n}^{\\#}(r)U(r, t_{0})^{\\ast },\\ \\ \\ t\\geq r\\geq t_{0}\n\\end{equation*}\nhas been used to get the last term on the right-hand side of Eq. (\\ref\n{dec11_7}). Since the system is asymptotically stable, sending $\nt_{0}\\rightarrow -\\infty $, Eq. (\\ref{dec11_7}) becomes \n\\begin{equation}\nb_{\\mathrm{in}}^{\\#}(t)=\\int_{-\\infty }^{t}g_{G^{-}}(t-r)^{\\#}U(r,-\\infty\n)b_{\\mathrm{in}}^{\\#}(r)U(r,-\\infty )^{\\ast }dr=\\int_{-\\infty }^{\\infty\n}g_{G^{-}}(t-r)^{\\#}U(r,-\\infty )b_{\\mathrm{in}}^{\\#}(r)U(r,-\\infty )^{\\ast\n}dr.  \\label{dec11_8}\n\\end{equation}\nThis, together with Eq. (\\ref{dec1_g_inv}), yields \n\\begin{equation}\nU(t,-\\infty )b_{\\mathrm{in}}^{\\#}(t)U(t,-\\infty )^{\\ast }\n=\n\\int_{-\\infty}^{\\infty }g_{G^{-}}(r-t)^{T}b_{\\mathrm{in}}^{\\#}(r)dr.  \n\\label{dec11_9}\n\\end{equation}\n\n\nNext, we switch to the Schr\\\"{o}dinger picture.  In the Schr\\\"{o}dinger picture, the joint system-field state at time $t\\geq t_0$ is $U(t, t_0)|\\phi \\Psi_{\\rm in}\\rangle$. Thus,  the steady-state output field state can be\nobtained by tracing out the system. That is, \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\langle \\phi |\\lim_{t_0\\to -\\infty, t\\to \\infty}U(t, t_{0})|\\phi \\Psi\n_{\\mathrm{in}}\\rangle .  \\label{dec13_1}\n\\end{equation}\nWe have\n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \\label{Dec11_10} \n\\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\left\\langle \\phi\n\\right\\vert U(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\int_{-\\infty\n}^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }dt_{1}\\cdots dt_{m\\ }\\psi _{\n\\mathrm{in}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t, t_{0})b_{\n\\mathrm{in,1}}^{\\ast }(t_{1})\\cdots b_{\\mathrm{in},m}^{\\ast }(t_{m})|\\phi\n0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n}\\int_{t_{0}}^{t}\\cdots \\int_{t_{0}}^{t}dt_{1}\\cdots dt_{m\\ }\\psi _{\\mathrm{\nin}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t, t_{0})b_{\\mathrm{\nin,1}}^{\\ast }(t_{1})\\cdots b_{\\mathrm{in},m}^{\\ast }(t_{m})|\\phi 0^{\\otimes\nm}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n}\\int_{t_{0}}^{t}\\cdots \\int_{t_{0}}^{t}dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{\nin}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t_{1}, t_{0})b_{\n\\mathrm{in,1}}^{\\ast }(t_{1})U(t_{1},t_{0})^{\\ast }  \n\\nonumber \\\\\n&&\n\\ \\ \\cdots U(t_{m},t_{0})b_{\\mathrm{in},m}^{\\ast }(t_{m})U(t_{m},t_{0})^{\\ast\n}U(t_{m},t_{0})|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n}\\int_{t_{0}}^{t}\\cdots \\int_{t_{0}}^{t}dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{\nin}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t_{1}, t_{0})b_{\n\\mathrm{in,1}}^{\\ast }(t_{1})U(t_{1},t_{0})^{\\ast }  \n\\nonumber \\\\\n&&\n\\ \\ \\cdots U(t_{m},t_{0})b_{\\mathrm{in},m}^{\\ast }(t_{m})U(t_{m}, t_{0})^{\\ast\n}|\\phi 0^{\\otimes m}\\rangle  \n\\label{dec11_11} \\\\\n&=&\n\\lim_{t\\rightarrow \\infty }\\int_{-\\infty }^{t}\\cdots \\int_{-\\infty\n}^{t}dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{in}}(t_{1},\\ldots\n,t_{m})\\left\\langle \\phi \\right\\vert U(t_{1},-\\infty )b_{\\mathrm{in,1}\n}^{\\ast }(t_{1})U(t_{1}, -\\infty )^{\\ast }  \n\\nonumber \\\\\n&&\n\\ \\ \\cdots U(t_{m}, -\\infty )b_{\\mathrm{in},m}^{\\ast }(t_{m})U(t_{m},-\\infty\n)^{\\ast }|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\int_{-\\infty }^{\\infty }\\cdots \\int_{-\\infty }^{\\infty }dt_{1}\\cdots\ndt_{m}\\ \\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi\n\\right\\vert \\int_{-\\infty }^{\\infty\n}\\sum_{j_{1}=1}^{m}g_{G^{-}}^{j_{1}1}(r_{1}-t_{1})b_{\\mathrm{in}\n,j_{1}}^{\\ast }(r_{1})dr_{1} \n \\nonumber \\\\\n&&\n\\ \\ \\cdots \\int_{-\\infty }^{\\infty\n}\\sum_{j_{m}=1}^{m}g_{G^{-}}^{j_{m}m}(r_{m}-t_{m})b_{\\mathrm{in}\n,j_{m}}^{\\ast }(r_{m})dr_{m}|\\phi 0^{\\otimes m}\\rangle  \n\\label{dec27_1} \\\\\n&=&\n\\sum_{j_{1}=1}^{m}\\cdots \\sum_{j_{m}=1}^{m}\\int d\\overrightarrow{r}\\;b_{\n\\mathrm{in},j_{1}}^{\\ast }(r_{1})\\cdots b_{\\mathrm{in},j_{m}}^{\\ast }(r_{m}) \\ \n\\boxed{\\int d\\overrightarrow{t}\\;g_{G^{-}}^{j_{1}1}(r_{1}-t_{1})\\cdots\ng_{G^{-}}^{j_{m}m}(r_{m}-t_{m})\\psi _{\\mathrm{in}}(t_{1},\\ldots\n,t_{m})} \\ |0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\sum_{j_{1},\\ldots ,j_{m}=1}^{m}\\int d\\overrightarrow{r}\\;b_{\\mathrm{in}\n,j_{1}}^{\\ast }(r_{1})\\cdots b_{\\mathrm{in},j_{m}}^{\\ast }(r_{m})\\ \\boxed{\n\\psi _{\\mathrm{out},j_{1}\\ldots j_{m}}(r_{1},\\ldots ,r_{m})}\\ |0^{\\otimes\nm}\\rangle  \n\\label{dec11_14}\\\\\n&=&\n\\psi _{\\mathrm{out}}\\circledcirc b_{\\mathrm{in}}^{\\#}\\ |0^{\\otimes\nm}\\rangle , \n\\nonumber \n\\end{eqnarray}\nwhich is exactly Eq. (\\ref{dec12_thm2}).  \nNotice that the following fact,\n\\begin{equation}\nU(t, t_{0})|\\phi 0^{\\otimes m}\\rangle =|\\phi 0^{\\otimes m}\\rangle ,\\ \\ \\\nt\\geq t_{0}  \\label{dec11_12}\n\\end{equation}\nupon a global phase, has been used to get Eq. (\\ref{dec11_11}) from the previous step.  In fact, Eq. (\\ref{dec11_12}) holds for general passive\nsystems, see, e.g., \\cite[Lemma 3]{PZJ15}. Eq. (\\ref{dec11_9})  has been used to derive Eq. (\\ref{dec27_1}). Finally, by the two terms highlighted in the two boxes above, it is clear that \n\\[\n\\psi _{\\mathrm{out},j_{1}\\ldots j_{m}}(r_{1},\\ldots ,r_{m}) = \\int d\\overrightarrow{t}\\;g_{G^{-}}^{j_{1}1}(r_{1}-t_{1})\\cdots\ng_{G^{-}}^{j_{m}m}(r_{m}-t_{m})\\psi _{\\mathrm{in}}(t_{1},\\ldots\n,t_{m}),\n\\]\nwhich is exactly Eq. (\\ref{dec23_2}).  $\\blacksquare $\n\n\n\\begin{remark}\\label{rem:SH}\nIn quantum mechanics, the Schr\\\"{o}dinger picture describes how quantum states evolve; on the other hand, the Heisenberg picture describes how operators evolve. Eq. (\\ref{dec12_thm2}) tells us how the input state $|\\Psi_{\\rm in}\\rangle$ evolves and becomes the output state $|\\Psi_{\\rm out}\\rangle$. That is, it is in the Schr\\\"{o}dinger picture. In the Schr\\\"{o}dinger picture, operators do not evolve. This is the reason why the input operator $b_{\\rm in}^\\#$ appears in Eq. (\\ref{dec12_thm2}).\n\\end{remark}\n\n\n\\begin{remark}\nWhen the input pulse is of a product form \n\\begin{equation}\n\\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m})=\\xi _{1}(t_{1})\\cdots \\xi\n_{m}(t_{m}),  \\label{july30_1}\n\\end{equation}\nthe input state $|\\Psi _{\\mathrm{in}}\\rangle $ in Eq. (\\ref{eq:aug23_10})\nbecomes a separable state \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\tprod\\limits_{k=1}^{m}\\mathbf{B}_{\\mathrm{in}\n,k}^{\\ast }(\\xi _{k})|0_{k}\\rangle ,  \\label{july30_2}\n\\end{equation}\nwhere the notation \n\\begin{equation}\n\\mathbf{B}_{\\mathrm{in},k}^{\\ast }(\\xi ) \\equiv \\int_{-\\infty }^{\\infty\n}\\xi (t)b_{\\mathrm{in},k}^{\\ast }(t)dt , ~~k=1,\\ldots ,m \n\\label{july31_B}\n\\end{equation}\nhas been used. In this case, by Eq. (\\ref{dec23_2})\nwe have\n\\begin{equation}\n\\psi _{\\mathrm{out},j_{1}\\ldots j_{m}}(r_{1},\\ldots\n,r_{m})=\\tprod\\limits_{k=1}^{m}\\int_{-\\infty }^{\\infty }\ng_{G^{-}}^{j_{k}k}(r_{k}-t_{k})\\xi _{k}(t_{k}) dt_{k}, ~~ j_{1},\\ldots\n,j_{m}=1,\\ldots ,m.  \\label{july30_3}\n\\end{equation}\nDefine \n\\begin{equation}\n\\xi _{\\mathrm{out},jk}(r)\\triangleq \\int_{-\\infty }^{\\infty\n}g_{G^{-}}^{jk}(r-t)\\xi _{k}(t)dt,\\ \\ j,k=1,\\ldots ,m.  \\label{july30_4}\n\\end{equation}\nThen, by Theorem \\ref{thm:passive_mm} and Eq. (\\ref{july30_3}),\n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\tprod\\limits_{k=1}^{m}\\mathbf{B}_{\\mathrm{in},j_{k}}^{\\ast\n}(\\xi _{\\mathrm{out},j_k k})|0^{\\otimes m}\\rangle\n=\\tprod\\limits_{k=1}^{m}\\sum_{j=1}^{m}\\mathbf{B}_{\\mathrm{in},j}^{\\ast\n}(\\xi _{\\mathrm{out},jk})|0^{\\otimes m}\\rangle . \n \\label{july30_5}\n\\end{equation}\n Interestingly, $|\\Psi _{\\mathrm{out}}\\rangle $ in Eq. (\\ref\n{july30_5}) can also be derived by means of \\cite[Theorem 5]{zhang13}.\nTherefore, Theorem \\ref{thm:passive_mm} generalizes one of the main results\nin \\cite{zhang13}.\n\\end{remark}\n\n\\begin{figure}[ptb]\n\\centering\n\\includegraphics[width=2.5in]{BS}\n\\caption{Schematic representation of a beamsplitter}\n\\label{fig:BS}\n\\end{figure}\n\n\\begin{example}\n\\label{ex:BS0} (beamsplitter.) A beamsplitter is a static device\nwidely used in optical laboratories, \\cite{Leo03}, \\cite{BR04}, \\cite{NJD09}%\n, see Fig.~\\ref{fig:BS}. In the $(S_{-},L,H)$ formalism, a\nbeamsplitter may be modeled by $L=0$, $H=0$, and \n\\begin{equation}\nS_{-}=\\left[ \n\\begin{array}{cc}\nR & T \\\\ \nT & R\n\\end{array}\n\\right] ,~~R,T\\in \\mathbb{C},~|R|^{2}+|T|^{2}=1.  \n\\label{july31_bs}\n\\end{equation}\nLet the 2-channel 2-photon input state be \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\int_{-\\infty }^{\\infty }\\int_{-\\infty\n}^{\\infty }dt_{1}dt_{2}\\;\\psi _{\\mathrm{in}}(t_{1},t_{2})b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})|0_{1}0_{2}\\rangle .\n\\label{eq:aug16_2}\n\\end{equation}\nBy Theorem \\ref{thm:passive_mm}, the steady-state output field state is\n\\begin{eqnarray}\n&&|\\Psi _{\\mathrm{out}}\\rangle  \\nonumber \\\\\n&=&RT\\int dr_{1}dr_{2}\\ b_{\\mathrm{in},1}^{\\ast }(r_{1})b_{\\mathrm{in},1}^{\\ast }(r_{2})\\psi _{\\mathrm{in}}(r_{1},r_{2})|0_{1}\\rangle \\otimes\n|0_{2}\\rangle +R^{2}\\int dr_{1}dr_{2}\\ b_{\\mathrm{in},1}^{\\ast }(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})\\psi _{\\mathrm{in}}(r_{1},r_{2})|0_{1}\\rangle\n\\otimes |0_{2}\\rangle  \\nonumber \\\\\n&&+T^{2}\\int dr_{1}dr_{2}\\ b_{\\mathrm{in},1}^{\\ast }(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})\\psi _{\\mathrm{in}}(r_{2},r_{1})|0_{1}\\rangle \\otimes\n|0_{2}\\rangle +|0_{1}\\rangle \\otimes RT\\int dr_{1}dr_{2}\\ b_{\\mathrm{in},2}^{\\ast }(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})\\psi _{\\mathrm{in}}(r_{1},r_{2})|0_{2}\\rangle , \n\\nonumber\n\\end{eqnarray}\nwhich is exactly \\cite[Eq. (6.8.7)]{RL00}.\n\\end{example}\n\n\n\\begin{example}\n\\label{ex:entangled} (optical cavity.) An optical cavity is a system\ncomposed of reflecting and\/or transmitting mirrors \\cite[Chapter 5.3]{BR04}, \n\\cite[Chapter 7]{WM08}, \\cite{GJ09}, \\cite{NJD09}. A widely used type of\noptical cavities is the so-called Fabry-Perot cavity. In the $(S_{-},L,H)$\nformalism, a single-mode Fabry-Perot cavity with two input channels, as\nshown in Fig. \\ref{fig:cavity}, can be modeled with parameters \n\\begin{equation}\n\\left( S_- =I_2, ~ L=\\left[ \n\\begin{array}{c}\n\\sqrt{\\kappa _{1}}a \\\\ \n\\sqrt{\\kappa _{2}}a\n\\end{array}\n\\right] , ~H=\\omega _{d}a^{\\ast }a\\right) .  \n\\label{eq:aug23_cavity}\n\\end{equation}\nHere, $\\kappa _{1}$ and $\\kappa _{2}$ are coupling strengths between the\ncavity and the external fields, and $\\omega _{d}$ is the detuning frequency between\nthe resonant frequency of the cavity and the external fields. (Here we\nassume that the two input light fields have the same carrier frequency.) \n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=2.5in]{cavity}\n\\caption{Schematic representation of a single-mode Fabry-Perot cavity with\ntwo inputs}\n\\label{fig:cavity}\n\\end{figure}\nBy Eq. (\\ref{sys_passive}) we have the following QSDEs \n\\begin{eqnarray}\n\\dot{a}(t) &=&-(\\frac{\\kappa _{1}+\\kappa _{2}}{2}+\\mathrm{i}\\omega _{d})a(t)-\\sqrt{\\kappa _{1}}b_{\\mathrm{in},1}(t)-\\sqrt{\\kappa _{2}}b_{\\mathrm{in},2}(t),  \\nonumber \\\\\nb_{\\mathrm{out},1}(t) &=&\\sqrt{\\kappa _{1}}a(t)+b_{\\mathrm{in},1}(t),  \\nonumber\n\\\\\nb_{\\mathrm{out},2}(t) &=&\\sqrt{\\kappa _{2}}a(t)+b_{\\mathrm{in},2}(t).\n\\label{july31_cavity}\n\\end{eqnarray}\nLet the input state be that given in Eq. (\\ref{eq:aug16_2}). In what\nfollows, we calculate the steady-state output field state. Define the\nfollowing two-variable functions \n\\begin{equation}\n\\Phi _{1}(r,t_{2})\\triangleq \\int_{-\\infty }^{r}dt_{1}\\ e^{-\\left( \\mathrm{i}\\omega _{d}+\\frac{\\kappa _{1}+\\kappa _{2}}{2}\\right) (r-t_{1})}\\psi _{\\mathrm{in}}(t_{1},t_{2}),  \\label{eq:aug13_8}\n\\end{equation}\n\\begin{equation}\n\\Phi _{2}(t_{1},r)\\triangleq \\int_{-\\infty }^{r}dt_{2}\\ e^{-\\left( \\mathrm{i}\\omega _{d}+\\frac{\\kappa _{1}+\\kappa _{2}}{2}\\right) (r-t_{2})}\\psi _{\\mathrm{in}}(t_{1},t_{2}),  \\label{eq:aug13_7}\n\\end{equation}\nand\n\\begin{equation}\n\\Phi (r,\\tau )\\triangleq \\int_{-\\infty }^{r}\\int_{-\\infty }^{\\tau\n}dt_{1}dt_{2}\\ e^{-\\left( \\mathrm{i}\\omega _{d}+\\frac{\\kappa _{1}+\\kappa _{2}}{2}\\right) (r+\\tau -t_{1}-t_{2})}\\psi _{\\mathrm{in}}(t_{1},t_{2}).\n\\label{eq:aug13_10}\n\\end{equation}\nBy Theorem \\ref{thm:passive_mm}, the steady-state output field state is\n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \n\\nonumber \\\\\n&=&\n\\sqrt{\\kappa _{1}\\kappa _{2}}\\int_{-\\infty }^{\\infty }\\int_{-\\infty\n}^{\\infty }dr_{1}dr_{2}\\;b_{\\mathrm{in},1}^{\\ast }(r_{1})b_{\\mathrm{in},1}^{\\ast }(r_{2})[\\kappa _{1}\\Phi (r_{1},r_{2})-\\Phi\n_{2}(r_{1},r_{2})]|0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n+\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty }dr_{1}dr_{2}\\;b_{\\mathrm{in},1}^{\\ast }(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})\\left[ \\psi _{\\mathrm{in}}(r_{1},r_{2})-\\kappa _{1}\\Phi _{1}(r_{1},r_{2})-\\kappa _{2}\\Phi\n_{2}(r_{1},r_{2})\\right.  \n\\nonumber \\\\\n&&\\ \\ \\ \\left. +\\kappa _{1}\\kappa _{2}(\\Phi (r_{1},r_{2})+\\Phi\n(r_{2},r_{1})) \\right] |0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&+|0_{1}\\rangle \\otimes \\sqrt{\\kappa _{1}\\kappa _{2}}\\int_{-\\infty\n}^{\\infty }\\int_{-\\infty }^{\\infty }dr_{1}dr_{2}\\;b_{\\mathrm{in},2}^{\\ast\n}(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})\\left[ \\kappa _{2}\\Phi\n(r_{1},r_{2})-\\Phi _{1}(r_{1},r_{2})\\right] |0_{2}\\rangle .\n\\label{eq:aug23_11}\n\\end{eqnarray}\nIn what follows we discuss two cases.\n\nCase 1) In the limit $\\kappa _{1}\\rightarrow 0$, the state in Eq. (\\ref{eq:aug23_11}) becomes \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\int_{-\\infty\n}^{\\infty }\\int_{-\\infty }^{\\infty } dr_{1}dr_{2}\\ b_{\\mathrm{in},1}^{\\ast\n}(r_{1})b_{\\mathrm{in},2}^{\\ast }(r_{2})[\\psi _{\\mathrm{in}}(r_{1},r_{2})-\\kappa _{2}\\Phi _{2}(r_{1},r_{2})]|0_{1}\\rangle \\otimes\n|0_{2}\\rangle .  \n\\label{july31_2}\n\\end{equation}\nIf the input field state is an entangled state, the output field state in Eq. (\\ref{july31_2}) is also an entangled state. Therefore, even though the system does not affect the first channel directly because of $\\kappa_1=0$, it does influence the first channel via its influence on the second channel.\n\nCase 2) The input state is a product state. Assume\n\\begin{equation}\n\\psi _{\\mathrm{in}}(t_{1},t_{2})=\\xi _{1}(t_{1})\\xi _{2}(t_{2}),\n\\label{eq:aug11_5}\n\\end{equation}\nthat is, the input is a tensor product state of two single-photon states,\none for each channel. In this case, there exists no entanglement between the two input channels. For this product state, Eqs. (\\ref{eq:aug13_8})-(\\ref{eq:aug13_10})\nreduce to \n\\begin{equation}\n\\Phi _{1}(r_{1},r_{2})=\\xi _{2}(r_{2})\\eta _{1}(r_{1}),  \\label{aug1_9}\n\\end{equation}\n\\begin{equation}\n\\Phi _{2}(r_{1},r_{2})=\\xi _{1}(r_{1})\\eta _{2}(r_{2}),  \\label{aug1_10}\n\\end{equation}\nand\n\\begin{equation}\n\\Phi (r_{1},r_{2})=\\eta _{1}(r_{1})\\eta _{2}(r_{2}),  \\label{aug1_11}\n\\end{equation}\nwhere\n\\begin{equation}\n\\eta _{i}(t)\\triangleq \\int_{-\\infty }^{t}e^{-\\left( \\mathrm{i}\\omega _{d}+\n\\frac{\\kappa _{1}+\\kappa _{2}}{2}\\right) (t-r)}\\xi _{i}(r)dr,\\ i=1,2.\n\\label{aug1_12}\n\\end{equation}\nAs a result, Eq. (\\ref{eq:aug23_11}) becomes \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle \n=\n\\left( \\mathbf{B}_{\\mathrm{in,}1}^{\\ast\n}\\left( \\xi _{1} - \\kappa_1 \\eta_1\\right)  -\\sqrt{\\kappa _{1}\\kappa _{2}}\\mathbf{B}_{\\mathrm{in},2}^{\\ast }\\left( \\eta _{2}\\right) \\right) \\left( \\mathbf{B}_{\\mathrm{in},2}^{\\ast }\\left( \\xi _{2} - \\kappa_2 \\eta_2\\right) -\\sqrt{\\kappa _{1}\\kappa _{2}}\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }\\left( \\eta _{1}\\right)  \\right) |0_{1}\\rangle \\otimes\n|0_{2}\\rangle   \\label{eq:aug13_5}\n\\end{equation}\n The state in Eq.\n(\\ref{eq:aug13_5}) is an entangled state. Therefore, the system entangled the initially separable input state. Sending $\\kappa _{1}\\rightarrow 0$\nin Eq. (\\ref{eq:aug13_5}) yields \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }\\left( \\xi\n_{1}\\right) |0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in,}2}^{\\ast }\\left(\n\\xi _{2}-\\kappa _{2}\\eta _{2}\\right) |0_{2}\\rangle ,  \n\\label{july31_1}\n\\end{equation}\nwhich is a product state. That is, if the coupling between the system and\nthe first channel is extremely weak, then the output fields are almost in a\nproduct state. This is reasonable: when the coupling strength $\\kappa\n_{1}= 0$, the first channel has no interaction with the system, so\nthe state of the first channel does not change if it is not initially entangled with the second channel. On the other hand, the pulse shape of the second channel has been transformed by the system from $\\xi_2$ to $\\xi_2-\\kappa_2 \\eta$.\n\\end{example}\n\n\n\\subsection{The passive case: the invariant set} \\label{subsec:invariant}\n\nDefine a class of $m$-channel $m$-photon states of the form \n\\begin{eqnarray}\n\\mathcal{F}_{1} &\\triangleq &\\Bigg\\{|\\Psi \\rangle =\\psi \\circledcirc ^{m}b_{\\rm in}^{\\#}|0^{\\otimes m}\\rangle \\ |\\mathrm{the~}m\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\textrm{-}}\\mathrm{way}~m\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\textrm{-}}\\mathrm{dimensional~tensor~function~}\\psi \\mathrm{~normalizes~}\n|\\Psi \\rangle \\Bigg\\}.  \\label{may1_1}\n\\end{eqnarray}%\n\nBy means of Eq. (\\ref{july31_5}), it is clear that the $m$-channel $m$-photon input field state defined in Eq. (\\ref\n{eq:aug23_10}) can be re-written as $|\\Psi_{\\rm in}\\rangle  = \\psi _{\\mathrm{in}}^{\\uparrow\n}  \\circledcirc ^{m}b_{\\rm in}^{\\#}|0^{\\otimes m} \\rangle$.  Therefore,   $|\\Psi_{\\rm in}\\rangle   \\in \\mathcal{F}_{1}$.  On the other hand, by Theorem \\ref\n{thm:passive_mm}, the steady-state output field state $\\left\\vert \\psi _{\\mathrm{out}}\\right\\rangle \\in \\mathcal{F}_{1}$ too. This motivates us to study more\ngeneral pulse shape transfer than that in Theorem \\ref{thm:passive_mm}.\n\nThe following is the main result of this  subsection.\n\n\n\\begin{theorem}\\label{thm:passive_invariance}\n Let the input state for the asymptotically\nstable passive quantum linear system (\\ref{sys_passive}) (initialized in the vacuum state) be an element $|\\Psi _{\\mathrm{in}\n}\\rangle \\in \\mathcal{F}_{1}$ with pulse shape parametrized by an $m$-way $m$-dimensional tensor function $\\psi _{\\mathrm{in}}$. Then, the steady-state\noutput field state\n \\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\psi _{\\mathrm{out}}\\circledcirc ^{m}b_{\n\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle   \\label{eq:pulse_transfer_mm}\n\\end{equation}\n is also an element in $\\mathcal{F}_{1}$, where the pulse shape is given by\n\\begin{equation}\n\\psi _{\\mathrm{out}}=\\psi _{\\mathrm{in}}\\circledast _{t}^{m}g_{G^{-}}.\n\\label{dec23_3}\n\\end{equation}\nAlternatively, in the frequency domain,\n\\begin{equation}\n\\psi _{\\mathrm{out}}=\\psi _{\\mathrm{in}}\\circledast _{\\omega }^{m}g_{G^{-}}.\n\\label{eq:pulse_transfer_mm_omega}\n\\end{equation}\nMoreover, we have\n\\begin{equation}\n\\left\\Vert \\psi _{\\mathrm{out}}(\\mathrm{i}\\omega _{1},\\ldots ,\\mathrm{i}\n\\omega _{m})\\right\\Vert ^{2}=\\left\\Vert \\psi _{\\mathrm{in}}(\\mathrm{i}\\omega\n_{1},\\ldots ,\\mathrm{i}\\omega _{m})\\right\\Vert ^{2}, ~~\\forall \\omega _{1},\\ldots ,\\omega _{m}\\in \\mathbb{R}. \n\\label{eq:norm}\n\\end{equation}\n\\end{theorem}\n\n\\textbf{Proof.~} By analogy with the proof of Theorem \\ref{thm:passive_mm},\nwe have \n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\left\\langle \\phi\n\\right\\vert U(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\int d\\overrightarrow{t}\\ \\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t, t_{0})b_{\\mathrm{in},j_{1}}^{\\ast }(t_{1})\\cdots b_{\\mathrm{in},j_{m}}^{\\ast\n}(t_{m})|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\int_{t_0}^t \\cdots \\int_{t_0}^t dt_1 \\cdots dt_m  \\ \\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert\nU(t_1,t_{0})b_{\\mathrm{in},j_{1}}^{\\ast}(t_1)U(t_1,t_{0})^{\\ast } \n \\nonumber \\\\\n&&\n\\cdots U(t_m, t_{0})b_{\\mathrm{in},j_{m}}^{\\ast}(t_m)U(t_m, t_{0})^{\\ast }|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\lim_{t\\rightarrow \\infty }\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\int_{-\\infty}^t \\cdots \\int_{-\\infty}^t dt_1 \\cdots dt_m \\  \\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t_1,-\\infty\n)b_{\\mathrm{in,}j_{1}}^{\\ast }(t_1)U(t_1,-\\infty )^{\\ast } \n \\nonumber \\\\\n&&\n\\cdots U(t_m,-\\infty )b_{\\mathrm{in},j_{m}}^{\\ast}(t_m)U(t_m,-\\infty )^{\\ast }|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\sum_{j_{1},\\ldots\n,j_{m}=1}^{m}\\int d\\overrightarrow{t}\\ \\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})\\int_{-\\infty }^{\\infty\n}\\sum_{i_{1}=1}^{m}\\int_{-\\infty }^{\\infty\n}g_{G^{-}}^{i_{1}j_{1}}(r_{1}-t_{1})b_{\\mathrm{in},i_{1}}^{\\ast\n}(r_{1})dr_{1}  \n\\nonumber \\\\\n&&\n\\cdots \\int_{-\\infty }^{\\infty\n}\\sum_{i_{m}=1}^{m}g_{G^{-}}^{i_{m}j_{m}}(r_{m}-t_{m})b_{\\mathrm{in}\n,i_{m}}^{\\ast }(r_{m})dr_{m}|0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\n\\sum_{i_{1},\\ldots ,i_{m}=1}^{m}\\int d\\overrightarrow{r}\\ b_{\\mathrm{in}\n,i_{1}}^{\\ast }(r_{1})\\cdots b_{\\mathrm{in},i_{m}}^{\\ast\n}(r_{m}) \n\\nonumber\n\\\\\n&&\\times \\  \\boxed{\\sum_{j_{1},\\ldots ,j_{m}=1}^{m}\\int d\\overrightarrow{t}\\\ng_{G^{-}}^{i_{1}j_{1}}(r_{1}-t_{1})\\cdots\ng_{G^{-}}^{i_{m}j_{m}}(r_{m}-t_{m})\\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m})} \\left\\vert 0^{\\otimes\nm}\\right\\rangle\n  \\nonumber \\\\\n&=&\n\\sum_{i_{1},\\ldots ,i_{m}=1}^{m}\\int d\\overrightarrow{r}\\ \\boxed{\\psi _{\\mathrm{out},i_{1}\\ldots i_{m}}(r_{1},\\ldots ,r_{m})} \\  b_{\\mathrm{in},i_{1}}^{\\ast\n}(r_{1})\\cdots b_{\\mathrm{in},i_{m}}^{\\ast }(r_{m})\\left\\vert 0^{\\otimes\nm}\\right\\rangle ,  \\label{may1_3}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\psi _{\\mathrm{out},i_{1}\\ldots i_{m}}(r_{1},\\ldots\n,r_{m})=\\sum_{j_{1},\\ldots ,j_{m}=1}^{m}\\int d\\overrightarrow{t}\\\ng_{G^{-}}^{i_{1}j_{1}}(r_{1}-t_{1})\\cdots\ng_{G^{-}}^{i_{m}j_{m}}(r_{m}-t_{m})\\psi _{\\mathrm{in},j_{1}\\ldots\nj_{m}}(t_{1},\\ldots ,t_{m}),  \\label{july18_1}\n\\end{equation}\nas highlighted in the boxes above.  In the compact form, Eq. (\\ref{july18_1}) gives Eq. (\\ref{dec23_3}).\nTherefore, Eq. (\\ref{eq:pulse_transfer_mm}) is established. Applying the $m$\n-dimensional Fourier transform (\\ref{eq:fourier}) to Eq. (\\ref\n{dec23_3}) gives Eq. (\\ref{eq:pulse_transfer_mm_omega}).\nBecause the system is passive, $G[\\mathrm{i}\\omega ]$ is a unitary matrix\nfor all $\\omega \\in \\mathbb{R}$. Consequently, Eq. (\\ref{eq:norm}) follows Eq. (\\ref{eq:pulse_transfer_mm_omega}) and \nLemma \\ref{lem:norm} immediately. $\\blacksquare $\n\n\n\\subsection{The non-passive case}\\label{subsec:nonpassive}\n\nA quantum linear system is said to be \\textit{non-passive} if $C_{+}\\neq 0$\nand\/or $\\Omega _{+}$ $\\neq 0$ in Eq. (\\ref{eq:ABCD}). Non-passive\nelements, such as optical parametric oscillators (OPOs), are key ingredients of\nquantum optical systems, \\cite{Leo03}, \\cite{BR04}, \\cite{NJD09}. In this\nsubsection, we study the output field state of a non-passive quantum linear system driven\nby the $m$-channel $m$-photon input field state $|\\Psi _{\\mathrm{in}}\\rangle $ defined in Eq. (\\ref{eq:aug23_10}). \n\nFirstly, we introduce some notation. Define operators\n\\begin{equation}\nb_{j}^{d}(t)\\triangleq \\left\\{ \n\\begin{array}{cc}\nb_{\\mathrm{in,}j}^{\\ast }(t), & d=-1, \\\\ \nb_{\\mathrm{in,}j}(t), & d=1,\n\\end{array}\n\\right. ,\\ \\ j=1,\\ldots ,m.  \\label{eq:aug15_4}\n\\end{equation}\nThen, define an $\\underbrace{m\\times \\cdots \\times m}_m \\times \\underbrace{2\\times\\cdots \\times 2}_m$ tensor\noperator $\\mathfrak{b}=\\mathfrak{b}_{j_{1}\\ldots j_{m}}^{d_{1} \\ldots\n  d_{m}}(t_{1},\\ldots ,t_{m})$, whose entries are\n\\begin{equation}\n\\mathfrak{b}_{j_{1}\\ldots j_{m}}^{d_{1}\\ldots d_{m}}(t_{1},\\ldots\n,t_{m})\\triangleq b_{j_{1}}^{d_{1}}(t_{1})\\cdots b_{j_{m}}^{d_{m}}(t_{m}),\\\n\\ j_{1},\\ldots ,j_{m}=1,\\ldots ,m,\\ d_{1},\\ldots ,d_{m}=\\pm 1.\n\\label{may1_6}\n\\end{equation}\nDenote\n\\begin{equation}\ng_{G^{d}}^{kj}(t)\\triangleq \\left\\{ \n\\begin{array}{cc}\ng_{G^{-}}^{kj}(t), & d=-1, \\\\ \n-g_{G^{+}}^{kj}(t)^{\\ast }, & d=1,\n\\end{array}\n\\right. ,\\ \\ j,k=1,\\ldots ,m.  \\label{eq:aug15_5}\n\\end{equation}\nThen, define an  $\\underbrace{m\\times \\cdots \\times m}_m \\times \\underbrace{2\\times\\cdots \\times 2}_m$\ntensor function $\\psi $ with entries \n\\begin{equation}\n\\psi _{j_{1}\\ldots j_{m}}^{d_{1}\\ldots d_{m}}(r_{1},\\ldots ,r_{m})\\triangleq\n\\int_{-\\infty }^{\\infty }dt_{1}\\cdots dt_{m}\\\ng_{G^{d_{1}}}^{j_{1}1}(r_{1}-t_{1})\\cdots\ng_{G^{d_{m}}}^{j_{m}m}(r_{m}-t_{m})\\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m})\n\\label{eq:aug15_6}\n\\end{equation}\n($ j_{1},\\ldots ,j_{m}=1,\\ldots ,m,\\ d_{1},\\ldots ,d_{m}=\\pm 1$).  \nFinally, define the following operation between tensors $\\mathfrak{b}$ and $\\psi $ \n\\begin{equation}\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle \\triangleq \\sum_{j_{1}, \\ldots, j_{m}=1}^{m}\\sum_{d_{1}, \\ldots, d_{m}=\\pm 1}\\int dr_{1}\\cdots dr_{m} \\  \\ \n\\psi _{j_{1}\\ldots j_{m}}^{d_{1}\\ldots d_{m}}(r_{1}, \\ldots, r_{m})b_{j_{1}}^{d_{1}}(r_{1}) \\cdots b_{j_{m}}^{d_{m}}(r_{m}).\n\\label{eq:inner}\n\\end{equation}\n\nThe following result shows how a non-passive quantum linear system processes\n$m$-channel $m$-photon input states.\n\n\n\\begin{theorem}\n\\label{thm:nonpassive} Let $G$ be an asymptotically stable non-passive  quantum linear\nsystem which is initialized in the vacuum state $|\\phi\\rangle$ and is driven by the $m$-channel $m$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle $ defined in Eq. (\\ref{eq:aug23_10}). The steady-state output\nfield state is \n\\begin{equation}\n\\rho _{\\mathrm{out}}=\\left\\langle \\mathfrak{b},\\psi \\right\\rangle\n\\left\\langle \\phi \\right\\vert \\rho _{\\mathrm{\\infty }}|\\phi \\rangle\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle ^{\\ast },\n\\label{eq:ss_nonpassive}\n\\end{equation}%\nwhere $\\left\\langle \\mathfrak{b},\\psi \\right\\rangle $ is given in Eq. (%\n\\ref{eq:inner}), and \n\\begin{equation}\n\\rho _{\\mathrm{\\infty }}\n\\triangleq\n\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty} U(t, t_{0})|\\phi 0^{\\otimes m}\\rangle\n\\left\\langle \\phi 0^{\\otimes m}\\right\\vert U(t, t_{0})^{\\ast }\n\\label{eq:rho_jan15}\n\\end{equation}\n is a zero-mean Gaussian\nstate for the joint system whose power spectral density matrix is given by Eq. (\\ref\n{eq:temp_gaussian}) in Lemma \\ref{lem:gaussian}.\n\\end{theorem}\n\n\\textbf{Proof. ~} When the system is initialized in the vacuum state $|\\phi\\rangle$ and is driven by a vacuum field $|0^{\\otimes m}\\rangle$, it is well-known that the steady-state joint system-field state $\\rho_{\\infty}$ in Eq. (\\ref{eq:rho_jan15}) is a zero-mean Gaussian state whose power spectral density matrix is given by Eq. (\\ref\n{eq:temp_gaussian}) in Lemma \\ref{lem:gaussian}, see, e.g., \\cite[Chapter 6]{WM10}.   Let the input state be the $m$-channel $m$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle$ defined in Eq. (\\ref{eq:aug23_10}). In this case, in steady state, the joint system-field state is \n\\begin{equation}\n\\rho _{\\rm ss} \\triangleq \\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}U(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\left\\langle \\phi \\Psi _{\\mathrm{in}}\\right\\vert U(t, t_{0})^{\\ast }.\n\\label{eq:temp_sept16_1}\n\\end{equation}%\nThe steady-state output field state, denoted by $\\rho _{\\mathrm{out}}$, is obtained by taking the partial trace\nof $\\rho _{\\rm ss}$ with respect to the system, that is, \n\\begin{equation}\n\\rho _{\\mathrm{out}} = \\langle\\phi| \\rho _{\\rm ss} |\\phi\\rangle  =\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}\\left\\langle \\phi \\right\\vert\nU(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle \\left\\langle \\phi \\Psi _{\\mathrm{%\nin}}\\right\\vert U(t, t_{0})^{\\ast }|\\phi \\rangle .  \\label{eq:temp7}\n\\end{equation}%\nBecause the system is asymptotically stable, according to Eq. (\\ref{eq:out_tf_3}),%\n\\begin{equation}\n\\breve{b}_{\\mathrm{in}}(t)=g_{G^{-1}}\\circledast \\breve{b}_{\\mathrm{out}%\n}(t)=g_{G^{-1}}\\circledast U(t,-\\infty)^{\\ast }\\breve{b}_{\\mathrm{in}}(t)U(t,-\\infty).\n\\label{july31_7}\n\\end{equation}%\nPre-multiplying Eq. (\\ref{july31_7}) by $U(t,-\\infty)$  and post-multiplying Eq. (\\ref{july31_7}) by $U(t,-\\infty)^{\\ast }$, we get\n\\begin{equation}\nU(t,-\\infty)\\breve{b}_{\\mathrm{in}}(t)U(t,-\\infty)^{\\ast }=g_{G^{-1}}\\circledast \\breve{%\nb}_{\\mathrm{in}}(t).  \\label{july31_6}\n\\end{equation}%\nThis, together with Eq. (\\ref{eq:G_inv}), yields%\n\\begin{eqnarray}\n&&\n\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}U(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle\n\\nonumber\n\\\\\n&=&\n\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}\\int dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{in}%\n}(t_{1},\\ldots ,t_{m})U(t, t_{0})b_{\\mathrm{in,}1}^{\\ast }(t_{1})\\cdots b_{%\n\\mathrm{in,}m}^{\\ast }(t_{m})|\\phi 0^{\\otimes m}\\rangle\n\\nonumber\n \\\\\n&=&\n\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}\\int dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m})U(t, t_{0})b_{\\mathrm{in,}1}^{\\ast\n}(t_{1})U(t, t_{0})^{\\ast }\\cdots U(t, t_{0})b_{\\mathrm{in,}m}^{\\ast\n}(t_{m})U(t, t_{0})^{\\ast }U(t, t_{0})|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \n\\\\\n&=&\n\\lim_{t\\rightarrow \\infty}\\int_{-\\infty}^{t}\\cdots \\int_{-\\infty}^{t}dt_{1}\\cdots dt_{m}\\ \\psi _{\\mathrm{\nin}}(t_{1},\\ldots ,t_{m})\\left\\langle \\phi \\right\\vert U(t_{1},-\\infty)b_{\n\\mathrm{in,1}}^{\\ast }(t_{1})U(t_{1},-\\infty)^{\\ast }  \n\\nonumber \\\\\n&&\n\\ \\ \\cdots U(t_{m},-\\infty)b_{\\mathrm{in},m}^{\\ast }(t_{m})U(t_{m},-\\infty)^{\\ast}\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty}U(t,t_{0})|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber \\\\\n&=&\\int dt_{1}\\cdots dt_{m}\\;\\psi _{\\mathrm{in}}(t_{1},\\ldots\n,t_{m})\\int_{-\\infty }^{\\infty }\\left[ \n\\begin{array}{cc}\n-g_{G^{+}}^{1}(r_{1}-t_{1})^{\\dagger } & g_{G^{-}}^{1}(r_{1}-t)^{T}\n\\end{array}\n\\right] \\breve{b}_{\\mathrm{in}}(r_{1})dr_{1}  \n\\nonumber\n \\\\ \n&&\n\\times \\cdots  \n\\nonumber\n \\\\\n&&\n\\times \\int_{-\\infty }^{\\infty }\\left[ \n\\begin{array}{cc}\n-g_{G^{+}}^{m}(r_{m}-t_{m})^{\\dagger } & g_{G^{-}}^{m}(r_{m}-t_{m})^{T}\n\\end{array}\n\\right] \\breve{b}_{\\mathrm{in}}(r_{m})dr_{m} \\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n} U(t, t_{0})|\\phi 0^{\\otimes\nm}\\rangle  \n\\label{jan9_1} \n\\\\\n&=&\n\\int dt_{1}\\cdots dt_{m}\\;\\psi _{\\mathrm{in}}(t_{1},\\ldots ,t_{m})\\left\\{\n\\sum_{j_{1}=1}^{m}\\int_{-\\infty }^{\\infty\n}-g_{G^{+}}^{j_{1}1}(r_{1}-t_{1})^{\\ast }b_{\\mathrm{in,}j_{1}}(r_{1})dr_{1}+\n\\sum_{j_{1}=1}^{m}\\int_{-\\infty }^{\\infty }g_{G^{-}}^{j_{1}1}(r_{1}-t_{1})b_{\n\\mathrm{in,}j_{1}}^{\\ast }(r_{1})dr_{1}\\right\\}  \\nonumber \\\\\n&&\\times \\cdots  \\nonumber \\\\\n&&\\times \\left\\{ \\sum_{j_{m}=1}^{m}\\int_{-\\infty }^{\\infty\n}-g_{G^{+}}^{j_{m}m}(r_{m}-t_{m})^{\\ast }b_{\\mathrm{in,}j_{m}}(r_{m})dr_{m}+\n\\sum_{j_{m}=1}^{m}\\int_{-\\infty }^{\\infty }g_{G^{-}}^{j_{m}m}(r_{m}-t_{m})b_{\n\\mathrm{in,}j_{m}}^{\\ast }(r_{m})dr_{m}\\right\\}\n\\nonumber\n\\\\\n&& \\times \n \\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n} U(t, t_{0})|\\phi 0^{\\otimes\nm}\\rangle  \n\\nonumber\n \\\\\n&=&\n\\sum_{j_{1},\\cdots ,j_{m}=1}^{m}\\sum_{d_{1},\\cdots ,d_{m}=\\pm 1}\\int\ndr_{1}\\cdots dr_{m}\\quad \\psi _{j_{1}\\cdots j_{m}}^{d_{1}\\cdots\nd_{m}}(r_{1},\\ldots ,r_{m})b_{j_{1}}^{d_{1}}(r_{1})\\cdots\nb_{j_{m}}^{d_{m}}(r_{m})\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n} U(t, t_{0})|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber\n \\\\\n&=&\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle \\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty\n} U(t, t_{0})|\\phi 0^{\\otimes\nm}\\rangle .  \n\\label{july31_8}\n\\end{eqnarray}\nNotice that in Eq. (\\ref{jan9_1}), $g_{G^\\pm}^j$ stands for the $j$th column of $g_{G^\\pm}$ for $j=1,\\ldots,m$. Substituting Eq. (\\ref{july31_8}) into Eq. (\\ref{eq:temp7}) we obtain \n\\begin{eqnarray}\n\\rho _{\\mathrm{out}} \n&=&\n\\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}\\left\\langle \\phi \\right\\vert\nU(t, t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle \\left\\langle \\phi \\Psi _{\\mathrm{in}}\\right\\vert U(t, t_{0})^{\\ast }|\\phi \\rangle  \n\\nonumber\n \\\\\n&=&\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle \\ \\lim_{t_0\\rightarrow-\\infty, \nt\\rightarrow\\infty}\\left\\langle \\phi \\right\\vert U(t, t_{0})|\\phi 0^{\\otimes m}\\rangle\n\\left\\langle \\phi 0^{\\otimes m}\\right\\vert U(t, t_{0})^{\\ast }|\\phi \\rangle\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle ^{\\ast }  \n\\nonumber \n\\\\\n&=&\n\\left\\langle \\mathfrak{b},\\psi \\right\\rangle \\left\\langle \\phi\n\\right\\vert \\rho _{\\mathrm{\\infty }}|\\phi \\rangle \\left\\langle \\mathfrak{b},\\psi \\right\\rangle ^{\\ast },  \n\\label{july31_9}\n\\end{eqnarray}\nwhich is exactly Eq. (\\ref{eq:ss_nonpassive}). $\\blacksquare $\n\n\n\\section{$N$ ($N\\geq m$) photons superposed over $m$ channels}\\label{sec:mn}\n\n\n\nIn this section, we study how a \\textit{passive} quantum linear system\nprocesses $N$ photons that are superposed over $m$ input channels, thus\ngeneralizing the results in Section \\ref{sec:mm}.\n\n\\subsection{State transfer}\\label{sec:mn_aug14}\n\n\n\nLet the input field be in a state where $N$ photons are superposed over $m$\ninput channels. Specifically, the input state considered in this subsection\nis defined to be \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n\\triangleq\n\\int d\\overrightarrow{t}\\;\\psi _{\\mathrm{in}}(t_{1}^{1},\\ldots\n,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots ,t_{k_{m}}^{m}) \\ b_{\\mathrm{in}\n,1}^{\\ast }(t_{1}^{1})\\cdots b_{\\mathrm{in},1}^{\\ast }(t_{k_{1}}^{1})\\cdots\nb_{\\mathrm{in},m}^{\\ast }\\left( t_{1}^{m}\\right) \\cdots b_{\\mathrm{in}\n,m}^{\\ast }\\left( t_{k_{m}}^{m}\\right) |0^{\\otimes m}\\rangle .\n\\label{eq:aug16_5}\n\\end{equation}\nIn Eq. (\\ref{eq:aug16_5}), the positive integers $k_{i}$ satisfy $\n\\sum_{i=1}^{m}k_{i}=N$. It is also implicitly assumed in Eq. (\\ref{eq:aug16_5}) that the $N$-variable function $\\psi_{\\rm in}$ normalizes the state $|\\Psi _{\\mathrm{in}}\\rangle$.\n\n\n\\begin{remark}\\label{rem:dec27_1}\n For the $i$th input channel $(i=1,\\ldots ,m)$, the creation\noperator $b_{\\mathrm{in},i}^{\\ast }$ appears $k_{i}$ times in Eq. (\\ref{eq:aug16_5}), thus there are $k_{i}$ photons in the $i$th input channel.\n\\end{remark}\n\nWith the notation introduced in Eq. (\\ref{dec3_dot}), Eq. (\\ref{eq:aug16_5}) can be re-written as \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\psi_{\\rm in} \\cdot\n_{k_{1}\\cdots k_{m}}^{N}b_{\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle.\n\\label{dec11_4}\n\\end{equation}\nMoreover, inspired by Eq. (\\ref{dec_up2}), update the $N$-variable function $\\psi _{\\mathrm{in}}(t_{1}^{1},\\ldots\n,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})$ to an $N$-way $m$-dimensional tensor $\\psi _{\\mathrm{in}}^{\\uparrow }(t_{1}^{1},\\ldots\n,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})$, whose elements are\ndefined as \n\\begin{eqnarray}\n&&\\psi _{\\mathrm{in,}i_{1}^{1}\\ldots i_{k_{1}}^{1}\\ldots i_{1}^{m}\\ldots\ni_{k_{m}}^{m}}^{\\uparrow }(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots\n,t_{1}^{m},\\ldots ,t_{k_{m}}^{m})  \\nonumber \\\\\n&\\triangleq &\\left\\{ \n\\begin{array}{ll}\n\\psi _{\\mathrm{in}}(t_{1}^{1},\\ldots ,t_{k_{1}}^{1},\\ldots ,t_{1}^{m},\\ldots\n,t_{k_{m}}^{m}), & \\mathrm{if\\ }i_{1}^{1}=1,\\cdots\n,i_{k_{1}}^{1}=k_{1},\\cdots ,i_{1}^{m}=\\sum\\limits_{j=1}^{m-1}k_{j}+1,\\cdots\n,i_{k_{m}}^{m}=N, \\\\ \n0, & \\mathrm{otherwise}.\n\\end{array}\n\\right.   \\label{aug1_14b}\n\\end{eqnarray}\nThen, Eq. (\\ref{dec11_4}) can be re-written as\n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\psi _{\\mathrm{in}}^{\\uparrow } \\odot _{k_{1}\\cdots k_{m}}^{N}b_{\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle ,\n\\label{jan9_psi_out}\n\\end{equation}\nwhere the operation $\\odot _{k_{1}\\cdots k_{m}}^{N}$ has been introduced in Eq. (\\ref{dec3_dot^2}).\n\nThe following result gives an explicit form of the steady-state output field state.\n\n\n\\begin{theorem}\\label{thm:aug23}\n\nIf the asymptotically stable \\textit{passive} quantum linear system (\\ref\n{sys_passive}) is initialized in the vacuum state and is driven by the $m$-channel $N$-photon input state $|\\Psi\n_{\\mathrm{in}}\\rangle $ defined in Eq. (\\ref{eq:aug16_5}), then the steady-state output field state is\n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle \n=\n\\psi _{\\mathrm{out}}\\odot _{k_{1}\\cdots k_{m}}^{N}b_{\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle \n\\label{eq:aug23_2}\n\\end{equation}\nwith the pulse shape  \n\\begin{equation}\n\\psi _{\\mathrm{out}}\n=\n\\psi _{\\mathrm{in}}^{\\uparrow } \\circledast\n_{t, k_{1}\\cdots k_{m}}^{N}g_{G^{-}}.  \n\\label{eq:aug23_1}\n\\end{equation}\nIn Eq. (\\ref{eq:aug23_1}), the tensor $\\psi _{\\mathrm{in}}^{\\uparrow }$ has been given in Eq. (\\ref{aug1_14b}),  and the operation $\\circledast\n_{t, k_{1}\\cdots k_{m}}^{N}$ has been defined in Eq. (\\ref{dec11_2}).\n\\end{theorem}\n\n\n\\textbf{Proof. ~} The proof is similar to that of Theorem \\ref{thm:passive_mm}, so is omitted.\n\n\\begin{example} \\label{ex:BS}\n Consider a beamsplitter\n\\begin{equation}\nS_{-}=\\left[ \n\\begin{array}{cc}\nR & -T \\\\ \nT & R\n\\end{array}\n\\right] ,~~R,T\\in \\mathbb{C},~|R|^{2}+|T|^{2}=1,  \\label{aug1_temp3}\n\\end{equation}\nand a 3-photon input state of the form \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\int_{-\\infty\n}^{\\infty }\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty\n}dt_{1}dt_{2}dt_{3}\\;\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\\;b_{\\mathrm{in,}1}^{\\ast }(t_{1})b_{\\mathrm{in,}2}^{\\ast }(t_{2})b_{\\mathrm{in,}2}^{\\ast\n}(t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle .  \\label{eq:aug21_14}\n\\end{equation}\nIn this case, there are two input channels ($m=2$). The total number of\nphotons is $N=3$. In fact, as explained in Remark \\ref{rem:dec27_1} above, there is one photon in the first channel ($k_{1}=1$) and two photons in the second channel ($k_{2}=2$). Simple\ncalculation yields\n\\begin{equation}\n\\left\\langle \\Psi _{\\mathrm{in}}|\\Psi _{\\mathrm{in}}\\right\\rangle \n=\n\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty\n}\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}dt_{3}\\;\\left( \\psi _{\\mathrm{in}\n}^{\\ast }(t_{1},t_{3},t_{2})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})+\\left\n\\vert \\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\\right\\vert ^{2}\\right) .\n\\label{aug1_14}\n\\end{equation}\nTherefore, the normalization condition requires that\n\\begin{equation}\n\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty\n}\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}dt_{3}\\;\\left( \\psi _{\\mathrm{in}\n}^{\\ast }(t_{1},t_{3},t_{2})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})+\\left\n\\vert \\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\\right\\vert ^{2}\\right)\n=\n1 .\n\\label{july23_2}\n\\end{equation}\nAccording to Theorem \\ref{thm:aug23}, the steady-state output field state is\n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \n\\nonumber \\\\\n&=&\nRT^{2}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},1}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle\n\\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n-T\\int d\\overrightarrow{t}\\;b_{\\mathrm{in}\n,1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{3})\\left[ R^{2}\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})+R^{2}\\psi _{\n\\mathrm{in}}(t_{1},t_{3},t_{2})-T^{2}\\psi _{\\mathrm{in}}(t_{3},t_{2},t_{1})\n\\right] |0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n-R\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{3})\\left[ T^{2}\\psi _{\\mathrm{in}}(t_{2},t_{1},t_{3})+T^{2}\\psi _{\n\\mathrm{in}}(t_{3},t_{2},t_{1})-R^{2}\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\n\\right] |0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n+|0_{1}\\rangle\n\\otimes R^{2}T\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},2}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3}) |0_{2}\\rangle .  \n\\label{july23_1}\n\\end{eqnarray}\nIn particular, if $R=T=\\frac{1}{\\sqrt{2}}$, Eq. (\\ref{july23_1}) reduces to \n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \n\\nonumber \\\\\n&=&\n\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},1}^{\\ast\n}(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes\n|0_{2}\\rangle  \n\\nonumber \\\\\n&&\n-\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{3})\\left[ \\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})+\\psi _{\\mathrm{in}}(t_{1},t_{3},t_{2})-\\psi _{\\mathrm{in}}(t_{3},t_{2},t_{1})\\right]\n|0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n-\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{3})\\left[ \\psi _{\\mathrm{in}}(t_{2},t_{1},t_{3})+\\psi _{\\mathrm{in}}(t_{3},t_{2},t_{1})-\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\\right]\n|0_{1}\\rangle \\otimes |0_{2}\\rangle  \n\\nonumber \\\\\n&&\n+|0_{1}\\rangle \\otimes \\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},2}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast\n}(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\n|0_{2}\\rangle .  \n\\label{july18_2}\n\\end{eqnarray}\nFurthermore, if $\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})$ is \npermutation-invariant, and \n\\begin{equation}\n\\left\\Vert \\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})\n\\right\\Vert \n=\n \\sqrt{\\int_{-\\infty }^{\\infty }\\int_{-\\infty }^{\\infty\n}\\int_{-\\infty }^{\\infty }dt_{1}dt_{2}dt_{3}\\  |\\psi_{\\rm in} (t_1,t_2,t_3)|^2}=\\frac{1}{\\sqrt{2}},\n\\end{equation}\n then by Eq. (\\ref{july23_2}), the state $|\\Psi_{\\rm in}\\rangle$ is normalized. Furthermore,  Eq. (\\ref{july18_2}) becomes\n\\begin{eqnarray}\n&&|\\Psi _{\\mathrm{out}}\\rangle  \\nonumber \\\\\n&=&\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\n\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},1}^{\\ast }(t_{3})\\psi _{\n\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle  \\nonumber \\\\\n&&-\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\n\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\n\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle  \\nonumber \\\\\n&&-\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\n\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\n\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle  \\nonumber \\\\\n&&+|0_{1}\\rangle \\otimes\\frac{1}{2\\sqrt{2}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},2}^{\\ast }(t_{1})b_{\n\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\n\\mathrm{in}}(t_{1},t_{2},t_{3}) |0_{2}\\rangle .\n\\label{july18_3}\n\\end{eqnarray}\nDefine states \n\\begin{eqnarray}\n\\left\\vert \\Pi _{30}\\right\\rangle\n&\\triangleq &\n\\frac{1}{\\sqrt{3}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},1}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle ,  \n\\nonumber \\\\\n\\left\\vert \\Pi _{21}\\right\\rangle \n&\\triangleq &\n\\int d\n\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},1}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle ,  \n\\nonumber \\\\\n\\left\\vert \\Pi _{12}\\right\\rangle \n&\\triangleq &\n\\int d\n\\overrightarrow{t}\\;b_{\\mathrm{in},1}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{1}\\rangle \\otimes |0_{2}\\rangle ,  \n\\nonumber \\\\\n\\left\\vert \\Pi _{03}\\right\\rangle \n&\\triangleq &\n\\frac{1}{\\sqrt{3}}\\int d\\overrightarrow{t}\\;b_{\\mathrm{in},2}^{\\ast }(t_{1})b_{\\mathrm{in},2}^{\\ast }(t_{2})b_{\\mathrm{in},2}^{\\ast }(t_{3})\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})|0_{2}\\rangle .  \n\\label{july18_4}\n\\end{eqnarray}\nIt is easy to show that all the states in Eq. (\\ref{july18_4}) are\nnormalized. Moreover, $\\left\\vert \\Pi _{30}\\right\\rangle $ is a $3$-photon\nstate for the first channel, $\\left\\vert \\Pi _{03}\\right\\rangle $ is a $3$-photon state for the second channel, and $\\left\\vert \\Pi _{21}\\right\\rangle \n$ and $\\left\\vert \\Pi _{12}\\right\\rangle $ are states where two channels\nshare three photons. With these new notations, Eq. (\\ref{july18_3}) can be\nre-written as\n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\frac{\\sqrt{6}}{4}\\left\\vert \\Pi\n_{30}\\right\\rangle \\otimes |0_{2}\\rangle -\\frac{\\sqrt{2}}{4}\\left\\vert \\Pi\n_{21}\\right\\rangle -\\frac{\\sqrt{2}}{4}\\left\\vert \\Pi _{12}\\right\\rangle\n+\\left\\vert 0_{1}\\right\\rangle \\otimes \\frac{\\sqrt{6}}{4}\\left\\vert \\Pi\n_{03}\\right\\rangle .  \\label{july18_5}\n\\end{equation}\nFinally, if \n\\begin{equation}\n\\psi _{\\mathrm{in}}(t_{1},t_{2},t_{3})=\\xi _{1}(t_{1})\\xi _{2}(t_{2})\\xi\n_{3}(t_{3}),  \\label{july19_4}\n\\end{equation}\nthen Eq. (\\ref{eq:aug21_14}) becomes\n\\begin{eqnarray*}\n|\\Psi _{\\mathrm{in}}\\rangle \n&=&\n\\int_{-\\infty\n}^{\\infty }\\xi _{1}(t_{1})b_{\\mathrm{in,}1}^{\\ast\n}(t_{1})dt_{1}|0_{1}\\rangle \\otimes \\int_{-\\infty }^{\\infty }\\int_{-\\infty\n}^{\\infty }dt_{2}dt_{3}\\;\\xi _{2}(t_{2})\\xi _{3}(t_{3})\\;b_{\\mathrm{in,}2}^{\\ast }(t_{2})b_{\\mathrm{in,}2}^{\\ast }(t_{3})|0_{2}\\rangle \n\\\\\n&=&\n\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }(\\xi\n_{1})|0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in,}2}^{\\ast }(\\xi _{2})\n\\mathbf{B}_{\\mathrm{in,}2}^{\\ast }(\\xi _{3})|0_{3}\\rangle .\n\\end{eqnarray*}\nThat is, the input is a product state, with one photon in channel 1 and two\nphotons in channel 2. Moreover, if $\\xi_1= \\xi_2 = \\xi_3 \\equiv \\xi$, then the normalization condition requires that $\\|\\xi\\|=\\frac{1}{\\sqrt[6]{2}}$.  In this case, Eq. (\\ref{july18_4}) reduces to \n\\begin{eqnarray}\n\\left\\vert \\Pi _{30}\\right\\rangle \n&=&\n\\frac{1}{\\sqrt{3}}(\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }(\\xi))^3|0_{1}\\rangle ,  \n\\nonumber \\\\\n\\left\\vert \\Pi _{21}\\right\\rangle\n &=&\n(\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }(\\xi))^2 |0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in,}2}^{\\ast }(\\xi)|0_{2}\\rangle , \n \\nonumber \\\\\n\\left\\vert \\Pi _{12}\\right\\rangle \n&=&\n\\mathbf{B}_{\\mathrm{in,}1}^{\\ast }(\\xi)|0_{1}\\rangle \\otimes (\\mathbf{B}_{\\mathrm{in,}\n2}^{\\ast }(\\xi))^2 |0_{2}\\rangle , \n \\nonumber \\\\\n\\left\\vert \\Pi _{03}\\right\\rangle \n&=&\n\\frac{1}{\\sqrt{3}} \n(\\mathbf{B}_{\\mathrm{in,}2}^{\\ast }(\\xi))^3 |0_{2}\\rangle . \n \\label{july31_10}\n\\end{eqnarray}\nThat is, all the states become product states. If we ignore pulse shapes and only count the number of photons in each channel, we\nmay identify $\\left\\vert \\Pi _{30}\\right\\rangle $ with $\\left\\vert\n3_{1}\\right\\rangle $, $\\left\\vert \\Pi _{03}\\right\\rangle $ with $\\left\\vert\n3_{2}\\right\\rangle $, $\\left\\vert \\Pi _{21}\\right\\rangle $ with $\\left\\vert\n2_{1}\\right\\rangle \\otimes \\left\\vert 1_{2}\\right\\rangle $, and $\\left\\vert\n\\Pi _{12}\\right\\rangle $ with $\\left\\vert 1_{1}\\right\\rangle \\otimes\n\\left\\vert 2_{2}\\right\\rangle $. Accordingly, the state in Eq. (\\ref{july18_5}) reduces to \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle =\\frac{\\sqrt{6}}{4}\\left\\vert\n3_{1}\\right\\rangle \\otimes \\left\\vert 0_{2}\\right\\rangle -\\frac{\\sqrt{2}}{4}\n\\left\\vert 2_{1}\\right\\rangle \\otimes \\left\\vert 1_{2}\\right\\rangle -\\frac{\n\\sqrt{2}}{4}\\left\\vert 1_{1}\\right\\rangle \\otimes \\left\\vert\n2_{2}\\right\\rangle +\\frac{\\sqrt{6}}{4}\\left\\vert 0_{1}\\right\\rangle \\otimes\n\\left\\vert 3_{2}\\right\\rangle .  \n\\label{july19_6}\n\\end{equation}\n\\end{example}\n\n\n\\subsection{The invariant set}\\label{sec:invariant_aug3}\n\nIn this subsection, we define a class of $m$-channel $N$-photon states and show that this\nclass of states is invariant under the steady-state action of a quantum\nlinear passive system. The discussions here generalize those in\nSubsection \\ref{subsec:invariant}.\n\n\n\nMotivated by Eqs. (\\ref{jan9_psi_out}) and (\\ref{eq:aug23_2}), define a class of $m$-channel $N$-photon states:\n\\begin{equation}\n\\mathcal{F}_{2}\\triangleq \\left\\{ |\\Psi \\rangle =\\psi \\odot _{k_{1}\\cdots\nk_{m}}^{N}b_{\\mathrm{in}}^{\\#}|0^{\\otimes m}\\rangle \\ |\\mathrm{the~}N\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\textrm{-}}\\mathrm{way}~m\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\textrm{-}}\\mathrm{dimensional~tensor~function~}\\psi \\mathrm{~normalizes~}|\\Psi \\rangle .\\right\\}  \n\\label{july19_8}\n\\end{equation}\n\n\nThe following result shows that the set $\\mathcal{F}_{2}$ is invariant under\nthe steady-state action of a passive quantum linear system.\n\n\n\\begin{theorem}\n\\label{thm:passive_invariance_general} \nThe steady-state output field state of the asymptotically stable \npassive quantum linear system (\\ref{sys_passive}), initialized in the vacuum state $|\\phi\n\\rangle $ and driven by an $m$-channel $N$-photon input state $|\\Psi _{\\mathrm{in}}\\rangle \\in \n\\mathcal{F}_{2}$ with pulse information encoded by an $N$-way $m$-dimensional tensor function $\\psi _{\\mathrm{in}}$, is another element $|\\Psi\n_{\\mathrm{out}}\\rangle \\in \\mathcal{F}_{2}$, whose pulse information is\nencoded by an $N$-way $m$-dimensional tensor function $\\psi _{\\mathrm{out}}$\ngiven by \n\\begin{equation}\n\\psi _{\\mathrm{out}}=\\psi _{\\mathrm{in}}\\circledast _{t, k_{1}\\cdots\nk_{m}}^{N}g_{G^{-}}.  \\label{eq:mar22_1}\n\\end{equation}\n\\end{theorem}\n\nThis result can be established in a similar way as Theorem \\ref{thm:passive_invariance}. So the proof is omitted.\n\n\n\\section{An arbitrary number of photons superposed over $m$ input channels}\n\n\\label{subsec:mN_general}\n\nIn all the previous discussions, we have implicitly assumed that the total\nnumber of photons is no less than the number of input channels. In this\nsection, we remove this constraint. More specifically, we study a class of $m\n$-channel $N$-photon states where $N$ can be an arbitrary positive integer.\n\n\n\\subsection{A class of $m$-channel $N$-photon input states}\\label{subsec:mN_input}\n\nIn this subsection, we present a class of $m$-channel $N$-photon input\nstates. Two illustrative examples are also given.\n\nLet a normalized $m$-channel $N$-photon input state be \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\\int_{-\\infty }^{\\infty }dt\\ \\psi _{\\mathrm{in},jk}(t)b_{\\mathrm{in},k}^{\\ast }(t)|0^{\\otimes m}\\rangle ,\n\\label{eq:temp1}\n\\end{equation}\nwhere $N$ is an arbitrary positive integer. The input state $|\\Psi _{\\mathrm{in}}\\rangle $ is\nparametrized by the pulse shapes $\\psi _{\\mathrm{in},jk}(t)$ ($j=1,\\ldots ,N$\nand $k=1,\\ldots m$). Clearly, different combinations of $\\psi _{\\mathrm{in},jk}(t)$ give rise to different $m$-channel $N$-photon states. By the\nnotation in Eq. (\\ref{july31_B}), the $m$-channel $N$-photon input state in\nEq. (\\ref{eq:temp1}) can be re-written as \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\\mathbf{B}_{\\mathrm{in},k}^{\\ast }(\\psi\n_{\\mathrm{in},jk})|0^{\\otimes m}\\rangle .  \\label{eq:temp1b}\n\\end{equation}\n\n\n\\begin{remark}\n\\label{rem:zhang13} A class of photon-Gaussian states has been defined in \n\\cite[Eq. (95)]{zhang13}. If the density matrix $\\rho _{R}$ there used is of\nthe form $\\rho _{R}=|\\phi 0^{\\otimes m}\\rangle \\left\\langle \\phi 0^{\\otimes\nm}\\right\\vert $, and moreover, $\\xi _{jk}^{+}\\equiv 0$, then the resulting\nstates are $m$-channel $m$-photon states. Actually, they form a special\nsubclass of the $m$-channel $N$-photon states defined in Eq. (\\ref{eq:temp1b})\n(with $N=m$).\n\\end{remark}\n\n\n\\begin{remark} \\label{rem:jan15_photon}\nAlthough the positive integer $N$ in Eq. (\\ref{eq:temp1}) is allowed to be arbitrary, the multi-photon input states defined in Eq. (\\ref{eq:temp1})  may not be able to include those multi-photon states studied in Sections \\ref{sec:mm} and \\ref{sec:mn} as subclasses. This can be easily seen by comparing the forms of multi-photon states in Eqs. (\\ref{eq:aug23_10}), (\\ref{eq:aug16_5}),  and  (\\ref{eq:temp1}). \n\\end{remark}\n\n\n\\begin{remark} \\label{rem:N}\n Eq. (\\ref{eq:temp1b}) provides flexibility for specifying multi-channel multi-photon states. \n \\begin{description}\n \\item[(i)] if for some $j_{0}$ ($1\\leq j_{0}\\leq N$) and $k_{0}$ ($1\\leq k_{0}\\leq m$), $\\psi _{\\mathrm{in},j_{0}k_{0}}\\equiv 0$, then the term $\\mathbf{B}_{\\mathrm{in},k_{0}}^{\\ast }(\\psi _{\\mathrm{in},j_{0}k_{0}})$ does not appear on the right-hand side of Eq. (\\ref{eq:temp1b}).\n \n\\item[(ii)]  As a special case of item (i) above, if for some $j_{0}$ ($1\\leq j_{0}\\leq N$), $\\psi _{\\mathrm{in},j_{0}k}\\equiv 0$ for all $k=1,\\ldots m$, then Eq. (\\ref{eq:temp1b}) reduces to \n\\begin{equation*}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\tprod\\limits_{j=1,j\\neq j_{0}}^{N}\\sum_{k=1}^{m}\\mathbf{B}_{\\mathrm{in},k}^{\\ast }(\\psi _{\\mathrm{in},jk})|0^{\\otimes m}\\rangle .\n\\end{equation*}\nIn this case, there are $\\mathbf{N-1}$ photons among $m$ channels.  Thus, the term  ``$N$-photon'' is a bit confusing. Nevertheless, the exact number of photons can be determined  easily from the context.\n\\end{description}\n\\end{remark}\n\nWe illustrate Remark \\ref{rem:N} with the following two Examples.\n\n\\begin{example}\\label{ex:12} \nWhen $N=1$ and $m=2$, by Eq. (\\ref{eq:temp1b}), the input\nstate is\n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle \n=\n\\mathbf{B}_{\\mathrm{in},1}^{\\ast }(\\psi _{\\mathrm{in},11})|0_{1}\\rangle \\otimes\n|0_{2}\\rangle +|0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast }(\\psi _{\\mathrm{in},12})|0_{2}\\rangle .\n\\label{july19_1b}\n\\end{equation}\n In what follows, we discuss two cases.\n\nCase 1): $\\psi _{\\mathrm{in},11}\\equiv 0$. In this case, as commented by  item (i) in Remark \\ref{rem:N}, Eq. (\\ref{july19_1b}) becomes \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =|0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast }(\\psi _{\\mathrm{in},12})|0_{2}\\rangle .  \n\\label{july19_1c}\n\\end{equation}\nThe normalization condition is \n\\begin{equation*}\n1=\\left\\langle \\Psi _{\\mathrm{in}}|\\Psi _{\\mathrm{in}}\\right\\rangle =\\int_{-\\infty }^{\\infty }\\left\\vert \\psi _{\\mathrm{in},12}(t)\\right\\vert ^{2}dt.\n\\end{equation*}\n In this case, the first channel is in the\nvacuum state and the second channel is in a single-photon state.\n\nCase 2): $\\psi _{\\mathrm{in},11}=\\psi _{\\mathrm{in},12}\\equiv \\xi $. Eq. (\\ref{july19_1b}) becomes \n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\boldsymbol{B}_{\\mathrm{in,}1}^{\\ast }(\\xi )\\left\\vert 0_{1}\\right\\rangle \\otimes \\left\\vert\n0_{2}\\right\\rangle +\\left\\vert 0_{1}\\right\\rangle \\otimes \n\\boldsymbol{B}_{\\mathrm{in,}2}^{\\ast }(\\xi )\\left\\vert 0_{2}\\right\\rangle .\n\\label{aug23_1}\n\\end{equation}\nThe normalization condition\n\\begin{equation*}\n1=\\left\\langle \\Psi _{\\mathrm{in}}|\\Psi _{\\mathrm{in}}\\right\\rangle =2\\int_{-\\infty }^{\\infty }\\left\\vert \\xi (t)\\right\\vert\n^{2}dt=2\\Vert \\xi \\Vert ^{2}\n\\end{equation*}\nrequires that $\\|\\xi\\|=\\frac{1}{\\sqrt{2}}$. Moreover, it can be readily shown that \n\\begin{equation}\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\left\\langle \\Psi _{\\mathrm{in}}|\\Lambda _{\\mathrm{in}}(t)|\\Psi _{\\mathrm{in}}\\right\\rangle =\n\\frac{1}{2}\\left[ \n\\begin{array}{cc}\n1 & 0 \\\\ \n0 & 1\n\\end{array}\n\\right] .  \\label{aug23_1b}\n\\end{equation}\nThat is, the photon is not localized in either of the two channels; instead,\nit is shared by two channels. This reveals the wave property of photons.\n\\end{example}\n\n\n\n\n\n\\begin{example}\n\\label{ex:23} Let $N=2$ and $m=3$. According to Eq. (\\ref{eq:temp1b}), the\ninput state is\n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =\\sum_{k=1}^{3}\n\\mathbf{B}_{\\mathrm{in},k}^{\\ast }(\\psi _{\\mathrm{in},1k})\\sum_{k=1}^{3}\n\\mathbf{B}_{\\mathrm{in},k}^{\\ast }(\\psi _{\\mathrm{in},2k})|0^{\\otimes\n3}\\rangle .  \n\\label{july19_2}\n\\end{equation}\nIf $\\psi _{\\mathrm{in},11} = \\psi _{\\mathrm{in},22}\\equiv 0$, then, as commented by item (i) in Remark \\ref{rem:N}, the\ninput state in Eq. (\\ref{july19_2}) becomes\n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{in}}\\rangle   \\nonumber \\\\\n&=&\n\\mathbf{B}_{\\mathrm{in},1}^{\\ast }(\\psi _{\\mathrm{in},21})|0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast\n}(\\psi _{\\mathrm{in},12})|0_{2}\\rangle \\otimes |0_{3}\\rangle +\\mathbf{B}_{\\mathrm{in},1}^{\\ast }(\\psi _{\\mathrm{in},21})|0_{1}\\rangle \\otimes |0_{2}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},3}^{\\ast }(\\psi _{\\mathrm{in},13})|0_{3}\\rangle  \n \\nonumber \\\\\n&&\n+|0_{1}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast }(\\psi _{\\mathrm{in},12})|0_{2}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},3}^{\\ast }(\\psi _{\\mathrm{in},23})|0_{3}\\rangle +|0_{1}\\rangle \\otimes |0_{2}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},3}^{\\ast }(\\psi _{\\mathrm{in},13})\\mathbf{B}_{\\mathrm{in},3}^{\\ast }(\\psi _{\\mathrm{in},23})|0_{3}\\rangle .  \n\\label{july19_2a}\n\\end{eqnarray}\nThat is, two photons are shared by three channels. If further $\\psi _{\\mathrm{in},21}\\equiv \\psi _{\\mathrm{in},13}\\equiv 0$, then, as commented by item (i) in Remark \\ref{rem:N}, Eq. (\\ref{july19_2a}) reduces to\n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =|0_{1}\\rangle\n\\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast }(\\psi _{\\mathrm{in},12})|0_{2}\\rangle \\otimes \\mathbf{B}_{\\mathrm{in},3}^{\\ast }(\\psi _{\\mathrm{in},23})|0_{3}\\rangle .  \\label{july19_2b}\n\\end{equation}\nIn this case, the first channel is in the vacuum state, and there is exactly\none photon in each of the second and third channels, respectively. Finally, if further $\\psi _{\\mathrm{in},23}(t)\\equiv 0$, the only existing pulse shape in Eq. (\\ref{july19_2}) is $\\psi _{\\mathrm{in},12}$, and therefore, as commented by item (ii) in\nRemark \\ref{rem:N}, we end up with a single photon state. Indeed, Eq. (\\ref{eq:temp1b}) reduces to\n\\begin{equation}\n|\\Psi _{\\mathrm{in}}\\rangle =|0_{1}\\rangle\n\\otimes \\mathbf{B}_{\\mathrm{in},2}^{\\ast }(\\psi _{\\mathrm{in},12})|0_{2}\\rangle \\otimes |0_{3}\\rangle .  \\label{july19_2c}\n\\end{equation}\nThat is, the second channel has one photon while both the first and third\nchannels are in the vacuum state.\n\\end{example}\n\n\n\\subsection{State transfer}\\label{subsec:mN_output}\n\nIn this subsection, we derive an analytic form of the steady-state output field \nstate of a passive quantum linear system driven by an $m$-channel $N$-photon input \nstate defined in Eq. (\\ref{eq:temp1}).\n\nThe following is the main result of this section.\n\n\\begin{theorem}\\label{thm:general} \nLet the asymptotically stable passive quantum\nlinear system (\\ref{sys_passive}) be initialized in the vacuum state and driven by the $m$-channel $N$-photon input $|\\Psi _{\\mathrm{in}}\\rangle $ defined in Eq. (\\ref{eq:temp1}). The steady-state output field state is another $m$-channel $N$-photon state of the form \n\\begin{equation}\n|\\Psi _{\\mathrm{out}}\\rangle \n=\n\\tprod\\limits_{j=1}^{N}\\sum_{l=1}^{m}\\int_{-\\infty }^{\\infty }dt\\ \\psi _{\\mathrm{out},jl}(t)b_{\\mathrm{in},l}^{\\ast }(t)|0^{\\otimes m}\\rangle ,\n\\label{eq:temp5}\n\\end{equation}\nwhere the output pulses are given by\n\\begin{equation}\n\\psi _{\\mathrm{out},jl}(t)\\triangleq \\sum_{k=1}^{m}\\int_{-\\infty }^{\\infty\n}g_{G^{-}}^{lk}(t-r)\\psi _{\\mathrm{in},jk}(r)dr,\\ \\ \\ j=1,\\ldots ,N,  \\  l=1,\\ldots ,m.  \\label{eq:temp4}\n\\end{equation}\n\\end{theorem}\n\n\\textbf{Proof.~} The proof is similar to that for Theorem \\ref{thm:passive_mm}; specifically,\n\\begin{eqnarray}\n&&\n|\\Psi _{\\mathrm{out}}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\left\\langle \\phi\n\\right\\vert U(t,t_{0})|\\phi \\Psi _{\\mathrm{in}}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\left\\langle \\phi \\right\\vert\nU(t,t_{0})\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\\int_{-\\infty }^{\\infty }dt_k\\\n\\psi _{\\mathrm{in},jk}(t_{k})b_{\\mathrm{in},k}^{\\ast }(t_{k})|\\phi\n0^{\\otimes m}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\lim_{t_{0}\\rightarrow -\\infty ,t\\rightarrow \\infty }\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\\left\\langle \\phi\n\\right\\vert \\int_{t_{0}}^{t}dt_{k}\\ \\psi _{\\mathrm{in},jk}(t_{k})U(t_{k},t_{0})b_{\\mathrm{in},k}^{\\ast\n}(t_{k})U(t_{k},t_{0})^{\\ast }|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\lim_{t\\rightarrow \\infty }\n\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\\left\\langle \\phi \\right\\vert\n\\int_{-\\infty }^{t}dt_{k}\\ \\psi _{\\mathrm{in},jk}(t_{k})U(t_{k},-\\infty )b_{\\mathrm{in},k}^{\\ast }(t_{k})U(t_{k},-\\infty )^{\\ast }|\\phi 0^{\\otimes\nm}\\rangle \n\\nonumber\n \\\\\n&=&\n\\tprod\\limits_{j=1}^{N}\\sum_{k=1}^{m}\n\\left\\langle \\phi \\right\\vert \\int_{-\\infty }^{\\infty }dt_{k}\\ \\psi _{\\mathrm{in},jk}(t_{k})\\int_{-\\infty }^{\\infty\n}\\sum_{l=1}^{m}g_{G^{-}}^{lk}(r_{k}-t_{k})b_{\\mathrm{in},l}^{\\ast\n}(r_{k})dr_{k}|\\phi 0^{\\otimes m}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\tprod\\limits_{j=1}^{N}\\sum_{l=1}^{m}\n\\int_{-\\infty }^{\\infty }dr_{k}\\ \\boxed{\\sum_{k=1}^{m}\\int_{-\\infty }^{\\infty\n}dt_{k}\\ g_{G^{-}}^{lk}(r_{k}-t_{k})\\psi _{\\mathrm{in},jk}(t_{k})} \\ b_{\\mathrm{in},l}^{\\ast }(r_{k})|0^{\\otimes m}\\rangle  \n\\nonumber\n\\\\\n&=&\n\\tprod\\limits_{j=1}^{N}\\sum_{l=1}^{m}\n\\int_{-\\infty }^{\\infty }dr_k\\ \\boxed{\\psi _{\\mathrm{out},jl}(r_k)}\\ b_{\\mathrm{in},l}^{\\ast }(r_k)|0^{\\otimes m}\\rangle ,\n\\label{eq:jan15_final}\n\\end{eqnarray}\nwhere the output pulse shapes $\\psi _{\\mathrm{out},jl}$ are those \nin Eq. (\\ref{eq:temp4}), as highlighted by the two boxes above. $\\blacksquare $\n\n\\section{Conclusion}\\label{sec:conclusion} \nIn this paper, we have studied the dynamics of\nquantum linear systems in response to multi-channel multi-photon states. We\nhave derived the intensity of the output field which can be used to\ninvestigate the influence of quantum linear systems on quantum correlations\nof multi-photon light fields. We have also presented the explicit formula of the\nsteady-state output field states when a quantum linear system is driven by three  classes of multi-channel\nmulti-photon input states. The results presented here are very general and\nhold promising applications in photon-based quantum coherent feedback networks. One of the future research directions is to study controller synthesis problem on the basis of the system analysis carried out in this paper,  for example, via the Lyapunov method \\cite{MvH07}, \\cite{KC08}.\n\n\\begin{ack}                              \nThe author wishes to thank the anonymous reviewers for their careful reading and constructive comments.\n\\end{ack}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn today's day and age of social media, there are ample opportunities for fake news production, dissemination and consumption. \\citet{rashkin2017truth} break down fake news into three categories, hoax, propaganda and satire. A hoax article typically tries to convince the reader about a cooked-up story while propaganda ones usually mislead the reader into believing a false political or social agenda. \\citet{burfoot2009automatic} defines a satirical article as the one which deliberately exposes real-world individuals, organisations and events to ridicule.\n\n\nPrevious works \\cite{rubin2016fake, rashkin2017truth} rely on various linguistic and hand-crafted semantic features for differentiating between news articles. However, none of them try to model the interaction of sentences within the document. We observed a pattern in the way sentences cluster in different kind of news articles. Specifically, satirical articles had a more coherent story and thus all the sentences in the document \\textit{seemed} similar to each other. On the other hand, the trusted news articles were also coherent but the similarity between sentences from different parts of the document was not that strong, as depicted in Figure \\ref{fig:example}. We believe that the reason for such kind of behaviour is the presence of factual jumps across sections in a trusted document. \n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n  \\includegraphics[width=3.5cm, height=2.6cm]{24_0_n.png}\n  \\subcaption{Trusted}\n\\end{subfigure}%\n\\begin{subfigure}{.2\\textwidth}\n  \\includegraphics[width=3.5cm, height=2.6cm]{263_1_n.png}\n  \\subcaption{Satirical}\n\\end{subfigure}%\n\\caption{TSNE visualization~\\cite{van2014accelerating} of sentence embeddings obtained using BERT~\\cite{devlin2018bert} for two kind of news articles from SLN. A point denotes a sentence and the number indicates which paragraph it belonged to in the article.}\n\\label{fig:example}\n\\vspace{-1\\baselineskip}\n\\end{figure}\n\nIn this work, we propose a graph neural network-based model to classify news articles while capturing the interaction of sentences across the document. We present a series of experiments on News Corpus with Varying Reliability dataset~\\cite{rashkin2017truth} and Satirical Legitimate News dataset~\\cite{rubin2016fake}. Our results demonstrate that the proposed model achieves state-of-the-art performance on these datasets and provides interesting insights. Experiments performed in out-of-domain settings establish the generalizability of our proposed method.\n\n\\begin{table*}[t]\n\\centering{}\n\\scriptsize\n\\begin{tabular}{c c c c c}\n  \\toprule\n Dataset & Trusted (\\# Docs) & Satire (\\# Docs) & Hoax (\\# Docs) & Propaganda (\\# Docs)\\\\\n  \\midrule\n\tLUN-train & GN except `APW' and `WPB' (9,995) & The Onion (14,047) & American News (6,942) & Activist Report (17,870)\\\\\n\tLUN-test & GN only `APW' and `WPB' (750) & The Borowitz Report, Clickhole (750) & DC Gazette (750) & The Natural News (750)\\\\\n\tSLN & The Toronto Star, The NY Times (180) & The Onion, The Beaverton (180) & - & -\\\\\n\tRPN & WSJ, NBC, etc (75) & The Onion, The Beaverton, etc (75) & - & -\\\\\n  \\bottomrule\n\\end{tabular}\n\\caption{Statistics about different dataset sources. GN refers to Gigaword News. \\label{tab:stats}}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=2.7cm,width=16cm]{framework.jpg}\n\\end{center}\n\\caption{Proposed semantic graph neural network based model for fake news classification.}\n\\label{fig:framework}\n\\end{figure*}\n\n\\section{Related Work}\n\nSatire, according to \\citet{simpson2003discourse}, is complicated because it occupies more than one place in the framework for humor, proposed by \\citet{ziv1988teaching}: it clearly has an aggressive and social function, and often expresses an intellectual aspect as well. \\citet{rubin2016fake} defines news satire as a genre of satire that mimics the format and style of journalistic reporting. Datasets created for the task of identifying satirical news articles from the trusted ones are often constructed by collecting documents from different online sources \\cite{rubin2016fake}. \\citet{mchardy2019adversarial} hypothesized that this encourages the models to learn characteristics for different publication sources rather than characteristics of satire. In this work, we show that our proposed model generalizes to articles from unseen publication sources.\n\n\\citet{rashkin2017truth} extends \\citet{rubin2016fake}'s work by offering a quantitative study of linguistic differences found in articles of different types of fake news such as hoax, propaganda and satire. They also proposed predictive models for graded deception across multiple domains. \\citet{rashkin2017truth} found that neural methods didn't perform well for this task and proposed to use a Max-Entropy classifier. We show that our proposed neural network based on graph convolutional layers can outperform this model. Recent works by \\citet{yang-etal-2017-satirical, de2018attending} show that sophisticated neural models can be used for satirical news detection. To the best of our knowledge, none of the previous works represent individual documents as graphs where the nodes represent the sentences for performing classification using a graph neural network.\n\n\n\\section{Dataset and Baseline}\nWe use \\textbf{SLN}: Satirical and Legitimate News Database~\\cite{rubin2016fake}, \\textbf{RPN}: Random Political News Dataset~\\cite{horne2017just} and \\textbf{LUN}: Labeled Unreliable News Dataset~\\citet{rashkin2017truth} for our experiments. Table \\ref{tab:stats} shows the statistics. Since all of the previous methods on the aforementioned datasets are non-neural, we implement the following neural baselines,\n\\begin{itemize}\n    \\item \\textbf{CNN:} In this model, we apply a 1-d CNN (Convolutional Neural Network) layer~\\cite{kim2014convolutional} with filter size 3 over the word embeddings of the sentences within a document. This is followed by a max-pooling layer to get a single document vector which is passed to a fully connected projection layer to get the logits over output classes.\n    \\item \\textbf{LSTM:} In this model, we encode the document using a LSTM (Long Short-Term Memory) layer~\\cite{hochreiter1997long}. We use the hidden state at the last time step as the document vector which is passed to a fully connected projection layer to get the logits over output classes.\n    \\item \\textbf{BERT:} In this model, we extract the sentence vector (representation corresponding to [CLS] token) using BERT (Bidirectional Encoder Representations from Transformers)~\\cite{devlin2018bert} for each sentence in the document. We then apply a LSTM layer on the sentence embeddings, followed by a projection layer to make the prediction for each document.\n\\end{itemize}\n\n\\section{Proposed Model}\nCapturing sentence interactions in long documents is not feasible using a recurrent network because of the vanishing gradient problem \\cite{pascanu2013difficulty}. Thus, we propose a novel way of encoding documents as described in the next subsection. Figure \\ref{fig:framework} shows the overall framework of our graph based neural network.\n\n\\subsection{Input Representation}\nEach document in the corpus is represented as a graph. The nodes of the graph represent the sentences of a document while the edges represent the semantic similarity between a pair of sentences. Representing a document as a fully connected graph allows the model to directly capture the interaction of each sentence with every other sentence in the document. Formally, \n\\begin{equation}\n    e_{ij} = Similarity(s_i, s_j)\n\\end{equation}\nWe initialize the edge scores using BERT \\cite{devlin2018bert} finetuned on the semantic textual similarity task\\footnote{Task 1 of SemEval-2017} for computing the semantic similarity (SS) between two sentences. Refer to the Supplementary Material for more details regarding the SS model. Note that this representation drops the sentence order information but is better able to capture the interaction between far off sentences within a document.\n\n\\subsection{Graph based Neural Networks}\nWe reformulate the fake news classification problem as a graph classification task, where a graph represents a document. Given a graph $G= (E,S)$ where $E$ is the adjacency matrix and $S$ is the sentence feature matrix. We randomly initialize the word embeddings and use the last hidden state of a LSTM layer as the sentence embedding, shown in Figure \\ref{fig:framework}. We experiment with two kinds of graph neural networks,\n\n\\subsubsection{Graph Convolution Network (GCN)}\nThe graph convolutional network \\cite{gcn} is a spectral convolutional operation denoted by $f(Z^l, E|W^l)$,\n\\begin{align}\n    Z^{l+1} &= f(Z^l, E|W^l)\\\\\n    f(Z^l, E|W^l) &= \\sigma(EZ^lW^l)\n\\end{align}\nHere, $Z^l$ is the output feature corresponding to the nodes after $l^{th}$ convolution. $W^l$ is the parameter associated with the $l^{th}$ layer. We set $Z^0 = S$. Based on the above operation, we can define arbitrarily deep networks. For our experiments, we just use a single layer unless stated otherwise. By default, the adjacency matrix ($E$) is fully connected i.e. all the elements are 1 except the diagonal elements which are all set to 0. We set $E$ based on semantic similarity model in our \\textit{GCN + SS} model. For the \\textit{GCN + Attn} model, we just add a self attention layer \\cite{vaswani2017attention} after the GCN layer and before the pooling layer.\n\n\\subsubsection{Graph Attention Network (GAT)}\n\\citet{gat} introduced graph attention networks to address various shortcomings of GCNs. Most importantly, they enable nodes to attend over their neighborhoods' features without depending on the graph structure upfront. The key idea is to compute the hidden representations of each node in the graph, by attending over its neighbors, following a self-attention \\cite{vaswani2017attention} strategy.\nBy default, there is one attention head in the GAT model. For our \\textit{GAT + 2 Attn Heads} model, we use two attention heads and concatenate the node embeddings obtained from different heads before passing it to the pooling layer. For a fully connected graph, the GAT model allows every node to attend on every other node and learn the \\textit{edge weights}. Thus, initializing the edge weights using the SS model is useless as they are being learned. Mathematical details are provided in the Supplementary Material.\n\n\\begin{figure*}[t]\n\\centering\n\\begin{subfigure}{.37\\textwidth}\n\\includegraphics[width=7.7cm, height=3.4cm]{trusted.png}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}{.50\\textwidth}\n\\includegraphics[width=7.7cm, height=3.4cm]{satire.png}\n\\end{subfigure}\n\\caption{Attention heatmaps generated by GAT for 2-way classification. \\textbf{Left:} Trusted, \\textbf{Right:} Satire.\n\\label{im:analysis}}\n\\end{figure*}\n\n\\subsection{Hyperparameters}\nWe use a randomly initialized embedding matrix with 100 dimensions. We use a single layer LSTM to encode the sentences prior to the graph neural networks. All the hidden dimensions used in our networks are set to 100. The node embedding dimension is 32. For GCN and GAT, we set $\\sigma$ as \\textit{LeakyRelU} with slope 0.2. We train the models for a maximum of 10 epochs and use Adam optimizer with learning rate 0.001. For all the models, we use max-pool for pooling, which is followed by a fully connected projection layer with output nodes equal to the number of classes for classification.\n\n\\section{Experimental Setting}\nWe conduct experiments across various settings and datasets. We report macro-averaged scores in all the settings.\\\\\n\n\\noindent\\textbf{2-way classification b\/w satire and trusted articles:\\label{2-way}} We use the satirical and trusted news articles from LUN-train for training, and from LUN-test as the development set. We evaluate our model on the entire SLN dataset. This is done to emulate a real-world scenario where we want to see the performance of our classifier on an out of domain dataset. We don't use SLN for training purposes because it just contains 360 examples which is too little for training our model and we want to have an unseen test set. The best performing model on SLN is used to evaluate the performance on RPN.\\\\\n\n\\noindent \\textbf{4-way classification b\/w satire, propaganda, hoax and trusted articles:} We split the LUN-train into a 80:20 split to create our training and development set. We use the LUN-test as our out of domain test set.\n\n\\section{Results}\n\n\\begin{table}[t]\n\\centering{}\n\\small\n\\begin{tabular}{l c c}\n  \\toprule\n Model & \\bf Precision &\\bf Recall\\\\\n  \\midrule\nCNN & 67.5&\t67.5\\\\\nLSTM  &82.2\t&81.4\\\\\nBERT  & 78.1&\t78.1\\\\\nSoTA* \\cite{rubin2016fake} & 88.0 & 82.0\\\\\n\\midrule\n\\multicolumn{3}{c}{Our Models}\\\\\n\\midrule\nGCN & 85.9\t&85.0\\\\\nGCN + SS & 86.4&\t86.3\\\\\nGCN + Attn & 87.1\t&86.9\\\\\nGCN + Attn + SS & 87.8\t&87.8\\\\\nGAT &  86.2\t&86.1\\\\\nGAT + 2 Attn Heads& \\textbf{89.1}&\t\\textbf{88.9}\\\\\n  \\bottomrule\n\\end{tabular}\n\\caption{2-way classification results on SLN. *n-fold cross validation (precision, recall) as reported in SoTA.}\n\\label{tbl:results:2way}\n\\vspace{-1\\baselineskip}\n\\end{table}\n\n\\begin{table}[t]\n\\centering{}\n\\small\n\\begin{tabular}{l c c}\n  \\toprule\n  \\bf Model & \\bf LUN-dev & \\bf LUN-test\\\\\n  \\midrule\nCNN & 96.48 & 54.04\\\\\nLSTM & 88.75 & 55.05\\\\\nBERT & 95.07 & 54.87\\\\\nSoTA* \\cite{rashkin2017truth} & 91.0 & 65.0\\\\\n\\midrule\n\\multicolumn{3}{c}{Our Models}\\\\\n\\midrule\nGCN & 96.76 & 65.0\\\\\nGCN + Attn & 97.57 & \\textbf{67.08}\\\\\nGAT & 97.28 & 65.51\\\\\nGAT + 2 Attn Heads & \\textbf{97.82} & 66.95\\\\\n  \\bottomrule\n\\end{tabular}\n\\caption{4-way classification results for different models. We only report F1-score following the SoTA paper.}\n\\label{tbl:results:4way}\n\\vspace{-1\\baselineskip}\n\\end{table}\n\nTable \\ref{tbl:results:2way} shows the quantitative results for the two way classification between satirical and trusted news articles. Our proposed GAT method with 2 attention heads outperforms SoTA. The semantic similarity model does not seem to have much impact on the GCN model, and considering the computing cost, we don't experiment with it for the 4-way classification scenario. Given that we use SLN as an out of domain test set (just one overlapping source, no overlap in articles), whereas the SoTA paper \\cite{rubin2016fake} reports a 10-fold cross validation number on SLN. We believe that our results are quite strong, the \\textit{GAT + 2 Attn Heads} model achieves an accuracy of 87\\% on the entire RPN dataset when used as an out-of-domain test set. The SoTA paper \\cite{horne2017just} on RPN reports a 5-fold cross validation accuracy of 91\\%. These results indicate the generalizability of our proposed model across datasets.\nWe also present results of four way classification in Table \\ref{tbl:results:4way}. All of our proposed methods outperform SoTA on both the in-domain and out of domain test set.\n\nTo further understand the working of our proposed model, we closely inspect the attention maps generated by the GAT model for satirical and trusted news articles for the SLN dataset. From Figure \\ref{im:analysis}, we can see that the attention map generated for the trusted news article only focuses on two specific sentence whereas the attention weights are much more distributed in case of a satirical article. Interestingly enough the highlighted sentences in case of the trusted news article were the starting sentence of two different paragraphs in the article indicating the presence of similar sentence clusters within a document. This opens a new avenue for understanding the differences between different kind of text articles for future research.\n\n\\section{Conclusion}\nThis paper introduces a novel way of encoding articles for fake news classification. The intuition behind representing documents as a graph is motivated by the fact that sentences interact differently with each other across different kinds of article. Recurrent networks are unable to maintain long term dependencies in large documents, whereas a fully connected graph captures the interaction between sentences at unit distance. The quantitative result shows the effectiveness of our proposed model and the qualitative results validate our hypothesis about difference in sentence interaction across different articles. Further, we show that our proposed model generalizes to unseen datasets.\n\n\\section*{Acknowledgement}\nWe would like to thank the AWS Educate program for donating computational GPU resources used in this work. We also appreciate the anonymous reviewers for their insightful comments and suggestions to improve the paper.\n\n\\section*{Supplementary Material}\nThe supplementary material is available\\footnote{\\url{https:\/\/github.com\/MysteryVaibhav\/fake_news_semantics\/blob\/master\/EMNLP2019_TextGraphs_Supplementary.pdf}} along with the code which provides mathematical details of the GAT model and few additional qualitative results.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBaryons from the large-$N_c$ point of view have been studied by quite a number of people who\nderived many algebraic relations between baryon observables, following from the requirement\nthat physical quantities should have ``natural'' behaviour in $N_c$, see\n{\\it e.g.}~\\cite{Manohar:2001cr,Goity:2005fj,Cohen:2005uv} and references therein. In this report,\nwe suggest a simple physical picture that results in those relations, and derive some \nnew ones. A longer paper has been recently published~\\cite{Diakonov:2012zz}, and a detailed report \nis currently in preparation.\n\nIf the number of colours $N_c$ is large the $N_c$ quarks constituting a baryon can be considered\nin a mean (non-fluctuating) field that is stable as $N_c\\!\\to\\!\\infty$~\\cite{Witten:1979kh}.\nAt the microscopic level quarks experience colour interactions, however gluon field fluctuations\nare not suppressed if $N_c$ is large; the mean field can be only `colourless'. An example\nhow originally colour interactions transform mathematically into interactions of dynamically massive \nquarks with mesonic fields is provided by the instanton liquid model, see {\\it e.g.}~\\cite{Diakonov:2002fq}.\nA non-fluctuating confining chiral bag model gives another example of a `colourless' mean field.\nA modern example is the 5- or 6-dimensional `gravitational', plus mesonic, background field\nin the holographic QCD models. They also need large $N_c$ for a justification.\n\nThe advantage of the large-$N_c$ approach is that at large $N_c$ baryon physics simplifies\nconsiderably. It becomes possible to take into full account the important relativistic\nand field-theoretic effects that are often ignored. Baryons are not just three (or $N_c$)\nquarks but contain additional quark-antiquark pairs, as it is well known experimentally.\nThe number of antiquarks in baryons is, theoretically, also proportional to $N_c$~\\cite{Diakonov:2008hc},\nwhich means that antiquarks cannot be obtained from adding one or two mesons to a baryon:\none needs ${\\cal O}(N_c)$ mesons to explain ${\\cal O}(N_c)$ antiquarks, implying in fact\na classical mesonic field.\n\nIf the mean field at large $N_c$ is a reality, one can consider quarks in that mean\nfield -- similarly to nucleons in the mean field of large-$A$ nuclei. Generally, quarks\nobey the Dirac equation in the background field, that may be in fact non-local. All intrinsic\nquark Dirac levels in the mean field are stable in $N_c$. All negative-energy levels should\nbe filled in by $N_c$ quarks in the antisymmetric state in colour, corresponding to the zero baryon\nnumber state. Filling in the lowest positive-energy level by $N_c$ `valence' quarks makes\na baryon, see Fig.~1, left. Exciting higher quark levels or making particle-hole excitations \nproduces baryon resonances~\\cite{Diakonov:2008rd,Diakonov:2009kw}, see Fig.~1, middle. \nThe baryon mass is ${\\cal O}(N_c)$, and the excitation energy is ${\\cal O}(1)$. When one excites \none quark the change of the mean field is ${\\cal O}(1\/N_c)$ that can be neglected to the first \napproximation.\n\nMoreover, if one replaces one light ($u,d$ or $s$) quark in light baryons by a heavy\n($c,b$) one, as in charmed or bottom baryons, the change in the mean field is also\n${\\cal O}(1\/N_c)$. Therefore, the spectrum of heavy baryons is directly related to\nthat of light baryons~\\cite{Diakonov:2009kw,Diakonov:2010zz}. In light baryons made of \n$u,d,s$ quarks, we first take the chiral limit of zero quark masses. It implies that \nall resonances appear as degenerate $SU(3)_{\\rm flav}$ multiplets that are later split \nby nonzero $m_s$.\n\nNext, we argue that the mean field in baryons has a definite symmetry, namely it breaks \nspontaneously the symmetry under separate $SU(3)_{\\rm flav}$ and $SO(3)_{\\rm space}$ \nrotations but does not change under simultaneous $SU(2)_{\\rm iso+space}$ rotations in \nordinary space and a compensating rotation in isospace~\\cite{Diakonov:2008rd,Diakonov:2009kw}.\nSimilarly, in nuclear physics the $SO(3)_{\\rm space}$ symmetry is broken spontaneously\nin the ground state since most of the heavy nuclei are elliptical-, not spherical-symmetric.\n\nSpontaneous breaking of $SU(3)_{\\rm flav}$ means, in particular, that one-quark levels\nfor $u,d$ quarks on the one hand, and for $s$ quarks on the other, are totally, 100\\% different\neven in the chiral limit $m_s\\to 0$. The latter levels are characterized by total angular\nmomentum ${\\bf J}={\\bf L}+{\\bf S}$, and parity, whereas the former are characterized by the\ngrand spin ${\\bf K}={\\bf J}+{\\bf T}$ where ${\\bf T}$ is the isospin, and parity. \n\nThe original $SU(3)_{\\rm flav}\\times SO(3)_{\\rm space}$ symmetry is restored by the (quantized)\nrotation of the mean field in flavour and ordinary spaces. It implies that each intrinsic \nquark state, be it the ground state or a one-quark excitation in the Dirac spectrum, \ngenerates a band of $SU(3)$ multiplets appearing as collective rotational excitations of \na given intrinsic state. The quantum numbers of those multiplets, their total number and \nthe ${\\cal O}(1\/N_c)$ splittings between them are unequivocally dictated by the symmetry \nof the mean field, see Fig.~1, right. Assuming the $SU(2)_{\\rm iso+space}$ symmetry of \nthe mean field, we obtain exactly the spin and flavour multiplets that are observed in Nature.\n\nAll properties of baryon resonances belonging to one band associated with a given one-quark \nexcitation are related by symmetry. For example, the splittings inside several $SU(3)$ \nmultiplets as due to the nonzero $m_s$, are related to each other: these relations are \nsatisfied with high accuracy, in some cases better than the Gell-Mann--Okubo relations \nfor separate $SU(3)$ multiplets.\n\nAt this time we do not consider any specific dynamical model but concentrate mainly on symmetry. \nA concrete dynamical model would say what is the intrinsic relativistic quark spectrum in baryons. \nInstead of calculating the intrinsic Dirac spectrum of quarks from a model, we extract it \nfrom the experimentally known baryon spectrum by interpreting baryon resonances as collective \nexcitations about the ground state and about the one-quark transitions. However, we show \nin Ref.~\\cite{Diakonov:2012zz} that the needed intrinsic quark spectrum can be obtained from \na natural choice of the mean field satisfying the $SU(2)_{\\rm iso+space}$ symmetry.\n\nIn summary, we show that it is possible to obtain a realistic spectrum of baryon\nresonances up to 2 GeV, starting from the large-$N_c$ limit. The lowest $SU(3)$ multiplets\nof charmed baryons also fit nicely into this universal picture~\\cite{Diakonov:2009kw,Diakonov:2010zz}. \n\n\n\\begin{figure}\n\\includegraphics[width=.34\\textwidth]{ground_state.eps}\\hspace{0.65cm}\n\\includegraphics[width=.34\\textwidth]{Lambda_transitions.eps}\\hspace{0.65cm}\n\\includegraphics[width=.19\\textwidth]{baryon_excitations_Nc.eps}\n\\caption{{\\it Left}: Filling in $u,d$- and, separately, $s$-quark levels for the ground-state \nlight baryons. {\\it Middle}: Examples of one-quark excitations. \n{\\it Right}: Hierarchy of scales in large-$N_c$ baryons.}\n\\label{fig1}\n\\end{figure}\n\n\\section{Baryons made of $u,d,s$ quarks}\n\nThe ground-state baryons are obtained from the level-filling scheme shown in Fig.~1, left.\nThe quantization of the rotations (restoring the original flavour and space symmetries) of the\n$K^P=0^+$ valence level gives rise to the lowest baryons multiplets, $({\\bf 8}, 1\/2^+)$\nand $({\\bf 10}, 3\/2^+)$. \n\nThe singlets, $({\\bf 1}, 1\/2^-)$ and $({\\bf 1}, 3\/2^-)$ (the $\\Lambda$ hyperons), are obtained\nfrom ``Gamov--Teller'' excitations of the $s$-quark levels, see Fig.~1, middle. In principle, these\ntwo transitions entail rotational bands, however the rotational splitting of higher rotational \nexcitations are not ${\\cal O}(1\/N_c)$ but ${\\cal O}(1)$~\\cite{Diakonov:2010zz}, therefore all states\nexcept the lowest-mass singlets must be ignored in this approach. \n\nThe parity-plus resonances are obtained from the $u,d$ one-quark excitations $0^+\\to 0^+$,\n$0^+\\to 1^+$ and $0^+\\to 2^+$, whereas parity-minus resonances can be obtained from\nthe $u,d$ transitions $0^+\\to 0^-$, $0^+\\to 1^-$ and $0^+\\to 2^-$, see ~\\cite{Diakonov:2012zz}.\nAll known baryon resonances below 2 GeV can be nicely fit into the rotational states stemming\nfrom those one-quark excitations. There are no resonances from the PDG tables that are left \nunaccounted for, and there are no extra states from the theory side, except the $\\Delta(3\/2^+)$\nresonance coming from the rotational excitation on top of the $0^+\\to 1^+$ transition. It\nis therefore our prediction.       \n\n\n\\begin{figure}\n\\includegraphics[width=.31\\textwidth]{parity_plus_transitions.eps}\\hspace{0.35cm}\n\\includegraphics[width=.31\\textwidth]{parity_minus_transitions.eps}\\hspace{0.35cm}\n\\includegraphics[width=.31\\textwidth]{s_quark_transitions.eps}\n\\caption{{\\it Left}: The origin of parity-plus light resonances. {\\it Middle}: The origin of parity-minus\nlight baryons. {\\it Right}: Pentaquarks are particle-hole excitations of a negative-energy $s$ quark.\nThe exotic parity-plus $\\Theta^+$ (recently seen again at JLab~\\cite{Amaryan:2011qc}) corresponds to a \nnon-diagonal or `Gamov--Teller' transition {\\it 2}.}\n\\label{fig2}\n\\end{figure}\n\n\n\\section{Splittings inside $SU(3)$ multiplets}\n\nProperties of baryon resonances, such as the couplings, partial decay widths, mass splittings, {\\it etc.}, \narising from one rotational band about a given one-quark excitation are related to each other by symmetry. \nTypically, the observables for resonances  from {\\em different} multiplets are expressed through a small \nnumber of constants, which leads to numerical relations between the observables, that can be tested \nagainst the data.\n\nFor example, in the linear order in $m_s$ the splittings inside an octet are determined by two constants\n$\\mu_{1,2}^{(8)}$ while those in a decuplet are determined by only one constant $\\mu^{(10)}$. We write the masses\nof individual members of the octets,\n$$\n{\\cal M}_{N}=M_8-\\frac{7}{4}\\mu^{(8)}_1-\\mu^{(8)}_2, \\quad  {\\cal M}_{\\Lambda}=M_8-\\mu^{(8)}_1, \n\\quad {\\cal M}_{\\Sigma}=M_8+\\mu^{(8)}_1 \\quad {\\cal M}_{\\Xi}\n=M_8+\\frac{3}{4}\\mu^{(8)}_1+\\mu^{(8)}_2,\n$$\nand the masses of the members of the decuplets,\n$$\n{\\cal M}_{\\Delta}=M_{10}-\\mu^{(10)}, \\qquad  {\\cal M}_{\\Sigma}=M_{10}, \\qquad {\\cal M}_{\\Xi}=M_{10}+\\mu^{(10)} \\qquad {\\cal M}_{\\Omega}\n=M_{10}+2\\mu^{(10)}.\n$$    \nFrom this parametrization one gets automatically the Gell-Mann--Okubo relations for the octets and decuplets\nseparately. However, the symmetry breaking pattern $SU(3)_{\\rm flav}\\times SO(3)_{\\rm space}\\to SU(2)_{\\rm iso+space}$\nthat we advocate enables one to relate the splitting parameters related to {\\em different} multiplets\nwith various spins. In particular, we obtain the following new relations for the different multiplets \nstemming from the $0\\to 1$ and $0\\to 2$  one-quark transitions, respectively,\n$$\n7\\mu^{(10)}\\left(\\frac{1}{2}\\right)+3\\mu^{(8)}_2\\left(\\frac{3}{2}\\right)=10\\mu^{(10)}\\left(\\frac{3}{2}\\right),\n$$\n$$\n5\\mu^{(8)}_2\\left(\\frac{5}{2}\\right)+11\\mu^{(10)}\\left(\\frac{3}{2}\\right)=16\\mu^{(10)}\\left(\\frac{5}{2}\\right)\n$$\n(in the parentheses we indicate the spin of the multiplet). Fitting the masses of individual multiplet members\nby the parameters $\\mu$ one can find them for various multiplets and then plug into the above relations.\nIt turns out that these very nontrivial relations are satisfied to an accuracy of 1-2\\%! It gives a strong\nsupport to the scheme suggested. Another support comes from the fact that the ${\\cal O}(1\/N_c)$ \nsplittings between the {\\it centers} of the $SU(3)$ multiplets are expressed through two moments of inertia,\nand phenomenologically many splittings can be described, to a fair accuracy, with the same moments of inertia.\nSurprisingly, approximately the same moments of inertia are needed to explain the splittings in heavy baryons.\n\n\n\\section{Heavy baryons}\n\nIf one of the light quarks in a light baryon is replaced by a heavy $b$ or $c$ quark,\nthere are still $N_c\\!-\\!1$ light quarks left. At large $N_c$, they form {\\em the same} mean field as\nin light baryons, with the same sequence of Dirac levels, up to $1\/N_c$ corrections. The heavy quark\ncontributes to the mean $SU(3)_{\\rm flav}$ symmetric field but it is a $1\/N_c$ correction, too. It means\nthat at large $N_c$ one can {\\em predict} the spectrum of the $Qq\\ldots q$ (and $Qq\\ldots qq\\bar q$) baryons from\nthe spectrum of light baryons. \n\nThe filling of Dirac levels for the ground-state $c$ (or $b$) baryon is shown in Fig.~3, left:\nthere is a hole in the $0^+$ shell for $u,d$ quarks as there are only $N_c-1$ quarks there, in an antisymmetric\nstate in colour. Adding the heavy quark makes the full state `colourless'.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.33\\textwidth]{ground_state_charmed.eps}\\hspace{0.3cm}\n\\includegraphics[width=.33\\textwidth]{charmed_transition.eps}\\hspace{0.3cm}\n\\includegraphics[width=.28\\textwidth]{antidecapenta.eps}\n\\end{center}\n\\caption{{\\it Left}: Ground-state charmed (or bottom) baryon with a hole in the valence $u,d$ level.\n{\\it Middle}: A particle-hole excitation of the $s$-quark from the negative-energy level. {\\it Right}:\nThe corresponding ${\\bf\\overline{15}}$-plet of heavy pentaquarks. The {\\em Beta} baryons in the upper\nline may be stable against strong decays.}\n\\label{fig3}\n\\end{figure}\n\nAs in the case of light baryons, the filling scheme by itself does not tell us what are the quantum numbers\nof the state: they arise from quantizing the $SU(3)_{\\rm flav}$ and $SO(3)_{\\rm space}$ rotations of the given\nfilling scheme. The quantization of the rotations about the ground state gives three $SU(3)$ multiplets:\n$({\\bf \\bar 3}, 1\/2^+), ({\\bf 6}, 1\/2^+)$ and $({\\bf 6}, 3\/2^+)$. These are exactly the lightest charmed\nmultiplets observed in Nature. The centers of the last two multiplets are split by $1\/N_c^2$ and $1\/m_c$ \ncorrections, and indeed the splitting is small (67 MeV), whereas the splitting between the centers of the ${\\bf 6}$'s\nand the ${\\bf \\bar 3}$ is ${\\cal O}(1\/N_c)$ and is in good accordance with the splitting between {\\em light}\nrotational states, thus supporting the main idea~\\cite{Diakonov:2009kw,Diakonov:2010zz}.  \n\nFinally, the hole in the ground-state $0^+$ shell can be filled in either by exciting a $u,d$ quark \nfrom lower lying shells, or by non-diagonal `Gamov-Teller' transition from the highest filled $s$ quark shell,\nsee Fig.~3, middle. In the second case the corresponding charmed baryon resonance will be a pentaquark exotic,\nactually belonging to a ${\\bf \\overline{15}}$-plet from the $SU(3)$ point of view, Fig.~3, right. \n\nThe masses of the exotic ``Beta baryons'', ${\\cal B}_c^{++}=cuud\\bar s,\\;{\\cal B}_c^+=cudd\\bar s$, and\n${\\cal B}_b^+=buud\\bar s$, ${\\cal B}_b^0=budd\\bar s$ have been estimated in Ref.~\\cite{Diakonov:2009kw,Diakonov:2010zz}\nto be quite light, $m_{\\cal B}\\approx 2420\\,{\\rm MeV}$, and hence they may be well stable against\nstrong decays! See Ref.~\\cite{Diakonov:2010zz} for the discussion of the ${\\bf \\overline{15}}$-plet of\nthe exotic charmed (or bottom) pentaquarks and of the possibilities of their observation at LHC\nand b-factories.\n\n\\section{Conclusions}\n\nWe have presented a new unified picture of baryon resonances, light and heavy, \nthat is in fact quite similar to that of nuclei at large $A$ whose r\\^ole is played by large $N_c$. \nWhile $N_c$ is only three in the real world, we see that the imprint of the large-$N_c$ picture is \nclearly visible. A by-product of this study is the prediction of a relatively light exotic charmed pentaquark, \n the {\\it Beta}.     \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n Chinese is a  logographic language. \nThe logograms of Chinese characters encode rich information of their meanings. Therefore, it is intuitive that\nNLP tasks for  Chinese should benefit from the use of the glyph information.\nTaking \ninto account logographic information should help semantic modeling. \nRecent studies indirectly support this argument: \nRadical representations  have proved to be useful in a wide range of language understanding tasks \\citep{shi2015radical,li2015component,yin2016multi,sun2014radical,shao2017character}.\nUsing the Wubi  scheme ---- a Chinese character encoding method that mimics the order of typing the sequence of radicals for a character on the computer keyboard ---- is reported to  improve  performances on \nChinese-English\nmachine translation  \\citep{tan2018wubi2en}. \\citet{cao2018cw2vec} gets down to units of greater granularity, and proposed  stroke n-grams for character modeling.  \n\n\n\\begin{figure}\n\\center\n\\includegraphics[width=2.5in]{newfigs\/1.png}\n\\includegraphics[width=2.5in]{newfigs\/2.png}\n\\centering\n\\caption{Many Chinese characters evolved from pictures. The image is copied from Wikipedia.}\n\\label{illustration}\n\\end{figure}\n\n\n\n\n\nRecently, there have been some efforts \napplying CNN-based algorithms on the visual features of characters.\nUnfortunately, they do not show consistent performance boosts \\citep{liu2017learning,zhang2017encoding}, and some even yield negative results \\citep{dai2017glyph}. \nFor instance, \\citet{dai2017glyph} \nrun CNNs on char logos\nto obtain Chinese character representations and used them in the \ndownstream\nlanguage modeling task. They reported that   the incorporation of glyph representations actually worsen the performance and concluded that CNN-based representations do not provide extra useful information for language modeling.\nUsing similar strategies, \n\\citet{liu2017learning} and \\citet{zhang2017encoding} test the idea on text classification tasks, and performance boosts were observed only in very limited number of settings.\nPositive results come from \\citet{su2017learning}, which found glyph embeddings help two tasks:\nword analogy and word similarity. Unfortunately,  \\citet{su2017learning} only focus on word-level semantic tasks and\ndo not extend improvements in the word-level tasks\n to higher level NLP tasks such as phrase, sentence or discourse level.  \n Combined with radical representations, \n\\citet{shao2017character} run CNNs on character figures and \nuse the output as auxiliary features in the POS tagging task. \n\n We propose the following explanations for negative results reported in the earlier CNN-based models \\citep{dai2017glyph}:\n (1) not using the correct version(s) of script: Chinese character system has a long history of evolution.\nThe characters started from being easy-to-draw, and slowly transitioned to being easy-to-write. Also, they became less pictographic and less concrete over time. \nThe most widely used script version to date, the {\\it Simplified Chinese}, is the easiest script to write, but  inevitably loses the most significant amount of pictographic information. \n For example,  \"\u4eba\" (human) and  \"\u5165\" (enter), which are of irrelevant meanings, are highly similar in shape in simplified Chinese, but very different in historical languages such as bronzeware script. \n  (2) not using the proper CNN structures: \nunlike  ImageNet images \\citep{deng2009imagenet}, the size of which is mostly at the scale of\n800*600, \ncharacter logos are significantly smaller (usually with the size of 12*12). \nIt\n requires a different CNN architecture to capture the local graphic features of character images;\n(3) no regulatory functions were used in previous work: \nunlike the classification task on the imageNet dataset, which contains tens of millions of data points, \nthere are only about 10,000 Chinese characters. Auxiliary training objectives are thus critical in preventing overfitting and  \npromoting the model's ability to generalize.\n\nIn this paper, we propose the GLYCE, the GLYph-vectors for Chinese character representations. We treat Chinese characters as images and use CNNs to obtain their representations. \nWe resolve the aforementioned issues by using the following key techniques:\n\\begin{tightitemize}\n\\item We use the ensemble of the historical and the contemporary scripts (e.g., the bronzeware script, the clerical script, the seal script, the traditional Chinese etc), along with the scripts of different writing styles (e.g, the cursive script) to enrich pictographic information from the character images.\n\\item We utilize the Tianzige-CNN  (\u7530\u5b57\u683c) structures tailored to logographic character modeling for Chinese.\n\\item We use multi-task learning methods by adding an image-classification loss function \n to increase the model's ability to generalize.\n\\end{tightitemize}\nGlyce is found to improve a wide range of Chinese NLP tasks.\nWe are able\nto obtain the SOTA performances on a wide range of Chinese NLP tasks, including\n tagging (NER, CWS, POS), \nsentence pair classification (BQ, LCQMC,  XNLI, NLPCC-DBQA), \nsingle sentence classification tasks (ChnSentiCorp, the Fudan corpus, iFeng),\ndependency parsing, and semantic role labeling.\n\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=6in]{newfigs\/CNN-TIANZIGE.png}\n\\centering\n\\caption{Illustration of the Tianzege-CNN used in Glyce. }\n\\label{TIANZIGE}\n\\end{figure*}\n\n\n\n \n\n\\section{Glyce}\n\\begin{table*}\n\\small\n\\center\n\\begin{tabular}{|c|c|c|}\\hline\nChinese & English & Time Period \\\\\\hline\n\u91d1\u6587&Bronzeware script& Shang dynasty and Zhou dynasty (2000 BC \u2013 300 BC) \\\\\\hline\n\u96b6\u4e66& Clerical script & Han dynasty (200BC-200AD)  \\\\\\hline\n\u7bc6\u4e66& Seal script & Han dynasty and Wei-Jin period (100BC - 420 AD) \\\\\\hline\n\u9b4f\u7891& Tablet script & Northern and Southern dynasties 420AD - 588AD \\\\\\hline\n\\multirow{2}{*}{\u7e41\u4f53\u4e2d\u6587} & \\multirow{2}{*}{Traditional Chinese} & 600AD - 1950AD (mainland China).  \\\\\n& & still currently used in HongKong and Taiwan  \\\\\\hline\n\u7b80\u4f53\u4e2d\u6587 (\u5b8b\u4f53) &Simplified Chinese - Song & 1950-now \\\\\\hline\n\u7b80\u4f53\u4e2d\u6587 (\u4eff\u5b8b\u4f53) & Simplified Chinese - FangSong & 1950-now \\\\\\hline\n\u8349\u4e66&Cursive script& Jin Dynasty  to now  \\\\\\hline\n\\end{tabular}\n\\caption{Scripts and writing styles used in Glyce.}\n\\label{script}\n\\end{table*}\n\n\\subsection{Using Historical Scripts}\nAs discussed in Section 1, pictographic \ninformation is heavily lost in the simplified Chinese script. \nWe  thus propose using scripts from various time periods in history and also of different writing styles.\nWe collect the following major historical script with details  shown in Table \\ref{script}.\n Scripts from different historical periods, which are usually very different in shape, help the model to integrate \npictographic\nevidence from various sources;\nScripts of different writing styles  help improve the model's ability to generalize. \nBoth strategies are akin to widely-used data augmentation strategies in computer vision. \n\n\n\n\n\\subsection{The Tianzige-CNN Structure for Glyce}\n\\begin{comment}\n\\begin{table}\n\\center\n\\small\n\\begin{tabular}{ |c|c|c| }\\hline\n layer & kernel size & feature size\\\\ \n input &  & n$\\times$ 12$\\times$ 12 \\\\\n conv & 5 & 1024$\\times$8$\\times$8 \\\\ \n relu &  & 1024$\\times$8$\\times$8 \\\\ \n maxpool & 4 & 1024$\\times$ 2$\\times$2 \\\\\n 8-group conv& 1 & 256$\\times$2$\\times$ 2 \\\\\n 16-group conv & 2 & 1024$\\times$1$\\times$ 1 \\\\\\hline\n\\end{tabular}\n\\caption{The tianzige-CNN structure in Glyce.}\n\\label{cnn_structure}\n\\end{table}\n\\end{comment}\nDirectly using deep CNNs  \\citet{he2016deep,Szegedy_2016_CVPR,ma2018shufflenet} in our task results in very poor performances because of (1) relatively smaller size of the character images:\nthe size of Imagenet images is usually  at the scale of\n800*600, while the size of Chinese character images is significantly smaller, usually at the scale of 12*12;\n and (2) the lack of training examples:\n classifications on the imageNet dataset utilizes tens of millions of different images. In contrast,\nthere are only about 10,000 distinct Chinese characters. \nTo tackle these issues,\nwe propose the Tianzige-CNN structure, which is tailored to Chinese character modeling\nas illustrated in Figures \\ref{TIANZIGE}. \nTianzige (\u7530\u5b57\u683c) is a traditional form of Chinese Calligraphy. It is a four-squared format (similar to Chinese character \u7530) for beginner to learn writing Chinese characters. \nThe input image $x_{\\text{image}}$ \nis first passed through a convolution layer with kernel size 5 and output channels 1024 to capture lower level graphic features.\nThen a max-pooling layer of kernel size 4 is applied to the feature map and reduces the resolution from $8\\times 8$ to the tianzige $2\\times 2$, .  \nThis $2\\times 2$ tianzige structure presents how radicals are arranged in Chinese characters and also the order by which Chinese characters are written. \nWe find the tianzige structure of significant importance in extracting character meanings. \nFinally, we apply convolutional operations to map tianzige grids to the final outputs.\nInstead of using conventional convolutional operations, we use group convolutions \\citep{krizhevsky2012imagenet,zhang2017interleaved}.\nGroup convolutional filters are much smaller than their normal counterparts, and thus are less prune to overfitting. \nIt is fairly easy to adjust the model from  single script to multiple scripts, which can be achieved by simply changing the input from 2D (i.e., $d_{\\text{font}}\\times d_{\\text{font}}$) to 3D (i.e., $N_{\\text{script}}\\times d_{\\text{font}}\\times d_{\\text{font}}$), where $d_{\\text{font}}$ denotes the font size.\n\n\\subsection{Image Classification as an Auxiliary Objective}\nTo further prevent overfitting, we use the task of image classification as an auxiliary training objective. \nThe glyph embedding $h_{\\text{image}}$ from CNNs\nwill be forwarded to an image classification objective to predict its corresponding charID.\nSuppose the label of image $x$ is $z$.\nThe training objective  for the image classification task $\\mathcal{L}(\\text{cls})$ is given as follows: \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}(\\text{cls}) & = -\\log p(z|x)\\\\\n& = -\\log \\text{softmax} (W\\times h_{\\text{image}})\n\\end{aligned}\n\\end{equation}\nLet  $\\mathcal{L}(\\text{task})$ denote the task-specific objective for the task we need to tackle, e.g., language modeling, word segmentation, Machine Comprehension, etc. \nWe linearly combine $\\mathcal{L}(\\text{task})$ and $\\mathcal{L}(\\text{cl})$, making the final objective training function as follows:\n\\begin{equation}\n\\mathcal{L} =   (1-\\lambda(t)) \\ \\mathcal{L}(\\text{task}) + \\lambda(t) \\mathcal{L}(\\text{cls})\n\\end{equation}\nwhere $\\lambda(t)$ controls the trade-off between the task-specific objective and the auxiliary image-classification objective. \n$\\lambda$ is a function of the number of epochs $t$:\n$\\lambda(t) = \\lambda_0 \\lambda_1^{t}$, where $\\lambda_0\\in [0,1]$ denotes the starting value,   $\\lambda_1\\in [0,1]$ denotes the decaying value. \nThis means that the influence from the image classification objective \n decreases as the training proceeds, with the intuitive explanation being that at the earlier stage of training, we need more regulations from the image classification task. \nAdding image classification as a training objective mimics the idea of multi-task learning.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=5.5in]{newfigs\/overview_combine.png}\n\\centering\n\\caption{Combing glyph information with BERT.}\n\\label{overview_combine}\n\\end{figure*}\n\n\\subsection{ Combing Glyph Information with BERT}\nThe glyph embeddings can be directly output to downstream models such as RNNs, LSTMs, transformers. \n\nSince large scale pretraining systems using language models, such as BERT \\citep{devlin2018bert}, ELMO \\citep{peters2018deep} and GPT \\citep{radford2018improving}, \n have proved to be effective in a wide range of NLP tasks,\n we explore the possibility of combining glyph embeddings with BERT embeddings.\n Such a strategy \n  will potentially endow the model with  the advantage of  both glyph evidence and large-scale pretraining. \n  The overview of the combination is shown in Figure \\ref{overview_combine}.\nThe  model consists of four  layers: the BERT layer, the glyph layer, the Glyce-BERT layer and the task-specific output layer. \n\n\\paragraph{BERT Layer}\nEach input sentence $S$ \nis concatenated with a special \\textsc{cls} token denoting the start of the sentence, and a \\textsc{sep} token, denoting the end of the sentence. \nGiven a pre-trained BERT model, the embedding for each token of $S$ is computed using BERT. \nWe use to output from the last layer of the BERT transformer to represent the current token. \n\n\\paragraph{Glyph Layer} the output from tianzege-CNNs.\n\n\\paragraph{Glyce-BERT layer}\nPosition embeddings are first added to the glyph embeddings. The addition is then concatenated with BERT to obtain the full Glyce representation. This marks the end of Glyph and BERT combination. \n\\paragraph{Task-specific output layer}\nGlyce-BERT representations are used to represent the token at that position, similar as word embeddings or Elmo emebddings \\citep{peters2018deep}.\nContextual-aware information has already been encoded in the BERT representation but not glyph representations. We thus need additional context   models to encode contextual-aware glyph representations. \nHere, we choose multi-layer transformers \\citep{vaswani2017attention}.  \nThe output representations from transformers are used as outputs to the prediction layer.  \nIt is worth noting that the representations for the special \\textsc{cls} and \\textsc{sep} positions are maintained at the final task-specific embedding layer. \n\\begin{figure*}\n\\center\n\\includegraphics[width=5.5in]{newfigs\/cls.png}\n\\centering\n\\caption{Using Glyce-BERT model for different tasks. }\n\\label{cls}\n\\end{figure*}\n\n\n\\section{Tasks}\nIn this section, we describe how glyph embeddings can be used for different NLP tasks. \nIn the vanilla version, glyph embeddings are simply treated as character embeddings, which are fed to \n models built on top of the word-embedding layers, such as RNNs, CNNs or more sophisticated ones. \nIf combined with BERT, we need to specifically handle the integration between the glyph embeddings and the pretrained embeddings from BERT in different scenarios, as will be discussed in order below:\n\n\\subsection{Sequence Labeling Tasks}\nMany Chinese NLP tasks, such as name entity recognition (NER), Chinese word segmentation (CWS) and part speech tagging (POS), can be formalized as character-level sequence\nlabeling tasks, in which  we need to predict a label for each character. \nFor glyce-BERT model, \nthe embedding output from the task-specific layer (described in Section 2,4)\n is fed to the CRF model for label predictions. \n\\subsection{Single Sentence Classification}\nFor text classification tasks, a single label is to be predicted for the entire sentence. \nIn the  BERT model, the representation for the \\textsc{cls} token in the final layer of BERT is output to the softmax layer for prediction. \nWe adopt the similar strategy, in which the representation for the \\textsc{cls} token in the task-specific layer is fed to the softmax layer to predict labels. \n\\subsection{Sentence Pair Classification} \nFor sentence pair classification task like SNIS \\citep{DBLP:conf\/emnlp\/BowmanAPM15}, a model needs to handle the interaction between the two sentences and outputs a label for a pair of sentences. In the  BERT setting, a sentence pair $(s_1, s_2)$ is concatenated with one \\textsc{cls} and two \\textsc{sep} tokens, denoted by \n[\\textsc{cls}, $s_1$, \\textsc{sep}, $s_2$, \\textsc{sep}]. The concatenation is fed to the BERT model, and the \nobtained\n\\textsc{cls} representation is then fed to the softmax layer for label prediction. \nWe adopt the similar strategy for Glyce-BERT, in which [\\textsc{cls}, $s_1$, \\textsc{sep}, $s_2$, \\textsc{sep}] is subsequently passed through the BERT layer, Glyph layer, Glyce-BERT layer and the task-specific output layer. The \\textsc{cls} representation from the task-specific output layer is fed to the softmax function for the final label prediction. \n\n\n\\section{Experiments}\nTo enable apples-to-apples comparison, we perform grid  parameter search for both baselines and the proposed model on the dev set. \nTasks that we work on are described in order below.\n\\subsection{Tagging}\n\\paragraph{NER}\nFor the task of Chinese NER, we used the widely-used OntoNotes, MSRA, Weibo and resume datasets.\nSince most datasets don't have gold-standard segmentation, the task is normally treated as \n a char-level \n  tagging task: outputting an NER tag for each character. \nThe currently most widely used non-BERT model is Lattice-LSTMs \\citep{DBLP:journals\/corr\/abs-1810-12594, DBLP:conf\/acl\/ZhangY18}, \n achieving better performances than CRF+LSTM \\citep{DBLP:conf\/acl\/MaH16}. \n \\paragraph{CWS}:\nThe task of Chinese word segmentation (CWS) is \nnormally treated\nas a char-level tagging problem. \nWe used the widely-used PKU, MSR, CITYU and AS benchmarks from SIGHAN 2005 bake-off for evaluation.\n\n\\paragraph{POS}\nThe task of Chinese part of speech tagging is normally formalized as a character-level sequence labeling task, assigning labels to each of the character within the sequence. \nWe use the CTB5, CTB9 and   UD1 (Universal Dependencies) benchmarks to test our models. \n\n\\begin{table}[!ht]\n \\small\n\\caption{Results for NER tasks.}\n\\label{NER}\n\\centering\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{OntoNotes}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM & 74.36 & 69.43&71.81 \\\\\nLattice-LSTM  & 76.35& 71.56& 73.88  \\\\\nGlyce+Lattice-LSTM&82.06& 68.74&74.81  \\\\\n&&&{\\bf (+ 0.93) }\\\\\\hline\nBERT& 78.01& 80.35& 79.16 \\\\\nGlyce+BERT & 81.87 & 81.40 & 80.62 \\\\\n&&&{\\bf (+1.46)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{MSRA}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&92.97&90.80&91.87\\\\\nLattice-LSTM  & 93.57& 92.79& 93.18  \\\\\nLattice-LSTM+Glyce& 93.86&93.92 & 93.89\\\\\n&&&{\\bf (+0.71)}\\\\\\hline\nBERT& 94.97 & 94.62 & 94.80 \\\\\nGlyce+BERT & 95.57 & 95.51 & 95.54 \\\\\n&&&{\\bf(+0.74)} \\\\\\hline\n\\end{tabular}\n\n\\columnbreak\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{Weibo}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&51.16&51.07&50.95\\\\\nLattice-LSTM&52.71&53.92&53.13 \\\\\nLattice-LSTM+Glyce&53.69&55.30& 54.32\\\\\n&&&{\\bf (+1.19)} \\\\\\hline\nBERT& 67.12 & 66.88 & 67.33 \\\\\nGlyce+BERT & 67.68& 67.71& 67.60 \\\\\n&&&{\\bf (+0.27)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{resume}\\\\\\hline\nModel & P & R & F \\\\\\hline\nCRF-LSTM&94.53 & 94.29&94.41 \\\\\nLattice-LSTM  & 94.81& 94.11&  94.46  \\\\\nLattice-LSTM+Glyce&95.72&95.63& 95.67 \\\\\n&&&{\\bf (+1.21)} \\\\\\hline\nBERT& 96.12 & 95.45 & 95.78 \\\\\nGlyce+BERT & 96.62 & 96.48 & 96.54 \\\\\n&&&{\\bf(+0.76)}\\\\\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\nResults for NER, CWS and POS are respectively shown in Tables \\ref{NER}, \\ref{CWS} and \\ref{POS}. \nWhen comparing with non-BERT models, Lattice-Glyce performs better than all non-BERT models\n across all datasets in all tasks. \nBERT outperforms non-BERT models in all datasets except Weibo. \n This is due to the discrepancy  between the dataset which BERT is pretrained on (i.e., wikipedia) and weibo. \nThe Glyce-BERT model outperforms BERT and sets new SOTA results across all datasets, manifesting the effectiveness of incorporating glyph information. \nWe are able to achieve SOTA performances on all of the datasets using either Glyce model itself or BERT-Glyce model. \n\n\\begin{table}[!ht]\n\\center\n\\small\n\\caption{Results for CWS tasks.}\n\\label{CWS}\n\\centering\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{PKU}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&- &96.3\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 96.1\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &96.6\\\\\nBERT&96.8 & 96.3 &96.5 \\\\\nGlyce+BERT&97.1&96.4&96.7 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{MSR}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&-&97.5\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 98.1\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&-&97.9\\\\\nBERT& 98.1 & 98.2 & 98.1\\\\\nGlyce+BERT&98.2 & 98.3 &98.3 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\end{tabular}\n\n\\columnbreak\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CITYU}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&- &96.9\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 97.2\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &97.6\\\\\nBERT& 97.5 & 97.7 & 97.6\\\\\nGlyce+BERT &97.9 &98.0 &97.9\\\\\n&&&{\\bf(+0.3)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{AS}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{DBLP:conf\/acl\/YangZD17}&-&-&95.7\\\\\n\\citet{DBLP:journals\/corr\/abs-1808-06511}&-&- & 96.2\\\\\n\\citet{DBLP:journals\/corr\/abs-1903-04190}&-&- &96.6\\\\\nBERT& 96.7 &96.4 &96.5 \\\\\nGlyce+BERT& 96.6 &96.8 & 96.7 \\\\\n&&&{\\bf(+0.2)}\\\\\\hline\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\n\n\n\\begin{table}[!ht]\n\\small\n\\caption{Results for POS tasks.}\n\\label{POS}\n\\begin{multicols}{2}\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CTB5}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&93.68&94.47&94.07\\\\\n\\citet{shao2017character} (Ens)&93.95&94.81&94.38\\\\\nLattice-LSTM  & 94.77& 95.51& 95.14 \\\\\nGlyce+Lattice-LSTM&95.49&95.72& 95.61\\\\\n&&&{\\bf (+0.47)} \\\\\\hline\nBERT& 95.86 & 96.26 & 96.06 \\\\\nGlyce+BERT &  96.50 & 96.74 & 96.61 \\\\\n&&& {\\bf(+0.55) }\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{CTB6}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&-&-&90.81\\\\\nLattice-LSTM  & 92.00& 90.86&91.43  \\\\\nGlyce+Lattice-LSTM& 92.72& 91.14&  91.92\\\\\n&&&{\\bf (+0.49)} \\\\\\hline\nBERT& 94.91 & 94.63 & 94.77 \\\\\nGlyce+BERT &  95.56 & 95.26 & 95.41\\\\\n&&&{\\bf(+0.64)}\\\\\\hline\\hline\n\\end{tabular}\n\n\n\\begin{tabular}{|llll|}\\hline\n\\multicolumn{4}{|c|}{CTB9}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&91.81&94.47&91.89\\\\\n\\citet{shao2017character} (Ens)&92.28&92.40&92.34\\\\\nLattice-LSTM  & 92.53& 91.73& 92.13 \\\\\nLattice-LSTM+Glyce&92.28 &92.85 &92.38 \\\\\n&&&{\\bf (+0.25)} \\\\\\hline\nBERT& 92.43 & 92.15 & 92.29 \\\\\nGlyce+BERT & 93.49 & 92.84 & 93.15 \\\\\n&&&{\\bf(+0.86)}\\\\\\hline\\hline\n\\multicolumn{4}{|c|}{UD1}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{shao2017character} (Sig)&89.28&89.54&89.41\\\\\n\\citet{shao2017character} (Ens)&89.67&89.86&89.75\\\\\nLattice-LSTM&90.47&89.70& 90.09\\\\\nLattice-LSTM+Glyce&91.57 & 90.19&90.87 \\\\\n&&&{\\bf (+0.78)} \\\\\\hline\nBERT& 95.42 & 94.17 & 94.79 \\\\\nGlyce+BERT & 96.19 & 96.10 & 96.14 \\\\\n&&&{\\bf (+1.35) }\\\\\\hline\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\\subsection{Sentence Pair Classification}\nFor sentence pair classification tasks, we need to output a label for each pair of sentence.\nWe employ the following four different datasets: (1) {\\bf BQ} (binary classification task) \\citep{DBLP:conf\/emnlp\/BowmanAPM15}; \n(2) {\\bf LCQMC} (binary classification task)  \\citep{liu2018lcqmc},\n(3) {\\bf XNLI} (three-class classification task) \\citep{williamsmulti}, and\n(4) {\\bf NLPCC-DBQA}  (binary classification task) \\footnote{\\url{https:\/\/github.com\/xxx0624\/QA_Model}}.\n\n\n\nThe current  non-BERT SOTA model is based on the  bilateral\nmulti-perspective matching model (BiMPM) \\citep{DBLP:conf\/ijcai\/WangHF17}, which specifically tackles the subunit matching between sentences. \nGlyph embeddings are incorporated into BiMPMs, forming the Glyce+BiMPM baseline. \nResults regarding each model on different datasets are given in  Table \\ref{pair}. \nAs can be seen, BiPMP+Glyce outperforms BiPMPs, achieving the best results among non-bert models. \nBERT outperforms all non-BERT models, and\nBERT+Glyce performs the best, setting new SOTA results on all of the four benchmarks. \n\n\n\\begin{table}[!ht]\n\\small\n\\caption{Results for sentence-pair classification tasks.}\n\\label{pair}\n\\begin{multicols}{2}\n\\begin{tabular}{|lllll|}\\hline\n\\multicolumn{5}{|c|}{BQ}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & 82.3 & 81.2 & 81.7 & 81.9  \\\\\nGlyce+BiMPM & 81.9 & 85.5 & 83.7 & 83.3  \\\\\n&&&{\\bf (+2.0)}&{\\bf (+1.4)}\\\\\\hline\nBERT & 83.5 & 85.7 & 84.6 & 84.8\\\\\nGlyce+BERT & 84.2 & 86.9 & 85.5 & 85.8\\\\\n&&&{\\bf (+0.9)}&{\\bf(+1.0)}\\\\\\hline\\hline\n\\multicolumn{5}{|c|}{XNLI}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & - & - & - &67.5 \\\\\nGlyce+BiMPM & - & - & - & 67.7 \\\\\n&&&&{\\bf(+0.2)}\\\\\\hline\nBERT & - & - & - & 78.4\\\\\nGlyce+BERT & - & - & - & 79.2\\\\\n&&&&{\\bf(+0.8)}\\\\\\hline\n\\end{tabular}\n\n\\begin{tabular}{|lllll|}\\hline\n\\multicolumn{5}{|c|}{LCQMC}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM & 77.6 & 93.9 & 85.0 & 83.4  \\\\\nGlyce+BiMPM & 80.4 & 93.4 & 86.4 & 85.3  \\\\\n&&&{\\bf (+1.4)}&{\\bf (+1.9)} \\\\\\hline\nBERT & 83.2 & 94.2 & 88.2 &87.5 \\\\\nGlyce+BERT & 86.8 & 91.2 & 88.8 & 88.7\\\\\n&&&{\\bf(+0.6)}&{\\bf(+1.2)}\\\\\\hline\\hline\n\\multicolumn{5}{|c|}{NLPCC-DBQA}\\\\\\hline\nModel & P & R & F & A \\\\\\hline\nBiMPM &78.8&56.5&65.8&-\\\\\nGlyce+BiMPM & 76.3& 59.9&67.1&-  \\\\\n&&&{\\bf(+1.3)}&-\\\\\\hline\nBERT&  79.6&86.0&82.7&-\\\\\nGlyce+BERT &81.1&85.8&83.4&-\\\\\n&&&{\\bf(+0.7)}&-\\\\\\hline\n\\end{tabular}\n\\end{multicols}\n\\end{table}\n\n\n\\begin{table*}[!ht]\n\\small\n\\center\n\\caption{Accuracies for Single Sentence Classification task. }\n\\label{single}\n\\begin{tabular}{|cccc|}\\hline\nModel & ChnSentiCorp & the Fudan corpus & iFeng \\\\\\hline\n LSTM & 91.7& 95.8&84.9\\\\\nLSTM + Glyce & 93.1 & 96.3 & 85.8  \\\\\n& {\\bf (+ 1.4) } & {\\bf (+0.5)} & {\\bf (+0.9)} \\\\\\hline\nBERT & 95.4 & 99.5 & 87.1   \\\\\nGlyce+BERT & 95.9 & 99.8 & 87.5   \\\\\n&{\\bf(+0.5)}&{\\bf(+0.3)}&{\\bf(+0.4)}\\\\\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Single Sentence Classification}\nFor single sentence\/document  classification, we need to output a label for a text sequence. The label could be either a sentiment indicator or a news genre. Datasets that we use include: (1) ChnSentiCorp (binary classification); (2) the Fudan corpus (5-class classification) \\citep{fudan}; and \n(3) Ifeng (5-class classification). \n\n\nResults for different models on different tasks are shown in Table \\ref{single}. \nWe observe similar phenomenon as before:  \nGlyce+BERT achieves SOTA results on all of the datasets. \nSpecifically, the Glyce+BERT model achieves an almost perfect accuracy (99.8) on the the Fudan corpus. \n\n\\subsection{Dependency Parsing and Semantic Role Labeling}\nFor dependency parsing \\citep{chen2014fast,dyer2015transition}, we used the\nwidely-used\n Chinese Penn Treebank 5.1 dataset for evaluation. Our implementation uses the previous state-of-the-art Deep Biaffine \n model \\citet{dozat2016deep} as a backbone. \nWe\n replaced the word vectors from the biaffine model with Glyce-word embeddings, and exactly followed its model structure and  training\/dev\/test split criteria. \n We report scores for\nunlabeled attachment score (UAS)\nand\nlabeled attachment score (LAS).\nResults\nfor previous models\n are copied from \\citep{dozat2016deep,ballesteros2016training,cheng2016bi}.\nGlyce-word pushes  SOTA performances up by +0.9 and +0.8 in terms of UAS and LAS scores.\n\n\\begin{table}\n\\small\n\\begin{tabular}{|lcc|}\\hline\n\\multicolumn{3}{|c|}{Dependency Parsing}\\\\\\hline\nModel & UAS & LAS \\\\\\hline\n\\citet{ballesteros2016training}& 87.7 &86.2\\\\\n\\citet{kiperwasser2016simple}&87.6& 86.1\\\\\n\\citet{cheng2016bi}& 88.1 &85.7\\\\\n Biaffine & 89.3 & 88.2 \\\\\\hline\n \\multirow{ 2}{*}{Biaffine+Glyce}&  90.2&89.0\\\\\n & {\\bf (+0.9)}&{\\bf (+0.8)}\\\\\\hline\n\\end{tabular}\n\\begin{tabular}{|cccc|}\\hline\n\\multicolumn{4}{|c|}{Semantic Role Labeling}\\\\\\hline\nModel & P & R & F \\\\\\hline\n\\citet{roth2016neural}& 76.9 &73.8 &75.3\\\\\n\\citet{marcheggiani2017encoding}&84.6 &80.4 &82.5\\\\ \n\\citet{he2018syntax} &      84.2 & 81.5&  82.8 \\\\\\hline\n \\multirow{ 2}{*}{k-order pruning+Glyce}& 85.4&82.1 & 83.7  \\\\\n &{\\bf (+0.8)}&{\\bf (+0.6)}&{\\bf (+0.9)}\\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFor \nthe task of\nsemantic role labeling (SRL) \\citep{roth2016neural,marcheggiani2017encoding,he2018syntax}, we used the CoNLL-2009 shared-task. \nWe used the current SOTA model, the k-order \npruning algorithm \\citep{he2018syntax} as a backbone.\\footnote{Code open sourced at \\url{https:\/\/github.com\/bcmi220\/srl_syn_pruning}}\nWe replaced word embeddings with Glyce embeddings.\nGlyce outperforms the previous SOTA performance by  0.9 with respect to the F1 score, achieving a new SOTA score of  83.7.\n\nBERT does not perform competitively in these two tasks, and results are thus omitted. \n\n\n\\begin{comment}\n\\section{Ablation Studies}\nIn this section, we discuss the influence of different factors of the proposed model. \nWe use the LCQMC dataset of the sentence-pair prediction task for illustration. \nFactors that we discuss include training strategy,  model architecture, auxiliary image-classification objective, etc. \n\\subsection{Training Strategy}\nSection \\ref{training-tactic} talks about a training tactic (denoted by BERT-first-glyce-second-joint), in which given task-specific supervisions,  we first fine-tune the BERT model, then freeze BERT to fine-tune the glyph layer, and finally jointly tune both layers until convergence. \nWe compare this strategy with other tactics, including (1)\nthe {\\it Glyph-first-Joint} strategy, in which BERT is not fine-tuned in the beginning: we first freeze BERT to tune the glyph layer, and then jointly tune \nboth layers until convergence; and (2) the\n{\\it joint-first} strategy, in which we directly jointly training two models until converge. \n\\begin{table}[!ht]\n\\center\n\\caption{Impact of different training strategy.}\n\\label{training-tactic}\n\\begin{tabular}{|lllll|}\\hline\nStrategy&P & R & F & A\\\\\\hline\nBiMPM & 77.6 & 93.9 & 85.0 & 83.4  \\\\\nGlyce+BiMPM & 80.4 & 93.4 & 86.4 & 85.3  \\\\\nonly BERT  & 83.2 & 94.2 & 88.2 & 87.5\\\\\\hline\nGlyph-first-Joint & 82.5 & 94.0 & 87.9 & 87.1 (-0.4) \\\\\njoint-first  & 81.5 & 95.1 & 87.8 & 86.8 (-0.7) \\\\\nBERT-first-glyce-second-joint & 86.8 & 91.2 & 88.8 & 88.7  (+1.2)\\\\\\hline\n\\end{tabular}\n\\end{table}\n\nResults are shown in Table \\ref{training-tactic}. \nIt shows the significant importance of the order by which different layers are trained. \nThe Glyph-first-Joint and joint-first models not only underperform the proposed tactic, but also the vanilla version of the BERT model. \nThe is due to the fact that\nthe BERT layer is pretrained but the glyph layer is randomly initialized. \nGiven the relatively small amount of training signals, the BERT layer could be mislead by the randomly initialized glyph layer at the early stage of training, leading to inferior final performances.  \n\n\n\n\\subsection{Structures of the task-specific output layer}\nThe concatenation of the glyph embedding and the BERT embedding is fed to the task-specific output layer. The task-specific output layer is made up with two layers of transformers. Here we change transformers to other structures such as BiLSTMs and CNNs\n to explore the influence.\nWe also try the  BiMPM structure \\citep{DBLP:conf\/ijcai\/WangHF17} to see the result. \n\n\\begin{table}[!ht]\n\\center\n\\caption{Impact of structures for the task-specific output layer.}\n\\label{structure}\n\\begin{tabular}{lllll}\\hline\nStrategy &P & R & F & A\\\\\\hline\nTransformers & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nBiLSMTs & 81.8 & 94.9 & 87.9 & 86.9\\\\\nCNNs & 81.5 & 94.8 & 87.6 & 86.6\\\\\nBiMPM & 81.1 & 94.6 & 87.3 & 86.2\\\\\\hline\n\\end{tabular}\n\\end{table}\n\nPerformances are shown in Table \\ref{structure}. \nAs can be seen, transformers not only outperform BiLSTMs and CNNs, but also the BiMPM  structure, which is specifically built for the sentence pair classification task. \nWe conjecture that this is because of the consistency between transformers and BERT. \n\\subsection{The image-classification training objective}\nWe also explore the influence of the  image-classification training objective, which outputs the glyph representation to an image-classification objective. \nTable \\ref{image} represents its influence. As can be seen, this auxiliary training objective given a +0.8 performance boost. \n\n\\begin{table}[!ht]\n\\center\n\\caption{Impact of the auxilliary image-classification training objective.}\n\\label{image}\n\\begin{tabular}{lllll}\\hline\nStrategy&P & R & F & A\\\\\\hline\nW image-cls  & 86.8 & 91.2 & 88.8 & 88.7\\\\\nWO image-cls & 83.9 & 93.6 & 88.4 & 87.9\\\\\\hline\n\\end{tabular}\n\\end{table}\n\n\\end{comment}\n\n\\section{Ablation Studies}\nIn this section, we discuss the influence of different factors of the proposed model. \nWe use the LCQMC dataset of the sentence-pair prediction task for illustration. \nFactors that we discuss include training strategy,  model architecture, auxiliary image-classification objective, etc. \n\\subsection{Training Strategy}\nThis section talks about a training tactic (denoted by BERT-glyce-joint), in which given task-specific supervisions,  we first fine-tune the BERT model, then freeze BERT to fine-tune the glyph layer, and finally jointly tune both layers until convergence. \nWe compare this strategy with other tactics, including (1)\nthe {\\it Glyph-Joint} strategy, in which BERT is not fine-tuned in the beginning: we first freeze BERT to tune the glyph layer, and then jointly tune \nboth layers until convergence; and (2) the\n{\\it joint} strategy, in which we directly jointly training two models until converge. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy&Precision & Recall & F1 & Accuracy\\\\\\hline\nBERT-glyce-joint & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nGlyph-Joint & 82.5 & 94.0 & 87.9 & 87.1\\\\\njoint  & 81.5 & 95.1 & 87.8 & 86.8 \\\\\nonly BERT  & 83.2 & 94.2 & 88.2 & 87.5\\\\\\hline\n\\end{tabular}\n\\caption{Impact of different training strategy.}\n\\label{training-tactic}\n\\end{table}\n\nResults are shown in Table \\ref{training-tactic}. As can be seen, the BERT-glyce-joint outperforms the rest two strategies. \nOur explanation for the inferior performance of the {\\it joint} strategy is as follows: \nthe BERT layer is pretrained but the glyph layer is randomly initialized. \nGiven the relatively small amount of training signals, the BERT layer could be mislead by the randomly initialized glyph layer at the early stage of training, leading to inferior final performances.  \n\n\\subsection{The image-classification training objective}\nWe also explore the influence of the  image-classification training objective, which outputs the glyph representation to an image-classification objective. \nTable \\ref{image} represents its influence. As can be seen, this auxiliary training objective given a {+0.8} performance boost. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy&Precision & Recall & F1 & Accuracy\\\\\\hline\nW image-cls  & 86.8 & 91.2 & 88.8 & 88.7\\\\\nWO image-cls & 83.9 & 93.6 & 88.4 & 87.9\\\\\\hline\n\\end{tabular}\n\\caption{Impact of the auxilliary image-classification training objective.}\n\\label{image}\n\\end{table}\n\n\\subsection{Structures of the task-specific output layer}\nThe concatenation of the glyph embedding and the BERT embedding is fed to the task-specific output layer. The task-specific output layer is made up with two layers of transformers. Here we change transformers to other structures such as BiLSTMs and CNNs\n to explore the influence.\nWe also try the  BiMPM structure \\cite{DBLP:conf\/ijcai\/WangHF17} to see the result. \n\\begin{table}[!ht]\n\\center\n\\begin{tabular}{lllll}\\hline\nStrategy &Precision & Recall & F1 & Accuracy\\\\\\hline\nTransformers & 86.8 & 91.2 & 88.8 & 88.7 \\\\\nBiLSMTs & 81.8 & 94.9 & 87.9 & 86.9\\\\\nCNNs & 81.5 & 94.8 & 87.6 & 86.6\\\\\nBiMPM & 81.1 & 94.6 & 87.3 & 86.2\\\\\\hline\n\\end{tabular}\n\\caption{Impact of structures for the task-specific output layer.}\n\\label{structure}\n\\end{table}\n\nPerformances are shown in Table \\ref{structure}. \nAs can be seen, transformers not only outperform BiLSTMs and CNNs, but also the BiMPM  structure, which is specifically built for the sentence pair classification task. \nWe conjecture that this is because of the consistency between transformers and the BERT structure. \n\n\\subsection{CNN structures} \nResults for different CNN structures are shown  in Table \\ref{CNN}. As can be seen, the adoption of tianzige-CNN structure introduces a performance boost of F1 about +1.0. Directly using deep CNNs in our task results in very poor performances because of (1) relatively smaller size of the character images: the size of ImageNet images is usually  at the scale of 800*600, while the size of Chinese character images is significantly smaller, usually at the scale of 12*12; and (2) the lack of training examples: classifications on the ImageNet dataset utilizes tens of millions of different images. In contrast, there are only about 10,000 distinct Chinese characters. We utilize the Tianzige-CNN  (\u7530\u5b57\u683c) structures tailored to logographic character modeling for Chinese. This tianzige structure is of significant importance in extracting character meanings. \n\n\\begin{table}[!ht]\n\\small\n\\centering\n\\begin{tabular}{llll}\\hline\n &Precision & Recall & F1 \\\\\\hline\nVanilla-CNN & 85.3 & 89.8 & 87.4 \\\\\n\\citet{he2016deep}& 84.5&90.8& 87.5  \\\\\nTianzige-CNN & 86.8 & 91.2 & 88.8  \\\\\\hline\n\\end{tabular}\n\\caption{Impact of CNN structures.}\n\\label{CNN}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we propose Glyce, Glyph-vectors for Chinese Character Representations. Glyce treats Chinese characters as images and uses Tianzige-CNN to extract character semantics.\nGlyce provides a general way to model character semantics of logographic languages. It is general and fundamental. Just like word embeddings,  Glyce can be integrated to any existing deep learning system. \n\\end{CJK*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplementary Information}\n\n\n\\subsection{Theory}\n\nIn Eq.\\ (4) of the main text, we give the result of $\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  =  \\int_0^{2\\pi} d\\theta \\, \\mathbf{\\Omega}_\\mathrm{SO} (\\mathbf{k} + \\mathbf{k}_\\mathrm{dr}) \\, f(\\theta)$ for the special case that drift occurs along the $y$-direction ($k_{\\mathrm{dr},x} = 0$) and detection at $x = 0$ ($k_{\\mathrm{di},x} = 0$). \nHere, we provide the result for a general case:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  = \\langle \\mathbf{\\Omega}_\\mathrm{SO,1} \\rangle  + \\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle \\, ,\n\\end{equation}\n\nwith a term proportional to $\\beta_1$:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,1} \\rangle  =  \\frac{2 \\beta_1}{\\hbar} \n\\begin{pmatrix} \nk_{\\mathrm{di},y} + k_{\\mathrm{dr},y} \\\\\nk_{\\mathrm{di},x} + k_{\\mathrm{dr},x}\n\\end{pmatrix} \\, ,\n\\end{equation}\n\nand one proportional to $\\beta_3$:\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle  =  \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \n- k_{\\mathrm{di},y} -2 k_{\\mathrm{dr},y} + \\frac{2}{k_\\mathrm{F}^2} \\left[  k_{\\mathrm{di},y} k_{\\mathrm{dr},x}^2 + 2 k_{\\mathrm{di},x} k_{\\mathrm{dr},x}  k_{\\mathrm{dr},y} +  k_{\\mathrm{dr},x}^2 k_{\\mathrm{dr},y} - 3 k_{\\mathrm{di},y} k_{\\mathrm{dr},y}^2 -  k_{\\mathrm{dr},y}^3 \\right] \\\\\n- k_{\\mathrm{di},x} -2 k_{\\mathrm{dr},x} + \\frac{2}{k_\\mathrm{F}^2} \\left[  k_{\\mathrm{di},x} k_{\\mathrm{dr},y}^2 + 2 k_{\\mathrm{di},y} k_{\\mathrm{dr},y}  k_{\\mathrm{dr},x} +  k_{\\mathrm{dr},y}^2 k_{\\mathrm{dr},x} - 3 k_{\\mathrm{di},x} k_{\\mathrm{dr},x}^2 -  k_{\\mathrm{dr},x}^3 \\right]\n\\end{pmatrix}  \\, .\n\\end{equation}\n\nFor simplicity, we assumed $\\alpha = 0$ in the above expressions. \nWe now move to the special case where the electric field is applied along the $y$ direction, such that $k_{\\mathrm{dr},x} = 0$, and obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO,3} \\rangle  = - \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \nk_{\\mathrm{di},y} \\left[1 + 6\\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] + 2 k_{\\mathrm{dr},y} \\left[1 + \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] \\\\\nk_{\\mathrm{di},x} \\left[ 1 - \\left(\\frac{2  k_{\\mathrm{dr},y}}{k_\\mathrm{F}} \\right)^2 \\right]\n\\end{pmatrix}  \\, .\n\\end{equation}\n\n\nTo describe a measurement where spins are tracked along the drift direction $y$ and at $x = 0$, we additionally set $k_{\\mathrm{di},x} = 0$ and obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle  =   \\frac{2 \\beta_1}{\\hbar} \n\\begin{pmatrix} \n k_{\\mathrm{di},y} + k_{\\mathrm{dr},y} \\\\\n0\n\\end{pmatrix} \n- \\frac{2 \\beta_3}{\\hbar} \n\\begin{pmatrix} \n k_{\\mathrm{di},y} \\left[1 + 6\\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] + 2 k_{\\mathrm{dr},y} \\left[1 + \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right]  \\\\\n0\n\\end{pmatrix}  \\, .\n\\end{equation}\n\nTranslating into a $(y, t)$ coordinate system via $y = \\frac{\\hbar}{m^*}t (k_{\\mathrm{dr},y} + k_{\\mathrm{di},y})$, we obtain\n\n\\begin{equation}\n\\langle \\mathbf{\\Omega}_\\mathrm{SO} \\rangle   t = q y + \\omega t \\, ,\n\\end{equation}\n\n\\noindent with\n\n\\begin{equation}\nq = \\frac{2 m^*}{\\hbar^2} \\left(  \\beta_1 - \\beta_3 \\left[  1 + 6 \\left(\\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}}\\right)^2 \\right] \\right)\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n\\omega = -\\frac{2 m^*}{\\hbar^2} v_\\mathrm{dr} \\beta_3 \\left[ 1 - 4 \\left( \\frac{k_{\\mathrm{dr},y}}{k_\\mathrm{F}} \\right)^2  \\right] \\, .\n\\end{equation}\n\nThe higher-order terms are not observed in the experiment and therefore not discussed in the main text. However, they might be observable for larger drift vectors or smaller electron sheet densities.\n\n\n\n\n\n\\subsection{Simulations}\n\n\n\\begin{figure}\n\\includegraphics{SI_fig1.pdf}\n\\caption{Validity of the model. (a) to (c) Simulation data of $S_z(y,t)$ for $\\alpha = 0$~eVm, $\\alpha = 1.5 \\times 10^{-13}$~eVm and $\\alpha = 3 \\times 10^{-13}$~eVm, respectively. In all cases, $\\beta^* = 3 \\times 10^{-13}$~eVm, $\\beta_3 = 0.7 \\times 10^{-13}$~eVm and $v_\\mathrm{dr} = 38$~km\/s (violet dashed line). We find good agreement between the simulation and the model (green solid lines) for the entire parameter range.\n\\label{SI_fig1} }\n\\end{figure}\n\nIn the main text, we discuss the validity of the model for cases away from the PSH symmetry, i.e., away from $\\alpha = \\beta^*$, by comparing the model with spin-precession maps obtained from numerical Monte-Carlo simulations. We state that, as long as drift occurs along $y$, we obtain good agreement between simulation and model. \nIn Fig.\\ \\ref{SI_fig1}, we show the corresponding simulations for three different cases between $\\alpha \/\\beta^* = 0$ (isotropic) and $\\alpha \/\\beta^* = 1$ (PSH). The model of Eqs.\\ (5) and (6) of the main text (green solid lines) correctly predicts the simulated spin dynamics for the entire parameter range for drift along $y$.\n\n\n\n\n\\subsection{Fitting}\n\nEquation (1) in the main text contains six independent fit parameters. Suitable starting values for the fitting are obtained in the following way. For the amplitude $A_0$ we choose the value of $S_z (t=0, y=0)$. The drift velocity, $v_\\mathrm{dr}$, is defined by the shift of the spin packet in time and its starting value is estimated manually. The spin diffusion constant, $D_\\mathrm{s}$, is determined by the broadening of the Gaussian envelope function and we start with a typical value for samples from the same wafer. For the dephasing time, $\\tau$, we use 1~ns as a starting value. The most important parameters for the presented study are $\\omega$, the temporal precession frequency, and $q$, the spatial wavenumber. Both quantities are little affected by the other fit parameters. Starting values for both of them are obtained from a line-cut through the data at a fixed time (a fixed position) for $q$ (for $\\omega$). Before calculating the mean-squared error between Eq. (1) and the measured $S_z (y, t)$, we perform a one-dimensional convolution of Eq. (1) with the Gaussian intensity profiles of the pump and probe laser spots along $y$. This step is very important, because its neglect distorts particularly the value of $\\omega$. \n\n\nAll fit parameters are then constrained to a reasonable range. To determine each parameter's fit value and confidence interval, we vary that parameter in small steps through its full range. At each step, all other parameters are optimized to minimize the mean-squared error between the data and Eq. (1) by a Nelder-Mead simplex search algorithm. The value of the parameter with the smallest error defines the fit value. For all fit parameters, we find a single minimum. The confidence interval, as shown in Fig. 2 in the main text, is then defined by an increase of the mean-squared error by 5~\\% from its minimal value. The mean-squared error is taken over approximately 3000 data points (typically 35 steps of $t$, 85 steps of $y$ or $x$). \n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:I}\nThe behavior of charged quantum particles inside a uniform magnetic field in \\linebreak \\mbox{Minkowski spacetime} displays quantized energy levels ---\\,the so-called Landau levels\\,--- that were first derived by Rabi based on the relativistic Dirac equation \\cite{Rabi} and by Landau based on the Schr\\\"odinger equation \\cite{Landau} (see e.g., Ref.\\,\\cite{LandauLifshitz} for a textbook introduction). This discovery led to many physical applications in condensed matter physics, such as the explanation of the Landau diamagnetism \\cite{Landau}, of the Shubnikov\u2013de Haas effect \\cite{Hadju}, of the de Haas-van Alphen effect \\cite{Shoenberg} and of the integer quantum Hall effect \\cite{QHEBook}. Remarkably, the Landau levels have even found astrophysical applications. In fact, the physics of neutron stars and other highly magnetized stellar objects is expected to be governed to some extent by the discrete nature of the energy levels occupied by the charged fermions moving under the influence of the intense magnetic field of such astrophysical objects \\cite{BookNeutronStar,WhiteDwarfs,WhiteDwarfs3,Mass-Radius,WhiteDwarfs4,Broderick,Chamel1,Chamel2,Yakovlev, NeutronStarMatter}. When we recall that such highly compact astrophysical objects are also sources of very intense gravitational fields, it becomes extremely important to investigate the fate of Landau levels under the influence of a combination of a magnetic field and the gravitational field of a spherical object. Such a study has recently been conducted in Refs.\\,\\cite{GravityLandauI,GravityLandauII,QHGravity} based on the Klein-Gordon and Schr\\\"odinger equations by considering the gravitational interaction of the particle with a massive sphere as a small perturbation compared to the interaction of the particle with the magnetic field. Although relying on the Schr\\\"odinger equation and treating gravity perturbatively as done in these studies might be sufficient for laboratory \\mbox{applications \\cite{Landry1,Landry2,COWBall,Josephson},} when seeking astrophysical applications, implementing a general-relativistic and a non-perturbative approach and appealing to the relativistic Dirac equation are highly recommended.  \n \nThe Dirac equation in a curved spacetime caused {\\it purely} by an intense magnetic field--known as the Melvin universe \\cite{Melvin}---has been solved exactly in Ref.\\,\\cite{Nunes} where the quantized energy levels were extracted. Contemplating curved spacetimes caused purely by a magnetic field is motivated by the well-known fact that the strong magnetic fields of neutron stars could reach up to  {$10^{12-15}$}\\,G  \n on the surface of magnetars \\cite{Magnetars}, and up to $10^{17}\\,$G in their interior (see, e.g., Refs.\\,\\cite{InsideNS1,InsideNS2} and the references therein). On the other hand, neutron stars are made of neutrons, as well as of a plasma of electrons, protons and muons, the energies of which are affected by such strong magnetic fields. The aim of the present paper is therefore to include the general-relativistic effect of the intense gravitational field of such stellar objects by studying the dynamics of charged fermions moving {\\it inside} a massive spherical object in the presence of a uniform magnetic field. To the best of our knowledge, such a study has not appeared in the literature before. Unlike the purely magnetic spacetime considered in Ref.\\,\\cite{Nunes}, however, we consider in the present work the curved spacetime caused purely by the distribution of matter inside the core of the stellar objects rather than by the uniform magnetic field. The contribution of the latter to spacetime curvature will thus be considered negligible. The justification behind such an assumption will be discussed in Section \\ref{sec:II}. \n\nThe remainder of this paper is structured as follows. In Section \\ref{sec:II}, we derive the Dirac equation in a general static and spherically symmetric curved spacetime in the presence of a uniform magnetic field. We discuss the various symmetries of the equation and the complications one faces when attempting to solve the equation exactly. In \\mbox{Section \\ref{sec:III}}, we apply the equation obtained in Section \\ref{sec:II} to the case of the interior Schwarzschild solution. We then extract the quantized energy levels of charged fermions moving along the equator inside a massive spherical object by keeping only the leading terms of the differential equation. In Section \\ref{sec:IV}, we briefly discuss the ways to overcome the limitations we encountered in solving our equation in Section \\ref{sec:III}. In Section \\ref{sec:V}, we apply our results to the study of the magnetization of neutron stars. We conclude this paper with a short discussion section in which we summarize our findings. \n\n\n\n\\section{Curved-Spacetime Dirac Equation in a Uniform Magnetic Field}\\label{sec:II}\nSince our aim in this paper is to consider the Dirac equation in the static and spherically symmetric interior Schwarzschild solution, let us start by deriving in this section the equation for a general static and spherically symmetric curved spacetime in the presence of a uniform magnetic field. For that purpose, let us write our spacetime metric in the following general form:\n\\begin{equation}\\label{GeneralMetric}\n    {\\rm d}s^2=-\\mathcal{A}^2(r)\\,c^2{\\rm d}t^2+\\mathcal{B}^2(r)\\,{\\rm d}r^2+r^2({\\rm d}\\theta^2+\\sin^2\\theta{\\rm d}\\phi^2),\n\\end{equation}\nwhere the radial functions $\\mathcal{A}(r)$ and $\\mathcal{B}(r)$ are everywhere regular except, maybe, at the origin $r=0$.  In addition, let us assume the constant and uniform magnetic field $\\bf B$\nto be parallel to the $z$-direction. A convenient gauge for the vector potential would then be $A_{\\mu}=(0,0,0,\\frac{1}{2}{Br^2\\sin^2\\theta})$. The Dirac equation of a particle of mass $m$ and of charge $e$ (that we take to be the negative charge of the electron for definiteness) moving inside a curved spacetime and minimally coupled to a Maxwell field $A_\\mu$ reads \\cite{Pollock,BookDiracInCS},\n\\begin{equation}\\label{GeneralDirac}\ni\\hbar e^\\mu_a\\gamma^a\\left(\\partial_\\mu+\\frac{1}{8}\\omega_\\mu^{\\;bc}\\,[\\gamma_b,\\gamma_c]-\\frac{ieA_\\mu}{\\hbar }\\right)\\psi=mc\\psi.\n\\end{equation}\nThe quantities $e^\\mu_a$ are the inverse of the spacetime tetrads $e^a_\\mu$, whereas $\\gamma^a$ are the gamma matrices, $\\omega_\\mu^{\\,bc}$ is the spin connection and $[\\gamma_b,\\gamma_c]$ is the commutator of the gamma matrices. The spin connection is antisymmetric, $\\omega_\\mu^{\\;bc}=-\\omega_\\mu^{\\;cb}$, and it is expressed in terms of the tetrad fields and the Christoffel symbols $\\Gamma_{\\mu\\nu}^\\lambda$ of the spacetime by $\\omega_\\mu^{\\;bc}=e^b_\\nu\\partial_\\mu e^{c\\nu}+\\Gamma_{\\mu\\nu}^\\lambda e^b_\\lambda e^{c\\nu}$. \n\n\n\\newpage\nOn the other hand, two choices for the four curved-spacetime gamma matrices $\\gamma^\\mu=e^a_\\mu\\gamma^a$ are possible. One may choose either to work with the tetrads' axes parallel to the $r$, $\\theta$ and $\\phi$ coordinates, or choose to orient the tetrads' axes parallel to some rectangular coordinate system \\cite{BrillWheeler}. When combined with the standard representation of the constant flat-spacetime gamma matrices $\\gamma^a$, given explicitly by\n\\begin{equation}\\label{DiracMatrices}\n\\gamma^0=\n\\begin{pmatrix}\n\\mathbbm{1} & 0\\\\\n0 & -\\mathbbm{1}\n\\end{pmatrix},\\qquad\n\\gamma^k=\n\\begin{pmatrix}\n0 & \\sigma_k\\\\\n-\\sigma_k & 0\n\\end{pmatrix},\n\\end{equation}\nwhere $\\mathbbm{1}$ is the $2\\times2$ unit matrix and $\\sigma_k$ (for $k=1,2,3$) are the three Pauli matrices, the curved-space gamma matrices $\\gamma^\\mu$ take on a simpler form in the first case than in the second \\cite{BrillWheeler}. The spinor wavefunction $\\psi$ in the two cases will be different, but physical quantities as well as the radial wave equation will all be the same as the two representations are simply related by a unitary transformation \\cite{BrillWheeler}. However, to deal with the mixed cylindrical and spherical symmetries imposed, respectively, by the magnetic and gravitational fields, it is easier to adopt the spherical-coordinates representation of the gamma matrices, in which the three Pauli matrices take the form\n\\begin{equation}\\label{PauliM}\n\\sigma_r=\n\\begin{pmatrix}\n\\cos\\theta & \\sin\\theta e^{-i\\phi}\\\\\n\\sin\\theta e^{i\\phi} & -\\cos\\theta\n\\end{pmatrix},\\qquad\n\\sigma_\\theta=\n\\begin{pmatrix}\n-\\sin\\theta & \\cos\\theta e^{-i\\phi}\\\\\n\\cos\\theta e^{i\\phi} & \\sin\\theta\n\\end{pmatrix},\\qquad \n\\sigma_\\phi=\n\\begin{pmatrix}\n0 & -ie^{-i\\phi}\\\\\nie^{i\\phi} & 0\n\\end{pmatrix}.\n\\end{equation}\nWe now choose the spacetime tetrads for the metric (\\ref{GeneralMetric}) to be $e^a_\\mu={\\rm diag}(\\mathcal{A},\\mathcal{B},r,r\\sin\\theta)$. With these tetrads, we easily evaluate the nonvanishing components of the contracted spin connection with the commutator of the gamma matrices to be:\n\\begin{align}\n    \\tfrac{1}{8}\\omega_0^{ab}[\\gamma_a,\\gamma_b]&=\\frac{\\mathcal{A}'}{2\\mathcal{B}}\\gamma^0\\gamma^1,\\qquad \\tfrac{1}{8}\\omega_2^{ab}[\\gamma_a,\\gamma_b]=\\frac{1}{2\\mathcal{B}}\\gamma^1\\gamma^2,\\nonumber\\\\ \\tfrac{1}{8}\\omega_3^{ab}[\\gamma_a,\\gamma_b]&=\\frac{\\sin\\theta}{2\\mathcal{B}}\\gamma^1\\gamma^3+\\frac{\\cos\\theta}{2}\\gamma^2\\gamma^3.\n\\end{align}\nPlugging these, together with the vector potential $A_\\mu$, into Eq.\\,(\\ref{GeneralDirac}) and then multiplying the whole equation on the left by $\\beta=\\gamma^0$ and using the definition $\\alpha^k=\\beta\\gamma^k$ of the Dirac alpha matrices, the resulting equation reads,\n\\begin{equation}\\label{ExplicitGeneralDirac}\n      \\left[\\frac{i\\partial_t}{\\mathcal{A}c}\\psi+\\frac{i\\alpha^1}{\\mathcal{B}}\\left(\\partial_r+\\frac{1}{r}+\\frac{\\mathcal{A}'}{2\\mathcal{A}}\\right)+\\frac{i\\alpha^2}{r}\\left(\\partial_\\theta+\\frac{\\cot\\theta}{2}\\right)+\\frac{i\\alpha^3}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieA_\\phi}{\\hbar}\\right)\\right]\\psi=\\frac{mc}{\\hbar}\\beta\\psi.\n\\end{equation}\nWe are interested in this paper in stationary states of energy $E$, for which we have $\\partial_t\\psi=-\\tfrac{i}{\\hbar}E\\psi$. On the other hand, we may simplify further Eq.\\,(\\ref{ExplicitGeneralDirac}) by setting $\\psi=\\Psi\/(r\\sqrt{\\mathcal{A}\\sin\\theta})$. With these two replacements, the resulting equation in $\\Psi$ takes the form\n\\begin{equation}\\label{SimplifiedGeneralDirac}\n    \\Bigg[\\frac{E}{\\mathcal{A}\\hbar c}+\\frac{i\\alpha^1}{\\mathcal{B}}\\partial_r+\\frac{i\\alpha^2}{r}\\partial_\\theta+\\frac{i\\alpha^3}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)-\\frac{mc}{\\hbar}\\beta\\Bigg]\\Psi=0.\\\\\n\\end{equation}\n\nThis is the general equation obeyed by fermions under the influence of the uniform magnetic field. Unlike the well-known case of the Dirac equation in a spherically symmetric potential, i.e., in a central potential (see, e.g., Ref.\\,\\cite{Greiner}), our Eq.\\,(\\ref{SimplifiedGeneralDirac}) may not be separated in the radial variables $r$ and the angular variables $\\theta$ and $\\phi$. This is due to the mixture of the spherical symmetry imposed by the gravitational field and the cylindrical symmetry imposed by the vertical uniform magnetic field $\\bf B$.\nOne might then be tempted to switch to cylindrical coordinates $(\\rho,\\phi,z)$ to benefit from the cylindrical symmetry pertaining to the magnetic field. Unfortunately, even doing so would not make Eq.\\,(\\ref{SimplifiedGeneralDirac}) easier to solve. In fact, by making the substitutions $\\rho=r\\sin\\theta$ and $z=r\\cos\\theta$, Eq.\\,(\\ref{SimplifiedGeneralDirac}) would contain operators of the form $(\\rho^2+z^2)^{-\\frac{1}{2}}z\\,\\partial_\\rho$ and $(\\rho^2+z^2)^{-\\frac{1}{2}}\\rho\\,\\partial_z$. It is then evident that even if we neglect the $z$-variation of $\\Psi$ by assuming $\\partial_z\\Psi\\approx0$, i.e., by neglecting the vertical momentum of the particle, we would still be left with an equation that is not separable in the variables $\\rho$, $z$ and $\\phi$. \n\nIt seems then that the only choice left is to set $z=0$ as well. Physically, this is due (in the {\\it absence} of any other interaction of the particle) to the unstable nature of any position $z\\neq0$, caused by the transverse gravitational attraction that pulls the particle away from any plane other than the equator. This observation can also be reached by \nreferring to the variable $\\theta$ in Eq.\\,(\\ref{SimplifiedGeneralDirac}). In fact, the only way of removing the $\\theta$-dependence in that equation is to set $\\theta=\\frac{\\pi}{2}$, i.e., to restrict ourselves to particles moving along the equatorial plane. This is because setting $\\theta=\\theta_0\\neq\n\\frac{\\pi}{2}$ would prevent us from letting $\\rho=r\\sin\\theta_0$ vary and have at the same time planar motion. If $\\theta_0$ is set to a constant other than $\\frac{\\pi}{2}$, then for $\\rho$ to change, $r$ must change too, which makes the particle leave the plane $\\theta=\\theta_0$. Only along the equatorial plane $\\theta_0=\\frac{\\pi}{2}$ does $\\rho=r\\sin\\frac{\\pi}{2}=r$ become a variable describing \\mbox{planar motion}. \n\nWe will come back to the possibility of considering a plane other than the equator in Section \\ref{sec:IV}. Before setting $\\theta=\\frac{\\pi}{2}$, however, we should first extract the general single differential equation from the coupled Dirac spinors.   \n\n\n\\section{The Quantized Energy Levels}\\label{sec:III}\nTo extract a quantization condition, if any, on the energy $E$ from Eq.\\,(\\ref{SimplifiedGeneralDirac}), we need first to decompose the Dirac spinor into a pair of $2$-spinors as follows\n\\begin{equation}\n    \\Psi(r,\\theta,\\phi)=\n    \\begin{bmatrix}\n    \\Phi(r,\\theta,\\phi)\\\\\n    \\Theta(r,\\theta,\\phi)\n\\end{bmatrix}.\n\\end{equation}\nSubstituting this ansatz into Eq.\\,(\\ref{SimplifiedGeneralDirac}), the latter splits into the following coupled first-order differential equations:\n\\begin{align}\\label{SplitEqsSpinors}\n\\left(\\frac{E}{\\mathcal{A}\\hbar c}-\\frac{mc}{\\hbar}\\right)\\Phi+\\left[\\frac{i\\sigma_r}{\\mathcal{B}}\\partial_r+\\frac{i\\sigma_\\theta}{r}\\partial_\\theta+\\frac{i\\sigma_\\phi}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)\\right]\\Theta=0,\\nonumber\\\\\n\\left(\\frac{E}{\\mathcal{A}\\hbar c}+\\frac{mc}{\\hbar}\\right)\\Theta+\\left[\\frac{i\\sigma_r}{\\mathcal{B}}\\partial_r+\\frac{i\\sigma_\\theta}{r}\\partial_\\theta+\\frac{i\\sigma_\\phi}{r\\sin\\theta}\\left(\\partial_\\phi-\\frac{ieB}{2\\hbar }r^2\\sin^2\\theta\\right)\\right]\\Phi=0.\n\\end{align}\nIt is clear from these coupled differential equations that given the $\\rho$-dependence of all the coupling terms, extracting one of the 2-spinors from one equation and substituting it into the other would result in a very complicated single equation. This is unlike what happens with the standard Dirac equation for the case of the central potential of the hydrogen atom \\cite{Greiner}. It would be natural then at this stage to attempt to solve this system of equations by applying the approach developed in Refs.\\,\\cite{Alhaidari1,Alhaidari2,Alhaidari3}, which consists of applying a gauge transformation followed by a unitary transformation to turn one of the coupling terms in one of the two equations into a constant. Indeed, such a method considerably simplifies the resulting final single equation one arrives at. Unfortunately, such an approach cannot be consistently applied in our case because that approach is specifically designed only for a certain type of interactions of the particle with the gauge field, which is not the case for our present physical system\\footnote{FH is grateful to Prof. A.\\,D. Alhaidari for the helpful discussion on this subtle point.}. The other possibility would then be to attempt to rather adopt the approach developed in Ref.\\,\\cite{Alhaidari4} for the Dirac equation in curved spacetime. Unfortunately, that approach is not helpful to us either as it becomes rapidly involved even when the particle is not coupled to a gauge potential. For this reason, such an approach has been applied in Ref.\\,\\cite{Alhaidari4} only in $1+1$ dimensions. We are thus left only with the old strategy for dealing with coupled differential equations of the type (\\ref{SplitEqsSpinors}).\n\nExtracting the 2-spinor $\\Theta$ from the second line of Eq.\\,(\\ref{SplitEqsSpinors}) and plugging it into the first, we obtain the following single equation for the 2-spinor $\\Phi$:\n\\begin{align}\\label{2-SpinorEq}\n&\\frac{\\Phi_{,rr}}{\\mathcal{B}^2}+\\left(\\frac{2}{\\mathcal{B}r}-\\frac{\\mathcal{B}_{,r}}{\\mathcal{B}^3}+\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,r}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\theta\\theta}}{r^2}+\\left(\\frac{i\\left[\\mathcal{B}-1\\right]\\sigma_\\phi}{\\mathcal{B}r^2}+\\frac{\\cot\\theta}{r^2}+\\frac{i\\sigma_\\phi}{r}\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\theta}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\phi\\phi}}{r^2\\sin^2\\theta}\n+\\left(\\frac{i\\left[1-\\mathcal{B}\\right]\\sigma_\\theta}{\\mathcal{B}r^2\\sin\\theta}-\\frac{ieB}{\\hbar}-\\frac{i\\sigma_\\theta}{r\\sin\\theta}\\frac{\\mathcal{A}_{,r}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\phi}\\nonumber\\\\\n&+\\Bigg(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}-\\frac{e^2B^2r^2\\sin^2\\theta}{4\\hbar^2}-\\frac{\\left[\\mathcal{B}+1\\right]eB\\sigma_\\theta\\sin\\theta}{2\\hbar\\mathcal{B}}+\\frac{\\sigma_reB\\cos\\theta}{\\hbar}\\nonumber\\\\\n&\\qquad-\\frac{\\mathcal{A}_{,r}EeBr\\,\\sigma_\\theta\\sin\\theta}{2\\hbar\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\Bigg)\\Phi=0.\n\\end{align}\nWith this result, we can now proceed to restrict the motion of the particle to the equatorial plane by setting $\\theta=\\frac{\\pi}{2}$. Therefore, we also need to perform the replacement $r\\sin\\theta=r\\equiv\\rho$ everywhere in this equation. On the other hand, for the $\\theta$-derivatives, we use the following identities valid for motion restricted to a plane: $\\partial_\\theta=r\\cos\\theta\\partial_\\rho$ and $\\partial_\\theta^2=-r\\sin\\theta\\partial_\\rho+r^2\\cos^2\\theta\\partial_\\rho^2$. Therefore, for $\\theta=\\frac{\\pi}{2}$ we should discard the partial derivative $\\partial_\\theta$ and perform the replacement  $\\partial_\\theta^2\\rightarrow-\\rho\\,\\partial_\\rho$. The above equation then becomes\n\\begin{align}\\label{Plane2-SpinorEq}\n&\\frac{\\Phi_{,\\rho\\rho}}{\\mathcal{B}^2}+\\left(\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}-\\frac{\\mathcal{B}_{,\\rho}}{\\mathcal{B}^3}+\\frac{\\mathcal{A}_{,\\rho}E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\rho}\\nonumber\\\\\n&+\\frac{\\Phi_{,\\phi\\phi}}{\\rho^2}\n+\\left(\\frac{i\\left[\\mathcal{B}-1\\right]\\sigma_z}{\\mathcal{B}\\rho^2}-\\frac{ieB}{\\hbar}+\\frac{i\\sigma_z}{\\rho}\\frac{\\mathcal{A}_{,\\rho}E}{\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi_{,\\phi}\\nonumber\\\\\n&+\\left(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}+\\frac{\\left[\\mathcal{B}+1\\right]eB\\sigma_z}{2\\hbar\\mathcal{B}}+\\frac{\\mathcal{A}_{,\\rho}EeB\\rho\\,\\sigma_z}{2\\hbar\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\right)\\Phi=0.\n\\end{align}\nWe have used here the fact that according to Eq.\\,(\\ref{PauliM}), the spherical-coordinates Pauli matrix $\\sigma_\\theta$ reduces to $-\\sigma_z$ for $\\theta=\\frac{\\pi}{2}$ (as expected), where $\\sigma_z$ is the usual third Pauli matrix in Cartesian coordinates. \n\nOn the other hand, according to expressions (\\ref{PauliM}) of the Pauli matrices, we see in Eq.\\,(\\ref{Plane2-SpinorEq}) that all the terms are diagonal and that the equation does not involve the angular variable $\\phi$ explicitly. Therefore, without loss of generality, we can write the two independent solutions $\\Phi_+(\\rho,\\phi)$ and $\\Phi_-(\\rho,\\phi)$ of the equation in a basis made of the spin-eigenstates along the $z$-direction with eigenvalues $s=\\pm1$, respectively. We thus introduce arbitrary radial functions $f_+(\\rho)$ and $f_-(\\rho)$ such that\n\\begin{equation}\\label{Sigma3Basis}\n\\Phi_+(\\rho)=e^{i\\ell\\phi}\n\\begin{bmatrix}\nf_+(\\rho)\\\\0\n\\end{bmatrix},\\qquad \n\\Phi_-(\\rho)=e^{i\\ell\\phi}\n\\begin{bmatrix}\n0\\\\f_-(\\rho)\n\\end{bmatrix}.\n\\end{equation}\nImposing the periodic condition along the equator, $\\Phi(\\rho,\\phi+2\\pi)=\\Phi(\\rho,\\phi)$, we learn that the quantum number $\\ell$ must be an integer number, which we take here to be nonnegative in accordance with the negative sign we chose for the charge $e$. Plugging these ansatzes into Eq.\\,(\\ref{Plane2-SpinorEq}) yields the following second-order differential equation in $\\rho$:\n\\begin{align}\\label{fEq}\n&\\frac{f_s''}{\\mathcal{B}^2}+\\left(\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}-\\frac{\\mathcal{B}'}{\\mathcal{B}^3}+\\frac{\\mathcal{A}'E}{\\mathcal{A}\\mathcal{B}^2\\left[E+\\mathcal{A}mc^2\\right]}\\right)f'_s\\nonumber\\\\\n&+\\Bigg(\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\mathcal{B}\\ell^2+\\left[\\mathcal{B}-1\\right]s\\ell}{\\mathcal{B}\\rho^2}+\\frac{\\left[\\mathcal{B}+1\\right]eBs}{2\\hbar\\mathcal{B}}\\nonumber\\\\\n&\\qquad-\\frac{\\mathcal{A}'Es\\left[2\\hbar\\ell-eB\\rho^2\\right]}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}\\Bigg)f_s=0,\n\\end{align}\nHere, and henceforth, a prime denotes a derivative with respect to the variable $\\rho$, and the function $f_s(\\rho)$ stands for the two cases $f_\\pm(\\rho)$ of $s=\\pm1$, respectively. Next, we introduce the following ansatz: \\begin{equation}\\label{FirstAnsatz}\n    f_s(\\rho)=\\left(\\frac{\\mathcal{B}}{\\mathcal{A}}E+\\mathcal{B}mc^2\\right)^{\\frac{1}{2}}F_s(\\rho),\n\\end{equation}\nfor some radial function $F_s(\\rho)$. Please note that only the convergence of the radial function $F_s(\\rho)$ is required henceforth, for the argument of the square root in this ansatz converges for both large and small values of $\\rho$ as we shall see below after introducing the explicit components of the spacetime metric. Plugging now this ansatz into Eq.\\,(\\ref{fEq}), the latter takes the following form:\n\\begin{multline}\\label{SimplifiedfEq}\n\\!\\!\\!\\frac{F_s''}{\\mathcal{B}^2}+\\frac{2-\\mathcal{B}}{\\mathcal{B}\\rho}F'_s+\\bigg[\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\mathcal{B}\\ell^2+(\\mathcal{B}-1)s\\ell}{\\mathcal{B}\\rho^2}\\\\\n+\\frac{(\\mathcal{B}+1)eBs}{2\\hbar\\mathcal{B}}-\\frac{\\mathcal{A}'Es(2\\hbar\\ell-eB\\rho^2)}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}(E+\\mathcal{A}mc^2)}-\\frac{1}{2\\mathcal{B}^2}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}[E+\\mathcal{A}mc^2]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)'\\\\\n-\\left(\\frac{\\mathcal{A}'E}{2\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{2\\mathcal{B}^2}\\right)^2-\\frac{2-\\mathcal{B}}{2\\mathcal{B}\\rho}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)\\bigg]F_s=0.\n\\end{multline}\nFinally, the following ansatz,\n\\begin{equation}\\label{FinalAnsatz}\n    F_s(\\rho)=G_s(\\rho)\\exp\\left[\\int\\frac{(1-\\mathcal{B})^2}{2\\rho}{\\rm d}\\rho\\right],\n\\end{equation}\nfor an arbitrary radial function $G_s(\\rho)$, allows us to convert Eq.\\,(\\ref{SimplifiedfEq}) into the following more useful form:\n\\begin{align}\\label{FinalfEq}\n&G''_s+\\frac{G'_s}{\\rho}+\\bigg[\n\\frac{E^2}{\\mathcal{A}^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB\\ell}{\\hbar}-\\frac{e^2B^2\\rho^2}{4\\hbar^2}-\\frac{\\ell^2}{\\rho^2}+\\frac{(1-\\mathcal{B})s\\ell}{\\mathcal{B}\\rho^2}+\\frac{(1+\\mathcal{B})eBs}{2\\hbar\\mathcal{B}}\\nonumber\\\\\n&-\\frac{\\mathcal{A}'Es\\left[2\\hbar\\ell-eB\\rho^2\\right]}{2\\hbar\\rho\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{1}{2\\mathcal{B}^2}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)'-\\left(\\frac{\\mathcal{A}'E}{2\\mathcal{A}\\mathcal{B}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{2\\mathcal{B}^2}\\right)^2\\nonumber\\\\\n&-\\frac{2-\\mathcal{B}}{2\\mathcal{B}\\rho}\\left(\\frac{\\mathcal{A}'E}{\\mathcal{A}\\left[E+\\mathcal{A}mc^2\\right]}-\\frac{\\mathcal{B}'}{\\mathcal{B}}\\right)-\\frac{\\mathcal{B}'(1-\\mathcal{B})}{\\mathcal{B}^2\\rho}-\\frac{(1-\\mathcal{B})^4}{4\\mathcal{B}^2\\rho^2}\\bigg]\\mathcal{B}^2G_s=0.\n\\end{align}\n\nIt is clear that Eq.\\,(\\ref{FinalfEq}) would not be easy to solve exactly. However, we can clearly see now which terms of the equation could safely be dropped out even without making any prior assumption on the orders of magnitude of the various physical quantities involved, such as the energy of the particle and the strengths of the gravitational and magnetic interaction terms. In Section \\ref{sec:IV}, we shall come back to this point to discuss the order of magnitude of the correction that would have been brought to our final result for the energy levels of the particle if we had kept those extra terms.\n\nNow, while Eq.\\,(\\ref{FinalfEq}) seems analytically challenging to solve even when keeping only the dominant terms, the fact that the metric components $\\mathcal{A}^2$ and $\\mathcal{B}^2$ of the interior Schwarzschild solution depend on $\\rho^2$ instead of $\\rho$ will greatly simplify our task as we shall see shortly. For a spherical body of mass $M$, of radius $R$ and of uniform density, the interior Schwarzschild solution describing the gravitational field inside the body is given by the metric (\\ref{GeneralMetric}), with \\cite{Synge}\n\\begin{equation}\\label{InteriorSchwar}\n    \\mathcal{A}(\\rho)=\\frac{3}{2}\\left(1-\\frac{r_s}{R}\\right)^{\\frac{1}{2}}-\\frac{1}{2}\\left(1-\\frac{r_s\\rho^2}{R^3}\\right)^{\\frac{1}{2}},\\qquad \\mathcal{B}(\\rho)=\\left(1-\\frac{r_s\\rho^2}{R^3}\\right)^{-1\/2},\n\\end{equation}\nwhere $r_s=2GM\/c^2$ is the Schwarzschild radius of the massive body. For convenience, we introduce the dimensionless parameter $\\eta=\\frac{3}{2}(1-\\frac{r_s}{R})^{\\frac{1}{2}}$ and the inverse length squared $\\lambda=\\frac{r_s}{R^3}$. Please note that by choosing this metric, we have ignored the possible geometric effect of the magnetic field on the spacetime. We will come back to this point in Section \\ref{sec:IV} as well.   \n\nLet us now show the convergence of the square root in our ansatz (\\ref{FirstAnsatz}). Please note that for very small values of $\\rho$, the functions (\\ref{InteriorSchwar}) approximate to $\\mathcal{A}\\sim\\eta-\\frac{1}{2}+\\frac{\\lambda}{4}\\rho^2$ and $\\mathcal{B}\\sim1+\\frac{\\lambda}{2}\\rho^2$, from which we see that the leading-order $\\rho$-dependent term in the argument of the square root in Eq.\\,(\\ref{FirstAnsatz}) is proportional to $\\rho^2$. This establishes thus the convergence of that square root for very small $\\rho$. On the other hand, for $\\rho\\geq R$, the functions (\\ref{InteriorSchwar}) lead, by continuity, to the metric components of the {\\it exterior} Schwarzschild solution, for we have then $\\mathcal{A}=\\mathcal{B}^{-1}=(1-\\frac{r_s}{\\rho})^{\\frac{1}{2}}$. Therefore, for very large values of $\\rho$, the functions $\\mathcal{A}$ and $\\mathcal{B}$ in the argument of the square root in our ansatz (\\ref{FirstAnsatz}) approximate to $\\mathcal{A}\\sim1-\\frac{r_s}{2\\rho}$ and $\\mathcal{B}\\sim1+\\frac{r_s}{2\\rho}$. This implies that the leading-order $\\rho$-dependent term in the argument of the square root is proportional to $1\/\\rho$. This establishes thus the convergence of that square root for large values of $\\rho$ as well, whence we deduce the convergence of that square root for all values of $\\rho$. \n\nNext, we use the functions (\\ref{InteriorSchwar}) to expand $1\/\\mathcal{A}$ and $1\/\\mathcal{A}^2$, up to the first order in the parameter $\\lambda$, as follows:\n\\begin{align}\\label{ABPowerSeries}\n    \\frac{1}{\\mathcal{A}}&\\sim\\frac{2}{(2\\eta-1)}-\\frac{\\lambda\\rho^2}{(2\\eta-1)^2},\\nonumber\\\\\n    \\frac{1}{\\mathcal{A}^{2}}&\\sim\\frac{4}{(2\\eta-1)^2}-\\frac{4\\lambda\\rho^2}{(2\\eta-1)^3}.\n   \n   \n\\end{align}\nExpanding only up to the first order in $\\lambda$ is justified by the order of magnitude of $\\lambda$ (even for neutron stars as we will see in Section \\ref{sec:IV}). On the other hand, we also have $1\/\\mathcal{B}\\sim1-\\frac{\\lambda}{2}\\rho^2$ and $\\mathcal{B}^2\\sim1+\\lambda\\rho^2$. With these expansions, Eq.\\,(\\ref{SimplifiedfEq}) takes, up to the first order in the parameter $\\lambda$, the following form:\n\\begin{align}\\label{1stOrderFinalfEq}\n&\\frac{{\\rm d}^2G_s}{{\\rm d}\\rho^2}+\\frac{1}{\\rho}\\frac{{\\rm d}G_s}{{\\rm d}\\rho}+\\Bigg[\n\\frac{4E^2}{(2\\eta-1)^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB(\\ell+s)}{\\hbar}+\\frac{\\lambda eBs}{4\\hbar}-\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&-\\frac{2\\lambda(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}-\\frac{\\ell^2}{\\rho^2}-\\Bigg(\\frac{e^2B^2}{4\\hbar^2}+\\frac{\\lambda m^2c^2}{\\hbar^2}+\\frac{8\\lambda E^2[\\eta-1]}{\\hbar^2c^2[2\\eta-1]^3}+\\frac{\\lambda eB[2\\ell+s]}{2\\hbar}\\nonumber\\\\\n&+\\frac{\\lambda eBs}{\\hbar(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}\\Bigg)\\rho^2-\\frac{\\lambda e^2B^2}{4\\hbar^2}\\rho^4\\Bigg]G_s=0.\n\\end{align}\nThis equation has the form\n\\begin{equation}\\label{GEquation}\n    G_s''+\\frac{G_s'}{\\rho}+\\left(\\varepsilon-\\frac{\\ell^2}{\\rho^2}-\\kappa^2\\rho^2-\\tau^2\\rho^4\\right)G_s=0,\n\\end{equation}\nwhere a prime denotes a derivative with respect to $\\rho$ and the constants $\\varepsilon$, $\\kappa^2$ and $\\tau^2$ are straightforwardly read off from Eq.\\,(\\ref{1stOrderFinalfEq}). Setting $G_s=\\chi_s\/\\sqrt{\\rho}$ in Eq.\\,(\\ref{GEquation}), the latter takes the form\n\\begin{equation}\\label{Anharmonic}\n    \\chi''_s+\\left(\\varepsilon-\\frac{\\ell^2-\\frac{1}{4}}{\\rho^2}-\\kappa^2\\rho^2-\\tau^2\\rho^4\\right)\\chi_s=0.\n\\end{equation}\nThis is a Schr\\\"odinger equation with a centrifugal barrier term and an isotropic quartic anharmonic oscillator potential term. Please note that (i) in the case of a weak gravitational interaction compared to the magnetic one, i.e., for $\\lambda\\ll |e|B\/\\hbar$, and\/or (ii) in the case of charged fermions moving very close to the center of the spherical mass, i.e., for $\\rho\\ll R$, the purely quartic term $\\tau^2\\rho^4$ can be dropped out. In that case, Eq.\\,(\\ref{Anharmonic}) reduces to the one that has already been solved exactly in Ref.\\,\\cite{GravityLandauI}, where the quantized eigenvalues $\\varepsilon_{n,\\ell}$ have been extracted. We will come back to these special cases in Section \\ref{sec:IV-1}. Here, we will extract the eigenvalues from Eq.\\,(\\ref{Anharmonic}) by keeping the quartic term.\n\nIt is shown in Ref.\\,\\cite{JMP(1986)} that even though  Eq.\\,(\\ref{Anharmonic}) is not solvable exactly, one may still extract an approximate analytical expression for its eigenvalues using the Jeffreys\u2013Wentzel\u2013Kramers\u2013\\\\Brillouin (JWKB) approximation. To the fourth order in the approximation, the result reads \\cite{JMP(1986)}\n\\begin{equation}\\label{QuantizedEpsilon}\n\\varepsilon_{n,\\ell,s}=(2 \\tau)^{\\frac{2}{3}}\\sum_{k=0}^8\\left[\\frac{3\\sqrt{\\pi}\\,\\Gamma(\\tfrac{3}{4})}{\\sqrt{2}\\,\\Gamma(\\tfrac{1}{4})}\\left(2n+1+\\ell\\right)\\right]^{\\frac{4-2k}{3}}N_k\\left(\\kappa,\\tau,\\ell\\right),\n\\end{equation}\nwhere the explicit expressions of the nine terms $N_k(\\kappa, \\tau,\\ell)$ are given in Appendix \\ref{sec:App}. It is clear from this expression that the usual flat-space Landau levels of a charged fermion moving under the influence of a uniform magnetic field are dramatically altered. In fact, instead of the splitting of the levels that occurs due to a weak gravitational field \\mbox{outside \\cite{GravityLandauI}} or inside a spherical mass \\cite{QHGravity}, what happens here is a complete redistribution of the levels. Both the separation of the latter and their strengths are redefined for each principle quantum number $n$. \n\nFor the sake of illustration and simplicity, we display here explicitly the expression we obtain when keeping only the first term of the sum in Eq.\\,(\\ref{QuantizedEpsilon}). By substituting the constants $\\varepsilon$, $\\kappa$ and $\\tau$, as well as the term $N_0=1$ (as given in Appendix \\ref{sec:App}) into \\mbox{Eq.\\,(\\ref{QuantizedEpsilon}),} we find\n\\begin{align}\\label{QEpsilonN0}\n   E_{n,\\ell,s}&=\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2c^4-\\hbar c^2eB\\left(\\ell+s+\\frac{\\lambda s}{4}\\right)+\\frac{\\lambda\\hbar^2c^2(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n   &+\\frac{2\\lambda\\hbar^2c^2(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}+\\hbar^2c^2\\left[\\left(2n+1+\\ell\\right)\\frac{3\\,\\Gamma(\\tfrac{3}{4})}{\\Gamma(\\tfrac{1}{4})}\\right]^{\\frac{4}{3}}\\left(\\frac{\\pi eB\\sqrt{\\lambda}}{2\\hbar}\\right)^\\frac{2}{3}\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nThis does not, in fact, look at all like the familiar Landau levels. It must be noted here that this formula does not reduce to the flat-space result when setting $\\lambda=0$ because the expansion in terms of the JWKB integrals is about the pure quartic oscillator levels, not the other way around \\cite{JMP(1986)}. Therefore, to achieve a high accuracy in the Formula (\\ref{QuantizedEpsilon}), $\\kappa^2$ should not be larger than $\\tau^2$. To properly recover the flat-space case, one needs to set $\\lambda=0$ in Eq.\\,(\\ref{1stOrderFinalfEq}). Similarly, when the gravitational interaction is weaker than the magnetic interaction, one needs to extract the corresponding Landau levels by starting from Eq.\\,(\\ref{1stOrderFinalfEq}) as we shall do now.\n\\subsection{Weak Quartic Term}\\label{sec:IV-1}\nWhen the gravitational interaction is weak compared to the magnetic interaction and\/or when the charged particles move very closely to the center of the spherical mass, the quartic term $\\tau^2\\rho^4$ in Eq.\\,(\\ref{GEquation}) can safely be dropped out so that the equation takes the form,\n\\begin{equation}\\label{GEquationWithoutQuartic}\n    G_s''+\\frac{G_s'}{\\rho}+\\left(\\varepsilon-\\frac{\\ell^2}{\\rho^2}-\\kappa^2\\rho^2\\right)G_s=0.\n\\end{equation}\nThe solution to this equation is easily found to be given in terms of the confluent hypergeometric function as follows (see Ref.\\,\\cite{GravityLandauI} for the details of the derivation):\n\\begin{equation}\n    G_s(\\rho)=C\\rho^{\\ell}e^{-\\frac{|\\kappa|}{2}\\rho^2}\\,_1F_1\\left(\\frac{1+\\ell}{2}-\\frac{\\varepsilon}{4|\\kappa|};\\ell+1;|\\kappa|\\rho^2\\right).\n\\end{equation}\nThis solution is one of the two independent solutions of Eq.\\,(\\ref{1stOrderFinalfEq}) that is finite at the origin $\\rho=0$. $C$ is one of the constants of integration, and$\\,_1F_1(a;b;\\xi)$ is the confluent hypergeometric function of a variable $\\xi$ for any values of the parameters $a$ and $b$, except when $b$ is a negative integer for which case the function has a simple pole \\cite{HandBook}. Therefore, to guarantee a square-integrable character for the wavefunction, the following condition must be imposed for an arbitrary positive integer $n$:\n\\begin{equation}\n    \\frac{1+\\ell}{2}-\\frac{\\varepsilon}{4|\\kappa|}=-n,\n\\end{equation}\nfor which case, the confluent hypergeometric function becomes indeed a polynomial of finite degree $n$. By substituting into this condition, the values of $\\varepsilon$ and $\\kappa$ as read off from \\mbox{Eq.\\,(\\ref{1stOrderFinalfEq})}, we arrive at the following condition involving the energy $E$ of the \\mbox{charged fermion:}\n\\begin{align}\n&\\frac{4E^2}{(2\\eta-1)^2\\hbar^2c^2}-\\frac{m^2c^2}{\\hbar^2}+\\frac{eB(\\ell+s)}{\\hbar}+\\frac{\\lambda eBs}{4\\hbar}-\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&-\\frac{2\\lambda(s\\ell+1)}{(2\\eta-1)[2+(2\\eta-1)\\frac{mc^2}{E}]}=(2n+1+\\ell)\\Bigg[\\frac{e^2B^2}{\\hbar^2}+\\frac{4\\lambda m^2c^2}{\\hbar^2}+\\frac{32\\lambda E^2(\\eta-1)}{\\hbar^2c^2(2\\eta-1)^3}\\nonumber\\\\\n&+\\frac{2\\lambda eB(2\\ell+s)}{\\hbar}+\\frac{4\\lambda eBs}{\\hbar(2\\eta-1)(2+[2\\eta-1]\\frac{mc^2}{E})}\\Bigg]^{\\frac{1}{2}}.\n\\end{align}\nWe may solve this equation for $E$ by keeping again only the first-order terms in the parameter $\\lambda$, to arrive at the following quantization condition on the energy $E$ of \\mbox{the particle}:\n\\begin{align}\\label{FinalQuantization}\nE_{n,\\ell,s}&=\\hbar c\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{\\frac{m^2c^2}{\\hbar^2}-\\frac{eB(\\ell+s)}{\\hbar}-\\frac{\\lambda eBs}{4\\hbar}+\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\nonumber\\\\\n&\\quad+\\frac{\\lambda(s\\ell+1)}{(2\\eta-1)}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}+(2n+1+\\ell)\\Bigg[\\frac{e^2B^2}{\\hbar^2}+\\frac{4\\lambda m^2c^2}{\\hbar^2}\\nonumber\\\\\n&\\quad+\\frac{2\\lambda eB(2\\ell+s)}{\\hbar}+(\\eta-1)\\frac{8\\lambda(m^2c^2+[2n+1-s]\\hbar eB)}{\\hbar^2(2\\eta-1)}\\nonumber\\\\\n&\\quad+\\frac{2\\lambda eBs}{\\hbar(2\\eta-1)}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}\\Bigg]^{\\frac{1}{2}}\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\n\nWe readily note that for $M=0$ (i.e., by setting $\\eta=\\frac{3}{2}$ and $\\lambda=0$), the result (\\ref{FinalQuantization}) reduces to the usual quantized energy levels of a charged fermion inside a uniform magnetic field in Minkowski spacetime: $E_{n,s}=[m^2c^4+\\hbar c^2(2n+1-s\\frac{e}{|e|})|e|B]^{\\frac{1}{2}}$. In addition, we also note, by setting $B=0$, that even in the absence of the magnetic field the gravitational field induces quantized energy levels. Furthermore, we neatly recognize a general-relativistic correction to the pure relativistic Landau levels arising from the magnetic field. Such a correction arises from the factor $(\\eta-\\frac{1}{2})$ multiplying the right-hand side. \n\nWe are now going to discuss the ways to overcome the limitations we encountered in this section and what modifications would be brought to the results we obtained so far.\n\n\\section{Refinements and Prospective Extensions}\\label{sec:IV}\nTwo limitations, we could not avoid when deriving our results in the previous section, were (i) the fact that we considered only the possible motion of the particle along the equator inside the massive sphere and (ii) the fact that we ignored the geometric effect that might be caused to spacetime by the magnetic field. The two other limitations we imposed on our analysis were (iii) the uniform mass density we assumed for the interior of the spherical mass and (iv) the first-order approximation in $\\lambda$ we limited ourselves to when writing Eq.\\,(\\ref{1stOrderFinalfEq}) to be able to solve the latter exactly. Our aim in this section is to address these limitations and argue that they do not actually alter much the actual physical conclusions we would have reached had we not adopted those various approximations.\n\nThe first limitation consists of the restricted number of degrees of freedom we imposed on the charged fermion. By restricting the motion of the particle to the equatorial plane inside the massive sphere, we implicitly assumed the particle's motion to be confined to such a plane due to the collective interaction of the particles among themselves. However, it is actually possible, for the same reason, to also consider the possibility of motion along a plane other than the equatorial plane. In fact, the vertical pull of gravity is then balanced by these other interactions so that only the horizontal variation of the gravitational potential matters. For such a case, however, we need to switch entirely to cylindrical coordinates. The interior Schwarzschild solution needs to be expressed in such coordinates and we need to set $r^2=\\rho^2+z_0^2$ for some vertical distance $z_0$ from the equator in the resulting differential equation that would be similar in form to Eq.\\,(\\ref{2-SpinorEq}). As the differential operator $\\partial_\\theta$ would be replaced by $z_0\\partial_\\rho$ and the operator $\\partial_\\theta^2$ would be replaced by $z_0^2\\partial_\\rho^2-(\\rho+z_0)\\partial_\\rho$, the equation would still be a second-order differential equation in the single variable $\\rho$. A first-order approximation in the parameter $\\lambda$ might then be adopted. Although such an equation would certainly not be as easy to solve as Eq.\\,(\\ref{1stOrderFinalfEq}), we expect the energy of the particle to be only shifted by a constant while obeying a quantization condition similar to our results (\\ref{QuantizedEpsilon}) and (\\ref{FinalQuantization}). A further refinement consists then of considering the case $\\partial_z\\Psi\\neq0$ and a non-constant $z$. Unfortunately, the fact that two variables would then be involved, no simple procedure could be expected. A separate future work should specifically be devoted to solving that case using cylindrical coordinates. Indeed, the problem of solving our very general Eq.\\,(\\ref{2-SpinorEq}) without restricting the latter to the equatorial plane might be very important for other astrophysical applications besides neutron stars. We believe that the problem might very well be tackled by adapting to our case the existing numerical methods that have already proved fruitful elsewhere for solving such equations (see, e.g., the recent work \\cite{Numerical} and the references therein).\n\nThe second interesting refinement that should be discussed here is the possibility of adding to the metric components, given by Eq.\\,(\\ref{InteriorSchwar}), the spacetime curvature caused by the magnetic field. The geometric effect of combining the external gravitational field sourced by a spherical mass with a uniform magnetic field $B$ gives rise to the so-called Schwarzschild\u2013Melvin spacetime \\cite{Ernst}. Therefore, an analogue modification to the interior Schwarzschild metric components given by Eq.\\,(\\ref{InteriorSchwar}) is also expected when including the spacetime curvature caused by $B$. However, since the corrections brought to the familiar exterior Schwarzschild solution by the magnetic field are of the order of $\\epsilon_0GB^2\/c^2$, where $\\epsilon_0$ is the vacuum permittivity constant \\cite{Galtsov}, we expect the correction that would be brought to the parameter $\\lambda$ in our result (\\ref{FinalQuantization}) to be of the order of $\\sim$$\\epsilon_0GB^2\/c^2$ as well. For a neutron star of mass $\\sim$$1.4\\,M_\\odot$ and of radius $10$\\,km, the Schwarzschild radius is about $\\sim$$4.1\\,$km for which we find that $\\lambda\\sim0.004\\,$km$^{-2}$. On the other hand, for such a neutron star with an interior magnetic field as strong as $\\sim$$10^{17}\\,$G, we have $\\epsilon_0GB^2\/c^2\\sim7\\times10^{-9}\\,$km$^{-2}$. The expected correction is thus quite insignificant.\n\nThe third limitation arises from having discarded in Eq.\\,(\\ref{1stOrderFinalfEq}) terms of the second order and higher in the parameter $\\lambda$. We can justify now such an approximation by the fact that, as we just saw for a typical neutron star, $\\lambda\\sim4\\times10^{-3}\\,$km$^{-2}$. This entails that if we had kept those higher order terms the expected correction to be brought to the quantized energy levels (\\ref{FinalQuantization}) can be obtained using time-independent perturbation theory in the manner worked out in Refs. \\,\\cite{GravityLandauI,GravityLandauII} for the case of Landau levels in the exterior Schwazschild solution. \n\nAnother refinement that should be taken into account is the fact that matter density inside neutron stars is not really uniform as we have assumed it in our calculations in the previous section. The interior matter density increases with increasing depth. However, the core of a neutron star makes up the largest part of the star and may be subdivided into outer and inner parts, each characterized by slowly varying mass densities with depth \\cite{NSDensity}. For this reason, although considering a uniform mass density is only a simplifying model for the interior of a neutron star, we expect that the conclusions that would be derived by assuming a slowly varying $\\rho$-dependent mass density would not differ much from those obtained here. Unfortunately, an analysis based on the interior Schwarzschild solution with a $\\rho$-dependent mass density is beyond the scope of the present paper and should be considered in future works as well. It must also be noted here that numerical methods would certainly be beneficial for such a task.\n\nThe last two extensions of this work we would like to mention here is the possibility of studying the effect of the interior Schwarzschild solution on (i) charged fermions when the magnetic field is nonuniform and (ii) on the neutron star matter when the latter is taken to be a superfluid \\cite{NSSuperfluid}. For the former case, one needs to use again numerical methods (as in Ref.\\,\\cite{NonUniformB}) to solve the resulting equation, whereas for the latter case, one needs to consider, instead of the Dirac equation, the Gross\u2013Pitaevskii equation coupled to the magnetic field inside the interior Schwarzschild solution.     \n\n\\section{Applications}\\label{sec:V}\nDespite the limitations we discussed above, it is still interesting to apply our results to some systems that rely on the Landau energy levels of charged fermions. We will first examine the charged fermions inside a neutron star, and then we examine charged fermions inside a laboratory-made spherical mass.\n\n\\subsection{Inside a Neutron Star}\nWe propose to use here our results (\\ref{QuantizedEpsilon}) and (\\ref{FinalQuantization}) to examine the effect of gravity on the magnetization $\\mathscr{M}$ inside a neutron star.  The magnetization $\\mathscr{M}$ is given by the following general formula \\cite{Broderick}:\n\\begin{equation}\\label{GeneralM}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left(\\frac{\\partial \\epsilon_i}{\\partial B}-\\mu^i_f\\frac{\\partial n_i}{\\partial B}\\right),\n\\end{equation}\nwhere the summation is over the electrons, muons and protons, respectively; $\\epsilon_i$ are the energy densities of those charged fermions and $n_i$ are their number densities. The magnetic field $B$ in this formula is the original seed field (magnetizing field) that is thus gradually modified by the magnetization into $H=B+4\\pi\\mathscr{M}$, in CGS units \\cite{LLPTextbook}. A strong magnetization has been suggested by some authors to play a role even in giving rise to the original (primal) magnetic field itself \\cite{Dong2013}. However, despite considerable research on the topic, there is in the literature no consensus yet on the origin of such strong seed magnetic fields at the surface nor in the core of such compact stars. For a quick guide through the various proposals put forward in the literature, see, e.g., the reviews \\cite{ReviewEoS2015,Universe2021}. \n\nConsidering the magnetizing field $B$ to be constant in time and uniform over the radius of the star, as we do in this paper, is also only an approximation. However, given that the relaxation times of the star's magnetic field decay is of the order of $\\sim$$10^3$ \\mbox{years \\cite{Universe2018}}, our present assumption of a constant magnetic field fits amply within the early stages of the magnetic evolution of the star. Similarly, assuming a slow variation of the magnetic field with distance inside highly magnetized stars as presented, for example, in Ref.\\,\\cite{BDistribution} is still consistent with our present purpose of providing a preliminary study on the effect of gravity on the magnetic properties of such compact stellar objects.\n\nFirst, recall that when ignoring the contribution of the gravitational field, the quantities $\\epsilon_i$ and $n_i$ are given by \\cite{Broderick}\n\\begin{align}\\label{EnergyAndNumberDensity}\n    \\epsilon_i&=\\frac{|e|B}{4\\pi^2}\\sum_s\\sum_{n=0}^{n_{max}}\\left(\\mu^i_fk^i_{f,n,s}+\\tilde{m}^{i2}_{n,s}\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{\\tilde{m}^i_{n,s}}\\right|\\right),\\nonumber\\\\\n    n_i&=\\frac{|e|B}{2\\pi^2}\\sum_s\\sum_{n=0}^{n_{max}}k^i_{f,n,s},\n\\end{align}\nwhere\n\\begin{align}\\label{mTilde}\n    \\tilde{m}^{i2}_{n,s}&=m^{i2}+\\left(2n+1-\\frac{e}{|e|}s\\right)|e|B,\\nonumber\\\\\n    k_{f,n,s}^{i2}&=\\mu_f^{i2}-\\tilde{m}_{n,s}^{i2}.\n\\end{align}\nThe Fermi energies $\\mu^{i}_f$ are fixed by the chemical potentials and the maximum integer $n_{max}$ in the summations represents the integer preceding the value of $n$ for which $k_{f,n,s}^{i2}$ becomes negative. As the equations are more involved, we will work in this subsection in the natural units $\\hbar=c=1$. Formula (\\ref{GeneralM}) then gives \\cite{Broderick} \\begin{equation}\\label{FlatMagnetization}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left[\\frac{\\epsilon_i-\\mu^i_fn_i}{B}+\\frac{B}{2\\pi^2}\\sum_{s}\\sum_{n=0}^{n_{max}}\\left(n+\\frac{1}{2}-\\frac{s}{2}\\right)\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{\\tilde{m}^i_{n,s}}\\right|\\right].\n\\end{equation}\n\nWe will adapt these general formulas to our case. Since the formulas to be developed here are more relevant for very strong magnetic fields, we also extract from Eq.\\,(\\ref{FinalQuantization}) the following approximation for the quantized energy levels:\n\\begin{align}\\label{StrongBApproximation}\n    E_{n,\\ell,s}&=\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2\\!+\\!\\left(2n+2\\ell+1-\\frac{e}{|e|}s\\right)|e|B\\!-\\!\\frac{\\lambda eBs}{4\\hbar}\\!+\\!\\frac{\\lambda(2\\ell^2+s\\ell-2)}{2}\\!+\\!\\frac{\\lambda(s\\ell+1)}{2\\eta-1}\\nonumber\\\\\n&\\quad+(2n+1+\\ell)\\Bigg[\\lambda (2\\ell+s)+(\\eta-1)\\frac{4\\lambda(2n+1-s)}{2\\eta-1}+\\frac{\\lambda s}{2\\eta-1}\\Bigg]\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nNote that in the absence of gravity, i.e., when $\\lambda=0$, this formula reduces to the first identity in Eq.\\,(\\ref{mTilde}). Let us then start with the consequences of this formula which is valid either (i) for weak gravitational fields and\/or (ii) for fermions very close to the center of the star. Since the terms proportional to $\\lambda$ in this expression de not contain $B$, it is straightforward to compute the $B$-derivatives in Eq.\\,(\\ref{GeneralM}). The result for the magnetization remains thus exactly the same as in Eq.\\,(\\ref{FlatMagnetization}), but with $\\tilde{m}_{n,s}^{i2}$ replaced by $E_{n,\\ell,s}$ as given by Eq.\\,(\\ref{StrongBApproximation}):\n\\begin{equation}\\label{CurvedMagnetization}\n    \\mathscr{M}=\\sum_{i=e,\\mu,p}\\left[\\frac{\\epsilon_i-\\mu^i_fn_i}{B}+\\frac{B(2\\eta-1)}{4\\pi^2}\\sum_{s}\\sum_{n=0}^{n_{max}}\\left(n+\\frac{1}{2}-\\frac{s}{2}\\right)\\ln\\left|\\frac{\\mu^i_f+k^i_{f,n,s}}{E^i_{n,\\ell,s}}\\right|\\right].\n\\end{equation}\nWe neatly see the general-relativistic correction factored out in the second term of the sum. More important, however, is the fact that the {\\it general expression} of the magnetization remains unaltered in this case.\n\nOn the other hand, for the case of a {\\it non-negligible} gravitational field the quartic term in Eq.\\,(\\ref{1stOrderFinalfEq}) should be taken into account, leading to the result (\\ref{QEpsilonN0}) for $E_{n,\\ell,s}$ at the first-order term of the JWKB approximation. As the Landau levels are completely destroyed, we see that even at this order in the approximation the magnetization is completely different from the expression (\\ref{CurvedMagnetization}). Actually, even Formulas (\\ref{EnergyAndNumberDensity}) and (\\ref{mTilde}) do not hold anymore, for such formulas were derived by assuming the linear dependence (\\ref{mTilde}) of $\\tilde{m}_{n,s}^{i2}$ on the integer $n$. As expression (\\ref{QEpsilonN0}) of $E_{n,\\ell,s}$ in terms of $n$ displays no such linearity, and since that expression (\\ref{QEpsilonN0}) is itself only an approximation for the highly nonlinear full \\mbox{expression (\\ref{QuantizedEpsilon}),} we conclude that the Landau-levels-based formalism for computing the magnetization $\\mathscr{M}$ of neutron stars does not generally apply to the charged plasma in the core of highly dense neutron stars. The condition for the applicability of the formalism is to have the quartic anharmonic oscillator potential term in Eq.\\,(\\ref{Anharmonic}) negligible, i.e., to have $\\lambda\\ll |e|B\/\\hbar$ or to apply the formalism only to fermions close to the vicinity of the star's center. In other words, we conclude that only for a magnetic neutron star of mass $M$, of radius $R$ and with a magnetic field $B$ such that $|e|BR^3c^2\\gg GM\\hbar$, that the usual formalism generally applies to the charged matter inside the core of the star.\n\nMagnetization in neutron stars affects both the equation of state of the stars' matter and the stars' structure and properties, such as their masses and transport properties. Indeed, both the pressure anisotropy and the softening of the equation of state of the stars matter is believed to be related to magnetization and the distribution of the charged fermions as they occupy the Landau levels. The equation of state determines, in turn, the mass-radius relation of neutron stars (see, e.g., the recent work \\cite{EOS2020} and the \\mbox{references therein}).  \n\nIt is believed that the net magnetization in neutron stars remains negligible until around $10^{16}$\\,G, beyond which it starts increasing and becomes oscillatory under the de Haas-van Alphen effect. Since we just saw that the Landau levels become dramatically altered only for non-negligible gravitational fields compared to the magnetic interaction, we conclude that such oscillations will not be much affected in typical stars. For weaker magnetic fields, however, the general-relativistic correction we found in Eq.\\,(\\ref{CurvedMagnetization}) for the magnetization might leave an observable signature on the equation of state of the star's matter and, hence, on the mass-radius relation of the star as well. More work dedicated specifically to such an interesting investigation is required though.\n\\subsection{At a Laboratory Level}\nAnother application of our results is to consider physical systems at the laboratory level. In this case, an interesting system would be the free electrons moving inside a massive metallic sphere put inside a uniform magnetic field. For this case, the gravitational contribution is much weaker than the magnetic interaction, so Eq.\\,(\\ref{FinalQuantization}) is what we need to apply here. Expanding that expression up to the first order in $\\lambda$, we obtain\n\\begin{align}\\label{FinalQuantizationWeakG}\nE_{n,\\ell,s}&\\approx\\left(\\eta-\\frac{1}{2}\\right)\\Bigg\\{m^2c^4+\\hbar c^2\\left(2n\\!+\\!2\\ell\\!+\\!1\\!-\\!\\frac{e}{|e|}s\\right)|e|B-\\frac{\\hbar c^2\\lambda eBs}{4}+\\frac{\\hbar^2 c^2\\lambda(2\\ell^2\\!+\\!s\\ell\\!-\\!2)}{2}\\nonumber\\\\\n&\\quad+\\frac{\\lambda\\hbar^2c^2(s\\ell+1)}{(2\\eta-1)}\\frac{\\sqrt{1+(2n\\!+\\!1\\!-\\!s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n\\!+\\!1\\!-\\!s)\\frac{\\hbar eB}{m^2c^2}}}+(2n+1+\\ell)\\hbar^2c^2\\lambda\\Bigg[\\frac{2 m^2c^2}{\\hbar eB}+2\\ell+s\\nonumber\\\\\n&\\quad+(\\eta-1)\\frac{4(m^2c^2+[2n+1-s]\\hbar eB)}{\\hbar eB(2\\eta-1)}+\\frac{ s}{2\\eta-1}\\frac{\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}{1+\\sqrt{1+(2n+1-s)\\frac{\\hbar eB}{m^2c^2}}}\\Bigg]\\Bigg\\}^{\\frac{1}{2}}.\n\\end{align}\nThese are the energy levels of the electrons inside the massive sphere. The $\\lambda$-terms are the corrections brought to the familiar levels observed in the absence of gravity. This expression is valid for both relativistic and non-relativistic fermions, and it reduces to the familiar relativistic Landau levels in the absence of gravity. This result offers novel ways for exploiting the Landau levels of fermions in the presence of gravity at the \\mbox{laboratory level.}\n\n\\section{Summary}\\label{sec:VI}\nWe have obtained the Dirac equation for a charged fermion in a general static and spherically symmetric curved spacetime in the presence of a static and uniform magnetic field. We applied the general equation we obtained to the case of fermions in the {\\it interior} Schwarzschild solution describing the gravitational field inside a massive sphere of uniform density, and we derived the quantized energy levels of the charged particles. We found that our result reduces to the familiar flat-spacetime relativistic Landau levels of charged fermions inside a uniform magnetic field only when the gravitational interaction is weaker than the magnetic interaction and\/or when the charged particles move very closely to the center of the spherical mass. \n\nWe then discussed, based on these results, the consequences on the physics of neutron stars. We found that the magnetization of the core of the latter would dramatically be altered for an extremely high gravitational field compared to the magnetic interaction and\/or when not focusing solely on the fermions in the vicinity of the center of the star. We arrived at the conclusion that in the general case, the usual formalism for extracting the magnetization of neutron stars applies only when the latter satisfy the condition $|e|BR^3c^2\\gg GM\\hbar$. Although we arrived at such a conclusion by relying on a first-order approximation in the gravitational parameter $\\lambda$, the generality of our results is not affected. In fact, such a first-order approximation remains valid for any relative strengths of the gravitational and magnetic interactions experienced by the charged particles inside typical neutron stars. Any extra correction to those levels would be obtained simply using time-independent perturbation theory.\n\nIn Section \\ref{sec:IV}, we outlined a few refinements that might be brought to our model and the prospective outlook on future improvements that will allow us to go beyond the limitations we imposed on our present study. However, even with the simple model dealt with here, we have already glimpsed with new insights into the contribution of gravity in shaping the magnetization of highly compact neutron stars and into novel ways of exploiting the dynamics of charged fermions in the presence of gravity at the laboratory level. \n\n\n\n\n\\section*{Acknowledgments}\nThe authors are grateful to Patrick Labelle for the helpful discussions and comments, and to the anonymous referees for their constructive comments that helped enrich our presentation. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant No. RGPIN-2017-05388 and by the Fonds de Recherche du Qu\u00e9bec - Nature et Technologies (FRQNT). PS acknowledges support from Bishop's University via the Graduate Entrance Scholarship award.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}Minimal surfaces of general type with $p_g(S)\n= 0$ have invariants\n$p_g(S)  =  q (S)= 0, 1 \\leq K_S^2  \\leq 9$, and, for each value of $\ny \\in \\{1,2, \\dots, 9\\}$, such surfaces\nwith $ K_S^2 = y$  yield  a finite number\nof irreducible components of the Gieseker moduli space of surfaces of\ngeneral type $\\mathfrak M^{can}_{1,y}$.\n\nAfter the first surfaces of general type with $p_g = q = 0$ were\nconstructed in the 1930' s by   Luigi Campedelli and by Lucien\nGodeaux (cf. \\cite{Cam},\n\\cite{god}) there was  in the 1970's a big revival of interest in the\nconstruction of these surfaces,\nas documented by a long and influential survey paper  written by\nDolgachev (\\cite{Dolgachev}). \n\nThe Bloch conjecture and\ndifferential topological questions raised by Donaldson Theory were a\nfurther reason for the remarkable and ongoing  interest\nabout surfaces of general type with\n$p_g = 0$, and  we refer to\n\\cite{survey} for an update  about   recent  important progress on\nthe topic, and about  the state of the art.\n\n\nLooking at the tables 1--3 of \\cite{survey} one finds  it striking\nthat for the value $K_S^2= 7$ there is only one  known family of\nsuch surfaces of general type. This family was constructed by Inoue\n(cf. \\cite{inoue}).\nFurther interest concerning this family comes from the problem raised\nin \\cite{survey}, and concerning the finiteness or not\nof the fundamental group of  surfaces with $p_g = 0$. Indeed this paper was\nmotivated by the observation that Inoue's surfaces have a\n``big'' fundamental group. In fact, the fundamental group of an Inoue\nsurface with $p_g = 0$ and $K_S^2 =7$ sits\nin an extension ($\\pi_g$ denotes the fundamental group of a compact\ncurve of genus $g$):\n$$ 1 \\rightarrow \\pi_{5} \\times \\mathbb{Z}^4 \\rightarrow \\pi_1(S)\n\\rightarrow (\\mathbb{Z}\/2\\mathbb{Z})^5 \\rightarrow 1.\n$$\n\nThis extension is given geometrically, i.e., stems from our observation that\nan Inoue surface $S$ admits an unramified\n$(\\mathbb{Z}\/2\\mathbb{Z})^5$ - Galois covering $\\hat{S}$ which is an\nample divisor in $E_1 \\times E_2 \\times D$, where\n$E_1, E_2$ are elliptic curves and $D$ is a compact curve of genus\n$5$; this made us believe\n   that the topological type of an Inoue surface determines an irreducible\nconnected component of the moduli space (a phenomenon similar to the\none which was already observed   in \\cite{keumnaie},\n\\cite{burniat1} \\cite{coughlanchan}).\n\nThe following is  one of the main  results of this paper:\n\n\\begin{theo}\\label{main}\\\n\\begin{enumerate}\n\\item Let $S'$ be a smooth complex projective surface which is\nhomotopically equivalent to an Inoue surface (with $K^2 = 7$ and\n$p_g=0$). Then $S'$ is an Inoue  surface.\n\\item The connected component of the Gieseker moduli space\n$\\mathfrak M^{can}_{1, 7}$ corresponding to Inoue surfaces is\nirreducible, generically smooth, normal and unirational  of dimension  4.\n\nMoreover, each Inoue surface $S$ has ample canonical divisor\\footnote{This is proven by Inoue, \nsee page 318 of \\cite{inoue}.} and the base $\\Def (S)$  of the Kuranishi family of $S$  is smooth.\n\\item\nFinally, the first homology group of an Inoue surface equals $$\\ensuremath{\\mathbb{Z}}\/4 \\ensuremath{\\mathbb{Z}} \\oplus (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4.$$\n\n\\end{enumerate}\n\\end{theo}\n\nSince this theorem is similar in flavour to other results that we \nmentioned above,\nthe main purpose of this paper is not only to give a more general proof, but also to  set up the\nstage for  the investigation and search for a new class of varieties, \nwhich we propose to call\nInoue-type varieties.\n\n\\begin{defin}\nWe define a complex projective manifold $X$ to be an {\\bf Inoue-type manifold} if\n\\begin{enumerate}\n\\item\n$ dim (X) \\geq 2$;\n\\item\nthere is a finite group $G$ and a Galois unramified covering $\n\\hat{X} \\ensuremath{\\rightarrow} X$ with group $G$,\n(hence $ X = \\hat{X} \/ G$) so that\n\\item\n$ \\hat{X}$ is an ample divisor inside a $K(\\Ga, 1)$-projective manifold\n$Z$ (hence by Lefschetz $\\pi_1 ( \\hat{X}) \\cong \\pi_1 (Z)\n\\cong \\Ga$) and moreover\n\\item\nthe action of $G$ on $ \\hat{X}$   yields a faithful action on $\\pi_1\n( \\hat{X}) \\cong \\Ga$:\nin other words the exact sequence\n$$ 1 \\ensuremath{\\rightarrow} \\Ga  \\cong \\pi_1 ( \\hat{X}) \\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1$$\ngives an injection $ G \\ensuremath{\\rightarrow} Out (\\Ga)$, defined by conjugation;\n\\item\nthe action of $G$ on $ \\hat{X}$   is induced by an action on $Z$.\n\\end{enumerate}\nSimilarly one defines the notion of an  {\\bf Inoue-type variety}, by\nrequiring the same properties\nfor a variety $X$ with canonical singularities.\n\\end{defin}\n\n\nWe should warn the reader that our approach was inspired by, but is different from the\noriginal construction of Inoue, who considers\nhypersurfaces and complete intersections (of not necessarily ample\ndivisors) in a product of elliptic curves.\nIn fact, the change of point of view in the particular case  of  an Inoue surface with \n$K^2 = 7, p_g = 0$ produces a different realization: instead of Inoue's   original\nrealization as\n a complete intersection of two hypersurfaces of multidegrees $(2,2,2,0)$ and\n$(0,0,2,2)$ in a product of 4 elliptic curves,\n  we view the same surface as a hypersurface of multidegrees\n$(2,2,4)$ in the product\n$E_1 \\times E_2 \\times D$ of two elliptic curves with a curve $D$ of genus $5$.\n\nOne can see that our definition, although imposing a strong restriction on $X$, \nis not yet satisfactory in order to obtain some\nweak rigidity result (of the type of theorems 4.13 and 4.14 of \n\\cite{isogenous}, amended in \\cite{annals}, theorem 1.3).\nSome hypotheses must be made on the fundamental group $\\Ga$ of $Z$,\nfor instance the most interesting case\nis the one where $Z$ is a product of Abelian varieties, curves, and other locally symmetric varieties with\nample canonical bundle.\n\n\\begin{defin}\\label{SIT}\n   We shall say  that an  Inoue-type manifold $X$  is \n\n\\begin{enumerate}\n\\item\n  a {\\bf SIT : = special Inoue type manifold}\nif  moreover\n$$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) \\times (M_1 \\times \\dots \\times\nM_s)$$ where each\n$A_i$ is an Abelian variety,  each $C_j$ is a curve of genus $ g_j \\geq 2$,\nand $M_i$ is  a\ncompact quotient of an irreducible bounded symmetric domain of \ndimension at least 2 by a\ntorsion free subgroup.\n\\item\n  a {\\bf CIT : = \nclassical Inoue type manifold}\nif  moreover\n\n$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) $ where each\n$A_i$ is an Abelian variety,  each $C_j$ is a curve of genus $ g_j \\geq 2$.\n\\item\na special Inoue type manifold  is said to be\na \n\n{\\bf diagonal SIT manifold : = diagonal\nspecial   Inoue type manifold }\nif  moreover:\n\\begin{itemize}\n\\item the action of $G$ on $ \\hat{X}$   is induced by a  diagonal \naction on $Z$, i.e.,\n\n$$\n(I) \\ \\  G \\subset    \\prod_{i=1}^r \\Aut(A_i)\n\\times\n\\prod_{j=1}^h\\Aut(C_j) \\times \\prod_{l=1}^s\\Aut(M_l)\n$$\n\n\\item\nand furthermore:\n\n(II) \nthe  faithful action on $\\pi_1 ( \\hat{X}) \\cong \\Ga$,\ninduced by conjugation by lifts of elements of $G$ in  the exact sequence\n\\begin{equation}\\label{prodFG}\n 1 \\ensuremath{\\rightarrow} \\Ga= \\Pi_{i=1}^r (\\Lambda_i) \\times   \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l))\n\\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1\n\\end{equation}\n(observe that each factor $\\Lambda_i$, resp. $\\pi_{g_j}, \\pi_1 (M_l)$ is \nnormal), has the Schur property\n$$(SP)  \\Hom (V_i, V_j)^{G} = 0,\\forall i \\neq j,   $$\nwhere $V_j : = \\Lambda_j\\otimes \\ensuremath{\\mathbb{Q}}$ (it  suffices then to verify  that \nfor each $\\Lambda_i$ there is a subgroup $H_i$ of $G$ for which $ \\Hom (V_i, V_j)^{H_i} = 0, \\forall j \\neq i  $).\n\\end{itemize}\n\\item\nsimilarly we define a {\\bf diagonal CIT manifold : = diagonal\nclassical  Inoue type manifold }\n\n\\end{enumerate}\nWe can define analogous  notions for    Inoue  type varieties\n  $X$ with canonical singularities.\n\\end{defin}\n\n\nProperty (SP) plays an important role in order to show that an\nAbelian variety  with such a $G$-action on its fundamental group\nmust split as a product.\n\nThere is however  a  big difference between the curve and locally symmetric\nfactors  on one side and the\nAbelian variety factors on the other.  Namely: for curves we have weak rigidity, i.e., \nthe action of $G$ on $\\pi_{g_j}$ determines a connected family of\ncurves; for compact free quotients of bounded symmetric domains of \ndimension $\\geq 2$ we have strong rigidity, i.e., \nthe action on the fundamental group determines uniquely the \nholomorphic action; \nfor Abelian varieties it is not necessarily so.\n\nHence, in order to hope for weak rigidity results, one has to \nintroduce  a further invariant, called Hodge type (see section 1).\n\nWe can now state our main general result:\n\n\n\\begin{theo}\\label{special diagonal}\\\n  Let $X$ be a diagonal SIT (special  Inoue type) manifold, and let $X'$ be a\nprojective manifold\nwith the same fundamental group as $X$, which moreover either\n\n(1)\nis homotopically equivalent to $X$;\n\n  or satisfies the following weaker property:\n\n(2) let $\\hat{X'}$ be the corresponding unramified covering of $X'$ and\nidentify both\nfundamental groups $\\pi_1 (\\hat{X'}) = \\pi_1 (\\hat{X})= \\Ga $ . Then \nwe require that\n\n {\\bf (SAME HOMOLOGY)} there is an isomorphism $\\Psi : H_*(\\hat{X'}, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $\nof homology groups which is compatible with the homomorphisms\n$$ u \\colon H_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})   ,  u'   \\colon\n H_*(\\hat{X'}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  ,   $$\ni.e., $u \\circ \\Psi = u'$.\n\n\nUnder these assumptions, we have, setting $ W : = \\hat{X'}$, that\n\n\\begin{itemize}\n\\item\n$X'=  W \/ G$ where $W$ admits a generically finite\nmorphism \n$f : W \\ensuremath{\\rightarrow} Z'$, and where\n\\item\n$Z'$ is also a $K(\\Ga, 1)$ projective manifold,\nof the form $ Z' = (A'_1 \\times \\dots \\times A'_r) \\times\n(C'_1 \\times \\dots \\times  C'_h) \\times (M'_1 \\times \\dots \\times\nM'_s)$.\n\nMoreover here $M'_i$ is either $M_i$ or its complex conjugate,\nand the product decomposition corresponds to the product decomposition\n(\\ref{prodFG}) of the fundamental group of $Z$.\n\\item\nThe  image cohomology class  $f_*([W])$ corresponds, up to sign,\nto the cohomology class of $\\hat{X}$.\n\\item\nThe morphism $f$ is\nfinite if $n = dim X$ is odd, and moreover it is\ngenerically injective if \n\n(**) the cohomology class of $\\hat{X}$ is indivisble,\nor if every strictly submultiple  cohomology class cannot be represented by an effective\n$G$-invariant divisor  on any pair  $(Z', G)$ homotopically equivalent to $(Z,G)$.\n\\item\n $f$ is an embedding if moreover $K_{X'}$ is ample and\n\n(***)  $\\ \\ \\ K_{X'}^n = K_{X}^n$.\\footnote{ This last property for\nalgebraic surfaces follows automatically from homotopy invariance.}\n\\end{itemize}\nIn particular, if $K_{X'}$ is ample and (**) and (***) hold, also $X'$ is a diagonal SIT (special  Inoue type) manifold.\n\nA similar  conclusion holds under the alternative assumption that  the isomorphism of homology groups sends the canonical class of $W$\nto the one of $\\hat{X}$: then  $X'$ is a diagonal SIT (special  Inoue type) variety.\n\n\\end{theo}\n\nHypothesis  (1) in the previous theorem allows to derive the\nconclusion that also\n$W : = \\hat{X'} $ admits a holomorphic map $f'$ to a complex manifold $Z'$\nwith the same structure as $Z$, while\n hypotheses  (2) and following ensure  that the morphism is birational to its image,\nand the class of the image\ndivisor $f' (\\hat{X'} )$ corresponds to $\\pm$ the one of $\\hat{X} $  under\nthe identification\n$$ H_*(Z' , \\ensuremath{\\mathbb{Z}}) \\cong  H_*(\\Ga , \\ensuremath{\\mathbb{Z}}) \\cong   H_*(Z , \\ensuremath{\\mathbb{Z}}).$$\n\nSince $K_{X'}$ is ample, one uses (***) to conclude that $f'$ is an isomorphism with its image.\n\nThe next question is weak rigidity, which amounts to the existence of a\nconnected complex manifold\nparametrizing all such maps (or the  complex conjugate).\nHere  several ingredients come into play, namely, firstly the Hodge type,\nsecondly a fine analysis of\nthe structure of the action of $G$ on $Z$, in particular concerning the existence of hypersurfaces on which $G$ acts freely. Finally, one would have to see\nwhether the family of the invariant effective divisors\nthus obtained is parametrized by a connected family: this also\nrequires further work which we do not undertake here except that for the case of Inoue surfaces.\n\nIt would  take  long to analyse here the most general situation, yet\n there is an even  more general\nsituation worth to be investigated. This is the case of {\\bf orbifold\nInoue type varieties},\nwhere the action of $G$ is no longer free: this situation is\nespecially appealing from the point of view of the construction of new\ninteresting examples.\n\n\nThe paper is organized as follows:  in the first section we deal with general Inoue type manifolds,\nestablish the first  general properties\nof Inoue type manifolds, and prove our main theorem \\ref{special diagonal}. Further, more complete, results\ndealing with weak rigidity will be given elsewhere.\n\nSection two is devoted to preliminaries, for instance  on curves of \ngenus 5 admitting symmetries by  $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$.\nThis is an important background for the construction of Inoue \nsurfaces with $K^2_S = 7$ and $p_g (S)= 0$,\nwhich is explained in detail in section three.\nThe end of section three is then devoted to a new result, namely, the calculation of the\nfirst homology group of an Inoue surface, which is shown to be equal to the  group $(\\ensuremath{\\mathbb{Z}}\/4 \\ensuremath{\\mathbb{Z}})\\oplus (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$.\n\nSection four proves the main result on Inoue surfaces with $K^2_S = \n7$ and $p_g (S)= 0$.\n\nFinally, section five is devoted to showing that the the moduli space \nof Inoue surfaces is generically smooth: this is achieved\nby looking at another  realization as  bidouble covers of a four nodal cubic.\n\\section{ Inoue type manifolds and varieties}\n\nAssume that $X = \\hat{X} \/ G$ is an Inoue-type manifold, so that \nthere is an isomorphism\n$\\pi_1 (\\hat{X} ) \\cong \\pi_1 (Z )= : \\Ga $, by virtue of Lefschetz' theorem.\n\nIn general, if $W$ is another K\\\"ahler manifold with $\\pi_1 (W  ) \n\\cong \\pi_1 (Z )= : \\Ga $,\nwe would like to be able to assert that there exists a holomorphic \nmap $ f : W \\ensuremath{\\rightarrow} Z'$ where $Z'$ is another $K(\\Ga,1)$ manifold and \nwhere  $f_* : \\pi_1 (W  ) \\ensuremath{\\rightarrow} \\pi_1 (Z' ) \\cong \\Ga$ realizes the above isomorphism.\n\n This is for instance the case if $Z$ is a\ncompact quotient of an irreducible bounded symmetric domain of \ndimension at least 2 by a\ntorsion free subgroup; this follows by combining the results of Eells and Sampson  (\\cite{es})\nproving the existence of a harmonic map in each homotopy class of maps $ f : W \\ensuremath{\\rightarrow} Z$, since $Z$\nhas negative curvature, with the results of Siu (\\cite{siu1} and \n\\cite{siu2}), showing the complex dianalyticity of the resulting harmonic map  (i.e., the map $f$  is holomorphic or antiholomorphic)\nin the case where $f_* : \\pi_1 (W  ) \\ensuremath{\\rightarrow} \\pi_1 (Z' ) \\cong \\Ga$ is an isomorphism, since then the differential of $f$ has rank $\\geq 4$\n(as a linear map of real vector spaces).\nObserve that in this case $Z'$ is  $ Z$, or the complex conjugate $\\bar{Z}$.\n \nAnother case is the case where $Z$ is a compact curve of genus $g \n\\geq 2$. In this case,\nafter several results by Siu, Beauville and others (see \\cite{albgentype}), a\n  simple criterion was shown to be the existence of a surjection $\\pi_1 (W  ) \\ensuremath{\\rightarrow} \n\\pi_1 (Z )$  with finitely generated kernel (see\n\\cite{barlotti} and \\cite{cime2}, Theorem 5.14), an assumption which holds true in our situation.\n\nIt is on the above grounds that we restricted ourselves to special Inoue type manifolds\n(the diagonality assumption is only a simplifying assumption).\n\n\n\nLet us prove the first general result, namely, theorem \\ref{special diagonal}.\n\n{\\it Proof. } {\\em of Theorem \\ref{special diagonal}.}\n\n{ \\bf Step 1}\n\nThe first step consists in showing that  $W : = \\hat{X'}$ admits a holomorphic\nmapping to a manifold $Z'$ of the above type $ Z' = (A'_1 \\times \n\\dots \\times A'_r) \\times\n(C'_1 \\times \\dots \\times  C'_h) \\times (M'_1 \\times \\dots \\times\nM'_s)$, where $M'_i$ is either $M_i$ or its complex conjugate.\n\nFirst of all, by the cited results of Siu and others (\\cite{siu1}, \n\\cite{siu2},\\cite{barlotti},  \\cite{cime2}, Theorem 5.14), $W$ admits a \nholomorphic map to\na product manifold $$Z_2'\\times Z'_3 = (C'_1 \\times \\dots \\times \nC'_h) \\times (M'_1 \\times\n\\dots\n\\times  M'_s).$$\n\nLook now at the  Albanese variety $\\Alb (W)$ of the K\\\"ahler manifold \n$W$, whose fundamental\ngroup is the quotient of the Abelianization of $\\Ga = \\pi_1 (Z)$ by its torsion subgroup. \n\nWrite the fundamental group of\n$\\Alb (W)$ as the first homology group of $A  \\times  Z_2  \\times Z_3$,\ni.e., as \n$$ H_1 ( \\Alb (W)) = \\Lambda \\oplus H_1 ( Z_2, \\ensuremath{\\mathbb{Z}}) \\oplus (H_1 ( Z_3, \\ensuremath{\\mathbb{Z}})\/ \\Torsion),$$ \n($\\Alb (Z_2)$ is the product of Jacobians\n$J : = (\\Jac(C_1) \\times \\dots \\times  \\Jac(C_h))  $).\n\nSince however, by the universal property,  $\\Alb (W)$ has a holomorphic map to \n$$B' : = \\Alb (Z'_2) \\times \\Alb (Z'_3),$$\ninducing a splitting of the lattice  $H_1 (\\Alb (W), \\ensuremath{\\mathbb{Z}}) = \\Lambda \\oplus \nH_1 (B', \\ensuremath{\\mathbb{Z}})$,\nit follows that $\\Alb(W)$ splits as $A' \\times B'$.\n\nNow, we want to show that the Abelian variety $A'$ ($W$ is assumed to \nbe a projective\nmanifold) splits as desired.  This is in turn a consequence of assumption (3) in definition \\ref{SIT}.\nIn fact, the group $G$ acts on the Abelian variety $A'$ as a group of \nbiholomorphisms,\nhence it acts on $\\Lambda \\otimes \\ensuremath{\\mathcal{R}}$ commuting with multiplication by \n$\\sqrt{-1}$.\nHence multiplication by $\\sqrt{-1}$ is an isomorphism of $G$ representations, and then (3)\n implies  that $\\Lambda_i\\otimes \\ensuremath{\\mathcal{R}}$ is stable by  multiplication by \n$\\sqrt{-1}$;\nwhence $\\Lambda_i\\otimes \\ensuremath{\\mathcal{R}}$ generates a subtorus $A'_i$. Finally, $A'$ splits because $\\Lambda$ is\nthe direct sum of the sublattices $\\Lambda_i$. We are through with the \nproof of step 1.\n\n{ \\bf Step 2}\n\nConsider now the holomorphic map $ f \\colon W \\ensuremath{\\rightarrow} Z'$. We shall show \nthat  the image\n$ W' : = f (W)$ is indeed a divisor in $Z'$.\n\nThis follows from the assumption  (SAME HOMOLOGY).\n\n\\begin{lemma}\\label{cohalg}\nAssume that $W$ is a K\\\"ahler manifold, such that\n\\begin{itemize}\n\\item[i)] there is an isomorphism of\nfundamental groups $\\pi_1 (W) = \\pi_1 (\\hat{X})= \\Ga$, where $ \n\\hat{X}$ is a smooth ample\ndivisor in a $K(\\Ga, 1)$ complex projective manifold $Z$;\n\\item[ii)] there exists a holomorphic map  $ f : W \\ensuremath{\\rightarrow} Z'$, where $Z'$ is another \n$K(\\Ga, 1)$ complex  manifold,  such that $f_* : \\pi_1 (W) \\ensuremath{\\rightarrow}  \\pi_1 (Z') = \\Ga$ is an \nisomorphism, and moreover\n\\item[iii)]  {\\bf (SAME HOMOLOGY)} there is an isomorphism $\\Psi : H_*( W, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $\nof homology groups which is compatible with the homomorphisms\n$$ u \\colon H_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})   ,  u'   \\colon\n H_*( W, \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow}   H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  ,   $$\ni.e., $u \\circ \\Psi = u'$.\n\n\\end{itemize}\nThen $f$ is a generically finite  morphism of $W$ into $Z'$, and \nthe  cohomology class  $f_* ([W])$  in\n$$ H^*(Z', \\ensuremath{\\mathbb{Z}}) = H^*(Z, \\ensuremath{\\mathbb{Z}})\n=H^*(\\Ga, \\ensuremath{\\mathbb{Z}})  $$ corresponds to $\\pm 1$  the one  of $ \\hat{X}$.\n\\end{lemma}\n\n{\\it Proof of the Lemma.}\n We can identify under $\\Psi : H_*(W, \\ensuremath{\\mathbb{Z}}) \\cong\nH_*(\\hat{X}, \\ensuremath{\\mathbb{Z}})  $ the homology groups of $W$ and $ \\hat{X}$, \nand then the image\nin the homology groups of\n$ H_*(Z', \\ensuremath{\\mathbb{Z}}) = H_*(Z, \\ensuremath{\\mathbb{Z}}) =H_*(\\Ga , \\ensuremath{\\mathbb{Z}})  $ is the same.\n\nWe apply the above consideration to the fundamental classes of the oriented manifolds $W$ and $ \\hat{X}$,\nwhich are generators of the infinite cyclic top degree homology groups $H_{2n}(W, \\ensuremath{\\mathbb{Z}})$,\nrespectively $H_{2n}( \\hat{X}, \\ensuremath{\\mathbb{Z}})$.\n\nThis implies a fortiori that $ f \\colon W \\ensuremath{\\rightarrow} Z'$ is generically finite:\nsince then the homology class  $f_* ([W])$ (which we identify to a cohomology class  by virtue of Poincar\\'e duality)\nequals  the class of $ \\hat{X}$, up to sign.\n\n\\hspace*{\\fill}$q.e.d.$\n\n\n\n{\\bf Step 3}\n\nWe claim that  $ f \\colon W \\ensuremath{\\rightarrow} Z'$ is generically 1-1 onto its image $W'$.\n\nLet in fact  $d$ be the degree of $ f \\colon W \\ensuremath{\\rightarrow} W'$.\n\nThen   $f_* ([W]) = d [W']$, hence if the class of \n$\\hat{X}$ is indivisible, then obviously $d=1$.\n\nOtherwise, observe that the divisor $W'$ is an effective $G$-invariant divisor and use our assumption.\n\n\n{\\bf Step 4}\n\nWe  established that  $f$ is birational onto its image $W'$, hence it is a \ndesingularization of $W'$.\n\n\nWe use now adjunction. We claim that, since $K_W$ is nef, there exists an effective \ndivisor $\\mathfrak A$,\ncalled adjunction divisor, such that\n$$K_W = f^* (K_{Z'} + W') - \\mathfrak A.$$\n\nThis can be shown by taking the Stein factorization\n$$ W \\ensuremath{\\rightarrow} W^N \\ensuremath{\\rightarrow} W', $$\nwhere $W^N$ is the normalization of $W'$.\n\nLet $\\sC $ be the conductor ideal  $\\mathcal{H}om (p_*\\ensuremath{\\mathcal{O}}_{W^N}, \\ensuremath{\\mathcal{O}}_{W'} )$\nviewed as an ideal $\\sC \\subset \\ensuremath{\\mathcal{O}}_{W^N}$; then the Zariski  canonical divisor \nof $W^N$ satisfies \n$$K_{W^N} =   p^*   (K_{W'}) - C =   p^*   (K_{Z'} + W') - C$$\nwhere $C$ is the Weil divisor associated to the conductor ideal (the equality on the Gorenstein locus\nof $W^N$ is shown for instance in \\cite{bucharest}, then it suffices to take the direct image from the open set to\nthe whole of $W^N$).\n\nIn turn, we would have in general \n$K_W = h^* (K_{W^N } ) - \\mathfrak B $, with $\\mathfrak B $ not necessarily effective; but, by Lemma 2.5 of \\cite{alessio},\nsee also Lemma 3.39 of \\cite{KollarMori},\nand since $- \\mathfrak B $ is h-nef, we conclude that $\\mathfrak B $ is effective.\nWe establish the claim by seting $\\mathfrak A : = \\mathfrak B + h^* C$.\n\n\nObserve that, under the isomorphism of homology groups, $f^* (K_{Z'} + W')$\ncorresponds to $(K_{Z} + \\hat{X})|_{\\hat{X}}= K_{\\hat{X}} $, in \nparticular we have\n$$ K_{\\hat{X}}^n = f^* (K_{Z'} + W')^n = (K_W + \\mathfrak A)^n . $$\nIf we assume that $K_W$ is ample, then $ (K_W + \\mathfrak A)^n \\geq (K_W)^n $,\nequality holding if and only if $  \\mathfrak A = 0$.\n\nUnder assumption (**), it follows that  $$K_{\\hat{X}}^n = |G| K_X^n = \n|G| K_{X'}^n= K_W^n,$$\nhence  $  \\mathfrak A = 0$. Since however $K_W$ is ample, it follows \nthat $f$ is an embedding.\n\nIf instead we assume that $K_W$ has the same class as $ f^* (K_{Z'} + W')$,\nwe conclude first that necessarily $\\mathfrak B =0$, and then we get that $C= 0$.\n\nHence $W'$ is normal and $W$ has canonical singularities.\n\n{\\bf Step 4}\n\nFinally, the group $G$ acts on $W$, preserving the direct summands of \nits fundamental group\n$\\Ga$. Hence, $G$ acts on the curve-factors, and the locally symmetric factors.\n\nBy assumption, moreover, it sends the summand $\\Lambda_i$ to itself, \nhence we get a well defined linear action  on each\nAbelian variety $A'_i$, so that we have a diagonal linear action of \n$G$ on $A'$.\n\nSince however the image of $W$ generates $A'$, we can extend the \naction of $G$ on $W$ to a compatible  affine\naction on $A'$.\n\nOur final step consists in showing that \n\n\\begin{lemma}\\label{affine}\nGiven a  diagonal special Inoue type manifold, the real affine type of the action of $G$ on the\nAbelian variety $ A = (A_1 \\times \\dots \\times A_r)$ is determined by the fundamental group\nexact sequence\n\n$$ 1 \\ensuremath{\\rightarrow} \\Ga= \\Pi_{i=1}^r (\\Lambda_i) \\times   \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l))\n\\ensuremath{\\rightarrow} \\pi_1 ( X) \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.$$ \n\\end{lemma}\n{\\em Proof.}\nDefine as before $\\Lambda : = \\Pi_{i=1}^r (\\Lambda_i) = \\pi_1 (A)$; moreover, since all the summands in\nthe left hand side are normal in $ \\pi_1 ( X)$, set \n$$\\overline{G} : =  \\pi_1 ( X) \/ ( \\Pi_{j=1}^h (\\pi_{g_j}) \n\\times \\Pi_{l=1}^s\n(\\pi_1 (M_l)) ).$$ \n\nObserve now that $ X$ is the quotient of its universal covering \n$$\\tilde{X} = \\ensuremath{\\mathbb{C}}^m \\times \\prod_{j=1}^h\\mathbb{H}_j \\times \\prod_{l=1}^s\\mathcal{D}_l$$ by its fundamental group,\nacting diagonally (here $\\mathbb{H}_j$ is a copy of Poincar\\'e's upper half plane,while $ \\mathcal{D}_l$ is an irreducible  bounded symmetric domain\nof dimension at least two), hence we  obtain that $\\overline{G}$ acts on $\\ensuremath{\\mathbb{C}}^m$ as a group of affine transformations.\n\nLet $\\sK$ be the kernel of the associated homomorphism $$ \\alpha : \\overline{G} \\twoheadrightarrow \\im (\\alpha) =: \\hat{G} \\subset \\Aff (m, \\ensuremath{\\mathbb{C}}),$$\nand let $$\\overline{G}_1 : = \\ker (  \\alpha_L : \\overline{G} \\ensuremath{\\rightarrow} \\GL  (m, \\ensuremath{\\mathbb{C}})).$$\n\n$\\overline{G}_1 $ is obviously Abelian, and contains $\\Lambda$, and maps onto a lattice $\\Lambda' \\subset \\hat{G}$.\n\nSince $\\Lambda$ injects into $\\Lambda'$, $\\Lambda \\cap \\sK = 0 $, whence $\\sK$ injects into $G$, therefore $\\sK$ is a torsion subgroup; since $\\Lambda'$ is free, we obtain\n$$\\overline{G}_1 = \\Lambda'  \\oplus \\sK,$$\nand we finally get $$  \\sK = \\Tors (\\overline{G}_1 ), \\ \\  \\hat{G} =  \\overline{G}  \/ \\Tors (\\overline{G}_1 ).$$\n\nSince our action is diagonal, we can write $\\Lambda'  = \\oplus_{i=1}^r (\\Lambda'_i) $, and the linear action of\nthe group $G_2 : = G \/ \\sK$ preserves the summands. \n\nSince $ \\hat{G} \\subset \\Aff (\\Lambda')$, we see that $$ \\hat{G} = (\\Lambda')  \\rtimes G'_2 ,$$\nwhere $G'_2 $ is the isomorphic image of $G_2$ inside $ \\GL ( \\Lambda'))$. \n\nThis shows that the affine group  $ \\hat{G}$ is uniquely determined.\n\nFinally, using the image groups $G_{2,i}$ of $G_2$ inside $ \\GL ( \\Lambda'_i))$,\nwe can  define uniquely groups of affine transformations of $A_i$ which fully determine\nthe diagonal action of $G$ on $A$ (up to real affine automorphisms of each $A_i$).\n\n\n\\hspace*{\\fill}$q.e.d.$\n\n\n\\QED {\\em  for Theorem \\ref{special diagonal}}\n\\bigskip\n\n\nIn order to obtain weak rigidity results, one has to \nintroduce  a further invariant, called Hodge type, according to\nthe following\ndefinition. We shall return to this question  in a  sequel to this paper.\n\n\n\\begin{defin}\n   Let $X$ be an  Inoue-type manifold (or variety) of special diagonal \ntype, with\n\n$ Z = (A_1 \\times \\dots \\times A_r) \\times  (C_1 \\times \\dots \\times\nC_h) \\times (M_1 \\times \\dots \\times\nM_s)$\n\nThen an invariant of the integral representation $ G \\ensuremath{\\rightarrow} Aut (\\Lambda_i)$\nis its {\\bf Hodge type},\nwhich is the datum, given the decomposition of $\\Lambda_i \\otimes \\ensuremath{\\mathbb{C}}$\nas the sum of\nisotypical components $$\\Lambda_i \\otimes \\ensuremath{\\mathbb{C}} = \\oplus_{\\chi \\in \\Irr (G)}\nU_{i, \\chi}$$\nof the dimensions   $$\\nu (i, \\chi): = dim_{\\ensuremath{\\mathbb{C}}} U_{i, \\chi}  \\cap\nH^{1,0}(A_i) $$\nof the Hodge summands for non real representations.\n\\end{defin}\n\n\n\n\n\n\n\\section {Genus $5$ curves having a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$-action}\\label{section2}\n\nThe following is well known (see however section 1 of \\cite{burniat1}).\n\n\\begin{lemma}\\label{genus1}\nLet $E_1$ be a compact curve of genus $1$  and assume $ G_1: = (\\ensuremath{\\mathbb{Z}}\/ 2\n\\ensuremath{\\mathbb{Z}})^n \\subset Aut (E_1)$. Then $ n \\leq 3$, and, for $n=3$,\n$ E_1\/ G_1 \\cong \\ensuremath{\\mathbb{P}}^1$ with quotient map  branched on exactly\n$4$ points $P_1, \\dots, P_4$.\nThe covering  $ E_1 \\ensuremath{\\rightarrow}  E_1\/ G_1$ factors through multiplication by \n2 in $E_1$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{genus5}\nLet $D$ be a compact curve of genus $5$  and assume $ H : = (\\ensuremath{\\mathbb{Z}}\/ 2\n\\ensuremath{\\mathbb{Z}})^n \\subset Aut (D)$.\nThen $ n \\leq 4$, and, if $ n = 4$, $ D \/ H \\cong \\ensuremath{\\mathbb{P}}^1$ and the quotient map is branched on exactly\n$5$ points $P_1, \\dots, P_5$.\n\\end{lemma}\n\\proof\nBy the Hurwitz' formula, one has, setting $ D \/ H = C$, and setting\n$h$= genus ($C$),\n$$ 8 = 2^n (2h-2 + \\frac{m}{2}) \\Leftrightarrow  2^{n-4} (4h-4 + m) = 1$$\nwhere $m$ is the number of branch points $P_1, \\dots, P_m$. Hence\n$n \\leq 4 $. If $n=4$, then $h=1$ is not possible,\nsince in this case  the  abelianization of  $\\pi_1 (C \\setminus {P_1}) $ would equal $\\pi_1\n(C)$, and one would have $m=0$,\na contradiction.\nHence $h=0$ and $m=5$.\n\n\\hspace*{\\fill}$q.e.d.$\n\nThe following geometrical game is based on the fact that the 15\nintermediate double covers of\n   $ D \/ H = \\ensuremath{\\mathbb{P}}^1$ are 5 elliptic curves (branched each on 4 of the 5\nbranch points) and 10 rational curves\n(branched each on 2 of the 5 branch points).\nLet $A_i$ be the elliptic curve branched on all the five points with\nexclusion of $P_i$: then\n$D \\ensuremath{\\rightarrow} D \/ H $ factors as $$D \\ensuremath{\\rightarrow}  A_i \\ensuremath{\\rightarrow} A_i \\ensuremath{\\rightarrow} D \/ H $$\nwhere the middle map is multiplication by 2, and $D \\ensuremath{\\rightarrow}  A_i$ is the\nquotient by an involution with\n   fixed points; the number of  fixed points is exactly 8, since, if $g\n\\in H$, the fixed set $\\Fix(g)$ is an $H$-orbit, and has\ntherefore cardinality equal to a multiple of $8$. The other 10\ninvolutions have no fixed points, hence they\nyield each an unramified covering of a curve $C_j$ of genus $3$.\n\nWe try now to stick to Inoue's original notation, except that we refuse to\nuse the classical symbol for  the {\\em Weierstrass $\\wp$-function} to denote\nthe Legendre function $\\sL$;   $\\sL$ is a  homographic transform of\n  the Weierstrass function, but not equal to the Weierstrass function.\n\n  The Legendre function satisfies the quadratic relation (see \\cite{burniat1})\n  $$ y^2 = (\\sL^2 - 1) ( \\sL^2 - a^2) .$$\n\nLet $E_1$, $E_2$ be two complex elliptic curves. We assume $E_i = \\ensuremath{\\mathbb{C}}\n\/ \\langle 1, \\tau_i \\rangle$. Moreover, we denote by\n$z_i$ a uniformizing parameter on $E_i$.\n\n  Then $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 =\n\\langle \\gamma_1, \\gamma_2, \\gamma_3 \\rangle$ acts on\n$E_i$ by\n\\begin{itemize}\n   \\item[-] $\\gamma_1(z_i) = -z_i$;\n\\item[-] $\\gamma_2(z_i) = z_i + \\frac 12$;\n\\item[-] $\\gamma_3(z_i) = z_i + \\frac{\\tau_i}{2}$.\n\\end{itemize}\n\n We consider the Legendre $\\sL$-function for $E_i$ and denote it by \n$\\sL_i$,  for $i=1,2$:\n$\\sL_i$ is a meromorphic\nfunction on $E_i$ and $\\sL_i \\colon E_i \\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{P}}^1$ is a double cover \nramified in\n$\\pm 1, \\pm a_i \\in \\ensuremath{\\mathbb{P}}^1 \\setminus \\{0, \\infty\\}$.\n\nIt is well known that we have (cf. \\cite{inoue}, lemma 3-2, and also \ncf. \\cite{burniat1}, pages 52-54, section 1 for an algebraic \ntreatment):\n\\begin{itemize}\n   \\item[-] $\\sL_i(\\frac 12) = -1$, $\\sL_i(0) = 1$,\n$\\sL_i(\\frac{\\tau_i}{2}) = a_i$,  $\\sL_i(\\frac{1+\\tau_i}{2}) = - \na_i$;\n\\item[-] let $b_i:= \\sL_i(\\frac{\\tau_i}{4})$: then $b_i^2 = \na_i$;\n\\item[-] $\\frac{\\rm{d} \\sL_i}{\\rm{d} z_i} (z_i) = 0$ if and \nonly if \n$z_i \\in \\{0, \\frac 12,\\frac{\\tau_i}{2}, \\frac{1+\\tau_i}{2} \n\\}$, since $\\frac { d x_i}{ \\sL_i} = dz_i$.\n\\end{itemize}\n\nMoreover \n$$\n\\sL_i(z_i) = \\sL_i(z_i+1) = \\sL_i(z_i+\\tau_i) = \\sL_i(-z_i) = \n-\\sL_i(z_i+ \\frac 12),\n$$\n$$\n\\sL_i(z_i + \\frac{\\tau_i}{2}) = \n\\frac{a_i}{\\sL_i(z_i)}.\n$$\nWe consider now the vector space \n$V_i:=H^0(E_i, \\ensuremath{\\mathcal{O}}_{E_i}(2[0]))$ \n(for $i=1,2$), and note that $V_i \n\\cong \\ensuremath{\\mathbb{C}}^2$ with basis\n$1, \\sL_i$, since \n$$div (1 - \\sL_i) = 2 [0] \n-  {\\it Poles} (\\sL_i) .$$ .\n\nObserve that $[- \\frac 14] + [\\frac \n14]$ is a $(\\ensuremath{\\mathbb{Z}}\/ 2\\ensuremath{\\mathbb{Z}})^2 =\\langle \n\\gamma_1, \\gamma_2 \\rangle$ - \ninvariant divisor, hence $V_i$\nis a $(\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2$ - module and \nsplits in its isotypical components as\n$$ V_i = V_i^{++} \\oplus \nV_i^{+-},\n$$ since $1$ is invariant under $(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^2$ and $\\sL_i$ \nis invariant under $\\gamma_1$ and is an eigenvector with\neigenvalue \n$-1$ of $\\gamma_2$.\n\n\nIf $c \\in \\ensuremath{\\mathbb{C}} \\setminus \\{ \\pm 1, \\pm a_i, \\pm \na_1a_2 \\}$, then the divisor\n$$ D_c:=\\{ (z_1,z_2) \\in E_1 \\times E_2 \n\\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c \\}\n$$\n\nof bidegree $(2,2)$ is a \nsmooth curve of genus $5$. More precisely,\n$$\n\\ensuremath{\\mathcal{O}}_{E_1 \\times \nE_2}(D_c) \\cong p_1^* \\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}(2[0]).\n$$\n\nConsider the product action of $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \n\\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ \non $E_1 \\times E_2$.\n\\begin{rem} 1) It is easy \nto see that $D_c$ is invariant under the subgroup $H \n\\leq (\\ensuremath{\\mathbb{Z}}\/2 \n\\ensuremath{\\mathbb{Z}})^3 \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ given by\n$$ (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \\cong H:= \n\\langle (\\begin{pmatrix} 1\\\\ 0 \\\\ 0 \n\\end{pmatrix},\\begin{pmatrix} \n0\\\\0\\\\0\n\\end{pmatrix}),(\\begin{pmatrix} 0\\\\ 0 \\\\ 0 \n\n\\end{pmatrix},\\begin{pmatrix} 1\\\\0\\\\0 \n\\end{pmatrix}),(\\begin{pmatrix} \n0\\\\ 1 \\\\ \n0\n\\end{pmatrix},\\begin{pmatrix} 0\\\\1\\\\0 \\end{pmatrix}) \n\\rangle,\n$$\n\nwhere the coordinates are taken with respect to the \nbasis $\\gamma_1, \n\\gamma_2, \\gamma_3$ on each factor.\n\n2) Moreover, \nif we choose $c:= b_1b_2$, then we see that for \n$(z_1,z_2) \\in \nD_{b_1b_2}$:\n$$\n\\sL_1(z_1 + \\frac{\\tau_1}{2})\\sL_2(z_2 + \n\\frac{\\tau_2}{2}) = \n\\frac{a_1a_2}{\\sL_1(z_1)\\sL_2(z_2)} \n=\n\\frac{a_1a_2}{b_1b_2} = b_1b_2,\n$$ whence $D_{b_1b_2}$ is invariant \nunder\n$$ (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4 \\cong G:= H \\oplus \\langle (\\begin{pmatrix} \n0\\\\ 0 \n\\\\ 1 \\end{pmatrix},\\begin{pmatrix} 0\\\\0\\\\1\n\\end{pmatrix}) \\rangle \\leq (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3 \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3.\n$$\n\\end{rem}\n\nWe want to show that the converse holds. More precisely, we prove the following\n\\begin{prop}\\label{genus5,2}\n   Let $f \\colon D \\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{P}}^1$ be the maximal $G:= (\\ensuremath{\\mathbb{Z}}\/2\n\\ensuremath{\\mathbb{Z}})^4$-covering branched in $5$ given points $p_1, \\ldots , p_5 \\in\n\\ensuremath{\\mathbb{P}}^1$. Then there are two elliptic curves $E_1$, $E_2$ such that $D\n\\subset E_1 \\times E_2$ is a $G$-invariant divisor with\n$$\n\\ensuremath{\\mathcal{O}}_{E_1 \\times E_2}(D) \\cong p_1^* \\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^*\n\\ensuremath{\\mathcal{O}}_{E_2}(2[0]).\n$$ Choosing appropriate coordinates we can moreover assume that $D=\n\\{ (z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2) = b_1b_2 \\}$.\n\\end{prop}\n\\begin{proof}\n   Let $e_1, e_2,e_3,e_4$ be a basis of the $\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}$ - vectorspace\n$G$ and let $D \\rightarrow \\ensuremath{\\mathbb{P}}^1$ branched in $p_1, \\ldots\np_5$ be given by the {\\em appropriate orbifold homomorphism}\n$$\n\\varphi : \\mathbb{T}(2,2,2,2,2) := \\langle x_1, \\ldots , x_5 |\n\\prod_{i=1}^5 x_i, x_1^2, \\ldots, x_5^2 \\rangle \\rightarrow\n(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^4,\n$$ where $\\varphi(x_i) = e_i$ for $1\\leq i \\leq4$, $\\varphi(x_5) =\ne_5:=e_1 +e_2+e_3+e_4$. Then Hurwitz' formula shows that\n$D$ is a smooth curve of genus $5$.\n\nNote that the only elements of $G$ having fixed points on $D$ are the 5 elements \n$e_i$ ($1 \\leq i \\leq 5$), and each of them has exactly $8$\nfixed points on $D$. Hence, $E_i:=D\/\\langle e_i\\rangle$ is an\nelliptic curve.\n\nWe get therefore $5$ elliptic curves (as intermediate covers of $D\n\\rightarrow \\ensuremath{\\mathbb{P}}^1$), all endowed with a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^3$ - action.\n\nChoose two of these elliptic curves, say $E_1, E_2$, and consider the morphism\n$$ i \\colon D \\rightarrow D\/\\langle e_1\\rangle \\times D\/\\langle\ne_2\\rangle = E_1 \\times E_2.\n$$\n\nThen $D \\cdot E_i =2$ and $i$ is an embedding of $D$ as a $(\\ensuremath{\\mathbb{Z}} \/ 2\n\\ensuremath{\\mathbb{Z}})^4$ - invariant divisor of bidegree $2$.\n\nWe fix the origin in both elliptic curves so that $D= \\{s=0 \\}$, where $s \\in H^0(E_1 \\times E_2,\np_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^G$.\n\nIt remains to show that we can assume $D$ to be of the form $\\{\n(z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) =\nb_1b_2 \\}$.\n\nFor this we show that\n$$H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^H \\cong \\ensuremath{\\mathbb{C}}^2.$$ In fact,\n\n\\begin{multline*}\n   H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))) \\otimes\np_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0]))) = V_1 \\otimes V_2 = \\\\ = (V_1^{+++} \\oplus\nV_1^{++-}) \\otimes (V_2^{+++} \\oplus V_2^{++-}),\n\\end{multline*} whence\n\\begin{multline*} H^0(E_1 \\times E_2, p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0])))\n\\otimes p_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0])))^H = \\\\ = (V_1^{+++}\n\\otimes V_2^{+++}) \\oplus (V_1^{++-} \\otimes V_2^{++-}).\n\\end{multline*}\n\nTherefore we have a pencil of  $H$-invariant divisors $D_c:=\\{\n(z_1,z_2) \\in E_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c\n\\}$. It is now obvious that $D_c$ is $G$ - invariant iff $c = \\pm b_1b_2$.\nThe change of sign for $b_i$ is achieved by changing the point $\\frac{\\tau_i}{4}$\nwith $\\frac{\\tau_i}{4} + \\frac{1}{2}$.\n\\end{proof}\n\nConsider now for the moment  the action of $(\\ensuremath{\\mathbb{Z}} \/\n2\\ensuremath{\\mathbb{Z}})^2 = \\langle \\gamma_1, \\gamma_2\\rangle$ on $E_1$ and $E_2$, and\nthe induced product action on $E_1 \\times E_2$. Assume $H \\cong (\\ensuremath{\\mathbb{Z}}\n\/ 2 \\ensuremath{\\mathbb{Z}})^3 \\leq (\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2 \\times (\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2$,\nsuch that\n\\begin{enumerate}\n  \\item on each factor $E_i$ it induces the action \nof $(\\ensuremath{\\mathbb{Z}} \/ 2\\ensuremath{\\mathbb{Z}})^2= \n\\langle \\gamma_1, \\gamma_2 \\rangle$;\n\\item there \nis an $H$-invariant pencil in $\\ensuremath{\\mathbb{P}}(V_1 \\otimes V_2)$, \ni.e., $\\dim \n(V_1 \\otimes V_2)^H = 2$.\n\\end{enumerate}\n\nThen it is easy to see \nthat we can choose $(\\gamma_1, 0)$, \n$(0,\\gamma_1)$, $(\\gamma_2, \n\\gamma_2)$ as basis of $H$.\n\nTherefore\n$$ (V_1 \\otimes V_2)^{+++} = \n(V_1^{+++} \\otimes V_2^{+++}) \\oplus \n(V_1^{++-} \\otimes \nV_2^{++-}),\n$$ and the $H$- invariant pencil is given by $D_c = \n\\{(z_1,z_2) \\in \nE_1 \\times E_2 \\ | \\ \\sL_1(z_1) \\sL_2(z_2) = c \n\\}$.\n\nConsider now $G:= H \\oplus \\langle (\\gamma_3, \\gamma_3) \\rangle \n\\cong \n(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$. \n\nThen $D:= \\{(z_1,z_2) \\in E_1\n\\times E_2 \\ | \\ \n\\sL_1(z_1) \\sL_2(z_2) = b_1b_2\\}$.\n\n\n\n\n\\begin{rem} \\\n\\begin{enumerate}\n \n\\item The restriction map  $$ H^0(E_1 \\times E_2, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}) \\rightarrow H^0(D, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes\np_2^* \n\\ensuremath{\\mathcal{O}}_{E_2} | D)\n$$ is an isomorphism of $H$-modules.\n\\item There is a \npencil of such $H$- invariant divisors of degree 4 \non $D$. But there \nis no one which is invariant under $(\\ensuremath{\\mathbb{Z}} \/2\n\\ensuremath{\\mathbb{Z}})^4$.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{proof}[Proof \nof rem. ( 1).] Let be $S:= E_1 \\times E_2$ and for \nsimplicity we \nwrite\n$$\n\\ensuremath{\\mathcal{O}}_S(2,0) := p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0]) \\otimes p_2^* \n\\ensuremath{\\mathcal{O}}_{E_2}.\n$$ Then we consider the exact sequence:\n$$ 0 \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(-D) \\otimes  \\ensuremath{\\mathcal{O}}_S(2,0) \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(2,0) \\rightarrow \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D \\rightarrow 0.\n$$ By K\\\"unneth's formula \nwe get\n\\begin{itemize}\n  \\item[i)] $h^0(\\ensuremath{\\mathcal{O}}_S(-D) \\otimes \n\\ensuremath{\\mathcal{O}}_S(2,0)) = \nh^0(\\ensuremath{\\mathcal{O}}_{E_1})h^0(\\ensuremath{\\mathcal{O}}_{E_2}(-2) = 0$;\n\\item[ii)] \n$h^1(\\ensuremath{\\mathcal{O}}_S(-D) \\otimes  \\ensuremath{\\mathcal{O}}_S(2,0)) = 2$.\n\\end{itemize} \nTherefore\n$$ r \\colon H^0(S, \\ensuremath{\\mathcal{O}}_S(2,0)) \\cong \\ensuremath{\\mathbb{C}}^2 \\rightarrow \nH^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D)\n$$ is injective. Since $D$ is not \nhyperelliptic it follows by \nClifford's theorem that $h^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) \\leq 2$.\nThis implies that $h^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) = 2$ and $r$ is \nan isomorphism (of \n$H$-modules).\n\\end{proof}\n\n\\begin{rem}\n  This implies that\n$$ H^0(S, \n\\ensuremath{\\mathcal{O}}_S(2,0) \\otimes \\ensuremath{\\mathcal{O}}_D) \\cong \\ensuremath{\\mathbb{C}}^2 = V^{+++} \\oplus \nV^{++-}.\n$$\n\\end{rem}\n\n\n\\section{Inoue surfaces with $p_g=0$ and \n$K_S^2 = 7$}\n\nIn \\cite{inoue} the author describes the construction \nof a family of \nminimal surfaces of general type $S$ with $p_g= 0$, \n$K_S^2\n= 7$.\n\nWe briefly recall the construction of these surfaces \nand, for lack of \nreference, we calculate $K_S^2$ and $p_g(S)$.\n\nFor \n$i \\in \\{1,2,3,4\\}$, let $E_i := \\ensuremath{\\mathbb{C}} \/ \\langle 1, \\tau_i \\rangle$ \nbe \na complex elliptic curve. Denoting again by $z_i$ a\nuniformizing \nparameter of $E_i$, we consider the following five \ninvolutions on \n$T:=E_1 \\times E_2\\times E_3 \\times E_4$:\n\\begin{itemize}\n\\item[] $ g_1(z_1,z_2,z_3,z_4) = \n(-z_1+\\frac 12, z_2 + \\frac 12, z_3,z_4)$,\n\\item[] $g_2(z_1,z_2,z_3,z_4) \n= (z_1, -z_2 + \\frac 12, z_3 +\\frac 12,-z_4+ \\frac 12)$,\n\\item[] $g_3(z_1,z_2,z_3,z_4) = (z_1+\\frac 12, z_2, -z_3 + \\frac 12,-z_4  + \n\\frac 12)$,\n\\item[]  $g_4(z_1,z_2,z_3,z_4) = (z_1, z_2, -z_3,-z_4)$,\n\\item[] $g_5(z_1,z_2,z_3,z_4) = (z_1+\\frac{\\tau_1}{2}, \nz_2+\\frac{\\tau_2}{2}, \nz_3+\\frac{\\tau_3}{2},z_4+\\frac{\\tau_4}{2})$.\n\\end{itemize}\n Then $G:= \\langle g_1,g_2,g_3,g_4,g_5 \\rangle \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5$.\n\nConsider\n$$\n\\hat{X} := \\{ (z_1,z_2,z_3,z_4)) \\in T \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2)\\sL_3(z_3)  = b_1b_2b_3, \\ \\sL_3(z)\\sL_4(z_4) =\nb_3b_4\\}.\n$$ Then\n\\begin{itemize}\n   \\item[-] $\\hat{X}$ is a smooth complete intersection of two\nhypersurfaces in $T$ of respective multidegrees $(2,2,2,0)$ and\n$(0,0,2,2)$;\n\\item[-] $\\hat{X}$ is invariant under the action of $G$, and $G$ acts\nfreely on $\\hat{X}$.\n\\end{itemize}\n\n\nThe above equations show that  $\\hat{X}$ is the complete intersection of a $G$ -\ninvariant divisor $X_1$ in the linear system\n$$\n\\ensuremath{\\mathbb{P}}(H^0(E_1 \\times E_2 \\times E_3 \\times E_4, p_1^*\\ensuremath{\\mathcal{O}}_{E_1}(2[0])\n\\otimes p_2^*\\ensuremath{\\mathcal{O}}_{E_2}(2[0]) \\otimes\np_3^*\\ensuremath{\\mathcal{O}}_{E_3}(2[0]) \\otimes p_4^*\\ensuremath{\\mathcal{O}}_{E_4})),\n$$  with a $G$-invariant divisor $X_2$ ($\\cong E_1 \\times E_2 \\times\nD$), where $D$ is a curve of genus $5$, a Galois cover\nwith group $(\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}})^4$ of the projective line  ramified in 5\npoints) in the linear system\n$$\n\\ensuremath{\\mathbb{P}}(H^0(E_1 \\times E_2 \\times E_3 \\times E_4, p_1^*\\ensuremath{\\mathcal{O}}_{E_1} \\otimes\np_2^*\\ensuremath{\\mathcal{O}}_{E_2} \\otimes p_3^*\\ensuremath{\\mathcal{O}}_{E_3}(2[0])\n\\otimes p_4^*\\ensuremath{\\mathcal{O}}_{E_4}(2[0]))).\n$$\n\n\n\\begin{rem}\n   It is easy to see from the above explicit description that $G$ acts\nfreely on $\\hat{X}$. Note that the action of $G$ has fixed points\non $T$, and even on $X_2$.\n\\end{rem}\n\n\\begin{defin} A smooth projective algebraic surface $S:= \\hat{X} \/ G$ as above\nis called an {\\em Inoue surface} with $K_S^2 =7$ and\n$p_g=0$.\n\n\\end{defin} \n \\begin{rem}\n   It is immediate to see that the involution $g_4$ acts freely on  $ \\hat{X} $\n   and trivially on $(E_1 \\times E_2)$.\n   \n    It follows therefore that,\n    setting $ \\hat{X}_2 : =  \\hat{X} \/ g_4 $,\n    $S$ has another\n   representation as $ \\hat{X}_2 \/ (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4$, where\n   $ \\hat{X}_2$ is a divisor in the product $(E_1 \\times E_2 \\times C)$,\n  and $C$ is a smooth curve of genus $3$.\n   \nThis was the representation of $S$ announced in \\cite{survey}, and it follows from our results that\nthis representation is also unique.\n\n\\end{rem}\n\n\n\nWe will show in the next lemma that $S$ is a minimal\nsurface of general type with $K_S^2 = 7$ and $p_g=0$.\n\n\n\\begin{lemma}\\label{invariants}\n  Let $S=\\hat{X} \/G$ be as above. \nThen $S$ is a minimal surface of \ngeneral type with $K_S^2 = 7$ and \n$p_g=0$.\n\\end{lemma}\n\n\\begin{proof}\n\nSince $\\hat{X}$ is of general \ntype, also $S$ is of general type being \nan \\'etale quotient of \n$\\hat{X}$ by a finite group.\n\n\nWe first remark that $\\hat{X}$ is a \n$G$-invariant hypersurface of \nmultidegree $(2,2,4)$ in $W:=E_1 \n\\times E_2 \\times D$,\nwhere $D \\subset E_3\\times E_4$ is a smooth \ncurve of genus 5 given by \nthe equation\n$$\n\\{ (z_3,z_4))\\in E_3 \n\\times E_4 \\ | \\ \\sL_3(z)\\sL_4(z_4) = b_3b_4\\}.\n$$\n\nBy the adjunction \nformula, the canonical divisor of $\\hat{X}$ is the \nrestriction to \n$\\hat{X}$ of a divisor of multidegree $(2,2,12)$\non $W$.\n\nTherefore \nwe can calculate (denoting by $F_i$ the fibre of the \nprojection of \n$W$ on the $(j,k)$-th coordinate (with $\\{i,j,k\\} \n=\n\\{1,2,3\\}$).\n\\begin{multline*} K_{\\hat{X}}^2 = ((K_W + \n[\\hat{X}])|S)^2 = \n(2F_1+2F_2+12F_3)^2(2F_1+2F_2+4F_3)=\\\\ = \n(8F_1F_2+\n48F_1F_3 + 48 F_2F_3)(2F_1+2F_2+4F_3)= \\\\ = 32F_1F_2F_3 + \n96F_1F_2F_3+96F_1F_2F_3 = 224 = 7 \\cdot 2^5.\n\\end{multline*}\n\nSince $G$ acts freely on $\\hat{X}$, we obtain\n$$ \nK_S^2 = \\frac{224}{|G|} = \\frac{224}{2^5} = 7.\n$$\n\nMoreover,  \nconsider the exact sequence\n\\begin{equation}\\label{adjunction}\n  0 \n\\rightarrow \\ensuremath{\\mathcal{O}}_W(K_W) \\rightarrow \\ensuremath{\\mathcal{O}}_W(K_W + [\\hat{X}]) \n\\rightarrow \\omega_{\\hat{X}} \\rightarrow 0.\n\\end{equation}\n\nUsing \nK\\\"unneth's formula and Kodaira's vanishing theorem, we \nget:\n\\begin{itemize}\n\\item[-] $\\dim H^0(W, \\ensuremath{\\mathcal{O}}_W(K_W)) = \n5$,\n\\item[-] $\\dim H^0(W, \\ensuremath{\\mathcal{O}}_W(K_W+ [X])) = 32$,\n\\item[-] $H^i(W, \n\\ensuremath{\\mathcal{O}}_W(K_W+ [X])) = 0$, for $i=1,2,3$,\n\\item[-] $\\dim H^1(W, \n\\ensuremath{\\mathcal{O}}_W(K_W)) = 1 +5+5 = 11$,\n\\item[-] $\\dim H^2(W, \\ensuremath{\\mathcal{O}}_W(K_W)) = \nq(W) (= q(\\hat{X})) = 7$.\n\\end{itemize} Therefore, by the long exact \nsequence associated to \n(\\ref{adjunction}) we get:\n$$ p_g(\\hat{X}) = \nh^0(\\hat{X}, \\omega_{\\hat{X}}) = 32+11-5 = 38.\n$$ \nTherefore\n$$\n\\chi(\\ensuremath{\\mathcal{O}}_{\\hat{X}}) = 1+p_g(\\hat{X}) -q(\\hat{X}) = 1 + \n38 -7 =32.\n$$\n\nThis implies that $\\chi(\\ensuremath{\\mathcal{O}}_S) = 1$. In order to show \nthat $p_g(S) = \n0$, it suffices to show that $q(S) =0$. Using the \nfact that\n$$ i^* \\colon H^0(W, \\Omega^1_W) \\rightarrow H^0(\\hat{X}, \n\\Omega^1_{\\hat{X}})\n$$ is an isomorphism and that\n$$ H^0(W, \n\\Omega^1_W) = H^0(E_1, \\Omega^1_{E_1})\\oplus H^0(E_2, \n\\Omega^1_{E_2}) \\oplus H^0(D, \\Omega^1_D),\n$$ it is easy to see that \n$H^0(W, \\Omega^1_W)^G = H^0(S, \n\\Omega^1_S)=0$.\n\\end{proof}\n\n\\subsection{The torsion group of Inoue surfaces with $K_S^2 =7$}\nThe aim of the section is to prove the following\n\\begin{theo}\\label{homology}\nLet $S$ be an Inoue surface with $K_S^2 = 7$. Then \n$$\nH_1(S, \\ensuremath{\\mathbb{Z}}) \\cong \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4.\n$$\n\\end{theo}\nIt is clear from the construction that the fundamental group of an Inoue surface sits in an exact sequence\n\\begin{equation}\\label{fundex}\n1 \\rightarrow \\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\pi_1(S)\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n\\end{equation}\n where $\\pi_5$ denotes the fundamental group of a compact curve\n of genus five.\n \n Observe that after moding out by $\\ensuremath{\\mathbb{Z}}^4$ in the exact sequence (\\ref{fundex}) we obtain the orbifold exact sequence (plus a summand $\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}$) of the maximal $(\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^4$-covering of $\\ensuremath{\\mathbb{P}}^1$ ramified in 5 points:\n \\begin{equation}\\label{orbifold}\n 1 \\ensuremath{\\rightarrow} \\pi_5 \\ensuremath{\\rightarrow} \\pi_1^{orb} \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}  \\ensuremath{\\rightarrow} G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^4 \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} 1,\n \\end{equation}\n where $\\pi_1^{orb}:=\\pi_1^{orb}(\\ensuremath{\\mathbb{P}}^1 \\setminus\\{p_1, \\ldots , p_5\\};2,2,2,2,2)$.\n \n The proof of the above theorem will be divided in several steps, and we shall first \n prove  some auxiliary results.\n\\begin{lemma}\\label{action}\nAssume that there is an exact sequence of groups\n$$\n1 \\ensuremath{\\rightarrow} \\Lambda \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nwhere $\\Lambda$ and $G$ are abelian. \n\nAssume moreover that\n\\begin{itemize}\n\\item[($\\#$)]  $\\Lambda$ admits a a system $L$ of generators with the following property:\n $ \\forall h \\in L$  $ \\exists g \\in G$ such that \n$$\nghg^{-1} = -h.\n$$\n\\end{itemize}\nThen $2 \\Lambda \\subset [\\Gamma', \\Gamma']$ and, in particular, we have an exact sequence \n\\begin{equation}\\label{lem1}\n\\Lambda\/ 2 \\Lambda \\ensuremath{\\rightarrow} \\Gamma'^{ab} \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $h \\in \\Lambda$, $g \\in G$ be such that $ghg^{-1} = -h$. Then $[g,h] = -2h$, whence $2h \\in [\\Gamma', \\Gamma']$. \nSince this holds for all $h \\in L$ and $L$ generates $\\Lambda$ the claim follows.\n\n\\end{proof}\n\n\\begin{rem}\\label{reduction}\ni) Assume that there is an exact sequence of groups\n$$\n1 \\ensuremath{\\rightarrow} H \\ensuremath{\\rightarrow} \\Gamma \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nwhere $G$ is abelian. Defining $\\Gamma' := \\Gamma \/ [H,H]$,  we have an exact sequence\n\\begin{equation}\\label{exrem}\n1 \\ensuremath{\\rightarrow} \\Lambda:= H^{ab} \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n\\end{equation}\nand we have\n$$\n(\\Gamma')^{ab} = \\Gamma^{ab}.\n$$\nSuppose now that assumption ($\\#$) of lemma \\ref{action} is satisfied for the exact sequence (\\ref{exrem}). Then by lemma \\ref{action} we have an exact sequence \n\\begin{equation}\n\\tilde{\\Lambda} :=  \\Lambda\/2 \\Lambda \\ensuremath{\\rightarrow} \\Gamma^{ab} \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n\\end{equation}\n\n\\noindent \nii) Define $\\Gamma'':= \\Gamma' \/ 2 \\Lambda$. Then we have an exact sequence\n$$\n1 \\ensuremath{\\rightarrow} \\tilde{\\Lambda} \\ensuremath{\\rightarrow} \\Gamma'' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $(\\Gamma'')^{ab} = (\\Gamma')^{ab} = \\Gamma^{ab}$.\n\nChoose generators $(g_i)_{i \\in I}$ of $G$ and choose for each $g_i$ a lift $\\gamma_i$ to $\\Gamma''$. Moreover, let $(\\lambda_j)_{j \\in J}$ be generators of $\\Lambda$ and denote their images in $\\tilde{\\Lambda}$ by $\\tilde{\\lambda}_j$. Then obviously $\\Gamma'' = \\langle (g_i)_{i \\in I}, (\\tilde{\\lambda}_j)_{j \\in J} \\rangle$.\n\nAssume also that \n\\begin{itemize}\n\\item[($\\# \\#$)] $\\gamma_i \\lambda_j \\gamma_i^{-1} = \\pm \\lambda_j$, $\\forall i \\in I, \\ \\forall j \\in J$.\n\\end{itemize}\nThen (since $2\\tilde{\\Lambda} = 0$) we have $[\\gamma_i, \\tilde{\\lambda}_j] = 0$ (in $\\Gamma''$), $\\forall i \\in I, \\ \\forall j \\in J$.\n\n In particular this implies that \n$$\n\\Gamma'' \/ \\langle [\\gamma_i, \\gamma_j], i,j \\in I \\rangle = (\\Gamma'')^{ab} = \\Gamma^{ab}.\n$$\n\\end{rem}\n\nSince the genus 5 curve $D$ is an ample divisor $D \\subset E_3 \\times E_4$, we have by Lefschetz' theorem a surjective morphism\n $$\n\\varphi:  \\pi_5 \\cong \\pi_1(D) \\rightarrow \\pi_1(E_3 \\times E_4) \\cong \\ensuremath{\\mathbb{Z}}^4.\n $$\nDefining  $K:= \\ker \\varphi$ and  $\\Lambda:= (\\pi_5 \\times \\ensuremath{\\mathbb{Z}}^4)\/ (K \\times \\{0\\}) \\cong \\ensuremath{\\mathbb{Z}}^4 \\times \\ensuremath{\\mathbb{Z}}^4$, we get the exact sequence\n\\begin{equation}\\label{prop1}\n1 \\rightarrow \\Lambda \\rightarrow \\pi_1(S) \/ K \\rightarrow (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5 \\rightarrow 1.\n\\end{equation}\n\n\\begin{prop} \n$P:=(\\pi_1(S) \/ K)^{ab} \\cong  \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4$.\n\\end{prop}\n\n\\begin{proof}\nWe first verify condition ($\\#\\#$) for the exact sequence (\\ref{prop1}). \n\nFor this we recall the description of  the action of $G:= \\langle g_1,g_2,g_3,g_4,g_5 \\rangle \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5$ on \n$T:=E_1 \\times E_2\\times E_3 \\times E_4$:\n\\begin{itemize}\n\\item[] $ g_1(z_1,z_2,z_3,z_4) = \n(-z_1+\\frac 12, z_2 + \\frac 12, z_3,z_4)$,\n\\item[] $g_2(z_1,z_2,z_3,z_4) \n= (z_1, -z_2 + \\frac 12, z_3 +\\frac 12,-z_4+ \\frac 12)$,\n\\item[] $g_3(z_1,z_2,z_3,z_4) = (z_1+\\frac 12, z_2, -z_3 + \\frac 12,-z_4  + \n\\frac 12)$,\n\\item[]  $g_4(z_1,z_2,z_3,z_4) = (z_1, z_2, -z_3,-z_4)$,\n\\item[] $g_5(z_1,z_2,z_3,z_4) = (z_1+\\frac{\\tau_1}{2}, \nz_2+\\frac{\\tau_2}{2}, \nz_3+\\frac{\\tau_3}{2},z_4+\\frac{\\tau_4}{2})$.\n\\end{itemize}\n \n Observe that in this situation $\\Lambda = \\pi_1 (T) \\subset \\ensuremath{\\mathbb{C}}^4$.\n \n\nWe claim now that for each generator $\\lambda \\in  \\{\\tau_1 e_1, \\tau_2 e_2, \\tau_3 e_3, \\tau_4e_4, e_1,  e_2,e_3, e_4 \\}$ there is a $g \\in G$\n such that $g \\lambda g^{-1} = - \\lambda$.\n \n  For this it suffices to find, for each $k \\in \\{1,2,3,4\\}$, an element $g \\in G$ whose linear action has $e_k$ as $-1$ eigenvector.\n  \n But this is obvious (e.g. $(g_1g_3, g_1g_2,g_4,g_4)$ for $k=(1,2,3,4)$). \n \n Lemma \\ref{action} implies now that, setting $P':=\\pi_1(S)\/ K$, then  \n $$\n 2\\Lambda \\subset [P', P'].\n $$\nDenoting by $\\gamma_1, \\ldots , \\gamma_5$ lifts to $P'$ of the generators $g_1, \\ldots, g_5$ of $G$, we know that\n$$\nP' = \\langle \\gamma_1, \\ldots , \\gamma_5, \\lambda_1, \\ldots , \\lambda_4 \\rangle.\n$$\nMoreover, $[\\gamma_i, \\lambda_j] = \\pm 2\\lambda_j$, whence by remark \\ref{reduction}, we have an exact sequence \n$$1 \\ensuremath{\\rightarrow} \\Lambda \/2 \\Lambda \\ensuremath{\\rightarrow} P'':=P'\/2 \\Lambda \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $P = (P')^{ab}=(P'')^{ab} = P'' \/ \\langle [\\gamma_i, \\gamma_j], 1 \\leq i,j \\leq 5\\rangle$.\n\nA straightforward computation shows the following:\n\n$[\\gamma_1,\\gamma_2] = \\begin{pmatrix}\n0\\\\1\\\\0\\\\0\n\\end{pmatrix}$, $[\\gamma_1,\\gamma_3] = \\begin{pmatrix}\n-1\\\\0\\\\0\\\\0\n\\end{pmatrix}$, $[\\gamma_1,\\gamma_4] = 0$, $[\\gamma_1,\\gamma_5] = \\begin{pmatrix}\n- \\tau_1\\\\0\\\\0\\\\0\n\\end{pmatrix}$,\n\n$[\\gamma_2,\\gamma_3] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\0\n\\end{pmatrix}$, $[\\gamma_2,\\gamma_4] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\1\n\\end{pmatrix}$, $[\\gamma_2,\\gamma_5] = \\begin{pmatrix}\n0\\\\-\\tau_2\\\\0\\\\-\\tau_4\n\\end{pmatrix}$, $[\\gamma_3,\\gamma_4] = \\begin{pmatrix}\n0\\\\0\\\\1\\\\1\n\\end{pmatrix}$,\n\n$[\\gamma_3,\\gamma_5] = \\begin{pmatrix}\n0\\\\0\\\\-\\tau_3\\\\-\\tau_4\n\\end{pmatrix}$, $[\\gamma_4,\\gamma_5] = \\begin{pmatrix}\n0\\\\0\\\\-\\tau_3\\\\-\\tau_4\n\\end{pmatrix}$.\n\nThis immediately implies that \n$$\n\\tilde{\\Lambda} \/  \\langle [\\gamma_i, \\gamma_j] \\rangle \\cong \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}}\n$$\nhence we get the exact sequence \n$$\n1\\ensuremath{\\rightarrow} \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} P \\ensuremath{\\rightarrow} G = (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^5 \\ensuremath{\\rightarrow} 1.\n$$\n\nSince \n$$\ng_5^2 = \\begin{pmatrix}\n\\tau_1\\\\ \\tau_2 \\\\ \\tau_3\\\\ \\tau_4\n\\end{pmatrix},\n$$\nit follows that $P \\cong \\ensuremath{\\mathbb{Z}}\/4\\ensuremath{\\mathbb{Z}} \\times (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^4$.\n\n\\end{proof}\n\nFinally we prove\n\\begin{prop}\nThe natural surjective morphism \n$$H_1(S, \\ensuremath{\\mathbb{Z}}) = \\pi_1(S)^{ab} \\onto P\n$$ is an isomorphism.\n\\end{prop}\nThis implies theorem \\ref{homology}.\n\n\\begin{proof}\nLet $\\Gamma:= \\pi_1(S)$. Then from the exact sequence\n$$\n1 \\rightarrow H:=\\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\Gamma\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n$$\nwe get by remark \\ref{reduction} the exact sequence \n$$\n1 \\ensuremath{\\rightarrow} H^{ab} \\cong \\ensuremath{\\mathbb{Z}}^{10} \\times \\ensuremath{\\mathbb{Z}}^4 =: \\Lambda_1 \\oplus \\Lambda_2 \\ensuremath{\\rightarrow} \\Gamma' \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1,\n$$\nand $\\Gamma^{ab} =(\\Gamma')^{ab}$. \n\nBy lemma \\ref{action} we get an exact sequence \n$$\n\\ensuremath{\\mathbb{Z}}^{10} \\times (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4 \\ensuremath{\\rightarrow} \\Gamma'' \\rightarrow G \\rightarrow 1,\n$$\nwhere again $(\\Gamma'')^{ab} = \\Gamma^{ab}$. \n\nNote that $g_1$ acts trivially on the curve $D$ of genus 5, whence it acts trivially on $H_1(D) \\cong \\ensuremath{\\mathbb{Z}}^{10}$. \n\nMoreover, we have seen that the commutators $[\\ga''_1, \n\\ga''_i]$, $2 \\leq i \\leq 5$, span a subspace $V$ of rank 3 in $\\Lambda_2 \/ 2 \\Lambda_2 \\cong (\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^4$. Therefore we get (after moding out by $V$) an exact sequence\n$$\n \\ensuremath{\\mathbb{Z}}^{10} \\times \\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}} \\ensuremath{\\rightarrow} \\Gamma''' \\rightarrow G \\rightarrow 1,\n$$\nwhere $(\\Gamma''')^{ab} = \\Gamma^{ab}$. After dividing by $\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}}$, we obtain the exact sequence \n$$\nH_1(D, \\ensuremath{\\mathbb{Z}}) \\rightarrow \\pi := \\Gamma'''\/(\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})  \\ensuremath{\\rightarrow} G \\ensuremath{\\rightarrow} 1.\n$$\nWe finally get the following commutative diagram with exact rows\n\\begin{equation}\\label{diagram}\n\\xymatrix{\n\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}}\\ar[r]&\\Gamma'''\\ar[d] \\ar[r] & \\pi\\ar[d] \\ar[r]&1\\\\\n\\ensuremath{\\mathbb{Z}}\/ 2 \\ensuremath{\\mathbb{Z}}\\ar[r]&(\\Gamma''')^{ab}\\ar[r]&(\\pi)^{ab} = (\\pi_1^{orb})^{ab} \\oplus \\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}} \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \\ensuremath{\\mathbb{Z}})^5\\ar[r]&1,\n}\n\\end{equation}\n(cf. the exact sequence (\\ref{orbifold})). \n\nThis implies that $H_1(S) = \\Gamma ^{ab} = (\\Gamma''')^{ab}$ has cardinality at most $2 \\cdot 2^5 = 2^6$. Since $|P| = 2^6$, \nthe surjective morphism $H_1(S, \\ensuremath{\\mathbb{Z}}) \\onto P$ is then an isomorphism. \n\\end{proof}\n\n\n\n\\section{Weak rigidity  of Inoue surfaces}\n\nIn this section we  shall prove the following:\n\n\\begin{theo}\\label{homotopy}\n    Let $S'$ be a smooth complex projective surface which is\nhomotopically equivalent to an Inoue surface (with $K^2 = 7$ and\n$p_g=0$). Then $S'$ is an Inoue  surface.\n\nThe same consequence holds under the weaker assumptions that $S'$ has the same fundamental group of an Inoue surface,\nand that (2) (SAME HOMOLOGY) of theorem \\ref{special diagonal} holds.\n\n\\end{theo}\n\n\\begin{rem}\n   It is clear from the definition that  $S$ is a diagonal CIT (classical Inoue   type) manifold.\n   \n   We can therefore apply theorem \\ref{special diagonal} to this special case.\n    In this special case, we are going to see that the groups $\\hat{G}_i = \\Lambda'_i \\rtimes G_{2,i}$, $i=1,2$, are obtained simply by taking $  \\Lambda'_i : = \\frac{1}{2}  \\Lambda_i$, and $ G_{2,i} : = \\{ \\pm 1\\}$.\n    \n     We shall indeed view things geometrically, as follows.\n   \n  Recall  that there is an exact sequence\n$$ 1 \\rightarrow \\mathbb{Z}^4 \\times \\pi_5 \\rightarrow \\pi_1(S)\n\\rightarrow G \\cong (\\mathbb{Z}\/2 \\mathbb{Z})^5\n\\rightarrow 1,\n$$ where $\\pi_5$ denotes the fundamental group of a compact curve\n of genus five.\n\\end{rem}\n\n\\noindent Let $\\hat{X} \\subset E_1 \\times E_2 \\times D$ be the\n\\'etale $(\\ensuremath{\\mathbb{Z}} \/2\\ensuremath{\\mathbb{Z}})^5$-covering of $S$. Observe that $H_1:=\n\\langle g_2,g_3,g_4,g_5 \\rangle$ acts trivially on $H^0(E_1,\n\\Omega^1_{E_1})$. This shows that $q(\\hat{X} \/G) \\geq 1$, and\n$$ q(\\hat{X} \/H_1) = 1 \\ \\ \\iff \\ \\ D\/H_1 \\cong \\mathbb{P}^1.\n$$ But this is obvious, since $D\/G = D\/H_1$.\n\nNow, consider instead $H_2 := \\langle g_1,g_3,g_4,g_5 \\rangle$, which\nacts trivially on $H^0(E_2, \\Omega^1_{E_2})$.\nTherefore\n$$ q(\\hat{X} \/H_2) = 1 \\ \\ \\iff \\ \\ D\/H_2 \\cong \\mathbb{P}^1.\n$$\n\n$D\/H_2 \\cong \\mathbb{P}^1$ follows from the following lemma\n\\ref{quot}, since $H_2$ contains three elements having fixed\npoints on $D$ (in fact, the elements  $(-z_3 + \\frac 12,-z_4 + \\frac\n12)$, $(-z_3 + \\frac{\\tau_3}{2},-z_4 +\n\\frac{\\tau_4}{2})$,$(-z_3 + \\frac 12 +\\frac{\\tau_3}{2},-z_4 + \\frac \n12+\\frac{\\tau_4}{2})$).\n\nTherefore we have \nseen:\n\n\\begin{prop}\\label{irreg}\n  Let $S:=\\hat{X} \/ G$ be an Inoue \nsurface. Then there are subgroups \n$H_1, H_2 \\leq G$ of index 2, such \nthat $q(\\hat{X}\/H_1) =\nq(\\hat{X}\/H_2) = 1$.\n\nLet $S'$ be a smooth \ncomplex projective surface which has the same \nfundamental group as \n$S$. Then, denoting by $\\hat{X}'$ the\ncorresponding \\'etale \n$G$-covering of $S'$ we have:\n\\begin{itemize}\n  \\item there is a \nsmooth curve $D'$ of genus 5 and a surjective \nhomomorphism $\\hat{X}' \n\\rightarrow D'$;\n\\item there are two index two subgroups $H_1, H_2$ \nof $G$ such that \n$q(\\hat{X}\/H_1) = q(\\hat{X}\/H_2) = \n1$.\n\\end{itemize}\n\n\\end{prop}\n\n\n\\begin{lemma}\\label{quot}\n\nLet $D \\rightarrow \\mathbb{P}^1$ be the maximal $\\Gamma \\cong\n(\\mathbb{Z} \/ 2 \\mathbb{Z})^4$ -covering branched in five\npoints. Then:\n\n1) there are exactly 5 subgroups $H \\leq \\Gamma$, $H \\cong\n(\\mathbb{Z}\/2 \\mathbb{Z})^3$ containing exactly one element\nhaving fixed points on $D$ ($\\implies$ $g(D\/H) = 1$);\n\n2) there are exactly 10 subgroups $H \\leq \\Gamma$, $H \\cong\n(\\mathbb{Z}\/2 \\mathbb{Z})^3$ containing exactly three elements\nhaving fixed points on $D$ ($\\implies$ $g(D\/H) = 0$).\n\\end{lemma}\n\n\\begin{proof} Let $D \\rightarrow \\mathbb{P}^1$ be the maximal $\\Gamma\n\\cong (\\mathbb{Z} \/ 2 \\mathbb{Z})^4$ -covering\nbranched in five points $p_1, \\ldots p_5$, which determines a\nsurjective homomorphism\n$$\n\\varphi \\colon \\pi_1(\\mathbb{P}^1 \\setminus \\{p_1, \\ldots p_5 \\})\n\\rightarrow \\Gamma.\n$$\n\nWe denote $\\varphi(\\gamma_i)$, where $\\gamma_i$ is a geometric loop\naround $p_i$ by $e_i$. Then $e_1, e_2,e_3, e_4$ is a\n$\\mathbb{Z}\/ 2\\mathbb{Z}$ - basis of $\\Gamma$ and $e_1 + e_2 +e_3+e_4 = e_5$.\n\n\\begin{claim}\n   Let $H \\cong (\\ensuremath{\\mathbb{Z}} \/2 \\ensuremath{\\mathbb{Z}})^3 \\leq \\Gamma$. Then either there is a\nunique $i \\in \\{ 1, \\ldots , 5 \\}$ such that $e_i \\in H$ (and\n$e_j \\notin H$ for $j \\neq i$, or there is a subset $\\{i,j,k \\}\n\\subset \\{ 1, \\ldots , 5 \\}$ such that $e_i, e_j, e_k \\in H$ (and the\nother two $e_l$'s are not in $H$).\n\\end{claim}\n\n{\\em Proof of the claim.} It is clear that $H$ contains at\nleast one of the $e_i$'s, otherwise $D \\rightarrow D\/H$ is\n\\'etale. By Hurwitz' formula, we get\n$$ 8 = 2g(D) -2 = |H|(2(g(D\/H)-2) = 8(2g(D\/H)-2),\n$$ a contradiction. Since any four of the $e_i$'s are linearly\nindependent, $H$ can contain at most three of them.\n\nAssume now that there are $i \\neq j$, such that $e_i, e_j \\in H$.\nW.l.o.g. we can assume $e_1, e_2 \\in H$. Then $H = \\langle e_1,\ne_2, \n\\lambda_1e_1 +\\lambda_2e_2 +\\lambda_3e_3 +\\lambda_4e_4 \\rangle$, \n\n$\\lambda_i \\in \\{0,1\\}$. Note that $e_1, e_2,\n\\lambda_1e_1 \n+\\lambda_2e_2 +\\lambda_3e_3 +\\lambda_4e_4$ are linearly \nindependent \nif and only if $(\\lambda_3,\n\\lambda_4) \\neq (0,0)$. \nMoreover,\n\\begin{itemize}\n\\item $e_3 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (1,0)$,\n\\item $e_4 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (0,1)$,\n\\item $e_5 \\notin H  \\iff \n(\\lambda_3,\\lambda_4) \\neq (1,1)$.\n\\end{itemize} This shows: if $H$ \ncontains two of the $e_i$'s then \nalso a third one.\n\n\\QED\n\nNow, \nif $H$ contains three of the $e_i$'s, say $e_i, e_j,e_k$, then \n$H = \n\\langle e_i, e_j, e_k \\rangle$, and there are exactly\n$\\binom{5}{3} = \n10$ such subgroups.\n\nThe remaining $5$ subgroups $H \\cong (\\ensuremath{\\mathbb{Z}} \/ 2 \n\\ensuremath{\\mathbb{Z}})^3$ of $\\Gamma$ \ncontain therefore exactly one of the \n$e_i$'s.\n\nAssume now that\n$$ H_1 := \\langle e_1, e_2, e_3 \\rangle, \\ \n\\ H_2:= \\langle e_1+e_2, \ne_1+e_3, e_1+e_4 \\rangle.\n$$\n\nThen it \nremains to show that $g(D\/H_1)=0$ and $g(D\/H_2) = 1$.\n\nObserve that \n$H_2 = \\langle e_5 \\rangle \\oplus \\langle e_1 +e_2, \ne_1+e_3 \n\\rangle$, and $H':=\\langle e_1 +e_2, e_1+e_3\n\\rangle$ acts freely on \n$D$. Therefore $g(D\/H') = 2$ and $e_5$ acts \non $D\/H'$ having two \nfixed points. By Hurwitz' formula this\nimplies that $g(D\/H) \n=1$.\n\nNow, $g(D\/ \\langle e_1 \\rangle) = 1$ and $H_1\/ \\langle e_1 \n\\rangle \n\\cong \\langle e_2, e_3\\rangle$ acts with fixed points on\n$D\/ \n\\langle e_1 \\rangle$. This shows that $D\/H_1 \\cong \n\\ensuremath{\\mathbb{P}}^1$.\n\n\\end{proof}\n\nNow we are ready to finish the proof of  \\ref{homotopy}.\n\n\\begin{proof}(of thm. \\ref{homotopy})\n\nLet $S'$ be a \nsmooth complex projective surface which is homotopically equivalent \nto an Inoue surface $S$ with $K_S^2 =7$ and\n$p_g=0$. \n\nIn particular \n$\\pi_1 (S) \\cong \\pi_1(S')$ and we take the \\'etale $G:=(\\ensuremath{\\mathbb{Z}} \/ \n2\\ensuremath{\\mathbb{Z}})^5$- covering $\\hat{X}'$, which\nis homotopically equivalent to \n$\\hat{X}$ (the corresponding covering \nof the Inoue surface $S$).  By \nproposition \\ref{irreg} we\nknow that $\\hat{X}'$ admits a morphism to \na curve $D'$ of genus 5, \nand there are subgroups $H_1, H_2 \\leq G$ \nof index 2 such\nthat $X_i:=\\hat{X}'\/H_i$ has irregularity \none.\n\nTherefore there are elliptic curves\n$E_1', E_2'$ and \nmorphisms\n$$\n\\hat{X}' \\rightarrow X_i \\rightarrow E'_i.\n$$ By the \nuniversal property of the Albanese map we get a commutative \ndiagram\n\n\\begin{equation}\\label{diagr}\n\\xymatrix{\n\\hat{X}'\\ar[r]\\ar[d]&E_1' \n\\times E_2' \\times \\Jac(D')\\\\\n\\Alb(\\hat{X}')\\ar_{\\psi}[ru]&\\\\ \n}\n\\end{equation}\n\n\\begin{lemma} Let $E_i' = \\mathbb{C}\/\\Lambda_i'$ \nand denote by \n$\\Lambda_i:= 2\\Lambda_i'$. Then $\\psi$ corresponds to \n$$\nH_1(D', \\ensuremath{\\mathbb{Z}}) \\times \\Lambda_1 \\times \\Lambda_2 \\subset H_1(D', \n\\ensuremath{\\mathbb{Z}}) \n\\times \\Lambda_1' \\times \\Lambda_2'.\n$$\n\nIn particular $E_i:= \n\\mathbb{C}\/\\Lambda_i = E_i'$, $\\psi$ \nrestricted to $E_i$ is \nmultiplication by 2, and\n$\\Alb(\\hat{X}') = \\Jac(D') \n\\times E_1 \\times E_2.$\n\n\\end{lemma} \n\n By theorem \\ref{special diagonal} we have that\n$\\varphi \\colon \\hat{X}' \\rightarrow \n E_1 \\times \nE_2 \\times D' $ is a birational morphism onto its image \n$W'\n\\subset  E_1 \\times \nE_2 \\times D'  : = C_1 \\times C_2 \\times C_3$.\n\n In fact, since the fundamental group of $X'$ is isomorphic to the\none of a Inoue surface, it follows by lemma \\ref{affine} that $G$ acts on $Z'$ as for an Inoue surface, hence\n there is no effective divisor $\\De$ of \nnumerical type $(1,1,2)$ which is invariant by the action of the group $G$, as it is easy to verify.\n\nTherefore  $W'$ has homology class $2F_1 + 2F_2 \n+ 4F_3$, where $F_i$ is the fibre over a point in the i-th curve, $C_i$\nand $W'$ has rational double points as singularities.\n\n\nThe linear equivalence class of $W'$ is invariant for the group action. It is the sum of three classes $\\xi_i$ which\nare respective pull backs from the projection\nonto the i-th curve $C_i$. Hence each  class $\\xi_i$ is invariant for the action of $G$ on $C_i$,\nhence $\\xi_i$ is the pull back from the quotient of $C_i$ by the group $G_i$, projection of $G$ into the automorphism group\nof $C_i$.\n\nBy our lemmas, \\ref{genus1} and \\ref{genus5}, these quotients are rational curves, hence\nwe conclude that the linear equivalence class of the divisor $W'$ is the same as the one for\nan Inoue surface $S$.\n\nIt remains to show that $W'$ is given by Inoue's equations, i.e., if we consider the genus 5 curve $D'$ as a hypersurface \n$$\nD' := \\{ z_3,z_4)) \\in E_3 \\times E_4 \\ |  \\ \\sL_3(z)\\sL_4(z_4) =\nb_3b_4\\}, \n$$\nthen \n\\begin{multline}\nW' := \\{ (z_1,z_2,z_3,z_4)) \\in E_1 \\times  \\ldots \\times E_4 \\ | \\ \\sL_1(z_1)\n\\sL_2(z_2)\\sL_3(z_3)  = b_1b_2b_3, \\\\\n \\sL_3(z_3)\\sL_4(z_4) =\nb_3b_4\\}.\n\\end{multline}\nFor this consider the subgroup $H:=\\langle g_1,g_2,g_3,g_4 \\rangle \\leq G$. Then there is a $H$-invariant divisor with class $2F_1+2F_2 +4F_3$ in $Z':=E_1 \\times E_2 \\times D'$. Therefore $H^0(Z', \\ensuremath{\\mathcal{O}}_Z'(2F_1+2F_2+4F_3)) \\cong \\ensuremath{\\mathbb{C}}^8$ is an $H$-module which decomposes by K\\\"unmeth's formula and the results in section \\ref{section2} as follows:\n\n\\begin{multline}\nH^0(Z', \\ensuremath{\\mathcal{O}}_Z'(2F_1+2F_2+4F_3)) \\cong \\\\\n\\cong H^0(E_1 \\times E_2 \\times D', p_1^*(\\ensuremath{\\mathcal{O}}_{E_1}(2[0]))\n\\otimes p_2^*(\\ensuremath{\\mathcal{O}}_{E_2}(2[0]))\\otimes p_3^*(\\ensuremath{\\mathcal{O}}_{D'}(2,0))) \\cong \\\\\n\\cong V_1 \\otimes V_2 \\otimes V_3 \\cong (V_1^{++++} \\oplus V_1^{-+-+}) \\otimes  (V_2^{++++} \\oplus V_2^{--++}) \\otimes (V_3^{++++} \\oplus V_3^{+--+}) \\cong \\\\\n\\cong V^{++++} \\oplus V^{+--+} \\oplus V^{--++} \\oplus V^{-+-+},\n\\end{multline}\nwhere each of the four summands in the last line is isomorphic to $\\ensuremath{\\mathbb{C}}^2$. In fact, we have\n\n$$V^{++++} = (V_1^{++++} \\otimes V_2^{++++} \\otimes V_3^{++++} ) \\oplus (V_1^{-+-+} \\otimes V_2^{--++} \\otimes V_3^{+--+} ),\n$$\n$$V^{+--+} = (V_1^{++++} \\otimes V_2^{++++} \\otimes V_3^{+--+} ) \\oplus (V_1^{-+-+} \\otimes V_2^{--++} \\otimes V_3^{++++} ),\n$$\n$$V^{--++} = (V_1^{++++} \\otimes V_2^{--++} \\otimes V_3^{++++} ) \\oplus (V_1^{-+-+} \\otimes V_2^{++++} \\otimes V_3^{+--+} ),\n$$\n$$V^{-+-+} = (V_1^{-+-+} \\otimes V_2^{++++} \\otimes V_3^{++++} ) \\oplus (V_1^{++++} \\otimes V_2^{--++} \\otimes V_3^{+--+} ).\n$$\nThe equations of the hypersurfaces in the above pencils are then:\n$$\nW'_1(c) := \\{ c = \\sL_1(z_1) \\sL_2(z_2) \\sL_3(z_3) \\},\n$$\n$$\nW'_2(c) := \\{ c = \\sL_1(z_1) \\sL_2(z_2) \\frac{1}{\\sL_3(z_3)} \\},\n$$\n$$\nW'_3(c) := \\{ c = \\sL_1(z_1) \\frac{1}{\\sL_2(z_2)} \\sL_3(z_3) \\},\n$$\n$$\nW'_4(c) := \\{ c = \\frac{1}{\\sL_1(z_1)} \\sL_2(z_2) \\sL_3(z_3) \\}.\n$$\n\nThis shows that after possibly replacing one of the elliptic curves $E_i$ with parameter $a_i$ by the elliptic curve $E'_i$ with parameter $\\frac{1}{a}$ we can w.l.o.g. assume that the pencil of $H$-invariant hypersurfaces in $Z'$ is given by the equation $\\{ c = \\sL_1(z_1) \\sL_2(z_2) \\sL_3(z_3) \\}$.\n\nNow, we consider $g_5$. It is easy to see that $g_5(W'_1(c)) \\equiv W'_1(c)$, and if $W'_1(c) = div(s)$ for $s \\in V^{++++}$, then also $g_5(W'_1(c)) = div(s')$ for $s' \\in V^{++++}$. Therefore $g_5$ is an involution on $\\ensuremath{\\mathbb{P}}^1 := \\ensuremath{\\mathbb{P}}(V^{++++})$, which is obviously non trivial,  whence $g_5$ has exactly two fixed points. Therefore there are exactly two $G$-invariant divisors in the pencil $W'_1(c)$. On the other hand,\n$W'_1(b_1b_2b_3)$ and $W'_1(-b_1b_2b_3)$ are $G$-invariant. \n\nThis shows that $W'$ is of the desired  form, hence $X'$ is the canonical model of an Inoue surface.\n\n\\end{proof}\n\n\n\\section{Inoue surfaces as bidouble \ncovers and $H^1(S,\\Theta_S)$}\n\nThe aim of this section is to show the \nfollowing \n\\begin{theo}\nLet $S$ be an Inoue surface with $K_S^2 = 7$. \nThen:\n$$\nh^1(S, \\Theta_S) = 4, \\ \\  h^2(S, \\Theta_S) = \n8.\n$$\n\\end{theo}\n\nTo prove this result we resort to a result of \n\\cite{mlp}, where   Inoue surfaces are constructed as bidouble covers of the \nfour nodal cubic.\n\nWe briefly recall their description here, for \ndetails we refer to \n\\cite{mlp}, example 4.1 (we keep their notation, even if slightly inconvenient).\n\nWe consider a complete \nquadrilateral $\\Lambda$ in $\\ensuremath{\\mathbb{P}}^2$  and \ndenote the vertices by $P_1, \\ldots,\nP_6$.\n\nWe have labeled the \nvertices in a way that\n\\begin{itemize}\n  \\item the intersection point \nof the line $\\overline{P_1P_2}$ and the line \n$\\overline{P_3P_4}$ is $P_5$,\n\\item the \nintersection point of $\\overline{P_1P_4}$ and $\\overline{P_2P_3}$ is \n$P_6$.\n\\end{itemize}\n\nLet $Y \\rightarrow \\ensuremath{\\mathbb{P}}^2$ be the blow up in \n$P_1, \\ldots , P_6$, \ndenote by $L$  the total transform of a line in \n$\\ensuremath{\\mathbb{P}}^2$, let $E_i$, $1\n\\leq i \\leq6$, be the exceptional curve lying over \n$P_i$.\nMoreover, we denote by $S_i$, $1 \\leq i \\leq 4$, the strict \ntransforms on $Y$ of the sides $S_i : = \\overline{P_i P_{i+1}}$ for $ 1\\leq i \\leq 3$,\n$S_4 : = \\overline{P_4 P_{1}}$, of the quadrilateral $\\Lambda$.\n\nThe geometry of the situation is that the four (-2) curves $S_i$ come from the resolution\nof the 4 nodes of the cubic surface $\\Sigma$ which is the anticanonical image of $Y$,\nand the curves $E_i$ are the strict transforms of the 6 lines in $\\Sigma$ connecting pairs of nodal points.\n\nThe surface $\\Sigma$ contains also a triangle of lines (joining the midpoints of opposite edges\nof the tetrahedron with sides the lines corresponding to the curves $E_i$. These are the 3 strict transforms \n $\\Delta_1$, $\\Delta_2$,\n$\\Delta_3$ of the three diagonals of the complete quadrilateral $\\Lambda$. $\\Delta_1$\nis the strict transform of $\\overline{P_1P_3}$, $\\Delta_2$ of $\\overline{P_2P_4}$ and $\\Delta_3$\nof $\\overline{P_5P_6}$.\n\nFor each line $\\De_i$ in the cubic surface  $\\Sigma$ we consider the pencil of planes containing\nthem, and the base point free pencil of residual conics, which we denote by $| f_i|$. Hence we have\n$$| f_i| = |(-K_Y) -  \\De_i| , \\ \\  \\De_i + f_i \\equiv (-K_Y) .$$\n\nIn the plane realization we have:\n\\begin{itemize}\n  \\item $f_1$ is the strict \ntransform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_1$ through\n$P_2,P_4,P_5,P_6$,\n\\item $f_2$ is the strict transform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_2$ through $P_1,P_3,P_5,P_6$,\n\\item $f_3$ is the strict transform on $Y$ of a general element of\nthe pencil of conics $\\Gamma_3$ through $P_1,P_2,P_3,P_4$.\n\\end{itemize}\n\nIt is then easy to see that each curve $S_h$ is disjoint from the other curves $S_j$ ($j\\neq h$), $\\De_i$, and $f_i$\nif $f_i$ is smooth. Moreover, \n$$\\De_i \\cdot f_i = 2,  \\  \\  \\De_i \\cdot f_j = 0, i \\neq j, \\ \\ f_i ^2 = 0 , \\ \\  f_j f_i = 2,  i \\neq j . $$\n\n\n\n\\begin{definition}\\label{Inouedivisors}\n   We define the {\\em Inoue divisors} on $Y$ as follows:\n\\begin{itemize}\n\\item $D_1:= \\Delta_1 + f_2 + S_1 + S_2$, where $f_2 \\in |f_2|$ smooth;\n\\item $D_2:= \\Delta_2 + f_3$, where $f_3 \\in |f_3|$ smooth;\n\\item$D_3:= \\Delta_3 + f_1 +f_1'+ S_3 + S_4$, where $f_1, f_1' \\in\n|f_1|$ smooth.\n\\end{itemize}\n\n\\end{definition} \n\nLet $\\pi \\colon \\tilde{S} \\rightarrow Y$ be the\nbidouble covering with branch divisors $D_1, D_2, D_3$ (associated to the \n3 nontrivial elements of the Galois group.\n\n By the previous remarks we see\nthat over each $S_i$ there are two disjoint $(-1)$-curves.\nContracting these eight exceptional curves we obtain a minimal surface\nwith $p_g=0$ and $K_S^2 = 7$. By \\cite{mlp} these are exactly the Inoue surfaces.\n\n\\begin{rem}\\label{ratvar}\nWe immediately see that there is an open dense subset  in the product \n$$\n|f_1| \\times |f'_1| \\times |f_2| \\times |f_3| \\cong (\\ensuremath{\\mathbb{P}}^1)^4\n$$\nparametrizing the family of Inoue surfaces.\n\\end{rem}\n\n\n\\begin{rem}\n   The non trivial character sheaves of this bidouble cover are\n\\begin{itemize}\n   \\item $\\mathcal{L}_1 = \\ensuremath{\\mathcal{O}}_Y( - K_Y + f_1 - E_4)$;\n\\item $\\mathcal{L}_2 = \\ensuremath{\\mathcal{O}}_Y( - 2 K_Y  - E_5 - E_6)$;\n\\item $\\mathcal{L}_3 = \\ensuremath{\\mathcal{O}}_Y( - K_Y + L - E_1 - E_2  - E_3)$.\n\\end{itemize}\n\n\\end{rem}\n\n\n\\begin{lemma}\\label{inv} \\\n\\begin{itemize}\n    \\item $\\dim H^1(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} = \\dim H^1(S,\n\\Theta_S)^{inv} = 4$, \n\\item  $\\dim H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} = \\dim H^2(S,\n\\Theta_S)^{inv} = 0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nIt is well known (cf. e.g.  \\cite{cime2}) that  $H^2(\\tilde{S}, \\Theta_{\\tilde{S}})$ of the $(\\ensuremath{\\mathbb{Z}}\/2\\ensuremath{\\mathbb{Z}})^2$-covering  $\\pi \\colon \\tilde{S} \\ensuremath{\\rightarrow} Y$ decomposes as a direct sum of character spaces\n$$\nH^2(\\tilde{S}, \\Theta_{\\tilde{S}}) \\cong H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^{inv} \\oplus \\bigoplus_{i=1}^3 H^2(\\tilde{S}, \\Theta_{\\tilde{S}})^i,\n$$\nand that the dimensions of the direct summands can be computed as the dimensions of global sections of logarithmic differential forms on the basis $Y$. In fact, we have:\n\\begin{multline}\\label{chars}\nh^0(Y, \\Omega^1_Y(\\log D_1, \\log D_2, \\log D_3)(K_Y)) = h^2(\\tilde{S},\\Theta_{\\tilde{S}})^{inv} = h^2(S,\\Theta_S)^{inv}; \\\\\nh^0(Y, \\Omega^1_Y(\\log D_i)(K_Y + \\mathcal{L}_i))) = h^2(\\tilde{S},\\Theta_{\\tilde{S}})^i = h^2(S,\\Theta_S)^i, \\forall i \\in \\{1,2,3\\}.\n\\end{multline}\nNote that $|-K_{Y}| = |3L - \\sum_{i=1}^6 E_i|$ \nis non empty \nand does not have a fixed part. Therefore there is a \nmorphism\n$\\ensuremath{\\mathcal{O}}_{Y}(K_{Y}) \\ensuremath{\\rightarrow} \\ensuremath{\\mathcal{O}}_{Y}$, \nwhich is not \nidentically zero on any component of the \n$D_i$'s.\n\n\n\nWe get the \ncommutative diagram with exact \nrows\n\\begin{equation}\\label{diagram2}\n\\xymatrix{ 0\\ar[r]& \n\\Omega^1_{Y}(K_{Y}) \\ar[d] \n\\ar[r]&\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})\\ar[d]\\ar[r]\n&\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}) \\ar[d] \\ar [r]&0\\\\ \n0\\ar[r]&\\Omega^1_{Y} \\ar[r]&\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3) \n\\ar[r] &\\oplus_{i=1}^3 \\ensuremath{\\mathcal{O}}_{D_i}  \\ar [r]&0.}\n\\end{equation}\n\n From \nthis we get the commutative \ndiagram with injective vertical arrows\n\\begin{equation*}\n\\xymatrix{\n\\ensuremath{\\mathbb{C}}^2 \\oplus 0 \\oplus\\ensuremath{\\mathbb{C}}^2 \n\\cong&H^0(Y,\\oplus_{i=1}^3\n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))\\ar[r]^{\\delta}\\ar[d]^{\\psi_2}\\ar[dr]^{\\varphi}& \nH^1(Y,\n\\Omega^1_{Y}(K_{Y})) \\ar[d] \\\\\n\\ensuremath{\\mathbb{C}}^4 \n\\oplus \\ensuremath{\\mathbb{C}}^2 \\oplus\\ensuremath{\\mathbb{C}}^5 \\cong &H^0(Y,\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i})\\ar[r]^{\\psi_1} &H^1(Y,\n\\Omega^1_{Y}) \n.}\n\\end{equation*} \n\nA usual argument shows that (see \\cite{catjdg}) $\\delta$ is injective. In fact, the Chern classes of $S_1,S_2,S_3,S_4$ are linearly independent, hence  $\\varphi$ is injective, which implies that also\n$\\delta$ is \ninjective. Therefore\n$$ h^0(\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})) \n= \nh^2(\\Theta_{\\tilde{S}})^{inv} = h^2(\\Theta_S)^{inv} = \n0.\n$$\n\nTherefore\n\\begin{multline}\n  h^1(\\Theta_{\\tilde{S}})^{inv} = - \n\\chi(\\Omega^1_{Y}((\\log D_1, \\log D_2, \\log D_3)(K_{Y})) = \\\\\n=\n-(\\chi(\\Omega^1_{Y}(K_{Y}) + \\chi(\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))).\n\\end{multline}\n\nAn easy calculation shows now that \n$\\chi(\\oplus_{i=1}^3 \n\\ensuremath{\\mathcal{O}}_{D_i}(K_{Y}))) = 0$, \nwhereas\n$\\chi(\\Omega^1_{Y}(K_{Y})) = - 4$.\n\\end{proof}\n\nWe prove now\n\n\\begin{prop}\\label{h1theta}  \\\n\\begin{enumerate}\n\\item $h^0(Y, \\Omega^1_Y(\\log D_1)(K_Y + \\mathcal{L}_1)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^1 = h^2(S, \\Theta_S)^1 \\leq 2$;\n\\item $h^0(Y, \\Omega^1_Y(\\log D_2)(K_Y + \\mathcal{L}_2)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^2 = h^2(S, \\Theta_S)^2 \\leq 3$;\n\\item $h^0(Y, \\Omega^1_Y(\\log D_3)(K_Y + \\mathcal{L}_3)) = \nh^2(\\tilde{S}, \\Theta_{\\tilde{S}})^3 = h^2(S, \\Theta_S)^3 \\leq 3$.\n\\end{enumerate}\nIn particular, we get $h^2(S,\\Theta_S) \\leq 8$.\n\\end{prop}\n\nFrom this result, we easily obtain the following:\n\\begin{cor}\\label{cortheta}\nLet $S$ be an Inoue surface with $K_S^2 =7$, $p_g=0$. Then:\n$$\nh^2(S, \\Theta_S) = 8, \\ \\ h^1(S, \\Theta_S)= h^1(S, \\Theta_S)^{inv} = 4.\n$$\n\n\\end{cor}\n\n\\begin{proof}[Proof of the cor..]\nWe have by proposition \\ref{h1theta} \n$$\n8 - h^1(S,\\Theta_S) \\geq h^2(S, \\Theta_S) - h^1(S,\\Theta_S) = \\chi(\\Theta_S) = 2K_S^2 - 10 \\chi(S) = 4,\n$$\nwhence $h^1(S, \\Theta_S) \\leq 4$. On the other hand by lemma \\ref{inv} we know that\n$$\n4 = h^1(S, \\Theta_S)^{inv} \\leq h^1(S, \\Theta_S)\n$$\nand the assertions of the corollary follow.\n\\end{proof}\n\\begin{proof}[Proof of proposition \\ref{h1theta}.]\n1) Recall that by definition \\ref{Inouedivisors} $D_1$ is the \ndisjoint union of the four curves $\\Delta_1, f_2, S_1, S_2$, and\n$$\nK_Y + \\mathcal{L}_1 \\equiv  f_1 -E_4.\n$$\n\nWe consider the exact sequence (cf. e.g.  \\cite{esnaultviehweg}, p. 13)\n\\begin{multline}\\label{exseq}\n0 \\ensuremath{\\rightarrow} \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow} \\Omega^1_Y((\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2)(f_1-E_4)) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow} \\Omega^1_{f_1} (\\Delta_1 +f_2 + S_1+S_2+f_1-E_4) \\ensuremath{\\rightarrow} 0.\n\\end{multline}\n\nSince\n$$\n(\\Delta_1 +f_2 + S_1+S_2+f_1-E_4)f_1 = 2+2-1,\n$$\nwe have $ \\Omega^1_{f_1} (\\Delta_1 +f_2 + S_1+S_2+f_1-E_4) \\cong \\ensuremath{\\mathcal{O}}_{\\ensuremath{\\mathbb{P}}^1}(1)$, whence \n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log D_1)(K_Y+ \\mathcal{L}_1)) = h^0(Y, \n\\Omega^1_Y((\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2)(f_1-E_4)) \\leq \\\\\n \\leq h^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4)) + 2.\n\\end{multline}\nConsider the long exact cohomology sequence of the short exact sequence\n\n\\begin{multline}\n0 \\ensuremath{\\rightarrow} \\Omega^1_Y \\ensuremath{\\rightarrow}  \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1) \\ensuremath{\\rightarrow} \\\\\n\\ensuremath{\\rightarrow}  \\ensuremath{\\mathcal{O}}_{f_1} \\oplus \\ensuremath{\\mathcal{O}}_{f_2} \\oplus \\ensuremath{\\mathcal{O}}_{S_1} \\oplus \\ensuremath{\\mathcal{O}}_{S_2} \n\\oplus \\ensuremath{\\mathcal{O}}_{\\Delta_1} \\ensuremath{\\rightarrow} 0\n\\end{multline}\n\nSince $H^0(Y, \\Omega^1_Y)=0$, \n$H^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1))$ is the kernel of the connecting \nhomomorphism\n$$\n\\delta \\colon H^0(Y, \\ensuremath{\\mathcal{O}}_{f_1} \\oplus \\ensuremath{\\mathcal{O}}_{f_2} \\oplus \\ensuremath{\\mathcal{O}}_{S_1} \\oplus \\ensuremath{\\mathcal{O}}_{S_2} \n\\oplus \\ensuremath{\\mathcal{O}}_{\\Delta_1}) \\ensuremath{\\rightarrow} H^1(Y, \\Omega^1_Y).\n$$\nBy \\cite{catjdg} the image of $\\delta$ is generated by the Chern \nclasses of $ \\Delta_1, f_2,  S_1, S_2, f_1$. \n\n\n\\begin{claim}\n$\\dim \\im (\\delta) = 5$.\n\\end{claim}\n\\begin{proof}[Proof of the claim.] \nAssume that\n\\begin{equation}\\label{zero1}\n\\lambda_1 f_1 + \\lambda_2 f_2 + a_1 S_1 + a_2 S_2 + \\mu \\Delta_1\\equiv 0,\n\\end{equation}\nwhere $\\lambda_1, \\lambda_2, a_1, a_2, \\mu \\in \\ensuremath{\\mathbb{C}}$.\nIntersection with $E_4$ gives $\\lambda_1 =0$, whereas intersection with $S_i$, $i=1,2$ yells $-2a_i=0$. The equation \\ref{zero1} has become\n$$\n\\mu \\Delta_1 + \\lambda_2 f_2 =0.\n$$\n\nIntersection with e.g. $E_5$ gives $\\lambda_2 =0$, whence also $\\mu = 0$.\n\\end{proof}\n\nIt follows now that\n$$\nh^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)(-E_4)) \\leq \n$$\n$$\n\\leq h^0(Y, \\Omega^1_Y(\\log \\Delta_1, \\log f_2, \\log S_1, \\log S_2, \\log f_1)) = 0.\n$$\n\nTherefore we have proven\n$$\nh^0(Y, \\Omega^1_Y(\\log D_1)(K_Y+ \\mathcal{L}_1)) \\leq 2.\n$$\n2) By definition \\ref{Inouedivisors} $D_2$ is the disjoint union of \nthe two curves $\\Delta_2, f_3$, and\n$$\nK_Y + \\mathcal{L}_2 \\equiv  -K_Y -E_5-E_6.\n$$\nSince\n$$\n(K_Y + 2 \\Delta_2 +(- K_Y -E_5-E_6)) \\Delta_2 = (2 \\Delta_2  -E_5-E_6) \\Delta_2 -2 < 0,\n$$\nby  lemma 5.1 of  \\cite{burniat3} we have\n$$\nH^0(Y, \\Omega^1_Y(\\log D_2)(K_Y+ \\mathcal{L}_2)) \\cong H^0(Y, \n\\Omega^1_Y(\\log f_3)(K_Y + \\mathcal{L}_2 + \\Delta_2)).\n$$\nNote that\n$$\nK_Y + \\mathcal{L}_2 + \\Delta_2 \\equiv -K_Y -E_5 - E_6 + \\Delta_2 \n\\equiv S_1 + S_2 +S_4 + S_3 + E_1 + E_3.\n$$\n\nAgain by lemma 5.1 in \\cite{burniat3} we see that\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log f_3)(K_Y + \\mathcal{L}_2 + \\Delta_2)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4)(E_1 + E_3)).\n\\end{multline}\nSince $E_1(f_3 + S_1 +S_2 + S_3 +S_4 + E_1 +E_3) = E_3(f_3 + S_1 +S_2 \n+ S_3 +S_4 + E_1 +E_3) = 2$, we see by the same arguments (using the analogous exact sequence \\ref{exseq}) as in case \n1) that\n\n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4)(E_1 + E_3)) \\leq \\\\\n\\leq h^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log \nS_4, \\log E_1, \\log E_3)) +2.\n\\end{multline}\n\\begin{claim}\n$\\dim \\langle f_3, S_1, \\ldots , S_4, E_1, E_3 \\rangle =6$.\n\\end{claim}\n\\begin{proof}[Proof of the claim.]\nNote that \n$$\nS_1+S_4-S_2-S_3 \\equiv -2E_1 + 2E_3.\n$$\nTherefore we have proven the claim, if we show that the Chern classes of $f_3, S_1, \\ldots , S_4, E_1$ are linearly independent.\n\nAssume that \n\\begin{equation}\\label{zero2}\n\\lambda f_3 + a_1 S_1 + a_2 S_2 + a_3 S_3 +a_4 S_4 + \\mu E_1\\equiv 0,\n\\end{equation}\nwhere $\\lambda, a_1, a_2, a_3,a_4, \\mu \\in \\ensuremath{\\mathbb{C}}$.\n\nIntersection with $S_2, S_3$ gives $a_2 = a_3 =0$, whereas intersection with $\\Delta_1$ yields $\\mu = 0$. We are left with the equation $\\lambda f_3 + a_1 S_1 +a_4 S_4 \\equiv 0$. Intersection with $E_5$ resp. $E_6$ implies that $a_1=0$ resp. $a_4=0$, and we conclude that also $\\lambda=0$.\n\n\\end{proof}\nTherefore we get\n$$\nh^0(Y, \\Omega^1_Y(\\log f_3, \\log S_1, \\log S_2, \\log S_3, \\log S_4, \n\\log E_1, \\log E_3)) = 1,\n$$\nand we have shown 2).\n\n3)  $D_3$ is the disjoint union of the five curves $\\Delta_3, f_1, \nf'_1,S_3, S_4$, and\n$$\nK_Y + \\mathcal{L}_3 \\equiv L -E_1-E_2-E_3.\n$$\nSince\n$$\n(K_Y + 2 \\Delta_3 +(L -E_1-E_2-E_3)) \\Delta_3 = -2 < 0,\n$$\nby \\cite{burniat3}, lemma 5.1, we have\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log D_3)(K_Y+ \\mathcal{L}_2)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4))(K_Y \n+ \\mathcal{L}_2 + \\Delta_3)).\n\\end{multline}\nNote that\n$$\nK_Y + \\mathcal{L}_3 + \\Delta_3 \\equiv S_1 + S_2  + E_2.\n$$\n\nAgain by lemma 5.1 in \\cite{burniat3} we see that\n\\begin{multline}\nH^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4))(K_Y + \n\\mathcal{L}_2 + \\Delta_3)) \\cong \\\\\n\\cong H^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log \nS_1, \\log S_2)(E_2)).\n\\end{multline}\nSince $E_2(f_1+f'_1+S_1 +S_2 + S_3 +S_4 + E_2) =  3$, we see by the \nsame arguments as in case 1) that\n\n\\begin{multline}\nh^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log S_1, \n\\log S_2)(E_2)) \\leq \\\\\n\\leq h^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log \nS_1, \\log S_2, \\log E_2)) + 2.\n\\end{multline}\n\n\\begin{claim}\nThe Chern classes of $f_1, S_1, \\ldots , S_4, E_2$ are linearly \nindependent.\n\\end{claim} \n\n\\begin{proof}[Proof of the claim.]\n\nAssume that \n\\begin{equation}\\label{zero3}\n\\lambda f_1 + a_1 S_1 + a_2 S_2 + a_3 S_3 +a_4 S_4 + \\mu E_2\\equiv 0,\n\\end{equation}\nwhere $\\lambda, a_1, a_2, a_3,a_4, \\mu \\in \\ensuremath{\\mathbb{C}}$.\n\nIntersection with $S_3, S_4$ gives $a_3 = a_4 =0$, whereas intersection with $E_1$ yields then  $a_1=0$.\n Intersection with $E_3$ instead gives  $a_2 =0$.\nFinally,  intersection with $E_4$ yields $\\lambda =0$, whence also $\\mu =0$. \\end{proof}\nTherefore\n$$\nh^0(Y, \\Omega^1_Y(\\log f_1, \\log f'_1, \\log S_3, \\log S_4, \\log S_1, \n\\log S_2, \\log E_2)) = 1,\n$$\nand we have shown 3).\n\n\n\\end{proof}\n\n\\begin{proof}[Proof of theorem \\ref{main}]\n(1) has been proved in theorem \\ref{homotopy}.\n\n(2) By \\cite{inoue}, page 318, $K_S$ is ample and by corollary \\ref{cortheta} the tangent space $H^1(S, \\Theta_S)$ to the base $\\Def(S)$ of the Kuranishi family of $S$ consists of the invariants for the action of the group $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$. Therefore all the local deformations of $S$ admit a $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$-action, hence are $(\\ensuremath{\\mathbb{Z}}\/2 \\ensuremath{\\mathbb{Z}})^2$ Galois coverings of the four nodal cubic surface (the anticanonical image of $Y$).\n\nFurthermore, the dimension of $H^1(S, \\Theta_S)$ is equal to the dimension of the Inoue  family containing $S$ in the moduli space $\\mathfrak{M}^{can}_{1,7}$, whence the base of the Kuranishi family of $S$ is smooth.\n\nSince the quotient of a smooth variety by a finite group (in our case, the automorphism group  $\\Aut(S)$) is normal, it follows that the irreducible connected component of the moduli space $\\mathfrak{M}^{can}_{1,7}$ corresponding to Inoue surfaces with $K_S^2=7, p_g = 0$ is normal and in particular generically smooth.\n\nThe family of Inoue surfaces is parametrized by a smooth (4-dimensional) rational variety (cf. e.g. remark \\ref{ratvar}), whence  unirationality follows.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn elastic shell may be defined as a three-dimensional elastic body with one dimension that is much thinner than the other two \\citep{libai1998}. This enables us to consider the deformation of only the mid-plane of the elastic body in this thinner direction. A membrane may then be defined as a shell which has negligible resistance to bending \\citep{libai1998, humphrey2003}.  Elastic membranes are commonly found in biological and engineering contexts, where they may span relatively large areas despite having little volume or weight \\citep{humphrey1998}. \n\nThere are two different classes of non-linear elastic membrane theories and we refer to \\citet{haughton2001} for a more complete comparison of the two theories, but give a brief description here. The first of these membrane theories may be called a \\lq membrane-like shell\\rq \\citep{libai1998}, which takes the shell theory of three-dimensional elasticity and introduces the membrane assumption of no stress in the direction normal to the membrane. In this formulation the thickness is included in the derivation, as the principal stretch in the thickness direction, $\\lambda_n$, appears in the governing equations, which explicitly allows the membrane to get thinner to conserve mass when the constraint of incompressibility is imposed. This is a common method of treating membranes, and a comprehensive derivation may be found in \\citet{libai1998} and \\citet{steigmann2007}. A variational treatment may also be considered, for example see \\citet{ledret1996}.\n\nThe second class of membrane theories occurs from considering the mid-plane of the membrane as a two-dimensional sheet of elastic material embedded in  three-dimensional space, entirely neglecting thickness effects through the membrane \\citep{li1995}. \\citet{nadler2009} call this a \\lq simple membrane', and a consequence of this reduction is that the stretch through the membrane is not included in the formulation. The deformation gradient is then two-dimensional, and there are only two strain invariants and two principal stretches, rather than the three that arise in the three-dimensional theory,  see \\citet{steigmann2001,steigmann2009} for details.  \n\nIt has been shown that the simple membrane and membrane-like shell approaches give the same governing equations to leading order \\citep{naghdi1972, haughton2001, steigmann2007}, and it is the specification of the constitutive behaviour which varies between the two theories. It is possible to define two-dimensional strain-energy functions which have no counterpart in the three-dimensional theory, as discussed in \\citet{haughton2001}, as well as to use three-dimensional strain-energy functions in the two-dimensional theory. Further details of such matters may be found in \\citet{libai1998} and  \\citet{steigmann2007}.\n\nWhen considering deformations of membranes using either of the above theories, it is important to ensure that the membrane is in a state of tension, with both principal stresses remaining positive throughout the deformation \\citep{libai1998}. If compressive (negative) stresses occur then the membrane may wrinkle, a local buckling event \\citep{miyamura2000} where alternating crests and troughs appear parallel to the direction of principal tension \\citep{libai2002}, at which point additional refinements are required. In particular, \\citet{andra2000} show that equilibrium states with distributions of infinitesimal wrinkles are possible when bending stiffness is neglected, due to loss of convexity of the strain-energy function. \\citet{pipkin1986} showed how to modify the constitutive behaviour of the material in regions where compressive stresses exist, such that a pseudo-surface is considered which is in simple tension in these regions, neglecting the local details of the wrinkled structure in favour of global results; this is called tension field theory. For further details on treatments of wrinkling see for example, \\citet{pipkin1986}, \\citet{steigmann1990}, \\citet{epstein2001ani} and \\citet{libai2002}. A physical example of such wrinkles may be seen when a pointed object is pushed into a fixed curved piece of the plastic wrap, causing straight crests to radiate from the point of contact.\n\nThe indentation of an axisymmetric elastic membrane by a rigid body has been considered previously, although mostly through the use of linear elasticity. Here we are interested in large deformations, and therefore use non-linear elasticity theory to model the deflection of the membrane. Physical applications\/occurrences of such indentations include puncture of rubber gloves by medical needles \\citep{nguyen2009a, nguyen2009b}, stones embedding into rubber tyres \\citep{yang1971} and our specific interest of seed germination, which we shall discuss later.\n\n\\citet{yang1971} considered the indentation of a flat circular hyperelastic membrane by a rigid sphere, in the large deformation regime, using the three-dimensional Mooney-Rivlin strain-energy function. \\citet{liu2001} estimate the elastic modulus of a specific elastomeric film by comparing numerical simulations of the model introduced by \\citet{yang1971} with their experimental observations. \\citet{steigmann2005} and \\citet{nadler2006} consider the indentation and penetration of a material described by a compressible two-dimensional Varga strain-energy function by a spherical-tipped cylindrical indenter, and assume that the membrane will break if a punctured state, with a discontinuous membrane, is energetically favourable. \\citet{selvadurai2006} compares finite element simulations of indentation with a spherical indenter with experimental results, and provides a thorough literature review on rubber membranes, including a discussion of friction effects.\n\nThis rigid-body indentation is closely related to the adhesion behaviour of a rigid punch attached to a membrane, which exhibits stretching as the rigid body is pulled away. This has been considered by, for example, \\citet{nadler2008}, who show that non-linear elastic effects are important; they also find that the imposition of pre-stress on the material has a strong effect on the force required to separate the punch from the membrane.\n\n\\citet{haughton1996} consider the response of planar and hemispherical elastic membranes to filling with a specific volume of incompressible liquid, using a relaxed strain-energy function when the membrane becomes wrinkled. They find an instability whereby the depth of the liquid increases very rapidly with only a slight increase in volume. Depending on the strain-energy function this unstable behaviour may continue indefinitely or re-stabilise, corresponding to the bifurcation points of a uniformly inflated spherical membrane. \\citet{haughton1996} find that the plane membrane is a special case in that it remains tense throughout the deformation, whereas a hemispherical membrane contains a wrinkled section when the volume of liquid is small, which also reappears briefly after the instability.\n\nIn this article we shall generalise the previous work in three directions. Firstly, we allow the profile of the indenter tip to be of general axisymmetric shape rather than only considering spherical indenters, although we shall exclude sharply pointed tips by imposing mild continuity requirements. Secondly, we wish to investigate membranes which are initially curved, introducing this change of reference configuration into the elastic equilibrium equations. Lastly, we also wish to compare the effects of using different strain-energy functions to describe the constitutive behaviour of the membrane. \n\nOur primary motivation behind considering this deformation lies in seed germination, in particular the stretching and rupture of the endosperm by the root-tip in certain species of germinating seeds, specifically the two related Brassicaceae species \\textit{Arabidopsis thaliana} \\citep{holdsworth2008} and \\textit{Lepidium sativum} \\citep{muller2006}. These two species have approximately ellipsoidal seeds, including a thin, deformable tissue called the endosperm which surrounds the embryo. During the germination process the embryo grows, pushing the root-tip against the endosperm, stretching it before rupture occurs. The shape of the root-tip is approximately prolate spheroidal, as is the surrounding endosperm, hence our interest here in non-flat membranes and non-spherical indenter tips.\n\nThe endosperm itself is thin and highly curved, and rapidly dehydrates when separated from the rest of the seed. It is therefore difficult to use conventional techniques to measure the elastic properties of this tissue, and so an indirect method is required. We are particularly interested in determining the elastic properties of the endosperm as biological changes occur during germination which alter the cell wall microstructure.\n\nWe therefore wish to lay the mathematical groundwork before modelling biological puncture force experiments in a later work. In these experiments the seed is dissected and the embryo removed before a metal indenter is pushed through the endosperm, examples of such experiments are given in \\citet{muller2006}. The force and position of the indenter are measured during this experiment, and we then wish to estimate the elastic properties of the endosperm using this technique, particularly how these properties change over time. It has been shown in \\citet{muller2006} that the force required to puncture through \\textit{Lepidium sativum} endosperms decreases during the germination timescale. In preliminary experiments it has been shown that the vertical length of endosperm may double in size before rupture, clearly requiring non-linear elasticity to model this effect.\n\n\\section{Theory}\n\\subsection{Governing Equations}\nWe shall use the theory of non-linear elasticity as detailed in \\citet{ogden1997} to model the deformation. We assume that the midsurface of the undeformed membrane has symmetry about a particular axis, and may therefore be described by cylindrical coordinates with respect to the usual basis $(\\mathbf{e}_r,\\mathbf{e}_\\theta,\\mathbf{e}_z)$, \n\\begin{equation}\\label{undefcon}\n\\mathbf{X} = \\hat{R}(\\hat{S}) \\mathbf{e}_r(\\Theta) + \\hat{Z}(\\hat{S}) \\mathbf{e}_z, \\; 0 \\leq \\hat{S} \\leq \\hat{L}, \\; 0 \\leq \\Theta \\leq 2 \\pi, \n\\end{equation}\nwhere $\\hat{S}$ is the arclength along the reference midsurface, which will be used as the independent variable throughout. We shall take the origin of our coordinates to be the centre of the membrane, $\\hat{R}(0) = 0, \\hat{Z}(0) = 0$, as well as requiring that the membrane be smooth there, $\\hat{Z}'(0) = 0$.  If the membrane is initially flat we set $\\hat{R}(\\hat{S}) = \\hat{S}, \\hat{Z}(\\hat{S}) = 0$, in this case $\\hat{R}$ may be used as the independent coordinate as has been used in previous studies.\n\nWe are interested in axisymmetric deformations that map the reference configuration $\\mathbf{X}$ to the deformed configuration $\\mathbf{x}$, the midsurface of which is given by\n\\begin{equation}\\label{defcon}\n\\mathbf{x} = \\hat{r}(\\hat{S}) \\mathbf{e}_r(\\theta) + \\hat{z}(\\hat{S}) \\mathbf{e}_z, \\; 0 \\leq \\hat{S} \\leq \\hat{L}, \\; 0 \\leq \\theta \\leq 2 \\pi.\n\\end{equation}\nA point in the deformed surface is therefore given by two variables, $\\hat{r}$ and $\\hat{z}$, the three-dimensional shape being then given by rotation around the $z$ axis. Immediately, due to the imposed axisymmetry, we may identify $\\Theta = \\theta$. We also impose the same conditions as in the undeformed state at the axis, that is $\\hat{r}(0) = \\hat{z}(0)=0, \\hat{z}'(0)=0$.\n\nHere we shall solve the static problem, where we assume that there is no time-dependence involved in the deformation. This is appropriate to the quasi-steady cases in which we are interested, for which the indentation is slow compared to the inertia of the membrane.\n\nAppendix A outlines the derivation of the equilibrium equations for a curved  membrane, using the membrane-like shell approximation to the three-dimensional theory of elasticity, as discussed above. \nThe principal stretch ratios may be defined in the circumferential, radial, and normal directions in the form (see Appendix A for details),\n\\begin{equation}\\label{lambdas}\n\\lambda_s = \\dd{\\hat{s}}{\\hat{S}} = \\frac{\\sqrt{\\left(\\dd{\\hat{r}}{\\hat{S}}\\right)^2 + \\left(\\dd{\\hat{z}}{\\hat{S}}\\right)^2}}{\\sqrt{\\left(\\dd{\\hat{R}}{\\hat{S}}\\right)^2 + \\left(\\dd{\\hat{Z}}{\\hat{S}}\\right)^2}} , \\quad \\lambda_\\theta = \\frac{\\hat{r}(\\hat{S})}{\\hat{R}(\\hat{S})}, \\quad \\lambda_n = \\frac{\\hat{h}}{\\hat{H}}, \n\\end{equation}\nwhere $\\hat{h}$ and $\\hat{H}$ are the deformed and undeformed thicknesses and $\\hat{s}$ is the arclength in the deformed configuration. We assume that the membrane is made of a hyperelastic material with a strain-energy function given by  $\\hat{W}(\\lambda_s,\\lambda_\\theta,\\lambda_s^{-1} \\lambda_\\theta^{-1})$, where we have already assumed that the material is incompressible and therefore set $\\lambda_s \\lambda_\\theta \\lambda_n = 1$; see \\citet{ogden1997} for more details. The principal stress resultants per unit length in the deformed membrane are therefore defined as\n\\begin{equation}\\label{stressres}\n\\hat{T}_s = \\frac{\\hat{H}}{\\lambda_\\theta} \\pafrac{\\hat{W}}{\\lambda_s}, \\quad\n\\hat{T}_\\theta = \\frac{\\hat{H}}{\\lambda_s} \\pafrac{\\hat{W}}{\\lambda_\\theta},\n\\end{equation}\nwhich may be seen to be $\\hat{T}_\\alpha = \\hat{h} \\hat{\\sigma}_\\alpha$, where $\\hat{\\sigma}_\\alpha = \\lambda_\\alpha \\pafrac{\\hat{W}}{\\lambda_\\alpha}$ is the usual principal Cauchy stress in incompressible three-dimensional elasticity, and similarly for $\\hat{T}_\\theta$. Appendix A shows how the stresses only occur in the combination $\\hat{h}\\hat{\\sigma}_\\alpha$, and hence we introduce $\\hat{T}_\\alpha$ with dimensions of force per length.\n\nWhile the theory presented here is appropriate for any isotropic incompressible strain-energy function, we shall study some specific examples in order to make later comparisons. One commonly used such function is the Mooney-Rivlin strain-energy function,\n\\begin{equation}\\label{mooney}\n\\hat{W} = \\frac{\\mu}{2(1+\\alpha)}\\left[(1-\\alpha) \\left( \\lambda_s^2 + \\lambda_\\theta^2 + \\lambda_s^{-2} \\lambda_\\theta^{-2} - 3 \\right) + \\alpha \\left( \\lambda_s^{-2} + \\lambda_\\theta^{-2} + \\lambda_s^{2} \\lambda_\\theta^{2} - 3 \\right) \\right],\n\\end{equation}\nwhere $\\mu$ is the infinitesimal shear modulus of the elastic material and $0\\leq \\alpha \\leq 1$ is a parameter that controls the deviation from a Hookean response. An important special case of \\rr{mooney} is the neo-Hookean strain-energy function, which occurs when $\\alpha=0$. The $(1+\\alpha)$ in the denominator of \\rr{mooney} is necessary to ensure that $\\mu$ is the infinitesimal shear modulus of the material. \\citet{fung1993} introduced a strain-energy function designed for the modelling of biological soft tissues such as healthy arterial tissue, which commonly feature strongly  strain-stiffening behaviour,\n\\begin{equation}\n\\hat{W} = \\frac{\\mu \\left(e^{\\Gamma \\left(\\lambda_s^2 + \\lambda_\\theta^2 + \\lambda_s^{-1}\\lambda_\\theta^{-1} -3\\right)} - 1\\right)}{2 \\Gamma} \\label{Fung},\n\\end{equation}\nwhere $\\Gamma$ is a positive parameter representing the degree of strain stiffening. In the limit as $\\Gamma$ approaches zero, \\rr{Fung} also recovers the neo-Hookean strain-energy function.\n\nAs shown in Appendix A, the equilibrium equations in the tangential and normal directions are respectively given by,\n\\begin{subequations}\\label{equib}\n\\begin{gather}\n(\\hat{r} \\hat{T}_s)' -\\hat{r}' \\hat{T}_\\theta =0, \\label{equib1}\\\\\n\\hat{\\kappa}_s \\hat{T}_s + \\hat{\\kappa}_\\theta \\hat{T}_\\theta = \\hat{P},\\label{equib2}\n\\end{gather}\n\\end{subequations}\nwhere $\\hat{P}(S)$ represents the pressure difference across the membrane in the normal direction, and the principal curvatures $\\hat{\\kappa}_s,\\hat{\\kappa}_\\theta$ are defined in \\rr{curvatures}. The system \\rr{equib} contains terms involving $\\hat{r}, \\hat{r}',\\hat{r}'',\\hat{z}',\\hat{z}''$, and is therefore a third order differential system, as $\\hat{z}$ does not appear explicitly. Rearranging \\rr{equib1} for $\\hat{T}_\\theta$ and using Codazzi's equation, \\rr{codazzi2}, we may integrate \\rr{equib2} to find,\n\\begin{equation}\\label{intequib}\n\\hat{r}^2 \\hat{\\kappa}_\\theta \\hat{T}_s = \\frac{\\hat{F}}{2 \\pi} + \\frac{\\hat{P} \\hat{r}^2}{2} - \\frac{1}{2} \\int \\dd{\\hat{P}}{\\hat{S}} \\hat{r}^2 d \\hat{S},\n\\end{equation}\nwhich may be used, provided $\\hat{P}$ is constant, to reduce the differential order of the system by one. We note that this integral exists for a membrane of any undeformed shape, and corresponds to the resultant in the $\\hat{Z}$ direction. The integration constant $\\hat{F}$ represents the total transverse force acting on the membrane \\citep{nadler2006}.\n\n\\subsection{Wrinkling}\nUsing \\rr{equib} we may calculate the deformed shape of the membrane, assuming that the stress resultants both stay positive over the region. If either stress resultant becomes negative then the theory as outlined above is not appropriate, as a true membrane can not support compressive stresses. The regions with such a negative compressive stress are assumed to have wrinkled, with the crests of the wrinkles perpendicular to the compressive stress direction; this is a local buckling instability \\citep{jenkins1991}. We therefore introduce tension-field theory to consider a smoothed-out \\lq pseudo-surface' that is in equilibrium, with the corresponding stress exactly zero everywhere. \n\nFor the strain-energy functions considered here, the minimum of $W$ with respect to $\\lambda_\\theta$, subject to both stretches being positive, occurs  when $\\lambda_\\theta = \\lambda_s^{-1\/2} \\equiv w(\\lambda_s)$, with an equivalent expression for $\\lambda_s$. This is termed the natural width of the strip in tension by \\citet{pipkin1986}. Following \\citet{pipkin1986} we introduce the tension-field theory by defining a relaxed strain-energy density function as,\n\\begin{equation}\nW^{*}(\\lambda_s,\\lambda_\\theta) =  \n\\left\\{\n\t\\begin{array}{lcc}\n\\hat{W}(\\lambda_s,\\lambda_\\theta) & \\lambda_s \\geq 1, & \\lambda_\\theta \\geq 1 \\\\\n\\hat{W}(w(\\lambda_\\theta),\\lambda_\\theta) & \\lambda_s \\leq w(\\lambda_\\theta), & \\lambda_\\theta>1 \\\\\n\\hat{W}(\\lambda_s,w(\\lambda_s)) & \\lambda_s>1, & \\lambda_\\theta \\leq w(\\lambda_s) \\\\ 0  &\\lambda_s<1, & \\lambda_\\theta<1\n\\end{array}\n\\right.\n\\end{equation}\nwhich is the quasiconvexification of $\\hat{W}$. Using this definition of $W^{*}$ we recover the normal behaviour in the taut region while resolving the wrinkled regions. When both stretches are less than unity, the membrane becomes slack, with no tension in either direction. If this occurs here then \\rr{intequib} implies that $\\hat{F}=0$ and thus there is no indentation, when $\\hat{P}=0$.\n\n\nWe find numerically that the wrinkled region, if it exists, occurs when $\\hat{T}_\\theta = 0$. We therefore have the following system of equations in the wrinkled region,\n\\begin{equation}\\label{equibwrink}\n\\hat{\\kappa}_s \\hat{T}_s = \\hat{P}, \\;\\; \\left(\\hat{r} \\lambda_s^{1\/2} \\hat{W}_1(\\lambda_s,\\lambda_s^{-1\/2})\\right)'=0.\n\\end{equation}\nWhen the pressure $\\hat{P}$ is zero, \\rr{equibwrink}$_1$ implies that either the membrane is slack (and thus there is no indentation) or $\\kappa_s =0$. We may therefore integrate \\rr{equibwrink} to give,\n\\begin{equation}\\label{equibwrink2}\n\\frac{\\hat{z}'}{\\hat{r} \\sqrt{\\hat{r}'^2 + \\hat{z}'^2}} = \\text{const.}, \\;\\; \\hat{r} \\lambda_s^{1\/2} \\hat{W}_1(\\lambda_s,\\lambda_s^{-1\/2}) = \\text{const.},\n\\end{equation}\nwhere the constants are determined by the values of $\\hat{r}$ and $\\hat{z}$ at the point at which the membrane first becomes wrinkled, $\\hat{S}=\\hat{S}_1$. \n\nThe first order differential equation in \\rr{equibwrink2} may then be integrated until $\\tilde{\\lambda}_\\theta = \\lambda_\\theta$, at which point $\\hat{T}_\\theta$ given by \\rr{stressres} becomes zero again, this is $\\hat{S}=\\hat{S}_2$. This process may then be repeated if the membrane became wrinkled again, if required. Clearly, in the pseudo-surface the lines in the $s$-direction must be straight as the associated curvature $\\hat{\\kappa}_s$ is zero. \\citet{libai2002} present a general algorithm for dealing with axisymmetric wrinkling, and this approach conforms to their technique.\n\n\\citet{haughton1995} and \\citet{haughton1996} state that the extent (in terms of $\\hat{S}$) of the wrinkled region is determined solely by the tense theory, and suggest that if you use the tense theory even when $\\hat{T}_\\theta <0$, then $\\hat{T}_\\theta = 0$ at $\\hat{S}_2$. However, this is contrary to what we find numerically here, and so we integrate the wrinkled region exactly.\n\n\\subsection{Indenter Profile}\nWe assume that the indenter is a cylinder with a curved tip, with a specified axisymmetric profile where the components in the $\\mathbf{e}_r$ and $\\mathbf{e}_z$ directions are given parametrically in terms of an angle $\\phi$ from the negative $z$-axis. We therefore assume the indenter surface is described by \n\\begin{gather}\\label{indenter}\n\\psi(\\phi) = \\left\\{ \\begin{array}{lc}\n \\hat{\\rho} A(\\phi) \\mathbf{e}_r + \\hat{\\rho} B(\\phi) \\mathbf{e}_z, &  0 \\leq \\phi \\leq \\pi\/2, \\\\\n \\hat{\\rho} \\mathbf{e}_r + (\\hat{\\rho} B(\\pi\/2) + \\hat{\\rho} \\phi)\\mathbf{e}_z,  & \\phi >\\pi\/2 \n\\end{array}\n\\right.\n\\end{gather}\nwhere $\\hat{\\rho}$ is the radius of the cylindrical part of the indenter to which this curved tip is attached. We require that the axisymmetry around the $z$ axis is maintained and hence $A(0)=0$, and also that the indenter tip smoothly connects to the cylindrical part, $A(\\pi\/2) = 1, A'(\\pi\/2)=0$. In addition, we require the smoothness condition of $B'(0)=0$, which prevents the consideration of perfectly sharp pointed tips, although we may approach these. We shall also impose the restriction that the indenter profile is convex, although for some non-convex profiles replacing the profile by its convex hull may give the appropriate shape, although the calculation of the load exerted on the membrane may need to be adjusted.\n\nIf the indenter tip is spherical, then $A(\\phi) = \\sin \\phi, B(\\phi) = - \\cos \\phi$, where the negative sign merely ensures that we consider the  indentation to occur in the negative $\\mathbf{e}_z$ direction. Spheroidal tips may be considered by setting $B(\\phi) = - \\gamma \\cos \\phi$, where $\\gamma > 1$ for a prolate spheroid and $0<\\gamma < 1$ for an oblate spheroid. As $\\gamma \\to 0$, the case of the indentation by a flat cylindrical punch is approached, while $\\gamma \\to \\infty$ approximates indentation by a sharp tip. \n\nWe note that it is possible to use an indenter which consists of just the \\lq tip' by modifying \\rr{indenter}, for instance when a solid sphere is allowed to deform the membrane under gravity as in the experiments of \\citet{liu2001}. For a given applied force this may be modelled by a spherical-tipped cylindrical indenter, provided that the deformed surface does not touch the sides of the cylinder. Indeed, it has been stated by \\citet{selvadurai2006} and \\citet{nadler2006} that the deformed surface never makes contact with the cylindrical part prior to membrane penetration, although our preliminary experimental results and the second figure in \\citet{nguyen2004mech} suggest otherwise, so perhaps that statement is a specification to their equations. In particular for the Varga strain-energy function we agree with \\citet{nadler2006}, but not for general strain-energy functions. Hence we do allow the membrane to touch the cylindrical part of the indenter, where we set $\\hat{r}(\\hat{S}) \\equiv \\hat{\\rho}$ and solve for $\\hat{z}$, although these regions will be dashed on the load-indentation graphs presented in Section \\ref{results}.\n\n\\subsection{Solution Procedure}\nWe now non-dimensionalise with respect to the undeformed radius, $\\hat{R}_L= \\hat{R}(\\hat{L})$, and the infinitesimal shear modulus, $\\mu$,\n\\begin{gather}\n(\\hat{S},\\hat{R},\\hat{Z},\\hat{H},\\hat{r},\\hat{z}, \\hat{\\rho},\\hat{L}) = \\hat{R}_L (S,R,Z,H,r,z,\\rho,L), \\;  (\\hat{T}_s, \\hat{T}_\\theta) = \\mu \\nonumber\n\\hat{R}_L H (T_s,T_\\theta), \\\\\n\\hat{P} = \\mu P, \\; \\hat{F} =  \\mu \\hat{R}_L^2 H F, \\; \\hat{W} = \\mu W.\n\\end{gather}\nFor flat membranes $\\hat{R}_L \\equiv \\hat{L}$, but for a curved membrane this is not generally true. This rescaling leaves the previously stated equations for the stretches, stress resultants and curvatures unchanged, apart from dropping the hats from all variables and removing $\\hat{H}$ from \\rr{stressres}. The parameter $\\rho$ is now a non-dimensional ratio relating the radius of the indenter to the radius of the undeformed membrane.\n\nThe problem may be divided into two distinct regions of $S$, that is the region in the undeformed geometry that is now in contact with the indenter, $0 \\leq S \\leq S_c$, and the outer region where the membrane is free, $S_c < S \\leq L$. For a flat membrane, $S_c$ is the radius of the circle in the undeformed membrane which is in contact with the indenter in the deformed configuration, while for a curved reference surface $R(S_c)$ is the radius of the corresponding circle in the undeformed membrane. If wrinkling occurs in  the free region there will be four distinct regions instead, as we find numerically that the membrane returns to biaxial tension by $S=L$ in the examples considered here.\n\nWhile the membrane is in contact with the indenter, we assume that the deformed shape follows the surface of the indenter and we therefore have,\n\\begin{equation}\nr(S) = \\rho A(\\phi(S)), \\; z(S) = - \\delta + \\rho B(\\phi(S)),\n\\end{equation}\nwhere $\\delta$ is the depth of indentation to be found as part of the solution. Therefore in the contact region,\n\\begin{equation}\\label{indentstretch}\n\\lambda_s = \\rho \\frac{\\sqrt{\\left(\\dd{A}{\\phi}\\right)^2 + \\left(\\dd{B}{\\phi}\\right)^2}}{\\sqrt{R'^2 + Z'^2}} \\dd{\\phi}{S}, \\; \\lambda_\\theta = \\rho \\frac{A(\\phi(S))}{R(S)},\n\\end{equation}\nwhere we have used $\\phi'(S) >0$. We may therefore evaluate the first equilibrium equation, \\rr{equib1}, to find $\\phi(S)$ given appropriate boundary conditions. The second equilibrium equation enables us to calculate the pressure $P$ exerted by the indenter on the membrane after calculating the membrane deformation, but is auxiliary to computing the deformation itself in the contact region.\n\n\\subsection{Boundary Conditions}\nWe now specify the boundary conditions in order to integrate the equations above. The requirement of axisymmetry implies that $\\phi(0)=0$, while using the restrictions on $A$ and $B$ we find $\\phi'(0) = \\rho^{-1} \\lambda_s(0)\/A'(0)$, and we can therefore integrate \\rr{equib1} to find $\\theta(S)$ for a given stretch at the pole $\\lambda_s(0)$.\n\nIn the region $S_c<S \\leq L$, the membrane is governed by \\rr{equib} with $P=0$, which gives us a third order system for $r$ and $z$, or $\\lambda_s$ and $\\lambda_\\theta$. We require continuity of the membrane shape, and so we prescribe $r(S_c),r'(S_c),z'(S_c)$ to be those given from the inner region.\nWe allow for the membrane to be stretched prior to indentation, imposing a pre-stretch of $\\lambda_p \\geq 1$ on the material, that is $\\lambda_p = r(L)$.\nThis pre-stretch corresponds to moving the circular boundary $S=L$ outwards in the $\\mathbf{e}_r$ direction, inducing a deformation in the membrane; if the membrane is not initially flat then this will have the result of changing the shape of the initial profile by decreasing the vertical \\lq height' of the membrane, as the material moves towards the level of the boundary circle.\n\nUtilising a shooting method we specify the stretch at the pole, $\\lambda_s(0)=\\lambda_\\theta(0) = \\lambda_0$, and then guess a value of $S_c$. We then iterate on $S_c$ to satisfy the condition $\\lambda_\\theta(L) =  \\lambda_p$. Having found the appropriate value of $S_c$ for a given $\\lambda_0$ and $\\lambda_p$, we may then calculate the value of the central displacement of the membrane which is given by,\n\\begin{equation}\\label{zint}\nz_h=\\int_0^L \\dd{z}{S} \\; dS  = \\int_0^{S_c} \\dd{z}{S} \\;dS + \\int_{S_c}^L \\dd{z}{S} \\; dS,\n\\end{equation} \nwhere the solutions in the appropriate regions are substituted into each integral of \\rr{zint}. From this we calculate the indentation depth, $\\delta = z_h - Z_h$, where $Z_h = Z(L)-Z(0)$ is the original depth of the membrane. Having done this, we may vary $\\lambda_0$ and repeat the process in order to solve for a variety of indentation depths, or equivalently for values of the applied force $F$.\n\n\nWe therefore require the following parameters to be specified: the membrane prestretch $\\lambda_p$, the radius of the cylindrical part of the indenter $\\rho$, the undeformed membrane shape $(R,Z)$ and the shape of the indenter tip $(A,B)$. In addition, the strain-energy function must be specified, along with the value of any parameters included therein. The choice of strain-energy function is not a trivial matter in the large indentation regime, and we will show how the force required to indent the membrane is strongly dependent on the form of the strain-energy function used, past the initial linear regime.\n\n\\section{Results and Discussion}\\label{results}\n\\begin{figure}[!htbf]\n \\psfrag{aa}{$r$}\n \\psfrag{bb}{$z$}\n\\centering \\epsfig{figure=indent1.eps,width=.6\\textwidth}\n\\caption{Deformed configuration after indentation of a flat membrane by a spherical indenter for, from top to bottom, $\\lambda_0 = 1.05,1.1,1.2,1.4,1.6,2$, with $\\lambda_p = 1$ and $\\rho = 0.2$. The strain-energy function used is the Mooney Rivlin with $\\alpha=0.1$.}\n\\label{sphereindent}\n\\end{figure}\nFigure \\ref{sphereindent} shows how the deformed configuration changes with increasing $\\lambda_0$ for a spherical indenter and a flat membrane, reproducing Figure 2 of \\citet{yang1971}. It may be seen that the deformed shape goes from having positive curvature while in contact with the indenter to a negative curvature in the free region. This is as expected as equation \\rr{equib2} implies that an odd number of the quantities $T_s,T_\\theta,\\kappa_s,\\kappa_\\theta$ must be negative in the free region, and therefore in unwrinkled parts of the membrane one of the curvatures must be negative. When the membrane is initially flat wrinkling is never observed, as found by \\citet{haughton1995} for the fluid-loaded case.\n\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=varyshapecolour2.eps,width=.6\\textwidth}\n\\caption{\n a variety of spheroidal indenter tips with $A(\\theta) = \\sin \\theta, B(\\theta) = - \\gamma \\cos \\theta, \\lambda_p=1,\\rho=0.2$. From top to bottom, $\\gamma = 0.005,0.125,0.25,0.5,1,2,4,6,8,200$}\n\\label{varyshapecolour}\\label{dispindent}\n\\end{figure}\n\n\nHaving calculated the deformed shape of the membrane we may evaluate the transverse force $F$ in the free region, this corresponds to an indentation displacement $\\delta$. Figure \\ref{dispindent} shows a variety of non-dimensional force-indentation curves for spheroidal indenter tips of varying shape. As is intuitively obvious, blunter indenter tips require more force to achieve the same indentation depth. The two extremal curves in Figure \\ref{dispindent} approach indentation with a flat punch and a point load respectively. In Figure \\ref{dispindent}, and following figures, dashed lines represent where the membrane has come into contact with the cylindrical part of the indenter, which occurs sooner with flatter tips.\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\psfrag{lab1}{\\tiny neo-Hookean}\n\\psfrag{lab2}{\\tiny Mooney-Rivlin, $\\alpha=0.1$}\n\\psfrag{lab3}{\\tiny Mooney-Rivlin, $\\alpha=0.5$}\n\\psfrag{lab4}{\\tiny Fung, $\\Gamma = 1$}\n\\centering \\epsfig{figure=indent3.eps,width=.8\\textwidth}\n\\caption{Force-indentation curves for the indentation by a spherical indenter of a flat membrane modelled using the neo-Hookean, Mooney-Rivlin and Fung strain-energy functions with $\\lambda_p=1,\\rho=0.2$. }\n\\label{sevary}\n\\end{figure}\n\nFigure \\ref{sevary} shows the load-indentation curves for a variety of strain-energy functions, where it may be seen that for this set of parameters up to $\\delta \\approx 0.4$ the material responses are identical, which is consistent to the results of \\citet{pujara1978} regarding the inflation of circular membranes. Past this range the varied stress-strain behaviour of the strain-energy functions give different loads for the same indentation, and thereafter the specification of the strain-energy function becomes important. The Fung material model, generally used for biological tissues such as arteries and muscles, predicts much higher loads than the other models for the same amount of indentation, as expected due to the strain-stiffening behaviour the Fung model encapsulates.\n\nThe neo-Hookean material model gives a horizontal asymptote in Figure \\ref{sevary}, with a constant $F$ corresponding to an infinite range of indentation. This is a consequence of using the neo-Hookean strain-energy function outside its range of validity.\nThe Mooney-Rivlin material gives linear growth of $F$ with $\\delta$, depending on the value of $\\alpha$, while the Fung material shows an exponential growth.\n\nIn particular, when the membrane is touching the cylindrical portion of the indenter we have $r' = 0$, and therefore evaluating \\rr{intequib} at $S=S_c$ gives,\n\\begin{equation}\\label{Fcyl}\nF = \\left. 2 \\pi R(S_c) \\pafrac{W}{\\lambda_s} \\right|_{S=S_c},\n\\end{equation}\nwhich explains how the response is dominated by the behaviour of the stress-strain relationship, as \\rr{Fcyl} indicates that force is proportional to the stress.\n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=flatvaryrho.eps,width=.6\\textwidth}\n\\caption{Force-indentation curves for the indentation of a flat Mooney-Rivlin membrane by a spherical indenter of varying size, from bottom to top, $\\rho=0.1,0.2,\\ldots,0.9$}\n\\label{rhovary}\n\\end{figure}\nFigure \\ref{rhovary} shows how the size of the indenter affects the load behaviour, although the quantitative response behaviour is not affected. For large indentation depths the curves are parallel to each other, both before and after the membrane begins to touch the sides of the indenter. \n\nIn all the force-indentation curves for flat membranes presented above, in the linear elastic limit the relation $F \\propto \\delta^3$ may be seen, corresponding to the Schwerin-type solution in linear elasticity \\citep{komaragiri2005}. \\citet{begley2004} derive an approximate solution with such a cubic relationship for a finite-sized spherical indenter and a flat membrane, but we do not agree numerically with their proportionality constant. \n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\\centering \\epsfig{figure=linearlimitsph.eps,width=.6\\textwidth}\n\\caption{Close-up of the force-indentation curve for a spherical indenter and a flat membrane with the Mooney Rivlin strain-energy function with $\\alpha=0.1$, $\\lambda_p=1$, $\\rho=0.2$. Curve is given by $F=4.7292 \\delta^3$.}\n\\label{sphcubic}\n\\end{figure}\n\n\\begin{figure}[!htbf] \n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$}\n\\centering \\epsfig{figure=initspher.eps,width=.6\\textwidth}\n\\caption{Deformation of an initially spherical surface $R(S) = \\sin S, Z(S) = - \\cos S, \\rho = 0.4, \\lambda_p=1$ with a spherical indenter and a Mooney-Rivlin strain-energy function with $\\alpha=0.1$. From top to bottom: $\\lambda_0 = 1.2,1.4,1.6,2$. Dashed line shows the undeformed shape of the membrane, dotted sections of the deformed surface are wrinkled.}\n\\label{initspher}\n\\end{figure}\n\nFigure \\ref{initspher} shows the indentation of an initially spherical membrane by a spherical indenter of smaller radius. It can be seen that the deformed shape does not follow the curvature of the reference surface, which may be expected from the previous comment that $\\kappa_s$ or $T_\\theta$ must be negative, and from the fact that the governing equations do not include a term involving the reference curvatures, i.e. \\rr{curvatures} but with $r$ and $z$ replaced by $R$ and $Z$. In order to include such a term in the governing equations, we would need to consider a shell theory where the energy depends on the curvature of the shell in addition to the stretching, as discussed in \\citet{steigmann2001} for example. This increases the order of the governing equations from third to seventh order, and also introduces the question of determining how the strain-energy function depends on a further set of invariants; an additional two which depend only on the curvatures and three coupling terms between the curvatures and the stretches.\n\n \n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \\psfrag{lab1}{$\\zeta = 0$}\n\\psfrag{lab2}{$\\zeta = 0.125$}\n\\psfrag{lab3}{$\\zeta = 0.25$}\n\\psfrag{lab4}{$\\zeta = 0.5$}\n\\centering \\epsfig{figure=varyindentcurved.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation of an initially curved membrane with $R(S) = S, Z(S) = -\\cos (\\pi S\/2)\/4$ by various spheroidal indenter tips, from top to bottom: $\\gamma = 0.25,0.5,1,2,4,8$ with $\\lambda_p=1,\\rho=0.2$}\n\\label{RZvary}\n\\end{figure}\nFigure \\ref{varyindent2} shows the load-indentation curves for initially curved surfaces of the form, $R(S) = S, Z(S) = - \\psi \\cos ( \\pi S \/2), 0 \\leq S \\leq 1$. As $\\psi \\to 0$ we recover the flat membrane, and we see that as the reference surface departs from this shape the required load for greater indentation increases. \n\nFor pronouncedly curved reference surfaces, $\\psi \\approx 0.5$, we find that the theory used here gives a vanishingly small load for a finite indentation displacement. As we take the numerical limit $\\lambda_0 \\to 1$, the calculated load tends towards zero but there is a non-zero displacement, $\\delta \\approx 0.09$ in the membrane shown in Figure \\ref{varyindent2}. We believe that the reason this occurs is that the membrane wrinkles in the indentation region, where we are prescribing the displacement to be that of the indenter.\n\n\nThis limitation does not prevent us from solving for larger values of $\\lambda_0$ due to the quasi-static solution procedure used here. Figure \\ref{varyindent3} shows the same curved membrane as Figure \\ref{varyindent2}, but with the Fung strain-energy function instead of the Mooney-Rivlin. The linear elastic behaviour of the two graphs is the same, but at large displacement or load the impact of the strain-energy function may be seen. \n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \n\\centering \\epsfig{figure=varycurvedxx.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation with a spherical indenter of an range of initially curved membranes with $R(S) = 2 S\/ \\pi, Z(S) = -\\psi \\cos S$, from top to bottom: $\\psi=0.5,0.25,0.125,0.1,0.05,0$ and $\\lambda_p=1,\\rho=0.2$.}\n\\label{varyindent2}\n\\end{figure}\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\frac{\\hat{\\delta}}{\\hat{R}_L}$}\n\\psfrag{bb}{$\\frac{\\hat{F}}{2 \\pi \\mu \\hat{R}_L \\hat{H}}$} \n\\centering \\epsfig{figure=varycurvedxxFung.eps,width=.6\\textwidth}\n\\caption{Load-indentation curves for the indentation with a spherical indenter of an range of initially curved membranes with $R(S) = 2 S\/ \\pi, Z(S) = -\\psi \\cos S$, from top to bottom: $\\psi=0.5,0.25,0.125,0.1,0.05,0$ and $\\lambda_p=1,\\rho=0.2$, using the Fung strain-energy function.}\n\\label{varyindent3}\n\\end{figure}\n\n\n\nWhen the membrane is flat, both the stress resultants stay positive and wrinkling does not occur. For curved membranes, it is possible to increase the radial pre-stretch $\\lambda_p$ or the indentation distance $\\delta$ to ensure that the membrane remains in tension in the deformed configuration. This finding agrees directly with that of \\citet{haughton1996} for fluid filled membranes. For instance, if a spherical indenter nearly fills a spherical membrane, i.e. $\\rho$ is close to one, then wrinkling is only found for very small levels of $\\lambda_0$ with no pre-stretch, and eliminated entirely for slight amounts of pre-stretch, such as $\\lambda_p \\geq 1.1$. However, the imposition of the pre-strain will change the shape of the non-flat membrane prior to indentation, requiring knowledge of the shape of the membrane prior to the pre-strain, which may not be available in experimental situations.\n\n\\begin{figure}[!htbf]\n\\psfrag{aa}{$\\delta$}\n\\psfrag{bb}{$S_c$}\n\\psfrag{lab1}{\\tiny NH}\n\\psfrag{lab2}{\\tiny MR, $\\alpha=0.1$}\n\\psfrag{lab3}{\\tiny MR, $\\alpha=0.5$}\n\\psfrag{lab4}{\\tiny Fung, $\\Gamma = 1$}\\centering \\epsfig{figure=scvary.eps,width=.6\\textwidth}\n\\caption{Position of contact point $S_c$ as indentation depth changes for a indentation of a flat membrane with a spherical indenter, $\\lambda_p=1, \\rho=0.2$. The neo-Hookean (NH), Mooney-Rivlin (MR) and Fung strain-energy functions are shown.}\n\\label{scvary}\n\\end{figure}\n\nAs the indenter is pushed into a flat membrane the contact point $S_c$ initially increases as more of the indenter comes into contact with the material, before reaching a maximum value and then decreasing, as shown in Figure \\ref{scvary}. This later decrease is due to the fact that, at larger indentation depth, a comparatively small area of the reference membrane is highly stretched over the indenter tip. Once the membrane begins to touch the cylindrical sides of the indenter the behaviour of $S_c$ is governed by the strain-energy function in the same manner as $F$. This behaviour is found by \\citet{nadler2006} for the two-dimensional compressible Varga strain-energy function, although in this case the membrane never comes into contact with the cylindrical part of the indenter. \n\n\\section{Conclusions}\nWe have shown how to extend the previously presented non-linear elasticity theories of indentation with a finite radius indenter to include the effects of different strain-energy functions, non-spherical indenter tips and initially curved membranes. The importance of  correctly specifying the strain-energy function is shown clearly, along with the effects that changing the shape of the indenter has. \n\nIndentation tests are a useful method of determining the properties of an elastic membrane at large strain, and with experimental data from a wide variety of shapes and sizes of indenter a suitable strain-energy function may be fitted to the data, giving more information than merely spherical indenter tips. One difficulty may be in identifying where the start of the force-indentation curve occurs experimentally, due to the detection limits of the equipment used, which may mean that the beginning of the curve is missed.\n\n\nAs mentioned in the introduction, we plan to compare with experimental data from puncture force experiments on germinating seeds, with the aim of estimating the elastic properties of the endosperm using the theory presented here. The experiments will result in load-indentation curves, which we may then use to fit to the load-indentation curves resulting from the method here. In this case, the unknown parameters are the infinitesimal shear modulus and the strain-energy function of the material, and we shall attempt to characterize these through the  indirect indentation method.\n\n\\section*{Acknowledgements}\nThis work has been funded by an ERANET-PG grant for the vSEED project (BBSRC grant BBG02488X1). JRK additionally acknowledges the support of the \nRoyal Society and the Wolfson Foundation.\n\n\\section*{Appendix}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nIn this appendix we shall provide a sketch of the derivation of the equilibrium equations given in \\rr{equib}, using the membrane-like shell approach. For full details including proofs and error estimates of the derivation here see \\citet{haughton2001} or \\citet{haughton1978a}. Where possible we shall use upper case letters to describe the undeformed configuration and lower case letters to describe the deformed configuration. In addition, Greek indices vary over $(1,2)$, and the summation convention for repeated indices is used unless otherwise noted.\n\nWe consider a thin shell with position vector,\n\\begin{equation}\n\\mathbf{X} = \\mathbf{Y}(\\xi^1, \\xi^2) + \\xi^3 \\mathbf{N}(\\xi^1,\\xi^2),\n\\end{equation}\nwhere $(\\xi^1,\\xi^2)$ are curvilinear coordinates describing the midsurface of the shell, $-H\/2 \\leq \\xi^3 \\leq H\/2$ is a thickness coordinate in the positive normal direction $\\mathbf{N}$ and $H$ is the thickness of the undeformed shell. We assume that the shell is thin enough such that if $\\eta$ is the minimum principal radii of curvature of the midsurface then,\n\\begin{equation}\n\\epsilon = \\frac{H_{\\textrm{max}}}{\\eta} \\ll 1,\n\\end{equation}\nwhere $H_{\\textrm{max}}$ is the maximum thickness throughout the shell.\n\nWe assume that the position vector $\\mathbf{x}$ of the deformed membrane may be decomposed in the same way into a surface vector $\\mathbf{y}$ and a thickness component in the deformed normal direction $\\mathbf{n}$,\n\\begin{equation}\n\\mathbf{x} = \\mathbf{y}(\\zeta^1,\\zeta^2) + \\zeta^3 \\mathbf{n}(\\zeta^1,\\zeta^2),\n\\end{equation}\nwhere $(\\zeta^1,\\zeta^2)$ are coordinates describing the deformed surface of the shell and $-h\/2 \\leq \\zeta^3 \\leq h\/2$ expresses the distance through the thickness direction of the membrane. \\citet{haughton2001} show that, correct to within $\\mathcal{O}(\\epsilon)$, the surface $\\zeta^3=0$ corresponds to the midsurface of the reference membrane $\\xi^3=0$, so that $\\mathbf{y}$ may be used as the midsurface of the deformed membrane. Expanding $\\zeta^3$ around $\\xi^3$, we may also find that $h= H \\zeta^3_{,3}(\\xi^1,\\xi^2,0)$ to the same order. The tangent vectors to the undeformed and deformed surfaces may be written as,\n\\begin{equation}\n\\mathbf{G}_\\alpha = \\mathbf{Y}_{,\\alpha}, \\; \\mathbf{g}_\\alpha = \\mathbf{y}_{,\\alpha},\n\\end{equation}\nand we also define dual vectors,\n\\begin{equation}\n\\mathbf{G}^\\alpha = G^{\\alpha \\beta} \\mathbf{G}_\\beta, \\; \\mathbf{g}^\\alpha = g^{\\alpha \\beta} \\mathbf{g}_\\beta,\n\\end{equation}\nwhere $G^{\\alpha \\beta} G_{\\alpha \\beta} = \\delta^{\\alpha}_{\\beta}, G_{\\alpha \\beta} = \\mathbf{G}_\\alpha \\cdot \\mathbf{G}_\\beta, g^{\\alpha \\beta} g_{\\alpha \\beta} = \\delta^{\\alpha}_{\\beta}, g_{\\alpha \\beta} = \\mathbf{g}_\\alpha \\cdot \\mathbf{g}_\\beta$. The deformation gradient \\citep{ogden2001} may be given by \\citep{haughton2001},\n\\begin{equation}\n\\mathbf{F} = \\pafrac{\\mathbf{x}}{\\mathbf{X}}=\\pafrac{\\mathbf{y}}{\\mathbf{Y}} + \\zeta^3_{,3}(\\xi^1,\\xi^2,0) \\mathbf{n} \\otimes \\mathbf{N},\n\\end{equation}\nfrom which we conclude that $\\zeta^3_{,3}(\\xi^1,\\xi^2,0)$ is a principal stretch. We may write the two-dimensional deformation gradient as,\n\\begin{equation}\\label{defgradx}\n\\pafrac{\\mathbf{y}}{\\mathbf{Y}} = \\mathbf{g}_\\alpha \\otimes \\mathbf{G}^{\\alpha},\n\\end{equation}\nwhich can then be directly computed. Following \\citet{haughton2001} we now define an orthonormal set of basis vectors $(\\mathbf{a_i})$ by,\n\\begin{equation}\n\\mathbf{a}_\\alpha = \\frac{\\mathbf{g}_\\alpha}{\\left| \\mathbf{g}_\\alpha \\right|} \\;\\;\\;\\; \\textrm{no sum},\n\\end{equation}\nwhich enables us to dispense with the distinction between covariant and contravariant tensors when we use $\\mathbf{a}_\\alpha$ as a basis \\citep{fu2001}.\nReferred to this basis, the general non-linear elasticity equilibrium equation $\\textrm{div} \\, \\hat{\\mbox{\\boldmath $\\sigma$}} = \\mathbf{0}$ becomes at zeroth order in $\\epsilon$, \\citep{haughton1978a, haughton2001}, \n\\begin{subequations}\\label{equilx}\n\\begin{gather}\n\\hat{\\sigma}_{\\mu \\kappa, \\mu} + \\hat{\\sigma}_{\\mu \\kappa} \\mathbf{a}_{\\mu,\\nu} + \\hat{\\sigma}_{\\mu \\nu} \\mathbf{a}_\\kappa \\cdot \\mathbf{a}_{\\mu, \\nu} + \\hat{h}_{, \\mu} \\hat{\\sigma}_{\\mu \\kappa}\/\\hat{h} = 0, \\quad \\kappa=1,2 \\label{equilx1}\\\\\n\\hat{\\sigma}_{\\mu \\kappa} \\mathbf{a}_3 \\cdot \\mathbf{a}_{\\mu, \\nu} + \\hat{P}\/\\hat{h} =0\\label{equilx2}\n\\end{gather}\n\\end{subequations}\nwhere $\\hat{\\sigma}_{\\alpha \\beta}$ is the $(\\alpha,\\beta)$-th element of the stress tensor $\\mbox{\\boldmath $\\sigma$}$, $\\hat{P}$ is a hydrostatic pressure applied to the inner surface and due to the use of the orthonormal basis vectors we have the additional definition $()_{,\\mu} = \\left| \\pafrac{\\mathbf{y}}{\\zeta^\\mu} \\right|^{-1} \\pafrac{()}{\\xi^\\mu}$. The third principal stress, $\\hat{\\sigma}_{33}$, is of order $\\epsilon$ at most, and therefore does not appear in \\rr{equibeqn}. This is often called the membrane assumption, $\\hat{\\sigma}_{33} = 0$.\n\nIt was shown by \\citet{naghdi1977} that the equilibrium equations presented above may be shown to be equivalent to those derived by considering the two-dimensional surface theory, under certain assumptions on $\\hat{P}$ and $\\hat{\\sigma}_{ij}$. This may be verified by a direct calculation of the equilibrium equations using the membrane theory detailed by \\citet{steigmann1999} or \\citet{steigmann2001} for instance. See \\citet{haughton2001} for more details on the comparison between the two theories.\n\n\nWe now use the coordinate systems introduced in \\rr{undefcon} and \\rr{defcon} to describe the undeformed and deformed configurations respectively. We may therefore associate the two curvilinear coordinate systems $(\\xi^\\alpha),(\\zeta^\\alpha)$ with the arclength and polar angle,\n\\begin{equation}\n\\xi^1 = A \\Theta, \\xi^2 = S, \\;\\; \\zeta^1 = A \\theta, \\zeta^2 = S,\n\\end{equation}\nwhere $A$ is a representative length. The orthonormal basis then becomes,\n\\begin{equation}\\label{adef}\n\\mathbf{a}_1 = \\mathbf{e}_\\theta, \\; \\mathbf{a}_2 = \\frac{\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z}{(\\hat{r}'^2+\\hat{z}'^2)^{1\/2}}, \\; \\mathbf{a}_3 = \\mathbf{a}_1 \\wedge \\mathbf{a}_2  = \\frac{\\hat{z}' \\mathbf{e}_r - \\hat{r}' \\mathbf{e}_z}{(\\hat{r}'^2+\\hat{z}'^2)^{1\/2}},\n\\end{equation}\nwhere a prime represents differentiation with respect to $\\hat{S}$. It may be seen that these vectors are unit vectors in the meridional, tangential and normal directions of the deformed surface. A similar orthonormal basis $\\mathbf{A}_\\alpha$ may be described with respect for the undeformed configuration with upper case letters replacing all equivalent lower case ones. The deformation gradient \\rr{defgradx} is then given by,\n\\begin{align}\n\\mathbf{F} &= \\frac{\\hat{r}}{\\hat{R}} \\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta + \\left(\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z \\right) \\otimes \\left(\\frac{\\hat{R}' \\mathbf{e}_r + \\hat{Z}' \\mathbf{e}_z}{\\hat{R}'^2 + \\hat{Z}'^2}\\right) + \\frac{\\hat{h}}{\\hat{H}} \\mathbf{n} \\otimes \\mathbf{N} \\nonumber \\\\\n&=\\frac{\\hat{r}}{\\hat{R}} \\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta+\n\\frac{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}{\\sqrt{\\hat{R}'^2+\\hat{Z}'^2}} \\left(\\frac{\\hat{r}' \\mathbf{e}_r + \\hat{z}' \\mathbf{e}_z}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right) \\otimes \\left(\\frac{\\hat{R}' \\mathbf{e}_r + \\hat{Z}' \\mathbf{e}_z}{\\sqrt{\\hat{R}'^2+\\hat{Z}'^2}}\\right) + \\frac{\\hat{h}}{\\hat{H}} \\mathbf{n} \\otimes \\mathbf{N}.\\label{defff}\n\\end{align}\nwhich shows that the principal directions are the meridional, tangential and normal directions respectively. As the deformation gradient can be written in a diagonal form, the stress tensor must also be diagonal with respect to the same basis, and so the only non-zero terms of $\\hat{\\mbox{\\boldmath $\\sigma$}}$ relative to this basis are $\\hat{\\sigma}_{11}$ and $\\hat{\\sigma}_{22}$. Additionally, using this basis we may immediately see the principal stretches in the three directions as the coefficients in \\rr{defff}.\nUsing \\rr{adef} in \\rr{equilx}, along with some rearrangement and noting that $()_{,2}$ in \\rr{equilx} becomes $()'\/\\sqrt{\\hat{r}'^2 + \\hat{z}'^2}$, we find the equilibrium equations, \n\\begin{gather}\\label{equibeqn}\n\\pafrac{(\\hat{h} \\hat{r} \\hat{\\sigma}_{11})}{\\theta} =0, \\\\\n\\pafrac{(\\hat{h} \\hat{r} \\hat{\\sigma}_{22})}{S} =  \\hat{h} \\hat{r}' \\hat{\\sigma}_{11},\\\\\n\\hat{\\sigma}_{11} \\frac{\\hat{z}'}{r \\sqrt{\\hat{r}'^2+\\hat{z}'^2}} + \\hat{\\sigma}_{22} \\frac{\\hat{z}'' \\hat{r}' - \\hat{r}'' \\hat{z}'}{(\\hat{r}'^2+\\hat{z}'^2)^{3\/2}} - \\frac{\\hat{P}}{\\hat{h}} = 0.\n\\end{gather}\nThe first equation in \\rr{equibeqn} is immediately satisfied due to the restriction of axisymmetry in the deformed configuration. The remaining two equations are the governing equations for the deformation, and represent force balances in the tangential and normal directions respectively. In \\rr{equibeqn}$_3$ the coefficients of the stresses are actually the principal curvatures in the deformed configuration, \n\\begin{equation}\\label{curvatures}\n\\hat{\\kappa}_s =  \\frac{1}{\\hat{r}'}\\left(\\frac{\\hat{z}'}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right)' = \\frac{-1}{\\hat{z}'}\\left(\\frac{\\hat{r}'}{\\sqrt{\\hat{r}'^2+\\hat{z}'^2}}\\right)' , \\; \\hat{\\kappa}_\\theta = \\frac{\\hat{z}'}{\\hat{r} \\sqrt{\\hat{r}'^2+\\hat{z}'^2}},\n\\end{equation}\nwhere $\\kappa_s, \\kappa_\\theta$ are the principal curvatures in the circumferential and radial directions respectively. These curvatures may be found from the components of the embedded deformed curvature tensor $\\mbox{\\boldmath $\\kappa$} = \\kappa_{\\alpha \\beta} \\mathbf{G}^{\\alpha} \\otimes \\mathbf{G}^{\\beta}$ using the definition $\\kappa_{\\alpha \\beta} = \\mathbf{n} \\cdot \\mathbf{g}_{\\alpha, \\beta}$, and then referred to the appropriate basis by $\\kappa_\\theta = \\kappa_{11} g^{11}, \\kappa_s = \\kappa_{22} g^{22}$.\n\nWhen the undeformed surface is flat or spherical we have $\\sqrt{\\hat{R}'^2 + \\hat{Z}'^2} = 1$ and the radicals in equation \\rr{curvatures} may be expressed neatly including the principal stretch $\\lambda_s$, but for a general surface this is not the case. We also note that the curvatures are always related by Codazzi's equation,\n\\begin{equation}\\label{codazzi2}\n(\\hat{r} \\hat{\\kappa}_\\theta)' = \\hat{r}' \\hat{\\kappa}_s. \n\\end{equation}\nwhich enables us to integrate one of the equilibrium equations under certain conditions, see \\rr{intequib}.\n\n\\clearpage\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\nProduced by Roger Frank and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n[Illustration: He shot his hand across an' pulled his gun quick as a\nflash; but Horace didn't move, he just sat still, with a friendly\nsmile on his face]\n\n\n\n\n                               FRIAR TUCK\n\n                  BEING THE CHRONICLES OF THE REVEREND\n                  JOHN CARMICHAEL, OF WYOMING, U.S.A.,\n\n                    AS SET FORTH AND EMBELLISHED BY\n                         HIS FRIEND AND ADMIRER\n                             HAPPY HAWKINS\n\n                          AND HERE RECORDED BY\n                         ROBERT ALEXANDER WASON\n\n                               AUTHOR OF\n                             HAPPY HAWKINS,\n                        THE KNIGHT-ERRANT, ETC.\n\n                             ILLUSTRATED BY\n                            STANLEY L. WOOD\n\n                                NEW YORK\n                            GROSSET & DUNLAP\n                               PUBLISHERS\n\n\n\n\n                            Copyright, 1912\n                     By Small, Maynard and Company\n                             (Incorporated)\n\n                      Entered at Stationers' Hall\n                     Published, September 7, 1912;\n                     Sixth edition, November, 1912\n\n\n\n\nMany there are who respond to the commonplace, monotonous call of\nDuty, and year after year uncomplainingly spend their lives on the\ntreadmill of Routine; but who still feel in their hearts the call of\nthe open road, the music of the stars, the wine of the western wind,\nand the thrilling abandon of a mad gallop out beyond speed limits and\ngrass signs to where life has ceased to be a series of cogs and--a man\nis still a man.\n\nTo the members of this fraternity, whose emblem, hidden behind deep\nand steadfast eyes, is often missed by man, but always recognized by\ndogs and horses, I dedicate this book, in the hope that for an hour or\ntwo it may lift the pressure a little.\n\n                                                              R. A. W.\n\n\n\n\nJUST BETWEEN YOU AND ME\n\nReviews are not infrequently  by a temporary elevation of the\ncritic's mind (or a temporary depression of the critic's liver),\nadvertisements are not invariably free from bias; so, perhaps, a few\nwords of friendly warning will not be considered impertinent.\n\nWhosoever is squeamishly sensitive as to the formal technique of\nliterary construction will save himself positive irritation by\navoiding this book. It is a told, rather than a written story; and\nthis is a compromise which defies Art and frankly turns to the more\nelastic methods of Nature.\n\nIt is supposed to be told by an outdoor man in those delightful\nmoments of relaxation when the restraint of self-consciousness is\ndropped, and the spirit flows forth with a freedom difficult to find,\noutside the egoism of childhood. This general suggestion is easily\ntossed out; but the reader must supply the details--the night camps\nwith the pipes sending up incense about the tiny fires, the winter\nevenings when the still cold lurks at the threshold or the blizzard\nhowls around the log corners; or those still more elusive moments when\nthe riding man shifts his weight to a single thigh, and tells the\ninner story which has been rising from his open heart to his closed\nlips for many a long mile.\n\nNor will these details suffice to complete the atmosphere in which,\nbit by bit, the story is told. The greatest charm in the told story\ncomes direct from the teller; and, toil as we will over printed pages,\nthey obstinately refuse to reproduce the twinkle of bright, deep-set\neyes, the whimsical twist which gives character to a commonplace word,\nthe subtile modulations of a mellow voice, the discriminating accent\nwhich makes a sentence fire when spoken, and only ashes when written;\nor, hardest of all, those eloquent pauses and illuminating gestures\nwhich convey a climax neither tongue nor pen dare attempt.\n\nHappy Hawkins is complex, but the basic foundation of his character is\nsimplicity. His audience is usually a mixed one, men of the range and\nan Easterner or two, fortunate enough to find the way into his\nconfidence. Occasionally he amuses himself by talking to the one group\nover the heads of the other; but even then, his own simplicity is but\nthinly veiled. The phases of life which he holds lightly are exploited\nwith riotous recklessness; but whoever would visit his private shrines\nmust tread with reverent step.\n\nHis exaggerations are not to deceive, but to magnify--an adjunct to\nexpression invariably found among primitive people. A brass monkey is\nreally not sensitive to variations of temperature; and yet, even among\nthe civilized, a peculiarly vivid impression is conveyed by stating\nthat a particular cold snap has had a disintegrating effect upon the\nintegrity of a brass monkey. There is a philosophy of exaggeration\nwhich is no kin to falsehood.\n\nHappy has an eager, hungry, active mind, a mind worthy of careful\ncultivation; but forced by circumstances to gather its nourishment\nalong lines similar to those adopted by the meek and lowly sponge. A\nsponge is earnest, patient, and industrious; but, fixed to a submerged\nstone as it is, it is hampered by limitations which no amount of\npersonal ambition is quite able to overcome. As Happy himself was fond\nof saying: \"The thing 'at sets most strangers again each other, is the\nfact that each insists on judgin' everything from his own standpoint.\nA cow-puncher gets the idee that because an Eastener can't sit\ncomfortable on a bronco when it's sunfishin' or twistin' ends, he jes\nnachely ain't fit to clutter up the surface o' the earth; while the\nEastener is inclined to estimate the puncher an' his pony as bein' on\nthe same intellectual level. If they'd just open up an' examine each\nother impartial, they'd mighty soon see 'at the difference in 'em came\nfrom what they did, instead o' the choice o' their lines o' business\ndependin' on their natural make-up. I once had a no-account pinto\nwhich refused to squat back on the rope, and I rejoiced exceeding when\nI got seventy-five bucks for him; but the feller I took advantage of\nclipped his mane, docked his tail, introduced him into swell-society,\nand got three hundred for him as a polo pony; which all goes to\nshow--\" (The finish of this is an expansive wave of the hand, a tilt\nof the head to the right, and an indescribably droll expression.)\n\nThe above is a fair sample of the leisurely way in which Happy Hawkins\ntells a story. This is not the proper way to tell a story. A story\nshould travel an air-line and not stop at the smaller stations, while\nHappy prefers to take his bed along on a spare horse and camp out\nwherever the mood strikes him. The reader who delights in a story\nwhich speeds along like a limited, will probably be disappointed in\nthis book; while, on the other hand, the reader who enjoys the\nintimate association which is lighted with the evening camp fire, runs\na risk of finding some relaxation in taking another little trip with\nHappy Hawkins.\n\n                                                              R. A. W.\n\n\n\n\nCONTENTS\n    CHAPTER ONE--THE MEETING\n    CHAPTER TWO--THE BETTIN' BARBER O' BOGGS\n    CHAPTER THREE--ABOVE THE DUST\n    CHAPTER FOUR--TY JONES\n    CHAPTER FIVE--THE HOLD-UP\n    CHAPTER SIX--A REMINISCENCE\n    CHAPTER SEVEN--HORACE WALPOLE BRADFORD\n    CHAPTER EIGHT--A CASE OF NERVES\n    CHAPTER NINE--TREATING THE CASE\n    CHAPTER TEN--INJUNS!\n    CHAPTER ELEVEN--BENEFITS OF FASTING\n    CHAPTER TWELVE--A COMPLETE CURE\n    CHAPTER THIRTEEN--AN UNEXPECTED CACHE\n    CHAPTER FOURTEEN--HAPPY'S NEW AMBITION\n    CHAPTER FIFTEEN--TENDER FEELINGS\n    CHAPTER SIXTEEN--THEMIS IN THE ROCKIES\n    CHAPTER SEVENTEEN--KIT MURRAY\n    CHAPTER EIGHTEEN--TESTING THE FRIAR'S NERVE\n    CHAPTER NINETEEN--OTHER PEOPLE'S BUSINESS\n    CHAPTER TWENTY--QUARRELING FOR PEACE\n    CHAPTER TWENTY-ONE--PEACE TO START A QUARREL\n    CHAPTER TWENTY-TWO--A PROGRESSIVE HUNT\n    CHAPTER TWENTY-THREE--A LITTLE GUN-PLAY\n    CHAPTER TWENTY-FOUR--NIGHT-PROWLERS\n    CHAPTER TWENTY-FIVE--THE TRADE-RAT'S CHRISTMAS-GIFT\n    CHAPTER TWENTY-SIX--A CONTESTED LIFE-TITLE\n    CHAPTER TWENTY-SEVEN--A STRANGE ALLIANCE\n    CHAPTER TWENTY-EIGHT--THE HEART OF HAPPY HAWKINS\n    CHAPTER TWENTY-NINE--THE LITTLE TOWN OF BOSCO\n    CHAPTER THIRTY--TY JONES GETS A WOMAN\n    CHAPTER THIRTY-ONE--JUSTICE UNDELAYED\n    CHAPTER THIRTY-TWO--THE FRIAR GOES ALONE\n    CHAPTER THIRTY-THREE--THE FRIAR GIVEN TWO WEEKS\n    CHAPTER THIRTY-FOUR--A CROSS FOR EVERY MAN\n    CHAPTER THIRTY-FIVE--THE FRIAR A COMPLICATION\n    CHAPTER THIRTY-SIX--A SIDE-TRIP TO SKELTY'S\n    CHAPTER THIRTY-SEVEN--PROMOTHEUS IN THE TOILS\n    CHAPTER THIRTY-EIGHT--OLAF RUNS THE BLOCKADE\n    CHAPTER THIRTY-NINE--SKIRMISHES\n    CHAPTER FORTY--AN IRRITATING GRIN\n    CHAPTER FORTY-ONE--THE NIGHT-ATTACK\n    CHAPTER FORTY-TWO--HAND TO HAND\n    CHAPTER FORTY-THREE--THE GIFT OF THE DAWN\n    CHAPTER FORTY-FOUR--TY JONES NODS HIS HEAD\n    CHAPTER FORTY-FIVE--THE LITTLE GUST O' WIND\n    CHAPTER FORTY-SIX--THE FINAL MOVES\n\n\n\n\nCHAPTER ONE\n\nTHE MEETING\n\n\nIt's a curious thing--life. Ya might just as well ask a kitten to\nchase her own tail or a dog to bay at the evenin' star, or a\nperiodical spring to run constant, as to ask a feller right out to\ntell a story. Some things can only be done spontaneous.\n\nFriar Tuck used to say 'at whenever he could cut it, he allus got on\nthe lee side o' human nature and let it blow down on him natural; and\nmy way o' gettin' to the lee side o' human nature in story-tellin' is\nnot to ask for a story, but to start tellin' one myself. And it's a\ngood plan not to put over too good a one either; 'cause if it seems as\nthough a feller is short run on stories, some listener is likely to\ntake pity on him and fit him out with a new assortment so as he won't\nbe such bad company for himself when he's alone again. This is the way\nI've picked up most o' my stories.\n\nThen again, it's allus hard for me to tell what is the true beginnin'\nof a story. It's easy enough to tell cream from milk--after the milk\nhas stood long enough for the cream to rise to the top; but the great\ntrouble is, that a man's own recollections haven't stood long enough\nfor him to skim out just what part he might be in need of.\n\nWithout meanin' the least mite o' disrespect to any one, it does seem\nto me that if I was able to plan out any sort of a memory at all, I\ncould have made a few improvements on the ones we now have.\n\nMy own memory is as stubborn as a mule and as grippy as a bulldog.\nWhat it does remember, it calls up in the shape o' pictures; and I see\nold things just as plain as livin', breathin' beings; but try as I\nwould, I never could keep my memory from loadin' herself down with so\nmany trifles that sometimes I've had to spade it over as many as six\ntimes to turn up some important item which I was actually in need of.\nWhen my memory's in a good humor, I like to start a pipe and lean back\nand just watch old scenes over again, the same as if I was in a\nthe-ater; and I can see every twinkle in a pair o' well-known eyes,\nwhich have been lookin' up through six feet of earth for this many a\nlong year, and I can hear--actually hear--the half tones ripplin'\nthrough voices which have no more part in my to-day than the perfume\no' last year's flowers; and then, like as not, my memory'll lay her\nears back and refuse to confide what I did with my shavin' soap.\n\nWhen I look back at my own life and compare it with others, it seems\nlike a curious, patch-worky sort of affair, and not much more my own\nthan the lives o' those others with which I compare it. I allus liked\nmy work, and yet it never attracted my attention much. Side-trips and\nsuch-like stand out plain as figures in a hand-painted picture, such\nas I've seen in hotels down at Frisco; but the work part is just a\nblotchy, colorless sort of smudge, the same as the background o' one\no' these pictures.\n\nWhen I first took on with Jabez--every one called him ol' Cast Steel\nJudson at this time--they wanted to know if I could ride. I was\nnothin' but a regular kid then, so I handed in a purty high average as\nto my ridin' ability; though, truth to tell, I wasn't no bronco buster\nthose days. They gave me a genuwine mean one as a starter, and told me\nto ride him clean or step off and walk.\n\nAt that time I didn't even know how to discard a hoss when I couldn't\nstand the poundin' any longer; so when I felt my backbone gettin'\nwedged too far into my skull, I made a grab for the horn. My luck was\non the job that day and I got the quirt, instead. At his next pitch,\nmy hand went up as natural as ever, and I slammed down the quirt as\nhard as I could. It landed on a ticklish spot and before he had time\nto make up his mind, the cayuse had started to run, me whalin' him at\nevery jump and givin' thanks between 'em. I rode him good and out as\nsoon as he started to stampede, and they all thought I was a real\nrider. Well, this gave me a lot o' trouble--tryin' to live up to my\nreputation--but that's a good sort o' trouble for a kid to have.\n\nNow I can feel all the sensations o' this ride as plain as though it\nwas this mornin'; but the's a thousand rides since then which have all\nmelted an' run together. The same with most o' the rest o' my work: I\nallus aimed to do my bit a little quicker and cleaner 'n the rest; but\nas soon as I learned all the tricks of it, it fell into a rut, like\nbreathin' and seein'. Easteners seem to have an idee that our life\nmust be as carefree and joyous as goin' to a different circus every\nday in the year; but it ain't: it's work, just like all other work.\nWe're a good bit like our ridin' ponies: when we're in the thick of it\nwe're too busy to take notice; and when we're through, we're\nhungry--and that's about the whole story.\n\nJabez Judson was a high peak, and once a feller knew him, he never ran\nany risk o' gettin' him mixed up with any one else. He was the settest\nin his ways of any man I ever had much doin's with; but he didn't\nchange about any--if he faced north on a question one day, he faced\nnorth on it always; so a feller could tell just how any action would\nstrike him, and this made livin' with him as accurate as workin' out a\nproblem in multiplication, which I claim to hold qualities o' comfort.\n\nHis daughter, Barbie, was a little tot when I first took on; and she\nwas the apple of ol' Cast Steel's eye; an' his curb bit, and his spurs\nas well. Barbie and I were pals from one end o' the trail to the\nother, and this explains a lot o' my life which otherwise wouldn't\nhave any answer. My ordinary work at the Diamond Dot wasn't\nout-standin' enough to give me any special privileges; but I happened\nto come back one time when the Brophy gang was about to clean things\nout, and Jabez gave me credit for savin' Barbie's life; so 'at he\ndidn't check up my time any and I did purty much as I pleased, only\nquittin' him when I couldn't put up with his set ways any longer. I\naimed to play fair with Jabez, and he with me; but once in a while we\nlocked horns, though not often, takin' everything into account.\n\nIt was shortly after ol' Cast Steel had bought in the D lazy L brand,\nan' we was still pickin' up strays here an' there. Whenever he bought\nup a brand he allus put the Diamond Dot on the stuff as soon as he\ncould, his mark commandin' more respect than some o' the little\nfellers'.\n\nWhen I'd get tired o' loafing about the home place, I'd take one o'\nthe boys an' we'd start out to look for stray hosses. Spider Kelley\nwas with me this time, an' we had meandered here an' there until we\nhad picked up a big enough string to stand as an excuse for our trip,\nand were about minded to start back.\n\nWe had just forded a little crick when we heard a man's voice singin'\noff to the right. The' was a mess o' cottonwoods between us, an' we\nstopped to listen. Now I had never heard that voice before, an' I had\nnever seen the man who was running it; but right then I was ready to\nbelieve anything he had a mind to tell me. It was a deep, rich voice;\nbut mellow an' tender, an' a feller could tell that he was singin'\nsimply because he couldn't help it.\n\nSpider looked at me with his face shinin', an' I could feel a sort o'\npleasant heat in my own face. The' was a lift an' a swing, and a sort\nof rally-around-the-flag to this voice which got right into ya, an'\nmade you want to do something.\n\n    \"'T is thine to save from perils of perdition\n      The souls for whom the Lord His life laid down;\n    Beware, lest, slothful to fulfill thy mission,\n      Thou lose one jewel that should deck His crown.\n        Publish glad tidings; tidings of peace;\n        Tidings of Jesus, redemption and release.\"\n\n\"That feller can sing some,\" sez Spider Kelley; but just then the\nponies turned back on us an' by the time we had started 'em on again,\nthe singer had passed on up the trail, so I didn't make any reply.\n\nI was tryin' to figure out whether it was the words or the tune or the\nvoice, or what it was that had made my whole body vibrate like a\nfiddle string. As I said before, I see things in pictures an' I also\nremember 'em in pictures: a sound generally calls up a picture to me\nan' it ain't allus a picture anyways connected with the sound itself.\nThis song, for instance, had called to my mind a long procession of\nmarchin' men with banners wavin' an' set faces, shinin' with a glad\nsort o' recklessness. There ain't no accountin' for the human mind: I\nhad never seen such a procession in real life, nor even in a picture;\nbut that was what this song out there on the open range suggested to\nme, an' I hurried out o' the cottonwoods eager to measure the singer\nwith my open eyes.\n\nWhen we climbed up out of the woods, we saw him goin' up the pass\nahead of us with our ponies followin' behind as though they was part\nof his outfit. We could just catch glimpses of him; enough to show\nthat he was a big man on a big roan hoss, an' that he was a ridin' man\nin spite o' the fact that he was wearin' black clothes made up Eastern\nstyle. He was still singin' his song, an' I straightened up in my\nsaddle, an' beat time with my hand as though I held a genuwine sword\nin it; which is a tool I've never had much doin's with.\n\nWe scrambled on up the trail, an' when we reached the top we found a\nlittle park with the grass knee high an' a fringe o' spruce trees\nabout it. The song had come to a sudden end, an' we found the singer\non foot with a noose about his neck an' nine rather tough-lookin'\ncitizens holdin' a parley with him. We came to the same sort of a stop\nthe song had, an' Spider Kelley sez in a low tone, \"What do ya suppose\nthis is?\"\n\n\"I don't know,\" sez I, touchin' my pony, \"but I'm with the singer\"; so\nme an' Spider rode on down to 'em.\n\nI purty well sensed what it was: the' was a heap o' rebrandin' bein'\ndone at that time, an' stringin' a man up was supposed to be the only\ncure; but I was willin' to bet my roll that this singer wasn't a\nrustler. The feller in charge o' the posse was an evil-lookin' cuss,\nan' if he'd 'a' had the rope around his neck, it wouldn't have looked\nso misplaced. He was ridin' a Cross brand hoss; so I guessed him to\nbelong to the Tyrrel Jones outfit. Most o' the others in the posse was\nridin' the same brand o' hosses an' wearin' the same brand of\nexpressions. It was a tough-lookin' bunch.\n\nWe came up to 'em an' they looked our ponies an' us over an' nodded.\nWe nodded back an' I asked 'em what seemed to be the trouble.\n\n\"We've finally got the feller who has been doin' the rustlin' out this\nway,\" sez the leader, whose name was Flannigan, Badger-face Flannigan.\n\n\"That's good,\" sez I; \"but he don't look the part.\"\n\n\"He acts it all right,\" growls Badger-face, showin' his fangs in what\nwas meant for a grin. \"He's ridin' one of our hosses, an' leadin' a\nstring o' D lazy Ls.\"\n\n\"Leadin' 'em?\" sez I.\n\n\"Yes, he's got some sort of a charm in his voice. Whiskers, here, saw\nhim go up on foot an' rope this colt an' lead him off the same as a\nplow hoss.\"\n\n\"Did Whiskers, here, see him charm the loose string, too?\" I asked.\n\n\"No, he came in an' collected the posse, an' we decided that this\nwould be a good place to try him; so we cut up the other pass an'\nwaited for him. When he came up, this bunch o' ponies was taggin'\nafter him.\"\n\nI looked at the man with the noose about his neck, an' he was grinnin'\nas easy an' comfortable as I ever saw a man grin in my life. He was\nwearin' a vest without buttons an' a gray flannel shirt. He had a\nrifle on his saddle an' a sixshooter on his right hip. He had big gray\neyes set wide apart under heavy brows, an' they were dancin' with\nlaughter. I grinned into 'em without intendin' to, an' sez: \"Well, I\ndon't really think he charmed these loose ponies intentional. Me an'\nSpider was takin' 'em in to the Diamond Dot an' we had a hard time\nmakin' 'em ford the crick. I'm some thankful to him for tollin' 'em up\nthe pass.\"\n\nBadger-face scowled. \"Well, anyhow, he charmed the beast he's ridin,\nall right; an' he has to swing for it.\"\n\n\"Are you all done with tryin' him,\" sez I.\n\n\"What's the use of a trial?\" snarled Badger-face. \"Ain't he ridin' a\nCross brand hoss, ain't the brand unvented, don't every one know that\nwe never sell a hoss without ventin' the brand, an' can't any one see\n'at this hoss was never rode before?\"\n\n\"Got anything to say for yourself, stranger?\" I asked.\n\n\"Not much,\" sez the prisoner. \"I have an appointment to keep at\nLaramie; my hoss gave out; so I just caught a fresh one an' started\non.\"\n\n\"What more do you want?\" asked Badger-face of me.\n\n\"Well, now, the' ain't any particular hurry; an' I'm kind o' curious\nto learn a little more of his methods,\" sez I impartial. \"Don't ya\nknow 'at this is what they call hoss-stealin' out this way?\" I asked\nof the stranger.\n\n\"No, this is not stealin',\" he replied. \"I turned another hoss loose\nthat I had picked up a hundred miles or so farther back; and I should\nhave turned this one adrift as soon as he had tired. They allus wander\nback to their own range.\"\n\nThis wasn't no unheard-of custom to practice out our way; but it was a\nnew sort o' defence for a man with a noose about his neck to put up,\nan' I see that some o' the others was gettin' interested. The big man\nhad a smile like a boy, an' steady eyes, an' a clear skin; an' he\ndidn't look at all the kind of a man to really need stretchin'.\n\n\"What's your plan for earnin' a livin'?\" I asked.\n\n\"I am a kind of apostle,\" sez he, \"an' I live on the bounty of\nothers.\"\n\n\"Do you mean 'at you're a preacher?\" asked Badger-face.\n\n\"Yes,\" the stranger replied with a smile.\n\n[Illustration: We found the singer on foot with a noose about his neck\nan' nine rather tough-lookin' citizens holdin' a parley with him]\n\n\"Well, I never see a preacher with as short hair as yours, nor one who\ncarried so much artillery, nor one who made a practice o' pickin' up a\nfresh hoss whenever he felt like it. Where'd you learn to ride, an'\nwhere'd you learn to rope?\"\n\n\"Eastern Colorado. I lived there four years, an' travelled on\nhossback,\" sez the stranger.\n\n\"I'll bet you left there mighty sudden,\" sez Badger-face with an evil\nleer.\n\n\"Yes,\" replied the stranger, with a grin, \"an' I also left on\nhossback.\"\n\n\"Well, ya satisfied now?\" grunted Badger-face to me.\n\nLivin' out doors the way I had, I naturally had a big respect for\nbrands. It's mighty comfortin' to feel that ya can turn your stuff\nloose an' know that it's not likely to be bothered; so I was up\nsomething of a stump about this new doctrine. \"Where'd you get your\ncommission from to pick up a hoss whenever you feel like it?\" sez I to\nthe stranger.\n\nHe had a little leather sack hangin' from his saddle horn, an' he\nreached into it an' fished out a small book with a soft leather cover.\nThe feller 'at was holdin' his hoss eyed him mighty close for fear it\nwas some sort of a gun; but the stranger ran over the leaves with his\nfingers as ready as a man would step into the home corral an' rope his\nfavorite ridin' pony.\n\n\"Here's my commission,\" sez he, as self-satisfied as though he was\nholdin' a government document; an' then he read aloud with that deep,\nmellow voice o' his, the story of the time the Lord was minded to let\nhimself out a little an' came into Jerusalem in state. He read it all,\nan' then he paused, looked about, holdin' each man's eyes with his own\nfor a second, an' then he read once more the part where the Lord had\nsent in a couple of his hands after the colt that no man had ever\nbacked before--an' then he closed the book, patted it gentle an'\nshoved it back into the leather bag. I looked around on the posse, an'\nmost of 'em was rubbin' their chins, an' studyin'. I've noticed that\nwhile the earth is purty well cluttered up with pale-blooded an'\npartially ossified Christians, the's mighty few out an' out atheists\namong 'em.\n\n\"That don't go,\" sez Badger-face, after he'd taken time to pump up his\nnerve a little.\n\nNo one said anything for a space, an' then the stranger put a little\nedge on his voice, but spoke in a lower tone than before: \"That does\ngo,\" he said. \"No matter what else in life may be questioned, no\nmatter how hard and fast a title may stick, it must crumble to dust\nwhen one comes and says, 'The Lord hath need of this.' It may be your\nlife or it may be your property or it may be the one being you love\nmost in all the world; but when the Lord hath need, your own needs\nmust fall away.\n\n\"Now, boys, I love the West, I glory in the fact that I can lay\nsomething down and go on about my business an' come back a month later\nand find it just where I left it; and if I was takin' these hosses to\nsell or trade or use for my own selfish ends, why, I wouldn't have a\nword to say again' your stringin' me up. I brought my own hoss into\nthis country and when it gave out I didn't have time to barter an'\ntrade for another one; so I just caught one, and when it grew weary, I\nturned it adrift. I don't claim the hosses I ride; I don't want to own\nthem; I simply borrow them for a while because my Lord hath need of\nthem. I treat them well, and when they weary, send 'em back to their\nown range with a pat, and pick up another. The next fellow who rides\nthat hoss will find it a little less trouble than if I hadn't used it,\nand there's no harm done at all. I'm working with you, I'm going to\nmake your own work easier out here by raisin' the respect for brands,\nnot by makin' property rights any looser; and you are goin' to work\nwith me--whether you want to or not. Now then, how much longer are you\ngoin' to keep this fool noose about my neck?\"\n\nThat posse wasn't easy minded, not by a jugful. This stranger was\nspeakin' as though he had power an' authority an' public opinion all\non his side, and they felt consid'able like the tenderfoot who'd roped\nthe buffalo--they was willin' to quit any time he was.\n\nThe Cross brand boys were purty sullen an' moody; but four o' the\nposse belonged to another outfit, an' they couldn't stand the strain.\nOne of 'em, a grizzled old codger with one lamp missin', lifted the\nnoose from the prisoner's neck, an' sez most respectful: \"Parson, I'm\nan old man. I ain't heard a sermon for forty years, an' I'd be right\nobliged to ya if you'd make us one.\"\n\nBadger-face, he snorted scornful; but the rest of the posse was\nscattered all the way from repentance to sheepishness, an' the\nstranger he stepped to a little rise an' he certainly did speak us a\nsermon. First off, he sang us St. Andrew's hymn--I got to learn a good\nmany of his songs after this, but o' course at that time I was as shy\non hymns as the rest o' the crowd.\n\nI tell you it was wonderful up in that little park, with the lush\ngrass for a carpet, the spruce trees for panelin', the bare peaks\nstickin' out for rafter-beams, the blue sky above for ceiling, and\nthat soft, deep voice fillin' the whole place an' yet stealin' into a\nfeller's heart as easy an' gentle as a woman's whisper. He sort o'\nbeat time as though playin' on an instrument, until before he was\nthrough we were all hummin' in time with him--an' then he preached.\n\nHe told us about the fisher folks an' how they lived out doors under\nthe stars the same as we did; and that this was probably why the Lord\nhad chose 'em first to follow him. He said that city folks got to\nrelyin' on themselves so much 'at they was likely to forget that the\nwhole earth was still held in the hollow of the hand which had created\nit; but that men who lived with nature, out under the sun and the\nstars, through the heat and the cold, the wind and the rain, the\nchinook and the blizzard, felt the forces and the mysteries all about\nthem and this kept 'em in touch, even when they didn't know it\nthemselves, with the great central Intelligence back o' these forces\nand mysteries. Then he told 'em how grand their lives might be if they\nwould only give up their nasty little habits of thought, and learn to\nthink broad and free and deep, the same as they breathed.\n\nHe told 'em 'at their minds could breathe the inspiration of God as\neasy as their lungs could breathe the pure air o' the mountains, if\nthey'd only form the habit. Then he talked to 'em friendly an'\nconfidential about their natural devilment. He didn't talk like a\nsaint speakin' out through a crack in the gates o' Paradise, like most\npreachers do. He called the turn on the actual way they cut up when\nthey went to town, and just how it hurt 'em body an' soul; and his\nface grew set and earnest, and his eyes blazed; and then he said a few\nwords about mothers an' children and such, and wound up with a short\nprayer.\n\nWell two o' those fellers owned up right out in public and said that\nfrom that on they was goin' to lead a decent sort of life; and one\nother said 'at he didn't have any faith in himself any longer; but he\ninsisted on signin' the pledge, and said if that worked, why, he'd go\non an' try the rest of it.\n\nThe preacher shook hands with 'em all around--he had a grip 'at\nwouldn't be no disgrace for a silver-tip--an' then he sez that if any\nof 'em has the notion that bein' a Christian makes a weakling of a\nman, why, he's willin' to wrastle or box or run a race or shoot at a\nmark or do any other sort of a stunt to show 'at he's in good order;\nbut they size him up and take his word for it.\n\n\"Now, boys,\" sez he, \"I hope we'll meet often. I'm your friend, and I\nwant you to use me any time you get a chance. Any time or any place\nthat I can serve one of you, just get me word and I'll do the best I\ncan. It don't matter what sort o' trouble you get into, get me word\nand I'll help--if I can find a way. And I wish 'at you'd speak it\naround that I'm hard on hosses, so that the other fellows will\nunderstand when I pick one up, and not cause any delay. I'll have to\nhurry along now. Good-bye; I'm sorry I've been a bother to ya.\"\n\nHe swung up on the big roan, waved his hand and trotted out o' the\npark; and just as he went down the pass on the other side, it seemed\nthat he couldn't hold it in any longer; so he opened up his voice in\nhis marchin' song again, an' we all stayed silent as long as we could\nhear the sound of it.\n\n\"Well we are a lot of soft marks!\" sez Badger-face at last.\n\n\"That there is a true man,\" replied old Grizzly, shakin' his head,\n\"an' I'll bet my boots on it.\"\n\nThis seemed to be the general verdict, an' the Cross brand fellers\nwent off discussin' the parson, an' me an' Spider Kelley collected our\nponies an' went along to the ranch, also discussin' him.\n\nThat was the first time I ever saw Friar Tuck; I made up my mind about\nhim just from hearin' his voice, an' before I ever saw him; but I\nnever had to make it up any different. New lead an' new steel look\nconsid'able alike; but the more ya wear on lead, the sooner it wears\nout, while the more you wear on steel, the brighter it gets. The Friar\nwas steel, an' mighty well tempered.\n\n\n\n\nCHAPTER TWO\n\nTHE BETTIN' BARBER O' BOGGS\n\n\nYes, this was about the time I got interested in the bettin' barber\nover at Boggs. He hasn't anything to do with this story I'm about to\ntell ya, except that it was him 'at give the Friar his name; so I'll\njust skim through this part as hasty as possible. When a feller is\ntellin' me a story, I want him to stick to the trail of it; but it\nseems like when I try to tell one, myself, some feller is allus askin'\nme a question 'at takes me clear out o' range.\n\nAll barbers are more or less different, except in what might be called\nthe gift o' gab. This one came out to Boggs station, an' started a\nshop. His name was Eugene, an' he was a little man with two rollin'\ncurls to his front hair, which he wore short behind. A curious thing\nabout little men is, that they don't never find it out. A little man\nproduces more opinions 'n airy other kind, an' being small, they\nhaven't no place to store 'em up until they get time to ripen. A\nlittle man gives out his opinion an' then looks savage--just as if\nhe'd get a switch an' make ya believe it, whether you wanted to or\nnot.\n\nEugene had come from every city the' is in the world, an' he used to\ntell scandalous tales about the prominent people who lived in 'em\nwhose hair he had cut. He was also familiar with the other things\nwhich had happened since they've begun to write history, an' if any\none would doubt one of his statements, he'd whirl about holding up his\nrazor, an' say: \"I'll bet ya a dollar I can prove it.\"\n\nAll of us fellers used to go in as often as we got a chance to get our\nchins shaved an' our hair shampooed--just to hear Eugene get indignant\nabout things which wasn't none of our business. We used to bet with\nhim a lot, just for the fun o' makin' him prove up things; which he\ndid by writin' letters to somebody an' gettin' back the answers he\nwanted. We didn't have any way to prove our side; so Eugene got the\nmoney an' we had the fun.\n\nOl' man Dort ran the general store and kept a pet squirrel in a\nwhirlabout cage, which was the biggest squirrel I ever see, an' had\nits tail gnawed off by a rat, or something, before Eugene came. Ol'\nman Dort had a reputation for arguin', which spread all over our part\nof the earth. We had made a habit o' goin' to him to get our\ndiscussions settled an' when we began to pass him up for Eugene, he\nfoamed about it free an' frank.\n\nHe wore a prodigious tangle o' hair and a bunch o' grizzled whiskers,\nabout as fine an' smooth as a clump o' grease-wood. He used to brag\nthat razor nor scissors hadn't touched his hide for twenty years, an'\none of us boys would allus add, \"Nor soap nor water, neither,\" an' ol'\nman Dort would grin proud, 'cause it was a point of honor with him.\n\nEugene used to send out for his wearin' an' sech, so ol' man Dort\ndidn't get a whack at him in his store; ol' man Dort batched, an'\nEugene boarded, so they didn't clash up at their meals; an' finally\nol' man Dort swore a big oath that he was goin' to be barbered. The\nnews got out an' the boys came in for forty miles to see the fun--an'\nit was worth it.\n\nWe went early to the shop an' planted ourselves, lookin' solemn an'\nnot sayin' anything to put Eugene on his guard. When at last ol' man\nDort hove in sight with his brows scowled down an' his jaws set under\nhis shrubbery, we all bit our lips; an' Eugene stopped tellin' us\nabout the hair-roots o' the Prince of Wales, an' stood lookin' at ol'\nman Dort with his mouth gapped wide open.\n\nThe ol' man came in, shut the door careful behind him, glared at\nEugene, as though darin' him to do his worst, an' said: \"I want my\nhair shamped, an' my whiskers shaved off.\"\n\n\"If you expected to get it all done in one day, you should ought to\nhave come earlier,\" sez Eugene soberly, but tossin' us a side wink.\n\n\"Well, you do as much as you can to-day, an' we'll finish up\nto-morrow,\" sez ol' man Dort, not seein' the joke.\n\nOl' man Dort peeled off his coat, rolled up his sleeves, an' climbed\ninto the chair as if he thought it was liable to buck him off. Then he\nsettled back with a grunt, an' Eugene tucked the bib in around his\nneck, combed his fingers through ol' man Dort's hair a minute, an'\nsez; \"Your hair's startin' to come out. You should ought to use a\ntonic.\"\n\n\"Tonic, hell!\" snaps the ol' man. \"My hair sheds out twice a year,\nsame as the rest o' the animals.\"\n\n\"Then you should ought to comb it,\" sez Eugene. \"I've got some hair\nhere in my hand which was shed out two years ago. Leavin' dead hair\nan' such rubbish as that layin' around on your scalp is what kills the\nhair globules.\"\n\n\"It don't either; it acts like fertilizer, the same as dead grass\ndoes,\" sez ol' man Dort. He had made up his mind to take the contrary\nside of everything 'at Eugene said, an' it was more fun than a dog\nfight.\n\nEugene started in by mowin' away the whiskers, an' it was a long an'\npainful job; 'cause it was almost impossible to tell where they left\noff an' ol' man Dort began, an' then they was so cluttered up with\ngrit an' dead hair and kindry deb-ris that his scissors would choke up\nan' pull, an' then ol' man Dort would bob up his head an' yell out a\nbunch o' profanity, and Eugene would stand back an' say that he was a\nbarber, not a clearer of new ground, an' that the job ought to be done\nwith a scythe and hoe, not with scissors an' razor. Eugene wasn't\ncovetous of ol' man Dort's trade an' didn't care whether he insulted\nhim or not.\n\nThe most fun came, though, after Eugene had got down to where he could\ntell the outline of ol' man Dort's face. First he soaked it with\nlather, combin' it in with a comb, an' puttin' hot towels on it to\ndraw out the alkalie grit an' give his razors some show.\n\nOne of ol' man Dort's manias was, that a man ought to pay his debts,\nwhether it killed him or not; so as soon as Eugene had him steamin'\nunder the towels we begun to talk about a man's first duty bein'\ntoward his kin, an' that if he couldn't pay his debts without bother,\nhe ought to let the debts go an' show his relatives a good time while\nthey was still on earth an' able to enjoy themselves.\n\nOl' man Dort couldn't stand it, an' tried to answer back from under\nthe towels; but got his mouth full o' suds, an' choked on the corner\nof a towel until Eugene said that if he couldn't sit still an' behave\nhimself he could go out to some alfalfa farmer to get his tonsoral\nwork completed.\n\nIt wasn't the ol' man's fault--he simply couldn't help it. Touch him\nup on a ticklish subject, an' he just had to come back at ya, same as\na rattler. Finally, however, Eugene had the stubble wore down an'\nsoftened until he decided that he stood a chance again' it, an' then\nhe lathered an' rubbed, an' lathered an' rubbed, until nothin' stuck\nout below ol' man Dort's eyes except the peak of his nose; an' then us\nboys pulled out our trump card an' played it strong. We began to talk\nabout red squirrels.\n\nNow, we didn't know anything professional about squirrels, except what\nol' man Dort had told us; but we slewed his talk around this way an'\nthat as if it was our own private opinions; an' the ol' man began to\ngroan audible. He gritted his teeth, though, an' bore up under it like\na hero, until Eugene begin to chip in with what he knew about\nsquirrels.\n\nEugene was never content to just speak of a thing in a general\nway--his main method of convincin' us was to allus fall back on his\nown personal experience; so this time he began to tell of squirrels\nwhat he had been full acquainted with. He called 'em by name an' told\nhow they would run to meet him an' climb up on his shoulders an'\nchatter for nuts, an' so on; until the ol' man's ears turned red with\nthe strain he was under. And then, we got to discussin' the size o'\nsquirrels.\n\nWe told about squirrels we had heard about, an' contested again' each\nother to see which had heard o' the biggest one; but we never even\nmentioned ol' man Dort's squirrel. Eugene had shaved his way down to\nbelow the lobe of ol' man Dort's right ear, slippin' in a side remark\nto our talk every minute or so; an' purty soon he sez 'at he knows a\nsquirrel by the name o' Daniel Webster back in Montpelier, Vermont,\nwhich was a full half inch longer 'n airy red squirrel we had spoke\nof. The ol' man couldn't stand this. His head bobbed up, cuttin' a\ngash on the crook of his jaw, and as soon as he could blow the foam\nout of his mouth, he sez, \"I'll stake my life, the' ain't another\nsquirrel in this country as big as my own Ben Butler.\"\n\nEugene put his hand on ol' man Dort's forehead an' pushed him back\ninto the headrest. \"You lie there,\" sez he, \"until I get done shavin'\nya. Then, I'll bet ya a dollar that I can produce a livin' squirrel\nwhich'll out-stand, outweigh, an' out-fight your squirrel--an' I ain't\nnever seen your squirrel.\"\n\n\"A dollar!\" snorts the ol' man, flickin' up his head. \"I wouldn't\nbother wakin' Ben Butler up for a measly dollar. I'll bet ya ten\ndollars.\"\n\n\"Get back on that headrest,\" orders Eugene. \"Ten dollars looks a heap\nsight better to me than one, an' I'll be mighty glad to accommodate\nya.\"\n\nEugene took his fire-stick an' burned the ol' man's cut, an' the ol'\nman had to scruge up his shoulders with the pain of it; but he did it\nwithout noticin', 'cause his mind was on squirrels. \"What breed o'\nsquirrels is yours?\" he asked.\n\n\"If you don't keep your head where I put it, I'll throw up the job an'\nlet you go forth lookin' like the lost Goog o' Mayhan,\" sez Eugene,\nraisin' his voice. Ol' man Dort was a whalin' big man, an' it tickled\nus a heap to see little Eugene givin' him directions, like as if he\nwas nothin' but a pup dog.\n\nOl' man Dort settled back with a sigh, an' Eugene leathered up his\nrazor without sayin' anything for a minute or two. Then he sez, as he\nbegins shavin' again: \"That squirrel I have in mind for ring contests\nis the short-tailed grizzly ground-squirrel; and it's the biggest\nbreed of squirrels the' is.\"\n\n\"The' ain't no such a breed of squirrel as that!\" yells ol' man Dort,\nspringing erect in his chair, an' dullin' Eugene's razor by the\noperation.\n\nEugene stepped back an' looked at the blood flowin' from the fresh\ncut, an' he sez slow an' sarcastic; \"If it don't make any difference\nto you whether you have any skin on your face or not, why I'll just\npeel it off an' tack it on a board to shave it; but hanged if I'm\ngoin' to duck around tryin' to shave you on the jump. The' is too\ngrizzly ground-squirrels.\"\n\nWell, that's the way they had it back and forth: every time they would\nsettle down to business an' Eugene would get a square inch o' the ol'\nman's face cleared up, one of us boys would speak something in a low\ntone about there bein' rumors of an uncommon big squirrel out at some\nranch house a hundred miles or so from there. Eugene would ask what\nbreed of squirrel it was, an' then decide that it couldn't be a\npatchin' on a genuwine short-tailed grizzly ground-squirrel, an' then\nol' man Dort couldn't stand it no longer an' he would forget what he\nwas doin', bob up in his chair, an' lose some more of his life fluid.\n\nEugene scraped down both sides o' the ol' man's face, givin' all of\nhis razors a chance to take part in the job, an' then he set his lips\nan' started in on the chin.\n\n\"What does short-tailed grizzly ground-squirrels eat, Eugene?\" asked\nSpider Kelley, as innocent as an infant pigeon.\n\n\"They eat chickens,--\" began Eugene, but ol' man Dort flew clean out\no' the chair an' stood over Eugene shakin' with rage.\n\n\"Chickens?\" he roars. \"Chickens! The' never was a squirrel foaled into\nthis world what et chickens.\"\n\nEugene looked at ol' man Dort, an' then he wiped his razor an' sat\ndown on a chair, so full of disgust that he could hardly breathe.\n\n\"I wish you'd take off that apron an' bleed into the spittoon,\" he\nsaid as calm as he could. \"I've got customers whose patronage is what\nmakes up my living expenses; an' I don't want 'em to come in here an'\nsee the whole place a welter of gore.\n\n\"What do you think this shop is, anyway?\" yelled Eugene springing to\nhis feet an' entirely losin' his patience. \"Do you think that I make\nmy livin' by grubbin' down wire grass which has been let grow for\nfifty years, an' educatin' ignoramuses in the knowledge of squirrels?\nI don't care whether you believe in short-tailed grizzly\nground-squirrels or not; but if you don't let me tie your head down to\nthat chair, I won't shave another sprout off your chin. I take some\npride in my profession, an' I don't intend to have no man go out o' my\nshop leavin' a trail o' blood which will draw all the dogs for miles\naround. Now, you can take your choice.\"\n\nOl' man Dort had to give in that this was reasonable enough; so he\nclimbed back into the chair, an' Eugene tied down his head an'\nfinished him off without any more trouble. As soon as he had stopped\nthe bleedin' an' put on the perfume an' oil an' powder, he sez: \"Now,\nwhat I am goin' to do is to get some nourishment to recuperate back my\nstrength, an' if you want the waste products washed out o' your hair,\nyou come back here at one o'clock prompt.\"\n\n\"I want to settle on that bet first,\" said ol' man Dort, who was just\nas pernicious as Eugene, once you got him riled up.\n\n\"I'll make that bet with you after dinner,\" sez Eugene, \"but first off\nI got to have food; I'm faint with weakness. Now, I'm goin' to lock up\nmy shop.\"\n\nAfter Eugene had marched off to his boardin' house, we all gathered\naround ol' man Dort, an' complimented him on his improved appearance,\nthough to be strictly honest, the' was considerable doubts about it.\nHe had two teeth out in front, an' the tobacco habit; and now, with no\nshrubbery to catch the spray, he spluttered terrible when he tried to\ntalk fast. He said, though, that as long as he had started in he\nintended to take the full course, an' was comin' back, as soon as he'd\nhad a bite to eat, to get his hair laundried an' trimmed up some\naround the edges; an' then he was goin' to make that bet about the\nsquirrels.\n\nIt was some amusin' to see the ol' man get his hair sluiced out, but\nnot near as much fun as seein' him shaved. Whenever Eugene found any\nstray product, he'd call us all over an' show it to us, an' this riled\nthe ol' man up considerable; but the best joke was when Eugene found a\nwoman's hairpin.\n\nThe ol' man vowed an' declared an' carried on somethin' fierce; but\nthere was the hairpin, an' we made him pay for three rounds on the\nstrength of it. As soon as Eugene was all through, the ol' man settled\nthe bill, payin' for a full day's work like a regular sport, an' not\ntryin' to beg off at the ordinary retail price; and then he hardened\nhis face an' sez: \"Now I bet you ten dollars, that you can't bring\nforward a squirrel as big as my Ben Butler.\"\n\n\"I'll take that bet,\" sez Eugene, \"but you got to give me time to\nlocate a short-tailed grizzly. It's the scarcest breed the' is, an'\nit'll probably cost me twice the sum to get one, but I don't care\nabout that. What I want is to vindicate myself. I'd like to see that\nsquirrel o' yours.\"\n\n\"You come right along,\" sez ol' man Dort, glowin' with pride. \"I\nreckon when you see him, you'll just hand over the money at once--That\nis, if you know anything at all about squirrels.\"\n\nWe all marched around to the general store, an' ol' man Dort pounded\non the cage. When Ben Butler sat up an' looked around to see what was\nup, the ol' man waved his hand at him, looked down at Eugene, an' sez:\n\"Well?\" He said it just like that: \"Wu-el?\"\n\nBen Butler was rollin' fat, an' he certainly did look like some\nsquirrel to us; but Eugene merely glanced at him, an' sez: \"Hum, what\nwe call a dwarf red squirrel, up in Nova Scotia. They have tails,\nthough, up there.\"\n\nThe ol' man spluttered till we had to pound him on the back. \"Dwarf?\"\nhe chokes out. \"Dwarf! You produce a squirrel to match him, will ya,\nor else you pack up your truck an' move on. I don't intend to have\nno--\"\n\n\"See here, ol' man,\" sez Eugene, pointin' a finger at him the same as\nif he'd been a naughty child. \"A short-tailed grizzly ground-squirrel\nis from two to four times as big as this one, so if you want to\nsidestep the bet, you can do it; but if you want to have some show for\nyour money, I bet you fifty to ten that I can get a squirrel three\ntimes as big as this one. I own up that for its kind, this squirrel is\nof fair, average growth; but--\"\n\n\"I'll take that bet!\" yelled the old man. \"We'll put up our money with\nIke Spargle this minute; but I don't want your odds. I'll bet you even\nmoney.\"\n\nEugene shook his head as if he pitied the ol' man, an' he sez,\n\"Haven't you never travelled none, or seen a zoological garden?\"\n\n\"Yes, I've travelled some, an' I've seen all kinds o' gardens,\" flares\nback the ol' man; \"but what I want now is to fix up this bet.\"\n\n\"Who'll be the judges?\" sez Eugene.\n\n\"I don't care a snap. Any man who can see through the holes in a\nladder'll be able to decide between the claims o' two squirrels. Ike\nSpargle an' Bill Thompson can be the judges.\"\n\n\"There has to be three,\" sez Eugene. \"We'll have Dan Stedman be the\nother.\"\n\nSo they put up the money an' Eugene was to have six weeks to get his\nsquirrel; an' from that on we begun to divide up into rival camps.\nThe' wasn't any tree squirrels out in that neck o' the woods, an' we\nhad all forgot what wild squirrels really was like. We knew the' was\nground-squirrels, red squirrels, gray squirrels, an'\nflyin'-squirrels--although an argument was started about there bein'\nflyin'-fish all right, but no flyin'-squirrels, which would have ended\nin warfare if Eugene hadn't been handy to settle it.\n\nYou wouldn't think that a little thing like a bet about the size of a\nsquirrel would take the way it did; but Eugene was so confident on his\nside, an' ol' man Dort was so dead sure of Ben Butler, that the rest\nof us split up an' we each had a little side bet on the outcome. It\nseemed a tarnation long time while we was waitin'; but in a little\nover a month, Eugene got a big box which he took into his back room\nwithout lettin' even the fellers who had backed his squirrel get a\npeep at it.\n\nFrom that on we got shaved twice a day an' our heads washed till the\nhair started to change color; so that Eugene's trade was so improved\nthat even if he lost the bet, he was money ahead; but he scoffed the\nidy o' losin' the bet, even after his squirrel arrived; and as he was\nthe only man who had seen both the contestants, he had the whole\ncountry up in the air.\n\nOl' man Dort had made his squirrel run around the wheel four hours a\nday, pokin' him up with a stick when he got lazy; an' this gave Ben\nButler sech a prodigious appetite that the ol' man had to set up late\nat night to give him an extra meal. As the day o' settlement came\ncloser, the ol' man tapered off on the exercise, an' doubled up on the\nfeed, until Ben Butler looked a full size larger, an' us fellers who\nhad our money on Eugene's squirrel began to get shaky. If it had been\njust an even race, it would have been a fair deal; but to have to show\na squirrel three times larger than Ben Butler seemed an impossibility.\n\nEugene had been fussin' over his entry too, an' we used to sneak up\nbehind his shop at nights to listen to him. We could hear him snippin'\nwith scissors and pullin' stoppers out o' bottles and when he was\nthrough he'd say: \"Stand up there, Columbus\"--which was the name of\nhis champion, an' then he would seem to pass in a bunch o' feed, an'\nsay--\"Good boy, Columbus! that dwarf red squirrel can turn a double\nhandspring in your shadder.\"\n\nThis used to hearten us up again, and we'd lay a little more money on\nEugene's squirrel. Ike, an' Bill, an' Dan--the judges--said that they\ndidn't claim to know anything about the breeds o' squirrels, an' all\nthey was to judge on was the size, which would be settled by weight if\nthe' was any dispute. They got kind o' nervous toward the end, 'cause\nthe fellers were all on edge, an' a rank decision meant trouble in\nbunches.\n\nWhen the final day o' settlement arrived, Boggs was seven deep with\nfellers on edge to see the outcome. Most of us had all we could spare\nhung up in bets; but the' was still a lot o' coin in the crowd, and a\ncrew came over from Cheyenne to take charge of it.\n\nThey had a game which certainly was attractive, I'll say that much for\nit. It was a round board full o' numbers, and up the middle was a\ntower with slopin' sides covered with nails. A marble was dropped into\na hole at the top and bobbled on the nails until it went into a row of\nholes at the bottom, and came out in a groove leadin' to one o' the\nnumbers. Some o' these numbers doubled the player's money, some of 'em\npaid it over to the table; but most of 'em was neutral, and a feller\nhad to double what he already had up, in order to stand a show. It was\nan innocent-appearin' game, but deceptive. When a feller had up all he\ncould raise, some stranger would offer him two bits for his chance,\nput up the doublin' money--and win. This was a capper o' course; but\ncrowds don't have any sense when they start gamblin', and this crew\nwas cleanin' us out until, all of a sudden, I heard a clear, low-toned\nvoice say: \"If one o' you boys would upset that table, you'd see the\nlever which controls the marble.\"\n\nI glanced up, and there was the Singin' Parson, as cool as a frozen\nfish. Ol' Tom Williams, commonly known as \"Tank,\" had just lost six\ndollars, and he upset the table and saw just how tight braced the\nblame game was. Then he unlimbered his gun, and suggested that he\nwould feel calmer if he had the six dollars back, and the Cheyenne\ngambler looked into Tank's free eye, which was pointin' at the\nceilin', and he seconded Tank's motion. After this the rest o' the\nboys collected what they felt was due 'em, and the Cheyenne crowd had\nto fall back on charity for their noon lunch.\n\nJust about one o'clock, the head crook saw the Singin' Parson standin'\nclose to Eugene's barber shop. The shop was locked, and the crowd\naround was lookin' at it. The crook didn't want to attract any\nattention; so, instead o' usin' a gun, he struck at the Parson with a\nclub. He miscalculated, and hit the shoulder instead o' the head. The\nParson whirled, grabbed the club with his left hand, and the crook's\nshirt collar with his right. The crook started to pull; but we settled\ndown on him, and were all ready to serve out justice, when the Parson\ninterrupted to say that it was none of our business, and if we'd just\nform a ring, he'd settle it to everybody's satisfaction. He said he\nexpected to live among us for the rest of his life, and this would be\na good time to introduce his methods.\n\nWe took off the crook's weapons, and then formed a big ring. The\nParson was smilin' a business-like smile, while the crook was palin'\nup noticeable. \"I am convinced that a man must settle some things,\nhimself, in a new country,\" sez the Parson. \"I am larger than you, so\nit is fair for you to use this club; but I warn you in advance that I\nunderstand how to guard again' clubs, so do your best. I'm ready,\nbegin.\"\n\nIt was quite eddifyin' to behold: the crook made a vicious smash at\nthe Parson's head, the Parson bent his arm at the elbow, muscle out,\nso the bone wouldn't get bruised, stepped in, and hit the crook a\nswing in the short ribs. Some say it lifted him ten feet, some say\nonly eight; but any way, when he lit, he gave a grunt like an empty\nbarrel, and the Parson had no trouble in layin' him over his knee and\ngivin' him the most liberal spankin' with that club I ever was\nspectator to; while the crowd howled itself hoarse in the throat.\n\nNow the Parson wasn't angry, he grinned all the way through, and when\nhe had taken as much exercise as he felt was good for him, he set the\ncrook on his feet, and talked fatherly advice to him as sober an'\ndignified as was possible--considerin' the fact that the crook was\ndancin' about like a spider on a hot skillet, and rubbin' the part\nwhich had got most intimate with the club.\n\nEugene had seen it all through his window, and when it was over, he\ncame out and shook the Parson's hand, and said he was just the kind\nneeded in such an ungodly community, and that he reminded him for all\nthe world of Friar Tuck in Robin Hood. Now, we hadn't none of us heard\nof Friar Tuck up to that time; but it was a name well fitted to the\ntongue, and from the way Eugene said it, we elected it was a\ncompliment; so we gave it to the Singin' Parson on the spot, and it\nsoaked into his bones, and he hasn't needed any other since.\n\nThis little incident kept us all in a good humor until three o'clock,\nwhich was the fatal hour for the squirrel-contest.\n\nThen ol' man Dort marched to the center o' the street, carryin' his\ncage as though it was full o' diamonds; an' Ben Butler sat up an'\nchattered as if he was darin' the whole race o' squirrels to bring\nforth his equal.\n\n\"I don't reckon a squirrel could get three times as big as him without\nexplodin',\" sez Spider Kelley, who also had his money on Eugene's\nsquirrel.\n\n\"Here comes Eugene with Columbus,\" sez I, not carin' to waste breath\non an opinion I had backed up with good money.\n\nEugene came down the street carryin' one end of a box, with Doc Forbes\ncarryin' the other. The box was covered with a clean apron, an' Eugene\nwasn't lookin' down in the mouth or discouraged.\n\n\"From the size o' that box, we're goin' to have a run for our money,\"\nsez Spider. \"If Columbus just looks good enough to make 'em settle by\nthe scales, I haven't any kick comin'.\"\n\nWell, as Eugene drew closer, that crowd fell into a silence until all\na body could hear was Ben Butler braggin' about all the nuts he had\net, an' what a prodigious big squirrel he was; but Eugene never\nfaltered. He walked up an' set his box down careful, motioned Doc over\nto the side lines, made a graceful motion to ol' man Dort, an' sez:\n\"As yours is the local champion you introduce him first, an' make your\nclaim.\"\n\nOl' man Dort removed his tobacco, wiped his forehead, an' sez: \"Feller\ncitizens, I make the claim that Ben Butler is the biggest full-blooded\nsquirrel ever sent to enlighten the solitude of lonely humanity. This\nis him.\"\n\nThe ol' man looked lovin'ly down at his squirrel, an' we every one of\nus gave a rousin' cheer. It was all the family the ol' man had, an' it\nmeant more to him 'n a body who hadn't never tried standin' his own\ncompany months at a time could realize. Ol' man Dort thrust some new\ntobacco into his face, bit his lips, winked his eyes rapid, an' bowed\nto us, almost overcome.\n\nThen Eugene stepped a space to the front, bowed to the crowd in\nseveral directions, an' sez: \"Gentlemen, an' feller citizens--From\nIceland's icy mountains to India's coral strands an' Afric's sunny\nfountains, every nation an' every clime has produced some peculiar\nproduct o' nature which lifts it above an' sets it apart from all the\nother localities of the globe. When you speak of the succulent banana,\nthe golden orange, or the prickly pineapple, Nova Scotia remains\nsilent; but when you speak of varmints, she rears up on her hind legs\nand with a glad shout of triumph, she hands forth the short-tailed\ngrizzly ground-squirrel, an' sez, 'Give me the blue ribbons, the gold\nmedals, an' the laurel crowns of victory.' I have the rare pleasure\nan' the distinctive honor of presenting to your notice Columbus, the\nhugest squirrel ever exhibited within the confines of captivity.\"\n\nWe was so took by Eugene's eloquence that we hardly noticed him slip\nthe apron from in front of his cage; but when we did look, we could\nhardly get our breath. I was standin' close to the Friar; and at first\nhe looked puzzled, and then his face lit up with a regular boy's grin;\nbut he didn't say a word.\n\nColumbus was certainly a giant; he stood full two feet tall as he sat\nup an' scrutinized around with a bossy sort of grin. He was dappled\nfawn color on the sides with a curly black streak down the back an'\nsort o' chestnut-red below, with a short tail an' teeth like chisels.\nHe won so blame easy that even us what had bet on him didn't cheer.\n\nOl' man Dort give a grin, thinkin' Ben Butler must have won, an' then\nhe stepped around an' looked into Eugene's cage. He looked first at\nColumbus, an' then at Ben Butler, then he looked again. \"That damned\nthing ain't alive,\" he sez. \"It's made up out o' wool yarn. Poke it up\nan' let me see it move.\"\n\n\"Poke it yourself,\" sez Eugene. He was one o' these cold-blooded\ngamblers who ain't got one speck o' decent sentimentality; an' he was\nmad 'cause we hadn't cheered.\n\nOl' man Dort took a stick an' poked Columbus, an' Columbus give a\nthreatenin' grin, chattered savage, an' bit the stick in two. \"Give\nhim the money, Ike,\" sez ol' man Dort. \"I own up I never was in Nova\nScotia, an' I never supposed that such squirrels as this grew on the\nface o' the whole earth. What'll you take for him?\" he sez to Eugene.\n\n\"It ain't your fault that you didn't know about him,\" sez Eugene,\nthawin' a little humanity into himself. \"I don't want to rub it in on\nnobody; and I'll give you this here squirrel free gratis, 'cause I\nadmit that you know more about squirrels 'n anybody else what ever I\nmet; an' you have the biggest red squirrel the' is in the world.\"\n\nThen we did give Eugene a cheer, an' everything loosened up, an' we\nall crowded into Ike Spargle's so that them what won could spend a\nlittle money on them what lost.\n\nAfter a time, ol' man Dort got up on a chair, an' sez: \"I want you\nfellers to know that Columbus won't never be my pet. Ben Butler has\nbeen the squarest squirrel ever was, an' he continues to remain my\npet; but I'll study feedin' this condemned foreign squirrel, an' give\nhim a fair show; so that if any outsiders come around makin' brags, we\nwill have a home squirrel to enter again' 'em an' get their money.\"\n\nEugene led the cheerin' this time, which made Eugene solider than ever\nwith the boys, an' when Spider an' me got ready to ride home, he an'\nol' man Dort had their arms around each other tryin' to sing the Star\nSpangled Banner.\n\nSpider talked about Columbus most o' the way home, but I was still.\nThe' was somethin' peculiar about the Friar's grin when he first\nsighted Columbus, and the' was somethin' familiar about that squirrel,\nan' I was tryin' to adjust myself. Just as we swung to the west on the\nlast turn, I sez to Spider: \"Spider, I don't know what I ought to do\nabout this?\"\n\n\"About what?\" sez Spider.\n\n\"About this bet?\"\n\n\"Well, it was a fair bet, wasn't it? Columbus is full four times as\nbig as Ben Butler.\"\n\n\"Yes,\" sez I, \"but he ain't no squirrel.\"\n\nSpider pulled up to a stop. \"Ain't no squirrel?\" he sez. \"What do you\ntake me for, didn't I see him myself? What is he then?\"\n\n\"He's a woodchuck, that's what he is,\" sez I. \"He's a genuwine ground\nhog with his hair cut stylish and died accordin' to Eugene's idy of\nhigh art. I remember now that I used to see 'em when I was a little\nshaver back on my dad's farm in Indiana.\"\n\nSpider give a whoop, an' then he laughed, an' then he sobered up, an'\nsez: \"Well, you can't do nothin' now, anyway. The judges have decided\nit, ol' man Dort has give it up, it ain't your game nohow, an' if you\nwas to try to equal back those bets after they have been paid an'\nmostly spent, you'd start a heap o' blood-spillin'; an' furthermore,\nas far as I'm concerned, I ain't right sure but what a woodchuck, as\nyou call it, ain't some kind of a squirrel. We'll just let this go an'\nwait for a chance to put something over on Eugene.\"\n\nSo that's what we made up to do; but this gives you an idy of how fine\na line the Friar drew on questions o' sport. He knew 'at we weren't\nfull fledged angels, and that we had to have our little diversities;\nbut when any professional hold-up men tried to ring in a brace game on\nus, he couldn't see any joke in it, and he upset the money-changers'\ntables, the same as they was upset that time, long ago, in the temple.\n\n\n\n\nCHAPTER THREE\n\nABOVE THE DUST\n\n\nI'm only about twice as old as I feel; but I've certainly seen a lot\no' changes take place out this way. I can look back to the time when\nwhat most of us called a town was nothin' but a log shack with a\nbarrel of cheap whiskey and a mail-bag wanderin' in once a month or\nso, from goodness-knows-where. I've seen the cattle kings when they\nset their own bounds, made their own laws, and cared as little for\ngovernment-title as they did for an Injun's. Then, I've seen the sheep\nmen creep in an inch at a time until they ate the range away from the\ncattle and began to jump claims an' tyrannize as free and joyous as\nthe cattle men had. Next came the dry farmer, and he was as comical as\na bum lamb when he first hove into sight; but I reckon that sooner or\nlater he'll be the one to write the final laws for this section.\n\nWe're gettin' a good many towns on our map nowadays, we're puttin' up\na lot o' hay, we're drinkin' cow milk, and we're eatin' garden truck\nin the summer. The old West has dried up and blown away before our\nvery eyes, and a few of us old timers are beginnin' to feel like the\nlast o' the buffalo. The's more money nowadays in boardin' dudes 'n\nthe' is in herdin' cattle, an' that's the short of a long, long story.\n\nBut still we hammered out this country from the rough, and no one can\ntake that away from us. The flag follers trouble, an' business follers\nthe flag, an' law follers business, an' trouble follers the law; but\nalways the first trouble was kicked up by boys who had got so 'at they\ncouldn't digest home cookin' any longer and just nachely had to get\nout an' tussle with nature an' the heathen.\n\nThey're a tough, careless lot, these young adventurers; an' they're\nalways in a state of panic lest the earth get so crowded the' won't be\nroom enough to roll over in bed without askin' permission; so they\nkill each other off as soon as possible, and thus make room for the\npatienter ones who follow after. From what I've heard tell of history,\nthis has been about the way that the white race has managed from the\nvery beginning.\n\nAs a general rule it has been purt' nigh a drawn fight between the\ndark-skins an' the wild animals; then the lads who had to have more\nelbow-room came along, and the dark-skins and the wild animals had to\nbe put onto reservations to preserve a few specimens as curiosities,\nwhile the lads fussed among themselves, each one tryin' to settle down\npeaceable with his dooryard lappin' over the horizon in all\ndirections. Room, room, room--that was their constant cry. As soon as\none would get a neighbor within a day's ride, he'd begin to feel shut\nin an' smothered.\n\nTyrrel Jones was one o' the worst o' this breed. He came out at an\nearly date, climbed the highest peak he could find, and claimed\neverything 'at his gaze could reach in every direction. Then he\ninvented the Cross brand, put it on a few cows, and made ready to\ndefend his rights. The Cross brand was a simple one, just one straight\nline crossin' another; and it could be put on in about one second with\na ventin' iron, or anything else which happened to be handy. Tyrrel\nthought a heap o' this brand, an' he didn't lose any chances of\nputtin' it onto saleable property. His herd grew from the very\nbeginning.\n\nHis home ranch was something over a hundred miles northwest o' the\nDiamond Dot; but I allus suspicioned that a lot of our doggies had the\nCross branded on to 'em. Tyrrel was mighty particular in the kind o'\npunchers he hired. He liked fellers who had got into trouble, an' the\ndeeper they was in, the better he liked 'em. Character seeks its\nlevel, the same as water; so that Tyrrel had no trouble in gettin' as\nmany o' the breed he wanted as he had place for. They did his\ndevilment free and hearty, and when they had a little spare time, they\nused to devil on their own hook in a way to shame an Injun.\n\nThe sayin' was, that a Cross brand puncher could digest every sort o'\nbeef in the land except Cross brand beef. Tyrrel used to grin at this\nsayin' as though it was a sort of compliment; but some o' the little\nfellers got purty bitter about it. When a small outfit located on a\nnice piece o' water, it paid 'em to be well out o' Ty's neighborhood.\nNo one ever had any luck who got in his road; but his own luck boomed\nright along year after year. He allus kept more men than he needed;\nan' about once a month he'd knock in the head of a barrel o' whiskey,\nan' the tales they used to tell about these times was enough to raise\nthe hair. Ty would work night an' day to get one of his men out of a\nscrape; but once a man played him false, he either had to move or get\nburied. He wasn't a bad lookin' man, except that he allus seemed keyed\nup an' ready to spring.\n\nHis men all had to be top-notch riders, because he hadn't any use for\na gentle hoss; he didn't want his hosses trained, he wanted 'em\nbusted, an' the cavey he'd send along for a round-up would be about as\ngentle and reliable as a band o' hungry wolves. If a man killed a\nhoss, why Ty seemed to think it a good joke, an' this was his gait all\nthe way along--the rougher the men were, the better they suited him.\nHe kept a pack o' dogs, and the men were encouraged to kick an' abuse\n'em; but if one of 'em petted a dog, he was fired that instant--or\nelse lured into a quarrel. The' didn't seem to be one single soft spot\nleft in the man, an' when they got to callin' him Tyrant Jones instead\nof Tyrrel, why, it suited him all over, an' he used it himself once in\na while.\n\nThe next time I saw Friar Tuck, he recognized me at first glance, an'\nhis face lit up as though we had been out on some prank together an'\nwas the best pals in the world ever since. He wanted to know all I\nknew about the crowd that had started to string him up; and when I had\nfinished paintin' 'em as black as I could, what did he do but say that\nhe was goin' up their way to have a talk with 'em.\n\nI told him right out that it was simply wastin' time; but he was set\nin his ways, so I decided to ride part way with him. He had two hosses\nalong this trip, with his bed an' grub tied on the spare one; and on\nthe second day we reached a little park just as the sun was setting.\nIt was one o' the most beautiful spots I ever saw, high enough to get\na grand view off to the west, but all the rest shut in like a little\nroom. He jumped from his hoss, had his saddle off as soon as I did,\nand also helped me with the pack. Then he looked about the place.\n\n\"What a grand cathedral this is, Happy!\" he sez after a minute.\n\nI didn't sense what he meant right at first, and went on makin' camp,\nuntil I happened to notice his expression. He was lookin' off to the\nwest with the level rays of the sun as it sank down behind a distant\nrange full in his face. The twilight had already fallen over the low\nland and all the hazy blues an' purples an' lavenders seemed to be\nfloatin' in a misty sea, with here an' there the black shadows of\npeaks stickin' out like islands. It really was gorgeous when you\nstopped to give time to it.\n\nIt had been gruelin' hot all day, an' was just beginnin' to get cool\nan' restful, and I was feelin' the jerk of my appetite; but when I\nnoticed his face I forgot all about it. I stood a bit back of him,\nhalf watchin' him, an' half watchin' the landscape. Just as the sun\nsank, he raised his hands and chanted, with his great, soft voice\nbooming out over the hills: \"The Lord is in His holy temple--let all\nthe earth keep silence before Him.\"\n\nHe bent his head, an' I bent mine--I'd have done it if the'd been a\nknife-point stickin' again' my chin. I tell you, it was solemn! It\ngrew dark in a few moments an' the evening star came out in all her\nglory. It was a still, clear night without a speck in the air, and she\nwas the only star in sight; but she made up for it, all right, by\nthrowing out spikes a yard long.\n\nHe looked up at it for a moment, and then sang a simple little hymn\nbeginnin', \"Now the day is over, night is drawing nigh; shadows of the\nevening steal across the sky.\" It didn't have the ring to it of most\nof his songs; it was just close an' friendly, and filled a feller with\npeace. It spoke o' the little children, and those watchin' in pain,\nand the sailors tossin' on the deep blue sea, and those who planned\nevil--rounded 'em all up and bespoke a soothin' night for 'em; and I\nventure to say that it did a heap o' good.\n\nThen he pitched in an' helped me get supper. This was his way; he\ndidn't wear a long face and talk doleful; he was full o' life an'\nboilin' over with it every minute, and he'd turn his hand to whatever\ncame up an' joke an' be the best company in the world; but he never\ngot far from the Lord; and when he'd stop to worship, why, the whole\nworld seemed to stop and worship with him.\n\nWe had a merry meal and had started to wash up the dishes when he\nhappened to glance up again. He had just been tellin' me a droll story\nabout the first camp he'd ever made, and how he had tied on his pack\nso 'at the hoss couldn't comfortably use his hind legs and had bucked\nall his stuff into a crick, an' I was still laughin'; but when he\nlooked up, my gaze followed his. It was plumb dark by now, an' that\nevening star was fair bustin' herself, and the light of it turned the\npeaks a glisteny, shadowy silver. He raised his hands again and\nchanted one beginning: \"Praise the Lord, O my soul, and all that is\nwithin me, praise His holy name.\"\n\nThe' was a part in this one which called upon all the works o' the\nLord to praise Him, and I glanced about to see what was happenin'. A\nfaint breeze had sprung up and the spruce trees were bowin' over\nreverently, the ponies had raised their heads and their eyes were\nshinin' soft and bright in the firelight as they looked curiously at\nthe singer; and as I stood there with a greasy skillet in my hand,\nsomething inside of me seemed to get down on its knees, to worship\nwith the other works o' the Lord.\n\nIt was one o' those wonderful moments which seem to brand themselves\non a feller's memory, and I can see it all now, and hear the Friar's\nvoice as it floated away into the hills until it seemed to be caught\nup by other voices rather than to die away.\n\nWell, we sat up about the fire a long time that night. He didn't fuss\nwith me about my soul, or gettin' saved, or such things. I told him\nthe things I didn't understand, and he told me the things he didn't\nunderstand; and I told him about some o' my scrapes, and he told me\nabout some o' his, and--well, I can't see where it was so different\nfrom a lot of other nights; but I suppose I'd be sitting there yet if\nhe hadn't finally said it was bedtime.\n\nHe stood up and looked at the star again, and chanted the one which\nbegins: \"Lord, now let thy servant depart in peace\"; after which he\npulled off some of his clothes and crawled into the tarp. I crawled in\nbeside him about two minutes later; but he was already asleep, while I\nlay there thinkin' for the best part of an hour.\n\nNext mornin' he awakened me by singin', \"Brightest and best of the\nsons of the morning\"; and after that we got breakfast, and he started\non to Ty Jones's while I turned back to the Diamond Dot. I didn't\nthink he'd be able to do much with that gang; but after the talk I'd\nhad with him the night before, I saw 'at they couldn't do much to him,\neither. I had got sort of a hint at his scheme of life; and there\nisn't much you can do to a man who doesn't value his flesh more 'n the\nFriar did his.\n\n\n\n\nCHAPTER FOUR\n\nTY JONES\n\n\nTy stood in his door as the Friar rode up, and he recognized him from\nthe description Badger-face had turned in. Badger-face had been purty\nfreely tongue-handled for not havin' lynched the Friar, and Ty Jones\nwas disposed to tilt his welcome even farther back than usual; so he\nset his pack on the Friar. He had six dogs at this time, mastiffs with\na wolf-cross in 'em which about filled out his notion o' what a dog\nought to be.\n\nThe Friar had noticed the dogs, but he didn't have an idee that any\nman would set such creatures on another man; so he had dismounted to\nget a drink o' water from the crick, it havin' been a hot ride. The\npack came surgin' down on him while he was lyin' flat an' drinkin' out\no' the crick. His ponies were grazin' close by, and as soon as he saw\n'at the dogs meant business, he vaulted into the saddle just in time\nto escape 'em.\n\nThey leaped at him as fast as they came up, and he hit 'em with the\nloaded end of his quirt as thorough as was possible. He was ridin' a\nline buckskin with a nervous disposition, and the pony kicked one or\ntwo on his own hook; but as the Friar leaned over in puttin' down the\nfifth, the sixth jumped from the opposite side, got a holt on his arm\njust at the shoulder, an' upset him out of the saddle. In the fall the\ndog's grip was broke an' he and the Friar faced each other for a\nmoment, the Friar squattin' on one knee with his fists close to his\nthroat, the dog crouchin' an' snarlin'.\n\nAs the dog sprang, the Friar upper-cut him in the throat with his left\nhand and when he straightened up, hit him over the heart with his\nright. He says that a dog's heart is poorly protected. Anything 'at\ndidn't have steel over it was poorly protected when the Friar struck\nwith his right in earnest. The dog was killed. One o' the dogs the\npony had kicked was also killed, but the other four was able to get up\nand crawl away.\n\nThe Friar shook himself and went on to where Ty Jones and a few of his\nmen were standin'. \"That's a nice lively bunch o' dogs you have,\" sez\nhe, smilin' as pleasant as usual; \"but they need trainin'.\"\n\n\"They suit me all right,\" growls Ty, \"except that they're too blame\nclumsy.\"\n\nThe Friar looked at him a minute, and then said drily, \"Yes, that's\nwhat I said; they need trainin'.\"\n\nTy Jones scowled: \"They don't get practice enough,\" sez he. \"It's most\ngenerally known that I ain't a-hankerin' for company; so folks don't\nusually come here, unless they're sure of a welcome.\"\n\n\"I can well believe you,\" said the Friar, laughin', \"and I hope the\nnext time I come I'll be sure of a welcome.\"\n\n\"It's not likely,\" sez Ty shortly.\n\nThe Friar just stood and looked at him curiously. He didn't believe\nthat Ty could really mean it. The' wasn't a streak of anything in his\nown make-up to throw light on a human actin' the way 'at Ty Jones\nacted; so he just stood and examined him. Ty stared back with a sneer\non his face, and I'm sorry I couldn't have been there to see 'em\neyein' each other.\n\n\"Do you really mean,\" sez the Friar at last, \"that you hate your\nfellow humans so, that you'd drive a perfect stranger away from your\ndoor?\"\n\n\"I haven't any use for hoss-thieves,\" sez Ty.\n\nThe Friars face lighted. \"Oh, that's all right,\" sez he in a relieved\ntone. \"As long as you have a special grievance again' me, why, it's\nperfectly natural for you to act up to it. It wouldn't be natural for\nmost men to act up to it in just this way, but still it's normal;\nwhile for a man to set his dogs on a total stranger would be\nmonstrous. I'm glad to know 'at you had some excuse; but as far as\nhoss-stealin' goes, that roan is back with your band again. I saw him\nas I came along.\"\n\nTy was somewhat flabbergasted. He wasn't used to havin' folks try out\nhis conduct and comment on it right to his face; and especially was he\nshocked to have his morals praised by a preacher. He knew 'at such a\nreception as had just been handed to the Friar would have taken the\nstarch out o' most men an' filled 'em with a desire for revenge ever\nafter; but he could see that the Friar was not thinkin' of what had\nbeen handed to him, he was actually interested in himself, Ty Jones,\nand was honestly tryin' to see how it was possible for such a\ncondition to exist; and this set Ty Jones back on his haunches for\ntrue.\n\n\"For all time to come,\" he sez slow and raspy, \"I want you to leave my\nstuff alone. If you ever catch up and ride one of my hosses again,\nI'll get your hide; and I don't even want you on my land.\"\n\nThen the Friar stiffened up; any one in the world, or any thing, had\nthe right to impose upon the Friar as a man; but when they tried to\ninterfere with what he spoke of as his callin', why, he swelled up\nnoticeable. The Friar's humility was genuine, all right; but it was\nabout four times stiffer an' spikier than any pride I've ever met up\nwith yet.\n\n\"I shall not ride your hosses,\" sez he, scornful, \"nor shall I tread\nupon your land, nor shall I breathe your air, nor drink your water;\nbut in the future, as in the past, I shall use for the Lord only those\nthings which belong to the Lord. The things which are the Lord's were\nHis from the beginning, the things which you call yours are merely\nentrusted to your care for a day or an hour or a moment. I do not\ncovet your paltry treasures, I covet your soul and I intend to fight\nyou for it from this day forward.\"\n\nThe Friar spoke in a low, earnest tone; and Ty Jones stared at him. Ya\nknow how earnest an insane man gets? Well, the' was something o' this\nin the Friar when he was talkin' business. You felt that he believed\nthat what he was sayin' was the truth, and you felt that if it was the\ntruth, it was mighty well worth heedin', and you also felt that in\nspite of its bein' so everlastin' different from the usual view o'\nthings, it might actually be the truth after all and a risky thing to\npass up careless.\n\nAfter waitin' a minute without gettin' a reply, the Friar turned on\nhis heel to walk away, stumbled, and slipped to the ground, and then\nthey noticed a pool of blood which had dripped from him as he stood.\nHe had forgotten that the dog had torn him, an' the men had looked\ninto his eyes, as men always did when he talked, and they had forgot\nit, too. Now, when he fell, Olaf the Swede stepped forward to help him\nup.\n\nOlaf was the best man 'at Ty Jones had, from Ty's own standpoint. Ty\nhad happened to be over at Skelty's one night when Skelty was givin' a\ndance. Skelty had six girls at this time, an' he used to give a dance\nabout once a week. Along about midnight, they got to be purty lively\naffairs. This night Skelty had bragged what a fine shot he was, an'\nthe boys were kiddin' him about it, because Skelty wasn't no shot at\nall as a rule. It was a moonlight night, and while they was sheepin'\nSkelty about his shootin', two strangers rode up, tied their hosses to\nthe corral, an' started up the path toward the door.\n\nSkelty looked at 'em an' sez, \"Why, if I had a mind to, I could pick\none o' those fellers off with this gun as easy as I could scratch my\nnose.\" He pulled his gun and held it over his shoulder.\n\nAll the boys fair hooted, an' Skelty dropped his gun an' shot one o'\nthe strangers dead in his tracks. The other came along on the run with\nSkelty shootin' at him as fast as he could pop; but he only shot him\nonce, through the leg, and he limped in an' made for Skelty with his\nbare hands. Skelty hit him in the forehead, knocked him down an'\njumped on him. He kept on beatin' him over the head until the stranger\nmanaged to get a grip on his wrists. He held one hand still, an'\nputtin' the other into his mouth, bit off the thumb.\n\nThe's somethin' about bein' bit on the thumb which melts a man's\nnerve; and in about five minutes, the stranger had Skelty's head\nbetween his knees, and was makin' him eat his own gun. It must have\nbeen a hideous sight! Some say that he actually did make Skelty eat\nit, and some say that he only tore through the throat; but anyway,\nSkelty didn't quite survive it, and Ty Jones hired the stranger, which\nwas Olaf the Swede.\n\nOlaf was one o' those Swedes which seem a mite too big for their\nskins. The bones in his head stuck out, his jaws stuck out prodigious,\nhis shoulders stuck out, his hands stuck out--he fair loomed up and\nseemed to crowd the landscape, and he was stouter 'n a bull. When he\nlet himself go he allus broke somethin'; but he had a soft streak in\nhim for animals, an' Ty never could break him from bein' gentle with\nhosses, nor keep him from pettin' the dogs once in a while. Olaf\nhadn't no more morals 'n a snake at this time, an' when it came to\ndealin' with humans, he suited Ty to the minute; but he just simply\nwouldn't torture an animal, and that was the end of it. Olaf wasn't a\ntalkin' man; he never used a word where a grunt would do, and he was\nmiserly about them; but he certainly was set in his ways.\n\nThe Friar hadn't fainted, he had just gone dizzy; so when Olaf gave\nhim a lift he got to his feet and walked to his horse. He allus\ncarried some liniment an' such in his saddle bags, an' he pulled off\nhis shirt and cleaned out the wound and tied it up, with Olaf standin'\nby and tryin' to help. Now, it made something of a murmur, when the\nFriar took off his shirt. In the first place, the dog had give him an\nawful tear, and for the rest, the Friar was a wonderful sight to\nbehold. He was as strong as Olaf without bein' bulgey, and his skin\nwas as white and smooth as ivory. He was all curves and tapers with\nmedium small hands and feet, and a throat clean cut and shapely like\nthe throat of a high-bred mare. Olaf looked at him, and nodded his\nhead solemnly. Badger-face hated Olaf, because Olaf had a curious way\nof estimatin' things and havin' 'em turn out to be so, which made Ty\nJones put faith in what Olaf said, over and above what any one else\nsaid.\n\nAs soon as the Friar had finished tyin' up the wound, he turned and\nwalked up to Ty Jones. \"Friend,\" he said, \"I don't bear you a grain o'\nmalice, and nothing you can ever do to me will make me bear you a\ngrain o' malice. I know a lot about medicine, and perhaps I can help\nyou that way sometime. I want to get a start with you some way; I want\nto be welcome here, and I wish 'at you'd give me a chance.\"\n\n\"Oh, hell!\" sneered Ty Jones. \"Do you think you can soft-soap me as\neasy as you did the boys? You're not welcome here now, and you never\nwill be. I've heard all this religious chatter, and there's nothin' in\nit. The world was always held by the strong, by the men who hated\ntheir enemies and stamped them out as fast as they got a chance; and\nit always will be held by the strong. Your religion is only for\nweaklings and hypocrits.\"\n\nThe Friar's face lighted. \"Will you discuss these things with me?\" he\nasked. \"I shall not eat until this scratch is healed, I have my own\nbed and will not bother you; won't you just be decent enough to invite\nme to camp here, give me free use of water, and grass for my hosses,\nwhile you and I discuss these things fully?\"\n\n\"I told you I didn't want you about, and I don't,\" sez Ty. \"The's\nnothin' on earth so useless as a preacher, and I can't stand 'em.\"\n\n\"Let me work for you,\" persisted the Friar. \"All I ask is a chance to\nshow 'at I'm able to do a man's work, and all the pay I ask is a\nchance to hold service here on Sundays. If I don't do my work well,\nthen you can make me the laughin' stock o' the country; but I tell you\nright now that if you turn me away without a show, it will do you a\nlot more harm than it will me.\"\n\nTy thought 'at probably the Friar had got wind o' some of his\ndevilment, and was hintin' that his own neck depended on his men\nkeepin' faith with him; so he stared at the Friar to see if it was a\nthreat.\n\nThe Friar looked back into his eyes with hope beamin' in his own; but\nafter a time Ty Jones scowled down his brows an' pointed the way 'at\nthe Friar had come. \"Go,\" sez he, stiff as ever. \"The' ain't any room\nfor you on the Cross brand range; and if ya try anything underhanded,\nI'll hunt ya down and put ya plumb out o' the way.\"\n\nSo the Friar he caught his ponies and hit the back trail; but still it\nhad been purty much of a drawn battle, for Ty Jones's men had used\ntheir eyes and their ears, and they had to give in to themselves 'at\nthe preacher had measured big any way ya looked at him; while their\nown boss had dogged it in the manger to a higher degree 'n even they\ncould take glory in.\n\nAs the Friar rode away, he sagged in his saddle with his head bent\nover; and they thought him faint from his wound; but the truth was,\nthat he was only a little sad to think 'at he had lost. He was human,\nthe Friar was; he used to chide himself for presumptin' to be\nimpatient; but at the same time he used to fidget like a nervous hoss\nwhen things seemed to stick in the sand; and he didn't sing a note as\nlong as he was on the Cross brand range--which same was an uncommon\nstate for the Friar to be in, him generally marchin' to music.\n\n\n\n\nCHAPTER FIVE\n\nTHE HOLD-UP\n\n\nThis was the way the Friar started out with us; and year after year,\nthis was the way he kept up. He was friendly with every one, and most\nevery one was friendly with him. Some o' the boys got the idea that he\npacked his guns along as a bluff; so they put up a joke on him.\n\nThey lay in wait for him one night as he was comin' up the goose neck.\nI, myself, didn't rightly savvy just how he did stand with regard to\nthe takin' of human life in self-defence; but I knew mighty well 'at\nhe wasn't no bluffer, so I didn't join in with the boys, nor I didn't\nwarn him; I just scouted along on the watch and got up the hill out o'\nrange to see what would happen.\n\nHe came up the hill in the twilight, singin' one of his favorite\nmarchin' songs. I've heard it hundreds of times since then, and I've\noften found myself singin' it softly to myself when I had a long,\nlonely ride to make. That was a curious thing about the Friar: he\ndidn't seem to be tampin' any of his idees into a feller, but first\nthing the feller knew, he had picked up some o' the Friar's ways; and,\nas the Friar confided to me once, a good habit is as easy learned as a\nbad, and twice as comfortin'.\n\nWell, he came up the pass shufflin' along at a steady Spanish trot as\nwas usual with him when not overly rushed, and singin':\n\n    \"Guide me, O Thou great Jehovah!\n      Pilgrim through this barren land;\n    I am weak, but Thou art mighty;\n      Hold me with Thy powerful hand.\"\n\nHe came up out of the pass with his head thrown back, and his boy's\nface shinin' with that radiatin' joy I haven't ever seen in another\nface, exceptin' it first caught the reflection from the Friar's; and\nthe notion about died out o' the boys' minds. They were all friends of\nhis and wouldn't have hurt his feelin's for a lot; but they had itched\nabout his weapons for such a spell that they finally had to have it\nout; so when he rounded a point o' rock, they stepped out and told him\nto put his hands up.\n\nThey were masked and had him covered, and his hands shot up with a\njerk; but he didn't stop his singin', and his voice didn't take on a\nsingle waver. Fact was, it seemed if possible a shade more jubilant.\nHe had reached the verse which sez:\n\n    \"Feed me with the heavenly manna\n      In this barren wilderness;\n    Be my sword and shield and banner,\n      Be the Lord my Righteousness\";\n\nand as he sang with his hands held high above his head, he waved 'em\nback and forth, playin' notes in the air with his fingers, the way he\ndid frequent; and it was one o' the most divertin' sights I ever saw.\n\nThose blame scamps had all they could do to keep from hummin' time to\nhis song; for I swear to you in earnest that the Friar could play on a\nman's heart the same as if it was a fiddle. He kept on an' finished\nthe last verse while I crouched above 'em behind a big rock, and\nfairly hugged myself with the joy of it. Ol' Tank Williams was a big\nman and had been chosen out to be the leader an' do the talkin', but\nhe hadn't the heart to jab into the Friar's singin'; so he waited\nuntil it was all over. Then he cleared his throat as though settin'\noff a blast of dynamite, and growls out: \"Here, you, give us your\nmoney.\"\n\nTen six-shooters were pointin' at the Friar, but I reckon if he had\nknown it would of exploded all of 'em, he'd have had to laugh. He\nthrew back his head and his big free laugh rolled out into the hills,\nuntil I had to gnaw at a corner o' the stone to keep from joinin' in.\n\"My money!\" sez he as soon as he could catch his breath. \"Well, boys,\nboys, whatever put such a notion as that into your heads. Take it,\ntake it, you're welcome to it; and if you are able to find more than\ntwo bits, why, I congratulate you most hearty; because two bits was\nall I could find this morning, and that will only be a nickle apiece,\nand five cents is small pay for robbin' a volunteer missionary.\"\n\nOl' Tank Williams was a serious-minded old relic, and he was feelin'\nso sheepish just then that it seemed to him as though the Friar had\nimposed on him by lurin' him into such a fix; so he roars out in\nearnest: \"If you ain't got no money, why the deuce do ya tote those\nguns about with ya all the time?\"\n\n\"Would you just as soon tie me to a tree, or take some other measures\nof defence?\" asked the Friar politely. \"My arms are gettin' weary and\nI could talk more comfortable with 'em hanging' down.\"\n\n\"Aw put 'em down, and talk on,\" sez George Hendricks.\n\n\"Thank you,\" sez the Friar. \"Well, now, boys, the man who doesn't take\nthe time to put a value on his own life, isn't likely to make that\nlife very much worth while. He mustn't overvalue it to such an extent\nthat he becomes a coward, nor he mustn't undervalue it to such an\nextent that he becomes reckless--he must take full time to estimate\nhimself as near as he is able.\n\n\"I don't know that I can allus keep from judgin' my fellow men; but I\nam sure that I would not judge one to the extent of sayin' that my\nlife was worth more than his, so I should never use a gun merely to\nsave my own life by takin' away the life of another man--much less\nwould I use a gun in defence of money; but I am a purty good shot, and\nsometimes I can get a man interested by shootin' at a mark with him.\nThis is why I carry firearms. Do you want the two bits?\"\n\n\"Aw, go on,\" yells ol' Tank, madder at himself 'n ever. \"We didn't\nintend to rob ya. All we wanted was to hear ya sing and preach a bit\";\nand he pulled off his mask and shook the Friar's hand. All the rest o'\nthe boys did the same; and I clumb up on my rock, flapped my wings,\nand crowed like a rooster.\n\nWell, we sat on the ground, and he sang for us; and then he sobered\nand began to talk about cussin'. It used to hurt the Friar to hear\nsome o' the double-jointed swear words we used when excited. He tried\nnot to show it, because he didn't want anything to shut us away from\nhim at any time; but whiles his face would wrinkle into lines of\nactual pain.\n\n\"Now, boys,\" he began, \"I know, 'at you don't mean what you say in a\nprofane way. You call each other terrible names, and condemn each\nother to eternal punishment; and if a man said these things in\nearnest, his life would be forfeit; but you take it merely as a joke.\nNow, I do not know just how wicked this is. I know that it is\nforbidden to take the name o' the Lord thy God in vain; so it is a\ndangerous thing to be profane even in thoughtlessness; but I have\nheard the Lord's name used by the perfectly respectable in a way which\nmust have hurt his tender nature more.\n\n\"Once in the crowded slum district of a large eastern city, I saw a\nfreight car back down on a child and kill it. The mother was frantic;\nshe was a foreigner and extra emotional, and she screamed, and cursed\nthe railroad. A man had come to comfort her, and he put his hand on\nher arm and said, 'My dear woman, you must not carry on this way. We\nmust always bow our heads in submission to the Lord's will.'\n\n\"For years the poor people o' that neighborhood had begged protection\nfor their children; and I cannot believe that it was the Lord's will\nthat even one o' the least of 'em should have been slain in order to\ndrive the lesson a little deeper home; so, as I said before, I am not\ngoing to talk to you of the wickedness of swearing--but I am goin' to\ntalk about its foolishness, its vulgarity, and its brutality.\"\n\nHe went on showin' that swearin' was foolish because it wasn't givin'\na man's thought on things in a man's way; but merely howlin' it out\nthe way wolves and wild-cats had to, on account o' their not havin' a\ncivilized language with which to express the devilment which was in\n'em. He showed how it made a feller lazy; because instead of tryin' to\nsort out words which would tell exactly what he meant, he made a lot\nof noises which had no more real meanin' than a bunch o'\nfire-crackers.\n\nThen his voice got low and serious, and he said 'at the worst thing\nabout cussin' was, that it led a feller into speakin' lightly about\nthe sacred things of life. \"When you speak the word 'son,'\" he said,\n\"you are bound to also call up the thought of 'mother'; and I want to\nsay to you right now that any one who can be coarse and nasty in\nthinkin' or speakin' about maternity, is not a man at all--or even a\ndecent brute--but has some sort of soul-sickness which is more\nhorrible than insanity. Always be square with women--all women, good\nand bad. I know your temptations, and I know theirs. Woman has a heavy\ncross to carry, and the least we can do, is to play fair.\"\n\nThen he sprang some of his curious theories on us: told us how the\nbody was full of poisons and remedies; and it depended on our plan of\nlivin', whether we used the one or the other. He said he allus cut out\nfood and tobacco on Fridays, and if he didn't feel bright and clear\nand bubblin' over with vitality, he fasted until he felt able to eat a\nrubber boot, and then he knew he had cleaned all the waste products\nout of him, and could live at top speed again. He finished up by\ntellin' of a cross old doctor he once knew, who used to say 'at cattle\nand kings didn't have to control themselves; but all ordinary men had\nto use self-denial, even in matters of pleasure.\n\nIt was more the way the Friar said things than what he said; his voice\nand his eyes helped a lot; but the thing 'at counted for most was the\nfact 'at you knew it wasn't none of it put on. He loved to joke when\nit was a jokin' matter; but he was stiff as stone with what he called\nthe foundations of life. A man, you know, as a rule, is mighty timid\nabout the things which lie close to his heart, no matter how bold and\nfree he'll talk about other things; but the Friar was like a little\nchild, an' he'd speak out as bold and frank as one, about the things\nhe loved and hated, until he finally put a few drops o' this queer\nbrand o' courage into our own hearts.\n\nOf course we didn't get to be troubled with wing-growth or anything\nlike that; but a short time after this fake hold-up, ol' Tank Williams\nwent in to fill up with picklin'-fluid, and he started in on Monday\nand kept fightin' it all that week until Friday. Then he said that he\nwouldn't neither eat, drink, nor smoke on that day; and they couldn't\nmake him do it. He started in on Saturday to continue what had started\nout to be one o' the best <DW12>s he had ever took; but the first\nquart made him sick as a dog, and he came out to the ranch and said\n'at the Friar had made him a temperate man, and for the rest of his\nlife he intended to set aside one day a week in the Friar's favor.\n\nAfter the boys had started for the ranch, the Friar invited me to\nspend the night with him; so we unpacked his bed from the lead-hoss\nand we built a little fire and had a right sociable time of it. Me and\nhim was good pals by this time. He had said to me once: \"Happy, you do\nmore general thinkin' than some varsity men I've known.\"\n\n\"I reckon,\" sez I, modest as I could, \"that a man who has bossed a\ndozen men and ten thousand cattle through a three days' blizzard, has\nto be able to think some like a general.\"\n\nThen he explained to me that general thinkin' meant to think about\nstars an' flowers an' the human race an' the past an' the future, an'\nsuch things, and not to be all the time lookin' at life just from the\nway it touched a feller himself. This was another thing I liked about\nhim. Most Easteners is so polite that they haven't the heart to set a\nfeller right when he has the wrong notion; but the Friar would divvy\nup on his knowledge as free as he would on his bacon or tobacco; so I\nopened myself up to him until he knew as much about me as I did\nmyself.\n\nHe didn't have much use for the shut-eye this night, nor he wasn't as\ntalky as common; so we sat smokin' and lookin' into the fire for a\nlong time. Once in a while he'd speak a verse about some big deed a\nman had done years ago, or else one describin' the mountains or\nsomething like that; until finally I asked him how it came that a man\nwho loved adventure an' fightin' an' feats of skill, the way he did,\nhad selected to be a preacher.\n\n\"We don't select our lives, Happy,\" sez he. \"You're surely philosopher\nenough to see that. As far as we can see, it is like that gamblin'\ngame; we roll down through a lot o' little pegs bobbin' off from one\nto another until finally we pop into a little hole at the bottom; but\nwe didn't pick out that hole. No, we didn't pick out that hole.\"\n\nSo I up and asked him to tell me somethin' about his start.\n\n\n\n\nCHAPTER SIX\n\nA REMINISCENCE\n\n\nI pity the man who has never slept out doors in the Rocky Mountains.\nSwingin' around with the earth, away up there in the starlight, he\nfills himself full o' new life with every breath; and no matter how\ntough the day has been, he is bound to wake up the next mornin' plumb\nrested, and with strength and energy fair dancin' through his veins.\nFor it to be perfect, a feller has to have a pipe, a fire, and some\none close and chummy to chat with. This night me an' the Friar both\nwent down to the crick and washed our feet. We sat on a log side by\nside and made noises like a flock of bewildered geese when we first\nstuck our feet into the icy water; but by the time we had raced back\nand crawled into his bed, we were glowin' all over.\n\nWe didn't cover up right away, because the Friar just simply couldn't\nseem to get sleepy that night; and after a minute he put some more\nwood on the fire, filled his pipe again, and said: \"So you want me to\ntell you about my story, huh? Well, I believe I will tell you about my\nboyhood.\"\n\nSo I filled my pipe, and we lay half under the tarp with our heads on\nour hands and our elbows on our boots, which were waitin' to be\npillows, and he told me about the early days, talkin' more to himself\nthan to me.\n\n\"My mother died when I was six years old, my father divided his time\nbetween cleanin' out saloons, beatin' me, an' livin' in the\nwork-house,\" began the Friar, and it give me kind of a shock. I'd had\na notion that such-like kids wasn't likely to grow up into preachers;\nand I'd allus supposed 'at the Friar had had a soft, gentle youth. \"I\nwas a tough, sturdy urchin,\" he went on, \"but I allus had a soft heart\nfor animals. I used to fight several times a day; but mostly because\nthe other kids used to stone cats and tie tin cans on dogs' tails. I\nused to shine shoes, pass papers, run errands, and do any other odd\njob for a few pennies, and at night I slept wherever I could. I had a\nbig dry-goods-box all to myself for several months, once, and I still\nlook back to it as being a fine, comfortable bedroom.\n\n\"One morning I was down at the Union Depot when a farmer drove up a\nbig Norman hoss hitched to a surrey. Some o' the other kids joshed\nhim, called the hoss an elephant and asked where the rest o' the show\nwas. The man was big, well fed, and comfortable lookin', same as the\nhoss, and he didn't pay any heed to the kids except to call one of 'em\nup to hold the hoss while he went into the depot. The kid wanted to\nknow first what he was goin' to be paid, and he haggled so long 'at\nthe farmer beckoned to me to come up. 'Will you hold my hoss for me a\nfew minutes?' he asked.\n\n\"That big gray hoss with the dark, gentle eyes seemed to me one of the\nmost beautiful things I had ever seen, and I was mighty anxious to\nhave charge of him, even for a few minutes; so I sez, 'You bet I\nwill.'\n\n\"The other kids roasted me and made all manner o' sport; but they knew\nI would fight 'em if they got too superfluous, so after a bit they\nwent on about their business. The's somethin' about man's love for a\nhoss that's a little hard to understand. I had never had no intimate\ndealin's with one before, yet somethin' inside me reached out and\nentwined itself all about this big, gray, velvet-nosed beauty left in\nmy charge. I reckon it must be in a man's blood; that's the only\nexplanation I can find. All the way back along the trail o' history we\nfind the bones of men and hosses bleachin' together in the same heap;\nand about every worthwhile spot on the face o' nature has been fought\nover on hossback, so it's small wonder if the feel of a hoss has got\nto be part of man's nature.\n\n\"The farmer had had a woman and a little girl in his care, to see off\non the train, and he was gone some time. I had a few pennies in my\npocket, and I bought an apple an' fed it to the hoss, gettin' more\nenjoyment out of it than out of airy other apple I'd ever owned. I can\nfeel right now the strange movin's inside my breast as his moist nose\nsniffed at my fingers and his delicate lips picked up the bits of\napple, as careful an' gentle as though my rough, dirty little hand had\nbeen made o' crystal.\n\n\"I was so interested in the hoss that I gave a start of surprise when\nthe farmer's voice behind me sez: 'You seem to like hosses, son.'\n\n\"'I hadn't no idee 'at a great big one like this could be so smooth\nan' gentle,' I said, with my hand rubbin' along the hoss's throat. 'I\nthink he's a wonder.'\n\n\"'Do you like other animals?' asked the farmer.\n\n\"'I reckon I must be an animal myself,' sez I, 'because I allus get\nalong well with them, while I have to fight a lot with humans.'\n\n\"'What do you want for tendin' to this hoss?' he asked me.\n\n\"'I don't want nothin',' sez I. 'We've got to be friends, an' I don't\ncharge nothin' for doin' favors for a friend. Besides, he's got so\nmuch sense, I doubt if he needs much watchin'.'\n\n\"The farmer grinned, looked into my eyes a long time, and gave me a\ndollar. 'Now tell me how you'll spend your dollar,' sez he.\n\n\"Well, I was purty well floored. I had never owned a dollar before in\nmy whole life, my father havin' taken away every cent he had ever\nfound on me; and I stood lookin' at the coin, and hardly knowin' what\nto do. The farmer stood lookin' down at me with his eyes twinklin',\nand after a minute, I handed the dollar back to him. 'This is too\nmuch,' I sez. 'A dime would be plenty for the job, even if I didn't\nlike the hoss; but if my old man would find a dollar on me, he'd give\nme a beatin' for hidin' it from him, take it away, get drunk, and then\ngive me another beatin' for not havin' another dollar.'\n\n\"So he asked me all about my father; and I told about him and about my\nmother bein' dead, and the twinkle left his eyes and they grew moist,\nso 'at he had to wink mighty fast.\n\n\"He told me that his own boy was dead and his girl married, and that\nthe' wasn't any children out at the big farm, and asked me if I\nwouldn't like to come and live with him. He told me about all the\nhosses an' the cows an' the pigs, an' that I could have a clean little\nroom to sleep in, an' plenty o' food and clothes, and could go to\nschool. It sounded like a fairy tale to me, and I sez, 'Aw go on,\nyou're just joshin' me'; but he meant it; so I got on the seat beside\nhim, and as soon as we got out o' town he let me drive the big gray\nhoss--and I entered into a real world more wonderful than any fairy\ntale ever was.\n\n\"When we drove up the shady lane and into the big barn lot, a little\nold lady with sad eyes came to the door, and sez: 'Now, John, who is\nthat with you?' and my heart sank, for I thought she wasn't goin' to\nstand for me; but he took me by the hand and led me up to the door,\nput his arm about the little woman's shoulder, and sez with a tremble\nin his voice: 'This here is a little feller I've brought out to be\ncompany for ya, mother. He hasn't any folks, and he is fond of\nanimals, and, and--his name is John, too.'\n\n\"At first she shook her head and shut her lips tight; but all of a\nsudden the tears came to her eyes, and she put her arms about me--and\nI had found a real home.\n\n\"Those were wonderful years, Happy, wonderful; and I have the\nsatisfaction o' knowin' that I did them about as much good as they did\nme. Their hearts had been wrapped up in the boy, and he must have been\na fine feller; but just when he had been promoted out o' the grammar\ngrade at the head of his class, he had took the scarlet fever an'\ndied. I wasn't used to kindness when I went there; so I never noticed\n'at they kept me out o' the inner circle o' their hearts at first. I\ncalled the little woman Mrs. Carmichael for some time; but one day\nafter I'd brought home a good report from school, I called her this,\nand she spoke to me sharp--I never knew any soft-hearted person in the\nworld who got so much solid satisfaction out of actin' cross as she\ndid. Well, she spoke to me sharp, and sez: 'John Carmichael, why don't\nyou call me Mother?'\n\n\"I looked into her face, and it didn't look old any longer, and the\nsad look had left her eyes, and they were black and snappy an' full o'\nlife; so I tried it; and we both broke into tears, but they were tears\no' joy; and then he insisted that I call him Dad, and we became a\nfamily; and about the happiest one in the world, I reckon.\n\n\"I rode the hosses bareback, shot hawks with my rifle, picked berries,\ndid a lot o' chores, and worked hard with my books. It was a full,\nround life with lots of love and happiness in it, and I grew, body and\nmind and spirit, as free and natural as the big oak trees in the woods\npasture.\n\n\"Mr. Carmichael had looked up my blood father and had done what he\ncould for him; but it was no use, and one winter's morning he was\nfound frozen in an alley. I didn't learn of it until the next June\nwhen he took me down to the city cemetery where my father and mother\nlay side by side. I did feel downcast as we all do in the presence of\ndeath; but it wasn't my real father and mother who were lyin' there\nbeneath the quiet mounds. Fatherhood and motherhood are somethin' more\nthan mere physical processes. The real fathers and mothers are those\nwho put the best part o' their lives into makin' the big, gloomy world\ninto a tender home for _all_ the little ones; and after my visit\nto the graveyard I felt drawn even closer to Dad and Mother than I had\nbefore.\n\n\"Children ought to have dogs and hosses and plenty of air and soil\nabout 'em, Happy. We don't learn from preachin', we learn from\nexample; and we can learn a heap from the animals. We talk about our\nsanitary systems; but we allus mean the sanitary systems outside our\nbodies. Now, the animals have sanitary systems, but they are inside\ntheir own skins, where they rightly belong. Look at the beautiful\nteeth of a dog--These come from eatin' proper food at the proper time\nand in proper quantities. If a dog isn't hungry, the dog won't eat. If\na child isn't hungry, it is fed candy in a lot o' cases, and this is\ndownright wicked. Of course the animals find it hard to live, crowded\nup the way man allus fixes things; but as a rule animals are temperate\nand clean, patient and honest, wise and strong; and I wish we'd use\n'em more as instructors for the young. Most mothers think a dog's\ntongue is dirty--Why, a dog's tongue is chemically clean, and healin'\nin its action; while the human mouth is generally poisonous--ask a\ndentist.\n\n\"And a cow's breath, after she has rolled in with sweetly solemn\ndignity from the clover field--Ah, that's a pleasant memory! I'll\nventure to say 'at mighty few monarchs have been as worthy o' bein'\nkissed before breakfast, as Nebukaneezer was while he was undergoin'\npunishment for his sins. I had gone to that farm with my soul all\nstunted and gnarly; but it straightened out and shot its little stems\nup toward the blue, the same as the stalks o' corn did.\n\n\"All I had as a start was a love of animals; and this is why I allus\ntry to find the one soft spot in a man's nature--Even if it's a secret\nvice, it is something to work on. This is what makes such a problem of\nTyrrel Jones. I can't find out a single soft place in him; but I'm\ngoin' to get into the heart of him yet, if I can find the way.\n\n\"Well, Dad and Mother passed away within a week of each other a short\ntime after I had been graduated. I had made up my mind to stay on the\nfarm with 'em as long as they stayed; although all sorts of voices\nwere callin' to me from the big outer world; but their daughter lived\nin the city, and had been weaned away from the farm, so she sold it,\nand I started on my pilgrimage.\n\n\"They had left me an income of three hundred and fifty dollars a year;\nand I determined to go to college. When I thought of how rich and full\nmy own life had been made, after its stunted beginning, I wanted to do\nall I could to make the whole earth like that farm had been, and it\nseemed to me that the best way was to become a priest of the Lord. I\ntried my best; but I have been consid'able of a failure, Happy. Now, I\nhardly know where I stand. I am sort of an outcast now, and just doing\nwhat seems best on my own hook.\n\n\"A lot of my ideals have been lost, a lot of my hopes have faded, a\nlot of my work has seemed like sweeping back the waves of the sea; but\nfor all I have lost, new things have taken their place, and I have\nnever lost my faith in the Lord. Now, I am weak in doctrine and a\nstranger to dogma; and the things for which I fight with all my soul\nand heart and strength, are kindliness and decency.\n\n\"As long as one bein' in the world is cold or hungry or diseased,\nevery other bein' is liable to become hungry and cold and diseased.\nWhat I am fighting for is a world without poverty. Most o' the ills of\nlife spring from poverty, and poverty is the result of selfishness and\ngreed. The earth is reeking with riches, but its bounty is not divided\nfairly.\n\n\"Happy, if I could only hold up the Lord, so that all men might see\nthe beauty and fullness of Him, the glory and grandeur of His simple\nlife and His majestic self-sacrifice, the fleeting cheapness of\nmaterial things would sink to their real value, and we would all\nbecome one great family, workin' together in peace and contentment.\nNow, go on to sleep.\"\n\nIt was purty late by this time sure enough, and I fell asleep soon\nafter this; but I awakened durin' the night and found myself alone. It\nwas cold when I stuck my nose out from under the tarp, but it was a\nwonderful night, clear and still, with the stars swingin' big and\nbright just above my reach.\n\nAs I lay there, I heard Friar Tuck singin' softly to himself out where\nthe trail dipped down into the valley:\n\n    \"The night is dark, and I am far from home,\n        Lead Thou me on!\n    Keep Thou my feet: I do not ask to see\n    The distant scene,--one step enough for me.\"\n\nI had never heard his voice so wonderfully beautiful before; but, my\nstars, the sadness of it made me choke! It wasn't just a song, it was\na cry; and I knew that it came from a lonely, bleedin' heart. I put my\nhead under the covers again, puzzlin' over what was on his mind; but\nfirst thing I knew I was awakened by the glad voice of the old Friar\nTuck, singin' his favorite mornin' hymn: \"Brightest and best of the\nsons of the morning\"; so I cooked breakfast, and he went his way, and\nI went mine.\n\n\n\n\nCHAPTER SEVEN\n\nHORACE WALPOLE BRADFORD\n\n\nThe Diamond Dot, while it was about the idealest ranch in the West\nfrom most standpoints, was run a little loose. Jabez didn't have any\nluxurious tastes, and he wasn't miserly; so he didn't strain things\ndown to the last penny--not by a whole lot. All he asked was to have\nhis own way and be comfortable; and so he allus kept more punchers 'n\nhe had actual need of, and unless they got jubilant over imposin' on\nhim, he just shut his eyes and grinned about it.\n\nTakin' his location and outfit into account, and he just simply\ncouldn't help but make money; so we all had a fairly easy time of it\nand grew tender feelin's, the same as spoiled children; which is why\nwe sometimes quit, for we never had any other excuse for it.\n\nBarbie was a notice-takin' child, if ever the' was one; and she stood\nout for company as a general and standin' order. Company didn't affect\nol' Cast Steel one way or the other; they were just the same to him as\na couple o' hundred head o' ponies, more or less; and so the news got\nout that we allus had a lot of extra beds made up and any one was\nwelcome to stretch out in 'em who wanted to. The result o' this was,\n'at we drew visitors as easy as molasses draws flies. I lived at the\nhome house on account o' bein' Barbie's pal, and so I got into the\nhabit o' bein' a sort of permanent reception committee. Some o' these\nvisitors was a plague to me; but Jabez didn't like to run any risk of\nhavin' 'em ruined beyond repair, so it was generally understood that I\nhad to use ex-treme caution when I started in to file the clutch off\ntheir welcome.\n\nThis spring 'at I have in mind, we had as visitor one o' the\neasternest dudes I was ever tangled up with. He came out for his\nhealth, which is the excuse most of 'em gives; but this one took more\nways of avoidin' health 'n airy other of 'em I ever saw. He smoked\ncigars all day long, big black ones, strong enough to run a sawmill,\nhe ate fattenin' food from mornin' till night, and when he drove out\nin the buckboard to take his exercise, he suffered from what he called\nfatigue. He used to sit up as wide awake as an owl till along about\nten every night; and half the time he didn't crawl out until near\nseven in the mornin'. He certainly was a pest!\n\nWhat he complained of most, was his nerves; and he'd sit for hours,\ntalkin' about 'em to anything 'at had ears. He said the worst of it\nwas, he couldn't sleep nights. I had, of course, heard o' nerves\nbefore ever I saw him; but I had never heard of 'em turnin' to and\ndevilin' a man, the way his did; so at first I was honestly\ninterested, and asked him all I could think up about 'em; but after a\nday or so, I'd 'a' been perfectly willin' to put up the coin out o' my\nown pocket to have him go to a dentist and have every last one of his\nnerves pulled.\n\nI don't begrudge sympathy to any afflicted individual; but the more I\nsympathized with this feller, the more affectionate toward me he got;\nand he used to trot about after me, warbilin' out dirges about his\nnerves until I was tempted to tie a stone around his neck and lose him\ndown the cistern.\n\nHe ran to language, too, this one did. His conversation was so full of\nit that a feller could scarcely understand what he was tryin' to say.\nHe was ferociously interested in the ancient Greeks; and if a man\nsucceeded in wedgin' him away from his nerves, he began immediate to\ndiscourse about these ancient Greeks. Now, I didn't have a single\nthing again' any o' these ancient Greeks before this Dude struck us,\nnone of 'em ever havin' crossed my trail before; but they sure did\nhave a rotten outfit o' names, and they were the most infernal liars\n'at ever existed. Three-headed dogs, and women with snakes for hair,\nwere as common in their tales as thieves among the Sioux. Barbie\ndidn't have any use for this Eastener either; so I decided to fit him\nout with a deep-rooted desire for home influences.\n\nI took ol' Tank Williams into my confidence, he bein' the most\ngruesome lookin' creature we had in our parts. He was a big man of\ncurious construction and he had one eye which ran wild. Tank never\nknew what this free eye was up to; and while he would be examinin' the\nground, the free eye would be gazin' up at a tree as intent as though\nhe had set it to watch for a crow. Durin' his younger days, Tank had\nformed the habit of indulgin' in gang fights as much as possible, and\nall of his features had been stampeded out o' their natural orbits;\nbut this free eye beat anything I ever see.\n\nThey had him down on his back one time, and he was gnawin' away\ncontentedly at some feller's thumb, when the feller reached up his\ntrigger finger and scooped out Tank's eye. The shape and color weren't\nhurt a bit; but some o' the workin' parts got disconnected, so that he\ncouldn't see with it; but it appeared to be full as good an eye as the\none he looked with.\n\nAll the sleep Tank ever wanted was six hours out o' the twenty-four,\nand he didn't care how he got 'em--ten minutes at a time, or all in\none lump. He could sleep sittin' up straight, or ridin', or stretched\nout in bed, or most any way. I think he could sleep while walkin,'\nthough I was never able to surprise him at it. He agreed to back me\nup, and Spider Kelley also said he was willin' to do everything in his\npower to furnish our guest some pleasant recollections after he'd gone\nback to a groove which fitted him better.\n\nAs soon as I began to plan my trip, I started to rehearse curious\nsecrets about Tank to the Eastener, whose name was Horace Walpole\nBradford. I told Horace that Tank had a case o' nerves which made his\n'n seem like a bundle of old shoe-laces; and that if something wasn't\ndone for him soon, I feared he was goin' to develop insanity. I said\nthat even now, it wasn't safe to contrary him none, and that I'd be a\nheap easier in my own mind if Tank was coralled up in a cell\nsomewhere, with irons on.\n\nI didn't tell Tank what sort of a disposition I was supplyin' him with\nfor fear he'd overdo it. Tank didn't know a nerve from an ingrowin'\nhair; but when he and Horace paired off to tell each other their\nsymptoms, I'll have to own up that his tales of anguish an' sufferin'\nmade Horace's troubles sound like dance music.\n\nI told Horace that a trip through the mountains would soothe and\ninvigorate him, until he'd be able to sleep, hangin' by his toes like\na bat; but the trouble was to find something which interested him\nenough to lure him on the trip. There was a patent medicine almanac at\nthe place, and I studied up its learnin' until I had it at my tongue's\nend, and I also used a lot o' Friar Tuck's health theories; so that I\ngot Horace interested enough to talk my eardrums callous; but not\nenough to take the trip.\n\nI didn't know much about nerves; but I was as familiar with sleep as\nthough I had graduated from eleven medical colleges, and I knew if he\nwould just follow my directions, it would give him such an appetite\nfor slumber that he'd drop into it without rememberin' to close his\neyelids. Ol' Jabez happened to mention an Injun buryin' ground with\nthe members reposin' on top o' pole scaffolds, and this proved to be\nthe bait. Horace wanted to see this, and it was a four days' drive by\nbuckboard; so I heaved a sigh o' relief and prepared to do my duty.\n\nWhen all was ready, we packed our stuff in the good buckboard, putting\nin an extra saddle for the accident we felt sure was goin' to happen.\nSpider started as driver, while I rode behind, leadin' a horse with\nTank's saddle on, though Horace thought it was Spider's. We had told\nhim that it made our backs ache to ride in a buckboard all day, so we\nwould change off once in a while. Horace wanted to do the drivin'\nhimself; but we pointed out that he wasn't used to our kind o' roads,\nand consequently favored the little hills too much. He was inhumanly\ninnocent, and it was almost like feedin' a baby chalk and water.\n\nWe trotted along gentle, until the rear spring came loose goin' down a\nlittle dip to a dry crick bed, about ten miles out. We talked it over\nand decided 'at the best plan would be for Spider to drive back and\nget the old buckboard; so after unloadin' our stuff, I took the tap\nout o' my pocket, fixed the spring, tied a rope about it to deceive\nHorace, and Spider drove back for the old buckboard which had been\ndiscarded years before, but which we had fixed up for this trip and\npainted until it looked almost safe to use.\n\nBefore long we saw the buckboard comin' back; but much to our\nsurprise, Tank Williams was drivin' it, an' givin' what he thought was\nthe imitation of a nervous man. He would stand up an' yell, crack his\nmule-skinner, and send the ponies along on a dead run. He came up to\nus, and said that he had had an attack o' nerves, hadn't slept a wink\nthe night before; and when Spider Kelley had refused to let him go in\nhis place, he had torn him from the seat an' had trampled him.\n\n\"I trampled him,\" sez Tank solemnly, his free eye lookin' straight\ninto the sun. \"I hope I didn't destroy him; but in my frenzy I\ntrampled him.\"\n\nHorace looked worried. \"Tank,\" sez I soothin'ly, \"we don't really need\nany one else along. You just help us to load, an' then go back, like a\ngood feller.\"\n\nTank stood up on the seat, an' held the whip ready. \"My life depends\non me takin' this trip!\" he yelled. \"My life depends on it; it depends\non it, I tell you. My life depends on me takin' this trip!\"\n\nHe went on repeatin' about his life dependin' on his takin' that trip,\nuntil I made a sign to Horace, and said 'at we'd better let him go\nalong. Horace wasn't ambitious to be trampled; so he concluded to\nconcur, an' climbed into the seat beside Tank. Any one else would 'a'\nnoticed that it was Tank's saddle on the hoss I was leadin'; but\nHorace never noticed anything which wasn't directly connected with his\nown body. He didn't even have any idee that the sun had set habits in\nthe matter o' risin' an' settin'--which was another fact I had took\ninto account.\n\nWe were drivin' four broncs to the buckboard, an' they was new to the\ngame and in high spirits. Tank was also in high spirits, an' we went\nat a clip which was inspirin', even to sound nerves. We did our level\nbest to give Horace somethin' real to worry about, an' from the very\nstart his nerves was so busy handin' in idees an' sensations that his\nmind was took up with these instead of with the nerves themselves as\nwas usual.\n\nWell, we sure had a delightful ride that afternoon: every time 'at\nHorace would beseech Tank to be more careful in swingin' around\ndown-hill curves, Tank would seize him by the arm with his full\nsqueezin' grip, an' moan: \"It's my nerves, my pore nerves. This is one\no' the times when I'm restive, I got to have action; my very life\ndepends on it! Whoop, hit 'em up--Whee!\" an' he'd crack his\nmule-skinner about the ears o' the ponies, an' we'd have another\nrunaway for a spell.\n\nHorace hadn't the mite of an idee in which direction he was travelin';\nall he did was to hang on and hope. The confounded buckboard was\ntougher 'n we had figured on, and it didn't bust until near dark. As\nthey went up the <DW72>, I could see the left hind wheel weavin' purty\nrapid, an' as they tore down the grade to Cottonwood Crick, things\nbegan to creak an' rattle most threatenin'. We had decided to camp on\nthe crick, an' Tank swung up his team with a flourish. The hind wheel\ncouldn't stand the strain, an' when it crumbled, Horace, an' the rest\no' the baggage, whip-crackered off like a pinwheel. Of course when one\nwheel went, the others dished in company, an' the whole thing was a\nwreck.\n\nThe ponies were comfortable weary, an' after I had roped one an' the\nrest had fallen over him, we soothed 'em down without much trouble,\nan' started to make camp. Horace was all in, an' was minded to sit on\nhis shoulder blades an' rest; but this wasn't part o' the plan, an' we\nmade him hustle like a new camp-boy. As soon as supper was over, he\nlit a cigar, an' prepared to take a rest. We had decided that those\nbig, black cigars wasn't best for his nerves, so we had smuggled out\nthe box, an' had worked a little sulphur into all but the top row. He\nlit his cigar and gave us one apiece, but he was so sleepy he couldn't\nkeep his on fire; and it was comical to watch him.\n\nEvery time he'd nod off, Tank would utter an exclamation, an' walk up\nan' down, rubbin' his hands an' cussin' about his nerves. Horace was\ndead tired from bein' jounced about on the buckboard all day; but he\nwas worried about Tank, an' this would wake him effectual.\n\nAbout ten o'clock I sez: \"Tank, what happened that night when you got\nnervous up in the Spider Water country?\"\n\n\"Oh, don't ask me, don't ask me,\" sez Tank, gittin' up an' walkin' off\ninto the darkness.\n\n\"I wish to glory he hadn't come along,\" I sez to Horace. \"I fear we're\ngoin' to have trouble; but chances are that a good night's rest'll\nquiet him, all right.\"\n\nPurty soon Tank came back, lit his pipe, an' sat facin' Horace with\nhis lookin' eye, an' everything else in the landscape with his free\none. \"You know how it is with nerves,\" he sez to Horace. \"You perhaps,\nof all them I have ever met up with, know how strained and twisted\nnerves fill a man's heart with murder, set his teeth on edge and put\nthe taste of blood in his throat; so I'm goin' to tell the whole o'\nthat horrid experience, which I have never yet confided to a livin'\nsoul before. Have you got a match?\"\n\nTank's pipe allus went out at the most interestin' times; and he\ncouldn't no wise talk without smokin'. We all knew this; so whenever\nTank got headed away on a tale, we heaved questions at him, just to\nsee how many matches we could make him burn. He'd light a match and\nhold it to his pipe; but he allus lit off an idee with the match, and\nwhen he'd speak out the idee, he'd blow out the match. Or else he'd be\nso took up by his own talkin', he'd hold the match until it burnt his\nfingers; then, without shuttin' off his discourse, he'd moisten the\nfingers on his other hand, take the burnt end of the match careful,\nand hold it until it was plumb burnt up, without ever puttin' it to\nhis pipe. I didn't want to waste matches on this trip so I told Horace\nto hand Tank his cigar. Horace had already wasted two cigars, besides\nthe ones he had given us; and I wanted him to get to the sulphur ones\nas soon as convenient.\n\nTank's mind was preoccupied with the tale we had made up; so he took\nHorace's fresh cigar, lit his pipe by it, threw the cigar into the\nfire, and said moodily: \"He was unobligin'. Yes, that cross-grained\nold miner was unobligin'. Of course, I wouldn't have done it if I\nhadn't been nervous; but I say now, as I've allus thought, that he\nbrought it on himself by bein' unobligin'.\"\n\nTank's gloomy tones had wakened Horace up complete; and as he started\nto light another cigar, I got ready for bed. \"You two have already got\nnerves,\" I sez to 'em; \"but I don't want to catch 'em, so I'll sleep\nalone, and you can bunk together.\" I unrolled my tarp close to the\nfire and crawled into it, intendin' to take my rest while I listened\nto Tank unfold his story.\n\nIt was a clean, fresh night, just right for sleepin'; and it almost\nseemed a shame to put that innocent little Eastener through his\ntreatment; but it was for his own good so I stretched out with a sigh\no' content, and looked at the other two by the fire.\n\nHorace was short and fat around the middle with stringy arms and legs.\nHe wore some stuff he called side-burns on his face. They started up\nby his ears, curved along his jaws and were fastened to the ends of\nhis stubby mustache. He kept 'em cropped short and, truth to tell,\nthey were an evil-lookin' disfigurement, though he didn't seem to feel\na mite o' shame at wearin' 'em. His face was full o' trouble, and yet\nhe was so sleepy he had to hitch his eyebrows clear up to his hair to\nkeep his eyes open. Tank's face never did have what could rightly be\ncalled expressions. His features used to fall into different kinds o'\nconvulsions; but they were so mussed up it was impossible to read 'em.\nI looked at these two a minute, and then I had to pull my head under\nthe tarp to keep from laughin'.\n\n\n\n\nCHAPTER EIGHT\n\nA CASE OF NERVES\n\n\n\"I was all alone,\" sez Tank. \"I had been up in the Spider Water\ncountry lookin' for a favorite ridin' pony; but my hoss broke a leg,\nand I packed my saddle and stuff on my head until my nerves began to\nswell. Then I threw the stuff away and hunted for a human. I roamed\nfor weeks without comin' across a white man, and my nerves got worse\nan' worse. You know how it is with nerves; how they set up that dull\nache along the back o' your spinal cord until you get desperate, and\nlong to bite and scratch and tear your feller-bein's to pieces--well,\nI had 'em worse this time 'n ever I had 'em before; and they loosened\nup my brain-cells until my self-control oozed out and I longed to\nfling myself over a cliff. Have you got a match?\"\n\nHorace passed over his fresh cigar, and Tank lit his pipe and tossed\nthis cigar into the fire also. Horace looked at it sadly for a moment;\nbut he was game, and lit another.\n\n\"Finally,\" sez Tank, \"I came upon a lonely cabin at the bottom of a\ngorge; and in it was a little man who was minin' for gold. He was\nabout your build, except that toilin' with pick and shovel had\ndistributed his meat around to a better advantage, and he wore his\nwhiskers complete, without any patch scraped off the chin. It was just\nnight when I reached the cabin, and he invited me in to eat; which I\nam free to say I did until I was stuffed up to my swaller, and then we\nprepared to sleep.\n\n\"Now, a feller would nachely think I'd 'a' gone right to sleep; but\ninstead o' this, my nerves began to twist an' squirm an' gnaw at me\nuntil I was almost beside myself; and after fightin' it for several\nhours, I woke up the miner, and asked him as polite as a lady, if he\nwouldn't rub my brow for a few minutes. Seems like when I'm nervous,\nthe' won't nothin' soothe me so quick as to have my brow rubbed; but\nthis little coyote refused pointblank to do it.\n\n\"I finally got down on my knees and begged him to; but he still\nrefused. He said he had fed me six meals at once and given me shelter,\nand this was as far as he'd go if my confounded nerves exploded and\nblew the place up. I was meek about it, I tried my best to ward off\ntrouble; but just then a nerve up under my ear gave a wrench which\ntwisted me all out o' shape, and I lost patience. I seized that little\ncuss by the beard and I yanked him out on the floor, and I said to\nhim--\"\n\nTank had once been unusual gifted in framin' up bright-\nprofanity, but he had been shuttin' down on it since the night he had\nhelped to fake the hold-up on the Friar, and I thought he had lost the\nknack. This night, though, he seemed to find a spiritual uplift in\ntellin' to Horace exactly what he had said to the lonely miner. Before\nhe finished this part, he had used up all of Horace's good cigars, as\nlighters, and the Eastener's face had turned a palish blue. I'd be\nwillin' to bet that Tank made the swearin' record that night; though\nof course, the' ain't any way to prove it.\n\nWhen Tank couldn't think of any new combinations, he covered his face\nand broke into tears. Horace sat and looked at him with his eyes\npoppin' out. \"Don't you think you could go to sleep?\" he asked after a\nbit.\n\n\"Sleep!\" yelled Tank. \"Sleep? I doubt if I ever do sleep again. I feel\nworse right now 'n I did that night in the gorge.\"\n\n\"What did you finally do that time?\" asked Horace.\n\n\"I hate to think of it,\" sez Tank; and he put his elbows on his knees,\nhis chin in his hands, and stared into the fire as though seein'\nghosts.\n\nHorace watched him a while, and then he lit a cigar out of the second\nlayer. He took one puff and then removed the cigar and stared at it.\nHe tried another puff, and then threw it into the fire, where it\nspluttered up in a blue flame. He tried six more, and then said\nsomethin' I couldn't quite catch and threw the whole box into the\nfire; while Tank continued to stare into it as though he had forgot\nthe' was any one else on earth.\n\n\"Let's go to bed,\" sez Horace.\n\n\"Have you got a match?\" sez Tank, lookin' around with a start. Horace\ntook a burnin' stick from the fire, and Tank lit his pipe with it; and\nfrom that on Horace kept a lighted stick handy.\n\n\"How in thunder did you get to sleep that night in the gorge?\"\ndemanded Horace, who was gettin' impatient.\n\n\"Well,\" sez Tank, \"after I had told this unobligin' little cuss\nexactly what I thought of him, he pulled out a gun and tried to shoot\nme--actually tried to shoot me in his own cabin, where I was his\nguest. My feelin's were hurt worse 'n they'd ever been hurt before;\nbut still I tried to calm myself; and if it hadn't been for my nerves,\nI'd have gone out into that gorge in the dead o' night, and never set\neyes on his evil face again; but I couldn't get control of myself, so\nI took his gun away from him and knocked him down with it. When he\nregained consciousness, he was in a repentant mood; and he consented\nto rub my head.\n\n\"He rubbed my head a while an' I sank into a dreamless, health-given\nrepose; but as soon as I was asleep, the traitorious sneak crept out\nan' started to run. I fled after him as swift as I could, an' caught\nhim about two A. M. I had to twist his arms to make him come back with\nme; but when I had once got him back to the shack, I tied him good an'\ntight, an' made him rub my brow again. When he'd rub slow an' gentle,\nI'd sleep peaceful an' quiet; but the minute he'd quit, why, I'd wake\nup again; so he rubbed an' rubbed an' rubbed\"--Tank smoothed his left\nhand gentle with his right, an' spoke slow an' whispery--\"an' I slept\nan' slept an' slept an'--\"\n\nThe darn cuss said it so soothin' an' natural, that hanged if I didn't\nfall asleep myself, though the last I remember, I was bitin' my lips\nso I could stay awake an' see the fun. I must have been asleep full an\nhour before I was woke up by Tank's voice, raised in anger. I stuck my\nnose out o' the tarp, an' there was Tank kneelin' straddle o' the\nother bed which he had rolled up in the shape of a man. Horace was\nstandin' close by with his hands on his hips an' lookin' altogether\ndroopy.\n\n\"I raised his head from the floor, like this,\" said Tank, illustratin'\nwith the bed, \"an' then I beat it down on the planks o' the floor; an'\nthen I raised it up again, an' then I beat it down, an' then I raised\nit up--\"\n\nI had to stuff a corner o' the soogan into my mouth to keep from\nlaughin' out loud at the expression in Horace's eyes; but Tank kept\nraisin' that poor head an' beatin' it down again for so long that I\nfell asleep again without intendin' to.\n\nThe next time I woke up Horace was speakin'. He was so earnest about\nit that at first I thought he had been weepin'; but he was simply\ntryin' to make his voice winnin' an' persuadish.\n\n\"I'll rub it,\" he sez. \"I'll rub it soft an' gentle, just like you say\nyou want it rubbed. Come on, let me rub it.\" I looked at Tank with his\nfree eye rollin' about as though it was follerin' the antics of a\ndelirious mosquito; and I'd just about as soon have rubbed the brow of\na porcupine; but Horace was all perked up with sympathy.\n\n\"No,\" sez Tank, sadly. \"You're a guest, an' it wouldn't be polite. If\nyou was a stranger, now, why, I'd choke your heart out but what I made\nyou rub it; but not a guest. No, I couldn't do that. I'd wake Happy up\nan' make him rub it; but he allus sleeps with a gun under his head,\nan' he's apt to shoot before he's full awake.\"\n\n\"Well, just let me try it a while,\" sez Horace.\n\n\"I'm feared to,\" sez Tank, beginnin' to weaken. \"If you was to start,\nan' I was to fall asleep, an' you was to quit, I might dream 'at you\nwas that unobligin' man which betrayed me back in the lonely shack;\nan' I might strangle you or somethin' before I came to my senses.\nNope, the best plan is just to sit an' chat here till daylight. My\nnerves is allus better after sun-up.\"\n\n\"I don't think I can stay awake much longer,\" sez Horace, almost\nwhimperin'.\n\n\"What?\" sez Tank in surprise. \"You claim to have nerves, an' yet you\ncan talk o' fallin' asleep at this time o' night. Great Scott, man,\nyou ain't got no nerves! You are as flebmatic as a horn toad. Oh, I\nwish I could just fall sleepy for one minute.\"\n\n\"Let me try rubbin' your brow,\" sez Horace, whose eyes were blinkin'\nfor sleep, but whose face was all screwed up into lines of worry at\nwhat was goin' to happen to him after he had finally give in an'\ndrifted off.\n\n\"Well,\" sez Tank, \"I'll let you try; but if you're already sleepy, I\ndoubt if any good comes of it. You sit there at the head o' the bed,\nan' I'll lay my head in your lap, an' you rub my brow soft an' gentle.\nIf I do get to sleepin' natural, why o' course the' won't be no harm\ndone in you takin' a few winks; but for the love o' peace, don't sleep\nsound.\"\n\nI blame near choked while they were gettin' settled, 'cause Horace was\none o' those finicky cusses, an' Tank's head looked like a moth-eaten\nbuffalo robe. Finally, however, Tank stretched out with the covers up\naround his neck an' his head pillowed in Horace's lap, and then Horace\nbegan to rub his brow as soft an' gentle as he knew how.\n\n\"You don't do it clingy enough,\" sez Tank. \"You want to just rest your\nfingers lightly, but still have 'em draw along so 'at they'll give a\nlittle tingle. There, that's better. Now then, I'll lay as quiet as I\ncan, an' try to go to sleep.\" Tank was doin' such an earnest job, he\nhad plumb fooled himself into believin' it was mostly true.\n\nHe gave a start after layin' quiet for five or ten minutes, an' this\nput Horace on edge again; but Tank didn't wake up. Horace had a saddle\nblanket around his shoulders; and the last I saw just before I fell\nasleep, myself, was Horace gently rubbin' Tank's brow, an' lookin'\ndown careful for a change of expression. They made a curious sight\nwith the firelight back of 'em.\n\nIt was grayin' up for the dawn next time I woke up; and I'd had my\nsleep out, but when I stuck my nose out from under the tarp, I found\nit purty tol'able frosty. I knew it was my duty to roust out an' keep\nHorace from gettin' more sleep 'n my treatment for his nerves called\nfor; but I was too comfortable, to pay much heed to the still, small\nvoice of duty. At the same time I was curious to see what my boon\ncomrades was up to, so I stretched my neck an' took a look at 'em.\n\nHorace had keeled over so that his elbow rested on Tank's chest an'\nhis head rested on his hand; but the other hand was still on Tank's\nbrow, an' I reckon Horace must have rubbed until he didn't care\nwhether it was sleep or death he drew, just so he got rid o' keepin'\nawake. Tank had reached up one hand so it circled Horace's waist; and\nthey made the most lovable group a body ever see.\n\nWhile I was still watchin' 'em, Horace's arm gave out, an' he settled\ndown on top o' Tank's nose. In about two minutes Tank came to with a\njump, an' heaved Horace to the foot of the bed. Tank was really\nstartled, an' he came to his feet glarin'. \"You blame little squab,\nyou!\" he yelled. \"What are you tryin' to do--smother me?\"\n\nHorace staggered to his feet, but he couldn't get his eyes open more\n'n a narrow slit. \"I didn't do it on purpose, Mr. Williams,\" he\nblubbled like a drunk man. \"I rubbed until I thought my hand would\nfall off at the wrist; but I reckon I must 'a' dropped asleep. Lie\ndown again, an' I'll rub you some more.\"\n\n\"Too late,\" sez Tank, \"too late, too late. I never can sleep while\ndaylight's burnin'; but still, my nerves don't get so dangerous until\nafter nightfall; so we'll just turn to an' get breakfast.\"\n\nWell, I got up after yawnin' a few times; and after askin' if they had\nhad a restful night, I started to get breakfast. Horace staggered\nabout, gettin' wood an' water an' doin' what he was able to, while\nTank wrangled in the hosses.\n\nAfter breakfast, which I must say for Horace, he et in able shape, we\nstarted to saddle up, puttin' the spare saddle on the hoss I had rode\nthe day before. \"Which one o' you is goin' back after the other\nbuckboard?\" asked Horace.\n\n\"Why, we ain't goin' back at all,\" sez I. \"It's full fifty miles, an'\nwe can't keep switchin' buckboards every day on a trip like this.\nWe'll just ride the ponies the rest o' the way.\"\n\n\"Ride?\" sez Horace. \"Ride!\"\n\n\n\n\nCHAPTER NINE\n\nTREATING THE CASE\n\n\nHorace started to enlarge on how much he didn't know about ridin'; but\nTank breaks in with a plea for his nerves. \"Look here,\" he said,\nscowlin' at Horace with his good eye, while the free one rove around\nwild in his face, \"your nerves are a little out o' fix, an' mine is\nplumb tied into knots. This here outin' will be the best thing we can\ndo for ourselves, an' you got to come along. No matter which way you\ngo, you got to ride; so the' ain't no sense in makin' a fuss about it.\nWe'll mount you up on as gentle a cayuse as the' is in the West; an'\nwe won't tell no one if you hang on to the saddle horn goin' down\nhill.\"\n\n\"That's right, Mr. Bradford,\" sez I respectful. \"You'd have to ride\nback anyway, so you might as well come on with us an' have a pleasant\nouting.\"\n\n\"Besides,\" sez Tank, \"up there in the Wind River country we stand a\nchance o' gettin' somethin' for our nerves, if the Injuns happen to be\nin a good humor. Those Injun doctors know all about hurbs an' which\ndiseases they grow for, an' when they're in a good humor, they'll sell\nya some.\"\n\n\"What'll they do if they're not in a good humor?\" asked Horace.\n\n\"Well, that's the beatin'est question I've yet heard!\" sez Tank. \"How\ndoes any one know what an Injun'll do when he's not in a good humor? I\ndon't reckon any one ever tried to learn the answer to that question.\nWhen an Injun's not in a good humor, either you've got to kill him or\nhe'll kill you. If we hear tell 'at they're out o' humor, we'll simply\nscurry back at the first hint, an' don't you forget it.\"\n\nHorace wasn't resigned yet; so he kept sawin' away with his questions\nall the time we were tyin' on the beds an' grub. The grass had been\npurty brown down below, but it was fat an' green up above, an' the\nponies felt fine. We had picked out good ones, an' it took some time\nto get 'em wore down to where they was willin' to pack; but by seven\no'clock we were ready to start, an' then Tank lifted Horace into the\nsaddle, while I held the pony's head. We had chose a steady old feller\nfor Horace, because we didn't want any serious accidents. Ol' Cast\nSteel was dead again' sheepin' the Easteners, an' I knew they'd be\ndoin's about what we'd done already, let alone havin' any sort of a\nmishap.\n\nWe told Horace just what to do to save himself, an' we fixed his\nstirrups to just fit him; but he took it purty hard. It takes a\nridin'-man a couple o' weeks to harden up after he's laid off a spell;\nbut when a man begins to do his first ridin' at forty, it comes\nex-tremely awkward. Horace was the first feller I ever saw get\nsea-sick on hossback; but he certainly did have a bad attack. I\nsuppose it was the best thing 'at could have happened to him, an'\nafter he was emptied out, he rode some easier. We only covered about\nthirty miles that day altogether, an' Tank had plenty o' time to get\nall the sleep he could use; but when he came to lift Horace down from\nthe saddle, Horace couldn't make his legs stiff enough to stand on.\n\nWe let him stretch out while we were makin' camp; but he fell asleep,\nso we had to wake him up to help get supper. I was beginnin' to feel\nsorry for him, but he had pestered us regardless about his nerves, an'\nI knew 'at pity for him now would be the worse for him in the long\nrun.\n\nAfter supper, Horace spent consid'able time in bewailin' his fate\nbecause he had got disgusted an' thrown his whole box o' cigars into\nthe fire. \"I've got an extra pipe, if you'd like to try that,\" sez\nTank. \"It's lots better for the nerves than cigars--though from what I\ncan tell o' you, you ain't bothered much with nerves. I wish to glory\nI was in your skin.\"\n\n\"Oh, man,\" sez Horace, \"you can't imagine how I suffer. I ache like a\nsore tooth all over, an' it gives me a cute pain just to sit here on\nthe grass.\"\n\n\"Sit on the saddle-blankets,\" sez Tank, sympathetic. As soon as Horace\nhad piled up the blankets an' sat down on 'em, groanin' most bitter,\nTank sez with feelin': \"Gee, how I envy you. You have nothin' but a\nfew muscle-aches and chafed skin an' such, while my nerves is\nbeginnin' to threaten me again. I'm not goin' to bother either o' you\nfellers, though. I'm goin' to have you tie me to a tree to-night if I\ncan't sleep.\"\n\nHorace filled the pipe, which was an ancient one, bitter as gall; but\nwhen he began to smoke, his face became almost satisfied. The pipe was\npurty well choked up, so that he had some bother in keepin' it goin',\nbut after we'd run a grass stem through it, it worked purty well, an'\nwe was right sociable until along about nine o'clock, when I got\nsleepy, myself. Then Tank began to worry about his nerves. Horace had\nabout forgot his own nerves, he was sufferin' so from Tank's.\n\nWhen we see that Horace couldn't keep awake any longer without bein'\ntortured, Tank began to carry on fiercer. He rumpled up his hair, gave\nstarts an' jerks, but the thing 'at worked best, was just to sit an'\nlook at his fingers, an' pick at 'em. He'd form a circle with his left\nthumb and forefinger, then poke his right finger through this circle\nand try to grab it with his right hand before it could back out. It\nwas the craziest thing I'd ever seen; but before long Horace got to\ntryin' it himself. While Tank was lookin' at his fingers with his good\neye, the free one rambled around, an' half the time it rested on\nHorace, an' fair gave him the creeps; but when I couldn't stay awake\nmyself, I gave Tank the sign, an' he got delirious.\n\n\"I can't sleep,\" he wailed, \"I can't sleep! My nerves, oh, my nerves!\nOne minute they're like hot wires, an' the next they're like streaks\nof ice. You'll have to tie me up, boys, you certainly will have to tie\nme up.\"\n\nI argued again' it as bein' inhuman; but Tank begged so that finally I\ngave in, an' we tied him to a down pine tree. Horace helped to tie\nhim, an' he sure did his best to make a good job of it. I was a little\ndoubtful, myself, about Tank gettin' loose; but he had blowed up his\nmuscles, an' he coughed me the all-right signal, so me an' Horace\nturned in.\n\nHorace groaned consid'able while stretchin' out; but he began to snore\nbefore I had got through findin' the soft place. When I first go to\nbed, I like to roll about a bit, an' stretch, an' loosen up my\nmuscles--I like to stay awake long enough to feel the tired spots sink\ndown again' the earth, an' sort o' ooze into it; and before I had\ndrifted off, Horace was buzzin' away at a log in great shape.\n\nI must 'a' slept an hour when I was wakened by a bright light, an'\nlookin' out, I saw Tank Williams standin' with his back to the fire\nan' glowerin' down at Horace. \"As soon as this log burns off, I'm\ngoin' to get you,\" sez Tank between set teeth.\n\n\"What are you goin' to get me for?\" asked Horace. \"You asked me to tie\nyou to it. I didn't want to tie you to it, but you insisted. I'll\nuntie you if you want me to, and rub your brow again.\"\n\n\"It's too late,\" muttered Tank. \"It's too infernal late. Nothin' could\nput me to sleep now. As soon as this log burns off, I'm goin' to get\nyou. You was the one which brought back my nerve trouble, an' you are\nthe one what has to suffer.\"\n\nTank hadn't been able to free himself from the pine tree; so he had\ndragged it in an' across the fire. It wasn't such a big one as trees\ngo; but it was a mighty big one for a man, tied to it as he was, to\ntote along. Horace reasoned with him a while longer, an' then when he\nsaw that the trunk was about burned through, he got purty well off to\none side, an' threw a chunk at me. I popped out of bed on the instant,\nan' began to shoot about promiscuous; so as to live up to my\nreputation.\n\nWhen I'd emptied my gun, I looked at Tank, as though seein' him for\nthe first time, an' sez: \"What in thunder da you mean, by raisin' all\nthis havoc?\"\n\n\"My nerves,\" sez Tank, \"my pore nerves. I can't sleep, an' I can't\nkeep my senses if I'm left tied to this tree any longer. It's all his\nfault, an' as soon as this log burns up, I'm goin' ta hunt him down.\"\n\nTank an' I argued fierce as long as we could think of anything to say;\nan' just as the dead pine was gettin' too hot for Tank to stand it any\nlonger, Horace calls in from the darkness, \"Don't you want me to rub\nyour brow a while an' see if that won't put you to sleep?\"\n\n\"Come in here,\" I sez, cross. \"This man is liable to kill himself, an'\nyou know more about nerves 'n I do.\"\n\nHorace crawled out from behind a big rock, came in, shiverin' with the\ncold; an' we untied Tank from the log. He had managed to get his feet\nloose; but his hands had been tied behind him an' when they got cold,\nhe couldn't make a go of it. \"Well,\" sez I, as soon as Tank was free,\n\"what are you goin' to do now?\"\n\n\"I move we get up the hosses, an' start at once,\" sez Tank. \"I don't\ntrust myself any longer, an' we can ride faster at night. My one hope,\nis to get to an Injun doctor, or else get so tired out that I can fall\ninto a dreamless sleep.\"\n\n\"Why don't you ride alone?\" demanded Horace with a sudden burst of\nintelligence. \"Why don't you ride alone; an' then you could ride as\nfast as you wanted to, an' if you found the Injuns out o' humor, you\ncould come back an' let us know.\"\n\nThis set us back for a minute: we had been playin' Horace for bein'\nutterly thought-loose; but he had figured out the best plan the' was,\nan' his eyes were bright an' eager.\n\n\"Take the hoss that's fastened on the rope here,\" Horace went on; \"an'\nwe can take the manacled hosses in the mornin' and foller ya. Yes,\nthat's the best plan.\"\n\nYou see the fact was, we were only twenty or twenty-five miles from\nthe ranch house. We had been circlin' an' zig-zaggin' through the\nhills, an' at night we hung up Horace's pony on a picket an' put\nhobbles on the balance. Bein' fooled on direction wasn't any sign of\nHorace bein' a complete lunkhead; I've known a heap o' wise ones get\nballed up in the mountains.\n\nTank stood puzzlin' over it with his free eye trottin' about in a\ncircle; but he couldn't think any way out of it. \"All right,\" sez he,\n\"if you two can get along without me, why, I'll risk my life by bein'\na scout.\"\n\n\"Nonsense,\" sez Horace; \"the Injuns haven't riz for years, an' they're\nnot likely to again.\"\n\nTank only winked his lookin' eye, an' proceeded to fling the saddle on\nthe picketed hoss. Horace was smilin' purty contented with himself,\nuntil I sez: \"Which hoss are you goin' to ride to-morrow, Mr.\nBradford?\"\n\nThen his face went blank as he recalled the blow-up we'd had that\nmornin' gettin' the pack ponies contented with their loads. \"By Jove,\nI can't ride any of them!\" he exclaims. \"It would kill me to have a\nhoss buck with me. I'm so sore now I can hardly move.\"\n\n\"You don't look as nervous as you did, though,\" I sez to him for\ncomfort.\n\nHe didn't pay me no heed. \"Here, Williams,\" he calls, \"you can't take\nthat hoss. He's the only one I can ride, and you'll have to catch\nanother.\"\n\n\"You ort have thought o' that before,\" sez Tank, goin' on with his\narrangements, but movin' slow.\n\n\"Well, you two straighten it out among yourselves,\" sez I. \"I'm goin'\nback to bed. No wonder you're nervous. It would make a saw-horse\nnervous to jibe around the way you two do.\"\n\nI went off grumblin', an' I went to sleep before they settled it; but\nTank stretched it out as much as he could, an' Horace didn't oversleep\nany that night. Next mornin' when I looked out, I saw him tied up with\nhis back again' a tree, an' Tank's head in his lap. He was swathed in\nhis slicker an' saddle-blanket to keep warm, an' was sound asleep. He\nlooked purty well hammered out, but hanged if he didn't look a lot\nmore worth while 'n he did when he started to take my treatment.\n\nIt seemed a shame to do it, as it was just gettin' into the gray; but\nI woke him up, an' asked him in a whisper what he was doin'. He sat\nan' blinked at me for a full minute before he remembered what or where\nhe was, an' then he told me that he finally induced Tank to try havin'\nhis head rubbed again, by lettin' Tank truss him up so he couldn't\nkeel over on him. \"Gee, but I'm cold an' stiff,\" he sez in a husky,\nraspin' voice. \"I don't see how it can be so hot daytimes, an' so cold\nnights.\"\n\n\"This'll do you a world of good, Mr. Bradford,\" sez I. \"You see, you\nswell up with the heat daytimes, an' crimp down with the cold nights;\nan' this will goad on your circulation, fry the lard out o' ya, an'\ngive your nerves a chance to get toned up.\" I quoted from the patent\nmedicine almanac occasional, just so he wouldn't forget he was takin'\ntreatment.\n\n\"I can't possibly ride, to-day,\" he sez, shakin' his head. \"Honest,\nI'm in agony.\"\n\n\"That's just 'cause you're stiff,\" sez I, kindly. \"That'll all wear\noff when the sun softens up your joint-oil. Why, man, you'll look back\non this trip as one o' the brightest spots in your whole life.\"\n\n\"I got hit in the back o' the head with a golf ball once,\" he flares\nback real angry; \"an' that showed me a lot o' brightness, too. I don't\nwant no more brightness, an' I don't intend to ride to-day.\"\n\nI was especial pleased at the human traits he was displayin'. He\nhadn't acted so healthy an' natural since he'd been with us, an' I was\nencouraged to keep on with the treatment. \"You will have to ride with\nus, even if we have to tie you on,\" I sez. \"We are now close to the\nInjun country, an' we're responsible for you. O' course the' ain't any\ndanger from regular war parties; but Injun boys is just as full o'\ndevilment as white boys, an' they haven't as many safety valves.\nThey're all the time sneakin' off an' playin' at war, an' they play a\npurty stiff game, too, believe me. If a dozen o' these voting bucks,\neighteen or twenty years old, was to stalk us, they'd try most earnest\nto lift our hair.\"\n\n\"I'd as soon be killed one way as another,\" he sez. \"I can't stand it\nto ride, an' that's all the' is to it.\"\n\nHere was a queer thing: the little cuss actually wasn't afeared of\nInjuns, which I had counted on as my big card. Nerves or no nerves,\nHorace Walpole Bradford wasn't no coward; 'cause we are all afeared o'\ncrazy folks, an' he thought Tank was crazy. If Tank had had two good\neyes, chances are he wouldn't 'a' feared him; so I kicked Tank in the\nside an' woke him up.\n\n\n\n\nCHAPTER TEN\n\nINJUNS!\n\n\nWell, we sure had a hard time gettin' Horace in the saddle that day.\nHe was some like a burro, small but strong minded. Finally he agreed\nto try it if we would put the saddle-blanket on top the saddle instead\nof underneath.\n\n\"The hoss don't need it as bad as I do,\" sez he; \"'cause he's covered\nall over with hoss-hide an' has hair for paddin' besides; and\nfurthermore, the saddle is lined with sheepskin underneath, while it's\nas hard as iron on top; and I'm just like a boil wherever I touch it.\"\n\nWe told him that a hard saddle was lots the easiest as soon as a\nfeller got used to it; but he broke in an' said he didn't expect to\nlive that long, an' that we could take our choice of leavin' him, or\nputtin' the saddle-blanket on top. The's lots of folks with the notion\nthat a soft saddle or a soft chair or a soft bed is the easiest; an'\nit ain't much use to argue with 'em, though the truth is, that if a\nfeller lived on goslin' down, he'd get stuck with a pin feather some\nday an' die o' loss of blood; while if he lived on jagged stones, he'd\nfinally wear into 'em until he had a smooth, perfect fittin' mold for\nhis body. Still, the truth is only the truth to them 'at can see it;\nso we put the blanket on top, an' perched Horace astride it.\n\nHe stood it two hours, an' then said it was stretchin' his legs so 'at\nhe was afeared a sudden jerk would split him to the chin; an' then we\nput the saddle on right, an' he found it full as easy as it had been\nthe day before. The best way, an' the easiest an' the quickest, to\ntoughen up, is just to toughen up. The human body can stand almost\nanything in the way o' hardship. After it has sent up word, hour after\nhour, that it is bein' hurt, an' no attention gets paid to it, why, it\nsets to work to remedy things on its own hook. In order to ride\ncomfortable, a lot of muscles have to loosen an' stretch. Most o' the\npain in ridin' comes from ridin' with set muscles. A feller can't\nbalance easy with set muscles, it's just one strainin' jerk after\nanother, an' the trick o' ridin' is to move with the horse. Just as\nsoon as ya get to goin' right along with the hoss, loose an' rubbery,\nyou take the strain off o' both you an' him; but while you're bumpin'\nagain' him, it's painful for both.\n\nWe rode about forty miles that day; and at the end of it Horace wasn't\ncomplainin' any worse 'n at the start. Well, he couldn't, as far as\nthat goes; but his body had already begun to find the motion o' the\nhoss. Of course he hadn't learned to balance, an' he still rode rigid;\nbut we had give him an easy-gaited old hammock, an' when we drew up to\nmake camp, he sat on his hoss without holdin' to the horn, an' said he\nwas beginnin' to like it. When Tank lifted him down, though, his legs\nwobbled under him like rubber an' he squashed down in a heap,\ngroanin'. We let him sleep where he lit while we were gettin' supper;\n'cause we was sure he would need it before mornin'. He wasn't nervous\nany longer; all he wanted was food, sleep, an' a lung full o' tobacco\nsmoke. I felt rather proud o' my treatment.\n\nTank had to boot him about purty freely to waken him up enough to take\nhis vittles; but he took a good lot of 'em, an' I was glad of it,\n'cause this was the night the Injuns were goin' to attack us, an' he\nwasn't scheduled to have any more solid nourishment until we got back\nto the ranch house. After supper he went to his pipe like a young duck\nto a puddle o' water. He hadn't learned to handle his moisture while\nsmokin' a pipe, an' when the pipe began to gargle, he muttered a\nlittle cuss-word under his breath. H. Walpole Bradford was comin' out\nwonderful.\n\nThe stiffenin' had all blew out o' the rim of his hat, givin' the sun\nfull swing at him, an' his nose looked like a weakly tomato flung in a\nbed o' geraniums. He had wrinkled up his face around where his glasses\nfit, an' now with the sun gone down his skin had loosened up again,\nshowin' the unburned wrinkles like painted marks. He sure did look\ntough! He was wearin' a gray suit with a belt around the middle an'\ncanvas leggins.\n\nAlong about nine o'clock he nodded over into the fire, right at the\nmost excitin' part of an Injun tale which Tank was makin' up for his\nespecial benefit. We fished him out an' shook him awake; but he came\nto as cross as a hornet, an' swore he was goin' to sleep right where\nhe was with all his clothes on.\n\n\"You're a wise pigeon to sleep with your clothes on, to-night,\" sez\nTank; \"'cause this is the Injun country, an' ya can't tell what'll\nhappen; but the best plan for us to do is to divide up an' keep watch\ndurin' the night.\"\n\n\"Keep watch!\" yells Horace, glarin' at Tank. \"I wouldn't keep watch\nto-night if I was bound to a torture stake. You can keep watch if you\nwant to--an' it wouldn't discommode you no more 'n if you was an owl.\nYour dog-gone, doubly condemned nerves won't let you nor any one else\nsleep--but I'm goin' to get some rest if I die for it.\"\n\n\"You're a nice one, you are!\" sez Tank. \"This here expedition was got\nup just on account o' your nerves, an' now that we've come to the most\nimportant point of all, why, you flam out an' put all the risk on us.\"\n\n\"You make me tired,\" sez Horace, scowlin' at Tank as fierce as a\ncornered mouse. \"If you're so everlastin' feared o' the Injuns--what\nya got this bloomin' fire for?\"\n\n\"We don't intend to sleep near the fire, Mr. Bradford,\" sez I,\nsoothin'. \"We intend to roll up our beds like as if we was in 'em an'\nthen sneak off into the bushes an' sleep. We don't want any trouble if\nwe can avoid it. If you'll notice, you'll see we haven't turned the\nhosses out to-night.\"\n\n\"These here Injuns is livin' on a reservation,\" sez he, \"an' I don't\nbelieve 'at they'd dare outrage us.\"\n\nI was indignant with the little cuss for not bein' afeared of Injuns.\nMy theory was, 'at nerves was a lot like hosses: keep a hoss shut up\nan' he'll get bad an' kick an' raise Cain; but take him out an' ride\nhis hide loose, an' he'll simmer down consid'able. I wanted to give\nHorace's nerves such a complete stringin' out that they wouldn't worry\nhim any more for a year; an' here he was, not carin' a hang for\nInjuns. \"Beliefs is all right to the believers,\" sez I, stiffenin' up;\n\"but facts is facts whether you believe in 'em or not. Every Injun\noutrage since the Civil War was planned on a reservation, an' we can't\ntake no chances.\"\n\nWhile he was studyin' over this with a pouty look on his face, Tank\nsez: \"It's time we fixed up an' moved out into the dark\"; so we put\nrolls o' brush in the beds, an' went on up the side o' the rise where\nthe' was a level spot I knew of, Horace stumblin' an' grumblin' every\nstep o' the way. We were about two hundred yards from the fire an' it\nlooked cozy an' cheerful, dancin' away beside the tarps. I was half a\nmind to join in with Horace, an' go on back; but our plans were all\nlaid, an' besides, I had a little bet up with Spider Kelley, that I'd\nreturn Horace in such fine condition that he'd be willin' to drink\nblood or milk a cow calf-fashion.\n\n\"You go to sleep first,\" sez Tank to Horace; \"I'll watch till I get\nsleepy an' then I'll call Happy, he'll watch two hours, an' if it\nain't dawn by that time, he'll call you. I may not get sleepy at all,\nbut you know how nerves is. I stayed awake ninety-six hours once, an'\ncouldn't get a speck sleepy. Then I decided to stay out the even\nhundred an' see how far I could jump after stayin' awake a hundred\nhours. I went to sleep in ten minutes an' didn't wake up for two\ndays--so I'm liable to be took sleepy to-night.\"\n\nWe had brought the slickers up, an' Horace rolled up in one, under a\nlow evergreen, and began to snore in half a minute. As soon as he had\ngot to wrastlin' with his breath in earnest, I went to the head o' the\ntrail an' whistled for Spider Kelley. He an' four others were there,\nan' I told 'em it was all right to start in an hour, an' then I came\nback to Horace chucklin'. Spider enjoyed anything like this, an' he\nhad fixed up the boys with feathers an' fringe an' smears o' chalk an'\nraspberry jam, till they looked as evil-minded as any Injuns I'd ever\nseen.\n\nWe set Horace's watch ahead five hours. Tank curled up an' went to\nsleep, an' then I started to wake Horace up. It took so long; to get\nhim to consciousness that I feared the hour would be up; but he\nfinally got so he remembered what he was, an' then I told him not to\nmake any fuss if he saw any Injuns, but to just wake us up. I tried to\nget him to take one o' my guns, but I didn't wear triggers on 'em an'\nhe didn't savvy snap-shootin', so he took a club in his hand an'\nstarted to parade.\n\nHe looked at his watch while I was stretchin' out in his warm spot,\nan' he looked at it again before I was through loosenin' up my\nmuscles. It beats the world how slow time crawls to a man on watch. I\nwas sleepy myself, but I'd have bit out my tongue before I'd have give\nin. I lay half on my right side with my hat drawn down, watchin'\nHorace. After about ten minutes, he pulled out his watch again an'\nlooked at it. He pulled out the snap to set it ahead, in order to fool\nus, but he was troubled with too much morality, so he snapped it shut\nan' spoke to himself between his set teeth for several moments.\n\nI reckon he must have kept on his feet for twenty minutes, an' then he\nsettled down with his face to the fire, which I had fed up on my way\nback from seein' Spider, an' said loud enough for me to hear: \"This is\nall damn foolishness.\"\n\nHe said it so slow an' solemn an' earnest, that I purt nigh choked;\nbut I kept still, he kept still, an' the fire kept dancin' before him.\nHis breathin' grew deep an' steady, his nerves was all coiled up\ncomfortable; and tired muscles don't make a feller wakeful. Purty soon\nHorace began to gargle his palate, an' then I was ready for Spider\nKelley.\n\nThe plan was for him to come up close so as to entertain Horace while\nhis braves sneaked on to the dummies in the tarps; but the' was no\noccasion for sneakin'. Horace had turned over the camp to fate, an' he\nwasn't worryin' his head about what was goin' to happen to it.\n\nFinally, Spider got disgusted an' he went down an' joined the others,\nan' they sure raised a riot; but all the time, Horace slumbered on.\nSpider caught up our hosses, put our saddles an' packs on 'em, threw\nsome pieces of old canvas he brought along on the fire; and he an' the\nrest raised a wild warwhoop and galloped away; but Horace was too busy\nto pay any attention. Spider an' the boys had to work next day, an'\nthey was some put out not to have a little more fun for their trouble.\nIt was all Spider could do to keep 'em from sneakin' back an'\nkidnappin' Horace, but this was liable to give the whole thing away,\nso he talked 'em out of it. As soon as the noise had died down, I set\nHorace's watch back five hours, an' then I went to sleep myself. It\nwas purty chilly, and I wasn't quite sure who the joke was on.\n\nWhen Tank woke up, he started in on Horace; but his noise wakened me\nup first. When Horace saw what had happened to the camp, he was about\nwordless; but after we had called him down about it for five or ten\nminutes, he flared up an' talked back as harsh as we did. He said 'at\nhe had kept guard for over three hours, fightin' off sleep by walkin'\nback an' forth; and hadn't sat down until it had started to lighten in\nthe sky. He stuck to this tale, and I'm sure he believed it himself.\nHe'd been so sleepy the night before that he couldn't have told a\ndream from an actual happenin', so when he began to get excited, we\ndropped it.\n\n\"All right,\" sez Tank at last; \"you've put us into a nice fix, but\nthe' ain't no use tryin' to pickle yesterday. What we've got to do is\nto hoof it back, an' we might as well begin. We're in a nice fix:\nnothin' to eat, not a single cabin on the road back, an' for all we\nknow the's a pack of Injuns watchin' us this blessid moment.\"\n\n\"How do ya know it was Injuns?\" sez Horace.\n\n\"Look there, an' there, an' there,\" sez Tank, pointin' at moccasin\nprints an' feathers. \"Then besides, no white men would 'a' burned up\nthe tarps.\"\n\n\"Do you mean to say 'at we got to walk all the way back?\" sez Horace.\n\n\"All the way, an' without no grub,\" sez Tank.\n\nHorace sat down on the end of a charred log. \"Well, I'll die right\nhere,\" sez he. \"This spot suits me as well as any other.\"\n\n\"You don't have to die at all,\" sez I. \"A body can go forty days\nwithout food, an' it does more good than harm.\" Friar Tuck had told me\na lot about fastin', an' I was keen to try it out on Horace. From all\nI could see from the theory o' fastin', it was just what was needed\nfor Horace's nerves.\n\n\"Look at me,\" sez Horace, pullin' at the waist of his clothes. \"I bet\nI've lost twenty pounds already, on this fool trip. Twenty pounds more\nwould make me a corpse, an' I'd just as soon be made one here as\nanywhere. As soon as I rest up a little, I'm goin' to begin to yell\nuntil I draw those blame Injuns back, an' have 'em finish the job in\nshort order.\"\n\nHe wasn't bluffin', he was simply desp'rit. \"You'll have to walk with\nus,\" sez I; \"come on.\"\n\nTank took one arm, an' I took the other, an' we started forth. For the\nfirst hour he hung back, and then he began to step out on his own\nhook. When we rested at noon, he was the freshest one of us. Tank an'\nI had ridin' boots, an' ridin' muscles; while he had walkin' shoes,\nan' no muscles at all worth mentionin'. \"I can play at this game as\nwell as any one,\" sez Horace, chewin' a blade o' grass, an' lookin'\nproud of himself.\n\nTank was purty well fussed up; he wasn't workin' out any theories, he\nhad just come along to help pester Horace an' have a little amusement;\nbut it began to appear to him that his fun was comin' high-priced.\n\nBy nightfall we was all tol'able hungry; but Horace was so set up over\nbein' able to put over a full day's walk on nothin' to eat that he was\npurty speechy, an' it was nine o'clock before he went to sleep. As\nsoon as he had dropped off, I went down to meet Spider Kelley an' get\nthe grub he had brought out for me 'n' Tank. He said 'at the other\nboys wasn't braggin' none about their trip the night before; but they\nwere all ready to roast me an' Tank as soon as we got in. We'd had it\nfixed that Spider an' the rest was to take turns worryin' Horace on\nthe back trip; but Spider said that it looked to him as if I'd win the\nbet anyway, so he intended to play neutral from that on. As soon as me\nan' Tank had eaten, we turned in, an' all of us slept like logs.\n\n\n\n\nCHAPTER ELEVEN\n\nBENEFITS OF FASTING\n\n\nThe next day Horace walked easier 'n any of us. Now I'm tellin' this\nto ya straight 'n' you can believe it or not just as ya please; but\nthat little cuss stepped right along, began to notice the scenery, an'\neven cracked a few jokes now an' again; while me an' Tank just plodded\nwith our minds fixed on the meal we were goin' to get that night.\nHorace had give up all thought o' meals, so they didn't pester him\nany.\n\nAt the end of the third day Horace had lost his appetite complete.\nFriar Tuck had swore that hunger didn't worry a man more 'n three\ndays, an' sure enough, it didn't. Horace didn't care whether he ever\net again or not. He'd get a little dizzy when he'd start out, an' once\nin a while he'd feel a bit fainty; but as far as bein' ravenous went,\nme an Tank had him beat a mile.\n\n\"Where is the joke o' this fool trip?\" growled Tank to me on the\nevenin' of the fourth day as we were eatin' the supper Spider Kelley\nhad brought out. \"He ain't a human at all, Horace ain't; he's a\nreptile, an' can live without food.\"\n\nSpider was tickled a lot, and said he didn't care if he did lose his\nbet, that it was worth it to find how everlastin' tough a little\nhalf-hand like Horace could be when drove to it. I'd been thinkin' it\nover all day, but I didn't say anything.\n\nFriar Tuck had said it was a question of will power, more 'n anything\nelse: that if a man just held his thoughts away from food it wouldn't\nbother him; but if he kept thinkin' of it, the digestin' juices would\nflow into his stomach an' make him think he was starvin'; so I was\nminded to try a new plan next day.\n\n\"Spider,\" I sez, \"you put a cow an' calf up in Nufty's Corral\"--which\nwas the name of a little shut-in park we would go through the next\nafternoon. \"Put 'em there in the mornin', a cow with an off brand, if\nyou can find one, an' trim their hoofs down close, so they won't go\nback to the bunch. Remember 'at we're on foot, an' trim 'em close\nenough to make it hurt 'em to walk. I'm goin' to make Horace hungry if\nI can.\"\n\n\"I hate to play again' him and my own bet,\" sez Spider; \"but I'll have\nthe cow there, just to see what you're up to. If you're goin' to\nbutcher it, though, I don't see why a young steer wouldn't be better.\"\n\n\"I'll count on you havin' it there,\" sez I; an' then Spider rode back\nto the ranch house, an' me an' Tank went to sleep.\n\nNext mornin' me an' Tank put the cartridges out of our belts into our\npockets. As soon as we started to walk I began to talk about my\nhunger, an' weakness, an' the empty feelin' in my head an' stomach. At\nfirst Horace didn't pay any heed; but from the start, ol' Tank\nWilliams caught every symptom I suggested; until I feared he'd curl up\non the trail an' die o' starvation. Finally, though, Horace began to\npay heed to my suggestions, an' to sigh an' moan a little. What\nfinally got him was my gnawin' at my rope an' gauntlet. Tank an' I had\nsaved our ropes, 'cause we expected to have need of 'em; and when noon\ncame an' I sat with a stupid look in my face, chewin' first the rope,\nan' then the wrist o' the gauntlet, Horace began to have some of the\nsymptoms I was fishin' for. Finally he borrowed one o' my gauntlets,\nan' after he had munched on it a while, he was as hungry as any one\ncould wish.\n\n\"I can't go another peg,\" he sez when I got up to start on again.\n\n\"How does that come?\" I asked him. \"When we stopped to rest you was\nfeelin' more chipper 'n any of us.\"\n\n\"I'm dyin' o' hunger,\" he replied, solemn. \"I've got a gnawin' pain in\nmy stomach, an' I'm all in. I fear my stomach is punctured or stuck\ntogether or somethin'.\"\n\nI had had a lot o' discussions with Friar Tuck about the power o'\nsuggestion; but I had never took much stock in it. I could see now,\nthough, that it actually did work. As long as Horace was tellin'\nhimself that everything was all right, why, it was all right. Then\nwhen I suggested 'at we were dyin' of hunger, why, he actually began\nto die of hunger; an' it was wonderful to see the change in him. He\nshowed us how he had ganted down; and the fact was, his bones had\nbecome purty prominent without any help from suggestin'. He didn't\nhave any more belly 'n a snake; but his eyes were bright, an' his skin\nclear, except that it was peelin' off purty splotchy, from sun-burn.\n\nWe finally left him an' started on; and after we'd got some distance,\nhe staggered after us; but he was just goin' on his nerve now, an' not\ngettin' much joy out of existence.\n\nAbout four in the afternoon, we reached Nufty's Corral, a fine little\npark with only a narrow entrance at each end. Horace was up with us by\nthis time, an' we were all ploddin' along head down. Suddenly Horace\ngrabbed us by the arms. \"Hush!\" he sez.\n\n\"What's up?\" sez I, lookin' at him.\n\n\"Look,\" he whispers, pointin' at the cow an' calf; \"there's food.\"\n\nWe drew back an' consulted about it. \"The great danger after a fast,\"\nI sez in warnin', \"lies in overeatin'. All we can do is to drink a\nlittle blood for the first few hours.\"\n\n\"Why can't we broil a steak over some coals?\" sez Horace.\n\n\"It would kill us to eat steak now,\" sez I.\n\nHe held out for the steak; but I finally sez that if he won't promise\nto be temperate an' eat only what I tell him, I'll drive off the cow;\nand then he comes around, and agrees to it.\n\n\"You sneak around to the far openin', Tank,\" I sez, then I pauses, an'\nlooks at him as though shocked. \"Where's your cartridges, man?\" I\nasked.\n\nTank felt of his belt, and seemed plumb beat out, then he looked at\nmine, an' yelled, \"Where's yours?\"\n\nWe both sat down on stones an' went over what we had done every minute\no' the time since we had started out; until Horace became frantic, an'\nsez: \"What's the difference what became of 'em? Your revolvers are\nloaded. You can sure kill one cow out o' twenty-four shots.\"\n\n\"Twenty shots,\" I corrected. \"We allus carry the hammer on an empty\nchamber; an' I'm so bloomin' weak I doubt if I could hit a cow in ten\nshots.\"\n\nHorace turned loose an' told us what he thought of us, an' it was\nedifyin' to hearken to him--he hit the nail on the head so often.\nFinally I sez: \"Well, a man can do no more than try--Go ahead, Tank,\nbut don't let her get by you, whatever happens.\"\n\nThe cow, which was a homely grade-whiteface with a splotch on her nose\nwhich made it look as if most of the nose had been cut off, stood in\nthe center of the park, an' she was beginnin' to get uneasy, although\nthe wind was from her way.\n\nAs soon as Tank got to his entrance he shot in the air; an' she came\nchargin' down on me. I shot over her, an' she charged back. We kept\nthis up until Horace lost patience an' called me a confounded dub.\n\"Here,\" sez I, \"the's two cartridges left. You fire 'em, I won't.\"\n\nAt first he refused, but he was desperate, and finally after I'd told\nhim to use both hands, he took a shot. The cow was standin' closest to\nus, but lookin' Tank's way, an' Horace nicked her in the ham. Instead\nof chargin' Tank, like a sensible cow, she came for us head on. Now,\nwhen a bull charges, he picks out somethin' to steer for, then closes\nhis eyes, and sets sail; but a cow keeps her eyes open, an' she don't\naim to waste any plunges either. Horace stood out in the center of the\nentrance an' banged away again, strikin' the ground about ten feet in\nfront of him.\n\n\"Run!\" I yells to him, jumpin' back behind a big rock, \"Run!\"\n\nHe forgot all about bein' hungry, an' he started to backtrail like a\nscared jack-rabbit. The cow had forgot all about havin' had her hoofs\npared, an' she took after him like a hungry coyote. As she passed me,\nI roped her, took a snub around the rock, an' flopped her; but she did\njust what I thought she'd do--rolled to her feet an' took after me.\nShe was angry. I'd have given right smart for a tough little pony\nbetween my knees.\n\n[Illustration: The cow had forgot all about havin' had her hoofs\npared, an' she took after him like a hungry coyote]\n\nThe rock was too big to get a half hitch over, so I just ran at right\nangles from her, hopin' to stretch out more rope 'n she could cover. I\ndid it by a few feet; but she swung around into my rope head on, an'\nthis flung me up again' her side. I managed to hang on to the rope,\nhowever, an' this fixed her, 'cause she'd have had to pull that rock\nover before she could 'a' come any farther. Horace had stopped an' was\ngappin' at us from a safe distance; but Tank arrived by this time an'\nput another rope on her an' we had her cross-tied between two big\nrocks by the time Horace arrived.\n\n\"What ya goin' to kill her with?\" he asked, his eyes dancin' like an\nInjun's at the beef whack-up.\n\n\"My cartridges are all gone,\" sez Tank.\n\n\"Mine too,\" sez I.\n\n\"Can't you use a knife, or a stone?\" sez Horace, the dude.\n\n\"You can try it if you want to,\" sez I; \"but hanged if I will.\"\n\nHe took a big stone an' walked to the head of the cow, but his nerve\ngave out, an' he threw down the stone. \"What in thunder did you tie\nher up for, then?\" sez he.\n\n\"I beg your pardon,\" sez I, \"but I thought perhaps she might be a\nlittle vexed with you on account o' your shootin' her up. She was\nheaded your way.\"\n\nHe sat down on a stone an' looked at the cow resentful. Suddenly his\nface lit up. \"Why don't you milk her?\" sez he. \"We can live on milk\nfor weeks.\"\n\nIt's funny how much alike hungry animals look. As Horace sat on the\nstone with his anxious face, his poppin' eyes, his mussed up\nside-burns, an' the water drippin' from his mouth at thought o' the\nmilk, he looked so much like a setter pup I once knew that it was all\nI could do to hold a straight face.\n\n\"Do you know how to milk, Tank?\" I sez.\n\n\"I don't,\" sez Tank; \"nor I don't know what it tastes like.\"\n\n\"Go ahead an' milk her, Mr. Bradford,\" I sez. \"You're the only one\nwhat knows how to milk, or who cares to drink it. What you goin' to\nmilk it in?\"\n\n\"I never milked in my life,\" sez he; \"but I saw it done once when I\nwas a boy, an' I'm goin' to try to milk in my hat.\"\n\nHe had a bad time of it; but he only got kicked twice, an' both times\nit was short, glancin' blows, not much more 'n shoves. Finally, he\ncame over to where me an' Tank was settin' an' flopped himself down\nbeside us. \"Can't you strangle her with those ropes?\" he sez, in what\nmight well be called deadly earnest.\n\nWe shook our heads, an' continued to sit there lookin' at the cow as\nthough we expected she'd point the way out of our trouble. Presently\nthe calf remembered his own appetite, an' rushed up an' gave a\ndemonstration of what neat an' orderly milkin' was. Horace sighed.\n\"Gee, I bet that's good,\" he said, the water drippin' from his lips\nagain. He had been four days without food, walkin' all that time\nthrough the mountains, sleepin' out doors with no cover but a slicker;\nand he had about burned up all his waste products, which Friar Tuck\nsaid was a city man's greatest handicap. His eyes got a little red as\nhe watched the calf, an' I saw that he meant to slaughter it; so I sez\nto him: \"That's the way to milk, Mr. Bradford. Why don't you sneak up\non the other side an' try it that way, the same time the calf is?\"\n\nHe studied a moment, an' then shook his head. \"No, she could tell me\nfrom the calf,\" he said sorrowful. \"Our foreheads are shaped\ndifferent, an' I'd have to get down on my hands and knees. She'd tell\nme in a minute, an' I don't want to be on my hands an' knees when she\nkicks me.\"\n\n\"We could throw an' hog-tie her,\" sez Tank; \"and you could get it easy\nan' comfortable. Would you want us to do that, Mr. Bradford?\"\n\nHorace jumped to his feet an' shook his fist in Tank's face. \"Don't\ncall me Mister again,\" he yelled. \"I'm plumb sick of it. If I ever\nlive to get another bath an' back East where the's food in plenty,\nwhy, I'll take up the Mister again; but now that I've got to a point\nwhere I have to suck milk from a hog-tied cow, you call me Horace, or\neven Dinky--which was my nickname at school. Yes, for heaven's sake,\ntie the cow. I have to have milk, an' that's the only way I see to get\nit.\"\n\nWell, Tank an' I was so full o' laugh we could hardly truss up the\ncow; but we finally got her on her back so 'at she couldn't do nothin'\nbut snap her tail, an' then Horace threw his hat on the ground, an'\nstarted in. I was entirely joyful: I knew 'at Spider Kelley, an' as\nmany o' the boys as could sneak away, were watchin' us from up on the\nhill, an' this was the grand triumph of my treatment for nerves.\n\nHorace approached the cow with consid'able caution, as she was in an\nawkward position. The calf had been interrupted in his meal, before he\nhad squenched his thirst, an' he was still prospectin' about on his\nown hook.\n\n\"Here,\" said Horace, givin' him a push, \"this is my turn.\"\n\nYou know how a calf is: a calf ain't afeared o' nothin' except hunger.\nHere was his food-supply bein' robbed, right when he was needin' it.\nHe blatted down in his throat, an' tried to nose Horace out of the\nway. Horace was findin' that milk the best stuff he had ever tasted,\nan' he fought off the calf with his right hand, while he steadied\nhimself by puttin' his left on the hind leg o' the calf's mother, an'\ngot a nice coat o' creamy froth in his side-burns. He was so blame\nhungry he didn't see a speck o' humor in it; but me an' Tank nearly\ndied.\n\n\"Say,\" sez Horace, raisin' his head, the milk drippin' from his lips,\n\"can't one o' you fellers fend off this calf till I finish?\"\n\nTank held the calf while I advised Horace to be temperate, an' after a\nbit he gave a sigh an' said, that that was all he could hold just\nthen, but not to let the cow escape. We loosened her, left one o' the\nropes on for a drag picket, an' took off the other. She was purty well\nsubdued; but we refused to give Horace any more milk that night, an'\nhe went to sleep before we had a fire built. Spider Kelley was\nwabblin' with laughter when he brought us our supper. He had been the\nonly one who could stay after bringin' up the cow; but he said he\nwouldn't 'a' missed it for three jobs.\n\n\n\n\nCHAPTER TWELVE\n\nA COMPLETE CURE\n\n\nNext mornin' we fed Horace all the milk he could hold, an' tried to\ndrive the cow along with us; but her hoofs had been pared so thin that\nit made her cross an' we had to give that projec' up.\n\n\"How far are we from the ranch house?\" asked Horace.\n\n\"About sixty miles,\" sez Tank.\n\n\"That's what I thought,\" sez he. \"Now, I can't see any sense in all of\nus hoofin' that distance. I'd go if I knew the way; but one of you\ncould go, an' the other stay with me an' the cow. Then the one which\nwent could bring back food on the buckboard, and it would be as good\nas if we all went.\"\n\nNow this was a fine scheme; but neither Tank nor I had thought of it.\nWe had intended to follow our own windin' circle back every step o'\nthe way; but when the milk set Horace's brain to pumpin', he fetched\nup this idee which saved us all a lot o' bother.\n\n\"I shall go myself,\" sez Tank; \"weak as I am, I'll go myself.\"\n\nIt was only about fifteen or twenty miles by the short cut, an' this\nwould get him back to regular meals in short order; so he left me his\nrope an' set out. Horace helped me with the cow that night, an' he\nproved purty able help. He was feelin' fine, an' the milk had filled\nhim out wonderful. He said he hadn't felt so rough 'n' ready for\ntwenty years; but Spider Kelley failed to arrive with my meal that\nnight, and I went to bed feelin' purty well disgusted. Tank had met\nhim before noon that day, an' he had gone in for a hoss; and they had\ndecided that it would be a good stunt to give me some o' my own\ntreatment.\n\nNext mornin' I felt as empty as a balloon; so after Horace had enjoyed\nhimself, I took a little o' the same, myself; but I didn't take it\nlike he did. I held my mouth open an' squirted it in, an' it was\nmighty refreshin'.\n\n\"Huh,\" sez Horace, \"you're mightily stuck up. The calf's way is good\nenough for me.\"\n\n\"I got a split lip,\" I sez, half ashamed o' myself.\n\nThey left us there three days to allow for the time it would have\ntaken Tank to walk if it had been as far as we claimed it was; and\nthen Tillte Dutch drove out the buckboard. He said 'at Spider an' Tank\nhad quit and gone into Boggs for a little recreation; but after I had\neaten my first meal out o' the grub he brought, I didn't bear 'em any\nill will. The joke was on me as much as it was on Horace; but I'd 'a'\ngone through twice as much to test that theory, an' I'd had the full\nworth o' my bother. Horace was a new man: he was full o' vim an' snap,\nan' he gave me credit for it an' became mighty friendly an'\nconfidential.\n\nHe stood up in the buckboard an' made a farewell speech to the cow\nwhich lasted ten minutes. He also apologized to the calf, an' told him\nthat when he got back East, he would raise his hat every time he\npassed a milk wagon. He sure felt in high spirits, and made up a\nramblin' sort of a song which lasted all the way back to the house. It\nhad the handiest tune ever invented and he got a lot o' fun out of it.\nIt began:\n\n    \"Oh we walked a thousand miles without eatin' any food,\n    An' then we met a cow an' calf, an' gee, but they looked good!\n    Her eyes like ancient Juno's were so in-o-cent an' mild,\n    We couldn't bear to take her life, we only robbed her child.\n    She strove to save the lactual juice to feed her darling boy;\n    So we had to fling her on her back to fill our souls with joy.\n    Now Tank an' Happy were too proud to compete with a calf,\n    So they sat them down an' dined on wind, while they weakly\n      tried to laugh.\n    I'm but a simple-minded cuss, not proud like one of these;\n    So I filled myself so full of milk, I'm now a cottage cheese.\"\n\nHorace was as proud o' this song as though it was the first one ever\nsung. He used the same tune on it that blind men on corners use. I\nreckon that tune fits most any sort of a song; it's more like the\n\"Wearin' of the Green\" than anything else but ten times sadder an'\nmore monotonous. He said he had once wrote a Greek song at college but\nit wasn't a patch on this one, and hadn't got him nothin' but a medal.\nI used to know twelve or eighteen verses, but I've forgot most of it.\nIt was a hard one to remember because the verses wasn't of the same\nlength. Sometimes a feller would have to stretch a word all out of\nshape to make it cover the wave o' the tune, an' sometimes you'd have\nto huddle the words all up into a bunch. Horace said that all high\nclass music was this way; but it made it lots more bother to learn\nthan hymns.\n\nThe verse which pleased me the most was the forty-third. Horace\nhimself said 'at this was about as good as any, though he liked the\nseventy-ninth one a shade better, himself. The forty-third one ran:\n\n    \"A cow-boy does not live on milk, that's all a boy-cow'll drink;\n    But the cow-ma loves the last the most, which seems a funny think,\n    I do not care for milk in pans with yellow scum o'er-smeared.\n    I like to gather mine myself; and strain it through my beard.\"\n\nI never felt better over anything in my life than I did over returnin'\nHorace in this condition. It was some risk to experiment with such a\ntreatment as mine on a feller who regarded himself as an invalid; but\nhere he was, comin' back solid an' hearty, with his shape shrunk down\nto normal, an' full o' jokes an' song.\n\nTillte Dutch had been one o' the braves in Spider's Injun party; so\nwhen we got in, about ten in the evenin', he lured the rest o' the\npack out to the corral, an' we agreed not to make the details of our\ntrip public. The ol' man wouldn't have made a whole lot o' fuss seein'\nas it had turned out all right; but still, he was dead set on what he\ncalled courtesy to guests; and he might 'a' thought that we had played\nHorace a leetle mite strong. Barbie noticed the change in Horace and,\no' course, she pumped most o' the story out o' me.\n\nHorace himself was as game a little rooster as I ever saw. He follered\nme around like a dog after that, helpin' with my chores, an' ridin'\nevery chance he had. He got confidential, an' told me a lot about\nhimself. He said that he hadn't never had any boyhood, that his mother\nwas a rich widow, an' was ambitious to make a scholar out of him; that\nshe had sent him to all kinds o' schools an' colleges an'\nuniversities, and had had private tutors for him, and had jammed his\nhead so full o' learnin' that the' wasn't room for his brain to beat;\nso it had just lain smotherin' amidst a reek of all kinds o' musty old\nfacts. He said that he never had had time for exercise, and had never\nneeded money; so he had just settled into a groove lined with books\nan' not leadin' anywhere at all. He said that since his mother's death\nhe had been livin' like a regular recluse, thinkin' dead thoughts in\ndead languages, an' not takin' much interest in anything which had\nhappened since the fall o' Rome; but now that he had learned for the\nfirst time what a world of enjoyment the' was in just feelin' real\nlife poundin' through his veins, he intended to plunge about in a way\nto increase the quality, quantity, and circulation of his blood.\n\nYa couldn't help likin' a feller who took things the way he did--we\nall liked him. He told us to treat him just as if he was a\nfourteen-year-old boy, which we did, an' the' wasn't nothin' in the\nway of a joke that he wasn't up against before the summer was over;\nbut he came back at us now an' again, good an' plenty.\n\nTank an' Spider tossin' up their jobs had left me with more work on my\nhands 'n I generally liked, so I had to stick purty close to the line\nuntil they went broke an' took on again. Then one day me an' Horace\ntook a ride up into the hills. We had some lunch along and about noon\nwe sat down in a grassy spot to eat it. We had just finished and had\nlighted our pipes for a little smoke when we heard Friar Tuck comin'\nup the trail. I hadn't seen him for months, an' I was mighty glad to\nhear him again. He was fair shoutin', so I knew 'at things was right\nside up with him. He was singin' the one which begins: \"Oh, come, all\nye faithful, joyful an' triumphant,\" and he shook the echoes loose\nwith it.\n\nHorace turned to me with a surprised look on his face; \"Who's that?\"\nhe sez.\n\n\"That's Friar Tuck,\" sez I, \"an' if you've got any troubles tell 'em\nto him.\"\n\n\"Well, wouldn't that beat ya!\" exclaimed Horace, an' just then the\nFriar came onto our level with his hat off an' his head thrown back.\nHe was leadin' a spare hoss, an' seemed at peace with all the world.\n\nWhen he spied me, he headed in our direction, an' as soon as he had\nfinished the chorus, he called: \"Hello, Happy! What are you hidin'\nfrom up here?\"\n\nI jumped to my feet, an' Horace got to his feet, too, an' bowed an'\nsaid: \"How do ya do, Mr. Carmichael?\"\n\nA quick change came over the Friar's face. It got cold an' haughty;\nand I was flabbergasted, because I had never seen it get that way\nbefore. \"How do you do,\" he said, as cheery an' chummy as a\nhail-storm.\n\nBut he didn't need to go to the trouble o' freezin' himself solid;\nHorace was just as thin skinned as he was when it was necessary, an'\nhe slipped on a snuffer over his welcomin' smile full as gloomy as was\nthe Friar's. I was disgusted: nothin' pesters me worse 'n to think a\nlot o' two people who can't bear each other. It leaves it so blame\nuncertain which one of us has poor taste.\n\nWell, we had one o' those delightful conflabs about the weather an'\n\"how hot it was daytimes, but so cool an' refreshin' nights,\" an', \"I\nmust be goin' now,\" an' \"oh, what's the use o' goin' so soon\"--and so\non. Then Horace an' the Friar bowed an' the Friar rode away as silent\nan' dignified as a dog which has been sent back home.\n\n\"Well,\" sez Horace, after we'd seated ourselves again, \"I never\nexpected to see that man out here. I wouldn't 'a' been more surprised\nto have seen a blue fish with yaller goggles on, come swimmin' up the\npass.\"\n\n\"Oh, wouldn't ya?\" sez I. \"Well, that man ain't no more like a blue\nfish with goggles on than you are. He's ace high anywhere you put him,\nan' don't you forget that.\"\n\n\"You needn't arch up your back about it,\" he sez. \"I haven't said\nanything again' him. I gave up goin' to church on his account.\"\n\n\"That's nothin' to brag about,\" sez I. \"A man'll give up goin' to\nchurch simply because they hold it on Sunday, which is the one day o'\nthe week when he feels most like stackin' up his feet on top o'\nsomethin' an' smokin' a pipe. A man who couldn't plan out an excuse\nfor not goin' to church wouldn't be enough intelligent to know when he\nwas hungry.\"\n\n\"You must 'a' set up late last night to whet your sarcasm!\" sez\nHorace, swellin' up a little. \"Why don't you run along and hold up a\nscreen, so 'at folks can't look at your parson.\"\n\n\"How'd you happen to quit church on his account?\" sez I.\n\n\"He was only a curate, when I first knew him,\" sez Horace.\n\n\"He's a curate yet,\" sez I. \"I tried one of his cures myself, lately;\nan' it worked like a charm.\" I turned my head away so 'at Horace\nwouldn't guess 'at he was the cuss I had tried it on.\n\n\"A curate hasn't nothin' to do with doctorin',\" sez Horace. \"A curate\nis only the assistant of the regular preacher which is called a\nrector. The curate does the hard work an' the rector gets the big\npay.\"\n\n\"That's the way with all assistants,\" sez I; \"so don't bother with any\nmore details. Why did you quit goin' to church?\"\n\n\"I quit because he quit,\" sez Horace.\n\n\"What did he quit for,\" sez I; \"just to bust up the church by drawin'\nyour patronage away from it?\"\n\n\"He quit on account of a girl,\" sez Horace; an' then I stopped my\nfoolishness, an' settled down to get the story out of him. Here I'd\nbeen wonderin' for years about Friar Tuck; an' all those weeks I had\nbeen with Horace I had never once thought o' tryin' to see what he\nmight know.\n\n\n\n\nCHAPTER THIRTEEN\n\nAN UNEXPECTED CACHE\n\n\nHumans is the most disappointin' of all the animals: when a mule opens\nhis mouth, you know what sort of a noise is about to happen, an' can\nbrace yourself accordin'; an' the same is true o' screech-owls, an'\nguinea-hens an' such; but no one can prepare for what is to come forth\nwhen a human opens his mouth. You meet up with a professor what knows\nall about the stars an' the waterlines in the hills an' the petrified\nfishes, an' such; but his method o' bein' friendly an' agreeable is to\nsing comic songs like a squeaky saw, an' dance jigs as graceful as a\nstore box; while the fellow what can sing an' dance is forever tryin'\nto lecture about stuff he is densely ignorant of.\n\nThe other animals is willin' to do what they can do, an' they take\npride in seein' how well they can do it; but not so a human. He only\ntakes pride in tryin' to do the things he can't do. A hog don't try to\nfly, nor a butterfly don't try to play the cornet, nor a cow don't set\nan' fret because she can't climb trees like a squirrel; but not so\nwith man: he has to try everything 'at anything else ever tried, an'\nhe don't care what it costs nor who gets killed in the attempt.\nSometimes you hear a wise guy say: \"No, no that's contrary to human\nnature.\" This is so simple minded it allus makes me silent. Human\nnature is so blame contrary, itself, that nothin' else could possibly\nbe contrary to it. To think of Horace knowin' about the Friar, an' yet\ndoggin' me all over the map with that song of his, was enough to make\nme shake him; but I didn't. I wanted the story, so I pumped him for\nit, patient an' persistent.\n\n\"I never was very religious,\" began Horace. Most people begin stories\nabout other people, by tellin' you a lot about themselves, so I had my\nresignation braced for this. \"I allus liked the Greek religion better\n'n airy other,\" he went on. \"It was a fine, free, joyous religion,\nfounded on Art an' music, an' symmetry--\"\n\nI was willin' to stand for his own biography; but after waitin' this\nlong for a clue to the Friar's past, I wasn't resigned to hearin' a\njoint debate on the different religions; so I interrupted, by askin'\nif him believin' in the Greek religion was what had made Friar Tuck\nthrow up his job.\n\n\"No, you chump,\"--me an' Horace was such good friends by this time\nthat we didn't have any regard for one another's feelin's. \"No, you\nchump,\" he sez, \"I told you he quit on account of a girl. I don't look\nlike a girl, do I?\"\n\n\"Well,\" sez I, studying him sober, \"those side-burns look as if they\nmight 'a' been bangs which had lost their holt in front an' slipped\ndown to your lip; but aside from this you don't resemble a girl enough\nto drive a man out o' church.\"\n\nI allus had better luck with Horace after I'd spurred him up a bit.\n\n\"You see, Friar Tuck, as you call him, was a good deal of a fanatic,\nthose days,\" sez Horace, after he'd thrown a stone at me. \"He took his\nreligion serious, an' wanted to transform the world into what it would\nbe if all people tried their best to live actual Christ-like lives. He\nwas a big country boy, fresh from college, an' full of ideals, an'\nfeelin' strong enough to hammer things out accordin' to the pattern he\nhad chose.\n\n\"It was his voice which got him his place. He had a perfectly\nmarvelous voice, an' I never heard any one else read the service like\nhe did. This was what took me to church, and I'd have gone as long as\nhe stayed. You see, Happy, life is really made up of sensations an'\nemotions; and it used to lift me into the clouds to see his shinin'\nyouth robed in white, an' hear that wonderful voice of his fillin' the\ngreat, soft-lighted church with melody an' mystery. It was all I asked\nof religion an' it filled me with peace an' inspiration. Of course,\nfrom a philosophical standpoint, the Greek religion--\"\n\n\"Did the girl believe in the Greek religion?\" I asked to switch him\nback.\n\n\"No, no,\" he snapped. \"This Greek religion that I'm speakin' of died\nout two thousand years ago.\"\n\n\"Then let's let it rest in peace,\" sez I, \"an' go on with your story.\"\n\n\"You understand that this was a fashionable church,\" sez Horace. \"They\nwas willin' to pay any sum for music an' fine readin' an' all that;\nbut they wasn't minded to carry out young Carmichaels plan in the\nmatter of Christianizin' the world. They was respectable, an' they\ninsisted that all who joined in with 'em must be respectable, too;\nwhile he discovered that a lot o' the most persistent sinners wasn't\nrespectable at all. His theory was, that religion was for the vulgar\nsinners, full as much as for the respectable ones; so he made a\nround-up an' wrangled in as choice a lot o' sinners as a body ever\nsaw; but his bosses wouldn't stand for his corralin' 'em up in that\nfashionable church.\n\n\"He stood out for the sinners; an' finally they compromised by gettin'\nhim a little chapel in the slums, an' lettin' him go as far as he\nliked with the tough sinners down there through the week; but readin'\nthe service on Sundays to the respectable sinners in the big church.\nThis plan worked smooth as ice, until they felt the need of a soprano\nsinger who could scrape a little harder again' the ceilin' than the\none they already had. Then Carmichael told 'em that he had discovered\na girl with a phe-nominal voice, an' had been teachin' her music for\nsome time. He brought her up an' gave her a trial--\"\n\n\"An' she was the girl, huh?\" I interrupted.\n\n\"She had a wonderful voice, all right,\" sez Horace, not heedin' me;\n\"but she wasn't as well trained as that church demanded; so they hired\nher for twenty-five dollars a Sunday on the condition that she take\nlessons from a professor who charged ten dollars an hour. She was\ngame, though, an' took the job, an' made good with it, too, improvin'\nright along until it was discovered that she was singin' weeknights in\na cafe, from six to eight in the evenin', an' from ten to twelve at\nnight.\n\n\"The girl had been singin' with a screen o' flowers in front of her;\nand some o' the fashionable male sinners from the big church had been\ngoin' there right along to hear her sing; but they couldn't work any\nplan to get acquainted with her, and this made her a mystery, and drew\n'em in crowds. Finally, as her voice got better with the trainin',\ncritics admitted 'at she could make an agreeable noise; and the common\nsinners was tickled to have their judgement backed up, so they began\nto brag about it. The result o' this was, that one ol' weasel had to\nswaller his extra-work-at-the-office excuse, and take his own wife to\nhear the singer. Then the jig was up. The woman recognized the voice\nfirst pop; and within a week it was known that Carmichael had been\ngoin' home with her every night.\n\n\"Now, you may be so simple-minded that you don't know it; but really,\nthis was a perfectly scandalous state of affairs, and the whole\ncongregation began to buzz like a swarm of angry bees. Carmichael was\nas handsome a young feller as was ever seen; but he had never taken\nkindly to afternoon teas and such-like functions, which is supposed to\nbe part of a curate's duties; so now, when they found he had been\ngoin' home nights with a girl 'at sang in a cafe it like to have\nstarted an epidemic of hysteria.\n\n\"They found that the girl lived in a poor part o' the town, and\nsupported her mother who was sickly, that they were strangers to the\ncity, and also not minded to furnish much in the way o' past history.\nThey insisted upon her givin' up the cafe-singin' at once; and from\nwhat I've heard, they turned up their noses when they said it.\n\n\"Carmichael pointed out that she was givin' up twenty a week for\nlessons which they had insisted upon; and asked 'em if they were sure\na girl could be any more, respectable, supportin' a sickly mother on\nfive a week, than if she added fifteen to it by singin' in a cafe. He\ngot right uppish about it and said right out that he couldn't see\nwhere it was one bit more hellish for her to sing at the cafe than for\nother Christians to pay for a chance to listen to her.\n\n\"This tangled 'em up in their own ropes consid'able; but what finally\nsettled it was, 'at their richest member up and died, and they simply\nhad to have a sky-scrapin' soprano to start him off in good style; so\nthey gave her twenty a week and paid for her lessons. The cafe people\nsoon found what a card she'd been and they offered her fifty a week;\nbut she was game and stuck to the agreement.\"\n\n\"How did you find out all this, Horace?\" I asked.\n\n\"A friend o' mine belonged to the vestry,\" sez Horace; \"and he kept me\nposted to the minute. This was his first term at it, and it was his\nlast; but he was a lucky cuss to get the chance just when he did. I\nhave since won him over to see the beauty o' the Greek religion.\"\n\n\"What became o' the girl?\" sez I with some impatience, for I didn't\ncare as much as a single cuss-word for the Greek religion.\n\n\"Carmichael was a gentle spoken young feller,\" sez Horace, \"but for\nall that, he wasn't a doormat by inheritance nor choice, and he kept\non payin' attention to the girl, and got her to sing at his annex in\nthe slums. Night after night he filled the place with the best\nassortment o' last-chance sinners 'at that locality could furnish; and\nhe an' the girl an' the sinners all pitched in and offered up song\nmusic to make the stars rock; but St. Holiernthou wasn't the sort of a\nparish to sit back and let a slum outfit put over as swell a line o'\nmelody as they were servin', themselves; so they ordered Carmichael to\ncut her off his list. He tried to get 'em to hire another curate, and\nlet him have full swing at the annex; but they told him they'd close\nit up first.\n\n\"Next, a delegation o' brave an' inspired women took it upon 'emselves\nto call on the girl. They pointed out that she was standin' in the way\no' Carmichael's career, that, under good conditions, his advance was\ncertain; but that a false step at the start would ruin it all. They\nwent on and hinted that if it wasn't for her, he might have married an\nheiress, and grow up to be one o' the leadin' ministers o' the whole\ncountry.\"\n\n\"What did she do, Horace?\" sez I.\n\n\"The girl was proud; she thanked the delegation for takin' so much\ninterest in her--and said that she would not detain 'em any longer;\nbut would think it over as careful as she could. Then she walked out\no' the room; and the delegation strutted off with their faces shinin'\nlike a cavey o' prosperous cats. The girl vanished, just simply\nvanished. She wrote Carmichael a letter, and that was the end of it.\nSome say she committed suicide, and some say she went to Europe and\nbecame a preemie donner--a star singer--but anyway, that was the end\nof her, as far as that region was concerned.\"\n\n\"She was a fine girl,\" sez I; \"though I wish that instead of slippin'\noff that way, she had asked me to drown the members o' that delegation\nas inconspicuous as possible. I wouldn't put on mournin', if the whole\noutfit of 'em was in the same fix your confounded Greek Religion is.\nWhat was her name, Horace?\"\n\n\"Janet Morris,\" sez he.\n\nI said it over a time or two to myself; and it seemed to fit her. \"I\nlike that name,\" sez I. \"Now tell me the way 'at the Friar cut loose\nand tied into that vestry. I bet he made trade boom for hospitals and\nundertakers.\"\n\n\n\n\nCHAPTER FOURTEEN\n\nHAPPY'S NEW AMBITION\n\n\nOl' Tank Williams allus maintained that I had a memory like the Lord;\nbut this ain't so. What I do remember, I actually see in pictures,\njust like I told you; but what my memory chooses to discard is as far\nout o' my reach as the smoke o' last year's fire. I've worked at my\nmemory from the day I was weaned, not bein' enough edicated to know\n'at the proper way is to put your memory in a book--and then not lose\nthe book. I've missed a lot through not gettin' on friendly terms with\nbooks earlier in life; but then I've had a lot o' fun with my memory\nto even things up.\n\nThis part about the Friar, though, isn't a fair test. Horace's\nvestry-man friend was what is known as a short-hand reporter.\nShort-hand writin' is merely a lot o' dabs and slips which'd strain a\nChinaman; but Horace said it was as plain to read as print letters,\nand as fast to write as spoke words. Hugo took it down right as it was\ngiven; and Horace had a copy which I made him go over with me until I\nhad scratched it into the hardest part o' my memory; and now it is\njust the same as if I had seen it with my own eyes--me knowin' every\ntone in the Friar's voice, and the way his eyes shine; yes, and the\nway his jaws snap off the words when he's puttin' his heart into a\nthing.\n\nHorace sat thinkin', before he started on with his tale; and I sat\nwatchin' his face. It was just all I could do to make out the old\nlines which had give me the creeps a few weeks before. Now, it had a\nfine, solid tan, the eyes were full o' fire, and he looked as free\nfrom nerves as a line buckskin. The Friar sez we're all just bits o'\nglass through which the spirit shines; and now that I had cleaned\nHorace up with my nerve treatment, the' was a right smart of spirit\nshinin' out through him, and I warmed my hands at it. He simply could\nnot learn to roll a cigarette with one hand; but in most things, he\nwas as able a little chap as ever I took the kinks out of.\n\n\"I'm sorry I didn't belong to that vestry,\" sez Horace, after a bit.\n\"When I look back at all the sportin' chances I've missed, I feel like\nkickin' myself up to the North Pole and back. From now on I intend to\nmix into every bloomin' jambaree 'at exposes itself to the vision of\nmy gaze. I'm goin' to ride an' shoot an' wrestle an' box an' gamble\nan' fight, and get every last sensation I'm entitled to--but I'll\nnever have another chance at a vestry-meetin' like the one I'm about\nto tell you of.\n\n\"You saw how toppy Carmichael got this afternoon; so you can guess\npurty close how he looked when he lined up this vestry.\"\n\n\"Oh, I've seen the Friar in action,\" sez I; \"and you can't tell me\nanything about his style. All you can tell is the details. So go to\n'em without wastin' any more time.\"\n\n\"How comes it you call such a man as him Friar Tuck?\" asked Horace,\nwho allus was as hard to drive as an only son burro.\n\n\"Well, I don't approve of it,\" sez I, \"and I kicked about it to the\nFriar; but he only laughed, and said 'at one name was as good as\nanother. A bettin' barber over at Boggs give it to him for admonishin'\na gambler from Cheyenne.\"\n\n\"Was he severe?\" asked Horace.\n\n\"Depends on how you look at it,\" sez I. \"He took a club away from the\ngambler an' spanked him with it; but he didn't injure him a mite.\"\n\n\"Humph,\" sez Horace, \"I guess the name won't rust much while it's in\nhis keepin'. He took other methods at this vestry meetin', though I\ndon't say they were any more befittin'. Hugo--such was the name of my\nfriend--said it was the quietest, but the most dramatic thing he ever\nsaw.\n\n\"They started in by treatin' him like the boy he was, gave him a lot\no' copy-book advice, especially as to the value o' patience, how that\nPaul was to do the plantin', Appolinaris, the waterin'; but that the\nsize an' time o' the harvest depended on the Lord, Himself; and that\nit was vanity to think 'at a young boy just out o' college could rush\nthings through the way he was tryin' to.\n\n\"The' was a hurt look about Carmichael's eyes; but the hurt had come\nfrom the letter, not from them, so he sat quiet and smiled down at 'em\nin a sort of super-human calmness. They thought he was bluffed\nspeechless, so they girded up their loins, an' tied into him a little\nharder, tellin' him that his conduct in walkin' home nights with a\ncafe-singer was little short of immoral, although they wouldn't make\nno pointed charge again' the woman herself. Then they wound up by\nsayin' 'at they feared he was too young to spend so much time amid the\nenvirons o' sin, and that they would put an older man in charge o' the\nannex, and this would leave him free to attend strictly to cu-ratin'.\n\n\"When they had spoke their piece, they were all beamin' with the\nupliftin' effect of it; and they settled back with beautiful smiles o'\nsatisfaction to listen to Carmichael's thanks and repentance. He sat\nthere smilin' too--not smilin' the brand o' smiles 'at they were, but\nstill smilin'. It would strain a dictionary to tell all there is in\nsome smiles.\n\n\"Presently he rose up, swept his eyes over 'em for a time, and said in\na low tone: 'Then I am to understand that I am to follow in the\nMaster's footsteps only as far as personal chastity goes?' said he.\n'That I may respectably pity the weak and sinful from a distance; but\nmust not dismount from my exalted pedestal to take 'em by the hand an'\nlift 'em up--Is that what you mean?' sez he.\n\n\"They still thought he was whipped, so one of 'em pulled a little\nsarcasm on him: 'Takin' the weak an' sinful by the hand an' liftin'\n'em up is all right,' said he; 'but it's not necessary to go home with\n'em after midnight.'\n\n\"Carmichael bit his lips; he tried to hold himself down, he honestly\ntried for some time; but he wasn't quite able. His hands trembled an'\nhis lip trembled while he was fightin' himself; but when he kicked off\nhis hobbles an' sailed into 'em, his tremblin' stopped an' the words\nshot forth, clear an' hot an' bitish. Hugo sat back in a corner durin'\nthis meetin', without speakin' a single word; and he was glad of it.\nIt saved him from gettin' his feelin's kicked into flinders about him,\nan' interferin' with the view; and it gave him a chance to take his\nnotes.\n\n\"'As a matter o' faith,' said Carmichael, 'we believe that Jesus never\nsinned; but we cannot know this as a matter of fact. Yet we can know,\nand we do know, as a matter of history, that He mingled an' had\nfellowship with the fallen, the sinful, the outcast, and the\ndisreputable. With these He lived, and with these and for these He\nleft the power and the life and the glory of His religion--and you say\nthat I must live in a glass case, may only look in holy dignity down\nat the weak and sinful; but that I mustn't go home with 'em after\nmidnight. With God, a thousand years is but as a day--and yet it would\nbe wrong for me to be in a sinner's company after midnight!'\n\n\"Carmichael paused here to give 'em a comeback at him; but their\nmouths were dry, and they only hemmed an' hawed. 'Every Sunday, in the\nservice of this refined an' respectable church, hunderds of you admit\nthat you have no health because of your sins--and yet, because of my\nyouth, you say I must remain with you where sin is robed in silk and\nbroadcloth, and not risk my soul where sin is robed in rags.'\n\n\"He paused again, and this time his eyes began to shoot\njerk-lightning, an' when he started to speak his deep voice shook the\nroom like the low notes of a big organ. 'No,' he said, 'I am not\ncontent to walk with the Lord, only on the day of His triumph--The\nvery ones who strewed the pathway of His majesty with palms, and\nfilled the air with hosaners, deserted Him at the cross--but I must\nwalk with Him every step of the way. I do not pray that my earthly\ngarments be spotless, I do not pray that my sandals be unworn an' free\nfrom mud; but I do pray that when I stand on my own Calvery I may\nstand with those who bear crosses, not with those who have spent their\nlives in learnin' to wear crowns.'\n\n\"Carmichael had discarded that entire vestry by this time, and he\ndidn't care a blue-bottle fly what they thought of him. He towered\nabove them with his face shinin', and his voice rolled down over 'em\nlike a Norther sweepin' through the hills. 'Many there were,' he went\non, 'who cried to Him, Lord, Lord; but after the tomb was sealed, it\nwas the Magdalene whose faith never faltered, it was to her He first\nappeared; and on the final resurrection morning, I hope the lesser\nMagdalenes of all the ages, and from all the nasty corners of the\nworld into which man's greed has crowded 'em, will know that I am\ntheir brother, and, save for a lovin' hand at the right moment, one of\nthem to the last sordid detail.'\n\n\"Carmichael stopped after this, and the room was so quiet you could\nhear the consciences o' that vestry floppin' up and down again' their\npocketbooks. When he began again his voice was soft, an' the\nbitterness had given way to sadness. 'The old way was best, after\nall,' he said. 'When you pay a priest a salary, you hire him and he\nbecomes your servant. The custom is, for masters to dictate to their\nservants; it is an old, old custom, and hard to break. I think I could\nsuit you; but I do not think I shall try. The roots of my own life\nlead back to the gutter, and through these roots shall I draw strength\nto lift others from the gutter. I do not value my voice as a means to\namuse those already weary of amusement: I look upon it as a tool to\nhelp clean up the world. You are already so clean that you fear I may\ndefile you by contagion. You do not need me; and with all your careful\nbusiness methods, you have not money enough to hire me.\n\n\"'What you need here, is a diplomat; while I yearn to be on the firm'\nline. I care little for the etiquette of religion, I want to get down\nwhere the fightin' is fierce an' primitive--so I hereby resign.\n\n\"'This girl whom you have driven out of my life, needs no defence from\nme or any man. I have known her since she was a little child; poverty\nwas her lot, and self-sacrifice has become her second nature. We are\nforbidden to judge; so I judge neither her nor you; but I will say\nthat often I have stood silent before the beauty of her character, and\noften my face has burned at the tainted money you have put on the\nplate. Part of this money comes from the rental of dives. I have seen\nthe dives themselves, I have seen their fearful product; and I cannot\nbelieve that profit wrung from a helpless slave can find its way to\nGod--even on the contribution plate.\n\n\"'I love the music an' the service an' the vestments o' this church;\nand I hope I need not give them up; but my heart is in rebellion, and\nfrom this time on I take the full responsibility of my acts. I shall\nnot choose my path; but will go as the spirit moves me; and if ever I\nfind one single spot which seems too dark for the Light of the world\nto enter, then shall the soul in me shrivel and die, and I shall\nbecome a beast, howling in the jungle.'\"\n\nHorace said that after the Friar had left the room, those vestry\nfellers sat in a sort of daze for some time, and then got up an'\nsneaked out one at a time, lookin' exceeding thoughtful; while Hugo\nhad hustled around to his room to read off his notes.\n\nWe sat there on the hill until dark, me tryin' to pump him for more\ndetails, but he didn't have 'em. He said the Friar had started to work\nin the slums; but was soon lost sight of, and the first he had heard\nof him for years was when he had come up the pass, singin' his\nmarchin' song. Course, I'd liked it some better if the Friar had\nknocked their heads together; but still, takin' his eyes an' voice\ninto consideration, it must 'a' been a fine sight; and if ever I get\nthe chance, I'm goin' to take on as a vestry-man, myself, for at least\none term.\n\n\n\n\nCHAPTER FIFTEEN\n\nTENDER FEELINGS\n\n\nMe an' Horace was regular chums after this. I had got to likin' him\nafter he had showed up good stuff under treatment; but I never took\nhim serious until he got enthusiastic about Friar Tuck. This proved\nhim to have desirable qualities and made him altogether worth while. A\nman never gets too old to dote on flattery; but the older he gets the\nmore particular he is about its quality. It's just like tobacco an'\npie an' whiskey an' such things: we start out hungry for 'em an' take\na lot o' trouble to get 'em in quantity; but after a time we'd sooner\ngo without altogether than not to have a superior article; an' it's\njust the same way with flattery.\n\nI took Horace into my most thoughtful moods as soon as I found out\nthat he was as sound as a nut at heart, an' that it wasn't altogether\nhis fault that he had been a pest to me at first. The human mind is\nlike new land, some of it's rich an' some poor. Facts is like manure,\nidees is like seed, an' education is like spadin' up an' hoein' an'\nrakin'. Rich soil is bound to raise somethin', even if it's nothin'\nbut weeds; but poor soil needs special care, or it won't even raise\nweeds. Now, manure can be put on so thick it will turn ground sour,\nan' seeds can be sowed so thick they will choke each other, an' a\ngreen hand will sometimes hoe up the vegetables an' cultivate the\nweeds; but the soil ain't to blame for this.\n\nPoor Horace's mind had been bungled to an infernal degree; an' it kept\nme busy rootin' up sprouts o' Greek religion. I'd have stood this\nbetter if the Greek gods an' godduses had had Christian names; 'cause\nI own up 'at some o' his tales of 'em was interestin'; but I couldn't\nkeep track of 'em, an' so I made him discard 'em in his conversations\nwith me; an' the way he flattered me was, to reform himself accordin'\nto what I demanded.\n\nI was teachin' him how to shoot, an' he was enjoyin' it a lot. He had\nplenty o' money, and took pleasure in spendin' it. This was good,\n'cause it costs a lot o' money to become a good shot. I'm glad I don't\nknow what it cost me to learn how to shoot a man through both ears\nafter doin' the double reverse roll. I never had but one fit chance to\nuse this, an' then I shot Frenchy through his ears without rememberin'\nto use the roll. I allus felt bad about this, 'cause I had a good\naudience, an' nothin' saves a man from the necessity o' shootin' his\nfellows, so much as havin' it well advertised that he is thoroughly\nqualified to do it in proper style. I kept up my own practicin' while\nteachin' Horace, an' we had right sociable times.\n\nHe could throw up a tin can with his left hand, pull his gun and,\nabout once out o' ten shots, hit the can before it fell; which is\npurty fair shootin'; but he was beginnin' to suspect that he was a\nregular gun-man; which is a dangerous idee for any one to get into his\nhead. I tried to weight down his head a little to keep him sensible,\nbut instead o' thankin' me he went off with Tank, who shot up a lot of\nhis cartridges at target practice; and in return, puffed up the\ntop-heavy opinion Horace already had of himself.\n\nHe took Horace down to a warm canyon where the' was a lot o'\nrattlesnakes, claimin' it was necessary to test him out an' see if he\nhad nerve on a livin' creature. He shot off the heads o' three snakes,\nhand-runnin', an' it nearly broke his hatband.\n\nWhen he told me about it, I let him know 'at Tank was only workin'\nhim. \"A rattlesnake will strike at a flash, Horace,\" sez I; \"an' it\nwas the snake's eyes which were accurate, not yours.\" This cut him up\nan' made him a little offish with me for a few days, until he found I\nhad told him the truth. Ol' Tank Williams wasn't no fancy shot; but\nI'd rather have tackled Horace with a gun, cocked in his hand, than\nol' Tank, with his gun asleep in its holster.\n\nAfter Horace had made the test of shootin' at dead snakes an' had\nfound that he couldn't pop off three heads hand-runnin', he simmered\ndown a little an' paid more heed to what I told him; but after I had\nproved that I told him straighter stuff 'n Tank did, I decided it\nwould be necessary to punish him a little. I didn't get downright cold\nwith him, because I didn't want to exaggerate his vanity any more 'n\nit already was; but I made it a point to do my loafin' with Spider\nKelley. Horace was crazy to go bear-huntin'; but I didn't seem\ninterested, an' I recommended ol' Tank Williams as bein' some the best\nbear-hunter the' was in existence. I wasn't jealous of Horace goin'\noff shootin' with Tank; but still if a feller chooses to dispense with\nmy company, I allus like to show him 'at I can stand it as long as he\ncan.\n\nQuite a string o' years had slipped away since the bettin' barber o'\nBoggs had strung ol' man Dort; so I reminded Spider 'at we had agreed\nto help even that up sometime; and Spider, he said he was ready to do\nhis part, whatever it happened to be; so we planned idees out among\nourselves, while Horace hung around lookin' wishful.\n\nWe had never given it away about the woodchuck not bein' a regular\nsquirrel; so the boys still used to congregate together purty often at\nol' man Dort's to marvel at the way Columbus had filled out an' took\non flesh. He had got rough an' blotchy soon after he had won the\ncontest from Ben Butler, the red squirrel, an' it was plain to all\nthat Eugene had done some high-toned barberin' on him before the day\no' the show.\n\nOl' man Dort didn't have no affection for Columbus--fact is, he sort\no' hated him for bein' bigger 'n Ben Butler; but he kept him fat an'\nfit so as to be ready to enter in a contest the minute any feller came\nalong with a squirrel he thought was big enough to back up with a bet.\nThe trouble was, that mighty few fellers out that way owned any\nsquirrels, an' as the years dragged by without him gettin' any pastime\nout o' Columbus, ol' man Dort's affection for him grew thinner an'\nthinner. Some o' the boys discovered him to be a woodchuck; but no one\ntold of it for fear the old man would slaughter Eugene.\n\nThe old man kept on gettin' barbered, so as to have the chance o'\nclashin' with Eugene about every subject which came up; but finally he\ngot so he could be shaved in a decent, orderly manner without havin'\nhis head tied down to the rest. Him an' Eugene was the most\nantagonistic fellers I ever met up with; but it was a long time before\nme an' Spider could think up a way to get 'em fairly at it again.\n\nOne day Spider came ridin' in from Danders, bubblin' over with\nexcitement, and yells out--\"Pete Peabody's got a freak guinea-pig.\"\n\n\"That's glorious news,\" sez I. \"Let's get all the boys together an'\nhold a celebration.\"\n\n\"I guess a freak guinea-pig's as worthy o' bein' commented on as airy\nother kind of freak,\" sez Spider, stridin' off to the corral, purty\nwell pouted up.\n\nHe hadn't more 'n reached it before an idee reached me, an' I ran\nafter him. \"What is the' freakish about this guinea-pig, Spider?\" sez\nI.\n\n\"He's got a tail,\" snapped Spider.\n\n\"Ain't they all got tails?\" sez I.\n\n\"You know they ain't,\" he sez. \"You remember what that feller from the\nEast said last spring--if you hold up a guinea-pig by the tail, his\neyes fall out, an' then when we didn't believe it, he told us they\ndidn't have no tails. Pete sez that this guinea-pig is the only one in\nthe world what has a tail.\"\n\n\"Do you reckon he'd sell it?\"\n\n\"He'd sell the hair off his head,\" sez Spider.\n\n\"Well, you go back there an'--But say, has Pete got any others?\"\n\n\"He had ten when I left, an' no knowin' how many he's got by this\ntime. Pete sez 'at guinea-pigs is the prolificest things the' is,\" sez\nSpider.\n\n\"You buy three of 'em, Spider,\" sez I; \"a male one an' a female one,\nan' this here freak.\"\n\n\"What do I want with 'em?\" sez Spider.\n\n\"I'll pay half, an' show you how to make money out of 'em,\" sez I.\n\n\"I don't want to tinker with no such cattle as them,\" sez Spider.\n\n\"You get a fresh pony, an' it won't take you no time at all,\" sez I.\n\nSo Spider got the pony an' went off grumblin'. When he brought 'em\nback he had 'em in a small box an' they certainly was curious lookin'\ninsects. \"I paid four bits apiece for the male an' the female,\" sez\nSpider, \"an' twenty-five real dollars for the freak.\"\n\n\"If that's the way prices run,\" sez I, \"it ain't no wonder that\nguinea-pigs what are ambitious to be popular, are willin' to give up\nthe luxury o' tails.\"\n\n\"Now then, what in thunder are we goin' to do with 'em?\" sez Spider.\n\n\"Get a fresh pony,\" sez I, \"an' we'll go on over to Boggs.\"\n\n\"You go to the equator!\" yells Spider. \"I ain't had no sleep for a\nweek.\"\n\n\"Sleep,\" sez I, \"what's the use o' botherin' about sleep? You keep on\nlosin' your strength this way, an' in about a year they'll be\ntrundlin' you around in a baby cart. All right then, you stay home an'\nbe company for the freak. We'll hide him up in the attic so the rats\ncan't get him.\"\n\n\"Oh I could stand it to go without sleep, if I saw any sense in it,\"\nsez Spider; \"but hanged if I'm goin' to ride my bones through my skin\njust to please you.\"\n\n\"Suit yourself,\" sez I. \"We'll put the freak in the tin cake-box an'\npunch a few holes in it to give him air. I'll do that while you're\nmakin' up your mind about goin' along to Boggs.\"\n\n\"What you goin' to do with the male an' the female?\" sez Spider as I\nstarted away.\n\n\"I'm goin' to sell 'em to Eugene,\" I calls back over my shoulder, an'\nthen I knew I'd have company.\n\n\"I thought you was goin' to Boggs,\" sez Spider as soon as we had\nsettled into a travelin' trot. I allus find that I get along easier\nwith people if I just leave 'em one or two items to puzzle over.\n\n\"Webb Station is closer,\" sez I; \"an' if this deal causes any hard\nfeelin' it will be just as well not to be mixed up in it ourselves.\"\n\n\"I thought you was goin' to sell these to Eugene?\" sez Spider.\n\n\"If you'd just go to sleep, Spider,\" sez I, \"it would save your brain\nthe trouble o' thinkin' up a lot o' thoughts which ain't no use\nanyhow. I'm goin' to let Shorty take 'em over this evenin' an' sell\n'em to Eugene.\"\n\n\"How do you know he wants 'em?\"\n\n\"'Cause I know Eugene,\" sez I. \"I'll fix up Shorty's tale for him.\"\n\nWell, we explained to Shorty the bettin' principle of guinea-pigs, an'\ngave him the pigs, tellin' him he could have all he won from Eugene on\nthe first bet; but to then sell 'em to Eugene without lettin' any o'\nthe other fellers know anything about it, an' to make Eugene think\nthat he had picked 'em up from a train passenger, not from us.\n\nShorty said that he'd go over that afternoon as soon as the passenger\nhad gone--Shorty was the telegraph operator--so Spider an' I came\nback, he sleepin' all the way.\n\n\"Where do we come in on this deal?\" sez Spider next day.\n\n\"We'll give Eugene a chance to cut their hair a new way, an' then\nwe'll go over to Boggs an' line things up.\"\n\n\"I'm beginnin' to see how it could be worked out,\" sez Spider,\ngrinnin'.\n\nIn about a week we went over to Boggs, an' found the town purty well\ndeserted. We dropped into ol' man Dort's to compliment Columbus some\nan' sympathize with Ben Butler a little, while tryin' to hear if\nEugene had made his play yet. The ol' man was gloatin' over the fact\nthat Eugene wasn't havin' much trade, but he didn't mention anything\nabout guinea-pigs.\n\n\"You don't seem rushed, yourself,\" sez I.\n\n\"Course I ain't,\" he flares back. \"Most o' the fellers are still\nroundin' up, an' the rest are out huntin' for Red Erickson.\"\n\n\"Red been gettin' thoughtless again?\" sez I. Red Erickson was a big\nDane who had the habit o' runnin off stock an' shootin' any one who\ndisagreed with him.\n\nThe ol' man merely pointed to a paper pinned up on the wall offerin'\nfifteen hundred dollars for Red, dead or alive. He hadn't been\noperatin' on Diamond Dot stuff, so we hadn't paid much heed to him.\n\nWe strolled on over to Eugene's an' found him sittin' down an' talkin'\nabout the peculiar custom o' guinea-pigs; so we knew that he had\nswallered the bait; but he didn't offer to bet with us.\n\nThen we went back an' asked ol' man Dort if he believed that a\nguinea-pig's eyes would fall out if he was held up by the tail.\n\n\"It's all rot!\" sez the ol' man, indignant. \"Any one who sez such\nnonsense never studied the way eyes is fastened in. The tail ain't got\nnothin' to do with it.\"\n\n\"What kind o' tails has guinea-pigs got?\" sez I.\n\n\"Why they got--?\" sez the ol' man, an' then stopped an' looked blank.\n\"What kind o' tails have they got?\"\n\n\"They haven't got any,\" sez I. \"Now listen; would you be willin' to\nrisk a little money to even up with Eugene?\"\n\n\"I'd risk every thing I got, down to my very hide,\" sez the ol' man,\nearnest to a degree.\n\n\"Well, then, you play careful an' we'll provide you with the cards,\"\nsez I. \"Eugene has some guinea-pigs, an' he is plannin' to string you\non a bet. You come right along just as though you was as ignorant as\nyou look, have a day fixed to decide the bet, let us know, an' for the\nsmall sum of fifty dollars we'll provide you with a guinea-pig which\nhas a tail.\"\n\n\"I'll make a pauper out of him,\" sez the ol' man. \"I haven't had a\nchance to get a bet on Columbus since I owned him.\"\n\n\"You just land Eugene,\" sez I, \"an' that'll be sport enough for one\nwhile.\"\n\n\"I got shaved twice to-day,\" sez the ol' man feelin' his chin, \"'cause\nwe got into a discussion about comets; but I reckon I can stand\nanother to-morrow.\"\n\nThe next day the old man asked Eugene what all kind o' game grew in\nAfrica. \"Elephants, hippopotamusses an' guinea-pigs,\" sez Eugene.\n\n\"Guinea-pigs?\" sez the ol' man.\n\n\"Yes, they're the most curious animals the' is in existence,\" sez\nEugene.\n\n\"How big are they?\" asked ol' man Dort. He hadn't an idea in the\nworld, an' was beginnin' to think that if they sized up with elephants\nan' hippopotamusses, he didn't want to have to lift one by the tail to\nwin his bet.\n\n\"They ain't any bigger 'n young rabbits,\" sez Eugene, stroppin' his\nrazor; \"but the curious part of 'em is that if you hold up one by the\ntail, his eyes'll drop out.\"\n\n\"I'll bet a hundred dollars they wouldn't do it,\" sez the ol' man.\n\n\"That's a safe enough bet,\" sez Eugene, calm an' easy. \"They're worth\nall the way up to five hundred dollars a pair, an' it ain't likely\nthat a man would invest that amount in something, just to win a\nhundred-dollar bet.\"\n\nThey sparred back an' forth for a couple o' days until finally Eugene\nbet nine hundred in cash--all he had in the world--an' his shop an'\nfixin's, again' eleven hundred dollars, that the old man couldn't lift\na guinea-pig by the tail without his eyes fallin' out. If the ol' man\ndidn't lift one by the tail, he lost the bet. They set the date for a\nweek ahead, an' the ol' man bet Eugene three hundred dollars that he'd\nwin the bet, takin' Eugene's promissory agreement for his end of it.\n\nWe brought in the freak the day before the contest an' the ol' man's\neyes lit up when he see the tail. It wasn't much of a tail at that;\nbut it was a sure enough tail an' plenty long enough to lift him by,\nan' strong enough too, an' the' was regular bones in it, just like any\ntail.\n\nThe' was only a fair sized crowd of us on hand to see the test; but\nEugene went through all the preliminaries, an' then took the cover off\nhis box an' pointed to the guinea-pigs. He had shaved the parts of 'em\nwhere tails naturally belong, an' when the boys see that they didn't\nhave no tails, they howled with laughter an' began to hoot ol' man\nDort; an' Eugene confided to 'em the plans he had for spendin' the\nmoney he'd won.\n\nOl' man Dort, he walked calmly up to the box, examined the\nguinea-pigs, an' sez: \"These here is not the full-blooded guinea-pigs.\nThe full-blooded ones live in a mountainous? country an' use their\ntails to steer with when they jump from rock to rock; while this kind\nlive in swamps an' the young alligators keep on eatin' off their tails\nuntil they don't have any. I'll go get a thoroughbred an' do my\nliftin' on him.\"\n\nWell this set 'em back a good ways; an' as the ol' man was walkin' off\nto get his own speciment, a good many bets was put up, but Eugene\ndidn't take any.\n\nPurty soon, back come the ol' man; an' hanged if he hadn't clipped the\nhair off o' his one's tail too. He reached in his hand an' stroked the\nlong-faced little duffer, an' sez: \"Gently, George the Third, gently.\"\nThen he put on an anxious look an' picked up the guinea-pig by the\ntail, holdin' his other hand underneath to catch any eyes what\nhappened to spill out. They didn't none drop out, an' the crowd give a\ncheer; but Eugene was all in.\n\nHe was a bad loser was Eugene, an' he didn't join in the festivities\nany. He just took up his two guineas an' went back to his shop, while\nthe rest of us celebrated a few. After a time me an' Spider went to\nconsole with him a little. He was so infernally down in the mouth that\nI began to get a little conscience-struck. Eugene said he had been\nsavin' up his money to pay off the mortgage on his birthplace; an' he\nmade a purty sad story out of it. Fact was, that he made so sad a\nstory out of it that I decided to get him back his tools and give him\na new start.\n\n\n\n\nCHAPTER SIXTEEN\n\nTHEMIS IN THE ROCKIES\n\n\n\"How much money you got, Spider?\" I sez.\n\n\"I reckon I got sixty dollars,\" sez Spider.\n\n\"I don't mean just what you got with ya, I mean how much cash do you\npossess in the world.\"\n\n\"I suppose I could raise a hundred an' fifteen,\" sez Spider, after\nthinkin' a while. \"What do you want to know for?\"\n\n\"We got to give Eugene a start,\" sez I.\n\nSpider looked at me until he saw I was in earnest, an' then he talked\nout loud. \"What's the matter with you?\" he yells. \"We haven't adopted\nEugene, have we? Why-for do we have to give him a start? Didn't he\nlose at his own game. Great Snakes! You make me tired!\"\n\n\"That was a low-down trick we played,\" sez I.\n\n\"It wasn't no lower down 'n him ringin' in a woodchuck on the old man;\nand all we did it for was to square things up.\"\n\n\"Yes,\" sez I; \"but it took us some several years to square it up, and\nI don't intend to have Eugene's moanful voice surgin' through my ears\nuntil I'm able to think up a come-back for him. I'm goin' to give him\na start, and if you don't feel like riskin' your money, I'll do it\nalone.\"\n\n\"Do you mean 'at you're just goin' to pay over the price of his tools,\nan' let it go at that?\" sez Spider.\n\n\"That wouldn't be any fun,\" sez I. \"I'm goin' to get the tools; but I\nintend to get 'em for as little expense as possible, and if I can have\na little fun out of it, I don't intend to pass it up.\"\n\nSpider studied it over a while. \"Well, I'll risk fifty,\" he sez after\na bit; so we went back to Eugene's.\n\n\"Would you be willin' to do a stunt to get back your tools?\" sez I.\n\nHe raised a pair o' weepy eyes to me an' sez: \"Aw, the' ain't no show.\nI've a good mind to kill myself.\"\n\n\"Please don't do that,\" sez Spider, who never could stand a bad loser.\n\"When you lose your money, you allus stand a chance to win more money;\nbut when you lose your life, why, the' ain't nothin' left except to go\nup an' find out what reward it earned for you.\"\n\n\"Aw hell,\" muttered Eugene.\n\n\"Ye-es,\" agreed Spider, talkin' through his nose, like a missionary\npreacher, \"I reckon that is about what you'd draw, if you was to cash\nin now; but if you stick around an' do your duty, you run the risk o'\nhavin' better luck later on.\"\n\nAfter Spider had insulted Eugene until he began to sass back a little,\nI broke in and sez that if Eugene will agree to do what I tell him,\nI'll agree to get him back his outfit; so then he wants to know what I\nhave in mind.\n\n\"Are you willin' to disguise yourself as a genuwine mountain trapper?\"\nsez I.\n\nWhen I sez this, Spider exploded a laugh which would 'a' hurt the\nfeelin's of a sheep, and Eugene tied into us as wordy as a fox\nterrier; but I soothed him down an' told him I was in earnest. \"I'm\nwillin' to do most anything to get my tools back,\" sez Eugene; \"but I\ndon't see how I can make myself look like a genuwine trapper.\"\n\n\"Have you got any false wigs and beards?\" sez I.\n\n\"No, I haven't,\" sez he; \"but I saved up the stuff I reaped off o' ol'\nman Dort, and I reckon I could make some.\"\n\n\"The very thing!\" sez I. \"You fix up a rig that'll make you look to be\na hundred years old; and we'll hunt up clothes for ya. All you'll have\nto do will be to guide a green Eastener out to shoot a bear, and we'll\nhave the bear and everything ready for ya.\"\n\n\"No, ya don't,\" sez Eugene. \"I don't fool around no bears.\"\n\n\"I thought you was tired o' life,\" sez Spider.\n\n\"Well, I'm not so tired of it that I'm willin' to have it squeezed out\no' me by a bear,\" sez Eugene.\n\n\"This won't be a real bear,\" sez I; \"and anyhow, they'll be a ravine\nbetween you and it. You claimed once to be a show actor, and all\nyou'll have to do will be to pertend 'at you're actin'.\"\n\n\"I once was a genuwine amateur actor,\" sez Eugene, \"and if you'll make\nit clear to me that there ain't no danger, I'll take the job.\"\n\nThen I explained just what he had to do; and after this me an' Spider,\nwho was now keen for the outcome, went around to dicker with ol' man\nDort. He was bumpin' around among the clouds, so we didn't have any\ntrouble in buyin' back Eugene's stuff on time. When I asked him what\nhe'd charge for Columbus, the woodchuck, he gave a snort, and said\nhe'd throw him in for good measure; so I told him to just keep him out\no' sight for a few days, and we started back to Eugene's.\n\n\"What do you want with that dog-gone woodchuck?\" asked Spider.\n\n\"I want him to take the part of a grizzly bear,\" sez I.\n\nSpider stopped an' looked at me. \"This is goin' too far,\" sez he.\n\"It's bad enough to try to fool some one into believin' 'at Eugene's a\ngenuwine trapper; but you couldn't make a rag doll believe 'at\nColumbus was a grizzly bear.\"\n\n\"You go borrow that squaw dress from Ike Spargle, an' then we'll see\nhow much like a trapper Eugene'll look,\" sez I.\n\nI went on an' found 'at Eugene had done a master job o' wig makin',\neven fixin' false eyebrows, an' when he put on ol' man Dort's\nhair-crop he locked older 'n the human race. As soon as Spider came in\nwith the squaw dress, we put it on Eugene; and while he didn't look\nlike anything I'd ever seen before, he looked more like the first man\n'at ever started trappin' than like anything else, an' Spider Kelley\nnearly had a convulsion.\n\nWe bunked with Eugene that night; but he kept us awake bemoanin' his\ncruel fate until Spider threatened to drown him head first in a bucket\no' water and after that we had a little go at slumberin'. I routed 'em\nout about two an' drilled 'em up to the high ground above Spear Crick,\nwhere we waited until sun-up. Eugene was wearin' his trapper riggin',\nand in the starlight, he sure was a ghastly sight.\n\nJust across from us on the other side o' the crick was Sholte's Knoll,\nand when the sun rose, I lined us up to be just in a direct line with\nit across the knoll. Both Eugene, and Spider bothered me with\nquestions and discouragin' kicks; but I felt purty sure my scheme\nwould work, and only told 'em what was really for their good.\n\nThe crick ran south in a gorge, and just below us it ran into Rock\nRiver, which came from the east and made a sharp turn to the south\njust where Spear Crick ran into it. After the sun was up, we climbed\ndown a circlin' trail until we came to Rock River. Eugene refused to\ntry to ford it; but Spider and I went across and up to Ivan's Knoll.\nRock River was bigger than Spear Crick, and Ivan's Knoll was bigger\nthan Sholte's Knoll; but not one tenderfoot in a million could have\ntold 'em apart, and Spider got gleeful at the plan--except that he\nkept at me to know who I was tryin' to land. Back of Ivan's Knoll was\na round hole about ten feet across, called the Bottomless Pit, because\nthe' was no bottom to it. After examinin' this place, we went on and\ncrossed Rock River again until we came out at Sholte's Knoll across\nfrom where the shootin' was to be done.\n\n\"What you are to do, Spider,\" sez I, \"is to be at this place before\ndawn with Columbus tied by a stout cord. Tie him to the rock at the\nsouth end of the knoll by a weak cord, then pass your stout cord up\nover that jag o' rock at the top, and just as soon as the sun hits the\nknoll, pull hard enough to break the weak cord, lead him gently up the\n<DW72> until he has been shot at several times, then--\"\n\n\"Is Eugene, that genuwine, ancient trapper goin' to do the shootin'?\"\ninterrupted Spider.\n\n\"He is not,\" sez I. \"If Columbus gets shot, all you'll have to do will\nbe to wind around to Boggs and meet me there. If he don't get shot,\nyou can either turn him adrift, kill him yourself, or pack him back to\nol' man Dort's, accordin' to the dictates o' your own conscience. I'll\nbring the party 'at does the shootin' up to Ivan's Knoll, an' make him\nthink the bear has fallen down the Bottomless Pit after he was shot.\"\n\n\"Happy,\" sez Spider, \"hanged if I believe it'll go through; and I\nwon't be a sucker unless you tell me who is to do the shootin'.\"\n\n\"Horace,\" sez I, \"Horace Walpole Bradford.\"\n\nSpider's face changed expression a half dozen times in two moments;\nbut he didn't have any more kicks; so we went back to Eugene, and took\nhim up to a deserted cabin, where he was to stay until needed. I left\nhim and Spider to fix up the cabin, while I went back to the Dot to\nfix up Horace. Horace had a lot o' money; but it did go again' me to\nmake him pay for Eugene's outfit by puttin' up a practical joke on\nhim. Still, I felt called upon to square it up with Eugene, and this\nseemed the fairest way.\n\nWhen I reached the Dot, Horace came forth to meet me; and he was so\nglad to see me 'at I purt' nigh gave up the scheme; but I had gone too\nfar to back out now, so I acted cool, and cut him short with my\nanswers.\n\nAfter supper I got Tank started on bear. He saw I had something up my\nsleeve, so he talked bear until Horace's mouth began to water. \"I'd\ngive a hundred dollars, just to get a shot at a bear,\" sez Horace.\n\n\"This ain't the time o' the year to hunt bear,\" sez I. \"Food's so\ncommon at this season that a bear spends most of his time loafin'; and\nit's hard to get sight o' one. Course, if you was to go to a\nprofessional hunter, he'd know where bears were spendin' their\nvacation; but it might take a month for one of us to root one out.\"\n\n\"Do you know of any professional hunters?\" sez he.\n\nI didn't say nothin', and Tank told of some he knew several hundred\nmiles off. After Tank had talked himself out, I mentioned careless\nlike that old Pierre La Blanc was livin' less 'n twenty miles away;\nbut that I doubted if he'd take a bear-huntin' job. I went on to state\nthat he had money saved up, and it would take a sight o' coin to tempt\nhim.\n\n\"I'd give five hundred dollars for a shot at a real grizzly,\" sez\nHorace.\n\n\"Did you ever use a rifle?\" sez I.\n\n\"Ask Tank,\" sez Horace.\n\nTank told about Horace havin' borrowed ol' Cast Steel's\nforty-five-seventy, and that he had learned to hit a mark with it in\nable shape. Before we turned in that night, I had let Horace tease me\ninto takin' him over to Pierre's next day.\n\nWe reached the old cabin next afternoon, and found it lookin' purty\ncomfortable. Eugene had soiled his hands and what part of his face\nshowed; and he certainly did look outlandish. He could act some, I'll\nsay that for him; and he pertended so natural that it took Tank a half\nhour to tell who he was. He didn't talk much, but when he did he used\nbroken French, and he made a contract with Horace to get the five\nhundred as soon as he had showed him the bear, Tank to hold the check.\n\nEugene couldn't get food through his whiskers; so he said most of his\nteeth were gone, and et his supper in private. After supper, I stole\ndown the gulch and found Spider waitin'. He promised to be on hand the\nnext mornin' and we turned in early.\n\nNext mornin' we started at three, and took up our place at the mark I\nhad made across from Sholte's Knoll. Horace thought it perfectly\nwonderful that the old trapper would know exactly where a grizzly bear\nwould be at sun-up; and he chattered constant in a hushed voice. We\ntold him it was a full quarter across to the knoll, and he had a\nregular ecstasy about how deceivin' the atmosphere was--which was rank\nlibel, the atmosphere bein' about the least deceivin' member o' that\nparty.\n\nPresently, I caught the smell o' dawn, and I told Horace to keep his\neyes glued on Chimney Peak, a little over twenty miles to the west. He\ndid so, and in about five minutes, a gob o' rich crimson splashed on\nit, rippled down the sides, and poured along the foothills at the\nbottom. Horace gave a gasp. You don't see such a dawn as that with\nyour eyes alone; you see it with somethin' inside your bosom; and when\nI saw the gleam in Horace's eyes, it made me feel ashamed of what I\nwas up to; but I couldn't stop just for this; so I nudged Eugene, and\nthat hoary old trapper growled out to Horace to watch the knoll, or\nhe'd miss his chance.\n\nHorace was surprised to see the east still in a black shadow. He\nstarted to speak words about it, but just then the sun, lookin' like\nan acre of red fire, jumped up from behind Sholte's Knoll like a\nsacred jack-rabbit.\n\nThe knoll was consid'able higher than us, and just as the sun was\nhalf-circle behind it, a gigantic form started to walk across it from\nsouth to north. I knew, positive, that this was Columbus the\nwoodchuck; but it was just all I could do to believe it, myself, and\nHorace thought it was the biggest silver-tip in creation. I didn't\nthink the woodchuck ran much risk of gettin' shot; but Horace didn't\nlose his nerve a particle. He banged away, Columbus gave a lurch, took\na snap at his side, and rolled out o' sight behind the knoll, as\nnatural as a fried egg.\n\nHorace jumped up and down, hugged himself, slapped us on the back, and\nalmost knocked the aged trapper's fur off; but if he had, I doubt if\nhe would have noticed it, he was so eager to get to his first bear.\n\nWe wound down the path, and he complained about it bein' so much\nfarther 'n he had expected; but I spoke a few words about the\natmosphere, and he was soothed. When we struck Rock River, he was\nsurprised to see how much wider it was than it looked from where he'd\nshot; but he didn't falter none about goin' in; while I purt' nigh had\nto twist off the seasoned trapper's arm before he'd get his feet wet.\nThe water was purty high, and Tank and I had our hands full gettin'\n'em across.\n\nWe climbed the trail on the other side to Ivan's Knoll. This was about\na mile south o' Sholte's Knoll, and naturally I didn't expect to find\nany game on the other side of it; so you can judge my feelin's when we\ngot around to the other side, and saw that woodchuck's carcass, lyin'\nflat on its back with its front feet folded across a piece o' paper.\n\nHorace saw it, too; but he wasn't interested at first, and dove all\nabout, lookin' for his bear. He was plumb wild; but finally he picked\nup the piece o' paper, and read what was wrote on it in scrawly\nletters, which I knew to be the work o' Spider Kelley: \"Before I was\nshot I was a grizzly bar but it made me feel so small to get shot by a\ntender-foot that I have shrank to what you see befor you.\"\n\nThat confounded Kelley hadn't been able to resist workin' the joke\nback on me; so he had toted Columbus down from Sholte's Knoll, and\nthen skipped. I knew I wouldn't see him for some time--but I also knew\nI wouldn't forget what was comin' to him when I did.\n\nHorace read the note through in silence, then he looked at the remains\nof the woodchuck, then he read the note again, and his face got like a\nsunset. He read the note once more, and then he leaped through the air\nfor that veteran trapper, and grabbed him by the beard. The beard and\nwig came off in his hands, and Eugene started to flee, with Horace a\nclose second, kickin' the seat o' that squaw dress at every jump.\nHorace was in able shape, and Eugene was flimsy; so when he tripped\nand rolled over, Horace got him by the ears, and proceeded to beat his\nhead on a stone, the way Tank had told about doin' to the unobligin'\nold miner.\n\nI pulled Horace off to save Eugene's life, and then Horace pulled out\na gun and tried to take my life. It took us two solid hours to cool\nHorace down below the boilin' point; and then he started off alone\nwith his lips set and his eyebrows pulled down to the bridge of his\nnose. I liked him better 'n ever. He was as game as they made 'em, and\nhad even forgot the check 'at ol' Tank Williams was still holdin'; but\nI was honestly worried about Eugene.\n\nPart of it may have been due to havin' his head beat mellow on a\nstone; but still he allus did lack sand when he was losin', and now he\nsat tuggin' at his real hair an' swearin' he was ruined, and would\ntake his own life the first chance he had. It was partly my fault; so\nI made Tank help me tote back Eugene's needin's from the deserted\ncabin to his shop, Eugene goin' along in a stupor and repeatin' to us\nconstant that he intended to drink his own heart's blood.\n\nI sent Tank back to the Dot to see what he could do toward pacifyin'\nHorace, and then I returned the squaw dress to Ike Spargle. He broke\ninto a side-split when I stepped into his place, and fairly deluged me\nwith liquor; but I wasn't in no mood for it. Ike told me 'at Spider\nhad gone out to the Dot to notify that he had quit temporary; and then\nhe was goin' out to hunt down Red Erickson for the bounty. Ike was\nequally willin' to talk about bears or Red Erickson; but I wasn't\nconversational, so I went back to Eugene's.\n\nHe had his door locked, and at first refused me admittance; but\nfinally he let me in, and I told him I would let him have his outfit\non time. He wouldn't scarcely listen to me; so the best I could do was\nto get his promise that he wouldn't slay himself inside the house, as\nthe boys were superstitious again' it, and would burn it down. As it\nwas again' my credit at ol' man Dort's, I felt more agreeable toward\npayin' for a standin' house, than for just the ashes of one.\n\n\"When I'm gone, Happy,\" sez Eugene, \"I want you to send my watch back\nto Sommersville, Connecticut. That's all I ask of ya. You've been as\nnear a friend to me as any one in this ungodly community has, and I\ndon't bear ya no ill will. If I could just have paid off that\nmortgage--\"\n\nI shook hands with him and went outside, where I settled myself\ncomfortable and made ready to keep watch on him until he started to\ndrink. I felt sure that if he'd once get to elevatin' a bottle, it\nwould take his mind off suicide; but he paced up and down inside his\nroom until I was purt' nigh out o' my own head.\n\nIt must have been nine in the evenin' when he stole out his side door\nwith a forty-five under his coat; and started up the ravine which\nopens west o' town, and I follered like a coyote.\n\nHe went up it about a mile, an' then he stopped an' I flattened out\nan' crept closer an' closer. I knew he would make a few remarks first,\neven though he was alone, an' I judged I could wriggle up close enough\nto grab him in the act.\n\nHe fished out his gun, an' I see that he didn't savvy the use of it,\nwhich put a little uncertainty into my end o' the game.\n\n\"Farewell, cruel world,\" he muttered mournfully, usin' his gun to\ngesture with. \"Farewell, sweet dreams of childhood; farewell ambition\nan' love an' dear tyranic duty; farewell moon an' stars an' gentle\nbreezes, farewell--\"\n\nEugene would probably have gone on sayin' farewell to each particular\nthing in the world until he talked himself to sleep, but just then a\npebble slipped from the side o' the ravine and rolled to his feet, and\nhe stopped with a jerk an' listened. Then he straightened himself an'\nsez in a determined tone: \"Nobody can't prevent me. I shall end it\nnow.\"\n\nBefore I could move, he placed the muzzle to his forehead an' fired,\nrollin' over on his back. I heard a sort of cough, like when a man\nhits his best with an ax, an' somethin' came plumpin' down the ravine\nlike an avalanche.\n\nI rushed up, lit a match, an' there on his back was Eugene, a small\nred welt on his forehead, but looking calm and satisfied, while almost\non top of him lay a man in a heap. I straightened him out, lit another\nmatch, an' looked at the stranger. His hair was flamin' red an' you\ncould have tied his red mustaches around the back of his neck. He was\nshot through the forehead an' plumb dead.\n\nI saw how it was in a flash: Eugene had almost missed himself, but had\nshot Red Erickson, who had been hidin' up the side of the ravine\nbehind him. I slipped Red's empty gun into his hand, emptied Eugene's\ngun; an' then I tore for town, gathered up the boys an' told 'em that\nEugene had gone up the ravine bent on mischief. We got a lantern and\nhurried up the ravine where Eugene was just comin' back to genuwine\nconsciousness again.\n\nHe sat there with his head in his hands tryin' to cheer himself with\nsome o' the mournfullest moanin' ever I heard. I held the lantern to\nRed's face a moment an' bawled out: \"Boys, this is Red Erickson! Him\nan' Eugene has been duelin', an' they have killed each other.\"\n\nThis gave Eugene his cue--an' a cue was all Eugene ever needed. He\npulled himself together, took plenty o' time to get the lay o' the\nland; an' then he gave us a tale o' that fight which laid over\nanything I ever heard in that line.\n\nWe carried 'em back to town, an' Eugene was a hero for true. He got\nthe reward all right, paid off his debts, an' kept addin' details to\nthat fight until it was enough to keep a feller awake nights. His\nreputation picked up right along until even ol' man Dort had to admit\nthe' was more to Eugene than he had allowed.\n\nNext day when I got back to the Diamond Dot, I found Horace all packed\nup for leavin'; and it made me feel mournful to the bones o' my soul.\nI didn't know how much I thought of him until he started to pull out;\nand I felt so ashamed at what I had done, that I offered to let him\nkick me all about the place if he'd just forget about it and stick\nalong.\n\nBut Horace had a stiff neck, all right, and he wouldn't give in. Tank\nhad had all he could do to get Horace to take the check back; and now,\ntry as I would, I couldn't get him to stay. I drove over to the\nstation with him, and we had a long talk together. He was in a good\nhumor when he left, and I could see he was wishful to stay; but havin'\nmade up his mind, he stuck to it. He said he had had more fun while\nwith us than durin' all the procedure of his life; and that if we had\njust kept the joke among us Dotters, he wouldn't have felt so cut up\nabout it. I told him he had acted just right and that I had acted dead\nwrong, although it was him takin' Tank's word above mine which had\nfirst made me sore.\n\nThis was new light to him, and he softened up immediate. Fact was, we\ngot purt' nigh girlish before the train pulled out with him wavin' his\nhandkerchief from the back porch.\n\nI still feel some shame about this episode; and if any o' you fellers\nask any more questions to lead me into tellin' of my own silly pranks,\nwhy, I'll drive you off the place, and then get my lips sewed shut.\n\n\n\n\nCHAPTER SEVENTEEN\n\nKIT MURRAY\n\n\nHorace had left, I felt purty lonely for a while. It's hard for me to\nlook back and keep things in regular order; because the different\nlines cross each other and get mixed up. Always, little Barbie's\naffairs came first with me; but I reckon most of you have heard her\nstory, so I'm keepin' shy of it this time. First of all there was my\ninnermost life, which would have been mostly mine no matter where I'd\ngone; then there was the part of my life which touched Barbie's, and\nthis was the best and the highest part of it; and then there was the\npart which touched Friar Tuck an' a lot of others, each one of which\nhelped to make me what I am; but back of it all was my work; so it's\nnot strange if I find it hard to stick to the trail of a story.\n\nAnyway, it was while I was feelin' lonesome about Horace leavin' that\nthe Friar first began to use me as a trump card, and called on me for\nwhatever he happened to want done. I was mighty fond o' bein' with the\nFriar; so I lent myself to him whenever I could, and we got mighty\nwell acquainted. He loved fun of a quiet kind; but the' was allus a\nsadness in his eyes which toned down my natural devilment and softened\nme. The' was lots o' things I used to enjoy doin', which I just\ncouldn't do after havin' been with the Friar a spell, until I had give\nmyself a good shakin', like a dog comin' up out o' water.\n\nFor several quiet years about this time, I used to act as scout for\nhim, now and again, goin' ahead to round up a bunch when he had time\nto give 'em a preachin'; or goin' after him when some one who couldn't\nafford a doctor was took sick. We talked about purt' nigh everything,\nexcept that some way, we didn't talk much about women; so I was never\nable to pump his own story out of him, though he knew exactly how I\nfelt toward Barbie, long before I did myself.\n\nDurin' these years, the Friar tried his best to get on terms with the\nTy Jones crowd; but they refused to get friendly, and the more he did\nto make things better in the territory, the more they hated him.\n\nIt was right after the spring round-up that I first heard the Friar's\nname mixed up with a woman. This allus makes me madder 'n about\nanything else. When a man and a woman sin, why, it's bad enough, and\nI'm not upholdin' it; but still in a way it's natural, the same as a\nwolf killin' a calf. It's the cow-puncher's business to kill the wolf\nif he can, and he ought to do it as prompt as possible. This is all\nright; but gossip and scandal is never all right.\n\nGossip and scandal is like supposin' the wolf had only wounded the\ncalf a little, and a posse would gather and tie the two of 'em\ntogether, the wolf and the wounded calf; and take 'em into the center\nsquare of a town and keep 'em tied there for all to see until they had\nstarved to death; and then to keep on stirrin' up the carrion day\nafter day as long as a shred of it remained.\n\nThe Friar was allus a great one to be talkin' about the power of\nhabits. He said that if folks would just get into the habit of lookin'\nfor sunshiny days, an' smilin' faces an' noble deeds, and such like,\nthat first thing they knew they'd think the whole world had changed\nfor the better; but instead o' this they got into the habit of lookin'\nfor evil, and as that was what they were on the watch for, o' course\nthey found it. He said it was like a cat watchin' for a mouse. The cat\nwould plant herself in front of the mouse hole and not do anything\nelse but just watch for the mouse. While she would be on guard, a king\nmight be assassinated, a city might fall in an earthquake, and a\nship-load o' people go down at sea; but if the mouse came out and the\ncat got it, she would amuse herself with it a while, eat it and then\ncurl up before the fire and purr about what a fine day it had been,\nall because she had got what she had been lookin' for; and the's a lot\nin this.\n\nNow, when I came to think it over, I hadn't heard the Friar express\nhimself very free on women. I had heard him say to allus treat 'em\nkind an' square, the good ones and the bad; but when ya come to ponder\nover this, it wasn't no-wise definite. Still I couldn't believe ill of\nhim; so I took a vacation an' started to hunt him up.\n\nThe feller who had told me didn't know much about it, but the feller\nwho had told him knew it all. When I found this feller, he was in the\nsame fix; and he sent me along to the one who had told him. They were\nall a lot alike in not knowin' it all; but I finally found out who the\ngirl was.\n\nShe was a girl named Kit Murray, and she allus had been a lively young\nthing with a purty face, an' could ride an' shoot like a man. She had\ntook part in a couple o' frontier-day exhibitions, and it had turned\nher head, and she had gone out with a show. When she had come back,\nshe had put on more airs 'n ever, and naturally the boys were some\nwild about her--though I hadn't seen her myself.\n\nNews o' this kind travels fast, and I heard buzzin' about it\neverywhere; but it was just like all other scandal. Most people, when\nthey gossip, believe an' tell the story which comes closest to what\nthey'd 'a' done if they'd had the same chance; and what I figured out\nto be true was, that Olaf the Swede and another Cross-brander by the\nname o' Bud Fisher had scrapped about the girl, Olaf near killin' the\nkid and the girl runnin' off to the Friar. Now, all the good deeds 'at\nthe Friar had done hadn't caused much talk; but this news spread like\nwild-fire; and a lot o' those he had helped the most turned again' him\nand said they wished they could find out where he was hidin'.\n\nI took it just the other way; I knew the Friar purty well, and what I\nfeared most was, that he wasn't hidin' at all, and that Olaf would\nfind him before I could give him warnin'. It was two weeks before I\nfound the Friar; but once I came upon Olaf, face to face, and we eyed\neach other purty close. This was the first time I ever noticed his\neyes. They were the queerest eyes I ever saw, a sort of blue; but a\ndeeper blue, a bluer blue 'n anything I had ever seen outside a\nflower. The's a flower on the benches in June just the color of his\neyes, a soft, velvety flower; but Olaf's eyes weren't soft and velvety\nthe day we met, and they gave me a queer, creepy feelin'. I hope I\ndidn't show it any; but I did feel relieved after I'd passed him.\n\nFinally I found the Friar, just as I might have expected--by the sound\nof his voice. I had got clear over into the Basin and was crossin'\nthrough Carter Pass when I heard his voice above me, singin' one of\nhis marchin' songs. I was mightily rejoiced to find him; but I had\nthat all out of my face by the time I had wound around up to him. He\nwas totin' a log on his shoulder, and struttin' along as jaunty as\nthough the whole earth was simply his backyard.\n\n\"Here,\" I growls to him, indignant, \"what do you mean by makin' such a\nnoise? Haven't you got a grain o' gumption!\"\n\nHe looked up at me with the surprise stickin' out from under his grin.\n\"Well, well, well!\" sez he. \"Who are you--the special officer for the\nprevention of noise?\"\n\n\"I ain't no special officer of anything,\" I answers; \"but the's people\nlookin' for you, and you ought to have sense enough to keep quiet.\"\n\n\"And I'm lookin' for people,\" sez he, grinnin' like a boy; \"and the\nbest way to find 'em is by makin' a noise. The' ain't any rules again'\nwalkin' on the grass up here, is there?\"\n\n\"Olaf the Swede is after you on account o' the gal,\" I blunted; \"and\nhe ain't no bluffer. He intends to do away with you for good and all;\nand you'd better be makin' your plans.\"\n\n\"Goin' to do away with me for good an' all,\" he repeats, smilin'.\n\"Well, Olaf the Swede is a gross materialist. The worst he can do will\nbe to tear off my wrapper and leave me free to find out a lot of\nthings I'm deeply interested in. Why, Happy, you're all worked up!\nYou've lost your philosophy, you've become a frettish old woman. What\nyou need is a right good scare to straighten you up again. This Olaf\nthe Swede is part of Ty Jones's outfit, isn't he?\"\n\n\"He is,\" I replied, shakin' my head in warnin', \"and the whole gang'll\nback him up in this.\"\n\n\"Good!\" sez the Friar, smackin' his hand. \"I've wanted an openin'\nwedge into that outfit ever since I came out here. Of a truth, the\nLord doth move in a mysterious way, his wonders to perform.\"\n\n\"Well, he certainly will have to perform some mysterious wonders to\nget you out of this scrape,\" I said. I was put out at the way he took\nit.\n\n\"Don't be irreverent, Happy,\" sez he, the joy-lights dancin' in his\neyes. \"We are all merely instruments, and why should an instrument\ntake it upon itself to question the way it is used. Where is this\nOlaf?\"\n\n\"I met him yesterday; and for all I know, he's been followin' me.\"\n\n\"Fine, fine!\" sez the Friar. \"Now, you go on back to the Diamond Dot,\nand I'll go back over your trail and save Olaf as much bother as\npossible.\"\n\n\"I'm goin' along with you,\" I sez.\n\n\"No,\" sez he.\n\n\"Yes,\" sez I.\n\n\"It'll make folks think 'at I'm afraid for my skin, and have you along\nfor protection,\" sez he, gettin' earnest.\n\n\"If you had good judgment, you would be afraid for your skin,\" sez I.\n\"I tell you that Olaf is after your blood. He's one o' the worst; he\nkills with his bare hands when he gets the chance.\"\n\n\"Fine, fine!\" sez the Friar again, his eyes glowin' joyous. \"I'd have\na right to defend myself with my hands, Happy. I would have a right to\ndo this, for the sake of Olaf, you see--to prevent him from risking\nhis own soul by committin' murder. This is a great chance for me,\nHappy; now, please, please, go on back like a good fellow.\"\n\nI was secretly tickled at the argument the Friar had put up for a\nchance at physical warfare--and a barehand fight between him and Olaf\nwould have been worth goin' a long way to see--but I was as obstinate\nas either of 'em; so I just said 'at I was goin' along.\n\n\"Well, you're not goin' with, me,\" sez the Friar, as pouty as a\nschoolboy. \"I'll not speak to ya, and I'll not have a thing to do with\nya\"; and he threw down his log and glared at me.\n\nI took a certain amount o' pride because the Friar lived up to his own\nstandards; but I also found a certain deep-rooted amusement in havin'\nhim slip out from under 'em for a spell and display a human\ndisposition which was purty much kindred to my own. \"What do you\npurpose doin' with that club, Friar?\" I asked, pointin' to the log he\nhad flung down.\n\nHe pulled in his glare and looked to be a little discomposed. \"Why\nI--I'm livin' in a cave I got back there.\"\n\n\"Are you dead set again' havin' a little company?\" sez I, slow an'\ninsinuatin', \"or are ya livin' alone?\"\n\nFirst off, he was inclined to be resentful, then he grinned,\nshouldered his log again, and said: \"Come and see.\"\n\nI follered him back into the hills until we came to a little park in\nwhich his ponies were grazin', and then I hobbled mine, cached my gear\nalongside his, and trailed after him again. His path turned a crag and\nthen skirted along the edge of a cliff as straight up and down as the\nreal truth. The path kept gettin' narrower, until every time the Friar\nturned a corner ahead of me, I expected to see him walkin' off in the\nair with the log still on his shoulder.\n\nPresently I turned a corner around which he had disappeared, and there\nwasn't a soul in sight. The ledge still led along the cliff; but it\nhad got thinner than a lawyer's excuse, and a worm couldn't have\nwalked along it without hangin' on. While I stood there puzzlin' about\nit, a hand reached out o' the side of the cliff, and the Friar's voice\nsaid mockingly: \"Take my hand, little one; and then shut your eyes for\nfear you might get dizzy.\"\n\nThen I saw a jag of rock stickin' out just above my head, I grabbed it\nwith my left hand, and swung around into what was the mouth of a cave.\nIt was nothin' but a crack about eighteen inches wide, and the far\nside was sunk in enough to keep it hid from where I was standin'. The\nFriar was standin' a few feet back in the entrance with his log\nleanin' up again' the side. \"I know not what other animals may have\nsought shelter here,\" he said, \"but for the past three years this has\nbeen my castle, and, Happy Hawkins,\"--here the Friar bowed\nlow--\"obstinate and unreasonable as you are, I offer you a hearty\nwelcome.\"\n\nThe Friar said this in fun, but the' was an undertone to it which\ntightened the laces around my heart consid'able. Well, that cave was a\nsure enough surprise; he had three or four pelts and a couple of Injun\nblankets on the floor, he had a couple o' barrels fixed to catch snow\nwater, he had some cookin' tools; and books! Say, he must have had as\nmany as a hundred books, all of 'em hard-shells, and lookin' so\nedicated an' officious that I had to take off my hat before I had\nnerve enough to begin readin' the titles.\n\nAfter I'd taken everything in, I sat down in an easy chair he'd made\nout o' saplin's and rawhide, and looked all about; but I couldn't see\nany signs of their bein' any other rooms to this cave; and then I\njumped square for the mark, and sez: \"Friar, the's a lot o' talk about\nyou havin' run off with Kit Murray. Now I want the straight of it.\"\n\nHis face went grave and a little hurt. \"It's strange,\" he said after a\ntime, \"how hard it is for a man to believe in his own guilt, and how\neasy for him to believe in the guilt of his neighbor. Have you had any\ndinner?\"\n\n\"Yes,\" sez I. \"I didn't know just where I was headin'; so I et three\ndifferent times this mornin' to make sure of havin' enough to run on\nin case of emergency.\"\n\n\"It's a fine thing to be an outdoor animal,\" sez the Friar, smilin'.\n\"Well then, I've made up my mind to take you to see Kit Murray.\"\n\nHe didn't waste any time askin' me not to talk about what was other\nfolks' affairs; he just went to the door, grabbed the jag of rock,\nswung around to the ledge, and I follered after.\n\nWe saddled up, rode down a windin' path 'at I'd never heard of before,\nand then rode up again until we came to a little clump o' swamp\nshrubbery, backed up again' the north face o' Mount Mizner. We\nfollered a twisty path through this and finally came out on an open\nspace in which stood a fair-sized cabin. He whistled a five-note call,\nand the door was opened by an old woman who was a stranger to me.\n\"Mother Shipley, this is Happy Hawkins,\" sez he. \"How's Kit?\"\n\nThe old woman gave me a gimlet look, and then her sharp features\nexpanded to a smile, and she bobbed her head. \"Kit's gettin' hard to\nmanage,\" sez she.\n\nWe went into the cabin, and found Kit with a bandage around her ankle,\nsittin' in a rockin' chair, and lookin' patiently disgusted. She was a\nfine-lookin' girl, with a fair streak of boy in her, and she had never\nhad enough practice at bein' an invalid to shine at it. Her face lit\nup at the Friar; but her gaze was mighty inquirin' when she turned it\nat me.\n\n\"You know Happy Hawkins, don't ya?\" sez the Friar. She nodded her\nhead, and he went on. \"Well, he's one o' the fellers you can trust, if\nyou trust him entire; but he's got such a bump of curiosity that if\nyou don't tell it all to him in the first place, he can't do no other\nwork until he finds it out on his own hook. He's my friend, and he'll\nbe your friend; so I want you to tell him just how things are, and\nthen he'll be under obligations to do whatever we want him to.\"\n\nSo Kit cut loose and told me her story. Her father, ol' Jim Murray,\nhad got crippled up about ten years before, and since then had become\na professional homesteader, nosin' out good places, an' then sellin'\nout to the big cattle outfits. He also made it his business to find\nways to drive off genuwine homesteaders; and in addition to this he\nwas a home tyrant and hard to live with. He allus had plenty o' money,\nbut was generally dead broke when it came to pleasant words an'\nsmiles--which was why Kit had gone off with the show.\n\nWhile she was away, she had married a low-grade cuss, who had misused\nher beyond endurance; so when he had skipped with another woman, she\nhad come back to the old man. She didn't want folks 'at knew her to\nfind out how bad hit she'd been; so she had tried to bluff it out; but\nthe young fellers kept fallin' in love with her and wantin' to marry\nher. She hadn't meant no harm; but she had played one again' the\nother, hopin' they'd soon have their feelin's hurt and let her alone.\nThis was a fool notion, but she had been honest in it.\n\nBud Fisher, the Texas kid in the Ty Jones outfit, had got daffy about\nher; and then one night at a dance she had shot some smiles into the\neyes of Olaf the Swede. She said he was such a glum-lookin' cuss she\nhad no idee he would take it serious; but he had stood lookin' into\nher eyes with his queer blue ones, until she had felt sort o' fainty;\nand from that on, he had declared war on all who glanced at her.\n\nBud Fisher thought it a fine joke for Olaf to fall in love, and he had\nteased him to the limit. This made a bad condition, and all through\nthe spring round-up, each had done as much dirt as possible to the\nother; but Ty was mighty strict about his men fightin' each other; so\nthey hadn't come to a clash.\n\nFinally the kid brags that he is goin' to elope with Kit; and then\nOlaf kicks off his hobbles an' starts to stampede. The kid was wise\nenough to vamoose; so Olaf rides down to ol' man Murray's, and reads\nthe riot act to him. Kit was hidin' in the back room and heard it all.\nHe told the old man that he would slaughter any one who eloped with\nKit or who had a hand in it; and then he had gone back to hunt the kid\nagain.\n\nThe ol' man turned in and gave Kit a complete harrowin' as soon as\nOlaf had left and she had told him pointedly that she'd eat dirt\nbefore she'd eat his food again; so she saddled her pony and started\nto ride without knowin' where. Her pony had slipped on Carter Pass and\nshe had sprained her ankle so bad she couldn't stand. Just at this\njunction, the Friar had come along, and had put her up on his horse\nand held her on with one arm about her, because the pain in her ankle\nmade her head light. On the way they came smack up again' the kid, and\nhe gave 'em a grin, and went out without askin' questions.\n\nHe went straight to Olaf, and told him that Kit had eloped with the\nFriar. The Friar had brought her up to Shipley's, they havin' been\nfriends of his in Colorado. They had a daughter livin' up in Billings,\nMontana; and as soon as her ankle could stand it, Kit was goin' up to\nlive with the daughter, she havin' three little children and a\nrailroad husband who was away from home more 'n half the time.\n\nThis was the whole o' the story; but you can easy see what a fine\nprospect it made for gossip, and also what a fine time a young imp\nlike Bud Fisher could have with a sober feller like Olaf. Olaf\nwouldn't have just grounds for makin' away with Bud for doin' nothin'\nexcept grin, so long as the Friar remained alive with the girl in his\nkeepin'. It was a neat little mess; and from what we found out\nafterwards, the kid was as irritatin' as a half-swallered cockle-burr.\n\nBig, silent fellers like Olaf are just like big, new boilers. A little\nleaky boiler fizzes away all the time, but when it comes to explode,\nit hasn't anything on hand to explode with; while a big, tight boiler,\nwhen it does go off, generally musses up the landscape consid'able;\nand when Olaf started to stampede he made more noise in a week 'n Bud\nFisher had in his whole life.\n\nWhen Kit had finished tellin' me the story, I shook hands with her,\nand said that while she hadn't used the best judgment the' was, she\nhad probably used the best she had; and that it was more the men's\nfault than hers, so she could count on me as far as I could travel.\nThen I went outside while the Friar and ol' Mother Shipley fixed up\nher ankle.\n\nThey all seemed pleased about the way it was healin', and after it was\ntied up, Kit stood on it and even took a few steps. It twisted her\nface a time or two at first; but after she'd gone across the room and\nback a few times, she said it felt better 'n it had for years. This\nmade us all laugh, 'cause fact was, she hadn't been housed in near up\nto the average of a sprained ankle. The Friar allowed 'at she'd be fit\nto travel day after the next; so it was planned to start in the\nevenin', and for both of us to go with her. Then we had an early\nsupper an' started home.\n\nOn the way, I complained about the foolish way in which Kit had acted,\nfor the sole purpose of drawin' the Friar out and gettin' his views on\nwomen. Nearly always when I got him started, I was able to pick up\nsome little sayin' which furnished me with more thought-food than his\nblocked-out sermons did.\n\n\"Of course Kit was foolish,\" he admitted; \"but what show has she ever\nhad? Her father never was fit to bring her up; and he didn't even do\nthe best he could. A woman has more vital strength than a man, because\nthe future of the race depends on her; but she also has more emotions,\nso 'at the wear an' tear is greater. Man, on the other hand, has more\nmuscle 'n woman, and more brutality. Foolin' man has been the best way\na woman had to fight for a good many centuries; and this was the way\npoor Kit tried to fight. The plain, simple truth generally works best;\nbut it takes wisdom to see this, and wisdom is seldom anything more\nthan the dregs o' folly. The' was no one to teach Kit wisdom; so she\nhas had to strain off her own folly; but she is a fine, brave girl,\nand I think she will profit by experience.\"\n\nNow this was a new thought to me, about wisdom bein' nothin' but the\ndregs o' folly; but it's a good tough thought, and I've had a heap o'\nchewin' on it since then; so I feel repaid in havin' took sides again'\nKit and lurin' the Friar into heavin' it at me.\n\nIt was dark when we reached his twistin' path along the ledge, and I\nstepped as cautious as a glow-worm in a powder-mill; but as soon as we\nhad our pipes an' the fire goin', I wouldn't have swapped seats with\nthe fattest king in the universe.\n\n\n\n\nCHAPTER EIGHTEEN\n\nTESTING THE FRIAR'S NERVE\n\n\nAs soon as we had eaten breakfast next mornin', the Friar sez: \"You,\nbein' one o' the earth animals, have never had much chance to see a\nview. Yesterday your curiosity was itchin' so 'at I doubt if you could\nhave told a mountain peak from a Mexican hat; but now that you have\ntemporarily suppressed your thirst for gossip, had a good sleep, and a\nbetter breakfast, drag yourself out to the front porch and take a\nbird's-eye view of the world.\"\n\nWell, it was worth it, it certainly was worth it! What he called the\nfront porch, was the ledge after it had flipped itself around the\njutting; and when a feller stood on it, he felt plenty enough like a\nbird to make it interestin'. The Big Horns ran across the top o' the\npicture about a hundred an' forty miles to the north, and gettin' all\nblended in with the clouds. On the other two sides were different\nmembers of the Shoshone family, most o' which I knew by sight from any\nangle; and down below was miles an' miles of country spread out like a\nmap, but more highly .\n\n\"Friar,\" I sez, \"you're a wealthy man.\"\n\nThis tickled him a lot, 'cause he was as proud o' that view as if he'd\npainted it. \"I am, Happy,\" he said, \"and I have yielded to a wealthy\nman's temptations. Any one who comes here will be welcome; but I own\nup, I have kept this place a secret to have it all to myself.\"\n\n\"A man like you needs some quiet place to consider in,\" sez I.\n\n\"Get thee behind me, Satan, get thee behind me,\" cried the Friar. \"I\nhave been on far too friendly terms with that excuse for many a long\nmonth. But I do enjoy this place; so I am going to let you help me lay\nin my winter's supply of wood, and then make you a joint member in\nfull standing.\"\n\nWe packed wood along that spider thread of a path all morning; and\nfinally I got so it didn't phaze me any more 'n it did him. He sang at\nhis work most of the time, and I joined in with him whenever I felt so\nmoved, though it did strike me 'at this was a funny way to keep a\nplace secret; and my idee is that he sang to ease his conscience by\nshowin' it that he wasn't sneakin' about his treasure.\n\nI remember him mighty plain as he walked before me on the ledge,\ntotin' a big log on his shoulder, and singin' the one 'at begins,\n\"Hark, my soul! It is the Lord!\" This was one he fair used to raise\nhimself in, and it seemed as if we two were climbin' right up on the\nair, plumb into the sky. When he'd let himself out this way, he'd fill\nme so full of a holy kind of devilment, that it would 'a' given me joy\nto have leaped off the cliff with him, and take chances on goin' up or\ndown.\n\nWe had about filled his wood place, and were goin' back after the last\nload when just as he swung around a corner, I saw his hand go up as\nthough warnin' me to stop; and I froze in my tracks. He hadn't been\nsingin' this trip, for a wonder; but the next moment I heard a sound\nwhich purt nigh jarred me off. It was a low, deep growl which I\ninstantly recognized as belongin' to Olaf the Swede. Olaf didn't talk\nwith much brogue, though when he got excited he had his own fashion\nfor hitchin' words together.\n\n\"Where is the girl?\" he asked with quiet fierceness, and for a space I\nwas sorry my parents hadn't been eagles. There wasn't room to fight\nout on that ledge, the Friar didn't have a gun on, I couldn't possibly\nshoot around him; and Olaf was seven parts demon when he laid back his\nears and started to kick.\n\n\"Where she cannot be bothered,\" sez the Friar, full as quiet but\nwithout any fierceness. The' was a little bush about eight feet up,\nand I felt sure it would hide me, so I stuck my fingers in the side o'\nthe cliff and climbed up; but the' was no way for me to get out to the\nbush, and I had to drop back to the ledge and stand there with the\nsweat tricklin' down between my shoulders until I felt like yellin'.\n\n\"I intend to kill you,\" said Olaf, as calm as though talkin' about a\nsick sheep.\n\n\"It would be a foolish waste of time,\" replied the Friar, as if he was\nadvisin' a ten-year-old boy not to fish when the Blue Bull was high\nand muddy. \"It wouldn't do any good, and I shall not allow it.\"\n\n[Illustration: \"I intend to kill you,\" said Olaf, as calm as though\ntalkin' about a sick sheep.\n\n\"It would be a foolish waste of time,\" replied the Friar, as if he was\nadvisin' a ten-year-old boy not to fish when the Blue Bull was high\nand muddy. \"It wouldn't do any good, and I shall not allow it.\"]\n\nI got out my gun, and made ready to do whatever the angels suggested;\nbut for some time the' was silence, and durin' this time I was keyed\nup so tight my muscles began to ache. I knew they were lookin' into\neach other's eyes, and I'd have given a finger off each hand to see\nhow the Friar's steady gray eyes handled those queer blue ones of\nOlaf.\n\n\"Is she all right?\" asked Olaf, and all the threat had left his voice,\nand it had just a glint o' pleadin' in it. I wouldn't have been one\nbit more surprised to have seen a prairie-dog come flyin' up the\ngorge, blowin' a cornet with his nose.\n\n\"She has sprained her ankle; but aside from this has no physical ill,\"\nsez the Friar. \"You men have caused her a lot of worry, and her soul\nis sick; but her body is well.\"\n\nAfter another silence, Olaf said slowly: \"Yes, yes; I can tell by the\nlight that you speak true. What do you intend to do with her?\"\n\n\"I intend to cure her,\" sez the Friar. \"I intend to help and\nstrengthen her; and I want you to help her, too. Olaf, she has had a\nlot of trouble, and her wild gaiety is only a veil to hide the wounds\nin her heart. I want you to help her.\"\n\n\"I know, I know she is honest,\" said Olaf, and blamed if his voice\ndidn't sound like a new boy talkin' to the boss; \"but she made me love\nher. Yes, I do love her. I must marry her. Yes, this is so.\"\n\n\"She cannot marry you, or any one else, now,\" sez the Friar, kindly.\n\"This is why she has gone from one man to another--to disgust them all\nand make them leave her alone.\"\n\n\"That is a damn devil of a way,\" cried Olaf in anger. \"Why should she\ngo to dances, and out ridin', and so on, if she wants men to leave her\nalone?\"\n\n\"She was foolish, she knows that now; but her father is not the right\nsort of a man, and her home was not pleasant,\" said the Friar.\n\n\"I told him I kill him, if she marry any one but me,\" said Olaf. \"I\nknow he is not honest; but he is afraid of me, and he will not bother\nher now. I go to see him again purty soon, and tell him some more.\nWon't you tell me where she is?\"\n\n\"I want to be your friend, Olaf,\" said the Friar gently. \"I tell you\nhonest that she cannot marry now. When I see her again, I shall tell\nher of meetin' you, and what you have said. I have no desire except to\ndo the best for all of you, and if you love her truly, all you will\nwant will be to do that which is best for her.\"\n\nThe Friar paused, and I pulled my ear clear to the edge o' the rock,\nso as not to miss a word. \"Olaf,\" he went on in a low, sorrowful\nvoice, \"the love of a man for a woman is a wonderful thing, a terrible\nthing, a soul-testing thing. Don't let your love become common for men\nto talk over. In believing what men have told you of me you have\ninsulted her, by admitting that such a thing is possible. Go back to\nyour work, kill no man for what he says of her; but keep her pure in\nyour own heart, and this will be the best way to keep her pure before\nthe world. Silence the gossips by living above them; and if it becomes\nnecessary for you to take your own love by the throat, then do it, and\ndo it for love of her. I shall do all I can to make her worthy of\nyou.\"\n\nYou should have heard the Friar's voice when he was sayin' this. I\nstood on the little ledge, just breathin' enough to keep my lungs\nventilated, and lookin' out across the landscape--mountains on all\nsides of me, and down below the broken ground and the benches, with\nthe green strips along the cricks lookin' like lazy snakes in the hot\nsunshine. I couldn't see a livin' creature, I felt like the last man\non earth; and that deep, musical voice seemed comin' to me from\nsomewhere out beyond the limits of life. I didn't have any more fear\nnow: the' wasn't anything in the shape of a human who could have done\nviolence to the Friar after hearin' him say the words I'd just heard;\nso I put up my gun, and listened again.\n\n\"Can't ya tell me why she can't marry me?\" asked Olaf, and the' was a\ntremble in his voice, almost as though it flowed up from a sob.\n\n\"I think I can trust you to keep her secret,\" sez the Friar. \"She is\nmarried already. The man was a beast and deserted her; but he is still\nalive, and she cannot marry again.\"\n\nI heard Olaf make a queer, animal sound with his breath, and then he\nsaid: \"Yes, you speak true--I can tell by the light; but she loves\nme--I can tell that also by the light. Will you tell me when she can\nmarry?\"\n\n\"I will,\" sez the Friar, and his voice was a pledge. \"There's my hand\non it.\"\n\nThey brought their hands together with a smack I could hear, and then\nOlaf turned on the narrow ledge, with the Friar holdin' him on, an'\nstarted off. The Friar went along with him, and I sneaked after,\nkeepin' a turn between us. Olaf mounted his hoss and rode away without\nlookin' back, which, as a matter o' fact, was his way o' doin' things;\nand when he was out o' sight, I joined the Friar.\n\nThe' was still a look of sadness in the Friar's face; but back of it,\nand shinin' through it, was a quiet satisfaction. He was full o' the\nscene he had just gone through; and presently he turned an' said:\n\"That was a glorious victory he gained over himself, Happy. That man\nhas a good heart, and who knows but what he will yet be the means of\nbringin' me an' Tyrrel Jones together.\"\n\n\"What do you reckon he meant by the light tellin' him that you were an\nhonest man?\" I asked. This was the most curious part of the whole\nthing to me.\n\n\"How can I tell,\" he sez. \"Life is so crowded with wonders that I have\nquit wonderin' about 'em; but I always feel a thrill when I see the\nstubborn spirit of a strong man melt and run into the mold the Master\nhas prepared for it.\"\n\n\"I'll own it was about the weirdest thing I ever saw,\" sez I; \"but I'm\nwillin' to bet that whatever else Olaf's spirit has molded itself\ninto, it's not a doormat with 'welcome' wrote on it; as the first\nfeller 'at fools with that girl is likely to find out.\"\n\n\"Never doubt the power of the Lord, Happy,\" sez he. \"The hand that\npiled up these hills can easy shape even so stubborn a thing as the\nhuman will.\"\n\n\"Yes,\" I agreed; \"but it generally takes just about the same length of\ntime to do it, and a man don't usually last that long.\"\n\n\"Time!\" sez he; \"what do you know about time? It may have taken ages\nto form these hills; and then again, it may have been done in the\ntwinklin' of an eye. From the way the streaks tilt up, I'm inclined to\nthink it was done sudden.\"\n\nI looked at the lines along the faces o' the hills, and I was inclined\nto believe it, too; so I dropped that subject, and we sat down close\ntogether and looked off down the trail where Olaf had vanished.\n\nWe sat in silence a long time, me thinkin' o' what sort of a light\nOlaf had seen to make him know 'at the Friar was honest; and of the\nway the Friar's voice had gone through me when he had talked of love.\n\nThis was a new idee to me, and one o' the biggest I had ever tried to\ngrapple with. Before this, my notion o' love was, for a man to get the\ngirl any way he could; and it took me some time to see the grandness\nof a man takin' his own love by the throat for love of a woman. I knew\n'at the Friar had done this himself; but it never was clear to me\nuntil I heard the heartache moanin' through his voice as he laid out\nthis law for Olaf, and Olaf bowed his stiff neck and accepted it.\n\nI'm purty sure that if I'd 'a' known that day, that a few years later\nI would have to take my own love by the throat for the sake of little\nBarbie, I wouldn't 'a' had the nerve to go on playin' the game--but\nthis is life. We pick up a stone here, and another there, and build\nthem into our wall until the flood comes; and then if the wall isn't\nhigh enough to turn back the flood, all the sting and bitterness comes\nfrom knowin' that we haven't made use of all the stones which came\nrollin' down to our feet.\n\nThat night we had an uncommon fine fire in the cave. I used to enjoy\nthese evenin' fires with the Friar, as much as a dog likes to have his\nears pulled by the hand he loves best. He would tell me tales of all\nthe ages 'at man has lived on the face of the whole earth, and I'd sit\nand smoke my pipe, and make up what I'd 'a' done, myself, if I'd been\none o' these big fellers. These chummy little fire-talks used to\nbroaden me out and make me feel related to the whole human race, and\nit was then 'at I came to know the Friar best--though the' ain't no\nway to put this into a story.\n\nAlong about nine o'clock the Friar began to lecture me again' the use\no' violence, pointin' out that war nor gunfightin' nor any other sort\no' violence had ever done any good; and endin' up with the way he had\nhandled Olaf as illustratin' how much better effects spiritual methods\nhad.\n\n\"Humph,\" sez I, \"so you're tryin' to put that over as an ordinary\ncase, are ya? Did you ever before see such eyes in a man's head as\nwhat Olaf has?\"\n\n\"Now that you mention it,\" sez he, \"I did notice they were peculiar.\"\n\n\"I ruhly believe you're right,\" sez I, sarcastic. \"When he said he saw\nlight he wasn't speakin' in parables. He can see things 'at you nor I\ncan't see--though I doubt if he understands 'em himself.\"\n\n\"Still, violence would have spoiled everything,\" persisted the Friar,\nwho was as human as a raw bronco when you tried to make him back up.\n\n\"Now, don't forget anything,\" sez I. \"It wasn't my face 'at lit up\nwhen I said 'at he did his killin' with bare hands; nor it wasn't me\nwho gloated over this as furnishin' an excuse to use my bare hands in\ndefendin' myself.\"\n\n\"Oh, Happy, Happy,\" sez he, with one o' the bursts 'at made ya willin'\nto go through fire and water for him. \"I'm the entire human race:\nthere isn't a single sin or weakness which hasn't betrayed me at one\ntime or another, and yet the wicked pride of me persists in stickin'\nup its head an' crowin' every time I take my eyes off it.\"\n\n\"Well, I like your pride full as well as any other part o' ya,\" sez I;\n\"and before you wrangle it into its corral again, I want to say 'at no\nother man in the world could 'a' told Olaf what you told him this\nmornin', and lived to talk it over around this fire to-night--unless,\nhe had used the best and the quickest brand o' violence the' is, in\nthe meantime.\"\n\n\"Now, that you have succeeded in flatterin' both of us, we'll go to\nsleep,\" sez the Friar, and the' was a deep twinkle in his eyes which\nallus rejoiced me to call up.\n\nNext night soon after dark, we started out with Kit Murray. She rode\nlike a man and could tick out her fifty or sixty a day right along,\nwithout worryin' her pony. As soon as she was safe located in\nBillings, I turned back to the Dot, while the Friar rounded up some\nstray sheep he had near the border, and as far as I can recall we\ndidn't meet again all that summer.\n\n\n\n\nCHAPTER NINETEEN\n\nOTHER PEOPLE'S BUSINESS\n\n\nOlaf's theories concernin' violence didn't harmonize complete with the\nFriar's; but his method for discouragin' scandal was thorough to a\ndegree. He silenced the gossipers all right, though so far as I heard,\nmost of 'em recovered; and the outcome was 'at the Friar stood higher\nafter the scandal 'n he had before.\n\nThe Cross brand outfit was a good deal like a pack o' dogs: they each\nsought Ty Jones's favor, and they were all jealous of each other. Olaf\nstood high on account of his mysterious insight; so Badger-face, the\nforeman, backed up Bud Fisher to devil Olaf as far as possible without\ngivin' Olaf what Ty would judge a fit excuse for unscrewin' the kid's\nneck; and from the talk I heard, their outfit trotted along as smooth\nan' friendly as seven he bears hitched to a freight wagon; but our\ntrails didn't cross frequent, so it was all hearsay.\n\nThe winter before had been so fierce 'at a lot o' small outfits\ncouldn't winter through their stock. Towards spring, ol' Cast Steel\nhad bought in the Half Moon brand for a hundred an' fifty dollars; and\nthat summer me an' Spider Kelley put in our spare time huntin' strays.\nSpider had come back, flat broke and full o' repentance; so after I'd\nstood him on his head in a buffalo-wallow full o' mud, I forgave him\nfree and frank, and this summer we rode together most o' the time.\n\nOl' Cast Steel was as lucky as a hump-back cat, and this summer the\ngrass was fatter 'n ever I'd seen it. We rounded up over five hundred\nhead o' ponies, and over sixty cows, which was just like bein' caught\nout in a gold storm without your slicker on; so we didn't sympathize\nany with the old man, but prospected around for pleasure whenever we\nfelt like it.\n\nOne afternoon after the fall round-up, me an' Spider found ourselves\nin a mighty rough bit o' country on the north <DW72> o' the Wind River\nrange. We had been herdin' six or eight Half Moon ponies before us for\nseveral days, devilin' a parcel of Injuns into thinkin' 'at we was out\ntradin'; but we had got weary o' this, an' were just foolin' around\nand wishin' 'at somethin' would turn up to amuse us.\n\n\"Aw, let's go on back home,\" sez Spider, not knowin' he was speakin'\nwisdom. \"I'd sooner work at work than work at huntin' up somethin' to\namuse myself with.\"\n\n\"Well,\" I sez, \"we'll finish out this afternoon, an' then if nothin'\nturns up, we'll go back, draw our pay an' go into Boggs.\"\n\nWe saw our ponies start around a butte ahead of us an' stop to examine\nsomethin'. We followed 'em around the butte, and there below us on a\nlittle level, was a bunch of men--seven of 'em. We drew up an' gave\n'em a look-over.\n\n\"What do you make out?\" sez I.\n\n\"Olaf the Swede with a rope around his neck, an' Badger-face Flannigan\nholdin' the other end o' the rope,\" sez Spider. \"What do you reckon\nthey're goin' to do to him?\"\n\n\"Comb his hair, or fit a new sun-bonnet on him,\" sez I, sarcastic.\n\"What else do they put a man's neck in a noose for? Let's go down an'\nsee what happens.\"\n\n\"A feller's not sure of a welcome at such times,\" sez Spider.\n\n\"No,\" I agreed; \"but I want to see Olaf's eyes again, and this may be\nmy last chance.\"\n\n\"It may be your last chance to see anything,\" sez Spider. \"The best\nthing we can do is just to back-track. We interrupted 'em once before;\nand I don't want 'em to get the idee that we spend all our time\ndoggin' their footsteps for a chance to spoil their fun. This ain't\nany of our business.\"\n\n\"We won't spoil their fun,\" sez I. \"If they get suspicious, we can\ntake a hand in it, an' that will fix it all right. Olaf ain't nothin'\nto us; and I don't intend to risk my fat for him, just 'cause he's got\ncurious eyes.\"\n\n\"No, I'm not goin',\" sez Spider.\n\nI looked across at the group again, an' there comin' up the trail\nbehind 'em was Friar Tuck, ridin' a round little pinto, an' leadin' a\nbig bay.\n\n\"Well, you just stay here, an' be damned to you,\" sez I to Spider.\n\"I'm goin' on down.\" So me an' Spider rode down together, an' arrived\nat just the same time as the Friar did.\n\nBadger-face looked first at us, an' then at the Friar. \"What the hell\ndo you fellers want this time?\" he sez to us in welcome.\n\n\"We just happened along,\" sez I. \"What's goin' on?\"\n\n\"You're goin' on yourselves, first thing,\" sez Badger-face. \"That's\nwhat's goin' on.\"\n\n\"I guess 'at you ain't got neither deeds nor lease to this land,\" sez\nI. \"We haven't any intention of interferin' with you; but we don't\nintend to be sent where we don't want to go. We've got business here,\nhuntin' up stray hosses, an' I reckon we'll just stick around.\"\n\n\"You got business here, too, I suppose?\" sez Badger-face, turnin' to\nthe Friar.\n\n\"Yes,\" sez the Friar calmly. \"I came here entirely by accident; but\nnow it is my business to inquire into why you have a rope about this\nman's neck. You recall havin' put me into a similar perdicament, Mr.\nFlannigan.\"\n\n\"Yes, an' the only thing I regret is, that I was interrupted,\" growls\nBadger-face. \"But this time, the' ain't any chance to change the\nprogramme, so you might just as well poke on into some one else's\naffairs.\"\n\n\"What's the matter, Olaf?\" asked the Friar.\n\nBefore Olaf could reply, Badger-face gave a jerk on the rope. \"You\nshut up,\" sez he.\n\n\"Surely you will give the man a chance to speak,\" cried the Friar,\nindignant.\n\n\"It won't do him no good to speak,\" sez Badger-face. \"He's committed a\nmurder, but of course he denies it. Now, get out o' here, all three of\nya.\"\n\n\"Listen,\" sez the Friar, as steady an' strong as the sweep of a deep\nriver, \"I care more for justice 'n I do for law. I know that hangin' a\nman has never done any good; but it is usually regarded as a legal\nform of punishment, and the prejudice in its favor is still too strong\nfor one man to overcome. If you convince me that this man would be\nhung by a court, why, I shall never say a word about it; but if you do\nnot convince me, I shall stir up all the trouble I can. I have quite a\nnumber of friends, Mr. Flannigan.\"\n\nBadger-face studied over this a moment; and he saw it had sense. \"All\nright,\" sez he, \"we'll try him fair an' square; and then you three\nwill have to help string him, an' I guess that'll keep your mouths\nshut.\"\n\n\"Tell your story, Olaf,\" sez the Friar.\n\n\"Well,\" sez Olaf, \"we came up short on the round-up, an' the old man\nraised Cain about it, an' sent us out to hunt for strays. Badger-face\nsplit us into pairs, an' made me an' Bud Fisher work together. We saw\nsome cows up on a ledge where we couldn't ride to; so we left the\nhosses below, an' climbed to see if they had our brand. If they had,\nwe intended to ride around and get 'em. If not it would save half a\nday. Bud Fisher had a rifle along, hopin' to get a mountain sheep, an'\nhe insisted on takin' it with him. He climbed up on a ledge, an' I\npassed up the rifle to him. It was a long stretch, an' I passed it\nmuzzle first. The hammer caught on a point of rock, an' shot him\nthrough the stomach. I didn't bear him any ill will any more--I ran\ndown to the hosses, an' brought up the saddle-blankets an' the\nslickers, an' made him as comfortable as I could. Then I hunted up\nBadger-face an' told him. When we got back he was dead. This is the\ntruth.\"\n\n\"I think it is,\" sez the Friar.\n\n\"Aw rot!\" sez Badger-face. \"Come on, now, an' finish it. Every one\nknows how they hated each other; and it's plain enough that when the\nSwede here got the chance, he just put Bud out o' the way, an' Bud was\none o' the finest boys the' ever was in the world--always full o' fun\nan' frolic; while Olaf has allus been sour an' gloomy.\"\n\nMost men are as sappy as green grain, an' they bow whichever way the\nwind blows. The Cross brand punchers all looked extremely sad when\nBadger-face spoke o' what a royal good feller Bud Fisher had been, an'\nwhen he stopped, they all glared at Olaf as friendly as wolves,\nespecially a skinny feller by the name of Dixon, who had the neck and\ndisposition of a snake.\n\n\"If you thought 'at Olaf an' Fisher hated each other, why did you make\n'em work together?\" asked the Friar; and the Cross brand punchers\npricked up their ears an' looked pointedly at Badger-face.\n\n\"I thought they had made it up,\" sez Badger-face, surprised into\ntakin' the defensive.\n\n\"I have noticed that you are likely to jump hasty at conclusions,\" sez\nthe Friar, speakin' with tantalizin' slowness. He was a fisher of men,\nall right, the Friar was; and just then he was fishin' for those Cross\nbrand punchers. \"Did Bud speak before he died, Olaf?\" he asked\nimpartially.\n\nOlaf hung his head: \"All he said was, that she hadn't never cared for\nhim, an' that he didn't know one thing again' her,\" said Olaf.\n\n\"Aw, what's the use o' stringin' it out,\" sez Badger-face. \"Let's hang\nhim and have it over with.\"\n\n\"Hanging a fellow-bein' is a serious matter, Mr. Flannigan,\" sez the\nFriar. \"I am a party to this now, and shall have to assume my share of\nthe responsibility. I shall never consent to swingin' a man on such\nevidence as this. Let us go and examine the spot. The hammer may have\nleft a scratch, or something. If you convince me that Olaf committed\nthe murder, I pledge to assist in hangin' him. That's certainly fair,\nmen,\" he sez to the Cross-branders, an' they nodded their heads that\nit was.\n\nSo we clumb up to the spot where Olaf claimed to have handed the gun;\nbut the' wasn't any scratch on the rock. \"Did he fall from the ledge\nwhen he was shot?\" asked the Friar.\n\n\"No,\" sez one o' the punchers. \"He fell on the edge an' hung on.\"\n\n\"Did the bullet go clean through him?\" asked the Friar.\n\n\"Yes, it went clear through,\" sez the feller.\n\n\"Point with your finger just where it went in, an' just where it came\nout,\" sez the Friar.\n\nThe feller pointed with one finger in front, an' one behind. The Friar\ntook a rope an' had me hold it behind the feller at just the level of\nthat finger an' then he made Spider stretch the rope so that it passed\non a line with the finger in front. The whole crowd was interested by\nthis time. \"Now, then,\" sez the Friar, \"where could Olaf have stood to\nshoot such a line as that. He could not have shot while he was\nclimbin' up, nor he couldn't have reached high enough while standin'\nbelow.\"\n\n\"He could, too,\" sez Badger-face, \"for Bud would have been leanin'\nover, reachin' for the gun.\"\n\n\"If he had been shot while he was reachin' over, he would have fallen\nfrom the ledge,\" flashed the Friar.\n\n\"Maybe he did,\" snapped Badger-face, just as quick. \"Olaf here is as\nstrong as a horse, an' maybe he put him back on the ledge. He had\nblood on his hands an' you can still see it on his shirt. A man don't\nbleed much when shot in the belly.\"\n\nOlaf's queer blue eyes turned from one to the other, but his face\ndidn't change expression much. He had about give up hope in the first\nplace, an' his face had the look of a hoss, after he's been throwed\nfour or five times an' just keels over on his side an' sez to himself:\n\"Well, they've put the kibosh on me, an' I don't intend to make a fool\nof myself any more by tryin' to break loose.\" The rest of us was more\nexcited about it than Olaf was himself.\n\n\"Which one of us is the nearest size to Bud Fisher?\" asked the Friar.\n\nThey all agreed that Spider Kelley was; so the Friar had him <DW53> up\non the ledge. Then he had Olaf take the empty rifle just as he had\nheld it when he passed it up; but made him give it to Badger-face\nhimself to pass up. Badger-face passed it up, Spider Kelley reached\nfor it, took it, and started to straighten up--The hammer caught on\nthe precise knob that Olaf had said it had, an' snapped hard enough to\nset off a cartridge. \"There,\" sez the Friar, sweepin' his hands wide.\nWe could all see that the bullet would 'a' gone through just where it\ndid go.\n\n\"Hand back the rifle, an' I'll show ya how he passed it up,\" said\nBadger-face. Spider passed it down, an' we all watched intent. It had\nbecome like a real court o' law; we had forgot what the case was\nabout, we was so interested in seein' the scrap the lawyers were\nputtin' up.\n\nBadger-face cocked the rifle so slick we didn't see him, called out to\nSpider to catch it, an' tossed it up to him. It came just short o'\nSpider's hand; and without thinkin' o' what he was doin', Spider\nreached for the gun. This brought him squattin' just the time the gun\ndropped back into Badger's hands, and quick as a wink, he pulled the\ntrigger--and hanged if that bullet wouldn't have traveled through the\nsame hole the first one had made.\n\nI never saw circumstantial evidence give such a work-out before. If we\nhad all been fair-minded, it would have puzzled us; but as it was, we\nsided accordin' to our prejudices; an' the Cross brand fellers chose\nBadger-face to Olaf, Badger-face bein' foreman. The Friar saw he was\nstumped.\n\n\"Are there any marks up there?\" he asked of Spider.\n\n\"There's some blood streaks on a stone,\" sez Spider.\n\n\"Did you notice 'em?\" asked the Friar of Badger-face.\n\n\"Yes,\" sez he; \"but they don't mean nothin'.\"\n\n\"Let's go up an' look at 'em,\" sez the Friar, so we all clumb up.\n\nThey pointed out just where Bud Fisher had laid when they found him;\nand close beside him was a smooth white stone with blood marks on it.\nThe Friar examined the lay o' the ledge; but it didn't tell nothin',\nso finally he got down on his knees an' studied the blood-stained\nstone.\n\nPresently he nodded his head and straightened up. \"Examine that\nstone,\" he said, pointin' with his fingers. We all crowded about an'\nstudied it. The' was finger an' thumb prints all over it; but if you\nlooked close, you could make out the rude image of a man pullin' up a\ngun which had exploded on the edge of a ledge. It was a smudgey,\nshakey affair, but if ya looked just right you could make it out. Yet,\neven this didn't floor Badger-face.\n\n\"The Swede there did that himself,\" he growled; \"and this makes him\nout sneakier 'n we thought him. Let's hang him, and get rid o' this\nfoolishness.\"\n\n\"Flannigan,\" sez the Friar in cold, hard tones, \"you have gone too far\nthis time. If you had hung Olaf at first, you might have done it from\na proverted sense o' justice; but to do it now would be murder; and\nyour own men wouldn't help. Do any of you men chew tobacco?\"\n\nIf he had asked for a can o' face-paint, we wouldn't 'a' been more\nsurprised; but to show the hold the Friar had gained over that crowd,\nevery feller there but Badger-face held out his plug to him.\n\n\"Make some tobacco juice, Olaf,\" he said.\n\nOlaf bit off a hunk the size of a walnut from his own piece, an'\nproceeded to make juice, as though his life depended upon the amount\nof it. \"Wet your thumb and fingers with it, and make marks on the\nwhite stone,\" commanded the Friar.\n\nOlaf did so; and when we saw the difference in size and shape, we\nsavvied the game.\n\n\"Olaf took Bud's hand and made the marks with Bud's own blood,\" sez\nBadger-face.\n\n\"Did any one here ever try to handle a dead man's hand?\" asked the\nFriar; and that settled it. We all nodded our heads, except\nBadger-face, an' he had sense enough to see 'at he had lost the deal,\nso he didn't say nothin'.\n\n\"What I can't see is, why he didn't write,\" sez the Friar.\n\n\"He couldn't write,\" chirps up two punchers at once, an' then they\ntook the rope off Olaf's neck.\n\nThey talked it over and decided that the best thing to do was to bury\nBud Fisher right there in the canyon. The' was a little cave on the\nledge back o' where we were standin' so two o' the punchers went down\nwhere they had him laid out under the slickers, an' brought him up. We\nhad to hoist him on ropes, an' the Friar looked a long time into his\nface.\n\nIt was just a lad's face: not bad nor hardened; just the face of a\nmischievous boy, weary after a day's sport. We all took a look, an'\nthen put him in the little cave an' heaped clods over him an' piled\nstones on until the door was blocked shut again' varmints.\n\nThe Friar sat down on a big rock--he had worked as hard as any of\nus--and sat thinkin' with his chin in his hand. The Cross brand\nfellers muttered among themselves for a moment, an' then one of 'em\ntook off his hat, an' sez, \"Don't ya think ya'd ought to speak\nsomethin' over him, parson?\"\n\n\"Do you want me to?\" asked the Friar. And they all nodded their heads.\n\nSo the Friar, he took off his battered hat and stood up before us an'\nspoke a sermon, while we took off our hats, an' sat around on stones\nto listen.\n\nI'm convinced 'at the Friar's long suit lay in the fact 'at he allus\npreached at himself. Most preachers have already divided the sheep\nfrom the goats; and they allus herd off contented with the sheep on\ngreen pastures, and preach down at the goats on the barren rocks; but\nif the Friar made any division at all, he classed himself in with the\ngoats.\n\nYou see, in agreein' to help string Olaf should he be convicted, the\nFriar had bet his soul on the outcome; and this braced him up in that\ncrowd as nothin' else would; for they knew that if he had lost, he'd\nhave pulled harder on the rope 'n any one else.\n\nIt's child's play to put out a funeral talk over some old lady who has\nhelped the neighbors for seventy or eighty years; but to preach the\nneed of repentance to the livin', and then to smooth things out for\n'em after they've died in their sins, in such a way as it will jolly\nup the survivors and give 'em nerve to carve cheerful tidings on the\ntombstone, is enough to make a discriminatin' man sweat his hair out.\n\nThe Friar stood with his hands clasped in front of him, and his eyes\nfixed sort o' dreamy-like on the distance. It was a perfect day, one\no' those days 'at can't happen anywhere except in our mountains in the\nfall o' the year, and my mind drifted off to some lines the Friar was\nfond of rehearsin', \"Where every prospect pleases, an' only man is\nvile.\" Then I saw a change come to the Friar's face, and he began to\nchant the one which begins: \"Lord, let me know mine end, and the\nnumber of my days.\"\n\nHe chanted slow, and the words didn't mean much to us; but the solemn\nvoice of him dragged across our hearts like a chain. One line of it\nhas haunted me ever since. It seems to suggest a hundred thoughts\nwhich I can't quite lay my hand on, and every time I get sad or\ndiscouraged, it begins to boom inside me until I see 'at my lot ain't\nso much different from the rest; and I buck up and get back in the\ngame again: \"For I am a stranger with Thee and a sojourner as all my\nfathers were.\"\n\nThe Friar didn't preach us a long talk, and most of it circled about\nhis favorite text, that a man's real children were those who inherited\nhis character, rather than those who inherited his blood. Once he\nraised his finger and pointed it at us and sez: \"You were fond o' this\nboy; but did you love him for his good, or did you love him for your\nown selfishness? I knew him not save through the dark glass of\nreputation; yet after looking into his dead features, to-day, I think\nI know him well. Death tells, sometimes, what Life has hid away. I did\nnot see in his face the hard, deep lines of stealthy sin; I saw the\nopen face of a child, tired out after a day wasted in thoughtless and\nimpulsive play; but comin' home at nightfall to have his small cares\nrubbed away by a lovin' hand--and then, to fall asleep.\"\n\nO' course, the Friar landed on us good and plenty; but this was the\npart of his talk which stuck to us after the scoldin' part was all\nforgotten. When he was through he said a short prayer, and sang in a\nlow tone the one beginnin', \"One sweetly solemn thought.\" His eyes\nwere glistenin' through a mist when he finished this, and he climbed\ndown from the ledge, hurried over to his pinto, and rode off without\nsayin' another word.\n\nWe all sat silent for quite a spell, and then Spider and I got up and\nnodded good day to 'em. The Cross-branders also got up and shook\n'emselves, and started down with us--all except Olaf. He sat there on\na stone with his fingers run into his hair, and his face hid in his\nhands. Olaf had had regular religion when he was a child; and it had\ncome back to him up there on the ledge. They say it's worse 'n a\nrelapse o' the typhoid fever when it hits ya that way. I know this\nmuch, Olaf was doubled up worse 'n if he'd had the colic; and from\nthat time on, the Ty Jones outfit looked mighty worldly to him.\n\nEven Spider Kelley was savin' of his nonsense until we got in sight of\nthe Diamond Dot again.\n\n\n\n\nCHAPTER TWENTY\n\nQUARRELING FOR PEACE\n\n\nWe had a visitor once, which was a business man. One of his chief\ndiversities was to compare sedentary occupations with what he called\nthe joyous, carefree outdoor life. He said 'at sedentary came from\nsedan-chair, and meant to sit down at your work. I rode the range next\nspring until I felt more sedentary 'n an engineer; and sometimes at\nnight it used to strain my intellect to split the difference between\nmyself an' my saddle.\n\nI got out o' humor an' depressed and downright gloomy. Fact is, I was\non the point o' rollin' up my spare socks and givin' Jabez a chance to\nsave my board money, when I heard a sound 'at jerked me up through the\nscum and gave me a glimpse o' the sky again. I was ridin' in about\ndusk, and I had hung back o' the dust the other fellers had kicked up,\nso I could be alone and enjoy my misery, when I heard this inspirin'\nnoise.\n\nOl' Tank Williams once tried to learn to play on a split clarinet a\nfeller had give him, and at first I thought he had found where we had\nburied it, and had resumed his musical studies; but this outrage came\nfrom an instrument a feller has to be mighty cautious about buryin'.\nIt was a human voice, and these were the words it was screechin':\n\n    \"Fair Hera caught her wayward spouse\n      With a mortal maid one dawn.\n    Zeus charmed the maid into a cow,\n      To save himself a jaw'n'.\n    This seemed to me a liber-tee\n      To take with poor I-oh;\n    But now I find that he was kind,--\n      'T was I who did not know.\n    For girls use slang and girls chew gum,\n      And drape their forms in silk;\n    While cows behave with de-co-rum,\n      And furnish us with milk.\"\n\nWell, I gave a whoop and threw the spurs into my pony. This was the\nseventy-ninth verse of Horace's song, and it was his favorite, because\nit was founded on the Greek religion. I found him perched up behind a\nrock, and he kept on slammin' chunks of his song up again' the welkin\nuntil I shot some dirt loose above his head; and then he climbed down\nand reunioned with me.\n\nHe was lookin' fine, except that some of his waist products had come\nback, and we talked into each other until the air got too thin to\nbreathe. Then we suppered up and began talkin' again. He had tried all\nsorts of gymnastical games back East, from playin' golf to ridin'\nhossback in a park, but it didn't have the right tang. Folks thought\nhe'd gone insane an' lost his mind, the air didn't taste right, he got\nparticular about how his vittles were cooked; until finally, his\nendurance melted and began to run down the back of his neck. This\ndecided him 'at he'd had full as much East as was good for him; so he\nloaded up a box with firearms, tossed some clothin' into a handbag,\nand he said his grin had been gettin' wider all the way out until it\nhad hooked holes through the window lights on both sides o' the train.\n\nWe were all glad to see him, an' he dove into ranch life like a\nbullfrog into a cream jar; and he got toughened to a hard saddle in a\nmighty short time for a feller who had got used to upholstery back\nEast. He said 'at the only thing 'at had kept life in him had been to\nsing his song constant; but he denied 'at this was his main excuse for\nfleein' from his own range.\n\nHe didn't seem to bear a mite o' malice for the joke I had put up on\nhim; but still, I have to own up 'at he half pestered the life out of\nme with his song. He had what he called a tenor voice; but it was the\ndolefullest thing I ever heard, and the more he sang, the more his\nnotes stuck to him until I coveted to hear a love-sick hound\nserenadin' the moon. When he saw it was riskin' his life to drag out\nany more o' the song, he would pause temptingly, and then begin a\nlecture on the Greek religion. He got me all mussed up in religion.\n\nOf course, I knew 'at the Injuns had a lot o' sinful religious idees,\nand I was prepared to give the other heathens plenty o' room to swing\nin; but not even an Injun would 'a' stood for as immoral a lot as the\nGreek gods an' goddusses--especially the top one, which Horace called\nZeus an' Jove an' Jupiter.\n\nThis one didn't have as much decency as a male goat, and yet he had\nunlimited power. He was allus enticin' some weak-minded human woman\ninto a scrape; and when his wife, who was called Hera and Juno, would\nget onto his tricks, Zeus would snap his fingers, say \"Flip!\" and\ncharm the human woman into some sort of an animal. It was a handy\nscheme for him, true enough; and he didn't care a scene how\nembarrassin' it was for the human women.\n\nHe turned one of 'em into a bear, and, like most other women, she was\nfeared o' bears an' wolves an' snakes, an' the rest o' the company she\nwas forced to associate with. She led a perfectly rotten existence\nuntil her own son went bear huntin', and was just on the point of\njabbin' a spear into her, when even Zeus himself admitted 'at this\nwould be carryin' the joke a leetle too far; so he grabs 'em up and\nsticks 'em into the sky as a group o' stars.\n\nHorace tried to argue 'at this proved Zeus to be merciful; but as far\nas I can see it's as idiotic as havin' the law hang a man for murder.\nSupposin' some feller had murdered me--would I feel any happier\nbecause this feller who couldn't put up with me in this world, is sent\nover to pester me in the next? Course I wouldn't; but if one o' my\nfriends was murdered, and I had a chance to slay the feller 'at did\nit, this would give me a lot o' satisfaction an' joy an'\npleasure--though I don't say it would be just.\n\nPuttin' the woman an' her son up in the sky didn't square things in\nHorace's religion, neither; 'cause he said 'at Hera got jealous of\nZeus for elevatin' the woman and she went to her foster parents who\nhad charge of the ocean, and made 'em bar this woman and her son from\never goin' into it, the same as the other stars did, and he could\nprove it any clear night. I told him that he might get away with such\na tale as that back East among the indoor people; but that he couldn't\nfool a day-old child with it out our way.\n\nWe started this discussion the day after the fall round-up was over,\nHorace had toughened up before it began, and he had rode with me all\nthrough it, and takin' it all in all he was more help than bother,\nexcept that he shot too much. When he had come out before, he had been\nso blame harmless he couldn't have shot an innocent bystander; but\nthis trip, he was blazin' away at every livin' thing 'at didn't have a\ndollar mark on it, and when these wasn't offered, he'd waste\nammunition on a mark.\n\nI had some details to tend to after the round-up, so we didn't get a\nchance to settle the bet for several days. It was only a dollar bet;\nbut when the time came, I picked out a couple o' good hosses, bein'\nminded to look at the stars from the top o' Cat Head.\n\nWe reached it about dark, made some coffee, an' fried some bacon. Then\nwe smoked an' talked until it was entirely dark before we ever looked\nup at the stars. \"Now, bluffer,\" sez I, \"show me your woman-bear.\"\n\nHe looked up at the sky, an' then moved on out o' the firelight, an'\ncontinued to look at the stars without speakin'. \"Don't seem to see\n'em, do you?\" I taunted.\n\nHe turned to me an' spoke in a hushed voice: \"Man,\" he said, \"this is\nwonderful. Why, the way those stars seem to be hangin' down from that\nvelvet dome is simply awe-inspirin'. I've looked through three good\ntelescopes, but to-night, I seem to be viewin' the heavens for the\nfirst time.\"\n\n\"I thought you wasn't much familiar with 'em, or you wouldn't have put\nout that nonsense about a bear-woman,\" I sez.\n\n\"That,\" sez he, pointin' to the best known group o' stars in the sky,\n\"is Ursa Major.\"\n\n\"That,\" sez I, \"is the Big Dipper, an' you needn't try to fool me by\ngivin' it one o' your Greek names.\"\n\nHe didn't argue with me; but came back to the fire an' fixed some\nstones in the shape of the Big Dipper stars, then drew lines with a\nstick, an' sez 'at this made up the Great Bear. I looked him between\nthe eyes, but he held his face, so I knew he was in earnest. \"All\nright,\" I sez. \"I'll take you huntin' some o' these days, an' if we\nchance to come across a silver-tip--a real grizzly, understand, and\nnot a pet varmint backed up again' the risin' sun--you'll change your\nmind about what a bear looks like. If that was all your fool Greeks\nknew about wild animals, I wouldn't waste my time to hear what they\nhad to say about gods an' goddusses. I'm goin' to start back, an' you\ncan come or not, just as you please.\" This was the first time I had\nhinted about the woodchuck; but I was disgusted at his nonsense. He\ntook it all right, though, which proves he was game.\n\nI rode some comin' back, an' he kept tryin' to square himself; but I\ndidn't heed him. Just before we reached the foothills, we saw a fire,\nan' when we reached it, the Friar was just finishin' his supper. He\nan' Horace bowed stiffly to each other, an' I was just put out enough\nby Horace's star-nonsense to feel like roastin' some one; so I decided\nto roast 'em both.\n\nI sat on my hoss an' looked scornful from one to the other. \"Here is\ntwo religious folks,\" I said, impersonal to the pony, but loud enough\nfor all to hear. \"Here is two genuwine religious folks! One of 'em is\nworkin' for universal brotherhood, an' the other is peddlin' Greek\nreligion which he claims to be founded on beauty an' love an' harmony.\nThey meet in the mountains, an' bow as cordial as a snow-slide. I\nthink if ever I pick out a religion for myself, I'll choose the\nInjun's.\"\n\nI couldn't have asked for any two people to look more foolish 'n they\ndid. Neither one of 'em seemed to have anything to say; so I said to\nmy pony: \"Don't you worry none, Muggins, I got a match o' my own, an'\nif we want to set by a fire, why, we can ride on to some place where\nwood is free, an' build us one.\"\n\n\"Will you not dismount an' rest a while at my fire?\" sez the Friar, in\na tone meant as a slap at me.\n\n\"No, thank you,\" sez Horace, \"we must be goin'.\"\n\n\"Yes, Friar,\" I sez hearty. \"Me an' Horace has a bet up, an' you can\ndecide it. Also, you owe him somethin' on his own hook. You drove him\nout o' your religion an' into the Greek religion; an' if that don't\ngive him a direct call on you, why then you don't realize what a pest\nthe Greek religion is.\"\n\nThey were so embarrassed they were awkward an' spluttery; but I was\nsure 'at this was good for 'em, so I got off, threw the reins on the\nground, an' warmed my hands at the fire; while Horace apologized for\nme not knowin' any better, an' the Friar assured him coldly that\neverything was all right, an' he was rejoiced to have a little\ncompany.\n\nWell, for as much as ten minutes, we sat around enjoyin' what I once\nheard a feller call frapayed convivuality, an' then I took pity on 'em\nan' loosened things up by tellin' the Friar about the trip me an' Tank\nan' Horace had took into the mountains to pacify our nerves, just\nbefore he had stumbled on Horace that other time. O' course I didn't\ntell it all, as I didn't want Horace to know any more about it than he\nknew already; but I told what a seedy little windfall Horace had been\nwhen we started out, an' how he had come back crackin' jokes an'\nsingin' the infernalest song 'at ever was made up. I finally got\nHorace to sing ten or fifteen minutes o' this song, an' he droned it\nout so unusual doleful that he fetched a chuckle out o' the Friar, an'\nthen we were feelin' easy an' comfortable, like outdoor men again.\n\nThen I told the Friar what our bet was, expectin' o' course that he'd\nback me up; but what did he do but say 'at Horace was right as far as\nthe stars was concerned. This tickled Horace a lot, an' he began to\ncrow over me until I concluded to test the Friar; so I sez to Horace\nthat his religion havin' been endorsed by the Friar himself, I'd\nbecome a Greek the first chance I had.\n\nHorace didn't take any trouble to hide his satisfaction, an' he began\nto expound upon the beauty, an' the art, an' the freedom of the Greek\nreligion at a great rate.\n\n\"They certainly was free,\" I sez, \"an' easy too, an' I don't deny 'at\nthey might 'a' been some weight in art an' beauty; but, confound 'em,\nthey didn't know as much about bears as I know about e-lectricity. I'd\njust like to see Zeus himself go up into the Tetons in the early\nspring, to hunt for Big Dippers. I'll bet the first hungry grizzly\nhe'd come across would set him right on the bear question.\"\n\nThis was a good opener, an' in about two shakes, the Friar an' Horace\nhad locked horns. Horace was a crafty, sarcastic, cold-blooded little\nargufier; while the Friar was warm an' eager an' open as the day. It\nwas one o' the best gabbin' matches I have ever started.\n\nThey dealt mostly in names I had never heard of before, although once\nin a while they'd turn up one a little familiar on account of Horace\nhavin' told me some tale of it. The Friar knew as much about these\nthings as Horace did; but he called 'em myths, an' said while they\ndidn't mean anything when took literal, they had great historical\nvalue when regarded merely as symbols. He said that I-oh--the human\nmaid which Zeus had turned into a cow--was nothin' but the moon, an'\nthat Argus of the hundred eyes was simply the sky full o' stars; and\nthat the old god which ate up his children was nothin' but time.\n\nI didn't really understand much of what they said; but I did enjoy\nwatchin' 'em bandy those big words about. We all use a lot o' words we\ndon't understand; but as long as they sound well an' fill out a gap it\ndon't much matter. These two, though, seemed to understand all the\nwords they used, an' I was highly edified.\n\nAs they talked, an' I kept watchin' the Friar's face, I learned\nsomethin': the Friar had been mighty lonesome with only us rough\nfellers to talk with, an' had been hungerin' for just such a confab as\nthis to loosen up his subsoil a little.\n\nEvery now an' again, I'd cast an eye up to the stars; an' while I\ndidn't know the religious names of 'em, I knew how to tell time by\n'em; an' I knew 'at those two would have a turn when they remembered\nto look at their watches. It was full one o'clock when the\nconversation came to its first rest, an' then the Friar recalled what\nI had said when I had dismounted; so he up an' asked Horace\npoint-blank what he had had to do with makin' Horace quit the church.\n\nHorace was minded to sidestep this at first by intimatin' that I was\nnot responsible for what I said; but he finally came across and told\nthe Friar that he had give up that church for about the same reason\nthat the Friar himself had. This set the Friar back purty well on his\nhaunches, an' put him on the defensive. He had hammered Horace freely\nbefore, but now when he conscientiously tried to defend the gang he\nhad left, and also excuse himself for leavin', he had some job on his\nhands.\n\nI thought Horace had him when he compared the Golden Age of Greece an'\nPlato's Republic with the Dark Ages, which was a stretch of years when\nthe Christian religion about had its own way; but the Friar admitted\nthat what he called economical interests had put a smirch on the\nchurch durin' the Dark Ages, an' then he sailed into the Golden Age of\nGreece, showin' that slavery was the lot of most o' the decent people\ndurin' that period. When I fell asleep, they were shakin' their fists\nfriendly at one another, about Plato's Republic, which I found out\nafterwards was only a made-up story.\n\nBein' edicated is a good deal like bein' a good shot in a quiet\ncommunity--once in a long while it's mighty comfortin', but for the\nmost part it's nothin' but shootin' at a target.\n\n\n\n\nCHAPTER TWENTY-ONE\n\nPEACE TO START A QUARREL\n\n\nIt was broad day when I woke up--that is, the sun was beginnin' to\nrise--an' the fire had dwindled to coals, the breeze had begun to stir\nitself, an' I was consid'able chilly. I saw the Friar's nose stickin'\nout o' one side of his tarp an' Horace's nose stickin' out the other,\nan' I grinned purty contentedly.\n\nMy experience is, that quarrelsome people usually get along well\ntogether an' make good company; but sad, serious, silent, polite folks\nis about the wearin'est sort of an affliction a body can have about.\n\nI once heard a missionary preach about what a noble thing it was to\ncontrol the temper. He must have been a good man, 'cause he was\nunusual solemn an' wore his hair long an' oily; but he only looked at\none side o' the question. I've known fellers who had such good control\no' their tempers that after they'd once been put out o' humor over\nsome little thing, they could keep from bein' good tempered again for\na year. And then again, when a feller keeps too tight a holt on his\ntemper, his hands get numb, an' his temper's liable to shy at some\nsilly thing an' get clear away from him.\n\nWhat I liked about both the Friar an' Horace was, 'at they hadn't\nfroze up all their feelin's. It was possible to get 'em stirred up\nabout things, an' this allus struck me as bein' human; so I was glad\nto see Horace warmin' his feet in the small o' the Friar's back, an' I\nwhistled a jig under my breath while gettin' breakfast.\n\nThey grumbled consid'able when I rousted 'em out; but by the time they\nhad soused their heads in the crick, they were in good humor again;\nan' hungry! Say! Ever since I'd give him his treatment, Horace had had\nan appetite like a stray dog; while the Friar allus was a full hand at\nclearin' tables, except on his one off-day a week. I gave the Friar a\nwink just as Horace splashed into his third cup o' coffee, an' sez:\n\"Friar, you should have seen this creature when he first came out\nhere. His muscles had all turned to fat, so that he could hardly\nwobble from one place to another, an' he was so soft that when he'd\nlie down at night, his nerves would stick into him an' keep him awake.\nNow, if it wasn't for that fringy thing he wears on his face, he'd\nlook almost exactly like a small-sized human.\"\n\nThe only come-back Horace made was to start to sing with his mouth\nfull o' cornbread an' bacon. This was more 'n any one could stand, so\nI tipped him over backward, an' asked the Friar which way he was\nheadin'.\n\nThe Friar's face went grave at once; and then he began to post me up\non Olaf the Swede. I had heard some rumors that summer, but hadn't\npaid much heed to 'em. It now turned out that the Friar and Olaf had\nstruck up friendly affiliations; so he was able to give me all the\ndetails.\n\nBadger-face had a disposition like a bilious wolf, and when he was\ndenied the satisfaction o' jerkin' Olaf out o' this world, he had\nturned to with earnest patience to make Olaf regret it as much as he\ndid. Olaf could stand more 'n the youngest son in a large family o'\nmules, but he had his limitations, the same as the rest of us; so when\nhe saw that Badger was engaged in makin' the earth no fit place for\nhim to habitate, he began to feel resentful.\n\nWhen a boss is mean, he is still the boss and he don't irritate beyond\nendurance; but a foreman is nothin' but a fellow worker, after all; so\nwhen he gets mean, he's small and spidery in his meanness; and I\nreckon 'at Olaf was justified in tryin' to unjoint Badger-face,\nthorough and complete.\n\nO' course, Ty had to back up Badger for the sake o' discipline; but he\ndidn't wreak any vengeance on Olaf when he tendered in his\nresignation, which proves 'at Ty still was full o' respect for Olaf.\nBadger was groanin' on his back when Olaf left; but he called out that\nhe intended to get square, if he had to wear all the curves off his\nown body to do it.\n\nOlaf had the gift o' sensin' men, all right; but his judgment wasn't\nsuch as to make a yearlin' bull willin' to swap, and what he did was\nto take the Pearl Crick Spread as a homestead. It was only about\nfifteen miles from the Cross brand ranch house, and it was one o' the\nchoicest bits in the whole country. This act was on a par with an\ninfant baby sneakin' into a wolf den to steal meat. The Friar put the\nfinishin' touch by sayin' that Olaf had bought the old, run-down T\nbrand, and then I lost patience.\n\n\"Does Olaf sleep with a lightnin' rod connected to the back of his\nneck?\" I asked as sober as a boil.\n\n\"What do ya mean?\" asked the Friar, who was innocent about some\nthings.\n\n\"Well, that looks like another good way to attract trouble,\" sez I.\n\n\"Olaf does not want any trouble,\" sez the Friar with dignity. \"All he\nwants is an opportunity to work his claim in peace. He has more\nself-control 'n airy other man I've ever known.\"\n\n\"It's a handy thing to have, too,\" sez I, \"providin' a feller knows\nhow to use it. Why, ya could change a T brand to a Cross quicker 'n a\none-armed Mexican could roll a cigarette. Ty Jones'll get more o' that\nbrand 'n ever Olaf will. How is Kit Murray gettin' along?\"\n\n\"She is a fine girl,\" sez the Friar, his face lightin'. \"She has cut\nout all her wild ways, and Mother Shipley sez her daughter thinks as\nmuch of her as if they was sisters. I got word last week 'at her\nhusband died in a hospital; and I hope she'll marry Olaf some day.\"\n\n\"Well, I'll bet the liquor again' the bottle 'at she never does it,\"\nsez I. \"In the first place, she's got too much style, and in the\nsecond, she's got too much sense. Ty's already got more stuff 'n he\ncan take care of through a dry summer, and the next one we have, he is\ngoin' to need Pearl Crick Spread. A grizzly traffics along without\nbein' disturbed, until he gets the idee that he owns consid'able\nproperty, and has legal rights. Then one day the' don't seem to be\nanything else demandin' attention, so out go a parcel o' men and\nharvest the grizzly. That's the way it'll be with Olaf.\"\n\n\"I advised him to move,\" sez the Friar; \"but he's set in his ways.\"\n\n\"Self-control,\" sez I. \"I was workin' in a mine once with a mule and a\nHungarian; and both of 'em had an unusual stock o' self-control. One\nday right after a fuse had been lit, the mule decided to rest near the\nspot; an' the Hun decided to make the mule proceed. We argued with 'em\nas long as it was safe; but the mule had his self-control an' all four\nfeet set, and the Hun was usin' _his_ self-control an' a shovel.\nAll we ever found was the mule's right hind leg stickin' through the\nHungarian's hat, and we buried these jus' as they was.\"\n\nThe Friar sighed, pursed up his lips, and sez: \"I wish I could help\nhim.\"\n\n\"Help him all you can, Friar,\" sez I; \"but after the fuse is burnin',\nyou pull yourself out to safety. Ty Jones could easy spare you without\ngoin' into mournin'.\"\n\nThe Friar rode on about his business, an' me an' Horace went back to\nthe ranch, him pumpin' me constant for further particulars about Olaf\nan' Kit. \"Horace,\" sez I finally, \"did you ever see these folks?\"\n\n\"I never did,\" sez he.\n\n\"Then,\" sez I, \"what you got again' 'em 'at you want 'em to marry?\"\n\n\"Marriage,\" sez he with the recklessness common to old bachelors, \"is\nthe proper condition under which humans should live--and besides, I\ndon't like what you tell about Ty Jones.\"\n\nFrom that on, Horace began to talk hunt; and when Horace talked\nanything, he was as hard to forget as a split lip. He had brought out\nsome rifles which the clerk had told him would kill grizzlies on\nsight, and Horace had an awful appetite to wipe out the memory o' that\nwoodchuck.\n\nI admit that no one has any right to be surprised at anything some one\nelse wants to do; but I never did get quite hardened to Horace Walpole\nBradford. When ya looked at him, ya knew he was a middle-aged man with\nside-burn whiskers; but when ya listened to his talk, he sounded like\na fourteen-year-old boy who had run away to slaughter Injuns in\nwholesale quantities.\n\nAll of his projecs were boyish; he purt' nigh had his backbone bucked\nup through the peak of his head before he'd give in that ridin' mean\nones was a trade to itself; and the same with ropin', and several\nother things. It ground him bitter because his body hadn't slipped\nback as young as his mind, an' he worked at it constant, tryin' to\nmake it so.\n\nHe wore black angora chaps, two guns, silver spurs, rattlesnake\nhat-band, Injun-work gauntlets, silk neckerchief through a silver\nslip, leather wristlets, an' as tough an expression as he could work\nup; but the one thing of his old life he refused to discard was his\nside-burns. Sometimes he'd go without shavin' for two weeks, an' we'd\nall think he was raisin' a beard; but one day he'd catch sight of\nhimself in a lookin'-glass, an' then he'd grub out the new growth an'\nleave the hedge to blossom in all its glory.\n\nWe were long handed for the winter as usual, an' the' wasn't any\nreason why we couldn't take a hunt; so Tank an' Spider egged him on,\nan' I wasn't much set again' it myself. Horace agreed to pay us our\nwages while we were away, an' offered Jabez pay for the hosses; but o'\ncourse he wouldn't listen to it; and for a few days he even talked\nsome o' goin' with us, though he didn't ever care much for huntin'.\n\nFinally we started out with a big pack train an' enough ammunition for\nan army. Besides me an' Horace, the' was Tank, Spider Kelley, Tillte\nDutch, an' Mexican Slim. Slim was to do the cookin', an' the rest of\nus were to divvy up on the other chores all alike, Horace not to be\ntreated much different simply because he was payin' us our wages; but\nhe was to have the decidin' vote on where we should go an' how long\nwe'd stay. It was fine weather most o' the time, though now an' again\nwe'd get snowed up for a day or so in the high parts.\n\nI had allus felt on friendly terms with the wild creatures; an' I had\ntold him before we started that I wouldn't have no part in usin'\nhosses for bear-bait, nor shootin' bears in traps, nor killin' a lot\no' stuff we had no use for; but Horace turned out to be as decent a\nhunter as I ever met up with, an' after the second day out he did as\nlittle silly shootin' as any of us. He wasn't downright blood-thirsty,\nlike a lot of 'em who get their first taste too late in life. He cared\nmore for the fun o' campin' out an' stalkin' game than he did for\nkillin'. We only got one silver-tip, most of 'em havin' holed up; but\nwe found all the other game we wanted. Horace killed the grizzly,\nwhich was a monster big one, and this wiped the woodchuck off his\nrecord, and inflated his self-respect until the safety valve on his\nconceit boiler was fizzin' half the time.\n\nWe made a permanent camp not far from Olaf's shack, an' it didn't take\nme long to see 'at the foxy Horace was more interested in Olaf an' his\nwar with Ty Jones than he was in huntin'. As soon as we had our camp\narranged, he got me to take him over to Pearl Crick Spread to call on\nOlaf. I told him that Olaf wasn't what you'd call sociable; but he\ninsisted, so we went.\n\nWe found Olaf in an infernal temper, an' some tempted to take it out\non the first human he met; but this didn't phaze Horace. He thought he\ncould start Olaf by tellin' him that Kit Murray was a widow; but the\nFriar had already told him and Olaf wouldn't thaw worth a cent. He\nkept on askin' questions, even when they wasn't answered, until Olaf\ngot hungry an' asked us in to eat dinner with him. After we had eaten,\nwe sat around the fire smokin', an' Horace looked as contented as a\ncat. He kept at his questionin' until he got Olaf to talkin' freer 'n\nI had supposed he could talk.\n\nHorace tried him out on all sorts o' things, an' when Olaf snubbed\nhim, why, he just overlooked it an' tried somethin' else. Finally he\ntried his hand at religion, an' this was what loosened Olaf up. Now\nOlaf was actually religious, and called himself a Christian, but the'\nwas a heap o' difference between his brand o' it an' the Friar's.\n\nOlaf's God took more solid satisfaction in makin' hell utterly\ninfernal than a civilized community takes in a penitentiary; an' Olaf\nwas purty certain as to who was goin' there. When he got to talkin'\nreligion in earnest, his face grew hard an' his eyes bright, an' he\ngloated over the souls in torment till he showed his teeth in a grin.\nThe' wasn't any doubt in his mind that Ty Jones was goin' to be among\nthose present, an' this led him into tellin' what had put him so far\nout o' humor before we'd come along.\n\nHe had found another one of his cows shot an' only a couple o' steaks\ncut off. He fair frothed at the mouth when he told us this, an' he\ndidn't make any bones of givin' Ty the credit for it. He cut loose an'\ntold us a string o' things 'at he knew about Ty, an' ya couldn't blame\nhim for feelin' sore. He talked along in a rush after he got started,\ntellin' o' the way 'at Ty changed brands an' butchered other fellers'\nstock an' wasn't above takin' human life when it stood in his way. \"He\nmade me as big a devil as he is,\" sez Olaf; \"an' now he knows 'at I\ncan't get any backin'; so he is just persecutin' me; but some o' these\ndays, I'll get a chance at him.\"\n\nHorace had dropped into a silence while Olaf was talkin'; but now he\nraised a finger at me, an' said: \"I'll tell you what we'll do: instead\nof huntin' ordinary wild beasts, we'll just keep watch on Olaf's\nstuff, an' when any one bothers it, why, we'll take 'em into some town\nwith a jail.\"\n\nOlaf shook his head, an' I told Horace that the' wasn't any law for\nbig cattle men; but Horace was all worked up, an' after we'd left Olaf\nan' started for camp, he didn't talk of anything else. He put it\nbefore the boys; but they were all again' it, an' told him a lot o'\ntales about fellers who had tried to buck the big cattle men. Horace\ncalled us all cowards; but we only laughed at his ignorance an' let\nhim carry on as far as he liked. He sat up way into the night broodin'\nover it, an' from that on he did a lot o' scoutin' on his own hook. We\nused to keep an eye on him, though; so after all he had his own way\nabout it, an' Olaf's stuff was watched purty close.\n\nThe boys was proud of Horace, just as they'd have been proud of a\nfightin' terrier; but they was worried about him, too, in just about\nthe same way.\n\n\"I tell you, that little runt would shoot to kill if he got a chance,\"\nsez Tank Williams, one night while Horace was away.\n\n\"Aw ya can't tell,\" sez Spider. \"He thinks he would; but he's never\nbeen up against it yet, an' ya can't tell.\"\n\n\"Well, what if he did shoot,\" sez Slim, \"we wouldn't have to mix in,\nwould we?\"\n\n\"You know blame well we'd mix in,\" sez Tank, \"an' you can't tell where\nit would end. If Horace had 'a' come out here when he was a kid, he'd\n'a' turned out one o' the bad men for true. It's in his blood. Look at\nhim! when he came here first, he didn't have no more get-up 'n a sofy\npiller; but look what he's gone through since. I saw him, myself,\nmarch along without food for four days, an' when we came up with that\ncow, he was willin' to help kill her with a rock or strangle her to\ndeath, an' he didn't make no more bones o' calf-milkin' her than a\ncoyote would. He started out in life with more devilment in him 'n any\nof us, an' what he's achin' for now is a mix-in with the Cross brand\noutfit. That's my guess.\"\n\n\"An' that's my guess,\" I chimed in; but just then we heard two shots\nclose together, then a pause an' three more shots. We jammed on our\nhats an' guns an' rushed outside. It was a moonlight night, an' we\nhustled in the direction o' the shots. Before long we made out Horace\nan' Tillte Dutch comin' towards us, an' Horace was struttin' like\nCupid the bulldog used to walk, after he'd flung a steer. It was the\nfirst time I'd ever noticed this, but I noticed it plain, out there in\nthe moonlight.\n\n\"What's up?\" I asked.\n\n\"I reckon 'at somebody knows by now that Olaf's stuff is havin' a\nlittle interest took in it,\" sez Horace.\n\nWe came back into the old log cabin where we was campin', an' Dutch\ntold about how Horace had got him to walk with him, an' had sat down\non a rock where they could see Olaf's little bunch o' cattle grazin'.\nHe said 'at Horace sat with his rifle across his lap and kept watch\nlike an Injun scout.\n\nAfter a time they saw two men creep out of a ravine not far from where\nthey was sittin' an' sneak down on the bunch o' cows. One of 'em had\nshot a cow, an' Horace had shot him, bringin' him down, but not\nkillin' him. The two had run for the ravine, an' Horace had tried to\ncut 'em off, an' he had gone along 'cause Horace had; but the two had\ngot to their hosses first. Each o' the two had taken one shot, an'\nHorace had shot back but none o' these last shots had hit anything,\nan' the two had got away.\n\n\"I'll bet they haven't got so far away but what we'll hear from 'em\nagain,\" sez Tank.\n\n\"The thing for us to do is to start back to the Diamond Dot,\" sez I.\n\n\"We shall stay here, an' see what happens,\" sez Horace, lightin' his\npipe. His eyes were dancin' an' he was all puffed up. I didn't say any\nmore. I just looked at him. He was the same old Horace, side-burns an'\nall; but still the' was enough difference for me to begin to regret\nhavin' give him the treatment. I had cured his nerve so complete it\nseemed likely to boss the whole crowd of us into trouble.\n\n\n\n\nCHAPTER TWENTY-TWO\n\nA PROGRESSIVE HUNT\n\n\nThe Friar sez it's all rot about men bein' better for havin' sowed\ntheir wild oats when young. He sez 'at it's utter foolishness to sow\nany crop ya don't want to harvest; but I dunno. I don't mind havin' a\ncolt try to turn himself inside out with me on its back; but I'm some\nprejudiced again' an old hoss which is likely to pitch when I've got\nother business to attend to. When a young hoss is mean, why, ya can\nreason it out of him; but when an old hoss turns bad, you might just\nas well put the outlaw label on him an' turn him adrift.\n\nWe couldn't do a thing with Horace after he'd taken his shot at the\nfeller who potted one of Olaf's cows. Ol' Tank Williams was huge in\nsize an' had a ponderous deep voice which rumbled around in him like a\nbulldog croakin' in a barrel; an' he decided that it was his duty to\nbe firm with Horace, seein' the way 'at he had bluffed him when we\nwent on that trip for the nerves; so the follerin' mornin' he put a\nscowl on his face, grabbed Horace by the chest of his shirt, lifted\nhim so 'at nothin' but the tips of his toes touched, an' sez: \"Look\nhere, you little whippersnapper, we agreed to go where you said an'\nstay as long as you said; but we meant on a game-huntin' trip. You\nhaven't any idee what you're up again' out here, an' you got to give\nin an' come back with us.\"\n\nTank's free eye rolled about in his head, runnin' wilder 'n I'd ever\nseen it; but Horace wasn't as much phazed as if a fly had bit him. He\nscowled down his eyebrows, an' piped out in his squeaky tenor: \"Take\nyour hand off me, Tank--and take it off now.\"\n\n\"I've a notion to raise it up an' squash ya,\" sez Tank.\n\n\"Yes,\" sez Horace, without blinkin' a winker, \"you've got notions all\nright; but they lie so far to the interior of ya that they generally\nweaken before they find their way out. Take your hand off me.\"\n\nWell, Tank was beat. He gave Horace a shove, but Horace was light on\nhis feet, an' he never lost his balance. He just danced backward until\nhe had his brakes set, an' then he fetched up in front o' the fire,\nput his fists on his hips, an' stared up at Tank haughty.\n\n\"Ignorance,\" sez he, \"is the trouble with most people. The ignorant\nallus judge by appearances. If body-size was what really counted, why,\nwe'd have an elephant for an emperor. Instead of which we use 'em to\npush logs around. Goliath did a lot o' talkin' about squashin' David,\nbut as soon as David got around to it, he fixed Goliath all ready for\nthe coroner. Napoleon was of small size, an' fat, an' nervous, but he\ndidn't count it a fair day's work unless he had presented one of his\nrelatives with a full-sized kingdom. Where are the buffalos--where are\nthey--the big clumsy brutes! They're shut up out o' harm's way, that's\nwhere they are; but where are the mosquitoes? Why the mosquitoes are\ntakin' life easy at all the fashionable summer resorts. If you feel\nlike freightin' your big, fat carcass back to where it don't run any\nrisk o' bein' bumped into, why go ahead; but I'm goin' to stick around\nhere an' see what happens.\"\n\nWell, there we were: we didn't none of us have the courage to own up\n'at we were afraid of anything 'at Horace wasn't afraid of; so we\ndecided to stick with him, but that he had to take the blame. It was\nTillte Dutch who said this, an' Horace looked at him an' grinned.\n\"Take the blame?\" sez he. \"Why you big chump, it's the small-sized men\nwho allus take the blame. The big boobs rush about, makin' a lot o'\nnoise; but they only do what the small-sized men tell 'em to. I'll\ntake the blame all right, an' if you back me up, you'll be right\npleased to have a share in the kind o' blame the's goin' to be. This\nTy Jones outfit is nothin' but a set o' cowardly bullies who sneak\naround in the dark doin' underhanded work; but I intend to let the\ndaylight in.\"\n\n\"I'll bet the daylight will be let in, somewhere,\" sez I; \"but I'm\njust fool enough to stick with ya.\"\n\nTank was still smartin' from the way it had been handed to him. \"Say,\"\nsez he, \"p'raps you don't know it; but that David you was cacklin'\nabout a while ago wasn't nothin' but a sheep-herder.\"\n\n\"That don't change no brands,\" sez Horace, who didn't have any more\nuse for a sheep-herder 'n we did. \"He was a small-sized man, an' he\njust drove sheep a while to help his father out. Sheep-herdin' wasn't\nhis regular trade. Bossin' men an' fightin' an' bein' a king was his\nnatural line o' business. It allus seems to me 'at big, overgrown men\nought to be sheep-herders, so they could drive about in house-wagons,\nan' not wear down so many good hosses.\"\n\nOl' Tank slammed about, makin' a lot o' noise; but he had lost this\ndeal, an' it was plain to see.\n\n\"I'm goin' to ride over to Olaf's, an' tell him about what happened\nlast night, an' say 'at we'll keep an eye on his stuff if so be he\nwants to take a little trip to Billings,\" said Horace; and when he\nstarted I went along with him. At first Olaf was so white-hot about\nhavin' another cow killed that he couldn't think; but finally he\nlooked at Horace a long time, an' said: \"You have very brave flame,\nan' you speak true. I shall go to Billings, an' trust everything with\nyou.\"\n\nI was flabbergasted clear out o' line at this; but Olaf packed some\nstuff on one hoss, flung his saddle on another, an' set off at once.\nNow, I knew Olaf to be slow an' stubborn, an' I couldn't see through\nthis.\n\nAfter Olaf had rode out o' sight to the north, Horace sez: \"Has he\nallus been crazy?\"\n\n\"He's not crazy,\" sez I.\n\n\"Then what did he mean by sayin' I had a very brave flame an' that I\nspoke true?\" sez Horace. \"Course he's crazy. Didn't you notice his\neyes.\"\n\n\"Yes,\" I sez, \"I've noticed his eyes a lot; but I don't think he's\ncrazy--except in thinkin' 'at Kit Murray'll marry him. Why, she would\nas soon think o' marryin' a he-bear as Olaf.\"\n\n\"Well, I think they have drove him crazy,\" sez Horace; \"but I'm goin'\nto bestir myself in his favor.\"\n\nHe took himself as serious as if he had been Napoleon an' David both;\nan' I could smell trouble plain. We decided to move our camp down to\nOlaf's, an' wrangle his herd into the Spread every night. Pearl Crick\nSpread was as fine a little valley as a body ever saw; filled with\ncottonwoods an' snugglin' down out o' the wind behind high benches.\nThe crick came in through a gorge, an' went out through a gorge; an'\nit was plain to me that the Spread was worth fightin' for.\n\nWhen we got back to the camp we found that a couple o' Cross brand\nboys had happened along, by accident, of course, an' were tryin' to\nswap news o' the weather for news o' the neighbors. Our crowd hadn't\nloosened up none; and as soon as we came back the Cross-branders left.\n\nHorace looked pleased. \"I bet I got one of 'em last night,\" sez he,\nshakin' his head.\n\nWell, we all grinned, we couldn't help it. \"I bet you get another\nchance at 'em, too,\" sez Slim. Our outfit had been peaceable for so\nlong that the prospect of trouble actually made us feel nervous enough\nto show it.\n\nWe moved down to Olaf's, and each night we fetched in his little bunch\no' cows, an' allus kept up some hosses in the corral. The\nCross-branders used to wander by our place purty frequent, but allus\nin the matter o' business.\n\nOne day, after we'd been livin' at Olaf's about a week, Badger-face\nFlannigan, an' a pair of as mean-lookin' Greasers as ever I saw, came\nridin' along. Me an' Horace had been up in the hills after some fresh\nmeat, an' we see them before they saw us. They were ridin' slow an'\nsnoopin' about to see what they could pick up, an' when they saw us\nthey looked a bit shifty for a moment.\n\nThen Badger wrinkled up his face in what was meant for a friendly\ngrin, an' sez: \"Hello, fellers. Have you-un's bought Olaf out?\"\n\n\"Nope,\" sez I. \"We're just out here for a little huntin'; an' Olaf got\nus to look after his stuff for a few days while he went visitin'.\"\n\n\"Wasn't the' any huntin' closer to home?\" sez Badger-face, a little\nsarcastic.\n\n\"Not the kind o' huntin' we prefer,\" sez Horace, sort o' dreamy like.\n\nBadger-face drilled a look into Horace, who had put on his most\nno-account expression. \"What's your favorite game,\" sez he, \"snow-shoe\nrabbits?\"\n\n\"Oh, no,\" drawled Horace as if he felt sleepy, \"silver-tips an' humans\nis our favorite game; but o' course the spring is the best time--for\nsilver-tips.\"\n\n\"Where might you be from?\" asked Badger-face.\n\n\"I might be from Arizona or Texas,\" sez Horace; \"but I ain't. I'm a\nregular dude. Can't you tell by my whiskers?\"\n\nBadger-face was so puzzled when Horace gave a little rat-laugh that I\nhad to laugh too; and ya could see the blood come into Badger's\ncheeks, but still, he couldn't savvy this sort o' game, so he couldn't\nquite figure out how to start anything.\n\nHorace had practiced what he called a muscle-lift, which he said he\nused to see the other kids do on parallel bars; and now he slipped to\nthe ground an' tightened his cinch an' cussed about the way it had\ncome loose, as natural as life. Then he put one hand on the horn an'\nthe other on the cantle an' drew himself up slow. He kept on pushin'\nhimself after his breast had come above the saddle until he rested at\narm's length. Then he flipped his right leg over, an' took his seat as\nthough it was nothin' at all. Any one could see it was a genuwine\nstunt, though it was of no earthly use to a ridin' man.\n\nNow, just because the' was no sense to this antic, it made more of an\nimpression on Badger-face than the fanciest sort o' shootin' or ropin'\nwould 'a' done; an' he puzzled over what sort of a speciment Horace\nmight be, till it showed in his face.\n\n\"Come on down an' have supper with us,\" sez Horace. \"You can see for\nyourself what the prospect for fresh meat is; so you can be sure of a\nwelcome.\"\n\n\"No, we can't very well come this evenin',\" sez Badger-face.\n\n\"Why not?\" sez Horace. \"You look to me like a man who was gettin'\nbilious for the want of a little sociability. Come on down an' we'll\nswap stories, an' have a few drinks, an' I'll sing ya the best song\nyou ever hearkened to.\"\n\n\"No, we got to be goin',\" sez Badger-face; an' he an' the Greasers\nrode off while Horace chuckled under his breath as merry as a magpie.\n\n\"That's what you call a bad man, is it?\" sez he. \"I tell you that\nfeller's a rank coward.\"\n\n\"Would you have the nerve to pick up a horn-toad?\" sez I.\n\n\"No,\" sez he; \"cause they're poison.\"\n\n\"They ain't no more poison 'n a frog is,\" I sez; \"but most people\nthinks they are, an' that is why strangers are afraid of 'em. Now,\nBadger-face ain't no coward. He's a shootin' man; but he can't make\nyou out, an' this is what makes him shy of ya.\"\n\n\"Well,\" sez Horace, \"I'd rather be a free horn-toad than a mule in\nharness. Come on, let's go eat.\"\n\nThe next afternoon Horace went along to help bring in the bunch o'\ncattle; an' some one up on the hill took a shot at him. He couldn't\nride up the hill, so he hopped off the pony, an' started up on foot.\nMexican Slim was closest to him, an' he started after; but the feller\ngot away without leavin' any trace. Horace was wonderful pleased about\nit, an' strutted more than common.\n\n\"There now,\" sez he after supper; \"do you mean to tell me 'at that\nfeller wasn't a coward? Why the' ain't enough sand in their whole\noutfit to blind a flea!\"\n\nWe just set an' smoked in silence. When a feller as little as him once\nbegins to crow, the's nothin' to do but wait till his spurs get\nclipped.\n\n\n\n\nCHAPTER TWENTY-THREE\n\nA LITTLE GUN-PLAY\n\n\nIt's curious how hard it is, sometimes, to get trouble started. We all\nknew 'at the Cross-branders was ready to clean us out, an' itchin' for\nthe job; but the's one curious little holdback in the make-up of every\nhealthy animal in the world. Every sane animal the' is wants\nself-defence as his excuse for takin' life. I admit that now and again\nbeasts an' men both get a sort o' crazy blood-lust, an' just kill for\nthe sake of it; but it's the rare exception.\n\nOne of us allus made it a point to go along with Horace; an' most\ntimes when we'd meet up with any o' the Cross-branders, they'd never\nmiss the chance to fling some polite smart talk at him; but the little\ncuss could sass back sharper 'n they could, an' I reckon they was\nsuspicious that he wouldn't 'a' been so cool if he hadn't had bigger\nbackin' than was in sight. It was perfectly natural to think 'at he\nhad been sent out as a lure by some big cattle outfit, or even the\ngovernment; so they went cautious till they could nose out the game.\n\nOne day Badger-face an' the two Greasers came along when Horace was\nout ridin' with Tillte Dutch. Dutch was one o' these innocent-lookin'\nGermans--big, wide-open eyes, a half smile, an' a sort of a leanin' to\nfat. He never had but one come-back to anything--which was to\nsplutter; but he was dependable in a pinch.\n\n\"Whatever made you so unspeakable little?\" sez Badger-face to Horace.\n\nHorace looked behind him, an' all about, an' then sez in surprise:\n\"Who, me?\"\n\n\"Yes, you,\" sez Badger-face. \"You seem to dry down a little smaller\neach day.\"\n\n\"Well,\" sez Horace, speakin' in a low secret-tellin' tone, \"I'll tell\nya; but I don't want ya to blab it to every one ya see. When I was a\nyoung chap, I used to go with a big, awkward, potato-brained slob,\nabout your size. I could out-shoot him, out-ride him, run circles\naround him, an' think seven times while he was squeezin' the cells of\nhis brain so they'd touch up again' each other; but one day he made a\nbet that he could eat more hog-meat 'n I could; an' he won the bet.\nWhen I found out that the' was one single thing 'at this big,\nloose-jointed galoot could beat me at, I felt so blame small that I\nnever got over it, an' this is why I disguise myself in these\nwhiskers.\"\n\nThe two Greasers couldn't help but grin, an' the fool Dutchman\nsniggered. This was more 'n Badger-face could stand. He shot his hand\nacross an' pulled his gun quick as a flash; but Horace didn't move, he\njust sat still, with a friendly smile on his face; an' Badger-face sat\nthere with his gun in his hand, scowlin' jerk-lightnin' at him.\n\nSpider an' Slim had gone after meat that day, an' they came into view\nwith the carcase of a doe, just as Badger drew his gun. Me an' Tank\nwas listed to wrangle in the bunch, an' we came ridin' along just\nafter the other two came into view. The Greasers gave a little cough\nan' Badger-face looked up an' saw us. It looked like a put-up job, all\nright; an' chariots of fire, but he was mad! Pullin' a gun on a man is\nthe same as shootin' at him. Badger-face had been tricked into givin'\nus just grounds to slaughter him, and he wasn't quite sure what move\nto make next. Our outfit had been purty well advertized, through\ncleanin' out the Brophy gang, me an' Mexican Slim were both two-gun\nmen an' known to be quick an' accurate, while Tank was ever-lastin'ly\ngettin' into trouble, owin' to his friendly feelin's for liquor. As we\ndrew closer we made our smoke-wagons ready, while his two Greasers\nkept their hands in plain view, and harmless.\n\nBadger had a trapped look in his face; but he didn't say anything, an'\nhe didn't cover Horace with his gun; he just held it ready. We did the\nsame with ours, an' it was the foolest lookin' group I was ever part\nof. Ol' Tank was the one who finally started things. \"Look here,\nBadger-face,\" he bellowed, \"if you so much as harm a hair o' those\nblamed ol' whiskers, why, we'll have to put ya out o' business.\"\n\nHorace turned an' looked at Tank in surprise. \"Aw, put up your gun,\"\nhe said. \"Badger-face ain't in earnest. We had an argument the other\nday: I said 'at a man lost time crossin' his hand to pull his gun, an'\nhe said it could be done quicker that way 'n any other; so to-day he\njoked me about bein' as small in the body as he is in the brain, an' I\ncame back at him, also jokin' in a friendly way; an' he took this\nexcuse to pull his gun on me, without any ill intent; but only to\nprove how quick he could do it. It stuck in his holster, though; an'\nif we'd been in earnest, I'd have had to kill him. I've had him\ncovered all this time; but you can see for yourselves 'at his gun\nain't cocked. Now put up your guns, and next time, don't be silly.\"\n\nI know 'at Horace didn't have any gun in his hand when we came up; but\nwhen he stopped speakin', he pulled his hand with a cocked gun in it\nout from under his hoss's mane, an' Badger-face was the most surprised\nof any of us.\n\n\"Come on down to supper, Badger-face, an' I'll sing ya my song,\" sez\nHorace. \"We allus seem to have fresh deer meat when you happen along.\"\n\nWe all put up our guns along with Badger-face, an' he mumbled some\nsort of an excuse an' rode away with the Greasers. O' course we'd\nought to 'a' killed him right then, 'cause he was more full o' hate\nthan a rattler; but the simple truth was, that Horace had gained\ncontrol over us complete, an' we let him have his way.\n\n\"When did you get that gun in your hand, Horace?\" I sez to him after\nsupper. \"You didn't have no gun when I rode up.\"\n\n\"That's what's puzzlin' Badger-face right this minute,\" sez Horace. \"I\ndidn't draw that gun until Tank made his talk; but at the same time I\nwasn't as defenceless as I looked. I have told you all the time 'at\nthat man didn't have the nerve to harm me. He's a coward.\"\n\n\"I reckon you'll be killed one o' these days, still believin' that,\"\nsez ol' Tank. \"How much fightin' experience have you ever had?\"\n\n\"How much did Thesis ever have?\" asked Horace.\n\n\"Never heard of him,\" sez Tank. \"Who was he?\"\n\n\"He was a Greek hero,\" sez Horace. \"He never had had a fight till he\nstarted out to go to his father; but he cleaned out all the toughs\nalong the way, an' when he reached his father, who was king of Athens,\nhe found 'em just ready to send out seven young men an' seven maidens,\nwhich they offered up each year to the Minnietor, which was a beast\nwith the body of a man, and the head of a bull, just like Badger-face.\nThesis volunteered, an' what he did was to kill the Minnietor an' end\nall that nonsense.\"\n\n\"Well, I never heard tell o' that before, an' I don't more 'n half\nbelieve it now,\" sez Tank; \"but I'm willin' to bet four dollars 'at\nthe Minnietor didn't know as much about gunfightin' as what\nBadger-face does. He'll get ya yet, you see if he don't.\"\n\n\"Tell ya what I'm game to do,\" sez Horace. \"I'm game to go right to Ty\nJones's ranch house alone. Do ya dare me?\"\n\n\"No, you don't do that,\" sez I. \"That's a heap different proposition.\nTy Jones wouldn't pull his gun without shootin'; and besides, he'd\nmost likely set his dogs on ya.\"\n\n\"Well, I own up 'at I don't want no dealin's with dogs,\" sez Horace,\nthoughtful. \"Dogs haven't enough imagination to work on. If they're\ntrained to bite, why, that's what they do; but give a human half a\nchance, an' he'll imagine a lot o' things which are not so. You\ncouldn't tell Badger-face a big enough tale about me to make him doubt\nit. I tell ya, I got him scared.\"\n\nWe didn't argue with him none; the' was some doubt about him havin'\nBadger-face fooled; but the' wasn't any doubt about him havin' himself\nfooled--which is the main thing after all, I reckon. Anyway, we let\nHorace sit there the whole evenin', tellin' Greek-hero tales which\nmust have blistered the imagination o' the feller 'at first made 'em\nup.\n\nAlong about nine o'clock we began to stretch an' yawn; but before we\ngot to bed, Mexican Slim said 'at he heard a noise at the corral, an'\nwe all looked at one another, thinkin' it was the Cross-branders; but\nHorace was the first one to get back into his boots an' belt; an' he\nalso insisted on bein' the first to open the door, which he did as\nsoon as we blew out the candle. Then we all filed out an' sneaked down\ntoward the corral; but first thing we knew, a voice out o' the dark\nwhispered: \"This is me--Olaf. Is everything all right?\"\n\nWe told him it was, an' he whistled three times. You could 'a' knocked\nme down with a feather when Kit Murray an' the Friar came ridin' up;\nan' then we turned the ponies loose an' went into the house. It only\nhad two rooms, countin' the lean-to kitchen, an' we made consid'able\nof a crowd; but we were all in good spirits, on account of Olaf\ngettin' the girl an' us bein' able to hand him back his stuff with not\none head missin'.\n\nIt had been some interval since I'd seen Kit Murray, an' I was\nsurprised to view the change in her. She didn't look so much older,\nbut all the recklessness had gone out of her face, an' it had a sort\nof a quiet, holy look about it. \"Kit,\" I sez, \"I wish ya all the joy\nthe' is; but I'd 'a' been willin' to have bet my eyes 'at you'd never\ntake Olaf. I was glad to see him go up after ya, 'cause gettin'\nknocked on the head is some better 'n bein' kept hangin' on a hook;\nbut you sure got your nerve with ya. This homestead is purty likely to\nget in some other folks' way.\"\n\nKit had as snappy a pair o' black eyes as was ever stuck in a face;\nand now they flashed out full power. \"I know it's goin' to be hard to\nhold this place,\" sez she, \"but I reckon I can help a little. I can\nride an' shoot as well as a man, if I have to, and you know it. I\ndon't want anything but the quietest sort of a life the' is; but I'm\nready to stand for any sort o' luck 'at comes along. As for Olaf, he's\nthe only man in the world for me. I saw something o' the big cities\nback east, an' Billings, an' the boys on the range here, and out of\n'em all, Olaf's my man. The thing I hope more 'n anything else is,\nthat we can die together.\"\n\nHer voice caused a hush to come to the room. I had meant to be jovial\nan' hearty; but the' was an undercurrent of earnestness in her voice\nwhich put a tingle into a feller. Kit Murray had changed a heap, but\nall for the better.\n\nOlaf cleared his throat, an' we all took a look at him. He had\nchanged, too. He had lost the chained-bear look he generally wore, an'\nthe' was a light o' pride an' satisfaction in his face which was good\nto look upon. \"Boys,\" he said, \"I've been purty tough an' unsociable,\nan' I don't see why you've took so much trouble for me; but I tell ya\nright here that I stand ready to square it in any way or at any time I\ncan. Now, it seems mighty funny 'at Kit Murray should love me, an' I\ncan't account for it any more 'n you can; but I knew right from the\nstart that she did love me--I could tell by the light. If ever the\ntime comes that she don't love me any more, I get out of her way,\nthat's all about that; but I'm not goin' to make her stay here any\nlonger 'n I have to. I sell out when I get the first chance. Friar\nTuck, he softened my heart, an' he watched over her. He's a man.\nThat's all I can say.\"\n\nWell, this was an all-around noble speech for a stone image like Olaf\nhad been, an' we cheered him to the echo; but Horace had sort o' been\njostled to the outside an' forgot. Now, he come forward an' shook Olaf\nby the hand an' congratulated him, an' sez: \"The's one thing I'd like\nmightily to know, an' that is--what the deuce do you mean by this\nlight you're allus alludin' to?\"\n\nOlaf was some embarrassed; but it never seemed to fuss Horace any when\nhe had turned all the fur the' was in sight the wrong way; so he just\nwaited patiently while Olaf spluttered about it.\n\n\"I don't know myself,\" sez Olaf. \"Always, since I was a little child,\nI have seen a floating light about people. I thought every one saw\nthis light an' I spoke of it when I was a child an' asked my mother\nabout it many times; but at first she thought I lie, an' then she\nthought my head was wrong; so I stopped talkin' about it; but always I\nsee it an' it changes with the feelings and with the health. All the\ncolors and shades I cannot read, but some I know. I knew that Kit\nMurray loved me before she knew it, and I knew that the Friar was a\ntrue man when they told me tales of him. Animals, too, have this\nfloatin' light about 'em, an' I can tell when they are frightened an'\nwhen they are mean. This is why I handle hosses without trouble. Now I\ndo not know why my eyes are this way; but I have told you because you\nhave been good friends to me. I do not want you to tell of this\nbecause it makes people think I am crazy.\"\n\n\"Course it does,\" sez Horace. \"It made me think you were crazy. I\nnever heard of anything like this before. Tell me some more about it.\"\n\n\"There is no more to tell,\" sez Olaf. \"When I see the flame I do not\nsee the people. The flame wavers about them, and sometimes I have seen\nit at night, but not often. I do nothing to make myself see this way.\nAlways my eyes did this even when I was only a baby.\"\n\n\"Well, you have everything beat I ever saw yet,\" sez Horace. \"What do\nyou think o' this, Friar?\"\n\n\"I never heard of such a case,\" sez the Friar; \"although it may have\nbeen that many have had this gift to some extent. I think it is due to\nthe peculiar blue of Olaf's eyes. I think that this blue detects\ncolors or rays, not visible to ordinary eyes. I wish that some\nscientist would study them.\"\n\n\"I'll pay your way back East, Olaf,\" sez Horace, \"if you'll have your\neyes tested.\"\n\n\"No, no,\" sez Olaf, shakin' his head. \"I don't want to be a freak.\nWhat is the use? I can not tell how I do it, so it cannot be learned;\nand I do not want things put into my eyes for experiments. No, I will\nnot do it.\"\n\n\"Tell me how Badger-face looks to you,\" sez Horace.\n\n\"Oh, he is bad,\" sez Olaf. \"He has the hate color, he loves to kill;\nbut he is like the wolf; he does not like the fight, he wants always\nto kill in secret.\"\n\n\"I bet my eyes are a little like yours,\" sez Horace, noddin' his head.\n\"I knew 'at Badger-face was this way as soon as I saw him.\"\n\n\"Oh, here now,\" sez the Friar. \"You are puttin' down a special gift to\nthe level of shrewd character-readin'.\"\n\n\"What sort of a flame does a dead person have, Olaf?\" sez Horace.\n\nA queer look came into Olaf's face, a half-scared look. \"A dead person\nhas no flame,\" sez he, with a little shudder. \"It is a bad sight. I\nhave watched; I have seen the soul leave. When a man is killed, the\nsavage purple color fades into the yellow of fear, then comes the\nblue, it gets fainter and fainter around the body; but it gathers like\na cloud above, and then it is silver gray, like moonshine. It is not\nin the shape of the body, it is just a cloud. It floats away. That is\nall.\"\n\n\"Well, that's enough,\" sez Horace. \"Can you see any flame about a\nsleeping person?\"\n\n\"Yes,\" sez Olaf, \"just like about a waking person; and there is marks\nover a wound or a sick place.\"\n\n\"Well, Mrs. Svenson,\" sez Horace to Kit, \"you'll have to be mighty\ncareful or your husband will find you out.\"\n\n\"I am perfectly willin',\" sez Kit with a proud little smile. She was\ngame, all right, Kit was.\n\n\"That is why I say it is all right,\" sez Olaf. \"She is young, she\ncannot know how she will change. If ever she no longer love me, I will\nnot bother her. That would be a foolishness; but so long as she love\nme, no other man will bother her. That would be devilishness!\"\n\n\"You certainly have a nice, simple scheme of life,\" sez Horace. \"If\never you change your mind, I'll put up the money to take you back\nEast, an' pay you high wages.\"\n\n\"No,\" sez Olaf, \"I hate circuses an' shows, an' such things. I not\ngo.\"\n\n\"You say you can tell sick places, an' fear, an' hate, an' honesty,\"\nsez Horace. \"Now, when I came out here, I was just punk all over. You\ngive me a look-over, an' tell right out what you see.\"\n\nAt first Olaf shook his head, but we finally coaxed him into it; an'\nhe opened his eyes wide an' looked at Horace. As he looked the blue in\nhis eyes got deeper an' deeper, like the flowers on the benches in\nJune, then when the pupil was plumb closed, the blue got lighter\nagain, and he said: \"You have not one sick point, you have good\nthoughts, you are very brave, you are too brave--you are reckless. You\nhave very great vitality, an' will live to be very old--unless you get\nkilled. I knew an old Injun--over a hundred years old he was--he had a\nflame like yours. It is strange.\"\n\nYou could actually see Horace swellin' up with vanity at this; but it\nmade ol' Tank Williams hot to see such a fuss made about a\nsmall-caliber cuss; so he rumbles around in his throat a minute, an'\nsez: \"Well, you fellers can fool around all night havin' your souls\nmade light of, if ya want to; but as for me I'm goin' to bed.\"\n\nKit insisted that we sleep on the floor just as we had been, while she\nan' Olaf bunked in the lean-to; but a warm chinook had been blowin'\nall day, an' it was soft an' pleasant, so we took our beds out in the\ncottonwoods. Horace an' the Friar got clinched into some kind of a\ndiscussion; but the rest of us dropped off about as soon as we\nstretched out. The moon was just risin', an' one sharp peak covered\nwith glitterin' snow stood up back o' the rim. I remember thinkin' it\nmight be part o' the old earth's shiny soul.\n\n\n\n\nCHAPTER TWENTY-FOUR\n\nNIGHT-PROWLERS\n\n\nWhenever the's anything on my mind I sleep purty light; an' the whole\nCross brand outfit was on my mind that night; so it's not surprisin'\nthat I woke up after a bit. The moon had climbed consid'able, an' the\nstars told me it was about two. I had been sleepin' alone; Horace\nhavin' decided to crawl in with the Friar so they could quarrel at\nshort range.\n\nThe Friar's tarp was next to mine, an' I raised myself on my elbow an'\nlooked at it. I could hear him breathin' natural, an' the bulk of him\nwas so large that Horace wouldn't have made much of a mound anyway; so\nat first I couldn't tell whether he was there or not. I crept out till\nI could sit up an' get a clear view; but Horace wasn't there, so I put\non my boots as quick as ever.\n\nI sneaked over to the Friar's tarp; but Horace's hat was gone, so I\nknew he was up to some mischief, an' started for the corral to see if\nhe had taken a hoss. What I feared was, that he had got to thinkin'\nabout what a super-wonderful flame he had, and had decided to give it\na fair work-out by sneakin' down to Ty Jones's on his own hook. I was\nworried about this because I knew they'd do for him in a minute, if\nthey'd catch him where they could hide all traces.\n\nOlaf had built a large square corral an' a smaller round one, to do\nhis ropin' in; and when I reached the near side o' the square one, I\nheard a slight noise near the gate of the round one. I peered through\nthe poles of the corral, but the dividin' fence got in the road so 'at\nI couldn't see, an' I started to prowl around. All of a sudden,\nHorace's squeaky tenor piped out: \"Halt\"; an' I flattened out on the\nground, thinkin' he had spotted me; but just then the' was a smothered\ncurse from the round corral, an' when I started to get up I saw\nBadger-face vault over the fence in the direction of Horace's voice.\n\nThen I saw Horace standin' behind a clump with his gun on Badger-face.\n\"Put up your hands,\" sez Horace.\n\nBadger was runnin' straight for him; but he put up his hands at this\norder, and came to a slow stop about five feet from Horace. The square\ncorral was still between me an' them, an' I drew my right gun an'\nstarted around, keepin' my eye on 'em as much as I could through the\npoles.\n\n\"I reckon I got ya this time,\" sez Horace, just as I reached the\ncorner.\n\n\"I reckon you have,\" sez Badger in a give-up voice; but at the same\nmoment he took a step forward, threw his body back, an' kicked the gun\nout of Horace's hand. Then he lunged forward an' got Horace by the\nthroat, flung him on his back an' straddled him--an' I broke for 'em\non the run. Just before I reached 'em, the' came a heavy, muffled\nreport, an' Badger-face fell on his side an' rolled over on his back,\nclutchin' at his breast.\n\nHorace rose to his feet, holdin' a toy pistol, put his hands on his\nhips, looked down at Badger-face, an' sez: \"If you'd 'a' just asked\nOlaf what kind of a light I give out, you'd 'a' stayed at home an'\nsaved your life.\" That's how nervous Horace was.\n\n\"Don't stand an' talk to a shot man,\" I sez. \"Allus get his gun\nfirst.\"\n\nHorace gave a jump at the sound o' my voice, an' covered me with his\npop-gun. \"Oh, it's you, is it?\" he sez. \"Well, then, you get his gun;\nbut I don't much think he can use it.\"\n\nBy the time I had lifted Badger's gun, the other boys were arrivin',\nan' when they found that Horace had gone out alone an' shot a hole\nthrough Badger-face, they certainly was some surprised. Purty soon Kit\nMurray came out with Olaf, an' then Horace told about not feelin'\nsleepy an' bein' so disgusted at the way we were snorin' that he had\ngot up to take a little stroll. He said he just went toward the corral\n'cause that was the least uninterestin' place he could think of, and\nthat Badger had sneaked down an' started to cut the stirrups off the\nsaddles right before his eyes.\n\n\"I gave him all the time he wanted,\" sez Horace, \"so 'at there\nwouldn't be any doubt as to his intentions. I reckon 'at cuttin' up\nsaddles in another man's corral is goin' about far enough, ain't it?\"\n\nJust then the Friar finished his examination of Badger, an' went after\nhis saddle bags for a bandage. \"Went clear through his lung,\" was all\nhe said as he passed us on the run.\n\nIt was purty chilly at that time o' night; and as the cold began to\neat in, it suddenly came over Horace that no matter how much justified\nhe was, he had shot an' most likely killed a feller human, an' he\nbegan to shake. He went over to Badger-face an' put his coat over him,\nan' sez: \"Great heavens! are ya goin' to let this man lie out here in\nthe cold till he dies? Ain't the' some place we can put him? This is\nhorrible.\"\n\n\"Bring him in the house,\" sez Kit. \"He don't deserve it; but we can't\nlet him lie out here--can we, Olaf?\"\n\n\"No,\" sez Olaf. \"If you say bring him in, in he comes.\"\n\n\"That's right, that's fine. I don't bear him any malice,\" sez Horace.\n\"I hope he gets over it an' lives to repent.\"\n\nWe packed him into the house an' Kit made a fire an' heated some\nwater. As soon as the water was hot, the Friar cleaned out the wound\nwith it an' some foamy stuff out of a bottle. Then he dissolved a drab\ntablet in some water an' tied up both openings. Horace sat in a corner\ndurin' this operation, with his head in his hands, shiverin'. The\nreaction had set in; an' all of us knew what it was, though I don't\nsuppose any of us had had the chance to give way to it as free as\nHorace did.\n\nBadger-face was all cut an' scarred when we stripped him; but he\nlooked as tough an' gnarly as an oak tree, an' the Friar said he had\none chance in a hundred to pull through. He didn't speak to us until\nafter the Friar had finished with him. Then he said in a low, snarly\nvoice: \"I don't much expect to get over this; but before I slip off, I\nwish you'd tell me who the little cuss who got me really is, an'\nwhat's his game.\"\n\nWe didn't hardly know what to say; but finally Tank sez: \"We don't\nfeel free to tell you who he is, Badger-face; but I'll say this much,\nhe ain't no officer of the law.\"\n\nI thought it would be the quickest way to straighten Horace up, so I\ntold him 'at Badger-face wanted to talk to him. Sure enough, Horace\ntook a deep breath an' stiffened his upper lip. Then he walked over to\nthe bed. \"How do ya feel, Badger-face?\" sez he.\n\n\"Oh, I been shot before,\" sez Badger; \"but it burns worse 'n usual\nthis time, an' I reckon you've got me. It grinds me all up to think\n'at a little runt like you did it, an' it would soothe me to know 'at\nyou had some sort of a record.\"\n\nHorace looked thoughtful: he wanted to comfort the man he was\nresponsible for havin' put out o' the game; but he could see that the\nwhole truth wouldn't in no wise do, so he put on a foxy look an' sez:\n\"I never worked around these parts none; but if you've ever heard o'\nDinky Bradford, why, that's me. I know just how you feel. You feel as\nmuch put out at bein' bested by a small-like man, as I would at havin'\na big feller get ahead o' me; but you needn't fret yourself. There's\nfellers right in this room who have seen me go four days without food\nan' then do a stunt which beat anything they'd ever seen. Don't you\nworry none. Now that you're down an' out, we all wish ya the best o'\nluck.\"\n\nMe an' Spider an' Tank had to grin at this; but it was just what\nBadger needed to quiet him, an' his face lit up when he asked Horace\nhow he had managed to shoot him.\n\n\"I used my auxilary armyment,\" sez Horace, but that's all the\nexplanation he'd make. I found out afterward that he had a thing\ncalled a derringer, a two-barreled pistol, forty-one caliber, which he\ncarried in his vest pocket. I told him 'at this sneaky sort of a\nweapon would give him a bad name if it was found out on him; but he\nsaid 'at he shot from necessity, not choice, and that when it came to\ngettin' shot, he couldn't see why the victim should be so blame\nparticular what was used--which is sensible enough when you come to\nthink it over, though I wouldn't pack one o' those guns, myself.\n\nBadger-face was out of his head next day, and for two weeks followin'.\nThe Friar an' Kit an' Horace took turns nursin' him, an' they did an\nable job of it. Water, plain water an' wind, was about all the Friar\nused in treatin' him. Kit wanted to give him soup an' other sorts o'\nfunnel food; but the Friar said 'at a man could live for weeks on what\nwas stored up in him; an' Horace backed him up. Kit used to shake her\nhead at this, an' I know mighty well that down deep in her heart, she\nthought they would starve him to death before her very eyes.\n\nWe tore up the old shack on the hill, snaked the poles down with\nOlaf's work team, an' set it up in the Spread; so 'at we'd be handy in\ncase we was needed. A couple o' the Cross-branders drifted by, an' we\ngave 'em the news about Badger-face an' Dinky Bradford havin' come\ntogether an' Badger havin' got some the worst of it; but they wouldn't\ngo in to see him, an' they quit wanderin' by; so 'at we didn't hardly\nknow what to expect.\n\nWe had hard work thawin' out the clay for chinkin', an' we didn't get\nthe cabin as tight as we'd 'a' liked; but we had plenty o' wood, so it\ndidn't much matter as far as warmth was concerned; but we had the\nblamedest time with a pack-rat I ever did have.\n\nI don't know whether pack-rats an' trade-rats is the same varmints or\nnot; but neither one of 'em has a grain o' sense, though some tries to\nstick up for the trade-rats on account o' their tryin' to be honest. A\npack-rat is about three times as big as a barn rat, an' fifteen times\nas energetic. His main delight is to move things. Horace said 'at he\nwas convinced they were the souls o' furniture-movers who had died\nwithout repentin' of all the piano-lamps an' chiny-ware they had\nbroke. A pack-rat don't care a peg whether he can use an article or\nnot; all he asks is the privilege of totin' it about somewhere.\n\nWe weren't at all sure 'at we wouldn't be routed out in the night; so\nwhen we went to sleep, we'd stack our boots an' hats where we could\nfind 'em easy. Sometimes the pack-rat would toil so industrious 'at\nhe'd wake us up an' we'd try to hive him; but most o' the time he'd\nwork sly, an' then next mornin' we'd find our boots all in a heap on\nthe table, or in the corner under the bunk or somewhere clear outside\nthe shack; until we was tempted to move the shack back where it was,\nthere not bein' any pack-rats up there.\n\nThen either the pack-rat reformed into a trade-rat, or else he sold\nout his claim to a trade-rat. Anyway, four nights after we'd been\nsettled, we began to get trades for our stuff.\n\nHorace was sleepin' this whole night with us, an' next mornin' he\nwakened before light an' started to dress so as to relieve the Friar.\nHe had put his boots on the floor under the head o' his bunk, an' when\nhe reached down for 'em he found one potato an' the hide of a rabbit.\nThe rabbit hide had been tossed out two days before, an' it had froze\nstiff an' had a most ungainly feel at that hour o' the mornin'. Horace\nscrooged back into bed an' pulled all the covers off Tank whom he was\nsleepin' with. When Tank awoke, he found Horace sittin' up in the bunk\nwith the covers wound around him, yellin' for some one to strike a\nlight.\n\nWe all struck matches an' finally got a candle lit. When Horace saw\nwhat it was, he was hos-tile for true, thinkin' it was a joke one o'\nthe boys had put up. We had had a hard time convincin' him o' the ways\no' pack-rats, an' now when we sprung trade-rats on him, he thought we\nwere liars without mercy; but when the Friar came out to learn what\nthe riot was, an' told Horace it was all so about trade-rats, he had\nto give in.\n\n\"Well, they've got a heap o' nerve,\" sez he, from the center o' the\nbeddin' which was still wound around him, \"to lug off a good pair o'\nhigh-heeled ridin' boots, an' leave an old potato an' the shuck of a\nrabbit in place of 'em!\"\n\nAfter this Horace took a tarp into Badger's room an' bedded himself\ndown in a corner, which was all around the most handy thing he could\ndo; but the rest of us had a regular pest of a time with that rat. We\ncouldn't find out where the deuce he got in; but he distributed our\nbelongin's constant, an' generally brought us some of Olaf's\ngrub-stuff in exchange. We couldn't trap him nor bluff him, an' it\ngenerally took a good hour mornin's, to round up our wearin' apparel.\n\nOne night we kept the fire goin' an' changed watchers every two hours.\nOl' Tank was on guard from two to four, an' he woke us up by takin' a\nshot. We found him on his back in the middle o' the floor, an' he\nclaimed he had been settin' in a chair an' had seen the rat walkin'\nalong the lower side o' the ridgepole with one o' Tillte Dutch's boots\nin his mouth. Dutch had the spreadin'est feet in the outfit, an' we\ncouldn't believe 'at a trade-rat could possibly tote it, hangin' down\nfrom the ridgepole; but Tank showed us a lot o' scratches along the\nridgepole, an' a bruise on his chin where the boot had hit him when\nthe rat dropped it. The' was also a hole in the boot where his bullet\nhad gone, but this didn't prove anything. Still, Tank stuck to his\nstory, so we had to apologize for accusin' him of lettin' his good eye\nsleep while he kept watch with his free one.\n\nWe stuffed burlap into the hole about the ridgepole, an' that night\nbein' Christmas eve, we all gathered in and held festivities. We\ndanced an' told tales an' sang until a late hour. None of us were\ninstrument musicians; but we clapped our hands an' patted with our\nfeet, an' Kit took turns dancin' with us, till it was most like a\nregular party. Mexican Slim bet that he could do a Spanish dance as\nlong as Horace could sing different verses of his song; but we\nsuppressed it at the ninety-first verse. Tank wanted to let him\nfinish, in the hope it might kill the trade-rat; but we couldn't stand\nany more, ourselves.\n\nThen the Friar taught us a song called, \"We three Kings of Orient\nare\"; an' we disbursed for the night. It was a gorgeous night, an' me\nan' the Friar took a little walk under the stars. One of 'em rested\njust above the glisteny peak up back o' the rim, an' he sang soft an'\nlow, the \"Star of beauty, star of night\" part o' this song. He allus\nlifted me off the earth when he sang this way. Then he sez to me:\n\"After all, Happy, life pays big dividends, if we just live it hard\nenough\"; an' he gave a little sigh an' went in to tend to Badger-face.\n\n\n\n\nCHAPTER TWENTY-FIVE\n\nTHE TRADE-RAT'S CHRISTMAS-GIFT\n\n\nTrade-rats haven't as much idee of real music as coyotes have.\nNinety-one verses of that infernal cow-song, sung in Horace's\nnose-tenor, was enough to drive bed-bugs out of a lumber-camp; but\nthat night the trade-rat worked harder than ever. We had hid our stuff\nan' fastened it down, an' used every sort of legitimate means to\ncircumvent the cuss; but he beat us to it every time, an' switched our\nstuff around scandalous.\n\n\"Merry Christmas!\" yelled Spider Kelley, holdin' up a rusty sardine\ncan.\n\nThe trade-rat had remembered us all in some the same way, but we\nrecalled what day it was an' took it in good part; until, all of a\nsudden, ol' Tank gave a whoop, an' held up a brown buck-skin bag. We\ncrowded around an' wanted him to open it up an' see what was inside;\nbut he said it most probably belonged to Olaf or Kit or the Friar; so\nwe toted it into the cabin an' asked the one who could identify it to\nstep out an' claim his diamonds.\n\nThen we had a surprise--not one o' the bunch could identify the bag!\nWe stood around an' looked at the bag for as much as five minutes,\ntryin' to figure out how the deuce even a trade-rat could spring stuff\non us none of us had ever seen before.\n\n\"This is a real trade, sure enough,\" sez Horace.\n\n\"I tell ya what this is,\" sez I. \"This is a Christmas-gift for the\nFriar. Go on an' open it, Friar.\"\n\nThe' was some soft, Injun-tanned fawn-skin inside, wrappin' up a\ncouple o' papers, an' two photographs, and an old faded letter. \"I\ndon't think we have the right to look at these,\" sez the Friar.\n\n\"How'll we ever find out who they belong to, then?\" asked Horace.\n\"Look at the letter anyway.\"\n\nIt was in a blank envelope, an' it began, \"My dear son,\" and ended,\n\"Your lovin' mother.\" The letter was just the same as all mothers\nwrite to their sons, I reckon: full of heartache, an' tenderness, an'\ngood advice, an' scoldin'; but nothin' to identify nobody by; so we\nsaid 'at the Friar should read the papers. One of 'em was an honorable\ndischarge from the army; but all the names an' dates an' localities\nhad been crossed out. It was what they call an \"Excellent\" discharge,\nwhich is the best they give, an' you could tell by the thumb print 'at\nthis part had been read the most by whoever had treasured it.\n\nThe other paper was simply a clippin' from a newspaper. It was a\ncolumn of items tellin' about Dovey wishin' to see Tan Shoes at the\nsame place next Sunday, an' such things. The Friar said 'at this was\nthe personal column, an' he sure labeled it; 'cause if a feller chose\nto guess any, some o' those items was personal enough to make a\nbar-tender blush; but they didn't convey any news to us as to where\nthe trade-rat had procured the buck-skin bag.\n\nThe photographs were wrapped in tissue paper an' then tied together\nwith pink string, face to each. The Friar balked a little at openin'\n'em up; but we deviled him into it. The first he opened was a cheap,\nfaded little one of an old lady. She had a sad, patient face, an'\nwhite hair. Horace was standin' on a chair, lookin' over the Friar's\nshoulder, an' he piped out that the photograph had been took in New\nYork, an' asked if we knew any one who lived there, which most of us\ndid; but not the subject of the photograph.\n\nThen the Friar opened the other one. He took one look at it, an' then\nhis face turned gray. \"This one was took in Rome,\" sez Horace. \"Does\nany one here have a list o' friends livin' in Rome, Italy?\"\n\nHe hadn't looked at the face on the photograph, nor at the Friar's\nface; but when we didn't answer, he looked up, saw that we had sobered\nin sympathy with the Friar, an' then he looked at the face on the\nphotograph an' got down off the chair. The face was of a beautiful\nlady in a low-necked, short-sleeved dress. Not as low nor as short as\nsome dresses I've seen in pictures, but still a purty generous\noutlook.\n\nThe Friar's hands shook some; but he gradually got a grip on himself,\nan' purty soon, he sez in a steady voice: \"This is a picture of\nSignorina Morrissena. Does any one here know of her?\"\n\nWell, of course none of us had ever heard of her; so the Friar wrapped\nup the package again an' put it back into the buck-skin bag. We had\nexpected to have some high jinks that day, an' Kit had baked a lot o'\nvinegar pies for dinner, we had plenty o' fresh deer-meat, an' we had\nagreed to let the Friar hold a regular preachin' first; but when we\nsaw how the picture had shook him up we drifted back to our own shack\nan' sat talkin' about where the deuce that blame trade-rat could\npossibly have got a holt o' the buck-skin bag. I was purty sure that\nit was a picture o' the Friar's girl, the extra trimmin's on the name\nnot bein' much in the way of a disguise, an' as soon as I got a chance\nto see Horace I questioned him, an' he said it was the girl, all\nright; but that she had developed a lot.\n\nThe Friar had taken a hoss an' gone up into the mountains, an' had\nleft word that he didn't want any dinner. We were as full o' sympathy\nwith him as we could stand, but not in the mood to sidestep such a\nmeal as Kit had framed up; so we ate till after three in the\nafternoon. We didn't want to do anything to fret him a speck; so we\nhardly knew what to do. Generally it tickled him to have us ask him to\npreach to us; but we couldn't tell how he'd feel about it now, and we\nwere still discussin' it about the fire when the Friar came back.\n\nHe looked mighty weary, an' we knew he had been drivin' himself purty\nhard, although it wasn't just tiredness which showed in his face.\nStill, the' was a sort of peace there, too; so after he'd warmed\nhimself a while, ol' Tank asked him if he wouldn't like to preach to\nus a bit.\n\nThe Friar once said that back East some folks used good manners as\nclothin' for their souls, but that out our way good-heartedness was\nthe clothin', an' good manners nothin' more than a silver band around\nthe hat. \"And some o' the bands are mighty narrow, Friar,\" I added to\ndraw him out. \"Yes,\" sez he, \"but the hats are mighty broad.\"\n\nYou just couldn't floor the Friar in a case like this. He knew 'at the\npoliteness an' the good-heartedness in Tank's request was divided off\nabout the same as the band an' the hat; and that all we wanted was to\nease off the Friar's mind an' let him feel contented; so he heaved a\nsigh and shook his head at Tank.\n\nWhen a blacksmith goes out into company, folks don't pester him with\nquestions as to why tempered steel wasn't stored up in handy caves,\ninstead of havin' nothin' but rough ore hid away in the cellar of a\nmountain; and a carpenter is not held responsible because a sharp saw\ncuts better 'n a dull one; but it seems about next to impossible for a\nhuman bein' to pass up a parson without insultin' him a little about\nthe ways o' Providence, and askin' him a lot o' questions which would\nmoult feathers out o' the ruggedest angel in the bunch.\n\nWe could all see 'at the Friar had been havin' a rough day of it; so\nTank began by askin' him questions simply to toll him away from\nhimself; but soon he was shootin' questions into the Friar as rough\nshod as though they was both strangers to each other.\n\n\"You say it was sheep-herders what saw the angels that night the Lord\nwas born,\" sez Tank. \"How come the' wasn't any cow-punchers saw 'em?\"\nTank had about the deep-rootedest prejudice again' sheep-herders I\never saw.\n\n\"The' wasn't any cow-punchers in that land,\" sez the Friar. \"It was a\nhilly land an'--\"\n\n\"Well I'd like to know,\" broke in ol' Tank, \"why the Lord picked out\nsuch a place as that, when he had the whole world to choose from.\"\n\nO' course the Friar tried his best to smooth this out; but by the time\nhe was through, Tank had got tangled up with another perdicament.\n\"Then, there was ol' Faro's dream,\" he said, \"the one about the seven\nlean cows eatin' the seven fat ones. I've punched cows all my life,\nand I saw 'em so thin once, when the snow got crusted an' the chinook\ngot switched off for a month, that the spikes on their backbones\npunched holes through their hides; but they'd as soon thought o'\nflyin' up an' grazin' on clouds, as to turn in an' eat one another.\"\n\nBy the time the Friar had got through explainin' the difference\nbetween dreams and written history, Tank was ready with another query.\n\"I heard tell once 'at the Bible sez, 'If thy eye offends thee, pluck\nit out.' Does the Bible say this?\"\n\n\"Well, it does,\" admitted the Friar; \"but you see--\"\n\n\"Well, my free eye offends me,\" broke in Tank. \"It never did offend me\nuntil Spike Groogan tried to pluck it out, and it don't offend me now\nas much as it does other folks. Still, I got to own up 'at the blame\nthing does offend me whenever I meet up with strangers, 'cause it\nallus runs wilder in front of a stranger 'n at airy other time. Now,\nwhat I want to know is, why an' when an' how must I pluck out that\neye--specially, when it sez in another place that if a man's eye is\nsingle his whole body is full o' light. My eye is single enough to\nsuit any one. Fact is, it's so blame single that some folks call it\nsingular; but the' ain't no more light in my body 'n there is in airy\nother man's.\"\n\nYou couldn't work off any spiritual interpretation stuff on Tank. He\nthought an allegory was the varmint which lives in the Florida swamps.\nWell, as far as that goes, I did, too, until the Friar pointed out\nthat it was merely a falsehood used to explain the truth; but Tank, he\ndidn't join in with any new-fangled notions, an' a feller had to talk\nto him as straight out as though talkin' to a hoss. The' was lots of\ntimes I didn't envy the Friar his job.\n\nBut after he had satisfied Tank that it wasn't required of him to\ndiscard either of his lamps, especially the free one, he drifted off\ninto tellin' us how he had spent the day--and then I envied him a\nlittle, for he certainly did have the gift o' wranglin' words.\n\nHe told about havin' rode up the mountain as far as he could go, and\nthen climbin' as far as he could on foot. He showed how hard it was to\ntell either a man or a mountain by the lines in their faces, and he\nwent on with this till he made a mountain almost human. Then he\nswitched around and showed how much a mountain was like life, ambition\nbein' like pickin' out the mountain, the easy little foothills bein'\nthe start, the summit allus hid while a feller was climbin', and each\nlittle plateau urgin' him to give up there and rest. He compared life\nand a mountain, until it seemed that all a feller needed for a full\nedication, was just to have a mountain handy. Then he wound up by\nsayin' that he hadn't been able to reach the peak. He had sat in a\nsheltered nook for a time, gazin' up at the face of a cliff with an\noverhangin' bank o' snow on top, the wind swirlin' masses o' snow down\nabout him, and everything tryin' to point out that he had been a\nfailure, and might as well give up in disgust. He stopped here, and we\nwere all silent, for, as was usual with him, he had led us along to\nwhere we could see life through his eyes for a space.\n\n\"After a time,\" sez the Friar as soon as he saw we were in the right\nmood, \"I caught my breath again and followed the narrow ledge I was on\naround to where I could see the highest peak stand out clear and\nsolitary; and from my side of it, it wasn't possible for any man to\nreach it. There was no wind here, the air was as sweet and pure as at\nthe dawn o' creation, and everywhere I looked I met glory heaped on\nglory. A gray cloud rested again' the far side o' the peak, and back\no' this was the sun. Ah, there was a silver and a golden linin' both\nto this cloud; and all of a sudden I was comforted.\n\n\"I had done all I could do, and this was my highest peak. Whatever was\nthe highest peak for others, this was the highest peak for me; and\nthere was no more bitterness or envy or doubt or fear in my heart. I\nstood for a long time lookin' up at the gray cloud with its dazzling\nedges, and some very beautiful lines crept into my memory--'The paths\nwhich are trod, by only the evenin' and mornin', and the feet of the\nangels of God.'\"\n\nThe Friar had let himself out a little at the end, and his eyes were\nshinin' when he finished. \"I guess I have given you a sermon, after\nall, boys,\" he said, \"and I hope you can use it to as good advantage\nas I did when it came to me up on the mountain. We all have thoughts\nwe can't put into words, and so I've failed to give you all 'at was\ngiven me; but it's some comfort to know that, be they big or be they\nlittle, we don't have to climb any mountains but our own, and whether\nwe reach the top or whether we come to a blind wall first, the main\nthing is to climb with all our might and with a certain faith that\nthose who have earned rest shall find it, after the sun has set.\"\n\nThis was one of the days when the magic of the Friar's voice did\nstrange things to a feller's insides. We knew 'at he was talkin' in\nparables, an' talkin' mostly to himself; but each one of us knew our\nown little mountains, an' it was darn comfortin' to understand that\nthe Friar could have as tough a time on his as we had on ours.\n\nWe all sat silent, each feller thinkin' over his own problems; and\nafter a time, the Friar sang the one beginnin', \"O little town of\nBethlehem!\" It was dark by this time, but the firelight fell on his\nface, an' made it so soft-like an' tender that ol' Tank Williams\nsniffled audible once, an' when the song was finished he piled a lot\nmore wood on the fire, an' pertended 'at he was catchin' cold. When\nKit called us in to supper, we all sat still for a full minute, before\nwe could get back to our appetites again.\n\n\n\n\nCHAPTER TWENTY-SIX\n\nA CONTESTED LIFE-TITLE\n\n\nThe bullet which had gone through Badger-face hadn't touched a single\nbone. It had gone through his left lung purty high up, but somethin'\nlike the pneumonie set in, an' he was a sorry lookin' sight when the\nfever started to die out after havin' hung on for two weeks. He had\nbeen drinkin' consid'able beforehand, which made it bad for him, an'\nthe Friar said it was all a question of reserve. If Badger-face had\nenough of his constitution left to tide him over, he stood a good\nchance; but otherwise it was his turn.\n\nHe didn't have much blood left in him at the end of two weeks on air\nand water, and he didn't have enough fat to pillow his bones on. We\nall thought 'at he ought to have something in the way o' feed; but the\nFriar wouldn't stand for one single thing except water. He said 'at\nfood had killed a heap more wounded men 'n bullets ever had; so we let\nhim engineer it through in his own way.\n\nWhen the fever started to leave, he got so weak 'at Horace thought he\nwas goin' to flicker out, an' he felt purty bad about it. He didn't\nregret havin' done it, an' said he would do just the same if he had it\nto do over; but it calls up some mighty serious thoughts when a fellow\nreflects that he is the one who has pushed another off into the dark.\nOn the night when it seemed certain that Badger-face would lose his\ngrip, we all went into his room an' sat around waitin' for the end, to\nsort o' cheer him up a little. Life itself is a strange enough\nadventure, but death has it beat a mile.\n\nAlong about nine o'clock, Badger said in a low, trembly voice: \"What'd\nyou fellers do to me, if I got well?\"\n\nHe didn't even open his eyes; so we didn't pay any heed to him. When\nhe first got out of his head, he had rambled consid'able. Part o' the\ntime he seemed to be excusin' himself for what he had done, an' part\no' the time he seemed to be gloatin' over his devilment; but the'\nwasn't any thread to his discourse so we didn't set much store by it.\nAfter waitin' a few minutes, he quavered out his question again, an'\nthe Friar told him not to worry about anything, but just to set his\nmind on gettin' well.\n\nBadger shook his head feebly from side to side an' mumbled, \"That\ndon't go, that don't go with me.\" He paused here for a rest, an' then\nwent on. \"I've been in my right mind all day, an' I've been thinkin' a\nlot, an' tryin' some experiments. I can breathe in a certain way which\nmakes me easier an' stronger, an' I can breathe in another way which\nshuts off my heart. I don't intend to get well merely for the pleasure\no' gettin' lynched; so if that's your game, I intend to shut off my\nheart an' quit before I get back the flavor o' life. It don't make\ntwo-bits difference with me either way. What d' ya intend to do?\"\n\nHe had been a long time sayin' this, an' we had exchanged glances\npurty promiscuous. We hadn't give a thought as to what we would do\nwith him, providin' he responded to our efforts to save his life; but\nit was purty generally understood that Badger had fitted himself to be\nstrung up, just the same as if he hadn't been shot at all. Now,\nthough, when we came to consider it, this hardly seemed a square deal.\nThere wasn't much common sense in chokin' a man's life down his throat\nfor two weeks, only to jerk it out again at the end of a rope, an' we\nfound ourselves in somethin' of a complication.\n\n\"What do ya think we ort to do to ya?\" asked Tank.\n\n\"Lynch me,\" sez Badger, without openin' his eyes; \"but I don't intend\nto wait for it. I don't blame ya none, fellers. I did ya all the dirt\nI could; but I don't intend to furnish ya with no circus\nperformance--I'm goin' on.\"\n\nHe began to breathe different, an' his face began to get purplish an'\nghastly. \"Can he kill himself that way?\" I asked the Friar.\n\n\"I don't know,\" sez the Friar. \"I think 'at when he loses\nconsciousness, nature'll take holt, an' make him breathe the most\ncomfortable way--but I don't know.\"\n\n\"Let Olaf take a look at his flame,\" sez Horace; so Olaf looked at\nBadger a long time.\n\nOlaf hadn't wasted much of his time on Badger. He wasn't long on\nforgiveness, Olaf wasn't; an' ever since the time 'at Badger had been\nso enthusiastic in tryin' to have him lynched for killin' Bud Fisher,\nOlaf had give it out as his opinion that Badger was doomed for hell,\nan' he wasn't disposed to take any hand in postponin' his departure.\nOlaf was the matter-o'-factest feller I ever knew. The' don't seem to\nbe much harm in most of our cussin', but when Olaf indulged in\nprofanity, he was solemn an' earnest, the same as if he was sayin' a\nprayer backwards.\n\n\"It don't look like Badger's flame,\" sez he after a time. \"It's\ngettin' mighty weak an' blue, an' the's a thick spot over his heart\nwhich shows plainer 'n the one over his wound.\"\n\n\"I move we give him a fresh start,\" sez Horace.\n\n\"He'd ort to be lynched,\" sez Tank. \"I don't see why we can't try him\nout now, an' if we find him guilty, why he can kill himself if he\nwants to, or else get well again an' we'll do it for him.\"\n\nNeither what Horace said nor what Tank said called out much response.\nWe knew the' wasn't any one could say a good word for Badger-face an'\nso he well deserved his stretchin'; but on the other hand, there he\nwas turnin' gray before our eyes, an' it went again' our nature to\ndiscard him, after havin' hung on to him for two weeks. The Friar left\nthe side of the bed an' retired into a corner, leavin' us free to\nexpress ourselves.\n\n\"I don't see how we can let him go free,\" sez Tank. \"He sez himself\n'at he ort to be lynched; an' when a feller can't speak a good word\nfor himself, I don't see who can.\"\n\n\"Badger-face,\" sez Horace, \"you're the darnedest bother of a man I\never saw. First you infest us until we have to shoot a hole through\nyou, an' then we have to nurse you for two weeks, an' now you're\ndiggin' your heels into our consciences. I give you my word we won't\nlynch you if you get well. We'll turn you over to the law.\"\n\nBadger's thin lips fell back over his yellow teeth in the ghastliest\ngrin a live man ever hung out. \"The law,\" sez he with bitter sarcasm,\n\"the law! Have you ever been in a penitentiary?\"\n\n\"No,\" sez Horace, \"I have not.\"\n\n\"Well, I have,\" sez Badger. \"I was put in for another feller's deed;\nan' they gave me the solitary, the jacket, the bull-rings, the\nwater-cure, and if you'll roll me over after I'm dead, you can still\nsee the scars of the whip on my back. I've tried the law, an' I'll see\nyou all damned before I try it again.\"\n\nBadger-face was as game as they generally get. As soon as he stopped\ntalkin' he began to breathe against his heart again. Horace stood\nlookin' at him for a full minute, an' then he lost his temper.\n\n\"You're a coward, that's what you are!\" sez Horace. \"I said all along\n'at you were a coward, an' another feller said so too, an' now you're\nprovin' it. You can sneak an' kill cows an' cut saddles in the dark,\nbut you haven't the nerve to face things in the open. Now, you're\nsneakin' off into the darkness o' death because you're afraid to face\nthe light of life.\"\n\nThis was handin' it to him purty undiluted, an' Badger opened his eyes\nan' looked at Horace. His eyes were heavy an' dull, but they didn't\nwaver any. \"Dinky,\" sez Badger-face, \"the only thing I got again' you\nis your size. I've been called a lot o' different things in my time;\nbut you're the first gazabo 'at ever called me a coward--an' you're\nabout the only one who has a right to, 'cause you put me out fair an'\nsquare. I wish you had traveled my path alongside o' me, though. You\nain't no milksop, but after you'd been given a few o' the deals I've\nhad, you'd take to the dark too. You can call me a coward if you want\nto, or, after I'm gone, you can think of me as just bein' dog tired\nan' glad o' the chance to crawl off into the dark to sleep. I don't\nwant to be on your conscience; that's not my game. All I want is just\nto get shut o' the whole blame business.\"\n\nHe talked broken an' quavery, an' it took him a long time to finish;\nbut when he did quit, he turned on his bad breathin' again. Horace had\nflushed up some when Badger had mentioned milksop; but when he had\nfinished, Horace took his wasted hand in a hearty grip, an' sez: \"I\ntake it back, Badger. You ain't no coward. I only wanted to taunt you\ninto stickin' for another round; but I think mighty well o' ya. Will\nyou agree to cut loose from the Ty Jones crowd an' try to be a man, if\nwe give you your freedom, a new outfit, and enough money to carry you\nout of the country?\"\n\nIt was some time before Badger spoke, an' then he said: \"Nope, I can't\ndo it. Ty knows my record, an' he's treated me white; but if I quit\nhim, he'll get me when I least expect it. Now understand, Dinky, that\nI don't hold a thing again' you, you're the squarest feller I've ever\nmet up with; but I'm not comin' back to life again. From where I am\nnow, I can see it purty plain, an' it ain't worth the trouble.\"\n\n\"You could write back to Ty that you made your escape from us,\" sez\nHorace.\n\n\"That's the best idee you've put over,\" sez Badger, after he'd thought\nit out; \"but I haven't enough taste for life to make the experiment.\nDon't fuss about me any more. I don't suffer a mite. I feel just like\na feller in the Injun country, goin' to sleep on post after days in\nthe saddle. He knows it'll mean death, but he's too tired out to care\na white bean.\"\n\n\"Have you ever been in the army?\" asked the Friar from his place in\nthe corner. We all gave a little start at the sound of his voice, for\nit came with a snap an' unexpected.\n\nBadger's lips dropped back for another hideous grin. \"Yes,\" he said,\n\"I've been in both the penitentiary and the army--and they're a likely\npair.\"\n\n\"Did you have a buck-skin bag?\" asked the Friar, comin' up to the bed.\n\nBadger-face tried to raise himself on his elbow, but he couldn't quite\nmake it. \"Yes, I did,\" sez he, droppin' back again. \"What became of\nit?\"\n\n\"I am keepin' it for ya,\" sez the Friar. \"Do you wish to leave any\nword in case you do not recover?\"\n\n\"No,\" sez Badger, \"the' ain't no one to leave word to. That letter was\nfrom my mother, an' that was her picture. She's been dead a long\nstring o' years now.\"\n\n\"There was another picture an' a newspaper clippin',\" sez the Friar.\n\nBadger-face didn't give no heed; an' after a time the Friar sez: \"What\nshall I do with them?\"\n\n\"Throw 'em away,\" sez Badger-face. \"They don't concern me none. I was\nmore took with that woman's picture 'n airy other I ever saw. That was\nall.\"\n\n\"Where did you get it?\" asked the Friar.\n\n\"I got it from a young Dutchy,\" sez Badger wearily. \"He killed a\nfeller over at Leadville an' came out here an' took on with Ty Jones.\nHe said she was an opery singer, an' got drugged at a hotel where he\nwas workin'.\"\n\nBadger-face was gettin' purty weak by now, an' he stopped with a sort\nof sigh. The Friar took holt of his hand. \"I am very much interested\nin this woman,\" he said, lookin' into Badger's face as if tryin' to\ngive him life enough to go on with. \"Can you tell me anything else\nabout her?\"\n\n\"Not much,\" sez Badger-face. \"She was singin' at what he called the\nWinter Garden at Berlin, Germany. Some Austrian nobility got mashed on\nher an' drugged her at the hotel. Dutchy was mashed on her, too, I\nreckon. They had advertised for him in a New York paper, an' when he\ngot shot, over at Little Monte's dance hall, he asked me to write\nabout it. His mother had died leavin' property, an' all they wanted\nwas to round up the heirs. I reckon they were glad enough to have\nDutchy scratched from the list. I don't know why I did keep that\nclippin'.\"\n\n\"Have you any idee how long ago it was 'at the woman was drugged?\"\nasked the Friar.\n\n\"I haven't any idee,\" sez Badger-face weakly. \"Carl was killed four\nyears ago this Christmas eve; so it had to be before that.\"\n\n\"Listen to me, Badger-face,\" sez the Friar, grippin' his hand tight.\n\"I want you to get well. I know that all these men will stand by you\nand help you to start a new life.\"\n\n\"How long is it since I've been laid up?\" asked Badger.\n\n\"Two weeks,\" sez the Friar. \"This is two days after Christmas.\"\n\n\"Who tended to me?\" asked Badger.\n\n\"We all did,\" sez the Friar, \"and we all stand ready to help you make\na new start.\"\n\n\"I had a good enough start,\" sez Badger; \"but I fooled it away, an'\nI'm too old now to make a new one.\"\n\n\"Is there any word you want sent to your friends at Ty Jones's?\" asked\nthe Friar.\n\nOnce more Badger skinned his face into the grin. \"Friends?\" sez he.\n\"When you trap a wolf, does he send any word to his friends? I haven't\ngot no friends.\"\n\n\"Swallow this milk,\" sez Horace holdin' some of it out to him in a big\nspoon. Kit had made Olaf start to milkin' a cow, 'cause she wanted to\nuse milk in cookin', and intended to make butter when she had the\ncream saved up. Badger put the milk in his mouth, an' then spit it out\nagain.\n\n\"Don't you put anything else in my mouth,\" he sez. \"I told you I was\ngoin' to die; an' by blank, I am goin' to die.\"\n\n\"Fellers,\" sez Horace, turnin' to us, \"do you think this man is goin'\nto die?\" We all nodded our heads. \"Then, will you give his life to me,\nto do with as I will?\" asked Horace; and we nodded our heads again.\n\nHorace took off his coat, an' rolled up his sleeves, an' then he came\nover an' shook Badger-face by the shoulder. \"Listen to me,\" he sez. \"I\nfought ya once before, for your life, and I'm goin' to fight you for\nit now. Do you hear what I say--I'm goin' to fight you for your own\nlife. I'm goin' to make you swallow milk, if I have to tie you an'\npour it in through a funnel. You can't hold your breath an' fight, an'\nI'm goin' to fight you.\"\n\nBadger-face opened his eyes an' looked up into Horace's face. He\nlooked a long time, an' the ghost of a smile crept into his face.\n\"Well, you're the doggonedest little cuss I ever saw!\" he exclaimed.\nHe waited a long time, an' then set his teeth. \"You beat me once,\" he\nmuttered. \"Now, see if you can beat me again.\"\n\nIt was after midnight; so when Horace dropped the hint that he\nwouldn't need any help except from me an' the Friar, the rest o' the\nboys dug out for the bunk shack. Then Horace took us over to the\nfireplace an' asked us what was the best thing to do.\n\n\"I do believe 'at you have stumbled on the right plan to save him,\"\nsez the Friar. \"He has no fever, the wound is doin' splendid, and he\nhas a powerful constitution. The trouble is that he does not will to\nlive. We must spur on his will, and if we can make him fight back,\nthis'll help. Also we must control him as much as possible through\nsuggestion. Have you any plan o' your own?\"\n\n\"No,\" sez Horace candidly. Horace didn't need anything for any\nemergency except his own nerve. \"I am determined that he must live,\nbut I have no plan.\"\n\n\"The first thing is to give him a little warm milk,\" sez the Friar.\n\n\"All right,\" sez Horace. \"You tell me what to do--by signs, as much as\npossible--but let me give the orders to Badger-face. My size has made\nan impression on him, and we can't afford to lose a single trick.\" The\nFriar agreed to this an' we went back to the bunk.\n\n\"Badger-face,\" sez Horace, \"I'd rather give you this milk peaceful;\nbut I'm goin' to give it to ya, an' you can bet what ya like on that.\"\n\nBadger opened his eyes again, an' they were dull an' glazy. \"This\nreminds me o' the water-cure at the pen,\" he said, an' then set his\nteeth.\n\n\"Hold his hands, Happy,\" sez Horace, as full o' fight as a snow-plow.\n\"Hold his head, Friar. Now then, swallow or drown.\"\n\nIt looked purty inhuman, but Badger had to swallow after a bit, an'\nwhen we had put as much milk into him as we wanted--only a couple o'\nspoonfuls--we let him go, an' he fell asleep, pantin' a little. We\nwoke him up in half an hour, an' put some more milk into him. When he\nslept, his breathin' was more like natural, an' the fourth time, I\ndidn't have to hold his hands; so I went to sleep myself.\n\nWell, Horace won this fight, too. In about four days, Badger-face\nbegan to have an appetite, an' then it was all off with him. He\ncouldn't have died if we'd left him plumb alone; but he hadn't give up\nyet. The Friar kept him down to a mighty infan-tile diet, sayin' that\na lung shot was a bad one, an' the pure mountain air was all that had\nsaved him; but even now fever was likely to come back on him.\n\nIt was close to the tenth o' January when Horace came in from a ride\none evenin', an' went in to see Badger-face, still wearin' his gun.\nQuick as a wink, Badger grabbed the gun; but Horace threw himself on\nBadger's arm, an' yelled for help. The Friar an' Olaf rushed in from\nthe lean-to, an' corraled the gun in short order.\n\n\"You blame little bob-cat, you!\" sez Badger. \"I didn't intend to use\nthe gun on you.\"\n\n\"I know what you intended to do,\" sez Horace; \"but you don't win this\ndeal as easy as all that.\"\n\n\n\n\nCHAPTER TWENTY-SEVEN\n\nA STRANGE ALLIANCE\n\n\nAfter this we tied Badger-face in bed an' kept watch of him. He kept\non gettin' stronger all the time, an' a good percent of his meanness\ncame back with his strength. Sometimes he'd spend hours tauntin'\nHorace an' the Friar; but they didn't mind it any more 'n if Badger\nhad been a caged beast. Then one night he concluded to try cussin'. He\nstarted in to devise somethin' extra fancy in the way o' high-\nprofanity; but he hadn't gone very far on this path, before Olaf came\nin as black as a thunder cloud.\n\n\"Do you want to be whipped with a whip?\" he demanded.\n\n\"Naw, I don't want to be whipped with a whip,\" sez Badger-face.\n\n\"Then you stop swearin',\" sez Olaf. \"We been to enough trouble about\nyou, and I don't intend to have my wife listen to any more o' your\nswearin'. If you don't stop it, I whip all your skin off. You say you\nwant to die--I whip you to death before your very eyes.\"\n\nBadger heaved at his ropes a time or two, an' then he realized his\nweakness, sank back on the bed, an' the tears rolled down his cheeks.\nHe fair sobbed. \"You're a set o' cowards,\" he yelled, \"the whole pack\no' you! You wouldn't let me die, and now you threaten to whip me to\ndeath. I dare any one of ya to shoot me--you yellow-hearted cowards!\"\n\n\"I care not for what you say I am,\" said Olaf. \"You know if I am a\ncoward, and you know if I keep my word. I say to you, slow an'\ncareful, that if you yell swear words again in my house, I whip your\nhide off.\"\n\nWell, this had a quietin' influence on Badger's conversation; but he\nfretted himself a good deal as to what we intended to do with him.\nFinally one day when he began to look a little more like a live man\nthan a skeleton, Horace sez to him: \"Badger, you said you didn't have\nany friends, an' it must be true, 'cause not one of your own outfit\nhas ever been to see you, not even Ty Jones.\"\n\n\"Ty Jones don't stay out here through the winter,\" sez Badger-face.\n\"If he'd been here, he'd have squared things up for this, one way or\nanother.\"\n\n\"Where does he go?\" asked Horace.\n\n\"I don't know,\" sez Badger-face.\n\nHorace asked Olaf about it, and Olaf said 'at Ty Jones allus pulled\nout in December, an' didn't come back until March.\n\nThen Horace came in and sat by Badger again. \"I've got a proposition\nto make to you,\" sez he, \"and you think it over before you answer. I\nhave plenty o' money; but I've wasted most o' my life, sittin' down.\nIf you are sick of livin' like a wolf, I'll pay your expenses and half\nagain as much as Ty Jones is payin' you, and all you'll have to agree\nto is to go along as a sort of handy-man for me. I think we can get to\nbe purty good friends, but that can wait. I intend to ramble around\nwherever my notions take me. If you'll give your word to be as decent\nas you can, I'll give my word to stand by you as far as I'm able. Your\nlife is forfeit to me, an' if you'll do your part, I intend to make\nthe balance of it worth while to ya. Now, don't answer me; but think\nit over an' ask all the questions you want to. I'll answer true what I\ndo answer; but I won't answer any 'at I don't want to.\"\n\nIf Horace had crept in an' cut off his two ears, Badger wouldn't have\nbeen any more surprised. Well, none of us would, as far as that goes;\nthough why we should let anything 'at Horace chose to do surprise us\nby this time is more 'n I know.\n\nHe an' Badger talked it over complete for several days, Horace\nagreein' that he wouldn't ask Badger to go anywhere the army or the\nlaw was likely to get him an' not to make him do any stunts 'at would\nmake him look foolish. He told Horace 'at he had served one enlistment\nan' got a top-notch discharge, an' had then took on again; but a\ndrunken officer had him tied on a spare artillery wheel because Badger\nhad laughed when the officer had fallen off his horse into a mud\npuddle. He said they had laid the wheel on the ground and him across\nit, the small of his back restin' on the hub o' the wheel, an' his\narms an' legs spread an' tied to the rim, an' had kept him there ten\nhours. He said that he had deserted the first chance he got; but he\nrefused to tell what had happened to the officer afterward.\n\nFinally Badger said he would take up Horace's proposition; an' Horace\ncalled Olaf in to see if Badger was speakin' true. This was the first\nBadger had ever heard about Olaf's eyes seein' soul-flames; but he\nsaid 'at this explained a lot to him he hadn't understood before. Olaf\nlooked at him careful; an' Badger held up his right hand an' said that\nas long as Horace treated him square, he would be square with Horace,\neven to the point of givin' up his life for him.\n\n\"He is speakin' true,\" sez Olaf; and from that very minute,\nBadger-face became a different man, an' Horace took off the ropes.\n\n\"You do look some like a badger with that bum beard on,\" sez Horace;\n\"but I don't like this name, and I want you to pick out a new one.\nPick out some Christian name, your own or any other; but now that you\nare startin' on a new life, it will help to have a new name.\"\n\nBadger-face studied over this a long time, but he couldn't root up any\nname to suit him so he told Horace to pick out a name, and he'd agree\nto wear it.\n\n\"Well,\" sez Horace, after he'd give it a good thinkin' over, \"I think\nI'll call you Promotheus.\"\n\nBadger looked at him purty skeptical. \"I don't intend to take no\nGreaser name,\" sez he. \"Is that Mexican?\"\n\n\"No,\" sez Horace. \"That's Greek; an' the original Promotheus was an\nall around top-notcher. He was a giant, so you couldn't complain none\non your size; he rebelled again' the powers, so you couldn't call him\na dog-robber; but the thing 'at you two are closest together in, is\nyour infernal stubbornness. They tried to break Promotheus down by\nchainin' him to a rock while the vultures fed on his liver, but they\ncouldn't make him give in. 'Pity the slaves who take the yoke,' sez\nhe; 'but don't pity me who still have my own self-respect.'\"\n\nBadger-face was so blame weak that his eyes filled up with tears at\nthis; an' the only way he could straighten himself up was to put a few\nflorid curses on his own thumby left-handedness; but Olaf had gone\nafter some wood, so it didn't start anything. \"I'll take that name,\"\nsez he, \"an' I'll learn how to spell an' pronounce it as soon as I\ncan; but you've diluted down my blood so confounded thin with your\ndoggone, sloppy milk diet that I'm a long way from havin' that\nfeller's grit, right at this minute.\"\n\nHorace stood over Badger-face, an' pointed his finger at him, fierce.\n\"Listen to me,\" sez he. \"The next time you heave out an insult to\nmilksops or milk diets, I'll sing you my entire song--to the very last\nword.\"\n\nWe set up a howl; but Badger-face didn't realize all he was up against\nwhen he took on with Horace, so he only smiled in a sickly way, an'\nlooked puzzled.\n\n\"I'll tell ya what I'm willin' to do, Dinky,\" said he, as soon as we\nstopped our noise; \"now that I've took a new name, I don't need to\nwear this sort of a beard any more, an', if ya want me to, I'll trim\nit up the same fool way 'at you wear yours; an' I'll wear glasses,\ntoo, if you say the word.\"\n\n\"We'll wait first to see how you look in a biled shirt,\" sez Horace;\n\"but in honor of your new name, I'm goin' to let you have some\ndeer-meat soup for your dinner, an' a bone to gnaw on.\"\n\nWe had a regular feast that day, and called Badger-face Promotheus\nevery time we could think up an excuse; so as to have practice on the\nname. The Friar did his best to take part; but I knew every line in\nhis face, and it hurt me to see him fightin' at himself.\n\nAfter dinner we took a walk together; but we didn't talk none until we\nhad climbed the rim, fought the wind for a couple of hours, an'\nstarted back again. It was his plan to think of some big, common chunk\nof life when he was in trouble, so as to take his mind as much as\npossible off himself; and he started to talk about Horace an'\nPromotheus. He even laughed a little at the combination which\nPromotheus Flannigan an' Horace Walpole Bradford would make when they\nsettled down on the East again.\n\n\"The more I think it over,\" said the Friar, \"the plainer I can see\nthat most of our sorrow an' pain and savageness comes from our custom\nof punishin' the crops instead of the farmers. Look at the\npossibilities the' was in Promotheus when he started out. He has a\nstrong nature, and in spite of his life, he still has a lot o' decent\nhumanity in him. Who can tell what he might have been, if his good\nqualities had been cultivated instead o' smothered?\"\n\n\"That's true enough,\" sez I; \"and look at Horace, too. They simply let\nhim wither up for forty years, and yet all this time he had in him\nfull as much devilment as Promotheus himself.\"\n\n\"Oh, we waste, we waste, we waste!\" exclaimed the Friar. \"Instead o'\nusin' the strength and vigor of our manhood in a noble way, we let\nsome of it rust and decay, and some of it we use for our own\ndestruction. The outlaw would have been the hero with the same\nopportunity, and who can tell what powers lie hidden behind the mask\nof idleness!\"\n\n\"Well, that's just it,\" sez I. \"A human bein' is like a keg o' black\nstuff. For years it may sit around perfectly harmless; and only when\nthe right spark pops into it can we tell whether it's black sand or\nblastin' powder. Even Horace, himself, thought he was black sand; but\nhe turned out to be a mighty high grade o' powder.\"\n\nWe walked on a while without talkin'; but the Friar was wrastlin' with\nhis own thoughts, an' finally he stopped an' asked me as solemn as\nthough I was the boss o' that whole country: \"If you had started a lot\no' work, and part of it promised to yield a rich harvest with the\nright care, and part of it looked as though it might sink back to\nworse than it had been in the beginnin'--is there anything in the\nworld which could make you give it up?\"\n\nThe Friar knew my life as well as I did; so I didn't have to do any\npertendin' with him. \"Yes,\" I sez, \"the right woman would.\"\n\nThe Friar didn't do any pertendin' with me either. He stood, shakin'\nhis head slowly from side to side. \"I wish I knew, I wish I knew,\" he\nsaid.\n\nWe walked on again, an' when we came in sight o' the cabin, I sez to\nhim, in order to give him a chance to free his mind if he saw fit:\n\"Horace told me what he knew about it.\"\n\n\"Yes, I know,\" sez the Friar; \"but no one knew very much. She was a\nsplendid brave girl, Happy. I had known her when she was a little girl\nand I a farmer boy. I was much older than she was, but I was allus\ninterested in her. There wasn't one thing they could say against\nher--and yet they drove her out o' my life. I thought she was dead, I\nheard that she was dead; so I buried her in my heart, and came out\nhere where life was strong and young, because I could not work back\nthere. I tried to work in the slums of the cities; but I could not\nconquer my own bitterness, with the rich wastin' and the poor starvin'\nall about me. I have found joy in my life out here; but she has come\nto life again with that picture, and once more I am at war with\nmyself.\"\n\n\"Well, I'll bet my eyes, Friar,\" sez I, \"that you find the right\nanswer; but I haven't got nerve enough to advise ya--though I will say\nthat if it was me, I'd pike out an' look for the girl.\"\n\n\"I wish I knew, I wish I knew,\" was all the Friar said.\n\nPromotheus didn't have any set-backs after this. We talked over\nwhether it would be better to have him go up to Ty's an' tell the boys\nsome big tale about Dinky Bradford, or to just pull out an' leave 'em\nguessin'; and we finally came to the conclusion 'at the last would be\nthe best.\n\nHe was still purty weak by the first o' February; but he was beginnin'\nto fret at bein' housed up any longer, so we began to get ready to hit\nthe back-trail. By takin' wide circles we could get through all right,\nat this season; but with Promotheus still purty wobbly, it wasn't\nlikely to be a pleasant trip, an' we didn't hurry none with our\npreparations. Horace insisted on payin' Olaf two hundred dollars for\nhis share o' the bother, an' I'm purty certain he slipped Kit another\nhundred. He wasn't no wise scrimpy with money.\n\nWe started on the tenth of February, Promotheus ridin' a quiet old\nhoss, an' still lookin' purty much like a bitter recollection. They\nwere consid'able surprised when we arrived at the Diamond Dot; but we\nonly told 'em as much of our huntin' as we felt was necessary.\n\nHorace intended to start for the East at once; but next day when he\nput on his dude clothes again, Promotheus purty nigh bucked on him.\nMost of Horace's raiment was summer stuff, nachely; but he had a long\nchecked coat 'at he wore with a double ended cap, which certainly did\nlook comical. He had cut some fat off his middle, an' had pushed out\nhis chest an' shoulders consid'able; so that his stuff wrinkled on\nhim; and it took a full hour to harden Promotheus to the change.\n\n\"Do I have to look like that?\" sez he.\n\n\"You conceited ape you!\" sez Horace. \"You couldn't look like this if\nyou went to a beauty doctor for the rest o' time; but as soon as we\nget where they sell clothes for humans, I'm goin' to provide you with\nsomethin' in the nature of a disguise.\"\n\nDisguise sounded mighty soothin' to Promotheus, so he gritted his\nteeth, an' said he wouldn't go back on his word. The fact was, that it\ndid give ya an awful shock to see Horace as he formerly was. We had\ngot so used to seein' him gettin' about, able an' free, that it almost\nseemed like a funeral to have him drop down to those clothes again.\n\nThe Friar went over to the station with us, and he an' Horace had a\nconfidential talk; and then Horace and Promotheus got on the train and\nscampered off East.\n\n\"I'm goin' to stick right here, Happy,\" sez the Friar. \"I have let my\nwork get way behind, in tendin' to Promotheus; but from now on I'm\ngoin' to tie into it again. I'd like to do something to put the cattle\nmen and the sheep men on better terms; but this seems like a hard\nproblem.\"\n\n\"Yes,\" sez I, \"that ain't no job for a preacher, and I'd advise you to\nlet it alone. The cattle men will put up the same sort of an argument\nfor their range 'at the Injuns did; but between you and me, I doubt if\nthey stand much more show in the long run.\"\n\n\"I can't see why there isn't room for both,\" sez the Friar. \"It seems\nto me that the cattle men are too harsh.\"\n\n\"Nope,\" sez I, \"there ain't room for 'em both, an' the's somethin'\nirritatin' about sheep that makes ya want to be harsh with all who\nhave dealin's with 'em. Hosses can starve out cattle an' sheep can\nstarve out hosses; but after a sheep has grazed over a place, nothin'\nbigger 'n an ant can find any forage left. Cattle are wild an'\ntempestus, an' they bellow an' tear around an' fight, and the men who\ntend 'em are a good bit like 'em; while sheep just meekly take\nwhatever you've a mind to give 'em; but they hang on, just the same,\nan' multiply a heap faster 'n cattle do. A sheep man is meek--like a\nJew. If a Jew gets what he wants he's satisfied, an' he's willin' to\npertend 'at he's had the worst o' the deal; but a cattle man is never\nsatisfied unless he has grabbed what he wanted away from some one\nelse, an' then shot him up a little for kickin' about it. It'll\nprobably be fifty or a hundred years yet, before the sheep men are\nstrong enough to worry the cattle men; but they'll sure do it some\nday.\" That's what I told the Friar that time at the station, an' I\nguessed the outcome close enough, though I didn't make much of a hit\nas to the time it was goin' to take.\n\nWell, the Friar, he rode away east to Laramie, and I went north to the\nDiamond Dot, and got things ready for the summer work.\n\n\n\n\nCHAPTER TWENTY-EIGHT\n\nTHE HEART OF HAPPY HAWKINS\n\n\nLate the next summer, I got a fine long letter from Horace--and blame\nif he didn't succeed in surprisin' me again. He wrote this letter from\nAfrica, which is about the foreignest parts this world is able to\nexhibit, I reckon. He told about the East not findin' favor with\nPromotheus, though he had done all he could for him, startin' out with\nhigh society and endin' up by takin' him down one night to a sailor's\nsaloon and lettin' him mix into a general fight; but that Promotheus\njust simply couldn't stand the tameness, and so they had gone to\nAfrica to hunt big game, and give the folks out our way a chance to\nforget there ever had been such a cuss as Badger-face.\n\nHe sent along some photographs, too, and they was as novel as a blue\nmoon--Horace, Promotheus, and a lot o' naked <DW65>s totin' packs on\ntheir heads. Horace was the funniest lookin' mortal a body ever saw;\nbut Promotheus had him beat a mile. They both wore bowls on their\nheads an'  glasses; but Promotheus with side-burns was sure\nenough to frighten a snake into convulsions! His gnawin' teeth stuck\nout through a self-satisfied grin; and I was willin' to bet that as\nsoon as the heathen saw him, they'd give up bowin' down to wood an'\nstone.\n\nThe next time I saw Friar Tuck, he told me about receivin' a letter\nfrom Horace who had gone to Berlin on his way to Africa, but hadn't\nbeen able to learn anything satisfactory. The singer had been the big\ncard at their concerts, an' there had been some talk about her gettin'\ndrugged by an Austrian who belonged to the em-bassy; but she had\ndisappeared complete, an' nobody could be found who seemed to know\nanything about it. The Friar kept himself goin' like a steam-engine\nthese days; but while he became a little more tender if possible, he\nlacked something of his old-time spirits. Before this, he used to come\nsweepin' along like a big cool breeze, an' a feller's spirits just got\nup an' whirled along with him, like dry leaves dancin' in the wind.\n\nHe said 'at since Promotheus had slipped out o' the country, the\nCross-branders hadn't bothered Olaf any; but I called his attention to\nthe fact that this was a wet spring, an' told him 'at when we had a\nlong dry spell, Ty Jones would just swallow Olaf like quicksand.\n\nThings drifted along purty steady in our parts for several years. Once\nin a while, the Friar would tell me something about Olaf or something\nabout Ty Jones; but for the most part, I was too much took up with\nother things to care much for even the Friar's doin's.\n\nI was takin' my own Moses-trip durin' these years; and I say now, as I\nallus have said, that it wasn't a square shake to show Moses the\npromised land, an' then not let him into it for even one meal o' milk\nan' honey. I've handled a small bunch o' men an' trailed cattle with\n'em for only three months at a stretch; but I don't mind tellin' you\nthat the' was times when I had to sit up till after midnight, sewin'\nup the rips in my patience--an' we didn't have any women an' children\nalong either. Moses had forty years of it in the desert; with a whole\nblame tribe of Israelites; and yet, instead o' praisin' him for\nhangin' on to his sanity with all the odds again' him, he was handed a\ntantalizer, simply because he said he couldn't see why somethin'\ndidn't happen in a natural, orderly way, once in a while, without\neverlastingly ringin' in some new kind of a miracle on him.\n\nIf I had to pilot a mob like that through a desert for forty years,\nfollerin' a cloud by day an' a pillar o' fire by night, havin' dressed\nquail an' breakfast-food tossed to me out o' the sky, gettin' my\ndrinkin' water by knockin' it out of a rock, an' tryin' to satisfy the\ntourists that it wasn't altogether my fault that we traveled so\neverlastin' slow--I'd 'a' been mad enough to bite all the enamel off\nmy teeth, and yet as far as I could see, Moses didn't do a single\nthing but show out a little peevish once in a while.\n\nStill, we didn't choose our natures nor the kind o' life to range 'em\nover nor the sorts o' temptations we'd prefer to wrastle with; an'\neven our own experiences are more 'n we can understand--to say nothin'\no' settin' back an' decidin' upon the deeds of others. My own test\nwasn't the one I'd 'a' chosen; and yet, for all I know, it may 'a'\nbeen the very best one, for me.\n\nLittle Barbie had finally grown up through childhood to the gates o'\nwomanhood--and as generally happens, she had found a man waitin' for\nher there. Through all the years of her growin', she had been sendin'\nout tendrils which reached over an' wound about my heart, and grew\ninto it an' through it, and became part of it. If it hadn't 'a' been\nfor Friar Tuck, I might 'a' married her, myself; for I could have done\nit, if all the men I'd had to fight had been other men--but the man I\ncouldn't overcome, was myself.\n\nThrough all the years I had known Friar Tuck an' rode with him an'\nworked with him an' slept out under the stars with him, he had been\nquietly trainin' me for the time when it would be my call to take my\nown love by the throat, for the sake of the woman I loved. It don't\nweaken a man to do this; but it tears him--My God, how it does tear\nhim!\n\nI, my own self, brought back the man she loved to her, and gave her\ninto his arms; and I've never regretted it for one single minute; but\nI doubt if I've ever forgot it for much longer 'n this either.\n\nI did what it seemed to me I had to do--an' the Friar thinks I did\nright, which counts a whole lot more with me 'n what others think. I\nwent through my desert, I climbed my hill, for just one moment I saw\ninto my promised land--and then I was jerked back, and not even given\npromotion into the next world, which Moses drew as his consolation\nprize. And yet, takin' it all around, I can see where life has been\nmighty kind and generous to me after all, and I'm not kickin' for a\nminute.\n\nThe great break in my life came in the fall, and it left ol' Cast\nSteel a more changed man 'n it did me. I wanted to swing out wide--to\nride and ride and ride until I forgot who I was and what had happened;\nbut the ol' man worked on my pity, an' I agreed to stay on with him a\nspell. Durin' the three years precedin', I had got into the handlin'\nof the ranch, more 'n he had, himself; so I spent the winter makin' my\nplans, an' goin' over 'em with him. He came out toward spring and was\nmore like himself; but when the first flowers blossomed on the\nbenches, they seemed to be drawin' their life blood out o' my very\nheart. All day long I had a burnin' in my eyes, everywhere I went I\nmissed somethin', until the empty hole in my breast seemed likely to\ndrive me frantic; an' one day I pertended to be mad about some little\nthing, an' threw up my job for good and all.\n\nThe ol' man was as decent as they ever get. He knew how I had been\nhit, an' he didn't try any foolishness. He gave me what money I\nwanted, told me to go and have it out with myself, an' come back to\nhim as soon as I could. I rode away without havin' any aim or end in\nview, just rode an' rode an' rode with memories crowdin' about me so\nthick, I couldn't see the trail I was goin'.\n\nThen one night I drew up along side o' Friar Tuck's fire, saw the\nsteady light of his courage blazin' out through his own sadness, the\nsame as it had done all those years; an' I flopped myself off my hoss,\nthrew myself flat on the grass, an' only God and the Friar know how\nmany hours I lay there with his hand restin' light on my shoulder, the\nlittle fire hummin' curious, soothin' words o' comfort, and up above,\nthe same ol' stars shinin' down clear and unchangin' to point out,\nthat no matter how the storms rage about the surface o' the earth,\nit's allus calm and right, if a feller only gets high enough.\n\n\n\n\nCHAPTER TWENTY-NINE\n\nTHE LITTLE TOWN OF BOSCO\n\n\nI hadn't done much eatin' or sleepin' on that trip, an' I was plumb\nbeat out; so after I fell asleep, the Friar put a soogan over me and\nleft me by the fire. He awakened me next mornin', gettin' breakfast,\nand it didn't take him very long to talk me into joinin' on to him for\ncompany. I had been avoidin' humans, for fear I might be tempted to\nstart trouble and find the easy way out of it all; but his plan was\njust the opposite--to dive so deep into humanity that I could catch a\nglimpse o' the scheme o' things.\n\nThe Friar held that we all had crosses comin' to us any way. If we\npicked 'em up an' put 'em on our own shoulders, we'd still be free,\nan' the totin' of our crosses would make us stronger; while if we\ntried to run away, we'd be roped an' thrown, an' the crosses chained\non us. I'd a heap sooner be free than a slave; so I decided to carry\nmine, head up, an' get right with myself as soon as possible.\n\nThe Friar didn't work off any solemn stuff on me, nor he didn't try to\nbe funny; he just turned himself into a sun-glass, an' focused enough\nsunshine on to me to warm me up without any risk of blisterin'. I got\nto know him even better those days than I had before. His hair was\ngettin' a bit frosty at the temples; but aside from this, he hadn't\naged none since the first day I had seen him. He was like some big\ntree growin' all by itself. Every year it seems a little ruggeder,\nevery year it seems to offer a little roomier shade; but the wind and\nthe rain and the hot sun don't seem to make it grow old. They only\nseem to make it take a deeper root, and throw out a wider spread o'\nboughs.\n\nHe told me o' some o' the scraps between the cattle men an' the sheep\nmen--the Diamond Dot was out o' the way of sheep at that time. Then I\nbegan to take a little more interest in things, an' after takin' note\nfor a day or so, I prophesied a dry summer; and this brought us around\nto Olaf.\n\nThe Friar warmed up at mention of him. He said 'at he had never seen a\nmatch turn out better 'n Olaf's. He said Kit had just what Olaf\nlacked, an' Olaf had just what Kit lacked, an' their boy was just\nabout the finest kid he knew of anywhere. We decided to head up their\nway an' pay a visit.\n\nAs we rode along we took notice of the way things were changin'. We\npassed several sheep wagons, five or six irrigation ditches, an' here\nand there, we found men who put more faith in alfalfa 'n they did in\nstock. The Friar had been well to the north when I happened upon him,\nand we traveled a sight o' country before we reached our destination.\nEverywhere folks knew him, an' he knew them; and when I saw their\nfaces light up at sight of him, I had to admit that he had done the\nright thing in stickin'.\n\nMostly he sang the \"Art thou weary,\" one for his marchin' song, now;\nand it got into my blood and did a lot to healthen me up again. I\ncan't rightly say 'at I ever got religion; but more 'n once religion\nhas got me an' lifted me up like the Crazy Water in flood, bearin' me\non over rocks an' through whirlpools, an' showin' me what a weak,\nuseless thing I was at the best. The's somethin' inside me 'at allus\nresponded to the Friar's music, an' made me willin' to sweep on over\nthe edge o' the world with him; but when he tried to reason out\nreligion to me, I have to own up 'at the' was a lot of it I couldn't\nsee into.\n\nWe passed Skelty's old place on our way in, an' found a red-eyed,\nblack-headed man runnin' it. His name was Maxwell, but they still\ncalled the place Skelty's. We went in an' had dinner, an' found five\nor six Cross-branders there. They were doin' plenty o' drinkin' an'\ncrackin' idiotic jokes with the girls; but they nodded friendly enough\nto us, an' we nodded back.\n\nAs soon as we finished, the Friar went outside for his smoke; but I\nleaned back right where I was for mine. One o' the Cross-branders, a\ntall, gaunt, squinty cuss by the name o' Dixon, was sittin' near me,\nand presently he turned an' sez: \"You're Happy Hawkins, ain't ya?\"\n\n\"That's me,\" sez I.\n\n\"Well, on the level,\" sez he, \"what became o' Badger-face?\"\n\n\"I've often wondered about that myself,\" sez I.\n\n\"We supposed he got killed,\" sez he; \"but two fellers claimed they saw\nhim goin' south in the spring with your huntin' party.\"\n\n\"What made ya think he got killed?\" sez I.\n\n\"'Cause he started over here one night, and never showed up again,\"\nsez he.\n\n\"I don't know what become of him,\" sez I. \"Dinky Bradford said he was\ngoin' to take him to Africa; but whether he did or not I can't say. I\nnever felt no call to pry into Dinky's business. Looks to me as though\nwe were goin' to have an extra dry summer.\"\n\n\"I say so too,\" sez Dixon. \"Who was this Dinky Bradford?\"\n\n\"That's bothered me a heap,\" sez I. \"He claimed to be a Greek hero,\nthough what sort o' business that is, I can't say. Finished your\nround-up yet?\"\n\n\"Just got through. Where is this Greek hero these days?\" sez he.\n\n\"Can't prove it by me,\" sez I. \"He's one o' these fellers no one seems\nto know anything about. I saw him go without eatin' for four days\nonce, an' he came out of it in better shape 'n he went in. Badger-face\nwas your foreman, wasn't he?\"\n\n\"Yes,\" sez he. \"Ol' Pepper Kendal is foreman now.\"\n\n\"I should think a foreman would have some load on his shoulders with\nthe boss gone all winter,\" sez I.\n\n\"The boss brought a woman back with him this time,\" sez Dixon.\n\n\"What!\" sez I. \"You don't mean ta tell me 'at Ty Jones has got him a\nwoman after all these years?\"\n\n\"That's what,\" sez Dixon. \"Somethin' queer about her, too. Ty has had\na new shack built for her up back o' the old house. They don't seem\noverly friendly for a bride an' groom.\"\n\n\"Ain't nothin' overly friendly with Ty, is the'?\" sez I.\n\n\"Oh, I dunno,\" sez he. \"Ty ain't as sticky as taffy, but he's a mighty\ngood man to work for.\"\n\n\"What sort of a woman did he get?\" sez I.\n\n\"She don't show herself much,\" sez he. \"She's tall an' shapely, an'\nright smart younger 'n Ty; but she spends most of her time in the new\nshack; and from all we can tell, she's froze up tighter 'n Ty is.\"\n\n\"Well, I guess we'll have to jog on. Good luck,\" sez I, and me an' the\nFriar rode on. He was as much beat out over Ty Jones gettin' a woman\nas I was; but first thing he thought of was, 'at this might have a\nsoftenin' effect on Ty, an' give him an openin'.\n\nWe reached Olaf's in time for supper, and found Kit bustlin' about as\nhappy as a little brown hen. The Friar hadn't sprung it none about the\nkid. He was a solid little chunk with a couple o' dimples and all the\nsigns o' health. I looked careful into his eyes. They were full o'\ndevilment, an' he scowled his brows down over 'em when I held him; but\nthey were brown like Kit's.\n\n\"Oh, he's too dirty to touch,\" sez Kit, beamin' all over with pride.\n\"I just can't keep him clean, try as I will.\"\n\n\"Be careful, Happy, and don't soil your hands on that baby!\" yells the\nFriar as though in a panic. \"Let me have him. I was dirty once,\nmyself.\"\n\nIt was plain to see 'at the kid an' the Friar were old cronies; and it\nwas a pleasant sight to see 'em together. The Friar got down on the\nfloor with him an' played bear an' horse an' the kid entered into it\nan' fair howled with merriment. Kit scolded 'em both an' took so much\ninterest in their antics she hardly knew what she was doin' to the\nsupper things.\n\nBefore long Olaf came in. He still took up all the space not otherwise\noccupied; but he had an altogether-satisfied expression which made ya\nforget how everlastin' ugly he really was. He took us out an' showed\nus the garden, an' the new wire fencin' an' the baby's swing, an' all\nthe rest of his treasures. Olaf didn't want any more changes to take\nplace in the world. If his vote could have made it, things would just\ncontinue as they were until the earth wore out. It made me feel a\nlittle lonely for a moment; but I entered in as hearty as I could.\n\nDurin' supper I sez to Kit: \"Well, Ty Jones has a woman, now; and if\nit improves him as much as it has Olaf, he may blossom out into a good\nneighbor to you yet.\"\n\n\"Ty Jones got a woman!\" exclaimed Kit. \"Well, I'd just like to lay my\neyes on the woman 'at would take Ty Jones.\"\n\n\"Oh, all women ain't so set on havin' a handsome man as you were,\" sez\nI.\n\n\"Well, I wouldn't have any other kind,\" sez Kit, an' she gave her head\na toss while Olaf grinned like a full moon.\n\nThey were both purty well beat out to think o' Ty Jones havin' a\nwoman, an' we all talked it over durin' the rest o' the meal. After\nsupper, Olaf took the kid on his lap and sat by the fire tellin' us\nhis plans, while Kit cleared up the dishes an' stuck in a word of her\nown now and again. It was plain to see 'at she did full as much o' the\nplannin' as he did, an' this was probably what made her so satisfied.\nThe kid regarded Olaf's mustache as some sort of an exercisin'\nmachine, an' Olaf had to fight him all the time he was talkin', but he\ncertainly did set a heap o' store by that boy.\n\nHe told us he had about sixty cows and a fair run o' two an' three\nyear olds with a high average of calves; but that he intended to sell\nthe whole run to the Double V outfit up on the Rawhide, and get a\nsmall band of sheep. This flattened me out complete; but he had a lot\nof arguments on his side. He was also experimentin' with grain seed\nwhich he had got from Canada, an' he already had a patch of alfalfa\nwhich was doin' fine. He was one o' those fellers who can't tire\n'emselves out, an' so just keep on workin' as long as the law allows\n'em to use daylight. He had a young Swede workin' for him, but just at\nthat time, he was off lookin' for the work hosses which had voted\n'emselves a vacation, an' had gone up into the hills.\n\nThe Friar wanted to go up into the Basin country next day, so we\nbedded down purty early. I lay awake a long time thinkin' over what a\nfright Olaf had once been, and how he had straightened out of it.\n\nNext mornin' we started soon after sun-up. The Friar had a couple o'\nwomen runnin' a Sunday School at Bosco, and he wanted to see how they\nwere gettin' along. They had belonged to his brand of church clear\nback in England, and he set a lot of store by 'em; but owned up that\nthey had their work cut out for 'em at Bosco; it bein' one o' the most\nungodly little towns in the whole country.\n\nWe nooned on Carter, slipped over Boulder Creek Pass, and reached\nBosco at sun-down. It allus surprised me to see how much travel the\nFriar could chalk up, takin' his weight into account; but he was less\nirritatin' to a hoss 'n airy other man I ever met up with. The more of\na hurry he was in, the more time he took on the bad hills; and he\nnever robbed a hoss by sleepin' an hour late in the mornin', an'\nmakin' the hoss even up by travelin' beyond his gait.\n\nThe husband of one o' these women ran a saloon, the husband of the\nother--the women were sisters--was the undertaker and also ran a meat\nmarket. I thought this about the queerest business arrangement I had\never been confronted against; but the man himself was full as peculiar\nas his business.\n\nI have a game I have played with myself all my life. I call it \"why,\"\nan' I suppose it has furnished me more fun 'n anything else has. I\ntake any proposition I come across an' say all the whys about it I can\nthink up an' then try to answer 'em. Why did anything ever happen just\nas it did happen just when it did happen? This is the joke o' life to\nme. I have played it on myself times without end; but only once in a\nwhile even with myself can I follow the line back to common sense.\n\n\n\n\nCHAPTER THIRTY\n\nTY JONES GETS A WOMAN\n\n\nBosco was a regular town with twenty or thirty houses, a post office,\ntwo general stores, three saloons, an' all such things; and right on a\ngood stage road runnin' north an' south. We stopped with the\nmeat-market undertaker, 'cause they didn't think it quite respectable\nfor the Friar to live off the profits of the liquor traffic; though\nthe Friar allus said 'at he had a heap more respect for a square\nsaloon-keeper 'n for a sneaky drygoods merchant.\n\nShindy Smith was the saloon-keeper, an' Bill Duff was the undertaker.\nDuff was the absent-mindedest man I ever got intimate with, an' about\ndrove his wife to distraction, she bein' one o' these hustlers who\nnever make a false move. He had the idee that bein' an undertaker took\naway his license to laugh, so he allus walked on his toes an'\ndisported as solemn a face as nature would allow; but nature had\nintended him for a butcher, an' had made his face round and jowly.\nWhenever he didn't have anything else to do, he used to sit down an'\npractice lookin' solemn. He'd fix his eyes on the ceilin', clasp his\nhands across his stomach, pull up his eyebrows, droop his mouth, an'\nlook for all the world like a man dyin' o' the colic.\n\nHe was so absent-minded that he'd raise his cup to take a drink of\ncoffee, forget what he had started to do, an' like as not pour it over\nhis flapjacks for syrup. He started to engineer a funeral once with\nhis butcher's apron on, and they told all sorts of stories about him\nwhich was shockin' to an extent; though his wife kept such a sharp eye\non him, that I don't believe more 'n half of 'em. Still it wasn't any\nsort o' business for an absent-minded man to be in.\n\nIt was an uncertain business. Of course all lines o' trade in a thinly\nsettled country go by fits an' starts; but his was worst of all.\nSometimes he'd have as many as three funerals a month, and at others\nit would take him six weeks to sell out a beef carcass. A feller who\nhad a spite again' him started the story 'at he soaked his meat in\nembalmin' fluid, an' then if they came an extra special rush in both\nlines of his business at the same time, he'd--but then his wife kept\nsuch a skeptical eye on him, 'at I don't believe a word of these\nstories, an' I'm not goin' to repeat 'em. The worst I had again' him\nwas that he was so everlastin' careless. I lay awake frettin' about\nhis carelessness till I couldn't stand it a second longer; and then I\nrolled up half the beddin' an' started to sleep on the side porch.\n\n\"Where you goin'?\" sez the Friar.\n\n\"This here Bill Duff is too absent-minded an' forgetful for me,\" sez\nI.\n\n\"What do ya mean?\" sez the Friar.\n\n\"Well,\" sez I, \"I don't want to make light o' sacred things, nor\nnothin' like that; but Bill Duff's got somethin' stored up in this\nroom which should 'a' been a funeral three weeks ago, and I intend to\nsleep outside.\"\n\nThe Friar chuckled to himself until he shook the whole house; but it\nwasn't no joke to me; so I shunted the beddin' out on the roof o' the\nporch, which was flat, and prepared to take my rest where the air was\nthin enough to flow into my nostrils without scrapin' the lid off o'\nwhat Horace called his ol' factory nerve.\n\nAs soon as the Friar could recover his breath, he staggered to the\nwindow, an' sez: \"That's nothin' but cheese, you blame tenderfoot.\nLimburger cheese is the food Bill Duff is fondest of, and he has four\nboxes of it stored in this room.\"\n\n\"Then,\" sez I, comin' in with the beddin', \"I'll sleep in the bed, an'\nthe cheese can sleep on the porch; but hanged if I'll occupy the same\napartment with it.\" I set the cheese out on the porch--it was the\nripest cheese in the world, I reckon--and it drew all the dogs in town\nbefore mornin'. After they found it was above their reach, I'm\nconvinced they put up the best fight I ever listened to.\n\nIt took a long time for the memory o' that cheese to find its way out\nthe window; and I lay thinkin' o' the Friar's work, long after he had\ndrifted off himself. He wasn't squeamish about small things, the Friar\nwasn't, and this was one of his main holts. When we had got ready to\neat that night, Mrs. Duff had tipped Bill a wink to ask the Friar to\nsay blessin'. Bill was in one of his vacant spells, as usual, so he\nlooked solemn at the Friar, and sez: \"It's your deal, Parson.\" Now, a\nlot o' preachers would 'a' gone blue an' sour at that; but the Friar\nnever blinked a winker.\n\nThen after supper, all the young folks o' that locality had swooped in\nto play with him. This winnin' o' young folks was a gift with the\nFriar, and it used to warm me up to watch him in the midst of a flock\nof 'em. He showed 'em all kinds o' tricks with matches an' arithmetic\nnumbers, an' taught 'em some new games, and then he put up a joke on\n'em. He allus put up one joke on 'em each visit.\n\nThis time he puts a glass of water under his hat, looks solemn, and\nsez 'at he can drink the water without raisin' the hat. They all bet\nhe can't, and finally he goes into a corner, makes motions with his\nthroat, and sez he is now ready to prove it. Half a dozen rush forward\nand lift the hat, and he drinks the water, and thanks 'em for liftin'\nthe hat for him so he could drink the water an' make his word good.\n\nSome folks used to kick again' him and say he was worldly; but his\nmethods worked, an' that's a good enough test for me. He took out the\nshyness an' the meanness an' the stupidity, and gave the good parts a\nchance to grow; which I take it is no more again' religion than the\npublic school is. Why, he even taught 'em card tricks.\n\nHe could take a deck of cards and turn it into a complete calendar,\nleap year and all; and then he could turn it into a bible, showin'\neasy ways to learn things, until a feller really could believe 'at\ncards was invented by the early Christians who had to live in caves,\nas some claim. All the time he was playin' with 'em, he was smugglin'\nin wise sayin's with his fun, pointin' out what made the difference\nbetween deceivin' for profit, and deceivin' for a little joke, tellin'\n'em how to enjoy life without abusin' it--Why, he even went so far as\nto say that if a feller couldn't be religious in a brandin' pen he\ncouldn't be religious in a cathedral--which is a two-gun church with\nfancy trimmin's.\n\nBy the time he had expanded the young folks and made 'em easy and at\nhome, the older ones had arrived; and then he held a preachin'. The\nwhole outfit joined in with the singin', and when he began to talk to\n'em every eye in the room was glistenin'. You see, he knew them and\ntheir life; and they knew him and his. He had nursed 'em through\nsickness, he had tended their babies, he had helped to build their\ncabins an' turn 'em into homes; so the words flowed out of his heart\nand into theirs without any break between. This was the Friar and this\nwas his work--but I can't put it into a story.\n\nThe' was a no-account cuss by the name o' Jim Stubbs who lived--if ya\ncould call it livin'--at Boggs; and the Friar induced him to go along\non one of his trips. When Jim came back he was a made-over man, and\nevery one asked him if he had religion. \"Hell, no,\" sez Jim, tryin' to\nbe independent, \"I ain't got religion; but a feller catches somethin'\nfrom the Friar the same as if he had the measles; and I don't covet to\nbe a bum no more.\"\n\nThis gives ya the best idy of the Friar that I can think of; and I\nfinally fell asleep there at Bill Duff's, with my mind made up to bury\nmy own heartache, keep the grave of it green, but live out my life as\nhard as the Friar was livin' his.\n\nWe had intended to projec about in the Basin next day to rustle up\nsome new trade in the Friar's line; but my pony turned up lame, so we\nheld over to get him shod. When the stage pulled in that evenin', me\nan' the Friar went down to see it. A little feller sat on the seat\nwith the driver. His hat was covered with dust an' pulled down over\nhis eyes, an' what ya could see of him was the color o' coffee; but\nthe moment I lay eyes on his side-burns, I grabbed the Friar's arm an'\nwhispered, \"Horace!\" and by dad, that's who it was. Promotheus was in\nthe back seat, an' he looked for all the world like an enlarged copy,\nexcept that his side-burns were red an' gray, while Horace's were\nmostly brown. But they were cut exactly the same, startin' from his\nears, runnin' across his cheeks an' lips, an' then curvin' down to the\ncrook of his jaw, close cropped an' bristly.\n\nHorace an' Promotheus hit the ground as soon as the stage stopped, an'\nme an' the Friar dropped back out o' sight inside the hotel. Horace\ngave orders about his two boxes an' started into the hotel. Just as he\ncame through the door, I stepped out an' gave him a shove. \"You can't\ncome in here,\" I growled.\n\nHe stepped back as fierce as a rattler. \"I can't, huh?\" he piped.\n\"Well, we'll see if I can't.\"\n\nThen he recognized me, an' we began to pump hands. He said 'at he and\nPromotheus had only reached home three weeks before; but they couldn't\nstand it, an' so had made a streak for the West. He said they had been\nin Africa an' India, until they had become plumb disgusted with\ntropical heat, an' so had come out the northern route, expectin' to\noutfit at Bosco an' ride down to the Diamond Dot.\n\nWe suppered with 'em an' next day they bought a string o' hosses,\npacked their stuff on 'em, an' said they were ready for some\namusement. Horace had got a little snappier in his talk an' his\nmovements; but that was about the only change. As soon as we told 'em\nabout Ty Jones havin' a woman, that settled it. Horace insisted upon\nseein' the woman, an' Promotheus echoed anything 'at Horace said,\nthough his face clouded a bit at the idee of foolin' around the Cross\nbrand ranch. The Friar didn't feel any call to go along with us; but\nit was more to my mind just then 'n his line was, so I jumped at the\nchance.\n\nHorace was also mighty glad to add me to his outfit. He had been used\nto havin' a lot o' Zulus an' Hindus waitin' on him, and hadn't\nadjusted himself to a small outfit yet. He said he had sent a lot o'\nhides an' heads an' horns and other plunder from London, England, to\nthe Diamond Dot; but had been too busy to write durin' the past few\nyears. He and the Friar had quite a talk together before we left; but\nI could tell from their faces 'at Horace didn't have any news for him.\n\nWe had high jinks when we reached Olaf's; but Horace didn't make any\nhit with the kid. The kid had a jack-in-the-box toy 'at looked\nconsid'able like Horace, an' the kid couldn't square things in his own\nmind, to see a big size one, out an' walkin' about like a regular\nhuman; but when he also got to studyin' Promotheus, he was all undone.\nOlaf tried to have him make up to Horace, but he wouldn't stand for\nit. He'd sit on Olaf's knee and look first at his jack-in-the-box,\nthen at Horace, and wind up with a long look at Promotheus. Promotheus\nwould try to smile kind an' invitin', and then the kid would twist\naround and bury his face in Olaf's vest. Horace nor Promotheus didn't\nmind it any; but as far as that goes, the kid was only actin' honest\nan' natural, accordin' to his lights, an' the jack-in-the-box had as\nmuch of a kick comin' as anybody.\n\nTy had been down there just the day after we had left, an' had wanted\nto buy Olaf's place; but only offered half what it was worth. He had\ndone this half a dozen times, an' allus insulted Olaf as much as he\ncould about it. Olaf had wanted to sell out at first; but Kit had been\nable to see 'at they had a homestead fit for any thing, and she had\nallus insisted that they get full price or hang on. Now, it was\nimproved way beyond common, an' they were both fond of it; so they had\ndecided to stick it out.\n\n\"This is goin' to be a dry summer,\" sez I.\n\nOlaf's face clouded up but he only shut his lips tighter. We told 'em\nwe were on our way up to try an' have a look at Ty Jones's woman, and\nOlaf said he'd go along if he didn't have to trail his cattle up to\nthe Raw Hide, this bein' part o' the deal he had made. He said it\nwould take him about ten days probably, an' wanted us to camp in the\nSpread, an' keep an eye on his stuff. Olaf clipped the first joint off\no' Promotheus's name, an' I was glad of it.\n\nWe chucked our stuff into the barn next mornin' an' started to stalk\nthe Cross brand neighborhood. Horace had a small field glass which was\na wonder, and we worked as careful as we could. It was only fifteen\nmiles across from Olaf's; but all we were able to do the first day was\nto find a little sheltered spot up back o' the ranch buildin's where\nwe could get a good view of 'em through the field glass.\n\nNext day Olaf an' Oscar started with the bunch o' cattle, an' we rode\nalong part way with 'em to give 'em a good start; but Olaf had handled\nhis stuff so gentle that it was no trouble, an' we turned back an'\ntook up our watch again. We watched for a week without seein' a thing,\nridin' in each night to sleep back of Olaf's shack. Me an' Theus--I\nhad seen Olaf's ante an' had raised him one--were gettin' purty weary\no' this sort o' work; but Horace was as patient as a spider. Finally\nthough, we got a little more risky, and leavin' our hosses up in our\nsheltered spot, we follered down a ravine to get nearer to the new\ncabin.\n\nWe had caught several glimpses of a woman to prove to us 'at the' was\none there; but that was about all, an' so we went down this ravine,\ntryin' to figure out what excuse we'd give if we came across any of Ty\nJones's men. Neither me nor The--Promotheus had said 'at we couldn't\nbe no politer 'n he could, so he had lopped off the last joint, and\nnow had as neat a workin' name as any one, although Horace still\ninsisted on usin' the whole outfit when he had occasion to address\nhim. Well, neither me nor The felt just easy in our minds at snoopin'\nabout Ty's when we hadn't any business to, especially The; but Horace\nwas as selfcomposed as though he was herdin' lions out o' tall grass,\nwhich it seems had been his favorite pastime durin' the last few\nyears.\n\nThe knew the ravine well; he said it ran full o' water in the spring,\nbut after that was dry all the year. We got about half-way down it,\nan' then we came to a path 'at was plain enough to see. The stopped\nan' wagged his head. \"No one ever used to use this,\" sez he.\n\n\"Well, some one uses it purty constant, now,\" sez I.\n\n\"The woman is the one who uses it,\" sez Horace. \"She's lonely, that's\nplain enough. The path climbs the opposite bank--let's cross an' go\nup.\"\n\nMe an' The bucked at this for some time; but Horace hung out; so we\nwent along with him. We finally came to a little glen with a spring in\nit, an' grass, and in a little clump o' small trees, we came across a\nbook lyin' face down on a Navajo blanket.\n\n\"That's gettin' close,\" sez Horace.\n\n\"Yes!\" sez we, in low tones.\n\nWe scouted all around; but no one was there, an' then we took a line\non the hill back of us, picked out a likely spot, and returned the way\nwe had come, this bein' the only direct way. We didn't meet a soul--at\nleast none wearin' bodies, though from the creepy feelin' I had part\nof the time, I won't ever be certain we didn't meet any souls.\n\nNext day, we circled the peak and got up to the spot we had picked\nout. We could see the clump o' trees plain enough; and along about\nthree in the afternoon, we saw the woman come up the path, walkin'\nslow an' actin' weary. She had two big dogs with her, and whenever\nshe'd stop to rest a bit, she'd pet 'em. \"Well,\" sez The, \"things has\nchanged a heap when ol' Ty Jones stands for havin' his dogs patted.\"\n\nWe couldn't get a good view of her face from where we were, but we\ncould get a fine view o' the ranch buildin's. The' didn't seem to be\nmuch work on hand, and we saw eight or ten men foolin' around an'\npretendin' to do chores. The recognized the two Greasers he had been\nridin' with the day he had pulled on Horace, and one or two others;\nbut most of 'em was strangers to him. He said the Greasers were about\nthe most devilish speciments he had ever herded with--an' Ty's whole\noutfit was made up o' fellers who had qualified to wear hemp.\n\nHorace was keen to go on down to her an' get a good look; but me an'\nThe took the bits in our teeth at this. We knew what those dogs were\nlike, an' refused pointblank to go a peg unless he could think up a\ngood enough excuse for us to give to Ty Jones--and we wouldn't let\nHorace go down alone.\n\n\"The best plan I can see,\" sez I, pointin' to a cluster o' big rocks\ndown the <DW72> to the left, \"is to circle back to those rocks. We can\nsee her face plain from there when she comes back the path.\"\n\nAfter examinin' this plan we decided it was the best; but when we went\nafter our hosses, Horace's had broke his reins an' gone back through\nthe hills. By the time me an' The had rounded him up, it was too late,\nso we had to wait till next day.\n\nNext day I left the other two at our first look-out and rode on to the\nnew one. As soon as she came in sight, I waved my hat to 'em and they\nsneaked down to the bunch o' rocks. I rode back an' left my hoss with\ntheirs, an' then joined 'em.\n\nShe didn't come into view till after five o'clock. When she reached\nthe edge of the ravine an' started down, she paused an' looked off\ninto the valley with her face in plain view. Horace looked at it\nthrough his glasses, gave a start, and then handed the glasses to The.\n\"Have you ever seen any one who looked like her?\" sez he.\n\nThe looked and broke out into a regular expression. \"That's the\noriginal of the photograph I had,\" sez he.\n\n\"That's the Friar's girl, sure as the sky's above us,\" sez Horace.\n\nI grabbed the glass and took a look. She did look like the picture,\nbut older and more careworn. Some way I had allus thought o' the\nFriar's girl as bein' young and full of high spirits, with her head\nthrown back an' her eyes dancin'; but just as I looked through the\nglasses, she pressed her hands to her head, and her face was wrinkled\nwith pain. She was better lookin' than common, but most unhappy.\n\n\"That devil, Ty Jones, is mean to her!\" I growled between my teeth.\n\n\"Dogs or no dogs, I'm goin' down to have a talk with her,\" sez Horace.\n\nHe started to get up, but I pulled him back to the ground. I had kept\nmy eyes on her, and had seen the two dogs turn their heads down the\nravine, and her own head turn with a jerk, as though some one had\ncalled to her. Horace looked through the glasses again, and said he\ncould see her lips move as though talkin' to some one, and then she\nwent down into the ravine. We couldn't see the bottom of the ravine\nfrom where we were, nor we couldn't see the ranch buildin's; so we\nhustled back through some washes to our look-out, and reached it just\nas she and Ty came out at the bottom.\n\nThey were walkin' side by side, but Horace, who was lookin' through\nthe glasses, said they seemed to be quarrelin'. \"It's moonlight\nto-night,\" sez Horace, \"and I'm goin' to sneak down and try to see\nher.\"\n\nWe argued again' it all we could, but he stood firm; so all we could\ndo was to sit there and wait for the lights to go out in the\nbunk-house. As she was a reader, we figured 'at she'd be the last one\nto turn in; normal habits an' appetites not havin' much effect on\nbook-readers.\n\n\n\n\nCHAPTER THIRTY-ONE\n\nJUSTICE UNDELAYED\n\n\nHuman emotions are like clocks: some of 'em will run longer 'n others;\nbut they'll all run down unless they're wound up again every so often.\nEven fear will only run so long, as several late-lamented bullies have\nbeen forced to learn just before they passed over the Great Divide.\nAfter you've scared a feller as bad as he can get, it is well enough\nto let him alone. If you keep on addin' horror onto horror, his fear\nis likely to run down; and the chances are 'at he'll get irritated,\nand slaughter ya.\n\nI don't know whether or not patience can rightly be called an emotion;\nbut anyway, mine runs down a little easier 'n airy other o' my\nfaculties, and sittin' up in the chill an' waitin' for a lot o'\nfestive fools to go to bed, allus was just the sort o' thing to\ndisgust me. Those Cross-branders didn't seem to have any more use for\nshut-eye that night than a convention o' owls. Some of 'em rode off at\ndusk, but more of 'em arrived, and they held some sort of high jinks\nin the bunk-house, till I began to talk back at myself loud enough for\nall to hear. It was full moon an' we could see dogs loafin' an'\nfightin' down at the ranch, the light in the new cabin was the first\nto go out, an' for the life of me, I couldn't see where we had a\nsingle pair to stay on; but Horace seemed to accumulate obstinacy with\nevery breath he drew. The sided with me, but criticizin' Horace went\nagain' his religion, so he didn't make any more uproar than a gnat\nfight.\n\nFinally I calmed down until I could stretch each word out a full\nbreath an' sez in my doviest voice: \"Horace, will you kindly tell me\nwhat in hell you intend to do?\"\n\nHe studied the situation careful, and took all the time he needed to\ndo it. \"I'm goin' back to camp,\" sez he. \"To-morrow night they'll be\nsleepy, and we'll have the whole place to ourselves.\"\n\n\"Hurrah for hot weather! Greece has finally melted!\" I yelled, an' we\nhustled for our ponies.\n\nI have a buck-skin riggin' I put on the bridle of a hoss who gets into\nthe evil way of steppin' on his reins; and I had fixed one on Horace's\nhoss to bring him back to his senses should he attempt to play the\nsame trick he had worked on us the day before. When a hoss wearin' one\no' these contrivances steps on his reins it pinches his ears, down\nclose to his head where they're tender, and generally works a\nreformation in short order.\n\nWe forgot all about this, and when Horace jumped into his saddle, he\ngave a jerk on the reins--and got bucked into a clump o' cactus. The\nhoss didn't try any runnin', though, which proves he had learned a\nproper respect for trailin' reins. Still, Horace wasn't quite in the\nmood to see the beauty o' my method, so he insisted upon my swappin'\nhosses with him. It was a good two-hours' ride to Olaf's, and by the\ntime we had changed saddles, and I had convinced the pony that his\nidees of buckin' were childish an' fu-tile, and his show of temper had\nonly given him a hundred an' ninety pounds to carry instead of a\nhundred an' twenty, it was after nine o'clock.\n\nWe were hungry enough to call for speed; but still it was eleven by\nthe time we reached the Spread. We thought we had seen a horseman go\ninto it from the other direction; but the moon had ducked under a\ncloud and we couldn't be certain.\n\nWe didn't intend to waken Kit if we could help it; so we started to\nput the hosses into the corral as quiet as possible. Just as we had\nthrown our saddles over the top bar, we heard a commotion from the\ncabin, and started for it on the run.\n\nThere wasn't any light in the cabin; but we heard Kit screamin', and\nbefore we arrived, we saw a man rush around the corner just as the\ndoor was flung open, and two other men jumped towards him from the\ninside. These two had knives in their hands; and the man outside took\na step back. They rushed him, but he hit one with his right fist, and\nthe other with his left, and curled 'em both up again' the side o' the\nhouse in a way to make a feller's heart dance for joy. Then we saw it\nwas the Friar himself, and we gave a whoop.\n\nKit had banged the door shut, put up the bar, got a rifle and made\nready for what was to come next; but when she heard our whoop, she put\non her wrapper and opened the door. The two men 'at the Friar had\ncrumpled up were those same two Greasers 'at The had told us were the\nmeanest pair he had ever herded with.\n\nWe took 'em by the heels an' straightened 'em out, while Kit indulged\nin a few little hystericals. The Friar had allus been a great hand to\nexpound upon moral force an' spiritual force, and such items, and now\nwhen the two Greasers refused to come back an' claim their own bodies,\nhe got a little fidgetty.\n\n\"Friar,\" I sez, \"I give in to you. Your quiet way o' lettin' the right\nwork out its own salvation is the surest way I know; and in an\nemergency like this, it does full as well as violence.\"\n\nThe Friar wasn't in no mood for hilarity, though; so after gettin'\ntheir weapons an' tyin' 'em up, we soused the Greasers with water, and\nbrought 'em back to give an account o' themselves, Kit all the time\ntellin' us what had happened.\n\nIt seems 'at Kit had been hoein' in her beloved garden that day an'\nhad been purty tired at night; so after waitin' for us until she got\nexasperated, she had eaten her own supper, put ours on the table, an'\nturned in. Olaf had put up another cabin the same size as his first.\nHe had put 'em side by side with a porch joinin' at their eaves. In\none cabin was the dinin' room an' kitchen, all in one, and in the\nother was the bedroom an' settin' room.\n\nKit had heard a noise in the settin' room and had opened the door\nbefore she was full awake, thinkin' it was the dog or cat. The minute\nshe had opened the door they had grabbed her, and she had begun to\nscream. They shut off her wind a little; but they wasn't rough with\nher--quite the contrary. They leered into her eyes, and patted her on\nthe shoulders, and made queer, gurglin' noises in their dirty brown\nthroats; but they didn't speak to her, not one word.\n\nKit was strong, an' she had fought 'em to a standstill for what she\nthinks was twenty minutes, at least; but she was beginnin' to weaken.\nOne of 'em kept his arm about her neck, and whenever she tried to\nscream, shut off her wind. She had heard the Friar's hoss nicker when\nhe opened the first pole gate, and this provided her with enough moral\ncourage to sink her teeth into the wrist of the arm about her neck.\nThe feller had give a yell, and struck her; but at the same time, she\nhad opened up a scream of her own which loosened things all over the\nneighborhood.\n\nThe Friar had first put for the settin' room door; but they had locked\nthis door on the inside, intendin' to go out the side door. He savvied\nthis so he dove into the porch-way between the two cabins, and made a\nrattlin' on this door. They had paused at this; but he had to rattle\nseveral times before they took down the front bar. We had been fordin'\nthe crick about this time.\n\nThe Greasers had tried to get out the window once; but Kit had called\nout what they were up to; so they had turned on her an' choked and\nbeat her scandalous.\n\nThis was Kit's side, and by the time she had finished tellin' it, the\nGreasers had begun to moan an' toss. The Friar gave a sigh of relief,\nas soon as they came to enough to begin grittin' their teeth. I sat\n'em up with their backs again' the side of the cabin, and intimated\nthat we were ready to receive their last words.\n\nWe had to encourage 'em a bit, one way or another; but we finally got\nout of 'em that they had poisoned the dog, and then cut a crack in the\ndoor till they could raise the bar. They said 'at Ty Jones hadn't had\nno hand in plannin' their trip; but had offered 'em a hundred apiece\nif they could put Olaf in the mood of wishin' he had sold out\npeaceable.\n\n\"Well,\" sez I, as soon as they were through, \"shall we finish with 'em\nto-night, or give 'em till to-morrow to repent?\"\n\n\"We shall of course deliver them to the proper officials to be tried\nby due process of law,\" sez the Friar.\n\n\"What for?\" sez I. \"Ya never can tell how a trial will turn out; but\nwe know 'at they have forfeited the right to live; so we'll just give\n'em what they've earned and save all fuss.\"\n\n\"No good ever comes of men taking the law into their own hands,\" sez\nthe Friar firmly.\n\n\"How come, then, that you didn't run an' tell some justice o' the\npeace, 'at these two snakes was actin' disrespectful--instead of\nknockin' 'em up again' the logs?\" sez I.\n\n\"I should have done so if I had had time,\" sez the Friar with dignity.\n\n\"Well, you're better trained 'n we are,\" sez I; \"but it still takes a\nlittle time for you to make your hands mind your self-control, after\nyou've been het up. You can do it in ten minutes, say; but it takes us\nabout a week, and by that time the' won't be any need for the law.\"\n\n\"No,\" sez the Friar, \"I insist that we rely upon the law. We count\nourselves as of the better element; and the most vicious conditions\narise when the better element takes the law into its own hands. When a\nvicious man does illegal violence, it does not establish a precedent;\nbut when the decent man does the same thing, it tears away forms of\ncivilization which have taken centuries to construct.\"\n\n\"That sounds like sense,\" sez I; \"and after this is all over, I don't\nmind arguin' it out with you; but right now, it would seem to me that\nif we went to law about this, it would be because we wanted to\nshoulder onto the law the responsibility of doin' what we feel ought\nto be done, but which we haven't the nerve to do ourselves.\"\n\n\"If you attempt to lynch these men, I shall ride at once and give the\nalarm,\" sez the Friar.\n\n\"And when you came back, you would find 'em swingin' from a limb,\" sez\nI. \"I'm with you in most things, Friar, and if the' was a shred o'\ndoubt, I'd be with you in this; but it's too plain a case. I'm willin'\nto hold these two in secret until we can collect a posse o' twelve to\ngive 'em a jury trial; but this is the most I'll do. Ty Jones has got\nothers of his gang away from the law, but he don't get these two--not\nif I can help it.\"\n\nHorace sided with me, and so did The, though he didn't have much to\nsay. He was thinkin' of his own trip to pester Olaf, and it came back\nto him purty strong. The Friar finally had to agree not to notify the\nlaw until I'd had time to gather up a posse. I made Horace promise not\nto tell the Friar about our seein' the woman back at Ty's, saw that\nthe Greasers were planted safe in Olaf's log barn, and set out at once\nfor the Diamond Dot on a fresh hoss. I never want to eat none before\nstartin' a ride like this.\n\nI rode all that night through the moonlight; swingin' up over the\npasses, fordin' the rivers, and reachin' the Diamond Dot at noon the\nnext day. I didn't let on to Jabez 'at I was there at all; but I got\nSpider Kelley, ol' Tank Williams, Tillte Dutch, and Mexican Slim to\ntake a vacation and come on back with me. This gave five for the jury,\nas I didn't intend to have Horace or The sit on it, not knowin' how\nfar their prejudice might prevent 'em from executin' my idee of\njustice. We set out to return, about five o'clock, and rode into the\nSpread at seven the next mornin' with eight other fellers we had\nbrought along for good measure.\n\nOld Jimmy Simpson and his four grown sons were in this bunch, and I\nwas purty well acquainted with 'em. I knew 'at they had been amply\npestered by Ty Jones's outfit, and wouldn't be too particular about\nwhat book-law might have to say on the subject, though ol' man Simpson\nwas up on book-law. The other three were fellers they knew and were\nwillin' to guarantee. We were all a little sleepy, so we decided to\nhold the trial after dinner.\n\nThe Friar had spent as much time with the Greasers as they'd stand\nfor; but he hadn't made much impression on 'em. I knew 'at he was\nheart-whole in his attitude, an' I hated to cross him; but this was a\ncase o' principle with me, so when we got ready for the trial, I tried\nto get him to take a long walk, but he refused.\n\nWe held the trial in front o' the barn, and it was as legal as any\ntrial ever was, and as solemn, too. We untied the prisoners, and\ncalled Kit for the first witness. She told it just as she had told it\nto us, but her bruised face would have been all that was necessary.\nThen we called the Friar and he told his part, and we let him make a\nspeech in favor o' law and order; and cheered him hearty, too, when he\ngot through.\n\nI had just begun to give my part, when Olaf and Oscar rode up. Olaf\nsat on his hoss and looked at us a moment, at Kit with her bruised\nface, holdin' the boy in her arms, at the prisoners and us; and then\nhe asked the Friar what it all meant. The Friar was sunonomous with\ntruth, as far as Olaf was concerned.\n\nOlaf listened quietly, the dark red risin' in his cheeks bein' about\nthe only change in him. When the Friar finished, Olaf got off his\nhoss. \"The' won't be need of any more trial,\" sez he. \"Kit, you go to\nthe house.\"\n\nKit started for the house, and the Friar asked Olaf what he intended\nto do.\n\n\"Kill 'em,\" sez Olaf, \"with my two hands.\"\n\nHe unbuckled his belt and threw it on the ground, then kicked off his\nchaps, and stepped through the ring we had formed. \"Stop,\" said the\nFriar. \"Olaf, I forbid this.\"\n\n\"You had better go to the house, Friar,\" said Olaf with pleadin' in\nhis voice. \"Go in--please go in--an' comfort Kit.\"\n\nThe Friar made a rush, but we fended him off. The Greasers also tried\nto make a get-away; and between the three of 'em we were some busy;\nbut it didn't last long. When the Greasers saw they couldn't break our\nring, they turned on Olaf like cornered rats. They struck him and they\nchoked him; but not once did he speak, and whenever his grip closed on\ntheir flesh, he ruined that part forever. It was a horrid sight; but I\ncouldn't have turned my eyes away if I'd wanted to. In the end he\nbroke their necks, one after the other, and then he stood up straight\nand wiped his forehead. \"I take the blame,\" said he. \"I take all the\nblame, here and hereafter\"; which certainly was a square thing to do,\nthough we hadn't counted on it, any.\n\n\n\n\nCHAPTER THIRTY-TWO\n\nTHE FRIAR GOES ALONE\n\n\nThe Friar had been in earnest tryin' to get to Olaf; so 'at the four\nSimpson boys had finally been forced to throw, an' sit on him. As soon\nas it was over, they got up and apologized, offerin' to let him take\nout any spite on 'em he saw fit, and promisin' not to feel any\nill-will; but the Friar wasn't angry. He was hurt and sad to think 'at\nwe'd do such a thing; but he had no resentment towards us.\n\n\"I know most of you men well,\" said he; \"and I know you have done this\nbecause you felt it was right. I don't put you on one side and myself\non the other. I take my full share o' the blame. It merely proves that\nmy influence with you during the many years we have been together has\nnot been for the best, and I am very sorry to learn how poor my work\nhas been.\"\n\nHe turned and went up to the house; and we all felt nearly as bad\nabout the way he had taken it as though the confounded Greasers had\ngot away altogether. We talked it over and finally loaded their bodies\ninto Olaf's wagon, and hauled 'em up on the rim, where we buried 'em\nand heaped a lot o' stones over 'em. We began to feel better after\nthis, and shook hands all around, and the Simpsons and their three\nfriends rode away.\n\nThen we told the others about havin' seen the Friar's girl at Ty\nJones's and held a council as to how we should tell him. We finally\ndelegated Horace to do it, though he wasn't ambitious for the job. The\nFriar had told Kit that it was all over, and had left to take a walk\nwithout eatin' any supper. We still felt purty low-spirited, and we\ndidn't eat much ourselves; though we felt certain he wouldn't bother\nhis head much about a couple o' Greasers, as soon as he found out his\nown girl was Ty Jones's woman.\n\nThe boys had come light from the Diamond Dot, but Horace had outfitted\nway beyond his needs, intendin' to do consid'able campin' around, and\nOlaf also had a couple of extra tarps and plenty o' beddin'; so we\nfixed up our old bunk-shack which had been left standin', and settled\ndown as though the interval between our previous visit hadn't been\nmore 'n ten days.\n\nThe Friar came back about ten o'clock. He came into our shack as quiet\nas he could; but Horace was sittin' before the fire waitin' for him.\nIt was a warm night; but we had built the fire to make it a little\nmore cheerful, and had left the door wide open. Horace saw the Friar\nthe minute he reached the doorway, and he got up and went outside with\nhim.\n\nThey were gone nearly an hour, and then Horace sneaked in, and wakened\nme up. I follered him outside; and he said that the Friar intended to\nride down to see Ty Jones as soon as it was day, and that he insisted\non ridin' alone. The Friar was walkin' up and down in the moonlight,\nhis face was all twisted up, through his tryin' to hold it calm, when\nI took my turn at reasonin' with him; but it wasn't any use.\n\n\"Well, you'll not go alone,\" I said at last; \"and you can make up your\nmind to that now. We don't know how much Ty already knows about our\nputtin' the Greasers out o' the game, and we don't know how much of it\nhe'll lay to you; but we do know that he hates you, and would wipe\nyour name off the list the first good chance he had. I'm goin' along.\"\n\nThe Friar was hot; we stood there in the moonlight facin' each other\nand takin' each other's measures. He was a shade taller and some\nheavier 'n I was; and ya could see 'at he'd have given right smart to\nhave felt free to mix it with me. \"Do you think I'm a baby?\" he burst\nout. \"Do you think 'at I'm not fit to be trusted out o' your sight?\nYou take entirely too much on yourself, Happy Hawkins!\"\n\nI didn't want to taunt him to hurt him--I'd rather been kicked by a\nhoss than to do this--but I did want to arouse him to a sense o' the\ntruth. \"You have adjusted yourself to this locality purty well,\nFriar,\" sez I; \"but the's still a lot you don't quite savvy. Some\ncases must be settled by a man himself, but some must be left to the\nlaw. If this woman is the wife o' Ty Jones, he has the law on his\nside.\"\n\nHe turned from me and stamped off into the night with his hands\nclenched. He disappeared in the cottonwoods, and I was just beginnin'\nto wonder if I hadn't better foller him, when he came back again. \"Oh,\nI've been a fool, I've been a fool!\" he cried. \"All my life I have\ntried not to judge others, but all my life I have judged them. I have\ntried to put myself in their place, but allus I judged and condemned\nthem for giving way to temptations which I felt that I, in their\nplace, could have resisted. I have been a fool, and I still am a fool.\nI admit that you are right, and I am wrong--but, I am going to Ty\nJones's at dawn, and I'm goin' alone.\"\n\nWell, that settled it--me an' the Friar had to buck each other again.\nHe continued to stalk up an' down through moonlight and shadow; while\nI tried to plan a way to head him off. I was dead sleepy, but I went\naround and wakened up all the other fellers, and told 'em not to get\nup in the mornin' until called; next I got Tank to help me, and we\nwaited until the Friar had walked in the opposite direction, and then\nwe took the ponies out o' the corral and headed 'em toward the hills.\nThe farther we got, the rougher with 'em we got, and then we turned\nour own mounts loose, and sent 'em after the bunch. It was a big job\nto pack our saddles back on our heads, but we did it, and tore down\nthe fences to pertend 'at the ponies had vamoosed on their own hook.\nHorace was walkin' with the Friar now, arguin' the benefit of a little\nsleep, so 'at he'd be at his best. After a time the Friar did go to\nbed in Horace's tarp in the corner.\n\nI didn't wake up till after seven, myself, and all the fellers were\npertendin' to sleep as though it wasn't more 'n three. The Friar\ndidn't wake up till eight. He was beside himself when he found the\nponies gone; but he ate breakfast as calm as he could, and then set\nout with us to wrangle in some hosses on foot.\n\nGoin' after hosses on foot is sufficiently irritatin' to a ridin'\noutfit to make it easy enough to believe 'at this was all an accident,\nand we didn't come up with the ponies till nearly noon. When we\ncornered 'em up, I never in my life saw as much poor ropin', nor as\nmuch good actin'; but we finally got enough gentle ones to ride\nbareback, so we could wrangle in the rest; and after a quick lunch,\nthe Friar started to make his hoss ready.\n\nWe all started along with him. He stopped and faced on us, givin' us a\nlong, cold look-over. You can say all you want to again' swearin', but\nthe's times when it springs out of its own accord in a man, as natural\nand beautiful and satisfyin' as the flowers blossom forth on the\ncactus plants; and I haven't a shred of doubt that if the Friar had\nhanded us some o' the remarks that came ready-framed to his tongue\njust then, they'd have been well worth storin' up for future needs;\nbut all he did was to fold his arms, and say: \"Your methods are not my\nmethods. I am not goin' there to start trouble, and I do not even wish\nto give them the slightest excuse to start it of their own vo-lition.\nIf you are my friends, you will respect my wishes.\"\n\n\"Well, but you'll take at least one of us along, won't ya, Friar?\" sez\nol' Tank. \"Likely as not we wouldn't take it up, nohow; but still if\nthey made away with ya, we'd sort o' like to know about it as early as\npossible, in order not to feel suspensed any longer 'n was necessary.\"\n\n\"I should like to take one man along as a guide, as I am not entirely\nfamiliar with the trail from here,\" sez the Friar, still talkin' to us\nas though we were a lot of evil-lookin' strangers. \"If one of you were\nto go along until we came within sight o' the ranch buildin's--No,\nthey might see him and get the idee that he had gone back to join a\nreserve body, and I do not wish them to have the slightest grounds for\nresorting to force on their side. I shall have to go alone.\"\n\n\"I can see what you've been drivin' at, now,\" sez Tank, whose face was\nso muddled up that no one ever tried to read his thoughts in his\nfeatures, and so he could lie with impunity. \"Yes, I can see what you\nmean, now, and I got to own up 'at you're right about it. Still, you\nknow, Friar, we're bound to worry about ya. How long do you want us to\nwait before we start to projectin' around to get some news of ya?\"\n\nA look of relief came to the Friar's face: \"Why, if I don't come back\nwithin a week,\" sez he, \"I haven't any objections to your notifyin'\nthe legal authorities that you fear something has happened to me--but\ndon't make much fuss, for it doesn't really matter.\"\n\nWe all kicked about waitin' a week, but finally compromised on five\ndays as bein' about the right interval to allow before notifyin' the\nlegal authorities. Then we advised the Friar to go down by the ravine\nas it would take him to the ranch by the back way where he wouldn't be\nso likely to attract attention, especially from the dogs.\n\nHe asked Horace to ride with him until he could get a landmark; so\nHorace flung his saddle on a hoss an' started along, while the rest of\nus made ready to go trout-fishin', or take a snooze, or shake the\ncards, accordin' to the way we generally amused ourselves when\nloafin'. The Friar turned back once on the pretense that he wanted to\nget a good drink o' water before startin'; but he found us scattered\nout peaceful an' resigned, so he headed away at good speed.\n\nHorace took him the open road, while we went mostly through cuts, the\nway we had allus gone to our look-out. Our way was some the longer;\nbut we pushed our hosses a little more, and made the look-out just as\nthe Friar reached the point where the path went down into the ravine.\nHorace had agreed to do all he could to get the Friar to go up to the\nclump of bushes where the woman spent her afternoons, though he said\nhe doubted if the Friar would do it.\n\nI had the field glasses with me, and kept 'em on the Friar's face when\nhe paused to examine the spot and make sure he was right. He couldn't\nsee the ranch buildin's from where he was, nor the path leadin' to the\nclump of trees. I could see his face plain through the glasses, and he\nhad taken the guy ropes off and let it sag into just the way he felt.\nIt was filled with pain an' sufferin'.\n\nAs soon as Horace came, he and I sneaked down to the bunch o' big\nrocks from which we could see the path as it dipped from the opposite\nedge of the ravine, leavin' the rest of the boys to watch the ranch\nbuildin's. We could see them from where we were, and they could see\nus, and we had a signal for us to come back, or them to come to us;\nand another that the Friar was gettin' it bad down below, and to make\na rush for him. We hadn't seen any one about the buildin's, except the\nChinese cook. Our plan was to not rush the buildin's right away,\nunless we saw the Friar gettin' manhandled beyond his endurance.\nHorace said 'at the Friar had refused to go to the clump o' trees to\nsee the woman, as it might give the impression that she had sent word\nto him to meet her there, and he wouldn't cast the slightest suspicion\nupon her name.\n\n\"Horace,\" I said, as an awful fear struck me, \"supposin' after all, it\nain't the right woman!\"\n\nHorace's eyes stuck out like the tail lights on a freight caboose.\n\"Oh, I'm sure it's the same woman,\" sez he. \"Course she's changed\nsome; but we couldn't all three be mistaken.\"\n\n\"I still think it's the same woman,\" sez I; \"but as far as all three\nnot bein' mistaken, the's nothin' to that. Half o' the fellers who\nmake bets are mistaken, and most of us make bets. Still I think she's\nthe same woman.\"\n\nIn spite of this doubt, I was feelin' purty comfortable. The other\ntime we had been there, I hadn't been able to think up any excuse as\nto why; but this time I felt I was in right and it left me free to\nenjoy the prospects of a little excitement. I allus try to be honest\nwith myself; and when I'm elated up over anything, I generally aim to\ntrail back my feelin's to their exact cause. I'm bound to admit that\nwhen I'm certain that any trouble likely to arise will be thrust upon\nme in spite of my own moral conduct, I allus take a pleasant\nsatisfaction in waitin' for it.\n\nThe Friar slid his hoss down the bank o' the ravine, and disappeared\njust a few moments before we saw the woman comin' along the path from\nthe clump of trees. We kept glancin' up at the look-out now and again,\nbut mostly we glued our eyes on the woman. Horace hogged the field\nglasses most o' the time, but my eyes were a blame sight better 'n\nhis, so I didn't kick about it much.\n\nWhen she reached the edge o' the ravine, she paused and gave a little\nstart. \"Does she know him, Horace?\" I sez.\n\n\"She don't seem to,\" sez Horace. \"She's speakin' down at him; but her\nface looks as though she didn't know him.\"\n\n\"If it's the wrong woman,\" sez I, \"I'm goin' to start to the North\nPole to locate the fool-killer.\"\n\nWhile I spoke, she started down the path slow and matter o' fact; and\nme an' Horace scuttled back to the look-out to be in time to see 'em\ncome out at the bottom--providin' the Friar went on with her.\n\nWe didn't get there more 'n two minutes before they came out at the\nbottom; but it seemed a week. When they finally came into sight, the\nFriar was walkin' an' leadin' his hoss, and she was walkin' at his\nside about four feet from him with a big dog on each side of her. Just\nthen we saw six Cross-branders ride in toward the corral.\n\n\"It looks calm an' quiet,\" drawled ol' Tank, his free eye bouncin'\nabout like a rubber ball; \"but I'll bet two cookies again' the hole in\na doughnut that we have a tol'able fair sized storm before mornin'.\"\n\n\n\n\nCHAPTER THIRTY-THREE\n\nTHE FRIAR GIVEN TWO WEEKS\n\n\nAs Friar Tuck and the woman came out of the mouth of the ravine, Ty\nJones came out of the back door of the old cabin. He stopped a moment,\nlookin' at 'em, rubbed his eyes an' looked again. Then he walked\ntowards 'em. He spoke somethin' to the Friar, and the Friar answered\nit. The woman didn't pay any heed at all; but went around the new\ncabin to the door which was on the other side. Three more\nCross-branders rode in, and Ty Jones shook his fist at the Friar.\n\nOl' Tank was cussin' under his breath for comfort, but it didn't keep\nhim from gettin' fidgetty. \"Isn't the' no sort of a tool, Horace,\" he\nblurted out, \"that'll stretch out your hearin' the way these field\nglasses stretch out your eyesight? I'd be willin' to have one of my\nears run as wild as my free eye, forever after, if it could just hear,\nnow, what Ty Jones is a-speakin' to the Friar. I'm beginnin' to get\nnervous.\"\n\nWe all felt about the same way; but it was about two miles down to\nwhere they were, so all we could do was to watch.\n\nOlaf had come with us, leavin' Oscar with Kit, and now Horace turned\nto him and said: \"You and Promotheus know more about Ty Jones 'n the\nrest of us. I have never tried to pump Promotheus, but now I want you\nto tell us what you think he'll do with the Friar.\"\n\nThey said 'at Ty was generally purty cold blooded, and likely to take\nenough time in gettin' rid of a feller to make it purty hard to tell\njust how it had been done; but that when he once let go of himself, he\ndidn't care what happened, and if the Friar angered him about the\nwoman, the chances were 'at the Friar would never leave the ranch\nalive.\n\nThe shadows were beginnin' to fall, down in the valley; but Ty and the\nFriar kept on talkin', Ty wavin' his hands now and again, while the\nFriar stood straight with his hands hangin' easy at his side. I\ncouldn't stand it any longer.\n\n\"I believe 'at a feller could get almost to 'em without bein' seen, by\ngoin' along the edge o' the ravine,\" sez I; \"and I'm goin' to do it.\nIt'll be dark in a few minutes. If you want me to hustle to the Friar,\nwave a torch up and down; if you want me to come back here, wave it\nsideways.\"\n\n\"I'm goin', too,\" sez Horace.\n\n\"So 'm I,\" sez Olaf and The.\n\n\"Well, that's full enough,\" sez I, \"and the rest of ya keep a sharp\nwatch, and also keep the hosses ready, in case we need 'em.\"\n\nThe four of us started down the side o' the <DW72> at good speed. There\nwere only two places on the way down where we caught sight o' the\nranch buildin's; but just before we reached the top o' the cliff, we\nheard a sound down below in the ravine. Glancin' cautious over the\nside, I saw the Friar comin' back alone, on foot and leadin' his hoss.\n\nI drew back and whispered to the others, and we felt purty blame\ncheap. We hardly knew what to do, as the Friar was likely to see us if\nwe tried to run back to our look-out before he reached the place where\nthe path came up out o' the ravine, and most of all, we didn't want\nhim to know 'at we were follerin' him.\n\nHe had passed us by this time, so we looked over the edge o' the\nravine at him. He was walkin' slow with his head down, and his hands\nin his pockets. \"He'll ride home slow,\" sez I; \"and we can easy beat\nhim.\"\n\n\"Hush,\" sez The, draggin' us back from the edge, \"the's two fellers\nfollerin' him.\"\n\n\"Horace,\" I said, quick and firm, so as not to have any back-talk,\n\"you go about forty yards up the ravine, and keep your eyes on these\nfellers. Don't shoot 'em unless they try to pass you. Hurry, now! I've\ngiven you the most important post. If you shoot, shoot in earnest.\"\n\nHorace stooped over and ran to where a rock jutted out. \"Now, then,\"\nsez I, \"as soon as these fellers pass us, we'll try to bowl 'em over\nwith one stone each, and then drop back out o' sight. We don't want to\nshoot unless we have to.\"\n\n\"They're wavin' us to come back,\" whispered The, who had took a glance\nat our look-out.\n\n\"Never mind,\" sez I, lookin' down and seein' the two fellers crouched\nover and sneakin' after the Friar. \"Now then, throw and drop back.\"\n\nWe stood on our knees, threw one stone each, and dropped back. They\nrattled in the ravine below, and we heard a sharp yelp of pain. I had\nonly dodged away from the edge of the ravine and ran to where Horace\nwas.\n\n\"One feller was hit in the shoulder and knocked down,\" sez he; \"but he\ngot up again right away, and both of 'em ran back.\"\n\n\"What did the Friar do?\" I asked, not darin' to look over, lest he see\nme.\n\n\"He turned around and started back,\" sez Horace. \"I was afraid he'd\nsee my head again' the sky, so I pulled it back. I haven't heard him\nmove since those fellers started to run.\"\n\n\"Well, I don't believe 'at even the Friar would be daffy enough to go\nback,\" sez I; \"so we'll just lay here and listen. They signalled us\nfrom above a while back, but they've stopped again.\"\n\nWe waited some time without hearin' any one pass us, and then we\nsneaked up along the edge of the ravine. Before long we saw the Friar\ncome up the side. He paused on top and looked back, then mounted and\nstarted for Olaf's at a slow shuffle. As soon as he was well under\nway, we pushed for the look-out, and mounted.\n\n\"Slim, you and Tillte wouldn't be missed as soon as the rest of us; so\nyou trail the Friar, while we try to beat him home,\" sez I. \"If you\nneed us, shoot. Otherwise come in as unnoticeable as you're able.\"\n\nWe reached Olaf's, had our saddles off and the hosses turned loose\nbefore the Friar rode in. His face was white, but this was the only\nthing 'at showed what he was goin' through. We made a big fuss about\nhis gettin' back all right and asked him plenty o' questions, without\noverdoin' it enough to make him suspicious. He answered our questions\nright enough, but he didn't open up and talk free. Slim and Tillte\njoined us at supper without bein' noticed.\n\nAfter supper we gathered around the fire in Olaf's settin' room, and\nthe Friar gave us a purty complete account of what had happened. He\nsaid that it was his old girl all right; but he said that the' was\nsomethin' the matter with her, that she didn't recognize him even\nafter he had made himself known to her. He said she seemed dazed-like\nand not to take any interest in anything.\n\nHe said they had walked down the ravine together, and she had told him\nthat she was comfortable enough but not happy. That she had lost\nsomething which she could not find; but that she was getting stronger\nsince havin' come out to the mountains. He said 'at when Ty Jones saw\n'em together, he had carried on somethin' fierce, and had ordered her\ninto the house. Then he had turned on the Friar and told him that he\nwould give him two weeks to leave the state and after that his life\nwouldn't be safe in it. He said he had tried to reason with Ty; but it\nwasn't any use; so he had just come away.\n\n\"If he had set upon you, would you have shot him?\" asked Tank.\n\n\"I didn't have anything to shoot him with,\" sez the Friar. \"I was\ncareful to leave my weapons behind.\"\n\n\"Well, you didn't show much judgment in doin' it,\" sez Tank. \"He might\nhave sent a couple o' fellers after ya, and finished you out in the\ndark somewhere so 'at we never could 'a' proved it on him.\"\n\n\"I did think for a minute that some one was follerin' me,\" sez the\nFriar. \"I heard a rattle of stones and a cry a few hundred feet behind\nme in the ravine; but I think it was some animal slippin' down the\nside.\"\n\n\"Like as not,\" sez Tank. \"If it had been any o' Ty's gang, they\nwouldn't have give it up so easy; but another time we'll some of us go\nalong with you; so as to get your last words anyhow, if so be 'at\nyou're bent on suicide. What do you intend to do now?\"\n\n\"That's the worst of it,\" sez the Friar. \"I don't know what to do. She\nsaid she did not think she was married; but she was not sure; and Ty\nrefused to give me any satisfaction about it.\"\n\n\"Isn't the' any law out here, at all?\" sez Horace. \"Seems to me as\nthough there ought to be some way to get at Ty Jones.\"\n\n\"What would you charge him with?\" asked the Friar. \"She is not being\nabused or kept a prisoner, she says she is comfortable and gettin'\nstronger--I can't think of any way to bring him under the law. If you\nhad not taken the law into your own hands in regard to his two men, we\nmight have made the claim that he was behind them in this; but really,\nI do not see where we have any just grounds to go to law.\"\n\n\"That little matter o' the Greasers don't hobble us none,\" sez ol'\nTank. \"Don't you get the idee that you're bound in any way by this.\nThe whole country would uphold us; so if you want to use it as a\nlever, just make your claims again' Ty to the law officers, and we'll\ntell 'em 'at the Greasers confessed 'at Ty put 'em up to it.\"\n\nThis seemed to us like sage advice; and we all chipped in and urged\nthe Friar to act on it. Laws are all right, I haven't a word to say\nagain' laws. Fact is, I believe 'at we're better off for havin' a few\nthan not; but after all, laws come under the head of luxuries like\ndiamonds and elevators and steam heat. We all know there is such\nthings, and we haven't any objections to those usin' 'em who can\nafford it; but most of us have to wear cut-glass, pack in our own\nwood, do our climbin' on foot or hossback, and settle our troubles in\nour own way with as little bother as possible. When you figure it down\nto the foundation, laws depend on public opinion, not public opinion\non laws; and all the public opinion worth takin' into account would\nhave said 'at we had done the right thing with those Greasers. If\nthey'd 'a' tried to law us for a little thing like this, it would have\nstarted an upraisin' which would have let the law see how small a\nshadow it really does throw when it comes to a show-down.\n\nThe Friar didn't answer us right away, and when he did, it was in the\nmost discouraged voice I'd ever heard him use. \"I'm in the dark,\nboys,\" sez he, \"I don't know what to do. Even if I could find some way\nto take her away from Ty Jones, I do not know what to do with her. She\nis not herself, she needs care and protection--and I am not in a\nposition to supply them. I have an income of three hundred and fifty\ndollars a year, which is much more than enough for my own needs, for I\nlive mostly upon the hospitality of my friends as you well know\"--we\nalso knew 'at he spent most of his money in helpin' those who never\nsaw enough money to get on intimate terms with it; while all they gave\nhim in return was a little meal and bacon for savin' their souls and\ndoctor-bills. \"I don't know what I could do for her, even if I had the\nright to take her away from him,\" continued the Friar. \"My life has\nbeen a good deal of a failure; and I--\"\n\n\"For the love o' common sense, Friar!\" broke in Horace. \"You don't\nseem to have the smallest degree o' judgment. You know mighty well 'at\nI'm bothered to death to know what to do with my money. You get her if\nyou can, send her to any sort of a sanitarium you want to, and I'll\nfoot the bills. Don't you ever sit around and whine about money in my\npresence again. It worries and disgusts and irritates me--and I came\nout here for rest. You talk about faith and takin' no heed for the\nmorrow, and such things; but you act as though you were riskin' a\nman's soul when you gave him a chance to be of some little use in the\nworld.\"\n\nThe Friar was purty well overcome at this; but figure on it the best\nwe were able, we couldn't see just how to get a man's wife away from\nhim without provin' that he had abused her. It was a complication, any\nway we looked at it; so we all went to bed in the hope that one of us\nwould have a lucky dream.\n\nWe didn't have any more idees next mornin' than we'd had the night\nbefore; so after breakfast, the Friar took a walk and the rest of us\nsat around in bunches talkin' it over. About ten o'clock a feller\nnamed Joyce who lived about fifteen miles east of Olaf came by on his\nway for a doctor, his boy havin' been kicked above the knee and his\nleg broke. The Friar could patch up a human as good as any doctor; so\nwe went after him, knowin' that this would be the best way to take his\nmind off his own troubles, and the' was a look o' relief in the\nFriar's face when he rode away with Joyce.\n\nI never knew any feller yet who didn't spend a lot o' time wishin' he\nhad a chance to loaf all the laziness out of his system; but the fact\nof the matter is, that work gives us more satisfaction than anything\nelse. A wild animal's life is one long stretch after enough to eat;\nbut he's full o' health an' joy an' beauty. On the other hand, put one\nin a cage and feed it regular and it turns sick immediate. What we\nneed is plenty o' the kind o' work we are fitted for--this is the\nanswer to all our discontented feelin'; and what the Friar was best\nfitted for, was to help others.\n\n\n\n\nCHAPTER THIRTY-FOUR\n\nA CROSS FOR EVERY MAN\n\n\nThinkin', just plain thinkin', is about the hardest work the' is; and\nfor the next several days, we lay around doin' mighty little else. The\ntrouble was, 'at we couldn't devise a way to put Ty Jones out o'\nbusiness. He wasn't an outlaw; fact was, he stood high with the big\ncattle men; and we got light headed tryin' to scare up a plan which\nwould remove Ty in a decent manner, and leave the Friar free to take\nthe woman without causin' him any conscience-pains. We were the\nmournfulest lookin' bunch o' healthy men ever I saw; and finally I\ndecided to loaf with Kit and the kid, they not bein' expected to do\nany thinkin' and therefore havin' smooth an' pleasant faces.\n\nSometimes I wonder if women don't get along just as well without\nthinkin' as men do with it. I hadn't talked seven minutes with Kit\nbefore she suggested just what I would have thought up if I'd been\nable. She didn't even know she had suggested it; so I didn't call her\nattention to it for fear it might up-heave her vanity and give Olaf\nbother. I had a plan now and it was of such a nature that I was glad\nthe Friar wasn't there to mess into it.\n\nI found Promotheus an' Tank lyin' on the grass along the crick. They\nwere back to back, and their faces were so lined with genuwine\nthought, that they looked like a pair of overgrown nutmegs. I sat down\nbeside 'em lookin' worried.\n\nPresently Tank sez: \"What ya thinkin' about?\"\n\nI shook my head, and in about half an hour The asked the same\nquestion. I waited a minute, hove out a sigh, and sez: \"Gee, I wish I\nwas you.\"\n\n\"Why do you wish you was me?\" sez he.\n\n\"'Cause,\" sez I, \"you've got a chance to do the biggest deed I know\nof.\"\n\n\"What is it?\" sez he, examinin' my face to see if I was sheepin' him.\n\n\"No,\" sez I, shakin' my head; \"I ain't got any right to even think of\nit, let alone hint at it. You might think I was buttin' into your\naffairs, and then again--No, I refuse to suggest it. If it's your\nduty, you'll see it yourself; but I won't take the responsibility of\npointin' it out.\"\n\n\"What in thunder did you mention it at all for, then?\" sez The,\ngettin' curious an' exasperated.\n\n\"And then besides,\" sez I to myself, out loud, \"there's Horace. Like\nas not he wouldn't allow you to run your head into danger any more.\"\n\n\"What!\" yelled The. \"Didn't we run our heads into danger all over the\ntropics of the Orient, didn't we goad up danger an' search for it and\nroust it out of its hidin' places and--Why, confound you--\"\n\nIn about ten minutes I stopped him, an' sez in a quiet voice: \"Well,\nthen, if I was you, I'd go on down to Ty Jones's and take on with him\nagain.\"\n\nWe lay on the grass there, along Pearl Crick for some time without\nspeakin'. Up on the rim, the grass was burned to a crisp; but along\nthe crick it was still green. Promotheus pulled blade after blade of\nit and chewed 'em up in his mouth, while me an' Tank watched him.\n\n\"What you mean, is for me to take on with Ty Jones--and then to act\nspy on him. Ain't that what ya mean?\" sez The after a time.\n\nI'd 'a' sooner he hadn't put it into words--it did look rather raw\nwhen he stood it up before us naked. \"I don't mean nothin' in\nparticular, The,\" sez I. \"You and I are different, and what I could do\nwithout feelin'--\"\n\n\"That's all right,\" he broke in. \"The' ain't any need to treat me like\nan infant baby. Come right out with it--What you want me to do is to\nplay spy, ain't it?\"\n\n\"That's the only way I can see to help the Friar,\" sez I; \"but he\nwouldn't want you to do anything for him you didn't feel was right.\"\n\n\"I know, I know,\" he sez, lookin' down at his hands. \"Ty Jones is as\nmean as a snake, and I don't deny it; but he's been square with me,\nand once he saved my life. Then again, the Friar has been square with\nevery one, and if he hadn't nursed me night and day, Horace wouldn't\nhave had a chance to save my life. If Horace had killed me it would\nhave spoiled his life; so that the whole thing is held together in a\ntangle. I'm willin' to cash in my life for the Friar--it ain't\nthat--but I do hate to turn again' Ty Jones underhanded.\"\n\n\"Better just forget I mentioned it,\" sez I.\n\n\"No,\" sez Promotheus, \"I intend to lay the plan before Horace, and let\nhim settle on it.\"\n\n\"That's a good scheme, that's the best way out of it,\" sez ol' Tank,\nand I joined in with him.\n\nWe sat there on the bank a long time, thinkin' the thing over, and\nfinally just before supper, Horace hove in sight and started to josh\nus; but when he saw how sober we were, he settled down, and asked us\nwhat was up.\n\n\"Horace,\" sez The, \"what would you think of my takin' on with Ty\nJones, and playin' the spy on him?\"\n\n\"That would be madness!\" exclaimed Horace. \"He'd see through it and\nkill you first pop. I don't know though--you might fix up a tale--but\nthen it would be too infernal risky. Nope, don't you try it.\"\n\n\"If it could be done,\" persisted The, \"what would you think of it?\"\n\n\"Oh, it would be a great thing for the Friar,\" sez Horace; \"but,\nPromotheus, I don't like to have you take the risk.\"\n\n\"It ain't the risk I'm fussin' about,\" sez The. \"Ty was square to me\nin his own way. The Friar has been square to me also, and I know 'at\nhis way is the best; but at the same time--don't you think it would be\ndownright snakish for me to go back to Ty, tell him some excuse for my\nstayin' away, and then plot again' him while I'm eatin' his vittles?\"\n\nIt didn't sound good to us when Promotheus came out with it so\neverlastin' unpolluted; but he had worked up a sense of honesty since\nbein' with Horace, which wouldn't let him do any pertendin'. Horace\ndidn't answer, and he went on after waitin' a minute: \"I haven't any\nprejudices again' fightin' him in the open; but it does go again' my\ngrain to wear a dog hide when I'm playin' wolf, and Ty Jones was\nsquare to me.\"\n\n\"Well,\" sez Horace, \"I haven't the heart to advise you to do this,\nPromotheus. We'll sure be able to find some other way, and as long as\nit goes again' your grain the way it does, I don't want you to do it.\"\n\n\"Would you think any the less of me if I did?\" asked The, his eyes\ntakin' on a sad, hungry look, like a dog's eyes get when he's worried\nover what his master'll say about some trick he's been up to.\n\n\"Course I wouldn't think any less of ya,\" sez Horace without\nhesitatin'; \"but hang it, I'm afraid somethin' 'll happen to ya.\"\n\n\"Would the Friar think any the less of me?\" sez The.\n\n\"If the Friar heard about it, he wouldn't let ya go,\" sez Horace.\n\n\"I've puzzled more about the Friar 'n about airy other man I ever\nsaw,\" sez The, thoughtful. \"I wanted to lynch Olaf that time, guilty\nor not guilty; but the Friar straightened things out by riskin' his\nown soul. He hates lynchin', it goes square again' his grain; but he\nmade a bet with us to help stretch Olaf if we could prove him guilty;\nand this has stuck with me. This was a big thing to do, and I'd like\nto do somethin' big for the Friar--But I swear it would hurt me to spy\non Ty Jones!\"\n\nWe didn't have anything to say on the subject; so we just sat and\nchewed grass.\n\n\"I've been thinkin' about that old Greek feller, 'at you named me\nafter,\" sez Promotheus at last. \"He didn't ask no one else to take the\nresponsibility of tellin' him what to do. He just decided what was\nright, and then did it. If I go to Ty Jones, and he treats me right,\nmy own thoughts'll tear at me like vultures; but this here other\nPromotheus, he stood it, 'cause it was for man's good; and I'm game to\ndo the same.\n\n\"I don't intend to be any more sneakier 'n I have to be. All I intend\nto do is to find out what I can about the woman, and, if Ty ain't\ntreatin' her right, to help get her away from him; but I want it\nunderstood right now that I'm not goin' to work any tricks on Ty to\nget him into the law for what he's done in bygone days. Now then, I\ntake all the blame on my own shoulders; but we'll have to fix up a\ntale to fool a wise one, 'cause Ty won't be took in by chaff.\"\n\nWe talked things over a long time; but it seemed mighty unreasonable\nfor Promotheus to have pulled out without sayin' a word, and then to\ncome back without writin' in the meantime; and we couldn't quite hit\non it. Finally the idee came to me.\n\n\"They're goin' to graze the grass down to the roots, this summer,\" sez\nI; \"but still, the' won't be enough to go around. A lot o' cattle will\nhave to be sold off early, and some will be trailed up into Montana,\nand cow-punchers are goin' to be in demand. Ty is long on cattle and\nshort on grass, and he'll be glad to have extra help he can trust; so\nhe won't question ya too close. You tell him 'at Horace here was a\ngovernment agent, and that he arrested you as a deserter, and took you\nto prison where you was given a life sentence; that you broke out a\ncouple o' months ago, and have been workin' your way back as cautious\nas you could.\"\n\n\"My Lord, I hate to tell him that!\" sez The. \"It's too infernal much\nlike what I told him the first time.\"\n\n\"You got to make up a good story, or else give up your plan,\" sez I.\n\n\"Yes, that's so,\" he agreed. \"Ty'd believe that, too. What prison had\nI better say I've been in?\"\n\n\"Which one was you in?\" sez I.\n\n\"I never was in any government prison,\" sez he. \"I was in a state\nprison.\"\n\n\"Have ya ever seen a government prison?\" sez I.\n\n\"Yes, I've seen two, one in Kansas, and one in Frisco,\" sez he.\n\n\"Which would be the hardest to get out of?\" sez I.\n\n\"The one in Frisco; it's on an island,\" sez he.\n\n\"Choose that one,\" sez I; \"and make up your escape just as it might\nhave happened.\"\n\n\"Ty won't haggle me with questions,\" sez The sadly. \"He'll just\nbelieve me, an' this'll make it ten times as hard.\"\n\n\"You ought to be paler an' more haggard,\" sez I; \"but I doubt if the's\na way to do it.\"\n\n\"Keep soakin' his face in hot towels for a few days,\" sez Horace.\n\"That'll bleach him out.\"\n\n\"Are ya goin' foot or hossback?\" sez I.\n\n\"I stole a hoss down in Texas the last time I came,\" sez he, \"and\ntraded him off when he got footsore.\"\n\n\"We got some hosses with a Nevada brand, over at the Dot,\" sez I.\n\"I'll slip over an' get one while you're havin' your complexion\nbleached off. They broke out an' got with the herd before we finished\nbrandin' 'em, and we just let it go. The chances are they haven't been\nrebranded yet.\"\n\n\"All right,\" sez The. \"If I'm to do it at all, I want it to go\nthrough; but I have an idee 'at those vultures pickin' at my liver are\ngoin' to be mighty unpleasant company.\"\n\nMe an' Spider Kelley, Tillte Dutch an' Mexican Slim rode over to the\nDot and found two o' those Nevada hosses, still rangin' with their old\nbrands untouched; so we roped one, and came back with it, without\nhavin' word with any of the outfit. The Diamond Dot range was the best\nof any we rode over, and they had put up a lot o' hay that summer; but\nstill I felt sure 'at they would have to cut down purty close, though\nI knew 'at Jabez would hold as many as he could for a high price the\nfollowin' year.\n\nWe found The's complexion purty well stewed out and haggard, Kit\nhavin' put soda in the hot water; so I told him to play sick, and loaf\naround the house as long as possible. He agreed to it; but the' was a\nsettled look o' regret in his face which was a heap different from the\none he had wore when he dismounted from the stage at Bosco.\n\n\"Night and day,\" sez I, \"the'll be at least two of us at the look-out,\nand you come up with any news you have. Get into the habit of\nwhistlin' Horace's tune; so that if ever you'd want to warn us to\nvamose rapid, you can whistle it. You might ride that way with some o'\nTy's outfit, or somethin'.\"\n\n\"It's not likely,\" sez he. \"The's no range up that way, and no trail\nleadin' near it; but you fellers want to scatter your tracks all you\ncan, so as not to make a path.\"\n\nWe made plans for all the unexpected details we could think up; and\nthen he started forth one night, meanin' to circle to the southwest,\nand come in from that direction. He wore a red handkerchief under his\nnose as if to shut out the dust; but shaved clean, and pale as he was,\nmighty few would have recognized him either as Badger-face, or as the\nfeller what had come in with us a few weeks before. We all shook hands\nsolemn when he left, and promised to be at the look-out the followin'\nnight, and to be there steady from that on.\n\n\"What makes you fellers trust me?\" sez he just as he started. \"I came\ndown here to put Olaf out o' business, and then I turned over to your\nside. Now I'm goin' back to Ty's. What makes you think I won't turn\nagain' ya, if I get into a tight place?\"\n\nHorace went over and took his hand. \"Promotheus,\" sez he, \"I've been\nwith you through hot days and cold nights, I've been with you through\nhunger and thirst and danger; and I'd trust you as long as I'd trust\nmyself. You're not goin' to Ty's because you're a traitor. You're\ngoin' because you're a changed man, and the new man you've become is\nwillin' to risk his life for what he thinks is right. No matter what\nhappens, I'll trust ya; so take that along to think over.\"\n\nPromotheus winked his eyes purty fast, then he gave a sigh and rode\noff into the night. The' wasn't the hint of a smile about his lips,\nnor a glint o' gladness in his eyes; but somethin' in the straight way\n'at he held his back let ya know 'at the inside man of him was finally\nat peace with what the outside man was doin'--and if ya don't know\nwhat that means, the's no way to tell ya.\n\n\n\n\nCHAPTER THIRTY-FIVE\n\nTHE FRIAR A COMPLICATION\n\n\nWe all felt purty down-hearted after Promotheus had rode away, and we\nsat before the fire in Olaf's settin' room a good deal the same as if\nwe were holdin' a wake.\n\n\"Olaf,\" I sez, \"you can't have any finicky notions about treatin' Ty\nJones square, after all the persecutin' he's handed you. Do you know\nanything on him you could have him sent to prison for?\"\n\nOlaf shook his head. \"He's too clever to get caught in a trap,\" sez\nhe. \"He scarcely ever gave any orders to have things done. He'd just\nsay aloud as though talkin' to himself, that some one or other was in\nhis way; and then his men would begin to take spite on that feller. If\nthe calf tally showed a hundred percent increase, he would think that\nabout right, and no questions; but if ever it fell short o' what he\nexpected, we had it to make up some way. He'd send us out until we had\nbrought in enough to satisfy; but he'd never give us straight orders\nto rustle. He is a smart man. When one of his men got into trouble, he\ngot him out, no matter what cost; but he expected his men to do what\nhe wanted, without askin' questions. He has no fear, none at all. I\nknow, I have seen. He has no fear, and he is very strong. It is bad to\nbe at war with him; but I should like to have my hands at his throat\nonce, and none to interfere.\"\n\n\"Maybe you will, Olaf,\" sez I, \"maybe you will; and I don't mind\nsayin' that I hope to be on hand to see it.\"\n\nWe kept two men allus at the look-out with Horace's field glasses. It\nwas a queer sort o' summer, the air wasn't clear like it usually is,\nbut hazy, as though full o' dust; and in lots of places they were\nturnin' stock on the grass they generally aimed to save for winter.\nThere were only a few punchers around the Cross brand ranch houses;\nbut we saw Promotheus every day. He hobbled about with a stick part o'\nthe time, holdin' his hand on his back as though he had the rheumatiz,\nwhich was natural enough from bein' shut up in an island prison. Some\ndays we saw the woman; but she never came up the ravine path any more.\n\nPromotheus didn't make a report to us for about a week. Then he came\nout one night about eleven. He said 'at Ty hadn't doubted a word he'd\nsaid; but had done everything possible to make him comfortable,\ntellin' him to just loaf until he got in good order. He said 'at Ty\nand the woman didn't have much to do with each other and hadn't had\nsince she'd come out. He said 'at the woman was kind to all the\nanimals, in spite of everything 'at Ty could do, and the dogs was\ngettin' to act like regular, ordinary dogs. He said all but a few new\npups had remembered him, and one had even wagged his tail, though he\ncouldn't see any sense in this, he never havin' as much as spoke a\nkind word to the dog, so far as he could recollect.\n\nHe said he had held several talks with Ty, and Ty had asked him if he\nthought 'at Olaf was in league with any big outfits. He said 'at he\nhad told Ty that he was sure Olaf had been in league with 'em several\nyears before, but o' course, he couldn't know anything o' what had\nhappened since. Ty said he had come to the conclusion that Olaf was\nset out for a kind of bait to draw him into trouble, which was why he\nhad let him alone; but that he was short o' grass this season, and\nwanted Pearl Crick Spread bad. He also told The about the two Greasers\ndisappearin', though he wasn't sure what had happened to 'em. He knew\nabout us bein' over at Olaf's off and on, and The warned us to be\ncareful, as Ty expected to have Olaf's place watched as soon as he got\nthrough movin' several bands o' cattle.\n\nThe said 'at the woman had a soft spot for any dumb brute, or even a\nhuman in distress, and that he had touched her by hobblin' around with\nthe stick. He said she had cooked him some flabby invalid-food with\nher own hands, and that it was mighty captivatin'. He said she didn't\nspeak much; but he was tryin' his best to get on the good side of her.\nHe said 'at all the boys claimed 'at Ty treated her well; but didn't\nseem to care much for her. Horace didn't happen to be with us when The\ncame; but we said we'd move our camp higher up on the <DW72>, to be on\nthe safe side when Olaf's was watched, and would have Horace on deck\nsure the next time The came out; and we did this the next day.\n\nThe land was all slashed an' twisted around and broken, up west o' the\nCross brand ranch houses. The ravine leadin' down to 'em ran east and\nwest, the path leadin' up out of it to the trees where we had first\nseen the woman wasn't near so steep as the one comin' out of it on the\nnorth side toward the clump o' rocks. After the north path came out,\nthe ravine narrowed down until it wasn't more than a crack, the south\nside not risin' so high as on the north; so that soon the north side\nstood up like a cliff above the land leadin' down to the clump of\ntrees, and the only way we could get over to it was to go down the\nravine and up again on the other side.\n\nWe made our camp consid'able higher than our look-out had been, and it\nwas a well sheltered spot. An easy slopin' stretch led up to it from\nthe north, and a ledge skirted the face o' the cliff up back of it, to\nthe south. We examined this some distance; but it didn't seem to lead\nanywhere. We found several dips back in the hills where the snow water\nmade grazin' for our ponies, and we were as comfortable as it's ever\npossible to be while waitin'.\n\nI know what my plan would be for makin' a hell which would be\npunishment for any mortal sin, and yet not severe enough to make me\nhate all the peace out o' my own existence. I'd make the wicked sit in\nthe dark for a hundred years, waitin' to hear what their sentence was.\nThen, I'd let 'em into heaven, and I bet they would be in a fair way\nto appreciate it. I never met up with any one able to out-wait me\nwithout showin' it more 'n I did; but I'll wager what I got, that the\nsuspense was gorin' into me worse 'n into them, all the time.\n\nOne evenin', me an' Tank went up to camp after doin' our stunt at the\nlook-out, and as we went, we caught sight o' two riders headin' our\nway. We hastened along so as to be ready to move in case this was a\npair we didn't care to draw to; but by the time we reached camp, they\nwere close enough to recognize as the Friar and Olaf. The plan was to\nkeep the Friar in the dark as long as possible, and we waited their\ncomin' with consid'able interest.\n\nThe Friar had squeezed the whole thing out of Olaf, as we might have\nknown he would. You couldn't trust Olaf with a secret where the Friar\nwas concerned. Tank, now, would have sent the Friar off to Bosco or\nLaramie as contented as a bug; but just as soon as Olaf was backed\ninto a corner, he told the truth, and spoiled all our arrangements.\n\nThe Friar rode into our camp, dismounted, threw his reins to the\nground, and sez: \"Where is Promotheus?\"\n\nWe looked at Olaf, and he nodded his head as sheepish as the under dog\nat a bee-swarmin'. \"He's down at the ranch,\" sez Horace.\n\n\"Has he brought any news?\" asked the Friar. So we told him all 'at The\nhad reported. He took a few steps up and down, ponderin'.\n\n\"I can't permit this,\" he said after a minute. \"He is riskin' his life\ndown there, and I can't allow him to continue.\"\n\nThe rest all joined in and argued with him; but he was as obstinate as\na burro, once he got his back up; so I didn't say anything. I went off\nand started to eat my supper. When I was about half through, Horace\ncame over and said the Friar was bent on goin' down to Ty's himself.\n\"Well, let him go,\" sez I as cool as a snow-slide.\n\n\"Yes, but if he goes, Ty will kill both him and Promotheus!\" sez\nHorace raisin' his voice. I noticed the others headin' toward us so I\nonly flung my hands into the air, meanin' that it was none o' my\nbusiness.\n\n\"Do you mean to say 'at you back the Friar up in this?\" demanded\nHorace.\n\n\"Do I look like a fool?\" sez I. The Friar's eyes were on me, and I\nknew they were cold; but I pertended not to notice him.\n\n\"You don't look like a fool; but you act like one,\" sez Horace,\ngettin' riled.\n\n\"You can't blame me, Horace,\" I sez in my most drawly voice, \"because\nthe Friar cares more for havin' his own way than he does for human\nlife.\"\n\n\"What do you mean by that?\" demanded the Friar.\n\n\"Oh, nothin',\" sez I, \"except that if you go down there, it shows\nPrometheus up at once, we'd all have to go along to save Promotheus,\nand this would start a fight, with us to blame; and no one knowin'\nwhat the woman is, or how she stands in the matter. She seems\nperfectly satisfied with Ty Jones; and no matter how it turned out,\nall of us who survived would have to leave the country. I don't intend\nto argue with you, or to cross you in any way; but I do intend to\nstand by Promotheus, as it was me who first put the idee into his\nhead.\"\n\nI sympathized with the Friar, I knew that he wasn't himself. To find\nthe woman he loved in the hands of the man who hated him, after all\nthe years he had been in suspense about her was enough to tip any one\noff his balance; and I also knew the Friar. He had trained himself for\neternity so long that some of his earthly idees weren't sound, and the\nsurest way to bring him to himself was to let him bark his knees a\ntime or two. Some imported hosses carry their gaze so high they can't\nsee their footin' but after they've stepped into a few prairie-dog\nholes, they learn to take a little more interest in what they're\ntreadin' on.\n\nThe Friar came over and looked down at me. \"I shall wait until\nPromotheus comes up here, and then he can stay; and I shall go down,\"\nsaid the Friar in the voice a man uses when he thinks it's wrong to\nshow the sarcasm he can't help but feel. \"Have you any objection to\nthis?\"\n\n\"I have no objection to anything you choose to do, Friar,\" I said,\nfinishin' my supper.\n\n\"Do I understand that you approve?\" sez he.\n\n\"Certainlee not,\" sez I. \"Ty would see the connection between you and\nPromotheus at once. He knows 'at The was a deserter, and he would set\nthe law on him in one direction, and try to run him down on his own\nhook in the other. If you had been on hand while we were discussin'\nthe plan, you would have had the right to veto it; but now, it looks\nto me as though Prometheus was the one to consider.\"\n\nThe Friar sat down and ran his hands through his hair. \"I can't see\nany way out!\" he sez at last; \"but I'm forced to admit that since\nPromotheus has gone down there, it would put him in danger for me to\ninterfere.\"\n\n\"Well,\" sez ol' Tank, \"here is The himself. Now, we'll know better\nwhat to do.\"\n\nWe looked up, and there was Promotheus with a bruise over his eye,\ncomin' into our little nook.\n\n\n\n\nCHAPTER THIRTY-SIX\n\nA SIDE-TRIP TO SKELTY'S\n\n\nWe all crowded around him, thinkin' 'at the bruise betokened some sort\nof trouble; but he said he'd got afraid they'd begin to suspicion him;\nso he had tried to ride a hoss that day, and had let it buck him off.\nHe said the' wasn't much lettin' needed, as it had been a mean one;\nbut he had got his forehead grazed, and had lain on the ground,\nclaimin' his back was hurt. It was only about eight o'clock, and we\nwondered how he had the nerve to come so early; but he said they were\nhavin' a drinkin' bout over havin' dogged a feller by the name o'\nBryce off his claim on Ice Crick, thus gettin' a new outlet to grass\nand water.\n\nHe said the woman had been mighty good to him after his fall; but that\nhe couldn't get her to talk about herself at all. \"Have you ever\nmentioned the name of Carmichael to her?\" I asked.\n\n\"No,\" sez he, \"why should I?\"\n\n\"That's the Friar's name,\" sez I. \"He used to preach in a regular\nchurch down east, and she sang in the choir. Next time you get a\nchance, try to draw her out about this.\"\n\nThe Friar told him a lot o' small details to ask her about; and went\npart way back with him, as he wouldn't stay long for fear o' bein'\nmissed. The Friar insisted on stayin' along with us, while Olaf went\nback to the Spread.\n\nTwo nights after this Promotheus came up to our camp again. He said he\nhad had several talks with her, and that she remembered the names and\nplaces, all right, but insisted that Carmichael was dead. She said he\noften came to her in her dreams; but that she knew he had died long\nago.\n\n\"Does she ever sing?\" asked the Friar.\n\n\"Never,\" sez The. \"She don't even talk much. She has some sort of a\npain in her head, and sometimes she seems to wander; but at other\ntimes she is perfectly clear.\"\n\n\"Is Ty Jones ever mean to her?\" asked the Friar.\n\n\"Never,\" sez The. \"Ty ain't mean to those about him. He has his own\nidees--he likes to have his men and dogs and hosses all fierce and\nnervy--but he's not mean to 'em. And all the boys treat her\nrespectful, too. Fact is, I don't see where we got any grounds to take\nher away.\"\n\n\"But she does not care for him,\" sez the Friar; \"she could not care\nfor him! He must have used trick or force to bring her here; and you\nmust find out the truth about it. It all depends on you, now.\"\n\n\"I'm doin' all I can, Friar,\" sez The; \"but it's a hard tangle to see\nthrough.\"\n\nWhen he left to go back, me an' the Friar and Horace went with him.\n\"Supposin' they should see you comin' back?\" sez the Friar.\n\n\"Well,\" sez The, \"Ty don't keep his men in prison, and I'd tell 'em I\nwas up takin' a little air after bein' shut away from it so long.\"\n\n\"Supposin' they got suspicious an' follered ya?\" asked the Friar.\n\n\"I try to be as careful as I can,\" sez The; \"but I own up I allus feel\na bit nervous till I get back to my bunk.\"\n\n\"The best plan is for one of us to wait where the path leads down into\nthe ravine each night at eleven,\" sez the Friar. \"We could go at ten\nand wait until twelve. If we went any closer, the dogs might get scent\nof us.\"\n\nWe agreed 'at this would be the best plan; and after this, two of us\nmade it a point to spend a couple of hours waitin' there, while the\nrest stayed at the look-out ready to hustle down if the' was any\nexcitement; but nothin' happened and we got purty fidgetty.\n\n\"Tank,\" sez I one afternoon, \"let's ride over to Skelty's. The's\ngenerally some Cross-branders there, and perhaps we can find a little\namusement.\"\n\nWe reached there about seven, and ordered supper. There were five\nCross-branders there already, eatin' and drinkin'; and one of 'em was\nthe tall feller by the name o' Dixon. I nodded to him when I sat down\nand he nodded back. It's funny the way a man feels when he goes into\nan unfriendly place to measure an' be measured. It's not like fear,\nthat is, not like panicky fear; but still I suppose it's something\nlike what a jack-rabbit feels when the hounds are strung out after\nhim. He knows well enough what'll happen if he can't run fast\nenough--but then he takes a heap of exhilaration in the thought that\nhe most certainly can run fast enough.\n\nAll those fellers knew something o' me an' ol' Tank; while Dixon was\nthe only one we knew, the rest bein' mostly young chaps who had taken\non with Ty durin' the last few years; but as most o' Ty's men were\ntrailed out o' some other state by a posse, it was a safe bet that\nthey had more or less rattler blood in 'em. They were all on friendly\nterms with the girls, and the girls called 'em by name, whenever they\ncouldn't think up some other term 'at suited their taste better. One\no' these young fellers still had a boy's eyes; but most o' their eyes\nwere purty hard an' chilly.\n\nI never did set any store on havin' a strange woman call me \"dear\";\nand neither did ol' Tank. With his eye runnin' wild, and his mussed-up\nfeatures, the term dear fitted him about as snug as false bangs an'\nface-powder would; but one o' these young hussies came over an' stood\nbehind his chair, and sez: \"Why hello, dearie, where have you been all\nthe time?\"\n\n\"I've been over teachin' my grandchildren how to play the pianer,\" sez\nTank. \"Have you got any pork an' beans?\"\n\nMost any girl knows 'at most any man'll stand for most anything; so\nthis one grabbed hold o' Tank's hair and gave it a pull; but she\nsavvied 'at he didn't have any love for her, so she brought in his\ngrub, threw it down in front of him, and went back to soft-soapin' the\nfeller with a boy's eyes. He was still young enough to feel flattered\nby it, and truth to tell, she wasn't a bad lookin' girl, except that\nshe drenched a feller so constant with her feminine charms that she\nwashed away any hankerin's for 'em he might have had to begin with.\n\nAny healthy woman has all the allurement she can possibly need, if\nshe'll just take care of it. I like to see a hoss full o' fire, and I\nlike to see a woman full of enticement; but I like to see both the\nfire an' the enticement kept under good control, and not made to show\nout unnecessary.\n\nOnce, when I was in Frisco, I saw a parade of the Friendly Order of\nHindu Cats, and the Grand Thomas Cat o' Creation rode in front on an\nold gray hoss. This hoss had feet like worn-out brooms, and the' was\nknots all over his legs. All he asked in the way of entertainment was\nto pass a peaceful day in a quiet stable, face to face with a bale of\nhay; but they had clipped his mane an' tail, hung a beaded belt across\nhis brisket, put a scarlet blanket on him, and jabbed him with spurs\nuntil he was irritated to a degree.\n\nThe feller ridin' him had learned to ride in a barber's chair; but he\nhad a heavy frown, and a lot o' gold lace, and a big canoe-shaped hat;\nand I have to admit that if they had tied him fast to the saddle, and\nput rubber spurs on him, he would have looked the part like a picture.\nEvery time he'd see one of his friends he'd stab the hoss on the off\nside, then jerk back on the curb, and smile benevolent, as though he\nintended to save the populace from that fiery steed or sprain every\nbone in his face.\n\nThe old gray was as forgivin' a hoss as I ever see; but he had his\nlimits as well as the rest of us. For the first ten or fifteen blocks,\nhe'd only swish his tail and prance when his rider jabbed him an order\nfor a little more fire; but finally his flanks got touchy, and his\nsense o' justice began to write the declaration of independence on his\npatience. This would have been the time an intelligent human would\nhave traded off his spurs for an apple or a lump o' sugar, or some\nother welcome little peace-offerin'; but just then the parade passed\nunder a window jammed full o' the Grand Thomas Cat's closest friends,\nand o' course, they had to see a little fire.\n\nHe straightened out his legs, and then clamped the spurs into the old\ngray's flanks. I had fought my way through the crowd for fifteen\nsquares just to see it happen, and it was well worth it. The gray was\nstiff and awkward, but in his youth he had taken a few lessons in\nbuckin', and what he lacked in speed and practice, he made up in\nearnestness. The Thomas Cat didn't know any more about balancing than\na ball, and the grip of his knees wouldn't have put a dent in a\npullet's egg; the' was no horn to the saddle, and the mane had been\nclipped, so all he had to hang on with was the spurs and the curb bit;\nand things certainly did happen.\n\nThe old gray pitched and kicked and reared and backed and snorted and\ngot mixed up with flags and citizens and umbrellas and red-lemonade\nstands and policemen; until finally he scraped off the Grand Thomas\nCat of Creation on an awning, and tore off home, jumpin' and kickin';\nwhile the population threw their hats in the air and yelled their\npalates loose. They threw fruit and popcorn and friendly advice at the\nGrand Cat as he hung from the awning; but friend or foe, the' wasn't a\nsoul in that crowd to help him get down; so as soon as he got calm\nenough to remember what he was, he dropped the three feet to the\nsidewalk, and ran into the store and hid.\n\nIf ya want to fill a crowd with content and satisfaction and joy and\nfelicity and such-like items, just have some terrible accident happen\nto a popular hero, and all the joy-wells'll overflow and gush forth\nlike fountains--But what made me think o' this little incident was the\nfact that this girl at Skelty's put the spurs to her feminine charms a\nleetle too continuous.\n\nDixon, the Cross-brander, was one o' these lean, skinny ones, and as a\nrule, I don't crave to make their acquaintance. His Adam's apple ran\nup and down in his neck like a dumbwaiter, and the' was plenty o'\ndistance for consid'able of a run. If ya looked at just the part of\nhim between his chin and his shoulders, he resembled an ostrich,\nchokin' on an orange; but I decided to be as friendly as possible; so\nas soon as I'd filled a cigarette paper, I offered him my sack o'\ntobacco. He took it, and while he was rollin' himself a cigarette, he\nsez: \"I see you've cut loose from your preacher.\"\n\n\"Nope,\" sez I, \"he cut loose from me.\"\n\n\"How come you fellers spend so much time out this way?\" sez he.\n\n\"Nice country and pleasant folks,\" sez I.\n\n\"I've heard tell 'at you got so familiar over at the Diamond Dot, that\nthe old man turned ya loose,\" sez he. \"Is the' anything to it?\"\n\nI didn't reply at once. My first impulse was to see if I couldn't pull\nhim and his Adam's apple apart; for this wasn't no accident. This was\na studied insult, and every one there was watchin' to see what would\nhappen; but the' was too much at stake; so I gripped myself until I\nhad time to put that remark where it wouldn't run any risk o'\nspoilin'; and then I sez: \"Well, I don't just like to have it put that\nway; but I will admit that you haven't missed it so terrible far.\"\n\n\"Lookin' for a job?\" sez he.\n\n\"Oh, I'm not carin' much,\" sez I. \"I'm thinkin' some o' takin' a\nhomestead, or buyin' some other feller out; but I ain't in any hurry.\nI may go on down into Texas, or take on again up here. Any chance for\na job with your outfit?\"\n\nDurin' the time I had been decidin' on what I'd say, Dixon had been\nwonderin' how I'd take it; and I don't doubt he was some relieved.\nAnyway, he thawed out a little. \"Nope, I hardly think so,\" sez he.\n\"We've been hard pushed for grass this season; but Ty bought a\nwater-right on Ice Crick, and things has smoothed out again. Another\nthing is, that Badger-face has come back.\"\n\nI gave a start as natural as life, and I didn't put it on, neither. I\nhad no idy he'd mention Badger-face without a lot o' pumpin'.\n\"Badger-face?\" sez I. \"Good Lord, I thought he was dead!\"\n\n\"Well, we thought so, too,\" sez Dixon. \"We hadn't heard a word from\nhim; but he showed up a while back, and as soon as he gets able, he'll\ntake to ridin' again.\"\n\n\"What's wrong with him?\" sez I.\n\n\"He's purty well played out,\" sez Dixon. \"He sez 'at that feller,\nBradford, is some sort of a government agent. Now, we ain't got\nnothin' again' the government out this way, so long as it minds its\nown business; but when it gets to interferin' with our rights, why it\ngenerally has to find a new agent. You were along with this feller,\nBradford, when he scooped in Badger-face; and I doubt if that has\nslipped Badger's mind yet. Badger's memory for such things used to be\npurty reliable.\"\n\n\"Well, if it comes to that,\" sez I, \"I'd rather have Badger-face on my\ntrail than Dinky Bradford; though I own up, I don't just know what\ngovernment position Dinky holds.\"\n\n\"Ol' man Williams there was along with ya, too, wasn't he?\" sez Dixon.\n\n\"Sure he was,\" sez I. \"We got a heap better paid, for that trip 'n we\nusually get.\"\n\n\"Yes,\" sez he, slow an' drawly, \"but a feller can never tell when he's\nall paid out for such a trip as that.\"\n\n\"A feller has to take chances in everything,\" sez I. \"I still got a\nlittle money left to amuse myself with.\"\n\n\"It don't seem to make ya reckless,\" sez he. Dixon had been drinkin'\npurty freely, and I rather liked the effect liquor had on him.\n\n\"Maxwell,\" I called, \"this is a dry summer. Set up the drinks for the\nhouse.\" Some saloon-keepers fawn on ya as if they'd melt the money out\no' your clothes while some of 'em are cold and haughty, as though it\nwas an insult to offer 'em money. Maxwell was one o' this kind. He\nglared his red eyes at me as if I'd been rude; but he set out the\ndrinks all right.\n\nTank had been shut away from drink for so long that I had plumb forgot\nhow he had happened to win his title; but as soon as I had give the\norder, I sensed that he was in the mood to sluice himself out\nthorough. The very minute we had cooled off from the drinks--Maxwell\nkept a brand o' poison which would eat holes in an iron kettle, if you\nlet it set five minutes--Well, the very instant the steam had stopped\ncomin' out of our mouths, Tank ordered a round; and before that had\ngot on good terms with the first drink, Spider Kelley had arrived.\n\nMexican Slim had guessed where we were headin' for, and Tank had owned\nup to it, and Slim had told Spider, and, o' course, Spider hadn't been\nable to stay behind; so when he stuck his nose in the door, Tank sez\n'at the drinks was always on the last-comer, and Spider ordered a\nround.\n\nI can journey about with a fair amount o' booze, without lettin' it\nsplash over into my conversation; but I was there on business, so I\ndrank as short drinks as would seem sociable. Tank, on the other hand,\nhad formerly been as immune to liquor as a glass bottle; but he was\nout o' practice without realizin' it; and he splashed into Maxwell's\nforty-rod as though he was a trout hurryin' back to his native\nelement. Spider was a wise old rat, and he played safe, the same as\nme. O' course, the Cross-branders couldn't stand by and see us\npurchase Maxwell's entire stock, without makin' a few bids themselves;\nso for a while, we peered at the ceiling purty tol'able frequent.\n\nThe young feller with the boy's eyes was chin-ful to begin with, the\nother three Cross-branders were purty well calloused to a liberal\nsupply o' turpentine; while Dixon would load up his dumb-waiter and\nsend it down as unconcerned as though his throat was a lead pipe,\nconnectin' with an irrigation ditch. He had reached the stage where he\nwas reckless but not thoughtless, and the' didn't seem any way to wash\nhim down grade any farther.\n\n\"Any more o' you fellers liable to drop in?\" sez he, lookin' at me. I\nwaved my hand towards Spider, as though he, bein' the last to arrive,\nwould have the latest news; and Spider sez: \"Nope, I reckon not.\nLeastwise, not so far as I know.\"\n\n\"Badger-face has come back and taken on with Ty again,\" sez I.\n\n\"The hell he has!\" exclaimed Spider, just as I knew he would.\n\n\"Yes,\" sez Dixon with an evil chuckle, \"he's come back, and I doubt if\nhe'd feel any sorrow at meetin' up with some o' you boys.\"\n\n\"As far as I remember,\" sez ol' Tank, bulkin' up as ponderous as a\njustice o' the peace, \"I don't recall havin' asked Badger's permission\nto do anything in the past, and I don't intend to begin now.\"\n\n\"Well,\" sez Dixon, \"I don't mind tellin' ya that Ty Jones ain't so\nsure o' Badger as he used to be; and nothin' would suit him so well as\nto see Badger cut loose and get some o' you fellers for helpin' to\nhave him railroaded.\"\n\nThis surprised me. Dixon didn't seem a shade worse 'n he'd been when\nSpider arrived, but he'd sure enough leaked out the news I was after.\nTy was suspicious o' Promotheus, and we'd have to finish our job as\nsoon as possible. I didn't want to start anything at Skelty's so I\nproposed a little friendly poker. The Kid was asleep in the corner; so\nthe seven of us played stud for an hour or so until Tank fell out of\nhis chair, and then we broke up for the night.\n\nTank was all in; so we had to put him to bed, and the Kid had to be\nput to bed, also; but Dixon and the other three took a final drink and\nstarted back to Ty's.\n\n\n\n\nCHAPTER THIRTY-SEVEN\n\nPROMOTHEUS IN THE TOILS\n\n\nTank weighed like a beef when he got liquor-loose, and it was all me\nand Spider could do to get him to bed. His legs were like rubber; but\nhe insisted on tellin' us what he thought about things. He begged us\nto start back and let him ride, sayin' that it was only the heat o'\nthe room, not the drink, which had upset him; but he was in no shape\nto ride a hay wagon, so we put him to bed.\n\n\"I think more o' the Friar than of airy other man I know,\" he sez to\nus at the head o' the stairs; \"but I own up 'at I don't take kindly to\nreligion; and I'll tell ya why. The's hundreds an' dozens of hymns to\nthe doggone sheep-herders; but the' ain't one single one to the\ncow-punchers. Now, what I sez is this, if ya want to round me up in a\nreligion, you got to find one 'at has hymns to cattle men.\"\n\nWe didn't bother to explain it to him, 'cause he wasn't in condition\nto know a parable from a pair o' boots. We dragged him along the hall\nand flung him on his bed. By chance we put him on the bed with his\nboots on the piller; but he went sound asleep the moment he stretched\nout; so we just hung his hat on his toe, folded the blanket over him,\nlocked the door, put the key in my pocket, and went across the hall to\nour own room.\n\nI didn't want to harbor that liquor any longer 'n I had to, so me an'\nSpider slipped down, got some salt an' mustard, soaked it in water,\ndrenched ourselves--and repented of havin' been such fools. Then we\nwent up to bed. It had been some time since we had stretched out on\nsprings, and we were cordial for sleep; so we mingled with it in short\norder.\n\nStill, I wasn't easy in my mind, and twice I woke up and went into the\nhall; but I couldn't hear anything, though I had a feelin' that the'd\nbeen some good cause for my wakin' up. I lay on the bed the last time\nwith my mind made up to watch. Skelty's had allus had the name o'\nbein' a tough joint, and this red-eyed Maxwell with his Injun hair\nwasn't of the kind to purify it to such an extent that the old\ncustomers wouldn't feel at home.\n\nAs I lay there, I saw the window rise, slow and careful. The' wasn't\nany moon; but I could see a hand in the starlight. I made up my mind\nto sneak out o' bed, grab the hand, pull it in to the shoulder, and\nthen throw all my weight on it, and yell for Spider. I got up as\nnoiseless as cider turnin' into vinegar--and then upset a confounded\nchair, which sounded like two houses runnin' together.\n\nThe window dropped with a bang; and at the same moment the' came a\nshriek from across the hall, followed by some scufflin' and the sound\no' broken glass. After this all we heard was Tank's voice tryin' to\nexplain his opinion o' that part o' the country and all its\ninhabitants. I had thought that Tank had discarded most of his\nprofanity; but by the time we had got our guns and crossed the hall to\nhim, I changed my mind. When I put the key in the lock, he suggested\nto us what was likely to happen to any unfriendly individuals who\nattempted to enter that particular room.\n\nI told him gently to stuff the piller into his mouth, if he couldn't\nfind any other way to stop his yappin'; and then I unlocked the door,\njust as Maxwell and his bartender came into the hall. The bartender\nhad one gun and one candle, and Maxwell had two guns.\n\nWhen we opened the door, there was Tank with the blood runnin' down\nhis leg, while he stood in a corner of the room holdin' his weapon up\nabove his shoulder. \"What's the matter with you?\" I sez, a little\ncross.\n\n\"I'm homesick, you blame ijiot!\" sez Tank. \"What else would likely be\nthe matter with me?\" Tank was about as far out o' humor as I ever saw\nhim get.\n\nMaxwell came in and looked at the pool of blood. \"Don't stand there\nand bleed on the floor,\" sez he.\n\nTank looked at him baleful. \"What do ya wish me to do--upset your\nrotten dive and bleed on the ceilin'?\" sez he. \"I didn't come here\ndetermined to smear up your place with my life blood; and I want ya to\nunderstand that I didn't punch this hole in myself simply to cool off.\nI know what you're afraid of--You're scared that some o' your liquor\nhas got into my blood, an' that it'll leak out and set your floor on\nfire.\"\n\n\"You run get a bucket for him to bleed into,\" sez Maxwell to the\nbartender.\n\n\"Yes,\" sez Tank, sarcastic; \"and be sure to get a big one, as I am\nminded to draw off all o' my blood, just to see how much I have in me\nat this time o' the year.\"\n\nSayin' which, Tank walked over an' sittin' on the bed, held out his\nboot for me to pull off. He had been stabbed through the leg, through\nthe thick part o' the calf, and a jet was spoutin' out of the top cut,\nand a steady stream oozin' from the bottom one. I put my finger\nknuckle above the top jet, and the palm of my other hand over the\nlower one, and then sent Maxwell after a small rope and some bandages.\n\nWhile he was gone, a couple o' the girls strolled down the hall to see\nwhat the excitement was; but Tank began to lecture about morals and\nmanners, and they didn't bother us long. We patched Tank up in good\norder, and made him lie down again. He said that he had been woke up\nwhen his leg got stabbed, and had grappled with a man; but the man had\ngot out the window again.\n\nSkelty had built his place on a side hill. The bar and dinin' hall was\nin front, and a small dance hall and kitchen back of it. Upstairs were\nbedrooms, and the ground sloped so, that the back rooms were only\nabout five feet from the ground. This made the downstairs easier to\nheat in winter--and it was also convenient for any one who wanted to\nget in through a window.\n\nMe and Spider ate breakfast next mornin'; but we wouldn't let Tank\neat, rememberin' the Friar's rules for wounds. When we started away,\nTank insisted on goin' along; so we had to ride slow. We went north,\ninstead of in the direction we wanted to go, for fear some one might\nbe spyin' on us. I was mighty sorry we had come, even though I had\nfound out that Promotheus was under suspicion; and as soon as we had\ncome to a pass where we could see a good distance in all directions, I\nsent Spider on a circle to tell the boys to bring things to a head as\nsoon as possible.\n\nTank's leg ached him consid'able; and we had to ride purty slow; but\nby noon we had come to the Simpsons' cabin. We told 'em that Ty Jones\nwas suspicious about the Greasers and intended to get square with all\nwho had took a hand in removin' 'em; so they agreed to stand with us\nwhenever we were ready to make a raid on Ty.\n\nI made Tank lie down all afternoon, and drink all the water he could\nswallow, but that night when I started to ride over to the look-out,\nhe insisted on goin' along. It was a hard ride, and I wanted him to\nwait until the next night, but he tagged along, so I had to ride slow.\nWe had figured out that the feller who had tried to get him had seen\nthe hat on his foot at the head o' the bed; and before he had had time\nto locate him proper, the noise the other one had made slammin' the\nwindow to my room had scared him, so he had taken his stab haphazard.\n\nThis must 'a' been the way, 'cause when drinkin', Tank was usually a\nregular long range snorer, and only a hurried man would have mistaken\nhis feet for his head. Tank insisted that he had seen the feller's\noutline again' the window, and that it had been Dixon. I doubted this;\nbut Tank insisted that the feller had had a neck like a beer bottle,\nand then I had to give in.\n\nWe didn't reach camp until sun-up, and then we found 'at Promotheus\nhad been there the night before, with word that he had had a long talk\nwith the woman, who had been in the most rational mood he had ever\nseen her in. He had drawn her into tellin' him all she could remember.\nShe had told him about havin' her head full o' pictures; but not bein'\nable to tell the real ones from those she had dreamed. She said she\nhad lost the key to them and could not understand 'em, that she\nremembered havin' sung on many different platforms, but could not tell\nwhere or when, and could not sing any more, though she sometimes\ntried. She said that whenever he said the name Carmichael, she saw the\npicture of a young man in white robes, but that he had died. When\nPromotheus had tried to make her understand that he was still alive,\nshe had become frightened, and told him never to speak the name again.\n\nHe asked her about the Winter Garden in Berlin, and she said 'at this\ncalled up the picture of a man with curled-up mustaches, and then she\nhad covered her eyes, and told him he must not mention this again,\neither. Horace was tellin' me all this; and when he finished, I sez:\n\"Well, if this is the most rational she has ever got, she'd be a nice\none to handle in her usual condition. I don't see what we're to do;\nbut we have to move fast, as Ty Jones is suspicious.\"\n\nThe next night the Friar and I were down at the head of the path\nleadin' into the ravine when Promotheus came. He said that Dixon had\ncome in with his face cut, and had told about seein' us over at\nSkelty's, and how we had bragged about gettin' him rail-roaded, and\nDixon and the others had told him they were ready to back him up any\ntime he wanted to go an' get even. He also said 'at Ty had been\nroastin' the whole gang of 'em for bein' afraid of Olaf, and advised\nus to warn Olaf to be on guard. He said the woman had told him that\nday that at all times she had a dull pain in the top part of her head.\nThe was beginnin' to get worried, this was plain to see, and he didn't\nstay very long.\n\nWhen we told the others what he had said, we decided it was our duty\nto go and tell Olaf that very night, so that he could send over the\nnext day and get a couple o' the Simpson boys to come over and help\nwatch his place at night, until we were ready to finish with Ty. We\nwanted to put it off as long as possible, as Ty would soon be in the\nfall round-up and there wouldn't be so many men at the home place.\n\nMexican Slim and Tillte Dutch started to ride to Olaf's; but I was\nrestless that night, so I rode along with 'em. Just before we reached\nthe Spread, we saw a bright light at the side o' the cabin. In a\nminute two other lights shot up, and we knew they were firin' brush at\nthe side of it. We threw in the spurs and rode, keepin' close watch.\nTwo men rode towards us, and we drew off to the side of the road. Just\nas they got opposite, we ordered 'em to halt; but they whirled and\nfired at us. We fired back, and started after 'em; but it was dark in\nthe cottonwoods, and they gave us the slip and got away.\n\nWhen we reached the cabin, we saw it was doomed. Piles o' brush had\nbeen heaped on all sides of it and fired one after the other.\nEverything was so dry that even the dirt on the roof would have\nburned, and there was nothing to do. Kit with the boy in her arms, and\nOlaf and Oscar beside her were standin' close by, watchin' it burn,\nand they felt mighty bitter. We told 'em why we had come, and advised\n'em to go and leave Kit with the Simpsons, and come to our camp the\nnext night. Then we rode back before daylight and told the others what\nhad happened. We were all purty hosstile. Settin' fire to a cabin with\na sleepin' woman inside wasn't no fair way o' fightin'.\n\nThat afternoon as we were watchin' the ranch through the field\nglasses, we saw the woman and Promotheus walkin' together toward a\nlittle open space in the cottonwoods where the' was some grass close\nto the edge o' the crick. Thick bushes was all about this place, and\nit was cool and pleasant in the heat o' the day. They hadn't been gone\nvery long when we saw two others sneakin' after them. I looked through\nthe glasses, and one appeared to be the skinny feller, Dixon, and the\nother, the Chinese cook. We saw 'em sneak into the bushes and\ndisappear close to where the woman and Promotheus were sittin'. Part\no' the time they talked together, and part of the time she read to him\nout of a book.\n\nWe fair ached to yell to 'em and put 'em on their guard; but all we\ncould do was to sit up above in our look-out, feelin' weak and\nuseless. I suppose we felt like a mother bird when she sees some\ninhuman human foolin' about her nest.\n\nAfter a time the Chink crept out and scurried along to the old house.\nHe bounced across the porch, all crouched over, and we knew he had\nsome evil tale to cheer up his yellow soul with. In half a minute, Ty\ncame out with him and follered him into the clump o' bushes. We could\nsee the woman and Promotheus plain, with our naked eyes. It was a good\nthing, too; for Horace hung on to his glasses as though they were life\npreservers.\n\nIn about ten minutes, the bushes parted, and Ty stepped into the open\nspace in front of 'em. Promotheus got to his feet slow, but the woman\nsat still, and didn't seem much interested.\n\nTy glared at Promotheus durin' the few minutes he was questionin' him,\nand then they all went back towards the ranch house. The woman went on\nto her own cabin, and Ty blew on the horn which hung at the side of\nthe door, and that sneak of a Dixon came on the run, as though he had\nno idee what was wanted. Actin' under orders from Ty, he took The's\ngun and then tied his hands behind him and shut him up in an out\nbuildin' near the stables. There didn't appear to be any one else\nabout the ranch, and I suggested that we make a rush and take\npossession right then.\n\nWhile we were debatin' it, we saw the punchers comin' in from the\neast, across the crick. There were about a dozen of 'em, strung out\nand ridin' hard the way they generally rode.\n\n\"They're likely to string him up this very night,\" sez I; \"and we'll\nhave to settle this business before sun-up.\"\n\n\"They are not likely to be in any hurry,\" sez the Friar. \"If we go\nto-night it will mean a lot o' bloodshed. To-morrow they will go out\non the range again, and we stand a good chance of rescuing him without\neven a fight.\"\n\nOlaf, of course, sided with the Friar, Horace sided with me, and we\nhad a purty heated discussion. The Friar argued that he had the most\nat stake and had a right to select the plan with the least risk. I\nargued that Promotheus had the most at stake, and we had no right to\ntake risk into account. We got purty excited, I usin' the word coward\nfreely, while the Friar stuck to the word folly and kept cooler 'n I\ndid. He finally won 'em over to a compromise. We were to go down close\nand keep watch durin' the night; but not to make a rush until we saw\nPromotheus actually in instant danger.\n\n\n\n\nCHAPTER THIRTY-EIGHT\n\nOLAF RUNS THE BLOCKADE\n\n\nTy Jones had been as wise as a fox when he located his ranch house. It\nsat on high ground, while back of it rose a cliff; so 'at the only way\nyou could get to it without ropes from the back, was through the\nlittle ravine. The cliffs circled around to the crick on both sides,\nand the crick was so full o' rocks that the' was only two places a\nhoss could cross. He had strung barb wire through the cottonwoods in a\nregular tangle along the crick, and the only places he had to watch in\ncase of an attack, were the ravine and these two fords. He could see\nfor miles in all directions by goin' to the head o' the ravine; and\nyou could hardly pick out a purtier place for a last-stand 'n the one\nhe had selected.\n\nThe new cabin for the woman was right in front o' the mouth o' the\nravine, the old cabin a hundred yards or so farther on, the cook-house\nand the Chink's quarters to the north o' this, the mess-hall for the\nmen to the east of this, the barn, wagon-sheds, workshop, and so on,\nsome distance to the south, and the bunk-shack a little to the north\nof the stables. He had several corrals back o' the barn and a pasture\nof about thirty acres shut in by a wire fence.\n\nAfter I had cooled off a little, I saw that the Friar was right. The\nthing we couldn't tell was, just how much they had forced Promotheus\nto confess. If they had simply got Ty jealous that he was tryin' to\nget the woman away, we might make it all the worse by chargin' down on\n'em; while on the other hand he might have told where we were, and Ty\nmight take it into his head to try to get us all. This last would have\nbeen the finest thing 'at could happen to us; but the' was no way to\ntell; so after eatin' supper, we went down to the edge o' the cliff to\nsee what we could see.\n\nWe were most of us surprised to see how far the cabin stood from the\ncliff. In lookin' down from our look-out, we had failed to take the\n<DW72> into account so it had looked as though we had been able to see\nthe woman the minute she had come out o' the mouth of the ravine,\nwhile the fact was the cabin stood several hundred feet from the\nmouth. If it hadn't been for the confounded dogs, we could have gone\ndown and found out what we wanted to know. We made some remarks about\nthose dogs which would have seared their hair off if they'd 'a' been a\nlittle closer.\n\nThe light was kept in the mess-hall long after time to finish eatin';\nand we guessed they were tryin' Promotheus, right while we were\nlookin' on from above. All of a sudden, Olaf struck his palm with his\nfist, and exclaimed: \"What a fool I have been! Those dogs remembered\nPromotheus, and he never patted 'em. I have patted 'em and spoke\nsoothin' words to 'em, and they would know me. I shall go down and\nlisten.\"\n\nNow this was a noble thought and we hadn't a word to say again' it; so\nOlaf went back to camp, shed his boots and put on moccasins. Slim was\na good shot with a rifle, so he staid with Horace, who had an elephant\ngun and a yearnin' to use it, up on the cliff above the mouth o' the\nravine. They had seven rifles of one kind and another, and they\nthought they could make a disturbance if Olaf started anything. The\nrest of us went down the ravine to the last curve. We tried to get the\nFriar to stay behind; but his blood was up, and he wouldn't heed us.\nWe had it made up to rope and tie him hand and foot, when we were\nfinally ready to wind things up with Ty Jones.\n\nOlaf left us with his big, hard face set into rigid lines. He had a\nlong score to settle with Ty Jones, and he had made a funny gruntin'\nhum in his throat every few steps as we had walked down the ravine. We\nwaited what seemed weeks; but the' was no uproar, and finally, he came\nout o' the gloom, and spoke to us in a whisper. We went back with him\nto the top o' the path before he told us what he had heard.\n\nHe said they were tryin' to make Promotheus confess who was back of\nhim; but that Promotheus had steadily refused. He said 'at Ty had told\nhim over and over that if he would tell him where he could lay hands\non either the Friar or Dinky Bradford, he would give him a month to\nget out o' the country himself; but Promotheus had stood firm, and\nthey had shut him up in the workshop again, tellin' him he would get\nnothin' but water until he did confess.\n\nThis made us some easier in our minds. Promotheus had acted so worn\nout and done up since his return, that he had fooled Ty; and Ty looked\nupon him as a broke-down man, and nothin' but a tool in the hands of\nsome stronger men. Olaf said 'at Ty acted as though he thought the\nFriar had sent in a report to the government, and had got Bradford to\ncome out here the time that Promotheus had disappeared; and in some\nway they had got word o' Horace comin' through Bosco this last time.\nDixon had told about seein' us at Skelty's, and a strange feller told\nabout bein' shot at, the night Olaf's cabin had been fired. They\nbunched all this together, and decided 'at the best thing to do was to\ntrade Promotheus for Horace or the Friar, if it could be done. I had a\nchuckle all to myself, when I pictured Horace as he had been when I\ntook him in hand, and now with the reputation he hadn't quite earned,\nbein' a worry to the Ty Jones outfit.\n\n\"I allus said they were cowards,\" sez Horace, as soon as Olaf had\nfinished his tale. \"A man's got an imagination, and as soon as he\nstarts to live like a wolf, this imagination fills the world with\nwatchdogs. Ty Jones never has fought in the open, and we'll have no\ntrouble with him as soon as we once get him on the run.\"\n\n\"Ty Jones has no fear,\" sez Olaf. \"I know; I have seen with my own\neyes. He is too clever to be trapped; but he has no fear.\"\n\n\"Well, wait and see,\" sez Horace.\n\nMe and Tank kept watch on the cliff until mornin' and then as nothin'\nhad happened, we went up to camp, and Slim and Dutch took watch at our\nregular look-out. As we sat down to breakfast, we noticed 'at the\nFriar was gone. Several spoke of him havin' been restless the night\nbefore and not turnin' in when the rest did. The Friar allus was\nunregular in his habits, especially at night; so we didn't pay much\nheed to him when he wrote by the fire, or went off by himself in the\nquiet starlight, to sing some o' the pressure off his heart; but at\nsuch a time as this, we anticipated him to be as circumspect as\npossible.\n\nWe started to hunt him up, but it didn't take long. Horace found a\nnote pinned to the Friar's tarp, and the note told us that he had\nthought it all over careful durin' the night, and had decided that his\nduty compelled him to go down and offer himself in exchange for\nPromotheus. He said that when things came to such a tangle that no\nhuman ingenuity could unmix 'em, it was time to put trust in a higher\npower; that it was for him that Promotheus had risked his life, and\nthat he felt he must take his place, as Ty had promised to let\nPromotheus go if he would betray him. He said that he could not see\nany way to help the woman, and that if he lost his life, for us not to\nthink of revenge, as it would all turn out for the best in some\nmysterious way. The Friar had gone through a lot durin' the last few\nyears, and it had finally undermined his patience. I knew just how he\nfelt: he wanted something to happen which would end his suspense, and\nhe didn't care much what it was.\n\nAs soon as Horace had finished readin'; we all sat around in complete\nsilence, gawkin' at each other. \"Things has finally come to a head,\"\nsez Spider Kelley, solemnly.\n\n\"There now, that's the Christian religion!\" exclaimed Horace. \"The\nChristian religion is founded on self-sacrifice and martyrdom, and all\nthose who get it bad enough spend the bulk o' their time on the\nlookout to be martyrs and sacrifice theirselves for something--and\nthey don't care much what for. Look at the crusades--the flower o'\nEurope was lured into the desert and dumped there like worn-out junk,\neven children were offered up in this sacrifice. Nothing but\nsentimentality, rank sentimentality. Now, when the ancient Greeks--\"\n\n\"The thing for us, is to decide on what we're to do next, not what the\nancient Greeks did a few thousand years before we were born,\" sez I.\n\"There is no use hidin' any longer. The strongest card we have up our\nsleeve is the fake reputation of Dinky Bradford, and what we must do\nis to make up the best plan to play it.\"\n\n\"Why do you say fake reputation?\" demanded Horace.\n\n\"Well, you're not a government agent, are ya?\" I asked.\n\n\"No,\" sez he; \"but at the same time--\"\n\n\"I didn't say 'at you was a fake, Horace,\" sez I in a soothin' voice.\n\"I merely intimated that the things Ty Jones most fears about you are\nthe things which were not so.\"\n\n\"I see what you mean,\" sez Horace, \"and it's all right. What's your\nplan?\"\n\n\"Well, as soon as we are sure 'at the Friar has reached Ty's,\" sez I,\n\"we'll send Ty word to deliver him back at once, and to appoint a\nmeetin' place to explain things to us. Not make any threats nor bluffs\nnor nothin'. Just a plain, simple statement of what we want done, and\nsign your name to it.\"\n\n\"I think it would be better to tell him we had his place surrounded,\"\nsaid Horace.\n\n\"Nope,\" said I, \"your old theory is best: let their imaginations\nsupply the details. If we put the government into their minds too\nstrong, they're likely to find some way to deliver Promotheus over to\nthe law. I have a sort of impediment that The was a little rough with\nan officer or two, after he deserted, and Ty knows all about him.\"\n\n\"How the deuce will we get word to Ty?\" sez Horace. \"As fast as we'd\nsend messengers, Ty would shut 'em up.\"\n\n\"One thing is certain, at least,\" sez I. \"Ty won't string 'em up as\nlong as he knows he's bein' watched. And another thing is, that all of\nTy's men are wanted for one thing or another, and the longer we keep\n'em in suspense, the sooner they'll weaken. We ought to send word to\nthe Simpson boys. They are at least two to one again' us as we stand\nnow.\"\n\nJust at this junction, Slim arrived with the news that the Friar was\nridin' up to the ford. I was purty sure 'at he wouldn't go down by the\nravine. The Friar might lack judgment in certain matters; but you\ncould count on him lookin' out for his friends, every time.\n\nWe hustled down to the look-out, and saw the Friar ride out into the\nopen, and hail the house. In a minute the' was a crowd about him and\nthey pulled him from his hoss and dragged him toward the mess-hall,\nactin' mighty jubilant. The dogs raised a consid'able fuss; but they\ndidn't let any of 'em get to the Friar this time. I don't know whether\nthey were tryin' to save the Friar or the dogs.\n\nThey took the Friar into the mess-hall, and kept him there a good long\ntime; but I felt sure he wouldn't tell more 'n he wanted to. Then they\nbrought him out and shut him up in the workshop with Promotheus.\n\n\"You don't see 'em turnin' Promotheus loose, do ya?\" sez ol' Tank.\n\n\"Ty Jones would cheat himself playin' solitaire,\" sez Spider Kelley.\n\n\"He didn't agree to turn Promotheus loose if the Friar surrendered,\"\nsez Olaf. \"He only said he would if Promotheus enticed the Friar into\na trap.\"\n\nTy Jones certainly did have what ya call personal magnetism. His men\nstuck up for him, even when they was willin' to help snuff him out.\n\nWe sent Oscar over to get the Simpson boys; and then we made our\nplans. The' was no way to get to our camp from above, and we could\neasy guard the two trails 'at led up from below. Nothin' would have\nsuited us better 'n to have Ty decide to come and get us; so we told\nOscar to make all the fuss he wanted when he came back.\n\nNothin' happened down at the ranch that day. The woman drifted about,\nthe same as usual, not seemin' to observe 'at the' was anything\ndifferent from ordinary, and the punchers all stayed in sight. A few\nof 'em rode up to high spots across the crick and took gappin's, and a\ncouple of 'em came up the ravine and examined the ground on top; but\nthey didn't seem to find anything to interest 'em.\n\nThat night Horace wrote an order on Ty Jones to release the Friar--we\nhad decided not to mention Promotheus--and Olaf started down with the\nmessage. We posted ourselves the same as we had done before; and after\nabout an hour, Olaf returned.\n\nHe said he had examined the workshop, which was of logs, the same as\nthe rest o' the buildin's, and had heard the Friar and Promotheus\ntalkin'; but hadn't ventured to say anything for fear they were\nwatched. He said 'at the Friar was holdin' out on the value o'\nfastin'; while Promotheus was speakin' in defence of ham an' eggs.\nThen he said he had crept up to the front door of the old cabin, and\nhad fastened up the order with a dagger.\n\nOlaf looked to me as though he had been enjoyin' himself a little more\n'n his tale gave reason for; so I pressed him, and finally he admitted\nthat there had been a man on watch at the mouth o' the ravine. He said\nhe had wriggled through it on his belly, thinkin' it too good a place\nto be overlooked since the Friar had put 'em on their guard; and after\nlyin' still a moment, he had heard the man move. He said he had snaked\nup to him, and had got him by the throat. He said he thought it was\nDixon because the' was so much throat to get hold of. Dixon had been\nperfectly resigned to havin' Olaf lynched that time and Olaf's memory\nwas not o' the leaky kind.\n\n\"What became of him, Olaf?\" I asked.\n\n\"Oh, he fought some,\" said Olaf.\n\n\"Did he get away?\" I asked.\n\n\"Un, yes--yes he got away,\" sez Olaf.\n\n\"Where did he go to?\" sez I.\n\n\"I think he went down--way down,\" sez Olaf.\n\n\"Down where?\" sez I. \"Why don't you tell us what happened to him?\"\n\nOlaf looked down at his right hand. It didn't resemble a hand much;\nbut it would 'a' been a handy tool to use in maulin' wedges into a\nlog. \"Why,\" sez he, \"he wriggled about, and started to squeak; and\nwhen I squeezed in on his neck to shut off the squeak, why his neck\nbroke. It was too thin to be stout.\"\n\nI held out my hand. \"Olaf,\" I sez, \"I want to shake the hand that\nshook his neck.\"\n\n\"Yes,\" sez Tank, \"and by dad, so do I!\" Tank's leg was still tender.\n\n\n\n\nCHAPTER THIRTY-NINE\n\nSKIRMISHES\n\n\nOscar arrived durin' the night with the whole four Simpson boys; and\nword that Kit and the kid were in fine shape, with ol' man Simpson\nkeepin' a sharp watch, and Kit ready to take a standpat hand any time\ntrouble crowded too close. We expected to keep Ty busy, and so didn't\nworry any about Kit. Before dawn we started the four Simpsons out to\nmake a circle and cross the crick, tellin' 'em to use their own\njudgment to some extent; but not to run any risk. We wanted 'em to act\nlike scouts and, if possible, to draw Ty into chasin' 'em, and then to\nlead him back to our camp. We could see all of the other side o' the\ncrick from our look-out.\n\nBy dawn the rest of us were down on the edge of the cliff, and we saw\n'em find Dixon's body. They were consid'able excited about it; so we\njudged they had also read the notice on the door.\n\n\"What shall we do, to-day?\" asked Horace.\n\n\"Shoot dogs,\" sez I. \"There ain't any call to play safe any longer,\nand those dogs are the worst bother we have.\"\n\n\"All right,\" sez Horace. \"This will be a good chance for me to see if\nI'm still in practice. I'm a purty good rifle-shot, Happy.\"\n\nI never could quite harden myself to Horace. The change in him was\nalmost as much as that between an egg and a chicken; but yet the' was\nstill a suggestion of what he had been at first--his side-burns, most\nlikely--and it allus jarred me to see him steamin' ahead with\nself-confidence fizzin' out of his safety valve. He took his elephant\ngun and trained it on one o' the dogs which was sniffin' around the\nplace where Dixon's body had lain. We were purty well off to the north\nof the ravine; but it was still a consid'able angle of a down-shot,\nand a good long one too.\n\n\"Remember,\" sez I, \"that when shootin' down grade, you are mighty apt\nto shoot too high.\"\n\nHe lowered his gun an' looked at me as though I had called him a girl\nbaby. \"I have shot from every angle the' is,\" sez he; \"and I've shot\nbig game, too.\"\n\n\"Ex-cuse _me_!\" sez I. \"Shoot now, and let's see what happens.\"\n\nYou had to take off your hat to Horace when it came to a cultivated\ntaste in firearms. The thing he had got Promotheus with had been small\nenough to conceal in your back hair, while his present instrument\nwasn't rightly a rifle at all, it was a half-grown cannon. It shot a\nbullet as big as your thumb which mushroomed out and exploded, as soon\nas it hit. The dog died a merciful death; but he left a mighty\ndisquietin' bunch o' remains.\n\n\"Good boy, Horace!\" I said, slappin' him on the shoulder. \"You keep on\nremovin' the dogs, and I'll go up the <DW72>, and pertect your rear,\nshould they try to come up the ravine.\"\n\nI heartily endorsed this slaughter o' the dogs; but I wasn't ambitious\nto see it done. I have been well acquainted with a large number o'\ndogs of all sorts and sizes, and I have deep feelin's for dogs. When\nit comes to livin' accordin' to a feller's own standard, a dog has us\nall beat. When a dog signs up, he don't whisper nothin' under his\nbreath. He signs up for the full trip, and he don't ask a lot o'\nquestions about how long the hours'll be, or what sort o' grub and\nquarters and pay he'll draw. He just wags his tail, and sez: \"This\nhere feller is my idea of exactly what a feller ought to be; and I'm\nfor him in all he does. If he wants me to fight, I'm hungry for it, if\nhe wants me to be polite and swaller a lot o' insults, I'll do it, or\nif the time comes when my death is worth more to him 'n my life, why,\nI don't know nothin' about future rewards or such truck; but I'm\nperfectly willin' to swap life for death in his name, and I'm proud to\ntake the consequences--so long as he gets the reward.\"\n\nI own up 'at a dog has no morality; he's only a reflection of his\nmaster. A decent man has a decent dog, a vicious man has a vicious\ndog--and this is why it would have hurt me more to watch Horace\ntestin' his aim on the dogs 'n it would if he had been minded to pot a\nfew Cross-branders themselves, especially Ty Jones.\n\nNow, the sound o' this gun, and the sight of the dead dog made things\nbuzz down below. The men peered out from all directions, but hardly\nknew what to do. I had sent Mexican Slim off to the right, just above\nthe ravine, to pick off any dogs 'at came in that direction, and soon\nafter Horace got his, Slim also got one; and Ty whistled the dogs to\ncome to the house. Here was where his method of treatin' a dog showed\nup bad. Any time before this, a dog which so much as set foot on the\nporch had been belted with anything capable of inflictin' pain, and\nnow they refused to go inside.\n\nThe Chink was able to whistle 'em to the cook-house; but that was as\nfar as they'd go; and while they were standin' in a bunch, Horace and\nSlim each got one. Ty was standin' near one o' the poles which upheld\nthe back porch, and Horace exploded a slab from this pole in such a\nway that it knocked Ty down. This put the whole bunch into a\nconsternation. Horace certainly could shoot some. It made me think o'\nthe poorhouse, when I reflected on what it had cost him to learn how.\n\nNothin' much happened that day. Horace and Slim stuck to their\nknittin', and the Simpson boys played their part well. They rode in a\nbunch, and when they'd come in sight o' the ranch house, one would\nhold the field-glass case to his eyes, as though lookin' through the\nfield glasses, and another would turn and wave his hands, as though\nsignallin' to some one up in the hills. Once, two punchers went to the\ncorral and saddled hosses; but Horace shot one o' the hosses, and both\nmen flew for the stable without waitin' to take off the saddles. They\nhad never seen such wounds as Horace's elephant gun created, and it\nput 'em in a mighty thoughtful mood.\n\nThe Simpson boys came in soon after dark; and we all held a council of\nwar while eatin' supper. I was purty certain that we had a better\nbunch o' men than those we were fightin'. It is no test of nerve to\nkill a man: a lot o' men who got the reputation o' bein' bad were\nnothin' but accidents or sneaks; but when you have to stick through a\nslow fight without knowin' the odds again' ya, it gives your nerve a\nmighty searchin' try-out. I had hopes that after a day or so, they'd\nbe certain that the hills on all sides of 'em were full of enemies,\nand they'd be mighty glad to settle on our terms. I didn't want to\nkill a single man more 'n was necessary. Horace also thought we could\nwear out their nerve; but Olaf shook his head.\n\n\"Some o' the punchers may desert in the night,\" sez he; \"but as long\nas a single one remains to stand back to back with Ty Jones, Ty\nJones'll stay and fight. He has no fear--I have seen.\"\n\n\"The question is this,\" sez I, \"if those fellers are the kind to get\nfiercer the longer they're kept in suspense, the thing to do is to\nraid 'em to-night; but, on the other hand, if they're the kind whose\nnerve evaporates when it is kept uncovered, the thing to do is to wear\n'em down. Let's vote on it.\"\n\nWe decided to do some more wearin'; so we kept a guard at the camp,\nand the rest of us went down to the cliff, and tossed over stones to\nwhere we thought they'd be hid, providin' they had put guards at the\nmouth of the ravine. We raised a yelp the first throw, and heard a\nrush o' men from the new cabin, though the shadow was so dense down\nbelow we couldn't see a thing. This showed us that some o' the dogs\nstill survived and were bein' used as guards, and also that there were\nmen quartered in the woman's cabin. This was a bother, as it would\nforce us to be careful until we found out where she was livin'.\n\nWe posted a guard at the top of the path leadin' up from the ravine,\nanother at our camp, and went to sleep, feelin' purty tol'able well\nfixed. Nothin' happened that night, and the next day, we made ready to\ndo about the same as we had done the day before; but when we reached\nthe cliff, the' wasn't a sign o' life below--not a single, breathin'\nthing in sight, not even a hoss in the pasture.\n\n\"They've got away!\" exclaimed Horace.\n\n\"Where to?\" sez Olaf. \"Ty Jones hasn't any more use for the law 'n we\nhave, and you'll never make me believe 'at he's pulled out and left\nall his belongin's for whoever wants 'em.\"\n\n\"That's so,\" sez I; \"but where the deuce are they?\"\n\nWe watched all mornin'; but not a sign, not a bit o' smoke from the\ncook-house, just the ranch buildin's settin' there as deserted as the\nGarden of Eden. The Simpsons were workin' their stunts across the\ncrick; so about ten in the mornin', Slim and Dutch rode over to tell\n'em to come in, as they would look mighty foolish, providin' they were\nmakin' signals to one of the hills where the Cross-branders themselves\nwere hid.\n\nAfter eatin' dinner, the rest of us went down to the lookout, Horace\nshoulderin' his elephant exterminator, and lookin' peevish and\nfretful, 'cause the' was nothin' to shoot at. \"Boys,\" sez I, \"do ya\nsuppose 'at poor old Promotheus has been goin' all this time on\nnothin' but water.\"\n\n\"He's gone longer 'n this on nothing but water,\" sez Horace; \"and so\nhave I. Over in Africa, once, we sent a tribe o' blacks around to beat\nsome lions out for us; but they fell in with another tribe who were\nnot friendly, and they just kept on goin'. Promotheus and I were lost\nfrom everything, and we got into a desert before we found a way out.\nWe went for I don't know how long without water. Anyway, we went long\nenough to get into that numb condition when the earth becomes molten\ncopper, and the sky a sun glass, and a man himself feels like another\nman's nightmare. That tender old Promotheus you're sympathizin' with,\ncarried me the best part of a day, or a century--time had melted\nentirely away--and when we came back to our senses we lay beside a\npool of water. He's tough, Promotheus is.\"\n\n\"At the same time,\" sez Tank, \"settin' cooped up in a log hut with\nnothin' to cheer ya but water, isn't my idy of havin' high jinks.\"\n\n\"Perhaps, too,\" sez Spider Kelley, who didn't have enough sense of\nfitness to change a nickel, \"those mongrel coyotes lynched both him\nan' the Friar before they vamosed.\"\n\n\"They wouldn't do that,\" sez Olaf; \"but I wish we knew what they had\ndone.\"\n\n\"Let's go and shoot at the old cabin or the bunk-shack,\" sez Oscar.\n\n\"I move we wait, and raid 'em to-night,\" sez I, and this was what we\ndecided to do.\n\nThe rest of us lolled about purty patient--as active men, an' beasts\ntoo, are likely to do when the's nothin' on hand--but Horace who had\nlived in a room most of his life, hadn't quite learned to turn off his\nsteam when he hadn't any use for it; so he kept bobbin' up and fussin'\nabout. All of a sudden, he gave a sort of gasp, and pointed up the\n<DW72>.\n\nWe looked and saw one man crouched over and runnin' along where the\nsouth trail to our camp swung around a crag; and we sprang to our\nfeet, and looked up at the camp. As we looked, the face of Ty Jones\nwith a grin on it, poked up over a stone and leered down at us most\nexasperatin'.\n\n\n\n\nCHAPTER FORTY\n\nAN IRRITATING GRIN\n\n\nNow, you can mighty easy understand that this was a fair sized,\nable-bodied, bite-and-kick consternation for us, if ever the' was one\nin the world. Our look-out was behind a ridge which sheltered it\ncomplete from below, but left it as open from above as the straw hat\nwhich Stutterin' Sam made the dude crawl through. Up above us, lookin'\ndown from the rocks in front of our camp was Ty Jones, grinnin' as\nself-composed an' satisfied as a cat which has just removed all\nevidence of there ever havin' been any Canary birds; and truth to\ntell, we felt as indiscriminate and embarrassed as a naked man at a\ndance party.\n\nAll we saw was just Ty and his grin. We knew the' was one other man\nwith him, but that was all we did know; while our strength was as\nplain to them, as Tillte Dutch was the time he fell in love and used\niodaform on his hair instead o' perfume. We just stood and looked up\nat Ty, and then we turned our heads and looked at each other, and I\nnever saw as many stupid expressions in one mess. We felt as though\nevery minute was liable to be our next.\n\nWhenever ol' Tank Williams was surprised or puzzled or wrastlin' with\nhis own thoughts, he allus put me in mind of a picture I once saw of a\nwalrus. The walrus was loungin' up on a rock, and he looked as solemn\nand philosophical as though some young snip of a school boy had tested\nhis intellect by askin' him what two times one made. I never saw Tank\nlook so much like the walrus as he did this time 'at Ty Jones\nsurprised us. O' course Tank's teeth was different, but his mustaches\nstuck down in much the same way, and when I looked at him, I busted\nout laughin', though I own up I was scared enough to stampede the\nmoment before. When I laughed, it seemed to break the charm, and\nbefore I buttoned up my lips again, Horace had pulled up his elephant\ngun, and taken a blast at Ty's grin. Ty pulled down his face behind\nthe stone as soon as Horace aimed at him; but the range was long\nenough to strain even such a devil-tool as this half-grown cannon, so\nnothin' came of it.\n\nAfter my chuckle, I began to think in streams. The ground to the right\nof us--as we looked up towards Ty--was broken, and it occurred to me\nthat he had been holdin' us with his grin so as to give some of his\nmen time to sneak down and cut us off, he and the balance were above\nus, the ravine to our left, and straight back of us the cliff. We\ncouldn't stick where we were again' odds, and there wasn't any water\nin the clump of rocks which faced the path where it come out of the\nravine. As I ran over these details in my mind, I had as little\ntemptation to laugh as I ever did have; but the second I thought of\nthe clump o' rocks facin' the path, I saw that the path itself was the\nanswer.\n\nThere was no reason to hurry, as far as I could see; they could not\ncome to us without exposin' themselves, and every moment we waited,\nthe closer would come Dutch, Slim, and the four Simpson boys. To the\nright of us, as I said, the ground was broken, and here was where they\nwould be most likely to sneak down on us. By goin' in a diagonal\ndirection, we could get to where we could see straight up the washes\nwhich made up this broken ground, and so know what we had to fight.\n\n\"Come on, fellers,\" sez I, climbin' up over the ridge.\n\n\"Where ya goin'?\" sez Horace.\n\nI sat down on top o' the ridge. \"Have you got any plan?\" sez I calmly.\n\n\"No,\" sez he, \"I haven't; but I'd like to know--\"\n\n\"If you're willin' to take charge,\" sez I, \"why, go ahead, and I'll\nobey orders; but I don't care how small the body is, it can't do quick\nwork with more 'n one head, as you ought to know better 'n any of\nus--it havin' been tried frequent in those Greek tales you're all the\ntime inflictin' us with.\"\n\nHorace put his back up a little. \"I'm willin' to agree to anything\nreasonable,\" sez he; \"but I don't see any sense in leavin' this spot\nuntil we know where we're goin'.\"\n\nI folded my fingers together, set my thumbs to chasin' each other, and\nbegan to whistle. I wasn't jealous of Horace; but it just occurred to\nme that I had handled men before he'd mustered up courage enough to\nstay out after seven o'clock P. M. without gettin' his mother's\npermission, and I wanted to test the others and see if they thought he\nhad picked up more craft in three years 'n I had in a lifetime; so I\nwhistled the tune to his song, and looked up at the clouds.\n\n\"What's your idee, Happy?\" sez ol' Tank. I had nourished Tank on\nthought-food for a good long session, and I knew he'd feel mighty much\nlike a lost calf if I left him to rustle up his own idees; so I just\ngave my hands a little toss and kept on with my whistlin'.\n\n\"Aw, don't be so blame touchy,\" sez Spider Kelley. I had pulled Spider\nthrough a number o' tight places, also, and I knew he'd soon begin to\nfeel trapped up and smothery, if I left him to sweat out his own idees\nfor a few minutes longer; so I gave him the same gesture I had\nbestowed on Tank.\n\n\"What do you think we'd better do, Olaf?\" sez Horace.\n\nOlaf looked all around but did not see anything. \"They have come up\nthe ravine, took the path up the other side, through the clump o'\ntrees, made a wide circle and got to our camp,\" sez Olaf. \"If we try\nto get away, they cut us off. If we stay here, we die for want of\nwater. If we rush up the hill, they shoot us from behind the rocks.\nAll I can see is to wait until night, and then make a rush for it.\"\n\n\"Well, that don't look like much of an idee to me,\" sez Horace. I kept\non whistlin'.\n\n\"I move we foller Happy,\" sez Spider Kelley.\n\n\"I second the motion,\" sez Tank.\n\n\"I'm willin' to,\" sez Olaf, and Oscar nodded his head. This was about\nall Oscar ever used his head for except to hang his hat on; but he was\na good boy and sizey.\n\n\"All right,\" sez Horace. \"Now then, Happy Hawkins, the responsibility\nis on you.\"\n\n\"Now, be sure you mean this,\" sez I; \"for my plan is a foolish one,\nand I don't care to explain each step. I don't claim 'at my scheme is\nthe best; but my experience has been, that a poor plan carried out\nbeats a good plan which never came in. Climb up here, and we'll walk\noff in that direction without lookin' behind us.\"\n\nThey couldn't see any sense in this; but they follered me without\nchatterin', and I was satisfied. Horace had the field glasses in his\npocket; so when we had reached the place I thought would do, I set him\nto lookin' across the crick careful to see if he could see anything.\nAll the others watched him, and I got behind and looked up the <DW72>.\nI saw several men hidin' in the washes, and I said in a low tone:\n\"Keep on lookin' across the hill, Horace. Now, you others get out from\nbehind him. Now, Horace, whirl and examine the washes up the <DW72> and\nsee how many men you can count.\"\n\nHorace whirled, as did all the rest of 'em, and we found seven fellers\nin sight. We figured 'at there must be at least fifteen Cross-branders\nin the neighborhood, and probably more, and the ones we were able to\nsee in the washes convinced me 'at Ty had staked everything on gettin'\nus cornered. They didn't have enough to split up, so I felt sure they\nwould leave the ravine open, not thinkin' it likely we'd try to go\ndown there.\n\n\"Now,\" sez I, \"let's go to that clump o' rocks and hide.\" They all\ncame along; but didn't seem enthusiastic, because the washes led down\nclose to the rocks--we, ourselves, havin' sneaked down 'em while we\nwere waitin' for the woman that day. We couldn't see the path the boys\nwould take in comin' up to our camp from across the crick, while the\nCross-branders could see 'em a good part o' the way, and this fretted\nme a lot; though I hoped they had heard Horace's elephant gun.\n\nAfter a time, Horace, through the glasses, saw a feller's head\nwatchin' us from our old look-out; so we knew they had crept up along\nthe back o' that ridge. Then we heard consid'able shootin' off to the\nright, and knew the boys had got back. There were several good places\nfor ambush, and we felt purty blue at what had most likely happened;\nbut they were on hossback, and would be on their guard after knowin'\n'at the Cross-branders were up to some trick; so we hoped for the\nbest.\n\nThis clump o' rocks we were in was composed of one big crag and a lot\no' little ones. The big one shut off our view, and finally Horace said\nit would be a good plan to get on top of it, as the chances were we\ncould get a good view in all directions. It was fifteen feet up to\nwhere the' was footin', and we didn't see how it could be done; but he\nsaid it was simple; so we let him try it. He made Olaf and Tank face\nthe rock, holdin' on to each other. Then I climbed to their shoulders\nand they passed up Horace. I handed him up as far as I could reach,\nand it was as simple as peelin' a banana. The signal was for him to\ndrop a pebble when he wanted to come down.\n\nIn about two moments a stone the size o' your fist fell on Oscar's\nhead; which was a good thing, for it might otherwise have hurt a head\nwe had more use for. We laddered ourselves again' the rock, and Horace\ncame down without missin' a single one of our ears. When he reached\nthe level, he put his finger on his lips, and said he had seen ten men\nsneakin' up toward the rock and only a few hundred feet away. Oscar\nwas still holdin' to the lump on his head, so Horace explained 'at\nthe' hadn't been any pebbles on top the crag.\n\n\"Now, what ya goin' to do?\" asked Horace to me.\n\n\"You, Olaf, and Oscar go around the rock to the left,\" sez I; \"and\nTank, Spider, and I'll go around to the right. Each fire only once,\nand then run around the rock again and make for the path leadin' down\ninto the ravine. Keep close together all the way.\"\n\n\"The ravine!\" exclaimed Spider.\n\n\"Sure,\" sez I.\n\n\"All right,\" sez Spider, draggin' out the \"all\" until it would do for\n\"I told ya so,\" in case we got pocketed.\n\nIt worked fine; we flew around, surprised 'em, shot a volley into 'em,\nmade 'em seek cover, and then we flew for the head o' the path. Ol'\nTank, with his damaged prop, was as nimble as a one-legged Norman\nhoss, and Horace was loaded down with elephant ammunition; so that it\nwas wise to have all the time we could get. Ty and five others jumped\nup from our look-out, and tried to head us off; but they had to go\ntwice as far as we did. Ty and two others had rifles, and they stopped\nand took shots at us, but nothin' came of it.\n\n\"Hurry on to the ranch buildin's,\" I called as we went down the path.\nThen I turned back, to see what they were doin'.\n\n\"Let me take a shot at 'em,\" sez Horace's voice at my elbow.\n\n\"Why didn't you go on with the rest?\" sez I. \"I can give you half way\nand beat you runnin'.\"\n\n\"Let me take just one shot,\" sez Horace, so I gave in and let him. Two\nfellers were runnin' at a long angle toward the mouth o' the ravine to\nhead us off, and get a shot from above; so I told him to try for one\no' them. He fiddled with his hind sight as calm as though shootin' for\na Christmas turkey, and hanged if he didn't topple one over. The other\nstopped, and then ran back with his head ducked low to the ground,\nwhile the wounded one crawled behind a rock.\n\n\"Now dust for the buildin's,\" sez I; \"and don't try any more nonsense.\nLet me carry the weapon, and you won't be so overloaded. I'll start\nafter you in a jiffy.\"\n\nWhen I looked back, I saw that all of 'em had slowed down consid'able,\nout o' respect to the elephant gun; but I could still count seventeen,\nso we hadn't seen 'em all before. When they started towards the head\nof the path again, I took a shot at Ty Jones; but I didn't savvy the\nrear sight, and all it did was to make 'em slow down once more. Then I\nslid down the path and hot-footed it down the ravine. I saw signs o'\nhosses, so I knew they had rode most of their trip, and would be in a\nposition to circle around all they wanted to.\n\nI soon caught up with the others, and Tank was puffin' purty freely.\nAll the rest were runnin' easy, and we came out o' the mouth o' the\nravine without seein' a single soul. Now, we hardly knew what to do.\nIt was about the same distance from the mouth o' the ravine to the\nfirst curve in it, as it was to the woman's cabin; so I told Spider to\nstay at the corner o' the cabin, and watch that curve.\n\nThen we went around and found the door locked. We called twice to the\nwoman, but the' was no reply; so Olaf picked up a big stone and\nknocked off the lock. We made a quick examination; but the' was no one\nthere. I posted Horace and Spider in this cabin to watch the mouth o'\nthe ravine through the window facin' it, and to shoot into 'em, should\nthey foller us close.\n\nWe next went to the big house, where we had more trouble as everything\nwas fastened with bars on the inside, except the front door which had\nan immense padlock on the outside. We finally broke it off, and out\ndashed three o' their confounded dogs. We killed 'em, and went inside;\nbut the' was no one else there. Next we went to the workshop, and\nafter breakin' off the padlock, we found the Friar and Promotheus\ngagged and tied. The Friar was sad, and Promotheus was mad. We sent\n'em up to the cook-shack to get on speakin' terms with food again, and\nrummaged the rest o' the buildin's; but could find neither the woman\nnor the Chink, and by the time we were through, it was gettin' along\ntowards dark.\n\nI set Tank to cookin' a meal while the rest of us carried logs and\npiled 'em in the mouth o' the ravine. It would be moonlight up to ten\no'clock, and after that I intended to have a fire to see by. We also\nset up some logs at each o' the two fords. After supper we divided\ninto two equal groups o' four each, to stand guard, each man to watch\ntwo hours, one at the window of the new cabin, the other from the\nporch of the old one, where a view across both fords could be had.\n\nThe Friar was purty downcast at our not bein' able to find the woman,\nand at our still bein' in a state o' war; but he didn't kick none. He\npromised not to go over and surrender himself any more, and said he\nwould stand guard careful, and warn us the first thing 'at happened.\nWe decided 'at they would probably attack us that night, and we\nfinally chose the old shack, as it had water piped into it from a\nspring a hundred yards above. I figured 'at they'd be most apt to come\ndown the ravine, so I picked out the Friar, Olaf, and Tank to help me\nwatch it, and the others to take turns watchin' the fords.\n\nAbout half past nine, we lit the fires and turned in, with Oscar on\nthe porch, and Olaf at the window of the new cabin. I thought they\nwouldn't come before two o'clock, and had it arranged so 'at the last\nford watches would be held by Spider and Promotheus.\n\n\n\n\nCHAPTER FORTY-ONE\n\nTHE NIGHT-ATTACK\n\n\nI wasn't sleepy, and lyin' stretched out is the worst cure for\nsleeplessness 'at ever I tried; so after twistin' about for a while, I\ngot up and took a look around. Oscar hadn't seen a thing, which I took\nto be a mighty encouragin' sign. Mostly, when you set a boy on guard\nhe rouses ya out to meet the enemy every fifteen minutes, and then\ngoes to sleep just before the enemy actually does arrive; but Olaf had\ntrained Oscar to do what he was told, as he was told--when he was\ntold--and then not to talk about it for a couple o' years afterward.\nOscar was reliable to a degree; but for conversational purposes, I'd\nsooner have been shipwrecked with a brindle bull pup.\n\nI didn't have any doubts of Olaf; but I dropped in to see what sort of\na view he had, now that it had got dark. The fire was burnin' high,\nand the ravine was as bright as day. Enough o' the fire would last\nuntil mornin' to give a good view, so I strolled down around the\nbunk-shack and stables. I saw a form movin' in the shadow o' the\ncottonwoods, and stalked it careful, finally gettin' close enough to\nmake out the Friar.\n\n\"Can't ya sleep, Friar?\" sez I.\n\n\"No, no, I can't sleep,\" sez he with a sigh. \"Where do you think she\nis, Happy?\"\n\n\"They probably took her with 'em; and left the Chink to guard her,\nback in the hills,\" sez I. \"No matter what happens, they're not liable\nto harm her.\"\n\n\"It's sore hard to be patient,\" sez the Friar. \"I am honestly opposed\nto all violence and bloodshed. I have allus believed that all wars\nwere useless and unnecessary; but it's sometimes hard for me to love\nmy enemies.\"\n\n\"You're just worried and can't see clear,\" sez I soothin'ly. \"It's\nplain enough if you just think it out--that's the best part o'\nreligion. One place it sez: 'Love your enemies.' In another it sez:\n'Foller the Lord's example.' In still another it sez: 'Whom he loves,\nhe chasteneth'--which you said meant to punish. Now then, you have it\nall worked out: the proper way to love your enemy is to punish him;\nand, accordin' to this rule, we're goin' to love the hide off o' one\no' your enemies, if so be we're able to do it.\"\n\nBut the Friar never would stand for havin' his religion doctored to\nsuit the taste, he had to take it as stiff and raw as alcohol, where\nhe was concerned, himself; so he turned in and explained things to me\nuntil from my standpoint, misery was the only religious excuse a\nfeller had for bein' happy.\n\nBy this, it was time to change watches, so the Friar relieved Olaf,\nwhile Horace and his elephant-pest went out on the front porch to\nwatch the fords, and I turned in. None of us took our boots off that\nnight; we had a little fire in the big room, and slept on the floor,\nholdin' our belts in our hands. I drowsed off quick enough this time,\nknowin' 'at Tank and Promotheus would be next on watch and certain not\nto let anything surprise them.\n\nSure enough, just about the time we had slept ourselves into complete\nforgetfulness, we were all jerked to our feet by the first shot Tank\nfired, and this one shot was followed by a bunch of others. The\nCross-branders had crept down the ravine, and a little after three\nwhen the fire had burned low, they had tried to get by unnoticed. Ol'\nTank only had one eye, but it was a workin' eye, if ever the' was one,\nand he shot two of 'em with one o' their own rifles, and when they\nrushed him in a body, spreadin' out wide, he retreated to the old\ncabin, accordin' to directions.\n\nThe old cabin had loopholes in it, and we had found three fairly good\nrifles, but not much ammunition. We didn't waste any shots while it\nwas still dark; but they fired at us now and again. They had brought\nthe five rifles we had left at our camp, and used 'em freely. Slim had\ntaken the other rifle with him.\n\nAll durin' that day they broke the monotony by takin' frequent shots\nat us; but the logs in the cabin had been matched up for just such a\npurpose, and not one of us was even scratched with a splinter. What we\nwere most afraid of was, 'at they would find some way to set fire to\nthe cabin, and we counted on that bein' one o' the night's\ndiversities.\n\nThere were three good sized rooms in the old cabin which was only one\nstory high. One big room occupied the full south half o' the cabin, a\nbedroom was in the northeast corner, and a library in the northwest\ncorner. Yes, sir, a regular library, and the Friar and Horace both\nsaid it was a choice collection o' books. Horace showed us one book\nwhich had a photograph of the original Prometheus chained to a rock\nwith the vultures peckin' at his liver, and he certainly must have\nbeen some man to stand it. This picture made The's eyes light up\nconsid'able.\n\nThe' was also some chromos of naked stone images on the wall, which\nthe Friar and Horace called mighty fine copies. They were purty well\ndumb-founded to find 'at Ty Jones didn't live as much like a bob-cat\nas they'd thought. Under the book shelves was a row o' locked drawers.\nThey stuck out farther than the shelves above 'em, and we wanted to\npry 'em open to see what was inside; but the Friar wouldn't let us.\n\nThat was a wearin' day, and we were all glad when it finally dragged\nitself to the lake o' darkness, and dove in. We had our minds made up\nfor a busy night, but waitin' for trouble is more crampin' to the soul\nthan bein' in the midst of it, so we felt cheerfuller as soon as night\nactually settled down.\n\nWe didn't dare have a fire in the fireplace, for fear it would show\n'em our loopholes, and we didn't care to advertise these any more 'n\nwas necessary; but we set a lighted candle far back in the fireplace,\nto see to load by. The fireplace was across the southwest corner o'\nthe big room. There were no loopholes in the library, but we feared\nthe light might leak through a chink in the window shutter, so we\ndidn't have any light there. We kept one man watchin' through\nloopholes in the bedroom, and two watchin' in the big room, and were\nable to cover the whole neighborhood.\n\nThe cook-shack was the nearest buildin', and only the two loopholes in\nthe north end o' the bedroom covered that; so we decided to fling the\nlibrary window open and fire through that, in case they made a rush\nfrom that direction. We knew they wouldn't be likely to start anything\nuntil after eleven, as the moon wouldn't set until then, so we\nstretched out on the floor, leavin' Oscar, Horace, and Spider on\nwatch.\n\nWhen a feller has been keepin' his attention wound up for several\ndays, his mainspring finally gets strained, and the cogs in his head\nget to cuttin' up regardless. I managed to get a purty fair dab o'\nsleep; but it seemed as though I dove straight out o' wakefulness into\na dream, and it was some the rottenest dream I ever had. I dreamed\nthat Ty Jones had come and stooped over me and asked me what I thought\no' the way he had conducted his life. In a dream a feller is apt to do\nthe foolest things imaginable, so I looked up into Ty's face and told\nhim my true opinion. I sez to him: \"Ty, if your brains were blastin'\npowder, they wouldn't make enough explosion to raise your hat.\"\n\nTy didn't take kindly to this opinion; so he jumped into the air and\nlightin' on my face, began to trample it with his heels. The\ndiscomfort of this wakened me; but at first I didn't know I was awake.\nSeveral men had been actually tramplin' on me, and the' was a general\nfight takin' place in that room which was hard to make head or tail\nof.\n\nIn the flickerin' candle rays, it was mighty bothersome to tell who\nfrom which; so the' was no shootin'. Aside from Ty and Pepper Kendal,\nwe averaged bigger 'n they did, except Horace and Spider. Spider had\nlength but he ran small in the arms and legs, while Horace was\ntwenty-two caliber any way you looked at him. They abused Horace some\nconsid'able, and he got kicked and trampled on purty liberal; but he\nwas of terrier blood, and the second or third time he got kicked into\na corner, he crawled out on his hands an' knees, picked out a pair o'\nlegs which was strange to him, wrapped his arms about 'em, and fetched\ntheir owner to the floor with a thump. I spared enough time to knock\nthe feller on the head; and then Horace played his trick over again.\n\nOlaf was a mad bull in a mix-up like this--Horace said he had\nbeershirker blood in him, and this must be good stuff for it made Olaf\ngrin when Horace accused him of it. O' course the' ain't much head or\ntail to such a fight, and in lookin' back on it, it's just like\nspurtin' the pages of a picture-book with your thumb and tryin' to\nobserve the pictures. I saw the Friar leanin' again' the mantel-piece\nwith a hurt look on his face; and it disgusted me.\n\nIn times o' peace, I respected his prejudice again' violence; but this\nwas no time for foolishness, and I recall mutterin' to myself a wish\nthat Horace might have the loan of his big body for the next half\nhour. I saw Olaf knock down two men with one blow, I saw The save ol'\nTank's life, just as a half-breed was about to knife him from behind;\nbut for the most part it was just about as orderly a mess as a\npopper-ful o' corn over a bed o' coals.\n\nThe fight didn't last more 'n five or ten minutes. They had banked on\nsurprisin' us; and when this failed they were ready to back out. I\nafterward found out that it was the Friar who had caught sight of 'em\nfirst, he not' bein' able to sleep.\n\nTy and Pepper Kendal were the last to leave the big room; and when\ntheir own men were out of it, they opened fire on us; we fired back,\nand when they backed into the library where the rest o' their gang had\ndisappeared, we made a rush for 'em. I supposed they had come in\nthrough the library window, and I called for a candle, hopin' to grab\nTy before he could get out.\n\nSpider Kelley had already picked up the candle, and he had it in the\ndoorway in a second. The big drawers at the bottom o' the bookcase\nwere swung back, showin' a stairway behind 'em, and Ty Jones stood at\nthe top with Pepper Kendal just behind him. I dove through the air,\ncatchin' Ty's wrist with my left hand and his throat with my right,\nPepper Kendal bent his gun on me, Olaf grabbed the gun which was fired\njust as The grabbed Pepper's arms. It looked to me as though the\nbullet must have gone into Olaf's head; but just then we tripped,\nrolled down the stairs and the imitation drawers swung to behind us.\n\nAll holts were broke on the way down, and when I reached the bottom, I\nlay as quiet as a frozen moonbeam. I heard steps runnin' away from me\nin the dark, and presently the legs of the man next to me moved, and\nhe got up. I rose to a crouchin' position, held my arm above my head,\nand whispered, \"Who is this?\"\n\nFor answer, I got a smash on the arm with the butt of a forty-five\nwhich drove it down again' my head hard enough to bring me to my knees\nand wake up my horse-sense. I might 'a' known they'd have a signal.\n\nI waited with my back again' the wall until the silence began to soak\ninto my nerve. One o' my guns had got lost durin' the mess upstairs;\nbut I still had the other, and when I closed my grip around it, it\nseemed like I was shakin' hands with my best friend. As far as I could\ndiscover I hadn't been shot; but several knife-cuts and bruises began\nto hum little tunes which wasn't in nowise cheerin'. I just simply\ndon't like to be kept waitin' in the dark!\n\nAfter a bit I reached my hand out cautious, and felt the heel of a\nridin' boot. I examined as careful as though the feller inside the\nboot was a disguised bear-trap; but the' was no need. His neck was\nbroke. I felt of his face, and it was soft and smooth. The face of the\nyoung feller with the boy's eyes, I had seen put to bed drunk that\nnight at Skelty's, flashed across me, and I gave a sigh; but I had too\nmuch on my mind to turn soft, so I began to feel around again.\n\nPresently my fingers struck the heel of another boot. I shut down on\nmy bellows until the breath didn't get down past the top inch o' my\nneck, and I was as gentle with the heel o' that boot, as though it was\na bitin' man's eyeball; because I sure felt a quiver in it. I slid my\nfingers up that boot a quarter inch at a time, and I didn't use no\nmore rudeness 'n a mouse would use in tryin' to sneak a cheese piller\nout from under a sleepin' cat. When my fingers finally struck\ncorduroy, I purt nigh gave a shout, for this was what Promotheus wore.\n\nIt allus embarrasses a man to be felt over in the dark, so I took my\ntime with The; but after locatin' both hands and his crooked mouth, I\ndiscovered he'd been knocked out complete. I rubbed his wrists until\nhe began to moan, and then I pinched his nose until he was able to\nnotice my name when I whispered. He had bumped his head in fallin',\nand it made him sick to the stomach; so while he was gettin' tuned up\nagain, I prospected around.\n\nI crawled up the stairs but couldn't hear a sound, I scratched with my\nfingers, knocked softly, and pushed until my eyes began to hurt; so I\nknew 'at the only way out for us was to follow the Cross-branders.\nThings had happened so sudden up above that I hadn't an idy as to how\nmany were fightin' us; but I was still purty certain that a fair sized\nbunch had run out the tunnel just as I dove into it, and I didn't\nchoose to bump into 'em in the dark.\n\nWhen I came down the stairs, The felt able again; so we started to\nprospect. We agreed that strikin' our teeth together would be our\nsignal, and then we made our examination. The right side o' the tunnel\nwas smooth, the way Nature works, the left side was rough, and\nindicated man's doin's. Aside from us two, the only other one in the\ntunnel was the boy with the broken neck; but the tunnel opened into a\nbig cave, and we didn't know what to do about it.\n\nFinally we started around the right hand wall, me crawlin' first, and\nThe's fingers touchin' my boot at every move. After goin' some\ndistance, a great, straggly gray form rose up from the floor o' the\ncave, and gave me a shock which stopped my entire works. I kept my\npresence o' mind all right; but I'd 'a' been mighty glad to swap it\noff for absence of body. This was a most ghastly lookin' form, and I\nnestled up again' the side o' the wall, and felt my hand back for The.\nHe crawled up alongside o' me, and when he spied it, he gave a start\nwhich made his teeth click. \"What's that?\" he whispered.\n\nIt's funny how the mind works. This form didn't resemble anything\nearthly; so I hadn't really tried to figure on it much; but when The\nthrew his question at me, I looked at the shape more careful, and grew\nashamed o' myself. Here was I, a feller who had spent consid'able time\naround mines, and yet had got all balled up over seein' things\nunderground.\n\n\"That's your old friend, daylight, comin' down through a hole, The,\" I\nwhispered so prompt that I doubt if he noticed any gap.\n\nHe gave a sniff through his nose, and then we crept on to where this\nlight was comin' in through the opposite tunnel. It was mighty weak\nand sickly lookin' light, but the outline o' the tunnel mouth soon got\nperfectly plain to us. Every few inches we stopped to listen; but we\ngot clear to the mouth without hearin' anything. Then we paused. Just\nat that time, I'd have given right smart to have had my eyes fastened\non like those of a lobster I once saw in a window down at Frisco. This\ninsect had his eyes fixed to the ends o' fingers which he could\nstretch out in any direction.\n\nTo be honest, I felt some reluctant to push my face around that\ncorner; but when I did there wasn't a thing in sight. The tunnel\nstretched ahead of us for what seemed miles, but we couldn't see the\nouter openin', although the light was strong enough to recognize each\nother by. The was a sight, for the bump on his head had leaked\ncontinuous; but it hadn't disabled him none, so we drew back to\nconsult a little.\n\nIf we had known whether they were ahead or behind us, it would have\nbeen easy to decide; but under the circumstances, we hardly knew what\nto do. Bein' in the dark was one thing; but bein' out where we could\nbe seen was still another; so we thought full and deep.\n\nAfter a few minutes I told The a little story about a feller I helped\nto pick up after he had jumped from a thirty-foot ledge onto a pile o'\nstone. \"Why did you do it?\" sez I. He blinked his eyes at me a time 'r\ntwo, hove a long sigh, an' said: \"The' was a purple dragon in front o'\nme, a lot o' long-legged yaller snakes back o' me, and the peskiest\npink jack-rabbit you ever saw kept swoopin' into my face an' peckin'\nat my eyes. If I ever drink another drop, I hope it'll drown me.\"\n\nThe considered this story careful, an' then we crawled out into the\ntunnel, rose to our feet, an' ran along crouchin'. The tunnel ran\nupward at a sharp incline, which was why the light came down it so\nfar. We kept to the right wall, and after goin' some distance, we came\nacross a small cave. In this we found another dead Cross-brander; but\nwe weren't enough interested in him to risk strikin' a light; so we\nsat down a moment to rest and listen.\n\nPresently we noticed some curious noises, but for some minutes we\ncouldn't decide on what they were. Suddenly The grabbed my wrist an'\nsaid: \"That's shootin'; that's what that is!\"\n\nIt was as plain as home-cookin' the minute he pointed it out; so we\nrose to our feet and made a rush for the mouth o' the cave. We came\nout about half way up the face o' the cliff; and for a moment we\npaused to admire Ty Jones's foxiness. This openin' couldn't be seen\nfrom below, nor noticed from above, and for the most part the whole\ntunnel was natural, only havin' been hand-widened in three or four\nplaces.\n\nThe fightin' was goin' on near the face o' the cliff between us an'\nthe mouth of the ravine; so we circled around until we caught sight of\n'em. The first feller we made out was Mexican Slim; so we knew our\nboys hadn't been ambushed up above, and this raised our spirits like a\nballoon. We crept up until we could get good angle-shots, hid\nourselves, gave the old Diamond Dot yell, and began to shoot. Ty's men\nhad been losin' their bullet-appetite for some time, and they took us\nto be genuwine reinforcements. They were well planted where they were,\nbut they started to retreat, and we crowded 'em close.\n\nThen it was that Ty made Olaf's word good: he exposed himself to\nshots, he rallied his men, and that wolf-grin never left his face; but\nstill the tide had changed, and he had to go back with the rest. The\nwoman, with her hands tied behind her, was in charge o' the Chink, who\nwas tall and heavy-set with a dark, evil, leathery face. He kept a\ngrin on his face, too, which reminded me most of a rattlesnake at\nsheddin' time. He used the woman as a shield, an' this checked our\nfire an' kept us dodgin' for new positions. Still, all in all, this\npart o' the fight was about as satisfactory as any I ever took part\nin.\n\nFinally they retreated to the dip where the tunnel came out, and we\nhad to skirmish up the rocks to keep our vantage. Soon we discovered\nthat Ty had lost control of his men. He, Pepper Kendal, and two others\nstood in the mouth o' the tunnel, and took a few shots at us before\ndisappearin'; but six of his men ran straight across the dip, and down\nthe other side toward the crick. Tillte Dutch was standin' close to\nme, and I asked him where the hosses were. He said they were tied\nacross the crick just above the upper ford; so I sent him for 'em full\nspeed.\n\nHorace and Tank stayed to watch the mouth o' the openin', while the\nrest of us wrangled the six Cross-branders through the cottonwoods.\nThey had a good start, and so had time to cut the wire and cross the\ncrick toward some broken land on the left. By this time Tillte had\ntied the reins and thrown 'em over the horns o' the saddles so as to\nlead a string, and he came lopin' into view.\n\nSlim, two o' the Simpson boys, Olaf, and myself mounted and cut off\nthe six Cross-branders, who were too weary to even scatter. They had\nhad enough and surrendered. We tied their hands, and herded 'em back\nto the old shack, where Oscar, Spider, and three disabled\nCross-branders were runnin' a little private hospital. We fixed up\nwounds as well as we could, sat the last six on a bench along the\nwall, and left Dick Simpson to guard 'em. Spider had been shot and cut\nconsid'able; but he was able to stagger around some, while Oscar had\nbeen punctured below the ribs, and things looked bad for him. Olaf had\nbeen shot in the head, all right, just as The and I dove down the\nstairway the night before, but his skull was bullet-proof, so nothin'\ncame of it.\n\nThe Friar had been ransackin' the locality, and had found one o' the\nSimpson boys dead, and one badly hurt. Badly crippled, as we were, we\ndidn't see any way to get at Ty except to starve him out. First off,\nwe made some coffee, and those who weren't hurt dangerous were given\nsome side-meat and corn bread; for, truth to tell, we were about once\nthrough. We spent the afternoon under a tree half way between the\nmouth o' the tunnel, and the old cabin, so as to be handy in case we\nwere needed. After talkin' it all over, we couldn't quite see why they\nhad split up, some of 'em tryin' to escape, and some stayin' with Ty.\n\nFinally I went to the cabin, durin' a time the Friar was on watch at\nthe cave mouth, and picked out the weakest lookin' of the prisoners. I\nbrought him down, and we tortured him with questions until he got\nfuddled and told us that the two who had stuck to Ty had been so bad\nhurt, they couldn't go any farther; but that neither Ty nor Pepper\nwere hurt to speak of.\n\nThe fact is, that in a general fight a feller loses his aim complete.\nWe had all aimed at Ty and Pepper the most, and here they were the two\nnot hurt at all. As darkness fell, the Friar couldn't hold himself in.\nAll afternoon he had done what he could for the wounded; but at\nthought of the woman spendin' another night in the cave with those\nmen, he became as wild-eyed as a bronc at his first brandin'. Durin'\nthe afternoon, Tank had stiffened until he couldn't do much travelin';\nbut I saw the Friar had his mind made up to take a plunge, so I tried\nto fix things to prevent it.\n\nOlaf, two o' the Simpson boys, Promotheus, Tillte, Slim, Horace, and\nmyself lined up as bein' still in workin' order; but while he was in\nthe act of claimin' to be all right, Slim doubled up in a faint, and\nwe found he had been bad hurt without even himself knowin' of it; so\ncountin' Horace who had two black eyes and a shot through the\nfore-arm, the' was seven of us able to get about purty nimble. Hid\naway in the cave, somewhere, were Ty Jones, Pepper Kendal, and the\nChink, unhurt so far as we knew, and two others, still probably able\nto help a little.\n\nWe placed a couple o' logs again' the fake drawers in the library, and\nleft Tank to take charge of the prisoners and the cabin. Then we\nrustled up some tarps from the bunk-shack, and prepared to camp near\nthe openin' with a man allus on guard, to prevent them from comin'\nout--and the Friar from goin' in. We kept a lantern lit under shelter\nof a rock, and made ready to rest up a bit.\n\nI had told all the fellers to watch the Friar close, for he just\nsimply couldn't get the upper hand of himself. He tried his best to\nsimmer down and go to sleep, but every few minutes he'd boil over\nagain. I lay awake in my tarp watchin' him for some time; but I was so\nsore and weary myself I could scarcely recall what business I was on,\nand first I knew I had drifted off--and been shook awake again.\n\nPromotheus was bendin' over me with the news 'at the Friar had decided\nto go into the tunnel, and they couldn't hold him back. I sprang up\nand started for the opening with the rest following me. Dan Simpson\nhad relieved The on watch and when he found what was in the Friar's\nmind, he had crept down and told The, who had awakened the rest of us.\n\nWe reached the Friar, just as he was goin' into the openin'. I called\nto him in a low tone; but he only shook his head. It was eleven\no'clock, and the shadow from the moon had already crept out from the\nbase o' the cliff almost to the openin'. I saw that the Friar had took\nthe bit; so I whispered to the others: \"I am goin' in there with him;\nbut more 'n this would be bad. We'd be in each other's way. Listen and\nwatch, but do not follow us in.\"\n\n\"I know the way as well as you, and we could keep side by side,\" sez\nPromotheus; but I shook my head.\n\nHe came over to the openin' and said in a low tone: \"I haven't time to\nmake you understand; but--but I just have to go in with you.\"\n\n\"If you come, the rest'll come too,\" sez I, exasperated.\n\n\"You fellers stay here,\" sez he to them in a pleadin' tone; \"but I\nhave reasons. I just have to go in.\"\n\nSo we shed our boots and started down the incline after the Friar,\nPromotheus touchin' my feet with his fingers at every step I crawled.\nI didn't want to be there, I couldn't see how we could do any good;\nbut the Friar had made my world for me, such as it was, and I\nunderstood better 'n the rest what was gnawin' at his heart; so I\nhadn't any choice. I had to go in, and somethin' inside Promotheus\ndrove him in also. The only crumb o' comfort I could find, lay in the\nfact that Horace had been winged, and so couldn't foller us, whether\nhe wanted to or not.\n\n\n\n\nCHAPTER FORTY-TWO\n\nHAND TO HAND\n\n\nAt first it was black as pitch; but I crawled as fast as I could in\nthe hope of catchin' up with the Friar. It is instinct with most men\nto follow the right wall when goin' through a strange place in the\ndark, though I never could see why. A man carries his weapon in the\nright hand and naturally ought to be as free with it as possible.\nStill, most men do it, so I follered the right wall, hopin' each time\nI put out my hand it would touch the Friar.\n\nAfter a time, I saw a faint glimmer o' light to the left, and I\nstopped and pointed it out to The. We came to the conclusion that they\nhad a candle lighted in the offset where we had come upon the body,\nand we discussed whether they were likely to be in there, or had gone\non farther back and left the light to see any one who tried to crawl\nafter 'em. I held out 'at they wouldn't expect any one to crawl after\n'em; but The said 'at Ty would be likely to go into just such a place\nhimself, and so would expect others to do the same. Ty certainly had\nthe way of impressin' his own men.\n\nWhen we got a little closer, I lay flat and scanned along the floor,\ntryin' to make out the Friar between me and the light; but I couldn't\nsee him, and we went on again. I hope I may never have to do any more\nsuch work as this. Creepin' along in the dark eats up a feller's nerve\nlike a forest fire.\n\nWhen we got so close 'at I could see my hands by the light, I sent The\nacross to the other side, remindin' him to knock his teeth should he\nchance upon the Friar, or in case we come together again, ourselves.\n\nThen I lay flat with my hat down low, and nudged myself along with my\nelbows and toes. I couldn't even make out The across the tunnel, which\nwas only about twelve feet wide, and just for the fraction of a second\nit came across me that he had formerly been a Cross-brander, himself;\nbut this thought didn't live long enough to draw its second breath.\n\nFinally I reached the spot where the light threw a splash on the walls\nand floor, and I made my gun ready and stuck out my neck in what was\nthe most breathless silence I ever tried to listen to. Across the\nsplash o' light in front of me, all was a solid wall o' darkness; and\nI'd have paid over quite a sum to know what eyes were lookin' out of\nit.\n\nFarther and farther I pushed myself into the light without seein' a\nthing; until finally I saw the candle, itself, and beside it--the\nFriar.\n\nI wriggled across the tunnel just as The crept into the room from his\nside, and we felt a little better to be in the light, together again.\nThe body still lay again' the wall, and The looked at the face; but he\ndidn't know it. The Friar hadn't seen or heard anything, either; and\nwe were up a tree to the top branches. We talked it all over, tryin'\nto imagine what we would do under the same circumstances, and finally\ndecided they had gone on down the tunnel, leavin' a man on guard just\nbelow the light, and that the man had gone to sleep.\n\n\"Well,\" sez I after we had discussed things around in a circle for a\nwhile, \"here we are holed up again, as cozy as a cavey o' rats with\ntraps set at all the openin's and en-thusiastic terrier dogs diggin'\ndown from above. If it's not bein' too inquisitive, Friar, what plan\ndid you have in comin' down here?\"\n\n\"I wanted to be close to her,\" sez Friar Tuck. \"I kept thinkin' o' how\nlonely it must be for her through the dark, and I hoped the' might be\nsome chance o' helpin' her to escape. I did not have any definite\nplan--only faith and hope.\"\n\n\"Like the shark which swallered the parasol,\" sez I, for I was\nconsid'able put out; \"he had faith in his digestion and hoped the\nparasol was some new sort o' health-food. But to get down to\nfacts--Have you any weapon with you, and are you willin' to fight?\"\n\n\"I have no weapon,\" sez the Friar; \"but I am willin' to do whatever\nseems best. I am trusting in the same power which upheld Gideon, and I\nask to see no farther than he saw.\"\n\nThis was the Friar all right, so I merely swallowed a couple o' times\nand didn't say anything. Whether he lived or died was the same to the\nFriar, as whether he lived in Idaho or Montana would be to another\nman; so I saved myself a certain amount of irritation by just thinkin'\nquietly as to what was best for us to try. Fact was, I didn't take, as\nmuch stock in Gideon just then as I did in Ty Jones.\n\n\"I'll tell you what I think is best,\" I sez after a bit; \"for me to\ncrawl down the hall in the hope that the watcher really has gone to\nsleep; while you two stand ready in this offset. If they chase me,\nI'll run up the tunnel, and you spring out and take 'em at a\ndisadvantage as they go by.\"\n\nO' course they both wanted to do the crawlin', but it was my plan, so\nI stuck out for it, and started. I was really glad to be out o' the\nlight again, and I crawled as gentle as though crossin' a bridge of\neggs. Before long my fingers struck a boot, and I felt of it\nex-treme-lee careful. If ever I go blind, my experience durin' those\ndays will help consid'able in transferrin' my eyesight to my fingers.\n\nThe feller had toppled over again' the right wall, and I crept up\nalongside, holdin' my gun by the barrel, and ready to swat his head as\nsoon as I had located it; but the' was no use--the man had already\ndied. He had been shot twice, but they thought he could last a while\non guard, and this was why we had been able to cross the lighted\nplace.\n\nJust beyond this, I came upon another offset, on the opposite side\nfrom where the candle was. We hadn't noticed it that mornin' 'cause we\nhad gone out along the other wall. I heard some heavy breathin' in\nhere; but I also heard some one tossin' about an' mutterin', and I\nhardly dared risk an examination. I looked back at the splash of\nlight, and it seemed mighty cheery and sociable, compared with the\ndarkness and company I was in.\n\nIt's astonishin' the way pictures fly across a feller's mind at such a\ntime: I saw the boy down at the foot of the stairs, I saw him as he\nmust have been, a few years before some quick, rash deed of his had\ndrawn a veil across the laughter in his eyes; I saw the feller in the\noffset, and wondered how much it had taken to turn the expression of\nhis face into that beastlike hunger for revenge, and then dozens of\nschemes and plans for capturin' Ty began to flash upon me; but each\ntime, the presence of the woman spoiled everything. They had used her\nfor a shield once, they would do it again, and I couldn't see a way to\nget around her.\n\nWe knew 'at Ty had vowed he would never be taken alive; and I couldn't\nsee what we would do with him even if we did take him alive; but I\ncould see that he would take pleasure in draggin' as big a bunch into\nthe next world with him as possible, and yet every scheme 'at came to\nme was blocked by the presence of the woman. Finally I crept a little\nway into the offset. My hand touched a piece of cloth, I felt over it\nwith nothin' except the ridges on my fingers touchin'; but just when I\nmade sure it was the Chink, he moved and sat up. I stopped breathin';\nbut after a minute, he sighed and settled back.\n\nI waited a little longer and then crawled back and told what I had\ndiscovered. \"If the' was only some way we could throw a light into\nthat offset,\" sez I, \"I think we could fix 'em.\"\n\nWe studied over this for some time before the Friar thought up a way\nwhich seemed worth tryin'. I said I'd go back and stay at the far side\no' the openin', and when they brought the rope back, to come right on\nwith it along the left wall, and I'd knock my teeth together to show\nit was me--provided I was still there and able. So the Friar pulled\noff his boots, and The kept watch in the offset while the Friar ran\nback. I thought it must be several days since we'd come in, but he\nlooked at his watch before startin', and it was only two o'clock.\n\nFrom where I was, I could make out the shape o' the feller they had\nput on watch, and knew I could keep cases on all within the little\nrock room. After an age, I saw two forms creep like ghosts out of the\ndark beyond the candle, and ooze into the offset without makin' a\nsound. Then in a moment, Promotheus came stealin' along the wall with\nthe end of the rope. I made my signal to him, and he went on down the\ntunnel, slowly pullin' the rope after him.\n\nI was mighty curious to see how they had fixed the lantern, which they\nwere to light with the candle in the offset, and it made me feel a lot\nbetter when it came out of the recess. Horace had done the fixin', I\nafterward found out, and it had nearly broke his heart not to come in\nwith it; but he realized that it was necessary to have an outer guard,\nso he had stayed with the two Simpson boys. He had put the lantern\ninto a box after nailin' a couple o' short pieces of rope on the\nbottom for runners; and now it came slidin' along without makin' a\nsound. He had sawed a piece out of the side, so that all the light\ncame up again' the ceilin', and onto the side where the openin' was.\n\nSlowly it came along, and I stood in the shadow watchin' it. Finally\nit fell on the face of the man lyin' near the openin', and I saw he\nwas one of those who had been at Skelty's that night--for all I know,\nit was his hand I had seen raisin' the window to my room. Next, it\nlighted up the openin', itself; and then The stopped pullin' and crept\nup opposite me. We heard 'em sighin' and groanin', in the recess, and\nfinally the woman's voice gave a weary moan as she came awake.\n\nIn a second, Ty's voice was heard, askin' what was the matter; and we\nall braced up our nerves. A weak, delirious voice started to babble,\nbut it was broken by a shot, and a bullet ripped through the box, but\nwithout puttin' out the light. I started across the hall; but The had\nalready seen it, and had taken the rope and ran down the tunnel with\nit. He turned the box, so 'at just the left edge o' the light touched\nthe openin', and then came across to my side. We weren't in a black\nshadow now; but still, with the light in their faces, it would have\nbeen hard to see us.\n\nA hand reached out of the openin', and fired in our direction, I\ndropped to my knee and aimed at the hand, but neither shot counted;\nand for the next few minutes, all we heard was that weak voice,\nbabblin' indistinctly. It hadn't worked out as I thought it would. I\nfigured that they'd be surprised when the light shone in their faces,\nand would rush out and give us a chance. Now that it was too late, I\nthought up half a dozen better schemes.\n\nEven while I was thinkin' up a perfect one, I saw a form come out from\nthe recess, and threw my gun up--but I didn't snap the hammer. It was\nthe woman, and behind her I could make out the shaved head o' the\nChinaman.\n\nWe all stayed silent for some time, an' then Ty's voice said: \"Well,\nwhat kind of a settlement do you fellers want?\"\n\nHe spoke as self-composed as though puttin' through a beef-dicker, and\nno reply was made for several seconds. Then, as no one else spoke, I\nsez: \"All we want is just the woman and what's left o' your outfit,\nTy.\"\n\n\"Who's that speakin'?\" sez Ty.\n\n\"He's generally called Happy Hawkins, Ty,\" sez I.\n\n\"Who's in charge o' your gang?\" sez he.\n\n\"Dinky Bradford,\" sez I after thinkin' a moment; \"but I'm delegated to\nspeak for him.\"\n\n\"Tell ya what I'll do,\" sez Ty; \"I'll trade ya the woman for Dinky\nBradford an' the Singin' Parson. Send those two in to me, and I'll\nsend her out to you.\"\n\nThis was the foolest proposition ever I heard of. The woman wouldn't\n'a' been any use to us without the Friar. \"Dinky Bradford is guardin'\nthe mouth o' the tunnel,\" sez I; \"but he wouldn't stand for any such\nnonsense, nohow.\"\n\n\"Is the preacher here?\" asked Ty.\n\n\"Yes, I am here,\" sez the Friar, steppin' out from the offset and\ncomin' toward us. Olaf, who was with him, caught his arm and kept him\nfrom exposin' himself.\n\n\"Damn you,\" sez Ty, slow an' deliberate. \"I hate you worse 'n any man\nin this territory. You're at the bottom of all this kick-up. You're\nthe one which has turned my own men again' me; and all I ask is a\nchance to settle it out with you.\"\n\n\"You're mistaken if you think that I advised this method,\" began the\nFriar; but Ty broke in, and said: \"Never mind any o' that\npreacher-talk. I know what's what, and I'm all prepared to have you\nhide behind your religion, after havin' started all the trouble. I'll\noffer you a plan which any man would accept--but I don't class you as\na man. The fair way to settle this would be for the men who are with\nus to empty their guns an' lay 'em on the floor, then you and me strip\nto the waist an' fight it out with knives. They haven't anything at\nstake; but I suppose you'll be true to your callin', and make them\ntake all the risk.\"\n\n\"I want to be true to my callin',\" sez the Friar; \"and fightin' with\nknives isn't part o' my callin'.\"\n\nTy laughed as mean as a man ever did laugh; and both Olaf and I\noffered to take the Friar's place; but Ty said he didn't have anything\nspecial again' us any more 'n he'd have again' the Friar's ridin'\nhoss; and then he offered to fight the Friar and Dinky Bradford at the\nsame time.\n\nHe kept on roastin' the Friar till I bet I was blushin'; but the Friar\njust stood out straight in the gloom o' the tunnel and shook his head\nno. Then the woman took a half step forward, an' the Chink jerked her\nback, twistin' her wrist and makin' her give a smothered scream.\n\nI had moved the box around to give us a little more light; and when\nshe screamed, I saw the blood rush up the Friar's pale face to his\neyes, where it burst into flame. Livin' fire it was, and in a flash it\nhad burned away his religion, his scruples again' violence, the whole\nouter shell o' civilization, and left him just a male human with his\nwoman in the power of another. \"Strip,\" he said, and his words rolled\ndown the tunnel like a growl of a grizzly. \"Strip, and fight for your\nlife, for I intend to destroy you.\"\n\nI can still hear the laugh Ty gave when the Friar said this. \"Destroy\nme?\" he said. \"Destroy me? That's a good one! Now, do your men agree\nto let us go free if I win?\"\n\n\"I do,\" sez The.\n\n\"I do,\" said I, after I'd taken another look at the Friar, who was\nalready unbuttonin' his shirt.\n\n\"I do--if you fight fair,\" said Olaf slowly.\n\n\"Then one of ya hold the lantern while we empty the guns,\" said Ty.\n\nI didn't like this part of it; but couldn't see any way out; so while\nThe held the lantern, one on each side emptied a gun and tossed it to\nthe center of the tunnel. We emptied all of ours, and they emptied all\nof theirs, and then while Ty was takin' off his shirt, I went up to\nthe Friar. When I saw the taut muscles ripplin' beneath his white\nskin, I felt comforted; but when I saw him holdin' his knife point\ndown, the way they do in the picture-books, I got worried again.\n\n\"Take your knife the other way, Friar,\" I whispered; \"and strike up\nunder the floatin' ribs on his left side. That's the way to his\nheart.\"\n\n\"I know how to fight with a knife,\" he snapped; so I didn't say any\nmore. Horace had become a gun-fighter, here was the Friar claimin' to\nknow the knife game, and if the woman had stepped out and challenged\nthe winner to a fight with stones, why, I was so meek I wouldn't 'a'\ngot het up over it.\n\nThen Ty Jones came out of the other offset, stripped to the waist also\nand holdin' his knife in his left hand. The woman had gone into the\nniche on our side, me an' Olaf leaned again' our wall, Pepper Kendal\nand the Chink leaned again' the wall opposite us, The held up the\nlantern, and for a full minute the only sound was the wounded\nCross-brander, babblin' out his delirium back in the cave-room.\n\nTy was a shade beefier 'n the Friar; but his skin was dull, and the\nmuscles didn't cut off into the tendons so sharp, nor they didn't seem\nquite so springy or well oiled; but there was half a dozen knife scars\non his chest, and he had come up our way from Mexico.\n\nThey walked toward each other, Ty's eagle eyes an' wolf-grin tryin' to\nbeat down the grim set to the Friar's face. They both crouched over\nan' circled about each other like a pair o' big cats. Ty made a few\nlunges, but the Friar parried 'em as simple as though it was a game,\nand purty soon Ty was forced to slip his knife to his right hand with\nthe blade pointin' up for a rip. When he did this, the Friar smiled,\nturned his own knife the same way; and I recalled the Friar havin'\ntold me about learnin' knife tricks from an I-talian he had helped\nback East.\n\nI don't like knife fightin', and I don't approve of it; but I will say\n'at this fight was the cleanest, quickest thing I ever saw. The Friar\nwas the best man, but Ty was the best posted; and time and again the\nFriar saved himself by foot work. The follered 'em close with his\nlantern, while Olaf and I kept a half watch on the two opposite us.\n\nThey kept movin' faster and faster and the' was a continuous spattin'\nas they parried with their left hands. Finally the Friar grabbed Ty by\nthe wrist, Ty grabbed the Friar's wrist at the same time, lowered his\nhead, and butted the Friar in the pit o' the stomach. It looked bad;\nbut the Friar had raised his knee and caught Ty on the chin; so they\nstaggered apart and breathed deep for a minute, before beginnin'\nagain.\n\nThe grin had left Ty's face, and it had settled into black hate. When\nthey began again, the Friar seized Ty's wrist every chance he got,\ntwistin' it, bendin' the arm, and tryin' to thrust with his knife; but\nTy was tough and wiry, and managed to twist out every time. At last\nthe Friar caught Ty's right wrist, dropped his own knife, ran his head\nunder Ty's right arm, caught the slack of his right pant leg, gave a\nheave and threw him over his head. It was a clean throw and the Friar\nstooped, picked up his knife and started for Ty before he had time to\nget to his feet. Ty rolled to his feet and dodged away as though to\nrun, whirled, took the blade of his knife between thumb and\nforefinger, and spun it through the air. It struck the Friar's\ncollarbone, cut a gash through his shoulder, and twanged again' the\nwall o' the tunnel.\n\nThe two men eyed each other for a moment, the calm of victory in the\nFriar's eyes, the red of baffled hate in Ty's. They were about eight\nfeet apart. \"Will you give up?\" asked the Friar.\n\n\"No,\" sez Ty. He doubled up his fists as though to spring, then\nwhirled and stepped into the offset behind him. In a moment, he came\nout with a gun in his hand.\n\nAs soon as he had said no, Pepper Kendal an' the Chink had made a dive\nfor the offset, and Olaf and I had made a dive for them. I got Pepper\nwho was old and stiff, and I managed to hit him in the center o' the\nforehead just as Ty came out with his gun. Olaf was havin' trouble\nwith the Chink, and I picked up a gun and tapped Pepper on the head\nwith it, and then turned to knock the Chink. Just as I turned, I saw\nthe woman walkin' slowly down the tunnel behind the Friar, and I saw\nTy bend his gun on him. Even then he had to pause a moment to enjoy\nhis deviltry, and I still see that picture in my dreams--the Friar\nstandin' silent and proud, with his head thrown back and his level\neyes full on Ty, while back of him stood the woman as unconcerned as a\nsnow-bird. About six feet beyond 'em stood Promotheus holdin' the\nlight above his head, while his face seemed frozen with horror.\n\nFor an instant they stood like stone images. Then The lunged forward\nand caught Ty's arm, the lantern went out, I heard one clear report,\nand one muffled one, and then I started for 'em. I bumped into a heavy\nform, two naked arms went around me in a bear-grip, and we rolled to\nthe floor. The candle in our offset had burned out; but I knew it was\nthe Friar, 'cause his was the only smooth face among us. \"This is\nHappy,\" I muttered, and we rose to our feet.\n\nA struggle was goin' on beyond us, and I thought it was Olaf and the\nChink; so I lit a match, knowin' that Ty would 'a' had plenty o' time\nto get away already. As the match burned up, I saw the Chink lyin'\nstretched out, and Olaf and Ty locked together. Olaf had his leg\nwrapped around Ty's, and was bendin' his back. Ty's eyes were stickin'\nout white an' gruesome, and he was gurglin' in the throat. Suddenly,\nsomethin' cracked and they both fell to the floor o' the tunnel just\nas the match went out.\n\nI heard hard breathin', and then Olaf's harsh voice came out o' the\ndarkness. \"Well,\" he said, \"I guess that squares things.\"\n\n\"What's happened, what's happened?\" asked a panting voice, and then I\nknew 'at Horace hadn't been able to stand it any longer, and had come\nin, game wing and all.\n\n\"We've settled up with Ty Jones--that's what's happened,\" said Olaf;\nand as we stood there in the gloom, the drip o' the dawn came rollin'\ncold and gray down the slant o' the tunnel; and I shuddered and turned\naway to find somethin' for my hands to do.\n\n\n\n\nCHAPTER FORTY-THREE\n\nTHE GIFT OF THE DAWN\n\n\nThe first thing I did was to light the lantern, for the daylight which\ncame down there was too much in keepin' with the conditions to suit\nme. Promotheus was doubled up an' holdin' his side; so the first thing\nI did was to ask him if he was bad hurt. The' was a smile on his lips,\na regular satisfied, self-composed smile, but I didn't just like the\nlook in his eyes.\n\n\"Nope, I don't ache at all, Happy,\" he said in a firm voice; \"but I\ncan't move much. Tend to the others first.\"\n\nIt seems 'at Ty's first shot had hit the woman in the head, and his\nnext had got The in the side--but The had managed to get the gun away\nfrom him, which is why the rest of us were spared.\n\nThe Friar had carried the woman into our offset, and was rubbin' her\nwrists and workin' over her, though the' didn't appear to be much use.\nShe was still alive; but that was just all, so I left them and\nexamined the rest. Ty was all twisted out o' shape, and lay with his\neyes open, glassy an' stary and horrible. Olaf hadn't had time to\nquite finish the Chink, and he was crawlin' down the tunnel when I\nnabbed him. Then Horace took the lantern while Olaf and I hog-tied\nPepper Kendal and the Chink.\n\nWe next examined the cave-room where Ty had made his last stand. It\nwas fair-sized an' well stocked, and also had half a dozen extra guns\nin it. When I saw these fresh guns, I gave a low whistle to think what\na lot o' suckers we'd been to discard our own trumps and set in a game\nagainst a marked deck; but as the Friar allus said: \"Wrong feeds on\ndeath and Right feeds on life; so the' can't be no doubt as to the\nfinal result, even though things do look blue sometimes.\"\n\nThere was a fine spring in the corner o' this room--the same spring\nwhich afterwards came out near the mouth of the ravine and was piped\ninto the old cabin. The wounded Cross-brander was still babblin', so\nwe fed him some water and eased him around a little.\n\nNext we went outside and nailed some pieces to a couple o' light\npoles, and we were mighty glad to have enough left to man this vehicle\nwhen it was finished, for we were all purt nigh used up, Tillte, the\ntwo Simpson boys, and myself carried the litter, while Horace ran the\nillumination, and Olaf tended to Pepper and the Chink.\n\nWe took 'em all out, even to the dead; and the one at the foot of the\nstairs turned out to be the boy, just as I'd thought. Next to the\nwoman, with the Friar walkin' beside her his head on his breast, this\ntrip with the boy cut me worse 'n any. Promotheus got off three\naverage good jokes while we were packin' him out, and cheered us up a\nlot; but we put Ty Jones down with the dead. As we straightened him\nout he gave a groan which made us all jump. The whole thing had become\na nightmare, and we staggered about like the ingredients of a dream.\n\nThe woman's head was shattered on top an' the' wasn't any hope for\nher; but still, it gave the Friar comfort to work over her, so we\nacted as though we thought she had a chance. The nearest doctor was at\nMeltner's stage station, a full day's ride. Tillte went after him,\nwhile Dan Simpson rode over to his father's to break the news and\nbring back Kit. What with the prisoners still on our hands, the dead\nto bury, and the wounded to wait on, we were in chin-deep; and the\nworst of it was, 'at we didn't want the news to get out. We had tried\nto settle things without botherin' the law, and we preferred to finish\nthat way if possible.\n\nWe buried the four Cross-branders across the crick and down stream\nfrom the lower ford, and we buried Tim Simpson just a little way above\nthe upper ford. The Friar went along and helped dig the graves and\ncarry them to it; but he didn't preach nor sing, and his face was\ndrawn with sorrow.\n\nBy evenin' we had got things to some system. Spider, Tank, Slim, and\nHorace were able to help quite a little; but Oscar, Tom Simpson, and\nPromotheus were in bad shape; while we had seven prisoners, countin'\nthe Chink, and seven wounded enemies to look after. The feller Horace\nhad shot, up on top, got out o' the country, I reckon. Anyway they\nleft him above with the horses, and we never heard of him again.\n\nOl' man Simpson, Kit, and the boy arrived durin' the moonlight, and we\nwere all mighty glad to see Kit, though we hated to face the old man.\nStill, he was game, and took it mighty well. Tillte had got a fresh\nhoss at Meltner's and had started right back with the doctor; so they\narrived a little after seven next mornin'. The doctor was purty young\nlookin' to me; but he had a bagful o' shiny instruments, and he made\nhimself at home without any fuss. He had been in a Colorado hospital\nfor two years, a minin' hospital, and he was as familiar with a\nfeller's insides, as a pony is with the range he was foaled on. He had\ntook a claim near Meltner's, and was able to talk a long time on why\nit was better for a young doctor to come west.\n\nHe praised the Friar's work to the skies--and then turned in and did\nit all over to suit himself. He said that all the wounded stood a good\nshow except the woman, Promotheus, and Ty Jones. We none of us thought\n'at The was in much danger; but the doctor shook his head. Ty's spinal\ncolumn had been unjointed near the base, and he was paralyzed from the\nhips down; but in all that skirmishin', he was the only one who hadn't\nlost a drop o' blood. The Friar, himself, had two flesh-wounds beside\nthe one Ty had give him.\n\nI was with the doctor when he started to work on the woman's head; but\nI couldn't stand it. I'm not overly squeamish; but I own up I couldn't\nstand this; so I backed out, leavin' the Friar with his face like\nchalk, to hand instruments while little old Kit held a basin. I hated\nto leave 'em; but I didn't take a full breath until I was beside\nPromotheus again.\n\nHis voice had got weaker, but the smile never left his lips, and it\nwas restful just to sit and watch him. Horace hovered over him like a\nyoung hen, and The drank so much water, simply to please Horace, that\nI feared his bones would dissolve. Horace had told the doctor he would\npay all the bills, and to go the full limit and not try to economize\nnone on his patch-work. We put the seven prisoners in the workshop,\nand slept in tarps around the door, which was fastened with a chain,\nso 'at if they got it open, a board would fall on these sleepin' next,\nand wake 'em.\n\nThe Friar was all for notifyin' the authorities; but old man Simpson\nhad been a notorious public, or some such official, back in Vermont\nand naturally he was up on all the twists and windin's of the law. He\nsaid it would take the Su-preme Court itself fifteen years to sift out\nthe actual legalities of our tangle; and even then he wasn't sure\nwhich side would get the worst of it, so he advised us to just work it\nout on our own hook, which we had decided to do anyway.\n\nFor three days, the woman lay in a stupor. Kit had told me that her\nskull hadn't been actually shattered--that she had been shot in just\nabout the same way that Olaf had, but that Nature had counted on Olaf\ngettin' into some such a fix, and had provided for it by givin' him a\nflint skull, while the woman's skull wasn't of much use except in\ntimes of peace. Kit said the doctor had taken out a few splinters of\nbone, and had fastened up the openin', but had said the' wasn't any\nshow for her.\n\nOn the other hand, Olaf had looked at her careful, and had said that\nall the vital part of her was workin' on just this point. He said that\nthe light about her body was the blue o' weakness; but that just at\nthis point, the' was a constant bulgin' out o' different colors in a\nway he had never before seen. The doctor heaved up his eyebrows at\nOlaf's verdict, and looked as though he thought perhaps Olaf's brain\nhad been shifted a little out o' line, in spite of his flint skull.\n\nOn the third night I was what the doctor called his orderly, and went\non duty at midnight. I was sittin' out on the porch of the old cabin\nwhen the Friar came out holdin' his hand across his eyes. We had moved\nthe wounded men over to the bunk-shack, and the woman was in Ty's\nbedroom. I didn't speak to him, and he stood leanin' against one o'\nthe posts for some time without seein' me.\n\nHe trembled all over, and his breath came quick and catchy. Finally he\nlooked up at the stars and said in a low tone, as though speakin'\npersonal to some one near at hand: \"Save me, oh God, from mockery! I\nhave spoken for others in my vanity; and now that my own hour has\ncome, oh save me from the rebellion of my flesh; and give me grace to\nsay in my heart, Thy will be done.\"\n\nAs he stood with his face upraised, the late moon crept out and shone\nfull upon it, and the agony in it struck me like a blow; but even as I\nlooked, the change came. Before my very eyes, I saw the sign of peace\nmade upon the Friar's brow. A moment before and it had been torn into\nwrinkles and covered with beads of sweat; but now it was smooth and\ncalm. He clasped his hands across his breast, closed his eyes, and\nthe' came a smile to his lips which drew a mist to my own eyes. I\ncan't be absolutely certain of it, because o' this blur in my eyes;\nbut I think, I actually and honestly do think, that I saw white forms\nhoverin' in the moonlight above him.\n\nHe drew a full breath and turned to go in, but saw me settin' with my\nback again' the wall o' the cabin, and came over and put a hand on my\nshoulder. I couldn't say anything. I wanted to say somethin' to\ncomfort him; but I couldn't speak a word, until he asked me how the\nothers were gettin' along. I told him they were all doin' fine, and\nthat even Ty had been restin' well. He turned to go in, and then I\nfound the nerve to ask him how things were inside.\n\n\"It is all over, Happy,\" sez he, without even a catch in his voice.\n\"Just before I came out here, the doctor said the pulse had stopped.\"\n\nHe caught his breath with a little gasp at this; but that was all.\n\"What did Olaf say?\" I asked.\n\n\"Olaf says that she still lives,\" he answered; \"but I fear that Olaf\nis not to be relied upon this time. He has a strange gift; but he does\nnot understand it himself, and while I know he would not deceive me, I\nfeel that the doctor must know best.\"\n\n\"Well, I'll not give up until Olaf does!\" I blurted.\n\nHe smiled again and put his hand back on my shoulder. \"Come in and\nlook at her,\" he said, \"she is very beautiful. The strange mask has\nfallen from her face, and she is once more as she was in those old,\nhappy days when we walked together through our own Garden of Eden.\nCome in, I want you to see her.\"\n\nI went in with him, though I didn't want to. I knew what love did to a\nman, and that I hadn't seen the same woman he had; but the' was\nanother face allus before my eyes, and no one else was beautiful to\nme. I didn't want to do any pertendin' to the Friar, even at such a\ntime as this.\n\nI follered him inside, feelin' out o' place and embarrassed; but when\nI looked down at the quiet face in the bed, I was glad I had come. She\ndidn't look like the same woman, not at all. All the weary, puzzled\nexpression had left her face, and in spite of its whiteness, it looked\nlike the face of a girl. I looked at her a long time and the thought\nthat came to me over and over was, what a shame she couldn't have had\njust a few words with the Friar before she was called on; just a few\nwords, now that her right mind was back.\n\nAfter a time I looked up. Kit sat near the head of the bed, leanin'\nover and holdin' a handkerchief to her eyes, Olaf sat near her, a\nstrange, grim set to his lips. His head was bandaged and he looked\nless like a human than usual, as he kept his eyes fixed on the white\nface o' the woman. The' was a lamp on the stand and I could see his\neyes. Blue they were, deep blue, like the flowers on the benches in\nJune, and they didn't move; but kept a steady gaze upon the white,\nstill face. The doctor sat in a corner, his eyes on the floor. At\nfirst I thought he was asleep, and goodness knows, he was entitled to\nit; but just as I looked at him he rubbed his fingers together a\nmoment and stood up.\n\nHe walked over and put his hand on the Friar's shoulder. \"You might as\nwell all go to sleep, now,\" he said, gently. \"There is nothing more to\ndo.\"\n\n\"Are you positive?\" asked the Friar.\n\n\"Positive,\" said the doctor. \"There is no heart action, and when I\nheld a mirror to her lips no vapor was formed.\"\n\n\"She is still alive,\" said the deep voice of Olaf, and we all gave a\nlittle start.\n\nThe doctor took a silver quarter and held it to the woman's nose for a\nminute, and then looked at it. A puzzled look came to his face, and he\nwent back and sat down in the corner again.\n\n\"Was it discolored?\" asked the Friar.\n\n\"No,\" sez the doctor slowly; \"but I am sure there is no life\nremaining. I have seen several cases of suspended animation, but\nnothin' like this.\"\n\n\"She lives, and the light is getting stronger,\" said Olaf.\n\nKit took the handkerchief from her eyes which were still full o'\ntears. She wiped them away, and looked first at the woman and then at\nOlaf, and then she gave a sigh. The Friar's hands were opening and\nshutting. He had fought his fight out on the porch; but the suspense\nwas beginnin' to undermine him again.\n\nI went back to the porch and stayed a while. When I went in again,\nthey were all as I had left them; and after a few minutes I made my\nrounds, found everything all right, and came back. I went into the\nroom several times, and just as I caught the first whiff o' the dawn\nbreeze, I went in once more, determined to coax the Friar to lie down\nand try to sleep.\n\nThey were still in the same positions. Not a line had changed in the\nwoman's face, the Friar was almost as white as she was but still stood\nat the foot o' the bed lookin' down at her; while the wrinkles on\nOlaf's set face seemed carved in stone.\n\nI had just put my hand on the Friar's arm to get his attention when\nOlaf rose to his feet, pressed his hand to his blinkin' eyes, and said\nwearily: \"The blue color is givin' way to pink. She will get well.\"\n\n\"Don't say it unless you're sure!\" cried the Friar, his voice like a\nsob.\n\nFor answer Olaf pointed down at the woman's face. A faint color stole\ninto her cheeks, and as we looked her eyes opened. The first thing\nthey rested upon was the Friar's face bent above her, and her lips\nparted in a wonderin' smile--a smile which lighted her face like the\nmornin' sun on ol' Mount Savage, and made her beautiful, to me an' to\nall who've ever seen her.\n\n\"Is it you?\" she whispered. \"Is it really you?\"\n\nA warm, rosy beam of sunshine slipped in through the window and fell\nacross the bed, and the rest of us tiptoed out, leavin' the Friar\nalone with the gift of life which the Dawn had brought back to him.\n\n\n\n\nCHAPTER FORTY-FOUR\n\nTY JONES NODS HIS HEAD\n\n\nIt was a week after this before Olaf could see properly again. The\ndoctor was wild to take Olaf back East and hold doin's with him; but\nOlaf wouldn't listen to it. He hated to have people take him for a\nfreak, and said it wasn't any fault of his that he saw the way he did.\nThe doctor said 'at what Olaf saw was called the aurora; he said that\nscience had been tryin' to locate it, but hadn't found any way to do\nit, and that it was some sort o' rays shootin' out from this which had\nput the inflammation into Olaf's eyes.\n\nOlaf had had one of his teeth filled when he was young, and ever since\nthat he'd been suspicious o' science; so he just clouded up his face\nwhen they tried to devil him into bein' an experiment, and they\ncouldn't do anything with him. The Friar might have been able to, but\nthe Friar would have sent his own eyes East by freight before he'd\nhave asked Olaf to do a single thing he didn't want to do. The\nignorant allus scoff at the idee of Olaf seein' the soul-flame; but\nthe edicated allus take a serious interest which seems mighty\nfunny--don't it?\n\nFrom the very moment Janet opened her eyes and smiled up at the Friar\nthat mornin' she continued to improve. The doctor listened to all that\nwas told him about her havin' pains in the top of her head and not\nbein' right intellectually, and he said she must have had a blow there\nat some former time which had probably formed a tumor on the brain or\nknocked off a few splinters of bone into it, and that in removin' the\npressure, she had been put into perfect order again.\n\nShe had the smoothest voice I had ever heard, and I just doted on\nhearin' her speak the Friar's name, John Carmichael. I had a legal\nright to use the name John, myself; but it allus had the feel of a\nstiff collar to me, so I was glad enough to have it forgotten. But\nwhen Janet spoke the words John Carmichael, why, it cleared up the\natmosphere and started a little breeze. She didn't recall how she had\ncome to Cross Crick, nor anything much which had happened to her since\nthe night in Berlin. She said she had took singin' lessons in a place\ncalled Italy, and had expected to reach grand opery.\n\nShe had sung for pay whenever she got a chance, in order to get money\nenough to go on with her studies, and was gettin' what I'd call mighty\nlucrative wages at the Winter Garden; but was all the time bothered by\na lot o' foreign dudes who had the desire to make love, but not the\ncapacity. She said her manager had introduced an Austrian count for\nadvertizin' purposes, and she had finally consented to eat a meal with\nhim; but had been taken sick and had fallen. This was when she had\nbumped her head and she never got clear in it again until that morning\nwhen she had hovered between goin' out with the night or comin' back\nwith the dawn.\n\nShe said she had a hazy, dreamlike remembrance of havin' tried all\nkinds o' work after this; but couldn't tell the real from the unreal;\nand she didn't have any recollection of how she had come to the ranch.\nWe never mentioned Ty Jones to her for she was comin' along like a\ncolt on grass, and we didn't want to risk any set-back. She said she\nstill had it on her mind that she had lost something precious; but she\ncouldn't make out what it could have been, and the Friar allus told\nher not to worry, but to just rest herself back to complete strength.\n\nOscar and Tom Simpson had turned the corner, and it was only a\nquestion of time when they'd be all right again--which was true of all\nthe others except Ty and Prometheus. Ty wouldn't speak to us at all,\nthough he didn't seem to suffer to amount to anything. The doctor said\nhe might live for years, or he might slip away at a moment's notice;\nbut either way, he was doomed to be paralyzed for the rest of his\nlife; while the' wasn't any hope for Promotheus at all.\n\nHe had been shot through the liver, which pleased him a lot as bein'\nso in keepin' with his name; but we couldn't see why a feller who had\nsurvived bein' shot in so many other places, should have to give in on\naccount of an extra hole in his liver. Horace divided his time between\nwaitin' on The and spurrin' up the doctor to try some new treatment.\nHe read aloud to The out o' Ty's books, and he seemed as fond o' those\nold Greek fellers as Horace was himself. He was also mighty pleased to\nhave the Friar read and talk to him, and it softened us all a lot to\nsee how patient and gentle Promotheus had become. Humanity is about\nthe finest thing the' is about a human; and all humans have a showin'\ngrowth of it, if ya can just scratch the weeds away and give it a\nchance.\n\nThe prisoners bothered us a heap; we feared they might have some\nleanin's toward revenge; so we didn't dare turn 'em loose until they\nshowed some decided symptoms of repentance. Finally we got to bringin'\n'em up two at a time to talk with The. At first it didn't do any good,\nas Ty sat propped up in a bunk, grinnin' scornful, while The lay flat\non his back lookin' mighty weak and wan; but after several trials at\nit, they seemed to pay more heed to what The told 'em. We figured that\nTy must have ten or a dozen men still out on the range somewhere; but\nthey never showed up.\n\nIn about two weeks, or it might 'a' been three, all the wounded were\nable to walk about except Promotheus, Ty Jones, and Oscar. Oscar was\ndoin' fine; but the noise of the other men bothered The a little at\nnight, though he denied it up and down. Still, we thought best to move\nhim and Ty to a couple o' cots at the east end of the mess-hall, which\nwas large and airy, with a big fireplace for cool nights. By this time\nJanet was able to take short walks, leanin' on the Friar's arm; but\nthe Friar hadn't come any closer to findin' out what it was she had\nlost, nor whether or not she was Ty's wife. The only reply Ty ever\nmade to questions, was to skin back his lips in a wolf-grin.\n\nThe used to lay with his eyes fixed on Ty's face and a look of\nhopeless sadness in his own. When we'd come and talk to him, his face\nwould light up; but as soon as we left him, he would look at Ty again\nwith a sorrow that fair wrung a feller's heart. I wanted to separate\n'em; but when I suggested this to The, he shook his head. \"Nope,\" he\nsaid, \"he may speak to me before the vultures finish with my liver;\nand if ever the mood crosses his mind for a second, I want to be so\nhandy 'at he won't have time to change his mind.\"\n\nI told The 'at what was worryin' the Friar most was that all the\nfightin' had been on his account; but that next to this, it was\nbecause he didn't know whether or not Ty was married to Janet.\n\nThat evenin' just when the thinky time o' twilight came along, I was\nsettin' by the fire in the mess-hall, where I could see Ty, and his\nface didn't have quite so much the eagle look to it as common. The's\neyes rested on Ty's face most o' the time, and he, too, noticed it\nbein' a little less fierce than usual.\n\n\"Ty,\" he said in a low tone, \"I was drove into turnin' again' ya. Not\nby force, ya understand, nor by fear; but by something which has crept\ninto me durin' the last few years, and which I can't understand,\nmyself. Horace and the Friar have been mighty good to me--they saved\nmy life, ya know, after I had forfeited it by raidin' 'em durin' the\nnight. I told 'em I wouldn't be a spy on you about anything else\nexcept the woman. You haven't much excuse to bear me any ill will,\nseein' as it was your own hand which shot the move-on order into me.\nI'm goin' to slip out yonder before long; but the's no knowin' how\nlong you'll have to sit penned up in a chair.\"\n\nThe's voice gave out here, and he stopped a few minutes to cough. Ty's\nface hadn't changed, and his eyes looked out through the south window\nto where the western sky was still lighted into glory by the rays o'\nthe sun, which had already sunk.\n\n\"I've been locked up in a stone prison, Ty,\" said Promotheus as soon\nas he had quieted down again; \"and I want to tell you that the minutes\ndrag over ya like a spike-tooth harrow, when you haven't nothin' to\nlook at but four gray walls and the pictures on your memory. A feller\nfeeds himself on bitter recollections in order to keep his hate lusty;\nbut all this pilin' up o' hate is just one parchin' hot day after\nanother--like we've had this summer. Everything green and pleasant in\na feller's nature is burned down to the roots, and in tryin' to hate\nall the world, he ends by hatin' himself worst of all. Every kindly\ndeed he's done seems like a soothin' shower, and counts a lot in\nkeepin' him from fallin' down below the level o' snakes and coyotes.\n\n\"I'm not preachin' at ya, I'm tellin' you just what I know to be so\nfrom actual experience. I don't bear you no ill will, Ty, whether you\ntell me what I want to know, or not; but you have it in your power to\ngive me more content than airy other man in all the world. Are you\nmarried to the woman, Ty?\"\n\nFor a moment Ty didn't move, and then his lips tightened and he nodded\nhis head. Promotheus gave a sigh and settled back. He stayed quiet for\nsome time and then said in a weak voice: \"Thank ya, Ty. I'm purty\ncertain that at such a time as this, you wouldn't deceive me. I'm\nsorry you are married to her--on the Friar's account, understand--but\nI'm mightily obliged to you for tellin' me the truth. The Friar is a\nsquare man, and he's a strong man. He'll be able to fight what he has\nto fight; but none of us can fight uncertainty, without losin' our\nnerve in the end. I wish you would talk to me, Ty. I thought more o'\nyou than of airy other man I ever knew, except Horace and the Friar;\nand I wish, just for old time's sake, you'd talk to me a little before\nI slip away. You can talk, can't ya?\"\n\n\"Yes, I can talk,\" sez Ty Jones, facin' The with a scowl; \"but I\nhaven't any talk I want to waste on traitors. If I was to speak at\nall, it would be to ask 'em to separate me from your sloppy yappin'.\nYou may think 'at you sound as saintly as a white female angel when\nyou whine about duty and forgiveness and such-like rubbish; but the\nmore oil you put on your voice, the more I know you to be a sneak, a\nhypocrite, and a traitor. I won't ask 'em to move me; because I'm not\nin the habit of _askin'_ any man. When I had two legs to stand\non, I gave orders. Now that I can't give orders, I don't speak at all;\nbut every time you try to speak like a hen-missionary, you can know\nthat I'm sayin' to myself--sneak, hypocrite, traitor!\"\n\nOne thing you'll have to say about Ty Jones, an' that is, that when he\nstarted north, he didn't wobble off to the east or west much, let what\nwould come in his path. The only reply The made was to sigh; but what\nI wanted to do, was to lull Promotheus into a deep sleep, and then to\nfasten Ty Jones's neck to a green bronco, and let them two settle it\nout between 'em which was the tougher beast. What I did do, was to\nsteal out and tell Horace what had been said, and I also told him not\nto separate Ty and Promotheus as I thought The would set him an\nexample which might finally soften him a little and make him more fit\nto die, when the time came 'at some quick tempered individual lost\npatience and tried to knock a little decent conversation out of him\nwith an ax.\n\nHorace, though, thought only o' The, and he hurried in and sat beside\nhim. I also went in and took my seat by the fire again. Horace took\nThe's hand in one of his and patted it with the other. Horace didn't\nhave any upliftin' words to match the Friar's; but he had some chirky\nlittle ways which were mighty comfortin' to The, and when Horace would\nbe with him, all the sadness would leave his eyes, and he would talk\nas free as he thought--which, to my mind, is the final test of\ngenuwine courage.\n\nMighty few of us can do it. I know I can't. Time and again, I have had\ndeep feelin's for some one in trouble; but when I'd try to put 'em\ninto words, the knees o' my tongue would allus knock together, and I'd\ngrowl out somethin' gruff, cough, blow my nose, and get into a corner\nas soon as possible. The Friar was the first man who ever showed me\n'at a feller could speak out his softness without losin' any of his\nstrength, and I have honestly tried to do it myself; but I generally\nhad to dilute it down over half, and even then, it allus sounded as\nthough I had wrote it out and learned it by heart.\n\nThe asked Horace to either move him or Ty, said he didn't feel quite\ncomfortable beside Ty, and made out that it was his own wish; but\nHorace vetoed the motion, and pertended to scold The for not havin' a\nmore forgivin' nature. The thought he had been as circumspect as a\nland agent, and when his request rebounded back on him, he found\nhimself without any dry powder.\n\nHe lay quiet for some time, and then spoke in so low a tone I could\nhardly hear him. \"I can understand the real Promotheus purty well,\nHorace,\" sez he; \"and I've tried to be as game as he was; but I can't\nquite understand the One the Friar tells about. I have thought of Him\na heap since I've been laid up this time; but I don't believe I could\nbring myself to forgive them who had nailed me on a cross for doin'\nnothin' but good--I don't believe I could do that.\n\n\"I can feel things clearer now 'n I ever could before; and when I\npicture my own self as hangin' from nails drove through my hands and\nfeet, it just about takes my breath away. I've been handled purty\nrough in my time, but allus when my blood was hot, and pain don't\ncount then; but to have nails drove--My God, Horace, that's an awful\nthought! That's an awful thought.\n\n\"Then, too, I don't feel that any one has ill used me lately. The\ntreatment I got in the army, and in the pen, was consid'able hellish;\nbut I haven't had much chance to try forgivin' any one for the last\nfew years. Horace, you can't imagine all the joy the last part of my\nlife has been to me. I didn't know what life really was, until you and\nthe Friar pointed it out to me. I've been so happy sometimes it has\nhurt me in the throat; and now that I'm goin' on, I don't want to\ncause any one any bother. I asked Ty to tell me if he was married to\nthe woman, and he did tell me. I'm sorry to say 'at he is married to\nher, Horace; but I'm thankful to Ty for tellin' me. He don't feel easy\nnear me; so I wish you'd move me back to the bunk-shack.\"\n\nIt was some minutes before Horace could speak, and when he did, he had\nto put on pressure to keep his voice steady. \"I don't care one single\ndamn what Ty Jones wants,\" sez he. \"Let him stay right where he is and\nlearn the meanin' of friendship from the best friend a man ever had.\"\nAfter which Horace gave The's hand a grip and hurried out of the room.\n\n\n\n\nCHAPTER FORTY-FIVE\n\nTHE LITTLE GUST O' WIND\n\n\nI have seen some mighty quick changes brought about by flood o'\ncircumstances breakin' on a man all of a sudden--ol' Cast Steel\nJudson, himself, had melted and run into a new mold the night o'\nBarbie's weddin'--but I never saw such a complete change as had took\nplace in The since I'd first seen him. He loved devilment then, like a\nbear loves honey; while now he had swung back with the pendulem clear\nto the other side, until he was more unworldly 'n the Friar himself.\nIt wasn't what he said 'at made a feller feel funny inside, it was his\neyes. His eyes were all the time tryin' to tell things 'at his tongue\ncouldn't frame up, and it acted like brakes on a feller's breathin'\napparatus.\n\nI asked the Friar about it one evenin' while we were walkin' back\nthrough the ravine. He walked along with his brows wrinkled a few\nminutes, and then said: \"You see, Happy, the whole human race is made\nup o' millions of individuals, and each one is some alike and some\ndifferent. A man goes through childhood, youth, his fightin' period,\nand old age; and the race has to do the same thing.\n\n\"Now, ages ago when the childhood o' the race began, folks were\ndownright primitive; they used stone axes, skins for clothing, and ate\nraw flesh. They were fierce, impulsive, passionate, just like children\nare if you watch 'em close enough; but they lived close to nature,\njust like the children do, and their bodies were vigorous, and their\nminds were like dry sponges, ready to absorb whatever fell upon 'em.\n\n\"The outdoor man of to-day is still primitive; he delights in his\ndissipations, and recklessness, but the grim, set face which he wears,\nis a mask. The rich, pure air is all the time washin' his body clean,\nhis active life keeps his nerves sound and accurate, and his heart is\nlike the heart of a little child--hungry for good or evil, and needin'\na guiding hand all the time.\n\n\"In the mornin' a child is so full o' life that words don't mean much\nto him; but when the play o' the day is over, he comes home, through\nthe twilight shadows, bruised an' disappointed an' purty well tired\nout. All day long he's waged his little wars; but now he is mighty\nglad to pillow his head close to his mother's heart; and then it is\nthat the seeds o' gentleness are easiest sprouted. This is the\ntwilight time for Promotheus.\"\n\nWe didn't have anything more to say on this walk; but we both had\nplenty to think of. It allus seemed to me that in some curious way,\nthe Friar, himself, was better 'n his own religion. His religion made\nbadness a feller's own fault; but after gettin' to know the Friar, it\nallus made ya feel more like takin' some share in the other feller's\nsin, than like pointin' your finger at him and sayin' he never was any\ngood, nohow.\n\nA couple o' days after this, the doctor told us that the sands were\nrunnin' mighty low in The's hour-glass, and it wouldn't be long to the\nend; but still we couldn't believe it. He didn't look bad, nor he\ndidn't suffer; and we had seen him come back from the grave almost,\nthat time at Olaf's when Horace had claimed his life, and had saved\nhim in spite of himself.\n\nThen again, the doctor had missed it on Janet, and we were all hopin'\nhe'd get slipped up on again; but The himself seemed to side with the\ndoctor, and Olaf took one long look, an' then shut his lips tight an'\nshook his head. The said he wanted to live, and had done all he could\nto get a clinch on life; but that it was slippin' away from him drop\nby drop, and he couldn't stay with us much longer.\n\nHe seemed to want us about him, so we dropped in and sat beside him as\nlong as we could keep cheerful. All through the afternoon he lay with\na serious, gentle smile on his lips, but the sadness was mostly gone,\neven from his eyes. I closed my own eyes as I sat beside him, and\ncalled up the picture o' Badger-face the day he had wanted to lynch\nOlaf. Then I opened my eyes and looked at the real Promotheus, and I\nunderstood what the Friar meant by bein' born again.\n\nI spoke o' this to ol' Tank Williams, and he fired up at me as though\nI had poured red pepper in the nose of a sleepin' <DW36>. \"You're a\nnice one, you are!\" sez he. \"I'd sooner fill myself with alcohol and\ndie in a stupor than to call up The's past at such a time as this. You\nought to be ashamed o' yourself.\"\n\nThe' was no way to make Tank see what I meant so I sent him in to set\nwith The a while, and took a little walk up the ravine. Every step I\ntook brought some memory o' the time The and Horace and I had first\nstarted to find out about the woman; and it wasn't long before I was\nready to turn back.\n\nJanet was quite strong by this time, though she still had to wear a\nbandage; and after supper, the Friar took her in to see Promotheus. He\nhad told her all about him, and she was mighty sorry to think 'at his\nend was near. She didn't recall havin' been kind to him when he was\nplayin' <DW36>; but the Friar had told her about this, too. Horace\nhad told the Friar about what Ty had said, and it had cut him purty\ndeep; but he had braced up better 'n we expected. We didn't any of us\nknow what effect bringin' Janet in sight o' Ty would have, and when\nshe came into the mess-hall, we watched purty close.\n\nTy sat propped up, with his clenched hand restin' outside the blanket,\nand an expression on his face like that of a trapped mountain-lion. He\nglared up at her as she came near; but she only looked at him with\npity in her eyes, and she didn't seem to recognize him, at all--just\nlooked at him as though he was a perfect stranger which she was sorry\nfor, and Tank, who was settin' next me, gave me a nudge in my short\nribs, which was about as delicate as though it had come from the hind\nfoot of a mule. \"Well?\" I whispered. \"What do ya mean by that?\"\n\n\"Couldn't ya see 'at she didn't know him?\" sez Tank.\n\n\"That's nothin',\" sez I. \"He knew her all right.\"\n\n\"Yes, but Great Scott,\" sez he, \"a man can't claim that a woman's his\nwife if she don't know him, can he?\"\n\n\"Pshaw,\" sez I, \"if you'd settle things that way, the' wouldn't be any\nmarried people left. The' ain't one woman in fifty 'at knows her\nhusband, and the' ain't any men at all who know their wives.\"\n\n\"You're just dodgin' the question,\" sez Tank. \"I claim that if a man\nmarries a woman when she's out of her mind, he ain't got any claim on\nher when she gets back into her mind again.\"\n\n\"Look here, Tank,\" sez I; \"you've never had much experience with the\nworld, 'cause every time you went where experience was to be had, you\ngot too intoxicated to take notice; but I'm tellin' you the truth when\nI say that if women didn't sometimes get out o' their right minds,\nthey wouldn't get married at all.\"\n\n\"Aw, shut up,\" sez Tank.\n\nJanet had gone over to Promotheus, and was smoothin' his forehead. She\nhad a beautiful, shapely hand, and it made me feel a little wishful to\nwatch her. The lay perfectly still, and his sensations must 'a' been\npeculiar. Ty Jones didn't even look at 'em. He kept his brows scowled\ndown and his gaze out the south window.\n\nPresently Janet turned and walked out to the porch. It was an\nunusually warm night, and she sat there alone, while the Friar came\nback to The. Horace had gone off by himself to get a grip on his\nfeelin's; but he came in about nine o'clock, and went up and took\nThe's hand. \"Well,\" sez he, \"have you finally got over your nonsense?\nI have a lot o' plans I want to carry out, and you know I can't have\nyou loafin' much longer.\"\n\nNothin' suited The so well as to have a little joke put at him; but he\ndidn't have any come-back to this. He caught at his breath a time or\ntwo, and then said: \"I can't do it, this time, Horace. I hate to\ndisappoint ya--I've been countin' on what a good time we were goin' to\nhave--up there in the hills--but I can't come back this time--I,\ncan't, quite, make it.\"\n\nHe ended with a little gurgle and sank back on the pillow. Horace\nshook him a little and then flew for the doctor, who was on the porch\no' the old cabin. They were back in half a minute, Horace pushin' the\ndoctor before him; and we all held our breaths when he felt The's\npulse. The doctor squirted somethin' into The's arm, and after a bit,\nhe opened his eyes with a long sigh, and when he saw Horace bendin'\nover him, he smiled.\n\n\"I mighty near slipped away that time,\" sez he. \"It's not goin' to be\nhard, Horace; and I don't want you to worry. I feel as comfortable as\nif I was sleepin' on a cloud, and there isn't one, single thing to\ngrieve about. I've been like one o' those hard little apples which\ntake so long to ripen. I've hung up on a high bough and the rains beat\non me, and the sun shone on me, and the winds shook me about, and the\nbirds pecked at me until at last just the right sort o' weather came\nalong and I became softer and softer, and riper and riper, until now\nmy hold on the stem begins to weaken. Purty soon a little gust'll come\nalong and shake me down on the green grass; but this is all right,\nthis is perfectly natural, and I don't want you to feel bad about it.\n\n\"I own up now, that I've been afraid o' death all my life; but this\nhas passed. I don't suffer a bit; but I'm tired, just that pleasant\nweariness a feller feels when his last pipe has been smoked, and the\nglow o' the camp fire begins to form those queer pictures, in which\nthe doin's o' that day mingle with the doin's of other days. I'm\nliable to drop off to sleep at any moment, now; and I'd like--I'd kind\no' like to shake hands with the boys before I go.\"\n\nWell, this gave Horace something to do, and he was mighty glad to do\nit. After we had all shaken hands with The, he marched up the\nprisoners, even to the Chink, and they all shook hands, too; and by\nthis time Prometheus was purty tired; but he did look unusual\ncontented. He glanced across at Ty; but Ty had turned his face to the\nwall, and The gave a little sigh, settled down into the pillow again,\nand closed his eyes. Horace backed around until The couldn't see him,\nand shook his fist at Ty, good and earnest.\n\nPurty soon a regular grin came to The's face, and he opened his eyes\nand looked at the Friar with a twinkle in 'em. \"Friar Tuck,\" sez he,\n\"I don't know as I ever mentioned it before, but I'll confess now that\nI'm right glad I didn't lynch you for stealin' those hosses.\" He lay\nthere smilin' a minute, and then held out his hand. \"Good-bye,\nHorace,\" he said in a firm voice.\n\nHorace had been doin' uncommon well up to now; but he couldn't stand\nthis. He threw himself on the bed, took both o' The's hands and looked\ndown into his face. \"Promotheus, Prometheus,\" he called to him in a\nshakin' voice. \"Don't give up! You can win if you fight a while\nlonger. Remember that day in the desert, when I wanted to lie down and\nend it all. You said you didn't take any stock in such nonsense; and\nyou picked me up and carried me over the molten copper, while queer\nthings came out o' the air and clutched at us. You reached the\nwater-hole that time, Promotheus, and you can do it again, if you just\nuse all your might.\"\n\nPromotheus opened his eyes and his jagged, gnarly teeth showed in a\nsmile, weak and trembly, but still game to the last line of it.\n\"Nope,\" he said so low we could hardly hear him, \"I'm Promotheus, all\nright. I hung on as long as I could; but the vultures have finished my\nliver at last, Horace--they have finally finished it. I hate to leave\nyou; but I'll have to be goin' soon. The's only one thing I ask of\nya--don't send a single one o' the boys to the pen. They don't know\nwhat the world really is; but shuttin' 'em out of it won't ever teach\n'em. If the's anything you can do to give 'em a little start, it would\nbe a mighty good thing--a mighty good thing.\" His voice was gettin'\nawful weak, an' he'd have to rest every few words.\n\n\"And Ty Jones, too,\" he went on, \"Ty was square with me in the old\ndays. Try to make him understand what it was 'at turned me again' him;\nand if the's any way to make things easier for Ty, I want you to have\nit done. Ty had a lot o' tough times, himself, before he turned all\nthe hard part of his nature outside. Don't bear him any malice,\nHorace. Seventy times seven, the Friar sez we ought to forgive, and\nthat many'll last a long time, if a feller don't take offence too\neasy. The's a lot o' things I don't understand; but some way it seems\nto me that if I could just go out feelin' I had squared things with\nTy, I'd be a leetle mite easier in my mind.\"\n\nHorace stepped to Ty's bed and shook him by the arm. \"Did you hear\nwhat he said?\" he demanded. \"You know he's achin' to have you speak to\nhim decent. Why don't ya speak to him?\"\n\nTy looked cold and stony into Horace's eyes, and then took his left\nhand and pushed Horace's grip from off his arm. Horace stood lookin'\nat Ty with his fist clinched. The turned and saw it and a troubled\nlook came into his face.\n\n\"Friar Tuck,\" he said, \"you meant it, didn't ya--that about forgivin'\nseventy times seven?\"\n\n\"I did,\" sez the Friar, his voice ringin' out clear and strong in\nspite of its bein' low pitched. \"Be at peace, Promotheus, the laws of\nman are at war with the laws of God; but they're bound to lose in the\nend. I want you to know that I forgive Ty Jones as fully as you\ndo--and I shall do everything in my power to square things up with\nhim.\"\n\nThe held out his hand to the Friar, and they clasped in a\ncomrade-grip. \"I can trust you,\" he said; \"and I know you'll do all\nyou can to make Horace see it that way, too.\"\n\n\"I forgive him, too, you big goose!\" cried Horace. \"I promise you that\nI'll do all I can for him--on your account. Though I must say--but no,\nI mean it, Promotheus. I forgive him from my heart, and I'll be as\ngood a friend to him as I can.\"\n\n\"Now, let the little gust o' wind come,\" sez The. \"I'm perfectly ripe\nand ready for it, now.\"\n\nThe' was silence for several minutes; and then Promotheus said in a\nfaint voice: \"Friar, I wish you'd sing to me. All my life I've longed\nto hear a cradle-song, a regular baby cradle-song. I know it's a\ndamn-fool notion; but I never had it so strong as I've got it now--and\nI wish you'd sing one to me. My mother was a widow, mostly. She\ncleaned out offices at night to earn enough to keep us alive. She\nsacrificed her life for me, but I couldn't understand this then.\n\n\"Night after night I used to creep in from the street through dirty,\nstinkin' halls, and cry myself to sleep. An achin' came into my heart\nthen which hasn't never quite left it; and it was this lonesomeness\n'at finally made me run away--leavin' her to face it out--all by\nherself.\n\n\"My blood has turned to water, I reckon, and I feel like a baby\nto-night. I don't suffer, understand; I feel as though I was a little\nchap again, and that my mother didn't have to work; but was holdin' me\non her lap. She did hold me that way once--the time the ambulance\nbrought my old man home--but she couldn't sing then. It seems to me\nthat if you'd just sing me a regular cradle-song--I could slip away\ninto pleasant dreams.\"\n\nThe Friar cleared his throat a time or two before he found his voice;\nand then he said in a low tone: \"I used to sleep in a store-box,\nPromotheus, when I was a lad--and I know exactly what you feel. I'll\nsing you a cradle-song, a song for little children of all ages. It is\na great privilege to be a little child, Promotheus, and--and I wish\nyou pleasant dreams.\"\n\nThen Friar Tuck drew a deep, full breath, and held it down until all\nthe quiver had gone from his lips. When he started to sing, his voice\nwas low an' soothin', and full o' tenderness; and after the first\nline, Promotheus gave a little sigh o' content, nodded his head, and\nshut his eyes.\n\nThe' was one tune we every last one of us liked. The Friar generally\nsang it to words which began: \"Guide me, O Thou great Jehovah\"; and he\nusually sang it with a swing which was like a call to battle; and this\ntime he sang the same tune, but soft and close and restful, and the\nwords he used began: \"Jesus, tender Shepherd, hear me.\" These words\nsound purty flat when ya give 'em cold; but they didn't sound empty to\nus, as we stood lookin' down at Promotheus. All alone, he had taken\nhis chance when he took on with Ty Jones; and now he was cashin' in\nthis chance and it made us mighty sober.\n\nThe Friar finished the first four lines alone, and then the angels\nseemed to join in with him. We had all been purty certain that the'\nwasn't nothin' in the shape of earthly melody fit to hold a candle to\nthe Friar; but just at this point a new voice joined onto the Friar's\nwhich sent a thrill through us and made us stop breathin'. A queer,\nhalf frightened look crossed the Friar's face for a second; but his\nvoice didn't waver for a single note. Instead, the' came a new tone of\nthanksgivin' and confidence in it which took all the sting out o'\ndeath and made it all right and pleasant, like the cool and\nrestfulness o' night, after the heat of day.\n\n    \"All this day Thy hand has led me,\n    And I thank Thee for Thy care;\n    Thou hast warmed me, clothed and fed me;\n    Listen to my evening prayer,\"\n\nwent on the song and the' came an expression of wonder and of joy into\nThe's tired face.\n\nThere are only three little verses to this one, and to fill out the\ntune they had to sing the first one over again, soft and low. The\ncandles threw a soft glow on The's face which hid the pallor of it and\nthe rough lines, but brought out all the kindly strength we had come\nto be so fond of; and when the music died away, we all sat still for\nfear o' disturbin' him.\n\nHorace had been settin' holdin' one of his hands, and after a bit he\nleaned forward and whispered, \"Was that what you wanted, Promotheus?\"\n\nBut the' wasn't any reply. The little gust o' wind had come with the\nsong--and fully ripe, and soft to the core of his big, warm heart,\nPromotheus had loosed his hold on the bough of life, and dropped off\nonto the soft, deep grass of eternity.\n\n\"Promotheus! Promotheus!\" cried Horace, and then covered his face with\nhis hands and dropped forward upon The's quiet breast.\n\n\"Badger-face,\" called a harsh voice, and we looked at Ty Jones and saw\nhim leanin' towards The. \"Wait, Badger-face, wait--I want to speak to\nya. I want to tell you that I lied to ya. Oh Lord, it's too late, it's\ntoo late!\" And Ty Jones pressed his hand across his eyes and sank\nback.\n\nHorace whirled to tell Ty what he thought of him; but the Friar placed\nhis big hand on Horace's shoulder, and pointed down to The's placid\nface. Horace gave a shudderin' sob, and settled back into his former\nposition.\n\nJanet Morris crossed the floor to the Friar just then and said to him\nin a low tone: \"I have found it again--my voice has come back to me.\"\n\nTy Jones took his hand down from his eyes and straightened up and\nlooked at her. All the eagle had gone from his face, and it looked old\nand haggard. \"Don't you really know who I am?\" he asked.\n\nShe looked at him and shook her head.\n\n\"I'm your half-brother,\" he said. \"I'm Tyrell Jones Morris. Your\nmother might have been a good woman, but she was not good to me--she\nwasn't fair; she prejudiced my father again' me. You were sellin'\ntickets at an elevated station in New York when I found you. You\nlooked a good deal like your mother, for you were weak and sickly. I\ndidn't know then, whether I brought you back with me because we had\nthe same blood in our veins, or because I hated you--and I don't know\nyet. I'm not tellin' you this now, because I care any thing for you,\nor the preacher; but Badger-face was square, and I know now 'at he'd\nnever have turned again' me if the rest of ya hadn't tampered with\nhim. I'm sorry I didn't tell him before he died--and that's why I'm\ntellin' you now.\"\n\nI winked my eyes to the boys, and we filed out and went over to the\nbunk-shack. We lighted our pipes and sat a long time smokin' in\nsilence. One by one they dropped off to bed until only me and ol' Tank\nWilliams was left. Tank sat with a sour look on his face, and so\ndeeply buried in thought that the burnt matches around his stool\nlooked like a wood pile. \"What are ya thinkin' of, Tank?\" I said to\nhim.\n\n\"I'm not kickin', understand,\" sez he; \"but it does seem to me that\nwhen all The asked for was a cradle-song, the Friar could 'a' thought\nup somethin' besides another one o' those doggone sheep-herder hymns.\nThe didn't have any more use for sheep-herders 'n I have.\"\n\nThis was the real Tank, all right. Once an idee took possession of\nhim, it rode him rough shod till he keeled over with his tongue\nhangin' out.\n\n\n\n\nCHAPTER FORTY-SIX\n\nTHE FINAL MOVES\n\n\nWe buried The by the side o' Tim Simpson. Horace insisted on makin' a\ncoffin for him--fact was, he wanted to have a regular funeral, but we\ntalked him out o' this; so he made a coffin himself and lined it with\nsilk which Ty Jones had brought out for Janet to make dresses of. The\nFriar held some short services, but he didn't sing or preach any. Some\nway, the' didn't seem to be any need of it. After we had covered him\nover we stood around talkin' for quite a while; and then only turned\naway because the first rain we had had for months came rattlin' down\nfrom the mountains.\n\n\"Do you see that, now?\" asked ol' Tank after we had reached the porch\nand were sittin' watchin' it come down in torrents.\n\n\"I'm not totally blind,\" sez I.\n\n\"Well, I'm not superstitious,\" sez Tank; \"but I'm bettin' that he's\nhad that tended to, himself. He wasn't one to forget his friends, and\nhe knew 'at what we needed most was rain--so he's called attention to\nit the first chance he's had.\"\n\nFact was, Tank was so everlastin' superstitious that he spelt Tomas\nwith an \"h\" in it to keep from havin' thirteen letters in his full\nname; but it did seem queer about this rain, because they wasn't any\nsane man in the world who would have expected a rain just at this\ntime. It's astonishin' how many curious things there is if a feller\njust takes notice of 'em.\n\nThe Friar and Ty had had a long talk the night 'at Promotheus slipped\naway, and the Friar had agreed to settle down at the ranch and do what\nhe could for Ty. Ty wasn't thankful; but he hadn't much choice, so he\nbehaved better 'n any one would have expected. The Friar wanted me to\nstay and be foreman for him; but I told him I had promised Jabez to\ncome back as soon as I had got a good holt on myself again; and I\nintended to leave for the Diamond Dot the minute things were right at\nthe Cross brand. The Friar didn't much trust Pepper Kendal for\nforeman; but the minute I thought it over, I saw that Olaf was the\nvery man, and this suited the Friar to a T.\n\nWe brought the prisoners up to Ty and he told 'em how things were and\nadvised 'em to adjust themselves to new conditions as fast as\npossible, and they all agreed to do it and went to work under Olaf.\nThe Friar knew a preacher at Laramie; so Horace gave Tillte Dutch the\njob o' goin' after him, and as soon as he came, the Friar and Janet\nwere married, and then I made plans to hit the trail for the Diamond\nDot.\n\nHorace had made up his mind to build himself a cabin up at our old\ncamp and he tried to hire me for life; but I had taken root at the\nDiamond, and when I explained things to him, he owned up I was right.\nI suggested to Horace that ol' Tank Williams was the very man for him,\nand he admitted, when he came to look it over, that Tank would suit\nhim a heap better for hired help 'n I would. He even went so far as to\nsay he never could understand how it came 'at a stiff-necked man like\nol' Jabez could put up with my independent ways. I told Horace the'\nwas a lot of things it wasn't necessary for him to understand, and\nthen I whistled to Tank, and he came over and joined us.\n\nTank rolled the notion about in his head a while, and then he sez:\n\"Horace, I'll take ya up. We both got cured up of our nerves on the\nsame trip, and ever since then I have to own that you've found favor\nin my sight; but the one thing 'at counts bigger 'n anything else, is\nthe fact that, come what will, you'll never have any more hankerin' to\nbe pestered by a lot o' sheep, than I will.\"\n\nOlaf started to get things ready for the round-up and us Diamond Dot\nboys, aside from ol' Tank, rode off home, where we found things in\nconsid'able of a muddle. Durin' the three years previous I had been\ntakin' more and more o' the responsibility onto my own shoulders, and\nol' Cast Steel found himself purty rusty. We turned to and\nstraightened things out, and then I settled down to the sober business\no' handlin' a big outfit with a view on the future.\n\nAfter this, I didn't do any more skitin' around than my peculiar\nnature seemed to insist on; but I did make out to pay the Cross brand\na visit every once in a while. The Friar only intended to stay long\nenough to get things to slidin' easy; and then he and Janet were to go\nback East and work among the city poor; but the chance never came.\n\nJanet grew perfectly strong and well again; but the city allus made\nher nervous to return to the mountains, and they were kept so busy on\nthe ranch that the years slipped away without bein' noticed.\n\nTy's backbone was all in one piece, and solid--except where Olaf had\nunjointed it--and it took years to wear him down to friendliness; but\nwhen the Friar's first baby got big enough to creep, the contrary\nlittle cuss took more interest in ol' Ty Jones, than in airy other\nthing the' was on the place. I never saw any one yet who didn't feel\nflattered at a baby's endorsement--though why a baby should be\nsupposed to actually have better judgment than grown folks has never\nbeen fully explained to me yet.\n\nHorace kept his word to The, and he did all he could for Ty. Ty didn't\nlike him and he didn't like Ty; but Ty was human, and it made him\nlonely to sit in one spot all the time, so that while he refused to be\nthankful, he gradually got to relyin' on Horace; and Horace was also\nhuman, and the more he did for Ty on The's account, the more fond he\ngrew of Ty on his own account. He got him a wheelchair first, and this\nwas a big help. Then he fixed up a trapeze for Ty to practice on. Ty\ngot mad about this and said that <DW36> though he was, no man could\nmake a monkey of him; but one night when he couldn't sleep he\npracticed on it, and it gave him a lot o' relief.\n\nThe name of the Chinaman was Yuen Yick, and he thought 'at Ty Jones\nwas some sort of a god, and fair worshipped him--every one o' Ty's men\nswore by him, even after he turned decent. Ty used to abuse the Chink\nall he could and it pleased 'em both; and the Chink saw that Horace\nmeant well by Ty, so he kept Horace posted on just what Ty did and\nthought; and Horace had Janet make some flannel bricks filled with\ncotton for Ty to throw at the Chinaman. Ty got a lot o' satisfaction\nout o' these bricks, and the exercise helped him too.\n\nNext, Horace had a wide porch built all around Ty's house, and he\nswung ropes with rings on 'em from the ceiling, an equal distance\napart; and Ty got so he could swing from ring to ring, and go all\naround the house, and climb ladders, and as the boy got big enough to\nbecome tyrannical, which was soon enough, goodness knows, he made Ty\ndo all manner o' stunts--throw balls and juggle 'em, tell stories,\ndraw pictures--Well, the fact was, that between 'em all, they kept Ty\nso active that first we knew, the devil had all been worked out of him\nand he was as civilized as any of us. One day when Horace was down\nvisitin' him, he sent in the Chink and had him bring out a set of\nivory figures, carved most beautiful and called chess-men; and he\ndared Horace to play him a game, and this was the final surrender of\nthe old Ty Jones.\n\nHe was a well edicated man, Ty was; and each winter when he had left\nthe ranch, he had gone to some big city where he had pertended to be a\nregular swell. No one ever found out just what had soured him so on\nthe world, for his nature was to be sociable to a degree. He said that\nno one knew the cause of it except ol' Promotheus, and it was mightily\nto his credit that he hadn't devulged the secret.\n\nTy strung out his surprises quite a while. It seems he was also an\ninventor, and had patents which brought him in a lot o' money. He had\nfound this cave and had just widened it where widenin' was necessary,\nand had built his cabin above it. The floor was double and filled with\nearth, and the fake drawers were also filled with earth, so 'at no\nsound would show that it was hollow underneath. The drawers swung on a\nsteel piller which could be worked from above by a rope which hung\nback o' his bookcase and from below by a lever.\n\nIt was a curious thing to see Ty Jones with his bristly eyebrows and\nhis eagle's beak of a nose, makin' mechanical toys for the Friar's and\nOlaf's children. They didn't put any limit on what he was able to do,\nand he used to grumble at 'em as fierce as a grizzly--and then\nback-track like an Injun, and do whatever they wanted him to.\n\nThe Friar never quite gave up his plot to go back and work among the\npoor; but the' was allus so many things imposed upon him by the home\nfolks that he was pestered with letters every time he left; and\nusually compromised by gatherin' up a bunch o' the poor as hasty as\npossible, and bringin' 'em back with him. His head was full of what he\ncalled welfare plans, and he settled the poor along all the likely\ncricks he found vacant, and bulldozed 'em into goin' to work. It's a\ncurious coincident; but most of 'em turned out well.\n\nThe' was a bilious feller out visitin' me once, which called himself a\nsosologist. I told him about some o' the Friar's projects; and he said\nthat the Friar was nothin' but a rank Utopian, and that this sort o'\nwork would never remove all the evils of the world.\n\n\"You can call him anything ya want to,\" sez I, \"so long as it's a word\nI don't understand; but the Friar's not tryin' to remove all the evils\nin the world. He only removes those evils he can find by spendin' his\nwhole life in huntin' for 'em; but he certainly does remove these ones\nin quick and able shape.\"\n\nAnother time, right after the Friar had brought about a settlement\nbetween some sheep and cattle men, a preacher dropped off to give his\nappetite a little exercise at the Diamond Dot. He belonged to the same\nherd that the Friar had cut out from, and I thought he would be\ninterested; so I told him consid'able about the Friar. He was a most\njudicious-lookin' man, but baggy under the eyes and chin. He got all\nfussed up when I spoke well o' the Friar, and said he was\nun-co-nonical, said he was unorthodox--Oh, he cut loose and swore at\nthe Friar in his own tongue 'til I about lost my temper.\n\n\"Look here,\" I sez to him, \"it would take me some months to tell you\nall the good deeds the Friar has actually done; but I'll just give you\none single example. If I was to live up to my natural disposition, I'd\nwring your neck, or shoot off your ears, or somethin' like that; but\nowin' to the Friar havin' taught me self-control, I'm not even goin'\nto snap my fingers again' your blue nose. Make yourself perfectly at\nhome here, and stay as long as the East can spare ya; but you'll have\nto excuse me for a while, as the Friar has just written me an order to\ngo over into the Basin to see what can be done for a young feller who\nhas been arrested for hoss-stealin'.\"\n\nHorace contributes liberally to the Friar's projects; but he don't\ntake a hand in the game, himself--except with the imported poor which\nare gathered at the Cross brand, waitin' to be transplanted. Every\nyear he seems to shrink about an eighth of an inch smaller, and get\nabout that much tougher. He lights out for a trip now and again, and\nol' Tank allus tags along, grumblin'. Tank thinks full as much of\nHorace as The did; but Tank's a different proposition. The easier his\nlot is the more he grumbles; but I like nothin' better than to have a\nchat with him over old times.\n\nOne night I was up visitin' Horace, and after supper we got a little\nrestless and started out for a walk. We sauntered down to our old\nlook-out and stood gazin' down at the lights of the Cross brand ranch.\nTy had rigged up a water power to manufacture e-lectricity, simply\nbecause the children had needed it to run some o' their idees, but\nthe' was plenty of it to light the whole place. In token of Ty's\nbrand, and also as a symbol of his own callin', the Friar had built an\nimmense cross on the cliff just above the mouth of the ravine, and on\nthe upright, and at each end o' the cross-piece were big electric\nlights. These could be seen for miles, and every one knew 'at whatever\ntroubles they had, there was allus welcome, cheery hospitality, and\nsound advice waitin' for 'em in the shadow of this cross.\n\nIt was a moonlight night, one of those crisp, bright nights, when it\nmakes a feller feel solemn just to get up high and look down at the\nbeauty of the old, hard Earth. We had been talkin' o' the old days as\nusual; but not talkin' much, for we each saw the same set of pictures\nwhen we looked down from here, and they didn't need many words.\n\n\"Life is like a game o' chess,\" sez Horace. \"The openin' is not so\nabsolutely vital; but after a time the' comes one little move which is\nthe keynote of all the balance of the game--and the same is true o'\nlife. The way things has turned out down yonder seems to be the very\nbest way they could have turned out; but it's hard to look back and\ntell just what was the keynote of it all. Of course\nPromotheus--Promotheus was the prime mover; but then all the way along\nyou can see the Friar's influence. What would you say was the keynote\no' this tangled game, Happy?\"\n\nI looked down at Horace: he was wearin' a battered old hat, rough\nclothes and leggins, and smokin' a corncob pipe. \"That's an easy one,\"\nsez I, tryin' to shake off a feelin' o' sadness which was beginnin' to\ncreep over me, in spite of all I could do; \"gettin' your nerves cured\nup, Horace, was the keynote of it all.\"\n\n\"That was a long time ago,\" sez Horace, \"a long, long time ago.\"\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Friar Tuck, by Robert Alexander Wason\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHere's what kids have to say to  \nMary Pope Osborne, author of  \nthe Magic Tree House series:\n\n_WOW! You have an imagination like no other._ \u2014Adam W.\n\n_I love your books. If you stop writing books, it will be like losing a best friend._ \u2014Ben M.\n\n_I think you are the real Morgan le Fay. There is always magic in your books._ \u2014Erica Y.\n\n_One day I was really bored and I didn't want to read... I looked in your book. I read a sentence, and it was interesting. So I read some more, until the book was done. It was so good I read more and more. Then I had read all of your books, and now I hope you write lots more._ \u2014Danai K.\n\n_I always read [your books] over and over... 1 time, 2 times, 3 times, 4 times..._ \u2014Yuan C.\n\n_You are my best author in the world. I love your books. I read all the time. I read everywhere. My mom is like freaking out._ \u2014Ellen C.\n\n_I hope you make these books for all yours and mine's life._ \u2014Riki H.\nTeachers and librarians love  \nMagic Tree House\u00ae books, too!\n\n_Thank you for opening faraway places and times to my class through your books. They have given me the chance to bring in additional books, materials, and videos to share with the class._ \u2014J. Cameron\n\n_It excites me to see how involved [my fourth-grade reading class] is in your books... I would do anything to get my students more involved, and this has done it._ \u2014C. Rutz\n\n_I discovered your books last year... WOW! Our students have gone crazy over them. I can't order enough copies!... Thanks for contributing so much to children's literature!_ \u2014C. Kendziora\n\n_I first came across your Magic Tree House series when my son brought one home... I have since introduced this great series to my class. They have absolutely fallen in love with these books!... My students are now asking me for more independent reading time to read them. Your stories have inspired even my most struggling readers._ \u2014M. Payne\n\n_I love how I can go beyond the [Magic Tree House] books and use them as springboards for other learning._ \u2014R. Gale\n\n_We have enjoyed your books all year long. We check your Web site to find new information. We pull our map down to find the areas where the adventures take place. My class always chimes in at key parts of the story. It feels good to hear my students ask for a book and cheer when a new book comes out._ \u2014J. Korinek\n\n_Our students have \"Magic Tree House fever.\" I can't keep your books on the library shelf._ \u2014J. Rafferty\n\n_Your books truly invite children into the pleasure of reading. Thanks for such terrific work._ \u2014S. Smith\n\n_The children in the fourth grade even hide the [Magic Tree House] books in the library so that they will be able to find them when they are ready to check them out._ \u2014K. Mortensen\n\n_My Magic Tree House books are never on the bookshelf because they are always being read by my students. Thank you for creating such a wonderful series._ \u2014K. Mahoney\n\n_A few years ago I visited County Galway in Ireland. I traveled through seaside towns along the rocky coast and took a boat out to the lonely Aran Islands at the mouth of Galway Bay. I loved the lush green sheep meadows of the countryside, the smell of peat fires wafting through misty rain, the cozy pubs where we had gingerbread and strong tea. Ever since that visit, I've loved Irish music and literature\u2014especially the folklore of leprechauns, fairies, and legendary Irish heroes and heroines._\n\n_So now I want to share Ireland with you. Get ready for a journey to the enchanted countryside of Galway, to a time a hundred and fifty years ago, when mysterious creatures still hid in the forests and hills. Be careful not to let them see you, or you might be turned into a skunk or a weasel! But don't be afraid\u2014just have a great time with Jack and Annie_.\n\nThis is a work of fiction. All incidents and dialogue, and all characters with the exception of some well-known historical and public figures, are products of the author's imagination and are not to be construed as real. Where real-life historical or public figures appear, the situations, incidents, and dialogues concerning those persons are fictional and are not intended to depict actual events or to change the fictional nature of the work. In all other respects, any resemblance to persons living or dead is entirely coincidental.\n\nText copyright \u00a9 2010 by Mary Pope Osborne  \nIllustrations copyright \u00a9 2010 by Sal Murdocca\n\nAll rights reserved. Published in the United States by Random House Children's Books, a division of Random House, Inc., New York.\n\nRandom House and the colophon are registered trademarks and A Stepping Stone Book and the colophon are trademarks of Random House, Inc. Magic Tree House is a registered trademark of Mary Pope Osborne; used under license.\n\nVisit us on the Web!  \nwww.magictreehouse.com  \nwww.randomhouse.com\/kids\n\nEducators and librarians, for a variety of teaching tools, visit us at www.randomhouse.com\/teachers\n\n_Library of Congress Cataloging-in-Publication Data_  \nOsborne, Mary Pope.  \nLeprechaun in late winter \/ by Mary Pope Osborne ; illustrated by Sal Murdocca \u2014 1st ed.  \np. cm. \u2014 (Magic tree house ; #43)  \n\"A Merlin mission.\"  \n\"A stepping stone book.\"  \nSummary: Jack and Annie travel back to nineteenth-century Ireland to inspire a young  \nAugusta Gregory to share her love of Irish legends and folktales with the world.  \neISBN: 978-0-375-89466-4  \n1. Time travel\u2014Fiction. 2. Magic\u2014Fiction. 3. Tree houses\u2014Fiction.  \n4. Brothers and sisters\u2014Fiction. 5. Gregory, Lady, 1852\u20131932\u2014Fiction.  \n6. Ireland\u2014History\u201419th century\u2014Fiction.  \nI. Murdocca, Sal, ill. II. Title.  \nPZ7.O81167Le 2010 [Fic]\u2014dc22 2009016591\n\nRandom House Children's Books supports the First Amendment and celebrates the right to read.\n\nv3.0\n_For Lillian Grogan Osborne Reynolds_\n\n _And with special thanks to Dan Ringuette_\n\n#\n\n Cover\n\n Title Page\n\n Dear Reader\n\n Copyright\n\n Dedication\n\n Prologue\n\n 1. A Beautiful Word\n\n 2. The Big House\n\n 3. Miss Augusta\n\n 4. What Are You Good For?\n\n 5. A Fireside Tale\n\n 6. A Late\u2013Winter's Daydream\n\n 7. Willy\n\n 8. The Hollow Hill\n\n 9. Skunks or Weasels?\n\n10. Fare-thee-wells\n\n11. Lady Gregory\n\n More Facts from Jack and Annie\n_\"They can grow small or grow large. They can_  \n _take what shape they choose.... They go by us in a_  \n _cloud of dust; they are as many as the blades of_  \n _grass. They are everywhere.\"_\n\n\u2014from _Visions and Beliefs in the West of Ireland_  \nby Lady Gregory\n\n#\n\nOne summer day in Frog Creek, Pennsylvania, a mysterious tree house appeared in the woods. A brother and sister named Jack and Annie soon learned that the tree house was magic\u2014it could take them to any time and any place in history. They also learned that the tree house belonged to Morgan le Fay, a magical librarian from the legendary realm of Camelot.\n\nAfter Jack and Annie traveled on many adventures for Morgan, Merlin the magician began sending them on \"Merlin Missions\" in the tree house. With help from two young sorcerers named Teddy and Kathleen, Jack and Annie traveled to places both mythical and real to do Merlin's bidding.\n\nOn their most recent missions, Jack and Annie found four secrets of happiness to help Merlin when he was in trouble.\n\nNow Merlin wants Jack and Annie to bring happiness to others\u2014by helping four creative people give their special gifts to the world. They have already helped the first two; now they are ready to find the third....\n\n#\n\nIt was a chilly afternoon in late winter. Annie was doing her homework on the computer in the living room. Jack sat on the couch and stared at a blank page in his small notebook. He heaved a sigh.\n\n\"What's wrong?\" said Annie.\n\n\"I have to write a story for school,\" said Jack, \"and I'm stuck.\"\n\n\"Well, you'd better get unstuck,\" said Annie. \"Mom and Dad said we have to get our homework done before we go to the theater with them tonight.\"\n\n\"I know,\" said Jack. \"But I can't think of anything to write about.\"\n\n\"Why don't you do what you love to do?\" said Annie. \"Go outside and write down some facts about what you see. Then turn them into a story.\"\n\n\"Hey, that's a good idea,\" said Jack. \"Thanks.\" He jumped up and grabbed his coat from the hall closet. Then, taking his pencil and notebook with him, he headed outdoors.\n\nThe early March weather was sunny but cold and windy. Jack looked around. Then he wrote down some facts in his notebook:\n\n_old snow in yard  \nsun sparkling on sidewalk_\n\nJack looked up again. Treetops swayed in the March winds. Jack started to write about them. But when he looked down at his notebook, he nearly dropped his pencil. On the page were two large, fancy letters:\n\n**_T K_**\n\n\"Oh, man!\" whispered Jack. He dashed back in the house and into the living room. \"Annie! Look!\" Jack held up his notebook. \"Look at _this_!\"\n\nAnnie stared at the page. \"Old snow... sun sparkling... Nice....\"\n\n\"No, not that!\" said Jack. \"The letters!\"\n\nAnnie looked at Jack like he was a little crazy. \"Uh... what letters?\" she said.\n\nJack looked back at the page. \"They're gone!\" he said. \"A big, fancy T and K!\"\n\n\"T and K?\" said Annie.\n\n\"Yes! For Teddy and Kathleen!\" said Jack. \"The letters just appeared on the page when I was outside! They were there! Really!\"\n\n\"I believe you,\" said Annie. She jumped up from the computer. \"Let's go.\"\n\n\"Wait, I have to get my backpack from upstairs,\" said Jack.\n\n\"Forget it! Come on! The tree house must be waiting for us!\" said Annie.\n\n\"Okay, okay,\" said Jack. He quickly shoved his notebook and pencil into a pocket of his coat.\n\nAnnie grabbed her jacket. \"Mom! Dad! We're going to take a little break from our homework!\" she called.\n\n\"Okay, but make it short! We have to leave for the theater by seven!\" their dad called from the kitchen.\n\n\"We will!\" said Jack.\n\nJack and Annie headed outside. They ran over the melting snow in their front yard and up the sun-sparkling sidewalk. They charged across the street and into the Frog Creek woods. They hurried between the windblown trees until they came to the tallest oak.\n\nHigh in the branches was the magic tree house. Their friends from Camelot, Teddy and Kathleen, were looking out the window.\n\n\"Hello!\" called Kathleen.\n\n\"Hi!\" shouted Annie, waving.\n\n\"Good trick with the magic letters!\" Jack called.\n\n\"We thought you'd like that!\" said Teddy. \"I just learned how to do it!\"\n\nAnnie grabbed the rope ladder and started up. Jack followed her. They climbed into the tree house and hugged the young enchanters.\n\n\"So what's up today?\" asked Jack.\n\n\"Where does Merlin want us to go now?\" asked Annie.\n\n\"Merlin wants you to go to Galway, Ireland,\" said Kathleen.\n\n\"Ireland? Cool!\" said Annie.\n\n\"Morgan sent us to Ireland once before\u2014to the ninth century,\" said Jack.\n\n\"Yes. Well, this time you will go to Ireland in the _nineteenth_ century,\" said Teddy. \"To 1862, to be exact. Your mission is to find an imaginative and creative girl named Augusta.\"\n\n\"Augusta doesn't know yet what her talents are,\" said Kathleen. \"She lives in a time when it is not easy for girls to explore their creativity. Your mission will be to inspire her, so she can give her gifts to the world.\"\n\n\"What does that word mean exactly?\" asked Annie. _\"Inspire?\"_\n\n\"'tis a beautiful word,\" said Kathleen, her sea-blue eyes shining. \"It means to breathe life into a person's heart, to make her feel joyful to be alive.\"\n\n\"That _is_ beautiful,\" said Annie.\n\n\"You may need some magic to help you,\" said Teddy. From the corner of the tree house, he picked up the magic trumpet that had helped them on their last journey. \"Only this time...\"\n\nTeddy handed the trumpet to Kathleen. She held the shiny brass instrument for a moment. Then she tossed it into the air. The trumpet spun like a whirlwind. There was a flash of blue light\u2014and the trumpet was gone! In its place was a thin silver pipe with six holes.\n\n\"What's that?\" breathed Jack.\n\n\"An Irish whistle,\" said Kathleen. She plucked the instrument from the air. \"When you face great danger, one of you must play it. It will make magical music. And anything the other one sings will come true.\"\n\n\"But remember,\" said Teddy, \"its magic will work only once.\"\n\n\"Right,\" said Annie.\n\n\"Thanks,\" said Jack. He took the Irish whistle from Kathleen and put it into his pocket. \"And did Morgan send a research book to give us information?\"\n\n\"Not this time,\" said Teddy. \"Morgan wants you to draw upon your own experiences in life to help you on this journey.\"\n\n\"No problem,\" said Annie.\n\nJack wasn't sure about that. He liked having a book of facts to help them.\n\n\"So how do we find Augusta?\" said Annie.\n\n\"It should be easy to find her,\" said Teddy. \"When you land in the county of Galway, Ireland, just ask anyone for directions to the Big House.\"\n\n\"Hold on.\" Jack pulled out his notebook and wrote:\n\n_County of Galway, Ireland_  \n _Augusta_  \n _Big House_\n\n\"Got it,\" said Jack. \"But how do we get to Ireland in the first place if we don't have a research book?\"\n\n\"Point to the notes you just made and make your wish,\" said Kathleen.\n\n\"And when you are ready to come home,\" said Teddy, \"use the Pennsylvania book as you usually do.\"\n\n\"Got it,\" said Jack.\n\n\"Go now, and help Augusta,\" said Kathleen. \"She needs you.\"\n\nJack pointed to the words _Galway, Ireland_ in his notebook. \"I wish we could go there!\" he said.\n\n\"Bye!\" Annie said to Teddy and Kathleen.\n\n\"Farewell!\" said Kathleen.\n\n\"Good luck!\" said Teddy.\n\nThe wind started to blow.\n\nThe tree house started to spin.\n\nIt spun faster and faster.\n\nThen everything was still.\n\nAbsolutely still.\n\n#\n\nA cold wind blew rain into the tree house. Jack shivered. He was wearing an old overcoat and ragged trousers. Annie wore a scarf, a shawl, and a long red wool dress. They both wore scuffed, worn boots.\n\n\"So where are we?\" said Jack. He and Annie looked out the window.\n\nThe tree house had landed in a tree at the edge of a green meadow dotted with woolly white sheep. Next to the meadow was a narrow lane. It ran uphill between low stone walls. Through the drizzle, Jack could see mist-covered mountains in the distance and a flash of silver sea.\n\n\"It looks like a scene in a fairy tale,\" said Annie.\n\n\"Yeah, a fairy tale with bad weather,\" said Jack.\n\n\"I wonder where the Big House is,\" said Annie.\n\n\"I don't know, but I'd like to get inside it now,\" said Jack.\n\n\"Me too,\" said Annie, shivering. \"Let's go.\"\n\nJack crammed his cold hands into the pockets of his torn coat. In one pocket, he felt his notebook and pencil. In the other, he felt the Irish whistle. \"I've got the whistle,\" he said.\n\n\"Good,\" said Annie. She held her red skirt and started down the rope ladder. Jack climbed down after her.\n\nAnnie pulled her shawl tightly around her shoulders. Jack turned up the collar of his coat. As they tramped through the wet meadow, he felt cold water seeping through holes in his boots.\n\nJack and Annie climbed over a stone wall onto the muddy lane. A horse-drawn wagon was rattling toward them down the hill. The wagon was filled with squealing pigs.\n\n\"Excuse me!\" Annie called to the driver. \"Can you tell us where the Big House is?\"\n\nAn old man with a tired, rugged face pointed back up the hill.\n\n\"Thanks!\" said Jack.\n\nThe large wooden wheels of the wagon rumbled past, splashing Jack and Annie with mud.\n\n\"Yuck!\" said Annie.\n\n\"Now we're cold and wet and _dirty_ ,\" said Jack.\n\n\"Yeah, we're going to look great when we get to the Big House,\" said Annie.\n\n\"So what do we do when we get there?\" asked Jack.\n\n\"When we find Augusta, maybe we tell her that Teddy and Kathleen sent us,\" said Annie, \"like we told Louis Armstrong in New Orleans.\"\n\nJust thinking about their adventure with Louis Armstrong made Jack smile. \"I don't know if that will work,\" he said. \"His world seemed so different from this world. There it was so noisy and busy. Here it feels lonely.\"\n\n\"Well, we won't know until we find Augusta,\" said Annie. \"Let's go.\"\n\nJack and Annie lowered their heads. They plodded up the lane, sloshing in and out of giant puddles. When they reached the top of the hill, they stopped. The muddy lane wound down, past more sheep meadows and some cottages, past a long stable and several barns.\n\nAt the end of the lane was a large open gate that led onto the grounds of a white mansion. Gray smoke rose from the mansion's chimneys.\n\n\"The Big House!\" said Annie.\n\n\"Maybe the people who live there will invite us inside to get warm and dry by a fire,\" said Jack.\n\nJack and Annie started down the lane. As they passed the sheep meadows, black-and-white dogs barked at them. When they walked by several boys hauling wet hay, the workers looked up and eyed them suspiciously.\n\nJack was relieved to get to the gate and head toward the Big House. When they reached the front door, Annie lifted the heavy knocker and let it drop.\n\nA moment later, the door opened. A pale teenage girl looked out. \"Who are you? Why are you here?\"\n\n\"Uh... well...,\" started Jack.\n\n\"Are you the ones the butler sent for?\" the girl asked.\n\n\"The butler?\" said Jack.\n\n\"Yes, we are!\" said Annie.\n\n\"Then you should go round to the _back_!\" the girl said. Before Jack or Annie could ask for Augusta, the girl slammed the door in their faces.\n\n\"Nice,\" said Jack.\n\n\"I hope _she_ wasn't Augusta,\" said Annie.\n\n\"Why did you tell her 'yes'?\" asked Jack.\n\n\"It's a way to get inside the Big House,\" said Annie. \"Come on.\"\n\nJack and Annie tramped through the mud to the back of the mansion. They stopped at a door beneath a large smoking chimney. Annie knocked again.\n\nThis time a young red-haired girl in a cap and apron opened the door. \"Yes?\" she said.\n\n\"Is your name\u2014\" started Annie.\n\n\"Who is it, Molly?\" someone called from inside.\n\n_Molly? So_ she's _not Augusta_ , thought Jack.\n\n\"Who are you?\" Molly asked them.\n\n\"We're the ones the butler sent for,\" said Annie.\n\n\"You?\" said Molly. She looked doubtful. \"Well, come in and see him then.\"\n\nJack and Annie stepped inside.\n\n\"He's in the kitchen,\" said Molly. She started down the hall.\n\nJack and Annie followed Molly to the doorway of a dimly lit kitchen. The kitchen smelled of fish and onions. Pots and pans hung from a long rack over a big wooden table. A stout older woman was bent over the table, rolling out dough.\n\n\"Cook, here are the ones the butler sent for,\" said Molly.\n\nThe cook looked up from her dough and squinted at Jack and Annie. \" _You're_ the ones he sent for?\" she said.\n\n\"Uh, yes, ma'am, that's us,\" said Annie.\n\nThe cook turned toward the fireplace. Next to the fire an ancient-looking man with white whiskers sat slumped in a chair, snoring. \"Mr. O'Leary!\"\n\nThe old man jerked and opened his eyes.\n\n\"The ones you sent for are here!\" the cook shouted, as if the man were hard of hearing.\n\nThe groggy butler peered at Jack and Annie. \"I sent for _you_?\" he growled. \"Not possible! I sent for a coach driver and a blacksmith.\"\n\n\"Really?\" said Annie. \"I guess there was a mistake. But maybe there are some other jobs we could do around here.\"\n\n\"Well, what are you good for?\" asked the butler.\n\n\"What do you mean?\" said Jack.\n\n\"Do you know how to sweep the inside of a chimney?\" said the old man.\n\n\"Um... no,\" said Jack.\n\n\"Pluck a chicken?\" the cook asked.\n\n\"No way,\" said Annie.\n\n\"What about rats?\" the butler said.\n\n\"What about them?\" asked Jack.\n\n\"They're all over the cellar,\" said the cook. \"Can you catch 'em?\"\n\n\"I\u2014I don't think so,\" said Jack.\n\n\"Then you're no good to us here!\" snorted the butler. \"Be on your way!\"\n\nAt that moment, Jack heard the back door open and shut. A girl about Jack's age stepped into the kitchen. She wore a red cape and carried two large, empty baskets. Her wet hair was parted neatly down the middle and pulled into a tight bun in the back.\n\n\"Ah!\" said the cook. \"Welcome back, Miss Augusta!\"\n\n#\n\nJack and Annie looked at each other. _Augusta!_\n\nThe girl put down her baskets and took off her wet cape.\n\n\"Did you deliver your cakes to the poor, Miss Augusta?\" asked Molly.\n\n\"Yes, Molly,\" said Augusta. \"I visited seven cottages today.\"\n\n\"Seven? In this weather? You're an angel, Miss Augusta!\" said Molly. \"Always so kind to the poor.\"\n\n\"It is my duty, Molly,\" the girl said, \"to help those less fortunate than myself.\" Her gaze rested on Jack and Annie. \"And who are _these_ poor children?\"\n\n\"They're looking for work, miss,\" said the cook. \"But I'm afraid they're sorry creatures, not good for anything. I was just sending them away.\"\n\n\"Oh, surely we must not turn them out so quickly, Cook,\" said Augusta. \"How tired and miserable they look.\"\n\nJack didn't think they looked _that_ bad.\n\n\"We _are_ tired and miserable,\" Annie said. Her voice sounded sad. Her shoulders sagged.\n\n_Oh, brother_ , Jack thought. Annie was really acting her part.\n\n\"My poor dears, you must both come into the parlor and rest a bit,\" said Augusta.\n\n\"We would like that,\" Annie said pitifully.\n\n\"Follow me,\" said Augusta.\n\n\"Miss Augusta, surely you're not taking those dirty children into the parlor!\" said the cook.\n\n\"We must always be kind to the poor, Cook, no matter how dirty they are,\" said Augusta. \"We should give them something to drink if they are thirsty, and something to eat if they are hungry.\"\n\n\"You are _too_ kind, Miss Augusta,\" said Molly, shaking her head.\n\n\"Well, at least make them take off their filthy boots,\" said the cook.\n\nJack and Annie pulled off their boots and socks and set them by the door. Their feet were red and raw-looking.\n\nAugusta took two peeled potatoes from a bowl and put them in her pocket. Then she picked up a lit candle from the hearth. \"Come, let me take you to the parlor,\" she said to Jack and Annie.\n\n\"Thanks, Augusta,\" said Annie.\n\n\"Show some respect!\" the cook called after Annie. \"Call her ' _Miss_ Augusta'!\"\n\n\"Sorry!\" said Annie. \"Thanks, _Miss_ Augusta.\"\n\nJack rolled his eyes. Why should he call her 'miss'? Augusta didn't look like she was any older than he was!\n\nHolding her flickering candle, Augusta led Jack and Annie out of the kitchen. The wooden floor creaked as they walked barefoot through a narrow hallway.\n\n_How are we ever going to inspire this strange, serious girl?_ wondered Jack. _She acts as if she's already a grown-up and treats Annie and me like babies_.\n\n\"We'll sit in here, children,\" said Augusta. She directed Jack and Annie into a large room with heavy curtains and dark furniture. The pale teenage girl who'd answered the front door sat on a sofa, knitting. Another teenage girl knitted beside her. They scowled when they saw Jack and Annie.\n\n\"What are you doing, Augusta?\" asked the pale girl. \"Why are you bringing those two into the parlor?\"\n\n\"I invited them to tea, Gertrude,\" said Augusta. She turned to Jack and Annie. \"Pay no attention to my sisters,\" she said. \"Please, sit down.\"\n\n\"Augusta, have you gone mad?\" said Gertrude. \"You cannot invite these two ragamuffins to sit in here!\"\n\n\"Mother will be furious,\" said Augusta's other sister. \"They're filthy! They're not even wearing shoes!\"\n\nJack looked down at his muddy clothes and cold red feet.\n\n\"Cook made them remove their muddy shoes in the kitchen, Eliza,\" Augusta said. \"I only wish I had nice, dry shoes to give them. Sit down, children,\" she said to Jack and Annie again.\n\nJack and Annie slowly sat down.\n\n\"You're going to get into trouble, Augusta...,\" said Gertrude.\n\n\"Mother will never approve,\" said Eliza. Both sisters shook their heads as they went back to their knitting.\n\nAugusta ignored her sisters and walked to a silver teapot on a sideboard. \"Would you like some hot tea, my poor dears?\" she asked Jack and Annie.\n\n\"Yes, Miss Augusta,\" said Annie.\n\nJack nodded. Hot tea sounded good. He still felt chilled from the cold wind and rain. There was a fireplace in the dreary parlor, but no fire was lit. Everything in the room seemed dark and gloomy, except for a few books on a table.\n\nAs Augusta poured tea into fancy china cups, Jack leaned closer to get a look at the books. One was titled _The Plays of William Shakespeare_. Another was called _The Tales of King Arthur_. Jack smiled to himself. Seeing those book titles made him feel a little more comfortable.\n\nAugusta carried cups of tea to Jack and Annie. Then she pulled the potatoes out of her pocket and gave one to each of them.\n\n\"Thank you, Miss Augusta,\" said Annie.\n\nJack took a sip of tea, but it was too bitter and hot to drink. He took a bite of his cold potato, but it was too hard to chew.\n\n\"So, Miss Augusta, what do you like to do around here?\" Annie asked. \"What inspires you?\"\n\nAugusta looked puzzled. \"I do not know what you mean,\" she said.\n\n\"What about reading books?\" said Jack. \"Have you read those books?\" He pointed to _The Tales of King Arthur_ and _The Plays of William Shakespeare_.\n\n\"Those books belong to my brothers,\" said Augusta.\n\n\"The time has not come for Augusta to read such books,\" said her sister Gertrude.\n\n\"Not until she is older,\" said her sister Eliza.\n\n\"Why?\" asked Jack.\n\n\"Mother says _The Tales of King Arthur_ and the plays of Shakespeare are not for young ladies,\" said Augusta.\n\n\"Really?\" said Jack.\n\n\"Yes. But I'm afraid I sometimes peek at my brothers' books,\" Augusta said to Jack and Annie in a low voice. \"I love stories. I remember every story I read or hear.\"\n\n\"I love stories, too,\" said Annie. \"And I love books.\"\n\nThe two older sisters smiled. \"Keep striving, my dear,\" said Eliza. \"Perhaps one day you will learn to read.\"\n\n\"I already know how to read,\" said Annie. \"Jack and I read lots of books.\" She pointed to the books on the table. \"In fact, we know tons about King Arthur, and we go see plays by Shakespeare with our parents. And one time, we even acted in a play by Shakespeare\u2014 _A Midsummer Night's Dream_.\"\n\n\"It was at our school,\" Jack broke in, before Annie could tell them that she and Jack had actually met Shakespeare himself!\n\n_\"A Midsummer Night's Dream?\"_ said Augusta. She looked surprised.\n\n\"Don't listen to them, Augusta,\" said Gertrude. \"I doubt these children have ever been to school\u2014much less acted in a play by William Shakespeare.\"\n\n\"I suspect you are quite right, Gertrude,\" someone said.\n\nA tall woman was standing in the doorway of the parlor. She wore a long black velvet dress and stood very straight. There was an icy look on her face as she stared at Jack and Annie.\n\n\"Oh! Mother!\" said Eliza.\n\n#\n\n\"Hi there!\" Annie said cheerfully.\n\nAugusta's mother did not reply. She was staring at Jack's bare feet. Her expression made him sink down in his chair.\n\n\"Do not blame Eliza or me, Mother,\" said Gertrude. \"These are Augusta's friends, not ours.\"\n\n\"I took pity on them, Mother,\" said Augusta. \"They were wet and miserable.\"\n\nHer mother finally smiled. \"Yes, daughter, I imagine they were. It's very nice to have pity for the poor, but dirty children should not be sitting in our parlor.\"\n\n\"They were hungry, Mother,\" said Augusta.\n\n\"Yes, and I see you have given them food,\" said her mother. \"So it is time to take them out of the house _now_.\"\n\nJack and Annie stood up. Jack was happy to leave. He felt like Augusta and her mother were talking about stray dogs or cats.\n\nBut Augusta sat very still and just stared at her mother.\n\n\"Go on\u2014get them out of here, Augusta,\" said Gertrude. \"They are not clean! They might even have bugs in their hair.\"\n\nThe mere mention of bugs made Jack's scalp itch. He and Annie both scratched their heads.\n\n\"See!\" said Gertrude.\n\n_\"Augusta...,\"_ her mother said in a stern voice.\n\n\"Oh, all right! All right!\" said Augusta, standing up. \"I was trying to be kind! Come with me, please,\" she said to Jack and Annie. \"I'll lead you down the lane a bit, at least past the sheepdogs.\"\n\nAs Augusta started out of the room, her mother stopped her and pinched her shoulders. \"Carry yourself straight, daughter,\" she said.\n\nJack couldn't imagine how the girl could carry herself any straighter.\n\nAugusta led Jack and Annie back down the dark hallway, through the fish-smelling kitchen, past the ancient butler sleeping by the fire and the three kitchen maids and the cook. Without a word, she grabbed her red cape and pulled it around her.\n\n\"Where are you going, Miss Augusta?\" asked Molly.\n\n\"I have been ordered to send these poor children back out into the storm,\" Augusta said.\n\nJack and Annie forced their feet into their stiff, wet socks and boots. Augusta held the door for them, then followed them outside, slamming the door shut behind her.\n\nEven though it was still rainy and windy, Jack felt much happier outside the Big House than inside it. He and Annie followed Augusta past the gates and out to the lane. Augusta walked stiffly, leading them like a mother duck.\n\n\"What are we going to do about her?\" Annie whispered to Jack.\n\n\"I don't know,\" whispered Jack. \"She doesn't seem very creative or imaginative to me.\"\n\n\"Well, we have to _inspire_ her! Come on!\" said Annie. She and Jack hurried to catch up with Augusta.\n\n\"Miss Augusta!\" said Annie, walking alongside her. \"Do you like to sing? Dance? Paint? Play a musical instrument? Anything creative like that?\"\n\n\"No,\" said Augusta. She sounded angry. Jack figured she must be mad at her mother for kicking them out of the Big House.\n\n\"Well, what about nature?\" said Annie.\n\n\"What about it?\" asked Augusta.\n\n\"Walking in the woods?\" said Annie. \"Trees? Birds? Does anything like that inspire you?\"\n\n\"Not anymore. I _was_ close to nature once,\" Augusta said. \"I used to roam the woods with my younger brothers. They said I was like a robin with the eye of a hawk. I knew where to find the caves of the otters. I knew where to find the nests of wild birds.\"\n\n\"That's so cool,\" said Annie.\n\n\"I knew where the deer lay down to sleep,\" said Augusta. \"I knew the names of every tree: oak, beech, elm, hazel, larch, pine....\" Augusta's voice grew a little wobbly, as if she might cry. \"But I'm not allowed to roam the woods with my brothers anymore. Mother says it's not proper for a young lady.\"\n\n\"That's so sad!\" said Annie.\n\n\"Never mind,\" said Augusta, lifting her chin. \"Let us not talk about me anymore. Let us try instead to help the two of you. Cook said you were not good for anything. Why would she say that?\"\n\n\"They asked us if we were good for cleaning chimneys, plucking chickens, or catching rats,\" said Annie, \"and we said no.\"\n\n\"Then you must find other ways to make yourselves useful,\" said Augusta, \"or you will never find your way in the world. Can you shear sheep?\"\n\n\"We've never tried it,\" said Jack.\n\n\"Milk cows? Churn butter? Weave a shawl?\" Augusta asked impatiently. \"Hunt rabbits with hounds?\"\n\n\"Oh, never that!\" said Annie. Jack laughed.\n\nAugusta frowned. \"This is nothing to laugh about. Every day, you must ask yourselves: what am I good for?\"\n\nActually, that was a good question, Jack thought. What _was_ he good for?\n\n\"And I would like to give you a further piece of advice,\" said Augusta. \"Never make up stories about yourselves that aren't true.\"\n\n\"What do you mean?\" said Jack.\n\n\"You never acted in a play of Shakespeare's, did you? Tell the truth now,\" said Augusta.\n\n\"We did,\" said Jack. \"My sister _was_ telling the truth. We were both in _A Midsummer Night's Dream_.\"\n\n\"We played wood fairies,\" said Annie. \"We had green costumes, and Jack gave a little speech and I danced and sang.\"\n\nAugusta shook her head. \"You poor dears,\" she said. \"I know you only make up these wild stories because your real lives are so miserable, but\u2014\"\n\n\"Wait a minute. Stop,\" said Jack. \"What's wrong with you? Why do you act so snobby?\"\n\n\"Snobby? _Me?_ \" Augusta looked confused.\n\n\"Jack\u2014\" said Annie.\n\n\"No, I'm serious,\" Jack said to Annie. \"She thinks she's better than us.\"\n\n\"No, I don't!\" Augusta said, stunned. \"I'm not like that at all! Each day I walk several miles to town to give cakes and clothes to poor children like yourselves.\"\n\n\"That's nice,\" said Jack. \"But you think you're better than those poor children, don't you? You'd never want to be real friends with them, would you?\"\n\n\"What you say about me is _not_ true!\" Augusta said to Jack. \"I love the poor! Why, my favorite friend in all the world is quite poor and has never been to school. Some say she's even a little cracked in the head, but I love her dearly!\"\n\n\"Who's that?\" asked Annie.\n\n\"Mary! Mary Sheridan, our old nursemaid,\" said Augusta. \"I'll take you to meet her. Mary will tell you the truth about me! Come along!\"\n\nAugusta ran from the lane and across the muddy grass, her red cape flying in the wind.\n\n\"Um... I don't think you inspired her,\" Annie said.\n\n\"I know, I'm sorry,\" said Jack. \"I just couldn't take her attitude any longer.\"\n\n\"Well, get over it,\" said Annie. \"We're supposed to help her, not annoy her.\"\n\n\"She was annoying _me_!\" said Jack.\n\n\"Yeah, I know,\" said Annie. \"Me too. But we've got a mission. Come on.\"\n\nJack and Annie followed Augusta across the grass to a small white cottage with a straw roof. Augusta banged on the door, scaring away birds eating crumbs by the front steps. \"Mary! Mary! It's me, Augusta!\" she called.\n\n\"Come in, my dear,\" a voice answered.\n\nAugusta lifted the latch and led Jack and Annie inside.\n\nWrapped in a brown shawl, Mary Sheridan was stroking an orange cat by an open fire. She had ragged white hair and bright blue eyes. Her crooked smile revealed a few missing teeth.\n\n\"One and twenty welcomes on this _wonderful_ winter day!\" said the old woman.\n\n#\n\n\"Hello, Mary!\" said Augusta. She kissed the old woman on her wrinkled cheek.\n\nWith the firelight on her face, Mary seemed to glow. Her warm, snug cottage was the opposite of the Big House. It smelled of damp leaves and moss, bread and chocolate. Firelight danced on the earthen floor and stone walls. Rain dripped through the roof, _pinging_ into a couple of tin buckets.\n\n\"And who do we have here, Miss Augusta?\" Mary asked.\n\n\"Two poor children from town,\" said Augusta. \"I want you to tell them about me\u2014how I truly love the poor and try to help them.\"\n\nMary smiled. \"Please, sit down first,\" she said.\n\nJack, Annie, and Augusta sat down on three rickety wooden chairs.\n\n\"Would you children like some hot cocoa?\" Mary asked.\n\nJack and Annie nodded eagerly.\n\n\"Yes, please, Mary,\" said Augusta. \"But would you tell them\u2014\"\n\n\"Yes, I will tell them all about you,\" said Mary. She picked up a pot sitting on the hearth. She poured steaming cocoa into three mugs and handed them to Jack, Annie, and Augusta.\n\nThe cocoa smelled delicious. Jack took a sip and licked his lips. \"Yum,\" he said. His insides felt warm for the first time all day.\n\n\"Now, Mary?\" asked Augusta.\n\n\"In time, my child,\" Mary said to Augusta. \"Tell me, what have you been doing today?\"\n\n\"I delivered cakes in town,\" Augusta said proudly. \"Then I found these poor children in our kitchen looking for work. I've tried to help them. But they say I am snobby. I brought them here so you could tell them the truth about me.\"\n\n\"Ah, I see, Miss Augusta. How did you try to help them?\" said Mary.\n\n\"I've tried to discover what they are good for,\" said Augusta. \"But it appears they are good for nothing.\"\n\n\"Really?\" Mary fixed her twinkling eyes on Jack and Annie. \"Well, let us start with this, children: tell me something you like to do. No, wait\u2014what do you _love_ to do?\"\n\n\"Uh... well, I love to read,\" said Jack.\n\n\"And write,\" said Annie.\n\n\"Read and write?\" said Augusta. \"I don't think so.\"\n\nMary ignored Augusta and kept looking at Jack and Annie. \"What do you like to read and write?\" asked Mary.\n\n\"Facts mostly,\" said Jack. \"True stories.\"\n\n\"Jack writes facts down all the time,\" said Annie.\n\n\"Honestly, Mary,\" said Augusta. \"Soon they'll be telling you that they are actors, too\u2014and have performed in a play by William Shakespeare. Can we talk about me now?\"\n\n\"Actually, that's _true_ ,\" Jack said to Mary. \"We were in _A Midsummer Night's Dream_. Annie and I were fairies. I had stage fright, but Will\u2014\"\n\n\"William Shakespeare _himself_ ,\" said Annie.\n\n\"He helped me get over it,\" said Jack. He looked straight at Augusta.\n\nAugusta rolled her eyes.\n\n\"Will was _so_ nice,\" said Annie.\n\n\"And smart,\" said Jack.\n\n\"Of course he was!\" said Mary. \"You can tell that from his stories.\"\n\n\"Oh, please stop. Don't tell Mary those ridiculous things!\" Augusta said. \"What about me, Mary?\"\n\n\"Wait, child, I have a question for them,\" Mary said. She leaned forward and spoke in a whisper. \"Where is summer? Can you answer me that?\"\n\n_Mary's question doesn't make sense_ , Jack thought.\n\n\"I don't know. Do you know where summer is, Mary?\" asked Annie.\n\n\"Summer is hiding with the Shee!\" said Mary, laughing.\n\n\"The Shee? What's that?\" said Annie.\n\n\"Surely, you must know the Shee,\" said Mary. \"That's what we Irish call our fairies. In the winter, the Shee steal all the warmth and sunshine, leaving us to suffer with the cold and rain!\"\n\nAnnie laughed, too.\n\n\"So you've played the parts of fairies in a play by Shakespeare?\" said Mary. \"They're just like our Shee. Have you seen the Shee here in Ireland?\"\n\n\"Mary!\" Augusta said impatiently.\n\n\"Not yet,\" said Annie.\n\n\"That's a shame,\" said Mary. \"I have seen them. This is a true story\"\u2014she looked at Jack and smiled\u2014\"with facts. You might want to write them down.\"\n\n\"Oh. Sure,\" said Jack. He pulled his notebook and pencil out of his pocket. Augusta looked surprised.\n\nMary leaned close to Jack and Annie again. Her eyes were shining and her voice was hushed. \"One day long ago, a lonely young girl took a walk in an old forest,\" said Mary. \"All was still, until joyful music began coming from a hidden world....\"\n\nJack loved Mary's way of telling a story. He wrote down:\n\nold forest, all still  \njoyful music, hidden world\n\nAugusta frowned. \"So, I guess you _can_ write,\" she muttered.\n\n\"Suddenly there came a spinning wind,\" said Mary, \"and a cloud so bright, and a beam of light poured over a river!\"\n\nJack quickly wrote:\n\nspinning wind, bright cloud beam of light, river\n\n\"Then they came, rumbling and thundering!\" exclaimed Mary.\n\n\"Mary,\" said Augusta. She sounded impatient.\n\nBut Mary kept talking. \"Some with wings, some on horses of white! Queens and kings! In robes and gowns the colors of summer, fall, winter, and spring!\" Jack wrote:\n\nsome with wings  \nwhite horses  \nqueens, kings\n\n\"They galloped in a circle, a blinding swirl!\" said Mary. \"They swept up that lonely girl and carried her across the river to their secret hollow hill! Had she gone inside, she would have become very small and seen many wondrous sights!\"\n\nJack wrote:\n\ntake lonely girl  \nhollow hill of Shee  \nwondrous sights\n\nJack looked up from his notebook, waiting for Mary to go on. When she spoke again, her voice was very soft. \"But the girl grew afraid and ran home instead.\"\n\nMary sat back in her chair and closed her eyes. The only sound in the cottage was the crackling of the fire and the _pinging_ of rain into the tin buckets.\n\n\"Mary?\" Annie said softly. \"Are you the girl in the story?\"\n\nMary opened her eyes. \"I will never tell,\" she said.\n\n\"Oh, Mary,\" said Augusta, \"such tales!\" She turned to Jack and Annie. \"Mary still believes in the impossible.\"\n\n\"Aye, I do, I surely do,\" said Mary. \"Every night I leave a bit of milk on my windowsill for the Shee. I leave crumbs at my door. They eat them, too.\"\n\n\"Mary, the _birds_ eat the crumbs!\" said Augusta.\n\n\"Yes, the birds are hungry also,\" said Mary. \"But the Shee pick over the crumbs first! At twilight, they steal across the river from their hidden hollow hills. Just ask the old fishermen of County Galway. Ask the farmers and nursemaids.\"\n\nAugusta shook her head sadly. \"Mary, only simple-minded folk still believe in such things,\" she said. \"Educated people know what is true and what is not true.\"\n\n\"No, child,\" said Mary. \"They only know what they _think_ is true....\"\n\nAugusta straightened her shoulders. \"Well. We should be going now, Mary,\" she said. \"So could you please tell these children the truth about me now?\"\n\n\"Yes,\" said Mary. She turned to Jack and Annie. \"Do you children have names?\"\n\nJack smiled. This was the first time today anyone had asked them their names.\n\n\"Yes. Our names are Jack and Annie,\" said Annie.\n\n\"Well, Jack and Annie, thank you for coming to visit me today. I can tell that you are very special,\" said Mary.\n\n\"What about _me_ , Mary?\" Augusta asked. \"Am I special?\"\n\n\"Yes, child, you are,\" said Mary. She turned to Jack and Annie. \"Augusta is special, too, but in a different way.\"\n\n\"How am I different, Mary?\" asked Augusta.\n\n\"You try very, very hard to be good, and you are very smart. But you\u2014\" Mary stopped.\n\n\"What, Mary? I\u2014what?\" said Augusta.\n\n\"You are not happy,\" said Mary. \"And that breaks my heart.\"\n\nAugusta's eyes filled with tears.\n\n\"Oh, Augusta, don't cry,\" said Annie. She reached out to take Augusta's hand, but the girl stepped back.\n\nAugusta wiped her eyes. \"That's silly. I'm happy enough. I know I've never seen the Shee and I never will. But I don't care anymore. And if you like these miserable children more than me, Mary\u2014well, that's fine!\"\n\nAugusta ran to the door and opened it. The damp air swept inside as she rushed out of the cottage. Through the open doorway, Jack and Annie could see Augusta's red cape flying behind her.\n\n#\n\nJack sighed. Their mission really seemed hopeless now.\n\n\"We'd better go find her,\" said Annie.\n\n\"She won't go far,\" said Mary. \"My poor Augusta... she has a fine mind and a brave heart. But she is so unhappy.\"\n\n\"Why is she so unhappy, Mary?\" asked Annie.\n\n\"Yeah, what is her problem?\" said Jack.\n\n\"More than any of her brothers and sisters, Miss Augusta loved my stories,\" said Mary. \"Remembered every one of them, she did.\"\n\n\"Really?\" said Jack.\n\n\"Yes, she would repeat them back to me, word for word,\" said Mary.\n\n\"That's amazing,\" said Annie.\n\n\"She loved the stories so much that she grew desperate to see the Shee for herself,\" said Mary. \"At night she would carry a lantern across the fields, calling for them. By day, she poked and prodded every part of the farm. Why, she even used a magnifying glass, scouring the earth for tiny footprints! But I'm afraid she never found them.\"\n\n\"Why?\" said Annie.\n\nMary sighed. \"Because she looked for them with her head and not her heart,\" she said. \"Eventually she gave up and stopped searching. She didn't even want to hear the stories anymore. She's been a dutiful\u2014but sorrowful\u2014child ever since.\"\n\n\"That's terrible,\" said Annie. \"What can we do to help her?\"\n\n\"There is only one thing you can do,\" said Mary.\n\n\"What?\" breathed Jack.\n\nMary leaned forward in her chair. Her blue eyes seemed to stare right through Jack and Annie. \"You must show her the magic,\" she said.\n\n_What?_ thought Jack. _Does Mary know about the magic tree house?_ \"What do you mean?\" he asked.\n\n\"I know that you children are like me\u2014you see things that others don't,\" said Mary. \"Help Augusta see them, too. Help her find the magic in the fields and in the forest.\"\n\nFor a moment, Jack and Annie didn't say anything. The wind blew through the open door. The fire crackled.\n\nThen Annie took a deep breath. \"Okay,\" she said. \"We know exactly what to do.\"\n\n\"We do?\" said Jack.\n\n\"Yep, we'll talk about it outside,\" said Annie. \"Thanks, Mary. We'll find Augusta and take care of everything.\"\n\nJack and Annie stood up to go.\n\n\"One and twenty fare-thee-wells on this wonderful winter day,\" said Mary.\n\n\"One and twenty to you, too,\" said Jack. Then he and Annie left the cozy cottage, scattering the winter birds by the front door.\n\nIt had warmed up a little outside. The rain had stopped, but fog hung heavily over the sheep fields. The ground was soggy with mud.\n\nJack could barely make out Augusta's red cape through the fog. She was across the lane, sitting on a stone wall at the edge of the sheep meadow. \"Mary was right, she didn't go far,\" Jack said. \"So how do we show her the magic, Annie?\"\n\n\"Easy,\" said Annie. \"We play our magic whistle.\"\n\n\"No, we can't do that,\" said Jack. \"We're supposed to save the whistle for a moment of great danger.\"\n\n\"That moment is _now_!\" said Annie. \"Come on.\"\n\n\"Hold on,\" said Jack. \"What great danger are we facing right now?\"\n\n\"Not us. _Augusta_ ,\" said Annie. \"She faces the great danger of losing all hope and happiness and being bored and sad for the rest of her life\u2014and never being inspired and never sharing her gifts with the world! It's almost too late already!\"\n\n\"Okay, okay,\" said Jack. \"But are we just going to go up to her and start blowing the whistle and singing? That seems pretty weird.\"\n\n\"Hmm... yeah, it does,\" said Annie.\n\n\"How about this?\" said Jack. \"We'll tell Augusta that we want to put on a play for her.\"\n\n\"A play?\" said Annie.\n\n\"Yep,\" said Jack. \"We can tell her we want to prove that we weren't lying, that we really were in a play by Shakespeare.\"\n\n\"Oh, cool,\" said Annie. \"Then what?\"\n\n\"We play the magic whistle,\" said Jack. \"We sing about the Shee. We make them appear\u2014like in Mary's story\u2014galloping and thundering! Augusta sees them. She gets inspired. Our mission's done.\"\n\n\"Perfect!\" said Annie. \"Let's go!\"\n\nJack and Annie hurried across the lane to the stone wall. \"Excuse us, Miss Augusta,\" said Annie. \"We just had a really great idea! Want to hear it?\"\n\nAugusta didn't answer. She kept staring at the ground.\n\n\"How would you like to see a play?\" said Jack.\n\nAugusta looked up. \"A play?\" she said.\n\n\"We want to put on our own play for you,\" said Annie.\n\n\"Why?\" said Augusta.\n\n\"Because it's really good,\" said Jack. \"And maybe it will prove to you that we really _were_ in a play by Shakespeare.\"\n\nAugusta looked doubtful.\n\n\"Come on, you'll love it,\" said Annie. \"Do you know a quiet spot where no one can bother us?\"\n\nAugusta bit her lip and looked around. Then she stood up. \"All right,\" she said. \"The river near the old forest. I used to go there with my brothers.\"\n\n\"Great!\" said Annie.\n\nAnnie and Jack followed Augusta through the rain-soaked, misty meadow. They walked past grazing sheep, then down a slope toward a wide, rushing river. The river separated the sheep's meadow from an old forest. Jack could barely see the trees through the ghostly fog.\n\nAugusta stopped on a low ridge above the riverbank, near some large rocks. \"Here,\" she said.\n\n\"Good, those rocks can be our stage,\" said Jack.\n\nJack and Annie climbed the pile of small boulders and stood on a large, flat rock.\n\n\"Okay,\" said Annie. \"The name of this play is _A Late-Winter's Daydream_.\"\n\n_Not bad_ , thought Jack.\n\n\"And this is what's going to happen,\" said Annie. \"Jack will be the narrator. I'll play the Irish whistle. And Jack will sing a song that tells the story.\"\n\n\"What?\" said Jack. \"Excuse us a minute, Augusta.\" Jack turned to Annie. \"Why _me_ sing?\" he whispered. \"Why not _I_ play and _you_ sing?\"\n\n\"No, I want to play,\" said Annie. \"You took notes at Mary's, right? So just say a few words to describe the scene. Then use your notes about the Shee to make up a song. You can do that, can't you?\"\n\n\"I guess...,\" said Jack.\n\n\"Cool,\" said Annie. \"Give me the whistle.\"\n\nJack reached into his pockets and pulled out his notebook and the Irish whistle. He gave the whistle to Annie.\n\n\"Sorry, Miss Augusta,\" said Annie. \"We're almost ready.\" She whispered directions to Jack. \"Okay. Say your introduction. I'll start to play. Then you'll start to sing. Then\u2014\"\n\n\"I've got it,\" said Jack. \"Let's just start.\"\n\nJack and Annie turned to face Augusta. Jack cleared his throat. Then he spoke in a loud voice:\n\n_All is still in an old forest\u2014  \nuntil music sounds from a hidden world...._\n\nJack nodded to Annie.\n\nAnnie raised the magic Irish whistle to her lips and began to play.\n\nStrange, sweet music came from the whistle. The music was both sad _and_ happy. It was full of beauty and hope, pain and sorrow. Like the fog over the river, the music seemed to blend everything together.\n\nFor a moment, the whistle music was so powerful that Jack couldn't sing. He felt like crying and laughing at the same time. Finally he looked down at his notes and began to sing:\n\n**_In the spinning of light\u2014_**  \n ** _in a cloud, like a dream_** ,  \n ** _a bridge appeared over_**  \n ** _a wide, flowing stream_**.\n\nJack surprised himself. He thought his words for the song sounded pretty good.\n\nA long, fluttering high note burst from Annie's whistle. Bright, dancing light flowed across the river. The light arched through the fog toward Jack, Annie, and Augusta.\n\nAugusta gasped. Jack looked back down at his notes and sang:\n\n**_Some came on horses_** ,  \n ** _some came with wings_**.  \n ** _From an enchanted world_**  \n ** _little queens, little kings!_**\n\nThe wind began to blow. Jack looked up. Leaves and grass and twigs were flying everywhere.\n\nThe whistle music grew wilder.\n\nThundering and rumbling sounds came from the old forest. A herd of very small white horses galloped out of the mist. On their backs were proud and lovely riders\u2014men in gold helmets, women with long hair floating on the wind. Their capes and gowns were the colors of nature\u2014the pale rose of a spring dawn, the green of summer hummingbirds, the blue of winter twilight, and the gold of autumn oak leaves.\n\n\"The Shee!\" cried Augusta.\n\n#\n\nHundreds of Shee swarmed over the bright bridge. Behind the galloping riders more Shee came fluttering on wings like butterflies.\n\nAugusta stood up as if in a trance, her hands clasped over her heart.\n\n\"Oh, wow!\" breathed Annie.\n\n\"Keep playing!\" shouted Jack.\n\nAnnie blew into the whistle again.\n\nThe Shee flew and galloped along the wide, grassy bank. Their horses, fleet as the wind, had arched necks and flaming eyes. They moved faster and faster, swirling into a blinding circle of light and color.\n\nThe soaring sounds of the music inspired Jack to soar with his words, too. He looked at his notebook and sang:\n\n**_In a swirl they leave_** ,  \n ** _so wild, so free_** ,  \n ** _with a lonely girl_**  \n ** _to the hill of the Shee!_**\n\n\"Jack, no!\" shouted Annie.\n\nA mighty blast of wind nearly blew Jack and Annie off their rock. They crouched down and covered their heads. When the wind grew calm, they stood up\u2014just in time to see the swirling cloud of light vanish back into the old forest.\n\nThe Shee were gone. Once again, sky blended with water and earth in a veil of silver mist.\n\n\"Whew,\" Jack said breathlessly. \"That was amazing!\"\n\n\"Jack! Do you know what you just did?\"\n\n\"Yeah, I made the Shee appear and disappear,\" Jack said. \"I used my notes from Mary's story, just like we planned.\"\n\n\"Yes, you did that,\" said Annie. \"And you made _Augusta_ disappear, too!\"\n\n\"What?\" said Jack.\n\n\"The Shee took Augusta!\" said Annie. \"I tried to stop you, but it was too late! You'd already sung those words\u2014\"\n\n\"What words?\" said Jack.\n\n\"You sang: _In a swirl they leave, so wild, so free, with a lonely girl to the hill of the Shee!\"_ said Annie.\n\n\"I was just reading from my notes!\" said Jack. \"They took Augusta? Are you sure?\"\n\n\"Yes! She's gone!\" said Annie.\n\n\"Oh, no!\" said Jack. He and Annie ran to the ridge above the river. \"Augusta!\" he shouted.\n\nThere was no sign of Augusta's red cape anywhere.\n\n\"See? She's gone,\" said Annie.\n\n\"It's all my fault!\" said Jack.\n\n\"You couldn't help yourself,\" said Annie.\n\n\"But I'm responsible!\" said Jack. \"We have to get her back!\"\n\n\"Maybe Mary can help us,\" said Annie. \"She\u2014\"\n\n\"Wait, listen,\" said Jack. \"What's that sound?\"\n\nSqueaky noises were coming from behind the pile of small boulders. It sounded as if someone was trying to play the magic whistle. But the sound was definitely not magical.\n\n\"The whistle! I must've dropped it!\" said Annie.\n\n\"Maybe it's Augusta!\" said Jack.\n\nJack and Annie ran back and looked over the rocks.\n\nA man no higher than Jack's knee was blowing into the whistle. The man wore a green jacket and a three-cornered red cap with a white feather. He had big ears, a bushy red beard, skinny little legs, and silver buckles on his shoes.\n\n\"Oh, man,\" said Jack.\n\n_\"A leprechaun!\"_ whispered Annie.\n\nJack and Annie just stared at the leprechaun as he blew into the whistle. His small, bony fingers danced over its six holes. But only squeaks, chirps, and hollow tweeting sounds came out.\n\nThe leprechaun blew harder and harder. Then he stopped. He turned the whistle over and looked at it closely. He shook his head and frowned.\n\n\"Hello!\" said Annie.\n\nThe leprechaun jumped and looked up. \"Well, hello yourself!\" he said, grinning. \"You startled me! Here, take this back. It's no use to me, as you can plainly hear.\"\n\nThe little man held up the whistle. Jack reached over the rock and took it. He slipped it back into the pocket of his coat.\n\n\"Listen, we need your help,\" said Jack. \"Our friend\u2014\"\n\n\"Oh, you humans! Always in such a hurry!\" the leprechaun said.\n\n\"Sorry,\" said Jack, \"but we really need you to help us. You see\u2014\"\n\n\"First of all, who are you?\" the leprechaun asked.\n\n\"I'm Jack. She's my sister, Annie,\" said Jack. \"Our friend\u2014\"\n\n\"Ah, well then, Jack and sister Annie, I'm Willy,\" said the leprechaun. \"Now, let's get a few things straight right away\u2014never call me Little Willy or Tiny Willy. I don't like it. And never, _ever_ call me Wee Willy. That's the one I hate the most.\"\n\n\"Okay, fine!\" said Jack. \"But\u2014\"\n\n\"Jack, let me handle this,\" said Annie. She turned to the leprechaun. \"Willy, why didn't you go back over the river with the Shee?\"\n\n\"Ask me why I came over the river in the first place, and I'll tell you the answers to both questions,\" said Willy.\n\n\"Okay, why did you come over the river in the first place, Willy? _And_ why didn't you go back across with the Shee?\" said Annie.\n\n\"Answer number one: I was having a bit of a nap in the reeds when I heard your whistle playing,\" said Willy. \"Before I knew it, I was moving with the Shee across the bridge. Probably you didn't see me. Lost in the swirl of things, I was.\"\n\n\"Excuse me\u2014\" started Jack.\n\n\"I've heard some good whistle playing,\" Willy said. \"But yours, missy, was like none I've ever heard from a human\u2014not in the nine hundred years of my life. So! Answer number two: I didn't go _back_ over the river because I wanted to find out the secret of your playing. Well?\"\n\n\"That's easy,\" said Annie. \"The whistle played itself. It wasn't me.\"\n\n\"Ah, you're a modest girl,\" said Willy. \"And you like to keep the secrets of your talents to yourself.\"\n\n\"Not really\u2014\" said Annie.\n\n\"Listen, Willy!\" Jack broke in. \"Can you help us? We lost our friend, Augusta. It was my fault. I sang about a lonely girl. Then she was taken by the Shee\u2014\"\n\n\"Yes, I saw that,\" said Willy. \"Whisked away, she was. So I'm guessing now you want me to help you find her. Is that it?\"\n\n\"Yes! That's it!\" said Jack.\n\n\"We were going to ask Mary Sheridan to help us,\" said Annie. \"But you probably know the way better. And\u2014\"\n\n\"Wait a minute,\" said Willy. \"Did you say Mary Sheridan?\"\n\n\"You know Mary?\" said Annie.\n\n\"Know her?\" said Willy, grinning from ear to ear. \"Why, if I were four feet taller, I'd have married Mary Sheridan years ago.\"\n\n\"Really?\" said Annie.\n\n\"Oh, yes, indeed, we're very good friends,\" said Willy. \"You see, I live in what you might call the _In-between_. I have one foot in the magical world of the Shee. And I have one foot in the mortal world of humans, like Mary. Ah, lovely Mary...\"\n\n\"Yeah, Mary's great,\" said Jack. \"But now\u2014\"\n\n\"You want me to guide you to your missing friend,\" said Willy.\n\n\"Right, right!\" said Jack.\n\n\"I can do that, but what will you give me for it?\" said Willy.\n\n\"What do you mean?\" asked Jack.\n\n\"What will you give me for helping you find your friend? My time's very valuable, you know,\" said Willy.\n\n\"We don't really have anything,\" said Jack.\n\n\"We're poor and miserable,\" said Annie.\n\n\"I can see that,\" said Willy. \"All right then, how about this: I'll lead you to your friend, and you'll teach me how to play the whistle the way you play it. Fair enough?\"\n\n\"Sorry, I\u2014\" said Annie.\n\n\"Deal!\" Jack broke in.\n\n\"Jack?\" said Annie.\n\n\"Annie, it's a deal,\" Jack said. He was ready to promise anything to save Augusta!\n\n\"Good!\" said Willy. \"I'd love to play like that for Mary someday. Now, the path I'm about to show you is very, very secret. You must never show another living soul.\"\n\n\"Of course not,\" said Jack.\n\n\"Then come with me,\" said Willy, \"and I'll lead you across the water to the home of the Shee!\"\n\n#\n\nThe leprechaun scrambled down the steep bank of the river.\n\nJack started to follow, but Annie grabbed him. \"Jack, the whistle won't work for Willy!\" she said.\n\n\"I know,\" said Jack. \"We'll worry about that later. Right now, we just have to save Augusta! Come on!\"\n\nJack and Annie hurried after Willy to the wide, rushing river.\n\n\"How do we get across?\" Jack asked.\n\n\"Follow me along the river,\" said Willy, \"and I'll show you. Step lightly.\"\n\nWilly skipped ahead of them along the riverbank. Jack and Annie followed. Jack tried to step lightly, but it was impossible. Slopping through the mud, his boots filled with ooze and made squishing sounds with every step.\n\nThe river grew narrower and narrower, twisting and turning like a snake. The mist grew thicker, until Jack could hardly see a thing. He bumped into Willy, almost knocking him over.\n\n\"Careful, laddie!\" said the leprechaun.\n\n\"Sorry,\" said Jack.\n\n\"All right,\" said Willy. \"Here's where we cross. Hop from stone to stone after me.\" Willy disappeared into the mist, crossing the river. Annie followed him.\n\nJack started across, too. He tried to hop carefully from stone to stone, but the stones were slippery. On his third hop, his boot slid off a mossy rock, and he splashed into the freezing water!\n\n\"Jack, are you okay?\" Annie called in a loud whisper.\n\n\"Yeah, yeah, I'm fine!\" Jack said. He scrambled up. His clothes were soaked and heavy. He really was miserable now. He waded to the other side of the river and joined Annie and Willy.\n\n\"Ah, you fell in,\" said Willy. \"That happens sometimes. Come with me now, into the forest.\"\n\nJack, Annie, and Willy walked under the trees. They passed old oaks and maples that creaked in the wind. Jack shivered in his soaking wet clothes.\n\nA big black crow called from a high, bare branch.\n\nWilly jumped, then laughed. \"Ah, 'tis probably Patrick Daly,\" he said. \"Rumor is the Shee changed him into a crow for stealing a pair of the High Queen's shoes.\"\n\n\"Changed him into a crow?\" said Jack.\n\nWilly nodded. \"One of the Shee's favorite tricks,\" he said, \"changing humans who displease them into animals.\"\n\n\"Oh,\" said Jack.\n\nA white rabbit scampered through the forest. \"That bunny over there is no doubt the former Mrs. Shan McCartie. She was rude to the High King,\" said Willy.\n\n\"Rude to the High King?\" said Jack. He was starting to feel like he was trapped in a nightmare.\n\n\"The Shee cannot bear the rudeness of humans,\" said Willy. \"Look there.\" He pointed to a fawn peeking out from behind a fir tree.\n\n\"Aww, it's so cute,\" said Annie.\n\n\"Cute? Maybe. Maybe not,\" said Willy. \"Could easily be old John Foley. I heard he was changed just for being a grouch! Come along.\"\n\nWilly led Jack and Annie through the forest until they came to a tangle of briars and brambles. \"The hollow hill of the Shee lies just beyond this thicket,\" he said to them. \"Do you still want to go and find your friend?\"\n\n\"Yes,\" Jack and Annie both whispered.\n\n\"Then off you go, and good luck to you,\" said Willy, tipping his hat.\n\n\"What? Aren't you coming with us?\" said Jack.\n\n\"Oh, my, no,\" said Willy. \"The Shee would be furious if they knew I'd shown humans the path to their secret hiding place. And I certainly don't want to live the rest of my life as a weasel.\"\n\n\"But won't they be angry at _us_ for finding their secret hiding place?\" said Jack.\n\n\"Possibly,\" said Willy. \"Here's what I recommend you do: politely tell the High King and the High Queen that you've just come looking for your dear friend to take her back to her loving family. The Shee place very great value on friendship and family.\"\n\n\"Okay,\" said Jack. \"Friendship and family...\"\n\n\"And remember: be simple, direct, and honest at all times,\" said Willy.\n\n\"Simple, direct, honest,\" repeated Jack.\n\n\"And polite, that's the most important\u2014very, very polite,\" said Willy.\n\n\"Very polite,\" said Annie. \"Got it.\"\n\n\"Another thing,\" said Willy. \"In the world of the Shee, the old tales still live. So do not be afraid if you see odd sights floating about. They're just bits and pieces of the old stories. Now go. Be simple, direct, honest, and polite, and save your dear friend before she's lost forever.\"\n\n\"Thanks, Willy,\" said Annie.\n\n\"Best of luck,\" said Willy.\n\n\"See you later,\" said Jack.\n\n\"Aye, I'll be waiting for you,\" said the leprechaun.\n\nJack and Annie crouched down and started through the thicket. Thorns and briars pulled at Jack's wet coat. They scratched his hands and got caught in his hair.\n\nJack battled his way through the tangled brush until he caught up with Annie. They both pushed their way out of the thicket and stepped into a glade.\n\n\"Wow, it's like Mary said,\" whispered Annie. \"It _is_ like summer here.\"\n\nNo rain fell. No wind blew. Warm sunshine shone on the emerald-green glade. In the middle of the glade was a large grassy mound. At the foot of the mound was a small doorway framed by stones.\n\n\"That must be it\u2014the hollow hill,\" said Jack, \"the secret home of the Shee.\"\n\n\"And _that_ must be someone from an old story,\" whispered Annie. She pointed to a woman floating above the green mound.\n\nThe woman wore a wreath of flowers on her head and carried a branch with silver apples. She vanished in the sunny haze.\n\n\"Whoa,\" whispered Jack.\n\nA small sailing ship then appeared in the air. It had white sails and flags flying. The ship, too, vanished. Then an old woman spinning at a spinning wheel appeared... then a wispy dragon... then a knight with a sword. The parade of images faded into the sunshine like wisps of smoke.\n\n\"Whew,\" said Jack.\n\n\"Listen,\" said Annie.\n\nThe sound of drumming was coming from the doorway of the grassy mound. \"Come on, let's look inside,\" said Annie.\n\nJack and Annie snuck close to the small doorway. It was no higher than Jack's waist. He and Annie knelt down and peeked inside.\n\nThe hollow hill was filled with a pale-green light. Very small dancers, none more than eight inches tall, were bathed in the light. As drummers pounded tiny drums, the dancers danced together in rows. They held their arms straight by their sides and kicked their legs and turned around and around to the rhythm of the drums.\n\nA small shining couple sat on high golden chairs watching the dancers. They wore golden crowns.\n\n\"They must be the High King and the High Queen,\" whispered Annie.\n\nJack and Annie watched until the rows of dancers parted. Then they could see a third person watching the dance. She was no taller than the dancers. Draped around her shoulders was a red cape.\n\n_\"It's Augusta!\"_ said Annie.\n\n#\n\n\"Augusta!\" whispered Jack. \"She's\u2014she's tiny!\"\n\n\"They must have shrunk her!\" whispered Annie.\n\n\"Hide!\" whispered Jack. He crawled away from the door. Annie crawled after him. They pressed their backs against the grassy mound.\n\n\"Why are we hiding?\" asked Annie.\n\n\"We can't let them see us!\" said Jack. \"They might shrink us, too!\"\n\n\"But how are we going to save Augusta?\" asked Annie.\n\n\"I don't know,\" said Jack.\n\n\"Hey, I just remembered something,\" said Annie. \"Didn't Mary say the girl in her story would have become small _if she'd gone inside the hollow hill_?\"\n\n\"Yeah, she did,\" said Jack. \"So maybe that means you can only get shrunk if you go inside.\"\n\n\"Right,\" said Annie. \"So let's go back to the doorway and call from outside. We'll be simple, direct, honest, and very polite, like Willy said. We'll tell the king and queen we've come to take our friend back to her family.\"\n\n\"But wait a minute,\" said Jack. \"How can she go back to normal life with her family if she's only eight inches tall?\"\n\n\"Good point,\" said Annie.\n\n\"This is so weird,\" said Jack.\n\n\"Let's worry about her size later,\" said Annie. \"For now, we just have to help her escape.\"\n\n\"Right,\" said Jack. \"Let's try it.\"\n\nJack and Annie crawled back to the entrance of the hollow hill and peeked inside.\n\nIn the pale-green light, a very small Augusta stood watching the dancers.\n\n\"Excuse us, please!\" Annie called.\n\nThe drumming stopped. The dancers froze. All eyes turned to Jack and Annie. The High King and High Queen looked startled.\n\n\"Who are you?\" the king called. \"How did you find us?\"\n\n\"That's not important!\" said Annie. \"We are _very_ sorry to bother you! But we've come to get our _very_ dear friend Augusta! We have to take her home to her _very_ loving family!\"\n\n\"Please! Thank you!\" added Jack, trying to sound polite.\n\nBefore the king or queen could speak, Augusta rushed forward. \"No! I don't want to go home!\" she screamed in a high little voice. \"I don't want to leave the Shee!\"\n\n\"Whoa,\" said Jack. That was a surprise. Maybe Augusta was under a spell!\n\n\"Leave here at once!\" the High King commanded Jack and Annie in a squeaky voice. \"You were not invited! You have no business here!\"\n\n\"Yes, we are leaving right away. But Augusta has to come with us!\" Annie called. \"Thank you!\"\n\n\"She's right!\" said Jack. \"Augusta has to come with us, please. Thank you!\"\n\n\"No! I want to stay here!\" said Augusta. \"I'm not good for anything at home! Mary was right, I'm not happy there!\"\n\nJack was amazed\u2014Augusta _wasn't_ under a spell. She really wanted to stay with the Shee!\n\n\"The girl will stay with us!\" the High King shouted. \"Go, now! _Now!_ \"\n\n\"No way!\" Jack blurted out. \"We won't leave without Augusta!\"\n\nThe crowd gasped.\n\n\"What?\" roared the High King.\n\n\"Sorry, sorry,\" said Jack. \"I meant\u2014\"\n\n\"You will pay for this rudeness!\" the king yelled.\n\nBefore Jack and Annie could get away, the king thrust out his arm. Sparks flew from the tips of his tiny fingers. Suddenly Jack and Annie couldn't move their arms or legs!\n\nThe king kept pointing at them. \"Skunks?\" he shouted at the crowd. \"Or weasels?\"\n\n\"Neither, please! Thank you!\" shouted Jack. He was glad he could still talk!\n\nBut the crowd began chanting in strange, high voices, \"Skunks! Skunks! Skunks!\"\n\nThe king nodded and raised both arms into the air. Jack was desperate. He didn't want to live the rest of his life as a skunk!\n\n\"Wait! Please!\" Jack shouted. \"I'm sorry I was rude! Before you change us, I really, really need to tell Augusta some things! Thank you!\"\n\nThe king looked at Jack for a long moment. Then he lowered his arms, and the crowd grew silent.\n\n\"Thank you!\" said Jack. \"Augusta, listen to me! You should go back home. You're good for lots of things back there! You're very kind! And Mary said you have a brave heart and a fine mind! Those are really good things to have!\"\n\nTears rolled down Augusta's cheeks. She shook her head.\n\n\"Listen to me, please!\" Jack went on. \"Mary says you're not happy. But some things do make you happy, Augusta. You said you feel close to nature! You said you love simple folk like Mary. And I know you love stories, too! Mary said you remembered every story she told you! You used to tell them yourself, word for word! You have a great memory!\"\n\n\"He's right, Augusta!\" yelled Annie. \"Those are your gifts! You need to give your gifts to the world!\"\n\nAugusta was still for a moment. Then she shook her head. \"I want to stay!\" she said.\n\n\"You have had your say!\" the High King shouted at Jack and Annie. \"Now prepare to become skunks!\" He raised his arms again.\n\n_Oh, no!_ thought Jack.\n\n\"Wait, Finvara!\" said the High Queen.\n\nThe High Queen stepped closer to Augusta. The queen wore a silver cloak that glittered with diamonds. She had jewels in her long red hair that shone like stars. Her high, clear voice rang like a bell. \"I am Queen Aine of the Shee,\" she said. \"The boy said you love stories\u2014and that you remember every story you hear. Is this true?\"\n\nAugusta nodded.\n\n\"Then listen now to _our_ story,\" said Queen Aine.\n\nAll the Shee were very still, watching Augusta and their queen.\n\n\"In the morning of time, out of a rosy sky and a windy light, we came,\" said Queen Aine. \"We were tribes of a supernatural people called the Tuatha D\u00e9 Danann. Strong, fearless, and noble we were. Five roads carried our armies through the wild, wooded lands of Ireland, and for eons we ruled the Irish world.\n\n\"But when the humans came, the wild woods gave way to villages and pastures. Our tribes hid in the hollow hills, in ruined forts, and under the sea. Over time we made ourselves smaller and smaller so we could more easily hide from humans.\n\n\"Eventually we became known as the Shee, and we were mocked as the Wee Folk of Faerie. But, in truth, we are a tribe of the great Tuatha D\u00e9 Danann, and we live in enchanted places like this one, protected by what is left of our magic. Do you understand?\"\n\n\"I do, yes,\" breathed Augusta.\n\n\"Our stories were passed down for centuries in the old language of Ireland. But as the old language was replaced with English, the stories began to fade away,\" Queen Aine said.\n\n\"Go home now with your friends, human child. Go back, for our sake. Seek out the old storytellers, and ask them to tell you the tales of my people. Learn the old language. Read the old manuscripts. Write our stories down and share them before they are lost completely. Share them with all the people of Ireland and all the world. Will you do that for us? Will you use your gifts to tell our stories and restore our dignity?\"\n\nAugusta's eyes shone. \"Yes,\" she said, \"yes, I will, yes.\"\n\n\"Good. Then I will send you all swiftly home,\" said Queen Aine. She beckoned to a small dancer, who stepped forward with a tiny silver chalice.\n\n\"Sip the honey nectar of the Shee,\" said the High Queen. She took the chalice and held it out to Augusta. Augusta took a sip.\n\nA second later, Jack felt cold wind and rain. He and Annie were standing on the bank of the river. Augusta was standing beside them. She was her normal size again.\n\n#\n\n\"Well, _that_ was simple and direct,\" said Annie.\n\n\"Yeah....\" Jack felt dazed.\n\nAugusta looked at Jack and Annie. \"I saw them,\" she said, her eyes wide. \"I finally saw the Shee. I really saw them!\" She burst out laughing. Her laughter was so full of joy that Jack and Annie started laughing, too.\n\n\"I saw them! I saw them!\" Augusta kept repeating. \"I saw the Shee! And now I have important things to do!\"\n\n\"Yes, you do,\" said Annie.\n\n\"I must learn the old language! And I must start gathering stories at once, that's what Queen Aine told me!\" said Augusta. \"I can't wait to visit the old storytellers! There's Mary Sheridan and Biddy Early, too! I'll start with Mary! Let's go see her right now! Hurry! Will you come with me? Hurry!\"\n\n\"Sure,\" said Annie, \"but just a second.\" She turned to Jack. \"What about Willy?\"\n\n\"Who's Willy?\" asked Augusta.\n\n\"A friend of ours,\" said Jack, looking around. \"He said he'd wait for us. But where was he going to wait? Willy!\"\n\n\"Willy!\" called Annie.\n\n\"I guess he's still somewhere on the other side of the river...,\" said Jack.\n\n\"That's too bad,\" said Annie.\n\n\"Yeah,\" said Jack.\n\n\"Come, let us go see Mary now!\" said Augusta, grabbing Annie's hand and pulling her along.\n\nJack looked around for Willy one last time. He was sad about not seeing the leprechaun again. But he was a little relieved, too. He knew they couldn't keep their part of the deal: teaching Willy how to play the magic whistle.\n\n\"Come on, Jack!\" cried Augusta.\n\nJack followed the two girls through the wet meadow, over the stone wall, down the dirt lane, and across the muddy field to Mary Sheridan's cottage.\n\n\"Mary! Mary!\" Augusta called. She dashed ahead of Jack and Annie. She didn't even stop to knock. She threw open the door to the cottage. \"Oh!\" she said, freezing in her tracks.\n\nJack and Annie caught up with Augusta and looked inside.\n\nMary was sitting in front of her fire. Next to her was a small man wearing a green jacket and a three-cornered red cap with a white feather.\n\n\"Willy!\" cried Annie.\n\nAnnie and Jack hurried past Augusta into the cottage.\n\n\"You're here!\" said Jack.\n\n\"Of course! I said I'd wait for you!\" said the leprechaun. \"I see you found your dear friend and brought her home. Good work!\"\n\nAugusta stood in the doorway, gaping at Willy. \"Who are _you?\"_ she asked.\n\n\"Name's Willy. Just plain Willy,\" said the leprechaun. \"I usually leave before you visit Mary. But now that you've seen the Shee, I suppose there's no point hiding from you anymore.\"\n\n\"Are\u2014are you real?\" asked Augusta.\n\n\"Who knows?\" said the leprechaun. \"Maybe _I'm_ real and you're _not_! Depends on which one of us is asking the question!\"\n\nEveryone laughed. \"Good point,\" said Annie.\n\n\"Well, we'd better be getting home,\" said Jack. He wanted to leave before Willy asked Annie for his whistle lesson.\n\nJack took Annie's hand and started backing toward the door. \"Thanks for helping us, Willy,\" he said.\n\n\"Yeah, and thanks, Mary, for everything!\" said Annie, waving. \"Good-bye, Augusta! Good-bye, everyone!\"\n\n\"Wait just a minute, my friends,\" said Willy.\n\n_Uh-oh_ , thought Jack.\n\n\"Have you forgotten our deal?\" said Willy. \"My fingers are itching to make beautiful music for Mary.\"\n\n\"Well, you see... there's a problem with that,\" said Jack, squirming.\n\nA frown crossed Willy's face. \"A problem?\" he said. \"How could there be a problem? Find your friend, teach Willy to play the whistle. Doesn't get any simpler than that.\"\n\n\"Right,\" said Jack.\n\n\"So what's the problem?\" asked Willy.\n\n\"Well, the whistle...\" Jack didn't know how to finish.\n\n\"I'll answer that question,\" said Annie. \"Willy, I'm going to be simple and direct and honest with you.\"\n\n\"Yes...?\" said the leprechaun.\n\n\"Merlin the magician gave us the whistle to help us on our missions,\" said Annie. \"The whistle is magic, and the magic only works once. Without the magic, I don't really know how to play. So I can't teach you. There you have it.\"\n\n\"Ah,\" said Willy. He looked at the floor and shook his head. \"Then I'm afraid I'm going to have to turn you both into chipmunks.\"\n\n_\"What?\"_ said Jack.\n\nWilly burst out laughing. \"Joking! Only joking!\" he said. \"Merlin! You should have told me you were friends with Merlin in the first place!\"\n\n\"Do you know Merlin?\" asked Jack.\n\n\"Oh, yes, we spent a great deal of time together. Must be about eight hundred years ago now,\" said Willy. \"How is he?\"\n\n\"He's happy,\" said Annie.\n\n\"Good!\" Willy turned back to Mary and Augusta. \"I first met the master magician on the Isle of Merlin in the Irish Sea. He\u2014\"\n\n\"Wait, wait, please, Willy,\" said Augusta. \"Mary, do you have pen and paper?\"\n\n\"No, my dear, I'm afraid I don't,\" said Mary.\n\n\"I can help,\" said Jack. He took his pencil and small notebook out of his pocket. He tore some clean pages out of the notebook. Then he gave the pencil and pages to Augusta. \"There,\" he said.\n\n\"Oh, thank you, Jack!\" said Augusta. She turned back to Willy. \"Continue, please.\"\n\n\"Well,\" said Willy, \"Merlin was only a few centuries old then, and I was a mere lad....\"\n\nAs Willy told his story, Augusta began to write. Jack put his notebook back into his pocket and nodded to Annie. She nodded back, and the two of them started toward the door.\n\nJust as they were about to leave, Annie called out to the others, \"Bye!\"\n\n\"We have to go home now,\" said Jack.\n\n\"Trip lightly!\" said Willy.\n\n\"Thank you for everything!\" said Augusta.\n\n\"One and twenty fare-thee-wells!\" said Mary.\n\n\"Same to you guys,\" said Jack. Then he and Annie slipped out of the cottage.\n\nThe rain was pouring again, and the wind was blowing hard.\n\n\"I think we inspired Augusta,\" said Annie.\n\n\"Yep, we accomplished our mission,\" said Jack. \"Now let's get out of here.\" He couldn't wait to go home and get warm and dry.\n\nJack and Annie ran against the wind. They climbed over the stone wall, then hurried down the lane, slipping and sliding in the muck. They ran across the soaked field. By the time they arrived at the rope ladder, their clothes were caked with mud.\n\nJack and Annie climbed into the tree house. Annie found the Pennsylvania book in the corner. As rain blew through the window, she pointed to a picture of the Frog Creek woods. \"I wish we could go there!\" she said.\n\nThe wind blew harder.\n\nThe tree house started to spin.\n\nIt spun faster and faster.\n\nThen everything was still.\n\nAbsolutely still.\n\n#\n\n\"A hh, sunshine,\" said Jack. He closed his eyes and felt the sunshine streaming through the tree house window.\n\n\"And clean, dry clothes,\" murmured Annie. She placed the Pennsylvania book back in the corner.\n\nJack took the magic whistle out of his pocket and placed it next to the book. \"There. Let's go home now,\" he said. \"I want to look on the Internet for information about Augusta.\" He started down the rope ladder.\n\n\"Great idea,\" said Annie, following Jack. \"We can find out what happened to her.\"\n\nJack and Annie ran through the chilly Frog Creek woods. They crossed the street and hurried up the sparkling sidewalk to their yard. They tramped over old snow up to their porch. Annie opened the front door and led the way inside.\n\n\"Hi!\" Jack called. \"We're back!\"\n\n\"Hi!\" their mom called from the kitchen. \"Did you have a nice break?\"\n\n\"Yes, we did!\" said Annie.\n\n\"Good. Get back to your homework now,\" said their mom. \"So you can finish in time to go to the theater.\"\n\n\"Okay!\" called Annie. She went to the computer desk and sat down. \"What should I type?\" she asked Jack.\n\nJack pulled up a chair and sat beside her. \"Well, we don't know her last name,\" he said. \"So try _Galway... Augusta..._ and _Irish stories_.\"\n\nAnnie typed these words on the keyboard, then hit _enter_. There were lots of choices for different Web sites. Annie clicked on the first one.\n\nOn the screen was a black-and-white photograph of a woman. The caption under it said:\n\n_Lady Augusta Gregory_\n\n\"Look! It's _her_!\" said Annie.\n\nThe woman on the screen was middle-aged, but she still looked like Augusta. Her hair was parted neatly down the middle.\n\nJack read aloud from the screen:\n\n**Lady Augusta Gregory was born into a wealthy family in Galway, Ireland, in 1852. She wrote over forty plays and many poems and essays. She was a co-founder of the Abbey Theatre, the national theater of Ireland. Lady Gregory also learned the old language of Ireland and became well known for collecting Irish stories and legends and sharing them with the world**.\n\n\"Wow!\" said Annie. \"Augusta _did_ have a brave heart and a fine mind! And she must have liked our play, since she wrote forty of her own and started her own theater.\"\n\n\"Yeah,\" said Jack. \"She really turned out great.\" This reminded him of a question he'd asked himself earlier. \"I wonder what I'm good for? I didn't know how to do anything on that Irish farm.\"\n\n\"Me neither,\" said Annie. \"But hardly any kids today know how to do that kind of stuff.\"\n\n\"So what would we do if all our machines and computers broke down?\" said Jack.\n\n\"We'd have to figure out how to grow potatoes and make our own clothes and milk cows,\" said Annie.\n\n\"I'd probably read some kind of instructions first, then give it a try,\" said Jack.\n\n\"I'd probably give it a try first,\" said Annie, \" _then_ read the instructions.\"\n\nJack laughed.\n\n\"I know some stuff we're good for,\" said Annie.\n\n\"What?\" said Jack.\n\n\"First, we're good for helping each other,\" said Annie.\n\n\"Yeah, but\u2014\" said Jack.\n\n\"No, really. We help each other all the time,\" said Annie.\n\n\"That's true,\" said Jack.\n\n\" _And_ we're good for helping Augusta,\" said Annie, \"and helping Louis Armstrong, Mozart, and Leonardo da Vinci. We put the smile on the Mona Lisa's face, remember?\"\n\nJack nodded. \"Yep,\" he said.\n\n\"And we're good for saving an orphan penguin, a huge octopus, and the cities of Tokyo, Venice, and New York,\" said Annie. \"We're good for rescuing a baby gorilla from a leopard, and schoolkids from a twister. We're good for helping Shakespeare, Clara Barton, and George Washington. We're good for rescuing two kids from a tsunami, a Lakota boy from a buffalo stampede, and a baby kangaroo and a koala from a forest fire. We're good for\u2014\"\n\n\"Wait, stop,\" said Jack. \"Stop.\"\n\n\"But that's not even half of it,\" said Annie.\n\n\"I know,\" said Jack. \"But that's plenty. I'm inspired. I'm ready to write that story for homework. I'll use my own experience. I have a little more than I thought.\"\n\n\"Cool,\" said Annie. She went back to reading about Lady Gregory on the computer.\n\nJack grabbed a pencil and pulled out his notebook. He moved to the couch and sat down. As late-winter light slanted into the living room, he began to write.\n\n#\n\n**Irish Fairies**\n\n\u2022 There was a time when many people in Ireland believed in fairies who lived inside mounds of earth. The name for Irish fairies is _S\u00ed_. In Ireland, the word _S\u00ed_ is pronounced _Shee_. To avoid confusion in my story, I refer to the fairies as the Shee.\n\n\u2022 Leprechauns are a type of male Irish fairy. In folktales they often work as shoemakers or tailors and are thought to have hidden pots of gold. Leprechauns are featured on St. Patrick's Day, a national holiday in Ireland celebrated on March 17.\n\n**Lady Gregory**\n\n\u2022 When she was a child, Lady Gregory's full name was Isabella Augusta Persse, though everyone called her Augusta. After she grew up, she married Sir William Henry Gregory. As the wife of a knight, she was called Lady Gregory.\n\n\u2022 Ireland's most famous poet is William Butler Yeats (YATES). He was a very good friend of Lady Gregory's. Together in 1904, they founded Dublin's Abbey Theatre, the national theater of Ireland.\n\n\u2022 William Butler Yeats often accompanied Lady Gregory when she visited cottages and collected Irish folklore. It was said that Lady Gregory had a natural genius for remembering the direct speech of storytellers. \"In her many years of traveling, listening, transcribing and publishing, Lady Gregory... gave value to the stories, to the mind and the imagination of Irish country people\" (Lucy McDiarmid and Maureen Waters, introduction to _Lady Gregory: Selected Writings)_.\n\n**Irish Language**\n\n\u2022 From the 1600s to the early 1900s, the language of Ireland, called _Irish Gaelic_ , was replaced by English in many parts of Ireland. During the struggle for Irish independence in the twentieth century, the desire to learn the old language became very strong in Ireland. In 1922, Irish Gaelic became the official language of the country along with English. Today it is taught in Irish public schools. Lady Gregory's knowledge of Irish Gaelic helped her when she visited cottages and collected stories.\nAre you a fan of the Magic Tree House\u00ae series?\n\n**Visit our**\n\n**Web site**  \n **at**\n\nMagicTreeHouse.com\n\nExciting sneak previews of the next book.  \nGames, puzzles, and other fun activities.  \nContests with super prizes.  \nAnd much more!\n**Guess what?**\n\n_Jack and Annie have a musical CD!_\n\nFor more information about  \nMAGIC TREE HOUSE: THE MUSICAL  \n(including how to order the CD!),  \nvisit www.mthmusical.com.\n\n**Discover the facts  \nbehind the fiction with the**\n\n**Magic Tree House \u00ae Books**\n\n#1: DINOSAURS BEFORE DARK\n\n#2: THE KNIGHT AT DAWN\n\n#3: MUMMIES IN THE MORNING\n\n#4: PIRATES PAST NOON\n\n#5: NIGHT OF THE NINJAS\n\n#6: AFTERNOON ON THE AMAZON\n\n#7: SUNSET OF THE SABERTOOTH\n\n#8: MIDNIGHT ON THE MOON\n\n#9: DOLPHINS AT DAYBREAK\n\n#10: GHOST TOWN AT SUNDOWN\n\n#11: LIONS AT LUNCHTIME\n\n#12: POLAR BEARS PAST BEDTIME\n\n#13: VACATION UNDER THE VOLCANO\n\n#14: DAY OF THE DRAGON KING\n\n#15: VIKING SHIPS AT SUNRISE\n\n#16: HOUR OF THE OLYMPICS\n\n#17: TONIGHT ON THE _T ITANIC_\n\n#18: BUFFALO BEFORE BREAKFAST\n\n#19: TIGERS AT TWILIGHT\n\n#20: DINGOES AT DINNERTIME\n\n#21: CIVIL WAR ON SUNDAY\n\n#22: REVOLUTIONARY WAR ON WEDNESDAY\n\n#23: TWISTER ON TUESDAY\n\n#24: EARTHQUAKE IN THE EARLY MORNING\n\n#25: STAGE FRIGHT ON A SUMMER NIGHT\n\n#26: GOOD MORNING, GORILLAS\n\n#27: THANKSGIVING ON THURSDAY\n\n#28: HIGH TIDE IN HAWAII\n\n**Merlin Missions**\n\n#29: CHRISTMAS IN CAMELOT\n\n#30: HAUNTED CASTLE ON HALLOWS EVE\n\n#31: SUMMER OF THE SEA SERPENT\n\n#32: WINTER OF THE ICE WIZARD\n\n#33: CARNIVAL AT CANDLELIGHT\n\n#34: SEASON OF THE SANDSTORMS\n\n#35: NIGHT OF THE NEW MAGICIANS\n\n#36: BLIZZARD OF THE BLUE MOON\n\n#37: DRAGON OF THE RED DAWN\n\n#38: MONDAY WITH A MAD GENIUS\n\n#39: DARK DAY IN THE DEEP SEA\n\n#40: EVE OF THE EMPEROR PENGUIN\n\n#41: MOONLIGHT ON THE MAGIC FLUTE\n\n#42: A GOOD NIGHT FOR GHOSTS\n\n#43: LEPRECHAUN IN LATE WINTER\n\n#44: A GHOST TALE FOR CHRISTMAS TIME\n\n**Magic Tree House \u00ae Research Guides**\n\nDINOSAURS\n\nKNIGHTS AND CASTLES\n\nMUMMIES AND PYRAMIDS\n\nPIRATES\n\nRAIN FORESTS\n\nSPACE\n\n_T ITANIC_\n\nTWISTERS AND OTHER TERRIBLE STORMS\n\nDOLPHINS AND SHARKS\n\nANCIENT GREECE AND THE OLYMPICS\n\nAMERICAN REVOLUTION\n\nSABERTOOTHS AND THE ICE AGE\n\nPILGRIMS\n\nANCIENT ROME AND POMPEII\n\nTSUNAMIS AND OTHER NATURAL DISASTERS\n\nPOLAR BEARS AND THE ARCTIC\n\nSEA MONSTERS\n\nPENGUINS AND ANTARCTICA\n\nLEONARDO DA VINCI\n\nGHOSTS\n\nLEPRECHAUNS AND IRISH FOLKLORE\n\nRAGS AND RICHES: KIDS IN THE TIME OF CHARLES DICKENS\n\n**More Magic Tree House \u00ae**\n\nGAMES AND PUZZLES FROM THE TREE HOUSE\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# _But Not In Shame_\n\n**by John Toland**\n\n**March, 1942.** With ponderous grace the Flying Fortress touched down at Alice Springs, Australia, where the American general was met by reporters. When they asked him for a statement, he quickly scribbled a few lines on the back of a used envelope:\n\n> \"The President of the United States ordered me to break through the Japanese lines and proceed from Corregidor to Australia for the purpose, as I understand it, of organizing the American offensive against Japan, a primary object of which is the relief of the Philippines.\n> \n> _\"I came through and I shall return.\"_\n\n\u00a9 Copyright, 1961 by John Toland\n\nAll rights reserved under International and Pan-American Copyright Conventions.\n\nLibrary of Congress Catalog Card Number: 61-12162\n\nISBN 978-0-345-24285-3\n\nEbook ISBN 978-1-101-96929-8\n\nBALLANTINE BOOKS\n\nA Division of Random House, Inc.\n\nv4.1\n\na\n**TO TOSHIKO**\n\n# _Table of Contents_\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nAUTHOR'S NOTE\n\nPROLOGUE: _\"With Dangerous and Dramatic Suddenness\"_\n\nPART I TIMETABLE FOR CONQUEST\n\n1 \"Climb Mt. Niitaka\"\n\n2 \"Tora...Tora...Tora\"\n\n3 \"Tallyho! Bandits Over Clark!\"\n\n4 \"So, Shoot to Sink\"\n\n5 \"It Was Such a Beautiful Ship\"\n\nPART II THE DEFENSES CRUMBLE\n\n6 The Death Watches\n\n7 End of Wake\n\n8 Panic\n\n9 Dark Christmas\n\nPART III BATTLE FOR BATAAN\n\n10 Roads to Bataan\n\n11 Action at Abucay\n\n12 Retreat\n\n13 The Points and the Pockets\n\nPART IV DEATH OF TWO EMPIRES\n\n14 This Naked Island\n\n15 \"These Stout-Hearted Dutchmen\"\n\n16 The Battle of Java Sea\n\n17 \"Good-bye Till Better Times\"\n\nPART V \"THE BATTLING BASTARDS OF BATAAN\"\n\n18 \"...And I Shall Return\"\n\n19 The Good Friday Attack\n\n20 \"I Would Rather Die a Thousand Deaths\"\n\n21 The Death March\n\nPART VI FROM HUMILIATION TO VICTORY\n\n22 \"But Not in Shame\"\n\n23 Strange Surrenders\n\n24 \"Go in and Get a Hit\"\n\n25 The Turn of the Tide\n\n_Image Insert_\n\nACKNOWLEDGMENTS\n\nNOTES\n\n_List of Maps_\n\nThe Six Months After Pearl Harbor\n\nRetreat to Bataan\n\nThe Abucay Line\n\nThe Points and the Pockets\n\nBattles for Java\n\nThe Death March\n\n# _Author's Note_\n\nPerhaps the most controversial six months in American history started on December 7, 1941. Even now, after almost twenty years, many issues of those days are still alive and many others have remained buried because of their highly controversial nature: Why did Japan declare war on a nation as powerful as the U.S.? Did Roosevelt use Pearl Harbor as bait to lure the Japanese into war? Why was half of MacArthur's air force caught on the ground at Clark Field by Japanese bombers ten hours after Pearl Harbor? Why did Singapore's vaunted defenses fall apart so abruptly? What caused the crushing Allied defeat in the Battle of Java Sea? Did Roosevelt and Marshall stab MacArthur in the back in the Philippines? What were the true facts behind the two greatest surrenders in American military history, at Bataan and Corregidor? Did the Japanese plan the murder of thousands of American and Filipino prisoners in the \"Death March\"? What were the mistakes that brought disaster to the Japanese at Midway?\n\nThis epochal six months also gave birth to a number of myths. Many Americans still believe Captain Colin Kelly sank a Japanese battleship, that thousands of Japanese spies infested Hawaii in 1941, that Major Jimmy Devereux radioed from Wake: SEND US MORE JAPS.\n\nNo segment of World War II was more colored by American and British politics than this period of steady, alarming defeats in the Pacific. Consequently the truth was often clouded over by polemic and political expediency. This book attempts to tell what actually happened, to record the stories\u2014some of them untold\u2014of America's most humiliating six months. It is based primarily on hundreds of interviews in eight countries with generals, privates, admirals, seamen, civilians. These included Admirals Kichisaburo Nomura, Raymond Spruance, Chester Nimitz and Frank Jack Fletcher; and Generals Albert Jones, Clifford Bluemel, Minoru Genda and Carlos Romulo. General Akira Nara, after years of silence, told the inside story of the first Japanese breakthrough in Bataan. Former President Sergio Osme\u00f1a disclosed at last the mystery of Philippine collaborationists. General Bradford Chynoweth and Colonels John Horan and Jesse Traywick uncovered the details of the \"strange surrenders\" in the Philippines.\n\nThis book is further based on many new documents and manuscripts. Mrs. Masaharu Homma, for example, finally allowed her husband's diary and last letters to be read. Historians in many parts of the world generously contributed their knowledge. Officials of all the countries involved also helped unravel the tangled web of facts. This book could not have been written without the full cooperation of the U.S. Departments of Defense, Army, Navy and Air Force; the governments of the Republics of the Philippines and China; and Japan's official war historians.\n\n# _Prologue_\n\n# \"WITH DANGEROUS AND DRAMATIC SUDDENNESS\"\n\nOn December 6, 1941, official Washington circles were waiting for the Tokyo reply to Secretary of State Cordell Hull's strong note of November 26. It could mean continued uneasy peace\u2014or sudden war between Japan and the United States.\n\nTension was highest at the Japanese Embassy on Massachusetts Avenue where the following three-part message had just been received from Foreign Minister Shigenori Togo:\n\n> 1. THE GOVERNMENT HAS DELIBERATED DEEPLY ON THE AMERICAN PROPOSAL OF THE 26TH OF NOVEMBER AND AS A RESULT WE HAVE DRAWN UP A MEMORANDUM FOR THE UNITED STATES CONTAINED IN MY SEPARATE MESSAGE NO. 902B.\n> \n> 2. THIS SEPARATE MESSAGE IS A VERY LONG ONE. I WILL SEND IT IN FOURTEEN PARTS AND I IMAGINE YOU WILL RECEIVE IT TOMORROW. HOWEVER, I AM NOT SURE. THE SITUATION IS EXTREMELY DELICATE, AND WHEN YOU RECEIVE IT I WANT YOU TO PLEASE KEEP IT SECRET FOR THE TIME BEING.\n> \n> 3. CONCERNING THE TIME OF PRESENTING THIS MEMORANDUM TO THE UNITED STATES, I WILL WIRE YOU IN A SEPARATE MESSAGE. HOWEVER, I WANT YOU IN THE MEANTIME TO PUT IT IN NICELY DRAFTED FORM AND MAKE EVERY PREPARATION TO PRESENT IT TO THE AMERICANS JUST AS SOON AS YOU RECEIVE INSTRUCTIONS.\n\nThe steps leading to this fateful day began in the late summer of 1940. Germany had overrun Belgium, Holland and France with ridiculous ease and apparently would soon conquer England. On the other side of the world, Japan was bogged down in her seemingly endless undeclared war on China. Only two great powers in the world were at peace, America and Russia.\n\nThe U.S. was widely split. The interventionists, led by President Franklin D. Roosevelt, were convinced their country's future and ultimate safety depended on helping the democracies crush the aggressor nations. Supporting them were the \"Bundles for Britain\" group and national minorities whose European relatives had suffered at the hands of Hitler and Mussolini.\n\nTheir more numerous anti-war opponents included strange bedfellows: the \"America Firsters\" of Charles Lindbergh, Senator Borah and the German-American Bund; the \"American Peace Mobilization\" of the American Communist and Labor Parties; and the traditionally isolationist Midwest which, though sympathetic to Great Britain and China, wanted no part of a shooting war.\n\nWhen Roosevelt, on September 3, traded fifty old destroyers to the beleaguered British for bases, the more rabid isolationists claimed this was merely a stratagem to lead America into war through the back door. The situation worsened on September 27, when Japan formally joined the Axis. A Tripartite Pact was signed, recognizing \"the leadership of Japan in the establishment of a New Order in Greater East Asia,\" and Hitler and Mussolini's \"New Order in Europe.\" Each promised to help if one of the others was \"attacked by a Power at present not involved in the European war or in the Chinese-Japanese conflict.\"\n\nThe pact, by its veiled threat of a two-ocean war, was designed to keep the U.S. neutral. It had the opposite effect. Many Americans hovering indecisively between isolationism and intervention were now forced to agree with Roosevelt that these newly united aggressors were a direct menace to the United States. By March 10, 1941, Roosevelt had gained enough new supporters to pass the Lend-Lease Act. America was at last committed to giving unlimited aid, \"short of war,\" to the enemies of the Axis. She was to be the Arsenal of Democracy.\n\nLittle more than three months later, on June 22, Hitler shook the world, including his Axis partners, by suddenly invading Russia. This move wrecked the already greatly weakened isolationist movement in the U.S. Instantly the American Peace Mobilization, basically sympathetic to Russia, died, its followers becoming more interventionist than Roosevelt overnight.\n\nThe attack also caused a great commotion among Japan's ruling circles. One group favored an immediate attack on Siberia, but the Army disagreed. Although most of its key moves in the preceding five years had been dominated by fear of the growing strength of Communism, General Hideki Tojo, war minister in the Konoye Cabinet, felt this was a dangerous adventure. He pressed for a drive toward Southeast Asia\u2014the fabulous storehouse of tin, rubber and oil.\n\nWhile Hitler was amazing the world with his early victories in Russia, Japan suddenly seized Indo-China on July 25, in a bloodless coup. Now there was only one great power at peace\u2014America. And on July 26, she took a bold step up to the very brink of war when Roosevelt, against the advice of the Navy's planning chief who feared it might lead to early hostilities, froze all Japanese assets in the United States. It was an economic blitzkrieg. At one stroke, the bulk of the flow of oil, the lifeblood of battle, was shut off from Japan.\n\nThe reaction there was bitter. Japan was a dynamic country of 74,000,000 people crammed into islands whose total area was less than the size of California. Just as every dynamic country before her she felt she must either expand or deteriorate into a poverty-stricken second-class power. Why, argued her leaders, was America being so self-righteous about the China Incident when the Western world, including herself, had been setting the example of plunder in the Orient for a century?\n\nSpurred by extremists on both sides, relations between the two countries were quickly approaching a dangerous point. The Japanese military leaders felt the negotiations for agreement being discussed in Washington by Ambassador Kichisaburo Nomura and Secretary of State Hull had less than an even chance of success. They insisted that a definite date of war with America be set. Japan's oil reserves were shrinking dangerously.\n\nTo solve this problem, an Imperial Conference was called in Tokyo on September 6. At the beginning of the session, Admiral Osami Nagano, chief of the Naval General Staff, solemnly said, \"Japan is facing shortages in every field, especially in materials. In a word, Japan is becoming emaciated, while her opponent is growing stronger.\"\n\nBaron Yoshimichi Hara, president of the Privy Council and a leader of Japanese conservative thought, worriedly asked the High Command to clarify the apparent subordination of diplomacy to preparations for war.\n\nSuddenly Emperor Hirohito, who had been sitting in traditional silence, spoke. He regretted the Army and Navy hadn't made their attitude fully clear. As the others listened in shocked silence he read an ode written by his grandfather, the Emperor Meiji:\n\n> When all the earth's oceans are one,\n> \n> Why do the waves seethe and the winds rage?\n\n\"I have always read and appreciated this poem, and kept in my heart the Emperor Meiji's spirit of peace. It has been my wish to perpetuate this spirit.\"\n\nThere was a long silence. Nagano rose. \"I feel trepidation at the Emperor's censure of the High Command. I assure Your Majesty that the High Command places major importance upon diplomatic negotiations and will appeal to arms only at the last moment.\"\n\nThen an Outline of National Policy was placed before the Conference: Japan should continue to exhaust diplomatic measures to attain her demands; but if these negotiations dragged on inconclusively, war should be declared on the U.S. and Britain. In other words, hope for peace but prepare for war before the end of 1941. The fuse was lit. Only a diplomatic miracle could snuff it out.\n\nThe talks in Washington between Nomura and Hull continued inconclusively for another month. Hull kept insisting that Japan break with the Axis and withdraw troops from China. In desperation the moderates in Japan suggested a compromise\u2014withdrawal of troops over a period of years. The militarists were adamant. Their spokesman, General Tojo, said, \"We can accept no compromise on principle. After all the sacrifices we have made in China the Army won't agree to any withdrawals. Army morale would not survive it.\"\n\nPremier Hidemaro Konoye, a harried man, could no longer control the situation. On October 12, he called his key ministers to an emergency meeting at his home in Tekigaiso. \"What hopes do you have of bringing war with the United States and Britain to a close once you begin it?\" he asked General Tojo. America, he pointed out, was obviously superior in resources.\n\n\"There was no certainty of victory in the war with Russia in 1904,\" replied Tojo, a dynamic and dedicated samurai warrior. \"The Premier of Japan should have enough courage to jump off the veranda of the Kiyomizu Temple!\" Since this Kyoto temple stood at the edge of a cliff, he meant Prince Konoye should have the courage to take a chance.\n\nIn four days the ineffectual Konoye resigned. And on October 18 a man of decision was made the new prime minister\u2014Hideki Tojo. The spirit of nationalism never blazed higher. The people of Japan began to believe it was their duty to build a new world based on moral principles. The damage done by Western concepts of individualism and materialism must be undone. It was, they were told over and over again, Japan's destiny to return Asia to the Asians.\n\nTwo weeks later Joseph Grew, U.S. ambassador to Japan, warned Hull of the explosive atmosphere. Because of their emotional character, he wrote, the Japanese might chance an \"all-out, do-or-die attempt, actually risking national hara-kiri....While national sanity dictates against such action, Japanese sanity cannot be measured by American standards of logic....Action by Japan which might render unavoidable an armed conflict with the United States may come with dangerous and dramatic suddenness.\"\n\nOn November 26, little more than three weeks after getting this letter, Hull handed a note to Admiral Nomura and Saburo Kurusu, a diplomat recently sent to help the ambassador in the delicate negotiations. It was an answer to a note from Tokyo offering to remove troops from Indo-China if the U.S. would unfreeze Japanese assets and hand over a required quantity of oil.\n\nAlthough Hull offered economic concessions that might well have eventually given Japan everything she needed for national prosperity, he insisted categorically that all troops be withdrawn from China as well as Indo-China.\n\nAfter reading Hull's note, the two Japanese were dismayed. \"When we report your answer to our government,\" said Kurusu, \"it will be likely to throw up its hands.\" After a futile argument with the American, Kurusu said dejectedly, \"Your response to our proposal can be interpreted as tantamount to meaning the end. Aren't you interested in a _modus vivendi_ , a truce?\"\n\n\"We have explored that,\" said Hull.\n\n\"Is it because other powers would not agree?\" asked Kurusu. It was common knowledge in diplomatic circles that China and Great Britain were strongly advising America to take an uncompromising stand with Japan.\n\n\"I've done my best in the way of exploration,\" said Hull.\n\nThe interview was over. The two Japanese returned to their embassy. Nomura knew Tokyo would regard the note as an ultimatum, even though it was by no means that. It would mean the failure of his mission to America.\n\n\u2014\n\nThis was the situation on December 6, 1941, as informed circles in Washington awaited Tokyo's answer to Hull's \"ultimatum.\"\n\nEarly that afternoon, while most of the Japanese staff was attending a luncheon party at the Mayflower Hotel, the long-awaited message began to come in on the Tokyo-Washington circuit. Since this, like all other Japanese messages, was obviously being monitored by U.S. agents, it was in the diplomatic \"Purple Code.\"\n\nBy 3:00 P.M. in another part of Washington, men from the Communications Security Group of the U.S. Division of Naval Intelligence were studying the same message. Theirs was the greatest secret in American military history. Four months previously a team of Army cryptanalysts, led by a retired lieutenant colonel, William F. Friedman, had broken the Purple Code.\n\n# PART\n\n# 1\n\n# TIMETABLE FOR CONQUEST\n\n# 1\n\n# _\"Climb Mt. Niitaka\"_\n\n**1** At the Japanese Embassy on Massachusetts Avenue, the Tokyo-to-Washington circuit was silent. Thirteen parts of the long message had come in, but the last, the fourteenth, would not arrive until the next morning, December 7. It was now late Saturday afternoon and the decoders decided to quit work although only part-way through the job of deciphering. They figured the note wouldn't be handed to Secretary Hull until Monday.\n\nFirst Secretary Katsuzo Okumura was personally typing out the decoded parts. It was too secret for any stenographer. When he finished the copy on hand, he went to the basement to relax. It was a typical dead Saturday afternoon and the subterranean playroom was almost deserted. Two correspondents were playing ping-pong. One of these, Masuo Kato from Domei, laid down his paddle. He wanted to pump the first secretary about the liner, _Tatsuta Maru_ , which had left Yokohama on December 2, and was due to reach Los Angeles on the fourteenth. Most of the small Japanese community expected to sail home on this ship since trouble was sure to come soon.\n\n\"I'll bet you a dollar the liner never gets here,\" said Okumura mysteriously.\n\nThey shook hands to seal the bet. Kato ate at a Chinese restaurant near Union Station and then decided to check in at the Domei news agency. While going up to his office in an elevator, an INS reporter said, \"Did you know the President sent a message to the Emperor appealing for peace?\"\n\nDetail left\n\nDetail right\n\nKato thought the American was joking, but a moment later he learned the story was true. He was worried. The situation was even worse than he had suspected. He remembered the strange bet with the first secretary. Then there were the persistent reports all day of Japanese troop convoys heading toward the Gulf of Siam. This could well mean an attack on Singapore.\n\nHe typed out a dispatch on the Roosevelt message and sent it to Tokyo.\n\n\u2014\n\nAt the Navy Department, the Communications Security Group was much more industrious than the Japanese Embassy decoders that Saturday. It finished deciphering and typing the first thirteen parts of the Tokyo message by 8:30 P.M. The chief, Lieutenant Commander Alwin Kramer, realized this was an important message, for the language was appreciably stronger than earlier notes. It indicated a definite probability that the Japanese were breaking off negotiations.\n\nKramer began to telephone those who should get copies of the message. \"I have something important that I believe you should see at once,\" he told Secretary of the Navy Frank Knox; he also called the head of the Far East Section of the Division of Naval Intelligence, the Director of War Plans Division, the Director of Naval Intelligence, and the White House. One man on his list could not be reached. Chief of Naval Operations Admiral Harold Stark was not at his quarters on Observatory Circle on Massachusetts Avenue.\n\nA copy of the message for the President was placed in a letter pouch and locked. Copies for the other recipients were put in folders and then into a second pouch and locked. A little after 9:00 P.M. Kramer left his office and was driven by his wife to the White House grounds. He walked to the mailroom in the office building near the White House and handed the smaller pouch to the man on duty, Lieutenant Lester Schulz.\n\nSchulz took the pouch in to the White House. Here he got permission to go to the President's study on the second floor. Roosevelt was sitting at his desk, talking to his chief adviser, Harry Hopkins. Schulz unlocked the pouch and handed a sheaf of clipped papers to the President. It took Roosevelt about ten minutes to read Japan's detailed reasons why it could not possibly accept Hull's note of November 26 as a basis of negotiation. He silently handed the papers to his adviser. When Hopkins finished reading, Roosevelt said, \"This means war.\"\n\nWhile Schulz waited they talked of the deployment of Japanese forces. Roosevelt mentioned a radiogram he had sent earlier that day to Emperor Hirohito requesting the withdrawal of troops from Indo-China. A troop convoy was now heading from there to the Gulf of Siam. Where was this invasion force bound?\n\n\"Since war is undoubtedly going to come at the convenience of the Japanese,\" said Hopkins, \"it's too bad we can't strike the first blow.\"\n\n\"No, we can't do that,\" said Roosevelt. \"We are a democracy and a peaceful people.\" He raised his voice. \"But we have a good record.\" Then he said he was going to call Admiral \"Betty\" Stark. He reached for the phone and asked to be connected with the Chief of Naval Operations. The White House operator said the Admiral could be reached at the National Theatre. Roosevelt put the phone down. \"I'll call Betty later; I don't want to cause public alarm by having him paged in a theatre.\"\n\nThe President returned the papers to Schulz, who left the room. It was the most grievous dilemma of Roosevelt's entire career. The Japanese were about to attack British or Dutch possessions. What should\u2014or could\u2014be done? The British and Dutch were too weak to defend themselves. Without U.S. intervention the Japanese could carve out an empire from the Aleutian Islands to India. But the American people were in no mood for war. The Draft Bill had passed four months previously by the margin of a single vote. He remembered what he'd recently told Churchill when asked to get into the war. \"If I were to ask Congress to declare war, they might argue about it for three months.\"\n\nAmericans, he was positive, would not go to war to save Singapore or Java or even Australia.\n\n\u2014\n\nAdmiral Stark was trying to get some relaxation after months of tension. He was watching _The Student Prince_. But it made little impression on him. Later he wouldn't even remember where he'd been on the night of December 6. His mind was still on the Far East crisis. On November 27, although he had not yet read Hull's strong note of the previous day, he was so sure Japan would strike in retaliation he had sent an unprecedented message to commanders in Hawaii and the Philippines:\n\n> THIS DISPATCH IS TO BE CONSIDERED A WAR WARNING. NEGOTIATIONS WITH JAPAN LOOKING TOWARD STABILIZATION OF CONDITIONS IN THE PACIFIC HAVE CEASED. AN AGGRESSIVE MOVE BY JAPAN IS EXPECTED WITHIN THE NEXT FEW DAYS. EXECUTE AN APPROPRIATE DEFENSIVE DEPLOYMENT PREPARATORY TO CARRYING OUT THE TASKS ASSIGNED IN WPL-46 [THE WAR PLAN].\n\nWhat puzzled him, as well as his opposite number in the Army, General George Marshall, was where the attack would come. The troop convoy nearing the Gulf of Siam suggested Singapore, but it could come in the Philippines or even the Panama Canal. In Hawaii, at least, American defenses were formidable and prepared. In fact, Stark had been so pleased with the Joint Army-Navy Hawaiian Defense Plan for protection of the Pearl Harbor base against a surprise Japanese air attack that he had sent it to all his district commanders as a model.\n\nJust before midnight, Commander Kramer, still chauffeured by his wife, drove up to the Arlington home of Rear Admiral T. S. Wilkinson, Director of Naval Intelligence. The Admiral was entertaining Captain Beardall, Roosevelt's naval aide, and Major General Sherman Miles, Chief of Army Intelligence. All three read the message. They agreed that it certainly looked as though the Japanese were terminating negotiations.\n\nKramer, his messenger duties over, now returned the copies of the message to the safe in his office. Then he asked the watch officer if the fourteenth part had come in. When he was told that nothing even looking as if it might be the fourteenth part had been intercepted as yet, the industrious Kramer at last started for home.\n\nIt was just before 1:00 A.M., December 7. Many high officials were still awake in Washington, wondering when the Japanese would jump, and where. Not one\u2014Roosevelt, Hull, Stimson, Knox, Marshall or Stark\u2014expected it could be Pearl Harbor.\n\n**2** It was then almost 7:30 P.M., December 6, in Hawaii. Here, as in Washington, there was fear of an early war. One of the main topics of dinner conversation was Roosevelt's unprecedented appeal to Emperor Hirohito. The _Star-Bulletin_ had two conflicting front-page headlines. One read: JAP PRESS ASKS FOR WAR and the other: NEW PEACE REPORT URGED IN TOKYO. Even if war came, few in the islands had doubt of the issue. Everyone agreed with the United States Senator who, on page ten of the same paper, was quoted as telling an AP reporter that, \"The United States Navy can defeat the Japanese Navy any place, any time.\"\n\nLike Marshall and Stark, the Army and Navy commanders of Hawaii were not at all worried about an air attack on Pearl Harbor. Lieutenant General Walter Short, commanding the Hawaiian Department, had received a message from Marshall similar to the war warning Stark had sent Admiral Husband E. Kimmel, commander of the Pacific Fleet. It wasn't as sharp as the Navy message but it did warn of possible \"hostile action at any moment.\" At the same time, Short received another message, this from Army Intelligence.\n\n> JAPANESE NEGOTIATIONS HAVE COME TO PRACTICAL STALEMATE. HOSTILITIES MAY ENSUE. SUBVERSIVE ACTIVITIES MAY BE EXPECTED.\n\nTo Short this meant one thing: sabotage from Hawaii's 157,905 Japanese civilians. He reported to Washington that he was alerting the Army for sabotage. And when he got no answer, assumed he had taken all action necessary.\n\nAt the moment the general was on the _lanai_ of his home at Fort Shafter holding an emergency meeting with his intelligence officer, Lieutenant Colonel Kendall Fielder, and his counterintelligence officer, Lieutenant Colonel George Bicknell. The latter had just brought the transcript of a telephone conversation monitored by the FBI from a local Japanese dentist to a Tokyo paper, _Yomiuri Shimbun_. Its editor wondered about things in general in Hawaii: planes, searchlights, weather, even the flowers. In reply to this last question the dentist-correspondent had said the hibiscus and poinsettia were in bloom.\n\nThe three officers were puzzled. Was it code? If so, why talk so openly about obvious military objects? Yet, why spend money talking about hibiscus? They had already probed the matter for almost an hour. And all this time the wives of the general and Colonel Fielder were waiting impatiently outside in a car. Short finally said nothing could be done until morning and he and Fielder joined the ladies. They would have to hurry. It was 15 miles to the Schofield Barracks Officers Club, which was putting on a special benefit show that Saturday night.\n\nMost of the high-ranking naval officers were also spending an evening of relaxation. Vice Admiral Fairfax Leary was giving a dinner party at Honolulu's \"House Without a Key.\" Kimmel was there but, as usual, his chief of staff, Vice Admiral W. W. \"Poco\" Smith couldn't get him to relax. Kimmel was a dynamic, energetic man who was only content when working. At 9:30 P.M. Kimmel excused himself, after drinking his customary single cocktail and making the necessary small talk. He wanted to get to bed. Though there was some gossip that he and his Army counterpart, General Short, were not on speaking terms, they had a date to play golf the next morning. It would be one of the rare Sundays Kimmel didn't spend at his desk. He knew the mighty Pacific Fleet was the keystone of American defenses in the Far East. The responsibility weighed so heavily on him he had left his wife to whom he was devoted on the mainland so he could concentrate completely on work.\n\nBoth Kimmel and Short were faced with the same dilemma. Hawaii was not only an outpost but a training command. If they ordered a state of constant alert, men and material would be exhausted. Besides, the warnings from Washington hadn't specifically stated or implied that an air attack on Pearl Harbor was even a remote possibility. Influenced by this reasoning, both men decided to compromise. Kimmel was prepared for submarine attacks; Short was ready for saboteurs. But the Joint Army-Navy Hawaiian Defense Plan\u2014the one so admired by Admiral Stark\u2014was not in effect on the night of December 6. In fact, normal peacetime liberty had been granted to men and officers that evening.\n\nNothing but routine and limited air patrols were being planned for the next morning; Army and Navy anti-aircraft batteries protecting Pearl Harbor were lightly manned. Moored at this great base was the principal obstacle of any further Japanese invasion: the Pacific Fleet, 94 ships including 8 battleships and 9 cruisers. Most of the men, except the watch crews, were getting ready for bed. It was just another beautiful but uneventful tropical evening.\n\nThe only extraordinary precautions taken that night were at the Army airfields. Planes were bunched together neatly on runways so they could be protected by cordons of guards. Actually there was not the slightest danger from saboteurs. In all Hawaii there was only one Japanese Navy spy, a brusque, hot-tempered, young ensign, Takeo Yoshikawa. Except for one Nisei girl\u2014and her help had proved worthless\u2014the large Japanese community in the islands had refused to cooperate with him. To his surprise they considered themselves loyal Americans. It didn't make sense to him; many still worshipped at Buddhist temples and Shinto shrines and had contributed generously to the Imperial Army's relief fund.\n\nGossiping with American sailors had been just as fruitless. They talked a lot but said nothing. What information he got was by simple, unexciting methods. He walked almost every day through Pearl City to the end of the peninsula and scanned Battleship Row; he swam at every available beach, observing underwater obstructions and noting the tides; and most important, he sat on straw _tatami_ mats with geisha girls\u2014sometimes Shimeka, sometimes Marichyo\u2014in the Shunchoro, a Japanese restaurant located on a hill overlooking Pearl Harbor, and drew diagrams of the ships in the big base. Once he allowed the geishas to persuade him to take them on a sightseeing air tour over Oahu. The pilot never noticed Yoshikawa\u2014dressed in loud aloha shirt and flanked by pretty, excited girls\u2014as he observed atmospheric conditions and took photos of military airfields.\n\nHe was getting no outside help at all. Once he had given a Nazi agent using the cover name of Karama (his real name was Bernard Kuehn) a final payment of $17,000 to signal information to Japanese submarines. But as yet the German had not transmitted a single message. All the information sent to Tokyo so far had cost only $600\u2014the amount given Yoshikawa for expenses when he first arrived.\n\nThat night, he was working late at the Japanese Consulate in Honolulu. Here he posed as a vice-consul named Tadashi Morimura. Earlier he had radioed Tokyo in diplomatic code that there still were no barrage balloons over Pearl Harbor. Now he was at his desk writing another message in pencil: ENTERPRISE AND LEXINGTON HAVE SAILED FROM PEARL HARBOR.\n\nHe buzzed for the radio-room code clerk, gave him the message and then went for a stroll around the Consulate grounds. In the distance he could see a bright haze over Pearl Harbor. He could hear no patrol planes. Though he was restless, he started for home. He wanted to get up in time to hear the Sunday morning broadcast from Radio Tokyo. If the weather forecast, _higashi no kaze, ame_ (east wind, rain), appeared both in the middle and the end of the broadcast, this meant Japanese-American relations were in danger, war was coming.\n\nThe _Tatsuta Maru_ , the liner scheduled to land in Los Angeles on December 14, was supposed to be near Honolulu at that moment. But it had suddenly, and without explanation to its passengers, sharply swung off course. It was returning to Japan.\n\n**3** The lone-wolf spy in Hawaii had sent the correct information to Tokyo. Both of Kimmel's carriers were at sea. Task Force Twelve, including the _Lexington_ , was delivering planes to Midway. Task Force Eight\u2014the carrier _Enterprise_ , 3 cruisers and 9 destroyers\u2014was about 500 miles west of Hawaii. Commanded by Vice Admiral William Halsey, Jr. (\"Bull\" to the newspapers but \"Bill\" to his friends), it had delivered 12 Grumman Wildcats and their Marine fighter pilots to Wake Island three days before and was now heading for home. The _Enterprise_ was a day behind schedule because of bad weather. Many of the men were griping because they'd missed Saturday night in Honolulu. Instead of enjoying themselves in port they were watching a movie about World War I, _Sergeant York_. Thought of a new war was far from most of their minds, even though Halsey had told the crew, \"At any time, day or night, we must be ready for instant action.\"\n\nUp in the sea cabin, Halsey was waiting for a war he knew was about to explode. Admiral Kimmel had impressed on him when he left Pearl Harbor on November 28 how important the planes were for the defense of Wake. And they had to be delivered with absolute secrecy. It was imperative that the Japanese be kept ignorant of the move.\n\n\"How far do you want me to go?\" Halsey had asked.\n\n\"Goddamit,\" replied Kimmel, \"use your common sense.\"\n\nAs soon as they were at sea, Halsey issued Battle Order No. 1 putting the _Enterprise_ under war conditions. He ordered warheads placed on torpedoes, all planes armed with bombs or torpedoes, and pilots to sink any shipping or plane sighted.\n\nWhen his operations officer, Commander William Buracker, saw this order he asked incredulously, \"Admiral, did you authorize this thing?\"\n\n\"Yes.\"\n\n\"Goddamit, sir, you can't start a private war of your own. Who's going to take the responsibility?\"\n\n\"I'll take it. If anything gets in my way, we'll shoot first and argue afterwards.\"\n\nAfter dinner that same night, Major Paul Putnam, commander of the 12 Grummans, had cornered Halsey. \"I know I'm on my way to Wake, but what in the hell am I supposed to do when I get there?\" He only knew that he had been ordered to fly half of Marine Fighting Squadron 211 onto the _Enterprise_ and then proceed to Wake. It had to appear like a routine weekend exercise and most of his men carried only an extra pair of skivvies (shorts) and a toothbrush.\n\n\"Putnam, your instructions are to do what seems appropriate when you get to Wake. You're there under my direct and personal orders and will not report for duty to the island commander.\" Halsey held out his hand. \"Have a pleasant cruise.\"\n\nNow, 1500 miles west of the Hawaii-bound Halsey task force, Putnam and his fliers were on Wake Island. Here, because of the international date line, it was already the evening of December 7. The Marine commander had learned that his 12 planes were the only air defense of an extremely important island bastion. Yet he and his men had had only a few hours flying time in the stubby Wildcats.\n\nAlso on the island were 70 employees of Pan-American Airways, 1146 civilian construction workers, 69 Navy men, an Army Communication detail of 6, and 388 combat Marines, under Major James Devereux. In addition to the handful of planes, the island's only other defenses were 6 5-inch guns, 12 3-inch anti-aircraft guns, a few machine guns and Browning automatic rifles\u2014and about 400 rifles.\n\nMajor Devereux had been told the mission of his Marines was merely to stand off a \"minor raid.\" Today, for the first time in two months, he had given the men a holiday. They fished, swam and gambled. As Devereux walked to his tent to turn in he remembered the previous January when sudden overseas orders had arrived. He had told his brother, Ashton, that he guessed he was bound for \"some little spitkit of an island.\" And when Ashton asked what would happen to him, Devereux had said, \"Your guess is as good as mine\u2014but I'll probably wind up eating fish and rice.\"\n\n**4** That moment in Manila it was 5:00 P.M., December 7. It had been a hot, clear day. Here, as in the rest of the Philippines, there was a lively awareness that war was imminent. Only the time of attack was in question. Admiral Thomas Hart, commander of the Asiatic Fleet, insisted it might come any day. General Douglas MacArthur, commander of the United States Armed Forces of the Far East (USAFFE), was hoping hostilities would not start before April, 1942. By this date he had been assured that all necessary reinforcements from the United States would have arrived. MacArthur refused to be panicked, even though strange aircraft had been reported over nearby Clark Field, the main bomber base, four nights in a row.\n\nHe also discounted a warning which had come from Master Sergeant Lorenzo Alvarado, a United States secret service agent since 1917. Alvarado, fifteen days before, had attended a secret meeting of the \"Legionarios Del Trabajo,\" an underground organization, at the Triangulo Studio on Rizal Street. Shiki Souy, the Japanese owner of the studio, told his fellow conspirators that 100 Japanese ships and many planes were then in Formosa and would soon invade the Philippines.\n\nThat night Admiral Hart was more than ever convinced that a crisis was approaching. For months the Asiatic Fleet chief had been telling his commanders that war was on the way. That was why his small fleet\u20141 heavy cruiser, 1 light cruiser, 13 World War I four-stack destroyers and 29 submarines\u2014was ready for action. Ammunition was in the racks, warheads were on the torpedoes, Manila and Subic Bays were mined, and the fleet was dispersed from Manila Bay to Borneo.\n\nThe previous day Vice-Admiral Tom Phillips, commander of the British Far East Fleet, based at Singapore, had conferred in Manila with Hart and MacArthur. The three were worried about the Japanese convoy, sighted off the coast of Indo-China and then lost in fog. Was it heading for a direct attack on Malaya and Singapore or merely landing at Siam?\n\nAlthough MacArthur agreed that the situation was critical, he said that by April, 1942, he would have a trained army of about 200,000 men and a powerful air force of 256 bombers and 195 pursuit planes. This great strength would assure the defensibility not only of the Philippines but the entire Southeast Pacific area.\n\n\"Doug, that is just dandy,\" interposed Hart. \"But how defensible are we right now?\" The answer to that was painfully obvious. MacArthur did have about 130,000 men in uniform, but almost 100,000 of these were poorly equipped Philippine Army divisions with only a few months training in close-order drill. Though they were enthusiastic and willing and worshipped MacArthur, they could do little but salute properly. Most had never even fired their ancient Enfield rifles. The Air Force was in worse shape. On paper MacArthur had 277 airplanes, but of these only 35 Flying Fortresses and 107 P-40s were modern combat craft.\n\nAfter MacArthur left, Phillips\u2014nicknamed Tom Thumb because he was only five feet four, an inch shorter than Napoleon\u2014made one specific request of Hart. He wanted four U.S. destroyers to accompany his fleet on a sortie from Singapore up the east coast of Malaya as a countermove to the threatening Japanese armada. His two great ships, the battle cruiser _Repulse_ and the battleship _Prince of Wales_ , had only four destroyers as escort.\n\nHart agreed to send four of his own overage destroyers then anchored at Balikpapan. At 6:00 P.M., as everyone stood to leave, a messenger came in with a dispatch for Phillips. It was read aloud. Planes from Singapore had sighted the Japanese convoy off the Siamese coast. Course was 240 degrees.\n\n\"Admiral,\" said Hart to Phillips, \"when did you say you were flying back to Singapore?\"\n\n\"I'm taking off tomorrow morning.\"\n\n\"If you want to be there when the war starts, I suggest you take off right now.\"\n\nThe following night, December 7, the 27th Bombardment Group\u20141200 strong but without a single plane\u2014was giving a party at the Manila Hotel in honor of Major General Lewis Brereton, commander of MacArthur's air force, who had arrived little more than a month before. It was a gala affair long to be remembered by those who attended as being \"the best entertainment this side of Minsky's.\" But the guest of honor's mind was on war and his sadly inadequate air force. During the party he talked with Rear Admiral William Purnell, Hart's chief of staff. \"It's only a question of days or perhaps hours until the shooting starts,\" said Purnell. A few minutes later Brigadier General Richard Sutherland, MacArthur's chief of staff, said the War Department believed hostilities might begin at any time.\n\nAlarmed, Brereton went to a phone and called his own chief of staff, instructing him to put all airfields on combat alert. He was thankful heavy air reinforcements were coming. The _Pensacola_ convoy, now somewhere in the South Pacific, was scheduled to arrive January 4. Its 7 transports carried 52 dive bombers as well as two regiments of artillery and large amounts of badly needed ammunition and supplies. In addition, 30 Flying Fortresses, which would almost double his bomber force, should arrive in a few days. Twelve had already taken off from California and were due to land in Hawaii at dawn.\n\nAt Clark Field, 50 air miles to the northwest, 16 Flying Fortresses were lined up ready for flight. Three others were in the hangars being camouflaged or repaired. The flat field, rimmed only by a few trees and waist-high cogon grass, was honeycombed with revetments, foxholes and slit trenches. It looked weird and unworldly in the moonlight.\n\nIn one of the nearby wooden barracks, Staff Sergeant Frank Trammell of the 30th Bombardment Squadron was vainly trying to contact his wife, Norma, in San Bernardino, California, by ham radio. Every Sunday night for the past month he had talked with her. It was queer. The air was dead. The only thing he could raise was a city he was forbidden to talk to\u2014Singapore.\n\n\u2014\n\nThe 220-square-mile island of Singapore was about 1600 miles to the southwest, almost the same distance and direction as a flight from New York to New Orleans. Connected to the southernmost point of Asia, the Malay Peninsula, by a causeway, it was the keystone of the Allied defense system. If it fell, not only Malaya would be lost but the entire Dutch East Indies, with its oil, tin and rubber.\n\nThat night the sky over Singapore was ablaze with the probing fingers of searchlights. Huge 15-inch guns protected the sea approaches. And in the great Naval base\u2014built in twenty years at a cost of 60,000,000 pounds\u2014were moored two of the most powerful war ships afloat, the _Repulse_ and the _Prince of Wales_.\n\nAt every military installation there was a sense of alert excitement. The code word, \"Raffles,\" had just been signalled throughout the Malayan Command and all ranks were standing to arms ready for immediate action. British, Australian and Indian soldiers were prepared and confident. Singapore, they felt, was an impregnable fortress.\n\n\u2014\n\nNorth-northeast of Singapore about 1650 miles was Great Britain's other fortress in Southeast Asia, Hong Kong. Also an island, it was a few minutes ride by ferry from the mainland of southern China. That Sunday night the Colony of Hong Kong was practically on a war footing. Although ambassadorial telegrams from Tokyo had been extremely moderate lately, Major-General C. M. Maltby, military commander of the Colony, had alerted his 11,319 men.\n\nBy midnight, except for its patched regatta of ketches, proas, junks and sampans, the great harbor was almost barren. The previous night pages had gone through the bars and ballrooms of hotels telling all Merchant Navy seamen and officers to report to their ships immediately. Here, word of the Japanese convoy in the Gulf of Siam had meant only one thing: the balloon had gone up. But Hong Kong was ready and confident.\n\nFrom Washington to Hong Kong it was apparent that Japan might strike within hours. But in many places \"readiness\" was only a word. Few were actually prepared for the brutal reality of war. None were aware of a detailed Japanese plan that was already developing that moment step by step.\n\n**5** A week earlier the Imperial Conference had convened in Room 1 of the Imperial Palace in Tokyo. Present were the military and civil leaders of Japan and the Emperor himself. Everyone realized it was probably the most important meeting ever held in Japan. Out of it would come war or peace. There was a hush as Prime Minister Tojo\u2014an intense, nervous man who every day smoked at least fifty cigarettes and drank a dozen cups of coffee\u2014stood up.\n\n\"With the permission of the Emperor, I will take charge of the proceedings today.\" Everything had been done, he said, to adjust diplomatic relations with America but Hull's note of November 26 was clearly an ultimatum since it demanded withdrawal of troops from China without offering a single concession. \"Submission to these demands would not only deprive Japan of her authority and forestall her efforts for the successful settlement of the China Incident, but would also jeopardize her very existence.\" He reviewed the situation and then concluded grimly, \"Matters have now reached the point where Japan, in order to preserve her Empire, must open hostilities against the United States, Great Britain and the Netherlands.\"\n\nAdmiral Osami Nagano, representing the High Command, then explained the military situation. The Western powers were increasing their strength in the Far East. Soon it would be too late to strike. There was now almost no danger of an attack from Russia because of German successes. \"The officers and men of the Army and Navy are in high spirits and are burning with desire to serve their Emperor and their country, even at the cost of their lives.\"\n\nAfter optimistic reports on the nation's financial condition and food supply, Baron Yoshimichi Hara, president of the Privy Council, the supreme advisory body to the Emperor, arose. This was the voice of Japanese conservatism and his words were eagerly awaited. \"The Japanese Empire can concede nothing further. If she does, the gains obtained from the Sino-Japanese war in 1894 will be completely nullified. It is to be regretted that we are compelled to fight another war after having been engaged in the China Incident for more than four years, but the conclusion of diplomatic negotiations with the United States is hopeless, in spite of everything we have done....\n\n\"As there is no alternative I am compelled to give my consent to this plan. We must have confidence in our soldiers, be convinced of victory in war and, at the same time, leave no stone unturned to maintain the nation's unity over a long period.\"\n\nTojo assured Hara the Government was ready to call off the war plans at the very last moment if America would negotiate \"just terms.\" Then, seeing that there were no dissenting votes, the general said, \"In closing, I wish to say that the Japanese Empire is now on the threshold of progress or collapse.\"\n\nAll present except the Emperor, who had not uttered a word, signed the documents. Later that day the papers were presented to Emperor Hirohito. For the past three days he had been convening with the \"Senior Statesmen,\" former prime ministers, in a desperate effort to avoid war. They had assured him that if Japan bowed to America's \"unrealistic and unjust\" demands she would instantly become a second-rate power. And, feeling that further resistance to the advocates of war would be against the inevitable will of the people, the mild-mannered ruler reluctantly sanctioned the documents.\n\nThe next afternoon, December 2, the Imperial Navy Headquarters was told of the Emperor's decision. At 5:30 P.M. a radio message was sent to the powerful Pearl Harbor Striking Force, 32 ships including 6 carriers, already midway between Japan and Hawaii. It read: CLIMB MT. NIITAKA. It meant: Begin the War.\n\n**6** At 3:30 A.M. December 7, American minesweepers and destroyers were patrolling the entrance to Pearl Harbor, unaware that under them were five midget Japanese submarines, each manned by a crew of two. Released from their mother submarines near midnight, they were maneuvering toward the entrance to the harbor.\n\nAt 3:42 A.M. the minesweeper _Condor_ was also bound for Pearl Harbor. One thousand yards from the entrance, a lookout shouted. The ship had almost run into some object apparently heading into the harbor.\n\nThe _Condor_ , startled and alerted, searched for a few minutes and at 3:58 A.M. sent out a blinker signal to a nearby destroyer, the _Ward_. A suspicious object west of her sweep area had been sighted, said _Condor_. She believed it was a submarine.\n\nLieutenant William Outerbridge, the _Ward_ 's skipper, ordered his ship to general quarters. For half an hour the old four-stacker combed a wide pattern with all topside hands on the lookout. Nothing was seen and at 4:43 A.M. everyone except those on watch went back to bed. Neither _Condor_ nor _Ward_ reported the incident to higher headquarters.\n\nThe anti-torpedo net across the entrance to Pearl Harbor slowly began to open at 4:47 A.M. Eleven minutes later the gate was wide open. Soon the minesweeper _Crossbill_ entered. But the gate remained open. Since the _Condor_ was scheduled to follow soon, those operating the net decided it was too much trouble to close it for such a short time. At 5:32 A.M. _Condor_ passed through. Once more the net operators decided to keep the gate open. In forty-five minutes the tug _Keosanqua_ was due to leave Pearl Harbor.\n\nAnd so the mouth of Pearl Harbor was wide open for the five Japanese submarines. They were only seventy-nine feet long but each carried two full-sized torpedoes.\n\n\u2014\n\nThirteen minutes after the _Condor_ entered Pearl Harbor, the war in the Pacific started by mistake. According to the grand plan of the Imperial High Command the first blow was to fall on Pearl Harbor at 8:00 A.M. Then Singapore, Hong Kong, the Philippines, Guam and Wake were to be bombed in succession. These were but the first steps in a master plan to conquer all Southeast Asia in six months.\n\nThe mistake was the result of overeagerness. The great Japanese convoy which had been seen entering the Gulf of Siam by the Allies two days earlier was carrying troops scheduled to conquer Singapore. A section of this convoy arrived at its destination halfway down the Malay Peninsula too soon. British defense troops on the beaches of Kota Bharu happened to spot these invasion ships and opened fire at 12:45 A.M., Singapore time, with land artillery. The war in the Pacific was on and the first shot ironically had been British.\n\nAt that moment it was 1:15 A.M., December 8, in Japan. Aboard his flagship, _Nagato_ , in the Inland Sea, the commander-in-chief of the Combined Fleet, Admiral Isoroku Yamamoto, would not know for some time that the war had already started accidentally. He had been up all night waiting for word from Vice-Admiral Chuichi Nagumo, commander of the Pearl Harbor Striking Force. Nagumo's six carriers were presently about 250 miles north and slightly east of their target. In fifteen minutes, 6:00 A.M., December 7, Hawaiian time, the first wave\u201440 torpedo planes, 51 dive bombers, 49 high-level bombers\u2014would start taking off. And two hours later the first bombs would be dropping on the U.S. Pacific Fleet.\n\nYamamoto (his first name, \"Fifty-six,\" was the age of his father at the admiral's birth) was not waiting impatiently or nervously. A man of remarkable energy and purpose, his face showed no emotion. Unlike others in his own headquarters and in Tokyo, he was not worried about the Pearl Harbor attack. He had planned it carefully and prepared it efficiently. If Nagumo followed his instructions, he was sure the attack could not fail.\n\nIt was ironic that his strategy was being used for he had strongly opposed a war with the United States and Britain, feeling Japan would have no chance of victory. Overruled, he had then devoted his time to proving he was wrong. After study, he concluded that Japan's only chance was to cripple the Pacific Fleet with a single blow. This would checkmate America's main strength in the Pacific and, he hoped, dampen her fighting spirit.\n\nThere had been sharp opposition to Yamamoto's plan from his own subordinates as well as the Naval Staff. Not until October 20, when Yamamoto had threatened to resign, was the Pearl Harbor attack finally approved.\n\nDespite his optimism, he still hoped there would be a last-minute cancellation of the Climb Mt. Niitaka order. \"Admiral Nomura is a very capable man,\" he had recently told friends. \"Perhaps he will save the United States-Japan situation.\" But as time ran out that evening, he realized this was now a forlorn hope.\n\n\u2014\n\nIn Tokyo, American Ambassador Joseph C. Grew was just presenting Roosevelt's personal plea to the Emperor to Foreign Minister Shigenori Togo. It had been held up more than ten hours by the Tokyo Central Telegram Office by order of an officer of the General Staff.\n\nGrew asked Togo to arrange an Imperial audience in spite of the late hour. Togo agreed, and read the message. It recalled the long friendship between Japan and America, suggesting that if Japan would withdraw from Indo-China, no other country would invade that area.\n\nThough this was by no means a new proposal, Togo felt the Emperor should see it. If a last-minute peace were to be arranged, there was no time to lose. Late that afternoon he had sent off the fourteenth and final part of the long reply to Hull's \"ultimatum.\" With it he had also sent Nomura instructions to deliver the entire memorandum at exactly 1:00 P.M., December 7, Washington time, thirty minutes before the Hawaiian raid. The Third Hague Convention had prescribed no minimum time which should elapse between declaration of war and attack and to Togo's legalistic mind this half-hour warning would prevent the United States from later accusing Japan of a \"sneak attack.\"\n\nTogo telephoned Marquis Koichi Kido, lord keeper of the privy seal. An immediate audience with the Emperor was granted, but first a translation of the telegram had to be made.\n\n**7** In Washington it was then Sunday morning, December 7. Lieutenant Commander Kramer, though up late the night before delivering messages, was at his office in the Navy Department shortly after 7:30 A.M. He learned the anxiously awaited fourteenth part of the Tokyo message had finally come in at 4:00 A.M. and was almost decoded. In a few minutes Kramer had the message. It was short but its last paragraph was ominous.\n\n> THE JAPANESE GOVERNMENT REGRETS TO HAVE TO NOTIFY HEREBY THE AMERICAN GOVERNMENT THAT IN VIEW OF THE ATTITUDE OF THE AMERICAN GOVERNMENT IT CANNOT BUT CONSIDER THAT IT IS IMPOSSIBLE TO REACH AN AGREEMENT THROUGH FURTHER NEGOTIATIONS.\n\nThe entire fourteen parts were assembled, put in folders and once more Kramer began making his delivery rounds.\n\nGeneral Marshall was just then eating breakfast at his home in Virginia. Although copies of the same thirteen-part message delivered to Roosevelt and the Navy officials the previous evening had been automatically routed to the Army, the chief of staff knew nothing about the vital memorandum. It was odd because in addition to other Army officers even Colonel Bedell Smith, secretary of the General Staff, had received it Saturday night.\n\nAt any rate, Marshall, ignorant of a message that had meant war to the President, finished reading the papers and then about 8:30 A.M. began his usual Sunday horseback ride. He headed at a lively gait toward the government experimental farm, the site of the future Pentagon Building.\n\n\u2014\n\nAcross the Potomac River, Kramer was still delivering the full fourteen-part message. By 10:20 A.M., his rounds completed, he was back in his own office. Another decoded Japanese message was waiting for him, this of an even more alarming nature. It was a note from Foreign Minister Togo to Nomura marked \"Urgent\u2014Very Important.\"\n\n> WILL THE AMBASSADOR PLEASE SUBMIT TO THE UNITED STATES GOVERNMENT (IF POSSIBLE TO THE SECRETARY OF STATE) OUR REPLY TO THE UNITED STATES AT 1:00 P.M. ON THE 7TH, YOUR TIME.\n\nWhile it was being put in folders, Kramer hastily made a time circle. He wanted to get a picture of how the one o'clock time tied in with the movement of the big Japanese convoy in the Gulf of Siam. It would be, he learned, 1:30 P.M. December 8 in Kota Bharu; not a particularly significant time of day to start an attack. He then figured the time in Hawaii. There it would be 7:30 A.M. Since he had spent two years at Pearl Harbor he knew this was the normal time for the piping of the crew to Sunday breakfast\u2014a very quiet time indeed.\n\nHe hurried down the corridors of the sprawling Navy Building and by 10:30 A.M. reached Admiral Stark's office. Stark, just arrived from a long, leisurely walk around the grounds and greenhouses of his quarters, was busy reading the fourteen-part message. While waiting in the outer office for the Admiral to finish, Kramer chatted with his own chief, Commander Arthur McCollum, head of the Far East Section. Kramer pointed out the possible significance of the one o'clock time with reference to Malaya and Hawaii.\n\nAt last Stark finished the long memorandum and read the \"one o'clock\" note. It was suggested he send an alert to Navy commanders. Stark vetoed the idea. He felt his \"war warning\" of November 27 was enough to keep everyone on his toes.\n\nOut on northwest Massachusetts Avenue at the Japanese Embassy, the officials in charge of incoming telegrams, adapting themselves to leisurely American Sundays, had reported late and Admiral Nomura was just then reading Togo's instructions to deliver the memorandum at 1:00 P.M. Nomura immediately telephoned the office of Cordell Hull. He asked for an interview at 1:00 P.M.\n\nThe secretary of state had a luncheon appointment.\n\nNomura asked to see the undersecretary of state. There was a pause. Since Ambassador Nomura insisted it was a matter of the utmost importance, Mr. Hull would see him at 1:00 P.M.\n\nAs soon as Nomura hung up, he asked to see the rest of the fourteen-part message. He had only read the first few sections before leaving the Embassy the day before. To his dismay he learned that the decoders had quit work early on Saturday and would need two or three hours to finish. Nomura was frantic. It was almost 11:00 A.M.\n\nGeorge Marshall, finally alerted by his office that an important message had come in, was just then hurrying from Virginia to the War Department Building. By a few minutes after 11:00 A.M. he was reading the fourteen-part message. Some sections he read twice. Finally he reached the end. Underneath was the \"one o'clock\" note. Instantly he was alarmed. This was much more critical and significant. He was very much taken aback because of the time he had spent on the preceding lengthy message.\n\nHe immediately telephoned Admiral Stark. They discussed the possible significance of the \"one o'clock\" note. \"What do you think about sending the information concerning the time of presentation to the Pacific commanders?\" suggested Marshall.\n\n\"We've sent them so much already, I hesitate to send any more. A new one will be merely confusing.\"\n\nMarshall hung up. In spite of Stark's opinion, he still thought a warning to the Pacific was necessary. He wrote out in longhand on a ruled sheet:\n\n> Japanese are presenting at 1 P.M. Eastern Standard Time today what amounts to an ultimatum. Also they are under orders to destroy their code machine immediately. Just what significance the hour set may have we do not know but be on alert accordingly.\n\nHis phone rang. It was Stark. His voice was concerned. \"George, there might be some peculiar significance in the Japanese ambassador calling on Hull at 1:00 P.M. I'll go along with you in sending that information to the Pacific.\" Stark then offered the Navy's transmission facilities.\n\n\"No thanks, Betty, I feel I can get it through quickly enough.\"\n\n\"George, will you include instructions to your people to inform their Naval opposites?\"\n\nMarshall agreed, adding a sentence to that effect on the message. He marked it \"First Priority Secret,\" and then ordered it rushed to the message center for transmission to San Francisco, the Panama Canal, the Philippines and Hawaii. Worried about the time factor, he sent an officer several times to find how long it would take to encipher and send the message by radio. The final answer was reassuring\u2014within thirty minutes. Marshall didn't even consider using the direct scrambler telephone. Since this was an ordinary commercial line, and easily tapped, the Japanese might possibly deduct that their Purple Code had been broken.\n\nThe handwritten warning was rushed to the message center. Soon it was encoded and by a few minutes after noon San Francisco, the Panama Canal and the Philippines were warned. But Hawaii could not be raised. There were still two other direct radio communications to Hawaii available; those of the Navy and FBI. The message center signal officer, failing to realize how precious every minute was, sent it by Western Union. It wasn't even marked urgent.\n\n**8** Five days previously Captain Eddie Layton, Kimmel's radio intelligence officer, had reported he had no information on Japanese Carrier Divisions 1 and 2. They had mysteriously disappeared from Hitokappu Bay a week before and hadn't been sighted since.\n\nKimmel's face was stern but there was a twinkle in his eye when he said, \"Do you mean to say that they could be rounding Diamond Head this minute and you wouldn't know?\"\n\n\"I hope they would be sighted by now, sir.\"\n\nBut even by 6:30 A.M. Sunday, December 7, Nagumo's Pearl Harbor Striking Force was still unsighted. And the first wave of attackers had been airborne half an hour on their approximately two-hour trip to Hawaii.\n\nOutside the mouth of the great naval base, Lieutenant William Outerbridge, the young skipper of the destroyer _Ward_ , had just been roused from his bunk by his gunnery officer, Lieutenant Oscar Goepner. Wearing glasses and a Japanese kimono, Outerbridge peered off the port bow in the murky light of dawn at the target ship _Antares_ which was towing a raft into Pearl Harbor. Behind the raft was a strange object.\n\n\"We've been watching it, sir,\" said Goepner excitedly, \"and we think it's moving.\"\n\nIt looked like a conning tower to Outerbridge, and it apparently was following the _Antares_ into the harbor. \"Go to general quarters,\" he shouted. At that moment the _Antares_ blinkered a confirming message. A small sub was 1500 yards off her starboard quarter!\n\nThe midget submarine was seen also by Ensign William Turner, pilot of a passing PBY Catalina flying boat, but in the dim light of dawn he thought she was an American vessel in distress. Seeing the _Ward_ speed toward the submarine he decided to help the \"rescuers\" and dropped two smoke bombs.\n\nThe _Ward_ , now only 100 yards from the submarine, began firing. At point-blank range, No. 1 gun fired and missed. A moment later No. 3 gun fired, hitting the conning tower. The midget began to sink. While the gunners were still cheering, Outerbridge shouted, \"Drop depth charges!\" The destroyer's whistle blasted four times and four depth charges rolled off the stern.\n\nWhen Ensign Turner in the PBY saw the _Ward_ attack, he decided he'd made a mistake and dropped several bombs. Then he wondered if he'd been right in the first place. What if he had sunk an American sub?\n\nSome on the _Ward_ also began to wonder if they had just blasted an American boat, but Outerbridge was sure he'd done the right thing. At 6:51 A.M. he radioed the Fourteenth Naval District:\n\n> WE HAVE DROPPED DEPTH CHARGES ON SUB OPERATING IN DEFENSIVE AREA.\n\nA moment later he decided this message wasn't strong enough. He was almost positive he'd sunk the midget. And so at 6:53 A.M. he sent a second message:\n\n> WE HAVE ATTACKED FIRED UPON AND DROPPED DEPTH CHARGES UPON SUBMARINE OPERATING IN DEFENSIVE SEA AREA.\n\nA few minutes later there was another alarm aboard the _Ward_. A white sampan was sighted in the forbidden waters. Outerbridge gave chase and as the _Ward_ approached, the skipper of the sampan, a Japanese fisherman, waved a white flag.\n\nOuterbridge immediately radioed:\n\n> WE HAVE INTERCEPTED A SAMPAN. WE ARE ESCORTING THIS SAMPAN INTO HONOLULU. PLEASE INFORM COAST GUARD TO SEND CUTTER TO RELIEVE US OF SAMPAN.\n\nBecause of delay in decoding by a yeoman, Outerbridge's second message about attacking the submarine didn't reach Captain John Earle, chief of staff of the Fourteenth Naval District, until 7:12 A.M.\n\nIt took Earle three minutes to reach the commandant, Admiral C. C. Bloch. After passing through so many hands the tone of Outerbridge's message did not appear so urgent.\n\n\"What do you know about it?\" asked Bloch.\n\nEarle was dubious. \"We get so many of these false sightings. We can't go off half-cocked.\"\n\n\"I agree,\" said Bloch. In the past few months there had already been nearly a dozen such sub warnings, all false. \"Ask this to be verified.\"\n\nA moment later any alarm at Fourteenth Naval District headquarters was allayed when Outerbridge's report of the sampan arrived. If the _Ward_ had actually attacked a sub, everyone agreed, she never would have left her post to proceed to Honolulu.\n\nAt this same moment another warning of a different nature was being reported by telephone to the Army Aircraft Warning Service. At the Opana outpost on the northernmost tip of Oahu, Private George E. Elliott, Jr. of the 515th Signal Aircraft Warning Service had just seen a large blip on his radar unit. He was not supposed to be watching the radar since it was customarily turned off at 7:00 A.M., but Elliott, a recent transfer from the Air Corps, wanted more experience and had been given permission by his platoon leader to operate beyond the scheduled period so that Private Joseph Lockard, his partner, could instruct him in the operation of the oscilloscope.\n\nWhen the blip had appeared at about 7:06 A.M. Lockard shoved the novice aside and began operating the radar. It was the largest group he had ever seen on the scope. It looked like two main pulses. He began to check the machine, certain there was something wrong with the equipment, but soon he too decided it was a large flight of planes.\n\nBy now Elliott had already located the blip on the plotting board: 137 miles to the north, 3 degrees east. He excitedly suggested they telephone the reading to the Information Center at Fort Shafter. Lockard disagreed. Their problem, he said, was over at 7:00 A.M.\n\nElliott persisted, arguing that the Information Center might be able to make some sense out of their reading. Finally Lockard relented and let his assistant make the call.\n\n\"There's a large number of planes coming in from the north, three degrees east!\" reported Elliott.\n\nThe switchboard operator at the Information Center, Private Joseph McDonald, dutifully wrote down the data. But he told Elliott no one was around and _he_ didn't know what to do.\n\n\"Well, get someone who does know and let him take care of it,\" persisted Elliott and hung up.\n\nMcDonald finally spotted an officer at the plotting table, Lieutenant Kermit Tyler. The operator handed the message to Tyler, then asked, \"Do you think we should do something about it, sir?\"\n\nTyler, a pilot on duty here only for the training, wasn't excited. He figured the blip was either the big flight of Flying Fortresses coming in from the mainland or planes from a Navy carrier.\n\nMcDonald returned to the phone and called Opana. Lockard answered. By now he was as excited as Elliott. The blips were bigger and their distance from Oahu was less than 90 miles. Lieutenant Tyler now came to the phone. He listened to Lockard but was still convinced the oncoming planes were friendly. \"Don't worry about it,\" he said and hung up.\n\nThis skepticism quieted Lockard's momentary alarm. When he started to shut down the unit and go off the air, Elliott protested. The novice wanted more practice. At 7:30 A.M. he and Lockard watched the flight approach to within 47 miles. Nine minutes later the distance had shrunk to 22 miles. Then it suddenly faded out, lost in the blacked-out area caused by nearby hills. The two men shut down the station and went to breakfast.\n\nBy now the 183 planes of the first wave of attackers from the Pearl Harbor Striking Force were already racing down the northwestern coast of Oahu. A minute later, at 7:40 A.M., Commander Mitsuo Fuchida, the flight leader, fired his signal pistol once. Dive bombers, horizontal bombers and torpedo planes quickly orbited into attack position.\n\nIronically, they were coming because an American naval officer, Commander Matthew Perry, had forced his way into Tokyo Bay with a squadron of warships in 1853. For 250 years previously no Japanese had been allowed to travel abroad, no foreigners were admitted to Japan. Perry's warships, loaded with commercial samples, forcibly opened the doors of feudalistic Japan, whose warriors still used bows and arrows and wore armor, to the wonders of civilization. Now eighty-eight years later, they were returning Perry's visit with vastly improved samples of the products of civilization.\n\n**9** In Tokyo, Foreign Minister Togo was talking to the dynamic premier, General Tojo, about the Roosevelt message which he was about to take to the Emperor. After discussing the reply that should be sent by Emperor Hirohito, Togo said jokingly, \"It's a pity to run around disturbing people in the middle of the night.\"\n\n\"It's a good thing the telegram arrived late,\" said General Tojo cryptically. \"If it had come a day or two earlier we would have had more of a to-do.\"\n\nTogo hurried to the Royal Palace. He read the contents of the telegram to Marquis Kido, the keeper of the privy seal.\n\n\"That's no use, is it?\" replied Kido. \"What's Tojo's opinion?\"\n\n\"The same as yours.\"\n\nTogo was then received by the Emperor. The foreign minister read the telegram, explaining it was quite similar to a former proposal, and presented the draft of the reply he and Tojo had written.\n\nThe Emperor approved the answer: a polite refusal. At exactly 3:15 A.M. Togo withdrew from the Emperor's presence, deeply moved by his countenance. On it he had read \"a noble feeling of brotherhood with all peoples.\"\n\nGuided by a court official, Togo solemnly passed down the long corridors of the Palace. Emerging at the Sakashita Gate he looked up at the bright stars. The Palace plaza was dead silent. As he drove away the only noise in sleeping Tokyo was the crunching of gravel under his car. In a few minutes, he thought, one of the most momentous days in the history of the world would begin. He was convinced that Japan was taking the only possible course.\n\n# 2\n\n# _\"Tora...Tora...Tora\"_\n\n**1** Banks of cumulus clouds collected around the peaks of the mountain ranges east and west of Pearl Harbor on Sunday morning. But over the great naval base, lying in the valley between, were only a few scattered clouds. Visibility was good and a wind of 10 knots blew in from the north.\n\nAt 7:45 A.M. several civilian pilots were lazily circling over the area. There wasn't a single military ship visible. Eighteen planes approaching from the carrier _Enterprise_ were scheduled to land at Ford Island within the hour.\n\nThe only Army Air Corps planes aloft in the vicinity were the 12 Flying Fortresses from California earmarked for MacArthur. They were due to land at Hickham Field, several miles south of Ford Island, in about an hour. But of the Oahu-based Army planes, not one was on patrol. Still on four-hour notice, they were all tightly bunched together wing tip to wing tip for security against saboteurs at Hickham, Bellows and Wheeler Fields. So were the Marine planes at Ewa.\n\nOf all the military planes in Hawaii, only 7 Navy PBY's were on patrol. And these were many miles to the southwest. Anti-aircraft defense was almost as lax. The Pacific Fleet in the harbor had about a quarter of its 780 anti-aircraft guns manned. Of the Army's 31 anti-aircraft batteries, only 4 were in position, and these had no ready ammunition. It had been returned to depots after practice since it was \"apt to disintegrate and get dusty.\"\n\nAbout 25 miles to the northwest Japanese pilots in the leading attack planes were marvelling at the peaceful green scene below them. The entire island seemed to be lazing luxuriantly in the early sun. Not even a trace of smoke was coming up from the motionless mass of ships in Pearl Harbor.\n\nAt 7:49 A.M. Commander Fuchida from his high-level bomber gave the attack signal in Morse code, \"TO...TO...TO.\" Four minutes later the great naval base was spread out below him like a huge relief map. It looked exactly as he had imagined. Still no fighters were climbing up to challenge; nor was there a single mushroom explosion of anti-aircraft fire. It was unbelievable. They had achieved complete surprise.\n\nEven before a single bomb dropped he now radioed: \"TORA...TORA...TORA\" (Tiger). The repeated word was heard by Admiral Nagumo. It was also heard directly on board the _Nagato_ , at Combined Fleet Headquarters in Japan. When the message was brought to Yamamoto he said nothing, his face betrayed no emotion. The other officers spontaneously cheered when the laconic message was read aloud. The _Nagato_ was engulfed in excitement. The message decoded meant: \"We have succeeded in surprise attack.\"\n\nStill no bomb had fallen. Except for the roar of approaching planes all was quiet in the Honolulu area. At the RCA office, Tadao Fuchikami, one of the messengers, was checking over a handful of cables he was to deliver. One was addressed simply to \"Commanding General\" and was obviously for General Short at Fort Shafter. This was the same message Marshall had written so excitedly in pencil almost two hours previously, but since there was nothing on the envelope such as _Secret_ or _First Priority_ or even _Rush_ , Fuchikami decided to make other calls first.\n\nAt that same moment, near the center of the island of Oahu, Japanese fighters and bombers began to dive on the Army's Wheeler Field, adjacent to Schofield Barracks.\n\nSecond Lieutenant Robert Overstreet, of the 696th Aviation Ordnance Company, was sleeping in the two-story wooden BOQ. He was awakened by a terrific noise. At first he thought it was an earthquake. \"Looks like Jap planes,\" he heard someone shout. \"Hell, no,\" said someone else. \"It's just a Navy maneuver.\"\n\nOverstreet's door opened and an old friend, Lieutenant Robert Skalwold, looked in. His face was white, his lips trembling. \"I think Japs are attacking.\"\n\nOverstreet looked out the window, saw planes circling overhead. They seemed to be olive drab. One dove on the barracks, coming so close he could see the pilot and a rear gunner. On the fuselage and wing tips were flaming red suns. He finished dressing as he ran out of the barracks and headed for his organization. Soon he came onto a group of fighter pilots.\n\n\"We've got to get down to the line and tag some of those bastards,\" shouted one, Lieutenant Harry Brown. Another pilot pointed to the burning hangars and the ramp. There the closely grouped planes were already ablaze.\n\n\"Let's go to Haliewa,\" said Brown. This was an auxiliary field on the north coast where a few P-40's and P-36's were kept. Brown and several other pilots piled into his new Ford convertible and left. Lieutenants George Welch and Kenneth Taylor followed in the latter's car.\n\nHundreds were milling around in shocked confusion as bombs fell and buildings erupted. Overstreet weaved his way through the mob toward the permanent quarters area. On the Circle he saw Brigadier General Howard Davidson, the fighter commandant, and Colonel William Flood, the base commander, standing by their front doors in pajamas, staring at the sky, their faces aghast.\n\n\"Where's our Navy?\" said Flood. \"Where're our fighters?\"\n\n\"General,\" shouted Overstreet, \"we'd better get out of here. Those planes have tail-gunners.\" He ran toward the ordnance hangar. To his horror it was in flames. Inside were a million rounds of machine-gun ammunition ticketed for Midway Island. Suddenly the hangar began to explode, like an endless row of huge firecrackers.\n\nFifteen miles to the south, just below Pearl Harbor, Jesse Gaines and Ted Conway, aircraft mechanics at Hickham Field, were walking toward the flight line. They'd gotten up early that morning because they knew B-17's were due from the States and they had never seen a Flying Fortress.\n\nAt 7:55 A.M. a V-formation of planes suddenly appeared from the west. As they began to peel off, Conway said, \"We're going to have an air show.\"\n\nGaines noticed something fall from the first plane. He guessed in alarm that it was a wheel.\n\n\"Wheel, hell, they're Japs!\" cried Conway.\n\nAs Gaines said, \"You're crazy,\" a bomb exploded among the neatly packed planes on the field. The two men began to run toward the big three-storied barracks, \"Hickham Hotel.\" Gaines saw some gas drums and dove behind them for protection. Fighters were now diving in a strafing attack, their machine guns spitting orange flames. Gaines felt himself being kicked in the rear.\n\n\"Don't you know better than that,\" shouted an old sergeant. \"Those damn drums are full!\" Gaines scrambled to his feet and headed away from the ramp. He looked up and saw bombs falling. All seemed to be heading for him. He ran in terror first one way, then another.\n\nTwo miles to the north, right in the middle of Pearl Harbor, the first bomb was falling on the naval air station at Ford Island. Ordnanceman Third Class Donald Briggs was sitting in a PBY. He thought a plane from the _Enterprise_ had spun in and crashed. Then the sky seemed to cave in as a dozen more explosions followed.\n\nThe Japanese plan was simple but efficient. First, to prevent an air counterattack, the airfields were being systematically wiped out. In the first few minutes the Navy bases, Kaneohe and Ford Island; the Army bases, Wheeler, Bellows and Hickham; and the lone Marine base, Ewa, were all but crippled.\n\nA moment after the first bomb fell, the Pearl Harbor signal tower alerted Kimmel's headquarters by phone. Three minutes later, at 7:58 A.M., the message heard around the world was broadcast by Rear Admiral Patrick Bellinger from Ford Island:\n\n> AIR RAID, PEARL HARBOR\u2014THIS IS NO DRILL.\n\nClosely on its heels, at 8:00 A.M. Kimmel's headquarters radioed Washington, Admiral Hart in the Philippines and all forces at sea: AIR RAID ON PEARL HARBOR. THIS IS NO DRILL. Even as the messages were going out, torpedo planes were diving on the main target, Battleship Row.\n\nAdmiral C. C. Bloch was shaving at his quarters in the Navy Yard. He thought workmen were blasting in the nearby stone quarry. When the explosions continued he told his wife, \"I'm going outside and see what that noise is.\" He ran out the front door. Overhead he saw a plane in flames. He went back into the house. \"The Japanese are bombing us. I've got to get to the office. Don't stay down here.\"\n\nAt the naval housing unit adjacent to Hickham Field, First Class Metalsmith Lawrence Chappell was in bed. A plane roared overhead.\n\n\"What are those planes?\" asked his wife, starting toward the window. \"It's too late for the Bomber Patrol.\"\n\n\"Probably stragglers.\"\n\n\"The Rising Sun! The Rising Sun! JAPANESE!\" cried Mrs. Chappell.\n\n\"You're foolish, go back to bed.\" Another plane roared over and Chappell went to the window. A torpedo plane swept by, so close he could see the pilot turning around, unconcerned. He hurriedly dressed and ran outside. Now he heard anti-aircraft fire and saw flames and billows of black smoke rising from Pearl Harbor.\n\nKimmel was watching the torpedo attack from the hill at Makalapa near his quarters. Short was standing on the _lanai_ of his home near Fort Shafter watching the billows of smoke in the west and wondering what was going on at Pearl Harbor.\n\nThe smoke was rising from Battleship Row, on the east side of Ford Island where seven battleships, the heart of the Pacific Fleet, were moored. They were not protected from aerial torpedoes by nets because of Pearl Harbor's 40-foot depth. This matter had been discussed many times by Kimmel and Stark. Even the British had been consulted. Everyone agreed a minimum depth of 75 feet was necessary for torpedoes.\n\nThis unanimous conclusion was surprising since the British themselves had made a successful plane attack on the Italian fleet at Taranto the previous year with specially rigged torpedoes. The Japanese bombers diving on Battleship Row were proving clever as the British. They were dropping torpedoes with ingeniously constructed wooden fins, specially designed for shallow water.\n\nNot far from Battleship Row, Yeoman C. O. Lines of the oil tanker _Ramapo_ was in the crew's quarters. Boatswain's Mate Graff rushed down the ladder. \"The Japs are bombing Pearl Harbor!\" he yelled.\n\nThe men in the room looked at him as if he were crazy.\n\n\"No fooling,\" he said.\n\nSomeone gave a Bronx cheer.\n\n\"No crap. Get your asses up on deck!\"\n\nLines hurried topside to the fantail. He thought Graff was ribbing as usual. Then he heard a dull explosion and saw a plane dive toward the battleship _California_.\n\nShe was the last of the seven big vessels in Battleship Row. Two torpedoes hit her almost simultaneously. The ship took an 8-degree list and began to settle. Her fractured fuel tanks began to flood an entire lower deck. Bombs now fell and half a dozen fires flared. In minutes oil gushing from the ruptured ship burst into flame. She was surrounded by a wall of fire. The word was passed: Abandon ship.\n\nAhead, in tandem formation, were the _Maryland_ and _Oklahoma_. A torpedo couldn't hit the _Maryland_ because she was berthed inboard, next to Ford Island, and was protected by her mate. But the outboard ship, the _Oklahoma_ , was hit by four torpedoes within a minute. As she listed to port, Commander Jesse Kenworthy, senior officer aboard, ordered the ship abandoned over the starboard side. He calmly walked up the ship's side over the blistered ledge and then over the bottom. Soon the ship settled, its starboard propeller out of the water. Below more than 400 men were trapped in the rapidly filling compartments.\n\nNext in Battleship Row came another pair, the _Tennessee_ and _West Virginia_. Like the _Maryland_ , the _Tennessee_ was inboard and safe from torpedo attack. On the _West Virginia_ 's battle conning tower, Lieutenant Commander T. T. Beattie, the ship's navigator, heard Captain Mervyn Bennion groan.\n\n\"I've been wounded,\" said the skipper, doubling up. A fragment, probably from an armor-piercing bomb that had just hit the nearby _Tennessee_ , had torn into his stomach and part of his intestines were protruding. The captain sank to the deck. Beattie loosened his collar and sent a messenger for a pharmacist's mate. But the captain knew he was dying and only wanted to know how his ship was being fought. Some fires swept toward the bridge. There was no escape except by swinging hand over hand along a fire hose.\n\nNext in line came the _Arizona_. The first torpedo planes struck at the _Arizona_ and missed. A moment later high-level bombers attacked. Five bombs hit. One of these dropped through the forecastle into the second deck, starting a fire. About 1600 pounds of blackpowder, the most dangerous of all explosives, were stored here against regulations. Suddenly the blackpowder exploded, igniting hundreds of tons of smokeless powder in the forward magazines.\n\nTo those on nearby ships, the _Arizona_ seemed to leap out of the water amidst a tremendous blast of fire and debris. The great 32,600-ton ship apparently broke in two and then quickly settled. Sheets of flame and clouds of black smoke swirled around the _Arizona_. It didn't seem possible that a soul could have survived.\n\nAhead of the _Arizona_ was the leading ship in Battleship Row, the _Nevada_. She was down several feet by the head from a torpedo in her port bow and a bomb in the quarterdeck.\n\nOn the other side of Ford Island, torpedo bombers were diving on the thirty-three-year-old battleship, _Utah_. Now a target ship, her decks had been stripped and covered with timber and from above she looked like a carrier. Dive bombers were swarming all over her. The first torpedo hit at 8:01. Four minutes later the _Utah_ was listing to port at 40 degrees. As fighter planes swept down to strafe the tilting deck, the word was given to abandon ship. At 8:12 the tired old ship flopped over, keel sticking out of the water. There was a momentary quiet and men in a recently dug trench on Ford Island heard a faint knocking inside the ship. People in the ship's bottom were still alive.\n\nEven by 8:10 A.M. only one ship in the entire harbor was under way. This was the destroyer _Helm_ and she was trying to escape. At 27 knots she raced through the channel toward the mouth of the harbor.\n\nHere, the anti-torpedo net, opened hours earlier for the _Condor_ , was still unclosed. A Japanese midget submarine, its gyro-compass out of order, was trying to stab its way blindly into this opening and raid Pearl Harbor. The commander, Ensign Kazuo Sakamaki, surfaced to get his bearings. Through the periscope he saw columns of black smoke. He excitedly called to his aide, Second Class Petty Officer Kiyoshi Inagaki, \"The air raid! Wonderful! Look at that smoke. Enemy ships burning. We must do our best too, and we will.\"\n\nAt 8:15 A.M. he sighted the _Helm_ racing out of the harbor. It was so close he could see the white uniforms of the sailors aboard. Guessing correctly it was only a destroyer, he didn't fire. His two torpedoes were marked for bigger game, battleships. He quickly submerged and again blindly aimed at the mouth of the harbor. He hit a reef, backed away, tried again. This time he ran up on the reef and his conning tower stuck out of the water.\n\nAt 8:18 A.M. he heard a terrific explosion and figured the destroyer had seen him. The little boat shook violently. Something hit Sakamaki's head and he briefly lost consciousness. When he came to, the tiny inner chamber was filled with white smoke. He felt dizzy, sick. He reversed the motor to back off from the reef but the boat didn't budge. There was only one solution. He wormed his way on his stomach up the narrow passage to the bow and began the painful job of transferring 11-pound ballast weights to the stern. Finally to his relief he felt the submarine move.\n\nOn the surface, the _Helm_ was still firing at the bobbing midget with no success. Suddenly its target slid off the coral, vanished, SMALL JAP SUB TRYING TO PENETRATE CHANNEL, radioed the destroyer.\n\nWhile this new alarm was being signaled from ship to ship, another midget was slowly rising to the surface just west of Ford Island and preparing for attack. She had slipped through the open mouth of Pearl Harbor earlier that morning and hidden on the bottom.\n\nA moment later, at 8:30 A.M. the _Breese_ spotted its conning tower. Several ships quickly opened fire. The midget launched her two torpedoes. One detonated against a dock, the other against the shore. Then the destroyer _Monaghan_ slammed into the submarine. As the midget sank, depth charges from the destroyer finished the job.\n\nBy now action from the sky had slacked off momentarily. Admiral \"Poco\" Smith, Kimmel's chief of staff, who had been shaving in his apartment near the Royal Hawaiian Hotel when he learned about the raid, now drove up to the submarine base. Outside Kimmel's office he saw a group of Marines shooting rifles into the empty sky.\n\nInside Fleet headquarters, Kimmel was calm. He was talking to Vice Admiral William Pye, commander of Battle Force. Pye, too, was calm, though covered with oil. He had just escaped from his flagship, the _California_. Smith, after some persuasion, convinced Pye and Kimmel to get in different rooms so one bomb wouldn't kill both of them.\n\nA few minutes later, a little after 8:40 A.M. the sky above Pearl Harbor was again dark with raiders. This was Nagumo's second wave; 80 dive bombers, 54 high-level bombers and 36 fighters. This time they came from the south and east hitting Battleship Row and Drydock No. 1, where the _Pennsylvania_ , the eighth battleship, was berthed. Soon 18 dive bombers from the southeast joined the attack on Battleship Row.\n\nA principal target was the _Nevada_ , moving slowly past the blazing _Arizona_. Her gun crews were shielding their ammunition from the intense heat with their own bodies. In a few minutes the _Nevada_ , already suffering from one torpedo hit, drew up to the toppled _Oklahoma_. Several men stood up on the sides of that ship and cheered as the _Nevada_ headed for open water.\n\nBut the attackers were finding the range. In minutes six bombs hit. The bridge and forestructure of the battleship burst into flames. The ship turned to port and, with the help of two tugs, was beached not far from the _Pennsylvania_ 's drydock.\n\nThe 12 Flying Fortresses from California began to arrive during this attack. The first squadron of six, under Captain Ted Landon, bewildered by the heavy traffic over Oahu, headed uncertainly for Hickham Field. Four landed safely and one was shot in half by ground troops as it touched down. The sixth plane turned north and crash-landed at Bellows.\n\nAs the second squadron of six approached Waikiki Beach, Captain Richard Carmichael, the squadron commander, was pointing out the sights to Captain Jim Twaddell. When Carmichael saw planes ahead, he figured it was some Navy maneuver. Then he saw flames leaping from Hickham Field and knew it was an attack. He called the tower, asking permission to land.\n\n\"Land from west to east,\" said Major Gordon Blake. \"Use caution. The field is under attack.\"\n\nAs Carmichael lowered his wheels violent anti-aircraft fire broke out from ships in Pearl Harbor. He quickly turned north toward Wheeler. This field, too, was under heavy attack. Hangars and barracks were burning as dive bombers strafed the line.\n\nCarmichael flew around the Koola Mountain Range. Only half an hour's fuel was left and he had to find a home quickly. He started for Bellows Field, keeping near the coast line. But Kaneohe Naval Air Station was also under attack. Suddenly he remembered Haliewa, a sod field for P-40's. He went north of the mountains and in a few minutes touched down on the short grassy fighter field. It was only 1200 feet long and by the time the big Flying Fortress skidded to a stop he had used every foot of it. All 6 of his planes landed safely: 2 at Haliewa, 1 at Kanuka Golf Course and 3 at Hickham.\n\nOn the Hickham flight line, Second Lieutenant William Welch, and other eager young pilots of the 11th Bombardment Squadron, were squabbling to see who would fly the one bomber still intact. A bomb landed 50 yards away, knocking them all flat.\n\n\"Let's go across the field,\" suggested an old master sergeant. Welch, eager to do anything, followed him. As they reached the parking area, three Zeros skimmed by at tree level, their machine guns spitting. Bullets sprayed over their heads into a group of pilots just standing and looking up curiously. Welch turned and fired his .45 pistol at the last plane. It was his only shot of the day.\n\nJust then several Flying Fortresses from the States began to land. Welch ran out to disperse them. Two sprucely dressed captains stepped out of the first plane, looking shocked. \"Get your ammo, load up and get ready to go,\" said Welch.\n\nThe captains, too surprised to move, told Welch they were in no shape for battle. All their guns were packed in cosmolene and would take hours to clean.\n\nAt Wheeler there was the same feeling of shock. The men were still groggy from the first attack when the second hit. Lieutenant Overstreet was arguing with a sergeant from the Base Ordnance Office. Overstreet wanted rifles and pistols.\n\n\"I doubt if I'm authorized to give you any without a hand receipt,\" said the reluctant sergeant as bombs exploded nearby.\n\n\"Hell, man, this is war,\" said Overstreet angrily. He got the guns.\n\nAt Ford Island Field there was also frustration. Not a single Navy fighter had gotten into the air. By now all the planes were destroyed or inoperable. There was nothing to do. Six pilots, including Ensign Elbert Cain, were hiding behind palm trees and shooting at the invaders with their pistols.\n\nOnly about 30 Army Air Force fighters managed to get into the air that morning and these could do little to hinder the rain of bombs and torpedoes. Two pilots, Lieutenants Kenneth Taylor and George Welch, were credited with seven of the 11 Japanese planes shot down.\n\nBy 10:00 A.M. the skies above the smoke-bound harbor were suddenly empty. The second attack was over. The stink of burning oil was overwhelming. Battleship Row was a shambles. The _Arizona_ , the _Oklahoma_ and the _California_ were sunk at their berths. The _West Virginia_ , hopelessly ablaze, was sinking. The _Nevada_ was aground. The other three battleships\u2014the _Maryland, Tennessee_ , and drydocked _Pennsylvania_ \u2014were all damaged.\n\nIn nearby Honolulu, Takeo Yoshikawa, the lone Japanese Navy spy, was eating breakfast when windows rattled and several pictures dropped to the floor. Then he heard great explosions coming from Pearl Harbor. He ran outside in his slippers and saw Consul General Nagao Kita standing in front of his official residence. The two looked at each other but couldn't say a word. Other consular officials poured out into the courtyard, dazed and unaware of what was actually going on.\n\nKita and the spy went inside and tuned in Radio Tokyo. They heard \"east wind, rain,\" repeated twice. War with America! They hurried to the consul general's office and feverishly burned codebooks and secret instructions. By the time the police arrived at 8:30 A.M. everything except a half-finished sketch of Pearl Harbor had been destroyed.\n\nThen the FBI arrived; Yoshikawa knew it was the end of the war for him. He whispered to himself, \"Good-bye to the days of my youth\u2014forever.\" But he had accomplished his mission. The smoke boiling up from Pearl Harbor was in great part due to the information he had gathered for $600.\n\nThere was no panic in Honolulu even though explosions had hit many parts of the city. None of these were Japanese bombs, but strays from American guns. Even so, many people had no idea what was going on a few miles away. The first radio report came just as the second attack started, but when it announced that, \"A sporadic air attack has been made on Oahu,\" many listeners thought \"sporadic\" meant a mock attack.\n\nOthers in the city lived out the day as any ordinary Sunday. At the height of the attack Hawaiian girls in costume appeared as usual at the Pan American dock, arms loaded with leis to bid Aloha to departing Clipper passengers. They had to be told it was the end of traditional ceremony for a long, long time. They turned away bewildered and sad.\n\n**2** Kimmel's \"This Is No Drill\" broadcast was picked up by Mare Island Navy Yard in San Francisco and instantly relayed to Washington.\n\nSecretary of the Navy Frank Knox was in his office in the Navy Department on Constitution Avenue. It was long past noon and he was getting hungry. He was about to order lunch when Admiral Stark burst in with the message from Kimmel.\n\n\"My God,\" said Knox, \"this can't be true! This must mean the Philippines.\"\n\nStark assured him grimly it did mean Pearl Harbor. Knox picked up the phone connected directly with the White House. It was 1:47 P.M. He asked for the President. In a moment Roosevelt, lunching in the Oval Room with Harry Hopkins, answered. Knox read the dispatch.\n\nRoosevelt couldn't believe it. Nor could Hopkins. Surely Japan would never attack Honolulu. Roosevelt talked at some length of his efforts to keep America out of war, then said soberly, \"If this report is true, it takes the matter entirely out of my hands.\"\n\nAt 2:05 P.M. Roosevelt called Hull, advising the secretary of state to receive Nomura and Kurusu but not to mention that he already had the news of Pearl Harbor. He should be formal, cool, and bow them out.\n\nThen Roosevelt called Stimson. \"Have you heard the news?\" he excitedly asked the secretary of war.\n\n\"Well, I've heard the telegram which came in about the Japanese advances in the Gulf of Siam.\"\n\n\"Oh, no, I don't mean that. They have attacked Hawaii! They are now bombing Pearl Harbor!\"\n\nA little later Roosevelt saw Dr. Hu Shih, the Chinese ambassador, and suggested a telegram be sent to Chungking requesting that there be no outward signs of joy because Pearl Harbor had suddenly made America a potent ally.\n\nBack at the Navy Department, Stark, after getting the President's permission, was already sending a message to all commanders in the Pacific area and Panama:\n\n> EXECUTE UNRESTRICTED AIR AND SUBMARINE WARFARE AGAINST JAPAN.\n\nA few doors away, Knox was talking on the telephone to Admiral Bloch at Pearl Harbor. The navy secretary was amazed at Bloch's coolness as he described the damage which he could see through the window. It was a great disaster, Bloch said, but it could have been greater. \"The _Oklahoma_ 's badly hit. Also the _Arizona_. But the _Pennsylvania_ and _Tennessee_ are only superficially damaged and we can raise the _California_ without too much trouble. Fortunately there's no damage to the Navy Yard and oil reserves.\"\n\nKnox hung up, hurried to the White House to pass on the information. There the President had just called a special meeting of the Cabinet to work out the war address Roosevelt would deliver the next day to Congress.\n\n\u2014\n\nJust an hour after the first bomb fell on Hawaii, Ambassador Nomura and Special Envoy Saburo Kurusu were waiting in the outer office of Cordell Hull at the old-fashioned Department of State Building. Twenty minutes earlier, at 2:00 P.M. the two men had arrived with the fourteen-point message which was supposed to have been delivered at 1:00 P.M. Maddening delays in decoding and typing had caused the hour's tardiness.\n\nNomura, who had lost his right eye in the bombing of a parade at Shanghai in 1932, was embarrassed. Not only was he late, but the hastily typed message contained several minor typographical errors. First Secretary Okumura had wanted to do a fresh draft but the impatient Nomura had snatched it away from him. In fact the admiral still hadn't had time to read the note carefully. But it was apparent from a cursory glance that the talks with Hull had reached a dead end.\n\nAt 2:21 P.M. Nomura and Kurusu were finally ushered into Hull's office. \"I was instructed to hand this reply to you at 1:00 P.M.,\" said the admiral apologetically, holding out the note.\n\nHull's face was set, stern. \"Why should it be handed to me at 1:00 P.M.?\"\n\n\"I do _not_ know why,\" confessed Nomura, puzzled that his friend should be so upset just because he and Kurusu had been an hour late.\n\nThe secretary of state, without explaining the reason for his open anger, seized the note abruptly, scarcely glanced at it and said bitterly, \"I have never uttered one word of untruth during the last nine months. I have never seen a document so crowded with falsehoods and distortions!\"\n\nThe two Japanese statesmen, bewildered by Hull's antagonism, left the building. When Nomura reached the Embassy, Okumura said, \"Our planes have bombed Pearl Harbor!\" Now the admiral understood.\n\nIt was only then that the American public learned of the attack. At 2:26 P.M. WOR interrupted its broadcast of the Giant-Dodger football game with the first news flash. No war announcement was made at the Polo Grounds itself where Brooklyn was just kicking off after scoring the game's first touchdown in the second quarter against the already crowned champions of the Eastern Division. But a little later, when Colonel William J. Donovan was paged by Washington over the P.A. system, there was an ominous buzz.\n\nMuch of America heard the news a moment before the 3:00 P.M. CBS broadcast of the New York Philharmonic concert. In Washington Admiral Chester Nimitz, chief of the Bureau of Navigation, was just sitting down to enjoy the Artur Rodzinski concert when the flash came. He jumped out of his chair and soon was on his way to the Navy Building with his assistant, Captain Shafroth.\n\nA few blocks away Masuo Kato, correspondent for Domei, was in a taxi heading for a funeral chapel to attend services for Colonel Kenkichi Shijo, the assistant military attach\u00e9 who had died of pneumonia.\n\n\"God damn Japan,\" said the driver when the flash came over the cab radio. \"We'll lick the hell out of those bastards now.\"\n\nMost Americans were caught completely unprepared. There were no scenes of excitement or panic, but strangers on the streets began to look at each other with a new awareness. Personal problems were overshadowed by national catastrophe. The bitter wrangles between the interventionists and the America Firsters suddenly had no meaning. With almost no exceptions 130,000,000 Americans instantly accepted total war.\n\n**3** In England it was already Sunday evening. At his country residence, Chequers, Prime Minister Winston Churchill was tuning in the nine o'clock news on the portable radio given him by Harry Hopkins. He still knew nothing about Pearl Harbor. In fact, he was ignorant of the much earlier attack on Malaya. A BBC announcer calmly told of fighting on the Russian front, in Libya and then mentioned an attack on Hawaii by the Japanese.\n\nAverell Harriman and United States Ambassador John Winant, Churchill's house guests, sat up straight, shocked.\n\n\"It's quite true,\" said the butler, Sawyer. \"We heard it ourselves outside. The Japanese have attacked the Americans.\"\n\nThere was silence. Then Churchill strode to his office. He picked up the phone and asked to speak to President Roosevelt. Winant, assuming the prime minister was going to declare war on Japan, as he had recently promised \"within the moment,\" said, \"Good God, you can't declare war on a radio announcement!\"\n\n\"What shall I do?\"\n\n\"I will call up the President and ask him what the facts are.\"\n\n\"And I shall talk to him too.\"\n\nA moment later Winant was conversing with Roosevelt. \"I have a friend who wants to talk to you,\" said Winant cryptically. \"You will know who it is as soon as you hear his voice.\"\n\nChurchill picked up the phone. \"Mr. President, what's this about Japan?\"\n\n\"It's quite true,\" said Roosevelt. \"They have attacked us at Pearl Harbor. We are all in the same boat.\"\n\n\"This actually simplifies things. God be with you.\" Churchill couldn't help feeling the greatest joy, with the United States now at his side. Britain had won after all. The Commonwealth of Nations and the Empire would live! He recalled Edward Grey telling him thirty years previously that the United States was like a gigantic boiler. \"Once the fire is lighted under it, there is no limit to the power it can generate.\"\n\nSaturated with emotion, he went to bed and slept soundly.\n\n**4** The rising sun was just lighting up Tokyo. At Imperial Navy Headquarters there were waves of excitement as details of Pearl Harbor came in directly from planes returning to their carriers. Even to the most conservative it was obvious the Pacific Fleet had been crushed. It was also obvious that Nagumo's force, already on the way back to Japan, would escape scot free. The Americans were searching to the south, in the wrong direction. Yamamoto's brilliant, controversial scheme had succeeded beyond all expectations. With the main instrument of Allied power in the Pacific paralyzed, the conquest of Southeast Asia could now continue.\n\nThe next step was to crush enemy air power in the Philippines.\n\n**5** Almost four hours after the first bomb fell on Pearl Harbor, Tadao Fuchikami, the RCA messenger, passed through the gates of Fort Shafter on his Indian Scout motorcycle. He had been delayed, quite understandably, by the attacks. The message from Marshall warning Short to be on the alert was finally delivered. Since no one at Shafter was particularly interested in advice from Washington at the moment, it wasn't decoded and delivered to General Short until a moment before 3:00 P.M.\n\nWhen the general read it, he threw it disgustedly in the wastebasket. It was seven hours late.\n\nBecause of an almost incredible series of errors, mischances and coincidences the Pacific Fleet\u2014the main deterrent to Japan's conquest of the Southwest Pacific\u2014was crushed. Eighteen ships had been sunk or badly damaged; 188 planes destroyed and 159 damaged; 2403 Americans killed.\n\nThe cost had been minuscule. The Japanese lost 29 planes and 5 midget submarines. Fifty-five airmen had died and 9 submariners. One, Ensign Kazuo Sakamaki, was captured.\n\nThe reasons for the disaster would be debated bitterly for years. Stripped of politics, they were simple. The American high command had never dreamed the Japanese could even mount an independent carrier striking force. They also never imagined the Japanese would be \"stupid enough\" to attack Pearl Harbor.\n\nIn a deeper sense, every American would have to accept a share of the blame. Like many other great military disasters, it was caused by a national weakness of character. Americans had refused in the large to face the facts of twentieth-century world politics and economy until the festering war exploded.\n\nNow the American people were presented with an even more important test. How would they react, after this sudden catastrophe, to the prospect of a long, expensive war?\n\n# 3\n\n# _\"Tallyho! Bandits over Clark!\"_\n\n**1** As the last raider wheeled and left the wreckage of Pearl Harbor, other Japanese planes were hitting Singapore. An hour and a half later the Philippines and Guam were lightly bombed. Then Hong Kong's airfield was attacked, its tiny air force wiped out.\n\nTwo hours after this air raid, the strategic atoll of Wake, surprisingly, was still untouched. At a few minutes before noon, in the radio trailer of the small Army communications detail, Sergeant James Rex was busy relaying messages from the Philippines to the States, since direct contact had been lost. All morning, he had been trying to pass on news of Pearl Harbor to operators in Australia and New Guinea, but they kept telling him to stop kidding and he'd finally given up.\n\nAt the Pan-American base on Peale, the tiny island just off the northwest tip of the main island, the Clipper had just been emptied of passengers and mail. At dawn it had taken off for Guam but returned when the skipper, John H. Hamilton, learned of war. Hamilton was almost ready to take his ship on a voluntary patrol. He was talking by phone to Major Paul Putnam, the commander of Marine Fighting Squadron 211, who was five miles away in a tent at the edge of the airstrip on the main island.\n\nPutnam hung up the phone, stepped out of the tent and looked across the bald, narrow 3000-foot-long strip. Eight planes were scattered at 50-yard intervals. Six other planes were being warmed up, two to escort the Clipper and four to relieve the patrol then in the air. Even though there had been a constant air watch since dawn Putnam was worried. Other equipment had arrived recently, such as a garbage truck, but they still had no listening device, no radar.\n\nAt 11:58 A.M. Putnam happened to look to the south where a low-lying rain squall was sweeping in. He saw planes only 1500 feet up, gliding noiselessly, eerily out of the squall.\n\n\"Take cover\u2014bombers,\" he shouted. Most of the men were paralyzed a few seconds as they stared at the 34 medium bombers. Others reacted instantly and ran toward their foxholes. There was no foxhole near Putnam, but 100 yards away was the head. Before the first fragmentation bomb fell he was running. In high school he had once raced 100 yards in 10.2 seconds. Today he knew he would break 10 seconds. Out of the corner of his eye he saw bombs falling, about 12 from each plane. Still short of the head he dove when he heard machine guns sputter. He slid on his belly into the open latrine, head first, his legs still dangling in the air. His hands and nose were covered with filth but he was glad to be there.\n\nIn five minutes it was all over. Most of the raiders disappeared but ten turned and raked the Pan-American station. The hotel was blasted, several buildings completely destroyed, fuel tanks set afire. Ten employees, all Guamanians, were killed.\n\nThe airfield on Wake was an inferno. Seven of the parked planes and the two 12,500-gallon aviation gasoline tanks were blazing. Parts of bodies were scattered; the wounded moaned. Putnam, himself grazed by a bullet in the back and dazed by concussion, refused first-aid treatment as he tried to bring organization from chaos. His air force was crippled. Half of his men were killed or wounded and there were only four undamaged planes left to protect the island.\n\nAt Peale, workers feverishly patched the many bullet holes in the Clipper. The passengers, all unwounded, were loaded. Then, after throwing off two Guamanian stowaways, as many white employees as possible boarded. At 1:00 P.M. the big plane took off, heading for Midway.\n\nOn the island of Kwajalein, about 650 miles south, the main body of the Wake Invasion Force was almost ready to leave. Tokyo hadn't given Rear Admiral Sadamichi Kajioka, the over-all commander, a definite deadline for conquest. It was too simple a job. As the last of the Japanese invaders were climbing aboard, word came from returning bombers that all but four planes on Wake had been destroyed. Kajioka's mission now seemed ridiculously easy. Wake should be taken in two days.\n\n**2** Earlier that morning, Major General Lewis Brereton, commander of MacArthur's Far East Air Force, reported to No. 1 Victoria Street in the Walled City section of Manila. It was almost 5:00 A.M., December 8, two and one-half hours after the first bomb had fallen on Hawaii. Unable to see MacArthur, then in conference with Admiral Hart, he was talking with General Sutherland, so often the buffer between the commanding general and his subordinates.\n\n\"I'd like to mount all the B-17's at Clark Field for missions to Formosa,\" said Brereton. He wanted his Flying Fortresses to leave immediately after daylight and bomb Takao Harbor in southern Formosa, about 600 miles to the north.\n\nSutherland told him to go ahead and make preparations, but warned him to make no final commitment until approval by MacArthur was granted. Brereton left. There was nothing now for him to do but wait, since he had already alerted his headquarters at Nielson Field to prepare the Flying Fortresses for the mission to Formosa. For Brereton, eager to hit the Japanese before they hit him, the next hour crawled with painful slowness.\n\nA few minutes later, dawn and the Japanese reached the Philippines simultaneously. On Mindanao, the southernmost of the major islands, Japanese Navy dive bombers swept in from an island to the east, Palau, and attacked the seaplane tender, _William Preston_ , and two PBY's in Davao Harbor. The ship evaded the bombs but the two planes were sunk and the first American in the Philippines, Ensign Robert Tills, was killed. At about the same moment Army Zeros from Formosa hit the radio station in Aparri, a good-sized town on the north coast of Brereton's own island, Luzon, the northernmost of the major islands.\n\nBut the Far East Air Force chief knew nothing about these attacks so close to his own two main bomber bases\u2014Clark Field in central Luzon and Del Monte Field in northern Mindanao\u2014as he impatiently returned to No. 1 Victoria Street at 7:15 A.M. Brereton hurried into Sutherland's office. Across the hall, Colonel William Morse, one of MacArthur's staff officers, could hear the two men talking through open doors.\n\n\"I want to go up and bomb Takao,\" Brereton said insistently.\n\n\"I'll ask the general,\" said Sutherland. Morse saw the chief of staff walk into MacArthur's office and return a moment later. \"The general says no. Don't make the first overt act.\"\n\nMorse then heard Brereton protest that bombing Pearl Harbor was certainly an overt act. Sutherland was unmoved. Besides he felt that Brereton knew very little about the targets on Formosa since there had been almost no reconnaissance. Such a raid on Formosa would just be a wild stab. Brereton's role for the present, he said, was defense.\n\nHalf an hour later Brereton reached his headquarters, Nielson Field. On the southern outskirts of Manila, it was on a barren plain, protected by no natural cover. The hangar roof, painted in bold black and yellow squares, was an inviting target.\n\nHis staff was waiting eagerly. Captain Allison Ind, the intelligence officer, had assembled target folders on Formosa and although he had no maps indicating the best approaches and bomb release lines, the information was as good as it would be for a long time to come.\n\nBrereton, face pale, entered his office. \"What's your decision about bombing Formosa?\" he asked the staff.\n\nThey had been arguing about the bombing ever since dawn but had now agreed the B-17 Flying Fortress at Clark Field should set out immediately for Formosa.\n\n\"No,\" said Brereton, his jaw hardening. \"We can't attack till we're fired on.\" He explained that Sutherland had ordered him to prepare the B-17's but to take no offensive action. The staff officers were incredulous. If they didn't strike now, they probably wouldn't get a second chance.\n\nThe problem was much more complex than the airmen realized. Although the Philippines were an American possession, they were a Commonwealth and, in preparation for the full freedom which had been planned for three years hence, had their own elected government.\n\nMacArthur, also unaware of the two minor air raids on the islands, was positive the Philippines would soon be attacked but many Filipino authorities had high hopes the Japanese would consider them an independent nation and spare them. If he made a wrong move now, MacArthur knew his whole delicately constructed plan of local self-defense might collapse. He also remembered Marshall's recent orders: \"If hostilities cannot be avoided, the United States desires that Japan commit the first overt act.\"\n\nAs Brereton was ordering several planes to Formosa on photo-reconnaissance in case approval from MacArthur came through, he was called to the phone. It was Major General Henry \"Hap\" Arnold, commanding general of the Army Air Forces in Washington. Arnold warned him how everyone had been \"caught napping\" at Pearl Harbor. Every precaution should be taken in the Philippines. He didn't want Brereton's air force destroyed the same way.\n\n\u2014\n\nIn western Formosa, Japanese Navy officers of the 11th Air Fleet were just as disgusted and disappointed as the American fliers. Heavy fog surrounded their fields. They were to have taken off before dawn for strikes at the American fighter bases\u2014Nielson, Nichols and Iba\u2014and the main target, Clark Field. As the hours passed, worry grew that the Clark-based Flying Fortresses would suddenly appear and wipe out their own planes, gassed and lined up neatly on the runways.\n\nEast of the Navy fields the fog over the Japanese Army bases had lifted at dawn. One flight had already bombed Aparri. Other flights were nearing their targets: Baguio, the former summer capital of the Philippines about 85 air miles north of Clark, and Tuguegarao, 50 air miles south of Aparri. But the 11th Air Fleet men were not at all cheered by the better weather of their Army brothers. They knew the Army was bombing merely tactical targets. The fate of the entire Philippines campaign depended on the success of their own mission: quick destruction of the main body of MacArthur's fighters and bombers. If the weather didn't clear soon, all might be lost.\n\nEven at 9:00 that morning most Filipino villages were still completely unaware that war had come to their country. Hundreds of barrio fiestas were starting all over the islands, for today was the Feast of the Immaculate Conception and\u2014except for the Moslem-worshipping Moros of Mindanao, the Igorots and Negritos of Luzon and other tribes untouched by civilization\u2014most Filipinos had adopted the religion of their first conquerors, the Spaniards.\n\nBut in Manila, almost every one of its 600,000 people had heard the news by breakfast because of the newspaper extras and Don Bell's frequent news broadcasts over KMZH. Even these reports didn't completely convince many Americans. Although everyone had been sure war was coming, very few were mentally prepared to accept it. Many remembered the panic caused by Orson Welles' broadcast on the invasion from Mars.\n\nIn the city's suburbs at Nielson Field, the Air Warning Room was receiving reports of bombers heading for Lingayen Gulf. These were the Japanese Army bombers bound for Baguio but everyone at Nielson thought their target was Clark Field. The B-17 base was warned and soon all but one of the Flying Fortresses took to the air bombless, in fear of being caught on the ground. At the same time 18 P-40's from Clark and 18 from Nichols Field, on the outskirts of Manila, headed for Lingayen Gulf to intercept the invaders.\n\nBack at Nielson scores of reports were pouring in: enemy battleships sighted off the northern coast of Luzon; three Japanese flying boats seen over the tiny islands north of Luzon. As messages piled up, all civilian secretaries were abruptly ordered evacuated in case of a raid.\n\nAt the height of the ensuing confusion, at 9:25 A.M., two even more startling reports came in. Luzon itself had been bombed at Baguio and Tuguegarao. Now there could be no argument about an \"overt act.\" Brereton picked up the phone and told Sutherland about the attacks. \"If Clark Field is attacked, we won't be able to operate on it.\" He again begged for permission to bomb Formosa. His face fell and he hung up. Turning to Lieutenant Colonel Eugene Eubank, commander of the B-17's, he shook his head. Permission not granted. Eubank started back for Clark Field at 10:10 A.M.\n\nFour minutes later a call came from No. 1 Victoria Street. Offensive air action against Formosa was authorized. But it seemed too late to hit Takao Harbor as planned. Brereton's staff hastily plotted new missions.\n\nIn Formosa activity was just as frenzied at 10:10 A.M. The fog had cleared. On the Japanese Navy fields 196 planes were warming up for take-off. Suddenly sirens all over the island screamed. American planes were reported approaching Formosa. Gas masks were hurriedly passed out. The horde of medium bombers and Zero fighters hastily took to the air on two separate missions. Fifty-three bombers and 53 fighters were to wipe out the fighter field at Iba; 54 bombers and 36 fighters were to hit the main target, Clark Field.\n\nJust as the great armada was leaving Formosa the approaching planes which had alerted the entire island were spotted. Fighters left the Navy formation and sped to the attack. Just in time red suns were seen on the \"Americans' \" wing tips. They were Japanese Army bombers returning from the Baguio and Tuguegarao missions.\n\nIn the Philippines the big Flying Fortresses milling aimlessly around Mt. Arayat in fear of being caught on the ground by these same enemy Army bombers were now getting permission to return to Clark Field. The alert was over; false alarm. As soon as they landed and lined up neatly, the P-40's flying their cover came in for gas and lunch. By 11:00 A.M. the terrific tension of the morning slackened. Calm had returned to Clark Field.\n\nForty miles west of Clark Field, across the rugged Zambales Mountains, and on the South China Sea, was Iba, a village consisting of little but nipa huts. Next to it was a single grass landing strip, bordered on its west side by a sandy beach. After Nichols it was the most important fighter field in the Philippines. In addition to its 18 combat-ready P-40's, it had the only working radar in MacArthur's command.\n\nThe Iba pilots were dog-tired. Since 2:00 A.M. they'd been chasing phantoms seen on the screen in the half-buried radar shack. Now they were sitting in their planes, smoking cigars, waiting for the next alert. Some were kidding the squadron commander, Lieutenant Henry Thorne. They felt he should think up a memorable fighting phrase for that first day of war. Someone suggested, \"Damn the torpedoes and take to the hills.\"\n\nAt 11:27 A.M. the radar screen was again alive. A big formation was coming in across the China Sea. Thorne ordered his men to start their motors. The chase was on once more. Three minutes later Iba was telling Air Warning at Nielson Field that a large formation was presumably heading for Manila. Immediately 18 fighters from nearby Nichols were sent to patrol Bataan and Manila Bay.\n\nIn Manila air raid sirens began to wail, but there was still no panic on the streets. Alarms, by now, were coming to suburban Nielson by telephone and telegraph from towns all along the northwest coast of Luzon. Some spoke of 27 planes, apparently fighters, others of 54 heavy bombers. The plotters tried to make sense of the tangled information. Apparently one big group was heading for Manila and several for Clark Field.\n\nThe base of the Flying Fortresses was still calm. No sirens screeched. At Fort Stotsenberg, a mile away, cavalrymen, were eating lunch. So were mechanics and air crews at Clark.\n\nAt 11:45 A.M. Colonel Alexander Campbell, the aircraft warning officer at Nielson, sent a teletype to Clark. But the message was not getting through. Radio was then tried. Still Clark could not be raised. Evidently the operator was having lunch. Colonel Campbell finally got a faint telephone connection with Clark. A junior officer assured Campbell he would pass on the news to the base commander or operations officer immediately.\n\nIt was 11:56 A.M. While sirens shrieked and all except key personnel were trying to find cover on barren Nielson Field, Brereton was telephoning Sutherland. The new bombing missions had been drawn up. The Flying Fortresses at Clark would hit known fields in southern Formosa at dusk; the other 14 B-17's, now 550 miles to the south at the Del Monte pineapple plantation in Mindanao, would fly up to Clark, arriving after dark, and prepare for raids on Formosa the next dawn. Sutherland approved.\n\nBy 12:10 P.M. all fighter pilots on Luzon were either in the air or waiting for orders to intercept the oncoming attackers. All that is, but the fighters in the Clark Field area. Forty minutes earlier, the 18 P-35's at Del Carmen, 14 miles south of the bomber base, had been ordered to fly cover over Clark but the message never arrived. And at Clark itself the junior officer who had talked with Colonel Campbell had not yet passed on the warning.\n\nEven the alerted pursuit planes were becoming hopelessly tangled because of poor communications and confused orders. Twelve of Iba's 18 planes were still circling above their own field. But six planes of B Flight, losing radio communication, had sped to Manila looking for the Japanese. Suddenly their radios cleared up. They heard, \"Tallyho!\" Then a voice shouted, \"Bandits over Clark!\" The six P-40's streaked, full-throttle, to the north.\n\nThe 12 fighters hovering over Nichols heard the same call. As they headed for Clark, the squadron commander, Lieutenant William Dyess, radioed Nielson. Brereton's headquarters told Dyess to come back and patrol Manila Bay.\n\nBy now the six planes of Iba's errant B Flight were in sight of Clark. All was peaceful. Another false alarm. They turned west, heading for home.\n\nIt was now 12:25 P.M. and not a pursuit plane was flying cover over the parked Flying Fortresses at Clark. At that moment 27 new-type \"Betty\" bombers, with flaming red suns on their wings, roared over Tarlac, only 20 miles to the north. They were heading directly for Clark. An excited Filipino plane observer ran to the telegraph office to make his report.\n\nAt Clark Field many of the ground crew were walking unconcernedly from the mess halls to the flight line. Ordnance men were loading bombs on the big, unpainted Flying Fortresses. Pilots of the 18 P-40B's, under Lieutenant Joe Moore, were sitting in their planes at the edge of the field near their empty fuel drum revetments. Many of the men were still eating. At the 30th Squadron mess hall mechanics and Flying Fortress crewmen were listening to Don Bell broadcasting the news.\n\n\"There is an unconfirmed report,\" said Bell, \"that they're bombing Clark Field.\"\n\nSince the only noise was the usual clatter of talk, dishes, knives and forks, Bell's words were greeted with derision and laughter. Many of the men still refused to believe the Pearl Harbor reports, thinking it was probably just some harebrained general's idea of putting everyone on the alert.\n\n\u2014\n\nThe first of the two flights of Japanese bombers were then within sight of Clark. In fact their crews had spotted the mass of B-17's, shining in the bright sun, from Tarlac. Their prey was ridiculously obvious, sitting out in the great, unprotected central Luzon plain. As an extra guidepost, Mt. Arayat, 15 miles east of the field, stuck up like a huge traffic marker. No one could miss this 3867-foot lonely peak standing in the middle of miles of plain, its cone dented, according to native lore, when Noah's ark landed.\n\nNow, the neat outlines of Fort Stotsenberg became distinct to the bomber pilots with its white buildings, lines of acacia and mango trees and the big polo field. A mile east stretched Clark Field. Not a single protecting pursuit plane was above it. At first the Japanese pilots couldn't believe what they saw: rows and rows of parked P-40's and Flying Fortresses. It was incredible luck. Almost ten hours after Pearl Harbor every plane based at Clark was a helpless, sitting target.\n\nThe second flight of 27 old-type \"Nell\" bombers swung in behind the leading Betties. Suddenly 36 Zero fighters were hovering above the bombers like shepherds. Everything was working with precision. It was 12:35 P.M.\n\nGeorge Setzer, who had the taxi concession at Fort Stotsenberg, was just driving out the gate. Hearing a growing roar, he stopped the car. He and his daughter, Stella, got out. They saw a mass of silver planes coming from the northwest.\n\n\"It's about time they came to help us,\" he said joyfully.\n\nOn the perimeter of the airfield, New Mexican National Guardsmen of the 200th Coast Artillery were having lunch or loafing around their 37-mm. and 3-inch anti-aircraft guns. They, too, thought the approaching bombers were friendly. At the cry, \"Here comes the Navy,\" Sergeant Dwaine Davis, of Carlsbad, grabbed the movie camera bought from company funds and began taking pictures.\n\n\"Why are they dropping tin foil?\" asked someone.\n\n\"That's not tin foil and those are goddam Japs!\"\n\nJust then there was a sound as of rushing freight trains.\n\nAt the west end of the field, a crew chief of the 20th Pursuit Squadron was standing in front of the operations tent. He looked up. \"Good God Almighty,\" he shouted. \"Yonder they come!\"\n\nHearing this, Lieutenant Joe Moore, the squadron commander, raced for his P-40. Followed by six others, he quickly taxied into position. He shot into the air, swung wide and started a maximum power climb. Two others got into the air, but the last four planes were hit by bombs.\n\nCorporal Douglas Logan, a B-17 gunner and cameraman, was at Headquarters Building watching Major Birrell Walsh conduct the briefing for the photographic mission over Formosa at a blackboard. Just left of Logan stood Colonel Eubank, the bomber commander. Logan saw him look absently out the window at the sky and turn away. The corporal looked out and saw men running frantically on the field.\n\nEubank, realizing what he had seen, jumped back to the window, then yelled, \"Take cover, men! Here they come!\"\n\nAs everyone headed for the rear door a stick of bombs exploded. The last to reach the door was Logan. Suddenly the floor pitched. Logan flopped. As he rolled toward a corner he instinctively put his hands over his face.\n\nThe air raid siren was now wailing. Someone shouted to get out of the hangars. Men began to saunter out. Above they saw planes in a great V. It was such a beautiful formation, they were thrilled\u2014until they realized bombs were falling on the P-40's and B-17's at the far end of the field. They dove for the trenches, recently built by the base commander, Lieutenant Colonel Lester Maitland, and till now chidingly referred to as \"Maitland's Folly.\"\n\nAt the west end of the runway, Corporal Durwood Brooks, a combat radio operator, was sprawled on his bunk when he heard the first bombs fall. He sprang up and ran into the latrine. Then he wheeled and ran outside toward a row of slit trenches. They were filled with white-clad cooks. He looked up and saw three V's, forming one great V. It was so perfect it was beautiful. He watched in fascination until a bomb hit 100 yards away. He was shocked. Someone was trying to kill him. He ran to the library and hid behind a heavy wooden piling. Every time a bomb hit he seemed to rise a foot in the air.\n\nExcept for the bombs dropped on the pursuit planes about to take off, the raiders were concentrating on hangars, shops and buildings. Anti-aircraftmen of the New Mexico National Guard were shooting 37-mm. and 3-inch guns at the passing formations. It was the first time most of the men had fired live ammunition. Much of their training in the United States had been with broomsticks and boxes or wooden models. And even if their bursts were exploding far below the targets, it was satisfying and somehow exhilarating to shoot finally in earnest.\n\nAt Fort Stotsenberg, cavalrymen were standing under mango trees with their horses. They were proud of their mounts. Not one had panicked even though the ground shook from reverberations and flak from Clark Field clattered down on all sides.\n\nMajor General Jonathan \"Skinny\" Wainwright was watching the bombing from outside his headquarters. His houseboy ran to him, eyes big with terror. He was wearing the general's steel helmet. \"Mother of God, General, what shall I do?\"\n\n\"Go get me a bottle of beer,\" yelled the general above the din. A moment later his aide, Captain Tom Dooley, drove up.\n\n\"Tom, you damn fool,\" shouted Wainwright angrily. \"You didn't drive past Clark during this bombing, did you?\"\n\n\"You sent me orders to report as fast as I could.\"\n\nWainwright walked into his headquarters and wrote Dooley an order for a Silver Star.\n\nAbruptly the bombing of Clark stopped. Corporal Brooks dazedly walked toward the flight line. The idea of war was new and terrifying. Bodies and parts of bodies were lying all over. In a slit trench he saw two Filipino pin boys. They were good friends, killed by someone who hadn't even seen them. Brooks couldn't figure it out. Then he saw another friend, a Polish boy of nineteen. By some freak he was blown up like a balloon by an explosive bullet and to Brooks looked almost transparent.\n\nOthers staggered from the trenches. In the sudden silence the groans of the wounded could be heard. Many of the buildings were burning. The big oil dump blazed, sending dark rolls of smoke across the field.\n\nAs Colonel Eubank started a hurried inspection of the Group, the two B-17's being painted were taxied unharmed out of their burning hangar. Eubank learned only a few of the Flying Fortresses parked on the field were damaged. His bombers had been saved by a miracle.\n\nLieutenant Joe Moore was now about 20,000 feet above Clark in his P-40 with Lieutenant Randall Keator on his tail. Trailing half a mile behind and 3000 feet below was Lieutenant Edwin Gilmore, the third of the pursuit pilots to take off successfully from Clark. Nine Zeros suddenly swooped down on Gilmore. Moore and Keator jumped the Japanese. Almost instantly Keator shot down a Zero. It was the first American kill over the Philippines. A moment later, Moore found a Japanese in his sights. As Moore pulled the trigger, he wondered how it would feel to shoot live ammunition. Up until now it had been forbidden because of its scarcity. A Zero blew up in his face. He dove on another target. There was a second explosion.\n\nPilots of the 34th Pursuit Squadron in Dell Carmen, 14 miles to the south, saw smoke rising from Clark Field. Without orders they took off in their worn P-35's. Before they reached Clark several Zeros confidently attacked the much larger American group. Easily out-maneuvering the outdated 35's, the Japanese drove them off.\n\nAt 12:40 P.M. the 12 P-40's of Flights A and C were still circling over their field at Iba, looking anxiously for raiders. Suddenly over their radios they heard a hysterical voice shouting, \"All pursuits to Clark Field! All pursuits to Clark! Enemy bombers overhead!\" Then they heard the crash of bombs in their headsets. They raced to the east.\n\nStanding near the strip at Iba, Second Lieutenant Glenn Cave watched the P-40's disappear. A moment later at 12:44 P.M. another pilot said, \"Look at that pretty formation of B-17's.\"\n\nCave looked to the west. He counted 52 planes in perfect formation at 13,000 feet. \"You're crazy,\" he said, \"there aren't that many B-17's in the Philippines.\" Black objects started wobbling from the planes. The two pilots dove into one of the few foxholes on the beach. Cave landed first, the other man on top of him. As bombs exploded and the earth shook he was glad he was the man on the bottom.\n\nAt that moment the six Iba planes of B Flight which had flown to Manila and returned when they heard the mysterious false alarm from Clark were innocently approaching their home field for landing. The control tower frantically called the incoming planes, warning of the bombers high overhead, but the radios of the P-40's were jammed.\n\nAs the first plane touched down, Second Lieutenant Andy Krieger, pilot of the plane flying cover for the other five, saw the field explode in a great blinding flash. Krieger climbed away so fast he had to level off at 10,000 feet to let his engine cool. Looking down he saw what looked like a squadron of P-35's circling the field. They were crazy to try and land on that bombed-out strip. Then he saw they had big red spots on the wings\u2014Japanese! He dove into their circle, and shot at the plane ahead. When tracers suddenly zipped past him, he turned and saw three Zeros on his tail.\n\nOn the field below, Cave also thought the planes swooping down were P-35's. Red lights on the wings were blinking. Cave was puzzled. Suddenly bullets spattered in the sand. He finally realized he was being strafed. He heard a deep rumble. Another flight of bombers was making its run. This time the other man got to the bottom of the foxhole first. Cave looked at his legs sticking up and wondered if they'd be hit.\n\n\u2014\n\nHalfway to Clark, the other Iba pursuit planes heard Lieutenant Krieger call, \"All 3rd Pursuit to Iba.\" Half of the 12 planes turned back. But Lieutenant Fred Roberts and five other Iba pursuit pilots flew headlong into the Zero fighters preparing to dive onto Clark Field. Almost out of gas, three Americans were quickly shot down. To Roberts' amazement the Japanese planes were faster, more maneuverable and climbed at a terrifying rate. How could these possibly be Japanese? He and other pilots had been told there was no such thing as a good Japanese fighter plane.\n\nHe didn't know, of course, that exact data on these Zeros had been sent to the War Department by the brilliant, unorthodox Claire Chennault in the fall of 1940. The chief of the Flying Tigers had also revealed how the heavier P-40 could master the faster Zero, but this information, which could have saved the lives of bewildered American pilots dying that moment, had been filed and forgotten. Chennault was not at all popular with Air Corps commanders.\n\nHoles appeared in Roberts' wings. Suddenly a cable at his feet was shot out. He felt a sting in his leg, then numbness. The needle of his gas gauge wobbled near empty. He pulled sharply to the west, heading for Iba.\n\nAs he crossed the Zambales Mountains smoke was rising from his home field. Tiny objects, obviously Japanese strafers, were diving and circling, but he had to come in. As he approached the field he tried to lower his wheels. Something was wrong. As he swung in behind a two-place Japanese fighter, the man in its rear began blinking a red light at him. Roberts loosed a burst and turned out to sea. With only ten gallons of fuel, he decided to beach his plane. He headed toward the water at 120 miles an hour, then realized he'd misjudged the distance. He was too high. He quickly nosed down, crashing into the surf 50 yards from shore.\n\nHe swam to the beach. The barracks was burning. So were other buildings, and a gas truck. He heard a pig squeal and looked toward what had been the village of Iba. All the nipa shacks were blazing; palm trees were mowed down; carts were tipped over; horses lay dead, feet in the air. Children screamed in terror. Filipinos were moaning, \"Help me, Joe.\"\n\nThe strip was a mass of craters. Several P-40's were crackling. The radar shack was a shambles, its operators killed. The control tower was riddled with bullets, its crew of four dead.\n\nDisaster had struck suddenly. Iba was completely destroyed and most of the survivors were in a state of shock. The coolest man was not a combat pilot but the young flight surgeon, Lieutenant Frank Richardson. Taking charge, he commandeered a bus, quickly loaded the wounded and headed for Manila.\n\n\u2014\n\nBack at Clark, as the survivors stared unbelieving at the wreckage, there was a cry, \"Here come the strafers!\"\n\nThe Zeros which had been circling far above the bombers were diving. Their main targets were the big Flying Fortresses and P-40's parked on the field. Soon they were joined by 44 of the 53 Zeros from Iba, now looking for new targets. One by one the parked Fortresses exploded with great roars as tracers ignited their gas tanks.\n\n\"Get in the woods,\" shouted an officer to men in the big ditch near the main hangar. Several dozen men scrambled out of the deep straight ditch and headed for the nearby woods as three Zeros swept over, riddling the area.\n\nA score of white-coated Chinese dashed out of Charlie Corn's PX restaurant. A Zero swooped and the ground was instantly white with their bodies. Nearby, Corporal Brooks was looking helplessly for quick shelter. He threw himself at a shallow ditch, digging frantically with hands and feet as two Zeros dove at him, their bullets squealing past. He ran to a deeper ditch and jumped onto a master sergeant.\n\n\"Excuse me,\" he said.\n\n\"Never mind,\" said the sergeant. \"That's just one more it has to go through to get at me.\"\n\nIn a nearby trench an antique water-cooled .30-caliber machine gun was spitting bullets at the strafers. A youthful mechanic was pumping it to keep it cool. After the first wave passed, the young man still kept pumping frantically as if in a trance.\n\nA truck swung by, blood dripping out of the sides. It was filled with wounded heading for the Fort Stotsenberg Hospital. Other wounded were walking dazedly, eyes blank.\n\nThe Zeros now had little opposition except from the machine guns of the 200th Field Artillery, and the 192nd and 184th Tank Battalions. When the water-cooled barrels of the artillerymen's old Brownings burned out, the gunners grabbed rifles and shot at the swooping Zeros.\n\nAs suddenly as it started the attack was over. Great black clouds of smoke covered the field. All the parked P-40's and 30 medium bombers and observation planes were burning. All but three of the Flying Fortresses were completely destroyed. In one raid the Japanese Navy airmen had knocked out half of MacArthur's Far East Air Force. Too late America had learned what the lessons of war in Europe should have taught: a heavy bomber force without adequate fighter protection, air warning and anti-aircraft guns is helpless, useless.\n\nThe returning Japanese planes were checking in. Not a bomber had been lost and only 7 fighters were missing. The claimed results were so fantastic\u201425 planes shot down and 71 destroyed or severely damaged on the ground\u2014that conservative officers at Formosa were dubious. Yet as claim piled on claim it was obvious that, even allowing for the natural optimism of all fliers, complete surprise had been achieved. It was a second Pearl Harbor.\n\nAt Imperial\u2013Navy Headquarters in Tokyo the news was received with another enthusiastic demonstration. In half a day, two of the three most powerful deterrents to complete success in Southeast Asia had been cancelled: the Pacific Fleet and MacArthur's air force. The third was British Admiral \"Tom Thumb\" Phillips' powerful striking force. According to the last reconnaissance report the battleship _Prince of Wales_ and the battle cruiser _Repulse_ were still in Singapore harbor\u2014too shallow for their conventional aerial torpedoes and well protected by anti-aircraft batteries.\n\nIf only the two big ships could be lured into the open sea.\n\n# 4\n\n# _\"So, Shoot to Sink\"_\n\n**1** It was dusk, December 7, in Pearl Harbor. Smoke still rose from the shattered fleet. The stink of spilled oil, burning and death was thick and nauseating. It was drizzling.\n\nHawaii was in a state of shock. The seventy-one-year-old governor, Joseph E. Poindexter, after a soothing telephone talk with Roosevelt, had been persuaded to declare martial law a few hours earlier. There were wild rumors from all sides. Eight Jap transports were seen rounding Barbers Point. Gliders and paratroopers had dropped at Kaneohe. Other paratroopers were coming down in sugar cane fields southwest of Ford Island, still others in the Manao Valley. At 11:46 A.M. the Navy reported: ENEMY TROOPS LANDING ON NORTH SHORE. BLUE COVERALLS, RED EMBLEMS.\n\nFifth Columnists, saboteurs and spies were reported everywhere: driving taxis, waiting on tables, tending gardens, selling groceries. They had cut arrows in cane fields and ringed Oahu with sampans to direct the Japanese raiders to their targets; they had driven milk trucks down airstrips, methodically knocking off the tails of parked American planes; they had poisoned reservoirs and caused other damage and mischief. Takeo Yoshikawa, now in the hands of the FBI, would have been amused.\n\nThe Navy was in a state of alert. A new grim joke was going around. \"On this day even the chiefs worked.\"\n\nAt the Navy Yard, Admiral Bloch was having more than his share of alarms. A telegram from the Office of Naval Intelligence in Washington warned him of a Japanese-operated radio in the basement of the Marine Barracks. Nothing was found but the usual mops, brooms and supplies. Then Bloch discovered that some public-spirited but misinformed person had put up signs on the yard hydrants: DON'T USE THIS WATER, POISONED!\n\nWith darkness it was unsafe to be abroad. Every moving object was a potential target for wide-awake riflemen. At Wheeler Field, someone heard a pil\u00f2t, obviously referring to the fuel in his plane's tank, mention gas. In minutes the gas alarm was spread. At Hickham Field, barracksless men were sleeping on cots under the sky. Suddenly a guard saw a mysterious dark shape sneaking toward him. It was a friend returning from the latrine. But the guard fired several rounds. Instantly there was wild confusion and shouts of, \"Japs landing! Bombing! Attack coming!\" Anti-aircraft guns burst out on all sides in a fantastic Fourth of July display. Every man with machine gun, rifle or pistol shot into the air. Falling flak and bullets sent more to the hospital.\n\nAt 7:30 P.M. Lieutenant Fritz Hebel was calling Ford Island tower. His flight of six planes from the carrier _Enterprise_ was just off Diamond Head, and he requested landing instructions. They were returning from a fruitless search to the southwest for Nagumo's carriers.\n\n\"Break formation with landing lights on,\" was the answer.\n\nAs six planes skimmed over the sand of Waikiki Beach, the pilots could see spots of light all over the island in spite of the new blackout rule. Pearl Harbor itself was dark except for ghostly red glows from still-smoldering ships. The planes roared along Battleship Row, passing through great clouds of smoke. The pilots relaxed. It had been a rough day and they were glad it was over.\n\nThey formed in a neat right echelon at 100 feet, their red and green wingtip lights on as instructed. As Hebel dropped his wheels, a single string of tracers shot up from the Navy Yard. Then a second.\n\nHebel saw Herb Menge's plane explode. He banked to the right, but he too was hit. He headed in a crash dive for Aeia. Eric Allen was already parachuting through a hail of fire, toward the oil-covered water.\n\nThe other three pilots flicked out their lights. Gayle Herman dove for the Ford Island runway. He just made it, his plane peppered with eighteen holes. The other two pulled up wheels and scooted for the sea. One, Jimmy Daniels, waited for the firing to stop and then came back with lights out, low and fast, to land safely. The second, Dave Flynn, parachuted into a cane field at Barbers Point.\n\nThe anti-aircraft score was almost perfect. Of six planes, four were destroyed and one damaged. This time Pearl Harbor had not been caught napping.\n\nSoon the area was again comparatively quiet. The only lights came from fires on the _Tennessee_ and a few other burning ships. There were also flares from the overturned _Oklahoma_ where men with acetylene torches were frantically trying to cut a hole in the hull and rescue ten suffocating comrades inside.\n\nOther trapped men were inside the _West Virginia_ , lying flat on her keel at the bottom of the harbor. A huge pocket of air, sealed inside the ship when she sank, was keeping alive some of the 66 still aboard. The survivors were tapping on the sides of the ship to attract attention. But no one heard them.\n\n**2** In Manila it was then only late afternoon. At his headquarters MacArthur was reading a radiogram from Marshall. It assured him he had \"the complete confidence of the War Department,\" and promised he could expect \"every possible assistance within our power.\" It was reassuring after months of squabble for more men and material.\n\nIronically, at almost that very moment the _Pensacola_ convoy bound for Manila with planes and men was being diverted to the Fiji Islands. Four other big troopships bound for Manila, the _President Johnson, Bliss, Etolin_ and _President Garfield_ , had already been ordered back to San Francisco.\n\nMacArthur had not yet been informed of these moves nor was he aware of a secret British-American military agreement that had been reached long before Pearl Harbor, which recognized Hitler as the chief threat to the democracies and specified that the bulk of all supplies and men must go to Europe. The main strategy in the Far East, it had been agreed, would be defensive. It was a logical plan, unanimously approved by both high commands.\n\nJust after dusk the aircraft tender _Langley_ , the light cruiser _Boise_ and two oilers eased out of Manila Bay. Admiral Hart had ordered his striking force south where it would be safe from bombing. The following noon those on the _Langley_ bridge saw a silver object overhead. It looked like a twin-engined plane diving through the moving clouds.\n\n\"It has all the appearance of an unfriendly aircraft,\" said the officer of the deck. \"Tell all guns to stand by for a dive.\"\n\nChief Petty Officer Gus Peluso, the captain's talker, instantly broadcast the alarm. Anti-aircraft fire burst out from every 3-inch gun on the ship. The _Boise_ joined in. After half an hour a message by portable tube blinker came to the _Langley_ from the navigator of the oiler _Trinity_ : \"Your convoy is shooting at a fixed position millions of miles away\u2014the star, Venus.\"\n\nBy nightfall of December 8, except for submarines, two destroyers being repaired at Cavite, and a few supply repair ships, the Navy had left the defense of the Philippines to MacArthur's ground and air forces.\n\nAt Iba there was no longer a fighter base. The pockmarked strip was useless. The burned-out barracks and buildings were completely deserted. Part of the 3rd Pursuit was hurrying south along the highway to Manila. Part was tramping in terror across the Zambales Mountains. According to rumor, the Japanese were landing on the coast near Iba. One pilot recalled the motto jokingly suggested that morning: \"Damn the torpedoes and take to the hills.\" Now it wasn't funny.\n\nThere was the same feeling of near-panic at Clark Field. Most of the men were hiding out in the cogon grass, fearful of a new bombing. Some were miles away in the hills. There were constant rumors of parachuters and saboteurs. Many slept that night next to their guns, gas masks clamped to their faces. On the flight line, Colonel Eubank, who reminded some of Will Rogers, climbed up on a tractor. \"Gentlemen,\" he said to those who grouped around him, \"we're just like a good bird dog that's been shot too close to on its first time out.\"\n\nEubank's words had a strangely calming effect. Word passed that the old man was still full of fight. The sense of shock that made many worry more about their footlockers than war itself began to leave Clark Field.\n\nIn Manila, MacArthur was assessing his forces. Soon he would have no navy, probably no air force, but he still had his ground forces\u2014and Marshall's promise of every possible assistance.\n\n**3** In the United States, people were reading their Monday morning papers. Most Americans were completely stunned by the greatest military disaster in U.S. history. For the most part, Washington officials were in as great a state of shock as the man in the street. The smashing of the Pacific Fleet and the destruction of the heart of MacArthur's air force had altered in a single day the entire concept of war in the Pacific. The most immediate fear of the War Department was another Japanese carrier attack\u2014this time on the locks of the Panama Canal or on aircraft factories on the California coast, or a new attack on Pearl Harbor and its still undamaged Navy Yard.\n\nAs the truth of the first losses began spreading around official Washington, a wave of hysteria swept over many of the most eminent government officials. One even telephoned the White House, shouting that the West Coast was no longer defensible and demanding that battle lines be established in the Rocky Mountains.\n\nThe public response in general was quite different. It was a war born in shame. But the humiliating defeats acted as a spur to the people. Telegrams and letters flooded the White House pledging full aid and cooperation. Yamamoto had made a colossal blunder. Instead of striking fear into American hearts, the crippling of the Pacific Fleet had insulted the people, filling them with fury and fight. They would never forget Pearl Harbor.\n\nA little after noon, senators walked down the long corridor of the Capitol, through the rotunda. They entered the House chamber behind Majority Leader Barkley of Kentucky and Minority Leader McNary of Oregon. Then came the justices of the Supreme Court in black robes, sitting to the left of the rostrum. Also down front were Stark and Marshall. In the packed gallery was Mrs. Roosevelt. Near her at the President's request was the widow of another war president, Woodrow Wilson.\n\nAt 12:28 P.M. the Cabinet entered. A moment later, Speaker Sam Rayburn rapped his gavel for silence and said, \"The President of the United States!\"\n\nThere was great applause as Roosevelt slowly entered on the arm of his eldest son, James, who was wearing the uniform of a Marine captain. After a brief prayer by the chaplain, the President opened a black loose-leaf notebook and in the hush began to read: \"Yesterday, December 7, 1941\u2014a date which will live in infamy\u2014the United States of America was suddenly and deliberately attacked by naval and air forces of the Empire of Japan.\"\n\nTension rose as Roosevelt scornfully described the attack on Pearl Harbor even as peace talks were going on. It built as he recited tersely the other points of attack. \"Japan has, therefore, undertaken a surprise offensive extending throughout the Pacific area. The facts of yesterday and today speak for themselves. The people of the United States have already formed their opinions and well understood the implications to the very life and safety of our nation.\n\n\"As commander-in-chief of the Army and Navy, I have directed that all measures be taken for our defense, that always will our whole nation remember the character of the onslaught against us.\"\n\nThere was a sudden cheer as the mass tension was suddenly released. Roosevelt looked up from the notebook, smiled and waved his hand in acknowledgment. By now everyone was standing, applauding loudly.\n\nThe speech, often interrupted by applause, continued for several minutes. Finally he said, \"With confidence in the armed forces, with the unbounding determination of our people, we will gain the inevitable triumph. So help us God.\n\n\"I ask that the Congress declare that since the unprovoked and dastardly attack by Japan on Sunday, December 7, 1941, a state of war has existed between the United States and the Japanese Empire.\"\n\nThe room was filled with applause, whistles, shouts and rebel yells. Roosevelt closed the notebook, lifted his arm in greeting. Then he took the arm of his son and, with the great room still in an uproar, left the dais.\n\nFor the first time as President, Roosevelt had spoken what was in every American's heart. People of all political convictions and beliefs were today welded into one great angry voice. Partisan politics, for the moment at least, were forgotten. America was at total war.\n\nIt was already Monday evening in Italy. Foreign Minister Count Galeazzo Ciano was called to the phone. It was Joachim von Ribbentrop. The German minister was overjoyed about the Japanese attack. He was so happy that Ciano, himself very dubious of its long-range advantages, became infected with the German's enthusiasm. That is, until Ciano put down the phone and began to worry again.\n\n\u2014\n\nMost Japanese were then eating breakfast, for it was early morning, December 9, in Japan. By now the average Japanese had recovered from the shock of realizing his country was at war with America and Great Britain. Four years of the China Incident had prepared him to accept the idea of total war. Restrictive measures had been applied so subtly, so gradually, that he accepted as necessary and almost normal what would have seemed impossible four years before. Great Britain and America were obviously to blame for the desperate situation. They had plundered the Orient and now selfishly refused to allow Japan even the oil necessary to run her ships.\n\nThe Japanese citizen had been trained since childhood to accept authority and to devote his life to service to the living god, Emperor Hirohito. And he and his sons were imbued with Japanese _seishin_ , spirit, making them fanatics in battle.\n\nThe first full day of war, with its amazing and sweeping victories, had brought all Japan together. Even the most conservative accepted war with calmness and quiet determination. Even the most liberal were still Japanese and though many had done their best to avoid war, they now realized it was impossible to shape the future course of events. Some reformers even believed that war would help bring about their long-sought domestic dream: equality. As in America, all conflicting groups instantly became united. Japan was fighting a holy war of righteousness and self-defense. Every Japanese was ready for total sacrifice.\n\n**4** The citizens of Singapore first learned of the war in the Pacific when they were awakened by the crash of bombs at 4:00 A.M., December 8, just two hours after the Pearl Harbor attack. Though the Fighter Control Operations Room got a report of unidentified aircraft 140 miles from Singapore half an hour earlier, no one answered its repeated telephone calls to the Civil Air Raid Headquarters.\n\nConsequently the lights of the city guided the pleasantly surprised invaders to their target. In fact the lights stayed brightly lit during the entire raid since the man with the keys of the master switch could not be found.\n\nAlthough 63 were killed and 133 injured and not a single Japanese plane was downed, there was no sense of real alarm in Singapore. The great majority were soothed by the Order of the Day issued later that morning by Air Chief Marshal Sir Robert Brooke-Popham, commander-in-chief in the Far East.\n\n> We are ready. We have had plenty of warning and our preparations are made and tested....We are confident. Our defences are strong and our weapons efficient....What of the enemy? We see before us a Japan drained for years by the exhausting claims of her wanton onslaught on China....Confidence, resolution, enterprise and devotion to the cause must and will inspire every one of us in the fighting services, while from the civilian population, Malay, Chinese, Indian, or Burmese, we expect that patience, endurance and serenity which is the great virtue of the East and which will go far to assist the fighting men to gain final and complete victory.\n\nA few, like Yates McDaniel, the American representative of Associated Press, were not at all reassured by Sir Robert's words. McDaniel knew the Brewster Buffalo fighter planes protecting Singapore were slow and cumbersome. He knew there wasn't a single tank in Malaya; that almost every one of the great fixed guns of Singapore was pointing to the sea, useless in case of a land attack down the peninsula; that the troops in Malaya had no jungle training; and that the native groups had been excluded from any participation in the defense of their homes.\n\nOn the morning of that first day of war, about the same time Japanese fliers were ending their devastating attack on Clark Field, the telephone in the newsman's room at the Cathay Building rang. It was McDaniel's good friend, Vice-Admiral Sir Geoffrey Layton.\n\n\"We're sending out two capital ships under Tom Thumb Phillips,\" he said. By the tone of his voice McDaniel knew Layton strongly disapproved. \"Would you like to go along?\"\n\n\"How long will they be out?\"\n\n\"Five or six days.\" He explained that Phillips was determined to sail north, up the east coast of Malaya, and attack the great invasion convoy which was still landing Japanese troops at two points. These troops, obviously, were going to drive south down the peninsula and assault the island of Singapore by the back door.\n\nMcDaniel was tempted. It sounded like a good show, but since he was the only AP man in town and big news was liable to break at Singapore, he refused.\n\nCecil Brown of CBS was eating his dessert, ice cream, in the dining room of the famed Raffles Hotel when he was called to the phone.\n\n\"Do you want to go on a four-day assignment?\" asked Major C. R. Fisher, in charge of facilities for correspondents.\n\n\"What is it?\"\n\n\"I can't tell you what it is, or where we are going, but I must have an immediate yes or no and you must leave at once.\"\n\nBrown hesitated a moment, then said, \"All right, I'll take it.\"\n\nBy this time McDaniel was talking face to face with Layton. The newsman's guess had been right. Layton was very upset by Phillips' decision to sortie out of the relatively safe harbor with little or no air cover and attack an unknown force with the _Prince of Wales_ , the _Repulse_ and four destroyers.\n\nLayton said he had objected strenuously but, even though he was senior to Phillips, he didn't have the authority to override him. When McDaniel left he was uneasy. Layton was an old China Station hand who had spent many years in these waters and was a keen student of Japanese capabilities. The newsman had met and admired Phillips, obviously a brilliant man of great decision and courage. He had been struck by the strangely heroic figure of the little admiral standing on a box so he could look over the bridge. But this was his first major sea command. McDaniel remembered the black cat of the _Prince of Wales_ sitting in President Roosevelt's lap at the signing of the Atlantic Charter. It gave him a feeling of foreboding.\n\nJust before sailing, Phillips asked Air Vice-Marshal C. W. Pulford what air cover the fleet could get on its sortie. Pulford, a former Navy man, was eager to cooperate but his airfields in north Malaya were already reportedly knocked out. He promised to give Phillips air reconnaissance the next day, December 9. But he doubted if he could spare any planes at all on December 10.\n\nPhillips then boarded the _Prince of Wales_. Captain L. H. Bell, captain of the fleet, noticed the admiral's uneasiness.\n\n\"I'm not sure,\" said Phillips, \"that Pulford realizes the importance I attach to fighter cover over Singora on the tenth. I'm therefore going to send him a letter stressing this point again and asking him to let me know as soon as possible what he can do for certain.\"\n\nThe admiral wrote the letter. He handed it to Bell who gave it to a messenger on the jetty. As Bell stepped back on board the _Prince of Wales'_ securing wires were cast off. The sun was setting as the fleet steamed out of Singapore. The _Prince of Wales_ led, followed by the _Repulse_ and the destroyers. As they passed Changi Signal Station at the east end of Singapore Island, Phillips was handed a radiogram from Pulford:\n\n> REGRET FIGHTER PROTECTION IMPOSSIBLE.\n\nPhillips shrugged his shoulders. \"Well,\" he told his staff, \"we must get on without it.\" After the blaze of publicity about the two great ships since their arrival in Singapore, it would have been unthinkable, he felt, to do nothing now. The two big ships and their escorting destroyers continued on their northern course.\n\n\u2014\n\nHidden by rain and clouds, the _Prince of Wales_ and the _Repulse_ were halfway to their goal by noon the next day, December 9. They would be ready to attack the Japanese convoy the following morning. Cecil Brown, the CBS correspondent, was talking to Captain William Tennant, skipper of the _Repulse_. Brown was not at all happy about being on the _Repulse_ instead of the flagship. \"We can't even mention that the _Repulse_ is out here,\" he said.\n\n\"That's nothing,\" said Tennant, laughing. \"The _Repulse_ hasn't been mentioned for as long as I remember.\" In two and a half years of war the ship had never been in action and its men were keen for something to happen.\n\nAt 1:45 P.M. the Japanese submarine I-56 sighted the two warships and their destroyer escort speeding north. The commander excitedly radioed the news. But the reception was bad. The submarine's radioman couldn't make himself understood. He tried again and again.\n\nIn Saigon, Rear Admiral Sadaichi Matsunaga, of the 22nd Air Flotilla, was just starting a special staff conference. Two \"Babos,\" reconnaissance planes, had falsely reported sighting the _Prince of Wales_ and the _Repulse_ in Singapore harbor. The problem was how to attack. Pilots were alerted and told to study the depth of the harbor and the best directional approach.\n\nAt 3:00 P.M. another message was handed to Matsunaga. It was from the I-56. Two men-of-war and four destroyers were sailing north at 14 knots near Procondor Island. Which report was right? After discussion the staff agreed the submarine report was more logical. Torpedo planes were ordered to prepare for attack.\n\nAt the bases of Matsunaga's two air groups, while torpedoes were being hastily loaded and unpinned, excited Army officers drove up. Somehow the word had mysteriously spread that the Navy had tracked down the two great British ships. Each plane took off amidst cheers of enthusiasm and hope.\n\n\u2014\n\nOn the _Repulse_ a notice was posted just as Cecil Brown came to the wardroom for tea at 3:30 P.M. \"All officers and ratings from dawn tomorrow are to be at action stations and are to wear clothing to resist burns from flashes of exploding bombs and shells.\"\n\nThen a message came from Admiral Phillips:\n\n> WE HAVE MADE A WIDE CIRCUIT TO AVOID AIR RECONNAISSANCE AND HOPE TO SURPRISE THE ENEMY SHORTLY AFTER SUNRISE TOMORROW, WEDNESDAY. WE MAY HAVE THE LUCK TO TRY OUR METAL AGAINST SOME JAPANESE CRUISERS OR SOME DESTROYERS IN THE GULF OF SIAM. WE ARE SURE TO GET SOME USEFUL PRACTICE WITH HIGH-ANGLE ARMAMENT, BUT WHATEVER WE MEET I WANT TO FINISH QUICKLY AND SO GET WELL CLEAR TO THE EASTWARD BEFORE THE JAPANESE CAN MASS TOO FORMIDABLE A SCALE OF AIR ATTACKS AGAINST US. SO, SHOOT TO SINK.\n\nQuiet excitement spread over the _Repulse_. At last the ship was to see some action. After a dinner of hot soup, cold beef, ham, meat pie, pineapple and coffee, a group of officers in the wardroom mulled over Japanese capabilities.\n\n\"The Japs are bloody fools,\" said one. \"All these pinpricks at widely separated points is stupid strategy.\"\n\n\"The Japs can't fly,\" said another. \"They can't see at night and they're not well trained.\"\n\n\"They have good ships,\" said a third. \"But they can't shoot.\"\n\n\"You British are extraordinary,\" said Brown at last. \"You always underestimate the enemy. You did it time and again, in Norway and France, in Greece, in Crete.\"\n\nThe officers were shocked at such open criticism. \"We're not overconfident,\" one told Brown. \"We just don't think the enemy is much good. They couldn't beat China in five years and now look what they are doing out here, jumping all over the map instead of meeting at one or two places. They cannot be very smart to be doing that.\"\n\nThen at 9:05 P.M. Captain Tennant's voice came over the loudspeaker of the wardroom. He announced the fleet had been detected by three enemy aircraft and Admiral Phillips was sure the Japanese convoy at Singora would now scatter.\n\n\"I know that you all share with us the disappointment in not engaging the enemy at this time,\" said Tennant regretfully, \"but I am sure you will agree with the commander-in-chief's judgment. We are, therefore, going back to Singapore.\"\n\nThe officers groaned in disappointment and bitterness. Brown hurried to a sailor's mess to get their reactions. Men were sitting around long tables, some with tears in their eyes.\n\n\"How do you fellows feel about this?\" asked the newsman.\n\n\"This always happens to the _Repulse_ ,\" said one.\n\n\"We're just an unlucky ship,\" said another.\n\nThe three planes which had just forced Phillips to turn back were not Japanese, but Allied. They either did not see the British Fleet or neglected to report it.\n\n\u2014\n\nBack in Saigon it was midnight and Rear Admiral Matsunaga was getting discouraged. None of the scores of planes he'd sent to find and attack the _Prince of Wales_ had seen anything but Japanese ships. The first report must have been true. The big British battleships were probably safely tied up at Singapore. He decided to recall the planes.\n\n\u2014\n\nOn the _Prince of Wales_ , Admiral Phillips was handed a message from his chief of staff, Admiral Sir Arthur Palliser, still in Singapore. It read:\n\n> ENEMY REPORTED LANDING AT KUANTAN.\n\nThis was a point on the east coast of Malaya midway between Kota Bharu and Singapore. It wouldn't be too far off the track on the return home. Earlier in the evening Palliser had reported that the airport at Kota Bharu had finally fallen. This made Kuantan even more important. It had to be defended.\n\nAt 12:52 A.M., December 10, the _Prince of Wales_ altered its course and speeded west at 25 knots toward Kuantan, where in fact not a single invader was landing. The Japanese submarine I-58 was heading for the same general area. Their paths slowly converged. Little more than an hour later, at 2:10 A.M., the I-58 spotted the two big British ships at \"Fumoro 45,\" naval code for the position 140 miles off Kuantan. Excitedly the commander maneuvered his submarine till the _Repulse_ was in his sights. Two torpedoes were fired. Then four more. All missed.\n\nThe big battle cruiser, unaware of her narrow escape, slipped by.\n\n# 5\n\n# _\"It Was Such a Beautiful Ship\"_\n\n**1** Action on the third day of war started at Guam Island. This American possession, about midway between Wake and Manila, was almost within sight of the Japanese mandated island of Tinian. In spite of this strategic position Congress had thought it a waste of money to build fortifications or even dredge a harbor.\n\nConsequently Navy Captain George McMillin, governor of the island, commanded only a token force. His 430 Marines and Navy men and 180 native Insular Guards were armed with 13 World War I Lewis machine guns, 15 BAR's and 170 Springfield rifles.\n\nAt 2:00 A.M. on December 10, Seaman Second Class Juan Perez of the Insular Guards was on beach patrol at Tamuning, several miles east of the capital, Aga\u00f1a. He and his three mates could see great ominous shadows on the water. Then they heard splashing as landing boats approached. Perez loosed a burst from his Browning automatic rifle at the first boat. Then he ran down the road toward Aga\u00f1a. He had to warn Captain McMillin that the invasion had started.\n\nThe boats nearing shore contained 400 Japanese Marines. Landing at three other points on the island were three battalions of Colonel Masao Kususe's 144th Regiment. In all, 5400 men from 9 transports\u2014escorted by 4 destroyers, 1 seaplane tender and 1 mine layer\u2014were about to storm an unfortified island.\n\nAt 4:30 A.M. Perez, now manning the Palace grounds at Aga\u00f1a with 100 other guards and Marines, saw a figure steal across the Plaza. He fired. A large group of figures ran for cover. A fierce fire fight broke out.\n\nAs first daylight came, a bugle blew retreat. Perez and the other defenders ran back into the Palace.\n\n\"Fight for your lives!\" shouted someone. Perez put down his empty BAR and grabbed a rifle.\n\nAt 5:35 A.M. an auto horn blasted three times. American fire ceased.\n\nAs Commander D. T. Giles walked out of the Palace, Perez sneaked out the back and crawled toward the swamps. In front of the Palace, Commander Hiromu Hayashi was assuring Giles in sign language that the civil rights of the natives would be respected and prisoners treated according to the laws of war.\n\nThese were laws of war new to Americans. As soon as Captain McMillin came out to surrender, Japanese soldiers tore off his outer clothing, leaving him clad only in shorts. The Battle of Guam was over. Seventeen Americans and Guamanians were dead. One Japanese was killed.\n\n**2** By dawn of that same day, two convoys lay off Luzon, the main island of the Philippines; one at Aparri on the north coast and one at Vigan on the northwest coast. About 400 of the Tanaka Detachment of 2000 men began to transfer from transports to landing craft. The detachment's mission was to set up a flying field near Aparri. It was a miracle, thought Colonel Toru Tanaka. Since leaving Formosa three days before, he had been constantly expecting an enemy air attack. So far not a shot had been fired at them.\n\nIn Aparri a young American reserve officer, Lieutenant Alvin C. Hadley, was watching this formidable array of ships off shore. To defend the town he had about 200 untrained Filipinos. The 400 Japanese approaching the docks looked like an army to Hadley. He ran to the telegraph office, wired regimental headquarters that 10,000 Japanese were landing. The answer soon came back: Attack immediately and drive them into the sea.\n\nYoung Hadley did the sensible thing. He packed up his half-dozen worn machine guns, and by sign language and pidgin English, managed to convey to his men, who spoke half a dozen different dialects, that they were to withdraw south.\n\nAt Vigan the Kanno Detachment, also 2000 men, and also detailed to set up an advance airfield, didn't have even young Hadley to deal with. There wasn't a defender in town. Only the heavy seas battled the Japanese. But Lieutenant Grant Mahony, flying a P-40 on a lonely night reconnaissance, spotted this cluster of ships at 5:13 A.M. and radioed back the alarm. Forty-seven minutes later, five Flying Fortresses, recently flown up from Del Monte, left Clark Field. It was the first U.S. bombing mission of the war and the pilots were eager. Major Cecil Combs led his squadron up the middle of Luzon, then turned west, coming over Vigan at 12,000 feet. Transports sprouted white wake. Farther out to sea the decks of several Japanese light cruisers seemed ablaze with ack-ack fire.\n\n\"Christ almighty, there they are,\" shouted someone excitedly.\n\nThe squadron roared over, dropping bombs. It turned and made a second run. Now P-40's darted out from land and dove on the transports. They dropped fragmentation bombs and sprayed the landing barges with machine bullets.\n\nFinally worn-out P-35's from Del Carmen arrived, their hungry pilots groggy from lack of sleep. Sixteen had started but, because of engine failure, only seven had made it. As the cruisers and destroyers covering the landing moved inshore to defend, the seven old planes, led by Lieutenant Sam Marrett, made a series of low runs. Several barges were sunk and three transports set afire. Marrett, nicknamed \"The Rat\" because of his pointed nose and small face, continued to rake the largest transport with his machine guns. He wouldn't leave it alone, going back time after time. He dove again, guns blazing at mast level. Suddenly the transport blew up, tearing off the wing of Marrett's plane.\n\nIn spite of the spirited American air attack the bulk of the Kanno Detachment landed near Vigan by noon. At that same time wave after wave of Japanese medium bombers from Formosa began passing overhead, southward-bound.\n\nTwenty minutes later, whistles began to blast in the suburbs of Manila, 210 miles to the south, at Nielson Field. Men ran through the corridors of Brereton's headquarters shouting, \"All out! All out! Air raid!\"\n\nCaptain Allison Ind, the intelligence officer, was getting his first sleep in two days at his desk. \"They're coming fast,\" shouted his assistant, Lieutenant Lou Bell. \"And they're headed straight for this layout.\"\n\nInd was instantly wide awake. \"Serious, Lou?\"\n\n\"This looks like it! There's a big formation of mixed stuff coming down from the north. They may smack Clark on the way, but the guess is that they keep rolling right down. We're going to catch it this time.\"\n\nIt was a good guess. The medium bombers from Formosa were approaching in three separate attack groups. Twenty-seven Betties were scheduled to strike at the fighter base at Nichols; 27 Betties at Del Carmen; and 27 Nells at the Cavite Naval Base. Flying cover slightly ahead of these bombers and now only about 40 miles north of Manila were 73 Zeros.\n\nJust before 1:00 P.M. the twenty-seven Betties bound for Nichols roared over Manila. A moment later their bombs fell on the barracks, offices and warehouses of the airfield. Within minutes Zero fighters dove, strafing the grounded planes. The American fighter pilots already aloft on patrol fought hard but could not stem the invaders. Almost every American was shot down or crashed for lack of gas.\n\nCaptain Robert Taylor, chaplain of the 31st Infantry (U.S.), watched the attack from Isaac Peral Street. At the first sound of bombs falling on Nichols Field there was a sense of crowd panic. Then above the noises of sirens, horns, and clatter of ponies drawing _calesas_ , the church bells of Manila started ringing. Taylor noticed the soothing effect on the people. It seemed to him the bells were saying, \"Be calm. Have faith. Use good sense. God is the same in hours of battle as in peace.\" It made him suddenly realize it was his job to minister to men in time of battle and keep their spirits high.\n\nAbout 10 miles to the northeast, Captain Jesus Villamor was in the officers' mess at Zablan Field when he heard the distant clanging of Manila's church bells. The commander of the 6th Pursuit Squadron of the Philippine Air Corps ran toward the hangars, followed by his pilots. He could see about six Zero fighters swoop down. Their bullets tore into trainers lined up on the parking apron.\n\nVillamor clambered into his plane. Two men laboriously wound the engine and it finally caught. Without warm-up, Villamor taxied the antiquated P-26 fighter down the field. It was practically a museum piece, with nonretractable landing gear. As he took off, he saw a Zero close in from behind. For the first time he realized how awful it was to be a flier. He was alone and scared. Knowing he had only seconds of grace, he suddenly dipped down in the valley at the edge of the field. The Zero followed. Villamor hedgehopped recklessly, then climbed steeply to the right. Though the Zero was still on his tail, Villamor's confidence had returned. He knew he was a hot pilot and with a few tight turns could get behind the Zero.\n\nAt an altitude of only 500 feet, he made the tightest turn of his career. The Japanese followed easily. Again and again Villamor turned, but he couldn't shake the Zero. He knew it was hopeless. Only something unexpected and unorthodox would save him.\n\nHe suddenly dove under a row of high-tension wires. Evidently assuming the P-26 was crashing, the Japanese climbed, looking for another victim. Villamor was safe\u2014for the moment.\n\n\u2014\n\nBy now the second group of 27 bombers was sweeping toward Manila. These had abandoned their run on Del Carmen because preceding Zero fighters had almost totally destroyed that field's remaining P-35's in strafing attacks. These Betties were now heading for their secondary target, the shipping in Manila Bay.\n\nU.S. High Commissioner Francis Sayre watched them approach from his terrace on Dewey Boulevard. It looked as if his last hour had come. But abruptly the planes turned west and began unloading on merchant ships in the harbor.\n\nTen miles across the Bay, the siren sounded at Cavite Naval Base. From the control tower, Cecil Browne, supervisor of tugs, barges and ferryboats, had seen the third group of bombers, the 27 Nells. They were coming in across Corregidor Island in a great V. As he was calling Operations, Browne saw the planes pass almost overhead and continue east without dropping a bomb. Then they abruptly turned and started back. At 1:10 P.M. their bombs began to fall on Cavite. The nine 3-inch anti-aircraft guns from the naval base opened fire but their bursts were far below the high-flying Nells.\n\nBombs exploded near the pier, missing the ammunition dump but hitting the lumberyard, powerhouse, dispensary, Commandancia and torpedo storage. Browne and the two others in the forty-foot tower were so stunned they didn't move during the bombing. He saw 27 more bombers approach. These were the Betties which had just bombed the shipping in Manila Bay. Suddenly Browne realized he was surrounded by glass. \"What the hell are we doing here?\" he shouted. The three men hurried to the ground and hid under the dock.\n\nThousands in Manila watched the destruction of Cavite in horror and fascination. From atop the Marsman Building in the Walled City, Admiral Hart, in spite of warnings from his staff, was a grim witness. Carl Mydans of _Life_ and Mel Jacoby of _Time_ watched from the shore of Manila Bay. Mydans, peering through glasses, was amazed at the precision of the Japanese. Not a single bomb seemed to be wasted. The bombing in the Russo-Finnish war, he told Jacoby, was never as effective as this.\n\nBack at Cavite, Lieutenant Malcolm Champlin\u2014the aide of Rear Admiral Francis Rockwell, commandant of the Sixteenth Naval District\u2014was racing in his car to the scene of action.\n\nAs Lieutenant Champlin approached the bridge leading to the Navy Yard, cars poured out, loaded with wounded. A truck just ahead of Champlin swerved to avoid hitting a group of running Filipinos and collided with a truck coming from the yard. Champlin jumped out of his car. The two Filipino truck drivers were shouting at each other, waving arms.\n\n\"Shut up,\" shouted Champlin. Seeing bumpers were hopelessly locked, he ordered the wounded in the first truck transferred to private cars. He himself jumped into the back of the truck and helped pass out the wounded. The last man was covered with blood, one eye missing. As Champlin gently carried him to a car, the Filipino opened his good eye. Seeing Champlin's officer insignia he tried to smile. \"Thank you, sir. Thank you, sir,\" he mumbled. He gurgled and suddenly slumped. He was dead.\n\nAs Champlin ran into the yard, fires were raging. Cars moved slowly to avoid the many bodies in the streets. He hurried to the small shelter near the Commandancia. There he saw Rockwell, hatless, blood over his shirt, carrying wounded.\n\nThe powerhouse was demolished, eliminating the fire-fighting system. Rockwell telephoned to Manila for all engines capable of pumping water from the sea wall and dock. The admiral's face was tense as he told Champlin they had to stop fires before they reached the ammunition dump. Abruptly the wind shifted and the flames rose to new heights. Champlin looked around him. The most important shops were ashes and twisted beams. And close to 500 men were dead.\n\n\u2014\n\nThe Filipino fighter commander, Jesus Villamor, was about 4000 feet above Zablan Field when he saw 27 Betties make a second devastating attack on Cavite and then turn leisurely and head north for home.\n\nHe climbed to his ceiling and just as the bombers were closing in he dove head-on at the leading plane. He loosed a short burst with his two puny .30-caliber guns. To his amazement, smoke spilled out of the bomber and it spun out of formation.\n\nThree other equally daring Filipino fighter pilots were right behind Villamor. One, Lieutenant Jose Gozar, tried three times to ram his plane into a bomber, but his P-26 was too slow. Except for the plane shot down by Villamor all the raiders escaped.\n\nThe fires at the naval base were still raging late that afternoon. Lieutenant Champlin and a pickup crew of volunteers were pulling cases of gas masks from a burning loft. They saved 700, escaping just as the building collapsed.\n\nChamplin thanked the men.\n\n\"I'm up for summary court-martial,\" said one. \"Will you testify for me?\"\n\nChamplin assured him he would.\n\nThen the rest of the crew piped up, \"Will you do that for all of us?\" They had all been released from the brig during the bombing.\n\nThe yard was a shambles. The submarine _Sealion_ had received a direct hit at its mooring and more than 200 torpedoes had been destroyed. By now Admiral Hart had assessed the damage and made up his mind. Manila was untenable as a naval base. He ordered 2 destroyers, 3 gunboats, 2 submarine tenders and 2 mine sweepers south. Almost nothing was left of the Asiatic Fleet in the Philippines except submarines.\n\nAs darkness came, rumors swept Manila. The Philippines were being abandoned by the Americans; Japanese paratroopers were landing all over; saboteurs were ready to strike at the water system; spies were sending up rockets to guide bombers.\n\nThe rash of paratrooper reports had started that morning. By now Fort McKinley was being combed for Japanese; airmen and cavalrymen were roving the slopes of Mt. Arayat. Terror spread with every hour. Not a Japanese paratrooper had landed. The parachutists had been American fliers escaping from their burning planes.\n\n**3** By far the most important action on this third day of war was taking place 1500 miles southwest of Manila, just off the Malay Peninsula. By midmorning, the 10 search planes and 96 bombers of Rear Admiral Matsunaga's 22nd Air Flotilla, alerted before dawn by the submarine I-58, were still searching in vain for the _Prince of Wales_.\n\nThe two ships were almost 100 miles west of the nearest Japanese planes, lying off Kuantan. The destroyer _Express_ , sent to investigate that harbor, was signalling Admiral Phillips by lamp, \"All's as quiet as a wet Sunday afternoon.\" The report of a Japanese landing had been a false alarm.\n\nBefore returning to Singapore, Phillips decided to investigate a suspicious-looking tug and 4 barges sighted at dawn. The 2 big ships\u2014escorted by the destroyers _Express, Electra_ and _Vampire_ \u2014headed toward the tug at 9:00 A.M. The fourth destroyer, _Tenedos_ , had been sent home the night before because of low fuel.\n\nBy this time the 96 Japanese bombers and 10 search planes had about given up all hopes of finding the British. In fact, the search planes were on their way home. Then at 10:15 A.M., just as they were about to radio the bombers to return to their bases, one sighted 2 large battleships and 3 destroyers through the clouds. Excitedly, the word was radioed back to Saigon.\n\nAt 10:30 A.M., radio contact was finally made with the 27 Betty torpedo planes of the Kanoya Group and the new position of the British ships given. The Betties changed course, levelling off at 10,000 feet. Lieutenant Haruki Iki, leader of the 3rd Division of this group, forgot his exhaustion and hunger immediately. He was eager for action. He'd trained his nine-plane division so well they had been given the title of Champions of the Navy by the Combined Fleet. At 10:50 A.M. Iki saw what looked like a British Walrus observation plane through the clouds. The enemy fleet must be near!\n\nIt wasn't until then that Lieutenant Sadao Takai, leader of the 2nd Division of the Genzan Group, received a code message from Matsunaga pinpointing the British ships 70 nautical miles southeast of Kuantan. Takai radioed his own planes and headed north-northwest toward the enemy. The 1st Division, though it did not get the message, saw Takai's nine torpedo planes change course and followed. The 3rd Division had long since gone home, its nine bombs having ignominiously missed the _Tenedos_ earlier that morning as it was heading for Singapore.\n\nClouds began to fill the skies but patches of sea appeared occasionally. Every man strained to have the honor of being the first to see the enemy. Takai was so nervous his hands shook. He had a strange impulse to urinate. He remembered what Captain Kosei Maeda, the commander, had told them that morning just before take-off: \"Calm down and put your strength in your abdomen.\"\n\nThe British Fleet was also getting ready for battle. The Japanese search planes had been seen and Admiral Phillips had ordered first degree of High Angle readiness. Excitement was building on the _Prince of Wales_ , the _Repulse_ and the three destroyers. Action was coming at last.\n\nOn the _Repulse_ CBS correspondent Brown was taking pictures of a gun crew playing cards. As the ship zigzagged he snapped the _Prince of Wales_ half a mile ahead. Then at 11:07 A.M. he heard the loudspeaker announce, \"Enemy aircraft approaching. Action stations!\"\n\nHe looked to the south. Nine planes in single file were coming in at about 12,000 feet. Rooted to the port side of the flag deck as if hypnotized, he watched the falling bombs get larger. Suddenly 10 yards away a great geyser shot out of the sea, drenching Brown and his camera. As he instinctively hunched there was a dull thud and the ship shuddered.\n\n\"Fire on the boat deck! Fire below!\" blared the loudspeaker.\n\nGunner T. J. Cain of the destroyer _Electra_ had been watching these bombers with little apprehension, even though their formation was amazingly precise. In his long experience high-level bombing was usually more frightening than dangerous. Even with the much higher-rated Germans it hadn't proved very effective against moving ships.\n\nThen he saw bombs dropping. The pattern was impressive, horrifying. A series of bursts seemed to envelop the 794-foot-long battle cruiser. There was a great cloud of smoke and the _Repulse_ disappeared behind a giant wall of water. The men on the destroyer watched, shocked into silence. Suddenly the big ship reappeared, black smoke belching from her deck. The men on _Electra_ cheered spontaneously.\n\nThe battle cruiser's lamp blinked, \"One hit...some damage...but under control. Fighting efficiency unimpaired.\" As the message was translated the men of _Electra_ cheered again.\n\nIki saw the bombing from about seven miles' distance. His throat was dry. As his nine planes went into single file, he saw three Betties of his group's 1st Division and two from the 2nd dive on the _Prince of Wales_ and launch torpedoes. The other 13 planes were heading for the _Repulse_ so Iki decided to attack the _Prince of Wales_. He swept down, veins pounding with excitement. Five miles from the target he saw a terrific explosion in the rear of the ship. Then the front of the _Prince of Wales_ erupted. Two direct hits. Thinking quickly, Iki pulled up and turned north. No use killing a cripple. He was going after the _Repulse_.\n\nThe _Prince of Wales_ , listing 13 degrees to port, was already weaving uncertainly at 15 knots. Both port shafts were out of action and her steering gear wouldn't work.\n\nThe two divisions of Genzan Group now approached the battle area. Lieutenant Takai heard the group commander say, \"Form assault formation.\" Then, \"Go in!\" The 1st Division swept forward ahead of Takai and started a gradual dive. Takai followed. It was standard practice for the 1st Division to hit the largest vessel; he was supposed to take the next largest.\n\nTo his amazement there were no enemy fighters even though they couldn't be much more than 125 miles from Singapore. Anti-aircraft fire seemed to engulf the 1st Division but there was none near him. Through binoculars he studied a large ship giving off a narrow plume of white smoke, moving straight, flanked by three destroyers. It was the _Repulse_ , but to Takai it looked exactly like the battleship _Kongo_. His blood ran cold as he saw the 1st Division approach. Were they attacking their own ship? Takai called the observer who answered shakily, \"It looks like our _Kongo_ to me, too.\"\n\nTakai was now at 1700 feet, the position ideal, but he still couldn't make up his mind. Finally he got a close look. It was definitely _not_ the _Kongo_. He turned into the clouds to confuse the enemy and changed course. Suddenly he darted out of the clouds. He was a mile and a half from his target.\n\nOn the _Repulse_ a bugle blew. Over the loudspeaker came a roar, \"Stand by for barrage!\"\n\nEvery gun on the battle cruiser blasted as Takai's nine torpedo bombers swept in.\n\n\"Look at those yellow bastards come,\" said someone next to Brown. The newsman could do nothing but look. He saw paint blisters on the guns as big as tennis balls. The gunners' faces were eager, excited, streaked with sweat. He saw torpedoes drop one by one. The big ship dodged each one with heavy grace. The attack was abruptly over.\n\n\"Plucky blokes, those Japs,\" said the man next to the correspondent. \"That was as beautiful an attack as ever I expect to see.\"\n\nTo Brown's surprise the men seemed ecstatically happy, with no hate for the enemy. On the bridge Captain William Tennant had just sighted \"not under control\" balls hoisted from the _Prince of Wales_. He asked the flagship what damage she had suffered but Phillips did not answer. He signalled the admiral, \"We have dodged 19 torpedoes thus far, thanks to Providence.\" He added that all damage from the bomb hit was under control. The _Repulse_ was fit for action. But he still got no answer from the listing _Prince of Wales_. He wondered if Phillips had told Singapore of the attack. To be on the safe side he radioed home:\n\n> ENEMY AIRCRAFT BOMBING.\n\nThe message from Tennant was received in Singapore at 12:04 P.M. It was the first word they had received of the fleet since a message early that morning saying Phillips would be home sooner than expected. Six Brewster Buffalo fighters were sent to the rescue at 12:15 P.M.\n\nThree minutes later Captain Tennant asked Phillips if his wireless was out of order. Getting no answer, he reduced the _Repulse_ 's speed to 20 knots and moved closer to the flagship to see if he could be of any assistance. Just then nine more torpedo planes headed toward the _Repulse_. Six of these abruptly turned left and headed for the _Prince of Wales_ , but the other three dove straight at the _Repulse_. Tennant started to swing his ship to starboard. One torpedo dropped, heading for the _Repulse_. Tennant helplessly watched it for a minute and a half. If he turned he would be hit by at least two other torpedoes. It struck amidships. \"Order all possible men to starboard,\" he said as the battle cruiser listed to port.\n\nEven with this hit the _Repulse_ was still in no serious trouble. It could maneuver at 25 knots. But another much more powerful attack was already developing. Lieutenant Iki's nine torpedo planes were nearing the battle cruiser. Iki dropped quickly to 1300 feet, below the clouds. Pompom bursts blossomed on all sides. He had an impulse to pull up but he knew he had to get in much closer. Now he was skimming 125 feet above the water. Ahead was a wall of fire from the _Repulse_. He had to get through it. Nineteen hundred feet from the ship he yanked his release. He had it broadside!\n\nHoles appeared in his wings. He pulled sharply to the left. As he did the ship turned to starboard and momentarily the two were parallel. Below he saw sailors in what appeared to be raincoats lying down on the deck. He saw faces looking up as his crew peppered them with machine guns. There was an orange blaze as the plane behind Iki, piloted by Chief Petty Officer Toshimitsu Momoi, burst into flame. The third plane, Petty Officer First Class Yoshikazu Taue's, exploded and pinwheeled into the ocean. Iki saw two quick explosions on the bow of the _Repulse_. Of his first three planes, two had made direct hits. As he climbed 3200 feet to wait for his six remaining planes, he saw another torpedo hit.\n\nThe men on the _Electra_ saw the torpedo drop from Iki's plane and skim toward the _Repulse_. It exploded near her bows, shooting a great column of water above her upper deck. Before the water fell another huge spout rose a few yards astern. The big ship veered crazily, like a blind man staggering helplessly.\n\nWhen Iki's torpedo hit near the gunroom the rudder jammed. The _Repulse_ was out of control. Captain Tennant, on the bridge, knew his ship could not survive. Then three more torpedoes\u2014two on the port side, one on the starboard\u2014smashed home.\n\nThe decision to cease all work below is a hard one for a ship's captain to make, but Tennant didn't hesitate. To save the men in the hold he ordered everyone to come on deck. Carley floats were cast loose.\n\nIn a minute the ship listed 30 degrees to port as men poured onto the deck wearing their life belts. Tennant looked over the starboard side of the bridge and saw about 250 men filing up from below in an orderly manner. His men, long eager for battle, were responding as he had known they would. \"Prepare to abandon ship,\" he said over the loudspeaker in a cool voice. He told the men how well they had fought the ship, paused and said, \"God be with you.\"\n\nOn the flag deck, Brown was amazed by the lack of panic. It was incredible. Then one young sailor tried to shove to the head of the line.\n\n\"Now, now,\" quietly scolded an equally young sub-lieutenant, \"we are all going the same way, too.\" The young sailor stopped pushing.\n\nIt was 12:32 P.M. The big ship hesitated as if deciding whether to roll over completely or not. Ahead, almost hidden in smoke, Brown could see the _Prince of Wales_ settling. She belched steam and water and seemed to lie helpless in the water.\n\n\u2014\n\nMen were jumping off the tilting _Repulse_. Brown saw an eighteen-year-old Australian, Midshipman Peter Gillis, dive 170 feet from the control tower at the top of the mainmast. Another dove from another tower but crumpled as he hit the side of the ship. A third dove straight down the smokestack. A dozen Royal Marines leaped from the deck near the stern and were quickly sucked into a propeller.\n\nMany men scrambling up from the hold were trapped by the rising water. A group of 42 climbed up frantically inside the dummy smokestack but their way to freedom was barred by a wire screen. They were caged.\n\nThe ship was listing 70 degrees as Captain Tennant told his staff, \"Well, gentlemen, you had better get out of it now.\"\n\nWhen one asked if he was coming too he said, \"Off you go now. There's not much time.\"\n\nThey pleaded when he would not leave his ship. Several pushed him through the doorway onto the deck. He struggled to stay aboard but officers and men seized him, forcing him to leave the bridge.\n\nAt 12:33 P.M. the ship rolled over. Brown was hanging indecisively to the hull of the ship. He watched someone slide past him and then dive gracefully into the sea. Brown stood up and jumped, his camera swinging from his neck. As soon as he hit the warm oily water he automatically looked at his watch. It was 12:35 P.M. Eyes burning from oil, he saw the bow of the _Repulse_ sticking up like a church steeple, its underplates a gruesome red.\n\nFive thousand feet above, Iki couldn't believe what he saw. The great ship was pointing straight at him. Then it dove out of sight. It was not possible. \"Banzai, Banzai!\" he screamed, throwing up his hands. The bomber, with no hands on the wheel, dipped.\n\nIki's crew of seven were screaming as wildly as he was. He ordered wine passed. They all drank a toast. Below he saw hundreds of dots in the water. Two destroyers were darting about picking them up.\n\nBrown was struggling in the oily water, his only support a small table. \"I've got to remember all this,\" he said to himself. Then he thought, \"What the hell's the use? I will never be able to report this story.\" Almost exhausted, he drifted to a Carley float, but it was so full he didn't even try to get aboard. A young Marine, Morris Graney, pulled the half-drowned correspondent aboard. \"My Lord,\" said the Marine, \"do you still have your camera?\"\n\n\"Yes.\"\n\n\"Let's all sing,\" suggested Graney. He and a few others sang, \"When Irish Eyes Are Smiling.\" Then Iki's planes roared overhead. \"They're coming over to machine-gun us,\" someone shouted.\n\n\"Dive under the water!\"\n\nBrown knew if he dove he'd never come up again. He just watched. But the planes ignored them, flying on north. Many in the water thought the Japanese were merely out of bullets. It had never occurred to Iki, who had plenty of ammunition, to dive and strafe the survivors. The British had fought gallantly, in the tradition of Bushido. War was still new, chivalrous. Both sides would soon learn an enemy spared today might kill tomorrow.\n\nIt was 12:41 P.M. The _Prince of Wales_ , already hit by five torpedoes and settling badly, was barely steaming forward. Nine high-level bombers approached. The ship's remaining five .25-inch guns and pompoms sent up a heavy barrage but the planes came on. At 12:44 P.M. bombs wobbled down. Four straddled the ship but one hit the catapult deck.\n\nThe 35,000-ton battleship staggered. The destroyer _Express_ darted alongside the starboard side of the quarter-deck. Captain Leach ordered all wounded disembarked. Then he told those not required to fight the ship to board the destroyer.\n\nAdmiral Phillips, on the bridge, was still determined to save the flagship. \"Tell the _Express_ to signal Singapore for tugs to tow us home.\"\n\nBut the _Prince of Wales_ had received her death blow. A few minutes later the great ship began to founder. Her beams were almost awash. Leach ordered all hands to abandon ship, while he and Phillips stood together on the bridge waving to their departing men.\n\n\"Good-bye,\" called Leach. \"Thank you. Good luck. God bless you.\"\n\nAt 1:19 P.M. the _Prince of Wales_ heeled over quickly to port and the little admiral and Captain Leach disappeared. A minute later the famous battleship sank from sight, almost dragging with it the _Express_ , still faithfully standing by her side.\n\nAt that moment the six lumbering Buffaloes from Singapore arrived. Not a Japanese plane was in sight. Flight Lieutenant T. A. Vigors looked down, shocked. He swung low over the hundreds of men struggling in the water. Men waved and held up their thumbs. Vigors had seen disaster and courage at Dunkirk and in London night raids but he never encountered such an indomitable spirit. He came so low he could see the faces of the survivors. Any minute they might drown or be strafed, yet they waved and joked as if they were holiday-makers at Brighton.\n\nThe men of the _Electra_ were accustomed to tragedy. They had seen the _Bismarck_ sink the _Hood_. The little destroyer was again senior ship in a force struck by major disaster. Commander May quickly passed the news to Singapore. But why, he asked himself as he saw the Buffaloes overhead, hadn't Admiral Phillips called for air cover earlier?\n\nPerhaps the admiral thought such a request couldn't be met, for he had been told that fighter cover would not be given off Singora. Phillips also knew that all the northern airports had become untenable because of the Japanese advances down the Malay peninsula. Friends of Phillips guessed he might not have wanted to give away his position by breaking radio silence. The admiral had been noted for his extreme sensitivity to this danger. In any case, the answer would remain a mystery. All the senior officers on the bridge of the _Prince of Wales_ went down with Phillips and Leach.\n\nLieutenant Iki was now approaching his home base near Saigon. His excitement was marred as he thought of his two good friends, Momoi and Taue. Although he was reasonably sure his own torpedo had hit the _Repulse_ , he reported that the first two hits had been made by Momoi and Taue. It was the least he could do for them.\n\nWhen the division landed, excited pilots and mechanics crowded around each plane. Iki and the others were dragged out and thrown up in the air. Then the group commander, Captain Naoshiro Fujiyoshi, his face beaming with happiness, embraced Iki. He had told his men, just before they took off, \"Let us meet each other at the Yasukuni Shrine in Tokyo. Let us fight to the death.\" His fliers had vowed to fight so as to leave no regrets. They had done so.\n\nA few minutes later one of Iki's pilots came to him. \"As we dove for the attack,\" he confessed, \"I didn't want to launch my torpedo. It was such a beautiful ship, such a beautiful ship.\"\n\n\u2014\n\nIn Singapore the _Express, Vampire_ and _Electra_ were unloading the 2081 saved from the two ships. Eight hundred and forty would not come back. Already a fantastic rumor was sweeping the city\u20141000 Japanese planes from carriers had suddenly appeared, completely destroying the _Prince of Wales_ and the _Repulse_.\n\nThe news of the loss shocked the colony. At one stroke the fleet was gone. Most naval authorities were sure the ships had been sunk by carrier-based planes. Some, still doubting Japanese capabilities, claimed the _Tirpitz_ , brother ship of the _Bismarck_ , had helped.\n\nAir Vice-Marshal Pulford was shaken by the news. Many of his boyhood friends had gone down with the two ships. When he saw Captain Tennant he said, \"My God, I hope you don't blame me for this! I had no idea where you were.\"\n\nAt the request of the over-all commander in the Far East, Duff Cooper\u2014Churchill's special representative\u2014made a radio broadcast that evening to the people of Singapore. He pointed out that the loss of the _Prince of Wales_ and the _Repulse_ must not lead to despondency but merely to a determination to fight all the harder and so avenge the loss.\n\nThe words meant little to the people in Singapore. The unsinkable had been sunk. Were the island's defenses really \"impregnable\" as the authorities boasted?\n\nTokyo was just as shocked by the news. To old Navy men it was impossible. Great battleships under way simply couldn't be sunk by planes. Younger Navy airmen were exultant. Their theory was now a fact. And only four planes had been lost.\n\nIn England it was only the morning of December 10. Winston Churchill was in bed, opening his dispatch boxes before rising. The phone rang. It was the first sea lord, Sir Dudley Pound. \"Prime Minister, I have to report to you...\" Pound coughed and gulped \"...that the _Prince of Wales_ and _Repulse_ have both been sunk by the Japanese\u2014we think by aircraft.\"\n\n\"Are you sure it's true?\" Churchill was astounded. This was the most direct shock of the war.\n\nThe next dawn Lieutenant Iki flew over the graves of the _Repulse_ and _Prince of Wales_. As he passed the sunken hulls of the two ships he dropped bouquets of flowers.\n\n# PART\n\n# 2\n\n# THE DEFENSES CRUMBLE\n\n# 6\n\n# _The Death Watches_\n\n**1** On December 11, at 1:50 A.M., 11 Japanese ships cautiously approached the subtropical atoll of Wake, America's lonely mid-Pacific outpost. They drew within 8000 yards, then stopped. This was the Wake Island Invasion Force\u2014the light cruiser _Yubari_ , 6 destroyers, 2 patrol craft, 2 transports and a landing party of 560 infantry-trained sailors. They had come from the Japanese-mandated Marshall Islands to the south. Each patrol craft began lowering a single landing boat holding 80 armed men. Though the wind was brisk, they were safely launched. Now boats were carefully lowered from the two transports, but they banged dangerously against the steep sides of the old-fashioned ships and were quickly hauled back aboard. Word of this difficulty was flashed to the flagship, _Yubari_ , and the over-all commander, Rear Admiral Sadamichi Kajioka, postponed the landing until after sunrise.\n\nA little before 3:00 A.M. the _Yubari_ and 4 destroyers began to move toward the island. According to the latest intelligence report, the daily bombings of Wake had crippled at least half of the coastal guns and completely wiped out the air force. Kajioka was determined to bombard the remaining guns to rubble. There was still no sign of life ashore. The enemy, the admiral assumed, was asleep as usual.\n\n\u2014\n\nBut the Americans were well aware of the enemy off their southern shore. Major James Devereux was watching from the beach through night glasses. The problem of defending Wake was complicated by its peculiar geography. The visible top of an inactive underwater volcano, it is an atoll shaped like a V with the open end to the northwest. Each arm is about five miles long. Narrow channels cut off the tips of each arm so that there are actually three islands. The tiny island that is the end of the right arm is called Peale; the one at the end of the left arm, the southern side, is Wilkes. The main island is Wake. The land area of all three is only two and a half square miles, about that of New York City's Central Park.\n\nDevereux was faced with an almost bald defense position. The atoll's highest point was 21 feet and except for some dwarf trees and shrubbery there was no cover. He could use only four of his six 5-inch guns to defend the arm of the atoll now about to be invaded and all of these had been damaged partially by the bombings. Neither the two guns of Lieutenant Clarence Barninger at the apex of the V-shaped atoll, Peacock Point, nor the two of Lieutenant J. A. McAlister, on tiny Wilkes Island, had range finders.\n\nAs Devereux walked back to his dugout command post, which was about a mile inland of Peacock Point, he worked out his plan of defense. He telephoned McAlister and Barninger: \"Don't fire until I give the word.\" Then he called Major Putnam of Marine Fighting Squadron 211 who was at the narrow airstrip. This was about 400 yards south of Devereux and ran east-west for a little more than a mile along the southern arm of the atoll. \"How many planes do you have in commission?\" asked the soft-spoken Devereux.\n\n\"Four.\"\n\n\"Don't take off until I open fire. I'm trying to draw them in, and the planes would give the show away.\"\n\n\"Okay. Good luck.\"\n\nJust as the first faint daylight appeared, the _Yubari_ headed directly for the apex of the atoll. When it was about 7000 yards off Peacock Point, it swung to the west closely followed by the 4 destroyers. All 5 ships opened fire about 5:30 A.M.\n\nWhen _Yubari_ reached Wilkes, the end of the atoll's left arm, she reversed, coming in a few hundred yards closer, and continued the bombardment of the southern shore. At Peacock Point she again reversed and moved in to 6000 yards. When there still was no answering fire, Kajioka was convinced the Wake guns were harmless.\n\nOn Wake, Devereux's phone was clamoring. Barninger and McAlister kept begging for permission to fire. \"Under no circumstances fire until I give the word,\" he told his talker, Corporal Robert Brown.\n\n\"Hold your fire till the major gives the word,\" Brown relayed to the gunners.\n\n\"What does that little bastard want us to do?\" one answered. \"Let 'em run over us without even spitting back?\"\n\nBut the major still refused to give the word as the Japanese moved closer with each pass. At 6:10 A.M., the invaders were only 4500 yards from Barninger's guns on Peacock Point.\n\n\"Commence firing,\" ordered Devereux.\n\nThere was a roar from Barninger's battery. The first salvo went over. The _Yubari_ , stunned by the close miss, zigzagged out to sea. The next salvo from Barninger straddled the cruiser.\n\nDestroyers darted wildly, trying to lay a smoke screen to protect the flagship. The two guns from Wilkes opened fire on the destroyer _Hayate_. After the third salvo a great cloud of smoke covered the ship. When the smoke cleared there was no sign of the _Hayate_.\n\nMcAlister's gunners yelled and cheered.\n\n\"Knock it off, you bastards,\" shouted Sergeant Henry Bedell. \"Get back on the guns. What d'ya think this is, a ball game?\"\n\nBy 6:15 A.M., Putnam and three other Marine pilots were 15,000 feet above Wake, waiting for an expected attack by carrier planes.\n\n\"Well, it looks as if there are no Nips in the air,\" said Putnam on the two-way radio. \"Let's go down and join the party.\"\n\nPutnam's Wildcat broke through the overcast. Anti-aircraft fire burst all around but he kept coming. At 3500 feet he began a steep dive. At 1500 feet his two 100-pound fragmentation bombs, slung under the plane with homemade lugs, dropped. They missed a destroyer by 200 yards.\n\nPutnam dove on another destroyer. The crew had neglected to take down the glass from the bridge. Sun glistened as Putnam came head-on at bridge level. He got all four guns on the glass. The shattered splinters made a rainbow. Bombs and ammunition gone, Putnam headed back to reload.\n\nKajioka's force was wildly zigzagging trying to avoid the shells and bombs. A Wildcat swept down on the _Yubari_ , its machine guns raking the bridge and narrowly missing Kajioka himself. The admiral ordered withdrawal. The flotilla turned south, but Fighting Squadron 211 followed.\n\nAt 7:37 A.M. Putnam, now on his fourth flight, saw a destroyer, the _Kisaragi_ , lagging behind the main body. He hesitated, wondering if he should go after the cripple or follow the cruiser. Just as he decided to make a pass at the destroyer, the _Kisaragi_ suddenly became a ball of fire. Moments later, at 7:42 A.M., Putnam could see no wreckage below. Just empty, sparkling ocean. Although some thought he had sunk the ship, Putnam guessed a bomb dropped earlier by Captain Henry \"Baron\" Elrod had started a fire that finally reached the magazines.\n\nThe first battle for Wake was over. The southern horizon was clean of ships. Back at Devereux's command post all was quiet. Reports of casualties came in: three men slightly wounded by shrapnel. Devereux was puzzled. Why hadn't the Japanese brought air cover? Why hadn't they bombarded thoroughly before moving in? Why hadn't they sent assault troops ashore in landing craft?\n\n\"It's been quite a day, Major, hasn't it?\" said Corporal Brown to Devereux.\n\nThe little major nodded absently. He was already thinking of the next attack.\n\n**2** There was excitement that same morning in Manila. Rumors of a great battle at Lingayen Gulf the previous night raced through the city: the main Japanese landing had at last come but the 21st Division (Philippine Army) had sunk most of the invasion vessels; the bay was filled with floating bodies and the beaches strewn with Japanese dead.\n\nWhen _Life_ photographer Carl Mydans heard that the wild battle was still raging, he drove quickly to the battleground. Arriving at Lingayen Bay near noon he found no bodies on the beaches, no sign of battle. Except for Filipino soldiers resting beside their weapons, the great expanse of sand was empty.\n\n\"Looking for bodies?\" asked an American major, smiling.\n\n\"Well, that's what headquarters reported,\" said Mydans. \"I heard there was a big battle here.\"\n\nThe major explained that several dark shapes had been seen approaching the mouth of the Agno River the previous midnight. A battery of the 21st Field Artillery opened fire. A moment later Lingayen Gulf was ablaze as every gun in the area, from 155-mm. guns to pistols, opened fire.\n\nThe cause of \"The Battle of Lingayen Gulf\" was a small Japanese motorboat on a reconnaissance mission. Not hit, it reported that the main landing, actually not scheduled for eleven more days, should be made 30 or 40 miles farther north, where there were almost no beach defenses.\n\n\u2014\n\nTwo days later MacArthur received heartening news from Washington. At General Marshall's instigation, the _Pensacola_ convoy, originally bound for Manila and then turned back to Hawaii, was again being rerouted. Its desperately needed guns, planes and men were headed once more to the Philippines, this time by way of Brisbane, Australia.\n\nMacArthur, delighted at the change in atmosphere in Washington, immediately conferred with Admiral Hart. It was not a happy conference. The two men had been in conflict ever since MacArthur took command of the U.S. Army forces in the Far East the previous July. Their differences of opinion were sharpened by the sudden disasters of the past week. The admiral threw cold water on MacArthur's plan to bring the _Pensacola_ convoy up from Australia. The Japanese, he surmised, would throw a complete blockade around the Philippines before the convoy could leave Brisbane. Hart gave MacArthur the impression that the islands were doomed.\n\nNothing could have incensed the general more. Upset by what he considered defeatism, he wired Marshall:\n\n> IF THE WESTERN PACIFIC IS TO BE SAVED IT WILL HAVE TO BE SAVED HERE AND NOW.\n\nNo American knew the temper and ultimate capabilities of the Filipinos better than MacArthur. His father, General Arthur MacArthur, had been the first military governor of the Philippines. The son had served ten years in the islands, as a lieutenant, a brigadier general, commanding general of the Philippines Department and as military advisor to President Quezon.\n\nMacArthur was a highly controversial figure. Though Filipinos, almost to a man, regarded him as the greatest man alive, most of those who served under him had widely divergent opinions. Close associates, like Colonel Hugh Casey, his engineer, claimed MacArthur had a photographic mind that worked with the precision of a calculating machine; most men on the staff felt he was the outstanding military leader of his day. Detractors\u2014and most of these had little personal contact with the general\u2014insisted he was an egomaniac surrounded by an inferior staff whose main purpose was to build a legend around their chief; they ridiculed MacArthur's specially designed marshal's cap, his carefully combed but thinning hair, the clenched corncob pipe and the jutting jaw.\n\nMacArthur told Marshall that as soon as the Filipinos discovered the U.S. had abandoned hope, the entire structure would collapse around him. \"The Philippine theatre of operations,\" he radioed, \"is the locus of victory or defeat.\" He begged Washington to review its entire strategy and send him immediately air power to delay the Japanese advance. Saving the Philippines, he argued, would justify \"the diversion here of the entire output of air and other resources.\"\n\nThe next day, December 14, tension in the Philippines increased. A reported Japanese landing at Legaspi on the southern end of Luzon was confirmed. But MacArthur guessed correctly that this, like the previous landings on the northern end of the island, at Aparri and Vigan, was a minor probing. Again he held back, waiting for the main landing that was bound to come soon. \"The basic principle of handling troops,\" he told reporters, \"is to hold them intact until the enemy has committed himself in force.\"\n\nMany in Manila thought this main landing had already come, for the papers were still headlining eyewitness stories of the great conflict at Lingayen Gulf. The \"battle\" was even confirmed by Major LeGrande Diller, MacArthur's press chief. He gave reporters a release stating that an enemy landing had been attempted in the Lingayen area but was repulsed by a Philippine Army division.\n\nWhile other reporters were wiring their papers and magazines of the great victory, Carl Mydans buttonholed Diller. \"Pic,\" he protested, \"I've just been to Lingayen and there's no battle there.\"\n\nDiller pointed to the communiqu\u00e9 he had just read aloud. \"It says so here.\"\n\nThis, thought Mydans, was a startling indication of how poorly the war was being covered. He alone of all the reporters had gone to the scene of action. The others had stayed in town, satisfied to pass on press handouts. It wasn't the correspondents' fault primarily. There was no organization and little pressure from back home to get out and see what was really going on.\n\nThe Japanese were already laughing at a communiqu\u00e9 of December 12 claiming that Captain Colin P. Kelly, Jr., had put the battleship _Haruna_ out of commission during the landings at Aparri. On the day she was supposed to have been sunk, December 10, the _Haruna_ was 1500 miles away in the Gulf of Siam. There hadn't been a battleship near the Philippines. In fact nothing had been sunk or even heavily damaged near Aparri that day.\n\nThe Kelly story was already a news sensation, its facts becoming distorted with every telling. The most popular version, the one many Americans would retain through the years, was that Kelly won the Medal of Honor by diving his plane into the smokes ack of the _Haruna_ , becoming the first suicide pilot of the war.\n\nThe facts were far different. On the morning of December 10, a Japanese carrier was reported operating just north of Aparri. Two Fying Fortresses were sent from Clark Field to attack. One, piloted by Kelly, flew north and as it approached Aparri at 20,000 feet, its occupants continued north almost to Formosa but no carrier was sighted. It was almost noon when Kelly turned back toward the Philippines.\n\nWhen the B-17 approached Aparri, still at 20,000 feet, the crew saw a \"big ship\" just off the coast and was sure it was a battleship of the _Haruna_ class. The Americans were puzzled when the ship took no evasive action, only continuing its course up the coast. It didn't even fire its antiaircraft guns at the Flying Fortress.\n\nSince there wasn't a warship of any size in the entire Aparri area, it was probably a large transport, which explains why it didn't evade or fire ack-ack. Corporal Meyer Levin, the bombardier, using the touted Norden bombsight, began his run ten minutes from the target. The three 600-pound bombs were dropped in train. All those in the plane except Kelly looked from the windows, following the course of the bombs. They saw the first hit some fifty yards short. The second was a near miss alongside. The third appeared to go down the ship's smokestack. Smoke poured from the ship's stern. Though Kelly circled twice, the smoke became so thick, it was impossible to see the exact damage, but almost everyone aboard the B-17 was confident the ship had been sunk.\n\nThe plane, its crew in a celebrating mood, now sped south toward its home base, Clark Field. The rugged mountains of North Luzon were crossed and Kelly began to lower the big bomber. As Second Lieutenant Joe M. Bean, the navigator, leaned forward to check the altimeter, there was a burst of fire and his instrument panel suddenly shattered. The ship was under attack by a Zero fighter piloted by Saburo Sakai. The first round from Sakai's cannon had also wounded PFC Robert E. Altman and beheaded the radio operator, Staff Sergeant William J. Delahanty, who was manning the left waist gun.\n\nA moment later the left-wing gas tanks, which were not self-sealing, broke into flames. Sakai fired another burst with his cannon. This severed the B-17's elevator cables and the big plane headed into a steep dive.\n\nFlames spread rapidly and Kelly ordered his men to bail out. The bottom emergency hatch was pried partway open by Bean and Levin. The bombardier was the first man out. Then Sergeant James E. Halkyard, PFC Altman, PFC Willard L. Money, Bean and Second Lieutenant Donald D. Robins, the copilot, jumped. The ship suddenly blew up. A moment later it crashed on a dirt road about two miles west of Mt. Arayat, almost hitting a group of men from the 26th Cavalry led by Major (now Lieutenant General) Thomas J. H. Trapnell, trying to catch some reported Japanese paratroopers. Close by was the body of Colin Kelly. His chute was unopened. He had sacrificed his life so his crew could live.\n\nBy now Brereton's headquarters at Nielson Field was getting fragmentary reports of Kelly's bombing mission off Aparri. A heavily armored battleship had been badly hit and probably sunk. Captain Allison Ind excitedly raced to his office and opened a copy of _Jane's Fighting Ships_. While he was studying it the phone rang. It was Colonel Francis Brady, Brereton's chief of staff.\n\n\"It's either the _Haruna_ or her sister!\" said Brady. \"At least she must be one of that class.\"\n\nInd compared notes. He too was positive the victim had been of the _Haruna_ class. The news swept headquarters. Everyone was wildly exultant. At last American air power had asserted itself. The news was passed on to MacArthur's headquarters. Later the six survivors were closely questioned and a detailed report submitted to Brereton stating that Kelly's bombs had opened the seams of the battleship \"more on the starboard than the port side\" and forced it to be beached. There was no doubt. A Japanese battleship had been wrecked.\n\nWhen MacArthur read this report he accepted it at face value and issued the December 12 communiqu\u00e9: \"One spectacular instance was the feat of Captain Colin P. Kelly, Jr., of Madison, Florida, who successfully attacked the battleship _Haruna_ , putting that ship out of commission.\"\n\nAmerica now had its first great hero of World War II, for the story of the Irish pilot and Jewish bombardier was a copywriter's dream. Kelly's heroic action certainly deserved the posthumous D.S.C. he was awarded. But the fanfare was bad for the morale of other American air units whose pilots had accomplished more that same day over the Philippines\u2014and without a line of public acclaim. Major Emmett \"Rosie\" O'Donnell, Jr., and Lieutenants G. R. Montgomery and George Schaetzel all had more fruitful bombing missions and Lieutenant Sam \"The Rat\" Marrett's daring attack at Vigan in his outmoded P-35 was already a legend among American airmen.\n\nOn the morning of December 14, the story of \"The Battle of Lingayen Gulf\" broke in newspapers all over the United States. The New York Sunday _Times_ banner headline read: JAPANESE FORCES WIPED OUT IN WESTERN LUZON. Lingayen Gulf, reported the _Times_ , had been retaken from the Japanese in a great victory.\n\nThe United Press stated there had been a fierce three-day fight at Lingayen Beach. One hundred and fifty-four enemy boats had been blasted by the defenders. Not a one had reached shore.\n\nWhile American readers were being bolstered that Sunday by the paper victory in the Philippines, a recently promoted brigadier general, Dwight D. Eisenhower, was reporting to General Marshall in Washington. Two days previously Eisenhower had been chief of staff to the commanding general of the Third Army. He had no idea what his new assignment in Washington was going to be.\n\nMarshall outlined the situation in the Western Pacific. Then he abruptly asked, \"What should be our general line of action?\"\n\nEisenhower had spent four years in the Philippines as MacArthur's senior military assistant and he realized the problem was almost unlimited in its complexity. For the next few days he concentrated on Marshall's question. The facts as outlined were depressing. The Pacific Fleet didn't dare sail to the Philippines. Major reinforcements could be sent only when the Navy was rehabilitated. There was only one solution: set up Australia as a base for offensive operations.\n\nEisenhower reported back to the chief of staff on December 17. \"General,\" he said, \"it will be a long time before major reinforcements can go to the Philippines, longer than the garrison can hold out with any driblet assistance, if the enemy commits major forces to their reduction. But we must do everything for them that is humanly possible. The people of China, of the Philippines, of the Dutch East Indies will be watching us. They may excuse failure but they will not excuse abandonment. Their trust and friendship are important to us. Our base must be Australia....We must take great risks and spend any amount of money required.\"\n\n\"I agree with you,\" said Marshall simply and then added, \"Do your best to save them.\"\n\nThe previous day, Admiral Chester Nimitz, chief of the Bureau of Navigation, had entered the office of Secretary of Navy Knox. Nimitz had no idea why he'd been summoned. Like other Navy officers he had been stunned by Pearl Harbor but since then he had been so busy alleviating the troubles of survivors that he hadn't had much time to study the desperate situation in the Pacific.\n\nKnox, who had just returned from a personal inspection of Pearl Harbor, explained that Kimmel and his staff had, understandably, been badly shaken by the attack. So shaken that their recent estimate of the situation completely revised the strategy of a Pacific war. \"The loss of battleships,\" they wrote, \"commits us to the strategic defensive until our forces can be built up.\"\n\nA new commander must be sent to Hawaii, said Knox, one who could dispel the gloom and sense of fear that permeated Pearl Harbor and inspire a bold offensive.\n\nNimitz agreed. Defensive tactics never won a war.\n\n\"How soon can you get ready to travel?\" asked Knox. \"You're going to take command of the Pacific Fleet.\"\n\nNimitz was startled. It had never occurred to him that he would be the one to relieve Kimmel. There were twenty-eight flag officers senior to him. \"Sir,\" he said, \"first I'll have to get someone to relieve me. I want Rear Admiral Randall Jacobs.\"\n\n\"You can't have him; FDR doesn't like him.\"\n\n\"I have to have him.\"\n\n\"Get him.\"\n\nNimitz said Jacobs was presently at sea but he would send for him. As Nimitz walked out of Knox's office, he was surprised to see Admiral Jacobs in the corridor.\n\n\"Randall, where did you come from? You're just the person we're looking for. From now on you're chief of the Bureau of Navigation.\"\n\n**3** On the morning of December 13 the last Indian, Scotch and Canadian troops on the mainland of China had been evacuated across the narrow bay to the island of Hong Kong. The arrival of these defeated soldiers caused a near panic for few had realized how desperate the military position actually was. Life had been going on normally with shops open, buses running on schedule and night clubs filled to capacity.\n\nLater that morning, at nine o'clock, a launch bearing a white flag of truce put off from Kowloon, the mainland city just across the bay. It docked at Hong Kong's Victoria Pier and a Japanese officer, Lieutenant Colonel Tokuchi Tada, stepped out. He had a letter addressed to His Excellency the governor, from the commander of the Hong Kong Assault Force, Lieutenant General Takashi Sakai, advising the governor to surrender. This was, wrote Sakai, a humanitarian measure, taken to spare the lives of innocent noncombatants. If refused, severe artillery fire and aerial bombardment would follow.\n\nGovernor Young politely but categorically refused the offer. The siege of Hong Kong\u2014a rocky, mountainous island only half again larger than Manhattan\u2014was on.\n\nThe facts of the sudden fall of the vaunted defense line on the mainland\u2014its name, ironically, was Gin-drinkers Line\u2014were now common knowledge. The key point in the line, the Shingmun Redoubt, had been captured by a few Japanese wearing sneakers. They had merely crept up a mountain to the concrete redoubt, where the defenders were locked inside in apparent safety, and dropped hand grenades down the vents.\n\nOn December 14 the island's 10,000 troops were heartened by a message from Winston Churchill addressed to Governor Young and the \"Defenders of Hong Kong.\"\n\n> WE ARE ALL WATCHING DAY BY DAY AND HOUR BY HOUR YOUR STUBBORN DEFENSE OF THE PORT AND FORTRESS OF HONG KONG. YOU GUARD A LINK BETWEEN THE FAR EAST AND EUROPE LONG FAMOUS IN WORLD CIVILIZATION. WE ARE SURE THAT THE DEFENSE OF HONG KONG AGAINST BARBAROUS AND UNPROVOKED ATTACK WILL ADD A GLORIOUS PAGE TO BRITISH ANNALS.\n> \n> ALL OUR HEARTS ARE WITH YOU IN YOUR ORDEAL. EVERY DAY OF YOUR RESISTANCE BRINGS NEARER OUR CERTAIN FINAL VICTORY.\n\nThe following day morale rose sharply when the Colony's single-sheet newspaper issued a bulletin announcing that the 7th Chinese Army was only 30 miles from the border and would soon relieve Hong Kong.\n\nThat night a number of Japanese put off from Kowloon in junks, small rafts and rubber boats. But the British were ready. Four boats were sunk and the landing thwarted.\n\nThree days of almost intolerable tension followed. Then at dusk of December 18, the \"death watch\" ended as Japanese big guns began a heavy bombardment of the northeast coast of the island. Soon hundreds of launches and rubber boats quietly pushed off from the mainland. They came in three main groups, each a regiment of Major General Tadayoshi Sano's veteran 28th Division.\n\nOvercast skies and frequent showers hid the invading armada of small boats. As they neared the northeastern coast of Hong Kong, shells from the mainland hit oil tanks near North Point. Great black billows of smoke covered the water front, almost blinding the defenders.\n\nIn minutes the coastline between North Point and Lyemun\u2014the area closest to the mainland\u2014was alive with Japanese. The landing was so confusing and the fighting so deadly silent that few defenders realized Hong Kong had been invaded. In an hour most of the hill at Lyemun was in Japanese hands, yet 100 yards away the sentry at No. 2 Gun at Lyemun Fort reported, \"Nothing unusual has occurred.\"\n\nAt 10:00 P.M. the men of the fort heard someone shout, \"We Japanese have captured Saiwan Hill. It is useless to resist.\"\n\nBy dawn the situation was desperate. General Maltby, realizing the entire eastern half of the island was now in danger of being cut off, ordered the men who manned the great artillery pieces on D'Aguilar Peak to destroy their guns and retreat to Stanley Peninsula. Near panic hit the troops as they blew up their guns and then dashed down the narrow unprotected cliffside road toward Stanley. They left behind all their ammunition, chocolate, extra clothing, Christmas puddings and gin.\n\nAt 11:30 A.M. Governor Young visited Maltby in his underground headquarters, Battle Box. Shaken by the wild reports of sweeping Japanese victories, he told Maltby that no matter how bad the military outlook might be, the battle must be fought to the end. Every day gained was a direct help to the Empire war effort.\n\nBy noon the next day the Japanese had taken half of Hong Kong Island, penetrating to Repulse Bay on the southern side. Even so, General Sakai was disappointed. He had assumed that once landings had been made, the British would surrender. Instead the defenders, in spite of heavy losses, were fighting tenaciously for every foot they gave up.\n\nMaltby was still hopeful of aid from China. The previous day the Chinese had wirelessed that 60,000 troops were at the frontier, ready to attack. Maltby passed on this encouraging news to every unit. \"All ranks,\" he said, \"must therefore hold their positions at all costs and look forward to only a few more days of strain.\"\n\nBut at 4:00 P.M. the military attach\u00e9 in Chungking wired that the main Chinese attack could not start before January 1. Even the most optimistic at Battle Box now knew Hong Kong was doomed. The end would come any hour.\n\n**4** Early that same morning, December 20, it looked as if the \"death watch\" in the Philippines was also approaching its end. At 2:00 A.M., MacArthur's headquarters had received a report that a large convoy of 80 transports had been sighted by a submarine 40 miles north of Lingayen Gulf.\n\nLater that morning MacArthur learned more bad news. Before dawn 5000 assault troops in 14 Japanese transports, escorted by a cruiser squadron and the carrier _Ryujo_ , had approached Davao in southern Mindanao. A single Filipino machine gun squad of the 101st Division held up the storming of Davao until it was knocked out by a Japanese shell. By 3:00 P.M. the city and airfield were occupied. Before dark half of the assault troops were already being organized for the next step, the capture of nearby Jolo Island. Since Japanese troops were already in North Borneo, the capture of strategic Jolo would fulfill Admiral Hart's gloomy prediction: MacArthur would be completely cut off from Australia.\n\nThe next day, the submarine _Stingray_ saw the great Japanese convoy approaching Lingayen Gulf. When MacArthur received this report he knew this was the main invasion. He rushed tanks to the north and ordered Brereton to collect every bomber in the area for a morning strike at Lingayen. The beach defenders were warned to expect heavy landings momentarily.\n\nAlthough the Americans were forewarned, they were fooled. They had expected the Japanese to land near the mouth of the Agno River where the bulk of the artillery was emplaced. The 85 transports of the vast armada were planning to land about 40 miles up the coast where there were almost no defenses. At 1:00 A.M., December 22, the convoy was only a mile or two off the beaches of northern Lingayen Gulf.\n\nThe transports carried the main body of the Philippine invasion force, the 14th Army of Lieutenant General Masaharu Homma. The general, a heavy-set man almost six feet tall, was leader of the pro-British-American minority in the Army and was an open enemy of the German-trained General Tojo.\n\nHomma had spent eight years with the British, including service in France in 1918 with the B.E.F. After the fall of Nanking he had publicly stated that \"unless peace is achieved immediately it will be disastrous.\" Later when the Koiso Cabinet was formed he begged General Koiso to stop the war. The enmity between Tojo and Homma reached its climax when General Muto asked Homma what he thought of Tojo as a possible minister of war. Homma had objected strenuously. Now Tojo was not only head of the Army but Japan's prime minister.\n\nFew of Homma's men knew where they were. Five days before they had been secretly loaded at Formosa and the Pescadores amidst confusion and uncertainty. Only a few officers knew they were going to the Philippines and these were given only the vaguest instructions.\n\nHomma had been warned by the Imperial General Headquarters to expect bombings. As the days passed and the convoy neared its goal without being attacked neither Homma nor his staff could believe their luck. Where was the United States air force, where the Navy?\n\nAt 1:10 A.M., the pilot ship dropped anchor. Too late the skipper discovered he had been fooled by the darkness and was eight miles south of his intended goal. By this time the long line of transports behind had also begun to weigh anchor in a 15-mile line to the north. Now every landing craft would have to travel an extra eight miles to its designated beach.\n\nAt 2:00 A.M., the first of the 43,110 men of the 14th Army started going over the sides of the lead transport. Homma nervously waited for beach batteries to open up, but the only noise was the banging of landing craft against the sides of the transports. The high seas almost swamped the first boats, tossing them about like chips.\n\nIt wasn't until 4:30 A.M. that two battalions of infantry and one battalion of mountain artillery were loaded and ready to assault the beaches. Forty-seven minutes later the first boat hit the beach just below Agoo. It couldn't report because its radio, like every other radio coming ashore, had been ruined by the salt water. Already General Homma was out of communication with his men.\n\nMany of the landing craft, tossed high on the beach by the roaring breakers, were overturned. Some hit the beaches so hard they were out of commission for the rest of the day. The landing schedule was soon such a shambles that Homma ordered the heavy artillery and other heavy units to wait for calmer weather.\n\nFortunately for Homma not a single rifle shot welcomed the first invaders at Agoo. Not a single plane appeared overhead to drop a bomb. In spite of this incredible luck, he was consumed with worry. What if a pack of submarines was suddenly loosed among his 85 transports?\n\nFive American submarines were now roaming the Lingayen waters. At 6:00 A.M. the S-38, an old-type submarine, raised its periscope and found it was surrounded by dozens of Japanese transports. They looked to the skipper, Lieutenant Wreford \"Moon\" Chapple, like cars in a parking lot.\n\nAdmiral Hart had warned submarine commanders at the outbreak of the war not to be foolhardy. \"We have no idea what Japanese anti-sub capabilities are now. It will be a long war with plenty of glory for all.\" But Chapple, formerly a champion boxer at Annapolis, decided this was no time for caution. He slipped past escorting destroyers and carefully lined up four targets. The first torpedo missed. So did the next three. Assuming he'd misjudged the draft of the transports, Chapple ordered the submarine to submerge for a reload. He didn't know that all Mark 10 torpedoes ran four feet deeper than set and some were not even detonating on contact. Due to pre-war economy the submarine commanders had never been allowed to test-fire these torpedoes. At 7:58 A.M., though now chased by destroyers, Chapple moved in close to another transport. He fired two torpedoes. Thirty seconds later the submarine rocked from an explosion. He had hit the _Hayo Maru_. In minutes the 5445-ton transport disappeared. Only one other Japanese ship had been sunk by American submarines since Pearl Harbor.\n\nThe few planes left in Brereton's air force did even less damage. Not a bomber could respond to his call for a morning strike. Most were in Australia. A few were now bombing Davao in southern Mindanao, which was already in Japanese hands. Several P-40 fighters strafed the transports for a few minutes but soon the sky was full of Zeros. In the brief attack, Lieutenant \"Buzz\" Wagner, who had become America's first ace of World War II a few days previously, was knocked out of action when a shell hit his wind screen putting a glass splinter in his eye.\n\nBy mid-morning Homma's first wave landed. The only opposition along the entire 15-mile beach area came at the north end, near Bauang. Here, a Filipino battalion of the 11th Division fought well, holding off the invaders with several machine guns. But despite heavy casualties the Japanese pushed ashore, drove off the Filipinos and headed north to meet the 4000 men who had landed in northern Luzon on December 10 and were now moving south down the coast. At 11:00 A.M. contact was made. The beachhead was consolidated.\n\nIn Manila, General MacArthur was anxiously awaiting news from Lingayen Gulf. The next few hours were the critical ones. His desperate need was air and naval support. He radioed Marshall asking for any inkling of the strategic plans for the Pacific Fleet, suggesting that carriers could bring fighter planes within radius of the Philippines:\n\n> CAN I EXPECT ANYTHING ALONG THAT LINE?\n\nMarshall replied that the Navy insisted it was impossible. As for planes, he would have to rely on those in the _Pensacola_ convoy, just then docking at Brisbane.\n\nBy 5:00 P.M. Homma, still on an anchored transport, could see heavy surf and high seas raising havoc all along the beaches. The infantry and half the tanks had been landed, but as yet they had almost no artillery support. For all he knew the men ashore could be walking into a trap. Even so, he decided to adhere to the original plan. He ordered them to advance south down Route 3, the paved highway which ran along the beach.\n\nBoldness, as usual in battle, paid off. By dark the Japanese had pushed south five miles, easily routing Filipinos of the 11th Division and a platoon of light tanks. Now they headed due east up a hill. At the top of the rise were less than 500 experienced Philippine Scouts of the elite 26th Cavalry.\n\nAt 8:00 P.M. the tankers supporting the Scouts claimed they had orders to fall back. This unit, Company C, 192d Tank Battalion, began to retreat east down the hill toward the town of Rosario. A moment after they left, Japanese tanks came out of the darkness from the opposite direction. The mounted Scouts, thinking they were also American, let them pass through their lines. Suddenly the tanks began to fire into the mass of horses, mules and men.\n\nRiderless horses, mules and men raced down the steep slippery highway in panic. Major Thomas Trapnell tried to stop the chaotic rout but it was impossible. He ran to the bottom of the hill and set fire to a wooden bridge. For several hours, at least, the Japanese advance was halted.\n\n**5** At 10:30 P.M. on that same night, Major Devereux was standing on top of his Wake command post, a half-buried concrete igloo. He was watching strange flashes in the sky far to the northwest. At first he thought they were signals but when they increased in number and intensity, he decided a naval battle must be going on.\n\nMen all over the three islets that made up the Wake atoll were watching and arguing. Most guessed it was Task Force 14, the relief force that was coming to bring reinforcements and evacuate the 1200 civilians. Though everyone was exhausted from the almost daily bombings since the first invasion attempt, though Devereux's firepower had been bombed into half its effectiveness, the flashing lights kindled hope and enthusiasm. Help was coming.\n\nThe lights came from Japanese ships. A special unit, including 2 carriers, had been detached from Admiral Nagumo's returning Pearl Harbor Striking Force to support the second invasion of Wake. Planes from these carriers had shot down the last of Putnam's Wildcats earlier that day.\n\nAt that moment Wake was almost completely ringed by Japanese warships. Ten miles to the south 9 blacked-out ships were slowly approaching the lee side of the atoll. Admiral Kajioka was determined to regain face lost when he had been forced to turn back from Wake on December 10. About 150 miles to the east, also in support of Kajioka, was a third Japanese force: 4 heavy cruisers, 2 light cruisers and 2 destroyers.\n\nThe American relief force so eagerly awaited by the men of Wake was still some 500 miles east of Wake. On his flagship, the carrier _Saratoga_ , Admiral Frank Jack Fletcher, commander of Task Force 14, was just finishing one of the most frustrating and exasperating days of his long career. A Medal of Honor winner at Vera Cruz in 1914 and destroyer commander in World War I, Fletcher was known throughout the fleet as a man who could be both genial and salty. Today he was salty. Because of rough seas he had been able to refuel only four destroyers since morning. Several tow lines had parted in the long cross swell. Seven fuel lines had broken.\n\nMore important, Vice Admiral William Pye, who was taking Kimmel's place as fleet commander until Nimitz's arrival in Pearl Harbor, couldn't seem to make up his mind. First Pye had sent word to proceed to within 200 miles of Wake and launch planes to search out and attack the enemy. This order was countermanded and the seaplane tender _Tangier_ was sent ahead to evacuate Wake's personnel. Then that order too had been countermanded. Fletcher wondered what was going on. Fleet Headquarters knew a lot more than he did. Perhaps the great force that had laid waste to Pearl Harbor was closing in to attack him. Looking out from the admiral's sea cabin he could see almost nothing that dark night\u2014nothing but a threatening enemy sea.\n\nThe two and a half square miles of Wake's coral rubble were in the minds of most Americans that day. With little else to be thankful for in the vast Pacific battlefield, the island had become a symbol of United States resistance. Here a handful of Marines were showing the Japanese how Americans could fight. Here were men who, when asked what they needed, had answered, \"Send us more Japs.\"\n\nWhen the men of Wake heard a news commentator from America eloquently embellish the story of this heroic remark, they jeered. Devereux swore he hadn't sent the message; nor had Commander Winfield Scott Cunningham, the island commander. No one on that bleak pile of rubble would ever have sent such a message. The general feeling was: send our share of Japs somewhere else.\n\nThe famous words were not, as many later thought, the brain child of some public relations officer. They were sent; but only as nonsensical padding in a coded report. The sender never thought any intelligent decoder would possibly take it at face value.\n\nAt 11:00 P.M. a rain squall swept past and the flashes in the northwest grew dim, then faded completely. As Devereux climbed down from the top of his igloo, the roar of the surf sounded like artillery. He hurried into his command post, lay down on his cot and was almost immediately asleep.\n\nBy now Kajioka's invasion fleet was only five miles to the south. First came three destroyers, then the light cruiser _Yubari_ , with Admiral Kajioka aboard. At 11:30 P.M. Patrol Craft 32 and 33 slid past the _Yubari_ and slowly headed for Wake. These were old destroyers, rebuilt into troop transports. They were to be purposely grounded as close to shore as possible. Aboard, assault sailors were given last-minute instructions.\n\nAn hour later, at 12:30 A.M., December 23, the first of Kajioka's 830 assault troops, in PC 32, neared a reef almost in the middle of the southern arm of the atoll's V.\n\n\"Shore ahead,\" shouted the ship's commander. The men dropped to the deck. There was a loud crunching sound as the transport piled up on the reef. Only one landing boat could be lowered from the grounded ship because of the turbulent surf. Eighty men piled in. At 12:40 A.M. it beached at Peacock Point, the apex of the V. The men slowly crept across the coral beach. To their amazement not a shot greeted them.\n\nTen minutes later the other transport, PC 33, crashed onto the reef not far to the left of PC 32. From it two landing boats were lowered but no one could board them because of the crashing waves.\n\nBy now a destroyer two miles southwest of Wake was also lowering two boats, each carrying 80 invaders. These 160 men were to storm Wilkes, the tiny island at the extreme end of the atoll's southern arm, but halfway to the shore one of these boats lost its bearings and drifted to its right toward the main island of Wake.\n\nIt was now almost 1:00 A.M. and the defenders still were unaware that the second invasion was under way.\n\n# 7\n\n# _End of Wake_\n\n**1** At Devereux's igloo command post, Corporal Robert Brown was sleepily listening to the air raid warning phone. This was a network kept open for simultaneous communication among all positions on the atoll. Just before 1:00 A.M., December 23, Brown sat up, startled. He shook the sleeping Devereux. \"The enemy are reported on Toki Point, sir,\" he said excitedly.\n\nDevereux was instantly wide-awake. Toki was the extreme point of the atoll's right arm, the end of Peale Island. \"Any confirmation?\" he asked.\n\n\"No, sir.\"\n\nDevereux now phoned Lieutenant Woodrow Kessler at Toki Point. \"Any boats beached?\"\n\n\"Negative,\" answered the commander of Toki's two 5-inch guns.\n\nDevereux thought any landing on Toki would be just a feint. The main landing would probably come from the opposite direction, the south, since it was the lee side. This was where he'd placed his heaviest defenses. Even so the four-and-a-half-mile-long stretch was held by only about 200 Marine, Navy and civilian infantrymen. The rest of Devereux's entire defense force, about 250 Marines and 100 volunteers, were manning anti-aircraft and coastal guns. It was not a reassuring prospect.\n\nOf the 200 infantrymen defending the atoll's southern arm only 70 were located on Wilkes. At 1:15 A.M. Gunner Clarence McKinstry, commander of the 3-inch battery on the tiny island, thought he heard something off shore. McKinstry was a big man\u2014six foot three, weighing 260 pounds\u2014and wore a long red beard and bristling mustache. He told his men to pipe down. He listened again. In spite of the booming surf he heard a motor turning over. He called the over-all commander of Wilkes, Captain H. M. \"Cutie\" Platt, an easy-going man from South Carolina.\n\n\"Are you sure it's a motor, Mac?\" asked Platt. McKinstry was sure. \"Then fire.\"\n\nAt 1:20 A.M. McKinstry's .50-caliber machine guns fired. Tracer bullets lit up a boat coming into the beach. Two volunteers, Sergeant Bedell and PFC William Buehler, ran across the rough, coral-strewn beach. As the craft grated in the sand, the two men lobbed grenades among its occupants.\n\n\u2014\n\nA mile north of Peacock Point, the apex of the atoll's V, Devereux was getting a dozen reports at his igloo command post as Japanese began scrambling ashore at four places on the main island.\n\nBy now the 5-inch guns couldn't be lowered enough to shoot at the grounded landing craft. But a mile west of Peacock Point there was an unmanned 3-inch anti-aircraft gun on the dunes just above the beach. Lieutenant R. N. Hanna, commander of the machine guns at the airstrip, volunteered to man this gun. Corporal Ralph Holewinski and three civilians followed him as he ran south a quarter of a mile through the rough underbrush toward the beach.\n\nJapanese sailors were beginning to wade ashore from the two grounded ex-destroyers by the time Hanna and his crew reached the gun. There were no sights and Hanna had to peer through the bore to line up one of these ships. The first shot hit the bridge of PC 33. Flames shot up, momentarily lighting the area.\n\nLess than a mile northeast of Hanna, at the airstrip, Major Putnam was talking on the phone to Devereux. The commander of Fighting Squadron 211 was ordered to form a protective infantry line between Hanna's gun and the beach. As Putnam's 11 officers and 11 men started from the airstrip, 22 civilians who had been helping repair the Wildcats fell in.\n\nThese were part of the few hundred construction workers who, with their supervisor, Dan Teters, had volunteered for any job since the first bombing. The majority of civilians had done little but sit in tunnels or hide in the brush, only coming out at night to scavenge for food.\n\n\"If you're captured in combat your chances are mighty poor,\" Putnam told the 22 workers. \"You can't go with us.\"\n\nJohn P. Sorenson, leader of this group of civilians, pushed forward. Although twenty years older than Putnam he was bigger and more rugged. \"Major,\" he said, \"do you think you're really big enough to make us stay behind?\"\n\n\"I'm proud of you,\" said Putnam. \"I'd be glad to have you as Marines. But take off. Join the other civilians.\" As Putnam and his men zigzagged through the brush toward Hanna's gun, Sorenson and his men followed, loaded with extra ammunition. The wind and surf were so loud, Putnam could hear nothing else as he placed his men in a skirmish line in front of Hanna's gun and then along the dunes a hundred yards toward Peacock Point. He sat at the angle of his line and waited tensely. Where were the Japanese?\n\nEighty enemy sailors were a mile to his left near Peacock Point. Still unseen, they were creeping north toward Devereux's command post. They wondered where the Americans were. Not a shot had yet been fired at them.\n\nThe main body, however, was concentrated on a mile-long stretch of beach just to Putnam's right. Here were the remaining 550 Japanese sailors from the two grounded landing craft and the 80 who had planned to land on Wilkes but drifted to the east.\n\nDevereux, guessing this would be the main landing point, walked over to Lieutenant Arthur Poindexter, who was standing by with two truckloads of men\u201420 Marines and 14 civilians\u2014on the main road. These 34 men comprised Devereux's entire mobile force.\n\n\"Take your people and set up between Camp One and the west end of the airstrip, Art,\" ordered Devereux. Camp One was about three miles west of Peacock Point just short of the narrow channel between the main island and Wilkes.\n\nThe two trucks rushed west, skirting the airstrip, as fast as they could move in the dark. Young Poindexter stopped his trucks at the western end of the airfield about half a mile to Putnam's right and set up two .30-caliber machine guns on each side of the road. He put the rest of his men in a skirmish line facing southeast.\n\nAlthough a few Japanese mortar shells were falling, Poindexter couldn't determine where they were coming from. He told his men to hold fire until they had a definite target. He felt rising excitement for he was one of those rare individuals who liked to fight.\n\nSuddenly firing broke out to Poindexter's right along the shore south of the water tower near Camp One. Leaving Sergeant T. Q. Wade in charge of the line, Poindexter jumped into a truck and hurried farther west. The four machine guns on the beach near the water tower were manned by inexperienced sailors. He figured someone had opened fire at a shadow and everyone had joined in.\n\n\"Hold up,\" he called to the nearest gunner. Then Poindexter saw there were real targets. Two large landing craft, apparently barges, had run aground a few hundred yards up the beach. The first was the boat which had drifted east from Wilkes. The second was PC 33, one of the remodeled transports.\n\nBy this time, probably, both were empty, but Poindexter had no way of knowing this. He told the gunners to fire again. Tracers splattered against the ships and then ricocheted off. Only grenades would be effective. He quickly selected Boatswain's Mate First Class Barnes, Mess Sergeant Gerald Carr and a civilian, Raymond R. Rutledge, an officer in World War I.\n\n\"Cap,\" Poindexter told the civilian, \"take Carr and get one barge. We'll get the other.\"\n\nThe four men, each armed with six grenades, ran from the brush to the water's edge. They pitched their grenades at the grounded vessels. All fell short. They ran back to cover. Poindexter ordered the four machine guns to open fire again. After a few bursts he told the gunners to suspend fire.\n\n\"Let's try again and get closer this time,\" he said. Rearmed with grenades he ran to the beach. The other three followed. Several veteran Marines watching couldn't make up their minds about Poindexter. They figured he was either \"crazy as a bedbug or the bravest guy alive.\"\n\nThis time Poindexter and Barnes waded into the water. After 10 yards they reached a coral reef and lobbed grenades into the first landing craft.\n\nSuddenly the area was brilliantly lit by a spotlight from the right, and Poindexter could see that no one was coming over the sides of either vessel. In fact, he couldn't see a single Japanese anywhere. As he and Barnes scampered back to the machine gun nests, the light abruptly went off.\n\nThe illumination had come from an American spotlight on Wilkes Island and only hit the two grounded craft near Poindexter by chance. Captain Platt had ordered it turned on his own beach, in front of McKinstry's 3-inch guns.\n\nThe light was shot out by the Wilkes Island invaders in less than a minute. But in that time McKinstry and his men saw a landing craft beached right in front of them. Japanese with fixed bayonets poured over the sides. These 80 men were the only ones to reach Wilkes of the 160 scheduled for landing.\n\nMcKinstry didn't have riflemen to stop the attackers. \"Cut the fuses at zero,\" he shouted. He looked through the barrel of a 3-inch gun and fired point-blank. The attackers fell back but then started flanking McKinstry from both sides. As grenades began to fall in his position, McKinstry decided to withdraw and set up a new skirmish line. He told his men to remove the firing locks from the big guns and follow him. He headed east, toward the main island of Wake, through the rough underbrush. The beach here was empty. There was nothing to do but wait.\n\nIt was now 2:30 A.M. A moment after the searchlight on Wilkes was shot out, Devereux had lost communications with that isle. He knew little more about the real situation on Wake itself. At Peacock Point, Barninger reported machine-gun fire sweeping his position from a distance but he had no targets for his 5-inch guns. Then there was silence.\n\nPoindexter near Camp One reported he was being flanked. Then silence. There was no word from Putnam.\n\nDevereux ordered his executive, Major George Potter, to set up a last-ditch defense line 100 yards south of the igloo command post. Potter scraped together thirty clerks and communications men and dug them in across the road leading to the airfield.\n\n\u2014\n\nAbout four miles to the North, near the channel separating Peale and the main island, Commander Cunningham was sending a message to Admiral Pye:\n\n> ENEMY APPARENTLY LANDING.\n\nThen he sent a message to the submarine _Triton_ to attack enemy ships south of Wake.\n\nAt 3:19 A.M. Cunningham got an answer. It came from Pye.\n\n> NO FRIENDLY VESSELS IN YOUR VICINITY NOR WILL BE WITHIN THE NEXT 24 HOURS.\n\nBy 4:00 A.M. rain was lashing the three islands. The fighting had become completely confused. No one, Japanese or American, knew exactly what was going on. Poindexter and the defenders of Camp One were cut off. Mortar shells were falling all along the line he had set up at the edge of the airstrip. Several civilians had been killed, half a dozen Marines wounded.\n\n\"They're all over, on all sides,\" Sergeant Wade whispered to him. Poindexter ordered the men to withdraw west toward Camp One and the Wake-Wilkes channel by sections.\n\n\u2014\n\nWhen Peale Island reported all quiet, Devereux ordered all their available men to hurry south by truck to back up Potter's defense line. Just then a terrified civilian staggered in. One of Poindexter's men, he had seen Japanese bayoneting the machine-gun crews at the edge of the airstrip. \"They're killing 'em all!\" he cried hysterically.\n\nDevereux looked at his watch. It would soon be dawn. He wondered if the men from Peale would arrive in time. Corporal Brown, monitoring the air raid warning network, said, \"I've got something, sir.\" Someone was whispering over and over into a phone, \"There are Japanese in the bushes.\"\n\nSomeone else on the network suddenly broke in, \"For Christ's sake, where are you?\"\n\n\"There are Japanese in the bushes. Definitely. They are definitely Japanese.\" Suddenly there was a burst of sound, then silence.\n\n\"I guess they got him,\" Brown said.\n\nAt 5:00 A.M. Cunningham, after hearing a report on the general situation from Devereux, sent another message to Pearl Harbor:\n\n> ENEMY ON ISLAND\u2014ISSUE IN DOUBT.\n\n**2** At Pearl Harbor Admiral Pye was conferring with Rear Admiral M. F. Draemel and Captain C. H. McMorris. After much discussion all three concluded it was too late to evacuate or relieve Wake. The question now was: Should Fletcher be allowed to push west with Task Force 14 and engage the enemy assault force?\n\nA little later the answer to this question was radioed to the _Saratoga_. When Admiral Fletcher read Pye's order he said, \"We're called back to Pearl Harbor.\"\n\nThere was open dismay and indignation on the bridge. Several staff officers urged Fletcher to disobey. When he refused, the talk became so bitter that Rear Admiral Aubrey Fitch, who sympathized with the malcontents, left the bridge. Marine fliers in the ready room cursed upon learning the news; some wept.\n\nFletcher ignored the comments. He wasn't running a popularity contest. He was as eager to go in as the others\u2014but the only time to disobey orders was when you knew more than headquarters. He didn't know a damn thing. Apparently Pye did know something.\n\nPye had been governed not by special information but by overcaution, a common trait in Hawaii since the Pearl Harbor disaster. If Fletcher had been allowed to continue west he would easily have caught Kajioka's invasion fleet lying off Wake. Unprotected by carrier aircraft these ships would have been sitting ducks for the _Saratoga_ planes.\n\nAt the Officers' Club in Pearl Harbor there was bitter talk about the recall of Task Force 14. It wasn't helped when Tokyo Rose jibed, \"Where, oh where, is the United States Navy?\"\n\nPye was radioing an explanation to Admiral Stark:\n\n> THE USE OF OFFENSIVE ACTION TO RELIEVE WAKE HAD BEEN MY INTENTION AND DESIRE. BUT WHEN THE ENEMY HAD ONCE LANDED ON THE ISLAND, THE GENERAL STRATEGIC SITUATION TOOK PRECEDENCE AND THE CONSERVATION OF OUR NAVAL FORCES BECAME THE FIRST CONSIDERATION. I ORDERED THE RETIREMENT WITH EXTREME REGRET.\n\n**3** Putnam was on his stomach under a bush. His muscles were stiff from the long, tense wait. So far he hadn't seen anything. Suddenly at first light of day two silhouettes loomed over him. He emptied his pistol. One Japanese fell sideways, another forward, his helmet colliding with Putnam's. The American hauled the two Japanese toward his position. Both were dead. Now firing broke out to the right. It was a wild, hand-to-hand melee. His thin line slowly fell back toward the gun they were protecting. More Japanese jumped out of the bushes. Putnam saw the civilian Sorenson leap up to meet them, throwing rocks until he was cut down. Then Captain Hank \"Baron\" Elrod, Putnam's hottest pilot and a wildman in the air, jumped up shouting, \"Kill the sons of bitches!\" He raked the invaders with his tommy gun. The attack was stopped momentarily.\n\nThe Japanese re-formed and pressed forward again. Putnam told his men to fall back all the way to the 3-inch gun. He and four others made it. There they found Lieutenant Hanna and two men. As mortar shells began to fall, they all hid under the gun's perforated steel platform. A bullet creased Putnam's cheek and went through his neck. He didn't realize he'd been hit, only wondered why he was suddenly so sleepy. He passed out. A moment later he came to, cursed violently and passed out again.\n\nAt 6:00 A.M. Kessler called Devereux from Toki Point on Peale. He said he could see three destroyers closing in on Wilkes. They were within range. Devereux told him to fire, then called Cunningham, informing him Jap flags had been seen all over Wilkes. The island had evidently fallen, and there was still no word from Putnam or Poindexter on the southern part of the main island. Japanese were already firing on Potter's defense line near the igloo and were less than 300 yards away.\n\n\u2014\n\nAt 7:30 A.M. Devereux again called the island commander. \"It looks to me,\" he told Cunningham, \"as though the Japs have secured Wilkes, Camp One, the channel, the airstrip and probably Peacock Point. That's the best I can judge the picture on the dope I've got.\"\n\nThere was a pause and then came the reply, \"Well, I guess we'd better give it to them.\"\n\nDevereux hadn't thought of surrender. \"Isn't any help coming?\"\n\nNot even submarines were coming.\n\n\"Let me see if there isn't something I can do down here,\" said Devereux. He asked for more riflemen, but Cunningham had only the five Army communicators.\n\nDevereux, too, decided the situation was hopeless. To hold out any longer would only be a waste of lives. He said tightly, \"I'll pass the word.\"\n\n\u2014\n\nBy now most of the Japanese on Wilkes were concentrated at McKinstry's 3-inch guns. Gunner McKinstry was trying to retake this position and had already started back west with his men. Several hundred yards from the guns he met Lieutenant McAlister. Combining their thirty men they began to move toward the entrenched Japanese. But the advance was soon stopped by three invaders hiding behind a large rock.\n\n\"Clean out behind that big rock,\" ordered McAlister. As the big red-bearded McKinstry started forward, the lieutenant said, \"No, detail a man for the job.\"\n\n\"I've got 'em, Gunner,\" said Corporal Halstead, a small, agile young man. Halstead quickly, quietly ran up the rock. He emptied his Springfield into the three surprised Japanese crouching below him. Without a word, he waved for his comrades to follow. When they saw the three dead bodies behind the rock, their spirits rose.\n\nMcAlister and McKinstry decided to split the attack. The lieutenant took his small detail down the beach. McKinstry formed a skirmish line with his twenty-five men and pushed straight forward through the brush. The gunner led the way throwing grenades as fast as he could and shouting in a bull voice to the others to keep their flanks up. Two civilians kept feeding grenades to the big man as he plowed forward. Seeing a supposedly dead Japanese suddenly rise and try to bayonet one of his men in the back, McKinstry shouted a warning. The \"dead\" man was shot. \"Be sure the dead ones are dead,\" he yelled.\n\nCaptain Platt, with two machine-gun crews and eight riflemen, was attacking the 3-inch gun position from the opposite side. But McAlister and McKinstry didn't know this. By now McKinstry's right flank had run into a machine-gun nest. \"Get going,\" he bellowed when he saw his men pinned down. \"Move on! You don't want to stay here and die of old age.\" The right flank moved up.\n\nNow McKinstry could see a group of Japanese in his old ammunition hole. He threw a grenade. A fire blazed and then ammunition exploded. The gunner rushed forward, Halstead at his side. There was a wild burst of fire from the 3-inch battery emplacement. Halstead dropped dead at McKinstry's feet.\n\nCorporal Lee started around the first gun. A \"dead\" Japanese suddenly leaped to his feet. Lee and the Japanese lunged at each other with bayonets. Both missed, pulled back and lunged again. This time both found the mark, both were dead.\n\nMcKinstry lobbed a grenade into the emplacement. There was an explosion. After a momentary silence, McKinstry saw a figure rise on the other side of the battery. It was Captain Platt, the irrepressible South Carolinian, grinning as usual. \"Mac,\" he called out, \"that ugly voice of yours sounded like an angel's when you came through.\"\n\nAs the three American forces met at the battery position, five Japanese escaped in the brush. The rest were dead.\n\n\"Bring back one alive, Mac,\" shouted Platt to McKinstry, who was already starting after the escapees. Suddenly dive bombers from the two Japanese carriers northwest of Wake swept down on Wilkes. Platt's men hugged the almost bare ground. It was 7:40 A.M. At the 3-inch position, one of McKinstry's men climbed up to pull down a Japanese flag. \"Leave it there a minute,\" shouted someone.\n\nJust across the channel, on the main island, Poindexter was still holding the Camp One area at 8:00 A.M. His line of thirty-four riflemen and ten machine guns was in a semicircle facing seaward and to the southeast. Like Platt, he too had long since lost contact with Devereux but the young lieutenant was feeling hopeful about the situation. His line had held so well the Japanese were only probing occasionally. He decided it was time to counterattack east toward Peacock Point. He called a runner, telling him to go back and round up every able-bodied man at Camp One who had a weapon.\n\nA few minutes later the runner dove into the shelter at Camp One. It was crammed with a searchlight section, a few truck drivers, supply men and the gun-shed crew. \"The lieutenant says for everyone with a rifle to move up to the line on the other side of the water tower,\" shouted the runner. No one moved.\n\nSuddenly a small eighteen-year-old with glasses, a clerk named Fish, jumped up and yelled, \"What are we sitting on our asses here for?\" He grabbed his rifle and charged out of the shelter; the others lumbered dazedly after him.\n\nAt 8:30 A.M. Poindexter turned and saw young Fish scampering toward him across the coral. He was followed by some twenty other reinforcements. Poindexter took a quick count. He had about fifty-five armed men. More than enough. He quickly formed a platoon of three squads. \"All right,\" he said after issuing orders for the attack, \"let's move out.\" He squirmed forward through the thick brush.\n\n\u2014\n\nOutside Devereux's igloo, Gunner John Hamas was shouting to cease fire. \"We're surrendering! Major's orders.\"\n\nDevereux, thin-faced, haggard, appeared in the doorway of his command post. \"It's not my order, God damn it!\"\n\nHe was called to the phone. The line to Barninger's battery on Peacock Point had finally been repaired. Barninger reported he was ready for attack.\n\n\"Cease firing,\" said Devereux. \"Destroy all weapons. The island is being surrendered.\"\n\nThe order was passed along Potter's line and to the gun positions not yet overrun. Men took bolts from their rifles and threw them away. Blankets were crammed into muzzles and guns fired. Cables were cut, directors and height-finders shot with pistols. Weapons destroyed, the men now brought out the little food they had been hoarding. They sat and ate and waited for the Japanese.\n\nSergeant Bernard Ketner came into Devereux's command post. Seeing the major sitting dejectedly before the phone, he held out his hand. \"Don't worry, Major,\" he said. \"You fought a good fight and did all you could.\"\n\nThey shook hands.\n\nDevereux told his phone man to call the hospital, located in another concrete igloo, about 400 yards toward the Japanese. Dr. Kahn was to raise a white flag and surrender to an officer. But the hospital didn't answer.\n\n\"Rig a white flag you can carry,\" said Devereux. \"We'll have to go down there.\"\n\nSergeant Donald Malleck tied a white rag on a swab handle. Then he and Devereux started down the road. Near the hospital igloo a Japanese rifleman slowly rose and cautiously came forward. Devereux was brought to the steel door of the hospital.\n\n\"Well, we are surrendering,\" said Devereux to an officer wearing a sword.\n\nThe Japanese offered the major a cigarette and told him he had attended the San Francisco Fair in 1939. Both heard an excited shout. A United States truck had been stopped on the road. Commander Cunningham, dressed in formal blue uniform, came forward.\n\n\"Who Number One?\" asked the Japanese officer.\n\nDevereux pointed to Cunningham. \"While you arrange the formal surrender,\" he told the island commander, \"I'll go around the island with Malleck to be sure that everybody gets the word to surrender.\"\n\nAt 9:30 A.M. the surrender party reached Hanna's gun. Devereux pushed through the crowds of Japanese surrounding the gun and climbed onto a revetment. \"This is Major Devereux,\" he shouted. \"The island has been surrendered.\" When no one answered, Devereux called again. A few men staggered out. Of the thirteen defending the gun, only Captain Tharin was not wounded.\n\nPutnam was awakened. Weak from loss of blood and still unaware he'd been wounded, he looked up at Devereux, his face a red smear. \"Jimmy, I'm sorry. Poor Hank is dead.\" Elrod had been killed as he was throwing a grenade. Putnam staggered to his feet. He was relieved his men didn't have to shoot any more. He didn't yet know that 80 per cent of his squadron were casualties; that half of his twenty-two civilians were dead.\n\nDevereux, Malleck and their Japanese guards began moving slowly west toward Camp One and the tip of the main island. Firing could be heard ahead and they progressed cautiously.\n\nAt 11:00 A.M. Poindexter was advancing east toward the surrender party. He was elated. Although his attack had been held up on the left flank for an hour, he had finally regained all the ground lost the night before. The Japanese were falling back slowly but steadily and he figured he should join up with the Marines at Peacock Point by nightfall. Then he saw a small group walking toward him down the road with a white flag.\n\nHe assumed the Japanese were surrendering. \"Cover me,\" he told his riflemen. He cautiously moved out to the center of the road. He was a fearsome sight as he approached Devereux's surrender party. His face was smeared with black ointment as medication for flashburns. He wore a pistol belt with a .45 in the holster. The breast pockets of his khaki shirt bulged with grenades and he carried a Springfield at the ready.\n\nAs he neared the group he grinned in triumph. Then he heard one of them shout, \"Drop your rifle.\"\n\nPoindexter was startled. Finally he recognized Devereux, who was not much taller than the Japanese. He dropped his rifle and moved forward automatically. When Devereux kept shouting, he dropped his pistol belt. He approached Devereux with a look of complete consternation on his face.\n\nAs he dejectedly took the grenades out of his pockets the Japanese, apparently thinking he was about to commit suicide, scattered in alarm. He dropped the grenades with safety levers intact on the road.\n\n\"Have your men stand up and leave their weapons on the deck,\" said Devereux.\n\nAs soon as Poindexter's men did as commanded, the Japanese who had been fighting them all morning jumped up. Yelling \"Banzai!\" they dashed forward with fixed bayonets. The Japanese surrender officer stepped in front of the Americans and stopped the charge.\n\nThe surrender party continued west toward Camp One. An American flag was still flying from the water tower. Japanese, cheering, ran forward. One started up the tower.\n\n\"Hold it,\" shouted Devereux to his Marines as they glared at the climbing Japanese. \"Keep your heads, all of you.\"\n\nOn the other side of the tower at Camp One, Sergeant Dave Rush saw the Japanese taking down the flag. He drew a perfect bead with his machine gun. As he was about to fire he saw Devereux. He pushed the gun into the pit, pulled out the firing pin and threw it in the ocean.\n\nOthers at Camp One still couldn't see the surrendering party. A plane suddenly roared down. Sergeant John Cemeris, near the channel between Wake and Wilkes, opened up with his .30-caliber machine gun. Black smoke shot out of the plane. It was fine marksmanship but Poindexter and Devereux were afraid it might touch off a massacre. They hurried forward. \"Stand up,\" called Poindexter. \"Put your hands over your head.\"\n\n\u2014\n\nAcross the channel on Wilkes Island Platt's men had now cleaned out the few surviving Japanese. Three had been captured. At noon a destroyer moved within 2000 yards. Behind were several other ships. Platt ordered McAlister to fire with his 5-inch guns, but word came back that all pieces had been knocked out by the carrier dive bombers.\n\nWatchers reported that a number of small Japanese boats were moving in toward the channel. Platt ordered everyone to advance to the east and repel them. The little force was halfway to the channel when a rifleman shouted, \"Someone's coming down the road.\"\n\nMcKinstry ordered his men to hide behind rocks near the beach. He saw a white flag, and thought it must be a trick. Then he heard someone shout, \"This is Major Devereux. The island has been surrendered. Don't try any monkey business.\"\n\nMcKinstry was unable to believe it until he saw Devereux and Poindexter. Dazed, he stripped his pistol and threw away the parts.\n\nPrisoners were being rounded up all over the three islands. Devereux was taken back to the Japanese headquarters located near Hanna's gun. He felt weary and dead inside. He was full of despair. Just then prisoners in skivvies were marched by. They looked filthy, exhausted and hopeless. Their leader, Technical Sergeant E. F. Hassig, a barrel-chested man with a bushy mustache, turned and glared at the men. \"Snap outa this stuff. Goddam it, you're Marines!\"\n\nThe men's heads came up. They marched proudly in perfect cadence, past their commander.\n\nNear the channel Gunner McKinstry was under guard. Afraid that one of the three Japanese survivors of Wilkes would recognize his long red beard, he was looking for something with which to cut it off. Finally he found a butcher knife and began hacking.\n\nPoindexter was a few hundred yards away, looking through the brush for wounded Americans. A Japanese lieutenant, using a Samurai sword as a cane, was accompanying him. They came upon a Japanese officer, shot in the face. The Japanese lieutenant pulled a small flag from his pocket. He put it over the dead man's chest, tucking one end under the belt and the top corners under the shoulder straps.\n\nThe two enemy lieutenants and two United States Marines carried the dead officer down a path. Poindexter saw a can of pears on the ground. He reached down and picked it up.\n\nThe body was tossed on a truck. Poindexter sat beside the corpse and began to eat the pears. The Japanese lieutenant jumped up on the truck and joined Poindexter. When they had finished the pears, the Japanese pulled out a round tin of cigarettes. He gave a handful to the American.\n\nLater that afternoon, Admiral Kajioka, wearing whites, medals and dress sword, came ashore to take formal possession of Wake. It was renamed Bird Island.\n\n# 8\n\n# _Panic_\n\n**1** About the time Devereux was walking toward the Japanese with a white flag, a plane landed at Washington, D.C. Here it was only late afternoon of December 22. A short, stocky man, easily recognizable by his bowler hat and big cigar, stepped out of the plane and walked briskly to a waiting limousine. He shook the strong hand of the smiling man in the back seat. Churchill had arrived for the conference known as \"Arcadia.\"\n\nRoosevelt and his guest were driven to the White House. Here Mrs. Roosevelt installed the prime minister in the big bedroom across the hall from Harry Hopkins' room. The next few days the three men would detour the mounds of Christmas presents in the hallway to visit each other's rooms at all hours, while they plotted the ultimate destruction of the Axis and discussed the immediate fate of the Southwest Pacific.\n\nChurchill was one of the few men the President thought worth listening to; Roosevelt was one of the few men the prime minister let talk at any length. During this conference their intimacy was so great, Hopkins told friends, that once Roosevelt was wheeled into Churchill's room as the prime minister came from the shower stark naked. When Roosevelt apologized and started to leave, Churchill said, \"The Prime Minister of Great Britain has nothing to conceal from the President of the United States.\"\n\nIt was evident by midnight of the first evening that Arcadia would be a harmonious success\u2014even though Churchill later denied Hopkins' story. \"I never received the President,\" he said, \"without at least a bath towel wrapped around me.\"\n\n**2** In the Philippines, as at Wake, it was already December 23. That dawn the remnant of Brereton's air force had finally attacked the great Japanese convoy in Lingayen Gulf. Four Flying Fortresses dropped their 100-pound bombs and headed south for Australia.\n\nGeneral Homma was given an anxious moment but little damage had been done. His ground troops were having just as little trouble. The main force had finally rebuilt the bridge burned by Major Trapnell and pushed down to Rosario. By noon they had advanced another three miles to an important junction, the southern terminus of the 20-mile mountain road leading down from the old summer capital, Baguio. Thus all MacArthur's troops to the northeast\u2014the men of Baguio's Camp John Hay and a regiment of the 71st Division\u2014were cut off.\n\nLieutenant Colonel John Horan, commander of Camp John Hay, had been keeping MacArthur informed by radio of Homma's progress. The previous day, Horan had asked permission to retreat south down this winding, precipitous road from Baguio to the junction but MacArthur's headquarters, thinking he was \"unduly perturbed,\" notified him to stay until ordered out. Still waiting for word to retreat, he was peeved and anxious. Since a subordinate had destroyed the radio and code book, he had to send an ambiguous message to MacArthur in the clear:\n\n> MY RIGHT HAND IN A VISE, MY NOSE IN AN INVERTED FUNNEL,...MY REAR WIDE OPEN, MY SOUTH PAW OPEN. REQUEST ORDERS.\n\nHe figured some intelligence officer would figure it meant he was cut off in Baguio and could escape only to the north or east.\n\nEarly in the afternoon Homma's men attacked the Filipinos holding the main highway four miles south of the Baguio junction. These troops, the two remaining regiments of the 71st Division, had had only ten weeks of training. Few had learned how to shoot their outdated Enfield rifles. The great majority could only drill and salute.\n\nThe inevitable happened. Like their untrained brothers of the 11th Division, they broke and fled to the rear leaving their artillery unprotected.\n\nWhen Wainwright learned of the rout, he realized the situation was critical. Further defense of the Lingayen beaches, he phoned MacArthur, was impractical and he requested permission to withdraw behind the Agno River. This, the first formidable natural obstacle, was about 45 miles south of the first landing point on Lingayen Gulf and only 90 air miles above the final goal, Manila. From here he could counterattack \u2014if MacArthur would give him the well-trained Philippine Division.\n\nIn Manila, MacArthur was dismayed by the collapse of the two green divisions. Yet it was not surprising. They were fighting an experienced, tank-equipped enemy, with rifles and old-fashioned, water-cooled machine guns that easily jammed.\n\nWith no air force or navy, MacArthur had to abandon his cherished strategy of holding the enemy at the beaches. He was now forced to consider War Plan Orange-3. This was the plan formulated by his predecessors in the Philippines, and its primary mission was the defense of Manila Bay. If enemy landings on Luzon could not be prevented, and pressure became too great, the defenders would slowly withdraw to the Bataan Peninsula. Here a last-ditch defense could be made for about six months. By then, it was hoped, the Navy could send in supplies and reinforcements. MacArthur had shelved WPO-3 upon assuming command of the Philippines. It was, he felt, a defeatist move that might well lose him the support of Filipinos. Now he had no other choice. He called in his staff. \"Put WPO-3 into effect,\" he ordered.\n\n\u2014\n\nIn spite of the quick advance of Homma and the collapse of the 11th and 71st Divisions, there was still no panic among the other Philippine units.\n\nThe 21st Division, which had been defending the lower part of Lingayen Gulf, was retreating south in good order, with morale unimpaired. Outside of their midnight \"battle\" with the single Japanese reconnaissance boat on December 10, they had yet to meet the enemy.\n\nThird Lieutenant Antonio Aquino, the elder son of the speaker of the Philippine Assembly and recently admitted to the bar, had been sent to Manila to commandeer eight trucks for this division's retreat. He had sent back the trucks and returned the previous night to his family's great sugar plantation near Tarlac, about 30 miles south of Wainwright's Agno River line, so he could speak to the workers.\n\n\"Attorney,\" their apprehensive, puzzled spokesman had said, \"what are we fighting for? Why do not America protect us?\"\n\nDespite the sweltering heat, Aquino was nattily dressed in olive drab woolens, tailored hunting coat and steel helmet. He told the tenants that America was going to send help but Filipinos would have to do most of the fighting in the meantime. \"We are fighting with America because we believe in the American way of life.\"\n\nHe told them the Japanese were advancing from the north. They should evacuate to the mountains with their families. \"Wait for me if we lose this war. I shall escape and we shall be guerrillas like the time Grandfather hid in Mt. Arayat.\" This was General Aquino who had fought against the Americans with Aguinaldo. \"We shall not give up the fight. America will come back and we shall not lose faith with her. Are we all agreed?\"\n\nThe tenants cheered. They no longer had doubts.\n\nThe next day, about the time MacArthur ordered WPO-3 in effect, everyone had left the vast plantation. Aquino went out to the fields and set fire to the standing sugar cane and rice. He got into his brand-new yellow convertible and with his \"man Friday,\" Eladio, beside him, set off to rejoin his division. After a mile he stopped and looked back at the roaring fire and the rising columns of black smoke. He hoped his father would approve his \"scorched earth\" tactics.\n\nIn spite of his words to the tenants, Tony Aquino secretly believed, as almost everyone else in the Philippines believed, that the war would be over in two weeks or a month at the most. By that time convoys filled with soldiers, planes and guns from San Francisco would arrive. He had plenty of supplies in the back of his car to last until then: a case of Scotch, a complete kitchen, hunting equipment, a radio, a portable table and chairs, a large bag of ping-pong balls, a tuxedo complete with shoes and studs, and, in case of emergency, $1000 in pocket money.\n\nThat night, Wainwright was in his tent on the bank of the Agno River planning a counterattack. It was based on the hopeful assumption that MacArthur would lend him the elite Philippine Division, for of his 28,000 men only 3000 were trained.\n\nThe phone rang. It was USAFFE, MacArthur's headquarters. Perhaps he was getting his old division.\n\n\"WPO-3 is in effect,\" said Colonel Constant Irwin.\n\nWainwright couldn't talk. He knew this meant MacArthur had abandoned his pet theory of stopping the Japanese at the beaches. It also meant that the situation was desperate.\n\n\"You understand?\" said Irwin when Wainwright didn't answer.\n\n\"Yes, Pete,\" said the other quietly, \"I understand.\" Like many other senior officers he had always felt WPO-3 was the only practical defense of the Philippines. But fifteen precious days had already been lost. Was it too late? It was up to him, he realized, to stem the Japanese tide until the rest of MacArthur's troops could escape into Bataan.\n\nThe following morning, MacArthur learned that what he had feared most had happened. During the previous night 24 Japanese transports had landed in Lamon Bay, 60 air miles southeast of Manila. He was now caught in a giant pincers. Not a single big gun and less than two infantry battalions protected this east coast. Before mid-morning 9500 Japanese of the 16th Division were pushing toward Manila in three columns. At 10:00 A.M. Major General George Parker got orders from MacArthur to slowly withdraw the two divisions of his South Luzon Force to Bataan. The battle in the south was over before it started.\n\nAt 11:00 A.M. MacArthur's staff was called to a conference at No. 1 Victoria Street. General Sutherland informed them headquarters was being moved to Corregidor Island that evening. Men were to take field equipment and one suitcase or bedroll.\n\nNearby in the Marsman Building, Admiral Hart was telling Admiral Rockwell that he had decided to move his headquarters south to Borneo so he could be with the operating fleet. Rockwell would take over supreme command of the remaining naval forces.\n\nAs the sixty-four-year-old admiral was telling Rockwell to destroy all fuel, Diesel oil and aviation gas stored in Manila, planes roared overhead. Bombs began to fall in the Walled City. One hit the Marsman Building but Hart continued the conference. Subs, he said, should operate from Manila Bay as long as possible. Another bomb hit. Captain Dessez should blow up everything at Sangley Point and the Cavite Navy Yard. As a third bomb rocked the building, he told Rockwell to turn over the 4th Marines to MacArthur.\n\nThe port area was in flames, a chaotic shambles. Huge craters pockmarked the streets; streetcars were flung into buildings, the tracks behind curling up fantastically. Clouds of dust from pulverized cement and stone mixed with black billows of smoke to hang a pall over the entire Pasig River section.\n\nA new Packard pulled into the Walled City, then was stopped by the jumbled streets. Its occupant, General Brereton, walked into No. 1 Victoria Street, wondering why MacArthur had sent for him. The air chief had argued a few days before that he and his staff be sent south. There was nothing they could do in the Philippines without planes. USAFFE headquarters was alive with activity as Brereton was taken into MacArthur's office. The general, obviously distraught, was terse. He told Brereton to take his headquarters south. When Brereton now offered to remain personally and serve MacArthur in any capacity, the general said, \"No, Lewis, you go on south. You can do me more good with the bombers you have left and those you should be receiving soon than you can here.\"\n\nBrereton rose.\n\n\"I hope,\" said MacArthur, \"that you will tell the people outside what we have done and protect my reputation as a fighter.\"\n\n\"General,\" said Brereton, shaking his hand, \"your reputation will never need any protection.\"\n\nAt Malaca\u00f1an Palace, President Manuel Quezon was talking to his executive secretary, Jorge Vargas, Vice-President Sergio Osme\u00f1a, and Jos\u00e9 Laurel. Quezon's voice was choked. He, Osme\u00f1a and a few other officials were about to leave for Corregidor. The president, wearing a United States uniform and helmet, made them all promise never to reveal what he was about to say. Then he said to Laurel and Vargas, \"You two will stay here and deal with the Japanese.\"\n\nLaurel protested that people would think he was a collaborator. Quezon agreed but insisted it was Laurel's duty.\n\nLaurel burst into tears. He begged to be taken to Corregidor.\n\n\"Someone has to protect the people from the Japanese,\" said Quezon.\n\nLaurel at last sadly agreed. He knew that to his dying day only those in this room would know the truth. Perhaps the truth would never be revealed and he would go down in history as a traitor.\n\nA few minutes later Quezon's limousine left the Palace grounds, heading for Manila Bay. Bombs were still falling on the port area as it passed the Walled City.\n\nQuezon, his wife Aurora, his daughters Maria Aurora and Zeneida, and his son, Manuel, Jr., stepped onto the pier next to the Manila Hotel. While servants carried family belongings, including a parrot, from a truck, Quezon never stopped talking to those who quickly gathered on the dock. Always a flamboyant speaker, he was never more eloquent. He hated to leave, he shouted, but it was his duty as president.\n\nCalling out farewells, he walked slowly with Vargas toward the float. The two shook hands emotionally. Quezon knew he was dying of tuberculosis. Already his body was skeletal, his face emaciated. There was so much to do for the Philippines before he died. He stepped into his private speedboat. With a roar it headed into the bay.\n\nAshore, correspondent Arch Gunnison of _Collier's_ asked Vargas, \"Well, what's next?\"\n\n\"I wish I knew. But whatever it is, it looks like it's going to be for a long time.\" As he watched Quezon's speedboat approach the inter-island steamer _Mayon_ , he added a little bitterly, \"Well, some of us have to stay. I've got more to do now than I know how.\"\n\nBy this time, the streets of Manila were crowded with Army trucks and commandeered squat, fat-bellied Pambusco buses filled with soldiers and supplies. Traffic crept impatiently in the hot sun. Everything was going north\u2014to Bataan.\n\nMilitary installations in the Manila area were in a state of wild confusion as personnel hastily loaded into trucks, cars and buses. At Air Headquarters, recently transplanted to Fort McKinley, General Brereton was telling his staff of their retreat to Australia. The move was to be kept secret even from other headquarters officers. The big problem was how to get there. There wasn't a Flying Fortress in the Philippines. The Navy was telephoned. Brereton could have four seats on a PBY leaving for Java that night. The general picked three men to accompany him and the four hurried out to pack. The others, slated to follow soon in a transport, also left to pack.\n\nAir Headquarters soon became utterly disorganized. No one knew what was going on. Brereton had disappeared, so had his personal officer. Colonel George couldn't be found. No one knew, in fact, who was in command.\n\nNielson Field was just as chaotic. It was the wildest and most fantastic day in the history of the 27th Bombardment Group. Early that afternoon they had been told to move to Bataan. The only explanation from Air Headquarters was that Manila was to be declared an open city to save it from further bombing. The greatest haste was urged. When Fort McKinley was called back for further instructions there was no answer. Air Force Headquarters, apparently, had just quit work and disappeared.\n\nThe 17th Pursuit Squadron at Nichols Field, presently commanded by Second Lieutenant David Obert, was to be at the Manila docks by 6:00 P.M. By midafternoon Obert knew he'd never make the deadline so he decided to send the men ahead with whatever they had already packed. He hurried to McKinley to get approval. There, to his dismay, he too found Headquarters had evaporated. People were scurrying around excitedly burning papers and collecting personal equipment. It looked to him as if all the units were on their own.\n\nAt Zablan Field, the eighteen remaining planes of the Philippine Army Air Corps were being burned. It was a crushing blow to the Filipino pilots but the orders had come from some excited officer at Air Headquarters. Their planes destroyed, they began to set fire to hangars and barracks.\n\nOne hundred air miles to the north, Homma's advance troops were smashing at Binalonan, about ten miles above Wainwright's main line of defense on the Agno River. For the first time they had been stopped. Since 5:00 A.M. only 450 men, the tired remnants of the 26th Cavalry, had held off heavy attacks on the town.\n\nBy 3:00 P.M. the cavalrymen's situation was desperate. Japanese tanks were rolling toward their roadblock and they had no anti-tank guns. Captain Barker, commander of Troop B, called his platoon leaders together and asked for volunteers to knock out the oncoming tanks. Only one, Private Soria, raised his hand. As the rest of Troop B fell back to prepared foxholes, Soria hid in the ditch with grenades and three Coca-Cola bottles filled with gasoline.\n\nAs the first tank rumbled by, Soria leaped on the rear. He tapped on the hatch with his pistol hoping the turret would open. Instead the turret gun swung around, sweeping Soria into the road. The tank backed off and its 37-mm. gun roared. Machine-gun bullets raked the ditch. Soria, grazed in the shoulder by the big gun, jumped out of the ditch and ran to the rear. The tank hesitated and then, apparently scared off by Soria, retreated.\n\nWord came from Wainwright to pull back; the cavalrymen had held the Japanese long enough for the Agno River line to be manned. The troopers ran from their foxholes to their horses. Trotting five minutes, then walking five minutes, they headed for the Agno River.\n\nA rout had been saved by a few Coke bottles and the 26th Cavalry. Wainwright, an old cavalryman himself and perhaps a bit prejudiced, reported it had been \"a true cavalry delaying action, fit to make a man's heart sing.\" It also proved what MacArthur had always insisted: the Filipino if well trained made an excellent, courageous soldier.\n\nIn Manila, Major Carlos Romulo, MacArthur's press and radio chief, sat at his desk watching his commander in the adjoining room. He saw the general answer a phone. Some field commander obviously was reporting directly. MacArthur hung up and began pacing the floor, hands clasped behind him, head slightly bowed, his hawklike face lined with bitterness. Abruptly he stopped pacing and put on the gold-braided cap of field marshal of the Philippine Army. He glanced quickly at his flags, the beautiful oriental art objects collected over the years, the V-for-Victory poster, then strode into Romulo's room. It was exactly 6:00 P.M.\n\nQuietly he told the ex-newspaperman that he was going to the island of Corregidor to direct the Battle of Bataan. Romulo was to remain in the rear echelon. \"It's more important than ever,\" said the general, \"that I be kept in touch with the press while I'm in the field.\" He held out his hand. \"I'll be back, Carlos.\"\n\nMacArthur and most of his staff headed for the docks and boarded the _Don Esteban_. It was beginning to get dark when the steamer started across the bay toward Corregidor, less than 30 miles away. The moon was shining and the men of USAFFE headquarters sat in their shirt sleeves on the decks talking in hushed tones as they watched flames rise from Cavite's oil dump. It was a strange Christmas Eve for Americans used to cold and snow.\n\nColonel William Morse was watching his chief. MacArthur sat, head in hands.\n\nThe dark streets of Manila were jammed with creeping civilian cars. Occasionally there would be a shout of \"Ilao (Lights),\" then a rifle report as a shaded headlight, approved by the city's blackout committee, was shot out. Already innocent pedestrians and drivers had been killed by trigger-happy volunteer local guards, many recruited from the notorious, gangster-ridden Tondo slums.\n\nOn the east coast of Bataan, the first evacu\u00e9es from Manila were already setting up Hospital No. 1 at Limay. Second Lieutenant Hattie Brantley had been told she was coming to a 1000-bed hospital, completely equipped. She had mentally pictured beautiful buildings, surrounded by flowers. She had found thirty long nipa (palm) buildings, completely empty and covered with dust. Medical supplies in a nearby tin warehouse were wrapped in 1918 newspapers. Some of the cots crumbled to pieces when they were unfolded. Now the nurses were being issued new clothing. Lieutenant Brantley was given a size 42 pair of coveralls and heavy GI shoes. Then the fifty nurses were led into their quarters, a one-story wooden shack. There were six cots.\n\n\"Do we sleep on the floor?\" asked someone tiredly.\n\n# 9\n\n# _Dark Christmas_\n\n**1** Christmas was a night of hell in Hong Kong. The Japanese held three-fourths of the mountainous island's 32 square miles. The British forces were split in two. In the south, 2500 were jammed into Stanley Peninsula. The main force, about 5000, held the populous northwest corner, including the capital of Victoria. With no air or sea power and few anti-aircraft guns or mortars, defense lines everywhere were crumbling. Ammunition was running low and with most of the reservoirs captured there was only water for another day or two.\n\nThe defenders were exhausted. The three mainland battalions had been fighting, withdrawing and fighting for two weeks with almost no rest. The other three battalions and the Hong Kong Volunteer Defence Corps had been fighting three days without pause for sleep or food. Up until now morale had been high, though losses were heavy, but there were increasing murmurs of despair among the rank and file. Why postpone the inevitable by sacrificing their lives for a few hours of possession?\n\nOn the mainland the resistance had been disappointing. The stand on the barren, rugged mountain mass that was Hong Kong Island had been stubborn. Much of the credit belonged to the 1759 men of the citizen volunteers. At first disdainfully called \"playboy soldiers\" by the regulars, this conglomerate mixture of local British, Eurasians, Chinese and Portuguese had fought as bravely and well as any of the other troops. In fact some regular officers felt the citizen soldiers' determined defense of their home city had put steel into the better trained men.\n\nEven the Japanese had been amazed by the gallant defense of Hong Kong Island. To them the desperate fight made no sense.\n\nChurchill's message asking for a last-ditch stand made sense to Governor Young and General Maltby. Every day, every hour, they knew, was vital to the Empire war effort, but even they realized by midnight of Christmas Eve that the end was near. The Japanese held the power station and there were no fresh vegetables or meat. The 1,750,000 civilians of Victoria were beginning to suffer severely. With only a driblet of brackish water to drink, none could be spared for washing. Water closets were stopped up and the most luxurious hotels stank.\n\nIn the south, most of the men cut off at narrow Stanley Peninsula were dug in near the fort, on the high ground at land's end. Only a few men held the first defense line which ran through Stanley Village where the isthmus was barely 700 feet wide. And by 1:00 A.M., Christmas morning, these had been driven back to the second line, the low ridge beyond St. Stephen's College.\n\nThis put the main building of the college, presently an emergency hospital, between two fires. At dawn Private T. J. Cruz, a Portuguese volunteer stretcher-bearer, was looking out a downstairs window. To his horror he saw Japanese, bayonets at the ready, creeping toward the \"hospital.\"\n\n\"What'll we do?\" he asked his officer, a second lieutenant.\n\n\"Every man for himself!\"\n\nCruz and the other stretcher-bearers grabbed their rifles as the attackers outside began shrieking \"Banzai!\"\n\n\"No,\" shouted someone, \"throw away your rifles. Depend on the Red Cross armbands!\"\n\nThe sixty-five wounded patients in the main room, which was lit only by a hurricane lamp, stared in horror, some ducking under their blankets as the shouts grew louder. Private J. Jones (this is not his real name) had been brought by coincidence that afternoon to the hospital where his young wife was a nurse. She now shoved him under his bed and crawled next to him.\n\nCruz and the other stretcher-bearers ran up to the second floor as the front door burst open. Dirty, unshaven, wild-eyed soldiers stampeded through the doorway. To Cruz they looked like Formosans or Koreans and they acted drunk. Peering down in terror, he saw them run into the main room and with savage shouts begin bayoneting the men in the beds. These Japanese had seen rifles and tommy guns stacked in the anteroom and could have thought those in bed were unwounded soldiers trying to hide.\n\nJust then Lieutenant Colonel G. D. R. Black, a prominent Hong Kong doctor and head of the hospital, came from a rear room holding up a white sheet. Behind was his assistant, Captain Whitney.\n\n\"Look, you can't do that,\" cried Black. Followed by Whitney, he ran forward to block the doorway. The two men were shot, then stabbed many times. The attackers continued frenziedly up and down the rows of beds bayoneting the wounded. As they headed for adjoining classrooms containing another thirty patients, Japanese officers ran in and stopped the massacre.\n\nUpstairs Cruz and five others were hiding in a classroom. They could hear the piteous screams of the stabbed, the triumphant yells of the bayoneters. Then there were footsteps. A foot crashed against their door; the glass shattered. A man holding a tommy gun burst in shouting. Cruz came forward, hands up. The others in the room followed. The Japanese hit Cruz with the butt of his gun and motioned him to enter the next room. When forty survivors were crowded into the small room, the door was slammed. Outside the sound of battle continued.\n\nTwo hours later the Japanese combat troops returned to the emergency hospital. They broke into a small room where four Chinese and seven British nurses were confined. The Chinese girls, all wearing the St. John's ambulance badge, were dragged to another room. They were raped over and over and thrown into another room. Next the three youngest British nurses, including the wife of Private Jones, were taken out of their room. They, too, were raped many times.\n\nPrivate Cruz and the others heard the screams of the women. Then their door was opened. \"You were stupid; you should have surrendered,\" said a Japanese in English. \"You will be slowly killed four at a time,\" he added and closed the door. A few minutes later agonized cries came from the next room. Cruz wondered when their turn would come.\n\n\u2014\n\nThe main British force defending the capital, Victoria, was also being overwhelmed. Japanese were breaking into buildings in the suburbs. At 9:00 A.M. Major C. M. Manners, a retired officer, and a civilian, A. L. Shields, were escorted from the Japanese lines under a white flag. They told General Maltby of the incredible number of Japanese guns they had seen. In their opinion it was useless to continue the struggle. The Japanese, they said, promised not to fire for three hours while the British made up their minds. Maltby called a special defense meeting.\n\nThough the situation was hopeless and Japanese were even in some of the air raid tunnels, it was decided to continue fighting. At noon, the three-hour truce was over; Japanese artillery resumed fire. By 3:15 P.M. Maltby knew he'd reached the end. He advised Governor Young that no further useful military resistance was possible, then ordered his commanders to surrender.\n\nRumor of the surrender spread through the city. There was a sense of immense relief in spite of the humiliation of defeat. The useless slaughter had stopped. At the Gloucester Hotel someone suggested that all the liquor in the many luxury hotels in the neighborhood should be destroyed. Thousands of bottles were dumped down the bathroom drains. Since these were clogged, champagne, gin, bourbon and Scotch began running down staircases. But many of the bottles were consumed. It was a last, glorious, desperate binge.\n\nBelow the fort, at St. Stephen's College, Private Cruz was still waiting in the emergency hospital with his forty captured comrades. Though they had heard cries of agony about every half-hour from the next room\u2014now called \"the torture room\"\u2014none of them had yet been taken. They knew their luck couldn't hold much longer. When the door opened at 5:30 P.M. Cruz felt it was his last moment. A Japanese officer came in and said, \"You boys are very lucky. Hong Kong has surrendered. Now we will no longer have to kill you.\"\n\nThe prisoners were told to clean out the next room. Cruz almost retched when he looked inside. The seven British and Chinese nurses who had been raped were gone but blood covered the floor like a thick claret carpet. Mattresses and cushions were soaked red. Cruz splashed through the blood and picked up a cushion. It was soggy and heavy. The prisoners stacked the mattresses and cushions in the yard. Bodies\u2014some with ears cut off, all bayoneted\u2014were carried on stretchers from the building to the yard. The pile of mattresses was ignited. Three bodies were flung on top of the flames. It would take many more fires for at least seventy inside the hospital had been slaughtered. The flames cast fantastic shadows as more bodies were heaped on the pyre.\n\nThis was the end of Christmas on Hong Kong Island.\n\n\u2014\n\nDecember 26, Boxing Day, was a bitter one for the British.\n\nLieutenant Bill Stoker of the 2nd Scottish Company, still uncaptured, had heard rumors of rapings at St. Stephen's College. Late that morning with two volunteers he drove into Stanley Peninsula. The Japanese paid no attention to him as he entered the emergency hospital. He found the four older British nurses exhausted and near hysteria. As he boldly escorted them past the guards, Private Jones called out from a group of captured soldiers, asking if Stoker had seen his wife. Stoker and his men looked but none of the seven missing British and Chinese nurses could be found.\n\nA Japanese officer, Captain Kawai, pitying the frantic Private Jones, volunteered to help. He went outside. A moment later he called Jones. The British citizen-soldier ran outside, saw a pile covered with a coat belonging to his wife. Private Jones hesitantly pulled back the coat. Underneath were three dead women. One was his wife. He collapsed.\n\nIn Aberdeen Harbor, famous for its elaborate floating restaurants, every junk, proa and sampan was flying a Japanese flag. In Victoria, Japanese troops were making a formal triumphant entry as planes roared overhead, leaving a fluttering trail of pamphlets proclaiming the Co-Prosperity Sphere of East Asia. Thousands of Chinese lined the streets waving Rising Sun flags. To the Japanese it was a \"never-to-be-forgotten ceremony.\" European onlookers also would never forget it. The lead car, carrying a smiling Japanese general, had recently belonged to a Dutchman. A bright sticker on the windshield read:\n\n> HOLLAND SHALL RISE AGAIN.\n\n**2** It had also been a dark Christmas in the Philippines. On the morning of December 25 MacArthur was reviewing the deteriorating military situation at his new headquarters on Corregidor. This was an island in the shape of a polliwog that sat in the mouth of Manila Bay, not 30 miles from the city, and easily visible on any clear day.\n\nWhoever held Corregidor controlled Manila Bay, for it stuck in its throat like a bone. To MacArthur it had an even more important value. Less than two and a half miles to the north was Bataan, the mountainous peninsula which was to become the main battlefield of the Philippines.\n\nCorregidor, aptly nicknamed The Rock, was perfectly designed by nature for defense. The body of the polliwog was 519 feet high and covered with foliage. Here, dug in rock and well camouflaged, were the fort's main big guns. The middle of the island, where the tail began, was almost at sea level. Then after a few hundred yards, there rose another hump of solid rock, Malinta Hill. Through it had been bored a great tunnel with many laterals, so ingeniously constructed that ten thousand could live in its dank catacombs indefinitely without fear of bomb or shell. This was Malinta Tunnel.\n\nMacArthur and his family lived in a small cottage half a mile beyond the eastern mouth of the tunnel, further out on the polliwog tail. On Christmas morning he sat in this cottage confronted by apparent instant disaster. Without air or sea support his basic plan for aggressive defense of the Philippines had collapsed and he had been forced to the humiliating strategy of holding out on Bataan until help arrived.\n\nThe chances of successful withdrawal to the peninsula seemed hopeless to many that morning. One effective bombing would cripple the mass movement. Manila Bay was dotted with slow-moving barges, pleasure steamers, launches and tugs shuttling men, munitions and food to Corregidor and Bataan. Highway 3 leading north out of Manila was jammed with trucks, 155's on their carriages, naval guns on trucks, buses, cars, cartelas and oxcarts. One bomb on the Calumpit Bridge over the wide, unfordable Pampanga River, 30 miles north of Manila, would cripple the entire withdrawal plan.\n\nTen miles beyond this bridge the line of vehicles turned left at the busy town of San Fernando, sugar center of the Philippines, and until recently overhung with the nauseatingly sweet smell of burning sugar cane. Here was a chaotic traffic snarl, for almost as many trucks and buses were pouring in from the north. This southbound traffic was the van of Wainwright's main body, fleeing to Bataan while the covering force delayed the onrushing men of Homma. Many of these Filipino troops had no transportation and were marching barefoot, their outsize GI shoes tied to rifle barrels. They trudged along the hot unshaded roads in blue denims and pith helmets.\n\nThe road from San Fernando westward to Bataan was so narrow that traffic had backed up into town by 11:30 A.M. Ambulances, tank trailers and guns were parked on every side street, on the sidewalks and even the fields near the railroad station and cockfight arena. Those trapped in the town kept looking nervously into the clear sky. One dive-bomb attack would not only destroy hundreds of vehicles but tie up this key junction for days.\n\nThe same fear haunted men for the next 42 miles on the narrow stretch of road to Orion, halfway down the east coast of Bataan. It was lightly tarred but the crust had been crumbled by the first heavy vehicles in the line. Dust rose from San Fernando to Orion scribbling swirling plumes in the sky. Climbing hundreds of feet, this great spoor was visible for many miles, a luring target.\n\nJust before noon the rumble of many motors could be heard to the north. As it grew louder, some vehicles stopped and men scrambled into sugar fields or rice paddies; some vehicles kept going, racing madly to avoid the attackers. In San Fernando there was nothing to do but hide under the flimsy buildings and nipa shacks or in ditches.\n\nSoon a large formation of Japanese planes flew above the crowded town. Not a bomb dropped. Not a plane peeled off for attack. The attackers flew serenely south, oblivious of the panic below. They passed Calumpit Bridge without dropping a bomb and continued over jammed Highway 3 all the way to the city. At noon they were over Manila. Finally their bombs fell, hitting the port area and the Walled City, evacuated by MacArthur only the day before. Then the fliers headed back for Formosa, reporting that their objectives had been hit, mission accomplished.\n\nWhile the smoke was still rising from the wreckage in the port area, the great exodus was already in motion again. But there was no sense of relief. Other raiders must be getting ready to dive or make their bomb runs. The long winding target from Manila to Bataan was too obvious to miss.\n\nPFC John Connor, leading a convoy of twenty trucks loaded with C rations, felt a sense of shame as he drove north past crowds of staring Filipinos. Some would smile, hold up fingers in a V for Victory sign and shout, \"Hello, Joe.\" Most just stared. They seemed to say, \"You're running off and leaving us.\"\n\n\u2014\n\nBy dusk the Japanese still hadn't bombed the road to Bataan. In the north Wainwright had successfully drawn behind the Agno River, the second defense line of WPO-3, and a formidable natural barrier about 90 air miles above Manila.\n\nBut in the south, three columns of Japanese were steadily moving northwest from Lamon Bay and were only 55 air miles from the capital. Early that morning the 1st Infantry Regiment, which had been holding back the column nearest Manila with a stiff defense, suddenly withdrew by mistake when orders were confused. Brigadier General Albert Jones, the new commander of the South Luzon Force, was eating his Christmas dinner when a motorcyclist drove up to his command post. The messenger said the 1st Infantry was coming back over the mountain.\n\nJones, an energetic officer of Welsh descent, angrily jumped into his staff car and headed for the front. MacArthur had chosen Jones to command in the south because of his well-known aggressive spirit and his firm belief that the Japanese could be whipped. Jones ran into the retreating troops near Luisiana. When the senior American instructor of the 1st Infantry stepped forward, Jones said, \"Just what the hell do you mean pulling back?\"\n\n\"I was ordered to pull back by the commander of the South Luzon Force, sir.\"\n\n\"Well, I'm the new commander of the South Luzon Force and you turn your people around in a hurry.\" As an example Jones told his own driver to head down the road toward the enemy. He figured there wasn't enough time to persuade an advance guard to precede him. He knew he'd eventually meet the Japanese but that would take care of itself. By now it was dusk. At a bend in the narrow dirt road lined with bamboo and cocoanut trees, Jones came to a half-track astride the road. Written in chalk on its side was _Miss Oregon_.\n\nThe Americans in the half-track aimed their automatic weapons at the general's car. As they were about to shoot, Jones asked them who they thought they were aiming at.\n\n\"We thought you were Japs,\" apologized the sergeant in charge of the half-track.\n\n\"Where are the Japs?\"\n\nThe sergeant pointed east. \"Down the road about three kilometers.\"\n\n\"I'm going ahead and reconnoiter,\" said Jones. He waved the half-track to precede the staff car. The sergeant jumped into the half-track as it went down the dark road. After passing three kilometer posts it squealed to a stop.\n\nJones saw figures jumping out from behind cocoanut trees.\n\n\"Halt,\" shouted the sergeant from the half-track. The figures didn't answer. \"Halt!\"\n\nThere was a roar as a concealed Japanese gun at the bend of the road ahead hit the radiator of the half-track. Then machine-gun fire opened up from the right side of the road. Jones ran from the staff car, trying to get into the half-track, but machine-gun bullets had found the mark and were bouncing off the sides. The general dove into the left-hand ditch. As he landed in the mud, his driver flopped on top of him. Machine-gun fire chattered from the left side of the road, 25 yards away.\n\nThe men in the half-track were firing back with their machine gun and Jones wondered how they could live in the hail of fire. It looked as though a stream of bullets were passing through their bodies. Jones heard a terrific crack over his head. He thought he'd been hit. A bullet had struck his driver's pistol, knocking it out of his hand. Abruptly there was silence.\n\n\"Let's get the hell out of here,\" said Jones. His driver quickly turned the staff car around. Jones and the men from the half-track jumped into the car and it speeded up the road to the rear, lights on. Shots came from both sides of the road.\n\n\"Kill the lights,\" said Jones.\n\nThe lights went off but it was too dark to drive. \"Put them on again,\" said Jones, \"and drive like hell.\"\n\nAfter a few miles they met 1st Infantry troops returning to battle. Jones ordered them to fight a delaying action until forced to withdraw. They needed no urging to keep moving forward. They were more afraid of the angry general than of the Japanese.\n\n\u2014\n\nAt 4:00 A.M., the next day, December 26, Admiral Rockwell's barge silently left the dock at the middle of Corregidor, where the polliwog's tail began. Manned by an American coxswain and a Filipino crew of three, its sole passenger was a lieutenant junior grade, Malcolm Champlin. Six months ago he had been an FBI agent in Baltimore. Called back to service in the Navy, he had protested strenuously, but by now he was reconciled to his fate. As Admiral Rockwell's aide, his work was varied and interesting. This morning, for example, he was to steal back to Manila, which might fall at any minute, and give Captain Morsell instructions to blow up the vast oil dumps in the city. Morsell was to locate the American, British, Dutch and French owners of the oil and persuade them by diplomacy if possible to destroy their own property. Rockwell impressed on him the vital importance of the mission. It was still dark as the admiral's barge swung into the channel between Corregidor and Bataan. Then it turned right toward Manila and soon approached the mine sweeper guarding the channel.\n\n\"Boat, ahoy,\" called someone from the sweeper. When the admiral's barge swung in close, Champlin could see three large black hulks behind the sweeper. \"Can you lead these ships through the mine fields?\" shouted the sweeper captain from the wing of his bridge.\n\n\"No,\" promptly answered Champlin. He had been through the field only once months before when the _President Harrison_ brought him to Manila. Besides he had his own mission to worry about.\n\n\"Well, you can spot the buoys, can't you? These ships have Army on them and can't stay here during daylight. They'll be sunk by Jap planes. I want you to take them through to Bataan if you possibly can.\"\n\nSince the sweeper captain outranked him, Champlin said, \"Aye, aye, sir. Tell the leading ship to follow me and to stay well behind. I'll find the buoys but I don't want the heavy ships to follow too closely in my wake.\"\n\nThe captain thanked Champlin, gave him a compass bearing, and left. As the young lieutenant led the way through the mines he couldn't help thinking of the steamboat _Corregidor_ , blown up two weeks earlier almost at this very spot with the loss of many lives. He glanced behind and saw the first ship, its rail lined with troops. Instead of staying well behind, the three ships were closing in fast, as if eager to get the suspense over.\n\nFinally a Filipino in the bow of the barge spotted a little spray. Champlin guessed it was the buoy marking the end of the mine field. He changed course and passed clear of the spray on the open channel side. He was right. The barge and three ships slipped safely between the last buoys. Champlin waved to the soldiers on the decks and headed back through the mine field toward Manila.\n\nThe city was confused with rumors. Some said the Japanese were at the outskirts. Others said they were hundreds of miles away. Champlin borrowed a car and hurried to Captain Morsell's office. It was abandoned. Everyone had gone to Bataan. He hurried through crowded lines of vehicles heading north toward the Marsman Building. Hart's headquarters was also deserted. The admiral had left early that morning on the submarine _Shark_ for parts unknown. The others were on their way to Corregidor or Bataan. Finally Champlin found one officer, Commander R. L. Dennison. \"Do you have any instructions about the oil supplies and their destruction?\" asked Champlin.\n\nDennison knew nothing about oil. He was destroying radios and making a last checkup of the building. Champlin realized he had inherited the mission, but he had no idea what to do. Where were the oil executives? And if he found them how could he persuade them to destroy their own property?\n\nEarly that afternoon at the Army-Navy Club, Champlin finally learned that the man who controlled most of the oil in Manila was a Mr. Rock of the Standard Oil Company who was probably near by at the University Club. Champlin hustled along Dewey Boulevard for half a block. He found Rock at the University Club and the oil man promptly agreed to help.\n\n\"How much oil is there in Manila?\" asked Champlin. He was informed there were enough petroleum products in the Pandacan area to operate the Asiatic Fleet for two years. \"Admiral Rockwell told me Manila is going to be made an open city some time today. You and your workmen have got to help me destroy this petroleum before the Japs come.\"\n\n\"You might burn down the whole city.\" Rock explained that burning oil might flow into the Pasig River and set fire to every dock, ship and warehouse. The only solution would be to bleed oil off into the ground before igniting it, but 100 policemen would have to cordon the area. Sparks from a hobnail boot or a dropped cigarette could set off everything prematurely.\n\n\"We'll get every fire engine and policeman in Manila to help us,\" Champlin promised.\n\nRock agreed to burn his petroleum products, then added, \"You may have trouble with the French and British oil companies.\"\n\nThe two men called a meeting of the oil executives involved. All but the British representatives soon arrived and Rock explained the problem. Next Champlin told the oil men how important this was to the Allied war effort. They were not enthusiastic. After an hour's futile discussion, Rock picked up a telephone. \"Gather forty workmen,\" he told his foreman. \"Destroy all Standard Oil products in Manila beginning with alcolate.\"\n\nRock's example tipped the scales. When the other oil men had completed similar telephone calls, Champlin arranged to have eight fire engines and 100 policemen report to the Pandacan area. While he and Rock relaxed with a round of Scotch whisky, the oil man warned him there would probably be trouble with the two British executives controlling the Shell Oil holdings.\n\nChamplin recalled the recent report that Kowloon oil, not set afire by the British during the battle for Hong Kong, was now being used by Japanese tanks rolling steadily down the Malay Peninsula toward Singapore. \"In that case,\" he suggested, \"why don't you instruct your foremen to be ready to destroy the Shell holdings at a single word from you?\"\n\nRock called his foreman and made arrangements. A few minutes later the two British oil men, carrying tennis rackets and wearing long shorts, strolled into the University Club. Champlin explained that Japanese advance parties were probably nearing the Pandacan area.\n\n\"We are mere agents,\" said one, a Mr. Crawford. \"We would have to obtain permission from our principals in Singapore before destroying their property.\"\n\n\"This is also a British war,\" said Champlin. \"We can presume your principals would give an affirmative answer.\"\n\nThe British weren't so sure. The argument went on. Finally Champlin said, \"Would you rather see this oil in the hands of the Japanese than violate your instructions as agents?\"\n\n\"Yes,\" replied Crawford. \"We cannot violate our instructions.\"\n\nChamplin nodded to Rock, who picked up a phone. \"Go ahead as planned,\" Rock told his foreman, and put down the phone.\n\nChamplin brought the British oil men drinks. Suddenly there was a distant, hollow explosion. \"Gentlemen,\" said Champlin when the British finished their drinks, \"your oil is now in flames.\"\n\nPillars of smoke climbed as more explosions followed. Great tongues of flame from the Pandacan area lit up the early evening darkness. Electric lights began to pop on all over Manila. Don Bell, official news commentator for the Civilian Emergency Administration, was announcing over KZRH that to save the city from destruction MacArthur had just declared Manila an open city. The blackout was over.\n\nBut the confusion continued all through the night as self-appointed air raid wardens shot out windows showing light. No one knew exactly what was going on. The mood in the oppressive, muggy heat was bizarre and frightening. With the black swirls of smoke clouding the skies, the distant explosions from Cavite, and the occasional hollow booms of self-destruction from Fort McKinley and Nielson Field, it seemed to some as if the end of the world was coming. Yet many night clubs were open. This might be the last hour of freedom.\n\n\u2014\n\nThe hastily organized exodus to Bataan had become a nightmare. Since few military police controlled the traffic, terror and selfishness caused many wrecks and tie-ups.\n\nOn Bataan, hundreds of men and vehicles checked in every hour. Many had no idea where their units were. There were few road signs or markers. The Filipinos in the barrios and villages that suddenly became dumping grounds for thousands stared in wonder as incredible numbers of trucks, cars and guns roared past, coating their bamboo houses with thick layers of dust.\n\nAccording to WPO-3, the civilians on Bataan were to be evacuated. None had been. Moreover, thousands of terrified civilian refugees in the path of Homma's 14th Army were streaming toward Bataan in oxcarts, cartelas and old cars. No one stopped these people as they flooded from the north into the future battleground. The plan also specified defense positions were to be dug by 20,000 civilian laborers, under direction of Army engineers. The trenches and fortifications were still only on paper.\n\nBut food was the biggest problem. WPO-3 called for a food supply of six months. The quartermaster, Brigadier General Charles Drake, since getting his orders on the morning of December 24, had now stocked the island of Corregidor with enough food for 10,000 men for six months. Except for 3000 tons of canned meat and fish already stored on the peninsula, little else was in Bataan. More food was on its way by water, rail and highway, but how many more days would the road to Bataan be open? How many hours?\n\nAccording to MacArthur's plan, Wainwright would delay Homma's main attack, holding the key town of San Fernando until January 8. This would give the men already on Bataan time to dig defenses and Jones time to retreat with his South Luzon Force up through Manila and into Bataan.\n\nBut before midnight the Japanese had broken across the Agno River in a dozen places. The last formidable natural fortification between Homma and San Fernando was breached. It was now obvious that Wainwright's poorly trained, poorly equipped and exhausted troops could not possibly hold until the eighth. Or even the seventh or sixth. In fact, front-line commanders were wondering if they could hold until the first.\n\n**3** On Christmas morning a flying boat approached Hawaii from the northeast. In it was Admiral Chester Nimitz, the man chosen to relieve Kimmel. He was a trim-looking man with hair turning white and piercing light blue eyes. He wasn't too happy about the assignment, preferring a sea command. Down below he could see, even in the rain, hundreds of boats in the great harbor. As soon as the big four-engine plane touched the surface, a launch pulled alongside. Nimitz got in and shook hands with Rear Admiral Patrick Bellinger.\n\n\"What are those boats doing?\" asked Nimitz as they slowly pushed through the oily muck that still covered the harbor.\n\n\"They're full of drowned sailors and Marines,\" explained Bellinger.\n\nNimitz soon found what he had feared\u2014too much pessimism. Morale, since the recall of the Wake Island Relief Force, was at rock bottom. He learned from the fleet surgeon that the shock of the December 7 attack had literally turned some commanders' hair white. In fact the staff generally had become so jittery, the surgeon had been forced to feed some of them sedatives.\n\nNimitz, who had brought with him only a flag secretary, called together his inherited staff. \"There will be no changes,\" he said in his gentle, courteous manner. \"I have complete confidence in you men. We've taken a terrific wallop but I have no doubts as to the ultimate outcome.\" Later he told Kimmel, the man he had relieved, \"You have my sympathy. The same thing could have happened to anybody.\"\n\nIn his Academy classbook, Nimitz had been described as a man \"of cheerful and confident tomorrows.\" From this first day his calm air of serenity and trust was infectious. The complete rehabilitation of spirit at Pearl Harbor, however, would take much longer. It would be several months before the Pacific Fleet would be ready to strike back in force at the Japanese.\n\nOn that Christmas morning, all tapping from the sunken battleship _West Virginia_ had stopped. The last three survivors, clad in blues and jerseys, now lay lifeless on the lower shelf of storeroom A-111. Nearby was a calendar with an X marked on every day from December 7 through the twenty-third.\n\n# PART\n\n# 3\n\n# BATTLE FOR BATAAN\n\n# 10\n\n# _Roads to Bataan_\n\n**1** At 4:00 P.M., December 30, a few men sat on a wooden platform in a clearing outside Malinta Tunnel. As the audience of 150 watched solemnly, Virginia Bewley played \"Hail to the Chief\" on a hand organ. This bizarre ceremony was the second inaugural of President Manuel Quezon and Vice-President Sergio Osme\u00f1a. Chief Justice Jos\u00e9 Abad Santos administered the oaths. Then Quezon, his gray hair long and tangled, began to speak. Although now confined to a wheel chair, he spoke as a young man, his eyes bright with their old fire. The reason for this youthful enthusiasm was evident when Quezon read aloud a message he had recently received from President Roosevelt. \"News of your gallant struggle against the Japanese aggressors has elicited the profound admiration of every American....I give to the people of the Philippines my solemn pledge that their freedom will be redeemed and their independence established and protected. The entire resources in men and materials of the United States stand behind that pledge.\"\n\nFrancis Sayre, United States high commissioner, then spoke. And finally MacArthur stood up, looking to some spectators like a tired hawk. The day before he had narrowly missed death when bombs levelled his headquarters. As a bomb crashed through his bungalow, MacArthur had been standing on a hill counting the 72 raiders. His Philippine orderly, Sergeant Domingo, took off his own helmet and put it over MacArthur's unprotected head. The general pushed the helmet away. Domingo stubbornly pushed it back just as a piece of shrapnel hit it, nicking the sergeant's hand.\n\nThe close brush with death had acted as a tonic to MacArthur and today he, too, had been encouraged by the Roosevelt message. Perhaps, after all the delays and excuses, substantial help was really coming. His quiet, slow voice was a dramatic contrast to Quezon's high-pitched excitement and his audience had to strain to hear.\n\n\"The thunder of death and destruction, dropped from the skies, can be heard in the distance. Our ears almost catch the roar of battle as our soldiers close on the firing line. The horizon is blackened by the smoke of destructive fire. The air reverberates to the roar of exploding bombs.\n\n\"Such is the bed of birth of this new government, of this new nation. For 400 years the Philippines have struggled upward toward self-government. Just at the end of its tuitionary period, just on the threshold of independence, came the great hour of decision. There was no hesitation, no vacillation, no moment of doubt. The whole country followed its great leader in choosing the side of freedom against the side of slavery.\n\n\"We have just inaugurated him, we have just thereby confirmed his momentous decision. Hand in hand with the United States and the other free nations of the world, this basic and fundamental issue will be fought through to victory. Come what may, ultimate triumph will be its reward.\"\n\n\u2014\n\nThe brave new hopes on Corregidor were not reflected on the roads to Bataan. Though the first few days of bumper-to-bumper traffic had ended, the highways were still busy with convoys and single vehicles fleeing into the peninsula ahead of the invaders, their occupants frantically eyeing every plane that approached. But Calumpit Bridge still stood. San Fernando, the key escape town, was still intact.\n\nThe Japanese had completely misunderstood the mass migration to Bataan. General Homma and his staff thought it was a frantic, disorganized flight of unimportant elements of the Philippine Army. Only Homma's chief of staff, Major General Masami Maeda, disagreed with this theory. A student of the Spanish-American War, he had predicted since October that MacArthur would pull his main forces to Bataan. But the Imperial General Staff, the Southern Army leaders and the rest of Homma's staff insisted Manila was the key to the campaign. Once the capital was taken the Philippine campaign would be over, although MacArthur would probably hold out on Corregidor and the end of Bataan for a few weeks.\n\nAnd so that day the Japanese main forces kept driving at Manila from north and south, paying little attention to the lines of vehicles disappearing into the jungles of Bataan. The relationship of the peninsula to Corregidor, lying little more than two miles from its tip, was ignored.\n\n\u2014\n\nIn French Indo-China at Nahtrang, Admiral Nobutake Kondo, commander of the Second Fleet and the staff of General Count Hisaichi Terauchi, commander of the Southern Army, were holding a conference to decide the future of the Southeast Asia campaign.\n\nHomma's rapid march toward Manila and Yamashita's steady drive down the Malay Peninsula toward Singapore had exceeded all expectations. Why not invade Java a month ahead of schedule? This would mean complete occupation of all Southeast Asia while the Allies were still off balance.\n\nEveryone agreed. The first step would be the early transfer of the 48th Division from Homma to the Java invasion force. It was obvious this crack unit was no longer needed in the Philippines.\n\n\u2014\n\nLate that night Brigadier General Jones was returning from the South Luzon front to his rear echelon headquarters at Fort McKinley near Manila. He was tired but in a good mood. The day had been spent in setting up an ambush about 40 miles south of Manila. His men were in position on a ridge and would soon catch the approaching Japanese completely by surprise.\n\nAs Jones approached Fort McKinley he noticed lines of cars heading north. His headquarters troops were apparently being evacuated. Angrily he hurried into his command post. It was almost deserted. Here to his consternation he learned that orders had been changed without his knowledge. South Luzon Force was to retreat immediately to Bataan. MacArthur wanted all troops immediately available sent to Plaridel, a little town north of Manila, to try and stop the main Japanese forces which were heading straight for Manila. Jones was to hold Plaridel with these few troops until the rest of his men swung in behind him and escaped into Bataan.\n\nJones realized something very bad must have happened in Wainwright's North Luzon area. The withdrawal to Bataan was a week ahead of schedule. Jones now followed his troops that had gone ahead, driving north out of Manila on Highway 3. After 21 miles he came to a Y junction. Highway 3, continuing to the left, was the road to Bataan. Nine miles farther north was the important Calumpit Bridge, spanning the wide Pampanga River.\n\nThe right arm of the Y junction, going due north, was Highway 5. Homma was coming down this single road to Manila with 90 per cent of his troops. Jones told his driver to head up Highway 5. In three miles they came to Plaridel. This was a strategic point, for a good road ran six miles due west directly to the Calumpit Bridge on Highway 3.\n\nIt was still dark when Jones reached this little town. Just before dawn he found his new command post already established in a schoolhouse. It took only a quick look at the map to see why MacArthur wanted him to hold Plaridel. Homma's troops had broken through the two Philippine divisions defending Highway 5. A strong Japanese infantry-tank advance force was already only 20 miles north of the town and coming fast.\n\nHomma had two alternatives once he reached Plaridel. If he turned west, he would reach the Calumpit Bridge in six miles and Jones' remaining men in the south would be cut off from escape to Bataan. If Homma continued straight south three miles, he would come to the Y junction and accomplish the same purpose. The answer was simple to Jones: hold Plaridel.\n\nTo help Jones defend it were the retreating 71st and 91st Divisions of Wainwright's North Luzon Force. How much help they would be, he didn't know. By daybreak Jones had put a battalion of his own 51st Division astride the road just north of Plaridel. Then he got into his staff car and told his driver to head farther north. He wanted to reconnoiter. In four miles he ran into exhausted, terrified stragglers from the 91st and 71st Divisions. It looked like the beginnings of a rout. Something had to be done to stop them.\n\nA mile farther north he came to Baliuag, a town of rambling houses and nipa huts.\n\nSoon Brigadier General Luther Stevens, of the 91st Division, passed through. Jones ordered Stevens to take command of the first delaying position so he could return to his command post at Plaridel and devote himself to the over-all situation.\n\nAt 10:00 A.M. General Sutherland, MacArthur's chief of staff, telephoned Jones. Because of the chaotic situation, he said, he was putting Jones in charge of all troops east of the Pampanga River. \"Hold the bridge until the 1st Brigade (Philippine Constabulary) passes over.\" This was the rear guard of Jones' own South Luzon Force. \"Remember, all troops will have to be west of the Pampanga River by 0600, January 1. That's when the bridge will be blown.\"\n\nBy now Homma's advance infantry-tank force had reached Baliuag, only five miles north of Jones' command post, and were organizing an all-out assault, spearheaded by tanks. Though he didn't know it, complete disaster threatened Jones because of a strange confusion in command. Unfortunately Wainwright had not yet been told that his two divisions, the 71st and 91st, had been transferred to Jones. Thinking he still commanded these troops, Wainwright ordered them to retreat immediately to Bataan.\n\nAnd so at 2:00 P.M., just as the Japanese advance infantry-tank detachment was deploying its forces for an attack on Baliuag, the defenders of the town began pulling out in buses and speeding toward the Calumpit Bridge and safety. A few minutes later General Wainwright walked into Jones' schoolhouse command post at Plaridel. \"Take up positions for a close-in, perimeter defense of the Calumpit Bridge,\" he ordered.\n\n\"No, sir,\" said Jones. This would mean abandoning Plaridel and writing off all his troops coming up from the south. \"My orders are to defend the bridge until all my people come through. If I fell back to the river I wouldn't have enough space to slow down the Japs.\"\n\n\"I'm giving you a direct order,\" said Wainwright angrily.\n\nJones, shorter than the angular Wainwright, looked up at him defiantly. It was an unpleasant situation. Wainwright had two stars, he had only one. \"I've been ordered by Sutherland to take command of all troops below the Pampanga River.\"\n\nWainwright was astounded.\n\n\"I'm not going to withdraw,\" said Jones.\n\nOne of Jones' staff officers rushed in breathlessly, excitedly. \"The 71st Division has pulled out of Baliuag!\" he said.\n\nJust then General Stevens walked in.\n\n\"Stop them!\" Jones angrily told Stevens to put the 71st in position between the Japanese and the Calumpit Bridge. Then, as Wainwright left, still upset by the mix-up in command, Jones ran to his staff car. He soon discovered the 71st had already passed through Plaridel and was halfway to Calumpit Bridge. He ordered his driver to follow, but after several miles he realized the chase was hopeless and returned in disgust to the schoolhouse.\n\nWord from Baliuag was alarming. By now Japanese tanks were massing in the east part of town. Jones ordered ten tanks of Company C, 192nd Tank Battalion, to rush north and hit the Japanese before they could launch their own attack. He told Lieutenant Colonel David Babcock to support the American tanks with half a dozen 75-mm. self-propelled guns.\n\nAt 5:00 P.M. after a barrage by Lieutenant Colonel Halstead Fowler's 71st Field Artillery, the only unit which had stayed in Baliuag, 10 tanks commanded by Lieutenant William Gentry moved into the town. As dusk fell American tanks crashed through the nipa huts, tumbling them over and scattering Japanese infantry. American and Japanese tanks mixed in a wild free-for-all in Baliuag's narrow streets, firing at point-blank range. After a brief, savage fight 8 Japanese tanks were knocked out. Gentry pulled back. Now Bab-cock's 75's opened fire, leveling what remained of the town and completely disrupting the Japanese infantry-tank force.\n\nTanks, derided up until now by many infantry officers who knew little of their capabilities or limitations, had paid for themselves that day. They had given Jones the few extra hours he needed. Now there was nothing to do but wait and hope. He went outside the schoolhouse and lay on flagstones near a fountain. The moon was bright, the evening fragrant with the exotic scent of frangipani. He could hear the distant rumble and rattle of trucks and buses coming from the south and heading toward Calumpit. Was there enough time for his rear guard to reach the bridge?\n\n**2** As hundreds of vehicles rolled into Bataan that New Year's Eve, the _Mayon_ , the same interisland steamer Quezon rode to Corregidor, was being loaded at the southern end of the peninsula. Six hundred and fifty men of the 19th Bombardment Group were bound south for the island of Mindanao. They were to rebuild and defend the airfields at the Del Monte pineapple plantation.\n\nAs the steamer left Mariveles Harbor, the airmen's commander, Major Emmett \"Rosie\" O'Donnell, thought caustically of the brief disappointing fight made by the air force. No real money had been spent on Clark Field. It was merely a sod field with run-down hangars and little else. The B-17 had no tail gun and was helpless in a rear attack. When war broke out there were no supplies, no radar, no housing. There were too many sunshiners in the service. Hundreds had taken off in panic after the first bombing. The withdrawal to Bataan had been an unholy mess, a bad, humiliating memory. Now they were probably headed for some other damn foul-up.\n\nThe rock silhouette of Corregidor soon became clearly etched in the moonlight as the _Mayon_ headed south. Then it passed a launch pulling a barge slowly toward Mariveles, the port at the tip end of Bataan.\n\nOn the barge were Lieutenant Colonel Allen Stowell and fifteen men. They were laying a submarine cable of 100 circuits from Corregidor to Bataan. MacArthur had given Stowell a month to do the job. So far the Japanese hadn't discovered what he was doing.\n\n\u2014\n\nAt 1:00 A.M., New Year's Day, Jones rose from his flagstone bed in Plaridel and drove west across the six-mile connecting road to Highway 3 and the Calumpit Bridge. The last elements of the 51st Division were just crossing the bridge. Only his rear guard, left on the other side of Manila to hold off any attack from the south, was missing. When they arrived his entire South Luzon Force would have escaped.\n\nAn hour later the ten tanks of Lieutenant Gentry, which had scored such a brilliant victory at Baliuag, slowly crossed the long bridge. Soon Jones heard a low rumble from the south. The noise increased and in a few minutes trucks and buses approached the long, three-span bridge. Jones, standing at the north end of the bridge, anxiously watched them draw near. It was his rear guard, the Constabulary Brigade of Brigadier General Simeon de Jesus. Every one of his units had been safely pulled out of South Luzon. Now he could call in his last troops, the small covering force holding Plaridel.\n\nStanding near Jones was General Wainwright. He too felt great relief to watch the last of the troops from South Luzon pass by. He walked up to his engineer, Colonel Harry Skerry, and asked if everything was ready for the bridge's demolition.\n\nSkerry was prepared but MacArthur's engineer, Colonel Hugh \"Pat\" Casey, had ordered him to hold off the demolitions until 5:00 A.M. to permit the last elements from the south to clear. There was still a platoon of engineers under Major N. L. Manzano and other demolition groups destroying bridges and key facilities between Manila and Calumpit.\n\nCasey himself was still at MacArthur's old headquarters in No. 1 Victoria Street, Manila. He was at the phone, talking to one of the several hundred civilian mining engineers he had recently commissioned. The man was south of Manila with a few assistants.\n\n\"Calumpit's going to be blown pretty soon,\" said Casey. \"You won't be able to make it in time over the road. Better come up here to Manila and try to get a boat to Corregidor or Bataan before the Nips get here.\"\n\nCasey looked at his watch. It was almost 3:00 A.M. He decided to make a last quick tour of the city and then proceed to Corregidor before it was too late. Casey and his driver left the deserted headquarters building and started slowly through the rubble of the Walled City. Manila was shrouded by great black columns of smoke. The Pandacan area was ablaze. A few hours earlier his men had set off charges on each of the big oil tanks that had survived the fire set by Navy Lieutenant Champlin a few days earlier. The flames still roared so loudly that conversation within a few hundred yards was impossible. Pylons of smoke were also rising from Nichols Field, Fort McKinley and the entire port area. And across the bay the remaining facilities of Cavite Naval Base blazed.\n\nThere was nothing else Casey could do. He and his driver stepped into a launch and headed into Manila Bay. On all sides were smoke and flame. His engineers had done a thorough job. He was satisfied. Not far away, lights from the Manila Hotel twinkled and the nostalgic, distant music of a dance band could be heard.\n\nThirty miles to the north, the last Constabulary truck of Jones' rear guard was just then clearing the Calumpit Bridge. Jones telephoned his chief of staff. \"Get over to Plaridel,\" he said. \"Withdraw the covering force immediately.\"\n\nThis order came none too soon. As the covering force, commanded by Lieutenant Colonel Loren Stewart, pulled out of Plaridel in their trucks, rifle fire came from the north. Homma's advance force broke into the town just in time to see Stewart's trucks disappear west on the road to Calumpit Bridge. At 5:00 A.M. the last of these trucks crossed the bridge and passed a small group of officers\u2014four generals and a colonel.\n\n\"Are all your units safely across?\" asked Wainwright. Stevens, Jones and Brigadier General James Weaver, provisional tank group commander, all replied in the affirmative.\n\n\"All right, Skerry,\" said Wainwright, turning to his engineer. \"Blow it.\"\n\nSkerry asked for a delay. Major Manzano's demolition crew still hadn't arrived and no large explosions to the south had been heard.\n\n\"All right,\" said Wainwright. \"Wait an hour.\"\n\nSkerry made a last-minute check of the four tons of dynamite on the highway bridge. Then he checked the three tons of explosives on the railway bridge which ran parallel 50 yards to the west.\n\nAt 5:45 A.M. Wainwright extended the time of demolition to 6:15. He and Jones, both groggy from lack of sleep, went to a commercial truck overturned in the ditch and got a bottle of champagne. They toasted the New Year and the successful withdrawal of Jones' troops.\n\nDawn was now beginning to break. Scattered rifle fire could be heard coming from Plaridel. Every minute it grew louder. \"The situation is getting serious,\" Wainwright told the other generals. He adjusted his field glasses. In the dim light he saw a Japanese patrol advancing and turned to his engineer. It was almost 6:15 A.M.\n\n\"Skerry,\" he said regretfully, \"we cannot wait any longer. Blow it now.\"\n\nSkerry walked to the demolition group tensely standing at the abutment. \"Blow the highway bridge,\" he told Lieutenant Derrick. \"Then the railroad bridge.\"\n\nEveryone took cover. At exactly 6:15 A.M. there was a terrific rolling roar. Then a second. The sky lit up as debris from the two parallel bridges rained down. Wainwright and the demolition crew went back to the riverbank. In the swirling waters of the wide, deep, unfordable Pampanga River they saw tangled, curled masses of metal. Wainwright's flank was now safe, but he sighed. \"It's too bad we had to destroy such costly and important structures.\"\n\nJones was on his way to Bataan, his South Luzon Force already dissolved. He was again only a division commander. He drove through San Fernando marveling that such a key junction was still undestroyed. Then he headed due west toward Bataan. Soon he caught up to the tail end of his 51st Division. It was moving slowly, painfully, held up by the line of traffic ahead, which wound all the way to the middle of Bataan.\n\nAll morning light bombers of the Japanese 5th Air Group crossed and recrossed the crowded highway, sometimes high, sometimes low, but still none bombed or strafed the retreating troops. Near noon, several planes appeared over Jones. They circled, obviously preparing for attack. The line of vehicles kept crawling forward. There was no other place to go; nothing to do but march and pray. Bombs fell but all exploded in cane fields and rice paddies. It was a miracle, thought Jones, as the planes turned and disappeared.\n\n**3** In Manila stores were boarded up. Thousands were moving out to the hills and villages with cars and carts piled high with possessions. Thousands were moving back into the city after a few days in the country. Many just sat on their heels, waiting stoically for the conquerors.\n\nHomma's troops were converging on the city from two directions. Those from the south were still about 40 miles away because Casey's dynamiters had destroyed so many highway and railroad bridges. In fact, the commander of this force, Lieutenant General Susumu Morioka, had been walking all day just like the infantrymen and was hot and frustrated.\n\nThe main body, under Homma himself, was making better progress from the north. With Jones' covering force at Plaridel withdrawn, there was nothing to stop the drive straight down to Manila, but when they were 17 miles from their goal, Homma halted his men. He wanted them to spruce up and reorganize before entering the city. The general knew that soldiers filthy from combat might pillage. Those in clean uniforms were more likely to parade in orderly fashion.\n\nNear Manila's dock area, Arch Gunnison of _Collier's_ and Carl Mydans of _Life_ were watching citizens looting stores and warehouses. Everything from automobiles to unexposed moving picture film was being carted away in an orderly, systematic, good-natured mass operation involving about 40,000 civilians.\n\nGunnison noticed a man sitting by a pile of straw hats that completely encircled him. \"You've got a lot of hats there,\" he said.\n\n\"You like them, seer?\" The man smiled. \"You like, you take.\"\n\nWhen Gunnison protested the man said, \"You take, seer. Plenty hats. Plenty for you. Plenty for me. Plenty for everyone. I can get plenty more. You take these.\" He started to leave.\n\n\"You'd better not go away,\" warned Gunnison. \"Someone is likely to take these hats.\"\n\n\"This is my pile. No one will touch.\" He pointed to other piles that dotted the streets. One simply did not steal another person's loot.\n\nWhen Mydans returned to his hotel his wife, Shelley, showed him a cable from _Life_. It requested:\n\n>...ANOTHER FIRST-PERSON EYEWITNESS STORY BUT THIS WEEK WE PREFER AMERICANS ON THE OFFENSIVE.\n\nShelley showed him her answer:\n\n> BITTERLY REGRET YOUR REQUEST UNAVAILABLE HERE.\n\nAbout 150 air miles to the south the _Mayon_ was hiding in a cove of the island of Mindoro. At dark it would continue its dangerous trip down to Mindanao. The 650 men of the 19th Bombardment Group were loafing, gambling and griping. All were wondering where they were going and why.\n\n\"They're sending us to Frisco,\" someone told Sergeant James Holcomb. \"To make us into chaplains.\"\n\nAbout noon a four-engine flying boat roared by. It circled, as if to attack.\n\n\"One of our seaplanes thinks we're Japs,\" cried someone.\n\nSergeant Frank Trammell looked up and saw a red sun on the fuselage. \"Meatball!\" he yelled.\n\nAn excited young lieutenant ran along the deck shouting, \"Don't jump in the water if we're bombed. The concussion will kill you.\"\n\nThe flying boat lazily circled, coming lower and lower. A bomb dropped, missing the ship by 50 yards. The excited lieutenant, forgetting what he had told the others, jumped into the water. A dozen men followed. Corporal Durward Brooks, wearing a helmet, thought his head was going to be jerked off when he hit the water. His pistol, canteen and ammunition belt dragged him down. He unbuckled his belt and took off his shoes and finally popped to the surface.\n\nThe second bomb fell short. The third hit the railing, bounced into the water and exploded harmlessly. Major Rosie O'Donnell watched the lone bomber make a leisurely dry-run overhead, then turn back toward the ship. He grabbed a mattress and put it over him. Another copied him and shouted as the bomber approached, \"Drop it, you Jap son of a bitch.\"\n\nFive more bombs fell, narrowly missing the ship. The captain of the _Mayon_ , a Spaniard, shouted, \"Release oil. Make them think we're hit.\" A minute later black smoke poured from the stack and oil spilled into the water. The bomber circled once more and, completely fooled, disappeared.\n\nOn Corregidor, MacArthur was reading a radiogram from Marshall suggesting that President Quezon come to the United States and set up a government in exile. MacArthur quickly drafted a reply. Evacuation of Quezon was too hazardous. Besides, he warned, his departure \"would be followed by the collapse of will to fight by the Filipinos.\" He pointed out that most of his army were Filipinos. \"In view of their effort the United States must move strongly to their support or withdraw in shame from the Orient.\"\n\nAt 8:00 P.M. he asked Quezon to come to his cottage near the mouth of Malinta Tunnel. He showed Quezon the radiogram from Marshall and his own reply. Then he asked the president if he was going to the United States.\n\n\"I'll have to consult my War Cabinet first,\" said Quezon. He immediately called a meeting and read the telegram from Washington. The vote was unanimous to evacuate Quezon from the Philippines.\n\n\"But wouldn't my departure dishearten the Philippine Army?\"\n\nVice-President Osme\u00f1a and the others thought it would have the opposite effect. In America Quezon could get more help. But when the president told them what MacArthur's reply was going to be they urged Quezon to refuse to go. MacArthur knew best.\n\nThe president, not sure what should be done, now wrote a letter to MacArthur. \"I am willing to do what the government of the United States may think will be more helpful for the prosecution of the war.\"\n\nBut he was troubled and his growing doubts were inflamed by his sickness. Roosevelt's message offering the full resources of the United States had thrilled him but when was that help coming? Could he ask his soldiers to keep fighting against hopeless odds unless definite commitments were made?\n\n\u2014\n\nAcross the bay, only 30 miles away, the Japanese were at the gates of Manila.\n\nGreat clouds of smoke covered the city. Flames still soared from the Pandacan oil fields. Fort McKinley, Nichols and Nielson Fields and Cavite were still burning amidst occasional explosions. Manila, a beautiful city a few weeks before, was filthy with the ashes and soot of destruction.\n\nAt 5:45 that afternoon, Major General Koichi Abe led three battalions of his 48th Division into the city from the north while General Morioka, who had borrowed a car from Japanese civilians, led the way in from the south. Japanese freed from internment cheered wildly as Morioka and Abe passed. The lines of Filipinos only stared stolidly, sullenly.\n\nFrom their rooms in the Bayview Hotel, Carl and Shelley Mydans and Arch and Marjorie Gunnison watched two companies of Japanese sailors and a company of infantry form ragged lines on the lawn in front of High Commissioner Sayre's residence, just across the street. The American flag was lowered from a flagpole. Three small cannon fired as the flag fluttered to the ground. A sailor unhooked it, stepped on it and fastened in its place the emblem of the Rising Sun. As the Japanese flag slowly rose, the band played the _Kimigayo_ , the national anthem.\n\n# 11\n\n# _Action at Abucay_\n\n**1** On January 1, the Calumpit Bridge was the focal point of the American retreat into Bataan. Four days later action centered on another bridge. This, the Layac Bridge, was the only entrance to Bataan and was situated at the neck of the peninsula.\n\nAt 10:00 P.M., January 5, there were still more than 8000 Americans and Filipinos north of the bridge, waiting to escape into Bataan. On their heels, Japanese were pressing from two sides\u2014from the east along the main highway from San Fernando, from the north on a dirt road from Clark Field.\n\nThe greatest danger lay in the north. Holding off the Japanese on this dirt road was the 21st Division of Brigadier General Mateo Capinpin. Capinpin was now standing at Layac with his chief adviser, Colonel Ray O'Day. Both were worriedly watching a confused mass of trucks, guns, tanks and exhausted marching men from the 11th Division pour over the narrow bridge. Until these men passed, Capinpin's division, stretched out to the north on the dirt road for several miles, was at the mercy of Japanese planes or artillery\u2014or, worse still, a sudden infantry-tank attack.\n\nWainwright and his engineer, Colonel Skerry, now arrived at the bridge. A few minutes later, at 10:30 P.M., the last man and last vehicle of the 11th Division crossed.\n\nThere was a short pause. Then Capinpin's impatient division surged into Bataan as if they were entering the promised land. To O'Day it was a painful and tragic sight. His exhausted troops trudged along toting ridiculously large loads of equipment and personal effects. Many teams of two carried bundles of clothing, ammunition belts, helmets, even chickens on bamboo poles.\n\nAs they slowly dragged by, General Capinpin excitedly hurried up and down the columns kicking and driving the men to go faster. O'Day was amused: Matty was actually the kindest commander in the world.\n\nWainwright joined the two. \"How many men have you shot today, Matty?\" he joked.\n\nBy 1:00 A.M. all the foot troops and tanks from the north had crossed. Now word was sent for the platoon of tanks on the east road, the last covering force, to withdraw. O'Day reported to Wainwright. He held up his right hand and said, \"To the best of my knowledge and belief all units of the 21st Division and tanks have cleared the junction.\" Then O'Day and Capinpin walked across the bridge.\n\nWainwright and Skerry followed.\n\nA minute later the tank commander of the covering force reported to Colonel Skerry. He too raised his right hand and swore all his tanks had crossed. Suddenly there was a rumble in the darkness. A United States tank surged from the gloom and rattled across the bridge.\n\nWainwright turned to Skerry. \"Blow it,\" he ordered. Skerry signaled to Captain A. P. Chanco. At 2:00 A.M. there was a roar. The road to Bataan was severed. Wainwright shook his engineer's hand.\n\nSeveral miles south of the blown bridge, General Capinpin and O'Day were overtaking their toiling, trudging infantrymen. Capinpin, still worried and anxious, ran out of his staff car and again began cursing the men.\n\n\"They've had a hard march for almost twenty-four hours, Matty,\" pointed out O'Day.\n\nThe general got back in the car and as it passed the troops he now leaned out and solicitously called, \"The trucks will come back for you! You'll get your truck!\"\n\n\u2014\n\nGeneral Homma still had no suspicion that MacArthur had fooled him, escaping with the great bulk of his Luzon forces into Bataan. Even so, he was very disturbed. Higher headquarters, assuming the fall of Manila meant the virtual end of the Philippine campaign, had ordered Homma to ship at once the heart of his army, the 48th Division, to Java. The general knew the mopping-up operations would be difficult and the loss of his best division would place an extremely heavy burden on his remaining troops.\n\nThe Japanese chasing MacArthur into Bataan came from the about-to-be-relieved 48th Division. Their job was about to be inherited by the 65th \"Summer\" Brigade. Its commander, Lieutenant General Akira Nara, was surprised when he was told his 7500 troops were to go into combat. The brigade, mostly older men, was almost totally unprepared and unequipped for front-line fighting. It had been sent to the Philippines merely as an occupation force.\n\nNara, a well-known tactician and administrator, had been picked originally to command these garrison soldiers because of his many years in the United States. A graduate of the Fort Benning Infantry School, he had also been a classmate of President Coolidge's son at Amherst College.\n\nOn the night Layac Bridge was demolished, Nara\u2014a stocky, middle-aged man\u2014was walking to battle. He was about 10 miles behind the Japanese troops he was supposed to relieve.\n\nBehind him, stretching halfway back to Lingayen Gulf, were his footworn, forlorn troops already delayed for days by Colonel Skerry's engineers, who had left 184 destroyed bridges behind them. Unused to marching in the tropical heat, many were sick. Colonel Takeo Imai, commander of the 141st Infantry Regiment, was having trouble keeping his unconditioned men on the move.\n\nThe tropical evening was beautiful. The air was fragrant with the heavy sweetness of tropical flowers. Every so often Imai would pass bushes clustered with fireflies. They reminded him of Christmas trees. Suddenly he wondered what was delaying his troops up ahead and sent a runner forward. His men were bogged down in a stream. They had put their heavy equipment and supplies in carts drawn by sturdy Filipino carabao, usually so gentle that children often slept on their broad, armored backs. All had gone well for several miles. Then the water buffaloes, smelling the stream, had waded in, relentlessly dragging the carts with them.\n\nUnder harrowing conditions MacArthur had brought his troops safely into Bataan without losing a single major unit. Wainwright, in his difficult delaying fight, had lost 12,000 men but most of these were desertions. Jones had lost 1000.\n\nNow MacArthur had 15,000 Americans and 65,000 Filipinos in Bataan. Ten thousand were professional soldiers of the elite Philippine Division. The rest was a conglomerate ill-equipped group almost totally untrained. With this force he was supposed to hold out for six months.\n\nHis greatest asset was the rugged terrain of Bataan. As it was largely covered with mountains and thick jungles, defense positions could be hidden from air observation. The peninsula, 15 miles wide and 30 miles long, was almost completely occupied by two great extinct volcanoes, one in the north, one in the south. There were only two roads. The main highway from Layac Bridge, the gate to Bataan, led down the flat, swampy east coast, around the tip and two-thirds of the way back up the other side of the peninsula; a cobblestone road cut across Bataan like a belt through the valley between the two volcanoes.\n\nMacArthur had already ordered 5000 men to fight a delaying action a few miles below the Layac Bridge so the battle lines could be completed and manned. The main battle position, where the first real stand would be made, was about 10 miles south of Layac. It ran from Manila Bay across the northern volcano. The mouth of this volcano, dead for eons, had been eroded by time into four jagged peaks. The eastern and highest peak was rugged, precipitous Mt. Natib.\n\nThe inexperience of his troops didn't worry MacArthur as much as the serious shortage of food and clothing. In eight days his quartermaster, Brigadier General Charles Drake, had sent numerous launches, tugs and over 300 bargeloads of supplies from Manila to Bataan. In the last three days of December 200 American, British and Allied civilians, in response to a radio plea, had helped Colonel Frederick Ward load last-minute supplies on the docks of Manila in spite of bombings and rumors of approaching Japanese patrols.\n\nRelatively few supplies had come by highway. The shortage of trucks was only partially to blame. Because of red tape, stupidity and sheer panic, much food and clothing were being abandoned at Forts McKinley, Stotsenberg and other depots. The commander of the Tarlac Depot, Lieutenant Colonel Charles Lawrence, was forbidden to take 2000 cases of canned food and much clothing belonging to Japanese firms. MacArthur's headquarters said he had no right to commandeer personal property and threatened him with court-martial. But the greatest loss was at the Government Rice Central at Cabanatuan. Because of Commonwealth regulations forbidding rice to be transported from one province to another, 10,000,000 pounds of rice\u2014enough to last the Filipinos on Bataan for six months\u2014were left in the warehouses.\n\nThat night MacArthur checked the food inventory just prepared by General Drake. There were only enough unbalanced field rations for 100,000 men for thirty days. Desperate measures had to be taken. He ordered everyone on Corregidor and Bataan to be placed on half-rations.\n\nBy the morning of January 9 MacArthur's men were in position. Morale was high. The men of Bataan were tired of running. They wanted to stand and fight. MacArthur split the Main Battle Position in two, assigning the left, the western half, to Wainwright. His men, badly battered from their chaotic retreat from Lingayen, were in no condition for immediate combat.\n\nThe first Japanese attack, obviously, would come on the other side of Bataan, down the east coast road from the Layac Bridge. This eastern half was turned over to Major General George Parker and 25,000 men. Though Parker had reached Bataan on Christmas Eve, his defense positions were only partially constructed. His problems were varied. The east coast of Bataan was flat and swampy, with fish ponds and rice paddies extending inland for about two miles. Then came gradually rising hills covered with cane fields and patches of bamboo for another five miles. At this point rugged Mt. Natib, scarred by innumerable ravines and covered with huge trees and dense jungle growth, began to rise dramatically.\n\nParker's left flank ended here for he felt it was not only impossible to defend Mt. Natib but unnecessary. No military force could possibly come over that matted mass of precipitous crags, ravines and cliffs.\n\nGeneral Jones' 51st Division held this difficult left flank. One regiment extended up the slopes of Natib trailing off into scattered foxholes; to the right his other front-line regiment ran just in front of Abucay Hacienda, a small cluster of nipa shacks housing sugar-cane workers.\n\nThe middle of Parker's line, along the steep banks of the gorge above the Balantay River, was held by the untried 41st Division commanded by a Philippine general, Vincente Lim, a West Pointer and part-Chinese. Parker's extreme right flank, rice paddies and fish ponds bisected by the coastal highway, was given to a regiment of the elite Philippine Scouts, the 57th Infantry. Parker wanted his best troops here, the most obvious point of attack.\n\nThis was the Abucay Line. The Filipinos were anxious to show MacArthur that they deserved his faith and that the routs of the 91st, 71st and 21st Divisions on the hectic retreat from Lingayen Gulf hadn't been a fair test of the Filipino soldier.\n\nTheir American instructors, many now disgusted and frustrated with their charges, were not as sanguine. Here on the Abucay Line retreat would be difficult. It was fight or die.\n\n\u2014\n\nA few miles to the north, General Nara's 7500 men had relieved the veterans of the 48th Division and were getting ready to launch an attack on the Abucay Line. Less than half of Nara's overage soldiers had rifles; all were exhausted from the long march to Bataan. But what worried the general most was his almost complete ignorance of the enemy's whereabouts. To guide him he had only a road map and several 1\/200,000 scale maps. The 14th Army had given him no plan of attack. His instructions were merely to pursue the enemy in column down the highway.\n\nFor six years he had been a professor at the War College, teaching his pupils never to attack without accurate maps and proper preparations. A member of Homma's staff had assured Nara, however, that the job of capturing Bataan would be simple. Intelligence estimated there were at most 25,000 disorganized troops on the peninsula. These men would retreat almost immediately to Mariveles at the tip end, make a brief stand here and then withdraw to Corregidor Island.\n\nAlthough Nara protested, asking for time to make a survey, he was ordered to attack immediately. He hastily made plans. In addition to his own troops he had inherited two artillery regiments and the 9th Infantry Regiment of the 16th Division. He was delighted to get the 9th for it was commanded by an old and trusted friend, Colonel Susumu Takechi, an extremely aggressive and capable officer.\n\nNara's plan was simple; it had to be, with only a day for organization. Since the enemy was apparently weak, he ordered his own 141st Infantry, commanded by Colonel Takeo Imai, to attack straight down the coastal highway. Keeping his third infantry regiment in reserve, he told his friend, Takechi, to head inland with the 9th Infantry to the west, toward the slopes of Mt. Natib and circle behind the American positions, finally joining Imai at the highway.\n\nAt 3:00 P.M. there was a roar of Japanese artillery that seemed to Nara to shake the northern part of Bataan. When the hour-long barrage lifted, Imai headed down the highway and Takechi struck off into the wild jungle. Suddenly, to Nara's dismay, there was an even greater roar as tons of explosives from Parker's artillery fell on the highway in front of Imai. It was immediately obvious that the intelligence reports about American weakness were not accurate. What he still didn't realize was that Homma's intelligence officers had mistakenly placed the American defenders too far north. Nara's men wouldn't reach the Abucay Line for another three miles.\n\n\u2014\n\nThat night Wainwright and Parker both got a message from Corregidor. They were to assemble all general officers to receive an important visitor the next morning. At dawn a PT boat dashed from Corregidor to Mariveles. MacArthur and his chief of staff, General Sutherland, stepped out. They drove up the east coast road to Parker's headquarters where MacArthur gave an inspiring talk, promising planes and heavy reinforcements soon. Then he drove across the middle of Bataan on the winding 15-mile cobblestone Pilar-Bagac Road to inspect Wainwright's positions.\n\nMacArthur's Ford stopped in front of I Corps headquarters. The general walked up to Wainwright. \"Jonathan,\" he said cordially, \"I'm glad to see you back from the north. The execution of your withdrawal and of your mission in covering the withdrawal of the South Luzon Force was as fine as anything in history.\"\n\nWainwright wondered if he deserved that much praise, but said nothing.\n\n\"Where are your 155-mm. guns?\" asked MacArthur.\n\nWainwright suggested they walk over and see them.\n\n\"I don't want to _see_ them,\" replied MacArthur characteristically, \"I want to _hear_ them.\"\n\nSutherland told Wainwright he was worried by what he'd seen that day. There was a gap of more than five miles between the two corps. Mt. Natib was completely undefended. Wainwright did not agree. The terrain, he argued, was too difficult for major attack.\n\n\u2014\n\nBy the morning of January 11, Nara realized the defenders hadn't retreated and that Imai's regiment was only now running into the American Main Line of Resistance. To escape the heavy bombardment Parker's artillery was laying on the coastal highway, Imai shunted most of his regiment into the fields to the west. He ordered one battalion to hide in a tall cane field near the highway and launch an attack late that night.\n\nThe defenders of the highway, the well-trained Philippine Scouts of the 57th Infantry Regiment, felt secure. Out front were heavy barbed-wire entanglements; fields of fire had been cut through brush and bamboo clumps. They were in perfect position to stop anything\u2014infantry or tank\u2014coming down the road, but they had completely neglected the large sugar cane field just west of the road. It hadn't occurred to anyone to cut it down.\n\nAt 11:00 that night Japanese artillery fire began to fall on the left flank of the Philippine Scouts. Then rifle fire, surprisingly, came from the cane field. The front-line troops called for an artillery concentration, but as soon as American shells began to fall, a mass of dim figures leaped from the cane in the moonlight.\n\nYelling \" _Banzai!_ \" the first attackers threw themselves against the barbed wire in front of the Philippine Scout positions. Comrades, using them as stepladders, leaped over the wire like acrobats and then charged at the main line. Now they stopped yelling and the defenders in foxholes could only hear Japanese officers and noncoms calling orders.\n\nTracers seemed to come from every direction as infiltrating Japanese worked behind the Scout positions. Officers waving samurai swords led the attackers forward. Wave after wave of screaming Japanese flung themselves through the withering fire. The Scouts wavered, then broke.\n\nMajor Pete Wood, the executive officer of the battalion holding this sector, hurried to the rear to bring up his reserve company. He was an amiable, easy-going man but he thrived on combat. In a few minutes he reached the reserve company and told its commander, Captain Ernest Brown, to prepare his people for an attack.\n\nThe eager Wood called together Brown's noncoms and personally formed the reserve company in a skirmish line. As he started leading the men forward he noticed Brown sitting in a rice paddy smoking a cigarette. \"What the hell are you doing?\" he said.\n\n\"You're giving all the orders,\" said Brown. \"You take charge.\"\n\nWood grinned good-naturedly and called back the noncoms. \"Disregard what I said. Captain Brown will take over.\"\n\nBrown jumped to his feet. \"Follow me!\" he said, and ran as fast as he could across the open field in the gray sickly light of early dawn. Sporadic rifle fire came from the cane field. As Brown jumped over the first string of Scout barbed wire, he heard a rip. His canteen belt and holster were hanging on the wire. He grabbed the pistol, noticing dirt dancing around in the dust a few inches away. He leaped into one of the deserted Scout foxholes. He looked back and saw Wood, with drawn pistol, stalking a Japanese rifleman behind a truck. Wood dashed forward and fired point-blank at the Japanese. He missed and the Japanese ran around the truck, Wood chasing. During the second round trip Wood began to laugh. Brown started to laugh too. It was like a Mack Sennett comedy, until the third trip around the truck when a Filipino shot the Japanese.\n\nIn the full light of morning Brown could see about 250 dead Japanese in the field, but there were still Japanese infiltrators in the battalion position. While Brown was leading a mop-up party, he heard a cry for help. A badly wounded Japanese was holding his hands out entreatingly. Brown told his men to give any wounded enemy first aid. After the Japanese was bandaged, a presumably dead Japanese lying nearby grabbed a rifle and fired. A Philippine Scout dropped dead. It was an experience Brown would never forget. He decided that mercy could not be shown to a wounded Japanese, nor respect for their dead. They had to be bayoneted.\n\n\u2014\n\nSecond Lieutenant Alexander Nininger, Jr., a recent graduate of West Point, now came over to help Brown mop up. Recklessly he ran from foxhole to foxhole, wiping out dug-in Japanese with grenades. Finally a burst from a machine gun cut him down. Locating its position, he staggered forward. Just as he hurled a grenade, a burst of fire caught him in the face. He dropped dead\u2014and became the first to win the Medal of Honor on a World War II battlefield. Even as he died he had killed the machine-gun crew, completely clearing the area in front of the cane field.\n\n\u2014\n\nColonel Imai, about a mile and a half to the rear, had heard the fierce attack and counterattack. Now he was getting reports of the action. His battalion commander was dead and so were two-thirds of the attacking force. It had been a disaster.\n\nAt this moment he got a phone call from Nara. \"Move to the west,\" said the general. \"I want you to become the brigade right flank. I'm bringing up the reserve regiment to take your place on the left flank.\"\n\nThis unusual order was caused by the mysterious disappearance of Nara's entire right flank. Colonel Takechi's regiment had vanished in the heavy jungle of Mt. Natib. Nara didn't report this to Homma. Nor did he record it in his war diary or the brigade report. Takechi had been a classmate at the Academy and Nara wanted to protect his friend's reputation.\n\nBy dark that night, Imai had shuttled his regiment to the west and filled in the hole left vacant by Takechi's strange disappearance. He began to probe for a weak spot in the west flank of the Abucay Line. He couldn't make a frontal attack since he was down to about 500 men and half of these were not infantry and had no rifles. He would have to slip through a hole in the American lines and attack from the rear. At midnight groups of Japanese infiltrated through the Abucay Line near the junction of Jones' 51st Division and Lim's 41st Division. They had orders from Imai to make hit-and-run terror attacks, giving the impression of great numbers.\n\nBefore dawn Lim's division, to the right of Jones', was hit by one of these Imai attacks. Lim's left flank bent back. Seeing this withdrawal, the adjoining right flank of Jones' division also pulled back. No one knew what was happening in the deep ravines and heavy jungle. Terror spread like fire. Isolated Imai groups behind Jones' men began to pick off the defenders from the flanks and rear. Rumors that great masses of Japanese were on all sides spread from foxhole to foxhole.\n\n**2** That same day, January 13, a sudden dangerous rift threatened to split Fil-American relations. On Corregidor, Quezon was writing a radiogram to Roosevelt. Almost two weeks had passed since Roosevelt's pledge to help the Philippines, yet not a new plane had arrived, not a new company of infantrymen. In blunt terms Quezon asked Roosevelt to direct the full force of American strength against the Japanese without delay. Still burning with indignation, he wrote an explanatory note to MacArthur:\n\n> This war is not of our making....We decided to fight by your side and we have done the best we could and we are still doing as much as could be expected from us under the circumstances. But how long are we going to be left alone? Has it already been decided in Washington that the Philippine front is of no importance as far as the final result of the war is concerned and that, therefore, no help can be expected here in the immediate future, or at least before the power of resistance is exhausted? If so, I want to know, because I have my own responsibility to my countrymen....\n> \n> I want to decide in my own mind whether there is justification for allowing all these men to be killed when for the final outcome of the war the shedding of their blood may be wholly unnecessary. It seems that Washington does not fully realize our situation nor the feelings which the apparent neglect of our safety and welfare have engendered in the hearts of the people here....\n\nMacArthur was deeply disturbed by Quezon's abrupt change of heart but he felt that the Filipino was justified. Washington kept complaining of the difficulty of sending things up through the blockade the Japanese had thrown between the Philippines and Australia. MacArthur felt it was a paper blockade. A good determined push by a naval task force could break through and save the islands. The trouble, he believed, was that the Navy had been terrorized by Pearl Harbor.\n\nAnd so MacArthur forwarded Quezon's message. He hoped it would stir them up in Washington.\n\nOn the Abucay Line, Jones was in a dangerous position by January 15. His 51st Division was quickly becoming demoralized by the incessant behind-the-lines attacks of Colonel Imai.\n\nJones' best regiment, shot up on the retreat to Bataan, was in reserve. The two regiments on the line were green, and in Jones' opinion the outpost position was untenable. Jones now reported to Parker that his troops were perceptibly weakened and that unless he got reinforcements, he might have to withdraw his outpost.\n\nParker, in turn, asked MacArthur for reinforcements and as a result the remaining two regiments of the experienced Philippine Division were ordered to rush to the troubled spot, the west flank of the Abucay Line. Encouraged by this help, Parker decided to regain the ground lost by Jones, ordering him to counterattack the next morning at dawn. To Jones this was a ridiculous order. In the blunt language he used alike with subordinates and superiors, he told Parker his division had been seriously weakened from continuous combat losses the past month. Such a counterattack would be extremely hazardous. \"Moreover,\" he added, \"the present position is being held only with great difficulty.\"\n\nIn spite of this vigorous protest, Parker insisted. He was sending up a battalion to help Jones. It should arrive at 4:00 A.M. Angrily Jones prepared his exhausted and demoralized troops for attack. If Corps people came up to the front once in a while, he thought, they'd see what the real situation was.\n\n\u2014\n\nAll along the Abucay Line that night company commanders were reading a message from MacArthur designed to whip up the flagging spirits of the tired defenders, even though the commander-in-chief himself had doubts of Washington's intentions:\n\n> Help is on the way from the United States. Thousands of troops and hundreds of planes are being dispatched....No further retreat is possible. We have more troops in Bataan than the Japanese have thrown against us; our supplies are ample; a determined defense will defeat the enemy's attack....\n> \n> I call upon every soldier in Bataan to fight in his assigned position, resisting every attack. This is the only road to salvation. If we fight, we will win; if we retreat, we will be destroyed.\n\nSome of the men, mainly American noncoms, openly jeered. Ample supplies? They were already on half-rations. Their grenades were no good, only one in four or five exploding. Their old Stokes mortars were more dangerous to them than the enemy. Six out of seven rounds failed to detonate on landing; and too often the ill-fitting shells burst the barrels.\n\nBut to the great majority, the Filipinos, MacArthur's words were a bright hope and inspiration. The Filipinos were determined to prove they were men of courage, worthy to fight under the American flag.\n\n**3** About 150 miles north of the Abucay Line, in the mountains above Baguio, the old summer capital, Lieutenant Colonel John Horan, former commander of Fort John Hay, was walking toward Mankayan, home of the Surjac Gold Mine and the Lepanto Copper Mines. On Christmas he had led his 184 officers and men from cut-off Baguio in a march up and down precipitous trails. Now he was trying to organize a guerrilla army of Igorots, a proud, fierce tribe whose warriors still wore G-strings. Two weeks previously, he had made his decision to stay in the mountains and fight rather than try and break south to Bataan. Calling together all his officers, he told them he was going back into the mountains and conduct guerrilla warfare. It had been the hardest decision of his life but he felt that after taking salary for over twenty-five years, he owed it to the Army.\n\n\"You have my authority,\" he said, \"to go on your own and try to get through the enemy into Bataan. Or you can return with me into the mountains.\"\n\nSeveral elected to leave but 184 officers and men stayed with Horan. Now on the morning of January 16 he was holding a meeting with the miners of Mankayan.\n\n\"I want you to help me organize a guerrilla regiment of Igorots,\" he said. \"We will blow up bridges, roads and overhanging cliffs. We'll create avalanches over road fills on hillsides, ambush troop and supply columns and destroy warehouses and supply dumps.\"\n\nEvery man volunteered. Horan enrolled them into the 31st Infantry and promised to recommend them all for commissions as soon as he could make radio contact with MacArthur. After the meeting, Horan headed back toward Baguio to look for a miner named Walter Cushing, already a legend in the mountains for his daring raids on Japanese columns. Cushing would make a good assistant.\n\n\u2014\n\nOn dawn of the same day Jones started his counterattack, even though the promised reinforcements had failed to arrive. Heavy resistance was met but one of his regiments pressed forward so aggressively that the Japanese fell back. Before noon this regiment had pushed ahead of units on both sides.\n\nColonel Imai, realizing the Americans had formed a salient more dangerous to themselves than to him, quickly attacked the east side of the bulge. Just then, purely by chance, Colonel Takechi and his \"lost\" regiment burst out of the jungled slopes of Mt. Natib and found themselves on the other side of the Jones bulge. Takechi also attacked.\n\nJones' salient, assaulted from opposite sides, wavered, then began to crumble. By noon, when increased firing was heard near the rear, it completely collapsed. The green troops scattered in terror leaving a two-mile-wide hole in the Abucay Line.\n\nPart of Takechi's troops, still vague about their position, now began assaulting the western flank of Imai's regiment, but Takechi soon discovered he was fighting friends. He hurried to General Nara's field headquarters to make a personal report. The colonel\u2014uniform in tatters, his face lined with fatigue and hunger\u2014explained how he had become hopelessly lost in the jungles of Mt. Natib. Nara listened silently, sympathetically, then said, \"I'm putting you in reserve.\"\n\n\" _Hai_ , General,\" said Takechi. He saluted crisply and left.\n\nNara was greatly relieved. Now with Takechi in reserve and his rear protected, Nara could attack with confidence. He gave orders for increased artillery fire.\n\nBut Takechi was not heading north into reserve as ordered. Without pausing for supplies or rest, he was already leading his weary regiment back to the south. This time he was determined to take his men over Mt. Natib or die trying. To be ordered into reserve meant a great loss of face, and Takechi had mistakenly thought Nara had done this to punish him for getting lost.\n\nJones, as usual, was far up front. He didn't know that back at Division command post his staff had become alarmed by reports and ordered the regiment on the left to fall back before it was surrounded. By mid-afternoon that regiment had pulled out of its position and was hurrying to the southwest across the rugged slopes of Mt. Natib. The entire west flank of the Abucay Line had disappeared.\n\nOnly at dusk did Jones finally learn his division had evaporated. At a key junction of the Guitol Trail about two miles behind the Abucay Line, he grabbed terrorized men fleeing from the front and put them astride the strategic trail; but almost as fast as he would put them in place, they would sneak farther to the rear.\n\nAt dark Jones had only 100 men dug in. It was all that remained of his 51st Division; the smallest division in the world, he thought sardonically. And these 100 had to hold the entire left flank of the Abucay Line. From his new command post in a muddy, slimy hole just off the trail he telephoned Parker's headquarters far in the rear. After a long wait he reached the commander of II Corps. \"As far as I'm concerned,\" said Jones disgustedly, \"I'm through.\" He hung up. Obviously he'd be busted and relieved of command, but until then he'd keep fighting.\n\nNow he telephoned his aide, Lieutenant Pete Perkins, commanding the 100-man division. Perkins had no infantry training.\n\n\"They're infiltrating,\" said Perkins.\n\nJones, muffling the bell of the field telephone with toilet paper to prevent Japanese patrols from hearing its ring, advised Perkins how to handle each development. Finally he lay in the mud, telephone to his ear, and tried to rest.\n\nThe last three miles on the western end of the Abucay Line had dissolved. Except for Jones' 100 men on the trail, nothing was left to stop the Japanese, but at last luck was with the Fil-Americans. Colonel Imai, wary of being cut off himself, had turned his attack back east toward Lim's division. And Colonel Takechi was circling far west of the Abucay Line, in a quixotic dash to regain honor.\n\nBy midnight Lim's 41st Division was punch drunk, but the unseasoned Filipinos hung on in spite of casualties already exceeding 1200.\n\nIn the inky darkness, the two crack regiments of the Philippine Division were moving forward as fast as possible to plug up the giant hole. \"Come on,\" an ex-football star, Captain Jack Ellis, of the 31st Infantry (U.S.), the only all-American regiment in the Philippines, told those retreating, \"now you're going in with the big team.\"\n\nBut the other regiment, the Philippine Scouts of the 45th Infantry, was lost. They had no moon, no stars, no sign, and often no distinct trail to guide them. Their column stopped. Not far to the north Captain Louis Besbeck could hear the fast firing of artillery and see splashes of light in the blackness. Groups of Filipinos drifted past from the north, eyes shocked, yelling, \"The Japs are coming!\"\n\nBesbeck stopped one who was wearing the single star of a brigadier general and asked for road directions. \"It was awful,\" answered the general dazedly, obviously suffering from battle shock. \"They just faded away.\"\n\nBesbeck repeated the question.\n\n\"It was awful. They just faded away.\" Other figures came out of the darkness from the north. The general, still babbling, followed.\n\nAbout a mile to the left, another one-star general, Jones, was trying to get some sleep for the first time in three nights. He was deathly tired. The six men at his muddy command post were holding one another's ankles, ready to alert each other if a Japanese patrol appeared. For the past hour his aide, Captain Andr\u00e9 Soriano, had kept pulling Jones' ankle. Each warning had been a false alarm. Again Soriano yanked Jones' ankle.\n\n\"Damn it, Andr\u00e9,\" said the general, \"if you wake me up again, you'll have something to really worry about.\"\n\nA moment passed. Again came the jerk of warning. Jones leapt to his feet. \"Are there any Japanese sons of bitches out there?\" he shouted. \"If there are, let's get it over with.\" He yanked out his pistol and fired several times. \"Now they know damn well where we are, and we can get some sleep.\" He flopped down and went to sleep instantly.\n\n# 12\n\n# _Retreat_\n\n**1** Until now, midnight of January 16, very little had happened on the west side of Bataan, from Mt. Natib to the South China Sea. The reason was obvious. There wasn't a single road leading down into Bataan on this coast. Here the mountainous jungle continued all the way to the South China Sea.\n\nBut Wainwright's vacation was coming to an end. Five thousand Japanese were quickly closing in on his front-line positions by boat and trail. They had already reached the American front line at Moron, a town about a third of the way down the west coast. By late afternoon of January 17, Wainwright was pushed out of Moron and retreated to a ridge several miles behind the town.\n\nThe next morning the Japanese commander, Major General Naoki Kimura, arrived in Moron. When he learned that Wainwright's defense line extended only five miles inland from the sea to the western slopes of the great volcano, he decided to send a battalion of about 700 men around Wainwright's right flank and cut off the Americans from the rear. Lieutenant Colonel Hiroshi Nakanishi, well known for his aggressive spirit, was selected as leader. By dawn of January 21, Nakanishi had secretly circled around Wainwright's line. Then he turned sharply west and at 10:00 A.M. reached the west coast highway which ran from Moron to the tip end of Bataan. In one brilliant stroke he had cut off Wainwright's 5000 front-line troops.\n\nWhen word reached Wainwright at I Corps Headquarters that there was a Japanese roadblock between him and his front line, he quickly gathered the men available, only twenty, and drove north. After several miles of mountainous road the Wainwright party came to Nakanishi's roadblock. Here the general found a platoon of Filipinos, and led them, and his own twenty men, in a futile two-hour attack. While standing against a high bank on the east side of the road, his orderly, Sergeant Hubert Carroll, suddenly grabbed the seat of Wainwright's pants and yanked him down.\n\n\"God damn it, General,\" said Carroll in his Texas drawl, \"get down or you'll get your damn head shot off.\" Just then there was a burst of rifle fire and two men standing next to the crouching general fell dead.\n\nMeanwhile a company under First Lieutenant Beverly Skardon had been sent back from the front and was hitting Nakanishi from the north, but the well-dug-in Japanese threw back this attack too, determined to die rather than be dislodged.\n\n**2** That same night, January 21, Nara's constant attacks on the other side of Bataan had brought the defenders of the Abucay Line to the edge of their endurance. The two experienced regiments of the Philippine Division\u2014sent in to plug up the vanished left flank\u2014had become bogged down in the dense vegetation and rugged crevasses. The past five days they had attacked aggressively and fiercely but were learning that tying down a flank on a rough, unknown piece of terrain was impossible. Many units had had no food or water for several days nor could they evacuate their wounded.\n\nTo their right, General Lim's 41st Division was also groggy from constant fighting, hunger and exhaustion. When artillery wasn't falling, planes were strafing and bombing, or Imai's infantrymen were making slashing flank attacks. And to heighten the terror, the Japanese kept shooting firecrackers into the rear. When these exploded the cry would often go up, \"We're surrounded!\"\n\nThat night General Lim and his staff could hear small arms and machine-gun fire from their 41st Division command post. They conferred in whispers. \"It looks bad,\" said Colonel Malcolm Fortier, the division instructor. All reserves had been committed.\n\nA little later Captain Rigoberto Atienza, one of Lim's staff, heard the Filipino telephone operator mumbling in his sleep, \"My guardian angel is with MacArthur.\" No one paid any attention to the muttering. Suddenly the operator sprang up, shouting, \"You Americans, why are you standing back there? Why do we keep on defending? Are you afraid to attack? I'll lead the attack. My guardian angel is with MacArthur!\"\n\nWaving a pistol, he ran into the woods. A little later the operator came back, his eyes wild. \"Why didn't you follow?\" he screamed angrily. \"If you are afraid to lead, I'll lead. You cowards! That's why we're losing this war.\"\n\nAtienza and several others tried to soothe the man. They told him he was tired and should rest.\n\n\"I am not tired. But these goddam Americans are cowards and we're fighting for them.\"\n\n\"Shut up,\" said the assistant operations officer, \"or I'll slap you down.\"\n\n\"Sir, I do not quarrel with you. But these Americans are no good.\"\n\nA big figure came out of the shadows and said in a deep voice, \"I'm a full-blooded American. You want to quarrel with me? Or will you shut up?\"\n\nThe operator swung wildly at the American, then jumped at Colonel Fortier, who was listening to the telephone in the dugout. The Filipino officers grabbed the operator and dragged him away.\n\n\"Goddam American cowards,\" he screamed, struggling to break loose. Suddenly he slumped as if slugged and fell.\n\n\u2014\n\nGeneral Nara, who had been friendly with General Lim when both attended the Fort Benning Infantry School, was just as worried as the Filipino commander. Takechi and his regiment, instead of going to the rear in reserve, had again mysteriously disappeared on the jungled slopes of Mt. Natib. As before, Nara kept this information to himself.\n\nNara was being constantly needled by Homma's staff. Why hadn't he broken through and annihilated the enemy? Nara knew he could never battle his way directly down the coastal highway. The Philippine Scouts of the 57th Infantry had taught him that costly lesson. The only possibility for quick success was to launch an all-out attack against the disintegrating west flank of the Abucay Line. He hated to ask Colonel Imai to do this. The 141st Infantry had already lost most of its company officers and was groggy from constant attack.\n\n\"You are to drive the enemy southeastward,\" he told Imai, \"and annihilate them.\" The attack would start the next day, January 22, at noon.\n\nMacArthur's chief of staff, General Sutherland, happened to be in Bataan on the twenty-second when Nara's all-out attack began. He had come to get \"a clear picture of the situation.\" When he returned to Corregidor that evening, he told MacArthur the situation was so alarming that an immediate withdrawal should be made to a defense line behind the cobblestone road cutting across Bataan.\n\nAfter hearing the details, MacArthur agreed. Parker and Wainwright were telephoned. Starting at darkness the next day, January 23, a general retreat would begin to the next line, the last line.\n\n\u2014\n\nThat same evening a United States Navy motor torpedo boat was cruising up the west coast of Bataan on a routine mission. The torpedo squadron commander, Lieutenant John D. Bulkeley, wore two pistols at his side; he had a long, ragged beard; his eyes were bloodshot and red-rimmed from nightly missions and lack of sleep. Bulkeley noticed a dim light low in the water.\n\nThe torpedo boat, PT 34, headed toward the light. Soon Bulkeley could make out a queer-shaped vessel. From it came strange blinking dot-and-dash signals but Bulkeley told the men not to fire. These were American waters. They were 10 miles below Wainwright's front lines.\n\n\"Boat ahoy,\" he called through his megaphone.\n\nA stream of tracers spat out from the other boat as it headed for the American-held shore. Now Bulkeley could see it was a landing barge, armored on bow and stern, packed with Japanese. As the four .50-caliber machine guns of PT 34 showered the barge, it twisted and turned to keep Bulkeley from coming broadside. Bulkeley attacked head on, machine guns blazing. The barge began to sink and finally disappeared. Then American shore guns began to blast at Bulkeley.\n\n\"Half the time those dumb bastards don't know friend from foe,\" he told Ensign R. G. Kelly, the torpedo boat commander. A shell landed only 200 yards away and PT 34 pulled out, continuing up the coast on its mission. Before long Bulkeley met and attacked another invasion barge. He jumped aboard and before the barge sank, captured single-handed three Japanese and a dispatch case of secret documents.\n\nBulkeley didn't know there were many other Japanese barges heading for shore. Nine hundred men, led by Lieutenant Colonel Nariyoshi Tsunehiro, were about to land here, far behind Wainwright's lines. If they could reach the coastal road, about two miles inland, I Corps would be cut off from the main supply line to the rear.\n\nBulkeley's solo assault upset Tsunehiro's plans by splitting the flotilla of barges in two sections. Tsunehiro and the larger group of 600, confused by treacherous tides and poor maps, landed five miles south of their goal. The smaller group of 300 was even more confused, drifting almost to the end of Bataan.\n\nEarly the next morning, Tsunehiro's group found themselves on a narrow beach about halfway between the tip of Bataan and Wainwright's front lines. This was Quinauan Point. They heard movements and whispers on the cliff above but not a shot was fired. They climbed a precipitous path and plunged into a jungle of huge trees and dense underbrush.\n\nAmerican airmen of the 34th Pursuit Squadron, positioned on the cliff, had heard the Japanese. They were so startled they abandoned their machine guns without firing a shot and crept to the rear. About an hour later, at 2:30 A.M., the alarm reached Brigadier General Clyde Selleck. It was his difficult job to protect the rugged western coast line from Wainwright's rear to the tip of the peninsula with a conglomerate and poorly armed force of Filipinos and U.S. pilots, mechanics, sailors and marines. Most had no training and few had even fired a rifle.\n\nSelleck ordered a battalion of the Philippine Constabulary to drive the Japanese back into the sea, but by the time the former policemen reached the area, Tsunehiro was hidden in the dense jungle of Quinauan Point.\n\nFarther south, near the tip of Bataan, the smaller group of 300 was sighted at daylight by a naval lookout. Soon American blue jackets attacked the invaders, now on the slopes of the small mountain overlooking the harbor town of Mariveles. With their whites dyed a sickly mustard color, the Americans plunged toward the Japanese, crashing through the underbrush, smoking and talking loudly.\n\nThe Japanese were confused and shaken. They thought a new type of suicide squad, purposely dressed in brightly colored uniforms, was attacking and hurriedly pulled back into the thick woods.\n\n\u2014\n\nIt was now late afternoon of January 23. Nara's day-old, all-out attack on the Abucay Line was apparently making little headway. Not realizing that Imai's constant, terrorizing attacks already had the defenders on the verge of collapse, General Nara, ordinarily a calm man, was in a rage. Imai, according to front-line reports, was going nowhere and there was still no word from Takechi. Figuring rightly that his friend, against direct orders, must again be trying to cross Mt. Natib, Nara requested planes to drop bundles of food at various points.\n\nNone of this food reached Takechi. This time he had successfully crossed the slopes of the forbidding mountain; but again because of poor maps he had failed to cut east at the Salian River as originally ordered by Nara. Instead, he had continued on several miles to the next river, the Abo Abo. Here at last he turned sharply to the east and headed for the coastal highway. Unknown to him this was the road that was vital to the great American withdrawal starting that dusk. If Takechi could reach it in time, the men of the Abucay Line would be cut off.\n\nThe Japanese in Nara's all-out frontal assault were as exhausted as the Fil-Americans on the other side of the Abucay Line. At a forward battalion outpost, Second Lieutenant Hisamichi Kano of the Propaganda Corps was preparing his small group for a trip far forward to broadcast appeals to surrender with a small portable public address set. Kano, who had gone to school for seven years in England, New York and New Jersey, would talk a little, then play Philippine folk songs.\n\nThe battalion commander, a nervous man of forty, urged Kano not to go out that night. Following every Kano broadcast, a heavy barrage had saturated the area. Now, for the first time in days it was peaceful and the commander wanted to keep it that way. The phone rang. \"Yes, sir,\" he said, \"I'll send out patrols to find out what's going on.\" He hung up. Instead of forming patrols, he said, \"Tell the boys to fire a few rounds on the light machine guns.\" A minute later there was a short chatter, then silence. \"The enemy must have withdrawn,\" he remarked with relief. Of his company officers, only two who had started the battle were alive. Half his staff were dead.\n\nThe phone rang again. \"No, sir,\" lied the commander. \"My patrols haven't come back yet.\" Then he turned to his officers. \"They want us to start pursuit in an hour. We'll wait until daybreak.\"\n\nKano, who had heard only stories of fanatic Japanese courage in battle, was shocked. In spite of objections from the timid commander, he went forward with his four-man team as soon as darkness fell. After passing the last outpost, the propagandists crept even farther into the black jungle that was no man's land and set up their equipment.\n\n\"You are outflanked,\" said Kano, his voice booming toward the Abucay Line. \"The Japanese Army is driving down the main highway. A regiment has smashed through your left flank and is surrounding you. If you want to save your lives, surrender immediately or make a beeline for the south.\"\n\nFor the first time there was no answering fire. Kano played a sentimental Philippine record and then led the way back to the rear. Near the battalion command post he heard a burst of fire only a few yards behind him.\n\n\"It's the speaker, Lieutenant,\" said one of his men cheerfully. \"The Americans just filled it full of holes.\"\n\nKano knew his men had shot it up themselves but he said nothing. There was no replacement for the speaker. They would all have to return to Manila.\n\nKano had drawn no artillery fire that night because the first phase of the great Fil-American withdrawal had just begun. All service and artillery troops were heading south toward their new positions. The front-line troops also withheld their fire. They didn't want to incite any trouble. The next night, January 24, their own evacuation would begin.\n\n\u2014\n\nIn Malinta Tunnel, MacArthur was writing a radiogram to Marshall. With about 35 per cent of his entire force lost, he was retiring to a line behind the Pilar-Bagac Highway running across the middle of the peninsula.\n\n> I HAVE PERSONALLY SELECTED AND PREPARED THIS POSITION AND IT IS STRONG.\n\nIn conclusion he asked that Sutherland succeed him in case of death:\n\n> OF ALL MY GENERAL OFFICERS HE HAS MOST COMPREHENSIVE GRASP OF THE SITUATION.\n\nHe knew that what happened in the next twenty-four hours would probably determine the future of Bataan. If the Japanese broke through the thin protecting cover force, or bombed the coastal highway, a major disaster would follow, one that might be the end of USAFFE.\n\n\u2014\n\nBy 7:00 P.M., January 24, trucks, buses and men began pouring back from the Abucay Line. Just then violent Japanese small arms and machine-gun fire broke out all along the front. Though many thought some Japanese spy had found out about the great retreat, General Nara actually had no suspicion the American line was being abandoned. He was simply acting on insistent orders from Homma to finish the Battle of Bataan at once.\n\nIn spite of increasing pressure from Nara, the thin protective shell holding the front lines while their comrades escaped didn't break. Officers worked up and down this screen, encouraging the men. On the 45th Infantry sector, Major Dudley Strickler, armed with a pistol and wearing an old khaki hat instead of a helmet, was going from foxhole to foxhole, urging his men to hold just a little longer.\n\nBy midnight the trail to the rear was jammed with buses full of gaunt-faced Filipinos in blue denims and coconut helmets; trucks; command cars packed with haggard officers in filthy uniforms; and marching troops. There was no moonlight and everyone looked alike. Infiltrating Japanese, about the same height as the Filipinos, could easily have joined the march unnoticed.\n\nAs traffic neared an important trail intersection near Guitol, it slowed, then crept on again. Men and vehicles were jammed at the intersection. With no military police to regulate the flow to the rear, confusion grew. Units became separated in the nightmare chaos. Officers could do nothing but keep men and vehicles moving south and pray that no shells would fall.\n\nEven so there was little shouting, no hysteria. It was disciplined confusion. The two regiments of the Philippine Division, after continuous but fruitless attempts to plug up the chopped-off left flank, retreated thankfully. They knew they had been outflanked and would soon have been surrounded. The remaining stragglers of Jones' 51st Division also were slogging back, wondering how they had escaped almost certain destruction. But the men of General Lim's 41st Division were bewildered and resentful. They had held the center of the line and thought they were winning the battle of the Abucay Line. Now they were pulling out of positions that had cost the lives of many of their comrades. Why?\n\nAt 3:00 A.M. the men of the protective shell up front began leap-frogging to the rear. They, in turn, were being covered by the tanks and 75-mm. self-propelled guns of General Weaver's Provisional Tank Group. Soon the infantrymen passed through this protective line of tanks. They looked like walking dead men. Unwashed and unshaven for nine days, their starved faces were expressionless.\n\nA few minutes later Colonel Imai led a column in pursuit of the withdrawing Fil-Americans down the trail from the front. Imai knew he had them on the run at last. He remembered a book of tactics advising commanders to relentlessly pursue a fleeing enemy in spite of fatigue and, consequently, save heavy losses in the next fight.\n\nWeaver's men spotted Imai's column as it burst into the open. The tanks waited until Imai drew nearer. Then their 37-mm. guns roared. Completely surprised, Imai thought he'd come upon an enemy artillery position. His men scattered for shelter. While he was preparing an attack, isolated groups of Fil-Americans drifting back from the Abucay Line ran into the Japanese ranks in the pitch dark. Everyone was confused. Imai and his regimental headquarters dug in as bullets flew from all sides. It was like watching fireworks. Imai sent a message back to Nara: he was surrounded.\n\nThe withdrawal continued all through the next day. From dawn to dusk Japanese planes strafed and bombed the trails and coastal road without any opposition. As planes roared down, Filipinos looked up helplessly like sheep in a slaughter pen. They scrambled to the ditches, where there was little cover. American soldiers also scrambled, cursing the Japanese obscenely as they flattened themselves on the ground. Hundreds were killed but the movement to the rear never stopped.\n\nWeaver's tanks and 75's stayed in place, giving the retreating troops security from Imai's frontal attacks, but there was nothing at all to stop Takechi, coming from the east. Fortunately for the Americans he had veered somewhat to the south and was still several miles from the coastal road. If American luck held a few more hours, everyone would escape before the trap shut.\n\n**3** On the other side of Bataan, west of Mt. Natib, Wainwright's entire front line was also in full flight. By noon of January 25 most of the 5000 cut-off men had scrambled down rough trails to the narrow beaches. Their commander, Colonel K. L. Berry, had decided to withdraw down the beaches of the west coast. The road to the rear was still blocked by Colonel Nakanishi's 700 men. It would have been suicide, Berry felt, to try and break through such a strong position with his starving, exhausted men.\n\nIt was a gruelling march down the rugged coast in the pitiless tropical sun. Many men threw away their guns and took off their uniforms. The sharp rocks cut the men's sneakers to bits. By dawn almost everyone, including Berry, was barefoot. Then the long line turned, climbing laboriously up a cliff, and headed for the coast road.\n\nBerry halted his men. He wanted to go ahead into Bagac and find out what was happening. Still carrying a Springfield rifle, Berry trudged into the town. He saw a Packard coming down the cross-Bataan road. A tall, thin man got out and approached him. It was \"Skinny\" Wainwright. Berry wondered what the general would say about retreating without orders.\n\n\"Berry, I'm damned glad to see you,\" said the general.\n\n\"I've got all my men on the beaches and in the woods. I came up ahead to find out what's going on.\"\n\nWainwright pointed to the south. \"Keep going down the road to Trail 9. You'll take up position in there.\" Then his leathery face broke into a smile. \"I'll see that you're mentioned in orders.\"\n\n\u2014\n\nBy now, January 26, the new American line was almost manned. It lay in the valley between the two great dead volcanoes, just behind the cobblestone road running across Bataan's middle. Along this continuous line from Manila Bay to the South China Sea men were settling into foxholes, resting in dugouts from the arduous retreat and thanking God they'd survived the day's strafings and bombings.\n\nLate that morning Weaver's tanks and 75's were racing through Balanga, heading for the safety of the new line. Though many infantry officers had cursed the tankers for failing to come up to the Abucay Line, they now were grateful for the cover given the foot troops the past two days.\n\nAgain the Japanese had missed an opportunity to smash MacArthur with a few accurate artillery barrages and bombings. Though the men of Wainwright and Parker were weak, hungry and riddled with casualties, USAFFE was still intact, still ready to fight.\n\nIn his position on the new defense line, Lieutenant Henry G. Lee, Headquarters Company, the Philippine Division, was composing a poem about the Abucay withdrawal. Bataan had been saved, he wrote:\n\n>...saved for another day\n> \n> Saved for hunger and wounds and heat\n> \n> For slow exhaustion and grim retreat\n> \n> For a wasted hope and sure defeat...\n\nThe new American line was also divided in two commands. Once more MacArthur gave Wainwright the west half and Parker the east, but it was stronger and better coordinated. Behind it was an ingenious network of communication and supply trails, cut through the dense jungle just wide enough for a truck. Except for a few connecting trails, most ran from rear to front, south to north.\n\nOn the morning of January 26, Brigadier General Clifford Bluemel was walking up one of the most important of these, Trail 2. He wanted to go far front to inspect his sector, the center of Parker's corps. It was more than two and a half miles wide, and to defend it Bluemel had only his own 31st Division and the shattered remnants of Jones' 51st.\n\nBluemel, as was his custom, carried his Garand rifle like any infantryman. This was no affectation; Japanese snipers hiding behind the lines were gunning for officers. Behind Bluemel trudged his Philippine staff officers, for the general knew more about the trails behind the new line than they did.\n\nLike General Jones, Bluemel had been a terror to junior officers before Pearl Harbor. Convinced a war was coming in the Philippines, he was constantly criticizing with caustic tongue. Many who had disliked him most in days of peace now had a new regard for him. Their baptism in blood on the Abucay Line had made them realize Bluemel's peacetime needling had helped save lives.\n\nThe general didn't know he was heading toward a ghost front. Without consulting or even informing him, Parker had pulled out more than half of Bluemel's troops earlier that morning, transferring them to other positions miles to the east. This left a gigantic hole in the center of Bluemel's sector\u2014at the worst possible place. Both sides of Trail 2 were now neatly cleared of defenders.\n\nBluemel saw two officers coming down the trail from the front lines. Behind them was a column of infantrymen. When they approached, the general saw this was the 1st Battalion of his 31st Infantry Regiment. \"Where the hell are you going?\" he asked.\n\n\"We're looking for Colonel Irwin, sir.\"\n\nJohn Irwin commanded the area miles to the right. \"Hell,\" said Bluemel, \"he's on Manila Bay.\"\n\n\"We've been ordered to join him.\"\n\nBluemel exploded. \"Irwin's not commanding this damn division. You're not going to take those goddam troops out of the line.\"\n\nThe battalion commander protested that the order had come from General Parker.\n\n\"You're under my command,\" said Bluemel. \"And don't you dare move those men again without my personal orders.\" The general was incensed. If the Japanese attacked down Trail 2, the entire east half of Bataan could soon be lost. Half an hour later Bluemel discovered recently abandoned foxholes on the right of Trail 2. The 33rd Infantry Regiment was also missing!\n\nBluemel's anger reached new heights. He couldn't believe Corps Headquarters could be so stupid. More than half of his front was undefended. He scribbled a message to the commander of his remaining regiment, Colonel Johnson: \"You will have your reserve battalion occupy sector assigned to 33rd Infantry. That regiment has been taken away. They will move immediately.\" Then he turned to his intelligence officer, Major Salvador Villa, a man he knew was dependable. \"See that there's a man in every foxhole before you return to the CP tonight.\"\n\nNow he had to plug up the left side of Trail 2. He roamed the lines looking for strays. He ran into sixty-five men of a battalion headquarters battery of the 31st Field Artillery, presently acting as riflemen, and personally led the little group to the front.\n\n\"Here you stay and here you die,\" he told their commander, an eager, young Philippine lieutenant. \"I'm going to put in a battalion of the 32nd Infantry on your right with machine guns. Stop the Japs. If you don't, Bataan is finished.\"\n\n\u2014\n\nFortunately for the Americans, General Nara was giving his men a well-deserved rest. His 65th Brigade had been riddled by the fight on the Abucay Line. Of the 7500 overage men more than 2000 were casualties. The others were exhausted, stunned by their first terrible taste of battle.\n\nHe was just learning the whereabouts of his missing regiment. A liaison officer, attached to the 9th Infantry, reported that Takechi's men had crossed \"impassable\" Mt. Natib. Slowed down by the jungle, Takechi had never quite reached the coastal highway. \"We haven't eaten for a week,\" he said.\n\nNara handed over his personal supply of bread and cigarettes. Then, with amusement, he watched newspapermen interviewing the haggard officer. They would write that, in a brilliant maneuver, Takechi's men had marched over rugged Mt. Natib, completely surprising the Americans and causing the breakthrough of the Abucay Line.\n\nBack in San Fernando, Homma's staff, delighted at MacArthur's sudden retreat, was planning a new assault that would bring a quick end to the irritating Bataan battle. Pressure for immediate results was coming not only from General Terauchi of the Southern Army but also from Tokyo. Everyone wondered what could possibly have delayed such a minor mopping-up operation.\n\nNow at last it looked to Homma as if the embarrassing delays were coming to an end. Unexpected aid had come from the Americans themselves. A secret map of U.S. positions had just been found in an underground room in Manila showing the entire scheme of defense of Bataan. There was a line in red extending across the peninsula just behind the Pilar-Bagac Highway. This was MacArthur's new main line but it was drawn so sketchily that Homma's intelligence officer figured it was merely the outpost line. About five miles behind this line was another. The intelligence officer deduced that this was the new American main line.\n\nOn the basis of this false information Homma devised the attack. General Morioka's 16th Division would continue its successful assault on Wainwright's corps, driving down the west coast. On the east coast, Nara would sweep past the American \"outpost line\" along Trail 2 and then smash into the \"main line.\" The attack was to start as soon as possible.\n\nIt was midnight, January 26. Because of Bluemel's energy and foresight three battalions were thinly stretched across what had been a gaping hole that morning. Every foxhole on Trail 2 was filled and excited young Filipino officers crept from foxhole to foxhole passing on Bluemel's dramatic exhortation: \"Here you stay and here you die!\" Even so, this area should have been held by a division. Every one of these men would have to do the work of three.\n\nThe next morning General Nara got orders from Homma to attack down Trail 2. To Nara it was incredible to expect the 65th Brigade to resume the attack, but he drew up hasty plans and issued them to his regiments at 11:00 A.M. He ordered Takechi to attack at 3:00 P.M. down Trail 2 with Imai's regiment on the left.\n\nThe attack started on schedule. Takechi soon reached the cobblestone road that crossed Bataan. On the other side he found the northern terminus of Trail 2 and started down it. For a few hundred yards there was no opposition, then the Japanese were met with such violent fire from Bluemel's men that Takechi had to entrench in a bamboo thicket. When Nara learned this he was perplexed. Assuming Takechi had merely run into the outpost line, he ordered heavier assaults.\n\nAt dusk the next day, January 28, Nara's artillery opened up. An hour later it stopped and Japanese infantrymen crept forward, resuming the attack down Trail 2, soon reaching a low embankment. Here the Japanese ran into two strands of barbed wire strung along small trees several feet above the embankment. Bluemel's Filipinos suddenly dashed from their foxholes and bayoneted the invaders. Takechi's men fell back, but a dozen were left hanging on the wires. It was grim, deadly battle and it continued repetitiously till dawn.\n\nIn the light of day, the Filipinos holding Trail 2 counted ninety-seven dead Japanese in front of a narrow 150-yard strip. Twenty of these were only five yards from the Filipino foxholes.\n\nWhen Nara heard of this new setback he suspected the information given him by the 14th Army had been wrong, but he had no other choice. He called for further attacks. Takechi's regiment and Nara's own \"Summer\" Brigade troops, though discouraged and battle-weary, pushed forward again and again. Each time Bluemel's men threw them back. In three days the bamboo thicket near Trail 2 became a slaughter house.\n\nOn the morning of January 31 Nara felt he could not ask Takechi's exhausted men to continue. He ordered his friend to withdraw when dark came. The attack would be taken over by fresh troops. At 5:00 P.M. Japanese planes bombed and strafed Parker's II Corps artillery which had been playing havoc with Nara's entire offensive. An hour later the Japanese artillery preparation began, almost deafening General Nara. Fire was laid systematically along Trail 2 almost back to Bluemel's command post. After an hour and a half the barrage lifted and the infantrymen scheduled to relieve Takechi crept forward under cover of darkness.\n\nJust then Bluemel ordered his division artillery to lay fire on the ford over the river in front of his main line. U.S. shells exploded all along the riverbank. As the stunned Japanese stolidly came forward, machine guns dug in astride Trail 2 riddled the right section of the attack line. The new attackers turned and fled.\n\nNine Filipinos crept from their foxholes to see what had happened. After 75 yards they stumbled into a Japanese sapper trench extending toward their own lines. It was full of Japanese, long dead. The stink was so sickening the nine hurried back to their own foxholes. It was the end of the attack on Trail 2. Bluemel's thin, stubborn line had momentarily saved Bataan.\n\n# 13\n\n# _The Points and the Pockets_\n\n**1** On the day the battle for Trail 2 ended, January 31, Lieutenant Malcolm Champlin, Admiral Rockwell's aide, landed at the tip end of Bataan at Mariveles. He stepped into a battered sedan and was driven by a sailor up the dusty west coast road toward Wainwright's command post.\n\nThe general had asked Corregidor for a naval adviser. Four days earlier a second Japanese landing had been made behind his lines, reinforcing the troops that had already landed at Quinauan Point, about halfway between the American front lines and Mariveles. Wainwright feared new Japanese landings and he hoped Champlin's naval training might help him determine when and where they would come.\n\nOnly a company of Japanese had landed on the second attempt and they had yet to join up with the original 600 invaders who kept shifting position every night to evade the inexperienced conglomeration of grounded American airmen, Constabulary and Philippine Army troops who were trying to root them out.\n\nThe 300 Japanese who had drifted south, near Mariveles, were also still hiding in the dense slopes overlooking that harbor. Even fire from eight 12-inch mortars on Corregidor had failed to dislodge them.\n\nAfter an hour on the rough coastal highway, a Filipino soldier stepped onto the road and waved Champlin's car to a halt. Vehicles could go no farther; General Wainwright's headquarters were among the trees 300 yards to the right. Champlin walked up a hill into a grove of huge trees on the edge of a ridge. Surrounded by tents were several tables of split bamboo lashed with vines. He recognized the tall, thin man talking to several officers: his new boss, Wainwright.\n\nThe general looked up and smiled. \"I'm glad to have you here, son. Come over for a chat in the trailer as soon as you're settled.\" He got up, his loose figure seeming to unravel. He was not the strapping military man Champlin had expected. His face was thin and there was nothing that impressed the lieutenant except his eyes. These were quick and piercing.\n\nAfter settling in a tent Champlin unwrapped the package given him by Rockwell for Wainwright. It was a bottle of Scotch. \"Would the general be interested in this?\" he asked Wainwright's aide, Captain Tom Dooley.\n\n\"Would he!\" Dooley's eyes were popping. \"Just come with me...and handle that bottle with care.\"\n\nChamplin followed Dooley through jungle vines to a trailer. The lieutenant handed the bottle to Wainwright. \"General, Admiral Rockwell sends this to cement Army-Navy relations.\"\n\nWainwright took the bottle tenderly, held it up to the fading light and rotated it. \"Young man, do you realize that you have here the finest Scotch there is? And that I haven't had a drink for months?\"\n\nThe next morning Champlin rode up front with Wainwright and Dooley. While a corporal drove up the jungle trail, Sergeant Carroll stood on the running board scanning the sky for planes. Soon they came to the front. After inspecting company command posts and going forward to talk to squad leaders and privates, the general and his party started back home. Japanese artillery fire began to fall, sweeping back and forth across the trail. The car stopped and everyone ran into a clump of woods dotted with foxholes.\n\nWhile Champlin and the others impatiently waited for Wainwright to pick his foxhole, the general took a cavalry captain by the arm and sat down with him unconcernedly on a row of sandbags and talked.\n\nChamplin had never seen such coolness under fire. When the barrage stopped and the car started off again Champlin said, \"General Wainwright, why is it that you, commander of half the troops on Bataan, risk your life the way you did a few minutes ago? It seemed to me you were doing a very foolish thing.\"\n\nWainwright smiled slightly, then said in a slow drawl, \"Champ, think it over a minute. What have we to offer these troops? Can we give them more to eat? No. We haven't any more food. Can we give them any more ammunition? No, it's running low. Can we give them supplies or equipment or tanks or planes or medicine? No. Everything is running low. But we can give them morale and that is one of my primary duties. That's why I go to the front every day. Now do you understand why it's important for me to sit on sandbags in the line of fire while the rest of you run for shelter?\"\n\nThe following night, February 1, a bright full moon shone over Bataan lighting up a dozen large barges sailing down the west coast. In them were 700 men of Major Mitsuo Kimura's 1st Battalion, 20th Infantry. Their mission was to reinforce those who had already landed near Quinauan Point in the two previous landings. The combined force of 1500 would then drive south to Mariveles. With this town in Homma's hands, all supplies to Wainwright's corps would be cut off and the Battle of Bataan quickly ended.\n\nThe previous landings had completely surprised the defenders. Though the southernmost, only a few miles from Mariveles, had just been wiped out by Philippine Scouts, the other two beachheads near Quinauan Point (now called Quinine by the Americans) were still intact and inflicting heavy losses on other Scouts of the 57th Infantry.\n\nThis time a Navy signalman sighted the invasion flotilla long before it neared its goal and telephoned Champlin. The lieutenant hurried to Wainwright's trailer and told the general another landing attempt would probably be made near Quinauan Point. Then Champlin called Navy Headquarters on Corregidor recommending that Bulkeley's PT boats, already on patrol, be sent to the vicinity. In the meantime, Wainwright was rushing the 26th Cavalry to the area and MacArthur's Far East Air Force, four patched-up P-40's, was taking off from the strip near Cabcaben.\n\nAs the Japanese barges neared Quinauan Point, Battery D of the 88th Field Artillery and Battery E of the 301st opened up with their 75's and 155's. Machine guns emplaced on cliffs along the coast chattered. Then the four P-40's shot over Mariveles Mountains and found the invaders. After dropping 100-pound fragmentation bombs, they swooped down, strafing the barges with machine guns.\n\nA few minutes later two of Bulkeley's torpedo boats reached the scene. Suddenly MTB 32, carrying the bearded Bulkeley, was blinded by a large searchlight. Two shells screamed overhead, exploding only yards ahead of the boat. They came from a Japanese mine layer covering the landing. The torpedo boat headed directly for the mine layer. Avoiding more salvos, it loosed two of its torpedoes. Bulkeley was sure he had sunk the vessel but it was only slightly damaged.\n\nAmericans and Filipinos on the cliffs watched the combined sea, air and land assault on the flotilla in the brilliant moonlight. They were sure the landing had been completely smashed. Several of the barges were sinking, others fleeing south.\n\nBut Major Kimura was a man not easily discouraged. The importance of his mission had been accented by General Homma himself. Besides there were comrades ashore who needed help. Though half of his men were lost in the sea fight, Kimura led the rest to a beach about a mile north of Quinauan Point. There he was greeted by the 200 men of the second landing. These two forces consolidated and soon after midnight disappeared in the dense wooded jungle in search of the 600 original invaders. The Battle of the Points was on again.\n\n**2** About seven miles to the northeast, another unorthodox battle was developing in the middle of Wainwright's front lines. This was a densely jungled, hilly area almost four miles inland from the South China Sea and about a mile behind the American Main Line of Resistance.\n\nA week previously Homma had ordered the Japanese on west Bataan to attack Wainwright while Nara was launching his abortive drive against Parker in the east on Trail 2. Wainwright successfully threw back this attack at all places but one. Near the center of his line 1000 Japanese, commanded by Colonel Yorimasa Yoshioka, broke through before the Filipinos assigned to that area had finished setting up their defense line. The Japanese continued south for about a mile, completely undetected by the defenders. This was not too surprising.\n\nIt was easy for 1000 men to go unnoticed in the wild, bewildering jungle in the rolling valley between the two great extinct volcanoes. Tropical vines and creepers draped from huge mahogany and sprawling banyan trees made the hills a continuous woody screen; tall cane and thick clumps of bamboo matted the lower areas. Maps were so crude that officers couldn't even agree which was the Tuol and which the Cotar River.\n\nAnd so Yoshioka's force wasn't detected for three days. Even when the Japanese were discovered it was thought that only \"seventeen snipers\" had infiltrated the lines, but by the night of the third Japanese landing, February 1, Wainwright realized a strong enemy force was sitting at the junction of two important trails a mile behind his front. Already a battalion of Scouts sent to wipe out the \"seventeen snipers\" had been hard hit by casualties.\n\nTo complicate matters Yoshioka's 1000 men had become split in two. The Little Pocket, a company, was on top of a hill 600 yards behind the front line. The Big Pocket, the main force, was well dug in another 1000 yards farther south at the junction of Trails 5 and 7.\n\nAs a further complication Trail 7\u2014a main trail running north-south almost in the exact middle of Wainwright's line\u2014was the boundary between the Right Sector of Wainwright's line, commanded by Brigadier General William Brougher, and the Left Sector, commanded by General Jones. By February 2 little progress had been made by the Scouts trying to wipe out the pockets, even though they attacked time after time shouting their battle cry, \"Petay si la!\" (They shall die!)\n\nGeneral Wainwright, after narrowly missing death from an ingeniously camouflaged sniper with face painted green during a personal visit to the Big Pocket, ordered tanks to bolster the attack.\n\nWhen the four-tank platoon arrived a combined infantry-tank assault was planned by the tankers and Colonel Glen Townsend of the 11th Infantry Regiment. Sergeant Leroy Anderson would lead the tanks north up Trail 7 to an enemy gun emplacement at the base of a huge banyan tree. Lieutenant Robert Roberts volunteered to give support with a platoon of riflemen. Lieutenant Willibald Bianchi of Company D asked if he could go along.\n\nThe four tanks rumbled up the trail at 4:00 P.M. Anderson's lead tank quickly ran into heavy rifle and machine-gun fire from Yoshioka's camouflaged men. Bianchi ran behind Anderson's tank, firing a rifle with his left hand. As the tank neared the banyan tree, two bullets struck Bianchi's hand. He dropped the rifle but pulled out a pistol with his other hand. Fire had come from a well-hidden machine gun west of the big tree. He ran forward, threw two grenades. Machine gun and crew were knocked out.\n\nNow Anderson swung his tank toward the banyan tree. Soon he was within a few yards of the emplacement. When he tried to depress his 37-mm. gun to fire at the foot of the tree into the nest he found he was too close. Heavy fire rattled against the tank from the banyan tree.\n\nSeeing Anderson was in a vulnerable position, Bianchi leaped to the top of the tank. A Japanese machine gun in the nest chattered. Two bullets thudded into Bianchi's chest but he kept firing at the area around the tree. Suddenly there was an explosion near the turret. Seriously wounded, Bianchi toppled off the tank. Scouts pulled him back to safety.\n\nAnderson's tank engine stopped. There were muffled roars from the interior as ammunition exploded. The crew flung themselves out of the turret, their hair burning. Roberts and his riflemen pinned down Japanese in foxholes behind the banyan tree until the tankers escaped.\n\nThe three other tanks roared by, firing their 37-mm. guns. Soon the leader discovered another machine gun on the right of the trail. While the tank fired, pinning down these machine gunners, Lieutenant Edward Stewart crept forward with a Scout. \"Throw a grenade,\" he whispered. The Scout threw one. A few minutes later the same grenade came flying back out of the nest, exploding near Stewart. The Scout threw two more. Both came back.\n\n\"Must be a damned octopus in that hole,\" said Stewart. At a signal he and the Scout threw grenades simultaneously. There was a great muffled explosion. The machine gun was silenced. It was fortunate because the lead tank soon became caught on a stump. The crew had to crawl back to the rear.\n\nA few yards had been punched into the south side of the Big Pocket, but two tanks were lost. It was obvious that a large force somehow had filtered behind the lines. A long, deadly fight lay ahead.\n\nAs Bianchi, bloody froth on his lips, was being carried down the trail, he saw his friend, Major Adrianus van Oosten, and stopped the stretcher-bearers. Van Oosten wondered what important message his friend had for him. \"Van,\" said Bianchi, \"take care of my camera, will you?\"\n\n\u2014\n\nLate that afternoon Lieutenant Champlin returned to Corregidor. While he was reporting to Admiral Rockwell, Lieutenant Commander F. W. \"Mike\" Fenno entered the Navy tunnel. He had just brought in 70 tons of 3-inch anti-aircraft ammunition on the submarine _Trout_. Though Fenno had reloaded with fuel and torpedoes, his boat still was too light. Where could he find 25 tons of ballast fast?\n\nIt answered a problem that had been worrying Rockwell, Sayre, Quezon and MacArthur for days: the gold and silver they'd brought to Corregidor from the banks of Manila. In a few hours 2 tons of gold bars and 18 of silver pesos were flung into trucks and then down the hatch into the _Trout_ like so many sacks of potatoes.\n\nAt his house outside the east entrance of Malinta Tunnel, MacArthur was worrying about a much more important problem. The 70 tons of ammunition just brought in by the _Trout_ only accented the futility of such meager efforts. He needed large quantities to save Bataan. Washington had to be shaken from its lethargy. It would be a fatal mistake, he radioed Marshall, to continue the present Allied strategy. Instead of building up forces for Australia, the Japanese line of communications stretching over 2000 miles of sea should be attacked. There was only one way to defeat Japan: seek combat, not avoid it.\n\nHe wished that those safe and secure in America could realize time was fast running out in the Philippines.\n\n\u2014\n\nEarly on the morning of February 5, Lieutenant Reed of General Brougher's staff called out to the Japanese in the Big Pocket to surrender. The answer was heavy fire.\n\nWainwright by now realized the job needed special handling. He called a meeting at the command post of the 1st Division. At 10:00 A.M. Generals Wainwright, Jones, Brougher and Segundo conferred. The exasperating command problem had to be straightened out, said Wainwright. He was going to appoint one man, regardless of sectors, to command the operation. He asked Jones how he would tackle the problem.\n\n\"They have to be pinched out,\" said Jones. \"First I'd isolate the pockets and throw a cordon around each one.\" Only then would he launch the main attack. First against the Little Pocket. When that had been wiped out, he'd tackle the Big Pocket.\n\nThough Wainwright and Jones had occasionally clashed, they understood each other. Both believed in personal supervision by generals on the front line.\n\n\"All right, Honus,\" said Wainwright. \"You take charge.\"\n\n\u2014\n\nThe Battle for the Points was also grinding on grimly. Philippine Scout casualties were 50 per cent in some companies. Morale was falling because of hunger and continuous danger from camouflaged Japanese snipers tied to trees. There was no concentrated food; supplies moved slowly through the thick, deadly jungle near Quinauan Point. Commanders were learning that the ability to fight decreased as food decreased.\n\nTanks were brought up to help but in spite of aggressive efforts, the thick jungle made this a battle of riflemen. The Japanese would have to be rooted out one by one as they were pushed west into the China Sea. By February 7 the Scouts had isolated all the Japanese near Quinauan Point into an area 4000 yards wide. American Air Corps and Philippine Constabulary troops were rushed in to help finish the ugly mopping-up.\n\nAt the southern end of this line, on Quinauan Point itself, the end was in sight. The remnants of the 600 Japanese of the first landing had been shoved to the edge of the China Sea. They were now hiding in caves at the base of a sheer cliff.\n\nCaptain William Dyess, who started the war as a fighter pilot, was crouched on top of the cliff with seventy men of his planeless 21st Pursuit Squadron. On all sides were enemy dead\u2014snipers hanging from the trees, riflemen still crouched in their foxholes.\n\nSergeant Ray Hunt, Jr., formerly a mechanic, stuffed his nose with cotton so he could eat his dinner. He could see a dying Japanese tangled in a trench with two dead bodies, his wound crawling with maggots. The man had three watches on his right arm.\n\nHunt looked down. A Japanese had darted from a cave and was racing across the beach. Japan is one hell of a long way, he thought, as the man dove into the China Sea. A Scout machine gunner near Hunt slowly put his mess kit down, positioned himself comfortably behind his gun, took deliberate aim and squeezed the trigger. The swimming Japanese became motionless, his body engulfed in a growing circle of red. It was brilliant red, noted Hunt, against an emerald green. Vivid, beautiful and horrible.\n\nA mile to the south, two other Japanese had dived into the China Sea. Still unseen, they were swimming north. Each carried a bamboo pipe containing identical messages from Major Kimura to his commanding general. A bitter battle was going on, reported Kimura, the enemy attacking with infantry and tanks. \"The battalion is about to die gloriously,\" he concluded.\n\n\u2014\n\nThe end came the next day, February 8. Just after dawn twenty of Captain Dyess' grounded airmen left Mariveles in two whaleboats. Covering them were two armored naval motor launches, commanded by Lieutenant Commander H. W. Goodall and armed with 37-mm. and machine guns.\n\nIt was almost 8:00 A.M. when Dyess, aboard Goodall's launch, saw sheets hanging down from a cliff. Dyess pointed out caves under the sheets. These, he told Goodall, contained the last stubborn Japanese. The two launches began firing into the caves. Ten minutes later a sailor noticed a flight of Japanese dive bombers coming from the east.\n\n\"To hell with the airplanes, sailor,\" said Goodall. He turned to Dyess. \"Where do you want the next shot, Captain?\"\n\nThe four boats, all guns still firing, sped toward the beach as 100-pound fragmentation bombs burst on all sides. Soon a whaleboat was hit, then a launch. A moment later the second whaleboat was sunk. Just before beaching, Goodall's launch was upset and almost demolished by a near hit.\n\nDyess and his twenty men swam ashore. While they crept toward the caves from two sides, other American airmen on top of the cliff worked cautiously down through ravines. By noon the caves were cleaned of Japanese. Only one prisoner was taken.\n\nThe Battle of the Points was finally over.\n\n**3** That same day General Homma called an important conference at his command post in the sugar center of San Fernando. Here it was another muggy day with the heat above 95 degrees. Though he didn't show it, Homma was tormented with worries. The situation on Bataan was desperate, almost catastrophic. He had already lost almost 7000 men in combat and another 10,000 had been stricken with malaria, beriberi and dysentery.\n\nNot a word had yet been heard of the two battalions which had landed behind Wainwright near Quinauan Point. Another battalion had been swallowed up in pockets behind Wainwright's line. On the east coast Nara had been given a bloody nose at Trail 2.\n\nAnd still Homma was being pressed by Southern Army and Imperial General Headquarters to wind up the Bataan operation in a few days. How was it possible when they'd already robbed him of his only good division, the 48th? He had twice requested that the whole Philippine situation be reviewed. These weren't strong protests, of course, because even a lieutenant general rarely asks for reinforcements. He must do with what he has.\n\nIn addition to poor maps, he had been given intelligence reports indicating that the Filipinos were eager to turn on their American conquerors and completely misinterpreting MacArthur's capabilities and intentions. It was almost as though Tojo wanted him to fail. Their enmity was well known. These suspicions were never to be allayed. Three years later in his prison diary Homma would write, \"It is reported that Mr. Tojo will be here before long. There is nothing more unpleasant than this. To have to see his face every day will be more than I can bear....\" And two weeks later, \"At 1:00 P.M. Tojo arrived at the prison. The sight of his face is enough to make me feel disgusted. He came out saying, 'Hello, Your Excellency Homma!' et cetera but I merely nodded.\" Yet, a week later he wrote revealingly, \"Mr. Tojo is being shunned by most of the men and in a way I feel pity for him.\"\n\nThe conference began. General Maeda, the chief of staff, presided as chairman. Homma, as often was the custom of Japanese commanders, played a passive, listening role. Unlike the western military, it was Japanese policy to let staff officers do much of the thinking and planning. It was felt that otherwise they might lose their enterprise.\n\nHomma listened impassively as Maeda outlined the general situation. It was even worse than Homma had suspected. Because of the frightful casualties on the Abucay Line and the triple disasters at Trail 2, the Pockets and the Points, there were now only three infantry battalions on the entire Bataan line! If MacArthur knew, he could smash through and crush the 14th Army.\n\nOthers spoke. Major Moriya Wada, then Colonel Tatsuo Haki. The general attack on MacArthur's new line had bogged down, said Haki. To continue with the fast-dwindling forces would be disastrous. Colonel Motoo Nakayama, senior operations officer, disagreed. \"The offensive,\" he said, \"should be continued aggressively. However, the main effort should be made along the east coast, not the west.\"\n\nMaeda cleared his throat. He had first guessed MacArthur's intention to make Bataan a major point of resistance. Homma eagerly waited the views of his most brilliant officer. \"I believe the offense should be discontinued. Let us merely blockade Bataan while we occupy the rest of the Archipelago. By that time the men of General Matsukuasa (MacArthur) will be starved and ready to surrender.\"\n\nAs the conference continued in the sultry heat, Homma realized that he couldn't agree with either of the two major views. Maeda's argument for breaking off attack immediately and retreating to safer positions was irrefutable, but it was unthinkable to wait for the Americans to rot on the vine. Tokyo would never stomach such a face-losing tactic. He himself would have to be the one to lose face.\n\nHomma told them a new and much more powerful offense must be launched. To do this he would have to swallow his pride and ask for heavy reinforcements.\n\nThe others in the room saw tears running down the big general's face. To Lieutenant Colonel Monjiro Akiyama, Homma was a perfect human being, humanistic and just, who never raised his voice or criticized a man in public. To Wada, he was a humane and literate man who treated younger officers with consideration. As Homma's staff started to file out of the room a telegram was handed the general.\n\nHe opened the message. It was from Imperial General Headquarters. Tojo was displeased. Hong Kong had fallen and General Yamashita was about to score one of the greatest military triumphs in Japanese history: he would soon take Singapore. Everywhere there was victory except in the Philippines. Colonel Haki saw a look of supreme agony come over Homma's face. Then the general's head suddenly dropped to the table. Haki and two men carried the unconscious commander of the 14th Army into the next room.\n\n**4** Personal conflict between the fighting front and commanders in their homeland wasn't confined to the Japanese that day.\n\nOn Corregidor bitter rumors were circulating of abandonment and \"sellout\" by Washington. Some officers believed Roosevelt and Marshall were going to let MacArthur rot in the Philippines. It was common knowledge throughout the Army that Marshall and MacArthur were not on good personal terms. MacArthur's spectacular skyrocketing after World War I had made him a four-star general and chief of staff, while Marshall, only a year younger, remained a lieutenant colonel. Some said Marshall still blamed MacArthur for not making him a general during the former's regime as chief of staff.\n\nThat day MacArthur learned Roosevelt was making a radio speech telling of the thousands of aircraft that would soon be on their way to the battlefront\u2014Europe. Guessing that Quezon would hear it and would be infuriated, the general asked his intelligence officer, Colonel Charles Willoughby, to go at once to the Philippine president and try to placate him.\n\nQuezon was in a tent set up on a slope near the entrance of Malinta Tunnel so he could get fresh air. He was sitting in his wheel chair, angrily listening to the Roosevelt speech, as Willoughby entered. When it was over, Quezon poured out a torrent of violent denunciation in Spanish, a language Willoughby understood. Then the Filipino pointed to smoke rising from the mainland. \"For thirty years I have worked and hoped for my people. Now they burn and die for a flag that could not protect them. _Por Dios y todos los Santos_ , I cannot stand this constant reference to England, to Europe. Where are the planes this _sinverguenza_ is boasting of? _Que demonio!_ How American to writhe in anguish at the fate of a distant cousin while a daughter is being raped in the back room.\"\n\nWilloughby calmed the president, whose breath was coming in gasps, but later in the day Quezon insisted on seeing MacArthur. He was now in a deeply depressed mood. \"Perhaps,\" he said, \"my presence on Corregidor is not of value. Why don't I go to Manila and become a prisoner of war?\" He was afraid of Tojo's recent promise to give the Philippines independence with honor. \"I will defy the Japanese and this will solidify the opposition of all Filipinos.\"\n\nMacArthur thought this would be a mistake. \"I think the Japanese would allow you to go to Malaca\u00f1an Palace and then let no Filipino near you. They would give out statements coming from you urging Filipinos to surrender.\" The general also believed Quezon's surrender would be misinterpreted abroad.\n\n\"I don't care what outsiders think,\" flared Quezon, but then calmed down. MacArthur was his good friend, a man to be trusted. \"I'll think it over.\" He wanted to be another Apolinari Mabini, the great national hero, a paralytic prisoner who had refused to take an oath of allegiance during the Filipino-American War.\n\nAt this moment a young Filipino second lieutenant was swimming from Bataan to Corregidor to see Quezon. It was Antonio Aquino, elder son of Benigno Aquino, the sugar-cane king and speaker of the Philippine Assembly. Aquino had retreated with the rest of Capinpin's division into Bataan. The 21st Division was now holding a front-line position several miles to the left of strategic Trail 2. Concerned about the increasing tension between Filipinos and Americans, he had conferred with General Capinpin.\n\n\"The Americans are forever goddamming this blasted country for the war, the mosquitoes and the poor rations,\" he told the general. His Filipino enlisted men understandably resented the brusque American attitude. More important, they also felt Americans were getting better food. Aquino then asked permission to go to Bataan and inform President Quezon of the situation.\n\nCapinpin agreed and young Aquino immediately started south in his yellow convertible, already scarred by several shrapnel holes. He finally reached the east coast highway and continued to Cabcaben, a town near the eastern tip of the peninsula.\n\nJust before dusk Aquino started for Corregidor in a small fishing boat. A mile from the rock island, the fisherman refused to go any farther for fear of being fired at by trigger-happy American guards. Aquino stripped, covered himself with heavy oil and tied a bag filled with dozens of ping-pong balls on his back as a life preserver. He knotted his clothes in a ball, then jumped into the water.\n\nIn three and a half hours he finally reached the beach not far from the entrance of Malinta Tunnel. After drying his clothes he entered the tunnel and headed for the Quezon family quarters. In the lighted passageway he saw Quezon's two daughters, Nini and Baby. They failed to recognize him at first. He had lost twenty pounds and was tanned dark.\n\nThe two girls led Aquino into their father's room.\n\n\"Papa,\" said Nini, \"look who swam to Corregidor. Tony Aquino.\"\n\nAquino saluted, then, in respect to an elder, kissed Quezon's hand.\n\n\"Tony, sit down,\" said the president. \"Let me look at you. How is it in Bataan?\"\n\nAquino's fatigue instantly vanished. He swallowed to keep from choking. \"We are doing fine, Excellency. General Capinpin is sending you his regards and prayers for your health.\" Then he told Quezon of the antagonism between Americans and Filipinos.\n\nThe president took it calmly. \"You have to know the Americano to understand him. He is gruff and rough but it's just his way.\"\n\n\"There is one other thing, sir. We feel that we should have the same rations as the Americans. We eat only salmon and sardines. One can per thirty men, twice a day.\"\n\n\"What?\" Quezon was astounded.\n\n\"Yes, sir. One can of salmon for thirty, two gantas of rice and two cans of condensed milk for breakfast.\"\n\n_\"Pu\u00f1eta!\"_ exclaimed Quezon. \"I did not know that.\"\n\nAt that moment MacArthur walked in. Aquino stood at attention.\n\n\"General,\" said Quezon, \"this is Tony Aquino, the son of Benigno, my secretary of commerce and agriculture.\"\n\n\"How are you, young man?\" MacArthur shook Aquino's hand.\n\nQuezon asked Aquino to repeat his story. While the young Filipino was talking, he noticed several American officers standing behind MacArthur shifting in embarrassment.\n\nQuezon turned to MacArthur and said, \"I want you to improve the rations.\"\n\nThe general assented. \"You did a fine service, young man.\" He shook Aquino's hand and left.\n\n\"If I were forty years younger,\" Quezon wistfully told Aquino, \"I would be with you. I know Bataan, every nook and corner of it. I fought there during the Revolution. God will not forsake us and in the end we shall win. Do not lose hope!\"\n\nWhen he was alone Quezon mulled over what his friend's son had told him. Finally he called for his Cabinet. His mind was made up. \"I think I have the answer,\" he told them a few minutes later. \"I'll ask Roosevelt to let me issue a manifesto requesting the U.S. to grant immediately complete and absolute independence to the Philippines.\" Then he would demobilize the Philippine Army and declare the Philippines neutral. America and Japan would both withdraw their armies.\n\nManuel Roxas shook his head. He was afraid of the effects of such a proposition on Roosevelt. Vice-President Osme\u00f1a agreed with Roxas. \"I am sure that Roosevelt would misunderstand our motive.\"\n\nQuezon's voice rose excitedly in argument. He pointed out that food was running out on Bataan. The starved Filipino troops couldn't possibly hold out much longer.\n\nHacking coughs silenced Quezon. Osme\u00f1a and Roxas realized it was dangerous to try and argue the president out of his quixotic scheme. Further resistance to the plan might bring on a fatal attack. The Cabinet unanimously approved that the message be sent to Roosevelt.\n\nA little later MacArthur received the message. Although he had done his best to calm Quezon's growing fears that America was abandoning the Philippines, secretly he had his own doubts.\n\nHe had great personal respect for Roosevelt and knew the President respected him but he still vividly remembered their dispute in 1933 when he was chief of staff and Roosevelt ordered a cut in the Army budget. MacArthur had argued that Army morale would be seriously lowered at a time of national peril. Sharp words were exchanged in a decisive interview at the White House. At last the general stood up and said, \"Mr. President, if you pursue this policy which will lead inevitably to the destruction of the American army, I have no other choice but to oppose you publicly. I shall ask for my immediate relief as chief of staff and for retirement from the Army, and I shall take this fight straight to the people.\" The general saluted, turned on his heel, walked out of the White House and vomited on the lawn. A little later the proposed budget cut was dropped.\n\nMacArthur knew that Quezon could not be dissuaded from sending his alarming message. Perhaps it was for the best. He decided to take a gamble. In addition to Quezon's message, he would send one of his own, painting the real, tragic picture of Corregidor and Bataan. \"There is no denying,\" he wrote, \"that we are nearly done.\"\n\nThe Filipinos' attitude, he added, was one of \"almost violent resentment against the U.S.\" He concluded that militarily \"the problem presents itself as to whether the plan of President Quezon might offer the best possible solution of what is about to be a disastrous debacle.\"\n\nQuezon's scheme, obviously, was a wild one, yet perhaps its very wildness would shock Washington into action. MacArthur was risking his whole military career on these two messages, but he was willing to take the gamble.\n\n\u2014\n\nThe next morning, February 9, General Jones was still personally supervising the Battle of the Pockets. By now he was so weak from dysentery he could hardly stand, but he refused to leave the front. The climax of the strange battle was approaching. Colonel K. L. Berry and his 1st Division troops had pinched out the Little Pocket and were now joining in the final assault on the Big Pocket.\n\nThere was a new complication. Two days ago General Morioka, in a desperate attempt to relieve his surrounded troops, had ferociously attacked Wainwright's main line at Trail 7, 1400 yards north of the Big Pocket. Just when it looked as though this spirited Japanese drive would break through to the Pocket, Igorots of Colonel Glen Townsend's 11th Infantry counterattacked. Some, clad only in G-strings, rode on top of buttoned-up American tanks tapping with clubs to direct the drivers. The Japanese advance was stopped. By then, however, there was a 600-yard bulge in the line. It had already been nicknamed the Upper Pocket.\n\nWhile Townsend held off the attack on the main line, Jones was hammering at the Big Pocket from all sides. Now mortars were pounding the trapped men. In the silence after explosions Jones' men could hear the cries of the wounded Japanese. And when night fell they could hear the weird chanting of prayers, soon followed by the steady chopping of trees as the energetic defenders built more defenses and cut more fields of fire.\n\nInside the Big Pocket five captured Filipino soldiers were lashed to a huge nara tree. The past ten days had been an unbelievable nightmare as they watched the Pocket shrink and their captors get weaker from hunger. At first there had been plenty of water, dipped from the Tuol River at the edge of the Pocket. It was their job to sneak down at night under guard and carry water back to the center, but now the Tuol, actually a large stream, was in American hands. Its banks were already lined with the stinking corpses of Japanese who had been too thirsty for caution.\n\nThat night, February 9, the five Filipinos could sense a new attitude in the Japanese. It frightened them. They feared they would soon be executed.\n\nColonel Yoshioka had just received orders, dropped by plane, to retreat. It was a reprieve from death, for an unordered withdrawal was unthinkable. He took a count. Of the original 1000, less than 600 remained and 100 of these were wounded. He consulted with his few remaining officers. They would leave all their horses, bury their three small mountain guns and 47-mm. anti-tank gun and break out to the north the next night. The walking wounded would go with them. The others, naturally, would commit hara-kiri.\n\n**5** It was only 3:00 P.M., February 9, in Washington and Franklin Roosevelt, with Secretary of War Stimson and Sumner Welles, was drafting an answer to President Quezon. Roosevelt had been seriously shaken by the Filipino's plan to surrender to the Japanese, particularly since MacArthur and Sayre seemed to have made no effort to dissuade Quezon and had even given the proposal some support in accompanying messages.\n\nThe Battle of the Atlantic had just taken a dangerous turn. German U-boats were ravaging shipping along America's east coast. By the end of January, 31 ships totaling almost 200,000 tons had been sunk off the North Atlantic coast. The first week in February the toll had taken a sharp rise. If it continued, and there was no hope it would not, at least 70 ships would be destroyed before the month's end. Saboteurs, apparently, were as busy as the submarines. It was immediately believed, for instance, they were responsible for the mysterious burning of the French liner _Normandie_ that same day at her dock at the end of West 49th Street in New York City.\n\nThere was also a new disaster in North Africa. Rommel was again loose and threatening to drive the British back to Alexandria.\n\nThe President couldn't expect Quezon and MacArthur to agree with the war policy, universally accepted in Washington, that Hitler should be defeated first but they apparently doubted that everything possible was being sent them. The Southwest Pacific had already been sent more than its share of available men and supplies. During January three convoys and the Navy seatrain _Hammondsport_ had left San Francisco and New York for Australia. The _Queen Mary_ in Boston and the _Monterey_ and _Matsonia_ in San Francisco were being loaded with reinforcements. By the middle of March 79,000 troops would have left for the Pacific front, almost four times the number of troops heading for Europe. Most of the few available planes were also heading toward the Orient.\n\nRoosevelt realized he had to transmit all these facts to Quezon, but without threat or accusation. The immediate future of the Philippines rested on the right words.\n\nQuezon received the radiogram on February 10. In it Roosevelt declared bluntly that the U.S. would never agree to Quezon's proposal but added that no matter what Quezon did America would not abandon the Philippines.\n\n> SO LONG AS THE FLAG OF THE U.S. FLIES ON FILIPINO SOIL...IT WILL BE DEFENDED BY OUR OWN MEN TO THE DEATH. WHATEVER HAPPENS TO PRESENT AMERICAN GARRISON WE SHALL NOT RELAX OUR EFFORTS UNTIL THE FORCES WHICH ARE NOW MARSHALING OUTSIDE THE PHILIPPINE ISLANDS RETURN TO THE PHILIPPINES AND DRIVE THE LAST REMNANT OF THE INVADERS FROM YOUR SOIL.\n\nThere was not a single word of recrimination, not even a veiled threat. The effect on the sick man was overwhelming. He had long been Roosevelt's friend, ever since the President's days as Under-Secretary of the Navy, but now Quezon came to the conclusion that he was a great man. Roosevelt was placing the entire burden of defending the Philippines on the American people alone.\n\nQuezon swore to himself and God that as long as he lived he would stand by America regardless of the consequences to his people or himself. His worries and indecision were gone. Now he had his course of action.\n\nMacArthur was also reading a message from Roosevelt:\n\n>...THE DUTY AND THE NECESSITY OF RESISTING JAPANESE AGGRESSION TO THE LAST TRANSCENDS IN IMPORTANCE ANY OTHER OBLIGATION NOW FACING US IN THE PHILIPPINES....\n> \n> IT IS MANDATORY THAT THERE BE ESTABLISHED ONCE AND FOR ALL IN THE MINDS OF ALL PEOPLES COMPLETE EVIDENCE THAT THE AMERICAN DETERMINATION AND INDOMITABLE WILL TO WIN CARRIES ON DOWN TO THE LAST UNIT.\n> \n> I THEREFORE GIVE YOU THIS MOST DIFFICULT MISSION IN FULL UNDERSTANDING OF THE DESPERATE SITUATION TO WHICH YOU MAY SHORTLY BE REDUCED. THE SERVICE THAT YOU AND THE AMERICAN MEMBERS OF YOUR COMMAND CAN RENDER TO YOUR COUNTRY IN THE TITANIC STRUGGLE NOW DEVELOPING IS BEYOND ALL POSSIBILITY OF APPRAISEMENT. I PARTICULARLY REQUEST THAT YOU PROCEED RAPIDLY TO THE ORGANIZATION OF YOUR FORCES AND YOUR DEFENSES SO AS TO MAKE YOUR RESISTANCE AS EFFECTIVE AS CIRCUMSTANCES WILL PERMIT AND AS PROLONGED AS HUMANLY POSSIBLE.\n\nReading between the lines MacArthur knew the Philippines had been definitely, irrevocably written off, their only value now as a symbol of last-ditch resistance.\n\nThe next day, February 11, he replied to Roosevelt. He would abide by the President's decision. He and his family would remain in the Philippines and share the fate of the garrison. He would fight to destruction on Bataan and then Corregidor, making them names for Americans to remember forever.\n\n> I HAVE NOT THE SLIGHTEST INTENTION IN THE WORLD OF SURRENDERING OR CAPITULATING THE FILIPINO ELEMENT OF MY COMMAND....THERE HAS NEVER BEEN THE SLIGHTEST WAVERING AMONG THE TROOPS.\n\nMorale on Bataan had never been higher. Since the retreat from Abucay, the Japanese had been consistently stopped. Filipino soldiers entering Bataan as raw, frightened recruits had become tough, battle-scarred and dependable. Some American instructors couldn't believe these were the same men who had run on the hectic retreat from Lingayen Gulf; a number of junior officers were even admitting that some of the early disasters had been caused by their own faltering leadership.\n\nRiddled with dysentery and malaria, their uniforms in tatters, the half-starved men of Bataan were full of fight and confidence.\n\nAt the Big Pocket General Jones, under protest, was being carried away on a stretcher. He was so weak from acute dysentery he could no longer walk. He wanted to stay but he left knowing the savage battle was all but over. His men were punching into the Pocket from several sides, unaware that Colonel Morioka had slipped out the night before with less than 400 starved men.\n\nThe attackers were finding a grisly battlefield. There were hundreds of graves, some curiously outlined with upright cigarettes, but many bodies were unburied and the stink of tropical decay was overpowering. Some dead men were holding hunks of raw horse meat in their hands, their only sustenance. Huge bluebottle flies rose from the bodies in clouds. Commanders, seeing this ghastly charnel house, ordered it burned to the ground before disease could spread. Others like Major Crow tried to prevent some of their own starving riflemen from digging meat with bare fingers from sliced-open horses.\n\nOnly two live Japanese came out of the Pockets. One with a broken thigh was taken to Colonel Berry's command post. He was almost naked, his face skeletal. Berry gave him a pair of shorts, and a milk shake from the only food available, sugar and a can of milk. After gulping down the concoction the Japanese, to Berry's amazement, painfully tried to stand. He couldn't, bowed his head from a prone position in gratitude, then vomited.\n\n# PART\n\n# 4\n\n# DEATH OF TWO EMPIRES\n\n# 14\n\n# _This Naked Island_\n\n**1** The keystone of the British Empire in Asia was Singapore, the island at the southern end of the Malay Peninsula. Here East met West. Not far up the penninsula, East\u2014in the form of the 26,000 well-trained men of Lieutenant General Tomoyuki Yamashita\u2014was now surging toward the strategic island determined to drive West into the sea by March 1.\n\nOpposing Yamashita were 86,895 troops under the command of Lieutenant General A. E. Percival, a tall, thin man with two protruding, rabbit-like teeth. Although he was an outstanding staff officer of quiet charm, some observers felt Percival lacked the forcefulness to inspire the assorted British, Indian and Australian units under him.\n\nThe clumsy American-built Buffalo fighter in his air force never had a chance against the Zero. Within a week the air above Malaya was Japanese. On the ground Percival was almost as handicapped. There wasn't a single Allied tank in all Malaya because British experts had decreed armor was unsuited for jungle warfare. The Japanese experts felt differently and now their tanks were playing havoc with the defenders. The Indians in particular were terrified by the metal monsters that charged down the roads. Most had never seen a tank before.\n\nBy Christmas Yamashita had driven a third of the way to his goal. At 11:20 that morning CBS correspondent Cecil Brown began broadcasting another of his frank reports to America. \"This morning I talked with a number of people and it was not in their hearts to say or even think, 'Peace on Earth\u2014Good Will Toward Men.' The Japanese are too near for that, and most people know that this Christmas Singapore is reaping the terrifying fruits of wishful thinking and unpreparedness....On this Christmas, Singapore has the problem of rectifying almost overnight fifty years of a strange kind of administration of the natives, and one year of military apathy. During that year it was a conviction, almost a certainty, of the military, that Japan would not move, that there would be no war in the Far East. And that conviction carried down to three days before war actually came. That's why this is a grim Christmas in Singapore, because the British out here are getting ready for a war with the war going on, and the Japanese holding northwestern Malaya and dominating the skies over it. The tragedy of this Christmas in Singapore is felt on all sides, I've found, because the British have a capacity for dying with such bravery....\"\n\nIn spite of the situation many British officials and officers were still confident of victory. The Australians, they reasoned, had yet to see battle and would easily hold the line.\n\nThe Australian commander, Major General Gordon Bennett, didn't share such optimism. A brusque, red-haired man, Bennett looked more like a soldier than any other general in Malaya. Though he had won the DSC at Gallipoli, he was an accountant by trade. After several trips to the front he was convinced something drastic had to be done to stop Yamashita. \"I have seen a total absence of the offensive spirit,\" he wrote Australian Army Headquarters in Melbourne. He felt only counterattack and immediate air reinforcements could stop the Japanese.\n\nBennett's worries were passed on to the Australian prime minister, John Curtin. He, in turn, on that Christmas day cabled Roosevelt and Churchill, then attending the Arcadia Conference in Washington. He begged that reinforcements be sent to Singapore before it was too late. Then, fearing that these two leaders were too engrossed in Hitler to realize how grave the Singapore danger was, Curtin also wired his own minister in Washington:\n\n> PLEASE UNDERSTAND THAT STAGE OF SUGGESTION HAS PASSED...THIS IS THE GRAVEST TYPE OF EMERGENCY AND EVERYTHING WILL DEPEND UPON CHURCHILL-ROOSEVELT DECISION TO MEET IT IN BROADEST WAY.\n\nRoosevelt and Churchill responded by selecting General Sir Archibald Wavell as supreme commander of the entire Southwest Pacific area. As many planes, tanks and men as could be spared from the bitter fighting in the Middle East were to be sent to Singapore.\n\nWavell flew to Singapore Island on January 7 for a brief inspection tour. The previous night 15 Japanese tanks had burst through the front lines of the 11th Indian Division and crossed the key Slim River bridge. They were now less than 250 air miles from Singapore itself. The General rushed north by car. Here he found that the entire III Corps was disorganized, the 11th Indian Division completely shattered. He ordered a general withdrawal of nearly 150 miles to Johore Province where Bennett and his Australians would have to make the final stand on the peninsula.\n\nStill shaken by what he had seen on the mainland, Wavell returned to Singapore to inspect the defenses on the north side of the \"Fortress Island.\" He was shocked to find nothing. There weren't even detailed plans for defense against a land attack. He learned to his consternation that almost all the island's great guns faced the sea and couldn't be turned around to fire at the onrushing Japanese. Far from being impregnable, the island of Singapore was almost naked.\n\n**2** Six days later, on January 13, a convoy of large American vessels was approaching the harbor. The word spread: Singapore had not been abandoned; substantial help was finally arriving. Then there was a growing roar as a swarm of Japanese planes appeared in the north. Suddenly dark clouds closed in and before the Japanese could sight the oncoming ships, the storm broke. Many regarded it as a miracle. At any rate, reinforcements were soon unloading: the 53rd British Infantry Brigade, the British 6th Heavy and 35th Light Anti-aircraft Regiments, the 85th British Antitank Regiment, and 51 Hurricane fighters.\n\nThe wave of optimism increased the next day when control of the front passed to the aggressive General Bennett. Unlike Percival, the Australian thought Yamashita could be stopped by strong shifting lines and frequent ambushes. The purely defensive attitude of the British command, he thought, had to be replaced by strong counterattack methods. There were too many brilliant officers in Malaya, he wrote in his diary, too few aggressive fighters. At last Bennett would have the chance to prove his theories. About 120 miles northwest of Singapore, his Australians were hiding in the dense jungle on both sides of Gemencheh Bridge.\n\nAt 4:00 that afternoon a column of Japanese approached the bridge on bicycles, five and six abreast. To the hidden Australians they resembled a huge picnic party. After about 250 cyclists crossed the bridge there was a terrific explosion. Timber, bicycles and bodies were hurled in the air. Deadly fire from concealed Bren guns, Tommy guns and rifles swept the road.\n\nThe Japanese, caught completely by surprise, didn't have time to unstrap guns from their bicycles. In twenty minutes the 300-yard stretch of road was littered with dead and dying.\n\nNews of this first Australian action in Malaya kindled hopes in Singapore, already revived by the recent reinforcements. And the next morning when the Australians knocked out five Japanese tanks and disabled several others, confidence in ultimate victory returned. Singapore Radio declared the tide of battle was turning, \"with the A.I.F. as our sea wall against the vicious flood.\"\n\nBut hope of victory was short-lived. On the morning of January 17 the AP correspondent, Yates McDaniel, hearing ominous stories of a disastrous rout, drove up toward the front. Born in China of American missionary parents, he had been the first foreign correspondent to cover the Sino-Japanese war. Many of his fellow newsmen claimed, with good reason, that he knew more about Asia than any other Westerner. He was a slender man of thirty-five; his high forehead, light brown eyes and delicate, almost transparent skin gave him the aesthetic look of a poet or monk. Yet in spite of his fragile appearance and quiet manner, probably no other white man had traveled through as much of the rugged interior of Asia.\n\nJust before noon McDaniel found General Bennett at the edge of a rubber plantation, about to confer with Percival and Brigadier B. W. Key, the new commander of the re-formed 11th Indian Division. The red-haired Australian was critical of continued withdrawals. He told McDaniel that two days previously the inexperienced 45th Indian Brigade guarding the west coast road at Muar had been attacked and quickly overwhelmed. Australian reinforcements had been rushed to the danger point and were now fighting a stubborn but losing action. With the left flank gone, the entire line was endangered. Though it was the time for counterattack, said Bennett, retreat was still the fashion of the day.\n\nPercival now joined the group. To McDaniel he was a gallant, tragic figure, doomed to failure. Then Brigadier Key arrived. The three generals sat on fallen trees. Percival's high-pitched voice was calm. He was tactful and patient. Key said little, but Bennett spoke angrily, insisting on a more aggressive attitude. After an hour, the meeting broke up and the newsman returned with Bennett to the Australian's field headquarters. It was spartan, a small tent with a simple cot.\n\n\"We've got to make counterattacks,\" reiterated Bennett.\n\n\"We can't worry about a few of our people being cut off in night operations.\" His gloom deepened when new reports of further infiltrations on the left flank came in.\n\nOn the way back to Singapore, McDaniel stopped his jeep and talked to a group of bedraggled British soldiers. One who had come from Shropshire told of being captured the week before and later escaping. That evening McDaniel wrote a feature story of his trip to the front based on the escaped Tommy. \"Wandering along the road today,\" it began, \"I came upon a Shropshire lad.\"\n\nThe British censor refused to pass the story. \"You can't mention 'Shropshire,' Yates. It identifies the name of the regiment.\"\n\n\"There is no Shropshire regiment in Malaya.\"\n\n\"The name has to come out,\" insisted the censor.\n\n\"But it will kill the point.\" McDaniel had to explain it was based on one of Housman's famous poems.\n\nThe censor remained adamant.\n\nMcDaniel wasn't the only newsman suffering from censor trouble. Nine days previously Cecil Brown of CBS had been arbitrarily banned from broadcasting, not for what he had said over the air but for what he had wanted to say.\n\n\u2014\n\nThe steadily deteriorating Malayan battlefront was intensifying the bitterness between Churchill and Australian Prime Minister Curtin. The next day Curtin, in response to a hurt and testy message from Churchill, sent an equally testy reply:\n\n> WE HAVE CONTRIBUTED WHAT WE COULD IN LAND AND AIR FORCES AND MATERIAL TO THIS REGION (SINGAPORE) AND CONSISTENTLY PRESSED FOR THE STRENGTHENING OF THE DEFENCES, BUT THERE HAVE BEEN SUGGESTIONS OF COMPLACENCY WITH THE POSITION WHICH HAVE NOT BEEN JUSTIFIED BY THE SPEEDY PROGRESS OF THE JAPANESE....\n> \n> MY OBSERVATIONS ON CRETE AND GREECE IMPLY NO CENSURE ON YOU, NOR AM I PASSING JUDGEMENT ON ANYONE, BUT THERE IS NO DENYING THE FACT THAT AIR SUPPORT WAS NOT ON THE SCALE PROMISED....I HAVE STATED THIS POSITION FRANKLY TO THE AUSTRALIAN PEOPLE BECAUSE I BELIEVE IT IS BETTER THAT THEY SHOULD KNOW THE FACTS THAN ASSUME THAT ALL IS WELL AND LATER BE DISILLUSIONED BY THE TRUTH.\n> \n> NO ONE HAS GREATER ADMIRATION FOR THE MAGNIFICENT EFFORTS OF THE PEOPLE OF THE UNITED KINGDOM THAN THEIR KINSFOLK IN AUSTRALIA. NEVERTHELESS, WE MAKE NO APOLOGIES FOR OUR EFFORT, OR EVEN FOR WHAT YOU ARGUE WE ARE NOT DOING. THE VARIOUS PARTS OF THE EMPIRE, AS YOU KNOW, ARE DIFFERENTLY SITUATED, POSSESS VARIOUS RESOURCES, AND HAVE THEIR OWN PECULIAR PROBLEMS.\n\nIn London Churchill had just arrived from the U.S. in a flying boat and was examining his accumulated messages. When he read a report from Wavell revealing that Singapore was defenseless on its northern shore, he was incredulous. He was also astonished that no measures worth speaking of to build such defenses had been taken by any of the Singapore commanders since the war began. In fact they had not even mentioned to him that they didn't exist.\n\nLike so many others he had put his faith in Fortress Singapore. He blamed himself. He should have known. His advisers should have known and told him, but he had never asked about the matter, busy as he was, because the possibility of Singapore having no landward defenses was as ridiculous as launching a battleship without a bottom. There was no time for recriminations; the neglect had to be instantly repaired. He hastily dictated a note for the chiefs of staff:\n\n> I must admit to being staggered by Wavell's telegram of the 16th....It never occurred to me for a moment...that the gorge of the fortress of Singapore, with its splendid moat half a mile to a mile wide, was not entirely fortified against an attack from the northward. What is the use of having an island for a fortress if it is not to be made into a citadel?...How is it that not one of you pointed this out to me at any time when these matters have been under discussion? More especially this should have been done because...I have repeatedly shown that I relied upon this defence of Singapore Island against a formal siege, and never relied upon the Kra Isthmus plan....\n> \n> Not only must the defence of Singapore Island be maintained by every means, but the whole island must be fought for until every single unit and every single strong point has been separately destroyed.\n> \n> Finally, the city of Singapore must be converted into a citadel and defended to the death. No surrender can be contemplated.\n\nThe urgency long felt in Australia had finally reached 10 Downing Street.\n\n\u2014\n\nSingapore's 2,000,000 citizens were already getting a severe taste of what war was like. One the morning of January 22 the city, battered by two flights of 50 bombers, was digging out hundreds of dead and wounded from the ruins. Even so, morale was still high. Though thousands of leaflets snowed down from Japanese planes saying that \"the Japanese forces desire to refrain from seeing the city reduced to ashes,\" the general spirit of the conglomerate citizenry was expressed by the sign tacked on Government House: THEY CAN'T STOP OUR CLOCK.\n\nThe next day Cecil Brown prepared to leave Singapore. As he stepped in the airdrome bus two strangers recognized him.\n\n\"When you get out of here,\" said the first man, an American, \"I wish you would give this place the works.\"\n\nThe other man, an Englishman, agreed. \"What they did to you is disgraceful. The people of England should know what has happened here.\"\n\nAt the airdrome a censor examined his stories, broadcasts and diary, then said, \"I know all about you, Mr. Brown.\" The correspondent expected his diary would be confiscated, but the censor said under his breath, \"I hope you publish the whole story of Singapore. Publish everything about this country.\" He winked. \"Go ahead, close up your suitcase.\"\n\nLater that day the spirited exchange between Curtin and Churchill burst out again. The Australian representative in London, Sir Earle Page, had chanced to see a note Churchill had written to his chiefs of staff on January 21. In it Churchill wondered whether the docks and batteries of Singapore shouldn't be blown at once and everything concentrated on the defense of Burma and the Burma Road.\n\n> We may, by muddling things and hesitating to take an ugly decision, lose both Singapore and the Burma Road. Obviously the decision depends upon how long the defence of Singapore Island can be maintained. If it is only for a few weeks, it is certainly not worth losing all our reinforcements and aircraft.\n\nPage immediately radioed a copy of this note to Curtin. Now the outraged Curtin was radioing Churchill:\n\n> PAGE HAS REPORTED THAT THE DEFENCE COMMITTEE HAS BEEN CONSIDERING THE EVACUATION OF MALAYA AND SINGAPORE. AFTER ALL THE ASSURANCES WE HAVE BEEN GIVEN, THE EVACUATION OF SINGAPORE WOULD BE REGARDED HERE AND ELSEWHERE AS AN INEXCUSABLE BETRAYAL.\n\nWhen Churchill received this message he was faced with a difficult decision. Although he was morally certain further reinforcements to Singapore would be a waste, he ordered the 18th Division to continue to the beleaguered city as planned. Later he would write, \"The effect that would be produced all over the world, especially in the United States, of a British 'scuttle' while the Americans fought on so stubbornly at Corregidor was terrible to imagine. There is no doubt what a purely military decision should have been.\"\n\n**3** By January 23 all hope of holding any part of the Malay Peninsula was abandoned. At a conference in Bennett's headquarters Generals Heath and Key said their men couldn't stand. Bennett insisted his people could hold but since the rest of the line was buckling, he too felt withdrawal to Singapore was inevitable.\n\nThe retreat was uneventful and by midnight of January 31 almost all of Percival's troops had crossed the 70-foot-wide causeway that connected the peninsula to the island of Singapore.\n\nBennett watched the troops trudge across for a while, then retired to his new headquarters in a cottage on the island. \"Our duty now is to recapture Malaya at the earliest possible opportunity,\" he wrote in his diary just before retiring. \"We owe it to the nation! We owe it to ourselves!\"\n\nBy dawn, February 1, the Indians who had fought hard but to little avail had crossed. Then came the Australians and the Gordons. Finally a skirl of bagpipes could be heard and to the tune of \"A Hundred Pipers\" the battered remnants of the Argyll battalion, a mere 90 men, marched briskly onto the bridge. Bringing up the rear was their commander, Brigadier I. MacA. Stewart, the last man off Malaya. It was a sight that brought tears to many onlookers.\n\nDemolition squads began laying final depth charges on the causeway. Just after 8:00 A.M. the charges were touched off. There was a dull roar. After the smoke and din subsided, those on the island could see water rushing through a 70-foot gap. They thought their fortress was cut off from the onrushing invaders. Few realized the water in the gap was only four feet deep at low tide.\n\n\u2014\n\nTwo strategic islands had already been scenes of conflict\u2014Wake and Hong Kong. Singapore was much larger, ten times the area of Manhattan. It extended 26 miles from east to west, 14 miles from north to south. At the causeway it was 1100 yards to the mainland. On the west coast the distance across the Strait of Johore narrowed to 600 yards.\n\nMost of Singapore's population was crowded in the south and east. The rest of the island was covered with rubber plantations and jungle growth. There were a few hills but the highest was only 481 feet.\n\nLittle happened during the first few days of siege except bombings and intermittent Japanese shellings. Then on February 5 Australians defending the west coast of the island heard sounds of hammering and sawing from the mainland. About 15 miles to the east, in the heart of the city, Singapore's citizens were living without panic in spite of increasing bomb casualties. Although Churchill had called for every man and woman to dig defenses with shovels, almost nothing had been done before the troops arrived. Public concrete shelters had not yet been built, although more civilians had already been killed than in Malta's two years of bombing.\n\nGeneral Percival was taking personal charge of the island's defense. He had two choices: to hold at the beaches or to man the coast line thinly, keeping large reserves for a decisive battle on the island. Though the 200-square-mile island had a coast line of more than 70 miles, Percival decided to defend the beaches. He reasoned that dumps were dispersed and some would certainly be lost if the Japanese got a footing on the island. There was also the morale problem. If the Japanese landed the troops and civilians might panic.\n\nOn paper the situation looked better than it was. The Japanese would probably attack with 60,000 men. Now that the entire 17th British Division and 44th Indian Brigade had arrived, Wavell had 85,000, but 15,000 were noncombatants and many of the others were raw, untrained and poorly armed. There was another depressing factor. All but a single squadron of the newly arrived Hurricane fighters had been sent to Sumatra because only one of the four airdromes on the island was out of range of Japanese shells.\n\nArtillery fire from the mainland increased the following two days. On February 8 it rose to such a pitch on the west coast that General Bennett feared it might be a barrage preparatory to a landing. Early that evening he went to bed in his bungalow in Bukit Timah, a village on heights a few miles west of the city. He figured it might be his last good rest for a long time.\n\nHis troops had been dismayed when they walked onto the fortress island. Except for a half-finished tank trap not a trench had been dug in their swampy area. They were expected to defend more than 20 miles of the northwest coast, where the strait was narrowest, with about 2500 men.\n\nThe barrage increased. Bennett got out of bed and telephoned his duty officer, Major C. B. Dawkins. \"Ask 22nd Brigade Headquarters if it's had any reports from forward posts. Tell them to switch on their beach-lights.\" Bennett went back to bed, but couldn't sleep. By now the enemy bombardment was as rapid as drumfire. Again he got up. Just before 10:30 P.M. he drove to his operations room.\n\n\u2014\n\nJapanese barges, pontoons, and collapsible boats were at that moment nearing the northwest coast of Singapore. About 15,000 infantrymen were preparing to hit the beaches defended by Bennett's 2500 men.\n\nNow mortars, set at a fixed elevation on the Japanese barges, erupted. Their shells fell in front of the oncoming flotilla. As the barges drew closer, the barrage crept up, laying a screen of smoke. At 10:30 P.M. the first wave hit the beach at the end of Lim Chu Kang Road. Australians of the 24th Machine Gun Battalion sprayed the invaders, setting an ammunition barge afire. In the blazing light several barges and boats were sunk. Other landing craft beached on a nearby mangrove swamp area which was lightly defended.\n\nThe Australians fought hard all that night but they were unable to hold back the invaders. In the early morning hours scores of tanks landed and strong infantry-tank teams moved inshore. It reminded the Australians of fighting a bush fire in their own country, with flames encircling those who tried to put them out. By dawn, February 9, Yamashita had a firm foothold on the west section of the island, and was pushing aggressively toward the city of Singapore itself, only 10 air miles to the southeast.\n\nThe next day General Wavell flew from his headquarters in Java and broke through the Japanese air screen around Singapore. By now the situation was chaotic because of a series of errors of judgment. Someone had thoughtlessly issued Percival's secret and personal perimeter defense plan to all the field commanders. Many of these men, exhausted and distracted by the desperate fighting, were eager to assume the plan was an order to retreat and began falling back to the last-ditch perimeter positions.\n\nRumors were already spreading of angry scenes at headquarters: Wavell had severely criticized Percival because the Japanese had been allowed to land; then the ABDA (American, British, Dutch, Australian) chief had become so fed up with Bennett's comments that he told the Australian to \"Get the hell out\" and take his \"bloody Aussies\" with him.\n\nIt was certainly no rumor that all was not harmonious. When Wavell learned later that morning that many commanders were falling back to the perimeter he ordered a counterattack which promptly failed. He also issued a Churchill-inspired order of the day:\n\n> It is certain that our troops on Singapore Island greatly outnumber any Japanese that have crossed the Straits. We must defeat them. Our whole fighting reputation is at stake and the honor of the British Empire. The Americans have held out on the Bataan Peninsula against far greater odds, the Russians are turning back the picked strength of the Germans, the Chinese with almost complete lack of modern equipment have held the Japanese for 4 1\/2 years. It will be disgraceful if we yield our boasted fortress of Singapore to inferior enemy forces.\n> \n> There must be no thought of sparing troops or the civil population and no mercy must be shown to weakness in any shape or form. Commanders and senior officers must lead their troops and if necessary die with them.\n> \n> There must be no question or thought of surrender. Every unit must fight it out to the end and in close contact with the enemy....I look to you and your men to fight to the end to prove that the fighting spirit that won our Empire still exists to enable us to defend it.\n\nBy the following dawn, February 11, Yamashita's men held almost half the island. At 6:00 A.M. Percival was wakened by the chatter of machine guns. The general got up and learned that the Japanese were advancing down Bukit Timah Road and were nearing the Racecourse, about seven minutes by car from the Raffles Hotel.\n\nThe general drove up Bukit Timah Road to see what was happening. It was a strange sensation. The sky was covered by the pall of many fires. The great road, usually jammed with traffic, was deserted. Above, Japanese planes hovered, unopposed except for anti-aircraft, looking for targets. He felt naked in the only car driving up the wide road. \"Why,\" he asked himself, \"does Britain, our improvident Britain, with all her great resources, allow her sons to fight without air support?\"\n\nLater that morning\u2014the anniversary of Japan's greatest patriotic festival, the Foundation of the Empire\u2014planes dropped twenty-nine wooden boxes each about 18 inches long. Each contained a message from Yamashita addressed \"To the High Command of the British Army, Singapore,\" and read:\n\n> My sincere respect is due to your army which, true to the traditional spirit of Great Britain, is bravely defending Singapore which now stands isolated and unaided....But the development of the general war situation has already sealed the fate of Singapore, and the continuation of futile resistance would only serve to inflict direct harm and injuries to thousands of non-combatants living in the city, throwing them into further miseries and horrors of war, but also would add nothing to the honor of your army.\n\nStill remembering Wavell's final orders to \"fight to the end,\" Percival did not even send Yamashita a reply.\n\nIn spite of planes overhead and shells bursting nearby, civilians walked the streets and queued up outside the cinema in the Cathay skyscraper to see _The Philadelphia Story_. They ignored leaflets fluttering down marked EXTRA saying Roosevelt had just negotiated a separate peace with Japan and had asked the Japanese to declare Singapore a neutral city. Riller's Band still played at the Adelphi Hotel. The Raffles Hotel was jammed with staff officers with nothing to do but sit, drink and carp.\n\nThe _Straits Times_ , now a one-page newspaper, still pretended there was hope. On the top of each issue was blazoned: \"Singapore Must Stand; It SHALL stand\u2014H. E. the Governor.\" But it was obvious those in command knew the end was near when they ordered 1,500,000 bottles of wine and liquor and 60,000 gallons of samsu, Chinese spirits, destroyed.\n\nYates McDaniel was amazed at the calmness of the civilians in the face of the deepening rumble of guns. He was now the only foreign correspondent left in the city. His wife had sailed with the others but it had always been his rule to stay as long as communications held out. He passed a wall scrawled in chalk, ENGLAND FOR THE ENGLISH, AUSTRALIA FOR THE AUSTRALIANS, BUT MALAYA FOR ANY SON OF A BITCH WHO WANTS IT. Some Aussie, obviously, was expressing his opinion of the whole campaign.\n\nA little later McDaniel wrote what he thought might be his last story. As he walked toward the Cathay Building to file it, he saw many Europeans, military and civilian, hurrying toward the dock area. They were escaping on a fleet of small coast boats just collected. For a moment he felt impelled to join them, then decided to stay longer. The Singapore story was still alive.\n\nAt the Cathay Building he found the British censor packing. \"How can I do my job without a censor?\" protested McDaniel.\n\nThe censor cursed him in a friendly way, but hastily stamped CLEARED BY CENSOR on several blanks. \"Write anything you want. Send anything you want. I'm leaving.\"\n\nMcDaniel began to write: \"This will probably be my last message from this crumbling fortress....\"\n\n**4** The next morning the battle increased in ferocity. By 8:00 A.M. Yamashita's tanks were racing down Bukit Timah Road past the Racecourse. In half an hour they reached the Chinese High School on Singapore's outskirts, less than three miles from the city limits. Here they were finally stopped by mixed British and Australians armed only with rifles.\n\nHearing of the new threat, Percival again drove up to the front to see if this breakthrough was dangerous. One look convinced him to order the Indians and British guarding the northern and eastern shores to fall back and form a tight defensive arc in front of the threatened city.\n\nIn his room at the Cathay Building, Yates McDaniel of AP was just finishing a story:\n\n> The sky over Singapore is black with smoke from a dozen huge fires this morning as I write my last message from this once beautiful, prosperous and peaceful city. The roar and crash of cannonade and bursting bombs which are shaking my typewriter, and my hands which are wet with the perspiration of fright, tell me without need of an official communiqu\u00e9 that the war which started nine weeks ago 400 miles away is now in the outskirts of this shaken bastion of Empire.\n> \n> I am sure a bright tropic sun is somewhere overhead, but in my many-windowed room it is too dark to work without electric lights.\n> \n> Under the low rise where the battle is ranging I see relay after relay of Japanese planes circling, then going into murderous dives on our soldiers, who are fighting back in the hell over which there is no screen of our own fighter planes. But the Japanese are not completely alone in the skies. I just saw two Vildebeestes\u2014obsolete biplanes operating at 100 miles an hour\u2014fly low over the Japanese positions and unload their bombs with a resounding crash. It makes me ashamed, sitting here with my heart beating faster than their old motors, when I think of the chance those lads have of getting back. If ever brave men earned undying glory, those R.A.F. pilots have this tragic morning.\n> \n> There are many other brave men in Singapore today. Not far away are A.A. batteries. They are in open spaces because they must have a clear field of fire. (Please pardon the break in continuity, but a packet of bombs just landed so close that I ducked behind the wall, which I hoped would, and did, screen the blast.) But those gun crews keep on fighting and peppering the smoke-limited ceiling every time Japanese planes come near, which is almost constantly.\n> \n> The all-clear has just sounded\u2014a grand joke, because from my window I can see three Japanese planes hedge-hopping under a mile away. I heard a few minutes ago a tragic telephone conversation. Eric Davis, director of the Malayan Broadcasting Corporation, urged Sir Shenton Thomas, Governor, for permission to destroy the outlying broadcasting station. Sir Shenton demurred, saying the situation was not too bad. Davis telephoned the outlying station and instructed the staff to keep on the air, but to stand by for an urgent order. We tuned in to that station, and then, in the middle of a broadcast in the Malayan language, urging Singapore people to stand firm, the station went dead.\n\nMcDaniel hurried out of the Cathay Building to file the story at the cable office. He noted a new spirit in the great city. British, Chinese and Malayans were working side by side pulling out the wounded and dead from destroyed buildings. All races were flooding to hospitals to donate blood. Those British remaining were now doing the unthinkable: their villas were freely opened as places of refuge to all comers. Democracy had finally come to Singapore.\n\nIt was grimly amusing to hear some of his British friends at the Singapore Club and the Cricket Club who had gloomily predicted that the Asiatics of the city would panic and betray them to the Japanese, now praise \"their boys\" as loyal and brave. The stubborn fighting of hastily organized Asiatic defense teams like Dalforce\u2014a motley but formidable crew of Chinese college boys, rickshaw coolies, loyalists and Communists under the command of J. D. Dalley, a policeman\u2014was proving what a costly mistake British officials had made in not arming such native groups months ago.\n\nBlack clouds of smoke drifted from the north; the rumble of big guns was louder; shells were falling closer to the center of the city. As McDaniel hurried past the white granite buildings of the city, shored up with sandbags, he saw a dejected battalion of Indians sitting in front of their tents in a public park. It shook him.\n\nAt the cable office he soon convinced the operator to send his story as an \"official dispatch.\" As he came out, a Chinese girl of nineteen handed him a note. It was from his good friend Wong Hai Sheng, better known as \"Newsreel\" Wong, the most famous movie cameraman in the Orient, whose picture of a crying baby amidst the wreckage of Shanghai's railroad station had already become a classic. The girl was Wong's niece, Doris Lim. She had been refused passage on the ship that had evacuated \"Newsreel\" and other newsmen.\n\nMcDaniel assured the frightened girl he would get her safely out of Singapore. How was another question. The last convoy of refugees was due to leave any moment. The two hurried to the dock area. It was a hectic scene as 3000 civilians and military crowded aboard the fleet of pitifully ancient boats collected by Sir Robert Scott, head of the British Ministry of Information, Far East Section. Whenever a plane swooped down everyone froze, for almost anything in the air was Japanese.\n\nDeserters from the front were beginning to sneak to the docks, hoping by persuasion, bribe or force to get a passage to safety. The tension was becoming intolerable. These few boats seemed to represent the only escape.\n\nAs McDaniel was photographing the scene, Captain Henry Steel, an information officer, told him about an empty craft, the 1700-ton steamer _Kung Wo_ from the Yangtse River, which had been serving as a Navy mine layer. McDaniel, Steel and the girl searched the water front and finally located the old coal-burning vessel anchored about one mile off shore. The skipper, a reserve officer, was Scotch and stubborn. Though he promised to take the three when he sailed he refused to leave without definite orders. Another problem was coal. He had only enough for 10 miles.\n\nMcDaniel persuaded the Scotsman to steam to a nearby wharf piled high with coal. At the wharf McDaniel, Steel and some officers and men from the _Repulse_ and the _Prince of Wales_ began stoking coal by hand. The steamer's crew\u2014a strange mixture of soldiers, sailors, Chinese from the Yangtse, and petty officials\u2014was finally urged to help by the increasing roar of battle.\n\nAfter the coal was loaded McDaniel volunteered to return to the city, learn the military situation and get a technical excuse for the Scotch skipper to leave. Transportation was no problem. The docks were jammed with brand-new cars and tanks. He found a Ford with gas and, accompanied by Doris Lim, who wanted to pick up her belongings, he drove to the Cathay Building.\n\nHere he learned the defense perimeter was fast shrinking. It was only a matter of hours before the Japanese would break through. Finally he located Vice Admiral E. J. Spooner, in command of all naval forces. The admiral was burning files in preparation for flight that evening in a patrol boat. McDaniel hurried to his own room and grabbed a pack containing two bottles of German Rhine wine, two cameras, a pair of German binoculars, a few biscuits and four tins of Camels. Then he decided to add a paragraph to his last story. Hurriedly he typed:\n\n> We have less than fifty per cent chance of getting clear. I am leaving now by car. I swear before I embark to put the car in gear and head it straight for the Straits of Malacca. Do not expect to hear from me for many days, but please inform my wife, Mrs. McDaniel, Hotel Preanger, Bandung, Java, that I have left this land of the living, and the dying.\n\nA few minutes later he and Doris Lim were racing along rubble-strewn streets, littered with unburied dead, to the cable office. Here he filed his final message. As they drove to the docks in the fading light, there was a deep rumble. An ammunition dump, McDaniel guessed, had been blown up. Ominous, black clouds were rising from the fuel dumps on the off-coast islands. McDaniel stopped the car, took several pictures.\n\nAs McDaniel and the girl reached the docks, Japanese light bombers began to bomb and strafe everything that moved. Sir Robert Scott's last convoy of refugees was just beginning to move out. It abruptly stopped, hoping for protection from a nearby anti-aircraft battery. At the docks McDaniel explained the situation to Steel and the other officers. They decided first to try and destroy the rows of tanks and cars but a sudden burst of fire from the Kallang airfield less than a mile away discouraged the plan. The breakthrough might come at any moment.\n\nAt sunset the Japanese planes left and the Scott convoy slowly headed out of the harbor. The captain of the _Kung Wo_ refused to follow since McDaniel had not brought back definite orders to sail.\n\n\"But Admiral Spooner is leaving,\" argued McDaniel. The Scotsman shook his head. He was going to wait for orders. \"How can you possibly get orders,\" said McDaniel, \"when there's no one left to give them?\"\n\nThe captain, unconvinced, insisted on waiting. A few minutes later they could hear the individual cracks of rifles.\n\nAt 11:00 P.M. the _Kung Wo_ was still anchored, in spite of arguments by McDaniel and Steel and almost hysterical threats by some of the crew. The Scotch captain was stolidly signalling shore with blinkers. A few minutes after 11:00 P.M. an answer finally came: THE COMMODORE OF MALAYA HAS LEFT.\n\nEven with this confirmation of McDaniel's story, the captain was uncertain. He hated to leave without something more definite. After almost an hour of self-debate he reluctantly sounded bells. It was midnight when the old copper-lined steamer slowly headed toward the mine fields.\n\nThe ship headed south, threading its way through heavy mine fields and the small islands that hung down from Singapore for more than 100 miles like a string of beads. By dawn, Friday, the thirteenth, the _Kung Wo_ was midway through this \"1000 Island\" chain.\n\nEven the early sun was sweltering, enervating; in another 25 miles they would cross the equator. There were excited shouts. About 30 miles to their right, the west, the green hilly coast of Sumatra had been sighted. This was the first of a group of Dutch-owned long, thin islands, almost nose to nose, that formed a great barrier of about 3000 miles between the Japanese and Australia. The central and most strategic island in this lengthy barrier was McDaniel's goal: Java.\n\nSoon they could hear the roar of approaching Japanese planes, dispatched to sink the final flotilla of ships, launches and boats fleeing from Singapore. The skipper of the _Kung Wo_ quickly turned away from Sumatra toward the shelter of one of the \"1000 Islands.\" The little ship zigzagged at top speed, 10 knots, narrowly avoiding falling bombs. Finally one hit the engine room. Another exploded between the bridge and midships. The old ship heeled over and began to list dangerously to starboard as flames leaped up from the engine room. McDaniel saw the one seaworthy lifeboat being lowered over the starboard side. Contrary to British tradition, it was filled with twenty-five crew members. McDaniel alerted a bearded _Prince of Wales_ officer.\n\n\"By whose orders are you leaving this ship?\" shouted the officer.\n\n\"The hell with you, sonny boy,\" shouted a man in the boat. Others gave catcalls.\n\n\"I order you to stay aboard.\"\n\nThe lifeboat hit the water clumsily. The crew cut loose, completely fouling the lines of the pulley system, and rowed off.\n\nThe second lifeboat was hastily caulked. Then Doris Lim and all the passengers except McDaniel boarded and, with a junior officer in charge, it headed for a small island seven miles away. It was to return as soon as possible. The third lifeboat, even more rickety, was lowered and promptly sank. Life rafts were tossed over. They disintegrated at once. More planes swept over. One by one they descended, looked over the crippled ship and then, without wasting a bomb or bullet, proceeded to a better target. The afternoon hours passed slowly but still the lifeboat didn't return. Only the copper-lined bottom was holding up the _Kung Wo_.\n\nAt 5:00 P.M. McDaniel sighted the approaching lifeboat and suggested to the three officers who had remained that they celebrate. They found four crystal goblets and toasted each other, the British Empire, Singapore and finally the good ship _Kung Wo_ with McDaniel's two bottles of _Liebfraumilch_. All agreed that nothing had ever tasted better even though the wine had been heated by the equatorial sun to more than 90 degrees.\n\nThey didn't reach the small island until 7:00 P.M. As they waded ashore, the first lifeboat approached. The twenty-five crew members were blistered and almost raving from the heat. They had thought to take along food and liquor, but no water. They were greeted with absolute silence.\n\nIt was a night of tension on the malaria-infested island. A minor British official sleeping on the sand next to McDaniel and Doris Lim kept trying to steal the pooled food. He stopped only when threatened with shooting. He then turned his attention to the girl. Again McDaniel and Captain Steel had to threaten shooting.\n\nThe next morning, February 14, was hazy and muggy with an ominous heavy mist hanging over the sea. They heard the familiar roar of Japanese bombers. Then through the overcast they saw bombs falling on the smoldering _Kung Wo_. After two direct hits the old ship turned over and sank. The Scotch captain buried head in hands and wept. A moment later someone noticed a patrol launch approaching. The officers held a whispered conference. The Scotch captain said, \"If it's Jap and they hail us, I'm surrendering.\"\n\nMcDaniel and Steel asked if they could hide. The Scotsman gave them permission and while the other 130, including Doris Lim, prepared for surrender, McDaniel and Steel crawled into the brush. They peered down at the beach as the launch grew bigger.\n\nTwo formations of Japanese bombers flew over McDaniel's little island, completely ignoring the scattered refugee ships below. They kept heading south, soon reaching the coast of Sumatra. The Japanese were not waiting for the fall of Singapore to continue to their next destination on the master schedule of conquest. At a point about 150 miles south of McDaniel these planes debouched 360 picked paratroopers. They drifted down to Palembang, one of the most important oil refinery centers in the Southwest Pacific.\n\nThe combined Dutch-British force defending Palembang fought bitterly and by midafternoon it looked as if the paratroopers might be wiped out, but Japanese reinforcements were only 100 miles north of Sumatra in eight transports.\n\nCovering these invasion transports was the 7th Cruiser Squadron of Vice Admiral Ozawa. It was just sighting the remnants of the flotilla fleeing south from Singapore. Five cruisers began to finish the job on the refugees started by the bombers. By dusk more than 20 crowded ships had been sunk with heavy loss of life.\n\n**5** In the meantime, Singapore was nearing its end.\n\nThe day before, on Friday the thirteenth, at a morning conference, Percival's commanders had unanimously agreed that further resistance was hopeless. Even the newly arrived British 18th Division was \"Done,\" according to its commander. Ammunition was rapidly disappearing. The troops were falling back on every front. Morale was low. It was sadly decided to send a message to Wavell in Java urging him to agree to an immediate surrender.\n\nBy the next morning, February 14, all of the city was within artillery range of the Japanese. Gangs of armed deserters\u2014mostly from administrative units\u2014wandered the streets, their shirts stuffed with cigarettes and tinned food.\n\nThe suffering of civilians was almost intolerable. The green lawns of the Singa General Hospital were now pocked by graves. Inside, the stink of blood and entrails was nauseating. The dead and dying were on beds, under beds, between beds, in the corridors. The operating theatres were butcher shops; doctors and nurses were bathed in blood. Since water was short, instruments were sterilized in dirty water. Nurses had to wash their hands in bottled mineral water.\n\nPercival had just received an answer from Wavell.\n\n> YOU MUST CONTINUE TO INFLICT MAXIMUM DAMAGE ON ENEMY FOR AS LONG AS POSSIBLE BY HOUSE-TO-HOUSE FIGHTING IF NECESSARY. YOUR ACTION IN TYING DOWN ENEMY AND INFLICTING CASUALTIES MAY HAVE VITAL INFLUENCE IN OTHER THEATRES. FULLY APPRECIATE YOUR SITUATION, BUT CONTINUED ACTION ESSENTIAL.\n\nKnowing the situation was hopeless, Percival again radioed Wavell, telling him of the desperate water situation. The answer was soon received:\n\n> YOUR GALLANT STAND IS SERVING A PURPOSE AND MUST BE CONTINUED TO THE LIMIT OF ENDURANCE.\n\nSurprisingly, some of Yamashita's staff had become so impressed by the accurate British artillery that they were advising their general it was hopeless to keep attacking: there should be a withdrawal to a safer position. Yamashita listened, then ordered the assault quickened.\n\nThe next morning, February 15, Percival learned that the entire water system had broken down, with most of the scanty supply running to waste because of constant pipe breakage. After attending a Communion service at Fort Canning, he called a conference of his area commanders at 11:00 A.M.\n\nPercival told them there was almost no petrol, field gun or Bofors anti-aircraft gun ammunition. Water wouldn't last much more than twenty-four hours. There were only two choices: attack and regain the reservoirs or surrender. Each of his commanders said a counterattack was out of the question.\n\nIt was time for Percival to make the hardest, most momentous decision of his life. He made it quickly. He said he would ask the Japanese to cease fire at 4:00 P.M. and send a deputation into the city to discuss terms of surrender.\n\nNow he was handed a message from Wavell.\n\n> SO LONG AS YOU ARE IN POSITION TO INFLICT LOSSES TO ENEMY AND YOUR TROOPS ARE PHYSICALLY CAPABLE OF DOING SO YOU MUST FIGHT ON. TIME GAINED AND DAMAGE TO THE ENEMY OF VITAL IMPORTANCE AT THIS CRISIS.\n> \n> WHEN YOU ARE FULLY SATISFIED THAT THIS IS NO LONGER POSSIBLE I GIVE YOU DISCRETION TO CEASE RESISTANCE. BEFORE DOING SO ALL ARMS, EQUIPMENT AND TRANSPORT OF VALUE TO ENEMY MUST OF COURSE BE RENDERED USELESS. ALSO JUST BEFORE FINAL CESSATION OF FIGHTING OPPORTUNITY SHOULD BE GIVEN TO ANY DETERMINED BODIES OF MEN OR INDIVIDUALS TO TRY AND EFFECT ESCAPE BY ANY MEANS POSSIBLE. THEY MUST BE ARMED. INFORM ME OF INTENTIONS. WHATEVER HAPPENS I THANK YOU AND ALL TROOPS FOR YOUR GALLANT EFFORTS OF LAST FEW DAYS.\n\nRelieved to get official approval, Percival sent a surrender party up the Bukit Timah Road toward the lines of the Japanese 5th Division. Then he dispatched his last message, a telegram to Wavell:\n\n> OWING TO LOSSES FROM ENEMY ACTION, WATER, PETROL, FOOD, AMMUNITION PRACTICALLY FINISHED. UNABLE THEREFORE TO CONTINUE THE FIGHT ANY LONGER. ALL RANKS HAVE DONE THEIR BEST AND ARE GRATEFUL FOR YOUR HELP.\n\nYamashita and his chief of staff, Lieutenant General Suzuki, were watching the battle from the heights near Bukit Timah. Suzuki pointed out a Union Jack fluttering in the breeze atop Fort Canning. It would take a week of hard fighting to capture that hill, he said. Behind Singapore was a small fortified island. To the left was Fort Changi. To seize them would be no simple matter. All in all it would take many days to break through the last defense line.\n\nJust then a front-line commander telephoned that the British were sending a flag of truce.\n\nLieutenant Colonel Ichiji Sugita, who had written the surrender message of February 11, drove forward and met the British surrender team. \"We will have truce,\" he said in Japanese, \"if the British Army agrees to surrender. Do you wish to surrender?\"\n\nCaptain Cyril H. D. Wild, the British interpreter, was tall, blue-eyed, the son of the Bishop of Newcastle. He said, \"We do.\"\n\nSugita told Wild to bring Percival and his staff to the same place for a meeting with General Yamashita. The British agreed, returning to their lines.\n\nAt about 4:45 P.M. Sugita again drove to the front lines. Here he met Percival, Wild and several other British officers. In two cars the party proceeded toward the Ford factory just north of Bukit Timah village. Sugita sat next to Percival and introduced himself in broken English. \"We fought for two months. Now we're coming to the end. I compliment you on the British stand.\"\n\nPercival said little. These were probably the most bitter moments in his life. In faraway England his daughter, Margery, was celebrating her twelfth birthday.\n\nAt the factory the surrender party dismounted from the two cars. With Percival carrying a white flag, the group entered the new Japanese headquarters. A few minutes later Yamashita arrived.\n\n\"We have just received your reply,\" said Yamashita in Japanese. \"The Japanese Army will consider nothing but surrender.\"\n\nA Japanese, formerly a university professor, interpreted falteringly in his poor English. Colonel Sugita and Captain Wild, both of whom had picked up some of the other's language, also tried to help.\n\n\"I fear we shall not be able to submit our final reply before 10:30 P.M.,\" said Percival. His thin face was red, his eyes bloodshot.\n\nYamashita, annoyed, raised his voice. \"Reply to us only whether our terms are acceptable to you or not. Things have to be done swiftly. We're ready to resume firing.\"\n\nThe vague, disjointed discussion continued with Sugita now acting as interpreter. Partially because of Sugita's imperfect English and Wild's poor command of Japanese, and partially because of Percival's natural reluctance, no agreement could be reached. To Sugita, the whole scene, though tense and critical, was somewhat comical.\n\nFinally Yamashita said impatiently, \"Unless you do surrender, we will have to carry out our night attack as scheduled.\"\n\nWhen Wild translated this a look of amazement came over Percival. \"Cannot the Japanese Army remain in its present position? We can resume negotiations again tomorrow at 5:30 A.M.\"\n\n\"What!\" Yamashita again raised his voice. \"I want the hostilities to cease tonight and I want to remind you that the question is strictly a matter of this.\"\n\n\"We shall discontinue firing by 8:30 P.M.,\" said Percival almost in a mumble. \"Had we better remain in our present positions tonight?\"\n\nYamashita told him to keep his troops in their positions. All firing would cease at 8:30 P.M. Then the British would disarm all but 1000 men to keep order in the city. \"You have agreed to the terms,\" continued Yamashita, still annoyed by Percival's vagueness, \"but you have not yet made yourself clear as to whether you have agreed to surrender or not.\"\n\nPercival cleared his throat and nodded.\n\n\"If you've accepted our terms,\" said the exasperated Yamashita, \"we want to hear 'yes' or 'no' from you! Surrender or fight!\"\n\n\"Yes,\" said Percival faintly. \"I agree.\" Then he added, \"I have a request to make. Will the Imperial Army protect the women and children and British civilians?\"\n\n\"We shall see to it. Please sign this truce agreement.\"\n\nAt exactly 7:50 P.M., after fifty nerve-racking minutes, Percival signed.\n\nBoth generals left the factory. The two exhausted interpreters remained to iron out the details. Before they started work Sugita gave Captain Wild, with whom he now felt a friendly bond, two packages of food.\n\nAs agreed, that night at 8:30 the roar of battle abruptly ceased. The sudden silence was strange, uncanny. Singapore\u2014the City of the Lion\u2014was dead. At St. Andrews Cathedral wounded were lying in the nave and the aisles. Doctors and orderlies moved about their work murmuring softly. When the Right Reverend John Leonard Wilson, Bishop of Singapore, heard of the surrender he led a short service. In conclusion a medical corps major played the organ while everyone sang \"Praise My Soul, the King of Heaven.\"\n\nThe Battle of Singapore was over, the most publicized fortress in the world captured. It was the worst disaster and largest capitulation in British history. In seventy days Yamashita, at the cost of 9824 battle casualties, had punched down the Malay Peninsula 650 miles. The British had lost 138,708, including more than 130,000 prisoners. They had also lost an empire.\n\n\u2014\n\nThat night the waters below Singapore were dotted with launches, patrol boats and assorted small craft carrying off several thousand last-minute escapees, including General Gordon Bennett, Brigadier Paris of the 12th Indian Brigade and V. G. Bowden, the Australian Government representative. Farther south the remnants of Scott's convoy were desperately making for the north coast of Sumatra or hiding in the \"1000 Islands.\" Already the majority were captured or dead.\n\nThe 132 from the _Kung Wo_ were still alive, still uncaptured. The patrol launch had not been Japanese, but a cabin cruiser piloted by a Scotch planter. He had promised to send a rescue boat to the little island the following night if possible.\n\nNow on the night of February 15 McDaniel and the others were anxiously waiting. Finally just before 10:00 P.M. the newsman saw a flashlight signal\u2014two long, two short. It seemed to be about a mile away.\n\n\"Can you swim?\" he asked Doris Lim. She couldn't, but with the help of McDaniel and Steel, she thought she could make it. These three and less than 60 others began wading toward the signal over the sharp coral. After three hours of wading and swimming McDaniel and his two companions and about half of those who'd risked the trip finally reached a dingy ferry launch and were hoisted aboard, completely exhausted. The Malayan skipper waited another half-hour for laggards, then headed southwest.\n\nAbout three hours later they crossed the equator. At sunrise the little launch approached the lush green hills of the northern coast of Sumatra. When the skipper found the mouth of a river, the Indragiri, he headed inland to the southwest. He told McDaniel that the Japanese had landed on both ends of the long island. There was only one way to escape: cross to the southern side. They would have to go up river as far as possible by launch, then travel by foot, oxcart or car the rest of the way across Sumatra to Padang on the southern coast. Here they might be lucky enough to get passage on a ship going to Ceylon or Java.\n\nMcDaniel hoped it would be Java. His wife was there, waiting for him.\n\n\u2014\n\nThat morning the Japanese were triumphant. The leading paper, _Asahi Shimbun_ , headlined its story of Singapore: GENERAL SITUATION OF PACIFIC WAR DECIDED.\n\nIn an interview Colonel Hideo Ohira, chief of the Press Division, Imperial Headquarters, said, \"To seize Singapore Island in as little time as three days was only possible with our Imperial Army. Japan is the sun that shines for the world peace. Those who bathe in the sun will grow and those who resist it shall have no alternative but go to ruin. Both the United States and Britain should contemplate the 3000 years of scorching Japanese history.\n\n\"I solemnly declare that with the fall of Singapore the general situation of war has been determined. The ultimate victory will be in our hands.\"\n\nSingapore was given a new name, Shonan, \"Bright South.\"\n\n# 15\n\n# _\"These Stout-Hearted Dutchmen\"_\n\n**1** Yates McDaniel watched the dark shore line slip by rapidly on his left. It was almost dawn of February 19 and this was the southern coast of Sumatra. By a series of chances verging on the miraculous, Doris Lim and McDaniel were aboard a British destroyer speeding east toward Java. They had crossed Sumatra by cart, bus and car to Padang. Following a tip in a local bar, they had ridden a trolley car to the nearby port of Emmahaven and boarded the destroyer the previous afternoon. Now they would hug the coast until late dusk; then turn left, north, and try to break through the dangerously narrow strait between Sumatra and Java under cover of darkness. Their goal was Batavia, the Javan capital which lay on the north side of that slender 800-mile-long island.\n\nMore than half of the other escapees from Singapore were already dead or captured. V. G. Bowen, the elderly Australian Government representative, was captured, made to dig his own grave and shot; Admiral Spooner, formerly commodore of Malaya, and Air Vice-Marshal Pulford were shipwrecked and dying on a malaria-infested little island with sixteen others. Many had reached Padang, like McDaniel, only to be sunk en route to Ceylon or Java.\n\nMcDaniel was cheered by stories of the entirely different attitude in Java. Everyone in Sumatra had talked of the stubborn Dutch defense of their Indies empire. It was refreshing to hear of people who still believed the Japanese could be beaten, but there were also rumors of difficulties between the Dutch and the U.S. Asiatic Fleet. The Dutch admiral, Helfrich, wanted to fight but the American admiral, Hart, thought resistance was suicidal.\n\n\u2014\n\nMcDaniel was coming to another crumbling empire.\n\nGeneral Sir Archibald Wavell was in a gloomy mood at his headquarters in Bandung, 80 air miles from Batavia, high in the mountains of central Java. The ABDA commander had been told by Churchill and Roosevelt to defend the Malay Barrier. This was fast becoming a barrier in name only. The left flank, Singapore, had collapsed. The center, Java, was being deftly set up by the Japanese for a knockout blow.\n\nJava was already isolated. To its west, Sumatra was being swallowed up by paratroopers and a recently landed convoy. A few miles to its east, another advance invasion convoy had just anchored off the exotic island of Bali. And Wavell had no doubt that large invasion convoys were already heading for Java itself. He was still in pain and somewhat crippled from a recent accident. Returning to Java on February 10 from his last Singapore trip he had fallen from a quay in the dark and broken two small bones in his back.\n\nThe steadily worsening relations between the Dutch and the U.S. Navy hadn't improved his mood. He could see both sides. He admired Hart, a thin, wiry, elderly man who was calm and cool, reserved and frugal in words. Wavell also had great admiration for the outspoken and stubborn Helfrich, a short, rotund, balding man of intense energy. But since the Dutch admiral was determined to fight the Japanese before they landed on Java, Wavell supported Helfrich. Then, too, the Briton could never forget the ready and willing help the Dutch had given him in the defense of Singapore. Without hesitation they had weakened their own small forces to send submarines, planes and men to Malay.\n\nFortunately the understandable conflict between Hart and Helfrich was soon over. After much behind-the-scenes maneuvering in Washington President Roosevelt had been apparently convinced that the sixty-four-year-old Hart was too tired and unaggressive to command the ABDA Navy. Five days previously it had been officially announced that Hart had requested relief because of ill health. Helfrich was given the job and Hart was already on his way home.\n\nAt his quarters nearby, Vice-Admiral Conrad E. L. Helfrich still believed the Japanese could be beaten, but not by running away. Like Wavell, he admired Hart. What really had alarmed the Dutchman was Hart's assumption that the defense of the Dutch East Indies was already a lost cause. Helfrich's own little fleet had proved the Japanese could be hurt. His handful of submarines alone had already sunk more tonnage than the combined U.S. air, surface and underwater forces.\n\nHelfrich had been told the unimpressive American record was due in large part to explicit instructions to fight warily. In his opinion U.S. cruisers and destroyers had been sent too far south to fight the enemy properly. It wasn't until January 24, after much needling by Helfrich, that the American surface ships made their first attack. The heartening results of that raid in Makassar Strait\u2014three transports sunk in a brilliant and daring raid by four U.S. destroyers\u2014only proved his point. The Jap could be beaten. And the place to do it was not on the beaches of Java but in the waters to the north.\n\n**2** Japanese naval officers were just as amazed as Helfrich at their almost totally unopposed parade south past the Philippines. Except for stubborn but minor Dutch and Australian resistance they presently held all Borneo, the Celebes, and had strong footholds in Sumatra and New Britain.\n\nThe main goal in the Southwest Pacific was Java. Its fall would mean the end of the Indies campaign. Conquest had to come like lightning before the vast oil, tin and tungsten resources could be destroyed by the Dutch. Without the oil of the Indies the entire war machine could run dry.\n\nTwo great invasion convoys were converging on Java from both sides. The previous day the Western Force had left Indo-China in 56 transports, their destination the west end of Java. The Eastern Force\u201440 transports carrying the outfit taken from General Homma, the 48th Division\u2014was just leaving the southern Philippines for east Java. The two forces, each protected by powerful cruiser and destroyer units, were due to land February 27.\n\n\u2014\n\nHelfrich was willing to do anything to stop, or even delay, the invasion of Java. Half of his four-nation fleet, under his countryman, Rear Admiral Karel W. F. M. Doorman, was now steaming east along the southern coast of Java toward Bali. Helfrich had just learned of the Japanese landing on this famous island and was determined to try and wipe out the beachhead. If the Japanese could hold Bali, it would be a great step toward the conquest of Java. About the same size as Singapore Island, it was separated from the eastern end of Java by the narrow Bali Strait. There was an even more important channel on the other side of Bali. This was Lombok Strait, wide and deep, the main gate through the barrier of islands that lay between the Japanese and Australia.\n\nFor many centuries pirates had plundered these strategic waters. This \"Strait of the Red Peppers\" divided the two continents. Bali on the west was pure Asian. Lombok, only 25 miles to the east, had different trees and plant growth, different birds and animals. The skull-shaped island was a strange, transitional spot where Australian forms of life were already replacing the Asiatic.\n\nBy late afternoon Helfrich knew that if the vital Bali beachhead was to be wiped out, Doorman's force would have to do it that night. Java-based American bombers had made eighteen sorties on Bali that day, but only negligible damage had been done to the 2 Japanese transports and their 4 escorting destroyers.\n\nHelfrich and Doorman drew up a hasty, simple plan. Since every minute was vital, they figured there wasn't time to concentrate forces. The attack would have to be made in three waves. First 2 light cruisers and 3 destroyers would turn north, darting through Lombok Strait in a hit-and-run assault; then would come a light cruiser and 4 destroyers; and finally, to mop up anything still afloat, 5 torpedo boats.\n\n\u2014\n\nIt was deep dusk when the British destroyer carrying Yates McDaniel and other refugees from Singapore rounded the east end of Sumatra and began its dangerous dash north through the narrow channel between the two islands, Sunda Strait.\n\nAs it grew dark, tension rose. Just before 10:00 P.M. McDaniel saw a dramatic cone rising from the water ahead and to the left. This was the island volcano of Krakatao, whose spectacular eruption in 1883 had sent tidal waves all the way to Hawaii. In an hour they would be passing through the narrowest part of the strait, the 14-mile gap between Sumatra and Java. Aeons ago the two islands had been one. Then came a terrific volcanic explosion cutting it in half and giving birth to Sunda Strait.\n\nOn the other end of Java, the first wave of Doorman's Bali attack force was just turning up into the rips lacing the southern entrance of Lombok Strait. The light cruiser _De Ruyter_ , with Doorman aboard, led the pack, its main guns trained to starboard. Next came another Dutch cruiser, the _Java_ , her guns aimed to port. Three miles astern was a Dutch destroyer closely followed by 2 U.S. destroyers.\n\nAt 10:20 P.M. the _Java_ sighted enemy ships on the left. They were difficult to make out due to the background of the 10,000-foot cone of sacred Mount Bali but looked like 3 cruisers. Five minutes later the _Java_ opened fire.\n\nThe 3 \"cruisers\" laying off Bali were a Japanese Army transport, emptied of troops and just getting under way for Makassar, and 2 destroyers. These quickly swept the sea with their searchlights. Finally pinning down the 2 Dutch cruisers, they fired.\n\nThe _De Ruyter_ , its main guns facing the wrong direction, didn't fire a shot in return but the _Java_ scored a hit on the transport. As the 3 Allied destroyers raced north to join battle, the Dutch cruisers completely unharmed continued through Lombok Strait and turned left, heading home for Surabaja.\n\nThe 3 Allied destroyers were not as lucky. In less than an hour the Dutch ship was sunk and the 2 American destroyers turned around returning to the south.\n\nTwo hours later, at 1:15 A.M., February 20, the second Allied wave headed north into the strait. They sped in column\u20144 American destroyers and the Dutch cruiser _Tromp_ \u2014at 25 knots.\n\nThe original Japanese destroyers were now joined by 2 more. They raked the Allied column as it passed through the strait, heavily damaging the _Tromp_ and the U.S. destroyer _Stewart_. One Japanese destroyer was slightly damaged with seven men dead. Another suffered a serious hit in the engine room, killing 60, but steamed off safely under her own power. Based on damage alone it was a lopsided Japanese victory. More important, the Japanese beachhead on Bali was not even molested.\n\nEven so, Helfrich was already receiving optimistic reports. Doorman reported that the 5 torpedo boats of the third wave had seen nothing: Lombok Strait was clear. This news heartened Helfrich, but he was not overly jubilant. The Japanese still controlled Bali and its main airport. Soon it would be a base for air attacks on Java.\n\n\u2014\n\nIt was dark when the British destroyer carrying McDaniel cleared the strait between Java and Sumatra. It rounded the western tip of Java and proceeded up its north coast for 100 miles. It was sunrise by the time the ship docked at Batavia. The sun was already hot as he and Doris Lim walked to Des Indes Hotel, the AP headquarters. Here he learned his wife had gone on to Australia. He was also handed a note from \"Newsreel\" Wong, who had left the day before in a mass exodus of civilians and officials: \"Help Doris. On way to Ceylon, then Calcutta.\"\n\nMcDaniel, exhausted after the strenuous escape from Singapore, found a bed and fell asleep. Almost immediately he was awakened. It was a direct phone call from New York City. AP wanted a story of his adventures.\n\nNinety air miles inland, in mountainous Bandung, General Wavell was reading a radiogram from the Combined Chiefs of Staff in Washington ordering him to defend Java with the utmost resolution:\n\n> EVERY DAY IS OF IMPORTANCE. THERE SHOULD BE NO WITHDRAWAL OF TROOPS OR AIR FORCES OF ANY NATIONALITY, AND NO SURRENDER.\n\n**3** By the morning of February 22 the two Java invasion convoys were within six sailing days of their destination. As the moment for landing drew nearer, other powerful Japanese naval units were converging on the Southwest Pacific area: 4 battleships, 5 carriers, 9 heavy cruisers, 7 light cruisers and 52 destroyers.\n\nWavell didn't know the exact composition of the Japanese Java Invasion Forces, but he was certain his tiny ABDA fleet would be far outnumbered. That day he sent a realistic message to Churchill.\n\n> I AM AFRAID THAT THE DEFENCE OF A.B.D.A. AREA HAS BEEN BROKEN DOWN AND THAT DEFENCE OF JAVA CANNOT NOW LAST LONG. IT ALWAYS HINGED ON THE AIR BATTLE....ANYTHING PUT INTO JAVA NOW CAN DO LITTLE TO PROLONG STRUGGLE: IT IS MORE QUESTION OF WHAT YOU WILL CHOOSE TO SAVE....I SEE LITTLE FURTHER USEFULNESS FOR THIS H.Q....LAST ABOUT MYSELF. I AM, AS EVER, ENTIRELY WILLING TO DO MY BEST WHERE YOU THINK BEST TO SEND ME. I HAVE FAILED YOU AND PRESIDENT HERE, WHERE A BETTER MAN MIGHT PERHAPS HAVE SUCCEEDED....I HATE THE IDEA OF LEAVING THESE STOUT-HEARTED DUTCHMEN, AND WILL REMAIN HERE AND FIGHT IT OUT WITH THEM AS LONG AS POSSIBLE IF YOU CONSIDER THIS WOULD HELP AT ALL.\n> \n> GOOD WISHES. I AM AFRAID YOU ARE HAVING VERY DIFFICULT PERIOD. BUT I KNOW YOUR COURAGE WILL SHINE THROUGH IT.\n\nThe next day the situation in Java looked even worse. The damaged _Stewart_ , improperly braced on keel blocks by Dutch workmen, had rolled over in drydock and was ordered scuttled. The oil situation was also getting desperate, even in the heart of one of the world's great oil-producing centers. Most storage facilities were inland. Those on the coast were inadequate and by now the native crews had fled to the hills because of incessant air raids.\n\nThe Allied air defense, never strong, was now so depleted it could offer little resistance. The Dutch Air Force, originally composed of 200 outmoded or obsolete bombers and fighters, was down to a few battered planes; there were a few British planes, leftovers from the disaster in Malaya. Of the 111 planes which America had rushed to Java, only 23 heavy bombers and a few fighters remained. First 11 Flying Fortresses, survivors of the Philippines campaign, were thrown into the defense of the Netherlands East Indies. For weeks they were the only heavy bombers operating against the invaders. Then 38 B-17E's and 12 two-engine bombers began arriving in driblets after being ferried halfway round the world from MacDill Field. Though these planes went on missions almost every day and did everything from bombing at 30,000 feet to strafing from 1500, to date they had sunk at most 5 transports and 2 tankers. This poor record was understandable: many of the crews and some of the pilots had had almost no previous experience with the Flying Fortresses or LB-30's; 26 bombers were lost on the ground because of poor anti-aircraft protection, air warning and fighter cover; because of the violent tropical storms and lack of intelligence only about half the bombers ever reached their assigned targets.\n\nThe story was almost the same for the American pursuit fliers. Of the 83 P-40's which flew up from Australia only 39 had arrived. (Several pilots had never before stepped into a P-40.) Though the survivors knocked down their share of Zeroes, they were too few to do anything but slow down the hordes of enemy planes.\n\n\u2014\n\nOn the evening of February 23, Yates McDaniel took Doris Lim to the docks at Batavia and put her aboard a ship bound for Ceylon. He had persuaded the British consul general that she was a British citizen. It was a fast modern passenger ship and Doris was delighted, even though it would mean running the Japanese gantlet without a convoy.\n\nNow McDaniel had only himself to worry about, but before leaving for Australia he had to make sure he wasn't missing a big story in Java.\n\n\u2014\n\nIn Bandung, Wavell was just getting word of another setback. Strategic Timor, at the eastern end of the long barrier of islands between the Japanese and Australia, had been captured. Both ends of the barrier were now in Japanese hands. Java, in the middle, was isolated. One of the richest prizes in the world, she lay completely exposed to conquest\u2014except for Helfrich's four-nation fleet.\n\nIn spite of the sounds of disaster on all sides of him, the admiral still refused to give up hope. He wired London that the British and Americans were much too pessimistic:\n\n> SUNDA AND BALI STRAITS ARE STILL OPEN. IT IS STILL POSSIBLE TO SUPPLY US BY SHIP. IT IS NOT TOO LATE. BUT WE MUST ACT WITH EXTREME SPEED, GRIM RESOLUTION AND TAKE ALL RISKS.\n\nGloom at Bandung deepened in the next few days, culminating on February 25 when Wavell dissolved ABDA and turned over the final defense of the Netherlands East Indies to the Dutch governor general. After bidding a solemn good-bye to his Dutch friends, Wavell went to his quarters to pack. His next stop was India, where he was to be commander-in-chief. Was he going off to command another sinking ship?\n\nHelfrich now was solely responsible for stopping the sea invasion. The decisive moment was approaching. At mid-morning word came that a great Japanese convoy with strong escort was moving south down Makassar Strait and would apparently reach eastern Java in two or three days. Doorman, defending this end of the island, would need reinforcements. At 11:25 A.M. Helfrich ordered all available fleet cruisers and destroyers in Batavia to steam east to Surabaja to bolster Doorman's heavy cruiser, 2 light cruisers and 7 destroyers.\n\nHelfrich knew he could not stop the Japanese, but the Combined Chiefs in Washington had ordered him to defend Java until the end and if he couldn't stop the Japanese, at least he could hurt them. It was his duty to kill as many Japanese soldiers as possible before they got ashore. He would fight on the Java Sea regardless of consequences.\n\nThat afternoon Yates McDaniel watched the 5 ships of war file out of Batavia's naval base, Tanjok Priok, to reinforce Doorman in what would undoubtedly be the last gallant defense of Java. They were a brave sight, their ensigns flapping: the big British cruiser _Exeter_ , famous for her battle with the _Graf Spee_ , the Australian light cruiser _Perth_ , and three trim British destroyers. They looked potent and confident as they steamed east, but McDaniel recalled how powerful the _Prince of Wales_ , the _Repulse_ and their destroyer escorts had looked sailing out of Singapore for their rendezvous with disaster.\n\nNow McDaniel headed for the commercial docks. The Java story was about over and he was leaving at sunset on a Dutch cattle boat. The chances of getting through safely were ridiculously small but McDaniel knew this was his last chance. He got only blank looks when he asked the ship's destination; he hoped it was going to Australia. If it did, he wondered if this would be an evil omen for the Allies. First he had seen Singapore die; now Java was falling. Would Australia be next?\n\n**4** By late morning of February 26 the two Japanese convoys were within striking distance of their destinations. The Western Assault Convoy of 56 transports, escorted by 2 light cruisers, 2 flotillas of destroyers, and covered by 4 heavy cruisers, was about 250 miles from the west end of Java.\n\nThe Eastern Assault Convoy of 40 transports, escorted by a light cruiser and 6 destroyers, was less than 200 miles from its goal, east Java. Additional protection was also nearby: two heavy cruisers, a light cruiser and 7 destroyers. Over-all commander was the able but cautious Rear Admiral Sokichi Takagi.\n\nHelfrich knew the Western Assault Convoy was also approaching but he had decided to concentrate almost his entire force at Surabaja under Doorman to try and stop the greater danger from the east. Just before noon these eastern invaders were sighted by two Allied planes about 175 miles northeast of Surabaja. When Helfrich got this report he radioed Doorman to sail out of port after dark with his entire force and attack. A decisive duel between Doorman and Takagi was inevitable and imminent.\n\nA few hours later Helfrich received a sharp jolt. An R.A.F. plane reported another large invasion convoy off the west end of Java. Though the convoy was apparently heading north, away from Java, Helfrich knew this was the Western Assault Convoy, ahead of schedule and marking time.\n\nHelfrich had almost nothing to throw against this second threat\u2014only the light cruiser _Hobart_ , 2 antique cruisers and 2 equally old destroyers\u2014but he ordered them to set sail from Batavia at dark and engage the enemy.\n\nA little later Helfrich was told by Lieutenant General L. J. van Oyen, the Dutch air chief and now head of the combined ABDA air fleet, that the bulk of his force, the Americans, were leaving in spite of his definite orders to fight to the very end. Already U.S. ground crews were on trains heading for Tjilatjap, a town on the southern coast of Java, where a freighter would take them to Australia. It almost amounted, said van Oyen bitterly, to desertion. Colonel Eugene Eubank, who had ordered this evacuation, was only following specific instructions from his chief, General Brereton, who had flown on to India.\n\nSoldiers of all colors and many nationalities\u2014English, Australian, Chinese, Dutch, Indian, as well as Americans\u2014were swarming into the escape port of Tjilatjap. It was obvious to almost everyone that Java was fast falling.\n\nMany miles from the frantic scene, Churchill was writing a last message to the commander of those British and Australian troops chosen to stay behind and fight the land battle of Java.\n\n> I SEND YOU AND ALL RANKS OF THE BRITISH FORCES WHO HAVE STAYED IN JAVA MY BEST WISHES FOR SUCCESS AND HONOUR IN THE GREAT FIGHT THAT CONFRONTS YOU. EVERY DAY GAINED IS PRECIOUS, AND I KNOW THAT YOU WILL DO EVERYTHING HUMANLY POSSIBLE TO PROLONG THE BATTLE.\n\nAt 6:30 that evening Doorman sailed out of Surabaja with his 14 ships. He was a tall, dark-haired man with slightly drooping shoulders. Helfrich esteemed him highly as a tactician with great presence of mind in a crisis. Like Helfrich he was ready to die for his queen. Like all the other Dutch in Java he was also fighting for home, family and future.\n\nThe shadowy column nosed north into the Java Sea through the violet light of early dusk. Then it turned due east, beginning its night sweep in search of Takagi's transports and warships.\n\nThe column was an inspiring sight, and most of those aboard the assorted ships felt reassured to be in such proud company, but it was a patchwork fleet. Each national group was actually a distinct and separate task force, with no common doctrine, philosophy or technique. And Doorman's last hurried instructions before sailing had been, necessarily, vague and incomplete. It was, thought Lieutenant (junior grade) H. S. Hamlin, Jr., of the _Houston_ , like eleven all-stars playing the Notre Dame football team without a single practice session together.\n\nAt 8:55 P.M. Helfrich cabled: YOU MUST CONTINUE YOUR ATTACKS UNTIL ENEMY IS DESTROYED. He wanted Doorman to know that he must attack at any price\u2014at once. Delay might be disastrous. Only in the early moments would the Allied fleet enjoy numerical advantage.\n\nDoorman saw nothing in his sweep along the coast. At dawn he radioed Surabaja for air protection. None was available. A few hours later enemy bombers harassed but did not damage the fleet. Exhausted and discouraged by the fruitless night search, Doorman headed back for home.\n\nThat morning the seas off the southern coast of Java were dotted with ships fleeing toward Australia. Only two were heading north and these, bound for Tjilatjap, were loaded with American fighter planes. The freighter _Seawitch_ carried 27 P-40's in her hold. The U.S. aircraft tender _Langley_ was bringing 32 ready-to-fly P-40's and 33 Army Air Force pilots. The two ships were traveling without a convoy in a desperate race to reach Java before it was too late.\n\nThe _Langley_ , in the lead, was due to dock that afternoon. Its crew knew that approaching Java in daylight was dangerous, but Helfrich, with Admiral Glassford's approval, had insisted the risk be taken. The hours saved might be critical. The ABDA fleet still had no air cover for the coming battle in the Java Sea.\n\nAt 8:00 A.M. a lone Japanese reconnaissance plane spotted the _Langley_ and alerted the newly captured airfield on Bali. At 9:55 A.M. 16 Betty bombers and 16 Zero fighters took off.\n\nSeven of the bombers attacked the _Langley_ at 12:10 P.M. but Commander R. P. McConnell, the aircraft tender's skipper, ordered hard right rudder and the bombs fell to port. When the remaining Betties struck, McConnell again avoided the bombs. At 12:15 the group of nine bombers sharply wheeled and once more attacked. This time the _Langley_ , built in 1922 as America's first aircraft carrier and later the mooring ship for the famed dirigible _Shenandoah_ , was hit five times. Planes on deck burst into flame.\n\nHalf an hour later the Zeros strafed the deck planes briefly, then left when they saw a great explosion. At 1:32 P.M. McConnell ordered the ship abandoned.\n\n\u2014\n\nOn the north side of Java, Admiral Doorman's 14-ship striking force was still en route back to its base. But at 2:27 P.M. just as his flagship, the light cruiser _De Ruyter_ , entered Surabaja Harbor, he got a message from Helfrich to attack an enemy force east of Bawean Island. This was a point some 90 miles north of Surabaja.\n\nThere was no time to make an operation plan. Since the fleet didn't have a common code of tactical signals, Doorman handed an order to an American liaison officer on the _De Ruyter_. The order was translated and transmitted to the _Houston_ , which relayed it over voice radio to the American destroyers. The British and Australians got the news by flashing light in plain English and signal flags. It was:\n\n> FOLLOW ME, THE ENEMY IS 90 MILES AWAY.\n\n# 16\n\n# _The Battle of Java Sea_\n\n**1** Doorman's fleet quickly turned around and headed out to sea again. There was rising excitement in every ship. The 3 British destroyers, screening abreast, led the way to the northwest. Then came Doorman in his flagship, _De Ruyter_. Behind in column were the famous British heavy cruiser _Exeter;_ the American heavy cruiser _Houston_ , the host of Roosevelt on four cruises; the Australian light cruiser _Perth;_ and finally the Dutch light cruiser _Java_. To the left of the cruisers was a second column: 2 Dutch destroyers followed by a quartet of ancient American four-stack destroyers.\n\nSeamen and officers of all four nations were thankful at least to have something on the water to fire at. The past month they had all been pounded by bombs. Soon would come the chance to fight back. Men rushed to their battle stations. There was no time to eat. The enemy might be just over the horizon.\n\nDoorman had no planes to catapult from his cruisers to search out the enemy. They had been left ashore the day before since he'd expected only a night action. He would have to grope his way forward in a deadly game of blindman's buff. Once more Doorman radioed to Surabaja for air cover. Helfrich, who had previously asked in vain for command of all air forces, added his weight to Doorman's request, but it was denied. There were only eight Dutch fighters available, asserted the air commander, and these were already assigned to accompany bombers on a late-afternoon attack on Takagi's approaching transports. Doorman could either turn back or steam ahead into battle without air protection. The 14 ships continued northwest.\n\n\u2014\n\nAdmiral Takagi had two advantages. He wasn't blindfolded. Three float planes had already reported that Doorman was heading for him. He also had a slight numerical edge\u201417 ships to 14. Both admirals had 2 heavy cruisers but Takagi's 13 destroyers gave him a decided edge against Doorman's. The Dutchman's only advantage was in light cruisers, 3 to 2.\n\nAs Doorman headed toward him, Takagi guessed correctly that the Dutchman's primary purpose was to sink the transports. He ordered the 40-ship convoy, which was heading south for the Java coast, to turn away. He placed Rear Admiral Shoji Nishimura's Destroyer Squadron 4\u2014the light cruiser _Naka_ and 6 destroyers\u2014south of the convoy as a screen. Just east of Nishimura were Takagi's own 2 big cruisers, the _Nachi_ and _Haguro_ , his knockout punch. And farther east was Rear Admiral Paizo Tanaka's Destroyer Squadron 2, the light cruiser _Jintsu_ and 7 destroyers. By 3:15 P.M. Takagi was ready, waiting for the groping Doorman.\n\nEven so, the report that a strong enemy fleet was racing north to the attack had taken Takagi aback. According to Air Intelligence, the ABDA fleet had been battered to bits by incessant air raids and was too badly damaged to fight.\n\nIt was a clear, bright day. On Takagi's big cruiser, _Haguro_ , some of the men thought they could smell the fragrance of nearby Java. They had read much of this exotic land and were eager to see it firsthand. Midshipman Hachiro Kimura, a graduate of the Naval Academy only twenty-three days before Pearl Harbor, stood on the bridge. Like everyone else on the _Haguro_ he was excited and tense. The men, dressed in white fatigues and steel helmets, visited the ship's shrine and tied _hachimaki_ tightly around their foreheads. The officers, wearing white dress uniforms and baseball caps, were straining to be the first to see the enemy.\n\nBut Tanaka's force, a few miles to the east, was nearer. At 4:00 P.M. a lookout on his flagship, _Jintsu_ , sighted mastheads about 17 miles to the southeast. Word spread through the ship. Everyone was in high spirits. Victory again would be on the side of Japan and the Emperor.\n\nA minute later those on _Haguro_ also got their first glimpse of the enemy. Midshipman Kimura could see only the lofty masts of the _De Ruyter_. It looked terrifying with its strange, odd-shaped superstructure, like something from Mars. Kimura had never before experienced such a weird feeling of fright. He knew the _Haguro_ was faster and had 10 great 8-inch guns, but the _De Ruyter_ began taking the shape of some prehistoric monster as it grew bigger.\n\n\u2014\n\nThe _Electra_ , the middle of the 3 British destroyers bird-dogging in front of the Allied cruisers, sighted Tanaka's destroyer flotilla at 4:12 P.M. She reported to Doorman:\n\n> ONE CRUISER, UNKNOWN NUMBER LARGE DESTROYERS BEARING 330\u00b0 SPEED 18, COURSE 220\u00b0.\n\nThose on _Electra_ couldn't see that a few miles behind and a little to the west of Tanaka were Takagi's 2 big cruisers. And several miles farther west was the destroyer flotilla of Admiral Nishimura. All these Japanese forces now turned and began speeding almost at right angles toward Doorman's two parallel columns like three great arrows.\n\nIn a few minutes Doorman guessed that the oncoming enemy was trying to cross the bows of his formation and ordered the cruisers to flank speed of 26 knots. The 2 Dutch destroyers, unable to make this speed, fell back. Behind, the 4 American destroyers also slowed when Doorman ordered them to stay in line.\n\nTakagi's 2 big cruisers, _Nachi and Haguro_ , closed to within 28,000 yards of the Allied column. At 4:15 P.M. their 8-inch guns blasted. Geysers exploded harmlessly in front of their targets, _Exeter_ and _Houston_.\n\nAt 4:16 P.M. the 2 Allied heavy cruisers opened fire with almost a single crash. It was an unequal duel. The 2 Japanese ships each had 10 8-inch guns. _Houston_ 's No. 3 Turret had been wiped out and 46 men killed in the bombing of February 4, reducing her big guns to 6. Thus with _Exeter_ 's 6, Doorman had only 12 big guns against 20.\n\nAs the big guns of the _Houston_ roared, shaking the ship violently, Lieutenant Hamlin, turret officer of No. 1, looked through his periscope. He saw splashes falling around their target, a heavy cruiser. A second salvo came down on the enemy. Hamlin saw the dull red glow of bursting shells, so different from the orange flash of gunfire. From the waist of the Japanese ship a cloud of smoke burst up, and then, back aft, a bright orange flash that looked like a burning turret.\n\nHe turned to the voice tube leading to the gun chamber. Inside they were working like a team of perfectly drilled acrobats and had no time to watch the show. \"We've just kicked the hell out of a ten-gun Jap cruiser!\" he shouted.\n\nThe men in the gun chamber cheered.\n\n\u2014\n\nNow Tanaka in the light cruiser _Jintsu_ led his 7 destroyers in an attack at the head of the Allied line. _Electra_ was straddled, geysers rising on both sides. Commander C. W. May coolly twisted the ship, avoiding another near miss. It was almost as if he'd been a mind reader. Again he turned and shells burst in fantastic greens, yellows and reds where the destroyer had been a moment before.\n\nGunner T. J. Cain felt exhilarated and didn't know why. Others on _Electra_ shared his feeling. In a bombing they sometimes got the jitters, But this shelling had put everyone in a kind of holiday mood. It was quite extraordinary.\n\nBehind the British destroyers, Doorman's 5 cruisers kept on course with the _Houston_ and _Exeter_ , the only ones with guns big enough to reach the Japanese, providing all the fire power. Next to last in this line, on the _Perth_ , Paymaster Lieutenant-Commander P. O. L. Owen was watching the battle from the after end of the quarter-deck. He was in the battle by accident. Assigned to the _Hobart_ , he had missed boarding that ship in Batavia because of an air raid. Captain Waller of _Perth_ , an old friend and former commander, had invited him aboard as a temporary guest.\n\nJapanese salvos began to straddle _Perth_. Though he had undergone much bombing in the Mediterranean, this was Owen's first taste of shell fire and his reaction was quite different from that of Gunner Cain. He remembered Waller, a veteran of two fierce surface engagements, once telling him, \"I'd rather 1000 bombing attacks than to stand one enemy shelling.\" Now Owen knew what he meant. It was awesome. He and his three-man machine-gun crew stood watching in tense fascination as tight salvos lobbed around their ship, first 25 yards short, then 25 yards over. He could see the black streaking track of each shell as it winged toward them: for a split second the shell would become a black image just before it struck the water. It was a phenomenon not seen, he guessed, by many men.\n\nIt was extremely taxing to be a mere spectator. The guns of _Perth_ were silent, out of range. Owen became incensed that Doorman didn't order the light cruisers to turn into the enemy and get in close enough to fight.\n\nAhead, on the _De Ruyter_ , Doorman wanted to do just that since his only superiority\u20143 to 2\u2014was in light cruisers; but he had a more immediate problem. Takagi's 3 forces were closing in. The 2 heavy cruisers, particularly, were gaining fast. Doorman could see that if he continued his course, Takagi would soon pass across the head of his column, \"crossing his T.\" By this classic maneuver the Japanese would bring their broadsides to bear on the column of Allied cruisers who could retaliate only with their forward guns. Consequently, at 4:21 P.M. Doorman swung his cruiser column 20 degrees away from the Japanese to the left. His destroyers followed suit.\n\nThe Japanese also turned. Soon the two great enemy fleets were traveling almost parallel, heading west with Doorman in between Takagi and the Java coast line. Observing this spectacular scene from grandstand seats were Americans in 3 A-24 dive bombers and 10 escorting P-40's. They were on their way to bomb the 40 transports Takagi was presently trying to shield from Doorman. Their leader, Captain Harry Galusha, instead of skirting the action, became so fascinated by what he saw below that he led the formation right across the battlefield. Several of his men began to complain vocally that he had more nerve than brains, but Galusha, though he was supposed to be after transports, was looking for a carrier. If he found one, he was going after it. He counted 12 Japanese warships. Then to the north 6 cruisers. And beyond them, farther north, was clustered a great flotilla of transports, the Eastern Invasion convoy.\n\nFinding no carriers, Galusha turned north and attacked the transports. Though his bombers claimed 3 victims, not a single Japanese ship was sunk or even badly damaged. It was unfortunate for the Allies that the P-40's accompanying Galusha had not been assigned to Doorman, as he and Helfrich had requested. They could well have won the battle of Java Sea for the Dutchman, if only by knocking down the 3 persistent Japanese float planes that were giving Takagi's gunners such tremendous advantage by correcting fire in the increasingly smoky battle area.\n\nA few minutes later Takagi ordered Admiral Nishimura's destroyer flotilla to launch a torpedo attack. Nishimura, in the light cruiser _Naka_ , dashed toward Doorman's column followed by 6 destroyers. At 4:33 P.M. _Naka_ loosed torpedoes from a distance of 16,000 yards, then swung sharply away from the enemy. The 6 Japanese destroyers behind did the same and a swarm of long-range torpedoes headed toward the Allies. These, newly designed, had the astounding range of 30,000 yards and since they were propelled by oxygen, left no tell-tale trail of bubbles.\n\nTwo minutes after this school of torpedoes was launched, Doorman abruptly turned toward the enemy, trying to reduce distance for the benefit of his light cruisers. But by the time the move was executed all Doorman's attention was diverted to self-preservation. The torpedoes recently launched by Nishimura began appearing on all sides. The Allies, who had no torpedoes of such range, were sure a wolf pack of submarines had suddenly attacked. Many of these torpedoes exploded prematurely, giving rise to excited reports of sinkings by both Japanese and Allied observers, but none actually found a target.\n\nAt 5:00 P.M. Admiral Takagi sighted smoke from his transports. Realizing the battle had drifted dangerously near the invasion convoy, he ordered everyone to attack simultaneously, at once, even though his big cruisers were more than 20,000 yards from the enemy. In the next few minutes almost 50 torpedoes were launched by _Haguro_ and the two destroyer divisions. The distance was too great. All missed.\n\nThe area was now black with smoke from Japanese destroyers laying a screen between the two opposing cruiser columns. Doorman was blinded; only _Exeter_ had radar. But Takagi could still look over the smoke screen with his float planes.\n\nTanaka ordered his flag torpedo officer, Commander Yasumi Toyama, to launch another torpedo attack. Toyama signaled the order. One destroyer answered cockily by flag, \"It's simple, exactly like a maneuver.\" They all felt the same. There was nothing to it. At 5:07 P.M. Tanaka's squadron launched their torpedoes. At the same moment the _Haguro_ began shooting with its 8-inch guns at the biggest target in the line, the _Exeter_.\n\nEvery sea battle has its moment of decision. In the Battle of Java Sea it came at exactly 5:08 P.M., a sudden stroke of bad luck for the Allies, good luck for the Japanese. One of _Haguro_ 's shells crashed through an anti-aircraft mount on the _Exeter_ , pierced the ship's decks without detonating and landed in the boiler room. There was a tremendous explosion and in one stroke 6 of _Exeter_ 's 8 boilers were cut out. The big cruiser, her speed halved, lurched and turned hard left so the _Houston_ just behind wouldn't pile into its stern.\n\nCaptain Rooks of _Houston_ , assuming Doorman had ordered a simultaneous turn of all cruisers to avoid torpedoes, also quickly swung to port. The lurch almost sent Lieutenant Hamlin to the deck. His No. 1 Turret rumbled as it trained around trying to keep on target. Six hundred yards ahead he saw _Exeter_ moving at only five knots. Something drastic must have happened. It gave him a sick feeling.\n\n_Houston_ strained as she heeled sharp over to avoid ramming the British ship, the screws shaking her as she went astern. A few hundred yards behind the American ship, Captain Waller of _Perth_ also assumed Doorman wanted a left turn to let torpedoes comb the line and was sharply moving to port. So was the _Java_.\n\nHamlin grabbed the phone, heard the gunnery officer, Commander A. L. Maher, say calmly, \"Check fire. _Exeter_ has been hit.\" Just then _Perth_ swept by again, this time to starboard, a billowing cloud of dense white smoke steaming from her funnel.\n\nShe was laying a protective screen around _Exeter_. As she charged past Hamlin, he saw a beautiful snow-white \"bone in her teeth\"\u2014the foam at her bow\u2014and from the yardarms and the gaff, three huge battle flags streamed straight behind. She was firing rapidly. It was one of the finest sights he had ever seen.\n\nAhead, Doorman in _De Ruyter_ was steaming alone and oblivious. When he finally saw the melee behind, he too turned left. By now the entire Allied fleet\u2014except the damaged _Exeter_ \u2014was steaming south in disorder to get clear of the smoke-engulfed area.\n\nSuddenly the water was crisscrossed by the wakes of many torpedoes. The cruisers began zigzagging and dodging as destroyers rushed about like bird dogs. Many believed they'd run into another nest of submarines. These were the long-range torpedoes launched by Tanaka's destroyer squadron eight minutes previously.\n\nAt 5:15 P.M. there was a great explosion in the middle of the milling Allied ships. The Dutch destroyer _Kortenaer_ had caught one of the torpedoes in the waist. Hamlin saw her roll over on the port beam, break in two like a jackknife. The stern section stuck up, the propellers slowly turning over. Men were clinging to the rudder.\n\nCommander T. H. Binford, in charge of all four American destroyers, saw the disaster from his flagship, _Edwards. Kortenaer_ 's skipper, Lieutenant Commander A. Kroese, was a friend. He exclaimed, \"Oh, my God, there's my friend 'Cruiser'!\" In less than a minute there was nothing except debris and an oil slick. As the _Houston_ was passing the wreckage, several saw a Dutch sailor, clinging to a stanchion, wave his arm in cheerful salute.\n\nDoorman knew he had to re-form his cruiser line before other disasters struck. \"ALL SHIPS FOLLOW ME,\" he signaled at 5:20 P.M. and headed southeast. Five minutes later he altered course to northeast so the line could pass between the smoke-encircled _Exeter_ and the Japanese and gave the crippled ship a chance to escape.\n\nThough _Exeter_ was momentarily protected from the Japanese heavy cruisers by smoke, Doorman realized the enemy destroyers were maneuvering for another torpedo attack on the wounded ship. He ordered the three British destroyers to dash forward through the smoke and try to stop the enemy destroyers before they got started.\n\nAt 5:25 P.M. the men of _Electra_ heard the calm voice of their skipper, Commander May, come over the phone. \"The Japanese are mounting a strong torpedo attack against the _Exeter_. So we are going through the smoke to counterattack.\"\n\nThe other two British destroyers were far behind and May felt he could not wait. _Electra_ instantly darted northwest through the smoke. Soon she was in the open, alone and naked to the approaching _Jintsu_ and her seven destroyers.\n\n_Electra_ fired rapidly at _Jintsu_. Her aim was good and _Jintsu_ was hit, with one dead and four wounded, but the odds were too great. In seconds the water around _Electra_ was spouting with near misses and straddles. The little destroyer rocked. The stink of Japanese explosives fell on the crew with the spray. Still she fired at _Jintsu_. The crew saw two hits spark on the Japanese flagship's hull. They yelled but their triumph was cut off by a deafening explosion. Their own ship staggered from a hit below the bridge. All communications from bridge to ship were cut. Then a second and third shell hammered home. Escaped steam screeched from the after-boiler room. _Electra_ shivered, stopped at 5:30 P.M. She was a floating target for an entire destroyer division.\n\n_Electra_ began to sink but her gun crews stayed at their stations and continued to fire. Finally a message was passed from the bridge: \"Prepare to abandon ship.\"\n\nWith _Electra_ knocked out of the way, _Jintsu_ and two destroyers dashed through the smoke screen to attack the _Exeter_. In a wild melee the Dutch destroyer _Witte De With_ and the remaining two British destroyers rushed forward to protect the damaged British cruiser.\n\nIt was 5:40 P.M. The late afternoon was dark from smoke of battle in spite of the brilliantly forming sunset. Doorman ordered _Exeter_ to retire to Surabaja with _Witte De With_ as escort. In the confusion the two ships turned south and escaped. Now Doorman had only 10 ships to match Takagi's 17. Worse, he had only _Houston_ 's 6 big guns to duel with Takagi's 20 8-inchers.\n\nUnder cover of the smoky battlefield, Doorman formed a new line with his remnants. Behind _De Ruyter_ were _Perth, Houston, Java_ and the 6 destroyers. A few minutes later the line broke out of the smoke into the clear. He could see Takagi's big cruisers, _Nachi_ and _Haguro_ , steaming parallel but in the opposite direction about 19,500 yards away.\n\nFortunately for the Allies, Takagi had no idea the _Exeter_ was out of action and that he enjoyed overwhelming superiority. Even so he was pleased with the action. Not one Japanese ship had yet been sunk or even seriously damaged. He radioed the invasion convoy, standing off not far to the north, that it was now safe to head for Java.\n\nMany of his junior officers, like Midshipman Kimura, were not at all happy. They felt Takagi should have moved in much closer and wiped out the Allies. They also felt much of the fault should fall on Takagi's senior staff officer, Captain Kou Nagasawa, the one in charge of the whole operation. Stories were already circulating about Nagasawa's behavior under fire: he was so terrified he hung onto the compass every time a near miss from one of _Houston_ 's crimson-dyed shells shot up a terrifying blood-red geyser.\n\nEven so Nagasawa was now advising Takagi to move in closer. The 2 Japanese cruisers closed in slightly on Doorman and fired. Two shells hit the _Houston_. Both were duds. The American cruiser was forced to answer with slow fire. All ammunition in Turrets 1 and 2 was gone. Men were now having to transfer heavy 8-inch shells from disabled Turret 3 as the ship zigzagged and careened.\n\nAt about 5:50 P.M. Takagi saw Doorman's line turning in a counterclockwise circle and ordered his 2 big cruisers to launch a gun and torpedo attack. Five minutes later Tanaka's destroyer flotilla emerged from the smoke and also sighted Doorman turning. Within three minutes torpedoes were launched.\n\nThe Dutch admiral saw these torpedoes coming and at 6:00 P.M. ordered everyone to swing south. The torpedoes churned past harmlessly. By now the primitive Allied radio communication system had completely broken down and Doorman was reduced to visual signals which were almost useless in the smoke and haze. To make matters worse, the direct voice contact of both _De Ruyter_ and _Houston_ with the American destroyers was gone.\n\nAt 6:06 P.M. Commander Binford finally saw a flashing signal from Doorman: COUNTERATTACK. While Binford on _Edwards_ was leading his 4 American destroyers to the north, he got another signal from Doorman: CANCEL COUNTERATTACK. Then a third: MAKE SMOKE.\n\nIt was just 6:09 P.M. and the sun was setting. In the churned-up battlefield behind Binford, _Electra_ was beginning to sink. Gunner Cain, wounded in the leg, calmly limped to his cabin. His wife had recently sent him photos and he'd need them for company if he were captured. Once he had the pictures, he noticed the tilting walls of his cabin and became seized with sudden panic. Terrified of being trapped, he dashed up to the deck. There he met Steward Gretton, who held out a half-bottle of whisky. \"Sorry about the shortage of glasses, sir, but I thought you'd care to join me in a drink.\"\n\nCain refused. They'd soon be in the sea and Scotch would only add to their thirst. The steward looked at Cain's wounded leg. \"Doesn't seem much else left to do, sir,\" he said. \"Don't you think it's time we left?\"\n\nThe water around the foundering ship was soon dotted with swimmers and men on floats. A shout suddenly went up near Cain. It turned into a cheer. The skipper, Commander May, was leaning over the bridge waving.\n\nThe men in the water shouted. \"Come on, sir, jump!\" _Electra_ shuddered.\n\n\"She's going,\" shouted several. \"For God's sake jump!\"\n\nMay waved again then calmly turned away.\n\n\"He's going to leave from the port side,\" said Cain.\n\n\"But he's leaving it too bloody late,\" cried Gretton. The ship plunged, her stern momentarily rising steeply. She disappeared, her white ensign flying from the gaff. Cain wept.\n\nThe battle was still on not far away. At 6:15 P.M. Doorman signaled Binford: COVER MY RETIREMENT. The American destroyer commander wasn't sure what Doorman wanted but he decided the best thing was to attack. Leading with _Edwards_ , Binford put his 4 old destroyers in column, and steamed between the Allied battle line and the _Nachi_ and _Haguro_. By flag he ordered those following to prepare for torpedo attack. Two almost collided in the smoke, but the 4 ships finally got into the clear as Binford led them head-on toward the big Japanese cruisers. Then he swung the column parallel to the cruisers for a broadside attack.\n\nAt 6:22 P.M. Binford ordered all starboard torpedoes fired even though it was 10,000 yards to the Japanese cruisers. The column reversed and a few minutes later the rest of Binford's torpedoes, those on the portside, sped to the north.\n\nTakagi saw the wakes of the approaching torpedoes and dodged. Now in sight of the Surabaja lighthouse, he began worrying about mines and submarines. To be on the safe side he again radioed the convoy, 30 miles northwest of him and presently heading for Java, to reverse course. Half an hour earlier he had thought Doorman was shattered, yet the Dutchman was still leading his ships in good order. What were the Allies going to do next? Return to Surabaja for fuel? Or swing around and try to hit the convoy? Takagi broke the engagement and began retiring to the north. He would wait until darkness\u2014the time preferred by Japanese naval commanders to attack.\n\nDoorman saw this retirement dimly through the smoke and at 6:30 P.M. radioed Helfrich in Bandung, ENEMY RETREATING WEST. WHERE IS CONVOY? Helfrich radioed back that he had no current information; Doorman would have to make a guess.\n\nAt 6:31 P.M. Doorman signaled, FOLLOW ME, and headed to the northeast, but nine minutes later changed his mind and turned to the northwest. Hours earlier American fliers had sighted the Japanese convoy and radioed back its position. The information was never passed on to Helfrich or Doorman. Helfrich later was to term this state of affairs a \"scandalous lack of coordination.\"\n\n**2** The present lull was a welcome relief to those on _Houston_. The crew was served tomato juice and cold Vienna sausages. The men were deafened; their faces pocked with powder burns. Excitement of battle was passing and many realized they were on the point of exhaustion, but there was also a sense of exhilaration. _Houston_ had done well, scoring several hits. She herself had only been hit by two duds. The captain had pulled them through again, handling the cruiser like a destroyer in the heat of battle. The men below were near collapse. More than 70 had passed out from heat exhaustion in the torrid fire rooms during the fight.\n\nTwo destroyers, _Electra_ and _Kortenaer_ , had been sunk and _Exeter_ and her escort, _Witte De With_ , were out of action but the men felt the enemy had been hurt worse. They agreed, generally, that Doorman was right in tenaciously hunting the convoy.\n\nOn the _Jintsu_ a civilian servant was serving tea to Admiral Tanaka. The destroyer commander was pleased with the battle. At least 4 or 5 Allied ships must have been sunk. The Japanese losses were remarkably low. Not a ship had been sunk, and only the destroyer, _Asagumo_ , badly damaged.\n\nThe men were eating canned bamboo sprouts; fresh meat and vegetables had long disappeared. They were anxious to get the battle over so they could pick up fresh fruit and vegetables in Java.\n\nOn the heavy cruisers some junior officers were still criticizing Tagaki for being influenced by Nagasawa's extreme caution. Takagi, they argued, should have remembered the old saying of the Japanese Navy, \"Let the enemy cut the skin, if you can cut the meat.\"\n\n\u2014\n\nAt 7:26 P.M. Doorman was still blindly groping for the convoys to the west-northwest. His aim was almost perfect. The 40 Japanese transports were only 20 miles away but a minute later he sighted _Jintsu_ and 3 destroyers on the port beam. At the same time, a plane from _Jintsu_ dropped a flare behind the Allied column. The Dutchman, guessing these ships were a protective screen between him and the convoy, decided to make a wide circle around the _Jintsu_ group. He turned south toward Java.\n\nTempers began to fray as the tension built again. At 9:00 P.M. _De Ruyter_ reached shoal water and swung right to parallel the coast of Java. The other cruisers followed. So did the 2 British destroyers, _Encounter_ and _Jupiter_.\n\nThe 4 American destroyers hesitated. They were low on fuel and out of torpedoes and Binford had to make a sudden and hard decision. \"I'm not going in there after Doorman,\" he told _Edwards'_ conning officer, Lieutenant William Giles. \"That Dutchman has more guts than brains.\" He told Giles to turn left and lead the four destroyers to Surabaja, only 50 miles to the west, where they could refuel. Since he couldn't contact Doorman directly, Binford radioed shore and asked them to relay the message.\n\nDoorman didn't know he was now leading his line dangerously close to a mine field laid only that afternoon by the Dutch themselves. On every ship men were straining their eyes looking through periscopes, range finders and binoculars for the forest of masts that would mean a convoy. Nothing could be seen. Then suddenly at 9:25 P.M. there was a great explosion at the end of the line. The British destroyer, _Jupiter_ , was enveloped in flames. Some guessed she had been torpedoed by a submarine. She had most likely hit a drifting Dutch mine.\n\nDoorman turned north in hopes he was heading for the Japanese convoy. A feeling of anxiety spread as the ships plunged into the dark unknown. What had hit the _Jupiter_ might hit them any moment. This feeling turned to terror at 9:50 P.M. when a single parachute flare floated down and brightly lit the entire column. It was a strange game. Doorman, stalking the convoy, was himself being stalked by an enemy specifically trained and psychologically suited for night battle and whose planes could easily follow his phosphorescent wake. Two minutes later six more flares, looking like hobgoblins, straddled Doorman's line at right angles.\n\nMany felt they were steaming straight into a trap as _De Ruyter_ kept doggedly heading north. At 10:17 P.M. _Perth_ heard men in the water shouting in a strange tongue, perhaps Japanese. Just behind, _Houston_ also heard the foreign cries but to be on the safe side dropped a raft and flare.\n\nThe shouting men were survivors from the jackknifing _Kortenaer_ , sunk in the afternoon battle. A few minutes later _Encounter_ , drawn by the flare, stopped and picked up 113 survivors, including Binford's friend, Commander Kroese.\n\nFortunately for Doorman, Takagi was now also in the dark, for his float planes had left. The Japanese commander quickly altered course to south, hoping to throw himself between Doorman and the convoy.\n\n\u2014\n\nIt was almost 11:00 P.M. The wounded _Exeter_ and its escort, _Witte De With_ , had safely anchored in Surabaja. Other survivors of the great battle, the 4 American destroyers of Commander Binford, were just entering the harbor, when a radio order came from the Dutch shore station, relayed from Doorman, to return to Batavia for fuel and torpedoes. Binford turned his column west. Then he stopped. It made no sense. He held a conference by voice radio with the skippers of the other ships.\n\nCommander Edward Parker, considered one of the Asiatic Fleet's ablest and most aggressive destroyer commanders, said, \"It'll be suicide going to Batavia. We'd meet Jap destroyers and cruisers in the daylight.\"\n\n\"I know there's no oil there,\" said Binford. \"Or torpedoes.\" He suggested they fuel to capacity in Surabaja and clear the docks by daylight. \"Do any of you have any more comments or suggestions?\"\n\nAll agreed. Binford radioed Doorman he would proceed south as ordered after he had refueled in Surabaja. The column again reversed course and steamed back into port.\n\nAt this same time the rest of Doorman's fleet was about 100 miles to the northwest in the middle of the black Java Sea. It was in column formation with _De Ruyter_ in the lead, heading north still in search of the Japanese convoy. A few moments after 11:00 P.M. lookouts on _De Ruyter_ saw the 2 big Japanese cruisers on their port beam heading in the opposite direction. The Dutch cruiser fired. A moment later _Perth, Houston_ and _Java_ fired. Takagi's cruisers answered. The sky was bright with bursting star shells.\n\nFiring stopped abruptly and there was an awesome silence. The 4 Allied cruisers continued to steam north in column at high speed. In _Perth_ , immediately behind Doorman's flagship, Commander Owen was peering into the darkness from the quarter-deck. He could see nothing and wondered if they'd been firing at phantoms. It was eerily quiet. All he could hear was the swish of water, the throb of the screws. The deck pulsated with life and it gave him a feeling of hope. He wondered if the first light of day would bring them in sight of the Jap convoy. Like everyone else on _Perth_ he was eager to tear into the transports. Java was the last stand. If it fell, their homeland, Australia, would probably be conquered.\n\nThe Allied sailors did not realize that the _Nachi_ and _Haguro_ had just reversed their course and were now running a little behind and parallel to Doorman's column. They put on speed, quickly drawing abreast of the Allies. At 11:22 P.M. from 10,000 yards _Nachi_ loosed 8 torpedoes and _Haguro_ 4.\n\nDoorman's cruisers steamed on, oblivious of the torpedoes speeding toward them. Lieutenant Hamlin standing at his battle station on _Houston_ saw a sudden blaze ahead at 11:36 P.M. It came with the terrifying abruptness of a huge cigarette lighter. The _De Ruyter_ was on fire! A moment later rockets shot up from the stricken ship. Fire had reached her pyrotechnic locker.\n\nFrom his station in the after director, it looked like a Fourth of July celebration gone crazy to Lieutenant Joseph Dalton. It was impressive, horrifying. It didn't seem possible any human could survive such a holocaust.\n\nIn _Perth_ , yards behind the _De Ruyter_ , Owen was still dazed by the mighty column of incandescent flame that had suddenly shot up ahead. It was fantastically brilliant, a solid, gigantic mass of white flame\u2014the most frightening thing in his life. He was thrown to his feet as _Perth_ violently sheered off. From the deck Owen could see they had just barely missed hitting the blazing ship.\n\nJust behind _Perth_ , Captain Rooks had been waiting to see what Waller of _Perth_ would do. Making a lightning decision, he too ordered a sharp turn. The _Houston_ reared and shook as men were thrown against the forward bulkheads. The ship heeled dizzily from the turn and \"crash stop\" but it skimmed safely past the _Perth_.\n\nThen at 11:40 P.M. there was another tremendous explosion, this time just behind the _Houston_. The other Dutch cruiser, _Java_ , was burning furiously. In minutes she stood vertically with bow up. Hundreds of her crew were falling or jumping into the dark sea. With frightening abruptness the ship plunged out of sight. There was momentary quiet and the almost 500 seamen struggling in the water were heard shouting three cheers for Her Majesty the Queen.\n\nThe survivors of the blast on _De Ruyter_ were assembled on the bow. There was another great explosion and the huge superstructure that had so terrified Midshipman Kimura vanished as if by magic. With _De Ruyter_ went the tenacious fleet commander, Karel Doorman, and 366 officers and men.\n\nCaptain Waller of _Perth_ was now senior officer of the Allied fleet. One of Doorman's last orders had been to leave any survivors \"to the mercy of the enemy.\" He made his decision quickly. Ordering _Houston_ to follow, Waller raced to the southeast. Takagi gave chase but when the 2 Allied cruisers abruptly turned toward Batavia, the Japanese were fooled and didn't follow.\n\nThe Battle of Java Sea, the greatest surface engagement since the Battle of Jutland, was over. It was a complete Japanese victory. Takagi lost no ships, suffered comparatively little damage and successfully protected his convoy. Doorman lost 3 destroyers, 2 light cruisers and his life.\n\nHe had also lost Java.\n\n# 17\n\n# _\"Good-bye Till Better Times\"_\n\n**1** Doorman's tragic seven-hour-long battle did accomplish two things: it delayed both invasion convoys twenty-four hours. By dawn of February 28, however, the Western Convoy of 56 transports was closing in on the western tip of Java, preparing to land late that night. The Eastern Convoy, although badly worried the day before by the nearby rumble of the great sea battle and harassed by Allied bombers, still had all 40 transports afloat. They were heading south toward the central coast of Java, expecting to land about midnight.\n\nThe 10 battle-worn Allied ships surviving the classic sea fight were concentrated that morning at Batavia and Surabaja. Their crews were haggard and exhausted.\n\nAll the Allied destroyers and the crippled _Exeter_ were at Surabaja. Here Commander Binford, in charge of the American destroyers, was talking on Green Line, the secret telephone to Admiral Glassford's headquarters in Bandung. \"I've got four ships and 700 men. The bottom's dropped out. I want to get out of here and go to Australia. If we stay twenty-four hours, it will probably be too late to escape.\"\n\n\"I'll tell the boss and let you know,\" replied Glassford's operations officer.\n\nBinford returned to the _Edwards_. By this time he had acquired another destroyer, the _Pope_. She had missed the battle because of minor repairs but was now ready to sail. The afternoon wore on with no word. Anger and resentment against the mountain-based Asiatic Fleet headquarters of Admiral Glassford rose. To men like Lieutenant William J. Giles, Jr., of the _Edwards_ , it appeared that the Allied ships in Surabaja were being hemmed in like rats in a trap, while the headquarters people, safe in their mountain hideout, twiddled their thumbs.\n\nFinally Binford went ashore and again called Bandung.\n\n\"We sent you a signal,\" was the answer. \"We told you to go to Australia. But the _Pope_ has to wait and help escort the _Exeter_.\"\n\n\"If she does, she's a goner.\" With half her boilers out, _Exeter_ had little chance of escape.\n\n\"That's too bad. _Pope_ wasn't in the battle and still has her torpedoes. She has to escort _Exeter.\"_\n\nAt 5:00 P.M. _Edwards_ , with Binford aboard, led the other 3 U.S. destroyers out of Surabaja. As they passed the moored _Exeter_ , the crews lined the decks, lustily cheering their allies on the crippled cruiser for their gallant fight the day before. Then, with Lieutenant Giles navigating, the 4 slender ships turned right, to the east. Under cover of darkness, they would soon make a dash for freedom through the narrow Strait of Bali.\n\nTwo hours later _Exeter_ escorted by _Pope_ and the British destroyer _Encounter_ began their attempted escape. First they headed north toward Borneo. _Exeter_ was too big to follow the Americans through Bali Strait and the trio was trying to circle around the Japanese Eastern Convoy, now about to land 100 miles west of Surabaja. The next night the 3 ships would turn left and try to get through Sunda Strait at the west end of Java.\n\n\u2014\n\nNear this end, at Batavia, the other 2 survivors of the battle, _Perth_ and _Houston_ , were just leaving the naval base. Like _Exeter_ they were going to try and run the gantlet of Sunda Strait. They left behind a city of the dead. The docks of Batavia were still smoking from bombings. Sheds, godowns, ships in the harbor were gutted. There was no sign of movement.\n\nThe _Houston_ was far from the spick and span ship so loved by President Roosevelt. The admiral's cabin, occupied by the President on his four cruises, looked like the scene of a drunken brawl: furniture was scattered, mirrors cracked; large hunks of soundproofing torn from bulkheads and overhead were lying on the deck. In every cabin and wardroom, concussions from the 8-inch guns had emptied drawers, spewed clothing from lockers, torn pictures and clocks from walls.\n\nThere was more serious damage. Water seeped into the hull from plates badly sprung and leaking from near misses. Yet most of the men had that strange feeling enjoyed by most sailors. They felt their ship was immortal. Already reported sunk twice by the Japanese, the _Houston_ had been nicknamed \"The Galloping Ghost of the Java Coast.\" The rumor was they were bound for California. Soon they would see their families.\n\nAhead of _Houston_ was the _Perth_. Her skipper, Captain Hector Waller, was stocky, with heavy shoulders and thick legs; a dour, unemotional man with a high forehead. His crew felt he would get them through safely if anyone could.\n\nAt 8:00 P.M. the mine fields covering the port were cleared and a voice came over the intercommunication system. \"This is the captain speaking. We are sailing for Sunda Strait and will shortly close up to the first degree of readiness relaxed. Dutch air reconnaissance reports that Sunda Strait is free of enemy shipping. But I have a report that a large convoy is about 50 miles northeast of Batavia moving east. I do not expect, however, to meet enemy forces.\"\n\nUnlike the men of _Houston_ , many on this ship had a feeling of foreboding in spite of Waller's reassuring words. The ship's mascot, a cat named Red Lead, had tried to jump ship at Batavia, sneaking down the gangway to the godowns three times. The men remembered other bad omens. Three days previously, during an air raid at Batavia, the portrait of the Duchess of Kent, who had renamed the ship _Perth_ in 1939, had been knocked to the deck. Then someone recalled they had two chaplains aboard. One was bad enough.\n\nAs the 7000-ton _Perth_ zigzagged toward the strait at 22 knots, Paymaster \"Polo\" Owen, wearing white shirt and shorts and a blue Mae West, walked to his action station, a .5 machine gun, in the stern. He was uneasy. He felt like a pawn moved by an unknown force. By chance he had gone through the Battle of Java Sea on the wrong ship. Now what was coming? Certainly the Japanese, sweeping down on Java from two sides, wouldn't let them escape.\n\nThe narrowest point of Sunda Strait, where only 14 miles divide Java from Sumatra, was already alive with Japanese warships: 2 light cruisers and 8 destroyers. And a few miles to the north were the aircraft carrier _Ryujo_ , 4 heavy cruisers and several destroyers. This powerful aggregation was protecting the 56 transports of the Western Convoy, just now dropping anchor at the western tip of Java in Bantam Bay.\n\nSoon the 2 escaping Allied cruisers would round the tip of Java, turn left into Sunda Strait and continue on a course which would take them between the moored Japanese transports and this formidable battle escort.\n\nThe sea was calm, the air still and a full moon shone from a clear sky. It was another hot, sensuous tropical night. At 10:39 P.M. a patrolling Japanese destroyer sighted the 2 Allied ships as they neared the western tip of Java. In a few minutes 6 Japanese destroyers and a light cruiser were streaking toward Bantam Bay to protect their anchored transports.\n\n_Perth_ and _Houston_ , unaware they were steaming into hostile waters, were now turning left, heading south into the narrow strait. Suddenly at 11:06 P.M. Captain Waller sighted a vessel five miles ahead. \"Challenge,\" he ordered, but he still wasn't worried. \"It's probably one of our corvettes patrolling the strait.\" The chief yeoman signaled with his Aldis lamp.\n\nThe mystery ship was one of the Japanese destroyers guarding the transports in nearby Bantam Bay. She answered with her pale green lamp. No one on the _Perth_ could understand the message.\n\n\"Repeat the challenge,\" said Waller.\n\nIn response the Japanese destroyer turned away and began to lay a smoke screen in front of the transports massed in Bantam Bay.\n\n\"Jap destroyer,\" said Waller, recognizing her silhouette. \"Sound the rattles. Forward turrets open fire.\"\n\nA minute later Polo Owen, asleep at his station in the stern of _Perth_ , was wakened by a far-off popping noise. He looked up and saw two rockets, one chartreuse, one scarlet, falling lazily. Seconds later the 6-inch guns far forward blasted. Owen jumped to his feet, donned his anti-flash gear.\n\nSuddenly the two big guns only 10 yards forward swivelled and aimed dead astern just over his head. There was a mighty roar, and as fire from the two bores licked out Owen was blown to the deck. He flattened, face down, to avoid the simmering heat as salvo after salvo lashed out overhead.\n\nThe 2 Allied cruisers were now dashing south past Bantam Bay and its transports, their guns blazing at 4 enemy destroyers.\n\nAt 11:26 P.M. _Perth_ was hit in the forward funnel. Six minutes later a shell exploded near the flag deck, but as the cruiser swung to the right toward Sumatra, damage control reported her wounds were only superficial.\n\nIn Bantam Bay there was alarm in the crowded transports. On board the transport _Ryujo Maru_ , headquarters for General Hitoshi Imamura of the 16th Army, the general's aide was startled from his reading by the crash of guns. Only a few minutes before he had felt disappointed that the landing had been made without the firing of a single gun or rifle.\n\nMore Japanese warships were already racing to help the embroiled destroyers. From the south 2 destroyers met the Allied cruisers head on, launching torpedoes at 11:40 P.M. From the northwest the light cruiser _Natori_ and her 2 destroyers rapidly approached. Soon _Houston_ and _Perth_ had so many targets that unengaged destroyers began to dart in to very close range and light up the Allied ships with blue searchlights.\n\nAt 11:48 P.M. ominously large geysers shot up all around _Perth_. Two heavy cruisers, _Mogami_ and _Mikuma_ , were also converging from the northwest, their salvos dropping dangerously close. Two minutes later _Perth_ 's luck ended. A shell smashed into the ordinary seamen's mess from the starboard side near the waterline. Within ten minutes the gunnery officer told Waller there was almost no 6-inch ammunition left. The captain made an instant decision. He would make a run for it through Sunda Strait. As the _Perth_ headed full speed south, a Japanese destroyer raced in, threw a searchlight on the cruiser.\n\n\"For God's sake, shoot that bloody light out,\" called Waller. _Perth_ 's 4-inch guns spoke. The light vanished. Suddenly at five minutes after midnight the deck heaved. A torpedo had ripped into the starboard side near the forward boiler room. The ship rapidly began to lose life. A moment before she had been almost human, bounding and swaying. Now she was sluggish, as if dead tired.\n\nNear the stern at his machine-gun position, Polo Owen heard the great crash forward. Then terrifying silence. He ran to the port torpedo space. Empty. There was nothing he could do. It was bloody awful. He went back to his gun. It was torn and twisted. Suddenly a second torpedo hit. The ship seemed to go to pieces. It was, thought Owen, like standing on a match box.\n\nOn the bridge they were picking themselves up from the deck. The second torpedo, they guessed, must have hit on the starboard side almost under them. Water and oil gushed onto the bridge, bowling over several men.\n\n\"Christ,\" said Waller. \"That's torn it. Abandon ship.\"\n\n\"Prepare to abandon ship, sir?\" asked the gunnery officer.\n\n\"No. Abandon ship.\"\n\nPolo Owen and three sailors were trying to untie stacked rafts. The knots, tightened by the blast of the big guns, were like iron. \"Anyone got a knife?\" he asked.\n\nThe sailors ran forward. Owen followed them. _Perth_ was moving slowly, her screws thumping, one almost out of water. From below he could hear the grinding tear of objects sliding across the heeling ship. He saw escaping steam ahead. He looked over the side, then climbed the rail and dove. In mid-air he suddenly realized he'd forgotten to inflate his Mae West. Treading water, he bent his head and tried to blow into the valve. He kept sinking and finally ripped off the life preserver in disgust. The water was warm, peaceful, pleasant. He took off his shirt and shorts and emptied his bowels. It was a great relief.\n\nA few miles to the east, there was panic among the transports huddled together at Bantam Bay. Four were already sinking. Eight torpedoes aimed by the cruiser _Mikuma_ at the _Houston_ had sped by their target and exploded against friendly hulls. One of these unlucky transports was _Ryujo Maru_ , headquarters ship of the 16th Army. General Imamura, his aide and hundreds of soldiers leaped into the warm water. The general and his aide grabbed pieces of wood, for neither wore life jackets. Suddenly the aide heard a grinding noise. He looked up and saw _Ryujo Maru_ listing badly. The grinding increased as automobiles, freight and tanks rolled off the deck into the water with a dreadful sound.\n\nTo the northwest, the cause of all this, the heavy cruiser _Mikuma_ was firing her big guns at _Houston_ , which was already damaged by a torpedo. At 12:15 A.M. a salvo ripped into the U.S. cruiser's after engine room, scalding to death the entire force. The ship slowed down as steam geysered through gaping holes in the deck.\n\nIn Turret 1 Lieutenant Hamlin was looking through his periscope for targets. There was a tremendous jar. His head hit the periscope.\n\n\"Turret 2 is hit,\" said a voice over the phone, then added, \"Fire in Turret 2!\"\n\nWith one eye Hamlin tried to estimate the range to a blinding searchlight. His guns roared, the salvo fell short.\n\n\"Fire in Turret 2 magazines,\" said the telephone voice. If her ammunition were touched off, Turret 2 would be ripped out of the ship.\n\n\"Flood Turret 2,\" said the gunnery officer, Commander Al Maher, calmly.\n\nHamlin fired another salvo from Turret 1, this one 1000 yards longer.\n\n\"Fire in the small arms magazine,\" a voice on the phone told him\n\nHamlin was too busy to be worried. He had his target in a 1000-yard bracket. He split the difference and fired again.\n\n\"Fire in Turret 1 magazine,\" announced someone from Plotting Room in a strained voice.\n\nAt the same time Hamlin heard his own talker say tightly, \"Fire in the magazines!\" Then he heard Maher's calm voice: \"Flood Turret 1.\"\n\n\"One, Aye, Aye,\" answered Hamlin. In a split second he decided to flood the magazine five stories below him but keep the upper powder room and shell room dry so he could shoot six more salvos.\n\n\"Flood the magazines,\" he told his talker.\n\nIn the confusion, he was misunderstood and sprinklers also began to spray the upper powder and shell rooms. Now there was only one salvo left. Hamlin carefully aimed at the searchlight and fired. It blinked out.\n\n\"Turret 1 out of action,\" Hamlin told Maher.\n\n\"Very well, stand by.\"\n\nIt was just 12:25 A.M., March 1. Fires from Turret 2 were licking high, illuminating the entire area.\n\nSeveral miles away _Perth_ was writhing as if in pain. She righted herself for a second, then rolled over and sank.\n\nThe _Houston_ was also nearing the end of her life. The ship had slowed to 6 knots. All engines were quiet. On the bridge someone asked, \"Are we abandoning ship?\"\n\nCaptain Rooks nodded. His executive called out, \"Abandon ship!\" At 12:33 A.M. the bugle signal sounded. Officers began filing down from the conning bridge, Rooks in the rear.\n\nIn the forward engine room, Lieutenant Robert Fulton had felt the shock of the two torpedo hits. They were much lighter than the near bomb miss of February 4 which had broken the gauges, yet the engine order telegraph which signaled the speed to be made was now swinging crazily, moving without apparent sense. Though he was making full speed, the order from the bridge was constantly changing. The signaled speed from the control engine room aft also changed erratically. He wondered what was wrong, but no communication to the after engine room was possible, for all four telephone circuits were dead.\n\nThen an order came to abandon ship. It didn't make sense to Fulton. The forward engine room was unharmed; they were making full speed. \"Call back to the bridge and verify,\" he told the talker. \"Tell them we cannot contact the after engine room, but we are in good shape.\" Soon confirmation of the order to abandon ship came. Fulton ordered the fires under the boilers to be killed. The throttles of the two engines were left open so that the 400 pounds of steam pressure would be bled from the system. Finally the engineers filed calmly up through two hatches to the mess hall above the engine room, and through the starboard catapult tower to the quarterdeck.\n\nHamlin and his crew were abandoning Turret 1. As he walked aft down the slanting deck, he was amazed to see the men either standing around quietly drawing life jackets or lining up without confusion at their stations, in spite of the rain of shells dropping all over the foundering ship.\n\nRooks stood at the head of a ladder leading to the main deck. He was bidding good-bye to his men, wishing them good luck. Suddenly there was a blazing glare a few feet aft of the captain, as a 5-inch shell exploded. Rooks was dead.\n\nWord of the captain's death passed quickly. Ensign C. D. Smith saw Rooks' Chinese cook, Tai Chi-sah, nicknamed Buddha, sitting in front of his cabin and told him to jump overboard. The Chinese shook his head. \"Captain die, ship die. Now Buddha die.\"\n\n\u2014\n\nThere were only three men on the _Houston_ 's forecastle. The list was getting worse and the ship was rolling peculiarly, so Hamlin left the ship. Once in the water he swam as fast as he could, for he was sure she would plunge any moment. One hundred yards away a torpedo whirred past. Then there was a blinding flash as it hit the side of the _Houston_. Hamlin doubled up from a blow in the stomach. He couldn't breathe. A shell burst nearby. Another painful shock.\n\nLieutenant Joseph Dalton was near the incinerator trying to get men to take their shoes off and leave before it was too late. The men were calm and there was no shouting; many seemed reluctant to leave the sinking ship.\n\nBy now it was only 10 feet to the sea. He jumped. It felt like a warm bath but the oily crust was nauseating. Since he had no life jacket, he struck out for a floating mattress. Clusters of men, most without jackets, were struggling around him. Spotting a raft, he began herding men toward it. Shells began falling nearby. He lifted his chest high out of the water to relieve the concussion. Even so it felt like a mule kicking him. He drifted on the mattress through fuel oil, then through a pool of gasoline which was dissolving the oil. He finally reached the raft and began ripping the mattress to pieces to make a sail. In the glare of searchlights he saw _Houston_ listing, blazing.\n\nHamlin, still gasping for breath, turned around for a last look at the _Houston_. She lay dead in the water, her sides riddled with shell holes. Her guns stuck out crazily at odd angles. A bright orange flame sizzled and crackled halfway to the top of the foremast. He remembered the ship when he had first joined her, just after a cruise with President Roosevelt. She shone then with new paint, her brass and steel glistened.\n\nThe ship rolled gently to her side. The crackling flames went out with screaming hisses. Hamlin turned and swam as hard as he could to get out of the suction. Others, farther away, watched the _Houston_ 's yardarms dip near the water. The cruiser paused and the Stars and Stripes seemed to wave defiantly from the mainmast. Then the big ship shuddered and slid out of sight. It was 12:45 A.M.\n\nThere was sudden, terrifying quiet. The battle was over. Survivors of the two Allied cruisers were swimming for shore. Of _Houston_ 's 1000 men, less than half were still alive and many of these were drowning. About half of _Perth_ 's crew of 680 were also alive but some of these were choking to death in the oily sea.\n\nJapanese survivors of the torpedoed transports were crawling onto the sandy beaches of Bantam Bay. General Imamura's aide, exhausted and dispirited, was looking for his chief. At last he saw the general, his face black from oil, sitting on a pile of bamboo. The aide hurriedly limped to the general. \"Congratulations,\" he said, \"on the successful landing.\"\n\n**2** On the other, the eastern, end of Java, Lieutenant Giles was plotting _Edwards'_ course as she led the 3 other American destroyers through the narrows of Bali Strait at 28 knots. He hugged the shores of Bali so the ship wouldn't stand out so clearly in the bright moonlight. When heavy clouds suddenly shrouded the full moon Giles knew his mother was praying for him. At 2:05 A.M., just as _Edwards_ was clearing the strait and entering the Indian Ocean, he sighted a destroyer on the port bow, then another.\n\nCommander Binford also saw them. \"There are two ships on our port bow about 700 yards,\" he announced over voice radio to the three ships behind. \"They may be destroyers. I don't know if they've seen us but I'm not opening fire. I'm trying to get out undetected. But if they shoot, return fire immediately.\"\n\nThere was silence for fifteen minutes, then the Japanese began firing, their shells passing over the American destroyers and hitting the beaches of Java.\n\nThe Americans returned fire.\n\n\"Breakers ahead,\" cried a lookout on _Edwards_.\n\nGiles called out, \"Full left rudder!\"\n\nBinford, hearing this, called to the helmsman, \"Full left rudder!\"\n\nThe _Edwards_ came hard left, heading for the enemy ships, with the 3 American destroyers following. Several minutes later her skipper, Commander H. E. Eccles, brought _Edwards_ 30 degrees to the right. Again Giles called for hard left rudder, again the ship missed the reef by yards.\n\nBinford ordered smoke. As Japanese gunfire ceased, the 4 ships headed south for half an hour, then turned sharply to the west as if bound for Tjilatjap. Binford now stopped the smoke and headed south for Australia. Soon the last of the smoke blew away. No enemy was in sight. The old destroyers were safe.\n\n\u2014\n\nFor several years the families of those on _Houston_ and _Perth_ had no idea what had happened to their loved ones on the night of February 28, 1942. They knew only that the two ships had mysteriously disappeared somewhere off Java with all personnel listed as \"missing in action.\"\n\nMost United States naval authorities secretly concluded Captains Rooks and Waller and all their men had died, but by dawn of March 1, almost half were still alive. Some were swimming in the warm, oily water east toward nearby Java; some were heading south toward the tiny islands in the middle of Sunda Strait; some were just hanging on grimly to floats.\n\nThe oldest man on _Houston_ was clinging to the spare airplane pontoon with a dozen others. This was Commander G. S. Rentz, the portly, gray-haired chaplain. When the pontoon began to sag dangerously, Rentz, near exhaustion, quietly swam away. A seaman brought him back. After two more attempts to leave were frustrated, he gasped, \"You men are young with your lives ahead of you. I'm old and have had my fun.\"\n\nHe said a prayer for the men and then, before anyone could stop him, dropped out of sight. He had left his life jacket on the pontoon. Seaman First Class W. L. Beeman put it on.\n\nLieutenant Fulton, the engineer, was swimming toward a beach with three others. None of them knew it was Bantam Bay. Lieutenant Dalton's raft was still half a mile from the same bay. In the daylight he could see dozens of anchored transports, surrounded by many landing craft. One of these approached and started towing the raft to a larger ship. While the _Houston_ men sat in the bobbing raft wondering what their fate would be, officers in the Japanese craft began to argue. Then the raft was suddenly cut loose and it began drifting southwest in a heavy current. Dalton saw a man in the Japanese ship's fantail lift his automatic rifle as if to shoot them. It was disconcerting but the Americans were now too tired to try to hide. The Japanese didn't shoot.\n\nThough Dalton and the others paddled furiously to reach Java, the current was too strong. Several tried to swim toward shore and had to be hauled back.\n\nThe _Perth_ survivors were even farther south. Polo Owen, in almost exactly the middle of the strait, was approaching Toppers Island. He and Petty Officer Tyrell were hanging onto a wood and metal recreation seat, once on the deck of the _Perth_. The morning sun made Owen's eyes sting. His eyelashes were stiff with oil. The seat began breaking up as it headed for the little mound-like island. The two men saw a plank and swam for it.\n\nThey drifted past Toppers, heading for the larger island of Sangiang, and Owen could see a white beach and palm trees. Then a fresh current began swinging them away from this island past men floating face up in their life jackets, knees drawn up as if in sleep. All dead.\n\nNow Owen could see the blue haze of conical Krakatoa 20 miles away. Tyrell was muttering something and Owen thought, I can't last much longer. He wondered what drowning would feel like. After the horrors of the past few days it would be pleasant. He let go the plank and sank. Something rammed against his chin. Owen came up and saw Tyrell jerking the plank frantically and shouting, \"For God's sake, don't leave me!\"\n\nAll Owen could think to say was, \"Sorry.\" He saw Tyrell's head slowly disappearing. He jerked the plank. Tyrell popped to the surface and said slowly, \"I couldn't help it.\"\n\nOwen wondered who the man was. What was his name? Then he thought, this is madness. He prayed for shelter from the blistering sun, asked God to restore him to his family in Australia.\n\n\u2014\n\nOn the north coast of Java, the Western Invasion Force was disembarking 100 miles west of Surabaja. Ever since midnight the broken-down remnants of the ABDA air force had been making sporadic raids on these transports. The efforts of the American, Dutch, New Zealand and Australian fliers were gallant but useless.\n\nAt dawn of March 1, the U.S. 17th Pursuit Squadron, composed of veterans of the hopeless Philippines campaign, was making its last Java mission. Nine battered P-40's\u2014with blistered tires, worn brakes, shaky engines\u2014approached the coast flying low. They saw 30 transports along the beach, with many small boats ferrying back and forth. Ashore lines of men, like rows of ants, were heading into the hills. This was the same division that had landed at Lingayen Gulf in the Philippines.\n\nHeavy anti-aircraft fire from ships and beach caught the nine planes in a deadly cross fire. Three were hit instantly. One, piloted by Lieutenant Cornelius Reagan, burst into flame and headed for the beach. Lieutenant Robert McWherter, on his wing, signaled Reagan to follow him to the beach and bail out. Reagan waved. He rolled back the canopy of his P-40, reached forward, lit a cigarette on the flaming motor and put it to his mouth. Then he crashed.\n\nA few barges were sunk but only six planes returned\u2014a typical Java mission.\n\nOn the southern coast of the long island, the _Seawitch_ was unloading its 27 crated P-40's at Tjilatjap. When a group of stranded American airmen, veterans of the disillusionment of the Philippines, saw this, they laughed derisively. According to the rumor of the moment, the _Queen Mary_ was Java-bound, loaded to the gunwales with P-40's. It was the same old story\u2014too little, too late. Even the 27 P-40's were useless. There was no time to assemble them, and the pilots to fly them would soon be leaving for Australia. They were dumped, still crated, into the bay.\n\nWhile Japanese were landing at Bantam Bay and on the north coast of the island, Java was already disintegrating. In Bandung the British admiral, A. F. E. Palliser, Helfrich's chief of staff, telephoned Commodore Collins at Batavia, telling him to send all British warships to Tjilatjap at once and then leave Batavia himself. At 8:30 A.M. the British naval base staff left the city in a truck and car convoy and headed for the south. Half an hour later Palliser and the American commander, Glassford, came to Helfrich's headquarters. They asked him to cancel his order to fight to the end.\n\nHelfrich doggedly insisted he was only following the explicit orders of the Combined Chiefs of Staff. \"I am going to fight as long as there are ships. A large concentration of submarines in the Java Sea could still be successful. The enemy will make another attempt to land tonight near Rembang. Even if he succeeds I'll attack the next wave of transports.\"\n\n\"I have instructions from the Admiralty,\" said Palliser, \"to withdraw His Majesty's ships from Java when resistance will serve no further useful purpose. This time, in my judgment, has come.\" He was sending his ships to Ceylon at once.\n\n\"Do you realize you're still under my orders?\" said Helfrich angrily.\n\n\"I do, of course. But in this vital matter I cannot do other than my duty as I see it.\"\n\n\"Have you forgotten the enormous support I gave the British cause in Malaya? All of my fighting fleet\u2014my cruisers, my destroyers, my submarines, my air\u2014all of it was placed at your disposal. And to our great loss. I did much more to defend Singapore than the British Fleet has done to defend the Netherlands East Indies. Now when I ask you for the same sacrifice I made, when my newest submarines went into that hell on the east coast of Malaya and the west coast of Borneo and when my surface vessels protected your convoys...now you can refuse!\"\n\n\"I am grateful for your support but I cannot alter my decision.\"\n\nHelfrich turned to Glassford. \"What do you intend to do?\"\n\nAlthough the American agreed with Palliser that it was useless to fight, he said, \"My instructions are to report to you for duty. Any order you give me will be obeyed at once.\"\n\nHelfrich sighed heavily. \"You may give your ships any orders you wish, Admiral Palliser,\" he said. Then he turned to Glassford. \"You will order your ships to Australia.\" The Dutchman thanked the American effusively for his help.\n\n**3** At this same moment the last 3 ships to survive the Battle of Java Sea were about 60 miles south of Borneo. The damaged _Exeter_ and her 2 destroyer escorts were slowly heading northwest hoping to escape south through Sunda Strait that night. But Captain O. L. Gordon of _Exeter_ already realized this was a forlorn hope. A few minutes previously, at 9:35 A.M., the masts of 2 enemy heavy cruisers had been sighted to the south.\n\nThe Japanese cruisers belonged to the victor of the Battle of Java Sea, Admiral Takagi. They were _Nachi_ and _Haguro_ and, with 2 destroyers, were moving in for the kill. Another powerful force was coming from the other side. Alerted by Takagi, Vice Admiral Ibou Takahashi, commander-in-chief of the Third Fleet, was approaching from the north with 2 heavy cruisers and 2 destroyers.\n\nIn between these two forces, _Exeter_ and her 2 escorts writhed trying to escape. At 10:10 A.M. Captain Gordon turned the _Exeter_ to the southeast while the destroyers _Pope_ and _Encounter_ laid down a smoke screen. Speed was increased to maximum, 25 knots, but the 4 big Japanese cruisers closed the trap and at 10:20 A.M. opened fire. _Exeter_ replied while _Pope_ and _Encounter_ began firing at 4 converging Japanese destroyers.\n\nBy 11:00 A.M., though no real damage had been done, the _Exeter_ was hopelessly trapped. Five minutes later she loosed torpedoes in a desperate strike at Takahashi's 2 cruisers. The range was too great.\n\nSalvos from the Japanese cruisers were falling closer. Soon _Exeter_ was bracketed. _Pope_ and _Encounter_ darted around the slow ship, trying to distract attention. Lieutenant Commander Welford Blinn of _Pope_ launched 4 torpedoes at Takahashi's cruisers, then swung around and aimed 5 more at Takagi's force.\n\nEven so, the Japanese moved closer, and at 11:20 A.M. an 8-inch shell parted the one remaining steampipe of _Exeter_. Almost immediately the big ship slowed to 4 knots. _Pope_ and _Encounter_ , realizing nothing could now be done, raced to the east to save themselves.\n\n_Exeter_ 's main engines had stopped, so had the dynamos. Her turrets were frozen, with silent guns pointing in different directions. The steering gear was useless. All power had failed. The inside of the ship filled with smoke as the forward boiler room burst into flame from escaping fuel oil.\n\nThe order was given to sink the ship. Magazine flood valves were opened; the engine rooms were allowed to flood through the condensers; and all watertight doors opened. Down below the fire raged and the ship began to list as black smoke poured out from her funnels.\n\nTo Lieutenant Commander George Cooper, the first lieutenant, standing at the rear of the bridge, she still seemed defiant, like a stag at bay. Men were cutting down Carley floats; timber was being cast adrift. Shells were falling all over the ship from close-range fire. Cooper saw the after superstructure catch fire as projectiles whined overhead sounding like the song of the Valkyries.\n\nThe ship was beginning to heel over and she was only making slight headway. Men began going over the side in groups, drifting astern. The executive officer, calmly walking around the ship, came up to Cooper on the quarterdeck. \"This is it,\" he said in an impassive voice. \"Best of luck, Number One.\"\n\nCooper threw his binoculars on the deck and jumped from the quarterdeck into the sea. He saw a destroyer closing on the starboard beam of _Exeter_. It fired a torpedo. It was a good shot, hitting her right amidships. The big cruiser shuddered a bit and almost instantly flopped to starboard until her funnels and masts were horizontal. The white ensign was still flying from the masthead.\n\nAs _Exeter_ put up her forefoot as a final sign of defiance and began to plunge, everyone in the sea spontaneously cheered. Then there was only a great swirl of water, smoke and steam.\n\nTen miles to the east the other British ship, _Encounter_ , was also trapped and sinking. As men dropped over her sides, the _Pope_ raced toward a heavy rain squall just ahead. At 11:50 the old American four-stacker plunged into protective darkness, but she too was doomed. Ten minutes later she was sighted by a float plane from one of the cruisers and at 12:30 P.M. 6 dive bombers from the nearby aircraft carrier _Ryujo_ appeared overhead.\n\nCommander Blinn zigzagged his ship, avoiding 10 dive-bombing attacks. On the eleventh drop, a bomb just missed the side of the ship abreast No. 4 torpedo tube. It exploded underwater, ripping a large hole below the waterline and knocking the port shaft seriously out of line.\n\nSix high-level bombers from _Ryujo_ began attacking from 3000 feet. While Blinn was maneuvering to avoid the second level bombing attack, his damage control officer reported flooding was beyond control. Blinn ordered watertight doors and ports opened. After the wounded were put in a boat, Blinn ordered demolition charges set off in the engine rooms to insure sinking the destroyer. Finally the ship was abandoned.\n\nAt about 1:00 P.M., shells from Takagi's rapidly approaching cruisers fell all around the ship. A salvo hit _Pope_. It disappeared, stern first, in fifteen seconds. Now there wasn't a single Allied ship afloat on the Java Sea.\n\nGeorge Cooper of _Exeter_ was just being hauled aboard a Japanese destroyer. The decks were crowded with 300 survivors. When the destroyer steamed away Cooper and others pointed to the hundreds of men still struggling in the water, but the officers on the bridge took no notice. It was terrible to see messmates being left to drown but Cooper knew there was nothing he could do. He looked up at the Japanese skipper. He was smoking, occasionally giving an order; otherwise he was completely impassive and never looked at his 300 prisoners. He showed no signs of jubilation, relief or even fatigue. He might have been on a pleasure cruise, having just stopped to give his crew a bathe.\n\n\u2014\n\nThe sea carnage had also spread to the waters south of Java. By the end of the day the Southern Striking Force of Admiral N. Kondo owned the waters below Tjilatjap and was relentlessly cutting down transports and warships fleeing Java. The U.S. Navy oiler _Pecos_ , carrying all the survivors of _Langley_ , was sunk 400 miles south of Java. Although the U.S. destroyer _Whipple_ raced in to pick up 232 men from the sea, the rest had to be left to their death. _Whipple_ escaped but two other four-stackers, _Pillsbury_ and _Edsall_ , were caught by Kondo's battleships and carriers that afternoon and sent to the bottom.\n\nBy midnight the last flyable American plane was getting ready to leave the dying island of Java. It was an OB-30\u2014the British export version of the B-24\u2014and it was dangerously loaded with 35 passengers. At 12:30 A.M., March 2, the plane with its load of airmen left the Jogjakarta (now known as Jockstrap) field. The frustrating, hopeless American air effort in Java was over.\n\nThat same dawn, a flying boat left a lake near Bandung for Ceylon. In it was Admiral Helfrich. He took with him only a small valise. Behind lay his family, friends and possessions. Ahead was an unknown future. He felt like a raw ensign.\n\nBy midmorning Japanese of the Western Invasion Force were rapidly converging on Batavia and Bandung from two sides. The Eastern Force was already halfway to Surabaja and well on the way to Tjilatjap. Enemy resistance was being wiped out with little trouble. It puzzled General Imamura. He had expected a long, arduous campaign. Was he walking into a trap?\n\nCommander Shukichi Toshikawa of the 5th Destroyer Flotilla was talking to Imamura's chief of staff. He had been chosen, he explained, to apologize to the general for torpedoing his four transports and dumping him in Bantam Bay.\n\n\"Don't tell General Imamura,\" said the chief of staff in alarm. \"He thinks the American cruiser _Houston_ did it. Let her have the credit.\" To this day official records have done just that.\n\nThe survivors of _Perth_ and _Houston_ were still scattered around Sunda Strait. Polo Owen and Tyrell, each hanging desperately to the end of a plank, were near the end of their endurance. Then Owen saw something black to the left. Tyrell blew his shrill whistle. He blew until he was winded.\n\nFinally he cried, \"I can see it there. It looks like a boat. Christ, it's a boat. Now it's gone...gone...\" His head started going under water.\n\n\"Don't go. We'll be picked up. Hang on, for God's sake,\" cried Owen. As Tyrell's head rose from the water, Owen grabbed the whistle. He blew it over and over again.\n\nFinally an Australian voice shouted, \"For Christ's sake, stop blowing that bloody flute. We 'eard you the first time.\" In a moment Owen was in the bottom of a boat with Tyrell beside him. Owen sighed with contentment when he felt something solid between him and the sea, then passed out.\n\nThe largest group of _Houston_ survivors had been captured on the beach of Bantam Bay. Since landing, Lieutenants Hamlin and Fulton had been toting boxes and cases from the beachhead to assembly points. Fulton noticed many of the cases had soda bottles from Saigon tied to them. Now in the darkness the _Houston_ men were pulling rickshaws filled with ammunition and dogcarts loaded with luggage up a road. Most of the men had lost their shoes and their feet were cut and bleeding. They were heading toward Serang, on the road to Batavia, but they didn't know where they were going or what their fate would be when they got there.\n\n**4** That day General Gordon Bennett, the red-haired Australian commander, arrived in Melbourne after a thrilling escape from Singapore by native boat, launch and plane. When he called on Lieutenant General V. A. H. Sturdee, the Australian chief of general staff, his reception was cold.\n\n\"Your escape was ill-advised,\" said Sturdee. He returned to his work, leaving Bennett standing in the middle of the room.\n\nBennett had returned to a far different Australia. Eleven days previously the war had landed squarely on her front door. In a devastating daylight raid, 81 planes from the 4 carriers of Admiral Nagumo, of Pearl Harbor fame, had almost wiped out Darwin, the main seaport in north Australia. The U.S. destroyer _Peary_ and 8 other ships had been sunk, 9 damaged and 22 Allied planes destroyed. Only 5 Japanese had been shot down.\n\nAt one frightening stroke the Australian people were suddenly forced to realize they were wide open to invasion. New Britain was gone. Half of New Guinea was gone, the rest apparently going. Java and Timor were being gobbled up.\n\nFearsome posters pictured a brutal Japanese soldier coming across the sea out of a rising sun with one hand grabbing the map of Australia. Across it was written: THE WORD NOW IS MUST.\n\nPeople were confused and divided. \"People are unhappy about Darwin,\" wrote Sir Keith Murdock in the Sydney _Sun_. \"They know that workers were holding stop-work meetings a day or two before the raid. They know that ships that should have been cleared with proper appliances and labor are still lying in the harbor exposed to bombs. Australia has pegged out a tremendous claim upon the world. She asks now that she be allowed to hold a vast land for her seven million white people. Let us then seize the chance, for if we cannot rise to this occasion, then assuredly we can never rise to any occasion.\"\n\nNot long after this Yates McDaniel landed in Australia where he was reunited with his wife in Melbourne and made chief of bureau of the AP for the Australian area. One morning, while still recovering from his narrow escapes, he was awakened by an insistent knock. It was \"Newsreel\" Wong. Hearing that McDaniel had arrived in Australia, he had immediately flown from Ceylon.\n\n\"Where is Doris?\" he asked.\n\nMcDaniel became concerned. Her ship should have long since reached Ceylon or India. He told Wong the name of the ship. The two men went downtown and checked with shipping experts. There was no news at all. They wouldn't learn until after the war that Doris Lim's ship had been sunk soon after leaving Java. Not a passenger survived.\n\n\u2014\n\nBy the morning of March 3, the waters below Java were a graveyard of Allied ships as Admiral Kondo made this a Japanese sea. In addition to merchant ships, the British destroyer _Stronghold_ was sunk and the convoy being escorted by the Australian gunboat _Yarra_ was being hemmed in. Yet in spite of the shocking toll, many ships were stealing past the mighty Japanese force to Ceylon and Australia.\n\nThe land battle for Java was coming to a sudden humiliating end, too. The Allies were scattered and disorganized. Some groups battled desperately; other groups did almost nothing. Nobody knew what was going on as the two great invasion forces converged on the key cities of Java. On March 5, at a conference of senior Allied officers at Bandung, Lieutenant General H. ter Poorten, commander of the ABDA army, said guerrilla warfare was impossible. The Indonesians were too hostile to the Dutch. Three days later ter Poorten broadcast a message telling everyone to lay down arms.\n\nThe last message to the outside world came from a dispatcher at Bandung's commercial radio station. Just before it was destroyed he said, \"We are shutting down. Good-bye till better times. Long live the Queen!\"\n\nJava was gone. Like all other fortresses cut off by sea\u2014Guam, Wake, Hong Kong and Singapore\u2014she had fallen. The final sea and land battles had cost the Japanese 1 mine sweeper, 4 transports, a number of barges, a few damaged warships and less than 1000 lives. The easy victory, however, had its pitfalls. The Japanese had learned little, many top commanders concluding that their present equipment and tactics were good enough for future operations.\n\nThe Allies had suffered a humiliating and devastating defeat. Defense had been hopeless from the start, but it had not been a completely useless, senseless gesture of bravado. The Americans, British, Dutch and Australians had kept faith with each other. Abject surrender, as in France, might have demoralized the entire Southwest Pacific. In spite of bitter arguments and recriminations, there was still an underlying spirit of unity among the members of defeated ABDA and now all their eyes hopefully turned toward the only remaining bastions inside the new Japanese Empire\u2014Bataan and Corregidor.\n\n# PART\n\n# 5\n\n# \"THE BATTLING BASTARDS OF BATAAN\"\n\n# 18\n\n# \"... _And I Shall Return\"_\n\n**1** The stalemate in Bataan continued through February. In Tokyo, Prime Minister Tojo was becoming more impatient with each passing day. Militarily Bataan was of little importance, but its propaganda value for the enemy was constantly rising. It had to be liquidated quickly. About a month earlier Homma had informed higher headquarters that the Bataan campaign could not be concluded successfully without heavy infantry and artillery reinforcements. Although these were already arriving, and training for the final attack was under way, Tojo doubted Homma's ability to draw up a plan that would bring the quick success so sorely needed.\n\nThe prime minister called in his secretary, Colonel Susumu Nishiura, one of his young advisers. Tojo explained the problem, pointing out that he didn't want to violate procedure by speaking directly to the army chief. \"Perhaps,\" suggested Tojo, \"you could convey my worries to Colonel Hattori?\" This was Takushiro Hattori, army operations officer for Imperial General Headquarters. As prime minister and minister of war Tojo had no direct voice in actual combat operations. Even suggestions had to be made subtly. He knew he could trust Hattori, who had once been his private secretary. Hattori was not only one of the most brilliant young officers in the Army but one of the ablest military diplomats.\n\nThat day Nishiura visited Hattori. They had studied together at military school when both were youths and had always remained close friends. Hattori, already worried about Bataan, put his entire staff to work. Within a week his desk was piled with detailed staff reports and plans. After assimilating all this material, he decided that the most unexpected place of attack was Mt. Samat, a rugged hill 1920 feet high sitting just behind the center of the American line. Hattori's plan was simple: a concentrated air and artillery bombardment on a two-and-a-half mile sector in front of Mt. Samat, followed by a full-scale infantry drive through the hole. According to his calculations, resistance would be soon crushed and the Americans would surrender.\n\nThe plan was approved by his chief, General Sugiyama. Now the Hattori plan had to be presented to General Homma's staff so subtly they would think it was their idea and thus save face. Hattori called in Lieutenant Kumao Imoto, briefed him on the plan, and ordered him to fly to the Philippines. Then Hattori checked his personal schedule. He was due to make an inspection of the Southwest Pacific in early April with Sugiyama. He would juggle the schedule so he could be in the Philippines during the attack. For once he'd like to see one of his plans in operation.\n\n**2** After the elimination of the Pockets, the American and Filipino defenders had had little to do except patrol, train for future operations, and strengthen the long line across the peninsula. For a time morale continued to be high. The men had been heartened by reports of Japanese collapse. A communiqu\u00e9 from Corregidor stated that Homma had commited hara-kiri because of his failures in Bataan; Yamashita, the \"Tiger of Malaya,\" had taken his place. This false story became embellished with details: Homma had chosen for the suicide MacArthur's apartment in the Manila Hotel, where funeral rites had also been held.\n\nFood, however, had become the main interest in Bataan. Many front-line troops were getting only a third of a ration a day. They were walking skeletons, coming closer each day to actual starvation. The efforts to bring supplies to Corregidor and Bataan through the Japanese sea blockade had failed. Ships and submarines had brought in little more than 1000 tons of ration, only enough to feed Bataan's 80,000 soldiers and 26,000 civilians for four days.\n\nBy now most of the carabao (water buffalo) had been eaten. Early in March, while Navy Lieutenant Champlin was talking to Wainwright, a cavalry captain informed the general that the supply of fodder for the remaining 250 cavalry horses and 48 pack mules was almost gone.\n\n\"I knew this was coming, Captain,\" said Wainwright. One of the horses was his own prize jumper, Joseph Conrad. \"We have a lot of men who also are short of food and horse meat is not so bad.\" He paused, then said, \"You will begin killing the horses at once. And I want you to kill Joseph Conrad first.\" As he turned away and started back to his trailer, his eyes were filled with tears.\n\nThe hungry men were vulnerable to the most debilitating ailments. Quinine was almost gone, and they were now located in one of the most malaria-infested areas in the world. Over 500 were hospitalized in the first week of March alone. An epidemic of catastrophic proportions threatened to break out momentarily. Already base hospitals, collecting stations, field stations and aid stations were overflowing. It was a common sight, General Clifford Bluemel wrote his wife, to see men stricken with malaria lying helpless and untended beside their foxholes.\n\nWeakened by their constant fight against hunger, malaria, dysentery, beriberi, dengue and scurvy, most of the men became concerned only with the struggle for existence. Their minds were dangerously idle, making them prey to doubts, fear and enemy propaganda. Until recently most had believed the optimistic reports from MacArthur's headquarters that an enormous supply of food, planes, ammunition and reinforcements would soon arrive. These official reports to the front-line troops had given rise to a series of fantastic rumors. Some men actually believed the convoy was bringing an entire cavalry division of Negro soldiers who would gallop into Bataan on snow-white horses. Others believed it was bringing a special group of sharpshooters commanded by the World War I hero, Sergeant York.\n\nBut by now only the most na\u00efve believed the \"mile-long\" convoy was coming. And Sergeant York and his squirrel-shooters had become a standard joke. Such rumors had been displaced by disillusionment. Americans wrote disparaging poems about Roosevelt and \"Dugout Doug\"; many Filipinos were asking themselves what they were doing on Bataan, away from their loved ones. All the front-line troops griped about the rear echelons. Those in the service areas, they complained with justification, had more steel helmets and food than the fighters. Only one in four at the front had a blanket, shelter half or raincoat and about 20,000 were shoeless.\n\nEveryone on Bataan griped about those on Corregidor who were \"living off the fat of the land.\" This came to a climax when a waybill for three anti-aircraft batteries which were receiving Corregidor-type rations was made public. Among the items for this relatively small group were: a case of ham and bacon, 24 cans of lard substitute, peas, tomatoes, corn, peaches, 24 bottles of catsup, 50 cartons of cigarettes and 600 pounds of ice. The luxury items grew with every telling as the story spread all over Bataan. To men existing on rice and perhaps a little monkey or iguana, it made those responsible for such inequities worse than the Japanese. The 50 cartons of cigarettes alone were more than an entire infantry division was getting. A man was lucky to get one cigarette a day at the front.\n\nBy now the Japanese propagandists had changed their early tactics and were no longer trying to turn the Filipinos against their American \"masters.\" It was obvious that the great majority of Filipinos were fond of Americans and the American way of life. The appeal was now aimed at hunger, home and sex. Colored pictures of food were dropped. Then came menus from the Manila Hotel, the Panciteria Antigua and the Maritima Steamship Company. Stamped across the menus in red were the words: DON'T YOU WISH YOU COULD ENJOY THIS?\n\nOther leaflets pictured a tearful mother calling to her son or a surrendered soldier enjoying a sumptuous meal surrounded by his smiling wife and children.\n\nThe crudest leaflets featured sex. The most effective of these was a \"striptease\" series. First, the picture of the face of a beautiful woman was dropped from planes. Next came a view of the same woman from the waist up, with a shawl just covering her large breasts. The third showed the woman, full length, draped seductively with a shawl. In the fourth picture, the shawl was gone. The final picture showed the sex act.\n\nAlthough most of the propaganda leaflets were thrown away or used in the latrines, they had an undeniable psychological effect. By the first week in March morale on Bataan had dropped dangerously. The verse of correspondent Frank Hewlett, who had spent much time up front with Wainwright, expressed what most men on Bataan\u2014Filipinos and Americans alike\u2014were thinking:\n\n> We're the battling bastards of Bataan:\n> \n> No momma, no poppa, no Uncle Sam,\n> \n> No aunts, no uncles, no nephews, no nieces,\n> \n> No rifles, no guns or artillery pieces,\n> \n> And nobody gives a damn.\n\n**3** On the morning of March 10, Wainwright drove up front in a bantam car to see how the front-line troops were getting along. It was clear and hot. A perfect day for strafing, thought Lieutenant Champlin, who sat next to the general. But when no planes appeared. Champlin relaxed and put on sunglasses. As Wainwright began quizzing his two aides, Pugh and Dooley, on cavalry tactics, Champlin was thinking of the off-the-record talk he and the general had had the night before. After Champlin had promised not to repeat their conversation, Wainwright said, \"The Army high command has made three major military mistakes here in the Philippines. Ranking officers failed to visit troops in the front lines when all we have to offer our people is morale. Second: we shouldn't have had to rely on half-trained Filipinos. More troops should have been sent from the U.S. before the war.\" The last mistake he blamed on MacArthur. \"We should never have discarded the Grunert plan of retreating into Bataan for that grandiose scheme of getting the Japs at the beaches with a paper army.\"\n\nChamplin happened to look into the glare of the sun. For a moment he couldn't move or speak. A black speck was hurtling directly toward the jeep. The speck grew larger. It developed wings which began to dip from side to side.\n\nChamplin finally found his voice. \"Get the hell out of the car! Quickly!\"\n\nThe other three officers looked at Champlin as if he were crazy. It reminded the Navy man of a slow-motion movie. He yelled again, then released the safety strap across the general's seat, seized him by the scruff of the neck and leaped out of the car. The two tumbled down a ditch into a thorny bush.\n\nChamplin looked up at the road and saw bullets spattering into the empty car. Aiming a Garand rifle he had borrowed that morning from Wainwright, he fired at the strafing plane.\n\nAfter several passes the plane left. Heads popped up from vines and bushes. \"Jesus,\" said Dooley, looking at Champlin with wide eyes, \"that was a close one.\"\n\nChamplin looked at Wainwright, sitting calmly on the ground with an amused expression on his face, his eyes twinkling. \"You kind of like that gun, don't you, son?\"\n\n\"Yes, General, I guess I do.\"\n\n\"It's yours. Take it and thanks. He'd have gotten us if you hadn't spotted him coming out of the sun.\"\n\n\"But, General,\" said Champlin, \"this gun is Government Issue.\"\n\n\"Who's fighting this war? The pencil-pushers in Washington or you and me?\"\n\nThat noon, Wainwright and his aide, Major Johnny Pugh, were on a motor launch heading for Corregidor. MacArthur had asked to see the commander of I Corps immediately. Soon the two men were in MacArthur's office inside Malinta Tunnel. \"General MacArthur is going to leave here and go to Australia,\" said his chief of staff, General Sutherland. \"He's at the house now and wants to see you. But I'll give you a fill-in first.\"\n\nWainwright listened intently as Sutherland explained how Roosevelt had been trying to persuade MacArthur to evacuate the Philippines since February 22. \"Until yesterday the general kept refusing. He plans to leave tomorrow evening by torpedo boat for Mindanao.\" A Flying Fortress would then take him to Australia. \"You will be placed in command of all troops on Luzon,\" continued Sutherland. \"If it's agreeable to you, General Jones will get another star and take over your I Corps.\" After explaining that Colonel Beebe would be promoted to general and made MacArthur's deputy, carrying out his orders from Australia, Sutherland said, \"You look hungry, Skinny. Have some lunch and then we'll go up to the house.\"\n\n\"Nope, I think not.\" Wainwright jerked his thumb toward Bataan. \"We only eat twice a day over there.\"\n\nThe two generals walked out the east end of the tunnel, then through the scrub jungle for half a mile down the island's polliwog tail to a small gray house. MacArthur came out on the porch, grinned and held out his hand. \"Jonathan,\" he said as they shook hands, \"I want you to understand my position very plainly.\" He was leaving, he said, only because of insistent, repeated orders from Roosevelt. At first he had told his staff he would refuse, but they convinced him that defying the President's direct order would bring disciplinary action. \"I want you to make it known throughout all elements of your command that I'm leaving over my repeated protests.\"\n\n\"Of course I will, Douglas,\" said Wainwright.\n\n\"If I get through to Australia, you know I'll come back as soon as I can with as much as I can.\" Then he warned of the necessity of greater defense in depth. \"And be sure to give them everything you've got with your artillery. That's the best arm you have.\"\n\nThe two were quiet for a moment. In the distance the dull rumble of battle from Bataan could be heard. Wainwright was thinking of the dwindling ammunition and food supply, his air force of two battered P-40's, of the spreading malaria and dysentery and lack of medicine. He said, \"You'll get through.\"\n\n\"And back,\" MacArthur added with determination. He gave Wainwright a box of cigars and two large jars of shaving cream. \"Good-bye, Jonathan.\" They shook hands warmly. \"If you're still on Bataan when I get back, I'll make you a lieutenant general.\"\n\n\"I'll be on Bataan if I'm alive.\" Wainwright turned and slowly started back to his launch.\n\nAs darkness fell the following day, March 11, MacArthur stepped onto the porch of his house and approached his wife. Even those on Corregidor who disliked MacArthur thought she was modest, charming and courageous. \"It's time to mount up, Jeannie,\" he said quietly. MacArthur, Mrs. MacArthur, their four-year-old son, Arthur, a Chinese nurse and Major Charles Morhouse stepped into a car and were driven to the north Mine dock. To Morhouse it was like a dream. The day before he had been surgeon of the Provisional Air Corps Regiment on Bataan. Though he had never met MacArthur before, the general had chosen him to be the attending physician on the rigorous trip to Australia.\n\nThe MacArthur party walked along the wide, concrete pier, then up a gangway to PT Boat 41. Its commander was the colorful, bearded Lieutenant John Bulkeley. Following Mrs. MacArthur, the Chinese nurse, Ah Cheu, carried Arthur aboard. MacArthur shook hands with Major General George Moore, chief of harbor defenses. \"George, keep the flag flying. I'm coming back.\" MacArthur stepped aboard. Then came Major Morhouse, General Sutherland, Navy Captain Harold Ray. The last man aboard was MacArthur's aide, Lieutenant Colonel Sidney Huff. He waved to Colonel Frederick Ward and called out, \"We're coming back up here, Freddy.\"\n\nThe PT boat slowly pulled away from the pier. MacArthur was standing, looking back at Corregidor. At that moment, the loudspeaker near Ward crackled. It was Carlos Romulo, the \"Voice of Freedom,\" broadcasting to troops all over the Philippines. MacArthur took off his familiar field marshal's cap and raised it in farewell to the few on the pier.\n\nIt was about 8:00 P.M. by the time PT 41 reached the mine field protecting Corregidor. An hour later three other PT boats, carrying Admiral Rockwell and twelve more of MacArthur's staff, met Bulkeley's boat at the edge of the field. Suddenly, with Rockwell's craft leading, the four boats sped south with a great roar. A searchlight from Corregidor blinked on, lit up the boats and suddenly was doused.\n\nRockwell's boat, PT 34, piloted by Lieutenant Robert B. Kelly, exchanged places with Bulkeley when clear of the mine fields, then soon fell behind the leader. \"Don't you think we're getting a little far apart?\" said the admiral. Kelly tried to close up the gap, but the engine was full of carbon.\n\n\"Damn it,\" said Rockwell as the gap grew, \"let's close up.\" Kelly sent a whispered order to the engineers to disconnect the throttle and push the carburetors up by hand. In a moment, PT 34 roared wildly past Bulkeley's boat. Rockwell, worried by this erratic performance, looked at Kelly dubiously. It was only then that the young lieutenant revealed the poor condition of his boat.\n\n\"My God,\" said the admiral softly. He then noticed Kelly sighting along his fingers as they passed an island. \"Don't you have a pelorus?\" Kelly did not. \"I suppose Bulkeley's boat has better means?\" It didn't apparently. \"How in hell do you navigate?\" asked Rockwell sharply.\n\n\"By guess and by God, sir,\" said Kelly with a smile.\n\nBy midnight all four boats had become widely separated. Just before dawn the lieutenant junior grade in command of PT 32 scanned the horizon with his glasses. \"Jap destroyer,\" he cried excitedly, pointing behind them at something approaching fast in the murky light. \"She has 5-inch guns. If we resist it means the lives of all the men in the boat.\"\n\n\"We're damned well going to resist,\" said Brigadier General Hugh Casey, MacArthur's engineer, and one of the boat's five passengers. The excited young skipper shouted commands to man machine guns and prepare torpedoes for launching, then himself cut the lashings and pushed overboard all the reserve gas drums.\n\nBy now the other vessel was closing in from behind. Brigadier General Spencer Akin, the signal officer, was about to drop overboard a laundry bag filled with code devices. The sailors at the machine guns and torpedo tubes were preparing to attack.\n\n\"That's one of our boats,\" cried out Casey.\n\nA moment later the \"destroyer\" they had narrowly missed attacking pulled alongside. It was Bulkeley's boat. MacArthur, wearing gold cap and a fieldjacket, was standing, soaking wet. Beside him were his wife, also wet but smiling, and Ah Cheu holding Arthur. The boy, though wet and seasick, was not crying. With jaw set, he was a small imitation of his father.\n\nMacArthur asked Casey to come aboard to discuss their immediate future. They were still far from the first rendezvous, a small island of the Cuyo group, and it would soon be broad daylight. After a short conference, it was agreed that though it was risky to continue, it was even more dangerous to wait until dark.\n\nThe two boats didn't reach the island rendezvous until 4:00 that afternoon. Waiting in a cove in the sweltering sun was Rockwell's boat. Here a second conference was held. MacArthur wondered if he and his family should make the rest of the trip by submarine. One was scheduled to appear at this same spot the following day just in case its services were needed.\n\n\"We'd better get the hell out of here fast,\" advised Rockwell. There was even the chance, he said, that the submarine wouldn't appear. Sutherland agreed.\n\nBulkeley warned them that the next lap would be even rougher than the first, since the weather might worsen. Rockwell, anxious to get moving, assured MacArthur the weather was going to be good.\n\nThe general turned to his chief of staff. \"Dick, I can't do anything to Rockwell. But if it's rough tonight, I'll boil you in oil.\"\n\nThe men in the boat which had dumped its spare fuel were distributed between the other two boats and an hour before sunset, with Rockwell leading, they started toward Mindanao. Within an hour a Japanese cruiser loomed up in the northeast. The two boats turned, hiding in the glaring rays of the setting sun. Half an hour later, with tropical suddenness, it was pitch-dark. Rockwell led them back to the east and they slipped along the coast line of Negros Island.\n\nA wind sprang up, lightning zigzagged across the sky. Rockwell was steering through the narrow passage in the Mindanao Sea without charts, guided only by instinct and the smell from nearby islands. Bulkeley stayed behind letting the other boat run interference. He navigated by bubble sextant and bearing, using the only charts of the little flotilla. Fortunately both boats broke into the Mindanao Sea without running aground. Here huge 15-foot waves crashed against the frail craft.\n\nIn the first boat, Rockwell stayed on the bridge with Kelly even though foaming waves smashed over the cockpit, soaking everyone. As the waves grew in size he remembered his advice to MacArthur. \"The general's going to give me hell for this in the morning.\"\n\nAt dawn, March 13, the island of Mindanao was sighted. \"Good navigation, Kelly,\" said Rockwell. \"I wouldn't have believed it possible.\"\n\nThe two craft roared along the coast at top speed. At 7:00 A.M. Bulkeley's boat, now in the lead, headed for the Bugo dock, near the Del Monte pineapple factory.\n\nColonel William Morse, waiting on the dock with a guard of U.S. soldiers, saw a tall figure in the bow of the first boat. It reminded him of the picture of Washington crossing the Delaware. The sun glistened on the gold-braided visor of the man's cap. There was no other cap like this in the world.\n\n\"Hello, Morse,\" called MacArthur in an unconcerned voice. His face was pale, his eyes dark-circled. Obviously dead-tired, he squared his shoulders and stepped off the boat. Excited Filipinos began to appear magically. The general turned to Bulkeley. \"Commander, I'm recommending a Silver Star for all members of your crew. You've taken me out of the jaws of death and I won't forget it.\" Then he turned to Morse and asked where he could relieve himself.\n\nTall, bespectacled Major General William Sharp, commander of the Mindanao forces, drove MacArthur up the mountain road to the south. In an hour they reached the vast, sprawling Del Monte pineapple plantation. Four Flying Fortresses were scheduled to be waiting here for the general's party.\n\nSharp explained with some embarrassment that the planes had not as yet arrived. The following day a lone, patched-up Flying Fortress landed at Del Monte. Four had left Australia. Two had turned back with engine trouble, one had crashed in the sea off Bugo. These had been the best planes the U.S. Army commander in Australia, Lieutenant General George Brett, could scrape up. He had tried to borrow four of the dozen brand-new Flying Fortresses assigned to Vice Admiral Herbert Leary, commander of the Australia-New Zealand Force.\n\n\"I'd like to help you, Brett,\" Leary had said, \"but it is quite impossible. We need those planes here and can't spare them for a ferry job, no matter how important it is.\"\n\nWhen MacArthur saw the only B-17 that had safely reached Del Monte he was incensed. A veteran of the Philippine campaign, it was obviously worn-out. Its turbo superchargers wouldn't operate; the expander tube on the right brake had ruptured from age. The general was also not at all impressed with its pilot, Lieutenant Harl Pease, Jr. He told Sharp he flatly refused to ride in a brokendown crate piloted by an \"inexperienced boy.\" (Pease, in spite of his youthful looks, was already well-seasoned by combat and regarded by fellow fliers as one of the best pilots in the Southwest Pacific.)\n\nMacArthur sent a blistering radiogram to General Brett asking for other planes. He also radioed General Marshall:\n\n> THE BEST THREE PLANES IN THE UNITED STATES OR HAWAII SHOULD BE MADE AVAILABLE WITH COMPLETELY ADEQUATE EXPERIENCED CREWS. TO ATTEMPT SUCH A DESPERATE AND IMPORTANT TRIP WITH INADEQUATE EQUIPMENT WOULD AMOUNT TO CONSIGNING THE WHOLE PARTY TO DEATH AND I COULD NOT ACCEPT SUCH A RESPONSIBILITY.\n\nWhen Brett got this message, he once more called on Admiral Leary, this time determined to be more forceful. To his amazement, Leary didn't raise a single objection. A radiogram from Washington had seen to that, evidently. Brand-new bombers were instantly made available, but it took another day to prepare them for the long flight.\n\nThe resultant delay was causing increasing nervousness at Del Monte. By now everyone in north Mindanao knew MacArthur was there. Since Japanese were already entrenched in the southern part of the island, General Sharp was expecting enemy raids momentarily.\n\nHeavy guards surrounded the homes and clubhouse of the Del Monte compound. By the evening of March 15, the new plans from Australia still had not arrived and the tension was becoming intolerable. Captain Allison Ind, formerly Brereton's intelligence officer, went out for a walk that night. He was a small man with an alert, inquisitive mind. After some argument he persuaded a sentry from Texas to let him go outside the ring of American guards. It was quiet and peaceful in the strange Cagayan countryside. Suddenly he heard the crunch of a boot on gravel. Against the stars he saw the black figure of a man wearing a cap. He knew there were no Americans outside the ring of guards; Filipinos would wear straw hats. The cap could only belong to a Japanese. He silently dropped to the ground, pulled out his long-barreled pistol. He put the gun under his left armpit to muffle the sound and softly cocked it. The figure was now standing stockstill as if listening. Slowly Ind raised the gun until the sights were in position. His target was the crown of the cap. He began to squeeze the trigger, already filed down to a hair release. He knew he couldn't miss at this short distance.\n\nThen a second figure appeared and he heard a woman say, \"I don't hear it now.\" Ind recognized the voice of Jean MacArthur. He felt the strength drain from him. \"It's Ind, General,\" he said weakly.\n\n\"Where are you, Ind?\" asked MacArthur.\n\n\"In the plantation, sir. I mistook you for a Japanese infiltration party. I almost shot your ears off.\"\n\nMrs. MacArthur gasped when she saw the gun, but the general only laughed softly. \"Well, you better get up here and we'll decide who's going to escort whom back to the compound.\"\n\nThe next night, March 16, three new B-17's arrived from Australia. At 9:30 P.M. Captain William Monay, maintenance officer in charge of the flight, telephoned the commander of the Del Monte fields, Lieutenant Colonel Elsmore. \"Okay, Ray, we're all set up,\" he said. A few minutes later blacked-out cars left the compound with the MacArthur party, heading for Airfield No. 1.\n\nMonay escorted MacArthur and his family to a plane. He checked their luggage. Everyone, regardless of rank, was allowed 35 pounds. MacArthur did not ask for an extra allowance. One of the engines coughed, refused to start. MacArthur and the others filed out of the plane. The general strolled toward an officer standing nearby smoking.\n\n\"Have you another cigar? I left mine in my quarters at the plantation.\"\n\nHe lit up a cigar and sat on a bench next to Colonel Morse. The general talked to the little group of his hopes. He said there was a great army of Americans waiting for him in Australia. \"I'm going to get help for you.\"\n\nCaptain Monay approached, told MacArthur he would have to transfer to Captain Frank Bostron's plane. The general nodded and soon his party was aboard. Several enlisted men brought out a mattress and put it aboard the plane for Mrs. MacArthur and Arthur. Then Bostron taxied the plane to the end of the field. The engines roared and the Flying Fortress rumbled down the short runway. Suddenly at the end of the plateau it dipped out of sight. Monay stopped breathing. But seconds later the big plane rose and began climbing. Monay sighed, \"Thank God!\"\n\nBefore the second plane was in the air, rumors were spreading. Someone, hearing an enlisted man remark that the mattress he'd helped put aboard MacArthur's plane was heavy, started a story that it was filled with thousands of gold pesos. In a few hours there were a few men willing to swear they saw chests of drawers and even a huge refrigerator loaded on the general's plane. These groundless stories were soon to be whispered all over America by MacArthur's detractors.\n\nAt 9:00 the next morning the two big planes approached Batchelor Field, 45 miles south of Darwin, Australia. They set down with ponderous grace. MacArthur had broken through the Japanese blockade. Now all that remained was the long, uncomfortable trip across the northern wasteland of Australia. \"It was close,\" he said to those anxiously waiting on the runway. \"But that's the way it is in war. You win or lose, live or die\u2014and the difference is just an eyelash.\"\n\nMajor Morhouse got something to eat, then went to the office of the base commander. As he entered, MacArthur, wearing only long underwear, shirt and shoes, was walking up and down haranguing the base commander.\n\n\"What's the matter?\" asked Morhouse.\n\n\"They're just too damned lazy to do what I want,\" said MacArthur angrily. \"I want to take a train to Daly Waters and drive overland to Alice Springs. Mrs. MacArthur is tired of flying.\"\n\n\"General,\" said Morhouse, \"I'll do everything I can to keep Arthur alive but I'm not sure he'll survive that 800 miles of desert.\"\n\nMacArthur stopped pacing. \"Do you mean that, Doc?\"\n\n\"Yes.\"\n\n\"Get the plane ready,\" he said quietly.\n\nIt was none too soon. As the MacArthur party was loading on Major R. H. Carmichael's B-17 the red-alert sirens went off. Zero fighters were approaching the field. Carmichael took off quickly, keeping the plane at tree-top level until he was out of range of the Japanese planes.\n\nSoon the MacArthur plane was over the desert. It looked to Major Morhouse like an artist's drawing of Mars. He noticed MacArthur looking thoughtfully over the side at the desolate wilderness. Three hours later the plane touched down at Alice Springs. MacArthur put a hand on Morhouse's shoulder. \"Thank you, Doc,\" he said gratefully.\n\nHere the general was met by enthusiastic reporters. When they asked him for a statement, he quickly scribbled a few lines on the back of a used envelope.\n\n> The President of the United States ordered me to break through the Japanese lines and proceed from Corregidor to Australia for the purpose, as I understand it, of organizing the American offensive against Japan, a primary object of which is the relief of the Philippines.\n> \n> I came through and I shall return.\n\nSince his family was too sick and exhausted to go farther by plane, MacArthur now transferred to a train. Even this was by no means modern, running on narrow-gauge tracks, but the next leg of the journey was through far less arduous country. MacArthur ordered his deputy chief of staff, Brigadier General Richard Marshall, to fly ahead to Melbourne and learn the true state of affairs in Australia.\n\nNear sunset of the third day of the long trip, the MacArthur train finally reached the wide-gauge tracks where a luxurious private car was waiting. MacArthur's deputy chief of staff was also waiting with a detailed report. Marshall told the general there were only 25,364 Americans in Australia. And not one of these was an infantryman. Further, there were no tanks and many of the 250 planes were practically wrecks. But this wasn't the end of bad news. There were only about 7000 trained Australians on hand to defend their country from an invasion that could come momentarily.\n\nThe general's knees buckled as if he had been struck. His face drained of blood, his lips twitched as if in pain. \"God, have mercy on us!\" he whispered. He had left three times as many troops on Bataan alone and had assumed he was coming to command a great army.\n\nHe paced the corridor of his car all night.\n\n**4** President Quezon was still in the Philippines even though he had escaped from Corregidor by submarine almost three weeks earlier than MacArthur. The president was reluctant to leave his people. It wasn't until the night of March 18, after weeks of pleading by his own staff and the worried local American commanders, that he consented to let Commander Bulkeley take him by PT boat to the island of Mindanao, but here again Quezon dallied. Instead of driving directly to Del Monte and taking a plane for Australia, he insisted on making a visit to the picturesque Moro town of Dansalan on the shore of Lake Lanao. Finally, after another week's persuasion from his most steadying influence, Vice President Osme\u00f1a, he agreed to motor to the pineapple plantation.\n\nAt Del Monte, General Sharp was a badly worried man. MacArthur had impressed on him the importance of getting Quezon safely out of the country. The Japanese already controlled the southern part of the island and might seize Del Monte in a surprise attack at any hour. When the general learned that Quezon was finally coming, he ordered Colonel Morse to move out with two truckloads of Air Corps troops and escort the president on the last lap of his trip.\n\nOn the night of March 23, Morse was waiting in the blacked-out town of El Salvador. To his amazement he saw approaching from the west a long line of brilliantly lit-up cars. It was the Quezon party, but the president was missing. He had stayed behind to rest. Half an hour later a lone car appeared carrying the exhausted president. He swore angrily at the roads, demanding that the district engineer be dragged out of his bed so he could personally fire him. He refused to go another foot on such a miserable highway. While the even-tempered Osme\u00f1a and his staff tried to persuade him to continue, the argument grew so loud that lights went on all over El Salvador. Townspeople sleepily crowded out to the streets to look at their president.\n\nAt last Quezon consented to proceed slowly. Each bump was agonizing to the suffering man. After 20 miles the caravan reached Cagayan. Quezon again refused to go farther. He would sleep at the Bishop's palace. But Father Edralin, pastor of the cathedral, told Quezon that Bishop Hayes was away and he had no authority to allow anyone to stay at the palace.\n\n\"In that case,\" said Quezon, \"we will occupy your quarters.\"\n\nEdralin regretfully informed him church law permitted no women in his quarters. Again a loud argument broke out. Again lights popped on and citizens gathered. Finally Colonel Morse persuaded the president to continue to Del Monte. The roads, said Morse, were much better from this point on and General Sharp had prepared very comfortable quarters for the president and his family at the plantation.\n\nOn March 25, the Flying Fortresses for the Quezon party had still not arrived from Australia. That noon Morse got an excited call from General Sharp. He rushed to Sharp's headquarters near the plantation clubhouse. \"Quezon has disappeared,\" said the agitated general. \"He's taken his wife and son and daughters.\"\n\nMorse finally found the Quezon family far in the hills at the home of Mr. Crawford, manager of the plantation. Since the president refused to return to the compound, Morse sent for several truckloads of Air Corps men so a cordon could be placed around the house. The colonel spent the rest of the afternoon talking to the president. \"Everyone gets help from Roosevelt except the Philippines,\" said Quezon with asperity. \"If I can get to Australia perhaps I too can get some help.\"\n\nOn the night of March 26, three B-17's landed at Del Monte. Colonel Morse accompanied Quezon and his family to Airfield No. 1. The dying man, still reluctant to leave his people, was helped aboard a plane. To many American enlisted men it appeared as if he were being forcibly pushed into the Flying Fortress. They would have given anything for a place. This was to be Quezon's first ride in a plane and the idea of flying terrified him. In addition, he had heard alarming stories about the difficulty of breathing at high altitudes.\n\nAs with the MacArthur flight, Captain William Monay was in charge. Ironically, history repeated itself. One of the engines of Quezon's plane wouldn't start. The presidential party was transferred to another plane. To Monay's relief all four motors of this one started, but a moment later there was a sputter and then silence.\n\nColonel Elsmore and Monay ran to the plane as passengers started to disembark and learned that Quezon had ordered the pilot to cut the motors. The president couldn't find his chaplain and refused to fly until he was aboard.\n\nElsmore explained that Father Ortiz was aboard in another compartment. Quezon confirmed this and the plane finally started down the runway. As it took off and its wheels slowly retracted, Elsmore and Monay looked at each other with relief.\n\nThe volatile president lived to reach Australia but he was destined to die in America.\n\n**5** Wainwright was now on Corregidor. The War Department had made him commander-in-chief of all forces in the Philippines, promoting him to lieutenant general.\n\nWhen MacArthur learned this on March 21 he radioed Washington that he had already planned to divide the Philippines in four separate commands, giving Wainwright jurisdiction only over Bataan. MacArthur himself would control the islands from his headquarters in Australia through a deputy chief of staff on Corregidor. The general explained this plan was based on \"special problems\" because of \"the intangibles of the situation in the Philippines.\"\n\nSome of MacArthur's staff, including Sutherland, were privately more outspoken. In their opinion, Wainwright was not qualified to take over-all command of the islands.\n\nThe following day, Marshall sent his reply. Conciliatory in tone, there was no specific criticism of MacArthur's plan. Yet Marshall made it quite evident that Wainwright would retain over-all command unless MacArthur objected strenuously. MacArthur did not. Accepting the inevitable with good grace he radioed:\n\n> HEARTILY IN ACCORD WITH WAINWRIGHT'S PROMOTION TO LIEUTENANT GENERAL. HIS ASSIGNMENT TO PHILIPPINE COMMAND APPROPRIATE.\n\nIn spite of this surface harmony, command trouble was already brewing. A few days later Wainwright radioed Marshall directly that there was only enough food on Bataan to last until April 15. If supplies didn't reach him by then, he said, \"the troops there will be starved into submission.\"\n\nWhen MacArthur read a copy of this message he sent a critical comment to Marshall on April 1.\n\n> IT IS OF COURSE POSSIBLE THAT WITH MY DEPARTURE THE VIGOR OF APPLICATION OF CONSERVATION MAY HAVE BEEN RELAXED.\n\nMost of the men on Bataan would have agreed with Wainwright. They were starving and they knew it. By now they had been told that MacArthur was in Australia. Among Americans, opinion was divided. Some thought he had done the only practical thing, yet many enlisted men and junior officers felt he had abandoned them. Sarcastic poems were passed from foxhole to foxhole. One of the favorites was sung to the tune of \"The Battle Hymn of the Republic.\"\n\n> Dugout Doug's not timid, he's just cautious, not afraid,\n> \n> He's protecting carefully the stars that Franklin made.\n> \n> Four-star generals are rare as good food on Bataan.\n> \n> And his troops go starving on.\n\n> Dugout Doug is ready in his Chris-Craft for the flee\n> \n> Over bounding billows and the wildly raging sea.\n> \n> For the Japs are pounding on the gates of old Bataan.\n> \n> And his troops go starving on.\n\nWhile many Americans on Bataan deeply resented MacArthur's already widely publicized remark, \"I shall return,\" regarding it as extremely egotistic, others joked about it good-naturedly. In the Provisional Air Corps Regiment, a standard joke was, \"I'm going to the latrine. But I shall return!\"\n\nFilipinos, almost unanimously, still regarded MacArthur as the greatest man alive. \"I shall return\" to them was a personal guarantee from the American they most trusted that the Philippines would eventually be freed.\n\nBy April 1 it was obvious to everyone on Bataan that the long lull was coming to an end. General Homma had recently dropped beer cans, festooned with red and white ribbons, containing a demand for surrender.\n\nAddressed to His Excellency, Major General Jonathan Wainwright, they read:\n\n> We have the honor to address you in accordance with the humanitarian principles of \"Bushido,\" the code of the Japanese warrior....Your Excellency, you have already fought to the best of your ability. What dishonor is there in following the defenders of Hong Kong, Singapore and the Netherlands East Indies in the acceptance of honorable defeat?...The joe [sic] and happiness of those whose lives will be saved and the delight and relief of their dear ones and families would be beyond the expression of words....\n\nWainwright didn't answer. He knew heavy enemy reinforcements had landed; patrols were becoming more aggressive; skirmishes were flaring up along the outpost line; and Japanese planes, absent for a month, were appearing in increasing numbers.\n\nWainwright realized this was to be a powerful, all-out attack and his front-line strength was at a new low. He could do nothing about that. Disease and starvation were taking a fantastic toll. A thousand men a day were entering hospitals. Almost 75 per cent of the front-line troops had malaria. Perhaps even more serious, men were surly, physically exhausted and almost numb from privation. And there was no hope that any more food would reach Bataan.\n\nThe night of April 2, Good Friday Eve, was calm and peaceful along the Fil-American front. To the north 50,000 Japanese, including 15,000 fresh troops from the 4th Division and the Nagano Detachment, were massed for attack. Behind them 150 heavy guns, howitzers and mortars were ready to lay down the most devastating barrage of the campaign. Colonel Hattori's plan would be tested the next day.\n\n\"Our four groups have been brought into line and on a front of 25 kilometers ten flags are lined up,\" wrote General Homma in his operational diary that night. \"Artillery is plentiful....There is no reason why this attack should not succeed.\"\n\nOpposing Homma were about 80,000 Americans and Filipinos. But only 27,000 of these were even listed as \"combat effective.\" And of the 27,000 \"effectives\" about three-fourths were weak from malaria; all were on the edge of starvation.\n\nIn the Fil-American front lines some men were reading a poem just written by Lieutenant Henry G. Lee of the Philippine Division. Of all those writing down their thoughts the past few months, Lee's poems dug deepest.\n\n> I see no gleam of victory alluring\n> \n> No chance of splendid booty or of gain.\n> \n> If I endure\u2014I must go on enduring.\n> \n> And my reward for bearing pain\u2014is pain.\n> \n> Yet, though the thrill, the zest, the hope are gone,\n> \n> Something within me keeps me fighting on.\n\n# 19\n\n# _The Good Friday Attack_\n\n**1** The next morning dawned clear and warm. It was April 3, the opening day of the great Japanese attack. It was also a Japanese national holiday, the anniversary of the death of the first Emperor, Jimmu.\n\nGeneral Nara of the 65th Brigade looked south at a rugged jungle-covered hump called Mt. Samat. American and Filipino soldiers were stretched out between this hill and Nara. Soon the greatest artillery barrage of the campaign would rip up the two and a half miles in front of Samat. Then his soldiers, led by tanks, would attack. If lucky he would be standing on top of Samat in a week or ten days.\n\nNara's face was gaunt. Since landing on the Philippines his weight had tumbled from 150 pounds to 110. He prayed that this great offensive would be the last. General Homma had told him if all went well, they would completely crush the Filipinos and Americans in three or four weeks. If the offensive lasted longer, his already battered brigade would probably collapse.\n\nHe again carefully scanned the northern slope of Mt. Samat. This rugged part of the jungle line was held by the troops of General Vincente Lim. It was ironical. He knew Lim well and liked him. They had been classmates in 1928 at the Fort Benning Infantry School.\n\n\u2014\n\nLim's 41st Division woke up that morning little suspecting that their entire division line and a portion of the outfit to their right, Capinpin's 21st Division, was the two-and-a-half-mile front so carefully selected by Colonel Hattori in Tokyo.\n\nThe center of this stretch, where the weight of Nara's attack would come that afternoon, was held by the 42nd Infantry Regiment. Major Rigoberto Atienza, until a few days ago Lim's inspector general, had recently arrived to take over as regimental commander.\n\n\"We can expect an overcast sky this afternoon,\" he told the regimental supply officer. \"This is Good Friday. In my childhood, my grandmother used to tell me that it becomes cloudy in the afternoon in token of the world's sympathy for the sufferings of Christ. I've observed it since and found it to be so.\"\n\n\"That's superstition,\" said the supply officer.\n\nSuddenly at 9:00 A.M. a shell burst on the forward slope of the ridge only 50 yards ahead. Fragments rained down. Other shells fell thick and fast as Atienza and several of his staff huddled in a dugout. After half an hour, Atienza telephoned his three front-line battalions. \"Prepare for an attack,\" he told them, assuming this was a preparation barrage.\n\nThe Japanese artillery was only registering. At 10:00 A.M. they began firing for effect. Shells seemed to explode on top of each other. The Filipinos had never seen anything like this devastating barrage. To those Americans who had served in World War I, it seemed as intense as the heaviest German barrages.\n\nBombers of the 22nd Air Brigade roared overhead, dumping load after load on the tiny selected area. Defenses painstakingly built in the long lull were churned to rubble. The men huddled in their foxholes. It was terrifying but they were safe. Then, near noon, other planes passed, dropping clusters of sticklike objects. When they hit, dry leaves and bamboo burst into flame. Incendiaries. At first some of the men joked about the burning brush and lit cigarettes on the flaming bamboo. It was fun, a welcome change from the weeks of boredom. More incendiaries dropped. Since the rainy season was yet to start, the fires spread rapidly. Suddenly the men found they were surrounded by licking flames. The heat was intolerable, the flames leaped higher.\n\nThe men jumped from their foxholes and ran back to the second line of defense. Here their terror increased. Cover had been blasted away until most of the ground was as barren as No Man's Land of World War I. Shells again shrieked, exploding on all sides. They leaped into the bald craters. But soon, whipped by a sudden breeze, the fire leaped over the barren stretches to the lush jungle growth beyond. Hundreds were cremated. The others ran farther to the rear like frenzied animals, spurred by the smell of burning leaves, wood, clothes, leather and flesh.\n\nMasked by this confusion of smoke and flame, the Japanese infantry and tank attack started at 3:00 P.M. Nara's 65th Brigade smashed directly at Atienza's line. Already his regiment and the regiment to its right were running pellmell to the rear.\n\nLate that afternoon, Atienza prepared to move back to a safer command post. As he was throwing papers and one change of clothing into a musette bag, dazed men began streaming into the area. They were led by a captain.\n\n\"No more, sir,\" cried the captain. \"We're licked!\" His eyes were wild. His arms waved like a windmill.\n\n\"What's happened?\" asked Atienza.\n\n\"There's no more use. Our lines are gone! Our men have been roasted alive or hit directly by shells!\" His eyes were crazy with fright. His clothes were torn, burned; his hair askew.\n\n\"Shut up!\"\n\n\"No more, sir. No use, I tell you.\" Now the captain was surrounded by officers and men.\n\nAtienza knew he had to prevent a panic. He drew his .45 and pointed it at the captain. \"Shut up, goddam!\"\n\nThe other only cried, \"You can kill me if you want but we're beaten!\"\n\nAtienza slapped him. The captain reeled. Atienza drew back his hand again as the other shook his head vigorously.\n\n\"I'm sorry, sir,\" he finally said in a wavering voice.\n\n\"So am I,\" said Atienza. He took the captain by the arm, led him into a tent and gave him a cigarette.\n\n\u2014\n\nJust before dusk, General Parker, still commander of the corps on the east half of Bataan, learned that a great hole, almost three miles wide, had been torn in his lines just in front of Mt. Samat. He had only about 600 men in reserve and he ordered them to plug up the gap.\n\nHomma didn't get details of the Filipino rout until that evening. Delighted and surprised, he ordered Nara to take advantage of the sudden breakthrough and sweep west of Mt. Samat the next morning. Fresh troops from Shanghai were told to circle east of the mountain.\n\nApril 4 opened with another fierce Japanese artillery and air bombardment. By dusk Nara had cut past Mt. Samat on the west and the Shanghai men on the east. The Japanese were already more than a day ahead of schedule.\n\n\u2014\n\nWhen Wainwright went to Corregidor, he had turned over command of all forces on Bataan to Major General Edward King, Jr., a modest man, an intellectual. He was an artilleryman of wide experience; an extremely able soldier, reasonable and realistic. King was well-liked by junior officers and enlisted men for he was as courteous to a second lieutenant or private as to a fellow general. He looked and acted more like a professor than a military leader, giving orders in a quiet, undramatic way.\n\nIt was this modest man's unhappy job to try to patch the tremendous hole in the east half of the Bataan line. His first step was to turn over the most carefully hoarded unit in Bataan, the all-American 31st U.S. Infantry, to General Parker.\n\nOn the night of April 4, the eve of Easter Sunday, this well-trained American outfit moved forward through the jungles to help plug up the hole. Lieutenant Robert Taylor, the regimental chaplain, walked with the men. As the line wound up the east side of Samat in the darkness, a sergeant stopped ahead of Taylor and pointed to the sky. There was the Southern Cross, shining brilliantly. It was startling and beautiful.\n\n\"If more people in the world would march under the Cross,\" said the sergeant, \"we'd have less wars.\" Then he turned and headed north toward the quiet battleground.\n\nEaster dawned hot. While many Americans and Filipinos worshipped at dawn services, shells and bombs began to fall. Then at 10:00 A.M. an assault on Mt. Samat itself began. In less than three hours the Japanese planted the Rising Sun on top of the little mountain. At last they had a commanding view of the shattered American lines. More important, the 41st Field Artillery, which had been slowing the attack with its accurate fire, was now forced to evacuate the southern slopes of the mountain. The artillerymen rolled their guns over the cliffs, destroyed equipment and then ran to the rear. The Japanese were not far behind and by mid-afternoon had taken almost all of the strategic mountain.\n\nLess than seven air miles to the southeast, a jeep was wildly careering up the coastal road toward the town of Limay. The driver was Captain Tom Dooley. Next to him sat General Wainwright. The jeep stopped in a cloud of dust on a wooded hill. Then Wainwright climbed a trail to General Parker's II Corps headquarters.\n\nParker and his officers, hollow-eyed from lack of sleep, were poring over maps. Wainwright saw that a great wedge had been driven in the lines. Two divisions were practically destroyed. Parker's staff decided that only a strong counterattack would restore the ground. A plan was worked out. Tomorrow morning a general counterattack would be launched.\n\nWainwright approved the plan but he had misgivings. It was a plan born of desperation. The men were dead on their feet, starved and depressed. It was, however, the only chance. A few minutes later Dooley and Wainwright headed back for Cabcaben and the boat to Corregidor. Dooley drove so hard he broke an axle.\n\nThe American High Command still didn't know the Japanese had already stormed over Mt. Samat and were coming down the other side.\n\n**2** April 6 was to be the day of decision for everyone on Bataan.\n\nThe desperate Fil-American counterattack was a gallant conception. It seemed incredible that any of the men would allow themselves to be led back into what was obviously a hopeless battle; yet in the first minutes of April 6, units were actually moving out to attack, if not with enthusiasm, at least with a grim determination to do what had to be done.\n\nThe counterattack west of Mt. Samat soon collapsed. The result was utter confusion. By midday this entire left half of Parker's corps had disintegrated. Nothing remained to stop Nara from sweeping south to the end of the peninsula except the rugged terrain itself.\n\nThe counterattack east of the little mountain started under even worse auspices. Unknown to Parker the left flank of the troops scheduled to make this attack, Capinpin's division, had been crushed the night before and the general himself captured. An hour before dawn three exhausted officers from Capinpin's division staggered into the lines of the unit to their right, the 31st U.S. Infantry Regiment. The three Filipinos told Lieutenant Colonel Jasper Brady, the commander of the American unit, that the 21st Division had completely disintegrated.\n\nBrady had only 800 men. Realizing that even if his attack was successful he didn't have enough troops to hold the entire rear line of a division, he tried to call Sector headquarters by phone. The line was dead. He called the unit on his right, the 31st Division of General Clifford Bluemel. Telling Bluemel his troops were still at the line of departure, he asked the general to confirm his action.\n\n\"Like hell,\" was Bluemel's tart answer. \"Why didn't you attack as ordered?\"\n\nBrady explained.\n\n\"The preparatory and supporting fire by my artillery is coming as scheduled,\" said Bluemel. \"And my 51st Combat Team is ready to accompany your advance.\"\n\nBrady again asked Bluemel to confirm his action.\n\n\"Not by a damned sight. Don't pass the buck to me,\" retorted Bluemel. \"I advise you to launch your counterattack immediately. If you refuse to attack, I'm going to report your action to your Sector commander.\"\n\n\"Sir, I've tried but I can't raise General Lough's headquarters.\"\n\n\"Then I advise you to report your failure to counterattack to General Parker.\" Bluemel hung up. He had his own problems. He had asked Parker twice the day before to let his outflanked troops fall back to the banks of the San Vicente River. Twice Parker had refused.\n\nLate that morning Bluemel finally got a line through to II Corps. He told Parker of Brady's failure to attack. (Neither general knew that by this time Brady had finally contacted his own Sector and been ordered to cancel the counterattack and hold his present position.)\n\nAgain Bluemel asked if he could fall back to the southern bank of the San Vicente and dig in. The answer was no. Instead he would form a line on the second ridge in front of the river. Bluemel hung up and swore. This had been Brady's line of departure for the abortive counterattack. It was a poor position with no field of fire; there had been no reconnaissance or preparatory work. It was typical of Corps, he thought, to give orders without having personal knowledge of the terrain.\n\nBluemel set up his line, personally going from position to position. At 4:00 P.M. the peppery general got a call from Corps ordering him to form a new line on the south side of the San Vicente River. Bluemel was justifiably irate. It was too late to lay out any preparatory work. It would soon be dark. Several officers were rushed back to the river to make a hurried reconnaissance. Finally Bluemel started his troops to the rear, patrolling the columns to see that order was maintained.\n\nAt 5:00 P.M., while making a reconnaissance south of Trail 2, he found a small group of disorganized, demoralized soldiers from the 31st U.S. Infantry. He talked to them, explaining the new mission, and placed them in the position their regiment was supposed to occupy. Then he started back toward his command post. Half a mile south of the San Vicente River he met bedraggled men from his own division, moving farther to the rear. Gesturing belligerently with his rifle, Bluemel shooed them to the front. As he drove them up the trail like cattle, other retreating men were met. They just stared dumbly when Bluemel asked what outfit they were from. When he called for officers, no one moved forward. Pointing with his rifle toward the enemy, he ordered them all to turn around and join the movement back to the north.\n\nBy midnight, after frantic efforts by Bluemel, his signal officer, Captain Matt Dobrinic, and several others, a ragged but recognizable line was formed on the southern bank of the San Vicente. At last the general lay down to rest.\n\n\u2014\n\nColonel Takushiro Hattori, the planner of the offensive, had been watching the day's action from the heights of Mt. Samat. West of the mountain he saw Nara's troops streaming through, bypassing scattered American units, pushing relentlessly to the south. On the other side of the mountain he saw the main attack of the day, the assault of the Shanghai troops on General Bluemel's assortment of defenders. The Fil-Americans here were falling back across the San Vicente River. It was a sensational victory already with everything working exactly as he had imagined. It was a parade.\n\nThe day of decision had turned into a day of disaster for the Americans. By nightfall Homma's men had punched in the entire left half of Parker's Corps and were threatening to drive the east half into Manila Bay. Only the hurried line set up by Bluemel on the San Vicente stood between Homma and victory.\n\n**3** April 7 was another hot, clear day.\n\nJapanese bombers ranged the American rear that morning. Lieutenant Hattie Brantley, a nurse at General Hospital No. 1, watched planes circling over the great white cross of sheets that marked the hodgepodge hospital area. It was located in a field at the top of the steep, zigzag road near the southern end of Bataan in a settlement vaguely reminiscent of the former summer capital and thus called Little Baguio. The white cross didn't reassure her. A week ago a bomb had actually hit the sheets.\n\nAbout 10:00 A.M. there was a weird whistling noise. A bomb landed on an ammunition truck on the road in front of the hospital. There was a frightening roar, then the spatter of shrapnel, pebbles and earth on the tin roofs.\n\nIn the Orthopedic Ward nurses and corpsmen quickly cut traction ropes so the wounded men could roll out of bed. Father Cummins went to the middle of the room and raised his arms. He asked everyone to repeat the Lord's Prayer.\n\nA moment later other bombs hit the mess area, and the doctors' and nurses' quarters. There was a banshee shriek of tearing tin and wood. Iron beds doubled, breaking jaggedly like matchsticks. The patients began screaming.\n\nNear by, Nurse Brantley heard another bomb fall. This one was even closer, shaking the earth and raising a cloud of dust and smoke. Hearing the scream of patients, she realized a ward set up in a large open space and roofed by burlap had been hit directly; dozens must be dead, more dying. She ran for help into the Orthopedic Ward. Here open panic was threatening to break out. Then she saw Father Cummins climb onto a nurse's desk. Even in the roar of planes his voice could be heard in prayer. The patients quieted. A feeling of comfort came over Lieutenant Brantley. It was now out of her hands, she realized. She began to cry. Others were crying too.\n\nA moment later, the priest climbed down from the desk. \"Somebody take over,\" he said quietly. \"I'm wounded.\"\n\n\u2014\n\nUp front the entire left half of Parker's front was gone. But the right, the east half, though weak and shaky, still held. General Bluemel only slept an hour after setting up the San Vicente line. Then with two Filipino staff officers, he headed back to the front for another inspection. Less than a mile from the river he met large groups of men from his own division shuffling to the rear. They told him they were going to Lamao.\n\n\"There's no food in Lamao,\" said Bluemel. The general and the two Filipino officers herded the men back to their positions. After looking over the positions of the all-American unit, Bluemel walked back to his command post and slept an hour. When he awoke, he once more headed back north to the line. And once again he met retreating men of the 51st Division.\n\n\"Why did you desert?\" he asked. No one answered. He waved them back to the river. They turned and disconsolately trudged to battle.\n\nA truck came from the front at high speed. Bluemel stood in the middle of the trail, signaled it to halt. Without slowing it roared toward the general. He jumped back just in time. A minute later a truck column approached. Again Bluemel signaled the leading truck to stop. It slowed and an American of the 31st U.S. Infantry leaned out and shouted, \"The San Vicente Line has broken!\"\n\nBluemel never thought he'd see American soldiers running like that. Soon great crowds of Filipinos surged toward him. He gestured with his rifle. \"Form a line on both sides of the trail.\" Shells began to explode along the road. The men, shouting in terror, ran past Bluemel. He grabbed at them, but no one paid any attention to the irate general.\n\nBluemel walked back to his command post, ordered it moved to the rear and dispatched an officer to Parker's Corps with the bad news. He hopped in a jeep and returned north toward the disintegrating San Vicente Line. If he hurried perhaps he could rally some of the units and put up some kind of resistance. Trail 2 was filled with demoralized, sick, retreating men. Most had thrown away their arms and equipment. Shells again hit the trail and Bluemel parked the jeep, heading forward on foot. Soon he found a fairly intact battalion and told its commander to place it along a low ridge. He hurried toward the ridge to select the actual positions. As soon as Bluemel finished putting the battalion in place, shells began to fall once more. When the smoke cleared Bluemel discovered his newly placed battalion had vanished. It was the last known unit of his own 31st Division. The general remembered how courageously they had stopped Nara at the other end of Trail 2 two months before. Now through lack of medicine, food and clothing they had become an uncontrollable mob. More than half were sick with malaria and dysentery. He realized he could expect no more from the shattered Philippine Army units.\n\nBluemel helped load wounded on shelter halves. They were carried to the last division bus and driven off. His jeep and driver had disappeared. He was alone. Except for a few men of his staff he was all that remained of the 31st Division.\n\nRifle fire broke out to the right rear. Japanese were obviously converging from two directions. There was nothing else to be done at the San Vicente River. He turned and walked down Trail 2, following the rest of the retreating mob.\n\nHe met more of his staff officers. Something had to be done to prevent a complete rout. He told them to quickly reorganize the nearby Americans of the 31st U.S. Infantry Regiment and have them move through the underbrush in a column of files a few yards from each side of the trail. They were to guide on him as he moved slowly down the trail. This would prevent the column from attracting planes.\n\nIt was done and the movement to the rear continued in good order until an open space in the trail was reached. Here one battalion of Americans hurried across the open spot and proceeded down the trail. The next battalion followed, and before Bluemel could catch up to them, both battalions cut to the left through underbrush, and disappeared into an overgrown stream bed.\n\nBluemel angrily sent his Filipino staff officers to the stream. They called over and over again, \"31st Infantry!\" No American answered.\n\nNow only Bluemel and four Filipino officers were left to stop the oncoming Japanese on Trail 2. The angry general hurried south with Japanese patrols only a few hundred yards behind. Soon he saw ten American enlisted men of the 31st U.S. Infantry coming out of the brush. Bluemel collared them, ordering them to beat the brush on both sides of the trail. After 500 yards they flushed an American major.\n\n\"I'm going to the junction of Trails 2 and 10,\" Bluemel told the embarrassed major.\n\n\"Don't,\" the other warned. \"It's open country and can be bombed. Go across country through the underbrush and avoid the junction.\" Bluemel pulled out his map and began to examine it. When he looked up the major was gone. So were the ten enlisted men.\n\nBluemel and his four Filipinos continued toward the junction in spite of the major's advice. Soon they saw a scout car approaching from the south. It stopped. \"Do you know where I can find General Bluemel?\" called a cavalry officer.\n\n\"Right here,\" said the general.\n\nThe newcomer said he was Major William Chandler of the 26th Cavalry. \"Colonel Vance is waiting at the junction of Trails 2 and 10 with the 26th Cavalry. We have orders to report to you, sir.\"\n\nThis was the first good news Bluemel had heard in two days. \"Tell Vance to stay where he is.\" He motioned Chandler to get moving. \"We'll follow on foot.\"\n\nBluemel quickened his pace down the trail. He crossed a stream. There, to his amazement, he found the two missing battalions of the 31st U.S. Infantry and the ten enlisted men. He told them in terse, direct language to report immediately to the trail junction.\n\nBefore Bluemel reached the junction, rifle shots broke out behind him. The Japanese had caught up and now occupied the ridge just behind Bluemel's retreating troops. Late that afternoon Japanese attacked the junction in force. Bluemel knew his few hundred men couldn't hold long. He looked at the map. About a mile to the rear was the Mamala River. This would have to be the next delaying point.\n\nThe trail to the Mamala was jammed with masses of Filipinos and Americans and trucks of the 26th Cavalry. Suddenly shells exploded and Japanese dive bombers swooped down. One bomb hit an ammunition truck. The exploding ammunition momentarily blocked the retreat. As Bluemel hugged the dirt, he thought his ears would burst from the explosions. Then came an even louder roar. Fragments flew past Bluemel killing several men. Something walloped his leg.\n\n\"Are you hurt, General?\" asked Vance.\n\nBluemel pulled up his trouser leg. \"No blood. I'm all right.\"\n\nThe withdrawal became a panic. He thought, What do you do with a mob?\n\n\u2014\n\nThat afternoon Wainwright, on Corregidor, telephoned King at his Bataan headquarters near Hospital No. 1. The two desperately tried to figure some way to stop the Japanese breakthrough in the east half of Bataan.\n\nWainwright pointed out that the Fil-American front lines on the west half of the peninsula were completely intact. In fact there was still almost no activity at all on this side of Bataan. Why, he suggested, shouldn't these relatively fresh troops turn right and attack to the east? Perhaps they could even form a new line all the way across Bataan.\n\nKing was reluctant but Wainwright could see no other solution. He ordered the desperate plan carried out. King, in turn, telephoned Albert Jones, the new commander of Wainwright's former corps, now recovered from dysentery and promoted to major general.\n\nThe outspoken Jones promptly and bluntly told King that the attack ordered by Wainwright was senseless. It could not possibly be launched. His men were so weak from hunger and sickness, they couldn't even climb the steep cliffs of the Pantingan River, let alone fight their way up. Besides, said Jones, how could he possibly mount such an attack in time? A three-way telephone connection was then set up and the discussion continued among Jones, King and Wainwright. Finally Wainwright said with some exasperation that he was leaving the decision up to King and hung up.\n\nIt didn't take the Bataan commander long to agree with Jones. King ordered Jones to pull back his corps in four phases so he wouldn't be outflanked by the breakthrough on the other side of the peninsula.\n\nLate that afternoon King asked his chief of staff, Brigadier General Arnold Funk, to take a boat to Corregidor and tell Wainwright that surrender might come any moment. A few hours later Funk, a big man, wearily entered Wainwright's office in Malinta Tunnel. He told of the physical deterioration of the men, of the collapse of Parker's entire left wing, then said, \"General King sent me here to tell you he might have to surrender.\"\n\nFunk's face, thought Wainwright, was a map of the hopelessness of the Bataan situation. He knew as well as King and Funk of the suffering of the men of Bataan; he knew they were on the verge of starvation. But on his desk was a radiogram MacArthur had sent three days before.\n\n> I AM UTTERLY OPPOSED UNDER ANY CIRCUMSTANCES OR CONDITIONS TO THE ULTIMATE CAPITULATION OF THIS COMMAND. IF FOOD FAILS, YOU WILL PREPARE AND EXECUTE AN ATTACK UPON THE ENEMY.\n\nHe also had orders from Roosevelt forbidding surrender \"so long as there remains any possibility of resistance.\" Wainwright studied the exhausted Funk, then said in his slow drawl, \"General, you go back and tell General King he will _not_ surrender. Tell him he will attack. Those are my orders.\"\n\nFunk hesitated. Among fellow officers he was well known as capable, outspoken. There were tears in his eyes when he said, \"General, you know, of course, what the situation is over there. You know what the outcome will be.\"\n\n\"I do.\"\n\nFunk slowly, wearily turned and walked out.\n\n**4** The only general anywhere near the front lines in Parker's crumbling corps that April seventh was Clifford Bluemel. Assuming command of all troops in the area, he ordered a general retreat starting at 9:00 P.M. to the Alangan River, which was only seven air miles from the end of Bataan. He wrote an order to Colonel Vance, still holding along the Mamala River: the 26th was to cover the entire movement, pulling back as soon as the last of Bluemel's varied units departed.\n\nAt 9:00 P.M. the mass retirement began. It was so dark Bluemel ordered that each man hang onto the shirt of the soldier ahead. \"Move slowly,\" he said. \"Fifty yards at a time, then stop.\"\n\nIt was a slow, punishing march. Most of the men hadn't eaten for two days. The sick and wounded, unable to keep up, dropped by the side of the trail, groaning in agony, yet knowing they had to be left behind. Occasionally rifle shots burst out from the north and men pushed frenziedly against those ahead, trying to make them go faster.\n\nBy dawn, Bluemel had set up a defense line along the Alangan River for several miles. On paper it was a formidable force but actually he had only about 1400 men. On his right there was a big gap of 1500 yards and the defense line didn't resume until it reached the east coast highway. Here were 1200 assorted troops commanded by Colonel John Irwin, supported by the 21st Field Artillery, a few fixed naval guns, and the last three 155-mm. guns of the 301st Field Artillery.\n\nBluemel was dead tired, as was everyone else on the line. In addition he felt the weight of the entire defense on his shoulders. If he relaxed even momentarily it might collapse. He finally made telephone contact with Parker's headquarters and told II Corps' operations officer that there was a gaping hole to the right. He urged the immediate dispatch of Beach Defense units to fill in the hole.\n\n\"Troops will be sent up after dark.\"\n\n\"That'll be too late. You can be damned sure the enemy will find that big gap before dark and rush through.\" He was told of Irwin's 1200 men guarding the East Road to his right. The entire line was now under Bluemel's command, said Parker's staff officer. Bluemel was all that was left of II Corps. What would he need to assist him? \"I need four staff officers, communication personnel and equipment from II Corps.\"\n\n\"Use your own staff.\"\n\n\"I have only two Filipino staff officers. Like me they've been without food since breakfast of 6 April and have had about one and a half hours' sleep a night. I have no communication personnel or equipment. The 31st Signal Company disappeared.\"\n\n\"Well, you'll have to take staff officers from the troops with you. Use their communication equipment and personnel.\"\n\nBluemel was disgusted. There were at least thirty American officers on the II Corps staff who'd been eating and sleeping regularly, yet not a damned one could be spared for the fighting line. He said, \"Since you're not sending me any troops until after dark to fill the gap, I can't possibly hold the line. What line do you want me to hold next?\" His tone became sharper. \"This time I'd like to send out officers from every unit to make a daylight reconnaissance.\"\n\nThe answer was brusque. \"Hold your present line.\"\n\n\"Damn it, I told you it was impossible. Not unless the gap is filled immediately.\"\n\n\"Hold _that_ line.\"\n\nBluemel angrily banged down the phone and went out to inspect his Alangan River line in daylight. The south bank was a steep slope covered with cogon grass almost three feet high, with patches of underbrush and a scattering of trees. When Bluemel saw Japanese observation planes buzzing overhead, he guessed his men had been seen digging new positions. He again telephoned Parker's headquarters stressing the importance of filling the 1500-yard gap to the right. He repeated his request for staff officers and communication equipment. Nothing was available.\n\nAt 11:00 A.M. planes suddenly swooped down on the Alangan dropping incendiary bombs. The dry cogon grass and bamboo thickets burst into flames. Bluemel's men jumped from their foxholes, beat out the flames, and resumed their positions.\n\nA few miles to the right, other bombers hit Colonel Irwin's men on the East Road just as they were digging foxholes. The men fled and had to be rounded up by Irwin's officers. Another wave of bombers appeared. Again the starved, exhausted men ran to the rear. Again they were brought back to the front, some at pistol point. More attacks followed. Each time fewer men returned to the front lines. By about 3:00 P.M., though the Nagano Detachment had not yet even appeared, Irwin's line was completely deserted.\n\nAt the same time, the troops from Shanghai were already probing defenses on the Alangan River. As the general had predicted, they soon found the great gap and began filtering behind Bluemel's lines.\n\nWhile Bluemel was standing on Trail 20, a bus approached from the rear, filled with men of the 2nd Platoon, Company G, 4th Philippine Constabulary.\n\n\"We are to report to General Bluemel,\" said a sergeant.\n\n\"I'm General Bluemel.\"\n\nThe men in the bus were silent. \"Well, who the hell is in command?\" asked Bluemel. Finally the sergeant said he was and the men started reluctantly filing out with their barracks bags.\n\n\"You won't need those,\" said Bluemel.\n\nAt that moment, Lieutenant Colonel Smyth of the Provisional Tank Group arrived. \"Can you use any tanks?\" he asked Bluemel.\n\nBluemel cheered up. After telling the new platoon to form up and wait for him, he took Smyth forward to locate places for the tanks.\n\nAs the two officers were returning to the bus, small arms broke out from the front and the right. The Japanese began to close in. The 2nd Platoon, which had just debussed and formed a line, broke as one man for the rear. Bluemel chased them but they were too fast. He returned to Smyth.\n\n\"I'll send three tanks immediately,\" said the tanker and left.\n\nThe fight continued until dusk. No reinforcements arrived as the Japanese began to encircle the 26th Cavalry. An American officer reported from the front that the 31st U.S. Infantry had now fallen back and the Philippine Scouts of the 57th, enveloped on both flanks, were also having to withdraw.\n\nAt that moment five Japanese tanks rumbled down Trail 20 toward the embattled 26th Cavalry. The cavalrymen and the Philippine engineers of the 14th Battalion didn't panic, keeping up such heavy fire that those in the Japanese tanks didn't dare come out to remove the overturned trucks forming a roadblock.\n\nThough the tanks were momentarily stopped, Bluemel knew he had to pull back the rest of his people quickly or be completely surrounded. He ordered the stubborn engineers to withdraw. The 26th was to cover them and then follow.\n\nIt was pitch-dark as Bluemel and his staff, now mostly cavalrymen, started toward the rear. The general headed down Trail 20, hoping to find other units he could use in a new defense. Just after 9:00 P.M. the exhausted little group reached a stream. Here the general found the 57th Infantry and the all-American unit. He was told General Parker wanted to talk to him on the phone.\n\nIt was Parker's operations officer. \"Form a line on the Lamao River.\"\n\n\"Where the hell is the Lamao River?\"\n\n\"It's the stream near your phone.\"\n\nBluemel was not too tired to be angry. \"Why in the devil didn't you designate this position this morning as I asked? I could have had a daylight reconnaissance. Now it's too damned dark. None of my people know a thing about this terrain. Why in hell didn't your staff of thirty make a reconnaissance for us? Why aren't they here to guide my troops into position or at least give us some advice?\"\n\nThere was a pause. \"It was not known how you desired to put the troops in.\"\n\n\"I'm willing to put them in any damn way as long as they get in. The least you can do is send me four of your staff officers and help me get my people into position.\"\n\n\"I'm sorry, we can't spare anyone.\"\n\nBluemel blew up. \"We can't hold a line here! It's an impossible situation you've given me. I'm nearly all in and so are my officers and men. And I've told you before we haven't eaten for almost three days. If you want any fighting up here you'd better send me 1600 'C' rations and some small arms ammo.\" He hung up, disgusted. His muscles ached as he slowly, wearily walked to the little river. He took off his shoes and stuck his feet in the cool water. It was a wonderful feeling. He leaned over, scooped up water with his hands and bathed his dirt-encrusted face.\n\n# 20\n\n# _\"I Would Rather Die a Thousand Deaths\"_\n\n**1** That night the Japanese were steadily pressing down the eastern half of Bataan. A few hours earlier Colonel Takeo Imai, one of Nara's regimental commanders, had planted a large flag on top of Mount Limay. Now from the same heights he was looking south to the end of Bataan at occasional flashes of light. The enemy was probably blowing up equipment and munitions in desperation. Beyond he could even make out the dark outline of Corregidor Island. Occasionally angry spits of fire came from its heights where giant cannon were trying to stop the advance by interdicting the east road.\n\nFleeing in front of the relentless Japanese columns, Americans and Filipinos poured out of the jungles in hundreds of places and were being jammed into the toe of the peninsula. They came by trail, across the rugged valleys, by the main road. No one knew what was going on in the mass confusion and desperation. A strange dull terror drove many of the men. They were numb, exhausted, terrified and yet listless. Many didn't realize they were on the edge of starvation and delirium. Many didn't know they had malaria or dengue. Those with dysentery, of course, knew it. The humiliation of their affliction had long passed, in the general agony.\n\nSergeant Frank Bernacki and twenty-eight men of the Provisional Air Regiment were guarding the four bridges between Lamao and Cabcaben on the east road. Men suddenly streamed across the first bridge, kept going south. They were Americans, Philippine Scouts, Philippine Army men. All were frantic.\n\n\"All hell's broken loose!\" an American sergeant shouted to Bernacki. \"The lines have broken just down the road! Get out!\"\n\nBernacki hesitated. This meant the Japanese advance guard was rolling down the east coast highway toward him. He finally got an officer on the phone. \"What should I do?\" he asked.\n\n\"Grab your sack and take off.\"\n\n\"Shouldn't we fight?\"\n\n\"Hell, no. Save your tail. I'm going to try and get to Corregidor.\" Bernacki gathered his twenty-eight men. They drove several buses onto the road, tipped them over. Then with the help of a dozen civilians, they built roadblocks with logs. Men fleeing from the front continued to come through but though occasionally he could hear the chatter of a machine gun, no Japanese appeared.\n\nBernacki put his men in a truck and drove south to Cabcaben. Here the road turned due west. He followed it past Hospital No. 2 and up hills to Hospital No. 1, sitting on the right of the road at Little Baguio. The road became more and more jammed as it suddenly dipped down a harrowing zigzag. Finally Bernacki reached the last town on Bataan, Mariveles. It was a madhouse. A few boats were preparing to take the privileged to Corregidor. Other vessels were being towed out in the bay and sunk. Great mobs of disorganized troops from a dozen units clustered at the sides of the road.\n\nBernacki saw a brigadier general halting the path to the docks. \"No one can go to Corregidor,\" he shouted. \"We've had it,\" he told Bernacki wearily. \"We're waiting here for the Japs to kill us or capture us. If you've got guns, tear them apart.\"\n\nOut of the bay, the diesel tug _Manapala_ was towing the submarine tender _Canopus_ to its final rest. The battered old ship's valves were opened and _Manapala_ headed back toward shore. Many felt a personal loss as they watched _Canopus_ slowly sink. The \"Old Lady\" had been one of the bright spots in the Bataan campaign. Here a man could enjoy a real shower bath and a drink of cold water and eat a good meal on white linen.\n\nThe remnants of Lim's division were heading south on foot, by truck and bus. Major Atienza was lucky. He was in a bus. Near Mariveles it reached the division's final assembly point. Near a clump of three mango trees he and his companion, Captain Manuel Tinio, spread blankets. The two lay down to sleep, but Atienza soon awakened. He felt a chill. Thinking it was the night wind, he buttoned his shirt collar and put on his khaki cap. He grew more chilled.\n\nTinio awoke and felt the major's brow. \"You're burning with fever, sir,\" he said and wrapped his towel around Atienza.\n\nThe major began to shake. He knew he had malaria. Fires were burning down below in Mariveles, reddening the sky. He could hear the dull thuds of explosions. Suddenly the earth began to shake violently.\n\nIt was a severe earthquake, felt all over the southern half of Bataan. Some in their dazed condition thought this was the end of the world.\n\nAt the Cabcaben airstrip, the earthquake was rocking the last American plane on Bataan just as its motor was being warmed up.\n\n\"Quit shaking this plane,\" the pilot, Lieutenant Roland Barnick, shouted to the five passengers huddled on the bare floor behind him.\n\nThe plane, a navy amphibian, had been hauled from the bottom of Mariveles Harbor several months previously. Since then Captain Joe Moore, a former P-40 pilot, had used it to fly in supplies, medicine and mail. Since he always included several boxes of candy on each trip, it became known as \"The Candy Clipper.\"\n\nThe rickety Grumman Duck creaked as it waddled down the bomb-pocked strip. Finally it lifted heavily and a moment later was skimming a few feet above Manila Bay.\n\nBarnick turned and shouted at Carlos Romulo, who had been broadcasting the \"Voice of Freedom\" from Corregidor until Wainwright ordered him to leave. \"Last man out of Bataan, eh, Colonel?\"\n\nJapanese searchlights now picked up the lumbering plane and bullets began ripping through the wings.\n\n\"Our own guns are shelling us,\" shouted one of the crew, looking out a small window at the disappearing outline of Bataan.\n\nBarnick raised the ship to 70 feet. When it refused to go any higher, he hurriedly penciled a note and handed it back to the passengers. They instantly began throwing overboard all baggage, steel helmets, sidearms and lastly parachutes. \"The Candy Clipper\" rose another 50 feet and headed for the south.\n\n**2** On Corregidor, the distraught Wainwright was getting only disjointed scraps of information, but he knew from the chaotic reports that his eastern corps was collapsing. He remembered MacArthur's definite instructions to launch a last desperate attack. At 11:30 P.M. he telephoned General King.\n\nThough he must have known his orders could not be followed, Wainwright said, \"Launch an offensive with Jones' I Corps northward toward Olangapo.\" This, he explained, would take the pressure off Parker's disintegrating corps on the east.\n\nWainwright now radioed MacArthur in Australia. The troops on Bataan, he said, were fast folding up and the weakened men no longer had the power of resistance.\n\nNear the tip end of Bataan at his headquarters near Hospital No. 1, General King was mulling over Wainwright's recent order. It placed King in an impossible situation. If he obeyed, the resultant attack would end in a suicidal slaughter. The red-haired general called Jones. \"I have just received orders to launch an immediate attack with your Corps.\"\n\nJones said he was in the midst of a withdrawal to the Binuangan River as ordered. \"Any attack is ridiculous, out of the question. My men are too weak.\"\n\nKing didn't argue. He told Jones to forget the attack and hung up. It was now time to make the final ugly decision. The situation was chaotic and helpless. Any more fighting would lead to needless killing. He called for his chief of staff, General Funk, and his operations officer, Colonel James Collier.\n\nThe three men conferred at midnight in the narrow, flimsy headquarters building. They reviewed the tactical situation, considering all possible lines of action. At last King asked the key question: \"Will I be able to stop the Japanese from reaching the high ground above Mariveles?\"\n\nThe other two shook their heads. No matter what resistance was put up, they said, the Japanese would reach Mariveles by the next evening.\n\nKing hesitated. Wainwright was hamstrung by MacArthur's explicit order to attack until the end and Roosevelt's \"no surrender\" message. It was up to King to make the decision on his own, with the full knowledge that he was disobeying orders.\n\n\"I have decided to surrender Bataan,\" he said. If he ever got back to the States, he was sure he'd be court-martialed, but the lives of his 78,000 soldiers were far more important.\n\nIt was not a surprising decision yet it hit Collier \"with a terrible wallop.\" Tears came to his eyes. He saw the other two men were also crying.\n\n\"I have not communicated with General Wainwright,\" said King, \"because I do not want him to be compelled to assume any part of the responsibility.\"\n\nNow began the bitter mechanics of surrender. Two bachelors, Colonel E. C. Williams and Major Marshall Hurt, volunteered to drive to the Japanese lines as emissaries. Unit commanders were warned that surrender would take place at 6:00 the next morning. Parker heard the news almost at once for his headquarters was practically next door to King's. He telephoned Bluemel.\n\n\"If the Lamao River line can be occupied,\" Parker said, \"it'll only be necessary to hold it until morning.\"\n\n\"What's going to happen then?\" asked Bluemel.\n\n\"A car carrying a white flag will go through the lines on the east road at daylight.\"\n\n\"Do you mean we're surrendering?\"\n\n\"Yes. There must be no firing after the car passes through the lines. Did you get the ammunition and rations?\"\n\n\"I got the ammo. No rations.\"\n\n\"I'll check personally.\"\n\nThe nurses at both hospitals were being warned to pack. They would leave in half an hour for Corregidor. At Hospital No. 1, in Little Baguio, the nurses didn't want to leave their patients.\n\n\"The patients will follow you,\" said their commandant, chubby Colonel Duckworth. \"And so will the doctors. I'll see you all in San Francisco soon.\"\n\nThe two dozen nurses piled into a bus and soon were heading down the dangerous zigzag to Mariveles. Traffic was growing thicker and they seemed to creep. Dust filled the bus, irritating their throats. On all sides were glares of fire and rumbles of explosions. It reminded Lieutenant Brantley of Dante's Inferno. Strangely she didn't feel a part of the hell around her. She was numbed.\n\nThe nurses at Hospital No. 2, located several miles east of Little Baguio, also protested. What would happen to the patients? In surgery alone there was a long line of dying men waiting for operations. The doctors had to force the nurses to leave. Lieutenant Lucy Wilson had another reason for wanting to stay. The next day she was supposed to marry Lieutenant Dan Jopling of the 200th Coast Artillery. She and three other nurses got into a garbage truck and it slowly started west toward Hospital No. 1 on the way to Mariveles. It stalled. A half-track pushed until it started again. As they approached Little Baguio and the other hospital, there was a tremendous roar. The earth shook. Rubble rained on all sides. The sky lit up fantastically. Other explosions followed. The line of traffic stopped. Lieutenant Wilson was dazed. There were flames all over. The whole world seemed to be on fire.\n\nThe TNT warehouses were being destroyed. The sky was fantastic with bursting shells, colored lights and pillars of rainbows as several million dollars' worth of explosives and ammunition erupted.\n\nGeneral King, in his headquarters not far away, had just received a telephone call from General Jones.\n\n\"For crying out loud, Ned,\" said Jones, \"what's going on?\"\n\n\"The ammunition dump is blowing up,\" said King calmly. He didn't bother to say that the roof had just literally been blown off overhead.\n\n\"Hell, I can feel the ground shaking all the way up here. It must be an earthquake.\"\n\n\"I hate to tell you this, Honus, but I'm surrendering at 6:00 A.M. They're shelling the hospital. Parker's Corps is gone. There's nothing else I can do. Put white flags all along your line. You'll have to destroy your artillery and machine guns and stand by for further orders.\"\n\n\"I don't see what else you can do,\" agreed Jones. \"I'll spike the artillery, but I'm saving the machine guns. At the last minute we can throw the bolts in the jungle.\"\n\n\"Use your judgment, Honus,\" said King in a hopeless voice.\n\nJones wasn't the only one who thought the great explosion at 2:00 on the morning of April 9 was another earthquake. Many on southern Bataan could feel the earth rocking sickly.\n\nClifford Bluemel felt the tremor. He was so tired he hoped the earth would open and swallow everyone and end the chaotic mess. His phone rang. It was Brigadier General Lewis Beebe from Corregidor who said, \"I'm trying to get in touch with King and Parker.\" The explosion had temporarily severed all wire communications in the Little Baguio area. Bluemel could hear Wainwright shouting in the background. Skinny, he knew, rarely used the phone because of partial deafness.\n\nThe providential call gave Bluemel a chance to pour out his troubles. He said he'd asked for staff officers and got none. He also explained how impossible it would be to hold the Lamao River line. After a slight pause he heard Wainwright shout, \"Tell Bluemel to use his own judgment and take whatever action he deems best. Whatever he does will be approved by General Wainwright.\"\n\nWainwright was visibly concerned about the counterattack he had ordered King to make; King hadn't yet reported what was going on. When the conversation with Bluemel ended, Beebe again tried in vain to make contact with King. Wainwright told him to try Jones directly. Jones answered the phone and told Beebe that the order to attack had never been given by King.\n\n\"Well, stand by,\" said Beebe. \"You'll probably receive instructions to attack at any minute.\"\n\nNot long after this, King learned about the call to Jones and realized he would have to face the situation squarely. At just 3:00 A.M. he called Corregidor. Beebe again answered for Wainwright.\n\n\"I want a definite answer as to whether or not General Jones will be left in my command _regardless_ of what action I may take,\" said King. Beebe passed on this question to Wainwright.\n\n\"Tell him he's still in command of all forces on Bataan,\" answered the distraught Wainwright. For some reason he never asked King about the attack Jones was supposed to make.\n\nAnd King still did not tell his superior he was going to surrender the next morning, that two emissaries were now almost ready to leave for the front.\n\nThree hours later King called Corregidor and at last revealed what he had done. Lieutenant Colonel Jesse T. Traywick, Jr., night duty officer at Corregidor, hurried to Wainwright's tunnel quarters.\n\n\"Sir,\" he said, \"General King is going to surrender!\"\n\nWainwright was speechless.\n\n\"He's sent an officer to the Japanese to arrange terms.\"\n\n\"Go back and tell him not to do it!\" While Traywick hurried back to the office, Wainwright waited anxiously.\n\nTraywick returned, his face sober. \"It's too late,\" he said.\n\n\"They can't do it! They can't do it!\" Then Wainwright got hold of himself and went to his office, where he sent a message to MacArthur.\n\n> AT 6 O'CLOCK THIS MORNING GENERAL KING...WITHOUT MY KNOWLEDGE OR APPROVAL SENT A FLAG OF TRUCE TO THE JAPANESE COMMANDER. THE MINUTE I HEARD OF IT I DISAPPROVED OF HIS ACTION AND DIRECTED THAT THERE WOULD BE NO SURRENDER. I WAS INFORMED IT WAS TOO LATE TO MAKE ANY CHANGE, THAT THE ACTION HAD ALREADY BEEN TAKEN.\n\nBy dawn many assorted small boats were approaching Corregidor from the smoldering, defeated peninsula. In all about 2000 were escaping. The nurses from Hospital No. 1 had already landed and were anxiously waiting for their friends from Hospital No. 2. What had happened to them?\n\nDelayed by the spectacular explosions at the ammunition dump, most of these nurses were just loading into a motor boat at Mariveles, but Lieutenant Lucy Wilson and three others were still stranded. Their garbage truck wouldn't move and they had been forced to walk several miles to a beach. Here, a passing interisland boat had just seen them and pulled in to the shore. Soon they, too, were on their way to Corregidor. Lieutenant Wilson watched Bataan disappear. This was the day she was to have married Dan Jopling and she didn't even know if he was still alive.\n\n\u2014\n\nKing's chief emissary, Colonel Williams, was then passing through the American front lines. His jeep proceeded north on the coastal road displaying part of a white sheet attached to a bamboo pole. In Williams' pocket was a typewritten, signed letter of instructions from King ordering him to try and arrange a meeting between Homma and King. If this was not possible, Williams himself was empowered to surrender all the Bataan forces.\n\nA patrol of about thirty Japanese suddenly rushed on the road, bayonets flashing. The jeep stopped. The Japanese hesitated, then started to rush the car with fixed bayonets. Their commander, a noncom, abruptly changed his mind and ordered the others to withdraw. He approached the jeep with a hand grenade. As he was about to pull the pin, a Japanese lieutenant burst out of the jungle and ordered him to stop. After a hurried conference, the lieutenant and half a dozen men climbed onto the jeep and Williams' driver was told by sign language to head north toward Lamao.\n\nSoon the two Americans were brought to Major General Kameichiro Nagano, commander of the detachment attacking down the east coast highway.\n\n\"I've been sent forward by General King to contact General Homma to discuss surrender terms,\" Williams told a Japanese interpreter.\n\nNagano agreed to arrange a meeting between King and Homma at Lamao at 11 A.M.\n\n\"I'll report back to General King and tell him of the arrangements,\" said Williams.\n\nNagano shook his head. Williams was to be held prisoner. The driver would take back the message. After his driver left with oral instructions, Williams became more worried about the letter from King authorizing him to surrender Bataan. He kept his hand in his pocket and surreptitiously began tearing it into fragments. The first time he was alone he was determined to eat the pieces.\n\nA little after 9:00 A.M. the stocky King, wearing his last clean uniform, headed up front in a jeep. With him were his two aides, Majors Achille Tisdelle and Wade Cothran. One hundred and fifty yards ahead another jeep carried Major Hurt and Colonel Collier. Though both cars displayed sheets tied to bamboo poles, several planes swooped down, bombing and strafing. The jeeps stopped and the Americans flung themselves in a ditch. The surrender team started again. Just before reaching Lamao another plane dove, but at the last moment its pilot saw the white flags. Instead of firing, he wagged his wings. The Americans continued 100 yards to Lamao bridge. Here they saw Japanese infantry drawn up in columns of twos on the other side of the bridge.\n\nAs Japanese guides escorted the Americans to the Experimental Farm Station, King recalled Lee had surrendered to Grant at Appomatox on that same day. He even remembered what Lee had said, just before the ceremony: \"Then there is nothing to do but to go and see General Grant, and I would rather die a thousand deaths.\"\n\nAt last he knew exactly how Lee had felt.\n\nHe saw a Japanese general sitting at a long table with Colonel Williams, in front of a small building. It was Nagano. The Japanese commander motioned King to sit down, explaining through an enlisted interpreter who knew little English that he wasn't authorized to conclude surrender arrangements. A representative from General Homma would soon arrive.\n\nIn a few minutes a shiny Cadillac drew up. Tisdelle recognized it as belonging to his friend, Juan Elizalde. Colonel Motoo Nakayama, Homma's senior operations officer, stepped out with his own interpreter. King rose to greet the newcomer but Nakayama ignored him and sat at the head of the table. King again sat down, hands in front, back erect. Tisdelle had never seen his general look more like a soldier.\n\nThe new interpreter spoke to Nakayama, then turned to King and said in a German accent, \"You are General Wainwright?\"\n\n\"No. I am General King, commander of all forces on Bataan.\"\n\nPuzzled, the interpreter conferred with Nakayama, who was also puzzled, and finally said, \"You must go and get General Wainwright.\"\n\n\"I don't represent General Wainwright, only myself. I have no means of getting in touch with him.\"\n\nThere was another conference. \"For what purpose have you come?\"\n\n\"To secure terms for the force in Bataan.\"\n\nNakayama was confused and displeased. \"You'll have to get General Wainwright,\" said the interpreter. \"The Japanese cannot accept surrender without him.\"\n\nPatiently King repeated he represented only those on Bataan and could not communicate with Wainwright. He knew the lives of many depended on his words and manner. \"My forces are no longer fighting units. I want to stop further bloodshed.\" He asked that he be permitted to return to headquarters so he could send couriers to all his units. He also requested an armistice of twelve hours and permission to evacuate his men as prisoners of war in his own trucks and cars to their prison camp.\n\nNakayama spoke sharply to his interpreter. \"Surrender must be unconditional,\" said the interpreter.\n\nKing kept his temper. \"Would you consider these provisions? Give me permission to notify forward elements. It would save lives on both sides if the Japanese stayed in position. The movement of prisoners of war would be facilitated by using our trucks and cars.\"\n\n\"Surrender must be unconditional.\"\n\n\"Will our troops be well treated?\"\n\n\"We are not barbarians. Will you surrender unconditionally?\"\n\nKing knew further delay would mean more unnecessary deaths. He nodded.\n\n\"Please hand over your sabre,\" said the interpreter.\n\n\"I left it behind in Manila.\"\n\nAfter a brief, concerned conference the interpreter said a pistol would be acceptable. King placed his pistol on the table. So did the other Americans.\n\nNakayama rose, got into the Cadillac and was driven off. The American officers were taken in their jeeps another 17 miles farther north to Balanga, the capital of Bataan. Behind the sprawling Balanga Elementary School, King was led to some chairs and tables under a large tree. Japanese newsmen snapped many pictures of the captured general. Then Colonel Nakayama began to ask questions.\n\nHow many Japanese prisoners of war did King have? When King replied only about sixty, the Japanese were surprised there were that many. Talk switched to Corregidor. King insisted he didn't know how many troops were there. Or how many guns.\n\n\"Where are the guns placed?\"\n\n\"I don't know.\"\n\n\"Why does a general officer have so little information?\"\n\n\"I may give you any information you ask concerning my former command, but I am unable to give you any information concerning the defenses of Corregidor.\"\n\nNakayama pressed the point. \"Does this mean you don't know anything about Corregidor or refuse to answer?\"\n\n\"It means that I will not give any information about the defenses of Corregidor.\" King calmly crossed his leg, drew out a cigarette and lit it.\n\nThe Japanese interpreter leaned over the table, knocked the cigarette out of King's hand and yanked his leg down. \"Sit at attention,\" he ordered. King did.\n\nA black and white map of the lower tip of Bataan and Corregidor was laid on the table.\n\n\"Show where the tunnel leads from Mariveles to Corregidor,\" asked a Japanese staff officer.\n\nKing was amused. \"There's no such tunnel.\"\n\n\"There must be.\" King shook his head. \"Well, I am positive you have caverns near Mariveles where are stored all the large reserves of artillery.\" King truthfully denied their existence. \"There are such caverns,\" insisted the Japanese officer, pointing to the cliffs on the eastern side of Mariveles. \"Do not lie. Your artillery has been destroyed many times yet you kept bringing out additional artillery.\" King again shook his head.\n\nThe Americans were brought to a little house across the road. Two privates standing at the door as guards grinned. One came forward and offered King a pack of cigarettes. Half an hour later an amiable officer took off his cap, bowed and introduced himself as Colonel J. Takasaki. \"Ah so, now for you the war is over and we are all friends.\"\n\nKing nodded politely. \"Yes, for us the war is over but we are not friends. We are your prisoners.\"\n\n\"Oh, no. We are friends. Have you eaten?\" When the colonel learned they hadn't, he spoke to an orderly. In a minute the man returned with heated evaporated milk. Warm San Miguel beer and more cigarettes were also brought in.\n\nTakasaki rose. \"Shortly I will come back and you and you,\" he said indicating King and Williams, \"will come with me. I am going to take you for a ride.\" He smiled pleasantly and went out.\n\n**3** Though almost everyone else in II Corps was waiting to be captured, Bluemel was still fighting. He had pulled back his dwindling forces from the Lamao River line. Now, at noon, he was somewhere near Trail 20. Though lost, he did know the Japanese were on two sides. He ordered his people to deploy and hold off the attackers.\n\nSeveral worried and annoyed American officers approached Bluemel. \"General,\" said one, \"the High Command surrendered at daylight. Here it is noon and you're starting a fight. Everyone else has surrendered but you.\"\n\nBluemel told them they could do what they wished. He was going to keep fighting. A moment later one of the officers pointed at several white flags flying over Bluemel's own lines. The general looked at them, benumbed from exhaustion. His uniform was ripped. He wore only one star. The other had been torn off his shoulder. He had done everything he could. He threw his rifle on the ground and waited for the Japanese to come.\n\nMen were waiting all over the southern end of Bataan in huddled groups. Many were weeping unashamedly, Filipinos and Americans alike. Tears came mostly from the humiliation of surrender but partly from the relief of thinking that the torture was finally over.\n\nSergeant James Madden and 300 other American airmen were sitting at the side of the road halfway up the steep zigzag going up to Little Baguio. Their commander told them all sick men should get into a parked six-by-six truck. Madden helped a friend, whose ankles were painfully swollen from beriberi, into the truck just as Japanese soldiers appeared. Shouting and swinging their rifle butts, they chased everyone from the six-by-six.\n\nMadden's commander, somewhat shaken, told his men to line up in columns of twos. When the group started up toward Little Baguio, the Japanese shouted angrily and made them head back toward Mariveles. The airmen were puzzled. Rumors flew. They were going to be shipped to Japan. They walked down the steep zigzag road, finally reaching the airfield near Mariveles. By now there were about 3000 captives packed in the dusty field.\n\nThey sat in the hot sun, puzzled, fearful and confused. After a long wait Japanese soldiers approached and began stripping the captives of blankets, watches, jewelry, razor blades, mess equipment, food and even toothbrushes. When the looting was over, the prisoners were herded onto the road. Four and five abreast they started once more up toward Little Baguio.\n\nMen of the 200th Coast Artillery were also being looted in another part of Mariveles. A Japanese enlisted man took an amethyst ring from the man Lucy Wilson was supposed to marry that day, Lieutenant Dan Jopling. Another looter took a ring from the man standing next to Jopling just as a Japanese officer passed. The officer grabbed the ring from the soldier, saw it bore the University of Notre Dame insignia.\n\n\"Whose is it?\" he asked.\n\nThe Notre Dame man, a former football player named Tonelli, said it was his. The officer smashed the enlisted man in the face with his fist and politely handed the ring back to Tonelli. \"What year did you graduate?\" he asked.\n\n\"1935.\"\n\n\"I graduated from Southern California in '35,\" said the Japanese amiably. He noticed the white mark on Jopling's finger. \"Which one took it?\" Jopling pointed out the looter. The officer walked over, struck the man several times and returned with the amethyst ring. \"If you have anything of personal value,\" he advised, \"hide it.\"\n\nGroups were being rounded up all over the tip end of the peninsula. Lieutenant Colonel Allen Stowell was still at the communication center of Parker's headquarters watching a puzzled Japanese officer examine the switchboards. The Japanese rang every line, got no answers, and finally said to Stowell, \"Code, code.\"\n\nWhen Stowell said nothing an enlisted man approached, held a pistol to the American's head. Stowell scratched his head. \"Code? Oh, yes.\" He wrote out the International Code.\n\nThis partially satisfied the Japanese officer. \"More, more,\" he insisted.\n\nStowell pulled out an obsolete code book, Division Field Code No. 4. The Japanese officer was so delighted he ordered the enlisted man to lower the pistol.\n\nAbout five miles farther east, Lieutenant Tony Aquino of Capinpin's division was sitting on a tree trunk near the Cabcaben airstrip. He still had his yellow convertible Buick, his birthday gift. Now it had no fenders or mudguards, was splotched with green paint; its top was ripped off, headlights smashed. It reminded Aquino of a skull with the eyes gouged out, only the sockets remaining.\n\nTo the south he heard dull metallic clanking, then a burst of machine-gun fire. Tanks! He and the others braced themselves. They were unarmed, their weapons stacked or thrown away. Without warning Zeros dove, roaring down at treetop level. Amazingly, no one moved for cover. The men waved their white flags.\n\nThis was a mistake. The last plane made a steep climb, turned back and dove, its guns blazing. Aquino jumped behind a tree trunk. He pulled out his rosary and began to pray automatically, \"Oh, my God, I am heartily sorry....\"\n\nThe plane left. Several bodies littered the ground. Now Aquino again heard the roar of the approaching tanks. Dots of dust sprang upon the road. He saw they were machine-gun bullets.\n\n\"Let's get the hell out of here,\" someone shouted. Aquino flung himself down the decline at the side of the road. He jumped wildly across a 12-foot-wide stream. His heart pounding, he felt like a hunted deer. He wanted to do something but his mind was blank with fear.\n\nA loudspeaker blared out, \"Hiripino sordiars! Come hya and surrenda. We are youar frengs. Japang wirr nota kirr you. Come hya!\"\n\nAquino saw a friend come out of a thicket waving a white flag and expected to hear gunfire, but there were no shots. The Filipinos were like children, at first fearful of a stranger with candy. Seeing there was no danger, they trustingly approached. Aquino still held back, suspecting the Japanese were waiting for a large number to appear before opening fire with the tanks.\n\n\"Hiripino sordiars,\" shouted a Japanese officer standing in the turret of a tank, \"Japan come to riverate you from the Americang oppressors. Go home now. Manira go.\"\n\nAt the word \"home,\" everyone crowded around the tank. A Filipino field grade officer stepped forward, saluted the Japanese officer as a token of surrender. The Japanese saluted and everyone knew all was over. The Filipinos cheered wildly. The Japanese officer smiled and waved his arms. The Filipinos waved back. That was it. The game of war was over. They jumped with joy and embraced each other.\n\nAquino and his friends piled into the battered Buick and started east toward Cabcaben. Jubilantly they made plans for the future. Aquino said everyone was welcome to spend the rest of their lives at his hacienda. They passed another group of Japanese and waved happily, expecting another friendly welcome, but a sentry shouted at them to stop and advanced with a rifle. The Japanese motioned them to get out of the car, pointing to the nearby Cabcaben airstrip, already jammed with prisoners. Aquino took a last sad look at the Buick, and marched with the others toward the field. His fear was returning.\n\n**4** On the west half of Bataan, in Wainwright's former corps, the front lines were still intact, but General Jones had spiked his big guns and placed white flags all along his front. Even so he had ordered his men to return any Japanese fire. They were to keep fighting in spite of the white flags, until definite arrangements were made with a Japanese surrender team.\n\nIt was now dark and Jones hadn't been able to reach the Rock by phone or radio all day. Communications with King at the tip end of Bataan were also out. He had no idea what he was supposed to do. An American general from one of his units ran wild-eyed into the command post crying, \"All is lost!\" Jones put the general to bed and sent General Fidel Segundo to take over the leaderless men.\n\n\u2014\n\nOn Corregidor, Wainwright still had no idea what was going on in Bataan. He was somewhat cheered by a message from Roosevelt giving him belated permission to surrender.\n\n> AM KEENLY AWARE OF THE TREMENDOUS DIFFICULTIES UNDER WHICH YOU ARE WAGING YOUR GREAT BATTLE. THE PHYSICAL EXHAUSTION OF YOUR TROOPS OBVIOUSLY PRECLUDES THE POSSIBILITY OF A MAJOR COUNTERATTACK UNLESS OUR EFFORTS TO RUSH FOOD TO YOU SHOULD QUICKLY PROVE SUCCESSFUL. BECAUSE OF THE STATE [OVER] WHICH YOUR FORCES HAVE NO CONTROL I AM MODIFYING MY ORDERS TO YOU....\n> \n> MY PURPOSE IS TO LEAVE TO YOUR BEST JUDGMENT ANY DECISIONS AFFECTING THE FUTURE OF THE BATAAN GARRISON....I FEEL IT PROPER AND NECESSARY THAT YOU SHOULD BE ASSURED OF COMPLETE FREEDOM OF ACTION AND OF MY FULL CONFIDENCE IN THE WISDOM OF WHATEVER DECISION YOU MAY BE FORCED TO MAKE.\n\nWainwright radioed Roosevelt that since all communications with Bataan had been cut he had no knowledge of the terms arranged by General King. But he concluded:\n\n> OUR FLAG STILL FLIES ON THIS BELEAGUERED ISLAND FORTRESS.\n\nIn Melbourne, MacArthur was reading a prepared statement to reporters: \"The Bataan Force went out as it would have wished, fighting to the end its flickering, forlorn hope. No army has done so much with so little, and nothing became it more than its last hour of trial and agony. To the weeping mothers of its dead, I can only say that the sacrifice and halo of Jesus of Nazareth has descended upon their sons, and that God will take them unto himself.\"\n\nThe fighting, except for flurries of defense on the western half, in Jones' area, was all over on Bataan, but it was not to be the end of suffering for the 76,000 sick and starving Fil-Americans.\n\n# 21\n\n# _The Death March_\n\n**1** King's surrender of over 76,000 men including 12,000 Americans, was the greatest capitulation in U.S. military history. Except for the patients in Hospitals No. 1 and 2, all these men were now starting or about to start on the long trip to their prison camp at Camp O'Donnell, a few miles north of Clark Field. The first part of this odyssey, the 55-mile stretch between Mariveles at the tip end of Bataan and the sugar center of San Fernando, would soon be recorded in the minds of Americans and Filipinos as one of the major atrocities of the war. It would be called the Death March.\n\nMany of the 70,000 participants began the march at the airstrip near Mariveles on the morning of April 10. Even though groups of prisoners were leaving at intervals, the airfield kept getting more congested, for thousands of other Americans and Filipinos were streaming into Mariveles from another direction, the road coming down the west coast.\n\nAdding to the confusion were 26,000 civilians, crowded like sheep into the tip end of the peninsula since early January. Japanese guards soon gave up all hope of organization. Their officers had told them it would be a simple operation. They were to round up about 25,000 prisoners and bring them back by foot to Balanga. There, trucks would be waiting to take them to their prison camp. The guards' orders had been hurried and brief, for King's surrender had come as a surprise to the high command. Homma had figured the final attack would take about a month.\n\nHomma had assigned the job of handling prisoners to Major General Yoshikata Kawane, his transportation officer. Kawane had presented his plans to Homma ten days before the attack began. It was divided in two phases. Colonel Toshimitsu Takatsu was responsible for the first, assembling the prisoners at Balanga. No transportation was planned since the farthest point, Mariveles, was only 19 miles away, a day's march to Japanese soldiers. It wouldn't even be necessary to issue food to the prisoners since they could use their own rations. Kawane told Homma he would personally supervise the second phase: the trip from Balanga to the prison camp. He could spare only 200 trucks but these would be enough to shuttle the prisoners the 36 miles to San Fernando. Freight trains would then take the men to Capas, a little town about 13 miles north of Clark Field. From there they would be marched 8 miles to their new home, Camp O'Donnell.\n\nThe feeding arrangements were no problem, said Kawane. The prisoners would eat the same rations as Japanese troops. They would be fed not only at Balanga and San Fernando but at two points between, Orani and Lubao.\n\n\"What about the sick and wounded?\" Homma had asked.\n\nKawane said Major Hisashi Sekiguchi was setting up two field hospitals, one at Balanga and the other at San Fernando. If the staff and equipment they were momentarily expecting from Japan arrived, a third hospital would be set up in between. At any rate medical units, aid stations and \"resting places\" would be established every few miles.\n\nHomma approved.\n\nUnfortunately the plan was based on several fallacies. Homma knew Wainwright's men were low on food but he had no idea they were on the point of starvation. He also was ignorant of the defenders' alarming disease toll since the malaria rate in south Bataan was many times greater than in the Japanese lines, only a few miles north. He also miscalculated how his own men would react as guards of this defeated horde. But his greatest miscalculation was numerical. Homma's staff had assured him there were only 25,000 or 35,000 Fil-Americans in Bataan.\n\n\u2014\n\nIn Mariveles civilians and soldiers were separated into two columns.\n\nA Japanese officer was shouting reassurances. \"The papers lie about prisoners being shot. The Imperial Japanese Army will take good care of you. You will get good food and treatment. At Balanga you will be picked up by trucks. We will abide by the Geneva Pact.\"\n\nGroups of 300 were started off. Some had no guards. Other groups had as many as four. Some merely had one guard leading the way on bicycle.\n\nSergeant Frank Bernacki's group was halted near the airstrip in front of a small building flying an American flag. \"Face the flag,\" shouted a Japanese officer. The group, mostly U.S. airmen, turned and watched the Stars and Stripes lowered and the Rising Sun raised. Then the march continued.\n\nNot far behind, Colonel Ray O'Day, instructor of General Capinpin's division, had already thrown away his bed roll, and was carrying only a musette bag. Soon Japanese from the north began passing. These were rougher than their captors, looting as they passed. Two saw O'Day's gold Scottish Rite ring and motioned for him to hand it over.\n\n\"Wife, wife,\" he said, but they insisted. He had to suck his finger till the ring slid off. A few minutes later he was robbed of his personal papers and half a dozen ration cans. When O'Day protested, the looter threw him back five cans but kept the letters and pictures.\n\nAnother Japanese, seeing this, looked at O'Day sympathetically and said, \"So sorry, so sorry,\" and passed by.\n\nThe line of prisoners wound up the zigzag road toward Little Baguio. The ditches were littered with burned trucks, battered self-propelled mounts, rifles and abandoned equipment. O'Day stopped to rest. An American soldier plodded by and said, \"How you making it, Pop?\" For the first time he felt old.\n\nNow they passed Hospital No. 1, sitting at the top of the hill. Over it flew the Rising Sun. Guards were patrolling the area, keeping other Japanese from molesting or looting the patients. A tank commander, finding that Colonel Duckworth had given medical treatment to forty-two Japanese prisoners, had kept his promise to treat the hospital staff with consideration.\n\nThe columns continued, passing King's former headquarters, then the side road to Hospital No. 2. Here there was no protective and grateful commander. It was already commandeered by a Major Sekiguchi. When the hospital commander requested more food for his staff and patients, Sekiguchi had said, \"Your own forces didn't feed you adequately and I don't choose to feed you any more than you've been receiving.\" Several truckloads of food and medicine were at that moment being driven away.\n\nAmerican doctors rushed to Sekiguchi, protesting this mass looting. \"Do you have receipts for the food and medicine?\" he asked. When they answered in the negative he said, \"Ah, so, since you don't have a receipt, it is quite obvious the Japanese Army didn't take it.\"\n\nA small group of Filipino doctors and a detachment of unwounded Philippine Scouts began to file out of the hospital area, up the road. They had been ordered to join the march to the north. Immediately a rumor spread around the sprawling wards under the trees: the Japanese were freeing all Filipinos.\n\nLieutenant Colonel Jack Schwartz, chief of surgery, hurried from ward to ward begging the wounded Filipinos to stay, telling them it was a hoax. Japanese guards, evidently eager to rid themselves of responsibility, began encouraging the patients to start up the trail. Some guards actually ordered patients to leave, removing casts from the wounded, pushing them out of the area. Five hundred Filipinos, too weary to get out of bed, stayed, but about 5000 others, infected by mass hysteria, hurried down the dusty trail. Amputees chopped off limbs from trees for crutches and hobbled away, their dressings unravelling. Men with critical belly wounds staggered toward freedom. Soon the main highway was clogged with the maimed and wounded. Within a mile their hysteria dissipated and they realized they had been misled, but it was too late to return. Before long the ditches were lined with dead and dying.\n\nThe march continued east toward Cabcaben. Sergeant James Madden was in one of the first groups, having left Mariveles the night before. His throat ached from dust. Near the town his group saw a stream and rushed for it. The guards let the prisoners drink and wash themselves and their clothes. An hour went by and still they weren't herded back to the road. A feeling of gaiety passed from man to man. They talked of the war and agreed it would all be over in a year at the most. Abruptly they were shocked to silence by a series of roars. Shells from big Japanese guns emplaced just behind them were flying over their heads toward Corregidor. Puffs of smoke appeared on that rock island as the shells exploded. The gaiety was over.\n\nA few minutes later Madden heard a sound like milk trucks\u2014hundreds of bottles rattling in wire cases. These were mortar shells from Corregidor. Madden and the others dropped, seeking cover as explosions tore up the ground.\n\nAmerican shells were also landing around Hospital No. 2. While Colonel Schwartz tried to calm his remaining patients, there was a tremendous explosion. Dust, foliage, trees flew into the air. Several shells had landed directly on or near Ward 14. Schwartz headed through the cloud of dust toward a chorus of terrified screams and moans.\n\nAt Cabcaben guards hastily reorganized the marching groups. Here the road turned north and headed straight up the east coast of Bataan. Ordinarily this was a scene of beauty. On the left was towering Mt. Bataan, the peaks of its time-eroded crater usually wreathed in cool-looking clouds. On the right were the blue-green waters of Manila Bay. Everywhere was lush green tropical growth: banana trees, each topped by its purple phallus; occasional cocoanut trees bent in modest grace; nipa palms whose long leaves swayed slightly like beckoning fingers.\n\nNow there was no beauty. The colorful foliage was covered with a heavy coat of chalk. The road itself was a continuous, swirling dust cloud as an almost steady line of Japanese artillery, tanks and trucks came down the peninsula from the opposite direction. Many of the infantrymen in trucks jeered at the prisoners. Some began a cruel game, swinging at the marching men with long bamboo poles, knocking off their helmets and hats. Only occasionally a Japanese, usually an officer, would stop the sport and apologize to the conquered. Sergeant Leon Wolf, of the 14th Ordnance, saw a Japanese officer run up to an American tank commander and hug him. They had been classmates at UCLA. They chatted a moment, then went off in different directions.\n\nAnother motorized column of Japanese appeared. Somewhere its occupants had found golf clubs and with shouts of laughter began swinging them at the hapless marchers. With each mile treatment grew more brutal as Japanese combat troops thinned out and rear-echelon units took their place. Sun and dust made the marchers' thirst almost unbearable. Lieutenant Tony Aquino of Capinpin's division made a sign to one of his guards, cupping the palm of his hand.\n\nThe guard grunted and made a motion like slapping a fly from his nose. Aquino knew it meant \"No.\" In the middle of the afternoon, the group rested near an irrigation ditch. At last the guard motioned permission to drink. The Filipinos scrambled along the ditch ducking heads into the running water.\n\nWhen Aquino had drunk his fill he looked up to see the guards laughing, and pointing at the ditch a few yards upstream. A moment later the Filipino saw what had amused them. In the stream lay a naked corpse, intestines spilling out, face swollen, covered with maggots. Aquino vomited three times.\n\nOn this same dust stretch some Americans and Filipinos were hit by gun butts, wrenches and clubs when they tried to drink from ditches, streams or even the footprints of carabao. Yet some few lucky ones were actually stopped by Japanese foot soldiers going in the opposite direction and given water from their canteens. There appeared to be no rhyme, reason or consistency to the actions of the Japanese. Only one thing was clear to the marchers. Things were steadily growing worse.\n\n**2** As the men of Parker's corps headed slowly up the east coast of Bataan, the great majority of Jones' men on the other side of the peninsula were still waiting in their jungle bivouacs or foxholes. It wasn't until dark that a Japanese car displaying a white flag approached Jones' headquarters.\n\nJones issued orders for his men to stack arms. He felt relieved, as if a tremendous load had been lifted from his shoulders. At 9:00 the next morning Jones and his chief of staff, Colonel William Maher, were driven south, back toward Mariveles. He was brought to Lieutenant General Susumu Morioka, his old antagonist of the 16th Division. They conversed pleasantly for several minutes.\n\n\"Who's going to win the war?\" asked Morioka.\n\n\"We are, of course.\"\n\nMorioka smiled. _\"You_ won't.\"\n\n\"Not me,\" said Jones. \"But I have four boys at home who will.\"\n\nMorioka laughed.\n\nJones' front-line troops were now heading over jungle trails and on the cobblestone road bisecting Bataan toward the east coast town of Balanga. Those in the rear were following their commander south toward Mariveles down the west coast road. Among these were Brigadier General Luther Stevens, commander of the Left Subsector, I Philippine Corps, and his chief of staff, Colonel Edwin Aldridge. Great numbers of Japanese troops were streaming toward them from Mariveles.\n\nTheir staff car was stopped by a shouting Japanese officer. When Aldridge opened the door, the Japanese kicked his right leg. Then he kicked the American in the face and yanked him out of the car. As the dazed Aldridge turned to get his possessions he was booted in the rear.\n\nAt that moment a truck carrying Colonel Don Bonnett and his officers of the 71st Philippine Regiment drove up and stopped. The Japanese officer seized a long bamboo pole and began to swing viciously at the Americans, splitting open Colonel Bonnett's head. Everyone was shoved into the truck. About 10 miles from Mariveles, the vehicle was stopped and the twenty-two American officers led up a side road. They were told to sit on the ground. They were tied together, two by two. Soon a Japanese captain who spoke some English noticed the group. He released Stevens and his aide; but as Stevens was untying his officers, another officer ordered Stevens and his partner retied.\n\nIn the meantime the main march up the other side of Bataan on the east road was growing. Most groups made the march from Mariveles to Balanga in two or three days. Attempts at bringing order out of chaos failed and with each mile the guards became more confused and irritated\u2014and, consequently, more brutal. As the starving, disease-ridden prisoners staggered slowly up the dirt road in the blistering sun toward the capital of Bataan, guards urged them to hurry so the schedule could be maintained.\n\nA young American soldier next to Sergeant Floyd Grow was panting desperately, unable to keep up the pace. Finally he fell out of ranks, shouting wildly, \"I want him to shoot me!\" The guard pushed him back into line. Again the youngster fell back. Again the guard pushed him in place. The American stumbled, fell and couldn't get up. The guard shot the young soldier.\n\nGrow passed an American sitting in a culvert, his hands extending toward him in supplication. He started for him, then saw his countryman was stiff with rigor mortis.\n\nThere was little shade for the marchers on the long stretches between towns. Chalk from the road, stirred by the marchers, clung to the men's sweating bodies, stung their eyes, turned their damp beards to a dirty white. When Japanese motor columns rumbled down the center of the road, clouds of dust swirled blindingly.\n\nThe countryside was a bleak, terrifying desert. Thousands of bombs and shells had stripped the rolling hills of trees and foliage, leaving scarred, blackened stumps. Dirty clouds of smoke still rose to the left where woods were smoldering from the cataclysmic Good Friday bombardment. The afternoon heat was deadly and men threw themselves, when guards permitted, into each filthy stream.\n\nThe first group stumbled into Balanga at 2:00 P.M., April 11. Then at irregular intervals, other groups of soldiers and civilians approached the cool-looking Talisay River on the outskirts of the city. Some guards let their prisoners rush to the water, drink, bathe and rest on the banks. Others stolidly drove their men past, even though they piteously cried for water.\n\nOnce in the suburbs, there was shade at last from the sun. Palm and banana trees formed a green comforting arch. Fantastically beautiful tropical flowers\u2014reds, purples, blues\u2014brightened the dust-covered roadway.\n\nMany of Jones' men were also pouring into Balanga from the cross-peninsula cobblestone road and the capital of Bataan had already become a confused mass of shouting Japanese guards and milling prisoners. No one knew what was going on or who was in charge. Orders were given, then countermanded or forgotten.\n\nHere ended Phase One: a dismal failure, completely disorganized, uncoordinated, inadequately supervised, with an alarming lack of discipline. Now Colonel Takatsu's responsibility ended. General Kawane's began.\n\nA semblance of organization began to appear. For the first time in three days an attempt was made to feed the prisoners. Even so confusion continued and in some ways became more intolerable for the marchers. Some groups were fed a handful of rice and salt as they left Balanga as well as given water and marched in good order with proper rests. Others missed this first meal, were shooed away from every artesian well and marched either too fast or too slowly. A few guards were kind; more were indifferent or cruelly careless; others were outrightly sadistic.\n\nRank made little difference in the treatment at Balanga. While Colonel Harrison Browne, chief of staff of the Philippine Division, was being looted, he heard a groan. Turning, he saw his commander, Brigadier General Maxon Lough, on his knees. A Japanese lieutenant was viciously swinging a four-foot hardwood club at the general's head.\n\nBrowne and Captain Joseph Sallee, Lough's aide, helped the general to his feet and dragged him under a mango tree. An American doctor patched Lough's wound and Browne gave his commander a drink of whisky from a shampoo bottle.\n\nGeneral Jones was driven to the Balanga schoolhouse about this time. He wasn't beaten up but late on that afternoon of May 11 he was placed at the head of a column marching to the next point, Orani, about 11 miles due north. Jones protested. He wanted his men to ride. The guards paid no attention to him. The transportation situation was already going to pieces. Prisoners were swamping Balanga and it was obvious there would be more than twice the 25,000 expected plus thousands of civilians. There were less than 200 trucks for the job and many of these were being repaired. In fact in all the 14th Army that day there were only 230 trucks in running order.\n\nScores of U.S. trucks and cars were being picked up every hour but most of these were transporting Japanese soldiers to the tip end of Bataan in preparation for the amphibious invasion of Corregidor.\n\nAnd so, for the first time in history, numbers of American generals were walking toward a prison camp. General Jones led his column. Ahead of him a guard was pedalling a bicycle. After about 200 yards the guard would stop, motion Jones to hurry up his men and pedal another 200 yards. The pace was painfully fast.\n\n\"Slow down,\" shouted someone from the rear. \"We can't walk that fast. Men are dropping out.\"\n\nJones slowed the march. The guard pedalled angrily back and motioned the general to hurry. It was pitch-dark by the time they reached Abucay. In a few miles they passed through the burned-out village of Mabatang, its charred ruins still giving off a faint acrid smell. Many instinctively looked to the left at the rising hills toward rugged Mount Natib. Here on the Abucay Line they had received their first real baptism of battle.\n\nJones' party didn't reach Orani until after midnight. The general was held aside while the other prisoners were pushed into a rice paddy enclosed by barbed wire. Finally Jones was led to a corner in the crowded enclosure. Men were packed so tightly they had to draw up their knees. The stink was overpowering. Near Jones was an open latrine, crawling with maggots. The whole area was covered with feces. It was, he thought, another Andersonville.\n\nThere was water but no food. An American enlisted man gave the general a handful of uncooked rice. \"Keep several grains on your tongue,\" he advised.\n\nThe next morning Jones and the others were fed lugao, rice mush. It was sticky and tasted like paste but no one left a particle. They were lined up in the scorching sun and driven like cattle toward the next main station, Lubao, 16 miles away. This march was much harder than the previous night's for it was under the blazing tropical sun. Men dropped out, falling into the ditches. Every time the group passed through a village, the marchers called, \"Water!\"\n\nSometimes civilians would bring out cans of water from the open artesian wells, but the guards shoved them aside, spilling the precious water to the road. Water became an obsession.\n\nJones heard someone calling his name from a truck as it drew alongside. A Japanese officer motioned the general to get into the truck. Jones and his chief of staff, Colonel Maher, hopped into the back. After a mile the truck drew up to a nipa hut, brilliant with purple bougainvillaea. Inside were Generals King and Brougher and several high-ranking Filipinos. Guards led King and Jones to the rear where photographers were waiting.\n\n\"Take off your sunglasses and helmet,\" ordered an officer.\n\nJones looked at the Japanese levelly. He was tired of following their orders and said bluntly, \"No.\" He expected a beating, but the Japanese merely shrugged his shoulders and told the photographers to snap their pictures.\n\nOther groups following Jones into Orani were not so lucky, the enclosure growing more and more filthy. Half of the men had dysentery. So far there had been only isolated cases of brutality but as the foot traffic piled up, it became more general.\n\nSergeant Floyd Grow's chief guard was lining up his men at the side of the road, not far from Orani. \"You Americans are soft,\" he said. \"I was born and raised in San Francisco, I know. Take off your clothes. I'm going to give you the sun treatment.\" The men obeyed. \"Now stand at attention.\" The men stood in the blistering sun, sweat rolling down their naked bodies.\n\nThe roadside began to be dotted with the dead. They soon became swollen to monstrous size by the heat. Crows tore open these cadavers with their beaks, often fighting each other for the nauseating meat while buzzing hordes of fat, green flies covered the remnants. Most of these men had died of disease, hunger or exhaustion. Some had been murdered.\n\nLieutenant Tony Aquino's group had been marching for hours without rest or water. Their legs were swollen, their eyes burned. An American ahead of Aquino fell, the sharp stones of the road cutting his face. A guard ran up, kicked him in the ribs to make him get up. The American slowly rose on all fours, then dropped exhausted. The guard kicked harder. The prisoner\u2014blood dripping from his mouth, face covered with bruises\u2014again tried to get up. He extended his right hand to the Japanese in a pleading gesture. The guard slowly, deliberately placed the tip of his bayonet on the side of the American's neck, and quickly jabbed.\n\nFor a moment, the American squatted in the road suspended like a fly on a pin but when the bayonet was suddenly yanked free, he fell inertly. It was a picture Aquino would never be able to erase from his memory. The Japanese sank his bayonet into the limp figure once more and angrily motioned the horrified onlookers to continue their march.\n\nSergeant Jack Cape of the 31st U.S. Infantry had a sadistic guard who only let his charges drink from ditches or carabao wallows. Near Orani one man collapsed from dysentery. Two American comrades dragged the man along, but soon were too weak to continue. The man was dropped. Cape saw the Japanese guards hand the two Americans shovels, motioning them to give their fallen friend the coup de gr\u00e2ce. When they refused, the guard raised his pistol. The men hit their comrade with shovels.\n\nA mile later two other Americans were staggering dazedly. They fell. Cape saw guards kick them to make them walk, and when this was useless, run them through with bayonets.\n\nColonel Edwin O'Connor saw a guard push a staggering Filipino off the road, and shoot him. A little later another Filipino began to scream crazily. A guard led him to a bush. O'Connor heard a shot, then saw the guard emerge from the underbrush alone. The man walking next to O'Connor was a big American. One of the guards took a violent dislike to him, jabbing him constantly with his bayonet and once cutting him across the face.\n\nSergeant Calvin Graef of New Mexico saw two American airmen fall out from exhaustion. Guards beat the two fallen men with bamboo clubs until their heads were pulp.\n\nBy now occasional beheaded bodies could be seen in the ditches. With every few miles, their number increased. Lieutenant Colonel Allen Stowell could tell only by size if they were American or Filipino since the bodies were turning from brown to black. After counting twenty-seven headless corpses, Stowell told himself, \"You've got to cut this out.\" He began marching with eyes straight ahead.\n\nAlthough many were too dazed and crazed to do anything but scramble toward the nearest water, whether in ditch, well or wallow, others figured ways to keep their bodies from dehydrating. Colonel Edwin Aldridge found a five-gallon oil can in a roadside house and half-filled it with water. He and nine comrades were taking turns, two at a time, carrying the can.\n\nThose in Sergeant Charlie James' group were making water bottles of bamboo sections. James himself drank only from wells so he wouldn't get dysentery. To combat thirst he kept a can-opener key under his tongue. At one stream he saw a Filipino soldier sitting, looking vacantly. Blood was running from bayonet wounds in his sides.\n\nIronically the ranks of the marchers were being swollen by escaping soldiers sneaking down to the coastal road from the mountains. Many joined the long march not realizing they were heading for prison camp.\n\nThough most civilians were being trucked out, others were walking. Mothers had covered the faces of older daughters with dirt to make them unattractive. Even though it was known that General Homma punished rapists severely, Japanese soldiers crept up to the civilian resting places after dark and tried to drag girls back to their camps.\n\n\u2014\n\nThe atrocities that began on the Balanga-Orani stretch increased during the 16-mile lap to Lubao. The road, pocked by bombing, made walking difficult.\n\nGeneral Clifford Bluemel was now marching next to General Stevens. They were at the head of the column, eating dust, a common humiliation for officers. A truck approached from the north. As it passed, a Japanese soldier leaned out and swatted Stevens in the face with a bamboo stick. Bluemel saw his comrade's glasses falling, caught them. He grabbed the staggering Stevens and led him to the side of the road. They sat talking, while Stevens tried to clear his head. A guard came up, pointed a revolver at Bluemel's chest, motioning him to get up. When Bluemel started to argue, he saw the hammer go back and the gun turned to Stevens. Bluemel helped Stevens to his feet. When Bluemel realized his friend couldn't continue he led him toward a rice paddy. Stevens protested, knowing Bluemel was risking death. Just then another guard ran at the two men with fixed bayonet, thinking they were trying to escape. When the guard saw the blood on Stevens' head he merely prodded Bluemel back into line with the bayonet.\n\nStevens crawled into the irrigation ditch at the side of the road and hid beneath some undergrowth. He lay motionless as the column disappeared. But for Bluemel's courage, he knew he might be lying on the road with a bullet in his head.\n\nNot far behind, Sergeant Robert Franklin was watching a Filipino soldier dart from the road to a guava tree. As the man climbed to pick the fruit, there was a shot. The Filipino fell, dead.\n\nResting not far away at the side of the road without shade was a mixed group of Filipinos and Americans. Their guards told them to get up but the Filipino next to Captain Joseph Revak was too weak to rise. A guard prodded him. Though he tried, the man couldn't get up. The Japanese pulled out his pistol and, to Revak's horror, shot him. They marched and then, since the line straggled out, again stopped for a rest. This time it was Revak who couldn't get up. A guard motioned with a rifle but the American didn't care. \"Go ahead and shoot,\" he panted.\n\nThe guard stood over him. Revak was unable to move a muscle, but when he saw the rifle swing toward him, he felt a surge of strength and staggered to his feet. Half-conscious, he wobbled up the road.\n\nAt another resting place, a few miles farther north near the base of Bataan, Corporal Roy Castleberry watched two civilians dig a hole in the hard, dry ground. Beside the hole lay a delirious American captain. The captain passed out as he was laid in the hole. Suddenly he came to, realized he was being buried alive and began to crawl desperately out of his grave. A guard ordered the Filipinos to hit the American with their shovels. They refused. Then the guard raised his rifle as if to shoot them. The Filipinos, faces twisted in agony, hit the American, rolled him into the hole, and covered him with dirt. A moment later a hand rose from the grave. The fingers moved feebly, entreatingly.\n\nThe marchers crossed the strategic bridge at Layac and turned east toward San Fernando. At last they were leaving Bataan peninsula. This stretch of road was straight, unshaded. Hundreds dropped in their tracks. Some, maddened by thirst, risked their lives to run to the adjoining sugar cane fields. They chewed the cane, then threw the pulp away. Others behind, not willing to take the chance of getting shot, picked up the chewed cane and tried to get its last moisture. The men were so dehydrated that most were unable to urinate. To those who could, the few drops burned as if a hot iron had been shoved up their penis, yet at the same time brought unspeakable relief.\n\nIt was on this long stretch that Captain Ed Dyess, formerly a fighter pilot, saw six Filipinos in his group make a dash for an artesian well. As they neared the water, the guards loosed a rifle volley. Four fell dead. Two, badly wounded, crawled, eyes glassy, toward the bubbling well, hands outstretched. The guards fired again. Now all six were dead.\n\nA few miles later the Dyess group reached the outskirts of Lubao, a spraddling city of 30,000. Dyess saw a weird object draped over a barbed-wire fence. When he drew closer, he realized it was a Filipino soldier, abdomen slit open by a bayonet. His bowels were draped like grayish purple ropes along the wire.\n\nSoon they were passing in tragic parade through crowds of waiting Filipinos. Civilians risked their lives to throw out boiled eggs, fried chicken wrapped in banana leaves or pieces of panocha, brown hard sugar. Others placed pots of rice and cans of water in the road. Some guards did nothing. Others kicked over the rice and water vats and swung rifles at those throwing food.\n\nA few civilians were selling food but the great majority wouldn't even take the money offered them. The streets were lined with weeping people. Though Japanese guards hurried up and down the lines, old women swathed from crown to ankle would pull staggering marchers from the line, stand over them with their long skirts. When the guards had passed, the exhausted American or Filipino would be hidden in a nipa hut. Some Filipino soldiers, their tattered uniforms looking civilian, would dart into the crowd, grab a girl, posing as a husband, or seize a baby, holding it as a father.\n\nThe men passed the large church near the edge of town and then were herded into a large corrugated tin building, a rice mill. Inside were several rotting bodies and piles of feces. There was one water spigot for the several thousand men jammed into the 150-by-70-foot building but only those nearby could worm their way through for a drink. There was room only to sit, huddled. Doors and windows were shut and ventilation had to come from cracks in the walls. Men often found themselves next to dying or even dead comrades.\n\nWhen the mill was packed, the overflow was formed in groups outside. Here also there was a single water spigot. The water dribbled in a continual thin stream as men, desperate for water, crowded into line for a few drops. The guards here treated everyone badly. Colonel K. L. Berry saw an American and a Filipino arguing at the spigot. Guards rushed at the American with bayonets. He dodged but was soon caught. So was the Filipino. Both were tied to a tree and shot. Later Berry saw a guard beat a Filipino over the head with a pickhandle until his brains oozed out. He wondered what his \"crime\" had been. Colonel Aldridge knew. He had seen the Filipino glare at the guard a moment before.\n\nSergeant James Madden was at the spigot when a young American, not even twenty, was dragged into the area by two guards.\n\n\"Why did you try to get away?\" asked the interpreter.\n\n\"I was trying to get into the hills and join the guerrillas. I know what's going to happen to Americans.\"\n\nHe was led across the road, tied to a tree with hands behind his back. Two guards knelt down, aimed their rifles and fired. The bullets hit the tree. They aimed again. This time their guns jammed. An officer ran up, shouting and waving a sabre. The two guards stood at attention as the officer slapped them with the palm of his hand. The guards loosened the American's hands and made him kneel. Murmurs of shock came from the prisoners across the street but there was not a single sound of protest. The officer forced the American to get off his knees and walk into the bushes. A moment later the Japanese returned, his sabre dripping blood. He walked across the street, pushed aside Madden and the others at the spigot and washed his red sabre.\n\n**3** Most of the marchers stayed overnight at Lubao, but some remained two days and a few three or four. There were usually two meals daily, a handful of rice and a dash of salt. Those outside the corrugated rice mill crowded together, victims of the suddenly cool nights and ferocious mosquitoes. At least they had air.\n\nThe final lap to San Fernando was the shortest, about nine miles, and the cruelest. There was almost no cover. Here the asphalt, churned up by tanks and trucks, was melting in the sun. Those thousands without shoes, or with shoes in tatters, felt as if they were walking over hot coals. By now a mile was endless to these diseased, dehydrated, starving men. They passed field after field of sugar cane; many glanced longingly to the northwest at the rugged Zambales Mountains, symbols of freedom.\n\nSeveral thousand Filipinos escaped on this stretch. For Americans it was a much more difficult choice. Few knew the country or could speak Tagalog. The Filipino could quite easily become assimilated in the next village. The American, even with tanned face, was conspicuous by his size, manner and voice.\n\nSergeant Frank Bernacki, formerly of the Air Corps, was one American lured by the Zambales. Tanks with _Singapore_ scrawled on their sides approached. While the guards were waving at the tankers, Bernacki whispered to other Americans around him, \"I'm leaving. Anyone want to go with me? I hear Thorp has $50,000 with him.\" There were rumors that a Captain Claude Thorp had escaped from Bataan two months earlier with a large sum of money given him by MacArthur to set up guerrilla groups.\n\nNo one wanted to go. Bernacki crawled alone into the snake-infested cane. The leaves were sharp. It was hot, dusty, almost suffocating as he crawled on all fours through tangled vines. He passed out.\n\nNearby, another airman, Sergeant Ray Hunt, was driven to escape by desperation and hatred. Normally 160 pounds, malaria and starvation had lowered his weight to 100 pounds. As the marchers crossed a bridge, he suddenly dashed into a deep ditch and hid in thick foliage. He heard someone say, \"Don't look! Do you want him to get shot?\" Finally the footsteps of his group faded out. He crawled along the ditch and to his amazement saw two more Americans, face down, frozen in terror. Hunt touched one of the men on the leg, then softly spoke. It was a corporal named Chatum. The other man was an artilleryman, a Captain Jones.\n\nHunt wanted to leave the ditch. The others argued. Even so, Hunt rose slowly and called to several Filipino farmers across the river. After a few minutes one approached. \"Are there any Japs around?\" whispered Hunt.\n\n\"Stay down,\" the other warned and left. He returned soon and led the three Americans to a bamboo house on stilts, a buhay. Here was another American, a Lieutenant Kerry. The four could watch the march through the weave of the bamboo walls. They saw an American bayoneted.\n\nFilipinos visited the men several times a day bringing water, rice and a crude sugar. One boy offered to hide Hunt in a nearby fishpond. \"I will make you well again,\" he promised. \"I'll keep you safe until the Americans return.\"\n\nAnother boy told them of an American civilian still in a nearby hacienda and offered to guide them there. The four Americans were loaded into a cart with solid wooden wheels and covered with hay. All that day a carabao pulled them toward the northwest, toward the jagged Zambales Mountains.\n\nMarchers were staggering the last mile of the 55-mile trek into the sugar center of San Fernando. At the outskirts of town, trucks lined the road, forcing the men to pass by one at a time. It was a gantlet. Japanese soldiers in the trucks swung their rifle butts as the Filipinos and Americans staggered past.\n\nFarther on, the streets were lined with civilians. Although guards tried to hold them back, many defiantly ran to the passing men with jars of water and baskets of food. A Filipino girl threw Sergeant David Duran a ball of rice. He saw a guard rush toward her, batter her in the face with his rifle butt. This ball of rice was his first food since joining the march.\n\nPeople from all over Luzon were looking for loved ones, hoping to find a relative alive in the ragged, pitiful parade. The great crowd moaned and wept in public sympathy as the skeleton army dragged by. Occasionally a wife, a child or father would spot a Filipino soldier and rush to his side. Even though guards would push these civilians away, there was little open brutality.\n\nMost of the men were fed rice balls. Here also were water and crude medical care. The men were penned in various places: in a pottery shed, at the \"Blue Moon\" dance hall, in empty lots, in old factories, in school buildings and yards, and in the large cockpit arena near the railroad station. Colonel Aldridge, sitting at the edge of the jammed cockpit arena with General Stevens, watched a group of young airmen stumble toward the entrance as if it were the gates of heaven. One reached the threshold, then fell in, dead. The others in the same group were at the point of death. All had dysentery.\n\nLieutenant Aquino's group was taken to an old vinegar factory, a great concrete-block structure of Spanish architecture with a galvanized roof. Aquino, 150 pounds when he left for war in his new yellow convertible, now weighed 89. He flopped on a matting of straw and slept fourteen hours. When he awoke he was pulled out of the stifling factory by a guard, marched into a Japanese barracks, and led upstairs to a room. Inside were his father and a Japanese colonel. Father and son embraced.\n\n\"Thank God you are alive,\" said the elder Aquino after a long pause.\n\n\"How are Mother and the kids, Papa?\"\n\n\"Everyone is alive. We were so worried.\" The father, one of the leaders of the new \"Quisling\" Laurel government, introduced the officer as Colonel Ota of the _kempeitai_ , the secret police.\n\n\"Mr. Aquino is a good friend of Japan,\" said Ota in a British accent. \"He will help form your government with Japanese guidance. To thank him, we the benevolent Japanese people tell him that his son can go home with him.\"\n\nAt first Lieutenant Aquino wondered if his release was some sort of pay-off. \"Colonel Ota,\" said the young man, \"I thank you for your kindness and your esteem for our family but I have to refuse to go home. I cannot desert my men. If you really mean to help me, give all of us food and medicine.\"\n\n\"Your father was right,\" said the colonel. \"He said you would refuse. Please accept my apologies for the way you all have been treated.\" He left.\n\nNow Benigno Aquino quietly told his son that President Quezon had ordered Laurel and other leading Filipinos to appear to collaborate with the Japanese. The young man looked at his father and was convinced he could never be a traitor. The elder Aquino said that he and other top native officials were working with General Homma for early release of all Filipinos from prison camp.\n\n\"Hurry, Papa, we are dying like flies.\"\n\nThe two drank tea and ate cookies. When Colonel Ota returned the elder Aquino said, \"Good-bye, son. I will tell your mother to come and see you.\" They embraced. As young Aquino was leaving, his father handed him a package of food wrapped in banana leaves. Lieutenant Aquino returned to the vinegar factory, gathered his men and passed around the food; each got only a mouthful. A few minutes later the doors of the factory were opened and Japanese guards entered with wheelbarrows full of rice.\n\n\u2014\n\nMost stayed one or two days in teeming San Fernando, but a few were marched directly to the railroad station and, without being given food or water, loaded into small boxcars 7 feet high, 33 feet long and 8 feet wide, similar to the French 40 and 8. One hundred to 115 were shoved into each car, then both doors closed. Men were forced to stand in the steamy interior. The stink of the disease-ridden was unbearable as the trains slowly headed north on the three-to-four-hour trip to Capas.\n\nWhile Sergeant James Madden's car was still in the San Fernando yards, suffocating men in the rear kept shouting, \"Open the door!\" The shouts turned into weeping, screaming. Suddenly there was a crash as the engine backed into the line of boxcars. One of the doors sprung open. A guard with a gun ran over when he heard the moaning. He climbed into the packed car but kept the door open. Even with one door open those in the rear could barely breathe as the train headed north.\n\nWithout discussion or plan, the strong ones voluntarily moved to the rear of the car, allowing the sicker ones to breathe the sweet, fresh air. It was a miracle, thought Madden, that men who had suffered so much could retain their humanity. As the hours passed, men rotated in the jammed car without argument.\n\nMost of the cars were kept locked and many died. Sergeant Charlie James was in the back of his car. He was lucky. Near his nose was a small crack in the hot steel wall.\n\n\"I'm smothering,\" moaned someone.\n\n\"Shut up, we can't do anything about it,\" said someone else.\n\n\"Save your breath,\" said a third. \"You're using up oxygen.\"\n\nMen with dysentery were unable to control themselves; others vomited on their comrades. Some fainted from the smell. Some died but were still kept erect by the pressing mob. At every stop, the more friendly guards opened the locked doors. Filipinos crowded around, passing out bottles of water, tomatoes, bananas, rice, eggs, coffee, sugar cane. Those Americans who had previously regarded Filipinos as inferiors began to appreciate their courage and humanity. Occasionally a prisoner would make a break from one of the cars, mingling with the crowd. There were so few guards and the crowds were so belligerent, that not many of these escapees were caught or even chased.\n\nAt the town of Capas, the men were unloaded. The fresh air was like elixir. Here again hundreds of civilians were waiting with food and water. Some guards allowed food to be passed out; others kicked over baskets of food. It was about eight miles over a shadeless, dusty road to the prison camp, O'Donnell. Yet the men were happy to be again in the open. They were buoyed too by the tremendous surge of friendliness they had received all the way from San Fernando. The men staggered toward the barren plains. For a change there were no brutalities. Many of the guards were sympathetic on this stretch and even helped the weaker marchers.\n\nAt last the prisoners saw a maze of tumbledown buildings, some only half-built, sitting out on a great plain in loneliness. This was Camp O'Donnell, where perhaps the misery would be over.\n\nThey passed through a gate flanked by towers armed with machine guns. A long barbed-wire wall extended in both directions. They were marched up a hill and seated on the ground in the blazing sun in front of a building flying the Japanese flag. After a long wait, the commandant of the hastily established camp would speak to each group through an interpreter.\n\n\"The captain, he say Nippon has captured Javver, Sumatter and New Guinyah,\" a fat interpreter told Captain Ed Dyess' group. \"Captain, he say we soon have Austrayler and New Zealyer. The captain, he say America and Nippon enemies. Always will be war until America is Nippon's. Captain, he say you are not prisoners of war. You will be treated like captives. He say you do not act like soldiers. You got no discipline. You do not stand at attention while he talk. Captain, he say you will have trouble from him.\"\n\nEach group learned in the first minutes at Camp O'Donnell that their sufferings would continue indefinitely, or until death.\n\n\u2014\n\nAs the great sprawling march was nearing its end, the Manila Sunday _Tribune_ of April 19 printed many pictures of the masses of prisoners and civilians coming out of Bataan. There was also a Japanese-inspired story:\n\n> The task of making observations upon the tragic aspect of marching war prisoners from the Bataan front, where they surrendered on April 9, to San Fernando, Pampanga, previous to their entrainment to their permanent concentration camp is a sad one; hence, our effort to avoid details about the whole episode.\n> \n> So the public would not get the wrong impression from such an enigmatic remark, however, we make it plain that the Imperial Japanese Forces, whose business is clearly to prosecute the present war to its successful termination, are going well out of their way to feed and help 50,000 men who once were their enemies beyond most reasonable men's expectations.\n> \n> If, in spite of the humane treatment the Japanese are giving these prisoners, the latter are too weak to reach their destinations, we have only the high command of the American forces to blame for surrendering when many of their men had already been terribly weakened by lack of food and by diseases.\n\nAlthough General Homma now knew there were more than double the expected number of prisoners, he was not told by General Kawane that the general health of the captives was so dangerously low that they were near starvation.\n\nAssuming General Kawane could handle the march of prisoners, Homma was presently devoting his time and attention to the assault on Corregidor. In fact he was so involved with invasion plans, he was not even aware of the thousands of lives lost on the road to San Fernando. During these days he was isolated in his headquarters at Balanga and his only trips were south to the tip end of the peninsula.\n\nHe didn't learn until two months later that more Fil-Americans had died on the long march than on the battlefields of Bataan. Of the approximately 70,000 men who started, only 54,000 reached Camp O'Donnell. No one will ever know the exact death toll since many of those unaccounted for escaped. Even so, according to a consensus of responsible survivors, 7000 to 10,000 men died on the march from malaria, exhaustion, starvation, beatings or execution. Of those dead, about 2330 were American.\n\nMost survivors were convinced the march was cruelly, purposefully planned and executed.\n\nThere had been little plan at all. About half of the prisoners rode in trucks from Balanga to San Fernando and suffered little. Some who walked saw almost no brutalities and were fed, if not well, at least occasionally. Yet others a mile behind were starved, beaten and killed by brutal guards.\n\nBrutality to the Japanese enlisted man was a way of life. He was used to being slapped and beaten by his officers. He, in turn, slapped and beat from habit. He had been indoctrinated to obey swiftly and without question. When Americans and Filipinos failed to understand his orders or were too weak to follow them, he often resorted to violence and even murder. There was almost a complete failure of guidance and leadership from above.\n\nThe Japanese soldier despised those Fil-Americans who had surrendered. \"Do not fall captive even if the alternative is death,\" read his soldier's manual. \"Bear in mind the fact that to be captured not only means disgracing the army, but your parents and family will never be able to hold up their heads again. Always save the last round for yourself.\"\n\nAnother motivation for brutality was revenge. This act had always transcended other human ties in Japan, justifying any action.\n\nWhatever the causes, no man who marched would ever forget the horrors and torture. Few would ever forgive. The Death March was to become a focal point of hate and revenge to Americans and Filipinos.\n\n# PART\n\n# 6\n\n# FROM HUMILIATION TO VICTORY\n\n# 22\n\n# _\"But Not in Shame\"_\n\n**1** About dawn on the morning of April 18, some 650 miles northeast of Japan, a reconnaissance plane flew low over the U.S. carrier _Enterprise_ and dropped a beanbag containing a message to Admiral Bill Halsey, commander of Task Force 16. An enemy patrol ship was 42 miles ahead and the pilot was sure he had been sighted. Halsey swore and signalled the bad news to a nearby carrier, the _Hornet_.\n\nA harsh alarm immediately sounded on the _Hornet_ and sailors ran to their battle stations. Awakened by the noise, Army pilots and crewmen soon learned that Task Force 16 had been sighted by a Japanese picket ship and they would probably have to take off in their sixteen B-25 Army bombers at once instead of that evening, as planned, when they would be within 500 miles of Japan.\n\nIt was a jolt for the fliers. Everything had been precisely planned, to the last gallon of gas. They were to bomb Tokyo and three other cities that night, then continue west to China where they would land at dawn.\n\nFinally Colonel Jimmy Doolittle, leader of the fliers, came down from the bridge after a conversation by blinker signals with Admiral Halsey on the _Enterprise_. \"Come on, fellows, let's go,\" he said. Surprise was gone; they would have to bomb in daylight and 150 miles had been tacked on to their route. Just then a voice from the _Hornet_ 's bullhorn boomed, \"Army pilots, man your planes!\"\n\nAs Commander John Ford, the famous movie director, and his crew took pictures, mechanical donkeys pushed and pulled the twin-ruddered B-25's into position. The deck was wet, green water breaking over the carrier's ramps.\n\nTen extra five-gallon cans of gasoline were loaded into each plane. Then the main tanks were topped. Though gauges read \"Full,\" Navy men rocked the bombers in case any bubbles had formed in the wing tanks. Even an extra quart might mean the difference between life and death.\n\nDoolittle was to be the first to go. At 7:24 A.M. he gave his engines full throttle. They roared until some of his anxiously watching pilots were afraid he'd burn them up. Blocks were pulled from the plane's wheels and it rolled forward until the left wheel was resting on a white line that ran down the port side of the deck. The left wing of the bomber was far over the side of the carrier as it clumsily wobbled forward into the heavy wind, engines roaring, flaps down.\n\nThe other pilots watched tensely. Doolittle would have to time his take-off perfectly. If he didn't leave the deck at the apex of the _Hornet_ 's heavy pitch he would probably plummet instantly into the sea. The B-25 gained speed and took off just as the bow of the carrier was lifted high by the heavy sea.\n\nThere was a cheer from the Navy as the Doolittle plane circled and passed low over the _Hornet_. It headed straight for Tokyo; there wasn't enough gas to wait and fly formation.\n\nThe other bombers started taking off, each \"sweated\" into the air by every man in Task Force 16. All went well until the last plane slowly headed toward the starting line. Suddenly Aviation Chief Machinist Mate Thomas Respess and others holding down its nose, were horrified to see one of their group blown like a tumbleweed by the preceding plane's blast into the spinning left propeller of their plane. The man's left arm was chopped off by one blade. Another hit him in the rump knocking him clear.\n\nThe pilot felt the jar and glancing back saw the injured man lying on the deck. Rattled, he put his flap control lever back in retract instead of neutral. As the plane roared down the short runway, the flaps slowly came up. Consequently when the plane left the deck it dropped down out of sight.\n\nRespess was sure it was going to crash. Then he saw it skimming only feet above the waves. Ponderously it rose, turned and followed the other planes.\n\nThirteen planes were to drop their four bombs apiece on strategic targets in Tokyo. Single planes were to hit Nagoya, Osaka and Kobe. Doolittle reached Tokyo first. At noon he buzzed so low over a ball park, the game was broken up. There was no effective opposition from fighters or anti-aircraft as plane after plane swept over the city.\n\nOnly moments before Doolittle arrived, a mock raid had been staged and the citizens of Tokyo thought the American attack was just the realistic climax. Children in schoolyards and people in the crowded streets waved at the passing planes, mistaking their circular red, white and blue markings, similar to those used by the Allies in World War I, for the Rising Sun.\n\nFor more than fifteen minutes no air alarm was sounded. The picket ship _Nitto Maru_ , just before it was sunk by Halsey that morning, had warned of the approaching carriers but Imperial Headquarters had not expected an air raid until the next day. Not one American plane was shot down and soon fifteen were roaring toward China. A single plane was heading across the Sea of Japan toward Vladivostok.\n\nSince little damage was done, the raid caused no panic. To the people it was only as if a meteor had hit Japan, but their leaders were worried. It proved, argued men like Foreign Minister Togo, the falsity of the military assurances of the inviolability of the imperial capital. Four fighter groups were assigned to protect Japan from future raids. The High Command ordered the China Expeditionary Army to cease other operations and root out enemy bases in the Chekiang area. More important, they began to be obsessed by the island of Midway, the probable base of the attack. It had to be taken.\n\nThe effect in the Allied world was just as important. With the fall of Bataan had come the lowest point in morale and world standing. Doolittle's epic flight, although causing little physical damage to Japanese objectives, was a pledge that America would soon attack. Allies on every battlefield, in every prison camp were roused with new hope. American newspapers blazed the story in their biggest headlines, with the Los Angeles _Times_ composing the most original: DOOLITTLE DID IT. Naturally the Navy's key role in this raid was not revealed, Roosevelt merely stating that Army planes had taken off from Shangri-La.\n\nOne plane, piloted by Lieutenant Edward York, landed safely in Vladivostok where it was impounded by the Russians. The other 15 either crashed in China or were abandoned in mid-air. Of the 80 Doolittle fliers, two were drowned and one was killed when his parachute only partially opened. Eight were captured.\n\nThe men who had sent bombers over Chungking, Singapore, the Philippines and hundreds of unprotected cities ordered these eight brought to trial for their \"inhuman\" acts. The trial\u2014conducted solely in Japanese and lasting less than half an hour, during which the eight men were not advised of the charges, given interpreters or even informed it was a court-martial\u2014found all eight guilty. Five were given life imprisonment, one eventually dying of malnutrition. Lieutenants Hallmark and Farrow and Sergeant Harold Spatz were executed. But first, according to popular rumor in Japan, these three were used in medical experiments.\n\n**2** While Doolittle was bombing Tokyo, a radiogram was received by Lieutenant Colonel Orin Grover at Del Monte on the island of Mindanao. Corregidor was badly in need of medicine. A plane had to be sent immediately. For the past few months a small band of fliers\u2014including Captains Harold Slingsby, Dick Fellows, Joe Moore, Hervey Whitfield and Bill Bradford\u2014had been operating a taxi service, \"The Bamboo Fleet,\" between Bataan, Corregidor and Del Monte in worn-out military and private planes.\n\nNow there was only one plane left, the ancient Bellanca once belonging to the Philippine Air Taxi Company. Captain Bradford, nicknamed \"Jitter Bill,\" knew it well for he had been general manager and senior pilot of the organization. Seven years older than the legendary Paul \"Pappy\" Gunn and a veteran of World War I, he was the most experienced pilot in the Philippines. Five weeks previously he had logged his five thousandth hour over the islands when he took Colonel Arthur Fischer out of Corregidor in the desperate attempt to fly live cinchona seed to the United States and save quinine for the Western world. And only two days before, he had snatched Colonel Carlos Romulo and two others out of Iloilo the morning the Japanese landed on the island of Panay.\n\nBecause of his experience Colonel Grover asked Bradford what he thought were the chances of flying up to Corregidor. \"Exactly zero,\" was the reply. In addition to Panay, the Japanese now occupied most of Cebu. For practical purposes, the entire Visayan Group was Japanese. Grover wired this information to Corregidor. The answer came the next day: send a plane regardless.\n\nSomeone would have to fly the slow, unarmed, ten-year-old Bellanca up through enemy territory to the Rock. After the few qualified pilots were rounded up, Captain Benny Putnam handed Bradford a deck of cards. \"Shuffle and cut,\" he said. The low man would have to make the trip. Bradford shuffled, cut. He drew a card and put it in his pocket. The others drew and showed their cards. Then Bradford slowly pulled out his card, the deuce of diamonds. He was low man.\n\nCaptain William Monay, an interested onlooker, drew Bradford aside. \"I saw you stack the deck,\" he whispered.\n\nBradford denied this, pointing out that only a fool would willingly take on such a mission. \"But,\" he added, \"the others there wouldn't have a chance of getting into Corregidor. I know where to make that last dogleg turn and find it in the dark.\"\n\nSeveral hours later Bradford took off with old Number Nine heavily loaded with medicine. The first lap to the island of Negros, sandwiched between Japanese-occupied Panay and Cebu, was suspenseful but uneventful. His big worry as he touched down at Bacolod was who owned the island. Fortunately Americans ran out to refuel his ship but the Japanese were expected any day. Early the next morning he took off for the last, dangerous lap. It was only a three-hour trip to Corregidor and he wanted to land just at dawn.\n\nAs he approached Luzon clusters of twinkling lights marked villagers arising in the pre-dawn, but as he passed by, the lights blacked out in alarm. The coastline was shrouded in white stratus, distorting familiar landmarks. He fanned the Bellanca away from shore for he was ahead of schedule and was without lights or radio. He now had to mark time until dawn could bring enough visibility for recognition and a landing.\n\nWhen light came with tropical abruptness, he found himself half an hour from Corregidor. He headed northeast, at full throttle, flying just above the rollers with a thin layer of mist a few feet overhead half covering him. When he was still several miles from the Rock the plane abruptly burst into sunlight. He sped along the torn-up southern coast of the polliwog-shaped island of rock expecting to be shot down by friendly guns. No shots came. He nervously glanced behind, half expecting to see a Nip fighter on his tail. It didn't seem possible that he had lumbered through enemy territory at 80 miles an hour and was still in one piece. He flared old Nine up in a climbing turn over the island's tadpole tail, took another hurried look around for bandits and then settled down on the rough, scarred surface of Corregidor's tiny airstrip, Kindley Field. As his wheels touched, a man ran out, frantically waving him to a revetment. Bradford guided the patched-up old plane to her resting place, cut the switch and sat shakily in silence.\n\nBy the time he was in Malinta Tunnel and the precious medicine unloaded, air raid signals were shrieking. He reported to Wainwright.\n\n\"Brad,\" said the general, shaking his hand, \"I thought you'd get through. Glad you made it. Congratulations.\"\n\nBradford was startled by the change in the general and the others he knew since his last trip. Their faces were gaunter, more drawn. Everyone had the look of doom.\n\nThe shelling had done it. Until the fall of Bataan, the bombing had been tremendous yet caused relatively few casualties. The shelling was churning the entire island into a no man's land.\n\nThough morale was still fairly high, few of the 13,000 on the island had hopes of being saved. The men's favorite theme song was \"I'm Waiting for Ships That Never Come In.\" Others sarcastically asked if the chalked V's on many of the men's helmets stood, not for Victory, but for Victim.\n\nOn the day Bradford arrived, a new threat began to menace those living outside the tunnels. Japanese 240-mm. howitzers had been dragged from Cavite to Bataan. Now their high-angle fire was blasting Corregidor's hitherto unreachable 12-inch mortar pits. The number of dead and wounded rose alarmingly.\n\nJapanese fire increased, coming to a tremendous climax on April 29, Emperor Hirohito's birthday. In addition the 260th air alarm was sounded that day and bombers began to hit Malinta Hill. The combined attacks continued through the afternoon. Grass fires sprang up, two ammunition dumps were blown to shreds. By dark, most of the island was covered with thick clouds of smoke and dust.\n\nA few hours later two Navy PBY's broke through the blockade, landing on the bay several miles south of Corregidor. After medicine and 740 mechanical fuses were unloaded, 50 people were selected to fly south to safety: 30 nurses, 3 civilian women and 17 men. \"Jitter Bill\" Bradford was one of the men. His Bellanca had crashed in an attempted night take-off with three passengers. He shook hands with Wainwright, who had come to the dock to say goodbye, and said, \"Wish you were coming with us.\"\n\nWainwright smiled wryly, \"I couldn't.\" A pretty nurse, Lieutenant Juanita Redmond, embraced the gaunt, weary general and kissed him. \"Oh, thank you, General,\" she said.\n\nWainwright stood on the dock and watched the two heavy flying boats skim across the water, slowly rise and disappear in the south. Then he walked back to Malinta Tunnel.\n\nThose who thought the bombardment on the Emperor's birthday was the greatest the Japanese could muster were mistaken. Each day the pounding increased. General Homma knew he had to knock out the big mortars of Batteries Geary and Way before his invasion fleet could sail from Bataan.\n\nBy the morning of May 2, Battery Way had only two guns but Geary was intact. Near noon, Private Ralph Houston, an aid man, was sitting in Geary's projectile room. Its thick, squat guns were silent for the moment and he decided to stretch his legs. He walked through the open steel doors. The underbrush was torn up, the terrain unrecognizable. Suddenly there was a terrific roar. He was momentarily in total darkness. Dazed, he turned around. What had been the powder room was now a great hole, steaming smoke. He ran to the wreckage. Behind the crater he heard voices. Men were entombed. He shouted that he was getting help, then, in spite of the bombardment, ran from the heights of Topside down the hill toward Malinta Tunnel.\n\nEveryone on Corregidor knew something terrible had happened. The explosion had rocked the island like an earthquake. It was the end of Battery Geary. The barrels of its eight 10-ton mortars had been tossed about like matchsticks, one landing 150 yards away on the pock-marked golf course. Corregidor now had little except its beach defenses to hold off the landings everyone knew were imminent.\n\nLife outside Malinta Tunnel was hazardous and rugged, but at least there was fresh air and light. Those in the tunnel suffered from an intolerable tension. The 1000-bed hospital was packed and fresh casualties were being placed in aisles. Its odor, the smell of death, pervaded the laterals. Dust and dirt, loosened by every hit on Malinta Hill, showered the thousands of inhabitants. When the blowers were off during air attacks, the air soon became fetid, the heat almost unbearable. Huge black flies, roaches and other insects infested the place.\n\nMany, suffering from \"tunnelitis,\" refused to step foot outside the tunnel. These were nicknamed \"tunnel rats,\" and according to sarcastic comrades had become shelter-shocked. There was still food for six to eight weeks but the two meals a day were meager, giving little real nourishment. By now private hordes of food and liquor had been consumed. The hundreds of bottles of wine and whisky recovered from Quezon's sunken yacht were almost gone. Some of the men at the batteries were making their own \"raisinjack\" and \"jungle juice\" from raisins and prunes.\n\nIn the tunnel, tempers were short; arguments sprang up over trifles. Most felt life had to be lived day by day. Gambling was widespread, with fantastic stakes. Money was only paper. Jam sessions attracted big crowds. While some turned to religion in what they knew were their last free moments, a few, lucky enough to find willing women, took them outside during the night. In spite of bombing and shelling, there was still occasional love-making on Corregidor.\n\nThe next day, May 3, Wainwright was told the water supply was dangerously low. He wired MacArthur:\n\n> SITUATION HERE IS FAST BECOMING DESPERATE.\n\nThat afternoon, in the hospital latrine, Lieutenant Lucy Wilson was talking with an amputee patient, Lieutenant Louis Lutich. He was a friend of the man she was supposed to have married on Bataan, Dan Jopling. She asked Lutich if he knew whether Jopling was alive or dead. Lutich didn't know. While they were talking she was told to report to the chief nurse, Captain Maud Davison, and guessed it was because she had recently told off an amorous colonel. She was wrong. She had been selected to leave in the last transportation to freedom.\n\nAfter dark Lucy Wilson, ten other Army nurses, one Navy nurse, a civilian wife, six Army colonels and six naval officers were taken in a boat toward the edge of the mine field. Suddenly there loomed up directly ahead an indistinct shape which, after some hesitation, was identified as a submarine, the _Spearfish_.\n\nIts commander was Lieutenant James C. Dempsey, who on February 8, 1942, had commanded the old S-37 in a night surface attack against a division of four Japanese destroyers; that night S-37 became the first U.S. submarine ever to sink an enemy destroyer. This was Dempsey's first patrol in command of the _Spearfish_ , and so far at least two Japanese freighters had been sunk. Enroute to the base in Australia from the South China Sea, Dempsey only knew that he was to rendezvous that night at a point just off the mine fields about three miles from Corregidor.\n\nShortly after twilight _Spearfish_ had surfaced and commenced charging batteries while lying-to at the rendezvous. Now Dempsey, who was not even certain that the Rock was still American, saw a boat approaching. This was not the appointed rendezvous time and the boat was flashing an incorrect identification signal. Was it manned by Japs or friends? Was this a trap? Should he open fire on the boat and attempt a getaway on the surface? The storage batteries had been under charge only ten to fifteen minutes and Japanese patrol craft had last been sighted a few miles distant from the rendezvous point. Then he heard an unmistakably American voice, southern-accented: \"Submarine Ahoy,\" and saw American women among the boat's passengers. They wore khaki slacks and shirts or old dresses and no make-up, but after thirty-five days on patrol, they looked to Dempsey and his crew like beautiful girls from the Ziegfeld Follies.\n\nWhile Dempsey anxiously scanned the horizon, the twenty-five manifested passengers and two later-discovered civilian stowaways, who were in the boat to assist in transferring the cargo to the submarine, filed down the narrow hatch. Dempsey, sensing danger, warned everyone to hurry. Finally passengers and cargo were aboard. _Spearfish_ headed out on the surface. It was none too soon. As the full moon burst through the clouds, shells began to fall on Corregidor. Explosions and fires lighted up the entire area so brightly that Japanese patrol craft could be seen heading toward _Spearfish_. Dempsey sounded the diving alarm. The _Spearfish_ submerged, rigged for \"silent\" running and commenced maneuvering to evade the enemy.\n\nThere were no manuals to guide Dempsey in the complex problems so many women presented. The incident would later become the basis for the movie _Operation Petticoat_. When bunks were assigned in the cramped submarine, one sailor, thinking Lucy Wilson was a child because of her pigtails, patted her paternally and said, \"You can sleep with your daddy.\" But being the last on the manifest, she got no bunk. She slept on the floor.\n\n**3** Homma's artillery attack reached new heights the next day, May 4. In an almost continuous drumfire, 16,000 shells exploded on Corregidor in twenty-four hours. At times it sounded like fire from gigantic machine guns.\n\nIn the little whitewashed office he had inherited from MacArthur, Wainwright was writing a message to Marshall answering Washington's request for a frank estimate of the situation.\n\n> IN MY OPINION THE ENEMY IS CAPABLE OF MAKING ASSAULT ON CORREGIDOR AT ANY TIME.\n> \n> SUCCESS OR FAILURE OF SUCH ASSAULT WILL DEPEND ENTIRELY ON THE STEADFASTNESS OF BEACH DEFENSE TROOPS. CONSIDERING THE PRESENT LEVEL OF MORALE, I ESTIMATE THAT WE HAVE SOMETHING LESS THAN AN EVEN CHANCE TO BEAT OFF AN ASSAULT. I HAVE GIVEN YOU, IN ACCORDANCE WITH YOUR REQUEST, A VERY FRANK AND HONEST OPINION ON THE SITUATION AS I SEE IT.\n\nThe situation the following morning was critical. Morale was low. Those in the tunnels were psychologically battered, physically debilitated. Almost everyone was overwhelmed by the psychosis of doom. Those on the outside had different problems. Many had a bitter unreasoning hatred of the \"tunnel rats,\" and could not see why they had been forced to stay outside for weeks in shallow foxholes, serving no apparent purpose. As usual in a fighting situation, the combat troops got less than those they protected.\n\nThe wonder is that there was any morale at all among the beach defense troops. Of the 4000 in number at the fall of Bataan, there were today little more than 3000 effectives because of the heavy shell casualties. Of these about 1300 were well-trained fighters from the 4th Marine Regiment. The rest was a conglomerate force of Filipino fliers and artillerymen and American refugees from Bataan.\n\nIn spite of the tremendous shelling of the past few weeks, the justified gripes, and the obvious weakness of their own forces, many of these men thought they could hold Corregidor. Even so, there were some who had to be regularly dug out of the bushes or chased out of the tunnels and put back in their foxholes. That most men stayed in place was a tribute to the competent leadership of Marine Colonel Samuel Howard and his officers and the fighting spirit of his men.\n\n\u2014\n\nAcross North Channel in Bataan, General Homma was giving final instructions for the storming of Corregidor. He was already several weeks behind schedule. A malaria epidemic had suddenly struck his troops in the infested river valleys of southern Bataan, soon exhausting the quinine supply.\n\nThe Corregidor invasion troops were hardest hit by malaria, their strength dropping as much as two-thirds. When it looked as if the entire operation might have to be indefinitely postponed, 300,000 quinine tablets arrived from Japan by air and the epidemic was halted.\n\nThat evening Homma watched anxiously as his landing craft slowly left Lamao carrying 2000 men of the 61st Infantry and several tanks of the 7th Tank Regiment. They were to land in two waves on the north beach of Corregidor's polliwog tail and push west toward Malinta Tunnel. The next night reinforcements of about 3000 men and tanks would land on the north shore of the polliwog's body, Topside. The two forces were supposed to converge, crushing everything between them.\n\nAll went well until the fleet neared Cabcaben. Here the commander, Colonel Gempachi Sato, who was with the first wave, the 1st Battalion, discovered to his dismay that the 2nd Battalion, the second wave, had somehow drifted to his right instead of staying on the left. He ordered the 2nd Battalion to hang back, and then move over to the right to its proper place.\n\nThe men of the 1st Sea Operations Unit, who were handling the boats, began to get confused. These high-spirited veterans of Singapore had imagined this would be a simple operation and knew little of the tides and currents that were presently raising havoc with their boats. Since the tide had flowed west when they left Bataan, they assumed the current off Corregidor would also be west. Now they were discovering it went east. Colonel Sato and the 1st Battalion were approaching Corregidor half a mile east of their destination. Lagging far to the rear, the 2nd Battalion was innocently heading for a point almost a mile east of its beach.\n\nA heavy Japanese covering barrage began to fall just west of the intended landing points. The defenders dug in and waited, suspecting something would soon happen. At 11:00 P.M. they saw the dim, bulky shapes of boats slowly nearing North Point. Every gun on the tail end of Corregidor loosed a deadly barrage on the already confused invaders. Two 75-mm. guns, commanded by 1st Lieutenant Ray Lawrence and saved for such an emergency, finally revealed their position east of North Point, sinking boat after boat. Joining in the carnage were 37-mm. guns. Though the few remaining Corregidor searchlights were quickly shot out by Bataan guns, tracers lit up the entire north shore like a Fourth of July display.\n\nThe slaughter was sickening even to tensely watching beach defense officers. Half of Sato's 1st Battalion never reached the shore. Finally just before midnight, his tardy 2nd Battalion began to land near North Point. The moon had risen and the clouds cleared. Artillery from Topside began to dig up the beach area, then the last 12-inch mortar of Battery Way spoke. It was also light enough for men at Fort Hughes, on nearby Caballo Island, to open up with their mortars, 3-inchers and 75's.\n\nThe Japanese of the 2nd Battalion crouched helplessly as their craft headed for the beach. To them it seemed as if a hundred big guns were raining steel on their heads. Most of the boats were sunk or leaking badly. Off shore 90 yards, a lieutenant gave his men the signal to jump into the water and wade to the beach. They leaped overboard, but the water was still deep and many, weighed down by almost 100 pounds of equipment, were dragged underwater and drowned. When he reached the shore the lieutenant counted his men. Seven of every ten were missing.\n\nIn spite of the terrible losses, Colonel Sato began to lead the remnants of his two battalions yard by yard toward the east mouth of Malinta Tunnel.\n\n\u2014\n\nAt midnight, a U.S. Marine runner burst into the tunnel office of Major George Moore, commander of Harbor Defenses, reporting breathlessly that probably 600 Japs had landed.\n\nMoore telephoned Wainwright. \"The Nips are landing out near North Point!\" Then Moore ordered the men manning the island's big guns to report to the tunnel as infantry reinforcements. Not long after 3:00 A.M., Wainwright was informed the Japanese had seized Battery Denver, a Marine anti-aircraft gun pit scarcely a mile east of the tunnel. A few minutes later the general was handed a radiogram decoded on a ruled sheet of paper.\n\n> DURING RECENT WEEKS WE HAVE BEEN FOLLOWING WITH GROWING ADMIRATION THE DAY-BY-DAY ACCOUNTS OF YOUR HEROIC STAND AGAINST THE MOUNTING INTENSITY OF BOMBARDMENT BY ENEMY PLANES AND HEAVY SIEGE GUNS.\n> \n> IN SPITE OF ALL THE HANDICAPS OF COMPLETE ISOLATION, LACK OF FOOD AND AMMUNITION YOU HAVE GIVEN THE WORLD A SHINING EXAMPLE OF PATRIOTIC FORTITUDE AND SELF-SACRIFICE.\n> \n> THE AMERICAN PEOPLE ASK NO FINER EXAMPLE OF TENACITY, RESOURCEFULNESS, AND STEADFAST COURAGE. THE CALM DETERMINATION OF YOUR PERSONAL LEADERSHIP IN A DESPERATE SITUATION SETS A STANDARD OF DUTY FOR OUR SOLDIERS THROUGHOUT THE WORLD.\n> \n> IN EVERY CAMP AND ON EVERY NAVAL VESSEL SOLDIERS, SAILORS, AND MARINES ARE INSPIRED BY THE GALLANT STRUGGLE OF THEIR COMRADES IN THE PHILIPPINES. THE WORKMEN IN OUR SHIPYARDS AND MUNITIONS PLANTS REDOUBLE THEIR EFFORTS BECAUSE OF YOUR EXAMPLE.\n> \n> YOU AND YOUR DEVOTED FOLLOWERS HAVE BECOME THE LIVING SYMBOLS OF OUR WAR AIMS AND THE GUARANTEE OF VICTORY.\n\nFRANKLIN D. ROOSEVELT\n\nWainwright carefully folded the message. He would always treasure it. Then he wrote a reply:\n\n> YOUR GRACIOUS AND GENEROUS MESSAGE OF MAY 4 HAS JUST REACHED ME. I AM WITHOUT WORDS TO EXPRESS TO YOU, MR. PRESIDENT, MY GRATITUDE FOR THE DEEP APPRECIATION OF YOUR GREAT KINDNESS....AS I WRITE THIS AT 3:30 A.M. OUR PATROLS ARE ATTEMPTING TO LOCATE THE ENEMY POSITIONS AND I WILL COUNTERATTACK AT DAWN TO DRIVE HIM INTO THE SEA OR DESTROY HIM. THANK YOU AGAIN, MR. PRESIDENT, FOR YOUR WONDERFUL MESSAGE WHICH I WILL PUBLISH TO MY ENTIRE COMMAND.\n\nAt 4:30 A.M. Colonel Howard sent his last infantry reserves, the 500 untrained sailors of the 4th Provisional Battalion, into the battle just east of the tunnel. These men had been waiting at the tunnel's mouth for several hours, watching a steady parade of wounded men brought back from the front. It was a sight to shake experienced troops.\n\nLed by their commander, Major Francis Williams, the 500 sailors, armed only with rifles, filed toward the dark battleground. One of their officers, Army Captain Harold Dalness, had been among the 2000 to escape Bataan the night before its surrender. He felt relieved to get out of the gloomy, depressing tunnel where every trouble was enlarged. The noise was suddenly terrific. At first Dalness thought it was the change from tunnel to open air. Then he realized their avenue of approach was being heavily bombarded.\n\nShells dropped among the men causing scores of casualties. There was momentary confusion as sailors scrambled for shelter, but in ten minutes Williams and his officers reassembled the sailors. Shaken by their sudden baptism in blood, the men allowed themselves to be led forward. Dalness was astonished. Experienced infantry troops might have rebelled. These untested Navy men were showing unbelievable courage.\n\nWhen they were 200 yards from the thin American line, Williams ordered the two leading companies to form a line of skirmishers. They moved forward briskly to the left until they reached the front line. The next two companies moved to their right through the confusing darkness.\n\nIn spite of chaotic and deadly fighting, the newly reinforced line held. A few minutes before dawn, Williams went from company to company, encouraging the men. The whole line, he said, would attack at dawn. At exactly 6:15 A.M. Williams' sailors, together with the Marines of Headquarters and Service Company, moved forward. The attack completely surprised the dug-in Japanese, who were waiting for plane and tank support. They fell back on both flanks, but men on a small hill in the center held, their machine gun taking a heavy toll of Americans.\n\nOne of Williams' company commanders, Navy Lieutenant Bethel Otter, crept forward with five volunteers. When the six men were within 25 yards of the gun position, they threw their grenades. Several hit the target directly, knocking out the machine gun, but before the position could be occupied, other Japanese ran forward and killed Otter and four of his men.\n\nAfter the first quick advances, the Americans could make little more headway for there were no supporting weapons to assist their attack. By 9:00 A.M., after gaining 300 yards, the entire line was pinned down. Williams sent a runner to the tunnel to get reinforcements and artillery support. Neither was available. For an hour the Americans could do nothing but hold. Every time a squad moved, heavy artillery fire from the big Japanese guns on Bataan would fall.\n\nThen at 10:00 A.M. Dalness heard the ominous rumble of tanks. Three pushed forward. The awesome sight panicked some of the defenders. A few started toward the rear. Noncoms and officers stopped them and a rout was prevented.\n\nWhen Wainwright heard Japanese tanks were moving against men with no anti-tank defenses, he knew he had to make up his mind fast. He went over the position in his mind: most of his seacoast guns were destroyed; 46 of the 58 beach defense 75-mm. guns were knocked out; communications were gone.\n\nIn his imagination he could see a tank nose into the tunnel and attack the wounded and nurses. Leaning on the stick he carried because of a game leg, he walked into his headquarters and called for Generals Moore and Beebe.\n\n\"We can't hold out very much longer,\" he told them. \"Maybe we could last through this day but the end certainly must come tonight. It would be better to clear up the situation now, in daylight.\"\n\nIt was 10:15 A.M. when he ordered Beebe to broadcast a previously prepared surrender message. \"Tell the Nips,\" he said in a voice choked with emotion, \"that we'll cease firing at noon.\"\n\nThe men of Corregidor were ordered to destroy all arms greater than .45 caliber. Now Wainwright radioed Major General Sharp, commander of the Visayan-Mindanao Force, releasing to his command all the Philippines except Corregidor and its three neighboring islands. Sharp was told to report immediately to MacArthur for any further orders. \"I believe you will understand the motive behind this order,\" he added. By doing this, Wainwright hoped he could confine the surrender to the four fortified islands in Manila Bay.\n\nAt 10:30 A.M., Beebe, in a tired but clear voice, broadcast the surrender message in English, over the \"Voice of Freedom\" radio station. Then it was broadcast in Japanese.\n\nWhile guns were being spiked, codes burned, and radio equipment smashed and more than 2,000,000 pesos in paper money cut up with scissors, Wainwright was composing his last message to Roosevelt.\n\n> WITH BROKEN HEART AND HEAD BOWED IN SADNESS BUT NOT IN SHAME I REPORT TO YOUR EXCELLENCY THAT TODAY I MUST ARRANGE TERMS FOR THE SURRENDER OF THE FORTIFIED ISLANDS OF MANILA BAY....\n> \n> THERE IS A LIMIT OF HUMAN ENDURANCE AND THAT LIMIT HAS LONG SINCE BEEN PAST. WITHOUT PROSPECT OF RELIEF I FEEL IT IS MY DUTY TO MY COUNTRY AND TO MY GALLANT TROOPS TO END THIS USELESS EFFUSION OF BLOOD AND HUMAN SACRIFICE.\n> \n> IF YOU AGREE, MR. PRESIDENT, PLEASE SAY TO THE NATION THAT MY TROOPS AND I HAVE ACCOMPLISHED ALL THAT IS HUMANLY POSSIBLE AND THAT WE HAVE UPHELD THE BEST TRADITIONS OF THE UNITED STATES AND ITS ARMY.\n> \n> MAY GOD BLESS AND PRESERVE YOU AND GUIDE YOU AND THE NATION IN THE EFFORT TO ULTIMATE VICTORY.\n> \n> WITH PROFOUND REGRET AND WITH CONTINUED PRIDE IN MY GALLANT TROOPS I GO TO MEET THE JAPANESE COMMANDER. GOOD-BYE, MR. PRESIDENT.\n\n# 23\n\n# _Strange Surrenders_\n\n**1** At 11:00 A.M., in the Navy tunnel, Commander Melvyn McCoy wrote out his final dispatch and handed it to a radioman. \"Beam it for Radio Honolulu, and don't bother with code.\"\n\n> GOING OFF AIR NOW. GOOD-BYE AND GOOD LUCK. CALLAHAN AND MCCOY.\n\nThe Army radio operator, Corporal Irving Strobing, was still maintaining constant contact with WTJ in Honolulu. He was afraid to let them go for even a second. Since concussion from the shelling had knocked out most of the lines, Strobing was now keying directly at the transmitters near the west mouth of the tunnel.\n\n> NOTIFY ANY AND ALL VESSELS HEADED TOWARD THIS AREA TO RETURN TO THEIR HOME PORTS.\n\nAfter this radiogram was sent out in the clear Strobing was told there would be no more official messages. Wanting to keep in touch with the outside world as long as possible, he began sending out his personal account of the last moments of Corregidor.\n\n> WE ARE WAITING FOR GOD ONLY KNOWS WHAT....LOTS OF HEAVY FIGHTING GOING ON. GENERAL WAIN-WRIGHT IS A RIGHT GUY AND WE ARE WILLING TO GO FOR HIM BUT SHELLS WERE DROPPING ALL NIGHT FASTER THAN HELL. TOO MUCH FOR GUYS TO TAKE....CORREGIDOR USED TO BE A NICE PLACE BUT IT'S HAUNTED NOW....\n> \n> THE JIG IS UP. EVERYONE IS BAWLING LIKE A BABY. THEY ARE PILING DEAD AND WOUNDED IN OUR TUNNEL....I KNOW NOW HOW A MOUSE FEELS, CAUGHT IN A TRAP AND WAITING FOR GUYS TO COME ALONG AND FINISH IT UP.\n\nAt noon all American guns were ordered to stop firing. Colonel Paul D. Bunker went outside and lowered the post flag, twice shot down and replaced under fire in the past few weeks. As a white flag went up, Bunker burned the Stars and Stripes. Half an hour later, when Japanese fire continued, Beebe repeated his radio broadcast.\n\nFire persisted and Wainwright was forced to send Marine Captain Golland Clark, Jr., forward with an interpreter, a flag bearer and a musician. When the four men approached the front lines at about 1:00 P.M., the musician blew his bugle while the flag bearer waved a white sheet. Firing finally ceased, and the surrender team walked into the Japanese lines. Soon they met Colonel Motoo Nakayama, Homma's senior operations officer, who insisted that Wainwright meet him.\n\nWithin an hour Wainwright, General Moore, Colonel Pugh, Major Tom Dooley and Moore's aide, accompanied by Captain Clark, were riding east in a Chevrolet. At Denver Hill they got out of the car and advanced on foot past the dead and dying. Colonel Nakayama, preceded by a private and an English-speaking lieutenant, came from the other side.\n\nThe two parties met near the top of the hill. There was a brief silence, then the Japanese private snatched the field glasses that were around Colonel Pugh's neck.\n\n\"We will not accept your surrender unless it includes all American and Filipino troops in the whole archipelago,\" said the Japanese lieutenant.\n\n\"I do not choose to discuss surrender terms with you,\" said Wainwright. \"Take me to your senior officer.\"\n\nColonel Nakayama came forward.\n\nWainwright said he was only surrendering the four islands in Manila Bay. When this was translated, Nakayama replied with an angry torrent of Japanese. He had explicit orders from Homma to bring Wainwright to Bataan only if the American agreed to surrender all his troops.\n\nBack in Malinta Tunnel, Corporal Strobing was tapping out another message to Honolulu.\n\n> MY NAME IS IRVING STROBING. GET THIS TO MY MOTHER, MRS. MINNIE STROBING, 605 BARBEY STREET, BROOKLYN, NEW YORK....MY LOVE TO PA, JOE, SUE, MAC, CARRY, JOYCE AND PAUL. ALSO TO ALL FAMILY AND FRIENDS. GOD BLESS 'EM ALL. HOPE THEY WILL BE THERE WHEN I COME HOME. TELL JOE WHEREVER HE IS TO GIVE 'EM HELL FOR US. GOD BLESS YOU.\n\nStrobing added ZZA, STANDBY, but it was the last word from Corregidor. WTA\/WVDM was finally off the air.\n\nAt WTJ Honolulu, the man who had been listening, Arnold Lappert, wept over his key.\n\nIn his headquarters on Bataan, Homma was in agony, believing the entire invasion had failed miserably. Then a staff officer reported that a white flag was waving over Corregidor. Homma at first refused to believe the good news. Only an hour previously he had learned that 31 boats had been sunk during the first invasion and the second landing, consequently, would have to be cancelled since there were only 21 landing craft left. When the general was convinced the white flag was not an American trick, he was so relieved he radioed Nakayama to disregard former orders and bring Wainwright to Bataan at once.\n\nAt 4:00 that afternoon, a Japanese assault boat scraped against a small dock in Cabcaben. Wainwright, leaning on his cane, stepped onto Bataan. It was a strange, bitter feeling to be once again on this peninsula of tragic memories.\n\nTwo cars drove the Wainwright party up the road, so recently the scene of the agonizing march of American and Filipino prisoners. Soon the cars stopped and Wainwright was led to the home of Silvestre Castillo. It was a small house, painted blue, surrounded by a luxuriant growth of mangroves. Pocked with fragment holes, it was the only standing structure in sight.\n\nWainwright, Beebe and the four other Americans\u2014Colonel Pugh, Major Dooley, Major William Lawrence and Sergeant Hubert Carroll\u2014waited on the open porch. To the south, out in Manila Bay, they could see Corregidor still erupting with shell bursts. A strong wind from the bay whipped up dense clouds of sand and dust. They waited tensely. A Japanese orderly brought cold water which they gratefully drank. After half an hour a group of Japanese newspapermen and photographers arrived.\n\nThe Americans were lined up on the lawn in front of the house and their pictures taken. At 5:00 P.M. a Cadillac came around the sharp curve from the north and stopped with a jerk near the house.\n\nGeneral Homma, looking crisp and military in olive-drab uniform, stepped out of the car. His shirt, brilliantly white, was open at the collar. On his chest were rows of decorations. At his side hung a sword.\n\nWainwright was a striking contrast to the barrel-chested, heavy-set Homma. Over six feet tall, he weighed only 160 pounds and looked as thin as a crane to the newsmen. His uniform, his best, was simply a khaki shirt and trousers. He wore no decorations or sword.\n\n\"Welcome to Cabcallo (Cabcaben),\" said Homma through his interpreter, a lieutenant. Though he could speak English, Homma wanted his staff to understand the conversation. \"You must be very tired and weary.\"\n\n\"Thank you, General Homma,\" said Wainwright.\n\nThey went to the porch and sat around a long table, Homma in the middle, facing Manila Bay. On his right was Major General Takaji Wachi, who had relieved Maeda as chief of staff, and Colonel Nakayama on his left. Behind the latter stood the interpreter and behind Homma was newsman Kazumaro Uno.\n\nAcross the table sat Wainwright with Beebe and Dooley to his left, Pugh and Lawrence to his right. The American commander handed a signed surrender note to Homma. The Japanese commander, without glancing at it, passed the note to his interpreter, who read it aloud. The note surrendered only the four islands in Manila Bay.\n\n\"I can not accept your surrender unless it includes all American and Philippine troops on the archipelago,\" said Homma.\n\n\"I can only surrender my men on Corregidor and the three other fortified islands. The troops in the Visayan Islands and Mindanao are no longer under my command. They are commanded by General Sharp, who in turn is under General MacArthur's high command.\"\n\nHomma was annoyed. Did Wainwright take him for a fool? He turned to his interpreter. \"Tell him we've heard broadcasts from Washington confirming his position as commander-in-chief of American forces in the Philippines.\"\n\nWainwright played out the game, insisting he only commanded the harbor defenses.\n\nHomma was shrewd. \"Ask him when he released Sharp from his command.\"\n\n\"Several days ago,\" replied Wainwright. \"Besides, even if I did command General Sharp's troops I have no means left for communicating with them. I have destroyed my radio equipment.\"\n\n\"Send a staff officer to Sharp,\" answered Homma. \"I will furnish the plane.\"\n\nBut Wainwright refused, insisting he had no authority over Sharp.\n\nHomma lost all patience. He banged the table with his fists, then said slowly in a cool, controlled voice, \"At the time of General King's surrender in Bataan, I did not see you. Neither have I any reason to see you if you are only the commander of a unit of the American forces. I wish only to negotiate with my equal, the commander-in-chief of American forces in the Philippines. Since you are not in supreme command, I see no further necessity for my presence here.\" He started to rise.\n\n\"Wait!\" said Pugh in alarm. He quickly conferred with Wainwright and Beebe.\n\nWainwright, defeat etched deeply on his face, slowly nodded in assent. \"In face of the fact that further bloodshed in the Philippines is unnecessary and futile, I will assume command of the entire American forces in the Philippines at the risk of serious reprimand by my government following the war.\"\n\nHomma was not convinced of Wainwright's sincerity. He said sternly in Japanese, \"You have denied your authority and your momentary decision may be regretted by your men. I advise you to return to Corregidor and think this matter over. If you see fit to surrender, then surrender to the commanding officer of the regiment on Corregidor. He in turn will bring you to me in Manila. I call this meeting over. Good day.\" He nodded slightly and walked back to his Cadillac.\n\nThe Americans' confusion was compounded by a vague translation of Homma's final statement. Wainwright nervously chewed the cigarette in his mouth to a pulp; Beebe was speechless.\n\n\"General,\" said Dooley earnestly to Wainwright, \"you'll have to arrange something. At your command all troops on Corregidor and the other harbor islands disarmed this afternoon. The Japs will slaughter our unarmed people.\"\n\nWainwright, leaning heavily on his cane, walked over to Colonel Nakayama, who had been detailed to take the Americans back to Corregidor, and asked, \"What do you want us to do now?\"\n\nThe Japanese interpreter translated this to Nakayama then replied, \"We will take you and your party back to Corregidor, and you can do what you damn please.\"\n\nIn the meantime, Colonel Pugh had buttonholed a friendly young Japanese lieutenant who formerly taught school in Osaka, and asked him what Homma had really said. \"Wainwright should return to Corregidor,\" explained the Japanese, \"and either resume fighting or surrender to the Japanese commander there.\"\n\nPugh hurried toward Wainwright, who was angrily walking up and down the yard, still burning from the rude treatment. After Pugh explained the situation, both men approached Colonel Nakayama.\n\n\"General Wainwright will surrender the entire American forces to General Homma unconditionally,\" said Pugh. \"Take us to General Homma and General Wainwright will dispatch me to Mindanao to instruct General Sharp to comply with his demands.\"\n\nFortunately newsman Uno, who had been raised in Utah and was sympathetic, was standing next to Nakayama. He translated. It took a moment for Nakayama to make up his mind, then he said, \"I shall go with you to Corregidor and safely turn you over to the commanding officer there. You will stay for the night. First thing tomorrow morning you will go to General Homma with a new surrender and an understanding to contact the other American forces in the Philippines.\"\n\nThe Wainwright party, Nakayama and Uno were driven to the Cabcaben pier in a car and a beer truck. It was getting dark as they headed for smoldering Corregidor in the Japanese assault boat. Uno, noticing an American lying on the heap of luggage, asked, \"What's the matter?\"\n\n\"I'm not feeling well.\" It was General Beebe; he was seasick.\n\nColonel Nakayama turned to Uno. \"This is what the Japanese soldier eats in place of candy. Give some to the Americans.\"\n\nUno passed the candy. Beebe didn't want any. Wainwright took several. So did Pugh, who ate one and said, \"Not bad.\" Uno didn't agree. They tasted like dog biscuits to him.\n\nFinally they approached the looming outline of Corregidor. The boat ground to a halt near North Point. Sergeant Carroll and another American soldier jumped into the chest-high water and started carrying Wainwright to shore. When they tripped, Wainwright waded the rest of the way. Beebe, deathly sick, was also carried to shore. Seeing he was too weak to stand, Nakayama ordered a halt while the general rested. Finally the group started up a steep bank toward the road, Beebe helped by two soldiers.\n\nThey walked up the road with Nakayama in the lead. Next came Uno with a flashlight. The stunted woods were only charred wood. The earth was churned by shells. Bodies lay in grotesque positions. It was a frightening scene.\n\nWainwright was shocked to see many campfires all over Corregidor and guessed the Japs had landed reinforcements. He was even more shocked to find the main Japanese line within 100 yards of the eastern entrance of Malinta Tunnel. He was led around Malinta Hill to the destroyed market place of San Jose in the middle of the island. Here he was introduced to Colonel Sato. The commander of the invasion force told the general that the Japanese controlled the area between the west entrance of Malinta and Morrison Hill on Topside. The tunnel had already been occupied and cleared of all Americans except those in the hospital. Now Colonel Sato was about to attack Topside.\n\nWainwright, convinced most of the men on Corregidor would be slaughtered if he didn't surrender unconditionally at once, agreed to sit down with Sato and draw up a document based on Homma's demands. At midnight Wainwright signed in the feeble light. He felt drained of energy as he was taken to the west entrance of Malinta Tunnel, past solemn groups of Americans and Filipinos. The weight of surrender was heavy. Many of his men reached out to seize his hand. Some patted his shoulder and said, \"It's all right, General, you did your best.\"\n\nHis eyes were full of tears by the time he entered the tunnel. It was an added blow to find the laterals filled with Japanese. He went to General Moore's office and said he had surrendered totally. Moore agreed nothing else could have been done.\n\n\"But I feel I have taken a dreadful step,\" said Wainwright gloomily. He and Pugh went to his quarters, escorted by a sentry. The two flopped on their cots and slept for the first time in two days.\n\n\u2014\n\nThe next morning Homma's intelligence officer, Lieutenant Colonel Hikaru Haba, and an interpreter entered Wainwright's quarters while the general was dressing. \"I am here to discuss the details of your surrender,\" said Haba.\n\nWainwright was convinced that the Japanese would massacre the men on Corregidor unless all his commanders in the field also surrendered immediately. The man who commanded the greatest number, General Sharp, would have to be convinced first. A messenger had to be sent to Mindanao with an explanatory letter. For this important and dangerous mission Wainwright decided to use his operations officer, Colonel Jesse Traywick. He sent for Traywick and told him to deliver the following letter personally to Sharp:\n\n> To put a stop to further useless sacrifice of human life on the fortified islands, yesterday I tendered to Lieutenant General Homma, the surrender of the four harbor forts in Manila Bay.\n> \n> General Homma declined to accept my surrender unless it included the forces under your command. It became apparent that the garrisons of these said forts would be eventually destroyed by aerial and artillery bombardment and by infantry supported by tanks, which have overwhelmed Corregidor.\n> \n> After leaving General Homma with no agreement between us I decided to accept in the name of humanity his proposal and tendered at midnight, night 6-7 May 1942, to the senior Japanese officer on Corregidor, the formal surrender of all American and Philippine Army troops in the Philippine Islands. You will therefore be guided accordingly, and _will_ repeat _will_ surrender all troops under your command both in the Visayan Islands and Mindanao to the proper Japanese officer. This decision on my part, you will realize, was forced upon me by means beyond my control.\n> \n> Colonel Jesse T. Traywick, Jr., G.S.C., my assistant chief of staff, G-3, who will deliver this to you, is fully empowered to act for me. You are hereby ordered by me as the senior American Army officer of the Philippine Islands to scrupulously carry out the provisions of this letter, as well as such additional instructions as this staff officer may give you in my name.\n> \n> You will repeat the complete text of this letter, and of such instructions as Colonel Traywick may give you, by radio to General MacArthur. However, let me emphasize that there must be on your part no thought of disregarding these instructions. Failure to fully and honestly carry them out can have only the most disastrous results.\n\nBy now Japanese were also occupying the other three Manila Bay islands: Forts Hughes, Drum and Frank. At nearby Hughes, ex-Navyman Cecil Browne and his mates were sitting tensely in rows. A Japanese major was walking up and down in front of them. He was furious. Some American or Filipino had stolen a can of food. The prisoners all knew the culprit was an American officer but no one spoke.\n\n\"Imperial Japanese food has been stolen,\" stormed the major. \"Someone will have to account for it and be executed or you will all suffer.\"\n\nThere was silence. When the guilty American officer made no move, a small, thin sailor stepped forward and said, \"I stole it.\"\n\nTo everyone's amazement the Japanese major beamed. \"You are like the brave Japanese soldier.\" He unsheathed his sword dramatically and tapped the puny American gob on the shoulder. Then he called to one of his men, who returned with a six-pound can of corned beef. The major presented it ceremoniously to the seaman.\n\n\u2014\n\nAt 5:00 that afternoon, Wainwright and five of his staff officers were led by Colonel Haba out of Malinta Tunnel's west entrance. Once again the general passed groups of his men. They looked hot, weary and hungry. All rose; some stood at attention and saluted; others took off their hats and held them against their chests. Wainwright, eyes welling with tears, raised his hand to his worn sun helmet.\n\nBut as the little party headed for the dock, a scrawl on one of the wrecked buildings indicated that Americans had surrendered everything but their sense of humor. It read: CORREGIDOR STILL STANDS! UNDER NEW MANAGEMENT.\n\nFar to the south, in Australia, MacArthur was solemnly penciling on a pad:\n\n> Corregidor needs no comment from me. It has sounded its own story at the mouth of its guns. It has scrolled its own epitaph on enemy tablets. But through the bloody haze of its last reverberating shots, I shall always see a vision of grim, gaunt, ghastly men, still unafraid.\n\n**2** Wainwright and his five officers were taken by assault boat to Lamao on Bataan. Here they waited for two hours, then ate rice and bony fish, their first food for two days. While another hour dragged on Wainwright reiterated his instructions to Colonel Traywick.\n\n\"Jesse, I'm giving you complete authority to handle this situation.\" Mindanao, he said, was his chief worry. Sharp must be made to comply with the surrender orders with absolutely no delay. In addition to the letter of instructions was another which Traywick was to use only in case Sharp refused to obey. It was a note permitting Traywick to place the Mindanao commander under arrest.\n\nThe general was completely broken up. His final words were, \"Jesse, I'm depending on you to carry out these orders.\"\n\nIt was dark when the group headed north in a car. They followed the route of the Death March to Balanga, Orani and San Fernando. Then they turned southeast, arriving in the completely forsaken streets of Manila just before 11:00 P.M. The car drove up to radio station KZRH, where the party was met by Lieutenant Hisamichi Kano of the Propaganda Corps. Kano, who had been schooled in New York and New Jersey, greeted the general affably and gave the Americans some fruit.\n\nWainwright went over the prepared speech he was soon to broadcast. It was in stilted English and when Kano saw Wainwright was having trouble reading it, he said, \"General, I can straighten it out for you.\"\n\nThe young Japanese officer took the statement into his office, rewording it in more colloquial English.\n\nAt 11:43 P.M. Wainwright sat down at a small round bamboo table in front of a microphone. In the monitor room was Kano, ready to switch off the general if he varied the script.\n\nWainwright spoke every word as written, in a voice husky with suppressed emotion:\n\n> This is Lieutenant General J. M. Wainwright. Message for General William F. Sharp, commanding the Mindanao and Visayan Forces. Repeat for General Sharp, commanding the Mindanao and Visayan Forces. Anyone receiving this message please notify General Sharp at once. The message: By virtue of the authority vested in me by the President of the United States, I, as Commanding General of the United States Forces in the Philippine Islands, hereby resume direct command of Major General Sharp, Commander of the Visayan-Mindanao Forces and of all troops under his command. (Repeat message.)\n> \n> I now give a direct order to William Sharp. I repeat, those receiving it please notify General Sharp at once. Subject: Surrender! To Major General William F. Sharp, Jr., commanding the Visayan and Mindanao Forces. The message: To fully stop the further useless sacrifice of human life on the fortified islands, yesterday I tendered to Lieutenant General Homma, the commander in chief of the Imperial Japanese Forces in the Philippines, the surrender of the four harbor defense forts of Manila Bay. General Homma declined to accept my surrender unless it included the forces under your command. It became apparent that the garrisons of these forces would be eventually destroyed by aerial and artillery bombardment and by infantry supported by tanks, which have overwhelmed Corregidor. After leaving General Homma, with no agreement between us, I've decided to accept, in the name of humanity, his proposal, and tendered at midnight of 6-7 May, 1942, to the Senior Japanese officer on Corregidor, a total surrender of all American and Filipino Army troops in the Philippine Islands. You will therefore be guided accordingly and will, I repeat, _will_ , surrender all troops under your command to the proper Japanese officer.\n> \n> This proceeding on my part, you will realize, was forced upon me by means entirely beyond my control. Colonel Jesse T. Traywick, Jr., General Staff Corps, my assistant chief of staff (G-3), who will deliver this letter to you personally is truly empowered to act for me. You are hereby ordered by me as a senior American Army officer in the Philippine Islands to scrupulously carry out the provisions of this letter as well as such additional instructions as this staff officer will give you. You will repeat the complete text of this letter and such other instructions as Colonel Traywick will give you by radio to General MacArthur. However, let me emphasize that there must be on your part no thought of disregarding these instructions. Failure to fully and honestly carry them out can have only the most disastrous results.\n\nWainwright, his voice growing even more choked, now broadcast instructions to Colonels Horan and Makar in north Luzon. After adding other explicit instructions to Sharp, Wainwright said:\n\n> The Japanese Army and Navy will not cease their operations until they recognize the faithfulness of execution of these orders. These orders must be carried out faithfully and accurately, otherwise the Imperial Japanese Army and Navy will continue their operations. If and when such faithfulness of execution is recognized, the commander in chief of the Japanese forces in the Philippine Islands will order that all firing be ceased.\n\nThe general coughed and was momentarily speechless. Then he said:\n\n> Taking all circumstances into consideration, and...\n\nThere was another long pause. Then the announcer, Marcela Victor Young, broke in with a blunt announcement and signed off. It was 12:20 A.M., May 8.\n\nThe ordeal over, Kano invited the emotionally drained Wainwright and the accompanying officers into his office. He poured them drinks from a hoarded bottle of Scotch while the Americans tried to comfort their stricken commander.\n\n\u2014\n\nThe speech was heard by many Americans and Filipinos on Mindanao. Those who didn't know Wainwright personally insisted it was a fake. Friends recognized his voice, even though it was huskier than usual. They felt he was broadcasting under duress.\n\nGeneral Sharp was in a quandary. That morning he'd received a radio from Wainwright relinquishing command of the Visayans and Mindanao. Later he'd received this message from Melbourne:\n\n> WAINWRIGHT HAS SURRENDERED. FROM NOW ON COMMUNICATE ON ALL MATTERS DIRECT WITH ME. HAVE YOU COMMUNICATED WITH CHYNOWETH?\n> \n> MACARTHUR.\n\nSharp radioed MacArthur the gist of the Wainwright broadcast, asking for further instructions. His problems were complicated by the deteriorating military situation in his vast command. The islands of Panay and Cebu were occupied. His own headquarters, transferred from Del Monte to Malaybalay, 40 miles farther inland, were now being threatened. On April 29 a strong Japanese detachment had landed at Cotabato and, against spirited but useless resistance, had driven north past Lake Lanao. Another detachment had landed in Cagayan and Bugo and was driving down the Sayre Highway. Advance units had already reached the Del Monte airfields and threatened to break through the last roadblocks in front of Malaybalay.\n\nSharp's escape route to the south was also cut off. Some of the invaders landing at Cotabato had driven east to Pikit, the southern terminus of the Sayre Highway. There they were being joined by other Japanese coming from the southeast coast of Mindanao.\n\nWhen MacArthur received Sharp's message, he radioed Washington: I PLACE NO CREDENCE IN THE ALLEGED BROADCAST BY WAINWRIGHT. He was extremely annoyed. The chaotic surrender situation would never have occurred if Washington had let him retain control of the Philippines. If Wainwright had not been given over-all command, the Japanese would have been forced to accept surrender of the four fortified islands in Manila Bay.\n\nAfter careful thought and much rewriting, MacArthur and his staff composed a radiogram they felt would be a guide to Sharp. At 4:45 A.M. it was sent.\n\n> ORDERS EMANATING FROM GENERAL WAINWRIGHT HAVE NO VALIDITY, IF POSSIBLE SEPARATE YOUR FORCE INTO SMALL ELEMENTS AND INITIATE GUERRILLA OPERATIONS. YOU, OF COURSE, HAVE FULL AUTHORITY TO MAKE ANY DECISION THAT IMMEDIATE EMERGENCY MAY DEMAND. KEEP IN COMMUNICATION WITH ME AS MUCH AS POSSIBLE. YOU ARE A GALLANT AND RESOURCEFUL COMMANDER AND I AM PROUD OF WHAT YOU HAVE DONE.\n\nWhen Sharp got this message, he felt his course of action was by no means clarified. Though he now had complete authority to act on his own judgment and had been relieved of responsibility to Wainwright, he could not ignore the latent threat of the Manila broadcast. On the brink of military disaster himself he decided to delay his final decision until he talked to Wainwright's emissary, Colonel Traywick, who reportedly had left Manila the morning after the broadcast.\n\nIn the meantime, he sent messages to his island commanders releasing them from his command. He also told his own officers that surrender was imminent; they could escape into the mountains and fight as guerrillas if they wished.\n\nSergeant Frank Trammell, formerly a bomber radioman and now Sharp's chief radio repairman, was getting ready to leave with a group planning to hide out in the rugged mountains of east Mindanao. Before leaving he radioed a last message to his wife, Norma, in Escondido, California.\n\n> PLEASE DON'T WORRY IT IS ALWAYS DARKEST BEFORE THE DAWN LOVE FRANKLIN TRAMMELL\n\nThen Trammell and the others headed southeast over the grassy plains.\n\n\u2014\n\nJesse Traywick was already on Mindanao, having arrived by plane the day before with Colonel Haba. After the two were driven to Cagayan, the headquarters of Major General Saburo Kawamura, commander of the detachment attacking south down the Sayre Highway, Traywick had suggested he simply be driven along this highway to Sharp's headquarters.\n\nKawamura had a different plan. Hundreds of letters were written by hand, signed by Traywick, and dropped by plane on the afternoon of May 9.\n\nTO COMMANDING OFFICER OF THE FRONT LINE:\n\n> I, having a very important message from Lieutenant General J. M. Wainwright, must deliver it personally to Major General William F. Sharp.\n> \n> When any one who receives this letter, he should raise a white flag visible to the Japanese Army and stop firing and the Japanese Army will do the same.\n> \n> When both sides stop firing, you send a U.S.A. officer to the front line with a white flag. I will meet him.\n> \n> In order to deliver my message to General SHARP as soon as possible you should be prepared to furnish a guide and a car for me.\n> \n> JESSE T. TRAYWICK, JR.,\n> \n> COLONEL. G. S. C.\n> \n> UNITED STATES ARMY\n\nMay 9, 1942\n\nTraywick protested in vain that this procedure was unrealistic and had little chance of success. At dusk a Japanese officer drove him south toward Del Monte on the Sayre Highway. Finally they dismounted and walked toward the Japanese front lines. Traywick was given a small map, a canteen of water and pointed southeast. The Japanese officer told him American trucks were 60 kilometers ahead.\n\nTraywick insisted it was a ridiculous mission, but argument was useless. The American headed down a jungle trail in the pitch dark. He forded a river and climbed a steep ravine. Within an hour the trail ended and he pushed into dense undergrowth. He waded and swam across several small rivers and climbed more ravines until, at 2:00 A.M., May 10, he came to a nipa hut. He shouted. There was no answer. He walked into an empty house, lay down on a cot and tried to sleep. Just as he was dozing off a kitten jumped on the bed, curling up by his neck.\n\nThe two slept until dawn. When Traywick awoke he discovered he was in hilly, jungle country with mountains to the west. He started south and soon saw a Japanese machine-gun nest. He shouted. Five Japanese soldiers turned, looked at him in astonishment as he advanced with a piece of white sheet on a stick.\n\nThe weary, tattered colonel was driven back to Cagayan. This time he insisted on going up the Sayre Highway.\n\n\"You can't go far,\" said Kawamura through an interpreter. \"The bridges are blown.\"\n\n\"I'll go as far as I can, then walk.\"\n\nAccompanied by Colonel Haba, Traywick was again driven toward Del Monte. When they came to a blown bridge, the two men crossed on foot and continued down the Sayre Highway. It was a hot trip. Once they rested near three Japanese bodies. After they rose to leave, Traywick faced the dead soldiers and saluted. Colonel Haba bowed and patted the American approvingly on the shoulder.\n\nFinally, that afternoon, they reached the American lines and were driven to Malaybalay. As he approached the building housing Sharp's headquarters, Traywick rehearsed what he would say. If none of his arguments worked he would have to arrest the general.\n\nHe walked up to the tall, thin Sharp, handing him the letter of instructions. To Traywick's surprise, Sharp made no protests but readily agreed to go forward and surrender to Kawamura. First he sent radiograms to his island commanders:\n\n> AS I HAVE NOT YET SURRENDERED, THE INSTRUCTIONS GIVEN YOU YESTERDAY RELEASING YOU FROM MY COMMAND ARE WITHDRAWN. I RESUME COMMAND AND DIRECT YOU TO CEASE ALL OPERATIONS AGAINST THE JAPANESE ARMY AT ONCE. YOU WILL RAISE A WHITE FLAG AND AWAIT THE ARRIVAL OF MY STAFF OFFICER WHO WILL MAKE THE TERMS OF THE NEGOTIATIONS FOR SURRENDER OF THE FORCES UNDER YOU. THIS IS IMPERATIVE AND MUST BE CARRIED OUT IN ORDER TO SAVE FURTHER BLOODSHED. ACKNOWLEDGE.\n> \n> SHARP COMMANDING\n\nAt 7:15 P.M. he sent another message, this one to MacArthur:\n\n> I HAVE SEEN WAINWRIGHT'S STAFF OFFICER AND HAVE WITHDRAWN MY ORDER RELEASING COMMANDERS ON OTHER ISLANDS AND DIRECTED SURRENDER. DIRE NECESSITY ALONE HAS PROMPTED THIS ACTION.\n\nMen all over Mindanao began to surrender. Others were heading for the hills. Corporal Durward Brooks, former radio combat operator of the 28th Bombardment Squadron, was with a group of nine escapees. They had tried to get their lieutenant to join them but his answer was, \"I couldn't live in the mountains. I'm from Boston.\"\n\nIn Washington, General Marshall was reading a message from MacArthur:\n\n> I HAVE JUST RECEIVED WORD FROM MAJOR GENERAL SHARP THAT GENERAL WAINWRIGHT IN TWO BROADCASTS ON THE NIGHT OF THE 7\/8 ANNOUNCED HE WAS REASSUMING COMMAND OF ALL FORCES IN THE PHILIPPINES AND DIRECTED THEIR SURRENDER GIVING IN DETAIL THE METHOD OF ACCOMPLISHMENT. I BELIEVE WAINWRIGHT HAS TEMPORARILY BECOME UNBALANCED AND HIS CONDITION RENDERS HIM SUSCEPTIBLE OF ENEMY USE.\n\n**3** John Horan didn't hear Wainwright's broadcast. Since the fall of Bataan, when he was promoted to full colonel and given authority by Wainwright to change his Casual Regiment to an official one, Horan and his staff had kept on the move, going from village to village in the wild mountains north of Baguio.\n\nOn May 10 he telephoned his regimental headquarters in the Batang Buhay Gold Mine to tell Captain R. H. Brown that since there could be no more effective fighting until after the rainy season, he and his staff were going to hide out in the Kalinga area for several months. Brown told him of the emotional Wainwright broadcast, demanding that he and Colonel Nakar, who was operating similar guerrilla detachments to the north, surrender immediately.\n\nHoran was amazed, seeing no reason at all to surrender. Most of his ammunition was gone but he had safely dispersed his Bontoc fighters to their homes. He and his staff knew they would be safe with the faithful Kalinga tribe, and he was confident MacArthur would return as soon as the rains stopped.\n\nHoran debated with himself four days. He knew he had the nucleus of a powerful guerrilla force. Already his men had become proficient in blowing up bridges, hijacking Japanese supplies and conducting deadly hit-and-run raids. His Bontocs made excellent guerrillas and several officers showed promise of developing into reliable leaders. One, Walter Cushing, a half-Mexican, half-Irish civilian mining engineer, was already a legend in the mountains, his sudden raids keeping the Japanese constantly terrorized. With storybook daring he even raided warehouses, carting away scores of cases of ammunition through enemy lines.\n\nBut Horan kept thinking about Wainwright's orders to put a stop to the further useless sacrifice of human lives. This meant, he guessed, that the men of Corregidor were being held as hostages.\n\nAfter the third successive sleepless night, Horan decided, on the morning of May 14, to hike up the mountain to Lubuagan and surrender. He was consoled by the thought that the Japanese were getting very little except a few American officers. A month before he had made Cushing second in command, and told him, \"In case I'm captured or surrender, ignore any orders I might give to surrender. You will carry on.\"\n\n**4** Surrender in the Visayan Islands was even more complicated. Most of the commanders of the various islands heard the Wainwright broadcast, yet felt it was made under duress or was an enemy trick. The situation was further complicated by the strained relations between Sharp and many of his island commanders. Some of these younger men thought the general was too old to command fighting troops. They also felt he was unrealistic in his dealings with Filipinos and enlisted men. On an inspection trip early that year Sharp had reportedly asked a Filipino, \"Whose corporal are you?\" The man, who knew little English, was confused. \"You are not your company commander's corporal,\" continued Sharp. \"Not your battalion commander's corporal; not your regimental or even division commander's corporal. You are _my_ corporal.\" From that day on the corporal refused to obey his sergeant, his captain or even his colonel, telling them he was only supposed to take orders from General Sharp. Soon the entire regiment was out of control.\n\nWhen Colonel Albert Christie, commander of Panay Island, and an exceptionally forceful and blunt man, received Sharp's radiogram instructing him to surrender, he promptly replied that he didn't see \"even one small reason\" why he should surrender just because \"some other unit has gone to hell or some Corregidor shell-shocked terms\" had been made. He questioned Sharp's authority to order his surrender, adding:\n\n> TO SATISFY ME I MUST HAVE MACARTHUR'S OKAY: OTHERWISE IT MAY BE TREASON.\n\nThe next day, May 11, Sharp again ordered Christie to surrender at once, assuring him that MacArthur had been informed of his actions and that Lieutenant Colonel Allen Thayer was flying to Panay with instructions. He warned:\n\n> YOUR FAILURE TO COMPLY WILL PRODUCE DISASTROUS RESULTS.\n\nChristie, whose immediate commander was General Chynoweth on Cebu, still refused to obey. Months previously Chynoweth had made definite plans for guerrilla resistance in just such a case. Panay was geographically suited for guerrilla operations; its forces were fairly well trained and though short of weapons and ammunition, had sufficient food supplies. Christie replied:\n\n> YOUR RADIO SURRENDER OF MY FORCES SOUNDS TOTALLY UNNECESSARY. I STRONGLY URGE YOU TO HAVE THE APPROVAL OF THE WAR DEPARTMENT THROUGH MACARTHUR. I AM CONSULTING MY IMMEDIATE COMMANDER, GENERAL CHYNOWETH. IN THIS DELICATE SITUATION PLEASE DO NOT ISSUE ME ANY PEREMPTORY ORDERS THAT WILL EMBARRASS OR GET US INTO MUTUAL CONFLICT. RATHER DO I FEEL A FREE HAND IN CARRYING OUT MY MISSION UNINFLUENCED BY ANY HYSTERIA INHERENT IN LOCAL ACTION. NO ARMY SURRENDERS PORTIONS STILL FREE, INTACT, AND HAVING A GOOD CHANCE OF HELPING THE GENERAL MISSION. MAKE ME INDEPENDENT. DO NOT PUT ME ON THE SACRIFICE BLOCK.\n\nThe exasperated Sharp didn't answer this strong message. He had already sent Colonel Thayer with a letter of instructions and a personal letter. These, he felt, would convince the stubborn colonel.\n\nOn that same day MacArthur, extremely distressed by the unhappy situation, radioed Marshall the details of Sharp's surrender. He concluded:\n\n> I HAVE HAD NO WORD FROM GENERAL CHYNOWETH SINCE APRIL 12. I NOW HAVE NO MEANS OF COMMUNICATION WITH THE PHILIPPINES.\n\nWhat happened in the Visayans was out of his hands.\n\n\u2014\n\nBrigadier General Bradford Chynoweth was hiding with his staff, a few hundred Filipino soldiers and about sixty Navy officers and civilians in the dense mountainous forests north of Cebu City. When he learned about Wainwright's broadcast, he held a council of war with his chief of staff. They concluded the order was not binding since it was obviously issued under duress.\n\nHe decided to carry on, writing messages to Christie on Panay, Colonel Roger Hilsman on Negros and Colonel Theodore Cornell on Leyte. But on May 12 his trusted adviser, Dr. Emilio Osme\u00f1a, son of the vice-president, told him the Filipino troops were so demoralized they were leaving for their homes.\n\nBecause of this Chynoweth decided to take his staff to Leyte and then Panay, making that island headquarters for Visayan guerrilla operations. While completing plans for the dangerous trip the next day a Christie message to Sharp was heard faintly on the radio:\n\n> WHERE IS GENERAL CHYNOWETH? TELL HIM TO COME BACK OVER HERE AND COMMAND US.\n\nChynoweth was cheered to hear the blunt, buoyant words of his subordinate. A few minutes later a message from Sharp came over the radio in the clear again ordering everyone in the Visayans to surrender. Chynoweth by now was suspicious of anything coming from Mindanao and he told his message center to accept no more radios from Sharp. Just then a courier arrived from Negros Island with a message from Colonel Hilsman. It stated that Sharp's order to surrender was authentic and that Sharp was sending a staff officer to negotiate the surrender of Chynoweth's forces.\n\nChynoweth now was utterly confused. He realized that Sharp was still in contact with General MacArthur and apparently MacArthur had approved the surrender. Even so Chynoweth hesitated, deciding to wait until the evening radio broadcast. Perhaps then MacArthur himself would clarify the situation.\n\nThat evening Chynoweth's aide, young Tom Powell, dejectedly returned from the radio. \"The only word over KGEI,\" he said, \"was a statement of General MacArthur that all contact with the Philippines was now cut off.\"\n\nThis sounded final, thought Chynoweth. MacArthur had crossed them off his book. The next few hours were torture. It was agony to determine what to do. He had gone through all the Army schools up to the War College, but had never been taught how, when, or when not to surrender.\n\nHe ordered all his scattered people to gather at the White Horse Inn in Sudlon Forest. Then he sent a message to the Japanese in Cebu City that he was expecting a staff officer from Sharp. He and Osme\u00f1a talked most of the night. The vice-president's son was in a difficult position. \"You can leave us and return quietly to your home,\" said Chynoweth, \"or you can go with us if we have to surrender.\" Osme\u00f1a was afraid the Japanese would force him to take an administrative job in the occupied government. \"Someone will have to continue government,\" said the general. \"They're your people. It would be better for you to do it than someone else.\"\n\n\"But I'm afraid the Japanese will ask me to do things inimical to the American forces. I won't do that.\"\n\nThe next morning they walked to the White Horse Inn. There they met Captain Gray, the emissary from Mindanao, who had written orders from Sharp and Wainwright. The Japanese, said Gray, were holding American troops on Corregidor, ready to turn their guns on them in case the surrender didn't go through.\n\nChynoweth gathered his men and told them he was surrendering. Any who wished could hide out in the hills. The following morning, May 16, the general led the sad march from White Horse Inn. With him were Dr. Osme\u00f1a, several staff officers, a few naval officers and fifty Filipinos. A day later, Japanese came up the road from Cebu City to meet them and accept the surrender. Japanese newsmen took pictures while several movie cameramen recorded the scene.\n\nThe chief administrative officer of Japanese forces on Cebu, Colonel Kawakami, a tall, dark, somber man, came forward. Chynoweth was too overcome with emotion to say a word. Kawakami turned to Colonel Irvine Scudder, the general's chief of staff, and negotiations began.\n\nChynoweth, tongue-tied and miserable, did nothing as his staff officers carried on the necessary business. Then the prisoners were put in cars and taken to the large white administration building in Cebu City. But first they were toured around the burned buildings and docks of the city.\n\n\"Don't you think it was a terrible thing to destroy a city and cut off the water supply?\" asked Kawakami through a stern-faced interpreter.\n\n\"I regret the horrors of war, too,\" said Chynoweth. \"But I don't consider that we were the ones to initiate these horrors.\"\n\nKawakami said the auxiliary pumping machinery in the water line had been put out of action and the key was missing. Where was it?\n\n\"I cannot give you any information to help any of your activities,\" replied the general.\n\nKawakami turned to Scudder. The colonel told him where to find the key but refused to answer other questions. The courteous Japanese commander now told Chynoweth he was putting him in a separate house as a hostage. \"You will be given an armband and the freedom of the city if you promise not to escape.\"\n\n\"If I ever see any chance of escaping, I will do so,\" said Chynoweth belligerently. Just then a radio in a nearby room burst out with \"The Stars and Stripes Forever.\" It was too much for the general. Again he was so choked he couldn't talk.\n\n\"I know just how you feel,\" said Kawakami. He ordered tea.\n\n\u2014\n\nChristie was still unconvinced. The day after Chynoweth's surrender he radioed Sharp asking what MacArthur's final answer had been. By now Sharp had lost all patience and replied curtly:\n\n> NO FURTHER COMMENTS FROM YOU ARE DESIRED. ACKNOWLEDGE THIS MESSAGE AND STATE ACTIONS TAKEN AT ONCE.\n\nThat same day, May 18, Sharp's emissary, Colonel Thayer, finally landed on Panay Island. \"Wainwright's surrender of Corregidor,\" he told the recalcitrant Christie, \"is conditional on the surrender of all forces in the Philippines.\"\n\nChristie was now confronted with the same problem faced by Horan and Chynoweth. Was holding Panay Island worth the lives of all the Corregidor prisoners? He reluctantly agreed to surrender.\n\n**5** These were days of gnawing worry to Wainwright as he tensely waited in Manila. He couldn't understand why some of his commanders were taking so long to surrender. Didn't they know they were endangering the lives of thousands of comrades by their delays?\n\nHis conqueror, General Homma, was not in a triumphant mood. Even though he finally had won the Philippines, he knew he was in disfavor with the High Command. General Count Hisaichi Terauchi, commander of the Southern Army, had recently inspected the Philippines and had been very unhappy with Homma's lenient treatment of Filipino civilians. Homma admitted he had ordered his troops not to regard the Filipinos as enemies, to respect their customs and traditions and religion. His men had been forbidden to burn, pillage or rape. He also confessed he had prevented a pamphlet describing the exploitation of the Philippines by the Americans from being distributed.\n\n\"We must face the facts,\" he said. \"The Americans never exploited the Philippines. It is wrong to make such false statements. They administered a very benevolent supervision over the Philippines. Japan should make better and more enlightened supervision.\"\n\nTerauchi also criticized the general for not occupying Malaca\u00f1an, the presidential palace.\n\n\"Even the United States didn't occupy Malaca\u00f1an,\" protested Homma. \"It would not be proper for the Japanese to take it over.\"\n\n\"You are much too lenient,\" said Terauchi sternly. He advised his subordinate to discontinue his soft methods.\n\nHomma knew his military career was in danger, but it was not in his nature to change his methods. He still believed the best way to win the Filipino people was not through beatings but with enlightened leadership. He went ahead with plans for the immediate release of all Filipino prisoners.\n\nIn the meantime he ordered a victory parade through the streets of Manila. As his conquering army passed in review, thousands of Filipinos stood silently on the sidewalks. With undisguised resentment they watched Vargas, Laurel, Aquino and other members of the new \"Quisling\" government march by carrying little Japanese flags. Then came a Filipino band. As it approached Homma's reviewing stand, it suddenly began playing a stirring march. For the first time the onlookers broke into enthusiastic cheers.\n\nThe band was playing, \"The Stars and Stripes Forever.\"\n\nNot long after this, Homma was relieved of his command and forced to retire in semi-disgrace because of his persistent leniency. Four years later he would be tried and executed as a war criminal by the man he defeated, MacArthur.\n\n# 24\n\n# _\"Go in and Get a Hit\"_\n\n**1** When the Netherlands Indies collapsed after the Battle of Java Sea, many Japanese naval leaders, led by Yamamoto, felt the first goals of conquest had been far too modest. Japan should push her destiny to its limit. The Army planners disagreed, insisting that Japan should now consolidate her successes and proceed cautiously.\n\nIt was a difference in strategic concept. The Army General Staff agreed that neutralization operations in certain areas were practical, providing they were on a small scale. Japan, they argued, had captured immensely rich resources. These should be held and developed. From now on, the enemy should be forced to do the attacking. Naturally it would mean a prolonged war, but this was the general plan agreed on before the war started. Hasty improvisations could only lead to catastrophe.\n\nThe Navy believed systematic operations should be started against Australia, Hawaii and India. These would naturally result in great naval battles. And, as at the Battle of Java Sea, the enemy would be destroyed. In proof, they asked their critics to look at the record. Instead of the anticipated 25 per cent naval losses a mere 25,000 tons had been sunk in conquering an empire. The biggest warship lost had been a destroyer. The Navy persisted in pressing for immediate action. Being on the defensive was disadvantageous. The enemy should be kept on the defensive. The next conquest should be Australia, which was obviously being built up as a springboard for the counterattack.\n\nColonel Takushiro Hattori, Tojo's former secretary and now Army chief of operations, opposed this venture. Australia was twice the area of China with a population of only 7,000,000. Its land communications were poor. The Australians would defend their homeland tenaciously. Such an invasion, he told Captain Sadatoshi Tomioka, his opposite number in the Navy, would require the main body of the Combined Fleet and twelve infantry divisions. The shipping for the Army alone would run to 1,500,000 tons.\n\nIf it went through, he said, the Army would have to strip its defenses against the USSR in Manchuria and cut down drastically on the China front. \"Total national power must be concentrated in the prosecution of a protracted war. It is vital at this point to develop resilience in national power and war potential under a sound plan.\"\n\nWhen Tomioka insisted, Hattori picked up a cup. \"The tea in this cup,\" he said, \"represents our strength.\" He dumped it on the floor. \"You see, it only goes so far. If the plan is approved I will resign.\"\n\nGeneral Sugiyama, chief of the Army General Staff, backed up Hattori. On March 4 a provisional compromise was reached. Three days later at a liaison conference a policy of future war guidance was adopted. The Australian scheme was dropped, but the Army agreed to extend its gains in less ambitious projects such as an amphibious operation on Port Moresby, a town only 400 miles north of Australia, on the southern coast of New Guinea, the second largest island in the world. Also approved by Hattori, after a staff study, was the conquest of three islands off the northeast coast of Australia: Samoa, Fiji and New Caledonia. This, he agreed, was a practical move for it would cut the lifeline between Australia and the United States.\n\nHattori didn't know it, but Admiral Yamamoto was pressing for a much more ambitious invasion: Midway and the Aleutians. This plan was being hotly disputed within the Navy itself. How could it be supplied, argued the Imperial General Headquarters Navy Section, even if won? How could an American counterattack be driven off?\n\nYamamoto refused to budge. Unlike many Japanese militarists he was well aware of the immense potential of American industry. \"We can carry through for one year, some way,\" he said, \"but after that I don't know.\" He insisted Japan's only possible chance of success was to lure the U.S. Pacific Fleet into the open and destroy it. And it had to be done before the great balance of naval power presently enjoyed by Japan was whittled down.\n\nIf the position of commander-in-chief of Combined Fleet had been selected by votes, Isoroku Yamamoto would have been the heavy choice among naval officers. Young officers admired him because he was a strong leader. Unlike many other Japanese admirals, brought up in the British tradition of gentlemanliness, he commanded his fleet with an iron hand. He was a fighting leader.\n\nHe was an expert poker and bridge player, following the instinctive hunches of a born gambler. He ran the fleet with the same bold, imaginative daring. Fliers swore by him because he was one of the few older officers who believed in aviation. In 1915, when he was young and unknown, an American writer had asked him what he considered the war vessel of the future. \"The most important ship of the future,\" he promptly answered, \"will be a ship to carry airplanes.\"\n\nUnder pressure the Navy Section finally bowed to the insistent Yamamoto and a directive to invade Midway and the Aleutians was issued on April 16. Yet it was far from a complete victory for Yamamoto. No date was set. The admiral begged for immediate action; time was running out. The Navy Section refused to be rushed. On April 18 a bolt from the blue, the Doolittle raid, gave Yamamoto the ammunition he needed.\n\nRecalling the panic that seized the inhabitants when a Russian naval force suddenly appeared near Tokyo Bay during the Russo-Japanese War, he insisted the Midway invasion be executed immediately. Unless Midway was captured promptly, he argued, air and sea patrols in front of the homeland would have to be heavily strengthened. Even so it would be impossible to defend Tokyo from future attacks.\n\nThe deadline for invasion, he said, would have to be June 7. A few days' delay would mean a month's postponement, since his carrier air groups needed pre-dawn moonlight for night action. Even the most violent opponents of the plan now had to admit the threat from the east was more immediate than that from Australia. All opposition had vanished because of Doolittle. The Naval General Staff readily agreed to postpone the Fiji-Samoa invasion and attack Midway on June 7.\n\nWhen the Midway project was submitted to the Army, Colonel Hattori was led to believe it was strictly a measure to defend the homeland. Since the occupation of the Aleutians would cut the line between America and Russia, he advised General Sugiyama to approve the Yamamoto plan.\n\n**2** While Yamamoto's Combined Fleet enthusiastically threw itself into preparation for the great Midway operation, the Port Moresby invasion started as scheduled. On May 3 Vice Admiral Shigeyoshi Inouye's Fourth Fleet seized Tulagi, a small island 25 miles north of Guadalcanal. The next day the Port Moresby Invasion Force left the northern New Britain port of Rabaul in 14 transports escorted by 1 light cruiser and 6 destroyers and covered by the light carrier _Shoho_ , 4 heavy cruisers and 1 destroyer.\n\nThe Japanese still didn't have the slightest suspicion that their \"Purple\" Code had been broken months before by William Friedman and a team of U.S. army cryptanalysts. Consequently they didn't know that Admiral Chester Nimitz, who had replaced Kimmel at Pearl Harbor, was aware of their proposed invasion of Port Moresby. In fact, Nimitz, knowing the Japanese operations would start on May 3, had already assembled a naval force consisting of 2 carriers, 6 heavy cruisers, 2 light cruisers and 11 destroyers in the Coral Sea off northeast Australia under Rear Admiral Frank Jack Fletcher.\n\nWhen Fletcher learned of the Tulagi landing he immediately launched an air attack from his flagship, the carrier _Yorktown_. In three strikes 99 planes hit Tulagi. Though the fliers reported they had sunk 2 destroyers, a freighter and 4 gunboats, and severely damaged other warships, they had done relatively little damage.\n\nThis attack gave the Japanese their first warning of the presence of an American task force. Vice Admiral Takeo Takagi, then some distance to the north with his Carrier Striking Force, immediately rushed south to engage the enemy with his 2 heavy carriers, _Zuikaku_ and _Shokaku_ , 2 heavy cruisers and 6 destroyers.\n\nThe arrival of Takagi in the Coral Sea gave Admiral Inouye an appreciable superiority over Fletcher: an extra light carrier, 2 destroyers and 7 submarines. In addition, close by was the group supporting the invasion transports: 2 light cruisers, a seaplane carrier and 3 gunboats.\n\nFletcher was well aware he was outnumbered and outgunned but when he learned from intelligence reports that the Port Moresby Invasion Force would probably round the eastern tip of New Guinea on May 7, he was determined to stop it. For the first time since Pearl Harbor, the U.S. Navy was steaming forth with capital ships to try and check the Japanese tide of conquest. Two of the four carriers in the Pacific Fleet were being put on the line. If these two great ships were sunk, as the _Prince of Wales_ and _Repulse_ had been in the last previous attempts to stop a Japanese convoy, victory in the Pacific could well be postponed indefinitely.\n\nFrom intelligence reports Fletcher learned that the Port Moresby Invasion Force had already left Rabaul and would probably round the eastern tip of New Guinea on May 7. It was his job to stop them. Since they were probably protected by three enemy carriers and several heavy cruisers, he knew he had to employ his weaker forces cautiously.\n\nOn the morning of May 7, the day after Wainwright's surrender, a Japanese search plane discovered the oiler _Neosho_ and destroyer _Sims_ and reported excitedly it had found an American carrier and cruiser. Soon high-level bombers struck in two waves. Then 36 dive bombers joined the attack. The _Sims_ sank soon after noon. The _Neosho_ , mortally wounded, was helplessly adrift.\n\nIn the meantime, Fletcher's task force was still unobserved. That morning he had ordered the support group under British Rear Admiral J. G. Crace to push ahead and attack the Port Moresby Invasion Force while he lagged behind to carry on an air duel with the Japanese carriers.\n\nAt 11:00 A.M., while the Japanese carrier planes were concentrating on the _Sims_ and _Neosho_ , 93 planes from the _Yorktown_ and _Lexington_ found the light carrier _Shoho_ and began an aggressive bomb and torpedo attack. About 160 miles away comrades of the attackers strained to hear this action at their battle stations, but the voices of the pilots were almost inaudible because of static. Suddenly the voice of Lieutenant Commander Robert Dixon, leader of a scout bomber squadron, came in strong and clear: \"Scratch one flat-top! Dixon to carrier. Scratch one flat-top!\"\n\nThe _Shoho_ went down at 11:36 A.M. For the first time in the five-month-long Pacific war, a Japanese ship larger than a destroyer had been sunk.\n\nBy this time Admiral Inouye had become so concerned he ordered his transports to turn back and wait until the seas had been cleared of Americans. That afternoon visibility decreased and squalls limited aerial observation. By midnight the two great naval forces had lost contact with each other.\n\nFletcher was determined to find the enemy again and attack his carriers. At 6:25 the next morning 16 search planes were launched from the _Lexington_. For almost two hours there was no news. Then at 8:15 A.M. a radio report came from Lieutenant J. G. Smith. He had sighted the Japanese striking force. Thirteen minutes later he radioed:\n\n> TWO CARRIERS, FOUR HEAVY CRUISERS, MANY DESTROYERS, STEERING 120 DEGREES, 20 KNOTS. THEIR POSITION 175 MILES, ROUGHLY NORTHEAST.\n\nFletcher immediately ordered both his carriers to launch air strikes. At 10:57 A.M. 39 planes from the _Yorktown_ sighted the carriers _Zuikaku_ and _Shokaku_ , heavily screened by heavy cruisers and destroyers. Lieutenant Commander Joe Taylor, Torpedo Squadron 5, led the first American attack on a large carrier. Though all torpedoes missed or failed to explode, two bombs hit the _Shokaku_ , starting several furious fires.\n\nLess than an hour later 24 planes from _Lexington_ found the same targets. They did little damage, scoring only one bomb hit on the already burning _Shokaku_.\n\nAt the same time, 70 planes from the two Japanese carriers were hitting _Lexington_ and _Yorktown_. An 800-pound bomb pierced the flight deck of _Yorktown_ , penetrating to the fourth deck, but fires were skilfully brought under control. _Lexington_ was not so lucky. Two torpedoes ripped into her port side, while two small bombs hit the main deck forward and the smokestack structure.\n\nThe first battle in history between aircraft carriers was over by noon. It was also the first naval engagement in which opposing ships had never seen each other or exchanged shots. At that moment it looked like a clear-cut victory for Fletcher. He had sunk a light carrier, a destroyer and three small vessels, while losing only a destroyer and an oiler. Then that afternoon two explosions rocked the wounded _Lexington_ , setting off raging fires. More and more explosions followed but the battle to save the \"Lady Lex\" continued all through the afternoon in spite of heavy casualties. Though everyone, including about 600 men on their first cruise, knew the ship's own ammunition stores might go off any moment, no one faltered. There was no panic.\n\nThe fight was hopeless. At 5:07 P.M. Rear Admiral Aubrey Fitch, commander of the carrier group, called down from his bridge to the _Lexington_ 's skipper, Captain F. C. Sherman, \"Well, Ted, let's get the men off.\"\n\nRafts were thrown overboard. The men neatly lined up their shoes on the flight deck and calmly began going over the sides. It was all as orderly as a drill. One group even raided the ship's service store for ice cream while waiting in line.\n\nOne of the last to leave the deck was Lieutenant Noel Gaylor, a fighter pilot. That afternoon he had knocked down 4 Zeros in a single flight to become the Navy's leading air ace with 8 victories. He dove 50 feet into the sea and swam for several minutes. Then he suddenly turned and soon was climbing up a rope to the _Lexington_ 's deck.\n\n\"What did you come back for?\" asked a friend.\n\n\"Oh, I got a bit lonely out there. I didn't know any of those guys. When are you fellows goin' to come?\"\n\nAfter everyone was off the ship\u2014not a man was drowned and even Captain Sherman's dog, Wags, was saved\u2014Admiral Fletcher ordered the destroyer _Phelps_ to sink the smoldering wreck. At about 8:00 P.M. four torpedoes drove into the carrier's starboard side. She shuddered and steam rose in great clouds.\n\nOn a nearby cruiser, newsman Stanley Johnston watched the big ship settle rapidly. She stayed upright until waves folded over her.\n\n\"There she goes,\" said an officer of the _Lexington_ standing next to Johnston. \"She didn't turn over. She is going down with her head up. Dear old _Lex_. A lady to the last.\"\n\nWith the sinking of the _Lexington_ , the Coral Sea battle became a tactical victory for the Japanese. The strategic victory went to Fletcher. Just before dusk Admiral Inouye had definitely postponed the Port Moresby invasion and recalled the transports. Fletcher's primary mission had been accomplished. For the first time in the war a Japanese invasion had been turned aside.\n\nWhen Yamamoto learned of the postponement he was furious. At midnight he ordered the cautious Inouye to continue the pursuit of Fletcher and destroy his \"remnants.\" The unharmed _Zuikaku_ and her escorts obediently turned southeast at 2:00 A.M., May 9, but it was too late. Fletcher had vanished.\n\nThe battle caused a sensation all over the world. The New York _Times_ on May 9 announced in headlines:\n\n> JAPANESE REPULSED IN GREAT PACIFIC BATTLE, WITH 17 TO 22 OF THEIR SHIPS SUNK OR CRIPPLED; ENEMY IN FLIGHT, PURSUED BY ALLIED WARSHIPS\n\nThere was jubilation and immense relief in Australia. The Australians felt the battle fought just off their northeast coast had saved their country. Americans in Sydney and Brisbane were treated as heroes. Nothing was too good for the Yanks.\n\nThe Japanese claims were just as wild, announcing they had sunk 2 battleships and 2 carriers. On May 9 the headlines of Japan's leading newspaper, _Asahi Shimbun_ , read:\n\n> DREAM OF COUNTERATTACK ON JAPAN LOST IN CORAL SEA ONLY 3 CARRIERS LEFT; SIGN OF FALL OF UNITED STATES\n\nThere was also jubilation in Germany. Hitler said, \"After this new defeat the United States warships will hardly dare to face the Japanese Fleet again, since any United States warship which accepts action with the Japanese naval forces is as good as lost.\"\n\nImperial Headquarters firmly believed their Navy had scored a tremendous victory. They were positive the _Lexington_ and _Yorktown_ had been sunk, leaving only two or three American carriers in the entire Pacific. Their optimism wasn't dimmed when _Zuikaku_ and _Shokaku_ steamed into port a few days later and it was learned both would have to be scratched from the Midway operation. _Shokaku_ would take a month to repair and _Zuikaku_ had lost too many planes and pilots.\n\nBy now even the former critics of the Midway plan were no longer worried. Japan was striking with the greatest sea force ever assembled: more than 200 ships, including 11 battleships, 8 carriers, 22 cruisers, 65 destroyers, 21 submarines and about 700 planes.\n\nThe seizure of Midway and the Aleutians was only of secondary importance. Yamamoto's main objective was to lure the remnants of Nimitz's fleet into the open where it could be completely destroyed. His plan was simple. On June 3 the Northern Force would raid the Aleutians. In addition to destroying enemy installations, this would protect Yamamoto's northern flank and momentarily draw attention away from the Midway area. The next day his most powerful unit, the 1st Carrier Striking Force of Admiral Nagumo, would launch a pre-invasion air strike on Midway, wiping out enemy planes. The landing itself would come on June 6.\n\nThis was the bait in the trap. Obviously Nimitz would react since Midway was so important and steam out of Pearl Harbor to do battle, never suspecting a force outnumbering him two or three to one was waiting in ambush. It would be a slaughter.\n\nUnfortunately for Yamamoto, his plan was based on a fallacy, surprise. As at Coral Sea, decoded Japanese messages had already warned American naval leaders of the coming operation. This intercepted information, however, was spotty. The Japanese messages only referred to the invasion point as \"AF.\" Washington thought \"AF\" stood for the island of Oahu. Nimitz disagreed. He was sure it meant Midway.\n\nThe admiral visited Midway and asked its commander what additional equipment and men were needed to stop a largescale amphibious attack. When he returned to Hawaii he sent back every plane he could spare: 16 Marine dive bombers, 7 Wildcat fighters, 30 Navy patrol flying boats, 18 Flying Fortresses and 4 B-26's. He also ordered additional anti-aircraft batteries installed, the garrison brought up to 2000, and three submarine patrol arcs set up. He did this knowing that if he was wrong, he might lose the entire fleet. He also knew it was a risk that had to be taken.\n\nIn the middle of May, Nimitz ordered Fletcher to leave the South Pacific area and return to Hawaii as fast as possible. The carriers _Hornet_ and _Enterprise_ were also recalled.\n\nSince there was still doubt as to the invasion point, Commander Joseph J. Rochefort of Nimitz's Combat Intelligence staff suggested Midway be ordered to send a fake, uncoded message complaining of the breakdown of its distillation plant. Nimitz agreed and the decoy message was sent.\n\nTwo days later the cryptanalysts in Pearl Harbor's top-secret \"Black Chamber\" intercepted a Japanese dispatch: AF was low on fresh water. At last Nimitz knew for certain that he'd been right. Midway was Yamamoto's target.\n\nEven with this information Nimitz realized he was still the underdog. His fleet would be greatly outnumbered. He had only two carriers ready for action. The third, the badly damaged _Yorktown_ , was limping toward Pearl Harbor from its battle in the Coral Sea and wouldn't arrive until May 27.\n\nHe knew Yamamoto's main purpose was to lure him into a fight, yet even against these tremendous odds he had to take the challenge.\n\n**3** Yamamoto's great plan moved into gear on May 26 when Rear Admiral Kakuji Kakuta's Second Carrier Striking Force sailed out of Ominato harbor at noon and headed east toward the Aleutians. His planes were to attack Dutch Harbor on June 3.\n\nThe following day the muscle of the Midway invasion, Vice Admiral Chuichi Nagumo's First Carrier Striking Force, left Hashirajima with 21 ships, including 2 heavy carriers and 2 light carriers. On this force depended ultimate success. Nagumo was fresh from his triumphs at Pearl Harbor and the Indian Ocean. Always noted for his fire and dynamism, some of his more perceptive junior officers were noticing that his once vigorous fighting spirit was gone. He had aged perceptibly.\n\nNagumo no longer took the initiative. This particularly worried his operations officer, Commander Minoru Genda, probably the most highly thought of and brilliant of the rising officers. It was he who had perfected the technique for the shallow-running torpedo attack at Pearl Harbor and had conceived the basic idea for the Midway operation. Lately Genda's plans were being approved by Nagumo almost without consideration. It worried Genda to find these plans coming out as formal orders, completely unchanged. He was self-confident, but not so self-confident he didn't realize anyone can make mistakes.\n\nThe following day Vice Admiral Moshiro Hosogaya, commander of the Northern Force, set sail with the rest of the Aleutian ships. And that same evening, far to the south, the fleet of transports carrying the 5000 troops scheduled to land on Midway embarked with its strong escort from the island of Saipan.\n\nA few hours later, on the morning of May 29, the last two units left: first the 16 warships of the Midway Invasion Force under Vice Admiral Nobutake Kondo and finally the 32 warships of the Main Force.\n\nFrom Yamamoto down to the last seaman there was complete confidence that the most decisive victory in naval warfare would soon be scored by the Emperor's Combined Fleet.\n\nTo stop Yamamoto, Admiral Nimitz had only 8 cruisers, 15 destroyers and 3 carriers. And one of these carriers, _Yorktown_ , was so badly damaged some experts estimated it would take about three months to repair. Nimitz called in his two commanders, Rear Admirals Frank Jack Fletcher and Raymond Spruance, a last-minute substitute for Halsey, who was suffering from a skin disease. Nimitz told them what to expect, detailing almost exactly Yamamoto's actual plans. Their orders were to \"inflict maximum damage on enemy by employing strong attrition tactics.\" In other words, they were to strike again and again from the air. Then Nimitz gave each commander a special Letter of Instruction:\n\n> In carrying out the task assigned...you will be governed by the principle of calculated risk, which you shall interpret to mean the avoidance of exposure of your force to attack by superior enemy forces without good prospect of inflicting, as a result of such exposure, greater damage on the enemy.\n\nTwo days before Yamamoto's last force left Japan, Spruance sailed out of Pearl Harbor with the carriers _Enterprise_ , _Hornet_ , 6 cruisers and 9 destroyers. Two days later, Fletcher steamed out with the _Yorktown_ , _2_ cruisers and 6 destroyers. Because of the remarkable efforts of more than 1400 workmen the damaged carrier had been patched up in two days.\n\n\u2014\n\nIn spite of bad weather, the transports carrying the Midway invaders were only 1000 miles west of their goal by May 31. In fact, they were advancing too fast.\n\nYamamoto's own 32 ships, the Main Force, were behind schedule, their fueling operations delayed by heavy seas and poor visibility. Yamamoto was not in a happy mood. In addition to stomach trouble, he had recently learned that Operation K, a plan to send two flying boats over Hawaii for reconnaissance, had bogged down. Now he wouldn't know what strength Nimitz had in Pearl Harbor.\n\nHe received another disappointment on June 1 when submarines failed to reach their positions on the cordon lines supposed to be laid northwest and west of Hawaii. They were two days behind schedule and the admiral was pushing toward Midway almost completely in the dark about Nimitz's strength or intentions. That same day a disturbing scrap of news came from Lieutenant Commander Yahachi Tanabe of the submarine I-168, who reported intensive flying patrols from Midway. The island seemed to be on a strict alert; many construction cranes were visible, indicating an expansion of defenses.\n\nSix hundred miles ahead of Yamamoto, Admiral Nagumo knew even less than his superior. Since there was radio silence, he had no idea Midway was already alerted. His Striking Force, unaided by radar, was shrouded in a mist so thick that ships could see only the dim, unearthly outlines of their closest neighbors.\n\nThe following day, June 2, was even worse. Nagumo and his officers stared anxiously from the bridge of _Akagi_. They were completely blanketed in heavy fog. It was as if they were alone. Another problem was worrying the admiral. The entire Midway plan had been hastily drawn up and pushed through without ironing out certain inconsistencies. One of these was the double tactical mission he had been given. His planes were to attack Midway on June 4 as a preparation for the landing two days later. His other mission was to contact and destroy Nimitz's fleet. How could he do both? The second mission required freedom of movement and secrecy. If he hit Midway, mobility and secrecy were gone.\n\nHe decided the time had come to settle the matter of priority and gathered his staff. Captain Oishi, his senior officer, spoke first. \"The Combined Fleet operation order gives first priority to the destruction of enemy forces. Cooperation with the landing operation is secondary. But if we do not neutralize the Midway-based air forces as planned, our landing operations two days later will be strongly opposed and the entire invasion schedule will be upset.\"\n\n\"But where is the enemy fleet?\" asked Nagumo.\n\nOishi admitted nobody knew, then said, \"Even if they are already aware of our movements and have sortied to meet us, they cannot be far out from base at this moment and certainly can't be near us. I think, therefore, that the first thing for us to do is to carry out the scheduled air strike on Midway.\" The others agreed.\n\n\u2014\n\nThe next morning, June 3, planes from Admiral Kakuta's carriers bombed Dutch Harbor in the Aleutians. Little damage was done. The raid's main purpose, to divert Nimitz's attention to the north, also failed since the American admiral had long since known where the main attack was going to fall.\n\nAt 9:00 that same morning Ensign Jack Reid, flying a Catalina on patrol from Midway, suddenly saw what looked like a mass of miniature ships in a backyard pool about 30 miles ahead.\n\n\"Do you see what I do?\" he asked his copilot.\n\n\"You're damned right I do!\"\n\nDodging behind clouds, Reid followed the ships, finally reporting at 11:00 A.M. that 11 ships less than 700 miles from Midway were heading east at 19 knots. It was, he thought, the main Japanese fleet.\n\nReid had seen only the invasion transports. They, in turn, had spotted Reid. This information was relayed to Yamamoto. Late that afternoon he received an even more alarming message: at 4:24 P.M. nine Flying Fortresses, apparently based on Midway, had bombed the Midway-bound transports but scored no hits.\n\nThe complete confidence of the day before deserted the bridge of Yamamoto's flagship _Yamato_. No one had ever imagined the transports would be discovered before Nagumo's air strike on Midway.\n\n\u2014\n\nThat evening Fletcher and Spruance, 300 miles east-northeast of Midway, were about 400 miles east of the position where Nagumo was planning to launch his planes the next morning for the strike on the island. When Fletcher, in over-all command of the two forces, had finally learned that Japanese ships had been sighted he correctly guessed they were only transports and their escorts, not, as reported by Ensign Reid, the main Japanese fleet. Figuring the enemy carriers would approach Midway from the northwest, he turned his ships to the southwest at 7:50 P.M. He wanted to be about 200 miles north of Midway. From here he could launch his planes against Nagumo's carriers the next day.\n\nThe Americans, from admirals to seamen, knew a great and decisive engagement was approaching. The day before, Spruance, by visual signal, had told his men:\n\n> An attack for the purpose of capturing Midway is expected. The attacking force may be composed of all combatant types including four or five carriers, transports and train vessels. If presence of Task Forces 16 and 17 remains unknown to enemy, we should be able to make surprise flank attacks on enemy carriers from position northeast of Midway. Further operations will be based on results of these attacks, damage inflicted by Midway forces, and information of enemy movements. The successful conclusion of the operation now commencing will be of great value to our country. Should carriers become separated during attacks by enemy aircraft, they will endeavor to remain within visual touch.\n\nThat night there was a feeling of tension and excitement in the wardrooms and mess halls. Rumors spread that the Japanese code had been broken and a trap was being set for the Midway invaders. The first big victory, long hoped for by every Navy man, could come the next day.\n\n\u2014\n\nAt 2:45 A.M., June 4, the loudspeakers of Nagumo's flagship, _Akagi_ , blared. Air crews tumbled out of bunks to the deafening roar of warming motors. The noise woke Commander Genda, Nagumo's young operations officer. He hurried from the sick bay, where he had been confined, to the bridge, his eyes feverish. \"I am very sorry, sir, to have been absent so long,\" he told Nagumo.\n\nThe admiral put an arm around his shoulder. \"How are you feeling?\"\n\n\"I have a slight temperature but am feeling much better now.\"\n\nEveryone on the bridge was buoyed by the sight of Genda. Still uninformed by Yamamoto that the invasion transports had already been sighted by the Americans, Genda ordered launching preparations for the first wave. The four carriers were now only 240 miles northwest of Midway and steaming full into the wind.\n\nAt 4:30 A.M. the wind gauge on _Akagi_ indicated the required velocity and Nagumo said, \"Commence launching.\" A Zero fighter sped down the flight deck and, as the crew shouted enthusiastically, waving hands and caps, took off into the black sky. Eight Zeros followed, then 18 dive bombers.\n\nAbout two miles to port, _Hiryu_ was launching. The illuminated decks of _Kaga_ and _Soryu_ were also being cleared of planes. In fifteen minutes a total of 108 were in the air and forming. A moment later they circled and those on the ships below could see neatly grouped red and blue lights heading toward Midway Island.\n\nAt the same time, reconnaissance planes were being catapulted from the battleship _Haruna_. Half an hour later, delayed by catapult trouble and balky engines, the rest of the search planes left the heavy cruisers _Tone_ and _Chikuma_.\n\nGenda watched them go with some misgivings. They were bound on a one-phase search. He knew it should have been two-phase but he was short of aircraft and wanted to throw as many into the Midway attack as possible. The flight deck of the _Akagi_ was now empty and strangely silent, but just before the sun rose at 5:00 A.M. the loudspeaker blared, \"Prepare second attack wave!\"\n\nInstantly the uproar began again as bells clanged and the planes of this wave were brought up by elevator and rolled into place. All 18 of the _Akagi_ planes carried torpedoes, as did the 18 of _Kaga_. The other two, the light carriers _Hiryu_ and _Soryu_ , were filling their decks with dive bombers. Thus Nagumo was ready, even if enemy ships should unexpectedly appear. Excitement and expectation swept the Striking Force.\n\n\u2014\n\nFifteen Flying Fortresses, commanded by Lieutenant Colonel Walter Sweeney, had already taken off from Midway before dawn with orders to attack the Japanese transports heading for the island.\n\nCatalina search planes had also left Midway, looking for the enemy's Striking Force. At 5:25 A.M. one of these planes, PBY-5, approached Nagumo's force. A moment later Lieutenant Howard Ady and his copilot, Lieutenant William Chase, were staring in awe at a mass of ships ahead of them. It was hard for Ady to believe what he saw. He said later it was \"like watching a curtain rise on the Biggest Show on Earth.\"\n\nENEMY CARRIERS, Ady radioed Midway. Then the big plane dodged behind clouds, circled and approached the Striking Force from the rear.\n\n_Enterprise_ intercepted this message at 5:34 A.M. Eleven minutes later came another:\n\n> MANY ENEMY PLANES HEADING MIDWAY BEARING 320\u00b0, DISTANT 150.\n\nThen at 6:03 A.M. those aboard the _Enterprise_ heard Ady's third report:\n\n> TWO CARRIERS AND BATTLESHIPS BEARING 320\u00b0, DISTANT 180, COURSE 135\u00b0, SPEED 25.\n\nThis information was relayed to Admiral Fletcher on the _Yorktown_. Though he realized this was the Japanese Striking Force, Fletcher decided not to attack until his own search planes returned and there was more explicit information. He radioed Spruance:\n\n> PROCEED SOUTHWESTERLY AND ATTACK ENEMY CARRIERS WHEN DEFINITELY LOCATED. I WILL FOLLOW AS SOON AS PLANES RECOVERED.\n\nTwo hundred miles to the west, air-raid sirens were shrieking on Midway. Radar had picked up the approaching 108 attackers from Nagumo's carriers. Planes of all types were scrambling. Bombers and search planes scattered. Fighters sped to the northwest to intercept the Japanese. Six Avenger torpedo planes and four Army Marauders, also carrying torpedoes, headed toward the enemy carriers.\n\nMarine fighters sighted the first Japanese bombers 30 miles out of Midway, coming in at 13,000 feet in a rigid V formation with Zero fighters above. The Marines climbed to 17,000 feet and dove on the horde of attackers. There was a savage battle but the Americans, outnumbered two to one, had no chance in their antique Brewster Buffaloes and outclassed Wildcats.\n\nThe Japanese bombers continued unopposed to their target. Climbing to 14,000 feet, they began releasing their bombs at 6:30 A.M. A few minutes later \"Val\" dive bombers swept in through heavy anti-aircraft fire, blasting buildings, the powerhouse on Eastern Island, oil tanks on Sand Island and the seaplane hangar.\n\nIt was over in twenty minutes. In the greatest Marine air loss of the entire war, 15 of the 25 fighter pilots had died. The 10 survivors were acrimonious. One, Captain Philip White, later said, \"It is my belief that any commander who orders pilots out for combat in an F2A (Buffalo) should consider the pilot as lost before leaving the ground.\"\n\nThe damage to Midway itself was not critical. Though many buildings were burning and the gasoline system was damaged, few men had been killed and the main purpose of the attack had failed. Midway was still in the airplane business. Five minutes after the attack, at 6:55 A.M., 16 U.S. Marine dive bombers took off and headed for Nagumo's carriers.\n\nThe Japanese raiders were returning with mixed feelings. They figured they had shot down 42 planes while losing only 5, but their leader, Lieutenant Joichi Tomonaga, realizing he had not knocked out Midway's air force, radioed Nagumo at 7:00 A.M.:\n\n> THERE IS NEED FOR A SECOND ATTACK.\n\nBy the time this message reached Nagumo his flagship, _Akagi_ , was in a state of alarm. The PBY of Ady and Chase had been seen. Fighters had gone after the persistent Catalina but it kept hiding in the clouds.\n\nFive minutes after Tomonaga's message from Midway was received, a bugle sounded Air Raid. Everyone on _Akagi_ 's bridge peered anxiously to the south. Though there were fairly heavy clouds at 6000 feet, the sky was clearing. It was going to be a beautiful day. Then a screening destroyer suddenly hoisted a flag signal: ENEMY PLANES IN SIGHT.\n\nThe 4 Army Marauders and 6 Avengers which had left Midway just after the attack on the island were coming in for their own raid.\n\nThose on the bridge of _Akagi_ saw 4 planes, apparently torpedo bombers, approach from 20 degrees to port. Japanese fighters swooped down, shooting down 3. The other fled to the east. A moment later a lookout shouted, \"Six medium land-based planes approaching, 20 degrees to starboard! On the horizon!\"\n\nDestroyers and cruisers fired. Then the battleship _Kirishima_ opened up with her main batteries. The 6 planes\u20144 Marauders and 2 Avengers\u2014kept coming in spite of the wall of fire. Now _Akagi_ 's guns fired and 3 Zeros dove on the stubborn Americans.\n\nWithin a minute 3 attackers were in flames. The other 3 kept boring in, finally dropping their torpedoes and swinging sharply to the right. Two escaped but the lead plane skimmed over the _Akagi_ , nearly grazing the bridge, then suddenly burst into flames and tumbled into the sea.\n\nThe _Akagi_ abruptly swerved and the torpedoes churned by harmlessly. This narrow escape convinced Nagumo that a second attack on Midway should be made. He ordered those planes of the second wave carrying torpedoes for attack on ships to de-arm and reload with bombs. Since _Hiryu_ and _Soryu_ carried only dive bombers for this wave, they had nothing to do but wait, but there was frantic action on _Akagi_ and _Kaga_. The torpedo planes on the flight decks had to be lowered to their hangars one by one for the arduous job of removing torpedoes and rearming with bombs.\n\nThe tension on the bridge of _Akagi_ finally abated. Then at about 7:40 A.M. Nagumo was handed a message from one of the tardy search planes from the cruiser _Tone_ :\n\n> TEN SHIPS, APPARENTLY ENEMY, SIGHTED. BEARING 010\u00b0, DISTANT 240 MILES FROM MIDWAY. COURSE 150\u00b0, SPEED MORE THAN 20 KNOTS. TIME 0728.\n\nThe enemy surface force wasn't supposed to arrive for two days. The American position was quickly plotted. Only 200 miles away! At 7:47 A.M. Nagumo curtly radioed the _Tone_ plane:\n\n> ASCERTAIN SHIP TYPES AND MAINTAIN CONTACT.\n\nThe rearming of the planes with bombs was instantly stopped since torpedoes were considered more effective against ships. Before the _Tone_ search plane could answer Nagumo's impatient message, enemy planes were sighted.\n\nThese were the 16 Marine dive bombers from Midway. At 7:55 A.M. the flight commander, Major Lofton Henderson, ordered his planes to make a glide-bombing attack since his inexperienced pilots knew little about dive-bombing. All headed straight for _Hiryu_.\n\nZeros quickly pounced on the Marines, knocking down 8. The others gallantly kept on course. When their bombs finally dropped _Hiryu_ was enveloped in smoke and splashes. Finally the smoke cleared and those on _Akagi_ saw to their relief that the _Hiryu_ was still steaming ahead, completely untouched.\n\nBy this time an identification finally came from the _Tone_ search plane: ENEMY SHIPS ARE FIVE CRUISERS AND FIVE DESTROYERS. There was great relief. Without carriers the enemy was of no immediate danger. There was no time for congratulations. Fifteen American land-based heavy bombers were over _Hiryu_ and _Soryu_. From 20,000 feet the big bombers dropped their loads. Great geysers shot up on all sides of the two carriers. Again no damage was done.\n\nThe bombers were the 15 Flying Fortresses which had left Midway before dawn to attack the invasion transports. They had been diverted en route and sent against these richer targets. As they started home unharmed they reported 4 hits.\n\nA few minutes later another message came from the _Tone_ search plane:\n\n> ENEMY FORCE ACCOMPANIED BY WHAT APPEARS TO BE AIRCRAFT CARRIER BRINGING UP THE REAR.\n\nSome on _Akagi_ doubted there was a carrier. If so why hadn't they already been attacked? At worst, what was there to fear? The three harmless attacks from Midway had proved the enemy level of attack efficiency was very low. Just then a screening destroyer signaled that more enemy planes had been spotted. The destroyer fired several rounds, then quickly stopped in embarrassment. The planes were Japanese, the first of the 108 raiders returning from the Midway strike.\n\nAt that moment, 8:30 A.M., the _Tone_ search plane reported two additional ships in the enemy force, probably cruisers. Nagumo now realized by the size of the force that it must contain at least one carrier. Genda agreed and recommended that an immediate attack be made on these ships. Since many planes had not yet been rearmed with torpedoes, Nagumo was in a quandary. Should those torpedo planes presently armed with 800-kilogram bombs be sent off at once with the 36 dive bombers of the two light carriers _Hiryu_ and _Soryu_?\n\nGenda agreed that, under the circumstances, bombs had to be used; they could still do heavy damage. What worried him was the absence of fighter protection. All the Zeros of the second wave had been sent aloft to intercept the attackers from Midway.\n\nWhile this discussion was going on, Nagumo received a message from Rear Admiral Tamon Yamaguchi, commander of the two light carriers and a possible eventual successor to Yamamoto.\n\n> CONSIDER IT ADVISABLE TO LAUNCH ATTACK FORCE IMMEDIATELY.\n\nNagumo hesitated and finally turned to Genda in perplexity. The young commander had a new worry. He was anxiously watching the hordes of planes returning from Midway hover above their carriers. Many were obviously down to their last few gallons of gas. Several were smoking from hits. The fighters were also low on fuel after their successful defense against the three American attacks. They would have to accompany the attackers as cover or the whole operation might fail. Even if they had enough gas to reach the American force and protect the bombers, it was a certainty not a one would have enough to get home safely.\n\nGenda pondered. Almost every one of the bomber and fighter pilots was a personal friend. He turned to Nagumo. \"I believe all our aircraft should first land and refuel. Then we can launch the attack.\"\n\nTaking his advice, Nagumo ordered the decks cleared so the waiting planes could land. One by one the planes on _Akagi_ 's deck were lowered to their hangar. When there was enough space on the flight deck the tired Midway attackers began landing. By 8:55 A.M. although recovery of the Midway planes was still incomplete, Nagumo impatiently ordered by blinker:\n\n> AFTER COMPLETING RECOVERY OPERATIONS, FORCE WILL TEMPORARILY HEAD NORTHWARD. WE PLAN TO CONTACT AND DESTROY ENEMY TASK FORCE.\n\nIt wasn't until 9:18 A.M. that the last planes landed. Then Nagumo raised speed to 30 knots and turned sharply to the east-northeast, away from Midway. In the meantime, the 4 carriers had been feverishly rearming and refueling for attack. In addition to fighter escort, 36 dive bombers and 54 torpedo bombers would be thrown at the U.S. Pacific Fleet. Excitement increased. The big victory was approaching.\n\n\u2014\n\nAt 6:07 A.M. Fletcher had ordered Spruance to attack the enemy carriers as soon as they were definitely located. At first, Spruance planned to make his strikes at 9:00 A.M. when he'd be approximately 100 miles from his targets. When reports from Midway told of the heavy Japanese attack on that island Captain Miles Browning, the chief of staff Spruance had inherited from Halsey, vigorously pressed for an earlier attack. Browning, a temperamental but extremely perceptive man, pointed out that this might well catch the Japanese carriers while refueling their aircraft.\n\nNo two commanders could have been more different than Halsey and the man who had replaced him. Spruance, already known in the Navy as a \"brain,\" was quiet, studious and thoughtful. He always projected himself into the future and was aggressive only when he thought it well worth the risk.\n\nHalsey, like his army counterpart, George Patton, knew the value of publicity. Spruance, like Courtney Hodges, had a distaste for the public eye and did not enjoy being interviewed by correspondents. Consequently, little was known of him. Even to most of those on the _Enterprise_ Spruance was enigmatic. He would pace the deck of his ship for several hours each day for exercise. In his cabin he would study charts covering areas of operations posted on the bulkhead.\n\nTo Spruance the Midway operation had resolved itself along these clear-cut lines: the Japanese had to be prevented from capturing Midway; since the enemy was probably much stronger, the Americans must retain the element of surprise and not be surprised; therefore, the Japanese carriers had to be hit at the earliest possible moment with everything available.\n\nSpruance's first important decision was to take Browning's advice. His second was just as important: he ordered every operational plane, except patrol craft, to make the attack. At 7:02 A.M. they began leaving the two carriers: 60 dive bombers, 20 fighters and 29 torpedo planes. They would have barely enough fuel to return home, if lucky. Spruance had taken the chance Nagumo feared to.\n\nPlanes from the third American carrier, _Yorktown_ , didn't start leaving until 8:38 A.M. By 9:06 A.M. 12 dive bombers, 6 fighters and 17 torpedo planes had left the deck of Fletcher's flagship and were winging toward Nagumo's Striking Force.\n\nIt was twelve minutes later that Nagumo made his abrupt turn to the north-northeast, away from Midway. Nagumo did this because he feared another air strike from Midway. He didn't realize, of course, that this sharp turn to the left was also turning the Striking Force away from the 144 American carrier planes now trying to find it.\n\nA few minutes later _Hornet_ 's 35 dive bombers and 10 fighters reached the point where they were supposed to intercept the Japanese carriers. The leader of these planes, Commander Stanhope Ring, saw only clouds to his right\u2014Nagumo was behind them. Ring concluded that the Japanese had raised their speed and continued on toward Midway. And so he turned southeast, away from the enemy carriers.\n\nThree groups of lumbering American torpedo planes were heading almost directly for Nagumo. The first to reach the Striking Force were the 15 unescorted torpedo planes from _Hornet_. They had not followed Ring toward Midway since their leader, Lieutenant Commander John Waldron, had a hunch the Japanese would do exactly what they had done.\n\nWaldron was a seamy-faced, square-jawed man from South Dakota. One of his great-grandparents had been a Sioux Indian. His men looked like a band of Mexican revolutionaries, each wearing a .45 pistol under the left shoulder, another at the hip and a hunting knife.\n\nThe night before Waldron had written his wife, Adelaide:\n\n> I believe that we will be in battle very soon\u2014I wish we were there today. But as we are up to the very eve of serious business, I wish to record to you that I am feeling fine. My own morale is excellent and from my continued observance of the squadron\u2014their morale is excellent also. You may rest assured that I will go in with the expectation of coming back in good shape. If I do not come back\u2014well, you and the little girls can know that this squadron struck for the highest objective in naval warfare\u2014\"to sink the enemy.\"\n> \n>....I love you and the children very dearly and I long to be with you. But I could not be happy ashore at this time. My place is here with the fight....\n\nThen he wrote another letter to his men:\n\n>....My greatest hope is that we encounter a favorable tactical situation, but if we don't, and the worst comes to the worst, I want each of us to do his utmost to destroy our enemies. If there is only one plane left to make a final run in, I want that man to go in and get a hit. May God be with all of us. Good luck, happy landings and give 'em hell.\n\nBelow him to the right eight miles away lay the enemy carriers in a box-like formation. It was obvious they had seen the glint of sunlight on the wings of Torpedo Squadron 8 for all ships were zigzagging to avoid attack.\n\nAnti-aircraft fire reached out from the ships screening the four carriers. Then about 30 Zeros streaked toward the unprotected torpedo planes. Waldron waggled his wings, signaling his men to follow, and headed forward at full speed. Soon a plane plunged into the water.\n\n\"Was that a Zero?\" Waldron asked his radioman in the rear seat.\n\nIt was American. Moments later a second torpedo plane went down. \"Let's go back and help him, sir,\" Radioman Robert Huntington suggested to his pilot, Ensign George Gay.\n\n\"To hell with that. We've got a job to do.\"\n\nA third and fourth American plane pinwheeled into the sea. Yet not one of the survivors wavered from his course, continuing through the wall of anti-aircraft fire, even though it was now apparent that chance of survival was almost impossible. The much faster Zeros were swooping down like hawks on chickens.\n\nSoon half of Torpedo Squadron 8 was dead. Still Waldron bored in, followed by his gallant remnants. The barrage from the ships became even heavier. The air was filled with acrid black smoke.\n\nSuddenly Gay saw his leader's plane burst into flames. Waldron stood up and tried in vain to get out of the flaming cockpit as it crashed into the waves.\n\nGay saw another comrade spin into the water and disappear. It reminded him of a day long ago when he was throwing orange peels from a speedboat. There were only two other planes left: one to the left, one ahead and just below the nose of his own plane. Gay lowered his plane's nose. There was nothing ahead. He looked to the left. Nothing. He was the only one left. He remembered Waldron's letter, the instructions for the last plane \"to go in and get a hit.\"\n\nHe heard his radioman say, \"Mr. Gay, I'm hit!\"\n\n\"You hurt bad? Can you move?\" When there was no answer Gay turned. Huntington's head hung limp. Gay felt a sharp jab in his upper left arm. There was a hole in his jacket sleeve. Shifting the stick to his left hand, he ripped the sleeve and pressed a bullet from the wound. When he tried to put it in his pocket as a souvenir, the safety belt held it shut. He put the bullet in his mouth.\n\nHe was approaching a carrier. It turned to starboard. Gay also swung to the right and pressed the button to release the white-nosed \"pickle\" under the belly of his plane. Nothing happened. Holding the stick between his knees, he pulled the emergency lever with his right hand and the torpedo dropped.\n\nLess than half a mile from the carrier now, he executed a flip-over, skimming 10 feet over the big ship's bow. As he climbed and tried to turn back, several Zeros dove. His left rudder pedal was hit and he crashed into the water about 400 yards beyond the carrier.\n\nAs his plane sank he ripped open the hood and popped out. When he got to the surface he grabbed a black seat cushion and held it over his head. Two Japanese cruisers steamed past. Then a destroyer came so close he could see white-clad sailors pointing at him. It sped past without firing.\n\nAt that moment the 14 torpedo planes from _Enterprise_ sighted the enemy carriers. Their leader, Lieutenant Commander Eugene Lindsey, circled, hoping to make a beam attack on _Kaga_. He, like Waldron, was also without fighter protection. With the same dedicated courage of Waldron's fliers, Lindsey's men ignored Zeros and anti-aircraft fire and hurtled directly at the Japanese carrier. The first ones didn't have a chance; in minutes Lindsey's plane and 9 others spun into the sea. This time 4 Americans broke through the barrage. They released their torpedoes but the carriers dodged them.\n\nWhile the Japanese were recovering from these close calls, _Yorktown_ 's torpedo squadron, under Lieutenant Commander Lance Massey, appeared. Before it reached the carriers its 6 fighter escorts were swamped by Japanese interceptors. Like the other 2 squadrons, VT-3 bored in relentlessly, regardless of the consequences, just skimming the waves. Massey and 6 mates were shot down before they reached their target, _Hiryu_. Five of his planes broke through and dropped torpedoes but only 2 planes survived.\n\nThe 5 torpedoes plowed toward the desperately dodging _Hiryu_. All missed. On _Akagi_ , Nagumo and his officers were limp with relief. Though the attacks had completely failed and 35 of the 41 torpedo planes had been shot down, the dauntless courage of the American pilots had kept the issue in doubt to the end. It had been a thrilling drama to watch but the sudden silence was welcome.\n\n\u2014\n\nThe Americans had thrown their heaviest punches at Nagumo. Ten torpedo planes, 27 dive bombers and 15 Flying Fortresses from Midway and 41 torpedo planes from the _Enterprise_ , _Yorktown_ and _Hornet_ had failed to score a single hit.\n\nNow only the carrier dive bombers were left. If they failed, it could mean the end of the Pacific Fleet. The 35 dive bombers of _Hornet_ , under Commander Ring, were already eliminated. The 17 dive bombers from the _Yorktown_ , under Lieutenant Commander Maxwell F. Leslie, were to the southeast. If Leslie pursued his present course, he, like Ring, would miss the Japanese carriers, passing to their south.\n\nThe 37 dive bombers of the _Enterprise_ , led by Lieutenant Commander Clarence McClusky, had taken off more than an hour before Leslie. When Nagumo made his abrupt turn to the left, McClusky, like Ring, had missed him, but when he continued and still could find nothing, McClusky decided to turn to the north.\n\nAt 9:55 A.M. he sighted the white trail of a Japanese destroyer, the _Arashi_ , speeding northeast. Correctly guessing that _Arashi_ was heading toward Nagumo's Striking Force, McClusky also turned. A moment later he heard an excited voice shouting over his radio telephone, \"Attack! Attack!\"\n\nIt was Captain Miles Browning, Spruance's chief of staff. It wasn't until a moment before that the _Enterprise_ had learned Nagumo's carriers had been definitely located.\n\n\"Wilco,\" replied McClusky. \"As soon as I find the bastards.\" He continued northeast for twenty minutes but nothing was in sight. His fuel was running low and in a minute he would have to give up the search and set course for home. It was 10:20 A.M.\n\n\u2014\n\nAt that same moment all of _Akagi_ 's planes were on the flight deck, their engines warming up. The bombers of the other three carriers were also in position, neatly lined up.\n\n\"Launch when ready,\" ordered Nagumo. The carrier turned into the wind. Genda and the others on _Akagi_ 's bridge felt much easier. In five minutes their planes would be in the air and winging toward the enemy carriers.\n\n# 25\n\n# _The Turn of the Tide_\n\n**1** Just before Genda gave the order to launch, Lieutenant Commander Leslie, leader of the 17 _Yorktown_ dive bombers, sighted smoke smudges on the horizon to his right. He signaled his squadron to turn to the northwest. For a few minutes the sea was blanketed by clouds. Then suddenly there was an opening and Leslie saw two Japanese carriers about 20,000 feet below.\n\nOne was relatively small, the _Soryu_ ; the other large. Since all the Japanese fighter planes had been drawn down to stop the three torpedo attacks, and hadn't yet regained altitude, Leslie had plenty of time to make his choice. Waldron and the others had not sacrificed their lives in vain. Leslie signaled his men by patting his head, then, with the bright morning sun at his back, pushed over in a steep dive on the bigger ship. It was probably the _Kaga_.\n\nLeslie's own 1000-pound bomb had unfortunately been dumped by accident an hour earlier, but he led his men toward the target at a 70-degree angle. He could see dozens of planes ready for take-off as he aimed at a large red sun painted on the yellow flight deck.\n\nLeslie began firing his machine guns at 10,000 feet. When they jammed at 4000 feet, he pulled up into a climb. Behind him, Lieutenant Paul \"Lefty\" Holmberg continued to dive. At 2500 feet, with the red sun directly in his sights, Holmberg pushed the bomb-release button, then instantly pulled back the emergency lever to make sure.\n\nAt that moment, McClusky's 37 dive bombers suddenly appeared from the southwest. Just as they were about to set course for home, they saw hundreds of square miles of ocean covered by carriers, battleships, cruisers and destroyers.\n\nIn addition to his own squadron, McClusky commanded those of Lieutenants Wilmer Earl Gallaher and Dick Best. McClusky saw two carriers turning into the wind to launch their planes. He ordered Best to attack the smaller one, the _Soryu_ , while he and Gallaher took the other. By mistake three of Best's planes followed Gallaher and McClusky as they dove on the 27,000-ton _Akagi_.\n\nGallaher sighted on the flaming Rising Sun at the forward edge of the flight deck. From the day he saw the _Arizona_ lying at her berth in Pearl Harbor, smoldering and smashed, he had dreamed of making a dive-bombing attack on a Japanese carrier. There was little anti-aircraft fire and no fighter resistance as he swept low. At about 1800 feet he released his bomb, then pulled up into a steep climb and kicked his plane around so he could watch the progress of the bomb. For months he had warned his pilots never to do this since it made the plane a sitting duck for anti-aircraft fire, but he was so sure his bomb was going to hit he couldn't resist the temptation. A moment later he saw it explode in the middle of the planes parked on the after part of _Akagi_ 's flight deck. It was a moment of exultation. He thought, _\"Arizona_ , I remember you!\"\n\n\u2014\n\nA minute after Genda's order to launch had been signaled to the _Kaga_ those on the bridge saw black dive bombers screaming out of the sun at them. This was Leslie's squadron. The first few bombs missed, then four struck in rapid succession, hitting the forward, middle and after sections of the flight deck. _Kaga_ was soon raging with uncontrollable fires. She was obviously doomed.\n\nThe _Akagi_ was also ablaze from several direct hits by the planes of McClusky and Gallaher. A huge hole had been blasted out of its flight deck. The amidship elevator was twisted like molten glass. Planes were standing rakishly, tails up, shooting out tongues of fire and black smoke. Flames jumped from plane to plane. Soon their torpedoes began to explode, driving away those trying to control the fire. The blazes spread to fuel and munition reserves carelessly stacked on deck. These exploded, tearing out great chunks of the ship and shaking the bridge.\n\nIn minutes the hangar area was in flames. As the fire spread toward the bridge, Rear Admiral Ryunosuke Kusaka, Nagumo's chief of staff, begged his chief to transfer his flag immediately to the light cruiser _Nagara_.\n\nNagumo, stricken by the sudden disaster, seemed dazed. \"Sir,\" continued Kusaka, \"most of our ships are intact. You must command them.\"\n\nThe admiral refused to budge from the bridge. The skipper of the ship, Captain Taijiro Aoki, also entreated him. \"Admiral, I will take care of the ship. Please, we all implore you, shift your flag to _Nagara_ and resume command of the Force.\"\n\nWhen he still refused, Kusaka ordered several officers to drag him by the hand. While the admiral was being half-pulled, half-persuaded, the flag secretary, Lieutenant Commander Nishibayashi, reported that all passages below were afire. Almost choked by clouds of smoke, Nagumo said goodbye to Captain Aoki and then, boosted by Nishibayashi, climbed out the bridge window and clambered down a rope to the deck.\n\nThe light carrier _Soryu_ , Best's target, was also in flames. At 10:25 A.M. a bomb had dropped through its flight deck, exploding in the hangar. Second and third hits straddled the amidship elevator, blowing the deck to shreds. Fires spread, quickly reaching gasoline tanks and munition storage rooms. By 10:30 A.M. the 688-foot-long carrier was a fiery hell. Explosions rocked the ship, the main engines stopped. _Soryu_ was soon helpless, its steering system out, its fire mains in shambles. By 10:45 A.M., seeing only a mass of flame, Captain Ryusaku Yanagimoto shouted, \"Abandon ship!\"\n\nBut he refused to leave his bridge. A Navy wrestling champion, Chief Petty Officer Abe, was picked to save the captain in spite of himself. Abe climbed up to the bridge through the black smoke. There he saw Captain Yanagimoto holding a sword and staring forward with resolution.\n\n\"Captain,\" said Abe, \"I have come on behalf of all your men to take you to safety. They are waiting for you.\" The captain acted as if he hadn't heard. \"Please come with me to the destroyer, sir.\" Abe approached the captain, hands outstretched.\n\nYanagimoto turned. His grim determination stopped Abe. The sailor, tears in his eyes, sadly turned around. As he started toward the deck he could hear Yanagimoto calmly singing \"Kimigayo,\" the national anthem.\n\nIn a few minutes 54 American pilots had completely wrecked 3 carriers and turned the tide of the Pacific war. Out of Nagumo's 4 carriers, only _Hiryu_ was left. It was up to the aggressive Admiral Yamaguchi, once a student at Princeton, to carry on the duel with the U.S. Pacific Fleet. He gave immediate orders to strike back. At 10:45 A.M. 6 Zero fighters and 18 dive bombers took off, led by Lieutenant Michio Kobayashi. The planes headed in the general direction of the American carriers at 13,000 feet. Soon Kobayashi spotted groups of American dive bombers apparently returning to their mother ship. He signaled his pilots to lay behind. They would follow the homing planes to their roost.\n\n\u2014\n\nThis was Fletcher's flagship, _Yorktown_ , the scarred veteran of Coral Sea.\n\nJust before noon, as Leslie's dive bombers, fresh from their great victory over _Kaga_ , were preparing to land, _Enterprise_ 's radar indicated 30 to 40 enemy planes about 40 miles to the west-southwest. Leslie's dive bombers were waved off and a combat patrol sent off to intercept the approaching raiders.\n\n\"The attack is coming in, sir,\" warned Admiral Fletcher's aide.\n\nThe admiral looked up from the chart he was studying. \"Well, I've got on my tin hat. I can't do anything else now.\" He returned to his work.\n\nThe Wildcats threw themselves so fiercely at Kobayashi's formation that only 8 Japanese dive bombers broke through. These raced toward the dodging _Yorktown_. Two were quickly shot down by anti-aircraft fire, but 6 dropped their bombs.\n\nThe recently repaired _Yorktown_ was hit 3 times. The first struck the flight deck, setting fires below. The sprinkler system and water curtains put out the blazes almost instantly, then a second bomb exploding in the smokestack started other fires and knocked out two boilers. The final bomb only ignited a rag-storage compartment.\n\nBy 12:20 P.M., with 5 of the ship's 6 boiler fires out, _Yorktown_ was at a standstill; but damage control parties worked so fast that the big ship was steaming at 18 knots by 1:40 P.M. As she was refueling her fighter planes, warning came of new raiders only 40 miles away.\n\nTwelve Wildcats sped out to do battle. They ran into more planes from Yamaguchi's _Hiryu_ : 10 torpedo planes and 6 fighters. The Zeros tied up the Wildcats while the 10 torpedo planes raced, unhindered, toward the _Yorktown_. At 2:32 P.M. they split up and began to make runs from various directions.\n\nThough 5 raiders were shot down, 4 torpedoes were dropped. The sluggish _Yorktown_ dodged 2, but 2 others tore into the damaged carrier. With rudder jammed and all power connections severed the big ship listed 17 degrees. When the list increased Captain Elliott Buchmaster was afraid his ship was going to capsize at any minute and just before 3:00 P.M. ordered, \"Abandon ship.\"\n\nThere were still 2 American carriers intact, Admiral Spruance's _Enterprise_ and _Hornet_. At 3:30 P.M. Spruance ordered dive bombers to make their second strike of the day. Since McClusky had been wounded in the arm on his trip home from the morning attack, Lieutenant Gallaher commanded the group. The 24 dive bombers took off without a single fighter and headed toward _Hiryu_. Spruance then informed Fletcher, who had transferred his flag to the heavy cruiser, _Astoria_ , what he had done and asked:\n\n> DO YOU HAVE ANY INSTRUCTIONS FOR ME?\n\nSince Spruance had 2 carriers and he had none, Fletcher decided to let his subordinate have a free hand. He replied:\n\n> NONE. WILL CONFORM TO YOUR MOVEMENTS.\n\nThe first word of the great battle had come to Yamamoto at 10:50 that morning when the Main Force was about 700 miles northwest of Midway. The radio message was wordlessly handed to the admiral by the chief signal officer, Commander Yushiro Wada:\n\n> FIRES RAGING ABOARD KAGA, SORYU, AND AKAGI RESULTING FROM ATTACKS BY ENEMY CARRIERS AND LAND-BASED PLANES. WE PLAN TO HAVE HIRYU ENGAGE ENEMY CARRIERS. WE ARE TEMPORARILY WITHDRAWING TO THE NORTH TO ASSEMBLE OUR FORCES.\n\nYamamoto couldn't believe such an immense disaster had struck so suddenly. Unable to speak, he groaned. Finally he said, \"Is Genda all right?\"\n\nThe stern-faced commander soon got control of himself. He knew there was only one hope: to gather all his forces and crush the Pacific Fleet by sheer power. He ordered Admiral Kondo to detach part of his powerful force to cover the invasion transports in their temporary retreat to the northwest. The remainder plus Kakuta's Second Carrier Striking Force, far north near the Aleutians, were to join Nagumo's depleted force as soon as possible. His own Main Force, spearheaded by the battleships _Yamato_ , _Mutsu_ and _Nagato_ , also steamed full speed toward the battle area. It was a potent force. The 63,700-ton flagship, _Yamato_ , was the world's mightiest warship. Her main battery of giant 18.11-inch guns alone could hurtle a broadside of 13 tons of steel.\n\nWith this force Yamamoto still thought he could destroy the U.S. Fleet and seize Midway.\n\nThis optimism was not felt on board _Nagara_ , Admiral Nagumo's new flagship. The admiral and his staff were still depressed by the morning's catastrophe. His operations officer, Commander Genda, was already analyzing the reasons, coming to the objective conclusion that he had made two mistakes. First, he should have sent out a two-phase search early that morning; second, and more important, he should have launched the strike on the American carriers immediately without waiting for the Midway planes to land. He had been thinking as a pilot, not a commander; he had been influenced by fear of condemning friends to death. The man who was destined to become Japan's postwar chief of the Air Self-Defense Force had learned there is no friendship in battle.\n\nNot far away on the carrier _Hiryu_ , all planes had returned from the second _Yorktown_ attack by 4:30 P.M. From pilots' reports Admiral Yamaguchi believed his 2 strikes had hit different targets and that 2 American carriers had been destroyed. Just before 5:00 P.M. he ordered his men to prepare for another strike. The airmen still on General Quarters were relieved and served sweet rice balls. As they began eating, Yamaguchi ordered a fast plane to search out the enemy's \"last\" carrier. The search plane was rolled into position at 5:03 P.M. Just then a lookout yelled, \"Enemy dive bombers!\"\n\nEveryone on the bridge looked to the southwest and saw a terrifying sight. Lieutenant Gallaher's 24 dive bombers were darting out of the sinking sun. As Gallaher's planes screeched in a steep dive, several Zero fighters swooped down, picking off one American. The others continued hurtling toward the _Hiryu_. Gallaher was sure he had the carrier cold. Then it made a sudden sharp turn in the opposite direction. Gallaher pulled out of his dive sharply in an attempt to throw his bomb at the dodging ship but it fell 50 feet astern. Those behind Gallaher had more time and soon four bombs exploded in rapid succession near the bridge. Fires quickly spread from plane to plane.\n\n\"Look at that bastard burn!\" said Gallaher over his radio.\n\nEvery carrier in Nagumo's Striking Force was ablaze and sinking. A large percentage of Japan's best fliers were dead\u2014and irreplaceable.\n\nGallaher himself was in trouble. He had injured his back in the abrupt pull-up and by the time he brought his group back to the _Enterprise_ , he was in such pain he couldn't reach down to lower the hook designed to catch in one of the retarding wires stretched across the carrier's flight deck. He turned over the lead to his number 2 wing man and circled while the others landed.\n\nHe knew he couldn't land aboard without the hook. If he ditched in the water he was afraid he wouldn't be able to get out of the plane. He forced himself to put down the hook, almost passing out from pain. He approached the _Enterprise_ as slowly as possible. Suddenly there was an explosion of pain as the hook caught and his plane jerked to a stop. He taxied into his parking spot. He was lifted tenderly out of the cockpit. It was the end of the battle for Gallaher.\n\nWith _Yorktown_ out of action, Nimitz put command of future American attacks completely in Spruance's hands. Although Spruance knew heavy damage had been dealt Nagumo, he wasn't sure how badly the 4 Japanese carriers had been hit. By evening he had intercepted a message from 12 Flying Fortresses which had bombed, and missed, the blazing _Hiryu_ about an hour after Gallaher's strike. They reported Zeros in the area. These were homeless _Hiryu_ planes fighting until their tanks ran dry but Spruance had to consider the possibility of a fifth carrier.\n\nHe remembered Nimitz's \"calculated risk\" instructions. Was he justified in risking a night encounter with possibly superior forces, knowing the Japanese were superior to Americans at this kind of action? If he were the skipper of this possible fifth Japanese carrier what would he do? He certainly wouldn't continue on the same course. He'd probably go west.\n\nSpruance decided not to risk a night action, figuring that this was probably what the Japanese high command was hoping he would do. Instead he slowed Task Force 16 to 15 knots to conserve fuel and headed back east. Later, since he wanted to be in air-supporting distance of Midway in case the Japanese decided to push their attack on that island, he would reverse his course.\n\n\u2014\n\nIt was now dark, the vast sea battleground a flaming graveyard. The _Soryu_ tilted. Watching from destroyers were her survivors. At 7:13 P.M. the carrier dove out of sight. With her went 718 corpses and one live man, Captain Yanagimoto.\n\nAbout 40 miles to the south another burning hulk was being torn by two great explosions. This was _Kaga_ , and a few minutes later she and 800 of her crew were swallowed up.\n\nWhile the water above the vanished _Soryu_ was still bubbling, Yamamoto was sending out a message to all his scattered forces designed to bolster sagging morale:\n\n> (1) THE ENEMY FLEET HAS BEEN PRACTICALLY DESTROYED AND IS RETIRING EASTWARD.\n> \n> (2) COMBINED FLEET UNITS IN THE VICINITY ARE PREPARING TO PURSUE THE REMNANTS OF THE ENEMY FORCE AND, AT THE SAME TIME, TO OCCUPY MIDWAY.\n> \n> (3) THE MAIN BODY WILL REACH CBJ \u00b0 08'N, 175 \u00b0 45'E AT 0300 ON THE 5TH. COURSE, 090 \u00b0; SPEED, 20 KNOTS.\n> \n> (4) THE CARRIER STRIKING FORCE, INVASION FORCE (LESS CRUISER DIVISION 7), AND SUBMARINE FORCE WILL IMMEDIATELY CONTACT AND ATTACK THE ENEMY.\n\nYamamoto and his staff still had not lost hope of crushing the Pacific Fleet, their main objective, but even the most eager of his officers was dismayed when a message from the shaken Nagumo arrived at 9:30 P.M.:\n\n> TOTAL ENEMY STRENGTH IS 5 CARRIERS, 6 HEAVY CRUISERS, AND 15 DESTROYERS. THEY ARE STEAMING WESTWARD. WE ARE RETIRING TO THE NORTHWEST ESCORTING HIRYU. SPEED, 18 KNOTS.\n\nWhen this inaccurate message was read Rear Admiral Matone Ugaki, the chief of staff, said with disgusted gloom, \"The Nagumo Force has no stomach for a night engagement.\"\n\nAs time passed it was obvious that the Americans were not steaming west but doing just what Yamamoto had prayed they wouldn't do\u2014backing away from a night action. By midnight all hope that Spruance's surviving carriers could be lured into an engagement was gone.\n\nNow Yamamoto's staff began considering the most desperate and impractical schemes and Captain Kameto Kuroshima, senior Fleet operations officer, even drew up a tentative plan to shell Midway with all battleships.\n\n\"The stupidity of engaging such shore installations with a surface force ought to be clear to you,\" was Admiral Ugaki's answer to this wild proposal. \"The battleships would be sunk by air and submarine attacks before they moved in close enough to use their guns.\" He suggested they postpone another air strike until the Second Carrier Striking Force from the Aleutians joined them. \"But even if that proves impossible and we must accept defeat in this operation, we will not have lost the war. There are still 8 carriers in the Fleet. So we should not lose heart. In battle, as in chess, it is the fool who lets himself be led into a reckless move through desperation.\"\n\nSome of the officers were willing to gamble anything to save face. \"How can we apologize to His Majesty for this defeat?\" asked one.\n\nYamamoto, silent until now, had made up his mind. He said sharply, \"Leave that to me. I am the only one who must apologize to His Majesty.\" Fifteen minutes after midnight he ordered Kondo and Nagumo to turn west and join the Main Force.\n\nA few hours later he faced another problem new to the Japanese Navy. The _Akagi_ , its fires long out of control, had to be scuttled. Its skipper, Captain Aoki, had already requested permission to sink her and then lashed himself to the carrier's anchor. Yamamoto reluctantly approved and soon 4 destroyers were sending torpedoes into the great ship.\n\nThe last carrier, _Hiryu_ , was also reaching the end of its career. Dead in the water, its engine rooms were raging with uncontrollable fires. The former Princeton student, the indomitable Yamaguchi, finally gave up hope and ordered all men topside at 2:30 A.M., June 5. Then he solemnly addressed the 800 survivors. \"As commanding officer of this carrier division I am fully and solely responsible for the loss of _Hiryu_ and _Soryu_. I shall remain on board to the end. I command all of you to leave the ship and continue your loyal service to His Majesty, the Emperor.\"\n\nThough his staff pleaded, he would not let them stay on board with him. They all drank a silent toast in water. The admiral said he had no farewell messages and gave his senior staff officer, Commander Ito, his black deck cap as a memento. He took from Ito a length of cloth. Everyone knew he would use it to lash himself to the bridge.\n\nCaptain Tomeo Kaku made a last appeal to Yamaguchi to leave the ship. The admiral only smiled and shook his head, but didn't protest when the skipper took his place beside him on the bridge.\n\n\u2014\n\nBefore dawn of June 5 misfortune continued to dog Yamamoto. The heavy cruisers, _Mogami_ and _Mikuma_ , collided. Not long after dawn a Midway-based Catalina sighted the 2 lagging ships and 6 Marine dive bombers and 6 Vindicators took off from Midway. They followed an oil slick and at 8:05 A.M. attacked the 2 cruisers. Anti-aircraft was so heavy and accurate that not a bomb went home but Captain Richard Fleming purposely dove his burning plane into one of _Mikuma_ 's after turrets.\n\nThat afternoon Ensign Gay, the only survivor of Waldron's torpedo squadron, was picked up by a Catalina. When a doctor asked how he had treated his wounds Gay replied, \"Soaked 'em in salt water for ten hours.\"\n\nThe next day, June 6, action resumed almost as soon as the sun rose on a cloudless sky. _Enterprise_ search planes sighted the two slow-moving cruisers which had helped sink _Perth_ and _Houston_. Spruance launched three attacks, finally sinking _Mikuma_. Though _Mogami_ was smashed by six bombs, damage control parties saved the badly wounded ship and she limped safely to the west.\n\nIn the meantime the Japanese were also striking. At about 11:00 A.M. Lieutenant Commander Yahachi Tanabe was standing on the bridge of his submarine I-168, searching the sea. Suddenly he saw a dot on the sea. He decided to approach on the surface and investigate. The speck grew larger. When he was six miles away a thrill went through him. It was the _Yorktown_! He'd received a message that it was drifting, listing to port, and he should find and destroy it. Though his boat was in poor condition with chlorine gas seeping from the damaged bow torpedo room, he decided to attack. He submerged and approached.\n\nThrough his periscope he could see about 7 destroyers darting around the big ship. He dove, passing under the carrier, then reversed and took another look. Now, though he heard many sounds on sonar, he could see no destroyers\u2014only the _Yorktown_ looming up like some huge, prehistoric monster. He slowly maneuvered into position.\n\nAt 1:30 P.M. he was ready. From a depth of 57 feet he cautiously approached the carrier. When he was 1300 yards away he launched 2 torpedoes. Three seconds later he fired 2 more. Without waiting, he dove to 100 feet. Suddenly everyone on the submarine felt three tremendous jars. The crew shouted, \"Banzai!\" They were wild with excitement. They had scored 3 hits.\n\n\"Shut up,\" shouted Tanabe. The slightest noise inside a submarine could be picked up. A sailor brought him cider.\n\nOne of Tanabe's torpedoes had hit the destroyer _Hammann_. She broke in two, sinking within four minutes. The other two hits were on the _Yorktown_. Her death was lingering. Hours later some of her crew were still hopelessly trapped in submerged compartments. An officer finally made contact by phone with one of these compartments. \"Do you know what kind of a fix you're in?\" he asked.\n\nA voice from under the sea answered, \"Sure, but we've got a hell of a good acey-deucey game down here. One thing though, when you scuttle her, aim the torpedoes right where we are. We want it to be quick.\"\n\nBefore she could be scuttled the _Yorktown_ rolled over on her port side and, just after dawn the next day, sank with all battle flags flying. As she slid out of sight, men on nearby destroyers stood by at attention with hats off.\n\n\u2014\n\nAn hour and a half after Tanabe launched his torpedoes, Yamamoto was still desperately trying to salvage victory. At 3:00 P.M. he ordered all ships:\n\n> COMBINED FLEET UNITS OPERATING IN THIS AREA WILL CATCH AND DESTROY THE ENEMY TASK FORCE WITHIN ATTACK RANGE OF AIR FORCES BASED ON WAKE ISLAND.\n\nIt was a useless gesture. The wily Spruance, the happiest choice as a replacement for Halsey that Nimitz could have made, refused to be lured into a battle in which he would be heavily outgunned. Yamamoto realized this the next morning, June 7, and dejectedly headed back toward Japan.\n\nOne of the greatest sea battles in history was over. It had been won by a unique combination of Japanese mistakes, a broken code, brilliant command decisions, and the courageous determination of men like Waldron, Gay, Leslie, McClusky and Gallaher.\n\nThe Japanese Navy had been dealt an unexpected defeat and with one stroke America had seized control of the Pacific Ocean. Though some criticized him for being premature, Admiral Nimitz was essentially right when he said in his communiqu\u00e9 of June 6:\n\n> Pearl Harbor has now been partially avenged. Vengeance will not be complete until Japanese sea power is reduced to impotence. We have made substantial progress in that direction. Perhaps we will be forgiven if we claim that we are about midway to that objective.\n\n**2** Coupled with the repulse of the Port Moresby invasion in the Coral Sea, the Midway triumph gave Australians their first real sense of security.\n\nThe victory put the spotlight on the Pacific. The period of withdrawal and holding in this area was over. For the first time since Pearl Harbor the Allies were in a position to attack.\n\nMacArthur, insisting that the tide had turned, pressed his demands for an all-out surprise attack on Rabaul. Now he had two fresh American divisions, the 32nd and the 41st, as well as the battle-tested Australian 7th. He was impatient to start on the road back to the Philippines.\n\n\u2014\n\nIn the Philippines many were preparing for his return. Spontaneously, hundreds of leaders\u2014Americans, Filipinos, civilian and military\u2014were forming guerrilla bands. Others were already organizing intricate spy rings.\n\nHigh in the Zambales Mountains, not far from Clark Field, two American civilians, William Fassoth and his twin brother, Martin, had set up an elaborate rescue camp for many American escapees of the Death March. Here came Sergeants Ray Hunt and Frank Bernacki. Here also came Major Russell Volckmann, Captain Donald Blackburn and more than 200 other American officers and men, half-dead from starvation and disease.\n\nThe Fassoths, sugar planters since 1913, had only one purpose: to bring their gaunt, sick charges back to health. But most of the Americans were not merely interested in hiding in the mountains until MacArthur returned. Men like Hunt, Volckmann and Blackburn were already making plans to form groups, steal arms from the Japanese and start their own private wars.\n\nTo the north, in the mountains near Baguio, two groups had been operating for months. One was commanded by a Filipino, Lieutenant Colonel Guillermo Z. Nakar. The other, originally organized by Colonel John Horan, was now led by the former miner, Walter Cushing. Striking like lightning, the redoubtable and volatile Cushing was the man most wanted by the Japanese in Luzon.\n\nFar to the south in the island of Cebu his brother, Jim, also an ex-miner as well as ex-boxer, was building his own legend as the most capable and daring of the many leaders in the Visayan Islands. Nearby Panay was being well organized by General Chynoweth's former operations officer, Macario Peralta, Jr. And farther south, in Mindanao, hundreds of Filipinos and Americans were beginning to form a guerrilla army of formidable proportions that would soon be led by one of General Casey's engineers, Colonel Wendell Fertig. One of the most effective of these groups was the band of fierce Moros under Major Salipala Pendatun, formerly a Moro lawyer.\n\nCivilians were also busy undermining the uneasy regime of General Homma. Dr. Romeo Atienza, medical director of the Philippine Red Cross, was risking his life daily to smuggle medicine, food and messages to the starving, disease-ridden prisoners of Camp O'Donnell. Here 370 Americans and Filipinos were dying every day. The tragic toll would have been much higher but for Atienza, his wife and their helpers.\n\nAtienza's principal support came from a group of men and women organized by a remarkable woman, Mrs. John Utinsky, wife of one of the prisoners. Using such _noms de guerre_ as \"Miss U\" (Mrs. Utinsky), Dr. A (Atienza), Morning Glory (Father Loler), High Pockets (Mrs. Claire Phillips), Fahny (Fanny Greenwell) and Sparky (Ramon Amusategui), this improvised organization brought life and hope to men who thought they had been abandoned and forgotten.\n\nThe islands were far from conquered. All over the Philippines men and women and even children were plotting against the Japanese, working and waiting for the coming of the man they knew would keep his word: MacArthur.\n\n**3** In Japan the real facts of the Battle of Midway were hidden by Imperial Headquarters from leading officials as well as the public. Survivors of sunken ships were kept isolated and the battle was being celebrated as one of the empire's greatest victories. On June 11 the _Asahi Shimbun_ carried this announcement from Imperial Headquarters:\n\nMIDWAY DETERMINES PACIFIC WAR SITUATION\n\n> As a result of the Battle of Midway, the United States has lost almost all her carriers and Japan has secured supreme power in the Pacific. Our suicide method of letting the enemy slice our flesh while we cleave them in two brought us success in this great sea battle of carrier against carrier....\n> \n> This well-fought battle helps eliminate our anxiety in the future. The situation of the Pacific War was indeed determined in this one battle.\n\nTo celebrate the great triumph enthusiastic citizens of Tokyo staged a flag procession and a lantern parade.\n\nThere was one Japanese who was not celebrating. This was Ensign Kazuo Sakamaki, the only survivor of the submarine attack on Pearl Harbor, now in a POW camp in Tennessee. (His midget submarine was being toured around the United States to spur the War Bond drive.) From the day Sakamaki had entered the Japanese Naval Academy every moment had been devoted to strict conditioning of mind and body, preventing him from understanding any human being except other fanatic servants of the Emperor.\n\nBut his few months of captivity had done a curious thing. In spite of himself he had gradually come to admire and respect Americans. This realization had come like a hammer blow against the heart of his whole past. The entire history and culture of Japan out of which he had been born had abruptly collapsed. It had been, he realized, the rebirth of reason.\n\nWhat he read about Midway in American newspapers, he believed. After the countless factories he had seen on his recent trip down the Mississippi from Camp McCoy, Wisconsin, he knew the United States was just starting to fight. He was sure that Midway was the beginning of the end of Japan's hope of conquest.\n\nIt was. Midway was more than victory. Like a boxer who has swung wildly with an all-out punch and missed, Japan was wide open, vulnerable. By now the \"Arsenal of Democracy\" was in full production. No longer would Allied pilots have to fight the Zero with obsolete planes. No longer would Japanese tanks have to be stopped with gasoline-filled Coca-Cola bottles, grenades or rifles.\n\nThe first six months of the war in the Pacific had been fought by poorly trained, ill-equipped and, in many cases, ineptly led soldiers and sailors. These men had been flung into battle without knowing the true nature of their enemies. They had been forced to fight with bullets, fuel and food strictly budgeted. That they were forced into a series of disastrous retreats and humiliating surrenders was no fault of theirs. It is true they often fought to no avail; but not in shame.\n\nNor did their military leaders deserve the calumny many critics were already heaping on them. The essential blame for the disasters of these six months should not have been laid on Kimmel, Short, MacArthur, Percival, Wavell or Helfrich; even less on the political leaders, Roosevelt and Churchill, who had been begging for arms long before battle came. The entire peoples of America and Great Britain were responsible for the scandalous state of unpreparedness in the Far East.\n\nAfter the Battle of Midway the crisis was over in the Pacific. The war here, however, would not turn into a simple parade of easy victories. The Allies would have to dig out a deeply entrenched and tenacious enemy in a series of costly battles of extermination. With Nimitz and MacArthur sharing command, the new-won Japanese empire would shrink island by island: Guadalcanal, New Georgia, Bougainville, New Britain, New Guinea, the Philippines, Tinian, Saipan, Iwo Jima. And finally in 1945 the Allies would produce the atom bomb.\n\n\u2014\n\nWhen Sakamaki learned of Hiroshima and Nagasaki he was overwhelmed\u2014and repentant. \"We fought and lost,\" he told himself. \"We cannot blame others. We cannot complain. It is all our fault. It was we who were weak, inadequate and ignorant. We must rebuild our country with our own hands in silence. That is the only way we can pay our debts.\"\n\nThe past must not be forgotten, or even forgiven\u2014only understood.\n\n(U.S. Navy Photo)\n\n_Battleship Row, Pearl Harbor, as seen from one of the first Japanese torpedo planes_.\n\n(Life Photo)\n\n_The battle that was never fought. Carl Mydans, of_ Life, _photographed this beach on Lingayen Gulf at noon on December 11, when the first major Japanese invasion of the Philippines was supposedly taking place. American newspapers reported 154 Japanese vessels were sunk and many of the enemy killed in a fierce three-day battle starting December 10_.\n\n(Wide World Photo)\n\n_Crew abandoning_ Prince of Wales _moments before it sinks, December 10, 1941_.\n\n(Wide World Photo)\n\n_Soon to be the first conqueror of a large American force, Lieutenant General Masaharu Homma, the controversial commander of the Japanese 14th Army, steps ashore at Lingayen Gulf, December 24_.\n\n(U.S, Navy Photo by Carl Mydans)\n\n_Rear Admiral Francis W. Rockwell_ (left) _and his aide, Lieutenant Malcolm Champlin, at Cavite Naval Base_.\n\n(Wide World Photo)\n\n_Wake Island\u2014where the Japanese timetable of conquest was first upset_.\n\n(Wide World Photo)\n\n_Acting on his own initiative, Lieutenant Champlin ordered these Pandacan oil tanks near Manila set afire on December 26 to prevent the Japanese from getting millions of gallons of fuel. For weeks Champlin wondered if he would be decorated or court-martialed_.\n\n(Wide World Photo)\n\n_Prisoners from Wake in Shanghai. Major James Devereux, Marine commander_ (left) _and Raymond R. Rutledge, civilian worker who threw grenades into Japanese invasion craft_.\n\n(U.S. Army Photo)\n\n_MacArthur inspects the Abucay line on Bataan, January 10, 1942. To his left, Brigadier General Albert Jones, whose division was reduced to about 100 men two weeks later_.\n\n_Colonel Susumu Takechi crossed \"impenetrable\" Mt. Natib and flanked the entire American Abucay Line. His commander, General Nara, recently revealed this was done against his direct orders_.\n\n_Lieutenant General Akira Nara, former classmate of Coolidge's son at Amherst College. His \"Summer Brigade\" crushed Jones' division and broke through the Abucay Line_.\n\n_President Manuel Quezon_ (left) _and Colonel Carlos Romulo. Rumulo later became Philippines Ambassador to the U. S_.\n\n_Brigadier General Clifford Bluemel, whose quick thinking prevented General Nara from immediately overrunning the second and final American defense line on Bataan at Trail 2_.\n\n(U.S. Navy Photo)\n\n_Lieutenant John D. Bulkeley, the colorful man of action whose PT boat carried MacArthur through enemy waters to Mindanao on March 11-13_.\n\n(U.S. Army Photo)\n\n_MacArthur and his chief of staff, Major General Richard K. Sutherland, in Malinta Tunnel, Corregidor, a few days before MacArthur made his bold, dramatic dash to Australia_.\n\n(U.S. Army Photo)\n\n_Japanese propaganda leaflets. Japanese \"Ticket to Armistice\" reads: \"The bearer of this ticket wishes to surrender. Killing or harming those surrendering is strictly forbidden. Commander, the Imperial Japanese Army.\"_\n\n(U.S. Army Photo)\n\n_Banzai! The 141st Infantry captures Mt. Limay, Bataan, in the final, overwhelming Japanese offensive starting on Good Friday. General Nara honored the regiment's commander, Colonel Takeo Imai_ (bottom center), _by renaming the peak Mt. Imai_.\n\n(Wide World Photo)\n\n_Escapees from Singapore shipwrecked on a small island near Sumatra. A. P. correspondent Yates McDaniel_ (right rear) _and 131 others were forced to abandon their bombed ship the_ Kung Wo, _the previous day, February 13. The girl is Doris Lim, whose flight to freedom would soon end in tragedy_.\n\n_Vice-Admiral Conrad E. L. Helfrich, the aggressive Dutch commander of the Allied naval forces in the Netherlands East Indies, who was determined to fight the Japanese Navy for Java even though outnumbered_.\n\n_America's first great humiliation in the Philippines. Major General Edward P. King, Jr. grimly discusses surrender of Bataan with Colonel Motoo Nakayama on April 9. Facing camera_ (left to right); _Colonel Everett C. Williams, General King, Major Wade Cothran and Major Achille C. Tisdelle_.\n\n(Wide World Photo)\n\n_The Death March: 72,000 captured Americans and Filipinos\u2014most of them diseased and near starvation\u2014heading for prison camp. Many were beaten and murdered en route. This picture was stolen by Filipinos from the Japanese during the occupation_.\n\n(Wide World Photo)\n\n_Five of the approximately 2330 Americans who died on the Death March. Picture stolen by Filipinos_.\n\n(Wide World Photo)\n\n_American and Filipino survivors carry comrades the last mile of the Death March into Camp O'Donnell_.\n\n_Americans surrender at the mouth of Malinta Tunnel, Corregidor, on May 6_.\n\n(U.S. Army Photo)\n\n_Misunderstanding between Lieutenant General Jonathan M. Wainwright and General Homma over surrender terms soon ended this first meeting on May 6 at the Castillo home in Cabcaben, Bataan. Facing Homma_ (left to right), _Wainwright, Brigadier General Lewis C. Beebe and Major Thomas Dooley_.\n\n_\"After leaving General Homma with no agreement between us, I've decided to accept, in the name of humanity, his proposal...\" In Manila, Wainwright was forced to recite prepared surrender instructions on the night of May 7-8 to those still fighting in the Philippines_.\n\n(U.S. Navy Photo)\n\n_In spite of raging fires the aircraft carrier_ Lexington, _mortally damaged in the Battle of Coral Sea, was abandoned May 8 with no panic. Some men waiting in line calmly ate ice cream before going over the side_.\n\n_Commander Minoru Genda_ (left), _who made the fatal Japanese decision at the Battle of Midway. He is now General Genda, Chief of Staff, Japan Air Self-Defense Force_.\n\n(U.S. Navy Photo)\n\n_\"If there is only one plane left to make a final run in, I want that man to go in and get a hit,\" wrote Lieutenant Commander John C. Waldron, leader of Torpedo Squadron 8, to his men on the eve of the Battle of Midway. Only one member of the squadron, Ensign George Gay, survived the gallant attack the next day_.\n\n# _Acknowledgments_\n\nTo gather material for this book it was necessary to travel more than 75,000 miles and visit eight countries. The majority of the almost 800 people interviewed do not appear in the book, but the accounts of such people were invaluable. Without knowing their stories, a full understanding of the events would sometimes have been impossible.\n\nFor example, I spent many hours in half a dozen interviews with Colonel William L. Osborne. He fought with the 51st Infantry on Bataan, then escaped in a small boat with Lieutenant Mabini Fontanilla across Manila Bay. After being hidden for months by Fontanilla and other Filipinos, Osborne and Captain Damon Gause sailed a small boat to Australia. Theirs was the first personal information MacArthur's intelligence officers had from the Philippines since the surrender of Corregidor. Yet, because of space limitations, not a line of Osborne's saga appears in this book. Air General Shu-ming \"Tiger\" Wang, Claire Chennault's intelligence officer, told how the Flying Tigers were often alerted promptly by three captured Japanese disguised as Chinese radio operators whenever enemy raiders approached. It is a fascinating inside story, never even revealed to Chennault himself, but had to be cut from the final draft.\n\nLibraries contributed greatly to the book: the Historical Archives in Alexandria, Virginia; the library of the Office of the Chief of Military History, Department of the Army, Washington, D.C.; the Library of Congress; the library of the Naval History Division, Washington, D.C.; the Air University Library at Maxwell Air Force Base; the Main Branch of the New York Public Library; the U.S. Army Infantry School Library at Fort Benning, Georgia (Miss Ruth Wesley); the Diet Library, Tokyo; and the Red Bank, N.J., Public Library.\n\nNumerous agencies, organizations and individuals made this book possible. To list them all would be impossible but here are a few:\n\nWashington, D.C.: Major General William Quinn, Brigadier General Chester Clifton, Colonel Sidney Williams, Lieutenant Colonel John Chesebro and Major R. F. Prentiss of the Office, Chief of Information, U.S. Army; Major James F. Sunderman and Alice Martin, Magazine and Book Branch of the U.S. Air Force; Major Ben Fern and Edith Midgette, Office of News Services, Department of Defense; Commander Russell L. Bufkins, Magazine and Book Branch, U.S. Navy; Rear Admiral E. M. Eller, Director, Naval History Division; Judge Israel Wice; Lieutenant Colonel and Mrs. Herbert Trattner; General Carlos Romulo; and Roger Pineau.\n\nPhilippines: President Carlos P. Garcia; Secretary of National Defense Alejo Santos; Lieutenant General Manuel Cabal, Chief of Staff, Armed Forces of the Philippines; Major General Pelagio A. Cruz, Vice Chief of Staff, Armed Forces of the Philippines; Brigadier General Isogami Campo, Chief of the Philippine Constabulary; Brigadier General Pedro Molina, Commanding General, Philippine Air Force; Commodore Jose Francisco, Philippine Navy; Brigadier General Tirso Fajardo, Commanding General, Philippine Army; Lieutenant Colonel Wilfredo Encarnacion, Constabulary Commander of Iloilo; Major Jos\u00e9 Reyna, SEATO; Lieutenant Hanover D. Gloria, PIO, Camp Crame; Lieutenant Fernando Edralin, Philippine Navy; Lieutenant Aurelio Repato, PAF Historical Officer; Major Samuel Quiwa, Camp Olivas, Luzon; Lieutenant Colonel Leo Gaffney, JUSMAG; Dr. W. T. T. Ward, Historian, 405th Fighter Wing, Clark Field; H. J. H. C. Hildreth, Shell Company of the Philippines.\n\nHong Kong: Forrest Edwards and Roy Essayan, AP; Leo Landau of Jimmy's Kitchen; Joe Lowe; Irving Hoffman.\n\nTaiwan: Dr. Sampson Shen, Yu Wei, and Robert C. H. Liao of the Government Information Office; Lieutenant General P. C. Yu, Vice-Chief of General Staff; Spencer Mousa, AP; Major Kenneth Weber and Technical Sergeant C. W. Curry, MAAG; H. S. \"Newsreel\" Wong.\n\nJapan: Lieutenant General Robert W. Burns, Commander, United States Forces in Japan; Colonel Ray Vandiver, Captain Ben Scarpero, Isaac Taira and Dr. Kazutaka Watanabe of the Office of Information Service, Fifth Air Force; CM Sergeant Thomas Rhone, Master Gunnery Sergeant Alan Sydow, Master Sergeant Art Hicks, A2C Stewart Diamond, Toki Koizumi (the grandson of Lafcadio Hearn) and Miss Yasuko Hoshiba of the Press Liaison Office at the Sanno Hotel; Sol Sanders, _McGraw-Hill World News_ ; Igor Oganesoff, _Wall Street Journal_ ; Count Ian Mutsu; Captain Sanematsu; Captain Tsunoda; Colonel Takushiro Hattori; Captain Susumu Nishiura, Chief of War History Office, Defense Agency; General Moriya Wada, Army Staff Office; General Takeo Imai; Tokiji Matsumura; Dr. Keigo Okonogi; Mrs. Saburo Kurusu; and Mrs. Masaharu Homma.\n\nHawaii: General Emmett O'Donnell, Jr., Commander in Chief, Pacific Air Force; Colonel C. B. Whitehead and Major Alfred J. Lynn of Cincpacaf; Commander Herbert Gimpel, Anne Kelly and the many others at Fourteenth Naval District who were so cooperative; Rear Admiral Ernest Holtzworth.\n\nGuam: Lieutenant Commander George York and Francisco Q. Cruz.\n\nWake: The anonymous Air Force captain who lent me his truck for a day to explore the atoll.\n\nOthers include Colonel Laurence H. Macauley and Dr. Albert Simpson of the Air University; Dr. Robert F. Futrell, Air Force Historian, Pacific; Colonel James Chesnutt, Presidio, San Francisco; Mr. and Mrs. Malcolm Champlin, Oakland, California; Florence Clark and Janet Ludwick of Los Angeles; Mr. and Mrs. Lawrence Dugan, Philadelphia; Charles Dornbusch, New York Public Library; Audrey Sinclair, London, England; Mr. and Mrs. Pete Upton, Lubbock, Texas; Colonel Ray O'Day, Seattle, Washington; Franklin Trammell, of the American Guerrillas of Mindanao, San Diego, California; Dr. Louis Morton, Dartmouth College; Stephan de Voogd, Netherlands Information Service, New York; and Paul R. Richter, of the American Ex-Prisoners of War, San Antonio, Texas.\n\nI would like also to thank my typists, Edith Lentz and Helen Toland, my chief interpreter and translator, Toshiko Toland, and John Jamieson and Professor Fred Stocking for their encouragement and advice. Finally, this book could not have been written without my editor, Robert Loomis, who gave me the basic idea, and my agent, Rogers Terrill.\n\n# _Notes_\n\nThe main sources for each chapter are listed later with explanatory details. Those books which proved of great value in numerous chapters, and will not be listed again, are: _The Fall of the Philippines_ by Dr. Louis Morton; _The Rising Sun in the Pacific_ by Samuel Eliot Morison; _Battle Report: Pearl Harbor to Coral Sea_ by Walter Karig and others; _The Japanese Thrust_ by Lionel Wigmore; _Grand Alliance_ and _Hinge of Fate_ by Winston Churchill; _They Fought with What They Had_ by Walter D. Edmonds; _General Wainwright's Story_ by Jonathan Wainwright; _The Good Fight_ by Manuel Quezon; and _MacArthur and the War Against Japan_ by Frazier Hunt.\n\nDr. Morton's book, a rare combination of scholarship and readability, was indispensable.\n\nJapanese documentary material came from a group of histories written by Japanese Army and Navy officers at the direction of the Intelligence Officer, U.S. Far East Command; the collection of the Allied Translator and Interpreter Section; numerous monographs prepared by the Japanese Research Division of the Military History Section, U.S. General Headquarters, Far East; and the 30 volumes of testimony of Homma's trial. More important were the 123 interviews with Japanese war veterans and civilians. Many of these were arranged by Captain Susumu Nishiura, Chief of the War History Office, Japanese Defense Agency. Conferences of experts and participants were held on every campaign. Captain Nishiura and Captain Tsunoda were present at all these as advisers and Captain Sanematsu acted as interpreter. Without the full cooperation of all these men many revealing stories of the war would still be untold. Captain Nishiura's office is presently working on an authoritative history of the war, a ten-year project. When finished it should be a major contribution to war history.\n\n## PROLOGUE\n\nThis was based primarily on interviews with Colonel Takushiro Hattori, former private secretary to Prime Minister Tojo; Mrs. Saburo Kurusu, widow of the Special Envoy; Admiral Kichisaburo Nomura; and the following books: _On Active Service in Peace and War_ by Henry L. Stimson and McGeorge Bundy; _Back Door to War_ by Charles Tansill; _Kishi and Japan_ by Dan Kurzman; _The Cause of Japan_ by Shigenori Togo; _Chief of Staff: Prewar Plans and Preparations_ by Mark Watson; _Japan Fights for Asia_ by John Goette; _The Road to Pearl Harbor_ by Herbert Feis; and _Political Strategy Prior to Outbreak of War_ by Rear Admiral Sadatoshi Tomioka.\n\n## CHAPTER 1\n\nOf the many interviewed on these fateful hours the most important were: Admirals Nomura, W. W. Smith, Harold Stark, C. C. Bloch, Colonel Hattori, Captain Nishiura, Sergeant Frank Trammell, Brigadier Generals Paul Putnam and James P. S. Devereux, Sergeant Lorenzo Alvarado, and Kazuo Sakamaki. The information concerning the lone Japanese Navy spy in Hawaii, Takeo Yoshikawa, came from Lieutenant Colonel Norman Stanford, USMC, Naval Attach\u00e9 at the U.S. Embassy in Tokyo. An article on Yoshikawa by Stanford appeared in the December, 1960, issue of _U_. _S_. _Naval Institute Proceedings_. Further information came from a series of articles by Yoshikawa himself in the Mainichi _Daily News_. He is now living on Shikoku Island and manages a gas station near Matsuyama. George Elliott's story of the radar contact came directly from him, his testimony at the Pearl Harbor investigation and an interview in the Red Bank, N.J. _Register_ , Dec. 7, 1959, by Art Kamin. Information concerning the visit of Admiral Phillips to the Philippines came from Rear Admiral W. G. Lalor and Vice Admiral F. A. Rockwell. The reactions of President Roosevelt to the message brought on the eve of Pearl Harbor are based on Kramer's testimony and Robert E. Sherwood's _Roosevelt and Hopkins_.\n\nBooks include: _On Active Service in Peace and War_ by Stimson and Bundy; _The Lost War_ by Masuo Kato; _Day of Infamy_ by Walter Lord; the 40 volumes of the U.S. Congressional investigation; _Admiral Kimmel's Story_ by Admiral Husband Kimmel; _What Happened at Pearl Harbor_? edited by Hans Louis Trefousse; _The Final Secret of Pearl Harbor_ by Rear Admiral Robert Theobald; _Admiral Halsey's Story_ by William Halsey; _The Cause of Japan_ by Togo; _This Is Pearl_ by Walter Millis; and _Sunk_ by Mochitsura Hashimoto.\n\n## CHAPTER 2\n\nThose interviewed included Robert Overstreet, Ordnanceman 3rd Class Donald Briggs, Admirals Bloch and Smith, 1st Class Metalsmith Lawrence Chappell, Yeoman C. O. Lines, Rear Admiral Ernest Holtzworth, Major General Richard Carmichael, Colonel William Welch, Ensign Wilbert Cain, Admiral Chester Nimitz, Sergeant Jesse Gaines and Dr. Wang Shih-chieh. To familiarize myself with the scene I flew over Oahu several times, following the routes of the invasion planes. In addition to touring Pearl Harbor, I was taken by destroyer through the harbor mouth to the scene of Lieutenant Outerbridge's experiences.\n\n## CHAPTER 3\n\nBecause of its controversial nature 85 people were interviewed on the Clark Field attack. A few of these were: Colonel William Morse, Brigadier General Joseph Moore, Stella Setzer, Major Douglas Logan, Major Durward Brooks, Colonel Tom Dooley, Major Glenn Cave, Colonel Fred Roberts, Commander Kenichi Tanaka and Colonel Allison Ind, author of _Bataan: The Judgment Seat_.\n\nAs at Oahu, I traced the attacks at Clark and Iba by air.\n\nThe most helpful printed material came from Walter Edmond's extensively researched book, a monograph compiled at the Air University, _Problems of Historiography: Pacific Theatre_ , _Initiation of Air Hostilities in the Philippines_ , _8 December_ , _1941_ , and _The Brereton Diaries_ by Lewis Brereton.\n\nThe Japanese historians helped solve to my satisfaction the disputed time element, providing times of departure and attack from Imperial Navy after-action reports. A probable solution to the disagreement about the number of planes to raid Iba and Clark came when the Japanese revealed, for the first time, the embarrassing attack on their own Army planes returning from the Philippines.\n\nThose who have read Wainwright's book may wonder why the phrase, \"I was more afraid of you than the bombs,\" is deleted from Tom Dooley's reply to the general's question: \"You didn't drive past Clark during this bombing, did you?\" When Colonel Dooley checked the portions about himself he wrote, \"General Wainwright told that story many times and I think believed it himself but I was never afraid of him or his reactions. He was a fine man and soldier and I cannot recall any time during our five or six years association that he said an unkind word to me, so no fear of him could have developed. The old man told the story and liked it so he continued it.\"\n\n## CHAPTER 4\n\nYates McDaniel, now with AP in Washington, D.C., provided the anecdote concerning Vice-Admiral Sir Geoffrey Layton. Other information came from Cecil Brown's book, _From Suez to Singapore;_ \"Operations in the Far East\" by Air Chief Marshal Sir Robert Brooke-Popham; _The Royal Australian Navy_ by G. Herman Gill; \"Loss of H. M. Ships _Prince of Wales_ and _Repulse_ \" by Vice-Admiral Sir Geoffrey Layton.\n\n## CHAPTER 5\n\nInterviewed: Juan Perez, Colonel Ind, Captain Masamichi Fujita, Lieutenant Haruki Iki, Cecil Browne, Admiral Rockwell, Carl Mydans, Robert Taylor (now a brigadier general and deputy chaplain of the Air Force), Jess Villamor, Malcolm Champlin. Champlin, a lawyer in Oakland, has written an unpublished book of his wartime experiences, \"One Man's Story.\" Jess Villamor, who lives in Washington, D.C., is presently writing a book of his adventures. Three useful books detailing different aspects of the sinking of the two great British warships were _Samurai_ by Saburo Sakai with Martin Caidin and Fred Saito, H.M.S. _Electra_ by Lieutenant Commander T. J. Cain as told to A. V. Sellwood and _Main Fleet to Singapore_ by Russell Greenfeld. Captain George McMillin checked the material on Guam.\n\n## CHAPTER 6\n\nAdmiral Nimitz told how he first learned he was to relieve Kimmel. The incredible story of \"The Battle of Lingayen Gulf\" came directly from Carl Mydans. Admiral Frank Jack Fletcher himself revealed why he did not relieve Wake Island. The Hong Kong sections are based on accounts given by 43 interviewees, including Tony Weller, Malcolm Swan, Sonny Capell, Bill Stoker, T. J. Cruz, General John Price, Francis Kendall, Dr. L. T. Ride, Alex Dinnen and Douglas Hunt. Leo Landau drove me all over the hilly island, pointing out the various battlefields. Transportation for a tour of the New Territories was provided by Joe Lowe. Other information came from _The Hong Kong Volunteer Defence Corps in The Battle For Hong Kong_ , _A Tear for the Dragon_ by John Stericker, \"Operations in Hong Kong\" by Major-General C. M. Maltby and _Japanese Land Operations_.\n\nThe scene between Eisenhower and Marshall came from the former's _Crusade in Europe_. Important background information was provided by _Strategic Planning for Coalition Warfare_ by Maurice Matlow and Edwin Snell, _United States Submarine Operations in World War II_ by Theodore Roscoe, and _Samurai_ by Sakai and Caidin.\n\n## CHAPTER 7\n\nThe Wake Island story is based on interviews with Lieutenant Colonels Clarence McKinstry and Arthur Poindexter, Brigadier Generals Putnam and Devereux, Sergeant Dave Rush, Captain Iwao Kawai and Commander Giichi Kikuchi. Books used were: Devereux's _The Story of Wake Island_ , _History of Marine Corps Aviation in World War II_ by Robert Sherrod, and _The Defense of Wake_ (Marine Corps monograph).\n\nI have known for several years of the claims of Rear Admiral Winfield Scott Cunningham, the Island commander, that he was in sole command and made the decisions at Wake. His book, _Wake Island Commander_ , written with Lydel Sims, presents many facts unknown to the general public, but the main fact cannot be altered. Devereux commanded the fighting troops and the fighting decisions were his. While it is true that Cunningham was technically in command of all the service personnel, he was a naval aviator, untrained in land warfare. It was agreed, quite logically, before hostilities that Devereux would coordinate the fighting efforts of the atoll.\n\nFor those who revisit old battlefields Wake Island is one of the most evocative. The ruined gun emplacements, the foxholes, the shell cases are still strewn over this coral island. Walking along the deserted beach between Peacock Point and Camp One at night, one can almost see the remodeled Japanese destroyers stealthily heading for shore while the surf booms.\n\n## CHAPTER 8\n\nI first listened to the Tony Aquino story at an artificial lake on his huge sugar plantation in the shadow of Mt. Arayat on a sweltering New Year's Eve. This story, among others, was lost when one of my bags was stolen in Hong Kong, but Mr. Aquino obligingly wrote it out in 48 pages.\n\nFormer President Osme\u00f1a was interviewed in his beautiful home on the island of Cebu. It was time at last, he said, that the world knew the hidden truth of the so-called collaborators. Now that the other principals\u2014Quezon and Laurel\u2014were dead, he felt he could break his promise not to reveal the facts of the important meeting on December 24.\n\nOthers who gave valuable firsthand information were Colonel John Horan, Admiral Rockwell, Carlos Romulo, author of _I_ _Saw the Fall of The Philippines_ , Colonel William Morse and Captain Hattie Brantley.\n\nArch Gunnison's book, _So Sorry_ , _No Peace_ , was helpful here and in subsequent chapters on Manila. The Roosevelt-Churchill anecdote came from Sherwood's _Roosevelt and Hopkins_.\n\n## CHAPTER 9\n\nSee Chapter 6 for Hong Kong sources.\n\nMajor General Albert Jones gave the information about his participation in the Philippine battles in two long interviews at the Presidio, San Francisco.\n\n## CHAPTER 10\n\nThe story of the Calumpit bridges came from Generals Jones, Weaver and Casey and Colonel Harry Skerry, whose voluminous papers give a detailed picture of engineer activities in Luzon. Among others interviewed were General Emmett O'Donnell, now Commander in Chief, Pacific Air Forces; President Osme\u00f1a, Colonel Allen Stowell, Sergeant Frank Trammell and Corporal Durward Brooks. The story of the looting in Manila came from Carl Mydans and Gunnison's _So Sorry_ , _No Peace_.\n\n## CHAPTER 11\n\nThe Layac Bridge account is based primarily on information from Colonel Skerry and Colonel Ray O'Day. O'Day's diary, written in prison camp, provided a fund of detail on the entire Philippine campaign.\n\nTwo of the most important Japanese contributors begin their stories in this chapter. Colonel Takeo Imai granted permission to use all material in his long article in the _Sunday Nippon_ , No. 44, \"Tragedy of Fukuyama Regiment: Was It Imperial Headquarters Mistake?\" The two lengthy interviews with Lieutenant General Akira Nara in the Sanno Hotel were invaluable. All the information about Colonel Takechi (now deceased) and most of the personal data have never before been revealed, not even to Imperial Headquarters. Other major interviewees were Major Ernest Brown, who now sells cars in San Antonio, General Jones and Colonel Frederick Ward.\n\n## CHAPTER 12\n\nSome of the most valuable new information in this and following chapters came from General Clifford Bluemel. Others who contributed greatly were Colonel Rigoberto Atienza, author of an unpublished book, \"Retreat to Samat\"; Captain John D. Bulkeley (now Commanding Officer, Clarksville Base, Clarksville, Tennessee); Hisamichi Kano; General Sadamu Sanagi; and Major General K. L. Berry (Adjutant General for the State of Texas).\n\nI traveled the entire length of the Abucay Line on Bataan by truck and foot. The long trench, reminiscent of World War I, is still there. Several farmers have built their nipa shacks directly over it.\n\nAccompanying me by foot up the rugged slopes of Mt. Natib were a dozen men of the Philippine Constabulary, armed with rifles, pistols and sub-machine guns. We followed the foxhole line until it disappeared, then continued up the dense slope, which was often steeper than 45 degrees. It seems incredible that Colonel Takechi and his men could have scaled such heights with equipment and mountain guns. Six months after leaving Bataan I learned from my host, Lieutenant Colonel Wilfredo Encarnacion, then Philippine Constabulary Chief of Iloilo, why such a large armed guard had been necessary. Mt. Natib was the hiding place of diehard Huks, Communist outlaws, and Colonel Encarnacion wrote that he and his men had just captured Linda Bie, the _nom de guerre_ of Silvestre Liwanag, Number 3 Huk chief.\n\nThe excerpt from Lee's poem \"Abucay Withdrawal\" came from his book, _Nothing but Praise_. The poems were buried in Luzon before Lieutenant Lee was transferred to Formosa and were recovered only after the war. The talented young man met a tragic death, killed like so many others by an American bomb that sank his prison ship en route to Formosa.\n\nDocumentary aids to the Bataan battles included _Armor on Luzon_ , The Armored School, Fort Knox, Kentucky, and monographs written at the Fort Benning Infantry School by Major Charles C. Underwood, Major Kary C. Emerson, Major Donald G. Thompson, Major John I. Pray, Major Everett V. Mead, Major Eugene B. Conrad, Major Henry J. Pierce, Major Louis Besbeck, Major John E. Olson, Major Ernest L. Brown, Captain Harry J. Stempin, Major William E. Webb, Major William R. Nealson, Major Brice J. Martin, Major Beverly N. Skardon, and 1st Lieutenant Sheldon H. Mendelson.\n\n## CHAPTER 13\n\nInformation for the Battle of the Pockets came from Colonel Glen Townsend, author of two unpublished manuscripts, \"Defense of the Philippines\" and \"Documents Pertaining to the War in the Philippines\"; Colonel Adrianus van Oosten, author of the monograph, \"Operations of the 1st Battalion, 45th Infantry (Philippine Scouts) in the Battle of the Tuol Pocket\"; Major Archie McMasters, author of \"Memoirs of the Tuol Pocket\"; General Berry, General Jones, and Colonel Saburo Watanabe.\n\nIt took two days to find the Tuol River and the Pockets. Today the valley between the two extinct volcanoes is nothing like the dense jungle area of the war. Great banana plantations now extend south of the cobblestone road across Bataan. The battle blasted off almost every tree. Even so, the hills and streams are so numerous that guides who had fought in the Pocket battle couldn't recognize the slaughter-ground. On the second day of the search, we finally found the nest of foxholes that Colonel Yoshioka's men had dug in the Big Pocket. Nearby was a pleasant stream, the Tuol River. Some of our party drank from it, but I remembered the bodies clogging the tiny river.\n\nMaterial on the Points came from Captain Ray Hunt, Jr., author of an unpublished book on his Philippine adventures; _The Dyess Story_ by William Dyess; _They Were Expendable_ by W. L. White; Colonel Saburo Watanabe; and Colonel Cecil Sanders, author of the monograph, \"The Operations of the 57th Infantry (Philippine Scouts) at Anyasan and Silaiim Points.\"\n\nSince the former road from Mariveles up the west coast is now impassable after about 10 miles, I was flown up and down the coast several times at low level in a small observation plane to see the formidable terrain the Battle of the Points was fought over.\n\nThe story of Homma's conference came from Colonel Tatsuo Haki, General Moriya Wada, Lieutenant General Monjiro Akiyama, the unpublished Homma diary, and documents used in the defense of General Homma at his trial. Eijiro Namikawa and Lieutenant General Eiichi Tatsumi gave valuable material concerning General Homma.\n\nThe scene between Roosevelt and MacArthur is based on _The Untold Story of Douglas MacArthur_ by Frazier Hunt. The Quezon scenes came from his own book, as well as _The Good Fight_ , _MacArthur 1941-1951_ by Major General Charles Willoughby and John Chamberlain, and from interviews with President Osme\u00f1a and Tony Aquino.\n\n## CHAPTER 14\n\nThis chapter is based mainly on more than a dozen interviews with Yates McDaniel in the Pentagon. Details of the Yamashita-Percival meeting came from General Ichiji Sugita, presently Chief of Staff, Ground Self-Defense Force of Japan and from Ronald McKie (author of _The Heroes_ , the suspenseful story of the two Singapore raids), who corresponded at length with the widow of Colonel Wild. Other information came from Newsreel Wong, the Rt. Reverend John Leonard Wilson, now Bishop of Birmingham, and these books and documents: _Suez to Singapore_ by Cecil Brown, _Fortress_ by Kenneth Attiwell, _Singapore Is Silent_ by George Weller, \"Operations of Malaya Campaign, from 8th December, 1941, to 15th February, 1942\" by Lieutenant-General A. E. Percival; and _Why Singapore Fell_ by H. G. Bennett. _The Royal Australian Navy_ by G. Herman Gill was useful in this and the next three chapters.\n\n## CHAPTER 15\n\nAdmiral Conrad E. L. Helfrich gave many of the details for this and the following two chapters. For a complete picture of the battle for the Netherlands East Indies, the two volumes of the Admiral's _Memoirs_ must be read. These contain a detailed and accurate picture of the complicated situation. Unfortunately they are in Dutch and probably were not studied carefully, if at all, by some American authors in their accounts of these contentious days.\n\nReaders of Samuel Eliot Morison's extremely readable _Rising Sun in the Pacific_ may wonder why the score of U.S. destroyers in Makassar Strait is recorded here as three, instead of four, transports. Japanese survivors of this American raid revealed that one of the transports, the _Tsuruga Maru_ , was sunk by a Dutch submarine four hours before the American action.\n\n## CHAPTER 16\n\nFor the Battle of Java Sea 89 men of four nationalities were interviewed. Chief among these were: Captain H. S. Hamlin, Jr., author of \"The _Houston_ 's Last Battles\" in the May 1946 issue of _Shipmates;_ Midshipman Hachiro Kimura; Commander P. O. L. Owen; Commander Yasumi Toyama; Vice Admiral Thomas Binford; Captain William Giles, Jr., author of \"War Log of the U.S.S. _John D_. _Edwards_ \"; Rear Admiral Edward Parker, now Chief Defense Atomic Support Agency; Captain Joseph Dalton; and Yeoman 3rd Class John Harrell. The book by Commander Cain, _H_. _M_. _S_. _Electra_ , was helpful as was the lengthy report by Rear Admiral Henry E. Eccles, commander of the _John Edwards_ , entitled, \"The Java Sea Battle.\"\n\n## CHAPTER 17\n\nFor the savage engagements following the Java Sea debacle main interviewees were Captain Robert Fulton, Lieutenant Commander Tsujimoto and Senior Lieutenant George Cooper. Rear Admiral Welford Blinn provided documentary information on the sinkings of _Exeter_ and _Pope_.\n\nMuch of the data on U.S. air activities in Java came from a long statement compiled by Lieutenant Colonel William A. Sheppard and Major Edwin Gilmore.\n\nThe story of the sinking of the four Japanese transports in Bantam Bay was told by Commander Shukichi Toshikawa, the man detailed to apologize to General Imamura for his unceremonious immersion. For details of the heroic last hours of the _Perth_ I am indebted to Ronald McKie for permitting use of material from his book, _Proud Echo_.\n\n## CHAPTER 18\n\nThe story of the Mt. Samat attack plan, based on interviews with Colonel Hattori and Captain Nishiura, is another first-told tale. The famous MacArthur PT escape from Corregidor was told by Brigadier General Morhouse, now Air Surgeon of Fifth Air Force, Captain Bulkeley, Colonel Frederick Ward, Admiral Rockwell and General Casey. Useful also were _They Were Expendable_ by W. L. White and _MacArthur_ , _His Rendezvous With History_ by Courtney Whitney.\n\nThe date of leaving Corregidor, March 11, is not a typographical error, although MacArthur's official report and even _Fall of the Philippines_ give the date as March 12. This error has been passed down from book to book. Admiral Rockwell and Captain Bulkeley both confirm the March 11 date from their records. Colonel Morse, who met the boats in Mindanao, asserts the arrival was on March 13, not March 14. For details of the flight from Del Monte to Australia 32 people were interviewed including Colonel Morse, Colonel Ind, Captain Monay and Major General Carmichael. Although one man insisted he helped load a refrigerator and a mattress full of gold pesos on the MacArthur plane, the 31 others insisted categorically that General MacArthur and his family took no more than the usual 35 pounds of luggage apiece.\n\nCaptain Bulkeley was helpful in correcting important details of certain stories that have gained general credence. He also added a comment on General MacArthur that deserves quotation. \"MacArthur was really a man without any fear. He never flinched and was inflexible in his purpose to get to Australia and continue the war against the Nips. A truly inspirational man in all respects and professionally competent in that he let the competent commander run his show if he was good and fired him if he was bad. Also a man of great loyalty and appreciation afterwards.\"\n\n## CHAPTER 19\n\nBased on information from General Bluemel, Jones and Nara, Colonel Atienza, Colonel Tom Dooley, Captain Matt Dobrinic and Captain Hattie Brantley. Official histories of the divisions involved and _I Served on Bataan_ by Juanita Redmond were also useful.\n\n## CHAPTER 20\n\nMaterial concerning the last hours of Bataan came from more than 100 men and women. Among them were: General Romulo, Imai, Nara, Jones and Bluemel, Sergeants Frank Bernacki and James Madden, Tony Aquino, and Lieutenants Dan Jopling and Lucy Wilson, who is now Mrs. Jopling.\n\nDetails of the surrender were based on testimony of General King, General Funk and Colonel Collier at the Homma trial. Japanese officers testifying for General Homma also gave many details of the events of April 9.\n\n## CHAPTER 21\n\nI walked over much of the 55-mile route of the Death March in the hottest part of the day. Along the way Bataan citizens, many of whom had witnessed the tragic parade, were interviewed. More than 100 survivors contributed to this chapter. They include: Colonel O'Day, General Jack Schwartz (now Commandant of Letterman General Hospital), Tony Aquino, Generals Jones, Bluemel, Stevens and Berry, Colonels Aldridge, Browne, O'Connor, Revak and Stowell, Sergeants Grow, Cape, Graef, James, Franklin, Bernacki, Hunt and Duran and Corporal Castleberry. The documents for defense of the Homma trial, Homma's papers and _The Dyess Story_ by William Dyess were helpful. Particularly revealing was the thesis written by Stanley Lawrence Falk for his Masters Degree at Georgetown University, 23 May, 1952, entitled, \"Bataan Death March.\"\n\n## CHAPTER 22\n\nInformation on the Doolittle raid came from General Doolittle, Aviation Chief Machinists Mate Respess, Captain Toshikazu Ohmae and several books, including _Thirty Seconds Over Tokyo_ by Ted Lawson. The Bradford story was told by \"Jitter Bill\" himself, one of the most remarkable fliers in aviation history. Colonel Arthur Fischer, who brought the cinchona seed to the U.S., was also interviewed. His story alone deserves a chapter.\n\nThe last days of Corregidor came from Private Ralph Houston, Rear Admiral Dempsey, Colonel Dalness, Lucy Wilson and the article by Colonel William C. Braly, \"Corregidor\u2014A Name, A Symbol, A Tradition,\" in the _Coast Artillery Journal_ , July-August, 1947.\n\nI flew over Corregidor several times, finally landing at Kindley Field on the tail of the polliwog. The grass on the old strip was about three feet high and we learned it was the first landing in many years. I spent some time in Malinta Tunnel, exploring the important laterals. The island itself is still a gold mine for scrap collectors. A company of Philippine marines attempts to keep pirates from stealing the great shells which remain in abundance.\n\n## CHAPTER 23\n\nThe surrender details came from Corporal Irving Strobing, Captain Melvyn McCoy, Colonel Jesse Traywick, Colonel Tom Dooley and the Homma defense papers. Also helpful was the book by newsman Uno, _Corregidor: Isle of Delusion_. Printed during the war, its objectivity is remarkable.\n\nI visited the Castillo home in Cabcaben. The son of the former owner showed me the remains\u2014a few concrete posts. A breeze was blowing in from Manila Bay. Corregidor, only a few miles away, was distinct in every detail. It must have looked the same that day in May almost 20 years ago, except that no columns of smoke were rising from the rock island.\n\nWainwright's radio speech on the night of May 6-7 over KZRH was taken from a transcript of the short-wave broadcast by the Foreign Broadcast Intelligence Service of the F.C.C. Certain unintelligible words were deciphered by studying a copy of the original draft of Wainwright's speech.\n\nThe details of Colonel Traywick's bizarre and historic trip to Mindanao were told by Traywick in San Antonio, where he now lives. This is the first time they have been published. Much of the material concerning the north Luzon area came from a long unpublished account written by Colonel Horan, \"The First Five Months of Guerrilla Warfare in Northern Luzon.\" This too appears for the first time. Additional new material came from General Chynoweth. In addition to a long interview, the general granted permission to use material from his unpublished book \"Strange Surrenders.\"\n\nJohn H. Skeen, Jr., now a lawyer in Baltimore, was appointed chief counsel for the defense at Homma's trial by authority of General MacArthur; Skeen had only been admitted to the bar in 1941, a year before entering the Army. He was allowed 15 days to prepare the defense, whereas preparation for the prosecution had been going on for some months and apparently had accumulated a great amount of evidence.\n\nSkeen said the trial was not conducted according to recognized legal procedures. Hearsay testimony and affidavits were accepted in evidence with no opportunity for cross-examination of deponents. He still feels it was wrong for Homma to be tried by authority of a man he had defeated on the field of battle and by whose authority both defense and prosecution personnel were appointed and any appeal would be reviewed. \"It was a highly irregular trial, conducted in an atmosphere that left no doubt as to what the ultimate outcome was to be.\"\n\nThis opinion was not his alone. The others on the defense staff signed a letter to General Homma stating they felt he had been unfairly convicted. Equally concerned were two Associate Justices of the U.S. Supreme Court, Wiley Rutledge and Frank Murphy.\n\n\"This nation's very honor, as well as its hopes for the future, is at stake,\" write Murphy. \"Either we conduct such a trial as this in the noble spirit and atmosphere of our Constitution or we abandon all pretense to justice, let the ages slip away and descend to the level of revengeful blood purges....A nation must not perish, because in the natural frenzy of the aftermath of war, it abandoned the central theme of the dignity of the human personality and due process of law.\" While Homma was waiting for sentence he wrote his wife, Fujiko, on January 25, 1946:\n\nTO MY WIFE,\n\nI have already written what I wanted and said almost all I wanted to say. There's no longer anything I want to leave in this world. No, there's a lot. But there's nothing I can do about it being the way I am now. Japan will recover unexpectedly soon. I believe in five years Japan will start a positive step forward.\n\nI'm not afraid of a death sentence. I'm prepared for it. I think I'll be able to face execution with little thought but just thinking the time has finally come. When I was in Omori Prison an army doctor told me it was a painless way to die to be shot to death. It is only that I feel I haven't expressed my thanks enough to my wife. In the twenty years of our married life we've had many differences of opinion and even violent quarrels. Those quarrels have now become sweet memories. You are a righteous person. Unusually righteous, and therefore, you are fearless and strong. You would speak far more irritating words than you meant with your sharp tongue. This used to start our fights. However, in my mind I had absolute confidence in you. In many ways you have offset shortcomings of my character and assisted me in the right sense.\n\nNow as I am about to part from you, I particularly see your good qualities, and I have completely forgotten any defects. I have no worry about leaving the children in your hands because I know you will raise them to be right and strong. I also have no doubt that you will take good care of my mother in Sado after I die. My mother will die a peaceful death attended by you and her grandchildren.\n\nTwenty years feels short but it is long. I am content that we have lived a happy life together.\n\nIf there is what is called the other world, we'll be married again there. I'll go first and wait for you there but you mustn't hurry. Live as long as you can for the children and do things I haven't been able to for me. You will see our grandchildren or great-grandchildren and tell me all about them when we meet again in the other world.\n\nThank you very much for everything.\n\nThe last words from Homma came in a letter to his children just before execution:\n\nI don't know if this letter will reach you. But I'm writing because they gave me the paper. I had never thought \"a death sentence,\" \"execution by shooting,\" etc. had any connection with our life, but they have now become the reality before my eyes. Of the 15 condemned prisoners here, only I and Sub-Lieutenant Okuda who is in a cell next to mine are to be shot, the rest will all be hanged. General Yamashita and three others were already executed several days ago. I consider myself luckier to be at least shot to death with the honor of a military man. Your mother as well as all our friends have done their best for my sake. There's nothing more to do but to leave it to fate. [All of you mustn't forget why and because of whom*1 your father was executed by such severe punishment.]*2 There are six men here who have been sentenced for life. It will be better to be shot to death like dying an honorable death on the battlefield than spending a disgraceful life in such a cage the rest of one's life. Don't lose courage, children! Don't give in to temptation! Walk straight on the road of justice. The spirit of your father will long watch over you. Your father will be pleased if you will make your way in the right direction rather than bring flowers to his grave. Do not miss the right course. This is my very last letter.\n\nMy address: Three- _jo_ (mat) room fenced all\n\naround with wire, about 30 miles\n\nsoutheast of Manila\n\n## CHAPTER 24\n\nThe story of Japanese operational planning after the Battle of Java Sea came mainly from Colonel Hattori. American planning for the Battles of Coral Sea and Midway was told by Admirals Nimitz, Spruance and Fletcher. The story of two of the mistakes that lost the Midway battle came from General Genda. He spoke of himself and what he should have done as a third person, with complete objectivity. Mrs. John Waldron checked her husband's story and gave permission to quote his last letter to her. Also helpful were: _Queen of the Flat-Tops_ by Stanley Johnston; _Climax at Midway_ by Thaddeus Tuleja; the articles, \"Never A Battle Like Midway,\" by J. Bryan, III, from the _Saturday Evening Post_ , March 26, 1949, and \"Torpedo Squadron 8,\" _Life_ magazine, August 31, 1942; _History of Marine Corps Aviation in World War II_ by Robert Sherrod; and a report by Admiral Soemu Toyoda. Much Japanese material came from the authoritative book, _Midway_ , by Mitsuo Fuchida and Masatake Okumiya, edited by Clarke Kawakami and Roger Pineau.\n\n## CHAPTER 25\n\nEarl Gallaher gave the details of his two strikes. His personal story has never been told before. Other firsthand information came from the three American admirals responsible for the great victory and Commander Tanabe, who sank the _Yorktown_. The identities and positions of the four Japanese carriers, long disputed, are based on Japanese information and the ingenious reasoning of Thaddeus Tuleja in his book on the battle.\n\nFor the section on Philippine guerrillas and underground workers, 64 men and women including Representative Pendatun, Jess Villamor, Wendell Fertig, Dr. Atienza, Fanny Greenwell, William Fossoth, Jr., Frank Bernacki, Lacsamana (brother of the Baluga king), Jim Cushing, Senator Peralta, Salvador Abcede, Colonel Ind, Julian Manansala and Colonel Marking were interviewed. Contrary to the opinion of many American observers who have spent their time almost wholly in the Manila area, today most Filipinos have a deep affection for America. They remember the deadly struggle, the agony of hunger and defeat they shared with us. Anti-American stories to the contrary, the Philippines remain a bridge to the continent of tomorrow, Asia.\n\n* * *\n\n*1 MacArthur\n\n*2 The bracketed lines were censored. Later a friend sent the censored portions to Mrs. Homma.\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9781101969298\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n\n## Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Table of Contents\n  6. Author's Note\n  7. Prologue: \"With Dangerous and Dramatic Suddenness\"\n  8. Part I - Timetable for Conquest\n    1. 1: \"Climb Mt. Niitaka\"\n    2. 2: \"Tora...Tora...Tora\"\n    3. 3: \"Tallyho! Bandits Over Clark!\"\n    4. 4: \"So, Shoot to Sink\"\n    5. 5: \"It Was Such a Beautiful Ship\"\n  9. Part II - The Defenses Crumble\n    1. 6: The Death Watches\n    2. 7: End of Wake\n    3. 8: Panic\n    4. 9: Dark Christmas\n  10. Part III - Battle for Bataan\n    1. 10: Roads to Bataan\n    2. 11: Action at Abucay\n    3. 12: Retreat\n    4. 13: The Points and the Pockets\n  11. Part IV - Death of two Empires\n    1. 14: This Naked Island\n    2. 15: \"These Stout-Hearted Dutchmen\"\n    3. 16: The Battle of Java Sea\n    4. 17: \"Good-bye Till Better Times\"\n  12. Part V - \"The Battling Bastards of Bataan\"\n    1. 18: \"...And I Shall Return\"\n    2. 19: The Good Friday Attack\n    3. 20: \"I Would Rather Die a Thousand Deaths\"\n    4. 21: The Death March\n  13. Part VI - From Humiliation to Victory\n    1. 22: \"But Not in Shame\"\n    2. 23: Strange Surrenders\n    3. 24: \"Go in and Get a Hit\"\n    4. 25: The Turn of the Tide\n  14. Image Insert\n  15. Acknowledgments\n  16. Notes\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. x\n  11. xi\n  12. xii\n  13. xiii\n  14. xiv\n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. \n  373. \n  374. \n  375. \n  376. \n  377. \n  378. \n  379. \n  380. \n  381. \n  382. \n  383. \n  384. \n  385. \n  386. \n  387. \n  388. \n  389. \n  390. \n  391. \n  392. \n  393. \n  394. \n  395. \n  396. \n  397. \n  398. \n  399. \n  400. \n  401. \n  402. \n  403. \n  404. \n  405. \n  406. \n  407. \n  408. \n  409. \n  410. \n  411. \n  412. \n  413. \n  414. \n  415. \n  416. \n  417. \n  418. \n  419. \n  420. \n  421. \n  422. \n  423. \n  424. \n  425. \n  426. \n  427. \n  428. \n  429. \n  430. \n  431. \n  432. \n  433. \n  434. \n  435. \n  436. \n  437. \n  438. \n  439. \n  440. \n  441. \n  442. \n  443. \n  444. \n  445. \n  446. \n  447. \n  448. \n  449. \n  450. \n  451. \n  452. \n  453.\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Table of Contents\n  5. Start\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nBegin Reading\n\nTable of Contents\n\nAbout the Author\n\nCopyright Page\n\nThank you for buying this\n\nSt. Martin's Press ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nTo Clara Barton, my very distant cousin, who was a far more interesting woman than I ever suspected\nAcknowledgments\n\nThis book was inspired by a trip to Boonsboro, Maryland, where I strolled around the small town and realized that the buildings on the main street had changed very little since the nineteenth century, except for a thin skin of later siding. It was easy to envision peeling off that skin to reveal the original town beneath.\n\nI am grateful to my editor, Hannah Braaten at St. Martin's Press, who saw the potential in this new series and whose comments made this a better book. Many thanks also to my tireless agent, Jessica Faust of BookEnds, who put me together with Hannah.\n\nI've been published for ten years now, and I never would have made it this far without the steady support and encouragement of the mystery writers community, and I can't begin to count the friends I have made there. The national organization Sisters in Crime, along with my local New England chapter, have been mainstays. The same goes for my blogger colleagues from Mystery Lovers' Kitchen, Killer Characters, and Wicked Cozy Authors, where we all share our writing and publishing ups and downs.\n\nAnd I have to add a nod to the Massacusetts Barton family from which I am descended. They have provided a wealth of interesting plots, and I simply borrow them and drop them into my books.\n\nI stopped in the doorway of the second-floor restaurant at the Oriole Suites Hotel in Baltimore, where I'd worked for the past five years, looking for Lisbeth. She'd called out of the blue and asked to meet me for lunch, but she'd been very secretive about why, which led me to think it was more than just a \"hey, how are you, what have you been doing for the past ten years?\" kind of lunch. Still, we'd been good friends in high school then sort of drifted apart. As far as I knew, she'd stayed in Asheboro ever since, going to the local community college, marrying a local guy, having a couple of kids. Unlike me: I'd headed out of state for college and never looked back. Not that there was anything wrong with Asheboro, Maryland. It was a nice small town\u2014with emphasis on the \"small\"\u2014but I'd always had bigger plans. Most of what I knew about Asheboro now had come from holiday cards from the few people I'd kept in touch with, like Lisbeth.\n\nShe spotted me standing in the doorway and waved enthusiastically. I smiled and wove my way through the tables until I reached hers. When she'd called me, I'd made the reservation for us, and made sure there was a nice view of Baltimore Harbor. Working for the hotel had a few perks.\n\n\"Katie, you look great!\" she exclaimed, standing and hugging me. \"Love your outfit. But then, you work in a city, so I guess you have to look good. I had to dig into the back of my closet to find something to wear that wasn't jeans or sweats. I'm so glad you could meet with me!\"\n\n\"It was great to hear from you,\" I told her, as we took our seats. \"How long has it been?\"\n\n\"I think you came to Jeffrey's christening. He's eight now.\"\n\n\"Wow, that long? But since my folks moved to Florida, I haven't had much reason to visit Asheboro.\"\n\n\"They bailed out early, didn't they? What are they, fifty-something?\"\n\n\"Yes. But they always figured they'd head south eventually, and they decided to go while they were still mobile enough to enjoy it. They didn't want to get stuck in one of those old-folks towers on the beach. They've got a nice small cottage that suits them, and I get to visit when I want.\"\n\n\"Sounds great!\" Lisbeth said, although I sensed a false note in her voice. \"Anything new with you? A guy? Or lady? Kids?\"\n\n\"No to all of those,\" I told her, smiling. \"I'm married to my job, and that's enough for me.\"\n\nA waiter appeared. I smiled at him; I knew Tony pretty well, since I often dropped into the restaurant to pick up something to eat. He handed us menus, and we ordered iced tea.\n\n\"What's good here?\" Lisbeth asked Tony, staring in awe at the multipage menu.\n\n\"If you want some local flavor, the crab cakes are great.\"\n\n\"Okay, I'll do those.\" She shut the menu with a firm slap and handed it to Tony. \"How long have you been working at the hotel, Katie?\"\n\n\"About five years. I don't remember when I last updated you, but you already know I went to college, worked for a while, decided I wanted to be in management so I got an MBA, worked at a big chain hotel in Philadelphia for a couple of years. Then this hotel opened up in Baltimore, and I snagged a really good job in administration, and here I am.\"\n\n\"I'm surprised you came back to Maryland,\" Lisbeth said. \"Didn't you want to travel, see other places? Los Angeles? Chicago? London?\"\n\n\"I thought about it, but this was too good a job to pass up and I was given a free hand, since the hotel was new.\"\n\nWaiter Tony delivered the tea and slipped away again. \"What exactly is it you do here?\" Lisbeth asked.\n\nSince I'd more or less created the position, it wasn't easy to explain. \"A little of everything. I'm in charge of customer satisfaction. I know, that sounds vague, but it's kind of a catch-all job about making sure that everything goes smoothly. No, more than that, really: that everything is comfortable but memorable. Appearance, cleanliness, responsiveness of the staff, even the quality of the food and the polish on the silverware and glassware. We want people to go home talking about this place, sending their friends here, coming back themselves.\"\n\n\"You always were a little OCD,\" Lisbeth said. When I looked startled, she spoke quickly to explain. \"In a good way, of course. I mean, you were always really well organized. You turned in your assignments on real paper, not just something torn out of a notebook. You always spell-checked everything twice. When we were on the soccer team, you always knew all the rules, and you even argued with the refs now and then. Now you just do it on a much bigger scale. I mean, you've got a whole hotel here!\"\n\nHad Lisbeth really seen me that way, years ago? \"Yes, but you can still break it down into smaller tasks, like service staff, restaurant, website, reservations, and so on. I won't bore you with it all. What about you? What have you been doing?\"\n\nTony brought our lunch dishes and refilled our water glasses, then disappeared discreetly once again. While we ate Lisbeth nattered on about her husband, and her kids, and how smart they were (all of them), and how wonderful, and what a busy life she led\u2014what with the PTA and the annual school fund drive, and her daughter's Brownie troop and her son's softball team. I listened with half an ear and tried to summon up my memories of Asheboro: small town, one main street, one stoplight in the center. Surrounded by pretty rolling country (although for all I knew it was nothing but strip malls now) and near a lot of Civil War battlefields, but most history buffs just passed right through, since there was no reason to stop in town. One elementary school, ending with eighth grade, and one high school\u2014or maybe now there was a regional high school\u2014I wasn't exactly on their email list for updates. One old factory that had been closed since long before we were born. Maybe it had fallen down by now. Parents had warned us to stay away from it, but there were always a few guys who couldn't resist a dare. Yet I couldn't recall that anybody had ever gotten hurt there. I had no recollection of what the old factory had made.\n\nMy hometown was sounding really dull, even to me\u2014and I'd been more or less happy growing up there. So I hurried to recall a few good points: no violence, no poverty, no pollution. It was a safe, quiet town, like so many others. It was also a place I hadn't wanted to stay, but that didn't mean that good people didn't go on living happy lives there. People like Lisbeth, who seemed to have thrived in that setting.\n\nI noticed that Lisbeth kept darting nervous glances at me. Did she have another motive that she hadn't mentioned? \"So, what brings you to Baltimore, Lisbeth? Don't tell me it's just to have lunch with me and catch up. Are you leaving your husband? Looking for a job?\"\n\nShe giggled nervously. \"No, no, nothing like that. Phil and I are fine. He's got a great job. The kids are healthy and happy. I keep busy with volunteering, like I told you, and we're doing okay financially.\"\n\nShe wasn't answering my question, so I pressed on. \"You wanted a mini-vacation? Come on, Lisbeth, there's something you aren't saying. Isn't there?\"\n\nShe sighed and looked out at the sparkling harbor, the clusters of people passing by, the boats on the water\u2014anywhere but at me. \"I have a favor to ask. Or we do. And it's a big one.\"\n\nWe? \"You mean, you and Phil? And what's big?\"\n\n\"No, actually, I mean the town.\" She took a deep breath. \"We want you to save Asheboro.\"\n\nWell, that was something I didn't expect to hear. \"What on earth do you mean?\"\n\n\"It's kind of complicated. You have time?\"\n\n\"After a request like that, I'll make time. Start at the beginning.\"\n\nTony appeared again. \"Would you ladies care for anything else?\"\n\n\"Just coffee for me, please,\" Lisbeth told him. I asked for coffee too, but I also requested one of the petit fours plates to share for dessert. The restaurant had a great chef and baking staff, and I'd asked that they create a plate of small bites for those patrons who didn't want anything heavy to end their meal. It had proved very popular, even for people requesting room service, and I was proud of it. Attention to detail in the hospitality business was important.\n\nMy mind was wandering. Was I stalling for time? Lisbeth was staring at me with an odd expression on her face, but she stayed silent until Tony reappeared with the pretty platter of goodies and set it carefully between us, then retrieved the polished coffee carafe and filled our cups. Then he retreated silently.\n\nI looked her straight in the eye. \"Have a strawberry tartlet and then tell me what the heck is going on, Lisbeth.\"\n\nShe reached tentatively for the silver-dollar-size pastry and took a bite\u2014and smiled. \"Ooh, this is good.\"\n\n\"I know it is. What is it you want me to do?\"\n\nLisbeth finished the tartlet with a second bite. \"I'm what I guess you'd call an emissary, although I was the one who brought your name up, and volunteered to come talk to you. When was the last time you visited Asheboro?\"\n\nI had to stop and think. \"I know I was there to help my folks pack for the move\u2014that must have been at least three years ago. Other than that I have no reason to go back. I haven't been to a high school reunion. Why?\"\n\n\"You heard about that big storm a couple of weeks ago?\"\n\n\"I saw it on the news, but they didn't mention Asheboro.\" Now she was beginning to worry me. \"Was the town in its path?\"\n\n\"No, not directly, but close enough. There was a lot of damage, mostly superficial rather than structural. You know, high winds blowing shingles off, or siding. But a lot of people had been putting off fixing things the past few years, so things were loose, you know? And people, mainly the store owners, were cutting back on their insurance, because it just cost too much. They may not be able to make all the repairs they'd like to, so they might just pack up and go somewhere else.\"\n\nI still didn't see where I fit in this. \"Can't the town help? There must be some municipal funds for emergencies.\"\n\n\"Well, yes, there would be, normally. But... you remember the old Barton place, outside of town?\"\n\n\"Sort of. Did it blow down?\"\n\n\"No, not at all. It was built to last. You probably don't remember the history of the place. It was built by some Civil War veteran who moved to Asheboro and made a mint with the factory in town, and poured most of it into making his house as grand as he could. But he never had kids, so when he died, nobody inherited. He created a trust to support it until a buyer came along, and the bank has been managing it ever since. But they couldn't manage to sell it\u2014too big, too expensive, too far out of town, who knows what. So that's where things rested for quite a while. More than a century, actually.\"\n\n\"So what's that got to do with anything?\"\n\nLisbeth helped herself to another pastry. \"Well, last year a member of our town council proposed that the town buy it from the bank and turn it into something useful. She was really enthusiastic, not to mention persuasive, so she sold the rest of the council on the idea. The bank was happy to go along with it to get it off their books, so they expedited the paperwork and set a really reasonable price on it, and the town bought the place outright. Which used up whatever surplus money we had. If it's used for any commercial purpose, the town will get some of that back, in taxes, but right now our account is just about empty, so we can't help the businesses in town.\"\n\n\"Ambitious,\" I said, nodding, \"but not a bad idea, if they can turn the place around. Which of course does not solve your immediate problem.\"\n\n\"It gets worse,\" Lisbeth said. \"Turns out that board member had her own ideas for what to do with it, but when she laid them out to the board, after the purchase had been voted on by the town and the title had been transferred, they pitched a fit and voted her idea down. She's royally pissed off. But nobody's got a better idea.\"\n\n\"Buyer's remorse, probably. And that's where you think I come in?\"\n\nLisbeth nodded. \"Yes. You're in the hotel business. You know how to run things. You know how to fix things up and make people want to visit them, like you said. We thought maybe you could come spend a little time in Asheboro and maybe come up with some new ideas?\" Lisbeth looked at me like an eager puppy.\n\nObviously if the town had no money, there wouldn't be anything like a consulting fee in it for me. Did I really want to bail out a place I had deliberately left behind? Could I even do it? From Lisbeth's very brief description of the issues, nobody had any money to throw at the problem\u2014and it was pretty clear that it would take money. And nobody had a clue which way to go. As I remembered the town, there really wasn't much there, and no way to dress it up\u2014and that had been before this storm had swept through. I had to admit I had barely seen the Barton mansion, because it was surrounded by a substantial piece of property that kept it invisible from the roads. But if in fact it had been sitting more or less empty for decades, there had to be some major problems with it, didn't there? Even if it was gorgeous, could it be brought back?\n\n\"Say something. Please?\" Lisbeth begged. \"We're at the end of our rope, and you're our last resort. I wouldn't have come to see you if there was any other way out of this mess.\"\n\nI thought about it for a moment. While I could see plenty of pitfalls, this project in Asheboro sounded like a real challenge. If I looked around and saw there was no hope for the town, I could tell them as much and walk away. Couldn't I? What the heck\u2014why not?\n\n\"All right. But I have to see what's going on\u2014what kind of damage the town has suffered, and what kind of shape that Barton place is in. Maybe talk to some of your town council. Will people be around if I come down this weekend?\"\n\nLisbeth looked like the weight of the world had been lifted from her shoulders. \"Of course! I'll make sure they are.\" She grabbed another pastry and ate half of it in a bite, beaming all the while. \"We'll set up a meeting for you.\"\n\n\"Don't get too worked up about it, Lisbeth\u2014I'm not promising anything. But I'll look things over and tell you what I think can be done. Or if it's hopeless.\"\n\n\"That's all I can ask. Thank you, Katie. You're a lifesaver.\"\n\n\"Don't say that until after I've looked at things.\" I glanced at my watch. \"Look, I've got a meeting at two and a foot-high stack of paperwork on my desk. But how about I drive over on Saturday and you can show me around and connect me with the people I need to talk to?\"\n\n\"Perfect. Thank you so, so much, Katie!\"\n\n\"I go by Kate, mostly, these days.\"\n\n\"Oh, right, of course. But I'll always remember you as Katie.\"\n\nI wondered just what I had let myself in for. Bailing out my old hometown? Yeah, right. But it was worth taking a look, if only to help an old friend. I hoped.\n\nAt the end of the day I headed back to my condo on the fringes of the city, and I found a message from my boss, George, on my answering machine. That was odd, since I'd been in the office all day except for my lunch with Lisbeth, and I was pretty sure he'd been somewhere in the hotel as well. In the message he asked to meet me early the next morning when I came in, but didn't leave any explanation. George and I got along well so I hoped it might be good news. I'd have to wait and see.\n\nThe next morning I was in my office at Oriole Suites early, in advance of the meeting with my boss. I still had no idea what it was about: as far as I knew, everything was going smoothly, and our stats looked good. A few minutes ahead of schedule I presented myself to George's executive assistant, who said quickly, \"He's expecting you. Go on in.\" Then she looked away.\n\n\"Good morning, George,\" I greeted him as I walked in. \"Is this about the refurbishment of the penthouse?\"\n\n\"Sit, please, Kate,\" he said, his expression somber. He sat back in his chair and steepled his hands. \"We've worked together for some time now, so I'll get right to the point: this hotel has been bought out by a Japanese conglomerate, one that owns quite a few hotels in multiple countries. As a business acquisition, it makes a lot of sense.\"\n\nI hadn't seen that coming. I felt gut-punched. \"And?\"\n\n\"They want to bring in their own management team to run the hotel. They've decided that I'd be redundant, so I've been let go.\"\n\n\"Oh, George, I'm so sorry to hear that. You've done a great job here.\" I knew that for a fact: the hotel was running like a well-oiled machine, and bookings and profits had been up for the past two years. Which was probably why it had been a tempting target for a buyer. I sat back and waited for the other shoe to drop, because I was pretty sure there had to be one.\n\n\"As have you, Kate. It's hard for me to tell you that they don't want to keep you on either. It's no reflection on your work or your abilities. It's purely a business decision. There will be a generous severance package\u2014at least a year's salary.\"\n\nIt took me a moment to wrap my head around what he'd just told me: I'd been fired. No matter how anyone dressed it up, I was out of a job. \"When will this happen?\" I finally managed to say.\n\n\"As soon as possible. The new managers are already onsite, and they'd like to speak with you.\"\n\nAnd ask for all my records and pick my brains about the hotel and the city, before they booted me out the door. But I was determined to stay cool and collected, so I said, \"I will certainly extend them all professional courtesies. How long have you known?\"\n\n\"I was told about this last week, but I was asked not to share the news. I've enjoyed working with you, and you've done an impressive job here. Even made me look good.\" He smiled, but his heart wasn't in it.\n\n\"We've made a good team. Have you had time to make any plans?\"\n\nGeorge shook his head. \"No, not yet. I'm a bit older than you, and I have kids in college, so I have to think through what my options are.\"\n\nGeorge and I reviewed some of the corporate details that needed consideration, and when we were finished, I made my way to my office, shut the door, and sat in my swivel chair, staring into space. Poor George. I didn't want to add \"poor me\" but I was pretty sure I could see the handwriting on the wall: after the new owners had pumped me for any information they needed I was out the door. I hadn't been unemployed since I got my MBA.\n\nI sat a moment longer to pull my thoughts together. Then I walked out of my office, told my administrative assistant that I had an unexpected appointment out of the office and wasn't sure how long I would be gone, but she could reach me on my cell with anything urgent. And then I retrieved my car and drove home.\n\nWhat was I supposed to do now? I had enough money in the bank to get by for a while, and George had promised I'd get a generous severance payout. I had friends and colleagues in the hospitality industry up and down the East Coast. Or, I could do what Lisbeth had reminded me of my old dreams\u2014I could take some time off and travel. I hadn't been to Europe since college, and now I could check out the high-end hotels instead of those ratty B&Bs of my early travels. I didn't have to make a decision any time soon, did I?\n\nAnd I had promised Lisbeth that I'd come back to Asheboro and see what the heck was going on there and whether I could be of any help to the town. I certainly had the time for that now. And it might be a good distraction while I tried to figure out what I wanted to do next with my life.\n\nI thought about cleaning out my office, but I rejected that because what I really wanted to do was go home and feel sorry for myself. When I did get home, what I found myself doing was cleaning my small apartment, something I'd seldom had time to do. I hoped chasing dust bunnies would keep me from thinking.\n\n* * *\n\nIt didn't. As I vacuumed under my bed I thought about Asheboro\u2014as I had known it years ago. Lisbeth and I had been good friends in high school. Looking back, I'd come to think of us as the Invisible Girls. We weren't cheerleaders, but we weren't pond scum either. We were part of that amorphous middle: nice kids who didn't make trouble or get in trouble, who kept their grades up, who volunteered for things like library fundraisers and litter cleanup days. I'd lost touch with most of that friend group\u2014in fact, all of them except Lisbeth. I'd left, made new friends in college (not that I was doing much better keeping in touch with them, but there was always the alumnae mag for that, and occasionally Facebook), and just kept moving.\n\nIt had to have taken a lot of guts for Lisbeth to come to Baltimore just to ask for my help. From what little I knew of her current life, she was pretty much a town mouse, next to my city mouse. I wondered just why she thought I was a miracle worker. Sure, I knew the town. Correction: I used to know the town, but a lot could have changed. I had no background in municipal management, for a town of any size. I could handle a hotel that held maybe a thousand people in full convention mode, but that was about my limit. Although, I reflected, maybe Asheboro didn't have more than a thousand people anymore.\n\nThe town wasn't unique in the problems it faced. Towns that had thrived in an earlier century, with industries and shops and agriculture, had been left behind by changing times through no fault of their own. Industries were farmed out to foreign countries. Ditto agriculture. New businesses, particularly in the tech industry, could be managed from anywhere and didn't necessarily need brick-and-mortar headquarters anymore, except for show. And I had real trouble visualizing Asheboro\u2014at least, the Asheboro I remembered\u2014as the hub of any sort of commercial enterprise. Its location was all right: near major cities, with an airport close by and highways for easy access, and moderate weather as well. But it was just a shade too far away to qualify as a bedroom community for Baltimore, although that boundary kept moving farther out. But then, Asheboro didn't have a lot of housing.\n\nFunny how when you live in a place you don't really see it. My mom and dad had lived in a modest post\u2013World War II colonial house, but I didn't harbor any particular nostalgia for it\u2014it had been bland and functional. The main street in town had had some basic shops, and there had been a supermarket on the fringes of town, along with a gas station or two. But nobody had ever built a mega-mall anywhere in the area, and since Internet shopping had ballooned, now a lot of malls were dying too. So, bottom line, the town now had a damaged main street that had been subpar to begin with, a white elephant of a crumbling mansion outside of town, a defunct factory, and no money. Not a whole lot to work with.\n\nI thought about all of this as I drove west out of the city to meet Lisbeth in Asheboro. I'd be the first to admit that it was pretty country\u2014I'd had little reason to travel west from the city recently, and I'd forgotten. It wasn't spectacular\u2014mostly low, rolling hills, with tidy farms tucked in the shallow valleys. The near-empty highways sported signs for towns even I recognized as historic, mainly for Civil War battles. I'd never taken much interest in history. In school, I'd memorized all the names and dates and done well on tests\u2014and then promptly erased all the information from my brain as soon as I got my diploma. It was hard for me to visualize armies numbering in the thousands crawling over these gentle hills and killing each other. But then, I'd seen some parts of Europe where centuries of bloody wars had been fought, and now they were serene and studded with delightful small hotels and restaurants, as if all that violence had never happened. Those towns had faced unimaginable horrors and moved past them. They'd survived for centuries. Why shouldn't Asheboro, whose problems were much less dramatic?\n\nI arrived in Asheboro before lunch. It always surprised me that it took me little more than an hour's drive to reach the place from my current Baltimore home, since it felt like traveling to a different world, or going through a time warp. I drove slowly down the one and only street\u2014creatively labeled Main Street\u2014and pulled into a parking space in the center. Lisbeth hadn't described the storm damage to me in detail, and it looked both better and worse than I had expected. All the buildings were still standing, though a few here and there sported broken windows; some of the street-level plate-glass windows were covered over with sheets of plywood. The roofs were all in place, although quite a few had lost some shingles. Modern-day aluminum and plastic siding hadn't fared well, peeled back like a second skin to reveal the brick-and-stone structure beneath. Heaps of trash were still stacked by the curb, waiting for pickup. Hadn't it been something like two weeks since the storm had done its damage? Nobody in town seemed to be in a hurry to fix things up.\n\nI pulled out my cell phone and called Lisbeth. She answered on the fifth ring. \"Katie! Please don't tell me you're canceling on me!\" She must have put her hand over the phone, because I heard muffled orders to various family members.\n\nWhen she came back to the phone, I said, \"No, I'm sitting in my car in the middle of Asheboro. You want me to meet you at your house?\"\n\n\"Uh, no, this place is a mess. I'll tell Phil to take Jeffrey to practice, and Melissa's got a play-date and sleepover with a friend, so I have to drop her off. I could meet you in half an hour?\"\n\n\"That's fine. I can look around a bit. Where do you want to meet?\"\n\n\"Is McDonald's okay? I know it's not what you're used to, but it's close and easy to find.\"\n\n\"Hey, I'm here to look at the town, not to sample fine dining\u2014I get plenty of that in the city. Where is it?\"\n\n\"About half a mile from the town center, on the west side. It'll be on your right.\" She turned away again, apparently, and I heard her yell, \"Melissa! Get your overnight stuff together! We're leaving! Phil, don't forget Jeff's glove!\" When she returned to me, she said, \"See you at Mickey D's.\" Then she hung up. I almost smiled: her crises and mine were so very different. She worried about misplaced baseball gloves, while at the hotel I had to track down international shipments of high-priority documents and make sure they reached the right people.\n\nSince I had a half hour to fill, and a five-minute drive to our lunch spot, I decided to walk around and get a feeling for the place. After a few steps I realized I felt like I was seeing double: I remembered what used to be there, and I saw what had replaced it, but I was kind of seeing both at once, one laid over the other. Once there had been a drugstore on the corner; now it was a store that sold outdoor wear. The genteel little gift shop next to it had morphed into a three-chair beauty salon. And so on. Had everybody moved on since I'd lived here? And who had moved in? As I strolled I started paying attention to the structure of the buildings themselves. I didn't see anything new\u2014most buildings looked like they'd been built in the later nineteenth century, maybe 1900 at the latest. They were uniformly two or three stories tall, and originally made of brick or stone. Fake siding had been slapped over most of them, in an effort to modernize, but it didn't look like it had been very successful, and now it had been trashed by the storm. The bank building looked pretty much as I remembered it. It wasn't large, hardly wider than the shops next to it, but the front fa\u00e7ade appeared to be granite. The old library at the end of the street was about the right vintage to be one of the many Andrew Carnegie libraries of the era, probably built around 1900, but it was pretty plain. At least it seemed to be functioning as a library still.\n\nThat took care of one side of the street, and the opposite side looked much the same. My overall impression was that the whole place looked shabby and neglected. I checked my watch, then turned around and headed for my car, picking up my pace. I arrived at McDonald's at the same time as Lisbeth.\n\n\"Did you beat me?\" she panted as she approached.\n\n\"Nope, just got here myself. Let's grab a table and some food and we can talk.\"\n\nThree minutes later we were settled in a corner, which gave us a little privacy. I was surprised at how empty the place was, given that it was a Saturday. \"Where is everybody?\" I asked, waving at the vacant seats.\n\nLisbeth shrugged. \"Games, practices, rehearsals, whatever. But to be honest, it's not very busy on the best of days. You know what it costs to feed an adult and two growing kids here?\"\n\n\"Can't say that I do.\" I tried to remember the last time I had eaten in a chain restaurant like this and failed.\n\n\"Too much. We bring our kids here as a treat, not as a regular thing. So, what do you think?\"\n\n\"About Asheboro?\" I paused, searching for the right words. Lisbeth still lived here, and she'd told me she was happy, so I didn't want to be too harsh. \"It kind of feels like whatever was happening here stalled about a century ago.\"\n\nLisbeth's mouth quirked. \"You've got that right. I think it peaked around 1915, with a small surge in housing after World War II.\"\n\n\"Why do you stay?\" I asked.\n\n\"In a nutshell, Phil has a good job, between here and Baltimore. The schools are decent, it's safe for the kids, and parents in the neighborhoods look out for their own and others. It's home. That probably doesn't make sense to you anymore.\"\n\n\"Not really. It was a fine place to grow up, but I didn't want to live here forever. I don't mean to criticize your choice.\"\n\nLisbeth smiled gently. \"I'm not offended. I know its flaws. But I don't want to see it just wither away and die. Sure, we could live somewhere else, but we wouldn't have the memories, the sense of belonging somewhere.\"\n\nWhich were things I'd given up almost without thinking. Did I regret not having that? I didn't think so. There were still plenty more places I wanted to see, wanted to spend a year or two in. I had deliberately chosen a career that gave me those opportunities. It wasn't for everyone, but it worked for me. \"What is it you think I can do for you, for Asheboro?\" I asked, grabbing up a French fry. And then another. Those things were good.\n\n\"I don't know.\" Lisbeth studied her Quarter Pounder, looking for the best bite. \"Other places have reinvented themselves, or found a way to attract new business, new people. Can we do that here?\"\n\n\"To be honest, based on my quick walk-through earlier, you don't really have anything here to attract people. Now if some huge company decided to move here and hire a couple of thousand people, you'd have a chance, but I don't see that happening. You have no natural wonders, like caves or waterfalls. You're near a lot of historic sites, of course, but so are a lot of other towns, and they've gotten their acts together faster to accommodate the visiting history buffs. Any famous people born here? Any gruesome murders in the past century?\"\n\n\"Like Lizzie Borden, you mean? Nope. Nothing important has ever happened here. It's never been more than an average town.\"\n\n\"What did the factory make, back in the old days? I don't remember anyone ever telling us. They just called it 'The Old Factory.'\"\n\n\"I don't know. You should talk to our town librarian\u2014I think there's a decent local history section.\"\n\nGood point\u2014if I decided to take on this seemingly bleak project and needed local resources. \"I think it's a safe bet that whatever they made isn't coming back again. Anybody useful in the cemetery?\"\n\nLisbeth shook her head. \"Not that I know of. Oh, and the storm took down the steeple of the church, but it was ready to fall anyway. Luckily nobody got hurt.\"\n\n\"Maybe we could build a fence around the town and put up a sign saying \"The Land That Time Forgot.\"\n\nLisbeth sighed. \"I was afraid you'd feel like that. I don't take it personally, but I hoped...\" She shook her head. \"I don't know what I hoped. That you'd swoop in and fix things, I guess. Not exactly realistic, I know.\"\n\n\"Was there anything else you wanted me to see while I'm here?\" I asked.\n\n\"Two things. One, you have to see the Barton place. I've gotten the key from the bank\u2014the bank manager, Arthur Fairchild, is a friend of mine, plus he's on the council\u2014so we could go this afternoon. Two, I asked the town council and managers to get together and sit down with us and kick around ideas, and that's on for tomorrow, since they're all free on Sunday. I want you to meet them. This isn't official\u2014it's more like a brainstorming meeting.\"\n\n\"Do I know any of them?\"\n\n\"Maybe. Some of them were a good few years ahead of us in school, so you wouldn't have known them back then. It's the older people and the retirees who have the time to dabble in community groups. And then there's Cordelia Walker.\"\n\nThat surprised me. \"Cordy's on the board? I would have thought she couldn't get out of Asheboro fast enough.\" Cordelia had been student council president for three of my high school years. I didn't begrudge her that honor, but mostly she used it to boss other people around, and I was pretty sure she'd run for the office so she could play it up on her college applications.\n\n\"She did for a while, but then she kind of slunk back a few years ago. I think she decided that being a big fish in a small pond was better than battling all those sharks out in the bigger world. And she's the one who pushed the town to buy the Barton house.\"\n\n\"Didn't you say that the board turned against her after that went through?\"\n\n\"It did, when she told them she wanted to open a swanky bed and breakfast in the place, and run it herself.\"\n\n\"What's wrong with that, as long as it brings in paying customers?\"\n\n\"The thing is, she already has a small B&B in town, and she's had trouble filling that. Now she wants to move it into the big house for a nominal rent.\"\n\n\"Has anyone proposed another idea?\"\n\nLisbeth shook her head. \"Not really. A museum? We don't have any kind of collection to show off, and the library can accommodate most of that anyway. A private hospital? It would need too much work to refit it. A hush-hush conference center? There are plenty of those around here already. So, the short answer is no, there's no idea on the table that anyone thinks will work.\"\n\nI wasn't surprised. But before I could consider any ideas, I had to see the place. \"Then let's go see the Barton house now. Maybe then I'll have a better idea of the possibilities.\"\n\n\"Works for me. You finished eating?\"\n\n\"I have.\" I pushed the few remaining fries away from me. \"Let's hit the road. We'll take your car, okay? Unless you have to be somewhere or pick someone up.\"\n\n\"Nope, I told Phil he'd just have to cope. My afternoon is free.\"\n\n\"So let's go.\"\n\nLisbeth turned off the main street and headed... I didn't know which direction. I had seldom had any reason to go out of town this way, because there was nothing there except the Barton place, and that was off-limits to us as kids. As in the case of the factory, there were probably brave souls who had ventured to explore, but I couldn't remember anybody reporting the details. Oddly enough, it wasn't said to be haunted, which abandoned Victorian heaps were often accused of.\n\nIt was a bright sunny spring day. The leaves on the trees were still a tender young green, and there were early wildflowers scattered around. Fields were marked off with sturdy picket fences, and we passed a horse or two on our way. I couldn't remember the last time I'd treated myself to a day in the country, if that's what this was. It was nice to let someone else do the driving, so I could admire the landscape. But didn't people get tired of looking at fields and trees and barns? Didn't they want at least a little excitement in their lives? Restaurants, museums, theater?\n\nAfter a couple of miles, we pulled up in front of a heavy pair of iron gates hanging slightly askew from a pair of massive stone pillars. A brick wall, whose height I estimated at eight feet, ran as far as I could see in both directions. Beyond the gate a paved driveway wound its way across a field and continued up a small rise before disappearing on the far side. If you didn't know it was there, you'd never think there was a house on the property.\n\n\"Just a sec,\" Lisbeth said, as she fished a large ring with an assortment of old-fashioned keys from her bag. One of them apparently opened the relatively modern padlock for the linked chain that held the gate shut; even from where I sat in the car I could tell the original lock was too rusty to function. Not a good introduction to the place. Lisbeth wrestled with the chain, then pulled back the panels of the gate. They moved smoothly and quietly, which told me that someone had oiled them fairly recently. She got back in the car and drove through, then got out again and pulled the panels together, but without reattaching the lock.\n\n\"Is there anything worth stealing?\" I asked.\n\n\"We're more worried about vandals. Funny how kids like to destroy things, just for the heck of it. Thank goodness the bank installed a basic alarm system awhile back.\"\n\nHow long ago was that? I wondered. Did the alarm system still function? \"Have there been problems?\"\n\n\"Nothing significant. There was a raccoon that decided to make itself at home inside, and that gave the alarm company a few anxious moments.\"\n\nIf a raccoon was the worst threat they had faced, the town was lucky. I settled back in my seat, waiting for the house to appear. Lisbeth was driving slowly\u2014trying to build up some drama?\u2014until she reached the crest of the low hill and stopped. \"There it is.\"\n\nI looked through the windshield. \"Holy crap! It's huge!\" The sprawling building lay spread out in the valley below, and from this distance it was hard to estimate the actual square footage. My first impression was that anywhere the builder could stick a gable or a turret or a chimney or a porch or a railing or a mansard roof, he did. It was absurd: high Victorian taste at its best. Or did I mean worst?\n\nI loved it on sight.\n\nLisbeth wisely said nothing and let me take it all in.\n\n\"Nobody wanted this?\" I said incredulously.\n\n\"Seriously? It's kind of big for most people. Eight bedrooms, not counting the servants' quarters in the attic. Two formal parlors, plus a library and a dining room. A kitchen, of course. The latest in indoor plumbing\u2014over a century ago.\"\n\nI shut my eyes, then reopened them, to see if the gorgeous monstrosity had disappeared. It hadn't. \"If we get any closer, will I see it's in abysmal condition? Or should we just sit here and admire the glories of yesteryear?\"\n\nI don't know what had come over me. I prized order and efficiency, and the Victorian style was anything but orderly. More like chaotic. I valued simplicity and precision, and in my condo in Baltimore I had created a spare precise space that worked well for me. No unnecessary clutter, and a place for everything. Serene and unencumbered with stuff. So why was I slavering over a Victorian heap?\n\n\"Had enough ogling?\" Lisbeth asked with a smile, after a minute or so.\n\n\"I guess. Are we going in?\" Would that make my instant obsession better or worse?\n\n\"Of course we are. And don't worry\u2014it's structurally sound. The bank made sure of that, over the years.\" She put the car back in gear and we rolled down the hill toward the house. In front there was a wide semicircular drive, and in its center stood a fountain that had seen better days, surrounded by what might have been flowers once upon a time. Lisbeth pulled up in front of the main door and turned off the engine, then got out of the car and retrieved the key ring from her purse. I climbed out more slowly, absorbing details. Granite stoop, fully ten feet wide. Front door needed some paint but looked solid, and the brass fittings appeared to be original. Tall windows down to the floor, in the front, and some odd configuration of panes\u2014did they open like doors?\n\nLisbeth finally found the right key and opened the front door. Like the gate it swung inwards silently. I noted original hinges, ornate, immense. The hall had to be twenty feet wide, with marble flooring. Ten-foot ceilings sat atop crown moldings a foot high. I felt like I had walked into an alternate universe. No, that's absurd, Kate. I looked around and realized I didn't see anything that dated from after 1900. How could that be?\n\nWhen I found my voice I said, \"There's got to be a story behind this, Lisbeth. I mean, nobody's touched it for more than a century, but it seems to be in good condition. That doesn't just happen by accident. Is this a portal to a time warp?\"\n\nLisbeth laughed. \"No, of course not. There's a resident caretaker, and always has been. There's a small apartment over the stable behind the house, and a succession of people have lived there in exchange for free rent. It's worked out well.\"\n\n\"But do they dust and polish and all that stuff? Turn the heat on in winter? Make sure the pipes don't freeze? Open the windows to air it out now and then?\"\n\n\"Yes, actually, they do. When the bank took it over originally, years ago, it put together a checklist of things that should be done regularly until the property was sold. I don't think they expected to wait this long, but as the saying goes, If it ain't broke, don't fix it. You can see the results\u2014it worked.\"\n\n\"And now the town owns it. They've kept up the same arrangement with a caretaker?\"\n\n\"So far. But it's only been a couple of months, and nobody's made any plans going forward. Yet.\"\n\nI stood in the middle of the hallway and made a 360-degree turn. Heavy, ornately carved wooden doors with intricate knobs led off on both sides. A staircase with intricate spindles and a wide mahogany rail led to the second story. \"What's the layout?\" I asked.\n\n\"Two parlors on the right, with pocket doors between. On the left, the library's in front, with a small sitting room beyond it\u2014that gets some nice sun. The dining room is behind the library, and the kitchen and the pantry are at the back.\"\n\n\"And you said eight bedrooms?\"\n\nLisbeth nodded. \"Well, one is pretty small\u2014we figured it was meant to be a nursery originally. There's only one closet per room, and they're all pretty skimpy. And there's only one bath on the second floor.\"\n\n\"For eight bedrooms?\" I asked, unbelieving.\n\nLisbeth nodded. \"Chamber pots. Remember, they had servants to empty them.\"\n\n\"Heating?\"\n\n\"Coal furnace originally, but the ductwork reaches only the first floor, with some passive heat by way of the staircase. There are small coal-burning stoves in the bigger bedrooms, which the servants were responsible for. No heat in the servants' rooms, of course.\"\n\n\"Well, of course,\" I said, then regretted my sarcasm. It had been a different way of life, and the \"girls\" had no doubt been poor immigrants who'd have been glad to have a job and a roof over their heads.\n\nWe drifted toward the kitchen at the back of the building, where there was a massive stove\u2014wait, was that a coal burner too?\u2014and deep slate sink and a wood-clad icebox. Farm to table had a whole different meaning in 1900. \"Lisbeth, when did the original owner die?\"\n\n\"Henry Barton? 1911, I think. He outlived his wife and they had no children, so there was no one to leave the place or its contents to. He had a couple of brothers, I think, but they were well established, around Boston if I recall, and they didn't want a place in Maryland, and neither did their offspring, not that there were many of those either. Maybe there was some fighting among them about what to do with this place, and in the end, they just signed away their rights. Hard to believe, isn't it? The bank has managed it ever since, thanks to a generous trust.\"\n\n\"What a waste,\" I said. But tastes had changed, and by the early twentieth century the ornate Victorian style had become pass\u00e9. \"Can we look at the bedrooms?\"\n\n\"Sure,\" Lisbeth said. \"Follow me. There's a servants' stairway back here, but we'll go up the grand staircase.\"\n\nI trailed behind her, gliding my hand over the smooth (and only slightly dusty) finish of the stair rail. On the second floor the trim was a bit less opulent, but it was still very, very nice. In one of the front bedrooms\u2014the master bedroom?\u2014there was an immense carved bed, the headboard standing at least six feet high. \"You couldn't hope to get it out of the room now, without taking it apart.\"\n\nLisbeth didn't comment. \"Let me show you the bath.\"\n\nI tore myself away from the enormous bed and followed her down the hall. The bathroom proved to be almost as large as the bedroom, but the fixtures were... old. And weird. \"Is that a metal bathtub?\" I asked.\n\n\"Sure is. Nice mahogany casing, though.\"\n\n\"Does all this work?\"\n\n\"I hope so. The caretaker's supposed to come through periodically and flush the toilet and run the taps. Probably cast-iron pipes in the basement, so I'd bet the water's kind of rusty.\"\n\nI had to agree, looking at the reddish stains left by decades of slow drips in the sink.\n\n\"What about the attic?\" I asked, after marveling at plumbing I'd never seen before, like a toilet with a tank (also mahogany-clad) mounted close to the ceiling above it, and a pull chain with a china handle. I'd read about those, but I'd never expected to meet one. I resisted pulling the chain just to see if it really worked.\n\n\"This way,\" Lisbeth said, leading the way down the hall to a door at the end. When she opened it, cool musty air rushed out, but I didn't smell anything like mildew or damp. The stairs were narrow\u2014naked wood. Nobody had wasted any fancy details up here. At the top of the stairs Lisbeth waited for me, then pointed. \"That side's attic storage, unfinished. There are two maid's rooms on the opposite side. Well, more likely a cook and a general servant.\"\n\n\"Show me.\" Lisbeth led me to the nearer room. It was small, with a dormer window on one side, and a row of hooks attached to the unpainted plaster wall. The rough board floor was covered with what I guessed was an early version of linoleum. There was still a single bed with a painted iron frame against one wall, and a small table beside it. Apparently nobody had bothered to clean out this space.\n\n\"The other room's pretty much the same.\"\n\n\"No heat up here, you said?\" I asked.\n\n\"I gather the hired help weren't supposed to spend much time up here\u2014they would be downstairs working all hours.\"\n\n\"Is that all?\"\n\n\"You want to look at the basement? There's not much there except the furnace.\"\n\n\"I know nothing about furnaces from any era, so I guess I'll pass. But it really is like everyone just walked away from the place. Or, well, when this Henry of yours died, there was no one who cared. I'm guessing he died in the house?\"\n\n\"Of course, although he had the best medical care available around here. Luckily his doctor stopped by once a day toward the end, so if you're wondering, he didn't lie here, uh, deteriorating for a while. And Henry had left everything regarding his estate in good order. I'd bet he was surprised when no one stepped up to claim the place. If only he'd left it directly to the town, our lives would be a lot easier.\"\n\n\"He wasn't involved in local affairs?\" I asked.\n\n\"Not that so far as I know. He managed the factory as long as he was able, so I guess that was enough.\"\n\nWe went back downstairs and toured the parlors and the sun room and the library, but by the end I was feeling overwhelmed by the sheer opulence of the place. Finally Lisbeth led me out the back door. The stable was set back behind the house about a hundred feet, with tall sliding doors in front.\n\n\"Anything inside there?\"\n\n\"Stalls. I think I remember an old carriage, but that might have fallen apart by now. Nothing so modern as an automobile. The caretaker has a small apartment above.\"\n\nI surveyed the landscape from this view: rolling hills again, trees, a fence or two. Not a sign of the modern world. \"Did they grow their own vegetables here?\"\n\n\"I think so. There's what looks like a fenced plot behind the stable\u2014wouldn't want to sully the view from the house, right? You'd need the fence because there are a lot of deer around here. Seen enough?\"\n\nI nodded. We strolled around the house, and I admired the ornate brickwork of the chimneys on the side. The chimneys and the foundation, also brick, looked like they could use some repointing, but overall they were in decent condition. Whoever had built this place had built it well and apparently had spared no expense.\n\nWe both looked up at the sound of another car approaching. When Lisbeth and I reached the front of the house, we saw a tall fortyish man emerging from a battered car parked in the driveway. Lisbeth leaned toward me and whispered, \"That's the caretaker.\"\n\nWhen I'd first heard him mentioned, I'd assumed it would be some down-on-his-luck local person happy for a free room, or maybe a college student. This man was clearly neither: even in casual clothes, he looked like a professional of some sort. How had he ended up here?\n\nHe approached us without hurrying. \"Lisbeth, what's up?\" He gave me a glance that I wouldn't call friendly. \"This isn't one of Cordelia's, uh, colleagues, is it?\"\n\nApparently there was no love lost between the man and Cordelia. I stepped forward. \"No. I'm Katherine Hamilton, and I grew up in Asheboro. Lisbeth and I go way back. She asked me here to see if I could help the town with its problems.\"\n\n\"And how is an old pal supposed to solve the town's financial crisis?\" he said.\n\nI didn't like his tone. \"I manage a hotel in Baltimore, so I know something about the hospitality industry.\"\n\n\"So you are one of Cordelia's flunkies,\" he said.\n\n\"No,\" I said more firmly than I might have intended. \"I don't know the full story, but from what little Lisbeth has told me, what Cordelia thinks she's planning would destroy the place.\"\n\n\"You disagree with her grand scheme?\" he said, his tone a shade less hostile.\n\n\"In principle,\" I replied. \"Look, I've only just heard about the whole mess, so I'm not up to speed.\" I decided not to mention that I'd known Cordelia most of my life, and I wasn't exactly a fan of hers. \"We may have gotten off on the wrong foot, so let's try again. Hello, I'm Katherine Hamilton, former resident of Asheboro, more recently a hotel manager in Baltimore, called in by Lisbeth and the town council to consult on what options are open to the town of Asheboro to revitalize its economy.\" I extended my hand and almost by reflex the man shook it. It was a start. \"And you are?\"\n\nHe almost smiled. \"Joshua Wainwright, history professor at Johns Hopkins, currently on sabbatical so I can finish a book on post\u2013Civil War industrialization. Henry Barton played a larger role in that than most people are aware of. Call me Josh\u2014when I hear Joshua I think of some character from the Bible.\"\n\nMy professional ears pricked up: maybe there was something in Henry Barton's story that could be useful. \"I'd love to hear more about that. So I take it that you oppose Cordelia's plans?\"\n\n\"I do. She has no sense of history, and what she wants to do would seriously devalue this place as an asset to the town, and to local historians.\"\n\n\"I agree, without even having seen her plans. But as I understand it, the council needs an alternate plan if there's any hope of stopping her from going forward.\"\n\nThe man's expression relaxed. \"I have some ideas about that. We should talk. Unfortunately I have a prior engagement. Do you have a card? We can set up a time later.\"\n\n\"Of course.\" I fished in my bag and found a business card, which I handed him. \"I stopped by today just to get a sense of the place. Let's wait a day or two\u2014I still have to process what I've seen and heard here today. But I think I see some possibilities.\" To my own surprise I meant it.\n\n\"Great.\" He smiled, and his face was transformed. \"I'll call you, then. Lisbeth, good to see you.\" With that he turned and strode toward what must be his lodgings over the former stable.\n\nI turned to see Lisbeth grinning at me. \"No, he's not married. Or gay,\" she said gleefully.\n\n\"And I'm not looking,\" I replied, a bit more tartly than necessary.\n\n\"Uh-huh,\" Lisbeth said, as she leaned against her car. \"You want to go find an early dinner, or do you need to go home?\"\n\n\"I'm thinking maybe I should go back home now and do some online research. We can have lunch tomorrow after the meeting, if that works for you. We may have things to talk about by then.\"\n\n\"Sounds good to me. I'll warn Phil that he'll be on kid duty for most of the day.\"\n\n\"I don't mean to disrupt your life, Lisbeth.\"\n\n\"Hey, this is important, and you're doing us a huge favor just looking at what we've got to work with in Asheboro and helping us figure out what we can and can't do.\"\n\nShe really seemed to mean it, and I hoped I could come up with some kind of good news to repay her faith in me. \"I'll do the best I can. We're meeting with the town council in the morning, right? We'll have time to talk afterwards.\"\n\nImpulsively she laid a hand on my arm. \"Does eleven work for you? That's after church around here, so the council members will be around. And remember, this isn't an official meeting, and nothing is binding. We're looking for ideas.\"\n\n\"Where's the meeting?\"\n\n\"Town Hall, in the conference room.\"\n\n\"How many people will be there?\"\n\n\"Eight or ten? Depends on how people feel about the old Barton place, and how committed they are to the town. I'll admit there are those who'd like to walk away entirely, although I'm pretty sure they have no idea what the outcome would be. I've heard of big cities wrestling with bankruptcy, but a small town? If everyone just packed up and left, what would happen?\"\n\n\"I'm not the person to ask, Lisbeth,\" I told her. \"Used to be that plenty of ghost towns happened that way, but I have no clue about the legal aspects. Does the town have a lawyer?\"\n\n\"Yes\u2014Jim something-or-other. I don't know him well.\"\n\nI racked my brain for other information I needed to ask about. \"Do you have a town utility, or are you on a grid?\"\n\n\"That's a story of its own. Local utility, but maybe not the best solution.\"\n\nI filed that away for the future\u2014if there was one. \"Water?\"\n\n\"The lake, same as always. Don't worry\u2014it's clean. We have a water purification plant for the town. We do test it regularly.\"\n\n\"Property taxes?\"\n\n\"Too low. We haven't done an assessment for over a decade. I'd worry about saddling homeowners with any more costs, but then, the property values may not have gone up at all. Kind of a lose-lose situation.\"\n\nShe painted a very bleak picture for Asheboro. From what I was hearing, maybe the best option was to let the town die a natural death. It might also be the least expensive, in the long run. If it was legal. That was outside my area of expertise. I sighed. \"It would be helpful if you gave me a quick rundown of who's who at this meeting, and where they stand. If you've got the time now.\"\n\n\"No problem. Let's sit on the steps, because this might take a few minutes.\" We sat, and once again I admired the view, which probably hadn't changed since the house was built. Lisbeth began, \"I already told you about Cordy. She's an elected member of the town council. She's miffed at everyone, but she can't just walk away. I don't think she would even if she could\u2014she wants to keep an eye on what everyone else is doing. There's the mayor, of course\u2014that's Skip Bentley\u2014and the deputy mayor, Howard Short, and three other elected members. The bank manager, Arthur Fairchild, is on the council, and then there's the town manager, Megan Conroy, and the town engineer, Brian Anthony, and that's all. I'll introduce you all tomorrow. Again, this is not a formal meeting and it's not binding. Nobody's taking minutes or voting on anything. We're just kicking around options. Feel free to be frank with us\u2014at least you can wash your hands of us after the meeting.\"\n\n\"I can handle it. I really am sorry the town is having such problems.\" I might have left it in my rearview mirror years ago, but I didn't wish it any harm. \"I'll help if I can, but don't get your hopes up.\"\n\n\"Don't worry\u2014we're scraping bottom. Thanks for helping us out, Katie. I really do appreciate it.\"\n\n\"Hey, it's my hometown. It's the least I can do.\" Did I believe that? Did it matter? It was an interesting problem, whether or not I had a personal stake in it.\n\n* * *\n\nI waved as Lisbeth pulled away, and then I drove home.\n\nWhich gave me too much time to think. Why did it have to be the Barton house? The place loomed large in my memories of Asheboro, but for the wrong reasons. I'd buried them deep, only to have them thrust in my face again now. Well, I was a big girl, so I could deal with it, right? But the situation was made worse by the role of my nemesis, Cordelia. Well, \"nemesis\" was probably too strong a word. I had barely made a dent in her consciousness back then.\n\nLisbeth and I had been best friends in high school. The last time we'd gotten together was when I'd been back to help my parents pack up and move a few years ago. Since high school, I could probably count the times we'd seen each other face to face on my fingers, leaving out the thumbs. Her wedding, of course, and the christening of her older child, Jeffrey. I couldn't even remember if I'd been invited to the christening of her second one\u2014a girl, I knew. What was her name? Melissa?\n\nBut I didn't think about Asheboro much.\n\nBut that mansion... offered some intriguing possibilities\u2014if I could just set aside what I remembered about it. The house had been spectacular in its day, and based on what little I'd seen, no expense had been spared in its decoration. Even in my brief walk-through, I'd seen light fixtures that screamed \"Tiffany Studios\" and wallpaper that could well be original William Morris patterns. If you liked Victorian style, it would be a treasure. Not my style, of course. My condo was kind of the antithesis of the Victorian style, wasn't it? I seemed to be a subscriber to the \"less is more\" philosophy. The Victorians preferred \"more is better.\"\n\nStill, the idea of Cordy and her team of renovators ripping down that wallpaper and tearing out the original woodwork made me sick to my stomach.\n\nI arrived back at my condo by nine o'clock, and entered my uncluttered space with a sigh of relief. The home I had created for myself: orderly, easy to manage, and everything in its place. I didn't have a pet or a plant because my job sometimes required quick trips to somewhere else on a moment's notice, not to mention industry conventions\u2014and pets and plants needed attention. I didn't bring back souvenirs from my various excursions, because that was just... stuff. I had pictures, didn't I?\n\nIt was neat and tidy and... totally lacking in personality. Why had I chosen to live like this? And why had I fallen for the Barton mansion on first sight, when it was everything I had convinced myself that I disliked?\n\nI told myself I was just tired and drained by wrestling with a past I thought I'd left behind.\n\nMaybe I shouldn't have driven home. Maybe I should have stayed and chatted about this and that with my old friend and her husband, whom I barely knew, and her two children, whom I'd hardly met. It would have been a great distraction, and I could have stayed in a crappy local motel after a bit too much wine and awakened bright and early, in time for lots of coffee and then the secret council meeting about saving Asheboro.\n\nWhy hadn't I done that? Because, well, like a wounded animal I wanted to crawl back into my cave and lick my wounds. I was trying to ignore the fact that I'd been fired, but the reality was that I'd been dumped without warning from a job that I loved and was good at, and it hurt. Even if my head was saying that things like that happen all the time, I was grieving. It was going to take some alone time to get over that.\n\nAnd there was that council meeting looming. I'd always been the good girl who got all her homework done, as Lisbeth had reminded me, and I wasn't going to walk into that meeting cold without having done a little research. Not that there was much I could do in a few hours, but I wanted to try, so at least I could present a general concept to the council. I knew there were other examples of places that had somehow transformed themselves into living history sites, but I wasn't sure whether that would fit Asheboro's needs. Not everyone in a small town wanted to dress up in quaint clothes and pretend to pull taffy or braid rugs, or whatever people in a small rural town did around 1900. Plus I wanted to find out why Henry Barton had been such an enigmatic figure, hiding in the shadows inside his opulent mansion. Was there anything from his story that I could use? Josh Wainwright had hinted there might be something there.\n\nBut my experience\u2014and my gut\u2014was telling me that the mansion alone, lovely and untouched as it might be, was not going to be enough to lift the town into financial stability. There had to be something more to the package in order to attract visitors. The factory was still an unknown quantity, but who would detour from visiting battle sites to wander through an antique widget factory? But somewhere lurking in my brain was a glimmer of an idea, if I could just unearth it.\n\nI reached home without any grand \"Aha!\" or \"Eureka!\" moments. I parked and unlocked the door to my uncluttered condo, and turned on a couple of lights. It was neat and tidy and... totally lacking in personality. Why had I chosen to live like this? And why had I fallen for the Barton mansion on first sight, when it was everything I had convinced myself that I disliked?\n\nI changed into some comfortable raggedy sweats, poured myself a glass of wine, and sat back on the oatmeal-colored couch and started at the ceiling. If the mansion was not enough, what could be added to make it more appealing? Cordy's proposal was still on the table, but I doubted that she could survive financially with only eight bedrooms, especially if she was going to have to add modern amenities like more plumbing, and it wouldn't really help the town\u2014just Cordy. Okay, if I was honest with myself I would really, really like to snatch the project away from her greedy hands and make something impressive out of it. But how?\n\nMentally I reviewed the sad and shabby Asheboro I had seen earlier in the day. It had no redeeming features. It looked like exactly what it was: a failing town with no hope for the future. But then I realized what I had seen in my brief tour. The modern town was pathetic, but from what little poking around I'd done, I guessed that the bones of the Victorian town were still intact under the cheap modern siding and windows and asphalt shingles. What if, what if... we made the whole town the attraction?\n\nI sat up straighter, thinking hard. Say instead of repairing the recent storm damage, we simply peeled off all the later ticky-tacky and repaired the underlying original structures? Made Main Street look like Main Street 1900? With era-appropriate shops, and maybe a stable\u2013slash\u2013harness maker, and an old inn, and... It could work. I could make it work, with help. It didn't have to be very expensive, and I was pretty sure I could find some outside funding for some parts of it.\n\nI looked at my watch and realized it was getting late. I didn't want to get too invested in my scheme to make the whole town an attraction, and I needed to do my homework. I could already see a lot of obstacles, but there were also a lot of plusses, if Lisbeth's feelings for the town were any indication. If there were more people like her in the town, they'd want to see their town survive, wouldn't they? Would the town council be willing to put some time and effort into making that happen? At least I could ask. Fine: I would put the idea to bed for now, and toss it on the table tomorrow at the council meeting and see who took the bait. If it flopped, they weren't any worse off than they were now. If it flew, it could be an exciting challenge. And it would be my project, not Cordy's.\n\n* * *\n\nSunday morning I drove back to Asheboro for the council meeting, with my nighttime ideas still churning in my head. Was the whole concept simply impossible? Would the people of the town get behind it? Did I have enough information, gleaned from a quick Internet search, to make a convincing case? And was I going to have to face off against Cordy, with a half-baked idea?\n\nIt wasn't hard to find the town hall, which was on Main Street, where it had always been. It was a squat Victorian cube built of red brick, with a few incongruous boxy air conditioners hanging precariously out of windows. It had a small parking lot on one side, and that's where I parked. There weren't many cars in the lot. Had the town leaders carpooled, or had they decided to skip this meeting?\n\nLisbeth was leaning against one of the cars, talking on her cell phone. \"I'll be home as soon as the meeting is over, Phil, and I think Kate and I are going to have lunch. I told you that. Surely you can handle the kids that long, can't you?\"\n\nI didn't need to hear the other end of the conversation to know that Phil wasn't happy about dealing with the kids on his own, for the second day in a row.\n\nLibby looked up and saw me approaching, and pocketed her phone. \"Hi, Katie. You ready for this? Any big ideas that came to you in your sleep?\"\n\n\"Actually, maybe, but it's pretty preliminary, and I want to find out if the council is interested before I dig into it. Do we go in now, or do you want to make a grand entrance once everyone is assembled? I warn you, I left my white horse at home.\"\n\n\"What? Oh, I get it. I'd like to see that, though. We might as well go in. If someone's not here by now, they're not coming.\"\n\nI followed her through a side door that was unlocked, and down a rather dim corridor with ancient brown linoleum on the floor. At the back of the building Lisbeth stopped and opened the door to what was clearly a conference room, with a long table surrounded by chairs. A few of them were occupied, and some other people were standing around chatting. Everyone stopped talking and turned to look at Lisbeth and me when we entered.\n\n\"Hi, everyone,\" Lisbeth began, sounding artificially chirpy. \"I bet a lot of you have met Katie in the past, but let me introduce you in case she doesn't remember you.\"\n\n\"Kate, please,\" I said firmly before she could go any further.\n\nLisbeth faltered. \"Oh, right, Kate. All right, at that end\"\u2014she waved vaguely to her left\u2014\"you've got Mayor Skip Bentley, and next to him is our deputy mayor, Howard Short. We have a total of five council members, but only four of them are here: Arthur Fairchild, the president of the Asheboro Savings and Loan, Janeen Logan, Barbara Metzler, and Bridgette Connors.\" Each nodded in turn. I wondered where the power lay.\n\nLisbeth was still talking. \"I also asked our town manager, Megan Conroy, to attend, since she knows the most about the day-to-day stuff of running the town. I didn't ask the tax collector or the water and sewer clerk to sit in, but that guy at the other end is Brian Anthony, who's our town engineer and handles planning and zoning.\" Brian gave me a curt nod. I returned it.\n\n\"So the people in this room are the ones who keep Asheboro going,\" I said, pasting on a polite smile. \"I'm happy to meet you all, and I apologize if we knew each other back in the day and I've forgotten you. But aren't we missing someone, Lisbeth? What about Cordelia?\" I asked.\n\n\"She doesn't seem to be here,\" Lisbeth said, looking surprised. \"Has anyone heard from her? Did she say she was coming?\"\n\nMost heads turned toward Mayor Skip, who said sheepishly, \"I don't know how much Lisbeth has told you about our current situation, but Cordelia is a strong advocate for one particular plan. We asked you here to explore other ideas, and I think we can go forward without her. This is not an official meeting, merely an exploratory one.\"\n\n\"She's hoping we all change our minds without looking at any other ideas,\" one of the councilwomen said bluntly. \"Not happening.\"\n\n\"I don't think we need to air our dirty laundry in front of Kate here,\" Town Manager Megan protested.\n\nI had to admit that secretly I was relieved, both that I wouldn't have to deal with Cordy today, and that at least some of the council members appeared willing to consider other ideas. \"I'm afraid you do,\" I said firmly. \"If there's anything to be done to keep this town alive, it's going to take everybody's participation, particularly the council's. If you send a mixed message, it's not going to work. So if there's something dividing you all, I'd rather hear about it now than be surprised later.\"\n\n\"I take it Lisbeth filled you in about Cordelia's scheme?\" the mayor asked.\n\n\"Only the barest details\u2014Skip, right? Look, let me tell you what I think I know, and then you can correct me and fill in the gaps. All right?\"\n\nAfter everyone nodded and we had all taken a seat, I proceeded to give a shortened version of what Lisbeth had told me over the last couple of days and what I'd seen for myself. It didn't take long: I wrapped it up in about five minutes. \"Is that a fair representation of the current status?\"\n\nMost of the people there were sitting in stunned silence. They must be masters of self-delusion: Had they never heard the case stated quite this baldly? No wonder the town was having problems. It boiled down to this: no money, plus no people, plus no businesses equals no town. Simple.\n\nFinally Skip the mayor spoke. \"Kate, I wish I could contradict what you've just said, but it's pretty much on target. I appreciate your honesty. You must think we're all a bunch of ostriches with our heads stuck in the sand.\"\n\nMuch as I agreed with him, I couldn't tell him so. \"And you have every right to think I'm a know-it-all who's got a swelled head after living in the big city. The question is, can anything be done about the situation? The answer is, maybe, and I have an idea I want to run by you. Look, I was here briefly yesterday. I took a look around at the town, and I have to say Asheboro looks a lot worse than I remember it, and I haven't been gone that long. Lisbeth also took me to see the Barton mansion, which I think we'll all agree is an asset\u2014but only if handled properly. So I've had some time to think about what I've seen, and what I think is possible, and here's what I've arrived at.\" I proceeded to give them a quick outline of the ideas I'd come up with, for both the mansion and the town. It didn't take long because I had few details to offer, just an ambitious vague concept\u2014and a growing enthusiasm, it seemed.\n\nWhen I finished, I looked around the table. Everyone seemed stunned, but I might have seen a hint of optimism in their expressions, which gave me hope. \"Lisbeth told me that Cordy\u2014Cordelia\u2014had presented a plan to you and you'd turned it down. Can you tell me why?\"\n\nOne council member who had remained silent snorted now. \"Because it benefited Cordelia first, last, and in-between. She wanted a free ride to run her business at the Barton place, and it wasn't clear that the town got anything out of it, except maybe some tax dollars down the road. And she wasn't open to negotiation.\" The woman looked at her companions before adding, \"Cordelia is pretty used to getting her own way. She does not like to hear the word 'no.'\"\n\nThat fit with what I remembered about Cordelia. \"I knew her in high school here, and I think I know what you mean. But you've got enough votes to override her, right?\"\n\n\"We do,\" Skip said. \"But we need a plan to vote on. Maybe I shouldn't say it, but Cordelia has made it clear that she has some big guns backing up her plan\u2014people with money\u2014and if her numbers work, we won't have an excuse to turn it down.\"\n\n\"Tell me why you said no, then,\" I prompted quickly.\n\n\"Because her vision of the place was not one we shared,\" Skip said, in a decisive voice. \"Look, you grew up here. You know it's a sleepy little town filled with decent people. We simply can't see a glitzy hotel for rich folk fitting here. Or maybe what I mean is, we don't see it as contributing anything positive to the town and we think that matters.\"\n\nI nodded. \"Good, because I agree with you. What's your time line?\"\n\n\"Fast. We need to make repairs to the damaged buildings in town sooner rather than later, but if we're going to be making changes to those buildings, we need to know going in\u2014you know, whether it's preservation or reconstruction. How fast can you turn a proposal around?\"\n\n\"A week?\" I was guessing but I'd done it before. \"You have a regularly scheduled council meeting coming up?\"\n\n\"We do\u2014a week from tomorrow. If you could pull something together by the end of the week and send us a draft, we'd be good to go\u2014we could review it over the weekend.\" He glanced at the people seated around the table, and most nodded yes.\n\nThey really did want to move fast. \"I think I can manage that. And if you don't like the ideas I propose, I won't complain. I know it won't be easy, but this could make a real difference to the town.\"\n\nSkip stood up, followed by the other members of the council. \"Kate, thank you for coming, and thank you for your efforts. I think I speak for the group when I say you've given us hope, and we appreciate it.\"\n\nThe meeting broke up quickly after that, with a lot of handshaking all around. If nothing else, I'd made this group of people happy, at least for a short time.\n\nLisbeth came up to me, beaming. \"Kate, that was terrific! And I think they like the idea.\"\n\n\"Don't get too excited yet\u2014there are a lot of hurdles, and a lot of things that could go wrong. Listen, I'm starving. Where should we go for lunch?\"\n\n\"Heck, let's celebrate. We can go to the hotel\u2014they've got a decent restaurant.\"\n\n\"There is no hotel in Asheboro.\"\n\n\"Exactly\u2014it's the next town over. Please? What with the kids, I don't get to go out for a nice meal very often.\"\n\n\"Sure, why not?\n\nWe took both cars, and I followed Lisbeth as she drove back toward Baltimore, pulling off the main road at a pleasant-looking but undistinguished modern hotel. We parked and went in, and Lisbeth guided me to the restaurant. It looked much as I would have expected, based on the exterior of the building, but Lisbeth seemed excited to be there, and I wasn't going to rain on her parade. We had to wait a few minutes for a table, but once we were seated and had ordered, Lisbeth turned to me. \"You think you could really make this happen?\"\n\n\"Frankly I don't know. If I didn't make it clear at the meeting, I'm not picturing a quaint theme park. Asheboro has to remain a living city, with people and jobs. But there's only the one main street, and I think it could be remodeled to meet both historic standards and modern needs. Of course, the good people of Asheboro would have to be involved, and it may not be for everyone. But then, they're not exactly thriving as it is, so they don't have much to lose. If some people are unhappy serving as re-enactors of a sort, I'd guess there are people who would like to step in. But right now there are more questions than answers.\"\n\n\"Well, I'm excited,\" Lisbeth said firmly. \"So let's pretend this is our mini-celebration and enjoy lunch.\" Which we did, accompanied by one glass of wine each, and wrapped up with a ridiculously rich dessert.\n\nAt the end of the meal Lisbeth said with regret, \"I hate to abandon you, but Phil will be going nuts by now, so I should head home. What are you planning to do?\"\n\n\"Don't feel guilty\u2014go. Me, I may walk around Asheboro a bit more, maybe look at it more critically. And I want one more look at the Barton place, to see how it could be connected with the town in some way.\"\n\n\"I gave the keys back to Arthur\u2014I didn't think you'd need them again.\"\n\nI grinned at her. \"I know the back way in.\"\n\n\"No!\" Lisbeth said. \"And did you...\"\n\n\"Almost,\" I told her. \"Once I get inside the fence, I'll look around for Josh\u2014he could let me in. But that's not important\u2014mostly I want to get a feel for the place.\"\n\n\"Let me get the check,\" Lisbeth said.\n\n\"No way!\" I told her. \"It's on me, and if things work out, it will be a business expense for me.\"\n\n\"Shoot, nobody ever talked about paying you for your time.\"\n\n\"Don't worry about it\u2014we can all talk about that later, when and if this project goes forward.\"\n\n\"Thank you, Kate. For all of it. It means so much to a lot of people.\"\n\n\"And to me, Lisbeth,\" I replied, surprising myself.\n\nI decided to skip looking at the town, and half an hour later I pulled up behind the Barton property, near the not-so-secret entrance that plenty of high school kids had known about. I hoped nobody had barred it yet. Heck, I could always go over a wall, although I was dressed for \"meeting with the council\" rather than climbing. But nothing ventured, nothing gained. Nobody had added any new houses on this side to observe my efforts at trespassing. I persuaded myself that I could have called Josh Wainwright and asked to be let in, but I didn't have his number, and he could have been anywhere. I was going to tackle this on my own.\n\nThe foliage had grown up since the last time I had attempted this. In high school. With Ryan Hoffman. It had been dark then, and we had been... in lust. He had brought a flashlight, and he had lent me a hand when I needed it\u2014not because he was a gentleman, but because he was really eager to get inside and get into bed. From the looks of things, nobody had come this way recently. I wondered briefly where kids these days went to \"do it.\" By now there must be some sort of alarm or security system. Henry Barton had had the foresight to leave a pile of money to take care of his place, although I doubted he had expected it would have to last for a century, but he hadn't anticipated the invention of surveillance equipment. As I clambered through the bushes\u2014damn thorns!\u2014I reminded myself that I ought to look at the details of the financial arrangements Henry had made. If there was any money left, after paying maintenance and most likely a stipend for the caretaker, what would happen to it? Would it make its way into some philanthropic organization's coffers? Or would there be some way to divert it to the ongoing care or even improvement of the property? Was I going to need to involve a lawyer in this, one for the estate rather than the town? One step at a time, Kate.\n\nI emerged from the shrubbery and dusted myself off, then set off toward the building. It had been carefully situated in a dell, which was why it couldn't be seen from any direction. Maybe Henry had valued his privacy, or maybe it was a strategic decision to minimize the impact of adverse weather, not that that was a large problem in this area. I realized again how little I knew about Henry's background beyond the fact that he had fought in the Civil War and decided to settle here. Was he trying to hide? Himself, or his obvious wealth? Where had he come from, before the war?\n\nI reached the top of the rise that had concealed the building, and paused to catch my breath\u2014obviously I didn't do much cross-country hiking these days. I spotted one car toward the back of the property, so most likely Josh was there. I decided I should gather my wits before going any farther, so I sat down and studied the scene laid out below, looking at it as a potential luxury hotel. Lovely countryside, more or less well maintained\u2014that endowment might not have stretched to a staff gardener, but at least the lawns near the house had been mowed. Only the one driveway, from the front. Limited parking: that could be a problem if there were going to be guests. How many bedrooms had Lisbeth said there were? Eight? Would that be enough to sustain a small luxury hotel? I couldn't see any way to add more space without spoiling the ambience. That small stable and presumably the caretaker's suite above it were too small to expand into bedroom space, and would be more useful for storage, or maybe a small spa. Maybe someone on staff could use the caretaker's rooms. No pool, no tennis court, no stable with horses\u2014that might eliminate athletic guests.\n\nI retreated to fantasizing about what I was beginning to visualize in my head: a true Victorian experience. Gas lighting inside. Fireplaces, kept lit and cleaned by servants. And that single bathroom that Lisbeth had mentioned\u2014that wasn't going to work, period. I'd have to figure out a way to shoehorn in the bare necessities for modern guests. Back in high school Ryan and I hadn't bothered to explore all the rooms, but had headed straight to the master suite with its huge bed. When Lisbeth and I had taken our quick tour on Saturday, I hadn't been focused on how the place could work as a hotel, but now I had to think of the place in a different light. There was a large and opulent dining room, and it would certainly be possible to prepare authentic meals in the original kitchen, if I could find a chef willing to work with antique appliances. Unlikely: maybe there would be a way to camouflage the modern counterparts, or put the real kitchen in the basement. Which, I reminded myself, I hadn't seen yet. Finding authentic china and silver and glassware would be no problem, if Henry's were long gone, but what about cookware? Maybe I should binge-watch Downton Abbey to get a feeling for the kitchens of the era.\n\nI jumped a foot in the air when a branch cracked behind me. I turned to find Josh Wainwright looming over me.\n\n\"What are you doing here?\" he asked, although he didn't sound exactly hostile\u2014more like curious.\n\n\"Oh! You scared me. I wanted to see the house again. I don't have any of the keys. I figured you would have yours, but I couldn't get through the front gate because it's locked, and I couldn't call you because I didn't have your number, so I came around the back. I was going to look for you\u2014I saw your car so I figured you were around somewhere.\" I took a deep breath to calm my breathing then went on the offensive. \"What the heck are you doing here, sneaking around the grounds? Aren't you supposed to be hunched in your garret writing weighty scholarly tomes about\u2014what was it, industrial history? Is patrolling part of your job description?\"\n\n\"Not exactly. Actually I wasn't looking for intruders\u2014I was probably doing what you were\u2014trying to see the place from a different direction. Why did Henry Barton set it in a shallow valley, rather than planting it on a hilltop for everyone to see?\"\n\n\"You know, I was wondering the same thing. I thought it could be due to weather conditions here, but this isn't exactly New England. Does it snow much? Big storms? You've been living here for a bit\u2014are there howling gales in the night?\"\n\n\"No. So maybe the man just wanted his privacy. Okay, your turn. What do you hope to learn?\"\n\nI realized he didn't know the details from the council meeting, so I filled him in. \"The town asked me here to find a way to save this building and the whole future of Asheboro. This place seems to be the biggest draw, although I don't think it's enough. It can be the centerpiece, but the whole project needs to be bigger. And I kind of proposed a larger project to them, or the kernel of one.\"\n\nHe raised one eyebrow. \"Which is?\"\n\n\"Promise you won't laugh\u2014I kind of sketched this out overnight. I proposed that Asheboro re-create itself as an authentic Victorian village.\"\n\nI sneaked a look at Josh's expression, but it remained neutral. \"I realized when I walked around on Saturday that most of the changes to the town since Henry's day were cosmetic. The town is going to have to spend money on storm repairs, so why not divert that money to restoration instead?\" I stared at him directly. \"Say something.\"\n\n\"I hope you're not thinking of Disneyland.\"\n\n\"Of course not. More along the lines of Old Sturbridge Village, but eighty years later.\"\n\n\"Ah. A good model. So why are you here rather than poking around in town?\"\n\n\"As I said, this is going to be the main draw. Plus I need to have ammunition to counter Cordelia's proposal, which is already on the table, and possibly with some outside funding in hand. I need to know more about Henry Barton\u2014where he came from, why he came to Asheboro, what he did while he was here. I've heard it said that his wife died young and there were no children, but was that a romantic myth? If it's true, did the man just wander around this spectacular house all by himself? Did he entertain? Did he have any lady friends? A staff of twelve who played poker with him on Saturday nights?\"\n\n\"So you want the story of Henry Barton,\" Josh said.\n\n\"Yes. Before you protest, I don't want to intrude on whatever your research is, although I can't imagine you writing that upon the premature death of his lovely wife he was devastated and spent the next year or five wandering around the echoing rooms, unwashed and unshaven, calling out her name. Were you planning to?\"\n\nHe finally smiled. \"God, no\u2014sounds like a cheap romance novel, except that if it was, some lovely, kind, sympathetic, and no doubt beautiful replacement would have to show up in the last third of the book and they'd live happily ever after, wallowing in Henry's riches and popping out six kids.\"\n\n\"So why are you interested in Henry?\"\n\n\"How much do you know about post\u2013Civil War history?\" he countered.\n\n\"Not a lot. Mostly what I learned in public school. I might have taken a few required history courses in college, but not about this particular era. Maybe you can give me a quick summary.\"\n\n\"For the whole war? Not right now. What you do need to know at the moment is that Henry and some of his brothers fought bravely in the war\u2014\"\n\n\"Hold on,\" I interrupted him. \"I don't actually know which side he fought for. North or South? I seem to recall Maryland was kind of mixed.\"\n\n\"North, as did his brothers. Most of them survived, and obviously Henry did. He'd taken part in some of the battles around here, so when the war was over he decided to settle here.\"\n\nTo me it seemed odd to return to a place where an army of men had tried to kill you, but I'd never been in a battle. \"Where'd he come from originally?\"\n\n\"I think the family originated in New England, although they'd lived in several other states.\"\n\n\"And where'd the money for all this come from?\" I waved my hand at the lush vista spread out below us, and the ornate mansion embedded in the middle of it.\n\n\"Now that is the interesting question. You know about the factory in town?\"\n\n\"I know it's there. Lisbeth and I were talking about how we grew up with it in the background, but we never knew what it produced. Was it enough to generate income to support all this?\"\n\n\"That, my dear Ms. Hamilton, is the key question. I haven't collected all the information yet, much less the original sources, but I'm beginning to find hints that Henry was a major player in the rebuilding of this region after the war. Possibly even more. But he was also one of the most self-effacing men I've ever come across. He really didn't like to draw attention to himself, publicly or privately.\"\n\n\"Interesting,\" I said. \"Which might explain why he built this fancy pile but hid it. But why build it at all? For his wife?\"\n\n\"Could be. She's about as enigmatic a figure as he is, although that's less surprising for a woman of her era.\"\n\n\"So, are you writing a survey of regional economic history, or a biography of Henry Barton?\"\n\n\"That is what I'm beginning to wonder.\"\n\n\"How long have you got?\"\n\n\"'Til classes start in the fall. For the research, at least.\"\n\n\"Ah. I haven't got anywhere near a plan for the town, with or without the mansion, but off the cuff I'd guess there's no way we could launch anything before fall, and then only in stages. So we're on the same time line. Are you and I competing for something?\"\n\n\"I can be of use to you, if I find some interesting stories. What can you do for me?\"\n\n\"You want to write a stuffy academic tome or a best seller?\"\n\nHe cocked his head at me. \"I'm an academic. What do you think?\"\n\n\"I think if you add some human elements, it might sell better on both fronts. Maybe not to the audience you originally intended, though.\"\n\n\"And where do you fit in this new and improved warm and fuzzy approach to my opus magnum?\"\n\n\"If I\u2014and the town\u2014go ahead with this ambitious project, we'll either go down in flames or we'll be a hot ticket in the preservation community for at least a few years. Either way there'll be a lot of publicity\u2014I've got contacts around the region and you can believe that I'll tap them all. You can ride on my coattails, or you can give sorrowful comments to the press about how you tried to tell Asheboro and me that this was a terrible idea but we wouldn't listen. You're covered either way.\"\n\nHe actually looked like he was considering the idea. But he didn't answer directly. \"Why don't we go take a look at the house? I know you went through it with your friend on Saturday\u2014\"\n\n\"And yesterday,\" I added.\n\nHe gave me an odd look but didn't comment. \"\u2014but you might not have been paying attention to the details. And you've had time to think about things. You may see things differently now.\"\n\n\"Is there a security system?\" I asked abruptly.\n\n\"You mean like cameras? No. The most the trustees would spring for was a motion-sensor alarm system. If there was any movement outside the building, a god-awful siren would go off, not that anyone except me would hear it\u2014it's not connected to the local police station. I suppose I should be flattered that they believed that I could handle determined intruders.\"\n\n\"Were there any?\" Certainly there hadn't been any alarms back when Ryan and I had been sneaking around the place, or somebody had figured out how to disable them.\n\n\"Not unless you count the occasional fox. I still had to drag myself over to the house and disable the noise each time.\"\n\nI struggled to my feet\u2014I was getting stiff. \"Well, let's go check the place out while it's still light. The power is on, right?\"\n\n\"It is, but the light bulbs are about as low wattage as possible. I suppose we shouldn't complain. If the bank trustees hadn't been so thrifty, the funds might have run dry a while ago, and where would you and I be then?\"\n\nSince we were already behind the building, we went in through the kitchen entrance at the rear. I thought briefly about making notes as we walked through, but decided I'd rather just get a general feeling for the place and worry about specifics later. We headed slowly through the dining room toward the main hallway. I paused long enough to admire (once again) the ornate built-in china cabinets, brimming with what appeared to be matched sets of bone china. For twelve. Did Henry and his wife entertain, or was this standard issue for the upper classes? \"Do you know, I've never learned what Henry's wife's name was\u2014she's always 'the wife.'\"\n\n\"It's Mary,\" Josh said.\n\n\"Ah. Have you spent much time in the house?\" I asked.\n\n\"Not really. Actually I've been remiss\u2014I should be examining whatever documents survive.\"\n\n\"There are documents in the house?\" I asked. \"Nobody ever cleared them out?\"\n\n\"There was nobody who cared. I supposed we should be thankful that nobody ever did buy the place, or they might have gotten rid of everything. So, short answer, there are boxes and trunks of I-don't-know-what in the attic. In my own defense, I arrived in the fall, and the attic is unheated, with about two hanging light bulbs, so it was not an appealing place to spend time in winter. I intend to delve into all that now that it's warmed up a bit.\"\n\n\"Can I help?\" I said. \"I did get a peek at it, with Lisbeth.\"\n\n\"Don't you have a day job?\" he countered.\n\nI supposed that I might as well confess\u2014I'd have to at some point. I cleared my throat and said formally, \"I was informed on Friday that the hotel has been acquired by an international chain that wants to bring in its own staff, so it was, Thank you very much, and here's your severance.\"\n\n\"You didn't see that coming?\" He seemed honestly concerned.\n\n\"No, I did not. My boss said he'd known for only a few days, and he's out too. So now my time is my own and at the disposal of the town of Asheboro, if the council approves my plan. But I haven't said anything to the town council about this yet. I wanted to get a better sense of the options\u2014and the costs\u2014before I agreed to anything. I may have a vision for the town, but that doesn't mean they can afford it, and I'm not about to start something and walk away leaving it half finished. If I decide it's hopeless, I'll tell them.\"\n\n\"I'm glad to hear that. So where do you want to start?\"\n\n\"Let's start with the first floor and work our way up. I've already done a couple of walk-throughs. What I want to know from you is what kind of shape the systems are in. You know\u2014plumbing, heating, that kind of stuff. You have kept an eye on those?\"\n\n\"Kind of,\" Josh said, \"but mostly to make sure nothing has blown up or stopped working. So, in the kitchen: slate sink, hand pump next to it. Icebox\u2014and that does mean ice, literally. A cast-iron coal-burning stove. There's also a pantry that could probably supply an army for a week, if it was filled. There's some old cookware on a few of the shelves, but I don't know if it's usable. Henry didn't splurge on the hired help.\"\n\n\"No wine cellar?\"\n\n\"Maybe there's something like it in the basement. I haven't been inspired to explore down there\u2014too many spiders. The best I've done is to check to make sure nothing is flooding\u2014we are in a valley of sorts here.\"\n\n\"Well water, I assume,\" I said, almost to myself. There couldn't possibly be municipal water out this far from town.\n\n\"There's a cistern system too, actually. To trap rainwater.\"\n\n\"Really? Is that what you've been using out back?\"\n\n\"Pretty much. My wants are few. Actually it's been a good experience\u2014it gives me more insight into the impact of creeping industrialization. Lights, running water, basic sanitation. Things we all take for granted now.\"\n\n\"How quickly we forget. So we saw the kitchen. Let's check out the other side of the hall.\"\n\nBefore we left the dining room, I looked longingly at the magnificent mahogany dining table that occupied the center of the room. Above it hung an elaborate crystal chandelier that was lit with gas. \"Where'd the gas come from?\" I asked.\n\n\"I can't say I've given it any thought. Besides, it was superseded by the electric lights you see now.\"\n\n\"Where'd the electricity come from?\"\n\nJosh looked bewildered. \"Uh, the town? I've never tried to rewire anything here so I haven't poked around. As long as my lights work I don't complain.\"\n\n\"You get wireless out back?\"\n\n\"Yes, but it's kind of patchy. But I don't use it a lot. When I hit a thorny research question, or when I start talking to the birds, I take myself off to the library in town and catch up on news and actually talk to people. Why all the questions?\"\n\n\"I'm trying to sort out the time line for the place, which means: how much is original, from the last quarter of the nineteenth century or earlier, and what Henry added, mainly later in the century. I'm sure I can find out when the town was electrified, but it's less clear when it would have extended out here. Although I suppose if Henry was rich enough to put all this together, he could have afforded to run his own line out this way\u2014it's only a couple of miles. Okay, next?\"\n\n\"Parlor one and parlor two. I've never quite understood the need for two. I suppose if you were a guest you could figure out where you stood with your host by which room he greeted you in. Nice original pocket doors between the two.\"\n\n\"Let's see them.\" I followed him into the nearer of the two rooms on one side. Again, high ceilings, ornate moldings, fancy hanging lights, plus a few gas-fed wall sconces. The carpets were long gone\u2014I wondered if someone had parked them in the attic, or if the moths had gotten to them\u2014but the floors seemed to be in good condition. How sad to think of two adults, and then only one, rambling around all this elegant space.\n\nJosh wrestled with the pocket doors and finally pried them open. The front parlor was similar to the back one, save that it had a big bow window in the front.\n\n\"What's on the other side?\" I asked.\n\n\"A very handsome spacious library\u2013sitting room combo, and a glassed-in sun room on the far side. A few of the panes there are cracked, but basically it's intact. It faces north and gets indirect sun, so it doesn't get too hot.\"\n\nI looked at him critically. \"You really like this place.\"\n\n\"I do. It's as if it's frozen in time, and that's a rare thing these days. I will admit that I shudder at the idea of Cordy taking it down to the studs and starting over. It seems like sacrilege.\"\n\n\"I can understand that. What do you think she is really after? Money? Prestige? Power over the town? She is on the council, and she has her own business.\"\n\n\"Hey, I'm not from around here, and I don't hear much gossip. From what little I've heard, nobody on the council likes her, and her little business doesn't make any money. I'd guess she couldn't stand to see it fail, so she was hoping to move on to bigger and better things before that became too obvious. Somehow she got fixated on this place. To tell the truth, I'm amazed the council stood up to her as long as they did.\"\n\n\"You seem to have collected quite a bit of information in a short time.\"\n\n\"I'd meet for a beer or two with one or another of the council members occasionally. After two, people tend to get chatty.\"\n\n\"Did Cordy know you were going behind her back?\"\n\n\"I'm not sure. I never saw much of her, and when I did run into her here, she treated me like the hired help. No one on the council went out of their way to hide our meetings, but we didn't invite her to join us either.\"\n\nI almost felt sorry for Cordy, until I reminded myself that she'd made her own bed, alienating people left and right for years. She knew what she wanted, but she apparently never learned the best way to get it. \"Upstairs?\"\n\nWe took the broad staircase to the second floor. The staircase was unquestionably gorgeous: four-foot-wide mahogany treads, ornate spindles, and a curving mahogany railing. The upstairs landing was wide, as was the hallway it led to. I was beginning to understand how there could be eight bedrooms up here. \"Maybe we'd better start with the bathroom,\" I said dubiously.\n\n\"Because that might be your worst problem? Don't worry, no major leaks\u2014but it's not exactly up to code.\"\n\n\"I know\u2014I checked it out yesterday. I was trying to visualize what it would take to convert this place into a historically correct but functional boutique hotel.\"\n\n\"And?\" Josh asked.\n\n\"We'd have to get very friendly with the building inspectors,\" I told him.\n\n\"Hmm. Bedrooms?\"\n\n\"Again, Lisbeth and I took a look. And we did check out the help's quarters upstairs, but we didn't go poking into corners.\"\n\n\"The light is lousy up there. If you really want to see what's up there, you should come earlier in the day, with some big lights, and prepare to get filthy.\"\n\n\"No treasures? Like a trunk full of gold certificates?\"\n\n\"To be honest, I really don't know. I'm not a treasure hunter, I'm a historian.\"\n\nHe hadn't been curious to see if Henry had left any documents up there? Personal or professional? An odd thought hit me. \"Do you know if Cordy ever went up there?\"\n\n\"I can't say for sure, because I'm not around all the time, and I can't see much from my rooms. The rare times I've been up there, I haven't noticed any signs of recent disturbance.\"\n\n\"Huh,\" I said intelligently. I was pretty sure Cordy wasn't interested in history, and she wouldn't have wanted to get her clothes dirty. I checked the time. It was after four now, and I still had an hour's drive home ahead of me, and it would be dark soon. \"Well, I guess I've seen enough. It's a start.\"\n\n\"May I make one request?\" Josh asked, almost sheepishly. \"If you do decide to go ahead with this project, may I stay, at least until my contracted time is up?\"\n\n\"I don't see a problem, although it may not be up to me. I don't think there's that much noisy work that needs to be done to the place, if it's planned right\u2014and pitched right to potential guests. But I can't promise anything. Besides, don't you have a home to go to?\"\n\n\"Not lately. I'm divorced, and my wife got the house. This was a handy fill-in while we sorted things out.\"\n\n\"Ah. Well, I'll let you get back to work, and I'll head home.\"\n\n\"When will you be back?\"\n\n\"I don't know. But I promised the council a draft proposal by the end of the week. Can we walk out the front? I want to see the main view again.\"\n\n\"Sure.\" We descended the grand staircase, and Josh sorted through the keys until he found the one he wanted, a large brass one. He opened the massive door and pulled it back, but then he stopped in his tracks without crossing the sill, and I more or less bumped into him.\n\nAnd then we were looking down at the body of Cordelia, and I was pretty sure she was dead. Fallen backwards on the stoop, draped over the massive granite slabs, her eyes staring at the sky. And a pool of blood under her head.\n\nFor a moment all I could see was flickering lights. I couldn't exactly feel my legs.\n\n\"Don't move,\" I heard Josh say, as if from a great distance. \"You've turned kind of green.\" I realized he'd grabbed my arms and was more or less holding me up.\n\n\"I wasn't planning on it,\" I told him, swallowing my nausea and taking a deep breath. \"Is she...?\"\n\n\"I'm pretty sure she is.\"\n\n\"How long?\"\n\nHe let me go, and I turned to face him. \"How long?\" he asked, his tone incredulous. \"Seriously? You think I know?\"\n\nMy vision was finally clearing. \"Well, no. I'm just wondering if we have alibis, or need them. Have you been here all day?\"\n\n\"Yes. I was working.\"\n\n\"Did you see or hear Cordy's car drive in?\"\n\n\"Car?\" He looked around, bewildered, until he saw it parked off to the side. \"No. I had my headphones on, and I was on my computer. What about you? I know when and where I caught you sneaking in, but you could have been there earlier.\"\n\n\"No way. I spent last night at my home in Baltimore, where I had dinner and did some Internet searching. I met Lisbeth just before eleven this morning, and then we spent an hour with the council, and then we had lunch together. So my morning is covered. The blood isn't dry yet, so she can't have been... gone for long. Unless you think I came to Asheboro at dawn, tracked Cordy down, killed her, and dragged her out here and dumped her before my meeting.\" I nodded toward the body. \"From the look of her, it would have been messy, so you'd have to give me time to change clothes too before I addressed the town council.\"\n\n\"Did you have a motive?\"\n\n\"Sure did. Most of the town probably knows about it. Cordy and I had a history. Okay, what about you? Where were you?\"\n\nHe stared at me for a long moment. \"Here. Working. If anybody in authority cares to look, he would find evidence of our computer activity, right?\"\n\nI shrugged. \"All I know is what I see on Law & Order.\" We both fell silent, until I finally said, \"This is ridiculous. I didn't kill the woman, and I don't think you did. But someone did\u2014she didn't just fall, hit her head, and bleed to death. I guess we should call the police. Who's in charge in Asheboro?\"\n\n\"For the town, it's the local police chief Jack Wilson. For a homicide, it's the county police, over in Pikesville. But I think we have to go through the procedures. I'm the official caretaker, so I guess it's my responsibility to make the call to the local cops, and if they decide it's a homicide, they would call in the county.\"\n\n\"And you know this why?\"\n\n\"The bank manager gave me a list of contact numbers when I signed on. It's posted on my fridge. I'll call now.\"\n\nWhile Josh made the necessary call, I sat gingerly on one of the spindly side chairs against the wall in the hallway, fearful of contaminating a crime scene or disturbing evidence (and maybe breaking the fragile chair). At least I couldn't see Cordy's body from there. So Cordy was dead\u2014that much was obvious. And I could think of a long list of people who might have wanted her dead, but that was from years ago. What had she gotten herself into since?\n\nWhen Josh returned, we sat in stony silence, waiting for the police. There really wasn't much to say, and my mind was blank. I'd never seen a dead body before, certainly not someone I had known\u2014and hated with all of my teenage heart. But I knew nothing about her death. A tiny part of me was mad because I hadn't had the chance to confront her over her plans for the mansion\u2014and win. Well, actually, I had kind of won, because she wasn't in the running anymore, but I wasn't going to point that out to the police. One old motive was plenty.\n\n\"Did you see a weapon?\" I asked, more to break the silence than because I cared.\n\n\"Uh, no. But I wasn't looking for one. You?\"\n\n\"Same here.\"\n\nWe went silent again. It was probably no more than fifteen minutes before the local police chief arrived, took one look at the body, and quickly pulled out his phone and called the county office. I wondered idly when there had last been a murder in Asheboro, and it was clear that the chief had little experience in that kind of investigation. An investigator from the county office arrived half an hour later, introduced himself, and ascertained my identity as well as Josh's. Since both of us looked thoroughly rattled and were not covered with blood, he told us we should expect to appear to give a full statement the next day at the county office in Pikesville. Then he said we could go home, which for Josh meant walking around to the other side of the building.\n\nWe moved away from the front. \"Your car's still at the other end of the property. I can drive you to your car, if you want.\"\n\n\"That would be great, thanks. All I want now is to go home and try not to think.\"\n\n* * *\n\nIt took only a few minutes to retrieve his car, two more to take me to where I'd left mine. When I climbed out I said, \"I assume I'll be seeing you again. I want to talk to you more about the project.\"\n\nHe looked at me as though I'd lost my mind. \"Really? You're thinking about that now?\"\n\n\"It must be important, if somebody was willing to kill Cordelia to stop her from moving forward.\"\n\nJosh was already shaking his head. \"You don't seriously think that's why someone killed her, do you?\"\n\n\"Do you have a better idea?\" I replied tartly.\n\nHe didn't answer my question. \"We'll talk later. Good night.\"\n\nI climbed into my car and tried to remember how to turn it on, until the fog in my head cleared, and I started it and turned toward home. I managed to get there on autopilot, without thinking, and when I arrived I fell into bed.\n\nI was awakened far too early by the ringing of my landline, and it took me a moment to collect my wits. Right: it was Monday, and Cordelia Walker was dead. Great way to start the week. I stumbled to the phone, and the caller ID said something about police. I answered, \"Katherine Hamilton.\"\n\n\"Ms. Hamilton? This is Detective Reynolds of the Maryland State Police Criminal Division. I'd like to talk with you about the death of Cordelia Walker. You were present when the body was discovered?\"\n\n\"Yes,\" I said. Keep it short.\n\n\"I understand you were acquainted with Ms. Walker?\" the detective asked.\n\n\"Yes, in high school. I hadn't seen her since then.\"\n\n\"Why were you in Asheboro yesterday?\"\n\nHere we go. I took a deep breath. \"I was invited by the town council to discuss ideas for the economic revitalization of the town.\" Wow, that was an impressive sentence I'd managed to string together with little sleep and no coffee.\n\n\"Why were you at the Barton house?\"\n\n\"My primary professional experience lies in hospitality management. There was some discussion of utilizing the house as a hotel. I wanted to take a critical look at it. Was Cordelia murdered?\"\n\n\"Yes.\" The detective did not elaborate, and an odd silence fell.\n\n\"What does this have to do with me?\" I finally asked.\n\n\"You were in Asheboro this weekend, according to Lisbeth Scott.\"\n\n\"Yes. We're old friends. She thought the town council should talk with me. We got together for a quick tour of the town on Saturday, and she showed me the house. I came back yesterday to talk with the council, then went back to the Barton house after that. Joshua Wainwright walked through the house with me, and then we discovered... Cordelia on the front steps, and called the police. When did Cordelia die?\"\n\n\"Ms. Walker was killed early in the morning on Sunday\u2014yesterday. I'm merely verifying the time line. You were acquainted with her, and you made several trips to Asheboro over the past few days. It's my responsibility to see if there is any connection between those two facts. Did you at any time see or meet with Ms. Walker?\"\n\n\"No.\"\n\n\"As I believe you were told yesterday, you need to come in and make a formal statement.\"\n\nI had nothing to hide. \"When?\"\n\n\"This morning, if possible. Is that convenient?\n\nHis tone suggested that I should make it convenient. And there was no need for me to rush to my former job in the city. \"Yes. Where?\"\n\n\"At our headquarters in Pikesville. I understand that's not too far from your home.\"\n\n\"I'm sure I can find it. Ten o'clock?\"\n\n\"Fine.\"\n\nDamn, here I was driving again, and there was just me and the highway on the way to Pikesville. I tried to figure out what I was feeling\u2014and thinking. No job, but I felt more sorry for George. He needed his job more than I needed mine. I was going to need a new job, but all options were on the table. I had enough money in the bank or would have with the severance pay, to take my time, survey the field, decide what I was really looking for. This Asheboro project had popped up at an ideal time, but I had to wonder how much Cordelia's death would complicate things. And now I had an appointment to talk with a police detective about Cordelia's murder. That would certainly be a new experience.\n\nI arrived at the Pikesville State Police Headquarters before ten, and sat in my car for a moment, lining up my facts. I'd first talked with Lisbeth on Thursday, and then spent most of the day on Saturday with her, though I'd left Asheboro before dinner. I hadn't seen Cordy at any time. I'd driven straight home, although there was no way to prove that\u2014I hadn't stopped to buy gas at some convenient gas station halfway between Asheboro and Baltimore, with a tidy date-stamped receipt. I suppose it was conceivable to a suspicious cop that I might have tracked Cordy down and killed her after I'd left Lisbeth. But I knew I hadn't harmed the woman, and as I'd already said, hadn't visited Asheboro in years. They might as well investigate Lisbeth\u2014maybe she'd killed Cordy out of loyalty to me, after we'd parted. Or maybe since I'd last seen her she'd turned into a psycho killer. But if Cordy had died in the early hours of Sunday, the time line didn't work.\n\nI gave a snort of laughter, climbed out of the car, and headed toward the building. Inside, I told the person at the desk that Detective Reynolds was expecting me, and she picked up the phone. I studied the interior: it was drab, neutral, and non-threatening, and gave nothing away. Which pretty much described the uniformed man who emerged from the back of the building and approached me. \"Ms. Hamilton?\"\n\n\"Yes, I'm Katherine Hamilton,\" I said cautiously. That seemed safe enough.\n\n\"Please, come with me.\" He turned and walked away, leaving me with no option but to follow him. He stopped halfway down a dimly lit hallway and opened a door, then stepped back to let me enter first.\n\n\"Have a seat,\" he said. I sat. \"Thank you for coming in so quickly. As you can guess, we're still collecting information, but the sooner we can nail down some facts, the sooner we can find who killed your friend.\"\n\nMy mouth opened before my brain was in gear. \"She was not my friend.\"\n\nHe looked at me. \"Acquaintance?\"\n\n\"I knew Cordelia in high school. We had some issues back then. I hadn't seen her since I graduated.\" Alive, at least, but I didn't add that.\n\n\"And that was fifteen years ago, right? I'm sorry, I seem to be doing this backwards, and there's always that dreaded paperwork.\" He pulled out a lined pad and a pen. \"Could you give me your full name, please?\"\n\n\"Katherine Eleanor Hamilton.\"\n\n\"Address?\"\n\nI recited my address. I knew he already had my phone number, thanks to Lisbeth.\n\n\"Employer?\"\n\n\"The Oriole Suites Hotel in Baltimore. For about another week.\"\n\nHe looked up briefly. \"Fired?\"\n\nWhy did he assume that? \"Let go. Corporate takeover.\"\n\n\"Ah. You've known Lisbeth Scott for how long?\"\n\n\"Since high school. We were best friends then.\"\n\n\"Why were you in Asheboro yesterday?\" He looked up and fixed me with a steely stare. Were his eyes gray or blue?\n\n\"That's a kind of complicated story,\" I said.\n\n\"I've got time.\" He sat back in his chair and waited for me to begin.\n\nI wondered what I was in for. Should I have called a lawyer before talking to the detective? Sure, this nice solid-looking man had implied over the phone that he just wanted some basic facts from me, since my presence in town on the day Cordelia died as well as the day before was kind of an anomaly. I had popped up in town just in time for her murder.\n\nWhat could I say about why I was in Asheboro over this weekend? I figured the town council would forgive me if I told Detective Reynolds why I had been invited to Asheboro. But when he found that that same council had taken up arms against fellow member Cordelia to stop her wily plans, and she ended up dead... Well, it did look a little suspicious. I amused myself by wondering just how much the good citizens of Asheboro would have offered me to off the woman.\n\n\"Ms. Hamilton?\"\n\nI gathered up my wool\u2014figuratively speaking\u2014and regarded him. \"Detective Reynolds, do you know Asheboro at all?\"\n\nHe shook his head. \"I can't say that I do.\"\n\n\"Well, this story is going to sound rather odd, and what's worse, I'm not sure what I'm going to say puts me in a very good light with regard to Cordy. Let's just say you won't find a shortage of suspects. Before I get into it, can you tell me how she died?\" I could make a pretty good guess.\n\n\"It may not be appropriate to share the details, although I'm sure the press will have this information soon. She was killed by a blow to the head with a blunt object\u2014most likely a rounded rock\u2014on the front steps of the building known as the Barton mansion outside the town.\"\n\n\"Obviously I know the place, since I was there yesterday. I grew up in Asheboro, and everybody knew about it. But that place was one of the reasons the town council invited me to Asheboro this week. To put it bluntly, the town needs help. They sank a lot of money into buying the mansion but they don't have enough to fix the place up, and now the town buildings need repairs as well. They wanted to know if I could recommend any new avenues for raising funds and bringing tourists to the town.\"\n\n\"And have you?\"\n\n\"Detective, I've spent very little time in the town since I first learned about the problem from Lisbeth last week. That's not exactly enough time to create a rescue mission for an entire town. I proposed a rather ambitious idea to them at an informal council meeting yesterday, but I need a lot more information, and I need to look at examples in other areas that have been successful.\" And I needed to think hard about whether I wanted to be involved at all.\n\n\"Hmm. Did Cordelia Walker have a role in this?\"\n\nTread lightly, Kate! \"She was a member of the town council. She also ran a bed and breakfast in the town. I've been told that she had proposed turning the mansion into a much bigger and better version of that and possibly running it herself.\"\n\n\"Is that a bad idea?\"\n\n\"I don't know yet, but I do know the council was resisting that plan. But as I understand it, Cordy was used to getting her way. She said if the council didn't get off the pot, so to speak, she had some other ideas, like corporate investors lined up who would step in and transform the place.\"\n\n\"And I'll ask again: is that a bad idea?\"\n\n\"Done right, not necessarily. However, the plan Cordelia presented, which involved outside investors, relied on gutting the place on the inside and modernizing it. There are those in the town who did not like that idea. They'd rather see the place restored to its original glory\u2014and believe me, it is glorious\u2014then use that to attract tourists.\"\n\n\"Ah. So there was conflict between Cordelia and the council?\"\n\n\"I think it's fair to say that. Would anyone on the council be willing to kill her to save a piece of the town's history? I don't know, having spent no more than an hour or two with the members of the council. I'd say it's possible but unlikely.\"\n\nThe detective nodded. \"Based upon my discussions with a few people in Asheboro, I've gotten the impression that Cordelia Walker was not exactly popular with all of the townspeople.\"\n\nAnother crossroads. Well, in for a penny, in for a pound. \"Let me be frank. Cordelia Walker was a bitch and a bully, back when I first knew her. She was the queen bee, the leader of the pack among the popular girls when we were both in high school. She ran everything, and she had her own clique of followers. From what little I heard about her yesterday, I gather that hadn't really changed. She went away to college, but then she came back\u2014I'd guess she decided she'd rather be a big fish in a small pond than one of many minnows in the big wide world. She opened the B&B, but I doubt she had the right personality to make visitors feel welcome.\" That at least came from my area of expertise, not personal hostility.\n\n\"Ah, yes.\" The detective consulted his notes. \"Were you aware that she was, until recently, married to a Ryan Hoffman?\"\n\nI felt like I'd been hit in the gut. \"Oh, crap.\" Lisbeth hadn't mentioned that\u2014she probably thought she was protecting me. Or conning me to get me to show up in Asheboro. When the detective looked startled, I hurried to explain. \"Sorry, but everything so far just seems to dig me in deeper and deeper. Ryan Hoffman and I thought we were in love, our senior year in high school. Cordy didn't like me, and I think even then she wanted him\u2014he was smart and funny and good looking. So she and her little band of bitches cooked up a way to break us up and make me look like a fool.\"\n\n\"And that would be?\"\n\nI had to tell him, or somebody else would. \"The Barton mansion used to be a place for high school kids to get together and hang out\u2014not for drugs, but usually for sex. We all knew how to get into the house the back way, and all the furniture was there. Including beds. So Ryan and I decided to...\" I fumbled for words, because nothing felt right. Get it on? Consummate our love?\n\nLuckily the detective got the message. \"I think I can fill in the blank. So what happened?\"\n\n\"Ryan and I went out there one night, and things were heating up quite nicely, when Cordy and her pack burst in with cameras flashing. Then they spread the evidence all over the school in the next few days. I kept my head down and my grades up for the rest of the year, and got out of town as fast as I could.\"\n\n\"Ah, so that's your motive? The mean girl embarrassed you more than a decade ago?\"\n\n\"I suppose you could see it as a motive, but I hope I've moved past the embarrassment.\" Not! \"And you're telling me she ended up marrying Ryan?\"\n\n\"So it would appear. I take it you two didn't remain a couple for much longer after the... incident?\"\n\n\"Not once Cordy got her claws into him. She usually got her way in the end.\"\n\n\"Not this time.\" He leaned back and studied me, and I wondered if I detected a hint of a smile. Finally he said, \"So you're saying you fled Asheboro in what you perceived as disgrace years ago. You returned for the first time this past weekend by invitation. You learned that the town was broke and the council hoped you could fix it. And your high school nemesis ends up dead shortly after your arrival. Does that about sum things up? Oh, and I forgot to include that the dead woman married your first crush. Adding insult to injury.\"\n\nThis was a smart man. \"Yup, I think that covers it\u2014although I didn't know that she and Ryan had gotten married until you just told me. What now? You going to arrest me?\"\n\n\"I think I might be laughed out of the department if I tried to arrest you on such circumstantial grounds.\"\n\n\"Well, that's good to hear.\" I was beginning to feel a lot more optimistic. \"Who else are you looking at?\"\n\n\"Ms. Hamilton, you must know I can't tell you that.\"\n\n\"Do you have a first name?\"\n\nHe looked startled at the unexpected question. \"What? Oh, well, yes, of course. Brady.\"\n\n\"Well, Brady, I'm Kate. Now that you've heard my most mortifying secrets, I think first names are appropriate.\"\n\n\"Buttering me up will not get you off a list of suspects. Let me ask you this, Kate: Are you going to work with the town on this project, or are you going to wash your hands of the whole thing?\"\n\nI took about four seconds to think about it. I was out of a job, but I had enough money to keep me going. I needed a distraction to keep me busy while I hunted for a new job. I was possibly a murder suspect, but since I knew I hadn't done it, the killer had to be somebody else in Asheboro, and if I stayed around Asheboro I could learn more about the town's current dynamics\u2014and who else had been provoked by Cordy into a murderous rage. And that's what it had to be\u2014a big rock was a weapon of opportunity and suggested that Cordy's death hadn't been planned, but that she'd pushed somebody's buttons once too often. And last but not least, the Barton mansion was magnificent, and if I could get the town to go along with my vision of the downtown portion, we might be able to inject some new life into struggling Asheboro. It was a win-win-win-win situation.\n\n\"If the town is still interested in my idea, I think I'm going to see if I can make it work. I like a challenge.\"\n\nDid Brady Reynolds actually look pleased? \"If I may suggest something\u2014strictly off the record\u2014it would be valuable in my investigation to have someone on the inside, in a manner of speaking. Someone who knows the town, but is not of it, if you know what I mean.\"\n\n\"Someone like me,\" I said bluntly. \"Does this mean I'm not a suspect and you trust me? And you want me to spy for you?\"\n\nNow he looked exasperated. \"Nothing like spying. And I would not ask you to betray any friendships you have in the town, or to jeopardize your professional activities, if you choose to help the town council. But I'd like to think you can be a more objective observer than people who are closer to this crime.\"\n\nI could count my Asheboro friendships on one finger, and I didn't think Lisbeth had killed Cordy. As for the professional side, I had no idea what the revitalization job would entail, if I decided to take it, or how long it would take, or how much it would cost. Once I dug into the details, I might decide it was hopeless\u2014even I couldn't work miracles. But the time it would take me to assess all those factors was probably more than enough to allow the state police to solve this murder. Plus I'd be in an ideal position to ask questions that would otherwise be considered nosy. Like how successful Cordy's B&B was. How she'd tracked down outside investors\u2014and whether she had called in personal favors. Or might these investors be interested in a modified project. And was there anything she knew that she hadn't shared with the town? Cordy looked out for Cordy, not Asheboro.\n\n\"Detective, I think we can work something out. I do have some fond memories of Asheboro, despite what happened. I'd hate to see it go down the drain because I walked away from it, now that I know about its problems. While I didn't like Cordelia Walker, I didn't wish her dead, if only for the town's sake. And, as it happens, I have some time off to really explore the options for the town. So, yes, I'd be willing to work with you.\"\n\n\"Excellent. What are your plans for the rest of the day?\"\n\nHe certainly was in a hurry. \"I think I'll go back to Asheboro,\" I said.\n\n\"Keep your ears open. If you notice something that seems out of place, let me know.\" He fished a business card out of his wallet. \"Use the direct line, on the bottom there.\"\n\n\"Fine. I'll let you know if I hear anything interesting.\"\n\n\"Thank you, Kate. I appreciate your help, and I look forward to hearing from you.\"\n\n\"Right.\" I stood up, then wavered, wondering if he was going to escort me out. Nope. Apparently he trusted me to find the front door, so I left.\n\nOutside I sat in my car and tried to figure out what had happened since breakfast and what I was supposed to do next. Cordelia was dead. I might have been a suspect for about a minute, but now apparently I was not. I had no job. And the state police wanted me to snoop in Asheboro, kind of. Interesting day so far, and it wasn't even lunchtime.\n\nWhat I really wanted to do was go back and find more information about the Barton house. For that I should talk to Josh Wainwright. The detective hadn't said he was a suspect, so that should not be an issue. I really needed to have a heart-to-heart with Lisbeth, who hadn't bothered to tell me that Ryan had married Cordelia\u2014although the marriage apparently hadn't lasted, and who was to blame for that? But I'd have to wait until her kids got home from school for that.\n\nI reversed out of the police parking lot and headed back toward Asheboro. I had my plan: talk to Josh first, if I could find him, and then Lisbeth. That left the rest of the drive free for me to fantasize about the Barton mansion. I had to confess: it was turning me into a Victorian junkie. The Victorians with money to spend reveled in acquiring and displaying stuff. Their rooms had moldings upon moldings on the ceilings and walls, and the wallpaper was ornate and elegant. The real wood floors often had elaborate inlaid patterns. Once the space was decorated, the owners started adding pictures to the walls, sometimes in two or three tiers, one above the other. The floors would be covered with richly colored (real) Oriental carpets. The windows would be draped with at least two layers of curtains, underneath real lace (of course) and over that usually velvet (with swags and tassels and fringe). The fireplace would be surrounded by ornately carved mantles in marble or wood, and patterned tiles to protect the floor in front. In short, every inch of space received attention and decoration. If you were worried about housekeeping it would be a nightmare to keep clean\u2014all that coal dust!\u2014but the rich owners didn't trouble themselves about it\u2014they had staff for that.\n\nThe final product was gorgeous\u2014a feast for the eyes and senses. If you wanted to be politically correct, you could say the homes were offensively opulent, built with deliberate thumb-your-nose ostentation. But part of me was glad that now and then we could enjoy what unlimited funds could achieve. Henry's house did not disappoint.\n\nI found myself wondering again what Henry's wife, Mary, had been like. Had she been sickly? How had she died? Had she had a hand in shaping the house, and had she lived long enough to enjoy it? I wanted to know more, even if it was only for my own satisfaction. If I took this project on, I'd be the one who had to find the financing. But I truly believed it would be worth it.\n\nI'd outlined the problems to Josh already, and I hadn't minimized or exaggerated them. He had a different perspective on the broad problem, but the information he had collected on Henry could be useful. The town had asked for a proposal by the end of the week. Would Cordy's death have given us all a reprieve, in terms of time, even though her proposal had been conveniently wiped out by her death? Did anyone in town know who her mystery investors were? What kind of promises had she made to them? And were they going to show up in Asheboro with some kind of documents in hand and demand satisfaction? Heck, did they even know she was dead? Did they believe she had had the council on her side with her ambitious plans? I wasn't going to poke that hornet's nest right now. If those people chose to come looking for her, we'd deal with that later.\n\nWhat did I need to do to put together this proposal? The Barton house was impressive, and in pretty good condition. Not a top priority for worry. I needed to walk through the town again and pay attention to details; to determine what changes had taken place over the past hundred years and whether they were reversible. I needed to talk with the current shopkeepers and building owners, to get a sense of their reaction to my still vague plan. Sure, I could let the council deal with them, but I wanted to see their faces, hear their concerns and objections. If they weren't on board with the plan, it wouldn't work.\n\nI needed to look at funding sources. I knew the town itself was tapped out. I had to check with the bank to see what was left of the original endowment for maintaining the Barton property and see how much of that could be repurposed. I needed to investigate external funding sources\u2014state and federal grants or foundations that supported historic preservation. I'd dipped my toe in those waters for the hotel in Baltimore, when we had wanted to preserve its original fa\u00e7ade, but no way could I call myself an expert. Nor could I hire a municipal financial consultant, because the town council had no money. It would be necessary to find a way to finance this entire project, which wasn't going to be easy. Maybe not even possible. I doubted that Asheboro could raise enough taxes to cover the costs, and that wouldn't make the good citizens of the town very happy. I'd have to put \"find money\" at the top of the priority list.\n\nWhere did Cordelia's death fit in all this? Was it personal or was it related to her plans for the Barton house? Hard to guess right now. Those of us who had been victimized by her as meek students in high school had never compared notes about how Cordy had treated us, and we'd certainly never banded together to try to stop her. I did wonder if she'd changed as she'd gotten older, or if she still ruled her little world the same way. I suspected the latter.\n\nAnd now the state police wanted me to snoop? Kate Hamilton, undercover police spy. In my old hometown. To help find the killer of a woman I once thought had ruined my life. Could I just go back to last week, when life was simpler?\n\nI needed to talk to Lisbeth first. I owed it to her to tell her what had happened since we'd seen each other yesterday, but if I was honest I had to admit that I was not going to do this for her benefit but for mine. Right now I needed a friend\u2014and she owed me an explanation about why she hadn't bothered to mention that my one-time high school love had somehow married the b-word who had split us up. Which kind of answered the question: that had been her goal all along. So before I could talk myself out of it, I pulled into a gas station on the edge of Asheboro, pulled out my cell phone, and hit her number.\n\nShe answered after five rings, and once again I could hear kid-type noises in the background. \"Kate! I was hoping I'd hear from you. What's up?\"\n\n\"Do you have time to talk? Because this is going to take more than a few minutes.\"\n\n\"Of course,\" her answer came quickly. \"But I may have to keep an eye on the kids and make sure they do their homework. What's going on?\" From the decline in background noise I deduced she had moved to a different room\u2014one with a door that closed. I was already practicing my sleuthing.\n\n\"I think this would go better face to face. Can I come over?\"\n\n\"You're in Asheboro? Sure, come right ahead.\"\n\nIt took only five minutes to reach her house. When she opened the door she gathered me into a hug. \"I heard about Cordelia. You were there?\"\n\n\"Josh and I practically tripped over her on the front step. Then we had to talk to the local police, and they handed us over to the state police, and I met with a detective this morning in Pikesville.\"\n\n\"Do they know who killed her?\"\n\n\"Not yet, and I think there may be a lot of suspects. That's one of the things I wanted to talk about. But not the only one. I got laid off on Friday,\" I began. Lisbeth made appropriate dismayed noises, and we gnawed on that together for a while. Then I segued into my conversation with Detective Brady Reynolds of the county police. Lisbeth hung in there until I was wrung out. \"So that's been my past few days, apart from the time we spent together. What do you think?\" I wrapped it up. I realized belatedly we'd been talking for well over an hour.\n\n\"I think my life is boring,\" she said, laughing. \"What are you going to do? Or is it too soon to ask?\"\n\n\"For once in my life I really don't know. I do want to spend some time in Asheboro, to see if what I envision for its revival is even remotely possible.\"\n\n\"You're not just using that as a distraction? Because if you get the council's hopes up and then walk away if something better comes along, nobody will ever forgive you.\"\n\nI hoped Lisbeth didn't believe I was capable to that kind of callous betrayal. \"I don't think that's what I'm doing, but thanks for the reality check. No, I think I'd like to do something positive for the town. Now that Cordy's gone, I won't have to spend my time watching my back. Is it terrible of me to say that?\"\n\n\"I know what you mean. She was not a nice person, and that's as far as I can go with kids in the house. I can fill you in.\"\n\n\"Actually I'd appreciate that. I'd really like to know how many other enemies Cordy had made in town.\"\n\n\"How much time do you have? It's not a short list. Cordy was completely tone-deaf when it came to human relationships, and she thought the world revolved around her. I'm surprised she survived as long as she did.\" Lisbeth hesitated. \"Are you going to talk to Ryan?\"\n\nFinally she'd given me the opening I'd been waiting for. \"How come you didn't happen to mention he and Cordy had gotten married?\"\n\nLisbeth looked embarrassed. \"Oh. Well, I didn't want to bring up the past, and I didn't want to upset you. Besides, I was pretty sure it wouldn't last, when they got together. And it turned out I was right.\"\n\n\"Why didn't it work out?\"\n\n\"For a start, because Ryan was kind of a trophy husband, you know? Smart, good-looking, successful. And apparently easy to manipulate. She set her sights on him, probably just to see if she could land him. I don't think they even made it to their fifth anniversary. No kids.\"\n\n\"I'm not carrying a torch for him. We were kids back then. He didn't exactly stand up for me at the time, you know, and I never heard from him after I left town. But he doesn't live around here anymore, does he?\"\n\n\"No, but I understand he's actually the owner of the B&B Cordy was trying to run\u2014probably part of the divorce settlement\u2014so I assume he's going to have to come back to take care of business. For all I know he's still the executor of her will\u2014I can't think of anyone else she would have asked.\"\n\n\"Good point. Well, I won't go hunting for him, but if we happen to cross paths, I'm not going to run away. You don't think he killed her, do you? Was she running up his credit cards? Bleeding him for money? Anything like that?\"\n\n\"Not that I know of, but I'm not really part of the 'ladies who lunch and gossip' crowd. I'm sure you could find out. After all, the B&B is part of the town, so you have to work out what you think could be done with it now. Will there be a need for a small operation like that, under your grand scheme?\"\n\n\"I haven't thought that far. I'm guessing a memorial for Cordelia is not an option?\"\n\n\"Not hardly. Oh, Kate, I'm so glad you called, and that we've gotten past the whole Ryan and Cordy mess. We'll all understand if you can't fix all our problems, but it's so good that you're willing to try.\"\n\nWith Cordy lying in the morgue, I wasn't so sure the mess was over yet. \"And I'm not Cordy, so I don't plan to trash the town's history.\"\n\n\"Exactly. Listen, are you going to commute? I know you don't live too far away, but it could get old going back and forth, and we don't have much to offer in the way of accommodations.\"\n\nA little idea blossomed in the back of my mind. \"I could camp out at the B&B?\" That idea was both appealing and repulsive at the same time.\n\nLisbeth reflected for a moment. \"Maybe, and that would put you right in the thick of things.\"\n\n\"Mainly I'm just looking for an inexpensive, convenient place to stay,\" I told her. \"Who should I ask?\"\n\n\"I don't know, but I can find out,\" Lisbeth said firmly. \"I'll see who's in charge of the place now\u2014I don't think Cordy was exactly hands-on about the day-to-day running of it\u2014and I'll let you know tomorrow if I can. Will you go home tonight?\"\n\n\"I think so, if I'm planning to spend more time here. I need more clothes and my computer. Thanks, Lisbeth. I don't say it often enough, but thanks for being there, and thanks for listening.\"\n\n\"That's what friends are for. Let me walk you out, and we can talk tomorrow.\"\n\n\"Sounds good to me. Goodnight, Lisbeth.\"\n\nTalking to Lisbeth had felt good. I didn't have any friends in Baltimore that I could talk to in the same way. Lisbeth had been right\u2014it made sense to be in Asheboro, at least for the next week, so I could assess the commercial possibilities and talk to people. Maybe before Cordy's death I wouldn't have considered it, but it would be a lot easier now. Which was sad in itself, because I'd have to admit I'd been driven out of my hometown for the past fifteen years by that one mean-spirited woman. And I'd let her do it.\n\nNo more. I was going to reclaim Asheboro as my own. Maybe not permanently, but I could leave my mark on it, in a good way. Maybe to offset what Cordy had done. Maybe the problem was too great to solve now, but I would give it my best effort.\n\nSo now what? Go home and pack a bag. Clear out the pitiful contents of the fridge. Take out the trash, in case I wasn't back for a few days. Make sure I took my laptop and assorted chargers\u2014and my all-important list of professional contacts (not all of which I'd confided to my computer), because I was pretty sure I was going to need them.\n\nAnd then come back to Asheboro and see if I could fix it.\n\nI was surprised that I slept really well that night. Not the sleep of emotional exhaustion, even though my life had been upended in a matter of days, but I slept with the sense that I was preparing myself for battle\u2014and was looking forward to that battle. Had I really grown so stale in a job I thought I liked?\n\nI was on the road before nine, and arrived in Asheboro by ten. I'd phoned Lisbeth before I left, to confirm that her children would be safely delivered to school by then and her husband would be out of the house. I didn't want to disrupt her established routine, but I needed to get myself set up somewhere, somehow. I couldn't exactly go to the town council and say \"help!\"\u2014that would definitely send the wrong message if I wanted to persuade them that I knew what I was doing.\n\nLisbeth greeted me at the door and led me back to the kitchen, where a fresh pot of hot coffee awaited me. \"My one vice,\" she admitted. \"I really need a cup of good coffee in the morning, so I treat myself to the primo blends.\"\n\n\"Believe, me, I understand.\" I accepted the mug she handed me, and then sat down at the kitchen table, after pushing a stack of kids' drawings and papers aside. \"You have anything new to report, Lieutenant?\"\n\n\"Ooh, I get a rank and title? Yes, sir, Captain, sir. Or ma'am, I should say. The good news is, I've cleared it for you to stay at the B&B, free of charge, for as long as you need it. The bad news is, you have to have lunch with Ryan to get the keys. He's always owned it but rented it to Cordy for, like, a dollar a year. Kind of in lieu of additional alimony.\"\n\n\"I can handle that. Or if I can't, I should leave now. Did Cordy have a staff? I can't see her scrubbing the toilets, even if it was her own place.\"\n\n\"She used to have a couple of cleaners, more or less full-time\u2014they took care of breakfast too. Then she started using local high school kids, probably to save money, and she graciously took over the breakfast part. So I'm not really sure what you'll find, but you'll be the only guest.\"\n\n\"For now. If I start beating the bushes for investors, I'll need a place to put them up, and the Barton house won't be ready for a while.\"\n\n\"Good point. You hungry? I've got muffins. Real ones, not store-bought.\"\n\n\"Sounds good\u2014I need to keep up my strength. You make them?\"\n\n\"No, we actually have a bakery in town that's pretty good.\"\n\nI took a bite of a raspberry muffin and before I finished chewing I said, \"Well, then, we'll just have to give them a reason to stick around, won't we? Yum.\" I swallowed then said, \"You have a list of the local businesses? The ones in town, I mean, not the chains or the ones outside the center.\"\n\n\"I think there's a list on the town website, but I don't know if anyone has updated it lately.\"\n\n\"It's a place to start. I'm going to put together a wish list\u2014the kinds of stores and such that I'd like to see in a Victorian village, which kind of eliminates anything high tech or even modern, and see where the existing stores fit. But at the same time I don't want to drive out anyone with a successful business\u2014we need all the help we can get. It may be a juggling act for a while. Can I have another muffin?\"\n\n\"Of course. Now I know where your weak spot is.\"\n\n\"Yup. Ply me with sweet carbs and I'm yours. Have you talked to anyone else since yesterday?\"\n\n\"Like you think I have time? No, but I've thought about the whole project. Right now I think it's important to get people here excited about the idea, before we get to the nuts and bolts of which store and what style and how much it will cost. You planning to ask the store owners to pony up at least some of the money?\"\n\n\"I haven't thought that far, and I know it could be a hardship for at least some of them. Who's running the bank these days?\"\n\n\"Arthur Fairchild\u2014you met him on Sunday. He took over from his father.\"\n\n\"Oh, right. Then I should talk to him sooner rather than later. He may not have the money in the bank to lend, or may not be willing to lend it even if he does, but he should have some ideas about where we could go looking. It never hurts to have a banker on your side, particularly a local one.\"\n\n\"Where do you want to start? I've asked Phil to pick up the kids after school so you have me for the day.\"\n\n\"Great. I'd like to be able to get into the B&B, but I guess that's got to wait until after lunch.\"\n\n\"You want me there for lunch?\"\n\n\"I have a feeling it would be easier if it was just Ryan and me. What does he do these days?\"\n\n\"Lawyer.\"\n\n\"Ooh. Local or city? Any particular specialization?\"\n\n\"Contract law. He's with a midsize firm but not in Baltimore, and that's all I know. The backstory is, he went off to college, as did Cordy, but she roped him in the year after they both graduated. Before he went off to law school. After a few years he saw the light and moved on. She went around with steam coming out of her ears for a while after that, and I think part of the settlement was the house and some seed money to get the B&B outfitted. He hasn't had much reason to come back here since they split up.\"\n\n\"He married again?\"\n\n\"Nope. Once bitten, twice shy? Unless he's been pining for you for the last decade.\"\n\n\"If he was, I don't know about it. He's never made an effort to get in touch with me. Maybe he's ashamed of what happened\u2014not that we got caught doing what a lot of kids were doing in those days, but because Cordy blew the story all out of proportion, made him look like a big stud while I was a lame loser. I assume he figured that out at some point.\"\n\n\"Let's hope so, the jerk.\"\n\n\"Ah, don't be too hard on him,\" I said. I was now old enough to be gracious. \"He was young, and the hormones were running high. And Cordy was an accomplished manipulator, even back then.\"\n\n\"Indeed she was. So back to Asheboro. We've got our one main street, which already has the library and town hall\u2014I assume you plan to keep those?\u2014and maybe fifteen or twenty shop fronts beyond that. What would a Victorian town have?\"\n\n\"There are plenty of old town directories available online these days\u2014we could pull up a few and look at what shops there were, and what their ads looked like.\"\n\n\"Ooh, great idea. What about eating places? Would there be a bar?\"\n\n\"Shoot, I don't know. Was Maryland dry in 1900? Could a store sell liquor by the bottle, or only by the glass? I remember once taking a flight that went through Kentucky, I think it was, and when I wanted a drink during the layover, I was informed that an unescorted woman could not sit at the bar but had to sit at a table.\"\n\n\"You're kidding,\" Lisbeth said.\n\n\"No, I'm not. Places are weird about liquor licenses. Too bad there's not a real hotel in town, with a sit-down restaurant, but it's too late to fix that. Maybe a tearoom? People are going to want to eat and drink somewhere, and I don't want to see soda machines lined up along the street. When did cars first appear in Asheboro?\"\n\n\"You think I know?\" Lisbeth giggled. \"I'm not even sure how to find out. Our old friend Henry Barton had the most money of anyone in town, and he lived a ways out\u2014maybe he bought one. There might be an article in the paper about that.\"\n\n\"Oh, right, that reminds me. I need to check out what the local library has in the way of resources. Josh said it's a pretty fair local history collection, all things considered. So there should be newspapers, although probably not scanned. Local histories. Maybe genealogies. There's a lot I don't know about how Henry ended up here. Who was his wife? Was she a local girl? Or did she arrive with him?\"\n\n\"Sorry, can't help you there.\"\n\n\"I didn't think you could\u2014I'm just thinking out loud. But I bet we can find out a lot of that online too. Wow, I'm beginning to wish I had a whole staff helping out. We need to find out about small-town commerce, the genealogy of Henry Barton and his wife, what the place looked like in 1900\u2014the library might be helpful there. And practical things, like, was there a doctor in town? Did he literally hang up a shingle on Main Street? Veterinarian? Pharmacy? Where was the earliest school? How many churches were there? Was there an influx of ex-soldiers after the Civil War? Even freed slaves? We're going to have to have some sort of diversity here if we hope for any public funding. I'd rather not have former slaves played by re-enactors. The children of freed slaves, maybe? I'm embarrassed about how little I know about that era and what actually happened in this town.\"\n\n\"Hey, I've lived here all my adult life, and I don't know much more. When do you want all this information?\"\n\n\"Yesterday. Well, it kind of depends on the council, and whether they're willing to push back the date of when they want my proposal. Their next meeting is less than a week away. But in any case, if I can identify potential funding sources, then we need to have a pretty professional-looking pitch assembled and ready in only a few days. One with some solid historical information, pictures\u2014whatever is available. Nobody's going to hand over six or seven figures just because we smile nicely at them and ask.\"\n\nLisbeth smiled. \"I'm tired already, just listening to you. Look, the council has been wrestling with this\u2014and I do mean wrestling, because Cordy was pulling in one direction and they were pulling in the opposite direction\u2014and now this project will either fade away, along with the town, or it will shift into a higher gear. What do you think is a reasonable time line?\"\n\n\"I don't know if I can give them a rough proposal by the end of the week, as they asked\u2014one more week would be helpful. Based on what I've seen, most of the work to restore both the Barton house and the town shops, at least outside, won't be too difficult to accomplish because not much has been changed in them. I'd tell them that we wouldn't be able to go public with even part of this for at least six months, which takes us into the fall. But there will be a catch-22 of a sort: we need the money up front to do the work, but nobody's going to give us money, or certainly not all of it, until they've seen that it's a good investment and that we can follow through.\"\n\nLisbeth looked deflated. \"And here I was just beginning to get excited about all this.\"\n\n\"Hey, I don't mean to burst your bubble. I'm just laying out the things we need to consider. And in fact, we can't promise success to the council, because we won't know if it works until we're open for business. Chicken and egg problem.\" Were there any other stale clich\u00e9s I could trot out? Cart before the horse? Aha, got one. \"But Rome wasn't built in a day, and we have to start somewhere. Do we have time to go to the library, or should we save that for after lunch?\"\n\n\"Why don't we stop in at the library and I can introduce you? Audrey closes from twelve to one like clockwork, but you aren't quite ready to start doing research, are you?\"\n\n\"No, this is just a fact-finding mission. That sounds like a good idea. And then I can trot over for lunch with my former dreamboat.\"\n\nLisbeth cocked an eyebrow. \"Dreamboat? Seriously? How old are you?\"\n\n\"You should know. I was always kind of out of the mainstream. Do I know the librarian?\"\n\n\"Audrey Perlaw. You might know her, because we overlapped in high school. But she was pretty much invisible.\"\n\n\"I can't seem to call up an image of her. Never mind\u2014maybe I'll recognize her when I see her. What do I need to know about Audrey?\"\n\n\"Raised here in Asheboro. Went to college, worked somewhere else for a while. Never married, then came back to Asheboro to take care of her aging parents\u2014really sad. Her mother had early-onset Alzheimer's, and as soon as she passed on, Audrey's father was badly injured in a car accident and then declined gradually for quite a while before passing away. But Audrey's still here, although if it had been me I'd have gone as far away as I could once my parents were gone. Anyway, Audrey knows everything there is to know about the library and its collections.\"\n\n\"You are very good at these thumbnail sketches of people, you know. Will she help?\"\n\n\"If you ask nicely and promise not to mess up her very tidy library. She must care about the town if she's stayed this long.\"\n\n\"All right, then.\" I stood up. \"Let's go pay a call on Audrey, and then I'll go to lunch, and we can regroup afterwards, if that's okay with you.\"\n\n\"Fine. I've got some errands to run. Want me to drop you off for lunch?\"\n\n\"Where am I going?\"\n\nLisbeth smiled. \"I told you, there's only one decent place to eat around here, and you've already seen it.\"\n\n\"I'll put it on my list of things to fix.\"\n\nI'd spent a lot of time in the library when I lived in Asheboro. It could be hard to snag a ride to the library because so few of my peers actually went there, preferring to hang out at one of the fast-food places on the edge of town. That suited me fine\u2014and went a long way toward explaining why I got A's in my classes (which did not make me any less unpopular). I was a walking clich\u00e9, which was probably why I fell so hard for Ryan when he began to show interest in me. He'd seemed like a nice enough guy\u2014until Cordy had threw a monkey wrench into our relationship, and it had imploded. No, that was too melodramatic. It had wilted and shriveled up and died a quiet death, except for a snigger or two from one of Cordy's blondes in a school hallway now and then.\n\nKids, especially teenagers, could be cruel, but that cut both ways. There wouldn't be bullies if people would stand up to them.\n\nAt midday on a Tuesday, half an hour before the lunchtime closing, there were maybe three people in the library, in addition to the librarian. I was beginning to remember her now from school, where she had been a couple of years ahead of me. Tallish, skinny rather than slender, hair obviously colored at home. Her clothes back then were selected to blend into the woodwork. In fact, if any of us had thought about her at all, we would have pinned the label \"Librarian\" on her long before she'd graduated. Now, of course, a librarian's job included a wider range of skills than just shelving and checking books out: there were computer responsibilities, tracking of the collections, offering the occasional course, and so on. It was the collections I was interested in, and now I remembered that the local history collection had its own room, in sight of the librarian's desk, in case anyone was thinking of swiping a 1922 phone directory. I wondered if the recent surge in interest in genealogy had been a boon, or if all that happened online now, bypassing original documents. Still, it seemed to me that small rural libraries were never going to have the money to scan all their collections, unless one particular individual in the town took an interest and found the funding to do it, so they still held unique information.\n\nLisbeth made a beeline for the desk, a smile plastered on her face. \"Audrey!\" she said, with a carefully calibrated pitch. \"I was hoping you'd be here. Do you remember Kate Hamilton? We were all at the high school together, although you were a couple of years ahead of us.\"\n\nAudrey put on her professional smile. \"Can't say that I do, Lisbeth. Welcome back\u2014Kate, was it? Mostly I kept my eyes on my feet back then and just tried to get through each day. You still live here? Moving back?\"\n\nOops. I had forgotten to put together an excuse. It was too early in the process to tip my hand about the town council's ambitious plan. \"No, I left for college and really haven't been back much. I live in Baltimore now. My parents stayed here awhile longer, but they moved to Florida a few years ago. Actually they asked me to look up a few things for them, mostly out of sentiment. They moved to Asheboro right after they got married, and I came along a few years later. So I should be looking at old newspapers, town directories, things like that, right? High school yearbooks, maybe?\"\n\n\"Yes, we've got a lot of those. Still on paper, though. For the time you're talking about, they'd probably still be in the history room, over there.\" She waved toward the back of the building. \"For really early stuff\u2014there's not a whole lot of that\u2014it's still on site here, but it's archived in the basement. Climate controlled, of course. But I'd have to know what you're looking for\u2014we don't let the public just go wandering around down there. What made your folks choose Asheboro?\"\n\n\"I'm not really sure. My dad was an engineer at a plant north of Baltimore, but I don't think they could afford city house prices, and this seemed like a pleasant community, with convenient highway access. Or maybe they had a friend or two who already lived here. I can ask them, next time we talk.\" Why had I never done that before? Why are children so clueless, assuming their parents existed in the world solely to take care of their children, with no prior history of their own? We'd had no near family here, so why had we ended up in Asheboro? That was a question for another day.\n\n\"You staying around here?\"\n\n\"I'm hoping she'll stay with me,\" Lisbeth spoke up quickly. \"We used to be best friends, but we don't get near enough time to just chat, you know?\"\n\nI noticed that Lisbeth hadn't mentioned the B&B plan. She was turning out to be a good co-conspirator. Although I wasn't sure why I was hiding anything from Audrey\u2014we needed her help.\n\n\"Yes.\" Audrey didn't elaborate. \"Kate, why don't you let me put together a list of our resources about the town, with date ranges, and then you can decide what you want me to pull out for you? You know nothing can leave the building, but I think you can work in the Local History Room without being disturbed.\"\n\n\"That sounds perfect.\" I tried to look cheery and enthusiastic. \"How long would you need to get that list together?\"\n\n\"How's tomorrow morning?\"\n\n\"Great. You open at nine?\" When Audrey nodded, I added, \"I'll be on your front step waiting. I appreciate your help, and I'm sure my parents will too.\"\n\nLisbeth pointedly looked at her watch. \"Kate, we have that appointment, remember? We'd better get going if we don't want to be late.\"\n\n\"Oh, right, of course. I'll see you in the morning then, Audrey.\"\n\nOnce we'd escaped to the library's front steps, I said. \"Wow, I didn't think this through, did I?\"\n\n\"I thought your story about the memory book was simply charming,\" Lisbeth said.\n\nI looked at her to see if she was kidding. \"It was the best I could do on short notice. But to tell the truth, I never did know why we ended up here rather than somewhere else.\"\n\n\"No family ties here?\"\n\n\"Not that I ever heard about. No cousins dropping in. The in-laws all lived a safe distance away and only showed up for alternating holidays. Maybe that was the point. Do you get along with your in-laws?\"\n\n\"I do\u2014they're great people. Of course, I may be the exception to the rule.\"\n\n\"You're lucky. But I should have planned ahead. I hope Audrey won't be angry when\u2014if?\u2014she finds out I'm poking around for the town council, although I'm not supposed to be spreading that fact around. At least the library is from the right period, so that's safe.\"\n\n\"I'm sure Audrey will understand. I'd better hustle you over to the hotel for your hot date.\"\n\n\"Give it up, will you? It's a meeting, period. Hey, will I recognize Ryan when I see him? He hasn't gained fifty pounds or gone bald, has he?\"\n\n\"No. He looks mature and distinguished, and he buys expensive suits. Business must be good.\"\n\n\"I thought you never saw him around here?\"\n\n\"Rarely. It's like sighting a white tiger in the rain forest or something like that. Maybe he was checking up on his investment. I'm pretty sure he wasn't yearning for his ex. He was AWOL a lot even when they were married. Cordy always put out the story that he was just so busy, first making partner, then being partner, that he kept a small apartment for himself for those late nights he just couldn't get home. And she, brave soul, had to take up the B&B just to keep herself busy.\"\n\n\"A B&B as a substitute for a husband? I can't say I've tried it, but I don't think that's the way it works. So what's your take? The marriage fell apart and she slunk home to Asheboro without telling anyone? It had to come out sometime? She couldn't exactly kill Ryan off in a convenient war far, far away.\"\n\n\"Alas, no, although I think she would have liked to. Then she got herself involved in local politics, and nobody had the nerve to tell her it was a bad idea and somehow she ended up voted into office, most likely because nobody else wanted to run. Luckily she missed more meetings than she attended.\"\n\n\"You aren't actually on the council, are you?\"\n\n\"No\u2014I don't really have the time. Maybe someday. But I help out when I can.\"\n\n\"Like calling me. Don't worry\u2014I won't hold it against you. Obviously the town needs some professional help, and I'm both an outsider with relevant experience and an insider with a history in Asheboro. Perfect match.\"\n\n\"I'll remind you of that later,\" Lisbeth said.\n\n\"Do I look all right?\" I asked, as we drove toward the hotel.\n\n\"You look fine. You nervous?\"\n\n\"No. Well, maybe. It'll be fine once we break the ice. You talked to him? What did he say?\"\n\n\"I told him you were here and why\u2014it didn't seem right not to mention that, since his ex-wife's death is tangled up with it\u2014and asked if you could stay at the B&B while you checked out the scene. He was fine with it.\"\n\n\"He's not staying there, is he?\"\n\n\"No, he's at the hotel tonight. I think the B&B isn't grand enough for him. But he probably thought he should stay close while the Cordy mess is sorted out.\"\n\nI wasn't sure how I felt about having him hanging around town. \"So at least I won't have to worry about running into him at the B&B.\"\n\n\"Nope, you're in the clear. You sure you don't want to stay with us?\"\n\n\"Maybe when the business side of this is over. Right now I need someplace closer to the center of town, and where I have some peace and quiet to work and get in touch with other people who might be useful. This isn't going to happen overnight. If it happens at all.\"\n\n\"That's okay, I get it. Our house can get rather chaotic. Here we are.\" She pulled into a parking space near the door of the hotel. \"Ryan will probably volunteer to take you to the B&B and let you in. But if you want a ride, just call me and I'll pick you up, okay?\"\n\n\"Will do. Thanks for setting this up, Lisbeth. I appreciate it.\"\n\n\"Let's see how you feel about it after lunch. Good luck!\"\n\nI watched her pull away and turned reluctantly toward the hotel, pulling my jacket down and smoothing my hair. Why was I nervous? I had handled multimillion-dollar budgets. I had welcomed CEOs and local politicians. Was I really worried about meeting a guy who I'd crushed on years earlier and had seldom thought of since? Well, now and then, maybe when one fledgling romance or another had crashed before it really got off the ground. But I might as well get it over with. I strode into the dining room attempting to radiate confidence, and stopped in the doorway to scan the crowd for my First Love. There he was, in the corner, at a table for two. Damn, he looked good. It wasn't fair. But, as I reminded myself, if he'd been dumb enough to fall for Cordy and actually marry her, his mental capacities must be lacking\u2014something I hadn't noticed all those years ago. I made my way through the mostly empty tables, waving off the assistance of the hostess, until I arrived in front of him.\n\nHe was busy texting on his cell phone. He didn't even look up. If it turned out he was tweeting, that would be a deal breaker.\n\n\"Ryan?\" I said, after about thirty seconds.\n\nFinally he looked up, then shoved his chair back and stood up. \"Kate! Oh, sorry, I didn't see you. Work stuff, you know.\" We wobbled through that awkward moment\u2014hug? kiss cheeks? shake hands? We settled on a half hug with plenty of air between us. \"Please, sit down. You look great!\"\n\n\"So do you,\" I said, in all honesty. He'd grown into himself. \"I hear you're a lawyer now.\"\n\n\"I am. Went to law school right after college. Look, I'm sorry we lost touch. Actually, I'm sorry about quite a few things.\"\n\nMight as well get it over with. \"Like marrying Cordelia?\"\n\nHe smiled ruefully. \"Yes, that's on the list. Are you in a hurry, or do we have time to talk?\"\n\n\"As long as you feed me lunch, I'm good. You don't mind if I stay at the bed and breakfast?\"\n\n\"Heck, no. Maybe you can hold an exorcism.\"\n\n\"That bad?\"\n\n\"It is. Was. You ready to order? Would you like wine or something?\"\n\n\"Coffee's fine.\" I wanted to keep a clear head, especially since I wasn't sure what I wanted from him, or what he might want from me. I also had to decide whether to discuss the town council's idea, then figured I might as well since his ex-wife had been part of that council, and the B&B she had been running was smack in the middle of town\u2014and that still belonged to Ryan.\n\nWe ordered lunch, and when the waitress had vanished, he said, \"So, Kate, what have you been doing for the last fifteen years?\"\n\nI gave him the short version: college, degree in business management with emphasis on the hospitality industry, the first couple of jobs, and the past five years at the Baltimore hotel. I didn't add that that job had just evaporated.\n\nWhen I wrapped it up, he said, \"You haven't been back here much, have you?\"\n\n\"Not since my parents left. I didn't have a lot of happy memories of the place. Well, except for Lisbeth. She's always been the one bright spot.\"\n\n\"Let me say now that I regret any role I might have played in keeping you away. I apologize for how I treated you our senior year\u2014or how I let Cordelia treat you. Please allow me to blame it on teenage hormones and the stupidity of youth. I hope I've outgrown both. Truce?\"\n\n\"Ryan, I gave up worrying about that years ago. Cordelia was a piece of work. How did you come to marry her?\"\n\n\"Condensed version? We both left for college, at different schools. During our last year she got back in touch with me and we started seeing each other. I don't think she was ever happy in college\u2014too big a pool for her to be noticed. When I got into law school she decided I was going places and latched on to me like\u2014what's a good analogy? A remora eel? The one that attaches itself to a shark and feeds off it.\"\n\nI had to laugh. \"Now, there's a lovely image. How long did that last?\"\n\nRyan smiled down at his still-empty plate. \"We got married after my first year of law school. I think we had a couple of good years, a couple of so-so years, and one year from hell, and then she came back to Asheboro and declared she wanted to have her own business.\"\n\n\"The bed and breakfast?\"\n\n\"That's what she decided on. I bankrolled it and bought the building for it, but she said she wanted to run it. She lost interest pretty quickly\u2014too much work, not enough admiration. I'm not sure why she hung on to the place, but I'd guess she couldn't think of anywhere else to go. And she didn't find another willing sucker in Asheboro who would take her away from all this.\"\n\nThe waitress appeared with our meals, and we fell silent as she distributed plates in front of us. Then I said, \"You knew she was on the town council?\"\n\n\"I did, although I wondered how she managed to get enough votes.\"\n\n\"I had the same thought. My guess is that there were still enough people in town who'd been bullied by her and didn't want to face that again.\" I paused for a moment. \"Did she tell you what's been going on with the town recently?\"\n\n\"No, we didn't talk. I paid the mortgage on the place, and she took care of operating expenses. No contact required. Why do you ask?\"\n\n\"This is still confidential. You're a lawyer, so you should understand that. Plus you still own a property right off Main Street. So here goes. The town is basically broke, since they bought the old Barton place outright.\"\n\n\"Really? Our Barton place?\" He added air quotes to the last question.\n\n\"That very one. It appears that Cordy had a yen to move up in the little Asheboro world. She persuaded the town to buy the place by dangling visions of a luxury hotel and high-end guests at the place. Run by her, of course.\"\n\n\"She was never short on ego. No, she hadn't told me anything about it. Still in the planning stages?\"\n\n\"The very distant planning stages. Buying the property, even at a bargain rate, took about all the money the town had. It's a familiar story\u2014small town, little or no industry, too far out to be a bedroom community, declining tax base, et cetera, et cetera. And then the storm hit, and the town realized they didn't have any money to fix anything.\"\n\n\"What's it going to take to restore the old place?\"\n\n\"That's a funny thing\u2014not much. The former owner, Henry Barton, loved the place and wanted to see it survive. When he died, over a century ago, he left an endowment to sustain the place physically while the bank was finding a buyer. Only they never did. Luckily there was enough money to maintain the property well. It really is a beautiful place.\"\n\n\"So where's the problem?\"\n\n\"I gather that Cordy got tired of waiting for the town to act, and she was shopping around for outside investors. She claimed she had some lined up, but they had a very different vision for the place\u2014kind of executive modern, rather than prime Victorian. The town was not happy with the idea.\"\n\nTo his credit, Ryan sat back and digested what I had just told him. \"And then Cordy was found dead. At the Barton house.\"\n\nI wondered if the police had shared any more details with him. \"Yes, the day after Lisbeth asked me, on behalf of the town, to find some way to preserve the house and to bring some money back to Asheboro.\"\n\n\"So you were here before she died.\"\n\n\"I was, but not at the time she was killed, it seems. Lisbeth filled me in a bit on the background with the council. I gather they dug their heels in and refused to cave to Cordy. That's why I was there\u2014to meet with the council and look at other options. We met on Sunday, but Cordy wasn't invited.\"\n\nRyan shook his head. \"I had no idea... Are you a suspect?\"\n\n\"You mean, do the police know about our, uh, relationship and how it ended and think it's a good motive for murder after all this time? I gave them the story, but more to the point, if they look carefully I can show that I was in Baltimore using my computer at the time of her death. I'm not sure whether any of the council members are under suspicion. Or if there's anyone else Cordy harassed at some point, who might have decided that this was the last straw.\"\n\n\"What a sad thing, to be so hated in your own town. But I'll be the first to admit she asked for it. I don't think she was ever happy about anything, and she took it out on other people.\"\n\n\"You are very charitable, Ryan. But you certainly knew her better than I ever did. You said you're the owner of the B&B property?\"\n\n\"Yes. Why?\"\n\n\"Buy me dessert and I'll explain.\" We ordered, and then I launched into part two of the discussion. \"I told you I saw the Barton place, and it is gorgeous. I can certainly see it as a high-end hotel, but it would be more interesting as a themed hotel\u2014a step back in time to the Victorian era, with all the best it had to offer. That's the easy part. But I don't see how that alone can sustain the town, no matter how many rich guests it brings in\u2014it's simply not large enough. I'm not sure even Cordy's corporate model would work, although I'd guess she was more interested in running the place than in making money for the town.\"\n\n\"I think that's a fair assessment. So? You have a plan of your own?\"\n\n\"I'm working on one, but please keep this under wraps. This week I walked around the town center. I haven't been back much, so I could look at it with fresh eyes. It's shabby-looking, and any changes in the shops have been for the worse. It's a pretty typical story for a lot of small towns since malls took over retail\u2014and now the malls are dying in the face of Internet competition. That's business. Then there's the recent storm damage. But in an odd way, that might be a plus. When I looked closely at Asheboro, with all the missing shingles and siding, I realized that the fabric of the place really hasn't changed since Henry Barton's day. The majority of the changes have been cosmetic and superficial. So I got to thinking, What if instead of repairing the shops in town, we stripped off those so-called improvements and converted the town center into a Victorian village? The Barton house would be the anchor, but people could stroll along Main Street and picture themselves in another century.\"\n\n\"Interesting concept. You'd try to divert some of the battlefield traffic?\"\n\n\"Exactly. And make it a destination and offer something different. Now, before you start arguing, I know it will cost money, and I know that it will take some sweet-talking to convince the shop owners to buy into the idea. But right now they haven't got much to look forward to, so I think they'd be willing at least to listen. Am I completely out in left field?\"\n\n\"Actually, no, I don't think so. But there would be a lot of work as well as some money involved. How would you plan to integrate the Barton house, which is, what, a few miles outside of town, with this bustling Victorian village?\"\n\n\"I haven't worked out many of the details yet. Horse-drawn trams? Carriages for hire? And as for Henry Barton\u2014do you have any idea what they made in that factory on the edge of town?\"\n\n\"I can't say that I do. It was closed long before our time.\"\n\n\"I want to do some research on Henry\u2014where'd he come from? Why was he here? Apparently he'd fought for the North in the Civil War and settled here after that and built his house. He married but had no children and outlived his wife. And he had lots of money\u2014from his family? Or from that mysterious factory? And, yes, I've already stopped in at the library, and Audrey is going to put together some resources for me.\"\n\n\"Did you tell her about the town's idea?\"\n\n\"No. I'm not ready for that. I'm not an employee or an elected official of the town, just an unpaid consultant so far. There's no proposal on the table yet. If anything about this idea trickles out, the town will get flack from all directions, pro and con. Conservationists will want to preserve the status quo. Developers will want more, sooner, and oh, by the way, wouldn't you like to add a strip mall, cheap? And housing for a few thousand people? Historians will no doubt have widely varying ideas of what happened here\u2014if anything. Do you know about the Barton house resident custodian\u2013slash\u2013historian?\"\n\n\"Uh, what?\" Ryan asked, looking confused.\n\n\"The Barton trust made provisions for someone to live rent-free on the property\u2014over the stable, not in the house\u2014to keep an eye on things, watch for leaks or vandals or whatever. Somehow that funding has survived to this day. The current custodian is a history professor who's researching and writing about post\u2013Civil War industrialization.\"\n\n\"Did he know Cordy?\" Ryan asked.\n\n\"I gather they crossed paths a time or two. They did not see eye to eye.\"\n\n\"Does that make him a suspect?\"\n\n\"Heck, I don't know. It's not like he knew her well. Would he have killed her because he disagreed with her taste in interior design?\"\n\n\"Seems unlikely. Look, is there anything I can do to help?\"\n\n\"Nice of you to offer. What kind of law do you practice?\"\n\n\"Mostly corporate litigation. No violent crime.\"\n\n\"Anything for nonprofits?\"\n\n\"Now and then. Why?\"\n\n\"If this goes forward, we might want to set up a nonprofit corporation in parallel with the town. We may want to solicit support from philanthropic donors, who would want to see us have nonprofit status.\"\n\n\"Lady, you don't mess around. You came up with this idea what, three days ago? And since then there's been a murder that might or might not be connected, and you're already talking about creating a nonprofit corporation. How much have you told the town council?\"\n\n\"Just a broad concept, so far. I haven't seen them since Cordy died. But I guess I'm getting excited about the idea. It's bigger than anything I've ever tried.\"\n\n\"Well, I'm happy to help.\"\n\nWe finished dessert quickly, and Ryan asked, \"You ready to go check out the B&B?\"\n\n\"Sure. You driving?\"\n\n\"Yes, of course. I haven't been inside the place myself for a while, and I want to make sure everything is all right in case I decide to sell. Or maybe your hypothetical corporation can buy it to use as a headquarters.\"\n\n\"Or maybe you and your rich buddies can gift it to the hypothetical corporation,\" I shot back with a grin. \"Nice tax deduction, right?\"\n\n\"I'll take it under advisement.\"\n\nI'll admit I was curious to see what Cordy's bed-and-breakfast establishment looked like. I'd driven past it, and from the outside it appeared to be a modest late-Victorian building painted in San Francisco Painted Lady colors\u2014not quite typical of this part of Maryland, but pretty in its own way\u2014and in better repair than most of the central Asheboro buildings. There was a tasteful sign planted in the midst of the small lawn in front. Ryan pulled up and parked.\n\n\"This is it,\" he said.\n\n\"Good location. How much parking?\"\n\n\"Five spaces directly behind the house. Otherwise street parking.\"\n\n\"I'm surprised Cordy didn't try to wangle the council to provide more street spaces for her.\"\n\n\"You think she didn't try?\" He grinned.\n\nI was not exactly surprised. \"How long did she have this place?\"\n\n\"About five years.\"\n\n\"And how many of those years did she make a profit?\"\n\n\"No comment.\"\n\n\"Great investment, Ryan.\"\n\n\"If it kept Cordy out of my face, it was worth every penny. Ready to go in?\"\n\n\"Sure. Did she have any bookings that needed to be cancelled?\"\n\n\"Nope. I think once the novelty had worn off, she got bored. She was kind of chained to the place. She couldn't go out to lunch with the girls as much as she would have liked and stir up trouble. She didn't put much effort into it over the last year or two, and the bookings fell off.\"\n\n\"How sad,\" I said. \"She never really changed, did she?\"\n\n\"I'd hoped...\" Ryan stopped.\n\n\"That marrying you would make things better?\" Or change her?\n\n\"I guess. I did love her, you know. At least at the beginning.\"\n\n\"Don't blame yourself, Ryan. She was who she was, and I don't think anybody could have changed that. Should we go in?\"\n\n\"Sure. Oh, before I forget, here's a set of keys for you.\" He handed me a key ring with several keys on it.\n\n\"Which does what?\" I asked. \"Is there an alarm system?\"\n\n\"Exterior alarm, but that operates from keypads. I'll give you the code number. You disarm the system then you use the key to open the door. The keys are for the doors\u2014front and back\u2014and your room. I've put you in the best room.\"\n\n\"Gee, thanks. Has the place been cleaned lately?\"\n\n\"Only Cordy would know. I can tell you that she hated housework.\"\n\nI wasn't surprised. The manicure took top priority. \"Can I use the kitchen while I'm here?\"\n\n\"Sure. Even if you burn the place down, the place is insured for guests, and you're a guest. How long do you think you'll be needing a room?\"\n\n\"Somewhere between one and a few hundred nights. Seriously, right now I'm operating day to day. If I find some positive things, I'll stick around for a few days. If it looks hopeless, I'll tell the council and then go home. Is that a problem? Or are you planning to sell it ASAP?\"\n\n\"No to that last question. I wouldn't have pulled the rug out from Cordy, unless she was really trashing the place. Truthfully I didn't think she'd stick it out much longer. But I'm nowhere near to putting in on the market. I may wait to see if your grand scheme works out.\"\n\n\"I wouldn't hold your breath, but we'll see. Did she leave a will?\"\n\n\"Yes\u2014we had them drawn up years ago, and we modified them when we split. As you know, the building is in my name. Cordy had very little to leave.\"\n\nI'd always thought her parents were reasonably well off\u2014but maybe that was what she wanted us all to think. \"What are the burial plans?\"\n\n\"Cordy's parents were buried here, just outside of town, and there's room for her cremains in that plot. I don't suppose anybody will object.\"\n\nI was reminded that I didn't know where Henry Barton was buried\u2014I should check. If it was local, that could be woven into his story. If it was somewhere else? Well, that might give me a clue to his origins. Surely there would have been an obituary in the local paper\u2014I could check in the morning.\n\nRyan disabled the front door alarm as I watched, and let me in using his own key. The interior was cool and dark. From a brief glance, it was like a Barton house in miniature\u2014central hallway and stairs, doors leading off to rooms on both sides. \"How many bedrooms?\"\n\n\"Four,\" Ryan said. \"Two share a bath, and the others are en suite.\"\n\n\"Kitchen at the back?\" When Ryan nodded, I went on. \"Neighbors?\"\n\n\"It's a mixed neighborhood, a combination of residential and home office\u2014a dentist, an accountant. That kind of thing. Nothing obviously commercial.\"\n\n\"Can I use whatever food I find?\"\n\n\"Sure. And I'd bet Cordy laid in some good wine, but I don't know where she kept it.\"\n\n\"Are the police finished with this place?\"\n\n\"They never even got started. Cordy was killed where she was found, they told me. End of story. There was no sign of anything disturbed here, so no reason to search the place.\"\n\nThat sounded surprisingly sloppy for Detective Reynolds. \"You went through the building?\"\n\n\"Only briefly. As I said, the police didn't seem interested, and I wasn't either.\"\n\n\"Any documents or souvenirs you want to take with you?\"\n\n\"I already have copies of the legal documents, like the lease, and I took along the preliminary proposals from her mystery corporate partners. I gave copies to the police. As for jewelry or mementoes... well, I don't particularly want anything to remember her by. If you stumble on something valuable, let me know, but as far as I'm concerned it's finders-keepers.\"\n\n\"Noted. Well, that's all I need for now. You're staying at the hotel?\"\n\n\"Yes, until the final arrangements are made. I could commute, I supposed, but it would be wasting time. Dinner?\" he asked.\n\nI didn't waste a lot of time thinking about it. \"Not tonight. I want to get my bearings, and I have a lot to think about. Maybe later in the week if I decide to stick around. Give me your cell number so I can reach you.\"\n\nHe looked a bit crestfallen, but he fished out a business card and scribbled his cell phone number on the back, along with the digits of the door code. \"I'll let you know if there'll be a service.\"\n\n\"I'd appreciate that.\" Much as I'd disliked Cordy, at least a memorial service would give me some final closure, and I was curious to see who else would show up. Still, Ryan's hesitation suggested that maybe there wouldn't be a service\u2014out of fear that no one would attend? \"I'll let you know if I make any decisions. Thanks for lunch, Ryan. It's been good to see you.\"\n\nHe wavered a moment, then went back down the hall and out the front. I wondered if he'd cherished some hope of rekindling what we'd had, but even if he did, now was not the time. My car was still parked at Lisbeth's house, so I needed to call her first.\n\nShe answered quickly, as if she'd been waiting by the phone. \"Hey, Katie. Where are you?\"\n\n\"At the bed and breakfast. Ryan dropped me off, and he just left. Listen, if you can come get me, we can take a look around this place, and then you can take me back so I can collect my car.\"\n\n\"Ooh, I get to snoop? Cordy was living there, right?\"\n\n\"Uh, I gather. Why, you want to paw through her underwear?\"\n\n\"Ew, no\u2014that's creepy. But I'll admit I'm curious. Listen, why don't you have dinner with us, and then you can drive back?\"\n\n\"Okay, sounds good. You haven't heard anything from the council, have you?\"\n\n\"Nope. I think they're waiting in deference to the late Cordelia. Or they're afraid to hear what you might tell them, like there's no hope.\"\n\n\"Don't give up yet. See you in a few.\"\n\nAfter we'd hung up, I wandered back to the kitchen. It wasn't intended to be a public space, but it was passably neat. I opened the fridge: as I expected, it was mostly breakfast fixings\u2014milk, butter, eggs, bread, and the like. The freezer held several bags of frozen muffins. So much for home-baked, not that I'd expected it from Cordy. I poked around in the cupboards, where I found some attractive mid-price matching china and cups, as well as glasses. Including wineglasses. I wondered where Cordy would have hidden the good stuff.\n\nI strolled out through the dining room and across the hall. The main parlor had a handsome fireplace, apparently functional, flanked by bookshelves holding real books, mainly mysteries and romances. Things that patrons had left? Or did Cordy have a taste for light fiction? I didn't care. If I had a book, a comfortable chair, a fireplace, and a glass of good wine, I'd be a happy camper for as long as I stayed.\n\nI wandered up the stairs. Ryan said he'd tidied up the \"best\" room, which I assumed was in the front, over the parlor. It was. The room was plain but clean, although the bed had been stripped and not remade. I wondered where the linen closet might be hiding, but I would look later. The adjoining bath had its original fixtures\u2014and clean towels. Back in the bedroom I tested the lights and made a mental note to get some light bulbs that were brighter than 25 watts. I peeked in the closet: empty, except for some bent hangers, an extra pillow and a couple of dust bunnies in the corners. I sensed that Cordy's heart really hadn't been into making this a welcoming place. What had she been thinking? Had she really believed she was a \"people person\" and could charm guests? That must have gotten old fast, but where else could she go?\n\nI heard a knocking downstairs and I went to open the front door for Lisbeth. \"Come on in,\" I said. \"Have you ever been inside before?\"\n\n\"I think Cordy once held a party for the council members here\u2014grudgingly. That's about it. Find anything good?\"\n\n\"I've only been here ten minutes, but the whole place feels generic, if you know what I mean. Kind of the bare minimum standard. Ryan hinted there might be a secret stash of wine, if you want to go looking for it.\"\n\n\"A treasure hunt! Have you checked out Cordy's room?\"\n\n\"No, I only got as far as the room where Ryan said I should sleep, and the bathroom. So she lived here too?\"\n\n\"I gather she didn't have a choice, financially.\"\n\n\"Ryan said the police weren't interested in this place, and he's taken all that he wants. Shall we go explore?\"\n\n\"Let's!\"\n\nCordy's room proved to be a small one at the back of the house. Unlike the rest of the place, this one was kind of messy, except for the clothes, which were hung neatly in the closet, on a standing rack\u2014and in any other possible place. I spied some designer labels. Lisbeth and I surveyed the room. I knew that I felt uncomfortable, prowling around the personal possessions of someone we knew, who had been murdered, even if we did have all the permissions we could want.\n\n\"This is creepy,\" I half-whispered.\n\n\"I know what you mean,\" Lisbeth replied, in a similar tone. More loudly she said, \"Are we looking for anything in particular?\"\n\n\"I don't really know. Ryan said he found the preliminary documents for that corporate deal she was hinting about.\"\n\n\"So it was real? I figured maybe she was making that up just to pressure the council.\"\n\n\"Apparently there was something in the works. And it makes sense, from her perspective. This place doesn't quite fit with her own self-image. Anyway, what kind of item would spur anyone to kill her? And where would she leave it?\"\n\n\"Nude photos?\" Lisbeth suggested. \"Maybe she was blackmailing someone. Or several someones.\"\n\nI had to smile at the idea. \"On one hand, I can see her enjoying that kind of power over other people. But on the other hand, I can't see her sleeping around for money. She thought she was better than that.\"\n\n\"Hmm. Maybe she was a drug addict and her dealer killed her?\"\n\nLisbeth's day-to-day life might be dull, but it appeared that she had a lively imagination. \"Easy enough to check the bathroom and drawers for her\u2014what do they call it? a kit?\u2014but I don't think so. She would want to keep her wits about her.\"\n\n\"Listen to you\u2014you've been reading too many old mysteries lately. But I get your drift. Your turn. Come up with an idea.\"\n\n\"Maybe she was harboring a secret passion for someone who wasn't interested, and she wouldn't let go and her lover had to kill her?\" I suggested.\n\n\"Possible. But I can't see Cordy being that needy. And she'd be more likely to be a blackmailer. You know, seduce some poor guy and threaten to tell his wife unless he pays up. She probably needed the money. But the guy\u2014or maybe one of many\u2014could have snapped and killed her?\"\n\n\"Let's keep that on the list. What if she was blackmailing someone and his wife found out and whacked her? If she was one of Cordy's earlier victims, that would be a twofer.\"\n\n\"Doesn't shorten the list much,\" Lisbeth commented. \"Cordy bullied half the girls in Asheboro. So we'd need to identify one of the many, who's married to a guy who Cordy was able to seduce.\"\n\nI sighed. \"I suppose it's possible. But we're getting ahead of ourselves, you know. We'd have to find the incriminating pictures first. I wonder how she took them. I doubt she'd have an accomplice with a camera.\"\n\n\"True. You see a computer here?\"\n\n\"No. But maybe Ryan took it with him\u2014I can ask him. It might have belonged to the business. There's not much else of value here, but if she was blackmailing someone, he'd want to find the evidence.\"\n\n\"If it exists,\" Lisbeth pointed out. \"This is just one theory. Got any more?\"\n\n\"She seduced one of the corporate partners she was wooing?\"\n\n\"Fits her self-image better. You could ask Ryan who was listed on those documents.\"\n\nI checked the time: three o'clock. \"You said your husband was dealing with the kids?\"\n\n\"He's picking them up from their afterschool sports things in a bit. Why?\"\n\n\"Well, if we can find the liquor cabinet, we have time for one respectable glass of something before we head for your house, right?\"\n\n\"One glass only\u2014that's legal, right?\"\n\n\"So I've heard, although I haven't tested it. Besides, you must be friends with the police department.\"\n\n\"Yes, but I'd like to keep it that way. Let's look anyway.\"\n\nWe found Cordy's secret stash of wines, and they were indeed fine. The bottles had been carefully concealed behind a wooden wall panel in the parlor. Had Cordy found it ready and waiting when she moved in, or had she created it for her own use? Didn't really matter now. The wines, nestled in racks that were obviously more recent than the house itself, bore labels that even I recognized.\n\n\"Glasses in the kitchen,\" I told Lisbeth. \"I'll get a couple. And a corkscrew.\"\n\n\"I'll while away my time in the front parlor,\" Lisbeth said, then giggled.\n\nI found two glasses appropriate to the vintage we had selected and returned quickly, bearing a corkscrew. \"Sorry, no snacks to be had. Ryan said there hadn't been many guests of late.\" I proceeded to demonstrate my expertise in opening good wine, thanks to plenty of practice at the hotel, and the cork slid out gracefully.\n\n\"Shall we test it?\" I asked.\n\n\"You just want to show off. Go right ahead\u2014I might need to know how, once the kids graduate from college.\"\n\nI did the expected sniff, taste, make silly slurping noises, and pronounced the wine fit to drink. \"Shall I pour?\"\n\n\"Please,\" Lisbeth replied.\n\nWe settled ourselves in the two richly upholstered chairs that flanked the unlit fireplace and sipped appreciatively. \"This is really nice,\" Lisbeth said.\n\n\"You're absolutely right. This is a treat.\"\n\n\"So,\" Lisbeth said, swirling her glass before taking another sip, \"what was it like seeing Ryan after all these years?\"\n\nI searched for an honest answer. \"Not exactly memorable. I was surprised he hadn't remarried, after Cordy, but we didn't talk about a lot of personal stuff, either at the restaurant or here, and I didn't want to ask. I gather he saw through Cordy fairly quickly once they were married, but I think he hoped she would change. We know how well that worked out. I'm pretty sure he was the one who left, which must have really pissed off Cordy. This place was Cordy's compensation, at least in part. Ryan didn't exactly look back, after he left.\"\n\n\"So no flickering embers of lost romance?\"\n\n\"Sorry to disappoint you, but no. He's a good-looking, successful man, and he's single, but there weren't any sparks. And this is just a guess, but I don't think he felt strongly enough about Cordy now to do her in. He was glad to wash his hands of her.\"\n\n\"Okay, just checking. Are you sorry?\" Lisbeth asked, her gaze on her wineglass.\n\nI took another sip of my own wine, relishing the way it glided down my throat. \"No. If it didn't work out then, there's no reason why it would now. He might be a good resource as a lawyer, though. I told him about what the council was talking about and why I was here. And I told him it wasn't public knowledge yet, so he won't spread it around.\"\n\n\"Good idea, on both counts. I didn't mean to be nosy\u2014my daily life is a bit lacking in grand romance.\"\n\n\"And you love it just the way it is. That's fine. I love my job, or did, at least. There's no man on the horizon, but I'm open to the idea, when it seems right. But I'm not hitting the bars or checking out online dating sites.\"\n\n\"Hey, you don't have to explain yourself to me. I'm glad you're here, and I trust your judgment about what can be done to salvage Asheboro. If anything. I am not playing matchmaker, not that there are a lot of options around here.\"\n\n\"Enough said,\" I declared. \"So, what else can we hope to learn from Cordy's home here? Oh, by the way, I asked Ryan to let me know if there would be a memorial service\u2014it might be interesting to see who shows up.\"\n\n\"Good point,\" Lisbeth said. \"As for this place? I don't see that she spent a whole lot of money or energy on it. In a way I feel sorry for her. She left town with high hopes, then came back with her tail between her legs. She married the hometown boy\u2014handsome, a lawyer\u2014and that turned out to be a disappointment, although she probably wouldn't ever admit that might have been her fault. She decided she wanted to run a business of her own and cajoled Ryan into buying her this place\u2014\"\n\n\"Which he still owns, by the way\u2014\"\n\n\"Ah. Maybe she could have made a go of it, but I'd guess she got bored fairly quickly, and she didn't like catering to other people. Then she decided to dabble in local politics, and promptly made a bunch of enemies around town. She tried to do an end-run around the council about the Barton house, and that didn't sit well with them.\"\n\nI interrupted. \"We're assuming that she wanted to run the place herself, which no doubt she would see as a step up.\"\n\nLisbeth nodded then went on. \"And then someone bashed her over the head with a rock and killed her. End of story.\"\n\n\"Except there's still a killer out there,\" I reminded her.\n\n\"And a boatload of suspects.\" Lisbeth drained her glass. \"I'm driving, so I guess I'd better stop now.\"\n\n\"Don't worry, you can take the bottle home with you and enjoy it after the kids are in bed. There's more where that came from.\"\n\n\"Lucky you! Anything else we need to cover, before we jump into the mob scene at my house? You have a game plan?\"\n\n\"Not really. We still don't know enough about the history of Asheboro, which is why I'm headed to the library tomorrow morning. Even if the records are in apple-pie order, there may not be anything in them on which to build a compelling story that will bring in plenty of tourists. I mean, there was never an important battle here. It was never a major center for anything. The train line that came through the region in eighteen-hundred-whatever skipped it entirely. The best we could hope for is the world's largest rutabaga, since I think the world's largest ball of string is taken.\"\n\nLisbeth nodded her agreement. \"So what we've got here is a pleasant little town in the middle of nowhere, that's dying a natural death.\"\n\n\"I'll do my best to come up with something, Lisbeth, but I can't make any promises.\"\n\n\"I understand, and I appreciate your coming here and trying. So, ready for dinner? Not gourmet\u2014I thawed a casserole this morning. I think the main ingredient is last Thanksgiving's turkey.\"\n\n\"Let's do this. You take the bottle,\" I said, shoving the cork back into the top. \"I've got to figure out how the door works, and that will take both hands. It's got keys and a keypad, but I haven't tried it yet.\"\n\nI left a light burning in the back parlor and turned on the porch light, and then I managed to get the door opened and closed without setting off any alarms. Once Lisbeth and I were pointed in the right direction, she said softly, \"Will you be okay all alone in the house?\"\n\n\"What, you think Cordy's going to come back and haunt me? Taunt me is more likely. Can't you hear her? 'What makes you think you can save this town, huh? You ran away as fast as your flat feet could carry you. You believe you can charm donors and charitable organizations? Heck, I stole your boyfriend out from under your nose and then I married him. Let me tell you, he was no bargain, but you couldn't even hold on to him. Loser!'\"\n\n\"Oh, Katie, is that really the way you feel?\"\n\n\"No, not anymore. Once, maybe. But I've got the perfect answer now for her: 'I'm alive and you're not.'\"\n\n\"Good one. Although it's barely possible that she killed herself, you know.\"\n\n\"By bashing herself over the head with a large rock? That I'd like to see.\"\n\nOur laughter and chatter took us to Lisbeth's house, and once inside I had to greet the kids\u2014remarkably well mannered, I thought\u2014and make small talk with Lisbeth's very pleasant husband and eat the substantial casserole. I turned down dessert. \"I'd really better get going before I fall asleep. I'll be at the library in the morning, and I'll let you know if I find anything good. By the way, should I tell Audrey what's going on? I'd hate it if she found out from someone else and was annoyed at me\u2014I need her help.\"\n\n\"What the heck, go ahead and tell her. It'll make her feel important, which she is to the town. You can blame the council for making you keep quiet about it.\"\n\nLisbeth walked me to the door and hugged me impulsively. \"I know we haven't solved a darned thing, but it's good to have you here. Happy hunting tomorrow.\"\n\nI drove back to the B&B carefully, even though I'd laid off the wine. I wanted to see the town after dark. Lisbeth's house lay in a near suburb, and the houses grew larger, as did their lots, as I approached the town center. Which didn't take long. It had never been a large town, or even a medium-size one. Just another ordinary American town. You could have dropped it in any number of different states and it would have been just the same.\n\nI pulled into the otherwise empty driveway and parked. I retrieved the keys from my bag and approached the door. For all my bravado, I found myself thinking of too many TV crime shows: this was the moment when some guy in a ski mask would leap out of the bushes and try to kill me. Since I was the heroine of this dark fantasy, I would of course overpower him, and after tying him up with the garden hose, I would have discovered in his pocket the single clue that would explain everything. Then I would call the local police and triumphantly explain to them that I had solved the case for them. Laughing, I disarmed the alarm, unlocked the door, and let myself in\u2014but I made sure to lock the door and reset the alarm, and I left the porch light on.\n\nInside I wandered aimlessly from room to room, looking for... what? A clue? Cordy hadn't died here. She'd barely lived here, and she'd left very little imprint of her own personality. What would a bullying, manipulative personality look like if it was a house? I amused myself trying to imagine it as I retrieved a fresh bottle of wine from the secret closet and took it to the kitchen to open. Every item in relentless order, and afraid to even twitch lest it be thrown out for insubordination? An interior that was uniformly bland, the better to showcase its colorful owner? Or an interior that was riotously colorful, as if the owner dared the surroundings to outshine her?\n\nRidiculous, I knew. I took my glass of wine and went back to the parlor and sat. Maybe I should try to conjure up Cordy's spirit, and then she would tell me who had killed her and we'd all live happily ever after. No, Cordy wasn't the type to share anything. She'd want to figure out a way to tell the police herself so she could get the credit. Except she was dead, which would make that difficult. Wherever she was now, I bet she'd hate it if I solved this mess.\n\nAs I sat in the near darkness, my mind drifted aimlessly from topic to topic. Henry Barton was the key to it all, I was convinced. What had drawn him back to Asheboro? The people? The views? The natural resources? (Were there any? I'd have to check.) The fact that it was some place that was not where he came from? Had he been fleeing from something, or looking for something new? A clean slate? And where the heck had all his money come from?\n\nI finished my single glass of wine and stood up. I still had to find sheets and make the bed, and figure out how the plumbing worked (please let there be hot water!), and make some sort of list for what to look for at the library in the morning. All in all, a busy day today, and more to come. Good thing I didn't have that pesky day job anymore, I thought to myself as I trudged up the stairs with my small suitcase.\n\nNobody murdered me while I slept. Nor did Cordy haunt my dreams, even though I was under what had been her roof. A decent start to the day, I figured.\n\nI hauled myself out of bed, trying to smooth the mismatched linens I'd found the night before. I took a quick shower and went down the stairs to hunt for breakfast. If there weren't provisions for making coffee somewhere in the kitchen, I'd have to kill Cordy all over again.\n\nThere was coffee and a large French press. I brewed myself a pot, nuked a frozen muffin, and sat myself down at the kitchen table to make a list. Should I start with Henry or with the town itself? Henry was no doubt the catalyst, but it would be more difficult to piece together a picture of what the town had looked like in 1900, and that was what I should be looking for at the library.\n\nAmericans have a kind of fixed image of small-town Main Street. It signals home, a simpler, gentler time, when more people walked places, and all the stores were in a central location and easy to reach on foot. What would be essential in a small town? Food, of course, but food wouldn't be available in a huge supermarket, with one-stop shopping, More likely food vendors would be scattered along the street, with a butcher shop\u2014did they also handle chickens and eggs?\u2014and a grocery store that sold canned goods, cleaning products, and the like. Then there were clothes. Many women would still have made their own, but they had to get the fabric and notions somewhere, so that was another store. Or they could have had a dressmaker in town. Things that could not be easily made, like hats, might be sold at a dedicated store or a so-called general store, along with gloves and shoes and\u2014gasp\u2014unmentionables. No handy mail order in those days, although I should check when the Sears Catalog first became popular. A supply and feed store, although that might have been on the fringes of town\u2014where farmers would buy tools and feed and harnesses. A doctor's office, although maybe Asheboro hadn't been large enough to warrant a full-time doctor. A pharmacy, of course.\n\nWas there a social hierarchy for the town? Was one end of the town classier than the other? What about side streets? And where had the factory fit into the whole picture? Somebody must have a history of that\u2014what I had taken to calling the Barton Widget Works, for want of a better title. I really needed to find the real name, and what it produced. It was a big building, so it must have employed quite a few people in its day.\n\nI made another heading for Henry Barton and his house. Henry had arrived, fresh from the war, with a new wife (or acquired said wife shortly afterward), and settled in Asheboro. He set up some kind of factory in the tumultuous postwar years, and by 1885 or 1890 he had a handsome factory in town producing lots of widgets and earning him buckets of money, and he could afford to fancy up his house and furnish it at the peak of current style. A decade or two later he had died and been forgotten. Poor Henry. I needed censuses, directories, local histories\u2014oops, look at the time!\n\nI needed to get my town clothes on and get my butt to the library or I'd be late, and I didn't want to insult Audrey Perlaw on my first day there.\n\nI made it to the library with five minutes to spare. Audrey was waiting inside and unlocked the door for me, then locked it behind me. We stared at each other for a long minute, and she didn't look happy. Finally she said, \"I remember you now.\" And stopped.\n\nAnd I immediately felt guilty about my mild deception. \"Look, there's something I've got to tell you. Can we sit somewhere?\"\n\n\"This way.\" She turned on her heel and marched toward what must be her office, a cubicle behind the checkout desk. She sat behind the desk and gestured toward a chair opposite. \"Why are you really here?\" she asked.\n\n\"I'm sorry I, uh, misled you about why I was here. If you remember me, you know my parents did in fact live here, and they moved to Florida a few years ago. I'm here because the town council invited me to talk with them.\"\n\n\"That Barton place, right?\"\n\nI should have realized that she would know what was going on, even if it wasn't public knowledge. \"Yes, at least in part. You live here, and you must know a lot of people, including the council members, or maybe their kids. So you must have a pretty good idea of the financial state of the town.\"\n\n\"In a word, lousy. So?\" She wasn't giving an inch, and I'd have to fight for her trust.\n\n\"I was told that the town bought the Barton property with its last dime, more or less. With voter approval, as you must know, although I'm pretty sure most of the voters here haven't pored over the town budget. The council had high hopes that it could be turned into a moneymaker of some sort, a high-end hotel or maybe a conference center.\"\n\n\"Why'd they think of you?\"\n\n\"In no particular order, I'm from here, I have a degree in the hospitality industry, and I've been managing a boutique hotel in Baltimore for the past five-plus years.\" Had she thawed a degree or two? I wasn't sure. I plunged ahead. \"Lisbeth Scott mentioned this to the council, so they called me in as a kind of hired gun, or a fixer. To see if their idea made sense, and if it did, did the numbers work. The first I heard of any of this was last week, so I told them I needed to do some research, get a feel for the place, particularly the Barton house, before I could begin to give them an opinion.\"\n\n\"Fair enough,\" Audrey said. \"And?\"\n\n\"I went home for the night, and by the next day Cordelia was dead.\"\n\n\"You have any idea why she died?\" Audrey's expression remained carefully neutral.\n\n\"You mean, who killed her?\"\n\n\"That too.\"\n\nI considered how to phrase my response to avoid looking like a whiny teenager\u2014again. \"In high school\u2014when I still lived here\u2014she made a lot of people's lives unpleasant, including mine. If she continued that sort of behavior as an adult, I'd guess she had a lot of enemies here by now. I can't begin to guess how many were angry enough to kill her. I'd like to believe that her plans for the Barton house were the last straw for someone\u2014they didn't want Cordy to trash the place and then get a job out of it. One that she probably believed would let her look down her nose at the rest of the town.\"\n\nAudrey nodded once. \"So you think it was someone who loves this town and its history and who hated Cordelia.\"\n\n\"Something like that. Do you disagree?\"\n\n\"Not necessarily. Although there might have been better ways to remove her from the project than killing her. Have you met Josh Wainwright yet?\"\n\n\"I have. He must know the house well by now.\"\n\n\"He will have told you he's an academic historian but his main interest is not period architecture. He's interested in Henry Barton's role in the industrialization of the South.\"\n\n\"He mentioned that.\" Which was one thing I really needed to explore, and for that I needed Audrey's help. \"Has he used your resources here at the library?\"\n\n\"He has. He's told me that we have a stronger collection in some ways than even Johns Hopkins,\" she said proudly. \"I'm not sure what kind of research you may have done in the past. Many people labor under the misconception that everything has been digitized and uploaded for computer access. But for small towns and local history, the library has custody of most of the documents. Or maybe a historical society, if there is one\u2014which there isn't here. Where do your interests lie? Whether the building can be converted to a hotel, and how many rooms and baths you can squeeze in?\"\n\nShe certainly was slow to warm up. \"Audrey, I hope I'm not as crass as that. And there are plenty of people the council could hire who could move walls around and update the plumbing and make it perfectly respectable. The problem is, people would have no reason to come here, or at least not in large numbers, just because there's a quaint hotel here. It's not enough of a draw. And I would have told the council that. But after I had walked around the town a bit, I started thinking about what I saw, and I think I've come up with a different approach with a bigger scope.\"\n\nWas that a flicker of real interest I saw in her eyes? \"And that would be?\"\n\nHere goes nothing. \"What if we make the Barton estate the centerpiece of a larger ensemble? From what I've seen on my walks, the changes made to the town since 1900 are largely superficial. We could make it look authentically Victorian quickly and without spending a lot of money. But what I would envision would be dignified and historically correct, not tacky and commercial. Think of places like Old Sturbridge Village or Plimoth Plantation in Massachusetts, Williamsburg in Virginia.\"\n\nAudrey nodded again, unsmiling. \"All of those required a lot of time and money to create that deceptively simple appearance.\"\n\n\"I know, but I was only suggesting them as models, and all of those are substantially larger. And those sites had to adhere to specific historic events. If we could take Asheboro back to an ordinary town, serving ordinary people, we wouldn't face some of those constraints. It would be informative but not oppressively so. Yes, we'll need ordinary modern businesses for local residents, but they're mostly concentrated outside the town center. And we'd need to get the people who have shops and businesses in the heart of town to participate as a group\u2014it has to be consistent throughout. What do you think?\"\n\nAudrey didn't answer immediately but gave it some thought. I kept my mouth shut, waiting. Her first comment surprised me. \"I notice you said 'we' throughout. Do you see yourself as an ongoing party in this\u2014well, I suppose one could call it a reverse-development? Or would you drop a glossy plan on the council members' laps and wave farewell?\"\n\nA surprisingly good question. \"To be honest, I haven't thought that far ahead. There are no guarantees that I can convince the council that it's a good idea, and I certainly haven't had time to draw up plans and cost out the improvements and talk to all the people who would be directly affected. I'm just throwing the idea out there.\"\n\n\"You didn't exactly answer my question. It's your vision. Would you stay on and be part of it?\"\n\n\"I... think I might.\" Wow, where had that come from? But I realized that I was more excited about the pie-in-the-sky plans I had thrown together in my mind than I had been about much of anything for a long time. It was a challenge. It could be fun. Hard work, of course, but I was confident I could do it. Not alone, of course\u2014there would be a need for people with a wide range of expertise, in fundraising, finance, history, authentic restoration, and more. And they'd have to agree to work for a pittance, or maybe pro bono.\n\nAudrey was still watching me, but I thought I detected a faint smile. \"If you think I'm crazy,\" I said, \"now's the time to tell me, before I get in too deep. You know this town: could it work?\"\n\n\"I don't really know, Kate. But we need something here to jump-start us. We've lost too many good people, who had to leave for purely practical reasons like a job. New people are not replacing them, and why should they? We don't have a lot to offer. Trying to return Asheboro to its Victorian state wouldn't necessarily be easy, but it could make a significant difference. So, if this goes forward, I'd like to help.\"\n\nI felt a surge of triumph: I'd won her over. \"Thank you! You're the one who's sitting on all the history here, and I'd need you. But we're still a long way from that. I have to sell the council on the idea, if I think it's a viable project, and for that I need at least a rough estimate of costs and a mock-up of how it could look. Things like that.\"\n\n\"When would you want this to happen?\"\n\n\"I think we could open the Barton house by September. The rest, maybe by next summer.\" There, I'd said it. It sounded insane. \"Kind of a phased opening. But I want to catch the fall season this year. Sort of a taste of things to come.\"\n\nAudrey almost laughed. \"Maybe you're crazier than I thought.\"\n\n\"Will you still help?\"\n\n\"With the planning phase, at least. After that, let's wait and see. And the town won't blow away before next spring.\"\n\n\"Fair enough. So, did you find anything helpful for me to look at, now that you've heard the whole story?\"\n\n\"Well, I might need to rethink a few things, but I've set out sort of a buffet for you\u2014old town directories, a few copies of the newspaper, some local histories actually written by local people. Will those help?\"\n\n\"Definitely. I need to familiarize myself with what life was like around 1900. Would you have any genealogy stuff? Family histories that never made it to print? Anything on Henry Barton? I suppose it's too much to hope for that he left a detailed journal.\"\n\n\"It certainly is. By all accounts he was a good man, but not particularly literate.\"\n\n\"You've been in the house?\"\n\n\"Yes, I was given a tour some time ago. I was concerned that if there were still books on the shelves, that they would be vulnerable to damage by mold or damp. I was pleased to see they were surviving well. The caretakers hired over the years have done an excellent job of stabilizing the place, which was their mandate.\"\n\n\"Do you know what Henry made in the factory?\"\n\n\"I can't say that I do. I was always a bit timid when I was younger, but I knew people who sneaked in to see if there was anything in there, and I won't swear that their intentions were entirely honest. But the valuables in the interior had been pretty well cleared out, except for the furniture and fixtures, or maybe what they found was too big to carry away in a pocket. The factory was certainly well built, or else it would be a heap of rubble by now. I suppose we should count it as a blessing that no eager developer decided to take on the property. But probably it wouldn't have been worth it.\"\n\n\"I'm sure I can find something constructive to do with it, as part of the whole. Do you know who has the keys to the factory building?\"\n\n\"I'd assume the town council, or the town's solicitor, or the bank manager. You can probably track down to whom Henry left the factory\u2014I don't know if it was linked to the house. That you would find among the county or state documents rather than here.\"\n\n\"I'll check out the property records.\" I looked through the glass at the top of the office door to see a few hardy souls clustering at the front door. \"It looks like you have customers, Audrey. I'll start going through the documents you've assembled and you can go on about your business. And I really do appreciate your help.\"\n\n\"That's what I'm here for, Kate,\" Audrey said primly, and escorted me out of the office.\n\nI made straight for the Local History Room at the back of the building, where a stack of materials was waiting for me.\n\nThree hours later I had barely made a dent in the stack of materials that Audrey had collected for me. The fact that most Victorians used extremely small fonts for anything printed (I had often wondered how anyone in those days could hope to read anything at all) didn't help, and the handwriting varied widely in those few documents that were not printed. This was not going to be a quick survey. I could tell myself that I didn't need to read everything\u2014only enough to get a sense of the character of the era on this first pass. But I couldn't wade through the research forever; the town council wanted to hear from me sooner rather than later. I needed to know myself just what I was thinking. Was this project going to consume the next six months of my life, or should I let the town down gently, pack up my bags, and go find another long-term job? I was still on the fence but I could certainly spare a few more days.\n\nI stood up and stretched, and realized I was hungry. There was next to nothing to eat back at Cordy's place, so I needed to do some basic food shopping, even if it was for only a short time. That meant going to the supermarket, which was too far to walk. I'd have to walk back to the B&B, retrieve my car, and go to the market. If I was lucky I could pick up a sandwich or something there, and then maybe head out to the Barton place to\u2014what? Commune with the dead? I hadn't seen the basement yet and barely glanced at the attic, although they would most likely never become public spaces, no matter what happened with the house. But I liked to be thorough. And if I saw evidence of a crumbling foundation and six inches of standing water in the basement, that would add a lot of zeroes to the cost estimate. Ditto the attic: if I could see birds flying between the roof shingles, that cash register would go ka-ching a few more times. Not that I doubted that the caretakers had done their job\u2014but I had to see for myself.\n\nSo, plan: walk to B&B, get car, find food, go to Barton house. Done. If I could find keys to the place. Which Josh should have, but where would I find him? Or Arthur at the bank. I should have asked somebody to give me a set, just for the week.\n\nThat simple plan worked for about five minutes. I rounded the corner from Main Street to the B&B and saw a state police car parked in front. Detective Reynolds was standing in front of the building. What now?\n\n\"Can I help you with something?\" I asked when I was close enough to be heard.\n\nThe detective turned in my direction, and after a few moments he recognized me. \"Ms. Hamilton? What are you doing here?\"\n\n\"I'm staying here for a few days, with the owner's permission. What are you doing here?\"\n\n\"I thought this place belonged to Cordelia Walker. I am informed that the police did not feel it necessary to search this place for any evidence of the possible killer, so I am remedying that. How can you be staying here?\"\n\n\"The building is owned by Cordy's ex-husband, Ryan Hoffman, and he said I could use it while I'm here. He told me the police hadn't even asked to look at it, and there was no crime scene tape. You may certainly look all you like, if my permission counts.\"\n\n\"Now that you've tramped through it. What parts of the building have you visited?\"\n\n\"Most of it. I'm staying in the big bedroom in the front, but I checked out the other rooms\u2014women staying alone in an unfamiliar place do that, you know. I've also used the kitchen, and I'm sure you'll see the evidence of that.\" Since I hadn't bothered to wash the dishes before I dashed off to the library.\n\n\"Did Mr. Hoffman accompany you here?\"\n\n\"Yes. He showed me around the place, plus how to manage the alarm panels at the doors, and gave me the keys. Why are you asking?\"\n\n\"Just trying to ascertain who has been in the building since the murder.\"\n\n\"Are you looking for anything in particular?\"\n\nDetective Reynolds appeared to consider his options. He wasn't under any obligation to tell me anything, but he had asked for my help, after all. He decided to speak. \"We were curious about Mrs. Hoffman's financial situation. Also any information she might have left about her preliminary negotiations with potential investors.\"\n\n\"She used the library on the first floor as her home office. But Ryan told me he took whatever legal or business documents he found, since no one had told him he shouldn't. He said he'd turned copies of them over to your people.\" Did I need to mention the wine?\n\nAs I watched his face, I wondered if members of his staff were going to get a lecture about investigation procedures and coordination. Still, it was unlikely that there would be another murder here any time soon, so Detective Reynolds might just decide to save his energy. \"When you first entered that room, did it appear that anything had been disturbed?\"\n\n\"It seemed neat enough to me, but I don't know how Cordy usually kept it. And Ryan had already been there. If someone was looking for something, he or she tidied up well.\"\n\n\"Did you find anything of interest?\"\n\n\"No. I didn't even see a computer or laptop. Cordy must have had one, to keep track of reservations, do accounting, that kind of thing. Ryan might have taken that away as well\u2014I haven't asked him. Have you met him by now?\"\n\n\"Yes, briefly. He has an alibi for the time of the murder, but from what little conversation we had, I got the impression that Cordelia Walker was a thorn in his side.\"\n\n\"He as much as said so to me too. So what is it you want from here?\"\n\n\"Nothing specific. Just following through.\"\n\n\"Will you be taking fingerprints? Because my friend Lisbeth Scott was here with me last evening. We opened a bottle of wine. There's a secret compartment for wine in that wall there.\" I waved vaguely toward the parlor. \"Good stuff, too. And probably expensive.\"\n\n\"You're suggesting that Cordelia was a drinker?\"\n\n\"No, not necessarily. But the stuff was well hidden, where guests were unlikely to find it. And if she was hard up for money, what I saw seems a bit above her bracket. A gift, maybe? Ryan said I was welcome to drink what I found.\"\n\nDetective Reynolds's mouth twitched.\n\n\"Still no idea who killed the woman?\" I asked, mainly to divert his attention from the wine.\n\n\"Given the height, direction, and depth of the fatal blow, I think it's safe to say it wasn't a child. Or a short person.\"\n\n\"What if she was sitting down?\" I pressed.\n\n\"On the front step of the building?\"\n\n\"Maybe she was rummaging through her purse trying to find the keys.\" I sighed. \"All right, maybe if it was a big purse with a dark interior and a lot of stuff in it. And if she wasn't in a hurry. What do your crime scene analysts say? Was she sitting down? Were there microscopic flecks of granite or marble on her butt?\"\n\nDetective Reynolds gave me an almost-smile. \"You do ask the oddest questions. As she was found, she did not topple over from a sitting position. She fell face forward, with her head away from the house. As you well know.\"\n\n\"So somebody sneaked up behind her from inside the house?\"\n\n\"So it would appear. Do you happen to have a convenient list of who had access to the building?\"\n\n\"What if it was someone who accompanied her to the building? Apart from that, I'd guess that at least one council member would have keys and could get in at will.\" I paused to think. \"Tell me this: why a rock? That's a pretty unsophisticated weapon. If someone had gone to the house with the intention of killing her, wouldn't they have brought their own weapon, rather than picking up something lying in front of the house? Did you find any footprints?\"\n\nThe detective ignored my first question. \"You've seen the approach. There's a brick pathway leading from the drive in front. It does not take footprints. You know, you seem to be enjoying this.\"\n\n\"I am, I guess.\" Was it wrong of me to feel that way? Especially because the detective knew I'd been the one to find the body. \"Nobody seems to miss her.\"\n\n\"A rather harsh judgment,\" he commented.\n\n\"You didn't know the woman. So now what? Do I need to let your team alone while they hunt for clues? Do I have to point out which items are mine? What I've touched? I can give you fingerprints if you need them for comparison.\"\n\n\"If you have some errands to run, we shouldn't take long, and you can come back after we're done.\"\n\n\"Right now I need lunch. Do you eat, Detective? Can I get you something? Because there's not much in the house, whether or not it's evidence.\"\n\n\"Maybe another time. You go ahead. We should be out of here by three at the latest.\"\n\n\"Fine.\"\n\nI left. Reynolds didn't want to share, or maybe he had nothing to share. I had no idea what he hoped to find in the house. He might do better to talk to Ryan, who'd been through the place before I had. But I wasn't about to tell a police detective how to do his job.\n\nSo, change in plan: lunch, then Barton house, then grocery shopping. I decided I could handle fast food and pointed the car toward the road leading out of town, where I found a Wendy's. I wasn't desperate enough to use the drive-through window, so I went inside, ordered something resembling a hamburger, and settled myself at a table. I consumed the whole mess in ten minutes and took to the road again\u2014and remembered the gate at the Barton house would still be locked. Clearly I had too much on my mind.\n\nI pulled over and called Josh's cell phone, and luckily he answered. \"Where are you?\" I asked.\n\n\"Working. At my place. Why?\"\n\n\"I want to get into the house, and I can't even get through the gate. Oh, have the state police been by?\"\n\n\"Why are you asking?\"\n\n\"Because they showed up at the B&B a little while ago. Apparently they dismissed the B&B a bit too quickly, and now they're hunting for something. They didn't share what.\"\n\n\"Well, if they come out here, I'll let them in. Although they've already checked out this place.\"\n\n\"Maybe they have a piece of information that points in a new direction.\"\n\n\"Or maybe they're getting desperate. Whatever. Come on over\u2014I'll meet you at the front gate.\"\n\nI was there in five minutes. When he saw me he unlocked the gate and let me pull my car in, then locked it behind me again. \"Isn't that kind of like locking the door after the horse has left the barn?\"\n\n\"Maybe. But I'm responsible for this place, and I'd rather be safe than sorry. What is it you're looking for today?\"\n\n\"I'm still trying to get a feel for the place. You gave me the basic tour the other day, but I haven't seen the basement or most of the attic.\"\n\n\"Why would you want to?\"\n\n\"Because I like to see everything with my own eyes. Because Henry is still a mystery to me. Maybe there's a gold mine in the basement, or maybe Henry buried twenty bodies down there. Think of what treasures you might be missing! Original documents that nobody's read for a century! And they're all yours.\"\n\n\"All right, I get it. You want to start with the basement or the attic?\"\n\n\"I think it will take less time to go through the basement, and I don't expect to find much down there. But I want to check. You don't have to come with me.\"\n\n\"And risk you doing serious damage to the boiler? Or yourself?\"\n\nI noted that he'd put the boiler ahead of me. \"Fine, you can come along.\"\n\n\"All right. Follow me back to the house and we'll park in front.\"\n\n\"You aren't exactly dressed for wandering around the basement, you know,\" Josh said.\n\n\"I wasn't planning to dig up the floor. I'll manage. Where's the entrance?\"\n\n\"There are two\u2014one outside, for big stuff, and one tucked under the main staircase.\"\n\n\"How big is it?\"\n\n\"It extends under the full house, so it's big. You having second thoughts?\"\n\n\"No, I'm just trying to wrap my head around it. Furnace?\"\n\n\"Coal originally, then the bank replaced it with an oil furnace awhile ago.\"\n\n\"How was the heat distributed?\"\n\n\"Passively. Grates in the floor on the first floor, and then the heat was supposed to migrate up to the second floor. Of course, there are small fireplaces in the bedrooms to take the chill off. The maids were supposed to lay those and light them. And dump the thunder mugs. And bring up hot water for the morning wash.\"\n\n\"Well, they had a job, right? And food and a bed? Tell me, O Industrial Historian, was that a better or worse job than the girls had in factories about the same time?\"\n\n\"Good question. Things were changing fast then. There was definitely a migration to the cities. Look, let's quit stalling and just get down there. I've got a chapter to finish.\"\n\n\"Lead the way.\"\n\nI followed him through the hallway and to a door set discreetly in the wall beneath the grand staircase. \"Is it locked?\"\n\n\"No. There's nothing down there worth stealing.\"\n\n\"Was there ever?\"\n\n\"There was a cold cellar for root vegetables and plenty of space for tools. Apart from that, I have no idea why they wanted so much open space down there.\"\n\nThe stairs were old and well worn; they dipped in the middle, the wood worn away by years of the hired help trudging up and down.\n\nWe reached the bottom of the stairs. I stopped and turned around slowly. Fieldstone foundation, between four and five feet high. Brick above that\u2014could use some repointing. The floor joists\u2014good heavens, they had to measure at least ten inches by three inches, something nobody ever saw anymore\u2014were low, and I could barely stand upright. The furnace was a hulking behemoth, sprouting ducts in all directions. The ducts disappeared into the flooring above.\n\nI was beginning to think that if I spent enough time looking\u2014really looking\u2014at how things had been put together back when the house had been built, I would come to understand how the people had lived in the house, both upstairs and downstairs. There was a logic to it that was no longer obvious now, but it made sense if you stopped to think about it. And took your time.\n\nThere was something odd about the space, the more I studied it. No, not the space, but the perimeter, the footings for the building. \"Hey, Josh?\"\n\nHe'd been pottering around at the far end of the room, poking at piles of what appeared to be junk that had been abandoned in the corners. \"What is it?\"\n\n\"There was already a house here before Henry built this place.\"\n\n\"What? Why do you think that?\"\n\n\"Come back and stand where I'm standing.\" He came up beside me and looked quizzically at me. \"See the changes in the stonework around the bottom? Not the brickwork\u2014that's consistent. But the stones and the way they're set varies, and there's a pattern to it.\"\n\nHe still didn't look convinced. \"Which is?\"\n\n\"I'm going to guess that there was a smaller house here before. Maybe a farmhouse. Henry was thrifty and reused as much of the foundation as he could. He had to use bigger stones for the new parts because he was building a bigger building. Do you see it?\"\n\n\"Damn, I think I do. You're saying you think maybe Henry bought a simple farmhouse when he first settled here, and when he made his money, he built over and around it?\"\n\n\"It's possible, isn't it?\"\n\n\"I guess so. I haven't looked at property records.\"\n\n\"Audrey said I should check out the country records, which include deeds,\" I said slowly. \"We can find out who lived here before Henry, and when Henry bought the place. Figuring out exactly when he started the bigger building will be more guesswork\u2014I don't know when they started issuing building permits locally, but we can go by materials and styles, which I'm guessing date to around 1880. Maybe even newspaper articles\u2014there might be some society news there, if they ever entertained. I wish I knew something about his wife. I don't suppose you've looked up anything about her?\"\n\n\"No, not really,\" Josh said. \"She's kind of a shadowy figure in all of this, but she's not exactly relevant to what I'm looking at, unless she brought one heck of a big dowry with her. You want to take Mary on, along with Henry's genealogy?\"\n\n\"I plan to\u2014I already alerted Audrey. I'm hoping there may be a story in it. You know, war veteran settles here, makes good, builds his dream house for his lovely bride. His life is touched with tragedy when the children they both so wanted never happened, so he built an orphans' home or something. You get my drift.\"\n\n\"Yeah, sounds like something out of a bad romance novel.\"\n\n\"Romance sells. If we want to attract people, we have to humanize this place. Was the mystery wife an invalid? A madwoman he kept in the attic? Did she pick out all the furnishings and decorate the place, or did he do it to please her, out of love? Things like this matter.\"\n\nI glanced at Josh to see if he was still laughing at me, but he looked oddly curious. \"You're really into this,\" he said, not unkindly.\n\n\"Yes, I guess I am. It's like creating, or re-creating, other lives, other times. Don't you find yourself doing something like that, when you talk about the changes that the factories brought in the nineteenth century? How much people's lives changed? How the whole demographic of the country shifted? And what was the impact on the national psychology? The country was redefining itself under everybody's eyes.\"\n\n\"I guess I hadn't looked at it like that,\" he said in a subdued tone. \"I'm a person who likes facts, order. My research is on how we became an industrialized nation after the war. But I will concede that that misses a lot of the human element.\"\n\nI smiled. \"Am I actually converting you to my point of view? Look, here's an example where our viewpoints overlap. Look at the height of the ceilings in this place, the large single-pane windows. How incredibly wasteful a place like this was to heat! But there was what must have seemed like endless fuel available, so nobody worried. Until the fuel got harder to find, and technology wasn't keeping up, and it got more and more expensive. And then they started putting up storm windows and making furnaces more efficient. Not to mention lowering the ceilings. You could say it was a period of conspicuous consumption, when the rich flaunted their riches, and then they couldn't do it anymore.\"\n\n\"So now you're seeing this place as a model of conspicuous consumption? Who's your target audience?\"\n\n\"Now you're making fun of me. But I'm trying to tell you what I mean about weaving a story around the place. There's the possible internal story\u2014two people who loved each other, who faced their own troubles together\u2014Henry's war experiences, their childlessness. Then there's the external story\u2014the rise of the factories and the changes that came with them. Did Henry see what was happening, and did it make him sad or proud? He must have taken pride in this place, but did that make him feel guilty, that he had so much more than most people? And all his money couldn't bring his wife back, and the children that never happened. I wonder if he had a dog...\"\n\nOkay, I had wandered off on a tangent, but I was really beginning to feel for poor Henry. Then Josh began to laugh. \"You are really something else.\"\n\n\"You're laughing at me,\" I said stiffly. I refrained from hitting him.\n\n\"Yes, but in a good way. You know, you're beginning to convince me that this is all possible. You'll probably wow the town council.\"\n\n\"Thank you\u2014I think. Now all I have to do is find the money to make it happen.\"\n\n\"Seen enough of the basement?\" he asked.\n\n\"For now, I guess. I wonder if Henry ever even spent time down here, or just sent the servants down to see to things like stoking the furnace. I wish there was a way to get to know the man!\"\n\n\"Well, unless you're into spiritualism, you'll have to settle for genealogy and local histories. It sounds to me like you'd better do a quick survey of all the things you mentioned, so you can pitch it to the town. You can get into the in-depth stuff once they're on board.\"\n\nI felt a bit deflated, but I knew he was right. I had to focus on the high points for now. \"But can we look at the attic? Just a quick pass through it?\"\n\n\"If you'll let me set a time limit and allow me to drag you away when time's up.\"\n\n\"Oh, all right. It'll be dark soon enough anyway.\"\n\nWe'd reached the main hallway when there was a pounding on the front door. I looked at Josh, and he shrugged. \"I'm not expecting anyone.\"\n\nOnce again I realized how isolated the place was, and I was glad that Josh was there. \"Better go see who it is.\" I made no move to go to the door, so Josh had to. Hey, it was part of his job, wasn't it? He pulled the door open to reveal... Ryan? \"Ryan, what the heck are you doing here?\" I demanded.\n\n\"Not my idea,\" he said, sounding surly. \"A state police detective called and told me to meet him here.\"\n\n\"How'd you get through the gate?\" Josh asked sternly.\n\n\"Cordy's keys.\" He held up a loaded key ring.\n\n\"Hold on,\" I interrupted. \"Do you two know each other?\"\n\n\"No,\" Josh answered first. \"But apparently you do. You want to fill me in, Kate?\"\n\nRyan wasn't going to stand for that. \"I'm Ryan Hoffman, Cordelia's ex-husband. I found her keys at the B&B she was running in town. I suppose I should return them to you?\"\n\n\"Yes. I'm the official resident caretaker, although my day job is history professor. I'm on sabbatical. Joshua Wainwright.\"\n\nThe guys shook hands more or less politely. I wondered why the state police wanted to talk to Ryan now, and why here. \"Josh, you said the police didn't call you today.\"\n\n\"Uh, no, but my cell is out in my apartment out back, so I wouldn't have heard it. They know about this watchman arrangement, and they already interviewed me, but I don't know if I'm supposed to be here to talk to them now.\"\n\n\"Well, they definitely don't know that I'm here.\" I did hope they'd let me stay, in any case. \"Ryan, did they say why they wanted to meet you here?\"\n\nHe shook his head. \"Just suggested that I be here. I have to assume it had something to do with Cordelia and her connection to the place.\" He turned to Josh. \"I know Kate here had a long history with Cordelia, but how about you, Josh?\"\n\n\"I think I first met her when I talked to the town council about taking this temporary position a few months back. Then she started dropping in, and sometimes she brought other people. She didn't bother to introduce me to them, but I assumed they were potential buyers or investors.\"\n\nI wondered briefly if Josh had been put off by Cordy treating him like the hired help. That must have been the period when Cordy was trying to pressure the council into doing what she wanted. Maybe talking down to the caretaker was supposed to demonstrate her power? \"Did any of them come more than once?\" I asked.\n\n\"I can't tell you. I wasn't always here. And she had the keys to get in.\"\n\nSimultaneously we heard the sound of another car approaching by the long front driveway. \"Damn. Did you lock the gate behind you, Ryan?\"\n\n\"I thought I did.\"\n\nJosh sighed. \"Well, either you messed up, or there may be more copies of the keys out there. If it's the police arriving now, I wonder where they got their set, if you have Cordy's set?\"\n\n\"And I have Lisbeth's set,\" I volunteered. \"Which she got from the bank manager.\"\n\n\"So half the town could have come and gone here whenever they wanted. That's going to make this investigation even more fun,\" Josh said, sounding frustrated. \"I should have demanded a list from the beginning.\"\n\nThe car stopped, a door slammed, and after a strong knock, Josh opened the door to Detective Reynolds. He scanned the group, noting my presence with a raised eyebrow, and nodding to the two men, whom apparently he'd already met. \"Mr. Hoffman, thank you for agreeing to meet with me. Mr. Wainwright, I'm glad I found you here, and you're welcome to join in the conversation. Ms. Hamilton, what are you doing here?\"\n\n\"I came to check out the parts of the house I hadn't seen before. For this possible hush-hush town project that I've told all three of you about. We just came up from the basement, and we haven't gotten to the attic yet.\"\n\n\"I see. Is there someplace we can all sit?\"\n\n\"The dining room,\" Josh said, and led the way, followed by Ryan, the detective, and me.\n\nWe made an odd group, seated around the table. I wasn't sure who was supposed to be in charge, beyond the fact that I certainly wasn't\u2014I was merely an observer. Detective Reynolds seemed unperturbed by any undercurrents among us, and he was the one who had asked to meet with Ryan\u2014and ended up with two extras. Ryan had assumed his lawyer demeanor and looked solemn. Josh looked wary and out of his depth. Me, I tried to look invisible.\n\nThe detective took his time before speaking, and I wondered if the delay was to rattle us. He couldn't rattle me because I didn't know anything so I couldn't possibly be hiding anything. Finally Detective Reynolds cleared his throat. \"You may be wondering why I asked Mr. Hoffman to meet me here.\"\n\nWell, duh, I thought. Josh didn't comment.\n\n\"The investigation of Cordelia's murder is under the jurisdiction of the state police. I was somewhat disappointed by the degree of thoroughness in the original inspection, so I am reviewing whatever has been collected to date. I know Mr. Hoffman here was formerly married to the victim and is the owner of the physical property where she was operating a small business. This suggests an ongoing connection between them.\"\n\n\"But... wait... I didn't\u2014\" Ryan sputtered.\n\nThe detective held up a hand. \"I'm not accusing you of anything. I'm only pointing out that certain aspects were not considered in the first round of investigation. And I think we\u2014I'm speaking for the police\u2014need to reexamine this Barton property more closely. It would appear that Cordelia had a strong interest in this place, for her own reasons. It was here that she was killed. One should consider whether there is a connection between those two facts,\" the detective said. \"Now, Mr. Wainwright, based on phone records, it appears that you and Cordelia exchanged quite a few phone calls. Business or pleasure?\"\n\nJosh looked at Detective Reynolds for a moment, then burst out laughing. \"Are you seriously asking if Cordelia and I were involved personally?\"\n\n\"Yes, that's exactly what I'm asking. Were you?\"\n\n\"Most definitely not. I could give you reasons why, but if I say something negative about the woman you could twist it to make it look like I had a reason to kill her.\"\n\n\"I've confirmed your alibi, Mr. Wainwright, and I'm not questioning it. You can speak freely.\" Detective Reynolds glanced briefly at Ryan. \"I hope that if you take offense at anything Mr. Wainwright might say about your former wife, you'll restrain yourself from commenting.\"\n\nRyan looked incredulously at the detective then he also laughed. \"I doubt he can say anything I haven't already thought about her. We'd been apart longer than we'd been together. Fire away, Josh.\"\n\n\"All right,\" Josh said. \"I told you people when you first interviewed me, but I'll say it again. Since I'm not part of the town of Asheboro, obviously I don't know the whole backstory, but I understand that the town was looking into making this house into a small elegant hotel. Cordy saw an opportunity to move up the food chain and made a bid to run it. The idea was greeted with what might be called muted enthusiasm. And there things sat for a while. Then, when things weren't moving fast enough to please her, she called in a backup team, some corporate group or a bunch of rich investors, and told them to put together a proposal for use of the place, which they did. The town council hated it. And talks kind of stalled. Keep in mind that this is mostly hearsay and gossip that I've picked up since I've been looking out for this place.\"\n\nI knew that Detective Reynolds had heard the same story from me, but he didn't mention that. I kept silent\u2014again.\n\n\"And where did you enter this equation, Mr. Wainwright?\" the detective asked.\n\n\"I'd already signed on as caretaker before all this came up. As I said, I'm an outsider, and I won't be around long. In any case, Cordelia made a play for me\u2014kind of threw herself at me. And she expected me to melt. I didn't.\"\n\n\"Was this a one-time effort or an extended campaign?\" I asked. Detective Reynolds sent me a warning look.\n\n\"Somewhere in-between,\" Josh answered. \"She didn't give up easily, and she seemed convinced that her feminine wiles would wear me down eventually. But for all I know she could have been trying to seduce half the town. Sorry, Ryan.\"\n\nAnd then an idea hit me between the eyes. \"You know, Josh, maybe she was making up to you because she knew she might need you.\"\n\nHe turned to me. \"Why on earth would she need me? Apart from any, uh, romantic reasons?\"\n\nI quickly laid out my reasoning. \"She had had access to this place for a while now\u2014she had her own keys. Maybe there's something here that she found, something that she thought might be valuable or useful or something. But she couldn't just pull it out and say, 'Gee whiz, look what I found!' I think she'd have to have someone with more expertise vet it, to add credibility. Like a respected history professor.\"\n\n\"And that would be me?\" Josh said. \"Well, even if she did find something, she didn't bring it to me. And see what you've done? Now you're going to have the police thinking something absurd, like she found something incredibly rare and desirable, and when she showed it to me I went crazy with lust for it\u2014not for her, mind you\u2014and killed her to get my hands on it.\" He turned back to Detective Reynolds. \"Do you like that scenario, detective?\"\n\nBrady Reynolds had a half smile on his face. \"Well, it's an angle I hadn't considered. Would she be selling it for personal gain, or did she want to use it to promote this place? Or maybe we should be looking at it from the other side. Maybe she planted something important here and was planning to pretend to find it, for the same end. And she'd still need you for your stamp of approval.\"\n\n\"That won't fly if she had to buy something appropriate to do it,\" Ryan interrupted. \"She had no money and no collateral to get any. And I can't recall any terribly valuable heirlooms from her family either. If she was trying to leverage an artifact of some sort, it would have to have come from this building.\"\n\n\"I knew I had come to the right place,\" Detective Reynolds said calmly. \"You've already given me several useful ideas. Anything else?\"\n\nAfter a pause, I said slowly, \"Detective Reynolds, much as I respect the abilities of your police force, can you see your people sifting through this place looking for some historic artifact that might be worth big money? Jewels, silver, all that kind of stuff, fine, but they're already in the bank vault, I understand. But documents or books? I'm not sure what they'd recognize. In any case, assuming this priceless whatever existed, then either nobody's found it, or Cordelia or someone with a key did find it, or Cordelia let someone else in, and they found it and then he killed her to keep it for himself.\"\n\nThe detective steepled his hands. \"Why do I think there's a suggestion coming?\"\n\n\"Because you're a smart detective,\" I answered promptly. \"Let Josh and me do the looking\u2014deputize us or authorize us or whatever. You know we both have alibis for the murder. Josh would most likely recognize this valuable thing, whatever it is. Or at least he would know who to call to verify it.\"\n\n\"And why do you need to be there?\"\n\n\"Two sets of eyes are better than one? No, seriously: I'm being selfish. I would love to find something of value, whether it's monetary or historic or sentimental, so I can use it to promote this place. It's not for me. In a sense it's for the town. Look, if you don't trust either of us, assign one of your men to keep an eye on us. Film the whole process. Whatever it takes.\"\n\n\"Interesting proposal. You have any idea what you're looking for?\"\n\n\"Not a clue,\" I admitted cheerfully. \"George Washington's cookie recipes. Nude pictures of Ulysses S. Grant in bed with a lady of the evening. Who knows?\"\n\nDetective Reynolds turned back to Josh. \"Mr. Wainwright, what do you think of this scheme?\"\n\nJosh was looking a little shell-shocked. \"I think it has possibilities. To tell the truth, Cordelia was in and out quite a bit\u2014more than her responsibility to the town might warrant, so it could well have been for other reasons. I didn't follow her around like a puppy dog to see where she went or what she looked at\u2014that wasn't part of my job description. I've been through the house, of course, because that's what I agreed to do, but haven't examined the attic in any detail, because it's cold and dark up there\u2014I was waiting for spring. So I can't tell you whether or not Cordelia was up there, once or many times. I will be happy to explore the contents of the attic now, with your permission.\"\n\nThe detective sighed. \"As far as I can tell, it can't hurt, although recruiting civilians is somewhat unorthodox. We certainly know she wasn't killed up there. Mr. Hoffman, did your ex-wife have any expertise in antiques or fine art or old documents?\"\n\nRyan shook his head. \"Only in her dreams. She never had any formal training in anything like that, and her taste was conventional at best. She wasn't an antiques person. In fact, she had a particular dislike of all things Victorian, and she was really annoyed that she couldn't change the d\u00e9cor of the bed and breakfast. Unfortunately for her, tourists like Victorian interiors.\"\n\n\"Detective,\" I said tentatively, \"she's the one who ended up dead, which means somebody killed her. But if our nice theory about some great find is true, it could be that she had a partner, who seized whatever it was she or they had found and took it away. Who wanted all the proceeds or the glory for himself? They could have arranged to meet here, so there might be no record of a phone call at the right time. And then he\u2014or she\u2014double-crossed her. Since none of us knows what we're looking for, that person could easily dispose of it somewhere else. Isn't that explanation possible?\"\n\nThis whole scene was absurd. Here we were, four outsiders, trying to figure out who might have been the killer, in a town none of us knew very well in the present. But, I realized, it might not have been someone from Asheboro. It could have been an art expert from somewhere else, whom she'd called in, and nobody would have known him. Or her. That might make sense, since Asheboro seemed a little short on art experts, but there were some major cities not far away that were overflowing with them.\n\nWe could speculate all day, but to me it made more sense to look around more closely to see if there were any signs of a search. \"So, do we have permission to look around?\"\n\nThe detective sighed. \"I don't see why not. It could be said to fall under Mr. Wainwright's purview as the official caretaker, so implicitly you also have the town's permission. Mr. Wainwright?\"\n\n\"Works for me,\" Josh said promptly. \"As I said, I've been planning to dig around in the attic anyway, to see if there are any historic documents, and now I have a great excuse. Since I haven't seen any signs that anything large has been removed from the house during my tenure here, I'm going to assume that whatever it is, it's smaller than a breadbox. Cordelia was not a particularly sturdy woman, although she could have had a stronger partner. In my estimation, the attic is the best place to start. I'll try to get some floodlights so we can see to work. Are you okay with all that, Kate?\"\n\n\"Sure,\" I agreed. \"But can we start tomorrow? This sounds like a grubby-clothes job.\"\n\n\"No problem.\"\n\nDetective Reynolds cleared his throat. \"In my official capacity, I need to tell you not to mention what you're planning to anyone in town. There are altogether too many people who we haven't been able to eliminate from our list of suspects.\"\n\n\"Got it,\" I said. Josh nodded in agreement, as did Ryan. I had to wonder what single item, or collection of items, would be worth killing for, and if Cordy's killer was involved somehow, whether he\u2014or she\u2014might come back to finish the search, as long as our plans remained secret.\n\nThe meeting seemed to be over. Everyone stood up. \"Josh,\" I said, \"you want me to meet you here in the morning? Say, nine o'clock?\"\n\n\"Fine,\" he said.\n\n\"Why don't you and I have dinner together tonight, Kate?\" Ryan volunteered. \"We still have a lot of catching up to do. Is there any place around here that does decent takeout? Because it's hard to talk freely in a restaurant, and I don't expect you to cook at the B&B.\"\n\n\"I'm sure we can find something,\" I said absently. \"Detective, are we free to go now?\"\n\n\"Yes, you are. Please keep me apprised of anything you find. And if you find anything, keep it safe.\"\n\n\"Of course. And thank you for letting us help.\" I had a feeling he was bending more than one rule in doing it.\n\nThe detective left first, then Josh said, \"I have to lock up so I'm going to ask you to clear out.\" Ryan and I followed Josh meekly out the front door. He said good-bye, leaving the two of us standing awkwardly on the front stoop.\n\n\"So, dinner?\" Ryan asked.\n\n\"Sure. Pizza all right? I don't think we've got a lot to choose from in town.\"\n\n\"It's not the same pizza place we used to hang out at, is it?\"\n\n\"Oh, no. But they made terrible pizza, at least in hindsight. This one couldn't be worse. Did I mention that Lisbeth and I found Cordy's secret wine trove?\"\n\n\"You did not. Anything good?\"\n\n\"I'm no expert, but there were bottles in there that I've drooled over but could never bring myself to actually splurge for.\"\n\n\"Interesting. Cordy was never really a connoisseur. I wonder if she had them to impress someone? Her corporate allies? Or her mystery partner?\"\n\n\"Possibly. The cache was well hidden, not where anyone would just stumble over it and snag a bottle. Can we drink one? It might do wonders for bad pizza.\"\n\n\"I'm executor of her estate, so I officially approve your request for one bottle of wine. Although if it's really good stuff, I should probably make a note for the inventory, at least for inheritance tax purposes.\"\n\n\"Whatever. Pickup or delivery?\"\n\n\"What? Oh, does this pizza place deliver?\"\n\n\"I have no idea. Remember, I haven't been around much for more than fifteen years, until this week. Let's go to the B&B and call. If they do deliver, I'll show you where to find the wine.\"\n\n\"Sounds like a plan,\" Ryan agreed amiably. \"I'll meet you there, since we've got the two cars.\"\n\n\"Great.\"\n\nHe waited for me to locate my car keys then we made our way to the front gate and headed back to the center of Asheboro. As I drove, I reviewed the impromptu plan I had come up with at the Barton house. That there might be something of value in the house somewhere seemed perfectly plausible to me, although I had no idea what it could be. Judging by the d\u00e9cor of the house, Henry Barton had plenty of money to spend, at least after 1880 or so, and he had chosen the best of the best at the time. Although, I amended, what I'd seen of the foundation in the basement suggested that the place had begun as a much simpler house when he bought it, which in turn suggested that when Henry had become wealthy, he had expanded it to reflect his new status. I'd have to check property records. And any society news in the local papers, I added to myself. Or maybe finding a site for his factory had dictated where he would settle. I made another mental note to see what history there was about the factory. I really needed another session at the library. Maybe Audrey could point me in the right direction.\n\nI parked behind the B&B. Ryan's car was already there\u2014did he know a shortcut, or was he a faster driver than I was? Probably the latter\u2014I was still reacquainting myself with the area so I took it slowly. He still had a key so he had let himself in, and he was already on the phone ordering something or other. I really didn't care what we ate, although I had to make a mental note to myself: if this pipe-dream of a town makeover actually happened, we'd need to attract at least a couple of better places to eat.\n\nI tossed my jacket on the coat rack in the hall. Cordy must have hated the thing, because it was about as Victorian as it could get: large mirror, marble-topped seat, spaces for umbrellas and walking sticks, hooks for hats and coats, and gingerbread everywhere that wasn't already occupied. I loved it. Ryan emerged from the kitchen, slipping his cell phone into a pocket. \"Fifteen minutes. So, where's the wine?\"\n\n\"I'll show you.\" I led him to the back parlor and knelt by the wainscoting, then showed him how the panel opened. \"There you go.\" I stood up and stepped back.\n\nHe knelt and peered at the space. \"That's good work\u2014it really is invisible if you don't know it's there. How'd you ever find it?\"\n\n\"Actually it was Lisbeth and me, working together. Plus persistence. We just assumed she'd hide the good stuff, so we started looking for hiding places.\"\n\n\"Yes, Cordy probably offered a glass of wine to her guests, but I doubt she would have shared something this good. Not that she was into fine wines herself, except maybe for their snob appeal.\"\n\n\"Maybe it was for her partner in crime?\" I suggested, as I went into the kitchen to get wineglasses and a corkscrew.\n\nRyan followed, carrying a bottle. \"You buying that theory? That she was working with somebody else, who may have double-crossed her?\"\n\n\"I don't really know what I think. I hadn't seen Cordy since high school. Everyone in town who knew her has been\u2014how shall I put it?\u2014very careful in referring to her. Like they aren't sure whose side I'm on.\"\n\n\"They're the ones who called you in, right? Are they paying you?\"\n\n\"It was Lisbeth who drafted me\u2014I think she's the only one who knows what I've been doing for the past few years at the hotel. As for the money, no. They're kind of desperate, and they're hoping I'll come up with a proposal out of the goodness of my heart, or for old times' sake. Or because I have a history with Cordy and they figured I might want a chance to wreak my revenge. Of course, if we come up with a plan for the town that would work, I'll expect them to pay me. But I've barely scratched the surface, and having my chief competitor and old nemesis found dead in the middle of it all, literally and figuratively, isn't making things any easier.\"\n\nRyan had opened the wine with a skill that suggested a lot of practice, and he filled two glasses. He handed me one, and picked up the other and raised it. \"What shall we toast to?\"\n\n\"To burying the past and bringing it back to life,\" I said, after a moment's consideration.\n\n\"I like that. Here's to your success.\"\n\nWe clinked glasses and I took a sip. Nice. Velvety on my tongue, with a hint of... oh, who was I kidding? I might not know a lot about wine, but I knew that this one tasted very good. And I should pace myself, especially on an empty stomach.\n\nWe sat across from each other at the kitchen table. \"So, you lived here in Asheboro when you and Cordy were married?\"\n\n\"Yes. I had my law degree by then and passed the bar, but I opted to work for a midsize firm outside of Baltimore rather that setting up here in Asheboro.\"\n\n\"You're still there?\" I felt a small pang: he had been in the same city as me for years, but he had never looked me up. Water under the bridge.\n\n\"Yes. It was the right choice, especially in light of what happened\u2014when Cordy and I split up.\"\n\n\"You know there are a lot of questions I'd like to ask about that, but I have no right to.\"\n\n\"Are you sure about that? Cordy pulled a really nasty trick on you back in high school so you might say I'd be repaying a debt by being honest with you.\"\n\n\"What she did to me\u2014to us\u2014was hard to forgive, and she wouldn't have let me forget it if I tried. Did you know what she was planning back then?\" I hadn't had the heart to ask back when it happened.\n\n\"No. Honestly. It was cruel, and I wouldn't have gone along with it. I was as surprised as you were.\"\n\nAnd as embarrassed? Unlikely. \"So knowing what she was capable of, how on earth did you two end up together?\"\n\nRyan studied the ornate ceiling moldings, avoiding my eyes. \"Pity? Stupidity? She could be very persuasive, you know. But you didn't see her when she first moved back here. She thought she'd go to college and be the queen bee there, but she figured out pretty fast that she was one of a whole lot of pretty blondes, and some of them were a lot smarter than she was. She was stuck in the middle of the pack, and she didn't like it.\"\n\n\"So she figured she'd come back and reclaim her throne?\"\n\n\"Maybe. But she was disappointed that it didn't happen. She hadn't realized that there was a whole crop of pretty girls that had come up after she left.\"\n\nThe doorbell rang, and Ryan got up to answer it. He returned a minute later with a pizza box that smelled like heaven. I must have been hungrier than I thought. I got up to fetch paper plates and napkins, and we settled again in our former seats. We both devoted the next minute or two to inhaling pizza. Ryan poured another glass of wine for each of us while we chewed.\n\nAfter the first piece had disappeared, I picked up the thread of the conversation. \"So how'd she get back together with you?\"\n\nRyan sighed. \"Looking back, I think she went through the roster of available men in town and she set her sights on me, although I only realized that later.\"\n\n\"And you went along with it?\"\n\n\"Well, I wasn't seeing anyone else at the time, and she was still pretty and blonde.\"\n\n\"So you proposed eventually?\"\n\n\"I did, God help me. Uh, it was when she told me she was pregnant.\"\n\n\"Was she?\"\n\n\"No\u2014it was her final ploy to snag me. Not a good start to a marriage. It didn't get any better. If I had to sum her up, I'd say she was basically an unhappy person. Nothing was ever good enough for her. Nobody recognized how talented and smart she was. She had no friends, once her high school posse had moved on with their lives. And I didn't seem to be able to give her what she wanted or needed.\"\n\nRyan's assessment was fair\u2014and kind. It made it a little easier to feel sorry for Cordy, but not much. Had she died because of one of her manipulative games? She would probably have been horrified to know that the murder weapon was a common rock\u2014if she'd been planning the event, she would have chosen something more fitting, like a silver chalice. I smiled at the idiocy of my own thoughts.\n\n\"What?\" Ryan asked.\n\n\"I was trying to imagine Cordy planning her own murder. She would not have liked that rock as a weapon. Very low class.\"\n\n\"I see what you mean.\" He smiled too, but only briefly. \"Kate, have you ever been married?\"\n\nI shook my head. \"Not even close.\"\n\n\"Was it because of, well, us, and what Cordy did?\"\n\n\"No, Ryan. I moved on. What you and I had then, well, it was sweet. Puppy love. But we probably would have broken up as soon as we'd graduated anyway. It was the sheer venom of what Cordy did that upset me the most. She really enjoyed making other people suffer. I can see why she would have felt frustrated at college if nobody took her seriously, or even paid attention to her.\"\n\n\"Are you seeing anyone now?\"\n\nI cocked my head at him. Why was he asking? \"No. Please don't ask if I think you and I should try again. I think 'You and Me: the Rerun' is a bad idea.\" And if he was thinking I was dessert, he was way off base.\n\nHe shrugged. \"So, what now?\"\n\n\"About the murder? Frankly this is not my area of expertise. I think Josh and I should check the attic to see if there's anything of interest\u2014or if it looks like Cordy or someone else has been rummaging around. If there was something valuable, it could be long gone now, and we'll never know what it was. I suppose as executor you could check Cordy's financial records and see if there are any unexplained deposits.\"\n\n\"You mean, like bribes or payoffs, or maybe the results of the sale of something important? Huh. Worth a try, I guess. She could have been tempted\u2014she was always short of money.\"\n\n\"What about staff at the B&B?\" I asked.\n\n\"I think I told you, there weren't many. She hated housework, but she knew she had to maintain certain standards for guests, no doubt grumbling all the way. She hired part-timers, short-termers, and I'm sure she loved bossing them around. Your pal Lisbeth would be more likely to know who they were than I would.\"\n\n\"She would have declared their salary as a business expense, wouldn't she?\"\n\n\"Oh, right. I'll look for her tax returns. Anyway, I doubt it will be a long list. Why do you think it matters?\"\n\n\"It probably doesn't, but you don't know what people around the house might hear or see or find, and maybe even talk about, so it can't hurt to ask. And after that I'm pretty much out of ideas.\"\n\nRyan paused for a moment, then said, \"You think you can pull off this Save the Town campaign?\"\n\n\"I really don't know. There are a lot of 'ifs,' and this murder won't make it any easier.\"\n\n\"I'm sure you'll try your best.\" Ryan stood up. \"Sorry, I'm getting sleepy, and I've still got a drive ahead of me, back to the hotel. Walk me out?\"\n\n\"Sure.\" I followed him to the B&B's front door, unlocked it, and pulled it open. But he wasn't quite ready to leave yet: he reached out and pulled me close and kissed me. I felt a twinge of sadness\u2014no sparks at all. And when I didn't respond\u2014at least, not the way he had hoped\u2014he stepped back and smiled.\n\n\"Sorry. I just wanted to be sure. We can be friends, right?\"\n\n\"Of course. I'll let you know if Josh and I find anything.\"\n\n\"And you can celebrate with the rest of Cordy's Secret Cellar. Good night, Kate.\"\n\nI watched as he walked down the front walk to his car. Then I shut the door\u2014and double locked it.\n\nI threw out the paper plates Ryan and I had used, rinsed out our wineglasses, and trudged up the stairs, wondering what I was going to wear in the morning to sift through an attic that hadn't been disturbed for more than a century.\n\nI hadn't thought to bring along anything to read, and apparently Cordy didn't cater to a literary crowd: I'd noted earlier that the books in the library were mostly light vacation reading. I could have boned up on Civil War history on my laptop, but I figured that would put me to sleep fast. I selected a tattered paperback romance from one of the library shelves, and that knocked me out about fifteen minutes after I'd crawled into bed.\n\nOf course I woke up early, since I'd fallen asleep so early. I had a few hours until I was supposed to meet Josh, and nothing I needed to do. It felt odd, not having to check in with the hotel to make sure everything was running smoothly and no crises had erupted overnight. There were slim pickings for breakfast, but I managed to make coffee and snagged another couple of the rather dry muffins from the freezer and ate them. That took all of fifteen minutes. After I'd finished the muffins, I refilled my coffee mug and wandered around the ground floor of the B&B. I should ask Ryan whether Cordy (using his money) had bought it complete with furnishings, or whether she'd gritted her teeth and furnished the place with Victoriana to please guests.\n\nI had trouble picturing Cordy as the gracious hostess of a small guest house of any sort. What made that kind of place work was someone who would welcome visitors and make sure they had a comfortable and pleasant stay. That meant looking after them\u2014assuring that the room was clean, that supplies were restocked regularly, that there was breakfast, and snacks and coffee and tea were available throughout the day. But it took more than that. Innkeepers had to make guests feel valued. Guests also needed to feel that the manager had gone the extra mile to make sure that they knew the best places to eat locally, where the local monuments were, and how to reach them with a minimum of fuss. Basically it meant putting the needs of others ahead of your own\u2014and I couldn't visualize Cordy doing that. No wonder she'd gone looking for something bigger and better\u2014a setting that featured her. I doubted that she would have found it if the Barton house had become a luxury hotel: she'd still have been a glorified servant.\n\nI wandered into her office\u2013slash\u2013library. The room seemed oddly sterile, and somehow I doubted that Cordy had spent much time in it. Lisbeth and I had snooped only until we'd found the wine stash in the parlor, but there could well be other hiding places in this house. This hypothetical valuable mystery item would definitely have to be smaller than a breadbox, if it was concealed in any potential hiding places in the library. If it was here, it had been carefully hidden.\n\nWhere would Cordy hide something, if she had found something she thought was important? The desk would be too obvious. There were books on the shelves, but I didn't see Cordy thinking of a book as the best place to hide anything. A safe? Would she have kept a place for visitors' valuables and cash receipts she hadn't had time to take to the bank? Possible. I scanned the perimeter of the room but didn't see anyplace large enough for a safe, old or modern. There was an elegant oriental rug covering most of the floor, so I dismissed the idea that something was hidden under it\u2014Cordy would have protected her manicure rather than drag a dusty old rug around.\n\nBut what I did see, in the clear light of morning, was a crack around one of the wainscoting panels that ran along the wall under the bookshelves. The crack was a bit larger than the rest of its mates. It looked a whole lot like the panel that hid the wine cache in the parlor. Could the original owners have built in not one but two hidey-holes? Only one way to find out. I crossed the room and felt gently around the edge of the panel, and in one corner I pressed it and the panel sprang open.\n\nUnlike its counterpart in the other room, this one had no wine rack. What it did have was a single large manila envelope lying on its floor. I sat back on my haunches and thought about it. It was clean, so it hadn't been sitting there for a hundred years. Would there be fingerprints on it? Cordy's or someone else's? If I touched it at all, would I be contaminating evidence?\n\nBut of course I was itching to know what was in it. Maybe if I could find a pair of rubber or latex gloves, and something flat and narrow to open it and at least peer inside? My other choices were, option one, to call Detective Reynolds and tell him about it, although it was hard to say that it was a top priority. What were the odds that Cordy had sat down at her desk and scribbled a note saying, \"If I am found dead, it will be So-and-So who killed me,\" and then stuffed it in an envelope and hidden it behind a panel in a place that nobody knew existed? Small. Reynolds would just tell me to leave it where it was and he'd send someone over when he had the time. Option two, I could ask Ryan, since technically it was his house, and Cordy's possessions were now his. But would he drop everything and come racing over? Unlikely. It was altogether possible that it was nothing more than pictures of Cordy's family and a regrettable one of her with glasses and braces, or maybe the envelope wasn't Cordy's at all.\n\nI decided to find gloves and take a peek. Luckily there were clean latex gloves on a shelf in the pantry next to the kitchen.\n\nWhen I returned to the office, suitably gloved and having added a pair of kitchen tongs so I could pick up the mysterious envelope without handling it, I reached in with the tongs, carefully grasped the envelope, and withdrew it from its hiding place into the light. So far, so good. Definitely a modern envelope, and nothing exotic about it. Hardly full: I could feel the outline of what seemed to be one letter or document, and it crackled when I pressed it gently. Old paper?\n\nI carried it over to the desk and sat down in the chair, placing the envelope on the surface in front of me. It wasn't sealed, so I reached in with my gloved hand and pulled out the contents. It turned out to be two smallish pages of old paper, folded in half, with no envelope. Even before I opened them, I could tell it was a handwritten letter. I unfolded the pages and laid them carefully on the desk in front of me, weighting the edges gingerly with a couple of books. And then I focused on what it said.\n\nThe handwriting was large and clear, so it didn't take long to read the letter. And reread it. When I was finished I laid down the letter again, and sat back in the desk chair to think.\n\nI believed what I had just read was a letter sent to Henry Barton not long after the end of the Civil War. It appeared that Henry had been searching for one of his brothers, who hadn't returned from the war, and no family member had heard from that brother since, and naturally feared the worst. He was asking for help. The letter was short and to the point: the writer had little to report but would keep Henry apprised of her progress and would be delighted to report upon the status of her investigations on behalf of her cousin Henry.\n\nThe letter was signed \"Clara Barton.\"\n\nI was at best a casual historian, and usually interested in only what concerned my work. But even I had heard of Clara Barton, founder of the American Red Cross and an extraordinary nurse who had ventured into battlefields as a single woman and performed nursing feats I couldn't begin to imagine, under the worst possible conditions. It had never occurred to me to connect Henry Barton with the much more famous Clara Barton, yet she had called him \"cousin.\"\n\nThere was only the one letter in the large envelope. Was this the only one\u2014a polite brush-off\u2014or were there more somewhere else? Had Clara Barton followed up? Where? This letter seemed to be in good condition, so if we were lucky and there were others, they should be equally well preserved. Could this be the tip of the iceberg of the hypothetical discovery that had gotten Cordy killed? It was kind of a stretch, but here was the letter, and there couldn't have been many people who knew about the spot where it had been hidden.\n\nWhere would Cordy have found this, and why would she have removed it and hidden it? I would have guessed that if she'd found it, she would have trumpeted her discovery to the press and the world. Kate, stop and think about this. If she had found the letter\u2014and its siblings, if there were any\u2014at the Barton house, she would have had no legal claim to them. But since no one knew they existed, she could have stolen them and pretended to find them somewhere else\u2014like in the house where I now sat. Right time frame, and Henry Barton hadn't started out rich. Maybe this had been a boarding house right after the war. Or Cordy could have taken word of her prize find to the corporate investors she was courting and said, Look what I've got, this could be a big draw for your new conference center, maybe you could interest the medical community in staying here. Or she could have approached the town committee and told them that she had found something important, but she wasn't going to reveal it until she got the green light to move ahead on her own version of the project. Or, heck, she could have just taken them and sold them on the open market. I had no idea what Clara's letters might be worth if sold openly, or if their value would be increased if there was an actual collection, not just a single letter.\n\nToo much I didn't know. But I did know, selfishly, that such a collection could be a real attraction for history and Civil War buffs who came to visit Asheboro. They were worth more than just money to me\u2014and to the town.\n\nI glanced at my watch and realized I'd better hustle if I was going to meet Josh on time, but at least now I could tell him I thought I knew what we should be looking for. I debated briefly about taking the letter with me, but it was old and fragile, and I didn't trust myself to transport it safely. If Josh really wanted to see it, he could come back with me later in the day. I replaced it carefully it the envelope in which I'd found it, put the envelope in the secret compartment, and pushed it shut firmly, then stepped back to make sure it was concealed once again.\n\nThen I darted upstairs to put on the only pair of jeans I'd brought, a faded sweatshirt, and my running shoes. I checked my bag to make sure all the keys I had were where they should be. Then I added a camera, just in case we found something important. I looked at all the doors downstairs to be sure they were locked, then took off in my car.\n\nJosh was waiting on the front steps when I pulled in. \"Am I late?\" I asked, when I opened the car door.\n\n\"No. I was just enjoying the spring weather.\"\n\n\"Have you stopped locking the front gates?\"\n\n\"Over the last few days, there's been a lot of coming and going\u2014you, the police, Cordy, a killer. It hardly seems worth it. It's more a psychological barrier than a practical one.\"\n\n\"So you don't get cruising kids out here anymore?\"\n\n\"Not that I've noticed. Now they tend to head for the city if they want to get rowdy.\" He stood up. \"Ready to dig in?\"\n\n\"Yes, but I've got something to talk to you about.\"\n\nJosh sighed. \"Why does that always sound ominous?\"\n\n\"This time it's good news. Maybe.\"\n\n\"Then let's go inside.\"\n\nI followed him to the opulent dining room and we sat at the table as we had the day before. I took a deep breath and launched into my discovery. Josh didn't interrupt. When I ran out of steam he asked, \"So you're guessing that this was Cordelia's big discovery? Clara Barton slept here?\"\n\n\"I don't know where it fits in the realm of big discoveries, but would Cordy have thought it was important? Or anybody else around here with whom she shared that information?\"\n\n\"I can't really say. I know Clara Barton was active for quite a while, but she was involved with humanitarian concerns, not industrialization, so she wouldn't have turned up in my own research. But I do know she was a very public figure, and I'm sure there's plenty of information on her available.\"\n\n\"And you don't know anyone in the business of buying and selling artifacts and memorabilia, I assume?\"\n\n\"That would be correct. I could ask around. Look, if it's just the one letter you found, it might not be worth a whole lot on its own. If it's part of a larger collection, that could be a different story. Either way you'd have to have it authenticated. And the title might be murky. If it was found at the B&B, it could have belonged to Cordelia, and then it would go to Ryan, since he still owned the underlying property. If it came from this house, we'd have to assume it was addressed to Henry, and he left everything to the bank, which sold it to the town, so now the town owns it. As well as any others that might exist, if they're from the same cache.\"\n\n\"Sure, I get that. But it could be worth something to the town beyond its monetary value. Anyway, we're theorizing ahead of our facts. Today we go up to the attic and we look to see if there are footprints, or if any of the items look like they've been disturbed or if anything has been removed. And if we're really, really lucky, we'll find that Clara Barton sent more than one letter to her dear cousin Henry, and there will be a shoebox full of them tucked away somewhere. By the way, what would we do if we did find them?\"\n\nJosh looked troubled. \"Normally I'd say leave it where we find it\u2014if we find it\u2014for the sake of provenance, but with all the turmoil lately that doesn't seem very safe. We could take it back to the B&B and hide it where you found the one letter.\"\n\n\"Yes, but the accomplice could already know about that,\" I pointed out.\n\n\"We could take it to Ryan. He must have a safe or something for client papers.\"\n\n\"Are we sure he's not involved with this somehow?\" Was I sure?\n\nJosh looked exasperated. \"Do you have any better ideas?\"\n\n\"Maybe stash it at Lisbeth's house? I trust her, and nobody would think to look there. We'd have to make sure to put it somewhere where her kids wouldn't get to it, though.\"\n\nJosh stood up abruptly. \"Why don't we just go see what's up there? If we don't find anything, it's moot. I'm sure we can find someplace to hide one piece of paper.\"\n\nJosh was right: I was putting the cart before the horse. We didn't need a hiding place until we knew we had something to hide. I was kind of hoping we would find something.\n\nI followed Josh to the second-floor door that led to the attic. Stacked outside of it were a couple of large lights on stands, plus a long heavy-duty extension cord. Josh was quick to say, \"I haven't been up there since we talked last night\u2014you were right about looking for any disturbances, including footprints, so I thought I'd wait for better light.\"\n\nGood call on his part. \"I've already told you that Lisbeth and I were up here before, but we didn't exactly search the place. As you say, it's cold and dark\u2014I pity those poor servants who slept up here. But I still feel kind of silly, making this all cloak-and-daggerish. If I hadn't found that one letter, maybe I would have said, don't bother. I don't think we have anything we could take to Detective Reynolds\u2014he'd just laugh at us.\"\n\n\"We don't have anything yet,\" Josh corrected me. \"But we've got a whole attic to explore. This is original research at its best\u2014nothing's been touched since it was shoved up here, as far as I know. It's like a slice of Victorian life, even if there's nothing more than stacks of old magazines. And we're about to open the treasure chest.\"\n\nI had to smile. \"You really are excited about this, aren't you?\"\n\nHe smiled sheepishly. \"Well, yeah. My own research can get kind of boring, and I've been doing a lot of reading lately. I haven't managed to write much beyond a general outline and some draft chapters. So rooting around in a place like this is the perfect antidote, and maybe it'll jump-start my thinking process. Does that make sense?\"\n\n\"Actually, it does. And now you've got me drooling over the possibility that the wife's good dresses, complete with accessories, are lying peacefully in a handsome steamer trunk just waiting for me to bring them back to the light of day.\"\n\n\"That's the spirit!\" Josh said.\n\n\"I wish I knew more about the family connections. Clara Barton referred to Henry as 'cousin', but what did that mean back in eighteen-sixty-something? Was it a courtesy term, or were they actually related? I've got lots of questions.\"\n\n\"Look, we can stand here all day and talk about the 'what ifs' we'd like to find, but we may as well get up there and start looking.\"\n\n\"Lead the way!\"\n\n\"Here, you can carry one of the lights.\" He thrust one at me.\n\nI grabbed it and juggled it. \"Are there plugs up there?\"\n\n\"There was minimal wiring added around 1900 or just before, hence the one lonely light bulb up there. I've plugged in the extension cord on this floor\u2014it should be long enough.\"\n\n\"I brought a camera,\" I volunteered. \"So we can record the process. And any footprints we find.\"\n\n\"Great. Here we go!\"\n\nJosh trudged up the uncarpeted wooden stairs, wrangling the awkward light fixture with its tripod, and paused a couple of steps short of the top of the stairs. I almost bumped into him. \"What?\" I asked.\n\n\"I haven't been up here in a while\u2014usually I just stick my nose in to make sure there aren't any mice or squirrels or birds. But from this angle, I can see footprints.\"\n\n\"Some of them are going to belong to Lisbeth and me, but we did not follow the crates and trunks all the way to the end. Maybe halfway. Not yours?\"\n\n\"Nope, I haven't been up here, past where I am now. What I see are smaller ones, like a woman's. One woman, alone. You and Lisbeth seem to have shuffled around and made a mess.\"\n\nI ignored his comment. \"A few? Lots? Where do they go?\"\n\n\"Slow down, Nancy Drew. Here, pass me your camera.\"\n\nI fished it out of my pocket and handed it to him. It took him three seconds to figure out how it worked, and he began taking pictures from a low angle, turning slightly between each shot. Then he climbed up the next two steps and did the same thing.\n\nFinally he said, \"Okay, in answer to your questions, it looks like you and your pal milled around at this end of the attic, and checked out the rooms on the side. But if you go past that, you find the sets of single tracks, presumably Cordy's. I'd say she made two, maybe three trips\u2014there's not a lot of prints, and not much overlap among them. One trip was a scouting mission\u2014there's a line of tracks that run from one end of the attic to the other, and back. The second time she was more focused and went straight to one part\u2014the prints are all jumbled there.\"\n\n\"So that's where we should start,\" I said firmly.\n\n\"Let me get the lights set up so we don't trip over them,\" Josh said.\n\nI managed to control my eagerness while he first set up the light he was carrying, then came back and took mine from me and set that up too. There was a hanging bulb on a cord in the center of the attic, over the staircase, but it shed about as much light as a candle. Anybody looking for something would have had to bring a flashlight. In fact, that person would probably have known she would need a flashlight, because she had known how dark it would be up here. One more strike against Cordy.\n\n\"Okay, all set. Watch the cord,\" Josh warned.\n\nI stepped cautiously up to the top of the stairs and surveyed the attic that ran from front to back of the house. There was a small, dirty window set low in the gable at each end, but no other outside light source. Along the one side, under the eaves, boxes and trunks and odds and ends of furniture were stacked, surprisingly neatly. A path had been left down the center. Naturally nothing was labeled: the owner\u2014or his servants\u2014was supposed to remember where they'd stored their possessions. I wondered if Henry had been a hoarder and whether there would be any evidence from the house before he had remodeled it so extensively. Something to think about later.\n\n\"So, where did Cordy start looking?\" I asked. \"Wait, we really haven't looked at why she started looking up here. I mean, she wasn't an attic kind of person, right?\"\n\n\"As far as I know. What are you asking?\"\n\n\"Well, I would have expected her to look for silver or jewelry or maybe art first.\"\n\n\"None of which are left in the house. The bank trustees weren't stupid\u2014they made an inventory of the valuables and moved the good stuff to their vaults. Of course, for this stuff they most likely wouldn't have bothered with details, so they would have peeked in and written 'chest full of letters' in the inventory and left it at that. And not bothered to move it because it didn't seem important to them.\"\n\n\"Good to know. But Ryan said Cordy wouldn't know what to look for in books and documents\u2014that's more specialized. And it's not pretty or sparkly.\"\n\n\"Your point is?\"\n\n\"Someone must have told her to look in the attic to see if there was anything there that might be valuable, or historic, or both. And Cordy would have pouted and said, 'Why? I don't even know what to look for.'\"\n\n\"Ah, I see what you mean. Somebody else would have sent her up here and said, 'Look for old papers or boxes or whatever.'\"\n\n\"Exactly. So Cordy started with the easy stuff that she could reach. And it looks like she started with that big trunk in the back. You can see it's been pulled out of line so she could get the top open. Is the lock broken?\" I asked.\n\n\"Not that I can see. Maybe it was unlocked, or maybe she found the keys somewhere in the house. Anyway, it's open now.\"\n\n\"And apparently she didn't look any farther so she must have found something!\" I said triumphantly.\n\nJosh seemed to consider that idea. \"All right, say she opened it and did find a stash of Clara Barton's letters in the trunk. What did she do next?\"\n\n\"She took one letter for proof, to show her partner. She left the rest right where they were\u2014after all, they'd been safe here for over a century.\" I hoped I was right. It felt right, knowing what I did about Cordy's character. The bigger question was, who was pulling her strings? How did Cordy expect to benefit?\n\n\"You know, Kate, this is still speculation. We could just open the trunk and see what's in there,\" Josh said wryly.\n\n\"I'm channeling Carly Simon. You know, 'Anticipation'?\"\n\n\"Just do it, will you?\"\n\n\"What, no gloves?\"\n\n\"I'm a historian, not a forensic investigator. Don't touch anything you don't have to.\" I rubbed my hands together, approached the trunk head on, and tugged at the lid. It rose easily, which was encouraging. It was one of those lined with richly patterned paper, and it had a tray lying across the top. And nestled in that tray were several stacks of letters, bound with blue satin ribbon. To my unskilled eye, the handwriting matched that of the one letter I'd seen. \"Bingo.\"\n\nJosh moved quickly to stand beside me. \"Seriously? I thought we were just spinning a story. There really are letters?\"\n\n\"Looks like it,\" I said, grinning like a fool. I estimated that the three stacks might total maybe twenty to thirty letters. It looked like Cordelia hadn't dug into them, merely pulled out the top one to take with her. And if there were in fact that many letters, it was likely that this was more than a polite brush-off\u2014sorry, couldn't find your relative, the war took a heavy toll, et cetera. This would indicate a more personal correspondence, which to me was good news. \"Wow,\" I said reverently. \"Makes you wonder what else is hidden up here.\"\n\n\"One thing at a time. We came up here to look for possible evidence of a motive for murder, and it looks like we've found it. We can save the rest for later.\"\n\n\"Some researcher is going to have a field day up here.\"\n\n\"Like me, you mean?\" Josh grinned.\n\n\"Well, it's not me\u2014I'm no scholar. Tell me, Josh, when you took this job, did anyone explain to you why nothing was touched here? Henry Barton died without heirs, right? But he did make careful provisions for the care and ultimate disposition of his property, at least in the near term. Except he never thought that no one would want it.\"\n\n\"Practically speaking what was up here was most likely simply personal correspondence, discarded books, maybe memorabilia,\" Josh said. \"Nothing that would have been valuable in itself, I'd guess. Presumably he assumed that whoever moved in was welcome to it\u2014let them worry about clearing the place out. He was old by then, and there was no one to help him. Or he didn't remember, or he didn't care.\"\n\n\"Poor Henry. The more I learn, the more I want to know more about the man. I've got to get back to the library and dig into his background, his genealogy.\"\n\n\"What do you want to do now?\"\n\n\"You're deferring to me?\"\n\n\"Your find, your call.\"\n\nI thought for a moment. \"The fact that we found the letter at the B&B and matched it to this trove here suggests strongly that it is somehow involved with Cordelia and may have led to her death. Even though she might only have been considering a little theft. So logically we should run right over to the state police headquarters and hand them to Detective Reynolds, with our explanation.\"\n\n\"But?\" Josh said, smiling.\n\n\"I guess I'm afraid he wouldn't believe us and would send us on our way. Or he'd nod and smile and stuff them all in an envelope and shove it into the evidence box or locker or whatever and get around to actually examining them a whole lot later. I'd like to suggest one intermediate step: we get the letters authenticated. For all we know, they're a collection of recipes from Henry's aunt Bertha, and maybe she liked to sign her letters Clara Barton because she nursed him through measles when he was a child, and he saved them out of sentiment.\"\n\n\"A sensible idea, but it's still an open murder investigation, so we can't afford to send them off to some expert and wait for his opinion.\"\n\n\"Ah, but I have an idea. I know someone in Philadelphia who's the head of a museum there, and if she doesn't have someone on staff who specializes in authenticating this kind of document, she'll know someone who can. I know her well enough to say that it's urgent.\" Well, I did, sort of. I'd met her when I worked in Philadelphia and we'd hit it off and kept in touch since, getting together at conferences now and then. From what I knew of her, I had the impression she would care about the history of the letters. \"I can call her.\"\n\nJosh didn't look convinced, but neither did he object\u2014or propose a better idea. \"We have to set a time limit. If this pans out the way we think it might, Reynolds will skin us alive for sitting on the information.\"\n\n\"Well, it's still only a guess. I'll call her this afternoon.\" And pray that she wasn't on vacation in Hawaii or something. I took a deep breath. \"So, back to our earlier discussion about where to hide them, now that we know that somebody else knows they're here? Or at least, in this house.\"\n\nJosh rubbed his chin. \"My first impulse is to leave them here, and I'll keep an extra-close eye on the place in case he or she comes back. But I don't know what keys this person might have, or how desperate he is, and I really don't relish the idea of wrestling with someone in the middle of the night. And most likely the same person knows you're involved and knows you're staying at Cordy's old place, so that's out. Which brings us back to your friend's place, which the person most likely doesn't know about. Have you been to her house since you got here?\"\n\n\"Yes, including after Cordy died. I didn't notice anyone following me around, but I certainly wasn't watching for that. You really think I should keep my eye out for a tail?\"\n\n\"Sounds ridiculous, doesn't it? Say these letters are what we think they are. Even on the open market or at auction they might bring in a hundred thousand tops, I'm guessing. To some people that would be a lot of money. And somebody may have killed Cordelia for them. So, short answer: yes, make sure you aren't followed. Tell your friend Lisbeth not to say a word to anyone about the letters, and to put them in a very safe and not-obvious space. And if we're lucky, your Philadelphia friend will hop in a car and race down here to see the letters tomorrow, and we can get them off our hands.\"\n\n\"That works for me. Are we done here? Because I'd rather talk to Lisbeth before her kids get home from school. And then call my friend in Philadelphia.\"\n\n\"Makes sense. So you should go now. But first we've got to wrap up these letters, and I don't think a plastic bag from the market is the best choice.\"\n\n\"Let's go back downstairs and look for something better.\"\n\nI'd driven about halfway back to town before I started getting paranoid. Josh and I had found a wonderful cache of Clara Barton letters. That was good. But so far only Josh and I knew about it, well, apart from Cordy and her mystery accomplice, assuming there was one. How much did I know about Josh? He'd said he was fairly recently divorced, which explained why a not-young tenured professor was camping out in an apartment above the garage in a remote part of the state. Maybe. Sure, it would be a nice quiet place to work, and with the Internet he could do research anywhere. But how badly did he need money? Not that these letters were worth what, say, an undiscovered Shakespeare folio might bring, but he'd guessed somewhere in the six-figure range. Tempting. Well, not to me, but I was ridiculously honest. Most people weren't. Was Josh?\n\nJosh had outlined the pitfalls of bringing anyone else into our little circle. Yes, the police might not see our find as important, and they might sideline it for who knows how long. I thought it would be safe to say that no one on the police force would try to sell the letters, although they might not handle them carefully. I didn't think that fingerprint powder would be good for antique paper.\n\nAnd what about Ryan? He would have a different interest in the letters. I didn't know what his financial picture was\u2014maybe he needed money. I also didn't know whether he'd been in on Cordy's plan, divorced or not. Maybe there had been more letters, from Clara or from other noteworthy figures, that Cordy had already fenced. Maybe she'd demanded a bigger share of the profits and he'd bashed her. No, I wasn't prepared to trust Ryan, especially after he'd tried making moves on me and I'd turned him down. Which, of course, might put me in his line of fire. But, wait\u2014he didn't know I knew about the letters. That I knew of.\n\nThis was getting far too complicated. Then my anxiety ramped up another notch: Did I want to drag Lisbeth and her family into this? Did I have the right to even consider it? Sure, I could be careful handing over the goods\u2014we could don big hats and dark glasses and make the handoff in a caf\u00e9 three towns over. Stop it, Kate\u2014this is starting to sound like a bad 1960s spy movie. And Josh now thought I was headed for Lisbeth's house. He'd told me to be careful. But if he was the bad guy, he'd head straight there. Crap.\n\nWhat other options did I have? I could call Nell Pratt and hand them over to her\u2014I hadn't mentioned her name to anyone. All I'd said was that I had a friend who might be able to help. But Nell might not be available, or might not be willing to drive down here and back, or she might just laugh at me and tell me I'd gone crazy. Scratch that idea, even though I had to get her down to Maryland to see the letters, in the event that nobody murdered me and stole them overnight.\n\nGet a grip, Kate! I was creating problems where there might be none. I was sounding paranoid even to myself. And in this small town, I didn't know of anywhere to hide.\n\nCome up with a plan. I could take the letters back to Baltimore and stash them in the very secure safe at the hotel I used to manage. A desperate person might search my home but wouldn't be able to get to the letters in the hotel. I could leave a message for Detective Reynolds: Dear Detective, If I die mysteriously, there's an important clue in the safe at the Oriole Suites in Baltimore. Right. Then he would think I'd lost my mind. So I backtracked. The only option that seemed to make sense was to call Nell Pratt and somehow convey the letters to her. She could take them back to her museum, where nobody would be likely to look for them. She could get them verified while those of us in Maryland tried to figure out all the rest, including the murder.\n\nI had reached town, but instead of going to the B&B I pulled into a parking lot behind Main Street and retrieved my cell phone. I didn't have Nell's personal number, but it was easy enough to find her work number. I sweet-talked my way through her assistant, and after about five minutes I was talking with Nell Pratt.\n\n\"Kate!\" she said enthusiastically. \"To what do I owe the pleasure of this call?\"\n\n\"It's complicated, but I need your help. Do you have time to talk?\"\n\n\"Sure. What's going on?\"\n\nIt took me twenty minutes to explain the weird chain of events that had led to my request. She listened and asked some pointed questions, and finally said, \"Wow, you don't mess around, do you! You've been in that town for less than a week and you're in the middle of a murder investigation and a possible high-dollar theft of historic items. Not to mention juggling a handsome stranger and an ex-boyfriend. I wish my life was half as interesting.\"\n\n\"I seem to recall you've run into a few criminal speed bumps of your own. And a dash of romance.\"\n\n\"That's true on both counts. So\u2014what is it you think I can do?\"\n\n\"A couple of things, I guess. First, I need someone to tell me if these letters are the real thing. Second, I need to get them to some safe place that isn't in or near Asheboro, and I'm not even sure where that would be now, because I'm not sure who else is involved. I think I've got to get them out of my hands in order to do that, somewhere that no one will think to look.\"\n\n\"Have you considered that you may be at risk, even if you hand off the letters?\"\n\n\"You mean, someone might come after me thinking I still have them? No, I hadn't thought about that. Thanks for a new dose of paranoia.\"\n\n\"You don't think you can go to the police?\"\n\n\"I could, but I don't think they'd see why a bunch of old letters would be worth killing someone over. They'd just pat me on the head and tell me not to worry. That's why I thought of you. You understand their significance.\"\n\nAfter a long pause, Nell said, \"Where are you?\"\n\n\"About an hour northwest of Baltimore.\"\n\n\"Which means more like three hours away for me, by car.\"\n\n\"Look, I'm willing to drive to Philadelphia, if that will help. I'd really like to get these letters out of Asheboro, if you know what I mean.\"\n\n\"If you're okay with the drive, fine. Actually I live in Chestnut Hill, and I've got plenty of bedrooms. I can meet you at the house at, say, five?\"\n\n\"Great. Just give me the address and I'll figure it out.\"\n\n\"Done. And I'll check with my staff about the best person to authenticate your letters.\"\n\n\"I can't thank you enough, Nell. This whole thing has got me spooked. Maybe a drive will help clear my head, and when I get there you can tell me I'm being silly.\"\n\n\"If your gut tells you something's wrong, it's better to listen to it than ignore it. Maybe you'll feel stupid, but at least you'll be safe. See you later!\"\n\nI debated about stopping at the B&B to pick up some clean\u2014or cleaner\u2014clothes, but I decided I could pick some up along the way. Surely I could manage with a man's XL T-shirt as a nightgown for one night, while I washed the rest. And I wasn't a girly girl who needed my very own special shampoo or moisturizer. I could cope. I pulled out of the parking lot, filled my tank at a gas station on the way out of town, and hit the open road. I knew how to get to Philadelphia, and Chestnut Hill was just outside it. I could look up the details when I got closer.\n\nI opted to go by way of Harrisburg to pick up the Pennsylvania Turnpike and stopped along the way for a quick lunch, the aforesaid T-shirt and a toothbrush, plus a map of the greater Philadelphia area. Then back to the open road. I found Chestnut Hill with little trouble but made a few wrong turns on local roads there before I found Nell's house. It was a handsome midsize Victorian with a spacious lawn around it. And a good-looking man I didn't recognize standing on the front steps.\n\nHe grinned at me when I climbed cautiously out of the car and stretched, keeping one eye on him. \"Are you Kate? Nell told me to expect you. I'm James Morrison, her\u2014hell, what is the popular term these days? Partner? Cohabitor? You can pick whichever you like. And I'm also a Special Agent with the FBI, so you should be safe.\"\n\nI relaxed just a bit. \"Good to meet you. Did Nell explain my problem?\"\n\n\"Only in the barest terms. Come on in and bring the letters.\"\n\nI retrieved the precious bundle from under the passenger seat where I'd hidden them and followed him into the house, and then toward the back into the kitchen.\n\n\"You want a drink?\" he asked.\n\n\"Sure. But better make it wine, if you have it, or I won't be coherent for more than an hour.\"\n\n\"No problem. So, how do you know Nell?\"\n\n\"I worked in Philadelphia a few years ago, and we kept crossing paths. Somehow we managed to keep in touch after I moved to Baltimore. She's the only person I know in the museum community, so I called her when all this... stuff started happening.\"\n\n\"You don't have any contacts in Baltimore?\"\n\n\"Not really. I managed a small boutique hotel there. The closest I came to art or anything like it was choosing the paintings for the walls. I'm way out of my depth now, but I know Nell's Society has a lot of document collections and I figured somebody there should be able to help. Of course, I worked that part out before I got nervous about someone trying to get the letters back, and I didn't know who to trust.\"\n\nJames busied himself with opening a bottle of wine and filling two glasses. Without looking at me, he said, \"Someone died?\"\n\n\"Yes. The town prima donna, my high school nemesis, self-proclaimed femme fatale, et cetera. She'd been running a small B&B in town\u2014the town where we both grew up\u2014but she had bigger plans. Look, maybe I'd better save the story until Nell gets here, so I don't have to tell it twice.\"\n\nHe handed me a glass of wine. \"Makes sense. How did the woman die?\"\n\n\"Somebody bashed her head in with a rock from the garden around the Barton manor house.\"\n\n\"Barton as in Clara?\"\n\nFunny that he went straight to that point. \"Maybe. Probably. The house was built by one Henry Barton, but I don't know for sure if there's a connection between them and I haven't had time to find out. But that's where Cordelia died.\" I took a sip of the golden wine, and then another one. \"Damn, this is good.\"\n\n\"You have any idea who killed her?\"\n\n\"Nope. There are some in town who might say I have a motive, but I didn't\u2014really.\" When he quirked an eyebrow at me, I waved my hand at him. \"Long story, but not relevant now. The best idea I've come up with was that she was working with someone who double-crossed her, possibly in some theft from the Barton mansion. She could be a very annoying person.\" I took another sip of wine. \"In fact, nobody in town liked her, so there are plenty of suspects.\" I drained my glass and held it out to him. \"More, please.\"\n\nHe filled my glass without comment, for which I was grateful. That was when we both heard the sound of keys in the front door, and James went out to the hall to greet Nell. I wondered if he was going to warn her that I was delusional and half-drunk, but I was too tired and frazzled to care.\n\nNell came into the kitchen and immediately said, \"So it takes a murder to get you back to Philadelphia?\"\n\n\"Well, I've been busy. But now I don't have a job, so I've got plenty of time. As long as I'm not arrested for murder.\"\n\n\"Most of that sentence will need explaining. James, can I have a glass of whatever you're having?\"\n\n\"Of course. Should I order food?\"\n\n\"Please.\" She accepted a filled glass from him, sat down at the high island in the kitchen, kicked off her shoes, and said, \"Okay, start at the beginning. Or no, wait. You brought the letters?\"\n\n\"Yup. Are you going to kill me for them and bury me in the backyard?\"\n\n\"They aren't worth that much, Kate. But Clara Barton was an amazing woman, in her day or in any era. I'd like to see them, and then we can put them somewhere safe.\"\n\nI pulled out the package I had slid into my tote bag, but before I could open it, Nell said firmly. \"No, not here. It's not clean. Let's wash our hands, then take them into the dining room and use the table there. Technically we should be wearing cotton gloves, but I think we can skip that this once.\"\n\n\"Yes, ma'am.\"\n\nWe dutifully washed and dried our hands, then went to the dining room and spread out the letters across the polished mahogany surface. Some had envelopes, but most did not. Nell scanned them as a group first. \"Definitely all the same general period. The paper looks right. And from what I can see, they were all written to one person\u2014a Henry Barton?\"\n\n\"Yes. It was his house where they were found, in a trunk in the attic. He bought the place sometime after the Civil War, built a factory in town, apparently made some good money, then remodeled the house. Now it belongs to the town, but they only acquired it recently, and nobody has been through it for years.\"\n\n\"Interesting. How much do you know about Henry Barton?\"\n\n\"Not much, even though I grew up in the town. He was kind of a recluse, and he's been dead for over a century. I was planning to do some genealogy digging, but I haven't had the time.\"\n\n\"Why were you there at all?\"\n\nI filled her in on the details of how the town had ended up owning the building and all the rest\u2014a story I was getting far too used to telling.\n\nNell nodded, but her eyes never left the letters. \"Have you read any of these?\"\n\n\"Only the first one that I found. It looks like Henry had asked Clara to help find his brother, who hadn't been heard from since the war.\"\n\n\"Are you familiar with the Search for Missing Soldiers? The organization in Washington organized by Clara Barton?\"\n\n\"Never heard of it. What is it?\"\n\n\"Something Clara Barton put together right after the war, with the support of Abraham Lincoln. Most people have never heard of it. But it was really something special, particularly in a time when women didn't usually hold jobs. She tracked down something like twenty-two thousand soldiers so she could tell their loved ones what happened to them, and she answered queries about a lot more. And although she was from Massachusetts, she helped a lot of families from the South as well as the North.\"\n\n\"And all anybody ever talks about is the Red Cross part?\"\n\n\"Exactly. And it looks like you have a capsule history of the search for one missing soldier.\"\n\n\"You think they're authentic?\"\n\nNell nodded firmly. \"I do. And I know who'd love to see them. Once we figure out who killed your not-friend.\"\n\nI crashed pretty quickly after supper. I felt bad about that for maybe three minutes, since Nell was being so kind to put me up for the night unexpectedly, but it had been a long and exhausting day. The wine might have helped a bit too.\n\nI woke up early the next morning. It was good to have an ally like Nell, one who had zero connection to Asheboro or any of the people in it. Plus she knew enough about history in the region, and documents, and authentication to be worth her weight in gold.\n\nI had not counted on James the FBI agent. I hoped that now that he knew what was going on, he didn't have some sort of legal responsibility to do something about it. This whole mess was complicated enough without bringing in another agency, which probably wouldn't make anybody happy.\n\nI rolled over and stared at the ceiling, admiring the moldings. The Victorians sometimes skimped on the fancy details on the second floor, saving all their efforts and money for the more public spaces, but not here. It was consistent throughout the house. What now? I'd left town in a hurry, without notifying anyone, including Detective Reynolds. But he'd already cleared me, right? He'd never said \"don't leave town.\" Lisbeth might be worried if she couldn't track me down, but she had my cell number. The first step for today was to hand off the documents to Nell or the colleague of her choice. They'd be safe, and they'd be authenticated. I didn't have to tell anyone where the letters were. Second... and there I stalled. Cordy's murder was still an open case, unless something wonderful had happened since I left town less than twenty-four hours earlier and it was all wrapped up with bows on it. I wanted to ask James if he had an opinion, but at the same time I didn't want to get him more involved than he already was. Plus as far as I knew, he had no jurisdiction in the matter. Unless investigating the theft or sale of stolen items of historical value was included in his job description?\n\nBut surely we could figure this out on our own. Asheboro was a small town, one of those where everybody knew everybody else. I hadn't been kidding when I said that nobody in town liked Cordy, but that didn't mean that a lot of people hated her enough to kill her, or would even know how to do it if they wanted to. Still, a rock was a pretty unsophisticated weapon\u2014good for a spur-of-the-moment crime, an unthinking act of passion or anger. And this wasn't something any person would carry around in his pocket, just in case the opportunity presented itself. It had been sitting there, handy, and someone had chosen to use it. Who in town would react that strongly? Man or woman? Had Cordy been seeing anyone lately? Would he admit to it if she had been, under the current circumstances?\n\nI was getting nowhere, unless depressing myself counted. I was relieved when I heard a quiet rapping on the door. \"Kate? You awake?\"\n\n\"Come on in, Nell,\" I told her. That giant T-shirt concealed all there was to conceal.\n\nNell slipped into the room. \"Just wanted to let you know that there's coffee and food, if you want it. We didn't have much time to talk last night, so I didn't know whether you were a morning or an evening person.\"\n\n\"Both, as needed\u2014you may have noticed that hotels never really sleep, especially if you're catering to rich and demanding clients. Left to my own devices, I'm more of a morning person. Coffee sounds great.\"\n\n\"Well, then, get yourself sorted out and come downstairs when you're ready.\"\n\n\"I will. Uh, Nell? I didn't know about James\u2014I mean, that you were in a relationship with an FBI agent. He doesn't have to jump in and take over, does he?\"\n\nNell shook her head. \"He's not that kind of guy. If you ask for help, he'll give it freely, but he won't insist on running the show. And he won't tell anybody else about it.\"\n\n\"That's good to know. If he doesn't mind, I'd welcome any suggestions he has. I am completely out of my depth.\"\n\n\"Been there, done that. See you downstairs.\"\n\nGetting dressed didn't take long since I had few choices. I washed my face and brushed my hair and declared that it was as good as it was going to get. Then I meandered downstairs to yet more lovely moldings, inlaid floors, brass sconces\u2014I was getting jealous.\n\nJames and Nell were seated at the island in the kitchen, and the coffeepot was sitting between them. \"Good morning,\" I said. (I had never claimed to be a brilliant conversationalist first thing in the morning.)\n\nJames's mouth was full of toast, but he raised his mug at me.\n\n\"Toast? Eggs?\" Nell asked.\n\n\"Toast is fine.\"\n\nNell got up and started retrieving bread and butter and jam and honey and whatever else looked good to her. Over her shoulder she said, \"James and I were talking last night, after you went upstairs.\"\n\n\"What did you come up with?\"\n\n\"Well, what I was hearing from you was that you wanted to keep the letters safe, first and foremost, which of course I applaud. There's one person I'd really like you to talk with. She doesn't work for me, but she did an internship at the Society a couple of years ago. She's working on a graduate degree at Penn on working women in the later nineteenth century, so of course she knows a lot about Clara Barton. I can connect you two if you want.\"\n\n\"Sounds great. And apart from confirming the letters really are from Clara, maybe she can put them in context, on where Clara was in her own career, and what the dating of the letters would be. You see, Henry Barton is still a mystery to me, and maybe your friend can point me in the right direction.\"\n\n\"You're more than welcome to use whatever resources the Society has.\"\n\n\"Thank you, and I may want to take you up on that at some point. Right now I feel that I need to be in Asheboro to help sort out Cordy's death. But assuming that can be accomplished fairly quickly, there's another project I'd really like your input on. Could you come down to the town for a day, maybe? I'm staying at what used to be Cordy's bed and breakfast, so I can put you up for the night.\"\n\n\"I'll have to check my calendar, but I think I could manage that. Are you going to give me any hints?\"\n\n\"Sure. I want to suggest to the council that they turn the central part of town back to its Victorian form, and market it as an authentic Victorian village. The good\u2014or bad\u2014thing is, it wouldn't take much work, since nobody's done any major building since 1900. Most of the original store fa\u00e7ades are just covered by later paint or siding, which would be easy to remove and wouldn't cost much.\"\n\n\"Interesting,\" Nell said thoughtfully. \"I'd love to kick ideas around. I take it nobody's got a lot of money to pour into this?\"\n\n\"Nope. They blew it all on buying the Barton house. They can't raise taxes because there's no tax base left\u2014people are moving away from the town to where the jobs are.\"\n\n\"Would that house be part of this project?\"\n\n\"I hope so. It's kind of the centerpiece of this project, even though it's not exactly in the middle of town. But I think Henry jump-started the town. He moved there after the Civil War and built a factory, and that kept the town going for a long time. But he died, and the factory went into a decline, and whatever glory days there were faded away.\"\n\n\"Let me guess\u2014you're hoping that connecting Clara Barton's story with Henry could give your project a boost?\" Nell said, smiling.\n\n\"I won't deny it's crossed my mind, but we only found the letters yesterday, so I'm not sure how that would fit. But my gut tells me it could be useful.\"\n\n\"I'll admit I'm intrigued. James, you've been awfully quiet. Are you thinking or just digesting?\"\n\nJames smiled at Nell. \"This really is your turf more than mine. I have no place in the murder investigation, and it's not clear whether there's a theft involved in the letters, since you said you found them where they appeared to have been for a century.\"\n\n\"True. I did find the first letter in a different location, and one could argue it was stolen, but I reunited it with the rest. I don't think anybody's going to accuse me of stealing the lot of them\u2014technically I think they belong to the town, and they more or less gave me the run of the place, but maybe Nell or her friend should give me a receipt, just in case. But one concern we did have was that maybe Cordelia and her invisible accomplice had been stealing documents from the attic for a while. After all, nobody really knew what was up there, so they couldn't tell if anything was missing. And\u2014Nell, you can correct me if I'm wrong\u2014as long as a person is discreet about selling these kinds of items, and not just dumping them all on eBay\u2014nobody is likely to notice. Is that accurate?\"\n\n\"Close enough,\" Nell agreed. \"It happens. But let's stick to the one batch we know about. I haven't read through them all, but they look to be a more or less complete series, from Henry's first contact with Clara to when she finally and regretfully declares there is no more she can do\u2014or she finds the missing brother. Do you happen to know?\" When I shook my head, she said, \"You know, you might do well to look up wills for the family.\"\n\n\"Great idea, Nell, but I don't know the brother's name, I don't know for sure what state he was born in or what his parents' names were. Henry had no heirs, so the story stops with him. But I'll get right on it.\"\n\n\"It was just a suggestion,\" Nell told me, buttering her toast.\n\n\"I'm sorry,\" I said, immediately contrite. \"I really am looking forward to getting to know Henry better. I feel guilty that I lived in Asheboro for years and never knew anything about the man.\" I turned back to James, who was absorbed in reading the paper. \"So, James, if I do find evidence that anything was stolen, do I get in touch with you?\"\n\n\"Sure. But get that murder cleared up first.\"\n\n\"That's my plan. Nell, what do you want to do next?\"\n\n\"How long have you got?\"\n\n\"Well, I kind of left town without telling anyone, so I should get back sooner rather than later before someone starts worrying about me.\" And who would that be? Lisbeth? Josh? \"Although if either of you has a hot lead, I could be persuaded to stay.\"\n\n\"Let me call Carroll and see when she's available.\" Nell stood up, grabbed her phone from a table, and walked toward the front of the house.\n\nI poured myself more coffee and asked James, \"Am I your typical houseguest?\"\n\n\"I think you're our first houseguest\u2014we haven't lived in this house for long. But your story is certainly interesting.\"\n\n\"I can't believe that all I did was say yes to an old friend, when she asked me to talk to the town council. And that was barely a week ago! And you know what? I feel kind of pissed off that I will never be able to resolve my conflict with Cordy. I haven't given you all the details, but let's just say that she made my senior year in high school very unpleasant, and I fled to college and never looked back. But as an adult, I would like to have had the chance to move past that.\" I had to wonder if somebody had had a worse time than I had with Cordy, back in high school, although it was hard to imagine.\n\n\"Maybe that's why you want to see her murder solved,\" James said. \"It would give you some kind of closure.\"\n\nI hadn't looked at it that way. \"Maybe,\" I agreed. \"Anyway, since my job evaporated, I can reexamine my life and decide where I want to be next.\"\n\n\"You seem very excited about this Victorian village concept.\"\n\n\"Do you know, I am. It just sprang into my head, but I can see it so clearly. And this Clara Barton angle plays so neatly into it. And I'd like to know more about the factory and what Henry was making, And...\"\n\nNell walked back in. \"Carroll can meet us for lunch today, if that works for you.\"\n\n\"Great. How much did you tell her?\"\n\n\"Not a lot\u2014I'll let her draw her own conclusions.\"\n\n\"Good. And will you take the letters so they'll be safe?\"\n\n\"Of course. I really don't think your detective would make much of them, and I shudder to think what kind of damage his crew would do if they looked for fingerprints.\"\n\n\"Then I'd better call my friend Lisbeth and let her know I'll be back in Asheboro in time for dinner.\"\n\nLike Nell before me, I located my cell phone and wandered toward the front of the house to call Lisbeth. She answered on the fifth ring, sounding breathless.\n\n\"Kate? You're not dead? I tried your phone and the B&B line, and you weren't answering.\"\n\n\"I'm sorry! I'm in Philadelphia, which W. C. Fields might have something to say about. I'm fine, and I'll explain when I get back. Has something happened?\"\n\n\"No, not really. Nobody's solved the murder yet, if that's what you're asking.\"\n\n\"I came up here to follow up on a new angle on that. I hope I didn't alarm you but I wanted to keep it kind of hush-hush until I knew more.\"\n\n\"You don't have to apologize to me\u2014I'm just glad you're all right. When will you be back? Or should I say, will you be back?\"\n\n\"What, you think that pesky murder turned me off? Don't worry about it. I hope to be back there in time for dinner.\"\n\n\"Are you angling for an invitation?\"\n\n\"Are you offering one?\"\n\n\"Of course, as long as you can deal with the kids.\"\n\n\"I'll manage. See you six-ish, if all goes well.\"\n\n\"Will you be able to talk about this lead you're following?\"\n\n\"I really don't know.\"\n\n\"Are you driving?\" I asked Nell. \"Because I usually get lost in Philadelphia\u2014all those one-way streets mess me up. When I worked in the city I took public transportation.\"\n\n\"Don't worry\u2014I told Carroll to meet us at the Society, and I can drop you back at your car when we're done.\"\n\n\"Sounds good.\"\n\n\"You ready to go?\"\n\n\"I guess so. You have the letters?\"\n\n\"I do, in an acid-free folder. No sense in exposing them to any harm now, after all these years.\"\n\n\"You're the expert.\"\n\nThe ride into the city was surprisingly quick. Nell pulled into the parking lot across the street from her building and parked. I saw a woman standing next to the door under the columned portico. \"That's Carroll waiting for us,\" Nell said.\n\nNell got out of the car and I followed her across the street and up the steps. \"Carroll, thanks for meeting with us on such short notice,\" Nell greeted her. \"Let me disable the alarm and get the door open and we can go in and get comfortable. This is Kate Hamilton\u2014she's the one who found the letters I told you about.\"\n\nCarroll Peterson turned out to be younger than I expected, but Nell had said she was working on a graduate degree. I stuck out my hand. \"I hope you can help. It's a kind of complicated situation, and the letters are involved in a weird way. I just wanted to make sure we know what we're dealing with, and also that they're safe.\"\n\nCarroll shook my hand. \"Hey, I was intrigued by what little Nell told me. And they're part of a murder investigation? That is so cool.\"\n\n\"I guess.\" I suppose I would think it was cool too, if I wasn't smack in the middle of it.\n\nNell held the door open. \"Ladies, come on in. I thought we could set up in the old boardroom. There's a big table and good light. But I want to dust the table first. I'll get some cotton gloves too. Kate, why don't you explain the background to Carroll? I've already heard that part of the story.\"\n\n\"Happy to,\" I said. Carroll led the way to the ground-floor room, behind an imposing staircase. It was a handsome space, although it looked as if it didn't get much use.\n\n\"Sit, please,\" Carroll said. \"So, what's the story?\"\n\nI led her through what little I knew about Henry Barton and his missing brother, and the possible link to Clara Barton. Carroll listened carefully.\n\n\"Wow,\" she said when I'd finished. \"Sounds almost like a movie\u2014the body on the steps, a slew of possible suspects who hated the victim, the unexpected discovery. Do you have any reason to believe anybody else knows about the letters? And whether there's anything else hidden in that attic?\"\n\n\"It's our guess that the victim\u2014Cordelia\u2014was working with an accomplice, although whether she found the letters and told him or her\u2014or whether that person suspected there might be something up there and sent Cordelia back to look for more is not clear. Either way, she was the kind of person who didn't like to get her hands dirty, but she probably would have liked the money if she could sell some unknown letters by somebody famous. Maybe she got greedy or threatened a bit of blackmail. We really don't know, but now at least we have the letters. Unless, of course, you tell us that they're fake.\"\n\n\"What did Nell tell you about what I'm working on?\"\n\n\"Not much\u2014we haven't had a lot of time to talk. She's the only museum professional I could think of on short notice, so I bolted out of town with the letters.\"\n\n\"Ooh, more cloak and dagger stuff. I love it! Anyway, I'm working on a degree in Museum Administration, and my primary research interest is the change of women's roles after the Civil War. Obviously Clara Barton jumps to the head of the line there, at least in terms of her visibility and ultimate success. She's still nearly a household name, which is more than you can say for a lot of Civil War generals and the like.\"\n\n\"I will plead ignorance. What do I need to know?\"\n\n\"She fought hard to be allowed to help the Union soldiers on the battlefield\u2014and I mean literally on the field. Seems like the North hadn't quite thought through what to do with thousands of sick and wounded\u2014it's a depressing story. Clara stepped up, identified the most pressing needs, and set about organizing and collecting supplies and the like. She met with a lot of resistance from the military and the government, but she was a very persistent woman. She made a real difference.\n\nAs you can imagine, as the war wound down she was kind of adrift, after years of intense work and effort, so she identified another need: finding what had happened to soldiers who had never come home, and there were quite literally thousands of them from both sides. So Abraham Lincoln authorized her to set up an office in Washington, staffed with women, and they spent a couple of years tracking down the missing men. She looked for something like sixty thousand in a very short time by any era's standards, and she managed to locate over twenty thousand of them. All this from a tiny third-floor office that's still there. It was only a couple of years ago people realized that, and now it's on the National Historic Register. I'm going to guess that the letters you found relate to your Henry Barton's request for help to locate his brother. It's just icing on the cake that Clara Barton may have been a relative.\"\n\nNell reappeared. \"Sorry it took so long\u2014would you believe I couldn't find the cleaning supplies? I've got some lint-free cloths here. Just dust off the tabletop and we can dig in\u2014carefully! Did you explain your interest to Kate, Carroll?\"\n\n\"I did, and her story\u2014or her guesses\u2014dovetails neatly.\"\n\nWe all started dusting. Funny how three professional women trying to solve a murder started out by cleaning house. \"I don't mean to be crass, but are these particularly rare or valuable?\"\n\n\"There's no simple answer to that,\" Carroll said. \"Clara was a prodigious letter writer, even on the battlefield, so there are many around. The Library of Congress has a large collection. They come up at auctions now and then and sell well. From what Nell said, it might be that your letters\u2014or rather, Asheboro's\u2014cover a single investigation, from start to finish. I'd have to see if such a complete set exists somewhere else. The personal connection might bump up the value more. But I hope that's not your primary concern?\"\n\n\"No, except on behalf of the town. But I'd much rather see them installed as some sort of display relating to the history of the house and the town. I think it would be a terrific draw if we develop the house and the town as a tourist destination.\"\n\n\"Don't display the originals! Replicas, please!\" Nell and Carroll spoke simultaneously.\n\n\"Gee, and here I thought I'd just tack them to a corkboard. Just kidding! Are we ready to look at them?\"\n\n\"Put your gloves on,\" Carroll said firmly. \"Then sort the letters by date, where possible. We can insert any undated ones later.\"\n\nWe spent a very quiet fifteen minutes arranging the letters in order\u2014altogether they took up more than the length of the conference table. It was hard to resist the urge to read them, but I hoped that would come later. Right now the need to avoid damaging them was foremost.\n\nFinally Carroll stepped back and said, \"I think that's it.\"\n\n\"Are they...\" I began.\n\n\"Original? They certainly appear to be. The paper and ink are right, and the handwriting looks like other Barton letters. Nell, I'm sure you can find someone to confirm that.\"\n\n\"Sure, no problem,\" she said quickly. \"What would you suggest we do now?\"\n\n\"Apart from putting them in a safe place and not telling anyone where they are?\" Carroll smiled. \"Not that I expect the mystery killer to track them down here and burgle the place, but it can't hurt to be careful.\"\n\n\"Are you, we, going to read them?\" I asked.\n\nCarroll looked pained. \"I know you're curious to know what they cover, and so am I, but I think we need to have a conservator handle them first. What is it you want to know about the contents?\"\n\n\"Nothing that's urgent, I guess,\" I admitted reluctantly. \"It's not like the whole remodeling of the town depends on whether Clara Barton spent the night there. And we're a long way from approving any plans, much less starting on them. But I would like to know what she found. Henry Barton is pretty much a mystery himself in the town.\"\n\n\"Do you know if that missing brother was ever found?\" Nell asked.\n\nI shook my head. \"Using all the willpower I possess, I did not tear into the letters and look at each one. I didn't want to risk damaging any of them.\"\n\n\"Good for you! So here's what I'd suggest,\" Carroll said firmly. \"Nell, you look after the physical preservation and conservation of the letters. Kate, you go back to Asheboro and comb the town records, and that attic of Henry's, and see what you can find out about the man and his family, including that brother. And I'll do some research on the Search of Missing Soldiers\u2014the time line, the funding, the results\u2014and see where your letters might fit. Does that make sense?\"\n\n\"I was going to do the genealogy in any case,\" I told her. \"I've already asked our local librarian to pull together any references she might have. Nell, maybe you can point me to some online resources I can use, if I come up dry.\"\n\n\"Of course.\"\n\nI turned back to Carroll. \"What do you think I should do about whatever else is in the attic? I mean, these letters were in the top section of the only trunk we looked at. There could be a lot more things in there, and I know there are a lot more boxes and trunks.\"\n\n\"How's your security system?\" Carroll asked.\n\n\"Pretty pathetic. Nobody ever thought there was anything worth stealing. All the silver and artworks and any other valuables are in a vault in the town bank. I don't think anyone had looked in the attic for a long time, before Cordelia. But we now know or guess that somebody else knows something, and that person may come looking.\"\n\n\"Exactly. Did you mention a caretaker?\"\n\n\"Yes, but he's an academic, not a guard. Henry left enough money to keep the place maintained, and the caretaker works in exchange for the use of the rooms over the stable or garage or whatever it was. Josh is pretty scrupulous about checking the perimeter, making sure things are locked and such, but that may not be enough now.\"\n\n\"What did you say his name was?\" Carroll asked.\n\n\"Josh Wainwright. He said he teaches at Johns Hopkins, but he's been on sabbatical this year.\"\n\n\"Midsize guy, fortyish, glasses, beard?\" When I nodded Carroll said, \"I think I know him.\"\n\n\"This may sound ridiculous, but do you know him well enough to trust him?\"\n\n\"I don't understand your question.\"\n\n\"Might he have any personal interest in pilfering whatever historic documents he might find up there? I mean, he has free access to the whole place, and there's no inventory at all. He says he's never been up to the attic, and the footprints in the dust back that up, but now that he knows what's up there, he might be tempted.\"\n\n\"Hmm, I see your point. All I can tell you is that he's well respected in his discipline, and I've never heard anyone say anything negative about him. But I don't know him well.\"\n\n\"Well, he has an alibi for the time of Cordy's death, but that doesn't mean he isn't involved in stealing documents. Although we came up with the whole theft theory together, and he suggested looking in the attic. Am I being paranoid?\"\n\n\"You want us to come down and see what's what?\" Carroll asked.\n\n\"You'd probably have a better handle on what questions to ask than I do. And I've got free use of Cordy's bed and breakfast, if you want to stay. Both of you. We can have a slumber party and read Clara's letters out loud.\" I must have been getting punchy\u2014I sounded ridiculous even to myself.\n\nNell smiled and said gently, \"Take a step back, Kate. You've been living with this for a few days now, right? You're stressed out, and you don't know where to turn. Why don't you go back, if you feel safe enough, and Carroll and I will go over what you've told us, and figure out what to do with the letters, and maybe then we could join you? But I don't think it benefits anyone for the three of us to go running around the countryside in a panic.\"\n\nI took a deep breath to calm myself. \"You're right. You know, it's a kind of weird feeling. I mean, Asheboro was my hometown, the place where I grew up. Now I realize there was a lot of stuff I never knew or even thought about. And I don't know who to trust. I'd like to think a peaceful night would help me see things more clearly, but it still troubles me that Cordelia may have had an accomplice who may also be a killer lurking around the town, and now through no fault of my own I could be in his sights.\"\n\n\"I can't say that I blame you, Kate,\" Nell said. \"But I'm not sure what more we can do today. I would urge you to talk to the police who are investigating, especially now that the letters\u2014which still seem to make up the only tangible proof\u2014are safely out of reach. If your detective is a decent man, I'm sure he won't accuse you of obstruction of justice, or anything like that.\"\n\n\"I hope so. All right then. Carroll, thank you for your information. I'll keep it in mind when I start digging into the Barton family history. Nell, you said something about taking me back to my car?\"\n\n\"Oh, right, of course. Carroll, I'll walk you out. Oh, wait\u2014let me lock away those letters.\"\n\n\"You're going to keep them in this building?\" I asked anxiously.\n\n\"Yes, and I won't tell you where I put them, just in case the bad guys kidnap you and try to force you to reveal their whereabouts.\"\n\n\"You are too kind,\" I said sarcastically. She'd just given me another reason to feel paranoid.\n\nNell disappeared up the stairs, leaving Carroll and me staring at each other in the lobby. \"Bet you didn't think old letters could be so interesting,\" Carroll said.\n\n\"Can't say that I did. Oh, can I ask one favor?\" When Carroll nodded, I went on, \"Henry was said to have made his fortune in the factory he built in town, but I know next to nothing about the place, except that the building is still standing. I know Josh is looking into that, but if you have anything in your research that might shed light on any part of his efforts, industrial or commercial, could you share it with me?\"\n\n\"Of course. And all you've got is Henry Barton who fought in the Civil War, then moved to this small town in Maryland and bought himself a house, married someone named Mary, then built a factory and got rich, then completely remodeled his house, and left it to the town when he died. Have I got that right?\"\n\n\"More or less. And could you start with what he made there? Lisbeth and I keep saying he made widgets, but they don't really exist, do they?\"\n\nCarroll smiled and shook her head. \"Not specifically. Before the computer age they were usually some kind of small mechanical things that nobody knew or could remember the name of.\"\n\n\"Well, that fits.\"\n\nOnce we had retrieved Nell's car and navigated the downtown streets, she asked, \"Did that help?\"\n\nI shook myself out of my lethargy\u2014I was suffering from a post-adrenaline slump now that I knew the letters were stashed away. \"Goodness, yes! And thank you for getting Carroll and me together. It's a big weight off my mind to know that the letters are safe.\"\n\n\"Maybe you should make it known that you don't have them anymore,\" Nell said. \"If you seriously think someone might come after you.\"\n\n\"I don't know what I think anymore. In the clear light of day it seems absurd that anyone would want to do me harm for a bunch of old letters. And then I remember that Cordelia is dead, and the letters are real. Not priceless, but worth something. And it's a town that's struggling to keep its financial head above water. No money in the bank, a declining tax base, no new people moving in. They as much as told me I'm their last best hope. And I really do want to help, but it seems to get more complicated with each day.\"\n\n\"Will you accept Carroll's judgment about Josh Wainwright?\" Nell asked. \"That he's one of the good guys, or at least that it's unlikely that he killed Cordelia?\"\n\n\"I guess so. The detective accepted his alibi. It's possible he's in financial straits since his divorce, although I'm just guessing. He's doesn't seem like the type who would do something like this just for the money. And I really have to trust somebody. I can't do this all alone.\" Wow, I was beginning to wallow in self-pity.\n\n\"Well, when you see him, tell him you met Carroll and see how he reacts,\" Nell said firmly. \"And tell him you met with me, so he knows that someone else knows... Well, you get the idea.\"\n\n\"Yes, I do. Thank you. And if we all get through this, I really would appreciate it if you, or you and Carroll, could come down and give me some ideas about how to revive the town by going backwards in time.\"\n\n\"I'd love that. A field trip where I don't have to give a speech and don't need to ask anyone for contributions? Heaven.\" She pulled up outside her house. \"Keep in touch, will you?\"\n\n\"Of course. Thanks for the ride, and, well, everything else. Thank James for me too, and let me know if he comes up with any insights. Must be handy to be in bed with the law.\"\n\nNell grinned. \"It has its moments. We'll talk.\"\n\nI climbed back into my car and turned toward Asheboro. I was still smiling. I'd done one thing right: I'd found some more allies.\n\nOn the way back I debated about stopping by and telling Detective Reynolds what I'd just done. I vetoed it because, one, I was afraid he would yell at me, and two, I wanted to know if there were any new developments in Asheboro\u2014like something that would mean I didn't have to talk to Detective Reynolds. Besides, it was a long drive back. Maybe tomorrow.\n\nI pulled into town about five thirty and headed for the B&B. I'd call Lisbeth from there, but first I wanted a shower and a change of clothes. I let myself in the front door, and then I kind of stopped. I wasn't even sure why. I thought I'd purged my paranoia on the drive back, but maybe I was kidding myself. Here I was walking into a large building that I didn't know well, and there was nobody else there. Right? I listened, but didn't hear anything beyond the creaking of an old wooden house as the temperature dropped for the evening. All normal. I'd been so proud of myself about letting Nell know where I was staying, thinking I was somehow protecting myself, but when I thought about that in practical terms, if I was found with my head bashed in on the floor here (after who knew how long?), why would anyone think to call a museum administrator in Philadelphia? How likely was that?\n\nLisbeth had invited me to dinner. To hell with the shower\u2014I was going to head over there, where there was a warm, normal, chaotic family and light and activity\u2014and I'd think about coming back to the B&B later. Or maybe never.\n\nI turned myself around and marched out the front door, pulling it shut behind me and making sure the lock had engaged. Then I pulled out my cell phone and called Lisbeth. When she answered I said, \"Hey, I'm back. Is that dinner invite still open?\"\n\n\"Sure, come on over. If the kids get too noisy I'll just lock them in a closet\u2014you won't call the cops on me, will you?\"\n\n\"Of course not. They'd arrest me too, for concealing evidence or something. I'll be there in five.\"\n\nWhen I parked in front of Lisbeth's house, I could hear the racket of children from the driveway. I led a quiet life, and much of my time over the past few years had been spent trying to ensure that our hotel guests didn't hear anything at all in their luxury suites. That much chaos coming from two less-than-full-size people was startling. It took a minute or two of determined knocking before Lisbeth finally opened the front door. \"Kate! I thought I heard something. Where the heck have you been?\"\n\n\"I told you\u2014Philadelphia. Quick trip, there and back again within twenty-four hours.\"\n\n\"Do I want to know why?\" Lisbeth asked.\n\n\"Probably not, but it was a useful trip.\" I followed Lisbeth into her kitchen, where every countertop was covered with assorted cooking utensils, both clean and dirty. \"I'll give you the short version.\" At least I'd kept her safely out of the mess so far, and I didn't want to put her or her family at risk.\n\n\"Find a seat, if you can. Dinner's in half an hour, more or less. You want wine or something?\"\n\n\"No, it would probably put me to sleep in minutes. I'm not even sure how far back to go in explaining.\"\n\n\"Keep it short before the kids descend like a swarm of locusts. If a swarm can be only two.\"\n\n\"Mom?\" a voice came from upstairs.\n\n\"What, Melissa?\" Lisbeth called back, loud enough to be heard upstairs.\n\n\"The cat's barfing on the bed.\"\n\nLisbeth sighed. \"The joys of the domestic life,\" she said to me. \"Which bed?\" she yelled.\n\n\"Jeffrey's.\"\n\n\"Is your father up there?\"\n\nThere was a silence of about thirty seconds. \"Yeah. He was taking a nap.\"\n\n\"Well, tell him to clean up the cat barf. I'm talking with my friend.\" She turned back to me. \"One more crisis averted. You were saying?\"\n\nOh, what the hell. She was already involved in this. I took a few moments to condense the events and theories of the past two days into a very short version and then dumped it all on Lisbeth in a rush.\n\nShe didn't interrupt, but when I finished, or at least ran out of steam, she sat down across the kitchen table and said, \"Wow. All this started in sleepy Asheboro? Not just murder, but also a possible theft of who knows how many historic documents worth who knows how much money, and you had dinner with an FBI agent and a museum president? Makes my life seem pathetic.\"\n\n\"No way. You lead a normal life. I seem to have wandered into some weird alternate universe.\"\n\n\"I notice you didn't mention our detective.\"\n\n\"Nope. I thought about updating him, really I did, but the theories Josh and I cooked up were so fragile that I thought he'd just dismiss us, or the letters would get lost or damaged in police storage. I know the letters are safe for now, and if I have to explain later, I'll cross that bridge when I come to it. At least we're pretty sure they're authentic. And Nell and her friend could be really helpful if this Victorian village idea goes forward. Anything new on that front?\"\n\n\"Not really. Everybody got so flustered by Cordy's murder that they more or less forgot about it for now. I'm going to guess they'll schedule a regular meeting this coming week. You can present whatever you have then, or nothing at all\u2014I'm sure the council members will understand.\"\n\n\"We'll see. But to tell the truth, the murder aside, I'm really getting excited about the potential for this project. Now that I'm back, the first thing I have to do is dig into Henry Barton's history, because he seems to be the key to all this. So come tomorrow, I'm going to vanish into the library and dig.\"\n\n\"Did Audrey come up with anything good?\"\n\n\"I don't really know. I met with her once, and I kind of explained what I was looking for\u2014the true version\u2014and she pulled some good stuff together, but I haven't had time to go back since. Anyway, whatever I don't find in the local collections I can hunt for online.\"\n\n\"So you're sticking around Asheboro for a while longer?\" Lisbeth asked, stirring a pot.\n\n\"At the risk of sounding pathetic, I don't really have any reason to go back to my place right now. I've got free board at the bed and breakfast, and a full kitchen there, so why not stay? Oh, did I mention that Ryan tried to put the moves on me?\"\n\n\"No! Really? Are you, uh, interested?\"\n\n\"In getting back with Ryan? No way. But I didn't want to burn any bridges. Actually I think he was just testing the waters, in case I was still into him, but I don't think his heart was in it. Which is fine.\"\n\nLisbeth extracted plates from a cupboard. \"Dinner's ready\u2014I'm going to go call the rest of the gang.\" She set the plates on the table and went out to the front stairs, where she yelled, \"Dinner's ready! And we've got company, so be polite, kids!\"\n\nLisbeth's husband, Phil, was the first to appear, and Lisbeth immediately asked, \"Did you take care of the cat, er, stuff?\"\n\n\"Hello, Kate. Yes, Lisbeth, I did; I gather the kids were playing with the cat and he got a bit overexcited.\"\n\n\"Thank you, Phil\u2014you are a prince,\" Lisbeth said sweetly.\n\nThe kids came clomping down the stairs, battling over who could arrive first. \"Wash your hands and say hello to my friend Kate, please. You've met her before.\"\n\n\"Hello, Kate,\" the children chimed in unison. Then Jeffrey said, \"You the lady who's poking around that spooky house where the body was?\"\n\nI glanced at Lisbeth, but she only shrugged. \"Yes, that's me.\" I decided it wouldn't be wise to mention that I'd found the body.\n\nJeffrey looked disappointed. \"Was there any blood left?\"\n\n\"I didn't look. Are you planning to be a detective or a vampire?\"\n\n\"Can I be both?\" he asked eagerly.\n\n\"I won't say no,\" I replied.\n\nLisbeth set plates at each place around the table. \"Milk? Water? No soda.\"\n\n\"Aw, Mom!\"\n\n\"You heard me.\"\n\nDinner progressed in what Lisbeth seemed to find a normal fashion, and as I watched I wondered if I was missing something by staying single. I'd had relationships, but none had really stuck. Was I too smart for the men I met? Too successful? Or just too boring? All I ever did was work, and yet here I was, out of a job. But on the plus side, I was, as Nell had noted, really excited about the idea of bringing Asheboro back to life somehow, and I was kind of intrigued that the whole town had somehow forgotten Henry Barton, who had in large part shaped the town. Why? Sometimes I wished I had taken a class or two on urban planning when I was in college\u2014it would have come in handy more than once.\n\nBy the time everyone had cleaned their plate, I was beginning to droop again. I accepted Lisbeth's offer of coffee, since I still had to drive back to the B&B. She sent the kids off to bed, with instructions to her husband to make sure they brushed their teeth; and he should read them a story if he wanted any peace. Finally it was just the two of us in the kitchen.\n\n\"Welcome to my world,\" Lisbeth said softly to me.\n\n\"Go on, you love it,\" I told her. \"I like your husband, and the kids are good kids. If you get bored, you can get a job or volunteer when they're a bit older. So don't you dare apologize for your life. Listen, if this village thing ever happens, I'd love to have your help on it. There might be money in it, although we're a long way from finding any, but at least you could set your own hours and have some flexibility.\"\n\n\"Let me think about that, because you're a long way from making it happen. But I do admire your enthusiasm. If it does go forward, would you stay around?\"\n\n\"Probably. At least until it gets up and running.\"\n\n\"I'd like that.\"\n\nI checked the clock on the stove. \"I'd better head back\u2014it's been a long day. Not that I have any idea what I'm doing tomorrow, beyond going to the library. But if I do that I need to be able to stay awake, so I'd better get some sleep.\"\n\n\"Let me know if you want to go play or something\u2014as long as it's during weekday school hours.\"\n\n\"Will do. Thanks for feeding me.\"\n\n\"Happy to.\"\n\nI walked out into the cool night and breathed deeply. That could have been my life in there, if I hadn't been so determined to get out of town and make something of myself. I couldn't see myself in her place though. But Lisbeth seemed content, so I wasn't about to criticize. The world is full of choices\u2014and roads not taken.\n\nI drove home carefully and parked on the street in front of the B&B. It was only when I got out of the car that I saw a man sitting on the front porch. Damn, I'd forgotten to leave the outside light on.\n\nI gathered up my courage and walked up the front pathway, until about halfway there I recognized Josh. \"What the hell are you doing here? Something wrong?\"\n\nHe stood up. \"No. When I didn't hear from you, I got worried.\"\n\n\"Oh. Well, come inside and I'll fill you in, if I can stay awake long enough.\"\n\nI fished out my keychain from my bag and fumbled to get the key in the lock. And when I finally did manage it, I hesitated, recalling the weird feeling I'd had earlier.\n\nJosh noticed. \"Problem?\"\n\n\"No. I don't think so. Well, maybe. I'm not sure.\" I was not exactly articulate.\n\n\"You want me to go in first?\" he asked. I nodded mutely, so he reached around me and turned the doorknob. The door swung in smoothly, and the inside of the house was as quiet as when I had left it. He stepped around me and walked into the center of the hall, where he stopped. \"Come on in\u2014the water's fine.\"\n\n\"I am acting like an idiot,\" I said, as I closed the door behind me.\n\n\"Why were you scared? Did something happen?\"\n\n\"No, nothing at all. I drove back from Philadelphia, and\u2014\"\n\n\"Philadelphia? Your museum friend?\"\n\n\"Yes. I spent last night at her house. It was easier for me to go up there than to have her come down here.\"\n\n\"You're going to have to back up. You went to Philadelphia to talk to your friend. You took the letters with you?\"\n\n\"Yes. All of them. And I left them there.\"\n\n\"And this is why?\"\n\n\"Because the more I thought about, well, everything, the more rattled I got. You and I more or less agreed that there was somebody working with Cordy who was probably somehow involved with the letters, but I have no idea who, and I didn't want to put the letters at risk. So I just grabbed them and ran. To Philadelphia.\"\n\nJosh took a deep breath. \"I get the feeling that this is going to take more than a couple of minutes. Is there any of that wine left?\"\n\n\"You mean, have I managed to drink up the whole supply of premium vintages in three days? Of course there's some left. You want some?\"\n\n\"I think you need some. You fetch the wine, and I'll find some glasses.\"\n\nHe disappeared toward the kitchen, and I popped open the secret panel in the parlor. The wine was still all there. Unless somebody had swapped out all the bottles for something poisonous and then carefully replicated the labels and added just the right amount of dust to the bottles and... Kate, get a grip! I pulled out a bottle, more or less at random, and marched to the kitchen. \"Here you go,\" I said, handing it to him. He looked at the label, and then he looked at me, but I didn't volunteer any information and he didn't ask.\n\nJosh opened the bottle and poured two glasses. \"I'm sorry it hasn't had time to breathe.\"\n\n\"I can live with that.\" I accepted the glass he offered me, and took a sip. \"Nice. Too bad I can't get used to this.\"\n\n\"So, sit down and enjoy it now.\" Josh waved a hand at the table. \"And fill me in.\"\n\n\"Where do I start? Oh, by the way, Carroll Peterson says hi.\" I watched his face to see how he reacted to her name.\n\nHe smiled, and his eyes lit up. \"You met her?\"\n\nNothing suspicious about his response, I decided. \"She's a friend of Nell's. Nell's the friend I went to see, and she invited Carroll to join us. Carroll said your areas of research overlapped somehow.\"\n\n\"They do, although she takes more of a feminist slant. It makes perfect sense that your Nell would think of her when she learned about the Barton letters. What did Carroll say about the letters?\"\n\n\"I think she was really excited, but she tried not to show it. We laid out the letters in chronological order, and she said she thought it was the complete story of Henry's search for his brother, which might be unusual\u2014having the whole series, I mean. Although there are plenty of Clara's letters around\u2014apparently she wrote to everyone, all the time, even in the middle of battles.\"\n\n\"And where are the letters now?\"\n\nI wasn't going to share all the details, even if I trusted Josh. \"Carefully locked away somewhere in the Society for the Preservation of Pennsylvania Antiquities, or whatever the heck they call the place. And you are now the only person, other than me, Nell, and Carroll, who knows where they are. So the letters are safe.\"\n\n\"Are you sure we are?\"\n\nFunny that he should zero in on that particular point. \"From our mystery murderer? I don't know. I worked myself up into a lather on the way to Pennsylvania, because I hadn't even thought about that earlier. Oh, did I mention that Nell's partner is an FBI agent? So he's heard our story too, and he knows about the letters.\"\n\n\"Did he have any comments?\"\n\n\"Not yet, but I gather the FBI has some rules about not getting involved in somebody else's investigation unless they're invited, and only in special cases. He was deliberately very noncommittal.\"\n\nI realized my glass was empty. Josh noticed and refilled it. His was still half full.\n\n\"So everything was fine in Philadelphia\u2014no men following you or lurking in the shadows?\"\n\n\"Not a one. I was beginning to feel stupid for worrying, but it all came back while I was driving back here. You might have noticed that I don't handle being scared very well. I got as far as this place, and I opened the front door, and I got spooked about being alone here, so I bolted to Lisbeth's house. I didn't want to put her or her family at risk, and I really hope nobody thinks I might have given her the letters to hold. Like you and I talked about. But she noticed I was pretty jumpy, so I ended up telling her everything. Josh, am I going crazy? You think anybody is at risk?\"\n\nJosh took a sip of wine before responding. \"Frankly, Kate, I don't know how to answer that. We now know the letters are probably authentic, and that they have some market value. We know Cordy had one of the letters, and was most likely the person who removed it from the others in that trunk, as some sort of proof of something. We've made the leap of logic that she was working with someone else, since she didn't have the smarts or the contacts to sell them herself, and that person is quite possibly her killer. That person still doesn't know who else knows about the letters, and clearly no arrests have been made for the murder. So it's altogether possible that he is quietly looking for the letters now.\"\n\nThat did not make me feel any better. \"Has anybody tried to get into the Barton house, or into the attic?\"\n\n\"Not that I've noticed, but I don't watch the place 24\/7. And if this person is careful, he could conceal the fact that he's ever been to the house. Or here. Do you want me to search this place?\"\n\nI was surprised by how much I welcomed that idea. \"Could you? I hate it that I'm acting like such a wimp, but I guess murder makes me nervous. But I'm coming with you. No way are you leaving me here alone to listen to the house creak. Even with a good bottle of wine.\"\n\n\"Fine. Just stay behind me.\"\n\n\"Ooh, you gonna protect me, Mr. Professor? How are your martial arts skills?\"\n\n\"Kate, I'm only trying to help. Just say the word and I'll go home.\"\n\n\"No,\" I said in a small voice. \"Please don't. I apologize. I'm not myself these days. Damn that Cordelia\u2014is she going to hound me for the rest of my life? Even when she's dead?\"\n\n\"Apology accepted. You want to start upstairs or down?\"\n\n\"I don't know. You decide. Oh, do you want a weapon?\"\n\n\"What did you have in mind?\"\n\n\"Maybe a fireplace poker, if there is one? Or a small frying pan? God, I am such a clich\u00e9. Is there a more modern version of impromptu defensive weapons for hapless females?\"\n\n\"I'd go with the poker. Right there.\" He pointed at the fireplace in the dining room.\n\n\"Right. Thanks.\"\n\n\"I vote for starting downstairs.\"\n\n\"Okay,\" I said cautiously. \"Right behind you.\"\n\n\"We know the kitchen is clear\u2014unless somebody's messed with the lock on the back door. I'll check.\" He stood up and walked over to the door, then bent down and peered closely at the knob and lock. \"Looks all right. No suspicious scratches. Now the dining room.\"\n\nWe must have looked like something out of the Three Stooges, tiptoeing in single file into the dining room, Josh leading, me carrying the sturdy brass-and-iron poker. \"I didn't see anything out of the ordinary in the wine cabinet,\" I whispered.\n\n\"Why are you whispering? If there's someone here, he already knows we're here.\"\n\n\"Oh. Right. Anyway, short of dusting for fingerprints, I can't tell if anyone has messed with that cabinet.\"\n\n\"Okay. Then we should check the one in the library, where you found the first letter.\" He strode across the hall and turned on the overhead light from the wall switch next to the door. As soon as the light flooded the room, I grabbed Josh's arm.\n\n\"What?\" he demanded.\n\n\"Look at the paneling there. The one where the letter was hidden. It isn't completely closed\u2014you can see the crack from here.\"\n\n\"You're right. Are you sure you closed it when you removed the letter?\"\n\n\"I think I did. I don't remember how tightly it fit, and as you know, there's no exterior latch. But there was nothing inside it.\"\n\n\"Let me take a look.\" He crossed the room and crouched by the panel and opened it with one finger. I clutched my poker with a death grip.\n\n\"Anything?\"\n\n\"Not that I can tell. Inconclusive. Is there a cellar here?\"\n\n\"Like I'm supposed to know?\" I asked, a bit shrilly. \"I'm a guest here, more or less. I have not gone down to the cellar to check on anything. Isn't that what gets someone killed in every slasher movie ever made?\"\n\n\"Only if the person going down the stairs is wearing a skimpy nightie and clutching a candle. I suppose we can save that for last. Upstairs, then.\"\n\n\"After you.\"\n\nHe gave me a dirty look and started up the stairs. Several of the treads creaked, but I was expecting that. I tried to imagine our villain removing each tread to see if Cordy had hidden her finds under one, but that would have taken a lot of work, not to mention destroying the staircase. So very not like Cordy to find such a complicated hiding place.\n\nUpstairs there were plenty of bedrooms, and I'd explored only the one I was using and had peeked in Cordy's. Everything in the other rooms looked untouched, even to the light coating of dust, which had not been disturbed, even by the police. I wondered if Cordy had tacked up the name of a local housecleaner somewhere, and whether Ryan would foot the bill for cleaning the place.\n\n\"Which room was Cordy's?\" Josh asked.\n\n\"Oh, that's at the back. Very unlike her, but I guess she figured that the rental income was more important than her pride. Follow the scent of her perfume\u2014it's that door at the end of the hall.\"\n\nJosh didn't spend long back there. \"All clear,\" he said as he reappeared through the door. \"Although I'm pretty sure I'm allergic to that perfume, whatever it is.\" He fished out a handkerchief from a pocket and blew his nose.\n\n\"Is that a clue to anything?\"\n\n\"I doubt it. Anyway, nothing looks like it's been touched back there. Did the police check it out?\"\n\n\"I think maybe Detective Reynolds and his crews checked, but I left while they were here.\"\n\nI got bored after the next bedroom, and sat down in the middle of the hall floor with my legs crossed, clutching my trusty poker, and listened to Josh move quietly around the rooms, opening and closing doors and apparently peering under beds, if the creaking of bedsprings was any indication. Victorian furniture didn't offer many places to hide, except for the beds and any armoires. I think he covered them all. He was certainly thorough.\n\nFinally he emerged. \"Nothing. You sure you didn't hear any odd noises when you came in earlier? Smell somebody's perfume or aftershave?\"\n\n\"No, nothing specific. It just felt wrong when I walked in. Look, I'll admit it might have been just my paranoia. But I'm used to living alone, and I'm not usually like this. Of course, it's been a really odd few days.\"\n\n\"I get it, Kate. Attic?\"\n\n\"What about it?\"\n\n\"Is there one?\"\n\n\"Please refer to the part about cellars. I have not gone into the attic since I arrived here. I have not opened the door to the attic. For all I know, there may be bats up there. Or squirrels. I am not in the mood to have a confrontation with a bat or a squirrel. Please feel free to investigate all you want. I will remain here and fend off attackers from the rear.\"\n\n\"That gives me great confidence,\" Josh said solemnly, and disappeared up the stairs. I continued to sit, listening to his footsteps moving around above my head and wondering what I would do if he met someone up there, or if someone tried to sneak up on me. I was kind of ashamed of myself: I'd worked for years in a hotel in a big city that had its share of crime, but I'd always been careful, and I'd relied on the security guards at the hotel for protection, even if it was only to walk me to my car in the parking garage late at night. I'd never taken a self-defense class in my life. My plan if I was accosted on a dark street was to run like hell and scream as loudly as I could. Right now that didn't seem like enough anymore.\n\nI heard Josh coming down the stairs, and in the dim light I thought he looked concerned. \"Is there something I need to know?\" I asked quickly.\n\n\"No. And nothing appeared disturbed. Not even the dust.\"\n\n\"Okay, so I am now officially paranoid without any basis. Thank you for checking, anyway. I appreciate it.\"\n\n\"Glad to help. I have a stake in this too, you know. If something was taken from the Barton house on my watch, I might be liable in some way since I'm the official caretaker, and I think I signed some documents for the bank. The sooner someone goes through that attic there to see if there is anything else of value, the happier I'll be. Here, let me help you up.\" He held out his hand, and I untangled my legs and took it.\n\nHe must be stronger than he looked, because I was on my feet in seconds, and then somehow my momentum carried me forward into his arms. He looked as startled as I was.\n\n\"Did you do that on purpose?\" I asked, noting that I hadn't moved away from him.\n\n\"Would that be a problem?\" he said, with a slight smile.\n\n\"Uh, actually, no, I don't think so.\"\n\nI woke up with the sun the next morning, with only a vague memory of the rest of the night before. I was willing to bet that Josh and I had finished that bottle of wine; I was a little less clear about whether we'd opened another one. Apparently the combination of stress, fatigue, and Burgundy, with a dash of relief after our search of the house, had done me in.\n\nI heard footsteps on the stairs, but before I could panic Josh's face peered around the corner. \"Oh, good, you're awake. What do you want for breakfast?\"\n\n\"You're here,\" I said intelligently.\n\n\"Brilliant deduction, Watson. Coffee?\"\n\n\"You spent the night,\" I said flatly.\n\n\"Yes. It was late.\"\n\n\"Did we, uh, do anything?\"\n\nJosh laughed. \"No, we did not engage in any unseemly behavior. We did, however, polish off that bottle of very good wine\u2014downstairs, I might add\u2014and then it appeared unlikely that you could navigate the stairs, so I assisted you and tucked you in. Then I went looking for another bed because I didn't trust myself to drive. No one attempted to break in here while we slept the sleep of the just. Or do I mean the innocent?\" He was still smiling.\n\nAnd I was suddenly very conscious that I had been wearing the same clothes for two days without benefit of a shower and was suffering from extreme bed head. \"Yes to the offer of coffee. Let me get dressed and I'll join you downstairs. You'll have to improvise as far as breakfast goes.\"\n\n\"Got it. See you soon.\" He disappeared down the stairs.\n\nI had intended to spring out of bed and perform my ablutions, but I needed a minute to think about what had happened\u2014or not happened\u2014the night before. I had not been at my best, and that was before we opened the wine. I was unmoored, camping out in an unfamiliar place, with a murderer on the loose somewhere in the neighborhood, and an ex-boyfriend hovering around the edges, and I'd just lost my job, and... now I was wallowing in self-pity. Focus, Kate! This was not the time to consider a relationship with anyone. Or hop into bed with a man\u2014that wasn't my style under any circumstances, whether or not wine was involved. Did I like Josh? Yes, well enough\u2014he was mature, reasonably attractive, obviously intelligent. And single. But only yesterday\u2014or was it the day before?\u2014I had been wondering if he might be a killer or a thief. Potential mate material or combination killer\/thief\u2014I couldn't have it both ways.\n\nAll right: Get up, get clean, get dressed, and go find coffee and food. Everything always looked better with coffee. And I needed to find more clothes\u2014I rejected the idea of rummaging through Cordy's wardrobe, even if we might be the same size, which left the options of making a fast trip back to my condo to pick up more, finding a local clothing store, or doing a very fast load of laundry. Oh, right, that might mean going to the basement. It couldn't be too bad by daylight. Could it?\n\nI was downstairs in fifteen minutes, clean but not exactly glamorous, but why would I need to be? Josh wordlessly handed me a mug of hot coffee. Smart man. After a few sips I said, \"Do we have an agenda for today, or are you going back to your regular work?\"\n\n\"I should be working,\" he said, as he sat down across from me with his own coffee, \"but I'll admit this whole Barton thing has got me hooked. I think a trip to the library is called for.\"\n\n\"I agree, and that was my plan for today,\" I told him. \"But there are multiple aspects to that research, like Henry's family background, and whether he and Clara were connected and how closely. The building of the factory and the development of the town, and then of course, when the house was built, or rather, expanded, which was clearly some years after the Civil War ended, which tells you when he made his money. That side is more your department. I'd be happy to do the genealogy side of things, if you want to do the history and the local economics.\"\n\n\"That works for me,\" Josh said amiably. \"But what about the criminal side? There's the murder, and there's the theft. Are they related? Or coincidental? Do we have any suspects that took part in both, or are we really dealing with two different criminals?\"\n\n\"You are far too coherent for this early in the morning,\" I told him. \"Besides, that's not something we're going to find at the library. You said something about breakfast?\"\n\n\"Oh, right. Slim pickings, as you said.\"\n\n\"I wasn't expecting to stay here, if you remember. And there hasn't been time to go food shopping. Are you saying you can't throw together a breakfast using only the ingredients you can find in the pantry and kitchen? May I remind you that until a week ago this was an operating bed and breakfast?\"\n\n\"Is that a challenge?\"\n\n\"If you like.\"\n\n\"Then I shall rise to the occasion.\"\n\n\"And I shall observe and toss in pithy observations while you work.\" Gee, this was kind of fun. The man actually appeared to have a sense of humor, in addition to his other sterling qualities. Plus, he might even know how to cook.\n\nJosh appeared to have whipped up some kind of quick bread and was sliding it into the preheated oven when I heard a knocking at the door. \"Can you get that?\" he asked.\n\n\"Sure. You weren't expecting anyone, were you?\"\n\n\"Uh, nobody knows I'm here, remember? So it must be for you. Or maybe a guest that Cordelia forgot to notify that she was dead.\"\n\n\"Oh. Right.\" I got up and made my way to the front door. When I pulled it open, I found Audrey standing there, with a tote bag full of books. \"Hi, Audrey. I was planning to stop by the library this morning. What brings you here?\"\n\n\"I thought I could get you started. I brought some published references over\u2014signed out, of course, in your name. Any original documents can't leave the library building, so you'll have to stop in to see those. Are you going to ask me in?\"\n\n\"Of course. Would you like some coffee? Something to eat?\"\n\n\"I'd appreciate the coffee. Let me leave the books on the dining table.\" Audrey seemed to know her way around the place and went straight to the dining room, where she carefully set down the bag and one by one pulled out a stack of shabby bound books that looked fairly old. Then she turned and walked toward the kitchen\u2014and stopped dead in the doorway. \"Oh,\" she said flatly. \"I didn't know you had company.\"\n\nOh, right, Josh was in the kitchen. \"Hey, you're not interrupting,\" I assured her. \"Sit down and have a cup of coffee, at least.\" Did she look pale?\n\n\"No, thank you. I need to go open up the library. I'll see you later.\" She turned on her heel and hurried toward the front door, and I had to scramble after her to open the door for her.\n\n\"Thanks for the books,\" I called out to her retreating back. She didn't reply.\n\nI went back to the kitchen feeling puzzled. \"What was that all about?\"\n\nJosh looked uncomfortable. \"What, Audrey? Well, to put it delicately, she kind of has a thing for me. Do you know her story?\"\n\n\"Lisbeth filled me in a bit when I first arrived. Unmarried, spent her best years taking care of her sick and aging parents, et cetera. Like Marian the Librarian in The Music Man, or maybe I'm thinking of Donna Reed in It's a Wonderful Life\u2014the part with the lousy outcome. Poor Audrey is almost a caricature. I take it there aren't a lot of eligible men in Asheboro?\"\n\n\"Not many under the age of seventy who aren't already married. I would have thought that once she didn't have to worry about her parents any longer, she could have sold the house and moved somewhere else, but maybe she believed it was too late. She has a niche here and she can't seem to dig herself out of it.\"\n\n\"And where did you come into the picture?\"\n\nAt the sound of a timer, Josh retrieved the loaf of whatever it was from the oven and set it on a rack to cool. It smelled wonderful. \"Let it cool for a few minutes before we cut into it,\" he said sternly. \"So, back to Audrey. When I first arrived, some six months ago, I checked out the library, just to see what there was to work with there. We academics are always hoping to find some long-lost documents that will rock the scholarly world, but no luck. Just a nice small library, although the local history section is pretty decent.\"\n\n\"And? She saw you across a crowded room and sparks flew?\"\n\n\"Not exactly. She's been professional throughout, but sometimes I've been in there and I've noticed her watching me. Trust me, I haven't encouraged her. I've never even had lunch with her.\"\n\nAn aging woman with a serious crush? \"Josh, I'm sure she treasures every scrap of encouragement that she thinks you've offered. I feel bad for her\u2014now she thinks we're an item.\"\n\n\"You want me to tell her we're not, or would you like to do the honors?\"\n\n\"It might be less awkward coming from me, I guess. I can simply ask whether she might have misinterpreted what she saw this morning.\"\n\n\"Two fully clothed adults eating breakfast? Why would she jump to any conclusions?\"\n\n\"Because, based on my thirty seconds of observation, I think she's got it bad for you. Just what we need\u2014one more complication. It's not easy to get things done around here, is it?\"\n\nHe looked like he wanted to debate what Audrey had really thought, but in the end he let it go. \"I can't really say\u2014I've spent most of my time out at the Barton place, and I rarely come into town.\"\n\n\"Smart man. Maybe I should have given Lisbeth a polite 'no' after my preliminary visit and gone on my merry way. Maybe I'm using this project as a diversion from the fact that I don't have a job and I don't even know what I want to do next.\"\n\nJosh moved the bread to a trivet he had found in a drawer, and handed me a plate, setting one in front of his seat as well. He then retrieved butter and butter knives, refilled our coffees, and sat. \"That's actually two problems, you know.\"\n\n\"Why do you say that?\" I mumbled, my mouth filled with crumbs. \"By the way, this is great.\"\n\n\"Thank you. I meant, the first problem was what to do about the town, which poses an interesting challenge. The second problem was the sudden and unexpected loss of your job. Your brain said something like, 'Oh, look, I have something that will keep me busy now that my job is gone, and I actually think it would be fun.' The issue is, do you want to take on this project? Are you really interested or are you only using it to distract yourself?\"\n\n\"You sure you aren't a shrink? Those are very good questions, and I've already been asking them myself. And here's what I think: to the best of my knowledge, and after a few days of reflection, I really do think this project to take Asheboro back to an earlier era is very promising and could make a real difference to the town. Especially now that I don't have to worry about dealing with Cordy\u2014not that I'd kill her just to avoid butting heads with her on the town council.\"\n\n\"I never thought you would.\" Josh grinned.\n\n\"Glad to hear that. Anyway, losing my job gives me the opportunity to choose whether I want to take this on for the town, while I sort out the rest of my life. You already know that the discussions are very preliminary and there's nothing like a plan in place. Even if I present the council with a glossy report with my recommendations, there's no guarantee that the council will accept it. Or the numbers might not work out. Right now it's only a couple of weeks out of my life. So, in case I haven't answered your question in all this rambling, it's not just a diversion for me.\"\n\n\"Got it. If I'm going to get sucked into it, I want to know I'm not just wasting my time.\"\n\nHe really did want to be part of this? \"I promise I'll pull my weight. What you choose to do is up to you.\"\n\nI realized we'd both stopped eating, so we turned our attention to the marvelous bread, and there might have been a second piece involved. \"I'll clean up,\" I volunteered.\n\n\"Good. I'd better go back to my place and shower and change, and grab my tablet. Want me to pick you up here in, say, an hour?\"\n\n\"No, I think I should go over alone and try to talk to Audrey woman to woman, without you lurking around. But if she's still casting yearning glances your way, it will be up to you to let her down gently.\"\n\n\"Fair enough. I'll be on my way then, and I'll meet you at the library.\"\n\nAfter Josh had left I tidied up the kitchen\u2014and thought. Had Cordy more or less shut down her own B&B operations to deal with the Barton letters or others? Then I wondered what, if anything, was happening or might happen between Josh and me. I was not looking for a romantic partner at this particular time. He was probably on the rebound from his broken marriage since he hadn't managed to find a long-term living arrangement yet. Put those together and you got a lousy situation in which to start anything new.\n\nBut... it had been a long time since there had been a man in my life. Maybe I didn't recognize flying sparks anymore. So why was I smiling as I washed the dishes?\n\nI'd been telling the truth when I said I thought working on the town project would be rewarding and interesting. I had few doubts of my ability to carry it off, but I still wasn't sure I could sell my vision to the town. I would still need outside help from people like Nell, and the town would need an infusion of outside money, at least at the start. Both were potentially high hurdles, and I wouldn't be doing anyone a favor if I tried to downplay them.\n\nBut I thought it was worth trying.\n\nAs I prepared to leave for the library, I had to ask myself: How had things gotten so complicated? I had come back to Asheboro at Lisbeth's request to scope out what potential there was to use the Barton estate to generate some money for the town. Then that simple plan had ballooned to include the whole town. Now all I needed was the town's approval\u2014and a big chunk of money.\n\nThen Cordy was murdered. And in poking around trying to find out why, Josh and I had uncovered a significant theft, and we still weren't sure if it was an isolated incident or part of a larger scheme that had been going on for a while. I made a mental note to ask Nell to check auctions and sales over the past year or two, to see if any collections had shown up unexpectedly. But the question remained: Had Cordy died because of her involvement in this hypothetical theft, or for some other reason?\n\nAnd that's where things stood. I was a hotel manager, not a detective. I'd been dumped into the middle of this mess purely by accident, unless I chose to believe in karma. I had no skills or expertise to bring to the crime side, apart from common sense and some insider knowledge of the town and some of its citizens that was years out of date. It was a bit more than Detective Reynolds had, as far as the town went, but he was a trained professional and it was his job to track down murderers. As for thieves? That was a little murkier, but I had faith that he could figure out the right person to handle that.\n\nSo what was I doing now, heading for the library? First: I wanted to soothe Audrey, for whom I felt sorry. I could understand her attraction to Josh, but he said he wasn't interested. But I was not the one to tell Audrey that. Besides, I needed Audrey's help to find the information I wanted.\n\nThen I had to see exactly what information the library had on Henry Barton, and if there was any, to determine whether that shed any light on his role in Asheboro, both personal and commercial. Then figure out if this Victorian village project was actually viable, and talk to Mayor Skip about presenting a coherent plan to the town council to consider and ultimately vote on. Once I had their decision, I could either settle in Asheboro, at least for a while, or start job hunting aggressively. And last but not least, decide if there was any possibility of a longer-term relationship with Josh, and if I thought there was, whether I should see where it went, or nip it in the bud?\n\nThat seemed like a long enough list for one day: solve a murder and a theft, figure out my romantic life, and plan how I was going to spend my next six to twelve months. First on the list was Audrey. To the library!\n\nI checked to make sure my cell phone was charged and tucked in my purse, then gathered up my laptop, pads, pens, pencils, and the keys to the B&B and let myself out. It seemed to be a nice day, so I decided I might as well walk to the library, which was only a few blocks away. After all, I had to work off that second piece of bread that Josh had made, which had been pretty darn good. As I walked I looked critically at the buildings I was passing. The B&B's location was well chosen for its current purpose: an attractive large old house, in good repair, close to town but not fronting on a busy, noisy street. If my plan went forward, it could still serve as a bed and breakfast, under new ownership. If my plan evaporated, there would be few guests, and Ryan might decide simply to sell the property as a home, or home plus office suite.\n\nAs I strolled toward the center of town, I paid more attention to the commercial buildings, and kind of let my imagination run wild again, visualizing what this approach might have looked like in 1900. Would Main Street have been paved? If so, with what\u2014brick, stone, macadam? When did asphalt become the norm? The bank, the post office, the library, and a couple of churches anchored the center of the town, and they had changed little over time. There was a fire station, but set a block or two over. Had it always been there? Or had it moved from the main street to accommodate larger fire equipment, and what had filled its slot? It could be fun to have an old horse-drawn pumper truck\u2014for display only, of course.\n\nStables. Had there been any in town? When had car sales begun? Had there been a tram or trolley that ran along Main Street, before automobiles had become widespread? I realized that I kind of liked this kind of fantasizing, and reminded myself to look at the library for old photos of the town\u2014after I nailed down who Henry Barton was. And then I found I had already arrived at the library.\n\nI peered in the window: Audrey seemed to be the only person inside, which wasn't surprising given that it was barely nine o'clock on a Saturday morning on a fine spring day. The door was unlocked, which kind of surprised me, so I went in. If Audrey noticed me, she didn't react, so I approached the desk and waited until she looked up.\n\nShe didn't smile\u2014her expression was all business. \"Oh, Kate. I left a batch of non-circulating materials for you on the table in the Local History Room. Remember, no pens and no photocopies.\"\n\nSo she was going to play it as though our earlier encounter had never happened. \"Audrey, we should talk.\"\n\nAudrey was shuffling papers in front of her, and she didn't look up. \"Why? What's the point? He's a grown man, you're a grown woman. What you do is your business, not mine.\"\n\n\"You're upset.\" I stated it as a fact, not a question.\n\nShe didn't answer right away. \"Yes, I guess I am. Stupid of me. So?\"\n\n\"Audrey, please, can we talk? We don't have to take long.\"\n\nFinally she looked up, not at me but at all the empty chairs in the library. \"I'm not expecting many people this morning. You want to talk. So talk.\"\n\nThere was no reason to dance around the subject. \"All right. I think you misread what you saw this morning. Josh and I are not involved, if that's what you're thinking. He stopped by last night because he knew I had gone to Philadelphia on a research errand, and he was eager to hear the results, and I got back late.\" That was true enough, even if it wasn't the whole story. \"We had a couple of drinks and talked, and by then it was late so we went to bed\u2014in separate rooms.\" Having covered the basic facts, I realized I had no idea where to go next. Apologize to Audrey for inadvertently encroaching on what she thought was her turf? How could I have known?\n\nAudrey's expression was neutral with a tinge of grim. \"You don't need to apologize to me. I have no claim on Josh Wainwright. I know he's not interested in me. I was just startled this morning because I wasn't expecting to run into him. Or anyone else,\" she amended quickly.\n\n\"Audrey,\" I began desperately. \"You don't have to explain, or to defend yourself, whatever you feel. Look, I'm single. I've never been married, or even close to it. My job is more or less my life\u2014I don't even own a houseplant. So I'm guessing we have something in common.\"\n\nAudrey stood up abruptly. \"You don't know the first thing about me!\" she spat. \"Oh, poor Audrey, still stuck in her old hometown\u2014a librarian, of all things! What a pathetic clich\u00e9! Doesn't she have a life?\" She snorted in a very unladylike way. \"At least you have choices.\"\n\n\"And you don't?\" I shot back. \"You're not even forty. You don't have anyone holding you here. You can go anywhere.\"\n\n\"And be a librarian? Who reads books anymore? In case you haven't noticed, libraries are cutting their staffs, and in some cases they're closing. I'm not exactly a valuable commodity on the job market. And I haven't kept up on my computer skills, because I haven't needed them here. Those few high school kids who come in know more than I do about those little electronic toys, and they do most of their research from home now.\"\n\nI debated saying something soothing and trying to cheer her up\u2014but I realized she was right. She'd picked a career that was contracting, not expanding. She'd stayed in this town for personal reasons, but that added nothing to her resume, and she'd already waited too long. \"She was kind to her parents\" wouldn't look very impressive on a job application. Plus her personality was not exactly warm and welcoming. Maybe it had been, a decade earlier, but right now she'd kind of shrunken into herself.\n\nI realized she was watching me, with an ironic half smile on her face. \"I can tell exactly what you're thinking. I don't want your pity. I don't even want your sympathy. You got out. I didn't.\"\n\n\"I got out for one reason,\" I said.\n\n\"Cordelia. That bitch.\"\n\nI was surprised at the anger in her tone. \"Yes, Cordelia. You were a couple of years ahead of me in school, right? Did everybody know about what happened?\"\n\n\"You and Ryan? Sure. She made sure everybody knew. If it happened today, there would probably be videos all over the Internet.\"\n\nIck. \"But she was younger than you.\"\n\n\"Yes. I missed the first reign of Queen Cordelia\u2014the high school years. But I had to move back to town, to take care of my parents, and then she moved back. She might have gotten away from here for a while, but she missed her adoring fan club, and she didn't do particularly well in college. She wasn't school smart, but she could read people well enough to figure out their weak spots, and she knew just how to stick it to them. You could almost admire it, unless you were her target.\"\n\n\"What the heck did she have against you? You never did anything to her.\"\n\n\"Didn't matter. She came back, she found her fan girls had moved on with their lives, and then she latched onto Ryan. When that didn't prove to be enough to keep her busy, she went back to her old ways. I was easy prey. I mean, here I sat in the library, day after day. Watching the kids I hung out with in school get married, have babies, find jobs. And I was stuck, watching my mother die by inches, and then my father. And she just couldn't resist twisting the knife. 'Oh, Audrey, you poor dear. How are your parents doing? Is there any hope for improvement? What a shame you had to give up your own life to take care of them.' If I'd had any guts I would have given her as good as she gave. 'How kind of you to ask, Cordelia. Have you had any guests at that little business of yours lately? It seems awfully quiet over there. Too bad you couldn't work things out with your husband\u2014I hear his law practice is doing very well.'\" Audrey sighed. \"I could have, but I didn't. I let her get away with it. And I knew I wasn't the only victim. Sitting here in the library cuts both ways. After she and Ryan split, I watched her carry on with half the men in town, married or not. I'd see her on the street with the new man of the moment, or in the market\u2014anyone could tell when she'd picked a new one. She'd be all over him, and then she'd drop him like a hot potato. The guy never saw it coming. That was how she amused herself\u2014torturing other people. I don't think there are many in this town who will mourn her passing.\"\n\nWhile I agreed with Audrey, I wasn't going to say so. \"I think I'll start looking at the materials you put together for me, if you don't mind.\"\n\n\"You go right ahead. You won't be disturbed.\" She turned to sort a stack of papers on her desk, signaling our conversation was over. I escaped to the empty Local History Room, where stacks of books and folders were neatly arrayed, waiting for me. I took a seat facing the main room\u2014somehow turning my back on it didn't feel right.\n\nI'd barely begun sorting through the piles, trying to decide where I wanted to start, when I saw Josh come in. There still were no other library patrons, and he went straight to Audrey's desk and leaned toward her. I had to guess he was following my suggestion, but I had to look away. Poor Audrey\u2014she simply couldn't catch a break.\n\nA couple of minutes later Josh slipped into the Local History Room and closed the door quietly behind him. \"You talked to her,\" he said.\n\n\"I did. What did she say to you?\"\n\n\"I apologized if I'd misread any signals she was trying to send. She said she was used to that. I fled, like the clueless coward that I am, and here I am. Shall we get down to business?\"\n\n\"Pick a pile. I've only just begun.\"\n\n\"Maybe we should talk more specifically about what we're looking for and where we're likely to find it,\" he suggested.\n\n\"You're the professional researcher\u2014you tell me. No, wait\u2014maybe there's something else we should be thinking about first. Like, who knew about the Barton letters, and when? Why did somebody decide to take them now? And we should verify that they really do belong to the town, even if they aren't specified in the will. That might point us to whoever had a motive to kill Cordelia.\"\n\n\"Making money from selling them isn't enough of a motive?\" Josh asked, not unreasonably.\n\n\"Maybe. I'm sure there are plenty of people in this town who could use some money. And as far as we know, the letters aren't listed in any sort of inventory for the property, so if they're gone, nobody would know. Maybe that tells us the 'why now.' If the Grand Barton Hotel or whatever Cordy wanted to call it was going to be developed and opened to the public, the attic would be cleared out, and the letters\u2014and whatever other goodies are up there\u2014would be discovered. So the thief had a narrow window to make off with them. Maybe Cordy's partner was getting nervous, what with the council dragging its feet, and then me getting called in to suggest other ideas.\"\n\n\"If they knew about you, and it's not clear that Cordelia did. Although some of the other council members must have,\" Josh said. \"But what you say makes sense. Somebody could have wanted to collect the good stuff sooner rather than later, and Cordy somehow got in the way. In any event, we'd better get started. We outlined areas of responsibility earlier. Do you want to change any of those?\"\n\nI shook my head. \"I don't know any more than I did. I think we boiled it down to 'I do the people, you do the things.' And we don't have a lot of time. I've still got the books that Audrey brought over to the B&B earlier today,\" I reminded Josh. \"I assume they aren't duplicates, and we can look at them later. So the new materials are here on the table.\"\n\nWe started searching through what Audrey had gathered. \"Should we split this up, physically?\" I asked. \"I mean, you take this end of the table and I'll take the other?\"\n\n\"Fine. Though I wish we'd had time to do some online searches before we start in, but I guess that's moot. You could do a basic genealogy for Henry and his family quickly on any of a number of websites, but we're here now, and there may be information in these documents that's not online. Cross your fingers and hope that there's something like a family tree lurking here. Plus, Audrey didn't have a lot of time to pull these together, so if we come up dry, maybe we can do a broader search in the library for anything connecting to any Bartons.\"\n\n\"Am I doing Clara?\" I asked.\n\n\"Save that for the computer. I'm sure there's plenty available for her.\"\n\nI looked at the sizable pile of materials in front of us, and I remembered the stack of books waiting for me back at the B&B, and I felt a surge of panic. \"Josh, I hate to bring this up, but our thief may decide to move faster now, with so much outside attention focused on Cordy and her death. He doesn't know we have the letters, and he may double his efforts to find them. Do you think it's risky to leave the Barton house unprotected?\"\n\nJosh looked defensive. \"I armed what pitiful alarms there are. I made sure everything was locked, including the gates. I wasn't hired as a security guard, you know\u2014I'm just a caretaker, there to report if the roof caves in or the furnace explodes. Not to fend off desperate thieves and potential killers.\"\n\n\"I know, I know. I'm not saying you should put yourself at risk. And I'm not any better equipped than you are, if somebody decides to ransack the B&B. All I meant to say was that we'd better get this worked out pretty fast.\"\n\n\"I agree.\"\n\nAnd so we began. After a while we each became so engrossed in our reading that we didn't notice whether anybody came into the library, and Audrey left us alone. I struck gold first, in the form of a lengthy obituary of Henry Barton from a local paper. \"Got something,\" I said.\n\n\"What is it?\"\n\n\"Henry's obituary. It says he was born in Massachusetts, and it gives the names of his parents and his surviving relatives. He was one of several brothers, two of whom survived him. The one Clara was looking for was named William, and he isn't mentioned in the obituary, so he must have been dead by 1911. In letters I saw, Clara had only just started looking for him. We still don't know if she found him.\"\n\n\"You didn't stop en route to Philadelphia and read them all, to see what Clara found?\" Josh said, smiling.\n\n\"No, I didn't want to risk damaging them. And I wanted to get them somewhere safe. We did lay them out in order in Philadelphia, and I'll admit I peeked to see who Clara was searching for. Carroll was going to go through them for conservation purposes, but I'm sure someone will send us copies soon enough.\"\n\n\"Wonder if she ever found William, or at least his remains?\" Josh mused. \"That is, if he's buried anywhere? I wish I had time to read up on how Clara went about tracking down people back then. There were no dog tags, and it was total chaos on the battlefields. What little I've read tells about body parts flying all over during the shelling, and then the bodies\u2014and the pieces\u2014were stacked in heaps after the battle. It's hard to imagine, isn't it?\"\n\n\"I'd rather not.\" I suppressed that image, which wasn't easy. Then I had a sudden brainstorm. \"That place where Nell works, the Society\u2014they have one of the best genealogy libraries in the country. We don't have a lot of time. I think I'll call her and see if she can sic one of her trained people on it. I'm pretty sure it wouldn't take long for one of them to collect what information there is, now that I've got Henry's parents' names and his birthplace.\"\n\n\"Makes sense, if she's willing. What information do we want?\"\n\n\"The official record of William's death, assuming he died in the war. Any wills the family left. Property records. When Henry arrived in Maryland. Military records. All that kind of stuff. Might as well go straight to the top. The worst Nell could do is say no.\"\n\nJosh nodded. \"And we work here with the original records for the house and the town and the factory that are not available there and also not online.\"\n\n\"Exactly.\"\n\n\"So call her!\"\n\nI got my cell phone out of my bag and retreated to a corner so I wouldn't distract Josh. I called Nell, crossing my fingers that she wasn't out to lunch or something inconvenient like that. I almost wept when she actually answered.\n\n\"Hey, Kate. I didn't expect to hear from you so soon. What's up?\"\n\n\"I have another big favor to ask.\"\n\n\"All right\u2014tell me.\"\n\n\"Josh and I are at the local library now, and we're trying to figure out more about Henry\u2014who he was, where he came from, how he made his fortune. And did his older brother, William, ever show up, or was he really dead? We're going through local records, but I would be eternally grateful if you could find someone at your place to look up all those family history details. I know they're available online, for the most part, but I don't have the skill or the time to do it. Can you help?\"\n\n\"Sure. It would probably take only a couple of hours to collect the basic information, so I could get it to you tonight or tomorrow.\"\n\n\"Thank you! I hate to sound like such a wimp, but it really bothers me that whoever may be looking for the letters is still at large, and may decide to wrap things up sooner rather than later if he sees his chance slipping away. He could be desperate. I'd certainly rather not find out.\"\n\n\"Believe me, I understand. Tell me what you need, specifically.\"\n\nI read off the pertinent details from the obituary, and listed the questions Josh and I wanted to answer quickly\u2014the rest could wait. Maybe someday the Barton House would attract guests from all over, and my staff (ha!) and I could put together a glossy pamphlet outlining Henry's career, but that was for later. Maybe.\n\n\"Got it,\" Nell said at last. \"I've got your email so I can send you the results. Oh, and Carroll is over the moon about the letters. She's already scanned them. I'll have her send you copies of them. Same email?\"\n\n\"Sure.\" I could always call them up on Josh's computer\u2014he must have a printer in his rooms at the estate. \"I can't thank you enough!\"\n\n\"Don't worry\u2014I'll find a way for you to thank me. Happy hunting, and I'll be in touch later.\"\n\nJosh looked up expectantly when I returned to the Local History Room. \"What did she say?\"\n\n\"I was right\u2014someone who knows what they're doing can get what we need in a couple of hours. She thinks we may have the information by tonight. And she's sending scans of the letters. Have you found anything good?\"\n\n\"I've been putting together a general time line, with a lot of guesses. We know Henry fought in the war, and we'll probably find out which unit he fought with, and which battles he took part in. I'm going to guess that it was somewhere around here, since he didn't go back to Massachusetts. Clara Barton was hunting for the lost soldiers for a relatively short time, before she went on to other projects like the Red Cross, so, based on the addresses on the letters, Henry must have contacted her shortly after the war ended. By the way, the addresses don't tell us much\u2014they were to Henry Barton, Asheboro, Maryland. Presumably it was such a small place then that anyone would have known where to find him.\"\n\n\"I'm with you so far. What else?\"\n\n\"We'd have to look at county records to find out when he did buy the property. What we do know is that the factory was up and running by 1870.\"\n\n\"What did they make?\"\n\n\"Shovels. Don't laugh. Shovels were important then. And after a couple of years it looks like he graduated to plowshares\u2014bigger metal items. And so on, for the next forty years or so. At its peak, the factory employed almost every male in town who wasn't a farmer or a shopkeeper.\"\n\nI pondered. \"So Henry buys a property outside of town, and we're guessing that it was a pretty simple place then\u2014your basic farmhouse. He starts up his factory in town, and it does well. Somewhere in there he marries. He's getting richer, so he builds his grand mansion\u2014based on the style that has to be 1880 or later. How long did the factory keep going after Henry had died?\"\n\n\"From what little I've seen, probably until the Depression.\"\n\nI nodded. \"And it was well on the way to becoming a ruin when I was growing up here. So, what else do we need to know?\"\n\n\"How did he fund building the factory in the first place? That would have taken some money. Of course, it could have been family money, although New Englanders preferred to keep their money close.\"\n\n\"I hate to say it, but I think we need to look in the attic again. There may be a lot of answers up there, like factory records.\"\n\n\"True. But it's getting kind of late today.\"\n\nI looked at my watch. He was right\u2014it was past four. Where had the day gone? \"I suppose we should adjourn and pick this up in the morning. We might have Nell's information by then.\"\n\n\"I, uh, I'm not sure how to say this,\" Josh began awkwardly, \"but I think it might be best if we stayed together. For protection. The thief doesn't know if Cordy hid anything at the B&B, and he'd probably target the mansion first. So I think we should stay at the Barton place tonight, just in case.\"\n\n\"That makes sense,\" I said. I hadn't thought about staying alone at the B&B overnight, alone. \"Can we stop by the B&B so I can pick up some stuff? And we should think about food, since we missed lunch completely.\"\n\n\"Sounds like a plan.\"\n\nWe left the library with no more than a nod to Audrey, who was busy with an older woman with a stack of mysteries. Josh drove me back to the B&B.\n\nIt was frustrating not knowing who Cordy might have roped into her moneymaking scheme\u2014or by whom she had been roped in? I doubted she had the brains to plan such a theft, but she did have access and she wanted money. And she had (or believed she had) the charms to wheedle some man to do what she wanted. Maybe that's why I kept thinking of Cordy's accomplice as a \"him\"\u2014her track record with female friends was kind of pathetic. It was still possible that she was being used, but she was going to get paid for it. Too bad she didn't live to collect that fee.\n\nEverything at the B&B looked peaceful, but that could be deceiving. Maybe the Bad Guy had gotten keys from Cordy and was inside even now, rummaging through drawers and closets and such.\n\n\"You want me to come in?\" Josh asked quietly.\n\n\"Damn it, how do you keep doing that?\" I demanded.\n\n\"Doing what?\"\n\n\"Reading my mind.\"\n\n\"It's not hard\u2014you have a very expressive face. I hope you don't play poker.\"\n\nHe'd actually been watching my face? I didn't know whether that pleased or alarmed me. Something to think about later. \"I don't have the time. Would you please come in with me? I'll admit I'm nervous.\"\n\n\"No problem.\"\n\nHe followed me as I unlocked the door and stepped into the hall. Nothing had changed, except that the air was beginning to smell a bit stale since the place had been closed up for a few days. \"Wait here,\" I told him, and scurried up the stairs. Nothing had changed up there either. I pulled clothes and some underwear from the dresser I'd been using, picked up my toiletries, stuffed them all into my bag, and hurried back downstairs. \"You have food at your place?\"\n\n\"Some. I wasn't planning on entertaining,\" Josh said.\n\n\"Then we're going to have to stop somewhere and pick up some.\"\n\n\"Fine.\"\n\nI set the alarm system and locked up.\n\nWhen Josh started the car and pulled out, he headed toward the supermarket just outside of town. While he drove, he said, \"What kind of food places do you envision for this Victorian town of yours?\"\n\n\"A grocer, of course. Maybe a butcher\u2013slash\u2013poultry shop, although I have mixed feelings about that. It's appropriate to the time and place, but I doubt a lot of tourists are going to stop and buy a chunk of stew beef or a dead chicken. A dry-goods store. A tearoom. Are you trying to distract me?\"\n\nJosh smiled without turning his gaze away from the street. \"Maybe. What's for dinner?\"\n\n\"You're asking me? It's your kitchen.\"\n\n\"Kitchenette at best.\"\n\n\"Does it have a microwave?\" I asked.\n\n\"Yes.\"\n\n\"Then you're good to go.\"\n\nAt the store we stocked up on a mish-mash of quick and easy microwavable foods and left quickly. It was already getting dark.\n\n\"Too bad I didn't think to snag a bottle of that wine from the B&B,\" I said.\n\n\"I've got some at my place.\"\n\n\"Do you have a printer?\"\n\n\"I do. All the comforts of home. You're thinking about printing out Henry's history?\"\n\n\"Of course I am. And the scanned letters. You know, if all this goes forward, I'm going to have to bone up on Maryland history. And the Civil War.\"\n\n\"There are worse problems to have,\" Josh said wryly. \"Do you think you'll do the project? Not just the proposal?\"\n\nIt was a fair question, but I hadn't really made up my mind. \"I don't know. This past week has been such a disaster that I haven't really had time to think of any of the details, much less any long-range plans. I need to put together a written plan for the council, with financials, if we have any hope of raising money.\"\n\n\"You said 'we.'\"\n\n\"Did I?\"\n\n\"You're identifying yourself with the town now. Dead giveaway.\"\n\nMaybe it was, again\u2014I hadn't done it consciously. Something else to think about. That list kept growing.\n\n\"Do you know how much money is available from the funds Henry left?\" Josh asked.\n\n\"Actually I don't. I can ask the bank president\u2014he's on the town council, so he should be willing to help.\"\n\n\"I would think so. What would it cost to pretty up Henry's house?\"\n\n\"If it were a modern rebuild, I'd have a better idea. Henry's house is a challenge. I'm guessing a lot of the interior details came straight from the Tiffany Studios, which is a big plus. But on the minus side, some of it is a bit shabby, just from age, so it becomes a question of restoring it or replacing it, and if the latter, with what? And what about the infrastructure\u2014plumbing, heating, wiring? Nobody's looked at that for a while.\"\n\n\"Despite your protestations, you have been thinking about it,\" Josh said.\n\n\"Yes, I guess I have.\"\n\nWe had arrived at the front gate, which was locked. Josh got out to unlock it, drove through, then locked it after us. \"No point in making it easy,\" he said.\n\n\"Heck, if I were a thief, I'd come in the back anyway, although it really doesn't matter from which direction they come in because there's nobody to see. Think Cordy has been replaced as an accomplice?\"\n\n\"Don't even go there!\" he said firmly.\n\nJosh parked in front of the stable, and we unloaded the bags of food. He went up the stairs and unlocked his own door, and I entered after him. The place was Spartan, to put it politely, but at least it was neat.\n\n\"Cute little place you've got here. Mind if I use the computer while you cook?\" I asked.\n\n\"Oh, now I'm cooking?\" he protested.\n\n\"Well, it is your place, so technically you're host. You do know how to use a microwave, right? Is the computer password protected?\"\n\nHe slipped past me to the other side of the table where the computer sat. \"Here, let me.\" He typed in a few commands then stepped back. \"All yours.\"\n\nI logged into my own account. As promised, Nell had delivered the information. Her cover email said only, \"Piece of cake. Let me know if there's something more you want.\"\n\n\"Josh, do you have a blank disk or a thumb drive? I want to save this stuff before I mess with it.\"\n\n\"Thumb drives in that box on your right,\" he said, intently reading the microwave instructions on the food we'd bought.\n\nI fished out a new thumb drive and inserted it in an empty port, then went back to Nell's attachments. There were quite a few, but as I downloaded them I saw that most were only a page or two long. Except for the file folder of Clara's letters. I decided to start with those. \"Mind if I print out the letters?\"\n\n\"You have them all? That was fast. Sure\u2014the printer's on but in sleep mode. Just open the files and print.\"\n\n\"Thanks,\" I said. As I knew, there were some thirty letters from Clara to Henry, but I'd never seen them fully before. I downloaded them then started sending them to the printer, one at a time. I wrestled with myself: read those letters right away, or be responsible and download all the other files and print them out as needed? Sometimes being a responsible adult could be a pain. \"Do you want to read them together? I can finish downloading while you cook.\"\n\n\"Cooking at this point consists of pushing about three buttons on the microwave. But go ahead and finish downloading and printing\u2014we might want to compare the information in the other items to what's in the letters.\"\n\n\"Good point.\" I returned to downloading, saving, printing, rinse and repeat. Actually it didn't take all that long, and by the time I'd finished Josh had filled two plates with food.\n\n\"Let's eat first,\" he suggested. \"It's been a long time since breakfast, and we should be clear-headed.\"\n\nAgain, he was right, but he was driving me crazy. Was I the only person in the room who was eager to see what Nell had collected for us? \"Okay. What is it?\"\n\n\"Some kind of chicken, some rice and a veggie that came in a plastic bag. Real gourmet stuff.\"\n\n\"It's food\u2014it'll do.\"\n\n\"Wine?\" he asked.\n\n\"Let's save that for later.\"\n\nWe dug in. After a few bites, I said, \"You know, I feel like a kid on Christmas morning. We've got some facts here about a guy who lived over a century ago, in the very place we're sitting now. He was a big part of this town in the nineteenth century, and then he was kind of forgotten, and now we're talking about bringing him back so he can save the town. How cool is that?\"\n\nJosh smiled. \"It is officially cool.\"\n\nWe finished eating quickly and cleared off the table, wiping it down to remove any grease.\n\nI was getting very antsy to get started. \"Oh, I do have one suggestion. I said before that we could read the letters first, but it occurs to me that we should look at his family tree, and check whether Henry and Clara are related, and if they might have known each other apart from the war.\"\n\n\"Good point. Have you got that?\"\n\nI called up that file and sent it off to the printer, which spit it out thirty seconds later.\n\nJosh picked it up and set it between us on the table. As best I could see, it looked like Henry and Clara were less than twenty years apart in age, but somehow that came out to two generations. Which didn't necessarily mean anything. If I was reading the tree right, their lines came together four generations back for Clara, six for Henry. Both families had lived in the central part of Massachusetts, but that didn't guarantee that they had crossed paths. So for practical purposes it might be safer to conclude they did not know each other, but somehow Henry had heard of Clara's efforts to find missing soldiers immediately after the war, and he had contacted her. And that should be reflected in the letters.\n\nIt was time to turn my attention to them. As I'd told Josh, in Philadelphia we'd sorted them chronologically, and then Carroll had taken them away to scan them by archival standards, to avoid damaging them. I needed to know more about Clara's history with the search for missing soldiers, and where these letters fit with that, but right now I just wanted to see what she had written to Henry. I stacked the copies up by date and started reading.\n\nIt didn't take long. While I was looking at only one side of the correspondence, from Clara to Henry, the outlines were clear. Henry was the second child in his family, and his older brother, William, had not come home from the war, while the two younger ones had, only somewhat damaged. Henry must have heard about Clara's efforts and reached out to her. Her first response acknowledged his letter as well as their kinship, which I'd suspected. His first letter must have been a brief inquiry, because she asked for additional information, which apparently he had provided in a following letter. There were a number of letters from Clara after that, confirming receipt of his letter, and then providing additional information about her search. They were short and polite, but businesslike rather than familiar.\n\nAs I read I found myself wishing that I knew more about the war. I tried to imagine Clara taking on the huge task of trying to track down literally thousands of men who remained unaccounted for. I'd read that soldiers had taken to carrying some form of identification on their persons for such eventualities. But reading about the horrific carnage that had taken place, I had to wonder if something as fragile as a slip of paper with an address could have survived. And much of the time the commanding officers were wounded or dead, so they couldn't account for all their own men.\n\nWhich often just left personal correspondence, during or just after the war. Clara had made a name for herself by the end of the war, and she was a rather manic letter writer. I was going to guess that she had looked up enlistment records, eliminated the known dead, read individual regiment reports, and then started working her way through the survivors searching for any news of the soldier she was looking for. It must have been largely hit or miss, trying to find the survivors or their kin, but clearly it had worked some of the time.\n\nBy the time I reached the end of the small stack of letters, I knew that Clara had found out what happened to William Barton, Henry's brother. But she didn't reveal it in the letter. Instead she asked Henry's permission to call upon him and convey her findings directly. So she had in fact visited Asheboro to meet with Henry.\n\nAnd the correspondence ended there. I might have snarled.\n\nJosh looked up from whatever documents he had been immersed in. \"What?\"\n\n\"Why was I snarling? Because there's a cliffhanger here, in the last letter.\"\n\n\"Which is?\"\n\n\"Clara Barton found what happened to Henry Barton's older brother, William. But instead of just writing to him, she decided to come visit and deliver the news to him personally.\"\n\n\"Here?\" he asked.\n\n\"Or wherever Henry was living in Asheboro at the time. And there's nothing after that.\"\n\n\"What was the date on her last letter?\"\n\n\"March 1865\u2014the day is kind of illegible. Why do you ask?\"\n\n\"What do you know about the Office of Correspondence with Friends of the Missing Men of the United States Army?\"\n\nCouldn't the man give me a straight answer instead of another question? \"You mean, her Missing Soldiers Office? Only what I've learned in the past few days, which is not much. Why? I mean, I know why it's important, but why does the date matter?\"\n\n\"President Abraham Lincoln appointed her to take on the search for missing prisoners of war on March 11, 1865. She recruited a few volunteers, including her own family members, and set up the organization in rented space in Washington and started collecting inquiries. She spread the lists of names as widely as possible for those days, in newspapers and post offices. Veterans could check the lists and send her any information they had. The group logged in over sixty-three thousand letters. The practice was carried over to the Red Cross years later. She was the first woman to run a government bureau, and she received payment through congressional appropriations. She made a final report to Congress in 1869, when she told them that maybe forty thousand soldiers were still unaccounted for and were most likely dead.\"\n\nJosh wasn't finished yet. \"The Treasury back then was reluctant to hand out back pay to the families if there was no evidence of death, or a date. Clara proposed to change that, to help the widows and orphans.\"\n\n\"Josh, this is a wonderful story, but what's it got to do with our Henry Barton?\"\n\n\"Bear with me. There are a couple of key points. One, look at the timing. That March Lincoln had barely created this group, and there was little money to be had in the government at that point in time. The President may have promised support, but Clara had to fund things at the beginning with her own money or contributions. Keep that in mind. Two, the Treasury didn't want to pay out if no one could prove a soldier was dead. I don't know much about William, so I can't say if he was owed anything, but the fact got me thinking. Henry was the second son. The first son, William, was missing. Henry was eager to find out what happened to William. What if his parents' will specified that their estate would go to their eldest son?\"\n\n\"Interesting logic, but we don't even know if the parents were dead in 1865.\"\n\n\"We can find out,\" Josh shot back.\n\n\"Okay. We don't know what's in their will.\"\n\n\"We can find that too, assuming there was one. Look, I've got a pal in Massachusetts who can access all this information and answer exactly these questions.\"\n\n\"What if there wasn't a will?\" I protested.\n\n\"Then he can look up the laws for inheritance without a will as of 1865\u2014maybe it was split evenly among the surviving children. And it might depend on which state they lived in. Again, easy enough to look up.\"\n\nNow he'd finally gotten excited by our research, just as I was about to dissolve into a lump. \"Josh, it's getting late. What's your point?\"\n\n\"This is just a theory, mind you. What if Henry needed to know what had happened to William because he needed the money, and could only inherit if he could prove that William was dead? Either all of it or a one-third share rather than a one-quarter share, assuming both of his younger brothers survived?\"\n\n\"Needed it? Or wanted it?\"\n\n\"Does it matter? Look, we know Henry moved here after his service in the war and was living here when Clara was writing to him in 1865. We don't know how much money was involved, but say he was already thinking about his factory, and that it was the perfect moment to start up a new business in this area. Property might have been cheap, and there would be a lot of returning soldiers who could use a job. He needed a good chunk of change to take advantage of the opportunity.\"\n\n\"And you can check local history for references to the start-up date,\" I said, almost to myself. \"Or ads. Or patent applications.\"\n\n\"Exactly!\" Josh said triumphantly. \"But there's one more thing. Clara Barton had approval from the President to start up her search group, but little or no money. Once she could show that William was dead, and Henry received his inheritance, what if he gave part of it to Clara, who then used it to start up her organization?\"\n\n\"So you're suggesting that Henry might have offered to provide the seed money for Clara's work if she could prove that William was dead?\"\n\n\"It's possible, isn't it? Providing we can find something like evidence. But a lot of her papers have survived. Think she was good at keeping accounts?\"\n\nI sat back and rubbed my stiff neck. \"Oh, Josh, don't tempt me. You've just put together a scenario based on a lot of wild guesses and maybes. I'd love to believe you\u2014it would be such a great hook to draw people to this house and the town, because it puts Henry and this place into the mainstream of the Civil War. But it has to be true for us to do that. So we've got more work to do. And I hate to burst your bubble, but we really need to find Cordelia's killer before we go chasing pretty theories. Right?\"\n\nJosh looked disappointed, but I knew I was right: murder first, nineteenth-century history later.\n\n\"I know,\" he said reluctantly.\n\nGod, I was tired, and I couldn't think straight. \"But one other point: Where does the theft of the letters fit in this? Was it coincidental, and all it meant to the thief was a juicy payday? Or was there something darker behind it? Who else would have an interest?\" I asked.\n\n\"That I can't tell you\u2014yet. Maybe things will look better in the morning. Look, like I said earlier, you need to stay here tonight, and maybe beyond, until we get this sorted out\u2014there's still that killer at large. You can take the couch.\"\n\n\"Gee, what a gentleman!\"\n\n\"Don't worry, it's comfortable. I've fallen asleep on it plenty of times. The bathroom is through the bedroom. I'll hunt down some pillows and a blanket.\"\n\nI went one way while he went another. The bathroom was surprisingly clean for a guy who lived alone. Right now we were partners in anti-crime, with a working theory about who had killed Cordelia, although we were kind of lacking a name for the person. Now if we could just nab the thief, maybe we could hang up a shingle as investigators.\n\nI went back to the main room, and Josh retreated discreetly to the bedroom and shut the door. I checked the time\u2014I hadn't realized how late it was. I had no idea what I or we would be doing tomorrow.\n\nI lay down, pulled the blanket up to my neck, and turned off the light. Wow, it was dark here. I should have thought about that. This was the country. There was next to no traffic on the road, no sounds of ringing phones or loud music. No lights, either on the house or along the road. I wondered what the electric and heating bills were like. Had Henry liked it out here? I wasn't familiar with where he had come from in Massachusetts. Was it like this? Or more of a town?\n\nSince it was so dark, it occurred to me that I ought to take a look at the stars, something I seldom had a chance to do where I lived\u2014too much light pollution. Not that I had a lot of time for stargazing, but I should indulge in it while I could. I tossed back the blanket and padded barefoot across the floor, carefully feeling my way around the table in the middle of the room where we'd laid out our documents, to look out the nearest window, one that looked toward the house. The house loomed like a big black hulk, only slightly darker than the space around it.\n\nExcept for one small light, moving from room to room on the second floor. Either it was another pair of hormonal teenagers, or our letter thief was back.\n\nI darted across the room to Josh's door without stumbling over the table, and crept in quickly. If the light snoring was any indication, Josh was asleep, so I put a hand on his shoulder and shook him gently. And then not so gently.\n\nHe finally woke up with a start, \"Wha? Oh, Kate. Bad dreams?\"\n\n\"There someone with a flashlight inside the big house.\"\n\nSuddenly he was alert. \"Who? No, scratch that\u2014stupid question. Did you hear a car?\"\n\n\"Nope, and I haven't been asleep.\"\n\n\"The gates are locked anyway, so whoever it is had to walk in from the road. He must have assumed I was alone, since only one car is here, and I'd turned off the lights. He doesn't know you're here.\"\n\n\"What're we going to do about it?\"\n\nJosh took a moment to think. \"If we call the cops, he'll hear them coming and clear out on foot before I can get the gates unlocked. I'll have to go over and check him out.\"\n\n\"Are you, uh, armed?\"\n\n\"You mean, do I have a gun? Of course not. I'm a history professor.\"\n\n\"So you're going to sneak up on a possibly armed intruder in the middle of the night and ask him politely, 'What are you doing here?'\"\n\n\"Look, lady\u2014you've vetoed the cops and you just vetoed me. What's left?\"\n\n\"We sneak up from two sides at once? One of us is bound to surprise him, since he's only expecting one person.\"\n\n\"You want to go with me?\" he asked, incredulous.\n\n\"It sure beats sitting here and worrying.\"\n\nWhat little I could see of Josh's face in the dark radiated dismay: he didn't want to take me along. Heck, I was pretty sure he didn't want to go himself. In his shoes, I probably wouldn't either. But here we were, two intelligent people. We were stronger together, weren't we? We should be able to outsmart one white-collar criminal, shouldn't we? \"I'm going to put some clothes on,\" I whispered. \"Maybe you should too.\"\n\nHe looked down at himself\u2014we were dressed pretty much alike, with ratty T-shirts and baggy pajama bottoms. Not the right outfits to confront a thief. Who might also be a murderer.\n\nJosh disappeared into the bathroom, while I pulled on what I'd been wearing earlier in the day. Josh emerged a minute later, and I almost giggled because our \"stalking the thief\" outfits matched as well as our earlier ones: shirt, jeans, sneakers, and a dark windbreaker. \"Just for the record,\" he said, \"I think this is a bad idea.\"\n\n\"Duly noted. By the way, what do we do when we get there?\"\n\n\"Split up, for a start. You made a good point, about him looking for only one person, not two. I guess from there it depends on who it is. If it's someone we know or recognize, that's one thing. If it's a stranger, that's different. If the guy is armed, either way, I think we should back off and call the cops ASAP.\"\n\n\"State or local?\" I asked.\n\n\"My first reaction is to call Reynolds or his office, But I think local has jurisdiction over thefts, and they're closer. Ready?\"\n\n\"I want a weapon,\" I said, sounding pathetic even to myself.\n\n\"I don't have one,\" Josh said, sounding testy.\n\n\"I know. I don't mean a gun\u2014just something to whack the guy with, to defend myself. Or you.\"\n\n\"I think there's an old shovel downstairs, outside the door\u2014it's heavy. Will that do?\"\n\n\"I guess.\"\n\n\"Fine, we'll pick it up on our way out. Don't drop it, drag it, or trip over it.\"\n\n\"If we don't get moving, I may use it on you. The guy's going to be in DC fencing the goods by the time we get there.\"\n\n\"So let's go.\"\n\nI hadn't brought a flashlight because I hadn't packed one, but Josh had a small but efficient one in his jacket pocket, or so he told me. We didn't dare turn it on yet. We tiptoed down the stairs, and he pointed at the shovel leaning against an outside wall. I had to wonder why we were tiptoeing, if, as we guessed, the thief was rummaging through the attic, looking for one last haul. But better to be safe than sorry, and we needed surprise on our side. We tiptoed across the rolling lawn until we reached the house. I mimed unlocking the door, and Josh nodded and pulled a key ring out of his pocket. Turned out it didn't matter\u2014the door was already unlocked\u2014not a good sign\u2014and it opened silently.\n\nI tried to figure out how to mime \"when do we split up?\" There were a few details we should have covered back at Josh's place when we could actually speak, but we were kind of new at this game. I tried to review what I knew of the house. There was only one door to the attic, if that was where our guy was currently, and it was on the second floor. So unless he jumped out a window, we could trap him up there. If he was on the second floor, he had more options, and I thought I remembered a back stairway, for the servants, of course.\n\nI leaned close to Josh and whispered in his ear. \"Can you hear him now?\"\n\n\"Only faintly\u2014I think he's still in the attic,\" Josh whispered back.\n\n\"Can we lock him in before he hears us?\"\n\n\"Maybe. Can't say how long the old lock would hold.\"\n\n\"There are two stairs from the second floor to the first. Our prowler might not know the back stairway to the kitchen is there, so maybe you should cover the main stair and I'll take the other one. Ready?\"\n\nWe tiptoed up the grand staircase, single file, placing each foot carefully. The higher we got, the more clearly we could hear our thief bumbling around the attic, believing that nobody was paying attention to him. Under ordinary circumstances he'd have been right. At the top of the stairs, Josh pointed to me and then to the door to the back stairs. I took a firmer grip on my shovel and retreated to stand guard, pondering which end of the shovel would do more damage. Josh's shadowy figure advanced toward the attic door, with the key already in his hand.\n\nAnd that's when our plan went wrong. As Josh was preparing to insert the key in the door, we both heard footsteps clomping down the attic stairs. Definitely male, I thought irrelevantly. Before Josh could react, the door swung open, shoving Josh back against the wall. The shadowy figure said, \"What the hell?\" and pulled the door away to confront Josh, who was armed only with a bunch of keys to defend himself. Luckily, as I had predicted, the guy from the attic was focused on Josh and had no idea I was there, lurking in the shadows. Before either of the men could react, I gave the loudest roar I could muster and rushed toward them with my shovel raised\u2014and brought it down as hard as I could on the guy's head. He toppled over onto the floor, and for a brief moment I wondered if I'd killed him. Apparently not, since he started trying to get up.\n\n\"Now what?\" I hissed at Josh, and then remembered we didn't need to whisper anymore.\n\nJosh kept his eye on the guy as he said, \"You keep your, uh, shovel cocked, and I'll take off his belt and tie his hands.\"\n\n\"Good plan. I'm ready.\" Now I was holding the shovel handle like a baseball bat.\n\nThe guy on the floor didn't protest when Josh stripped off his belt, rolled him over to pull his wrists together, and secured them with the belt. \"Lights?\" Josh said to me.\n\n\"Oh, right.\" I fumbled for the nearest light switch\u2014a classic push-button kind\u2014and managed to turn on the overhead hall light, whose single bulb gave off a feeble glow. Then I turned to take a better look at our captive and did a clich\u00e9d double-take. \"You? You're behind this? Josh, you recognize this guy?\"\n\n\"Uh... damn, I do. This is the bank president, Arthur Fairchild. He's the one that had me sign the contract when he picked me to keep an eye on this place. He's come by a couple of times since to see how things are going.\"\n\n\"Alone?\" I asked, trying to convey the rest of the question: was he with Cordy?\n\n\"As far as I know,\" Josh said. \"He might have made visits when I wasn't around.\"\n\nInconclusive. \"I met him at that town council meeting when I first got here. Oh, shoot\u2014he's the one in charge of the Barton trust that pays for this house. He could come and go any time he wanted\u2014he controls the keys. The plot thickens.\"\n\n\"It certainly does.\"\n\n\"Can I sit up?\" Arthur wheezed. \"You two scared the living daylights out of me.\"\n\n\"We'd be happy to let you sit up, as long as you don't try to make a break for it,\" Josh said. \"And you can tell us what's been going on here?\"\n\n\"What do you mean?\" Arthur asked, trying to sound innocent. He failed. He was a lousy actor.\n\nJosh pulled his cell phone out of a pocket. \"I'm calling the police.\"\n\n\"Wait, no, please!\" Arthur protested. \"I can explain.\"\n\n\"I'm sure you can,\" I told him, \"but I have a feeling that the police will want to hear your story. Let's let them decide what to do with you.\" Now Arthur looked like he wanted to cry.\n\nJosh had turned away to make his brief call then turned back to us. \"I called the local department. They're sending someone over now. They can decide whether to turn this guy over to the state police. I'll have to go unlock the gate. Kate, are you okay staying here with him?\"\n\n\"Sure, I'll be fine. After all, I have a loaded shovel. Go!\"\n\nJosh loped downstairs and out the front door, leaving me alone with Arthur. I wondered what the law would say if I asked a few pointed questions, just to satisfy my own curiosity. I decided to try a test question. \"How many letters?\"\n\nThe poor man was leaning against the wall, and he sort of collapsed into himself at my question. \"Clara's? Or others?\"\n\nBingo! He'd been in on it. \"Start with Clara's. By the way, they're all safe, including the one that Cordelia had. Were there any others taken?\"\n\nArthur shook his head vehemently. \"No! We picked that batch because we thought there'd be plenty of interest in them. Then Cordy... died, and I figured people would be paying more attention to this place and I'd better move fast. That's why I was here tonight, to get the rest of the letters.\"\n\nAnd probably anything else that looked tempting and easy to sell. \"So you and Cordy were working together?\"\n\n\"That's hard to answer. We had a... thing awhile back, and when it ended, she threatened to tell my wife unless I helped her.\"\n\n\"Who found the letters? Who knew about them? Are they included in the inventory of this place?\" I was on a roll, and I wasn't sure how much time I would have.\n\n\"No, or at least, not specifically. The inventory says only 'trunks and cases with papers.' It was Cordy who got curious when she was scoping it out for her so-called investors, and she started poking around in the attic. Seems like nobody had looked up there for a very long time. I can't tell you what all she found, but the name Clara Barton rang a bell. So she took one letter out and brought it to me and promised that there were a lot more like it. She said she'd checked on eBay and they were really popular, and we could make a lot of money by selling something that nobody in town knew existed.\"\n\nI suppressed an urge to strangle him. \"That's theft, you know. Maybe some bigger fancier title once the market value of the items is determined.\"\n\n\"But we never stole them! At least, I didn't, and Cordy took only the one. Borrowed, I mean. So she could verify what it was. I could say I authorized her to take it, which I can do because I'm the custodian of record for this place.\" Arthur was working hard to cover his ample derriere.\n\nHe could have a point. \"Are you sure?\"\n\n\"Well, that's what she told me.\"\n\n\"Let me get this straight, Arthur. You've lived in Asheboro most of your life, and you knew Cordy way back when. I speak from experience when I say that she was not a very nice person. How the heck did you get sucked into her plan?\"\n\nArthur couldn't meet my eyes and looked down at his knees. \"Dunno.\" He sounded like a kid at school who had swiped some other kid's lunch.\n\nI decided to switch tactics. \"Did you have a buyer lined up for the letters?\"\n\n\"I don't think so. I was going to let Cordy handle all that. I mean, in the beginning, I was the only person in town with free access to this house, and I could come and go without anyone questioning me. Then Cordy got this crazy idea about turning the house into something else, and she wangled a key for herself so she could show it off. Since the town owns the building, and she's on the town council, she could go in unsupervised. And I guess she could take whatever she wanted.\"\n\nOnce she'd wrapped poor Arthur around her little finger. Now for the fastball. \"Did you have anything to do with Cordy's death?\"\n\n\"Me?\" Arthur's voice almost squeaked. \"No! No way! I'm not a killer.\"\n\nI wasn't sure that I believed him. \"What was Cordy doing here the day she died?\" I snapped at him.\n\n\"I don't know. Maybe she'd decided to collect the letters while she still had a chance. She knew the town council didn't like her plan for this place, and she might not get too many more opportunities.\"\n\n\"Who might have wanted her dead?\" All right, I was being cruel to the man, but I figured he deserved it.\n\n\"Who didn't? She'd made a lot of enemies here.\"\n\nAs I knew only too well. \"But you stuck by her and this crazy idea about the letters?\"\n\n\"I was just waiting for the right time to end things. I figured when she sold the letters, I could say that we didn't want anyone looking too hard at us and we should end our, uh, collaboration. It might have worked.\"\n\nMaybe, unless Cordy had made off with the letters or other loot and pocketed all the money. I wondered if she would have stayed around, if she had been successful. I thought it was a possibility: she could think to herself smugly that she had a secret and she'd pulled one over on the whole town. She would have enjoyed that. But I had a feeling she had been aiming higher than Asheboro.\n\n\"Arthur, why were you planning to take the letters? Because Cordy wanted money and leaned on you to do it? Because you could? Because you thought autograph letters from Clara Barton belonged to the nation, and shouldn't be hidden away in a trunk in east nowhere?\"\n\nHe stared at his knees again. \"I needed the money.\"\n\n\"Why? Doesn't the bank pay well enough?\"\n\n\"No.\" Now he looked up at me, with tears in his eyes. \"Do you realize my salary hasn't gone up in the past five years? Business is down, but that's not my fault. Nobody needs mortgage loans around here anymore. Savings accounts have all but disappeared. The bank's in trouble. I was getting desperate.\"\n\nThen I was slammed by an unpleasant realization: the Barton fund or endowment or whatever they called it around here was the single largest account at the bank. Heck, maybe the only large account. And Arthur controlled it. \"Arthur, were you taking money from the Barton fund?\"\n\n\"Yes.\" He looked away again.\n\n\"How much?\" I demanded.\n\n\"About eighty thousand dollars,\" he said grudgingly.\n\nOuch. There were so many things I wanted to say, most of them obscene. There were so many other things I wanted to ask, like what the hell he had done with it (I hoped with all my heart that Cordy hadn't gotten her hands on any of it).\n\nI could hear the sound of cars approaching: the police had arrived. Shucks\u2014I wouldn't get the chance to pummel Arthur to a bloody pulp for what he'd done. Just as well\u2014I couldn't afford an arrest or worse on my resume if I needed to start looking for a new job.\n\nI heard voices outside. The door opened. \"Kate?\" Josh called out.\n\n\"Upstairs,\" I called out. \"Right where you left us.\"\n\nHe took the stairs two at a time. \"Everything all right?\"\n\n\"Just dandy. Arthur has been explaining a lot of things. I'll fill you in later.\"\n\n\"Great. I told the cops as much as I knew, and they're going to take him in for questioning. And hand him over to Detective Reynolds, if that's appropriate.\"\n\nThe cops followed a bit more slowly. Apparently Josh had had time to explain the basic story to them, so they simply hauled Arthur to his feet and told him they were escorting him to the police station for an interview. He didn't protest. He didn't even comment. Apparently it had never occurred to him that he could get caught for larceny and embezzlement. Had Cordy really had that kind of power over men? That reduced them to blubbering idiots?\n\nJosh and I promised we'd show up at the police station in the morning to give our statements to yet another law enforcement agency, and the police took off with their suspect. That left Josh and me alone in the Barton house.\n\n\"You okay?\" He asked. Not very original, but I appreciated the thought.\n\n\"I really don't know,\" I said to him.\n\nJosh didn't reply, but he didn't argue with me. \"We need to get some sleep if we're talking to the police in the morning.\"\n\n\"You've got that right. What time is it?\"\n\n\"Somewhere between two and three. You want to go back to the B&B? It's probably safe now that Arthur is in custody.\"\n\n\"No,\" I said quickly. \"Back to your place. I don't want to deal with this alone. I'll probably crash when the adrenaline stops flowing.\"\n\n\"Okay,\" he said gently, without asking any questions, and then he led me out the back door, and we walked across the dew-wet grass to his door, and then I kissed him, or he kissed me, and things got kind of blurry after that.\n\nI could think about that in the morning.\n\nSomewhere far away a phone was ringing. I refused to open my eyes, even though I could tell through my eyelids that the sun had risen. Wasn't my phone, nope. That was somewhere else. Wait, where was I?\n\nI grudgingly opened my eyes. First observation: I was not at my home or the bed and breakfast or the Barton mansion. My sluggish memory began to rake up fragmented memories for the night before. Police. Darkness. A shovel? Arthur? By a brilliant process of deduction I concluded I must have stayed over at Josh's place, even though whatever risk there might have been from the thief (Arthur? seriously?) was gone. So where was Josh?\n\nBehind me, struggling to get his ringing cell phone out of his jeans pocket and cursing. Quick readjustment in my head: yes, I was at Josh's place. Apparently I had slept on the sofa I had originally planned to sleep on before we went off to catch a thief. Somehow Josh had ended up sleeping on it too. A quick check verified that we were still fully clothed. Well, it had been a rather difficult day, and apparently neither of us had had the energy for anything romantic.\n\nThe phone had stopped ringing. I attempted to roll over to face Josh without falling off the sofa, and said, \"We have to stop meeting like this.\"\n\nHe snorted. \"Whatever. The phone call was for you. Sounds like your pal Lisbeth is freaking out, although I don't know how she found my number. You'd better call her before she sends in troops.\"\n\n\"Oh, right. What time is it?\"\n\n\"About eight, I think. She's awake even if you're not.\"\n\n\"She's got kids\u2014she has no choice. All right, all right\u2014I'll call her.\" Lisbeth would probably understand why I hadn't called her already when she heard how we'd spent the last\u2014goodness, less than twenty-four hours. I managed to get my feet under me and staggered off toward the bathroom. I'd now spent two nights with Josh, and they were quite possibly the least romantic nights I had ever spent with any man. Well, it had been a really nice kiss at the door, from what I remembered of it, but we'd both been so strung out on adrenaline and exhausted from the stress of hunting thieves and killers that neither of us had managed to pursue whatever sparks we'd struck. Hmm... maybe we needed to work on that.\n\nOn the other hand, our crime-solving relationship was going really well.\n\nMarginally clean and combed, I returned to the main room and rummaged through my bag for my cell phone, which I had conveniently turned off. I turned it on and called Lisbeth. She started shrieking before I had even had time to say hello.\n\n\"Where the hell have you been? The police arrested Arthur Fairchild for embezzlement.\"\n\n\"I know,\" I said in a calm and soothing manner.\n\nIt didn't soothe Lisbeth. \"What do you mean, you know? Where have you been?\"\n\n\"Well, Josh didn't think it was safe for me to stay alone at the B&B, so we came back to his place. And then we captured Arthur in the Barton house in the wee hours, where he was trying to find the rest of the letters that Clara Barton wrote our Henry Barton, and we handed him over to the local police to be charged with embezzlement of funds or something.\" I deliberately didn't mention the murder. \"It was a busy day, so we just crashed for what was left of the night.\"\n\nApparently I had finally stunned Lisbeth into silence. In the end she said, \"You have a lot of explaining to do, and talking on the phone will not be adequate. Lunch?\"\n\n\"Sure, if I've got time, after we give the local police department our statements about Arthur. Where?\"\n\n\"If what you have just told me is true, I would recommend not eating in town because everyone is going to want to know the details. The hotel?\"\n\n\"Fine. I'll meet you there at noon.\"\n\n\"Fine.\" Lisbeth hung up.\n\nJosh handed me a cup of hot coffee, and I took it gratefully. \"Well,\" I said, and stopped. I had no idea what to say.\n\n\"Exactly,\" he replied. \"Sit.\" He waved at the table.\n\nI sat, across from him. \"That was a very interesting day. Unique, I might say.\"\n\n\"It was,\" he agreed. \"Can we summarize what we think happened? Because we were both kind of busy. Then you had a chat with Arthur while I was guiding the police, and you promised to fill me in.\"\n\n\"Ah, yes. Do we need to write this down?\"\n\n\"I'll do it,\" he said, hauling a lined pad from a pile of stuff at the other end of the table. \"Speak.\"\n\n\"I don't think Arthur killed Cordelia.\"\n\nJosh didn't sputter or protest. He studied my expression for a moment then said neutrally, \"Why?\"\n\nI sat up straight and laid my hands on the table. \"Arthur told me a whole lot of stuff about what he and Cordelia were doing, and about the money and the letters, and most of that made sense, or it made other stuff make sense. Arthur needed money for personal reasons. He'd already gutted the Barton trust and who knows what other accounts at the bank, and he was desperate for money. He didn't get around to telling me if Cordy was involved in looting the bank accounts, but the bank records should show how long it had been going on. You and I figured that he and Cordy joined up to steal Clara's letters, again for money. But Cordelia had gotten what she wanted and was more or less dumping him, but she could use blackmail at any time\u2014you know, tell his wife, tell the town, get him fired\u2014and she was mean enough to do it. I'll admit that gives Arthur a motive. But the bottom line is, I think he's too much of a wuss to bash a woman in the head with a rock.\"\n\nJosh thought for a while before answering. \"I think that makes sense, Kate,\" he said calmly. \"From what little I know about the man, I can't see him killing Cordelia in a fit of rage.\" He paused before adding, \"I'm sorry.\"\n\nThat confused me. \"For what? You didn't do anything wrong. You just walked into this idiotic drama by accident.\"\n\n\"But this wasn't your fight either, Kate. Are you worried about what will happen to the town's project?\"\n\nHe'd hit the nail on the head. \"Well, yes. I know, so far it was mainly a figment of my imagination. Now I'd have to go back to the town council and tell them there's no money left because Arthur stole it, and there's little hope of finding any more money, so the project is dead and Asheboro is going to fade quietly away.\"\n\n\"There are still Clara's letters\u2014they can be sold on the open market, if the town approves,\" Josh pointed out. \"And there may be other letters or valuables in the attic. I'd suggest getting a reputable dealer or expert in to find what actually is up there.\"\n\nSuddenly I realized something else. \"I don't want to sell those letters! They are so intimately linked to Henry Barton's personal history and to the town. They belong here. We need to find something else to sell, something less personal.\"\n\n\"Spoken like a true historian. You know, you've really gotten invested in this. Maybe you were kidding yourself about your life,\" Josh said.\n\nI glared at him. \"Damn, stop doing that. How do you know what I'm thinking?\"\n\n\"I told you, it all shows on your face. Look, I don't know you well. But from what I've seen, you came here to help a friend. You agreed to help out the town, without asking for anything in return. In the midst of all this you were told you were fired from your day job, but I haven't noticed you grieving a lot about that.\"\n\n\"I've been kind of distracted by murders and thefts and things like that,\" I mumbled.\n\n\"But you never talked much about hotel management, and I never saw you excited about it,\" Josh pressed on. \"The only enthusiasm that you've shown has been for this historical project. Which leads me to think that matters to you.\"\n\nI was afraid he was right. The hotel job had matched my skills and paid enough for me to live quite well. Maybe once I'd entertained vague hopes of moving to a hotel in some exotic foreign location at some point, but I hadn't done anything to make it happen. And it had taken only a week in Asheboro to divert me from what I had thought was my life plan.\n\n\"Is there a cure for that?\" I asked feebly.\n\n\"You tell me. You can still walk away from Asheboro. Now you've got the perfect excuse\u2014someone took all the money. You at least will be covered with glory because you caught Arthur, and you can leave with your head held high, the thanks of the grateful citizens ringing in your ears.\"\n\n\"Don't be patronizing.\"\n\nHe gave me an odd look. \"I'm not, and I'm sorry if it sounded that way. You came back to help a friend, and you ended up solving a crime that nobody even knew was happening.\"\n\n\"I already knew the history of the town. That helped.\"\n\n\"Yes, it did.\"\n\n\"Why did you become a historian, Josh?\" I asked.\n\nHe looked bewildered by my abrupt change of subject but decided to answer anyway. \"Simple. I liked history. I excavated my parents' garbage can when I was five. I wanted a profession that let me focus on history, paid reasonably, and gave me a certain flexibility. Academia fit. And here I am. Obviously investigating murder and theft in this small town wasn't part of the plan, but both of those came about because of events that happened in the past. I came here to study regional industrialization. Turns out Henry Barton was a leading industrialist. But the more you and I poked around, the more we both realized that there was a personal element to his role, by way of his lost brother. And what he did in 1865 or 1880 led to the theft and the death in the here and now, separately or together.\"\n\nThat actually made sense to me. \"But the story included a dose of luck. Don't forget that.\"\n\nJosh cocked his head. \"What do you mean?\"\n\n\"If Lisbeth had invited me a month or so later, Cordelia might have already sold the letters and moved to Palm Beach. If Henry Barton hadn't fought in a battle around here and lost a brother and contacted distant cousin Clara, this town might not have survived into the twentieth century. Those are only two examples of that luck. Do you see what I'm saying?\"\n\n\"In a sense, yes,\" Josh replied. \"But to get back to the present, you know what we have to do now.\"\n\nIt was a statement, not a question. \"Yes\u2014find out who killed Cordelia.\" Which was somehow the part of my personal history.\n\n\"Exactly.\" I helped myself to more coffee, mostly to give myself time to think. \"We agree that Arthur didn't kill Cordelia.\"\n\nJosh nodded once.\n\n\"Good. So who's the next suspect?\"\n\n\"Ryan?\"\n\nI shook my head. \"I may be biased, but I think Ryan was over Cordy, both emotionally and financially. He seems to be making good money as a lawyer. He never had any interest in history. So what's his motive? And why would he choose the Barton house to kill her?\"\n\n\"Fewer witnesses?\" Josh suggested. \"Maybe Cordy invited him out there, to ask him to invest in her project for the house. Or maybe she tried to use her physical charms to do it\u2014it had worked before\u2014and Ryan got so fed up he simply grabbed the first thing he could find and bashed her. Every man has a breaking point.\"\n\n\"Creative solution, but I'm not buying Ryan as a killer. Who's next?\"\n\nJosh sat back in his chair and looked at me. \"I'm not an expert on Asheboro, but it's my impression that a lot of people disliked Cordelia\u2014including myself, based on limited exposure\u2014but I'd guess that few people hated her enough to kill her. I don't really have any other suggestions.\"\n\nI returned his look. \"I know what you mean about Cordy. She made my life hell for a time, and I've spent years getting past that, but I still can't imagine killing her myself. Or even hiring a hit man. Or woman. Let's go back to Arthur. Does he have a wife? Who might have found out about the affair? That could have made her angry enough to go after Cordelia.\"\n\nHow many motives were there for killing anyone, I found myself wondering. Hate\u2014but that could come from so many sources. Revenge? Jealousy? I had to believe that a brutal, unplanned murder like Cordelia's had to come from emotion, not calculation. Who in this small town had hated her that much?\n\nAnd then an unlikely thought crept into my mind. I turned it over and around, studying it. It was improbable, but was it possible?\n\n\"What about Audrey?\" I said carefully.\n\n\"Audrey?\" Josh stared at me incredulously. \"As Cordelia's killer?\"\n\nI had to admit that the idea of the patient drab librarian killing the town bitch Cordelia seemed ridiculous, but we were running out of candidates. What could Audrey's motive have been?\n\nJealousy. Audrey had fixated on Josh as her last best hope for love, or at least for escape from Asheboro, perhaps even knowing how unlikely it was that he was attracted to her. But the heart wants what it wants, and love is not rational. At least that's what romance novels said.\n\nIf Cordy had sensed Audrey's yearning, she would have pounced on it like a cat on a mouse. She would have tormented Audrey just because she could. And she wouldn't have let up. So Audrey would have grown angrier and angrier.\n\nWould there have been a trigger incident that could have pushed Audrey over the edge?\n\n\"Josh?\" I said slowly. \"You told me awhile ago that Cordelia had made a play for you. Did Audrey ever see you and Cordelia together?\"\n\n\"That's an odd question,\" he said.\n\n\"Just tell me\u2014did she?\"\n\n\"Probably. Every time Cordelia and I crossed paths in town, she was kind of all over me\u2014you know, the whole girly standing too close, touching me, et cetera. I didn't encourage her, but short of shoving her away there wasn't much I could do. I suppose Audrey could have seen us any one of those times and jumped to conclusions.\"\n\n\"And Cordy would have played it up, if she had an audience,\" I said, almost to myself. \"Did Audrey ever come out to the Barton house?\"\n\n\"Rarely. But there was one time when she said she wanted to check the condition of the books, and she came out and spent some time in the Barton library. That was about the same time that Cordy was in and out of the house with her so-called investors. They could easily have overlapped.\"\n\n\"Any reason why Audrey would have been there on Sunday?\"\n\n\"Kate, I don't like where you're going with this,\" Josh warned.\n\nBut I would not be diverted. \"Did she come out to see you, or to deliver something to you?\"\n\n\"No. I wasn't around for much of the day. But... Cordelia might have asked Audrey for something, asked her to bring it to the house. Let me get this straight: are you really saying you think Audrey might have killed Cordelia?\"\n\n\"I think it's possible. Do you?\" I shot back.\n\n\"I... don't know.\" Josh looked troubled as he wrestled with the question.\n\nI stood up. \"I need to talk to Audrey.\"\n\n\"I'll come with you,\" Josh volunteered quickly.\n\nI held up a hand. \"That's not a good idea. This will be hard enough without you there. If I'm wrong, she doesn't need to know that you know what I'm thinking.\"\n\n\"If she's killed once, she might go after you.\"\n\nI really couldn't picture that. \"You can wait close by, or peer in the window. But I can't see her attacking me.\"\n\n\"I don't like it,\" Josh muttered.\n\n\"Do you have any better ideas for the killer? Like, maybe a random stranger wandered into the estate and bashed Cordy because she happened to be in the wrong place at the wrong time?\"\n\nJosh sighed. \"No, that doesn't seem likely. But then, none of the other possible suspects, which would be half the people in town, seems likely.\"\n\n\"Then let me eliminate Audrey from the list, if I can. If I've guessed wrong, I'll admit it. I kind of like her, and I know I feel sorry for her.\"\n\n\"Kate, it's Sunday. You think she'd be at the library?\"\n\n\"I do. She really has nowhere else to be. Sad, isn't it?\"\n\n\"It is. So let's do it\u2014try to talk with her. At least it will move this mess one step forward. But I'm not letting you walk into this alone.\"\n\nJosh drove us into town and parked a block or so from the library. \"What do you plan to do?\"\n\n\"Just talk to Audrey. I don't intend to accuse her to her face. I hope I'm wrong, you know. She's not a bad person.\"\n\n\"Good people do bad things all the time,\" Josh reminded me. \"Be careful, will you? I'll be waiting outside.\"\n\nIt hadn't even occurred to me that the library might not be open on Sunday, but I knew Audrey had nowhere else to be. As I walked the block to the library, I remembered what Audrey and I had talked about earlier. I could understand Audrey feeling trapped by the demands of her family. But Cordy's veiled insults and contempt would have made things worse. Plus Audrey was not the type to stand up to Cordy. If I believed Audrey, she'd never responded to Cordy's taunts, which I was pretty sure must have pushed Cordy to go even farther.\n\nWhen I reached the front of the library building, I peered through the windows. There seemed to be no one except Audrey inside, and I was relieved both because I'd found her and because there's be no one to overhear what promised to be a very awkward conversation. I pulled myself together and opened the door, and then strode over to the main desk. \"Audrey, I need to talk to you.\"\n\nShe looked up without smiling. \"All right. What's this about? Henry Barton again?\"\n\nI shook my head. \"There's no easy way to say this, and you have every right to be angry with me and refuse to say anything, but I hope you'll give me an answer.\"\n\n\"Get to the point, will you, Kate?\" Audrey folded her arms across her chest.\n\n\"You told me you were, uh, interested in Josh Wainwright. Did Cordelia set her sights on him?\"\n\nAudrey studied my face, and I wondered if she was trying to make a decision about how much to say. \"What do you think? New in town, attractive, intelligent? Of course she did.\"\n\n\"Did that make you angry?\"\n\nI was surprised when she actually answered my tactless question. \"That's not exactly easy to say,\" Audrey said. \"Did I honestly expect that Josh would ever even notice me as a living, breathing woman? I knew perfectly well that he wasn't interested in me. That was a bit too much to hope for. But I guess I had dreams, or fantasies. Helps pass the time. But then Cordy spied him. You know, he was staying in that big place outside of town and he didn't come into town often, except to the library for his research now and then, so it took Cordy awhile to notice him at all, but when she did it was right here in the library. 'Who's that?' she asked me. I gave her a short answer\u2014'He's the caretaker at the Barton house.' But he was fresh meat and attractive and breathing, so she moved in. I got to watch the whole thing. To his credit, he was polite but he didn't take the bait. But you know how persistent Cordy could be.\"\n\n\"I do, unfortunately.\" And then I waited, because I was pretty sure there was more.\n\nAudrey looked past me, and I guessed she was revisiting her own memories. \"Then the town bought the Barton property, and Cordy decided to get herself involved with it. And, from what I hear, she thought the town wasn't moving fast enough with its plans to use the place, so she went out looking for other people to step up. And found some. She thought she was on her way to dumping her tacky little bed and breakfast and moving up in the world, or at least in the Asheboro universe. Of course she was out at the mansion a lot, showing the place off to investors and the like.\"\n\nI could see it. Cordy would sometimes by out there by herself. And she could just happen to run into Josh.\n\nIt was time to go all-in, and I knew I had to ask the question. \"Audrey, did you kill Cordelia?\"\n\nAudrey almost smiled. \"I wondered when you'd figure that out.\"\n\nWe stared at each other across the book-strewn desk, but she made no move to threaten me. There were plenty of times during high school that I had fantasized about killing Cordelia, or at least torturing her\u2014a lot of spiders and worms figured in those daydreams. But I'd left town and moved past that. Audrey had had to live with Cordelia just popping in at the library on a regular basis and saying snide things. And it seemed that Audrey had finally had enough.\n\n\"What happened, Audrey? Would you like to explain? Or, wait\u2014maybe I should go lock the door. I don't think this is the time to deal with people looking for books.\"\n\n\"You're right.\" Audrey reached into her pocket and fished out a heavy ring of keys, then handed it to me. \"The big one is for the front. We'd better stick up a sign, or people will start pounding on the door.\"\n\nI waited silently while Audrey made a sign with a broad black marker, reading CLOSED DUE TO PERSONAL EMERGENCY. Then I scurried to the front door and made sure it was locked, and taped the sign to the glass of the door. Josh was indeed waiting outside, near the street, and I gave him a quick thumbs-up so he wouldn't come barging in. Then I went back to the main desk where Audrey was waiting. I had now locked myself in the library with a woman who had just admitted to being a murderer.\n\nNow what? Part of me really wanted to call Detective Reynolds and tell him to get his butt over here and pick up his suspect. But I felt I owed it to Audrey to give her the chance to tell her story before the inevitable process of police procedures took over. I found a stool and pulled it behind the desk, so that Audrey and I could sit face to face.\n\n\"What happened?\" I said quietly.\n\nAudrey cocked her head at me. \"Look, Kate, let me say now, don't feel bad about ratting me out, which I know you're going to do. I probably would have turned myself in anyway. Even if I wanted to run, I have nowhere to go.\"\n\nI couldn't think of anything to say to that. \"Audrey, why were you at the Barton house that day?\"\n\nAudrey sat up straight on her stool and raised her chin. \"Cordelia asked me to meet her there. She wanted to know what to do with the books\u2014I don't think a stuffy old library fit into her grand plans for the place, but thank goodness she knew better that to toss them all in a Dumpster. I'd seen some valuable volumes there, and I wanted to make sure the town got a good price for them\u2014the town owns them, you know. I wish we had room for them in the library.\"\n\nI nodded. \"So what finally pushed your last button\u2014that was because she was making fun of you, right?\"\n\n\"Yes. Nothing I hadn't heard before, but enough was enough.\"\n\nI knew how she felt.\n\nAudrey interrupted my thoughts. \"Listen, I know you wanted information about Henry Barton, how he came to live here, who his family was, all that, and I put together all that I could find on that rolling cart over there. Once I'm arrested, I'm not sure whom they'll get to run the library\u2014they might even just shut it down. If it makes any difference, as one of my last official acts I give you and Josh permission to use whatever you like here, whenever you like. I doubt the town will object. There's a spare set of keys in the drawer here.\"\n\nAudrey was a librarian to her bones\u2014I hadn't even thought of that. She went on, \"You know, I haven't done a lot of research myself, but you'll find that Henry Barton was a very interesting character. There was more to him than you might think at first. But he was a private man as well, so you have to hunt for the trail.\"\n\n\"I'm just beginning to realize that. Look, Audrey, I'm sorry I figured out that it was you. You've had enough troubles in your life without this.\"\n\nAudrey smiled ruefully. \"Some people seem to be cursed, and I think I'm one of them. I'm proud of you, though.\"\n\n\"Me? Why?\"\n\n\"Because when Lisbeth called you to help Asheboro, you could simply have said 'sorry, no thanks,' and left it at that. Okay, maybe you didn't know then that Cordelia was still in town. But once you found out, which I'll wager didn't take long, you could still have told the town that you didn't think you could help and walked away.\"\n\n\"I could have, I guess, but that's not me. I needed to be a grown-up here in Asheboro and deal with those old issues. And I do think the project is challenging and worth following up on.\" An errant thought struck me. \"You didn't kill Cordelia just to get her out of the way to make sure I'd stick around and save the town?\"\n\nAudrey smiled ruefully. \"You give me too much credit, Kate. It wasn't planned at all. No, it was pure fury, distilled for years. And you know what? It felt really, really good to admit to myself how angry I was, and to act on it. I didn't think it through\u2014I just snapped. I grabbed a rock from the garden next to the door and hit her as hard as I could. Had I been in a rational state of mind, I could have plotted something much more subtle. But in the end, on her way out the door she said something like, 'I'll leave you to your dusty old books. If you see Josh Wainwright, tell him I want a word with him, will you?' And then she sashayed out the front door, turning her back on me. And that was it\u2014I'd had enough. She'd twisted the knife one time too many. I followed her to the front, and when she heard me coming she turned around. That's when I found that convenient rock in the garden next to the steps. I picked it up and raised it over my head\"\u2014Audrey closed her eyes for a moment, but she was smiling\u2014\"and I brought it down on her head, hard\u2014wham. And I looked down at her then, lying on the steps, bleeding, and all I felt was triumph. I'd finally taken charge of my own life.\"\n\n\"Oh, Audrey, I'm so sorry it had to come to that.\"\n\nI needed to do something. But what? The clear and obvious answer would be to tell the state police, right now.\n\nOr I could tell someone else and share the misery. Someone like Josh, who I could see approaching the front door of the library at that very moment. Audrey noticed at the same time. \"Oh, crap,\" she muttered. \"You brought him?\"\n\n\"Can he come in?\" I asked.\n\nAudrey smiled sadly. \"Why not? It's all over anyway.\"\n\nI went over to the front door and unlocked it. Josh looked past me at Audrey, still seated behind the main desk then looked at me. \"Yes?\" he asked quietly.\n\nI could only nod.\n\n\"Then I should go and call the detective.\" Without waiting for my response, he turned and walked toward the road until he was out of sight from inside.\n\nAudrey looked up at me after Josh had left and I'd carefully locked the door behind him. She looked oddly serene. \"He's calling the police now, isn't he?\" she asked.\n\n\"Yes, he is. Do you want a lawyer?\" I asked.\n\nShe sighed. \"I suppose I should have one, but the only one I know around here is the one who handled my parents' estates. He's about eighty, and I don't think he could deal with a murder trial.\"\n\n\"There's always Ryan, although I don't know if he handles... big things like this, and it probably wouldn't be appropriate anyway. But he could recommend someone at his firm. Actually, you might do better to save him for a witness\u2014he could testify why Cordy was such a pain in the ass. You could plead temporary insanity, or something along those lines.\"\n\n\"Good point.\"\n\nJosh came back, and I let him in again. \"He's on his way,\" he told me. \"I'll wait out front for him.\"\n\nAudrey and I sat in uncomfortable silence\u2014what was there to say? Twenty minutes later I looked up to see Josh leading Detective Reynolds and a couple of uniformed cops to the front door, so I hurried over to unlock it again. As the police walked into the library, Audrey remained where she was.\n\n\"Kate,\" Detective Reynolds said then turned to Audrey. \"And you're Audrey Perlaw? The librarian?\"\n\n\"I am. And you are?\"\n\n\"Detective Reynolds of the Maryland State Police. I understand you have something to tell us?\"\n\n\"I do. Please, sit down. This may take awhile, and I'd like to tell you the story in my own way.\"\n\n\"Fine.\" The detective glanced at Josh and me. \"Please wait outside.\" Josh and I didn't argue. Besides, I'd already heard the story, and I could fill Josh in.\n\nOnce outside we sat numbly on a bench in front of the library. I had no idea what to do. I watched as a number of people strolled by after church, then slowed, then tried to peek inside. It must have been the police car parked in front that had attracted them, but now they kept looking at Josh and me too, as if trying to put the pieces together.\n\nJosh leaned toward me. \"We can sit in my car if you want.\"\n\nI shrugged. \"Why bother? Is the detective going to want to interview us when he's finished with Audrey?\"\n\n\"I can't say\u2014I haven't been part of too many murder investigations. In fact, none.\"\n\n\"Same with me.\" I took a breath. \"Look, Josh, Audrey explained what happened, and it's pretty much the way we guessed. She didn't think you treated her unfairly.\"\n\n\"I suppose I should be happy to hear that, but that doesn't make me feel much better.\"\n\n\"I know what you mean. Poor Audrey.\"\n\nAfter a while I said, \"Do we need to tell the police about the Barton letters?\"\n\n\"I think not,\" Josh said. \"Audrey's motives for attacking Cordelia were purely personal and had nothing to do with the letters, and I think the local police will handle Arthur's thefts, so Reynolds won't be involved. I think we're safe in leaving that door closed.\"\n\nI nodded. \"That's what I was thinking. I don't think Audrey knew anything about them, and I doubt she'd mention them if she did.\"\n\nTime slowed to a crawl. The detective took his time interviewing Audrey, and when I got up to stretch my legs and just happened to glance through the front window, I saw the two other officers sitting mutely in the corners of the back room, taking notes. Audrey stayed calm and answered all of Detective Reynolds's questions, as far as I could see, so apparently she hadn't asked for a lawyer. I went back to the bench and sat again. Josh and I waited, and then we waited some more. Finally after an hour or so, one of the uniformed policemen came out of the building, and I went to meet him.\n\n\"The detective wants to talk to you next, then your friend.\"\n\n\"Fine.\" Inside, Detective Reynolds rose to meet me.\n\n\"Kate, this is not a formal statement\u2014we can deal with that later.\" He turned to one of the guys in uniform. \"Jenkins, please take Miss Perlaw out to the car.\"\n\nHe talked to me first, and I explained what little I'd learned before he arrived. The he sent me out to tell Josh to come in, and I went back to what seemed to have become \"our\" bench, and sat staring into space. When he had finished talking to Josh, the two men came out to join me.\n\nReynolds said quietly, \"I understand you have keys to the library?\"\n\n\"Yes, Audrey wanted us to have them, to continue some research we'd been doing. Is that a problem?\"\n\n\"I don't believe so,\" Detective Reynolds said. \"I think we can trust you.\"\n\n\"What happens now?\"\n\n\"We take Ms. Perlaw to our offices and arraign her. I'll be in touch with you two about giving an official statement. And thank you.\" With that he turned and left.\n\nSince I had no better ideas, I sat down on the bench again, and Josh joined me. The small crowd had vanished. \"Poor Audrey,\" I said to him. \"That could have been me, if things had turned out differently in my life. What do we do now?\"\n\n\"Are you forgetting Henry Barton?\" Josh asked.\n\n\"You mean our research, and where Clara's letters fit? We do have the keys to the library, thanks to Audrey. Are you seriously thinking of doing research right now?\"\n\nJosh looked down at his shoes. \"It might be better to use the library resources while we still can\u2014Reynolds or the town could change their minds and we won't be able to get back in.\"\n\n\"Audrey told me that she thought Henry Barton was a very interesting man. Maybe she was hinting she knew something. I think she'd expect us to follow up.\"\n\n\"Definitely.\"\n\nSo we went back into the library, locked the door, leaving the sign taped in place, and assembled all the files Audrey had collected for us, sat down at the long table in the Local History Room in the back, and got back to work.\n\nThe next time I looked outside, it was already getting dark. The afternoon had vanished while Josh and I sat in the empty library immersed in nineteenth-century Asheboro history and Henry Barton's life. I had stopped reading only because my brain was no longer accepting input.\n\nI stood up and stretched. Josh looked up and said hopefully, \"Quitting time?\"\n\n\"I vote yes.\" Then I stopped. It was time to leave, but where would I go? The B&B, I supposed, although I could certainly go back to my Baltimore condo, only an hour away. But I didn't want to go back to Baltimore tonight. I wanted to stay and celebrate in Asheboro... with Josh. Except I didn't know how to ask, exactly. There had been so many misunderstandings lately\u2014Josh missing Audrey's signals, Cordy completely oblivious to anybody else's signals, and so on. I didn't want to mess things up. But what did I want?\n\nOh, grow up, Kate! I thought there had been some electricity between me and Josh, and it would be better to know now rather than play silly games. \"Want to come back to the B&B with me? We can pick up some dinner, and I know for a fact that there's more of that good wine. Maybe we could exorcise Cordy's ghost from the place.\"\n\n\"If we drink a bottle of that stuff, I probably shouldn't drive back,\" Josh said carefully. His unspoken question hung in the air between us.\n\n\"Then why don't you plan to stay over?\" I said.\n\nHe smiled. I smiled back. No mixed signals here.\n\nWe went back to the B&B, stopping along the way to pick up an assortment of Chinese food. Maybe it didn't go well with fine wine, but we didn't care. The wine was perfect for other activities.\n\nMorning found us tangled up in the sheets of the big bedroom upstairs. As I rolled over to look at Josh, the irreverent thought flitted through my mind: maybe if we tried this in the master bedroom at the Barton mansion, it would put to rest any bad memories of that awful moment with Ryan, when Cordy and her army of giggling friends arrived with cameras flashing. Worth a try, anyway.\n\n\"Hey,\" Josh said. His eyes were open now.\n\n\"Hey, yourself. Are we okay?\"\n\n\"More than okay,\" he said, smiling.\n\nSometime later I asked, \"What day is this?\"\n\n\"I'm not sure. A weekday, maybe? I'd guess Monday.\"\n\n\"I hope so. I have to call Lisbeth and explain and apologize and all that stuff. And then I have another call to make.\"\n\n\"Town council?\"\n\n\"No, Nell Pratt, in Philadelphia. She's going to help us sort out the treasures in the Barton attic. I want her to see them sooner rather than later, so I know what to tell the council.\"\n\n\"So you've figured out what comes next?\" Josh asked.\n\n\"What part do you mean? For the Barton house? For Asheboro?\" For me? \"I've got some ideas. And I'm willing to present them to the town, in rough form, when the dust settles. This week or next. If they reject my solution, I need to know that up front.\"\n\n\"There's something else I could add, I think...\" He seemed to be struggling to find the right words. \"This is really speculative, but as we've been looking at the materials that Audrey assembled, and also checking out Internet sources, I'm beginning to see something that could be really interesting.\"\n\n\"Are you planning to share this with me?\"\n\n\"I'm not ready to, but let's say it falls within my area of academic expertise, which is to say, commercial history rather than personal.\"\n\n\"Why do you bring it up now?\"\n\n\"Because if I'm right, it could be really important, both historically, which benefits me, and financially, which could help you\u2014us save the Barton mansion and the town. Assuming you don't mind hanging around and working on the project for a while longer.\"\n\n\"I confess I've been thinking along the same lines. If Nell can find someone qualified to assess both the Barton letters and whatever else might be in the attic, it could provide some seed money for the project. And if it works out, I'd like to be a part of it, going forward. But don't you have a day job, when your sabbatical is over?\"\n\n\"Yes, I do. But if you look at a map, Kate, you'll notice that my job is not far away. And what I have in mind fits quite neatly with the research I've been doing for a while now. I'm sure my department would understand.\"\n\nI laughed. \"We really are a pair! After last night\u2014which was great, by the way\u2014here we sit kicking around ideas that might turn things around and help the town save and promote the Barton house. And we both seem to be on the same page about giving it its rightful place in history. Have I got that right?\"\n\n\"I think so.\"\n\n\"Josh, do you think we have a shot at pulling this off?\"\n\n\"I think it's worth trying,\" he said. \"I hope so.\"\n\nNell's train pulled into Baltimore's Penn Station on time, and I was there waiting. I was happy to see she was carrying a small travel bag\u2014so she was staying? goody, goody\u2014more time for me to convince her of the potential significance of the town project, including the Barton house, and to pick her brain on how to make it work. I pulled over to the curb and opened the passenger door for her. \"Welcome! And thank you for coming. I hope I didn't sound too crazy.\"\n\nShe buckled her seatbelt and turned toward me. \"Don't worry\u2014I was ready for a field trip anyway, and you've certainly got me curious. Leading with those letters was a good strategy\u2014you sucked me right in.\"\n\n\"I'm glad to hear that. And there's a lot more to talk about.\"\n\n\"Like crime solving?\" Nell grinned at me.\n\n\"Exactly.\" As we drove from the station and headed west, I filled her in on the house and Audrey and the louse of a bank president and the quandary the whole mess had put everyone in. When I finally ran out of steam, she said, \"Wow, you make my job look tame. Although I did find one body in the stacks. Welcome to the sisterhood of accidental sleuths.\"\n\n\"Before you make any judgments about the project,\" I said, \"I really do want you to see the house\u2014it's fabulous. It's as if it's preserved in amber. But it's still a big step to finding money to preserve it\u2014that's where finding anything of value that the town could sell becomes important.\"\n\n\"And you're hoping what might be in the attic can make a difference?\"\n\n\"I can hope, can't I?\"\n\n\"So you really have no idea what's up there?\"\n\n\"No. Josh and I don't know where to start. We figured we'd leave that to the experts.\"\n\n\"Josh?\"\n\n\"Long story. He's going to stay involved.\" I proceeded to tell Nell the story as we drove back to Asheboro.\n\nWhen I'd finished my G-rated summary, she said, \"It sounds as though he could be a real asset to the project.\"\n\n\"I think he's hooked. The Barton stuff dovetails beautifully with his own work.\"\n\n\"Selling the Clara Barton letters should be no problem. That market has held up well.\"\n\n\"That's good to know. But in a way I'd like to keep them and weave them into whatever story we put together to market the Barton place. Selfish, I know. If we have to sell them, so be it, but I'd like to know if there are alternatives first.\"\n\n\"I understand, believe me. We've done our share of de-accessioning at the Society, and it hurts me every time.\"\n\nWe reached the outskirts of Asheboro by mid-afternoon, and I told Nell, \"I'd like to take you by the house first, because that's the centerpiece of the plan, if not exactly the physical center. It won't take long to check out the town after that. Have you been to this part of Maryland before?\"\n\n\"Not often, and not lately. It's rather lovely, isn't it? Although I hate to think of what it must have been like during the Civil War.\"\n\n\"As far as I know\u2014and I'm no expert\u2014I don't think any important action took place here, just all around. And I assume there would have been regiments of soldiers and supply trains and, yes, medical tents and supplies and personnel tramping through. From what little I've had time to read, it sounds like such an awful war, although all wars are, aren't they?\"\n\n\"That they are,\" Nell said quietly.\n\nWe arrived at the property. Josh had left the front gate open for us, so I passed through it and drove slowly toward the house, still concealed in its dell. When we came over the crest of the hill, when the house first became visible, I heard Nell say quietly, \"Oh, my.\"\n\nI stopped the car and let her admire the view. Finally she nodded once, and I drove slowly forward to the front of the house, where I parked. \"You haven't even seen the inside yet.\"\n\n\"I can't wait. All the furnishings are intact?\"\n\n\"They are.\"\n\n\"Be still, my heart,\" Nell breathed.\n\nOnce we went in, I wasn't sure I'd ever get her out of the building. At one point she asked, \"Can I touch things?\"\n\n\"Why not?\" It was getting dark when I finally managed to tear Nell away from the building. We walked out the front door and admired the unmarred landscape in front of us.\n\n\"Wow,\" she said finally, in a reverent tone. \"I get it now, why you were so excited. It really is rare to find a place so untouched by time. And I wouldn't change it, apart from any necessary safety issues.\"\n\n\"But where's the money going to come from?\"\n\n\"We'll find it. I promise.\"\nALSO BY SHEILA CONNOLLY\n\nORCHARD MYSTERIES\n\nOne Bad Apple\n\nRotten to the Core\n\nRed Delicious Death\n\nA Killer Crop\n\nBitter Harvest\n\nSour Apples\n\nGolden Malicious\n\nPicked to Die\n\nA Gala Event\n\nSeeds of Deception\n\nA Late Frost\n\nMUSEUM MYSTERIES\n\nFundraising the Dead\n\nLet's Play Dead\n\nDead Letters\n\nFire Engine Dead\n\nMonument to the Dead\n\nRazing the Dead\n\nPrivy to the Dead\n\nDead End Street\n\nCOUNTY CORK MYSTERIES\n\nBuried in a Bog\n\nScandal in Skibbereen\n\nAn Early Wake\n\nA Turn for the Bad\n\nCruel Winter\n\nMany a Twist\n\nRELATIVELY DEAD MYSTERIES\n\nRelatively Dead\n\nSeeing the Dead\n\nDefending the Dead\n\nWatch for the Dead\n\nSearch for the Dead\nAbout the Author\n\nSHEILA CONNOLLY has published over thirty mysteries, including several New York Times bestsellers. Her series include the Orchard Mysteries, the Museum Mysteries, the County Cork Mysteries, and the newest, the Victorian Village Mysteries. She is a member of the Daughters of the American Revolution and the Society of Mayflower Descendants, and owns a cottage in West Cork. She lives in a too-big Victorian in southeastern Massachusetts with her husband and three cats. You can sign up for email updates here.\n\nThank you for buying this\n\nSt. Martin's Press ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nContents\n\nTitle Page\n\nCopyright Notice\n\nDedication\n\nAcknowledgments\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nAlso by Sheila Connolly\n\nAbout the Author\n\nCopyright\nThis is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously.\n\nMURDER AT THE MANSION. Copyright \u00a9 2018 by Sheila Connolly. All rights reserved. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.\n\nwww.minotaurbooks.com\n\nCover design by David Baldeosingh Rotstein\n\nCover illustration by Larry Jacobsen\/Shutterstock.com; boards \u00a9 Anna Volotkovska\/Shutterstock.com; birds \u00a9 studio23\/Shutterstock.com\n\nThe Library of Congress Cataloging-in-Publication Data is available upon request.\n\nISBN 978-1-250-13586-5 (hardcover)\n\nISBN 978-1-250-13587-2 (ebook)\n\neISBN 9781250135872\n\nOur books may be purchased in bulk for promotional, educational, or business use. Please contact your local bookseller or the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, extension 5442, or by email at MacmillanSpecialMarkets@macmillan.com.\n\nFirst Edition: June 2018\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Dedication\n  4. Acknowledgments\n  5. Chapter 1\n  6. Chapter 2\n  7. Chapter 3\n  8. Chapter 4\n  9. Chapter 5\n  10. Chapter 6\n  11. Chapter 7\n  12. Chapter 8\n  13. Chapter 9\n  14. Chapter 10\n  15. Chapter 11\n  16. Chapter 12\n  17. Chapter 13\n  18. Chapter 14\n  19. Chapter 15\n  20. Chapter 16\n  21. Chapter 17\n  22. Chapter 18\n  23. Chapter 19\n  24. Chapter 20\n  25. Chapter 21\n  26. Chapter 22\n  27. Chapter 23\n  28. Chapter 24\n  29. Chapter 25\n  30. Chapter 26\n  31. Chapter 27\n  32. Chapter 28\n  33. Chapter 29\n  34. Chapter 30\n  35. Chapter 31\n  36. Chapter 32\n  37. Chapter 33\n  38. Chapter 34\n  39. Also by Sheila Connolly\n  40. About the Author\n  41. Newsletter Sign-up\n  42. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n\n## Pagebreaks of the print version\n\n  1. Cover Page\n  2. iii\n  3. v\n  4. vii\n  5. viii\n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. ii\n  329. iv\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Chuck Greif and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net (This file was\nproduced from images available at The Internet Archive)\n\n\n\n\n\n\n\n\nTypographical errors have been corrected (see list following text0.) No\nattempt has been made to correct or normalize the spelling of place or\npersonal names. A letter symbolized in this manner [=D] indicates that it is\npictured in the text with a line over it. One symbolized in this manner [A=]\nindicates it is pictured in the text with a line under it. The caret character\n(^) indicates a superscript follows. (note of etext transcriber.)\n\n\n\n\n[Illustration: image of the book's cover\n\nSpanish Arts\n\nBY\n\nJUAN F. RIA\u00d1O.]\n\n\n\n\nFREDERICK P. PRITCHARD\n\nSOUTH KENSINGTON MUSEUM ART HANDBOOKS.\n\nSPANISH INDUSTRIAL ARTS.\n\n_This Volume, forming one of the Series of Art Handbooks issued under\nthe authority of the Lords of the Committee of Council on Education, has\nbeen prepared by_ SE\u00d1OR JUAN F. RIA\u00d1O, _of Madrid, whose Catalogue of\nArt Objects of Spanish production in the South Kensington Museum, issued\nin 1872, has proved of great value and interest._\n\n_April, 1879._\n\n\n\n\nTHE INDUSTRIAL ARTS\nIN SPAIN.\n\nBY\n\nJUAN F. RIA\u00d1O.\n\nWITH NUMEROUS WOODCUTS\n\n[Illustration]\n\n_Published for the Committee of Council on Education,_\n\nBY\n\nCHAPMAN AND HALL, LIMITED,\n\nLONDON.\n\n1890.\n\nRICHARD CLAY AND SONS, LIMITED,\n\nLONDON AND BUNGAY.\n\n\n\n\nCONTENTS.\n\n\n                                                                    PAGE\n\nGOLD AND SILVER WORK                                                   1\n\nIRON WORK                                                             58\n\nBRONZES                                                               68\n\nARMS                                                                  79\n\nFURNITURE                                                            109\n\nIVORIES                                                              126\n\nPOTTERY AND PORCELAIN                                                143\n\nGLASS                                                                228\n\nTEXTILE FABRICS                                                      250\n\nLACE                                                                 271\n\n\n\n\nILLUSTRATIONS.\n\n\nNO.                                                                 PAGE\n\n1. Votive Crown of King Swinthila, Madrid                              7\n\n2. Leaf of Triptych in Cathedral of Seville                           17\n\n3. Processional Cross, 15th century, South Kensington Museum          20\n\n4. Chalice, 16th century, S.K.M.                                      30\n\n5. Chalice, 17th century, S.K.M.                                      31\n\n6. Pax, 16th century, S.K.M.                                          33\n\n7. Jewel, 17th century, S.K.M.                                        35\n\n8. Silver Dish, 17th century, S.K.M.                                  36\n\n9. Breast Ornament set with emeralds, late 17th century, S.K.M.       37\n\n10. Breast Ornament set with diamonds, late 17th century, S.K.M.      39\n\n11. Moorish Keys in Cathedral of Seville                              60\n\n12. Nail heads, S.K.M.                                                66\n\n13. Moorish Lamp, Arch\u00e6ological Museum, Madrid                        72\n\n14. Tenebrarium in Cathedral of Seville                               75\n\n15. Monstrance, dated 1537, S.K.M.                                    77\n\n16. Sword of Boabdil, Villaseca Collection, Madrid                    85\n\n17. Moorish Dagger, 15th century                                      89\n\n18. Rapiers, 16th and 17th centuries, S.K.M.                          97\n\n19. Breastplate, formerly in Bernal Collection                        98\n\n20. Saddle of the Cid--Armoury, Madrid                                99\n\n21. Muskets                                                          100\n\n22. Botica de los Templarios, from Toledo, S.K.M.                    114\n\n23. Vargue\u00f1o Cabinet, 16th century, S.K.M.                           121\n\n24. Ivory cylindrical box, Moorish, 10th century, S.K.M.             128\n\n25. Ivory rectangular box, Moorish, 10th century, S.K.M.             129\n\n26. Ivory Casket, Moorish, 11th century, Cathedral of Pamplona       131\n\n27. Cross of King Ferdinand I., Arch\u00e6ological Museum, Madrid         136\n\n28. Samian Bowl                                                      144\n\n29. Plateau, (probably of Valencia), S.K.M.                          152\n\n30. Moorish Vase at the Alhambra                                     155\n\n31. Tile, (probably from the Alhambra)                               157\n\n32. Hispano-Moresco Vase, 14th or 15th century, S.K.M.               159\n\n33. Hispano-Moresco Plateau with raised ribs and studs,\n    15th or 16th century, S.K.M.                                     160\n\n34. Hispano-Moresco Salver, 15th or 16th century, S.K.M.             161\n\n35. Wine Jar, \"Tinaja,\" S.K.M.                                       164\n\n36. Brim of Well from Toledo, S.K.M.                                 166\n\n37. Earthenware Bowl, Talavera ware, S.K.M.                          174\n\n38. Earthenware Plaque, Alcora ware, S.K.M.                          194\n\n39. Marks and Signatures of Painters who worked at Alcora       204--210\n\n40. Room decorated with Buen-Retiro porcelain in\n    the Palace at Madrid                                             217\n\n41. Vase, Buen Retiro porcelain, S.K.M.                              220\n\n42. Marks used at the porcelain manufactory of Buen Retiro      221--223\n\n43. Glass Vase, 16th century, S.K.M                                  232\n\n44. Pilgrim's Bottle, Glass, 17th century, S.K.M.                    233\n\n45. Glass Bottle, modern, S.K.M.                                     237\n\n46. Glass Vessels, S.K.M.                                            238\n\n47. Glass Vase, Cadalso, 17th century, S.K.M.                        239\n\n48. Vase, Green glass with black handle and ribs,\n    17th century, S.K.M.                                             242\n\n49. Glass Vessels, S.K.M.                                            245\n\n50. Tapestry at Cathedral of Gerona, 11th century                    267\n\n_The Department of Science and Art is indebted to_ MR. JOHN MURRAY _for\nthe use of the Woodcuts Nos. 19, 20, 30, and 31. No. 50 is from a\nSpanish Woodcut._\n\n\n\n\nSPANISH INDUSTRIAL ART.\n\n\n\n\nGOLD AND SILVER WORK.\n\n\nThe Greek and Latin authors who have described the Spanish Peninsula,\nstate that the quantity of gold and silver ore found there was very\ngreat, and that hence the district became an important centre of\ncommercial activity of Ph\u0153nicians, Carthaginians, Greeks, and Romans.\nSome authors have gone so far as to assert that the Ph\u0153nicians made\nthe anchors of their ships of gold and silver, and that the\nCarthaginians were astonished to find in Andalusia, that the mangers and\nvases for holding wine and oil were made of the same materials. These\nreferences have been constantly mentioned in ancient Spanish authors.\nAmbrosio de Morales, in his \"Antiguedades de Espa\u00f1a,\" Alcala 1577,\nenters into every detail on this subject.\n\nI have seen a specimen of this period, a bowl of an earlier and\ndifferent style to Roman silversmiths' work, which belongs to a\ncollector in the province of Cordova. This bowl is of a conical shape:\nit is perfectly plain, and has an inscription in Iberian characters\nengraved on one of its sides: there are signs outside and at the bottom\nwhich indicate that this bowl was made on the wheel. Velazquez in his\n\"Ensayo sobre las letras desconocidas,\" Madrid, 1752, describes a silver\nbowl of a similar kind, which was found in Andalusia in 1618 full of\nIberian coins: this bowl weighed ten ounces. Several ornaments, chiefly\nconsisting of necklaces and earrings, may be studied at the Academy of\nHistory, and private collections, in Madrid; they have been classified\nby antiquarians as belonging to this uncertain period, and are similar\nin style to others which have been frequently found in England and the\nnorth of Europe. The most interesting objects of this kind which I have\nseen in Spain, are gold ornaments proceeding from Galicia; they were\nfound there by Sr. Villaamil, who gave a description of them in the\n\"Museo Espa\u00f1ol de Antiguedades,\" vol. iii. p. 545.\n\nThe information and remains which have reached us illustrating silver\nand gold work of the Roman domination are more important; we find,\nhowever, in this and similar artistic industries that, as a general\nrule, the Romans imposed their style and system on the inhabitants of\nthe countries which they conquered, and it is not easy to point out any\none example which can be given as an exception to this general rule. It\nis, undoubtedly, a fact that objects of all kinds in gold and silver\nwere used in Spain to a very great extent--for, notwithstanding the\ndestruction of ages, we still possess inscriptions which allude to\nsilver statues, and a large number of objects in the precious metals\nexist in museums and private collections. See \"Inscrip. Hisp. lat.,\" by\nDr. Emile H\u00fcbner, Berlin, 1869. Of these it will be sufficient to\nmention one of the most important. It is known in the province of\nSantander by the name of the dish of Ota\u00f1ez; and belongs to a gentleman\nwho lives in that locality. It was found at Ota\u00f1ez at the end of the\nlast century, buried in a stone quarry. This dish is made of silver, it\nweighs thirty-three ounces, and is covered with an ornamentation of\nfigures in relief, some of which are gilt, representing an allegorical\nsubject of the source of medicinal waters. In the upper part is a nymph\nwho pours water from an urn over rocks; a youth collects it in a vessel;\nanother gives a cup of it to a sick man; another fills with it a barrel\nwhich is placed in a four-wheeled car to which are yoked two mules. On\neach side of the fountain are altars on which sacrifices and libations\nare offered. Round it is the inscription: SALVS. VMERITANA, and at the\nback is engraved, in confused characters, the words: L. P. CORNELIANI.\nPIII....\n\nAnother very interesting silver dish may be seen at the Academia de la\nHistoria, Madrid. Although not of Spanish manufacture, it deserves\nspecial notice in a description of works of this style. This dish was\nfound in 1847, buried in a field at Almendralejo, province of\nEstremadura: it is 28-6\/8 English inches in diameter, and weighs 533\nounces. It is ornamented with fine figures in relief, representing the\nEmperor Theodosius appointing a magistrate. The emperor is seated on his\nthrone in the centre, at the sides are his sons Arcadius and Honorius,\nwith four soldiers of the guard, and the magistrate, who receives the\nvolume. In the lower part there is an allegorical representation of a\nnymph holding the horn of abundance, winged angels, and ears of wheat,\nprobably alluding to the abundance and plenty of the empire. Round the\nrim is the following inscription:--\n\n    D. N. THEODOSIVS PERPET.\n    AVG OB DIEM FELI CISSIMVM.\n\nAt the back, in points, are the following Greek words.\n\n    [Greek: POS N MET \u03c0\u03bfc \u2191\u039d \u039c\u03b5\u03a4]\n\nWe can form a very good idea of the jewel work of the Visigothic period\nfrom the discovery of the treasure of Guarrazar. These jewels were found\nin 1858 at the village of this name, six miles from Toledo, on the spot\nwhere, in the Visigothic period, a Christian sanctuary had existed. The\nobjects found, which may be seen at the Museum of Cluny, Paris, and\nArmeria Real, Madrid, constitute the most important collection in Europe\nof the jewel work of that period. No less than eleven votive crowns,\nsome of extraordinary magnificence; two crosses with inscriptions, and\na large number of fragments of all kinds, of gold and precious stones,\nwere found at Guarrazar. Unfortunately a great part of the treasure has\nbeen lost, for the labourers, who were the first to find it, sold\nseveral objects to the silversmiths at Toledo, who melted and destroyed\nspecimens of the highest artistic interest. Those that remain in Spain\nare--\n\nAt the Royal Armoury of Madrid: A gold crown of Swinthila inlaid with\nprecious stones, with pendent cross, and inscriptions of letters hanging\nfrom it: SVINTHILANVS REX. OFFERET. See woodcut on p. 7. A similar crown\nof Theodosius, with the inscription: OFFERET MVNVSCVLVM. SCO STEFANO\nETHODOSIVS. ABBA. The cross of Lucetius with the following inscription:\n+ IN NOMINE DNI. IN NOMINE SCI OFFERET LVCETIVS: E. Fragments of another\ncrown and of large crosses: several stones and fragments, and an\nemerald, on which is engraved _en creux_ the Annunciation of the Virgin.\n\nAt the Arch\u00e6ological Museum, and in several private collections in\nSpain, may be seen a large number of stones and pearls which were found\nat Guarrazar.\n\nAmong the objects at the Arch\u00e6ological Museum at Madrid, a small\nsection, which proceed from a different locality, have also been\nclassified as Visigothic. They consist of necklaces, earrings, and\nrings, and are less important in workmanship than those found at\nGuarrazar. The student will find further details in \"Museo Espa\u00f1ol de\nAntiguedades,\" vol. vi., p. 137.\n\nThe objects forming part of the treasure of Guarrazar now in the Museum\nof Cluny, Paris are: Crown of Recesvinthus, with pendent cross and\ninscription: + RECESVINTHVS REX OFFERET. A similar crown, without\ninscription. Four crowns formed to imitate basket work, with crosses\nhanging from the centres. Three crowns, the rims of which are formed of\n_repouss\u00e9_ work, only one of which is ornamented with precious stones,\nand without pendent crosses. The cross of Sonnicus with the following\ninscription: \u2720 IN [=D]I NOMINE OFFERET SONNICA [=S C E] MARIE IN SORBACES.\n\nSwinthila reigned from 624 to 631, A.D.; Recesvinthus from 650 to 672\nA.D.; hence these crowns belong to the 7th century, and the remaining\nobjects are certainly of the same period.\n\nMuch has been written on these Visigothic jewels by French and Spanish\nauthors, the most interesting and valuable studies are:--\"Descrip. du\nTr\u00e9sor de Guarrazar, par M. Lasteyrie. Paris. 1860.\" \"El arte Latino\nBizantino en Espa\u00f1a y las Coronas de Guarrazar,\" by Se\u00f1or Rios, Madrid,\n1861. Consult also his article in \"Monumentos Arq. de Espa\u00f1a.\" \"Coronas\nde Guarrazar que se conservan en la Armeria Real de Madrid,\" by Se\u00f1or\nRada. Museo Espa\u00f1ol, vol. iii. Madrid, 1874.\n\nThe importance of this goldsmith's work has led these authors to discuss\nseveral archaeological and artistic points which deserve attention. I\nwill also give my opinions on this subject, which in some respects\ndiffer from those hitherto advanced. In the Middle Ages the name of\ncrown was equally applied to those worn on the head, to the votive\ncrowns hung before altars, and to the pendent lamps which were also of\nthis circular form. The \"Etimologies\" of San Isidoro help us,\nunfortunately, but little on this subject; but Du Cange, in his\n\"Glossarium Medi\u00e6 et infim\u00e6 Latinitatis\" gives us sufficient details to\nunderstand the differences between them. I refer the student to two\nworks on this subject, which treat especially of crowns of light, which\nnone of the authors who have written on Guarrazar have consulted:\n\"Tratado de la Sagrada Luminaria,\" by Gonzalez Villar, Madrid, 1798. \"De\nLucernis Pensilibus,\" by Fanceulli, Maceratae, 1802.\n\nThe crowns found at Guarrazar were undoubtedly votive offerings, for\nthey all have chains by which they were suspended in the sanctuary in\nwhich they were found. Their size and the structure of their rims lead\nus to suppose that they may have been worn on the head. The same\nobservation applies to a similar crown at the Cluny Museum; if so, the\nchains, cross, and pendent letters were added when the crown was offered\nto the sanctuary.\n\nAlthough details which appear in the handwork of these jewels betray a\ncertain rudeness, not uncommon in the time they were made, their beauty\nand richness are truly surprising. These crowns are decorated round\ntheir rims with rosettes of pearls and sapphires, and a delicate\nornamentation of _cloisonn\u00e9_ work, which encloses a substance resembling\nred glass. From the upper part are attached four chains formed of leaves\n_perc\u00e9es \u00e0 jour_, these are united to a double gold rosette with pendent\nsapphires, in the centre of which is set a piece of rock crystal. In the\ncross of Recesvinthus the pendant is in the form of a capital, and from\nit hangs a small cross of the same style of work: from the lower rim of\nthe crown are suspended the letters, which are ornamented with the same\nvitreous paste, _cloisonn\u00e9_, and from each hang large single pearls and\nsapphires.\n\nThe border of some of these crowns is formed of a net-work of small gold\nmassive balustrades; having between them square spaces in which pearls\nand sapphires are set. Others are made simply of _repouss\u00e9_ gold, their\nchief interest consisting in the ornamentation which has been so\nartistically carried out. Some of them have inscriptions and a few\nstones set at intervals. The pendants from these crowns are inferior in\nrichness to the others already described. The pendent crosses are\nornamented in the same style, either with _cloisonn\u00e9_ work or _repouss\u00e9_\ninscriptions.\n\n[Illustration: VOTIVE CROWN OF KING SWINTHILA, MADRID.]\n\nOne of the most interesting results produced by the study of the\ntreasure of Guarrazar has been to show us the immense luxury which it\nrepresents, if we remember the period of decay and poverty of the\nVisigothic monarchy. We find this magnificence frequently alluded to by\nancient writers. The Arabs when they took possession of Toledo in the\n8th century, mention in different works the immense quantity of jewels\nof all kinds which they found and carried away. The gold and\nsilversmith's work of this period was everywhere in a very high state of\ndevelopment. We find it constantly alluded to in the works of Paul\nSilentiarium and other writers of the time of Justinian, and in the\ninventories and references given by Du Sommerard of the jewel work\nanterior to the Carlovingian period in Italy and France. The Visigothic\nkings, who copied from the Eastern emperors even their legal forms,\nfollowed this rule to a great extent in everything which bore relation\nto their daily life.\n\nThe most important question is to determine the origin and locality\nwhere these jewels were made. M. de Lasteyrie considers that jewellery\nornamented with red glass _cloisonn\u00e9_ work was only made by nations of\nthe Germanic race. To prove this he presents, among other arguments, the\ncomparison of a fragment of the ornamentation which appears on the crown\nof Recesvinthus, with a similar one on the Evangelistiarium of\nTheodolinda in the treasure of Monza, which he considers to be of German\nworkmanship. These theories are untenable. Sr. Rios is of a different\nopinion, and considers these jewels to have been made in Spain, owing to\nthe similarity of ornamentation with different specimens which occur in\nSpain in architectural remains of the Roman and Visigothic periods.\n\nBoth these writers give in my opinion the decorative elements of the\ncrowns of Guarrazar an importance which they do not possess. The\nornamentation which most frequently occurs consists of a combination of\ncircles, imbrication, and palms of such an elementary kind that it would\nbe difficult to ascertain its origin; it appears equally in mosaic work\nof the later Roman period, in vases and other objects of the best times\nof Greek art, and in Asiatic and archaic works. It may be affirmed that\nthe ornamentation of the goldsmith's work of the north and south of\nEurope are derived from a common origin; from the oriental civilization\nwhich in the first centuries of the Middle Ages penetrated into Europe;\nit may also be affirmed that its manufacture and technical proceedings\nare precisely similar. Later on, the ornamentation and manufacture of\nthese jewels received from the western nations some slight\nmodifications; but this artistic industry by no means proceeded from\nthem.\n\nOne example will be enough to prove this. Setting apart the famous\n_cloisonn\u00e9 pectorale_ at the Boulac Museum, Cairo, and other jewels of\nthe queen, Aah-Hotep, 17th century B.C.; the hawk of a similar period,\nLouvre Museum (Paris), and a number of analogous objects in the British\nMuseum, we have enough specimens of this kind to show that _cloisonn\u00e9_\nwork was known in the east from the very earliest period. I will draw\nespecial attention to the _plaque_ in the Museum of Wiesbaden, found at\nWolfsheim in 1870. It is formed of gold, with a circular and triangular\nornamentation and squares, _d\u00e9coup\u00e9s \u00e0 jour_, set with jacinths, or red\nglass paste. At the back may be read an inscription in Persian\ncharacters, which gives the name of a Sassanide king, Artachshaber, of\nthe 3rd century of our era. (See A. von Cohausen, \"Roemischer\nSchmelzenschmuck,\" Wiesbaden, 1873). I know no more ancient specimen in\nthe north of Europe of these jewels with coloured stones, nor can I give\na better example of their oriental origin. Those who may wish to make a\nmore profound study on this subject, can also examine the interesting\njewels with stones found at Petrossa, Wallachia, in 1835, now in the\nMuseum of Bucharest. They are attributed to the Visigothic King\nAthanaric, who lived towards the middle of the 4th century of our era,\nand are considered in the present day of undoubted Sassanide origin.\n\nThe artistic and technical origin of the jewels of Guarrazar must be\nlooked for in the East; their manufacture was most probably Spanish. On\npart of these jewels, inscriptions of the names of the _donatarios_\nappear, and it is highly improbable that they were made in another\ncountry. We cannot imagine the extraordinary magnificence of the\nVisigothic court, so similar to that of Constantinople and other\ncontemporary ones, without the presence at each of a group of artists\nwhose task was to satisfy these demands.\n\nThe Visigothic style continued to be followed in Spain at the court of\nthe Christian kings, until the 11th century, notwithstanding the Moorish\ninvasion, and the poverty of the kingdom. The specimens at the treasury\nof the cathedral of Oviedo, and others which will be described farther\non, will give a good idea of the manufacture of jewels of this period.\nAmong the relics which are kept in the shrine called the _Camara Santa_,\nat Oviedo, are two most interesting gold processional crosses studded\nwith stones, which are known by the names of _Cruz de los Angeles_, and\n_Cruz de la Victoria_ or _de Pelayo_. The _Cruz de los Angeles_ is\n16\u00bd inches high, by 16 inches wide, it is covered at the back with an\nornamentation in fine filigree work, set with different stones. In the\nfront are five medallions, and an inscription in the vacant spaces.\nAmong the gems there is a good cameo, and seven engraved stones in the\nGnostic style. (Consult \"Monumentos Arquitectonicos de Espa\u00f1a, Camara\nSanta\"). The inscription on this cross shows us that it was made, A.D.\n808.\n\n    \"Susceptum placide maneat hoc in honore Dei\n     Offeret Adefonsus humilis servus Xpi\n     Hoc signo tuetur pius\n     Hoc signo vincitur inimicus\n     Quisquis auferre presumpserit mihi\n     Fulmine divino intereat ipse\n     Nisi libens ubi voluntas dederit mea\n     Hoc opus perfectum est in Era DCCCXLVI.\"\n\nThe cross of Victory is supposed to have been the same wooden one borne\nby Don Pelayo when he began, early in the 8th century, his struggles\nagainst the Mahomedan invasion, it was decorated in the next century\nwith gold platings and precious stones. It is 36 inches high by 28\u00bd\ninches wide; the ornamentation is similar to the former one, and in the\nvacant spaces at the back appears the following inscription, by which\nwe learn it was made at the Castle of Gauzon, Asturias, A.D. 828.\n\n    \"Susceptum placide maneat hoc in honore Dei, quod offerent\n     Famuli Christi Adefonsus princeps et Scem\u00e6na Regina\n     Quisquis auferre hoc donoria nostra presumpserit\n     Fulmine divino intereat ipse\n     Hoc opus perfectum et concessum est\n     Santo Salvatori Oventense sedis\n     Hoc signo tuetur pius, hoc vincitur inimicus\n     Et opera us est in castello Gauzon anno regni nostri.\n             XLII. discurrente Era DCCCLXVI.\"\n\nAnother most interesting object at this shrine is Don Fruela's casket,\nwhich is ornamented with agates set in gold, and is similar in style.\nThe inscription, which appears outside, gives the date A.D. 910.\n\n    \"Operatum est Era DCCCCXLVIII.\"\n\nA diptych which belongs to the same shrine may also be mentioned. It was\ngiven by Bishop Don Gonzalo, who was bishop of Oviedo from A.D. 1162 to\n1175. Round it are the words--_Gundisalvus Episcopus me jussit fieri._\nThis diptych is 5 inches long by 7 wide: it is ornamented with ivory\nfigures, stones, crystals and engraved gems. Although I do not consider\nit to be of Spanish workmanship, it is necessary to mention the splendid\n_Arca Santa_, in which it is believed many of the relics were taken to\nOviedo. It is covered with silver plates, with _repouss\u00e9_ and chiselled\nwork representing different religious subjects: the Crucifixion, Our\nLord surrounded by the attributes of the Evangelists, saints, apostles,\nangels, and a fine ornamental border with letters, imitating Cufic\ninscriptions. This splendid casket is 3 feet 9\u00bd inches long, by 3 and\n3\u00bd wide, and 28\u00bd inches high, and appears to have been made\nbetween the 10th and 12th centuries. The figures are similar in style to\nthe paintings and sculpture of this period, and it is highly probable\nthat it may have been the work of Italian artists. The Cufic inscription\nis illegible, and is interrupted in the angles by the symbols of the\nEvangelists. This style of simulated inscriptions was frequently used by\nItalian artists.\n\nA cross of the same style as those already described exists at the\ncathedral of Santiago. It is made of wood covered with gold platings and\nprecious stones; some of these are old intagli, which are set in\ndelicate filigree work. Round it runs a long inscription, from which it\nappears that it was a present from _Adefonsus Princeps cum conjuge\nScemena Regina_, and that it was made in Era DCCCCA. Duodecima; A.D.\n874. The number of gems which have reached us, after so many centuries\nof ruin, the similarity of the different specimens, and the statement\nwhich appears on the cross of King Pelayo, that it was made at the\ncastle of Gauzon, prove that the goldsmith's industry had attained great\nimportance in Spain during this period.\n\nBy studying the different objects of silver and gold work which still\nexist in Spanish churches, we can form a good idea of the historical\nprogress of this industry in the following centuries of the Middle Ages;\nbut before we do so, it is well to make some observations upon objects\nof _orf\u00e8vrerie_, the work of the Moors. There is a constant connection\nbetween these objects and those made of ivory: the same characteristics\nexist which I shall hereafter point out as appearing in Moorish ivory\ncarvings. The art of ivory carving was imported from the East, the\nsubjects are much alike in ivory and metal when men and animals are\nrepresented, and the inscriptions and bands of ornamentation are similar\nin style. The main variations consist in the different systems employed\nin metal work, by which the work differs according to the proceedings\nadopted, of _repouss\u00e9_ or chiselling, filigree, niellos or enamels.\n\nA fine Casket belongs to this kind of oriental work which still may be\nseen on the high altar of the cathedral of Gerona, Spain. This casket is\n15 inches long by 9 wide and 10\u00bd high. It is made in the usual manner\nof wood covered with silver gilt platings with a heavy _repouss\u00e9_\nornamentation of leaves enclosed within circles of pearls. Round the\nrim of the cover runs the following Cufic inscription:\n\n[Illustration: Arabic]\n\n\"In the name of God. The blessing of God and happiness and prosperity\nand permanent joy for the servant of God, Alhakem Emir Amumenin\nAlmostanser Billah, because he ordered this casket to be made for Abdul\nwalid Hischem, heir to the throne of the Muslims. It was finished by the\nhands of Hudzen Ibn Bothla.\"\n\nAlhakem reigned in Spain from A.D. 961 to 976, in which year he was\nsucceeded by his son Hischem II. This casket belongs, therefore, to this\nperiod, and is especially interesting as giving the artist's name. Two\nother silver Arabian caskets may be seen at the Arch\u00e6ological Museum,\nMadrid, which were formerly at the shrine of San Isidoro of Leon, but\nthey possess less artistic interest than the casket at Gerona. One of\nthese is elliptical in form; it is ornamented with a good design of\nleaves and tendrils, and Cufic inscription; the whole of the casket is\nenamelled in black. The ornamentation belongs to the 12th century. The\ninscription only mentions the owner's name, Abdo Shakir. The other\ncasket is silver-gilt, square in form, and rather poorly ornamented. The\ntwo Cufic inscriptions which surround it are laudatory. At the cathedral\nof Oviedo there is another silver casket with a laudatory inscription\nand medallions with figures, in which from very early times, the remains\nof S^{ta.} Eulalia have been kept. I suspect that this casket and the\nformer one are not of Spanish Arab workmanship, for besides the\ncircumstance that their inscriptions can be applied to any owner, their\nornamentation is unlike others of the same kind. In the first casket it\nis insignificant, but on the shrine of S^{ta.} Eulalia the background\nof the medallions is covered with an imbricated pattern which I have\nnever seen repeated on any Arab or Moorish example in Spain. It is\nhighly probable that they were productions of Oriental industry and were\nimported commercially.\n\nSeveral specimens of the 14th and 15th centuries, the last period of the\nMoorish domination, exist in Spain. They consist of jewels and sword\nhandles. The most interesting trinkets are a bracelet and fragments of a\nnecklace and earrings which are at the Arch\u00e6ological Museum, Madrid.\nThey are made of gold, covered with a geometrical _repouss\u00e9_\nornamentation, and a delicate filigree pattern. There are specimens also\nat the Kensington Museum, Nos. 1455 to 1447, 70, consisting of a\nbracelet, silver-gilt, formed of seven alternate oval and rectangular\nplaques, with impressed pattern and applied filigree and bossed\nornament, and earrings made of gold, formed of clusters of united\ncircles and lozenges with filigree bosses. Other jewels of less\nimportance are known to exist, consisting of bracelets, amulets,\nearrings, and rings, mostly made of silver niello-work, these are\nornamented with geometrical patterns and inscriptions of little\nimportance.\n\nMoorish arms are most artistic; they are fully described in the article\nArms. The most important specimens are in the Royal Armoury, and noble\nhouse of Villaseca, Madrid; another fine example of a similar style is\nat the Generalife of Granada. The hilt and settings of the sheath are of\nsolid silver, gilt, and covered with geometrical patterns ornamented in\nhigh relief, parts of which are filled with translucid _cloisonn\u00e9_\nenamel. In some instances the hilt is made of ivory. It is impossible\n(see plate on p. 85) to find anything more beautiful than the\nornamentation of these swords, or greater perfection in every detail. It\nis evident, therefore, that this industry had reached a very high grade\nof perfection at Granada in the second part of the 15th century. The\nsword now in the possession of the Villaseca family belonged to Boabdil,\nthe last Moorish king; the one at Granada to one of Boabdil's nearest\nrelations.\n\nIn continuing our description of Christian silver-work in the 11th and\n12th centuries, we meet with two historical chalices of the highest\ninterest. One was made by the order of Saint Domingo de Silos [A.D.\n1045-1074] when abbot of this church. This chalice still exists there,\nwith the following inscription:--\n\n    _In nomine Domini ob honorem Sci Sebastiani Dominico abbas\n    fecit._\n\nIt is ornamented with fine filigree work, forming zones and horse-shoe\narches, in a similar style to that of the silversmiths' work of\nAsturias, which has never been completely abandoned in Spain. The author\nof the life of this saint, Fr. Juan de Castro, Madrid 1688, says, p.\n297, that he does not consider it was ever used for the sacrifice of the\nmass, owing to its great height [13 inches]. The other chalice might\nhave been seen until very lately at San Isidoro, Leon; it has been\ntemporarily concealed owing to political disturbances. The cup and foot\nare of agate, probably specimens of the classic period; the mounting,\nwhich dates from the time of D^{na}. Urraca, is studded with a profusion\nof precious stones and pastes. Some of the gems of the chalice and paten\nare antiques.[A] In the centre of the paten is set a splendid flat onyx.\nRound the lower part runs the following inscription:--\n\n    + _In nomine D[=n]i Urraca Fredi[=n][=a]di._\n\n[A] For further details of this interesting work of art, see \"Monumentos\nArquitectonicos de Espa\u00f1a,\" in which a good reproduction is given.\n\nDona Urraca, who was a sister of Alfonso VI., and was generally called\n_Urraca Fernandez_, bestowed many important gifts on the church of San\nIsidoro. She died A.D. 1101. Another interesting chalice of the same\nperiod, although not of the same importance as those just named, belongs\nto Cardinal Moreno, archbishop of Toledo. Round the stem are represented\nthe emblems of the Evangelists, and the inscription: _Pelagius abbas me\nfecit_; this formula appears so frequently that it must be understood\nin the sense of _fecit fieri_, ordered to be made.\n\nThe Santo Caliz at Valencia has been traditionally held as the cup used\nby Our Saviour at the Last Supper. This chalice consists of a circular\ncup hollowed out from a fine brown sardonyx which is tastefully moulded\nround the lip. The base is formed of another inverted sardonyx. These\nare united by straps of pure gold. The stem is flanked by handles, which\nare inlaid with delicate arabesque in black enamel. Oriental pearls are\nset round the base and stem, which alternate with rubies, sapphires and\nemeralds. This chalice is a work of the Roman imperial epoch, and the\nmounts are of a later date. Other specimens of jewellers' work of the\nRoman period might be mentioned which exist in Spain, but I do not find\nsufficient evidence to justify the opinion that they were made in that\ncountry. I consider those that I have described to be of Spanish origin,\nfor they keep to the same technical modes of workmanship as the jewel\nwork of Asturias, and the inscriptions which appear on them refer to\nhistorical personages. It would be difficult, considering all things, to\nsuppose they were imported.\n\n[Illustration: ONE LEAF OF TRIPTYCH IN CATHEDRAL OF SEVILLE.]\n\nWe can mention in the thirteenth century a specimen of Spanish\nsilversmiths' work which illustrates the transition to the new style,\nand the progress in the design of the figures owing to the Italian\nRenaissance--I refer to the interesting triptych at the Cathedral of\nSeville, known as the \"Tablas Alfonsinas,\" made by the order of Don\nAlfonso el Sabio for holding relics. It is of wood, covered inside and\nout with silver-gilt plates; it is 22 inches high by 39 wide when its\nthree leaves are open (the woodcut opposite represents the outside of\nleft leaf), and is divided inside into fifteen compartments full of\nminute ornamentation, among which are set a large number of capsules\ncovered with rock crystal containing relics, each one with an\ninscription of enamelled gold, _cloisonn\u00e9_. Several good cameos with\nsacred subjects appear near the edge of the side leaves. The outside of\nthis triptych is decorated with twelve medallions containing the arms\nof Castile and Aragon, and forty-eight others in which are repeated\nalternately the subjects of the Adoration of the Magi and Annunciation\nof the Virgin, which are also _repouss\u00e9_. In the centres are eagles,\nwhich S^{r}. Rios supposes to allude to Don Alonso's claim to be crowned\nEmperor, in which case it was made in the year 1274. (See Mus. esp. de\nAntig., vol. ii. p. 83.) The ornamentation which surrounds the panels\nbelongs to the 16th century. S^{r}. Rios suggests that the possible or\nprobable author of this interesting object of silversmith work was\nMaestro Jorge, a silversmith of Toledo, who is praised by Don Alonso in\nhis Cantigas--he also mentions the names of Don Lorenzo and Don Niculas\nas silversmiths of Seville who worked in this period.\n\nThe most important example of Spanish silversmith's work of the 14th\ncentury is the Retablo and Baldaquino of the cathedral of Gerona. Mr.\nStreet, in his Gothic Arch. in Spain, p. 326, describes this work of art\nin the following manner: \"The Retablo is of wood entirely covered with\nsilver plates, and divided vertically into three series of niches and\ncanopies: each division has a subject, and a good deal of enamelling is\nintroduced in various parts of the canopies and grounds of the panels.\nEach panel has a cinque-foiled arch with a crocketed gablet and\npinnacles on either side. The straight line of the top is broken by\nthree niches, which rise in the centre and at either end. In the centre\nis the Blessed Virgin with our Lord; on the right San Narcisso; and on\nthe left St. Filia. The three tiers of subjects contain figures of\nsaints, subjects from the life of the Blessed Virgin, and subjects from\nthe life of our Lord.\"\n\nAt the base of this Retablo may be read the words--_Pere Bernec me\nfeu_--Peter Bernec made me. Bernec was a silversmith of Valencia, and in\nanother document he was called Barners. It has been supposed that two\nother contemporary silversmiths, whose names appear in papers of the\ncathedral, worked also at the Retablo. Their names were Raimundo Andreu,\nand Master Bartolom\u00e9. Formerly in front of this altar there was a\nmagnificent silver and gold frontal studded with stones, a fine work of\nthe 11th century--which was unfortunately carried off by the French in\ntheir invasion of the Peninsula early in the present century, and was\nprobably with other innumerable priceless treasures melted by them. See\nfurther details in \"Viage Literario de Villanueva,\" vol. xii. p. 180.\n\nIn the Sala Capitular of the Cathedral of Gerona there are three\nsplendid processional crosses belonging to the 15th and 16th centuries;\none of them is of enamelled gold, and is undoubtedly one of the most\nartistic works of the kind in Spain.\n\nAmong Spanish art treasures of the 15th century of a historical style\nmust be mentioned the splendid silver throne of king Don Martin de\nAragon, d. 1410, still existing in the cathedral of Barcelona; it is\ncovered with chiselled ornamentation, and a band of velvet embroidered\nwith gold and completely studded with precious stones. This throne is\ncarried in the procession of Corpus Christi. The monstrance, a splendid\nwork of art in the Gothic style, ornamented with delicate pinnacles and\njewel work, is placed on a fine silver foot and carried on this day in\nfront of this throne. This monstrance is covered also with jewels of\ngreat value which almost conceal it, the gifts of royal personages. The\nfine Gothic silver-gilt cross must also be mentioned, known at Toledo by\nthe name of _Guion de Mendoza_; it was borne before the great Cardinal\nMendoza, and was the first cross placed on the highest point of the\nAlhambra Torre de la Vela on the day of the conquest, 2nd January, 1492.\n\nThe following woodcut represents a processional cross of Spanish work of\nthe beginning of the 15th century, in the South Kensington Museum, No.\n514-'73. It is of wood covered with plates of silver-gilt _repouss\u00e9_\nwork. On one side is a rood with the Virgin and St. John. Over the\nfigure of Christ is the word Inri. At the extremities angels in high\nrelief bear the emblems of the Passion. On each side of the figure of\nChrist are plaques of translucent enamel representing the penitent and\nimpenitent thief, at the foot of the cross the Resurrection and\nAdoration of the Magi, and above the figure of Christ the Nativity.\n\n[Illustration: SPANISH PROCESSIONAL CROSS. SOUTH KENSINGTON MUSEUM.]\n\nAt the back there is a figure in high relief of the Almighty; in the\nfour extremities the emblems of the Evangelists in high relief and\nenamelled plaques representing the Annunciation, Flight into Egypt,\nChrist's descent into Hades.\n\nThis cross, which is three feet in height, is marked in several places\nwith the name of\n\n    +-----+    +-----+\n    |     |    |     |\n    | PE  |    | MAR |\n    | DRO |    | TIN |\n    |     |    |     |\n    +-----+    +-----+\n\nA large number of images exist in Spain belonging to this period, and\neven to an earlier date, chiefly consisting of images of the Blessed\nVirgin; their garments are formed of silver platings, chiselled and\n_repouss\u00e9_ in the traditional Byzantine style. Among the most remarkable\nmay be mentioned those preserved in the cathedrals of Seville, Pamplona\nand Astorga. The following document undoubtedly alludes to one of these\nfigures. It is dated 12th May, 1367:--\n\n\"I Sancho Martinez Orebse, silversmith, native of Seville, inform you,\nthe dean and chapter of the church of Seville, that it was agreed I\nshould make an image of Saint Mary with its tabernacle, that it should\nbe finished at a given time, and that you were to give me the silver and\nstones required to make it.\"\n\nNotwithstanding the poverty of the Spanish monarchs, their personal\nornaments were rich and splendid. We find in \"Memorials of Henry the\nVIIth,\" edited by Gairdner, an interesting description by Machado, the\nherald or king-of-arms of Henry VII., of the embassy sent to Spain in\n1489, to ask for the hand of the Princess Catharine for the Prince of\nWales. The account he gives us of the jewels worn by Queen Isabel la\nCatolica is most interesting. They varied at every interview. In one of\nthese she wore \"a line of trimming composed of oblong bosses, of gold,\neach decorated with fine and valuable jewels, so rich that no one has\never seen the like. She wore round her waist a girdle of leather made in\na man's style; the pouch was decorated with a large balass ruby, the\nsize of a tennis ball, between five rich diamonds and other stones, the\nsize of a bean. The rest of the girdle was decorated with other precious\nstones. Round her neck she wore a rich gold necklace composed of white\nand red roses, adorned with jewels. Two ribbons were suspended from her\nbreast adorned with diamonds, balass and other rubies, pearls, and other\njewels of great value to the number of a hundred or more\" (p. 341.)\n\n\"After the King came the Queen, mounted on a fine mule, and all the\nharness of the said mule was adorned with pearls and other precious\nstones. She was dressed in a robe of a rich woven cloth of gold made in\nthe fashion of the kingdom, and over that a mantilla all spangled with\nlozenges of crimson and black velvet, and on each lozenge was a large\npearl. And with each of these pearls was a rich balass ruby the size of\na beech nut, the richest thing that could be seen, no man ever saw\nanything equal to it. She had on her neck a large necklace, adorned with\nlarge diamonds, balass rubies, carbuncles, large pearls, and a great\nnumber of other rich precious stones. She had upon her head-dress two\nbalass rubies as pendants, the size of a pigeon's egg, and at the end of\nthe said rubies a large pearl, which jewel was supposed to be worth\n12,000 crowns\" (p. 348.)\n\nIn the specimens described belonging to the Visigothic period, and many\nothers to which we might refer, we find constant similarity in form with\nsilversmith's work of other European countries. It is true that we\noccasionally meet with Moorish _orf\u00e8vrerie_, and some details, such as\nfiligree work, due to oriental influence; but in general Byzantine,\nRoman, and Gothic styles were adopted and copied in Spain; and the\ntechnical details were the same with those followed in other countries,\nwith the exception of _champlev\u00e9_ enamel, which appears to have been\nalmost exclusively used at Limoges in the middle ages. It is\ninteresting, however, and worthy of remark, that important objects are\nalso found in Spain decorated with _champlev\u00e9_ enamel, such as the\nsplendid altars of San Miguel in Excelsis (Navarre), and Santo Domingo\nde Silos (Rioja), and the image of the Virgin de la Vega at San Esteban\n(Salamanca), three specimens of the greatest importance.\n\nIn the Renaissance period, Spanish _orf\u00e8vrerie_ enters into its most\nbrilliant epoch, not only on account of the beauty of the form of the\nobjects produced, but also owing to its great richness. Among objects of\nthis period the most important are the _Custodias_ or monstrances of the\ncathedrals; these are exclusively peculiar to Spanish art. The almost\nincalculable quantity of silver-work produced at this period is\naccounted for by the reconquest of the Peninsula from the Moors, the\ndiscovery and possession of America, and other circumstances which\nincreased the power and wealth of Spain, and elevated the country to\ngreat importance.\n\nWe find frequent mention at this time of silversmiths, many of whom came\nfrom Germany, France, or Italy, attracted by the large number of works\nordered. Many settled in Spain, such as Enrique de Arphe, Jacome Trezzo,\nMateo Aleman, Hans Belta, and others. The Spaniards who joined them were\ngreater in number, and not inferior in merit. Silversmiths were already\nat that time divided into different groups, according to the technical\nproceedings which each one adopted: _plateros de la plata_, workers in\nsilver, was the name given to those who worked exclusively in _repouss\u00e9_\nand chiselled work, and imitated sculpture and architectural models;\ngoldsmiths, those who worked jewels with stones, enamels, and\nniello-work. Even within these groups were workers in filigree, and\nthose who decorated different objects with painted or mosaic work,\n_atauxia_, in the Moorish style. Almost all the most important Spanish\ntowns were large centres of these industries. Leon, Burgos, Valladolid,\nCuenca, Toledo, Cordova, and Seville rivalled each other in the number\nand quality of their productions. The Venetian ambassador, Navagiero,\nwho visited Valladolid in 1527, says, \"Sono in Valladolid assai artefici\ndi ogni sorte, e se vi lavora benissimo di tutti le arti, e sopra tutto\nd' argenti, e vi son tanti argenten quanti non sono in due altri terre,\nle prime di Spagna\" (\"Il Viaggio di Spagna,\" Vinegia, 1563, p. 35.)\n\nIn order to complete the study of this subject, it is necessary,\nbesides, to give some notice of the legal dispositions contained in the\nMunicipal Ordinances and in other laws of a more general character.\n\nIn Capmany's \"Memorias,\" vol. i., part 3, p. 88, are to be found several\nstatutes concerning the silversmiths of Barcelona from the 14th century,\nproving the importance of this guild in 1301. At this period three of\nits members formed part of the town Council. The introduction of a\nstatute of 1489 proves the excellence of the works which they made.\n\"Experience having shown us in past times, and proof existing at the\npresent time, that such clever silversmiths have existed and exist at\nBarcelona that their works are highly reputed by kings and great people,\nand held in great honour and estimation in the town itself.\" The\nSilversmiths' guild still preserve the folio volumes full of drawings,\nand the description of the different objects which they presented for\nthe approbation of the jury, during the 16th and 17th centuries. The\ndesigns contained in these volumes constitute a most interesting\ncollection of jewels, giving a good idea of the great height of this\nindustry in Barcelona.\n\nBaron C. H. Davillier is about to publish a volume on Spanish\nsilversmiths, in which etchings of several of these designs will be\nreproduced. In the list of artists which follows I give the names of the\nmost remarkable of those who worked at Barcelona.\n\nIn the Municipal Ordinances of Toledo of the year 1494, some laws\nrelating to silversmiths appeared; they are, however, uninteresting. The\nsame thing occurs with the Ordinances of Seville, which were re-compiled\nin 1526. The guild of silversmiths of Toledo must have been most\nimportant, for in 1423 they already formed a brotherhood or guild under\nthe protection of St. Eloy, in which they agreed to help the members of\nthe guild in every way. See \"Documentos Ineditos,\" published by Zarco\ndel Valle, p. 166. The Ordinances of Granada enter more into details\nconcerning the technical proceedings of silversmith's work. These\nOrdinances appeared in 1538; the work in the Moorish style is described\nin full detail; it appears to have been preserved in this locality more\nthan elsewhere.\n\nWe can judge of the enormous quantities of objects which were made by\nthose still to be seen in Spanish cathedrals and churches, having\nsurvived the French invasion of the present century. For full details of\nthe barbarous treatment of these works of art during the French\ninvasion, see Ford's \"Handbook of Spain.\" The expropriations of the\nSpanish government during the civil wars of 1833 to 1840, and the\ninjudicious law of expropriations of 1869 for the purpose of collecting\nartistic objects, have brought about the destruction and disappearance\nof numberless works of art in order not to give them up to the\ngovernment.\n\nThe same splendour and abundance of silver objects of every kind existed\nin the royal palaces and houses of the grandees. Madame d'Aunoy in her\n\"Voyage en Espagne, Lyon, 1643,\" p. 109, says: \"L'on ne se sert point de\nvaisselle d'\u00e9tain, celle d'argent ou de terre sont les seules qui soient\nen usage. Le duc d'Alburquerque est mort, il y a d\u00e9j\u00e0 quelque tems; l'on\nm'a dit que l'on avoit employ\u00e9 six semaines \u00e0 \u00e9crire sa vaisselle d'or\net d'argent. Il y avoit 1400 douzaines d'assiettes, 500 grands plats, et\n700 petits, tout le reste \u00e0 proportion, et 40 \u00e9chelles d'argent pour\nmonter au haut de son buffet, qui \u00e9toit par gradins comme un autel plac\u00e9\ndans une grande salle.\" The splendid silver table belonging to the\nMarquis of Villaseca at Madrid gives a good idea of the furniture of\nthis time. In the Inventories, a great number of which exist, we find\nnumerous details of silver objects of every kind. In one which was drawn\nup in 1574, of the effects of Princess Da Juana (MS. folio, Acad. de la\nHistoria), the sister of Philip II., we find mention, without counting\nthe jewels, of a silver balustrade, weighing 121 pounds, to be placed\nround the bed. The greater part of the kitchen utensils were also made\nof silver.\n\nAmong the most important objects of Spanish silversmith work are\nundoubtedly, as I have said before, the custodias.\n\nThe name of custodia is given in Spain, not only to the monstrance or\nostensoir where the Blessed Sacrament is exposed, but also to a sort of\ntemple or tabernacle, of large size, made also of silver, inside which\nis placed the monstrance, which is carried in procession on Corpus\nChristi day. In order to distinguish these objects one from another, the\nname of _viril_ is given to the object which holds the consecrated host;\nit is generally made of rock crystal, with a gold stem and mount\nornamented with precious stones. The small tabernacles are generally\nobjects of the greatest importance both from their artistic and\nintrinsic value. The description of one of them will be sufficient to\ngive an idea of their construction.\n\nAlthough a fine custodia existed formerly at Toledo, which we know\nweighed 164 pounds, Cardinal Ximenez de Cisneros wished a finer one to\nbe made, and caused the plan to be furnished by competition. Diego\nCopin, Juan de Borgo\u00f1a, and Enrique de Arphe presented designs; the one\nby Enrique de Arphe was selected. Arphe began his work in 1517, and\ncontinued exclusively employed in this, and without the help of other\nmaster silversmiths, until April, 1524, when he gave up the monstrance\nto the authorities of the cathedral. The silversmith, Lainez, finished\nin 1523 the gold and jewelled cross which is on the top. It represents a\nGothic hexagonal temple, 8 feet high, of three orders, with all the\nvariety and number of necessary architectural details such as\npilasters, arches, columns, pyramids, canopies, crest-work, &c., to the\nclosing of the vaulted roof; the whole is _perc\u00e9 \u00e0 jour_, and so\ndelicate that it looks like lace. From the roof hang bells and\nincense-holders of filigree work; in the key-stone are studded precious\ngems. Carvings in relief, representing passages of the life of Our\nSaviour, appear on the base of the six pedestals; they are admirably\ncarved. In the centre of the second order is a figure representing the\nResurrection of Our Lord. On the pilasters and brackets which appear in\nthe temple there are more than 260 statues of different sizes, all of\nwhich are executed with the same skill. This monstrance was mounted on\niron wires; and Archbishop Fonseca, wishing that the whole of it should\nbe made of silver, gave orders that Arphe himself should alter it, which\nhe did in 1525, when the total weight was found to be 388 pounds. The\n_viril_ was then placed inside it, this was made of the first gold\nbrought from America. It is completely covered with precious stones, and\nwas bought by Cisneros from Queen Isabel (the Catholic); it weighs 29\npounds of gold. The tabernacle was ordered to be gilt in 1595 by\nArchbishop Quiroga; this was done by the Masters Diego de Valdivieso and\nFrancisco Merino. This splendid work of art remains in this state, and\nmay be seen at the cathedral of Toledo; it was most fortunately saved\nfrom the rapacity of the French, by being sent to Cadiz during the war.\nIn 1513 the monstrance at the cathedral of Cordova was also made by\nArphe, it is similar in style and importance to that of Toledo. Before\nthis, he had also finished the splendid one formerly at Leon, which was\ndestroyed by the French, as was likewise a similar smaller one, also by\nArphe, formerly at the Monastery of St. Benito, at Sahagun.\n\nThe custodia made by Juan de Arphe in 1587, a Leonese artist, and\ngrandson of Enrique, for the cathedral of Seville, competes with that of\nToledo. It is formed in the same manner as a temple, but in the\nGr\u00e6co-Roman style, covered with an immense number of statuettes, some of\nwhich are upwards of a foot high, and reliefs of all kinds, and\ndelicate ornamentation, worked with the utmost skill. The chapter of the\ncathedral commissioned the theologian, Francisco Pacheco, to direct the\nsubjects which were to be represented, and when it was finished Arphe\npublished a full description of the monstrance, which he does not\nhesitate to call \"the largest and finest work in silver known of its\nkind.\" This opinion is hardly an exaggerated one if we look at this\nsplendid work of art. Its plan is circular, and measures 3\u00bd yards\nhigh, and weighs 1082 pounds of silver. For details consult Cean\nBermudez' \"Diccionario,\" Descripcion de la Catedral de Sevilla, Museo\nEspa\u00f1ol de Antiguedades, vol. viii., p. I.\n\nBesides these two celebrated silversmiths there was another of the same\nfamily, the son of Enrique, and father of Juan, Antonio Arphe, an artist\nalso of great merit, who made in 1554 the custodia which still exists at\nthe cathedral of Santiago.\n\nCean Bermudez says in his \"Diccionario,\" \"that in the same manner as the\ncity of Leon gave Spain three illustrious silversmiths, Cuenca gave them\nother three in the Becerrils,\" these were Alonso and Francisco Becerril\nbrothers, and Christoval, the son of Francisco. They all worked at the\nfamous and splendid custodia of Cuenca, and between them they produced a\nmost important series of works from 1528 to 1584.\n\nIt is extremely difficult to give in so small a space the description of\nthe works and names of the numerous artists on silver and gold work, who\nworked in Spain during the 16th century. At the present time,\nnotwithstanding the innumerable objects lost, a long list would remain\nof the specimens which have reached us, and their different forms and\napplications, still visible in the churches of Toledo, Seville,\nZaragoza, Palencia, Santiago, and others of the Spanish peninsula. Some\nidea may be gathered of the importance this art attained in Spain by\nlooking through the following list of artists who worked in silver and\ngold, upwards of 450 of whom I have added to the 95 given by Cean in his\ndictionary. It must be borne in mind that the objects on a large scale\nwhich reproduce an architectural model, adopt three styles during the\ncentury, all three of them admirable as regards beauty of form. The\nfirst is Gothic, a reminiscence of the former time, improved by the\nchange which had already taken place, in drawing and modelling. The\nsecond style is known by the name of _plateresque_, when applied to\narchitecture, and consists in copying the general structure of buildings\nin the classical style, and applying the orders and pointed arch, while\nkeeping to the profusion of decoration of the earlier period, and\nmodifying the general plan with the object of introducing the greatest\nquantity of ornamentation. The third style is the Greco-Roman; it is\nmore sober in decoration, and has a greater tendency to keep to the\nimitation of the classical school.\n\nBesides the objects described, which may be considered as original works\nand the most important examples of Spanish silversmiths' work, I must\nmention those which came from South America, chiefly from Mexico, which\npossess a certain aspect; they consist of carved and _repouss\u00e9_ work\nornamented with flora of the country adapted in an oriental style;\nothers consist of filigree work, double-headed crowned eagles are\nfrequently met with in the same style as those made at Cordova and\nSalamanca.\n\nThere are interesting specimens at the Kensington Museum which give an\nexcellent idea of Spanish silversmiths' work. Besides those already\ndescribed attention must be drawn to\n\nNo. 305-66. A silver-gilt cross ornamented with foliage, statuettes of\nsaints and the Evangelists with their emblems, Marked NOE\/M. About 1560.\nHeight 3 feet 2 inches.\n\nNo. 302-66. Silver-gilt chalice, ornamented with foliated scroll work\nand half figures beaten and chiselled. Marked Estorga. About 1540.\n\nNo. 132-73. A silver-gilt chalice, the bowl inscribed outside \"+ Sangvis\nmevs vere est potvs;\" the stem is of baluster form, in several tiers,\nornamented with brackets and large chatons set with crystal, and a band\nof cherubim. The foot is chased with masques, festoons, harpies, and\nbirds, and surrounded by eight semicircular projections, on which are an\narmorial shield and a cross set with emeralds and lapis lazuli. Engraved\nat bottom \"S. I. de Salinas.\" [See woodcut] With it is a paten dated\n1549.\n\n[Illustration: SPANISH CHALICE. SOUTH KENSINGTON MUSEUM.]\n\nNo. 481-75. Chalice, silver-gilt; the bowl chased in relief with the\ninstruments of the Passion; on the knop are ten applied figures of\nApostles on ground of translucent blue enamel; the foot, which has eight\nsemicircular projections, is _repouss\u00e9_ with representations of the\nEvangelists, cherub and other heads, the Crucifixion, and a shield with\nthe initials L. B. P. around a crown of thorns enclosing a heart. 17th\ncentury. [See woodcut, p. 31.]\n\nNo. 314-64. Silver-gilt pax of architectural design; in the centre is a\ngroup in full relief of the Virgin giving the chasuble to St. Ildefonso.\nAbout 1540-50. [See woodcut, p. 33.]\n\nNo. 1129-64. Incense holder, boat-shaped, of rock crystal mounted in\nsilver-gilt. Around the rim is a band of guilloche pattern, set with\namethysts and garnets; on the lids a band inscribed \"Oratio mea\ndirigatur sicut incensum.\" About 1540-50.\n\n[Illustration: SPANISH CHALICE. SOUTH KENSINGTON MUSEUM.]\n\nNo. 93-65. Silver triptych with suspending chain, the interior painted\nin oils, the exterior engraved in arabesque. About 1550.\n\nWe find that Spanish jewels were as magnificent in the 16th century as\nwere the large architectural objects for ecclesiastical use. One of the\nmost important in richness and artistic merit was the splendid crown\nbelonging to the Virgen del Sagrario at Toledo, which it is deeply to be\nregretted disappeared in 1868. This crown was made in 1556 by the\nsilversmith Hernando de Carrion; it then consisted of a gold circle with\nchiselled and enamelled ornamentation, set with pearls, emeralds and\nrubies. In 1574 Cardinal Loaisa wished to enrich it, and ordered a\nsilver worker called Alejo de Montoya to add to it an upper part, formed\nas an imperial crown, which Montoya agreed to do by a special agreement.\nThis addition was formed of small figures of angels of enamelled gold,\nin pairs supporting the side bands, which met in the upper part forming\na group of allegorical figures, upon which was placed a spherical\nemerald, without a flaw, 1\u00bd inches in diameter, which served as a\nbase to the cross. The bands were studded with precious stones and\nornamented inside with subjects of the life of the Blessed Virgin in\nenamel. The height of this crown was 10\u00bd inches by 8\u00bd wide.\nMontoya took 12 years to do this work--he finished it in 1586. The fine\nbracelets belonging to this crown, which have also disappeared, were\nmade at the same time by Julian Hernando.\n\nThe jewels worn by the Spanish kings and grandees were equally\nmagnificent. In the description of the gems which Prince Don Carlos, the\nson of Philip II, left to be distributed at his death, are included a\nsword the hilt of which was of solid gold enamelled in different\ncolours: this Don Carlos bequeathed to the Grand Master of the Order of\nSt. John:--a halberd composed of 27 pieces of enamelled gold in high\nrelief; and a sword with gold mount enamelled in colours with masks,\nmedals and festoons in the Roman style, made by Rodrigo Reynalti.\nConsult \"Coleccion de Documentos Ineditos para la Historia de Espa\u00f1a,\"\nvol. 27, Madrid, 1855. I owe to the courtesy of Count Valencia de Don\nJuan, the following description of arms made by _Toto platero de su\nAlteza_ in 1554. [Archives of Simancas legajo, No. 37.] These arms\nbelonged to Prince Don Carlos.\n\n[Illustration: SPANISH PAX. SOUTH KENSINGTON MUSEUM.]\n\nA gold sword, the cross of which is ornamented with masks of white,\ngrey, and black enamel.\n\nA gold dagger, and sword-belt belonging to the same sword, ornamented in\na similar manner.\n\nAlso a gold sword, belt and dagger ornamented with figures of children\nin solid gold and enamel.\n\nAlthough the greater part of these silversmiths were Spaniards, the\nMilanese artist, Jacome de Trezzo was very celebrated during the reign\nof Philip II. He made several jewels of great importance for the king\nand royal family. The splendid tabernacle which was taken by the French\nin 1810 from the Escorial was one of his finest works; they carried off\nat the same time the superb shrines, the gifts of kings and princes, and\neverything they could lay their hands on of gold and silver, loading ten\ncampaign carts. Consult, \"Historia del Escorial,\" by Quevedo. Madrid,\n1849, p. 220.\n\nAt the South Kensington Museum are several objects of this kind, which\nwill give an excellent idea of Spanish jewel work.\n\nNo. 334-70. A gold enamelled pendant, in form of a chained dog,\nsupported on a scroll from which small pearls depend, and suspended by\ntwo chains of alternate enamelled and plain links, united to a fastening\ncrowned by a bird.\n\nNo. 335-70. Enamelled gold pendant in form of a pelican and her young,\nenriched with a carbuncle and pearls, and suspended by pearl links.\n\nNo. 336-70. Enamelled gold pendant in form of a dog enriched with\njewels.\n\nNo. 337-70. Enamelled gold pendant, in form of a parrot, set with\nhyacinth, suspended by chains.\n\nNo.340-70. Enamelled gold pendant representing the Virgin of the\nImmaculate Conception.\n\nNos. 341 and 342-70. Pendants representing the Virgen del Pilar\nSaragossa, attended by saints. [See woodcut opposite.]\n\n[Illustration: SPANISH JEWEL, 17TH CENTURY. SOUTH KENSINGTON MUSEUM.]\n\nSpanish jewel work does not decrease during the 17th century, the\nnumber of artists who worked was very great, and the quantity of objects\nof all kinds which were made by them to enrich the shrines of churches,\nand the houses of grandees, was remarkable, although their artistic\nmerit was far inferior to the work of the 16th century. The general\ndecay of art, which produces in Europe the _barroco_ style, appears in\nSpain more exaggerated and to a greater extent than elsewhere. The\nobjects made during this period reproduce until the beginning of the\n18th century the lines and extravagant ornamentation which we meet with\nin architecture, the handiwork however continued to be excellent, and no\nexpense was spared to give an aspect of richness to the objects made.\n\n[Illustration: SILVER DISH. SPANISH, 17TH CENTURY. SOUTH KENSINGTON\nMUSEUM.]\n\nThe large quantity of objects of all kinds made of silver, and the\nquantity also used in wearing apparel, gave rise to constant\nprohibitions restricting its use from the reign of Ferdinand and Isabel,\nand even to a far greater extent at the beginning of the 16th century.\nIn a Decree, issued at Madrid in 1594, \"it is forbidden to make or sell\ncabinets, escritoires, caskets, brasiers, _chapines_ (clogs), tables,\ncommodes decorated with silver, either beat in _repouss\u00e9_, stamped,\ncarved or plain, and whoever makes, sells, or buys them, is to lose\nthem.\" Notwithstanding this and other restrictions which appeared in the\n17th century, this abuse can hardly have been checked, judging by the\nnumber of these objects which have reached us, not counting those\npreserved in the shrines of Spanish churches and cathedrals.\n\nThe luxury which was apparent in this century of great decay for Spain\nwill be found in the numerous descriptions which exist of different\nfeasts and ceremonies.\n\nA good idea may be had of this style of silversmiths' work from the\nsilver dishes in the South Kensington Museum. An engraving of one of\nthese appears on the preceding page.\n\n[Illustration: BREAST ORNAMENT SET WITH EMERALDS. SPANISH, LATE 17TH\nCENTURY. SOUTH KENSINGTON MUSEUM.]\n\nDuring the 17th century, jewellery underwent a complete\ntransformation--the proceedings and _renaissance_ forms of the Italian\nschool were abandoned, with all their richness of sculptures, enamels,\nand variety of stones. Instead of this, jewels were formed of emeralds,\ndiamonds or rubies in gold setting, _perc\u00e9s \u00e0 jour_, producing an\nexcellent effect. The exceptions to this rule are the objects in which\nenamelled work still predominates, a reminiscence of the former century.\nAmong the finest and best examples may be mentioned the gold crown of\nthe Virgen de los Desamparados of Toledo, made in Mexico in the 17th\ncentury. It is formed like a basket of flowers, of delicate tracery, and\nrichly studded with fine emeralds. Several most interesting specimens\nexist at Kensington of Spanish jewellery of this kind, bought at the\nsale which took place in 1870 of the jewels belonging to the Virgen del\nPilar at Zaragoza.\n\nNo. 325-70 is a breast ornament of gold scroll open work, with enamelled\nflowers, set with emeralds. [See woodcut on p. 37.]\n\nNo. 320-70, a gold breast ornament with five bosses and seven pendants,\nset with rose diamonds.\n\nNo. 406-73. Breast ornament, gold open strap work and floral filigree,\nthe lower part an oval pendant, set with table diamonds. [See woodcut on\np. 39.]\n\nThe following fine Spanish jewels of the 17th century in the Kensington\nMuseum are also worthy of attention:\n\nNo. 330-64. A pectoral cross, with medallions containing relics.\n\nNo. 298-66. Gold filigree cross, within which is an ivory crucifix.\n\nNo. 344-70. Enamelled gold pectoral cross set with amethysts.\n\nNo. 417, 417A.-69. A pair of earrings of gold open work, branches set\nwith white crystals.\n\nNo. 323, 323A.-70. Silver open work earrings set with rose diamonds.\n\nNo. 330 to 330C.-70. Four miniature ewers of silver filigree open work,\nthe bodies of Chinese enamelled copper.\n\nNo. 1224-71. Silver frame _repouss\u00e9_, with the Holy Dove, and a bleeding\nheart encircled with thorns.\n\nModels of the baroque or, as it is called in Spain, _Churrigueresque_\nstyles continued to be copied during the beginning of the 18th century,\nin the same manner as in the 17th century. At this time, as in the rest\nof Europe, a reaction begins in every branch of art, due in Spain to the\ninfluence of French and Italian artists who accompanied the family of\nBourbon. The Academy of Fine Arts of St. Fernando was founded by a king\nof that House towards the middle of the century, the teaching was\nreduced to copying Greco-Roman models, such as they were understood at\nthat time.\n\n[Illustration: BREAST ORNAMENT SET WITH DIAMONDS. SPANISH, LATE 17TH\nCENTURY. SOUTH KENSINGTON MUSEUM.]\n\nThe great centres which in the 16th century had produced such splendid\nworks of art had almost completely ceased. No great silversmiths\nremained at Valladolid, Leon, Toledo, or Seville. The only localities\nwhich have preserved even to the present day the traditional forms of\nthese ornaments are Salamanca, Cordova, Astorga, and Santiago. Madrid\nabsorbed from the middle of the 17th century the whole of this industry.\nIn Larruga's \"Memorias,\" Vol. IV., will be found every detail given on\nthis subject. Several important establishments for the object of making\nsilver work on a large scale were founded at Madrid, the most important\nbeing that of Tomas de Buenafuente, which passed after to Francisco\nNovi. Two Frenchmen called Isaac and Miguel Naudin established a\nmanufactory in 1772. In each the greater part of the work was machine\nmade. Others were founded to cut and polish precious stones, and mount\npaste stones. This was done with great skill by Antonio Martinez in\n1778, in a building fitted up for the purpose, which still exists\nopposite the Botanical Garden of Madrid. Martinez was pensioned by\nCharles III. in Paris and London to study the improvements in this\nindustry. The principal object of the manufactory was to teach the\ntechnical proceedings required in order to extend this industry in the\ncountry, and supply a school in which machinery, models, drawings, &c.,\nwere to be met with. Pupils of both sexes were admitted; machinery was\nmade in the workshops, and Martinez undertook to teach the manufacture\nof gold, doubl\u00e9, or steel trinkets, with or without enamel or stones.\nSword-hilts, buckles, snuff-boxes, needle-cases, handles for sticks,\nbrooches, necklaces, orders, and other different objects, were made\neither of open work or enamelled gold.\n\nInkstands, dishes, dinner services, chocolate stands, cruets, knives and\nforks, were made of silver in different styles, generally imitating the\nEnglish manner. [Consult Larruga, \"Memorias Politicas y Economicas.\"\nMadrid, 1789. Vol. IV., p. 116.]\n\nThe results obtained by the manufactory of Martinez were most\nsatisfactory; a large number of apprentices were taught there; but their\nwork was completely French in character; the manner and style of the\nSpanish school of silversmiths was completely forgotten.\n\nAs I have already mentioned, the traditional forms were preserved in\nsome localities which require to be mentioned here in order to finish\nthis account of this industry in Spain. An interesting and varied\ncollection of modern Spanish peasant jewellery exists at the South\nKensington Museum. Strange to say, although this collection was formed a\nvery few years ago, in 1870, it would be very difficult now to make\nanother; for owing to the means of communication having been of late\nyears so much improved in Spain, the peasantry are leaving off their\nnational costumes, and substituting in every detail modern fashions.\nAmong this peasant jewellery the silver gilt necklace and reliquaries of\nAstorga, No. 1114-73, deserve special attention. These necklaces were\nworn round the neck and part of the body. The neck ornaments of gold and\nseed pearls made at Salamanca, those of silver gilt of Santiago; the\nfiligree work of Cordova in the Moorish style, and the long earrings of\nCatalu\u00f1a with precious stones, are interesting reminiscences of older\ntimes.\n\n                     LIST OF SPANISH GOLDSMITHS AND SILVERSMITHS.\n\n\n                  10TH CENTURY\n\n  Years in which\n  they worked.                                                    Residence.\n\n  961.            Hudzen ben Bozla, a Moor. He made a silver\n                  casket which exists still at Gerona Cathedral     Gerona.\n\n\n                  13TH CENTURY\n\n  1279.           Maestre Jorge                                     Toledo.\n\n  1283.           Modova, Pablo de                                  Burgos.\n\n                  Niculas (Don)\n\n  1262.           Perez, Juan                                       Burgos.\n\n\n                  14TH CENTURY\n\n  1357.           Andreu, Raimundo de                               Gerona.\n\n  1358.           Barners, Pedro                                    Gerona.\n\n  1325.           Bartolom\u00e9, Maestro                                Gerona.\n\n                  Bernec, Pere, V. Barners\n\n  1382 to }       Capellades, Pedro                                Tortosa.\n  1393.   }\n\n  1378.           Fernai, Rodrigo                                   Oviedo.\n\n  1334.           Frau, Ramon                            Palma de Mallorca.\n\n  1367.           Martinez, Sancho                                  Sevilla.\n\n  1382 to }       Paris, Pedro de                                   Tortosa.\n  1393.   }\n\n  1373.           Perpi\u00f1a, Juan                                    Valencia.\n\n  1370.           Ponce, Bartolom\u00e9                       Palma de Mallorca.\n\n\n                  15TH CENTURY\n\n  1417.           Abello, Joan                                      Daroca.\n\n  1495.           Alca\u00e7ar, Juan de                                  Toledo.\n\n  1477.           Almerique                                        Barcelona.\n\n  1494.           Berenguer, Juan                                   Valencia.\n\n  1499.           Castellano                                        Toledo.\n\n  1454.           Castelnou, Juan de                                Valencia.\n\n  1460.           Castelnou, Jayme de, son of Juan                  Valencia.\n\n  1470.           Cetina, Mestre                                    Valencia.\n\n  1458 to }       Diez, Pedro el Cabalan                             Toledo.\n  1463    }\n\n  1494.           Diaz, Thomas                                       Toledo.\n\n  1417.           Diaz, or Diez Caro, Ferrando                       Daroca.\n\n  1418 to }       Garcia de Valladolid, Alfonso                      Toledo.\n  1426.   }\n\n  1438.           Garcia, Alonso                                     Burgos.\n\n  1442.           Garcia de Pielagos, Juan                           Burgos.\n\n  1477.           Gomez, Garcia                                     Valencia.\n\n  1424 to }       Gonzalez de Madrid, Juan                           Toledo.\n  1459.   }\n\n  1477.           Hance\n\n                  Lorenzo, Don\n\n  1425.           Medina, Juan de                                    Toledo.\n\n  1499.           Medina, Pedro de                                   Toledo.\n\n  1470.           Nadal Yvo, Maestre                                Valencia.\n\n  1493.           Narbona, Diego                                     Toledo.\n\n  1495.           Nu\u00f1ez, Alonso                                      Toledo.\n\n  1485.           Oviedo, Fernando de                                Burgos.\n\n  1487.           Pizarro                                           Guadalupe.\n\n  1457.           Rodriguez de Villareal, Alonso                     Toledo.\n\n  1459.           Rodriguez de Villareal, Anton                      Toledo.\n\n  1483.           Rodriguez, Gonzalo                                 Toledo.\n\n  1496.           Rodriguez, Geronimo                               Sevilla.\n\n  1459.           Rodriguez de Villareal, Lope                      Toledo.\n\n  1417 to }       Roiz, Pero                                        Darroca.\n  1423.   }\n\n  1489 to }       Ruby, Maestre                                     Toledo.\n  1491.   }\n\n  1416.           Ruiz de Astudillo, Alfonso                        Burgos.\n\n  1431.           Ruiz de Medina                                    Toledo.\n\n  1426.           Ruiz, Juan                                        Toledo.\n\n  1498.           Ruiz, Juan                                        Toledo.\n\n  1404.           Sanchez, Martinez                                Sevilla.\n\n  1424.           Sanchez, Anton                                    Toledo.\n\n  1417.           Sancho, Manuel Hernando                           Burgos.\n\n  1487.           Segovia, Fr. Juan de                            Guadalupe.\n\n  1418.           Valles, Juan                                      Toledo.\n\n  1484 to }       Vigil, Pedro de                                Valladolid.\n  1488.   }\n                  Yvo V. Nadal Yvo                                   ----\n\n\n                  16TH CENTURY\n\n  1586.           Abedo de Villandrando, Diego                      Madrid.\n\n  1531.           Aguirre, Pedro de                                 Toledo.\n\n  1515.           Aleman, Mateo                                    Sevilla.\n\n  ----            Aleman, Nicolas                                  Sevilla.\n\n  1596.           Alfaro, Francisco                                Sevilla.\n\n  1539.           Alonso, Juan                                      Toledo.\n\n  1552.           Alvarez, Baltasar                                Palencia.\n\n  1568.           Alvarez, Francisco                                Madrid.\n\n  1531.           Alvarez, Juan                                    Granada.\n\n  1560.           Alvarez, Juan                                     Madrid.\n\n  1538.           Alvear, Juan de                                   Burgos.\n\n  ----            Alvear, Nicolas de                                Burgos.\n\n  1520 to }       Andino, Cristoval de                              Burgos.\n  1521.   }\n\n  ----            Angel, Pedro                                      Toledo.\n\n  1565.           Anrique, Joan                                     Toledo.\n\n  1570.           Avila, Alonso de                                  Toledo.\n\n  1567.           Avila Cimbron, Diego de                           Toledo.\n\n  1544.           Arfe, Antonio de                                    Leon.\n\n  1595.           Arfe y Villafa\u00f1e, Juan de                        Sevilla.\n\n  1506 to }       Arphe, Enrique de                                   Leon.\n  1525.   }\n\n  1531.           Baeza, Francisco de                              Granada.\n\n  ----            Baeza, Juan                                      Granada.\n\n  1580.           Ballesteros, Fernando de                         Sevilla.\n\n  1560.           Baptista, Juan                                    Toledo.\n\n  1584 to }       Baroxa, Gregorio de                               Toledo.\n  1601.   }\n\n  1534.           Becerril, Alonso                                  Cuenca.\n\n  1575.           Becerril, Cristoval                               Cuenca.\n\n  1552.           Becerril, Francisco                               Cuenca.\n\n  1590.           Belta, Hanz                                       Madrid.\n                  Belthae, V.--see Hanz Belthae\n\n  1582.           Benavente, Juan de                               Palencia.\n\n  1546.           Benavente, Pedro de                               Toledo.\n\n  1556.           Borgo\u00f1es, Juan                                    Toledo.\n\n  1538.           Buentalante, Alonso de                           Granada.\n\n  1586.           Camps, Bernat                                   Barcelona.\n\n  1547 to }       Carrion, Hernando                                 Toledo.\n  1561.   }\n\n  1556.           Carrion, Fernando                                 Toledo.\n\n  1586.           Castro, Juan de                                   Madrid.\n\n  1531.           Castro, Luis de                                  Granada.\n\n  1505.           Cetina, Bernardo Juan                           Valencia.\n\n  1546.           Comes, Gabriel                                 Barcelona.\n\n  1553.           Conill, Antonio                                Barcelona.\n\n  1531.           Cordova, Anton de                                Granada.\n\n                  Cordova, Juan de                                 Granada.\n\n  1514.           Cota, Martin                                      Toledo.\n\n  1567.           Davila Cimbron, Diego                             Toledo.\n\n  1558.           Davila, Diego                                     Toledo.\n\n  1593.           Diaz, Francisco                                   Toledo.\n\n  1526.           Diaz, Hernando                                    Toledo.\n\n  1586.           Dominguez, Juan                                   Madrid.\n\n  1513.           Donanti, Juan                                    Sevilla.\n\n  1564 to }       Due\u00f1as, Alonso de                               Salamanca.\n  1598.   }\n\n  1531.           Due\u00f1as, Juan de                                   Granada.\n\n  1537.           Escripian, Luis                                    Toledo.\n\n  1531.           Fernandez, Diego                                  Granada.\n\n  1554.           Fernandez, Manuel                                 Sevilla.\n\n  1567 to }       Fernandez, Marcos                                  Toledo.\n  1593.   }\n\n  ----            Fernandez del Moral                                Madrid.\n\n  1531.           Flores, Diego                                     Granada.\n\n  1572.           Font, Joan                                       Barcelona.\n\n  1555.           Franc\u00e9s, Juan                                      Toledo.\n\n  1561 to }       Gallego, Miguel                                   Toledo.\n  1590.   }\n\n  1531.           Garcia, Thome                                    Granada.\n\n  1561.           Gallo, Gaspar                                     Toledo.\n\n  1555.           Goden, Juan                                       Toledo.\n\n  1509.           Gomez, Heros de                                 Valencia.\n\n  1573.           Gomez, Baltasar\n\n  1574.           Gonzalez, Pedro                                    Uveda.\n\n  1513.           Gueran, Ferrer                                 Barcelona.\n\n  1531.           Hermosilla, Bartolom\u00e9 de                         Granada.\n\n  1567.           Hernandez, Gonzalo                                Toledo.\n\n  1531.           Hernandez, Luis                                  Granada.\n\n  1567.           Hernandez, Marcos                                 Toledo.\n\n  1524 to }       Hernandez, Pedro                                  Toledo.\n  1544.   }\n\n  1531.           Herrera, Gonzalo de                              Granada.\n\n  1524.           Herreros y Manzanas                               Toledo.\n\n  1585 to }       Hanz, Belthae                                     Madrid.\n  1587.   }\n\n  1582 to }       Honrado, Julian                                   Toledo.\n  1599.   }\n\n  1531.           Jaen, Fernando de                                Granada.\n\n  1562 to }       Lainez, Juan Bautista                             Madrid.\n  1573.   }\n\n  1524.           Laynez, Rodrigo                                   Toledo.\n\n  1559.           Leon, Leoni                                       Madrid.\n\n  1597.           Leoni, Miguel                                     Madrid.\n\n  1531.           Lopez, Francisco                                  Granada.\n\n  1548 to }       Lopez, Juan                                       Toledo.\n  1570.   }\n\n  1530.           Lopez de Leon, Juan                               Toledo.\n\n  1508.           Madrid, Pedro de                                  Toledo.\n\n  1531.           Mar, Alonso de la                                 Granada.\n\n  1588 to }       Marques, Lorenzo                                  Toledo.\n  1614.   }\n\n  1522.           Marquez, Cristoval                                Toledo.\n\n  1531.           Martinez, Pero                                   Granada.\n\n  1544 to }       Martinez, Francisco                               Toledo.\n  1547.   }\n\n  1553.           Mas, Miguel                                      Barcelona.\n\n  1534.           Masanell, Joan                                   Barcelona.\n\n  1517.           Mayquez, Diego                                    Toledo.\n\n  1500 to }       Medina, Pedro de                                  Toledo.\n  1515.   }\n\n  1538.           Mendoza, Alonso de                               Granada.\n\n  ----            Mendoza, Blas de                                 Granada.\n\n  ----            Mendoza, Diego de                                 Burgos.\n\n  1564 to }       Merino, Francisco                                 Toledo.\n  1601.   }\n\n  1579 to }       Montoya, Alejo de                                 Toledo.\n  1586.   }\n\n  1598.           Morales, Luis                                     Madrid.\n\n  1592.           Morales, Tomas                                    Toledo.\n\n  1515 to }       Moran                                             Toledo.\n  1517.   }\n\n  1556.           Moran, Pedro Fernandez de                         Burgos.\n\n  1561.           Morel, Thomas du (an Englishman)                  Toledo.\n\n  1574 to }       Montoya, Alexo                                    Toledo.\n  1590.   }\n\n  1573.           Mu\u00f1oz, Alonso                                     Toledo.\n\n  1598.           Mu\u00f1oz, Antonio                                    Madrid.\n\n  1573.           Niebre, Matias de, a native of Burgundy,\n                       living at                                    Toledo.\n\n  1512.           Nu\u00f1ez, Alonso                                     Toledo.\n\n  1531.           Nu\u00f1ez, Francisco                                 Granada.\n\n  1532 to }       O\u00f1a, Antonio de                                   Burgos.\n  1560.   }\n\n  1531.           O\u00f1ate, Juan de                                   Granada.\n\n  1537 to }       Ordo\u00f1ez, Andres                                   Toledo.\n  1546    }\n\n  1507 to }       Ordo\u00f1ez, Pedro                                    Toledo.\n  1510    }\n\n  1537 to}        Orna, Juan de                                     Burgos.\n  1528.\n\n  1581.           Ortiz, Pedro                                      Madrid.\n\n  1529 to }       Oviedo, Diego de                                  Toledo.\n  1546.   }\n\n  1586.           Pan, Juan                                       Barcelona.\n\n  1594.           Pardo, Martin                                    Escorial.\n\n  1534.           Parra, Juan de la                                 Toledo.\n\n  1577.           Pastrana, Luis de                                 Madrid.\n\n  1595.           Pedraza, Diego de                              Estremadura.\n\n  1593.           Pedraza, Esteban\n\n  1559.           Perez, Francisco                                Barcelona.\n\n  1598.           Perez, Pedro                                      Madrid.\n\n  1535.           Pierres, a Frenchman, residing at                 Toledo.\n\n  1587.           Poch, Pero Juan                                 Barcelona.\n\n  1568.           Poggini, Juan Pablo                               Madrid.\n\n  1532.           Pons, Pere                                      Barcelona.\n\n  1574.           Portigniani, Juan Bautista                        Toledo.\n\n  1541 to }       Ramirez, Pedro                                    Toledo.\n  1554.   }\n\n  1590.           Reynalte, Francisco de                            Madrid.\n\n  ----            Reynalte, Juan de                                 Madrid.\n\n  ----            Reynalte, Pedro de                                Madrid.\n\n  1590.           Reynalte, Rodrigo de                              Madrid.\n\n  1531.           Rivera, Diego Lopez de                           Granada.\n\n  1590.           Rodriguez Bermudez, Gonzalo                   {Madrid and\n                                                                {  Toledo.\n\n  1557 to }       Rodriguez de Babia, Juan                         Toledo.\n  1586.   }\n\n  1596.           Rodriguez del Castillo, Melchor                 Segovia.\n\n  1551.           Rodriguez, Duarte                                Toledo.\n\n  1568.           Rodriguez Machado, Juan                          Toledo.\n\n  1567.           Ros, Felipe                                    Barcelona.\n\n  1537.           Rozas, Gregorio de                               Burgos.\n\n  1538.           Rozas, Jeronimo de                               Burgos.\n\n  1584 to }       Ruiz, Diego                                       Toledo.\n  1590.   }\n\n  1531.           Ruiz, Jeronimo                                  Granada.\n\n  1524.           Ruiz, Julian, lapidary                           Toledo.\n\n  1590.           Ruiz, Juan                                       Toledo.\n\n  ----            Ruiz, Juan, el Vandolino                  {Sevilla, Jaen\n                                                            {and Cordova.\n\n  1545.           Sabat, Benedicti                                Barcelona.\n\n  1566.           Salamanca, Antonio de                            Madrid.\n\n  1590.           Sanchez, Andres                                  Toledo.\n\n  1522.           San Roman, Pedro de                              Toledo.\n\n  1531.           Sevilla, Hernando de                             Granada.\n\n  1524.           Sig\u00fcenza, Pedro de                               Sevilla.\n\n  1595.           Tamarit, Andreu                                 Barcelona.\n\n  1590.           Tello de Moreta                                  Toledo.\n\n  1554.           Toto                                             Madrid.\n\n  ----            Trezzo, Jacome de, an Italian, who worked at     Madrid.\n\n  1589.           Trezzo, a nephew of Jacome                       Madrid.\n\n  1590.           Urbano, Juan                                    Cordova.\n\n  1580.           Usatigni, Juan Alonso                            Madrid.\n\n  1537.           Valdes, Antonio de                             Barcelona.\n\n  1589.           Valdes, Lucas                                   Cordova.\n\n  1564 to }       Valdivieso, Diego de                             Toledo.\n  1598.   }\n\n  1593.           Valdivieso, Lucas de                             Burgos.\n\n  1575.           Valla, Narcisco                                Barcelona.\n\n  1548.           Valladolid, Juan de                               Toledo.\n\n  1566.           Valle, Antonio del                                Madrid.\n\n  1524.           Valles, Hernando de                               Toledo.\n\n  1512.           Vargas, Gutierre de                               Toledo.\n\n  1563.           Vazquez, Baptista                                 Toledo.\n\n  1512 to }       Vazquez, Diego                                    Toledo.\n  1543.   }\n\n  1558.           Velasco                                           Toledo.\n\n  1580.           Velez                                             Toledo.\n\n  1569.           Vergara, Nicolas de                               Toledo.\n\n  1546.           Vicente, Mateo, lapidary                          Toledo.\n\n  1561.           Vida, Francisco                                 Barcelona.\n\n  1534.           Villagran, Juan de                                Toledo.\n\n  1590.           Villanueva, Juan Domingo de                       Toledo.\n\n  1552.           Villaseca, Alexo                                  Toledo.\n\n  1538.           Vitoria, Francisco de                            Granada.\n\n  1538.           Vitoria, Juan de                                 Granada.\n\n  1528.           Vozmediano, Diego de                             Sevilla.\n\n  1536.           Vozmediano, Juan de                              Sevilla.\n\n  1561.           Ximenez, Juan                                   Barcelona.\n\n  1537.           Ximenis, Rafael                                 Barcelona.\n\n  1599.           Zepes, Bartolom\u00e9 de                               Toledo.\n\n\n                  17TH CENTURY\n\n  1618 to }       Adeba, Jusepe                                     Madrid.\n  1627.   }\n\n  1677.           Alcario, Francisco de                            Sevilla.\n\n  1664.           Aleman, Juan                                      Toledo.\n\n  1623.           Alonso, Jeronimo                                  Madrid.\n\n  1676 to }       Alonso de Prado, Pedro                            Burgos.\n  1714.   }\n\n  1622.           Alvarado, Ambrosio de                             Madrid.\n\n  1635.           Alvarez Brizuela, Blas                            Madrid.\n\n  ----            Alvarez, Blas                                     Madrid.\n\n  1636.           Alvarez de Peralta, Pedro                         Madrid.\n\n                  Alvear, Andres                                    Burgos.\n\n  1640.           Alvear, Juan de                                   Burgos.\n\n  1617.           Alvear, Juan de                                   Madrid.\n\n  1691.           Arandas, Gaspar                                 Tarragona.\n\n  ----            Arfe, Joseph, a grandson of Juan de Arfe          Sevilla.\n\n  1613.           Avalos, Diego de                                  Madrid.\n\n  1604.           Avenda\u00f1o de Tudela, Luis                          Madrid.\n\n  1677.           Avila, Agustin de                                 Sevilla.\n\n  1620.           Barinci, Juan Bautista                           Escorial.\n\n  1624.           Baraona, Antonio                                  Madrid.\n\n  1638.           Barona, Pedro                                   Barcelona.\n\n  ----            Bellicis, V. Bardi\n\n  1650.           Bardi, Bellicis                                 Barcelona.\n\n  1604.           Belta, Pedro                                      Madrid.\n\n  1602.           Benavente, Pedro                                  Toledo.\n\n  1645.           Belorado, Antonio de                              Burgos.\n\n  1630.           Bonino, Pedro                                     Madrid.\n\n  1616.           Bracho, Alexandro                                 Toledo.\n\n  1628.           Brizuela, Gaspar Miguel de                        Madrid.\n\n  1622.           Calvo, Juan                                       Madrid.\n\n  1626.           Camanyes or Comanes, Aloy                        Tortosa.\n\n  1650.           Campo, Juan del                                   Burgos.\n\n  1646.           Campo, Juan de                                    Madrid.\n\n  1643 to }       Carcaba, Felipa, the wife of\n  1664.   }             Andres Salinas, worked at                    Toledo.\n\n  1621.           Carranza Alvear, Juan de                          Madrid.\n\n  1622.           Carranza, Martin de                               Madrid.\n\n  1615.           Castro, Juan                                      Madrid.\n\n  1651 to }       Ciga, Miguel de                                   Burgos.\n  1670.   }\n\n  1646.           Coma\u00f1es, Aloy                                    Tortosa.\n\n  1668.           Concepcion, Friar Juan de la                    Escorial.\n\n  ----            Cruz, Friar Eugenio de la                       Escorial.\n\n  1620.           Cuello, Amaro                                     Madrid.\n\n  1661.           Dandez, Agustin                                Barcelona.\n\n  1623.           Delgado Maldonado, Miguel                         Madrid.\n\n  1660.           Diaz, Juan                                       Segovia.\n\n  1619.           Dominguez, Luis                                   Madrid.\n\n  1613.           Duarte, Cristoval de                              Madrid.\n\n  1671.           Duran, Juan                                       Toledo.\n\n  1613 to }       Duran, Jusepe                                     Madrid.\n  1624.   }\n\n  1664.           Ebrart, Pedro                                     Toledo.\n\n  1660.           Elizes, Alonso de                                 Toledo.\n\n  1642.           Espinosa, Juan de                                 Madrid.\n\n  1640.           Espluga, Tomas                                  Barcelona.\n\n  ----            Estrada, Manuel de                               Sevilla.\n\n  1624.           Estrada, Sebastian                                Madrid.\n\n  1655 to }       Faneli, Virgilio                                  Toledo.\n  1678.   }\n\n  1630.           Franconio, Juan Bautista                         Sevilla.\n\n  1657.           Fernandez, Andres                                 Burgos.\n\n  1679 to }       Fernandez, Gamonal Joseph                         Toledo.\n  1694.   }\n\n  1631.           Fernandez de Angulo, Miguel                       Madrid.\n\n  1612.           Fernandez de Castro, Juan                         Madrid.\n\n  1618.           Fernandez, Martin                                 Madrid.\n\n  1627.           Fernandez de Tapia, Andres                        Madrid.\n\n  ----            Fernandez de Tapia, Juan                          Madrid.\n\n  1667 to }       Fortuna, Juan de                                  Toledo.\n  1670.   }\n\n  1629.           Gallo, Alonso                                     Madrid.\n\n  1615.           Gamonar, Mathieu de                               Madrid.\n\n  1600 to }       Garcia, Alonso                                    Toledo.\n   1635.  }\n\n  1622.           Garcia, Diego                                     Madrid.\n\n  1618.           Garcia, Francisco                                 Madrid.\n\n  1613.           Garcia, Martin                                    Madrid.\n\n  1600.           Garrido, Gaspar                                   Madrid.\n\n  1620.           Generino, Francisco                              Escorial.\n\n  1610.           Gomez, Antonio                                    Madrid.\n\n  ----            Gomez, Diego                                      Madrid.\n\n  1626.           Gomez, Juan                                       Madrid.\n\n  1610.           Gomez, Marcelo                                    Madrid.\n\n  1615.           Gonzales, Gonzalo                                 Madrid.\n\n  1613.           Gonzales, Enrique                                 Madrid.\n\n  1634.           Gonzales, Francisco                               Madrid.\n\n  1630 to }       Gonzales, Pascual                                 Madrid.\n  1640.   }\n\n  1654.           Gonzales, Rafael                                 Segovia.\n\n  1607.           Guardia, Melchor, a native of Milan             Barcelona.\n\n  1679.           G\u00fcergo, Domingo de                                Burgos.\n\n  1639.           Guigelmo, Eugenio                                 Madrid.\n\n  1627 to }       Guigelmo, Juan                                    Madrid.\n  1634.   }\n\n  1618.           Heredia, Juan de                                  Madrid.\n\n  ----            Heredia, Sebastian                                Madrid.\n\n  1631.           Hurtado, Luis                                     Madrid.\n\n  1694 to }       Jaranta, Juan de                                  Toledo.\n  1698.   }\n\n  1650.           Landeras, Jeronimo de                             Burgos.\n\n  1611.           Leal, Sebastian                                   Madrid.\n\n  1606.           Leoni, Pompeyo                                    Madrid.\n\n  1620.           Lopez, Domingo                                    Madrid.\n\n  1639.           Lopez, Francisco                                  Madrid.\n\n  1611.           Lopez de Alvarado, Antonio                        Madrid.\n\n  1676.           Lopez Barona, Andres                              Burgos.\n\n  1617.           Lorenzo, Juan                                  Valladolid.\n\n  1638 to }       Maestre, Luis                                     Toledo.\n  1646.   }\n\n  1611.           Marisanz, Pedro                                   Madrid.\n\n  1653.           Martinez, Juan                                    Burgos.\n\n  1673.           Mata, Francisco de                                Burgos.\n\n  1677.           Mata, Pedro de                                    Burgos.\n\n  1686.           Mayares, Miguel                                   Madrid.\n\n  1614 to }       Maym\u00f3, Bernardo                                  Tarragona.\n  1625.   }\n\n  1639.           Medina, Juan Bautista de                          Madrid.\n\n  1628.           Medina Requexo, Bautista de                       Madrid.\n\n  1641.           Mendez, Duarte                                    Madrid.\n\n  1614.           Mexia, Juan                                       Madrid.\n\n  1604.           Mimbre\u00f1o, Alonso Roman                            Madrid.\n\n  1641.           Mi\u00f1ano, Juan de                                   Madrid.\n\n  1626.           Miranda, Alonso de                                Madrid.\n\n  1611.           Moles, Juan                                       Madrid.\n\n  1639.           Montero, Francisco                                Madrid.\n\n  1630.           Morales, Francisco                                Madrid.\n\n  ----            Morales, Manuel de                                Madrid.\n\n  1695.           Mu\u00f1oz, Juan                                       Madrid.\n\n  1661.           Nadal, Andres                                    Barcelona.\n\n  1620.           Nadal, Martin                                     Madrid.\n\n  1621 to }       Navarrete Escarraman, Bartolom\u00e9 de                Madrid.\n  1629.   }\n\n  1613 to }       Navarro, Cristoval                                Toledo.\n  1623.   }\n\n  1624 to }       Navarro, Manuel                                   Madrid.\n  1628.   }\n\n  1626.           Nu\u00f1ez, Antonio                                    Madrid.\n\n  1630.           Oliva, Gregorio de                                Madrid.\n\n  1650 to }       Ortiz de la Revilla, Juan                         Madrid.\n  1675.   }\n\n  1615.           Oviedo, Martin de                                 Madrid.\n\n  1619.           Oviedo, Pedro de                                  Madrid.\n\n  1611.           Paiba, Vicente de                                 Madrid.\n\n  1665.           Palomo, Diego                                     Toledo.\n\n  1640.           Pallares, Juan de                                 Madrid.\n\n  1628.           Pancorvo, Cristoval de                            Madrid.\n\n  1624.           Paradi\u00f1as, Andres de                              Madrid.\n\n  1659.           Pedraza, Bernardo de                              Toledo.\n\n  1632.           Perez, Alonso                                     Madrid.\n\n  1652 to }       Perez, Antonio                                    Toledo.\n  1658.   }\n\n  1680.           Perez de Montalto, Antonio                        Toledo.\n\n  1685 to }       Perez de Montalto, Miguel                         Toledo.\n  1688.   }\n\n  1630.           Puig, Luis                                        Valencia.\n\n  1627.           Quero, Juan de                                     Madrid.\n\n  1632.           Quixano, Marcos                                    Madrid.\n\n  1617.           Ramirez, Agustin                                   Madrid.\n\n  1627.           Ramos, Juan                                        Madrid.\n\n  1628.           Reynalte, Juan                                     Madrid.\n\n  1660.           Riba, Manuel de                                    Toledo.\n\n  1624.           Ribera, Martin de                                  Madrid.\n\n  1617.           Rios, Esteban de                                   Madrid.\n\n  1626 to }       Roda, Agustin                                     Tortosa.\n  1646.   }\n\n  1626.           Rodriguez, Bartolom\u00e9                               Madrid.\n\n  1631.           Rodriguez, Eugenio                                 Toledo.\n\n  1645.           Rodriguez, Gabriel                                 Madrid.\n\n  1618.           Rodriguez, Diego                                   Madrid.\n\n  1613.           Rodriguez, Jeronimo                                Madrid.\n\n  1617.           Rodriguez, Juan                                    Madrid.\n\n  ----            Rodriguez Bermudez, Gonzalo                        Madrid.\n\n  1623.           Romano Valmaseda, Juan                             Madrid.\n\n  1607 to }       Ros, Pedro                                         Barcelona.\n  1625.   }\n\n  1617.           Rosales, Francisco de                                Madrid.\n\n  1611.           Ruiz, Eugenio                                        Madrid.\n\n  ----            Ruiz, Juan Bautista                                  Madrid.\n\n  ----            Ruiz de Valdevieso                                   Burgos.\n\n  1675.           Ruiz de Velasco, Juan                                Burgos.\n\n  1684.           Salazar, Francisco                                   Burgos.\n\n  1604 to }       Salinas, Andres de                                {Toledo and\n  1635.   }                                                            {Madrid.\n\n  1639 to}        Salinas, Francisco de                               Toledo.\n  1670.  }\n\n  1639 to }       Salinas, Vicente de                                 Toledo.\n  1680.   }\n\n  1616.           Sanchez, Francisco                                  Toledo.\n\n  1659 to }       Sanchez Ormachea, Pedro                             Toledo.\n  1661.   }\n\n  1614.           San Martin, Juan de                                 Toledo.\n\n  1610 to }       Santillana, Antonio de                              Madrid.\n  1617.   }\n\n  1642.           Santo Domingo, Hipolito                            Madrid.\n\n  1653.           San Vicente, Jacinto de                            Burgos.\n\n  1640.           Sarabia, Cristoval de                             Sevilla.\n\n  1668.           Segura, Juan de                                   Sevilla.\n\n  1632.           Serrano, Francisco                                 Madrid.\n\n  1640.           Soler, Baltasar                                    Madrid.\n\n  1629.           Soria, Francisco                                   Madrid.\n\n  1608.           Soria, Jeronimo de                                 Madrid.\n\n  1617.           Soria, Lucas de                                    Madrid.\n\n  1620.           Spagna, Juliano                                    Madrid.\n\n  1694.           Scase, Gaspar                                     Sevilla.\n\n  1662.           Suer, or Sueur, Esteban, a Frenchman who\n                       worked at                                    Toledo.\n\n  1671.           Tellez, Juan                                      Toledo.\n\n  ----            Tebres, Antonio                                   Toledo.\n\n  1677.           Tello, Mateo, a German, worked at                Sevilla.\n\n  1617.           Tofino, Alonso                                    Madrid.\n\n  1618 to }       Valera, Domingo de                                Madrid.\n  1623.   }\n\n  1619.           Valedomar, Angel de                               Madrid.\n\n  1626.           Valle, Juan del                                   Madrid.\n\n  ----            Valmaseda, V. Juan Romano                           ----\n\n  1624.           Vega, Juan de                                     Madrid.\n\n  1672.           Velasco Ganado, Pedro                             Toledo.\n\n  1625.           Velasquez, Juan                                   Madrid.\n\n  1627.           Villalta, Antonio                                 Madrid.\n\n  1600 to }       Villamayor, Baltasar                              Toledo.\n  1626.   }\n\n  1638.           Villarago, J.                                    Barcelona.\n\n  1622.           Villarroel, Andres de                             Madrid.\n\n  1614.           Villegas, Martin de                               Toledo.\n\n  1645.           Vivanco, Simon                                    Madrid.\n\n  1615.           Xabier, Antonio                                   Madrid.\n\n  1610.           Zabalza, Diego de                                 Madrid.\n\n  1615 to }       Zaldivia, Lucas de                                Burgos.\n  1633.   }\n\n  ----            Zoga, Miguel de                                     ----\n\n\n                  18TH CENTURY\n\n  1746.           Aguilar, Geronimo                                Cordova.\n\n  1794.           Alamo, Bernardo de                                Toledo.\n\n  1703.           Aragon, Diego de                                 Segovia.\n\n  1716.           Aranda, Alberto                                   Madrid.\n\n  1746.           Aviles, Manuel de                                Cordova.\n\n  End of     }    Balmet, Bartolom\u00e9                                 Madrid.\n  18th cent. }\n\n  1740.           Ballestero, Manuel                               Cordova.\n\n  1730 to }       Bargas, Manuel de                                 Toledo.\n  1752.   }\n\n  1702.           Bargas, Mateo de                                  Toledo.\n\n  1739.           Bargas y Machuca, Manuel de                       Toledo.\n\n  1789 to }       Bargas Machuca, Manuel Timoteo                    Madrid.\n  1797.   }\n\n  1753.           Bautista, Vicente                                 Toledo.\n\n  18th cent.      Benitez, Pedro                                  Salamanca.\n\n  End of     }    Buenafuente, Tomas                                Madrid.\n  18th cent. }\n\n  1746.           Bustamante, Franco                               Cordova.\n\n  1734.           Bustos, Floro de                                 Sevilla.\n\n  1784.           Calzado, Jos\u00e9, enameller                          Malaga.\n\n  1746.           Camacho, Lorenzo                                 Cordova.\n\n  1746.           Camacho, Rafael                                  Cordova.\n\n  1713.           Campo, Matias del                                  Soria.\n\n  1734.           Castillo, Matias del                        Aranda del Duero.\n\n  1746.           Castro, Damian de                                Cordova.\n\n  1748.           Castro, Diego                                     Toledo.\n\n  18th cent.      Chevalier, Luis Claudio, a Frenchman, residing at Madrid.\n\n  1728.           Colomes, Pedro, worked at the mounts of silver\n                       knives made at                               Alcora.\n\n  End of     }    Conde, Domingo                                     Madrid.\n  18th cent. }\n\n  1754.           Cros, Josef                                      Valencia.\n\n  ----            Cuerda, Feliz de la                               Toledo.\n\n  1738 to }       Cuerda, Jose de la                                Toledo.\n  1745.   }\n\n  1722 to }       Dominguez, Juan Antonio                           Toledo.\n  1748.   }\n\n  1776.           Elosua, Antonio                                   Burgos.\n\n  1776.           Elosua, Manuel de                                 Burgos.\n\n  1790.           Fernandez, Pablo                                  Burgos.\n\n  18th cent.      Fernandez Clemente, Melchor                     Salamanca.\n\n  1741.           Fernandez de la Fuente, Antonio                   Toledo.\n\n  1778.           Ferroni, Juan                                     Madrid.\n\n  1701.           Figueroa, Juan                                  Salamanca.\n\n  1688.           Fornaguera, Buenaventura                        Barcelona.\n\n  1790.           Gallardo, Vicente                                 Sevilla.\n\n  1710.           Gamonal y Guzman, Juan                            Toledo.\n\n  1746.           Garcia, Jose                                     Cordova.\n\n  1760 to }       Garcia Reyna, Manuel                              Toledo.\n  1766.   }\n\n  1753.           Garcia Crespo, Manuel                           Salamanca.\n\n  1772.           Gaudin, Isaac, a Frenchman, residing at           Madrid.\n\n  ----            Gaudin, Miguel, a Frenchman, residing at          Madrid.\n\n  1714.           Gomez, Pedro                              Alcala de Henares.\n\n  1746.           Hidalgo, Gabriel                                 Cordova.\n\n  1700 to }       Jaranta, Juan de                                  Toledo.\n  1717.   }\n\n  1743.           Jaranta y Zapata, Juan de                         Toledo.\n\n  1717 to }       Jaranta, Lorenzo                                  Toledo.\n  1732.   }\n\n  1746.           Jurado, Jose                                     Cordova.\n\n  1746.           Lara, Diego de                                   Cordova.\n\n  1790.           Lecaroz, Antonio                                 Sevilla.\n\n  1745.           Lopez, Bernardino                                 Toledo.\n\n  1784.           Lopez, Manuel                                     Burgos.\n\n  1746.           Luna, Andres de                                  Cordova.\n\n  ----            Luque, Juan de                                   Cordova.\n\n  1700.           Llanos, Manuel de                                 Burgos.\n\n  1746.           Madue\u00f1o, Francisco                               Cordova.\n\n  1790.           Mairod y Vassner, Jos\u00e9                           Sevilla.\n\n  1734.           Manrique, Franco Esteban                     Aranda de Duero.\n\n  1784.           Marti, Jose                                     Barcelona.\n\n  1778.           Martinez, Antonio                                Martinez.\n\n  1754.           Martinez, Estanislao                             Valencia.\n\n  End of     }    Martinez, Gregorio                                Madrid.\n  18th cent. }\n\n  1703.           Martinez del Valle, Jose                         Segovia.\n\n  1693.           Matons, Juan                                    Tarragona.\n\n  End of     }    Mendeluce, Fermin                                Navarra.\n  18th cent. }\n\n  End of     }    Molinez, Francisco                              Zaragoza.\n  18th cent. }\n\n  1722.           Monteman y Cusens, Lorenzo                     Salamanca.\n\n  1746.           Moreno, Manuel                                   Cordova.\n\n  1725.           Mu\u00f1oz de Amador, Bernardo                         Madrid.\n\n  1746.           Mu\u00f1oz, Francisco                                 Cordova.\n\n  ----            Navas, Fernando de                               Cordova.\n\n  1784.           Nieva, Antonio de                                 Malaga.\n\n  1770 to }       Ni\u00f1o, Jose                                        Toledo.\n  1776.   }\n\n  1767 to }       Ni\u00f1o, Jose Bernardo                               Toledo.\n  1777.   }\n\n  End of     }    Nivel, Juan                                       Madrid.\n  18th cent. }\n\n  End of     }    Novi, Francisco                                   Madrid.\n  18th cent. }\n\n  1786.           Olivares, Fermin                                  Madrid.\n\n  1746.           Palomino, Jose                                   Sevilla.\n\n  1784.           Pechenet, Juan                                    Madrid.\n\n  1734.           Perez de Oviedo, Francisco                       Granada.\n\n  1746.           Pineda, Manuel                                   Cordova.\n\n  ----            Quintero, Jos\u00e9\n\n  1760 to }       Reyna, Manuel                                    Toledo.\n  1777.   }\n\n  1743.           Reyno, Tomas                                     Toledo.\n\n  End of     }    Roche, Nicholas                                  Madrid.\n  18th cent. }\n\n  1715 to }       Rodriguez de Castro, Juan                        Burgos.\n  1757.   }\n\n  1784.           Rovira, Jos\u00e9                                   Barcelona.\n\n  1746.           Roxelio, Francisco                              Cordova.\n\n  1791.           Ruiz, Donata                                     Burgos.\n\n  1746.           Ruiz, Juan                                      Cordova.\n\n  ----            Ruiz, Miguel                                    Cordova.\n\n  1777.           Ruiz, Pedro                                      Burgos.\n\n  1746.           Sanchez, Francisco                              Cordova.\n\n  1731.           Sanchez Renentes, Tomas                         Sevilla.\n\n  1767 to }       Sanchez Ni\u00f1o, Jose                               Toledo.\n  1777.   }\n\n  1731.         Sanz, Pedro                                       Sevilla.\n\n  18th cent.    Sanz de Velasco, Toribio                         Salamanca.\n\n  1700.         Simancas, Francisco de                             Burgos.\n\n  1716.         Supuesta, Juan                                     Madrid.\n\n  1731.         Tamaral, Ignacio                                  Sevilla.\n\n  1746.         Torralvo, Sebastian                               Cordova.\n\n  1755.         Torrijos, Lucas de                                 Burgos.\n\n  1771.         Urqueza, Domingo                                   Madrid.\n\n  ----          Valadrez Romero, Luis                             Sevilla.\n\n  1746.         Vargas, Jose de                                   Cordova.\n\n  ----          Vega, Pedro de la                                 Cordova.\n\n  1754.         Vicente, Bautista                                 Valencia.\n\n  1728.         Vicente, Luis                                     Valencia.\n\n  1734.           Villa, Andres de                                 Sevilla.\n\n  1713 to }       Zurre\u00f1o, Antonio                                 Madrid.\n  1715.   }\n\n  1799 to }       Ximenez, Manuel                                  Toledo.\n  1800.   }\n\nI owe to the courtesy of Baron Ch. Davillier the names of the following\nartists, which appear in his \"Histoire des Principaux Orf\u00e8vres\nEspagnols,\" Paris, 1879.\n\n  Perez de las Cellas, Antonio, a native of Saragossa;\n  he worked at Rome in                                              1456\n\n  Ruiz, Alonso; he worked at Toledo in                              1431\n\n  Ferrandez, Garci; he worked at Toledo in                          1431\n\n  Pielagos, Juan Garcia, Burgos                                     1442\n\n  Fernai, Rodrigo, a Frenchman, who worked at Oviedo in             1368\n\n  Bells, Antonio, Barcelona                                         1458\n\n  Closes, Francisco, Barcelona                                      1464\n\n  Rodriguez de Villareal, Lope, Toledo                              1466\n\n  Sano, Salvador, Barcelona                                         1475\n\n  Zobarola, Francisco, Banet                                        1480\n\n  Jujuce, a Valencian jew, Pamplona                                 1356\n\n  Freset, Perrin,   } Frenchmen working at Olite, Navarre, in       1444\n  Rodez, Conrat de, }\n  Bonte, Daniel de, a German, working at Olite, in Navarre    about 1400\n\n  Valdubia, Maestro Ferrando, Rome                                  1525\n\n  Fuente, Gonsalvo de, Rome                                         1539\n\n  Alfonso, Rome                                                     1546\n\n\n\n\nIRON WORK.\n\n\nNumerous iron mines have existed in Spain, especially in the Cantabrian\nprovinces, and have been worked from very early times. They are\nmentioned by Pliny, lib. 34, cap. xii., and have given rise to the\ndevelopment of excellent metal works in the Spanish Peninsula: although\nin many localities this tradition has been continued until our days, the\nBasque Provinces have in all times excelled in this artistic industry.\n\nThe want of durability and the little care taken of objects of\niron-work, owing to the insignificant value of the material employed,\nprevent us from being able to mention any important examples of a very\nearly date. After drawing attention to some interesting specimens which\nhave reached us from the Spanish-Arabs, we must begin the history of\niron-work in Spain in the second half of the 15th century; it continues\nto progress in the 16th, and produced undoubtedly at that period works\nwhich were unrivalled in Europe. The _rejas_ or chancel screens\nenclosing chapels in the cathedrals of Toledo, Seville, Granada,\nSalamanca, and other churches in Spain, of which it is much to be\ndeplored no drawings or photographs exist, deserve the especial\nattention of those who follow this industry in the present day, owing to\nthe beauty of their forms, and the inexhaustible variety of models which\nthey present to the manufacturer.\n\nBesides specimens of Iron-work connected with arms which will be\ndescribed in a separate article, the most interesting examples of\nMoorish manufacture which have reached us are keys of most delicate\ntracery; their perfect state of preservation shows that they have only\nbeen used as symbols of cities or fortresses, which on given occasions\nwere offered to kings or great people. Even in the present day the\nceremony is still kept up of offering a key to the foreign princes who\nstay at the royal palace of Madrid, and in a similar manner as far back\nas the Middle Ages, keys have been presented to Spanish sovereigns on\ntheir visiting such towns as Toledo or Seville, and a ceremonial is gone\nthrough of swearing to uphold their privileges, a reminiscence probably\nof what occurred when these towns were conquered from the Moors.\n\nOne of these keys at Valencia, belonging to Count de Trigona, measures\n9\u00bd inches long, and was originally gilt; its handle or bow is closed,\nand covered with delicate work in relief. The wards are ornamented in\nthe same manner with a combination of several words written in Cufic\nletters of difficult interpretation. Round the handle we can read most\ndistinctly in Arabic the name of the artist: \"It was made by Ahmed\nAhsan.\"\n\nThis key appears to be of the 13th to 14th century. Two similar ones\nexisted in the town hall of Valencia of a most monumental character;\nthey were considered of great antiquity, but it has been ascertained\nthat they were made in 1632, by the locksmith Juan Marti.\n\nIn the shrine of the Cathedral of Seville there are two interesting keys\n(_see_ woodcut); one of them is of iron, the other of silver, of a\nsimilar style. For further details see \"Museo Espa\u00f1ol de Antiguedades,\"\nvol. ii., p. I. The first, which is represented to the right, is of\ngenuine Moorish workmanship; the wards are covered with letters in Cufic\ncharacters, which several oriental scholars have tried to interpret,\nwithout coming to a satisfactory conclusion, probably owing to the\nconfused, or double meaning of the letters. It is supposed with good\nfoundation that this was the identical key which was given to King St.\nFerdinand, the conqueror of Seville, in 1248, the day he took possession\nof the city. The silver key, to the left of the plate, has traces of\ngold and niello work, representing in the upper part, ships, castles and\nlions. Round the handle runs an inscription in Hebrew--\"The King of\nKings will open; the King of the whole Earth will enter.\" In the wards\nappear in delicate open work carving the following words in Spanish,\n\"God will open; the king will enter. Dios abrir\u00e1; rey entrar\u00e1.\" It may\nbe affirmed to have been made in the 14th century, and in that case,\nsymbolizes some event of the life of King Pedro the Cruel, a great\nprotector of the Jewish race. Five Moorish keys of a similar kind, but\ninferior merit, may be seen in the local Museum of Segovia, and the\nArch\u00e6ological Museum of Madrid.\n\n[Illustration: MOORISH KEYS IN THE CATHEDRAL OF SEVILLE.]\n\nThe objects of iron-work made by the Christian artists of this period in\nSpain must have been good, for although no specimens exist, we have at\nany rate historical information which confirms this. In the Consistorial\nOrdinances of Barcelona, Capmany, \"Memoirs,\" vol. i., we find that\nironsmiths formed an extensive guild in the 13th century; in 1257 four\nof its members formed part of the chief municipal council; this guild\nincreased in importance in the following centuries. The Ordinances of\nSeville of the 15th century, which were reformed in 1502, and those of\nToledo, also re-enforced in 1582, give an idea of what was done by\nworkers in bronze, the methods of workmanship and other details of\ninterest. The Ordinances of Seville mention _rejas_ made in Biscay, and\ngive a good idea of the styles adopted by the iron-masters there. The\nOrdinances of Granada repeat almost exactly the former prescriptions.\n\nThe Cathedrals and large churches in Spain lent themselves in an\nadmirable manner to the construction of objects of all kinds in iron\nwork, especially the railings enclosing the side chapels, or sepulchres,\nand the double screens required for the _Capilla Mayor_ and _Coro_,\nowing to the ancient Spanish custom of constructing the choir in the\ncentre of the principal nave of the cathedral. Hence we meet with names\nof the iron-masters, _rejeros_, mentioned in early documents as attached\nto the different cathedrals in the same manner as painters and\narchitects. We find Bartolom\u00e9 Morey working at Palma de Mallorca in 1389\nto 1397. He was succeeded by his son Juan, from 1401 to 1407. See\nPiferrer, \"Recuerdos y Bellezas de Espa\u00f1a.\" Maestro Basil worked at\nBurgos, and Maestro Pablo at Toledo and its dioceses towards the end of\nthe same century. Maestro Juan Franc\u00e9s worked at Toledo in 1482. By this\nsame artist is the beautiful _reja_ of the Capilla Mayor of the\n_Colegiata_ of Alcala de Henares; it is signed, \"Maestre Juan Franc\u00e9s,\nmaestro mayor de las obras de fierro en Espa\u00f1a.\" From this time the\nnames of iron-masters are well known and numerous; we may safely affirm\nthat those who lived in the 16th century have left us the most important\nworks.\n\nOne of the finest specimens of this artistic industry is the splendid\n_reja_ which divides the nave at the royal chapel of Granada. Its\nimmense size has enabled the artist to carry out a splendid\nornamentation in the \"plateresque style,\" combined with reliefs on a\nlarge scale of figures of apostles and saints, terminated at the upper\npart with a wide band of ornamentation of leaves and flowers, crowned\nwith a Rood, with the Virgin and St. John on either side. The splendid\nbalustrades and supports are forged with the hammer; the figures and\ncircular piers are formed of large plates, _repouss\u00e9_ and carved in the\nmost admirable manner, and give a good idea of the difficulties of this\nwork, which the artists of this time had overcome, long before the\nvarious technical facilities of the present day existed. The\nornamentation of this _reja_ was originally gilt, and the figures are\npainted in oils. It was made about the year 1520 to 1530; in the lock,\nformed as a Gothic pinnacle, is a small inscription, \"Maestro Bartolome\nme fec.\" This same artist worked at Jaen and Seville. Cean Bermudez\ncalls him \"sculptor and iron-master.\"\n\nTwo most important specimens of iron-work exist also at the cathedral of\nToledo, enclosing the _Capilla Mayor_ and _Coro_. The _reja_ of the\nCapilla Mayor is 42 feet wide by 19 inches high; it rests on a pediment\nof marble ornamented with masks and bronze work, upon which rises the\n_reja_, which is divided horizontally by means of a frieze of\nornamentation, and this again vertically in five compartments. In each\nvertical division there is a pilaster of four sides formed of _repouss\u00e9_\nplates, carved with a fine ornamentation in the renaissance style, this\nis again terminated with life-size figures in high relief of bronze. The\nsecond compartment rises upon the band which divides it in an horizontal\nsense; it follows the same decoration in its pilasters, and is\nterminated by a series of coats of arms, torches, angels, and a variety\nof foliage which finishes the upper part. Upon the centre, hanging from\na thick chain supported from the roof, is suspended a life-size Rood, of\nadmirable effect, which completes the decoration. In several spots there\nare labels with mottos in Latin; in one of them appears the following\ninscription, and the date of 1548 when this splendid work was finished:\n\"Anno MDXLVIII. Paul III. P. M. Carol. V. Imper. Rege. Joannes Martinez\nSiliccus. Archipiscopus. Tolet. Hispaniae. Primat.\" The railings of this\n_reja_ are silvered, and the reliefs and salient points gilt. The artist\nwho made it was Francisco Villalpando, a native of Valladolid; this\nmodel was chosen among those of several artists, who presented their\nplans in competition before the ecclesiastical authorities; it is\ncalculated that ten years elapsed before it was finally finished in\n1548. Villalpando was greatly distinguished likewise as a sculptor and\narchitect. In 1563 a book was printed after his death of a translation\nhe made of the work on architecture by Sebastian Serlio. Other\nremarkable bronze works of art, which will be described further on, were\nalso by him.\n\nThe _reja_ of the coro, which is placed opposite that of the Capilla\nMayor, is almost exactly similar in size. It is less rich in\nornamentation, but so pure and sober in its general lines, and its\nornamentation is so perfect, that it is perhaps superior to the other.\nThis _reja_ consists of a single architectonical body, divided in six\nvertical compartments, covered with bas reliefs of such delicate work\nthat they appear rather to be by the hand of a silversmith than by an\niron-master. An innumerable number of figures are combined with the\nornamentation in the base, terminations of the columns and coronation;\nin the same manner it is full of banderoles with Latin mottos, and gilt\nand silvered; it was finished in 1548, as we see by the inscription.\nThis _reja_ was made by Domingo de Cespedes, a native of Toledo, with\nthe help of his son-in-law, Fernando Bravo; the design was also chosen\nby competition, and it was finished in seven years. Both these admirable\nmodels of iron-work belong to the best productions of the renaissance\nschool, and contain models which might be adapted with great advantage\nto ironwork of every description in the present day.\n\nIt would be an interminable task to describe the multitude of railings\nsimilar in richness and good taste to these which happily survive in\nSpanish cathedrals and churches. It is sufficient to call attention to\nthe following. The _reja_ of the Capilla del Condestable in the\ncathedral of Burgos, by Christoval de Andino in 1523, has been\nconsidered one of the finest specimens of its kind, owing to the\nperfection with which every detail is carried out. A contemporary\nwriter in describing it says: \"Good workmen, and those who wish that\ntheir work may have authority and be blameless, must endeavour to be\nguided by ancient models, as your fellow-citizen, Cristoval de Andino;\nhis works are thereby more elegant and excellent than any others which I\nhave seen up to the present time; if not, judge of his work by looking\nat the _reja_ which he is making for your lord the Condestable, which is\nundoubtedly superior to all those which have hitherto been made in\nSpain.\" Sagredo--\"Medidas del Romano,\" quoted by Cean Bermudez. In the\ncentre of the upper part, towards the middle, appears the inscription:\n\"Ab. Andino, A.D. MDXXIII.\" The _reja_ of the _coro_ of the cathedral of\nSeville, made by the iron-master, Sancho Mu\u00f1oz in 1519, a native of\nCuenca, is also very remarkable; besides the finer ornamentation which\ncovers it, it has figures of kings and prophets representing the\ngenealogy of Our Lord. The _reja_ belonging to the _capilla mayor_ of\nthe same cathedral was the work of Friar Francisco de Salamanca,\n1518-1533; it is covered with an open-work ornamentation, figures of\nangels and bas-reliefs, and is terminated in the upper part with a\nrepresentation of the Entombment of Our Lord. The large _reja_ of the\ncathedral of Cuenca, made by Arenas in 1517, those at Palencia, by\nAndino, 1520, and Rodriguez, 1555; the one at the church of Sn. Juan de\nla Penitencia, Toledo; the beautiful railing at the chapel of\nPalenzuela, in the cathedral of Salamanca, 1524, and a number of objects\nof smaller size, such as iron pulpits in the cathedrals of Barcelona,\nand in some chapels at Burgos, Avila, Palencia, and Toledo, several of\nwhich are anterior to the 16th century, are all worthy of the student's\nespecial notice. Among these objects of smaller dimensions special\nreference must be made to the fine and picturesque _reja_ surrounding\nthe sepulchre of Archbishop Diego de Anaya, in the chapel of Sn.\nBartolom\u00e9, in the cloister of the cathedral of Salamanca. The beauty of\nits details and the open-worked inscription which surrounds it, place\nit at once among the finest specimens of ironwork of the 15th century.\nThe two window _rejas_ at the house called \"de las Conchas\" in the same\ntown--an admirable specimen of Burgundian architecture, although\nrestored--may be quoted as excellent models of ironwork applied to civil\nbuildings.\n\nIron pulpits have been made in Spain with great success. Two interesting\nexamples of the end of the 15th century still exist at the cathedral of\nAvila. They are hexagonal in shape, and supported by lions' claws; they\nwere originally gilt. One of these pulpits is ornamented in the\nFlamboyant style, the other in good Renaissance; consult Sir Digby\nWyatt's \"Architect's Note Book in Spain.\" Friar Francisco de Salamanca\nmade two interesting iron pulpits for the cathedral of Seville; they are\ncovered with bas-reliefs representing the Evangelists and subjects taken\nfrom the Acts of the Apostles and the Apocalypse. The pulpit at the\nparish church of S^{n.} Gil de Burgos must also be mentioned. Mr. Street,\nnotwithstanding his want of sympathy with every object not strictly of\nthe Middle Ages, says: \"It is of very late date, end of the 15th\ncentury, but I think it quite worthy of illustration. The support is of\niron, resting on stone, and the staircase modern. The framework at the\nangles, top and bottom, is of wood, upon which the ironwork is laid. The\ntraceries are cut out of two plates of iron, laid one over the other,\nand the ironwork is in part gilded, but I do not think that this is\noriginal. The canopy is of the same age and character, and the whole\neffect is very rich at the same time that it is very novel. I saw other\npulpits, but none so old as this.\"\n\nAmong the applications of iron to decorations of civil architecture must\nbe specially mentioned the nails and knockers on the doors of houses,\nwhich are so characteristic of Toledo and other old Spanish towns. This\nstyle was imitated from the Moors. Some doors still exist at the\nAlhambra, Granada, covered with enormous heads of nails of a\nhalf-spherical form with embossed pattern. These same nails are\nconstantly to be found on old Spanish houses, to which are added in the\nangles pieces of iron of a most artistic order. A large number of these\nnails from Toledo, are in the Kensington Museum. Examples of two are\ngiven.\n\n[Illustration: SPANISH NAIL HEADS. SOUTH KENSINGTON MUSEUM.]\n\nAfter the 16th century this industry loses its artistic interest and\nimportance. Ironwork becomes simply practical and useful, and ceases to\nreproduce the grand carved and chiselled works of former times. The\nartistic objects are reduced to specimens on a small scale: door-locks,\nof which an excellent example exists at the church of Calatravas,\nMadrid; a weighing machine, made by the locksmith Salinas for the Mint\nin the 17th century, which was exhibited at the Special Loan Collection\nof Scientific Apparatus at Kensington in 1876, the fine ornamentation of\nthe clock of the cathedral of Seville, made in the last century by Friar\nJosef Cordero, and the very remarkable iron locks made by Millan at the\nbeginning of the present century, for the rooms at the palace of the\nEscurial, called \"Piezas de Maderas finas.\" The art of inlaying iron\nwith gold and silver, constitutes in Spain a special industry; the\nprincipal artists who carry it out at the present time with great skill\nare Messrs. Zuluaga at Madrid, and Alvarez at Toledo. The splendid\nsepulchre of General Prim at the church of Atocha, Madrid, made entirely\nof damasquin\u00e9 ironwork, is worthy of the greatest commendation; the fine\nvases and shields of _repouss\u00e9_ ironwork inlaid with gold, made by\nAlvarez of Toledo, are artistic and excellent revivals of the Milanese\nwork of the 16th century.\n\n\nIRON MASTERS.\n\n  Year in which                                                Locality of\n  they worked.      Name.                                     their Residence.\n\n  1520.        Avila, Fr. Juan de                              Guadalupe.\n\n  1527.        Andino, Pedro                                    Seville.\n\n  1540.        Andino, Christoval                                Toledo.\n\n  1557.        Arenas, Hernando                                  Cuenca.\n\n  1523.        Bartolm\u00e9, El Maestro                         Jaen and Seville.\n\n  1555.        Barco, Alonso                                    Palencia.\n\n  1559.        Brabo, Hernando                                   Toledo.\n\n  1579.        Cela, Juan Tomas                                 Zaragoza.\n\n  1541.        Cesp\u00e9des, Domingo de                              Burgos.\n\n  1561.        Corral, Ruy Diaz del                              Toledo.\n\n  1604.        Celma, Juan Bautista                     Aragon y Galicia.\n\n  1692.        Conde, Sebastian                                  Seville.\n\n  1797.        Cordero, Fr. Josef                                Seville.\n\n  1537.        Delgado, Juan                                     Seville.\n\n  1540.        Domingo, Maestro                                  Toledo.\n\n  1518.        Elias, Joan de                                   Palencia.\n\n  1519.        Esteban, Maestro                                 Seville.\n\n  1494.        Frances, Juan                                     Toledo.\n\n  1555.        Herreros, Llorente                               Palencia.\n\n  1522.        Idrobo, Diego                                    Seville.\n\n  1524.        Juan, Fray                                       Seville.\n\n  1518.        Lopez, Juan do Urisarri                          Palencia.\n\n  1531.        Lemosin                                           Cuenca.\n\n  1518.        Mu\u00f1oz, Sancho                                     Cuenca.\n\n  1389.        Morey, Bartolomo                            Palma de Mallorca.\n\n  1555.        Moreno, Benegno                                  Palencia.\n\n  1565.        Pedro, Maestre                                   Palencia.\n\n  1533.        Palencia, Antonis de                             Seville.\n\n  1607.        Pe\u00f1afiel, Luis de                                 Toledo.\n\n  1510.        Prieto, Fernando                                 Seville.\n\n  1512.        Prelojero, Juan                                  Palencia.\n\n  1555.        Rodriguez, Gaspar                                Palencia.\n\n  1607.        Rodriguez, Bartolom\u00e9                              Toledo.\n\n  1607.        Silva, Francisco de                               Toledo.\n\n  1533.        Salamanca, Francisco de                          Seville.\n\n  1518.        Urisarri, Lopez                                  Palencia.\n\n  1561.        Villalpando                                       Toledo.\n\n  1518.        Yepes, Juan de                                   Seville.\n\n\n\n\nBRONZES.\n\n\nObjects of bronze of native origin are less frequently met with in Spain\nthan those of silver and iron. Many foreign artists worked at this\nindustry, the history of which only begins in the 16th century. The\nspecimens of bronze work made by the Moors will be described in their\nproper place. It is true that examples of bronze made by the Iberians\nand Romans are frequently met with in excavations; they consist chiefly\nof hatchets and other arms, bracelets, fibulas, etc., but they are\nsimilar in every respect to objects of the same kind found in other\nEuropean countries. During the Middle Ages, objects of enamelled gilt\nbronze were used on a very large scale in churches; specimens of a very\nhigh order may be seen in Spain, such as the splendid altar at San\nMiguel de Excelsis in Navarre; that at Santo Domingo de Silos, and the\nstatuette of the Virgin de la Vega at San Esteban, Salamanca. These\nobjects were, however, probably made at Limoges, or in some other\nlocality out of Spain. We hardly can trace any bronze of this period but\ncathedral bells. One was made at Puig-Valencia as early as A.D. 622,\nmelted in 1550, the only trace of the early work being the inscription\nand date, which are given by contemporary authors.\n\nAnother bell, about half a foot high, of A.D. 875, exists at the Local\nMuseum of Cordova, with the inscription: \"_Offert hoc munus Samson\nabbatis in domum Sancti Sebastiani martyris Christi, Era_ DCCCCXIII.\" We\nfind the name of \"_Joannes Calcena me fecit Anno Domini 1306_,\" on a\nbell at the cathedral of Valencia, and on one of Lerida was to be read:\n\"_Fecit factum per magistrum Joannem Adam Anno Dei 1418 in mense\nAprili_;\" the author appears to have been, by reference to the documents\nof the church, \"_de burgo Sanct\u00e6 Mari\u00e6 Turlensis diocesis regni\nFranci\u00e6_,\" for details see Villanueva, vol. ii., p. 147-152, xvi. 88.\n\nBefore entering into the renaissance period, I must mention some\nremarkable specimens which have reached us, the work of the Spanish\nArabs. Probably the most ancient and interesting is a sculpture\nrepresenting a stag, now at the Provincial Museum of Cordova, which\nevidently belonged to a fountain. It was found in the ruins of the\npalace of Medina Az Zahra, the construction of which corresponds to the\ntime of Abd er Rahman III., (961); it is undoubtedly work of this\nperiod, and probably belonged to this palace, so celebrated by Oriental\nwriters. The fragments of a bronze fountain and several lamps, at the\nMuseum of Granada, are of a later date; they were found in the\nexcavations of the early city of Illiberis, which was abandoned in the\n11th century, when its inhabitants chose the present site of Granada.\nThe fragments, which are supposed to have belonged originally to a\nfountain, consist of a small temple 22 inches high, of an hexagonal\nbase, with twelve small columns supporting bands of open work, frescoes,\ncupola, and turrets: in the angles are birds. There are six lamps, all\nof which are mutilated and incomplete; their form and object is\nindicated by the remains of chains which suspended them. Some of them\nare half melted, giving testimony to the conflagration which the mosque\nsuffered in the time of the Arabs. Other bronze fragments have been\nfound in the same locality. All these objects are artistic in their\ngeneral lines, but the workmanship is indifferent, and the ornamentation\nheavy and coarse.\n\nA bronze lion and a mortar, found some years since in the province of\nPalencia, are more artistic in form and general details. The mortar was\nfound near Monzon, and the lion not far from it. Remains still exist of\na castle, which was founded by the Arabs, in the locality, and which at\nthe beginning of the 11th century was in the hands of the Christians; it\nis highly probable that these objects belonged to its Moorish\ninhabitants. If we take into account how frequently we find Oriental\nremains in this locality, both these objects may be safely classified as\nbelonging to the 10th century, although there are details in their\nornamentation which appear rather to be of a later period. The lion is\n12\u00bd inches high by 14\u00bd inches long; an aperture in the lower part\nwhich communicates with the mouth of the animal, appears to suggest that\nit belonged to a fountain, the water of which issued, or not, owing to\nthe movement of the tail, which rotates and acts as a key. The shape of\nthis lion recalls those at the Alhambra; its forms are stiff and\nangular, in the conventional Oriental manner of reproducing animated\nbeings. The surface is covered with ornamentation, the mane is arranged\nin mannered and symmetrical curls, like those so common in Assyrian\nsculptures. On the back and two sides is the following inscription in\nCufic characters:\n\n[Illustration: Arabic \"Perfect blessing. Complete happiness.\"]\n\nThis object is similar in detail and the inscription to a bronze griffin\nat the cemetery of Pisa; it belonged to the collection of the painter\nFortuny, and at the sale in 1875 it was bought by Mr. E. Piot.\n\nThe mortar is circular in form, and is surrounded by twelve prismatic\nsections. Two lions' heads serve to support the hanging rings or\nhandles. The whole of the outside is most delicately carved with\narabesques, among which are to be met frequently figures of birds and\nquadrupeds. See \"El arte en Esp.,\" vol. iii., Madrid, 1864. The\nfollowing inscription in elegant Cufic characters runs round the upper\npart: it is repeated twice. \"Complete blessing, and ever-increasing\nhappiness and prosperity of every kind, and an elevated and happy\nsocial position for its owner.\" From the richness of the ornamentation\nof this object, it is probable that it was used to pound aromatic drugs.\n\nA bronze lamp of a similar ornamentation was found some years ago at\nCordova--it is not unlike in form and shape Roman lamps of the same kind\nmade of earthenware. It is engraved all over, and the subject of a dog\npursuing a hare appears often repeated. None of these objects of bronze\nwork are however equal in importance to a lamp made during the reign of\nMohammed III. of Granada, now at the Arch\u00e6ological Museum, Madrid [See\nPlate.] It is composed in its base of a body destined to hold the light,\nupon which is supported a large four-sided piece in the form of a\npyramid, which is completed with an octagonal body which surmounts it.\nFour graduated balls suspend it. The height of this lamp is 2\u00bd yards,\nand taking into account its general structure, it appears as if the\nchains or intermediate pieces are wanting, which probably existed on the\ninverted bell of the base. The manufacture and elegance of decoration of\nthis object is of a very high order, and may be compared with bronze\nwork of Damascus. The greater part of the pieces are covered with\npierced open work, and the motto of the Kings of Granada, \"There is no\nconqueror but God.\" In the lower part of the large pyramid, truncated in\nfour sides, is a long inscription in Arabian characters, stating the\nlamp to have been made by order of Mohammed III. year of the Hegira, 705\n(A.D. 1305).\n\nThe remaining objects of bronze work of Spanish-Moorish production are\nless important. They are interesting notwithstanding. The spherical\nperfume-burners which were used to roll on the pavement are highly\nartistic and might be adopted in the present day. One or two bronze\nbuckets of a well exist at the Madrid Arch\u00e6ological Museum, with some\nsmall objects of little importance found at Cordova. This artistic\nindustry has continued until the present day in Spain in the form of\nobjects of domestic use, such as brasiers, mortars and lamps, several of\nwhich preserve their ancient traditional form.\n\n[Illustration: MOORISH LAMP. ARCH\u00c6OLOGICAL MUSEUM, MADRID.]\n\nIn metal work as in architecture, a large number of instances exist in\nSpain in which the Christian and Moorish styles are blended--as an\nexample of this style may be mentioned the fine bronze gates of the\ncathedral of Toledo. These doors, which are 18 ft. high by 12 ft. wide,\nare covered on both sides with bronze plates; the outer side is\ndecorated with a geometrical Moorish design and small Arabic\ninscriptions, alternating with castles and a number of Arabic mottoes.\nIn one of the side bands and in the lower part may be read in Spanish\nthe following inscription:\n\n\"Estas puertas fueron acabadas en el mes de Marzo era de mil c ccc.\nsetanta e cinco a\u00f1os.\"\n\n\"These doors were finished in the month of March, in the era of 1375\nyears\" (A.D. 1337).\n\nThe fine doors of the cathedral of Cordova, del Perdon, are similar in\nstyle. They are made of wood and covered with bronze plating, and Gothic\nand Arabic inscriptions,--the word \"Deus,\" and\n\n[Illustration: Arabic]\n\n\"The empire belongs to God, all is His.\" Round these doors, alternating\nwith the arms of Castille and Leon, is the following inscription. \"Dia\ndos del mes de Marzo de la era del Cesar de 1415 a\u00f1os, (A.D. 1377.)\nReinante el muy alto et poderoso D. Enrique, rey de Castilla.\" These\ndoors were restored in 1539.\n\nThe Puerta del Perdon of the cathedral of Seville is similar in style,\nand a good example of moresque bronze work.\n\nReturning to the bronze work of the Christian artists of the renaissance\nperiod, the general rule which may be established with reference to\nstatues is that they have been made by foreign artists. This is the case\nwith the splendid groups of figures at the high altar and presbytery of\nthe Escorial by Pompeyo Leoni, the equestrian statue of Philip IV. by\nPedro Tacca, and others of a similar kind. One of the very few\nexceptions to this rule occurs in Bartolom\u00e9 Morel, a Spaniard, the\nauthor of the statue called the \"Giralda\" on the tower of the cathedral\nof Seville, the desk of the choir, and _Tenebrarium_ at the same church.\nThese objects were all by him, and are among the finest specimens of\nbronze work in Spain. The _Tenebrarium_ is described by Cean Bermudez in\nhis \"Descripcion artistica de la catedral de Sevilla,\" p. 129, in the\nfollowing manner: \"This object is better executed and more graceful in\ndesign than any of its kind in Spain. It consists of a triangular\ncandelabrum, which is used during matins in the three last days of Holy\nWeek with fifteen tapers, which are extinguished on reading each psalm.\n\n\"It was designed and executed by Bartolom\u00e9 Morel in 1562. Juan Giralte,\na native of the Low Countries, and Juan Bita Vazquez helped him to make\nthe statues which are at the head of this candelabrum, and Pedro\nDelgado, a sculptor of great renown, worked at the foot.\n\n\"It is 8 yards and a half high, and the triangular upper part is 3 yards\nwide: it is surmounted by fifteen statues which represent Our Saviour,\nthe Apostles and two other disciples or evangelists. In the vacant space\nof the triangle there is a circle ornamented with foliage, in the centre\nof which is a figure in high relief of the Blessed Virgin, underneath is\na medallion bust of a king. This centre is of bronzed wood, and is\nsupported by four small bronze columns, below which are four caryatides,\nresting on a nobly designed border ornamented with lions and other\nanimals in the renaissance style.\" See woodcut on next page.\n\n[Illustration: THE TENEBRARIUM IN THE CATHEDRAL OF SEVILLE.]\n\nVillalpando distinguished himself at the cathedral of Toledo in the same\nway as Morel at Seville. The splendid _reja_ at the Capilla Mayor with\nthe fine gilt bronze pulpits, the bas-reliefs of the doors of Lions,\n1564, the font, and railing surrounding the altar of the Virgin in the\ncoro were all made by him. The pulpits, made as it is asserted out of\nthe bronze sepulchre of Dn. Alvaro de Luna are octagonal, six of their\nsides are admirably decorated with bas-reliefs of exquisite work\ndivided by pilasters and terminated by a finely designed frieze. The\ndoor of Lions is covered in a similar manner with bronze plates\nornamented in the finest renaissance style; the knockers are models in\ntheir way. In the opposite door del Reloj, these reliefs have been\ncopied in 1713 by the silversmiths Zurre\u00f1o y Dominguez.\n\nNotwithstanding the great merit of these works, the stands for the choir\nbooks are even finer and more exquisite in detail. They are made of gilt\nbronze and represent subjects from the life of St. Ildefonso, the\nProphet David and the Apocalypse. They were modelled and designed by\nJuan Navarro in 1562, and carved by Nicolas de Vergara and his son some\nyears afterwards.\n\nTwo artists, natives of Aragon, were very famous during the same century\nfor their bronze work. One of these, Celma, made the pulpits of the\ncathedral of Santiago, which are finely ornamented with busts between\nthe columns, and bas-reliefs representing subjects of the lives of the\nsaints, and a fine design of leaves and flowers. In one of the\ninscriptions may be read: \"_Joannes Baptista Celma, Aragonentis patria\npingendi artifex salutis anno 1563. Compostell\u00e6 faciebat._\" The other\nartist was Cela, the author of the fine reja del coro of the church of\nthe Pilar at Saragossa, dated 1574-79.\n\nTwo gilt metal Monstrances of Spanish work of the 16th century are in\nthe South Kensington Museum. No. 4310, 57, represents an architectural\nshrine, decorated with strap and cartouche-work, columns, and arcades in\nthe renaissance style; it is dated 1537. See woodcut on next page. The\nother, No. 190, 66, has an ornamented stem, with knob, on which stands a\ntriangular shrine with kneeling angels; above is a smaller shrine\nsurmounted by a crucifix.\n\n[Illustration: SPANISH MONSTRANCE, DATED 1537. SOUTH KENSINGTON MUSEUM.]\n\nObjects of bronze work of the 17th century which are not by Fanelli,\nTacca, or other foreigners, are less numerous; most of them show signs\nof the decline of art at this period. The chiselled and openworked altar\nfrontal, at the burial-place of the kings of Spain at the Escorial, is\na very fine work of its kind. The shrine of the Sagrario de la Santa\nForma, also at the Escorial, is an excellent piece of workmanship--both\nthese objects were made by lay brothers of the convent. Friar Eugenio de\nla Cruz and Friar Juan de la Concepcion, silversmiths, worked during the\nreign of Philip IV. at chiselling these and other objects of bronze.\n\nThe exaggerations and bad taste, which were so common in every branch of\nartistic industry, were as prominent in bronzes--the bas-reliefs of this\nmetal which ornament the back of the high altar of the cathedral of\nToledo, at the \"Transparente,\" are generally mentioned as models of bad\ntaste. They are by Narciso Thom\u00e9; the architecture, painting, statues\nand carvings in marble, jasper and bronze are by the same person, as\nappears in the inscription which he placed there at its termination in\n1734.\n\nOn the accession of King Charles III. from Naples in 1759, these defects\nwere corrected, in part owing to the classical influence which became so\ngeneral in Europe, and continued to the early part of the present\ncentury. The silver manufactory of Martinez founded in Madrid under his\nauspices, has already been mentioned in treating of silver work. Objects\nof bronze of all kinds were made there in this classic style, and at the\nporcelain manufactory of Buen Retiro, where splendid tables were made of\npietre dure, mounted in bronze. The most important objects there\nproduced may be seen at the chapel of the royal palace of Madrid, and\nthe palaces of Aranjuez and the Escurial.\n\n\n\n\nARMS.\n\n\nThose who have a taste for the study of prehistorical monuments in\nSpain, will find a very extensive collection of arms and other utensils\nof this period at the Mus\u00e9o Arqueologico, Madrid, which have been found\nin different localities of the country. They chiefly consist of\nhatchets, knives, and lance and arrow points made of flint, and are\nsimilar in form to those which have been discovered in the north and\ncentre of Europe. There are therefore, constant analogies between the\nimplements used by the primitive Iberians, and those of other European\nraces of the prehistoric period.\n\nThe arms and weapons of the following age, known as the age of bronze,\nare less common in Spain, although a few specimens exist in the Mus\u00e9o of\nMadrid, and the \"Academia de la Historia.\" They chiefly consist of\nswords, daggers, and lance and arrow points. The hilts are formed of the\nsame material as the blade in some instances, in others they consist of\na blade fitting into a wooden or ivory handle. The blades are cut on\nboth sides; they are straight, and finished off in a point; in the\ncentre runs a thick rim made for the purpose of strengthening the blade,\nand that it should not bend. One of these specimens measures 30 inches\nlong. The daggers are about one-third smaller, with the same rim in the\ncentre; the bronze hilt is nailed to the blade, and the lance and arrow\npoints are of the same kind. These weapons are similar in style to the\nnumerous examples found in other countries, especially England. The\nhatchets are also the same, their manner of being fixed on to their\nhandles and their size and weight being identical with the English ones.\nOne of the few specimens which varies from this general rule, is a\ndagger, with a cylindrical horn-like hilt, belonging to S^{r}. Villaamil:\nit was found at Galicia with other interesting arms. (Consult Mus\u00e9o\nEspa\u00f1ol de Antiguedades, iv. 63.)\n\nThe probable antiquity of these weapons always remains a matter of\ndoubt; this is especially the case with stone implements. When on their\ndiscovery the geological structure of the soil has been studied, some\nevidence exists on which to ground an opinion; but this is very seldom\nthe case in Spain, and Messrs. Prado, Villanova, Botella, and\nMacPherson, are among the few and most trustworthy geologists who have\ninvestigated this subject. With reference to bronze implements, Spanish\nauthors have a tendency to attribute them to prehistoric times. I see no\nreason to justify this opinion, and it appears to me that when this\nsubject has been studied in a more satisfactory manner, it will appear\nthat these arms were used during the Roman domination at the same time\nas the iron ones.\n\nIn reaching the Iron period, already within historical times, we find\nthat the Spanish people, when once under the Roman rule, used the well\nknown arms and utensils which will be found reproduced in all elementary\nbooks on arch\u00e6ology. In this, as in every other sphere, the Romans\nimposed their civilization on the races they conquered. Some important\nexceptions, however, exist to this general rule. The Spanish swords must\nhave been excellent in quality, since the Romans adopted them after the\nCarthaginian war: they were however, never able to imitate the manner in\nwhich they were tempered. Suidas says: _Romani patriis gladiis depositis\nHannibalico bello Hispaniensium assumpserunt ... sed ferri boni atem et\nfabrica solertiam imitari non potuerunt._ The locality where the best\nswords were made in Spain was Bilbilis (Calatayud, province of Aragon),\nthe birthplace of the poet Martial, who, when praising the waters of the\nriver Jalon and the excellence of its quality for tempering metals,\nsays, _Salone qui ferrum gelat._ These swords were wide, and cut on both\nedges; their points were sharp, and all of them had in the centre signs\nof a central groove running down its length. Several specimens may be\nseen at the Museo Arqueologico, Madrid, which are 15 or 19 inches long.\nThe Spaniards used also swords of another form, which were known by the\nname of _falcata_, from falx, sickle; the blade was curved, and it has\nbeen supposed to be the genuine Spanish model. The blade was widened in\nproportion from the hilt to the point, which was very sharp: it cut like\na sickle in the interior curve, and only a small part was sharpened in\nthe opposite side. The best specimen, which exists in the Madrid Museum,\nis 22\u00bd inches long. The weapons made at Toledo must have been very\nfamous during the Roman period: Gracio Falisco, a poet of the time of\nJulius C\u00e6sar, says, in mentioning them: _Ima toletano pr\u00e6cingant ilia\ncultro_; the sword manufactory there attained afterwards great\nimportance.\n\nI must end by mentioning the _funda_, or slings, and leaden plummets\n(_glandes_), which were thrown so dexterously by the slingers of the\nBalearic Islands. The slingers of this province are mentioned with great\npraise by ancient authors. (V. Smith, Dictionary of Greek and Roman\nAntiquities. _Funda._)\n\nDuring the Visigothic domination, some modifications must have been\nintroduced in the form and number of the arms used, probably owing to\nthe tradition of the Gothic race, and the Byzantine or Oriental\ninfluences, which they accept in every sphere; but the names and\nexplanation of these weapons, given by San Isidoro in his \"Etimologies,\"\nlib. xviii., indicate that the Roman system was adopted as a general\nrule. The chief modification which may be established for the Visigoths,\nand the greater part of the European races, is that from their time the\nbreast-plates and metallic pieces, which served as a protection for the\nlegs and arms, fell into disuse, and were substituted by coats of mail\nor chain armour. This is deduced from San Isidoro's not describing\nthese metallic plates; although he gives numerous details on different\nsorts of coats of mail, including those made of coarse stuffs woven in\nSilisia. At this time, the use of planks or boards to protect the body\nwas abandoned, and did not prove acceptable until the end of the 13th\ncentury, when they again became very generally adopted in Spain.\n\nDuring the 13th and 14th centuries, the Spanish Christians continued to\nuse the same weapons as in other European states. The history of Spanish\narms, in like manner with its artistic history and civilization, is\nsimilar to that of other nations; it frequently occurs that objects of\nthis kind are classified with those of other countries, the only\nexceptions to this general rule being those which result from the\nMoorish conquest and influence.\n\nThe Arabs, after they took possession of Spain at the beginning of the\n8th century, imported, with their arts and industries, special arms and\nweapons, the greater part of which were copied from Persian models;\ntheir swords, helmets, and shields deserve special attention; in other\nweapons the difference is not so great; and, indeed, in the two first it\nis chiefly confined to their decoration.\n\nNone of the arms made by the Moors have been so justly celebrated, or so\ndelicately worked as their swords. An Arabic author, El Camus, says the\nArabs had as many as 1000 names to designate swords. In the first years\nof the Hegira, their historians praise the swords made at Yemen, and in\nIndia; later on, those of Syria; Damascus was, however, the great centre\nwhere arms of all kinds were made. Several interesting Arabic\nmanuscripts are known on this subject, such as the _Treatise on steel\nblades_ in the library of Gotha, that on _Different kinds of arms, with\nthe properties of lances, swords, and horses_, in the Library of Leyden,\nand numerous articles by Oriental scholars, which have appeared in the\n\"Journal Asiatique,\" and other reviews. The manufacture in Syria decayed\nin the 15th century; other centres gained in importance, specially those\nin Egypt, Morocco, and Spain, which had rivalled the East in this\nindustry during the Middle Ages.\n\nThe Arabs introduced their forms and manner of decorating arms when they\ninvaded the Peninsula; it is, however, highly probable that the\ntraditions still existed at Bilbilis and Toledo of the manner of\ntempering steel. It is known that Abd-er Rahman II. (A.D. 822-852)\nreformed the manufacture of arms at Toledo, and that in A.D. 965, Al\nHakem II. sent a rich present of specimens made in the locality to Don\nSancho, King of Leon. Notwithstanding the high reputation of the\nindustries of Cordova, the great centre and court of the Spanish Arabs\nduring the earlier period of their domination, \"it never became famous\nfor its working steel,\" as Fernandez Gonzalez tells us in his study on\nSpanish moresque swords, from which much of this information has been\nderived. (V. Mus. Esp. de Antiguedades, V. I. and V.). Almeria, Murcia,\nSeville, and Granada were greatly distinguished in this manufacture\nduring the domination of the Arabs in Spain.\n\nWe know that Almeria, during the 12th and 13th centuries, \"was also\nfamous for the fabrication of all sorts of vases and utensils of iron,\ncopper, or glass.\" (Mohamadan Dynasties in Spain, by Al-Makkari, vol. i.\np. 51.) Abou Said, in writing in the 13th century of Murcia, says,\n\"Objects are made there of latteen and ron, consisting of knives and\nscissors, with an ornamentation in gold. Other similar utensils, proper\nfor the outfit of a bride, or a soldier, in such large quantities, that\nthe mere thought of it confuses the imagination.\"\n\nThe same author, when speaking of the swords of Seville, says, \"The\nsteel which is made at Seville is most excellent; it would take too much\ntime to enumerate the delicate objects of every kind which are made in\nthis town.\" These industries must have continued in the hands of the\nMoors after the town was conquered by Spaniards in the 13th century. In\nthe following century we find in the will of King D^{n}. Pedro, \"I also\nendow my son with my Castilian sword, which I had made here in Seville,\nornamented with stones and gold.\"\n\nNo specimens of Hispa\u00f1o moresque swords exist previous to the 15th\ncentury. We possess some highly interesting examples of this period,\nall, or the most part, of which were made at Granada, the last centre of\ncivilization of the Spanish Arabs. The most important are the sword,\ndagger, and double-handed sword and knife, which, with the authentic\ncostume, belonging to Boabdil, the last king of Granada, are now in the\npossession of the Marquis of Villaseca, at Madrid. These objects were\ngained by an ancestor of the marquis, who took Boabdil prisoner in 1482\nat the battle of Lucena. In compliance with the chivalrous practice of\nthe time, the arms of the conquered king went to the conqueror, and have\nbeen kept as heirlooms since that time.\n\nThe sword of Villaseca is 39 inches long, this includes 12 inches of the\nhilt. (See woodcut.) \"The steel blade,\" says S^{r}. Fernandez, in his\narticle, vol. v. p. 395, \"is of a later date, and appears to have been\nadded to the sword after the older one had disappeared. It is a Toledo\nblade, marked with the letter S, similar to the one used by Alonso\nSahagun the elder, and with a hollow line in the centre.\"\n\nI am of opinion, however, that it may be the original blade, for the\nother sword, which still exists at Granada, of the same kind, has\nlikewise a blade marked T, Toledo. This coincidence appears to suggest\nthat those made there were preferred, and in both instances they exactly\nfit the sheath, which is the original one.\n\n[Illustration: Arabic]\n\n[Illustration: Arabic]\n\nThe hilt of this fine sword is formed of solid gold, enamelled in blue,\nwhite, and red. This decoration runs along the pommel and cross bars.\nThe axle is made of ivory carved with the utmost skill. Two octagons are\non each side, with the following inscription in semi-Cufic letters: (may\nyou) \"obtain your object.\" On the other side: \"in saving his life.\"\n\n[Illustration: SWORD OF BOABDIL. VILLASECA COLLECTION, MADRID.]\n\nFour shields surround in the upper part this axle, which contains the\nfollowing inscription in rather illegible characters:\n\n[Illustration: Arabic]\n\n\"In the name of God, power belongs to him, there is no other divinity\nbut he, happiness proceeds from God alone.\"\n\nIn the lower part in similar shields we read:\n\n[Illustration: Arabic]\n\n\"Miracles belong to God, for certainly the ignorant do not know God at\nfirst, for it is their habit to err.\"\n\nOn the pommel is the following inscription:\n\n[Illustration: Arabic]\n\nSay, \"He alone is God, Eternal God, who neither created, nor was\nengendered.\"\n\nUnder the pommel, on green enamel, appear the following letters:\n\n[Illustration: Arabic]\n\nSay, \"The only God, Eternal, not ...\"\n\n[Illustration: Arabic]\n\nOn the other side: \"was neither created, nor engendered, and has no\nequal.\"\n\nOn a band which appears under the axle, on enamel, are the following\nletters:\n\n[Illustration: Arabic]\n\n\"God is clement and merciful,\" and on the other side:\n\n[Illustration: Arabic]\n\n\"God is gifted with the best memory.\"\n\nThe learned archaeologist and orientalist, D^{n} Pascual de Gayangos, is\nof opinion that this sword was worn hanging round the neck, between the\nshoulder blades. A small bag, _tahali_, probably hung also from the\nleather girdle, which still exists in the collection of Villaseca.\n\nThe _montante_, or double-handed sword, has an iron cylindrical hilt\ninlaid with ivory. On it appears the motto of the kings of Granada:\n\n[Illustration: Arabic]\n\n\"God alone is the conqueror.\"\n\nThe blade, part of which is wanting, is marked with the crescent.\n\nThe dagger is superior in artistic merit. The hilt is made of iron,\nornamented with ivory delicately engraved in arabesques. The blade is\ndamascened in gold, with inscriptions on one side, repeating:\n\n[Illustration: Arabic]\n\n\"Health, permanent glory, and lasting happiness [belong to God\"].\n\nOn the opposite side:\n\n[Illustration: Arabic]\n\n\"It was made by Reduan.\"\n\nThe scabbard of this dagger is most beautiful, its chapes are made of\nsilver enamelled in green, and the remainder is of crimson velvet\nembroidered in gold, from which hangs a fine tassel of silk and gold\nthread. A small eating knife is fitted into this same scabbard, which\npossesses no artistic interest.\n\nThe woodcut on the next page will give a good idea of the blade of a\nMoorish dagger of the 15th century.\n\nA sword of a similar kind to the one already described may be seen at\nGranada at the _Administrador's_ of the Generalife. It belongs to the\nmarquis of Campotejar, a descendant of Sidi Jahia, a Moorish prince who\nwas converted to Christianity. The marquis of Vega de Armijo has an\ninteresting sword of the same kind, and two others exist at the Museo de\nArtilleria, Madrid, which belonged to Aliatar; Boabdil's sword is at the\nRoyal Armoury. The Hispano Arab sword, which for centuries had been in a\nsaint's hand at the church of San Marcelo, Leon, is now at the Museo\nArqueologico, Madrid.\n\nThe Spanish Moors used helmets similar in form to those of the\nChristians, though their manner of decoration was different; after the\nimportations made by the Crusaders, the similarity must every day have\nbeen greater. The _almofar_, which appears by its name to have been\noriginally oriental, and which is constantly named in Spanish documents\nfrom the poem of the Cid, was a protection for the head, of a similar\nform to those used in France and other countries, consisting of a hood\nmade of chain armour, covering the head and leaving the face free; upon\nit was placed the hood or helmet. Some helmets exist at the Royal\nArmoury of Madrid of Spanish Moorish origin; among them are two very\nremarkable ones, which have been attributed to Boabdil, the last king of\nGranada. (Nos. 2345, 2356, of the Catalogue.) They are decorated with\ngold filigree, niellos, and geometrical ornamentation in the best\noriental style, most admirably worked, and different in this respect to\nthe helmets used in this time. Another interesting helmet, which\nbelonged to this ill-fated monarch, exists in the province of Almeria.\n\n[Illustration: MOORISH DAGGER, 15TH CENTURY.]\n\nThe _adargas_, or shields, are more varied: they were frequently adopted\nby the Christians. They were generally round, with a salient point in\nthe centre, _ombilicus_, or a sort of iron grating made for the purpose\nof entangling the adversary's sword. These shields were of wood or thick\ncowhide, _vacaries_, and were decorated outside in a variety of ways,\nsometimes with pierced iron plates or bands of leather, forming\narabesques; and at other times with an ornamentation of iron, and\nleather embroidered with gold and silver, with rich hanging tassels and\npendants. Shields of a prolonged form were also very constantly used,\nterminating in a semicircle in the upper part, and in the lower by a\nsharp point or a semicircle at the top and bottom, as may be seen in the\npaintings of the Sala de la Justicia at the Alhambra. A good collection\nof round shields may be seen at the Armoury at Madrid: they are not\nearlier in date than the 15th century, the most interesting among them\nare Nos. 233, 253, 389, 595, and 607 of the Catalogue. Some of them are\nexquisite in work and detail. These shields, although belonging to the\nlatest period of the middle ages, were used before this time, for they\nappear on the ivory casket existing at the cathedral of Pamplona, [V.\nIvories, p. 130], dated A.D. 1005, and in miniatures of Spanish MSS. of\nthat date. One of the best examples, which may be mentioned to confirm\nthese indications, will be found in a MS. at the British Museum (Add.\nii., 695), which was painted during a period of twenty years in the\nmonastery of Silos, near Burgos, and finished and completed A.D. 1109.\nMr. Shaw, in his \"Dresses and Decorations,\" vol. v., reproduces some of\nthese figures, and says: \"The figures which form our plate represent\nSpanish warriors of the later part of the 11th century, and are\ninteresting on account of their remarkable resemblance to the\nAnglo-Norman soldiers on the celebrated Bayeux tapestry. This\nresemblance is observable in the style of the drawing, as well as in the\ncostumes. It is highly probable that the military habits of this period\nwere borrowed from the Saracens. This supposition is strengthened by the\nfact that Arabic inscriptions in Cufic letters are found among the\nornaments of the several robes still preserved which belonged to German\nand Frankish barons of the 10th and 11th centuries. One peculiarity of\nour Spanish warriors is the round shield with the elegant ornaments on\nthe disc.\"\n\nThese may be considered the chief varieties of Spanish arms in the\nMiddle Ages. In other instances the French and Italian forms are\nadopted. The manufacture of arms was not reduced then as at a later date\nto the monopoly of the Government or to determined localities. Besides\nthe arms made at Toledo, those of Seville, Granada, Valencia, Zaragoza,\nBarcelona, the Basque provinces, and Cuellar, were very famous. Readers\nof Shakespeare will remember Falstaff's _bilbo_, a rapier made at Bilbao\nin the Basque provinces. The principal merit of these arms consisted in\nthe manner in which the artist tempered his metal. It was generally done\nat night in order to distinguish in the darkness the exact colour of the\nheated steel in dipping it into the water. The swords known by the name\nof _perrillo_ were highly esteemed in Spain during the 15th and 16th\ncenturies; they were marked with a figure resembling a dog. Cervantes\nmentions and praises these blades in his \"Rinconete y Cortadillo\" and\n\"Don Quixote.\" They were made by a Moor of Granada, who it is stated had\nbeen swordsman of King Boabdil, and became a Christian under the name of\nJulian del Rey. His godfather was King Ferdinand, and besides working at\nGranada he did so at Zaragoza and Toledo. When an artist of merit\nexcelled in a given locality, it absorbed all the fame of this industry.\nDuring the Renaissance larger centres of this industry were established\nin large towns, and the fame of the objects they produced was\nconcentrated there. Toledo absorbed the importance in the industry of\nsword making.\n\nWe find that the Municipal Ordinances of the Middle Ages give very\nlittle information by which we can judge of the merits of the objects\nproduced. A Guild of Armourers existed at Barcelona as early as 1257,\nand of Sword makers from the 14th century; but the information given\nconcerning this subject is very slight. We find more details in the\nOrdinances of Toledo, Seville, and Granada, whence we learn that\nmanufactories of arms existed in these towns.\n\nThe sword manufactory of Toledo acquires its greatest importance during\nthe Renaissance period until the end of the 17th century, when it\nterminated. It was re-established again in 1760 under the patronage of\nthe Government, and continues to work in the present day. Don Francisco\nde Santiago Palomares wrote, in 1772, an \"Account of the Sword\nManufactory of Toledo,\" the manuscript of which exists at the Academia\nde la Historia (E. 41). The most interesting part of this study consists\nin the details given upon the manner of tempering the blades. The names\nhe has collected of the most remarkable artists, are taken from the\noriginal dies of their marks which existed at the Archives of the\nAyuntamiento at Toledo. These marks have been published in the \"Catalogo\nde la Armeria,\" Madrid, 1849, from which I copy them. It must be borne\nin mind that Palomares, after quoting the text by the Poet Gracio\nFalisco, which I have given at the beginning, does not establish any\ndefinite fact which enables us to study the historical progress of this\nmanufactory during the Middle Ages. The author says, p. 111: \"At the\nbeginning of this industry there was no centralization or monopoly in\nthis manufactory, some armourers formed a guild, placing each artist in\nhis proper place. The kings of Castille granted them privileges of\ndifferent kinds.\"\n\nBowles says, in his \"Introduction to the Natural and Geographical\nHistory of Spain,\" that the steel used at the manufactory of Toledo was\ntaken from an iron mine existing at a league from Mondragon, the only\none then known in Spain.\n\nThe celebrity of Toledo blades has excited the curiosity of many who\nwished to ascertain the cause of their great excellence and renown. Some\nsupposed the sword manufacturers of Toledo possessed a secret for\ntempering their arms: it was not so, however; their only secret being\nthe waters of the Tagus and the fine white sand on its banks. This sand\nwas used for cooling the steel: when the steel was red-hot, and began to\ngive forth sparks, it was uncovered a little and sprinkled with sand,\nand sent on to the forgers. As soon as the blade was ready it was\ntempered in the following manner: a line of fire was made and the blade\nplaced in it in such a manner that only four-fifths of its length should\ntouch the fire. As soon as the blade was red-hot it was dropped\nperpendicularly into a bucket of Tagus water; when cold, if it was found\nto be bent, a small portion of sand was poured on the yoke, the blade\nwas placed upon it, and beaten until properly straightened. After this\nthe fifth part of the blade was fired, and when red-hot was seized with\ntongs and rubbed with suet, which soon began to melt; after this the\nblade was sent to the grinding-stones, and finished by being polished by\nwooden wheels with emery powder.\n\nCharles the Third, a year after he became King of Spain, re-established\nofficially the manufactory of arms of Toledo. He placed it in a\nbuilding near the Miradero Alto, and the works began in 1761. The king\nsoon found the building too small for the purpose, and ordered his\narchitect, Sabatini to build the present one, outside the town near the\nriver Tagus, which was finished in 1783, and from this time has been\nunder the superintendence of the Royal Artillery. Not a single sword\nmaker existed in 1760 of any note who was competent to be placed at the\nhead of the works, and it was found necessary to bring a proper person\nfrom Valencia.\n\nPalomares, who was present, says: \"As soon as the building was ready and\ndisposed for working, Luis Calisto, a famous sword maker, began to work.\nCalisto was a native of Valencia, and more than 70 years of age when he\nwas appointed. Other artists were chosen at the same time by the\nDirector. In the short space of time in which that chief master armourer\nlived, he made most excellent weapons; he was most skilful, and was\nprobably imitated by his successor.\"\n\n\nNAMES OF THE SWORD MAKERS OF TOLEDO.\n\nThe numbers given correspond to the plates of their marks, given in\n\"Catalogo de la Real Armeria.\"\n\n     1. Alonso de Sahagun, el _viejo_, 1570.\n\n     2. Alonso de Sahagun, el _mozo_.\n\n     3. Alonso Perez.\n\n     4. Alonso de los Rios; he also worked at Cordova.\n\n     5. Alonso de Cava.\n\n     6. Andres Martinez.\n\n     7. Andres Herraez; he also worked at Cuenca.\n\n     8. Andres Munesten; worked at Calatayud.\n\n     9. Andres Garcia.\n\n     10. Antonio de Baena.\n\n     11. Antonio Gutierrez.\n\n     12. Antonio Gutierrez, a son of the former.\n\n     13. Antonio Ruiz, 1520; he used the initial letter of his name.\n\n     14. Adrian de Zafra; worked at Sn. Clemente.\n\n     15. Bartolom\u00e9 de Nieva.\n\n     16. Casaldo y Campa\u00f1eros; worked at Cuellar and Badajoz.\n\n     17. Domingo de Orozco.\n\n     18. Domingo Maestre, el _viejo_.\n\n     19. Domingo Maestre, el _mozo_.\n\n     20. Domingo Rodriguez.\n\n     21. Domingo Sanchez; called el Tigerero.\n\n     22. Domingo de Aguirre.\n\n     23. Domingo de Lama.\n\n     24. Domingo Corrientes; worked also at Madrid.\n\n     25. Fabrian de Zafra; hijo de Adrian.\n\n     26. Francisco Ruiz, el _viejo_, 1617.\n\n     27. Francisco Ruiz, el _mozo_.\n\n     28. Francisco Gomez.\n\n     29. Francisco de Zamora; worked also at Seville.\n\n     30. Francisco de Alcozer; worked at Madrid.\n\n     31. Francisco Lurdi.\n\n     32. Francisco Cordiu.\n\n     33. Francisco Perez.\n\n     34. Giraldo Reliz.\n\n     35. Gonzalo Simon, 1617.\n\n     36. Gabriel Martinez.\n\n     37. Gil de Almau.\n\n     38. Hortu\u00f1o de Aguirre, 1604.\n\n     39. Juan Martin.\n\n     40. Juan de Leizalde; worked at Seville.\n\n     41. Juan Martinez, el _viejo_.\n\n     42. Juan Martinez, el _mozo_, 1617.\n\n     43. Juan de Almau, 1550.\n\n     44. Juan de Toro.\n\n     45. Juan Ruiz.\n\n     46. Juan Martinez de Garata.\n\n     47. Juan Martinez Menchaca; he lived at the beginning of the 16th\n     century, and worked at Lisbon, Seville, and Madrid.\n\n     48. Juan Ros.\n\n     49. Juan Moreno.\n\n     50. Juan de Saludo.\n\n     51. Juan de Meladoria.\n\n     52. Juan de Vargas.\n\n     53. Juan de la Horta, 1545.\n\n     54. Juanes de Toledo.\n\n     55. Juanes de Algruniva.\n\n     56. Juanes de Muleto.\n\n     57. Juanes, el _viejo_.\n\n     58. Juanes de Uriza.\n\n     59. Julian del Rey, the Moor; he worked for Boabdil, King of\n     Granada, 15th century.\n\n     60. Julian Garcia; worked also at Cuenca.\n\n     61. Julian de Zamora.\n\n     62. Jos\u00e9 Gomez.\n\n     63. Jusepe de la Hera, el _viejo_.\n\n     64. Jusepe de la Hera, el _mozo_.\n\n     65. Jusepe de la Hera, el _nieto_.\n\n     66. Jusepe de la Hera, el _visnieto_.\n\n     67. Jusepe del Haza.\n\n     68. Ignacio Fernandez, el _viejo_.\n\n     69. Ignacio Fernandez, el _mozo_.\n\n     70. Luis de Nieves.\n\n     71. Luis de Ayala.\n\n     72. Luis de Belmonte.\n\n     73. Luis de Sahagun.\n\n     74. Luis de Sahagun.\n\n     75. Luis de Nieva; worked at Calatayud.\n\n     76. Lupus Aguado, 1567.\n\n     77. Miguel Cantero, 1564.\n\n     78. Miguel Sanchez.\n\n     79. Melchor Suarez; worked at Lisbon.\n\n     80. Nicolas Hortu\u00f1o de Aguirre, 1637.\n\n     81. Pedro de Toro.\n\n     82. Pedro de Arechiga.\n\n     83. Pedro Lopez; worked at Orgaz.\n\n     84. Pedro de Lerzama; worked at Seville.\n\n     85. Pedro de Lazaretea; worked at Bilbao.\n\n     86. Pedro de Orozco.\n\n     87. Pedro de Belmonte.\n\n     88. Roque Hernandez.\n\n     Sarabal; used no mark.\n\n     89. Sebastian Hernandez, el _viejo_, 1637.\n\n     90. Sebastian Hernandez, el _mozo_; he worked also at Seville.\n\n     91. Silvestre Nieto.\n\n     92. Silvestre Nieto.\n\n     93. Tomas de Ayala, 1625.\n\n     94. Zamorano, el Toledano.\n\nNos. 95, 96, 97, 98, and 99, are the marks of sword makers whose names\nare unknown.\n\n     Miguel y Manuel Fernandez; worked at Toledo in 1786.\n\n     Pedro de Barreta; at Bilbao at the end of the 16th century.\n\nThe following names may be added to this list--\n\n     Luis Calisto, 1760; and the four following artists are mentioned by\n     Palomares as working at Toledo.\n\n     Juan de Valladolid.\n\n     Sebastian Herb\u00e1s, 1617.\n\n     Manuel Ruiz, 1700.\n\n     Ignacio Fernandez, 1708.\n\n     Juan Orenga, 15th century; worked at Tortosa, and mentioned in the\n     \"Corvacho\" of the Arcipreste de Talavera, fol. xi.\n\n     Martin Garro, v. p. 120, \"Almanaque de la Industria.\"\n\n     Alonso Gutierrez, master of sword making at Madrid in 1625.\n\nThis artist, and the eleven following, are mentioned in original\ndocuments at the Academia de San Fernando.\n\n     Francisco de Borja, sword maker and gilder; Madrid, 1634.\n\n     Gaspar Martin, sword maker; working at Madrid in 1637.\n\n     Juan Mazon de Santorcas, sword maker and gilder; Madrid, 1636.\n\n     Juan de Medina; Madrid, 1620.\n\n     Lorenzo de los Rios; Madrid, 1585.\n\n     Miguel de Berrio; Madrid, 1575.\n\n     Pedro Casado, sword maker; Madrid, 1636.\n\n     Francisco de Elias, an apprentice of Pedro Casad\u00f3s.\n\n     Francisco de Salinas; Madrid, 1636.\n\n     Acheza----; Toledo.\n\n     Camilo, 1500.\n\nTwo fine rapiers are in the Kensington Museum, Nos. 626, '68, and 2214,\n'55, which belong to the 16th and 17th centuries, and give a good idea\nof this industry of Toledo. Woodcuts of these appear on the opposite\npage. One of the blades is marked with the name of Francisco Ruiz. A\nlarge and most important collection of rapiers, swords, and other arms\nexists at the Royal Armoury of Madrid.\n\nThe breastplates are unusually fine, but the finest among them are by\nMilanese artists. A good example, which has been considered Spanish, was\nformerly in the Bernal Collection (see woodcut on p. 98); it gives an\nexcellent idea of the artistic importance attained by this industry.\n\n[Illustration: SPANISH RAPIERS. SOUTH KENSINGTON MUSEUM.]\n\nThe saddles used in Spain were similar to those of other European\ncountries, for we find the same names adapted to the Spanish language.\nIn the poem of the Cid, 13th century, and other works of the Middle\nAges, _gallega_ and _barda_ saddles are mentioned which were peculiar to\nSpain. The _gallega_ saddles were called so from being made in Gallicia;\nthe _barda_ proceeded from the Moors. We find no details which explain\ntheir exact structure.\n\n[Illustration: BREASTPLATE, FORMERLY IN THE BERNAL COLLECTION.]\n\nA very remarkable collection of saddles of every description may be\nstudied at the fine collection of armour at Madrid; among them are two\nof special interest--No. 2311 of \"Catalogue of Armoury\"--traditionally\nsupposed to have belonged to the Cid, and No. 1310, which was used by\nKing James the Conqueror in the 13th century. The Cid's saddle (see\nwoodcut) is intended for a war-horse; its bows are covered with black\nmetal plating, ornamented with a design of leaves and pilgrims' shells,\npartially gilt, and the word \"Fides,\" considered to be the Cid's\ndevise. Much has been written on the origin and authenticity of this\nsaddle. Cervantes alludes to it in the 49th Chapter of \"Don Quixote,\"\nPart I. For further details see \"Catalogue of the Armeria.\"\n\n[Illustration: SADDLE OF THE CID. ARMOURY, MADRID.]\n\nThe greatest novelty which was introduced during the Renaissance period\nwas the use of fire-arms. The only names of ancient artists that I have\nmet with, who had any connection with the manufactory of large pieces of\nartillery were Pedro Burgues, an iron master of Barcelona, who lived in\n1393; Rodrigo de Almanza, and Pedro Colomer, who made fire-arms at\nBarcelona in 1413; and the Moor, Alfarax Darhin, an iron master of\nTarazona; V. Fernandez Duro. [Museo Espa\u00f1ol, vol. v. p. 18.]\n\nPortable fire-arms were introduced into Spain towards the middle of the\n15th century. Count Clonard, in treating this subject in the \"Memorias\nde la Academia de la Historia,\" ix. p. 214, says:--\"Alfonso de Palencia\ntells us the _espingarda_, or large hand musket (see woodcut), and\n_arbatana_ culverin, were used in the insurrection at Toledo in July,\n1467. He assures us they had been very recently invented and named. They\nhad been mentioned as early as 1449. In the siege of Toledo, Don Alvaro\nde Luna could not be kept back from the dangerous position in which he\nhad placed himself, notwithstanding the shower of stones, rockets,\narrows, and shots of _espingardas_ which were thrown into the town.\"\n\n[Illustration: SPANISH MUSKETS.]\n\nAt the same time as these portable fire-arms were used in Spain, the\nancient _ballestas_, crossbows, so generally used in the Middle Ages,\nwere very popular. Several artists became very famous in their\nmanufacture. Alonso Martinez de Espinar enters into many details of this\nimplement in his rare volume on the \"Arte de Ballesteria y Monteria,\"\nMadrid, 1644, 4to, from which I copy the following passage, p. 11:--\"The\ncrossbow is more secure and less dangerous than the arquebuse; for it\nhas never been known that a man's life has been lost by breaking the\nstring or cord, two things which are dangerous, but not to a\nconsiderable extent. The crossbow has many advantages over the\narquebuse: it kills, but does not frighten game; this cannot be done\nwith the arquebuse, for the sound it makes alarms and frightens the\ngame, and is heard everywhere. Once set, its shot is secure; which is\nnot the case with the arquebuse, which often misses fire.\n\n\"This weapon has been used in Spain from very early times, and has been\nmade by the best masters; the most famous among them were--\n\n    Azcoitia el Viejo.\n    Pedro de la Fuente.\n    Christoval de Azcoitia.\n    Juan Hernandez.\n    Juan Perez de Villadiego.\n    Juan Azcoitia.\n    Vzedo.\n    Hortega.\n\n\"The only famous maker of crossbows of the present time is Juan de\nLastra.\"\n\nEspinar does not mention the name of Miguel, a crossbow maker of\nZaragoza, or another celebrated maker who lived at Barbastro. The\nambassador Salinas alludes to them both in an hitherto unedited letter\nwritten to King Ferdinand of Hungary, the brother of the Emperor Charles\nV. He says:--\"Before I was able to find a lodging in this town of Monzon\n(Aragon), which was about five or six days, I went to Balbastro,\nBarbastro, and there occupied myself in making a pair of crossbows for\nyour Majesty. I believe they are so made that they will satisfy the\ndesires which were required; they were made in my presence and according\nto my wishes, and as your Majesty is annoyed when they do not go off as\nyou wish, I determined to make them in the following manner: One of them\nweighs l. ounces, which is considered here a very great weight. I,\nknowing your Majesty's wishes, had another made of lvi. ounces, that is\nto say 3\u00bd lbs. of Castille, carved and worked in such a manner that\nthe cords should not break; and that it be pleasant to shoot, I can\naffirm your Majesty that the maker is the best which exists in Spain,\nand yet he has never made so fine a crossbow as this one which is now\nsent to your Majesty; and if not to your liking, I do not expect you\never will be pleased with one. I have sent them to Victoria in order\nthat the screws may be made to fit the cords, and have ordered them to\nbe made without delay; and as soon as they are ready they are to be sent\nto Bilbao to be forwarded to Flanders to the Queen, to whom your Majesty\nmust write, in order that whatever you may wish may be done with them.\nWhen we leave here, it is said, we are to go to Zaragoza, where your\nMajesty will stay ten or twelve days. I will order two crossbows from\n_Maestro Michel_, who competes with the maker at Barbastro. I will do\nthe same with these as the others. I entreat your Majesty should provide\nthat the best of the two should not be lost. I am thoroughly satisfied\nwith it, and am sure it will please your Majesty. From Monzon, 27th day\nof August, MDXXXIII.\" (Copy of a letter by Martin de Salinas to\nFerdinand, King of Bohemia and Hungary, MS. fol. Acad. of History, c.\n71).\n\nMartinez de Espinar gives in his \"Arte de Ballesteria,\" p. 41, the names\nof the best Spanish arquebuse or musket makers, and says: \"The first\ncannon brought to Spain came from Germany; the artist who made them used\nthe following marks:--\n\n\"Two heads.\n\n\"A vase.\n\n\"A vase and pine.\n\n\"A pair of tongs.\n\n\"Flames.\n\n\"A knife.\n\n\"An excellent artificer worked formerly in Italy called Lazari Cominaz,\nbut many bad cannons have been attributed to him.\n\n\"The best Spanish masters were--\n\n\"Maestre Simon, el viejo.\n\n\"Maestre Pedro, his brother, who made at Madrid excellent cannon and\nlocks.\n\n\"They came over with the Emperor Charles V., and worked also for the\nKings Philip II. and III., and marked the pieces they made with three\nsickles.\n\n\"Maestre Simon had four sons; all of them followed his profession.\nFelipe and Simon Marcuarte have worked for Philip III. and IV. Simon\nMarcuarte is still alive. They mark with a sickle in a shield, and make\nalso excellent hunting-knives, archers' knives, halberds, and other\nthings. Pedro, their brother, is also an excellent artificer.\n\n\"Juan Salado worked in several localities, and died at Salamanca. He was\nan excellent arquebuse maker. His mark was the letter of his name and a\nhorse.\n\n\"Sanchez de Mirve\u00f1a, his son-in-law, followed him. He marked with a lion\nand his initials.\n\n\"Gaspar Fernandez was brought by King Ferdinand from Salamanca, and the\narms he makes are the best in Spain.\n\n\"Pedro Mu\u00f1oz worked at Seville. He was surnamed El Toledano, and marked\nwith all the letters of his name.\n\n\"Juan de Metola did the same.\n\n\"Leguizamo worked at Seville. He marked with his name and two stags.\n\n\"Francisco Hernandez used his full name.\n\n\"Andres Herraez was a native of Cuenca. He made every sort of arm, and\nmarked with an eagle and his initials.\n\n\"Maestre Cristoval de Ricla marked with an X.\n\n\"Pedro Palacios with P P.\"\n\nA fine musket exists by Palacios at the Royal Armoury of Madrid--No.\n427. By Cristoval de Ricla there is a cannon of worked iron, No. 2319,\nmounted on its gun-carriage, with the following inscription, \"Hizome en\nRicla Cristoval Frisleva a\u00f1o, 1565.\" A fine gun, also at the Armeria, is\nby the same artist.\n\nJuan de Espinar does not mention the following artists, who lived at his\ntime at Madrid. I find their names given in original documents in the\nLibrary of the Academia de San Fernando.\n\n  Bartolom\u00e9 de Orgaz                                                1643\n\n  Juan de la Cruz                                                   1629\n\n  Juan de Mazo                                                      1613\n\n  Juan de Pozo                                                      1625\n\n  Juan de Zuazo, armourer of his Majesty                            1645\n\n  Lucas de Ros, armourer of his Majesty                        1623-1628\n\n  Matias Suezo, was arquebusier of the guards of Seville in         1625\n\n  Rafael Villato in                                                 1625\n\n  Pero Matia appears in documents of the archives of Simancas as musket-maker\n  living at the Alhambra at the end of the 16th century.\n\nBy extracting the information on this subject which is given in the\nintroduction to the Catalogue of the Armoury, we find this list can be\nbrought down to our time:--\n\nGaspar Hernandez had two excellent pupils.\n\nDomingo Garcia and Juan Belen, who marked with a lion with a lifted paw.\nHe was appointed gunsmith to King Charles II. in 1699.\n\nAlonso Martinez was an excellent artist; he marked his work with the\nletters of his name.\n\nLuis Santos, 1739.\n\nNicolas Bis.\n\nMatias Baeza, gunsmith of King Philip V., 1739.\n\nAlonso Martinez, 1732.\n\nDiego Esquivel.\n\nJuan Fernandez, appointed in 1726.\n\nDiego Ventura, appointed gunsmith to Charles III., 1760.\n\nLuis Santos.\n\nMatias Baeza, 1740.\n\nFrancisco Bis.\n\nIgnacio Barcina.\n\nSebastian Santos, 1752.\n\nGabriel de Algora, appointed gunsmith to Ferdinand VI., 1746.\n\nJuan Fernandez.\n\nManuel Sutil, an excellent artist.\n\nJos\u00e9 Cano, 1740.\n\nJoaquin Celaya, 1749.\n\nJos\u00e9 Lopez.\n\nDiego Ventura.\n\nBenito San Martin.\n\nJuan Santos.\n\nFrancisco Lopez, a great artist, gunsmith of King Charles III., in 1761.\n\nJos\u00e9 Cano.\n\nDiego Alvarez, 1775.\n\nJoaquin Celaya.\n\nSalvador Cenarro, 1762.\n\nAntonio Gomez, 1762.\n\nPedro Ramirez.\n\nAgustin Bustindui.\n\nSebastian Santos.\n\nPedro Fernandez.\n\nGabriel de Algora.\n\nAgustin Ortiz, 1761.\n\nMiguel Cegarra, 1768.\n\nFrancisco Lopez.\n\nFrancisco Garcia, 1788.\n\nIsidoro Soler, 1792.\n\nFrancisco Targarona, 1792.\n\nGregorio Lopez, 1792.\n\nAgustin Ortiz.\n\nPedro Fernandez.\n\nCarlos Rodriguez.\n\nAntonio Navarro.\n\nDiego Alvarez.\n\nValentin Lopez.\n\nJuan de Soto.\n\nCarlos Montargis, 1783.\n\nManuel Cantero, 1792.\n\nHilario Mateo.\n\nAntonio Gomez.\n\nJuan Lopez.\n\nRamon Martinez.\n\nBasilio Escalante.\n\nManuel Soler.\n\nMelchor Alvarez, the first gunsmith who forged spiral cannon in Spain,\nand made double-barrelled guns.\n\nGregorio Lopez.\n\nAquilino Aparicio.\n\nRamon Zuloaga.\n\nEusebio Zuloaga.\n\n*** The marks used by these artists are given in plate 9 of \"Catalogo de\nla Armeria de Madrid,\" 1849.\n\nDaggers, knives, scissors, and other small arms made in Spain, have been\nfamous from early times. A good example of a dagger is No. 2238, 55 at\nthe Kensington Museum. A great number of towns have been distinguished\nfor this industry, such as Albacete, Murcia, Alcazar de San Juan, and\nGuadix, where even in the present day much cutlery is produced. The\ncutlery made in Spain preserves the traditional Moorish forms. Al\nMakkari says, in speaking of these objects, in his \"Moham. Dyn. in\nSpain,\" vol. i., p. 93:--\"Murcia was likewise famous for the manufacture\nof coats of mail, breast-plates, and steel armour inlaid with gold; all\nkinds of instruments of brass and iron, such as knives, scissors, and\nother trinkets, and especially weapons and other warlike instruments,\nwrought in such perfection as to dazzle with their brightness the eyes\nof the beholder.\" The translator adds, p. 393: \"Several towns in the\nprovince are renowned for this industry. At Albacete there are several\nmanufactures of well-tempered scissors, daggers, and knives, which, from\nthe shape of their blades, betray their Moorish origin. Since the\nexpulsion of the Moriscos, the Spaniards have kept up this manufacture,\nand daggers and knives of the end of the last century are often met with\nbearing Arabic inscriptions and verses from the Koran. I have seen one\nwhich on one side has the following inscription, 'I shall certainly kill\nthy enemies with the help of God,' and on the reverse, 'Fabrica de\nNavajas de Antonio Gonzalez, Albacete, 1705.'\"\n\nSr. Rico y Sinovas has published an interesting article on ironwork in\nthe \"Almanaque de el Museo de la Industria,\" 1872. He gives the names of\nthe following artists who were famous in this industry.\n\n     Aguas, Juan; he worked at Guadix in 1735.\n\n     Albacete, signed Cel, 18th century.\n\n     Ambrosio, worked at Mora in the 18th century.\n\n     Arbell, Ramon, worked at Olot, Catalu\u00f1a, in the 17th century.\n\n     Beson, Manuel, worked at Madrid.\n\n     Castellanos, el _viejo_, worked at Albacete in 1766.\n\n     Castellanos, el _mozo_, worked at Albacete in the 18th century.\n\n     Castello, Gregorio, 16th century.\n\n     Cerda, Miguel de la, worked at Madrid and Segovia in 1590.\n\n     Diaz, Pedro, worked at Albacete in the early half of the 18th\n     century.\n\n     Escobar, Cristoval, 16th century.\n\n     Escobar, Juan, 17th century.\n\n     Fernandez Manso de Payba, Jose, master cutler of the 18th century.\n\n     Garcia de la Torre, Teodoro, 18th century.\n\n     Garijo, master cutler of Albacete; he worked in 1771.\n\n     Gomez, Mateo, worked at Albacete in 1659.\n\n     Grande, Juan, 1643.\n\n     Gutierrez, worked at Chinchilla, 1701.\n\n     Herrezuelo, el _viejo_, worked at Baeza in 1643.\n\n     Herrezuelo, el _mozo_, worked at Baeza in 1643.\n\n     Horbeira, Angel, worked at Madrid in the second half of the 17th\n     century.\n\n     Lallave, Juan, locksmith of Madrid in 1820.\n\n     Leon, worked at Albacete early in the 18th century.\n\n     Llorens, Pablo, worked at Olot in 1699.\n\n     Moro, worked at Madrid late in the last century.\n\n     Ramirez, Juan, worked at Mexico in 1590.\n\n     Romero, worked at Albacete in 1769.\n\n     Rosel, worked at Mora.\n\n     San Jos\u00e9, worked at Jaen in 1673.\n\n     Selva, Juan, worked at Cartagena in 1780.\n\n     Segura, worked at Mora towards the end of the last century.\n\n     Sierra, Juan, worked at Albacete in 1771.\n\n     Sosa, Madrid, 17th century.\n\n     Torres, Albacete, 17th century.\n\n     Vicen Perez, worked at Albacete in 1674.\n\n     Vilarosa, Antonio, worked in the 17th century.\n\n     Vicen Perez, Julian, worked at Albacete in 1710.\n\n     Zervantes, Francisco, Toledo, 17th century.\n\nThe following artists' names may be added; they consist of master\ncutlers of Madrid, and are to be found in a bundle of original documents\nat the Academy of San Fernando.\n\n     Alcocer, Francisco, 1635.\n\n     Baltanos, Lucas de, 1611.\n\n     Castillo, Rafael del, 1625.\n\n     Cuenca, Francisco de, 1613.\n\n     Fuente, Pedro de la, worked at Mora and Madrid in 1628.\n\n     Garcia, Bartolom\u00e9, 1642.\n\n     Gonzalez, Marcos, 1624, 1625.\n\n     Heras, Antonio de las, 1611.\n\n     Luzon, Andres de, 1611.\n\n     Martin, Alonso, 1643.\n\n     Martinez de Machuca, Pedro, 1611.\n\n     Morel, Alonso, 1643.\n\n     Rodriguez de Qui\u00f1ones, Pedro, 1611.\n\n     Torres, Antonio de, 1622.\n\n     Valsarias, Lucas de, 1611.\n\nThe following names of artists are given in Calomarde's \"Historia\nPolitica de Aragon.\"\n\n     Ferrara, Andres, Zaragoza, 16th century.\n\n     Picado, Jeronimo, Calatayud, 1722.\n\n     Nieva, Luis, Calatayud, 17th century.\n\n     Munester, Andres, Calatayud, 18th century.\n\n\n\n\nFURNITURE.\n\n\nIt is difficult to give an account of the furniture used in Spain from\nthe earliest times, owing to the absence of specimens belonging to the\nRoman or Visigothic period. We are safe, however, in affirming that,\nduring the domination of the Romans, the same models were used in Spain\nas in Rome itself, the similarity between objects of other kinds which\nexists, and the continual habit of the Romans of imposing their culture\non the countries which they conquered, is sufficient to justify this\nidea. During the first period of the Christian era until about the 11th\ncentury, other objects which have reached us as gold and silversmiths'\nwork, seem to prove that all industrial art work, including furniture\nmust have been Classic, Byzantine, or Oriental in form, according to the\nperiod in which these styles predominated. The Roman or Classical style\nmust have lasted for a considerable time, if we judge by the texts given\nby San Isidoro, in his \"Etymologies,\" lib. xx., cap. xi. and xii. San\nIsidoro lived in the 7th century of our era. We find he mentions the\nwords, _de lectis et sellis_, _de vehiculis_, which refer to furniture\nof the Roman period; but we do not meet with any distinctive terms which\ncan be applied to the time in which he lived, on the contrary he\ncomments on phrases and alludes to the forms of objects in the same\nmanner as Rufus, Varrus, and other classical writers.\n\nAt the invasion of the Arabs in the beginning of the 8th century the\nByzantine element increased in its application to furniture. During the\nreign of the caliphs, contemporary authors tell us that luxury of\ndecoration of every kind was carried to a great extent. The description\nof the pulpit or _minbar_ of the mosque of Cordova will give an idea of\nits richness. Al Makkari, in his \"History of Mohammedan Dynasties in\nSpain,\" says, \"the length of the Mihrab was 8 cubits; its breadth, 7\ncubits; the height of its dome, 13 cubits. There stood on one side a\npulpit, constructed by Alhahem, equalled by none other in the world for\nworkmanship and materials. It was made of ivory and exquisite woods,\nsuch as ebony, sandal, Indian plantain, citron wood, aloe, and so forth,\nat the expense of 3,575 dinars: the steps by which it was ascended were\nnine in number. Another writer says it was formed of 36,000 pieces of\nwood fastened together by gold and silver nails, occasionally encrusted\nwith precious stones, and its construction lasted for seven years; eight\nartists being daily employed on it,\" vol. i. p. 222.\n\nEdrisi, who lived at Cordova at the beginning of the 12th century,\ndescribes this pulpit in the same manner. It was to be seen in the\ncathedral of Cordova as late as the 16th century, when it was destroyed,\nand its materials employed in the construction of an altar. An Arab\n_minbar_ is in the South Kensington Museum, which came from Cairo, and\nalthough belonging to the last centuries of the Middle Ages, it will\ngive an idea of this structure and its elaborate workmanship.\n\nThe oldest specimen of furniture which exists in Spain, belonging to\nChristian art, is the shrine of San Millan de la Cogulla, in the\nprovince of La Rioja, described in _Ivories_, p. 134. This casket or\nshrine was made by the orders of King Sancho el Mayor, in 1033, in order\nto preserve the relics of San Millan. It is of wood, covered with 22\ncompartments of highly interesting ivory plaques, representing episodes\nfrom the life of the Saint. The names of the artists, _Apparitio_ and\n_Rodolpho_, appear on it. In 1808 the French stripped the shrine of its\ngold and silver ornaments, but, most fortunately, left the casket and\nits ivory carvings behind.\n\nAnother interesting object, which is useful for illustrating furniture\nin Spain in the 12th century, is the Bishop's throne in the cathedral of\nGerona. It is of white marble of one piece, covered with a simple\nornamentation in excellent taste, in a similar style, and for the same\npurpose, as the throne of San Clemente in Rome. It is raised at the back\nof the altar, and is ascended by thirteen steps. The Bishop sits on this\nthrone after the first purification, and continues there until the\nOffertory, when he descends and finishes the mass at the altar; owing to\nthe existence of this chair, one of the most ancient rites of\nCatholicism has been preserved at Gerona.\n\nMention must also be made of the shrine in which the remains of Queen\nUrraca are kept in the cathedral of Palencia, and the Cofre del Cid in\nthe cathedral of Burgos, although they are more interesting for their\nantiquity than their artistic merit.\n\nDuring the 13th century furniture in Spain must have been similar to\nthat used in other countries, it was influenced by French and Italian\nforms; from which, as was the case in architecture, these models were\nconstantly copied. Some idea of this is given in the triptych known by\nthe name of the Tablas Alfonsinas, which is fully described in Gold and\nSilver Work. (See woodcut on p. 17.)\n\nThe principal novelty which we find in Spain in this industry consists\nin the combination of the Christian and Moorish styles applied to\nfurniture. A very remarkable specimen exists at the Academia de la\nHistoria, consisting of a reliquary, triptych or armoire of very large\ndimensions, formerly in the church of the Monasterio of Piedra in\nAragon. It is 13 feet wide and 8 feet high. When the two doors are\nclosed six compartments appear on each, surrounded by a border headed by\na wide cornice. This cornice is divided in sixteen spaces formed by\narches adorned with stalactite ornamentation in the Moorish style;\ninside each span is painted the figure of a saint. In the twelve\ncompartments which form the doors there are subjects from the life of\nOur Lord, beginning with the Annunciation, and ending with the Descent\nfrom the Cross painted in excellent taste. The border which surrounds\nthese doors is formed of a beautiful geometrical ornamentation, in the\npurest Oriental style; in the upper and lower part appears the following\ninscription in Gothic characters, which gives us the date in which this\nobject was made.\n\nTabernaculum hoc vocabitur aula Dei quia vere Dominis est in loco isto.\nFuit autem constructum ad honorem et reverentiam sacratissimi corporis\nDomini nostri Ihu Xpi et pasionis ejusdem nec non ad honorem == et\nreverentiam sanctissime genitricis ejusdem et totius celestis curie et\nsanctorum ... at fuit ... depictum anno MCCCXC. anima ordinatoris\nrequiescat ... sinu salvatoris. Amen.\n\nWhen this triptych is opened several compartments appear, formed by\ncusped Gothic arches, placed within others, in the form of a mitre\ndivided by pinnacles. The ornamentation which fills the vacant space\nbetween the arches, and the seven banderoles in the lower part are in\npure Gothic style, without a reminiscence of Moorish art. In the central\nbanderole appears the name of _Dopnus Martinus Poncii Abbas_, the abbot\nof the monastery where this reliquary was made, and the person by whom\nthe expense was probably borne. The inside of the doors is divided into\ntwo horizontal zones, which include half the total size. The upper zone\nis covered with geometrical tracery, similar to the outer borders. The\nlower one is divided into four compartments for each leaf of the door,\neight within two, which are formed by cusped Gothic arches; within these\nspaces are painted standing figures of angels holding musical\ninstruments in their hands and nimbi round their heads. The details and\nforms of the instruments and delicate ornamentation of the costumes and\ncarpets, constitute one of the most beautiful details of this triptych.\nThese details and several of the architectural lines of the Gothic part\nof this ornamentation denote an Italian influence, probably Sicilian,\nfor this province belonged then to the crown of Aragon. This triptych\nis richly gilt, and painted inside and out, and is one of the finest and\nmost remarkable specimens of the kind which exist in any country. A\nplate may be seen in \"Mus. Esp. de Antiguedades,\" vol. vi., p. 307.\n\nA very interesting specimen of this combination of styles is a recess or\ncupboard of the 14th century, which is at the Kensington Museum, No.\n1764, 71, [see woodcut]. It is 5 feet 4 inches high, by 8 feet wide; and\nis composed of an outer arch, with the tympani ornamented with fine\narabesques in relief, made of stucco,--representing vine tendrils,\nleaves, and flowers, similar to those which appear in the Cosa de Mesa\nand other Moorish houses at Toledo. The arch is surrounded by two Latin\ninscriptions in Gothic characters, of which only the following words are\nlegible + Autem transies per medium ilorum.... mente +. The first of\nthese is from St. Luke, iv., v. 30, a passage often quoted by\nalchemists.\n\nThis arch serves as a portal to the cupboard, which is about a foot\ndeep, and is divided into two compartments with two rows of shelves.\nEach of these is supported by a series of Moorish arches, the tympani of\nwhich are ornamented with tracery, or an ornamentation of leaves in a\ngeometrical and Oriental manner. On the rim of the upper part is\nrepeated the following inscription in African characters:\n\n[Illustration: Arabic]\n\n\"Felicity and Fortune.\"\n\nOn the lower shelf are the following words in Spanish in Gothic\ncharacters: + Dios: te: salve: estrella: de: la: mannana: medicina: de:\nlos: peccadores: reina + \"Hail! morning star; medicine of sinners;\nqueen.\"\n\n[Illustration: \"BOTICA DE LOS TEMPLARIOS.\" FROM TOLEDO. SOUTH KENSINGTON\nMUSEUM.]\n\nThis \"Alhacena,\" or cupboard, existed formerly in the court of an old\nhouse at Toledo, known by the name of \"Casa de la Parra,\" and is\nconstantly called \"Botica de los Templarios,\" the Templars' dispensary,\nprobably because the Templars occupied the parish of St. Michael, in\nwhich the house stands, and because the word \"medicine\" is mentioned in\nthe inscription.\n\nIn order to increase the list of these objects, which are unfortunately\nso rare, it is necessary to mention the organ case, decorated with fine\nMoorish tracery, in the chapel founded by Diego de Anaya, Archbishop of\nSeville in 1374, in the cloister of the cathedral of Salamanca.\n\nAnother specimen of furniture of interest of the 14th century is the\nshrine or casket containing the remains of San Isidoro, existing in the\nparish church of San Andres at Madrid. This casket is 7 feet 5 inches\nlong; the cover is formed of two plain surfaces, forming a roof, which\nleaves a triangular pediment at each end. The shrine is made of wood,\ncovered outside with a coat of painting in distemper, and is decorated\nwith a series of painted compartments simulating pointed tricuspid\narches let into others which are mitre-shaped; in the interior spaces\nare painted a series of representations of the life of the saint. This\nobject is extremely interesting: it is evidently of Spanish workmanship,\nalthough it has great resemblance with the Italian _cassoni_ of the same\nperiod. See vol. iv. \"Mus. Esp.\" p. 593, and \"Monumentos\nArquitectonicos\"--livraison 67.\n\nThe most important period of artistic furniture in Spain must be looked\nfor in the 15th and 16th centuries: we find in this time works decorated\nwith sculptured figures and most delicate and exquisite ornamentation.\nNothing gives a better idea of this than the choir seats of Spanish\ncathedrals. The cathedrals in Spain are different to those of France and\nthe north of Europe owing to the circumstance that the choir, or _coro_,\nis placed in the middle of the central nave, in the traditional form of\nthe primitive basilicas, such as San Clemente at Rome, with the\ndifference however, that towards the last centuries of the Middle Ages\nthe _pluteus_ or walls that surround it, which are elevated about one\nyard from the floor, were turned into an enclosure or wall four or five\nyards in height. A space is therefore set apart in the middle of the\nchurch, which, although spoiling the general effect of the nave,\nbecomes, as is the case in most Spanish cathedrals, a museum of artistic\nobjects of all kinds. The side facing the presbytery or high altar is\nenclosed by a _reja_. [See Iron and Bronze work.] In the three remaining\nsides stalls are placed against the wall, forming two rows, one near the\nfloor, and the other raised about a yard from the lower ones. The\ngeneral system adopted is a row of stalls separated from each other by\nthe arms; the backs of the lower ones serving as stands for the books\nwhich are used by those who sit in the upper range. The subsell\u00e6 are\nprolonged and form canopies, pinnacles, crest-work, and other varied\nornamentation resting on columns. They are usually made of walnut wood,\nand there is not a single vacant space which is left uncovered with\nfigures or ornamentation in relief: the effect produced by the whole is\nadmirable. The number of seats varies according to the size of the\nchoir, or importance of the church.\n\nIt is probable that the earliest wood carvers who worked in Spain came\nfrom Flanders or Holland, if we judge by the decided Germanic character\nof the figures and ornamentation, and frequent mention made of foreign\nmasters; but this industry very soon took root in Spain, and Spanish\nartists soon equalled or excelled their masters. The most important\nchoir stalls carved in the Gothic style in Spain belong only to the last\nyears of the 15th and beginning of the 16th century; those in the\nRenaissance style come immediately afterwards, and include the whole\n16th century.\n\nOne of the finest is that of Leon, it is in the purest German-Gothic\nstyle, and may be considered as a masterpiece in its way: the large\nfigures are as delicately carved as the small ones. The representations\nof the \"Last Judgment\" and \"Tree of Jesse\" are as beautiful as can be\nimagined. The choir at Zamora is also in the German style; the large\nfigures which ornament it are splendid in effect. These stalls are fully\ndescribed in \"The Ecclesiologist,\" in the following manner: \"The stalls,\nthough not remarkably early, are very fine. The bishops, as usual, at\nthe centre of the west nave; then three dignitaries on each side; then\nat each extremity of the west end a door into the nave; then on the\nnorth and south sides respectively twenty stalls with subsell\u00e6, which\nalso are returned. All the panelling is very fine: but that at the back\nof the subsell\u00e6 I will give at full, because the types of, and legends\nconcerning Our Lord are so remarkably well chosen, that they may be\nuseful as giving ideas to our own church builders, and might be\nintroduced with great advantage in stained glass.\" \"Ecclesiologist,\"\nvol. xiv., p. 363.\n\nThe stalls at the cathedral and church of Sto. Tomas of Avila, and\ncathedral of Burgos are also very fine. The _coro_ at Barcelona is\nextremely picturesque; at the backs of the stalls are placed the coats\nof arms of the knights of the Golden Fleece, in a similar manner to\nthose of St. George's Chapel at Windsor; among these we find the arms of\nHenry the VIIIth. In 1519 the Emperor Charles V. held at Barcelona a\nchapter of this Order, the only one which has ever been held in Spain.\n\nIt would take too much space to enumerate the choir seats in Spain\nworthy of notice; those at Toledo rank as the most important, as\nincluding the last period of the Gothic and best Renaissance styles.\nThis _coro_ contains 45 stalls in the lower, and 71 in the upper part.\nMr. Street fully describes the lower part in his \"Gothic Architecture in\nSpain,\" p. 252, and says: \"The lower range of stalls, fifty in number,\nare the work of Maestro Rodrigo, circa A.D. 1495; and the upper range\nwere executed, half by Berruguete and half by Felipe de Borgo\u00f1a, A.D.\n1543. The old stall ends are picturesque in outline, very large, and\ncovered with tracery, panels and carvings with monkeys and other animals\nsitting on them. The upper range of stalls is raised by four steps, so\nthat between the elbows of the lower stalls and the desk above them, are\nspaces filled in with a magnificent series of bas reliefs, illustrating\nthe various incidents of the conquest of Granada. They were executed\nwhilst the subjects depicted were fresh in the minds of the people, and\nare full of picturesque vigour and character. The names of the\nfortresses are inscribed on the walls; in some we have the siege, in\nothers the surrender of the keys; in others the Catholic monarchs,\naccompanied by Cardinal Ximenes riding in triumph through the gates.\"\nThe upper stalls, although considered by Mr. Street \"heavy dull\nPaganism,\" are among the most important artistic works existing in\nSpain. Their sculptures and ornamentation reproduce the Renaissance\nItalian style in its finest period. Columns of red jasper are combined\nwith the walnut-wood carvings, and the termination of the upper part\nwhich rests on the wall is of white marble carved by the same artists.\n\nChoir stalls, sharing the fate of other architectural works, followed\nthe general path of decay after the 16th century. Those stalls at the\ncathedral of Malaga, which were carved in the 17th century by Jos\u00e9\nMichael, Pedro de Mena, and Luis Ortiz, are worthy of mention: the\nfigures in high relief at the back are very fine. This falling off of\nartistic feeling is particularly observable at the cathedral of Cordova;\nthe stalls there, which were carved by Pedro Duque Cornejo in the 18th\ncentury, are admirable in richness of ornamentation, but the whole\ndecoration is in the worst possible taste. The fine lecterns which are\nplaced in the centres of the choirs are often good examples of carving.\nAn admirable model of this style exists at the cathedral of Zamora.\n\nIn treating of furniture of other kinds, the silver chair or throne of\nD^{n.} Martin de Aragon must figure in the first instance. It is\ndescribed in \"Gold and Silver Work,\" p. 19. In these and other objects\nwhich are not Oriental in style, constant reminiscences are observable\nof the similarity of ideas and forms with Italian or French objects. The\nlarge chests or _cassoni_ of different sizes which are constantly met\nwith in museums, or in private collections, are similar to objects of\nthe same kind which abound in Italy. Some are covered with fine Gothic\nor Renaissance carvings in the same taste as the choir stalls, and are\nornamented also with good iron-work. Some of them were made in the\nprovince of Catalu\u00f1a of inlaid ivory, imitating in a coarse manner\nFlorentine and Milanese work. These objects were of the end of the 15th\nand beginning of the 16th century. An excellent example exists at the\ncathedral of Toledo, near the Puerta de los Leones, consisting of an\niron coffer covered with carved and repouss\u00e9 work.\n\nA fine cupboard of walnut wood may be seen at the same cathedral which\nis unrivalled for its beauty. It is composed of an architectural order\nwith six pilasters, forming five vacant spaces completely covered with\nmedallions, groups of children, flowers, and a multitude of subjects of\nornamentation in reliefs, terminating with a frieze crowned with\ncandelabra and salient points. This piece of furniture was made in\n1549-1551, by the sculptor, Pedro Pardo, for holding vestments of the\nclergy. In 1780 a reproduction was made by Gregorio Lopez Durango: the\noriginal and the copy face each other in the room which is passed on\nentering the Sala Capitular. As Spanish carved wood work of the\nbeginning of the 16th century may be mentioned the specimens at the\nKensington Museum, Nos. 245 and 246-64, of two panels with armorial\nshields.\n\nDuring the 16th, but more especially the 17th, century it became very\nfashionable in Europe to use secretaires or cabinets to a great extent,\nwhich, if we calculate by those that still are found in private houses\nin Spain, have never been so numerous in any other country. A great\nvariety of every description may be seen at the South Kensington Museum.\nIn looking through old inventories at Madrid, the number of these\nescritoires which decorated the rooms of the Palace is almost\nincredible. Many of them came from Italy, Flanders, or Germany. In a\nmemorial drawn up by Pedro Gutierrez and presented to the King, begging\nfor protection for this industry, he says: \"The cabinets and\nescritoires, _contadores y bufetes_, which were worth 500, 600, and 700\nreales when brought from Germany, are now made in Spain for 250 and 300\nreales each.\" (Acad. de la Hist., N. 6, vol. iv. 371.) The quantity and\nimportance of those brought from Germany is to be inferred from an edict\npromulgated by Philip III., in Valladolid in 1603, in which \"cabinets of\nevery kind coming from Nuremberg are not allowed to enter the country,\"\nib. 518.\n\nThese cabinets were made in a great variety of ways. Some were, as we\nhave seen, copied in Spain from foreign models, but the greater number\nconstituted a characteristic industry of the Peninsula. Some were\ncovered with wood carvings in a similar style to the large armoires in\nthe Sala Capitular at Toledo. Others were known by the name of\n_Vargue\u00f1os_, because the great centre of their manufacture is supposed\nto have existed at the village of Vargas in the province of Toledo.\nThese cabinets are effective; their geometrical ornamentation is a\nreminiscence of the Oriental style, and the iron work outside is very\nstriking. A good specimen may be seen at the Kensington Museum, No.\n1073, '71; a woodcut of the upper portion is given on the next page.\n\nWe also meet with cabinets and armoires with glass doors covered with\ntortoise-shell and gilt bronze. A splendid series of inlaid work of\ntortoise-shell, ebony, mother-of-pearl and ivory is preserved at the\nSacristy of the Cartuja at Granada, made by a friar of the same convent,\nManuel Vazquez, at the beginning of the last century. Cabinets of ebony,\ninlaid and covered with repouss\u00e9 silver work, must have been very\ngenerally made in Spain; silver was used to so great an extent after the\nconquest of America, that a law was issued in 1574 prohibiting with the\nutmost rigour the making and selling of this kind of merchandise, in\norder not to increase the scarcity of silver. \"No cabinets, desks,\ncoffers, braziers, etc., shall be manufactured of silver.\"\n\n[Illustration: VARGUE\u00d1O CABINET. SPANISH. 16TH CENTURY. SOUTH KENSINGTON\nMUSEUM.]\n\nCabinets of inlaid ivory or different coloured woods which were\noriginally imported from Italy and Germany were constantly reproduced in\nSpain, as will appear by the following dialogue:-- \"How much has your\nworship paid for this cabinet? It is worth more than 40 ducats. What\nwood is it made of? The red one is made of mahogany from the Habana, and\nthe black one is made of ebony, and the white one of ivory. You will\nfind the workmanship excellent. Here you will find a finer cabinet.\nWhere was it made? It was brought with these chairs from Salamanca.\"\n(\"Dialogos Familiares,\" by J. de Luna, Paris, 1669, p. 111.) Among\ncabinets inlaid with bone or ivory there is a peculiar style in which\nfurniture brought from the Portuguese possessions in the East was\nimitated in a rough way. Some also have reached us in which the drawers\nare covered with embroidery in silks of different colours, exquisitely\nworked. In the 17th century furniture of different kinds was covered\nwith embroidered silks. At the Kensington Museum there is an interesting\ncoffer, which came from the Convent of Loeches, and was originally the\ngift of Count Duke de Olivares, Philip the IVth's minister.\n\nThe cabinets decorated with _pietre duri_, which are frequently met with\nin Spain were of foreign make. Madame d'Aunoy, in describing the house\nof a grandee of Spain in her \"Voyage d'Espagne,\" p. 56, vol. ii., Lyon,\n1643, says, \"et de grands cabinets de pi\u00e8ces de rapport enrichis de\npierreries, lesquels ne sont pas faits en Espagne: des tables d'argent\nd'entre eux, et des miroirs admirables tant pour leur grandeur que pour\nleurs riches bordures, dont les moins belles sont d'argent. Ce que j'ai\ntrouv\u00e9 de plus beau sont des _escaparates_; une esp\u00e8ce de petit cabinet\nferm\u00e9 d'une seule glace et rempli de tout ce qu'on se peut figurer de\nplus rare.\" Cabinets ornamented with paintings on glass, or copper, or\nenamels came from Flanders, and some rare specimens of damasquine work\nwere most probably Milanese manufacture.\n\nIn Cean Bermudez \"Dicc Historico de los Profesores de Bellas Artes\"\nappear among the sculptors' names, those of the artists who carved the\nmost important stalls of the cathedrals, and other works of art of a\nsimilar kind, but the names of those who made cabinets have never yet\nbeen published. I have been fortunate enough to be able to collect the\nfollowing from unpublished documents which exist at the Library of the\nAcad. de San Fernando at Madrid. They worked in the 16th and 17th\ncentury.\n\n  Aguayo, Urban de, wood carver                                     1623\n\n  Carpintero, Francisco, wood carver                                1630\n\n  Garcia, Marcos, wood carver of his Majesty                     1637-42\n\n  Gomez, Juan, carver in wood                                       1598\n\n  Gorostiza, Juan de, carver in wood                                1627\n\n  Higares, Nicolas de, carver in wood                               1625\n\n  Hispano, Francisco, carver in ivory                               1618\n\n  Hoz, Martin de la, carver in wood                                 1624\n\n  Lara, Benardino, de, carver in wood                               1612\n\n  Lozano, Pedro                                                     1622\n\n  Marcos, Juan, carver in wood                                      1636\n\n  Martinez, Andr\u00e9s, carver in wood                                  1622\n\n  Martinez, Dionisio, carver in wood                             1621-25\n\n  Martinez, Gabriel, carver in wood                                 1623\n\n  Murga, Tomas de, carver of his royal Highness                     1614\n\n  Osoz, Martin de, wood carver                                      1623\n\n  Parezano, Alonso, wood carver of his Majesty                      1623\n\n  Pelegrin, Joan, wood carver                                       1614\n\n  Pe\u00f1a, Jeronimo de la, wood carver                                 1622\n\n  Quero, Melchor de, carver in wood                                 1586\n\n  Radis, Francisco, master maker of cabinets in\n  ebony and ivory                                                   1617\n\n  Riofrio, Martin de, wood carver                                   1612\n\n  Riofrio, Tomas de, wood carver                                    1626\n\n  Rodriguez, Bernardo de, wood carver                               1624\n\n  Rodriguez, Domingo, wood carver                                   1633\n\n  Roxo, Domingo, wood carver                                        1630\n\n  Sanchez, Matias, wood carver                                      1565\n\n  Santana, Juan de, wood carver                                     1617\n\n  Sierra, Francisco de, wood carver                                 1634\n\n  Spano, Jeronimo, wood carver                                      1617\n\n  Torres, Juan de, wood carver                                      1658\n\n  Velasco, Lucas de, master in painting and\n  gilding cabinets                                                  1633\n\n  Zorrilla, Domingo                                                 1642\n\nLarge arm-chairs of a quadrangular form, with arms, back and seat of\nleather or embroidered stuffs, were used to furnish rooms at the same\ntime as these cabinets; tables, and frames inlaid with ivory,\ntortoise-shell, ebony, bronze and silver, were hung on the walls;\nside-boards, beds, and braziers were made of inlaid woods or silver.\nWomen during the 16th and 17th centuries sat on low stools on the\nground. The beds were made of rich brocades embroidered with gold,\n(_vide_ Madame d'Aunoy), trimmed with point d'Espagne, and on the\nsplendid carpets were placed silver braziers which burnt crushed olive\nstones. The walls were covered with tapestry and rich silks, and from\nvery early times stamped, painted or gilt leather _guadameciles_ were\nused in Spain to a very great extent. These _guadameciles_ were imitated\nin France and other countries in the 16th and 17th centuries; a very\nlarge quantity of this stamped leather is to be met with in England. The\nBaron Ch. Davillier has lately published an interesting notice on this\nsubject, from which I take the following information. (\"Notes sur les\ncuirs de Cordoue. Guadameciles d'Espagne,\" Paris, 1878.)\n\nThe word guadamecil applied to this leather comes from the village of\n_Ghadames_ in Africa, celebrated from the 12th century for this\nindustry. It was imported by the Moors into Spain, and Cordova became\nfrom the beginning of the Middle Ages the great centre of this\nproduction, although other towns, such as Seville, Granada, Toledo, and\nBarcelona, exported these leathers also. The chronicler Ambrosio de\nMorales, in his \"Las Antiguedades de las ciudades de Espa\u00f1a,\" Alcala,\n1575, says, p. 10, \"A great commerce is produced by the exportation of\nskins, and many have been enriched by it, those prepared at Cordova are\nso excellent, that now in Spain any goat hide prepared in any locality\nis called Cordovan. Guadameciles are made of leather, and are so well\nfashioned in Cordova that none can compete with them; they are exported\nto the Indies and all Europe. They produce much for the town, and\nbeautify the principal streets, for the hides are hung out to dry after\nthey are painted and gilt, and it is a fine sight to see the walls\ncovered with such variety of colour and form.\"\n\nA good collection of these guadameciles is in the South Kensington\nMuseum. See Nos. 471 to 485, '69, and 1651 to 1654, '71.\n\nAt the end of the 17th and first half of the 18th century, the Baroque\nor Churrigueresque architecture had such influence over furniture, that\nalthough exaggerated examples were very common, they reached in no\ncountry to such a pitch as they did in Spain. As specimens, the enormous\n_retablos_ over the high altars made of gilt carved wood may be\nmentioned, which are so frequently to be seen in Spanish churches. The\nexuberance of uncouth heavy ornamentation destroys the regularity and\nharmony of the general effect. Chairs and tables were made then in the\nsame style, and the _cornucopias_ or decorated mirrors which adorn\nSpanish churches and sacristy, belong to this period. The name of\n_cornucopia_ was given in allusion to the horn of abundance.\n\nThe French influence of the last century brought the same fashions for\nfurniture into Spain, and we also find there the styles known as Louis\nXVI. and Empire. Spain follows the general rule; porcelain plaques from\nthe manufactories of Alcora and Retiro were let into furniture, and gilt\nbronze mountings were very much used. The most important specimens of\nfurniture of the early present century are the splendid rooms inlaid\nwith metal work at the Palace, called Casa del Labrador at Aranjuez, and\nthose known by the name of \"Piezas de maderas finas,\" at the Palace del\nEscorial. The description given by J. Quevdo, in his \"Historia del Real\nMonasterio de San Lorenzo,\" Madrid, 1849, gives a good idea of their\nimportance; he says, p. 343, \"This series of four small rooms were\ndecorated at a cost of 28,000,000 of reales (\u00a3280,000). The pavement,\nfriezes, windows, and doors, are made of the most delicate inlaid work\nrepresenting landscapes, vases and festoons of flowers, which look as if\nthey were painted with a brush. King Charles IV. helped in this work,\nand they were finished in 1831 under the direction of Angel Maeso. The\nsplendid iron work was made by Ignacio Millan. It is of polished iron\ninlaid with gold, and most exquisite in workmanship.\"\n\n\n\n\nIVORIES.\n\n\nNo artistic industry in Spain has left behind so little historical\ninformation as ivory carving. Only a very small number of examples of\nthis art have reached us which are of undoubted Spanish manufacture, but\na group exists among them which has been but little examined or studied;\nit is worthy of the utmost attention, on account of its artistic\ncharacter, and the inferences which may be drawn from it.\n\nWe find no allusion in the works of Pliny or St. Isidoro to the\nexistence of the industry of ivory carving in Spain during the Roman and\nVisigothic dominations. An interesting example of ivory work of the\nRoman period, consisting of a consular diptych may be seen at the\ncathedral of Oviedo, which, although certainly not carved in Spain, must\nnot be overlooked here, in order that it may be included among the\nnumber of the carvings which have reached us of this period. The two\nleaves of this diptych are complete, and in a perfect state of\npreservation: they are 16 inches long by 6 inches wide; in the centres\nare two medallions in relief with a bust of the consul, who is\nrepresented in the act of throwing down with his right hand the _mappa_\nor handkerchief, and holding in his left hand a sceptre. The right\nangles are ornamented with masks, the rest of the surface is plain. The\ntwo following inscriptions run along the upper part of the leaves:\nFL\u00b7STRATEGIVS APION. STRATEGIVS APION V. ILL: COM. DEVV. DOMM. ET \u00b7 CONS\n\u00b7 OR.\n\nFlavius Strategius Apius. Strategius Apius, illustrious man, count of\nthe most fervent servants, and consul in ordinary.\n\nThis consul belongs to the period of Justinian, by which emperor he was\ninvested with this dignity in A.D. 539. We do not know how this diptych\ncame to Spain; it is generally supposed that it belonged originally to\nthe shrine of the cathedral of Toledo, from whence it was removed to\nAsturias, with other relics, to be concealed there during the invasion\nof the Arabs at the beginning of the 8th century. The student will find\nfurther details in \"Corpus Ins.\" by H\u00fcbner. \"Monumentos\narquitectonicos.\" \"Mus. Esp. de Antiguedades.\" Vol. i. p. 385.\n\nFrom the invasion of the Arabs, which began early in the 8th century,\nand on the foundation of the empire of the Caliphs of Cordova in the\nyear 756, an era of grandeur began for the Arabs in Spain, coinciding\nwith their independence from the Caliphs of Damascus, which lasted for\nmore than two centuries; during this time Cordova became the most\nimportant literary and scientific centre in Europe. The direct influence\nof the East and Constantinople may be traced without interruption from\nthis time on the culture of the Spanish Arabs. At times this culture was\ntransmitted by the objects of every kind which the Spaniards received\nfrom the East, at others by the influence exercised by the artists who\nestablished themselves on the territory of the Caliphate of Cordova. The\ngreater part of the industrial arts, which were imported at this time,\nbecame naturalized in the country, and we find them developed in every\nlocality of the Peninsula, and although we do not possess any positive\nhistorical information stating this fact, which distinctly bears on\nivory carvings, we are safe in affirming that the industry existed among\nthe Spanish Arabs in a very high state of perfection during the last\nyears of the Caliphate. The description and details of the most\nimportant ivories which have reached us of this period confirms this\nopinion.\n\nAt the South Kensington Museum there is a cylindrical box No. 217-'65\nwith rounded cover. I copy Mr. Maskell's description of this object, of\nwhich a woodcut is given. (Vide \"Ivories, Ancient and Medi\u00e6val, in the\nSouth Kensington Museum,\" London, 1872.)\n\n\"This beautiful box is carved throughout, except the bottom of it, with\ninterlacing narrow bands forming quatrefoils, in which on the cover are\nfour eagles. These have spread wings and stand erect; well designed and\nmost delicately executed. A small knob serves to lift the lid.\n\n[Illustration: IVORY BOX. MOORISH. 10TH CENTURY. SOUTH KENSINGTON\nMUSEUM.]\n\n\"Round the side, each quatrefoil is filled with a star having a leaf\nornament. The same decoration is repeated in the spaces between the\nlarger quatrefoils on the cover.\n\n[Illustration: Arabic]\n\n\"The whole is carved in pierced work, except a band which forms the\nupper upright portion of the box, round the side of the lid. This band\nhas an Arabic inscription:\n\n[Illustration: Arabic]\n\n\"A favour of God to the servant of God, Al Hakem al Mostanser Billah,\ncommander of the faithful.\" He was a Caliph who reigned at Cordova, A.D.\n961-976.\"\n\nAnother very interesting oblong box is preserved in the same Museum, No.\n301-'66. The cover and sides are carved with scroll foliated ornament:\nthe hinges and clasp are of chased silver inlaid with niello. Round the\nsides, immediately below the lid is the following Arabic inscription in\nCufic characters:\n\n[Illustration: Arabic]\n\n\"In the name of God. This (box) was ordered to be made by Seidat Allah,\nthe wife of Abdo-r-rahman, prince of the believers. God be merciful and\nsatisfied with him.\"\n\n[Illustration: IVORY BOX. MOORISH. 10TH CENTURY. SOUTH KENSINGTON\nMUSEUM.]\n\nThis inscription must allude to Abd er Rahman III. the first Caliph of\nCordova who bore the title of Emir, el Mumenin. The formula \"God be\nmerciful,\" &c., denotes that he was dead when it was written. He died\nA.D. 961. (See woodcut.)\n\nAnother casket, undoubtedly the most important in size which is known of\nthis period, proceeds from Sanguesa, in the province of Navarre, and is\nnow preserved in the treasury of the cathedral of Pamplona.\n\nThis splendid box, hitherto undescribed, is 15 inches long, by 9\u00bc\nwide. A woodcut is given opposite. It is completely covered with\ncarvings in relief, within circular cusped medallions, with figures in\nthe centres representing different subjects: men seated, hawking, or\nstruggling with wild beasts, and numerous single figures of lions,\nstags, and other animals. The intermediate spaces contain an\nornamentation of leaves and flowers which is accommodated to the\ngeometrical style of Saracenic art. Round the upper part of this box\nappears an Arabic inscription in fine Cufic characters.\n\n[Illustration: Arabic]\n\n\"In the name of God. The blessing of God, the complete felicity, the\nhappiness, the fulfilment of the hope of good works, and the adjourning\nthe fatal period (of death), be with the Hagib Seifo daula (sword of the\nState), Abdelmalek ben Almansur. This (box) was made by the orders (of\nthe said Hagib), under the inspection or direction of his chief eunuch,\nNomayr ben Mohammad Alaumeri, his slave in the year of 395,\" [A.D.\n1005].\n\nIn the centre medallion, on the opposite side to the lock, is\nrepresented the standing figure of a man who is attacked by two lions.\nHe holds on his arm a shield, upon which is engraved an inscription,\nwith the following religious formula: \"There is no God but God,\" or a\nsimilar one, for the characters are very illegible and confused. In the\ncentre of this shield may be read [Illustration: Arabic], \"made by\nHair,\" undoubtedly one of the artists who made the box. Another artist's\nname may be read with difficulty in a similar inscription which appears\non one of the medallions on the left side: it is written on the thigh of\na stag, which is attacked by a lion [Illustration: Arabic], \"it was made\nby Obeidat.\" Three other inscriptions of a similar character appear in\nother parts of this box, which probably give the names of other\nartists, but I have been unable to decipher them.\n\n[Illustration: MOORISH CASKET. 11TH CENTURY. CATHEDRAL OF PAMPLONA.]\n\nAmong other artistic objects in the shrine of the cathedral of Braga, in\nPortugal, there is an ivory box of the same period, and the inscription,\nwhich runs round the cover, mentions the same persons:\n\n\"There is no God but God, and Mahomad is his prophet. In the name of\nGod, a blessing, prosperity and fortune for the Hagib Seifo, d. daula,\nfor this work, which he ordered to be made by the hands of\"--here the\ninscription has been broken off--\"his principal eunuch.\" (Vide 'Artes e\nLetras,' No. 6, 3rd series, p. 94, Lisbon, 1874.)\n\nBoth these caskets were made for Hadjeb Abd el Melik, a minister of\nHischem the second.\n\nIt is necessary to add to them an ivory diptych, preserved at the\nProvincial Museum of Burgos, which, as we find by the following\ninscription, was also made for Abder Rahman III, (A.D. 912-961).\n\n[Illustration: Arabic]\n\n\"This was ordered to be made by the Iman servant of God Abder Rahman,\nprince of believers.\"\n\nTwo other boxes of the same artistic character, belonging to a private\ncollection, must also be mentioned. They are both cylindrical, and are\nterminated by a spherical cover. They measure 7\u00bd inches high, by\n4-5\/8 wide, and are covered with a profuse and splendid ornamentation of\nfigures and animals. Round the lid of one them is a band, with the\nfollowing inscription in fine Cufic character:\n\n[Illustration: Arabic]\n\n\"The blessing of God and his favours, joy, and prosperity, for\nAlmogueira, son of the Prince of the faithful, whom may God have\nforgiven. It was made in the year 357 [A.D. 967].\"\n\nAlmogueira was the son of the Caliph of Cordova, Abder Rahman III.\n\nOn the other may be read in similar characters--\n\n[Illustration: Arabic]\n\n(\"In the name of God, clement and merciful, blessing\")--this part of the\ninscription is missing--\"and prosperity and happiness for Riyadh ben\nAflah, captain of the superior guard. It was made in the year 359\" [A.D.\n969].\n\nThe style of the objects which we have hitherto described is undoubtedly\nOriental, and we must seek in Persia the origin of this industry. There\nis, however, every probability that these seven ivory boxes were made in\nSpain by Spanish Arabs, or artists who had settled there from the East.\nOn all these carvings the names of Spanish historical persons appear,\nand it is hardly possible that they were ordered in remote countries,\nespecially as some of these objects are small and comparatively\nunimportant. It must also be borne in mind that we find in contemporary\nauthors many details on the luxury and magnificence of this period of\nthe Spanish Arabs, and the great height which the arts and industries\nhad reached at that time.\n\nThe ivory carvings which I have described present all the\ncharacteristics of the Oriental school, which was copied by European\nChristian sculptors during the 11th and 12th centuries. We find in\nChristian productions of this period, too constantly to require any\nfurther comment, the same geometrical traceries, flowers, leaves,\nanimals, and birds. The subjects represented on monuments of Christian\nart have been erroneously interpreted by modern ecclesiologists who have\nendeavoured to demonstrate the symbolism of these figures in a purely\nChristian sense with only the unsatisfactory result of checking and\nleading astray art students of the Middle Ages. It is of the utmost\nimportance to go to the primitive sources from which this art is derived\nin order to illustrate this important theory. The Oriental school of\nsculpture was soon transmitted, either through the influence of the\nSpanish Arabs, or by other means, to Christian artists. As an\ninteresting example of this may be mentioned the shrine which King Dn.\nSancho ordered to be made, A.D. 1033, in which to deposit the bones of\nSan Millan, still preserved in San Millan de la Cogulla, in the province\nof La Rioja, Spain. This fine shrine is 4 feet 6 inches long, by 2 feet\n3\u00bd inches high. It is of wood, and covered with gold plates and\ninlaid stones and crystals. Between this metal work are placed 22\nplaques of ivory carved with subjects representing passages from the\nlife of the saint, and single figures of the princes, monks, and\nbenefactors who helped to defray the expense of this work of art. Among\nthem there are two small figures with the names of Apparitio\nScholastico, Ramirus Rex. These have been generally supposed to be the\nartists' names. A sculptor is also represented carving a shield, and\nnear him is one of his workmen. Underneath these figures ran formerly an\ninscription, of which the first part only remains, containing the name\nof the artist--\"(Magis) tro et Rodolpho filio.\"\n\nTwo other interesting specimens remain of Spanish Moresque art of the\n11th century. One is a casket at the South Kensington Museum, No. 10,\n66, which Mr. Maskell describes as \"richly carved in deep relief with\nfoliage and animals in scrolls interlacing one another, and forming\nlarger and smaller circles. The top and each side is a single plaque of\nivory; the sloping lid at the front and back has two panels. On the two\nare two animals, like does; a large bird stands on the back of each,\nattacking it with his beak. The sloping sides have, in the large\ncircles, men on horseback, and animals fighting. The intermediate spaces\nare completely filled with foliage, and smaller beasts. Similar\nsubjects are repeated in the circles on the panels forming the lower\nsides of the casket, and among them are two groups of men and women\nsitting; one blowing a horn, another playing on a guitar, another\nholding a cup in one hand and a flower in the other.\" There is no\ninscription on this casket, but in one of the medallions on the lid\nthere is a bust which is carried on the back of a horse, and which is\nprobably a representation of the prince for whom the casket was made.\n\nThe other example of this period is that known as the cross of Don\nFernando, at the Arch\u00e6ological Museum at Madrid. This cross is 20\u00bd\ninches by 14. On the front is a figure in high relief of Our Lord with\nthe inscription: IHE. NAZARENVS REX IVDEORVM. In the upper part is\nrepresented the figure of Our Lord at the moment of his Resurrection,\nand in the lower a symbolical figure of Adam; both these figures are in\nhigh relief. Underneath may be read in two lines,--\n\n    FERDINANDVS REX.\n    SANCIA REGINA.\n\nAt the back of the cross appears in the centre the Lamb of God: in the\nfour corners the emblems of the Evangelists. Besides these purely\nChristian symbols, the ground work is covered with foliage, circles\ninterlacing each other, figures of animals of different kinds, and men\nstruggling with wild beasts. The whole of the ornamentation corresponds\nto the style of decoration of the moresque objects which I have already\ndescribed. King Ferdinand I. died A.D. 1065, his wife Sancha A.D. 1071;\nand documents exist proving that in 1063 they gave this interesting work\nof art with other artistic objects to the Church of San Isidoro of Leon,\nwhere it remained until 1870, when it was given up by the authorities of\nthe church to the Museum at Madrid. The following woodcut represents the\nback of the cross:--\n\n[Illustration: BACK OF CROSS OF KING FERDINAND I. ARCH\u00c6OLOGICAL MUSEUM.\nMADRID]\n\nThese different specimens represent, in my opinion, the most\nsatisfactory view of the art industry of ivory carving during the 10th\nand 11th centuries. Several other examples of oriental carving in ivory\nstill exist in Spain in the shrines of different cathedrals; others of\nless artistic interest may be seen at the Arch\u00e6ological Museum at\nMadrid. These caskets are for the most part not ornamented with carvings\nin relief, but are decorated with inscriptions painted in gold and\ncolours. The most important are:--\n\nA large casket at the Arch\u00e6ological Museum of Madrid, decorated with\npainting in red and green, and a fine inscription in Cufic characters.\n\nA similar casket at the same museum, decorated with a design painted in\ngreen, red, and blue; an inscription in cufic letters runs round a band\nin the upper part.\n\n[Illustration: Arabic]\n\n\"Made by Mohammad Ben Assarag.\"\n\nAn ivory casket at the Real Academia de la Historia at Madrid, with an\nornamentation and inscription painted in the style of the former ones\nand the shield of arms of the Kings of Aragon. The inscription\nreproduces several Suras of the Koran, and the arms of the house of\nAragon were probably added when the casket came into the possession of\nsome person belonging to the family.\n\nA casket exists at the Church of Santo Domingo de Silos (province of\nBurgos), which merits a special mention on account of what has remained\nto us of the inscription.\n\nThis casket is 13\u00bc inches long by 7\u00bd inches wide and high. It is\nornamented in part with foliage and flowers in the moresque style,\nalternating with hunting subjects, men shooting with bows and arrows,\nriding upon lions, fantastic animals and leopards mounted on the back of\nbulls. The work is inferior in art to the caskets at Pamplona and South\nKensington. On a band which runs round the four sides of the lid is an\ninscription in Cufic characters, of which unfortunately the two longer\nsides have been destroyed, and have been substituted at a very early\nperiod by bands of cloisonn\u00e9 enamel, evidently belonging to other\ncaskets. On the two sides an inscription remains, upon which may be read\nthe year Hegira 417 (A.D. 1026); the name of the artist who carved it,\nMohammad Ibn Zeiyan, and the two first letters of the town in which it\nwas made (probably Cuenca)\n\n[Illustration: Arabic]\n\n\".....for its owner (may God lengthen his days.) It was made in the town\nof Cu[enca] in the year 417. (A.D. 1025.) By Mohammad--ibn Zeiyan, his\nservant. May God glorify him.\"\n\nIn the geography of Edrisi, an Oriental author who describes Spain in\nthe beginning of the 12th century, only two names of towns are mentioned\nwhich agree with this inscription, Coria and Cuenca [Illustration:\nArabic]. The first of these towns was always a less important centre\nthan Cuenca. Edrisi praises the woollen fabrics made at Cuenca, and\nthere is every probability that this casket was carved there, as there\nis also an ivory monstrance at the cathedral of Perpi\u00f1an, which has\nlikewise an inscription in Cufic characters, stating it was made for the\nHageb Ismail.\n\n[Illustration: Arabic]\n\n\"The blessing of God. Made at the town of Cuenca, for the Hageb Ismail.\"\n\nVillanueva in his \"Viage por Espa\u00f1a,\" Vol. 5, p. 144, mentions two large\nivory caskets with Cufic inscriptions which still exist at the cathedral\nof Tortosa, Catalu\u00f1a.\n\nAnother of a similar description is preserved in the Treasury of the\ncathedral of Bayeux.\n\nThis casket is, as M. Andr\u00e9 tells us in a pamphlet on \"Antiquites Arabes\nde la Normandie,\" Rennes, 1869, O^{m} 42 L. by O^{m} 28 W. and 13 H. It\nis decorated with bands of enamelled metal, and covered with a fine\nrunning design of peacocks and other birds. Round the lock runs the\nfollowing inscription in Cufic characters:\n\n[Illustration: Arabic]\n\n\"In the name of God, clement, merciful, the blessing and His benefits\ncomplete.\"\n\nThese ivory caskets were made originally to hold perfumes, jewels, or\nprecious stones. For besides the Arabic inscriptions which allude to\nthis, we find the idea distinctly expressed in an inscription in Cufic\nletters on a casket which came from Cordova belonging to the Caliphate.\nThis casket was exhibited at the Paris Exhibition in 1867; I do not know\nwhere it is at present.\n\nIt appears at first sight difficult to explain why the Spanish Moors\ndecorated these objects with the representations of animated beings,\nagainst the precepts of the Koran, and the reason why these objects of\nundoubtedly Moorish origin have been preserved until the present day in\nthe treasuries of Spanish cathedrals. The prohibition of the Koran to\nrepresent animated beings is, however, not so strict as is generally\nsupposed; it is reduced to the following sentences. \"O Believers! Wine\nand games of chance, and statues and the divining arrows, are only an\nabomination of Satan's work! Avoid them that ye may prosper.\" [Sura v.\nver. 22.] Later commentators on the Koran have added the severest\nprohibitions against painters and artists who represented animated\nbeings, but to very little effect, and we find in contemporary authors\nnumerous details of the specimens of sculptures and paintings which\nwere in the houses of Moorish magnates. Coins, textile fabrics,\nfurniture, and other objects which have reached us, leave no doubt that\nthe representations of animated beings were constantly used by the\nSpanish and Eastern Arabs from the first century of the Hegira.\n\nThe fact that these Moorish caskets should have been used for preserving\nthe relics of saints in Spanish churches, is explained by the custom\ncommon in the middle ages in Spain and other countries, of offering war\nspoils and treasures brought from long and distant peregrinations, and\neven objects of natural history, to the different churches. Alligators\nmay still be seen hanging in churches in Seville, Toledo, Valencia, etc.\nThe Moors did the same thing: the famous warrior Almanssor, the minister\nof Hischem II. at the end of the 10th century, carried off the bells\nfrom the cathedral of Cordova, and had them turned into lamps and used\nat the mosque of Cordova. We find in ancient writers frequent mention of\nthe custom of Spanish Christians of offering these ivory caskets to the\nchurches, as trophies taken in their warfares with the Moors. The\necclesiastical authorities probably placed them at the time with other\nvaluable objects in the treasuries of the churches, filling them with\nrelics, for such is the manner in which they are found; they have\nremained untouched from the earliest times and are constantly mentioned\nin local histories of the cathedrals. We must not suppose that the\necclesiastical authorities ignored their Mohamedan origin, for we find\nat every step during the middle ages the names of priests who knew and\ninterpreted the Arabic language. The principal reason why these objects\nof Oriental art have been preserved, is, that the hatred of race and\nbelief between Moors and Christians was by no means as great as has been\nsupposed by modern authors, and certainly never went so far as to\ndestroy objects of industrial and artistic interest. In the year A.D.\n1275, certain privileges were granted to Moorish workmen who were set\napart and ordered to repair the Mosque at Cordova, at that time already\nconverted into a christian cathedral. During the 13th, 14th and 15th\ncenturies a large number of parish churches were built in Spain in the\nMoorish style, either by Oriental architects, or Spaniards who had\nadopted their architecture, and numerous examples might be given of\ninscriptions and details of ornamentation which confirm most fully these\ntheories.\n\nObjects of ivory carvings of the middle ages, posterior to the 12th\ncentury, are frequently met with in Spain. Among the most remarkable is\nthe Virgin de las Batallas, in the cathedral of Seville. This image\nbelonged to St. Ferdinand, early in the 13th century, and the tradition\nexists that it was carried on the king's saddle in battle. The fine\nivory diptychs at the Escorial and Arch\u00e6ological Museum at Madrid must\nalso be mentioned, and a large number of ivory caskets, and fragments,\nexisting in the same Museum and in different Spanish churches.\n\nNotwithstanding, however, the numerous examples of ivory carvings which\nare still to be met with in Spanish churches and cathedrals, I find no\ninformation which enables us to affirm that this artistic industry\nexisted in Spain during the 16th, 17th, and 18th centuries. We find\nartists mentioned who carved in wood, iron, and silver work, and\nnumerous details of their work, but ivory carvers are never mentioned,\nif any existed, their number must have been comparatively small; and I\nam led, therefore, to suppose that the specimens existing in Spain were\nimported from Italy or France, and for this reason it is necessary to\nend at the Renaissance the history of ivory carving in Spain.\n\nThe inlaid ivory work so constantly used in Spanish furniture of the\n16th and 17th centuries, cannot be included in this notice on ivory\ncarvers, owing to its limited character, and the use to which it was\nemployed. One branch of sculpture must be mentioned representing sacred\nimages, which were carved in the 16th and 17th centuries by natives of\nthe Philippine Islands or the Portuguese Colonies. They are frequently\nmet with in Spain, and are remarkable for their bad and careless\nmodelling, a mannered unartistic style, combined with the exaggerated\nrigidity so common in Chinese and Indian productions. As examples of\nthis style of art may be mentioned the representations of St. Erasmus,\nand the Immaculate Conception (Nos. 9069, '63, 183, '64), in the South\nKensington Museum.\n\n\n\n\nPOTTERY AND PORCELAIN.\n\n     Roman and Visigothic.--Hispano Moresque earthenware.--Painted,\n     glazed and lustred pottery.--Terra-cotta.--_Azulejos_ (Tile\n     decorations).--Pottery made at Talavera, Valencia, Seville, Triana,\n     Zamora, Puente del Arzobispo.--Unglazed pottery.--Bucaros.--Alcora\n     ware and porcelain.--Buen Retiro porcelain.\n\n\nROMAN AND VISIGOTHIC.\n\nThe productions of Ceramic Art have constituted from the earliest times\na very important industry in Spain. Fragments of vases of\ngreyish-coloured paste, ornamented with bands or zones, are constantly\nfound in excavations in different localities. It cannot, however, be\ndetermined whether they were importations, or imitations made in the\nSpanish Peninsula. The earliest mention which we find of this industry\nin Spain is in Pliny (Lib. xxx., cap. xii., line 19, Edition of Paris,\n1526-7), who, in praising vases of pottery made in different countries,\nmentions those of Saguntum (Murviedro) near Valencia. An epigram by\nJuvenal (Sat. v. xxix.), and several by Martial (iv. 45, viii. 6, xiv.\n108) on the same subject, prove that the pottery from the eastern coast\nof the Mediterranean was very famous at that time.\n\nCount Lumiares, in his work on pottery of Saguntum (\"Barros Saguntinos,\"\nValencia, 1779, 8vo), mentions having examined more than 1500 specimens\nof pottery of different kinds, which he classifies in four groups: grey\npottery, cream-coloured pottery, yellow pottery and red glazed ware,\nwith ornamentation in relief (Samian ware); this ornamentation\nconstitutes, in my opinion, the only distinctive feature of the pottery\nmade at Saguntum.\n\nRemains of this pottery are very frequently found in the ruins of former\nRoman cities; a much larger number of specimens of all kinds have\nappeared since Count Lumiares wrote his book, none however differ\nmaterially from the terra-cottas of the Roman period found in Italy, or\nin other countries. Some of the marks are identical with those given by\nBirch and several authors who have written on Roman pottery; hence it\nwould appear that this industry was imported into Spain, or the forms\nand marks copied there to a very large extent.\n\n[Illustration: SAMIAN BOWL.]\n\nThe number of inscriptions and potters' marks met with on the vases and\nfragments found in Spain is very great. Students who take a special\ninterest in this subject will find full details in Dr. Emile H\u00fcbner's\nimportant work, \"Inscrip. Hispani\u00e6 Latin\u00e6,\" Berlin, 1869. This writer\nhas collected the greatest number of inscriptions. He mentions 43 which\nappear on bricks, 36 on vases and drinking vessels, 63 on lamps, 579 on\nvases of red pottery, 115 inscriptions of doubtful reading, 21 which are\nincomplete, and 56 written by hand. He gives us, besides, other marks\nwithout inscriptions, representing hands, horses, rabbits, butterflies,\nbees, flowers, etc. In an inedited history of the ancient city of\nEmporion (Catalu\u00f1a) [MS., Acad. of Hist., Madrid], the author, Dr.\nJoaquin Botet y Sis\u00f3, has collected 198 potters' marks. A large number\nare still unpublished which have been met with on fragments of pottery\nfound in Extremadura and Andalucia; we may therefore confidently assert\nthat the number amounts to upwards of 1500.\n\nDuring the Visigothic monarchy, after the downfall of the Roman empire,\nwhich lasted until the invasion of the Arabs in the 8th century, the\nsame style of ceramic industry, copied from the Romans, continued in\nSpain. No special study has been made of the pottery of this epoch, but,\njudging by the large amount of fragments of vessels for domestic use,\nwhich are constantly found in ruins of the Visigothic period, there can\nbe no doubt that pottery continued to be manufactured in Spain. What San\nIsidoro, who died in 636, says in his \"Etimologies, Book xx.,\" confirms\nthis opinion, and undoubtedly refers to vessels similar to those of the\nRoman period. (_De vasis Escariis, potoriis vinariis et aquariis,\noleariis, coquinariis et luminariorum._)\n\nAfter the Roman domination and Visigothic monarchy, ceramic art in Spain\nmay be divided into the following groups:--\n\n1st. Objects imported by the Moors into Spain; these consist of remains\nof pottery belonging to the first period of their invasion; lustred\nwares, manufactured in the Peninsula, which attained great importance\nduring the Middle Ages, and still continue to be made in the present\nday; terra-cottas, and green and white glazed pottery; and lastly tiles,\n_azulejos_, of bright colours in the Moorish style.\n\n2nd. Pottery of a distinct Italian style, made principally at Talavera;\nporous, unglazed, coloured pottery, _bucaros_; white, unglazed pottery,\nmade at Andujar and La Rambla.\n\n3rd. Pottery and porcelain made at Alcora, and the porcelain manufactory\nof Buen Retiro, near Madrid.\n\n\nHISPANO MORESQUE EARTHENWARE.\n\nThe Spanish Peninsula was invaded by the Arabs about the year 711 A.D.,\nand they absorbed for several centuries the industries of the country.\nCeramic Art attained great importance in their hands during the Middle\nAges and Renaissance period, for even while the pottery works\nestablished at Talavera, Seville, and other localities, manufactured\npottery to a very great extent, which was chiefly imitated from Italian\nmodels, the Moorish style still continued, and has never been\ninterrupted in the province of Valencia down to the present day.\n\nThe Arabs had as early as the beginning of the 12th century, if not even\nbefore, established the industry of metallic-lustred pottery in Spain.\nIt remains to be seen what were the leading characteristics of the\npottery of the period of the greatest importance of the Spanish Moors\nfrom the 8th to the 11th century.\n\nCordova, the capital of the independent Caliphs of Damascus, was the\ncentre from which works of art of all kinds of a high order were largely\nexported. The ruins of the palaces at Medina Az zahra have, it is deeply\nto be regretted, never been excavated, and Granada is the only locality\nwhere some fragments of Moorish pottery may be studied belonging to this\ntime.\n\nGranada was almost unknown in the 8th century; the ancient Roman town of\nIlliberis, about six miles from the present site of Granada, had alone\nany importance: it was one of the bishoprics of Andalucia. The Arabs\nfirst settled in the remains of the Roman city: in the 10th century they\nremoved to the spot occupied by the present town, and Illiberis was then\nabandoned. Roman remains are frequently found at Illiberis, as are also\nvestiges of the Moorish occupation, chiefly consisting of objects in\nbronze and fragments of pottery. These specimens are decorated with\narabesques in green and black on a whitish ground. Some interesting\nexamples may be studied at the small _Museo Provincial_ at Granada,\nwhich certainly belong to the 10th or beginning of the 11th century. One\nof these fragments has unreadable Arabic letters; another is decorated\nwith a stag; the most remarkable is a plate fourteen inches in diameter,\nin the centre of which is represented a falcon on a horse's back. The\nform and every detail of the horse, the plaited tail, ending in the\nform of a trident, all are identical with one on which is mounted a man\nholding a hawk in his hand, carved on the ivory casket described on page\n133 as dated 359 of the hegira (A.D. 969), and probably of Cordovese\nmanufacture. Both these objects possess a very decided Persian\ncharacter, and undoubtedly belong to the date ascribed; for besides the\ncircumstance of their having been found in ruins of this period, the\nshape of the horse is conclusive; its character changes soon after, as\nwe find in the miniatures of the MS. of the 13th century at the Bib.\nNationale de Paris, containing the \"S\u00e9ances de Hariri.\" It is almost\nimpossible to assert whether this pottery was made in, or imported into\nSpain. One argument in favour of its Spanish-Moorish origin is that the\nglaze and paste are similar to pottery which we know was manufactured at\nGranada. The common earthenware proceeding from that locality is\ndecorated in much the same manner.\n\nSoon after the fall of the Caliphate, metallic-lustred ware was made in\nSpain: Edrisi, the most remarkable Arabic geographer of the Middle Ages,\nin describing Calatayud, says: \"Here the gold-coloured pottery is made\nwhich is exported to all countries.\" (\"Descrip. de l'Afrique et de\nl'Espagne,\" Leyde, 1866.) Edrisi was born in the year 1100. He studied\nat Cordova, and finished writing his book in 1154. The circumstance of\nthis pottery being mentioned in the 12th century as excellent enough for\nexportation, certainly makes it appear probable that the _fabrique_\nalready existed at an earlier date, especially when we bear in mind that\nCalatayud was conquered in the year 1120 by the Christians, and it is\nimpossible to suppose that they established an Oriental industry there\nwhich was foreign to their culture, or to the contemporary Christian\nart. This text of Edrisi's has hitherto not been properly interpreted.\nIn Jauber's translation (Edrisi, Paris, 1836-40), he interprets the word\n_guidar_, [Illustration: Arabic], _lutum purum_ of potters, as the name\nof a gilt textile, perhaps because in his time it was difficult to\nbelieve in the existence of metallic-lustred pottery. In the Edit. of\nLeyden, by Messrs. Dozy and Goeje, the sense of the word is properly\ngiven; indeed, Jauber himself translates it by _porcelain_ in the\nchapter of this volume which refers to China.\n\nThis text of Edrisi's has never yet been quoted by writers on Ceramic\nArt; but once known it is impossible for a moment to state that the\nearliest manufacture of Moorish lustred ware was that at Malaga, an\nopinion which has been supported hitherto on the quotation from Ben\nBatutah's works, which will be given later on. I regret to have found no\nother allusion to the lustred ware of Aragon until the 10th century. No\nmention is made of it in the geographical texts published by Juynboll,\nAl Makkari, or other Arabic writers. From what we know of Aragon in the\n16th century, it was a great centre of this industry, and its\nproductions rivalled those of Valencia and Andalucia.\n\nThe next text which alludes to this manufacture is given by Ben Batutah,\na celebrated traveller, who, after travelling for twenty years in the\nEast, went from Tangiers to Granada, from 1349 to 1351. Passing by\nMalaga, he says: \"At Malaga, the fine golden pottery is made which is\nexported to the furthermost countries.\" (\"Voy. d'Ibn Batoutah,\" Paris,\n1853-58, vol. iv. p. 367.) This text has been constantly reproduced and\ncommented upon, since Baron Charles Davillier first drew attention to it\nin his interesting little _brochure_ on \"Fa\u00efences Hispano-Moresques.\"\n\nThe next time I find lustred pottery mentioned is in the 15th century.\nEximenus, in his \"Regiment de la cosa publica,\" Valencia, 1499, in\nspeaking of the excellent things made in the kingdom of Valencia, says:\n\"The twenty-seventh excellent thing is that some artificial objects are\nmade there which bring great renown to the country, for they are\nexcellent and beautiful, and are now to be found in other localities ...\nbut above all is the beauty of the gold pottery so splendidly painted at\nManises, which enamours everyone so much, that the Pope, and the\ncardinals, and the princes of the world obtain it by special favour,\nand are astonished that such excellent and noble works can be made of\nearth.\"\n\nLustred pottery had already attained great importance in Aragon early in\nthe 16th century. We find in a deed granted at Calatayud in 1507, that\n\"Muhamed ben Suleyman Attaalab, an inhabitant of the suburb of the Moors\nat Calatayud, and an artificer of lustred golden earthenware, engaged\nhimself with Abdallah Alfoquey of the same locality, to teach him the\nsaid industry, in the space of four years and a half, from the date of\nthe deed.\" (\"Estado social de los mudejares de Castilla, by Fernandez y\nGonzalez,\" Madrid 1866, p. 437.) At Muel, a village in the province of\nAragon near Zaragoza, this industry existed to a great extent in 1585.\nIn the travels of Henrique Cock (\"Relacion del viage hecho por Felipe\nII. en 1585,\" por Henrique Cock, publicado por Morel Fatio y Rodriguez\nVilla, Madrid, 1876,) we find the following interesting details of the\nmanner in which this pottery was made, p. 30:--\n\n\"Almost all the inhabitants of this village are potters, and all the\nearthenware sold at Zaragoza is manufactured in the following manner.\nFirst the vessels are fashioned of a certain ingredient the earth\nfurnishes them in that locality, in the shape they may require. Once\nmade, they bake them in an oven fitted for the purpose. They then remove\nthem to varnish with white varnish and polish them, and afterwards make\na wash of certain materials in the following manner: twenty-five pounds,\n_one arroba_, of lead, with which they mix three or four pounds of tin,\nand as many pounds of a certain sand which is to be found there. All\nthese ingredients are mixed into a paste like ice; it is broken into\nsmall pieces and pounded like flour, and kept by them in powder. This\npowder is mixed with water, the dishes are passed through it, and they\nare rebaked in the oven, and keep their lustre. Afterwards, in order\nthat the pottery may be gilt, they take very strong vinegar, mixed with\nabout two reales (a small coin equivalent to 6_d._) of silver in powder,\nvermilion, and red ochre, and a little wire. When all is mixed\ntogether, they paint with a feather on the dishes any decoration they\nmay like, rebake them, and then they remain gold-coloured for ever. This\nwas told me by the potters themselves.\"\n\nBut nothing can be compared in exactitude to the following receipt of\nthe manner of preparing this lustred ware, which I was fortunate enough\nto find in a manuscript in the British Museum. (Egerton, No. 507, MS.\nfol. 102).\n\nCount Florida Blanca wishing in 1785 to establish at Madrid a\nmanufactory of metallic-lustred ware, had the following report on the\nactual state of the industry sent to him from Manises with full details\nof the manner in which it was required to be carried out.\n\n\"After the pottery is baked, it is varnished with white and blue, the\nonly colours used besides the gold lustre; the vessels are again baked;\nif the objects are to be painted with gold colour, this can only be put\non the white varnish, after they have gone twice through the oven. The\nvessels are then painted with the said gold colour and are baked a third\ntime, with only dry rosemary for fuel.\n\n\"The white varnish used is composed of lead and tin, which are melted\ntogether in an oven made on purpose; after these materials are\nsufficiently melted, they become like earth, and when in this state the\nmixture is removed and mixed with an equal quantity in weight of sand:\nfine salt is added to it, it is boiled again, and when cold, pounded\ninto powder. The only sand which can be used is from a cave at\nBenalguacil, three leagues from Manises. In order that the varnish\nshould be fine, for every _arroba_, 25 pounds of lead, 6 to 12 ounces of\ntin must be added, and half a bushel of fine-powered salt: if a coarse\nkind is required, it is sufficient to add a very small quantity of tin,\nand three or four _cuartos_ worth of salt, which in this case must be\nadded when the ingredient is ready for varnishing the vessel.\n\n\"Five ingredients enter into the composition of the gold colour:\ncopper, which is better the older it is; silver, as old as possible;\nsulphur; red ochre, and strong vinegar, which are mixed in the following\nproportions: of copper three ounces, of red ochre twelve ounces, of\nsilver one _peseta_ (about a shilling), sulphur three ounces, vinegar a\nquart; three pounds (of twelve ounces) of the earth or scori\u00e6, which is\nleft after this pottery is painted with the gold colour, is added to the\nother ingredients.\n\n\"They are mixed in the following manner: a small portion of sulphur in\npowder is put into a casserole with two small bits of copper, between\nthem a coin of one silver _peseta_; the rest of the sulphur and copper\nis then added to it. When this casserole is ready, it is placed on the\nfire, and is made to boil until the sulphur is consumed, which is\nevident when no flame issues from it. The preparation is then taken from\nthe fire, and when cold is pounded very fine; the red ochre and scori\u00e6\nare then added to it; it is mixed up by hand and again pounded into\npowder. The preparation is placed in a basin and mixed with enough water\nto make a sufficient paste to stick on the sides of the basin; the\nmixture is then rubbed on the vessel with a stick; it is therefore\nindispensable that the water should be added very gradually until the\nmixture is in the proper state.\n\n\"The basin ready prepared must be placed in an oven for six hours. At\nManises it is customary to do so when the vessels of common pottery are\nbaked; after this the mixture is scratched off the sides of the basin\nwith some iron instrument; it is then removed from there and broken up\ninto small pieces, which are pounded fine in a hand-mortar with the\nquantity of vinegar already mentioned, and after having been well ground\nand pounded together for two hours the mixture is ready for decorating.\nIt is well to observe that the quantity of varnish and gold-coloured\nmixture which is required for every object can only be ascertained by\npractice.\"\n\nExcellent lustred ware was made at Murcia, and in several villages of\nthe province of Valencia; a good example exists of this pottery at the\nmuseum (see woodcut); none, however, was so important as the manufactory\nof Manises. Diago, in his \"Anales del reyno de Valencia, 1613-40,\"\nrepeats the words of Eximeno: \"That the pottery made at Valencia is\npainted and gilt with so much art, that it has enamoured every one, so\nmuch so that the pope, the cardinals, and princes send for it,\nastonished that things of such beauty can be made of earth.\"\n\n[Illustration: PLATEAU, PROBABLY OF VALENCIA. SOUTH KENSINGTON MUSEUM.]\n\nThis and similar texts relating to the exportation of Spanish pottery\nexplains the fact of these specimens being so frequently met with in\nItaly. Pottery of a dark copper-coloured lustre is made at the present\ntime at Manises, and the imitations and forgeries to be met with at the\ndealers' shops in Madrid are made in that locality.\n\nI have mentioned the different historical texts which allude to this\nindustry, and the technical receipts showing the ingredients employed to\ngive the lustre. It is now no longer possible to doubt that this pottery\ncame originally from the East. This opinion has gained ground of late\nyears; for among the objects discovered by Sir Henry Layard at Nineveh,\nand in more recent excavations in Ephesus and Asia Minor, fragments of\npottery have been found with metallic lustre. This and other important\nexamples of artistic culture undoubtedly arose in Persia. No examples\nwhich may be mentioned can give a better or more distinct idea of this\nthan a comparison between the Persian and Hispano-Moresque wares at the\nSouth Kensington Museum. The technical proceedings and effect produced\nby the metallic lustre on a white ground, with touches here and there of\nblue, are exactly the same in both cases. The principal difference\nconsists in a greater richness and variety of ornamentation which we\nfind in the specimens brought from Persia, especially those covered with\na turquoise blue lustre, those of a dark blue and metallic lustre, and\nthose on which gold is applied _en froid_, of which I know no similar\nexamples in Spain. It is possible they may have existed in the mosques,\nbut no traces remain of mosques built by the Spanish Moors during the\n13th, 14th, and 15th centuries. The mosques at Cordova and Toledo which\nstill exist are of an earlier date. If we judge by the mosque of Amru at\nCairo, and others of the same early period, tile decorations were not\nused in their ornamentation, and it appears probable that this industry\nonly became important after the 10th century. The similarity of objects\nused in Spain and Persia is constantly observed by travellers in that\ncountry. Ruy Gonzalez de Clavijo, who was sent there as ambassador from\nthe King of Castille from 1403 to 1406, gives a most interesting\ndescription of tile decoration, by which we find they were used in the\nsame manner as in Spain. (\"Historia del gran Tamorlan,\" Seville, 1582,\nfol.) Another Spanish traveller, Silva de Figueroa, in a MS. belonging\nto Don Pascual de Gayangos, in describing his embassy to Persia in 1618\nmentions the manner in which these tiles were used. I do not quote from\nmany other travellers, for their remarks are hardly so valuable as those\nof Spaniards, who were accustomed to see these objects in their native\ncountry.\n\nAfter mentioning the historical sources from which I have traced the\nmanufactory of Hispano-Moresque lustred ware in Spain, it is advisable\nto enumerate the most important examples which have reached us belonging\nto the 14th century, or to an earlier date.\n\nThe fine vase at the Alhambra, Granada. (See woodcut.)\n\nA vase of the same character which exists at the Arch\u00e6ological Museum of\nMadrid.\n\nA vase of the same kind, which belonged to the painter, Mariano Fortuny,\nand which was sold at the sale of his art objects in Paris, April 30th,\n1875. (_Vide_ \"Atelier de Fortuny,\" Paris, 1875.)\n\nA fragment of another vase, which belonged also to Fortuny, and was sold\nat the same time.\n\nA fragment of a vase at the Museo of Granada; it is similar to the\nothers in form and size, but without metallic lustre.\n\nA large plaque, or _azulejo_, which belonged also to Fortuny.\n\nThe _azulejos_, or tile decorations of the house known by the name of\nCuarto Real de Santo Domingo at Granada.\n\nSeveral dishes and bowls at the Kensington, Madrid, and Cluny Museums,\nwhich belong to this period.\n\nBaron Charles Davillier in his article on Hispano-Moresque lustred ware,\nin \"Atelier de Fortuny,\" Paris, 1875, says that interesting specimens of\nthis pottery exist also at the Museum of Stockholm, and the Cathedral\nof Mezzara in Sicily, but that he had not had the opportunity of\nstudying them.\n\n[Illustration: MOORISH VASE AT THE ALHAMBRA.]\n\nAll these objects have hitherto been considered to have proceeded from\nMalaga, writers on ceramic art derived their evidence from the text\nalready mentioned, given in Ben Batutah's travels, the only one which\nalluded to the existence of this industry in Spain before the 15th\ncentury; but since I have found in Edrisi's Geography a manufacture\nmentioned of lustred pottery which existed at Calatayud at the beginning\nof the 12th century, that is to say, more than _two centuries_ earlier\nthan Ben Batutah's travels, I consider it extremely difficult to\nclassify them.\n\nThe fine vase at the Alhambra is 4 feet 5\u00bd inches high by 8 feet\n2\u00bd inches in circumference. It is decorated in the centre with two\nantelopes, and a series of elegant traceries of knots, stems, and leaves\nwhich cover the body of the vase. The colours employed are brown and\nblue on a yellow ground, the metallic lustre is extremely pale, of a\nmother-of-pearl colour. The following inscriptions are repeated all over\nthe vase in African characters:\n\n[Illustration: Arabic\n\nFelicity and Fortune.]\n\n[Illustration: Arabic\n\nPermanent prosperity.]\n\nA similar vase was at the Alhambra until the beginning of the present\ncentury; it disappeared at that time, and its present whereabouts is\nunknown. Drawings exist of this vase which have been published in\nLozano's \"Ant. Arab. de Espa\u00f1a,\" and Murphy's Atlas of \"Arabian Ant. of\nSpain.\"\n\nThe fine vase at the Museo Arq. of Madrid is similar in style. It has\nbeen reproduced in a chromo-lithograph in Mus. Esp. de Ant. VI. p. 435.\nBoth these vases are decorated with colours which are disposed in quite\na distinct manner to all the other specimens of this pottery hitherto\nknown. The vase at Granada is ornamented in the centre with two\nantelopes, and from drawings which have reached us of the companion\nvase, we find birds are introduced in the decoration of the handles.\nAnimals combined with ornamentation are never met with in the immense\nnumber of Moorish traceries of all kinds which may be studied at\nGranada. This circumstance has induced me to consider these objects to\nhave been brought from Persia, for Ben Batutah tells us that several\nPersians of importance had settled at Granada; and it is highly\nprobable that a direct communication existed between these two\ncountries. It is fair, however, to mention a detail which is against\nthis argument. The arms of the Moorish kings of Granada appear on the\nvase which has disappeared from Granada, and although it may have been\nordered from Persia, it is also possible that it was made in the\nprovince of Andalucia.\n\nThe vase which belonged to Fortuny, and the large _azulejo_, Nos. 42, 44\n(_vide_ \"Atelier de Fortuny\"), are very different in style. They have no\ncolours, and their metallic lustre is very low in tone, a common\ncircumstance in Hispano-Moorish pottery. The following inscription,\n\n[Illustration: Arabic]\n\n[Illustration: TILE PROBABLY FROM THE ALHAMBRA; FORMERLY IN THE MARRYAT\nCOLLECTION.]\n\n\"Glory to Our Lord the Sultan Abul Hajaj,\" [A.D. 1333-1354.] occurs on\nthis tile, an inscription very frequently met with also on the walls of\nthe Alhambra. Abul Hajaj carried out works of restoration to a very\nlarge extent at the Palace. The two large tiles on either side of the\nentrance-door of the Cuarto Real de Santo Domingo at Granada, although\nnot so fine as the vase and tile which belonged to Fortuny, are similar\nin general character, and it is safe therefore to consider these\nobjects, and others of a similar kind, to have proceeded from the\nmanufacture of Malaga mentioned by Ben Batutah, or other pottery works\nof the same kind, which probably existed at that time in the province of\nGranada.\n\nMetallic-lustred dishes have at times an even surface without\nornamentation in relief, and sometimes are ornamented with ribs and\nconvex dots, which appear to suggest that they are meant to imitate the\nstructure of nails, bars, or other work common to metal vessels, in the\nmanner of Etruscan vases.\n\nThe finest specimens of Hispano-Moresque ware at the Kensington Museum\nare--\n\nNo. 8968-'63. A vase which belonged to the Soulages Collection. Mr.\nFortnum, in his \"Majolica, Hispano-Moresco, Persian, Damascus and\nRhodian Wares, at the Kensington Museum,\" London, 1873, describes this\nvase as spherical on a trumpet-shaped base, the neck of elongated funnel\nform, flanked by two large wing-shaped handles perforated with circular\nholes. The surface, except the mouldings, is entirely covered with a\ndiaper-pattern of ivory or briony leaves, tendrils and small flowers in\nbrownish lustre, and blue on the white ground. Spanish. 14th or 15th\ncentury. (See woodcut.)\n\nNo. 486-'64. Bowl, funnel-shaped, with representation of a ship in full\nsail, with the royal arms of Portugal, 15th century.\n\nThe following are also especially worthy of mention:--\n\nNo. 7659-'62. Bowl and cover, painted with a scroll diaper in alternate\ncompartments of gold lustre and blue, the cover surmounted by a\ncupola-shaped ornament in gold lustre. 16th century.\n\n[Illustration: VASE DIAPERED WITH IVY OR BRYONY IN GOLDEN LUSTRE.\nHISPANO-MORESCO, 14TH OR 15TH CENTURY. SOUTH KENSINGTON MUSEUM.]\n\nNo. 489-'64. Plateau with lustred arabesque ornaments in compartments.\nIn the centre are two simulated Arabic inscriptions. 15th century.\n\n[Illustration: PLATEAU, DIAPERED AND WITH RAISED RIBS AND STUDS, IN BLUE\nAND GOLDEN LUSTRE. THE ARMS OF LEON, CASTILE, AND ARAGON.\nHISPANO-MORESCO. 15TH OR 16TH CENTURY. SOUTH KENSINGTON MUSEUM.]\n\nNo. 1680-'55. Plateau, gold shield in the centre, with the arms of Leon,\nCastille, and Aragon. The diapered ground is curiously ribbed, and\ndotted with raised studs. (See woodcut.)\n\nNo. 243-'53. Plateau, in the centre an escutcheon of arms of Aragon,\nLeon, and Castile. (See woodcut.)\n\n[Illustration: SALVER, DIAPERED IN GOLDEN LUSTRE, WITH THE SHIELD OF\nARAGON, LEON, AND CASTILE. HISPANO-MORESCO. 15TH OR 16TH CENTURY. SOUTH\nKENSINGTON MUSEUM.]\n\nNo. 104-'69. Vase, with shield of arms, seemingly of Sicily and\nPortugal.\n\nSpecimens of a later date are distinguished by the redness of the\nlustre, such as No. 327-'66, a bowl, with feather-like ornament in the\ncentre; Nos. 326-'66 and 328-'66 are good examples of this lustred ware\nof the last and early present century made at Manises, Valencia.\n\nIt is almost impossible to classify the existing specimens of\nHispano-Moresque pottery. A few examples are marked with the monogram of\nManises, or possess some special sign, such as the Eagle of St. John,\nwhich would make it appear they were made in the province of Valencia,\nbut as the texts from ancient authors only praise this pottery and give\nno details of its general characteristics, it is very difficult to\nassign any given specimens to the manufactories of Aragon and Valencia.\nThe receipts which I have given prove that the ingredients used were the\nsame, the modification in lustre was owing to the different combinations\nof silver and copper. The system hitherto adopted of classifying this\nware by the localities in which it has been purchased is not a safe one,\nunless borne out by further evidence. All that can be done at the\npresent time is to point out the oldest specimens, and probable period\nof their manufacture. As a general rule the finest specimens are also\nthe oldest, for this industry decayed after the Moors were expelled from\nSpain, and Italian pottery was imported into that country in the 16th\ncentury.\n\nThe opinion which has been so generally accepted that gold lustred\npottery was made and exported from the Balearic Islands has been\ncontradicted by D^{n}. Alvaro Campaner y Fuertes in a letter addressed to\nBaron Davillier which appeared in the \"Museo Balear,\" Palma, 1875, in\nwhich he refutes the Baron's arguments on the pottery which he supposes\nto have come from that locality.\n\n\nTERRA COTTAS.--_AZULEJOS_, OR WALL DECORATIONS OF\nTILES.--EARTHENWARE.--BUCAROS AND UNGLAZED POTTERY.\n\nThe earliest historical notice which I have met with of Terracottas is\nin a grant made in A.D. 1222, by King Don Fernando el Santo, in favour\nof the manufacturers of bricks or tiles at Talavera (M. S. Bibl.\nNacional Madrid, Dd. 114, p. 17), and another document of a similar kind\nby his son, King Don Alonso el Sabio A.D. 1281, in favour of the\nCathedral of Cordova, \"that no pottery works should be allowed inside\nthe town, unless they were established in the same manner as in the time\nof the Moors.\" (Mem. Hist. Espa\u00f1ol, t. ii. p. 45). Although this\ninformation is not of great importance, it enables us to infer that in\nthe 13th century an industry existed in Spain which was closely\nconnected with pottery. Baron Davillier mentions a privilege granted by\nKing D^{n}. Jaime de Aragon to the inhabitants of Jativa, province of\nValencia, in which he commands, \"that any master potter who made jars,\npots, pans, tiles, and bricks, should pay a tribute of one _besant_\nyearly for each oven, and be permitted to carry out his industry with no\nother obligation or service.\" (Col. Doc. Ineditos, vol. xviii. p. 67.)\n\nThe details concerning this industry in Catalu\u00f1a are more interesting;\nespecially on account of the references given to the marks of the\nobjects proceeding from this manufacture. The potters, who were known by\nthe generic name of _olleros_ sent, in 1257, two individuals to form\npart of the town council. They formed at that time a guild; in 1314 they\nagree in their regulations as to the mark to be adopted for their\npottery, and the actual spot is mentioned on which every potter was to\nstamp his wares. This stamp was ordered to be very visible, and\ninstructions are given as to the quality of the clay, and its\nmanipulation, in order to produce pottery of a first-class order. We\nfind statutes in this sense up to 1355. But we cannot find in the \"Libro\nde Ordenanzas\" of the potter's guild that any laws existed touching the\ntechnical details of this manufacture; most of the regulations refer to\nthe management of the guild from 1304, which was composed of potters,\nbricklayers, jar-makers, and manufacturers of pottery for domestic\npurposes of Barcelona. (Capmany, Memorias Historicas, 1779-92, vol. i.\npart 3, p. 69.)\n\n[Illustration: WINE JAR--\"TINAJA.\" SOUTH KENSINGTON MUSEUM.]\n\nThese references are not, however, sufficient to identify the kind of\npottery made at that time in Catalu\u00f1a. It is highly probable that\nlustred ware was also made there to a great extent, not only because it\nwas the most characteristic produce of the Peninsula, but also because\nthis ware was exported to a great extent, and highly esteemed.\n\nThe most important examples of Spanish terra-cotta which have reached us\nare _tinajas_, large jars for holding wine and oil.\n\nOne of the best _tinajas_ which I know is at the South Kensington\nMuseum, No. 330-'66. (See woodcut.) It is a wine jar, amphora-shaped,\nand ornamented with incised pattern of vine leaves, and stamped diaper\nof a Gothic character. This and similar specimens have always been found\nin Andalusia and the province of Toledo, and hence we may infer that the\nchief centres of this industry existed at Toledo and Seville. They\ncontinue to be made there, and are used for the same purposes as in the\nMiddle Ages, for keeping stores of wine, oil, meat, grain, etc.\n\nBelonging to this section of objects of a large size, are some specimens\nof glazed earthenware made for brims of wells. I only know four or five\nexamples in the local museums of Toledo and Cordova; the font of the\nchurch of San Salvador, Toledo, belongs also to the same pottery. The\nbrim of a well, at the South Kensington Museum, No. 1763-'71, is of the\nsame manufacture. It was bought at Toledo for three guineas at a\nshoemaker's shop. It is made of glazed white and green earthenware, with\nornamental Cufic characters in high relief all round, which appear to be\nof the 14th century (see woodcut on next page). The inscription, which\nis repeated, is imperfect, and all that I can decipher are the words:\n\n[Illustration: Arabic\n\n\"The power, the excellence, and the peace.\"]\n\nThere is no difficulty in affirming that this form of earthenware was\nmade at Toledo, for Marineo Siculo, in his interesting volume treating\nof \"De las Cosas Memorables de Espa\u00f1a,\" Alcala, 1539, says, \"much coarse\ngreen and white pottery is made at Toledo.\"\n\n[Illustration: BRIM OF WELL FROM TOLEDO. SOUTH KENSINGTON MUSEUM.]\n\nThe manufacture of tiles, as was the case with so many other industries\nin Spain, was imported by the Arabs. In the Middle Ages it attained\ngreat importance, and has never ceased in Spain up to the present day.\nThe earliest tiles (_azulejos_) made in Spain are composed of small\npieces let into the wall, forming geometrical patterns. The proceeding\nis similar to that employed by the workers of Byzantine mosaic,--tile\ndecoration undoubtedly took the place of this mosaic work in Moorish\nbuildings. It is difficult to fix the precise period when they were\nfirst made, but it must have been posterior to the 10th century, when\nMoorish architecture underwent a radical change in its system of\ndecoration. The oldest tiles which exist in Spain are at the Palace of\nthe Alhambra, Granada. They belong to the 14th century. The great\nvariety of their design may be seen in Owen Jones' \"Plans and elevations\nof the Alhambra,\" London, fol. 1842, and the fine work published by the\nSpanish government, \"Monumentos Arquitectonicos.\" We can judge of the\nstyle adopted by two interesting fragments, both of which came from the\nAlhambra, No. 1104-'53, No. 324-'64, South Kensington Museum.\n\nThe earliest mention which I have found of this manufacture occurs in a\nletter from Do\u00f1a Juana de Mendoza, the wife of the Almirante de\nCastilla, which is addressed to the lady abbess of the nunnery of Santo\nDomingo at Toledo. She begs that a large number of \"azulejos\" of\ndifferent colours, black, white, yellow, and green should be sent to\nher. She alludes, in the same letter, to painted tiles, and says she was\nexpecting a master potter from Seville to place these tiles in their\nproper places. This shows us that it was only in the province of\nAndalusia that the art was known of cutting these tiles into geometrical\nsections and mosaic patterns. This letter is not dated, but it is bound\nup with other documents of 1422, and evidence exists to prove that both\nthe Almirante and his wife were dead in 1431. The locality for which the\ntiles were required was probably Palencia; the convent of Santa Clara\nwas built by them at that time, and they both were buried there. It was\nextremely difficult to cut and join together these tiles; the workmen\nwho did so had to pass through a regular examination in compliance with\nthe municipal ordinances drawn up in the 15th century; without this\nrequisite they were not allowed to exercise their trade.\n\nWe know, from documents existing at the Archives of the Alhambra, at\nGranada, that tiles were made there at the end of the 16th and beginning\nof the 17th century. We find the names of Antonio Tenorio, Gaspar\nHernandez, and Pedro Tenorio working there at that time as master\npotters.\n\nThe use of these _azulejos_ was not restricted to Moorish buildings;\nthey were largely employed to decorate the walls of Christian churches,\nconvents, palaces, and private houses. The style of work, however, soon\nchanged, for instead of being made of mosaic work, they were formed of a\nsingle piece like the ordinary tiles, but imitating the Moorish colours.\nIn the 16th century designs of the Renaissance style alternated with\nthese, although the system and colouring continued the same. This style\ncontinued in vogue until the expulsion of the Moriscos in 1610. A fine\ncollection may be seen at the South Kensington Museum of 200 tiles, in\nwhich upwards of 150 varieties of design will be found, Nos. 308-'66 and\n367-'66. Their chief characteristic is that only pure colours are used\nwithout gradations or half tints. Lustred reflets are very seldom met\nwith in tiles; the only examples I know are two beautiful plaques of the\n14th century of a pale lustre which are on either side of the entrance\ndoor of the Moorish saloon called Cuarto Real de S^{to}. Domingo, at\nGranada; some small polygons of copper-coloured lustre at the Tower of\nla Cautiva, Granada: some few details in the coats of arms which are let\ninto the tiles which adorn the walls of the Chapel at the Alhambra, and\nsome few tiles with the arms of the Mendozas at the Casa de Pilatos,\nSeville.\n\nA few dishes belong also to the 15th century, ornamented with the same\ncolouring as the tiles without metallic lustre, and generally decorated\nwith fantastic animals. They are very scarce. Two examples exist at the\nSouth Kensington Museum, Nos. 1459 and 1461-'70. They probably were made\nat Seville; all those I have seen come from there, and we know that the\npottery from Seville was much esteemed. In Pedro de Medina's \"Libro de\nGrandezas y Cosas Memorables de Espa\u00f1a,\" Sevilla, 1549, fol. Iii., he\nsays, \"In this town of Triana much excellent pottery of Malaga is made,\ncoloured white and yellow, and of different sorts and kinds. This\npottery is made in about fifty houses, and it is exported from here to\nmany localities. In the same manner excellent _azulejos_ are made, of\ngreat variety of colour and design. And likewise fine reliefs of men\nand other things. Great quantities of these _azulejos_ are taken to\ndifferent localities.\"\n\nAt the introduction of Italian forms of art in Spain, at the beginning\nof the 16th century, artistic pottery divided itself into two\nschools--one kept to the traditional designs and strong tones of colour,\nthe other used shaded colours, and especially yellow and blue tints, at\ntimes imitating Italian maiolica, but always following the decoration\ncharacteristic of the Renaissance style.\n\nThe Italian influence, and its finest results in Spain may be favourably\nstudied in Seville, in the tile decorations of the chapel at the\nAlcazar, the doorway of the convent of Santa Paula, and a remarkable\nsepulchre decorated with tiles at the church of St. Anne at Triana. All\nthese tile decorations are signed by the same person, and it would be\ndifficult to find any more remarkable.\n\nThe _azulejos_ at the Alcazar represent a picture in the manner and\nstyle of Pietro Perugino; they are admirable in colour and design. Those\nwhich decorate the sepulchre at the church of St. Anne have the\nfollowing inscription, \"_Niculoso Francesco italiano me fecit, en el\nagno del mil ccccciii._\" Some very interesting tiles also may be seen in\none of the rooms in the lower part of the Alcazar; they are marked in\nthe following manner: A.V.S.T.A.--A.V.G.W.S.T.A. 1577-1578. On the floor\nof the pavilion of the Emperor Charles V. in the garden there are also\nsome fine tiles which are signed Juan Hernandez, 1540.\n\nWe find mention in Cean Bermudez of a painter of _azulejos_ named Juan\nFlores, a native of Flanders; he painted tiles for the royal palaces of\nMadrid, the Pardo and Segovia; he was appointed Maestro Azulejero by\nPhilip II. in 1565.\n\nReturning to other examples of earthenware we meet with the ancient\nmanufactures of the Balearic Islands, although it is not easy to\ndetermine the kind of pottery which was really made there. Bernardo da\nUzzano, in 1442, made a commercial treaty, (vide Davillier, \"Fa\u00efences\nHispano Moresques,\" p. 26), in which he mentions the \"fa\u00efence\" made at\nMallorca and Minorca, which \"was exported to a very large extent.\" The\nroyal Ordinances for the island of Iviza, which have not been mentioned\nby either of these authors, state that the principal traffic and the\ngreatest industry of ancient times of the island of Iviza consisted in\nsome vessels of the finest earthenware, well baked and curiously worked,\nof which an innumerable quantity were sent off to Africa and other\nplaces, not only on account of the excellence of their worth, which was\nadmirable, but the speciality of the clay, which preserved from poison.\n(\"Reales Ordinaciones de Iviza,\" Palma, 1751).\n\nManufactures of pottery also existed in a great number of localities in\nSpain; the most important centres were at Talavera, Valencia, and\nTriana. Judging by the manner in which different authors praise the\nearthenware of Talavera, its productions were evidently the most\nesteemed. The earliest mention we find of Talavera pottery occurs in a\nMS. volume dated 1560--a history of this town, in which mention is made\nof \"white, green, blue and other coloured Talavera ware.\" In the\ninventory of the effects of D^{na}. Juana, a sister of Philip II., 1573,\nseveral objects of \"white earthenware of Talavera\" are mentioned. In a\nreport drawn up by order of Philip II. 1576, it is stated that Talavera\n\"produced fine white glazed earthenware-tiles, and other pottery, which\nsupplied the country, part of Portugal and India.\" Father Ramon de la\nHiguera in his \"Republicas del Mundo,\" 1595, mentions the ware of\nTalavera in terms of the highest praise. In a MS. history of Talavera\nwritten in 1651, (Bib. Nac. Madrid, G. 112,) the author, Father Alfonso\nde Ajofrin, says that \"the pottery is as good as that of Pisa, a large\nnumber of _azulejos_ are also made to adorn the front of altars,\nchurches, gardens, alcoves, saloons, and bowers, and large and small\nspecimens of every kind. Two hundred workmen work at eight different\nkilns. Four other kilns are kept to make common earthenware. Red porous\nclay vases and drinking cups are baked in two other kilns, in a thousand\ndifferent shapes in imitation of birds and other animals, also\n_brinqui\u00f1os_ for the use of ladies, so deliciously flavoured that after\ndrinking the water they contained, they eat the cup in which it was\nbrought them.\" In another MS. history of Talavera (Bibl. Nac. G. 187) we\nfind mention of \"perfect imitations of oriental china,\" and that the\npottery made there, \"was used all over Spain, and sent to India, France,\nFlanders, Italy and other countries, and was esteemed everywhere for the\nperfection of the colouring and brilliancy of the glaze.\"\n\nThis information may be increased by quoting another interesting MS. on\nthe history of this town (Bib. Nac. F. 142) in which the writer, Fr.\nAndres de Torrejon, who professed in the convent of Santa Catalina in\n1568, says:--\"The earthenware pottery made here has reached to a great\nheight of perfection; it is formed of white and red clay. Vases, cups,\n_bucaros_ and _brinqui\u00f1os_ are made of different kinds, dishes and table\ncentres, and imitations of snails, owls, dogs and every kind of fruits,\nolives, and almonds. These objects are painted with great perfection,\nand the imitations of porcelain brought from the Portuguese Indies are\nmost excellent. Every one wonders that in so small a town such excellent\nthings should be made. The varnish used for the white pottery is made\nwith tin and sand, it is now found to be more acceptable than coloured\nearthenware, so much so, that persons of importance who pass by this\ntown although they have in their houses dinner services of silver, buy\nearthenware made at Talavera, on account of its excellence. The sand\nwhich was used to make the white varnish was brought from Hita, it is\nnow found at Mejorada, near Talavera. This sand is as fine and soft as\nsilk.\"\n\n\"The red pottery made at Talavera is much to be commended, for besides\nthe great variety of objects which they make, the different medals which\nthey place upon them, they have invented some small _brinqui\u00f1os_ of so\nsmall and delicate a kind, that the ladies wear them. Rosaries are also\nmade of the same material. A certain scent is added in the manufacture\nof this pottery which excites the appetite and taste of the women, who\neat the pottery so frequently that it gives great trouble to their\nconfessors to check this custom.\"\n\nIn a volume of \"Relaciones Topograficas de los Pueblos de Espa\u00f1a hechas\nen 1576,\" (MS. in fol. t. ii.) we find, in speaking of Talavera, that\nthe author says: \"What is most excellent there is the white pottery, and\ntiles and other objects of this earthenware with which the kingdom and\npart of Portugal are provided.\"\n\nIn Larruga's \"Memorias Politicas y Economicas,\" (vol. x. Madrid, 1741,)\nwe find, p. 22, that \"the manufactory of fine earthenware of Talavera de\nla Reyna continued to make much pottery of importance until 1720: eight\nkilns existed then, which employed more than 400 persons, men, women and\nchildren. These manufactories possessed at that time large sums of\nmoney. From that time this industry decayed to such an extent, that in\n1730 only four kilns existed where pottery of a very inferior kind was\nproduced.\"\n\nThe industry was revived later on in the century, but the finest\nspecimens are of an earlier date. Larruga adds, in p. 17, vol. x., of\nhis \"Memorias,\" that \"Talavera ware is of a lighter clay than the\nimitations of this earthenware which were made in other localities.\"\n\nPottery of a similar kind was also made in different other localities of\nSpain. Mendez Silva says that at Puente del Arzobispo, near Toledo,\n\"fine pottery is manufactured in about 8 kilns, which produce more than\n40,000 ducats yearly.\" (\"Relacion General de Espa\u00f1a,\" Madrid, 1645, fo1.\n32.) Paton likewise in his \"Historia de Jaen,\" 1628, writes that \"the\nwhite unglazed earthenware made at these towns, is very remarkable for\nthe curious manner in which they imitate different figures of animals,\nsuch as porcupines, fish, syrens, tortoises, &c.\" Objects of a similar\nkind are still made in Spain.\n\nThe English traveller Clarke, in his \"Letters concerning the Spanish\nNation during the years 1760-1761,\" London, 1763, p. 263, says, \"the\npottery fabrics are very numerous and excellent, particularly\nTalavera.\" And in \"Annales de l'Espagne,\" by Alvarez de Colmenares,\nAmsterdam, 1741, we find, p. 187, in speaking of Talavera, that \"On y\nfait des ouvrages verniss\u00e9s d'une fa\u00e7on ing\u00e9nieuse, avec des peintures\nde bon go\u00fbt; on estime ces ouvrages autant que ceux de Pise et des Indes\nOrientales; on en fournit plusieurs provinces. Ce n\u00e9goce rend plus de\n50,000 ducats par an.\"\n\nUdal ap Rhys in his \"An Account of the Most Remarkable Places and\nCuriosities in Spain and Portugal,\" London, 1749, in speaking of\nTalavera says, \"It is noted also for a very curious kind of earthenware\nthat is made in imitation of China.\"\n\nAlthough we find by the remarks we have quoted from contemporary authors\nthat earthenware of every description was made at Talavera, the\nspecimens which are more generally met with may be divided into two\ngroups, which are painted on a white ground, either in blue, or in\ncolours in the manner of Italian maiolica. The most important examples\nwhich have reached us consist of bowls of different sizes, dishes,\nvases, tinajas, holy-water vessels, medicine jars, and wall decorations.\nBlue oriental china was imitated to a vast extent, the colouring was\nsuccessful but the design was an imitation of the baroque school of the\ntime, and the figures, landscapes, and decoration follow the bad taste\nso general in Spain in the 18th century. The imitations of Italian\nmaiolica are effective. The colours most commonly used are manganese,\norange, blue and green.\n\nSeveral specimens of this manufacture are in the South Kensington\nMuseum; among these may be noted--\n\nNo. 327-'76. A large vase painted in blue with cavaliers and men on foot\nin a hilly landscape, on the shoulder are the words \"Silva Coronel.\"\n\nNo. 1282-'71. Holy-water vessel, with raised flowers and scrolls of\nfoliage with polychrome decoration.\n\nNo. 1281-'71. Soup tureen and cover, white, with raised scrolls and\ngroups of painted flowers.\n\nNo. 351-'76. Another soup tureen with cover and stand, painted with\ngarlands and the arms of Portugal. On the cover is a group of fishes and\nshells in relief.\n\nNo. 1279-'71. A bowl decorated within with a bull-fight; outside are\nstorks and trees in green, orange, and manganese. (See woodcut.)\n\n[Illustration: EARTHENWARE BOWL. TALAVERA WARE. SOUTH KENSINGTON\nMUSEUM.]\n\nSeveral pottery works were established in Spain in the 18th century, all\nof them, in the same manner as the earlier fabriques, modified the\nsystem of decorating their wares. In some instances the colours and\ndesigns of Italian maiolica were imitated, others copy the blue\nfa\u00efences of Pisa, Genoa, and Savona, while others adopted the styles of\nMoustiers, Nevers, and Rouen, or English earthenwares.\n\nTen or twelve manufactories existed at Toledo in the 17th century which\nimitated Talavera ware; in the 18th they hardly produced anything of\nimportance. Ignacio de Velasco in 1735 founded one at great expense at\nToledo in which imitations of Genoese pottery were chiefly made. At the\ndeath of Ignacio in 1738, these works passed to his son George; in 1742\nFrancisco Hernandez directed them, and in 1747 imitated Japanese models.\nSeveral specimens proceeding from Toledo, at the South Kensington\nMuseum, painted blue on a white ground, in the style of Savona and\nJapan, belong to this period.\n\nIn 1755 thirteen pottery kilns existed at Puente del Arzobispo near\nToledo; they still worked in 1791, but their productions were very\ninferior in artistic merit.\n\nEarthenware pottery was made at Segovia from a very early period,\nchiefly for domestic use, until a manufactory was founded by two\nbrothers--Manuel and Tom\u00e1s Ledesma in 1752--they had seen some specimens\nwhich were made at Bolonia for Isabel Farnesio, the widow of King Philip\nV., and they endeavoured to imitate them. In 1774 they tried with a most\nunsatisfactory result to imitate English wares. This industry fell into\ndecay towards the end of the century, and only ware of a very common\ndescription was made there.\n\nTalavera ware was also imitated at Zamora. At the middle of the 18th\ncentury works existed there where pottery was made in the manner adopted\nat Alcora with few results, for soon after it was established the master\npotter, who was at the head of these works, left the locality.\n[\"Memorias,\" Larruga, Vols. 13 and 34.]\n\nThe further we advance into the 18th century, the more we find the\ntendency in Spanish ceramic art to imitate the pottery most in vogue in\nother countries. Francisco Cavalli, a potter of Ruidoms, won a prize at\nTarragona in 1787, for his excellent imitations of brown and white\nGenoese ware. [MS. fol. Bibl. de S. Magestad el Rey, S. 2, E. B. pt. 8.]\n\nThe efforts made by King Charles III. towards increasing industrial arts\nin Spain, contributed to the reproductions and efforts made to imitate\nforeign wares. When the king founded in 1768 the villages of La Carolina\nand La Carlota in Andalucia, he ordered that pottery works should be set\nup there. At the same time that he established at Madrid the important\nporcelain manufactory of Buen Retiro, he wished that earthenware works\nshould exist in the same locality, where specimens should be made in\nimitation of the best work produced elsewhere. The king was most anxious\nto revive to a great extent the almost extinct industry of metallic\nlustred pottery, and thanks to this we are able to know most accurately\nthe receipt and manner in which this lustre was applied. In the same\nvolume in which I found these documents, [Brit. Mus. MS. Egerton, 507],\nare two reports addressed to Count Florida Blanca in 1786, by Iriarte\nand Vargas, who were instructed to facilitate the development of this\nindustry. The first report contains information relating to the pottery\nworks of the county of Stafford; the second tells us that in the\nbuilding of San Isidro el Real, essays had been made to reproduce\nEnglish wares, and the lustred productions of Manises, with an idea of\nestablishing inside or outside of Madrid pottery works on a large scale,\nunder the protection of Count Florida Blanca. Iriarte and Vargas were of\nopinion that these works should be established far from Madrid,\nsuggesting as the best spot El Viso in La Mancha, owing to the excellent\nquality of the clay. Don Sebastian Schepers, a son or brother of\nCayetano Schepers, was at the head of these works. Cayetano was the\nchief modeller at the Retiro manufactory. Their imitations of English\nearthenware did not succeed; the varnish turned out badly, and they\ndetermined to bring out English workmen. Their imitations of gold\nlustred ware were eminently successful, so much so that competent judges\ndeclared it was equal to what was made at Manises. Pottery works where\nearthenwares of different descriptions were made existed also at this\ntime in Madrid; the best were those of Rodriguez and Reato, mentioned in\nLarruga's \"Memorias Economicas.\"\n\nAt the end of the 18th, and beginning of the 19th century, Valencia and\nAragon supplied the country with painted tiles. One of the finest\nexamples of this class which have reached us is the pavement of the\nchapter house of the cathedral of Saragossa, on which landscapes,\nmedallions, and animals are finely designed in the Italian renaissance\nstyle. In a shield may be read the following inscription:\n\n    Real^{s} Fb^{cas}\n         D E\n     D\u00aa Maria Salb\n        adora\n       Disdier\n      Bru  f^{t}\n      A\u00f1o 1808.\n\nValencia has been much renowned for its manufacture of painted tiles,\n_azulejos_, which continue to be made there in a very creditable manner\nat the present time. J. Townsend, in his \"Journey through Spain in the\nYears 1786-1787, London,\" 1792, says: \"I was most delighted with the\nmanufacture of painted tiles. In Valencia their best apartments are\nfloored with these, and are remarkable for their neatness and elegance.\nThey are stronger and more beautiful than those brought from Holland.\"\nIn a \"Nouveau Voyage en Espagne,\" Paris, 1789, p. 56, the author says:\n\"L'industrie des Valenciens tire d'ailleurs parti de toutes les\nproductions de leur sol. Il contient une esp\u00e8ce de terre, dont ils font\nces carreaux de fa\u00efence color\u00e9e connus sous le nom de azulejos, et qu'on\nne fabrique qu'\u00e0 Valence. On en pave les appartements, et on en rev\u00eat\nleurs lambris; on y peint les sujets les plus compliqu\u00e9s, tels par\nexemple qu'un bal masqu\u00e9, une f\u00eate de taureaux. La couleur rouge est la\nseule qui ne puisse \u00eatre fix\u00e9e sur cette esp\u00e8ce de fa\u00efence. Elle\ns'alt\u00e8re par la cuisson.\" In \"Voyage en Espagne, 1797-1798,\" Paris 1801,\nthe author says, p. 245: \"Les plats sont faits de fa\u00efence bleu\u00e2tre ou\ntoute autre couleur orn\u00e9 de figures d'oies.\"\n\nBefore we pass to describe another most important branch of Spanish\npottery the unglazed earthenware must be mentioned, which from a very\nearly period has constituted and still constitutes a most important\nbranch of its industry. This pottery, generally used for cooling water,\nconsists of white porous vessels of which a large modern collection may\nbe seen at the South Kensington Museum proceeding from Andujar and La\nRambla (Andalusia). This industry remains in precisely the same state as\nin the time of the Arabs.\n\nThe earthenware vessels called _Bucaros_ are similar to these. This\nporous pottery was made to a very large extent at Talavera. It was\nimported originally from America; the great centre existed at Mejico.\nThe paste of this ware is unglazed and whitish, black or red--when\npainted the colours chosen are generally red, black, and gold. It was\nmade in Spain as early as the 16th century, and we constantly find\n_Bucaros_ alluded to in documents of this period. In the inventory of\nthe effects belonging to D^{na} Juana, the sister of Philip the Second,\ndrawn up in 1573, _bucaros_ made at Lisbon, Estremoz, and Montemayor in\nPortugal, and those of Ciudad Rodrigo and Castille, are also mentioned.\nMadame d'Aunoy in her \"Voyage d'Espagne,\" Lyon, [MDCXCIII.], mentions\nthe habit of Spanish ladies of eating this porous clay. At the South\nKensington Museum there are several good specimens of red pottery of\nthis kind, Nos. 285 to 318--'72, which, as we have remarked were made at\nTalavera and Toledo.\n\n\nALCORA POTTERY AND PORCELAIN.\n\nDon Buenaventura Pedro de Alcantara inherited in 1725 the estates\nbelonging to the title of Aranda in the province of Valencia. Count\nAranda found that the inhabitants of the village of Alcora made coarse\nearthenware of every description, and that their vicinity to the sea\ncoast favoured exportation; he determined, therefore, upon establishing\nin 1726 a manufacture of pottery there, in which fine wares might be\nmade in imitation of those imported from Italy, Germany, France and\nEngland. The count's efforts were so successful that in less than two\nyears specimens of different kinds of Alcora pottery were exported to a\nvery large extent.\n\nNo account has hitherto been published which gives any idea of the\nimportance of this manufactory, nor have the names of the artists who\nworked there been known, or the works which they executed. Wishing to\nascertain this, I applied to the Duchess of Hijar, the present\nrepresentative of the house of Aranda, and permission was granted me,\nthanks to the kindness of the Duchess's _Apoderado general_, Sr. Robles,\nto look through the Archives, where the accounts, contracts, and details\nof the manufactory are kept. This has enabled me to give an idea of the\nimportance of this industry, and the names of the artists who worked\nthere, which have been ignored until the present time by writers on\nceramic art.\n\nCount Aranda spent in 1726 about \u00a310,000 in establishing the manufactory\nof Alcora, and in May, 1727, the first specimens appeared, consisting of\npottery made \"in the manner of China, Holland, and other localities.\"\nThe manufactory was at that time under the superintendence of Dn.\nJoaquin Joseph de Sayas and Joseph Ollery, a Frenchman, chief\ndraughtsman and carver, who was engaged at a good salary in 1726, and\nbrought to Alcora from Moustiers by the painter, Edward Roux. In 1728\nCount Aranda increased his salary owing to the \"excellent manner in\nwhich Ollery has worked at Alcora, the fine and numerous models which he\nconstructed, which have contributed to make my manufacture the first in\nSpain.\"\n\nFive painters and two modellers from Catalu\u00f1a and six Valencian painters\nand two modellers joined these French artists. The personnel of the\nfabrique was completed with eleven potters from the locality. The French\npainters, M. Pierre Maurissy and M. Gras, and the master of the\nmodellers, M. Sebastian Carvonel, were engaged in 1728 for two years to\nwork at the manufactory. Ollery only appears in the lists up to 1737.\nThe Count granted him a yearly pension of 500 francs besides his salary,\n\"for his especial zeal in the improvement of the manufactory, and his\ngreat skill in directing the construction of every kind of work.\" From\nthis date until the manufacture of porcelain in 1764, only Spanish\nartists worked at Alcora.\n\nThe Count was able from the year 1729 to circulate the pottery made at\nAlcora through the Spanish dominions, free of custom-house duties. The\ngovernment granted him several other privileges and the manufactory\ncontinued to improve, and spared no pains to import the foreign shapes\nand designs which were most acceptable. No Spanish pottery manufactory\ncould compare with Alcora in the excellence and beauty of its work.\n\nAmong the obligations of the artists engaged, whether Spaniards or\nFrenchmen, was that of teaching drawing and modelling to a certain\nnumber of pupils. A special Academy was created for this purpose, which\nat one time held more than one hundred pupils, who were constantly\nrenewed and increased with those who appear henceforward in the works at\nAlcora. In 1736 there were fifty-six painters, eleven masters, twenty\nworkers at the wheel, and twenty-five apprentices. In this same year,\n1736, specimens of pottery made at Alcora were sent \"to all the\ndominions of Spain, Rome, Naples, Malta, many Italian cities, Portugal,\nand some provinces of France.\"\n\nThe manufactory produced yearly about 300,000 specimens of different\nkinds. The ordinances are interesting which in 1732-1733 prescribe,\n\"that in our manufactory only pottery of the most excellent kind should\nbe made, similar to the Chinese, to be equally fine as to the earths\nemployed, that the models and wheels should be perfect, the drawing of a\nfirst-rate kind, and the varnish and colours excellent, and the pottery\nlight and of good quality, for it is our express wish that the best\npottery should only be distinguished from that of an inferior kind by\nthe greater or less amount of painting which covers it.\"\n\nMiguel Soliva, Christobal Cros, Francisco Grangel, Miguel Vilar,\nChristobal Rocafort, Vicente Serrania, and Joseph Pastor were the best\npainters at Alcora in 1743; they decorated a fine dinner service made\nfor the Tribunal of Commerce, and the large slabs for the Convent of Las\nDescalzas Reales at Madrid, representing the Virgin as the Divine\nShepherdess.\n\nPottery painted with metallic lustre was made in 1749. We find among the\nreceipts used in that year one brought from Manises for this object.\n\nWe find it also stated in the communications which passed between the\nTribunal of Commerce and the count in 1746, that \"the perfection of the\nearthenware of Alcora consisted in the excellent models which had been\nmade by competent foreign artists, the quality of the earth and receipts\nbrought at great expense from abroad.\" Joseph Ochando is mentioned in\nthat year as an excellent painter, and Juan Lopez as the best carver and\nmodeller. This document tells us \"that from the earliest period of the\nmanufacture pyramids with figures of children, holding garlands of\nflowers and baskets of fruits on their heads, were made with great\nperfection, likewise brackets, centre and three-cornered tables, large\nobjects, some as large as five feet high, to be placed upon them,\nchandeliers, cornucopias, statues of different kinds, and animals of\ndifferent sorts and sizes. The entire ornamentation of a room has also\nbeen made here; the work is so perfect that nothing in Spain, France,\nItaly or Holland could equal it in merit.\"\n\nThe objects which were made to a great extent at this time consisted in:\n\n     Vases of different shapes.\n\n     Small pots, Chinese fashion.\n\n     Teapots and covers, Chinese fashion.\n\n     Teapots and covers, Dutch fashion.\n\n     Cruets, complete sets, Chinese style.\n\n     Entr\u00e9e dishes.\n\n     Salt-cellars, Chinese style.\n\n     Escudillas (bowls) of Constantinople.\n\n     Barquillos (sauce bowls), Chinese style.\n\n     Bottles, in the Chinese manner.\n\n     Cups, plates, and saucers of different kinds with good painted\n     borders in imitation of lace-work (_puntilla_); some were designed\n     in the Chinese manner, and especial care was taken with\n     fruit-stands, salad-bowls and dishes.\n\n     Trays and refrigerators.\n\nIn 1750 Count Aranda passed the pottery works on to a private company,\nin whose hands they remained until 1766. We know the pottery continued\nto be excellent. Unfortunately almost all the details of this period are\nmissing from the Archives. One of the few documents remaining is a\ncontract drawn up in October, 1741, with Fran\u00e7ois Haly (the name of this\nartist is given by Baron Davillier), a Frenchman, in which he agreed to\nwork at the manufactory during a period of ten years with a yearly\nsalary of over 1000 francs, under the following conditions:\n\n\"That the travelling expenses of his wife and children should be given\nhim, and that his salary should be paid as soon as he made before the\nDirector and two competent judges the different kinds of porcelain which\nhe had undertaken to make.\" He agreed to give up his receipts, and it\nwas promised him that he should have two modellers and one painter\nworking by his side, and that if in one year Haly's porcelain was\nsatisfactory the Count undertook to make him a present of 1000 livres\n(_tornoises_).\n\nTowards the middle of the century, porcelain was made for the first time\nat Alcora. A contract was drawn up on 24th March, 1764, with a German,\ncalled John Christian Knipfer, who had already worked there in the\npottery section. By the original agreement, which exists at the\nArchives, we find he was to prepare works of \"porcelain and painting\nsimilar to those made at Dresden, during a period of six years, under\nthe following conditions:\"\n\n\"That the said Knipfer obliges himself to make and teach the apprentices\nthe composition and perfection of porcelain paste, its varnishes, and\ncolours, and whatever he may know at the present time, or discover\nduring this period of six years; he is not to prevent the Director of\nthe Works from being present at all the essays made.\"\n\n\"The said Knipfer offers to make and varnish porcelain, and to employ\ngold and silver in its decoration, and in that of the ordinary wares;\nlikewise the colours of crimson, purple, violet, blues of different\nshades, yellow, greens, browns, reds, and black.\n\n\"That Knipfer will give up an account of his secrets, and the management\nand manner of using them, in order that in all times the truth of what\nhe has asserted may be verified.\"\n\nFrom the original documents which exist we gather that Knipfer was\nchiefly famed for his excellence in the painting and decoration of\nporcelain.\n\nFran\u00e7ois Martin was engaged in 1774 for his skill in preparing different\npastes for manufacturing porcelain and pipeclay. He agreed to make \"hard\npaste porcelain, Japanese fa\u00efence, English paste (pipeclay), and\nlikewise to mould and bake it. The necessary materials were to be\nprovided by the Count of Aranda.\" His expenses were to be paid if the\nspecimens he presented to the competent authorities gave a satisfactory\nresult, and his salary was to be increased to 1200 francs a year.\n\nKnipfer and Martin greatly added to the importance of the works made at\nthe manufactory. Don Pedro Abadia, the Count's steward, an intelligent\nman, possessing great scientific knowledge, who had studied this subject\nin Paris and London, writes to the Count that the presence of both these\nartists was of absolute necessity at Alcora, \"until the workmen who were\nnear them perfected themselves.\" For owing to the carelessness of the\nmanagers of the porcelain works in 1776 Count Aranda wrote from Paris,\nduring his embassy there: \"My pottery of Alcora, notwithstanding every\neffort which has been made, the money spent, and foreign masters which\nhave been brought over, gets worse every day instead of improving.\"\nAbadia repeats this in his reports. Porcelain of other kinds decidedly\nimproved. He says also that the pipeclay which Martin had found at\nAlcora was the best in Europe.\n\nIn my opinion, a large number of unmarked white biscuit and\ndemi-porcelain figures which are so constantly found in collections\nbelong to this period of the manufactory of Alcora. They have hitherto\nbeen classified with very great difficulty, and attributed to the\nporcelain manufactory of Buen Retiro, without any reason which justifies\nthis opinion. For the help of collectors I will mention the subjects\nwhich they represent, which I have found in a document, dated 1777, of\nthe figures and groups and other objects made during that year.\n\n\nFIGURES OF DEMI-PORCELAIN.\n\nFigures of tritons.\n\n   \"    of soldiers, two sizes.\n\n   \"        \"        one-third _palmos_ high.\n\n   \"   of the four seasons (two sizes).\n\n   \"   of dancers.\n\n   \"   of tritons in form of children.\n\n   \"   with brackets.\n\n   \"   of different animals.\n\n   \"   of gardener and female companion in the Dresden style.\n\nDancing figures in the German style.\n\nFigures of Neptune.\n\nFigures of shepherd and shepherdess.\n\n   \"   of the Moorish king, Armenius.\n\n   \"   of the four parts of the world, two sizes.\n\n   \"   of peasant and his wife.\n\nSmall figures holding musical instruments.\n\nFigures representing different monarchies.\n\n   \"         \"       historical personages.\n\n   \"         \"       the history of Alexander the Great, two sizes.\n\n   \"         \"       Marius Curtius, two sizes.\n\n   \"    of elephants.\n\n   \"    of a man mounted on an elephant.\n\n   \"    representing Chinese figures.\n\n   \"    of Heliogabalus.\n\n   \"    of a general on horseback.\n\n   \"    of a grenadier supporting a candlestick.\n\nLarge figures representing Julius C\u00e6sar.\n\nFigures representing the different costumes worn in Spain, on brackets.\n\nGroups of Chinese figures.\n\nSnuff-boxes, sugar basins, inkstands.\n\nRabbits, horns, and pug dogs for holding scent.\n\nSmall scent bottles.\n\nNeedle cases.\n\nLarge vases with foot and cover.\n\nBrackets.\n\nWalking-stick handles.\n\nKnife handles.\n\nTea-spoons.\n\n\nFIGURES OF WHITE BISCUIT CHINA.\n\nFigures representing Spanish costumes, two sizes.\n\nGroups of two figures.\n\nLarge and small figures of the four parts of the world.\n\nFigures of the four seasons, two sizes.\n\nWe find also the following figures of painted and glazed porcelain:\n\nFour seasons, two sizes.\n\nGroups of two figures.\n\nFigure of a Moorish king.\n\n   \"   of musicians and huntsmen.\n\nFigure of peasants.\n\n   \"   of Chinese.\n\nSmall figures of a gardener and female companion.\n\nFigures of soldiers in the German style.\n\nIn 1780 four rival pottery works were established in the neighbourhood\nwhich copied and imitated the pottery made at Alcora. The two most\nimportant were at Rivasalbes and Onda, the other two at Alcora itself.\nMany of the artists who belonged to the works established by Count\nAranda worked at the rival factories; among them were Mariano Causada,\nJoaquin Ten, Francisco Marsal, Vicente Alvaro, Christoval Mascar\u00f3s,\nFrancisco and Miguel Badenas, and Nadal Nebot; some of these artists\nreturned to the Count's manufactory. In order to distinguish the genuine\npottery from imitations, orders were given, with the authorization of\nthe Tribunal of Commerce, that the pottery made there should be marked\nhenceforward with the letter A; no special mark had hitherto been used\nat the manufactory, the artists very often signed the specimens they\nmade with their monograms or signatures, of which those most frequently\nmet with will be found accompanying the list of artists' names, for the\nhelp of collectors; all of them have been copied from original\ndocuments. It is interesting for collectors to bear in mind, that all\nspecimens which are marked with the letter A are _posterior_ to 1784.\nThe pottery works founded in imitation of the manufactory belonging to\nCount Aranda came to an end before 1790, some by special agreement with\nthe owners themselves, and others by the express orders of the\nauthorities, in virtue of the privileges granted to the Count.\n\nMr. Martin died at Alcora in the month of May, 1786. Knipfer left soon\nafterwards, and was succeeded by a French artist, M. Pierre\nCloostermans, a skilful man, and well versed in the manufacture of\nporcelain pastes, as well as in painting and decorating them. According\nto his contract, which was drawn up in Paris in 1787, \"Pierre\nCloostermans, chemiste, natif de Paris, demeurant \u00e0 Paris, Rue de\nClery, au coin de celle Montmartre,\" agreed to \"live for thirty years at\nAlcora as director and workman, to make soft and hard porcelains, and\nall the necessary colours with which to paint and decorate it.\"\n\nHe also promised to make good pipeclay pottery and marbled wares which\nwere to be as excellent as those of Strasburg. The expenses of his\njourney were paid, his sons were to be employed in the works, and it was\nstipulated that if the works increased to a great extent, 500 pounds\n(Valencian money), was to be added to his yearly salary. It was against\nthe Count's express desire that pottery of an exclusively artistic\ncharacter should be produced at Alcora; his chief object was to improve\nthe industry itself. In one of his letters to Abadia, written in\nOctober, 1789, he says: \"I wish to export the porcelain of my\nmanufactory, but chiefly in common objects, such as cups of different\nkinds, tea and coffee services, etc. These may be varied in form and\ncolour, the principal point being that the paste should bear hot\nliquids, for we Spaniards above everything wish that nothing we buy\nshould ever break. By no means let time be wasted in making anything\nthat requires much loss of time. The chief object is that the pastes\nshould be of first-rate excellence and durability.\"\n\nCloostermans suffered much discomfort and annoyance from other workmen\nat Alcora, who were envious of his merit: they put every difficulty in\nhis path, insulted him daily with pasquins, accused him of not\nfulfilling his religious duties, and annoyed him incessantly. His\nletters are full of these complaints. Count Aranda treated him with\nevery consideration. During Cloostermans' stay at Alcora, the pottery\nmade improved greatly in artistic merit. Figures and groups of many\nkinds were attempted, and even Wedgwood jasper ware was creditably\nimitated. In 1789, among other pottery that was sent to Madrid were \"two\nhard paste porcelain cups, adorned with low relief in the English\nstyle.\" The most important one was moulded by Francisco Garc\u00e9s, the\ngarlands and low reliefs by Joaquin Ferrer, sculptor, the flowers on the\ncovers by an apprentice, helped by Cloostermans. The composition of\nthese objects was suggested by Abadia, who brought some specimens from\nParis which came from England.\n\nCloostermans sent the Count in 1789 a number of objects of different\nkinds made of porcelain. Among them the most interesting were \"a tea and\ncoffee service painted and gilt of glazed porcelain, and ten unglazed\nfigures. Those painted by Albaro are marked A, those by Escuder, E, and\nby Mas, M, and Cloostermans' son.\" A large and varied collection of\nmarbled wares and toys were sent at the same time. Cloostermans' marbled\nwares are pronounced superb.\n\nCount Aranda writes in July, 1790, to Diez Robles alluding to a large\ncollection of pots for plants, which were made at Alcora for the King,\ndecorated with the royal arms. Another series are still to be met with\nat the Royal Gardens; they are of pipeclay, and ornamented with rams'\nheads.\n\nIn 1784 the Count sent two potters, Christoval Pastor and Vicente\nAlvaro, to Paris to study the last improvements in porcelain. They\nreturned in 1789, and the porcelain they made was much commended and\nhighly approved. They write to the Count in September, 1789: \"We know\nthat Don Domingo has sent your Excellency 97 objects made by us, marked\nNo. 3, 1, No. 4, No. 3, with a dot, and No. 3 with the letter 'P'; No. 4\nwith the letter 'H,' all made by me in clay, and varnished with the\ngreatest care. Three flask-stands and two toothpick-stands were made by\nme, Pastor, before I went to Paris. The marble wares, Nos. 3 and 4, and\nother similar objects with gold lines, are also made by me.\"\n\nIn 1784, Mariano Garcia of Valencia made some experiments before the\nDirectors of laying on gold, and different shades of purples. A number\nof specimens were sent to the Count, but Knipfer did not approve of the\nplan adopted, and it was afterwards abandoned.\n\nMarbled wares of different colours were made at Alcora in vast\nquantities in 1790. A large dep\u00f4t was established in 1791 of Alcora ware\nin the Calle de Luzon at Madrid. The printed prospectuses which were\nissued give a long and detailed list of the different productions of the\nmanufactory, which chiefly consisted of dinner and tea services, and\nother objects of domestic use.\n\nIn 1792 nearly 100 painters and modellers existed at the manufactory; 45\nwere employed to work in porcelain and pipeclays, and 26 were\napprentices.\n\nCloostermans was forced to leave Alcora in April, 1793, owing to certain\ndisturbances which occurred at Valencia, when, owing to a proclamation\nof the Captain-Generals, he and other Frenchmen residing in the province\nof Valencia, were ordered to leave the country. The Count gives\ninstructions that Cloostermans should want for nothing on his journey;\nhe writes ordering that 3000 reales should be given him, and his yearly\npension of 1200 pounds (_tornoises_). Cloostermans left with his three\nsons after giving up the receipts and other documents connected with the\nmanufactory. In 1795 permission was given that Frenchmen might return to\nSpain, and he resumed his post in the manufactory.\n\nThe principal efforts at Alcora since Knipfer, Martin, and Cloostermans\nentered the manufactory, had been centred in making porcelain and\npipeclay wares of different kinds. A great number of essays with foreign\nearths were made; and all those of a suitable kind which were known in\nSpain. Count Aranda was always most anxious that Spanish materials\nshould be used in the manufactory; he says, in a letter written in 1790,\n\"the Kaolin of Catalu\u00f1a may be good or bad, but it is acknowledged to be\nKaolin, and if not used these works must be closed.\" This Kaolin had\nbeen found by Christobal Pastor and Vicente Albaro on their return from\nParis.\n\nBaron Davillier has been good enough to inform me that he has found\nmention of some objects of Alcora wares which Count Aranda sent as a\npresent to his friend Voltaire, at Ferney.\n\nFrom 1789 to 1797 the following kinds of pottery were made at Alcora:--\n\nHard paste porcelain (French).\n\nPorcelain of three different kinds called Spanish.\n\nPorcelain of pipe-clay (English).\n\nBlue pipe-clay porcelain.\n\nMarbled pipe-clay porcelain.\n\n_Bucaro_, painted and gilt.\n\nStrasburg Ware.\n\nPorcelain painted _en froid._\n\nMarbled and gilt wares, hitherto unknown.\n\n\nPORCELAIN (FRITA).\n\nPorcelain painted with gilt lines.\n\n  \"         \"      without gold.\n\nPorcelain (frita), canary colour.\n\nBoxes in relief.\n\n  \"    plain.\n\nPorcelain (frita), painted with marble wares.\n\nPlain boxes of the same kind.\n\nPorcelain (frita), of blue and brown ground.\n\nCups and saucers of a similar kind.\n\n\nBISCUIT PORCELAIN.\n\nFigures.\n\nVases.\n\nPedestals.\n\nWhite porcelain (frita) cups of different kinds.\n\n  \"       \"       ornamented and plain.\n\nBoxes with busts.\n\nBoxes with ornamentations in relief.\n\nFigures.\n\nVases for holding flowers, plates, etc.\n\nLarge figures of the Four Seasons.\n\nFlower vases with rams' heads.\n\nPlain boxes.\n\nBoxes with ornaments in relief.\n\n\nWHITE PORCELAIN.\n\nPlates, cups, etc.\n\nFigures of different kinds.\n\n\nPAINTED PORCELAIN.\n\nCups, saucers, plates, etc.\n\nCream pots.\n\nPlain snuff boxes, or in the shape of a dog.\n\nFruit stands in relief.\n\nIn 1799 we find mention made of partridges modelled by Christoval Mas,\nand Clemente Aycart much commended for his dogs, ducks, tortoises and\nfrogs.\n\nJoseph Ferrer writes to the Duke of Hijar in the same year, that \"he had\njust seen a bust of D^{n} Jos\u00e9 Delgado, a trifle smaller than the one\npreviously sent of your excellency.\" These busts and some medallion\nportraits were made of porcelain _frite_. Ferrer complains in the same\nletter that Delgado the manager had asked for a fine jug and basin of\nS\u00e8vres, which had been given by the King of France to the late Count,\nand that he had been obliged to hide it away with some English specimens\nwhich were in the warehouse.\n\nCount Aranda died in January, 1798, and was succeeded by his son the\nDuke of Hijar. Cloostermans died the same year; Vicente Prats is stated\nto be the best painter and decorator at Alcora at this date. In an\nextract drawn up of the state of the pottery works in 1798, it appears\nthat 200 workmen were employed, and pottery of every description was\nmade, common earthenware, pipeclays in imitation of the English ones,\nand porcelain in small quantities; common wares were made in large\nquantities; the pipeclays were pronounced superior to the English in\nbrilliancy, but were so porous that they were easily stained, a large\nnumber of snuff-boxes and other small objects belong to this period.\n\nIn 1800, the Duke of Hijar, who succeeded the Count of Aranda in the\nmanagement of the manufactory, writes to D^{n} Josef Ferrer, saying:\n\"As I do not know the authors of the pipe-clay porcelain or that of\nother kinds which is sent here, I beg you to order the master workmen of\nporcelain and common pottery to engrave, in making it, the initials of\ntheir names, as it will enable me to distinguish the good from the bad.\"\n\nTwelve porcelain baths with the arms of Spain in blue were made at\nAlcora for Queen Maria Louisa in 1800.\n\nThe Duke of Hijar ordered in 1800 that a dinner-service should be made\nthere for his use. He sent instructions that it should be of the same\nkind as a tea-service previously sent to his son the Duke of Aliaga. The\npainter chosen to decorate it was Mariano Alvaro, and the designs\nselected by Ferrer were taken from the Loggie of Raphael.\n\nIn the same year some cases of pottery were sent to the Duke; in one of\nthem was a fine soup tureen modelled by Josef Ferrer, Cloostermans' son,\nPierre, writes at this time, asking to be appointed to the post left\nvacant by his father's death; his petition however was not granted.\n\nWe have already seen that pottery and porcelain continued to be\nmanufactured in the present century at Alcora in the same manner as in\nthe 18th century, but owing to the French invasion of 1808 this industry\nsuffered the consequences of war, and the work done there greatly\ndiminished in excellence and quantity. Even before this the Director,\nD^{n} Jos\u00e9 Delgado complains in several documents of the bad state of\nthe manufactory, the Directors who replaced him, D^{n} Juan Bautista\nCabot and D^{n} Pedro Bezarco, write continually repeating the same\nthing. After the French were turned out of Spain, the industry was\nrevived and strengthened by fresh artists from the porcelain manufactory\nof Madrid. D^{n} Luis Poggetti was appointed drawing master there in 1815,\nand D^{n} Domingo Palmera master of ornamental art, both these artists\nhad worked at Buen Retiro, Poggetti as Director of pietre dure, and\nPalmera as second-class sculptor. After this time the manufactory of\npottery at Alcora ceased to produce artistic works, and limited itself\nto send out common wares for domestic purposes; this system continued\nuntil 1858, when the Duke of Hijar sold the manufactory to D^{n} Ramon\nGirona, who brought over English workmen from Staffordshire in order to\nimprove the wares. Many imitations of the older styles have also been\nmade at Alcora of late years.\n\nOne of the most important results of the present study is the necessity\nof changing the classifications of a great number of specimens which\nhave been believed to be manufactured at Moustiers and other localities\nin France, but which in fact were made at Alcora. It is sufficient to\nmention the names of Grangel, Cros, Soliva, and Vilar, which one of the\nbest informed writers on Ceramic industry, Baron Davillier, has\ndiscovered on different specimens of pottery, and which, in his \"Hist.\ndes fa\u00efences et porcelaines de Moustiers, etc., Paris, 1863,\" he\nconsiders to be the names of artists who worked in France, although all\nof them belonged exclusively to Alcora, as will be seen in the subjoined\nlist of artists' names. I am in hopes, also, that many errors may be\ncorrected in future, touching French and English pottery, which was\nimitated to a great extent, and with much success at Alcora.\n\nWe gather also by this information that an immense number of objects\nwere made of pipeclay porcelain, in imitation of English wares; and, in\nmy opinion, a great quantity of objects of white pipeclay porcelain\nwhich have been found of late years in Spain are of Alcora manufacture.\nThey have been hitherto classified by amateurs as Leeds pottery. We\nfind, in papers relating to Alcora, that a decided distinction is made\nbetween white and straw-coloured pottery. This indication may be\nsufficient to distinguish it from English wares.\n\nTownsend, in \"A Journey to Spain in the years 1786-1787,\" London, 1792,\np. 255, says: \"At Alcora, in the neighbourhood of Valencia, a\nmanufacture of porcelain has been successfully established by Count\nAranda, and deserves encouragement. I was much pleased with their\nimitations of gilding. It is very natural, and the manager informed me\nthat after many years' trial it was found to be durable.\"\n\n[Illustration: EARTHENWARE PLAQUE, ALCORA WARE. SOUTH KENSINGTON\nMUSEUM.]\n\nSpecimens exist of Alcora ware and porcelain at the South Kensington\nMuseum.\n\nNos. 1051, 1052-'71. Two fine plaques, painted with mythological\nsubjects of Pomona and Galatea, the borders raised in form of a frame,\nwith scroll ornaments [see woodcut of No. 1052].\n\nNo. 341-'76. A porcelain cup and saucer, blue ground, gilt; painted with\nflowers in white medallions.\n\nNo. 333-'76. A plate painted with sprigs, and containing models of fruit\nin full relief.\n\n\n     A LIST OF THE DIRECTORS AND ARTISTS WHO WERE EMPLOYED IN THE\n     MANUFACTORY OF POTTERY AND PORCELAIN OF ALCORA FROM ITS FOUNDATION,\n     1726, UNTIL THE BEGINNING OF THE PRESENT CENTURY.\n\nDIRECTORS.\n\n\nDr. Joaquin Joseph de Sayas, 1727.\n\nJoseph Ollery, 1727 to 1733.\n\nManuel de Molina, 1727 to 1735.\n\nCayetano Allue, 1727 to 1750.\n\nMarcial Guirandeta, 1778 to 1783.\n\nJuan Villalonga, 1789.\n\nPierre Cloostermans, 1789.\n\nDomingo Abadia, 1789.\n\nGabriel Berenguer y Cebrian, 1789.\n\nJosef Ferrer, 1799.\n\nJosef Delgado, 1800.\n\n\nARTISTS.\n\nAbella, Francisco, 1750, at Alcora.\n\nAlvaro, Cristobal, 1750.\n\nAlvaro, Joseph, paints pottery in 1743, 1750.\n\nAlvaro, Vicente, el mayor, works at the wheel in 1750.\n\nAlvaro, Vicente, painted with Knipfer in 1783; he was sent by Count\nAranda in 1784 to Paris to learn the making of porcelain; he returned to\nSpain in 1789, and continued to work at Alcora.\n\nAlvaro, Tiburcio, painted with Knipfer in 1783.\n\nAndr\u00e9s, Cristoval, modeller and carver in 1783.\n\nAndr\u00e9s, Francisco, modeller in 1783.\n\nAndr\u00e9s, Francisco, modeller, 1743 to 1750.\n\nAndr\u00e9s, Gabriel, painter, 1743 to 1750.\n\nAndr\u00e9s, Gabriel, figures as one of the leading painters in 1794.\n\nAndr\u00e9s, Jaime, painter, from 1728 to 1737.\n\nAndr\u00e9s, Mariano, paints from 1739 to 1750.\n\nAndr\u00e9s, Mariano, works in 1789.\n\nAndr\u00e9s, Miguel, painter, 1743 to 1750.\n\nAparicio, Manuel, painter, 1750.\n\nArqua, Vicente, 1750.\n\nAycart, Clemente, sculptor, worked at the porcelain works in 1789.\n\nAycart, Roque, worked at the wheel in 1783.\n\nAycart, Ventura, worked at the wheel in 1783.\n\nBachero, Vicente, painted porcelain in 1789.\n\nBadenas, Cristoval, painter, 1727 to 1750.\n\nBadenes, Francisco, establishes with Miguel Badenas pottery works, where\npottery was made in imitation of Alcora: it was put a stop to by\nagreement in 1789.\n\nBadenes, Miguel; see Francisco.\n\nBeltran, Pedro, retouched painting on porcelain in 1783 to 1789.\n\nBerenguer, Cristoval, painter, painted from 1727 to 1750.\n\nBlasco, Bautista, painted from 1727 to 1750.\n\nBlasco, Francisco, painted from 1731 to 1738.\n\nBlasco, Joaquin, painted in 1750.\n\nBlasco, Joseph, modeller, 1731 to 1735; he painted in 1750.\n\nBlasco, Manuel, painter, 1728 to 1750.\n\nBlasco, Vicente, painted 1727 to 1750.\n\nBuxad\u00f3s, Ildefonso, painted 1727 to 1750.\n\nBuxad\u00f3s, Manuel, painted common pottery in 1783.\n\nCalvo Perales, Joseph, painter from 1727 to 1750.\n\nCalvo, Manuel, painter of common pottery in 1783.\n\nCalvo, Ramon, painted common earthenware from 1750 to 1783.\n\nCampion, Juan, a Frenchman, worked at the wheel in 1743.\n\nCarnicer, Juan, worked at pipe-clay porcelain in 1789.\n\nCarnicer, Vicente, modeller from 1783 to 1789.\n\nCarbonel, Sebastian, a Frenchman; he modelled in 1728.\n\nCatal\u00e1, Cristoval, modelled in 1783.\n\nCatal\u00e1, Juan, 1750.\n\nCatal\u00e1, Manuel, 1750.\n\nCatal\u00e1, Pascual, painter from 1729 to 1750.\n\nCatal\u00e1, Pedro Juan, painted common pottery in 1783.\n\nCaussada, mayor, Jacinto, painted from 1727 to 1750.\n\nCaussada, menor, Jacinto, from 1727.\n\nCaussada, Joseph, a son of Jacinto, painted from 1743 to 1750. He ran\naway from the manufactory and went to work at Talavera, and was brought\nback to Alcora.\n\nCaussada, Mariano. It was proposed that he should be turned out of the\nmanufactory, owing to his having gone over to the works established at\nOnda, where he gave them the receipts of the colours and varnishes used\nat Alcora; he returned to Alcora in 1789.\n\nCaveta, Pascual, painter, 1743.\n\nChiva, Cristoval de, painted porcelain in 1789.\n\nChiva, Joseph, painted from 1727 to 1738.\n\nChiva, Manuel, painted in 1789.\n\nCloostermans, Pierre, a French artist; he entered the manufactory in\n1787, and continued to work there until his death in 1798.\n\nCloostermans, a son of Pierre's, was an excellent painter on porcelain\nin 1789.\n\nCorr\u00e1s, Ignacio, a native of Catalu\u00f1a, painted from 1727 to 1728.\n\nColl, Jaime, a native of Catalu\u00f1a, painted from 1727 to 1736.\n\nCros, Cristoval, painted from May, 1727 to 1743; he was one of the best\nartists who painted at Alcora.\n\nCros, Manuel, worked at the wheel in 1783.\n\nCros, Manuel, 1750.\n\nCros, Pascual, painted from 1727 to 1736.\n\nCros, menor, Vicente, painted porcelain from 1750 to 1789.\n\nCros, Vicente, 1735 to 1750.\n\nDatos, Gaspar, modeller, 1731 to 1750.\n\nDatos, Julian, modeller, in 1783.\n\nDatos, Ramon, painted in 1750.\n\nDatos, Vicente, 1750.\n\nEscuder, Vicente, painted with Knipfer from 1783 to 1789.\n\nFabra, Francisco, painted from 1730 to 1743.\n\nFabra, Vicente, painter, 1727 to 1735.\n\nFalco, Joseph, painter, 1727 to 1743.\n\nFalco, Pedro, painter, 1727 to 1735.\n\nFeliu, Vicente Tomas, painter, from 1727 to 1750.\n\nFeliu y Thomas, Vicente, painted common pottery in 1783.\n\nFerrer, Joaquin, a carver, who worked with Mr. Martin from 1783 to 1789.\n\nFerrer, Esteban, painted porcelain in 1789.\n\nFerrer y Carnicer, Vicente, worked at the wheel in 1783. In December,\n1789, he had an oven on his own account at Alcora, the only one which\nremained out of four which had been established four years previously\nthere.\n\nFerrer, menor, Vicente, 1750.\n\nFerrer, Vicente, painted from 1727 to 1743.\n\nFlor, Antonio, painted common pottery from 1750 to 1783.\n\nFores, Joseph, painter, 1727.\n\nFornench, Joseph, painted in 1739.\n\nFornench, Phelipe, painted common pottery in 1783.\n\nFornench, Francisco, worked at the wheel in 1783.\n\nFornench, Phelipe, painter, 1727 to 1750.\n\nFuste, Salvador, painter, 1727.\n\nGalvez, Juan, modeller, 1731 to 1735.\n\nGarc\u00e9s, Joaquin, painter. He worked with Joaquin Ter at the manufactory\nwhich he established at Alcora, and returned to the Counts in 1789.\n\nGarc\u00e9s, Francisco, worked at the wheel with Mr. Martin in 1783, and was\nconsidered by Cloostermans, in 1789, his most able workman.\n\nGarc\u00e9s, Joseph, painter, 1727.\n\nGarc\u00e9s, Pedro, 1743.\n\nGarcia, Mariano, a native of Valencia. He went to Alcora to try a system\nof gilding and purple which he had invented, which produced an\nunsatisfactory result.\n\nGasch, Agustin, painter from 1728 to 1750, although he was absent from\nthe works from 1735 to 1741.\n\nGasch, Bruno, a son of Juan's, worked at the wheel from 1743 to 1750.\n\nGasch, Cristoval, worked with the turners, 1731 to 1741, and joined the\npainters until 1750.\n\nGasch, Francisco, worked at the wheel from 1729 to 1750.\n\nGasch, mayor, Joaquin, worked at the wheel in 1783.\n\nGasch, Joseph, painter, 1731 to 1735.\n\nGasch, Juan, worked at the wheel from 1728 to 1750.\n\nGasch, Manuel, worked at the wheel in 1783.\n\nGasch, Miguel, the son of Juan, painted in 1743.\n\nGasch, Vicente, 1750.\n\nGasch, Correo Vicente, worked at the wheel in 1783.\n\nGardo, Juan, painter and modeller, 1731 to 1735.\n\nGil, Francisco, modeller, 1783.\n\nGiner, Manuel, painter, 1727.\n\nGomez, Cristoval, painter on common pottery, 1783.\n\nGomez, Francisco, modeller, 1731 to 1750.\n\nGomez, Vicente, painter, 1750, 1783.\n\nGomez, menor, Vicente, worked at the wheel and painted from 1783 to\n1789.\n\nGorris, Joseph, 1750.\n\nGranell, Cristoval, painter, 1729, 1750.\n\nGranell, Joseph, modeller, 1731 to 1735.\n\nGranell, Vicente, 1731 to 1750.\n\nGrangel, Francisco, pintor, painted from 1727 to 1783. In 1743 the\nfinest work was given to him.\n\nGrangel, Juan, painted from 1727 to 1750.\n\nGras, Monsieur, painter, 1728.\n\nHaly, Fran\u00e7ois, 1751.\n\nHerrando, Francisco, 1727.\n\nHerrando, Joseph, painter, 1727 to 1736.\n\nHerrando, Juan, painted from 1729 to 1735.\n\nHerrando, Manuel, modeller, 1783.\n\nHerrando, Pascual, worked at the wheel from 1728 to 1743.\n\nHerrando, Thadeo, 1750.\n\nHuguet, Cristobal, worked at the wheel in 1783.\n\nHuguet, Francisco, worked at the wheel and modelled in porcelain from\n1783 to 1789.\n\nHuguet y Mascar\u00f3s, Joseph, painter, 1783 to 1794.\n\nHuguet, Vicente, painter. He worked at the manufactory established by\nJoaquin Ten at Alcora, and returned to Count Arandas in 1789, and\ncontinued there in 1794.\n\nHuguet Serra, Joseph, modeller in 1783.\n\nIba\u00f1ez, Juan, painter from 1727 to 1735.\n\nKnipfer, Juan, a native of Saxony, 1783.\n\nLazaro, Joseph, painter, 1727.\n\nLopez, Julian, 1792.\n\nLlorente, Francisco, 1750.\n\nLloscos, Joaquin, painter, 1783.\n\nLlosca, menor, painter, 1783.\n\nMalanco, Nicolas, painter, 1727.\n\nMarin, Pedro, painter, 1727 to 1736.\n\nMarques, Miguel, painter, 1794.\n\nMarras, Francisco, painter and modeller in 1727.\n\nMarsal, Bautista, painter from 1727 to 1743.\n\nMarsal, Francisco. He worked at Alcora, and left it for the manufactory\nof Onda; in 1783 he returned to Count Aranda's works.\n\nMarti, Miguel, worked at the wheel in 1783.\n\nMartin, Francisco, modeller in 1783.\n\nMartir, Pedro, worked at the wheel in 1739.\n\nMas, Cristoval, modeller from 1783 to 1789.\n\nMas, Francisco, painted common pottery in 1783.\n\nMas, Julian, a skilful painter on porcelain in 1789.\n\nMas, menor, Manuel, painted common pottery in 1783.\n\nMas, Manuel, painted from 1727 to 1750.\n\nMas, Pedro, painter in 1743.\n\nMascar\u00f3s, Cristobal, painter from 1728 to 1750.\n\nMascar\u00f3s y Thomas, Cristoval, painted on pottery from 1783 to 1794.\n\nMascar\u00f3s, Francisco, 1743 to 1750.\n\nMascar\u00f3s, Joseph, painted porcelain in 1789.\n\nMascar\u00f3s, Jos\u00e9, painter in 1735 to 1736.\n\nMascar\u00f3s, Pedro Martin, painted from 1729 to 1736.\n\nMascar\u00f3s, Vicente, worked at the wheel from 1750 to 1780.\n\nMasso y Fabra, Vicente, painted in 1727.\n\nMass\u00f3, Joseph, 1750.\n\nMass\u00f3 y Fabra, Francisco, painter, 1739, 1750.\n\nMass\u00f3 Nadal, Vicente, painter from 1727 to 1750.\n\nMallol, Joaquin, modeller, 1783.\n\nMallol, Cristoval, modeller in 1783.\n\nMallol, Vicente, modeller, 1783.\n\nMaurisi, Pedro, a Frenchman, began to work in 1728.\n\nMezquita y Chiva, Francisco, worked at the wheel in 1783.\n\nMezquita, Francisco, painted in 1750.\n\nMezquita, Jaime, painter, 1731 to 1750.\n\nMezquita, Pascual, painter from 1727 to 1735.\n\nMiguel, Vicente, 1750.\n\nMiralles, Vicente, worked in clay from 1731 to 1743.\n\nMoliner, Cristoval, worked at the wheel in 1783.\n\nMoliner, Miguel, painted from 1728 to 1750.\n\nMoliner, Vicente, modeller in 1783.\n\nMonfort, Agustin, modeller in 1783.\n\nMonfort, Pablo, modeller in 1783.\n\nMontemenor, Joseph, modeller in 1783.\n\nMontolin, Jacinto, painter, from 1731 to 1750.\n\nMontolin, Juan, painter, the son of Jacinto, from 1735 to 1750.\n\nMontolin, Juan, varnisher and painter in 1783.\n\nMontolin, Joseph, painter on porcelain in 1789.\n\nMontolin, Vicente, painter of common pottery in 1783.\n\nMontolin, Vicente, painter, from 1727 to 1750.\n\nMoya, Crisostomo, retouched porcelain in 1789.\n\nNadal, Felix, 1727 to 1735.\n\nNadal, Juan, master at the wheel, painter and carver from 1727 to 1737.\n\nNadal, Miguel, 1783.\n\nNadal, Nebot, works from 1743 to 1783.\n\nNavarro, Cristoval, worked at the wheel in 1783, and painted porcelain\nin 1789.\n\nNebot, Bautista, painter, 1794.\n\nNebot, Cristoval, modeller, 1750.\n\nNebot, Cristoval, menor, 1794.\n\nNebot, Francisco, 1750.\n\nNebot, Francisco, painter in 1794.\n\nNebot, Joseph, worked at the wheel from 1728 to 1740.\n\nNebot, Joaquin, worked at the wheel in 1794.\n\nNebot, Juan, painter, 1750 to 1783.\n\nNebot, Miguel, painter, 1783 to 1794.\n\nNebot, Narciso, painter, 1794.\n\nNegre, Deodato, painter, 1727.\n\nNegre, Francisco, worked at the wheel from 1783 to 1794.\n\nNegre, Julian, worked at the wheel in 1794.\n\nNegre, Manuel, modeller, 1783 to 1794.\n\nNegre, Ramon, painter, 1783 to 1794.\n\nNegre, Vicente, modeller, 1727.\n\nNondedeu, Cristoval, modeller, 1729.\n\nNondedeu, Miguel, turner, 1731 to 1743.\n\nOchando, Joseph, draughtsman and carver, from 1727 to 1742.\n\nOlery, Joseph, director of the works from 1735 to 1737. In August, 1729,\nhe was appointed principal draughtsman.\n\nPacor, Bartolom\u00e9, painter, 1728 to 1735.\n\nPalau, Francisco, painter, from 1727 to 1750.\n\nPalmera, Domingo, master of ornamental art, 1815.\n\nPardo, Cristoval, turner, from 1727 to 1750.\n\nPardo, Francisco, 1750.\n\nPardo, Joseph, turner, from 1731 to 1750.\n\nPardo, Vicente, master at the wheel in 1728.\n\nPastor Bartolo, Antonio, painter, 1750.\n\nPastor Butoni, Antonio, painter, 1750.\n\nPastor, Bartolom\u00e9, a brother of Vicente's, from 1729 to 1750.\n\nPastor, Bautista, varnisher, 1783.\n\nPastor, Cristoval, worked at the wheel with Mr. Martin from 1783 to\n1789.\n\nPastor, Gaspar, 1750.\n\nPastor, Joseph, painted in common pottery and porcelain from 1783 to\n1789.\n\nPastor, Joseph, painter, from 1728 to 1750; his name appears in 1743\namong the best artists of Alcora.\n\nPastor, Vicente, painter, from 1728 to 1743.\n\nPastor, Vicente, painter, was pensioned in Paris by Count Aranda in\n1784; he returned to the works in 1789, and continued there for several\nyears afterwards.\n\nPe\u00f1a, Vicente, modeller, 1783.\n\nPerales, Ramon de, painter, 1750.\n\nPeriz, Joseph, modeller, from 1727 to 1731.\n\nPeriz, Joseph, mayor, painter, from 1750 to 1783.\n\nPeriz, Joseph, menor, modeller, 1783.\n\nPerpi\u00f1an, Cristoval, a son of Vicente, painter, from 1727 to 1740.\n\nPerpi\u00f1an, Vicente, modeller, from 1731 to 1743.\n\nPinazo, Andres, draughtsman and carver, 1727.\n\nPoggetti, Luis, drawing master, 1815.\n\nPorcar, Cristoval, painter, from 1727 to 1735.\n\nPorcar, Pascual, modeller, from 1729 to 1735.\n\nPrast, Antonio, painter of common pottery in 1783.\n\nPrats, Cristoval, modeller, 1783.\n\nPrats, Cristoval, turner, from 1731 to 1750.\n\nPrats, Francisco, modeller, from 1750 to 1783.\n\nPrats, Fulgencio, painter, 1783.\n\nPrats, Vicente, painter, from 1750 to 1794. In 1789 he ranked among the\nfirst artists at Alcora.\n\nQuerol, Vicente, 1743.\n\nQuerol, Joseph, a son of Vicente, painter, from 1727 to 1750.\n\nQuerol, Manuel, painter, from 1750 to 1783.\n\nRech, Joseph, a native of Catalu\u00f1a, painter, in 1727.\n\nRedolat, Joseph, painter on porcelain in 1789.\n\nRibot, Joseph, turner, from 1731 to 1735.\n\nRicart, Antonio, turner, from 1729 to 1735.\n\nRicart, Clemente, modeller, in 1783.\n\nRicart, Joaquin, modeller, in 1783.\n\nRobert, a Frenchman, painted in 1729.\n\nRocafort, Cristoval, painted from 1727 to 1750; he was one of the best\nartists at the manufactory in 1743.\n\nRoman, Juan, painted from 1731 to 1735.\n\nRoman, Manuel, painted from 1735 to 1750.\n\nRomualdo, Joseph, painter, from 1728 to 1750.\n\nRoux, Edouard, a French painter, who worked at Alcora, where he had been\nbrought by Ollery in 1728 until 1735.\n\nRules, Joseph de, turner, from 1731 to 1735.\n\nSaborit, Cristoval, modeller, in 1783.\n\nSaborit, Cristoval, painted from 1727 to 1736.\n\nSaborit, Joaquin, painter, 1783.\n\nSaborit, Joaquin, painted from 1728 to 1750.\n\nSaborit, Manuel, modeller, in 1783.\n\nSagao, Rafael, draughtsman and carver, 1727.\n\nSalvada, Joseph, 1789.\n\nSancho, Joaquin, painted in 1783.\n\nSerrania, Cristoval, 1750.\n\nSerrania, Vicente, painted from 1728 to 1743.\n\nSoliva, Joaquin, painted pottery in 1783, and retouched porcelain in\n1789.\n\nSoliva, Miguel, painted from 1727 to 1750. In 1743 he was considered the\nbest artist at Alcora.\n\nSoriano, Joseph, turner from 1731 to 1735.\n\nSoriano, Nicasio, turner from 1731 to 1735.\n\nSorolla, Manuel, worked at the wheel in 1783.\n\nTarazona, menor, Cristoval, modeller, 1783.\n\nTarazona, Cristoval, painted from 1727 to 1750.\n\nTarazona, Ramon, retouched porcelain in 1783.\n\nTarrag\u00f3, Joseph, painter from 1728 to 1735.\n\nTarrag\u00f3, Vicente, painter from 1727 to 1750.\n\nTen, Francisco, modeller in 1783.\n\nTen, Joaquin, painter from 1732 to 1750. In 1789 he closes the pottery\nworks which he had established in imitation of those of Alcora.\n\nTen, Joseph, 1750.\n\nTen, Jaime, painter from 1735 to 1750.\n\nTerra, Cristobal, modeller, 1783.\n\nTerra, Vicente, varnisher, 1783.\n\nThomas, Antonio, modeller, 1783.\n\nThomas, Gaspar, 1750.\n\nThomas, Joaquin, modeller in 1783.\n\nThomas, Jos\u00e9, 1750.\n\nThomas, Jos\u00e9, worked at the wheel in 1783.\n\nThomas, Juan, 1743, 1750.\n\nThomas, Mateo, 1750.\n\nThomas Feliu, Vicente, painter from 1729 to 1740.\n\nThomas, Vicente, 1783.\n\nTorres, Cristoval, painter from 1728 to 1750.\n\nTorres, Juan, from 1731 to 1735.\n\nVadenes, Cristobal, 1770.\n\nValentin, Pedro, painter from 1727 to 1743.\n\nVilar, Cristobal, painter from 1727 to 1750.\n\nVilar, Cristobal, sculptor, works in porcelain in 1789.\n\nVilar, Francisco, 1750.\n\nVilar, Joseph, modeller in 1783.\n\nVilar, Miguel, painter from 1727 to 1743.\n\nVilar, Pedro, worked in the ovens from 1735 to 1750.\n\nVilar, Ramon, retouched porcelain in 1789.\n\nVilar y Bordo\u00f1an, Cristobal, modeller in 1783.\n\nVilar y Bordo\u00f1an, Mariano, worked at pipe-clay porcelain in 1789.\n\nVilar Perpi\u00f1an, Cristobal, painted from 1739 to 1743.\n\nVilar Ricart, Cristobal, painter from 1727 to 1735.\n\nVilar Porcar, Cristobal, painter, 1727.\n\nVilar Ricart, Joseph, painter from January, 1731, to 1735.\n\nVilar Porcar, Joaquin, painter from 1727 to 1735.\n\nVilar Saboret, Joaquin, painter in 1729.\n\nYguet, Vicente, painted common pottery in 1783.\n\nZaragoza, Friar, worker in porcelain, 1799.\n\n\nMARKS AND SIGNATURES OF THE PAINTERS WHO WORKED AT ALCORA.\n\nFrom 1727 to 1784 no special mark was used at Alcora. In several\ninstances specimens were signed with the painter's name or initials.\n\nIn order to distinguish the qualities or sizes, coloured numbers were\nfrequently added.\n\nFrom 1784 the letter A in gold or colours was used to mark the pottery\nand porcelain made at Alcora.\n\nThe following marks were used after 1784. The a underlined, [A=],\nsometimes in blue.\n\nA number and letter, No. 4 A, G 8; No. 3 P. In 1799 the pottery made by\nFriar Joseph de Zaragoza was marked M. O. X.\n\n[Illustration]\n\n[Illustration]\n\n[Illustration]\n\n[Illustration]\n\n[Illustration]\n\n[Illustration]\n\n[Illustration]\n\n\nMADRID.--BUEN RETIRO PORCELAIN.\n\nIn 1759 King Charles III. came from Naples, having inherited the Spanish\ncrown on the death of his brother Ferdinand VI. Soon after his arrival\nhe determined to establish a porcelain manufactory at Madrid in the same\nstyle as one which in 1736 he had founded at Naples. The documents which\nexist relating to this manufactory at the archives of the Royal Palace,\nMadrid, Alcala, Ministry of Finance, etc., state that before the King\nleft Naples, he ordered the following letter to be written to the\nSecretary of State, Richard Wall, on September 11, 1759: \"Likewise the\nworkmen and utensils used at the royal manufactory of porcelain of Capo\ndi Monte must be embarked from Naples to Alicant, in the vessels\nprepared for that purpose, in order to continue from there the journey\nto Madrid. The necessary conveyances are to be provided, and the\nexpenses to be charged to his Majesty's account.\"\n\nCharles III. landed at Barcelona on the 17th of October, 1759, and we\nfind a letter written by order of the King, by the Marquis of Esquilace\nto Secretary Wall, in November of the same year, in which he says that\nthe King had heard of the arrival in Spain of the workmen from Capo di\nMonte, and gives orders that money should be supplied to the Director,\nDon Juan Thomas Bonicelli. Wall answers that he \"will give orders and\nhelp the workmen who are to establish the manufactory, and let them have\nevery facility to examine different sorts of earths and localities which\nmay suit them, and that the Director, Bonicelli, should have the money\nhe might require.\"\n\nIt appears also that 300 gold ducats were given to Giuseppe Gricci,\n\"modeller,\" in absence of Bonicelli. The King ordered Wall to be\ninformed that he understood that a place had been found near Madrid to\nestablish these works, and that he was anxious to see a plan of the\nexact locality. Giuseppe Gricci drew the plan of the spot chosen, and\nwas paid 100 gold doubloons for works connected with the manufactory.\nBonicelli sends the following list of workmen who had arrived from\nNaples to the King.\n\nCayetano Schepers, chief composer.\n\nPablo Forni.\n\nJoseph Gricci, principal modeller.\n\nCarlos Gricci.\n\nEsteban Gricci,        modeller.\n\nCayetano Fumo,              \"\n\nBasilio Fumo,               \"\n\nJoseph Fumo,                \"\n\nCarlos Fumo,                \"\n\nMacedonio Fumo,             \"\n\nJoseph Santorum,            \"\n\nJuan Bescia,                \"\n\nBautista de Bautista,       \"\n\nAntonio Morelly,            \"\n\nSalvador Nofri,             \"\n\nPhelipe Esplores,           \"\n\nAmbrosio de Giorgi,         \"\n\nPedro Antonio de Giorgi,    \"\n\nPablo Frate,                \"\n\n\nWORKMEN EMPLOYED IN THE KILNS.\n\nJenaro Bonincosa.\n\nNicolas Rocio.\n\nPasqual Rocco.\n\nJuan Frate.\n\nBaldo de Beneditis.\n\nVincenzio Frate.\n\nMatheo Mayni.\n\nGiorchino Amable.\n\nJoseph Esclavo.\n\nAntonio Aquaviva Esclavo.\n\n\nWORKMEN WHO POUNDED THE COLOURS.\n\nFrancisco Conte.\n\nNicolas Conte.\n\nAngelo Lionelli.\n\nJoseph Caramello.\n\nJoachim Pataroti, carver in pietri dure.\n\n\nWORKMEN EMPLOYED AT THE WHEEL.\n\nJoseph Grossi.\n\nNicolas Botino.\n\nJuan Remini, gold beater.\n\nPedro Chevalier, mounter.\n\n\nPAINTERS.\n\nJos\u00e9 de la Torre.\n\nJuan Bautista de la Torre.\n\nNicolas de la Torre.\n\nFernando Sorrentini.\n\nMariano Nani.\n\nJenaro Boltri.\n\nNicolas Donadio.\n\nAntonio Provinciale.\n\nJoseph del Coco.\n\nCarlos Remissi.\n\nFrancesco Simini.\n\nXavier Brancacio.\n\nJoseph Esclavo.\n\nFrancisco Esclavo.\n\nOn the 19th of December, 1759, Don Carlos de Borbon, the King's\narchitect, presented him with the plans of the porcelain manufactory.\nThe spot selected was inside the gardens of the Royal Palace of Buen\nRetiro. Ponz tells us in his \"Viage,\" Vol. VI. p. 108, that the building\nwas large and of regular architecture. We know it cost 179,130 reals.\n\nDon Carlos de Borbon was a black slave who had been captured with other\nblacks during the reign of Ferdinand VI. The Queen-mother sent them to\nNaples, and Charles III. gave them an artistic education.\n\nDon Carlos Antonio became the King's architect. On the 22nd of May,\n1760, the building was finished, the money then spent amounted to\nupwards of 145,000 reals, and Larruga tells us, in his Memorias, \"The\nKing spent in establishing this manufactory \u00a3115,000, with a yearly cost\nto keep it up of \u00a320,000.\"\n\nWilliam Clarke, in his \"Letters concerning the Spanish Nation during the\nyears 1760-1761,\" London, 1763, says, p. 262: \"At Madrid is lately set\nup a manufacture of porcelain in the gardens of the King's palace at the\nRetiro, wrought by artificers brought from Saxony.\" Documents exist\nproving that in 1760-1761, they were already working there. Townsend, in\nhis \"Journey through Spain in 1786 and 1787,\" London, 1792, says, Vol.\nII., p. 278:--\n\n\"I tried to obtain admission to the china manufacture, which is likewise\nadministered on the King's account, but His Majesty's injunctions are so\nsevere that I could neither get introduced to see it, nor meet with any\none who had ever been able to procure that favour for himself. I was the\nless mortified upon this occasion because, from the specimens which I\nhave seen, both in the palace at Madrid, and in the provinces, it\nresembles the manufacture of S\u00e8vres which I had formerly visited in a\ntour through France.\"\n\nIn the \"Nouveau voyage en Espagne, ou Tableau de l'\u00e9tat actuel de cette\nmonarchie,\" Paris, 1789, Vol. I. p. 233, the author tells us, how \"Le\nmonarque actuel \u00e0 \u00e9tabli dans leur interieur une fabrique de porcelaine,\ndont l'entr\u00e9e est jusqu\u00e0 present interdite \u00e0 tout le monde. On veut sans\ndoute que ses essais se perfectionment dans le silence, avant de les\nexposer aux yeux des curieux. Ses productions ne peuvent encore se voir\nque dans les Palais des Souverains, ou dans quelques Cours d'Italie,\nauxquelles il les envoie en presens.\"\n\nWe find in a \"Nouveau voyage en Espagne,\" Paris, 1805, p. 34, in\ndescribing this porcelain manufactory that the author says; \"Cet\n\u00e9tablissement tres couteux ne travaille que pour le Roi et a son compte;\nil en sort des vases d'une beaute et d'un fini qui ne le cedent point \u00e0\nceux de S\u00e8vres.\"\n\nCitoyen Alquier, in 1800, the envoy of the French Republic, was allowed\nto visit the manufactory.\n\nWe do not know the precise date when porcelain began to be manufactured\nat the Retiro. Clarke, writing in 1761, says the works had begun, and in\n1764 pupils attended the classes at the Academy of S^{n} Fernando.\nLarruga, in his \"Memorias,\" says that as soon as the building was\nfinished, china was made under the superintendence of Don Cayetano\nSchepers; the works, during his superintendence, proved very\nunsatisfactory, to his great astonishment, as the same process and\nworkmen were employed as at Naples. Schepers attributes it to squabbles\nbetween the Spanish and Italian workmen. Sebastian Schepers, from 1783,\na son of Cayetano's, tried various experiments with different clays of\nthe country.\n\nThe porcelain made at Buen Retiro was kept for the first thirty years\nfor the exclusive use of the royal family, or to be sent as presents to\nforeign courts. Nothing was offered for sale until January, 1789, after\nCharles III.'s death, 1788, when Charles IV. determined that the china\nmanufactured at Buen Retiro might be sold. Even in Spain the specimens\nof this china are very scarce; it is only at the palaces of Madrid,\nAranjuez, the Escorial and La Granja that an idea can be formed of the\nperfection of this manufacture.\n\nThe director at that time was Don Domingo Bonicelli, a son of Don Juan\nBonicelli. Don Domingo chose a room within the Retiro, which was\narranged at a cost of \u00a3350, in which to exhibit the objects for sale.\nAnother room was taken in the Calle del Turco, which is mentioned in\n\"Noticias varias y curiosas de Madrid,\" Valero Chicarro, 1762-1793,\nwhich we find was closed in 1800, as the \"objects manufactured at the\nRetiro were simply for ornament, and could only be bought by very rich\npersons.\" Southey in his \"Letters from Spain,\" London, 1797, p. 118,\nsays, \"The old palace of Buen Retiro is converted into a royal porcelain\nmanufactory; the prices are extravagantly high, but they have arrived at\ngreat excellence in the manufacture. The false taste of the people is\ndisplayed in all the vases I saw there, which though made from Roman\nmodels, are all terminated by porcelain flowers.\"\n\nEvery kind of porcelain was made at Buen Retiro, hard and soft paste,\nwhite china, glazed or unglazed, or painted and modelled in the style of\nCapo di Monte. A great many existed imitating the blue jasper ware of\nWedgwood, and they also made flowers, coloured and biscuit, groups, and\nsingle figures, and painted porcelain of different kinds. Great\nquantities of tiles for pavements were also made there, which may still\nbe seen at the Casa del Labrador at Aranjuez; they are mentioned in the\naccounts which exist at the Ministry of Finance for 1807 and 1808. We\nfind in these same accounts interesting details of the objects made\nmonthly. In January, 1808, a large number of figures were made,\nincluding 151 heads for the table centre which was made for the king,\n306 objects ornamented with paintings, 2,056 tiles, 577 objects of less\nartistic importance, such as dishes, plates, etc. The finest specimens\nwhich exist are in the Neapolitan style, and are two rooms at the\nPalaces of Madrid and Aranjuez of which the walls are completely covered\nwith china plaques and looking-glasses, modelled in the most admirable\nmanner with figures, fruits, and flowers. The room at Aranjuez is\ncovered with a bold ornamentation of figures in the Japanese style, in\nhigh relief, painted with colours and gold with the most exquisite\ndetails. The figures unite the fine Italian modelling with the Japanese\ndecoration. The chandelier is in the same style. Upon a vase on the\nwainscot to the right of the entrance door is the following inscription:\n\n      JOSEPH\n     GRICC^{I}\n    DELINEAV^{it}\n       ET\n     SCUL^{it}\n      1763.\n\nThis same date is repeated in the angles, and in some shields near the\nroof we find,\n\n    A\u00d1O\n    1765;\n\nprobably the year the work was terminated. Antonio Conca, in his\n\"Descrizione Odeporica della Spagna in cui spezialmente si da notizia\ndelle cose spettanti alle Belle Arti,\" Parma, 1793, Vol. III., p. 310,\nsays, \"Il Gabineto abbelitto di porcellana della Fabbricca del Retiro ha\nmeritato le bodi de curiozi viaggiatori.\" We also find in p. 119, \"Un\naltro Gabinetto vien chiamato della Cina pel sud principal ornata di bei\nputti, di bassi relievi, e di altre opere di porcellana della nuova Real\nFabbricca del Ritiro.\" Ponz, in his \"Viage de Espa\u00f1a,\" Madrid, 1782,\ndescribes the room at the Palace of Madrid, saying, \"it is covered with\nlarge plaques of porcelain made at Buen Retiro. In some are represented\nfigures of children copied from models, and between each compartment\nlooking-glasses are let in.\" (See woodcut.)\n\n[Illustration: ROOM DECORATED WITH BUEN RETIRO PORCELAIN, IN THE PALACE\nAT MADRID.]\n\nFrom the establishment of the manufactory in 1759 by Charles III.\nuntil 1803 the styles adopted at Capo di Monte had been followed. At the\nbeginning of this century Dn. Bartolom\u00e9 Sureda went to Paris to learn\nthe manner in which S\u00e8vres porcelain was made. On his return in 1803 he\nwas appointed director of the works at Buen Retiro and endeavoured to\nimitate the paste and brilliancy of decoration of S\u00e8vres. Two workmen\ncame over from Paris--Victor Perche, and Vivien.\n\nAmong the finest specimens of this period of the manufactory are a\nsplendid clock and four vases, two m\u00e8tres high, with porcelain flowers,\nwhich exist in one of the state rooms of the Palace of Madrid. The vases\nare placed in the four corners of the room. The clock is ornamented with\nlarge biscuit figures. A large number of vases exist at the royal\nPalaces of Madrid, Aranjuez, and Escorial, of Retiro china. They are\noften finely mounted in gilt bronze with muslin or porcelain flowers.\nThe blue of the imitations of Wedgwood is not so pure, nor is the\nbiscuit work so fine as the English. Gold is often added to these\nspecimens.\n\nWe find at the archives of the Ministry of Finance interesting details\nof a dinner service made in 1798 for Charles IV., and a centre-piece,\nwhich probably is that now in the Casa del Labrador at Aranjuez.\n\nWhen the French made their entry into Madrid in the spring of 1808 they\ntook possession of the position occupied by the royal manufactory. In\nJuly of the same year it continued in the hands of the French, who\nforced open the doors of the laboratory. Porcelain continued, however,\nto be made there during the reign of Joseph I.; we find in \"Travels\nthrough Spain and part of Portugal,\" London, 1808, p. 23, that, the\nauthor says, \"the gardens of the Buen Retiro are open to the public. In\nthe neighbourhood of these the royal porcelain manufacture is carried on\nin a large white building.\" Lord Blayney, in his \"Narrative of a Journey\nthrough Spain and France in 1810-1814,\" London, 1814, says that \"the\nroyal manufactures of tapestry and porcelain have declined since the\ndeath of Charles III. and have now entirely ceased.\"\n\nWe find in \"Paseos por Madrid,\" Madrid, 1815-8, p. 87, it stated that\n\"The English, at the second entry of our troops in Madrid, ruined this\nbuilding in order that it should not be used as a fortress by the French\ntroops.\"\n\nRichard Ford, in his \"Handbook for Travellers in Spain,\" London, 1845,\nsays, \"Everything was destroyed by the invaders, who turned the\nmanufactory into a fortification, which surrendered with 200 cannon,\nAug. 14th, 1812, to the Duke of Wellington. Ferdinand VII., on his\nrestoration, re-created La China, removing the workshops and ware rooms\nto the Moneloa.\"\n\nThe South Kensington Museum contains an interesting collection of Buen\nRetiro porcelain of different kinds, of which may be named:\n\nNo. 344-'66. A vase of biscuit porcelain, two-handled, with frieze of\nclassic dancing figures and flowers.\n\nNos. 333, 4-'66. Two small vases for tea, white porcelain, covered with\nflowers in relief.\n\nNo. 892-'75. A pair of vases painted with young bacchanals in rose\ncama\u00efeu, and gilt.\n\nNo. 893-'75. A pair of draped female figures, each holding a cornucopia,\nstanding on an altar-shaped plinth.\n\nNo. 894-'75. A clock case, white porcelain, of rock and scroll work,\nwith flowers and groups of amorini.\n\nNo. 332-'76. A group of Ariadne and the panther.\n\nNo. 1068-'73. A pair of tall vases, with gilt serpent handles, the necks\nfluted with gold, the upper part of the body painted with classic\ngroups, and with coloured scroll foliage in relief, the lower part\npainted with leaves and scrolls on white ground. (See woodcut on next\npage.)\n\n[Illustration: BUEN RETIRO VASE. SOUTH KENSINGTON MUSEUM.]\n\nMARKS USED AT THE PORCELAIN MANUFACTORY OF BUEN RETIRO.\n\n[Illustration]\n\nThe usual mark in blue.\n\n[Illustration]\n\nIn blue, and sometimes in violet and gold.\n\n[Illustration]\n\nCayetano or Carlos Fumo. The initials and date are graved in the clay\nunder the glaze; the fleur-de-lis is pencilled in blue. On a fine group\nof children playing with a goat.\n\n[Illustration]\n\nThis mark is graved in the clay, under the glaze, on a fine group\nmodelled by Salvador Nofri.\n\n[Illustration]\n\nOchogavia? graved in the soft clay on a figure.\n\n[Illustration]\n\n1798, Sorrentini? These marks are pencilled in red on a pink cup and\nsaucer, with landscapes painted _en grisaille._\n\n[Illustration]\n\nPedro Antonio Georgi? The initials P. G. are gilt; the M crowned in red;\nthe V and M graved in the clay; on a cup and saucer buff coloured.\n\n[Illustration]\n\nProvinciale? The letters Po are graved in the clay, the fleur-de-lis in\nblue; on two saucers, beautifully painted with children.\n\n[Illustration]\n\nProbably the initial of the king. On two jardini\u00e8res; the interlaced C's\ngraved in the clay, the fleur-de-lis in blue.\n\n[Illustration]\n\nOn two large vases imitating Wedgwood's blue and white jasper, with\nwhite biscuit flowers.\n\n[Illustration]\n\nOn a group of biscuit Retiro porcelain of two figures representing\nPainting. The same mark appears on a figure of Apollo, about one foot\nhigh. This mark is stamped on the porcelain; the letters are in relief\ntinted rose colour.\n\n[Illustration]\n\nAt the room decorated with Buen Retiro plaques at Aranjuez. It appears\non a vase in relief, which is placed on the basement to the right on\nentering. The date 1765 appears on the ceiling, probably the year when\nthe work was finished.\n\n[Illustration]\n\nGraven on the clay on a white soft paste bracket painted with coloured\nflowers belonging to Count Valencia de D^{n} Juan.\n\n[Illustration]\n\nOn a cup at the Museo Arqueologico, Mad., painted with landscapes. The\ninitials are of Pedro Antonio Giorgi, who painted from 1802 to 1808.\n\n[Illustration]\n\nEngraved on the clay on a plaque of blue biscuit porcelain imitating\nWedgwood jasper ware, representing a mythological subject.\n\n[Illustration]\n\nOn a dessert plate representing a vine leaf at the Museo Arqueologico.\nThe initials appear to be those of Felipe Gricci, a son of the first\nmodeller Jos\u00e9, who came with Charles III. from Naples.\n\n[Illustration]\n\nFleur-de-lis graven in the clay on a fine group of three figures.\n\n\nLIST OF DIRECTORS AND ARTISTS WHO WERE EMPLOYED IN THE ROYAL MANUFACTORY\nOF THE BUEN RETIRO FROM ITS FOUNDATION, IN 1759, UNTIL 1808:\n\nDIRECTORS.\n\nBonicelli, Juan Thomas, principal Director at the establishment of the\nmanufactory in 1759.\n\nBonicelli, Domingo. In 1786 he was Director; in 1796 he solicits his\nretirement, and died soon after.\n\nCristobal de Torrijos, appointed Director in 1797, after the death of\nDon Domingo Bonicelli.\n\nSureda, Bartolom\u00e9, Director in 1804, and continues in 1808.\n\n\nPRINCIPAL MODELLERS AND SUPERINTENDENTS, POSSESSING THE SECRETS OF THE\nFABRICATION (SECRETISTAS).\n\nSchepers, Cayetano, first Modeller in 1759.\n\nGricci, Carlos, son of Joseph Gricci, came to Spain, 1759. He appears in\na list of artists employed in 1764: he died 1795.\n\nGricci, Felipe, 1785. In 1802 he was first Modeller.\n\nForni, Antonio, second Modeller in 1802.\n\n\nSCULPTORS.\n\nAgreda, Esteban, born at Logro\u00f1o, 1759. He obtained several prizes at\nthe Academy of San Fernando; employed in 1797, and continued to work\nthere in 1808.\n\nAvila, Ceferino de, employed 1799, and continued there in 1808.\n\nAvila, Juan de, 1771, and continued there in 1808.\n\nBautista, Bautista de, 1759.\n\nBautista, Cayetano, 1785.\n\nBautista, Juan Lopez, employed from 1799 to 1808.\n\nBenedictis, Cayetano, 1785, 1802.\n\nBenincasa, Miguel, 1778, and continues to work in 1808.\n\nBenincasa, Vicente, 1785.\n\nBergaz, Alonso, 1764.\n\nBescia, Juan, 1759.\n\nBorbon, Geronimo, 1802\n\nBorbon, Genaro, 1784 and 1808.\n\nCaravielo, Miguel, 1785.\n\nChaves, Alonso, born at Madrid in 1741. In 1760 was appointed Modeller,\nand in 1763 obtained a Second Prize at the Academy, and a First Prize in\n1766.\n\nChaves, Justo, 1785.\n\nEsplores, Felipe, 1759.\n\nEstebe, Antonio, 1778 and 1808.\n\nFlores, Josef, 1785.\n\nForni, Pablo, 1759.\n\nFrancholy, Angel, 1776 to 1808.\n\nFrancholy, Jos\u00e9, 1804 to 1808.\n\nFrancholy, Luis, 1785.\n\nFrate, Carlos, 1785 to 1802.\n\nFrate, Josef, 1785.\n\nFrates, Juan, 1794 to 1808.\n\nFrate, Pablo, from 1759 to 1785.\n\nFrates, Mateo, 1797 to 1808.\n\nFrates, Mateo, born at Madrid in 1788. First Prize of the Academy in\n1805; in 1829 was appointed Director of the China establishment at the\nMoncloa.\n\nFrates, Francisco, 1764 to 1808.\n\nFumo, Basilio, in 1759 was Director of the China manufactory; in 1779 he\nwas appointed a Member of the Academy of San Fernando; died in 1797.\n\nFumo, Carlos, 1759.\n\nFumo, Cayetano, 1759.\n\nFumo, Joseph, 1759; died in 1799.\n\nFumo, Macedonio, 1759, and continues in 1802.\n\nFumo, Bernab\u00e9, 1802.\n\nGiorgi, Pedro Antonio de, 1759, and continues in 1785.\n\nGiorgi, Carlos, 1785 to 1808.\n\nGiorgi, Ambrosio de, 1759.\n\nGiorgi, Antonio, 1795 to 1808.\n\nGricchi, Joseph, 1759. In 1766 was appointed Honorary Director of the\nAcademy of San Fernando; died in 1769.\n\nGricci, Esteban, 1759.\n\nGuijarro, Dionisio, 1798 to 1808.\n\nLlorente, Manuel, 1764 to 1785.\n\nMorelly, Antonio, 1759 to 1785.\n\nNofri, Salvador, 1759 to 1785.\n\nNofri, Justo, 1778 to 1808.\n\nNofri, or Noferi, Juan, 1802.\n\nOchogavia, Manuel, 1764. Born in Galicia in 1744; in 1760 won a Second\nPrize of Sculpture at the Academy, and in 1763 a First Prize.\n\nPalmerani, Domingo, 1795, 1808.\n\nPalmerani, Angel, 1799 to 1808.\n\nPenaba, Joseph, 1793 to 1808.\n\nRodriguez, Antonio, 1797 to 1808.\n\nSancho, Dionisio, 1788. Born at Cienpozuelos in 1762; won a Prize at the\nAcademy in 1793; was appointed a Member of the Academy, 1796; in 1810 he\nwent to Mexico, where he died, 1829.\n\nSantorum, Joseph, 1759.\n\nSorrentini, Fernando, 1785 to 1808.\n\nSorrentini, Rafael, 1785.\n\nSorrentini, Francisco, 1802.\n\nValentin, Jos\u00e9, 1779 to 1808.\n\nValentin, Miguel, 1785.\n\n\nPAINTERS.\n\nAlonso, Francisco, 1764.\n\nBoltri, Genaro, 1756. Born in Naples in 1730; in 1759 he came to Madrid\nwith Charles III.'s household, and worked at the Retiro; died in Madrid\nin 1788.\n\nBrancasio, Xavier, 1759.\n\nBrancacho, Domingo, 1762 to 1803.\n\nBranga, Ignacio de, 1800, painter of figures; he continues there in\n1808.\n\nCamaron, Josef, 1802. Born at Segorbe in 1760; in 1776 he won a prize of\npainting at Valencia; he was pensioned to Rome, and appointed Painter in\nOrdinary to the King.\n\nCastillo, Fernando del, born at Madrid in 1740. He was appointed Painter\nat the manufactory, and worked there until his death in 1777.\n\nCoco, Joseph del, 1759.\n\nCruz, Mariano de la, 1807, 1808.\n\nDomeu, Carlos, 1785.\n\nDonadio, Nicolas, 1759.\n\nGiorgi, Pedro Antonio, 1802.\n\nMartinez, Antonio, 1764.\n\nMartinez, Pedro, 1796 to 1808.\n\nNani, Mariano, 1759. His wife received a pension from 1804, probably the\nyear of his death.\n\nPeshorn, Jorge, 1788, and continues working in 1802.\n\nProvinciale, Antonio, 1759 and 1785.\n\nQuir\u00f3s, Juan Jos\u00e9, 1802.\n\nRimini, Carlos, 1759.\n\nRomero, Juan Bautista, 1800. Flowers and fruit; appears in lists of\n1802.\n\nRubio, Joseph, 1799 to 1808.\n\nSemini, Francisco, 1759.\n\nSoriano, Joaquin, 1799. Landscape painter in 1800; continues in 1808.\n\nSorrentini, Josef, 1756, probably from Capo di Monte. In 1802 he asks\nfor a retiring pension.\n\nSorrentini, Fernando, 1759.\n\nSorrentini, Pablo, 1764 to 1808.\n\nSorrentini, Gabriel, 1769 to 1808.\n\nSorrentini, Manuel, 1785 to 1802.\n\nTorre, Joseph de la, 1759.\n\nTorre, Nicolas de la, 1759. In 1802 asks for a retiring pension.\n\nTorre, Raphael de la, 1759.\n\nTorre, Juan Bautista de la, 1759 and 1808.\n\nTorre, Josef de la, 1785 and 1802.\n\nTorre, Francisco de la, 1796 to 1808.\n\nTorre, Julian de la, 1802.\n\nVelasquez, Castor, 1807, and continued in 1808. Born in Madrid in 1768\nand obtained a prize at the Academy in 1787.\n\n\nVARIOUS ARTISTS EMPLOYED IN THE MANUFACTORY.\n\nAgreda, Manuel, Sculptor, a brother of Esteban Agreda; he superintended\nthe making of biscuit china; born at Haro in 1773; won prizes at the\nAcademy; and was employed at the Manufactory from 1805 to 1808.\n\nBautista, Juan, employed to make porcelain flowers from 1785 to 1808.\n\nBautista, Francisco, appears in 1802 as maker of porcelain flowers.\n\nBautista, Sebastian, appears in 1802 as a maker of porcelain flowers.\n\nChevalier, Pedro, mounter of snuff-boxes from 1759, and continued to\nwork at the Manufactory in 1763.\n\nEscalera, Josef, mounter of snuff-boxes from 1781, and continued to work\nat the Manufactory in 1808.\n\nPerche, Jaime Victor, French workman brought from Paris to prepare\nporcelain, from 1803 to 1809.\n\nVivien, French workman, brought from Paris to prepare porcelain, from\n1803 to 1809.\n\nAt the same time that porcelain was made at Retiro and Alcora, other\nmanufactories of a similar kind were established in Spain; but none of\nthem could compete with these. The most important was established at\nSargadelos, Galicia, in 1804. One of the finest specimens which have\nreached us of this manufactory is a large bas relief representing the\nmassacres of the Spaniards by the French in Madrid on the 2nd of May,\n1808.\n\n\n\n\nGLASS.\n\n\nSince the South Kensington Museum purchased the largest collection of\nold and modern Spanish glass in Europe, the taste for and study of this\nbranch of industry has increased to a very great extent. It is evident\nthat this manufacture attained great importance during the three last\ncenturies, and possibly existed at an earlier period; and that glass\nobjects made in Spain possess a special and distinct character,\ndifferent to those made in other countries. Before the objects at the\nKensington Museum were collected, the existence of this industry in\nSpain was comparatively unknown.\n\nThe earliest mention of glass works in Spain will be found in Pliny,\nwho, while explaining the proceedings which were employed in this\nindustry, says that glass was made in a similar manner in France and\nSpain: \"Jam vero et per Gallias Hispaniasque simili modo harenae\ntemperantur.\"--L. XXXVI. cap. 66.\n\nThe next allusion to this industry will be found in the works of San\nIsidoro. This eminent man lived in the 7th century, and after quoting\nthe observations of the Roman author, gives us to understand that this\nindustry existed before his time in Italy, France, and Spain: \"Olim\nfiebat et in Italia, et per Gallias, et Hispaniam arena alba mollissima\npila mola que terebatur.\" [Divi. Isid. Hisp. Etymologiarum, I. XVI. cap.\n16.] It is evident, therefore, from this passage, that glass was made to\na large extent in the Spanish Peninsula during the Roman period. This is\nconfirmed by the number of specimens which are constantly found in\nruins. We learn also that the manufacture had ceased to exist in the\nseventh century.\n\nGlass vessels of the Roman period found in Spain are similar in form and\nmanufacture to those which we know were made in France and Italy. This\nis not to be wondered at, if we remember that the Romans imposed their\nartistic forms on the countries they conquered. It is impossible to\nclassify the specimens of this industry into determined localities. The\nstudy of the glass paste may, at some future period, give materials for\nsuch a classification.\n\nOne special characteristic of Roman glass may be taken into account to\nbe applied to Spanish glass of a later period. We find ancient specimens\nconstantly ornamented with a sort of thread or line which runs all over\nthe vase. These lines are sometimes made of transparent glass, and\nsometimes of white opaque glass, termed in Italy _latticinio_ from its\nmilky whiteness. When the industry of glass making was revived in Europe\nduring the fifteenth and sixteenth centuries, classical forms were\ncopied in this as in other industries; this line ornamentation was\ncopied on a much smaller scale in Italy than in Spain, where it\nconstituted the chief and constant characteristic of glass making. It is\nan interesting fact that objects of a traditional Moorish form have the\ngreatest amount of lines of this style of ornamentation. We cannot,\nuntil this subject is more thoroughly investigated, do otherwise than\ninfer either that the tradition of this industry was preserved in Spain,\nor that the Arabs imported this style of decoration from the same\nlocalities from which it had been copied by the Romans centuries before.\n\nThe comparison of these different styles of glass making can be carried\nout in a most satisfactory manner in London by examining the fine\nspecimens of glass in the British Museum (Slade collection), and the old\nSpanish glass at South Kensington.\n\nWe have no specimens of glass of the Visigothic period. If, as is most\nprobable, glass was used by the Visigoths, they may have imported it\nfrom the East, for the text I have quoted from the works of St. Isidore\nseems to prove that this industry had ceased to exist in his time. The\nglass paste of different colours must, however, be mentioned, which is\nset in gold in the Visigothic crowns found at Guarrazar, near Toledo. It\nimitates precious stones, and was very generally used during the\nByzantine period; its occurrence here makes it appear probable that at\nany rate the tradition of this industry existed in Spain.\n\nFrom the 8th to the end of the 15th century, during the Mahomedan\ndomination, I infer that the industry of glass making became as\nimportant in Spain as that of pottery. No specimens, however, of the\nearlier period have reached us, and we must judge of what it was from\nthe glass vessels in the Kensington Museum, belonging to the Renaissance\nperiod, which preserve their Oriental form, and are of a different style\nto that of Venice and other localities in Europe.\n\nA most interesting fact, which confirms this theory, will be found in a\ntranslation made from Hebrew to Arabic of a work which treats of the\nvirtues of precious stones, _Lapidario_ [MS. Biblioteca del Escorial],\nquoted by Rico y Sinobas in \"Almanaque de la Industria,\" 1873. We do not\nknow exactly when the author Abolais lived, but in the prologue to this\nMS. it is mentioned that D^{n}. Alfonso el Sabio found this book at\nToledo, and gave orders to a Jew called Juda Mosca, and a priest, Garci\nPerez, to translate it into Spanish. The translation was terminated in\n1250. Technical details are given in this volume concerning the\nsubstances which are employed in glass making, and some of the minerals\nfound in Spain which are used in painting or enamelling it, but as this\nwork was written to explain the properties of minerals, as they were\nunderstood in the author's time, he does not enter into any interesting\ndetails, or describe the forms of these objects. Another allusion to\nglass making in Spain will be found in Al Makkari, the Arabian author of\n\"Mohamedan Dynasties in Spain,\" [London, 1840, 2 vols. 4to], who quotes\nan Oriental writer of the 13th century, who says, \"Almeria was also\nfamous for the fabrication of all sorts of vases and utensils, whether\nof iron, copper, or glass.\" (Vol. I. p. 51.)\n\nThe tradition of this industry has been undoubtedly preserved at Almeria\nuntil the present day, for in this province, and in the adjoining\nvillages of the province of Granada, we find that specimens are to be\nmet with, which possess a marked Oriental form, and are completely free\nfrom the influence of Italian models which existed in other localities.\nThe most characteristic specimens consist of jars of two, four and eight\nhandles, bowls with ribs and handles, pilgrim's bottles, etc., of which\ninteresting examples exist at the South Kensington Museum. Woodcuts of\nseveral of these are given.\n\nAll these objects are decorated with a serrated ornamentation of\nbuttons, trellis-work, and the lines to which I have already alluded\nwhich were placed there after the object was made, in the Roman style.\nThe paste is generally of a dark green colour, and when we find these\nsame features in vessels of white clear glass, we may affirm that they\nare contemporary imitations made at Cadalso or elsewhere, for they are\nvery seldom to be met with in the provinces of Almeria and Granada, and\nare generally found at Toledo and other localities; it is, moreover, a\ncommon condition of Oriental art that its general form complies with a\ngeometrical tracery, and we never find as in Italian works of art, forms\nand capricious ornamentations which interfere with the symmetry of the\ngeneral lines, and sacrifice them to the beauty of the whole.\n\n[Illustration: GLASS VASE.--SPANISH, 16TH CENTURY.]\n\nI have been unable to find the glass industries of this period mentioned\nby any contemporary author, but I owe to the courtesy of S^{r}. Romero\nOrtiz some interesting details which have been taken from the archives\nof certain villages, which although of a more modern date prove that\nthis industry existed in these localities. At about 14 miles from the\nPuebla de D^{n}. Fadrique, there is a locality called Pinar de la\nVidriera, where traces of ovens and scori\u00e6 exist belonging to an\nancient glass manufactory, which is likewise mentioned in documents of\nthe municipality as continuing to work in 1620. At Castril de la Pe\u00f1a, a\nmanufactory yet exists where glass is still made, and which has existed\nfrom time immemorial in the village. The building itself is decorated\nwith the escutcheon of the family of Hernando de Zafra, one of the\nsecretaries of Queen Isabel, late in the 15th century, who must have\npurchased it with other lands which he possessed there, towards the year\n1492. A gallery, one mile long, which exists at the entrance of the town\nfrom whence the sand has been extracted for this manufacture, gives an\nidea of the antiquity of this industry in this particular locality. It\nhas been calculated that about two tons of sand were used at these glass\nworks every month. At Royo Molino, in the province of Jaen, a very\nancient building still exists, now half in ruins, which has been used\nuntil very lately as a glass factory. At Maria, in the province of\nAlmeria, several glass manufactories have existed. The oldest, it is\nbelieved, was about two miles from the town, and is called traditionally\n\"del Campo,\" but I have not found mention of it in any document. Three\nglass ovens existed inside the town: one was established by Vicente\nBotia, towards the year 1750, which lasted until 1790. Juan Martinez\nestablished at about this period an industry of a similar kind near the\nformer one, which continued to work until 1854. The same green glass has\nbeen constantly made in these localities, the same forms have been\ncopied, the only difference between the older and more modern specimens\nconsisting in the coarser and heavier quality of the glass.\n\n[Illustration: PILGRIMS' GLASS BOTTLE.--SPANISH, 17TH CENTURY.]\n\nIn treating of the glass manufactures in Spain, where Italian models\nwere imitated to a very great extent, we find very few allusions in\ncontemporary authors of the Middle Ages; it may be because glass vessels\nwere chiefly used with wooden and pottery utensils by the poorer\nclasses, and metal utensils of all kinds by those who were able to\nafford them. At any rate, it is only towards the end of the Middle Ages\nthat we meet with information concerning this industry, which continues\nuninterrupted until the present day.\n\nBarcelona is one of the towns distinguished for the antiquity and\nexcellence of its glass. In a municipal edict of 1324, we find a special\nprohibition that the glass ovens should not be inside the city, owing to\nthe danger they might cause to the rest of the population. In 1455,\npermission was granted to the _vidrieros_, glassmakers, to form a guild\nunder the patronage of St. Bernardino, and from this period some of its\nmembers figure as holding municipal charges. Capmany, \"Memorias,\" Vol.\nI. p. 134. According to this author, the special Ordinances of this\ncorporation are not known, but only those which were given by the\nmunicipality in 1659. He adds that the master _vidrieros_ required six\nyears of apprenticeship and practice to be admitted to work.\n\nFrom the 15th century several authors praise the glass made at\nBarcelona. In a MS. by Jeronimo Paulo, who writes in 1491, a description\nin Latin of the most remarkable things at Barcelona, he says, \"they also\nsend to Rome and other places many glass vessels of different sorts and\nkinds which may well compete with those of Venice.\" In the account of\nPhilip le Bel's journey to Spain on his marriage with Queen Joanna we\nfind the following mention of the town of Barcelona: \"Et sont l\u00e0 faicts\nles plus beauls ouvrages de voires (verres) et de cire qui soient faicts\nau monde. Le Jordi (19 Janvier, 1503) Monsigneur alla au dehors de la\nville vioir ung jour o\u00f9 on faict voires (verres) de cristallin tr\u00e8s\nbeaus.\" [\"Collection des Voyages,\" Bruxelles, 1876, Vol. I., p. 257.]\nMarineus Siculus, who writes early in the 16th century says, \"the best\nglass made in Spain is that of Barcelona,\" and Gaspar Barreiros in his\n\"Chorographia,\" [Coimbra, 1569,] tells us, \"they made excellent glass at\nBarcelona, almost equal to the Venetian.\" At the beginning of the 17th\ncentury the authors, Jaime Rebullosa, in his \"Descripcion del mundo,\"\n[Barcelona, 1603, 8vo,] and Luis Nu\u00f1ez in \"Hispania,\" [Antverpi\u00e6, 1607,\n8vo, p. 279,] continue to praise the glass made at Barcelona, and from\nthat period we find its merit and the vast quantity which was exported\nconstantly alluded to. The fame of Spanish glass must have been\njustified, for in the \"Viage del Cardinal Infante,\" by Aedo, printed in\n1639, we find it stated that when the Infant Cardinal was at Barcelona,\nin 1632, he went with his galleys to Matar\u00f3, four leagues from\nBarcelona, to see the \"glass made there which was so abundantly sold all\nover the country.\" The Spanish translator of \"La Piazza universale di\ntutte le professioni,\" [Madrid, 1615,] adds the name of Barcelona to the\nItalian author's mention of glass objects made in Venice, saying, \"This\nindustry has reached such a degree of excellence at Murano and Barcelona\nthat nothing can be compared to it; there is nothing now which cannot be\nmade of glass and crystal, even cabinets have been made, and castles\nwith their towers, battlements, artillery, and fortifications.\" Mendez\nSilva in his \"Poblacion de Espa\u00f1a, [Madrid, 1654, p. 243,] repeats the\nsame idea when he says they made at Barcelona, \"fine glass which might\ncompete with the Venetian.\" This industry continued in Catalu\u00f1a to a\ngreat extent in the last century, and was praised by the following\nwriters. The author of the \"Atlante Espa\u00f1ol,\" [Vol. IV., Madrid,\n1778-1795,] tells us that \"they continued to make excellent glass at\nBarcelona, in imitation of the Venetian, with which it might compete,\"\nand that this industry was carried out to a great extent at Matar\u00f3,\nCervell\u00f3, and Almatret, all three towns of Catalu\u00f1a; he says that in\nthis last mentioned town the glass made was \"so excellent, and the\nnumber of workmen employed was so large, that an oratory had been built\nin order that the workmen should hear mass there.\"\n\nThe constant comparisons which we find between the glass made at\nBarcelona with that of Murano suggests two things--that Spanish glass\nmust have been of a first-rate order, and that the form of the glass\nvessels was similar to those made at Venice. It is highly probable that\na great part of the specimens of glass of different kinds which are\nclassified as Italian in several collections, are really Spanish,\nalthough it is extremely difficult to point out the difference. In this,\nas in other branches of industry, the mania for classifying has gone too\nfar, and comparative studies of a more concrete order are necessary;\nuntil these are made, the principal fact to be borne in mind in\nclassifying glass vessels is to compare them with objects of a similar\nmanufacture which have been made until very lately in Catalu\u00f1a, and of\nwhich an extremely interesting collection exists at the South Kensington\nMuseum, proceeding from Barcelona, Matar\u00f3 and Cervell\u00f3, (Nos. 149 to\n193) which will enable the amateur to see how the industry of\nglass-making has continued traditionally to be preserved in that\nprovince. See woodcut opposite representing a glass bottle, also the\ncentral vessel in woodcut on p. 238.\n\nAmong the specimens of old Spanish glass we may consider to have\nproceeded from Catalu\u00f1a the following examples, forming part of the\ncollection acquired in 1873. Bottles, Nos. 249 to 262, cups, Nos. 336 to\n339, and tumblers, Nos. 303 to 305, 312 to 314, and 328, '29. Among\nthese, there are some which are undoubtedly still made in the locality.\nThe most characteristic are the _arruxiad\u00f3s_, or _borracha_, a vessel\nspirally ribbed with several narrow spouts, used by the peasant girls of\nthe villages to sprinkle rose water in the festival of the patron saint\nof the district; the _porrones_, a bottle with a long spout, used by the\npeasantry for drinking wine; the _cantaro_, a jug, funnel shaped with\nwide mouth, handle and curved spout, and the _pilas de agua bendita_, a\nholy water vessel of different shapes.\n\n[Illustration: GLASS BOTTLE.--MODERN SPANISH.]\n\nVery few details have reached us of the famous glass works which existed\nat Cadalso, in the province of Toledo; we know, however, that the glass\nmade there was as excellent as that made at Catalu\u00f1a, and was compared\nin a similar manner to the Venetian. The unknown author of \"El\nCrotalon,\" published by the Spanish Bibliophiles, from a rare MS., in\n1871, writing in the time of the Emperor Charles V., mentions \"the fine\nglass made at Cadalso.\" Marineus Siculus also writes in 1517 in his \"De\nlas Cosas Memorables de Espa\u00f1a,\" [Alcala de Henares, 1539, fol., Vol.\nI.,] that \"Glass was made in several towns of Castile, the most\nimportant of them being Cadalso, which supplied the whole kingdom.\" It\nwould appear therefore that this glass manufactory was already\nestablished in that locality in the 16th century, and we frequently meet\nwith allusions to the excellence of its productions in contemporary\nauthors. The town itself has continually been called \"Cadalso de los\nvidrios.\" Mendez Silva in his \"Poblacion general de Espa\u00f1a,\" [Madrid,\n1645, p. 40] says: \"They make in three glass ovens fine glass of\nbeautiful colours and forms, which can compete with the Venetian.\" At\nthe South Kensington Museum, there are two fine specimens of glass made\nat Cadalso--No. 1068, '73, a drinking vessel of white glass, the upper\npart in form of the Spanish vessel called \"bucaro,\" ribbed and streaked\nwith colours; the stem pineapple shape, with remains of gilding (this is\nrepresented in the woodcut); and No. 1082, '73, a bowl of plain glass,\nthe lower part spirally waved, and decorated with scale pattern in gold\nand coloured dots. Both these objects proceed from a nunnery in a\nvillage near Cadalso.\n\n[Illustration: GLASS VESSELS FROM SPAIN, IN THE SOUTH KENSINGTON\nMUSEUM.]\n\nLarruga says in his \"Memorias,\" [Madrid, 1791, Vol. X., p. 53,] that\n\"two glass ovens existed at Cadalso, which belonged to the Marchioness\nof Villena; they are worked by the inhabitants of this town. One of the\novens has fallen into disuse from the beginning of the present (18th)\ncentury, and the fame of the glass made there, formerly so renowned for\nits clearness and variety of the objects made, has declined. In\nexamining the state of this industry during the reign of Charles II. we\nfind how much it had fallen off, for Dn. Antonio de Obando in 1692,\nundertook to reestablish it, and to make glass vessels, and window\nglasses as had been made there formerly. It appears that 200 dozens of\nobjects of different kinds were sent out from there yearly.\" The vase,\nNo. 333, '73 represented in the woodcut is an example of the coloured\nglass of Cadalso.\n\n[Illustration: GLASS VASE.--SPANISH (CADALSO), 17TH CENTURY.]\n\nGlass ovens existed from a very early period in the central provinces\nof Spain, which probably imitated from the 16th century the productions\nof Cadalso. We know from papers at the Archives of the Convent, of two\nof these which paid a yearly rent to the monastery of San Jeronimo de\nGuisando, from 1478 to 1480. One of these glass works was situated at\nthe Venta del Cojo on the borders of Escalona, and the other at the\nVenta de los Toros de Guisando.\n\nIt is evident that a glass manufactory existed at Toledo, from some\naccounts which I have found in the Archives of the Cathedral, in which\nBartolom\u00e9 Lopez, vidriero, is mentioned in 1546--Pedro Fernandez in\n1590, and Tomas Nu\u00f1ez in 1660; as furnishing glass vessels, principally\nlamps for the cathedral. Tomas Nu\u00f1ez is also paid for \"three crosses of\ncrystalline glass.\" In a letter written in 1690 to Count Gondomar, the\ncelebrated ambassador to James I., (Archives of the Palace at Madrid), I\nfind a glass manufactory mentioned which existed at Cebreros in the\nprovince of Segovia. The writer says, \"Your lordship knows we have a\nglass oven here; this week we have made the glass called crystalline, of\nwhich I send in a basket sixteen specimens for my lady Dna. Costanza.\"\n\nLarruga tells us in his \"Memorias,\" [Madrid, 1792, Vol. XVI., p. 222,]\n\"That in 1680 a glass manufactory was established at San Martin de\nValdeiglesias, where glass in the Venetian style was made--the glass was\nexcellent. This manufactory was under the direction of Diodonet Lambot a\nnative of Namur. He was succeeded by Santiago Bandoleto, who was by no\nmeans as skilful. Glass ovens were also established at the Torre de\nEstevan, Hambroz, under the direction of Guillermo Torcada.\" At\nRecuenco, in the province of Cuenca, glass works existed at the\nbeginning of the 16th century; but in 1722 Don Fernando Lopez de Aragon\nestablished a manufactory on a large scale. The glass made there was\nused by the king and the greater part of the rich people at Madrid. In\n1739 two other glass works existed there, which continued their\noperations during the rest of the century. In the same province several\nglass ovens existed towards the middle of the last century. The\nprincipal ones were established in the villages of Arbetota, Vindel and\nArmallones; only coarse specimens were made there, but at an earlier\nperiod their productions had been, Larruga tells us in the nineteenth\nvolume of his \"Memorias,\" very important. The glass works which we find\nmentioned in the \"Atlante Espa\u00f1ol,\" [Madrid, 1795,] were also of a very\ninferior quality; they were established at the end of the last century\nat Busot, in the province of Alicante, Hinojares and Carolina, province\nof Jaen. The \"Correo de Espa\u00f1a,\" [Vol. I. Madrid, 1771,] mentions an\nimportant glass manufactory which existed at Andorra near Alca\u00f1iz, and\none where glass of an inferior quality was made at Cabra, in the\nprovince of Cordova. Cabanilles, \"Historia y descripcion del reino de\nValencia,\" Barcelona, mentions the existence from a very early period of\nglass works at Olleria, a village of the kingdom of Valencia.\n\nThe only mention which I have found in which Seville appears as a centre\nof this industry, is an interesting allusion to the village of Cala,\nwhich was represented in a festival which was given to Philip II. in\n1578, \"with a crystal glass in its hand, and several others at its feet,\nbecause a famous glass oven existed there.\" [\"Recibimiento que hizo\nSevilla al Rey Don Phelipe.\" Por Juan de Mallara, Sevilla, 1570, p. 89.]\n\nIn the 17th century, during the reign of Philip IV., the glass made at\nValdemaqueda in the province of Avila was very celebrated. In a royal\nschedule, dated 1680, which states the price at which things were sold\nat Madrid, \"the glass made at Barcelona, Valdemaqueda, and Villafranca\"\nare mentioned. We find the glass of Valdemaqueda was sold for a higher\nprice than that made at the other localities.\n\nThe principal value of these extracts is to prove that the industry of\nglass-making existed in these localities; none of the authors mentions\nthe principal characteristics by which the vessels made may be\nclassified. Those made at Almeria and Barcelona are more easily known,\nfor they have kept until very recently their general character; while at\nCadalso and other localities of the centre of Spain, the tradition of\nthe Italian models has quite disappeared. In one fact every author\nagrees, namely, that the glass made in those provinces closely resembled\nthe Venetian, and was completely different to the system followed by the\nArabs. It is, therefore, highly probable, as I have already stated, that\na large number of objects of glass of different kinds, which have been\nhitherto considered Italian, were made in Spain.\n\n[Illustration: VASE OF GREEN GLASS, WITH BLACK HANDLES AND\nRIBS.--SPANISH, 17TH CENTURY.]\n\nThe following observations may serve as a guide in the difficult task of\nclassifying this group. The specimens of glass vessels most generally\nmet with in the neighbourhood of Cadalso are:--\n\nObjects of white transparent glass, sweetmeat dishes ornamented with\nribs, buttons, and reticulated rims with touches of gold.\n\nObjects of clear greenish glass of a paler hue than is made in the\nprovince of Almeria, decorated with blue, or some opaque colour (see\nwoodcut); in some instances part of the object is decorated with a rough\ncrackled surface.\n\nObjects of opaque glass with different colours, vases, glasses, cups and\nsaucers, of thin milk-white texture with blue spots, or imitations in\nthe Japanese style of different colours. Vases for holding flowers, and\nother objects, of dark blue glass, milk-white cups, glass and other\nobjects ornamented with lines of red or blue of a thicker paste than the\npreceding ones.\n\nAlthough these objects are copied from Venetian models, they are coarser\nin every detail, they are heavier and thicker, and the delicate and\nelegant ornamentation which we find on Italian specimens is almost\nalways wanting, we seldom find examples of the beautiful _millefiori_\nchalcedony or tortoise-shell paste, and the outline of these objects is\nsymmetrical and Oriental in style.\n\n       *       *       *       *       *\n\nThe royal glass manufactory at La Granja de San Ildefonso, was founded\non the remains of an important one which had been established some years\nbefore, under the protection of Philip the 5th, at the Nuevo Bastan, in\nthe province of Madrid.\n\nTowards the years 1712 to 1718, this king commissioned D^{n}. Tomas del\nBurgo and D^{n}. Juan B. Pomerague to establish glass-works at the Bastan;\ntwenty foreign workmen were brought over with this purpose, with their\nfamilies, and the necessary implements. These gentlemen did not carry\nout the undertaking to the king's satisfaction, and in 1720 his majesty\ngave D^{n}. Juan de Goyeneche special privileges that he might \"make every\nkind of glass manufacture up to the height of twenty inches, and have\nthese glasses worked and polished, embroidered and covered with metal;\nto make looking-glass and similar decorations, and every kind of glass\nvessels, and white glass for window-panes, and every sort of glass\nvessel of different kinds and forms which have been invented in the\npresent time, or likely to be invented in this art.\" In order to carry\nthis out, the king allowed them to have as many foreign masters and\nworkmen as they might require, with the sole condition that a fourth\npart of the workmen employed should be Spaniards. It was prohibited that\nany industry of a similar kind should be established in Spain for thirty\nyears, or that glass made abroad should be imported into the country.\n\nAfter a long series of annoyances of every kind, Goyeneche succeeded in\nmeeting with the king's approbation, but as fuel was very scarce at the\nBastan, he removed the glass works to Villanueva de Alcorcon, in the\nprovince of Cuenca. From the ruins of this glass manufactory, Larruga\nsays in his \"Memorias,\" [vol. xiii., p. 274], \"was founded the splendid\nglass manufactory of San Ildefonso.\"\n\nVentura Sit, a native of Catalonia, constructed an oven there, which\nworked from 1728 to 1736 with great success. Queen Do\u00f1a Isabel\nencouraged Sit, and the king ordered him to make some glasses which\nmight serve for mirrors. Those he made at first were small, but Sit\nbegan by making them 30 inches long and ended by making them as large as\n145 inches. The machine for polishing them was invented by a Catalan\ncalled Pedro Fronvila.\n\nThe section of glass vessels of a superior quality began to work in\n1771. Glasses, bottles, and objects of all kinds were made there under\nthe direction of a glass worker called Eder, a Swede, and Sivert, a\nFrenchman. Another section where the same quality of glass was made was\ndirected by Don Segismundo Brun, a native of Hanover, but who had been\nbrought up from a very early age at La Granja. It was this artist who\ninvented gilded glass _a feu_. A large number of foreign artists worked\nat this manufacture; they impressed their special style on the objects\nthey made, which were similar to those of the same kind manufactured in\nFrance, England, and Germany.\n\nFrom the time of Ventura Sit, towards the year 1734, the manufactory of\nglass at La Granja belonged to the crown, and continued under the\nprotection of the Spanish kings, who spared no expense to obtain its\ndevelopment. Early in the present century this industry began to decay.\nTowards 1828 it passed into private hands, and continued to work until\n1849. The manufactory is now closed, although there is some talk of\nreviving this industry.\n\n[Illustration: GLASS VESSELS FROM SPAIN, IN THE SOUTH KENSINGTON\nMUSEUM.]\n\nThe glass objects made at La Granja possess a very marked French style,\nwhich renders them liable to be mistaken for French and German\nproductions of a similar kind. The greater part of the objects which\nwere made were of white clear transparent glass, richly cut and\nengraved, or ornamented with gold (see woodcut). The cut glass is\ngenerally engraved with inscriptions, views, flowers, and devices, and\nuntil very lately has been made there in a rougher style. Coloured and\nenamelled glass has also been made there, but not to so great an\nextent. Glass for window panes and mirrors were ordered for the Royal\npalaces. Lustres of all kinds of white and coloured glass were also made\nthere in the Venetian style, richly ornamented with coloured flowers. A\nsmall chandelier of this kind is in the South Kensington Museum (No.\n998, '73), the only example I have seen of a lustre of latticinio glass.\nIn the collection of old Spanish glass at the Museum will be found a\nsufficient variety of specimens of this manufacture to give an idea of\nthe different styles made at La Granja. A fac-simile of one of the\nengraved mirrors and a fine cut glass will be found in \"Museo Espa\u00f1ol de\nAntiguedades, vol. IX.\"\n\nBesides glass vessels and objects of a similar kind, painted glass\nwindows have been made in Spain from a very early time, of which most\nimportant specimens exist at Toledo, Leon, and other Spanish cathedrals.\nThe proceedings employed in this industry are too well known to require\nrepetition in this volume. The industry was undoubtedly imported by\nforeign workmen from France and the north of Europe, and as was the case\nwith other industries it took root in the country, and a large number of\nSpanish artists followed and took part in this industry.\n\n       *       *       *       *       *\n\nIn order to convey an idea of the height of this industry in Spain, I\ngive a list of the painters who worked on glass which I have collected\nfrom the following works, to which I refer the student who may wish for\nfurther details on the subject. Cean Bermudez, \"Dicc. Hist. de los mas\nilustres Profesores de las Bellas Artes en Espa\u00f1a. Madrid, 1800.\"\nVillanueva, \"Viage \u00e1 las Iglesias de Espa\u00f1a.\" Piferrer, \"Recuerdos y\nBellezas. Mallorca.\" \"Documentos Ineditos.\" Zarco del Valle.\n\n\nGLASS PAINTERS IN SPAIN.\n\n  Years in which\n  they worked         Artists.                    Residence.\n\n  1682.            Alcalde, Francisco               Burgos.\n  1504.            Aleman, Cristoval                Toledo.\n  1458.            Aleman, Pedro                    Toledo.\n  1645.            Alonso, Francisco                Burgos.\n  1544.            Arce, Juan de                    Burgos.\n  1581.            Arce                             Burgos.\n  1613.            Argete, Luis de                   Leon.\n  1424.            Aragan, Maestro Joan              Leon.\n                   Arfian, Antonio                 Sevilla.\n                   Arteaga, Matias                 Sevilla.\n  1516.            Ayala, Francisco                Palencia.\n  1605.            Babel, Jorge                     Madrid.\n  1442.            Baldovin, Maestro                 Leon.\n  1519.            Bernal, Juan                    Sevilla.\n  1533.            Borgo\u00f1a, Jorge                  Palencia.\n  1439.            Bonifacio, Pedro                 Toledo.\n  1562.            Bruges, Carlos                   Cuenca.\n  1602.            Campo, Diego del                 Madrid.\n  1522.            Campa, Juan                      Toledo.\n  1674.            Chilberri, Pierres de           Segovia.\n  1509.            Cuesta, Juan de                  Toledo.\n  1513.            Cordova, Gonzalo de              Toledo.\n  1538.            Cotin, Gaspar de                 Burgos.\n  1459.            Cristoval, El Maestro            Toledo.\n  1566 to }        Dangles, Sebastian              Mallorca.\n  1573    }\n  1676.            Danis, Juan                     Segovia.\n  1565.            Diaz, Diego                     Escorial.\n  1418.            Dolfin, El Maestro               Toledo.\n  1485.            Enrique, Maestro                 Toledo.\n  1566.            Estaenheyl, Ulrrique             Madrid.\n  1565.            Espinosa, Francisco de          Escorial.\n  1565.            Espinosa, Hernando de           Escorial.\n  1551.            Ferrera, Rodrigo de               Leon.\n  1526.            Fernandez, Pedro                 Sevilla.\n  1557.            Flandes, Arnao de                Sevilla.\n  1494.            Fontanet, Gil                   Barcelona.\n  1459.            Frances, Pedro                   Toledo.\n  1571.            Galceran, Il Maestro           Escorial.\n  1518.            Gelandia, Bernardino de        Sevilla.\n  1608.            Guillermo, Maestro                Leon.\n  1571.            Guasch, Juan                 Tarragona.\n  1674.            Herranz, Francisco            Segovia.\n  1520.            Holanda, Alberto de            Burgos.\n  1548.            Holanda, Geraldo de            Cuenca.\n  1535.            Holanda, Nicolas de            Burgos.\n  1509.            Jacques, Juan                 Sevilla.\n  1427.            Juan, Maestro                  Burgos.\n  1510.            Juan, Hijo de Jacobo          Sevilla.\n  1458.            Juanico                        Toledo.\n  1428.            Loys, Maestro                  Toledo.\n  1600.            Ludeque, Diego de              Madrid.\n  1569.            Menandro, Vicente             Sevilla.\n  1538.            Ortega, Alonso de             Toledo.\n  1534.            Ortega, Juan de               Toledo.\n  1458.            Pablo, Fray                   Toledo.\n  1459.            Pedro, Maestro                Toledo.\n  1639.            Perez, Sebastian               Leon.\n  1559.            Pesquera, Sebastian de        Sevilla.\n  1600.            Pierres, Antonio               Madrid.\n  ----             Resen, Pelegrin                 ----\n  1565.            Resen, Renerio                 Madrid.\n  1500.            Rolando, Alberto de             Avila.\n  1500.            Rolando, Nicolas de             Avila.\n  1624.            Ruiz, Valentin                 Burgos.\n  1625.            Ruiz, Simon                    Burgos.\n  1368-- }         Sacoma, Francisco             Mallorca.\n  1369.  }\n  1420 to }        Sala, Antonio                 Mallorca.\n  1447.   }\n  1542.            Salcedo, Diego de             Palencia.\n  1512.            Santillana, Diego             Palencia.\n  1498.            Santillana, Juan de            Burgos.\n  1392.            San Amat, Juan                 Lerida.\n  1503.            Troya, Vasco de                Toledo.\n  1579.            Valerio, Octavio               Malaga.\n  1562.            Valdivieso, Diego de           Cuenca.\n  1497.            Valdivieso, Juan               Burgos.\n  1538.            Vergara, Arnao de              Sevilla.\n  1521.            Vergara, Nicolas de            Burgos.\n  1606.            Vergara, Nicolas de, El Mozo   Toledo.\n  1574.            Vergara, Nicolas de, el Viejo Toledo.\n  1590.            Vegara, Juan de               Toledo.\n  1518.            Vivan, Juan                  Sevilla.\n  1605.            Volui, Gil                     Leon.\n  1509.            Ximenez, Alexo                Toledo.\n  1458.            Ximeno.                       Toledo.\n\n\n\n\nTEXTILE FABRICS.\n\n\nThe silence of San Isidoro on the subject of textile fabrics in Spain\nwould lead us to suppose that this industry only became important during\nthe Arab domination; the Moors probably imported this industry into\nSpain from the earliest times of their conquest. A great number of\nprovinces were famed then for the excellence of their textile fabrics;\nthe most important of them, according to the testimony of Moorish and\nChristian writers, was Almeria. The Cordovese historian, Ash Shakandi,\nwho wrote at the beginning of the 13th century, says: \"Almeria is an\nopulent and magnificent city, whose fame has spread far and wide; the\ninhabitants are very elegant in their dress. Almeria is the greatest\nmart in Andalus, Christians of all nations came to its port to buy and\nsell, and they had factories established in it. From thence the\nChristian merchants who came to its port travelled to other parts in the\ninterior of the country, where they loaded their vessels with such goods\nas they wanted. Costly silken robes of the brightest colours are\nmanufactured in Almeria,\" [Moh. Dyn. in Spain, I. 52]. Almakkari adds,\n[p. 51]: \"But what made Almeria superior to any other city in the world\nwas its various manufactures of silks and other articles of dress, such\nas the _dibaj_ [silken stuff of many colours], a sort of silken cloth\nsurpassing in quality and durability anything else manufactured in other\ncountries; the _tiraz_ or costly stuff on which the names of sultans,\nprinces, and other wealthy individuals are inscribed, and of which no\nless than 800 looms existed at one time; of more inferior articles such\nas the _holol_ (striped silks) and brocades, there were 1000 looms, the\nsame number were continually employed in weaving the stuffs called\n_iskal\u00e1t\u00f3n_ (scarlet). There were also 1000 for weaving robes called Al\njorj\u00e1ni (Georgian), and another 1000 for weaving robes called Isbahani,\nfrom Isfahan, and a similar number for Atabi. The manufacture of damask\nfor curtains and turbans for the women, of gay and dazzling colours,\nemployed a number of hands equal to that of those engaged in the\nmanufacture of the above mentioned articles.\"\n\nAn oriental author asserts that Abd-ul-Rahman II. A.D. 825-852 was the\nfirst sultan of his race who introduced into Spain the use of the\n_tiraz_ [_ib._ Vol. II. 434]. Another explains that Atabi took the name\nof a suburb of Bagdad \"where were made the stuffs called _otabi_,\ncomposed of cotton and silk of different colours.\" [Consult Dozy, 'Dict.\ndes v\u00eatements.']\n\nAsh Shakandi also mentions Malaga as famous for its textile fabrics; he\nsays--\"Malaga is also famous for its manufactures of silks of all\ncolours and patterns, some of which are so rich that a suit made out of\nthem will cost many thousands; such are the brocades with beautiful\ndrawings, and the names of khalifs, amirs, and other wealthy people,\nwoven in them.\" [Moham. Dynast. Vol. I. 49.] Further on the same writer\nadds, \"As at Malaga and Almeria, there are at Murcia several\nmanufactures of silken cloth called _al washiu thalathat_, [variegated,\nof many colours.] It is likewise famous for the fabric of carpets called\n_tantili_, which are exported to all countries of the east and west; as\nalso a sort of mat, of the brightest colours, with which the Murcians\ncover the walls of their houses.\" [_ib._ p. 69.]\n\nArabic texts are frequently met with which allude to the manufactures of\ntextile fabrics of Granada, Seville, and other towns, in which their\nproductions are praised, and there is no doubt that they were very\nremarkable; for Christian authors of the middle ages refer to them so\ncontinually. Students who wish for a more extensive idea of this subject\nmust remember the quotations and information given by Francisque Michel\nconcerning Spanish stuffs, [Recherches sur le Comm., la Fabric. etc.,\ndes Etoffes, etc. Paris, 1852.] He tells us--\"Nous trouverons tout\nd'abord les pailles d'Alm\u00e9rie, ville de la c\u00f4te meridionale d'Espagne,\nqui jouissait d'une r\u00e9putation proverbiale pour la beaut\u00e9 et la finesse\nde ses tissus de soie, vant\u00e9s dans mille endroits de nos vieilles\nchansons de geste, de nos anciens po\u00ebmes\" [Vol. I. p. 284]. Further on\nhe continues [Vol. I. p. 286], \"\u00e0 l'\u00e9poque \u00e0 laquelle appartiennent les\ntextes, et m\u00eame auparavant, la culture et la fabrication de la soie\n\u00e9taient des plus prosp\u00e8res \u00e0 Alm\u00e9rie et en g\u00e9n\u00e9ral dans le royaume de\nGrenade. 'Du royaume de Grenade vient cire, soie, figues, etc.' ... ce\nqu'il y a de bien certain, c'est que les \u00e9toffes d'Espagne \u00e9taient\nc\u00e9l\u00e8bres d\u00e8s le 9^{e} si\u00e8cle. Anastase le Biblioth\u00e9caire en parle en\nquatre endroits sous le nom de _spaniscum_; et en placant ce tissu \u00e0 la\nsuite de _fundatum_ et du _stauracin_ il nous donne suffisamment \u00e0\nentendre qu'il \u00e9tait de grand prix, et de soie comme eux. Un biographe\nde Saint Ansegise, mort en 835, plac\u00e9 de m\u00eame une couverture d'Espagne,\n_stragulum_, _Hispanicum unum_, \u00e0 la suite de tapis ou teintures.\n\n\"Il est malaise de d\u00e9terminer la mati\u00e8re des \u00e9toffes qu'un \u00e9crivain\nbyzantin du 12^{e} si\u00e8cle nous dit fabriqu\u00e9es en Espagne et aux colonnes\nd'Hercule--tout nous permet de croire qu'elles \u00e9taient de soie de tout\nou en partie, ou du moins qu'elles \u00e9taient compt\u00e9es parmi les tissus\npr\u00e9cieux.... Saragosse produisait aussi des draps d'or--et nous trouvons\nau 13^{e} si\u00e8cle que le g\u00e9ographe Edrisi pouvait dire qu'il y avait dans\nle seul territoire de Jaen 3000 villages o\u00f9 l'on \u00e9levait des vers de\nsoie. Nous savons qu'\u00e0 Seville, sous la domination des Maures, elle\ncomptait 6000 m\u00e9tiers pour ces \u00e9toffes de soie. Les r\u00e8glements\nmunicipaux dont les royaumes de Grenade et de Seville ont \u00e9t\u00e9 l'objet\nsuffisaient pour nous apprendre que l'industrie de la soie fut\nflorissante durant le moyen \u00e2ge. Nous signalerons les draps d'or et de\nsoie que D. Pedro de Luna en 1327 apporta \u00e0 l'infante L\u00e9onore, s\u0153ur\nd'Alphonse XI., \u00e0 l'occasion de son mariage, et nous savons qu'un\nministre de Pedro le cruel poss\u00e9dait \u00e0 Toledo 125 coffres de drap d'or\net de soie.\"\n\nEdrisi in his \"D\u00e9scrip. de l'Afrique et de l'Espagne,\" [Leyde, 1866,]\nwriting in the 12th century, says p. 239: \"Alm\u00e9rie \u00e9tait la ville\nprincipale des Musulmans \u00e0 l'Espagne des moravides. Elle \u00e9tait alors\ntr\u00e8s industrieuse, et on y comptait entre autres 800 m\u00e9tiers \u00e0 tisser la\nsoie, o\u00f9 l'on fabriquait des \u00e9toffes connus sous le nom de _holla_, de\n_debady_, de _siglaton_, d'_espahani_, de _djordjani_: des rideaux orn\u00e9s\nde fleurs, des \u00e9toffes orn\u00e9s de clous, de petits tapis, des \u00e9toffes\nconnus sous les noms de _attabi_, de _mi djar_.\"\n\nTo complete this historical information, and especially in order to add\ndetails as to the technical character of this manufacture, the Rev.\nDaniel Rock's interesting introduction to \"Textile Fabrics, a\nDescriptive Catalogue, London, 1870,\" must be consulted; although,\nnotwithstanding the learned author's remarks, it is extremely difficult,\nin my opinion, to classify textile fabrics proceeding from Syria,\nSicily, or the Spanish Arabs, as all, whether imitations or originals,\nare similar in manufacture.\n\nM. Michel and Dr. Rock enter into numerous details concerning the\nancient names of stuffs, and in order further to illustrate this subject\nI venture to propose a different etymology than the one given by these\nwriters of the word Samit, Samitum, or Xamet, which, although slightly\ndifferent in orthography, is met with everywhere in Europe. My opinion\nis that it means stuff made at Damascus, for the word Sham\n[Illustration: Arabic writing] is applied to the name of this town as\nwell as to that of Syria, and the appellative Shami and Shamit\n[Illustration: Arabic writing] appears in the lexicon to express what\ncomes from Damascus or Syria.\n\nThe oldest specimen which I know of Spanish Arab textile fabric, is a\nwoollen fragment of very fine quality embroidered in colours, which is\nat the Real Academia de la Historia, Madrid. It is about 1\u00bd yards\nlong, by 18 inches wide. In medallions embroidered in silks are\nrepresented seated figures which appear to be a king, a lady, lions,\nbirds, and quadrupeds. In two borders occurs the following inscription,\nrepeated in cufic characters: \"In the name of God, clement and merciful:\nThe blessing of God and happiness for the caliph Iman Abdallah Hixem,\nfavoured of God, prince of believers.\"\n\n[Illustration: Arabic]\n\nHixem reigned from A.D. 979 until the first years of the 11th century.\nThis fragment was found inside a small casket on the altar of a church\nin the town of San Esteban de Gormaz, province of Soria, where it was\nprobably taken as a war trophy: it may be classified as a stuff called\n_tiraz_ [Illustration: Arabic], the skirt of a robe. An oriental author\ntells us that among the customs which contribute to give splendour to\nsovereignty is that of putting the name, or some other sign belonging to\nkings on the stuffs of their robes; that these inscriptions are woven\ninto the material with gold or coloured thread of a different hue to the\nground, and that royal robes were always made of _tiraz._ The caliphs of\nCordova had a place set apart in their palaces where this stuff was\nkept: this custom lasted until the 11th century when it disappeared, and\nwas re-established in the 13th century with the kings of Granada. (Moh.\nDyn. vol. i., pp. 356 and 397; Mus. Esp. de Ant. vol. vi. pp. 464 and\n465.)\n\nAnother most important object, although less ancient than the former\none, is an Arabian banner which is traditionally believed to have been\ntaken from the Almohades at the battle of Las Navas, A.D. 1212. This\nbanner, which is preserved at the monastery of Las Huelgas of Burgos, is\nten feet high by seven feet three inches wide. It is made of a crimson\nstuff, covered with an ornamentation woven and embroidered in gold and\ncolours. In the centre is a large circle, placed within a square, which\nis enlarged in parallel bands. The four largest bands, which are also\nthe last, are covered with inscriptions in blue African characters. The\nupper part of this banner is prolonged with several bands, one of which\nhas inscriptions of a similar kind to those already mentioned; other\nborders contain several sentences in small African letters. The lower\npart is also prolonged and is terminated by eight points cut in a\nsemicircle, inside which there are discs with illegible inscriptions.\nThree violet coloured lions, similar in form to those which appear on\nthe arms of Leon, may be seen on three of the sides of the square. In\nthe large circle in the centre of this banner, appears eight times\nrepeated the word \"The Empire,\" [Illustration: Arabic]. This inscription\nis embroidered or woven in Cufic characters, in a similar style to those\nat the Alhambra; it is a strange circumstance that the letters appear on\nthe wrong side, as if this was the back of the stuff; the word which\nseems to be required to complete this sentence, \"God,\" is also wanting.\nThe large inscriptions in blue of the five bands reproduce Suras of the\nKoran, and pious sentences in one of the small ones on the upper part.\nS^{r.} Fernandez, who has published an article on this banner in Mus.\nEsp. [vol. vi., p. 469], thinks he finds a date which corresponds with\nthe year A.D. 1140. I am not, however, satisfied with his interpretation\nof the inscription contained in the large centre circle, or the date he\ngives; for comparing the ornamentation of this specimen with other\ntextiles, and the designs used in Moorish architecture, it appears to me\nthat it must be considered as belonging to the 14th century. This may be\neasily accounted for by some historical mistake in the tradition\nrespecting it at the convent of Las Huelgas. It is highly probable that\nKing Alfonso XI. may have given this banner to the convent, and have\nbeen mistaken for King Alfonso VIII.; the conqueror of the battle of Las\nNavas.\n\nThe specimens of Spanish moresco stuffs in the Kensington Museum, will\nbe found numbered 51, 121, 124, 125, 152, 160, 180, 241, 244. (V. Dr.\nRock's Catalogue.)\n\nThe artistic industry of silk manufactures which was initiated in Spain\nby the Arabs, continued to flourish during the Middle Ages and a great\npart of the Renaissance. Malaga and Almeria were important centres, but\nlater on this industry was chiefly centred at Granada. The silk fabrics\nmade at Seville, Toledo, Murcia and Valencia were much esteemed. The\nMoorish style of ornamentation in embroideries and stuffs must soon have\nfallen into disuse. Owing to the materials of similar kinds imported\nfrom Italy, France, Flanders, and other countries, these manufactures\nwere imitated to a great extent, as may be gathered by the immense\nnumber of specimens which are still to be met with in Spanish churches.\nThe cathedral of Toledo is quite a museum of objects of this kind, but\nall the textiles there are woven in the European manner.\n\nThe Moorish style in stuffs was preserved at Granada longer than in any\nother town in Spain. In 1502, ten years after its conquest by the\nChristians, we find it stated in the \"Voyage of Philip le Beau,\" the\nfather of the Emperor Charles V.: \"Grenade est fort marchande,\nprincipallement de soyes, car les marchans y achattent la pluspart des\nsoyes que l'on maine en Italie, pour faires les draps de soyes. Le lieu\no\u00f9 on les vendt est nomm\u00e9 le Sacquatin. Aupr\u00e8s de ce lieu est une place\nappell\u00e9e l'Allecasserie, o\u00f9 on vendt les draps de soyes ouvr\u00e9s \u00e0 la\nMoresque, qui sont moult beaus pour la multitude des couleurs et la\ndiversit\u00e9 des ouvrages, et en font une grande marchandise.\" [\"Collect.\nde Voyages, par M. Gachard, Bruxelles,\" 1870, i. 205.] Navagiero, who\nvisited Spain twenty years after, tells us in his \"Viaggio fatto in\nSpagna\" [Vinegia 1563, p. 21]: \"One enters a place called Alcaiceria\nwhich is enclosed within two doors, and full of alleys where the Moors\nsell silks and embroideries of every kind\" [p. 29]: \"All sorts of cloths\nof silks are made there; the silks made at Granada are much esteemed all\nover Spain; they are not so good as those which come from Italy. There\nare several looms, but they do not yet know how to work them well; they\nmake good taffetas, sarcenet, and silk serges. The velvets are not bad,\nbut those that are made at Valencia are better in quality.\"\n\nDr. Rock considers the following specimens in the Kensington Museum to\nhave been made at Granada: Nos. 26, 27, 60, 65, 73, 128, 161, 166.\n\nLarruga tells us in his \"Memorias\" [vol. vii. 205], that the silk\nmanufactures established at Toledo consumed in 1480 about 450,000 pounds\nof silk: they decreased about forty years afterwards. In the ordinances\nissued from that town in 1494, which were confirmed by the Emperor\nCharles V. we find that the following stuffs were made at Toledo:\n\n\"Stuffs of gold and silver which are made in the same manner as satin.\n\n\"Satins woven with gold.\n\n\"Satins brocaded with silk and gold or silver flowers.\n\n\"Silver serges with double filigree.\n\n\"Silver and gold materials, which are made like _gorgoran_ or serge.\n\n\"Silver and gold stuffs which are made like taffetas, spring silver with\nsilk flowers.\n\n\"Embroidered stuffs.\n\n\"Embroidered stuffs called silver serge, or _berguilla_.\n\n\"_Lama_, cloth of silver, shaded with silver watering.\n\n\"Plain silk stuffs woven with silver or gold called _resta\u00f1o_.\n\n\"Silk stuffs woven with gold or silver called _relampagos_.\n\n\"Serges woven with gold and silver for church vestments.\n\n\"Plain filigree serges.\n\n\"_Velillo_ of silver.\n\n\"Satin woven with gold and silver.\n\n\"Brocades of different kinds.\n\n\"Church vestments.\n\n\"Silver _primaveras_.\n\n\"Serges for church vestments.\"\n\n(\"Ord. Ant. de Toledo,\" Tol. 1858, p. 223).\n\nUntil the middle of the 17th century, Larruga tells us in vol. vii. p.\n208 of his \"Memorias,\" the silk manufacture of Toledo was one of those\nmost highly esteemed in Spain; from this time it was superseded by the\nmanufactures of Seville, Granada, Valencia, and others. These cheapened\ntheir productions; but Toledo insisted on keeping to the old Spanish\nyard and weight in every kind of stuffs. In 1651 fifty looms belonged to\nthis manufacture, although most of them were established in the suburbs.\nIn this year there was a great decrease in the price of the coin, and\nthe Genoese introduced a large quantity of silk stuffs, so much so that\nonly twenty looms could be kept working at Toledo. After this, in 1663,\n2061 looms existed there.\n\nThe silk manufacture of Toledo continued to lose its importance in\nconsequence of the vast importations of foreign silks, but the\ntraditions of this industry have never disappeared from the city.\nLarruga, who enters into this subject at great length in the seventh and\neighth volumes of his \"Memorias,\" mentions the silk manufacture of D^{n}.\nMiguel Gregorio Molero, \"who made from the year 1714, under his\ndirection and that of Christoval de Morales, his father-in-law, a large\nquantity of silk fabrics of wide and narrow materials of every kind, and\nstuffs woven with silver and gold.\" It is an interesting fact that the\nmanufactory of Molero continues to work in the present day, and gold and\nsilver stuffs are made there for ecclesiastical purposes similar in\ndesign and manufacture to the early established one of 1714; specimens\nexist at the South Kensington Museum of the silks made at Toledo by\nMolero.\n\nTowards the middle of the 18th century the most important centre of\nartistic silks existed at Talavera, where it was established on a very\nlarge scale by King Fernando VI. and supported by the state. It was\nfounded in 1748 under the superintendence of a Frenchman called Jean\nRouli\u00e8re, who was educated at Lyon, a great mechanician, who entered the\nmanufactory with a yearly salary of \u00a3450, and 4 per cent. of the sale.\nFine church vestments were made there, and the richest stuffs woven\nwith silver and gold for decorating apartments and furniture, of which a\nvast quantity remain in the palaces of Madrid, Aranjuez, the Escorial\nand La Granja. In 1762 the Spanish government handed over this\nmanufactory to the firm of Uztariz and Company; it remained in their\nhands until 1780, and at this period it continued to be worked for five\nyears by the state; in 1785 it passed to the superintendence of the\nTribunal of Commerce entitled Cinco Gremios Mayores de Madrid. The\nFrench invasion of 1808 and general decay of the country contributed\ntowards the extinction of this industry.\n\nIn a similar manner to the silk manufactures established at Toledo,\nsilks of all kinds were made at Seville, Granada, Murcia, and other\nprovinces, to a very great extent during the 16th and 17th centuries.\nThis industry continues in the present day, and specimens exist at the\nSouth Kensington Museum, made in 1874, which recall the ancient style of\nthese stuffs.\n\n\nEMBROIDERIES.\n\nIt is probable that the history of embroidery in Spain followed the same\nsteps as that of gold and silver stuffs, owing to the great similarity\nwhich exists between these industries, as may be seen by the banner\ncalled de las Navas, which is composed of both these materials.\nOrnamentation in the Oriental style must have fallen sooner into disuse\nthan in woven fabrics owing to the fashion of introducing figures in\necclesiastical vestments, which much before the Renaissance period\nbecame so general in Europe. The first mention which I find of the\nexistence of this artistic industry appears in the Ordenanzas de\nSevilla, 1433; it refers to regulating this industry, and preventing\ncertain frauds, by which we may infer its existence from an earlier\nperiod. Similar dispositions appear in one of the Ordinances of Toledo,\ndated 1496, and as sumptuary laws were continually issued we have\nconstant references to the existence of this industry: [consult\n\"Sempere, Hist. del Lujo,\" vol. ii. 8vo]. These legal prohibitions did\nnot, however, reach church vestments; after the aggrandizement of the\nmonarchy by the conquest of the New World such enormous sums were spent\non these objects, that notwithstanding the ravages of time, wars,\ndisturbances of all kinds, and vandalisms and neglect, the cathedrals of\nToledo and Seville, and many other churches, are museums of this style\nof art.\n\nThe art of embroidery was imported by Italians and Germans early in the\n16th century, in the same manner as wood carving, metal work, and other\nindustries. The comparative study of embroideries of different kinds\nwhich are preserved in churches and museums in other countries confirms\nthis theory; some have been made in the north and others in the south,\nwhile their historical origin may be traced to Italy. In Spain itself\nthis comparison may be made, and the Germanic influence is most\napparent. At the cathedral of Burgos there is a fine series of\necclesiastical vestments which Bishop Alonso de Cartagena gave to the\ncathedral on his return from the council of B\u00e2le, 1431-43. Another\nseries of most important specimens is at Toledo, a present to Cardinal\nMendoza by the German emperor Frederic in 1489. In both these instances\nwe find that the embroidery is identical with what was made in Spain at\nthis period, until they changed the Gothic for the Italian Renaissance\nstyle.\n\nI must mention two splendid altar frontals of foreign workmanship which\nexist in Spain, one of them is at the chapel of the Disputation of\nBarcelona; it represents St. George slaying the dragon. This embroidery\nis in high relief, and is so perfect in every detail that it appears to\nbe chiselled. Another altar frontal of the same style is at the\nColegiata of Manresa, Catalu\u00f1a; on this one the Crucifixion is\nrepresented, with eighteen other subjects, taken from the Old and New\nTestaments. This frontal is signed--_Geri Lapi Rachamatore Me fecit in\nFlorentia._\n\nAlmost all Spanish towns of importance are distinguished for their\nembroidery, Toledo, Seville and Valencia are especially so; Ciudad\nRodrigo figures as an important centre of this industry during the 16th\ncentury. We find the sentence, \"obra de Ciudad Rodrigo\" applied to\nembroidery of gold thread, [Acad. de la Historia, MS. C. 122.] Father\nSiguenza tells us in his \"Hist. de la orden de San Jeronimo\" that this\nstyle of embroidery was exclusively Spanish, and probably the remains of\nthe Moorish influence. The principal localities where embroideries and\nartistic work of every kind can be studied are the cathedrals of the\ntowns we have mentioned, which were great centres of artistic industries\nin other times, and the monastery of the Escorial. The collection at the\ncathedral of Toledo alone is sufficient to illustrate this subject.\n\nAbout forty sets of splendid vestments exist at this cathedral which are\nembroidered with the most exquisite taste, belonging to the 15th and\n16th centuries. Each set generally includes a chasuble, dalmatic, cope,\naltar frontal, covers for the gospel stands, and other smaller pieces.\nThe embroideries on the orphreys, which are formed of figures of saints,\nare as perfect as the miniatures on illuminated MS. The _manga_, or case\nwhich hangs round the processional cross given by Cardinal Ximenez, is\none of the most splendid specimens of this collection. As a fine\nspecimen of embroidery on a large scale must be mentioned the _dosel_ or\ncanopy called the tent of Ferdinand and Isabel, also at the cathedral of\nToledo, which is interesting as being the same which was used in the\nreception of the English envoys Thomas Salvaige and Richard Nanfan who\nwere sent in 1488 to Spain to arrange the marriage of Prince Henry with\nthe Infanta D^{na.} Catalina. The ambassadors describe it in the\nfollowing manner: \"After the tilting was over, the kings returned to the\npalace, and took the ambassadors with them, and entered a large room:\nand there they sat under a rich cloth of state of rich crimson velvet,\nrichly embroidered with the arms of Castile and Aragon, and covered\nwith the device of the king, which is a ... (blank in original), and his\nmotto, written at length, which is 'Tanto Monta.'\" (\"Memorials of King\nHenry the Seventh,\" Gairdner, Lon. 1858, p. 348.)\n\nThe most remarkable specimen of embroidery which exists in Spain, not so\nmuch on account of its artistic merit as for the enormous value of the\nmaterials employed, is the mantle of the Virgin del Sagrario at Toledo.\nIt is completely covered with pearls and jewels forming a most effective\nornamentation. This embroidery was made in the beginning of the 17th\ncentury, during the lifetime of Cardinal Sandoval, who presented it to\nthe church. S^{r.} Parro in his exhaustive volume of \"Toledo en la\nMano,\" [vol. i., p. 574,] describes it in the following manner: \"It is\nmade of twelve yards of silver lama, or cloth of silver, which is\nentirely covered with gold and precious stones. In the centre there is a\njewel of amethysts and diamonds. Eight other jewels appear on each side\nof enamelled gold, emeralds and large rubies; a variety of other jewels\nare placed at intervals round the mantle, and at the lower part are the\narms of Cardinal Sandoval enamelled on gold and studded with sapphires\nand rubies. The centre of this mantle is covered with flowers and\npomegranates embroidered in seed-pearls of different sizes. Round the\nborders are rows of large pearls. Besides the gems which were employed\nin this superb work of art, no less than 257 ounces of pearls of\ndifferent sizes, 300 ounces of gold thread, 160 ounces of small pieces\nof enamelled gold, and eight ounces of emeralds were used.\"\n\nA fine altar frontal of a similar kind embroidered in corals may be seen\nin the same room where this mantle of the Virgin is kept at the\ncathedral of Toledo.\n\nWe find in Madame de Villars' letters, [p. 39, Paris, 1823,] the\ndescription of a similar embroidery. Writing in 1680 she says, \"Ce que\nj'ai vu de plus riche, de plus dor\u00e9, de plus magnifique, est\nl'appartement de la reine. Il y a entre autres meubles dans sa chambre,\nune tapisserie, dont ce qu'on y voit de fond, est de perles. Ce ne sont\npas des personnages, on ne peut dire que l'or y soit massif, mais il\nest employ\u00e9 d'une mani\u00e8re et d'une abondance extraordinaires. Il y a\nquelques fleurs: ce sont des bandes de compartimens; mais il faudrait\n\u00eatre plus habile que je ne suis pour repr\u00e9senter les choses, pour vous\nfaire comprendre la beaut\u00e9 que compose le corail employ\u00e9 dans cet\nouvrage. Ce n'est point une mati\u00e8re assez pr\u00e9cieuse pour en vanter la\nquantit\u00e9; mais la couleur et l'or qui paraissent dans cette broderie,\nsont assur\u00e9ment ce qu'on aurait peine \u00e0 d\u00e9crire.\"\n\nFrom the period when these embroideries were made until the middle of\nthe 18th century, Spanish embroideries lost much of their artistic\ncharacter, although the work itself continued equally excellent. During\nthe whole of the 17th century, a style of embroidery became very general\nin Spain, which appears to have been copied from eastern importations by\nthe Portuguese or the Spanish possessions in America. The specimens most\ngenerally met with in Spain are chiefly bed covers, the ground of which\nis either linen or satin, embroidered in chain stitch, with figures and\nexotic birds and animals. Their effect is very rich, and the\nornamentation is arranged generally in circles. Specimens exist in\nEngland of this kind of work; for example, a quilt, said to have been\nmade for an Archbishop of Toledo, lent by Lady Cornelia Guest to the\nSpecial Exhibition of Embroidery held at South Kensington in 1873.\n\nAt the Mus. Arq., Madrid, is a fine quilt of this kind embroidered with\nmaize-coloured silk on linen, with representations of figures and\nanimals.\n\nEmbroideries were made in Spain to cover furniture. Sedan chairs,\ncoffers, &c., were ornamented in this manner. At the Kensington Museum\nthere is an interesting example of a trunk with silver lock, covered\nwith embroidery, which was given towards 1680 by Count Olivares to the\nnuns at Loeches. Some fine embroidered tapestries belonging to the same\ncollection are at the Museo Arqueleogico, Madrid. A varied collection of\necclesiastical vestments of the kinds described may also be studied at\nthe museum. (V. Nos. 78, 79, 84, 673, 1194, 1195, 1250, &c., in Dr.\nRock's Catalogue.)\n\nThe sumptuary laws, and especially the Decrees issued by Philip IV. in\n1622, prohibited any display or ostentation in costume, and embroideries\nwere confined to ecclesiastical purposes. In the middle of the 18th\ncentury embroideries were introduced in the costumes of men and women\nwhen French fashions were imported into Spain. Although many of them\ncame from France, they were imitated in Spain to a large extent. A guild\nof embroiderers was formed in 1779, who submitted their statutes to the\nTribunal of Commerce. We find that Madrid was the principal centre of\nthis industry, and that French designs were universally copied, as was\nthe case in the whole of Europe. The splendid curtains and embroidered\nhangings for apartments which exist at the royal palaces of Madrid, the\nEscorial and Aranjuez, are admirable specimens of this industry.\n\nThe following list of Spanish embroiderers is taken from Cean's \"Dic. de\nProf de las Bellas Artes;\" Martinez, \"Hist. del Templo Cat. de Burgos;\"\nSuarez de Figueroa, \"Plaza Universal,\" etc.\n\n  Years in which\n  they worked.             Artists.              Residence.\n\n  1603.      Alcala, Fr. Fernando de              Escorial.\n\n  1526.      Alonso, Esteban                      Toledo.\n\n  1688 to }  Arroyo, Diego de                     Burgos.\n  1715.   }\n\n  1594.      Aspe, Simeon de                      Burgos.\n\n  1646.      Ansin, Francisco de                  Burgos.\n\n  1500.      Barcelona, Fr. Rafael de             Escorial.\n\n  1599.      Berrio, Francisco de                 Burgos.\n\n  1452.      Bilbao, Juan                         Burgos.\n\n  1514.      Burgos, Pedro                        Toledo.\n\n  1574 to }  Cami\u00f1a, Alonso                       Burgos.\n  1580.   }\n\n  1580 to }  Cami\u00f1a, Miguel, a son of Alonso      Burgos.\n  1593.   }\n\n  1580.      Castillo, Juan del                   Madrid.\n\n  1571.      Cordova, Fr. Francisco               Escorial.\n\n  1616.       Corral, Felipe de                     Toledo.\n\n  1514.       Covarrubias, Marcos de                Toledo.\n\n  1422.       Fernandez, Garcia                     Burgos.\n\n  1422.       Fernandez, Pedro                      Burgos.\n\n  1645.       Garcia de Jalon, Juan                 Burgos.\n\n  1500.       Gil, Francisco                        Madrid.\n\n  1688.       Gomez, Juan                          Sevilla.\n\n  1514.       Hernandez, Alonso                     Toledo.\n\n  1630.       Landa, Alonso de                      Burgos.\n\n  1625.       Martinez de Manurga, Sebastian        Burgos.\n\n  1521.       Monte, Cornieles de                   Burgos.\n\n  1576.       Montserrate, Fr. Lorenzo              scorial.\n\n  1580 to }   Ochandiano, Andr\u00e8s                    Burgos.\n  1593. }\n\n  1595.       Ortiz de Zarate, Pedro                Burgos.\n\n  1565.       Palenzuela, Francisco de              Burgos.\n\n  1591.       Palenzuela, Jeronimo                  Burgos.\n\n  1500.       Perez, Gabriel                        Madrid.\n\n  1580.       Perez, Juan                           Madrid.\n\n  1514.       Rica, Hernando de la                   Toledo.\n\n  1500.       Rosales                                Toledo.\n\n  1500.       Rosicler, Lu\u00eds de                      Madrid.\n\n  1514.       Ruiz, Martin                           Toledo.\n\n  1716 to }    Ruiz, Antonio                         Burgos.\n  1734. }\n\n  1585.       Rutinez, Diego                        Escorial.\n\n  1563.       Salas, Juan de                        Granada.\n\n  1718.       Santos Martires, Fr. Esteban de los   { Carde\u00f1a\n                                                    { Burgos.\n  1561.       Sarabia                                 Burgos.\n\n  1514.       Talavera, Juan de                       Toledo.\n\n  1500.       Toledo, Fr. Juan de                   Escorial.\n\n  1752.       Transmontana, Fr. F. Javier de          Burgos.\n\n  1684.       Valle, Antonio del                      Burgos.\n\n  1500.       Vega, Felicis de                        Madrid.\n\n  1563.       Villegas, Nicolas de                   Granada.\n\n  1502.       Xaques, el Maestro                      Toledo.\n\n\nTAPESTRIES.\n\nThe manufacture of tapestry or carpets was introduced into Spain by the\nArabs. The earliest historical information which I have been able to\nfind relating to this industry occurs in Edrisi, an oriental author of\nthe beginning of the 12th century, [\"Descrip. de l'Afr. et de l'Esp.,\ntrad. de Dozy el Goeje,\"] when speaking of Chinchilla in the province of\nAlicante, he says: \"On y fabrique des tapis de laine qu'on ne saurait\nimiter ailleurs, circonstance qui d\u00e9pend de la qualit\u00e9 de l'air et des\neaux.\" In mentioning Cuenca, he says: \"Les tapis de laine qu'on y fait\nsont d'excellente qualit\u00e9,\" p. 237. Although we find very few\ndescriptive details of this industry, it is undoubtedly the fact that in\na wide zone which comprehends from the kingdom of Valencia until that of\nGranada, carpets have been constantly made; for even in the present day\nthis industry is alive in a number of towns, where not only the old\ntechnical proceedings are continued, but also much of the primitive\ncharacter of the designs and colours. In confirmation of this, I find\nfrequent mention in books and MSS. of the 16th century of carpets of\nLetur, made in the kingdom of Murcia, Alcaraz, and Baeza.\n\nThe first time I find this industry mentioned by Christian authors is in\nFrancisque Michel's work on \"\u00c9toffes de Soie,\" vol. i., p. 292; he says:\n\"A une \u00e9poque qu'il nous serait difficile de pr\u00e9ciser un po\u00e8te Latin\nvantait les teintures pr\u00e9cieuses \u00e0 sujets, et les tapis d'Espagne.\"\n\n\"Tunc preciosa suis surgunt aulaea figuris.\n\n\"Ac in se raptis ora tenent animis.\n\n\"Tunc operosa suis _Hispana tapetia_ villis.\n\n\"Hinc rubras, virides inde ferunt species.\"\n\nAt the cathedral of Gerona an extremely interesting tapestry of the 12th\ncentury may be seen, which may have been made in the locality.\n\n[Illustration: TAPESTRY AT THE CATHEDRAL OF GERONA; REPRESENTING THE\nCREATION, &C.--SPANISH, 11TH CENTURY. _To face p. 267._]\n\nThis tapestry is about 4\u00bd yards wide by 4 yards high. The composition\nrepresents the Genesis. In the centre is a geometrical figure formed by\ntwo concentric circles. In the lesser circle is a figure of Christ\nholding an open book, on which appear the words, _Sanctus Deus_, and on\neach side _Rex fortis_, surrounded by the inscription, _Dixit quoque\nDeus, Fiat lux, Et facta est lux._ In the larger circle are the words,\n_In principio creavit Deus coelum et terram, mare et omnia qu\u00e6 in eis\nsunt, et vidit Deus cuncta qu\u00e6 egerat et erant valde bona._\n\nThe space between the two circles is divided by radiating lines into\neight portions, in which are represented the Mystic Dove, the angels of\nlight and darkness: the division of land from water, the creation of\nsun, moon, and stars, of birds, fishes, and beasts, and of Adam and Eve.\nIn the angles outside the larger circle are the four winds, and the\nwhole is surrounded by a border, imperfect in parts, containing\nrepresentations of the months, and apparently of certain scriptural\nincidents, too much defaced to be clearly made out. The accompanying\nengraving, though imperfect and inaccurate in some of its details, gives\na general idea of this work.\n\nAlthough this tapestry is embroidered with crewels on linen, it appears\nadvisable to include it in the description of Spanish tapestries, owing\nto its similarity and general aspect to other tapestries. It is easy for\nstudents of Spanish works of art of the Middle Ages to fix the period\nand style to which this example belongs. Several remarkable\nilluminations exist in Spain representing the Apocalypse which were\npainted in the X. XI. and XIIth. centuries, which have already been\nalluded to in other articles of this work. The figures of the tapestry\nof Gerona are disposed in a precisely similar manner to the miniatures\nof the 12th century. They are in the same style as the paintings on the\nceiling of the chapel of St. Catalina, at St. Isidoro of Leon, which\nwere painted at that period. There can be therefore no doubt that they\nbelong to that period and style. Several MSS. may be mentioned to\nfurther illustrate the subject. An excellent specimen is at the\ncathedral of Gerona, those at the National Library, Museo Arqueologico,\nand Academia de la Historia, Madrid, Escorial, Bibliotheque Nationale,\nParis--and especially the fine MS. at the British Museum (Add. II.\n695),--dated A. D. 1109, referred to in the chapter on Arms, page 90.\nThe illuminations of this volume are extremely similar to the design of\nthis tapestry. Plate 83, twelve inches by eight, within a circle,\nrepresenting Christ holding a book in his hand, may be given as an\nexample.\n\nI do not find any information of a later date which suggests the\nexistence of the manufacture of tapestries in Spain during the Middle\nAges. There are constant allusions to the splendid tapestries which were\nbrought from Flanders, many of which are at the Royal palace of Madrid.\nThe earliest mention I find is a memorial printed without date, in which\nPedro Gutierrez, tapestry maker of Salamanca, asks Philip II. to protect\nthis industry. He evidently gained his object, for in the documents\npublished by S^{r}. Cruzada in his \"Tapices de Goya,\" [Madrid, 1870,] it\nappears that in 1578 Queen Do\u00f1a Ana appointed him to work in her Camara\nas tapestry maker to make _reposteros_[B] and Philip II. in 1582\nconfirms this appointment. Gutierrez worked at this period, at Salamanca\nand Madrid. He was succeeded in his charge in 1625 by Antonio Ceron, who\nestablished this industry definitively in the Calle de Santa Isabel at\nMadrid. The fine picture by Velasquez, at the Madrid Gallery, \"The\nWeavers,\" which represents the interior of the tapestry manufactory,\nbelongs to this time. We find this industry soon fell into decay. In\n1694 a Belgian named Juan Metier tried to revive it, but without\nsuccess; the same thing occurred with Nicolas Hernandez, a tapestry\nmaker of Salamanca, in 1707.\n\n[B] Reposteros is the ancient name given to the hangings which are\nplaced outside the balconies on state occasions in Spain. Several\nsplendid examples of the sixteenth and seventeenth centuries may still\nbe seen at the houses of Spanish grandees, of which those belonging to\nthe Conde de O\u00f1ate and Marques de Alca\u00f1ices at Madrid are the most\nremarkable for their artistic design.\n\nThe tapestry manufactory of Santa Barbara at Madrid was founded soon\nafter; it was the most important of all, and continues to produce\nexcellent work in the present day. It appears that Jacobo Vandergoten of\nAntwerp was engaged expressly by the king; he began to work in 1720, and\ncontinued there until his death in 1724. He was succeeded by his sons,\nFrancisco, Jacobo, Cornelius, and Adrian, who worked in _basse lisse_\nlooms until 1729, when a Frenchman, Antonio Lenger mounted a _haute\nlisse_ one.\n\nOn the occasion of the temporary removal of the court to Seville, the\nking established a tapestry manufactory there in 1730; Jacobo\nVandergoten was put at the head of it, and the painter, Andres\nProcaccini, helped him in the direction of the manufactory. It only\nlasted three years, at the end of which the artists returned with their\nimplements to establish themselves in the old tapestry manufactory of\nthe Calle de Santa Isabel: and continued to work there until 1744, at\nthat time they again joined the works at Santa Barbara.\n\nThree of the brothers Vandergoten were already dead in 1774. Cornelius\nalone was alive. In the same year the Spanish artists, Antonio Moreno,\nDomingo Galan, Tomas del Castillo, and Manuel Sanchez were placed at the\nhead of these works, under the superintendence of Sanchez, who continued\nthere until his death in 1786, when he was succeeded by his nephew,\nLivinio Stuck. This manufactory was much neglected at the end of the\nlast century; it was destroyed by the French in 1808, and was\nrehabilitated in 1815 under the direction of a son of Stuck. This family\ncontinues to work there in the present day. The carpets made at the\nmanufactory of S^{ta.} Barbara are of the highest excellence and\nartistic order.\n\nVery fine tapestries have been made at this manufactory during the last\ncentury, some after original cartoons of excellent painters and others\nreproductions of ancient Flemish ones belonging to the royal\ncollections. During the years 1721 to 1724 the tapestries representing\npastoral subjects after Teniers were woven there, and a series of\nhunting subjects which still exist at the Pardo and palace of Madrid.\nThe excellent copy of the series of tapestries representing the conquest\nof Tunis, after cartoons by William Pannemaker were made partly at\nSeville and partly at Santa Isabel and Santa Barbara. The tapestries\nrepresenting scenes from the life of Don Quixote, designed by the\npainter Procaccini, were also made at the looms of Santa Barbara.\n\nFine carpets in the Turkish style were made there under the\nsuperintendence of Cornelius Vandergoten, who excelled in this special\nindustry. Carpets also in the French style of the Gobelin manufactory\nwere made at the same time. The best specimens of carpets and tapestries\nof the manufactory of Santa Barbara, will be found at the palaces of the\nEscorial, the Pardo, and Madrid. The series made from cartoons of\nTeniers and the Spanish painter Goya are extremely interesting. A\ncomplete collection of photographs of the foreign and Spanish tapestries\nat the palace of Madrid will be found in the Art Library of the South\nKensington Museum.\n\n\n\n\nLACE.\n\n\nIt is not easy to give a clear idea of the history of the manufacture of\nancient Spanish lace; the principal reason being that this industry was\ncarried out by private persons, who never formed themselves into Guilds\nor corporations, as was the case during the middle ages and Renaissance\nperiod with those who devoted their time to other arts and industries.\nWe find in the Royal Ordinances constant references to weavers, silver\nand goldsmiths, tailors, etc., but in none do we find the manufacture of\nlace mentioned. The most important Ordinances relating to Spanish\nindustries are those published at Toledo and Seville in the 15th and\n16th centuries, and at Granada in the 16th and 17th centuries, and in\nnone of them do we find lace even alluded to. In the innumerable laws\nwhich appeared from the middle ages until the 18th century for the\nobject of reforming costumes and checking their excessive luxury no\nmention is made of lace; when it is referred to, it is only when\n_puntas_ or _entredoses_, edgings or insertions, are mentioned in which\ngold and silver are introduced. We never find any allusion in\ncontemporary writers to the fine heavy thread lace, sometimes called\nPoint d'Espagne or Point de Venise, the origin of which has been a\nsource of so much doubt to collectors of the present day.\n\nThe only thing which we can do therefore at the present, until this\nsubject is more fully investigated, is to assert that lace of different\nkinds has been made from the very earliest times in Spain, and do our\nbest to classify the different kinds by the differences which we observe\nin those of other localities.\n\nFather Fr. Marcos Antonio de Campos in his \"Microcosmia y Gobierno\nUniversal del Hombre Cristiano,\" [Barcelona, 1592, p. 225,] says: \"I\nwill not be silent and fail to mention the time lost these last years in\nthe manufacture of _cadenetas_, a work of thread combined with gold and\nsilver; this extravagance and excess reached such a point that 100 and\n1000 of ducats were spent in this work, which, besides destroying the\neyesight, wasting away the lives, and rendering consumptive the women\nwho worked it, and preventing them from spending their time with more\nadvantage to their souls, a few ounces of thread and years of time, were\nwasted with so unsatisfactory a result. I ask myself, after this fancy\nhas passed away, will the lady or gentleman find that the chemises that\ncost them 50 ducats, or the _basqui\u00f1a_ (petticoats) that cost them 300,\nare worth half their price, which certainly is the case with other\nobjects in which the material itself is worth more.\" This quotation is\ninteresting as proving that lace was made during the 16th century. The\nfriar, in following the prevalent fashion of preaching against luxury\nand extravagance, shows us that this industry must have reached to a\ngreat height in order to be an object of censure.\n\nFrom the 16th century until the end of the 18th we find this manufacture\ncontinued in Spain, but that foreign lace of different kinds was\nlikewise imported to a great extent, although we may affirm that black\nlace especially was made in Spain during the whole period.\n\nWe find lace mentioned by ancient Spanish authors under the following\nnames: _punta_, _randa_, _cadeneta_, _entredos_, _red_, _deshilado_,\n_franja_, _blonda_, and _encaje_, which may be translated as edging, or\npoints, r\u00e9seuil, chain stitch, insertion, netting, ravelling, fringe,\nblonde, and lace.\n\nM. H. de Cocheris in his introduction to \"Patrons de Broderies,\" Paris,\n1872, says that \"gold and silver laces were made at Paris and Lyons.\"\nThe four books which are reproduced in this volume mention Lyons as\nbeing a great centre of this particular branch of industry. Gold and\nsilver lace was, however, also made and copied in Spain during the 17th\ncentury. We find in the \"Pragmatica y Nueva Orden cerca de los Vestidos\ny Trajes, asi de Hombres como de Mugeres,\" [fol. Madrid 1611, p. 5,]\n\"May we be allowed to border or edge the said silken materials with\nthread lace, which are not to be made of chain stitch, or gold or\nsilver, and when these laces are mentioned we should understand they are\nexclusively for women's use.\" We also find in an Ordinance of the time\nof Philip III., dated 1623, that gold and silver lace was prohibited,\nand that \"simple edgings, and curls, and fraises for women were enjoined\nfor women's use.\" In the \"Fenix de Catalu\u00f1a, compendio de sus Antiguas\nGrandezas y Medio para Renovarlas,\" Barcelona, 1683, by Narciso Feliu,\nthe author states, p. 75, that \"edgings of all sorts of gold, silver,\nsilk, thread, and aloe, is made there with greater perfection than in\nFlanders.\"\n\nMany interesting details have reached us of the costume of a Spanish\nlady in the 17th century. We will copy the description given by Madame\nd'Aunoy in her \"Voyage d'Espagne,\" Lyon 1628. \"Under the vertingale of\nblack taffety they wear a dozen or more petticoats, one finer than the\nother, of rich stuffs trimmed with lace of gold and silver to the\ngirdle. They wear at all times a white garment called sabenqua; it is\nmade of the finest English lace, and four ells in compass. I have seen\nsome worth 500 or 600 crowns, and so great is their vanity, they would\nrather have one of those lace subenquas than a dozen coarse ones, and\neither lie in bed till it is washed, or else dress themselves without\nany, which they frequently enough do.\" A number of portraits exist in\nthe Spanish galleries, especially by Velasquez and Carre\u00f1o, in which\nthese extravagant costumes are fully portrayed, but in very few Spanish\nportraits of the seventeenth century does thread lace of the kind known\nas Point d'Espagne or de Venise ever appear. Mrs. Palliser, in her\ninteresting \"History of Lace,\" p. 80, quotes a vast number of\ndescriptions of these gold and silver laces. The celebrated bed at\nVersailles, the interior lacings of the carriages, the costumes of the\ngentlemen and ladies of court, and wedding dresses, were all trimmed\nwith this silver and gold lace, either made in Spain, or called Point\nd'Espagne.\n\nWe copy again Madame d'Aunoy's description of a Spanish bed. \"It is of\ngold and green damask lined with silver brocade and trimmed with point\nof Spain. Her sheets were laced round with an English lace half an ell\ndeep. The young Princess of Monteleon bade her maids bring in her\nwedding clothes. They brought in thirty silver baskets so heavy four\nwomen could only carry one basket: the linen and lace were not inferior\nto the rest.\" Beckford, in describing the apartment of a Spanish lady,\nwrites: \"Her bed was of the richest blue velvet trimmed with point\nlace.\" Aarsens de Sommerdyck in his \"Voyage d'Espagne,\" [Paris, 1665,]\nwrites, in describing the costume worn by the men: \"Leurs collets ou\ncravates sont de grandes pointes, qui sans doute coustent beaucoup, bien\nqu'elles ne paroissent pas belles. La mode en est presque la mesme qu'en\nFrance, l'ayant prise de la princesse de Carignan quand elle estoit \u00e0\nMadrid, dont elles les nomment Valonas \u00e0 la Carignan.\" In the\ninteresting \"Journal du Voyage d'Espagne,\" Paris, 1669, l'Abb\u00e9 Bertaut\ntells us that on the 15th of October, 1659, on the occasion of his\nembassy to Madrid, the king of Spain sent eight postilions, and forty\npost-horses, the saddles and bridles of eight of which were covered with\ngold and silver lace.\n\nNotwithstanding the opinion of so competent an authority as Mrs.\nPalliser, I doubt the statement, finding no evidence to support it, that\nthread lace of a very fine or artistic kind was ever made in Spain, or\nexported as an article of commerce during early times. The lace alb,\nwhich is mentioned, to prove this, as existing at Granada, a gift of\nFerdinand and Isabel, is of Flemish lace of the 17th century. The\nchasuble, etc., splendid ecclesiastical vestments embroidered on crimson\nvelvet, were certainly the gift of these kings, and if Cardinal Wiseman\nofficiated in this vestment, it is probable he may have made a mistake\nand not separated the alb from the chasuble, which certainly may be\nworth 10,000 crowns. J. Barretti, in his \"Journey from London to Genoa,\nthrough Portugal and Spain,\" London, 1770, describing the fine church of\nLas Salesas, lately built, writes, \"The nuns showed me in the sacristy\nsome surplices for the mass priest, that are made of the finest Flanders\nlace. Each surplice has cost about 1000 doubloons.\" In \"Tassa de los\nPrecios a que se han de vender as Mercaderias,\" Madrid, 1628, we find\nthat the \"puntas or edgings made in Spain are to be sold at the same\nprice as those imported from Paris.\" Puntas from Flanders are estimated\nin this tariff at a much higher rate than those which were brought from\nFrance; we may calculate they were sold for double the price. The\n\"Ordenaciones\" made at Barcelona to settle commercial tariffs state in\n1704: \"As experience has shown us that most of the edgings or puntas\nmade in this principality are sent out of the country, and we do not\nfind them mentioned in the financial accounts, to the great disadvantage\nof the community, it is determined--\"\n\nWe find reference to lace brought from Flanders, Paris, and Lyon, in the\nfirst volume of \"Apendice a la Educacion Popular,\" [p. 61.] [Madrid,\n1775,] the author mentions the large sums of money which were taken out\nof the country by the importations of foreign lace, and he adds, [vol.\nii., p. 61,] \"Lace is employed to a very considerable extent; all the\nfine qualities come from foreign lands, and the greater varieties of the\ncoarser ones. Spanish matrons, among other branches of their education,\nare taught to make lace of different kinds, and many respectable people\nlive on this industry.\" Larruga, in his \"Memorias,\" Madrid, 1788,\nmentions, vol. ii., p. 149, a manufacture of gold and silver lace which\nhad been set up lately at Madrid; and in the 17th vol., p. 294, in\nmentioning lace made at La Mancha, \"the industry of lace has existed at\nAlmagro from time immemorial. D^{n}. Manuel Fernandez and D\u00aa Rita\nLambert, his wife, natives of Madrid, established in this town in 1766\na manufacture of silk and thread lace.\" This industry also existed at\nGranatula, Manzanares, and other villages of La Mancha. At Zamora \"lace\nand blonde were made in private houses. The finest kinds are sold to\nlace merchants.\" In Sempere, \"Historia del Lujo,\" [Madrid, 1788, 8vo,]\nwe find that in the Ordinance issued in 1723, the \"introduction of every\nsort of edgings or foreign laces was prohibited, the only kinds allowed\nwere those made in the country.\"\n\nCabanillas writes in his \"Observaciones sobre la Historia natural del\nReino de Valencia,\" Madrid, 1797, that at Novelda a third part of the\ninhabitants made lace, \"more than 2000 among women and children worked\nat this industry, and the natives themselves hawked these wares about\nthe country.\" Swinburne, in his \"Travels in Spain in 1775,\" also says,\n\"The women of the hamlets were busy with their bobbins making lace.\"\nLaborde writes that in 1809 the manufacture of blonde was almost\nentirely confined to Catalu\u00f1a. Barcelona is in the present day the great\ncentre of this industry in Spain; the black and white blondes which are\nmade chiefly for mantillas are very fine. The white laces made there are\nin the style of Lille and Buckinghamshire laces, Brussels and Duchesse\nlaces, and Chantilly is also made to a large extent. Gold and silver\nblondes were revived during the years 1830-40; and it may interest my\nreaders to know that the late Queen Mercedes, in her wedding trousseaux,\nhad a garniture de robe, mantilla included, of gold blonde.\n\nModern torchon laces are still made at Almagro to a very large extent.\nBed linen, even in the poorest houses, is elaborately trimmed with lace\nor embroidery. Valances for beds of ravellings, point coup\u00e9 and lace\nwork are still constantly found to decorate beds at weddings in the\nprovinces of Spain.\n\nTHE END.\n\nRICHARD CLAY AND SONS, LIMITED, LONDON AND BUNGAY.\n\n       *       *       *       *       *\n\nSOUTH KENSINGTON MUSEUM SCIENCE AND ART HANDBOOKS.\n\nHandsomely printed in large crown 8vo.\n\n_Published for the Committee of Council on Education._\n\nMARINE ENGINES AND BOILERS. By GEORGE C. 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With Illustrations and\nMarks. 2_s._ 6_d._\n\nHANDBOOK TO THE SPECIAL LOAN COLLECTION of Scientific Apparatus. 3_s._\n\nINDUSTRIAL ARTS: Historical Sketches. With Numerous Illustrations. 3_s._\n\nTEXTILE FABRICS. By the Very Rev. DANIEL ROCK, D.D. With numerous\nWoodcuts. 2_s._ 6_d._\n\nJONES COLLECTION IN THE SOUTH KENSINGTON MUSEUM. With Portrait and\nWoodcuts, 2_s._ 6_d._\n\nCOLLEGE AND CORPORATION PLATE. A Handbook to the Reproductions of Silver\nPlate in the South Kensington Museum from Celebrated English\nCollections. By WILFRED JOSEPH CRIPPS, M.A. F.S.A. With Illustrations.\n2_s._ 6_d._\n\nIVORIES: ANCIENT AND MEDIEVAL. By WILLIAM MASKELL. With numerous\nWoodcuts. 2_s._ 6_d._\n\nANCIENT AND MODERN FURNITURE AND WOODWORK. By JOHN HUNGERFORD POLLEN,\nM.A. With numerous Woodcuts. 2_s._ 6_d._\n\nMAIOLICA. By C. DRURY E. FORTNUM, F.S.A. With numerous Woodcuts, 2_s._\n6_d._\n\nTHE CHEMISTRY OF FOODS. With Microscopic Illustrations. By JAMES BELL,\nPh.D., &c., Principal of the Somerset House Laboratory. Part I.--Tea,\nCoffee, Cocoa, Sugar. &c. 2_s._ 6_d._ Part II.--Milk, Butter, Cheese,\nCereals, Prepared Starches, &c. 3_s._\n\nMUSICAL INSTRUMENTS. By CARL ENGEL. With numerous Woodcuts, 2_s._ 6_d._\n\nMANUAL OF DESIGN, compiled from the Writings and Addresses of RICHARD\nREDGRAVE, R.A. By GILBERT R. REDGRAVE. With Woodcuts. 2_s._ 6_d._\n\nPERSIAN ART. By MAJOR R. MURDOCK SMITH, R.E. With Map and Woodcuts.\nSecond Edition, enlarged, 2_s._\n\n       *       *       *       *       *\n\nThese typographical errors were corrected by the etext transcriber:\n\nGuadamecis are made of leather=>Guadameciles are made of leather\n\nThe French influence of the last century brought the same hions fasfor\nfurniture into Spain=>The French influence of the last century brought\nthe same fashions for furniture into Spain\n\ngilt bronze mounting were very much used.=>gilt bronze mountings were\nvery much used.\n\nIt is of polished iron inlaid with gold, and most exquisite in\nworkmanhip.=>It is of polished iron inlaid with gold, and most exquisite\nin workmanship.\n\nA wooodcut is given opposite.=>A woodcut is given opposite.\n\nRound the lid of of one them is a band=>Round the lid of one them is a\nband\n\nthere are some which are undoubtedly still made n the locality.=>there\nare some which are undoubtedly still made in the locality.\n\nsame collection are at the Museo Arquelogico, Madrid=>same collection\nare at the Museo Arqueleogico, Madrid\n\nthe chaper on Arms, page 90=>the chapter on Arms, page 90\n\nModels of the baroque or, as it is called in Spain,\n_Churriguesque=>Models of the baroque or, as it is called in Spain,\n_Churrigueresque\n\n       *       *       *       *       *\n\n\n\n\n\n\n\nEnd of Project Gutenberg's The Industrial Arts in Spain, by Juan F. Ria\u00f1o\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nATAC BRIEFING FOR AGENTS FRANK AND JOE HARDY\n\nMISSION: To finally get to the bottom of the ongoing deadly scares taking place on the set of the reality TV show Deprivation House.\n\nLOCATION: A huge villa in Beverly Hills, CA, without a single luxury left inside.\n\nPOTENTIAL VICTIMS: Every contestant on the reality show is in extreme danger.\n\nSUSPECTS: It's possible one of the new contestants has a devious agenda, or else someone who's been there all along is hiding a huge secret.\n\nThis mission requires your immediate attention. This message will be erased in five seconds.\n\nWatch out for our next case, Book One of the new Double Danger Trilogy: Double Trouble\n\n  ALADDIN PAPERBACKS  \nSimon & Schuster, New York  \nCover designed by Sammy Yuen Jr.  \nCharacter photograph copyright \u00a9 2008  \nby Michael Frost; background photograph  \ncopyright \u00a9 2008 by PictureQuest  \nAges 8\u201312  \nSimonSaysKids.com  \n0908\nIs their cover blown?\n\nShe can't know, I told myself. ATAC security is way too good. She can't have gotten past it.\n\nI completely believed that. In my head. In my stomach, I wasn't quite as sure.\n\nVeronica raised her eyebrows. \"Interesting choice,\" she told Georgina. \"And something of a bargain, because Frank's secret gives away something about Joe as well.\"\n\nMy palms began to sweat. A secret that was about me and Joe? How many things could that be?\n\nStrategy. I needed strategy. What should Joe and I do if we were outed right here and now?\n\nDeny it? Say it was some ratings stunt of Veronica's? Or something she and the other producers had come up with to create tension in the house? I don't watch a lot\u2014or actually any\u2014reality TV, but Joe says they all end up having lots of fights among the contestants. That was definitely turning out to be true on our show.\n\nOr should we admit to the truth and warn everybody that they were in danger? Just explain that Joe and I had been placed undercover as contestants because the death threats they'd received were being taken very seriously\u2014and for good reason, since several people in this house had already been victims of murder!\n\n#1 Extreme Danger\n\n#2 Running on Fumes\n\n#3 Boardwalk Bust\n\n#4 Thrill Ride\n\n#5 Rocky Road\n\n#6 Burned\n\n#7 Operation: Survival\n\n#8 Top Ten Ways to Die\n\n#9 Martial Law\n\n#10 Blown Away\n\n#11 Hurricane Joe\n\n#12 Trouble in Paradise\n\n#13 The Mummy's Curse\n\n#14 Hazed\n\n#15 Death and Diamonds\n\n#16 Bayport Buccaneers\n\n#17 Murder at the Mall\n\n#18 Pushed\n\n#19 Foul Play\n\n#20 Feeding Frenzy\n\n#21 Comic Con Artist\n\nSuper Mystery #1: Wanted\n\nSuper Mystery #2: Kidnapped at the Casino\n\n#22 Deprivation House\n\n#23 House Arrest\n\n#24 Murder House\n\nSpecial Ghost Stories\n\nEdition: Haunted\n\n**Available from Simon & Schuster**\n\nThis book is a work of fiction. Any references to historical events, real people, or real locales are used fictitiously. Other names, characters, places, and incidents are the product of the author's imagination, and any resemblance to actual events or locales or persons, living or dead, is entirely coincidental.\n\nALADDIN PAPERBACKS\n\nAn imprint of Simon & Schuster Children's Publishing Division\n\n1230 Avenue of the Americas, New York, NY 10020  \nwww.SimonandSchuster.com\n\nCopyright \u00a9 2008 by Simon & Schuster, Inc.\n\nAll rights reserved, including the right of reproduction in whole or in part in any form.\n\nALADDIN PAPERBACKS, THE HARDY BOYS MYSTERY STORIES, HARDY BOYS UNDERCOVER BROTHERS,  \nand related logo are  \nregistered trademarks of Simon & Schuster, Inc.\n\nDesigned by Sammy Yuen Jr.\n\nThe text of this book was set in Aldine 401 BT.\n\nFirst Aladdin Paperbacks edition September 2008\n\n10 9 8 7 6 5 4 3 2 1\n\neISBN-13: 978-1-4424-5347-0\n\nLibrary of Congress Control Number 2008920166\n\nISBN-13: 978-1-4169-6409-4\n\nISBN-10: 1-4169-6409-6\n\nChapter 1: Sabotage\n\nChapter 2: Everybody Has Secrets\n\nChapter 3: Accusations\n\nChapter 4: Worried?\n\nChapter 5: Deeply Disturbed\n\nChapter 6: Silent Girl Screams\n\nChapter 7: Ripped Apart\n\nChapter 8: Psycho Teammate\n\nChapter 9: Another One Down\n\nChapter 10: Targeted\n\nChapter 11: Twisted Twins\n\nChapter 12: So Much Blood\n\nChapter 13: Killer Pool\n\nChapter 14: Why Would She Do That?\n\nChapter 15: The Knife\n\nChapter 16: Game Over\n\n## Sabotage\n\nI figured the wheels of my Suzuki RM-Z450 had to be smoking as I wove through the orchard. But I didn't let up on the throttle. I had the lead, and I didn't want to lose it.\n\nI broke out of the trees and into the field behind the tennis courts. Mounds of dirt were heaped up all across it. I could maneuver the bike around them. But I'd lose time.\n\nThe three mounds almost dead ahead weren't so much different from the triple jump at Highland Park. I'd taken that one lots of times.\n\nOkay, this hill was a little higher, I decided as I started up it. I heard at least one bike coming up behind me. Fast. I moved into attack position. Standing on the dirt bike's pegs. Knees tightened around the tank. My elbows up and out.\n\nYeah, this hill was definitely higher than the first of the triple back home. I kept an even throttle as I approached the top. Here it came. I was about to go airborne.\n\nStay low over the jump, I reminded myself. It would buy me a few extra seconds. Not much. But a few seconds could be the difference between winning and losing. I was undercover as a reality show contestant, and that meant I should want to win more than anything. Somebody on the show wanted to win so badly he or she had sent death threats to all of the contestants.\n\nAs my bike came off the top of the hill, I pushed down on the right footpeg and pulled up on the left side of the handlebars. The bike laid out flat. Nice. The move had stopped the upward motion of the jump. That's where I picked up the extra seconds.\n\nI got on the throttle right before I hit ground to help the rear suspension take the impact of landing. Solid. Now I had to\u2014\n\nMy strategy was worthless. A rider came over the hill behind me kamikaze style, lost control of their ride, and slammed into my back wheel.\n\nWe both went down. And another bike was just going airborne. There was no time to do anything but curl my body into a protective ball. I felt the bike's heat on the top of my head as it cleared me. Next time I might not be so lucky.\n\nI scrambled to my feet and saw that it was Georgina who'd taken me out. \"Come on, we've got to get out of the way before someone else takes the jump!\"\n\nI grabbed my bike by the handlebars, got it up, then ran it down the side of the hill, out of the path of the most direct route to the grocery shelves at the end of the field. That's what this race was. A contest to see who could get the groceries on their list back to the mansion first.\n\n\"Are you okay?\" I asked Georgina.\n\nShe pulled off her helmet. \"Yeah. But the bike's messed up,\" she answered.\n\n\"I don't know if mine got damaged or not,\" I said.\n\n\"No, I mean mine was messed up before. The brakes stopped working,\" Georgina told me. She slapped the seat of her bright purple Kawasaki.\n\nThe dirt bikes we were using for the race were brand-new. That could mean only one thing.\n\nSabotage.\n\n## Everybody Has Secrets\n\n\"Everyone, everyone! Please gather round,\" Veronica called from her spot on the patio near the large fountain. She backed up a step as we headed toward her. I'm sure she was afraid we'd contaminate her. We were all sweaty and dirt smeared and grass stained.\n\nVeronica was perfumed and nail painted and color coordinated in her pale blue suit and pale blue spike heels. She should have looked like TV Host Barbie in all that perfect, with her white-blond hair smoothed back into one of those knot things that aren't buns but that I don't have the words to actually describe, having only one X chromosome.\n\nBut the smile she gave the group was pure, lipstick-coated nastiness. You'd never see Barbie looking so eeevil. And enjoying it so much.\n\n\"First, I want to announce the winner of this competition.\" She paused. Of course. When Deprivation House actually aired, there would probably be a commercial break right here. \"Brynn Fulgham got her groceries home before the rest of you. That means she will be deciding the next deprivation.\"\n\n\"Don't take away toiletries!\" Ripley called. Ripley Lansing was the group's celebrity, although she was only famous for having famous\u2014and rich\u2014parents. \"The boys smell bad enough now that they hardly ever shower. We won't survive if they lose deodorant privileges!\"\n\n\"Who can shower when there's no hot water anymore?\" Gail protested. Gail's parents definitely weren't rich. So not rich they didn't always have enough money to pay the heating bill. Sometimes Gail had to go to bed in her coat to stay warm.\n\n\"All the girls still shower,\" Ripley shot back.\n\n\"I don't care about deodorant or shampoo or any of that. I just want to be able to keep my drawing supplies,\" Hal said. Hal was always drawing plans for the planet around which he planned to design a computer game.\n\nI tried to figure out what Joe Carr would want most. That was my cover. A rich boy from Connecticut, adopted by a different family from my brother, Frank Dooley. It was hard to come up with anything. We'd already lost cell phones, iPods, junk food.\n\n\"Brynn, I'm begging you. Think of the odor,\" Ripley pleaded.\n\n\"Now, now. It's Brynn's decision,\" Veronica reminded everyone. \"And she doesn't need to make it until tomorrow night. Right now, I want to take some time for us all to get to know one another a little better.\"\n\n\"I know these losers as well as I want to,\" James Sittenfeld told her. \"I've been trapped in the house with them for almost a month.\"\n\nWe had been spending almost all our time together since our first day as contestants on Deprivation House, a reality show where teenagers saw how long they could last without luxury items. The network probably thought it would be fun for old folks to laugh at the younger generation freaking out because they were unable to go online, watch TV, text message, and talk on the cell at the same time.\n\nStill, I wouldn't mind learning some more about the other kids in the house. They were all suspects. See, my brother and I weren't really contestants. That was just our cover. We were actually here on an assignment for American Teens Against Crime, an organization our dad helped found. ATAC puts teens undercover in places where adult operatives would stick out too much. Like on a reality show where all the contestants are kids. Get me?\n\nFrank and I ended up here because all the contestants had received death threats. And since we'd shown up, a lot of nasty things had been happening. Some people had even died. None of them contestants. At least not yet.\n\nSo that's why I was up for finding out as much as possible about every kid in the house. Although I've got to say, if James wasn't a suspect and was just some guy, I'd be happy to never learn another thing about him. Like last night, I found out he clips his toenails\u2014with his teeth. I already knew he was maniacally competitive, and a bully. But I'm not biased or anything. Detectives have to keep open minds. And I'm a detective. Just because James is your basic rat dropping in human form doesn't mean he's a killer.\n\n\"George and Georgina only joined us very recently, and I thought some of you might be curious as to why they were allowed to join our game so late,\" Veronica continued.\n\n\"To get ratings,\" Olivia muttered.\n\nVeronica pretended not to hear her. \"The truth is, I got a letter from their father asking me to consider allowing them to come to Deprivation House. It was so touching that the network and I just couldn't say no.\" She gave a sweet smile. And when she smiles like that, somebody's about to get sucker punched.\n\n\"Let me read it to you,\" Veronica went on. I did a face check on George and Georgina. Not happy. And not happy. I miscalculated. Make that two somebodys were going to get a Veronica smackdown. She began:\n\n\"Dear Ms. Wilmont, I saw a commercial for your upcoming show, Deprivation House, and I had to write to you. You and your house may be the last hope for my fifteen-year-old twins, George and Georgina. All their lives, their mother and I have given them everything they've asked for. As an example, for their last birthday, they received a trip to Maui with eight of their friends, Georgina received a Tennessee walking horse, and George an Escalade, in addition to thousands of dollars' worth of smaller presents.\n\n\"It would have been our pleasure to give our children these gifts. We want them to enjoy all life has to offer. But they don't seem to understand that not everyone has the privileges they do. They have no concept of the work that has gone into earning the money to pay for their lifestyle. They show no appreciation. They show almost no awareness that a horse or a car are any different from the air or the ground, something that is simply there at any time for their use.\n\n\"That's why I'm begging you to make them a part of your show. Not so they have a chance to win a million dollars. That would only make the situation more nauseating. All I want is for George and Georgina to have the opportunity to live without what they consider necessities, but what the majority of the population considers luxuries. For a brief time, I want them to be deprived. I know that this is not your responsibility. It is their mother's and mine. But we have not been able to take control of the situation, and that is why we are asking for your help. No, begging.\n\n\"I know the deadline to enter the contest has passed, but please don't refuse us. Please give George and Georgina a chance to become better people.\n\nSincerely,  \nGeorge Taggart\"\n\nVeronica carefully refolded the letter. \"Of course, I couldn't say no.\"\n\nI did another George and Georgina face check. Very not happy. And very not happy. In fact, make that George red-faced, looking like he was about to go nuclear. And Georgina, face drained of color, looking like she was plotting somebody's long and painful death.\n\n\"If their parents don't want them to have the money, does that mean they're ineligible to win it?\" Olivia asked.\n\nI wasn't surprised that she was the first to get that question out. Olivia is the queen of strategy in the game.\n\n\"That would be completely uncool,\" George burst out. \"If we're here, and we have to follow all the same rules as everybody here, we should have the same chance to win the cash as anybody else.\"\n\n\"George and Georgina will follow the same rules and have the same privileges as the rest of you,\" Veronica answered. \"Including the chance to win the prize money.\"\n\n\"What would you even do with the cash?\" Gail protested. \"It sounds like you have two of everything that exists already.\" Gail was a latecomer to the House too, just not as late as the twins. She joined the group about a week and a half into the game as a surprise twist.\n\n\"Ever heard of legal emancipation?\" Georgina asked. Some of the color had returned to her cheeks, and her eyes were bright with excitement. \"That's what I would do with the money. Hire myself a lawyer. Cut all ties with my parents. As you might have guessed, they're complete control freaks, emphasis on the freaks. And use the rest of the cash to live on.\"\n\n\"My plan exactly,\" said George.\n\n\"Not until you just heard me say it,\" Georgina snapped. \"You never have any ideas of your own.\"\n\n\"It doesn't matter, because only one of us is winning the moola, and it's going to be me,\" George told his sister. \"Don't think I'm sharing, either. I'll be living solo, and you'll be home with the freaks.\"\n\n\"You've never won anything without my help and you know it,\" Georgina shot back. \"The only reason I didn't win today is that my bike was messed with. You couldn't\u2014\"\n\n\"Hey, don't fight,\" James cut in. He gave a smile every bit as nasty as Veronica's, but minus the lipstick. \"It's pointless. Neither of you is going to win. 'Cause I'm winning. Everybody else here has already accepted it. You'll just have to suck it up too.\"\n\n\"Did you forget that Brynn won today's competition?\" Olivia demanded.\n\n\"Competitions aren't everything,\" James reminded her. \"It's all about how you handle the deprivation. And I'm stone. Nothing gets to me.\"\n\n\"You practically meowed when your exercise equipment got\u2014\" Olivia began.\n\nVeronica clapped her hands. \"Enough of this bickering. It's very entertaining, but I think we've got plenty of footage. Now, I imagine it must have been somewhat humiliating for George and Georgina to have had that letter read aloud to all of you\u2014and of course to all the viewers when our show airs. So I think it's only fair that a few other secrets are revealed today. George, why don't you pick one of the other contestants, and I'll reveal a secret about them.\"\n\n\"Why does he get to pick?\" Georgina whined.\n\nVeronica ignored her. Veronica is very good at ignoring anything that displeases her.\n\n\"What secrets are you even talking about, anyway?\" demanded James.\n\nVeronica ignored him, too.\n\nI shot a fast glance at Frank. His face was carefully blank. But I knew we both had the same question James did. What secrets did Veronica know? She couldn't know the real reason Frank and I were here. Could she?\n\nIf she outed us, our mission was over. We wouldn't be able to protect anyone in the house if our cover was blown.\n\nGeorge glanced around the group, taking his time, enjoying making the rest of us sweat. What a guy. \"I pick . . .\" He did a dramatic pause. He'd learned something from Veronica. \"Ripley.\"\n\nOlivia snorted. \"Don't we know all her secrets already?\" she asked. \"She's in the paper practically every day. Even though all she's famous for is going to parties and spending her parents' money.\"\n\nOlivia really had a thing against rich people. It wasn't Ripley's fault that her dad was the drummer for Tubskull and her mom ran a cosmetics company and that they had a zillion dollars. I guess it was her fault that she'd been pretty obnoxious before she came to the house\u2014stomping on paparazzi cameras, getting waiters fired for pretty much nothing, and other brat stuff like that. But she'd been cool since we'd met her.\n\n\"She saved my life,\" I reminded Olivia. \"And Bobby T's.\" Bobby T was a contestant who had already been booted from the show.\n\n\"She saved your life and Bobby's so she wouldn't get her platinum American Express cut up by her parents,\" Olivia corrected me. \"We all know that Ripley's mom and dad got sick of all the bad publicity she was getting, and that they told her she had to start proving she could be nice\u2014and getting the good PR to prove it\u2014or they wouldn't give her any more money. She'd have to wait until she was thirty, when she got her inheritance, before she saw another dollar.\"\n\n\"What if they did?\" Ripley asked. The muscles in her neck tensed, but she forced a smile. I have to say, she was working pretty hard at the nice. \"Is it so wrong for them to want me to be a better person? Isn't that what your parents are supposed to do?\" She looked from George to Georgina. \"It's what your dad was trying to do by getting you on the show, right?\"\n\n\"Yeah, what a great guy,\" George answered with a sneer. \"I really respect him.\"\n\n\"We're getting off the subject of Ripley's secret,\" said Veronica.\n\n\"Is the secret that Ripley is the one who put Joe and Bobby T in danger in the first place?\" Olivia jumped in. \"Because I think it's way too convenient that she was right on the spot when two people needed rescuing.\"\n\n\"Frank was the one who took a chainsaw to the sauna when people got trapped in there,\" Ripley protested. \"Why aren't you accusing him of setting things up to look like a hero?\"\n\n\"It was Mitch who rigged the sauna. It was Mitch who put peanut oil in Bobby T's toothpaste to give him the allergic reaction. He's the one who fed jimsonweed to the dog who attacked Joe too,\" I reminded everyone. Mitch was a bank robber who'd hidden cash in the house before he'd gone to prison. He was trying to get us all out of there so he could collect it.\n\n\"Do you really know that for sure?\" asked Olivia. \"Stuff kept happening.\"\n\n\"Mary is the one who was sabotaging us. You know that,\" Joe said. Mary was a former fellow contestant who'd confessed to most of the sabotage before the police took her away.\n\n\"She said she didn't put the glass shards in the ice,\" Olivia continued. \"And she also denied setting that fire in Bobby's room, and putting the knives in Brynn's makeup bag. Why wouldn't she admit to those when she admitted to everything else? And there was that writing in the\u2014\"\n\n\"And somebody messed with my dirt bike,\" Georgina interrupted. \"I was completely sabotaged.\"\n\n\"Yeah, right,\" said George. \"She always has an excuse when she loses,\" he explained.\n\n\"I only lose when\u2014\" Georgina began.\n\n\"Enough!\" Veronica ordered. \"I can see that having to go without your usual luxuries is making some of you distrustful and unusually angry. That's something the other judges and I will have to look at when we review the tapes to decide who must leave this week.\"\n\nOlivia's mouth dropped open. Then she snapped it shut.\n\n\"All right, this is the secret about Ripley.\" Veronica's tongue flicked across her lower lip. \"It's something Ripley herself doesn't even know, since she has been deprived of TV and the Internet along with the rest of you.\" Veronica did another one of her pauses, then she turned toward Ripley. \"I'm sorry to tell you this, but your mother entered a rehab clinic in Malibu yesterday morning.\"\n\n\"Ooooh,\" James called out. Like we were in school and someone had just gotten called to the principal's office. Dillweed.\n\nRipley's face paled, but she just gave a short nod.\n\n\"You should really see this as a positive thing,\" said Veronica. She reached out and touched Ripley's arm. \"I just hope your mother won't try to leave too soon. You know what happened in this house.\" She gave a delicate shudder. \"I'm sure it could have been prevented if Katrina Decter had only stayed in rehab long enough to make a complete recovery.\"\n\n\"What is she talking about?\" Georgina asked Brynn.\n\nBrynn didn't seem to hear her. It didn't matter. Veronica was more than happy to answer. What had happened in this house ten years ago made good television.\n\n\"Katrina Decter was an actress on her way to superstardom when her husband killed her. Right in the great room upstairs,\" Veronica explained. \"She was just out of rehab herself, and she had had a relapse. She actually attacked her little daughter, Nina, who was only four. Her husband killed her to save Nina's life.\"\n\n\"I don't think I have to worry about my mom trying to off me, if that's what you're saying,\" Ripley said sharply.\n\nVeronica better back off, I thought. She was about to face off with Bad Ripley. And I had a feeling that if you put Veronica and Bad Rip into the cage together, Ripley might just walk out with Veronica's head in her hand.\n\n\"Aren't you sensitive,\" Veronica cooed. \"That must be one of your deep fears coming out. It wasn't my implication at all.\"\n\nUh, yeah it was. It so was.\n\n\"Whatever,\" muttered Ripley.\n\n\"Let's move on. Georgina, now it's your turn to choose a contestant, and we'll hear one more secret,\" Veronica said. Her eyes flicked around the group. \"I see some nervous faces,\" she commented. \"It makes me wonder if I know what each of you is hiding.\"\n\nVeronica strolled among us. \"Who will it be, hmmm?\" She tapped Brynn on the shoulder, and Brynn flinched. \"Twitchy. I bet she's got something good.\"\n\nShe moved on and stopped next to me. I used my ATAC training to keep my body relaxed and a pleasant expression on my face. \"And what about this all-American teen? He looks like his big secret is getting away with breaking curfew. But I know better.\"\n\nI didn't allow myself to look at Frank. She's just trying to build up the intrigue, I told myself. It's her job to make a good show. Make sure nobody gets busy with the remote during Deprivation House.\n\n\"It's true. I'm really a girl,\" I joked. Lame, I know. But it was all I could come up with in the moment.\n\nVeronica continued to walk from person to person. It was like some demented game of duck, duck, goose. \"So, who will it be, Georgina?\" she finally asked.\n\nGeorgina didn't hesitate. \"Frank.\"\n\n## Accusations\n\nShe can't know, I told myself. ATAC security is way too good. She can't have gotten past it. I completely believed that. In my head. In my stomach, I wasn't quite as sure.\n\nVeronica raised her eyebrows. \"Interesting choice,\" she told Georgina. \"And something of a bargain, because Frank's secret gives away something about Joe as well.\"\n\nMy palms began to sweat. A secret that was about me and Joe? How many things could that be?\n\nStrategy. I needed strategy. What should Joe and I do if we were outed right here and now?\n\nDeny it? Say it was some ratings stunt of Veronica's? Or something she and the other producers had come up with to create tension in the house? I don't watch a lot\u2014or actually any\u2014reality TV, but Joe says they all end up having lots of fights among the contestants. That was definitely turning out to be true on our show.\n\nOr should we admit to the truth and warn everybody that they were in danger? Just explain that Joe and I had been placed undercover as contestants because the death threats they'd received were being taken very seriously\u2014and for good reason, since two people in this house had already been victims of murder!\n\nVeronica smiled at me. \"Here's a secret about Frank and Joe,\" she said, breaking her long pause. \"Joe's father makes twenty-eight times the salary Frank's does.\" She pretended to do the math on her fingers, her red nails flashing. \"Let's just say there are quite a few dollars' difference.\"\n\nRelief washed through me. I struggled to prevent it from showing on my face. Frank Dooley wouldn't be relieved to hear Veronica blabbing about how little money his family had in front of everybody. He'd be outraged. But he'd also probably be trying not to show that Veronica had gotten to him.\n\nI jammed my hands in my pockets and locked eyes with Veronica. \"It's not exactly a secret that his family is richer than mine.\" I jerked my chin toward Joe.\n\n\"True,\" Veronica answered. \"But did either of you know there was that much of a difference? Every single year?\"\n\nI glanced over at Joe. He smirked at me and fingered the Diesel sunglasses he had stuck in his shirt pocket. He was enjoying himself a little too much. He'd better know he wasn't keeping those sunglasses after the mission was over. There would be no way to explain to Mom and Aunt Trudy how he'd gotten them.\n\n\"Dude!\" James gave my brother a congratulatory smack on the back. \"Guess it won't hurt you too bad when I win,\" he added.\n\n\"Joe drives a Corvette. Frank gets to borrow the family Toyota Corolla. When it's not in use. Or in need of repair,\" Veronica continued. \"Last year Joe went on three vacations\u2014Switzerland, the south of France, and Florida. Frank and his family went to visit his aunt Sharon in Boise, Idaho. For those of you who don't know it, and I imagine that's most of you, Boise's biggest attraction is, I believe, an extremely large potato.\"\n\n\"I think seeing a Super Spud sounds cool,\" said Olivia. I wasn't surprised that Olivia jumped in to take my side. Basically, Olivia thinks that if you're rich, you're a spoiled snob, and that if you're poor, you're a basically decent person. It was that black and white with her.\n\nVeronica clapped her hands. \"I'm off. The secrets should give you all something to talk about. Gossip is about the only type of entertainment you lambs have left, am I right?\"\n\nShe didn't wait for an answer, just turned and strode back into the mansion.\n\n\"What a complete witch,\" Gail said, staring after her.\n\n\"Cameras,\" Hal reminded her.\n\n\"Right. They can't film twenty-four\/seven because of the union rules about minors working on TV shows, but you know they had to have been filming when Veronica laid out all those secrets. And I'm sure they still are. They'd want our reactions.\"\n\nJoe snorted. \"I don't think you really have to worry about it, Gail. Veronica would probably consider being called a witch a compliment.\"\n\n\"At least you two didn't give her what she wanted,\" Gail told me and Joe. \"You didn't start throwing punches.\"\n\n\"I don't have anything to be mad about,\" Joe commented, shrugging. \"Am I supposed to be upset that my family has a lot more money than his?\"\n\n\"He's your brother,\" Olivia snapped.\n\n\"We just met each other,\" Joe shot back. \"I know him as well as I know you.\" He raked his hands through his hair. \"It's not like I took something from him.\"\n\n\"The situation is what it is,\" I said. \"I'm fine with the family I have.\"\n\n\"What about you, Ripley? Are you okay?\" asked Brynn. \"It must have been so hard hearing that about your mom, especially because we don't have our cells anymore. You can't even e-mail anyone to see if she's okay.\"\n\n\"You could drop out,\" James reminded her. \"There are still cash rewards left for the next two people who do. You could scoop up the thirty thou dropout bonus, go see your mom. You've already done what you came here to do, right? Proved how nice you are.\"\n\n\"You just think you know why I'm here,\" Ripley said.\n\nHuh. Interesting.\n\n\"And anyway, my mom will want her privacy,\" Ripley continued. \"She won't want to see me until she's through the program.\"\n\n\"Sounds like you're familiar with the drill,\" Olivia observed. \"I don't get how people with everything even get substance abuse problems. They have everything. If I had half of what your mom has, I'd be the happiest person on the planet.\"\n\nDid she even hear herself? Her prejudice against rich people made Olivia as much of a witch as Veronica sometimes.\n\n\"Tell that to the little girl whose mother tried to kill her,\" Brynn muttered. \"Both of her parents were pretty rich. Famous actor. Famous director. Living in this fancy house.\"\n\n\"You know what?\" said Gail. \"I'm starting to think this place really has been cursed since that director killed his wife here. Think about it: We had another accident during the competition today. Georgina's crash could have been a lot worse.\"\n\n\"The curse is something Veronica and the producers want us to believe in,\" Joe said. \"They're trying to make the show more dramatic. Like that message we found in the Deprivation Chamber. A curse doesn't write 'Death House' on a wall. A person does that. Or am I wrong?\"\n\n\"Did you forget the part about the demon?\" asked Hal. \"Nina's mother was supposed to have been possessed by a demon when she attacked Nina. That's why Nina's father had to kill her mother. A demonic spirit could write on a wall.\"\n\n\"I thought you were more into the science part of science fiction,\" I told him. Usually the only thing he talked about\u2014or seemed to think about\u2014was creating the specifications for L-62, the planet he was going to base a computer game on. He wanted to win the million-dollar Deprivation House prize to get the start-up money he needed for the game.\n\n\"There is scientific evidence that spirits exist. The energy they produce can be measured,\" Hal answered.\n\nBrynn turned around and headed inside. \"Where are you going, Brynn?\" Ripley called after her.\n\n\"We've had this conversation a million times already,\" Brynn answered without stopping. I wanted to follow her. I pretty much wanted to be with Brynn every minute I could. I wouldn't be able to see her after the mission was over and I was back to being Frank Hardy.\n\nBut I couldn't leave now. I was afraid I'd miss something key. Our suspects were all gathered in one place.\n\n\"She was just out of rehab,\" Georgina reminded the group. \"Duh. That's the only demon in the story. Her mom started using drugs again and went out of her head and attacked Nina. There's no demon and no curse.\"\n\n\"You don't know the number of bad things that have happened here,\" said Gail. \"Two people died before you got here.\" She was referring to two of the production assistants. Leo had been killed by Mitch, and Sylvia had died of an aneurysm after Mary hit her over the head with a piece of wood.\n\n\"We know that Mitch and Mary were the ones behind everything that happened,\" Hal told her.\n\n\"'House of Death' was written in the Deprivation Chamber after Mary was taken away,\" Olivia pointed out.\n\n\"That's not the same as trying to kill people,\" I replied. \"It's just a stupid prank.\" I wasn't at all sure that was true, but I wanted to chill everyone out.\n\n\"Disabling my brakes wasn't a stupid prank,\" Georgina said.\n\nSilent Girl's eyes widened. Her name was Ann, but we called her Silent Girl because she hardly ever talked. It was part of her strategy. She didn't want to give away anything about herself, in case it could be used against her somehow.\n\n\"If I find out who it is, they are going to be in a world of pain,\" James threatened.\n\n\"You're the most likely candidate,\" Joe told him. \"You're always bragging about how great you are and how you're going to trample the rest of us. But how many competitions have you won? And how loud did you whine when exercise equipment got taken away?\"\n\n\"Yeah, you have to be running scared,\" said George. \"I can see you giving yourself a little help.\"\n\n\"You don't know anything!\" James yelled. \"You just got here. You're never going to last. The rest of us have had things taken away little by little. You and your sister have to deal with tons of things being gone from day one. No hot food. No Internet. No TV. No hot water. No books or magazines. You're never going to make it. Just drop out now. Take the consolation money and go.\"\n\n\"It would give you fewer people to sabotage,\" George shot back.\n\n\"If the reason you think I'm sabotaging people is because I want to win, then everybody should be a suspect,\" James growled. \"You all want to win as bad as I do. You just aren't as honest about it.\" He strode into the mansion.\n\n\"Touchy,\" Olivia muttered.\n\n\"He's right, though,\" said Ripley. \"If someone is still sabotaging us now that Mary's gone, it could be any of us. And I'm voting for you.\"\n\n\"Me? Why me?\" Olivia protested.\n\n\"You've made it very clear you don't think any rich kids should be allowed to compete for the money,\" Ripley answered. \"Why not try to knock Georgina out of the competition? She doesn't deserve to win\u2014that's what you think, isn't it?\"\n\nOlivia blinked rapidly. She opened her mouth, but she couldn't find words quickly enough.\n\n\"Don't bother. We all know it is. You've pretty much said so every day since you got here,\" Ripley continued. She turned to Gail. \"And I know how much you need money. I almost wouldn't blame you if you were willing to cheat\u2014I mean murder\u2014to get it.\"\n\n\"I would never hurt anyone to win,\" Gail said, her voice low and hard. \"Never.\"\n\nRipley shrugged. \"I'm just saying, I can see it.\"\n\n\"Let's get out of here.\" Olivia grabbed Gail's arm, and they hurried inside.\n\n\"If I were you, I wouldn't be as okay as you act like you are. About my family having so much more money,\" Joe said to me. \"I'd be overheated. I might be willing to, uh, bend the rules to get some cash.\" Nice work on our cover.\n\n\"I'm not you,\" I told him.\n\nThere was a long silence. Even though we were outside, it felt like all the air had been sucked out from where we were standing.\n\n\"So, great,\" George finally said. \"What you're all telling me is that my sister and I are probably sharing a house with a killer.\"\n\n## Worried?\n\n\"I guess that is what we're saying,\" I told George. \"We're probably living in a house with a killer. Zoinks, as Shaggy would say.\"\n\nMy joke got a few half laughs. More than it deserved. Because it wasn't that funny. And because the living-with-a-killer thing was truly scary.\n\nRipley let out a sigh. \"So what should we all do for fun? That doesn't involve bleeding or poisoning or snakes or anything like that.\"\n\n\"Well, if you take away all that stuff . . . ,\" Hal said.\n\n\"That leaves us with practically nothing,\" said George.\n\n\"Charades?\" Ann the Silent Girl offered.\n\nThat got real laughs from everybody. \"Of course you want to play charades. You never speak. You'll probably kill at it. Uh, I mean, you'll probably beat us all,\" Ripley quickly corrected herself.\n\n\"I want to change these pants first,\" Georgina said. \"I got a grass stain on them, and I have to try to get it out. They're my friend Carly's.\"\n\n\"All you girls better watch yourselves,\" George warned. \"Georgina is always borrowing her friends' clothes, then wrecking them.\"\n\n\"No, I'm not. Don't listen to him. We share some DNA, that's all. He doesn't know anything about me,\" Georgina told us, rubbing at the grass stain on the knee of her purple jeans.\n\n\"Meet up in the great room?\" I asked.\n\n\"Where else?\" answered Hal.\n\nAs I led the way inside, I caught Frank checking the balcony that overlooked the fountain. Pretty much Brynn's favorite hangout.\n\nI let the others go ahead of me, and dropped into step next to Frank, who was bringing up the rear. I couldn't talk to him for too long. It would look suspicious, especially after Veronica pretty much invited us to hate each other. But I figured we could grab a second. \"Do you know where the Band-Aids are in the supply closet?\"\n\n\"I guess I can show you,\" Frank muttered. \"You're probably so used to having people wait on you that you're helpless.\"\n\n\"You know you want to go after her,\" I said once we were inside the walk-in closet. It was one of the few places we knew for sure there were no cameras. There and the bathrooms.\n\n\"Who?\" he asked.\n\n\"Did you just say who?\" I shook my head.\n\n\"Okay, I know who,\" Frank admitted.\n\nHere's the deal. I started crushing on Brynn from the first day we showed up at Deprivation House. And for once, a girl realized that I am truly the superior Hardy brother in looks, charm, and all other departments that matter. You got it: Brynn seemed pretty happy to hang with me. A lot.\n\nThe thing was, Frank was crushing on her too. Mad crushing. Like he should go out and write the girl a poem or something. And Frank's never that way. Usually he avoids girls, because he gets all blushy and stammery when he tries to talk to them. But for Brynn, he was willing to actually, you know, go up to her and try to speak. What could I do but step aside? I couldn't make Frank compete with my magnificence when his heart was practically breaking out of his chest whenever he saw Brynn.\n\n\"So go find her,\" I told him. \"I can keep watch by myself for a while.\"\n\n\"Are you sure?\" he asked.\n\n\"I know I'm the little brother, but I've had as much training as you\u2014\" I began.\n\n\"No,\" Frank interrupted. \"Are you sure it's okay with you . . . about Brynn . . . that I . . .\"\n\nSee, he even gets stammery when he talks about girls. \"I'm sure, I'm sure,\" I promised. \"Ripley actually seems to be getting into real nice mode instead of just camera nice mode. Maybe I'll let her experience the wonder of me.\"\n\nFrank laughed. I've been trying to teach him when things are funny and when they aren't. He still hasn't quite gotten the hang of it.\n\n\"Here are the Band-Aids.\" Frank tossed me the box.\n\n\"Sorry, I couldn't find any antiblush cream for you, bro,\" I said.\n\nHe didn't laugh. See?\n\nI walked out of the closet with a couple of Band-Aids clearly in my hand, in case the cameras were on and anybody cared. Then I headed up the S-shaped staircase and down the hall to the great room, this massive space that was like three or four living rooms combined. It used to have this awesome TV. Now, well, now it still had a great fireplace.\n\nI plopped down on one of the sofas next to Hal. He and George were the only ones there so far. \"Are we really going to have to play charades?\" he asked. The same way he'd ask, \"Are we really going to have to drink pickle juice?\"\n\nI shrugged. \"Nothing else to do. But it's not like it's a challenge or anything. Veronica's not going to come in here and make you.\"\n\n\"Hey, let's psych the girls out,\" said George. \"Let's say we want the teams to be guys against girls. We can write up the list of charade ideas right now and substitute them for whatever ones the girls team comes up with. Then we'll be able to guess the movie titles or whatever they are in seconds.\"\n\n\"Uh, but the girls will realize that we aren't guessing the stuff that they came up with. If you shout out a movie title that they didn't write down, they'll know we're cheating.\" Not that I was planning to cheat anyway. It was interesting that George was so up front about wanting to cheat, though\u2014since we were all in a competition together.\n\n\"Oh, okay. Then maybe we could\u2014\" He stopped abruptly as Georgina and Ripley came in.\n\nGeorgina narrowed her eyes at her brother. \"What were you just talking about?\" she demanded. \"He has his guilty face on,\" she told Ripley.\n\n\"Nothing,\" George said.\n\nI wasn't getting into this one. I wasn't on a mission to make sure there was no foul play in charades.\n\n\"You were trying to figure out a way to cheat,\" she accused. \"He is humiliatingly bad at charades. He never guesses anything,\" she added to the rest of us. \"I'm sure he was afraid of shaming himself. He actually started crying once because he didn't know what one of the words he was supposed to act out even meant.\"\n\n\"I was seven!\" George shouted. \"Will you stop telling people these stupid stories?\"\n\n\"When you stop doing stupid things,\" Georgina told him sweetly.\n\n\"I am going to be so happy to get away from you. As soon as I win the mil, I'm getting legally emancipated. I'm moving out. I'm never going to have to look at you again. Or Dad. That bloated control freak.\"\n\n\"Yeah, this is it. Dad's gone too far. Legal emancipation is the only option. I'm not even going home after I leave here.\"\n\n\"You'll have to,\" George told his sister. \"Because you won't have any money to live on, so you'll be at home, seeing the face every day, hearing the lectures. I'll be in my own place.\"\n\n\"We'll see,\" Georgina told him. The words came out sounding like, In your dreams.\n\n\"No, you'll\u2014\" George started.\n\n\"Keep at it!\" James called out as he entered the room. \"Go on and take each other out. It wasn't really fair that they brought you two in late anyway.\"\n\nThere was a serious edge of anger to his voice. Had James been so angry to have two new players in the game that he'd decided to try to take one out? He was almost lethally competitive.\n\nSUSPECT PROFILE\n\nName: James Sittenfeld\n\nHometown: Huntley, Wyoming\n\nPhysical description: 5'11\", 220 lbs., short hair with skull and crossbones cut into the back.\n\nOccupation: High school student\n\nBackground: Extreme sports fan, youngest of three brothers, average grades.\n\nSuspicious behavior: Very aggressive in presenting his desire to win.\n\nSuspected of: Sabotaging Georgina's dirt bike.\n\nPossible motive: Losing isn't an option.\n\n\"Worried?\" George asked. \"You should be. Georgina would have kicked everybody's behind in the race today. She might not look like it, but she's a total extreme spots fan.\"\n\nInteresting how George jumped to his sister's defense against an outsider. Also interesting how sure he'd been that she'd win today's competition. George badly wanted out of his house. Did he want it badly enough to sabotage his sister? To risk injuring her? Possibly killing her?\n\nSUSPECT PROFILE\n\nName: George Taggart\n\nHometown: Charlotte, North Carolina\n\nPhysical description: 5'11\", 170 lbs., blond hair, blue eyes.\n\nOccupation: High school student\n\nBackground: Spoiled child of rich parents; Georgina's twin brother.\n\nSuspicious behavior: Knows Georgina would have won dirt bike race.\n\nSuspected of: Sabotaging Georgina's dirt bike.\n\nPossible motive: Wants to win money to become legally emancipated.\n\n## Deeply Disturbed\n\nStrategy. I needed a strategy.\n\nIt's not that I can't think on my feet. I can improvise. I have to do it on missions all the time. But I'm definitely more comfortable with a well-thought-out plan. With maps and schematics and facts I can memorize before going into an unknown situation.\n\nTalking to a girl I liked . . . that was pretty much uncharted territory. I'd had some crushes, yeah. But I usually avoided them, to avoid the embarrassment of turning into a guy who lost 60 percent of his capacity to think, and 80 percent of his capacity to form words.\n\nAnd how I felt about Brynn . . . it wasn't a crush. I think Joe got that before I did. I don't know how to describe it exactly\u2014\n\n###   JOE\n\nJoe here to help my bro out. The way you describe it is loooove.\n\n###   FRANK\n\nGet out. This is my section. \"Loooove\" isn't the word I'd use. \"Loooove\" isn't a word at all. The word \"love\"\u2014which is a word\u2014isn't what I'd use either. But how I felt around Brynn . . . it was good. I should be able to say it better than that. All I can tell you is, the more I was around her, the more I wanted to be around her. And I wanted to be around her right now.\n\nI didn't bother checking the basement. There was nothing down there. The bowling alley was locked. The exercise equipment had been removed from the gym. The sauna had been hacked up by a chain saw. I'd had to do the hacking. Don't ask. There's been way too much badness at this house in the past few weeks to go through it all.\n\nI made a pass through the kitchen and the dining room, then decided to stop by the billiard room\u2014the former billiard room\u2014where the Deprivation Chamber was. The chamber was a booth with a camera in it where any of us contestants could go and vent about anything that was bugging us. I couldn't believe any of the real contestants here voluntarily went in there. You knew whatever you said was going to end up on TV.\n\nBut maybe Brynn had felt like venting. She'd seemed upset when she left the group.\n\nI stepped into the room, my footsteps echoing on the polished wooden floor. The big room felt empty. \"Brynn?\" I called out, just to be sure. No answer. As I turned back toward the door, I caught sight of something white out of the corner of my eye. A sheet of sketch paper lying on the ground.\n\nProbably one of Hal's drawings of L-62, I thought as I headed over to retrieve it.\n\nThe back of my neck prickled when I picked the piece of paper up. It wasn't one of Hal's drawings. It was some words written in what looked like a child's handwriting. It read HOUSE OF DEATH.\n\nExactly the same words that had been written on the chamber wall. The handwriting looked the same too. Was this a practice sheet?\n\nI carefully folded the sheet of paper and put it in my pocket. I needed to discuss this with Joe. But first I'd find Brynn. I wanted to make sure she was all right. And, okay, I just wanted to be around her for a few minutes. I went upstairs and down the hall to the room Brynn shared with Georgina and Olivia. Gail and Olivia sat on Olivia's bed, their heads close together. \"What are you doing in here?\" Gail asked.\n\n\"The door was open,\" I said. \"I was looking for Brynn.\"\n\n\"Haven't seen her,\" Gail told me. She shifted her body, shoving something behind her.\n\nWhich of course made me want to know what it was. I wouldn't be a detective if it didn't. I walked over and sat down on the bed across from them. \"It was pretty intense down there with Veronica, wasn't it?\"\n\n\"Veronica's not as bad as some other people,\" Olivia answered.\n\n\"Really? Veronica's pretty slimy,\" I said, not caring if this was getting filmed or not. I casually leaned back on the bed. I could almost see what Gail had hidden.\n\n\"Veronica didn't accuse us of murder today,\" Olivia reminded me. \"That was the newly nicest person in the world. Ripley.\"\n\n\"My brother pretty much said the same thing to me after you left,\" I told her. \"Everyone was accusing everyone. We're all freaked out.\" I rolled onto my side and propped my head on my hand.\n\nYes! I could see it now. Gail was hiding a sketchbook behind her. Could it be the sketchbook the sheet of paper with the note on it had come from?\n\n\"Hey, did you find Hal's sketchbook? He's been looking all over for it.\" A lie. But you've got to be a good liar when you're an ATAC agent. \"You know how obsessed he is with his drawings of that planet.\"\n\n\"This one's mine,\" Gail said quickly. \"Hal's not the only one who likes to draw. I take art for every elective I can. I even won a prize once for a poster I made. Got fifty dollars.\"\n\nGail wasn't such a good liar. She gave way too much detail. And she looked up as she talked, like she was trying to find the answers up in her brain.\n\n\"Cool,\" I answered. \"Can I see some of your stuff?\"\n\n\"No,\" Olivia answered.\n\nI laughed. \"Is she your manager?\" I asked Gail.\n\n\"I already asked her if I could see,\" Olivia explained. \"She told me she never shows anybody anything until it's completely finished, and she doesn't have anything done yet.\"\n\n\"Well, if you finish something, tell me,\" I said. I stood up. \"I'm going to go see if I can find Brynn. Everybody's going to play charades in the great room, if you guys are up for it.\"\n\n\"I'm not ready to hang with Ripley right now,\" said Olivia.\n\n\"I think I'll stay here too,\" Gail told me.\n\nI nodded and left the room. My brain was whirring as I mentally reshuffled Joe's and my suspect list. I decided I needed to bump Gail and Olivia up. There was something they didn't want me to see in that sketchbook. Was it where they practiced writing in that kidlike way? Were they teaming up to sabotage the rest of the contestants?\n\nSUSPECT PROFILE\n\nName: Gail Digby\n\nHometown: St. Louis, Missouri\n\nPhysical description: 5'9\", 145 lbs., sandy hair, brown eyes.\n\nOccupation: High school student\n\nBackground: Oldest of three kids raised by single mother. Grew up in severe poverty.\n\nSuspicious behavior: Trying to keep sketchbook hidden.\n\nSuspected of: Sabotaging Georgina's dirt bike; writing the message in the Deprivation Chamber.\n\nPossible motive: Winning the millions could change her life and the lives of her family members.\n\nSUSPECT PROFILE\n\nName: Olivia Gavener\n\nHometown: Homestead, Florida\n\nPhysical description: 5'7\", 140 lbs., red hair, freckles, brown eyes.\n\nOccupation: High school student\n\nBackground: Oldest of five kids, family on welfare, helps out with paycheck from fast-food job.\n\nSuspicious behavior: Contemptuous of rich people; always strategizing how to win.\n\nSuspected of: Sabotaging Georgina's dirt bike; writing the message in the Deprivation Chamber.\n\nPossible motive: Needs money to continue to help family and have a different life.\n\nI took a fast look in the great room to see if Brynn had rejoined the group. Nope. I continued down the hall, deciding to try the library next. I'd run into Brynn there once. That's where I had my first real conversation with her. And actually, I had no plan or strategy then, because I didn't know it was going to happen.\n\nNow that I thought about it, it had been easy to talk to her that day. Maybe that's what made what I felt for Brynn\u2014whatever I should be calling it\u2014different from a crush.\n\nThe library door was ajar, and I pushed it open. Brynn stood inside. For a second I felt like I was in free fall. Like I'd just done a bungee jump and the cord hadn't started to pull me back yet.\n\n\"This isn't a bad place to hang, even now that we've been deprived of all the books and magazines,\" I said from the doorway.\n\nBrynn didn't answer. She didn't turn toward me. She was staring at one of the walls. I didn't remember there being a painting on it or anything.\n\nI took a few steps toward her. Then I saw it. Someone had drawn in crayon on the pale wall-paper. Pressing down so hard in places that the paper had torn.\n\n\"What is that even supposed to be?\" I asked, hurrying over to Brynn. One figure in the picture was clearly a little blond girl, drawn the way a little kid would do it. Big head. Stick neck. Those hands that look like mittens. The other . . . it was some kind of creature. Blank eyed, with claws almost as long as its body.\n\nIt took me a moment to realize that Brynn hadn't answered. I pulled my gaze away from the drawing to her. She was still staring at the wall, her eyes dull and blank.\n\n\"Brynn! Are you okay?\" I touched her shoulder, and she started.\n\n\"Frank!\" She blinked rapidly. \"I didn't even hear you come in. I saw this\"\u2014she jerked her hand toward the drawing\u2014\"and it completely freaked me out.\"\n\n\"It is pretty creepy,\" I agreed.\n\n\"The way the demon is about to grab her.\" She shook her head hard.\n\n\"Demon? Is that what you think it's supposed to be?\" I asked.\n\n\"I guess it could be a basic under-the-bed monster,\" she answered. \"But with Veronica bringing up the story about Nina and how her father told her that her mother had been taken over by a demon . . . My mind just went there.\"\n\n\"Makes total sense. It's probably what whoever drew it wanted us to think,\" I said. \"Somebody's trying to scare us. Keep us thinking about the curse, about the house's history.\"\n\n\"Well, it worked. I'm officially scared,\" Brynn admitted.\n\n\"Maybe it's something Veronica and the producers came up with. I can see them trying to add extra excitement,\" I suggested.\n\nThe motivation made sense to me. But there were a lot of puzzle pieces that didn't fit with that theory. The sketchbook Olivia and Gail really didn't want me to see, for one. Georgina's sabotaged brakes, for another. Trying to make us think the house was cursed was a lot different from actually putting a player's life in danger.\n\nAnd there was still the glass that had been mixed with the ice in James's water. Mary had insisted she hadn't done that when the police came to take her away. She'd confessed to everything else. The e-mail threats. The dead bird. The wood rolling off the woodpile on top of me. Putting the rattler in Ripley's bed. Everything but the glass. Why bother to deny it, if she'd really done it? Still, the glass in the ice didn't seem like something Veronica and company would do.\n\n\"I don't see Veronica doing this,\" Brynn said.\n\n\"Really? Why?\" I was curious to hear her take on the situation at Deprivation House.\n\nBrynn reached down and flicked on a light switch at knee level on the wall. Weird place for it. The light made the drawing look even more unnatural.\n\nShe ran her fingers across it, then trembled as they snagged on a ripped piece of wallpaper. \"This wasn't done by somebody rational, somebody thinking of TV ratings.\" She turned to face me. \"Frank, whoever drew this is deeply disturbed. Truly, seriously unbalanced.\"\n\n## Silent Girl Screams\n\n\"I agree with Brynn,\" Olivia said. \"Whoever drew that is one messed-up individual.\"\n\nFrank and Brynn had called the rest of us down to the library. We were staring at the drawing Brynn had discovered.\n\n\"Either that, or one great strategy mastermind,\" Frank commented.\n\nWas that aimed at Olivia? She was the player most concerned with strategy. She'd even organized a secret alliance. I only knew about it because Frank had told me.\n\nAnn's deep into strategy too, I reminded myself. The whole reason she never talked was because she thought revealing any info about herself could give the competition an advantage.\n\n\"I'm sticking with my theory that the producers are behind all this demon curse stuff,\" I said.\n\n\"Yeah? I think one of you did it,\" James announced. \"One of you standing in this room right now, acting all innocent.\"\n\n\"Does that include you?\" I asked, interested to see what reaction I'd get. \"You came back into the mansion pretty fast after Veronica played her I-know-a-secret game. What did you do? Maybe a little drawing.\" I jerked my chin toward the wall.\n\nJames had some . . . let's call them anger management issues, and the way the crayon had ripped into the wallpaper seemed Jamesian to me somehow.\n\n\"Have I entered a time warp? Didn't everyone just say all this stuff?\" asked Hal. \"Scary thing, accusations.\" He shook his head. \"Clearly, talking about it isn't going to solve it. No one's going to confess. We're going to go around and around and around and get nowhere.\"\n\n\"You have something better to do, nerd boy?\" James demanded. \"I'm thinking you don't want to talk about it because you have something to hide.\" His eyes narrowed. \"Who have we all seen drawing away the whole time we've been here?\"\n\n\"Hal,\" said Olivia quickly.\n\n\"Those are schematics. Plans. They are highly detailed and precise,\" Hal protested. \"Nothing like what's on the wall. It looks like a kid drew it.\"\n\n\"Yeah,\" Frank agreed. \"To me it seems likely that whoever wrote that message in the chamber drew the picture. That writing looked like it had been done by a little kid, and Hal's right about the drawing. It definitely seems like it was made by a child.\" He glanced over at Olivia and Gail, who were standing close together. \"Or at least someone who wants to give that impression.\"\n\n\"So you're saying it could still be the nerd,\" James said. \"Only he changed his style to get us off track.\"\n\n\"I didn't exactly\u2014\" Frank began.\n\n\"You know what?\" Gail interrupted. \"I just realized James wasn't the only one who left the group early after our little meeting with Veronica. Brynn, you left early too.\"\n\n\"You think Brynn made the drawing and then pretended to find it? That's extra twisted,\" Olivia said.\n\n\"Why are we thinking that the drawing had to have been made today?\" I asked. \"Did anyone go into the library before Brynn?\"\n\n\"That's what we're trying to find out, boy genius,\" James answered. \"Who came in and did the drawing.\"\n\n\"I mean did anyone\u2014not the perp\u2014come into the library earlier today?\" I clarified.\n\n\"Perp. Now he's all hard-core.\" James sneered.\n\nShoot, I had to be careful not to be so obvious about my training.\n\n\"I didn't come to the library before now,\" Georgina answered.\n\nI got no's and head shakes from everybody else. \"So we don't know when the drawing was done. It could have been last night. That would make sense, since we're pretty sure the show doesn't waste the camera time it has filming us sleeping.\"\n\n\"So we don't know anything,\" said Gail.\n\n\"I still say it's someone in this room,\" James insisted.\n\n\"And I say he who smelt it dealt it,\" George answered.\n\nJames jerked his head toward George. \"What is that supposed to mean?\"\n\nGeorge raised his eyebrows. \"I guess they don't teach analogies in public school. What it means is, you're the one who started all the accusations about the drawing. That's the 'smelt it' part. Am I talking slowly enough? And I think you're the one who actually did the drawing. That would be the 'dealt it' part.\"\n\n\"Want to see what I learned in public school?\" James asked. He didn't wait for an answer. He launched himself at George and knocked him to the ground.\n\n\"Smash his smug rich-boy face in!\" Olivia shouted.\n\nI knew she had an attitude about rich people, but\u2014wow.\n\nJames pinned George with a knee to the stomach. \"I learned how to fight!\" James yelled. Then he slammed George in the jaw.\n\nGeorge pretty much just . . . squirmed.\n\nI figured as a rich boy myself\u2014at least as long as I was undercover\u2014I had to step in. I rushed over and locked one arm around James's thick neck. A couple of moves and he was on his back, with my knee on his chest.\n\n\"Rich boys also take martial arts,\" I told him.\n\n\"That's right!\" said George as his twin helped him off the ground.\n\nIdiot. I shot him a look that I hoped he read as I'm not going to save your pampered rear twice.\n\n\"So, we're in agreement. We don't know who did this?\" I asked the whole group the question, but I kept my eyes on James. He gave a tiny nod.\n\n\"I keep looking at the drawing,\" Olivia said. \"Is it just me . . . or does the little girl kind of look like Brynn?\"\n\n\"It's practically a stick figure,\" Hal protested.\n\n\"Same hair,\" Olivia pointed out. \"And the way the nose is drawn, it looks like it turns up at the end, the way Brynn's does.\"\n\n\"And Georgina's,\" her twin observed. \"She has the hair, too.\"\n\n\"I am now officially creeped out,\" Georgina announced.\n\nI looked over at Brynn to see how she was reacting. She was staring at the drawing, her eyes almost glazed. Then she gave her head a hard shake. \"I think I've looked at it long enough. I'm going to go take a special Deprivation House cold shower. I'm never taking hot water for granted again!\"\n\nAfter she left, Hal stepped closer to the drawing and ran his finger over the picture of the girl. \"I don't see it.\"\n\n\"If the girl is supposed to be Brynn or Georgina, are we supposed to think that demon beastie thing is real too?\" asked Gail.\n\n\"I hope so,\" James said. \"Charades is way too boring. And we don't have any TV. If there's a demon in the house, I hope it makes an appearance. Can't they do your bidding if you sell them your soul?\" He tilted his head back and threw out his arms. \"My soul is up for grabs if you could arrange for a\u2014\"\n\n\"Stop it!\" Ann shrieked.\n\nWe all stared at her. Ann hardly ever talked. Hearing her let out a scream like that was shocking.\n\n\"What's your impairment?\" James asked.\n\n\"You can't invite a demon to come to you like that!\" Ann cried. \"It's not some joke. Don't you know how dangerous it is? The demon will take over your body, your mind. It will make you do horrible things. And when it finally leaves, you'll be insane. If your body doesn't completely give out. Do you want to die?\"\n\nThose were more words than Ann\u2014formerly known as Silent Girl\u2014had spoken the entire time we'd been at Deprivation House.\n\n\"That's messed up,\" Olivia said softly.\n\nI was hit by what Olivia had said when she first looked at the drawing. Whoever drew that is one messed-up individual.\n\nI didn't know if I'd call Ann messed up. But she'd definitely shown that she had some deep feelings the rest of us hadn't known about. Disturbing feelings. She'd hidden them really well. What else had she been able to hide?\n\nSUSPECT PROFILE\n\nName: Ann Sommerfeld\n\nHometown: Plano, Texas\n\nPhysical description: 5'5\", 120 lbs., brown hair, brown eyes.\n\nOccupation: High school student\n\nBackground: Only child. Parents deceased. Raised by grandmother.\n\nSuspicious behavior: Hysterical over possibility of demon. Very competitive\u2014rarely speaks, so she won't give anything away that can be used against her.\n\nSuspected of: Sabotaging Georgina's dirt bike. Leaving message in Deprivation Chamber and drawing on the library wall.\n\nPossible motive: Desperate to win.\n\nAnn rubbed her face with both hands. \"Sorry. I didn't mean to go off that way. It's just that what James was doing was so perilous. It really scared me.\"\n\n\"Maybe you should sit down,\" suggested Frank. \"Do you want some water?\"\n\n\"No, I'm fine,\" Ann answered.\n\n\"Maybe we should go back to the great room,\" I said. \"We've seen what there is to see here.\"\n\n\"Yeah.\" Frank started for the door\u2014and froze as a long, terrified scream blasted down the hall.\n\n\"It's the demon!\" Ann cried. \"It answered the summoning!\"\n\n## Ripped Apart\n\nBrynn! I knew that scream had come from Brynn. I tore out of the library and down to the girls bathroom.\n\nShe stood outside with her back pressed tight against the door. Like she was trying to keep something trapped inside. \"Are you okay? What happened?\" I exclaimed.\n\n\"What's in there?\" Georgina asked. The rest of the group had reached Brynn only seconds after I did.\n\nBrynn opened her mouth to answer, then shook her head and simply stepped away from the door. \"It's safe to go in?\" I asked.\n\nShe nodded.\n\nI slowly opened the bathroom door. I sucked in my breath with a hiss when I saw what was inside.\n\n\"What's in there?\" Georgina asked again, her voice louder and higher, more scared.\n\nJames, Joe, and Olivia followed me into the bathroom. \"It's a bloody teddy bear,\" Olivia called. \"It's so gross, you guys.\"\n\nThe bear had been clawed open. Its stuffing lay all over the floor. One of its eyes was gone.\n\nAnn pushed her way inside. \"The demon did this. They start small sometimes, before they have full power in this world. But that's what it wants to do. To all of us.\"\n\n\"The red stuff is just ketchup,\" said Joe. \"You think the demon wants to cover us with ketchup?\"\n\nAnn didn't smile.\n\n\"Whose bear is it, anyway?\" I asked.\n\n\"It's Brynn's,\" Ann answered.\n\nOlivia's eyes widened. \"Maybe I was right about the girl in the drawing looking like Brynn. Maybe the demon really is coming after her, like in the picture.\"\n\n\"Do you really believe in demons?\" Joe asked.\n\n\"How can you doubt their existence after seeing this?\" Ann swept her arm toward the teddy bear carnage.\n\n\"Humans are capable of doing something like that too,\" Joe commented.\n\n\"Don't tempt a demon to show its power,\" Ann warned, then left the room.\n\nJoe raised his eyebrows at me. Then he gave a half smile. \"Maybe somebody should go check on Brynn.\"\n\n\"Practically everybody's out there with her,\" said Olivia.\n\n\"I'll go see how she's doing,\" I said, trying to sound like it was no big deal.\n\n\"You do that,\" Joe said, his smile widening.\n\nYou know, while he can be annoying a large percentage of the time, Joe is actually a good brother.\n\nBrynn hadn't gotten far. She was sitting next to the bathroom door, leaning against the wall. Everyone else stood around in clusters, talking quietly. It didn't look like another fight was going to break out tonight.\n\n\"Hey,\" I said to her. \"You want to maybe go out on the balcony, get some air?\" I knew the balcony was one of her favorite places.\n\n\"Air would be good. I think I forgot to keep breathing about half an hour ago,\" Brynn answered.\n\nI reached down and grabbed her hand so I could help pull her to her feet. I got this little jolt. Kind of like a static electricity shock. Except in a good way.\n\n\"Crazy night, huh?\" I asked as we started through the house.\n\n\"I really need to talk about something not here,\" Brynn answered. \"Tell me some random thing about you. Anything.\"\n\nI hesitated. I wanted to answer as Frank Hardy, not Frank Dooley. I wanted to be my real self with Brynn. But there'd always be this huge chunk of stuff I'd have to keep hidden. I couldn't talk about my actual family or town or school.\n\n\"Come on. Anything. I'm not asking for deep, dark secrets. This isn't a Veronica torture session,\" Brynn said. She opened the glass doors to the balcony, and we stepped out. \"You can tell me your favorite flavor of ice cream if you want. I don't care. As long as you say something completely unrelated to Deprivation House.\"\n\nI didn't see any reason why Frank Hardy and Frank Dooley couldn't have the same favorite ice cream, at least. We did share a tongue and taste buds and a stomach. \"Vanilla,\" I told her.\n\n\"That's it? Just vanilla?\" she asked. \"Not Vanilla CocoMocho or Vanilla Nutso?\"\n\n\"Just vanilla,\" I said. Joe's the one who likes his ice cream stuffed with candy and nuts and pretzel bits and all that. I'm always telling him there's no ice cream in his ice cream.\n\n\"I like my ice cream straight up too. Otherwise you can't really taste it.\" She braced her hands on the balcony railing and stared down at the fountain. \"I do enjoy the cone, though. I never eat ice cream from a dish if I can help it.\"\n\n\"Yeah.\" I realized I wasn't having any talking-to-girls anxiety. Maybe because conversations with Brynn were a little . . . odd. Or maybe because she made me feel like I'd already known her forever.\n\n\"So are you going to reveal your favorite flavor?\" I asked.\n\n\"Don't have one,\" Brynn answered. \"Or maybe I do, but I don't like to admit it.\" She lowered her voice to a whisper. \"I think it hurts the other ones' feelings.\" She laughed. \"You're going to think I'm as wackadoo as Ann. I heard all that demon stuff she was saying when she looked at the bear.\" Her face turned serious.\n\n\"I thought we were having a no-Deprivation-House conversation,\" I said.\n\n\"It creeps in,\" Brynn answered. She sighed. \"I end up thinking about the house and everything all the time. Like the demon story. I can't believe a dad would tell his little girl her mother was possessed by a demon.\"\n\nI looked down at the fountain for a moment. They hadn't turned it off as a deprivation yet. It was lit up at night, and right now the water was glistening under the beams.\n\n\"Maybe he thought it was a way for her to understand her mother's drinking and drug use,\" I finally said. \"Not that I think it was a good way. But maybe he was trying to get across the idea that when her mother was on those drugs, she wasn't the same person.\"\n\n\"You think the demon story started before the night of the murder?\" Brynn asked.\n\nI shrugged. \"I'm just trying to make sense of it.\"\n\n\"So the mom\u2014the actress\u2014had just come back from rehab, but she immediately had a relapse and went after the kid. And the father had to kill her,\" Brynn said. \"It's really hideous. Then the girl had to testify and everything. It's sick the way Veronica talks about it and makes sure it's all on camera.\"\n\nI nodded. \"If she and the producers are behind that drawing and what happened with your bear, it's even worse.\"\n\n\"Taking some family's tragedy and twisting it into entertainment. It's disgusting.\" Brynn threw her arms out wide. \"I hope this is all getting on camera. I hope it makes the cut!\" she added loudly.\n\n\"Did you know humans have forty-six chromosomes, peas have fourteen, and crawdads have two hundred?\" It was the first thing that came into my head.\n\n\"That's so cool! Crawdads must be hiding some truly superhuman powers,\" Brynn exclaimed. \"Thanks for the subject change, Frank. Let's see if we can last five whole minutes without even thinking of all the creepy stuff in this place.\"\n\nI was definitely up for trying. But I wasn't sure it was possible.\n\n\"Good morning, everyone,\" Veronica said as we sat around the table eating our cold cereal. We'd lost the right to hot food days ago. \"Eat well. It's a competition day!\"\n\n\"It's cereal,\" James grumbled. \"How well can we eat?\"\n\nI was sure that comment would end up in the footage we'd be viewing before Veronica told us who was the next person to be eliminated. Staying at Deprivation House was all about being able to deal with the deprivations without getting whiny.\n\n\"George, that bruise on your face looks so painful,\" Veronica commented. \"How did you get it?\" I couldn't decide if Veronica's question was sincere. She knew a lot about what went on in the house. But even she couldn't see everything.\n\n\"Walked into a door,\" George muttered. Probably too embarrassed to admit he got hit.\n\n\"Oh. I'm relieved to hear it,\" Veronica said. \"It looks more like an injury from a fistfight. And I'd hate to think there was any kind of friction between our contestants.\"\n\nNow that, I knew, wasn't sincere. Veronica would love any kind of friction.\n\n\"Especially with today's competition,\" Veronica continued. \"Your relationships with one another are going to be important.\" She smiled. \"But I don't want to say too much. You'll have to wait until this afternoon. Meet me at the fountain at three.\" And she was gone.\n\nAs soon as Olivia had downed her cereal, she signaled for me to meet her in the supply closet. She got Hal and Gail to meet us in there too. \"Okay,\" she said. \"I think today's competition is going to involve teams. That's what I got from Veronica. What about the rest of you?\"\n\n\"Possibly,\" Hal said. \"But not definitely.\"\n\n\"I'm not saying it's definite,\" Olivia snapped. \"I'm saying it's possible. Will you all agree with that?\"\n\nWe all quickly agreed. Olivia was in complete drill sergeant mode.\n\n\"What the 'alliance' needs to do\u2014oh, clearly Gail is part of the alliance now,\" Olivia began.\n\nI wasn't sure \"alliance\" was exactly the right word. Maybe dictatorship?\n\n\"What we need to do is make sure the rich kids don't win the competition. An alliance member needs to win. If that means your individual team losing, fine.\"\n\n\"So you're saying we should sabotage our teammate if they aren't in the alliance?\" Hal asked.\n\n\"Exactly,\" Olivia answered.\n\nWas Olivia behind Georgina's accident? Was she behind the drawing and the message in the chamber? Was she trying to recruit some help right now? I wasn't sure. But I was definitely sure this was a bad idea.\n\n\"What's the point?\" I said. Olivia narrowed her eyes at me.\n\n\"The point is what it's always been since we formed the alliance. To win. The whole enchilada. One of us right here is going to win the million, and we're going to split it four ways,\" Olivia explained impatiently.\n\nThe stress of the competition was getting to her. She hadn't been this irritable and bossy when she first asked me to join the alliance. Or maybe she has been, but she was just hiding it because she knew she needed to be nice until I said yes.\n\n\"I know the purpose of the alliance,\" I said, trying to keep annoyance out of my own voice. \"But it doesn't really matter who wins the competition. It's just for the show. It doesn't actually have anything to do with winning or losing the money, if you think about it.\"\n\n\"Winning the competition means controlling the next deprivation,\" Olivia told me, speaking slowly, exaggerating each word, like she was trying to teach me to talk. \"Controlling the deprivation gives us power. We can choose a deprivation that will really hurt the other side. I've been studying them. I know their weak spots. I know how to make them hurt.\"\n\nO-kay.\n\nHal cleared his throat. \"Uh, joining the alliance made logical sense to me.\" He did another throat clearing. He sounded nervous. \"It greatly improved my odds of winning a significant amount of money.\"\n\nI could hear the \"but\" coming. It seemed like Olivia could too. The muscles in her jaw had clenched.\n\n\"But,\" Hal went on, \"I didn't know that being in the alliance would involve cheating. I'm not . . . comfortable performing any kind of sabotage.\"\n\n\"What you're saying is that getting your computer game up and running isn't that important to you,\" Gail told him. Olivia shot her a proud smile.\n\n\"I didn't hear him say that,\" I jumped in. \"Have you thought about the fact that if we get caught cheating, we get booted? That means no money.\"\n\nI figured that argument might mean more to Olivia and Gail than talking about right and wrong.\n\n\"Even if we didn't get caught, I'm not willing to hurt somebody to win,\" Hal said, voice cracking.\n\n\"I didn't say hurt. I said make sure they don't win the competition,\" Olivia protested.\n\n\"And you think that can be done in a completely peaceful, nonviolent way?\" I shot back.\n\n\"Are you saying you want out?\" Olivia demanded.\n\n\"No. I'm just trying to get things clear,\" I told her. I definitely needed to stay in the alliance. The person Joe and I were looking for could be right here in this little group.\n\n\"What about him?\" Gail jerked her chin toward Hal.\n\n\"I think I do want out,\" Hal said.\n\nGood for him.\n\n\"Fine. Of course, we'll have to tell the others you were part of an alliance against them,\" said Olivia.\n\n\"What? Why? That will mess you up as much as me,\" Hal protested.\n\n\"It will mess us up. But not as much,\" Olivia answered. \"Because we'll still have each other. We'll still have the alliance.\"\n\nHal stared at the ground. Olivia, Gail, and I stared at him.\n\n\"Well?\" Olivia finally said.\n\n\"I'm in,\" Hal told her, without looking up.\n\n\"You did the right thing.\" Gail patted his shoulder, and Hal flinched. \"Those other kids have tons of money. They've had everything their whole lives. It would be so unfair if one of them got a million dollars.\"\n\n\"Yeah. Gail's family doesn't always even have money for food and heat. I have to contribute the money I make frying burgers to keep our household going,\" Olivia reminded him. \"It would be so wrong for somebody like Brynn to win over Gail. One of Brynn's shoes would probably pay Gail's family's rent for a month. I can't believe she lets Georgina borrow them.\" Olivia shook her head. \"Not the point. The point is that rich people don't need to win. Between Frank and Joe, it would be completely unfair for Joe to get that money.\"\n\n\"Completely,\" Gail agreed. \"We all heard what Veronica said. Joe's dad makes a million times more than Frank's. Frank probably would use the money for stuff like college.\" She looked over at me, and I nodded. \"What would Joe use it for? A second car? 'Cause you know he already has one.\"\n\n\"You know it,\" Olivia said. \"You must be so furious at him, Frank. Now that you know how he's been living all these years. I mean, don't you look at him sometimes and think, 'I would really like to kill my brother'?\"\n\n## Psycho Teammate\n\n\"What did you tell her?\" I repeated. \"I just want to know.\"\n\n\"That's not the point,\" Frank answered. He shifted uncomfortably on his seat at the edge of the tub. I'd scored the closed toilet seat. It was the most comfortable spot in our \"conference room.\"\n\n\"I'm giving you a heads-up,\" he continued. \"If you're paired up with Gail, Olivia, and maybe even Hal, you have to watch out for sabotage. I don't know if Hal's actually going to do anything. But Olivia and Gail were pushing hard, and Olivia is definitely going to be watching him.\"\n\n\"I just want to hear what you told her when she asked you if you wanted to kill me,\" I said. Sometimes I can't resist torturing Frank a little.\n\nHe sighed. \"You realize I'm undercover.\"\n\nI waved my hand in a hurry-it-up motion.\n\n\"I told her that when I couldn't fall asleep I did sometimes imagine different ways that I could off you and get away with it,\" he admitted.\n\nI laughed. \"I've been doing that for years about you! Many of my scenarios involve cleaning supplies, because everyone knows you're way into cleanliness and I'm a slob.\"\n\nFrank decided to take the mature route. Otherwise known as the boring route. \"Just be careful, okay?\" he said.\n\nI nodded. \"You have no idea what form this sabotage might take?\"\n\n\"None. It's up to the individual alliance member. We act alone. If we get caught, we get thrown out of the contest alone,\" Frank told me.\n\n\"You think Gail or Olivia wrecked Georgina's brakes?\" I asked.\n\n\"Definite possibility,\" Frank answered. \"I think they should be high on our suspect list for anything bad that's directed at one of you rich brats.\"\n\nIt's so cute when Frank tries to be funny.\n\n\"I'm even thinking that Georgina and Gail could be behind that drawing in the library and the message on the chamber wall,\" Frank went on. \"When I went looking for Brynn yesterday, I found a sheet of paper from a sketchbook. Same words as on the Deprivation Chamber wall. Same handwriting. I figured it was a practice draft.\"\n\n\"Sketchbook, huh?\" I said. \"That makes me think of Hal right off. He almost always has his with him.\"\n\n\"That was my first reaction too. But then I went upstairs to Brynn's room, and I saw Olivia and Gail talking. Gail hid something behind her back. I finally got a look at it\u2014\"\n\n\"And it was a sketchbook,\" I finished for Frank. \"It's interesting that Olivia's the one who pointed out that the girl in the picture looked like Brynn. That fits with the whole destroy-the-rich campaign.\"\n\n\"And it was Brynn's teddy bear that became a demon sacrifice. Freaking out Brynn would definitely throw her off her game,\" Frank added. \"Olivia didn't say anything about the drawing or the bear at the alliance meeting. You'd think she'd want to make sure that Hal and I didn't get spooked along with Brynn and the rest of you. She should want us focused.\"\n\n\"Maybe there's an alliance within the alliance,\" I suggested. \"Maybe the girls are playing the two of you guys. Maybe they're planning to use you to help eliminate some of the competition, but cut you out later.\"\n\n\"If Olivia wins, I think she's going to forget how to even pronounce the word 'alliance,'\" Frank predicted.\n\n\"Joe, your partner for this competition will be . . .\" Veronica smiled.\n\nI got this ate-way-too-many-hot-dogs-way-too-fast feeling in my gut.\n\n\"Olivia,\" she concluded.\n\nOf course. Perfect. The girl who wanted every rich person dead, dead, dead. Who thought my own brother probably fantasized about killing me.\n\nThis is a good thing, I told myself. You're a highly trained ATAC agent. You can handle whatever sabotage Olivia decides to throw at you. You wouldn't want her paired with someone who didn't have any advance warning\u2014or any skillz.\n\nFrank hates that word, just FYI. Skillz. He hates any word that has a z where he thinks there should be an s. I know way too many facts like this about my brother.\n\nVeronica kept on matching up the pairs as we stood around her on the front patio. Here's how it laid down. Frank and Ann. She showed up for the competition looking like a goth girl ready for a night out. She'd made some kind of weird charm out of twigs and string and hair. But after seeing Ann lose it last night, I knew she wasn't making a fashion statement. That stuff had to be antidemon protection. She was one scared weird girl.\n\nBrynn and Ripley got teamed up. Georgina and Hal. I wondered if Hal would try anything to sabotage Georgina. It sounded like Olivia\u2014with backup from Gail\u2014had put some real fear into him.\n\n\"That leaves James, George, and Gail,\" Veronica said. \"Hmmm. An odd number.\"\n\nJames's hand shot up. \"I'll go solo,\" he volunteered.\n\n\"Why am I not surprised to hear that?\" Veronica asked. \"Why do I suspect that the words 'does not play well with others' were featured on many of your elementary school progress reports?\"\n\nI noticed George's hand go up to the bruise on his jaw. I'm sure he was praying that Veronica would go ahead and let James play by himself.\n\nGeorge didn't know Veronica that well yet, or he wouldn't have bothered. She liked to do things in whatever way would make the most people unhappy. \"No, I think I'll give Gail the chance to prove what she can do on her own. Gail, you seem like something of a follower to me. This will be a good opportunity for you, I promise.\"\n\n\"It's not fair,\" Olivia muttered.\n\n\"There's nothing in the rules requiring fairness on the producers' side,\" Veronica stated, looking pleased with herself. She turned to James. \"That means you and George will be a team.\"\n\n\"Good,\" George said quickly.\n\nHe'd figured out that attitude was the difference between staying at Deprivation House and getting kicked out.\n\nJames rolled his eyes but managed not to say anything that the judges wouldn't like when they were deciding who had to hear the words, \"You are deprived of the chance to win one million dollars,\" during the next elimination.\n\nStill, I was pretty sure that if team James-George lost today, there'd be another fight to break up. Cameras on or not.\n\n\"Now, on to the competition,\" Veronica said. \"Perhaps those of you with families wealthy enough to hire help have heard that some house cleaners don't do windows. Perhaps even those of you whose families don't have so much in the bank haven't had to wash windows either. Maybe your parents take on that chore.\"\n\n\"Or they leave them dirty,\" George said under his breath. \"Poor people don't mind being dirty, do they? Or is it that they can't afford soap?\"\n\nI felt like punching the guy myself. Just because I was undercover as a rich kid didn't mean I couldn't think he was a complete wad. Did it?\n\nI reminded myself that I wasn't here to teach George Taggart manners. Or James. Or anybody else. I was here to stop a murder. The lives of every contestant had been threatened.\n\n\"As you may have noticed, we have lots of windows at Deprivation House,\" Veronica continued. She flicked one hand in the direction of the mansion. Oh, man. I can't believe I hadn't noticed before. It looked like somebody had sprayed down each of those windows with an oily, gray-green goop. I didn't especially want to think about what it was.\n\n\"We have so many windows that I guarantee each of the teams will get a chance to wash some,\" Veronica told us. \"At least those teams that want a chance at winning today.\" She clapped her hands. \"What are you waiting for? Go ahead and get started!\"\n\n\"With what?\" Georgina asked.\n\n\"Look around,\" said Veronica. \"There may be some useful items lying about.\"\n\n\"Forget that,\" said James. \"I'm getting to work right now.\" He raced toward the house, pulling off his T-shirt off as he ran. He used the shirt to scrub the glass of the closest window. All that happened was that he smeared the goo around\u2014and got his shirt covered with the gunk.\n\n\"Idiot!\" George shouted at him. \"You're already losing us time. Help me look for something that will actually work!\"\n\nThe rest of us were already looking. I spotted a bucket hanging from a branch of one of the trees. A branch very near the top. Of course.\n\n\"I'll go get that,\" I said to Olivia in a low voice. I gave my chin a quick jerk toward the bucket. I didn't want to give away its location to anyone else.\n\n\"I'll help,\" Olivia volunteered.\n\n\"That's okay. I got it,\" I said quickly. I definitely didn't want any \"help\" from Olivia when I was that far off the ground!\n\nI didn't wait for her to answer. I took off in the direction of the tree\u2014not going straight for it.\n\n\"Remember that the windows of the upper stories do count!\" Veronica called.\n\nI guessed I also needed to look for a hidden ladder. And some sponges. That should get us started.\n\nI made a sharp turn toward the tree. A second later I was climbing. I could see the branches shaking below me. Somebody else had spotted the bucket. I glanced down. Olivia!\n\n\"I think I can get this one solo,\" I called down to her. \"See if you can find some cloths or sponges or something. Or something we can use for a ladder.\"\n\n\"I think I see a sponge by the bucket,\" Olivia answered.\n\nI squinted up at the top of the tree. Didn't see anything spongy. \"I'll get it while I'm up there,\" I promised.\n\n\"Do you see where I mean?\" Olivia asked. She kept climbing. So did I.\n\n\"Uh, I think so.\" Lie. Which made us equal. I was pretty sure Olivia was lying about seeing any sponges up there. Either she was trying to slow us down, or she was trying to get a chance to shove me out of this tree.\n\n\"Just go! I'll find it when I'm up there, I'm sure.\" I was pretty far up there already.\n\n\"Hal, get that mini tramp!\" Georgina shouted from below. \"George is a trampoline fiend! He'll destroy us on that thing, bouncing up to wash the second-floor windows.\"\n\nI took a glance down. Bad strategy. It was a good idea to nab something the opposition could use to a big advantage. But Georgina had basically announced the location of the tramp to her brother\u2014and he was a lot closer to it than either Hal or Georgina was.\n\n\"George! I saw it first!\" Georgina shrieked. Like that mattered.\n\nI got my attention back on my own prize and kept climbing. There weren't too many great branches in this part of the tree. I picked one I thought would hold me, and jammed my foot on it. . . .\n\nAt the same moment Olivia decided to reach up and use it as a handhold. She jerked the branch back, and my foot was suddenly slamming down into empty space. A second later, my entire body weight was hanging by one hand.\n\n\"Look out!\" Olivia yelled.\n\nLook out. Yeah. For psycho teammates.\n\nI managed to wrap my other hand around the branch, then haul myself up until I could straddle it. Don't doubt that I was hugging the trunk with both arms at that point. The branch I was perched on wasn't much stronger-looking than the one Olivia had managed to bend back.\n\n\"Hey, I see a pair of stilts. Over in the empty pool. We can use them to reach at least some of the higher windows. I've totally got the bucket and whatever else is up there,\" I told Olivia. \"Or you can keep climbing and I'll go for the stilts.\"\n\nIt would look totally suspicious if she thought we should stick together now. Another team could get the stilts\u2014and a big advantage. I knew she wanted our team to lose. But she didn't know I knew, and she didn't want me to know. Got it?\n\n\"I'll go,\" Olivia said. And she started climbing down. A lot more slowly than she'd gone up. I liked the new sabotage method. Slowness. No pain involved for Mr. Joe.\n\nI scrambled up the next couple of branches. They trembled under my feet, but without any assistance from Olivia, they stayed steady enough to stand on. I grabbed the pail and did a sponge check. None. Knew she was lying.\n\nAs I started down the tree, I did a check of the area below me. Olivia hadn't made it to the stilts, but George was already bouncing on the trampoline, trying to get high enough to reach the second-floor windows. James was using a big palm frond to scrape gook off one of the bottom windows on the first floor. For two guys who hated each other, they were a decent team.\n\nGeorge came down on the tramp, and I noticed it sagging. Sagging way more than it should have with just one person on it. \"George!\" I shouted as loudly as I could. \"Get off the trampoline. There's something wrong.\"\n\nHe didn't hear me.\n\nI moved from branch to branch faster. I took time for another fast look at George. Couldn't he feel that something was wrong with the trampoline? I leaped the last five feet down and hit the ground at a run.\n\n\"George!\" I yelled. \"Get off of that!\"\n\n\"We've got him worried!\" James called up to George.\n\nGeorge grinned, did a one and three-quarter somersault, landed on his back, then let the bounce take him to his feet. The bottom of the trampoline had almost touched the ground on the last bounce.\n\n\"Show-off!\" Georgina yelled.\n\nGeorge grinned wider and started to do the move again. A somersault. A three-quarter. But when he landed on his back, he didn't bounce. The trampoline didn't hold his weight, and he fell straight to the ground.\n\nAnd lay still.\n\n## Another One Down\n\nAnn grabbed her charm in both hands. So hard I could hear some of the twigs it was made of snapping. \"It's the demon curse!\" she cried.\n\n\"Come on, Ann. It was an accident,\" I told her. Honestly, I wasn't sure about the accident part. But I was sure a demon wasn't involved.\n\nI rushed over to where James was lying. Ripley was crouched beside him, most of the rest of the group gathered in a loose circle around them. Georgina had her fingers pressed to her lips as she stared down at her brother. His face was pale, his eyes only half open.\n\n\"George, can you hear me?\" Ripley asked.\n\n\"Yeah,\" George croaked out.\n\n\"Don't try to sit up,\" she ordered. \"Can you move your fingers and toes?\"\n\nShe'd clearly had some training. She was launching into the check for a spinal injury.\n\nI watched George closely. He was able to move his toes and fingers as soon as Ripley asked him to. Good.\n\nThen a not-so-good thought struck me. Ripley had been the first one to come to the rescue. The same way she had when Bobby T had his allergic reaction and when the dog attacked Joe. Was she just trying to make herself look like a hero on the show? If she got some good PR, her parents would keep supplying her with cash and clothes and, I don't know, whatever girls want.\n\nRipley didn't plan the allergic reaction or the dog attack, I reminded myself. Mitch had been behind both of those. But if Ripley felt like she needed even more good PR, would she create the situations that would let her get it?\n\nI ran my eyes over the trampoline. The thing looked brand-new. Which made sense. Deprivation House should spring for new equipment, even though they were trying to make us live with as little as possible. It was a safety issue.\n\nSo why hadn't the trampoline held George's weight? A ten-foot tramp like that one should easily handle someone his weight. I had another not-so-good thought. A thought I'd been having too often lately: sabotage.\n\nSUSPECT PROFILE\n\nName: Ripley Lansing\n\nHometown: Malibu, California\n\nPhysical description: 5'10\", 140 lbs., straight brown hair, blue eyes.\n\nOccupation: High school student\/heiress\n\nBackground: Only daughter of rock star dad and cosmetics company CEO\/owner mom. One older brother, one younger. The wild kid of the three.\n\nSuspicious behavior: Often the first one to reach an injured person, winds up being the \"hero.\"\n\nSuspected of: Sabotaging contestants so she can save them.\n\nPossible motive: Needs to improve her bad-girl public image or she'll be financially cut off until she's thirty.\n\nBut had Ripley had time for sabotage? Had she managed to find out ahead of time what the challenge was going to be\u2014or at least that the trampoline was going to be involved\u2014and loosen the springs or distress the canvas in some key places?\n\nOr could she have found the trampoline first, done some fast damage, and left it for someone else to find and hurt themselves so she could come to the rescue?\n\n\"Ripley, out of the way, please. Let the medic take over,\" Veronica called as she strode toward us.\n\n\"I've got it,\" Ripley answered.\n\n\"Yes, I'm sure a high school girl who spends more time at movie premieres and concerts has the equivalent of a medical degree,\" Veronica shot back. \"Move, please.\"\n\nRipley stood up and backed away.\n\n\"Is he going to be okay?\" Georgina asked.\n\n\"It doesn't matter,\" Ann muttered. \"It's the curse. Bad things are going to keep happening until we're all dead.\" She stroked the charm.\n\nIs it impolite to say that I liked her a lot more when she wasn't talking? At least she wasn't one of the rich kids. I didn't have to try and make it look like I was sabotaging her to keep Olivia and Gail off my back. Even if I did something really mild, it could send Ann over the edge. She was so freaked already.\n\nThe medic carefully checked George out. \"He looks good,\" she told Veronica. \"Maybe a little aspirin. Some ice for any sore spots.\"\n\n\"So I can stand up?\" George asked. He didn't wait for an answer. Just shoved himself to his feet with a grimace.\n\n\"That's all for today, everyone,\" Veronica announced. \"This competition is canceled.\"\n\n\"Unfair,\" James told her. \"George and I were clearly going to win.\"\n\nThere were all kinds of \"untrue's\" from pretty much everyone in response.\n\n\"The competition had just gotten started,\" Olivia protested. \"Just because you were slightly ahead\u2014\"\n\n\"Slightly! Did you say slightly?\" James interrupted.\n\n\"Why can't we\u2014\" Gail began.\n\n\"This is not up for debate,\" Veronica told us. \"The competition is over. We will have a new competition tomorrow.\"\n\n\"Can't we just start right where we were tomorrow? With a new trampoline?\" George asked. \"I'll be fine.\"\n\n\"Everything is not all about you, George,\" said Veronica. \"Although after reading your father's letter, I'm not surprised you think it is. The element of surprise was key to this competition. We can't regain that. There will be a new competition tomorrow. I have nothing more to say.\" She turned and walked inside the mansion.\n\nAs soon as she was gone, George faced off with James. \"This wasn't enough for you?\" he asked, touching the bruise on his jaw.\n\n\"What are you talking about?\" James demanded.\n\n\"There is no way that trampoline wasn't messed with,\" said George. He started to crouch down to inspect it, but his sister got there first.\n\n\"Definitely messed with,\" Georgina announced. \"The canvas is cut away from the springs. About every other one.\" She walked over to her brother and stood close to him. \"You know what? I'm starting to think there are people here who aren't happy they got handed some new competition this late in the show. First I get a bike with death brakes. Then you get the killer trampoline.\"\n\nGood theory. Except how could anyone know that George was going to end up on the trampoline? I didn't say anything. It seemed like one of those situations where listening was the best strategy.\n\n\"I'm satisfied with the spanking I gave you last night,\" James told George. \"If you think I care more about handing down a little more punishment to some rich kid than winning a million dollars, you're insane. I would never betray a teammate. Never.\"\n\nYou know what? I believed him. James is competitive to the nth degree. I couldn't see him doing anything to hurt his chances of winning, no matter how much he hated somebody.\n\n\"He couldn't have known that George would end up with the trampoline, anyway,\" Hal put in. \"And James couldn't have sabotaged it once George had brought it over to the mansion to start on the windows. George would have seen him.\"\n\n\"Thank you, Hal,\" said James.\n\n\"Not taking sides,\" Hal muttered, eyes on the ground. \"Just being logical.\"\n\nHis logic matched mine.\n\n\"So somebody sabotaged the trampoline without caring who would end up being the victim,\" Olivia said. The girl was all about strategy. She was good at putting herself in the mind of the perp. Because she was the perp? And trying to work some fake-out on the rest of us by bringing it up?\n\n\"It's the demon!\" Ann burst out.\n\n\"The cuts are pretty neat,\" Ripley observed. \"They don't look like they were made with huge, hideous claws.\"\n\n\"You people have to stop mocking! Is somebody going to have to die before you stop?\" Ann cried. She dashed into the house.\n\n\"Maybe freak girl snapped and did it herself,\" James said.\n\n\"She's not a freak,\" Brynn told him.\n\nJames raised one eyebrow. \"What would you call her?\"\n\nBrynn hesitated, then shrugged. \"What I'm saying is, everybody's afraid of something.\"\n\n\"You know what I'm afraid of?\" Ripley asked. We all looked at her, waiting. \"I'm afraid of what's going to happen at tomorrow's competition.\"\n\n\"Good afternoon, and welcome to the replacement competition,\" Veronica said to us the next day. She'd had us all gather in the dinning room.\n\n\"What's with the chef's hat?\" James called.\n\nVeronica adjusted the tall white hat she was wearing. It was an odd match with her pale blue suit and spike heels. Even I've absorbed enough about fashion to know that.\n\n\"If you had better manners, you'd already know,\" Veronica told him. \"I was about to explain, before you interrupted, that today's competition involves cooking skills.\"\n\n\"Are we still having teams?\" Olivia asked.\n\nVeronica let out a small sigh. \"Yes. Perhaps you and James should ask some of your other housemates to review what they learned in charm school. I'm sure some of them were sent,\" she answered.\n\n\"Same teams as last time?\" George wanted to know.\n\n\"Even if it seems that they learned nothing,\" Veronica added. \"Yes, same teams. No more interruptions, please. Or I will simply cancel today's competition as well and choose the next deprivation myself.\"\n\nThe room went silent.\n\nVeronica smiled. \"Today you will be making pies. And, as a special treat, you will be able to eat the pie you make. I know many of you have missed your desserts.\"\n\nNobody spoke. I think everybody was still afraid Veronica would take over choosing the deprivation. She chose a lot of them anyway. We didn't only get stuff taken away after a competition. But with a competition, there was a chance you'd win and get to pick something to lose that you didn't care about too much.\n\n\"You'll find everything you need\u2014or at least everything you may use\u2014out on the back lawn,\" Veronica concluded. \"Go!\"\n\nWe went, racing through the house and out the back doors onto the sloping lawn. There were six tables with pie-making ingredients laid out on them.\n\n\"Is there strawberry? I would kill for a strawberry pie!\" Ripley exclaimed. \"Not really, you guys! You know what I mean.\"\n\nI didn't care about flavor. I just wanted to get started. I dashed to the closest empty table. Apple. \"This okay?\" I asked Ann.\n\n\"I don't care.\" She'd added a second charm. Her face was pale, and there were dark smudges around her eyes.\n\n\"George, you know I love peach. Come on! Chose another table,\" I heard Georgina cry.\n\n\"Fine, fine. I'm not getting between you and your stomach,\" her brother answered. \"Then I would end up dead.\"\n\n\"Why is everyone joking about it?\" Ann's voice cracked as she asked the question.\n\n\"Because they're scared too,\" I told her. \"It's a different way of dealing. You want to find the recipe for apple pie in there?\" I nodded toward the cookbook. Then I realized there were no measuring cups, no mixing bowls, nothing to stir with, no baking pans.\n\n\"What are we\u2014? Are we supposed to use our hands or what?\" Olivia called out.\n\n\"Finally, an intelligent question.\" Veronica had followed us out to the lawn at a slower pace. \"I'm sure you're all familiar with the green movement. Around the corner, in the formal garden, you'll find a variety of materials that, if recycled creatively, should provide you everything you need.\"\n\n\"Can we use the ovens in the house?\" asked Georgina. I hadn't even thought about ovens.\n\n\"No,\" Veronica answered. Just no. With one of her Veronica smiles.\n\n\"Guess we'll have to build a fire,\" I told Ann. \"I can deal with that. I was a Boy Scout.\" And a current ATAC agent with wilderness survival training. \"But first we should grab supplies before all the good stuff is gone.\" I followed the group running for the garden.\n\n\"It looks like they moved a whole garbage dump over here,\" Brynn commented, staring at the mound of . . . who even knew what. Whatever it was didn't stink, so at least it was clean garbage.\n\nOlivia started climbing straight up the heap. \"I'll find stuff. You stand guard over the pile we make,\" she called to Joe. \"We can definitely use this!\" She picked up a glass goldfish bowl and hurled it at Joe. If it had hit him in the head . . . But he caught it.\n\nI realized I was just standing there, keeping an eye on little brother. He could take care of himself. And Frank Dooley wouldn't be worrying about Joe Carr. Frank Dooley would be worrying about winning.\n\nSo I began circling the pile. I didn't think climbing on it was the best strategy. I figured it would be best to spot useful stuff from the ground. I did an Ann check. She had an iron poker in one hand. Okay, might be good for stirring if we were desperate. Could help with the fire. She was all right. I hadn't been sure she was going to be able to keep it together, for the competition or anything else. But she seemed all right.\n\nI gathered up a couple of Frisbees and some foil. Combining them might make pie pans. They'd be better as pizza pans, but hey. I grabbed a plastic doll head that looked like it would hold just about eight ounces. So there was our measuring cup. I took a wooden bird, a wooden bat, and a wooden chair. That would be more than enough for fuel.\n\n\"I'm making a trip back to the table,\" Ann said. \"I've got as much as I can carry.\"\n\n\"Looks like you found a bowl-like object.\" She had a plastic football helmet under one arm. \"Want to take my measuring cup and start mixing? I want to find a couple more things, then I think we're good.\"\n\n\"Sure,\" Ann answered. I tossed her the doll head, and she actually almost smiled. I turned my attention back to the garbage heap.\n\n\"I call that ceiling fan,\" shouted James.\n\n\"You can't call. There's no calling!\" Ripley shouted back. She was closer to the fan and grabbed it.\n\n\"Okay, keep it. I call that wagon wheel.\" Ripley got there first too. She staggered under the wheel's weight as she hauled it out from under a Styrofoam reindeer.\n\nJames laughed as she started rolling it back toward the tables.\n\n\"You didn't want that, did you?\" George asked.\n\n\"Nope,\" James answered. \"Didn't want the fan, either.\"\n\n\"My partner,\" said George admiringly. They slapped a high five.\n\n\"See, I am a good guy to\u2014\"\n\nHe was interrupted by a shrill scream of agony.\n\n## Targeted\n\nA girl screamed again. The sound sent a bolt of electricity through my body. I half scrambled, half slid down the pile of junk and raced toward the tables. That was the direction the scream had come from.\n\nI rounded the corner of the mansion, Frank and a couple of the others right behind me. Immediately, I saw Georgina sitting on the ground with both hands pressed over her mouth. She rocked back and forth in pain.\n\n\"Milk!\" Ripley yelled from her spot at Georgina's side. \"Somebody go get her some milk! She's been poisoned. We need to dilute it.\"\n\n\"I'll get it.\" Brynn started for the house at a run.\n\n\"Wait,\" cried Gail. \"Different kinds of poisons need different treatments. We could end up hurting her. Just wait. The medic is going to be here in a second.\"\n\n\"What happened?\" I cried, skidding to a stop in front of them.\n\n\"The demon!\" Ann shouted. She stood at her own table, not taking a step closer to the rest of us. \"The demon did this.\"\n\n\"Georgina ate a piece of peach from her table and it burned her tongue,\" Ripley answered. \"Show them.\"\n\nGeorgina lowered her hands. She let out a whine as she opened her mouth. Think of the worst burn you've ever gotten from eating pizza when you're too hungry to let it cool enough. One of those hot, oily cheese burns that takes off a layer of skin. Then triple that.\n\nEven standing several feet away from her, I could see that Georgina's tongue looked like raw meat. Oozy and puffy.\n\n\"That definitely looks like a corrosive poison,\" I said. I wasn't sure Joe Carr would have that kind of knowledge. He was more the guy who could tell the price of a pair of sunglasses with one glance. But this was important. And Ripley, who was the most privileged of us privileged rich kids, obviously knew something about first aid. \"That means we definitely don't want to make her throw up.\"\n\n\"Yeah, I accidentally drank some toilet bowl cleaner once,\" George said, staring down at his sister. \"They said throwing up would wreck my throat.\"\n\n\"How do you accidentally drink toilet bowl cleaner?\" asked James.\n\n\"I was five. I liked the color. I guess I drank it on purpose. But I didn't, you know, drink poison on purpose,\" George answered.\n\n\"Here's the milk!\" Brynn called.\n\n\"Don't give it to her,\" I said. \"I think there are some poisons where you shouldn't give the person anything.\" Actually, I was positive. \"We don't know what she took. We've got to wait for the medic.\"\n\n\"I'm here,\" the medic called. \"Back up and give me some room.\"\n\nWe all backed up a little, but stayed close. Except Ann, I mean. She still hadn't moved closer to the group.\n\n\"What did you take?\" the medic asked Georgina.\n\n\"Noting,\" Georgina answered, her voice thick and distorted. \"Ust peach.\" She flapped her hand toward the table where she and Hal were going to make their pie.\n\nHold up. Hal. Had Hal poisoned the peaches? From what Frank told me, Gail and Olivia had leaned on Hal pretty hard. They'd pretty much told him that if he got a rich partner and didn't do what he had to do to make sure the rich kid lost, they'd hang him out to dry.\n\nSUSPECT PROFILE\n\nName: Hal Sheen\n\nHometown: Coshocton, Ohio\n\nPhysical description: 5'8\", 150 lbs., straight brown hair, brown eyes.\n\nOccupation: High school student\n\nBackground: Mensa member, statewide science fair winner three years in a row.\n\nSuspicious behavior: Was Georgina's partner when she was poisoned.\n\nSuspected of: Sabotaging contestants who aren't part of the alliance.\n\nPossible motive: Threatened by Olivia to do sabotage for good of the alliance.\n\n\"I want to get Georgina to a hospital ASAP,\" the medic told Veronica. \"I don't think she ingested much of the substance. The abrasions don't continue deep into her throat.\"\n\n\"Pit it ou,\" Georgina said.\n\n\"She spit it out,\" George translated for his sister.\n\n\"Even so,\" said the medic. \"I don't want to take any chances.\"\n\n\"I've already called an ambulance,\" Veronica announced.\n\nAn ambulance. Again. Bobby T had needed an ambulance before he'd been booted from the show.\n\nAt least he'd come back alive. We'd had a couple of people, both crew members, leave Deprivation House in silent ambulances. No sirens because there was no rush. The patients inside were already dead.\n\nToday's ambulance ride wouldn't be one of those. But that didn't mean we'd had our last.\n\n\"What have we got?\" Frank asked. We were back in our \"office\" late that night. I'd managed to call toilet seat again. Yes! The edge of the bathtub is hard. And cold.\n\n\"We've got a poisoning, the sabotage of a trampoline, sabotaged brakes, a drawing of a demon, and a partridge in a pear tree.\" I sang that last part. Frank didn't even smile.\n\n\"All that since Mary was removed from the house.\" Frank shook his head.\n\n\"And the perps keep on comin',\" I said.\n\n\"Do we think this is the work of one person?\" Frank asked.\n\n\"Usually I'd say it would be really unlikely that more than one person in our little community had turned homicidal,\" I answered. \"But since we've already had to oust two wackos . . . Who's to say there's only one more?\"\n\n\"Hal seems like a possibility,\" said Frank.\n\n\"I was thinking the same thing. Do you think he would have gone that far to keep Olivia and Gail happy?\" I replied.\n\n\"They basically made it a choice between keeping them happy and losing the chance to split the million dollars with them,\" Frank mused. \"Hal really wants start-up money for his game.\"\n\n\"Poison, though. He could have done something a lot less hard-core, right?\" I asked. \"I mean, Olivia threw a goldfish bowl at my head and tried to help me out of the top of a tree. He could have done something medium-core like that.\"\n\n\"I agree,\" Frank answered. \"He's a possibility, but not top of the list, I don't think. Also, if he is behind the poisoning, then we definitely have two people at work. There would be no reason for Hal to seriously hurt George. That wasn't anything Olivia and Gail were expecting from him.\"\n\n\"No reason for him to have sabotaged Georgina the first time either. Olivia and Gail hadn't threatened him yet.\" A new thought hit me. \"Or else Hal's a lot more sly than we're giving him credit for. Maybe he wanted to take some of the competition out all along. Maybe he was working on it before Olivia and Gail even brought it up. Maybe he damaged the dirt bike's brakes, and tried to poison Georgina, and messed up George's trampoline.\"\n\n\"Why not just admit he sabotaged the brakes when Olivia brought up sabotage?\" asked Frank.\n\n\"Maybe he didn't want to give them any info to use against him. Maybe he's planning on playing the alliance somehow,\" I suggested. \"Maybe he even did the demon drawing. He's always carrying around a sketch pad, and the practice handwriting you found was on sketch paper.\"\n\n\"But Gail and Olivia were trying so hard to keep their sketch pad hidden from me. Why do that, if there was nothing to hide?\" Frank asked.\n\nI shrugged. \"Maybe they were trying to hide something else. Like . . . that they'd been drawing pictures of me! Yeah, they didn't want you to know that they think I'm the more handsome brother. They thought it would hurt so bad, since I'm already the rich one.\"\n\n\"Yeah, I'm sure that's it.\" Those were the words Frank said. But somehow they came out sounding like, Only in your delusional skull, my brother.\n\n\"I'm just saying that there's more than one reason to keep the sketch pad hidden. It doesn't mean they wrote the note or did the drawings,\" I clarified. \"So, we got Hal. Who else?\"\n\n\"James. The guy wants to win. We know that. He never lets anyone forget it,\" Frank answered.\n\n\"I think we'd need a two-perp scenario with James, too. Because I really don't believe he'd sabotage his teammate. Not that he cares about sportsmanship or anything like that. But, like you said, he wants to win,\" I said.\n\n\"Olivia and Gail are clearly good suspects. All the victims we've had since Mary left have been rich. Two attacks on Georgina, one on George. It was Brynn's teddy bear that got shredded, and maybe it was even supposed to be Brynn in the picture.\" Frank stood up. I could tell he wanted to do some pacing, but there wasn't space in the bathroom.\n\n\"I never even think of Brynn as being one of the rich kids,\" I said.\n\n\"I don't either. Olivia was saying stuff about her shoes. I guess they're expensive. I wouldn't know,\" Frank admitted.\n\n\"Don't look at me,\" I protested. \"I know my shades, yeah. And sneakers. But that's about it.\"\n\nFrank sat back down. \"So Olivia and Gail may be working together. Maybe either Olivia or Gail is working separately. Gotta say, Gail seems less likely. She seems to be a follower type.\"\n\n\"I'm with you. I think Olivia should be at the top of our list,\" I agreed. \"And we should be thinking about Ripley, too. She's been right there next to two of the accidents. Giving instructions on how to take care of Georgina. Checking George for spinal injuries. She wasn't right on the spot when Georgina crashed, but that would have been hard to accomplish. You and Georgina were flying. Ripley was near the back of the pack.\"\n\n\"Georgina. George. Georgina,\" Frank said slowly.\n\n\"It seems like their dad has kind of the ego,\" I commented. \"He basically named both his kids after himself.\"\n\n\"Georgina. George. Georgina,\" Frank said again.\n\n\"What about Ann? You were partnered with her. Has she snapped or what? Do you think she could\u2014\" I began.\n\n\"George thought maybe somebody was going after him and his sister because they were the new people,\" Frank interrupted. \"It is kind of strange that they've been targeted repeatedly.\"\n\nI tapped my head. Sometimes it helps me shake thoughts loose. Didn't work so much this time. \"None of the people we've talked about have a reason to focus on George or Georgina. Yeah, they're rich, and that makes Olivia and Gail insane. But there are other rich people around.\"\n\nI did another couple of taps. A thought came loose. \"And anyway, even though it seems like the G's were targeted, they couldn't have been.\"\n\n\"Why not?\" Frank asked.\n\n\"Because no one knew it was going to be George who ended up on the trampoline or Georgina who would end up with the peaches and the bad brakes,\" I explained.\n\nWe both sat in silence for a moment.\n\n\"You remember how you said you'd know how to kill me because I'm your brother?\" asked Frank.\n\n\"I don't think I said it exactly like that,\" I objected.\n\n\"Okay, but you said something like you'd make it look like an accident involving cleaning projects, because I like to clean and it wouldn't look suspicious,\" Frank went on.\n\n\"How did we go from talking about George and Georgina to talking about me and you?\" I asked.\n\n\"They're siblings, just like us,\" Frank answered.\n\n\"Not just like us. One of them is a girl,\" I said. \"Oh, well, maybe just like us.\"\n\nFrank rolled his eyes. \"What I mean is, you know the people you're related to really well.\"\n\nAnother thought came loose, and I didn't even have to tap my head. \"Like that your sister loves peaches. And the color purple. There was only one purple dirt bike, and she went right for it.\"\n\n\"Georgina would know what her brother loves too. Like trampolines,\" Frank added.\n\n\"When I was clinging to the tree like a scared monkey in the challenge yesterday, I heard Georgina yell for Hal to get the trampoline so George couldn't. I thought she was being stupid, because George was closer to the tramp than she or Hal was\u2014and she'd basically just pointed it out to him,\" I said. \"What if that's what she was trying to do?\"\n\n\"George led Georgina to the peach table in a way. He got there first. I heard Georgina begging him to choose another table because he knew she loved peach,\" Frank told me.\n\n\"So that places him near the dish of peach slices. He had the opportunity to poison them,\" I answered.\n\n\"And George even said he drank toilet bowl cleaner when he was little,\" Frank reminded me. \"He probably wasn't trying to kill Georgina, just take her out of the competition. They both want to get legally emancipated. Only one of them can win the money. . . .\"\n\n\"They've made it clear they aren't sharing,\" I added. \"I think we need to do an evidence sweep. I call heads.\"\n\n\"What are we calling for?\" asked Frank.\n\n\"Heads, I get to search George's stuff,\" I explained.\n\n\"Which means the loser has to search Georgina's,\" Frank said. \"Which means sneaking into one of the girls' bedrooms in the middle of the night.\"\n\nIt's probably a good thing I lost. If Frank had to sneak into a bedroom full of sleeping girls, he might lose his lunch. And the hurling sounds would wake them all up.\n\nI had a few butterflies myself as I softly swung open the door to Brynn, Olivia, and Georgina's room. It wasn't so bad I felt like I was going to puke them up, though.\n\nIf the cameras are on, this will probably be on the show, I thought. Won't Mom and Aunt Trudy be proud? I guessed I could tell them it was all part of the socioeconomic experiment. Frank and I'd given this story about how we were going to use being on the show as a project for school. It was the best explanation for why we'd be on TV with different names, pretending to be from different families. I guessed as part of the experiment one of us would have to like going through girls' personal belongings while they were asleep.\n\nI dropped to my knees and crawled into the room. I doubted any of the girls were awake, but this way I'd be harder to spot, just in case. I crept over to Georgina's bed and slowly pulled opened the bottom drawer of her dresser. It would be stupid of her to leave any evidence in there. But it's not like Georgina was a professional.\n\nClothes. Of the underwear kind. That's all I've got to say about it. I searched, trying not to touch anything.\n\nNo evidence.\n\nNext drawer. More clothes. Safe to touch.\n\nNo evidence.\n\nI was out of drawers. I should check the pockets of whatever clothes she had hanging in the closet. But first, I slid the suitcase out from under her bed. Not quite empty. I felt something shift inside.\n\nThe zipper sounded as loud as machine-gun fire as I slid it open. I paused and listened. All I heard was breathing and a little snoring from somebody. Good. No one had woken up.\n\nI lifted the lid of the case. A Teen World magazine. Oooh, forbidden. We'd already been deprived of all reading material. And that was it.\n\nNo evidence.\n\nI started to slide the suitcase back under the bed.\n\n\"Don't,\" a girl muttered.\n\nI froze.\n\n\"Please, don't. Don't hurt me. Please.\"\n\nI recognized the voice.\n\n\"Brynn, it's me, Joe,\" I said softly. \"I'm not going to hurt you. It's just . . . a stupid prank the other guys and I came up with.\"\n\n\"Don't!\" Brynn cried, her voice higher.\n\n\"I'll get out right now.\" I stood up. \"I'm going. See?\" I scanned the beds until I found Brynn.\n\nShe was asleep. She's having a nightmare, I realized. I also realized my heart was beating like somebody was using it as a bongo. Hard and fast.\n\n\"Please,\" Brynn said again. Then she rolled over on her side and went silent.\n\nAnn wasn't the only one who'd gotten freaked by what had been happening in the house. Clearly Brynn had too. Who was I kidding? The house was giving me the wiggins. The only one it wasn't bothering was the person\u2014or people\u2014behind all the badness.\n\nAnd that's why I was in here. To get some evidence about exactly what was really going on in Deprivation House.\n\nI crouched down and did another check under the bed. Other than the suitcase, the only thing I spotted was a pair of sneakers. Well, I wasn't necessarily looking for something big.\n\nI had to get down on my stomach and stretch my arm out until I practically dislocated my shoulder, but I was able to get the shoes without waking up Georgina. Right shoe.\n\nNo evidence.\n\nLeft shoe.\n\nSomething. Something smooth and cool. I slid it free. Did you know they make pink Swiss Army knives? Because that's what it was. I began opening the blades. The first one was rough all down one side, and not very sharp.\n\nIt took me a second to get it. Because of the Y chromosome. It was a nail file. The next blade was regulation. I cautiously ran one finger down the flat side\u2014and picked up a thread.\n\nI bagged it, put the Swiss Army knife back where I found it, and crawled out of the room. Frank was waiting for me in the hall. \"Let's go to the bathroom,\" he said.\n\nI could tell from his face that he'd found something. \"Look at this,\" he told me as soon as the bathroom door was safely closed behind us.\n\nIt was a baster from the kitchen. \"Smell it,\" Frank urged.\n\nI pulled off the squeeze top and sniffed. Something harsh and chemical. Something I suspected was the corrosive poison that had burned Georgina's tongue.\n\n\"I got something too. Off the blade of a knife I found in one of Georgina's shoes.\" I handed the small plastic bag to Frank.\n\n\"Looks like a piece of canvas thread,\" he commented. \"Think it'll match the trampoline?\"\n\n\"Yeah,\" I answered. \"Think Veronica will care that her contestants are trying to kill each other?\"\n\n## Twisted Twins\n\n\"What am I to do about these?\" Veronica tapped one crimson fingernail on her desk. The desk where the baster and the plastic bag lay.\n\nJoe and I exchanged a glance. Wasn't it obvious what she was to do? Hand George and Georgina over to the police. Immediately.\n\n\"I wasn't trying to hurt Georgina,\" said George.\n\n\"That's true,\" Georgina jumped in. \"George drank a mouthful of the same stuff, almost. His was toilet cleaner. He gave me drain cleaner. But he knew I'd be okay. And the dirt bike\u2014I ride hard. I'm always taking spills. Wiping out isn't anything.\"\n\n\"It's like how Georgina knows I've taken falls off a trampoline before,\" George added quickly. \"She probably just thought I'd break a leg or something. Worst case.\"\n\nGeorgina nodded hard. \"And he's always breaking stuff. He's already broken his arm twice snowboarding.\"\n\nThis was bizarre. Suddenly the two of them were a team.\n\n\"So you're admitting it?\" Veronica asked.\n\nBoth George and Georgina's mouths dropped open. It's like neither of them had realized they had an option. Lying, for example. Something like, \"I have no idea how that (fill in the blank) got in my (fill in the blank). Someone must have put it there.\"\n\n\"I\u2014\" George began.\n\n\"We\u2014\" Georgina started.\n\n\"So you're admitting it.\" Veronica cut them off. It wasn't a question this time.\n\n\"But I won't press charges!\" Georgina exclaimed triumphantly. \"And neither will George. So there's nothing you can do to us.\"\n\nGeorge grinned. \"I would never press charges against my twin.\"\n\nVeronica shook her head. \"Again, I'm staggered by the knowledge a private school education provides.\"\n\n\"Yeah. So do we get to stay on the show? Since we aren't prosecuting each other?\" George asked.\n\n\"If this gets out\u2014and I am certainly strongly considering leaking it to the police at the earliest opportunity\u2014it will be the district attorney who decides whether or not you will be prosecuted. And with this evidence\"\u2014again Veronica's nail tapped her desk\u2014\"I'm certain you both would be.\"\n\n\"Do you want money?\" asked Georgina. \"Call our dad. He'll give you whatever you want.\" She looked over at me and Joe. \"You guys too. This was no big. Just me and George messing around.\"\n\n\"You know what I think this is?\" Veronica said. \"I think this is truly compelling television, which means you'll both be allowed to stay at the house.\"\n\n\"That's it?\" I protested.\n\n\"That's it,\" Veronica answered. \"I appreciate you boys bringing this to my attention. And I assure you, if anything like this happens again, I will personally take George and Georgina directly to the police.\"\n\nShe flicked her hands at us. Clearly, we were dismissed.\n\n\"I can't believe she didn't throw you out of here!\" James yelled at George and Georgina as soon as we stepped out of Veronica's office. \"If I had done that, I'd be in jail already.\"\n\nThe whole group was gathered in the hall. Staring at the twins.\n\n\"You did the drawing, too, didn't you?\" Brynn demanded. \"The one of the demon? And mutilated my teddy bear?\"\n\n\"No way!\" exclaimed George. \"Georgina's the only person I wasn't sure I could beat.\"\n\n\"Yeah, I wanted to get George out of the running. I'm not worried about you guys,\" Georgina added.\n\nOne of the production assistants hurried up to the twins. \"Veronica wants both of you to have a session in the Deprivation Chamber. She wants to get all your thoughts and feelings on tape\u2014the two of you together.\"\n\n\"I'm next in line,\" James said as the three of them hurried away. \"I have some things I want recorded too.\"\n\n\"How did you guys find out what happened so fast?\" Joe asked.\n\nExactly what I was wondering. Joe and I had gone straight to Veronica in the morning, and she'd pulled the twins into her office while we were sitting there. No one had had time to talk to the rest of the group.\n\n\"Oh, you'll love this,\" Brynn answered. \"They put the plasma back and told us we were getting a special treat. We were being allowed to watch TV! And what comes on? Closed circuit of you and the psychos in Veronica's office. Of course, the camera crew was right in our faces. They wanted to get some good close-ups of our reactions.\"\n\n\"It really was truly compelling television,\" Olivia snarked.\n\n\"I think we should be asking you two how you found out what was going on with the twins so fast,\" James commented. He looked back and forth between me and Joe. \"You guys fingered Mitch, too. You knew what was going on with him before the rest of us. What I want to know is how.\"\n\nOlivia's eyes narrowed. \"James is right\u2014for once. There's something off here. You two didn't even know each other before you got to the house. How come you keep teaming up? Have you made some kind of deal? Are you teaming up to win or what?\"\n\nOlivia had alliances on the brain.\n\nJoe laughed. I have to admit, he does a good fake laugh. Mine always sounds a little . . . fake.\n\n\"Me team up with him? The guy hates me, just because I ended up with a family who has bucks. All that garbage Mitch pulled? I thought my brother over there was behind it. I started watching him. It turned out he was just as suspicious of me.\"\n\n\"Yeah. I was keeping an eye on you. I didn't like you,\" I said to Joe. \"I still don't, just for the record.\"\n\n\"Anyway, we were both watching each other, and we ended up catching Mitch,\" Joe explained. \"Then when more bad stuff started going down, I figured he\"\u2014Joe jerked his head toward me\u2014\"could probably help me figure out who was behind it. He was decent at it the last time.\"\n\n\"We decided to work together\u2014just on finding out who was pulling the sabotage,\" I said. I turned to James. \"Happy now?\"\n\n\"Whatever,\" James muttered.\n\n\"Nice attitude,\" I said.\n\n\"I'm going to go get breakfast,\" Olivia announced. \"I get headaches if I don't eat, and I want to be ready for whatever Miss Veronica decides to throw at us next.\"\n\nThe rest of us followed her into the dining room. The cereal bowls and silverware had already been set out by the morning's assigned cooking crew. Not that there was much actual cooking going on anymore, since we no longer had hot food privileges.\n\nBrynn, Gail, and Hal brought in the boxes of cereal and the milk, and all they had left to do was handle the dishes. Olivia, Ann, and I were scheduled to make lunch. It would take about ten minutes to slap peanut butter and jelly sandwiches together for everyone.\n\n\"Those two really should have gotten kicked out. Who cares that they only tried to hurt each other? They cheated!\" Ripley exclaimed as she shook some cereal into her bowl.\n\n\"Hang on. I thought you cared about everybody, princess,\" said James. His bowl was already full. You had to be fast to get food before that guy. \"Isn't that your new PR image?\" He grabbed the milk.\n\n\"Being nice doesn't mean you have to like everybody or that you're an idiot,\" Ripley exclaimed. \"Are you saying I'm being mean to feel like cheaters should be thrown out of the game?\"\n\n\"You're not in it to win it anyway, right?\" Olivia asked. \"You don't need the money. You have multiple millions, even though people are starving. And not just in other countries. Right here\u2014\"\n\n\"I get it. You hate me because I'm rich. You hate everybody who's rich. But obviously you want to be rich yourself, or you wouldn't be here trying to win a million dollars,\" Ripley snapped. I noticed a muscle in her neck twitch. Then she forced a smile in Olivia's general direction. \"But everyone is entitled to their own opinion, right?\"\n\n\"Right,\" said Hal softly.\n\nI wondered if he was thinking about his opinion that sabotage was unacceptable.\n\n\"I think we should ice them out. Not deal with them at all. They can cook for themselves, talk to themselves. I don't want to even look at either one of them,\" Gail said, taking the subject back to George and Georgina.\n\n\"What they do doesn't matter to me\u2014as long as they keep doing it to each other,\" Brynn commented.\n\n\"But they've already done stuff to you,\" Olivia reminded Brynn. \"That drawing of you with the monster. The teddy bear massacre.\"\n\n\"They said they didn't do that,\" Brynn told her.\n\n\"And you believed them?\" exclaimed Olivia.\n\n\"George admitted he poisoned his sister and sabotaged her brakes,\" Brynn said. \"Georgina admitted she did something that could have paralyzed her brother for life. I think if they'd gutted a stuffed animal, they would have confessed.\"\n\n\"Then who did do that other stuff?\" asked Gail.\n\nWas she putting on an act? Or had she and Olivia really been acting so weird about the sketchbook for some reason that had nothing to do with the investigation?\n\n\"The demon! It was the demon!\" Ann shouted. She jumped to her feet so fast she knocked over her chair. \"Every time one of you doubts its existence, it does something worse!\"\n\nI didn't bother asking her why a demon would use crayon drawings and words on a wall to scare people when it was a demon. Logic wouldn't work on Ann. I'd finally accepted that.\n\n\"Whoever did it\u2014including the demon,\" Olivia added when Ann gave her a death look, \"was doing stuff before the twins even got here. Mary said she didn't put glass in the ice.\"\n\n\"Whatever. I'm not worried about\u2014\" James began.\n\nAnn pointed her finger at him. \"Don't say it!\" Her whole arm trembled.\n\n\"Demons,\" James mumbled into his cereal bowl as he lifted it to drain the milk at the bottom.\n\n\"What's that?\" asked Hal.\n\n\"What's what?\" James answered.\n\n\"There's something stuck to the bottom of your bowl,\" Hal told him.\n\nJames felt around, then pulled a piece of paper off the bowl. He stared at it for a couple of seconds, then slapped it down in the middle of the table so we could all see it.\n\nEVERYONE DIES AT DEATH HOUSE. The words were written in the same childlike handwriting as the message on the Deprivation Chamber wall and the piece of paper I'd found on the floor of the billiard room\u2014the one that read HOUSE OF DEATH.\n\n\"That wasn't there when we set the table,\" Gail said.\n\n\"You looked at the bottom of each bowl?\" Joe asked.\n\nGail frowned. \"No,\" she admitted.\n\n\"I took the bowls out of the dish drainer,\" Hal told the group. \"I would have seen the note then if it was there. The bowls were drying bottoms up.\" He stood. \"I'm using the intercom to get Veronica.\"\n\n\"You're dropping out?\" James asked. \"Sweet.\"\n\n\"No. But I want to see the film of the kitchen and the dining room. From when I took the dishes out of the drainer until right now,\" Hal said.\n\n\"I want to see it too,\" Gail agreed.\n\n\"I'm going.\" Hal strode out of the room.\n\nI studied the faces of everyone around the table, trying to decide if anyone looked nervous. Like they were about to be exposed. The thing was, everyone did look kind of nervous. Joe did. I probably did too. Nervous was getting to be the usual state for everyone in the house. Brynn was so pale even her lips had mostly drained of color. She had her eyes locked on the note.\n\n\"Hey, Ann, you'd know the answer to this. Can you see demons on film?\" James asked. \"Or are they like vampires?\"\n\nAnn let out a shriek of rage and hurtled out of the room.\n\n\"Just leave her alone, all right?\" Joe said.\n\n\"No. It's not all right,\" James answered. \"I don't like being told what to do by you.\"\n\nIt looked like another fight might be about to start up, but Hal interrupted by walking back into the dining room, followed by George and Georgina.\n\n\"Well?\" Olivia asked.\n\n\"She said they weren't taping,\" Hal answered. \"It was part of the off hours.\"\n\n\"Uh-uh. No way.\" Ripley shook her head. \"Veronica would have to know we'd be talking about those two.\" She jerked her chin toward George and Georgina. \"She'd want to get all that on film.\"\n\n\"Yeah. Why would she choose now to kill the cameras?\" Joe asked. \"There's plenty of time when we're all sitting around staring at the walls. I watch a ton of reality shows. They never leave out the part where the whole group is hating on a few people.\"\n\n\"You were bad-mouthing us?\" George demanded.\n\n\"Ya think?\" asked Gail.\n\n\"This proves Veronica and the producers did the drawing and the notes. Has to be,\" Olivia said. \"That's why she said there was no film. The film would have shown a PA sticking that piece of paper on the cereal bowl. I'm positive. They're using the scary history of the house as part of the show.\"\n\n\"Half the time they lie to people about what the reality show they're on really is,\" Joe added. \"Maybe we're in Demon House, not Deprivation House.\"\n\n\"If that's true, I'm suing. I'll get more than a million,\" said James. \"Brynn, hand me the rice puffs. I'm ready for seconds.\"\n\n\"I guess it does make sense that Veronica is behind the drawing and note,\" Hal commented. \"It's way too coincidental that they weren't filming at the exact time when we knew something had happened.\"\n\n\"Brynn! Cereal!\" James commanded.\n\nI looked over at her. She was still staring at the note. I leaned across Gail, took the cereal box that was in front of Brynn, and handed it to James. \"Hey, want to go outside, Brynn? Are you finished?\" I asked.\n\nShe blinked a couple of times but didn't answer.\n\nGail touched her arm, and Brynn gave a jerk. \"Frank asked you something,\" Gail said. \"And James issued an order, which you were right to ignore.\"\n\nBrynn turned to me. \"Sorry. What did you say?\"\n\n\"I was just thinking if you're finished we could take a walk or something,\" I told her.\n\n\"Awww,\" James cooed.\n\n\"Sure.\" Brynn grabbed her light blue jacket off the back of her chair and hurried out of the room. I followed her. We ended up sitting next to the fountain in the courtyard.\n\n\"Getting intense, huh?\" I asked.\n\n\"Yeah.\" Her voice was flat, her eyes glassy.\n\n\"Should we make this a no demon, no murder zone?\" I suggested. \"No talking about any house stuff.\"\n\nBrynn nodded.\n\nI wanted to tell her right then how I felt about her. I'd been wanting to for a while. But the time never seemed right. And it still didn't. Brynn needed to chill. She was seriously stressed.\n\nShe pulled off her shoes and socks and stuck her feet in the fountain. \"Get yours in here!\"\n\n\"It's not exactly hot out,\" I said. \"You're wearing your jacket.\" The light blue material made her hair look even blonder.\n\n\"That's the price of hanging with me. Feet in or walk your feet somewhere else.\"\n\nI got my shoes and socks off fast, sat down next to her on the edge of the fountain, and plopped my feet into the water. Brynn kicked up and gave me a little splash. \"I love this. This was my favorite place.\"\n\n\"What?\" I asked.\n\n\"Oh, I was just remembering this park. When I was little I used to go there with my mom. It had a big fountain like this. It was my favorite place,\" Brynn explained. \"We used to splash around in it.\"\n\nHer mood seemed to have completely turned around. Getting outside away from everybody for a while had been a good idea.\n\nBrynn reached down into the fountain and flicked some water at me. Some of it landed in her blond hair and sparkled in the sun. Maybe this was the right time. Maybe I should tell Brynn everything I was feeling about her. . . .\n\nYeah, she's been feeling better for a whole minute, I thought. I decided not to say anything. Instead I splashed her back.\n\nI woke up before daybreak the next morning and couldn't go back to sleep. I kept thinking about Brynn. My mind kept jumping back and forth between fun thoughts, like hanging out with her at the fountain, and worries. Was Brynn being specifically targeted by whoever had done that drawing and ripped up her bear?\n\nIt's not like the crayon drawing had really looked like her. The blond hair reminded me\u2014and clearly Olivia\u2014of Brynn. But George had thought it looked as much like his sister. And really, it was pretty generic.\n\nI rolled over on my side and told myself to take some deep breaths and let myself drift back off. Sleep's important for a sharp mind, and I was going to need one to figure out the rest of the case with Joe.\n\nHow many people knew that bear was Brynn's?\n\nThe deep breathing didn't work. I definitely wasn't going to fall back to sleep. A run. That's what I needed. Sometimes a change of scene will give me new ideas. New angles to investigate.\n\nI got dressed fast and headed out of my room. As I started down the main staircase, I heard footsteps behind me. I didn't think anybody else was up. I jerked my head around\u2014and saw Joe. Wearing sneakers and sweats.\n\n\"Run?\" he asked.\n\nI nodded.\n\n\"Think Olivia and Gail will try to off you if you're seen with me?\" Joe said. \"We said we only teamed up to find the saboteur. We're not supposed to like hanging out together or anything.\"\n\n\"I'll just tell them I was trying to find a way to off you while we were jogging,\" I answered.\n\n\"Gotta say, I'm glad you're my brother and not George,\" Joe told me as we headed outside.\n\n\"Aww, that's so sweet. You prefer me to a deranged attempted murderer.\" I started to run. Joe matched me step for step. Neither of us likes to come in last.\n\n\"I couldn't sleep,\" said Joe. \"I kept thinking about the notes and the drawing and the teddy bear incident. I actually believed the Twisted Twins when they said they didn't do that stuff.\"\n\n\"Me too. You go with the group theory that the producers are behind everything that plays on the history of the house? The old murder and the demon story?\" I asked.\n\n\"Makes sense. Hal was right about the timing. It's a big coincidence that they weren't filming during the time we know that last note was put in place,\" Joe answered.\n\nWe left the path and entered the orchard. A few beams of light were just starting to filter through the trees. \"Would that mean it was the producers who sent everybody the death threats to begin with? To get everybody scared before we even showed up?\"\n\n\"We were thinking Mary had done that,\" Joe reminded me. \"But since she's probably in a juvenile detention somewhere and the demon is still playing with us, I guess not. Maybe the\u2014\"\n\nJoe grabbed my arm and yanked me to a sudden stop.\n\n\"What?\" I demanded.\n\n\"Over there,\" he said, his voice coming out choked.\n\nThen I saw what Joe had seen.\n\nBrynn lying facedown in the dirt. Her light blue jacket red with blood. So much blood.\n\n## So Much Blood\n\n\"Brynn!\" Frank shouted. I've never heard that much fear and pain in my brother's voice. Never.\n\nWe both tore over to the body. There was no way she could be alive\u2014was there? Not with so much blood.\n\n\"She's been stabbed,\" Frank cried as he dropped to his knees next to Brynn.\n\nHe gently laid his fingers against Brynn's neck. Her head shifted, turning sideways. He let out a long breath.\n\nIt wasn't Brynn. It was Georgina. They had that same blond hair, and Georgina must have borrowed Brynn's jacket.\n\n\"She's alive,\" Frank said. \"Her pulse is thready, but she's alive. I need to get a pressure bandage on her.\"\n\n\"You stay with Georgina,\" I told Frank. \"I'll run back to the house and get the medic and call an ambulance.\" I turned and dashed off. \"Cell phones,\" I muttered, thinking of the seconds ticking by, seconds that could save Georgina's life. \"They had to take away our cell phones.\"\n\n\"Georgina's been stabbed!\" I shouted as I pounded back into the mansion. My eyes darted back and forth as I searched for a PA. \"I need somebody with phone access! We need an ambulance!\"\n\nPeople appeared from all directions. The stairs. The kitchen. The dining room. The front door.\n\n\"What happened?\" Olivia cried.\n\n\"I don't know. All I know is she's lost a lot of blood and she needs help\u2014now!\" I answered.\n\n\"Veronica's on her way down. I'm calling an ambulance. We're tracking down the medic,\" a PA wearing a headset told us. \"Where exactly is Georgina?\"\n\n\"She's in the orchard. Back in the northeast corner. Frank's with her,\" I answered.\n\n\"Stabbed? As in stabbed?\" asked James.\n\n\"As in clawed with demon claws.\" Ann clutched her charm, snapping more of the twigs.\n\nVeronica strode into the large entryway. \"Where's George?\" she demanded.\n\nI looked around. I hadn't even taken in the fact that he wasn't with the others. Get in ATAC mode, I told myself. I locked down everything I was feeling about what I'd just seen. I needed to be on full alert. Cool and logical and ready for whatever went down.\n\n\"I think he's in the shower,\" Hal said.\n\n\"I want him out and down here,\" Veronica ordered the PA, who began talking rapidly into her headset.\n\nBrynn sat down on the floor. She pulled her long robe tighter around her, like she was freezing.\n\n\"I want to drop out,\" Gail blurted. \"I want to go home. I want to go home now.\"\n\n\"Fine.\" Veronica nodded at the PA.\n\n\"You should all go,\" Gail told us, voice trembling. \"It isn't safe here. You all know it. If you stay, you're all going to die! No amount of money is worth that.\"\n\n\"I'm not dying. I might kill somebody myself, but I'm not dying,\" James declared. He hitched up the cut-off sweats he slept in.\n\n\"I'm staying,\" Ripley said. \"I've gotten this far.\"\n\n\"I'm certainly not leaving.\" Olivia crossed her arms over her chest.\n\nHal shifted from foot to foot. \"I disagree that we're in danger. George is the one who\u2014\"\n\n\"Noooo!\" The howl carried all the way from the second floor to the entryway. Footsteps pounded toward us, and George appeared at the top of the stairs, two PAs behind him.\n\nHe scrambled down the stairs. Barefoot. Shirtless. Dripping water. \"Where is she? I want to see Georgina!\" he shouted.\n\n\"Georgina is or will shortly be on her way to the hospital,\" Veronica told him. \"And you will shortly be on your way to the police station.\"\n\n\"Is she okay? Is she going to be okay?\" George demanded. He lunged toward Veronica, and the PAs grabbed him and jerked him back. \"I want to go to the hospital with her. She'd want me to.\"\n\n\"I don't think having her attempted killer at her bedside would help your sister in her fight for her life,\" Veronica told him.\n\n\"You think I did it?\" It was like it had never occurred to him. Even though Veronica had already pretty much told him the cops were coming for him.\n\n\"Ask Georgina. She'll tell you. I'd never hurt her!\" George ran his hands through his wet hair.\n\n\"You tried to poison her yesterday, dude!\" James exploded.\n\n\"I wasn't trying to kill her!\" George shouted. \"Just ask Georgina. Or let me talk to her.\"\n\n\"Georgina has lost a great deal of blood,\" Veronica told him. \"She's in critical condition. She isn't able to speak to anyone.\"\n\n\"Critical condition? Like she could die?\" asked George, all the color draining out of his face.\n\n\"You stabbed her. What do you think?\" James shot back.\n\nGeorge's shoulders started to shake. It took me a second to realize he was crying. Sobbing. \"I would never do anything to hurt Georgina,\" he choked out.\n\n\"You should have seen George's face,\" I told Frank late that night. It was the first time we'd gotten the chance to talk by ourselves. By the way, I had to sit on the edge of the tub this time. \"He'd completely lost it. Crying so hard he could hardly even breathe. The police could barely get him out to the cruiser.\"\n\n\"Man.\" Frank shook his head.\n\n\"I'm not sure he did it,\" I said.\n\n\"I'm not either,\" said Frank\n\n\"Not just because of the crying . . . ,\" I began.\n\n\"Lots of guilty people cry,\" Frank agreed. \"But there are still all these pieces that don't fit. I'd really like to know for sure whether the producers are behind the drawing, and the warning on the wall, and note and the bear, for one thing. It couldn't be George. He didn't have the opportunity to place the note on the cereal bowl.\"\n\n\"Right,\" I said. \"Which means if it's not the producers, the person who almost killed Georgina is still in the house.\"\n\n\"Let's go through the list again. Who've we got?\" Frank asked.\n\nBefore we could start reviewing the suspects, someone screamed.\n\nFrank was on his feet instantly. \"That sounded like Brynn!\"\n\n\"Maybe a nightmare,\" I said as we dashed toward her room. \"She was having one the other night.\"\n\nIt wasn't a nightmare. I knew that the instant I saw Brynn's face.\n\nFour deep claw marks cut across her right cheek. Dripping blood.\n\n## Killer Pool\n\n\"Who did it?\" I burst out.\n\nBrynn shook her head. \"I don't know. I was asleep. Then I felt this ripping across my cheek.\"\n\n\"You didn't see anything?\" Joe asked. \"Or maybe hear or even smell something?\"\n\nBrynn lightly touched the scratches, then winced. \"Nothing. I just . . . nothing.\"\n\n\"I didn't see anything either,\" said Olivia. She was the only roommate Brynn had left, with Georgina in the hospital.\n\nI couldn't stop staring at Brynn's face. How had this happened? How had I let it happen? Joe and I were here to keep the other contestants safe. Now Georgina was in the hospital and somebody had gone after Brynn.\n\nIt shouldn't have mattered that it was Brynn. I should have felt the same way if it was anyone who was supposed to be under my protection. I should have felt that way if it was James curled up on his bed, trembling, with bloody scratches on his face.\n\nAnyway, just standing there, looking at her, feeling completely useless, was making me insane. I wanted to do something. Take some kind of action.\n\nI hurried into the bathroom and wet a washcloth with warm water, then rushed back to Brynn's side. I used the cloth to wipe the blood off her cheek.\n\n\"Aww, so sweet,\" James said. Yeah, James was there. By the time I returned with the cloth, everybody in the group was there. No one could have slept through that terrible scream of Brynn's.\n\n\"The demon. It's getting stronger,\" said Ann, eyes dark and solemn. For once, she wasn't shouting. And it made her words seem less crazy and more frightening.\n\n\"Do you think there could actually be a demon?\" Ripley asked. \"Or maybe the house is haunted by Katrina Decter. She died here. Just two rooms over. Maybe we should ask Veronica to have somebody come in and, I don't know, do a psychic scrub-down of the place.\"\n\n\"Like an exorcism?\" Hal asked.\n\n\"Yeah,\" Ripley answered. \"That might help, right, Ann?\"\n\n\"What do you think?\" asked James. With no sarcasm. It seemed like pretty much everybody was willing to at least consider Ann's demon theory now.\n\nI noticed Brynn's hands had blood on them too. I took one of them in mine, then used the washcloth to clean her fingers. For a second, I was hit by the fact that I was pretty much holding hands with a girl\u2014a girl I liked\u2014in front of a roomful of people. And I wasn't blushing or anything.\n\nMaybe fear overrides embarrassment. And I was afraid, I admit it. Not of demons or ghosts. But just of the possibility that Brynn\u2014or one of the others\u2014could die. And that I might not be smart enough or fast enough to stop it.\n\n\"Exorcisms don't always work,\" Ann told the group. \"This demon is strong. You finally all see that.\"\n\nThis wasn't good. Things were bad enough at Deprivation House without the whole group getting as demon crazy as Ann. \"You know what I see?\" I asked loudly. Nobody asked, \"What?\" but I told them anyway. \"I see blood under Brynn's fingernails.\" I held her hand up so everyone could see. \"I'm sure there's tissue under there too.\"\n\n\"You must have really fought back hard,\" Hal said. He was good at making connections. I could see him with ATAC. Unless he was the one who'd been trying to kill people. ATAC has a problem with hiring murderers.\n\n\"That's my point,\" I explained. \"Brynn probably left her attacker with a couple of claw marks of her own. That's where the blood and tissue under her nails came from. It's ordinary blood and tissue from another human. Not some supernatural creature.\"\n\n\"You don't know what forms demons can take,\" Ann told me. \"You don't have any reason to think it couldn't look like an ordinary human.\"\n\n\"I saw one of the old movies Nina's father directed once. The demons did all look like ordinary humans. Well, really hot ordinary humans,\" said Ripley.\n\n\"I didn't know he directed horror movies. Maybe that's where he came up with the idea to tell Nina her mom was possessed by a demon in the first place,\" Joe suggested.\n\n\"Or maybe it's because it was true!\" Ann shot back. \"How much more proof do you want?\" She flung her hand out at Brynn. \"Look at her.\"\n\n\"But like Frank said, we have to consider that whoever attacked Brynn left ordinary blood behind,\" Hal commented.\n\nI glanced around the room. All our suspects were there, with the exception of Veronica. If one of them had attacked Brynn, they would be showing some marks too. The thing was, heat was one of our deprivations, and it got cold at night. That meant we all slept with a lot of clothes on. Everyone was wearing long pajama pants or sweats or long johns, with sweaters or sweatshirts or something else with sleeves on top. A few people\u2014Olivia, Ann, and Hal\u2014were even wearing gloves.\n\nAnybody could have some wounds from Brynn's nails. Which one of them did? I had no way of knowing.\n\n\"Are you looking at me for some reason?\" Joe demanded, glaring at James.\n\n\"What's your damage?\" James asked.\n\n\"I saw you look at me when my brother over there was talking about how Brynn must have left some marks on her attacker.\" Joe took a step closer to James. \"You think I did it?\"\n\n\"I think you'd do anything to win. And you know a lot about how sabotage is done. I noticed that when we teamed up,\" I said to Joe. I thought I knew where he was going with this. And I liked it.\n\n\"Of course you'd side with him,\" Joe yelled at me. \"You're completely prejudiced against rich people. Something bad happens, and you assume it's one of us.\"\n\n\"What? You think rich people are special?\" Olivia demanded. \"You think nobody rich would ever cheat? Or hurt somebody?\"\n\n\"I know you need money a lot more than I do, Olivia,\" Joe shot back. \"You too, James. If somebody's trying to take Brynn out of the running, I think it's a lot more likely that it's one of you.\"\n\n\"I don't need to take anybody out of the running,\" shouted James.\n\n\"Yeah? Brynn won the dirt bike race,\" Joe told him.\n\nJames yanked off his sweatshirt. He pulled his sweatpants up above his knees. Then he turned in a slow circle, arms stretched wide. \"Anybody see any blood?\" he demanded. \"Anybody see any scratches from Brynn's fingernails?\"\n\nJoe had done it. He'd goaded James into showing the skin I needed to see to eliminate him. He didn't have any wounds. He couldn't have been the person who attacked Brynn.\n\n\"Your turn,\" James told Joe.\n\n\"Fine.\" Joe pulled off his sweatshirt and shoved his pajama pants up.\n\n\"It's not him,\" said Hal.\n\n\"Well, it's not me,\" I answered. I did my own strip show.\n\n\"I want to see everybody,\" James declared. Hal sighed and started unbuttoning his pajama top.\n\n\"I'm not showing you anything,\" Ripley said.\n\n\"The girls will go into the bathroom and check each other,\" Olivia stated. She looked over at Ripley. \"Unless there's some reason you don't want to clear yourself.\"\n\n\"Fine.\" Ripley headed for the bathroom, followed by Olivia and Ann. I trusted them to come back with an accurate report. One good thing about all the suspicion and tension and rich-poor garbage in the group: With all the girls in the bathroom, they'd never be able to agree on some kind of cover-up.\n\n\"I'm voting for the crazy chick,\" James said after Hal had proved he had no scratches or blood on him.\n\n\"What about Ripley?\" Hal asked. \"Maybe she wanted another chance to give first aid. Maybe she's been hurting everyone for PR.\"\n\n\"I could see Ripley doing it too,\" James agreed. \"And Olivia would probably eat her own foot to win.\"\n\n\"How would that possibly help?\" Hal replied.\n\nJames ignored him. \"What about you?\" he asked me. \"Who do you want in the Killer Pool?\"\n\nI glanced over at Brynn. She had her arms wrapped tight around her knees. Like she was afraid somebody was going to assault her again. \"Let's just wait and see, okay?\" I answered James. Brynn didn't need to listen to us turning what happened to her into a game.\n\nI turned toward the door. Was the case going to be closed when the girls came back in? I could actually come up with a scenario where Ripley or Olivia attacked Brynn. They both had motives. Ripley, the good PR she'd get for coming to Brynn's aid. Olivia might have decided to take out some of the non-alliance competition.\n\nAs for Ann . . . I didn't have a motive that made sense for her. But she'd been almost hysterical for days. What was going on with her? She'd been so quiet in the beginning that Joe and I hadn't been able to get very much information out of her.\n\nI heard footsteps in the hall. Was this it? Were we about to have absolute proof of the attacker's identity?\n\nThe door swung open. Olivia was the first one through. \"We're all clean,\" she announced.\n\n\"I'm getting Veronica down here,\" I said.\n\n\"Oooh. He's calling the principal,\" James called.\n\nActually, I was calling the only suspect who hadn't been checked for wounds.\n\n\"Why drag her into it?\" asked Olivia. \"What if she decides to shut the show down?\"\n\n\"Maybe she should,\" I answered. I didn't care if that happened\u2014as long as Joe and I found the perp first. I didn't want anyone leaving the house until we had the case locked.\n\nI hurried out to the intercom in the hall\u2014and saw Veronica striding toward me. Clearly, she'd been alerted by a PA\u2014or had been watching the camera footage live.\n\n\"Did you get who attacked Brynn on film?\" I asked.\n\n\"We didn't start the cameras until the scream alerted us that something of interest was happening,\" Veronica replied.\n\nOf interest. She had to be made of ice.\n\nIf she was the one who had attacked Brynn, it was very convenient that there was nothing on film. That was one advantage Veronica had that none of the other suspects did. She knew when the cameras were live. She controlled them.\n\nVeronica walked past me and into Brynn's room. I rushed after her. She was as perfectly dressed and groomed as always, even this late at night. Her short lavender skirt showed a lot of her legs, but her suit jacket could be hiding the scratches Brynn had given her assailant. How were Joe and I supposed to get it off her?\n\nAn idea slammed into my head. I didn't know if it would work, but it was worth a try. I turned around and slipped back out of the room as the rest of the group pounded Veronica with questions.\n\nWhen I got downstairs to the kitchen, I went straight to the fridge and studied the contents. Grape juice, I decided. I poured a glass, filling it almost to the rim, then I returned to Brynn's room.\n\n\"I thought maybe you'd want something to drink,\" I told Brynn.\n\n\"Isn't he sweet?\" James asked.\n\nI started toward Brynn, making sure that I took a path that led me past Veronica. Then I \"tripped.\" A second later, grape juice was splattered all over her.\n\nVeronica let out a little shriek. \"That's going to stain,\" Joe told her. \"Give it to me and I'll run it to one of the PAs. They can do . . . whatever you're supposed to do to deal with grape juice.\"\n\nShe narrowed her eyes at him. \"You think you're very clever, don't you?\" She turned to me. \"Both of you.\"\n\nWhat was that supposed to mean? Did Veronica know our secret? Had she somehow managed to find out that we were ATAC?\n\n\"I know you were checking one another for wounds,\" she continued. \"Clearly you think I might be the one who assaulted Brynn. Which is ridiculous.\" She slid off her jacket. The T-shirt\u2014it probably has another name because it had lace on it\u2014underneath showed her arms and shoulders and throat. No scratches.\n\n\"Happy?\" she asked, still looking at me.\n\nI nodded.\n\n\"I'll be charging you for the dry cleaning,\" she said. \"Now, is there anything else?\"\n\n\"What are you going to do about the demon?\" Ann burst out.\n\n\"What are you going to do about what happened to Brynn?\" cried Ripley. \"Look what somebody did to her.\"\n\n\"Deprivation House is clearly bringing out the worst in one of you. At least one,\" Veronica answered. \"It certainly affected George, and, of course, we can't allow the kind of behavior he exhibited.\" She gave a small shrug. \"But a scratch isn't the same as a knife in the back. Any of you are welcome to leave now. Otherwise, I suggest you all start sleeping with your eyes open.\" She left the room without a backward glance.\n\n\"Hard-core,\" Joe muttered.\n\n\"I'm not going to be able to sleep at all,\" Brynn said. \"I'm going to go to the great room and stare at the place where the TV used to be.\"\n\n\"I'll go with you,\" I immediately offered.\n\n\"I'm going to see what's in the kitchen. Maybe there's some celery or something,\" said Joe. \"We haven't been deprived of vegetables yet, right?\"\n\nThe three of us headed out of the room. \"You guys want anything? See, I can be polite to the poor,\" Joe added to Brynn. I could tell he was setting up an opportunity for us to talk.\n\n\"Maybe I'll get us some non\u2013junk food,\" I told Brynn. She nodded. \"I'd rather get it myself. I don't think my brother would be that good at serving,\" I added.\n\n\"What?\" I said to Joe when she had disappeared into the great room. \"I don't want to leave her alone.\"\n\nJoe stepped close to me and lowered his voice. \"Remember how we thought it was Brynn at first\u2014in the orchard?\"\n\n\"Because Georgina was wearing Brynn's coat,\" I answered.\n\n\"And because they have the same hair,\" Joe said. \"I was thinking\u2014what if someone else made the same mistake we did? What if that knife wasn't meant for Georgina? What if it was meant for Brynn?\"\n\n\"And they went after her again tonight.\" My gut tightened just thinking about it.\n\n\"It's possible,\" Joe answered. \"Watch her, okay? And watch yourself.\"\n\nI nodded and walked into the great room. Brynn looked so small in the huge space. \"Nothing good?\" she asked.\n\nIt took me a minute to realize she was talking about food. \"Uh-uh,\" I answered. \"Are you hungry?\"\n\n\"No. I may never eat again. I'll have a lot of time for hobbies with the no eating or sleeping.\" Brynn was trying to joke around, but her voice was flat.\n\n\"So what are you going to start doing? Knitting? Field hockey? What?\" I asked. I figured the best thing I could do for her was try and take her mind off the attack.\n\n\"I'm thinking . . .\" Brynn's words trailed off. \"My brain is gone,\" she admitted. \"Can we just sit here? Just sit.\"\n\n\"Sure,\" I told her.\n\nThen Brynn reached over and held my hand. I wanted to tell her that I'd never felt this way before. That she was the most amazing girl I'd ever met. But that would involve talking. And she didn't want talking. So it wasn't the time. Was it ever going to be the time?\n\nWhoever attacked Brynn had to have been as close to her as I am right now. The thought sent a shiver through me.\n\nShe had to have screamed as soon as her face was clawed. If the attacker was so close, how did he or she get out of the room without Olivia seeing the person? If he or she was close\u2014and they had to be\u2014why hadn't Brynn even gotten a glimpse?\n\nBrynn squeezed my hand. I looked down at where our fingers were twined together. I could see the dried blood under her nails.\n\nAt least she'd caused her attacker some pain. But why hadn't there been a mark on any of the suspects? Who were Joe and I forgetting?\n\nAnd then it clicked. I got it. I knew who had attacked Brynn.\n\nWhat I didn't understand at all was why.\n\n## Why Would She Do That?\n\n\"It makes sense, right?\" Frank asked as he paced around the bathroom.\n\n\"Yeah. It's actually the only thing that makes sense,\" I told him. \"It explains everything. Why neither Brynn nor Olivia got even a fast look at the attacker. Why we didn't see a mark on any of the suspects.\"\n\n\"It's the only thing that make sense.\" Frank sighed as he sat down on the edge of the tub.\n\n\"Are you okay?\" I asked.\n\n\"Yeah. Why?\" he replied.\n\n\"Why?\" I repeated. \"You know why. Because you like her. You might even love her. I wouldn't blame you if you weren't okay now that we know Brynn is behind the attack.\"\n\n\"I just want to understand it,\" Frank said. \"And I don't. Not at all. Why would she do that to her own face?\"\n\n\"I don't get it either. But it explains the blood under her fingernails. It was her own blood,\" I answered. \"And nobody saw her attacker, because when Brynn screamed, there was no one in the room except Brynn and Olivia.\"\n\n\"What else do you think she did?\" asked Frank. \"Did she rip up her own bear, leave the notes, do the drawings? Did she . . .\"\n\nHe couldn't finish, so I had to. \"Did she do all the sabotage that's happened since Mary was taken out of the house? Did she stab Georgina?\"\n\nFrank grimaced. \"Yeah. Do you think she did?\"\n\n\"With this new info, anything seems possible,\" I admitted.\n\n\"I keep going back to the why,\" Frank said. \"I want to understand. . . .\"\n\n\"Okay, so what do we know about Brynn?\" I asked.\n\n\"She likes to read. She has expensive shoes. She likes ice cream with nothing in it. . . .\" Frank's voice trailed off. \"It's not much. And I've spent more time talking to her than any other suspect.\"\n\n\"Me too. And I don't have anything good either. She made up this game where you have to come up with the best opposite for something else. And the fountain is pretty much her favorite place here.\"\n\n\"Yeah. She told me there was a fountain like it in this park she went to with her mom,\" Frank said.\n\nThen his eyes narrowed, and I could almost see a thought ripping into his head. \"What?\" I asked.\n\n\"At first she said the fountain was her favorite place. Then she corrected herself and gave me the story about the park and her mom.\"\n\n\"Story?\" I repeated. \"You think it was a lie?\"\n\n\"What if\u2014and this is kind of a crazy theory\u2014but what if Brynn was here, at the mansion, when she was a little girl? What if it's true that the fountain here was really her favorite place as a kid?\" Frank asked.\n\n\"Meaning that Brynn is actually Nina?\" I thought for a second. \"She's the right age. Nina was four when her mother was killed. That was ten years ago. Brynn's fifteen.\"\n\n\"And she knew exactly where a light switch in the library was. It was so low, she shouldn't even have noticed it,\" said Frank. \"The library used to be Nina's room.\"\n\n\"So why is she here? With a fake name and everything?\" I'm usually pretty good at motives, but nothing about this was making sense.\n\n\"I get why she wouldn't want everyone to know she was Nina,\" Frank answered. \"But I don't know why she'd ever want to come back here. Every night we all hang out in the room where her mother was killed.\"\n\n\"Maybe that's why she's always going out to the balcony and looking down at the fountain,\" I suggested. \"Maybe she can't stand being in that room. But then we're back to the question of why she came here in the first place.\"\n\n\"To stop the show from getting on TV?\" Frank said. \"Maybe she thought the show would bring attention back to the murder, and she didn't want that. After ten years, it's not something anyone talks about much. But with the show, that could change.\"\n\n\"Do you think Brynn would stab someone to shut down the show? Are we saying that she tried to kill Georgina?\" The thought of Brynn doing that made my stomach do a slow roll. How must Frank be feeling? He was in love with her. And he's not like me. I fall half in love with half the girls I see. Frank's not that way.\n\n\"I hope not,\" Frank admitted. \"But I can't let that get in the way of finding out the truth. And the truth is, she could have attacked Georgina.\"\n\n\"George is still a possibility,\" I said. \"We know for sure that George poisoned Georgina and sabotaged the brakes of her dirt bike. He admitted it.\"\n\n\"And Georgina admitted that she sabotaged the trampoline,\" Frank agreed. \"Let's think about the timing of the other events. Could Brynn have been behind them?\"\n\n\"She definitely could have torn up her teddy bear. She could have done it, then screamed like she'd just discovered it,\" I answered.\n\nFrank gave a sharp nod. \"Right. All the rest of us were still in the library, talking about the crayon drawing of the girl and the demon, when she 'found' the bear. She definitely had time to set it up, then yell for us.\"\n\n\"Brynn left during the conversation we were having about the murder after the dirt bike race,\" he added. \"She had the opportunity to do the drawing then. Or she could have done it earlier\u2014in the middle of the night.\"\n\n\"Now that we're talking about it, I can think of other times when Brynn left the room when the rest of us started talking about Nina's murder or the demon story Nina's dad had come up with to explain things to his daughter. To Brynn, if our theory is right.\"\n\n\"And a lot of times, after one of those conversations, something bad would happen. The bear. The drawing,\" Frank said. \"Maybe hearing people talk about what happened when she was little sets Brynn off.\"\n\n\"Maybe,\" I agreed. \"So I think what we have to do is set her off again.\"\n\n\"Right. Set her off again. She'll leave\u2014but this time we'll follow her.\"\n\n## The Knife\n\nBrynn and I sat in the great room late the next night. She couldn't sleep again. Joe and I had counted on that in our plan.\n\nMy part of the plan was to stir things up by talking about the murder of the woman who might be Brynn's mom. When she bolted\u2014which we were pretty sure she would\u2014Joe was in place to see where she went. Right now, he was positioned out of sight in the library. He had a small mirror that allowed him to keep watch on the hallway.\n\nSo go on, I told myself. There was no point in waiting.\n\nI opened my mouth, but I couldn't do it. I couldn't intentionally try to freak Brynn out.\n\nI was letting my emotions get in the way. That had never happened before. Because I'd never felt about another suspect the way I did about Brynn.\n\nMaybe I should have agreed to let Joe do this part. He'd offered. But I was the natural one to offer to sit up with Brynn. She and Joe were friends, yeah. But he hadn't spent most of last night holding her hand. The way I was holding her hand right now.\n\nDo it, I ordered myself. It's the plan. If she's the perp, you have to know it. You can't put everyone in this house at risk because you won't step up.\n\nI swallowed hard. My mouth and throat were as dry as if I'd been eating sand. \"Sometimes when it's really quiet in here, like now, I start thinking about the murder,\" I said, my voice coming out rough. \"I've never been in a room where somebody was killed before. At least not that I know of.\"\n\n\"Me either,\" Brynn said. Her fingers tightened on mine. They were cold. And trembling a little.\n\n\"Do you think it's true that emotions can get trapped in a place? I saw something like that on this show, Spectral.\" The only reason I'd ever watched it was because Joe was addicted\u2014even though the ghosts on the show were so clearly fake. \"People who are sensitives are supposed to be able to feel the fear and anger that's trapped wherever something violent has happened.\"\n\n\"You're starting to sound like Ann,\" Brynn told me. She pulled her hand away. Was she getting ready to take off?\n\nNot yet. I had to push harder.\n\n\"I'm not talking about demons,\" I answered. \"I'm just talking about psychic energy. A man killing his own wife in front of their daughter\u2014that's intense stuff. Maybe it does leave some kind of residue. Maybe it's my imagination, but when the room is almost empty, I think I can feel something.\"\n\nBrynn jerked to her feet. \"I think . . . I'm going to go to bed.\"\n\n\"You sure?\" I asked.\n\nShe didn't answer. Just left. The way she always seemed to leave when the murder came up.\n\nI forced myself to do a slow twenty-five count. Then I stepped into the hall. Joe was waiting for me.\n\n\"Did she go into her room?\" I whispered.\n\nHe shook his head. \"Down the stairs.\"\n\nThat's not what I wanted to hear. What you want is the truth, I reminded myself.\n\n\"Come on,\" said Joe. Keeping in the shadows, we crept over to the head of the stairs. Just in time to see Brynn heading out the front door.\n\nJoe and I rushed down the stairs. I strode over to the dining room window and peered outside. \"She's in the garden. She's walking fast, \" I told Joe.\n\nWe let her get a little more of a head start, then followed. Through the garden and into the orchard. The orchard where Georgina almost died.\n\nWhy was Brynn out here?\n\nShe started to run. She stumbled over a root but didn't slow down. Not until she got to the northeast corner. Then she dropped to her knees and began to dig. Only about twenty-five feet from where Georgina's blood had soaked into the ground.\n\nJoe and I froze. Watching.\n\nWhat was she doing?\n\nShe clawed at the ground. Frantic. I could hear her ragged breathing as she pulled something free. Something that glinted in the moonlight. It took me a second to register what it was.\n\nA knife.\n\nBrynn stood up and raised the blade high in the air. Aimed at her chest.\n\n\"No!\" I screamed. \"Brynn, stop!\"\n\n\"Don't!\" Joe shouted.\n\nBrynn started to bring the knife down. I launched myself at her. I hit the back of her knees with my shoulder, and we both hit the ground.\n\nWhere was the knife? Did she still have it? I didn't see it in her hand.\n\n\"Got it!\" Joe cried.\n\n\"Brynn, what were you doing?\" I exclaimed. I grabbed her by the shoulders and stared into her face. \"What were you trying to do?\"\n\nHer eyes were blank. I don't think she saw me at all.\n\nI gave her a light shake. \"Brynn!\"\n\nShe didn't answer. She didn't even blink. It was like I was holding a dead body.\n\n## Game Over\n\n\"It's over.\" Ripley set her suitcase down in the entry hall.\n\n\"I don't see why the show's canceled,\" James complained aloud. \"We know Brynn was the psycho.\"\n\n\"This time,\" said Hal. \"Maybe they figured three people trying to end the game meant it really shouldn't happen.\"\n\n\"If we want to keep going, they should let us,\" Olivia declared.\n\n\"Georgina did almost die. And Mitch killed a PA,\" Hal reminded us. \"Maybe the show should have been shut down a long time ago.\"\n\nOlivia raised her eyebrows. It was like even now she couldn't believe Hal was disagreeing with her in front of everyone.\n\n\"The taxis should be here shortly,\" Veronica announced as she joined us.\n\n\"I think we should each get some cash,\" James told her. \"The next person to drop out would have gotten paid. And we're being forced to drop out.\"\n\n\"Yeah, we should get something,\" Olivia agreed.\n\n\"You've all been given something very valuable,\" Veronica told us.\n\n\"What? I wasn't given anything,\" James protested.\n\n\"You were all given the chance to realize that you have more than you need. Every one of you. You've learned that you can get by with much less than you ever thought,\" Veronica explained. I could tell she was trying not to smile.\n\n\"Maybe we've all been given the chance to sue you and the other producers,\" Olivia shot back.\n\n\"You all signed releases when you agreed to be on the show,\" Veronica reminded us. \"You agreed not to hold the producers responsible for any injuries. And it was made clear that the show could be stopped at any time, and that it might not even air at all.\"\n\n\"Hey, Bobby T said the option on his blog has been renewed. He's getting a boatload of cash,\" Ripley announced, reading a text message on her phone. We'd gotten all our luxury items back an hour ago.\n\n\"He has a boatload of debt, too,\" James said. \"The loser probably didn't break even.\"\n\nI heard a car pull up in front. Then two more. It was time to go home.\n\nOlivia suddenly rushed over to us. Make that Olivia suddenly rushed over to Frank.\n\n\"Here.\" She thrust a large sheet of paper into my brother's hands. \"I know Gail would want you to have this\u2014even though she practically gave birth to a cow the day you almost caught her sketching it.\"\n\nFrank and I exchanged a quick glance. So that was why the girls had acted so freaky that day. They were trying to keep Frank from seeing the sketch book, but not for any crime-related reason.\n\nI turned my attention to the sketch. It was the worst drawing of Frank you could possibly imagine. The eyelashes on him. Step back. It was like he was the spokesboy for a new kind of man mascara or something.\n\nI started to snicker. Then it hit me. Olivia wasn't handing over a sketch of me.\n\nI smiled as something else hit me. Gail probably took the sketch she did of your hero home with her. She was probably sleeping with it under her pillow. Yeah. She'd wanted to keep the one she'd drawn of me. I'm pretty sure that's what the deal was. And I'm a detective, so I should know.\n\nOne of the cab drivers gave an impatient honk.\n\n\"I guess we'd better get going,\" Olivia said. \"Sorry if I got a little intense sometimes,\" she added to Frank.\n\n\"Everybody did,\" Frank answered.\n\n\"I just wish we'd realized what was going on with Brynn earlier,\" Frank said quietly as we stowed our suitcases in the trunk of the second taxi. \"Maybe we could have helped her.\"\n\n\"She's getting help now,\" I reminded him. \"The psychiatric hospital is where she belongs.\"\n\nFrank nodded, but I knew he was still running through scenarios where we were able to stop Brynn before she hurt anyone.\n\n\"Let it go, Frank,\" I told him as we got into the taxi together. \"We completed our mission. None of the contestants ended up dead.\"\n\nHe nodded again. But he was still thinking about her. I wasn't sure if he'd ever stop.\n\n###   FRANK\n\n\"I've got to do it,\" I told Joe. \"But you don't have to come. It's not an official mission.\"\n\n\"Are you kidding?\" Joe said. \"I'm not passing up the chance to wear my Diesels.\" He patted the jacket pocket where he'd stashed the sunglasses that had been part of his Joe Carr cover. He hadn't been able to wear them since we came back from Deprivation House a few months ago. There would have been waaay too many questions. Our parents are generous and everything. But they'd never shell out for three-hundred-dollar shades. And it's not like Joe has that kind of cash.\n\nWe mounted up on our motorcycles. Four hours later, we rode through the wrought-iron gates of Plainview Hospital. It was supposed to be the best in the country. It looked like a nice place. There were some patients sitting in the front garden, getting some sun. But it was still hard to think of Brynn here.\n\n\"You ready?\" Joe asked after we parked.\n\nI wasn't. But more time wasn't going to change that. \"Yeah,\" I told him. I led the way inside and gave our names\u2014our cover-story names\u2014to the woman at the front desk. \"Nina's doctor will take you back,\" she said. \"She wants to talk to you before you see Nina.\"\n\nWas Brynn\u2014I still thought of her as Brynn\u2014okay? She'd been in here for a couple of months. Had her doctor been able to help her at all? Would she be happy to see me? And Joe? Or would we just be part of a horrible memory?\n\n\"Hi, I'm Dr. Bastajian.\" A short woman with a ponytail walked over and shook our hands.\n\n\"How is Brynn\u2014I mean Nina?\" I asked in a rush.\n\n\"She's making progress,\" the doctor answered. \"But she still has a long way to go. She's just beginning to remember what happened when she was back in the house.\"\n\n\"She blanked everything out? Trying to stab herself? Stabbing Georgina?\" Joe asked.\n\n\"It's more complicated than that. Being back in the house where she saw her mother killed by her father was extremely traumatic. Nina began to dissociate, have blackouts.\"\n\n\"So she wasn't even aware she was doing things like putting pieces of glass in with the ice cubes?\" I asked. We'd figured out that that's when Brynn had started breaking down. Mary had insisted she hadn't been responsible for the glass that cut up James's mouth as well as the fire in Bobby T's room and the knives in Brynn's makeup bag, and it turned out she was telling the truth. Mary had been doing a lot of sabotage\u2014but nothing violent. Sending us e-mail death threats was more her style. But Brynn . . . She'd even done things to hurt herself!\n\n\"She absolutely wasn't aware,\" Dr. Bastajian said. \"I think in some way, she was trying to make the demon story that her father told her true. It was too horrific to accept that her father had simply killed her mother.\"\n\n\"So the demon story . . . ,\" said Joe. \"I mean, I know it wasn't true, but are you saying that Brynn's mother didn't actually try to attack her?\"\n\n\"No. Nina was beginning to remember what really happened that night,\" Dr. Bastajian explained. \"Her mother hadn't started using drugs or alcohol again. But she had decided she wanted to leave her husband. They fought, and he killed her.\"\n\n\"And he came up with that story about the demon and convinced Brynn\u2014Nina\u2014that that was what happened,\" I said.\n\n\"Exactly,\" the doctor agreed. \"Nina grew up with her grandparents in Indiana, living a quiet life away from all of this. Her grandparents helped her change her first and last name to Brynn Fulgham, so that she would be safe from nosy media hounds. Then, when Nina realized a reality show was going to be filmed in her old house, the house where the murder happened, she was determined to go back. She thought being there would somehow allow her to remember everything that really happened that night. The pieces she was starting to remember were haunting her.\"\n\n\"So are we supposed to not mention the old murder or the stuff Brynn did when she was having one of her episodes?\" Joe asked.\n\n\"Just follow her lead. Don't push. But if she wants to talk, it's fine,\" Dr. Bastajian said. \"Come on. I'll take you to her room.\" She led us through a maze of hallways, then tapped on a closed door. \"Nina, you have visitors.\" She swung the door open.\n\nAnd I saw Brynn. For the first time in months. She sat in front of a small desk in the corner of her room. And she looked . . . she looked like Brynn. Clear-eyed. A little smile on her face. \"Hi,\" she said, without quite looking at me or Joe.\n\n\"Hey, Nina,\" said Joe. I guess he could tell I was having trouble getting words out.\n\n\"You can keep calling me Brynn,\" she said.\n\n\"So, uh, how are you?\" I asked. It was like I had just learned English a few days ago. It was hard to find any words.\n\nBrynn lifted her hands, then let them fall in her lap in a helpless gesture. \"Let's talk about you guys instead. So I guess being in the house was good for you two at least.\"\n\nJoe and I glanced at each other.\n\n\"I mean, you two, you got to know each other. And you're here, together. So I guess you finally kind of became friends,\" Brynn explained.\n\n\"Yeah, I guess we finally kind of did. Getting winning or losing out of the picture helped,\" Joe told her.\n\nDr. Bastajian's cell beeped. She scanned the text message. \"Nina, your father is here. I'm assuming\u2014\"\n\n\"I don't want to see him,\" Brynn answered, her voice hard.\n\nThe doctor nodded. \"I'll go speak to him. Enjoy your visit.\"\n\n\"I can't believe he thinks I'd want to see his face!\" Brynn burst out. \"Now that I know what he did. And he'll never have to go to prison for it. He's already had his trial. He can't be tried again for the same crime. And it's my fault. It was my testimony that got him off.\"\n\n\"You were a little girl. You believed what your father told you,\" I said. \"Pretty much anyone would.\"\n\n\"I just hate that he gets to live his whole life as this man who killed his wife to protect his daughter. No one will ever know the truth.\" Brynn took a deep breath. \"Sorry,\" she said. \"I know I'm going off. I'm working on dealing with it. And dealing with what I did too. I wrote to Georgina. I tried to explain. She was actually really great about it.\"\n\n\"Did you hear the movie option on Bobby T's blog got renewed?\" Joe asked. \"It's getting an insane number of hits a day.\"\n\nI guess Joe was having trouble knowing what to say too. He was kind of babbling.\n\n\"If I wanted to get info out there, Bobby T's blog is where I'd want it,\" Joe continued.\n\nHe wasn't babbling. \"I'm sure Bobby T would be happy to put the real story about your dad on his site. At least then people would know the truth,\" I told Brynn.\n\n\"You think he'd do that?\" asked Brynn.\n\n\"Are you serious?\" Joe replied. \"He'll have it on there today if you want.\"\n\n\"I want,\" Brynn said. \"I want to tell what I did too. I want to try to explain. Apologize.\"\n\n\"You weren't even aware of what you were doing,\" I reminded her.\n\n\"I still did it,\" Brynn told me. \"I have to take responsibility. I'm not going to be like my father. I'm not going to try to come up with a story that makes me a hero. Or that just makes what I did seem okay somehow.\"\n\nShe was an amazing person. No wonder she was the first girl I fell in love with.\n\nBut I'd never be able to see her again. Frank Dooley didn't really exist. I'd only been able to be him for a couple more hours. To say good-bye.\n\n\"Are you okay?\" Joe asked me when we left.\n\n\"Yeah. What about you? Are you going to be okay?\" I climbed on my bike.\n\n\"Me? Yeah,\" said Joe.\n\n\"Are you sure? You know you're not going to be able to wear the Diesels again,\" I reminded him.\n\nJoe grinned. \"Even without them, I'm still the more happenin' Hardy. And someday a girl is going to realize that absolute truth!\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nBooks by V. M. Burns\n\nTHE PLOT IS MURDER\n\nREAD HERRING HUNT\n\nPublished by Kensington Publishing Corporation\nRead Herring Hunt\n\nV. M. BURNS\n\nKENSINGTON BOOKS\n\nwww.kensingtonbooks.com\n\nAll copyrighted material within is Attributor Protected.\nTable of Contents\n\nAlso by  \nTitle Page  \nCopyright Page  \nDedication  \nAcknowledgments  \nChapter 1  \nChapter 2  \nChapter 3  \nChapter 4  \nChapter 5  \nChapter 6  \nChapter 7  \nChapter 8  \nChapter 9  \nChapter 10  \nChapter 11  \nChapter 12  \nChapter 13  \nChapter 14  \nChapter 15  \nChapter 16  \nChapter 17  \nChapter 18  \nChapter 19  \nChapter 20  \nChapter 21  \nChapter 22  \nChapter 23  \nTeaser chapter\n\nThis book is a work of fiction. Names, characters, places, and incidents either are products of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nKENSINGTON BOOKS are published by\n\nKensington Publishing Corp.   \n119 West 40th Street   \nNew York, NY 10018\n\nCopyright \u00a9 2018 by Valerie Burns\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.\n\nKensington and the K logo Reg. U.S. Pat. & TM Off.\n\neISBN-13-13: 978-1-4967-1184-7   \neISBN-10: 1-4967-1184-X   \nFirst Kensington Electronic Edition: May 2018\n\nISBN: 978-1-4967-1183-0\n\nIn loving memory of Coco Chanel  \nfor 17 years of love and cuddles\nAcknowledgments\n\nSpecial thanks to my agent, Dawn Dowdle, at Blue Ridge Literary; Mary Lasher for capturing my fur babies so well and providing such great cover art; Kristen Lepionka for answering my S.O.S. and providing great promo support and social media graphics; and the wonderful staff at Kensington Publishing, John Scognamiglio, Paula Reedy, and Michelle Addo.\n\nThanks to Anthony Cameron (aka Trooper Tony) and Eric Haskin for answering my questions related to police procedures. Thanks to Dave \"Moose\" Mitchell (aka Missouri Man) for allowing me to pick your brain regarding college football rules and regulations; and Chris Siriano for assistance with the House of David. Special thanks to my friend, Cassandra Morgan, for taking pity on me and allowing me to benefit from your social media expertise.\n\nI appreciate all of the support I've received from '17 Scribes and fellow unicorn Kellye Garrett. Thanks to my SHU Tribe (June 2012) and especially Jessica Barlow, Lana Ayres, and Michelle Lane. I am so incredibly blessed to have such a fantastic training team (Deborah Hughes, Grace Dixon, Jamie Medlin, and Tena Elkins). Thanks for helping to make my days easier so I can spend my nights pursuing my dream. You guys rock!\n\nMy dream of writing a book and getting published would not be a reality without the love and support of friends, family, and really good friends who have become my family. This includes my dad, Benjamin Burns; my sister, Jacquelyn Rucker; my niece, Jillian Rucker, and nephew, Christopher Rucker; and good friends Shelitha Mckee and Sophia Muckerson. You have all been there throughout all of the ups and downs on this entire roller-coaster ride. Thanks for always having my back.\nChapter 1\n\n\"Did you see the getup that little floozy had on?\"\n\n\"Shhhh.\" I glanced around to make sure the \"little floozy\" was out of earshot. Tact wasn't Nana Jo's strong suit.\n\n\"Don't shush me. I've seen Sumo wrestlers wearing more fabric.\"\n\nNana Jo exaggerated, but not by much. Melody Hardwick was a supermodel thin, heavily made-up college senior who had attached herself figuratively and literally to my assistant, Dawson Alexander.\n\n\"Surely that boy knows she's nothing more than a little gold digger.\" Nana Jo had taken an instant dislike to Melody.\n\n\"You don't know she's a gold digger. You just don't like her.\" I locked the door to the bookstore. \"Besides, it's not like Dawson has any money.\"\n\n\"He may not have a pot to pee in now, but the boy has PEP.\" Nana Jo wiped down the counters and bagged the trash.\n\n\"What's pep?\"\n\n\"Potential earning power. That boy is the best quarterback MISU's had in at least a decade. They're undefeated and if things keep going like last week, they have a shot at a bowl game and maybe a championship.\"\n\nMy grandmother had always been a sports enthusiast, but ever since the Michigan Southwest University, or MISS YOU as the locals called it, quarterback started helping out in my bookstore, she became more of a fanatic.\n\n\"He was embarrassed. Did you see how she clung to him?\"\n\n\"Dawson's a big boy. He can make his own decisions.\"\n\nBased on the look she gave me, she wasn't convinced. Frankly, I wasn't convinced either. I was concerned about him too. School was a challenge for Dawson. At the end of his freshman year, he was placed on academic probation. Thanks to a lot of hard work and tutoring from me and Nana Jo throughout the summer, he'd raised his grades, avoided academic suspension, and turned his life around. He didn't have to work at the bookstore anymore. His football scholarship covered room and board. I never wanted to charge him for staying in the studio apartment I created in my garage, but student athletes had to pay the going rate for housing and get paid fair market wages for work.\n\n\"Girls like that ain't nothing but trouble. You mark my words. Just like Delilah, she'll come after him with a pair of scissors first chance she gets. That woman is nothing but trouble.\"\n\nNana Jo's words broke my reverie and brought back the worry I thought I'd eliminated. I tried to shake it off, but it lingered at the back of my mind.\n\nWe cleaned the store and then she hurried off for a date with her boyfriend, Freddie.\n\nI took a quick tour around the store. I looked at the books neatly stacked on each shelf. It was still hard for me to believe I owned my own mystery bookstore. Market Street Mysteries had been a dream my late husband and I shared for years. After his death over a year ago, I was finally living our dream. I walked down each aisle and ran my hands across the solid wood bookshelves that still smelled woodsy and fresh and shined with the oil polish Andrew, my Amish craftsman, gave me. After six months, the store was doing well and I still got a thrill walking through and realizing it was mine. My four-legged companions on these strolls trailed along behind, toenails clicking on the wood floors. Toy poodles, Snickers and Oreo, may not share my love of mysteries, but they definitely approved of the baked goods that made their way under tables and counters.\n\nThe back of the bookstore was enclosed to provide a yard for privacy and an area for the poodles to chase squirrels and bask in the sunlight. As fall hit the Michigan coastline, the weather had turned cool. The leaves were starting to darken from bright shades of yellowy green to deeper, rich hues of amber, burgundy, and russet. Lake Michigan was also undergoing a change from the deep blue calm of summer to the pale blue that blended into the horizon and was only discernible from the sky by the choppy white swells that danced across the surface and pounded the shore. Autumn was my favorite time of year, and I lingered outside and enjoyed the sunset until Snickers reminded me she hadn't been fed by scratching my leg and ruining my tights. I needed to remember to make an appointment with the groomers first thing tomorrow or give up wearing skirts.\n\nWhen my husband, Leon, and I dreamed of the bookstore, we planned to make the upper level into a rental unit to offset the cost. After his death, I sold the home we'd lived in and turned the upper level into a two-bedroom loft for me and the poodles. Nana Jo moved in after a dead body was found in the back courtyard, but she still had her villa at a retirement village. I never dreamed how much I'd enjoy living in the space.\n\nNext week would be one year since Leon's death. The pain was less crippling. The bookstore kept me busy during the day. But the nights were still difficult. I started writing to help occupy my time and my mind. Six months ago I'd finished the first draft of a British cozy mystery and spent the last few months editing. Nana Jo wanted me to send it out to an agent, but that would involve allowing someone besides me and my grandmother, who loved me, to read it. I wasn't ready for that type of humiliation and rejection yet. Besides, in the unlikely event that a publisher was interested in my book, they'd want to know what else I had. What if one book was all I had in me? The only way to find out would be to try again. So after dinner, I made a cup of tea and headed to my laptop.\n\nWickfield Lodge, English country home  \nof Lord William Marsh\u2013November 1938\n\nThompkins entered the back salon where the Marsh family was having tea and coughed. \"I'm sorry, but the Duke of Kingfordshire is on the telephone.\"\n\nLady Daphne was in her favorite seat by the window. She started to rise but was stopped when Thompkins discreetly coughed again.\n\n\"His Grace the duke asked to speak to your ladyship.\" He turned toward Lady Elizabeth.\n\nLady Elizabeth Marsh glanced at her niece Daphne, noting the blush that left her cheeks flushed. She placed her teacup down and hurried out of the room. In the library, she picked up the telephone. \"Hello, James dear, is there\u2014\"\n\n\"Thank goodness you're home. I'm sorry but I don't have time for pleasantries. Time is of the essence.\" Lord James FitzAndrew Browning, normally calm and composed, had a slight tremor in his voice, which reflected the urgency of his call even more than his words and lack of propriety. The duke took a deep breath and then rushed on. \"This is going to sound strange, but I need you to trust me. You're going to get a call from the Duchess of Windsor asking for permission to move her hunting party to Wickfield Lodge this weekend. It's vital she be allowed to do so.\"\n\nWhatever Lady Elizabeth expected, it hadn't been this. She stood frozen for a moment before recovering herself enough to respond. \"Well of course, James. We . . . we have no plans this weekend.\"\n\nJames released a huge sigh, and she could almost see him wiping his brow.\n\n\"James, you know we're happy to help any way we can, but you mentioned this was 'vital.' Vital to whom?\"\n\nJames hesitated a moment before responding. \"Vital to England. The Crown. Maybe the entire world.\"\n\nChapter 2\n\nSaturdays were busy days at the bookstore, and I was thankful my nephews, Christopher and Zaq, were home from college for fall break and helping out. The twins were invaluable in getting the bookstore up and running over the summer. The boys were twenty and while they were identical, their personalities were so different it was very easy to tell them apart. Both were tall and slender. Christopher was business oriented and preppy, while Zaq was technology inclined and edgier. Neither was a mystery lover, but they each had their own gifts, and I was thankful they were willing to spend time helping out their aunt and earn extra pocket money.\n\nNana Jo was a mystery lover and was great at helping match customers with authors and mystery subgenres like hard-boiled detective stories, cozy mysteries, or police procedurals.\n\nToday was a home football weekend for MISU and a bye week for the twins' school, Jesus and Mary University, or JAMU to the locals. When Dawson started working at the bookstore, I toyed with the idea of putting a television in the store so we could watch him play on Saturdays. However, a television in a bookstore seemed paradoxical. I compromised by foregoing the smooth jazz I normally piped in and tuned into the sports channel instead, at least for MISU and JAMU games. I expected complaints from people who liked to sit and read in peace and quiet. But so far the comments were all positive. I suspected the lack of protest was due to the customers' desire to support a hometown boy combined with their affection for Dawson's baked goods. They were willing to give up a little peace and quiet to support someone they knew.\n\nThankfully, Dawson and the MISU Tigers had today's game well in hand with a healthy lead of three touchdowns. Home team wins made for happy customers, and happy customers spent more money. As locals discovered Dawson lived and worked here, I'd noticed an increase in traffic. Many were football fans who wanted to congratulate him, talk sports, and get autographs for wide-eyed kids. The others were infatuated young girls who glanced shyly at him when he was working and then hid behind books, giggling whenever he looked at them. Regardless of the reason, the extra traffic was good for business.\n\nMISU won handily and I had a very good day in sales. The twins had dates and hurried out immediately at closing.\n\n\"You should go to the casino with me and the girls,\" Nana Jo said.\n\n\"Thanks, but I think I'll stay home. I want to get some writing done.\" We reshelved books and cleaned the store.\n\n\"Great. You started working on the next book in the series? You know, I'm really proud of you. But you still need to start sending your book out to agents. I hear getting published is a long process. I read somewhere Agatha Christie was rejected for five years before she got her first book deal.\"\n\n\"I know. I\u2014\"\n\nThe alarm system I'd installed this summer startled me and I dropped the books I was shelving. The alarm buzzed whenever a door or window was opened, even if the system wasn't armed. Nana Jo stepped around to see who had entered and I picked up the books I'd dropped.\n\nI placed the books on a nearby table and headed for the front of the store. I could have sworn I'd locked the door. Just as I came around the corner, I heard Nana Jo.\n\n\"We're closed.\"\n\n\"Oh, I know. I just thought I'd wait for Dawson.\"\n\nI struggled to recognize the voice. As I got to the main aisle, I saw Dawson's scantily clad girlfriend, Melody. Today's ensemble included more fabric than the one she wore yesterday, but not by much. A short black skintight miniskirt with a deep V-neck mesh cut top with fabric that barely covered her breasts and red, six-inch heels that Nana Jo's friend Irma called hooker heels.\n\n\"Lord have mercy. What're you wearing?\" Nana Jo stared openmouthed.\n\nThe shocked expression wasn't lost on Melody, who laughed and twirled to insure Nana Jo got the full effect. \"You like?\"\n\n\"Is someone watching your pole?\"\n\nMelody flushed and cocked her head and took a step forward as though she were about to say something insulting.\n\nYounger people often thought of the elderly as feeble and weak. However, my Nana Jo was close to six feet, two hundred pounds, held a green belt in aikido, and could shoot a bat off the top of a building at three hundred yards. Don't ask me how I know that. Despite the difference in their ages, in a fight, my money was on Nana Jo.\n\n\"Dawson isn't here and the store is closed.\" I stepped in between the two women. \"If you're looking for Dawson, I suggest you try campus.\"\n\nFor a moment, Melody looked at me as though I were gum she'd scrapped from the bottom of her shoe.\n\n\"What's going on?\"\n\nI was so intent on preventing an altercation between Nana Jo and Melody I hadn't heard Dawson enter through the back door.\n\nApparently, Melody hadn't either. \"Dawson. How long have you been there?\" She smiled big.\n\n\"Long enough.\" The chill in his words made me turn to look at him. His eyes were hard and his face was set like granite. \"What're you doing here, Melody? I told you we were finished yesterday.\"\n\nMelody kept her smile in place as she sauntered around me. \"I knew you couldn't really mean that. We both said things we didn't mean yesterday.\" She stood inches from Dawson and placed her hands on his chest and leaned close. \"Let's go up to your room and talk things over.\"\n\nDawson didn't move for several seconds, but I could see the vein in the side of his forehead bulge with each breath. Finally, he grabbed Melody by the wrist.\n\nShe winced in pain. \"Ouch. You're hurting me.\"\n\nDawson turned and walked out the way he came, dragging Melody by the wrist along with him.\n\n\"I guess he was smart enough to see through that little cheap hussy after all,\" Nana Jo said. \"I think that's the last we'll see of her.\"\n\nI hurried to secure the front door. Something in the way Melody looked and a flutter in my spine told me Nana Jo was wrong.\n\n* * *\n\nNormally, Sundays were spent with my mom. Church, lunch afterward, and girl time. This Sunday was no different. Today we were shopping in downtown South Harbor.\n\nUnlike North Harbor, South Harbor had a bustling downtown with picturesque cobblestone streets and brick storefronts that sold everything from fudge and truffles to overpriced coffee. Mixed between quaint soda shops and antique stores were clothing stores with shoes that cost more than a month of my salary when I was a teacher.\n\n\"Honey, isn't this cashmere sweater lovely? It would look great on you.\" My mom held up a bubblegum-pink garment that looked as though it might fit one of my thighs.\n\n\"Mom, I couldn't fit my imagination in that sweater.\"\n\n\"They have larger sizes, dear. I really think you need to upgrade your wardrobe. Everything you own is black or brown. You look like you're still in mourning.\" She placed the fluffy concoction up to my neck.\n\nI glanced at the tag and nearly choked. \"Are you joking? That sweater costs more than my house payment.\"\n\n\"You really should put more effort into your appearance. You've really let yourself go since Leon died. I think you're hiding behind your mourning and it's time you started living again, and maybe dating.\"\n\nI stared openmouthed. \"Not all of us can live the life of a princess. I don't have the time or money to waste getting my nails and hair done and buying overpriced sweaters. I have a business to run.\"\n\nThe salesclerk, who had walked up with a bright smile on her face, turned and walked away.\n\nMy mom sighed and replaced the sweater. She walked to the back of the store. That sigh spoke louder than any words could have. Obviously I had disappointed her again. I stood there for a moment and then sorted through the rack of sweaters, looking for one that would fit over my head without making me look like an overstuffed sausage. I could afford the sweater. That wasn't the problem. Finances had always been tight when Leon and I were working. A cook and an English teacher didn't buy cashmere sweaters. But I'd sold the house and used the insurance money to buy the building. The bookstore was doing well, not Fortune magazine worthy, but thanks to low overhead, frugal spending, and hard work, it was making a profit. One cashmere sweater wouldn't break me, and it would make my mom happy. But, as a grown woman in her mid-thirties, I shouldn't have to buy a sweater I didn't want to make my mom happy. I wished Nana Jo had come with us today. She would have understood and helped intercede between me and my mom.\n\nMy mom was so very different from Nana Jo; it was hard for me to imagine my grandmother gave birth to her. They were polar opposites. Josephine Thomas was tall and hardy. My mom, Grace Hamilton, was five feet, less than one hundred pounds dripping wet, and delicate. My mom was like a dainty porcelain figurine you keep on the tallest shelf behind a glass door, locked away from harm for fear of breaking it. Nana Jo blamed my grandpa, who always called my mom his little princess, for planting the \"princess seed\" in her head. In her mid-sixties, my mother had never had a job outside of the home. She'd never paid a bill until after my dad died. She was the princess.\n\nI dropped my mom off at her South Harbor condo and headed back over the bridge to North Harbor, where I belonged. I glanced at the pink shopping bag on the seat that contained a white cashmere sweater I would be too afraid of spilling anything on to ever wear and swung my car into the parking lot of a nearby liquor store. I glanced at my watch. Thankfully, it was after twelve, when alcohol could be purchased. I looked at the license plates of the cars parked in the lot, noting the majority were Indiana residents who had escaped across the state line into Michigan, where they could buy alcohol on Sunday. We were all escaping from something, but I didn't have the time or energy to figure out what at the moment. A bottle of wine would have to substitute for therapy for now.\n\n* * *\n\nDuring the summer, I saw quite a lot of Dawson. When the fall semester started, we barely saw each other, despite the fact he lived in the apartment over my garage. Twice daily football practices, weight training, and classes took up a lot of his time. But Dawson loved baking and he was really good at it. His apartment was a tiny studio with only a one-burner stove, which made it challenging to bake on a large scale. Dawson had gotten into the routine of using my kitchen to bake enough goodies to get us through the week at the bookstore. So, when I entered through the back door, I smelled a sweet, delicious aroma wafting down the stairs to greet me.\n\nI climbed the stairs without my normal escorts. Snickers and Oreo usually heard the garage door and bounded to the bottom of the stairs to greet me. However, the possibility of a cookie or treat dropping to the floor was a greater enticement than seeing me.\n\nI placed my pink bag on the counter with less care than I used for the bottle of wine. Dawson had his back to me as he lifted a tray of cookies out of the oven and placed them on a rack on the counter.\n\n\"What an amazing smell.\" I breathed deeply and allowed the smell of vanilla, almonds, and sugar to fill my senses.\n\n\"Thanks. You're just in time to try one.\" Dawson turned to face me.\n\n\"Oh my God! What happened to your face?\"\n\nHe didn't say anything, merely hung his head. I hurried around the counter and turned his face toward the light to get a closer look. Three red scratches trailed across both cheeks. There was a gash under his left eye and a bruise on his forehead. His eyes were bloodshot and dark circles underneath indicated he hadn't slept.\n\nHe tried to turn away, but I held his chin and forced him to look at me.\n\n\"What happened to you?\"\n\nWe stood like that so long I didn't think he would answer.\n\nEventually, the silence grew too much for him. \"I'm fine.\"\n\nI snorted. \"Well, you sure don't look fine.\"\n\nDawson shrugged. \"It's nothing.\" He forcefully, but gently, pulled my hands away and walked to the back of the kitchen. He leaned against the wall and folded his arms, providing a barrier.\n\nI took a deep breath and tried to steady my breathing. \"Was it your father? Is he out of jail?\"\n\nHe shook his head.\n\n\"Then who?\"\n\nHe hung his head. \"Let's just say Melody didn't take our breakup well.\"\n\n\"You should go to the doctor. Those scratches look deep, you\u2014\"\n\nHe was shaking his head before the words were out of my mouth. \"If I go to the doctor, the newspaper might find out.\"\n\nSad that at nineteen you had to be concerned about the newspapers running a story about a girl who lashed out when her boyfriend broke up with her. But this season the MISU Tigers were getting a lot of publicity, Dawson in particular.\n\nI went to the bathroom and got a cold compress and mercuric acid. He didn't balk when I made him sit at the dining room table and didn't say one word when I started to treat the cuts. \"Newspapers are the least of your worries. Wait until Nana Jo finds out!\"\n\nHe winced, but I wasn't sure if it was the mercuric acid or the thought of what Nana Jo would say.\n\n\"What an unusual request. James didn't have any other information?\" Lord William asked as he absent mindedly broke off a piece of his scone and fed it to Cuddles, the Cavalier King Charles Spaniel positioned at his feet.\n\n\"Not that he told me. Although, I'm sure he'll fill us in when he gets here.\" Lady Elizabeth picked up the knitting she kept nearby, which she said helped her think clearly.\n\n\"Is Lord Browning coming too?\" Lady Daphne Marsh picked at an imaginary string on her skirt and avoided making eye contact with her aunt.\n\n\"Well, I suppose so, although I didn't ask him. I just assumed he would.\" Elizabeth looked at her husband. \"You don't mind do you, dear?\"\n\n\"No. No. Of course not.\" Lord William tossed the remains of the scone down to the dog and pulled out his pipe. \"I'm sure James wouldn't have asked if it wasn't important.\"\n\n\"My thoughts exactly.\" Lady Elizabeth resumed her knitting.\n\n\"I don't suppose you know anything about this?\" Lord William asked his niece.\n\nLady Elizabeth Marsh sighed. Sometimes her hus band could be rather slow to read the signs or he would have noticed his niece, Daphne, had said very little since Lord James Browning's name was mentioned. The two met six months ago when he came to help out his friend and old classmate Victor Carlston, Earl of Lochloren, who was accused of murdering one of Daphne's beaux. At the time, Victor believed he was in love with Daphne and chivalrously stepped in to protect her by allowing the police to believe him guilty of murder. Lord James helped to reveal the true killer and ensured his friend's freedom. Victor was now living in wedded bliss with Daphne's sister, Penelope, down the road at his family estate, Bidwell Cottage. The Marshes hoped another announcement of marriage would be forthcoming as Lord James and Lady Daphne seemed destined for the altar. However, the duke's visits of late had been fewer and far between.\n\n\"No. I haven't spoken to James . . . ah, the duke in nearly two weeks,\" Daphne said almost in a whisper.\n\n\"I suppose you better tell Thompkins and the rest of the staff to prepare for guests,\" Lord William said.\n\n\"I would, but I think I want to wait until we're sure,\" Lady Elizabeth said. \"Technically, she hasn't asked yet. I don't even know how many people to expect.\"\n\n\"Do you suppose David will come too?\" Lord William asked.\n\nLady Elizabeth knitted. \"I have no idea. The last I heard, he was in France.\"\n\n\"I don't suppose there will be a problem with the Queen Mother and the rest of the family?\" Daphne asked.\n\n\"Well, I guess that depends on what type of problem you mean.\" Lady Elizabeth knitted silently for a few moments. \"Bertie and Elizabeth are still very angry and the Queen Mother is disappointed in David. I still feel rather badly that none of the family attended the wedding.\"\n\nLord William sputtered. \"But really, how could we attend? It would have been a sign the family agreed with his abdication to marry a divorced woman\u2014an American.\" Lord William waved his pipe while he spoke, flinging ashes across the sofa.\n\nLady Elizabeth looked up and shook her head. The sofa was starting to show bare patches from the maids brushing off tobacco. It would have to be recovered soon. \"Well, I don't know if the fact she was twice divorced or an American was the objectionable part. I might have considered attending if the wed ding were one day earlier or one day later.\"\n\n\"I agree. It was as though they were thumbing their noses at the family by getting married on King George's birthday,\" Daphne said. \"Really, his own father's birthday.\"\n\n\"Bad form.\" Lord William refilled his pipe.\n\n\"Regardless of the circumstances, David and Bertie are brothers, and I believe they'll work things out in the end,\" Lady Elizabeth said. \"Besides, James said it was vital to the Crown that she hosts her hunt ing party here. So, that must mean the king is at least aware of the event.\"\n\nLord William nodded and puffed on his pipe.\n\n\"At any rate, it doesn't appear we'll find out how the Crown feels about things. The duchess hasn't called. What if she's found another place to hunt?\" Daphne asked.\n\nThompkins entered the room silently and coughed. \"Her Grace, Wallis Duchess of Windsor is on the telephone for your ladyship.\"\n\nChapter 3\n\nNana Jo's response when she saw Dawson's face was loud and littered with old-fashioned words like \"floozy,\" \"harpy,\" \"tart,\" and \"shrew.\" When she calmed down, she mixed up a concoction with aloe vera gel, honey, vitamin E oil, and baking soda. Dawson looked like he had leprosy most of the day Sunday, and he had to fight off Snickers, who kept trying to lick off his mask, but Monday his face looked so much better, it was like night and day. The scars were still there. Only time would truly heal them, but the improvement was amazing.\n\n\"Mrs. Thomas, you're a miracle worker.\" Dawson kissed Nana Jo on the cheek.\n\n\"Well, you need about two more days before the scars will disappear completely.\" Nana Jo stared at her handiwork. \"But at least you don't look as though you've been in a catfight.\"\n\n\"You never cease to amaze me,\" I said after Dawson hurried off to campus.\n\nNana Jo and I sat at the breakfast bar and drank coffee.\n\n\"Where on earth did you learn to mix up your healing paste?\"\n\nNana Jo smiled as she sipped her coffee. \"I grew up on a farm. There was always some kind of accident that happened on a farm and most people were too poor to go riding off miles to a doctor. My grandmother used to be the local midwife and, well . . . medicine woman. She mixed all kinds of things up in her kitchen and grew herbs for healing everything from the croup to rheumatic fever.\"\n\n\"I never knew that.\" I stared at my grandmother. I'd known this woman all my life and she was still able to surprise me.\n\nNana Jo shrugged. \"I never thought it worth talking about. Most of those old remedies would be considered nothing more than old wives' tales nowadays.\"\n\n\"Scientists are discovering that a lot of those old remedies actually worked. I read an article recently that chicken soup really does help with a cold. Although scientists aren't sure if there is some ingredient in the chicken soup itself or if it's in the person's mind. Whatever the reason, it works.\"\n\nWe sat for a few moments and talked about poultices, plasters, and herbal teas. Then we went downstairs to the bookstore.\n\nI had a lot of fears when I quit my job as an English teacher and opened the bookstore. Would I be able to handle things alone? Would I be able to make enough to support myself? Did people still read books? The answer to all of those questions was yes. Recently, an old friend I hadn't seen in over twenty years asked if I found working in a bookstore monotonous and boring. I didn't even need to think before I answered. Market Street Mysteries was a lot of things, but monotonous and boring certainly wasn't one of them. New people came in every day. Boxes of books arrived weekly. Some boxes included books from writers I'd read for years, which were like old friends. Familiar series from Victoria Thompson, Emily Brightwell, Jeanne M. Dams, and Martha Grimes sent a thrill of excitement through my body as I gazed at the bright covers and anticipated the joy of figuring out whodunit. There was also the joy of discovering new writers and wondering which ones would be added to my list of favorites. On those rare moments when the store was quiet, I sometimes went for a walk in downtown North Harbor and stepped into shops owned by my neighbors. The bookstore had helped me through one of the worst times of my life, the death of my husband and best friend. I'd created a new life for myself with new friends and I hoped a new career as a writer, one day.\n\nA few doors down from Market Street Mysteries, a new restaurant had opened. I stood in front of the window and stared at the menu taped on the door. I looked at my watch and realized it was after two, and my stomach growled as I read the menu. I stepped inside and waited while my eyes adjusted to the darker interior.\n\n\"I'm glad you decided to come in.\" A man with salt-and-pepper hair and beard, cut close in the style worn by the military, soft brown eyes, and a big smile came out from behind the bar.\n\nI must have looked puzzled because he motioned to the window. \"I saw you outside.\"\n\n\"Oh. Yes. Sorry.\"\n\n\"No need to apologize. That's why I put the menu up. I was hoping it would entice people to come inside.\"\n\n\"Well it worked.\" I laughed.\n\n\"How about a nice table by the window?\"\n\nI nodded and took a seat in the chair he held out for me.\n\n\"I have a lovely white wine from a local vineyard.\"\n\n\"Oh, no. Just water with lemon, please.\"\n\nWhen he left, I looked around. The restaurant was clean and decorated with an urban edge. Exposed brick walls, stained concrete floors, and iron fixtures created a modern, hip atmosphere. Televisions lined the wall behind the bar. My waiter returned with a glass and a carafe of ice water with lemon.\n\nHe smiled as he placed the carafe of water on the table. \"You own the mystery bookstore a few doors down, don't you?\"\n\nI took a sip of water and nodded.\n\n\"I thought so. I've been trying to meet all of the other store owners around here. I'm Frank Patterson.\"\n\nHe held out his hand and we shook.\n\n\"Samantha Washington, but you can call me Sam.\"\n\n\"Sam, I'm pleased to meet you. How long have you been down here?\"\n\nI knew he was asking about how long my bookstore had been open. I looked at him and started to respond when my attention was caught by the picture on the television behind him.\n\nMelody Hardwick's picture filled the screen. Then it was replaced by pictures of a body covered by a blanket.\n\nI gasped.\n\nThe words that scrolled across the bottom of the screen said Melody's body had been found by early morning joggers. The police believed her death was the result of \"foul play.\" The picture that next filled the screen and nearly stopped my heart was of Dawson getting into the back of a police car.\n\n* * *\n\nI didn't remember the walk back to the bookstore. Nana Jo said I came in looking like a whirling dervish. I did remember marching into the South Harbor police station with Nana Jo. The brick two-story building was downtown and not far from my bookstore. North Harbor and South Harbor shared the same Lake Michigan coastline. The two towns were separated by the St. Thomas River that zigzagged through northern Indiana and southern Michigan for over two hundred miles and ended as it wrapped around North Harbor in a U and flowed into Lake Michigan.\n\nThe county police station and courthouse were attached and comprised a sprawling complex located on an area that sat on a small street in between North and South Harbor. Other than field trips as a child, I had only been to the complex as an adult when I was summoned for jury duty. My memory of the facility was prior to 911 and didn't include security cameras and metal detectors that would rival those at the nearby River Bend airport.\n\nI was so concerned about Dawson I didn't remember a number of things from the time I saw his face on the television to the moment I walked into the police station. However, the memory that would live with me until my death would be when Nana Jo set off the metal detectors and we were instantly swarmed by police officers with guns drawn, all shouting for us to raise our hands and lay down on the floor. I remember the officer who pulled my wrists behind my back and the feel of the cold metal handcuffs as he placed them on my wrists. I looked over at my grandmother as she lay by my side, also cuffed and on the ground. My heart raced and my blood pounded in my head. Yep. That was a memory that would stay with me forever.\n\nThankfully, my nephews hadn't been idle after we left the bookstore. One of them must have called their mother. Never had I been so happy to see and hear my sister, Jenna, as I was at that moment.\n\n\"What do you think you're doing?\" Jenna said in the cold, steely voice I dreaded. \"You have ten seconds to get my sister and grandmother off that floor or as God is my witness, I will sue every last one of you.\" Jenna was a criminal defense attorney well-known by the police for her tough, no-nonsense attitude. I'd once heard that the district attorney's office referred to her as a pit bull and I had to say, as her sister, I thought it was pretty accurate.\n\n\"You know these women?\" One of the officers stepped up from the pack.\n\n\"I just said that, didn't I? And you have two seconds to lower your weapons,\" Jenna said between clenched teeth. Then she turned her back to the officers and looked directly into the camera that was positioned over the door. \"As you can see, these imbeciles have my sister and elderly grandmother handcuffed and laying on the cold concrete. Obviously they aren't a threat, yet these officers continue to point their weapons.\"\n\nThe officers put away their guns. One of the officers helped me to a standing position. It took two of them to help Nana Jo to stand.\n\n\"Your 'elderly' grandmother set off the metal detector.\" He turned Nana Jo's purse upside down and all the contents flew across the floor. His cocky smile turned into a sneer as he looked at Nana Jo's iPad, phone, notebook, brush, holster, makeup, and about twenty other items lying on the ground. He kicked the empty holster and looked around the floor. But there was no weapon.\n\n\"Hey, Barney Fife, you break my iPad and you're buying me another one,\" Nana Jo said.\n\nJenna smiled and continued to address the camera. \"No weapon, just an empty holster. And if she had brought her weapon, you'd find her permit to carry in her wallet.\"\n\nThe smirk vanished as the officer looked at the wallet and found the permit. He returned the wallet and other belongings to Nana Jo's purse and nodded to the officers holding us and the handcuffs were removed.\n\n\"I guess it was the iPad that set off the detector. Anyway, she should have announced she had a weapon and shown her carry permit immediately,\" he said as though he were educating a child\u2014bad mistake.\n\n\"I'll keep that in mind the next time when I actually have a weapon. Of course, you would have known that if you nincompoops would have waited a minute before Wyatt Earp and the rest of the posse drew their guns like they were about to shoot it out at the O.K. Corral.\" Nana Jo snatched her purse away from the officer.\n\n\"I intend to subpoena the videotapes, so make sure nothing happens to them. If my sister or grandmother is injured due to this incident, you will be hearing from me.\"\n\nThe officer looked as though he wanted to say something, but the look in Jenna's eyes showed him silence would be his best defense.\n\n\"Now, where have you taken Dawson Alexander?\" Jenna's question brought me back to the reason for our visit.\n\nWe were escorted to a reception area. We signed in and were then led to a small conference room. Jenna was allowed to go with the police officer, but Nana Jo and I were left to wait. I wanted to protest, but the look in Jenna's eyes convinced me silence was my best defense as well.\n\nStill flustered from the experience of being handcuffed and having guns pointed at me, I was glad for an opportunity to sit down. I hadn't realized how nervous I was until I poured myself a cup of water from the pitcher on the table and my hands shook so badly I spilled most of it on the table.\n\n\"Are you okay?\" Nana Jo grabbed some napkins from her purse and helped me clean up the water.\n\n\"No,\" I answered truthfully. \"But I will be. That was scary.\"\n\nNana Jo smiled. \"That was pretty nerve-racking. I'm sorry, honey.\"\n\n\"I'm glad you thought to leave your gun at home.\"\n\nShe smiled. \"I didn't.\"\n\n\"What?\"\n\nShe shook her head. \"Honestly, I was as surprised as that policeman. I must have left it at home, but it was an oversight. I was so upset about Dawson I didn't even think about my peacemaker.\"\n\n\"Lucky for us.\" I smiled and leaned across the table. \"I'm pleasantly surprised to know you have a permit to carry.\"\n\nNana Jo grinned. \"Well, we can thank Jenna for that too. To be honest, I've been carrying a gun for more years than you've been alive. But after everything that happened in the summer, Jenna convinced me I needed a permit.\"\n\nWe waited for what felt like an eternity but was only twenty minutes. Jenna returned with Detective Bradley Pitt. Detective Pitt had been the lead investigator in the murder of Clayton Parker, a realtor who was found in the backyard of my building over the summer. He was an unpleasant man with a knack of jumping to the wrong conclusion, like when he thought I murdered Parker. Detective Pitt was short with a bad comb-over and polyester pants that were too short and a bright, flowered polyester shirt that was too tight. His certainty that I was a murderer forced me, Nana Jo, and her friends from the retirement village to become sleuths to figure out who murdered Parker.\n\n\"Stinky Pitt, I should have known you'd be behind this debacle. Once again you've got everything bass-ackward, upside down sideways.\" Nana Jo taught Detective Pitt in elementary school and loved to embarrass him by using his childhood nickname.\n\nDetective Pitt's jaw clenched and his ears got red. \"I wish you would remember that no one calls me that anymore.\" He glared at Nana Jo, who contrived to look innocent.\n\nShe loved to goad him and knew exactly what she was doing. But she was careful never to call him Stinky Pitt around other police officers.\n\n\"Look. Dawson Alexander is absolutely not a killer,\" I said. \"You've arrested the wrong person.\"\n\n\"We haven't arrested anyone yet. We just brought him in for questioning.\" Detective Pitt walked around the small conference table. \"But, that's not to say we won't be pressing charges. We've barely had time to talk to him.\"\n\n\"And from now on, you won't be talking to him without legal counsel present,\" Jenna said. \"Now, I want to talk to my client.\"\n\nDetective Pitt looked as though he wanted to comment but, to his credit, he kept silent. He merely shook his head and mumbled something about lawyers meddling or muddling, I couldn't really tell, under his breath as he left.\n\n\"Wow. You're tough,\" I said with more than a little awe.\n\nJenna laughed. \"They don't call me pit bull for nothing. I earned that title.\"\n\nWe waited for several minutes and then the door opened and Detective Pitt returned with Dawson.\n\nDetective Pitt looked as though he intended to stay and listen, but Jenna wasn't having it.\n\n\"Thank you, Detective. I'll let you know when we're done.\"\n\nHis only response was to turn and leave, but the door did seem to close with a bit more force than I remembered him using previously.\n\nDawson looked as though he'd aged ten years since this morning. Could it only have been a few hours? His eyes looked tired and the scars that seemed faded this morning looked more prominent now; although I might have been more attuned to them, given our current situation.\n\n\"Dawson, I've told Detective Pitt I'm your legal counsel. However, you don't have to accept me. If there's someone else that you\u2014\"\n\n\"No. I really want you to represent me. I was just too embarrassed to ask.\" He hung his head and looked up sheepishly. \"I don't know how I'm going to pay you.\"\n\n\"Pshaw. Don't worry about that. I don't charge family.\"\n\nDawson looked surprised and misty-eyed.\n\nI was a bit misty myself. Dawson was like family. Leon and I were never blessed with children, but Dawson had come into my life during a time when I was in need of someone to focus my attention onto. He'd allowed me to mother him and had slid into one of the holes that Leon's death had opened up.\n\n\"Now, I need to know, what have you told them?\" Jenna looked at me and Nana Jo. \"Normally, I wouldn't do this in front of you. Conversations between a client and an attorney are privileged.\"\n\n\"That means you can't tell anyone what he says to you, right?\" I asked.\n\n\"Pretty much. It's complicated, but that's the general idea. Legal counsel is charged with giving the best advice possible. However, if someone comes to a lawyer for advice, they need to feel free to communicate everything without fear anything he says will be used against him. If he holds back, because he's afraid, I can't give the best advice.\" She looked at Dawson. \"Do you understand?\"\n\nHe nodded.\n\n\"Now, understand, only the conversation between an attorney and a client is privileged. If Sam and Nana Jo stay, they could be compelled to reveal it. That's why I would recommend they leave.\"\n\nNana Jo and I both started talking at the same time, but Dawson overrode us. \"It's okay. I didn't kill her. I don't have anything to hide. They can stay.\"\n\n\"I don't care what the law says. I'd never tell them anything,\" Nana Jo said.\n\n\"Neither would I.\"\n\nJenna merely shook her head. \"Am I the only law-abiding citizen in this family?\" She smiled. \"My family would apparently lie under oath.\" She shook her head.\n\n\"For someone I care about, I'd lie like a rug,\" I said.\n\n\"Darned straight,\" Nana Jo agreed. \"That's why we keep you and your husband, Tony, around. Two lawyers in the family come in pretty handy.\"\n\nJenna shook her head, then turned her attention to Dawson. \"I need you to tell me everything you've said to the police.\"\n\n\"I didn't really say anything. They kept me waiting in a room for almost an hour. Then Detective Pitt showed up. He started asking me a lot of questions, but I'd had a lot of time to think when I was waiting.\" He paused and then shook his head. \"My dad didn't teach me much, but he always said, 'Never say nothing to no cops, boy, not without a lawyer. They've gotta give you a lawyer in this country. Ain't America great?' That's what he would say.\" He looked around at us. \"So, I didn't say anything. I just said I wanted a lawyer.\"\n\nJenna breathed a sigh of relief.\n\n\"Attaboy,\" Nana Jo said.\n\nDawson smiled. \"I didn't think they'd do it. Detective Pitt kept saying I didn't need an attorney because I wasn't under arrest. When that didn't work, he said innocent people didn't need attorneys. If I wasn't guilty of anything, then I should want to help them.\" He looked at Jenna. \"I have to admit, I was starting to crack. If you hadn't come, I probably would have started talking.\"\n\nJenna looked as though she wanted to spit nails. She got up and started pacing around the small room. \"Why, that no-good dirtbag. I'll have his badge,\" she mumbled. After a few minutes, she sat down again and smiled at Dawson. \"I'm sorry you had that experience. In this respect, your dad was right. If you ask for an attorney, they are supposed to stop and immediately call for a public defender.\" She sighed. \"However, now on to business. Do you have a dollar?\"\n\nDawson looked puzzled but didn't question Jenna's request. He pulled out his wallet and took out a dollar and handed it to her.\n\nShe took the dollar. \"Thank you. This is my retainer. That means you are retaining me to represent you and be your lawyer. I'll write you a receipt and have you sign a document to that effect. Okay?\"\n\nDawson nodded.\n\n\"Okay. Now, you are not to talk to the police at all unless I'm with you. Understand?\"\n\nDawson nodded.\n\n\"I called your coach on my way here. He's aware of the situation. He doesn't know the university's position yet. But I'm guessing the university will want to distance themselves from you until this whole mess is cleared up. There's been so much negative publicity about football players and other athletes getting arrested, I'm sure the university counsel will recommend they take a neutral stance. But we'll deal with that hurdle when we get there.\"\n\nJenna pulled a notebook, tape recorder, and pen out from her briefcase. \"Normally, I wouldn't do things this way. But I need to know everything. Start from how you met Melody to today.\" She turned on the tape recorder.\n\nHe paused and took a couple of deep breaths before beginning his tale. \"I met Melody on campus. One day she came up to me in the quad and asked for my cell phone. When I gave it to her, she put her number in my contacts and handed the phone back. She told me to call her.\"\n\nNana Jo whistled.\n\n\"Shush. I'm willing to let you two stay, but you need to be quiet and let him finish,\" Jenna said.\n\nWe nodded.\n\nShe looked at Dawson. \"Go on.\"\n\n\"Well, she was hot and she was a senior and . . . she was really hot.\"\n\n\"We get the picture. She was hot and easy and you got involved. Is that right?\" Jenna asked.\n\nDawson nodded. \"Yeah. We were involved.\"\n\n\"How long?\"\n\n\"Only a couple of months. The season started in late August, and she came up to me in September.\"\n\nNana Jo made a sound that sounded like, harrumph. \"Figures. MISU was on a winning streak, and your picture was on the front page of the River Bend Times. She saw her chance to latch on to a meal ticket and she took it.\"\n\n\"Nana Jo, please.\" Jenna looked irritated. \"Stop interrupting.\"\n\n\"It's okay. She's right. At first I was flattered. Guys looked at me different when I walked around campus with Melody on my arm. Girls too. But she didn't really care about me. She only wanted to be seen with me. She just wanted to go to parties and have her picture taken. She wanted me to move out of my apartment.\" He glanced at me shyly. \"She went to the owner of Harbor Point Apartments and convinced him to rent the penthouse apartment to me.\"\n\n\"Harbor Point?\" Despite Jenna's warning, I couldn't stop myself from interrupting. \"Those units are really expensive. They look right out on Lake Michigan. The penthouse must cost a fortune.\"\n\n\"Normally, they lease for three thousand a month.\"\n\nNana Jo whistled, and I nearly choked.\n\nEven Jenna seemed surprised. \"How could you afford that?\"\n\n\"I can't. The owner is a big MISU fan and a friend of hers. She said he was willing to lease it to me for two hundred a month.\"\n\nJenna looked as though she could barely believe her ears. \"Two hundred? That's ridiculous. Sam could get more than two hundred a month for the garage studio. Why would he do that?\"\n\nDawson shrugged. \"I don't know, but it seemed shady. I didn't like the guy. He seemed slimy. He was older and wore polyester shirts with all these gold chains. He had really dark chest hair and big fake-looking hair that looked like a toupee. I just didn't trust him.\" Dawson looked down.\n\nAfter six months, I knew Dawson pretty well. Nana Jo and I exchanged glances. There was more to this. Jenna hadn't spent nearly as much time with Dawson as we had, but her instincts must have kicked in because she remained quiet and waited. Most people didn't recognize the power of silence and tried to fill it in quickly. As a former teacher, I can honestly say silence generated more results than anything else I'd ever said or done. It worked this time too.\n\n\"He reminded me of my dad. I knew there would be something in it for him later. One night I was playing pool with some friends by the old HOD.\"\n\n\"The HOD?\" Jenna stared. \"That trailer park owned by the House of David?\"\n\nDawson nodded.\n\nThe House of David was a religious commune that flourished in North Harbor during the early twentieth century. At one time, the area thrived with an amusement park, baseball stadium, and fruit and vegetable market. Practically all their businesses had closed long ago when their founder, Benjamin Purnell, was tried for fraud and accused of child molestation. The molestation charges were never proved, but the rumors did plenty of damage. Purnell died not long after the trial and the House of David split into two factions.\n\n\"I saw him there. Virgil Russell was at the bar, drinking.\"\n\nEither Dawson had learned to read my mind, or I'd let my thoughts show on my face, because he hurried to add, \"I wasn't drinking. I just like to go there and shoot pool. It's quiet. No one there talks football or knows who I am.\" He hung his head.\n\nI hadn't realized how much pressure he was under. I thought his biggest worry was keeping his grades up and staying on the team. I hadn't realized how much the pressure of the media and the fans was weighing on him. As I scanned his face now, I saw what I hadn't before.\n\n\"So, you saw this Virgil Russell at the pool hall,\" Jenna continued. \"Did he see you?\"\n\n\"No. I snuck out as soon as I saw him. But, he looked like he was waiting for someone. I don't know why, but I waited in the car until he came out. When he came out, he wasn't alone.\"\n\nDawson's voice got very soft. He was obviously reluctant to continue, but he took a couple of deep breaths and plowed forward. \"That's when I saw them. Melody and Virgil were together.\"\n\nJenna looked at me.\n\nI shrugged. Obviously she knew Virgil since she introduced Dawson to him. Again, my face must have revealed my confusion.\n\nDawson fidgeted and refused to make eye contact for several seconds. Then he looked at me. \"Don't you see, they were together.\"\n\nFinally, it dawned on all of us at the same time.\n\n\"It was disgusting. He had to be old enough to be her father. He had his hands all over her, and she was wrapped around him like a . . . like a . . .\" Dawson struggled to find the right simile to describe what he'd seen.\n\nNana Jo didn't have any trouble conjuring up the right comparison. \"Like an octopus.\"\n\n\"Yeah. That pretty much says it all.\"\n\nI studied Dawson's facial expressions and body language, and I knew there was more. He was holding something back. But I refused to entertain the thought he murdered that girl.\n\nJenna took a couple of deep breaths. \"Okay, so you saw your girlfriend with another man. Did that make you angry?\"\n\nDawson seemed to think about the question before answering, \"Maybe for a few seconds. But, honestly, I think I felt relieved.\"\n\n\"Relieved? Why relieved?\" Jenna asked.\n\n\"I knew things wouldn't work between Melody and me. I knew she didn't care about me. This gave me the excuse I needed to end things with her. And that's what I did. I broke it off.\"\n\n\"When was this?\"\n\nHe was silent for a few moments. \"Friday night, right after the pep rally.\"\n\n\"How did she take it?\"\n\nDawson rubbed his face. \"Not so good. At first, she tried to deny it was her. Then, when that wouldn't work, she said I was mistaken about their relationship. She claimed they were just friends.\" Dawson shook his head. \"She must think I was the biggest hayseed on the planet. There is no way I mistook their relationship when he had his hands all over her. He was groping her like a . . . like a . . .\"\n\n\"Like a blind man at a produce stand.\" Nana Jo again came to the rescue.\n\n\"Nana Jo.\" Jenna was not amused, and her voice said, either be quiet or you'll have to leave.\n\nNana Jo used her hands to indicate she was zipping her mouth shut and throwing away the key.\n\nJenna turned back to Dawson. \"Okay, so three days ago, you broke up with her. She didn't take it well. What happened next?\"\n\n\"Saturday was the football game. Later that evening, she came by the bookstore. She said she wanted to talk. I took her up to my apartment. We got into an argument.\"\n\n\"Did you hit her?\" Jenna asked.\n\nDawson shook his head. \"No. I never hit her. Although she hit me several times. She scratched me. I had to grab her wrists to protect myself.\" He held up his hands to show how he had grabbed her. \"She kicked and spit and lashed out with everything she had. I held her down on the sofa until she calmed down. Then I picked her up and put her out. She screamed and cussed and beat on the door for a while. I was afraid she'd wake up the whole neighborhood, but I never opened the door. She finally must have gotten tired and left. That was the last time I saw her.\"\n\n\"Sounds like she really made a big racket.\" Jenna looked at me. \"Did you hear it?\"\n\nI shook my head. \"No. I didn't hear a thing.\" I turned to Nana Jo. \"Did you?\"\n\nNana Jo shook her head.\n\nJenna sat and stared. \"Dawson, is there anything else you want to tell me?\"\n\nHe paused and eventually shook his head. \"No. That's everything.\"\n\n\"What about yesterday? Did you see her Sunday?\"\n\n\"No. I stayed home. I baked.\"\n\n\"That's right. I was there with him,\" I said eagerly.\n\n\"But you weren't there all day. You went to church with Mom yesterday. I know because she told me. You went to church and dinner and then shopping.\"\n\n\"Well, yeah, but he was home baking when I got home.\"\n\nJenna didn't look relieved. \"What about you?\" She turned to Nana Jo.\n\n\"Ruby Mae's granddaughter sang a solo at their church. I went to hear her and then I went to brunch with the girls.\"\n\nThe girls were Nana Jo's friends from the retirement village. They were feisty, active, and sweet each in their own way.\n\nJenna frowned. Then she got up and paced. \"I'm going to be honest. It doesn't look good. We need to find out when she died. The police will most likely arrest you.\"\n\nDawson looked terrified and, I have to admit, something clutched my chest that seemed to be restricting my airway and forcing tears to my eyes. \"But he's innocent.\"\n\n\"I know, but he has a really good motive. Plus, he had an altercation with the deceased. It's just a matter of time before they find Dawson's skin cells under her fingernails. Even a fool like Detective Stinky Pitt could get a conviction with that.\"\n\nThe thing clutching at my heart made it hard to speak. With effort, I managed to squeak out, \"But what are we going to do?\"\n\n\"I'm going to work on a defense.\"\n\nNana Jo stood. \"And we're going to figure out who killed that floozy.\"\nChapter 4\n\nIt was dark when we finally made it home. Jenna kicked us out hours earlier. She wanted to go over Dawson's testimony with him again without interruption before they talked to the police. Nana Jo and I waited for them in the waiting room. Three hours later, we saw Jenna on her way to the restroom. She didn't think they were going to charge Dawson today, but there were still hours of more questions. She suggested we go home, and she promised to call if there were future developments. I hadn't eaten much more than stale peanut butter crackers from the vending machine, and I was starving. Neither Nana Jo nor I felt like cooking, so we left and picked up a pizza on the way home.\n\nChristopher and Zaq kept things going at the bookstore while we were out. When we arrived, everything was clean and locked up tight. Crumbs on the rug in the kitchen and no other surprises told me Snickers and Oreo had been fed and let out to take care of business. I needed to remember to give my nephews each a bonus.\n\nNana Jo and I finished off an entire pizza that included, according to the menu, everything except the kitchen sink, along with a bottle of wine. Afterward, the lethargy hit and neither of us felt like talking. Before Nana Jo went to bed, she sent a text to the girls. We were scheduled to have dinner on Tuesday night at Randy's Steak House.\n\nMy body was exhausted, but I couldn't shut off my mind. I tossed and turned so much even Snickers decided the tidal wave was too much and hopped off my bed in exchange for her cozy little dog bed in the corner. Oreo was too excitable to be allowed to roam free at night. If he stepped on Snickers one time too many, he'd learn her bite was as bad as her growl. For his own safety, I crated him at night. When I finally gave up trying to sleep and turned on the light, Oreo barely glanced at me. Snickers got up, circled her bed three times, and laid down in a tight ball with her back to me. I grabbed my laptop and decided to force my mind to wind down. Escaping to the British countryside would be just the trick.\n\n\"A shame 'er ladyship 'as to entertain the likes of that American upstart?\" Mrs. McDuffie pursed her lips as though she'd just tasted something sour.\n\nThompkins, the Marsh family butler, stood erect and, without saying a word, looked with disapproval at the stout freckled housekeeper.\n\n\"Her ladyship is not one to shirk her duty.\" Thompkins looked down his long bony nose at the housekeeper.\n\n\"Well, o' course not.\" Mrs. McDuffie's face turned almost as red as her hair at the very idea anyone would imply she was criticizing her ladyship. Mrs. McDuffie might not be the most highly educated of servants, but loyalty to the Marsh family, and her ladyship in particular, she had in abundance. \"I never said she did.\"\n\nThompkins had shared the news of the hunting party with the other servants and asked if anyone had questions.\n\n\"'Ow many is comin'?\" Gladys, the latest addition to the household staff, asked softly. She was still relatively new and a bit in awe of the butler.\n\n\"Her ladyship said we should prepare for a weekend hunting party to include between twelve and fifteen guests.\" Thompkins removed an invisible speck of lint from his immaculate sleeve. \"The duchess wasn't exactly sure.\"\n\nMrs. McDuffie snorted. \"Not sure? 'Ow can she not know 'ow many people are comin' to 'er own bloomin' party?\"\n\nGladys and the two footmen had the decency to suppress their laughter. Millie and Flossie were day help and didn't have the same training.\n\nThompkins could remember a time when servants were a noble breed who took pride in a life of service to the great families of Britain. In his early days with the Marsh family, there was a large staff of more than fifty servants. Most lived on the grounds in the servants' quarters. Now there were only seven full-time servants who were live-in. Himself, Mrs. McDuffie the housekeeper, Gladys the housemaid, Mrs. Anderson the cook and her daughter, Agnes, the undercook, and Frank and Jim the footmen all lived on the grounds. The term footman was rather misleading. The two men weren't footmen in the traditional sense of the word. Basically, they helped with heavy lifting and did whatever was needed around the house and estate. Frank was the son of the groundskeeper and lived with his dad in the grounds keeper's cottage at the back edge of the property. The two new housemaids, Flossie and Millie, came in daily from the village. The Marsh family didn't entertain often, so the smaller staff saw to the family's basic needs. Thompkins used a temporary staffing service when additional help was needed, which thankfully wasn't often.\n\nHe looked at the staff waiting for his instructions and cast aside thoughts of the old days. Thompkins wasn't delusional. He knew the \"good old days\" weren't good for everyone, especially servants. Many servants were ill treated by their masters and poorly paid. They worked fourteen or more hour days and retired to tiny shared rooms where they roasted in the summer and froze in the winter. He'd grown up serving the Marsh family, and while the previous duke was a kind man, the conditions were less than ideal. Fewer servants meant everyone at Wickfield Lodge had their own room, something that would have been given to only the privileged few in days past. The wages were decent, modern conveniences made the work easier, and the Marshes treated their servants with respect, allowing them to have lives outside of service and encouraging them to better themselves through education. No, Thompkins didn't long for the way things were so much as he longed for the work ethic and pride in a job well done and the respect.\n\nFlossie looked at Thompkins in wide eyed delight. \"Mr. Thompkins, sir, do you know if Her Royal Highness will be bringing any servants, like a lady's maid? I'd be more than happy to\u2014\"\n\nMrs. McDuffie snorted. \"Bloody 'ell. 'Er Royal 'Ighness indeed. I'll not be bobbing and curtsying to that little American piece of\u2014\"\n\n\"Mrs. McDuffie.\" Thompkins, the normally calm, cool, and collected butler prided himself on always maintaining an even demeanor and not allowing any hint of emotion to show, trembled with the effort to maintain his calm. He took several deep breaths and regained his composure. He turned to the maid. \"You will address the duchess as Her Grace, or Duchess of Windsor, or simply as duchess. The king has not awarded the duchess the title of Her Royal Highness. We therefore shall not refer to her as such.\" Thompkins looked sternly at all the servants but lingered on Flossie, who turned red and dropped her gaze.\n\nFlossie was a good girl, obedient and well-mannered. Her biggest flaw was an affection for the cinema and American actresses. He had once caught her dancing around the kitchen, pretending she was Miss Ginger Rogers. He'd told Mrs. McDuffie, who gave her a stern talking to. She took the correction well, without any sulking, and he'd never caught her dancing again. With any luck, this tendency toward star worship was a phase she'd outgrow in time. He looked kindly at her now. Her demeanor was composed and respectful, but her eyes still held a spark which indicated she was still starstruck.\n\n\"I believe the duchess will bring her own maid, but I do not know about any other staff. I will meet with her ladyship later this afternoon and find out what the staffing needs are.\" Thompkins turned to the cook. \"Mrs. Anderson, her ladyship will want to go over the menu with you for the weekend. I believe the Duchess of Windsor is very concerned about her weight and some . . . modifications may be requested.\"\n\nMrs. Anderson sniffed but nodded her head. She was a good cook but could sometimes be a bit touchy about anyone who interfered with her reign over the kitchen. Even Thompkins walked a fine line when it came to the cook. He took a deep breath, relieved she had not taken offense at the request.\n\n\"Is that all?\" Mrs. McDuffie looked at the butler as she rose from the table.\n\nThompkins nodded.\n\n\"Then come along, girls. We better get busy airing out the bedrooms.\"\n\nMrs. McDuffie and the maids got up and headed up the back staircase to make sure all of the guest rooms were properly aired and ready for their guests.\n\nFrank and Jim stood and turned to Thompkins. Frank, tall, blond, and good-natured, was solid, with an athletic build. He was well liked, especially by the local girls, but was getting a reputation for his prowess on the rugby pitch.\n\nJim was also tall and athletic, but with more grace and panache. He too enjoyed rugby but was also a very good bowler on the regional cricket team.\n\n\"Sir, I was wondering if you want me to fetch me dad.\" Frank asked.\n\nFrank's father, Hyrum McTavish, had been the Marsh gamekeeper for the past fifty years and lived in a cottage on the property.\n\nThompkins was grateful he wouldn't have to make the long trek out to find the gamekeeper, who could be anywhere on the large estate, although Lord William mentioned he believed the man was checking on an injured muntjac fawn. Just as he was about to respond, Lady Elizabeth entered the room.\n\n\"M'lady.\" Thompkins stood straight and bowed his head slightly. He was happy to notice the two footmen were also standing at attention in a respectful manner.\n\n\"Thompkins, I'm so glad I found you. I just got off the phone with the Duchess of Windsor. Our earlier conversation was so hurried I didn't get to ask very many questions about the guests, and Lord William was particularly curious about the hunting.\"\n\nLady Elizabeth was usually very refined, but Thompkins couldn't help noticing she seemed annoyed.\n\n\"Yes, m'lady.\"\n\n\"Well, it turns out the gentlemen won't be hunting after all.\" She sighed. \"They want to shoot.\"\n\nThompkins understood her ladyship's annoyance. \"Excuse me, madam, but I thought . . .\"\n\n\"Yes, so did I. Lord Browning definitely said a hunting party and the duchess also said hunting. When I called, however, she talked about how much Count Rudolph was looking forward to the pheasant. That's when I realized we weren't talking about the same thing.\" She sighed. \"I guess it's one of those language differences between America and England. Apparently, hunting and shooting is the same thing in the States.\"\n\nThompkins scowled slightly but quickly removed all traces of distaste from his countenance and nodded. \"Very good, your ladyship. I was just about to send Frank out to notify his father about the hunting.\" He turned to Frank. \"Please inform Mr. McTavish to prepare for a shooting party. He can hire beaters and whatever additional help he will need.\" Thompkins turned to her ladyship, who nodded her approval of the orders.\n\nFrank nodded. \"Yes, sir.\"\n\nFrank and Jim nodded to her ladyship. They turned and left.\n\n\"Thompkins, I do hate springing this on you with such short notice. Please call the staffing agency and get whatever additional help is needed.\"\n\nThompkins nodded. \"Yes, m'lady.\" Thompkins coughed delicately. \"May I ask if the duchess had a firm idea of the number of guests we should expect?\"\n\n\"Oh, yes. I almost forgot. That was one of the main reasons I called.\" Lady Elizabeth handed Thompkins a list of names. \"These are the people that were invited.\"\n\nThompkins looked over the list. When he was done, he looked up.\n\n\"Yes. I know what you're thinking. I was as surprised as you are.\" Lady Elizabeth's brow creased, and her normally happy expression was replaced by one of worry. \"Lord William is furious. He's upstairs smoking at a furious pace.\" She sighed. \"Neither of us likes to dabble in politics, especially now. Things are so uncertain in Europe and the slightest misstep could send England spiraling into another world war.\"\n\n\"Perhaps it would be better to decline . . .\"\n\nLady Elizabeth sighed. \"I wish I could, but I've already said we would do it.\" Lady Elizabeth stood up straight and pushed her shoulders back. \"Besides, Lord Browning will be here. I don't believe he would have asked us to host the party unless it was vitally important. We shall just have to trust in providence and Lord Browning to keep an eye on things.\"\n\n\"Yes. M'lady.\"\n\nChapter 5\n\nI don't remember exactly what time I went to bed, but it was late. Normally, I dreaded the sound of my alarm clock. However, this morning, I awoke before it went off, thanks to the amazing aroma that infiltrated my room.\n\nI yawned and reached over to turn off my alarm and shuddered as Snickers licked me at just the right moment. I used my blanket to wipe off my tongue and the inside of my mouth and opened my eyes to find her inches from my face, awaiting another opportunity to clean my face. I sat up and scrubbed off my early morning dog kiss and hopped out of bed, eager to get dressed. Thanks to the glorious aroma of cinnamon rolls and coffee coming from my kitchen, I knew Dawson was home and my breakfast would consist of more than coffee and cold cereal.\n\nI'd never timed myself, but I suspected I could complete my morning routine 50 percent faster when I knew bacon or coffee was waiting for me. Showered and dressed, I hurried to let the poodles out so I could dig into breakfast and find out how Dawson was doing.\n\nNana Jo beat me to the kitchen and was perched on a barstool enjoying cinnamon deliciousness. I think I heard her moan as I passed.\n\nDawson placed a plate on the counter, along with a mug of steaming hot coffee. \"You eat. I'll take the dogs out.\"\n\nI didn't argue.\n\nDawson grabbed two dog biscuits from the canister on the counter and Snickers and Oreo followed him downstairs like lemmings. My poodles were loyal, but they would sell their souls for dog treats.\n\nNana Jo and I ate in peace until Dawson and the poodles returned.\n\n\"You look like you've been rode hard and put away wet.\"\n\nI kicked Nana Jo. \"Why don't you tell him what you really think?\"\n\n\"I'm not sure I know what that means, but I get the general idea.\" He smiled.\n\n\"How are you?\" I asked, although if the dark circles under his eyes and the haggard look of his face were any indication, he hadn't slept.\n\nHe confirmed my thoughts. \"I couldn't sleep, so I gave up and decided to make cinnamon rolls.\"\n\n\"They're delicious.\" I licked icing off my fingers.\n\n\"Thanks.\" He took a deep breath. \"I'm going to pack a bag and stay on campus. Mrs. Rutherford said I'll probably be arrested soon.\" He hung his head.\n\nI gasped and tried to stop the tears that came to my eyes from falling. \"But they can't. You didn't do it. Jenna knows that.\"\n\nHe held up a hand to stop my tirade. \"She does know it, and it means a lot to me that you believe in me.\"\n\n\"We both do,\" Nana Jo said.\n\nHe nodded. \"Thanks. But she said the case against me is pretty strong. There's a lot of circumstantial evidence and the district attorney wants to show the community he's tough on crime.\"\n\n\"Why that two-bit, lily-livered, attention grabbing, snake oil salesman,\" Nana Jo sputtered. \"He'd arrest an innocent man just to get his name in the paper. I've got half a mind to march down there and give that young whippersnapper a piece of my mind. I\u2014\"\n\nDawson held up both hands. \"Whoa. Hold on there, Mrs. T.\" Dawson fought back a smile. \"I appreciate the sentiment, but I don't think making the district attorney mad is going to help.\"\n\n\"He's right. As much as I hate to admit it, he's just doing his job.\"\n\nNana Jo wasn't one to go down without a fight. \"No, he isn't. His job isn't to arrest innocent people. His job should be to get to the truth. I guess that's just too much for those pea brains.\"\n\nSomething Dawson said had been niggling at the back of my mind. \"Wait, why are you going to stay on campus? I don't understand.\"\n\nHe avoided eye contact. \"If I'm going to be arrested, I think it would be better someplace else.\"\n\nAt first I thought he was saying he would be embarrassed if Nana Jo and I were there to see him arrested by the police, but something in his eyes told me there was more to the story. \"Dawson, I can't lie and say it wouldn't make me sad to see you arrested. But knowing you're innocent makes it easier. If you're worried about me and Nana Jo, please don't be. We're a lot tougher than we look.\"\n\n\"Darned straight,\" Nana Jo added.\n\nDawson still avoided eye contact and shuffled his feet around. \"It wouldn't look very good for the bookstore to have one of your employees arrested.\"\n\nI got up and walked around the counter and lifted his head so he looked me in the eyes. \"Dawson, I appreciate your concern, but I don't give a flying fig what people think. This is your home and you don't have to leave it for fear of what people might say.\"\n\nHis eyes filled with water, and he looked like he wanted to say something but wasn't able to find the right words.\n\nI gave him a hug, and he put his head on my shoulder and wept.\n\nWe stood there for several moments and then he stood up straight and wiped his face with a dishcloth. \"Thank you, Mrs. W. Thank you both.\"\n\nNana Jo had been weeping into a handkerchief. She wiped her eyes and blew her nose. \"Now that we've gotten that off our chests, it's time to get busy. If the authorities won't figure out who killed that . . . girl, well, then, I guess we'll just have to do it for them.\" She took another sip of her coffee.\n\nAs much as I hated to admit it, Nana Jo was right. Jenna would do everything she could to protect Dawson legally. However, the police and Detective Pitt weren't going to look for another murderer when there was so much evidence against Dawson. Unless we stepped in and found the killer, an innocent man would go to jail.\n\nThe rest of the day went by in a daze. My mind was only half engaged on books. The other half of my brain scrambled to think through what I knew about Melody Hardwick. Several months ago, when the realtor who listed this building for sale was killed in the back courtyard and the police thought I was the murderer, Nana Jo and her friends from the retirement village helped me find the real killer. The girls used their vast collection of family and friends to get information to solve the crime. I wondered, was it merely luck that helped us put the pieces together or could we do it again. We were scheduled to meet for dinner tonight, so time would tell.\n\nTuesday was Senior Citizen's Day at Randy's Steak House. Half-priced meals, at least for my grandmother and her friends, had become a ritual. The host knew us by name. I was only in my thirties and didn't qualify for the discounted meal, but I racked up frequent diner points and was only one meal away from a free dinner.\n\n\"Alright, ladies. You know why we're here. Let's get this party started.\" Nana Jo took one last bite of her hot banana pudding and pulled her iPad out of her purse. \"Why don't I start?\" She looked around and saw no opposition. \"I called Freddie and filled him in on the situation.\"\n\n\"He's a hottie, that Freddie. I'd sure like to fill him in on a thing or two.\" Irma Starczewski broke into a coughing fit. Irma was a petite woman in her mid-eighties. Her voice was deep and raspy from decades of chain-smoking and she had a mouth like a sailor.\n\nNana Jo looked over her glasses and down her nose at Irma. \"I'll bet you would. Fortunately, he has better taste. Now, if I may be allowed to continue.\"\n\nIrma stuck her tongue out before another coughing spell hit. She looked around discreetly, then pulled a flask out of her purse, and took a swig.\n\n\"As you all know, Freddie's son, Mark, is on the state police. He said the little gold digger had a record a mile long, and Melody Hardwick wasn't her real name.\"\n\n\"Nana Jo. Just because you never liked Melody . . . or whatever her name was, doesn't mean we want to jump to conclusions. We have to try and be as open-minded and fair as possible.\"\n\nNana Jo looked at me with the innocence of a babe. \"I don't know what you're talking about. I'm merely reporting the facts.\"\n\nI smiled. \"So, you're telling me Mark really called her a 'gold digger'?\"\n\n\"Well, I might have added that bit myself, but you're missing the important part. Melody Hardwick isn't her name.\"\n\n\"I wonder if Dawson knew.\" I looked at Nana Jo, who shrugged.\n\n\"Are you going to tell us what her real name is? I've got a hot date tonight with a guy who used to work at MISU and I intend to ask about her.\" Dorothy Clark was six foot, three hundred pounds, and looked like a linebacker for the Chicago Bears. She was also a helpless flirt with a black belt in aikido. She had a way with men and they melted at her feet. She could give classes. If there was any information to be gotten out of her date, I had no doubt Dorothy would get it.\n\nNana Jo scanned her iPad. \"Elizabeth Mae Simpson.\"\n\nDorothy took a pen and wrote the name on a napkin and passed it to Irma and Ruby Mae.\n\n\"Apparently she used different aliases too. Beth, Lizzy, Bessie. Twenty-two, brown hair, brown eyes, and one hundred fifteen pounds.\"\n\n\"What did she get arrested for?\" Ruby Mae pulled her knitting out of the bag she always carried. Ruby Mae was my favorite of Nana Jo's friends. She was a soft-spoken African-American woman. I thought she was in her eighties, although her skin was so smooth it was hard to tell. She was born and raised in Alabama and she spoke with a southern accent, even though she'd spent most of her adult life in Chicago, where she single-handedly raised her nine children after her husband walked out on them.\n\n\"He called her a grifter,\" Nana Jo said.\n\nRuby Mae frowned. \"What's a grifter?\"\n\n\"I googled it and, I quote, 'a grifter is a con artist who swindles people out of their money through fraud.' \"\n\n\"Well, I'll be d\u2014\"\n\n\"Irma!\" everyone yelled.\n\n\"Sorry.\" Irma put her hand over her mouth and burst into a coughing fit.\n\n\"Sounds like that little girl was up to no good.\" Ruby Mae knitted. \"Looks like our boy, Dawson, dodged a bullet.\" Ruby Mae looked up from her knitting. \"I'm sorry. Poor choice of words.\"\n\n\"We know what you meant.\" Nana Jo patted Ruby Mae's hand. \"And for the record, I agree. He definitely dodged a bullet.\"\n\n\"What about the other guy you told us about?\" Dorothy asked.\n\nNana Jo put her reading glasses back on and went back to her iPad. \"Oh, yeah. Virgil Russell. Mark didn't have time to look him up.\"\n\n\"I've got a call into my great-grandson, Ernie. He's a realtor. He knew about this Virgil Russell person. He's a bad mother\u2014\"\n\n\"Irma!\"\n\n\"Sorry. Ernie says he's a slumlord.\"\n\n\"Did he have any details?\" Nana Jo typed into her iPad.\n\nIrma coughed. \"I'm having lunch with him tomorrow. He said he'd ask some of his associates. I'm sure he'll have some information for me.\"\n\n\"Great.\" I was always amazed at how quickly these ladies could spring into action and tap into their connections to get information.\n\n\"How is Dawson holding up?\" Ruby Mae asked.\n\nI got teary eyed, but now was no time for weakness. We had a job to do. I took a deep breath and told them what Dawson told us this morning.\n\nRuby Mae sniffed and dabbed her eyes. Irma swore under her breath before breaking into another coughing spell that required a swig from her flask. Dorothy looked stoic and determined.\n\n\"Well, things don't look good, but we can help. We need to get as much information as possible.\" Nana Jo spoke like a general rallying her troops.\n\n\"I'm going to call Jenna,\" I said. \"I'm hoping she can tell us whether the autopsy is done. There might be some useful information in there.\"\n\n\"Good idea.\" Nana Jo typed. \"I'm going to check with Freddie and see if there is anything else Mark can tell us about Virgil and his business dealings.\"\n\n\"My granddaughter, Jillian, goes to MISU,\" Dorothy said. \"I don't know if she knew Melody or Elizabeth Mae.\"\n\n\"I think it'll be easier if we continue to refer to her as Melody.\" I glanced around and received nods from everyone. \"It'll be less confusing.\"\n\n\"Good. I'll ask my granddaughter about Melody. If she didn't know her, maybe she knows someone who did,\" Dorothy said.\n\n\"Sam, I can see your wheels turning,\" Nana said.\n\nShe knew me so well. \"I'm going to the campus too. I want to talk to the athletic director about Dawson.\" I looked down. \"It would be horrible if he lost his scholarship because of this. I thought I'd go to the student center and look around too.\"\n\nNana Jo stared. \"What for?\"\n\nI took a moment to formulate my thoughts. \"Remember when that foreign exchange student drowned in the Thomas River last year? There was a memorial service on campus. Lots of students came together and set up a makeshift memorial with flowers and keepsakes.\"\n\n\"I remember. The newspapers and television cameras interviewed all those kids,\" Dorothy said. \"There were some really sad stories about his life in Kenya and how his family had such high hopes for him.\"\n\n\"I was thinking there might be something like that for Melody. If so, maybe I could talk to some of her friends, find someone who knew her really well.\"\n\nNana Jo nodded. \"Good idea.\"\n\n\"Well, it looks like we have a plan of attack.\" Dorothy started to rise. \"I've got to get ready for my date.\"\n\n\"Just one more thing,\" Nana Jo said. \"We've got to work fast on this one.\"\n\nWe all looked at Nana Jo with puzzled expressions.\n\n\"MISU has a bye week this Saturday, but if we're going to a bowl game, we need to clear Dawson before the next game.\"\nChapter 6\n\nAfter dropping Nana Jo and the girls at the retirement village, I swung by Jenna's house. My sister and brother-in-law were both successful attorneys who chose to live in North Harbor, rather than in a South Harbor McMansion. At one time, North Harbor was the place where the elite showed off their wealth, with large Victorian mansions set back from the street with expansive lawns behind wrought iron fences. When the lucrative automotive jobs moved south, so, too, did the wealthy. Those who stayed left their older homes and moved to working-class South Harbor.\n\nJenna and Tony bought an older, dilapidated Victorian mansion for one dollar from the city and lovingly and painstakingly renovated it. The house was large, at close to four thousand square feet but felt warm and cozy inside. My sister turned the front turret into her home office with a curved desk and one of the house's seven fireplaces; it was a cozy space which overlooked a lovely rose garden. When I pulled in front of the house, the light in the turret showed my sister at her desk. Jenna must have seen me. By the time I climbed to the top of the porch, she was waiting with the door open.\n\n\"Come in.\"\n\nI laid my purse on the table inside the door and followed my sister into her office. \"Where are Tony and the twins?\"\n\n\"Tony had to work late and the twins went to a poetry reading.\"\n\nI must have looked as surprised as I felt because Jenna laughed.\n\n\"I know. I was as surprised as you look. Quincy Troupe is doing a reading and they have dates.\"\n\n\"Ah . . . okay, that explains it.\"\n\nI didn't read a lot of poetry, but I was familiar with Quincy Troupe. He wrote a biography on jazz musician Miles Davis and a poem about former basketball hall of famer, Magic Johnson. If the twins went to a poetry reading, Quincy Troupe would be the poet.\n\nShe returned to her chair behind her desk, and I sat on the Victorian love seat we'd picked up at a yard sale and she'd reupholstered. Every time I saw that love seat, I smiled at the memory of Jenna and I trying to cram it into the back of my SUV without damaging it more than time and generations of wear and tear already had. The ornately carved mahogany wood with its overstuffed velvet seat, tufted back, and rolled arms wasn't my taste, but it fit the house and room perfectly.\n\n\"I would ask what brings you here, but I'm sure I can guess.\"\n\n\"I just came from dinner with Nana Jo and the girls. Any updates?\"\n\nShe looked at the papers on her desk. \"Things don't look good.\" She picked up a file. \"She was strangled.\"\n\n\"Do they know how?\"\n\n\"Based on the bruising, the killer probably used his hands.\"\n\n\"Which means . . . ?\"\n\nJenna nodded. \"It was probably a man or a very strong woman.\"\n\n\"When?\"\n\n\"The coroner believes her death occurred between midnight and two thirty Monday morning. Dawson claims she banged on his door until midnight, although the police won't take his word for that.\"\n\n\"Does the medical report say anything else useful?\"\n\nJenna shook her head. \"Not really. She'd eaten last around eight.\" Jenna scanned the document. \"She had sex with someone within the past day. The police are going to want a DNA sample from Dawson.\"\n\n\"He was her boyfriend. That doesn't mean he killed her.\"\n\nJenna looked down. \"No, but once we give the DNA sample, they'll know it's his skin cells under her fingernails and only Dawson's word for it she was alive afterward.\" She nodded. \"They'll arrest him.\"\n\nI took several deep breaths to steady my breathing.\n\n\"I'm going to take Dawson tomorrow for the cheek swab. We'll wait downtown. Most likely, they'll arrest him shortly afterward. The district attorney was a classmate of mine. He's not the sharpest tack in the box, but he's no fool and owes me a favor. We'll have a bond hearing tomorrow afternoon.\"\n\n\"Do you need any money?\"\n\nShe shook her head. \"No. I've got a bail bondsman I work with a lot. He's going to take care of everything.\"\n\nNo matter how much I blinked, I couldn't stop the tears from streaming down my face.\n\nJenna reached behind her desk and pulled a box of tissues from a shelf and brought them to me.\n\nI took the tissue, wiped my eyes, and blew my nose.\n\n\"Sam, I'm sorry. I know how much Dawson means to you, but I want you to be prepared. This is going to get worse before it gets better.\"\n\nI looked up. \"What do you mean?\"\n\nJenna paced. \"The media have already started calling. They'll show up at your bookstore and stick microphones and cameras in your face. Some may try to follow you. I can prevent them from entering, but I can't prevent them from setting up their cameras outside on the street. I recommend you don't make a statement. Let me handle all of that.\"\n\nI nodded. \"Is there anything we can do?\"\n\n\"Yeah. Find out as much as you can about her. Normally, I work with an investigator, but he's knee-deep in other higher-priority cases.\" Jenna smiled at me. \"Besides, I'm pretty sure even if I told you to stay out of it, you wouldn't.\"\n\nI smiled. \"You're right.\"\n\nJenna looked serious. \"Sam, be careful. There's a murderer out there. Don't go off on your own. Normally I don't condone guns, but Nana Jo should keep her 'peacemaker' ready. Whoever murdered Melody thinks he's gotten away with it. He won't take kindly to anyone who gets too close.\"\n\n* * *\n\nJenna's words stuck with me throughout my drive home. At home, I let Snickers and Oreo out and stood in the courtyard and watched while they sniffed for the appropriate places to do their business. I noticed Dawson's light on in his apartment. I found that light comforting and couldn't help wondering how many more times I'd see it. The tears started again, and I had to shake myself to prevent a breakdown.\n\nWhen I went back inside, my mind was restless and I knew sleep would be elusive. I avoided the struggle and distracted myself with a trip to the British countryside.\n\nThompkins entered the library of Wickfield Lodge and stood very stiff and straight in the doorway. He coughed and then announced, \"The Duke of Kingfordshire, Lord James FitzAndrew Browning.\"\n\nJames entered the library.\n\n\"Excellent. You're just in time for tea.\" Lady Elizabeth looked to the butler, who nodded, stepped back, and closed the door.\n\nJames was a freckled, fair haired, good natured man with broad shoulders and a stocky build. The Marshes met him six months ago when he came to help his longtime friend Victor Carlston avoid swinging for a murder he didn't commit. After the real killer was apprehended, he was best man at the wedding of Victor and Lady Penelope Marsh. Since then, he'd made many visits to Wickfield Lodge and had become very attentive to Lady Daphne.\n\n\"James, I'm so glad you've finally come. Now maybe you can tell us what this whole thing's about,\" Lord William announced in his loud, but good natured tone.\n\nLady Elizabeth shot a glare at her husband. Lord William frowned and pulled out his pipe.\n\nLady Elizabeth waited and watched.\n\nJames glanced at Daphne sitting in the window seat. He strode across the room to her and reached down for her hand.\n\nDaphne nodded, gave a brief smile that didn't reach her eyes, extracted her hand, and returned to the view out the window.\n\nLady Elizabeth noted her niece's cold greeting and added more warmth into her voice. \"James, we're all so glad you've arrived.\"\n\nThompkins rolled in a tea trolley filled with tea, sandwiches, scones, and tarts. He rolled the cart be side Lady Elizabeth and bowed.\n\n\"Wonderful, Thompkins. Thank you,\" Lady Elizabeth said to the retreating figure, then grabbed the silver teapot and poured. \"James dear, would you like tea or perhaps you'd prefer something stronger?\"\n\nJames accepted the teacup. \"No, tea will be fine. Thank you.\" He looked over at Daphne before taking a sip.\n\nOnce the tea was distributed, Lady Elizabeth turned to James. \"Now, maybe you can tell us what you've gotten us into?\"\n\nHe took a deep breath and placed his cup on the table. \"First, I want to thank you for agreeing to host the duchess's hunting party. We were at our wits' end trying to figure out what to do.\"\n\n\"Of course we're happy to help out, but it seems so strange,\" Lady Elizabeth said.\n\nLord James nodded. \"I dare say it does.\" He paused. \"I appreciate your trust, but I'm afraid I must ask you to trust me a little longer.\" He looked at Lady Daphne, who was engrossed in her cup of tea and didn't make eye contact with him.\n\n\"Unfortunately, I can't put forward an explanation, at least not yet. I hope to be able to explain at some point, but I can't right now. In fact, I must ask that you not even mention the fact I rung up to ask. It's imperative the Duchess of Windsor not know I had any involvement in this. As far as she or anyone else outside the family is concerned, I'm merely an invited guest, a friend of the family.\" James looked again at Daphne.\n\nLady Elizabeth stared in puzzled amazement. \"But how can she not know? I mean, I've never even met Wallace before. Surely someone had to suggest Wickfield Lodge as an alternative location. I can't believe she came up with the idea on her own.\"\n\n\"Ah . . . that. Well, all I can say is someone the duchess respects suggested the location. The fact that you're a favorite cousin of the king and could prove beneficial in smoothing the way with her royal relations was an additional enticement.\" James smiled briefly. \"However, I can't stress enough how important this is.\"\n\nLady Elizabeth looked to her husband, who puffed on his pipe and absentmindedly fed bits of his sandwich to Cuddles, sitting attentively at his feet.\n\nJames looked uncharacteristically nervous. The 15th Duke of Kingfordshire always exuded a casual, easygoing manner, common for young men of his station. A wealthy, privileged, aristocrat without a care in the world was the image he presented. However, closer inspection showed intelligence, a keen mind, a purposeful chin, and a determined jaw. If eyes were the window to the soul, then his soul showed a depth of conviction and courage.\n\nLady Elizabeth nodded her ascent.\n\n\"Dashed secretive.\" Lord William looked from his wife to his niece. After a pause, he nodded.\n\nAttention turned toward Daphne. Lady Daphne sat in her favorite seat in front of the large picture window that looked out onto the back terrace. Framed by the large window, she gazed outside. The sun shone through the window and illuminated her beauty with a brilliance that was dazzling. Golden hair and piercing blue eyes which sparkled like the sun on the ocean in the early summer. She was stunning. She turned her head toward James and lifted her long dark lashes. Her glacial gaze caused Lord James to gasp and Lady Elizabeth shivered involuntarily.\n\n\"What are you asking?\" Her words hung in the air like a cloud of smoke.\n\nLady Elizabeth placed her teacup on the table and stood. \"Darling, I think we need to have another talk with the servants.\" She picked up the knitting bag she kept at her feet.\n\nLord William stared after his wife.\n\n\"Are you coming, dear? I really think we both need to talk to them.\"\n\nAfter a few moments, Lord William's eyes registered the secret message his wife was sending and hoisted himself from the sofa, dislodging Cuddles, and followed his wife to the door, mumbling, \"Servants, yes. Have to talk to the servants. Tricky business.\"\n\n\"Wait,\" Daphne said. \"There's no need for you both to leave. His Grace can't have anything to say to me that can't be said in front of both of you.\"\n\nHis Grace looked as though he had a lot he'd like to say. However, he set down his teacup and walked to the seat next to Daphne.\n\nDaphne turned to face him and an invisible veil immediately closed over her eyes and provided a barrier to her thoughts.\n\n\"Daphne, I think you know I care about you deeply. The past few months have been rather busy and I apologize for not being more attentive. However, my feelings are\u2014\"\n\n\"Oh, dear.\" Daphne laughed. \"I think you've mistaken me. I wasn't asking you for a profession of love. We both know that would be phony.\"\n\nJames snapped back as though he'd been slapped.\n\n\"I was merely asking how far you want to go with this charade you've arranged. Are you merely an old family friend, my brother in law's dearest friend? Or are you supposed to mean something to the family.\"\n\n\"Daphne.\" Lady Elizabeth couldn't stop the shock that registered in her voice.\n\nDaphne looked innocently at her aunt. \"I'm sorry if I shocked you, Aunt Elizabeth. I was merely trying to determine how much of an actress my part required.\"\n\nJames's eyes flashed and his jaw clenched. \"I wouldn't want to tax your acting ability. Let's just say I'm a friend of the family.\"\n\n\"Fine with me.\" Daphne turned back to her window.\n\n\"Fine.\" James stood and took his pipe from his pocket and moved to the mantle.\n\nLady Elizabeth and Lord William looked at the two young people and then quietly slipped out of the room.\n\nVictor and Penelope walked the three miles from their home to join the family for dinner, which helped to remove some of the strain.\n\nDaphne and James were excessively kind on the rare instances when they spoke to each other, which increased, rather than decreased, the tension. Daphne also showed a deep interest in all of the details of Penelope's housekeeping dilemmas, even going so far as to offer to help interview maids. For his part, James remained in a daze. He was silent, only speaking when spoken to. After dinner, the family retired to the back parlor for coffee.\n\nLady Elizabeth pulled out her knitting and looked around. She gazed at her niece Lady Penelope and smiled. After nearly four months of marriage, it was obvious the couple was blissfully happy. Victor had believed himself to be in love with the fair and beautiful Daphne for most of his young life. Six months ago he'd realized Penelope was his heart's true desire. Lady Elizabeth glanced at her new nephew with fondness.\n\nVictor sat in a chair in front of the fireplace with his legs outstretched, completely at home. He smoked while Lord William reminisced about his days in His Majesty's service. Knowing glances were exchanged between Victor and James. Daphne retreated to her seat by the window and engrossed herself in a magazine. However, Lady Elizabeth was sure she hadn't seen her turn a page in nearly thirty minutes.\n\nPenelope leaned toward her aunt and whispered, \"What's with the Ice Queen?\" She inclined her head slightly toward her sister.\n\nLady Elizabeth sighed. \"She's hurt. I think she genuinely cares for James.\"\n\nPenelope glanced across at her younger sister. The two young ladies were so different; a casual observer wouldn't have guessed they were sisters. Daphne was petite, fair skinned with blond hair and blue eyes; traits she inherited from her father, Peregrine Marsh. Penelope looked more like their mother, with dark hair and eyes. In the summer, if she wasn't careful, her skin tanned to a dark brown, very similar to their mother, Lady Henrietta Pringle. The loving couple died in a car accident not long after Daphne was born. Lord William and Lady Elizabeth raised the two girls and couldn't help looking upon them as the daughters they never had.\n\n\"I believe James genuinely cares for her too.\" Penelope glanced at the duke. \"What happened?\"\n\nLady Elizabeth placed her knitting on her lap. \"He hasn't been around much in the past couple of months. Then he comes and, after arranging this elaborate weekend, he doesn't confide in her what's going on.\"\n\n\"Maybe he can't.\"\n\nLady Elizabeth looked at her niece. \"I was hoping he'd confided to Victor?\"\n\nPenelope shook her head. \"If he has, Victor hasn't said, but I'm sure he wouldn't have asked you to do it if it wasn't important. I've never seen his estate, but according to Victor, James's family home makes Buckingham Palace look like a nice cottage. If it was just a matter of needing a large estate for the shooting, he certainly could have hosted it himself.\"\n\n\"He doesn't want anyone to know he is anything more than an invited guest, a family friend.\" Lady Elizabeth frowned.\n\n\"What's the matter? You seem upset.\"\n\nLady Elizabeth knitted a few stitches. \"I just have a very bad feeling about this whole thing. After seeing the guest list, I have to say, if anyone other than James had asked me to do this, I would most certainly have declined. Your uncle hates getting involved in politics, and there's been so much negative publicity about people like the Cliveden set in the papers that I'm worried.\"\n\nPenelope scowled. \"The Cliveden set?\"\n\n\"You know, dear, that group Lady Nancy Astor invites to Cliveden House. The newspapers have been full of how they are trying to appease Hitler and, well, all kinds of things.\"\n\n\"I've read about them, but I'm rather surprised you have.\"\n\n\"One must stay up on current events.\" Lady Elizabeth knitted. \"It's so distressing. I don't know what's worse. To think we could be so close to war again\"\u2014she stopped knitting and looked at her niece\u2014\"or to sit back and do nothing while Germany takes over Europe. If he's left unchecked, he won't stop at the Channel.\"\n\nPenelope reached across and patted her aunt's hand. \"I pray it doesn't come to that. But, try not to worry. This is just a weekend shooting party and Victor and I'll be here as much as possible. We'll help in any way we can.\"\n\nTry as she might, Lady Elizabeth couldn't shake off the feeling in the pit of her stomach of trouble ahead. She could tell she was upsetting her niece, so she smoothed the frown creasing her forehead and smiled. \"Thank you, dear. I know it will be a tremendous help.\" She then turned to the others in the room. \"Would anyone care for more coffee?\"\n\nChapter 7\n\nI'd always loved Wednesdays, especially when I was a teacher. My coworkers started the week looking forward to Friday and the weekend. That meant five whole days of classes and grading papers. If I focused on Wednesday, instead of Friday, the week went much faster. This Wednesday changed that forever. This Wednesday, Dawson submitted to a DNA check and then waited patiently for the police to arrest him, which they did.\n\nI was determined to be strong and not permit myself to cry. I didn't want to make things worse for him by breaking down. I was the last thing he needed to worry about. I closed the bookstore so Nana Jo and I could go to the police station with Dawson and Jenna.\n\nFirst, Jenna took Dawson to the North Harbor Clinic, where he was escorted to a private room.\n\n\"Why are you having this done here? I thought the police took care of this,\" I asked while we sat in the waiting room.\n\n\"The state lab is backlogged. It'll take them at least six months before they get to us.\"\n\n\"Wouldn't that be better? I mean, they can't arrest him without the DNA evidence, can they?\"\n\n\"Oh, yes they can. Plus, I don't want to take the chance the state loses or contaminates the sample. They have some good people, but they're overworked, and when you're overworked, mistakes happen. Besides, I work with this lab all the time. They'll have the results back in a few hours.\"\n\n\"But in six months, we will have figured out who the killer is.\"\n\nJenna turned to face me. \"Sam, murder is a felony. If Dawson is arrested for a felony, he'll be suspended until his case is resolved. Trust me. I would prefer more time too, but ultimately the decision is his.\"\n\nHe returned before we could discuss things further. The entire process took less than ten minutes.\n\nI followed Jenna and Dawson to the police station and thought over what Jenna said. Six months wasn't a long time, but faced with less than two weeks to find a killer, it seemed like an eternity. Football meant a lot to Dawson. It was the vehicle that enabled him to get away from his abusive father and to get a good education.\n\nThis time Nana Jo had her gun, but she stopped at the metal detectors and held up her carry permit. She also announced loudly she was packing an iPad in her purse. The police officer manning the metal detector remembered us from two days earlier and didn't look amused. He looked through her purse carefully and handed back her wallet and purse and ushered us through. Our entry was unheralded and low-key.\n\nJenna notified Detective Pitt we were coming beforehand and he waited for us at the desk.\n\nMy sister was bossy, opinionated, and annoying. She was also hardworking, dedicated, and super organized. These skills came in handy as an attorney. To minimize the amount of time Dawson sat in a jail cell, she had reviewed the judge's schedule and talked to the district attorney. Dawson was arrested, fingerprinted, photographed, and arraigned all before three. The bail bondsman was good to his word and waited for us at the courthouse. By three thirty, we were back at the bookstore. Dawson hadn't slept well last night, so he made pie dough. Not my idea of fun, but to each his own.\n\nJenna, Nana Jo, Dawson, and I sat at one of the bistro tables in the back of the store, drank coffee, and ate apple pie \u00e0 la mode, drizzled with salted caramel. It was a delicious muddled mixture of warm spicy comfort which almost buried my sadness. For several minutes, we sat and ate in silence. It was a glorious few minutes.\n\n\"That pie was great. You could enter a baking contest.\" Nana Jo licked her spoon.\n\nThe tips of Dawson's ears were red. He still got embarrassed when people complimented him, even though he loved to watch people enjoying his creations.\n\n\"What happens now?\" I asked Jenna.\n\n\"Dawson's officially suspended from school.\"\n\n\"Looks like you'll have plenty of time to bake.\" Nana Jo patted his shoulder.\n\n\"Is there a way he can get his assignments?\" I asked Jenna. \"Nana Jo and I can tutor him so he doesn't fall behind when he goes back in a couple of weeks.\"\n\nJenna smiled at me. \"That's a great idea.\"\n\nDawson looked misty eyed. He looked like he wanted to say something but couldn't find the words. Eventually, he stared at his plate and mumbled, \"Thank you.\"\n\n\"So what are your plans for today?\" Jenna asked.\n\n\"I'm going to the campus.\" I looked at my watch. \"I better get a move on.\"\n\nJenna looked very serious. \"Sam, I know I don't have to tell you this, but there may be reporters there. It would be best if you leave all of the comments to the press to me.\"\n\n\"Don't worry. I have no intention of talking to anyone from the media.\"\n\nWe both turned to stare at Nana Jo.\n\n\"What're y'all looking at me for?\" She looked indignant.\n\n\"Nana Jo, this is important. I\u2014\"\n\n\"I know. I know.\" She zipped her lips and folded her arms over her chest.\n\nJenna sighed. \"I'm not trying to insult you. I know you would never do or say anything to deliberately hurt Dawson or our case. But the media can turn your words inside out and before you know it, we've got a big media mess on our hands. We need all of the goodwill and public opinion we can get.\"\n\nAt that moment, Jenna's phone rang. While she was answering, Nana Jo got a text message. Then Dawson's phone rang.\n\nJenna listened to the caller for a few seconds and then her face underwent a transformation. She went from relaxed to panicked in less than five seconds. \"Turn on the television. Quick!\"\n\nI didn't have a television downstairs in the bookstore, so we hurried upstairs.\n\nAt the top of the stairs, Jenna said, \"Turn on the news.\"\n\nI grabbed the remote and turned to one of the local news stations. I couldn't help but gasp at what I saw on the screen.\n\nStanding atop the courthouse stairs was A-squared, Alex Alexander, Dawson's father. There were at least three microphones shoved in his face. Greasy hair, dirty jeans I felt sure were once blue, and a faded-out, greasy shirt that read, BEAM ME UP, SCOTTY. THERE'S NO INTELLIGENT LIFE DOWN HERE.\n\n\"Turn it up,\" Jenna demanded.\n\nNana Jo and I exchanged glances, but I complied. We'd run into Dawson's father before when he broke into the bookstore. The last time I saw him, he was handcuffed and getting into the back of a police car.\n\n\"Police brutality and discrimination, that's what this is. My boy didn't kill that girl. The cops is trying to frame him. They don't like it when a local boy gets a chance to make it outta this hellhole of a town. They just wanna push you back down,\" A-squared ranted.\n\nOne of the reporters tried to steer the conversation into something more productive than the current tirade. \"Mr. Alexander, there's a lot of evidence against your son. How can you be so sure he didn't kill her? Have you talked to him?\"\n\n\"Nawh, I ain't talked to him. I been in jail. Just got out. Don't need to talk to him. I know he didn't kill nobody, specially not that little tramp. Fine piece of [bleep]. But that's all she was.\" He made a hawking noise and then spit something brown on the ground.\n\nI cringed, grateful the network sensors were working and bleeped out the foul language. This was a disaster, but try as I might, I couldn't stop staring at the television. Just like a train wreck, I couldn't turn away.\n\n\"She was asking for what she got.\"\n\nJenna groaned and put her hands to her face.\n\n\"Are you saying Melody Hardwick deserved to be murdered?\" one of the reporters questioned.\n\n\"[bleep] right. She wadn't no saint. I can tell you that. But it wadn't my boy that done killed her.\"\n\n\"How can you be so sure?\"\n\n\"'Cause I know who did kill her.\" A-squared stared into the cameras.\n\nThis sent the reporters into a frenzy. They flung out questions at the speed of light. But A-squared merely grinned at the cameras, tapped his forehead knowingly, and winked.\n\nDetective Stinky Pitt must have been standing nearby because the next thing we saw was him pulling A-squared away from the cameras.\n\nWe stared at the screen for what felt like an hour afterward, but he never reappeared. When it was clear nothing more would be shown, I flipped to the other news stations. The same A-squared circus played on all of the news stations, but none of them had more than what we'd already seen.\n\nJenna hopped up and began to pace around the room. I'd seen my sister mad before, but nothing compared to this. Her face was red, and she kept mumbling, \"Oh my God. Oh my God.\"\n\nDawson's jaw was set in such a way I was certain he was grinding his teeth. He clenched and unclenched his fists. He looked mad enough to spit nails, as Nana Jo would say.\n\nNana Jo was the only one who looked to be enjoying herself. There was a twinkle in her eyes and her shoulders shook in silent laughter. After a few seconds, she finally burst out laughing.\n\nJenna spun around and stared. \"What's so funny? God knows I need a good laugh right now. That moron just insulted women, implied Melody Hardwick deserved to be murdered, and announced to the world he knows who murdered her. Oh, and he just got out of jail. People will think like father, like son. We need people to like Dawson and to believe him innocent. Most people will be so repulsed by that idiot, they may convict Dawson in the public's eyes without a trial.\"\n\nNana Jo wiped a tear from her eyes. \"You were worried Sam or I would say something to the media to ruin your case. Looks like we're the least of your worries. We could hardly make a bigger mess than that fool just did. Beam me up, Scotty, indeed.\"\n\nWe stared at her and then each other.\n\nNana Jo smiled. \"Nothing you can do about it now anyway. What's done is done.\"\n\nDawson had been tight-lipped throughout the news report. \"What are you going to do now?\" he asked Jenna.\n\n\"I'm going home and have a nice long bath, a good dinner, and a glass of wine . . . maybe a bottle. Then I'm going to bed. Tomorrow, I'll figure out what to do about your dad. I need to figure out a way to distance you from him.\" She grabbed her keys. \"I'll talk to you in the morning,\" she said to Dawson.\n\nHe nodded.\n\n\"Please, don't make this any worse than it already is.\" She looked at me and Nana Jo.\n\nWe crossed our hearts and held up three fingers in the Girl Scouts' honor and watched her leave.\n\nMy stomach growled and Nana Jo and Dawson acknowledged they were hungry too, so we scanned the cabinets for something to eat. Dawson discovered my stand mixer not only mixed dough but could also make pasta. Leon had bought several of the attachments that made everything from ice cream to sausage. The company that manufactured the stand mixer was headquartered in North Harbor, so accessories weren't hard to come by. I promised myself one day, when things settled down, I'd take a class and learn how to use the attachments I already had. That day never came. Now, Leon was gone. I wiped away a tear and hoped Nana Jo would chalk it up to the onions I was chopping. Thankfully, there were videos for virtually everything on the Internet. Dawson watched a five-minute crash course on making pasta, then took flour, salt, and water and made spaghetti. Nana Jo made a spicy marinara sauce and I chopped and grated. The end result was delicious.\n\nSpaghetti, a salad, and garlic toast gave me the final push I needed to head out to the campus in search of answers. Surely there was someone else besides Dawson who wanted to see Melody Hardwick dead. I prayed I could find that person before Dawson paid for a crime he didn't commit.\nChapter 8\n\nI really should have gone to campus earlier or waited until tomorrow, but time wasn't on my side. I didn't have a plan as I parked in the visitor lot and walked to the student union.\n\nMISU was on a beautiful, sprawling campus with acres of rolling lawn manicured to perfection. The campus sat on the outskirts of North Harbor with views of Lake Michigan from its western side. Ivy-covered brick buildings dotted the landscape and presented a peaceful backdrop. The university was founded in the early twentieth century with an emphasis on agriculture and teaching. Later, as the automotive industry grew in North Harbor and Detroit, the university expanded to include engineering and finally, a liberal arts school was added. The large trees that provided shade and shelter were centuries old.\n\nI enjoyed walking across MISU's campus because there was so much beauty to be seen. I wasn't sure where to look for a memorial for Melody Hardwick, but I started at the student union.\n\nOn the first floor there was a bookstore that sold everything from textbooks to macaroni and cheese. There was a cafeteria and an entire wall set aside for event flyers, jobs, and protests. I spent fifteen minutes scanning the board. The only thing I found close to a memorial was a MISU Students for Peace candlelight vigil to be held next week to mark the anniversary of the Kristallnacht or Night of Broken Glass. Kristallnacht, November 9, 1938, was the date the Nazis torched synagogues, vandalized homes, schools, and businesses of Jewish citizens, and killed close to one hundred people. Students of all races, nationalities, and religions were invited to a peaceful vigil on campus to mark the date and pray for peace. I was touched that students in a small Midwestern university wanted to remember an event from more than seventy years ago. I felt a moment of nostalgia for my teaching days. As a high school English teacher, I met and, I hoped, influenced some of these caring people.\n\n\"Sam.\"\n\nI was so absorbed in my thoughts I hadn't seen or heard Dorothy Clark's approach and jumped when she touched my shoulder. \"Dorothy. I'm sorry. I didn't see you there.\"\n\nShe stood next to me, holding a large bouquet of flowers. \"I know. I called your name several times. What's got you so engrossed on that board? Or are you just woolgathering.\"\n\n\"I was looking for some type of notice about a memorial for Melody, but I can't find anything.\" I looked at Dorothy closely. \"You look nice. You must have a hot date?\" I smiled and sniffed her flowers.\n\n\"I had a date, but the professor had to cancel at the last minute.\" She frowned. \"If he'd given me a little more notice, I would have gotten a replacement. But, oh, well.\" She looked around. \"Hey, you wanna join me? My granddaughter is performing at the new Hechtman-Ayers Center for Performing Arts tonight. I've got an extra ticket so it won't cost you a thing.\"\n\nI protested, but Dorothy was persistent.\n\n\"Look, you're not going to find out anything that'll help Dawson tonight from that board, and it's too late to go wandering around campus by yourself, anyway.\" She grabbed me by the arm. \"You look as though you could use a little culture in your life, anyway. Besides, afterward, we'll take Jillian out for coffee and you can pump her for information.\"\n\n\"But I'm not dressed for it.\"\n\n\"You look fine. You won't be the only person there in jeans. You'll look just like the other students.\"\n\nDorothy's intention was to place my mind at ease, but her words reminded me of my mom's criticism of my attire earlier in the week. I thought I had shoved those thoughts into my internal sea of forgetfulness. Dorothy's words brought the feelings back to the surface. I looked at my jeans, black T-shirt, and black hoodie. It was a lot of black. I thought about what I'd worn yesterday and the day before and briefly thought of my closet. It was full of blue jeans and dark shirts. I could hear my mom's voice in my head, \". . . hiding behind mourning.\"\n\nI shook myself to get my mom's voice out of my head.\n\n\"Are you cold, dear?\" Dorothy asked.\n\n\"No. I'm just . . . never mind. I'm fine. I'd love to see your granddaughter perform.\"\n\nWe walked the short distance to the new performing arts building. Dorothy had excellent seats near the front. Once we were seated, I looked around. She was right. I wasn't the only person wearing jeans. However, I refused to head down that road again. Instead, I focused on the beautiful building. Unlike the majority of the buildings on campus, the Hechtman-Ayers Center wasn't the traditional ivy-covered limestone box but was a light and bright, modern, concrete and glass shell. The school now offered everything from sculpture and metal-smithing to painting and dance studios, along with three performance spaces.\n\nThe performance tonight was Giselle, a classic ballet. According to the program book, Giselle was a frail but beautiful peasant girl in love with a count pretending to be a peasant. The count was already engaged and when his duplicity was revealed, Giselle dropped dead and became a spirit who trapped men and forced them to dance to death. Dorothy's granddaughter had the lead role, and she looked and performed beautifully.\n\nWhen the performance was over, she received a standing ovation and there were very few dry eyes in the building. I hadn't attended very many ballets. I'd seen the Nutcracker and Swan Lake in person, but this was amazing.\n\nI walked around the art gallery in the lobby while Dorothy went backstage and congratulated her granddaughter. Dorothy's son and daughter-in-law were out of the country and she wanted to make certain Jillian had family present for her performance. The lobby was a multipurpose space. In addition to showing off the lovely architecture of the building, with its tall walls and glass ceiling, it was also a marquee for advertising future performances and an exhibition space for sculptures and artwork. I didn't understand all of the art exhibited, but I appreciated the creativity and intelligence needed to create it.\n\nDorothy and Jillian were back long before I'd finished perusing all of the art, but maybe I'd come back another day. The three of us walked to the student union. They had a small River Bend Chocolate Factory caf\u00e9 that served chocolate, treats, and beverages. Most of the ballet patrons must have had the same idea because the small shop was packed, but Jillian and I snagged a table in the corner while Dorothy ordered.\n\n\"You were amazing. How long have you studied ballet?\" I asked while we waited for Dorothy.\n\nJillian was tall with a slender, dancer's body. She had large expressive dark eyes, and her hair was pulled back in a tight bun at the back of her neck.\n\n\"Thank you. I started ballet when I was three.\" She beamed. \"I love dancing.\"\n\n\"Do you do other dancing besides ballet?\"\n\n\"Jazz, tap, hip-hop, ballroom, Latin, you name it.\"\n\n\"Wow. So, are you a dance major?\"\n\nShe looked sad. \"No. I'm majoring in business with a dance minor.\" She shrugged. \"My dad thought I needed to major in something that would provide a stable income.\"\n\n\"It's good advice, although I recommend studying what makes you happy and obviously for you, that's dance. But business skills are always handy.\"\n\n\"I agree. I don't dislike business; in fact, I've been thinking I might like to have my own dance studio one day, but business classes aren't the thing that gets me revved up every day. I have lots of interests. In fact, I think we have one of them in common.\"\n\nI smiled but must have looked puzzled.\n\nShe laughed. \"Grandma said you own a mystery bookstore. That must be cool.\"\n\n\"I do, and I enjoy it a lot. Do you like mysteries?\"\n\n\"I love to read. I read some mysteries, but mostly I like romance. Grandma said you're investigating who killed Melody Hardwick. That sounds pretty cool.\"\n\n\"Well, I'm not sure about cool, but I don't believe Dawson killed her and the police don't seem to be looking for anyone else.\"\n\n\"Of course. I totally get it. How can I help?\"\n\n\"Did you know Melody?\"\n\n\"She was supposed to be in my history class, but I don't think I ever met her. She was a senior and I'm just a sophomore, and we didn't run in the same circles. But I do know her roommate.\"\n\n\"Supposed to be?\" I asked.\n\nJillian looked away. She was trying to decide whether to share something. I remained quiet and waited to see what she did. Eventually, she continued. \"She was on the student list, and she gets grades . . . but I never met her.\"\n\nI must have looked confused.\n\n\"We can see the grades online, so we know how everyone did on their quizzes and tests. It helps to know how you're doing in comparison to the other students.\" She shrugged. \"Weird thing is, she's getting A's, but I've never seen her in class, not one time\u2014not even for quizzes.\"\n\n\"But she has an A in the class?\"\n\nJillian nodded.\n\n\"That is strange. What class is it?\"\n\n\"History of Cults. The professor is nice. He's a little odd, but he's British, so his accent is nice. I think he's some kind of expert on the House of David.\"\n\n\"Interesting.\"\n\nDorothy arrived with coffee for herself and tea for Jillian and me. There were also three slices of cheesecake, which looked divine. Jillian allowed herself one taste then declined the rest, stating if she ate the rest, her dance partner wouldn't be able to lift her. Dorothy and I were forced to eat her slice too.\n\nWe chatted until Jillian recognized some of her friends and said she had to get back to the dorm. Before she left, she agreed to meet me at the student union tomorrow at eleven, in between classes, and would introduce me to Melody's roommate.\n\nI offered to drive Dorothy back to the retirement village. She had taken a taxi here, which must have cost a small fortune. We walked to the parking lot and, even though it was dark, I felt safe. Whether my feeling of safety was due to well-lit walkways with emergency police call boxes placed along the path or the fact Dorothy had a black belt in aikido, I wasn't sure. I suspected it was probably due to Dorothy's presence. I hoped her six-foot stature would deter anyone with malicious intent. If not, I'd witnessed her martial arts skills up close and she was definitely no one to be trifled with.\n\nI drove to the retirement village.\n\nAs she got out of the car, she said, \"I should have some information for you tomorrow afternoon. Tell Josephine I'll call her.\"\n\nI promised to relay the message, waited until she was safely inside, and drove home. Since Dawson's studio apartment was immediately above my garage, he knew when I was home. As soon as my engine was off, he opened his door and Snickers and Oreo ran down to meet me. They took care of business and we went upstairs. Drained didn't even begin to tap the surface of how I felt. Every muscle in my body felt tense and wound as tight as an alarm clock. I needed to relax, so I took a shower and, while the hot water pelted my skin, I thought about why I felt like a rung-out dishrag. A lot had happened in one day. From Dawson's DNA test, arrest, and arraignment to the A-squared media debacle, it had been one crazy day. We'd gone through a lifetime worth of life's worst events in less than eighteen hours. I didn't want to upset Dawson, so I'd held my emotions in check all day. Without a release, I felt like I would explode. As the heat from the hot water massaged and loosened my muscles, I cried. I started out with just a few tears but ended up sobbing into a washcloth until I had no tears left to cry. When I was spent, I felt hollow and hungry. I dried off and put on my pajamas. When I opened the bathroom door, I noticed a light in the kitchen and smelled coffee and apple pie.\n\nIn the kitchen, Nana Jo sat on a barstool with a cup of steaming coffee and a plate of apple pie.\n\n\"The last slice of pie is in the fridge.\"\n\n\"I shouldn't eat this. I just ate one and a half slices of cheesecake with Dorothy a couple hours ago.\"\n\n\"You've had a shock. You're supposed to eat sugar when you're in shock. Isn't that what they always do in those British cozies you read?\"\n\nI laughed. \"They drink tea with sugar. They don't eat cheesecake and pie.\"\n\n\"To-may-to. To-mah-to.\"\n\nI took my pie to the microwave.\n\n\"Twenty seconds gets it hot enough to melt ice cream without making the crust rubbery.\"\n\nI punched the time on the microwave and got the vanilla ice cream out of the freezer. Apple pie and ice cream were great, but Nana Jo's coffee seemed like the perfect accompaniment. If I drank coffee this late at night, I'd be awake for hours. However, something in my body desperately needed that caffeine. So, I took an individual coffee packet and placed it in the single-serve coffee maker. The smell of the coffee, heated apples, and spiced, sugary goodness was divine.\n\nNana Jo was right. My ice cream melted atop my pie and created a swirl of vanilla bean with the spiced apples, and when I took a bite, I moaned.\n\n\"Told you so.\"\n\nWe ate in silence for a few moments.\n\n\"You feel better?\"\n\nWithout saying a word, I knew Nana Jo was referring to my breakdown in the shower. Either she had ears like a bat or I was a lot louder than I thought. My money was on bat ears. Teachers learned to hone their senses after decades in the classroom. She had the ability to hear a gum wrapper in the back of a large classroom.\n\n\"You heard me?\"\n\nShe nodded as she used her spoon to scrape the last bit of ice cream from her plate and then licked it clean before putting it in her coffee cup.\n\n\"Actually, I do feel better.\"\n\nShe nodded. \"Sometimes you just need a good cry and apple pie.\" She smiled. \"It helps to release the pent-up anxiety. Cry too much and you end up with a headache. Too little and you still feel tense.\"\n\n\"I guess it just all sunk in. I know Dawson didn't kill Melody. But unless we figure out who did, he could go to jail.\" I stared at my bowl. \"Or worse.\"\n\n\"Don't think about that. We have to stay focused. Now, Sherlock, what have you learned?\"\n\nI smiled. \"Well, Watson, I didn't find out much.\" I told Nana Jo what I learned from Dorothy's granddaughter, such as it was.\n\n\"Let's recap what we do know. It might help to get your 'little gray cells' working.\"\n\nNana Jo loved mysteries as much as I did, so I knew she knew the \"little gray cells\" were a reference to Hercule Poirot and not Sherlock Holmes. So, I didn't correct her.\n\n\"We know Melody was a con artist, and I don't believe leopards ever change their spots. Once a con artist, always a con artist.\"\n\n\"You think she was playing a con on Dawson?\"\n\n\"Yep. I think she planned to get that boy to marry her. I'll bet you my last plug nickel when she looked at him all she saw was dollar signs. That whole penthouse thing was just the start.\"\n\n\"But why?\"\n\n\"To get him away from us.\"\n\nI stared. \"What do you mean?\"\n\n\"It's what cults do. They have to get their victims away from friends and family so they can start the brainwashing.\"\n\n\"How do you know that?\"\n\n\"There was a lecture at the Village a month or so ago about the House of David. Ruby Mae and I went. It was pretty interesting stuff. I even bought a book on them since they were local.\"\n\nNana Jo and I talked for a while, until she said she needed her beauty rest if she was going out sleuthing tomorrow. I delivered Dorothy's message and she went off to bed.\n\nI was still pretty alert from the coffee and decided writing might help to organize my thoughts.\n\nThompkins was kept busy greeting the guests and showing them to their rooms. Lord Charles and Abigail Chitterly were the first to arrive at ten on Thursday morning. Lord Charles was a portly man with thinning hair who was known for a hardy appetite for wine, women, and rich foods. His wife was a mousy American whose only distinctive feature seemed to be her affection for jewelry. She looked like a walking jewelry advertisement, wearing rubies, emeralds, and diamonds in every shape and size. They were followed by the Polish ambassador, J\u00f3zef Lipski, a short, thin man with weak eyes, large spectacles, and a stammer. The French ambassador, Georges Brasseur, an arrogant, hawk nosed man was next. Thompkins showed both gentlemen to their rooms as the foot men, Jim and Frank, took care of the luggage.\n\nThe German emissary, Count Rudolph Heigel, was tall, with slightly thinning blond hair and vacant blue eyes. He was a bit of a dandy, with immaculate attention to attire. He was a model of Aryan fervor who wore his role as an aide to the German secretary of state, Joachim von Ribbentrop, like a medal. Geoffrey Fordham Baker, editor of the London Times was one of the last to arrive. Fordham Baker was the fourth son of Henry Fordham Baker, 2nd Viscount of Lampton. Short, fat, and bald, Fordham Baker dressed in a slovenly manner and left a trail of crumbs in his wake. Cuddles followed him like a living Hoover, cleaning up the trail. Thompkins escorted each of the guests to their room and made sure everyone knew what time lunch would be served.\n\nSince no one was exactly sure when the entire party would arrive, lunch was a cold buffet, allowing guests to eat whenever they arrived. This made serving a lot easier for the staff and Thompkins was able to keep an eye on the guests, without maintaining a constant presence in the dining room. Additional staff would arrive later once everyone arrived. The Marsh family wasn't on intimate footing with any of the early arrivals but did their best to make their guests as comfortable and welcome as possible.\n\nThompkins was grateful for the extra staff in the evening. Wallis, the Duchess of Windsor, arrived with her maid, Rebecca, and an American socialite, Virginia Hall, an hour before dinner. Rebecca was a dark haired, dark eyed vixen, who winked, smiled, and flirted with every male she encountered. Similar to Wallis, she was tall and slender but was curvaceous in areas where Wallis was not. The duchess had a vast amount of luggage, which required two extra footmen and the maid to organize. Virginia Hall was much more interesting, with less baggage. She was an intelligent, well spoken woman with an infectious laugh, thick dark hair, light green eyes, and a wooden leg. She spoke freely about her accident as Thompkins showed her to her room. A hunting accident in Turkey required the amputation of her left leg from the knee down. Despite the wooden appendage, she exuded an air of confidence and determination.\n\nLady Elizabeth held dinner, to allow the duchess time to freshen up and change. It was, therefore, after nine when the party sat down to eat. Chitterly and Fordham Baker were well on their way toward intoxication before they tasted their first bite of dinner. Fordham Baker made an indelible impression on Thompkins by requesting to have his port glass filled while the duck consomm\u00e9 was served. Thompkins halted momentarily but quickly recovered. A glance to Lady Elizabeth was acknowledged by a slight nod, and Thompkins was back to his stiff, proper self.\n\nWallis, the Duchess of Windsor, was neither beautiful nor brilliant, yet she carried herself in a way that made people almost believe she was. Her dark hair was pulled back in a severe bun at the nape of her neck and emphasized her gaunt frame. She barely ate more than a few spoonfuls of the delightful dishes Mrs. Anderson prepared, despite the hours Lady Elizabeth and the cook spent planning a menu that was satisfying to men who'd spent the day shooting and a woman reported to be overly conscious of her weight. Wallis had a raspy voice and a loud, obnoxious laugh. She flirted shamelessly with Victor, Count Rudolph, or Count Rudy, as she called him, and Brasseur.\n\nAfter dinner, the group retired to the parlor for coffee. Lady Elizabeth knitted and watched the spectacle. Lord William had been attempting to engage Geoffrey Fordham Baker in conversation, but the editor had fallen asleep while nibbling on a biscuit. Cuddles, who had just finished consuming crumbs from the cuff of his pants, attempted to climb into his lap to gain access to the biscuit. Thankfully, Lord William caught him and removed him from the room and further temptation.\n\nLady Abigail had taken a seat near Lady Penelope and was pretending to ignore her husband's attention toward Daphne.\n\nDaphne looked a bit strained as she removed Lord Charles's hand from her knee for the third time by Lady Elizabeth's count.\n\nVirginia Hall and Lord James laughed and talked amiably with the Polish ambassador, J\u00f3zef Lipski.\n\nWallis, Count Rudolph, and Brasseur were huddled in a corner near the fireplace. A word of French occasionally escaped the confines of their circle and was quickly followed by a laugh or wink from the duchess. The conversation was obviously meant to be private and no one ventured into their circle. Victor stood nearby but didn't attempt to infiltrate their conclave.\n\nThompkins silently entered the parlor and hurried to Lady Elizabeth. He whispered in her ear. She paused for a second but quickly nodded.\n\nThompkins left and, after a brief pause, he opened the door again and announced, \"Miss Rebecca Minot.\"\n\nThompkins stepped aside and in waltzed a slender raven-haired beauty that appeared to have just stepped off the screen of a Hollywood picture.\n\nThe maid smiled large as she entered the room. \"Ah, I am sorry for zee lateness. Zee butler he is so . . . how do you say\"\u2014she paused\u2014\"he is very proper.\" She stood very straight and stiff in a pose to impersonate Thompkins. She laughed. \"He must get zee permission for me to come. Ah . . . but I am here now.\"\n\nRebecca sauntered over to the duchess.\n\n\"Yes, Rebecca, you're here now. That's all that matters. I couldn't entertain all of these handsome men without you here to help me.\" The duchess glanced at Lady Elizabeth. \"You don't mind that I invited Rebecca to join us, do you? She's more like a companion than a maid, anyway.\"\n\nLady Elizabeth smiled. \"Of course I don't mind.\" \"I knew you'd understand. You're not the least bit stuffy and stuck-up like Cookie, your cousin.\" She tilted her head back and used a finger to push her nose up in the air. Then she looked down her nose and sniffed.\n\nRebecca and Wallis laughed.\n\nLord William blustered. He looked as though he wanted to speak but eventually pulled out his pipe and filled it with tobacco, dropping most of it on the carpet.\n\nLady Elizabeth couldn't hide her confusion. \"Cookie?\"\n\nWallis laughed. \"That's what I call her. Elizabeth looks just like a Scotch cook.\"\n\nLady Elizabeth took a few deep breaths to calm herself. \"No. I can't say that she does. I have always found Her Royal Highness to be elegant, graceful, and refined.\"\n\n\"I daresay we British must seem very silly to outsiders, but thousands of years of tradition are hard to forego.\" Daphne smiled. \"Did you really find Her Royal Highness stuffy and stuck-up? That's odd. I've never thought so. But then, she has made quite a few changes over the past year. I'm sure as she acclimates to her new position, she'll adapt.\"\n\nVictor nearly choked on his drink, and Penelope had to walk over and pat him on the back. \"I'm fine now. Thank you, dear.\"\n\nA brief flash in her eyes and flared nostrils were the only indication Wallis recognized the snub. She quickly recovered and gave a hollow laugh before pulling Rebecca over to join the French and German contingent.\n\nJames looked at Daphne with pride and respect in his eyes before returning to his conversation with Virginia Hall and the Polish ambassador.\n\n\"It's very dull here. How about zee music and dancing?\" Rebecca shimmied, which gained her the attention of Lord Charles and several other men.\n\nVictor flipped the switch for the wireless and tuned into the Seager's Good Mixers program featuring Oscar Rabin and his Romany band. Rebecca pulled Lord Charles to the center of the floor, where she tried to teach him the Lambeth Walk.\n\n\"Come on, dance with me, Rudy?\" Wallis turned to the count.\n\nRudolph Heigel's face was extremely red. His eyes bulged and he stood very erect. \"I will not defile myself with such things.\"\n\nThe duchess looked confused. \"But you dance so well.\" She turned toward the others. \"Rudolph used to be in pictures. He was an actor before he joined the military.\" She looked at the count. \"What's the matter?\"\n\n\"Oscar Rabin is a Jew,\" he spat.\n\nLord William's face turned beet red, and he sputtered, \"Abominable manners. Rude. Raised in a barn.\"\n\nLady Elizabeth managed to relay her displeasure in a single withering glance, which sent a flush up Count Rudolph's neck and made his ears look like beacons. A contrite look crossed his face briefly before he turned away toward Wallis.\n\nJ\u00f3zef Lipski watched near the fireplace. When he heard Count Rudolph's comments, his face grew red. He breathed heavily and clenched his hands into fists. The look in his eyes was one of murderous rage. He started toward Count Rudolph but was intercepted by Virginia Hall.\n\n\"Come on and dance with me and Cuthbert.\"\n\nJ\u00f3zef Lipski halted. He looked puzzled. \"Cuthbert?\"\n\n\"That's what I call my constant companion.\" She patted her wooden leg. \"Cuthbert isn't the best of dancers, but I think we can manage a slow twirl if you won't mind.\"\n\nLipski gave Count Rudolph a scowl then turned his attention to Virginia. \"I would be honored.\" He bowed and the two of them swayed to the music.\n\nDaphne looked relieved when Lord Charles abandoned her for the maid's company. For several minutes, she sat comfortably, without having to dodge unwanted hands.\n\nLady Elizabeth strolled over to her niece. A few seconds later, Penelope joined her sister in the seat vacated by Lord Charles, leaned over, and whispered, \"Well done.\"\n\nDaphne smiled. \"Thanks, but I doubt she even noticed the snub.\"\n\nLady Elizabeth smiled. \"Trust me. I've heard she and David are livid over being denied the title of 'Her Royal Highness.' She noticed.\" She raised her cup in salute. \"Nicely done.\"\n\nDaphne looked around. \"This all seems like such a bizarre farce.\"\n\n\"What do you mean, dear?\" Lady Elizabeth asked.\n\n\"The world is on the brink of another war. Representatives from all of the nations currently in conflict are assembled at one house, our house.\" She inclined her head toward Count Rudolph. \"Twenty years ago all of the nations represented here were at war. Now here we all sit, dancing, drinking, and making merry.\"\n\nPenelope looked at her sister with surprise. \"I never knew you were concerned with foreign af fairs.\"\n\nDaphne smiled. \"Surprised I care about more than my clothes and hair?\"\n\n\"Well, honestly, yes.\"\n\n\"I admit I've been pretty self absorbed and shallow, but things are different now.\" She glanced across the room at James. \"There's a lot at stake, not just for England, but for the world. I sure hope James knows what he's doing.\"\n\nLady Elizabeth stared at her niece. \"So do I, dear. So do I.\"\n\nChapter 9\n\nCoffee late at night, combined with too many hours spent writing into the wee small hours of the morning, left me sleep deprived and cranky Thursday morning. Not even the smell of bacon soothed the savage beast. I sat at the bar and drank three cups of coffee before I even spoke to anyone. It was the safest plan. That way, no one got hurt.\n\nDawson hadn't baked anything, which left me unreasonably irritated. He didn't have to bake every day. There were plenty of treats for the customers, even though we weren't charging for them. I hadn't bothered to get the certificate from the Health Department to allow us to sell food, but no permit was needed to simply offer them and leave a container for donations.\n\n\"You better now?\" Nana Jo asked after I finished my third cup of coffee and started to eat my bacon sandwich.\n\n\"Hmmm,\" I mumbled.\n\n\"Good. Jenna called. She'll be here any minute.\"\n\nRight on cue, the doorbell rang. Snickers and Oreo were momentarily torn between heading downstairs to bark at the newcomer and waiting for bacon to fall to the floor.\n\n\"I'll go.\" Dawson grabbed two biscuits from the jar on the counter and headed downstairs.\n\nThe poodles followed the biscuits downstairs.\n\nDawson let Jenna in and took the dogs out.\n\nJenna joined us upstairs. She had a cup of tea from one of those high-priced coffee shops that had sprung up all over town and a white box. She hopped onto the seat next to me and grabbed a piece of bacon from my plate.\n\n\"Ten minutes ago, that move might have cost you a limb,\" I growled.\n\n\"What's wrong with her?\" she asked Nana Jo.\n\nNana Jo shrugged. \"Woke up on the wrong side of the bed.\"\n\n\"I wish you would stop talking about me like I'm not here.\" Maybe three cups of coffee weren't enough. I poured myself another one and pretended I didn't see the looks Nana Jo and Jenna exchanged.\n\nDawson and the poodles made their way back upstairs.\n\n\"So, I've had some time to think about how we should proceed after the unfortunate events of yesterday.\" Jenna smiled at me and slid the box toward me.\n\nI knew my sister. If she brought me treats, she was working herself up to say something I wouldn't like.\n\nI stared at the box pointedly for several seconds and then looked at my sister and raised an eyebrow.\n\n\"Aren't you going to open it?\"\n\n\"Beware of geeks bearing gifts,\" I said.\n\n\"It's Greeks, not geeks.\" Jenna wasn't always the brightest bulb in the pack, but she eventually got the joke and groaned. \"Just open the box.\"\n\nI opened the box, which contained half a dozen strawberry tarts from my favorite bakery, A Taste of Switzerland. The tarts glistened with a sugary glaze under the lights in my kitchen, and I swore I heard them whisper my name.\n\nNana Jo reached in and grabbed a tart. \"Hmmm. This must be a biggie if you're softening the blow with ten dollar pastries.\"\n\nA Taste of Switzerland's pastries weren't quite ten dollars each, but they were pricey. I wanted to be strong and hold out until I heard what Jenna was leading up to, but after Dawson and Jenna each grabbed tarts, there were only three lonely looking tarts left in the box. My willpower fled. I grabbed one. I bit into the flaky crust and my eyes rolled back into my head as the sugary goodness seeped through my body. I might have moaned because I heard laughter and when I opened my eyes, Nana Jo, Dawson, and Jenna were all staring at me with silly grins on their faces.\n\n\"Shut up.\" I licked the gooey strawberry filling off my fingers.\n\n\"I guess that must have done the trick,\" Nana Jo said.\n\n\"I know my sister.\" Jenna wiped her hands on a napkin.\n\n\"Alright.\" I wiped my mouth and took a swig of my coffee, swishing it around in my mouth to make sure every delicious crumb made its way to my stomach. \"What do you want?\"\n\n\"I want to use the media to our advantage. I'd like to invite some of the media here to the bookstore and have them interview Dawson. I want them to see he's nothing like his dad. He isn't a killer. He's just a kind, honest kid who plays football, bakes, works hard, and goes to school.\"\n\n\"I thought you didn't want us talking to the media?\" I tried to avoid sounding whiney and deliberately flattened the scowl I felt developing on my forehead.\n\n\"Yeah, well, that was before Dawson's dad went on television and in one fell swoop insulted the police, the victim, and all women everywhere.\" She looked at Dawson. \"No offense.\"\n\n\"None taken,\" he mumbled.\n\n\"Here's what I want. One of Tony's fraternity buddies is the producer for WJMU. He's agreed to an exclusive interview. I get to see all of the questions in advance and I'll be right there the whole time. I can pull the plug if he goes off script and, trust me, I will.\"\n\nDawson looked uncertain. \"I don't know . . .\"\n\n\"You can't be shy. You've done interviews after football games. It'll be just like that.\"\n\n\"After a game, I talk about football. I can talk football all day. What am I gonna say to this guy?\"\n\n\"You'll tell him about growing up without your mom and how you were raised by an abusive, alcoholic father.\"\n\nDawson shook his head. \"I don't wanna talk about any of that.\"\n\nJenna looked down. \"Unfortunately, some of it's going to come out anyway. I talked to a friend at the River Bend Times, and some of the information from child protective services has already been leaked.\"\n\nNana Jo and I were indignant.\n\nI asked. \"Aren't juvenile records supposed to be confidential?\"\n\nJenna shrugged. \"Dawson's records are, but his father's aren't.\"\n\n\"Well, I'll be . . .\" Nana Jo said.\n\n\"This will be your chance to show the public who you are. There's been so much negative publicity in the news about athletes behaving badly and getting away with it, the public may not be supportive.\"\n\n\"Jenna's right.\" Nana Jo turned to Dawson. \"This is your chance to show them you're just a poor kid trying to pull yourself up by your jockstrap.\"\n\nI nearly choked on my coffee. \"I think the phrase is bootstrap.\"\n\n\"Whatever.\" She rolled her eyes. \"Show them how you bake to unwind. Heck, let them taste your cookies.\"\n\nDawson groaned. \"I'll never live this down. When the guys on the team hear I like to bake, they'll never stop raggin' me.\"\n\nNana Jo patted him on the back. \"Honey, I hate to break this to you, but the guys on the team are the least of your problems right now.\"\n\nJenna and Nana Jo were right, but I was still reluctant. Agreeing to this could expose Dawson to public ridicule, and there was no guarantee anyone would believe him. \"This seems risky to me.\"\n\nMy sister wasn't someone who took risks. There was more to this than she was revealing.\n\n\"What aren't you telling us?\"\n\nJenna sighed. \"Okay. I've talked to a friend in the DA's office. He wants to be seen as tough on crime, so he plans to come down hard on you, just to show how tough he is.\"\n\nNana Jo mumbled something that sounded like weasel.\n\n\"But what about my dad? He said he knew who the real killer is.\"\n\nJenna paused for a moment. \"Frankly, no one is taking him seriously.\"\n\nI exchanged a glance with Nana Jo.\n\nJenna took a deep breath. \"Your dad could be facing more jail time. He's been trying to extort money.\"\n\n\"From who?\" I asked.\n\n\"Anyone . . . everyone. The media, the police, the district attorney, you name it. He says he knows who the killer is, but he won't talk unless he gets a quarter of a million dollars.\"\n\nNana Jo whistled.\n\nDawson stared openmouthed but then dropped his head and looked away.\n\n\"My source at the police station doesn't believe he knows anything, but they'll investigate, provided he gives them something.\"\n\nWe discussed the interview for a while longer. Finally, Jenna looked at Dawson. \"Well, what do you think?\"\n\nDawson looked at Nana Jo and me. \"Do you think I should?\"\n\nNana Jo nodded. \"I do. I think it will be your chance to tell your story. Plus, if you want, I'll be there to support you.\"\n\nEveryone looked at me. I stared in my empty coffee cup, then took a deep breath. \"I think you should do it too. But, it's totally up to you. You have to follow your gut. What do you think?\"\n\nDawson took a moment before responding. \"I don't really want to do it\u2014\"\n\nI started to interrupt, but he held up a hand to stop me. \"But I think I should. Mrs. Thomas is right. It will give me a chance to tell my side of things.\"\n\nJenna exhaled. \"Great. I was hoping you'd agree, especially since I already told them to come around noon.\"\n\nDawson looked as though he was mustering up his courage to speak. Finally, he said, \"I'll do the interview.\" He turned to Jenna. \"But can you do me a favor, please?\"\n\nJenna said, \"Sure, if I can.\"\n\nDawson took a deep breath. \"Can you please help my dad? I know he's a big jerk and he's done some bad things, but he's still my dad.\"\n\nJenna stared at Dawson for a moment. \"Dawson, I don't know if I can help your father. He\u2014\"\n\n\"But he really needs someone\u2014\"\n\n\"Wait. It's not that I don't want to help. I may not be able to help. I have to be careful there's no conflict of interest. I'm your lawyer, and I have a responsibility to represent you to the best of my ability.\"\n\nDawson hung his head. \"I understand. I just thought maybe you could talk to him.\"\n\nJenna stared at him. \"Okay. I'll talk to him. But, I can't promise anything. I have to check with his PD. And as long as she's okay with it, I'll talk to him. Okay?\"\n\n\"Great. Yeah. That's great.\" He perked up and smiled. \"Thanks.\"\n\nWe talked for a bit and decided we didn't all need to hang around for the interview. Since I'd already made arrangements to go on campus and meet Jillian, I'd let Nana Jo and Jenna handle the media.\n\nJenna left not long afterward.\n\nI did some paperwork for the bookstore until the twins arrived to help. I would miss them when fall break was over and they went back to school. They were such a great help.\n\n* * *\n\nI met Jillian at the student union as arranged. She was dressed in a miniskirt with leggings and a sweater with ballet flats. Last night her hair had been slicked back into a bun. Today it was thick and wavy and loose.\n\n\"I like your hair.\"\n\n\"Thanks. It takes a lot to get it under control for performances.\" She laughed.\n\nShe took me to Melody's dorm. The door was slightly open, but we knocked and were instructed to come in.\n\n\"Mrs. Washington, this is Emma Lee.\" Jillian turned to Emma. \"This is . . . Mrs. Washington.\"\n\n\"Please, call me Sam.\" I extended my hand.\n\nThe dorm room was small, about the size of my bedroom at home. Small bedrooms are fine when it was just one person and all you needed to do was sleep in the room. However, this room was shared by two people. So there were twin beds, two desks, two dressers, and two closets. One side of the room had posters, family pictures, and a colorful comforter on the bed and looked homey. The other side was barren with nothing except a nondescript blue coverlet to prove anyone lived there.\n\nEmma Lee was petite. She might have been five feet and one hundred pounds if she wore weights. She was Asian with dark almond eyes and long dark hair, which she wore pulled back into a ponytail. She looked uncertain, but good manners always showed and she stepped forward and shook my outstretched hand.\n\nJillian went to Emma's bed and sat.\n\nEmma pulled out her chair for me and I sat. She sat in the other chair and waited.\n\n\"I wondered if you could answer a few questions about your roommate for me.\"\n\nEmma was silent.\n\nI intercepted a look between the two girls. Jillian nodded as if to say \"she's okay.\"\n\n\"Fine. But I'm not sure how much help I can be.\" Emma spoke with a distinct southern accent, which made me smile to see that southern belles came in all different shapes, sizes, and colors.\n\nI couldn't hide the surprise on my face and I heard it in my voice. \"As her roommate, you must have gotten to know each other pretty well.\"\n\nEmma shook her head. \"I hardly ever saw her.\"\n\nI looked at the bare side of the room.\n\n\"Melody rarely used this room.\" She went to the closet and opened it. \"She kept a few items of clothing here, but that's about it.\"\n\n\"But if she didn't stay here, where did she stay?\"\n\nEmma shrugged. \"Beats me.\"\n\n\"Do you know if she had any family?\"\n\nShe shook her head. \"I have no idea. When we met, I tried to ask the normal questions, like 'where're you from?' I tried to get to know her. But she shut me down so fast it made my head spin.\"\n\n\"How long were you roommates?\"\n\n\"Only since the start of the semester.\"\n\n\"Is there anything you can tell me about her?\" I got up and walked over to Melody's side of the room. \"Mind if I look?\"\n\n\"Help yourself, but there ain't much to see. When I realized she wasn't planning to stay, I took a peek.\" Emma got up and joined me.\n\nThere were three sweaters, a warm leather bomber jacket, and a pair of tennis shoes in the closet. The dresser had two pairs of underwear, a bra, a nightshirt, and a small bag of toiletries.\n\n\"Is that it?\" I closed the last drawer.\n\n\"Yes, ma'am.\"\n\nI stood in the room and looked around, mentally comparing Emma's side of the room with Melody's. \"But where are her school items?\" I walked over to the desk and opened the drawer. There wasn't even a pencil.\n\n\"Beats me.\" Emma shrugged.\n\nI stared at her. \"Did the police remove anything?\"\n\n\"No one from the police came by. Well, not while I've been here at least.\" She looked confused. \"That's odd, isn't it?\"\n\n\"Yes. I would have expected someone would have come by.\"\n\nA sudden thought made Emma gasp. \"Sweet Jesus, you think they notified her family?\"\n\n\"I'm sure someone from the school has if the police didn't,\" I said with more confidence than I felt.\n\nIt must have been enough to put Emma at ease, because she breathed a sigh of relief. \"Well, thank the Lord for that.\"\n\n\"Did you have any classes with Melody?\"\n\nEmma shook her head. \"No, ma'am.\"\n\n\"What was her major?\"\n\nShe shrugged. \"Beats me.\"\n\nI was confused. \"But she was a senior. That means she'd be graduating in a few months. Surely she mentioned what she was getting her degree in?\"\n\n\"All I know is she transferred here from the East Coast. Melody wasn't talkative.\"\n\nJillian mumbled, \"At least not to women.\"\n\n\"Help me understand.\" I looked from one of them to the other.\n\nEmma said, \"She didn't talk much, but from what I gathered, she was majoring in M-r-s.\"\n\n\"Excuse me?\"\n\nEmma smiled. \"You know, M-r-s.\" She pointed at her ring finger.\n\n\"You mean she was just here looking for a husband.\"\n\nThey nodded.\n\nI turned to Jillian. \"But aren't you in the same class?\"\n\n\"Supposed to be, but that's what's weird. I've only seen her there once all semester.\" Jillian pulled her laptop out of her book bag, typed something, and navigated for a few seconds. When she was finished, she turned the laptop to face me. \"Here's the online link to our class. We use a program called Canvas.\"\n\n\"I'm familiar with Canvas. I used to be a teacher.\"\n\nJillian smiled. \"Great. From here, anyone enrolled in the class can see the syllabus, the homework assignments, and exams, pretty much everything.\"\n\nI looked at the familiar site.\n\nJillian navigated to the grades section.\n\n\"I can see all of my grades because I'm logged into the system. But, if I click here\"\u2014she clicked on a link on the navigation bar\u2014\"I can see everyone's grades in the class. That way, I can see how I stack up against everyone else.\"\n\nI stared at the screen. \"But there are no names listed, only student identification numbers. How do you know which one is Melody's?\"\n\n\"They do that for privacy,\" Emma added.\n\n\"True. But, it doesn't take much to figure out who's who.\" Jillian looked sheepish. \"Process of elimination. Look. This is me.\" She pointed to a line on the screen. \"There are only fifteen people in this class and the rest of us talk to each other. \"\n\n\"They tell you their grades?\" I asked.\n\n\"Sometimes. But you can pretty much figure out who got what. Last month, Martin was complaining about getting a low grade. He said he had a perfect score up until then. So that has to be him.\" She pointed at a line on the screen. \"Eva had a family emergency and had to go home. So the incompletes for the last two assignments are hers.\"\n\nEmma added, \"Regina was telling everyone how she aced the last test, which made up for her getting low scores on the two previous ones.\"\n\n\"Based on what people have said, and through the process of elimination, I know this\"\u2014she pointed at one line\u2014\"has to be Melody. No one else has gotten a perfect score through the entire course.\"\n\nI stared at the screen and then at Jillian. \"You're amazing. You'll be a fine detective.\"\n\nShe grinned. \"Must run in my genes.\"\n\n\"I'll bet if we could find out her student number, we could confirm her grade,\" I said. \"Although, I'm not sure what that will tell us.\"\n\n\"She's getting a perfect score in a class she never attends,\" Jillian said.\n\n\"Seems fishy to me,\" Emma added.\n\n\"Me too.\"\n\nI offered to take the girls to lunch to thank them for their help. Thursday was Emma's short day for classes, but Jillian had a lab and wouldn't be free until after two. We agreed to meet at the student union at two fifteen.\n\nMISU wasn't a large campus, but it could be quite confusing to get around, so I had to ask for directions to the administration building. I hadn't held out much hope they'd provide information to a total stranger. The Family Educational Rights and Privacy Act of 1974, or FERPA, made it impossible for anyone to get student educational information without written permission from the student. Parents couldn't even get their son's or daughter's grades, so I wasn't surprised when they refused to provide them to me.\n\nAs I left the administration building, I ran into Peter Castleton, MISU's athletic director. He'd been very supportive of Nana Jo and me tutoring Dawson over the summer.\n\n\"How's Dawson?\"\n\nCastleton was about five ten, two hundred pounds. He was bald with the lean, muscular physique of a runner.\n\n\"As well as can be expected. He's worried, and he misses football.\"\n\nCastleton nodded. \"I'm sure, but the policy is very clear. Any athlete charged with a felony must be suspended from all academic and athletic activities until the situation is resolved.\"\n\n\"I know.\"\n\nCastleton stared at me pointedly. \"You understand I'm in a bit of a difficult situation here. I like Dawson. I think he's a good kid with a very bright future if . . .\"\n\n\"If he doesn't get convicted for murder?\"\n\nCastleton nodded. \"I'm really sorry. We try to warn the boys.\"\n\n\"Warn them?\"\n\nCastleton squirmed. \"Against honey traps.\"\n\n\"What's a honey trap?\"\n\nCastleton rubbed the back of his neck. \"Some young people enter college with the sole intention of finding a spouse, preferably a wealthy one or someone with the potential to become wealthy\u2014a meal ticket.\"\n\n\"You think Melody's only interest in Dawson was as a meal ticket?\"\n\nHe waved his hand. \"I don't know anything about her personally, and I don't want to speak ill of the dead, but some of the players talk.\"\n\n\"Can I talk to them?\"\n\nHe stared at me.\n\n\"We're trying to figure out who else might have wanted to kill her. The police are stuck on Dawson and aren't looking for anyone else. I'm hoping maybe some of the players may have seen or heard something that might help.\"\n\nCastleton seemed to think that through. \"The players will be watching a film at six tonight in the athletic center's media room.\" He pulled out a piece of paper and an envelope from his back pocket and wrote on it. \"Show this and your driver's license to the guard and he'll let you in.\" He handed me a note which granted me permission to enter, basically a permission slip.\n\n\"Thank you.\"\n\nI still had about thirty minutes before I was to meet Emma and Jillian at the student union, so I made my way to the history department. After asking around, I found a long, narrow closet with Professor Harley Quin's name on the door. The door was open.\n\nJazz drifted from inside, which made me stop. I peeked inside. A blue-eyed Sean Connery pretended to play the piano along with the music.\n\nI must have moved because he glanced my way, smiled, and turned down the music.\n\n\"I'm sorry. I didn't mean to interrupt.\"\n\n\"Caught in the act.\" He not only looked like Sean Connery, but he spoke with a British accent.\n\n\"That's one of my favorites.\"\n\nI could tell by the skeptical look on his face he didn't believe me. \"Really?\"\n\n\"I love David Benoit.\"\n\nSurprise and a small amount of awe played across his face. \"Ah . . . but what's the name of the song?\"\n\nI smiled. \" 'Linus & Lucy.' \"\n\nHe smacked his leg. \"Brilliant. Nine out of ten people would have said, 'Peanuts.' You really are a fan. Come in. Welcome to my closet.\" He made a grand sweeping gesture with his arm and pushed a pile of papers onto the floor. He had on brown tweed pants and a white shirt with a bow tie. He looked like Sean Connery in Indiana Jones and the Last Crusade.\n\nI walked into the room, which couldn't have been more than five feet wide. I could literally reach out and touch both walls at the same time. There were books and papers on practically every surface.\n\n\"Now, what can I do for you? You're not one of my students. I certainly would have remembered you.\"\n\nI grinned at the compliment. \"Well, I was hoping you could help me. I have some questions about one of your students.\"\n\nHe looked puzzled. \"I'll try.\" He turned in his seat so he faced me.\n\nI was finding it very hard to look him in the eyes without smiling like a lovesick schoolgirl. I needed to pull myself together. I took a deep breath. \"Can you tell me about Melody Hardwick?\"\n\nHe looked puzzled. \"Melody Hardwick . . . Melody Hardwick, I'm afraid I don't . . . you don't by any chance mean the girl who was strangled by that football player, do you?\"\n\nNothing could have cured my lovesick attitude quicker. I squinted my eyes and my blood pressure rose. \"Dawson Alexander is a fine young man, and he did not kill anyone.\" I enunciated when I was angry, and I was angry now. \"And, in this country, people are innocent until proven guilty.\" I rose and turned for the door.\n\n\"Whoa. Wait. I'm sorry.\" He jumped out of his seat and grabbed me by the arm. \"I'm sorry. I didn't mean to offend you. Please forgive me?\"\n\nHearing someone accuse Dawson of murder made me furious.\n\n\"Please. Won't you sit down?\" he begged.\n\nI took several deep breaths and returned to my seat.\n\n\"I'm very sorry. I didn't mean to insult you or Mr. Alexander. I was merely repeating what I'd heard. Now, how can I help you?\"\n\n\"Melody Hardwick was one of your students. I was hoping you could tell me something about her.\"\n\nHe shook his head. \"I wish I could, but I've only been here a semester and I'm afraid I don't really know many of the students yet. I saw her picture in the newspaper, but honestly, it didn't ring a bell with me. Are you sure she was in my class?\"\n\nI nodded. \"Yes. She was in History of Cults.\"\n\nHe thought for a moment and then shook his head. \"I'm sure you're right, but I don't remember her. I guess she didn't make much of an impression on me.\"\n\nHe smiled in a way that implied I was making an impression. I felt nervous and uncomfortable. It had been a long time since anyone flirted with me. I was out of practice and began to fidget so much I dropped my purse.\n\n\"I know one of the other girls in the class, and she tells me Melody never attended class but she had a perfect grade.\"\n\n\"Well, that's odd. Let's take a look.\" He turned to his computer and tapped away for a few minutes. \"Now I remember. It's the name that threw me. I have an Elizabeth Mae Simpson.\"\n\n\"Yes, that's her. She was using an alias.\"\n\n\"How odd.\" He scanned the computer screen. \"Miss Simpson is showing a perfect score. You say she never attended class? I'll certainly have to look into it. Must be a mistake.\" He turned to face me. \"This computer system is different than the one I'm used to back in England. I must have made a mistake.\" He smiled. \"I'm sure I would have found it before the end of the term.\"\n\nIt sounded reasonable.\n\n\"Now, is there anything else I can help you with?\"\n\n\"Actually, there is something else, but it's rather personal.\" My face heated.\n\n\"Now this sounds promising.\" He settled back and fixed an inquiring look on my face.\n\n\"I'm writing a book.\"\n\n\"Wonderful.\"\n\n\"Thank you. It's a mystery set in the British countryside. I was wondering if maybe you could help me with some research,\" I stammered.\n\n\"I'd love to. I'll be happy to answer any questions I can. Perhaps we can discuss it over dinner say, tomorrow night?\"\n\n\"Mrs. Washington?\"\n\nI turned, thankful to see Jillian standing in the doorway. \"Yes. Tomorrow night will be fine.\"\n\n\"Why don't you leave me your number and I'll call you with the details.\"\n\n\"Okay.\" I hurriedly wrote my cell number on a sheet of paper and rushed from the room, remembering at the last minute to turn and say, \"Thank you.\"\n\n\"No. Thank you.\"\n\nOutside, Jillian rushed to keep up with me. \"Hey, I'm sorry, Mrs. Washington. I didn't mean to interrupt. I had to drop off a book with one of my professors. I didn't mean\u2014\"\n\n\"It's okay. You didn't interrupt anything. He's just going to help me with some research for a . . . a book I'm working on.\"\n\n\"Granny told me you've written a book. That's great.\"\n\nDorothy Clark certainly had a big mouth. I'd have to be careful what I told her, but none of this was Jillian's fault. \"Come on. We better get over to meet Emma.\"\nChapter 10\n\nEmma was waiting for us at the student union. Both girls in tow, I headed to downtown North Harbor. They asked to see my bookstore and I was rather curious about Dawson's interview, so I headed home.\n\nThe interview had just wrapped up and the reporters were loading equipment into a large van in the back parking lot.\n\nDawson looked tired, but no other signs of distress were present. Nana Jo wasn't waving her peacemaker and there were no dead bodies, so I assumed all went well. I was a bit anxious to watch it.\n\nThere were a few patrons in the shop, but my nephews had things under control. The twins rushed to offer assistance to my two young friends. Boys would be boys.\n\nI walked over to Dawson. \"How'd it go?\"\n\nHe shrugged. \"Okay, I think.\" He rubbed the back of his neck. \"I hate talking about myself, but Mrs. Rutherford said it was good.\"\n\nI patted his shoulder. \"Then I'm sure it was fine.\" I stared at him. \"You look tired. Have you eaten?\"\n\nHe shook his head.\n\n\"Why don't you come with us?\" I looked around. \"Melody's roommate, Emma, and Dorothy's granddaughter, Jillian, helped me earlier, and I promised them lunch.\" I looked at my watch. \"Although it's almost dinnertime.\"\n\n\"Where are you going?\"\n\n\"I thought I'd try that new restaurant that opened up a few doors down. I started to eat there the other day, but I never actually got to eat.\" That was Monday, when I'd seen the police had picked up Dawson for Melody's murder and rushed to the police station. Definitely a day I'd like to forget.\n\nNana Jo joined us. \"He did a real fine job. Real fine.\"\n\nI could tell by the raspy sound in her voice, Nana Jo had fought back tears.\n\n\"I was just inviting Dawson to lunch.\"\n\n\"Good. I'm hungry enough to eat an entire heifer. Let me get my purse.\"\n\nNana Jo ran upstairs and by the time she returned, I'd arranged for Jillian, Dawson, Zaq, and Emma to go down and secure seating. Christopher agreed to stay and keep things going in the bookstore in exchange for a burger and fries.\n\n\"I called the girls while I was upstairs. They're on their way. We were supposed to meet for the book club.\"\n\n\"I totally forgot about the book club meeting today.\"\n\nThe Sleuthing Seniors Book Club formed over the summer, and Nana Jo and her friends met at the bookstore on the first Thursday of every month to discuss mysteries.\n\n\"Maybe I should go tell everyone there will be four more coming for lunch.\" I started toward the door.\n\n\"No worries. I already sent a text message to Jenna.\"\n\nNana Jo looked like she wanted to talk, so I waited.\n\n\"That boy has really had a tough time.\" She shook her head and swallowed hard in an effort to keep her emotions in check. \"I knew things were hard, but I never knew everything.\"\n\n\"He doesn't like talking about his dad or his life as a kid. Remember when we found him hiding out in the bathroom of the bookstore?\"\n\nNana Jo nodded. \"His father got drunk as a skunk and beat the crap out of him. He ran away with nothing but the clothes on his back.\"\n\nI patted Nana Jo and tried to keep from getting teary eyed as I thought back. \"I'm so glad he came here.\"\n\n\"Sometimes I wonder what would have happened if he hadn't been working at that country club.\"\n\nI laughed. \"And if I hadn't drank four glasses of champagne on an empty stomach and made a spectacle of myself at the funeral reception for Clayton Parker when I puked and needed help getting home.\"\n\nNana Jo laughed. \"Clayton Parker was an evil man, but if it weren't for him, we might not have Dawson with us now. Maybe he wasn't so worthless after all.\"\n\nClayton Parker's death wasn't a reason for rejoicing, but I understood what she was getting at.\n\nWhen the girls arrived, we walked to the restaurant. Two tables had been pushed together at the back of the restaurant. Dawson sat at the end with Jillian and Emma on either side of him. Zaq was next to Emma and Jenna sat next to Jillian. I sat next to Jenna and across from Dorothy. Nana Jo seated herself at the head of the table and Irma and Ruby Mae were on her sides.\n\nWhen the waitress arrived to take our drink orders, Nana Jo took a look at her watch and ordered a Wild Turkey, neat. She pulled out her iPad and put on her glasses. She waited until all the drink orders were placed. \"Now, we're under a time crunch, so I suggest we move forward with our meeting.\" She looked around.\n\nNo one disagreed.\n\n\"Who would like to start?\" She looked around the table.\n\nIrma coughed and raised her hand. \"I talked to my great-grandson, Ernie. He's a realtor.\" She coughed. \"He said Virgil Russell was a shady character and a slumlord. Apparently, he applied for a grant from the government to build that high-rise on the lake. Said he was building low-income housing for minorities and senior citizens so he got a super low interest rate. But then he made those units the smallest units he could. Ernie says he rarely rents to low-income folks unless they pay him under the table.\"\n\n\"What do you mean, pay him under the table?\" I asked.\n\n\"If they have a section eight voucher, they pay a reduced rate to their landlord and HUD pays the difference. The landlord is supposed to make repairs, but Ernie said Virgil Russell stalls on the repairs.\"\n\n\"What's HUD?\" Jillian asked.\n\n\"The Department of Housing and Urban Development,\" Ruby Mae added.\n\n\"Why would anyone put up with that?\" Dorothy asked.\n\nRuby Mae looked up from her knitting. \"They put up with it because they don't want to make trouble.\" She looked around. \"I found out quite a bit on that scoundrel, but I'll wait my turn. I don't want to interrupt.\" She nodded to Irma. \"Go ahead, honey.\"\n\nIrma continued, \"Ernie said Virgil Russell is a sleazy, lowlife slumlord who can't be trusted any farther than he can spit.\" Irma then turned and spit on the floor.\n\n\"Irma!\" everyone said.\n\nIrma coughed and then took a swig of the whiskey the server placed in front of her. \"Sorry.\"\n\nNana Jo turned to Ruby Mae. \"Perhaps you would like to go now.\"\n\nRuby Mae stopped knitting long enough to take a sip of her coffee. \"What I found out confirms what Irma said. Virgil Russell is a slumlord. My niece's boy, Robert Earl, works downtown at the County-City building. They've been trying to clean up a lot of the run-down housing in North Harbor. He said a lot of the worst properties are owned by absentee landlords, but Virgil Russell owns over one hundred rentals, basically shacks. Most of them need to be torn down.\"\n\n\"Roach motels is what Ernie called them.\" Irma burst into a coughing fit.\n\nDorothy smacked Irma on the back. \"You need to build up your immune system. You're just a wreck.\" She reached in her bag and pulled out a bottle of tablets. \"Here, you need to take these air immune tablets. They're wonderful.\" Dorothy handed two tablets to Irma.\n\nIrma looked at the tablets and then popped them both in her mouth and started to chew.\n\nDorothy looked stricken and then shouted, \"What are you doing? You don't just put them in your mouth. You have to dissolve them in water.\"\n\nIrma coughed and then began foaming at the mouth. She reached for a glass of water. With each gulp of water, more foam came from her mouth.\n\nRestaurant patrons began to stare.\n\nThe manager came over. \"Is everything okay? I'm trained on the Heimlich.\"\n\nIrma shook her head but then realized he was making his way toward her. Finally, she reached in her mouth and pulled out her teeth. She took her napkin and wiped her tongue and mouth.\n\n\"Blech. Dorothy, what the h\u2014\"\n\n\"Irma!\"\n\nIrma grabbed her drink and tossed it back, then grabbed Nana Jo's drink and took a long swig. Then she swished the liquid around in her mouth and swallowed.\n\nWe all stared for several seconds and then burst out laughing.\n\nThe manager looked around like he didn't know what to think of us. He eventually joined in the laughter too.\n\nWhen we finally pulled ourselves together, Jenna wiped her eyes. \"Oh my goodness.\"\n\nZaq and the younger crew continued laughing.\n\nIrma smiled and picked up her teeth and wrapped them in her napkin, which got everyone started laughing again.\n\nNana Jo said, \"Irma, what made you take your teeth out?\"\n\nIrma said, \"Once that stuff got in my teeth, I tried using my tongue to work it out. I guess it agitated the darned things with my saliva and I started foaming. Then when I drank the water, it got worse. Since I couldn't get it out of my teeth, I decided to get rid of the teeth.\" She grinned and everyone laughed again.\n\nNana Jo said, \"Ruby Mae, I think it's safe for you to continue now.\"\n\n\"Well, as I was saying, Virgil Russell is a slumlord.\"\n\nEmma tentatively raised a hand. \"Excuse me. I just think that's awful. Why do people put up with that?\"\n\nRuby Mae continued, \"The wait list for low-income housing is long, especially for more than two bedrooms. When I was in Chicago, after my husband left, I was on the wait list for five years for one of those Section Eight housing vouchers that helps low-income people afford decent housing. By the time I finally got to the top of the wait list, my cleaning business had taken off and I didn't qualify anymore, and I'm thankful for that. Lord knows I'm thankful. Some of those places were awful. I would have just as soon slept outside.\" She shook her head. \"There was a lady at my church who was living in one of Virgil Russell's run-down shacks. I've never seen such filth. She complained, and he made her life miserable. He wouldn't make repairs. Her toilet was out for almost six months, and he wouldn't repair it. He wouldn't collect the trash. It was bad.\"\n\nEmma looked appalled. \"But that's horrible. Why did she stay?\"\n\n\"Where else is she gonna go, baby? She had three small children and no other family. She didn't have the money to get the repairs done herself.\" Ruby Mae leaned in. \"But here's the worst part. You know that filthy little heathen had the nerve to tell her she could pay in other ways.\" Ruby Mae raised her eyebrows in a suggestive manner.\n\nEveryone got what Ruby Mae was talking about except Emma, who had a puzzled expression on her face.\n\nIrma leaned over and patted her hand. \"She's talking about sex, honey.\"\n\nEmma's face turned bright red, and her expression went from puzzled to shock to revulsion in less than ten seconds. \"Eww.\" She shook herself. \"That's just . . . eww.\"\n\n\"That's illegal, isn't it? How was he able to get away with doing that?\" Jillian turned to Jenna.\n\n\"Yes, it is illegal. Unfortunately, a lot of people won't report it for fear of repercussions, also illegal.\"\n\n\"But doesn't someone inspect these homes to make sure they're fit to live in?\" Jillian asked.\n\n\"They do. Sometimes that just plays into the landlord's hands. The government only pays so much for the apartment rent, whatever the average market rates are. The benefit to the landlord is they can get a low interest loan as long as they designate a certain amount of units for low-income residents. If the landlord accepts the Section Eight housing certificate, he's guaranteed to get his money from the government. Then he doesn't do repairs. The renter reports the landlord. He's given an ultimatum. Either do the repairs or he'll be removed from their list of approved rentals.\" Jenna paused and looked around. \"He doesn't do the repairs and is removed as a valid Section Eight landlord. They remove their tenants. The landlord can now do the repairs and raise the rents.\"\n\n\"And he's gotten the low interest loan.\" Nana Jo pursed her lips and scowled.\n\nJenna nodded.\n\n\"That sounds sordid, but . . .\" I tried to put my thoughts into words.\n\n\"But does it make him a killer?\" Nana Jo asked.\n\nI nodded.\n\n\"Unless, maybe Melody knew something about it and she threatened to turn him in?\" Jillian asked hopefully.\n\nI shook my head. \"No. More likely she knew about it and was helping him.\" I looked tentatively at Dawson, but he didn't seem to be bothered by talk of Melody.\n\n\"What happened to your friend?\" Emma asked Ruby Mae.\n\n\"Once our church found out what was going on, we packed up her stuff and moved her in with one of the members. Things worked out well, and now she's got one of those Habitat homes.\" She smiled. \"That's all I was able to find out.\"\n\nWe turned to look at Dorothy.\n\n\"My date did know a little bit about this Melody person, but he was a bit grabby, so I didn't get as much information out of him as I would have liked.\" She winked. \"I may have to go out with him again to see what else I can get out of him.\"\n\n\"Way to take one for the team,\" Nana Jo said sarcastically.\n\n\"I aim to please.\" Dorothy smiled, then got serious. \"Turns out Melody was in a special program in New Jersey for troubled teens or at-risk kids.\" She rummaged in her purse and pulled out a sheet of paper. \"Higher Ed Restart or some such thing. The way I understand it, some of these kids are really smart but keep getting in trouble. This program gives them an opportunity to go to college. The hope is they'll get a good education and redirect their energy so they use their brains for good rather than evil.\"\n\n\"I think I've read about this program,\" I said. \"When I was a teacher, there was a lot of information floating around about how it's cheaper to send kids to Harvard for four years than it is to prison. Plus, they can learn a skill that's useful and hopefully not end up back in their negative environment. But, I didn't know they were doing it here in North Harbor.\"\n\n\"That's dangerous, isn't it?\" Emma asked tentatively.\n\n\"Harvey said they had very strict criteria,\" Dorothy reassured her. \"No one with a history of violence. No rapists. No murderers. No arsonists. No child molesters.\" She continued to scan her paper. \"He said MISU only took two. Both were older than the traditional students. Melody was one and the other was a guy who was also one of those grifters.\"\n\n\"Seems like they should tell people, you know, warn them,\" Jillian said.\n\n\"They want to give the students the best possible chance for success. The hope is they'll blend in and adapt. Harvey says the program's very successful. There are students placed in universities all over the world.\"\n\nNana Jo tapped info into her iPad.\n\nWe waited until our server finished taking our orders. When all the orders were placed, I told them what I'd learned on campus. Emma and Jillian helped fill in the gaps. I told them about my conversation with the athletic director and plan to talk to the team later.\n\nWhen I was done reporting, I turned to Nana Jo.\n\n\"Freddie's son, Mark, found out that our friend Virgil Russell, spent time in prison for fraud, embezzlement, and\"\u2014she paused\u2014\"manslaughter.\"\n\nEveryone sat up in their chairs and fired questions at the same time.\n\n\"Hold your horses. Virgil was convicted of murdering his partner, a man named . . .\" She scrolled her iPad. \"Here it is, Max Simpson.\"\n\nI gasped. \"Simpson, that's Melody's last name.\"\n\nNana Jo nodded. \"Yep. Max had a daughter.\"\n\n\"Elizabeth Mae Simpson,\" we all said together.\n\nShe nodded.\n\nDawson looked shocked. \"But that's . . .\" He shook his head. \"That's just wrong. He had his hands all over her.\"\n\nI didn't understand that either. \"If he went to jail for murdering her father, it seems odd she'd be having a relationship with him.\"\n\nWe talked until our food arrived, and then we put aside talk of murder until we finished eating. Irma hadn't been able to clean her teeth, so she drank her dinner and had her food placed in a take-out container. She was pretty tipsy and abandoned our group to flirt with a couple of men at the bar.\n\nJenna looked amazed by all of the activity and whispered to me, \"Are they like this all the time?\"\n\nDorothy had joined Irma at the bar and both were drinking like fish. Emma and Jillian looked amused.\n\nI looked at Dorothy and Irma. \"This is nothing. You should see them when they really get revved up.\"\n\nJenna looked as though her eyes would pop out of her head. \"Seriously?\"\n\nI nodded. \"They can be a handful, but they're also amazing at using their connections to collect information.\"\n\nJenna nodded. \"I was impressed. They've found out a lot of information in a short period of time. Today is Thursday. It's only been four days.\"\n\nI thought about that. Had it only been four days? It sure felt like a lot longer. \"Well, Nana Jo gave us a deadline.\"\n\nJenna frowned. \"Deadline?\"\n\nI hid my smile. \"Yeah. She said we needed to have this figured out before next Saturday's game.\"\nChapter 11\n\nDorothy and Irma were too drunk to get home alone. Zaq volunteered to make sure Emma and Jillian arrived safely back on campus. I suspected Zaq had his own motives behind volunteering for taxi service. He and Emma were hitting it off well. I wasn't going to look a gift horse in the mouth, especially when it freed me up to get the girls back to the retirement village safely. Before they left, Jillian and Emma hugged me.\n\n\"Thank you for dinner and for including us in your investigation,\" Jillian said.\n\nI hugged them both. \"Thank you both. You were very helpful.\"\n\nJillian looked as though she was working up the courage to speak.\n\nEmma poked her in the ribs. \"Ask her.\"\n\n\"Ask me what?\"\n\n\"We wondered if it would be okay if... well, we were hoping maybe\u2014\" Jillian stammered.\n\n\"We wanted to know if we could continue to help investigate,\" Emma finished.\n\nI was surprised. \"Well, I'm not sure.\"\n\n\"Please,\" they begged.\n\n\"We really want to help,\" Emma said. \"After all, even though I didn't like her, she was still my roommate.\"\n\n\"And neither one of us believe Dawson killed her,\" Jillian said.\n\nI'd noticed a few looks between Dawson and Jillian while we were eating. Was something developing? However, I couldn't allow sentimental feelings to influence my judgment. \"Girls, I truly appreciate your willingness to help.\"\n\n\"Why do I sense a 'but' is coming?\" Emma asked.\n\n\"But this may seem like fun and games, but it's serious. Someone killed Melody. The killer may be on campus. If someone thinks you girls are asking questions and suspects you might figure this thing out, you might be in danger.\" I shuddered at the memory of what it felt like to be held at gunpoint during the summer. \"It's too risky, and I can't take the chance you might get hurt.\"\n\n\"But we'll be very careful. There has to be something we can do to help,\" Jillian said.\n\nI thought about it. \"Okay. Here's how you can help.\" I looked seriously at the girls and tried to help them see the gravity of the situation. I turned to Emma. \"As her roommate, maybe you could check with the university to see if anyone will be claiming her things. See if she has any family. Offer to pack up her belongings.\"\n\nEmma nodded eagerly. \"I can do that.\"\n\nI turned to Jillian. \"You can help by organizing a memorial.\"\n\nJillian frowned. \"A memorial?\"\n\nI nodded. \"Yes. Talk to the administration and put up flyers. Maybe a vigil of some sort.\"\n\n\"But I don't see . . .\"\n\n\"Once the memorial is scheduled, then we'll come and observe the people who show up. It's a long shot, but worth a try.\"\n\nJillian's face lit up. \"I get it. You're hoping the murderer will show up.\"\n\n\"Something like that. Actually, I'm hoping we'll run into people who knew her so we can talk to them.\" I looked at both girls. \"Can you do that?\"\n\nThey nodded eagerly.\n\n\"Good. But please be careful. This is very serious, and I don't want the killer to get suspicious.\"\n\nThey gave me their solemn promises to be careful and left with Zaq. I still felt nervous and wondered if I was doing the right thing by involving them. However, something in the back of my mind told me if I didn't give them a job to do, they might go out on their own, which could be more dangerous. This way I hoped to control and protect them, if needed. I shook off the doubt and began the process of corralling the girls. Irma took a bit of persuasion, but we finally convinced her.\n\nAs we were leaving, the manager returned. \"I hope you all enjoyed yourselves.\"\n\n\"Yes. Thank you. The food was very good,\" I said.\n\n\"I'm glad you liked it. I hope you'll come back again soon.\"\n\nEven though I was standing with Nana Jo and Jenna, he stared directly at me.\n\nHeat climbed up my neck. \"I'm sure we will. You're so close.\"\n\nFor some reason, I found it hard to make eye contact and focused on pretty much everything except his eyes.\n\n\"Hi. My name is Josephine Thomas.\" Nana Jo stuck out her hand.\n\n\"Frank Patterson.\" He shook it.\n\n\"Frank, we're glad to meet you. This is my granddaughter Jenna Rutherford, and my other granddaughter Samantha Washington.\"\n\nWe shook.\n\n\"Sam owns Market Street Mysteries down the street. You should swing by sometime and check it out.\" Nana Jo was clearly matchmaking, and the heat in my face intensified.\n\n\"Nana Jo! Everyone doesn't read mysteries,\" I said.\n\nNana Jo smiled. \"Do you like mysteries, Mr. Patterson?\"\n\nHe smiled. \"Actually, I do. I just haven't had much time to read, with opening the restaurant. I hope that changes soon.\"\n\n\"Perhaps your wife might enjoy mysteries?\" Jenna asked with a coy smile.\n\nSubtlety wasn't my family's strong suit. I tried to give Jenna a discreet pinch and prayed the ground would open up and swallow me whole.\n\nFrank Patterson laughed. \"I'm not married. \"\n\n\"Really? A nice-looking man like you? How on earth did you manage to escape?\" Nana Jo joked.\n\nFrank smiled. \"I guess I've been too busy to settle down.\"\n\n\"Sam here is single too. What a coincidence.\"\n\n\"Widow. I'm widowed. I'm sure Mr. Patterson doesn't want to hear about this. Oh my, look at the time.\" I grabbed my grandmother and sister by the arms and propelled them toward the door. \"Thanks for everything. We'll see you around.\"\n\nOutside on the street, I scowled at both of them. \"What exactly do you two think you're doing?\"\n\nThey smirked. \"Who, us?\"\n\n\"Yes. You.\"\n\nNana Jo smiled. \"We're just being neighborly.\"\n\nI huffed and stomped off toward the bookstore and ignored the laughter I heard from behind.\n\nAt the bookstore, the girls picked up a couple more books in the Mrs. Pollifax series, which they'd started reading over the summer. I was happy they enjoyed the series and decided to continue reading it. Although, Irma preferred more sex in her books and had also started reading J. D. Robb's In Death series. I preferred cozy mysteries, which tended not to have sex, violence, or bad language. Knowing J. D. Robb was the pseudonym for romance writer Nora Roberts, I thought there might be enough sex mixed in with the mystery to satisfy Irma. So far, she seemed pretty happy with my recommendation.\n\nAfter their purchases were made, I drove the girls back to the retirement village and headed back to MISU to talk to the football team.\n\nMISU wasn't a large university, so you wouldn't expect the athletic facility to be large either. However, you'd be wrong. The athletic and convocation center was a large facility which hosted not only athletic events for the university but was also a venue for concerts and other entertainment for the community. I remembered coming there as a child to see the circus. The main auditorium had more than five thousand seats. It wasn't as large as JAMU's stadium, which could hold almost ten thousand, but it was still bigger than Carnegie Hall.\n\nBehind the main facility was another smaller building, which was where the security guard directed me to go after I showed my note and driver's license. I parked and followed the signs to the media room.\n\nThe media room looked like a small movie theatre with a large screen that covered an entire wall, a projector, and about one hundred seats. Nearly every seat was taken when I entered. I was at the front of the room and felt like all eyes were directed at me. The lights were low, so I took a minute for my eyes to adjust and then hurried up the stairs to find a seat near the back. I was spotted just as I started to climb.\n\n\"Mrs. Washington?\" a voice boomed from the ceiling.\n\nI froze and looked around but couldn't tell where the voice originated. The lights suddenly came up, and at the top of the theatre, there was a glass booth, just like at the movie theatre. Peter Castleton was waving at me behind the glass and motioned to indicate he was coming down.\n\nI waited where I was until he and Coach Phillips came through a small door and descended the stairs. When they reached me, Peter Castleton did the introductions. \"Samantha Washington, you know Coach Phillips.\"\n\nWe nodded and shook hands.\n\nCoach Phillips was a little taller than me. I estimated his weight at one hundred fifty. I knew from the news that at thirty-seven, he was one of the youngest head football coaches in his division, but given MISU's success during his first season as head coach, he was getting a lot of attention. He always wore a baseball cap, which he tipped when we were introduced. I suspected the cap was an attempt to hide his receding hairline.\n\nThe players watched us in relative silence. Peter Castleton faced the group. \"Guys, this is Mrs. Washington. She's Dawson's friend and has been trying to help clear him. She'd like to ask you all some questions. I know Coach Phillips and I would really appreciate any help you can provide.\"\n\nCoach Phillips and Castleton left, along with some other older men I assumed were assistant coaches. I was facing a room full of large men and for a moment, I felt awkward. However, I took a deep breath and reminded myself of the many years, as a high school teacher, when I spoke to students who were a lot more dangerous and probably a lot less interested in what I had to say.\n\nBefore I could speak, a large guy who looked like a small tank raised his hand. He was at least four hundred pounds.\n\n\"Yes?\"\n\n\"How's Dawg?\"\n\nI scowled. \"Dog? You want to know about my dogs?\" Surely this guy wasn't asking about Snickers and Oreo.\n\nThe group laughed.\n\n\"No. Daaawg? You know, Dawson.\"\n\nReality dawned. \"Oh, I get it. Dawson is . . . hanging in there.\"\n\nHe beat his chest with his fist twice and then repeated the gesture. \"Tell him to stay strong. I don't believe he killed nobody.\" The small tank took his seat.\n\n\"I'm really glad to hear you say that. I don't believe he killed anyone either.\" The English teacher in me couldn't let the grammatical error slide, but it was really the least of my problems. \"That's why I'm here. I'm hoping one of you can tell me something that might help prove he didn't kill her.\"\n\nThey looked around at each other, but no one volunteered any information. Finally, my tank friend raised his hand again. \"How you gonna do that?\"\n\n\"Well, right now, I'm just looking for any information you can tell me about Melody or anyone that might have a reason to want to hurt her.\"\n\nA guy who was smaller than my tank friend, but bigger than a Volkswagen Beetle, said, \"Dat girl was a honey trap. Dawg should a stayed clear.\"\n\n\"Did everyone feel like she was a honey trap?\"\n\nThe guys mumbled amongst themselves. Most nodded.\n\nThe Beetle said, \"She should a stuck with B Ball.\"\n\nSeveral guys laughed\n\n\"What do you mean?\" I asked.\n\n\"She started with the basketball team but got clocked. Guess she decided basketball was too dangerous.\"\n\nThey laughed.\n\n\"Wait. Are you saying she used to date a basketball player?\"\n\n\"Yeah.\"\n\nOne guy who was sitting near the front said, \"Man, I forgot 'bout that.\" He laughed. \"She got whipped.\"\n\nI cringed but quickly wiped all judgment from my face. If I wanted them to be open and honest, I had to create an environment where they felt free to share openly. \"What happened?\"\n\nSeveral guys started talking at once, so I held up a hand. \"One at a time.\"\n\nTank said, \"Well, it must a been right after the winter basketball tournament. The men's team won the tournament and then the honeys started sniffing round. Next thing you know, Melody is hooked up with the star forward, Trammel Braxton.\"\n\nI pulled out a notepad and took notes as quickly as I could.\n\n\"So, I heard one day, Trammel and Melody were out at a party when Tray's girlfriend comes up and coldcocks Melody.\"\n\n\"Girlfriend? I thought Melody was his girlfriend?\" I said.\n\nTank laughed. \"Apparently, so did she. Unfortunately, Trammel forgot to tell his baby mama.\"\n\nThe players laughed.\n\n\"So, this other girl shows up and hits Melody?\"\n\nHe snorted. \"She beat Melody. She beat Trammel. She beat everybody who tried to stop her from beating Trammel and Melody.\"\n\nTank was an excellent storyteller, with great facial expressions and body movements to go along with his tale. He burst into laughter at one point. When he pulled himself together, he said, \"Man, I ain't never seen no woman fight like that. She whooped them like Muhammad Ali whooped Joe Frazier in the Thrilla in Manilla.\"\n\nInside I cringed at the violence. \"Anybody know this slugger's name?\"\n\nThey shook their heads. Tank responded, \"Nahw. You gonna need to get that from Trammel.\"\n\n\"Where might I find this Trammel Braxton?\" I asked.\n\n\"He was staying in those fancy apartments on the lake, but now his girl and baby moved up here, and they're staying in off-campus housing.\"\n\nOne young man who had been relatively quiet throughout most of the conversation finally spoke up. \"I can tell you where he lives.\" He was probably two hundred fifty pounds and about six feet. Compared to the other players, he looked like a shrimp.\n\nI asked a few more questions, but no one had any other suggestions of possible murderers. I wrote my name and e-mail address on a white board near the front of the room. The guys filed out and most sent a message to Dawson to hang tough, stay strong, or some other manly message as they left. Some simply pounded their chest like the tank had earlier in a Tarzan gesture.\n\nThe shrimp stayed until everyone left. He provided directions to Trammel's apartment.\n\nBy the time I got home, it was dark and I felt exhausted. However, I wanted to get some writing done, which would allow my subconscious to sift through the information I'd learned today.\n\nThompkins had mastered the art of silently entering and leaving rooms. He entered the servants' hall and watched Millie and Flossie unobserved for several minutes. Flossie could barely contain herself as she told Millie what she'd seen.\n\nThompkins had heard rumors for years about Lord Charles. He knew Lord Charles was a man given to excess. He ate in excess, drank in excess, and pursued women in excess. However, when he was seen by the maid leaving the room of the Duchess of Windsor in the early morning, this was excessive, even for him.\n\n\"Gawd, you don't say?\"\n\nMillie's shock and surprised expression were everything Flossie could have hoped.\n\n\"And that weren't all.\" Flossie looked around to make sure Mrs. McDuffie wasn't around. \"He was wearing pants, but his shirt was unbuttoned, and he was barefoot and carrying his shoes. He was skulking around like a thief in the night.\"\n\nMillie stared openmouthed. \"Oh my, poor Edward. I wonder if he knows.\"\n\nThompkins had heard enough. He moved forward and coughed.\n\nBoth girls were so engrossed in their conversation they failed to notice his approach.\n\n\"Knows what?\"\n\nHis question caused the girls to jump.\n\n\"Nothing, sir,\" Flossie said.\n\n\"Nonsense. If there's something going on, you need to tell me at once.\"\n\nThompkins had long ago learned the importance of a stern look. He applied it to good use. Flossie shared what she'd seen with the butler.\n\nHe frowned. \"You will not discuss what you've seen with anyone. What happens in this house stays in this house. Do you understand?\"\n\n\"Yes, sir,\" both girls said.\n\n\"Now, get back to work.\"\n\nThe girls turned and returned to their duties.\n\nThompkins went in search of Mrs. McDuffie. He didn't have to search long. As he passed the small room he used as an office, he heard the distinct voice of the housekeeper.\n\n\"Bloomin' 'ell that American upstart has a nerve.\"\n\nThompkins sighed, knocked briefly on the door, and then entered.\n\n\"Your ladyship. I'm sorry. I didn't realize.\" Thompkins started to back out.\n\n\"Good, Thompkins, I'm so glad you're here. Please come in,\" Lady Elizabeth said.\n\nThe butler entered and closed the door behind him and then stood near the wall.\n\n\"I was just sharing with Mrs. McDuffie the changes the Duchess of Windsor has requested.\"\n\n\"Changes, m'lady?\"\n\n\"Perhaps I should read the request.\" Lady Elizabeth sighed and picked up a sheet of stationary. \"'My dearest Elizabeth.'\"\n\nMrs. McDuffie snorted.\n\n\"'You've been a real doll for opening your home for my little gathering. As you can see from the guest list, there's a lot at stake, so I know you won't be offended if I make a few small changes.'\" Lady Elizabeth scanned the pages and then passed them to Mrs. McDuffie. \"Perhaps you had better read them yourself.\"\n\nMrs. McDuffie picked up the pages and read. \"She wants everyone's rooms changed.\" She scanned on. \"What bloody cheek. She wants that little tart of a maid moved to the blue room what looks out over the back garden.\" Mrs. McDuffie slammed the paper down. \"Well, I'll not do it. I'll not move 'Is Grace out of 'is large room for the likes of that little French strumpet.\"\n\nThompkins coughed gently. \"May I?\" He picked up the pages and read them. Then he gently placed the pages on the table and pulled out his hand kerchief and wiped his hands. He turned to Lady Elizabeth and coughed. \"If that is what your ladyship would like done, then, of course, we will honor your wishes, but . . .\"\n\n\"Yes, Thompkins?\"\n\n\"Well, in addition to the duke's room being slightly larger than the other rooms, it is also closest to the stairs. It is a large help to the staff not to have to carry His Grace's luggage down the hall.\" Thompkins's lips twitched. \"I'm not as young as I used to be.\"\n\nLady Elizabeth smiled. \"Thank you, Thompkins. I appreciate what you're trying to do.\" Lady Elizabeth sighed. \"I'm okay with permitting the maid to stay near the duchess, if that's what she wants. Do you think the other servants will mind?\"\n\nMrs. McDuffie snorted. \"No one will be missing that uppity little piece of baggage. You'd think she's the ruddy duchess, the way she carries on.\" Mrs. McDuffie stuck her nose in the air and looked down it.\n\nLady Elizabeth smiled at the housekeeper. She looked at the letter again, and her smile vanished. \"Well, that's good, but I agree James shouldn't have to sacrifice his room. He's practically family. I don't really care about the others.\"\n\n\"But it's highly unusual to have women in rooms immediately next to the men, especially when there are connecting doors,\" Thompkins said.\n\n\"I expect having men and women in separate wings is very old fashioned,\" Lady Elizabeth said. \"I suppose it's a sign of the changing times. We shall have to trust that everyone will behave themselves and there won't be any . . . inappropriate behavior.\"\n\nThompkins coughed discreetly and proceeded to tell Lady Elizabeth about the conversation he just had with the two housemaids.\n\nMrs. McDuffie stared openedmouthed. \"Cor blimey.\"\n\nLady Elizabeth looked from the butler to the housekeeper. \"Do you believe them?\"\n\nMrs. McDuffie's chest heaved and Lady Elizabeth held up a hand to stem her ire. \"I'm sure the girls are very honest and trustworthy. I just mean . . . It just seems so . . . I mean it isn't at all what I expected.\"\n\nMrs. McDuffie nodded. \"I see what you mean. Lord Chitterly isn't the type of man a woman would risk wrecking 'er marriage for?\"\n\n\"Something like that,\" Lady Elizabeth said.\n\nMrs. McDuffie nodded. \"Agreed. You mark my words that one isn't about to risk a king for the likes of tubby Lord Chitterly.\"\n\nLady Elizabeth suppressed a smile. \"I agree with your assessment.\" She picked up the letter and reread it. \"I won't move James. I'll come up with some excuse, but I see no good reason why we shouldn't do the other moves she requested. However, I want to talk to James before we do anything.\" Lady Elizabeth stood. \"So please hold off on the moves until I've had a chance to talk to him.\"\n\nLady Elizabeth found James in the library.\n\nHe read the note and scowled. \"Well, I don't see how it can do any harm, and I'll be happy to move if it\u2014\"\n\nShe held up her hand. \"Please don't feel you have to move. Actually, my housekeeper has flatly refused to move you out of the larger bedroom in favor of . . . 'that French tart.'\"\n\nJames laughed. \"I'll bet that got a reaction out of old Thompkins.\"\n\nLady Elizabeth smiled. \"Almost, but he maintained his composure. For the peace of my household staff, I shall have to insist you retain your bedroom.\"\n\n\"I want you to know how much I appreciate you agreeing to do this.\" James hesitated for a moment. \"I think you can see how important this is, not only for Great Britain but for the entire world.\"\n\n\"I can, but . . . well, I have to admit I'm slightly confused. I can't believe the king and Parliament are in favor of this . . . gathering.\"\n\nJames was silent for several minutes and then stood. \"If you're asking if this assembly is being done at the king's request, the answer is no.\"\n\n\"But then why?\"\n\n\"Britain's still a free country. We can't prevent people from attending house parties. The country's still recovering from the war and no one wants another one. If there is a peaceful way to prevent it, all the better.\"\n\n\"But you don't believe there is?\"\n\nJames thought for a moment and then shook his head. \"No. I don't. But, there are people like Mary Astor and the 'Cliveden set' who have been trying to negotiate peace.\"\n\n\"Peace, but at what cost?\"\n\n\"Exactly. Britain can't stand by helplessly with her hands behind her back while Hitler ravages Europe.\" James smoked. \"Plus, word is starting to trickle down about Hitler's true intentions.\"\n\nLady Elizabeth waited for him to continue, but he seemed far away. When he finally returned, he shook himself. \"But I hope you understand why this is so important. We needed to find out what was going on. That's why . . .\"\n\n\"That's why you needed them to come here.\"\n\n\"They never would have come to my estate. I'm a bit too close to the throne, I'm afraid. When we heard the duchess was planning this gathering, we needed it to be at a place where we could have access without seeming to condone it. We had heard Lady Emerald Cunard was looking for a country estate for the occasion, but thankfully, she was unable to secure it in time.\"\n\n\"I see. We have a large estate and we're cousins, so we're family, but not too close.\"\n\nJames nodded. \"Exactly.\"\n\n\"Plus, William isn't overly involved in politics, which means the duchess can take all of the . . . credit should a deal be made.\"\n\nLady Elizabeth folded the letter from the duchess. \"I had better instruct the staff to move forward with moving the guests' bedrooms. Everything should be done by the time you all return from shooting.\" She stood and headed to the door. One hand on the door, she turned back. \"James, please be careful. I have a bad feeling about this.\"\n\nEventually, the shooting party left. There were twelve in all. Geoffrey Fordham-Baker elected to remain in the library with a bottle of scotch. Lady Elizabeth and Lady Penelope also chose to forego the shooting. Thompkins brought the tea tray into the library. Geoffrey Fordham Baker snored in a chair in the cor ner while the ladies drank tea.\n\n\"I am rather surprised Daphne chose to go shooting,\" Penelope said.\n\nLady Elizabeth sipped her tea. \"Well, she is rather a good shot.\"\n\n\"I know. She was always a much better shot than me, but she never really seemed to enjoy the cold, the mud, or anything else about it.\"\n\nLady Elizabeth smiled. \"I know what you mean. She did buy a new outfit, and James hasn't seen how well she can shoot.\"\n\nPenelope smiled. \"True. You know, she isn't really as vain and frivolous as she seems.\"\n\n\"Well, she may be vain, but she has brains and I think she's found a reason to use them.\"\n\n\"What do you mean?\"\n\nLady Elizabeth set down her cup and took out her knitting. \"Well, I believe she is truly in love for the first time, and I think she realizes that a duke needs a wife who can be a credit to him in his career. James isn't just part of the idle rich. He's an important person with political aspirations, and I think she realizes his wife will need to do more than host tea parties and look pretty.\"\n\nPenelope stared at her aunt. \"I think you're right. I really do hope things work out between them.\"\n\nLady Elizabeth paused in her knitting. \"I just hope this weekend doesn't backfire.\"\n\n\"I've had an awful feeling about this whole thing. I know Victor is worried too. He thinks\u2014\"\n\nLady Penelope never got an opportunity to say what Victor thought because, at that minute, Thompkins abruptly entered the library.\n\n\"Excuse me, your ladyship, but there has been a terrible accident.\"\n\n\"What kind of accident?\" Lady Elizabeth asked.\n\n\"Someone's been shot.\"\n\nAll color left Penelope's face. \"Victor?\"\n\n\"No, m'lady. I believe it's the Duchess of Windsor.\"\n\nChapter 12\n\nI stayed up late writing and my brain wasn't functioning on all cylinders, even after two cups of coffee. At least that was my excuse. If I'd been well rested when my sister called and asked what I was doing today, I would have asked why before responding. I'd learned to be cautious with Jenna over the years. When the twins were younger, that question would have been followed up with a request to babysit while she went to the movies, a concert, or shopping. Rarely did the question what are you doing include an invitation to the interesting activity. Of course, now the twins were adults, her previous requests for babysitting were now replaced with requests to take her place with unpleasant activities she had committed to do with our mom; I have a really important court case I have to prep for and was wondering if you could take Mom to the license bureau\/dentist\/podiatrist \/etc. My sleep-deprived state was the reason I was now sitting at the police station. Part of her negotiations to get me to come included a large turtle caramel nut latte and a glazed donut from a gourmet coffee shop.\n\nI sipped my expensive coffee in the same room Nana Jo and I had sat in just five days earlier. \"I'm not sure why you need me here.\"\n\n\"I've never met this A-squared person. You have a relationship with him.\"\n\nI nearly choked. \"A relationship? He broke into my store and got into a fight with Dawson. I hardly call that a relationship.\"\n\n\"It's more interaction than I've had.\" She sipped her tea. \"Besides, he looks sleazy.\"\n\nI stared at her. \"You're at the police station. He's not likely to attack you.\"\n\n\"I'm not worried about him attacking me. I can take care of myself. I just don't want to be alone with him.\"\n\n\"If you're looking for a bodyguard, you should have invited Nana Jo. She's the one who's packing heat.\"\n\nWe both grinned.\n\nA police officer escorted A-squared into the conference room. \"You want me to stay?\"\n\nJenna shook her head. \"No, you can leave.\"\n\nHe nodded. \"You know the drill. Call if you need us.\"\n\nAlex Alexander, or A-squared, looked the same as he did on television. He was dirty. His clothes and hair were greasy. He smelled like a distillery, sweat, and vomit, which made the room seem even smaller.\n\n\"So, what can I do for you?\" He propped his feet on the table and leaned back in his chair with his hands behind his head. The maneuver released a musk which had been trapped under his armpits, and I fought the desire to gag.\n\n\"The first thing you can do is put your arms down. You stink,\" Jenna said.\n\nA-squared laughed and I thought he wasn't going to do it. I gave him my sternest schoolmarm stare, which did wonders with derelicts and delinquents. Jenna might not have been a teacher, but she had a fierce lawyer stare that caused criminals to cower in their boots. At least it made me want to cower. The combination seemed to work because he lowered his arms and chair and removed his feet.\n\n\"Thank you,\" Jenna said.\n\n\"Who're you?\"\n\n\"My name is Jenna Rutherford. I'm the lawyer representing your son. He asked me to talk to you.\"\n\n\"Yeah? What about?\"\n\n\"Well, he's concerned about you. He saw you on television and\u2014\"\n\n\"He saw that?\" He laughed. \"I was pretty good, huh? I'm a movie star. I saw him on television last night too. Not too flattering to his ole man, but . . .\" He shrugged. \"Everybody needs fifteen minutes of fame.\"\n\n\"Mr. Alexander, I\u2014\"\n\n\"Call me A-squared. Everybody does. Alex Alexander. Get it? A-squared.\"\n\nJenna was frustrated. \"Okay. A-squared, your son is concerned because you implied you know who murdered Melody.\"\n\n\"Yep.\"\n\n\"You need to tell the police. This is a dangerous game you're playing,\" she said.\n\nHe smiled. \"I'll be happy to tell the police everything.\" He paused and leaned forward. \"But there has to be somethin' in it for me.\"\n\n\"Mr. . . . ah, A-squared. If you know who killed Melody Hardwick, you need to tell someone. It's illegal to withhold information about a felony.\"\n\nHe shook himself. \"Oooh, I'm shakin' in my boots. What're they gonna do, arrest me?\"\n\n\"The police are not going to pay you to tell them who killed Melody Hardwick. What they're going to do is throw your butt in jail and leave you there.\"\n\n\"Won't be the first time.\" He stretched. \"Three hots and a cot.\"\n\nJenna's face could be very expressive. She stared at me with a \"can you believe this idiot\" look.\n\nI tried to get through to him. \"You have to see this is a dangerous business you're playing at. There's a killer out there who thinks you know who he or she is. You could be in danger.\"\n\n\"I can take care of myself. Been doin' it my whole life.\"\n\n\"The police believe Dawson killed her. If you don't tell them what you know, he could get convicted for murder.\" I doubted he possessed any parental emotions, especially considering how he'd beaten Dawson, but he was definitely concerned about money. \"And, if he's convicted for murder, there goes all hope of a professional football career.\"\n\nHe shook his head. \"Never happen. I'll never let my boy get hooked for a crime he didn't commit. That's why I went on TV. Besides, he's got you.\" He pointed toward Jenna. \"It's your job to get him off.\"\n\nJenna stared at him as though she'd like nothing better than to leap over that table and throttle him. Instead she said, \"Even the best lawyers can't guarantee an acquittal.\"\n\n\"He's not goin' to jail.\" He tapped his chest. \"I got a plan that'll set us up for the rest of our lives.\"\n\nJenna narrowed her eyes. \"I hope you don't mean what I think you mean.\"\n\nHe laughed. \"No idea what you're talkin' 'bout, but I think we're done with this little talk.\" He stood.\n\n\"Don't you want to know how your son is doing?\" I asked.\n\nA-squared turned to face me and grinned. \"How could he be anything but great with you two lookin' after him. Besides, you would a told me if he weren't okay.\"\n\nJenna buzzed for the police and they came promptly and removed A-squared.\n\nWe sat for several moments and stared at each other. Then we spontaneously burst into laughter.\n\n\"If I hadn't seen this with my own eyes, I wouldn't believe it,\" Jenna said.\n\n\"He's like a caricature of a real human being.\"\n\n\"That's why I wanted you here. I'm an attorney and I've dealt with some real lowlifes over the years. But this guy takes the cake.\" She paused. \"Let's get out of here.\"\n\nWe left the police station and went to a nearby coffee shop. Jenna needed a tea refill. We sat at a small table near the window and picked up our conversation where we'd left off.\n\n\"I've been thinking about why A-squared bothers me so much. I think this whole thing bothers me more because of Dawson.\"\n\n\"I know what you mean. It's different when you know the people personally.\"\n\nShe nodded. \"I can distance myself from my clients, but this is very different. Dawson is practically family. Heck, he is family.\"\n\nI allowed myself a moment of misty-eyed sentimentality and then my phone buzzed. It was a text message from Jillian informing me she had coordinated a memorial service for Melody for tomorrow.\n\nI told Jenna about the memorial service and immediately received another text. This one was from Emma. She'd received a text from a woman named Cassidy Logan claiming to be Melody's half sister.\n\nI read the text to Jenna.\n\n\"Half sister? Nana Jo didn't mention anything about a half sister.\"\n\n\"I know. I don't think Emma should meet with anyone alone. I'm going to tell her to hold off before responding.\" I typed the response and received an Okay.\n\n\"This is odd. I don't\u2014\"\n\nMy phone vibrated again.\n\n\"You're awfully popular,\" Jenna said.\n\nI looked at my phone and my heart skipped a beat. \"I totally forgot.\"\n\n\"Forgot what?\"\n\nI paused and took a deep breath before responding. \"I forgot I agreed to meet someone.\"\n\nI'm not sure if it was the blood I could feel rushing up my neck or the fact I was struggling to make eye contact that gave me away.\n\n\"Who did you agree to meet?\" Jenna asked with a smirk.\n\n\"Just a professor I met on campus yesterday.\" I hurried to add, \"He was one of Melody's professors, and I questioned him about her. He's British. So, I thought maybe he could help me with some of the details for my book. It's always good if I can add real details,\" I babbled.\n\n\"Hmm, a British professor. Interesting.\" Jenna grinned and sipped her tea.\n\n\"I barely know the guy. It's just research. That's all.\"\n\n\"'The lady doth protest too much, methinks.'\" Jenna quoted Shakespeare and took another sip of her tea.\n\nI stuck out my tongue.\n\nShe laughed but then leaned forward and stared. \"Where is he taking you for this . . . ah . . . research?\"\n\nI looked at my phone. \"I don't know.\"\n\n\"I hope you're going to do something about your hair.\"\n\n\"What's wrong with my hair?\"\n\n\"Do you want it straight or sugarcoated?\"\n\nI stared at my sister. \"Sugarcoated.\"\n\n\"Well, it's dull and lifeless. You've got split ends and the style does nothing for your face. Your eyebrows need to be arched. Most people have two. Yours are so thick and bushy you look like Oscar the Grouch. Plus, your makeup needs to come into the twenty-first century.\"\n\n\"If that's the sugarcoated version, I'd hate to hear it straight.\"\n\n\"The truth will set you free.\"\n\n\"The King James Version says 'the truth will MAKE you free,' not set.\"\n\n\"Whatever.\"\n\nMy feelings were hurt by Jenna's comments, but my sister was honest if nothing else\u2014brutally honest. I wanted to lash out and say something critical about the way she looked, but I couldn't. Her hair and makeup were always flawless. Jenna never wore much makeup, but she always looked polished and professional. With no way to relieve the sting of her words, I resorted to sulking. \"It's not a date.\"\n\n\"It doesn't matter whether it's a date or not. You don't have to do your hair and makeup to impress a man. You should do it for yourself. You used to dress better and take care of yourself, but ever since Leon died, you've just let yourself go.\"\n\n\"You've been talking to Mom.\"\n\n\"Doesn't mean it isn't true.\" She smiled kindly. \"Look, your birthday is coming up and I was going to suggest we do a spa day. Why don't we do things a little early and see if we can get a little pampering. My treat.\"\n\n\"Geez. I must really look bad if you want to send me on a spa day.\"\n\n\"I'm pleading the fifth.\"\n\nJenna picked up her phone and made a few calls. She got us both in for manicures and pedicures, plus a hair and makeup session for me and a massage for herself.\n\n* * *\n\nNorth Harbor Spa was a beautifully relaxing facility located atop the North Harbor Inn. North Harbor only had fifteen thousand people. The location on the shores of Lake Michigan and the addition of the new senior professional golf course had made the town a popular summer vacation spot, despite its economically depressed condition. Small boutique hotels dotted the coastline. North Harbor Inn was a newer building and this was my first time going inside.\n\nOn a rare instance when I splurged for a manicure, I went to a small beauty college. The technicians weren't licensed, but prices were low. Results weren't optimal, but I didn't expect much for five dollars. According to Jenna, the experience of being pampered was worth the extra money. Here, soft music played in the background. No televisions were tuned to soap operas or Jerry Springer, and the pedicure chairs weren't lined up against the wall like suspects in a police lineup.\n\nJenna exaggerated when she accused me of having the Oscar the Grouch unibrow, but it had been a long time since I'd last arched them. My brows were thick and in need of pruning.\n\nJenna introduced me to her hairdresser and gave her carte blanche to make me beautiful.\n\nMarika laughed at the look of terror in my eyes. \"Don't worry. I take good care of you. I won't do anything to you I wouldn't do to myself.\"\n\nI was even more terrified since her hair was electric blue and cut into an asymmetrical bob and shaved on one side. I decided to trust her. Jenna's hair looked nice. Marika permed, colored, conditioned, cut the split ends, and styled my hair. The dye hid my gray and she used caramel highlights. Between the brow waxing and hair change, my eyes looked huge. I barely recognized myself in the mirror and couldn't help smiling. My hair had bounce and shine and was so soft I kept touching it.\n\nBy the time the makeup artist came by, I was excited. She asked me a few questions about my daily makeup routine and then went to work. She explained everything as she went along and by the time she'd finished, I looked ten years younger. Neutral foundation and lip gloss with a smoky eye shadow that made my brown eyes pop.\n\nI looked like me, only better. Best of all, I felt beautiful. Both Jenna and I were pleased with the final results.\n\n\"You look awesome.\"\n\nI was in danger of bursting into tears, so I hugged my sister and whispered, \"Thank you.\"\n\nShe smiled. \"Happy birthday. Now, you're ready for your not-a-date research dinner.\"\n\nI felt guilty for taking a half day on pampering, but I kept looking at myself in the mirror and smiling as I drove to MISU.\n\nEmma and I were scheduled to meet at the student union. I found her sitting at a table by the window.\n\n\"Wow. You look amazing. I almost didn't recognize you.\"\n\nI laughed. \"Thank you.\" I sat down. \"I think.\"\n\n\"I'm sorry. I didn't mean to imply that you\u2014\"\n\nI held up my hand. \"No need to apologize. Now, tell me about this half sister, Cassidy Logan.\"\n\n\"I'm afraid I don't know much.\" Emma pulled out her phone. \"I did what you suggested and reached out to the university about her belongings. They gave her my number and she sent me a text.\" Emma pulled up the text message on her phone.\n\nI read the string. \"Doesn't sound like she and Melody were very close.\" I scrolled back and reread it, trying not to read tone into the sparse words.\n\n\"Nope. Sure doesn't. I mean if my sister was murdered, I'd be a basket case. I don't think I'd be asking for an inventory of her belongings so I can determine if it's worth my while to come pick them up.\"\n\n\"Well, she is a half sister,\" I said weakly.\n\nEmma shrugged. \"So.\"\n\n\"She said they weren't close.\"\n\n\"What do you want me to do?\" she asked.\n\nI pondered the question. \"She's in Chicago. That's only an hour and a half away. Tell her you have a friend who will bring the items to her if she'll send you the address.\"\n\nEmma picked up her phone and sent the message. We didn't have to wait long for the reply.\n\n\"She sent her address.\" Emma forwarded the message to me.\n\nWe went back to her dorm room and packed away Melody's few belongings. It seemed sad that all of her worldly goods fit into an old backpack Emma said she didn't need any longer. I thought about Melody and wondered where she kept her other belongings. Each time I'd seen her, she had on a different outfit and a lot of makeup. I put the backpack in my car. Since I was on campus, I decided to swing by the off-campus housing and pay a visit to Trammel Braxton.\n\nConstruction at MISU was never ending. Buildings were erected, renovated, or remodeled constantly. A few years ago, the off-campus housing for married couples and graduate students looked like a smaller version of Chicago's Cabrini-Green housing projects during the 1970s. The last of those buildings were demolished in 2011 and the MISU buildings met the same fate a couple of years ago. The new buildings looked like East Coast brownstone row houses with brick fronts and porches.\n\nI found the one I wanted, parked, and tried to come up with a plausible cover story. I finally came up with something and got out of the car. I was so focused on the story running through in my head I was oblivious to everything else. As I got to the sidewalk, I was blindsided by a child on a Big Wheel being chased by a golden retriever and a very pregnant woman.\n\nI avoided falling over by holding onto the Big Wheel. I held on until the child's mother caught up.\n\nShe was breathing heavily from hurrying down the street. \"I'm so sorry. I hope he didn't hurt you. He just broke away from me and in my condition, I couldn't catch him.\" She looked extremely young, not more than eighteen with a dark olive complexion and dark hair and eyes.\n\n\"No harm done.\" I smiled. \"He's adorable.\"\n\nThe boy was extremely cute. He had dark skin and jet-black curly hair, dark eyes, and chubby cheeks.\n\nShe smiled. \"Thank you.\" She turned to the boy. \"Now it's nap time.\"\n\nHe let out a howl and would have taken off again if I hadn't bent down and scooped him up.\n\n\"You're going to be a good boy for your mama,\" I said in a soothing voice, which might have worked if the mother hadn't reached to take him.\n\nHe immediately started to kick and squirm.\n\nI gasped, afraid he'd kick her stomach, but she must have been accustomed to this behavior because she turned to avoid a direct frontal kick just in time. She grabbed him by the waist and pulled, but my new companion must have also known what was coming because he wrapped both hands tightly around my neck and refused to let go, regardless of how much the mother pulled.\n\nI winced as he grasped my hair in addition to my neck. \"Maybe it would be better if I carried him in for you?\" I suggested. \"You seem to have your hands full, and I think two heads are better than one in this situation.\"\n\nThe little mother looked at me tentatively. However, I must have passed her scrutiny of not being a crazed child-napping serial killer because she nodded and gave a polite smile.\n\n\"If you're sure you don't mind?\" She reached down and picked up the golden's leash and the Big Wheel while she balanced a large diaper bag on her other shoulder.\n\n\"I don't mind at all.\" I looked around. \"Just lead the way.\" I stepped aside to allow her to get in front. I hid my surprise when she went to the wrought iron fence in front of me and walked up the stairs. This was the home of Trammel Braxton, just where I wanted to go.\n\nShe unlocked the door and stepped aside for me to enter. The living room was sparsely furnished with a cheap sofa in front of the window, two foldaway chairs, and a massive flat panel television with cords and controllers hanging off like tentacles on an octopus.\n\nShe put down the Big Wheel and released the dog from his leash and then tried again to get her son away from me. However, he still wasn't ready to let go and let out a bloodcurdling scream, which would have the neighbors believing a murder was occurring.\n\nAn embarrassed flush rose up her neck and she had a determined set to her eyes and chin which wouldn't bode well for the little guy if he didn't release me soon.\n\n\"Perhaps you'll let me put him to bed. I could read him a story.\" I tried to pull away far enough to look into his eyes. \"Would you like me to read you a story?\"\n\n\"Yes.\" He sniffed.\n\nShe nodded and led the way upstairs to a small nursery.\n\nThe nursery was decorated in a superhero theme with Superman soaring overhead through clouds which had been painted onto the ceiling. The room had a crib, dresser, rocker, and large rug. The floor was littered with toys. One wall had a small bookshelf and there were several books I remembered reading to my nephews when they were small. I turned sideways so my new appendage could see the books. \"Which book do you want me to read?\"\n\nHe looked up and pointed to a book which looked to have seen a lot of wear if the frayed corners and crayon marks were any indication.\n\n\"Panda Bear's Paint Box. I remember reading this to my nephews when they were little boys, just like you.\" I picked up the book and walked over to the rocker. I sat down. My companion turned so he could sit and see the pages of the book. He stuck his thumb in his mouth and leaned back against my chest as I read.\n\nIf the mother had hesitations about me, I think they evaporated as she watched me rock and read to her son. He was asleep before I finished the last page and I quietly got up and walked to the crib. I gently placed him in the crib and pulled the covers up over him.\n\n\"He's out like a light,\" I whispered.\n\nShe looked at her son and smiled. \"He'll sleep like the dead now,\" she said in her normal tone. She turned to me. \"Thank you.\"\n\n\"It was my pleasure.\"\n\nWe left the room and went downstairs.\n\n\"He's normally so good, but lately, he has fits and tantrums.\" She rubbed her belly. \"Especially now he knows I'm slower and can't react as fast as before.\" She looked at me. \"I do appreciate your help. I don't even know your name.\"\n\n\"My name's Samantha Washington, but, please, call me Sam.\"\n\nShe smiled. \"Thank you, Sam. I'm Mariana Braxton.\"\n\nShe walked to the door to let me out. \"I hope we didn't keep you.\"\n\n\"Actually, I was coming to see you and your husband, Trammel, that is.\"\n\nShe stopped and stared. \"Do we know you?\"\n\n\"No. I was hoping you could help me.\"\n\nShe looked skeptical and folded her arms across her chest. \"Well, Trammel isn't here at the moment. Maybe you'd prefer to come back when he is.\" Her voice and body language indicated all barriers were now up and in place, and if I didn't do something quick, I'd be outside in less than two seconds.\n\n\"Mrs. Braxton, could I sit down for one moment? I just want to ask you a few questions and then I'll leave. I promise.\"\n\nWhether it was the sincerity in my eyes or memories of me holding her son, it worked. She nodded and indicated I could sit on the sofa. She pulled over a folding chair and sat.\n\n\"Would you prefer\u2014\"\n\nShe waved away my protest. \"It's a lot easier for me to get up and down on this chair than it is on that sofa.\" She patted her stomach. \"Now, what questions do you want to ask me?\"\n\nI settled back down. \"Well, I'm a friend of Dawson Alexander.\"\n\nShe had a vacant look on her face that indicated she had no idea who Dawson Alexander was.\n\n\"He's the football player accused of killing Melody Hardwick.\"\n\nBased on the way she pursed her lips and rolled her eyes, it was clear she recognized Melody's name. \"What do you want from me, a medal?\"\n\n\"He didn't do it. Dawson didn't kill her. I was hoping you could help me figure out who might want her dead.\"\n\n\"Other than me, you mean?\" She rose from her chair. \"If you think I killed that gold digger, you've got another think coming.\"\n\n\"Please, Mrs. Braxton. I'm not accusing you of killing her.\" I pointed to her stomach. \"It's pretty clear that would be impossible in your condition.\"\n\nThat settled her down and she returned to her seat. I wasn't so naive as to believe a pregnant woman couldn't have killed someone, but Melody was in pretty good shape and would have put up a struggle.\n\n\"Well, I don't know how you think I can help.\"\n\n\"Just tell me what you know about Melody. I'm trying to understand her character and so few people know anything about her. No one was really close to her. I was hoping you . . .\" I looked down. \"Or your husband might be able to tell me something that might help prevent an innocent man from paying for a crime he didn't commit.\"\n\nShe settled back. \"I only saw her once. If you're here, someone told you what happened.\"\n\nI nodded. \"Yes, but I'd like to hear your side of the story, if you don't mind sharing.\"\n\nShe took a deep breath. \"Trammel and I dated in high school. That's when I got pregnant with our son. Tray was a star basketball player, the best in the state. He got offers all over the country, but he came to MISU because it was close by. I was two years behind him in school and my parents wanted me to finish high school. He came home to see me almost every weekend when he didn't have a game, especially after the baby was born. He loved little Tray.\" She smiled. \"We were going to get married as soon as he graduated. But his sophomore year he stopped calling and didn't visit as much. I got suspicious. He said everything was fine. He was just tired or studying. But he was never that into books. All he ever wanted was to play ball. So, I came to see for myself.\" She paused. \"At first his roommate didn't want to tell me where he was, but he finally did. They were together and she was all over him.\" Her face hardened. \"I snapped. I tried to beat the crap out of her. How dare she think she was gonna come and take my man.\" She took several seconds to recapture her composure. \"I'd invested years into that man. He wasn't about to leave me and our son for some no-account gold digger. Oh, no.\" Her chest heaved and her eyes flashed. \"So, yeah, I tried to kill her. It took Trammel and two other men to pull me off her.\"\n\n\"What happened next?\"\n\n\"I told Trammel if he thought he was about to leave me and our son for her, then he was next. I was so angry.\"\n\n\"I can understand that.\"\n\n\"Trammel apologized. He had a good thing and if he didn't want to lose me, he better straighten up and fly right.\" She smiled.\n\n\"That's when you moved here?\"\n\n\"He said he didn't want anyone but me, and we got married a couple weeks later. I figured it would be better if I was here on campus in case anyone else tried to get their claws on my man.\"\n\n\"And you never saw Melody again?\"\n\nShe shook her head. \"Probably a good thing too.\"\n\nI hesitated but decided I needed the truth. \"Do you know if Trammel saw her again?\"\n\nShe shook her head. \"I don't think so. I told him if I heard he so much as looked at her again, I'd beat both of them next time.\" She giggled.\n\n\"Do you know of anyone else that might have wanted to kill her?\"\n\nShe thought for a minute. \"Not unless she tried to take somebody else's man.\"\n\nI thanked Mrs. Braxton for her help and left. It was getting late and I still needed to get dressed for my non-date.\n\nWhen I returned to the bookstore, Chris and Zaq were closing up for the day.\n\nNana Jo stopped sweeping and stared. \"Sam. You look amazing. I love the hair.\"\n\nChris and Zaq seconded the compliments, which gave a huge boost to my ego.\n\nThere was only one customer in the store. When he turned around, I saw it was Frank Patterson, the owner of the restaurant down the street.\n\n\"Sam, you remember Frank.\" Nana Jo smiled as she guided Mr. Patterson toward me. \"Doesn't Sam look beautiful?\"\n\nI tried to kill my grandmother with my eyes, but I hadn't yet mastered that trick. So, I scowled at her and then plastered a fake smile on my face.\n\nHe had a hand full of books and juggled them to shake hands. \"Yes. She looks lovely.\"\n\n\"Are you a mystery lover, Mr. Patterson?\"\n\n\"Please, call me Frank,\" he said. \"Well, I don't know if I'd say I like mysteries as much as thrillers. Are those the same?\"\n\n\"Not always, although they can be. Every mystery doesn't have to be a thriller, but it can be.\"\n\n\"I tend to go for spy stories. I like Ian Fleming and John Le Carr\u00e9.\"\n\nI indicated the pile of books in his arms. \"I see Nana Jo has introduced you to a few new authors.\" I glanced at the titles of the books in his arms. \"Have you read Marc Cameron? I think you might like him.\"\n\nWe spent a little time talking about thrillers, and he purchased his books. Frank Patterson was a nice man. If I were interested in dating, I would consider him. However, I wasn't dating, so the point was moot. When he was gone and the store locked up, I hurried upstairs and quickly went through practically everything in my closet trying to find something to wear to my non-date. The restaurant Professor Quin had selected wasn't super fancy but it was nicer than most of my clothes. Jeans and a nice shirt would be appropriate or a dress. But, a dress might imply I was looking at this as more than just a research non-date opportunity. I didn't want to look too eager, nor did I want to look slovenly. My new hair and makeup made my clothes look shabby. However, I did have one nice pair of jeans, a gift from Jenna two birthdays ago. They fit beautifully, but I rarely wore them. I could tell they were expensive and I'd saved them for special occasions. Tonight qualified.\n\nI also had a pair of nice wedge shoes I loved because they were comfortable with a peep toe, which would show just enough of my pedicure and also raised me high enough that my jeans didn't drag the ground. My bottom half was set, but my shirts were faded out or stained. I was just about to give up and change to a dress when I noticed the pink bag at the back of my closet. I dug it out and found the cashmere sweater my mother shamed me into buying. I pulled the white cashmere sweater on and looked at myself in the mirror. The sweater looked fantastic with the dark-washed jeans and wedges. It was soft and felt like silk against my skin. It hugged my curves without being too tight and landed at just the right place on my hips.\n\nI went into the kitchen. Nana Jo and Dawson sat at the table. When Dawson came to live with me, he had been on academic probation. As a former mathematics teacher, Nana Jo tutored him in math while I tutored him in English.\n\nNana Jo whistled like a New Yorker hailing a cab. \"Wow, you look great. I don't think I've ever seen you wear that sweater before. Is it new?\"\n\n\"Yeah. Mom talked me into getting it last Sunday.\" Was it only a week ago that I'd gone shopping with my mom? A lot had happened in a short period of time.\n\n\"Well, she was right. That sweater looks divine. Is that cashmere?\" She rubbed it. \"Yep. You can always tell the real stuff.\"\n\n\"You look amazing Mrs. W,\" Dawson said.\n\nThe compliments boosted my courage. I left with a smile and headed to my non-date research dinner.\n\n* * *\n\nLake Michigan Grill was a South Harbor restaurant located near the beach. It was only about a mile from my building. We'd agreed to meet at the restaurant and he was there when I arrived.\n\n\"You look lovely,\" he said. \"I like your hair.\"\n\n\"Thank you. My sister thought I needed a makeover.\"\n\n\"Well, I like both versions.\" He smiled.\n\nThe hostess showed us to our table.\n\nHe ordered a glass of wine when the waitress came by. I wasn't much of a wine drinker, but Lake Michigan Grill served one of the local wines made just up the road. The climate and soil on the southwestern shore of Lake Michigan made it an ideal location for winemaking. One of our local wines was even served at the White House. I ordered the Classic Demi-sec.\n\n\"Professor Quin, I really appreciate your agreeing to help me with my research.\"\n\n\"Please, call me Harley.\" He smiled.\n\nI was thankful the lighting was dim because my face became heated. Thirty-something-year-old women shouldn't blush when they went out on non-dates.\n\n\"Thank you, Harley. Please call me Sam.\" I pulled a small notebook out of my purse. \"Now, I've been thinking about what questions to ask you.\"\n\n\"Certainly. Perhaps you should start by telling me a little about the book you're writing.\"\n\nSo I did. He asked a lot of questions and I found myself doing the majority of the talking. The waitress came and took our orders and he asked more questions. Normally, I didn't talk about my writing. It was still very private, but Harley asked the right types of questions. He knew a lot about mysteries and was an Agatha Christie fan. I could talk about mysteries for hours and found that I had. Several hours later, I looked over and noticed we were the last people in the restaurant and the staff was waiting patiently for us to leave.\n\n\"What time is it?\"\n\nHe looked at his watch. \"Eleven thirty.\"\n\n\"They close at eleven. I didn't realize how quickly the time passed.\"\n\nHe reached for the check, but I was quicker. \"This is on me. After all, you're helping me with my research. This is the least I can do.\"\n\nHe smiled. \"I'm glad to help.\"\n\nI pulled out my credit card and our waitress immediately came to take care of the check. I'm sure she wasn't allowed to leave until we left and was probably anxious to see the back of us. She returned promptly and I left a generous tip to compensate for her time.\n\nOutside, Harley walked me to my car.\n\n\"I very much enjoyed talking to you. I'm not sure I answered all of your questions, though. Perhaps we should try again,\" he said with a sly smile.\n\nI laughed. \"Perhaps we should.\"\n\nHe took my hand, bowed low, and kissed it.\n\nMy knees started to buckle the tiniest bit as I got into the car. I was grateful he didn't try to kiss me. I'd enjoyed the evening, but kissing a man other than Leon was something I wasn't quite ready for yet. Although, as I drove home, I thought maybe there might be a time in the near future when I might be ready.\n\nOnce I got home, I was still very excited. Maybe there was room for a James Bond who looked like Sean Connery in 1938 England. I decided a little writing would help me settle down before I went to sleep.\n\nLady Elizabeth and Penelope stared at the butler in shock.\n\nJames hurried past the butler into the library and closed the doors behind him. \"Quickly, there's very little time. I've already called the police. Has anyone else come by here?\"\n\nBoth ladies shook their heads.\n\nHe turned to Thompkins. \"Make sure no one leaves this house.\"\n\nThe butler nodded, turned, and left.\n\n\"Oh, James. What are we going to do? Someone has to break the news to David. He's going to be devastated. Is she going to live?\"\n\nJames looked surprised. \"David?\"\n\n\"Yes, her husband, Edward the VIII,\" Lady Elizabeth said with a slight frown. \"I thought you knew all the family call him David.\"\n\n\"I know, but Wallis isn't the one who's been shot. It was her maid, Rebecca.\"\n\nLady Elizabeth and Penelope stared at each other. \"Thompkins just told us it was the duchess who'd been shot.\" She sighed. \"I know I shouldn't be relieved, but I must confess I am. I certainly didn't fancy having to explain to the king his sister-in-law was shot at my home.\"\n\nPenelope stared. \"I wonder how Thompkins got things so wrong.\"\n\n\"Easy to do. The maid was wearing the duchess's clothes. Apparently she didn't have appropriate clothing for shooting, so the duchess gave her some of her things. They were dressed almost exactly alike. It wasn't until we got a close look at the body that we discovered the mistake.\"\n\nLady Elizabeth gasped. \"Will the maid be alright?\"\n\nJames looked grim. \"I'm afraid not. She's dead.\"\n\nThe local constables arrived to secure the scene and wait for Scotland Yard. The shooting party returned in groups. Daphne and Lord Charles and Lady Abigail Chitterly came back together. Lord Charles seemed to be especially shaken up and required a stiff drink immediately upon arrival. Lady Abigail was remarkably well composed and sat quietly in the library.\n\nDaphne looked a little pale, but she walked straight to the sofa where Penelope and Lady Elizabeth were seated. When she reached her aunt, she whispered, \"The duchess was so distraught she fainted and had to be carried in by Count Rudolph.\"\n\nLady Elizabeth rose immediately and left the room.\n\nVictor, Virginia Hall, and the Polish ambassador, J\u00f3zef Lipski, were the next to arrive. They stood quietly near the window and whispered.\n\nLord William was the last to arrive as he had stayed to talk to the gamekeeper and the police, but quickly left.\n\nThe gathering in the library was grim. Lady Penelope looked to Victor. He caught her eye and smiled.\n\nDaphne took her aunt's place on the sofa next to her sister.\n\nPenelope took her sister's hand. \"Are you okay?\"\n\nDaphne nodded. \"I'm fine. I mean, it's not as though I knew her, but it's awful that it happened here, again.\"\n\nPenelope nodded. She shuddered at the recollection of stumbling across Charles Parker's body, six months ago. Parker had been brutally stabbed. \"Was it horrible?\"\n\n\"No. It wasn't like . . . like before. She was shot in the back.\"\n\nThe sisters sat quietly for several seconds before Penelope asked the unspoken question on everyone's minds. \"It was an accident, wasn't it?\"\n\nDaphne paused before responding. \"I don't know. But, I wonder . . .\"\n\n\"What?\"\n\n\"I wonder if it was an accident, what kind of accident it was.\"\n\nPenelope scowled. \"I don't understand.\"\n\n\"Was the accident that a woman was shot? Or was the accident that the wrong woman was shot?\"\n\n\"I see. James did tell us she was wearing Wallis's clothes. But surely it was merely an honest mistake. Accidents happen during shooting parties. It could have been a bad shot.\"\n\nDaphne stared at her sister. \"True. Accidents do happen, but if it was an accident, why is no one stepping forward? No one would blame them.\"\n\n\"Do you mean no one knows who shot her?\"\n\nDaphne nodded.\n\n\"But surely loaders or the beaters saw who . . .\"\n\nDaphne shook her head. \"No one claims to have seen anything.\"\n\nPenelope stared aghast. \"That's impossible. You were all paired off, right? Surely someone was with her.\"\n\n\"She was paired off with Wallis, Count Rudolph, and Brasseur. They claim she said she was cold and wet and was heading back to the house.\"\n\n\"But where was she found?\"\n\n\"In the marsh.\"\n\nPenelope stared openmouthed. \"But that's the total opposite direction from the house. It's not surprising she was shot if she was in the marsh. That's the direction everyone would be shooting. Why, that's suicide.\"\n\nDaphne nodded. \"They said she started walking toward the house. No one knows why she changed direction or why she went toward the shooting. It's awful.\"\n\nJames entered the library and stood for a few moments. He looked around, made eye contact with Daphne, and then strode purposely toward her. \"Where's your aunt?\"\n\nPenelope rose to leave but James motioned for her to stay.\n\n\"She's seeing to the duchess. She fainted.\"\n\nHe looked around impatiently. \"Look, I've got to run up to London.\"\n\n\"What about the police?\" Daphne asked.\n\n\"If I wait to talk to the police, it'll be hours before I can go. I'm not sure . . .\"\n\nDaphne stared at him for several seconds. \"If you go down the back stairs, you can get out by the servants' entrance.\"\n\n\"Cut through the back to our house, and Victor's car is in the garage. He keeps the keys in the visor,\" Penelope added. \"I know he won't mind.\"\n\nJames squeezed Penelope's hand in thanks and absentmindedly kissed Daphne's forehead before hurrying out.\n\nMoments later, Lord William entered the library with a constable.\n\nThe constable stood at the door to the library. \"We appreciate everyone's patience. However, I'm going to need to ask you all to bear with us a little longer. Someone from the Yard will be here shortly to take your statements. Thank you.\"\n\nA low murmur started as guests whispered to each other.\n\nLord William walked over to his nieces. \"Bloody bureaucratic falderal. Chap won't tell me what's happening in my own house.\"\n\nDaphne and Penelope smiled at their uncle.\n\nDaphne stood. \"Would you like a glass of port? I think you deserve it.\"\n\nLord William smiled fondly. \"I could use a bit of a drink. Thank you, dear.\"\n\nHe looked around. His favorite chair was occupied by Fordham-Baker, who was nodding off with a nearly empty bottle of his best scotch on the table nearby.\n\nDaphne returned and handed the glass to her uncle.\n\n\"Thank you.\" He gulped down the liquid.\n\nThompkins entered the room with a tea cart. He looked around briefly and then rolled the cart to Daphne and Penelope. \"Her ladyship thought everyone might like some tea and sandwiches.\"\n\nDaphne and Penelope poured tea and offered sandwiches. Lord Charles looked as though he couldn't stomach the idea of eating but accepted a glass of port in lieu of tea. Lady Abigail, however, said she was famished and ate enough sandwiches and scones for both she and her husband.\n\nThe other guests declined food but accepted the tea graciously.\n\nThe atmosphere was strained and Penelope was just about to see if her aunt needed help with the duchess when the door finally opened. The constable returned with a tall man, who was lean and gangly with thick curly hair.\n\n\"Good afternoon. My name is Detective Inspector Covington from Scotland Yard.\" He looked around the room. He hurried over to Lord William, smiled big, and shook his hand. \"Lord William, I came as soon as I heard.\"\n\n\"It's good to see you again, although, well, I don't mean with another murder, but . . . oh, dash it all, man. This is bad timing.\"\n\nDetective Inspector Covington nodded. \"Quite. Quite. Where's the Duchess of Windsor now?\"\n\n\"Upstairs in her room. Fainted when she heard about her maid.\"\n\n\"Can you take me to her?\" He followed Lord William upstairs.\n\nLady Elizabeth came out of the bedroom and saw Lord William and Detective Inspector Covington.\n\n\"Detective Inspector Covington, how nice to see you again.\"\n\n\"I wish it were under better circumstances.\" The detective suppressed a smile and looked around cautiously. \"How is she?\"\n\n\"She's had a shock. Dr. Haygood just arrived. I believe he's given her a sedative. She'll rest and I'm sure she'll be fine.\"\n\nDetective Inspector Covington's shoulders relaxed and he released a sigh. \"Well, that's certainly good news.\" He looked around again. \"We don't have much time. Someone from the Home Office is coming down to handle this case personally. Is there any chance the maid was the intended victim?\"\n\nLord William pondered the question and shrugged. \"I don't know.\"\n\nAnother constable hurried down the hall to the small group. \"Sir, take a look at this.\"\n\nDetective Inspector Covington unfolded the sheet of stationery and read. When he finished, his eyes were large and his hands had a slight tremor.\n\nBefore the detective could say anything, Thompkins walked in, in a quiet, yet determined manner. He stopped at the group and said to Lady Elizabeth, \"Telephone, your ladyship. It's the king.\"\n\nDetective Inspector Covington's eyes looked as though they would pop out of his head. \"Sweet mother of God.\"\n\nChapter 13\n\nIt is a truth universally acknowledged that anyone who owns a pet and thinks they will get to sleep later on the weekend than on weekdays must be delusional, especially if said pet is a twelve-year-old pampered poodle. For some reason, ever since daylight savings kicked in, Oreo now woke at three every morning to go outside. Despite the fact I'd only gone to bed about an hour earlier, I got up and opened his crate and made the trek downstairs. Snickers's bladder wasn't on the same schedule. She slept through this excursion.\n\nHe found his favorite spot near the fence line and did his business and then trotted back up the stairs and was snoring by the time I got back in bed.\n\nSleep evaded me. I tossed and turned and turned and tossed and finally gave up trying. I wasn't in a mood for more writing, so I pulled out a notebook and tried to make sense of all of the information we'd collected about Melody Hardwick.\n\nI rolled the conversation with Mariana over and over in my mind. She was definitely a passionate individual. She'd gotten in a physical fistfight with Melody, so I knew she was capable of violence. However, I just wasn't sure her violent streak extended to murder. Maybe if she felt her family was in jeopardy, she might murder, but I doubted she would plan a cold-blooded murder. Of course, I could be wrong. She might have been lying. Maybe the relationship between Trammel and Melody wasn't over. However, my gut told me she was telling the truth, but I wasn't willing to bet Dawson's future on it.\n\nVirgil Russell was a slimy lowlife. He was, according to Dawson, in an intimate relationship with the daughter of the man he'd murdered. That was weird. Although, based on what Ruby Mae said, he had been guilty of sexual extortion. His relationship with Melody could have been extortion rather than consensual.\n\nMelody's half sister was another strange cog in the wheel. Where did she come from? No one knew anything about her, and she seemed to have popped out of nowhere. Chicago was only ninety miles away from North Harbor. Their relationship didn't seem that close, based on the tone of her text messages with Emma. Although, it wasn't fair to read tone into a text message. Maybe she was just a curt texter. I would assess her veracity more after our meeting in a few hours.\n\nFinally, did A-squared really know who the murderer was? He wasn't a reliable source. What was the likelihood he was in the exact location when the murder was committed, unless . . . unless he actually murdered Melody? I pondered the question, but my mind kept rejecting it. As much as I disliked him, I couldn't see him actually murdering her. Well, maybe I could. If he thought she would prevent him from reaping benefits from Dawson's professional football career, he certainly would kill her. But, would he let Dawson take the blame for it? Getting convicted of murder would prevent Dawson from making millions in the NFL. A-squared wasn't someone who thought things through to the end. His comment yesterday that Dawson was in good hands might imply he didn't worry about Dawson because he knew we would work to clear him. Surely he couldn't be that cold-blooded. Maybe if Dawson spoke to him? I wasn't sure Dawson and his father were on speaking terms, but he might know if his father was telling the truth or not. When Dawson was telling us about seeing Virgil with Melody, I got the feeling he was holding something back. What if he saw someone else there too, his father? I made a note to ask him.\n\nI reread my notes but none of it made any more sense at four than it had at three. I turned out my light and settled down. I lay perfectly still and took deep, relaxing breaths and meditated on the suspects in the hope a clear plan of action would reveal itself. Unfortunately, my mind refused to cooperate and kept intertwining Melody with my British cozy. I kept seeing Melody wearing a French maid's uniform while a tweed-clad Professor Harley Quin smoked a pipe near the fireplace and cleaned his shotgun. A sleazy character in polyester with lots of gold chains stood ominously over Melody's body while A-squared, who surprisingly still had on greasy blue jeans and a T-shirt, drank scotch and laughed.\n\nI awoke two hours later to the doorbell. It was barely six and I calculated I had gotten about three hours of sleep total. I looked around my room and Oreo and Snickers were still asleep.\n\n\"Some watch poodles you two turned out to be.\" I pulled on the jeans I'd worn the night before and grabbed a T-shirt from the hamper and slid it over my head.\n\nI hurried to the kitchen and looked over the rail and saw two uniformed police officers standing at my door. I was instantly awake. My heart raced and I hurried downstairs and opened the door.\n\n\"What's wrong?\"\n\n\"Does Dawson Alexander live here?\" one of the policemen asked.\n\nI tried to take deep breaths to slow down my heart. \"Yes,\" I whispered.\n\n\"May we speak to him?\" the same policemen asked.\n\nI closed my eyes and nodded. \"Follow me.\"\n\nI led them inside the garage. They seemed a bit hesitant about entering and looked around tentatively. I turned on the light and hurried to the door. I knocked loudly.\n\nIt took several minutes of me knocking to wake Dawson, who came downstairs wearing a MISU T-shirt, sweat pants, and flip-flops. He had bed hair and yawned when he opened the door.\n\nA look of terror crossed his face at the sight of me with two police officers. The blood drained from his face. His eyes had the startled \"deer in the headlights\" look that reflected the internal anguish he was feeling. He must have thought they were there to arrest him, again.\n\n\"Dawson Alexander?\" the officer asked.\n\nHe nodded.\n\n\"Your dad was involved in a car accident and was seriously injured. He's been flown to River Bend Memorial Hospital by Life Force.\"\n\nI didn't think he could get much paler, but he did.\n\nHe gulped. \"Is he gonna be okay?\"\n\n\"Unfortunately, all we know is his injuries are severe. You'll have to talk to the doctors.\"\n\nThe other officer had been quiet up until this point. \"Do you have someone who can take you to the hospital?\"\n\nI stepped forward. \"I can take him.\"\n\nThe officers looked at me and then back at Dawson, who nodded his acceptance.\n\nThey turned to leave, but before they got out the door, one of the officers turned back and said, \"I watched you play ball a couple of weeks ago. You're really good. I hope you get beyond all this.\"\n\nWe got in the car. I made the forty-mile drive in less than thirty minutes and dropped Dawson at the emergency room door. It wasn't until I pulled into the parking garage that I realized I'd left my purse and money at home.\n\nThankfully, I'd grabbed my cell phone and I called Jenna and asked her to swing by the house and grab my purse.\n\nTwo hours later, Jenna and Nana Jo walked down the hallway with my purse and a carrier from a nearby fast-food restaurant with four large beverages and a bag filled with greasy sausage biscuits. My mouth watered and my stomach growled at the sight of the coffee. Both Jenna and Nana Jo stared at me, but neither said a word as I snatched the coffee from the carrier and took a swig.\n\nJenna sat next to me. \"How is he?\" Jenna whispered as she glanced at Dawson, who was pacing up and down the hallway.\n\nI shrugged. \"Not good. A-squared's in intensive care. They let Dawson in for about ten minutes earlier but then alarms started going off and they kicked him out of the room. An army of doctors and nurses rushed in the room. A doctor eventually came out and said he'd gone into cardiac arrest. He's still alive, but . . . I don't think the outlook is good.\"\n\nNana Jo tsked. \"That poor kid. He's been through a lifetime of misery in a short period of time.\"\n\n\"We let the dogs out before we left, and Chris and Zaq said they'd take care of things at the store.\" Jenna sipped her hot tea.\n\nI was blessed to have my family. They were annoying and would result in my need for therapy, but they were here for me. I looked at Dawson and tried to guess how he must be feeling.\n\nJenna stared at Dawson and leaned close to me and whispered, \"I talked to Stinky Pitt this morning.\"\n\nI watched to make sure Dawson wasn't in listening distance. \"Did he say how it happened?\"\n\n\"Apparently A-Squared made two telephone calls. One was to Virgil Russell, who came and posted bail. They don't know who the other call was to.\"\n\n\"Virgil?\"\n\nShe nodded. \"A few hours later, they got a call someone was laying on the railroad tracks by Eden Springs trailer park.\"\n\n\"The railroad tracks?\" I shivered.\n\n\"Some kids were walking around out there and thought it was a heap of clothes on the rails. He'd been hit by a car and moved to the tracks.\"\n\n\"Oh my. Do the police have any ideas? Did they question Virgil Russell?\"\n\nShe paused as Dawson walked by before continuing. \"He claimed he dropped him at a liquor store near the HOD. A-squared said he needed to use the phone. He says that's the last time he saw him, but the police are getting a warrant for his car. They'll go over it with a fine-tooth comb.\"\n\nJenna turned to stare at me and then took several sniffs. \"Did you shower today?\"\n\n\"No. I didn't have time. Why?\"\n\n\"You stink.\"\n\nI lifted my armpit and took a whiff.\n\n\"Plus, your hair is sticking straight up in the back.\" She pulled a compact out of her purse and handed it to me.\n\nShe was right. My hair was sticking straight up in the back as though held up by electric current. I had crusts at the corners of my eyes and a dried trail of drool from one corner of my mouth. \"I can't believe no one said anything to me.\" I finger combed my hair into submission.\n\nMemorial Hospital was the biggest hospital in the area. It was a huge facility that had been added onto many times over the years and was now a winding maze with lots of twists and turns. Thankfully, someone thought to color code the building. The multi-striped legend was painted on the walls. I followed the blue stripe down hallways to the elevator and down to the gift shop. It always felt like a major accomplishment whenever I visited someone here and managed to make it out without stopping to ask for directions, and today was no exception.\n\nThe hospital gift shop had been renovated since the last time I was here. Instead of being a small closet containing flowers, get well cards, and overpriced snacks, it was now a shop that would give any downtown boutique a run for its money. I browsed the aisles until I found the toiletries. I bought a kit that included toothbrush, toothpaste, comb and brush, soap, deodorant, mouthwash, and shaving kit. I also bought underwear and a T-shirt. Since I was there, I also ordered flowers for A-squared.\n\nAt the intensive care nurses' station, I asked if there was a place where I could clean up. They provided me a washcloth and towel and took me down a back hall to the nurses' fitness area. There were showers and I cleaned up.\n\nWhen I returned, I felt better. Amazing what clean underwear and toothpaste could do. It was getting close to noon and Jenna said she needed to get home.\n\n\"You don't have to stay here with me. I'll be okay,\" Dawson said.\n\n\"I know I don't, but I want to stay.\"\n\nDawson paused. \"There's nothing you can do. I know you have things you need to do.\" He looked down.\n\n\"Is there something you need me to do?\"\n\n\"Yeah. I need you to find out who killed Melody and . . . who tried to kill my dad.\"\nChapter 14\n\n\"The girls hated missing out on our trip to Chicago, so I told them to take the South Shore and we'd meet them at the outlet mall later.\" Nana Jo looked sheepish. \"I also told them we'd swing by The Boat.\"\n\nThe South Shore commuter train ran between River Bend, Indiana, and Chicago. For less than ten dollars, the South Shore transported people from the airport in River Bend to downtown Chicago, without the stress of traffic or parking. I knew people who made the two-hour-and-twenty-minute trip daily for work. I'd taken the South Shore downtown to go to Cubs games, museums, or shopping. The train didn't go through Michigan, but North Harbor residents drove thirty minutes to Michigan City, Indiana, to get on the train.\n\nMichigan City was a small town on the Lake Michigan shoreline with nice beaches. It had a designer outlet mall, and The Boat. For most of my life, land-based gambling was illegal in the state of Indiana. Eventually, some creative developer came up with a work-around by turning a boat into a casino. When The Boat first opened, it cruised up and down Lake Michigan. Guests entered every two hours. Eventually, law-makers abandoned the requirement for the boat to move and people boarded or disembarked at will. A trip to The Boat meant I might not get home until the wee hours of the morning. Given that I'd only had a few hours of sleep, I wasn't happy.\n\nI followed the directions Cassidy Logan had sent to the West Inglenook area of Chicago. North Harbor was an economically depressed area with abandoned buildings, high unemployment, and crime. West Inglenook, at fifteen miles wide with three times the North Harbor population, took economic depression to another level. Most buildings were boarded up and covered with gang graffiti. The few shops still open had uninviting iron bars covering every piece of glass and brick walls which reminded me of a prison.\n\nI pulled up in front of an old brownstone with crumbling bricks, broken glass, and a tired-looking floral sofa from the seventies with no legs and sagging cushions in the yard. In the alley next to the house, a group of men were shooting craps. A couple of teenage boys with jeans down to their knees and bandanas tied around their heads hung on the porch.\n\nI turned to Nana Jo. \"Do you think my car is going to be safe here?\"\n\nShe looked at me like she'd never seen me before and then patted my knee. \"Honey, you can't be serious? This car is twelve years old and has close to two hundred thousand miles on it. The turn signal is attached to the steering column with duct tape and if you hit a bump, the lights turn off and on. The only thing of value is the radio, and it doesn't have any knobs.\"\n\n\"I have all of the knobs.\" I opened the ashtray and showed all of the knobs for the radio and air-conditioning, along with a pair of pliers I used to change channels. \"Besides, you'll hurt her feelings.\"\n\nNana Jo rolled her eyes. \"If I were you, I'd pay those guys fifty dollars to dispose of the car so the police don't find it.\"\n\n\"Don't listen to her, Martha.\" I patted the dashboard.\n\nWe got out of the car. I opened the hatch and picked up Melody's meager belongings. We walked toward the building. One of the young kids who looked to be about thirteen stared at me as though I were a piece of meat and then made a rude remark to his friend in Spanish. I always found it amusing when people thought you couldn't understand their language. When I responded in Spanish, the smirk immediately left his face. All outward signs indicated I should be afraid of this hooligan, but something in my gut told me not to be afraid. Years teaching in the public school system provided what I called a sixth sense. It had never failed me, so I trusted my gut.\n\n\"Would you keep an eye on my car?\" I asked.\n\nHe looked at my Honda CRV and laughed. \"That piece of junk? Lady, you'd have to pay somebody to take that piece of crap.\"\n\nTwo insults in less than five minutes was too much. I lifted my head and marched around him. Nana Jo came up behind me. Just as she climbed the porch, the kid came up on Nana Jo in a threatening manner. \"But you can give me your purse, old lady.\" He reached out his hand as if he thought she would just hand it over.\n\nI don't know if it was the idea of a thirteen-year-old boy trying to take her purse or the fact he called her \"old lady.\" Nana Jo spread her legs, crouched low in her aikido stance, and dipped her shoulder. She reached out, grabbed his outstretched hand with both of hers, twisted, and dropped to her knee in one motion. The kid was flipped onto his back and Nana Jo put her knee in his chest and held his arm in the air.\n\nHe yelled obscenities while he lay on the ground.\n\nHis friend stood openmouthed.\n\nAfter a few seconds, Nana Jo released the kid's arm and stood. \"Better watch who you're calling an 'old lady,'\" she said as she marched up the stairs.\n\nThere was no front door, so we walked into the building and headed up the stairs.\n\nWe walked up three flights of stairs in silence. When we got to the top floor, Nana Jo stepped aside for me to knock.\n\nI looked at her. \"You okay?\"\n\n\"Yep. That was just a basic throw down.\"\n\n\"Okay, Bruce Lee. Maybe you shouldn't antagonize the locals.\"\n\nShe smiled. \"Then the locals better stop provoking me.\"\n\nI knocked on the door.\n\n\"Come in.\"\n\nWe opened the door and walked inside.\n\nThe apartment was small and cramped, with oversized leather furniture that looked as though it had seen better days. One wall was dominated by the largest television I'd ever seen. It was massive and similar to the one I'd seen at Trammel Braxton's home with cords hanging down to video games. There was a playpen in the middle of the room with two toddlers inside wearing nothing but diapers. Sitting in a chair, smoking a cigarette, was a small woman. She was a petite Caucasian, fair-skinned, with thin red hair and green eyes. She was also about six months pregnant.\n\n\"You must be from that school?\"\n\n\"Well, actually, I'm just a friend of Emma's, your sister's roommate.\"\n\n\"Half sister.\" She took a long drag from her cigarette. \"You can drop the stuff anywhere.\"\n\nIf I didn't talk fast, we'd be out of here in less than two seconds. \"Sure, but let me introduce myself. My name is Samantha Washington and this is my grandmother, Mrs. Josephine Thomas.\"\n\nNana Jo frowned. \"You know smoking when you're pregnant can lead to birth defects.\"\n\nShe stared at Nana Jo and exhaled slowly. \"Thanks for the public service announcement.\"\n\nThis wasn't going well. \"May we sit for a few moments? Those stairs were a killer.\" I didn't wait for a response and plopped down on the sofa.\n\nUsing two fingers, Nana Jo removed some clothing and sat.\n\n\"Whew. Thank you so much.\" I tried to think of a way to ask my questions before she threw us out.\n\nNana Jo walked to the playpen. One of the kids was asleep. The other, a curly-haired boy, looked at her and immediately held up both arms to be picked up.\n\nNana Jo looked at Cassidy. \"Do you mind if I hold him?\"\n\nCassidy shrugged.\n\nNana Jo reached down and picked up the baby, who immediately threw back his head and laughed and reached for her earrings. Nana Jo had a way with babies, and she returned to her seat. The baby stood straight up. Nana Jo sang \"Row, Row, Row Your Boat.\" With each \"row,\" she rocked the youngster forward and back. He laughed hilariously each time he went backward.\n\nI watched them for several moments. \"Cassidy, when's the last time you saw your sister . . . ah, half sister?\"\n\nShe smoked silently for so long I thought she wasn't going to answer. \"It's been years. We weren't close.\"\n\n\"You both have the same father?\"\n\n\"Mother.\"\n\nNana Jo stopped singing long enough to ask, \"Did you know what your sister was involved in?\"\n\nCassidy smiled. \"She was always up to something.\" She finished her cigarette and put the butt out. \"That's how she ended up in that program. Go to jail or go to college.\" She laughed. \"Mel thought that was hilarious. She had access to an entirely new pool of marks.\"\n\n\"Did you know what scam she was running?\" I asked.\n\nCassidy looked at me suspiciously. \"You've got a lot of questions.\"\n\n\"We want to find out who killed her. We thought you might be able to help us.\"\n\n\"Why would I want to do that?\"\n\n\"Because she was your sister,\" I said\n\nShe shrugged. \"Everybody's gotta die of something.\"\n\n\"Maybe because, in spite of your attitude, you're basically a good person and want to do the right thing,\" Nana Jo said.\n\nI stared at my grandmother, who seemed determined to get us thrown out before we got information, and then turned and looked at Cassidy.\n\nCassidy was staring at Nana Jo but didn't say anything. Eventually, she wiped a tear from her eye. \"Look. I haven't seen Mel in years. She sent text messages sometimes. The last one was a month ago.\"\n\n\"Do you still have it?\" I asked.\n\nShe shook her head. \"Nawh. She was running a scam. Said if things worked out, she'd be set for life.\" She looked at her sleeping baby and the one on Nana Jo's lap. Then she looked at her stomach. \"Said she would get us out of this place.\" She wiped away an errant tear and got another cigarette out of the pack on the table and lit up. \"She talked a lot of crap.\"\n\n\"Did she mention anyone that wanted to hurt her?\" I asked.\n\nCassidy took a long drag of her cigarette and then she sat up. \"She didn't mention any names. But she said someone was trying to muscle in on her, but she had his number.\"\n\nWe asked a few other questions, but Cassidy didn't have any more answers. The sleeping baby woke up and started screaming. She hoisted herself out of her seat and picked him up. He looked exactly like the boy Nana Jo was bouncing on her lap.\n\n\"Twins?\" I asked.\n\nShe nodded.\n\nShe had her hands full, and it didn't seem as though we'd learn anything more. So, we rose to leave. Nana Jo handed over the baby.\n\nBefore we left, Nana Jo turned and asked, \"Did you know Virgil Russell?\"\n\nCassidy was jiggling a baby on each hip. \"Yeah. I know him. Why?\"\n\n\"It seemed strange to us that Melody was . . . involved with him,\" I said.\n\n\"Mel was always pretty close to Virgil. He taught her practically everything she knew about conning people. Virgil taught her how to dress and talk and act so she got richer marks.\"\n\n\"But he killed your father?\" Nana Jo said.\n\nCassidy shrugged. \"Her father, not mine. Besides, that wasn't a big loss. He was a drunk who beat all of us every chance he got. That's how he died. He beat Mel so bad we thought he'd kill her. Virgil tried to stop him. He pushed him. He hit his head on the end of the table.\" She lowered the toddlers back into the playpen. \"He and Mel got closer after that.\"\n\n\"Any chance he was the man trying to muscle onto her scam?\" I asked.\n\nCassidy shook her head. \"I doubt it. They worked together.\"\n\nWe left Cassidy and walked down the narrow stairway. Once outside, we saw the youngster Nana Jo had tossed earlier. He stood in the middle of the sidewalk in a threatening manner.\n\n\"Maybe I should call the police,\" I whispered to Nana Jo.\n\n\"No need.\" She stepped forward. \"You want a piece of me?\" she asked the kid.\n\nThe boy snarled. \"I've got something for you this time, old lady.\" He reached into his pocket and pulled out a switchblade, which he held up with his right hand.\n\nNana Jo smiled and reached into her purse and pulled out her gun and pointed it. She squinted, closed one eye, and fired. The bullet hit the blade of the knife and ricocheted off.\n\nThe startled hooligan dropped the knife, turned, and ran.\n\nThere was a loud burst of laughter from the gamblers in the alley. Five young men in jeans with tattoos and piercings whistled and laughed. One of the men walked forward. \"Hey, I've got two hundred dollars that says you can't hit that billboard?\" He held up several bills and grinned. Each of the other men also held up money.\n\nNana Jo looked at the billboard that stood above the building across the street and within an arm's distance of the elevated train track.\n\n\"Oh, no,\" I pleaded with my grandmother.\n\nThe grin on her face and the gleam in her eyes told me I was wasting my time.\n\nNana Jo extended her arm and pointed her gun at the billboard. She cocked her head to the side and pulled the trigger.\n\nWe all looked at the billboard. There was a large picture of a popular insurance company's spokesperson wearing a white apron with her black hair pulled back with a headband. The woman had a large smile on her face and a perfectly round hole in the center of her forehead.\n\nThe men hooped and laughed. One of them even bowed to Nana Jo as they all came and handed her their money.\nChapter 15\n\nI hustled my grandmother into my intact vehicle and hurried away.\n\nNana Jo laughed as she fanned herself with her money. \"Easiest thousand dollars I've ever made.\"\n\n\"You could have gotten in trouble. What if you'd missed?\"\n\nShe looked at me. \"I was Lauderdale County's sharpshooter three years in a row. Besides, that billboard was huge. You'd have to be blind to miss it.\"\n\nI gave up trying to shame my grandmother into reform and headed east. We drove in relative silence the forty minutes to the outlet mall in Michigan City. I found a parking space near one of the larger anchor stores.\n\nBefore I got out, Nana Jo took half of her winnings and handed it to me. \"Happy Birthday, Sam.\"\n\nI looked at the wad of cash with my mouth open. \"I can't take that.\"\n\n\"Why not?\"\n\n\"It's too much. Besides, you won the money fair and square. You should spend it on yourself.\"\n\n\"I intend to splurge and enjoy myself. However, I wouldn't have had the opportunity to win the money if it wasn't for you. Besides, I want you to get yourself some nice clothes. You never know when you might get another date or non-date.\"\n\n\"But really, I don't need\u2014\"\n\nShe waved her hand. \"Sam, will you please just take the money and say thank you?\"\n\nI reached over and gave my grandmother a big hug. \"Thank you.\"\n\nThe title outlet mall implied incredibly low prices, but discounted prices could be a relative term. Lighthouse Place Mall was a designer outlet mall. While the prices were discounted from what you paid in a regular mall, they were still higher than discount retail stores. In the past, I'd limited myself to three or four stores known for ridiculously low prices where I bought blue jeans for less than five dollars. Today I walked into the stores with the designers' names on the front. I bought several really good pairs of jeans, silk blouses, and even a few dresses.\n\nNana Jo bought new tennis shoes and workout attire and then splurged on a lovely dress with a plunging neckline. Laden down with shopping bags, we made our way to the car several hours later. The girls' train was due to arrive in a few minutes. I needed to pick them up from the train station. There was a trolley that took visitors from the train station to the outlet mall, but the girls preferred spending their money at The Boat.\n\nThey were waiting at the depot when I arrived. No need to park, I pulled up to the platform and loaded them in. I then headed for The Boat.\n\nI let everyone out at the front of the casino and parked. By the time I got to the lobby, I was tired. Lack of sleep and tons of walking at the outlet mall hit me at the same time. We'd agreed to eat first, so I went to the buffet and looked around until I found them.\n\nNormally we went to the Four Feathers Casino, which was closer to North Harbor. Trips to The Boat were rare, which meant the girls hadn't racked up enough perks for free food. One thing remained constant, regardless of where we went. Ruby Mae always ran into someone she knew. Today was no exception. When I sat down at the restaurant, she was being hugged by one of the chefs, and a waitress was nearby, awaiting her turn. The chef turned out to be a godson and the waitress was a great-niece. Ruby Mae's connections resulted in free dinner vouchers and dessert take-out boxes, which were usually prohibited at buffets. When we finished, I took the boxes to the car. By the time I walked back to the lobby, I had moved beyond tired to exhausted. We agreed to meet in two hours, so I found a comfy wingback chair in a quiet, cozy corner and took a nap. I set the alarm on my phone to wake me after an hour and a half. When I awoke and felt my face, the pattern of the chair's fabric was imprinted on the side of my face and there was a large wet stain on the chair's wing from drool. I was amazed how refreshed and energized I felt after my power nap.\n\nI walked around the casino and grabbed a coffee from a beverage station. I still had fifteen minutes before our meeting, so I put twenty dollars into a machine with a picture of Tarzan on the front. I only played penny slots and I was normally very conservative, but I tossed caution to the wind and bet the maximum, a dollar fifty. After only two spins, I went into a bonus. I was relatively new to slot machines, so I didn't know exactly what had happened, but the next thing I knew, Tarzan yodeled and swung across my screen. There was a lot going on. Wheels spun, Tarzan pounded gorillas, and drums beat. After the third spin, lights flashed, bells rang, my chair vibrated, and my screen was almost completely full of wilds. Coins started flying on the screen and the lady next to me started screaming and hitting me in the arm. A crowd of people gathered around my machine.\n\nTwo of the casino staff made their way through the crowd to my machine. One of them inserted a card into the machine, which stopped the flashing lights. The other one smiled and pulled out a clipboard. \"Congratulations. Can I see your driver's license, please?\"\n\nI was dazed and couldn't grasp what had happened. \"What just happened?\"\n\nHe smiled. \"You just won a lot of money.\" He took the driver's license I handed him and began copying information.\n\n\"How much did I win?\"\n\nHe looked up at the machine. \"You just won fifty-four thousand dollars.\"\n\nI nearly passed out. \"No way!\"\n\nThe casino worker smiled. \"Yes. Way!\"\n\nI stared at him with my mouth open. \"I think I need to go someplace so I can throw up.\"\n\nHe looked startled but helped me rise. I started to walk away but then turned back to the machine.\n\nThe other casino worker took me by the arm. \"It's okay. I'll take care of everything.\" He looked back at his coworker. \"I'll take her to the office.\"\n\nI followed the casino worker until we came to a small door behind a cash machine that I wouldn't even have noticed. He opened the door and we went inside. There was a desk, two guest chairs, and a small love seat. I sat and put my head between my legs.\n\n\"Can I get you some water?\"\n\nI nodded.\n\nHe left and came back with a bottle of water.\n\nI tried to open the lid, but my hands were shaking so badly I had to get the casino worker to open it for me. He untwisted the lid and gave me the bottle, and I took a long sip. My phone vibrated. Nana Jo and the girls must be wondering where I was. I couldn't get my phone out of my pants pocket fast enough. When I tried to dial, my hands shook so badly I kept missing her name. I turned to the casino worker. \"My grandmother and three other ladies are in the lobby waiting for me. Can you please get them? I have no idea how to get to them from here.\"\n\nHe said he'd be happy to go. I took deep breaths and tried to process the fact I had just won fifty-four thousand dollars. Every time I thought about the dollar amount, I felt light-headed and had to put my head back between my knees until the dizziness passed.\n\nSeveral minutes later Nana Jo and the girls came into the room.\n\n\"What happened? Are you okay?\" Nana Jo asked.\n\n\"Didn't he tell you?\" I pointed at the casino worker.\n\n\"He didn't tell us anything. He just said you weren't feeling well.\"\n\nThe room was small and with six people, it was claustrophobic. I looked at the casino worker.\n\nHe struggled to hide a smile. \"I thought you might prefer to tell them yourself.\"\n\nNana Jo's brow was wrinkled and a vein was pulsing on the side of her head. She was getting angry. \"Tell us what?\"\n\nI pointed at the casino worker. \"He said I won fifty-four thousand dollars.\"\n\nThere was dead silence for several seconds and then Nana Jo sank down onto a chair. \"Sweet mother of God.\"\n\nIrma's response was not holy and resulted in a sharp reprimand from the others before she burst into a coughing fit.\n\nDorothy and Ruby Mae congratulated me.\n\nThe worker slid out of the room and returned with more waters, which he passed around. Nana Jo sat next to me. Irma and Ruby Mae were in the guest chairs, and Dorothy propped on the desk.\n\n\"Son, can you prop that door open a bit.\" Nana Jo fanned herself. \"Either I'm having a hot flash or I'm going to pass out.\"\n\nHe propped the door open.\n\nThe casino worker who took my driver's license entered with a security guard. We were now packed like sardines in that room. He returned my driver's license and handed me some forms to fill out. \"These are for the IRS. Do you want us to subtract taxes or do you want to pay them yourself?\"\n\nI filled in the paperwork. \"Please subtract the taxes.\"\n\nWhen I was done, I handed the papers back.\n\n\"How do you want the money?\"\n\n\"Excuse me?\"\n\nThe casino worker smiled. \"Do you want a check? Cash? Or a combination of both?\"\n\nNana Jo whistled. \"Cash? You mean people actually want cash?\"\n\nThe woman nodded. \"You'd be surprised.\"\n\nI looked at the sheet she handed me, which indicated after taxes I would get over forty thousand dollars. \"Can I get five checks?\"\n\nShe smiled. \"You can have anything you want.\"\n\n\"I need five checks. One made out to each of us.\" I pointed around the room.\n\nThe girls began to protest. \"Oh, no. This is too much\u2014\"\n\nI held up my hand. \"We have a system. We always split the winnings. This is no different.\"\n\nWe argued for several moments, but ultimately I overruled them and made the arrangements with the casino. We would each get about eight thousand dollars. It took close to an hour for the casino to get all of the arrangements made.\n\nIrma asked for a whiskey. Our friendly casino worker took orders and brought drinks.\n\nNana Jo asked if someone could find our car and bring it to the door for us.\n\nI handed over my keys, along with directions on where I'd parked. By the time we were all settled out, the car was waiting at the front.\n\nI was still dazed but ecstatic to share this windfall with my grandmother and her friends.\n\nThe ride home was spent talking about how we'd each spend our money. Irma wanted plastic surgery to lift everything that had sagged over the years. Dorothy suggested she do something about her chronic coughing instead, but that suggestion fell on deaf ears.\n\nDorothy wanted to go on a cruise. Ruby Mae's church was going on a trip to the Holy Land and she wanted to go. Now she could. Nana Jo was very vague and refused to say what she planned to do.\n\nI didn't have plans for spending my share, but the girls were very vocal on how they thought I should spend it. The unanimous decision was I should use my share to buy a new car. I knew Nana Jo's views on my CRV, but the girls had never complained.\n\n\"Beggars can't be choosy,\" Irma said.\n\n\"Those newer cars have Wi-Fi and I would love to use my iPad or watch a movie,\" Dorothy said.\n\n\"My daughter has a Lexus and the ride is so smooth. You can barely tell the motor is running,\" Ruby Mae yelled from the back seat.\n\n\"'Et tu, Brute?'\" I looked at her in the rearview mirror.\n\nRuby Mae laughed. \"One thing I can say for this car is that my prayer life has improved since I've been riding in it.\"\n\nI stuck out my tongue.\n\nThey all laughed. I knew the joking was all in good fun. The girls weren't complainers. My CRV was old and loud and there were a number of aesthetic problems. But the engine was solid and the car was reliable. It had four-wheel drive and performed wonderfully in the snow.\n\nI dropped the girls at the retirement village and they asked me to get out of the car, which was unusual.\n\nI got out and each one of them hugged and thanked me. They had tears in their eyes, which made me cry too.\n\nI got back in the car and Nana Jo and I drove back to the store. When I pulled into the garage, Nana Jo stopped me.\n\n\"I want to thank you too, Sam. That was a very nice thing you did. You didn't have to. No one would have blamed you. That is a lot of money.\"\n\n\"I know, but I wanted to.\"\n\nShe nodded. \"I know you really love this car. It's served you well for more than a decade. However, I do think you should consider an upgrade. Not because the CRV is unreliable, but because you've been through so much over the years and I think it would be nice to treat yourself.\"\n\nI started to protest, but she held up her hand. \"You don't have to defend your decision. I just wanted to explain why you should do this. It's okay to have nice things, and it's not frivolous to spend money on yourself. You're a hard worker and you deserve nice clothes and a nice car. You're always thinking about others, and I want you to know it's okay to think about yourself too.\" She patted my hand and got out of the car.\n\nI sat in the car for several minutes and let the tears stream down my face. Leon and I had worked hard our whole lives, and we never had eight thousand dollars at one time. That was more money than we put down on our house. I was accustomed to scrimping and saving and making every dollar stretch as far as it could go. I wasn't good at spending money on things that weren't essential. I drove cars until they were basically no longer drivable. I looked at Martha and patted the dashboard. \"You've served me well, old friend, but maybe it's time for a change.\"\n\nI had been on an emotional roller coaster and felt tired. I finally got out of the car and went upstairs. Dawson had sent several text messages throughout the day. His dad was still in intensive care, but he was stable. The doctors had put him in a medically induced coma to decrease brain swelling. The twins had packed a bag for Dawson and took it to him at the hospital. They also had taken Oreo and Snickers home with them, which meant the house was quiet.\n\nOne pleasant surprise was a large vase of flowers on my kitchen counter. The card indicated they were from Professor Quin. I was glad Nana Jo was in her room because I couldn't stop smiling. The worst thing about trips to the casino was the smell of smoke that clung to your clothes and hair. I took a long hot shower and got ready for bed. Not surprisingly, I was too excited to sleep. Instead, I spent a couple hours looking at cars online. Eventually I decided writing therapy would be useful.\n\n\"Blasted doctor.\" Lord William beat his fist on the bed but winced as his leg teetered and nearly fell of the pillow where he had it propped.\n\nLady Elizabeth sat in a straight back chair near her husband's side. \"Now, dear, you know it isn't Doctor Haygood's fault your gout flared up. You have nobody to blame but yourself.\"\n\n\"You'd think he could do more than say, 'keep your leg elevated and watch your diet.' My God, it's the twentieth century. Medicine should have a cure for this by now. I'm surprised he didn't pull out leeches. It's archaic.\"\n\nLady Elizabeth smiled. She was familiar with her husband's rants and knew he was in a great amount of pain.\n\nPenelope paced. \"When the king called, did he say how he expected us to keep this out of the newspapers?\"\n\nDaphne looked up from her seat at the window. \"Especially with the editor of the London Times here in the house.\"\n\nVictor leaned against a large armoire. \"I doubt if Fordham-Baker even knows what day of the week it is. The man's been plastered all day.\"\n\nLady Elizabeth looked at Victor and then mumbled, \"I wonder . . .\"\n\n\"Maybe we can keep him that way,\" Penelope said.\n\nLady Elizabeth looked at her niece. \"That might not be a bad idea.\" She took a deep breath. \"The king wants us to solve this murder, like we did with Charles Parker's.\"\n\nPenelope stopped pacing and turned to stare at her aunt. \"But that was different. We were trying to save a friend.\"\n\nElizabeth sighed. \"Now we're trying to save a member of our family.\"\n\nPenelope looked sheepish. \"You're right. Whether we like it or not, Wallis is married to our cousin. She's family.\"\n\nDaphne sighed. \"Alright then, where do we begin? Remember, I wasn't involved in your other investigation.\"\n\n\"That's right. I'd forgotten.\" Lady Elizabeth frowned. \"Well, I think we divide and conquer. First, let's go through what we know and then we can come up with our plan of attack.\"\n\nLord William saluted his wife. \"Aye, captain.\"\n\nLady Elizabeth smiled.\n\nPenelope paced. \"We know she was shot in the marsh, but we don't know what she was doing there.\"\n\n\"Yes we do.\" Lord William sat up straight. \"In all the excitement and pain\"\u2014he patted his leg\u2014\"I nearly forgot. When I was talking to Detective Inspector Covington, one of the constables brought him a note.\" Lord William's lips twitched as he struggled to hold back a smile. \"I managed to get a look at the note when he was distracted.\"\n\n\"Well done,\" Lady Elizabeth said with pride and saluted her husband.\n\nHe beamed. \"The note was from Lord Charles, asking her to meet him in the trees by the marsh.\"\n\n\"That's very odd. I wonder why he chose that location,\" Lady Elizabeth mused.\n\n\"I think Detective Inspector Covington was going to find out,\" Lord William continued. \"He was heading to talk to Charles but got distracted by the call from the king and then the call from the chief inspector.\"\n\n\"Now that was odd,\" Penelope said. \"Why do you suppose he was called back to London?\"\n\nLady Elizabeth reflected, \"I suspect Bertie might have applied a little royal pressure. I think he's asked the Yard to back off for a few days to buy us time to figure this thing out.\"\n\nPenelope stared. \"And they listened?\"\n\n\"The king carries a lot of weight. There used to be a time when the aristocracy ruled supreme.\" Lord William shook his head. \"Those days are gone now.\"\n\n\"And rightly so,\" Lady Elizabeth said. \"Justice shouldn't be reserved for the nobility.\"\n\n\"So, we need to find out from Lord Charles why he wanted to meet with Rebecca,\" Lady Elizabeth said.\n\nDaphne smiled. \"I can tell you why he wanted to meet with her. It's because he's a dirty old man and was most likely having a fling with her, right under his wife's nose.\"\n\nEveryone stared.\n\n\"I think Lady Abigail was well aware of what Lord Charles was up to,\" Penelope added. \"I sat next to her the other night, and she may look like a mouse, but I could tell she was bothered.\"\n\n\"Bothered enough to kill?\" Victor asked.\n\nPenelope shrugged. \"I don't know.\"\n\n\"If she was, she certainly had the skill to do it,\" Daphne said. \"I heard her say she was a crack shot in America. Apparently she won an Annie Oakley shooting trophy in the States when she was a young girl.\"\n\n\"Very interesting.\" Lady Elizabeth turned to her husband. \"Do you feel up to doing a little investigating?\"\n\nLord William nodded. \"I can't go tramping around the grounds, but I can talk, and Charles and I are friendly enough.\"\n\nLady Elizabeth nodded. \"Wonderful. You talk to Lord Charles and see what you can find out.\" She turned to Daphne. \"Daphne, I'd like you to figure out who had the opportunity to shoot her. Where was everyone at the time of the shooting?\"\n\nDaphne frowned. \"You want to know who was unaccounted for?\"\n\nLady Elizabeth nodded. \"Yes. That's it exactly. Do you think you can do that?\"\n\nShe nodded.\n\n\"What would you like us to do?\" Penelope asked.\n\nLady Elizabeth looked at Victor. \"I know Bertie may want us to keep the police out of this, but there are things we need to know. I want you to work with Detective Inspector Covington to find out as much as you can about the gun that was used.\"\n\nVictor nodded.\n\n\"What about me?\" Penelope asked.\n\n\"I'd like you to talk to the servants. I'm sure they can tell you plenty about what was going on, plus, they've probably spent more time with Rebecca than anyone else.\"\n\nPenelope nodded. \"What are you going to do?\"\n\nLady Elizabeth hesitated. \"I'm going to try and make sense of this. If you stop and think about it, it really doesn't make any sense. Why would anyone want to kill the maid?\"\n\n\"You think the duchess was the intended victim?\" Victor asked.\n\nLady Elizabeth shook her head. \"I don't know. This whole thing seems like an elaborate charade.\"\n\nDaphne leaned forward. \"I know what you mean. There are diplomats from all over the world staying under one roof. The world is on the brink of war and the tension is thick enough to cut with a knife and the only casualty is a French maid.\"\n\n\"Exactly.\" Lady Elizabeth was surprised at her niece's perception. \"I feel this whole thing has been staged.\"\n\n\"But the duchess arranged the whole thing. Surely she must be the puppeteer behind the scenes,\" Victor said.\n\nLady Elizabeth frowned. \"Maybe. Wallis certainly has her own motives, but I'm not sure she is . . . well, to be completely honest, I don't think she's smart enough to be the mastermind behind this.\"\n\n\"Who then?\" Victor asked.\n\n\"I don't know, but I'm going to find out.\"\n\nChapter 16\n\nI woke up bright and early Sunday morning. Despite getting less than eight hours of sleep, I felt happy and alert. My good mood could have been attributed to the eight-thousand-dollar check in my purse. Or it could be related to the fact the twins had Snickers and Oreo and I was able to sleep past three. A night without my dogs was nice, but awkward. It felt weird not to have to look under my feet before I stood up or went down a flight of stairs, to make sure I didn't step on or trip over a lounging poodle.\n\nI showered and put on one of my new outfits, a denim dress that was fitted at the top and flared at the bottom. I had a cup of coffee and sniffed my flowers, which made me smile. I picked up my cell and sent a thank-you text message.\n\n\"What are you grinning at?\" Nana Jo came into the kitchen and hopped on a barstool.\n\n\"Nothing. I'm just very thankful.\"\n\nShe sipped the coffee I handed her. \"Well, I suppose you have eight thousand reasons to be.\"\n\n\"I got a text from Dawson. He said his dad is still in the coma. He's going to stay there with him.\"\n\nShe nodded. \"Maybe we can take him something to eat later today.\"\n\n\"That's a great idea.\" I looked at the time. \"You should come to church with Mom and me today. I'll treat you to dinner and we can pick up the girls and head to MISU for the memorial service. Afterward, we'll go check on Dawson.\"\n\nNana Jo agreed and got dressed.\n\nMy mom attended a large church located in downtown South Harbor. The building was brick but painted white with stained glass, a steeple, and bells. The church was one hundred and twelve years old and from the outside, nothing much had changed over the years. The changes were more noticeable inside. It still had a large pipe organ, but there was also a keyboard, a synthesizer, and drums. The pastor who baptized me and my sister as children died a few years ago. The new minister was very young. The congregation had grown substantially since Pastor Andy Timmons was installed. His messages were contemporary, as was the music. This change caused dissatisfaction from the older members, who were steeped in the traditional hymns and messages from decades gone by. To his credit, Pastor Timmons had done a fine job of finding the middle ground. The church now had two services. The first was traditional, with hymns and organ music. The second service was much more contemporary, with upbeat music.\n\nNormally, I found the traditional hymns boring and dreary. Today I felt comfort in the ritual of tradition and sang songs of thanks with a glad heart.\n\nAfterward, I treated Nana Jo and my mom to brunch at the Boulevard Hotel, one of the oldest and nicest hotels in South Harbor. When my mom learned we were going to River Bend to see Dawson, she insisted on picking up flowers for us to take with us. Nana Jo and I picked up extra flowers to take to the memorial service for Melody. We invited Mom to go along, but she declined with a vague excuse. I dropped her at home and headed to the retirement village and picked up the girls.\n\nAs arranged, I met Jillian and Emma at the student union. I was surprised to see Zaq there too, although I gathered he was there for Emma and not Melody. The memorial service was to be held at the campus chapel, a quaint building in the center of campus. The chapel had wooden pews, stained glass windows, and an ornately carved altar of dark mahogany. The building was small and only held about one hundred people, which didn't seem to be a problem. Only about twenty people showed up, despite the flyers Jillian placed all over campus and the announcement she put in the school newspaper.\n\nWe mingled among the few guests, but with a crowd this small, there wasn't much to find out. When I bought the flowers, I also picked up a memorial book for signatures. I planned to send it to Cassidy when this was over.\n\nI was disappointed by the low turnout until I saw Professor Harley Quin walk in. I smiled as he approached. I was glad I had on one of my new dresses.\n\n\"Hello, beautiful. You look amazing.\"\n\n\"What a lovely greeting,\" I said.\n\n\"I'll bet you say that to all the girls.\" Nana Jo smiled.\n\n\"Nana Jo, this is a friend, Professor Harley Quin. He's helping with my book.\" I turned to Harley. \"This is my grandmother, Nana Jo.\"\n\nNormally, Nana Jo was super friendly, but there was an edge to her voice. She held out her hand. \"Nice to meet you.\" She shook his hand. \"Please, call me Josephine.\"\n\nHeat rushed to my face. Harley raised an eyebrow but smiled.\n\nAn awkward silence followed. I struggled to think of something to fill the gap. I looked around. Virgil Russell had entered the chapel. \"Isn't that Virgil Russell?\"\n\nNana Jo looked but then turned back to Harley. \"Professor Quin, weren't you Melody's teacher?\"\n\n\"Quite so. Quite so. Although, I didn't know her. In fact, I'm not sure I actually ever met her personally. I gather she didn't attend many classes.\"\n\n\"But she was getting an A in your class. That's rather odd, don't you think?\"\n\nNana Jo was staring at Harley as if he were one of the students in her class. She was tall and could be very intimidating when she wanted to, and apparently she wanted to be today.\n\n\"I asked Harley about that. It turns out it was all just an error with the computer system,\" I said.\n\n\"Hmm . . . convenient,\" Nana Jo mumbled just loud enough for Harley to hear.\n\nHe laughed. \"I have to admit, I'm not great with computers, and the system they use here in the States is different from the one I used back home in England.\"\n\n\"Where exactly are you from?\" she asked.\n\n\"He's British,\" I said.\n\n\"I can tell by the accent.\" Nana Jo smiled. \"I meant, where in Great Britain?\"\n\nHe chuckled. \"It's a very small village. I doubt if you've ever heard of it\u2014\"\n\n\"Try me.\"\n\n\"It's a small town . . . Deering Vale. Ever heard of it?\"\n\nNana Jo hesitated. \"It sounds vaguely familiar.\"\n\n\"One English village is much like another. Small thatched roof cottages, a church, a pub, and a lot of sheep.\" He laughed.\n\n\"What brings you to North Harbor?\"\n\n\"I'm researching a book on the House of David. Fascinating stuff.\"\n\nI stood behind Harley and caught Nana Jo's gaze. I raised my eyebrows and flicked my head to the side to indicate she should scram.\n\nShe looked at the door. \"My God, it's Stinky Pitt.\"\n\nI turned. Detective Pitt was standing against the wall.\n\nHarley looked startled.\n\n\"Well, it's been a pleasure meeting you, Dr. Quin, but I've got to go and . . . talk to a friend. Please excuse me.\"\n\nWhen Nana Jo left, I exhaled, and Harley looked at me and laughed. \"Your grandmother is very protective.\" He pulled at his collar but smiled.\n\n\"I'm sorry. Usually she's very friendly.\"\n\nHe placed his arm around my back and whispered in my ear, \"Maybe she hasn't felt like you needed protecting before.\"\n\nMy stomach fluttered. My face heated and I knew I had a silly grin on my face, but I couldn't stop myself. His breath caressed my neck and my heart pounded in my ears. I turned my head slightly and looked in his eyes and giggled like a schoolgirl.\n\nHe smiled but didn't remove his hand from my waist.\n\nI was grateful when the minister stood and asked everyone to take their seats.\n\nWith his hand on my back, Harley steered me to a seat near the front of the small church on the opposite end of the pew from where Nana Jo was sitting with the girls.\n\nWe took our seats.\n\nJillian read a scripture from the Bible. Everyone recited the Lord's Prayer. The minister said a few words. It was clear he hadn't known Melody, but he talked about a young life taken too soon and the promise of eternity in heaven. He then asked if anyone wished to speak about Melody. No one moved, and I was concerned no one would get up. Imagine my surprise when Nana Jo rose and walked to the front.\n\n\"I only met Melody a couple of days before she died, but she was an intelligent girl who had experienced a great deal of trouble in her young life. Despite those obstacles, she found a way to escape. She knew a lot about people and mastered a number of skills in her young life. She used her skills and her assets to move ahead. She had tremendous plans. Unfortunately, someone thwarted her plans. I always feel it's such a shame when young people are taken from this earth without a chance to live and experience life.\"\n\nI was thoroughly amazed. Everything she said was the truth, but there was definitely more behind her words, much would only have meaning to a small few. Nana Jo was brilliant.\n\nOnce Nana Jo got the ball rolling, several others rose and said nice things, including Emma and Jillian. The last to speak was Virgil Russell. He seemed genuinely grieved. He spoke of Melody's beauty and her strength. He pulled out a handkerchief and hurriedly sat down.\n\nWhen no one else got up, the minister thanked everyone for coming and encouraged us to seek peace in the Word of God and then prayed and dismissed us. Irma immediately sidled up to Virgil Russell, while Nana Jo talked to Stinky Pitt. Dorothy talked to Jillian and a small group of students. Everyone was working and here I was, sitting with Harley. I felt guilty. My focus should be on finding Melody's killer.\n\n\"Thank you again for the flowers. They were lovely.\"\n\nHe leaned close and whispered, \"They were really a bribe.\"\n\n\"A bribe?\"\n\n\"I'm hoping you'll go out again with me.\"\n\n\"You don't have to bribe me to go out with you,\" I said. \"All you have to do is ask.\"\n\n\"Good. How about dinner?\"\n\n\"I'm sorry, I can't tonight. I have to check on Dawson. His dad's in the hospital.\"\n\nHarley looked surprised. \"Really? Is he going to be okay?\"\n\n\"We don't know. He was hit by a car. He's in intensive care. It doesn't look good.\"\n\n\"I'm very sorry to hear that.\" He did look sorry too, really troubled. \"Which hospital?\"\n\n\"Memorial in River Bend. How about tomorrow night?\" I asked.\n\nHe shook his head. \"Sorry. I can't tomorrow night. I've got a late class. Tuesday?\"\n\nI shook my head. \"Tuesday night is girls' night. But I'm free Wednesday.\"\n\n\"Wednesday it is.\"\n\nI thanked Jillian for all her hard work coordinating the memorial, and then we loaded into the car. We stopped once on the way to the hospital and picked up snacks for Dawson. There were several places to eat at the hospital, including a Subway and a River Bend Chocolate Factory. However, I'd seen him eat and knew it would be expensive for him. Nana Jo insisted on getting necessities like underwear, socks, orange juice, peanut butter, and Doritos; just the necessities.\n\nOn the ride to the hospital, Nana Jo filled us in on her conversation with Stinky Pitt. \"He thinks Virgil killed Melody and tried to kill A-squared. He's still waiting for the report on Virgil's car to come back to see if there's any trace evidence from the hit-and-run. In the meantime, he's keeping a close eye on him.\"\n\n\"He seemed genuinely upset about Melody,\" I said.\n\nIrma coughed. \"He was upset alright. I know real grief when I see it.\" She coughed again. \"He knew Melody had other lovers, but he said they had a 'special bond.'\"\n\n\"Yeah right. I'll bet they had a special bond. The dirty old coot,\" Nana Jo said.\n\n\"All I'm saying is he seemed sincere.\" Irma took a swig from her flask.\n\n\"Irma, are you sure that's what he said?\" I asked.\n\nIrma looked offended. \"Of course I'm sure. I'm not senile.\" She coughed.\n\n\"I just meant, are you sure he said Melody had other lovers, plural. Not that she had another love?\" I looked at her in the rearview mirror and saw the look of recognition in her eyes.\n\n\"He definitely said lovers, as in more than one.\" She coughed.\n\n\"But that could mean anything. Maybe he meant, through the years, she's had other lovers,\" Ruby Mae said. \"Nowadays, it isn't uncommon for girls to have had multiple lovers.\"\n\n\"Well, I always had more than one,\" Irma bragged. \"Sometimes it was hard to keep them straight, but boy was it fun.\" She laughed before breaking into a coughing fit.\n\nNana Jo stared at me. \"You're thinking she had another lover, other than Dawson and Virgil.\"\n\n\"Could it have been that Trammel Braxton?\" Dorothy asked.\n\nI shook my head. \"I really don't think so. I think that relationship was over.\"\n\nI drove to the hospital and let the girls out at the front while I parked the car. By the time I made my way to intensive care, Nana Jo and the girls were flocked around Dawson and were mothering him. He was hugged, kissed, fed, and loved. All of them shoved money into his jacket at one time or another. That much concentrated attention could be overwhelming. When I noticed his eyes glaze over, I suggested the girls go down to the gift shop.\n\nWhen they were gone, Dawson sat next to me and released a heavy sigh. \"Thank you. They're amazing, but I was starting to feel a little smothered.\"\n\n\"They really care about you.\"\n\nHe nodded. \"I know. I care about them too, but it's been pretty emotional. Nothing to do but think.\"\n\n\"You sleeping?\"\n\nHe shook his head. \"Some, but it's not easy. The nurses have been great. There's a nurses' lounge and they let me sleep in there. When he gets out of intensive care, then I can sleep on a cot in his room.\"\n\n\"Any idea when that will be?\"\n\nHe shook his head.\n\nWe visited a little longer and then the girls and I headed back to Michigan. I dropped Nana Jo and the girls at the retirement village and then swung by Jenna's to pick up Oreo and Snickers. Jenna claimed she wasn't a big dog lover, but her husband, Tony, was especially fond of Snickers.\n\nWhen I got to her house, Oreo was barking before I got to the front door. He met me at the door and barked nonstop until we left. Snickers, however, was curled up on the sofa with my brother-in-law, watching a football game and eating cheese crackers. She barely looked up to acknowledge my arrival. In fact, I had to pick her up and carry her to the car to get her to leave. Her affections were fickle and could be bought with snacks and belly rubs. She had received both from Tony.\n\nAt home, I let the dogs do their business before heading upstairs. The house felt very lonely without Nana Jo and Dawson. It was comforting to look outside and see a friendly light coming from his apartment. Nana Jo had her own house at the retirement village, but she had spent most nights here at the bookstore since the murder over the summer. I didn't realize how much I relied on her until she wasn't here.\n\nI tried to sleep but something kept nagging at my brain, and I couldn't figure out what it was. I sifted through the information from the day and tried to catch the clue, but it remained elusive. The best I could do was to narrow it down to something that happened at the memorial. However, the harder I tried, the further away it flew. Eventually, I gave up and decided to focus on other things. I got out my laptop.\n\nLady Penelope went down the back stairs to the servants' hall. It had been quite some time since she'd been down here. As she turned the corner at the bottom of the stairs, she couldn't help but glance at the place where she'd fallen as a child. She busted her lip and nearly split her head open. That was probably the closest she'd ever come to seeing Thompkins befuddled. He hurried to fetch the doctor, stopping only long enough to scold the housemaid for waxing the floors to such a dangerous state. Mrs. McDuffie had been the calm one. She had picked up the crying girl and rocked her to sleep, while holding a towel to the cut to staunch the bleeding.\n\nIn the hall, Thompkins and the other servants were just finishing their meal but saw Lady Penelope and immediately stood at attention.\n\n\"I'm terribly sorry. I didn't mean to interrupt your breakfast.\"\n\n\"Can we help you, Lady Carlston?\" Thompkins asked.\n\nPenelope had only been married a few months and still smiled every time she heard herself addressed as Lady Carlston. But she stifled the smile and looked at the servants. \"I wanted to talk to you all. Won't you please sit down?\" She took a seat at the large table, knowing the others would never sit while she stood.\n\nThe servants seemed confused and looked to Thompkins, who gave a slight nod indicating they could sit.\n\nPenelope looked around the table. \"I know you've all heard about the terrible accident yesterday.\"\n\nEveryone nodded.\n\n\"Well, we were hoping you could help us.\"\n\nThompkins coughed discreetly. \"We will, of course, be happy to assist in any way we can.\"\n\nShe looked around and saw confusion. \"Perhaps you could tell me anything you saw that seemed unusual.\" Penelope sat back.\n\nThe servants looked at each other. One of the housemaids squirmed in her chair.\n\nMrs. McDuffie smiled at Penelope. \"Well, I don't know much that'll 'elp. We didn't 'ave much to say to the little tart.\"\n\n\"Mrs. McDuffie,\" Thompkins said.\n\nPenelope smiled and help up a hand. \"It's okay, Thompkins. I hate to admit it, but we didn't like her very much either.\" She leaned toward Mrs. McDuffie. \"Please go on.\"\n\n\"Well, we didn't really get to know much about 'er. Thought she was too good for the likes of us.\"\n\n\"How do you mean?\" Penelope asked.\n\nFlossie leaned forward. \"It were the way she talked, ma'am.\" Flossie put her nose in the air and looked down it. \"Zis is the way Her Royal Highness prefers her coffee.\"\n\nThe others snickered.\n\nJim, the footman, joined in, \"I tried to talk to her when she first arrived, but she give me the cold shoulder and made it clear I wasn't good enough to shine her shoes.\"\n\n\"And our Jim is one of the best bowlers in England.\" Mrs. McDuffie sniffed.\n\nFrank chuckled. \"I don't think she cared much for cricket.\"\n\nFlossie poked Gladys in the ribs and gave her a stern look.\n\nGladys looked up and took a deep breath. \"I tried to talk to her too, just to be friendly. I offered to help her unpack her things, but she said she didn't need to unpack. Said she wouldn't be staying here long. She was going to be rich and moving into her own mansion where she would be the mistress and have her own servants.\"\n\n\"Pshaw.\" Mrs. McDuffie snorted.\n\n\"Did she really? I don't suppose she mentioned how she was planning to do this?\" Penelope asked.\n\nGladys shook her head. \"I'm sorry, m'lady. She didn't say how. She just said she'd been watching her mistress and copying her mannerisms. She\u2014\" Gladys's face flushed.\n\n\"It's okay, Gladys. You can tell me what she said.\" Penelope smiled.\n\n\"Well, she said if an American divorc\u00e9e could snare a king, then she should at least be able to catch a lord,\" Gladys said.\n\nThompkins gasped.\n\nMrs. McDuffie snorted again.\n\n\"I don't suppose she mentioned which lord she had in mind?\"\n\nGladys shook her head. \"No, ma'am.\"\n\nFlossie looked uncomfortable but took a deep breath. \"I think I know, miss.\" She looked at Thompkins, who nodded. \"Well, I told Mr. Thompkins about seeing Lord Chitterly coming out of the duchess's room.\"\n\n\"Yes. We know about that,\" Penelope said.\n\n\"But it weren't the duchess's room at all. It were her room, Rebecca. She and the duchess had switched rooms. I went in to clean and I saw her, Rebecca, I mean. She weren't wearing no clothes.\"\n\nMrs. McDuffie gasped. \"The little trollop.\"\n\n\"But I don't understand,\" Penelope said.\n\n\"She said the duchess wanted the room next to that German bloke, and her mistress didn't care that the other room was smaller. So, they switched.\"\n\n\"So Lord Chitterly had been with Rebecca and not the duchess?\" Penelope asked.\n\nFlossie nodded. \"Yes, m'lady.\"\n\nPenelope nodded. \"Very interesting.\"\n\nWhile Penelope was downstairs questioning the servants, Victor headed to Scotland Yard to talk to Detective Inspector Covington. He'd never been to Scotland Yard before and found the technology fascinating. Everywhere he looked there was something new and interesting. He made his way to the detective's desk.\n\nThey shook hands. \"My God, man, I had no idea half this stuff existed. It's no wonder the Yard has the reputation they do.\"\n\nDetective Inspector Covington smiled. \"The twentieth century technology is good, but nothing beats good old fashioned police work.\" He indicated Victor should sit.\n\nOnce both men were seated, he continued, \"But I don't think you drove all the way out here just to check out the latest technology at the Yard.\"\n\nOn the drive from the country, Victor had rehearsed several excuses to explain his presence. Face to face with the detective, none of them seemed believable. \"Actually, old man, I wanted to know if you've determined which gun killed the maid.\"\n\nDetective Inspector Covington looked around. \"I'm not working on that case. I've been pulled to work on a case of poaching in Shrewsbury.\"\n\nVictor stared. \"Poaching?\"\n\n\"Yes. Poaching. Serious business, poaching.\"\n\n\"But surely, murder takes precedence over poaching?\" Victor couldn't hide the astonishment in his voice.\n\nDetective Inspector Covington looked around again. He pulled a folder from a drawer and placed it on top of his desk. \"Well, you know poaching is a hanging offense. It threatens the very seat of the British government.\" He winked. \"Nothing more important than finding poachers.\" He tapped the folder and slid it toward him. \"Care for a spot of tea?\"\n\nVictor nodded. \"Yes. I believe I would.\"\n\nDetective Inspector Covington stood. \"Afraid the canteen is clear down to the other side of the building. It'll take me at least ten minutes to get the tea and get back. Think you'll be okay?\"\n\nVictor nodded.\n\nDetective Inspector Covington ambled down the hall.\n\nAs soon as he was gone, Victor grabbed the folder and opened it.\n\nVictor left before Detective Inspector Covington returned.\n\nWickfield Lodge was a large U-shaped mansion, with the Marsh family's bedrooms on one side of the U and the guest rooms on the other, with a long hallway in between. Lord William needed to talk to Lord Charles. In his present condition, there was no way he could make it to Chitterly's room. Daphne volunteered to help.\n\nShe stood in front of the guest room now occupied by the Chitterlys. Just as she raised her hand to knock, one of the nearby doors opened. She turned and saw James. Her first instinct was to smile until she remembered this wasn't the room set aside for the Duke of Kingfordshire. No, this wasn't his room. She thought for a moment and remembered the arrangements her aunt had shared. This was Virginia Hall's bedroom.\n\nHe saw her and had the decency to blush. \"Daphne. What are you doing here?\"\n\n\"Just getting Lord Charles. My uncle wants to talk to him.\" She knocked on the door.\n\n\"I can explain.\"\n\n\"No need.\"\n\nCharles Chitterly opened the door. \"Yes?\"\n\n\"My uncle was hoping you would join him for tea. He's had an attack of gout and is confined to his bedroom at the moment, but if you don't mind following me, I'll be happy to lead the way.\"\n\nLord Charles smiled. \"I would like nothing better, my dear.\"\n\nDaphne nodded at James as she walked arm in arm with Lord Charles down the hall.\n\nChapter 17\n\nMonday morning I had to get back to my routine. Fall break was over and the twins were back in school. Nana Jo promised to swing by at noon to give me a few hours break. The morning went by slowly, and I was soon missing both Nana Jo and Dawson a lot. Not just because of the help they provided, but they were also good company.\n\nWe'd gone through the stockpile of baked goods Dawson had left, so I ordered quite a few things from a South Harbor bakery that delivered. The baked goods were tasty but lacked the extra special oomph Dawson's had. Admittedly, I might be slightly biased.\n\nNana Jo arrived promptly at noon. I went to the bank and deposited the check I'd been carrying around all weekend. Then I swung by a fast-food restaurant and grabbed a sandwich. As I was driving back to the store, I spotted a bright red Ford Escape on the edge of the parking lot of a large car dealership. Before I realized what I was doing, I pulled into the lot. As soon as I got out of the car, salesmen swarmed around me like locusts. My personal feeling is there is a special level of hell where the inhabitants are continually haggling with salesmen and buying and selling cars. It's just above the level where your penance for a life of evil is spent buying and selling real estate.\n\nLeon had taken care of things like haggling with car salesmen. However, I was a big girl and refused to be intimidated.\n\n\"I'm just looking!\" I shouted. I pointed to one of the salesmen. \"What's your name?\"\n\n\"Bob.\" He stepped forward and reached out his hand. He opened his mouth to start his sales pitch but was halted when I held up my hand.\n\n\"Bob. I want to look at SUVs. I don't want to be bothered. Don't follow me. Don't help me. When I need you, I will come find you. Understood?\" I looked around at all of the salesmen. They all nodded and backed away.\n\nThe new models were on one side of the lot and the other side had a variety of used cars. The red Escape was last year's model and had thirty thousand miles, but it had a sunroof, leather seats, and tons of other bells and whistles I didn't even know existed. I spotted Bob about three aisles over and waved for him to join me.\n\n\"You were waving for me, right?\" he asked, out of breath from his quick jog.\n\n\"Yes. I have questions.\"\n\nBob answered all of my questions and went inside to get the keys so I could take a test drive. I showed him my driver's license, and he came back with a license plate, which he stuck in the back window and gave me a large key fob. I stared at it. There was no key.\n\nBob laughed and explained all I had to do was push a button and, as long as I had the fob within a certain range, the car would respond. The doors unlocked. The motor started. It was magic.\n\nI drove the car in a big loop around the block, which included a short jaunt on Interstate 94. I was amazed at how quiet the car was and how smooth the ride. Bob talked the entire test drive, explaining every feature he thought would sell me on the car. Little did he know, I was sold at intermittent wipers and automatic lights. I would have been happy with lights that dinged when left on. These lights actually turned themselves on and off, like magic.\n\nBack at the car lot, Bob asked if I had a trade-in. I escorted him to my SUV. I was insulted when he asked if I wanted to get it repaired before I got my trade-in estimate.\n\n\"If I wanted to get the car repaired, I wouldn't be buying a new one.\" I pointed out that all of the pieces were there and showed him the knobs in the ashtray.\n\nI looked at my watch. I needed to get back to the bookstore. Bob did everything he could to get me to stay, including a suggestion that I put a deposit down on the vehicle to make sure no one bought it before I returned. I declined. If it was gone in a few hours, then it wasn't meant to be.\n\nAt the bookstore, Nana Jo took one look at me and knew something was up. I had planned to surprise her when I arrived with the car, but she always knew when something was going on with me.\n\n\"It's about time,\" she teased when I told her about the car. \"You can go back. I can handle things here.\"\n\nThe store wasn't busy. In fact, she was reading an Agatha Christie collection of short stories, The Mysterious Mr. Quin. However, now that I'd told her, I wanted her to see it before I bought it. I wanted her approval.\n\nThe hours until closing passed very slowly. When it finally did arrive, we dashed off to the car lot. My red SUV was still parked where I'd left it, and I breathed a sigh of relief.\n\nNana Jo heartily approved. I'd completed some of the paperwork for financing before I left and when we returned, I learned the trade-in amount for my CRV and the rate the dealership offered for financing. If Bob thought I had been tough on him earlier, he learned I was a lightweight compared to Nana Jo. She pulled out her iPhone and went to a few websites. By the time she finished negotiating, I got the car for five thousand dollars less and 2 percent better interest rate. We transferred my belongings from the Honda, and my heart tugged the slightest bit as I took one last look. I smiled and gave her a pat. \"You served me well.\"\n\nI signed a ton of papers, shook hands with Bob, and drove my new SUV off the lot.\n\n\"Open the sunroof. Let's go pick up the girls and go for a spin,\" Nana Jo said.\n\nBefore we could head to the retirement village, I got a call from Dawson. The rings vibrated throughout the car's stereo system.\n\n\"What on earth . . .\" I said.\n\nNana Jo laughed. \"You have Bluetooth. I synched your phone with the car while you were talking to Bob. Touch that button there.\" She pointed.\n\nI pushed the button and Dawson's voice came through the speakers. \"Someone just tried to kill my dad.\"\n\n\"Oh my God. Are you okay?\"\n\n\"Yeah. I'm okay.\"\n\n\"Call the police. We'll be there in thirty minutes.\"\nChapter 18\n\nI didn't remember parking the car at the hospital. After disconnecting with Dawson, my next memory was giving him a big hug. Dawson was about a foot taller than me and probably a hundred pounds heavier, but when I put my arms around him, he bent down, put his head on my shoulder, and wept. I held him with everything I had. A few minutes, hours, who knew how long that embrace lasted, but I knew at that moment Dawson Alexander would forever be a part of my family, and I would do anything to protect him.\n\nWhen Dawson was composed, he straightened up and I got a good look at him. He looked haggard. There were dark circles under his eyes.\n\nNana Jo gave him a fierce hug, then pulled his face close. She stared into his eyes as though she were looking into his soul. \"You look terrible. You need a good meal and sleep.\"\n\nDawson kissed her on the cheek. \"I'm fine, Mrs. T.\"\n\n\"What happened?\" I asked.\n\nHe ran his hand through his hair. \"I was in the lounge, lying on a sofa watching television.\" He lowered his eyes. \"I should have been watching him.\"\n\n\"You can't blame yourself for this.\" I rubbed his shoulders. I hoped he felt reassured, although I doubted it. I'd gone through similar feelings of guilt when my husband died in the middle of the night. I should have been there. I should have single-handedly fought back death. I should have stayed by his side and watched him night and day. I should have forced him to go to the doctor for regular checkups years earlier. Regret was the survivor's burden. Like cancer, it got in your mind and ate away your common sense and reason. \"Your dad's still alive. Focus on that for now.\"\n\nDawson took a deep breath. \"One of the nurses came in and said I had a package at the front desk.\"\n\n\"That's odd. Don't they bring packages up to the room?\" I asked.\n\n\"I thought so too, but the nurse said he told her it was from you. He said the volunteers were gone for the day. If I came down to the lobby, I could pick it up now. Otherwise, it would be delivered tomorrow. I thought it was another pizza like the one you sent last night.\"\n\nI frowned. I hadn't sent Dawson a pizza, but apparently someone else had.\n\n\"I went down to the lobby. No one knew anything about a package.\" He looked up and pointed. \"That's when I saw him.\"\n\nWe followed where he was pointing. That's when we noticed Virgil Russell sitting at a desk in a small room off the nurses' station. He was covered in blood and was holding a bloody towel to his nose with his head tilted back.\n\n\"Virgil Russell?\" Nana Jo asked.\n\nDawson nodded. \"He claimed I sent him a note, but I didn't. So we rushed back up here and all hell had broken loose. Lights and alarms were going off in my dad's room. There were tons of people in there working on him.\" He looked at his hands. \"That's when I hit him.\"\n\n\"Good,\" Nana Jo said.\n\n\"Nana Jo, you shouldn't encourage violence.\"\n\nShe sniffed.\n\nThe elevator opened and Stinky Pitt and a uniformed policeman stepped out and walked toward us.\n\nOne of the doctors came out of the room and said a few words to the two officers. The officer took up his position in front of A-squared's door. Detective Pitt and the doctor joined us on the sofa.\n\n\"Your father had a really close call. If he hadn't been in a coma, he'd be dead now. In fact, I'm not sure why he isn't. He's a fighter. We'll know more in the next twenty-four hours. All we can do now is wait.\" The doctor left.\n\nDetective Pitt wanted to know what happened. Dawson repeated what he'd said to us. He asked a few questions and then Detective Pitt went to talk to Virgil.\n\nNana Jo wandered near the nurses' station outside the room where Detective Pitt and Virgil were talking. When I saw what she was doing, I told Dawson to wait there and I joined her at the desk. Unfortunately, when Detective Pitt turned and saw us standing nearby, he closed the door.\n\nOne of the nurses who had been sitting behind the desk looked up. She had a sympathetic face, and I'd seen her before when I visited. She winked at me, put a finger to her lips, and motioned for us to follow her. She led us behind the nurses' station, around a corner. There was a small room, barely bigger than a closet. There was a refrigerator and microwave and a small sliding door, which looked like a concession stand opening. The door was slightly ajar and we could hear everything in the room where Detective Pitt and Virgil Russell were sitting. We mouthed the words thank you. Our friendly nurse smiled and quietly returned to her desk.\n\n\"Come back to finish the job, did you?\" Detective Pitt asked.\n\n\"I never laid one hand on him,\" Virgil said in a muffled voice.\n\n\"What brought you here tonight?\"\n\n\"I got a note from Dawson to meet him here. He said he wanted to talk about Melody.\"\n\nNana Jo and I exchanged a glance.\n\n\"What note?\"\n\nWe heard fumbling and then the sound of a note being opened.\n\n\"Did you ask him about this?\"\n\n\"I never got the chance.\"\n\nWe heard the sound of a door opening and hurried back around the nurses' station. We took a roundabout route back to the sofa where Dawson was sitting. Detective Pitt stood nearby.\n\n\"Where have you two been?\" he asked.\n\n\"Ladies' room. I've got a weak bladder,\" Nana Jo said. \"It comes with old age.\"\n\n\"You got a weak bladder too?\" He looked at me.\n\n\"Do you have a weak bladder is the correct question.\" I stared at the detective. \"And, the answer is no. I do not.\"\n\n\"Women tend to go to the bathroom in pairs,\" Nana Jo added. \"Surely you've noticed, Stinky Pitt.\"\n\nA slight rise of color went up the detective's neck. Whether it was anger for the grammar lesson or frustration with Nana Jo for using his childhood nickname, I couldn't hazard a guess.\n\nHe turned to Dawson and held up a note. \"Did you send this?\"\n\nDawson reached out a hand, but Detective Pitt held it out of reach. \"Just look. Don't touch. Probably a slim chance we'll find any prints, but it's still evidence.\" He held the paper from the corners so Dawson could look.\n\nI strained to read the note, but Nana Jo was quicker and pulled out her iPhone and snapped a picture before Detective Pitt knew what was happening. She stretched the photo, and I looked over her shoulder and read the note.\n\nIf you want to know what happened to Melody,  \ncome round to the surgery at nine.\n\nThe note was on hospital stationery but didn't have a signature. Something about the note struck me as off.\n\n\"What's wrong?\" Nana Jo asked.\n\n\"I don't know. Something just seems . . . odd about that note.\"\n\n\"I know. I was thinking the same thing, but I can't figure out what it is.\"\n\n\"Well, I wish someone would tell me what it is,\" Detective Pitt muttered. \"Did you write this?\" He looked at Dawson.\n\nHe shook his head. \"Never seen it before.\"\n\n\"When did he get it?\" I asked.\n\nDetective Pitt looked puzzled, then turned and walked to the room where Virgil sat. He said a few words to him and then returned.\n\n\"Said he found it on the windshield of his car.\"\n\n\"But, I haven't left the hospital since my dad was admitted,\" Dawson said.\n\n\"How did he know A-squared was here? I haven't seen it in the papers.\" Nana Jo asked.\n\n\"Did you tell anyone?\" Detective Pitt asked Dawson.\n\n\"No. I haven't left the hospital.\" Dawson gritted his teeth.\n\n\"I might have mentioned it to a few people.\" I severely hoped I wasn't responsible for this attack.\n\n\"Who?\" Detective Pitt didn't look at all sympathetic.\n\n\"Well, I told Jenna and Chris and Zaq. I'm pretty sure I told Emma and Jillian. The girls from the nursing home and Professor Quin.\"\n\n\"Why didn't you just put an ad in the newspaper? It would have been faster.\"\n\n\"Stop. Can't you see she feels bad enough as it is?\" Nana Jo put a comforting arm around my shoulders.\n\nDetective Pitt grunted. \"Was Virgil Russell around when you told any of those people?\"\n\nI thought for a moment. \"He might have overheard when I told Professor Quin at Melody's memorial service.\"\n\nDetective Pitt asked a few more questions, then he got Virgil, and asked him to accompany him to the police station to answer questions. \"You can even file a report against Mr. Alexander for assault, if you'd like.\"\n\nNana Jo scowled at Stinky Pitt. I held onto Dawson's hand.\n\nVirgil Russell stared at Dawson, who didn't look contrite or repentant in the least. In fact, he looked like he'd like nothing better than to take another swing at him if given the chance. He and Dawson stared at each other like two boxers sizing each other up before a fight. During that exchange, something happened. Maybe Virgil realized Dawson was younger, bigger, and stronger than him. \"Nawh, I won't be filing no police report.\"\n\nStinky Pitt escorted Virgil to the elevator.\n\n\"What just happened there?\" Nana Jo asked as the elevator closed behind the two men.\n\n\"I have no idea.\"\n\nNana Jo and I offered to stay at the hospital, but Dawson urged us to go home. I was prepared to camp out, despite his assurances he would be fine.\n\n\"I appreciate you both for driving all the way here, but I'm fine. No one can try anything now. The hospital is on alert. There's a cop standing guard, and I'm not going to be fooled into leaving again.\"\n\n\"But we want to help,\" I said.\n\nDawson looked very intense. \"Then find out who killed Melody and who tried to kill my dad.\"\n\n* * *\n\nNana Jo and I tried to figure out what was bothering us about the note during the drive home. Neither of us could put our finger on it. She sent me the picture of Virgil's note so I could study it later. I tried to figure out what was nagging at my mind. Was it possible A-squared really knew who killed Melody? Whoever killed Melody tried to kill him. Why did he wait until now to make another attempt?\n\nIt was late or rather early morning by the time we made it home. Oreo and Snickers were knocked out. Not even turning on the lights woke them. I literally had to pick them up to take them out to take care of business. They immediately got back in their beds when they were done, but I didn't even try to sleep.\n\n\"Wallis claims she and Count Rudolph were talking all night.\" Lady Elizabeth pursed her lips as she continued knitting.\n\n\"Talking? Really, and they had to do this in his bedroom?\" Penelope asked.\n\n\"That's what she claims, dear.\"\n\nSomething in her aunt's voice made Penelope stare at her in disbelief. \"Don't tell me you believe her?\"\n\nLady Elizabeth knitted a few stitches and then paused. \"Actually, I do.\"\n\n\"But that's crazy. Why would she be in the count's bedroom? If they wanted to talk, there are over a hundred rooms in this house where they could have 'talked.' \"\n\n\"True, dear, but you have to remember, Wallis was trying to negotiate some type of agreement. They needed a place that was private.\" Lady Elizabeth looked at her family. \"I'm not saying it was wise, but I did believe her when she said she wasn't having an affair with Count Rudolph.\"\n\n\"Despite the fact that she's practically thrown herself at him the entire time she's been here?\" Penelope asked.\n\n\"Yes. Wallis isn't beautiful. She's a flirt and she knows how to use her sex appeal to attract men. But take a look at the type of men she attracts.\"\n\n\"I don't think I understand you, dear?\" Lord William looked at his wife as he sat in his bed with his leg propped up on pillows.\n\n\"I think I know what you mean,\" Daphne said from her seat near the window. She had a large drawing pad on her lap and appeared to be sketching. \"She seems to flirt with men who aren't very bright and who aren't . . . very manly.\"\n\n\"Exactly.\" Lady Elizabeth smiled at her niece. \"Let's face it, David isn't exactly known for his brains. He's always been a bit spoiled,\" she said thoughtfully. \"He liked women, dancing, drinking, and having fun. He was a playboy.\"\n\nVictor stood against the wall and smoked. \"He was the king. He could have had any woman he wanted.\"\n\nLady Elizabeth picked up her knitting. \"And who does he choose? A married woman with two living husbands. There's no way he believed he would be allowed to reign as the head of the Church of England.\"\n\nDaphne looked at her aunt. \"You think he married her to escape being king?\"\n\n\"Yes, I do.\"\n\n\"But why did she marry him?\" Penelope asked.\n\nLady Elizabeth knitted. \"I think Wallis is one of those women who's always attracted to the wrong type of men. She doesn't want a strong man. She wants a man she can manipulate and control, someone who will never love her, not in the real way a man loves a woman.\" Lady Elizabeth blushed slightly.\n\nLord William fumbled with his pipe. \"Well, I'll be.\"\n\n\"In David, she found someone in need of a mother figure. In Count Rudolph . . . well, I suspect she has found someone who is struggling with his own sexuality.\"\n\n\"My word,\" Lord William said.\n\n\"He is a bit of a dandy,\" Victor said.\n\n\"That explains why he never made a pass at me,\" Daphne said. \"Most men do, but Count Rudolph never did.\" She looked very serious.\n\nEveryone stared at Daphne for several seconds until they noticed the slightest twitch around her mouth before she broke into a laugh.\n\nEveryone laughed until a knock at the door caused them to stop and stare.\n\nPenelope looked to her aunt, who gave her a slight nod. She walked to the door and opened it slightly. When she looked out, she smiled and opened the door wider. \"James, at last. I thought you'd never get here.\"\n\nThe duke entered the room and Penelope closed the door behind him. Inside, he looked around. His eyes lingered longest on Daphne by the window. \"Sorry I'm late. Have you figured out who the murderer is?\"\n\n\"We were just waiting for Lady Elizabeth to tell us,\" Victor joked. \"Take a seat.\"\n\nLady Elizabeth shook her head and continued to knit. \"I have no idea who the murderer is. I can't even tell who the intended victim was.\"\n\nLady Penelope recapped her conversation with the servants.\n\n\"That goes along with what I learned from Lord Charles,\" Lord William said. \"Darned fool admitted he had a fling with the maid. Called it a bit of fun.\" He pursed his lips as though he had just tasted a bitter lemon. \"Got a nasty shock the next morning. The girl wanted him to divorce his wife and marry her.\"\n\n\"You mean she tried to blackmail him?\" Victor asked.\n\nLord William nodded. \"That's about the size of it. Threatened to tell his wife.\" He chuckled.\n\n\"Well, I don't think that's funny at all.\" Lady Elizabeth frowned at her husband. \"Looks like he had a motive to kill the girl.\"\n\n\"Except, Lord Charles had no intention of leaving his wife. She's got all the money.\"\n\nLady Elizabeth nodded. \"I thought so. His family has that large estate in Sussex, but no money. He went to America and came back with an American heiress.\"\n\n\"He'd have to go to America to find someone who'd marry him,\" Daphne added. \"His reputation in England was abominable.\"\n\n\"Yes. He was definitely a womanizer,\" Lady Elizabeth said.\n\n\"He could have killed her to keep his wife from finding out,\" Penelope said.\n\n\"She'd have to be deaf, dumb, and blind not to know her husband was a womanizer,\" Daphne said. \"And I think Lady Abigail is a very intelligent woman.\"\n\nEveryone stared at Daphne.\n\n\"I spent time with her after the shoot. She's very intelligent. Graduated from Smith College in the States.\"\n\n\"If she's so smart, why'd she marry an old bore like Charles Chitterly?\" Penelope frowned.\n\nLady Elizabeth smiled. \"Well, she's not very attractive. Maybe she didn't have a lot of other op tions.\"\n\n\"Exactly,\" Daphne added. \"She's very ambitious. I believe she plans to see that Lord Charles advances, politically.\"\n\n\"A woman with that type of ambition might not want to have her plans ruined by a French maid,\" James said.\n\n\"Didn't you say she was a crack shot who won shooting contests in the States?\" Penelope asked.\n\nDaphne nodded. \"Yes, but I don't think it was her.\"\n\n\"Why not, dear?\" Lady Elizabeth asked.\n\nDaphne looked around as though looking for the right word. \"I don't think she would shoot her in the marsh like that. She'd confront the girl and tell her to, pardon my language, bugger off.\"\n\nLady Elizabeth smiled at her niece. \"I think you're right. That's the impression I have of her as well.\"\n\n\"Well, I had a conversation with Detective Inspector Covington today. You know he's been pulled from the case to find poachers?\"\n\n\"Poachers?\" Penelope stared at her husband. \"You're joking, right?\"\n\nEveryone had a surprised look on their face, except James and Lady Elizabeth. \"Why do I get the feeling the two of you already knew about this?\"\n\nLady Elizabeth pulled yarn from a ball. \"I can't say I knew anything about poachers; however, I did suspect Bertie would do something.\" She sighed. \"The king doesn't want attention drawn to the case. I suspected he might use his considerable influence\u2014\"\n\n\"To delay justice,\" Penelope said.\n\n\"To buy us some time, dear,\" Lady Elizabeth said.\n\nJames stared at Lady Elizabeth with a look of respect. \"Three days. I just talked to Budgy, ah . . . I mean Chief Inspector Buddington. That's how long we have to solve this before the Met comes in with everything they've got.\"\n\n\"I looked through a report Detective Inspector Covington had on his desk. The maid wasn't shot with any of the guns belonging to Lord William.\"\n\n\"Thank God,\" Lord William said.\n\n\"But if it wasn't one of our guns, then whose gun was it?\" Lady Elizabeth asked.\n\n\"No idea. The constables were pulled off the case before they could search the place,\" Victor said.\n\n\"We could search for it,\" Penelope said.\n\n\"I'm afraid that gun will be long gone by now, dear,\" Lord William said.\n\n\"Agreed. I think it unlikely the killer would risk hanging onto the murder weapon.\" Lady Elizabeth shook her head. \"Pity we didn't think to have the guests searched sooner. But there might be something. . . Yes, we might be able to find out who had a gun before the killing and doesn't have one now.\" She knitted for a few moments. \"I'll have a word with Thompkins about it. Now, Daphne, were you able to make any progress on your assignment?\"\n\nDaphne walked over to her aunt, who was sitting in a chair next to her husband's bedside. Daphne stood in between her aunt and uncle and opened her drawing pad.\n\nThe others gathered close for a better view.\n\n\"I worked out who was where during the shoot.\" She pointed at the sketch pad, which had a timeline and circles indicating who was in each group at what time. \"J\u00f3zef Lipski and Virginia Hall are the only ones who are completely unaccounted for at the time the maid was shot.\"\n\nThey stared at the sketch pad and asked a few questions, which Daphne answered.\n\n\"Daphne, that's wonderful. You did a tremendous job.\" Lady Elizabeth smiled at her niece.\n\nEveryone congratulated Daphne. The only person who seemed less than pleased was James. In a brooding silence, he walked to the window and stared out.\n\n\"James?\" Lady Elizabeth asked. \"What's wrong?\"\n\n\"Nothing.\" James shook himself. \"Virginia Hall is beyond reproach. I know she didn't kill that maid.\"\n\nDaphne sniffed.\n\n\"How can you be sure?\" Lady Elizabeth stared at James.\n\n\"I just know. I need you to trust me.\"\n\nThey looked at each other, everyone except Daphne, who refused to make eye contact.\n\n\"Well, of course we trust you,\" Lady Elizabeth said. \"But can you tell us why you're so sure?\"\n\nJames stared at Daphne as though willing her to make eye contact, but she merely folded up her sketch pad and returned to her seat by the window. James shook his head. \"I'm afraid I can't at the moment, but I promise I'll explain soon.\"\n\n\"Well, if Miss Hall is out, that just leaves Lipski,\" Victor said.\n\nThe group agreed that Victor and James would question J\u00f3zef Lipski immediately following tea.\n\nJames released a heavy sigh.\n\n\"What's the matter?\" Penelope asked.\n\n\"Nothing. I was just thinking about what this will mean if it turns out J\u00f3zef Lipski is indeed the murderer.\" James looked out the window for several minutes. \"This incident could be twisted to justify German aggression toward the Pols and more German expansion in Europe. No one wants another war with Germany, but we won't be able to avoid it forever.\"\n\nEveryone stared in shocked silence.\n\n\"You think war is inevitable?\" Victor asked softly.\n\nJames nodded. \"I do, and I'm not alone. Other prominent people believe Poland is critical in this political chess game Germany's playing.\"\n\nLord William puffed on his pipe. \"Dash it all. I hope it doesn't come to that. I hope you're wrong. If it does, well . . . we beat the krauts once, we can do it again.\"\n\nJames lowered his head. \"The last war took a terrible toll on England and her allies. A lot has changed since the Treaty of Versailles. Advancements. Advance ments in weapons, airplanes, and submarines. Germany's stronger now.\" His eyes lingered on Daphne. \"This could be the match that lights the flame of war. And once that flame is lit, England will never be the same.\"\n\nChapter 19\n\nThe fact I had yet to go to bed did not prevent Oreo from needing to go potty at three thirty in the morning. Sure, I'd let him out just two hours earlier when I got home, but his biological clock was set. Snickers rolled her eyes and growled when I suggested she join us. I couldn't say I blamed her, so I let her sleep in peace.\n\nThis time I was tired enough to sleep. I must have slept through my alarm clock because the next thing I remembered was an eight-pound weight landing on my chest. Snickers was fourteen, and I've had her since she was six weeks old. You'd think by now I would have learned not to open my mouth. Unfortunately, I was a slow learner.\n\n\"Eww.\" I rolled over and dislodged the poodle who'd just stuck her tongue in my mouth. I used the blanket to wipe out my mouth.\n\nSnickers simply jumped off the bed and sat by my bedroom door.\n\nI glared at her. \"Do you know what time it is?\"\n\nShe yawned, stretched, and then began to lick herself.\n\nJust when I thought nothing could make me feel dirtier than I did already. I put a pillow over my head and contemplated ignoring her, but my bladder wouldn't cooperate. I stretched and then got up.\n\nI let the poodles out and then took care of business. By the time I got out of the shower, I got a whiff of my two favorite smells, coffee and bacon. My nana was my hero.\n\nThe day went by relatively quickly. We had a steady stream of customers and I didn't have much time to think about Virgil, A-squared, or Melody. Just before closing, Frank Patterson stopped by.\n\n\"Frank, what a pleasant surprise,\" Nana Jo said in a voice that sounded like she wasn't the least bit surprised.\n\nI should have guessed she was up to something when I noticed the goofy smile on her face when she came back from lunch.\n\n\"Finish all those books already?\" I asked.\n\n\"Not yet. I was wondering if maybe you'd be free for dinner.\"\n\nI smiled. \"Thank you, but I'm afraid I can't. Tonight I'll be hanging out with the girls. It's ladies' night.\"\n\n\"Pishposh,\" Nana Jo interrupted. \"Why don't you join us at the casino tonight?\"\n\nI stared at Nana Jo.\n\nFrank Patterson intercepted the look. \"Well, I don't want to interrupt.\"\n\n\"You won't be interrupting at all,\" Nana Jo said. \"Meet us at the Four Feathers Casino at eight.\"\n\n\"If you're sure it's okay.\" Frank looked at me.\n\nI smiled. \"Of course it's alright. Why don't you meet us in the lobby in front of the large fireplace?\"\n\nFrank agreed and left.\n\nI stared at Nana Jo, who seemed intent on dusting a bookshelf. \"Why do I suspect you arranged that?\"\n\n\"I have no idea what you're talking about.\" She turned to face me. \"But if you didn't want him to come, you could have said something.\"\n\nI smiled. \"You're incorrigible.\"\n\nTuesdays was ladies' day at a lot of local shops. The girls liked to stock up on paper towels, vitamins, and whatever else they could get at bargain prices. I normally would have gotten a half-priced oil change and car wash, but since I'd just bought my car yesterday, I wouldn't need to take advantage of that discount yet. We often dined at Randy's Steak House on Tuesday and then hit the bars for half-priced drinks. However, in light of my recent windfall, the girls were anxious to try their luck at the casino.\n\nThe girls were surprised and gushed appropriately at my new wheels. The hands-free back door was a huge help when loading up after a day of ladies' day shopping.\n\n\"We're ridin' in style now,\" Ruby Mae said.\n\nOn the ride to the casino, we filled the girls in on the latest from Dawson and Virgil Russell.\n\nIrma still doubted Virgil's guilt, but the others felt he was the most likely suspect.\n\nI dropped the girls at the front of the casino and then drove to the back of the parking garage. I parked as far away from potential scratches as possible. When I arrived in the lobby, Frank Patterson was waiting. He had a single rose, which made me smile.\n\n\"Where's Nana Jo?\"\n\n\"They said they'd get a table.\"\n\nFrank had on a pair of dark wash jeans and a sweater. He looked freshly groomed and smelled of shampoo. I liked that he wasn't wearing cologne.\n\nI was wearing one of my outlet mall finds, a silk blouse and bright red cardigan with jeans.\n\nWe walked to the buffet and I looked around until I spotted them. They were sitting at a small booth that could only hold four comfortably. The hostess was talking, but I was so surprised, I didn't hear a word she said. I made a beeline to their booth and stood looking at them.\n\nDorothy and Ruby Mae hopped up and hurried to the buffet. Irma said she needed a drink and flagged down a waiter. Nana Jo was suddenly really thirsty and drank an entire glass of water.\n\n\"Well?\" I stared at my grandmother.\n\n\"Sorry, dear, they didn't have any larger tables. I guess you two will have to find a table alone.\"\n\nI looked around at all of the empty tables and booths, which were clearly large enough for all six of us.\n\nThe hostess followed us and was at my side when I turned around.\n\n\"Come on.\" I followed the young girl to a table for two in a corner.\n\nOnce we were seated and beverage orders taken, I looked across at Frank.\n\nHis lips twitched and he did his best to hide a smile. Eventually, our eyes met and we both burst into laughter.\n\n\"Your grandmother is an exceptional woman.\"\n\n\"If by exceptional you mean meddling busybody, then you're right.\"\n\nWe filled our plates at the buffet and talked for hours. I was surprised at how much we had in common. There wasn't an awkward moment. Frank had a knack for telling interesting stories. He'd retired from the military earlier this year and was living his dream of living in one place and running a bar. \"Kind of like Cheers, a place where everyone knows your name.\"\n\nIt usually took a while before I opened up and shared about my personal life, but Frank was easy to talk to and he listened. He seemed interested in everything from Leon to Oreo and Snickers to Dawson.\n\n\"Your grandmother told me about your investigation. I'd like to help if I can.\"\n\nI stared. \"I'm not sure how you can. It looks like the police might have their killer.\" I shared what I knew about Virgil Russell and the incident at the hospital. Frank asked who knew about A-squared's accident and which hospital he was in.\n\n\"I have some friends who are very connected. Would you mind if I asked them to look up some of the names you gave me?\"\n\n\"Of course not.\"\n\nWe talked about Melody being a grifter, the special program she was in, her sister Cassidy, Virgil, and the mysterious person Melody thought was trying to muscle in on her mark.\n\n\"And you said Virgil mentioned he knew Melody had other lovers?\"\n\nI stared. \"You think the person who was trying to muscle in on her mark was her lover?\"\n\nFrank shrugged. \"Beats me. But it seems odd. She didn't make many friends, not even with her roommate. How else would someone get close enough to her to find out about her scam?\"\n\n\"That makes sense.\"\n\n\"What makes sense?\"\n\nI looked up.\n\nNana Jo was standing by our table. \"Are you two still in here?\"\n\nI looked at my watch. \"I can't believe it's almost midnight. Where's our bill?\"\n\nFrank smiled. \"I've already taken care of it.\"\n\nI started to protest, but he simply smiled. \"I'm old-fashioned. A gentleman always takes care of the bill.\"\n\nWe met the girls in the lobby and I told them I'd get the car and pick them up. I protested, but Frank escorted me to my vehicle and held the door open for me while I got in. He said he'd check with one of his buddies on some of the names I gave him, and he'd have something by tomorrow night.\n\nI drove to the front of the casino and picked up Nana Jo and the girls, who were surprisingly quiet for the majority of the ride back to the retirement village. I expected a lot of teasing about Frank but was pleasantly amazed when none came.\n\nNana Jo stopped in the bookstore and picked up the Agatha Christie book she'd been reading earlier and then went to her room.\n\nI let the poodles out, but something was nagging at my mind. When I went back upstairs, I got on the computer and tried to focus on the clues. I read an article about free writing, where you tried to turn off your conscious mind and let your subconscious write. However, with my eyes closed, my brain kept picturing Frank Patterson and my hands kept typing Quin and Deering Vale. Perhaps writing would help my subconscious grab hold of the elusive thread.\n\nEveryone except the Duchess of Windsor and Lord William assembled in the parlor for tea. Count Rudolph and Georges Brasseur stood near the fireplace.\n\n\"I have never come to understand the British obsession with tea,\" Brasseur said. \"In France we drink wine, or cocoa, but tea . . . it is like drinking dishwater.\"\n\nLady Elizabeth blushed slightly. \"How thoughtless of me. If you would prefer something else, I'll be more than happy to have the cook prepare you something. I'm sure she has cocoa.\"\n\nJ\u00f3zef Lipski stood near the window. \"Ah, but you have not tasted the cocoa until you have tasted Polish cocoa,\" he said proudly and puffed out his chest. \"It is a special blend.\" He pulled a small box from his inside pocket. \"Perhaps you would care to try some?\"\n\nBrasseur looked as though nothing would give him less pleasure than to try the Pol's special blend of cocoa.\n\n\"Thank you so much.\" Lady Elizabeth reached out her hand and Lipski brought the container to her.\n\nShe rang a buzzer that summoned the butler.\n\n\"Thompkins, would you please ask Cook to prepare a cup of cocoa for Monsieur Brasseur and . . . Mr. Lipski?\"\n\nLipski shook his head. \"No, I only drink cocoa before bed.\"\n\nThompkins took the tin and left.\n\nLady Elizabeth continued pouring tea for the rest of the guests. Fordham-Baker declined. He was drinking what appeared to be a new bottle of scotch. The only way she could gauge was by noting the current bottle contained more liquid than the one she'd seen him with earlier. If she weren't trying to keep the maid's murder out of the newspapers, she would talk to Thompkins about restricting his alcohol. However, in light of the current situation, perhaps she should allow the man to continue drinking.\n\nWithin a relatively short period of time, the butler returned with a steaming hot mug, which he presented to the French diplomat.\n\nBrasseur accepted the cup and nodded to Lipski. He placed the cup on the fireplace mantle and continued talking to Count Rudolph.\n\nJ\u00f3zef Lipzki's face turned purple. It was clear Brasseur didn't intend to drink his cocoa.\n\nThompkins walked to Lipski and returned the tin.\n\nLipski took the tin and replaced it in his pocket and turned his back to the crowd and walked to the window.\n\nCount Rudolph, normally very quiet and taciturn, surprised the group by speaking. \"Perhaps we can have more music, or is it not acceptable to play music during the British tea?\" he asked in a very theatrical manner.\n\n\"Well, certainly, you may have music,\" Lady Elizabeth said.\n\nCount Rudolph followed Daphne to the wireless. When she found a station playing music, he further surprised the group by asking her to waltz.\n\nCount Rudolph bowed dramatically, clicked his heels, and then took Daphne by the waist and waltzed her around the room.\n\nEveryone stared in shocked silence. Virginia Hall approached Lipski, who declined dancing.\n\nThe two stood in awkward silence.\n\nWhen the dance was over, Rudolph bowed to Daphne and returned to his place next to Brasseur.\n\n\"Ah, Monsieur Brasseur, you have not drunk your Polish cocoa. You will offend Heir Lipski. You must taste it and tell us if the Polish cocoa is superior to that of the French.\"\n\nBrasseur looked as though he would rather drink petrol; however, he picked up his cup and took a sip of the liquid. He pursed his lips and frowned. Within a few seconds, he was grasping at his throat and gasping for air. He fell to his knees and then lay prostrate on the ground.\n\nJames hurried to his side. He loosened his shirt and tie.\n\nEveryone looked on in stunned silence.\n\nThe French diplomat was seized by convulsions for several seconds. He foamed at the mouth. After one exceptionally violent shake, he lay silent.\n\nJames felt for a pulse. He looked to the crowd which had gathered around. \"He's dead.\"\n\nChapter 20\n\nI woke from a dream where James Bond was running from all of Agatha Christie's protagonists. Harley Quin was leading a pack which included Hercule Poirot, Miss Marple, Tommy and Tuppence, and Parker Pyne through an English village. At the last minute, Captain America, who bore a striking resemblance to Frank Patterson, dropped from a plane and hog-tied James Bond. Apart from confusion, I felt like I'd run a marathon. Given the knot my sheets were in, I guess I had.\n\nI showered and dressed with care. Tonight I was going on a date with Professor Quin. I had butterflies in the pit of my stomach and considered cancelling. Several times I actually picked up my phone to cancel but stopped myself. Nerves? Fear? Whatever the reason, it was time to put on my big girl panties and face my fears. Just as I made the decision to move forward, I got a call from Harley. His car was in the shop.\n\n\"Would you like to reschedule?\"\n\n\"Actually, I was hoping you would pick me up. I have an opportunity to tour Purnell mansion and I'm really keen on going.\"\n\n\"Okay, sure.\"\n\nWe arranged for me to pick him up around six on campus.\n\nNana Jo listened nearby with her lips pursed.\n\nI couldn't take the silence anymore. \"What?\"\n\n\"Nothing.\"\n\nI refused to probe deeper and we worked in silence most of the day, ignoring the giant elephant in the room. Between Nana Jo's silence, a dream that wouldn't go away, first date insecurity, and the nagging question of whether or not Frank would call, I was a nervous wreck by noon. I had just decided to skip lunch when Frank arrived with a tray.\n\n\"I hope you two ladies haven't had lunch yet.\"\n\nWe acknowledged that we hadn't eaten.\n\n\"Great. I thought about flowers but decided food might be more practical.\"\n\nThe spread he brought looked delicious. There was clam chowder and chicken salad, two of my favorites, and a single rose. \"Hmm . . . clam chowder is my favorite. How did you know?\"\n\nHe smiled. \"You mentioned it last night. I had to get up early to get to the market to get the clams or I would have been by earlier.\"\n\n\"I can't believe you went to so much trouble.\"\n\n\"No trouble at all.\"\n\n\"But there are only two servings?\"\n\nHe laughed. \"How hungry are you?\"\n\nI smiled. \"Aren't you going to join us?\"\n\n\"I wish I could, but I've got to get back to the restaurant. Lunch is my busiest time and one of my cooks called out today.\" He looked disappointed. \"Rain check?\"\n\nI smiled. \"Definitely, and thank you.\"\n\nHe hurried off and Nana Jo and I ate in shifts. However, the arrival of the food helped to eliminate the tension between us.\n\nThe only event of note was a text message I received from Emma in the afternoon asking if I would be on campus anytime in the near future. I responded that I was going to be on campus by five thirty and arranged to meet her at the student union.\n\nThe rest of the afternoon we worked in a companionable silence. When it was time to leave, I gave Nana Jo a hug.\n\n\"What's that for?\"\n\n\"Because I love you and I appreciate your help and everything you do.\"\n\nShe squeezed me. \"I love you too and I'm proud of you, regardless of your taste in men.\"\n\n\"Why don't you like Harley?\"\n\nShe shook her head. \"I don't trust him. Something isn't right with him.\"\n\nI thought about that for a moment. \"I know. I feel it too, but then I'm not planning to marry him. We're just going to dinner and then touring the Purnell mansion.\"\n\n\"The Purnell mansion? You mean that mausoleum built by the House of David?\"\n\nI nodded. \"I'm not excited about it either, but Harley's writing a book about the House of David and that's where he wants to go.\"\n\n\"Hmm . . . what about what you want?\"\n\nI laughed. \"I thought you'd be glad I was actually going on a date.\"\n\n\"I am proud of you, but I also don't want to see you get hurt.\"\n\nI gave her another squeeze. \"Honestly, I don't care about him like that. I'm flattered, but . . . that's it. Besides, it's just one date. I'll be fine.\" I pulled my purse strap onto my shoulder. \"I better get out of here. I'll see you later.\"\n\nDriving my new SUV put a smile on my face. I loved the light coming through the sunroof, the satellite radio, and the gauge that told me how many miles until empty. Today, I was disappointed by how quickly the drive to MISU was from downtown. I parked away from other vehicles and walked to the student union with a joyful heart.\n\nInside, Emma and Zaq sat together in a booth. They were sharing an order of onion rings and laughing with their heads together. I hesitated for a moment, not wanting to interrupt the moment.\n\nEventually, Emma looked up and smiled. \"There you are.\"\n\nI sat down and declined the onion rings they generously offered to share.\n\n\"I didn't want to bother you, but Zaq said I should tell you about a couple of weird things that happened.\"\n\n\"Of course you aren't bothering me. You can talk to me anytime,\" I said with sincerity. I learned long ago not to get attached to my nephews' girlfriends. Just when I got to know them, they were gone, but I liked Emma. I hoped we could be friends, regardless of what happened between her and Zaq.\n\nShe smiled. \"Well, first someone broke into my dorm room.\"\n\n\"Are you okay? Was anything taken?\"\n\n\"I wasn't there and the only thing missing was my laptop, best I can figure. It happened on Sunday. Zaq and I went to the movies after the memorial.\"\n\n\"Did you call the police?\"\n\n\"Campus police. They filed a report but said I'd probably never see it again. Good thing I back up everything. Plus, I paid for insurance, so I got a replacement.\" She patted her new MacBook.\n\n\"That's strange, but it isn't really that unusual. I think thefts are common on college campuses.\"\n\n\"The weird thing isn't that the laptop was stolen. The weird thing is that it was returned.\"\n\n\"Returned? You mean the thief brought your laptop back?\"\n\n\"Not exactly. They tossed it in a dumpster. It was kind of insulting, really, like my laptop wasn't good enough for them or something.\"\n\nZaq smiled. \"I don't think that's it.\" He looked at me. \"I think they thought it was Melody's, but when they looked through it, they saw it wasn't.\"\n\n\"What makes you think that?\"\n\nZaq shrugged. \"They didn't take the television or the microwave or any of her jewelry. Plus, you guys had already packed up all of Melody's stuff, but whoever broke in, opened all her drawers and cut open the mattress.\"\n\n\"But only the one on Melody's side of the room,\" Emma added.\n\n\"Aunt Sammy, I think the burglar was looking for something.\"\n\nEmma's phone vibrated. \"That reminds me about the other thing.\" She held up her cell. \"I got another text from Melody's sister.\"\n\nI read the screen.\n\n\"I tried to forward, but it wouldn't let me. Zaq said she must have sent it so it couldn't be forwarded.\"\n\n\"Can someone do that?\" I asked.\n\n\"Depends on their e-mail system, but yes. There are systems that prevent forwarding,\" Zaq said.\n\nThe text said she thought whoever was trying to get in on Mel's scam was someone she had been having an affair with.\n\n\"I guess this means Detective Pitt is right. It must be Virgil.\"\n\n\"Didn't Cassidy know Virgil? I thought you mentioned that at the memorial,\" Emma said.\n\nI nodded. \"She did know Virgil.\"\n\n\"Plus, Virgil was already in on the scam, wasn't he?\" Zaq asked.\n\nI didn't have answers. This was odd. I told them what Virgil told us the other night at the hospital.\n\nI looked at my watch. \"I better go. I've got to pick up Harley . . . ah, Professor Quin.\"\n\nEmma looked surprised. \"Are you dating Professor Quin?\"\n\n\"I wouldn't call it dating. He's helping me with some stuff for a book, being British and everything. How was your lab Monday night?\" I asked.\n\n\"What lab?\" Emma looked puzzled.\n\n\"I thought you had a lab Monday night for Professor Quin's class.\"\n\nEmma shook her head. \"Nope, in fact he cancelled class on Monday.\"\n\nSomething in the look that passed between Zaq and Emma made me stop. \"Why? What's up?\"\n\n\"Nothing.\" Emma was a horrible liar. Her face turned red and she couldn't make eye contact.\n\n\"You can tell me. I'm not in love with the guy. He's cute, but . . . oh, I don't know. There's just something odd about him.\"\n\nEmma breathed a sigh of relief. \"I'm so glad you aren't serious about him. I think the guy is kind of creepy. He hits on his students, and there was a rumor he was having an affair with one of them.\" Emma paused before making up her mind. \"I think it was Melody.\"\n\n\"Do you have any proof?\"\n\n\"No.\"\n\n\"Accusing a professor of having an affair with a student is serious business. You wouldn't want to make an accusation like that without proof,\" I cautioned. \"But maybe I can ask him some questions tonight.\"\n\n\"Be careful, Aunt Sammy,\" Zaq said.\n\n\"I will. Don't worry.\"\nChapter 21\n\nI met Harley outside his office.\n\n\"Is this a new car?\"\n\nI smiled. \"Yes. I've only had it for two days.\"\n\n\"You should have gotten one of those new Land Rovers. They have tons more room in the back.\" He proceeded to put his bicycle into the back of my new SUV. I tried not to grimace at the sight of the mud which dropped from the tires or the scratches he'd gotten on the molding.\n\n\"There's a really lovely vegan restaurant down the road. Why don't we grab a bite before we go to the mansion?\" He tossed his backpack on the back seat and got in and fastened his seat belt.\n\nI shook off the desire to tip my hat like a chauffeur and got in the car. I followed his directions and pulled into the parking lot of a very expensive restaurant which I'd heard mixed reviews about. The d\u00e9cor was extremely high-end. The service was awful. The food was beautifully plated, but the taste was average. Harley talked a lot and seemed very excited.\n\nNear the end of the meal, he finally seemed to notice I hadn't said much. \"You're very quiet tonight. Is anything the matter?\"\n\nI shook my head. \"No. I'm fine.\"\n\nThe waiter brought the bill and Harley sat very still. I, too, sat still. After an awkward few minutes, he smiled and pulled out his wallet. \"I'm afraid I didn't bring any cash with me.\"\n\nI smiled. \"It says on the window they take credit cards. You do take credit cards, don't you?\" I asked the waiter.\n\nHe nodded.\n\nHarley's smile seemed forced, but he kept it plastered on. He looked back in his wallet and pulled out a Diner's Club card.\n\nThe waiter frowned. \"We take Visa, Master Card, Discover, and American Express. We don't take Diner's Club.\"\n\nHarley looked at me. \"I'm afraid that's the only card I have. Would you mind covering this one? The next meal is on me.\" He smiled.\n\nI wondered briefly what would happen if I claimed to have forgotten my wallet. However, the waiter loomed over my shoulder and I wasn't up for making a mad dash to the car. Instead, I pulled out my credit card and promised myself this would be the last meal Professor Harley Quin would ever receive from me.\n\n\"Thanks, love.\" He smiled.\n\nI stared back. I hadn't noticed before how crooked his teeth were.\n\nOnce the bill was taken care of, I drove to Purnell mansion. The mansion was over a hundred and ten years old and it looked every day of its age. I could imagine the building in its heyday was grand and beautiful, with a front porch that covered the entire front of the building, leaded glass windows, and ornately carved moldings. However, the paint had chipped away long ago. The floorboards of the porch sagged and there was evidence of termite damage around the outside.\n\nHarley got out of the car and grabbed his backpack from the back seat and hurried up the sidewalk. He was halfway up on the porch before he seemed to realize I wasn't behind him. He turned and looked at me. \"You coming?\"\n\nPerhaps it was too much to expect him to open my door, but I couldn't help comparing my experience with Frank Patterson. I don't need someone to open doors for me, but it sure felt nice when they did. I sighed and got out of the car.\n\nAn older gentleman with a long white beard and thick glasses waited for us on the porch. He said his name was Josiah Templeton.\n\n\"You that young fella that called?\" Josiah stared long and hard at Harley.\n\n\"Yes. I'm Harley Quin, and this is Mrs. Washington.\"\n\nI smiled and shook his hand. \"I'm pleased to meet you.\"\n\nHe dragged his gaze away from Harley. He looked frail and his skin was almost translucent, but he smiled and patted my hand. \"Don't get a chance to see many pretty women.\"\n\nI smiled. \"I'll bet you say that to all the women.\"\n\nHe laughed. \"Don't get much chance to anymore. Just a handful of us left these days.\" He kept patting my hand. \"I'll be a hundred on Christmas Day.\"\n\n\"You don't look it,\" I lied.\n\nHe stared at Harley again. \"You look familiar. Have we met before?\"\n\n\"I just have one of those faces.\" Harley's smile seemed strained. \"Well, we appreciate you allowing us to tour the mansion, but . . .\" He stared at Josiah with sympathy. \"I'm afraid this tour might be a bit much for you, old boy. Look, why don't you just give me the key and we'll wander around and then I can lock everything up when I'm done.\"\n\nMy eyes narrowed and I would have berated him for insensitivity, but Josiah said he didn't mind and handed over the key. He pointed at a small building across the street. \"I have a cottage over there to the right. Just drop the key in the mail slot.\"\n\nHarley promised to do so and hurriedly unlocked the door. \"You coming?\" He turned to me.\n\n\"I'll be there in a few minutes. I'm going to make sure Josiah makes it home safely.\"\n\n\"Fine.\" Harley hurried inside.\n\nI offered an arm to Josiah and, between me and his walking stick, he hoisted himself up.\n\n\"Thank you. If you just help me down to the bottom of the stairs, I'll be fine from there. These old joints need a little boost to get going, but once I'm moving, you just watch my steam.\"\n\nWe walked down the stairs slowly and I mentally cursed Harley for dragging this nice man out here just to tour an old house. At the bottom of the stairs, Josiah refused further assistance. I continued talking and made sure he made it safely to his cottage.\n\nOnce inside, he thanked me and I turned and made my way back to the mansion. I contemplated not entering, but I knew my defiant gesture would be wasted. Instead, I went inside.\n\n\"Hello? Harley, where are you?\" I yelled from the foyer.\n\n\"I'm down here.\"\n\nI followed the voice past the elaborately carved staircase, high ceilings, and tapestries to a door that lead downstairs to the cellar. I was surprised, but thankful, the old house still looked relatively sturdy, although the cobwebs and musty smell indicated it had been many years since the house had been opened and aired.\n\nIn the cellar, I looked around and tried to ignore the creepy feeling there were more than spiders crawling around down there.\n\nHarley was on his knees with a flashlight, studying the cinder block walls.\n\n\"What are you looking for down here?\"\n\nHe continued scanning the walls. \"Legend has it there were secret tunnels under this building.\"\n\n\"You mean the tunnels where Benjamin Purnell allegedly abused young girls and conducted satanic rituals and where the commune stored his body in a hermetically sealed glass coffin?\"\n\nGrowing up in North Harbor, these rumors were spread widely among North Harbor children, especially around Halloween.\n\nHarley continued his search. I couldn't see his face, but I could hear the glee in his voice. \"Tunnels full of buried treasure.\"\n\n\"Oh my God. You're a treasure hunter.\" I didn't even try to keep the scorn out of my voice. This guy was a nutcase. I made up my mind to leave him there when my cell phone rang. It was Nana Jo. I'd barely gotten a word out when she started.\n\n\"Sam, you've gotta get away from that guy. He's not what he claims to be. Frank had one of his friends at Scotland Yard investigate him, and he has a record. He was a grifter too. He lied about where he was from. There is no Deering Vale\u2014\"\n\nI never got to hear what else she was going to say because Harley snatched the phone out of my hand. \"Your grandmother has a big mouth.\"\n\nI stared as Harley Quin turned off my cell phone and put it in his pocket. \"Deering Vale . . . Oh my goodness. I can't believe I missed it. That's a fictitious town Agatha Christie made up for The Mysterious Harley Quin.\" I hit myself on the head with my palm. \"Harley Quin? Oh my God, I can't believe I missed that.\"\n\nHe smiled smugly and pulled a gun from his pocket. \"Now, what am I going to do with you?\"\n\nSomething inside told me insulting him was a bad idea, but I didn't listen. \"You cheap murdering lowlife scumbag.\"\n\nHe hauled off and struck me across the face. The blow was so hard it sent me flying into the wall. I hit my head on the doorjamb and fell to the ground. I must have been knocked out for some time because when I came to, there was no light coming through the small basement windows.\n\nI felt a kick in my side. \"You awake?\"\n\nI groaned.\n\nHe leaned his face down within inches of mine. \"Now, what am I going to do with you?\"\n\nI had a few ideas of what I wanted to do to him. He must have read my thoughts because he laughed. \"You're a feisty one, aren't you? Too bad you had to be so smart. We could have had fun together.\" He leered and ran his finger along my cheek.\n\nI swatted his hand away. If I'd had any saliva in my mouth, I would have spit in his face. Unfortunately, I think the tofu and black bean mush I'd eaten earlier had absorbed all the fluid in my body.\n\n\"Few people come down here anymore. It'll be a long time before anyone finds your body. You and Benjamin Purnell can rot down here together.\" He stood and pointed his gun at me. \"Goodbye, Samantha.\"\n\nI needed to keep him talking, buy myself some time. \"Why did you do it? Why did you kill Melody?\"\n\nHe smiled. \"She got greedy. She had a good con going with that oaf Dawson, but she didn't want to cut me in.\" He sneered.\n\nI wondered how I had ever thought him attractive. In this cold, damp dark basement, the only thing I felt when I looked at him was revulsion. \"So it was about money?\"\n\n\"It was always about the money.\"\n\n\"Maybe she really cared about Dawson.\"\n\nHe laughed. \"You are so naive. She didn't care two bits for that dumb jock. He was just a mark. She said he was going to be her meal ticket out of the life.\" He paced. \"I didn't care if she wanted to believe she could stop and go the straight and narrow, as long as she gave me my cut.\"\n\nI shivered in spite of myself at the cold tone of his voice. \"So you killed her. Just like that?\"\n\nHe nodded. \"Just like that.\" He raised his gun.\n\n\"What about A-squared?\" I rushed to get in before he killed me.\n\n\"He saw us together at the HOD. Followed us.\" He shrugged. \"The crazy old fool tried to blackmail me. Me?\" He laughed. \"Can you believe that? He saw me kill her in the parking lot at the HOD, and he thinks I'm going to pay him money not to tell the police?\" He shook his head in disbelief. \"Now, enough talk. I don't like loose ends and you're just another loose end.\" He pointed the gun at my head and stared.\n\nI refused to close my eyes. Instead, I stared defiantly at him and awaited my fate.\n\nJust then, I heard a thunderous sound as someone leapt down the stairs. Harley turned just as a huge shadow came barreling toward him and knocked him to the ground. I slid back against the wall and felt arms coming around.\n\n\"Sam, are you okay?\" Nana Jo asked.\n\nI leaned against her arms and rested my head. \"I'll be fine. Who's that?\" I pointed to the tousling shadows.\n\nJust then, Dawson pulled Harley up by his shirt and punched him in the face. Harley went limp and Dawson reared back and hit him again. Dawson's face was red with fury. Just as he was about to hit him again, Zaq and Frank Patterson grabbed his arm and held onto him.\n\nTears streamed down Dawson's face. \"He killed Melody and tried to kill my dad.\"\n\nFrank put his arms around Dawson and held on to him. \"I know, but it's over now.\"\n\nZaq came over to me. \"You okay, Aunt Sammy?\"\n\nI nodded. \"I'm fine now.\"\n\nSirens blared. Footsteps trampled overhead.\n\n\"We're down here!\" Nana Jo yelled.\nChapter 22\n\nThe EMTs patched up my head. They wanted me to go to the hospital, but I declined. Nana Jo insisted I ride home with Frank, but I refused to leave my new car.\n\nI looked at Dawson. \"How did you get here?\"\n\nHe nodded toward Zaq. \"My dad came to. He told me it was Harley. He was taking a leak behind a bush at the HOD and he saw him kill her. They were kissing in the back seat of his car and then he strangled her, got in the front seat and drove away. He must have dumped her body by the river where the police found her.\" He looked sad. \"He left her like a piece of trash by the side of the road.\" He took a deep breath. \"My dad tried to get him to pay to keep quiet and he tried to kill him.\"\n\nI hugged him. \"Dawson, I'm so glad your dad is okay.\" I stared at him. \"But, how did you get here?\"\n\n\"Zaq and Emma came by to see me and I got a ride back with him.\"\n\nI looked around. \"Where's Emma?\"\n\nHe smiled. \"Zaq made her stay at the bookstore.\"\n\n\"Would you permit me to drive you home?\" Frank asked.\n\nWe were outside and Zaq circled my new car. It was the first time he'd seen it. \"Wow. Nice car, Aunt Sammy.\"\n\nI smiled and handed Frank my key fob. He held the door for me and I got into the passenger seat. However, before he closed the door, I hopped out.\n\n\"What are you doing?\" he asked.\n\nI opened the back door and pulled Harley Quin's bicycle out of the back and tossed it to the ground. I brushed the caked mud out and slammed the door. Then I went back and got in and buckled my seat belt.\n\nFrank drove Nana Jo and me home. Dawson rode with Zaq.\n\nDetective Pitt came by the bookstore later to get my statement. We were upstairs in my loft. Snickers and Oreo must have sensed something was wrong. They hadn't left my side from the moment I got home. Christopher and Jenna had been assigned taxi service. Jenna was, according to her, ordered by Dorothy to get to the retirement village and pick them up. Christopher had picked up my mom and picked Jillian up from MISU when she asked.\n\nSurrounded by friends, family, and Detective Stinky Pitt, I gave my statement.\n\n\"He's been singing like a canary,\" Detective Pitt said. \"Doesn't want to be extradited back to England.\" He chuckled.\n\nJenna pulled out her tape recorder. \"So Virgil and Melody were planning to get Dawson to go to the pros so they could try and hustle money from him. Is that right?\"\n\nI nodded. \"I think Melody hoped to trap Dawson into marriage to strengthen her bond.\"\n\nDawson shook his head. \"My dad knew about that. He thought he could get money out of me too.\"\n\n\"But Dawson broke up with her,\" I said. \"Virgil said she was dating someone else.\"\n\nEmma chimed in, \"That was Professor Quin.\"\n\nNana Jo scoffed. \"Professor, my big toe.\"\n\n\"He was the other person in that special program Dorothy told us about,\" I said.\n\n\"So if he wasn't a professor . . . ?\" Emma asked.\n\nI pointed to Frank.\n\nFrank said, \"He was a grifter. I have a friend who used to be in British intelligence. I had him run his name.\" He avoided eye contact with me, and I thought I detected a small amount of color on his ears.\n\n\"So, he was a con man, too, and he and Melody ended up at the same university together. Convenient,\" Detective Pitt said.\n\n\"Well, there were only a few universities willing to take them. I think if you check around, you'll find Harley Quin was originally sent to another school. I think he must have been in contact with Melody, who told him pickings were good at MISU, so he came here. I think he heard the rumors about the House of David and created an identity for himself.\" I looked around. \"That's why Melody didn't bother about classes. She knew she would get a good grade from Harley and just needed to hang around long enough to get an athlete with PEP.\" I glanced at Nana Jo.\n\n\"But what happened to Professor Quin? Or is there a real Professor Quin?\" Emma asked.\n\n\"He made up the identity and lied to the university. You'd be surprised how few places take the time to check references.\"\n\n\"Good for us he used the alias before or I might not have found him,\" Frank said.\n\n\"Thanks to Irma, we learned that Virgil Russell knew Melody had another lover.\" I turned to her. \"She found that out at the memorial.\"\n\nIrma smiled broadly before breaking into a coughing fit.\n\n\"Everything came to a head at the memorial service Jillian arranged.\"\n\nJillian looked shocked. \"Me?\"\n\n\"If you hadn't gotten the memorial together, Quin wouldn't have found out that A-squared was still alive and in intensive care.\"\n\nShe smiled.\n\n\"But then he made another attempt on my dad's life,\" Dawson said.\n\n\"Yes, but that's how he trapped himself. If Stinky . . . ah . . . I mean Detective Pitt hadn't asked who I'd told about your dad being in the hospital, I might not have put it together. We were so focused on Virgil, but someone had to have put the note on his windshield.\"\n\n\"He could have faked the note.\" Detective Pitt looked condescendingly at me. \"Real detectives would have thought of that.\"\n\nNana Jo grunted. \"A real detective arrested the wrong man.\"\n\nI chose to ignore Detective Pitt. \"The note bothered me and it took a long time before I figured out why.\" I stared at Nana Jo.\n\nShe nodded. \"It bothered me too.\"\n\n\"It was the language. It told Virgil to go to the surgery.\"\n\nDetective Pitt stared.\n\n\"Americans would say hospital or doctor's office, not 'surgery.' \"\n\n\"That's it? That was what tipped you off?\" Detective Pitt looked skeptical.\n\nI nodded. \"Not at first. It just bothered me. But, thanks to Nana Jo for catching on to the Deering Vale reference.\"\n\nDetective Pitt looked confused. \"Deering Vale? What's Deering Vale?\"\n\n\"It's one of the fictitious towns Agatha Christie created. That's where Quin said he was from when Nana Jo asked.\"\n\nNana Jo smiled. \"Thought he was so smart that one. I knew I'd read that name before. So, I started rereading the Agatha Christie books until I found it. Darned fool.\"\n\nFrank shook his head. \"That was pretty stupid. He should know better than to use a name and a village in an Agatha Christie novel to women who own a mystery bookstore.\"\n\n\"Well, it is a bit obscure. If he'd used something more obvious, like St. Mary Mead, I'd have caught on immediately. Most people haven't even heard of the Harley Quin or Parker Payne short stories. They aren't as famous as her other sleuths.\"\n\n\"Except to someone who owns a mystery bookstore,\" my nephews said simultaneously.\n\n\"It rang a bell, but it took me a while to place it.\"\n\n\"Why did he steal Emma's laptop?\" Zaq asked.\n\n\"I can answer that,\" Detective Pitt said. \"He thought it would help him find out what you were up to.\" He nodded toward me. \"He thought she had notes.\"\n\nWe talked for hours after Detective Pitt left until I was so tired I could barely keep my eyes open.\n\nI woke up when I smelled bacon and coffee. Nana Jo came in to check on me. I suspected she'd been in several times during the night because whenever I woke up, my blankets were always neat and Oreo never woke me to go potty. I looked around and didn't see them.\n\n\"I had Dawson keep them so you could sleep in peace.\" Nana Jo opened the curtains. \"You feel up to breakfast?\"\n\n\"Definitely.\"\n\nI was stiff and sore and there was an ugly red-and-blue bruise on my forehead. Otherwise, I felt pretty good. Despite my good appetite and assurances I could work in the bookstore, Nana Jo would have none of it. She was going to keep things going at the bookstore, with a little help from my mom and the girls. The thought of Irma hitting on my male clients struck fear in my heart for a moment, but I supposed the store and the mystery reading men of North Harbor could survive one day with Irma.\n\nI'd tried to go back to sleep, but it was hopeless. Eventually, I gave up and looked for other ways to occupy my mind.\n\n\"Dis is frame. Is Frame. Jestem wrobiony.\"\n\nJ\u00f3zef Lipski paced and alternated proclamations of innocence in broken English and Polish. Neither James nor Victor spoke Polish, but they were able to comprehend the gist of the diplomat's rants.\n\n\"Mr. Lipski, we aren't accusing you of murder. However, we do need to ask you some questions,\" James said.\n\n\"Questions. Questions. Always questions wit de English.\" Lipski's accent was thicker when he was emotional and made it difficult for Victor and James to understand him.\n\n\"Did you tell anyone about your special cocoa?\" James asked.\n\n\"Of course. I talk about de cocoa. I tell dem Poland has best cocoa.\" He puffed out his chest with pride. \"Is true.\"\n\n\"I'm sure it is,\" Victor said quietly as he looked around the room.\n\n\"Did you show the cocoa to anyone?\" James tried again.\n\n\"Show it? I tell dem. I no show.\"\n\nJames rubbed the back of his neck. \"The police are going to need to test the cocoa.\"\n\nLipski halted and turned to James. \"Goot. Goot. Your police test.\" He handed the cocoa to James. \"Test prove I did not kill.\"\n\n\"Can you think of anyone who would want to harm you?\" Victor asked.\n\n\"Me?\" Lipski stared. \"You dink poison was for me?\"\n\n\"We don't know.\"\n\nLipski was silent for several seconds. He walked to his bed and lifted a corner of the mattress, reached under, and pulled out a note. He took the note to James. \"I give you dis.\"\n\nJames opened the note and read it. He looked up at Lipski. When he reread the note, he handed it to Victor.\n\nVictor read the note and then stared at James.\n\n\"When did you get this?\" James asked.\n\nLipski walked to the door and leaned down. \"It appear on floor.\" He pointed down. \"Day French woman shot.\"\n\n\"Meet me by the Marsh at three. W,\" James read aloud. \"Did you meet her?\n\n\"No.\" Lipski paced. \"I shoot wit de American.\" He stopped pacing and faced James and Victor. \"We leave de shooting. But de leg. It get stuck in de mud.\" He pulled on his leg. \"I pull and pull, but . . . de leg it not come out. She take de leg off. Only den am I able to pull it out of de mud.\"\n\nJames's lips twitched as he watched Lipski's demonstration.\n\n\"So, once you got her leg out of the mud, what did you do then? Did you go to the marsh?\"\n\nHe shook his head vigorously. \"No. I help de American to de house.\" He hopped on one leg. \"I help. I not go to marsh. I not meet anyone. Later, she is killed.\"\n\n\"Why didn't you tell anyone?\" Victor asked.\n\nLipski shrugged. \"She is dead. Dat is how I know is frame. Jestem wrobiony.\"\n\nVictor joined the Marsh Family in Lord William's bedroom.\n\n\"What took so long?\" Penelope asked. \"And where's James?\"\n\n\"We were on our way here when we got stopped by Thompkins. James was wanted on the telephone.\"\n\nJames and Detective Inspector Covington rushed into the room and closed the door.\n\n\"I'm sorry, I don't have much time. I've been called back to London. I hope I haven't overstepped, but I invited Detective Inspector Covington. I think it'll be fastest if we just include him.\"\n\n\"How did you get here so quickly?\" Lady Penelope asked.\n\n\"I called him after Brasseur died,\" James said.\n\nVictor smiled. \"What about the poachers?\"\n\nDetective Inspector Covington grinned. \"I'm hot on their trail.\" Then he put his fingers to his lips and took a position near the window, where Daphne was seated.\n\nLady Elizabeth nodded. \"Of course, you're very welcome.\"\n\nThe detective smiled his thanks and quietly sat with a notepad on his lap.\n\n\"What's happened?\" Daphne stared at James.\n\n\"A German secretary, Ernst von Rath's, been assassinated in Paris.\"\n\n\"Good Lord. What's happening to the world?\" Lady Elizabeth whispered.\n\nJames paced. \"I'm afraid that's not all. A Polish teenager's been arrested for the murder.\"\n\nThe family looked shocked.\n\nLord William fumbled with his pipe. \"Here we are on the brink of war and now this.\"\n\nJames nodded. \"Exactly. And, somehow word has leaked out to The Week that an attempt was made on the Duchess of Windsor's life.\n\nDaphne looked pale. \"What's The Week?\"\n\nIt's an anti-fascist newsletter run by Claud Cockburn. The readership is small, but . . .\"\n\n\"But it could still cause trouble?\" she asked.\n\nJames nodded.\n\n\"So if word got out the French Ambassador was murdered by a Polish Emissary . . .\"\n\n\"It could be disastrous. I've got to run up to London. I'll be back as quick as I can.\" James hurried away, and the group sat in silence for a few moments.\n\nLady Elizabeth looked around. \"Well, we better get busy.\"\n\nThey filled Detective Inspector Covington in on what they had learned so far, including Victor's conversation with J\u00f3zef Lipski.\n\n\"So, Lipski couldn't possibly be responsible for the murder of the maid,\" Penelope said. \"If he was with Virginia Hall, then he has an alibi. Did anyone check with her?\"\n\nVictor nodded. \"Yes. As soon as we left Lipski, we questioned her. She confirmed everything he said.\"\n\n\"Do you trust her?\" Daphne asked.\n\n\"Yes. She was very credible. Both James and I agreed.\"\n\n\"He also couldn't have poisoned the cocoa,\" Lady Elizabeth said.\n\n\"Really? How can you be certain?\" Detective Inspector Covington asked.\n\n\"I talked to Mrs. Anderson.\"\n\n\"Who is Mrs. Anderson?\"\n\n\"Mrs. Anderson is the cook,\" Lady Elizabeth said. \"It took a bit of prodding to get her to admit it, but apparently, she was rather curious about this cocoa. She'd never had cocoa from Poland so . . . she made a couple of cups for herself and her daughter, Agnes.\"\n\n\"Well, I'll be,\" Lord William said.\n\n\"Yes. She said it tasted just like good old-fashioned English cocoa, and she and her daughter are perfectly fine.\"\n\nDetective Inspector Covington perched on the end of his seat. \"That's significant.\"\n\nLady Elizabeth nodded. \"I know. It means the poison wasn't in the cocoa.\"\n\nLord William stared. \"But how can that be? If the poison wasn't in the cocoa, how did the blasted fool get it?\"\n\n\"The poison has to have been put into his cup,\" Lady Elizabeth said.\n\n\"That's not the worst of it.\" Daphne looked around at her family.\n\nPenelope stopped pacing and stared at her sister. \"What do you mean?\"\n\n\"Unless you believe Thompkins or Mrs. Anderson put the poison in his cup, it had to have been put there by someone in the room.\"\n\nPenelope paled. \"You mean one of us.\"\n\n\"Exactly,\" Lady Elizabeth said.\n\nDetective Inspector Covington wrote furiously. \"You all are very good, very thorough.\"\n\nThere was a tentative knock on the door.\n\n\"Come in,\" Lady Elizabeth said.\n\nThompkins entered the room and closed the door. He moved to a corner and stood erect.\n\n\"Great, Thompkins, did you finish your assignment?\"\n\nThe butler coughed discreetly. \"Yes, m'lady.\"\n\nLady Elizabeth looked around the room. \"I asked Thompkins to come here and share the results from the search.\"\n\nDetective Inspector Covington looked confused. \"Excuse me, did you say search?\"\n\nLady Elizabeth nodded. \"Yes. We needed to find out if anyone had a gun, since none of the ones used in the murder belonged to us.\"\n\nDetective Inspector Covington stared at Lady Elizabeth with awe. \"Did you really? How did you get them to agree to a search?\"\n\nLady Elizabeth smiled. \"I didn't ask them.\"\n\nThompkins looked slightly uncomfortable but straightened his already straight back, pushed his shoulders backed, and lifted his head higher. \"As your ladyship instructed, I asked the maids, Flossie, Millie, and Gladys. The only one who remembered seeing a weapon of any type was Millie. She said Count Rudolph had a gun, but she hasn't seen it since. Under my supervision, they searched the room again, and there was no gun.\"\n\nLady Elizabeth sighed. \"That's what I was afraid of.\" She smiled at the butler. \"Well done, Thompkins. Please tell the staff I said thank you.\"\n\nThompkins nodded. \"Yes, m'lady.\" He paused for several moments.\n\n\"Is there anything else?\" Lady Elizabeth asked.\n\nThompkins coughed. \"There is just one other thing . . . I hesitate to bring it up, but you said you wanted to know about anything out of the ordinary.\"\n\nLady Elizabeth nodded. \"Of course.\"\n\nEveryone stared at the butler.\n\n\"Millie found this in the Duchess of Windsor's wastebasket.\" The butler held up a small piece of paper in the shape of a funnel. Everyone stared, but Detective Inspector Covington immediately hopped up and practically ran to the butler's side.\n\nThe detective grabbed the paper and stared at it and then whistled. \"I need to have this tested.\" He wrapped the funnel in a handkerchief and placed it in his pocket.\n\nThompkins nodded. \"I believe Her Grace is in the parlor with the rest of the guests.\"\n\nLady Elizabeth looked thoughtful.\n\n\"I know that look.\" Lord William sat up straight and stared at his wife. \"You've figured it out.\"\n\nEveryone stared at Lady Elizabeth. For several moments, she stared off into space. Eventually, she nodded and looked around and seemed to notice everyone was staring at her. \"I'm sorry. Did you say something, dear?\"\n\n\"You've figured out who the murderer is, haven't you?\" Penelope walked over to her aunt. \"Tell us.\"\n\nLady Elizabeth looked around at her family. \"Well, I believe I have.\"\n\nDetective Inspector Covington anxiously looked at her but stood very still.\n\n\"I'm afraid I've been terribly dense. I should have figured it out sooner. I believe Daphne commented on how 'theatrical' the murder seemed. I should have known then. Maybe that poor man wouldn't have died,\" Lady Elizabeth said softly. \"But, well, at least we can give him justice.\"\n\n\"Who did it?\" Detective Covington asked.\n\n\"Count Rudolph, of course. He used to be an actor before World War I. He even starred in some god-awful propaganda movie advocating the involuntary sterilization of the disabled and those with hereditary diseases.\" She shivered. \"He staged the murder. The setting, a British country home. The duchess, killed by the Polish ambassador.\"\n\nPenelope stared at her aunt. \"But the duchess wasn't killed.\"\n\n\"I think the count put the note in the duchess's pocket, sometime during the night when they were together. He wasn't to know she would have her maid wear her clothes.\"\n\n\"And Rebecca probably found the note in her pocket and thought the duchess wanted to meet her,\" Penelope said.\n\nVictor looked confused. \"But, J\u00f3zef Lipski has an iron-tight alibi. There's no way he could have killed her.\"\n\nLady Elizabeth nodded. \"But Count Rudolph wasn't to know about that. Virginia Hall is very casual about her wooden leg. She jokes and calls it Cuthbert, but I think she was embarrassed about getting it stuck in the mud. Neither she nor Lipski told anyone about the incident.\"\n\n\"But how did he kill her? Wouldn't he have been missed by the other members of his shooting party?\" Penelope asked.\n\nLord William reddened. \"Count Rudolph was with Georges Brasseur and Wallis. I suspect it would be pretty easy for a gentleman to . . . excuse himself for a few minutes and sneak off.\"\n\nDaphne stared at her uncle. \"Excuse himself?\"\n\nLord William squirmed uncomfortably.\n\nPenelope smiled. \"You mean to relieve himself?\"\n\nLord William's face reddened and he puffed on his pipe.\n\n\"But surely someone would have mentioned it,\" Daphne added.\n\n\"At the time it probably seemed normal. Later, after they'd had time to think about it, they might have suspected. That's probably why he killed Brasseur,\" Victor said.\n\nLady Elizabeth nodded. \"I suspect so.\"\n\n\"But surely the Duchess of Windsor would have said something,\" Daphne said.\n\nLady Elizabeth sighed. \"Perhaps, but the duchess may have found herself in an awkward situation. Remember, she had been seen coming out of his bedroom. She isn't exactly well loved by the British public at the moment. That type of negative publicity would have been the last thing she wanted.\"\n\n\"Where is he?\" Detective Inspector Covington asked.\n\n\"I believe Count Rudolph and the duchess are in the parlor,\" Thompkins said.\n\nDetective Inspector Covington got a constable and went to confront the count. Victor went along.\n\nAs the detectives entered the parlor, Count Rudolph glanced up at the approaching officers. A glimmer of realization flashed in his eyes and was quickly followed by panic. Before the detectives knew what was happening, Count Rudolph made a mad dash toward the door. He pushed the detectives aside and quickly closed the door behind him and turned the lock.\n\nWhen Detective Inspector Covington regained his feet, he pulled frantically at the doorknobs, but the heavy doors were locked.\n\nDetective Covington threw his body against the doors in an effort to force them open, but the sturdy oak doors didn't budge.\n\nVictor looked around for a way out. He glanced at the French doors. \"Quickly, the window.\"\n\nThey quickly unlatched the windows and rushed out.\n\nOutside, they split up. Detective Covington and Victor ran in one direction and the constable ran in the other.\n\nAround the back of the house, Victor spotted Count Rudolph making a run for the garage. \"There. He's going for the garage.\"\n\nThe footmen, Jim and Frank, were carrying a large basket of laundry outside for Gladys, who followed close behind.\n\n\"Stop him!\" Victor yelled.\n\nThe footmen looked where Victor pointed and dropped the basket and took off after the count.\n\nFrank caught up to Count Rudolph seconds before Jim. When they were mere feet from their target, both men took their stance, knees bent, feet about shoulder width apart, back straight, and heads up. In a few quick steps, they closed the gap between them and the count. They lowered their shoulders and wrapped both arms around him and took him to the ground, just as they had been taught to tackle ball carriers on the rugby field.\n\nCount Rudolph struggled briefly but quickly recognized the futility of his situation. Detective Covington arrived and relieved the footmen of their prey. \"Well done!\"\n\nFrank and Jim looked at Victor, who nodded further approval.\n\nThe Duchess of Windsor was so upset when told by Lady Elizabeth of Count Rudolph's actions, she immediately left and returned to Fort Belvedere; poor plumbing notwithstanding.\n\nJ\u00f3zef Lipski received a telegram recalling him to Poland. The assassination of the German ambassador and fear of escalating violence were cited as the reasons for his return. Lipski thanked the Marsh family profusely before leaving.\n\nLord and Lady Chitterly made discreet departures.\n\nFordham Baker was one of the last to leave.\n\nLady Elizabeth waited in the hall to say goodbye.\n\n\"Thank you so much for the wonderful hospitality,\" Fordham Baker said to Lady Elizabeth as he pre pared to depart.\n\nLady Elizabeth smiled. \"And thanks are due to you, too, I think.\"\n\nHe blinked and looked at Lady Elizabeth with surprise. \"To me, your ladyship? I don't know what you mean?\"\n\n\"I think you do.\" She smiled. \"I think you were not as . . . indisposed as you pretended to be. I also think you are the one responsible for feeding information to Claud Cockburn.\"\n\nHe blinked. \"I don't know . . .\" He chuckled. \"What gave me away?\"\n\n\"You were quite convincing.\" Lady Elizabeth smiled. \"But we were very concerned about your health. So, I had my butler substitute the scotch with a diluted mixture. Your reaction was the same to both.\"\n\n\"I should have guessed.\" He laughed. \"I take back all the evil thoughts I had about the quality of your cellars.\" He laughed until he had to wipe tears from his eyes. \"I find people are very reticent to speak freely in front of a newspaper editor unless he's in a drunken stupor. Then you would be surprised what they will say.\"\n\nLady Elizabeth bid him farewell.\n\nVirginia Hall looked for Lady Daphne before she left. She found her in the library, sitting on the window seat.\n\nDaphne looked up at her. \"Are you leaving? Well, it's been a pleasure meeting you,\" she said politely.\n\nVirginia Hall smiled. \"You British ladies are such well-bred liars. You haven't enjoyed meeting me at all.\"\n\nDaphne looked shocked. \"I don't know what you mean.\"\n\n\"Don't get your back up, dear. It's not your fault, and I don't blame you one little bit.\" She sat down on the window seat next to Daphne. \"I blame that handsome man of yours.\"\n\nDaphne's face heated and she looked down. \"I don't know what you mean.\"\n\n\"Yes, you do. Lord James FitzAndrew Browning.\" Virginia looked around the room. \"I told James he needed to come clean and explain everything to you, but he wouldn't do it.\"\n\nDaphne stood up abruptly. \"I don't know what you're talking about, but I don't think I want to hear it.\" She turned and took several steps away.\n\n\"There's nothing going on between James and me,\" Virginia said.\n\nDaphne stopped. \"What?\"\n\nVirginia got up and walked over to Daphne. \"Look. He should have told you himself, but he just kept saying you should trust him.\" She huffed. \"Men. They're impossible.\" She pulled a small card out of her purse and showed it to Daphne. \"I work for the American Secret Service. I was sent here to keep an eye on J\u00f3zef Lipski. The president believed the Germans might be up to some funny business and looks like he was right.\"\n\n\"You're a spy?\" Daphne asked.\n\nVirginia Hall nodded. \"Pretty good cover, right? Who'd believe a one legged woman could be a spy?\" She patted her leg.\n\n\"Is that real?\" She pointed to the leg.\n\n\"Yes. Cuthbert's real alright.\" She pulled up her skirt so Daphne could see the buckle. She put away her identification card and stared at Daphne. \"He really does care about you.\"\n\nDaphne shook her head. \"If he did, why didn't he tell me the truth?\"\n\n\"Because I was a fool,\" James said from the doorway.\n\nDaphne turned at the sound of his voice.\n\nVirginia Hall gave her a hug and whispered, \"I think you and I could be great friends. I hope so.\"\n\nDaphne squeezed her back and whispered, \"Thank you.\"\n\nVirginia nodded to James as she left.\n\nDaphne returned to her seat, and James walked over and stood next to her.\n\n\"I'm terribly sorry, darling. I should have told you the truth from the beginning.\"\n\n\"No, I'm sorry. I was just . . . jealous.\" She laughed. \"It's a new emotion for me. I don't think I've ever been jealous of anyone before.\"\n\nJames held her hand. \"You don't have any reason to be. You have my heart totally and completely.\" He looked down. \"I love you.\"\n\n\"I love you too,\" Daphne whispered.\n\nJames pulled Daphne toward him and kissed her passionately and thoroughly. Daphne couldn't have said whether they kissed for seconds or hours.\n\nWhen they finally pulled apart, James held her close and whispered, \"Darling, I plan to make an official request for your hand, but I have one favor to ask of you.\"\n\nDaphne snuggled next to him like a kitten. \"Anything.\"\n\n\"I would like to introduce you to my mother.\"\n\nDaphne smiled. \"I would like nothing better.\"\n\nLady Daphne had no idea how she would regret that remark.\n\n\"Hmmm. Maybe foreshadowing is a bit much here,\" I said aloud, even though there was no one to hear.\n\nI better leave that part out. I deleted the last sentence and read the last few lines.\n\n\"I would like to introduce you to my mother.\"   \nDaphne smiled. \"I would like nothing better.\"\n\nTHE END\n\nChapter 23\n\nThe Sleuthing Seniors Book Club met on Thursdays. That was the only reason I came downstairs. The bruise on my forehead was now a deep purple and it looked like I was getting a black eye. However, after a day alone, I was tired of my own company.\n\nDawson had been busy baking most of the day. Tomorrow, he would be allowed back at school and back on the football team. One of my favorite desserts was carrot cake, so he'd made a delicious carrot cake with cream cheese icing. It was moist and tasty and I ate more than I should, but I enjoyed every bite.\n\n\"Have you ladies decided on your next book?\" I asked.\n\n\"We're going for Raymond Chandler's The Big Sleep,\" Dorothy said.\n\n\"That's a good one,\" Nana Jo added.\n\n\"Is there sex in it?\" Irma asked.\n\nI thought back. \"It's been a while since I read it, but I don't think so.\"\n\nIrma looked disappointed.\n\n\"I've got something for you.\" I went to the back office and came back with three books which I placed in front of Irma. \"I ordered these for you.\"\n\nShe looked at them. They were romance novels by Brenda Jackson.\n\n\"A Brother's Honor is the first book in the series. A Man's Promise is the second, and A Lover's Vow is the third. Technically, they're romances, not mysteries, but there is definitely a mystery subplot. I thought you might enjoy them.\"\n\nIrma coughed and gave me a hug. \"Thank you, Sam. I can't wait to read them.\"\n\nFrank Patterson entered with a bottle of champagne, followed by Dawson, who had a tray of glasses. Nana Jo came over to me with a sly smile on her face and a mysterious envelope in her hand. \"Sam, I have a surprise for you.\"\n\nI looked at Nana Jo and tried to imagine what the surprise might be. I looked around at the girls, and they seemed to be a mixture of happiness and nervous anxiety.\n\nNana Jo took a deep breath. \"We know how . . . deeply private you are about your book and your writing. We all think it's wonderful.\"\n\nI could feel the heat rush to my face. It had been difficult to allow the girls to read my book, but they were all very complimentary. However, they were my friends and I knew they would never say or do anything to hurt me.\n\nDorothy said, \"Josephine, stop beating around the bush and tell the girl.\"\n\nI looked from Dorothy to Nana Jo. \"Tell me what?\"\n\nNana Jo hesitated for a moment. \"You know Ruby Mae has a second cousin whose daughter is a literary agent in New York.\"\n\nI suddenly felt a huge weight in the pit of my stomach. \"Yes.\"\n\n\"Well, we sent her a copy of your manuscript, and she loved it.\"\n\n\"You did what?\" My head was dizzy and I felt like I needed to puke.\n\nNana Jo looked shocked. \"We . . . I wanted to help.\"\n\nI put my head between my knees for several moments. The room was completely quiet.\n\nWhen I finally felt I could sit up without getting sick, I did. Tears streamed down my face.\n\nNana Jo hurried to my seat. \"I'm so sorry, Sam.\" She put her arms around me to comfort me. \"I never would have done it if I'd known it would upset you this much.\"\n\nI pulled away from her and stood. \"Upset?\" I stared at Nana Jo.\n\nHer expression was one of disappointment and remorse. I looked around at the girls and each of them looked ashamed.\n\nI took several deep breaths and collected myself. Then I threw both arms around my grandmother. \"Thank you.\"\n\nNana Jo pushed me away to look in my face. Seeing I was okay, she pulled me close and hugged me tightly.\n\nIrma took a swig from her flask. \"Jesus Christ, you scared the living daylights out of me.\"\n\nDorothy and Ruby Mae came over and hugged me. Frank popped the cork and poured champagne into glasses.\n\n\"I'm just in shock. I'm sorry. I just can't believe it.\" I stared at the letter Nana Jo handed me.\n\n\"You'd better believe it.\" Nana Jo smiled at Sam. \"I have one more surprise for you.\"\n\n\"I don't know if I can take any more.\"\n\n\"I decided how I want to spend my share of the casino winnings.\" She pulled an envelope out of her back pocket and handed it to me.\n\nI opened the envelope. Inside were brochures and flyers for a trip to England. I looked at my grandmother with tears in my eyes. \"Really?\"\n\n\"I thought it would help with your research . . . for your next book.\"\n\nI was dumbstruck. Tears streamed down my face. I grabbed Nana Jo and squeezed her. \"Thank you.\"\n\n\"You deserve it.\" Nana Jo wiped away a tear and took a drink from the glass of champagne Dawson handed her. \"Now, let's get this party started.\"\nPlease turn the page for an exciting sneak peek of  \nthe next Mystery Bookshop Mystery\n\nTHE NOVEL ART OF MURDER\n\ncoming soon wherever print and e-books are sold!\n\nChapter 1\n\n\"What the blazes do you mean I didn't get the part?\" Nana Jo's face turned beet red and she leapt up from her chair.\n\nI had never been so happy for a slow morning crowd at the bookstore as I was at that minute. My grandmother was about to blow a gasket, and while it might prove entertaining, I preferred keeping the drama contained to family and friends.\n\n\"Josephine, calm down.\" Dorothy Clark was one of my grandmother's oldest friends, which was probably why she was nominated to break the bad news to her.\n\n\"Don't tell me to calm down. I am calm. I'm always calm.\" Nana Jo pounded the table with her hand. The mugs shook and splashed coffee on the table. \"If I want to kick up a ruckus, I'll kick up a ruckus.\" She pounded the table again and then marched over to the counter and grabbed a dishcloth to wipe up the mess.\n\nRuby Mae Stevenson, another of Nana Jo's friends, shook her head and moved her knitting out of the way of the spills. \"I told you she wouldn't take it well.\"\n\n\"I've had the lead role in the Shady Acres Senior Follies for the past ten years. That role was created specifically for me. I don't just play the part of Eudora Hooper, retired schoolmarm dreaming about becoming a famous showgirl. I am Eudora Hooper.\" Nana Jo wiped up the spilled coffee.\n\n\"I know, and you've played the role splendidly.\" Dorothy's face reflected her sincerity.\n\nAmazingly, Dorothy didn't seem to be humoring my grandmother. Nana Jo's performance was inspired, and each year she got better and better.\n\nNana Jo looked at her three closest friends. \"Who got the part?\"\n\nRuby Mae put her head down and refused to make eye contact.\n\nIrma Starczewski reached for her mug, but it was empty, so she pulled a flask out of her purse and took a swig.\n\nNana Jo put her hands on her hips, narrowed her eyes, and stared at Dorothy.\n\nFor a large woman, almost six feet tall, Dorothy shrank as she stared at Nana Jo. \"Maria Romanov.\"\n\nI thought Nana Jo was red before, but the beet red coloring from earlier was nothing compared to the purple red that crept up her neck.\n\n\"Maria Romanov? That two-bit hack's only acting talent is in her ability to convince people she's a decent human being.\" Nana Jo pounded the table again, rattling the mugs.\n\nJust as quickly as the anger flared up, it vanished. Nana Jo flopped down in a chair. Nearly as tall as Dorothy, Nana Jo went through a transformation. Instead of the vibrant, active, five-foot-ten, sharpshooting, aikido-tossing woman I knew and loved, there was a seventy-something, old woman in her place.\n\nShe took a few deep breaths. \"If that's what Horace wants, then I guess I wasn't as good as I thought I was.\"\n\n\"Bull\u2014\"\n\n\"Irma!\" we shouted.\n\nIrma coughed and clamped her hand over her mouth. Years of heavy smoking, drinking, and hanging out with truckers, if Nana Jo was to be believed, had left her with a deep cough, a salacious sexual appetite, and a colorful vocabulary.\n\nI leaned over and gave Nana Jo a hug. \"Your performance was amazing and I'm not just saying that because you're my grandmother.\"\n\nShe absentmindedly patted my arm. \"Thank you, Sam, but Horace Evans is a top-notch director. He once directed Ethel Merman.\"\n\n\"He even won a Tony award. I've seen it. He keeps it in his bedroom.\" Irma smiled and then broke out in a fit of coughing.\n\nThe fact that Nana Jo didn't acknowledge Irma's quip about the location of the award was an indication of her state of mind. \"We've been fortunate to have someone with his experience and credentials at Shady Acres.\"\n\n\"Really? I didn't know he had a Tony award. They always run something about the Senior Follies in the newspaper, but they've never mentioned it.\"\n\n\"He likes to keep it low-key.\" Dorothy nodded. \"He worked on Broadway for more than twenty years.\"\n\n\"How in the world did he end up in Michigan?\" I asked.\n\n\"He wanted to be close to his family.\" Ruby Mae looked up from her knitting. \"I think his son was an engineer for one of the car companies.\"\n\nNorth Harbor used to have a lot of manufacturing plants that supplied parts for the Detroit automobile industry, but when the economy went south in the seventies, so, too, did most of the manufacturing jobs.\n\n\"I appreciate the kind words, but Horace is an expert. If he thinks Maria Romanov will make a better Eudora Hooper than me, I'll just have to accept his decision.\"\n\nWe tried to cheer Nana Jo up, but nothing we said had any effect. She smiled and continued to shrink. Only once did she perk up and demonstrate the flash of fire which characterized her personality.\n\nThe door chimed and a customer entered the bookstore.\n\nNana Jo rose from her seat. \"It's time to face the music. On opening night, I hope you all break a leg.\" She pushed her chair in and headed to the front of the store. \"And I hope Maria Romanov breaks her neck.\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nThere's a lot more to traveling with a horse than shoving him onto and yanking him off of a trailer. Indeed, if you've ever seen a horse man-handled onto a trailer, you've witnessed an approach so much more forceful than it needs to be \u2014 and so much less effective than it should be \u2014 that you may have decided that traveling with your horse simply isn't an option. If you can't ride Old Paint from your home barn to where you want to go, you're just not going.\n\nOr, you might have decided that you never want to own a trailer, load your own horse, or go to a show or to a state park for a trail ride on your own; instead, you'll call a horse-transport company and let _them_ load, transport, and unload your horse. The problem with that, of course, is that although the professionals may be quite adept at transporting your horse, they still have to do it, and you have to pay them for their services.\n\nIt's unfortunate that so many horse owners think in such clear-cut terms when it comes to trailering, because virtually _any_ horse can be trailered safely by his owner if the owner simply attends to a few details, the most critical being teaching the horse how to load.\n\nIn this text you'll learn the loading method taught by the late Dr. Edwin Goodwin, long-time equine professor at the University of Maryland. It's a good method because it works, almost without fail. It works quickly on willing green horses; it works fairly quickly on older horses that have always been led; it works a bit more slowly on horses that have had negative loading and\/or trailering experiences; and it can help your horse overcome fear, stubbornness, bad training, bad handling, bad nature, and bad horsemanship, at least insofar as loading is concerned.\n\n# Trailer-Training  \nYour Horse\n\nLaura Harrison McBride\n\n## CONTENTS\n\n**On the Road Again**\n\n**Travel Stress, Equine Style**\n\n**The \"Cue-On\" Loading Technique**\n\n**How to Cue-On Load**\n\n**Points to Remember**\n\n**Hauling Options**\n\n**Safety First**\n\n**Travel Equipment and Supplies**\n\n**Seasonal Strategies**\n\n**Documentation**\n\n**Rules for the Road**\n\n**When You Reach Your Destination**\n\n## On the Road Again\n\nIn today's highly mobile society, it's conceivable that in a single journey \u2014 especially as equestrian competitions continue to expand internationally \u2014 a horse will board a road vehicle, a railroad car, and an airplane. If the venue is particularly remote, boat travel may also be added to the mix. Still, despite our unmatched ability to transport our horses, often we don't give their journey a second thought, even though their safety and comfort during travel should always be a priority.\n\nThe mortality once associated with horse transport \u2014 the sort that occurred with unfortunate regularity when the New World was populated with European stock \u2014 is no longer an issue, the lower rates reflecting increasingly sophisticated techniques for managing travel stress. Good feed and management practices keep a horse in prime condition so that any losses of weight or muscle tone during transport are relatively inconsequential. In addition, vaccinations guard against germs a horse may encounter, and the design of trailers and protective equipment (such as leg wraps) gets better each year. Though sickness or injury can certainly add to a horse's level of stress during travel, it's not the main issue. Allaying a horse's fears of loading and unloading through proper training, and minimizing environmental stressors, should be our greatest concern, as these have the most significant impact on the horse's stress level and condition during and after transport.\n\n## Travel Stress, Equine Style\n\nStudies using heart rate as an indicator of stress show that a horse's heart rate is significantly elevated while he is being loaded onto a trailer. Studies also reveal that the steeper the grade of the loading ramp, the greater the horse's stress. There are several possible explanations for this reaction:\n\n  A horse instinctively fears dark, enclosed spaces \u2014 precisely the characteristics of the vast majority of trailers and vans.\n\n  Steep ramps are often covered with hemp matting that can slip. They also tend to be narrow, often only as wide as the horse, so the handler cannot enter with the horse. Uncertain footing and an absent handler increase the horse's anxiety.\n\n  A horse that has had a bad experience loading\/unloading or during a trip associates this negative experience with all similar situations and equipment.\n\n  A horse that has never been taught how to load properly and that is forcefully manhandled onto a trailer prior to a trip has negative expectations of travel.\n\nOnce a horse has been successfully loaded, environmental stressors associated with travel continue throughout the trip. The van may be too hot or too cold; the road may be rough or the driver inexperienced or careless; the hay might run out or the horse's neighbor may snap at him when he reaches for a wisp; the floor might be too hard or too slick; and the horse may simply wonder what's happening. In addition, he may not be accustomed to standing still for two hours or six hours or twelve hours. Each of these factors individually, and certainly any of them in combination, results in stress. The less stress a horse experiences during travel, the better.\n\n## The \"Cue-On\" Loading Technique\n\nDr. Goodwin is credited by many with inventing the cue-on loading technique, and he was certainly an early proponent in the United States. The basic technique, however, is nearly as old as horse transport and can be found in various guises in equine instruction books of antiquity. Dr. Goodwin spent more than thirty years teaching it to 4-H'ers, college equine science students, and private clients. And he taught it to me, his final graduate assistant.\n\n### _What You Need_\n\nIn the cue-on method, you use your voice, your body, and a cue to teach your horse to respond positively when you ask him to load onto a trailer. You'll use your knowledge of the process to condition your horse to do what you ask, when you ask. Be observant, heed your horse's body language, and be positive. Even the most willing, docile horse will think twice about strolling up a ramp into a dark little hole, never mind stepping up into one. When your horse agrees to do what you ask \u2014 which, in this case, goes against his instincts \u2014 he demonstrates infinite trust in you.\n\nPatience is key. Plan to invest time and to control emotion when teaching cue-on loading. Decide how much of the process you wish to accomplish successfully at each training session, and always try to end on a positive note. For an older horse that has come to believe that loading is a horrible thing, limit the number of cues you ask him to accept at each session. If your horse is a bad loader, having him approach the trailer on cue in a straight line may be enough work for one session; having him approach the ramp on cue might be enough for the next. Never give up at the horse's insistence, no matter how frustrated you or he may be. You are in charge, not your horse. Only stop working on your own terms.\n\nTo reiterate, break up the method into manageable parts that your horse can accomplish successfully at each session, be patient, and soon he will load willingly.\n\n### _The Horse's Motivation_\n\nThe cue-on method involves reward but not a food reward and not letting the horse off the hook when he has only partially mastered the goals you've set for the particular session. It is more cause-and-effect than carrot-and-stick or even perform-and-reward: simply put, you cue your horse to move forward and he does so to avoid repetition of the cue.\n\n### _Simple Tools_\n\nMany of Dr. Goodwin's students were self-proclaimed \"animal nuts,\" some of whom had limited equine experience. Of these, some objected to the simple tools \u2014 a lead rope with chain and a dressage whip \u2014 that Dr. Goodwin used to teach horses to load. The naysayers assumed that such tools were inhumane, not realizing that what would be severe for a housecat is gentle for a horse. In the cue-on method, cues are given with nothing more stringent than a three-inch bit of string, and this is done mainly to demand the horse's attention. It's also used to establish dominance.\n\n_The cue-on technique requires simple tools: a halter, a lead shank or a lead rope with chain, and a dressage whip_.\n\n**When to Seek Help**\n\nIf your horse strikes or kicks at any point during the training process, seek out the services of an experienced professional trainer for some good basic training for your horse and help with loading.\n\n## How to Cue-On Load\n\nThe cue-on loading technique is highly effective and humane. You might think that this exercise should be done in a fairly enclosed area, to keep the horse's attention, but that's not the case. It is far better to work in an open area where you cannot buy false cooperation from the horse and where he feels that he has an out and can flee if he really wants to. Giving him this sense of freedom will help put your horse at ease and help you gain his trust.\n\n_Equipment_\n\nLong lead rope with chain\n\nDressage whip\n\nTrailer, preferably a ramp type (a step-up trailer is slightly more difficult for the horse and may require an extra session or two of work)\n\n### _Getting Started_\n\nOne objective of a good training program is to establish a predictable routine that will put the horse at ease and help to reinforce the lessons learned. Before beginning each training session, take time to groom your horse. A thorough grooming will relax him and make him more receptive to the lesson that follows.\n\n_Grooming him before the training session begins establishes an enjoyable pre-lesson routine for your horse_.\n\n### _Walking and Halting on Cue_\n\nTo train your horse to load on cue, you must first teach him to walk and halt on cue. This may be the entire lesson for the first day, depending on the age of your horse, his attention span, and his prior loading experiences. Here's how.\n\n**1**. Pass the chain over the horse's nose, and hold him on the near side with the long lead rope. Pass the lead rope in front of your body and hold it in your left hand, just as you would when leading your horse; hold the dressage whip in your right hand.\n\n**2**. Say \"Walk,\" and simultaneously tap the horse on the hindquarters with the whip. If your horse doesn't move, give the command again and tap him on the hindquarters a bit more forcefully. When giving physical cues, always err on the side of too little force rather than too much. If the horse doesn't respond, repeat the command using slightly more force behind the cue. When he begins to walk, praise him warmly as you walk at his shoulder. An encouraging, enthusiastic \"Good\" is enough. Remember not to tug; simply walk beside him as he responds to the cue. _Note_ : When giving the Walk cue, be sure to tap your horse only on the hindquarters. If he feels the whip on his legs, he may kick.\n\n_Say \"Walk,\" and tap the horse on the hindquarters with the whip while walking beside him_.\n\n**3**. After walking several paces with your horse, bring him to a halt with the command \"Whoa\"and a fairly sharp tug on the lead. Don't pull so hard that the horse tosses his head in the air from pain or backs up to resist; instead, tug just enough to elicit the desired response. Praise him for halting.\n\n**4**. Repeat this sequence several times to reinforce the cues. Common wisdom suggests that a horse learns something after two repetitions, yet each horse is different. Observe your horse for increasing confidence or for signs of boredom; both are signals that the horse is ready for the next stage.\n\nAs your horse learns the cues, you will be able to tap less insistently and tug less forcefully. Eventually, a single tap or even the verbal cue alone may be enough. Conversely, if your horse is reluctant, you will need to increase the authority of the aids until you get the response you've requested. Once a positive response is received, back off by using a gentler request.\n\nAs your horse learns what each cue means and begins to respond consistently, gradually cut back and eventually eliminate the verbal praise. By doing so, the association between the cue and response becomes automatic: when you ask for the walk, your horse gives the walk; when you ask for the halt, he halts. If you were to continue rewarding your horse with the same level of praise after he responds to each cue, the praise would come to hold little meaning for him, and you would have difficulty providing positive reinforcement during the next phase of the maneuver, whatever it might be. When you are certain your horse knows the cues Walk and Whoa, it is time to approach the trailer.\n\n**Leading onto a Trailer?**\n\nYour horse is making progress. He now willingly walks and halts on the lead in response to a minor cue. So, you might wonder, why not just lead him onto the trailer? Won't that work every time?\n\nYes and no. If you choose to lead your horse onto a trailer, it will involve some degree of pulling on your part, and your horse might choose to resist, backing up and setting up the first failure to load, which would likely be followed by others. In addition, if you're in front of the horse, particularly if you are busy tugging and trying to hold him near the cross-ties or tie ring until someone snaps up the butt bar, you may have a difficult time getting out of the trailer.\n\n### _Loading onto a Ramp Trailer_\n\nPark your trailer on level ground, preferably grass or dirt, and choose to work at a time of day when the interior of the trailer is at its brightest. As mentioned earlier, horses are naturally wary of dark, enclosed spaces, so the brighter the trailer's interior the better. Before you begin, be sure the trailer is solidly parked and doesn't sway or tip. A trailer that isn't stable will undermine your horse's confidence.\n\nIt's useful to have a helper nearby for this lesson, particularly if your horse refuses to walk straight up the ramp (see page 10), but if you are a good horseman, have boundless patience, and know your horse well, a helper isn't absolutely necessary. You'll want to pay close attention to your horse's body language during this exercise, so you'll know if and when he might choose to resist or strike out.\n\nWhen training your horse to load, always end on a positive note within the horse's comfort zone, and reward him with lavish verbal praise. Training your horse to load will take at least two sessions of work and probably more.\n\n**1**. Groom your horse to relax him and make him receptive to today's lesson.\n\n**2**. Reinforce walking and halting on cue by revisiting steps 1\u20133 on pages 7\u20138.\n\n**3**. After your horse responds successfully to the Walk and Halt cues, give the Walk command and approach the ramp calmly and purposefully. Don't lead him to the ramp; instead, walk _with_ him, holding the lead rope as you cue him to walk forward by voice and a tap of the whip, if needed. Depending on the configuration of the trailer, you may have to lengthen the lead to let him walk forward alone into his space on the trailer. But that, too, is the beauty of this technique. Because he's responding to your cue to walk forward and isn't being led, this will be an easy transition.\n\n**4**. When the horse reaches the ramp, he will probably balk, walk forward at an angle, stop, or back up. _If he balks_ , gently nudge him back into position facing the ramp, and then tap his hindquarters with the dressage whip to cue him to walk forward. _If he walks forward at an angle_ , attempting to step off the side of the ramp, position a helper with a second dressage whip on the far side of the ramp. Back the horse, and ask him to approach again. Work together with your helper, alternately tapping the horse's hindquarters if necessary. This should keep him moving straight ahead. _If he stops_ , tap his hindquarters with the dres-sage whip and cue him again to walk forward. _If he backs up_ , back him until he stops. Then stand for a brief while, let him relax, and cue him again to walk forward. If he insists on backing \u2014 some horses have learned this technique to evade loading \u2014 swat his hindquarters with the whip, using sufficient force to compel him to walk forward. _Note_ : Horses don't like to back up very far and typically use the maneuver as a last-ditch evasion; choosing between backing into the sting of a whip and going forward is really no choice at all, and your horse will quickly comply.\n\n**5**. It may take five, six, or even more attempts for your horse to walk onto the ramp and into the trailer. (Some horses may even need to revisit an earlier step in the process before proceeding. This is to be expected.) Once he is in, praise him warmly and touch him lightly on the hindquarters, which cues him to move forward and away from the unsecured butt bar. If he cannot go forward any farther, he will take the cue to mean, \"Don't move back.\" You will have walked into the trailer with the horse as far as you are able, depending on the trailer configuration and the number of occupants.\n\n_Once he is loaded, touch your horse lightly on the hindquarters_.\n\n**6**. Once inside the trailer, the horse should appear fairly relaxed and be willing to stand still until you cue him to back out. Secure the butt bar and then the cross-ties. Remove the lead chain from the horse's halter.\n\n**Reminder**\n\nA horse that loads badly will eventually be put through far more trauma than you might inflict with a well-timed, well-intentioned swat of the whip during training.Your goal is to make your horse understand that he must walk up the ramp and onto the trailer at your cue. He also must learn that evading, by backing up or moving sideways, has unpleasant consequences.\n\n### _Backing Out of a Ramp Trailer_\n\nOnce the horse has loaded successfully onto the trailer, and you've praised him and secured the butt bar and cross-ties, let him stand quietly for a few minutes. Then prepare to back him out of the trailer. Enlist the help of an assistant.\n\n**1**. Unfasten the cross-ties, and reattach the lead chain to the horse's halter.\n\n**2**. Have your helper unlatch the butt bar and then stand at the side of the ramp, where the horse will back off.\n\n**3**. With your hand, touch your horse's flank gently, even with just the tips of your fingers. This is a \"guiding\" touch to help guide him back, keep him straight, and maintain his trust, rather than the stronger, more positive \"stop\" touch used earlier. Say \"Back,\" and praise him when he has come completely off the trailer and is standing quietly, awaiting further instructions.\n\n### _How Long Does It Take?_\n\nYou may think that this whole process will involve only two sessions of work for you and your extremely talented horse \u2014 one for walking and halting on cue and one for loading on cue. Anything is possible, but such swift success is likely only if your horse is a trusting, willing pupil, hasn't had previous negative experiences (loading _or_ training), and you have the right attitude and sufficient persistence to work patiently through each step until he performs it correctly.\n\nMost horses require more than two sessions to learn to load properly. Because a horse's attention span is fairly short, consider breaking down the training process into manageable mini-sessions such as this:\n\n  **Session 1** : Walk and halt on cue\n\n  **Session 2+** : Walk up to the ramp on cue (as many times as needed to instill confidence in your horse)\n\n  **Session 3+** : Walk up the ramp and into the trailer on cue (as many times as needed to instill confidence in your horse)\n\nPay special attention to your horse's body language during each session, never push him beyond his comfort level, and always end on a positive note. The horse may get halfway up the ramp, back off quickly, then turn sideways. In such a case, straighten him out and repeat the loading cue. If the horse is slightly more successful on the next attempt \u2014 say he pokes his head into the trailer before backing out, for example \u2014 repeat the loading cue and stop him at the place where he is still comfortable. Praise him with an enthusiastic \"Good boy,\"then quit for the day. Don't give food treats, as you'll be out of luck one day if you need to load your horse and don't have any food rewards handy.Verbal praise works best, and a pat or stroke is also appropriate.\n\nWhen resuming training, repeat what was done successfully during the previous session.Your horse's confidence and comfort level should improve with repetition during each session.\n\n### _Loading onto a Step-Up Trailer_\n\nBecause a step-up trailer is far more challenging for the horse to negotiate, both physically and mentally, than a ramp trailer, I recommend training horses to load onto a ramp trailer first, if possible. When your horse is confident and comfortable loading onto a ramp trailer, he'll be more willing and able to negotiate a foot-high step and walk into a step-up trailer. The training process for both trailer types is more or less the same, but, generally, training a horse to load onto a step-up trailer takes at least twice as long as training to load onto a ramp trailer, mainly because the horse may be thoroughly intimidated by the step. The basic training objectives are as follows:\n\n>   Cue the horse to walk forward.\n> \n>   Cue the horse to walk forward to the trailer.\n> \n>   Cue the horse to walk forward and into the trailer.\n> \n>   Praise horse warmly and touch him lightly on the flank.\n> \n>   Secure butt bar, then cross-ties.\n> \n>   Remove the lead chain from the horse's halter.\n\nIf you must begin training with a step-up trailer, be _absolutely_ certain that your horse will walk forward on cue, no matter what. Then introduce him to the trailer.\n\nEnlist the aid of two assistants to help you guide the horse on and off the trailer, but be certain both are horsepeople and thoroughly familiar with how quickly a horse can back away and wheel around. Instruct the helpers to remain alert, safe, and move out of harm's way if necessary.\n\n**1**. Because on a step-up trailer there is no ramp to define how far right or left the horse can move, position one helper on either side of the trailer to define the path you want the horse to take. In effect, the helpers create a human wall or chute for the horse's progress.\n\n**2**. Cue the horse to walk forward. Encourage your helpers to reach out and gently direct him onto the trailer.\n\n**3**. When the horse reaches the trailer, he may automatically step up in obedience to your cue to walk forward. More likely, though, he will stop. _Never_ lift a horse's front leg and place it on the trailer in an attempt to have him continue forward; this is a dangerous practice and will only confuse the horse. Instead, back him, walk away from the trailer, reinforce the Walk cue, and approach the trailer again.\n\n**4**. If you get tired after a few refusals and backings, instruct one of the assistants to restart the process for you.\n\n**5**. When the horse is successfully loaded, secure the butt bar, then the cross-ties. Remove the lead chain from the horse's halter.\n\n### _Backing Out of a Step-Up Trailer_\n\n_Once your horse is comfortable and confident, he will step up and walk right into the trailer at your urging, without help from assistants_.\n\nIf your horse loaded successfully onto a step-up trailer, next he must learn to back and step down. Because horses don't like to back, let alone back and step down blindly, this lesson can be quite stressful for the horse.\n\nYou'll begin as you would for backing from a ramp trailer, this time enlisting the aid of two helpers. As you did for loading, position your helpers on either side of the trailer to form a human chute. Take your position next to the horse as soon as possible, turning yourself so you face the trailer's exit.\n\n**1**. Unfasten the cross-ties, and attach the lead chain to the horse's halter.\n\n**2**. Have a helper unlatch the butt bar and then stand at the side of the trailer, where the horse will back off. Have the other helper stand on the opposite side.\n\n**3**. Position yourself at the horse's shoulder, give the Back command, and instruct the helpers to guide the horse back with encouraging words and a gentle touch on the hindquarters. Do not use reward words (e.g., \"Good boy,\"\"Good horse,\"etc.) until the horse is safely off the trailer.\n\n**4**. When the horse has backed and stepped down successfully, give lots of verbal praise and physical rewards, such as pats and neck rubs.\n\n**Good Footing**\n\nBecause it is difficult for a horse to exit from a step-up trailer, good footing is extremely important. Always park a step-up trailer on grass or dirt, not pavement. Horses respond best to soft, familiar surfaces when unloading, especially early on. If a horse puts his hind feet down on a slick, hard surface, he might scramble, hurting himself, a handler, or a bystander. When your horse is comfortable loading and unloading, it may be safe to unload on hard surfaces, but if you have a choice park on grass or dirt. (Note: A ramp trailer should also be parked on grass or dirt, especially at first, but this is not quite so critical as it is with a step-up trailer.)\n\n## Points to Remember\n\nCue-on loading is a straightforward training method, but lest you be overly optimistic, keep these points in mind.\n\n  **You must be patient**. Everyone who trains a horse in the cue-on loading technique knows that it's not a source of instant gratification. Learned successfully, however, it leads to a lifetime of good loading for your horse.\n\n  **It's not a quick fix**. Don't expect a so-called bad loader to learn cue-on loading in a day because you want to go to a show; and don't expect a good lead-on loader or a green-but-not-ruined horse to learn it in a day, either. Train your horse for loading as intensively as you would train him for any other essential skill _before_ you travel together.\n\n  **Always give the verbal command and physical cue simultaneously**. This allows your horse to associate the command with the desired response. If you were to give the physical cue before the verbal command, the horse might wonder why he's being hit and become agitated.\n\n  **Reward only the desired behavior**. During the training process, be careful not to talk nicely to or pet your horse when he becomes upset and does not do what you want. Such gestures of consolation only confuse the horse, causing him to think that he is being rewarded for not cooperating. Instead, use a firm, low tone, and don't yell. Think of this as your \"command\" voice. Repeat your cues insistently and calmly until your horse complies, no matter how upset he may seem to be. When he's performed the desired behavior, then you can reward him with verbal and physical praise.\n\n  **When loading, always fasten butt bar** _**before**_ **the cross-ties**.If the horse is cross-tied before the butt bar is up, he could back up quickly, break the cross-ties, and injure himself or a handler.\n\n  **When unloading your horse, always unfasten the cross-ties** _**before**_ **the butt bar**. If your horse is tied and the butt bar is down and something startles or frightens him, he may pull back, ripping the tie ring out of the trailer wall and injuring himself or you.\n\n  **Practice**. When you are fairly confident that your horse knows how to load and unload, practice with him several more times. Repetition reinforces learning. After reinforcing the skill, you can give it a rest until you're ready to load, particularly if you're traveling fairly soon, say within a week or two. If there are long time lags between trips, revisit the skill every few months, until the horse has been successfully loaded and hauled several times, at which point you'll be absolutely sure that knows what to do.\n\n  **Keep your dressage whip handy**. Always keep your dressage whip with you when traveling, particularly for the first few road trips with your horse, just in case an old fear or doubt should get the better of him. If, for reasons beyond your control, your horse has a bad loading experience or a bad trip after learning the cue-on loading method, you may need to do a short refresher course.\n\n## Hauling Options\n\nWhatever type of rig you choose and however many horses it can carry, the rig should be tall enough and wide enough to allow each horse to move his head about freely to maintain balance. If partitioned, each stall in the rig should be wide enough and long enough for the horse to move a step forward or backward and to shift his weight from side to side somewhat. If you can squeeze past your horse once he's loaded in the rig, the stall is probably a good size.\n\n### _Horse Vans_\n\nA British horse van, or permanent horse transport, might more aptly be described as a \"horse box\"\u2014 in short, a stall in which horses can move about with considerable freedom, as if at home. Studies have shown that most horses experience minimal stress when traveling this way. They are free to position themselves forward, backward, or sideways, as they wish and as the van's motion dictates, thereby minimizing discomfort and cramping. Horses can fall in a horse van if the ride is bad, but careful, predictable driving and good road conditions minimize its likelihood.\n\nThough horse vans generally offer a smooth, stable ride, their side ramps are typically steep. Horse vans are most commonly used in professional hauling, often for racehorses. At the racetrack, the ramp is extended, not from the truck's deck to the ground (which can be as much as three feet away) but to an earthen off-loading ramp that provides a gentle descent to the ground. Without that useful intermediary structure for loading and unloading, the ramp must be covered with a non-slip material, such as a hemp mat, to provide adequate traction so the horses can negotiate the ramp safely. Be sure the mat is fastened securely at top and bottom to provide solid footing. In the event of mechanical problems, horses must be off-loaded and then reloaded onto a replacement vehicle, which can be a major inconvenience.\n\n### _Horse Trailers_\n\nTrailers range in style from light all-aluminum carriers to heavy steel and plywood models, with a wide assortment of models in between.Your finances, tow vehicle, and horse should all be carefully considered before making a purchase. Determine how much you can reasonably spend, and then research what's most appropriate. The trailer's hitching device and configuration are also important considerations.\n\nAmong detachable trailers that require a separate towing vehicle, the basic choice is between a gooseneck and a bumper hitch, also called a _tagalong_. Each hitch type is available for various trailer styles, which you should consider carefully when choosing a trailer to suit your traveling needs. One advantage of a detachable rig is that if the towing vehicle has problems, the trailer can easily be reattached to a replacement vehicle, with no unloading of horses required.\n\n### _Hitch Types_\n\n_Gooseneck hitches_ are permanently mounted on the bed of a pickup with a large bracket. Goosenecks offer an excellent turning radius and are more stable on the road than bumper hitches. They are more expensive, however, and, pound for pound, require a heavier, more expensive towing vehicle than do bumper hitches.\n\ngooseneck hitch\n\nThe _bumper hitch_ attaches to a bumper or frame-mounted receiver on the towing vehicle. If the engine is strong enough and the tow vehicle's frame is sturdy and made of steel, pickup trucks, larger sports utility vehicles (SUVs), and even \"land yacht\" types of cars, such as old full-size Cadillacs, can haul trailers with bumper hitches. (If hauling by car, a oneor two-horse trailer is the largest size you'd want to risk. Confirm your vehicle's towing capacity with a mechanic.)\n\nbumper hitch\n\n### _Trailer Style_\n\nOnce you've selected a hitch type, you must consider what trailer style will be most appropriate for you and your horse. Any trailer type can be outfitted with a ramp or a step up.\n\n**Straight-load trailer**. A traditional trailer, the straight-load is available in several sizes and can accommodate from one to four horses, possibly even five. Horses generally enter side-by-side and face forward. A four-horse straight-load may place two pair of horses, one in front of the other, or in an extremely large trailer may place two horses facing the rear and two horses facing forward, with a single aisle between each pair from which the horses load and unload. If need be \u2014 and only if the flooring is sufficient and your tow vehicle is able to pull the extra weight \u2014 a pony can be hauled facing sideways in the aisle space.\n\n**Two-horse front-load trailer**. In the two-horse front-load, a variation of the straight-load, horses load from the front and stand facing _away_ from the direction of travel. The horses load through small ramps, on either side of the center line, that are angled to accommodate the tow bar and the tow vehicle. These trailers aren't common, but studies have shown that horses experience less stress when hauled facing away from the direction of travel, so it might be worthwhile searching for one, particularly if your horse doesn't travel well. _Warning: Never_ back horses into a straight-load trailer. The load distribution is not designed for rear-facing horses, nor are the ties and other equipment. Serious and needless injury could result.\n\n**Slant-load trailer**. In the slant-load, horses are loaded from the rear but stand at an angle, about 60 degrees off center, with their heads facing toward the left side of the rig. A metal partition or bar defines the space for each traveler. Although they may offer enough head room, slant-load trailers are often too narrow for warmbloods and large Thoroughbreds, unless the partitions are adjustable and can be rearranged to accommodate two larger horses in a space that might normally carry three Arabians or Quarter Horses, for example. Some slant-load trailers offer rear-facing travel (that is, 60 degrees off center, with the head facing toward the rear rather than toward the tow vehicle), but they are rare and difficult to find. Slant-loads usually have a high step up, as much as a foot from grade, in order to maintain the structural integrity required to carry so much weight in a relatively short trailer.\n\nslant-load trailer\n\n**Stock trailer**. In the United States, some people use stock trailers, originally intended to haul cattle, to approximate the \"loose-box\"\u2013style trailering of the British horse van.Though this is fine for shorter horses, such as Arabians and old-style Quarter Horses and ponies, a Thoroughbred, Saddlebred, racing-style Quarter Horse, or warmblood will not have sufficient headroom and, therefore, won't be able to maintain proper balance. If the tall horse cannot balance properly, he may fall down, resulting in serious injury. Also, if the horse is scared and flips his head up, even while wearing a head bumper (that is, a poll protector or guard), he may die.\n\nStock trailers offer good ventilation, the open-slat areas providing excellent air circulation. Some horsemen who use stock trailers partially cover the openings with Plexiglas to reduce drafts and to prevent rain from entering. Others fit the trailers with partitions. Some load as many as three horses side-by-side in a front row, then install a partition and load another row of three behind. Typically, horses in a stock trailer lean on each other without much protection, so it is especially important to wrap legs and use head bumpers.\n\nTrailers for two or four horses that have no permanent partitions are sometimes used in a similar manner when hauling a single horse. This approach can work quite well. The horse can then decide where he feels best balanced, and he probably won't upset the balance of the trailer when he adjusts his position. He also can stretch his legs and neck. The drawback is that a sudden stop could cause more serious injury to a loose horse than to one tied and held in place by leads, partitions, and a butt bar. Most typically, two horses are hauled in a two-horse trailer, three or four horses are hauled in a four-horse trailer, and so on, eliminating loose-box travel entirely.\n\n**Manger-style trailer**. In a manger-style trailer, the area in front of the horse's head is boxed as storage space.That may be nice if the trailer lacks a dressing room for storage, but it will restrict the horse's ability to stretch his head down and drain his sinuses, contributing to travel stress. If you have long trips planned, a walk-through or Thoroughbred-style trailer will be a better option for your horse, even though you'll have to store items in your vehicle. (If you're hauling with a car or an SUV and you need to carry lots of hay with you, you can mount a hayrack on the roof of the trailer. And by all means, cover the hay with a tarp if bad weather is predicted.)\n\n## Safety First\n\n_Before_ you buy, have a mechanic inspect the trailer, especially if it's used, to confirm that all mechanical and electrical parts are in good working order. Also, check the tire condition; tires are easily replaced if worn, so don't reject a trailer solely because of its tires. Then check, or enlist the help of a knowledgeable horseman to check, the trailer's structural integrity. The floor should be solid, not warped, damp, rusty, or soft. The ramp hinges should be in good working order, not corroded or sprung. The walls should be sturdy enough to withstand hoof impact. If you prefer a thin-walled trailer but have a big horse that kicks, hang stall mats on the walls for added insurance.\n\nBefore you haul or have your horse hauled by others, always check the tires for adequate inflation and good tread. Then do a through visual inspection of the trailer's interior. Make sure there are no nails, loose wood, bent metal parts, or other protrusions that might pierce your horse. Make sure everything is loaded securely, not loose or precariously positioned, as falling and rolling objects might frighten or injure your horse. Check the hay nets; they must be securely fastened in a place accessible to the horse. Check any side half-doors or drop-down windows to be sure they will stay closed during travel. Finally, be certain that when loaded, the horse and all his parts, including his head, will remain inside the trailer at all times during travel.\n\n## Travel Equipment and Supplies\n\nNo matter what type of trailer you choose or must use when traveling, preparing your horse for the trip and packing needed supplies can help make the trip as comfortable and as stress-free as possible for your horse.\n\n### _Protective Gear_\n\nWhenever your horse travels, he should be protected from bumps and cuts due to jostling and unexpected jolts in the trailer.\n\n  **Halter**. When trailering, _always_ use a leather or breakaway halter that has a nylon body and leather crownpiece. If your horse gets himself twisted, he will be able to struggle and break free of the halter.\n\n  **Lead rope with chain**. Use a lead rope with a chain to load the horse. When your horse has learned cue-on loading, you no longer need the lead, but because your horse became accustomed to it during the training process, continue with what he knows. Tie with cross-ties, then remove the chain. If the trailer isn't equipped with cross-ties, switch to a lead without a chain for tying.\n\n  **Head bumper**. A head bumper is a good idea, particularly for tall horses being transported in short trailers. Avoid hauling in a trailer not designed for your type of horse, and always take needed precautions. Horses generally don't rear in trailers, but they might fling their heads in the air or jump a little if they are frightened. Head bumpers are designed to protect the head and are made of either dense felt or foam covered with leather or leatherlike man-made material. The head bumper attaches to the halter.\n\nhead bumper\n\n  **Quilts and bandages**. If you have sufficient time and are able to apply them properly \u2014 that is, not so tight that they bow tendons nor so loose that they come off and entangle your horse's feet \u2014 you can use quilts and standing bandages to protect your horse's lower legs during travel. Applying quilts (or fleece or the newer no-bow, fleece-covered, foam-filled leg wraps) and bandages can be time-consuming, so if you choose this option apply them early on the day of the trip. (If you're new to bandaging, it may take 30 minutes or more for you to complete the job. Have an experienced person check that the bandages aren't too tight when you're done.) If the horse stands quietly, an experienced bandager can do the job in as little as five to ten minutes.\n\n  **Shipping boots**. For spur-of-the-moment trips or if your horse is being transported by someone who is not a proficient bandager, shipping boots are a good alternative. Typically, shipping boots, which reach from just below the horse's knee to the top of the hoof, incorporate a bell-boot section that protects the coronet band. (If you use wraps and standing bandages, begin with a bell boot, applying it before you begin to wrap, so the horse won't damage the hoof wall or coronet band if he scrambles.) If you plan to use shipping boots, put them on your horse and lead him around the property a few times to get him used to how they feel _before_ you ship.\n\n**1**. Wrap the quilt around the leg, keeping it low and fairly snug. Secure.\n\n**2**. Beginning in the middle, wrap the bandage in the same direction as you did the quilt. Leave some of the quilt visible.\n\n**3**. Wrap back upward, applying a fair amount of pressure. Finish at the top, and secure with tape.\n\n**4**. Wrapping the quilt and bandage low on the leg provides added protection.\n\n_Shipping boots feel strange to a horse. Get your horse accustomed to them_ before _you travel_.\n\n### _Other Provisions_\n\nIt goes without saying that when traveling you must load all of your horse's standard equipment and anything extra you think you might need at his destination or for the event in which you are participating. Here are a few essentials that I never travel without:\n\n  **Water**. Horses are often quite particular about their water, so stow some five-gallon jugs of their regular water, just in case. Adding Kool-Aid to strange water is said to make it palatable for horses, so you may want to bring some of that along as well. At home, an adult horse can drink 10 to 20 gallons of water per day. Generally, horses drink less during travel, but a minimum of 10 gallons per day per horse should be enough.\n\n  **Hay and feed**. Never change your horse's hay or feed immediately before or after a trip. If you are traveling to a new barn, bring along some of his regular hay and grain \u2014 enough for several days. Use his standard ration for the first day or two, then mix it gradually with what his new home offers. Changing food gradually in this way minimizes your horse's gastric distress. Before, during, and after travel, be sure your horse has good-quality hay to keep his gut active, something partially helped by daily exercise when he's not traveling.\n\n  **Hay nets**. For trips that take longer than an hour, and particularly if you're headed to a horse show where the horse may stand in or be tied to the trailer for most of the day, be sure you load hay nets to capacity, so he can nibble to pass the time and keep his gut active.\n\n  **Fly sheets, coolers, and sweat sheets**. These can be useful extras, particularly if you're traveling out of your immediate area.\n\n  **Treats**. Always have a generous supply of your horse's favorite treats available.\n\n### _Creature Comforts_\n\nWhen your horse can load successfully and you've packed all needed equipment, you've got about 90 percent of the trailering battle won, at least as far as your horse is concerned. But there are a few other considerations that deserve your attention, which may greatly improve your horse's comfort during travel.\n\n  **Ensure adequate ventilation**. Adequate ventilation protects against excessive stress during transport. Stuffy enclosed areas, such as a poorly ventilated trailer, not only contribute to your horse's physical stress, they also encourage the spread of airborne pathogens and can increase the risk for developing respiratory illness. If you must use different trailers, familiarize yourself with them before you travel, and consult their owners for advice regarding the best window\/vent arrangements.\n\nWhen making adjustments to a trailer's ventilation, angle the vents in such a way that the incoming air doesn't blow directly on your horse. One of the best ways to determine airflow is to adjust the vents, hop in the trailer (with no horses loaded), and have someone tow it around the farm property so you can feel for yourself how much air is entering and where the air travels. In addition, observe your horse after a few local trips. Has his mane blown to the opposite side? Is he trying to huddle against one side or other of the trailer?\n\n  **Allow sufficient slack in the lead**. Unless you must tie the horse's head close to the trailer wall \u2014 because he is next to a fighting horse, for example \u2014 don't run the lead short. When tied, your horse should be able to drop his head enough to allow for proper drainage of the sinuses, which helps to prevent buildup of harmful microorganisms and dust. If your horse can drop his head to the point of a grazing posture when tied, the lead is too long and must be shortened. A long lead is dangerous for the horse at all times when traveling, but especially during a sudden, unexpected stop.\n\n  **Provide sufficient bedding**. My very tall horse (16.3 hands high) lifts his tail and lets fly about five minutes down the road. I've always assumed that the trailer's motion relaxes him because he is a good traveler, but such ready defecation might just as easily be an expression of nerves. Whichever the case, be sure to provide sufficient bedding for your horse and monitor your horse's bodily functions closely during travel. Some horses simply refuse to urinate in a trailer; others urinate a lot. Both can be problematic for different reasons. Be vigilant during your first few road trips, carefully assessing your horse's condition before and after travel. If you suspect your horse is losing condition, even on short trips, consult your veterinarian for possible solutions.\n\n  **Try a special browband**. On a hot day, you might want to install a wettable visor on your horse's browband to keep him cool. Or, you might try a magnetic browband, which some equine massage therapists and acupuncturists swear by as effective tools for relaxing horses. It is believed that the magnetic field produced by the browband alters the way certain molecules important to cell function act within the brain, thereby stimulating the production of calming hormones such as melatonin. For best results, a magnetic browband should be applied at least an hour before travel.\n\n## Seasonal Strategies\n\nOn a trip of any length, plan to check the horses at least every three hours to be sure that they have hay and have not gotten untied or entangled in a fallen hay net. If you know that a trip will take longer than six hours, plan ahead and research a safe place to unload the horses to let them graze and stretch their necks for 15 to 20 minutes.\n\n**Summer travel**. In summer months, adequate ventilation is relatively easy to achieve. Just leave the windows and any top closure above a rear-facing ramp open. And keep the rig moving. Carry five-gallon containers of your horse's usual water, and offer it each time you stop, preferably in a familiar bucket. How many jugs you'll need depends on your horse's drinking habits, so observe them carefully before a trip, and load water accordingly.\n\nIf you stop to eat and refuel, try to park in the shade, open the tops of hay doors if you have them, and make your stop as short as possible.\n\n**Winter travel**. In winter, many people are inclined to keep a trailer or van tightly closed in an effort to make it warm and cozy for their horse. This is a mistake, because if your trailer is closed tightly you simply cannot provide your horse with adequate ventilation and he will be too warm. Dr. Goodwin was fond of telling his students that horses lived outside successfully, in all sorts of weather, for fifty-five million years without any help from humans, and he'd just as soon have seen all horses living outside virtually all of the time, unless they were sick or injured. And even then outdoor living had its benefits. So don't coddle your horse.\n\nIf you see condensation forming on the windows, you can be sure that the trailer or van is too hot and damp, and the horses are uncomfortable. By not providing adequate ventilation, you inadvertently create an environment that does more harm than good.\n\n**Tip for Winter Blanketing**\n\nIn winter, be careful not to blanket your horse too heavily during transport. Blanket your horse appropriately for his clipped or un-clipped state and the day's temperature. Then before loading your horse, physically stand in the trailer wearing proper attire for the weather outside. If you are slightly uncomfortable \u2014 perhaps a bit chilly \u2014 in the trailer or van, it's probably just the right temperature for your horse. If you're comfortable or warm, it's probably too warm for your horse and you should adjust the vents to allow in more air. Remember, he is always wearing a fur coat.\n\n## Documentation\n\nThere's nothing more unnerving than not having the papers you need when you need them most. When traveling with your horse, make it a point to bundle up these documents, just in case. Keeping all of your important papers in a flexible file is a good idea.\n\n  **Negative Coggins test**. When traveling with your horse, make sure that you or the hauling company (if your horse is being shipped professionally) has a current negative Coggins test on hand for your horse. The Coggins test determines whether the horse is a carrier of equine infectious anemia; a negative test indicates that the horse is not a carrier and poses no risk of spreading the disease.\n\n  **Current health certificate**. Some states require a current health certificate in addition to a negative Coggins, so check with the Department of Agriculture in your state before you cross state lines.\n\n**The Threat of Shipping Fever**\n\nThe incidence of strangles, or \"shipping fever\"\u2014 a highly contagious infectious disease caused by _Streptococcus equi_ that occurs most commonly in horses transported long distances (more than 500 miles in one trip) \u2014 has diminished greatly in the last quarter century. The decline clearly has much to do with vaccines and other preventive strategies, such as improved feed and disinfecting trailers after each trip.\n\nCharacterized by high temperature (103\u02da\u2013104\u02daF), nasal discharge, cough, difficulty swallowing, and swollen lymph glands that may abscess and burst, strangles is principally spread by direct contact with nasal secretions or pus draining from an abscess. Interestingly, _S. equi_ is found in just about every stool sample taken from horses, whether or not the horses show signs of disease. Stress, which negatively affects the immune system, may be all that is necessary to allow the incubating bacterium to take hold, successfully combating the horse's weakened immune system to cause obvious signs of disease. A lengthy, stressful road trip may increase your horse's susceptibility to _S. equi_. Treatment includes penicillin and excellent care, specifically clean water and feed, a dry stall, good sanitation, and peace and quiet.\n\n  **Insurance papers**. When hauling long distances, especially, you may want to bring your insurance papers \u2014 for horse and vehicle \u2014 along for the ride.\n\n  **Contact information and contingency plans**. In case of an emergency, particularly if you become injured or incapacitated during the trip or at a show, it's a good idea to carry with you contact information and contingency plans so people will know whom to contact and what to do.\n\n  **Recovery information**. If your horse had been tattooed or has had a chip implanted to prevent theft and to facilitate recovery efforts, you may wish to carry that information along with you, specifically the identification number itself and any documentation your veterinarian provided.\n\n  **Records and registration**. If you are going to a show and need proof of the horse's prior performance or size (pony papers, for example) or breeding (registration for a breed show), stow copies of that information, too.\n\n## Rules for the Road\n\nIn some states, a commercial driver's license is required if you plan to haul anything larger than a two-horse trailer. Check with your state's registry of motor vehicles before planning a cross-country trip with Sir Speedo. And always observe these simple rules of the road when hauling.\n\n  **Don't exceed your rig's hauling capacity**. Know what it is and the weight of all horses to be loaded _before_ you travel.\n\n  **Check tire pressure for adequate inflation**. Do this the day before you travel in case you must repair or replace tires.\n\n  **Load properly**. If you are loading one horse into a two-horse trailer, load him on the left so his weight is over the crown of the road, rather than on the lower shoulder at the edge of the road. If you're loading two horses, put the larger one on the left, again to keep the greatest weight toward the road's crown rather than toward its sloping shoulder. In the event of poorly maintained road surfaces or the need for defensive driving maneuvers, this may help to prevent a rollover.\n\n  **Drive well**. Remember that you're hauling precious cargo that's standing in the trailer behind you. Drive carefully, being considerate of your horse and other drivers. Make no jackrabbit starts or stops. As a general rule, leave at least one car-length in front of you for each ten miles of speed you are traveling. With a trailer adding pushing weight from behind, that's the minimum allowable distance you must maintain if you wish to make a good, safe stop, and even then a stop on demand will likely toss the horses a bit.\n\nWhen turning, slow down, and be sure you have sufficient room to make a wide turn. It's a good idea to practice turning and other basic maneuvers with an empty trailer _before_ hauling horses.\n\nWhen you're driving on the highway, realize that a semi that passes you will produce enough drag on your truck and rig to challenge your driving skills. Be prepared to compensate for such buffeting by skillful handling. Always keep two hands on the wheel.\n\n  **Monitor mirrors**. Large mirrors are essential for both sides of the towing vehicle. They allow you to see beyond the towing vehicle to the trailer itself and should be monitored regularly as you drive. You'll be able to see if a horse manages to get a body part outside the trailer, which can have tragic consequences.You'll also be able to see the semi bearing down on you and will be able to react accordingly.\n\n**More Travel Tips**\n\nA few final reminders:\n\n  Until you understand that when you back and steer left, your rig turns right, look for drive-through spaces at shows or rest stops when you park.\n\n  Carry water for horses to drink and water for people to drink.\n\n  Carry snacks for horses and people. Horses can eat their snacks (i.e., hay) as they travel, but for safety sake you should eat snacks only when you've stopped. Eating while driving increases the risk for accidents.\n\n  Carry first aid kits for horses and people.\n\n  Carry a cell phone, fully charged, but with a vehicle power cord as well. If you must use the phone, pull into a rest area and stop the vehicle. Using a cell phone while driving in a car increases the likelihood of an accident by 30 percent. Factoring in the challenges of managing a rig carrying horses, the risk only increases. Don't do it.\n\n**Don't Double Up**\n\nFor simplicity's sake, you might be tempted to bridle your horse and then slip the halter over the bridle before you travel. Don't do it.Your horse could catch the bridle inside the trailer and do serious damage to his mouth. The bridle also would interfere with hay consumption during the trip.\n\n## When You Reach Your Destination\n\nAfter safely reaching your destination, back your horse out of the trailer, let him graze on the lead, if possible, or walk him for at least fifteen minutes. Once your horse is comfortable and relaxed, you're ready to tack up and mount.\n\n### _Bolt-Free Bridling_\n\nSometimes it's impossible to bridle a horse in a trailer, and many horses don't cooperate if they must be bridled in an open space. By following these simple steps, you'll be able to bridle your horse with ease at any location, and at no time will he think he is free and able to escape.You'll loop the lead over his head in two places to \"secure\"him while you bridle him.\n\n**1**. Before you slip the halter off, prepare to apply a lead halter. Stand on the near side of the horse, at his head, close enough to reach under his jaw with your right hand.\n\n**2**. Unlatch or unbuckle the leather halter. Before you let it slip down onto your left hand, or before a helper grabs it, with your right hand pass the nonchain end of the lead away from you, under the horse's throatlatch, and back up over his poll. Rest your right hand on his nose, a few inches above the nostrils, grasping and holding the lead at that point. Basically, you are creating a figure eight around his poll and nose.\n\n**3**. Run the end of the lead away from you, over his nose, where the noseband of a halter would rest.\n\n**4**. Bring the end of the lead back under his nose, grasping it and the loose length of halter hanging from the near side of the horse's head with your right hand. The lead forms a loose figure eight that you now grasp in the middle.\n\n**5**. Slip the reins over his head with your left hand, grasping the crown-piece of the bridle at the same time. Alternatively, have a helper slip the reins over the horse's head while you hold the top of the bridle.\n\n**6**. Slide the bridle up as you normally do, positioning the bit under his mouth and slipping it in, all with your left hand. Slip the crownpiece over one ear, bringing the ear forward, and then prepare to let the temporary hand-held halter slip away as you use your right hand to slide his near-side ear through the bridle.\n\n**7**. Release the lead, which will drop to your horse's feet. Then latch or buckle the halter.\n\n### _Mounting_\n\nIf you are traveling to a show, always bring and use a mounting block. After loading and hauling your horse, and putting on his tack carefully with the least possible fuss and muss on his part, don't take any chances when mounting. Imagine your horse's confusion if you go through all that work, only to swat him up against a car bumper so you can clamber aboard. And you certainly wouldn't want to bounce down on his back after an inadequate leg up. If bystanders are pressed into groom service, you might even be foisted up and over \u2014 how embarrassing!\n\n\u2666 \u2666 \u2666\n\nIt's incredibly demanding to travel with your horse. But the rewards are worth the effort, particularly when your horse loads on cue, travels safely and comfortably with minimal stress, and maintains his condition.\n_The mission of Storey Publishing is to serve our customers  \nby publishing practical information that encourages personal  \nindependence in harmony with the environment_.\n\nEdited by Marie Salter  \nCover illustration by Joanna Rissanen  \nCover design by Carol J. Jessop (Black Trout Design)  \nText illustrations by Joanna Rissanen, except page 5 (right) by Jim\n\nDykeman and page 23 (top) by Alison Kolesar\n\nText production by Jennifer Jepson Smith and Deb Daly\n\nCopyright \u00a9 2001 by Storey Publishing, LLC\n\nAll rights reserved. No part of this bulletin may be reproduced without written permission from the publisher, except by a reviewer who may quote brief passages or reproduce illustrations in a review with appropriate credits; nor may any part of this bulletin be reproduced, stored in a retrieval system, or transmitted in any form or by any means \u2014 electronic, mechanical, photocopying, recording, or other \u2014 without written permission from the publisher.\n\nThe information in this bulletin is true and complete to the best of our knowledge. All recommendations are made without guarantee on the part of the author or Storey Publishing. The author and publisher disclaim any liability in connection with the use of this information. For additional information please contact Storey Books, 210 MASS MoCA Way, North Adams, MA 01247.\n\nStorey books and bulletins are available for special premium and promotional uses and for customized editions. For further information, please call the Custom Publishing Department at 1-800-793-9396.\n\nPrinted in the United States\n\n**Library of Congress Cataloging-in-Publication Data**\n\nMcBride, Laura Harrison.\n\nTrailer-training your horse \/ Laura Harrison McBride.\n\np. cm. \u2014 (A Storey country wisdom bulletin; A-279)\n\nISBN 1-58017-417-5 (alk. paper)\n\n1. Horses\u2014Transportation. 2. Horses\u2014Training. 3. Horse  \ntrailers. I. Title. II. Series.\n\nSF285.385. M33 2001\n\n636.1'083\u2014dc21\n\n2001049576\n\n## MORE COUNTRY WISDOM  \nBULLETINS YOU WILL ENJOY!\n\n_These and other Storey Country Wisdom Bulletins are available for $3.95 at your local bookstores, garden centers, farm stores, and gift shops. Use the order numbers listed under each bulletin to make your requests. You can also order directly from Storey Books by writing to us at 210 MASS MoCA Way, North Adams, MA 01247, or by calling 1-800-441-5700. For more information about our books and bulletins, visit our Web site atwww.storey.com_.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n_The Happy Lawyer_\nNANCY LEVIT & DOUGLAS O. LINDER\n\n# **The Happy Lawyer**\n\n_Making a Good Life in the Law_\n\nOxford University Press, Inc., publishes works that further  \nOxford University's objective of excellence  \nin research, scholarship, and education.\n\nOxford New York  \nAuckland Cape Town Dar es Salaam Hong Kong Karachi  \nKuala Lumpur Madrid Melbourne Mexico City Nairobi  \nNew Delhi Shanghai Taipei Toronto\n\nWith offices in  \nArgentina Austria Brazil Chile Czech Republic France Greece  \nGuatemala Hungary Italy Japan Poland Portugal Singapore  \nSouth Korea Switzerland Thailand Turkey Ukraine Vietnam\n\nCopyright \u00a9 2010 by Oxford University Press, Inc.\n\nPublished by Oxford University Press, Inc.  \n198 Madison Avenue, New York, New York 10016\n\nwww.oup.com\n\nOxford is a registered trademark of Oxford University Press\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.\n\nLibrary of Congress Cataloging-in-Publication Data Levit, Nancy.  \nThe happy lawyer : making a good life in the law \/ Nancy Levit and Douglas O. Linder.  \np. cm.  \nIncludes bibliographical references and index.  \nISBN 978-0-19-539232-6  \n1. Practice of law\u2014United States\u2014Psychological aspects.  \n2. Lawyers\u2014Job satisfaction\u2014United States. I. Linder, Douglas O., 1951\u2013  \nII. Title.  \nKF300.L485 2010  \n340.023\u203273\u2014dc22 2009047527\n\n9 8 7 6 5 4 3 2 1\n\nPrinted in the United States of America  \non acid-free paper\n_For Tim, Dylan, and Aaron  \nNEL_\n\n_For Cheryl, Kari, and Ann  \nDOL_\n_The logic of the rebel is to want to serve justice so as not to add to the injustice of the human condition, to insist on plain language so as not to increase the universal falsehood, and to wager, in spite of human misery, for happiness._\n\n\u2014Albert Camus\n\n## **Contents**\n\n_Foreword_\n\nCHAPTER 1 Are Lawyers Unhappy?\n\nCHAPTER 2 Happiness: A Primer\n\nCHAPTER 3 What Makes Lawyers Happy and Unhappy\n\nCHAPTER 4 The Happiness Toolbox\n\nCHAPTER 5 Preparing for a Satisfying Career: The Law School Years\n\nCHAPTER 6 What Law Firms Can Do to Make Lawyers Happier\n\nCHAPTER 7 Lawyers' Stories\n\nCHAPTER 8 Seeking Happier Ground\n\n_Acknowledgments_\n\n_Notes_\n\n_Index_\n\n## **Foreword**\n\nWE LIVE IN A NATION FOUNDED BY LAWYERS. THIRTY-four of the fifty-five men who gathered in Philadelphia in 1787 to draft our Constitution were lawyers, including the document's principal architect, James Madison. Lawyer Thomas Jefferson authored the Declaration of Independence (with the help of John Adams, another lawyer).\n\nAmong the inalienable rights Jefferson listed in his famous document of 1776, along with the rights to life and to liberty, was the right \"to the pursuit of Happiness.\" \"Happiness\" is not a word you expect to find in a formal document drafted and debated by sober eighteenth-century politicians. But there it is on parchment, one of the three most important rights of a free people.\n\nOf course, Jefferson couldn't promise we would find happiness, only that we had a right to pursue it. And pursue it we have, and still do\u2014with ever more intensity\u2014today. A quick Google search or perusal of the self-help section at your local bookstore will reveal the depth of our current interest in happiness. A whole movement in modern psychology called \"positive psychology\" has sprung up to help improve our odds of finding a greater measure of happiness.\n\nFor all our efforts, Americans are no happier today than we were a half century ago\u2014and, by most measures, we're less happy. We're richer than ever before, have more leisure time, and are awash in happiness literature, but we still cannot seem to get more satisfaction.\n\nFor lawyers, the picture is even bleaker. Many lawyers, as many as 70 percent in one poll, say they would not choose a legal career if they had to do it all over again. Half of all lawyers would discourage their kids from becoming lawyers. Over one-third of big firm associates leave their firms within three years of being hired. Students at Stanford Law School, dissatisfied with the nature of large firm practice, have launched a movement to change the way law is practiced.\n\nIt is an exaggeration, however, to describe law as a profession in crisis. We do not see headlines announcing: \"Lawyers leaving profession in droves.\" The number of law school applicants remains consistent, and the number of law schools is increasing. Despite the odd story of a lawyer leaving practice to start a vineyard or become a hot air balloonist, most in the practice say they intend to stick it out. Moreover, many say they are at least reasonably happy, even while admitting they'd like to be more so.\n\nThe story of the emotional state of American lawyers today is complicated and nuanced. Lawyers are smart people; they expect complications and appreciate nuance. If lawyers are not as happy as they'd like to be, there are many possible explanations for that. People generally don't become lawyers because they wake up one day and say, \"By God, I love the law!\" They might become lawyers because as college seniors they simply wanted to keep their options open and law school seemed a better place to do that than, say, dental school. Moreover, it might well be that the sort of person best suited to the practice of law is more likely to have a personality type somewhat prone to unhappiness. In law, unlike in many other fields, it often helps to be an anxious, pessimistic person who worries about all the things that can go wrong if, for example, a contract isn't written just so. If lawyers are slightly unhappier than the typical American, it may be because unhappy people are more likely to become lawyers rather than that the practice of law has made them unhappy.\n\nSorting out this complex story is one of the goals of this book. But we don't stop there. While recognizing the limits of self-help, we offer \"a happiness toolbox\" that we believe can lead to happiness boosts for lawyers who follow our suggestions. There are multiple paths to happiness, and a tool that works well for you might not for the lawyer in the next-door office. We also will offer tips for law firms and law schools that wish to improve levels of satisfaction among their ranks. Along the way, we'll share stories of both unhappy lawyers and happy lawyers, learning what we can from the insights they've garnered along their diverse professional trails.\n\nNow seems like the right time for a book about finding satisfaction in a law career. The past couple of decades have produced a torrent of research on happiness from the fields of neuroscience and positive psychology. Some of the research results suggest happiness boosts that are largely to be found outside the office, through activities ranging from playing tennis to meditating to eating chocolate. But many other of these research findings have important implications for the way law is practiced, if happiness really matters to lawyers\u2014and it obviously does. Perhaps more critically, such research also contributes to the better design of happiness strategies: we know, for example, that liking and trusting your co-workers contributes more to your long run satisfaction than snagging the corner office or a six-figure bonus.\n\nIn addition to the explosion of research on happiness, the practice of law has changed in a way that makes career satisfaction much more relevant to law schools than used to be the case. Law schools that previously told entering students, \"Look to your left and look to your right; one of the three of you won't be here at the end of the year\" now realize they have a stake in convincing prospective students that they can best help them achieve that satisfying law career they're looking for\u2014happy alumni are more likely to be generous and supportive alumni. This generation of students, more than any previous generation, ranks life satisfaction highest among their goals. Schools that can credibly promise a better shot at career happiness are in a better position to recruit the best and brightest students.\n\nLaw firms also have a growing interest in making associates happy. The old system that welcomed many to the firm but retained few of them (the \"up or out\" system) has given way to a process that places a premium on careful selection and retention. Firms have come to recognize the high costs of associate attrition. They now understand that demoralization among associates affects the quality of their work product as well as the law firms' bottom line. There is a growing consensus that trends adversely affecting happiness\u2014such as ever-increasing billable hour requirements and declining civility in the profession\u2014no longer can be ignored if firms hope to continue to attract the members of a new generation that places personal happiness on the top of life's wish list. Moreover, many older lawyers are coming to the same conclusion as a lot of newer ones: \"If I'm not happy doing what I'm doing, what's the point?\"\n\nIn short, lawyers, law schools, and law firms have a greater interest than ever in happiness. In the pages to come, we will provide tips and techniques, shaped by the new research, for all who have an interest in increasing satisfaction with the practice of law.\n\nOf course, as wise people have told us, happiness isn't everything. It's a very important thing, but so are other values that matter. There is even something to be said for melancholy\u2014and we will say it. Without unhappiness from time to time, would happiness lose its meaning? Would society lose an important source of creative inspiration, and would we lose a springboard to personal growth? We'll explore those questions.\n\nMainly, though, this is a book about forging a path to a better life for law students and lawyers. We cannot offer any promises\u2014lawyers, of all people, should understand that\u2014but it is our hope that the words that follow will help make your life as a lawyer more satisfying, fulfilling, and, well, happy.\n_The Happy Lawyer_\n\n## **CHAPTER 1\n\nAre Lawyers Unhappy?**\n\n_Lawyers have never made more money and never been so unhappy._\n\n\u2013Cordell Parvin, \"Career Happiness,\" September 24, 2008\n\nARE WE HAPPY BEING LAWYERS? BEING LAWYERS, WE tend to answer that question with other questions: What do you mean by \"happy\"? On a scale that runs from having root canals to a night of fine wines and sex on a tropical island, where does \"unhappiness\" turn into \"happiness\"? Do you mean \"happy\" right now as I write footnote 17 on this brief for Acme Investments or \"happy\" during the course of my ten-year legal career? If I'm happy one-quarter of the time, miserable one- quarter of the time, and somewhere in between half the time, how do I answer your question? Also, is anyone else going to know the answer that I give you?\u2013because if they do I'm going to inflate my happiness quotient.\n\nThose are all good questions and just the sort that bedevil happiness researchers. The fact is that happiness to one person might not mean the same thing as happiness to another person, yet that doesn't stop researchers from asking people whether or not they are happy. Nor are researchers stymied by the fact that happiness, when it happens, is a temporary state and your two-o'clock giggles might turn into three-o'clock tears. And then there is the \"Lake Wobegon problem\"\u2013just like the children in the mythical Minnesota town, nearly everyone\u2014in just about every category that matters (including happiness)\u2014considers themselves above average. _Well_ above average. When a researcher synthesized 916 surveys on happiness, conducted in forty-five countries and including more than one million respondents, he found that people, on average, ranked their own happiness as about a seven on a one-to-ten scale. A reviewer, summarizing recent literature on happiness research, concluded that \"claiming to be happy ... appears to be nearly universal, as long as you're not living in a war zone, on the street, or in extreme emotional or physical pain.\"\n\nSo, once again, are lawyers happy? Given that most humans are\u2014or say they are\u2014happy, it is not surprising that most of the 1.2 million of us (one of every 250 people in the United States), say we are. Yet it is revealing that lawyers, as a group, are decidedly less happy than are members of many other professions. Members of the clergy, travel agents, architects, scientists, engineers, airline pilots, physicians, financial planners, and detectives are all happier than lawyers. Even repair persons, housekeepers, and butlers report higher levels of happiness than do members of the legal profession. Still, it could be worse: lawyers do report more career satisfaction than either roofers or gas station attendants. Overall, 43 percent of American lawyers say they are \"very happy these days\"\u2014a statistic that is hardly alarming, but also suggests plenty of room for improvement. Not all lawyers are miserable, but neither is it a cheery profession.\n\nYou probably don't need convincing that the question whether lawyers are satisfied with their careers is an important one. Consider that you probably spend about one-third of your life and one-half of your waking hours on the job. How good can a whole life be if one-third of it is miserable?\n\nIn this chapter, we examine both empirical and anecdotal information about lawyers' happinesses and dissatisfactions. A wealth of research exists, but it seems conflicted: \"There are two literatures on lawyer satisfaction, and their findings differ so starkly that one might wonder whether they are studying the same phenomenon.\" We'll attempt to make sense of the seemingly contradictory data. We'll also look at who in the legal profession is happy and who's not, which may in turn shed more light on the lawyer happiness surveys.\n\n### MAKING SENSE OF LAWYER HAPPINESS DATA\n\nSurvey data, of course, depend on the nature of the question asked. The 43 percent of lawyers in the survey by the National Opinion Research Center at the University of Chicago who said they were \"very happy\" with their lives might make you think that there is a large contingent of happy lawyers. But when survey questions focus on _career_ , as opposed to _life_ , satisfaction, there is only about a coin-flip chance of being content. Results from an American Bar Association (ABA) survey revealed that only 55 percent of the lawyers responding said that they were satisfied with their job. A 2001 ABA survey of two thousand members of its Young Lawyers Division showed similar job satisfaction levels, with a majority of those responding saying that they were \"at least somewhat satisfied with both their current position and the practice of law generally.\" Yes, more than half reporting satisfaction is good\u2014but remember, seven on a one-to-ten scale is just par on the happiness course. Turning from satisfied lawyers to those who are dissatisfied (ignoring for now those who report neutral feelings about their careers), we find that career dissatisfaction numbers have remained fairly steady across two decades, according to several empirical studies conducted by the ABA Young Lawyers Division. In surveys conducted in 1984, 1990, and 1995, between 20 and 27 percent of lawyers stated that they were either \"dissatisfied or very dissatisfied with their work.\" Most of the young lawyers reporting dissatisfaction didn't hate their jobs: fewer than seven percent of respondents \"expressed _great_ dissatisfaction with either their career or the practice of law.\" The Young Lawyer Division's survey, sampling a population of joiners (who tend to be a disproportionately happy group), probably suggests more satisfaction with the practice than there really is.\n\nGrimmer is the 1992 _California Lawyer_ poll reporting that seven out often respondents would select a career other than law if they could make the decision over. A 1998 survey of Michigan lawyers echoed the California results, with 60 percent of lawyers saying \"they would not choose to become lawyers again if they had the chance to start their careers again.\" These studies of wishful do-over rates should be viewed with some skepticism. When surveys depend on voluntary participation from a wide swath of attorneys, ask about job dissatisfaction, and have a low response rate, they probably overstate dissatisfaction levels. Disgruntled lawyers are likely to be vastly overrepresented, as they are the lawyers most likely to take the trouble to respond to such a poll. For instance, the _California Lawyer_ survey\u2014which showed so many lawyers to be miserable with their careers\u2014was sent to potential respondents by fax; and it suffers from possible data collection flaws:\n\nBecause this survey does not use a random sample of lawyers, but instead relies on the voluntary participation of the magazine's readership, it is highly unlikely to provide a representative picture of the target population. Selection bias occurs in at least two ways: first, the readers of this magazine may not be representative of all California lawyers, and second (and more important), lawyers with an axe to grind may be much more likely to participate in a poll about job satisfaction. The magazine does not even report the total number of respondents to the poll, further diminishing its value as a measure of the attitudes of its target population.\n\nOn the other hand, a recent twenty-year longitudinal study of the University of Virginia (UVA) Law School class of 1987, in which 81 percent of the respondents said they were \"satisfied with their decision to become a lawyer, and 86 percent satisfied with their lives more broadly\" might overstate lawyer happiness. Perhaps Virginia law alumni are just a satisfied bunch, of course, but it is also possible that respondents to a poll sponsored by their alma mater might feel subtle pressure to help those who dedicated their lives to educating UVA law grads feel good about the effort they put in: \"See, you did well: I'm pretty happy being a lawyer.\"\n\nThe Virginia findings, however, are generally supported by a 2007 study of lawyers in firms of varying sizes from across the country. The study showed satisfaction levels only slightly below those of the UVA law graduates, with the vast majority of respondents either \"extremely satisfied\" (35 percent) or \"moderately satisfied\" (44 percent) with their decisions to become a lawyer. Happy and _proud_ , it seems. Eighty percent of those responding to a recent ABA survey were proud to be attorneys, and an equal percentage found legal practice to be intellectually stimulating.\n\nWhen lawyers are asked about their decision to become a lawyer\u2014reflecting on all the profession has meant and could mean to them in the future\u2014it is probably reasonable to expect a somewhat higher rate of satisfaction than when they are asked about how satisfied they are with their present law job, which might call to mind memories of long work hours, uncivil lawyers, and inadequate compensation\u2014especially when compared to the chum next door. It is not a stretch to say that while lawyers are only about as satisfied with their jobs as random workers off the street, they still tend to feel, deep down, that law is a career worth pursuing.\n\nGeneric surveys of lawyer happiness lump all lawyers together, obscuring the fact that some populations of lawyers are happier than others. In large law firms, for example, dissatisfaction runs high. Firms across the nation report high rates of attorney attrition\u2014\"a whopping 37 percent of associates at big law firms, defined by the study as those employing more than 500 lawyers, quit their firms by the end of their third years of practice.\" Firms are hemorrhaging their newer lawyers: in any single year, law firms will lose almost one-fifth of their associates, although most enter other jobs in law. A focus on large firm turmoil has led many to conclude that state of the profession is worse than it probably is. Law professor Douglas Litowitz capsulizes the state of despair: \"Lawyers are pathologically unhappy.\" A much-cited article in the _Cardozo Law Review_ , \"Why Lawyers Are Unhappy,\" notes what the authors call \"pervasive disenchantment\" in the legal profession. Robert Kurson, a 1990 Harvard Law School graduate, bemoaned the state of practice in an article in _Esquire_ entitled \"Who's Killing the Great Lawyers of Harvard?\" Kurson describes classmates either moving out of law firms or anxious to do so: \"One after another, those who have left law, especially law firms, seem happy. Those who have not are suffering, or worse, resigned. They talk about losing themselves.... More vow to leave the law with the next infusion of cash or gumption.\"\n\nTop law firms aren't begging law grads to apply. Associates are not abandoning jobs in droves and applying to art schools. Rarely do lawyers leap out of seventeenth-story windows. It is clear that portrayals of a profession in crisis are somewhat exaggerated.\n\nStill, there are other indications that all is not well within the profession. Lawyers suffer exceptionally high rates of depression, alcoholism, and suicide. Studies reveal that lawyers have the highest rate of depression among all occupational categories, with incidents of major depressive disorders at 3.6 times the rates for nonlawyers with similar sociodemographic characteristics. A Johns Hopkins University study of twelve thousand workers in 104 occupations found that while only 3 to 5 percent of the general population suffers from major depressive disorders, more than 10 percent of people in three job categories (secretaries, pre-kindergarten and special education teachers, and attorneys) suffer major depression. Lawyers also experience higher rates of drug and alcohol abuse than both the general population and other professionals. A survey of lawyers in the state of Washington estimated that one in five lawyers suffers substance abuse problems, a rate almost twice the national average.\n\nThe high rates of depression and suicide among lawyers is indisputable, but it is unclear what this tells us about satisfaction in the legal career generally. Depression is a qualitatively different state than merely being unhappy. Depressed and suicidal people are unhappy, to be sure, but clinical depression can be attributable to genetics, biochemistry, personality traits, and medical conditions, as well as situational or environmental factors, such as what is happening at work. Unhappiness rates and depression rates do not correlate precisely. Finland, a nation with one of the happiest populations on the planet according to surveys, also reports one of the highest suicide rates. Artists are, as a group, extremely happy with their chosen career, yet have one of the highest suicide rates among all occupations. (The same artist who cannot imagine being anything else might turn to alcohol or internal demons for inspiration.) Law might attract some seriously depressed people\u2014more so than primary education, let's say\u2014so we can hardly blame the practice when surveys later show that people who were depressed when they became lawyers are still depressed years later. At the least, however, the high rates of depression and alcoholism should be a caution. The practice of law\u2014with its accompanying time pressures, high stakes, adversarial positions, and win-loss outcomes\u2014doesn't seem to be a great cure for depression.\n\n### WHERE WE STAND\n\nWhat should we make of this decidedly mixed bag of data? Professor Kathleen Hull, a sociologist who specializes in qualitative methods, concludes that \"the most valid, well-designed research has produced little, if any support for the notion that lawyers are unhappy in their work.\" We come down a little differently. When we look at the cumulative picture painted by all of the studies, the results are mixed, but we are inclined to conclude that most lawyers fall somewhere near the middle of the happiness continuum. The more recent studies, and those with better sampling techniques and more sophisticated methodologies\u2014such as the emerging longitudinal research\u2014do indicate that the majority of lawyers are happy with their decision to become an attorney and generally satisfied with their lives. Nevertheless, even among lawyers who are pleased that they chose law, dissatisfaction, sometimes deep, exists regarding certain aspects of their jobs.\n\nFor most lawyers, things could be better. We believe lawyers could become happier with their work. Some lawyers are in the wrong type of practice\u2014their work does not align with their values, strengths, or interests. Law students often flounder, don't know what they want to do with their law degree, and surrender to traditional career paths or other people's expectations for them. Law schools could do a much better job of steering their students toward satisfying careers. But we will talk with you about all of those issues in the remaining chapters.\n\nWhether lawyers are happy is a question with an answer more complicated than \"yes\" or \"no.\" Some lawyers are happy; some are not. Let's see who is which category. In the next section, we also will consider what characteristics\u2014of both individuals and their jobs\u2014are more likely to lead to a lawyer's satisfaction.\n\n### WHO'S HAPPY AND WHO'S NOT\n\nGood news! If you are an attorney who is over fifty years old and work at a smaller firm, or work in-house or for the government, or work part-time, chances are you are among the happiest of lawyers. If you work part-time for a small branch of government, you could be in lawyer nirvana. On the other hand, if you are a mid-level associate at a large firm who is stuck in a library with fourteen crates of discovery documents, then you have been thinking about jumping ship, haven't you?\n\nProbing deeper into the data on reported career satisfaction, job happiness among attorneys is dependent in part on a number of variables: age, race, sex, firm size, and type of practice.\n\n#### WHAT DO LAWYERS DO?\n\n74% are in private practice.\n\n8% work in government.\n\n8% work in private industry.\n\n5% are retired or inactive.\n\n3% are in the judiciary.\n\n2% are in academic, consulting or accounting jobs.\n\n1% work in legal aid or public defender offices.\n\n#### TYPE OF PRACTICE\n\nMore than two-thirds (68 percent) of attorneys who work in the public sector report being satisfied with their career, the appreciation they receive, and their work-life balance. The least happy (although wealthiest) lawyers are those in large law firms, with only 44 percent reporting that they are satisfied. In between those two extremes fall solo practitioners and small firm lawyers, who generally report that their work autonomy results in greater career satisfaction. Lawyers who engage in routine or repetitive types of practice with little room for creativity, autonomy, or innovation\u2014and especially those who wear the straightjacket of billable hours\u2014are more likely to experience discontent. Attorneys working part-time report greater job satisfaction than those working full-time.\n\n#### AGE\n\nLawyers over the age of fifty are happier than younger lawyers. This comports with a study of more than two million subjects in eighty countries on happiness levels in the general population, which showed a somewhat U-shaped curve: young people and people over the age of fifty are generally happier than those in-between, with a slight trough of unhappiness for women around age forty and men around fifty. Somewhat related to age is a person's position or status in a firm. Much of the anecdotal evidence relates to disgruntled associates rather than partners. A survey of lawyers in practice between six and nine years showed that four out often were satisfied with their careers, compared to six out often who had been in practice for more than a decade. The reported increase in happiness as a career progresses may also have something to do with those who are least happy in law taking another job after several years of practice or finding their niche or gaining more control in their job over time.\n\n#### INCOME\n\nMoney matters, but maybe not in the way you'd think. You might assume that because law is a generally well-compensated occupation, its practitioners would not have economic dissatisfactions. But it isn't the absolute income levels that matter to lawyer happiness: far more important for lawyers is how their salaries compare to people they perceive as peers, as well as the pressure and uncertainty attached to keeping a steady flow of income. In 2006 the median earnings of all lawyers was $102, 470, although salaries varied by geographic region, sphere of work (public or private), practice size, and area of practice. Interestingly, the salaries of most lawyers are well above the amounts where research shows income matters much to happiness. While the difference in happiness levels between the wealthy and those below the poverty line is significant, studies show only a weak link between income and life satisfaction, particularly if the wage earner makes more than $50,000. \"Those with incomes over $90,000 were nearly twice as likely to report being 'very happy' as those with incomes below $20,000, although there is hardly any difference between the highest income group and those in the $50,000 to $89,999 bracket.\" So, if you earn more than, say, $70,000 (the midpoint of the $50,000 to $89,999 bracket), your absolute salary level should not matter much to your general life satisfaction\u2014you will be earning enough to live comfortably.\n\nBut, as we will discuss in chapter 4, relative income matters to people\u2014maybe even especially to lawyers, being the competitive bunch that we are. And, of course, lawyers just starting out and shouldering law school debt will have economic worries. Even for the well paid, uncertainty makes people unhappy, and uncertainty attaches to the pressures of keeping a steady flow of business coming in the door.\n\n#### SEX\n\nDuring the past thirty-five years, women have become less happy, both in absolute measures and in relation to men. Across all education levels and in most industrialized countries, whether women are young or old, single or married, working or staying at home, they report less happiness than women did three decades ago. One speculation is that work-family pressures have intensified for women, with women shouldering more economic responsibilities and no fewer family responsibilities. Another possible explanation is that increases in opportunities to succeed come with increased pressures to do so.\n\nFemale lawyers have distinctive patterns of happiness and dissatisfactions with the practice of law. Women are leaving the profession much more rapidly than their male counterparts. Attrition from private law firms is almost twice as high among female associates than among comparable male associates. Women do not leave law because they are dissatisfied with the substance of their work; in fact, female attorneys are generally happier than male attorneys with the type of work that they do. Rather, the problems women have with the practice of law relate to a lack of opportunities for professional development, concerns about their work environment, and the difficulties of juggling careers and families. A study of Massachusetts lawyers who worked at the one hundred largest firms in the state showed that for women who were leaving firms but not leaving the workforce, the reason most often cited was \"difficulty integrating work and family\/personal life.\" A twenty-year longitudinal study of University of Virginia law graduates showed two major differences between male and female lawyers. Males were more satisfied with their work-life balance than female lawyers, while women \"were far more likely than men to interrupt or forego full time employment (39% v. 1%), mainly in order to care for children, and were also more likely to have a spouse or partner employed full-time outside the home (77% v. 24%).\"\n\nProfessors Kenneth Dau-Schmidt, Marc Galanter, Kaushik Mukhopadhaya, and Kathleen Hull evaluated surveys from 12,000 University of Michigan Law School alumni spanning more than a quarter of a century, looking for the impact of gender on legal careers; what they found was that taking time away from law practice to care for children, more than gender, altered lawyers' career paths and affected their satisfaction. As Galanter explains:\" 'The big dividing line in terms of the experience attorneys have in private practice is not between men and women... It is between women with children and everybody else.'\" While the major differences in the aggregate appear between mothers and others, fathers who assume primary caretaking roles (a smaller group) suffer greater career setbacks. Attorneys who take time off from paid work for childcare, whether male or female, will experience reduced earnings and a reduced chance of partnership later in their careers, and \"male attorneys who perform childcare fare even worse than female attorneys who do the same.\" Despite the career costs, however, taking time away from work to do childcare turns out to be the right call for most lawyers: \"Both the men and women who take time away from paid work to do childcare, and who work fewer hours, are significantly happier with the balance of work and family in their lives as compared with any of the other groups.\"\n\nAlthough almost one-third of attorneys are female, and women have comprised 40 percent or more of law school graduates since 1985, women account for only 17 percent of \"partnership positions in law firms, despite graduating from law school and obtaining junior associate positions at an equal rate as men for the last twenty years.\" But not all partnerships are created equal. Almost three-quarters of larger law firms (defined as those employing more than seventy-five lawyers) have several tiers of partnership; women hold nonequity partner positions in disproportionate shares and collect smaller income shares upon attaining partnership status. In addition to the glass ceiling, female lawyers report several frustrations: a lack of networking opportunities, less desirable assignments, an absence of mentors, and instances of bias in the courtroom. Of course, men are not immune from discrimination, as law firms are much less hospitable to paternity leave than maternity leave.\n\n#### RACE\n\nAfrican Americans, Hispanics, Asian Americans, and Native Americans collectively constitute just less than 10 percent of all lawyers in this country, significantly below their representation in the population. An American Bar Foundation survey of law school graduates who had been in practice for three years or less found that Blacks expressed the highest level of satisfaction with their decision to become a lawyer, as well as the highest level of satisfaction\u2014among all racial groups\u2014with the substance of their work. Satisfaction levels regarding career choice were almost as high among Hispanics. More than 80 percent of Blacks and just less than 80 percent of Hispanics reported that they were either moderately or extremely satisfied with the decision to enter law. These reported rates of satisfaction may have to do with practice setting\u2014Black and Hispanic lawyers were much more likely to be working in nonprofit or government jobs. Interestingly, their satisfaction levels were higher despite having lower median salaries.\n\nAsians reported the lowest satisfaction levels for the substance of their work; but Asians and Whites expressed the highest satisfaction averages with their job setting. Both of those groups were more likely than members of other races to be working in business settings or at larger law firms. Yet members of the three largest minority groups (Blacks, Asians, and Hispanics), while satisfied with their decisions to become lawyers, \"were all more likely than their white peers to report that they were already looking for another position or intended to remain in their current position for less than a year.\" For Asians, the dissatisfactions may relate to the conditions at large law firms; Hispanics and Blacks expressed concerns about the absence of power track opportunities.\n\nAt the intersection of race, sex, and practice area, the unhappiest lawyers working for the nation's top firms are mid-level females of color. Thus, although Blacks and Hispanics may be happier than other races with their decisions to become lawyers, when all races are combined and the cohort is lawyers in practice at large firms, women of color are distinctly unhappy. In 2005, eight out often minority women associates left their jobs within five years. In one study conducted by the ABA, almost half of all women reported that they were subjected to demeaning remarks or harassment. Respondents in large firms gave negative report cards in areas such as the quality of work, the number of interesting projects, and the quality of professional development: \"white male lawyers graded their career satisfaction A, white women and minority men B and minority women B minus to a C plus.\"\n\n#### OTHER FACTORS\n\nVarious personal and demographic factors enter into lawyer happiness. Graduates of lower-ranked law schools report higher levels of happiness than graduates of top-ranked law schools. In one study, only 27 percent of graduates of the top-ten law schools reported that they were extremely satisfied with their career choice, compared to 43 percent of graduates from fourth-tier law schools. Six out often graduates of top-tier law schools intended to leave their employer within two years, while only four out often graduates from fourth-tier law schools reported any intent to move. Researchers considered the possibility that top-tier law school graduates were more likely to be working at large corporate firms and, therefore, might be dissatisfied simply by the nature of their practice. But when researchers controlled for that possibility by only comparing responses from lawyers working at the one hundred largest law firms, the differences became even more pronounced: \"26 percent of elites report extreme satisfaction, compared with almost half (49 percent) of those in the fourth tier.\"\n\nIndividual traits and circumstances can combine with firm policies or practices to produce unhappiness. Gay and lesbian lawyers at firms that afford no benefits to their partners may chafe at the arbitrariness. Lawyers whose religion recognizes Sabbath on Saturday may feel that they don't fit the culture of law firms in which attorneys are expected to show up for face time on Saturdays. Unattractive or obese lawyers may receive fewer assignments that involve public or court appearances. One law firm back in the 1960s did not hire men less than six feet tall because the firm thought that smaller men would not have a commanding presence in the courtroom. Life isn't always fair, and neither are the powers-that-be in law firms. When you feel that you have been the victim of discrimination, your happiness is bound to take a nose-dive.\n\n### THERE'S NO SORTING HAT\n\nAt the Hogwarts of J. K. Rowling's imagination, the Sorting Hat magically assesses the character and talents of first-year students and assigns them to the house\u2014Hufflepuff, Ravenclaw, Slytherin, or Gryffindor\u2014that best matches their individual strengths. Every student ends up where he or she belongs and stands the best chance of thriving.\n\nIn the muggle world\u2014our world\u2014there is no Sorting Hat. Choosing a career path requires doing our own assessments of our abilities, interests, and desires. Every year some 48,000 Americans perform that assessment, however imperfectly, and set down a path that they hope will lead to a satisfying career of practicing law. The fact that so many of them, perhaps 20,000 or so, will end up disappointed has many, varied causes. Some of the disappointed were never meant to be lawyers, their talents and passions pointing elsewhere\u2014perhaps in the direction of winemaking, printmaking, or teaching.\n\nOther unsatisfied lawyers embarked on their careers with unrealistic expectations about law practice. They considered themselves the best and the brightest, excelling in college and graduating from top law schools, and they believed their hard work entitled them to fulfilling and happy careers. High expectations and feelings of entitlement to great jobs might account for the higher levels of dissatisfaction among graduates of the nation's highest-ranked schools compared to those lower on the pecking order. People who go into sanitation work do not expect to be made deliriously happy by their jobs. They haven't dreamed since junior high of joining the sanitation force. \"It's a job isn't it?\" they might say, and they are pleased to have one. Once at work, however, they find some things about the job they like\u2014the camaraderie, the work in the sunshine, the sense that they are serving their community\u2014and, you know, they begin to feel pretty good about their job. When reality fails to match high expectations, however, as it probably does for many lawyers, the result can be a gnawing sense that a better career choice could have been made.\n\nFinally, one other point should be made about the surveys that place lawyers somewhere in the middle of the career satisfaction continuum. What it means to be, say, \"fairly satisfied\" with a job might be very different for different types of lawyers. For the trial lawyer, whose career is marked with the emotional peaks and valleys that result from our adversarial process, \"fairly satisfied\" might in fact reflect a perceived ratio of quite high highs and quite low lows. Trial work is a \"high amplitude\" career. On the other hand, a \"fairly satisfied\" transactional lawyer might be one who finds modest pleasures in the daily practice of law but keeps wishing for a bit more of an emotional charge. One person's \"fairly happy\" is not the same as another person's \"fairly happy.\"\n\nThe fact that more lawyers are happy with their decision to choose law than they are with their specific jobs suggests a sense among lawyers that careers could be better, either if they found another firm, found a new area of practice, or discovered better ways of coping with the frustrations of their existing work situations. We believe that they are right\u2014most law careers _could_ be better\u2014and that's really what this book is all about.\n\nNext, however, we turn to the question of what happiness really is, not just in one's career, but in one's life more generally. To become a happy lawyer it helps first to understand what makes a happy person. The answer turns out to be pretty complicated, involving a mix of genetics, circumstances, and our own choices.\n\n## **CHAPTER 2\n\nHappiness: A Primer**\n\nIT'S EASIER TO BECOME A HAPPY LAWYER IF YOU ARE already a happy person. We're not all that lucky, and you might be wondering why. Before exploring how to achieve happiness and meaning in a career, we consider in this chapter how to achieve a happy and meaningful life. We offer a brief survey of the scientific literature on the general question of happiness, leaving for the rest of the book the question of finding happiness within the practice of law.\n\nThe literature on happiness is voluminous and often contradictory, with more than 4,000 books published on the subject in 2008 alone, and runs the gamut from first-rate science to garden-variety self-help vacuity. We've done the hard work for you and read at least the best of it. Here's what we think you should know about happiness.\n\n### THREE MEANINGS OF HAPPINESS\n\n\"Happiness\" is one of those chameleon words whose color depends upon where you find it. It has a bright hue for the lawyer on the golf course, dance floor, neighborhood tavern, or ice cream shop. It has a different tone, somewhat paler usually, in a law office. You might be happy, yes, but the feeling is less intensely positive. Finally, \"happiness\" looks altogether different to the lawyer looking back at a career\u2014the happy times and not-so happy times merge in memory and, in the well-spent life, become a sort of soft glow.\n\nAs a concept, \"happiness\" really does triple duty and thus confuses our quest to achieve it. The fact is that maximizing one type of happiness, the giggles and pleasures of the present, might make it more difficult to achieve another type of happiness, the sense of satisfaction that comes later in life with a feeling that however hard life has been at times, you've come close to realizing your potential. In between these two types of happiness, short-term joy and the well-lived life, is an intermediate sort of happiness\u2014a variety we might also call contentment or satisfaction\u2014that is probably the type most relevant to you.\n\nAlthough intermediate-term happiness might be our primary goal, none among us would wish a life without its moments of pure joy (and maybe even a dash of ecstasy). Indeed, a life bereft of joyful moments hardly seems a strong candidate for satisfaction or contentment either. In this chapter, we lay out what researchers really know about achieving happiness in general before turning to what they can tell us about short-term pleasure and intermediate-term satisfaction. We leave for a later chapter some reflections on self-realization, or what it means to have lived life well.\n\n### SUDDENLY, STUDYING HAPPINESS IS IN\n\nGood lawyers understand the importance of facts. In the end, you cannot get around them. As John Adams said during his defense of British soldiers in the Boston Massacre trial, \"Facts are stubborn things.\" The facts about happiness\u2014and some of them are especially stubborn\u2014determine and limit our abilities to achieve greater satisfaction in our legal careers.\n\nAnd there are a great many facts. More pile up each day, with researchers competing for space in _The Journal of Happiness Studies_ or sending abstracts off to the Netherlands, home of the _World Database of Happiness_. Serious students of happiness are busy adding new bookshelves to their libraries or are seeking positions at the University of Pennsylvania's Authentic Happiness Center. It didn't always used to be this way.\n\nA couple of decades ago, this chapter on the science of happiness would have been briefer because, until recently, scientists generally ignored the subject. They considered it nearly impossible to evaluate and resistant to scientific methods. Definitional issues (\"What exactly is happiness anyway?\") and a sense that the whole subject, despite its obvious interest to nearly everyone, was unworthy of serious study added to the reluctance of scientific community to take up the cause. Until recently, the people talking about happiness were mostly pastors and self-help speakers, such as Reverend Rick Warren ( _The Purpose Driven Life_ ) and Norman Vincent Peale ( _The Power of Positive Thinking_ ).\n\nWhen the misgivings of researchers gave way, they really gave way. Now we are inundated with studies from numerous fields exploring this or that aspect of happiness. Happiness studies in the psychological and biological sciences generally are coming from three different, but related, fields. Neuroscientists, using new technology, study brain chemistry and evaluate how various firing patterns of brain cells are associated with happiness or other emotional states. Evolutionary psychologists develop and promote their theories as to how the reproductive advantages and disadvantages of different emotional reactions to events molded the brains of early humans and shaped the patterns of happiness and unhappiness we see today. Finally, members of the fast-growing field of \"positive psychology\" (encompassing cognitive and social psychology) use surveys and other research tools to explore everything from the genetics of happiness to what role geography, relationships, or careers play in determining happiness levels. Taken together, the lessons offered from these three fields of research give us a rich understanding of what happiness is, and what it takes to get there.\n\n### TRANSIENT HAPPINESS\n\nNeuroscientists and evolutionary psychologists generally have focused their research on transient states of happiness ranging from pleasure to exhilaration to euphoria. For neuroscientists, the reason is obvious: it's a lot easier to measure and observe brain changes in reaction to specific stimuli than it is to track changes in brain patterns over long periods of time and then try to explain them in anything resembling scientific terms. A neuroscientist can see what waving a one hundred dollar bill in front of our eyes does to our brains, but cannot conclude much about what ten years of practicing corporate law does for our happiness. Similarly, theories of evolutionary psychology offer little in the way of explanations for longer term satisfaction but instead provide insights into why sex or competitive success brings us pleasure.\n\n#### WHAT EVOLUTIONARY PSYCHOLOGY TEACHES US\n\nEvolutionary psychology tells us why we have emotions in the first place, as well as why we have the ones we do. As evolutionary psychologist David Buss explains it, \"Mechanisms of the mind are end products of a selective process, a sieve through which features passed that contributed to reproductive success.\"\n\nNature cares not for our happiness. From an evolutionary standpoint, being in a persistent state of contentment would not be a recipe for reproductive success; that is, it is more important to have a brain that sends out the message \"Danger!\" than to have one that says \"Be happy.\" Early humans who ignored poisonous snakes and saber-toothed tigers weren't around long enough to pass their genes on to future generations. As a result, nature has programmed us to better recognize the negative than the positive. When we see our mate sneaking off into a cave with another member of our tribe, we're designed to feel jealous or angry, not content. Humans whose genes might have predisposed them to stop and smell the roses when they should have been gathering berries and firewood for the harsh winter ahead left us little of their genetic heritage.\n\nOf course, it is also true that any early human who never took the time to eat or have sex would make no contribution to the gene pool, so it should come as no surprise that two of the activities most likely to cause our brains to send the \"okay, you're happy now\" message are eating and having sex. Evolutionary calculations are all about mate-finding and survival, not about achieving bliss or contentment. Most of what modern humans do with the bulk of their time has little to do with evolutionary biology. Drafting contracts just wasn't part of everyday life on the ancestral savannah, and nothing in our genetic programming is designed to turn on a happy button when we spend our time fine-tuning paragraph 17 on page 4 of a brief.\n\nReproductive success has been facilitated by having a variety of emotional responses, some positive but many negative. The ability to experience fright kept our ancestors from being eaten by lions or battered by invaders. The ability to experience envy or jealousy or sadness kept them searching for positions in their social group that would provide food, sex, and shelter. Pleasure had its place in primitive times, but it was only one of many emotions adapted to be triggered only under certain conditions. Evolution has made it impossible for us to maintain a constant state of happiness\u2014the best we can hope for is to grab more of it than we presently do.\n\n#### WHAT NEUROSCIENCE TEACHES US ABOUT OUR THREE BRAINS\n\nEvolution has left humans with three brains. The most primitive part of our brain is the brainstem, sometimes called the \"reptilian brain\" because its origins trace back that far in our evolutionary history. The brainstem controls heartbeat, respiration, body temperature, and other essential functions. The next part of our brain to develop was the limbic system, sometimes called the \"mammalian brain\" because it emerged more than one hundred million years ago at the beginning of the mammalian era and is something we share with other mammals, from rats to hippopotamuses. The limbic brain includes three main structures: the amygdala, the hippocampus, and the hypothalamus. The limbic brain is the source of our basic emotions, including love, fear, rage, jealousy, and\u2014yes\u2014happiness (at least of the transient variety). Our unconscious value judgments also are primarily the limbic brain's doing. Finally, the arrival of primates resulted in the construction on top of the brain stem and limbic system of a \"third brain\": the neocortex, where intellect and logic reside. The highly flexible neocortex, divided into a left and a right hemisphere, is the source of language, abstract thought, imagination, and consciousness. All three brains co-inhabit our skulls and, in the well-balanced individual, work in a sort of rough harmony with help from neural pathways that connect the limbic system to the neocortex.\n\nThe amygdala, part of our ancient limbic brain, plays a critical role in our emotional lives. When, for example, the amygdala is sending out \"be scared\" messages or \"be jealous\" messages, it is impossible to be happy. Many more neural circuits run from this manager of emotions to the prefrontal cortex, the section of the brain in charge of planning and reasoning, than run toward it. Without the amygdala and its constant emotional reporting to the prefrontal cortex, we would be unable to make decisions. The prefrontal cortex, confronted with an overload of information, ultimately relies on the amygdala to move us from inaction to action. We are, at bottom, and to the disappointment of some, still much more emotional creatures than we are intellectual creatures. Eduardo Punset, a Spanish lawyer, professor of neuroscience, and author of _The Happiness Trip: A Scientific Journey_ , bluntly describes the challenge presented by our evolutionary legacy: \"the problem with the search for happiness\" is \"despite our ardent wishes, we are anchored in a world of genetics and volatile, preset emotions.\"\n\nWith CAT scans and other brain-imaging and monitoring tools, neuroscientists have begun to learn what is going on inside brains that are experiencing transient states of happiness. A study of the brains of dogs being shown food, for example, revealed firing in the hypothalamus, which might be called the \"seeking circuit\" of the brain. Interestingly, researchers observed, the hypothalamus ceases firing as soon as a dog begins eating the food. (This probably comes as no revelation to many dog owners, who might have noticed that dogs wag their tails when they are expecting food but not during the act of eating itself.) This study suggests that happiness might often lie more in anticipation of events than in the actual event itself. Or, as Punset says, \"Happiness is hidden in the waiting room for happiness.\"\n\nResearch reveals that positive emotions are associated with high levels of brain activity in the left frontal cortex, while negative emotions are associated with active firing in the right frontal cortex. If the \"blue bird of happiness\" has a favorite roosting place, it is in the left prefrontal cortex. When subjects experience high levels of anger, depression, or anxiety, the area of the right prefrontal cortex (just behind the forehead) goes into overdrive. Watching an amusing movie clip produces enhanced brain activity in the left frontal cortex of test subjects, while a sad movie caused increased activity in the right. Interestingly, this is even true for babies.\n\nOf course, researchers peering into our brains are not actually _seeing_ happiness, which is a feeling and therefore lacks physical attributes. Neuroscientists cannot be confident that the same brain scan pattern produces the same _intensity_ of happiness in all subjects. Harvard University psychologist Jerome Kagan reminds us that a brain scan \"is no more equivalent to an emotion than a picture of an apple represents the texture and taste of the fruit.\" Ultimately, all claims of happiness come from the closed closet of an individual's point of view.\n\n#### THE SCIENCE OF PLEASURE: A CHEMISTRY LESSON\n\nUnderstanding transient emotions turns out to be a chemistry lesson as much as anything. Although the number of brain chemicals having effects on emotions is large, four in particular are worth mentioning here.\n\nCortisol, to oversimplify, is the stress chemical. When this hormone is coursing through the brain in significant quantities, we can't be happy. We are, instead, frightened or angry or jealous, just as we were designed to be in frightening or anger- or jealousy-inducing situations. Cortisol might not make us happy, but it has adaptive benefits, and even if we could somehow drain it from our systems entirely, we'd be better advised not to. Nonetheless, according to Richard Davidson of the University of Washington, the brain of a generally happy person is one with an \"adaptive pattern of cortisol release\" that allows effective regulation of the negative emotions we all experience from time to time. Davidson's research shows that cortisol levels are naturally lower at night than in the morning and that happier people tend to have lower nighttime levels of cortisol than do less happy people.\n\nIf cortisol causes stress, is there another substance in our brains that might be a \"happy juice\"? Things aren't quite so simple, it turns out. Various chemicals, including the neurotransmitters dopamine, oxytocin, and serotonin, play roles in producing good feelings, but each does many other things as well, and not all of those things make us happy.\n\nDopamine is the compact molecule, made of twenty-two atoms, that gets the most attention from researchers\u2014and deservedly so. When we anticipate a pleasurable event, dopamine flows along a specialized circuit of neurons above the brain. It is the _expectation_ of pleasure, not the pleasure itself, triggering the flow. Dopamine is a key component of the motivation system that nature has built in to reward us for searching for the food and sex critical to our reproductive success and survival as a species. It's also been called the brain's \"get-a-load-of-this device,\" the source of our attraction to the novel.\n\nDopamine is critical for our survival as a species, and it has the side benefit of producing boosts in moods and alertness. To call it a source of happiness, however, is misleading. The neurotransmitter is really all about survival, and dopamine neurons fire just as hard at the sight of an object we fear (a snake) as one that excites us in a positive way (a naked lover). Dopamine's goal is to keep us alive and produce offspring, and that means making us pay attention to threats as well as getting us to enjoy food highs and good sex.\n\nDopamine also is blamed for our addictions. As complex as the causes of addiction are, one aspect that they all share is dopamine. Addictions, from drugs to pornography to gambling, can be tied to this \"craving\" neurochemical. Dopamine, in effect, tells us to pay attention to whatever promises us short-term pleasure and to ignore whether our behavior might lead to longer term problems. Call it a design flaw, but our limbic systems are just not built to alert us when we're experiencing too much of a good thing.\n\nDopamine, as we said, is associated with sex\u2014and sex is an activity associated with happiness. Most people would rather not be asked as the moment of sexual climax approaches, but if they were, they would almost universally report being happy. In fact, sex is the activity that produces the highest average ratings of happiness among all activities measured in a recent survey. (Respondents gave sex an average ranking of 4.7 on a 5.0 scale, compared to a 3.8 for eating and a 2.7 for working.) So what is going on in our brains when we engage in sexual activity? In anticipation of sex, dopamine levels soar. Orgasm has been described as \"the biggest blast of dopamine (legally) available to us.\" Brain scans of people experiencing orgasm closely resemble scans of people experiencing heroin rushes. Alas, the happiness from sex is short-lived. After climax, dopamine levels drop off dramatically and the levels of neurochemical prolactin (telling us \"Whoa!\" and then asking us whether it's time for sleep) rise.\n\nStimulants also produce happiness boosts because they increase dopamine levels. Cocaine and amphetamines, for example, provide short-term pleasure by essentially hijacking the neural system for a joyride that evolution never intended. Cocaine works by blocking the dopamine transporter, \"a kind of janitor that picks up the used dopamine molecules and sweeps them back into the cells where they were born.\" With the transporter blocked, dopamine keeps banging its signals. Caffeine has a similar, if much milder, effect. That morning cup of joe increases dopamine flow and produces a short-term mood boost, but heavy caffeine consumption can also cause increased levels of aggression, anxiety, and sleep loss.\n\nMusic can make us happy\u2014even euphoric. Is the reason, as with sex and drugs, linked to dopamine levels in some way? The answer turns out to be \"yes.\" Neuroscientists have discovered that music increases dopamine flow (as well as levels of endorphins and oxytocin) and creates activity in the amygdala and prefrontal cortex. Harvard University psychologist Steven Pinker describes \"the direct effect of music\" as \"simply, the generation of meaningless pleasure.\"\n\nSex, drugs, and rock and roll are three happiness-producing activities associated with dopamine flows\u2014and the list could be made much longer. More generally, dopamine accounts for the pleasure we derive from new and stimulating environments. If variety is the spice of life, dopamine explains why.\n\nPracticing law generally is not associated with increased dopamine levels, but practice areas with surprises and novelty provide more frequent boosts than jobs where one day at the office is virtually the same as any other. Almost all lawyers, however, have the opportunity to inject a bit of variety into their days\u2014perhaps by checking out that new restaurant down the street or striking up a conversation with the new associate down the hall\u2014to perhaps get the dopamine flowing a bit. Laughter can have the same effect\u2014getting to know your office comedian can have its benefits.\n\nDopamine is responsible for our highest highs, but it is serotonin\u2014or rather, the lack thereof\u2014that is associated with our lowest lows. Research shows a clear link between low levels of serotonin and elevated rates of serious depression and anxiety disorders. Prozac, Zoloft, and other antidepressants work by steeping neurons longer in what little serotonin a depressed individual is able to produce, thus improving mood. When our levels of the critical neurotransmitter go up, we feel (depending on the situation) more empathetic, insightful, and uninhibited. For these reasons, serotonin is sometimes called \"the happy chemical\"\u2014a simplistic view that has been undermined by recent research showing that increased serotonin activity and pleasurable feelings do not always go hand in hand.\n\nThere are ways, both legal and illegal, to raise your serotonin levels. Falling in love does the trick. So does enjoying an intimate conversation with a good friend. Natural sunlight also increases serotonin turnover and, if you're stuck in Seattle in January, bright light therapy has the same effect. If you're more desperate, acupuncture seems also to increase serotonin. And if you're more desperate still for a serotonin rush, and won't let the law stand in your way, there's the recreational drug Ecstasy (MDMA). (While Ecstasy typically produces a relaxed and empathetic state, unfortunately for users of the party drug, the rapid deployment of serotonin results in a depletion of serotonin levels once the drug wears off, tumbling the user into a short-term depressed state).\n\nFinally, to complete our brief survey of the brain's emotion-regulating chemistry set, a word about the hormone oxytocin. Oxytocin is sometimes called \"the bonding chemical\" because it is closely associated with the deep connections of lovers or of mothers and babies. During sex, the hormone is released by both men and women, explaining the deep bonding often associated with that experience. Oxytocin is also released during childbirth and flows in a mother at the sound or sight of her young baby, triggering the release of breast milk. When the release of oxytocin is blocked, bad things happen. Experiments show, for example, that sheep and rats will reject their own young. (On the other hand, injecting oxytocin into a female rat causes her to bond with whatever other female rat's young might be nearby, protecting them just as if they were her own.)\n\nRecent evidence, however, indicates that oxytocin plays a far more extensive role than previously thought. According to Dr. C. Sue Carter of the University of Illinois, the hormone is suddenly \"very hot.\" Scientists have demonstrated, as Natalie Angier concludes in a recent summary of the research in the _New York Times_ , that oxytocin does nothing less than underlie \"the twin emotional pillars of civilized life, our capacity to feel empathy and trust.\" She cites a Swiss study finding that subjects given a nasal squirt of oxytocin were far more likely to trust strangers with their money than were subjects given a squirt of a placebo solution. A second study compared two groups of people, one group with a genetic code that made their oxytocin receptors more responsive to the effects of the hormone, and one group without that genetic variant. The study found that having the gene that increased responsiveness to oxytocin resulted in an improved ability to read faces, feel distress at another's hardship, and identify with characters in a novel. The enhanced receptor also made people less stress-prone. The results of the study left Dr. Sarina Rodriquez of the University of Oregon \"floored.\"\n\nDoes understanding the chemistry behind transient happiness make us better equipped to find it? We cannot change the chemistry sets our brains came equipped with, but if we know what activities and substances produce brain responses associated with pleasure, we are better able to obtain happiness boosts when we really need them. While some of the lessons of neuroscience reach beyond pleasure to what we've called satisfaction or contentment, the scientific research suggests that happiness is more easily achieved for the short-term than the longer term. There is no magic bean for contentment.\n\n#### THE SCIENCE OF SATISFACTION: TWO THINGS THAT WORK\n\nWhile temporarily increasing or decreasing the flow of various brain chemicals can bring about moments of pleasure, it does little to increase our intermediate-term happiness. To achieve happiness that stays with us for weeks rather than minutes or hours, it's not a simple matter of turning on this or that neurotransmitter faucet. Rather, the structures and the pathways of the brain need to be reshaped. An impressive body of growing evidence suggests two ways that can be done and produce long-lasting improvements in mood.\n\nThe first proven technique for increasing activity in the left prefrontal cortex is meditation. If you are inclined to dismiss mindfulness meditation as some sort of New Age mumbo-jumbo, you will have to ignore a number of studies that show meditation programs of eight-weeks or less reduce anxiety and depression, while increasing well-being and immune system responses. Most persons practicing regular meditation also report an enhanced ability to experience the present moment. Neuroscientists confirm that meditation and bio-feedback techniques can reprogram brains and re-direct brain activity from the right to left prefrontal cortex. As neuroscientist Richard Davidson puts it, happiness can be thought of \"as a skill, not fundamentally different than learning to play the violin or learning to play golf.\"\n\nOur second recommendation for improving moods over the longer term is to exercise. It's even a good idea if you're a rat. Princeton scientists recently discovered that rats that exercise create new brain cells that respond much better to stress than the brain cells of slothful rats. Exercise does nothing less than reshape your brain, making it more stress-resistant. Michael Hopkins, affiliated with Dartmouth's Neurobiology of Learning and Memory Laboratory concludes, \"It looks more and more like the positive stress of exercise prepares structures and pathways within the brain so that they're more equipped to handle stress in other forms.\" Hopkins calls the translation from the realm of the physical to the psychological \"pretty amazing, really.\" Don't expect, however, that thirty minutes on the treadmill will wipe away your problems. According to researchers at the University of Colorado, it seems to take at least three to six weeks of regular exercise to reap the psychological benefits.\n\n#### THE SLOPPY BUSINESS OF MEASURING SATISFACTION\n\nWhile researchers interested in the transient type of happiness can peer into brains or speculate about adaptation in early humans, they lack a tool that can objectively measure and compare the pleasure of one person to another. There is no \"hedonometer\" that can accurately measure how many \"hedons\" of happiness you are experiencing at the present moment and show for certain that it is more or less than the person sitting next to you. For a measure of how much pleasure a person is presently experiencing, researchers have to rely on self-reports of subjects (\"How happy are you feeling right now\u2014very happy, happy, indifferent, unhappy, or very unhappy?\").\n\nStudies of (the ultimately more important) intermediate-term happiness, such as measures of career or marital satisfaction, generally depend on self-reporting. Self-reporting is subject to variation over time and culture. For example, most studies suggest that the Japanese are generally less happy than Americans, but there is some reason to question that finding. It is difficult to determine how much the difference in reports of satisfaction might be attributable to the greater emphasis Western culture places on happiness. Because of that emphasis, it is possible that Americans might feel psychologically compelled to report higher happiness levels than do residents of Japan, who place a higher value on harmony and obligations. This could explain why moment-by-moment reporting (our make-do hedonometer) indicates that Asians and Americans have similar overall levels of transient happiness.\n\nThe vast majority of people (about 85 percent) claim above average satisfaction with their lives. \"Claiming that you're happy\u2014that is to an interviewer who is asking you to rate your life satisfaction on a scale from zero to ten\u2014appears to be nearly universal, as long as you're not living in a war zone, on the street, or in extreme emotional or physical pain.\" Are people really as happy as they say they are? Probably not. More likely, reported levels of satisfaction are skewed by the desire of people to \"manage impressions.\" An admission of unhappiness could be perceived today by many Americans as a sense of failure.\n\nAny measure of happiness is imperfect. Yet that is no reason not to study it, nor to draw lessons from those that have. As political scientist Alan Wolfe noted, \"The proper question... is which of the many imperfect measures of happiness we ought to rely on.\"\n\n#### THE THREE COMPONENTS OF SATISFACTION\n\nSatisfaction, or intermediate-term happiness, is determined by three things: genetics, circumstances, and our own internal decisions and actions. Our focus for most of this book will be on the third factor, those choices you can make to increase your own job and life satisfaction regardless of your genes or specific circumstances. Yet it is important to understand that happiness is not completely within your control. Understanding and accepting limitations on your satisfaction may actually prove helpful to your efforts to boost your own happiness. We are reminded of the wise words of theologian Reinhold Niebuhr's \"Serenity Prayer\":\n\nGod grant me the serenity  \nto accept the things I cannot change;  \ncourage to change the things I can;  \nand wisdom to know the difference.\n\n##### _Genetics: The Stubborn Reality of Happiness Set Points_\n\nThe most important insight about the nature of happiness that science has given to us is that, to a surprisingly large extent, our happiness levels are genetically determined. Genetics isn't fate\u2014how genes express themselves is not inevitable\u2014but genes provide strong behavioral momentum. Estimates as to the genetic contribution to happiness levels vary considerably (from about 40 percent to 80 percent), but nearly everyone recognizes the contribution to be substantial. The seminal study was a twin study conducted by David Lykken and Auke Tellegen. The two researchers asked pairs of identical twins to report on their happiness levels on two occasions nine years apart. They found that happiness levels reported on the first occasion predicted, to a large degree, happiness levels reported nine years later. In other words, they found happiness levels to be remarkably consistent over time. Second, and even more surprising, Lykken and Tellegen discovered that the happiness level of one twin could predict the happiness level of the other twin nine years later just as well as that twin's _own_ earlier happiness level. This turned out to be true even if the identical twins were separated at birth and raised in very different circumstances. The researchers concluded that inherited factors play large roles in shaping our perceived levels of happiness, much larger than the environments in which we live. The label that has been given to this genetic component of our happiness level is our happiness \"set point.\"\n\nYour happiness set point is the baseline level of happiness to which you _tend_ to return after periods of either unusual positive (falling in love, enjoying a dream vacation) or negative (getting fired, breaking a leg) emotion. We all know people who seem to keep a smile on their face even when circumstances would suggest otherwise. We know others for whom almost every day seems like a bleak Monday in November. When it comes to happiness, the genetic lottery produces both winners and losers. Happiness researcher Sonja Lyubomirsky puts it this way: \"[L]ike genes for intelligence or cholesterol, the magnitude of innate set points\u2014that is, whether it is high (a six on a seven-point scale) or low (a two) or in between (a four)\u2014governs to a large extent how happy we will be over the course of our lives.\" Lyubomirsky estimates that happiness levels are 50 percent determined by genetics, putting her squarely in the mainstream of happiness researchers. Regardless of your set point, as we discuss later, you still have a great deal of control over your own feelings of happiness.\n\nVarious personality traits have been linked to higher or lower happiness set points. People who are extroverted, optimistic, and take risks, for example, tend to be relatively happy. On the other hand, introverted, pessimistic, risk-averse people tend to be relatively unhappy. The least happy people of all, unsurprisingly, tend to be people who take worry and anxiety to a whole different level: neurotics.\n\nThe traits associated with higher or lower set points are not truly independent. Extroverts, for example, in addition to being sociable tend to take more risks, whether the risks involves sports such as downhill skiing, or behaviors such as excessive drinking and having multiple sex partners. In general, extroverts find more shops enticing than do introverts as both groups meander through the bazaar that is life.\n\nAlthough extroverts outnumber introverts by a three-to-one margin in the general U.S. population, there are more introverted than extroverted lawyers. According to a study of more than 3,000 lawyers, introverts outnumber extroverts by a 57 percent to 43 percent margin. The difference in ratios between the general population and population of lawyers is dramatic and suggests two things. First, the practice of law is a career that disproportionately appeals to introverts. As Michael Melcher notes in _The Creative Lawyer._ \"Introvert characteristics are more apt to be accepted and appreciated in the field of law than in the rest of the world.\" The work environments of lawyers are typically quiet, private, and place a premium on contemplation, preparation, and writing skills. (Trial lawyers, of course, rely more heavily on skills better suited for extroverts, such as face-to-face meetings and courtroom speech-making.)\n\nThe American Bar Association survey results suggest that when it comes to achieving happiness, most lawyers have one strike against them, their introversion, and other research suggests a second strike: pessimism. Psychologist Martin Seligman describes pessimism as \"a well-documented... major risk factor for unhappiness,\" and notes there is \"a surprising correlation between pessimism and success in law school.\" This correlation appears to be because a pessimist finds it much easier to conjure up the worst-case scenarios that it is a lawyer's job to guard against, whether in writing contracts or defending a client in court. Pessimism, in short, has a positive dimension that we often refer to as \"prudence.\" By spotting pitfalls ahead, the prudent lawyer brings clients safely into a happy harbor when a more optimistic sort might have left the client marooned by an unanticipated disaster. As helpful as this honed pessimism might be in the actual practice of law, it doesn't make for a happy worker in the office.\n\n##### _Circumstances: The 10 Percent Determinant of Happiness Levels_\n\n_Happiness depends, as Nature shows,  \nLess on exterior things than most suppose._\n\n\u2014William Cowper (1782)\n\nWhen asked to imagine a happy person, most people immediately put that person in fortunate circumstances. They imagine a person cruising around the Mediterranean on a yacht with her lover. They picture a fabulously wealthy person with power and fame to spare. Always, the persons imagined are healthy, good-looking, and well-short of old age. The reality of happy people is much different.\n\nAs we hope our discussion of set points has convinced you, circumstances have a lot less to do with our happiness levels than most people think. What is most surprising, however, is how little circumstances affect happiness at all. Millionaires are not much happier than middle-income folks. People in their twenties are no happier than people in their sixties. Men are about as happy as women, although women have a wider emotional range. Achieving fame barely moves the happiness register. In general, circumstances as a whole account for only about 10 percent of the difference in happiness levels.\n\nNor is well-being significantly affected by where you live in the United States. When the Centers for Disease Control (CDC) asked 1.3 million Americans between 2005 and 2008, \"In general, how satisfied are you with your life?,\" it turned out that residents in the happiest and least happy states differed by only one-tenth of a point on a 4-point happiness scale. Nonetheless, the CDC study provided plenty of fodder for researchers who offered their interpretations of the tiny differences. While the data suggested that residents of western states are, on the whole, slightly happier than residents of other states (the merriest five, in order, were Utah, Hawaii, Wyoming, Colorado, and Minnesota), when two researchers controlled for factors such as income, marital status, and age, they found that residents of southern states came out on top. In their study reported in the December 2009 issue of _Science_ online, economists Andrew J. Oswald of the University of Warwick and Stephen Wu of Hamilton College in New York concluded that the states that made people happiest were actually (generally poorer) southern states, with Louisiana, Hawaii, Florida, Tennessee, and Arizona ranking best using their methodology. Mississippians, for example, who ranked 48th in satisfaction in the raw averages, came in at an impressive number 7 on the satisfaction scale after Oswald and Wu made their statistical adjustments. Typically, the media overplays rankings of these sorts and readers are rarely informed as to how very modest the state-by-state differences actually are.\n\nNonetheless, there are a few circumstances that account for significant differences in reported levels of happiness. One of these is belonging to a church. People who describe themselves as \"very religious\" or \"religious\" are happier than those who don't. Why religious people are happier than their more secular brethren is somewhat of an open question. One theory is that religion gives the believer a sense of purpose and optimism about the future. A second possibility is that members within religious communities often develop deep bonds, and it is these relationships rather than the religious beliefs that account for the happiness boost. Finally, it is possible that the type of people who gravitate to religions tend to be happier people in the first place, rather than the religion making them happier. A secular person could say on Tuesday, \"I'll become religious as of today and get happier,\" but that is not a leap of faith most people could make even if they wanted to. (Interestingly, Ruut Veenhoven, director of the World Database of Happiness, reports that hedonists appear to be somewhat happier than nonhedonists. What this says about the higher levels of happiness reported by churchgoers is not clear. At the very least Veenhoven's finding suggests abstinence is not the answer to everything.)\n\nEnjoying a number of close relationships also makes people happier. This is one of those happiness determinants with both a circumstantial and intentional aspect to it, so it's hard to know exactly how to categorize it. We are social animals and interaction with others usually (we all have personal knowledge of many exceptions) makes us happy. Measured on a 5-point scale, with 5 being ecstatically happy, people report average happiness levels of 3.7 when they are interacting with friends, 3.3 to 3.4 when they are interacting with spouses or children, and 2.8 at times when they are interacting with clients or co-workers. By comparison, hanging out alone produces an average happiness rating of 2.7, although that's still better than time with the boss, which comes in at only 2.4.\n\nWhile married people are only slightly happier than single people, it is clear that ending a close relationship can have devastating short-term consequences for happiness levels. People going through a divorce are generally unhappy. Over time, most people get over the loss and move back toward their set points. Having intimate relationships appears to be a basic human need, and people who lack them rarely are as happy as those who have them. Moreover, the more such close friendships a person has, the happier he or she will be.\n\nOur social nature is reflected in recent studies that show happiness is contagious. In a study of nearly 5,000 residents of a Boston suburb, researchers examined how a sense of happiness moved through various social networks to which the study subjects belonged, including friends, family, neighbors, and co-workers. The study showed that happy people tend to cluster together and that happy people occupied the center of social networks. By contrast, researchers found unhappy people stranded on the periphery of networks, plugged into the center by only a few social lines. Neither of these results should surprise anyone. More interesting, however, was the conclusion that happiness is transmitted though social networks, weakening as it spreads out over the chain of relationships. For example, if Ann becomes happy and is a friend of Barb, then Barb is about 15 percent more likely to become happy herself; if Barb is a friend with Carol, Carol is about 10 percent more likely to benefit from spillover happiness that starts with Ann. Friends, the study suggested, are better transmitters of happiness than family, and family better transmitters than co-workers. The study found geographical distance key to the effectiveness of the transmission, with transmission most effective for friends living less than a half mile apart.\n\nWealth is the most overrated of all factors in people's guesses as to what will improve their happiness. Asked what single thing would be most likely to make them happier, a majority of Americans answer \"more money.\" Almost three-quarters of college freshmen in a recent survey said it was important to be \"very well off financially.\" Harvard University psychology professor Daniel Gilbert sums up the situation this way: \"'[W]e think money will bring lots of happiness for a long time, and it actually brings a little happiness for a short time.'\"\n\nWhat is true is that rich people are somewhat happier than poor people. About 45 percent of the wealthiest quarter of Americans describes themselves as \"very happy\" compared to 33 percent of the poorest quarter of Americans who describe themselves that way. (Still, 86 percent of the poorest Americans call themselves at least \"quite happy.\") Beyond lower-middle-income levels, however, increases in wealth give only the tiniest of happiness boosts. Some satisfaction seems to come from belonging to a higher social class, probably because membership in the higher class correlates with an increased likelihood of feeling of having \"control over life.\"\n\nAlthough geography makes little difference in happiness levels among Americans, it is worth noting that significant differences do appear when comparing citizens of different countries. Many factors, including standard of living and degree of individual freedom, seem to affect happiness differences among nations, yet no factor plays a bigger role than the level of trust people have in their government institutions and fellow citizens. Citizens of Denmark, a country where residents have a very high level of trust in their government, are much happier than citizens of Moldova, who think very poorly of their government. A map of world happiness shows high rates of happiness in the most democratic of nations, from Western Europe through the United States and Canada to Australia and New Zealand. On the other hand, the most autocratic of the old Soviet bloc countries, such as Belarus and Ukraine, as well as the governmentally challenged nations of Africa, tend to be home to the least happy of citizens.\n\nTrust in general seems a key component of happiness. The relationships that provide the biggest happiness boost are ones that are built on trust. Communities where residents trust each other are far happier than communities where they don't. Researchers dropped wallets on the streets of cities around the world. The wallets contained the name and address of a supposed owner. The nations with the highest return rate of wallets, such as Scandinavian countries, also turned out to be at the top of the happiness charts. The study showed a high correlation between return rates, a plausible measure of trust within a society, and happiness.\n\nUltimately, the circumstances that have the biggest effects on happiness are family relationships, employment status, health, and quality of government. Of these, the biggest drags on happiness are being recently separated (which makes a person less happy than either being divorced or never married), losing a job (almost, but not quite, as bad as separation), being in seriously bad health, and having the misfortune of growing up in a country such as Moldova. For people in these circumstances, their situation is a serious impediment to happiness. But for most people, all these situational differences hardly amount to a hill of beans on the hike up happy mountain.\n\n##### _Adaptation: Life on the Hedonic Treadmill_\n\nEven if circumstances matter little, a sudden change in circumstances might affect happiness levels\u2014at least for a brief time. When it comes to measuring our happiness, one thing we all know is that it's not a flat line. Some days (or moments) we're feeling relatively up, and then the next day (or moment), we're down in the dumps. Happiness set points don't determine our day-to-day levels of happiness; rather they are a measure of the happiness level we tend to gravitate back to over time. Our day-to-day, moment-to-moment happiness levels are strongly influenced by our circumstances and intentional activity, as we'll discuss below.\n\nThe tendency of our happiness levels to drift back toward our set points from our fluctuating highs and lows is called _adaptation_ , or life on \"the hedonic treadmill.\" If we get that hoped-for promotion, marry our sweetheart, or win the lottery, we will undoubtedly become happier in the short-term, but at some point in the future (be it days or months) we'll start moving back toward our set point. The same goes for negative experiences. If we get fired, end a relationship, or lose our fortune in a bear market, we'll undoubtedly experience some short-term unhappiness (or even depression), but the vast majority of us will see our happiness levels rise back in the long-term to near the point that they were before our misfortune. In general, events that we think will make us happy forever and ever (as the fairy tales suggest) rarely do, and even the most devastating of life's blows seldom leave us emotionally crippled for life. That is not to say, however, that there are not some circumstances to which we will never fully adapt. For example, chronic pain, life-threatening danger, and high ambient noise levels are all circumstances associated with long-term lower levels of reported happiness. Paraplegics are not as happy as other people. We adapt over time, but sometimes not completely back to where we started. Interestingly, we adapt less to minor annoyances and small pleasures than to more significant events. That too-long wait at work for the elevator grates every day because the negative experience is insufficient to trigger a response from \"our psychological immune system.\" Because adaptation is not well understood, you are likely to overestimate the amount of happiness that a positive events will bring you\u2014just as you underestimate your own ability to recover from one of life's many setbacks.\n\nAs it happens, people also generally do a rather poor job of predicting future levels of happiness. Most people (although not so much lawyers) have a bias toward optimism that causes them to overestimate future levels of happiness. Asked to predict how happy they will be five years from now, almost three-quarters of people surveyed expect to be happier and only about 5 percent predict that they will be less happy. In fact the best prediction of how happy you will be in five or ten years is precisely how happy you now are. Our mistake? Usually it is in thinking that the future will be more like the present than it actually will be. In our imagined futures, we continue to value the same experiences we do today, and we then adjust our happiness upward based on a few anticipated events, such as having kids, moving into a new home, or landing a sought-after promotion. The reality is that how we value various experiences continues to change over time and that life, almost always, pushes us in unexpected directions.\n\n##### _Internal Factors: The 40 Percent Determinant of Happiness Levels That Is within Your Control_\n\nWith genetics accounting for 50 percent of our happiness, and those circumstances that are largely out of our control a paltry 10 percent, that leaves 40 percent within the power of our own actions and ways of thinking to determine. In other words, well-chosen happiness strategies can boost happiness levels four times more than if somehow the circumstances of life that are largely beyond our control magically were to swing to the ideal. Of course, there is no single silver-bullet solution to improving happiness levels by 40 percent. Rather, the 40 percent figure represents the upper limit of the possible if you adopt all the approaches to improving happiness that happen to be best suited for your own circumstances and personality.\n\nWhat _are_ these happiness strategies that can give our happiness levels powerful boosts? They are many and they are varied.\n\n### GETTING MORE SATISFACTION\n\nThe \"forty-percent solution\" to happiness, focusing on those two-fifths of our overall happiness that we can control, is really the only solution we have. You cannot change the genes you've been dealt, and most of what we've labeled \"circumstances\" are either unchangeable (such as your age or health) or very difficult to change. The potential 40 percent gain in happiness from intentional strategies represents a ceiling, and one that is difficult to reach. Getting a full 40-percent boost might be as unrealistic as going from the ten push-ups you can do now to the hundred you'd like to do. It will involve a lot of focus and even then you might not make it. But if it isn't worth trying to get happier, then what _is_ worth trying?\n\nWe will discuss specific steps lawyers can take to increase career satisfaction in chapter 4, but the rest of your life matters too, so here we offer some strategies that can help boost overall life satisfaction. A happier person is usually also a happier lawyer, so adopting some of these strategies will probably make you feel better about your career, even though careers are not the focus of these strategies.\n\nFor the most part, the strategies we suggest are aimed at increasing satisfaction, not the more transient variety of happiness. Need we remind you, however, that the passing pleasures of life make a contribution to overall life satisfaction? So, in addition to trying out the strategies below, don't forget to treat yourself to life's pleasures. Once in a while, go for that double-fudge ice cream, grab the skis and head for the slopes, or indulge yourself with a Swedish massage.\n\nAnd don't forget the activities known to promote satisfaction, and not just short-term happiness. These activities include exercise, getting adequate and regular sleep\u2014if sleeping can be fairly called \"an activity\"\u2014and meditation. Finding enough hours in a day for exercise, adequate sleep, and meditation can be a challenge for lawyers. We will stress later the importance of a balanced life, but let this be an early warning of the costs associated with a seventy-hour-per-week job. Relax with your favorite guru or yoga instructor and enjoy that voyage of self-discovery.\n\n### BECOMING A THRIVING PERSON\n\n#### THE SIX KEYS TO LIFE SATISFACTION\n\nAcross all cultures, six experiences have been found central to making a person thrive\u2014and a thriving person is generally one who will be satisfied with his or her life. Those experiences are security, autonomy, authenticity, relatedness, competence, and self-esteem. It is within your power to either develop or erode these core elements of a thriving person. If you play your cards right, you'll be happier.\n\nSecurity is the odd duck on the list, in that it is more dependent on external conditions than the other five experiences, and is a prerequisite to a sense of well-being. A fearful, anxious existence is not consistent with thriving. Choices dictated by fear are not choices at all. People do not thrive in war zones when daily fears of bombs landing on their house crowd their minds. Most of us don't live in a war zone, but some of us do live in neighborhoods where security is an everyday concern. Others among us might worry about threats from someone living under our own roof. Still others may feel physically secure but be anxious about health or finances. When sources of deep anxiety are present, life satisfaction becomes impossible. Needless to say, then, anything that can be done to alleviate the anxiety should be done. Can you afford to move to a safer neighborhood? Can a psychologist or medical doctor or financial planner help you confront and control whatever it is that is the source of your insecurity? These feelings can be heavy and are really beyond the scope of this book\u2014but know that without facing these issues first, everything else we can offer will be of little help.\n\nAutonomy is the ability to make your _preferred_ choices and not have them dictated either by fear or imagined or real constraints. Put simply, an autonomous person can do the things he or she wants to do. It should surprise no one that autonomy is central to thriving. When it comes to autonomy, Americans start with an advantage over Belarusians or North Koreans\u2014at least we don't have a government severely constraining our choices. Nonetheless, we all sometimes have our choices limited. We might not have the money to take that dream vacation we want. Or we might have the money, but the demands of work make travel impossible. Or we have the money and the time, but we have an elderly parent whose needs we must care for. We can't always do what we want\u2014so what else is new? Accepting that not every door is open to us is not only part of growing up but is part of becoming a happier person.\n\nIn fact, what is perhaps most worth noting on the issue of autonomy is that our freedom is enhanced, not constricted, when our universe of choices shrinks to a manageable number. More people today suffer from having too many choices than from having too few. Having too many choices can lead to a paralysis that prevents us from making any choice at all or, if we finally do make a choice, can cause us to suffer regret (\"buyer's remorse\") over all of those other possible choices not made. In his wonderfully insightful book, _The Paradox of Choice: Why Less Is More_ , Barry Schwartz shows that people who overanalyze the vast array of choices modern life throws at us are less happy, less optimistic, and more depressed than people who are quicker to say \"good enough\" and make their choices. The overanalyzers, people Schwartz calls \"maximizers,\" pay \"a significant price in terms of personal well-being.\" Choice overload becomes a nightmare for maximizers but not for \"satisficers\" who understand that no choice is likely to be perfect and that the time spent trying to select from a set of reasonably good alternatives can better be spent in more satisfying ways.\n\nAuthenticity is being who you really are. An authentic person makes choices based on his or her true values. Being \"in the closet,\" whether on the issue of sexual orientation or political affiliation or spiritual preference, can be hazardous to your emotional health. When your goals are consistent with your deepest values and not the values someone else chooses for you, your life has a clarity and sense of purpose that makes achieving happiness much more likely. What this means for you and your law career will be a topic for a later chapter, but it probably is obvious to you already that if you're a social conservative, your prospects for career satisfaction aren't enhanced by joining a pornography defense firm. Authenticity, however, isn't just about the big choices, it is about the little ones too. Do you prefer a Budweiser to that Sonoma County pinot noir, but go with the wine because you think your friends will think less of you if you ask for a beer? Does your choice of furniture reflect your taste or that of an interior designer your mother recommended? Almost everyone hides some part of their true self, but the less you do it, the happier you'll become. You can take a step in the direction of authenticity, and in the direction of greater happiness, by identifying the activities that truly make _you_ (and not necessarily anyone else) happier\u2014and then spending more time doing those activities. If your friends leave you because you suddenly start playing volleyball at nudist camps or going to pyramid conventions\u2014well, they probably weren't the best of friends anyway.\n\nThe fourth key to thriving is relatedness, or feeling well-connected to others. People need people. People need to share, need to laugh, need support. Of course, creating deep connections takes time, and it usually involves a process of promise-making and promise-keeping that eventually leads to trust. The good news is that the very steps that can eventually create a close friendship often bring increased satisfaction along the way. Expressing gratitude frequently, for example, is one of the surest ways to increase your own life satisfaction. Forgiving others and moving on also increases satisfaction, while wallowing in and ruminating over past wrongs blocks happiness at the doorway. Acting happy tends to strengthen friendships and, interestingly, makes you happier than you were when you started acting. (Even a simple thing like forming a smile seems to bring a small amount of pleasure to the smiler.) Performing acts of kindness also increases satisfaction, as does joining with friends or relatives to reminisce over past events. We could come up with a much longer list here, but you get the idea.\n\nA fifth experience of the thriving person is a sense of competence. The person who masters challenges or accomplishes a difficult task feels capable, feels valued, and is likely to feel happier. As a lawyer, you have already mastered (more or less) a difficult skill. If you have practiced in a specialized area of law, and your skill is in demand, you probably already feel valued by clients or by colleagues. (If you have that mastery, and no one in your firm seems to give a damn, maybe it's time to get out?) Apart from your legal career, however, competence in other areas such as woodworking or tennis might also give you a happiness boost. In particular, by developing competencies, a person increases their opportunities for activities that thread that happy needle between boredom and anxiety and which are highly correlated with feelings of enhanced well-being.\n\nFinally, a thriving person has self-esteem. This experience, in fact, is so closely tied to satisfaction that the two terms are sometimes thought to be nearly synonymous. A person with self-esteem is satisfied with one's self. Self-esteem has been a large topic for many years, and we won't attempt to plow this field. Suffice it to say, adopting tools that increase self-esteem will also tend to make you happier. These tools include forgiving yourself, trying to imagine an ideal future for yourself, avoiding excessive rumination, and doing the things (such as practicing acts of kindness or meeting a difficult challenge) that could make you feel better about yourself.\n\nHow do you measure up on these six experiences of the thriving person? On a one-to-ten scale, which is your strongest suit and which your weakest? Focusing on these intrinsic (or self-development) goals produces more happiness than focusing on extrinsic goals, such as becoming rich or famous.\n\n#### DO THINGS GO BETTER WITH HAPPINESS?\n\nWe don't seek happiness because it will make us a better spouse or friend or lawyer; we seek happiness because we like being happy. \"People are strongly, perhaps even primarily, perhaps even single-mindedly, motivated to be happy.\" As it turns out, however, happiness is more than its own reward. One of the side benefits of happiness, as we've noted, is that the happy person spreads happiness to others. Happiness also allows us to be more receptive to information, either coming from others or from our environment. It makes us mindful: we see things that we miss when in a sour mood. We become aware of the loveliness of a garden, the sound of the oriole, or the smell from the freshly fallen rain. Just as significantly, we become better at reading the signals that those near us are sending: we become better listeners and become more capable of satisfying the desires of our lovers and our friends. Finally, happiness makes us healthier. It produces better immune systems and makes us more resilient when we suffer setbacks. Happy people live longer than unhappy people.\n\nIn this chapter, we've explored the components of happiness. We've learned that happiness has a significant genetic component, but it also can be strongly influenced by the choices we make and the strategies we adopt. In the next chapter, we will consider what aspects of a life in the law move us toward greater happiness and what aspects tend to pull us away from our goal. It's off to work we go.\n\n## **CHAPTER 3\n\nWhat Makes Lawyers Happy and Unhappy**\n\nWHEN A RESEARCHER AT THE UNIVERSITY OF Chicago's National Opinion Research Center (NORC) analyzed surveys of workers in 198 occupations, he discovered that the clergy ranked at the very top of the list in job satisfaction and roofers ranked at the very bottom. The study did not attempt to determine why the clergy were so happy or roofers so miserable, but what we've learned about the sources of happiness in general gives us some pretty good clues. We'll see where lawyers stand on the happiness scale shortly, but let's start by looking at which occupations are the most and least happy to give us a measuring stick.\n\nThe first thing to note about the clergy is not only are they very satisfied with their work, but they are also unusually happy people. In addition to reporting the highest level of job satisfaction (with 87.2 percent calling themselves \"very satisfied\"), the clergy also reported the highest level of overall personal happiness (with 67.2 percent pronouncing themselves \"very happy\"). These results, of course, raise a \"chicken-or-the-egg\" question: Are the clergy so happy doing what they do because of the nature of the job, or do people who enter seminaries tend to be happier people and then let their personal happiness translate into higher levels of job satisfaction? The answer, it seems, is both: those who join the clergy are happy people and they love their jobs. Happy people tend to look at glasses as being half full and report above average levels of satisfaction across multiple domains, ranging from marital satisfaction to satisfaction with where they live. It is quite likely that the typical seminarian is more purpose-driven, more extroverted, and more focused on relationships (all traits associated with increased levels of happiness) than the average person.\n\nThe evidence is compelling, however, that something in the nature of the work also makes the clergy so spectacularly satisfied compared to other workers. The number of clergy reporting themselves \"very satisfied\" with their work is a full 20 percent higher than the number of clergy calling themselves \"very happy\" in their personal lives. What explains this?\n\nOne factor is undoubtedly that the work of the clergy closely aligns with their own values. They identify paths to eternal life or personal fulfillment, and they then try to guide their flocks along that path. They comfort the afflicted and rail, when they wish, against the evils of the world. Happiness research makes it indisputably clear that people feel better about work when they think they are making a contribution to the public good than when they think their work either has no social value or actually undermines the public good. The clergy have a benefit that lawyers, for the most part, don't enjoy. They can choose to attend a divinity school or seminary that closely reflects their own personal values, whereas law schools, with a few exceptions, generally are not associated with teaching distinctive sets of values.\n\nOther factors help make the clergy so satisfied. Their work involves them in deep and meaningful ways in the lives of others, and deep relationships correlate with higher levels of happiness. Their job affords high measures of control. The clergy are usually free to shape church programs (\"What should be the theme of my sermon this week?\" \"What field trip should we send the youth group on this year?\"). They also have a fair amount of scheduling flexibility and the reasonable prospect of a good work-life balance. The clergy face creative challenges within their realm of competence and frequently interact with people less fortunate than themselves, both job characteristics associated with satisfying careers.\n\nCompare the clergy to our oh-so-miserable roofers: up there, laying asphalt shingles in 100-degree temperatures or 20-degree wind chills and not feeling entirely secure all the while. Probably worse, laying shingles affords almost zero opportunities for creativity\u2014the shingles are laid down in a preset order, and there's no opportunity for the roofer to let loose an inner Van Gogh. Control and personal interaction is limited at best; it's one long hour of tedium after another, often just you and your roof.\n\nJobs that align with values and involve helping others are those that rated at the very top of the NORC study. Following right behind the clergy in job satisfaction were physical therapists, firefighters, and education administrators (deans, principals, and school superintendents). Teachers came in sixth on the list, in-between two jobs offering bushel baskets of opportunities for creativity and personal control, artists and authors.\n\nOn the other hand, at the bottom of the list, just above roofers in reporting low levels of job satisfaction, were waiters, bartenders, packagers, and stock handlers. None of these jobs, with the possible exception of bartending, provides creative challenges or offers workers an opportunity to express their personal values through their work.\n\nGiven what the survey suggests about the most and least satisfying jobs, it should come as no surprise that lawyers rank slightly above the middle of the pack in terms of job satisfaction. Lawyers, with 52.4 percent reporting themselves \"very satisfied,\" are below doctors (57.9 percent satisfied) and in-between occupations such as editors (52.9 percent satisfied) and accountants (49.7 percent satisfied). The jobs that fall between the extremes of satisfaction and dissatisfaction, including the practice of law, are those that score well on some characteristics of satisfying jobs (say, providing challenge), not so well on others (say, allowing control), and in-between on still others (say, opportunities for interaction or alignment of work with values).\n\nLawyers report so-so satisfaction with their careers despite the fact that law is a high paying and prestigious job. Those qualities don't matter very much in terms of job satisfaction. It is also a helping profession, and that is generally a positive indicator of happiness, except that, unlike clergy, lawyers don't see clients at their happiest moments.\n\nAn American Bar Association (ABA) study of how the practice of law matched the expectations that lawyers had when they entered the field is revealing. The study showed that the \"intellectual challenge\" of law practice was by far the aspect of practice that most closely matched expectations. Law is a great field for people who love mental puzzles, and almost everyone in the field seems to know that. Fully 70 percent of lawyers report that the intellectual challenge of law practice \"very well\" matched their expectations. Compare that to what seems to be the great disappointment of lawyers, the failure of their job to contribute as much to the public good as they expected. Only 16 percent of lawyers, according to the survey, found that their jobs afforded the \"ability to contribute to social good\" as much as they had expected. A full quarter of lawyers reported, on this measure, that their expectations were \"not at all\" met in their jobs. Falling in-between these extremes of expectation and reality was \"financial remuneration,\" with 34 percent reporting their compensation \"very well\" matched what they thought it would be, while another 52 percent found that the financial rewards of law practice at least \"somewhat\" matched expectations.\n\nDepending on an attorney's specific job\u2014small firm practitioner in a cultivated niche, in-house counsel with a template of corporate rules, or large firm associate\u2014the attorney will have more or less control. But there are many things lawyers can't control, chief among them other lawyers, deadlines, and billing. According to a 2007 ABA study, 69 percent of lawyer respondents saw \"declining civility\" in their profession, and 90 percent of lawyers in the largest firms found \"competition between firms\" increasing, possibly forcing firms to abandon more traditional professional models in favor of more efficient models (i.e., more emphasis on billable hours and client development and less emphasis on building collegial relationships and doing quality work regardless of cost). Especially telling is the fact that one-half of all lawyers now say that pressure to increase billable hours is a \"very\" or \"somewhat\" important reason that would make them consider leaving a firm. A desire to have more time to spend with friends and family ranks well above the concern to earn more money or to advance within the firm's hierarchy.\n\nThese results clearly flag the trouble spots in a lawyer's search for happiness. Many lawyers, especially those in the private sector, are disappointed that their work doesn't more closely align with their values. Many are also dissatisfied with what they see as the overly competitive, zero-sum nature of much practice today. Finally, many lawyers are struggling, sometimes with limited success, to achieve a happy balance between their professional and personal lives. We look more closely at some of these issues in the next section.\n\n### WHAT ACCOUNTS FOR ATTORNEY UNHAPPINESS?\n\n#### WORK STRESSES, HIGH BILLABLE HOURS AND LOW QUALITY OF LIFE\n\nLying was a way of life\u2014a necessity and therefore a virtue.... Our corporate culture required the show of enthusiasm in all circumstances. A partner would come into your office and ask if you had any plans for the weekend. The correct answer was \"no.\" And you would then be given an assignment to fill your empty Saturday and Sunday. The first time I was asked the question, I mumbled something about having hoped to go to Vermont. The young partner, who was nicknamed \"Dave the Barracuda,\" looked at me with a combination of incredulity and sympathy, as if I had just confessed to a subnormal IQ. \"It's a rhetorical question,\" he explained with an exasperated sigh.\n\nMany lawyers are unhappy with the rigid billable hour requirements of firm life and the long hours that the profession demands. The nature of the work is stressful: clients may have hopelessly unrealistic expectations; deals fall through; running a business while bringing in business is a constant pressure; cases have numerous unalterable deadlines that require late-night work; expenses of taking a case to trial can mount; clients can be convicted\u2014and face life imprisonment or execution. Lawyers represent people who have been traumatized, people who are going through the worst crises of their lives, in sum, people who are unhappy.\n\nOverwork is a piece of the picture. In recent years, billable hour requirements have escalated. In the 1970s, associates at large law firms billed on average about 1,700 hours year (or thirty-four hours each week). In 2009, the national average billable hour requirement for firms of all sizes was 1,888 hours, but expectations for associates on large firm partnership tracks can amount to 1,900\u20131,950 hours each year (an average of roughly forty hours each week). A forty-hour week sounds manageable until you realize that forty billable hours means many more (some estimates range to 50 percent more) actual hours at the office. The billable hour meter typically does not run during firm administration work, pro bono cases, marketing and business generation activities, attorney and staff recruiting activities, training and mentoring sessions for newer associates, or continuing education classes let alone during office face time, coffee breaks, or socializing opportunities with colleagues. Compared to a median work week of forty hours for all full-time employees in America, lawyers work a median of fifty hours per week, and one-fifth of new lawyers work sixty hours or more each week.\n\nA 2009 study by NALP, the Association for Legal Career Professionals, offered billable hour estimates by firm size (see table 1).\n\nThe difference in billable hour requirements between small and large firms seems modest, although attorneys in larger firms put in substantially more nonbillable hour time, when, for example, they perform administrative duties or have face time. Associates work these strenuous hours, but through attrition\u2014both voluntary and involuntary\u2014only about 15 percent of them become partners.\n\nIt is not just that billable hour expectations are on the rise but that partnership and bonus systems are often specifically linked to billable hours. Slightly less than one-half of firms with more than one hundred attorneys used \"meeting fixed goals\" as a measure to determine bonuses. The responses received from various studies and surveys revealed frustration among lawyers that their worth was quantified, and their career advancement determined, by the number of hours they worked. One associate commented, \"devotion equals promotion. The more you work the higher you rise.\" Similarly, The Rodent (an anonymous associate at a large firm writing an underground newsletter) reported that he was often criticized for not billing enough hours until one month, when partners lavished him with praise for high billable hours even though he was busy with \"mostly clerical work\u2014proofreading, sorting documents and making copies.\"\n\nTABLE 1. Yearly Billable Hours by Firm Size\n\nEven some of the perks of large law firm life\u2014on top of starting salaries beginning at $160,000 for first year associates\u2014such as BlackBerrys, catered meals delivered to the associates' desks, and a chauffeur service to drive them home if the trains have stopped running\u2014are props for billable hour production. One law professor commented, \"Many associates feel they are working in nicely decorated sweatshops.\"\n\nTechnology can also contribute to lawyer unhappiness regarding mounting billable hours and less personal time. While today's lawyers may be more productive and efficient due to various technological advances, they are also always \"on call\"\u2014reachable by e-mail, cell-phone, or fax. One _American Lawyer_ survey respondent complained, \"Technology makes it possible for [partners] to encroach on all of your time, whether you're awake, asleep or on vacation.\" Making work more portable has many advantages, but it increasingly blurs the line between being at work and away from work, and this contributes to attorneys' dissatisfaction.\n\nThus, we see that some of the primary reasons for lawyers' discontent with their working lives center on quality of life issues: the overwhelming hours coupled with the idea that the value of an attorney is measured by the number of hours billed; the stressful nature of the work; and the gap between the ideals they had when they entered the profession and the repetitive nature of their daily tasks.\n\n#### LAW IS BECOMING MORE OF A BUSINESS, LESS OF A PROFESSION\n\nMoney is at the root of virtually everything that lawyers don't like about their profession: the long hours, the commercialization, the tremendous pressure to attract and retain clients, the fiercely competitive marketplace, the lack of collegiality and loyalty among partners, the poor public image of the profession, and even the lack of civility.\n\nThe practice of law has become more of a business and less of a profession. Law firms focus on maximizing profits. Money is equated with firm prestige and individual career success. These economic forces are not unique to law. The uptick in greed and competition can be found in most occupations, industries, and social life. Economists Robert Frank and Philip Cook call it the \"winner take all\" society.\n\nLaw firms afford lawyers much less job security than in previous generations. Firms are willing to abandon practice areas that are less lucrative. They also have few qualms about jettisoning lawyers. The early 1990s saw layoffs of both lawyers and bankers in response to the bank credit crunch and junk bond market collapse. In recent years, the cascade of economic problems resulting from the subprime mortgage crisis has prompted slower hiring, attorney and staff layoffs, summer hiring freezes, rescinding of job offers, and \"redeployments\" of real estate and finance lawyers into other practice areas.\n\nHigh rates of attorney turnover lead to reduced loyalty to firms. Within five years of graduating from law school, 80 percent of associates will have changed jobs at least once. This is up from two-thirds in the 1988\u201396 period, and compares even less favorably to the historical view of Traditionalists and Baby Boomers of a lifetime career at a single firm. Even when associates make partner, the relationship is no longer a permanent union. There are numerous different types of arrangements: typically equity (part-owners of the business) and nonequity partners, of counsel or senior counsel positions. These heavily stratified partnership structures create alienating hierarchies. In tiered partnerships, subordinate partners are often treated more like permanent employees than co-owners.\n\nLaw firm consolidations, dissolutions, and lateral hiring have always existed but increasingly have become routine. With the accelerating pace of consolidations has come a change in the nature of mergers. Law professor David Achtenberg says, \"It would have been virtually inconceivable twenty years ago for a firm to merge into another one knowing that doing so would create a conflict of interest that would force a significant number of partners in the firm to have to leave or would strip many of its partners of any meaningful control in the newly created entity. Today it is common.\" Mergers and acquisitions add to feelings of instability and undermine trust.\n\nLawyers pay attention to the financial bottom line and try to maximize profits; success is measured solely in quantifiable terms\u2014numbers of billable hours, hourly rates, fees, or favorable verdicts. Some lawyers may respond to economic pressures by doing what their clients want even if their behavior dances on the borderline of unethical conduct. Fee structures encourage continued combat. Examples abound of defense lawyers continuing litigation for years to protect their income stream of exorbitant hourly wage work.\n\nSome lawyers cross the line and participate in unethical conduct. Others engage in offensive and unprofessional behavior that ranges from rudeness to obstructionist tactics. We turn next to dissatisfactions related to lawyer incivility.\n\n#### INCIVILITY\n\n_She wants to play dirty, I can play dirty... I went to law school, people._\n\n\u2014Bride Wars\n\nOne of the largest dissatisfactions with lawyers' professional lives is other lawyers. Incivility is rampant. In any gathering of lawyers, you will hear complaints about lawyer incivilities. Competitive tactics have long been part of the profession, but incivility is escalating, particularly in larger cities. In smaller towns, everybody still knows everybody else; in more sprawling metropolitan areas, lawyers practice against opposing counsel they may see once and then never again. What is lost is not just a sense of a professional community but civility.\n\nThe adversarial system and the ethical obligation of zealous representation have spawned some mutant offspring. Some lawyers perceive that playing hardball brings a strategic advantage; others think litigation is a form of warfare. They may launch personal attacks and engage in insults, name-calling, threats of bar complaints, rudeness, and hostility. The unprofessional behavior extends from lawyer-to-lawyer contacts to depositions and into the courtroom. Examples of rudeness in reported cases include sarcastic and inappropriate language, diminutive names for female lawyers (e.g., \"little lady\"), references to defendants as, in one case, \"snake[s] in the grass,\" and in another, \"cowardly, dirty, low down dogs,\" and purely offensive and uncooperative behavior. In one survey of lawyers, 56 percent of those responding \"cited obnoxiousness as the most common, unpleasant quality they encounter among other lawyers.\" In a random sample of more than 4,600 respondents to an Eighth Circuit survey, nearly three-quarters of all female lawyers and one-half of all male lawyers reported that they had experienced general or gender-related incivility in the past five years. According to two other polls, 69 percent of lawyers thought that civility in the profession had declined over time, and 80 percent of judges had observed uncivil attorney conduct in their courtrooms.\n\nDeclining civility is due to an abundance of lawyers and an increase in competition for business, a decrease in mentoring of new lawyers, the absence of repeat interactions with the same small coterie of lawyers, and the rise of social incivility generally. Of course, part of the explanation may be a \"good old days\" problem. Older lawyers may remember practice thirty or forty years ago as more idyllic than it was. In 1932 Clarence Darrow wrote that \"trials were not being conducted in a dignified effort to find the truth but more like a prize-ring combat.\" (An interesting observation, coming as it does from one of the greatest courtroom prize-fighters of all time.)\n\nThe more contemporary scorched earth litigation tactics at times go beyond manipulation of facts, assertion of frivolous claims, hardball negotiation, disrespect for witnesses, ruthless cross-examination, discovery abuses, and unmeritorious appeals. State disciplinary boards have sanctioned lawyers for calling opposing counsel names like \"punk,\" \"boy,\" \"fool,\" and \"idiot\"; shoving other attorneys; and in one case threatening an attorney-defendant who refused to settle with \" 'the legal equivalent of a proctology exam.' \" The Supreme Court of South Carolina issued a public reprimand to a lawyer who told a \"deponent that he would like 'to be locked in a room with [her] naked with a sharp knife,' and that he needed 'a big bag' to put her in 'without the mouth cut out.'\" A popular YouTube video shows several prominent Texas lawyers devolving into colorful name-calling during a deposition.\n\nWith the constant friction, lawyers feel isolated and learn to mistrust each other. It is stressful to go to war (er, work) each day having to anticipate Rambo litigation tactics and personal attacks\u2014and not just from opposing counsel.\n\nThe competition and discontent also percolate into law firms. One of the first tensions is among peer associates in what Marc Galanter and Thomas Palay call the \"promotion-to-partner tournament\":\n\nOver a fixed period of time, the firm holds a contest in which all the associates in a particular \"entering class\" compete, with the prize of partnership being awarded to some fixed percentage of the \"top\" contestants. An associate's final standing in the tournament depends upon the size and quality of his production of two goods: (1) high-quality legal work; and (2) his own human capital, measured subjectively, not mechanically. After a specified period of time, the players in a particular class are ranked and those in the top stratum are declared \"winners.\" (This may vary from ten to ninety percent). The losers are told that they can remain employees but will never become partners; or they may be given consolation prizes, such as severance pay or help finding another job; or they may be unceremoniously dumped.\n\nPartners can play too, although the games are somewhat different. Some partners engage in client-hoarding. The lateral mobility of lawyers, particularly those who control business, has created substantial incentives for those who currently have close relationships with a client to be certain that no one else does. By not sharing work with other partners, but instead delegating work to subordinate associates, a lawyer retains bargaining power within the firm and has a book of business to take down the road to a potential acquiring firm. Lawyers, as mercenaries who are always ready to pack their briefcases and head down the road\u2014like a yuppie version of the _Grapes of Wrath_ \u2014also have little incentive to engage in meaningful social relationships with other people in their present firm.\n\nA side result of this declining civility is the elevation of marketing over craft. The internal economic forces in a firm that prompt attorneys to hoard clients also create disincentives for lawyers to share work with the most competent lawyer for a particular issue to avoid risks of client-poaching. Attention to revenue undermines professionalism.\n\nThe adversarial system can also take its toll on your personal relationships. Professor David Guenther tells this story of winning an argument, while a good thing in the courtroom, can be a bad thing for a relationship:\n\nSeveral years ago a male student thanked me for what he had learned in my course. \"For the first time,\" he said, \"I'm winning arguments with my girlfriend.\" Coincidentally, a few months later, a female student asked me for advice on how to get her boyfriend to stop arguing like a lawyer. I told her about the time I thought I was having a conversation with my wife when she suddenly retorted: \"Stop cross-examining me!\"\n\n#### KEEPING UP WITH THE JONES DAYS\n\nAnother dynamic that recently has entered the legal profession is the spread of information about co-workers' incomes, as well as the business-like calculation of partners' shares and associates' salaries and bonuses. The knowledge of relative wealth has exploded in the past twenty years. Accompanying this knowledge has been the conflation of wealth with worth.\n\nBefore 1975, when the Supreme Court held that the practice constituted price-fixing, local bar associations used to publish minimum fee schedules. Back then, few lawyers made astronomical amounts of money. Now there are more who do, whether it is through lucrative plaintiffs' practices or large draws by partners at firms. Not only has there been a surge in what the richest lawyers are making, but also there has been a vast change in the extent to which everyone knows all of the information.\n\nIn 1987 journalist and lawyer Steven Brill started _American Lawyer_. Before its publication, lawyers had a general understanding that it wasn't polite to talk openly about how much money they and their co-workers earned. Lawyers had a general sense of who was doing well, but it was not directly discussed. Brill's innovation, which made the magazine a huge success, was to blow away that inhibition on talking about what lawyers were being paid. The magazine published lists of starting salaries for new lawyers and of exactly what partners were making at all the big firms. That changed things, because lawyers became much more acutely aware of where they stood in the financial pecking order. Although _American Lawyer_ concentrated on the biggest firms, it trickled down because the focus on what those top-paid lawyers were making generated envy and insecurity.\n\n#### BIGGER IS NOT BETTER\n\nLaw firms are expanding in size and that has adverse consequences for lawyer happiness. In the United States, at least six hundred firms employ one hundred or more lawyers each. The 250 largest firms collectively employ more than 120,000 lawyers. Firm size can make an enormous difference in the types of relationships lawyers have with both their work and other lawyers. In a thirty-or forty-lawyer firm, every lawyer knows every other lawyer. In a firm with two hundred lawyers, that doesn't happen. Bigger may not be better when it comes to happiness in law firms. People feel uncomfortable when substantial control over their future and success depends on people they do not know.\n\nAs firms grow bigger, the slices of work done on cases by individual lawyers get smaller. Attorneys work on minor pieces of huge cases and engage in endless review of documents with little human contact. When they do communicate with each other, they do so by e-mail and conference calls and through Fed Ex drops. Lawyers become alienated from the nature of their work, and they do not see how their work matters.\n\nYet people are different, and for some lawyers the big firm is the perfect choice. Most highly specialized, sophisticated litigation is conducted by large firms, and many lawyers report enjoying the intellectual challenges presented in those cases. Other happy large-firm lawyers point to the high-quality training and mentoring they received, or to their delight in having so many smart, interesting colleagues from whom to draw their circle of friends. For introverted lawyers, large firms typically offer a variety of research or transactional work that may reduce the discomfort they would feel in more socially demanding small firm settings. Finally, of course, there is the money. Money can't buy happiness, but sometimes it can be a shield against unhappiness. Big firms generally offer compensation at the top of the salary charts, and for today's many debt-ridden law graduates, a generous starting salary means one less big stressor.\n\n#### FEELING UNLOVED\n\nQ: What's the difference between a lawyer and a bucket of pond scum?\n\nA: The bucket.\n\nQ: Why does California have the most lawyers and New Jersey the most toxic waste dumps?\n\nA: New Jersey got first pick.\n\nQ: What do you call six hundred lawyers at the bottom of the ocean?\n\nA: A good start.\n\nQ: What is the difference between a dead rattlesnake and a dead lawyer in the middle of the road?\n\nA: There are skid marks in front of the rattlesnake.\n\nQ: What's black and brown and looks good on a lawyer?\n\nA: A Doberman Pincher.\n\nOf course, the ultimate anti-lawyer joke is that lawyers don't think anti-lawyer jokes are funny and that everyone else doesn't think they're jokes.\n\nThe public has a dismal opinion of lawyers. In a 2006 Gallup poll survey of Americans' opinions about the levels of honesty and ethical standards in twenty-three professions, only 18 percent of respondents gave lawyers a \"very high\" or \"high\" ethics rating. Lawyers ranked well below nurses (84 percent), doctors (69 percent), clergy (58 percent), and police officers (54 percent) but ranked above members of Congress (14 percent), insurance salesmen (13 percent), and car salesmen (7 percent). An ABA survey revealed that nearly three-quarters of those responding believe that lawyers try to win no matter the costs and will \"manipulate both the system and the truth.\" Lawyers are perceived as crooked, unethical, greedy, aggressive, and untrustworthy: a combination of mistrust and dislike.\n\nWhy do people love to hate lawyers? A number of factors are at work. Lawyers inevitably suffer in reputation because they tend, just by the nature of their jobs, to be heavily involved in negative things. In other words, people don't need lawyers when things are going great, or at least they don't need them to be as visible. If a corporation is profitable, nobody is going to notice its lawyers, or think, \"gee, they must have an excellent legal team.\" Lawyers take a more prominent role when things are bad\u2014bankruptcies, accusations of crime, or divorces. People seek the services of lawyers when they are in trouble, so the \"public will always associate lawyers with some of life's worst moments.\" (Although clergy often step in during difficult situations too, they also celebrate the best of times with worshippers, and clergy don't absorb the blame when things go wrong.) It may be hard to avoid being tarnished by the negative associations, even if lawyers perform admirably and try to make the best of bad situations.\n\nLawyers are merchants of misery. They see people in times of stress or crisis and, unlike other professionals, can't always, perhaps not even often, make clients happy. As Professor Andrew McClurg observes, \"Litigated disputes, by their nature, have winners and losers. Each one leaves someone hurting financially and often emotionally.\" Aside from mediation, lawyers lose their cases about half the time, and even when they \"win,\" their clients may not obtain all the relief they want.\n\nWhile lawyers get dragged into the mud every time something bad happens, they rarely accumulate occupational brownie points for a job well done. Except for some fictional portrayals (such as Atticus Finch in _To Kill a Mockingbird_ ), civil rights cases, and actual innocence project cases, where lawyers championed the exoneration of incarcerated prisoners through DNA evidence, it is difficult to think of cases that prompted the public to react: \"Wow, good lawyering!\"\n\nLawyers must deal with clients' complaints about fees. The fee charged is often a large amount for the client and a small amount for an attorney. Very few clients feel the amount charged bears any relationship to the value received even when the lawyer is being responsible about fees.\n\nLawyers, particularly those who represent corporate giants or despicable criminal defendants, are perceived as hired guns whose values mirror those of their clients. A lawyer's function is not to be disinterested, impartial, or revealing; so, it is no wonder that the derogatory name \"mouthpiece\" has developed as a slang term for lawyer. The adversarial process encourages actions that may be perceived as lies, delays, and mindless posturing to create room for negotiation. \"Those initial positions... may make lawyers seem hostile, aggressive and bloodthirsty, more interested in vicious battle than peaceful and prompt resolution.\"\n\nThe public's attitude toward the profession is also formed by the media. Most Americans get the vast majority of their information about the legal system from the popular media, principally television. Hard news reports include spins about the \"increases in the volume, cost and duration of litigation, as well as the number of lawyers, and the size of their incomes.\" Much of the information about increases in the volume of litigation\u2014and especially the size of awards in personal injury cases\u2014is not true; but truth isn't the issue here. Sometimes the news and entertainment media blur together, resulting in the \"entertain-ification\" of the law. News reports about cases often come in sound bites from talking head legal analysts. Television personalities, such as Nancy Grace, urge the public to rush to judgment and portray lawyers (particularly criminal defense lawyers) as out to thwart justice at every turn. As a result, highly publicized circus trials occur, accompanied by \"media misinformation about what was actually going on in some of those trials.\" Every lawyer standing next to an unpopular client is tarnished by the association.\n\nPart of the negative public impression comes from the pop culture spin. TV and movie depictions are often of bad lawyers in a profession characterized by greed, incompetence, and cynicism. While the prosecutors on _Law & Order_ (especially Sam Water-ston as Jack McCoy) are heroic, the defense lawyers are smug and insufferable. In the movie _Jurassic Park_ , audiences cheered when the first human the dinosaur devours is the lawyer. In _The Devil's Advocate_ , Al Pacino plays the managing partner of a law firm who actually _is_ Satan. Being a member of a profession that faces a continual barrage of negative stereotyping from Hollywood and jokes from Jay Leno can be exhausting and saddening and can leave the human who holds the job \"feeling empty and disillusioned.\"\n\nThe cumulative effect of many of these dissatisfactions in the profession\u2014the hours, fierce competition, incivility, and public aversion\u2014may help explain why numerous people have had the same one-word reaction to the concept of happy lawyers: \"oxymoron.\" We like to think of the idea as not an oxymoron but as a possibility.\n\nWe sometimes forget there are many upsides of being a lawyer\u2014features of the job that are connected to attorney satisfaction. Let's remind ourselves of some of them.\n\n#### LAW IS A NOBLE PROFESSION\n\n[I]t is we lawyers who create the mechanism for people involved in serious conflicts to peacefully resolve their conflicts in a system that places a search for the truth as a centerpiece of dispute resolution.\n\nPerhaps you are one of the many attorneys who not only like your job but feel that you are pursuing a passion as a well as a career. Being a lawyer can involve doing rewarding work. Law is a career in which you can make significant contributions, whether by handling adoptions, defending the Constitution, fighting race and gender discrimination, drafting legislation, protecting a battered partner, advising business entrepreneurs, tackling corporate Goliaths, or simply by being a trusted advisor. It also opens possibilities to contribute to the architecture of society: \"Unfortunately, when one looks at a construction site or a building, one sees only the bricks and mortar, the workers, the cranes, the splendid architecture, and the engineering miracles. The myriad legal structures and devices are invisible.... Lawyers help build the world.\" Social activists\u2014those engaged in cause-lawyering\u2014really can change the world because they can change the rules. Lawyers' words can become laws.\n\nLawyers are public citizens; they are vital in a democracy. Almost every lawyer-disparager, it seems, likes to trot out the line from Shakespeare's _Henry VI_ : \"First thing we do, let's kill all the lawyers.\" Few people realize, however, that the character offering the anti-lawyer remark, Dick the Butcher, is an outlaw and that his suggestion is made in the hopes of making life easier for lawbreakers everywhere. A world without lawyers would be a world in chaos. Economists have found remarkably strong positive correlations between the availability of lawyers in a society and the existence of civil rights and civil liberties. Lawyers can engage in work that is socially significant and produces a relatively good income. But law is more about analytical challenges, intellectual stimulation, and the human dimensions of being a helping profession.\n\n#### LAWYERS MAKE A DIFFERENCE\n\nThe _Legal Underground_ blog tried to find out what lawyers enjoy about the profession, posing the question: \"What do you like best about being a lawyer?\" Some of the answers include:\n\nI like working with intelligent, thoughtful people.\n\nWhat I like best about being a lawyer has to do with the all-encompassing nature of the enterprise. Every day I am able to help people with disputes and issues that are beyond their own ability to handle.\n\nI enjoy counseling clients\u2014they trust me. They listen to me. I tell them what to do and they do it. I solve problems for them.\n\nI like being a lawyer because I have the ability, access and opportunity to change, make or clarify law.\n\nI like knowing the rules. More than anything else, I like that fact. I don't have to wonder where I stand legally with regard to much of anything. If I don't know something, I know where to find out about it.\n\nWhat I love about being a lawyer is that I give a voice to people that otherwise have no one to stand up for them.\n\nI like fixing problems. I am happiest when I am fixing a problem, regardless what kind of problem.\n\nI went to law school in large part because I saw the impact that some lawyers had in their communities. That's really what I want to do with my life, try to make the community in which I live a better place.\n\nA repeated theme was making a difference in people's lives. One child advocacy lawyer said, \"I love the work I do. It is an amazing feeling to be able to actually help people through such incredibly dark times.\" In a later chapter, we will return to the stories of lawyers who find the practice of law satisfying, who are passionate about their work, and who have attained personal fulfillment in their jobs.\n\nProfessor David Guenther argues that \"it is almost impossible to become a lawyer and not acquire a passion for justice.\" Exposure to people who have suffered injustice will cause you to \"feel what they feel\" and \"understand it is not a game.\" If you are a lawyer worth your salt, \"their cause will become your cause and you will become a passionate representative of their rights.\" Guenther maintains that this passion, if \"channeled properly,\" will \"bring out the best in you and inspire others.\"\n\nLaw, as a helping profession, can claim many heroes. John Adams, late in an accomplishment-filled life that included playing a key role in gaining our independence and serving as our second president, wrote that it was his work as a young lawyer defending British soldiers who had opened fire on a mob of Americans during the Boston Massacre in 1770 that he was proudest of: \"It was... one of the most gallant, generous, manly and disinterested Actions of my whole Life, and one of the best Pieces of Service I ever rendered my Country.\" From abolitionist lawyers in the eighteenth century who dedicated their careers to fighting slavery to lawyers of today's innocence projects who work tirelessly to free the wrongly convicted, each generation has produced lawyer heroes. Lawyers have helped bend \"the moral arc of the universe... toward justice.\" You, as a lawyer, have it within your power to help people, to make a corner of the world better, to vindicate rights, to do justice, to become a peacemaker. In the words of Pope Paul VI, \"If you want peace, work for justice.\"\n\nMany lawyers are passionate about their work\u2014they experience the nobility of law and truly enjoy making a difference in people's lives. Sometimes, though, when we are slogging through briefs and drafting documents late at night, we wonder if the rewarding parts of our jobs will outweigh the tedious ones in five years, or in ten.\n\n### THE ARC OF CAREERS\n\nThere is good news. Satisfaction with a career tends to increase over time. As the years go by, many workers manage to find ways to spend more time performing the tasks they enjoy most, while passing off to other\u2014usually younger\u2014workers some of the more annoying tasks. More experienced workers generally have achieved a higher level of competence and that makes for less job-related anxiety. They have a clearer understanding of what is expected from them. Finally, in the absence of excessive turnover, bad hires, or a repellant personality, they likely have accumulated a set of relationships that provide support and a source of fun.\n\nSurveys of lawyer satisfaction show a clear upward trend in job satisfaction over time. According to an ABA study, lawyers with more than a decade worth of experience reported career satisfaction that was 40 to 50 percent higher than those with less than ten years work as a lawyer. Atlanta career consultant Monica R. Parker offered this explanation for the happiness of more experienced lawyers: \"Probably they've become partners or senior counsel, and by that time they've decided that the law is for them; they're going to stay and they like it.\"\n\nIn addition to more satisfying work and more work product control, there is another reason for higher career satisfaction as the years go by. As we age, we learn to use more \"mature adaptations\" to deal with the inevitable setbacks and disappointments that come our way. Mature adaptations probably account for the higher rates of happiness reported by people in their sixties than in their twenties. When we are young and confronted by frustrations, we are more inclined to turn to repression, disassociation, projection, or passive aggression (consult your old Psych 101 text to jog your memory about these classic defense mechanisms). As we get older, we increasingly use healthier techniques, including altruism, humor, anticipation, suppression, and sublimation. For example, an older person is somewhat less likely to obsess about an office slight and more likely to shrug it off or laugh about it.\n\nThe central finding of George Valliant, in _Adaptation to Life_ , an account of a classic longitudinal study of'42, '43, and '44 male Harvard University graduates as they progressed from their early twenties through middle age and into old age, is that developing mature adaptations to setbacks is the single best predictor of a successful life. Of course, being mature in years is no guarantee of having developed mature adaptations, and many people\u2014including many in the Harvard study\u2014never mastered that singularly helpful ability. We all can think of a senior relative or colleague who still throws fits or turns sour and silent when things don't go his or her way.\n\nAs lawyers progress through their careers, they typically have a variety of work environments in which to practice their adaptations. By the time they hit midcareer, most lawyers have changed jobs more than once, and only 15 percent, according to a study of University of Virginia law grads, still hold the job they took when they graduated. Some clear trends emerge from longitudinal studies of lawyers. While income continues to increase as a career goes on, fewer and fewer lawyers remain in private practice. Many migrate into smaller firms or accept in-house counsel positions. Substantial numbers of law grads in their forties, perhaps a quarter or so according to the Virginia study, have left the regular practice of law altogether and have instead moved on to jobs in academia and banking or became CEOs and small business owners.\n\nWhile the story in many regards is the same for female lawyers as for male lawyers, there is one critical gender difference that emerges. Vastly more women than men interrupt their employment, usually to care at home for small children. The dramatic difference in this situation, although probably in decline, could hardly be overstated. For lawyers now in their forties, the percent of women who have at least temporarily left employment is about 40 percent while for men the figure is only about 1 percent. Despite the higher rate of interruptions and its effect on partnership tracks, the Virginia study found \"no significant gender differences on any dimension of job satisfaction.\" Women might complain more about \"work-life balance\" issues in law firms, but overall they still report levels of job satisfaction equal to that of males.\n\nThat lawyers change jobs, and that job-changing is occurring ever more often in our rapidly evolving economy, comes as no surprise. More often than not, the changes\u2014when they are voluntary\u2014seem to work out for the better. But it is important to remember that constantly being on the look out for a better job comes at a cost. Restless lawyers never seem to be able to enjoy what they've achieved or, more importantly, develop the sense commitment and deep collegial bonds that can lead to higher levels of career satisfaction. Psychologist Barry Schwartz observes, \"there comes a point at which opportunities become so numerous that we feel overwhelmed.... Time spent dealing with choice is time taken away from being a good friend, a good spouse, a good parent, and a good congregant.\"\n\nWhat about career's end? What about that time when all the job-changing is done and lawyers either put away their briefcase for good or began finding time amidst the cases for rounds of midweek golf? Going from full-time work to a dead stop poses some risks, both to mental acuity and to relationships, making part-time work an attractive option for many older lawyers. This plan works well for some; others find, however, that going part-time is not a real option and that the nature of the work keeps them plugging away as hard as ever, despite their intentions. Retirement, interestingly, seems to have almost no effect on levels of personal happiness. Almost exactly the same percentage of retirees (36 percent) and full-time workers (35 percent) report themselves as being \"very happy.\" The data are clear that busy people are happier than those with little to do: the most satisfied ex-lawyers are those who find hobbies, sports, and friendships to fill their free hours. Both working lawyers and retired lawyers, unsurprisingly, are considerably happier than the unemployed. Only 16 percent of unemployed men said they were \"very happy,\" less than half the percentage of employed men agreeing to the same statement. Daniel Nettle attributes the unhappiness of the out-of-work to a common feeling among this group that they have little control over their lives, not that they have lower incomes on average. Statistically speaking, Nettle notes, a sense of personal control is a twenty-times better predictor of happiness than income.\n\nLawyers often sprinkle the introductory phrase, \"But even if . . .\" into their briefs. You may be thinking, \"But even if... the arc of a career for most lawyers holds the promise of future satisfaction, will that apply to me?\" Maybe you feel you are too competitive, too analytical or too much of a worrier\u2014all qualities associated with lawyers (although not, of course, all lawyers)\u2014to ever be happy. You might wonder if lawyers have distinct personality traits that undermine their happiness.\n\n### IS THERE A \"LAWYER PERSONALITY\" THAT AFFECTS CAREER SATISFACTION?\n\n_I fear that happiness isn't in my line,... blaming the disposition that was given to me at birth._\n\n\u2014U.S. Supreme Court Justice Benjamin Cardozo\n\nDo lawyers become unhappy people or do unhappy people become lawyers? Many of the existing works on unhappy lawyers posit that the practice of law makes people unhappy. When assessing whether a profession breeds unhappiness, it is important to keep in mind the difference between correlation and causation. Do people inclined toward unhappiness select law as a profession or does the practice of law induce, or perhaps exacerbate, those feelings?\n\nThe NORC career study suggests that happier people gravitate to some careers more often than others. It is also true that different personality types are disproportionately represented in different careers. It is likely that shy, helpful people are overrepresented in our nation's libraries; jobs in sales are disproportionately filled by extroverts; and that rodeo performers, by and large, are less risk-averse than the general population.\n\nIs the practice of law disproportionately filled by people with certain personality traits? The answer is clearly \"yes.\" When compared to the general population on a standard measure of personality types, a distinctive picture of lawyers emerges. Lawyers, as a group, are more introverted, more doubt-ridden, and more cool and logical than most people. They are less open about their feelings and less inclined to live in the present than most people. Lawyers are competitive, confident (sometimes a kiss away from arrogant), aggressive, and achievement-oriented; they can be argumentative.\n\nWe're not talking about you, of course, but many lawyers\u2014including some very successful lawyers\u2014express \"high dominance\" personalities. High dominance personalities are associated with a strong competitive drive and are marked by frequently interrupting, controlling conversations, changing topics, offering unsolicited advice or instructions, and stating strong opinions. In the courtroom, a strong competitive drive can be the difference between winning and losing for your client, so this personality type can be a career positive at the same time that it might sour the personal relationships so essential to both personal and career happiness.\n\nMany lawyers are analytical problem-solvers who value thoughts and ideas, as opposed to feelings, in their interactions with other people. Lawyers may also be exacting and tenacious, which are useful qualities for their work, but those are behaviors that can be noxious in interactions with opposing counsel. In short, the personality traits most common among lawyers are not those usually associated with happy people. Of course, lawyers are a diverse group and some lawyers are as extroverted, confident, and inclined to live in the here-and-now as anyone, but the evidence is that many lawyers already have a strike against them in the quest for a happy career.\n\nMartin Seligman, the father of positive psychology, and colleagues published a law review article exploring the question \"Why Lawyers Are Unhappy,\" arguing that a major source of lawyer discontent stems from the fact that \"lawyers are selected for their pessimism, or \"prudence,\" and this generalizes to the rest of their lives.\" A study at the University of Virginia School of Law showed that pessimists outperformed their optimistic peers in terms of law school success, such as grade point average and participation on law review. People who choose law school are more often than not inclined to see glasses as half-empty rather than half-full, say the authors, and also more inclined to believe they are responsible for bad results, when they occur, and that those bad results will have lasting effects. These attitudes are not associated with good military leaders, insurance agents, or professional athletes, but they often do help people become better lawyers\u2014at least better lawyers of a certain type. Transactional lawyers, in particular, owe it to their clients to anticipate and worry over worst-case scenarios. Caution and skepticism can help clients if and when things do later go to hell in a hand basket.\n\nBut the \"downside\" of pessimism, if you will, is fairly obvious. As the authors note, \"the qualities that make for a good lawyer... may not make for a happy human being.\" When they are not worrying about the future of their clients, pessimistic lawyers are likely worrying about whether they will make partner, whether they are really contributing to the public good in any way, or whether an airplane will fly into their skyscraper. Of course, these traits are compounded by psychological stressors of the job: high pressure and for many lawyers, particularly newer associates, limited latitude to make decisions or control their working conditions. The interplay of attorneys' personality traits and the professional demands of the job can prove toxic.\n\nLaw professor Larry Krieger and psychology professor Ken Sheldon take issue with the hypothesis that law is the career of self-selection by unhappy people. Instead, they hold law schools responsible for turning happy students into unhappy graduates. Krieger and Sheldon studied classes from two different law schools from orientation through graduation and compared these students to a population of undergraduates. They concluded that the \"incoming [law] students were happier, more well-adjusted, and more idealistic\/intrinsically oriented than a comparison undergraduate sample,\" but that the educational cues in law school culture create graduates that \"are significantly different people from those who arrived to begin law school: They are more depressed, less service-oriented, and more inclined toward undesirable, superficial goals and values.\" In chapter 5, we examine the extent to which the teaching methods, implicit messages, and career steerage in law schools help mold lawyer personalities.\n\nTo achieve happiness for some lawyers means struggling against their natures and against the very traits that contribute to their success. Poet Edgar Lee Masters had a day job as partner in the firm of famous defense lawyer Clarence Darrow. He wrote sympathetically about Darrow's \"artist sense that drives him to shape his life to something harmonious, even against the schemes of God.\" It may well be, for many of us in the legal profession, the schemes of God do not include our happiness, and we must work to steal from life what satisfaction we can.\n\nIf you have been doing a psychological self-assessment while you read the last few paragraphs, mentally checking off the boxes for ambition, tenacity, and perhaps functional pessimism, remember that personality has some plasticity. These personality characteristics are not fixed. Abundant research shows that behaviors change over time and are influenced by social arrangements, and that optimism is an outlook that can be learned. If the psychological demographics of lawyer satisfaction do not work in your favor, there are things you can do within law to become happier.\n\nThis chapter has focused mostly on what makes lawyers unhappy. The next chapter offers a toolbox for achieving greater happiness. We turn next to the tools and tricks you can use to steal more happiness from your career as a lawyer.\n\n## **CHAPTER 4\n\nThe Happiness Toolbox**\n\nODDS ARE, IF YOU ARE READING THIS BOOK, YOU'RE not as happy with your job as you'd like to be. (If you've got the perfect job, congratulations\u2014pass this book on to a friend who can better use it.) You'd probably like to know what you can do to become a happier lawyer. You've come to the right chapter.\n\nThis chapter will give you the tools you can use to build a happier practice or repair an unhappy one. As is usually the case with a toolbox, some of the items in it might be well-suited to a particular job while others might not. Look through the collection of tools we offer, and see which of them seem promising for your own situation.\n\nThe source of career unhappiness can be the nature of your job, or it can be the nature of you, but it is almost always a combination of both. What might be a very good job for someone with a particular personality just might make you miserable. But also, as we'll see, there are characteristics of certain jobs that are virtually guaranteed to undo the smile of any human being with a pulse.\n\nPeople are more similar in their basic likes and dislikes than many people suppose. What makes you happy in a job most likely would also make your neighbor across the street happy. Your neighbor might play classical music while you prance around your living room to '80s disco music, but still, at your cores, similar things probably make you happy at the most basic level. You and your neighbor probably, according to consistent social science research, seek a measure of control in your life, enjoy being connected to other people, feel a bit better about your situations when you see people worse off than yourselves, like to \"get in the flow\" of an engaging activity, and want\u2014at some deep level\u2014to do work that aligns with your values. These core desires are not satisfied equally in all jobs, and many of these desires are not satisfied at all in some jobs, but the degree to which a job fulfills them goes a long way to predicting how happy you are in your career.\n\nOf course, if it were a simple matter of taking a checklist of near universal desires and applying it to a specific job opportunity in the field of law, we'd probably all have our checklists by now and be clamoring for the same jobs. Finding as much happiness as can be found in a legal career is not nearly so simple. It requires not only a deep understanding of a set of basic desires but also an understanding of how the various desires are weighted in your own hierarchy of values, plus an appreciation of what you can do to make a specific job do a _better_ job of meeting your basic desires. Jobs are not objects set in concrete; they are goal-directed, behavioral systems that can be restructured to make those in them happier creatures.\n\nSo let's open up our happiness toolbox to see what can be found that might make you a happier lawyer.\n\n### MATTERING MATTERS: THE JOY OF FEELING IN CONTROL\n\nIf you want happiness as a lawyer, get control of your life. \"Yeah, right,\" you say as you work frantically to meet the filing deadline for Acme's summary judgment motion while you speed dial your nanny to see if she can pick up your second-grader and get him to the baseball game that you're going to have to miss. Getting control of your life won't be easy, and there might be financial pain, but it has to be done for the sake of your emotional well-being. When we sense our lives are spinning out of control, we are very unhappy creatures.\n\n#### THE RAT RACE\n\nEduardo Punset argues that loss of control is \"the ultimate origin of unhappiness and depression.\" As he sees it, the brain hidden away in its \"dark room,\" removed from the actions it interprets, needs to feel that it is in charge. What is true for humans seems to be true as well for our animal brethren. Punset points to an experiment conducted in the late 1970s. Researchers put five rats in separate cages. The rats were subjected to strong electric shocks at random intervals. One rat, however, had a lever which, when pushed, ended the shocks for all five rats. By the end of six weeks, the immune systems and emotional systems of four rats had broken down and they died\u2014essentially of depression. The rat with the lever survived many months after the deaths of his fellow rats, even though all rats had received exactly the same doses of electric shocks.\n\nNumerous studies confirm that humans feel the same way about control as rats do. Studies with subjects ranging from college students to British civil servants to elderly nursing home residents consistently show that control is closely related to happiness. When, for example, nursing home residents were put in control of watering and tending a plants, they reported higher levels of happiness (and, remarkably, _half_ the death rate) compared to residents in a low-control group who were told that a staff person would take care of the plants. In another study, happiness levels among British civil servants turned out not to depend on their salary levels but on the degree to which the various jobs allowed them to exercise control.\n\nA sense of being in control is critical to happiness in virtually any domain, from relationships to bodily functions to motoring down highways. But we're interested here in jobs. Thinking about those unhappy experimental rats, Punset sees control as the most important key to job happiness. He writes, \"When my students... ask me for my opinion on possible job positions, I always suggest that they only accept posts with a control lever, however small, and that they never accept, even if offered a lot of money, a post where no one and nothing depends upon what they do.\"\n\nIt is a fact of life that most lawyers do not have the same degree of control in their jobs as do people in many other occupations. A shop owner controls when the shop opens, what products are sold and for how much, how the products will be displayed, and who will do the selling. A football coach has control over what plays will be run, how long practice will be, and who will start at quarterback. An artist wakes up each morning free to create anything within his imagination. But a lawyer? As a lawyer, you pretty much do the work a client needs done, whether it's your idea of an interesting project or not. True, you might have control over what arguments are made on a client's behalf or the exact form a contract will take\u2014but compared to some other occupations, that control might not seem like much. This lack of control might well account for the fact that lawyers had the highest rates of major depressive orders among surveyed occupations. Researchers suggest that lack of control is linked to depression and noted that lawyers and secretaries (two of the three highest risk groups) had relatively little autonomy.\n\nTo understand what might be done to better our lives, given that control is not one of our occupation's selling points, it is helpful to recognize what brings on a feeling of control. The feeling comes only when a number of things combine in a job. Control, it turns out, has several aspects.\n\n#### THE FACETS OF CONTROL\n\nWhat are these several facets of control? One comes when you feel that you've achieved an appropriate balance between the demands of your job and the demands that come from being a parent, spouse, relative, or friend. A sense of balance also requires that you take care of your own emotional needs, whether by simply making time for beers after work with colleagues, catching that new episode of your favorite television show, or fitting in an after-work golf game on an inviting May evening. How often on a late Friday afternoon have you seen your plans to catch your kid's school play or have dinner with friends all go up in smoke? That motion for a summary judgment has to be filed before the courthouse closes and you still have three voice messages and seven e-mails that need some sort of response before you're out the door. \"Work-life balance?\u2014you've got to be kidding!\" It's what everyone you know talks about, but no one in your office seems to have. We feel the demands of life are pulling us along and that our own internal compass is helpless to determine our direction.\n\nA sense of being out of control is commonplace today, especially in the large firms, where career dissatisfaction is the greatest. Seventy percent of lawyers in an American Bar Association (ABA) study said that \"wanting more time with family\" is a reason that might justify leaving one job for another. Nearly 20 percent of lawyers are \"not at all\" happy with the work-life balance in their present employment. Women, unsurprisingly, are far more likely than men to complain about the stress work is placing on their family life.\n\nA second key aspect of control in the workplace is job security. Peter Warr, a British professor who has extensively studied job happiness, identifies a sense of job security as one of the twelve most important keys to worker happiness. Worrying about whether the next slip to cross your desk will be pink, it seems, is incompatible with happiness. Nothing is really secure in this world except the prospect of death and taxes (and perhaps, therefore, the jobs of probate and tax attorneys), so all levels of job security are relative. Relatively speaking, however, lawyers in large firms may feel their jobs are less secure than those in smaller firms. Your job in a two-hundred-person firm might well depend upon the whim of a single supervising attorney, the decision whether your firm merges with another mega-firm, or whether an important account picks up and leaves. Although smaller firms also have job security issues, notably attracting enough business to stay afloat, colleagues who know you well are less likely to show you the door.\n\nA third aspect of workplace control depends on believing that your contribution matters. As Daniel Gilbert succinctly notes, \"Mattering makes us happy.\" Unfortunately, this sense of mattering through one's work is relatively rare today. Punset observes, \"What now prevails is a growing feeling of powerlessness to influence the product, the company, society, or what is going on in the rest of the world.\"\n\nA sense of mattering, in turn, has varied sources. It can come from supervisors or colleagues praising your work, or at least letting you know that what you did was important. It can come from your clients\u2014their sense of relief (and, one hopes, gratitude) from winning a favorable verdict, having a deal close successfully, or getting that speeding violation off of their motor vehicle record. Study after study shows job feedback to be critical to happiness on the job. Not all feedback is equally likely to produce happiness, of course. The most unambiguous feedback is generally the most appreciated. When we're doing something wrong, we prefer to know exactly what that is, so we have the best chance of making the necessary adjustments.\n\nNot all law jobs are equally likely to produce a sense of mattering. The ABA's 2007 survey of lawyer satisfaction found that public sector lawyers, the group most satisfied with their professional lives, were also the lawyers most likely to feel appreciated in the workplace. Seventy percent of all public sector lawyers felt that way.\n\nA sense of mattering can also have an internal source. You might know\u2014and no one needs to tell you, but it wouldn't hurt\u2014that what you do has real consequences for real people you care about. It's when we don't give a damn what happens to the people affected by our work, or don't see our work as affecting anyone in particular, that we grow dissatisfied. We want to feel our work has some purpose beyond simply bringing home the bacon. Some of us are more capable of realizing these internally generated rewards than others, who seem unhappy without recognition from other people.\n\nFinally, a sense of control can come from, not surprisingly, the number of opportunities in the workplace to control things. These things can relate to the content and timetable for a work product. Lawyers tend to feel happier when they can decide which file to pick up and which to put on the back burner, which paragraph should be deleted to reduce the brief to the maximum length allowed by the court, or who should be deposed this Tuesday and who the following week. People in first-chairs in the courtroom are generally happier than those in the third-chair.\n\nUnfortunately for lawyer happiness, our adversarial system of justice places the parties, and not you, in ultimate control of cases. You might want to settle and get the stinking file off your desk, but that's not acceptable to your client bent on a measure of justice. You might think a paragraph you wrote in a contract was a stroke of brilliance, but your client might think otherwise and want it out. You might want to spend a day leisurely exploring an interesting angle of your client's civil case, but your client might not have the deepest of pockets and wants to reign in your billable hours. The system is what it is, and you are never going to have the freedom of a novelist. Accept what you cannot change, as the serenity prayer urges.\n\nFinally, personal control also means the ability to alter one's work environment. Is it okay to put a valued family photograph on your wall? Will anyone complain if you bring in a comfortable chair, move your desk, or change the lighting in your office? The more power you are given to affect your own working environment, the happier you will be. One commentator notes, \"Small freedoms... are very good for satisfaction.\"\n\n#### GETTING CONTROL\n\nIf control in its many senses is so central to happiness, then giving lawyers more control over their work lives should be one of the highest priorities\u2014maybe _the_ highest priority\u2014for a firm. Work-life balance issues, according to ABA surveys, are a leading cause of stress among lawyers. More generally, the tilt toward long workplace hours in the United States may explain why happiness levels in this country have stagnated since 1975 while rising in increasingly laid-back Europe. While the average American puts in about 2,000 hours a year at work, the average Dane, Italian, or German works less than 1,700 hours a year. Working 15 percent fewer hours, and spending the additional time socializing or beachcombing, could give a much-needed happiness boost.\n\nFor some workaholics, however, dialing back the hours at the office might actually increase distress. As psychologist Jerome Kagan notes, once people establish an association between hard work and rewards, it is hard to break. For the person whose diligence paid off first with a good college experience and then with the landing of a high-paying job, the link between work and reward might have become hard-wired.\n\nOne good thing about law firms is that they are filled with smart people. People who are certainly bright enough to identify ways, beyond the first step of reducing the required number of billable hours, to give lawyers in the firm a better sense of control. Depending on the firm's goals and economic situation, a wide variety of changes might be considered. A partial list might include offering flexibility in hours and work location, allowing lawyers to bring children to the office, providing lawyers flexibility in choosing clients and work assignments, letting lawyers choose how they approach and complete work products, and encouraging lawyers to alter their personal work environments\u2014lighting, hangings, furniture\u2014to suit their own taste and work style.\n\nIf you are a lawyer whose firm seems unable or unwilling to give you the control you need, a change of jobs might be the best option. The firm down the street might accept your proposal to shift to part-time work or a more flexible schedule. A smaller firm might promise you greater control of your work product and work environment. Of course, a solo practice offers the maximum degree of control\u2014perhaps even too much control, if promotion and marketing is not your strong suit. In short, consider what aspects of control are most important to you, and then determine where you are most likely to gain it. Control of your life is just too important to give up without a fight.\n\n### IT COULD BE WORSE: THE JOY OF DOWNWARD COMPARISON\n\nMax, who knows barely half of what you do about corporate law, commands a gorgeous view of the Quatzawatamie River from his corner office. You look out over the backside of the Acme Building. Jane, who didn't even make law review at Middlebrow, earns $10,000 more per year than you\u2014you, the former articles editor at a law school that _U.S. News & World Report_ proclaims to be one of the five finest in the land. These things gnaw at you. The injustice of it all! How can the firm's managing committee be so clueless? You'd prefer to focus on the tasks at hand (that memo on _Snerk v. Google_ needs to get out soon), but your mind returns to that corner office or that extra ten grand that should be yours.\n\nYou are suffering the pains that come from upward comparisons.\n\nSometimes, it turns out, finishing worse is actually better. Researchers evaluating the happiness of Olympic medal winners discovered that bronze medalists generally were happier with their finishes than were silver medalists. Why? The silver medalists tended to compare themselves with gold medalists, the athletes who achieved the goal they long sought; bronze medalists, on the other hand, were just happy to be on the medal stand, thinking about how close they came to joining all those other hard-working athletes who failed to medal at all.\n\nThe acerbic social commentator H. L. Mencken observed, \"A wealthy man is one who makes $100 more than his wife's sister's husband.\" We seem to understand that happiness depends less on income than how we're doing compared to the Joneses. Asked whether they would prefer to have an income of $50,000 when the average is $25,000 or an income of $100,000 when the average is $250,000, most persons polled chose the first condition. Everyone knows that $100,000 will buy a better standard of living than half that much, but being _relatively_ well-off is seen as clearly superior to being _relatively_ poor. We are okay without that sixty-inch plasma screen so long as most of our neighbors don't have one either.\n\nAs we indicated in chapter 2, earnings have little to do with happiness once they rise above middle-income levels. When per capita incomes in the United States shot up 300 percent from 1970 to 1990, there was no increase in the happiness of Americans. Money really can't buy much happiness.\n\nRelative income, however, matters much more to people than absolute income. Being in a higher social class _does_ correlate with increased levels of reported happiness. The nonincome benefits of being in a higher social class, such as increased status in workplaces and leisure places, seem to make people happier. People who perceive themselves as being in lower social classes are people who most often make upward comparisons\u2014and get grumpy when they do. It grates to see their neighbors and colleagues doing much better financially than they are. In the terminology of British economist Richard Layard, the wealthy spew a \"social pollution\" that incites toxic envy among those around them. A _New Yorker_ cartoon, showing a mid-level executive in a supervisor's office, makes the point. Standing at his boss's desk, the employee is saying: \"O.K., if you can't see your way to giving me a pay raise, how about giving Parkinson a pay cut?\" It should be no surprise that the egalitarian nations of Scandinavia, where social class is largely a nonissue, consistently rank among the happiest countries on the planet\u2014with Scandinavian residents reporting significantly higher happiness levels than U.S. residents. (Proving Shakespeare wrong, Denmark is about the least melancholy place on earth\u2014it appears either at or next to the top of list in the most recent polls of happiness levels.)\n\nWhether the question is the amount of compensation, the size of offices, or plum case assignments, it is human nature to compare our situation with those around us. When we compare ourselves to persons better off than us, we are unhappy because, as Sonja Lyubomirsky observes, \"You can't be envious and happy at the same time.\" On the other hand, if we find our fate better than our comparison group we tend to be happier. So happiness turns in part on the group we choose for comparison. Bigger fish in smaller ponds really do turn out to be happier\u2014consider that the next time you are deciding where to go for a swim.\n\nOf course, it is no simple matter. You cannot just choose a group to downward compare against and then get happier. To some extent, our genetics and our upbringing will force comparisons\u2014either upward or downward\u2014on us. Nonetheless, we are not powerless to affect our choice of comparison groups. We tend to compare ourselves most against persons with whom we spend a lot of time, so by hanging around those less fortunate than ourselves, we set up more opportunities for downward comparisons.\n\nWe spend a lot of time working, especially Americans who work longer hours than people in just about any other country. Who we choose to work with, and who we choose to work for, are choices. We could have been undertakers or roofers or nurses or architects. We chose to be lawyers. Moreover, we decide to be small-town lawyers or big-city lawyers, immigration lawyers or securities lawyers, solo practitioners or associates in three-hundred-member firms.\n\nSurvey data suggest, as previously noted, we might well have been happier had we chosen one of the nurturing occupations: social work, nursing, physical or mental therapy, or Peace Corps work. The higher-than-average happiness levels reported by people with those jobs most likely has a lot to do with the opportunities those occupations afford for downward comparisons: \"I might have problems and my pay is low, but I'm not a single mom on drugs, quadriplegic, crazy, or stuck in a malaria-infested third-world nation for the rest of my life.\" People with those jobs are more inclined than the rest of us to see their own lives as half-full bottles, not half-empty ones.\n\nAll this suggests one way in which lawyers might increase job satisfaction: begin spending more time working for clients who have at least a few more problems than you do. Along with the positive feeling that comes with helping acts, you're likely to come home feeling better about where life has taken you. Yes, there is some status that comes with churning out the billable hours for zillionaires who want a few more zillions than the next zillionaire\u2014but odds are you'll be a little happier helping Farmer McDonald save the family back forty from the I.R.S., helping that well-intentioned Latin family obtain citizenship, or making sure that the little boy crushed by the falling anvil will get his medical expenses paid for by ABC Anvil Company (or whatever negligent bastard let loose the anvil.) On the whole, careers in elder law, social security law, personal injury law, and immigration law\u2014where you can make a direct contribution to other people's lives\u2014produce more job satisfaction than careers spent defending companies whose sole goal is to maximize profits. Somebody has to defend pyramid schemers and monopolists, but it doesn't have to be you.\n\nThere's another way to increase the likelihood of downward instead of upward comparisons. You could become the proverbial bigger fish in the smaller pool. Because size is relative, you'll feel bigger in the small pool and feel just a little happier when you think of yourself swimming along with all those smaller-finned creatures. Instead of taking that higher paying job at the prestigious firm of Thurston, Howell & Gilligan, how about accepting that slightly less lucrative offer from the seven-person firm in your hometown? If you really want to build a life of it in Metropolis, you still have an array of choices\u2014and you don't have to opt for the one that would most impress your fellow alums at the next law school reunion. You might be less smug once every five years, but happier day-to-day as the go-to lawyer in the somewhat less power-lunching firm. Sure, you might also have moments of daydreaming misery when you imagine the good life (which isn't really that good) of classmates in those high-paying prestigious jobs, but you're not getting corner-office envy every time you trot to the water cooler.\n\nThere's one last way of increasing the likelihood of downward comparisons, at least for those lucky enough to be in a firm that provides such opportunities. You can volunteer for pro bono work that will increase your contact with persons whose problems seem vastly larger than your own. Survey data show that increased pro bono work correlates with increased levels of happiness. Besides, the volunteer work you do is likely to be a departure from your typical tasks, and the data is clear that when we look at back at life from our rocking chairs a few decades hence, the odds are much greater that we will regret more the things we didn't do than the things we did. Do good, and be happy.\n\n### REAPING THE REWARDS OF RELATIONSHIPS: IT'S THE PEOPLE, STUPID\n\nWithout social connections it's next to impossible to be happy. In fact, according to one happiness expert, about 70 percent of our controllable happiness stems from relationships. We could debate whether control or connections is _the most important_ determinant of happiness levels, but there is little disagreement that they are the two biggies. We not only define ourselves in terms of our social ties, we also derive much of the meaning in our lives from those ties.\n\nIt should come as no surprise that your fellow human beings can be a source of happiness. People need people. Ever since writers have been putting words on papyrus, the praises of close relationships have been sung. \"Of all the things that wisdom provides to help one live one's entire life in happiness,\" wrote the Greek philosopher Epicurus, \"the greatest by far is friendship.\"\n\nNext to sex, socializing is the activity that makes us happiest. Interaction with friends gives us the biggest happiness boost, followed by time spent with spouses and children. We do not, as a general rule, enjoy interaction with co-workers and clients quite as much as our time with close friends and family, but it still beats\u2014by a wide margin, according to a survey of workers\u2014interaction with supervisors and, especially, time alone on the job. Most people like to be in contact with others much of the time, and for the 75 percent of Americans who are extroverts, this is especially true. Contact with others, both in quantity and quality, has been identified as one of the twelve factors most important to job happiness. This explains why one of the reasons that losing a job is so devastating: it severs the relationships with co-workers.\n\nAlthough socializing is our most common pleasure-inducing activity, working ranks near the bottom on the same scale. Working falls below exercising, eating, shopping, worshipping, watching television, and even (believe it or not) housework. Among daily activities involving significant amounts of time, only commuting makes us less happy than working. If socializing makes us very happy and working does not, one might wonder whether the workplaces that provide the most opportunities for interaction\u2014even if the interaction is less satisfying than raising toasts  with your buddies on a Friday night\u2014tend to be the happiest? It turns out that they are.\n\n#### CAN LAWYERS CONNECT?\n\nThe workplaces that produce the highest levels of happiness tend to be those where workers deal directly with other people. The deeper the connections between co-workers or people served on the job, the happier a worker generally is. When it comes to recruiting new associates, law firms understand the critical role that positive interaction with co-workers plays. As a summer intern, you are feasted, trotted out to luxury suites in ballparks, toasted in pubs, loaded into kayaks, and provided with a host of other bonding opportunities. Too often, unfortunately, when you join the associate ranks, the tool that worked so well to make you happy as an intern is forgotten.\n\nThe depth of job-related relationships may explain why, when 27,500 randomly selected people were asked to rate their jobs, the happiest of occupations turned out to be the clergy. Two of every three clergy reported being \"very happy\" as opposed to an average of just one out of three for all workers. One key reason for high levels of happiness among clergy is the opportunities the job provides for close personal connections. Jackson W. Carroll, a professor at the Duke Divinity School, had this explanation for the poll results: \"A pastor does get called on to enter into some of the deepest moments of a person's life, celebrating a birth and sitting with people at times of illness or death. There's a lot of fulfillment.\"\n\nLawyers rarely find it within their job description to celebrate a client's newborn or hold hands with a client on her deathbed. Clients often do, however, call on lawyers during periods of difficulty and crisis\u2014as any divorce, personal injury, probate, or criminal lawyer could attest. These encounters can sometimes be intense. They probably do not produce the same happiness benefits as the clergy experiences during important passages because the lawyer and the minister have different goals. The minister focuses on establishing a bond that can enhance the church member's sense of peace and well-being. The lawyer, in contrast, usually aims at meeting a more immediate, and often monetary, goal of the client. Those lawyers who develop personal bonds with clients, even while serving clients' pressing legal needs, tend to be happier lawyers than those who do not. Lawyers in public service positions report higher career satisfaction levels than other lawyers, and part of the reason might well be the greater client contact those jobs usually afford.\n\n#### EVERYBODY NEEDS SOMEBODY SOMETIME\n\nFor many who practice law, the deeper connections established in the workplace are likely to be with fellow lawyers rather than with clients. The quality of those connections is closely linked to our happiness. Everyone longs to belong. We want to be welcomed into a tribe and learn the secret handshake; we want to be accepted. It is not surprising, therefore, that we seek to spend time \"with people who like us and are like us.\" If your law firm colleagues share your interests and values, and they enjoy your company, you are more likely to be among the lawyers reporting themselves to be happy. The odds increase further if your firm is one that doesn't think building human connections is just for summer recruiting but instead provides numerous opportunities for interaction with colleagues, whether through collaborative work projects, brown bag conversations in the lunch room, or office parties and firm softball games. Robert Putnam observed in his important book, _Bowling Alone_ , that social connections are fraying in our cocooning modern world. The more a firm can do to rebuild some of these connections, the better it and its workers are likely to be for it.\n\nYou can do your part for the bonding process. Expressing gratitude has been shown to be especially rewarding, both for you and the person on the receiving end. Whenever a colleague offers helpful advice, performs a favor, or just makes a contribution to the firm's greater good, let that person know their gesture was appreciated. Handwritten notes are better than e-mails.\n\nAn ABA survey of lawyers suggests than smaller firms are happier places than large firms. This finding is consistent with other survey data involving nonlawyers and the size of workplaces. Smaller is usually better. But why?\n\nThe answer likely has a lot to do with trust. Survey after survey show trust is critical to happiness. Loss of trust in a personal relationship, such as might come with awareness that a spouse is having an affair, is devastating. Workers who trust their co-workers tend to be happiest.\n\nAmong all nations, the ones where citizens have the most trust in their government and their fellow citizens\u2014including Denmark and Switzerland, the world's two happiest countries\u2014are also the happiest. The quality of a society seems to have a much greater effect on personal happiness than an individual's specific role in that society. What goes for countries probably goes for workplaces as well. Where a sense of trust exists, workers are most likely to be happy.\n\nSmaller workplaces are more conducive to trust building than large workplaces. Trust is something that develops over time. It comes from positive engagement, especially when co-workers share stories and keep their promises. In a large firm setting, where you might not work on two projects in a year with the same colleague, trust can't develop as it can in a smaller workplace with more regular interaction between colleagues.\n\nObviously, however, small firms are not for everyone. If, for financial reasons or reasons relating to specialized expertise, only larger firms are viable options, then investigate the quality and quantity of interaction the firm will afford. Find out whether the firm has a high turnover rate\u2014high turnover makes it next to impossible to build trust. Try to determine whether you are likely to spend the majority of time in \"a firm within a firm\"\u2014working as a member of practice group with a specific mission and a nonrotating set of colleagues. Learn about firm culture: Do lawyers know each other's names? Do lawyers and staff hit it off? How common are water cooler conversations, retreats, collaborative work, group lunches, and social events? Does the firm provide a sense of play? How much emphasis is there on billable hours? Clearly, some big firm experiences provide better interaction than others.\n\n### COMMIT\n\nA firm with satisfying levels of interaction that promote trust will, over time, produce another great good. It will allow you to feel a sense of commitment to the larger enterprise. Without commitment, you're mostly just spending time. You cannot, it has been noted, love either a person or a place or a job with a foot out the door. Just as with marriage or other long-term love, career satisfaction over the long haul requires adjustments and compromises.\n\nLawyer career coach Hindi Greenberg deals often with mid-career boredom: \"The first couple of years, it's exciting to say, 'I'm a lawyer.' You carry a big briefcase and even the minutiae are new and exciting. But then the routine sets in.\" In addition to trying to do more of what they do like at work, Greenberg advises lawyers to make \"a change in attitude\": look at \"the glass as half-full\" and \"get over it.\"\n\nNo job is perfect. Lawyers spend a lot of time reviewing documents, filing motions, and pushing paper or electronic files. Newly appointed Supreme Court Justice Sonia Sotomayor reminds us, \"The vast majority of lawyering is drudgery work\u2014it's sitting in a library, it's banging out a brief, it's talking to clients for endless hours.\" Irritations sometimes crowd out satisfaction. When this happens, remind yourself what it is you really like about your job. Make a list, and see if that helps elevate your mood. Still, if you are like most people, there will be days when you think you'd rather be working somewhere else. But wondering whether the grass is greener on the other side of the hill (or office tower) comes at a cost. Besides diminishing the possibility of love, if that's a word you can wrap your head around in the job context, there are other costs associated with being in a perpetual job hunt. You can't really relax, for one thing. A multitude of job opportunities can leave you feeling overwhelmed\u2014\"So many to investigate, so little time.\" And all that time checking out other career options is time taken away from your family, friends, and favorite leisure activities.\n\nThere comes a time in life when you really should enjoy what you've achieved. Commitment is a big and serious word that we spend the first two or three decades of life running from\u2014but deep satisfaction comes to those who are able to embrace it.\n\n### FINDING THE FLOW\n\nWhen does time fly for you?\n\nThe sense of time flying comes when we are so absorbed in an activity that it crowds out the worries and self-consciousness that dog so much of our existence. The activities that absorb vary with our individual skills and interests. For some, time flies when they are rock climbing. For others, time whizzes by during a round of golf. For others still, minutes race during a foot-stomping guitar set with their neighborhood band. What all these activities provide, for someone with the right skills and interest, are stretching experiences\u2014or creative challenges.\n\nWhen everything is going just right, and the challenge is enough to test our skills and demand our attention but not so daunting as to produce anxiety, we are in what has been called in positive psychology circles \"a flow experience.\" The term \"flow\" is most associated with Mihalyi Csikszentmihalyi who defined flow as \"a dynamic state that characterizes consciousness when experience is attended to for its own sake.\" It is a state where \"action and awareness are merged.\" Athletes recall fondly their flow experiences, typically speaking of times they were \"in the zone.\" During a flow experience we are in the _now_ , not the past or the present\u2014and utterly mindful. Of course, over time, thoughts of the past (that critical comment a partner made about my last memo) and future (will I finish the brief in time to join the gang for lunch?) assert themselves and the flow ebbs. It is probably more accurate to say that we can experience various degrees of flow experiences, with true flow being their shortlived essence.\n\nIt is not entirely accurate to say we are _happy_ during flow experiences. We're so focused on the task at hand that where the dial is on our personal happiness meter seems irrelevant. Moreover, if we interrupt our flow experience to consider the question of how happy we are at the present moment, we almost certainly become markedly less happy and lose our sense of flow. When we are feeling the flow we are living actively in the present and living intensely. Is that being happy? Perhaps. Asked afterward to reflect on our flow experience, most of us would probably describe the activity as fulfilling, engaging, even fun\u2014but we'd pause before characterizing our feelings at the time because the one thing we know is that we were not thinking about our feelings. When you're intensely focused on making contact with that high-velocity serve of your opponent, you have no time to think about how happy or sad you are\u2014and that's the beauty of it.\n\nWhen asked about flow experiences, we are likely to draw examples from nonwork related activities such as athletics, outdoor recreation, hobbies, playing music, or writing poetry. Yet the characteristics of flow experiences can be found in workplace tasks as well.\n\nWhat are the features of an activity that might cause it to induce a flow experience? First, it has to be an activity that requires you to apply a skill that you have, or at least one you think you have. Self-delusion seems to be sufficient here. Second, the demands on the skill must not be so great as to produce anxiety yet must be sufficient to avoid the boredom that accompanies less-demanding tasks. As a colleague of ours notes, \"It's no fun to play tennis with someone who can't return your serve, but it's also no fun to play tennis with someone whose serve you can't return.\" Third, there must be clear goals so that the direction of your activity is not in doubt. Fourth, the activity should be one that frees you from the flights of mind and distractions that accompany the more humdrum experiences of life. It helps if there is a mystery or unresolved question that will be unraveled or answered by your efforts. The mind far prefers puzzles to certainties. Finally, the activity should be one that is not interrupted by phone calls, police sirens, bathroom breaks, daydreams, or sports score searches on espn.com. A flow activity has our full attention.\n\nFlow is not the first thing to come to mind when we consider our work as lawyers. Most hours in law firms are occupied with nonflow experiences such as answering e-mails, filling out billing forms, or digging around in files for an elusive document that suddenly has become important. It's worth asking, however, when work time does, more or less\u2014it's not a magic carpet ride\u2014fly. If you are a fire-in-your-belly trial lawyer, it might come when you cross-examine that key witness or reach the final flourish in your closing argument or ponder how to use your last precious preemptory challenge in jury selection. If you are a transactional attorney, it might be the eureka moment when you see a path to contract language that might spare your client costly litigation and months of potential worry. If you are a law professor, flow might come when classroom discussion gets buzzing and light bulbs go on amidst that sea of student desks. If you are a lawyer who takes pride in writing skills, flow can come in the simple task of putting one clear and significant sentence after another on a page. Flow is where we find it, given our own skills, interests, and goals.\n\nWe all want more flow. It might not be synonymous with happiness, but we recognize that it's pretty darn good. To get more flow, it helps to seek out more of those work experiences that have provided it the past. Take stock of those activities during your workday that give you a sense of flow and find a way to do them more often. If planning the firm retreat last year provided flow experiences, volunteer to do it again. Work more, if possible, with colleagues who you find engage your interest and challenge your intellectual skills. If trials do it for you, do more trials. Lose hours when writing appellate briefs? Then find a way to write more briefs. Needless to say, not all our hours will flow by like the Columbia (some chores are boring, but still have to get done), but improving the ratio of flow hours to nonflow hours should be every lawyer's goal.\n\nResearch tells us that some work conditions are more likely to produce flow experiences than others. Jobs with variety in tasks are better than jobs without. Offices and work spaces with views of outdoor life (especially with natural beauty) are more conducive to flow experiences than spaces without windows or with views of brick walls or vacant lots. Jobs with novelty are better than jobs without. Jobs without frequent interruptions, and work environments that are free from excessive noise, are better than work places with noise and other distractions.\n\nPuzzles are especially good at increasing the odds of a flow experience. Puzzles can come in many forms and still perform the trick. If we have a problem that offers no obvious solution, yet is a problem that we believe we have the skills to solve, there's a good chance that \"the flow\" will soon be with us. Puzzle-solving demands our attention. There's another bonus with puzzles, too: when we solve a puzzle, we garner a little happiness reward at the end\u2014that same sort of positive reinforcement that basketball players experience when their shots swish through the nylon net.\n\nThe practice of law, thankfully, affords numerous puzzles. How do we avoid enough estate tax to allow our client to keep the family farm in the family? How do we get a particularly helpful fact before the jury without exposing our witness to damaging impeachment? What case holds the key to convincing an appellate court to finding in my client's favor? The list goes on and on\u2014if you can't think of any puzzle in your work that gives you any pleasure to solve, you're undoubtedly in the wrong line of work.\n\nWe can become happy for good reasons, bad reasons, or neutral reasons. Helping others brings happiness for a good reason. Becoming happy because a romantic rival has just contracted a life-threatening disease is a bad reason. Becoming happy because you're passively consuming an enjoyable piece of fluff entertainment is a neutral reason. Flow, if it can be said to bring happiness (and we'd argue that it _does_ ), is almost always happiness for a good reason.\n\n### LEARNING FROM THE HAPPINESS (OR UNHAPPINESS) OF OTHERS\n\nYou think you know what will make you happy? You're most likely wrong. That is one of the principal lessons Gilbert offered in his bestseller, _Stumbling on Happiness_. Piling study on study until it can scarcely be doubted, Gilbert demonstrates how remarkably inept human beings are at predicting their own future levels of happiness. Our imaginations fail us when we try to picture what life would be like for us in new situations.\n\nWe think we will be as happy as a lark five years after we win the lottery. We won't be. We think life will be all downhill after we lose a limb in an automobile accident. It won't be. Far better than we imagine, we learn to adapt after setbacks. And far sooner than we think, the initial jolt of happiness brought on by a surprising success dissipates as we run on the hedonic treadmill.\n\nWe make numerous assumptions about happiness that aren't accurate. We think we'll be much happier if we move to California than Iowa, but it turns out that we aren't. We imagine how much we'll enjoy the mountains and beaches of the Golden State, but we forget about the horrendous traffic congestion and the higher cost of living. We think Iowa will be duller than cornflakes, but it turns out we'll find good friends and realize that there's some culture on the prairie after all. We fail to think about the vast majority of our time that won't be spent sunning in the sand or swooshing down slopes, including all that time stuck in traffic or groaning at the tab in an overpriced restaurant.\n\nOne cannot review Gilbert's mountain of research without gaining a little humility. We really don't have much of a clue as to what will make us happy. Asked what one thing could make them happier, Americans say \"money,\" according to a recent University of Michigan study. A worse answer could scarcely be imagined, from what a consistent body of happiness research shows. But there it is: Americans think more money is the key to future happiness.\n\nIf our imaginations are grossly deficient in predicting our future happiness, what can we do? Gilbert says he has the answer: \"Surrogation is the best way to predict if we'll be happy.\" We should \"observe how happy people are in different situations,\" he suggests. The best source of information about the future is other people who are presently experiencing events that we are contemplating experiencing. Chances are that our own happiness in those different situations will approximate those of the people we observe. Human beings are much more alike than they are different; we all tend to be made happy by the same sorts of things\u2014connections with other people, a sense of control, a sense of mattering. If a situation we observe has enough of those happiness contributors, it's likely to produce happy campers\u2014someone else today, perhaps us tomorrow.\n\nA recent experiment involving speed dating provides surprising evidence as to just how powerful a tool surrogation can be. Researchers set out to discover whether examination of a prospective date's personal profile and photograph was a better predictor of how much a subject would enjoy a five-minute speed date than the single report of another woman who previously participated in a speed date with the same prospect. Even though most of the women believed that they individually would be a better judge of how well they would enjoy the five-minute speed date based on the profile and photograph, they actually \"made more accurate predictions about how much they would enjoy a date with a man when they knew how much another woman in their social network enjoyed dating the man than when they read the man's personal profile and saw his photograph.\" The researchers concluded that although information from surrogates may produce a better affective forecast, people will tend to underutilize surrogation because they mistakenly think their own mental simulation will be more accurate.\n\nWhat works for dating also works for career shopping. \"Snooping in on other people's lives\" gives us our best shot at an accurate prediction of the workplaces and types of work that will make us happiest as lawyers. Our learning from the experience of others should come from observation. Asking another person if they like their job won't do. Most everyone says that they like their jobs, and their answers have virtually no bearing on how good their job actually is.\n\nObservation requires getting off your posterior. For practicing lawyers, this means taking mental notes on lawyers you interact with. If you're looking for a job, you might be better able to put yourself in a variety of workplaces. Visit courtrooms, law offices, judge's chambers, law schools, press conferences, administrative hearing rooms\u2014visit any place you can see lawyers acting as lawyers. Talk to any lawyers you can about their professional lives. How happy do they seem to be? Remember, of course, that happiness only rarely will be shown by laughs and big smiles. Happiness might be revealed by the focused attention of a lawyer engaged in a flow experience. The negative signs, the signs of unhappiness, will probably be at least as revealing as the positive signs. Boredom is hard to hide. Stress shows. So do anger and distraction.\n\nHaving said all this about there really being no substitute for observation, is there still something helpful that survey data can tell us about what workplaces produce the happiest lawyers? Yes, but the survey data are ambiguous. They come loaded with qualifiers. Make the best you can of it, and then go boldly out into the real world of lawyering with open eyes and an attentive mind.\n\n### KNOWING YOURSELF CAN MAKE YOU HAPPIER: IDENTIFYING YOUR PLEASURES AND YOUR STRENGTHS\n\n#### IDENTIFYING PLEASURES\n\nThe idea is so simple that you would think everyone would do it: Reflect on what things you like to do, then resolve to do them more often. Yet this most direct of all paths to a happier life is rarely taken. Instead, our daily activities are determined more by our _wants_ \u2014and wants turn out to have relatively little to do with happiness.\n\nAt first blush, it might seem that what we want is what is likely to make us happy. But evolution has designed us to be competitive creatures and to seek out the prizes that come from competition rather than activities that give us pleasure. We strive to get the promotion, to earn more money to get the biggest house on the block, or to acquire the firm's most high profile clients. We believe, more than ever before, that being financially well-off is an important key to happiness, despite study after study showing only a modest correlation between the two.\n\nMake a list of the activities that give you pleasure, large or small. Your list might include reading the morning paper in your bathrobe, playing doubles tennis or games of Scrabble by the fire with a couple of old friends, traveling to new places, or having beers with office mates after work on Fridays. Take special care to identify the small pleasures that occur each day. They may be something as seemingly insignificant as a moment of rest by a sunny window in the late afternoon or a cup of that really good coffee from the caf\u00e9. Now, if at all possible (it's hard to make more Fridays, but maybe you could also have lunch with co-workers on Tuesdays?), do the things on your list more often. What could be simpler?\u2014and you'll be happier. Recent research from the University of North Carolina backs this conclusion up. When people appreciate the daily \"micro-moments\" of happiness, those \"positive emotions blossom\"\u2014and help people develop resilience against adverse events. As Lyubomirsky counsels, \"Enjoy little things, for one day you may look back and realize that they were the big things.\"\n\nIn the context of your law job, unfortunately, things might _not_ be quite so simple. You might love making closing arguments to juries, but you cannot just decide to give more closing speeches and go do it. Giving more closing arguments requires taking more cases to juries\u2014and taking every case to a jury hardly serves your clients' interests. It might also mean doing more of something you do not like, such as pulling all-nighters preparing for examination of key witnesses. Still, there are likely to be certain pleasurable tasks on the job that can get more of your time without any obvious downside. Whatever it is you love to do\u2014whether it is \"rainmaking,\" searching LexisNexis, planning firm parties, deposing witnesses\u2014if you are any good at it, perhaps you can do more of it. Of course, you still must meet whatever other expectations your colleagues have for you, but the evidence is compelling that by doing more of what you most enjoy doing, you will become at least a little happier.\n\n#### IDENTIFYING STRENGTHS\n\nThree questions deserve your attention: What activities give you meaning? What activities give you pleasure? What activities allow you to exercise your personal strengths? The jobs with the greatest long-term satisfaction are those that provide a source for each of these three activities.\n\nIdentifying your strengths is no harder than identifying what gives you pleasure. If, ever since kindergarten, teachers have told you that you work well with others, you probably do. On the other hand, if every written project you've ever submitted came back filled with scribbled critical margin notes (\"What's this mean?,\" \"Poor word choice!,\" \"Bad organization!\"), writing is probably not your strength. The list of possible strengths is long and varied. It runs the gamut from being a great proofreader to having the empathy necessary to really connect with jurors in personal injury cases. List your strengths. Think hard about what jobs allow you to exercise those strengths. Happiness correlates with being good at what you do and having the feeling of control that comes with professional competence. Seligman argues that people who find jobs that allow them to use their strengths stand a greater chance of deriving the \"authentic happiness\" that is the most erosion-resistant of all forms of happiness.\n\nWhether law provides opportunities that lie at the intersection of your values, pleasures, and strengths is a question only you can answer. But it is a very important question. You should try to answer it and find the job (if one exists) that offers all three in the best proportions. You're almost guaranteed to be happier if you do.\n\n### FINDING A JOB THAT ALIGNS WITH YOUR VALUES: FOLLOWING YOUR HEART\n\nIf you could pick your clients' legal causes, what would they be? The ones you fight for now, or some entirely different set? If you're like most lawyers, many of the legal battles you wage on behalf of clients concern matters that are of little significance beyond the parties. Sure, an indifferent or bad cause pays as well (often better) than a good cause, but we'd be happier if we earned our bread in some other way. Poet, lawyer, and Librarian of Congress Archibald MacLeish wrote that he knew it was time for a different line of work when he lost the ability to care very much \"whether $900,000 belongs this way or that.\"\n\nThe happiest lawyers tend to be those who do work that they think make the world at least a marginally better place. Law professor Deborah Rhode, one of the nation's leading experts on the legal profession, states that attorneys experience \"the _greatest_ source of disappointment in practice\" when they feel that they are not contributing to the public good.\n\nUnlike some other sources of lawyer unhappiness, feelings about not serving a larger social purpose come from a choice you made. No one forced you to take the job with the fancy downtown firm with its list of well-heeled clients. You could have taken a job with the county prosecutor or the public defender (which one would have given you a sense of contributing to society's betterment depends upon your own views on law, order, and justice.) Maybe that dream job with the Justice Department or the Sierra Club wasn't going to happen, but there probably was _something_ out there that would have aligned better with your own values than the job you took. If you have sympathies that run to the little guy, perhaps a job in a personal injury plaintiff's. firm would have been a good fit. On the other hand, if you believe, as did President Coolidge, \"the business of America is business,\" corporate law or a job as corporate counsel might have been a surer route to a happier practice. What are _your_ values?\n\nIf you cannot readily answer the question that ended the last paragraph, make a list of people, causes, and things you pay attention to. (And, if you don't pay attention to _anything_ , get some professional help, because only attentive people have a clear shot at long-lasting happiness.) You might pay attention to some things that you'd rather not tell your mother about\u2014craps tables, pornography, bongs carved by hand. Cross them off the list. You might pay attention to other things that provide personal pleasure but hardly seem substantial in the big scheme of things\u2014your favorite football team or rock band. Cross them off too. Look at what's left on the list. It could be the environment, your family, gay rights, free trade, or the homeless. Now, think hard: What can lawyers do to aid your cause, and where do you find the lawyers who provide that aid? Go look for them. Beg to work for them. Honest passion sometimes pays off.\n\nIf you work at something you care about, you are much more likely to be happy. One reason for the higher satisfaction among public sector attorneys relates to their greater success in achieving a desired work-life balance, but another key factor is that they care about the causes they work for. A job that aligns with your values allows you to emotionally connect with your work. \"Without emotion,\" as Punset reminds us, \"no project is worth its salt.\"\n\nWe can't all work to save endangered species or defend civil liberties; most lawyers work in the private sector. In some private firms, pro bono opportunities provide a source of serving the public good that might not come from the firm's usual run of clients. Pro bono work has been clearly linked to increased self-esteem among lawyers. As one attorney sold on its benefits put it, pro bono work is an \"enormous morale booster for the entire firm.... No office parties or picnics could give you that.\" A side-benefit of pro bono work is that it improves the profession's decidedly poor reputation among the public. A survey suggested that two-thirds of persons surveyed would think more highly of lawyers if they engaged in more pro bono work.\n\nWhat is true for lawyers is true for persons in other lines of work. Professions that involve helping others rank high in general happiness. To be more precise, occupations that allow workers to _think_ they are helping others do well in happiness rankings. The fact that the help provided might seem rather modest in the eyes of those outside the profession is of little matter. Gilbert notes that often when a job doesn't provide an obvious sense of meaning, people will figure out how to add it. As an example, he points to hairdressers who derive satisfaction from seeing themselves as key confidants of their clients. There's a lesson here for lawyers: If you don't presently see your job as making the world a better place, perhaps you could give the matter a bit more thought.\n\nThe perfect job is the one that lies at the intersection of our deeply held values, our personal strengths, and our pleasures. Rarely do people stumble into such jobs; to find these jobs requires a great deal of reflection about what truly matters to us, as well as what sorts of tasks test our strengths and give us the most pleasure.\n\n#### NOT ALL HAPPINESS IS CREATED EQUAL\n\nWe suggested at the outset that happiness has many meanings. There is the happiness of the moment, what Seligman called \"the giggles and pleasures and joys of life,\" but there is also the happiness that comes from commitment. Aristotle used the word \"eudaemonia\" (\"good spirit\") to describe the feeling that accompanies a life well spent: one of engagement and immersion in activities that contribute to a better society. True happiness to Aristotle comes not simply from feeling good, but from feeling good for good reasons\u2014a feeling that generally comes from _doing_ good.\n\nThe type of happiness that results from doing work that you find meaningful, after giving serious reflection to that question, is an especially resilient form of happiness. Fulfillment through important work is less prone to \"set-point reversion\" than other forms of happiness, such as the shorter-term happiness boosts that come from pay raises or promotions or captures of coveted corner offices. Being engaged in the good work of building a better world brings the sort of happiness that tends to sticks with you for a good, long while.\n\nEnhance your happiness by either finding a legal career that aligns with your values or, at a minimum, finding one that allows you to believe that you are somehow improving the world. If your job involves strictly fights over money, money that you really do not care goes to whom, then you're not likely\u2014in the long run\u2014to enjoy much job satisfaction. Commit to a cause you believe in and be happy.\n\n#### IT'S A TOOLBOX, NOT A RECIPE\n\nIt's the rare construction project that requires every tool in a toolbox, and it's the rare lawyer who could successfully adopt every one of the suggestions we've made in this chapter. Consider each tool and its possible application to your career. Use the ones that make sense for your situation and your skills and personality. Now, get to work building the framework for a more satisfying life in the law.\n\n#### WAYS TO BECOME A HAPPIER LAWYER\n\n1. Make sure your job is one that matters to you.\n\n\u2022 choose meaningful projects over busy work\n\n\u2022 try to become a key player in your firm and legal community\n\n2. Think about the way your job positively affects other people.\n\n\u2022 identify how your work has bettered lives\n\n3. Strive for a comfortable work-life balance.\n\n\u2022 be willing to sacrifice income if necessary (it won't matter)\n\n\u2022 consider telecommuting or \"5 days work in 4\" options\n\n\u2022 discuss work flexibility with sympathetic partners\n\n4. Work to make your job more secure.\n\n\u2022 know and become friends with those who control your fate\n\n\u2022 meet or exceed firm expectations\n\n\u2022 develop expertise in noncyclical or countercyclical areas\n\n5. Take control of your work product and work space.\n\n\u2022 set, when possible, a timetable for finishing work\n\n\u2022 develop your own strategy for meeting goals\n\n\u2022 personalize your work space with photos, art, etc.\n\n6. Connect with people.\n\n\u2022 work on collaborative projects when possible\n\n\u2022 eat lunch with colleagues or clients\n\n\u2022 participate in firm social events\n\n\u2022 seek help and offer feedback\n\n\u2022 praise colleagues who do good work\n\n\u2022 remember birthdays and write personal notes\n\n\u2022 choose face-to-face work when possible\n\n7. If happiness seems possible in your job, commit to that job.\n\n\u2022 don't always look for greener grass (water your own)\n\n\u2022 remind yourself what you really like about your job\n\n\u2022 trust those who earn it and remember that building trust takes lots of interaction\n\n8. Increase the frequency of your \"flow experiences.\"\n\n\u2022 think about projects that have \"made time fly\"\n\n\u2022 identify common characteristics of those projects\n\n\u2022 look for tasks that challenge you but are within your abilities\n\n\u2022 avoid, when possible, tasks that are so easy as to bore you\n\n\u2022 find a work setting where distractions are minimized\n\n\u2022 try to include a variety of tasks within your work day\n\n\u2022 work in places with natural light and views of nature\n\n9. Avoid making upward comparisons.\n\n\u2022 focus on internal goals, not keeping up with colleagues\n\n\u2022 remind yourself that money has little to do with happiness\n\n\u2022 choose, when possible, projects that benefit the less fortunate\n\n10. Find out what experiences have made other lawyers happy.\n\n\u2022 remember that people are more alike than different\n\n\u2022 talk with other lawyers and ask about their experiences\n\n\u2022 observe what seems to make other lawyers happy\n\n\u2022 choose jobs and projects that have made other lawyers happy\n\n11. Know your strengths and what gives you pleasure.\n\n\u2022 identify tasks and events that give you pleasure, and do them more often\n\n\u2022 recognize your strengths and find ways to use them\n\n12. Align your work with your values.\n\n\u2022 identify your values and look for work consistent with those values\n\n\u2022 consider volunteering for pro bono work or work that you care about\n\n## **CHAPTER 5\n\nPreparing for a Satisfying Career: The Law School Years**\n\nTHE DAYS WHEN SOMEONE COULD BECOME A LAWYER by \"reading the law\" and apprenticing with an experienced attorney are largely in the past. Today, for most people hoping to practice law, the only road runs through law schools. It's a three-year-long stretch that can spell the difference between a happy career and an unhappy one.\n\nOur focus in this chapter is not on surviving law school. There are numerous good books available on that subject. This chapter is addressed to law students and would-be law students, but the focus is on how to approach law school with an eye toward developing a career in law that will be satisfying to you. Implicit in this chapter is the vision of an ideal law school\u2014one that is committed to its students' future career satisfaction.\n\n### IS LAW SCHOOL FOR YOU?\n\nNot everyone is cut out to be a lawyer. Every year, thousands of students decide that law school is not for them\u2014and thousands more have that decision made for them by law schools. Between 2002 and 2008, first year law school enrollment across the country remained consistently between 48,000 and 49,000 per year.\n\nIf you're thinking of going to law school, ask yourself why. If the answer is \"I want to make a lot of money\" or \"It will make my parents happy,\" we strongly urge you to reassess your options. There are a lot of good reasons to go to law school, but those aren't two of them.\n\nThink twice if you are going to law school expecting the glamorous life of lawyers in movies or television series, such as _Law and Order_ , _The Practice_ , or _Boston Legal_. \"Law in prime time media offers some combination of wealth, power, drama or heroic opportunities. Law in real time is something else.\" Through the cinematic lens, the profession looks dramatic and exciting: compelling stories, novel legal issues, exculpatory forensic evidence, riveting courtroom scenes, and flamboyant oratory. But the daily lives of most working lawyers do not offer the courtroom adrenaline rush and only sometimes, maybe even rarely, is there the opportunity to champion a victory for an underdog. Most lawyers spend their working days researching statutes, cases, or administrative regulations; reading the fine print in contracts; writing memoranda, motions, or briefs; preparing for or conducting depositions; and meeting with clients. \"Truly realistic portrayals of lawyers' work would hardly make for entertaining or interesting movies. Few, for instance, would line up to see a film titled _Adventures in Document Production_ or _The Man Who Did Due Diligence_\".\n\nCome to law school because you think you will enjoy learning about law and legal institutions. Come because you love problemsolving, critical thinking, and helping people.\n\nYou are most likely to fit well into law school if you are open to new ideas and approaches\u2014a trait that is one of the better predictors of both academic and career success. The single _best_ predictor of both academic and career success, by the way, is conscientiousness, defined as \"the degree to which a person is organized, persistent, and goal-directed.\" If you are open to new ideas, conscientious, and you like the activities that law school demands\u2014including tons of reading\u2014you not only are on course for a satisfying law school career, but a fulfilling professional career as well. Those qualities are \"a pretty good predictor of the ability to make a satisfying career out of the degree.\"\n\nThere's nothing necessarily wrong, if you have the time and money to spare, with giving law school a shot even when you have some uncertainty about your passion for practice or doubts about your ability to succeed. Many people attend law school as a generic grad school while they figure out what they want to do with their lives. Law school is often the default graduate program for smart college students. You'll have three years to make a decision\u2014and even after that there is time for a career correction. But go in with your eyes open. And yes, there are worse things that can happen than learning a thing or three about the law while falling short of a degree.\n\nWhen deciding whether you really want to be a lawyer, you should consider that the classic model of a lawyer working for a law firm, large or small, is on the wane. Only about two-thirds of lawyers are in private practice. The remaining third may work in corporate legal departments or state or federal government positions, but many of them use their law degree in fields outside of law. This is a significant demographic shift since 1950 when there were fewer than 250,000 lawyers (a quarter as many as there are now) in this country, and 80 percent of them were in private practice. These changing demographics mean that during the span of your career, you will likely be doing something other than serving clients in a law firm setting.\n\nKeep an open mind about the possibilities of using your law degree to pursue some path other than practicing law. Law degrees offer tremendous flexibility. If you go to dental school\u2014well, plan to spend years looking at teeth. A law degree, on the other hand, can open doors in politics, business, health care, journalism, law enforcement, and other fields where clear thinking and a knowledge of our nation's laws is valued. Don't limit your career vision to traditional law jobs. You can consult in an area of your expertise (such as business valuations); be a headhunter; go into arbitration or mediation; look at human resources jobs (such as affirmative action officer); develop real estate; write thrillers (think Scott Turow or John Grisham); become counsel for a school district; use the law degree to teach at the college, junior college, or paralegal level; become an agent in the entertainment or sports industries; manage a baseball team (Tony La Russa); coach football (Vince Lombardi); write poetry (Edgar Lee Masters); create crossword puzzles (Will Shortz); become a sportscaster (Howard Cosell) or broadcast journalist (Geraldo Rivera); become an actor (John Cleese); become a Presidential speechwriter and a game show host (Ben Stein); or even become a community organizer and then President of the United States (Barack Obama). In short, don't start law school thinking that the sole career outcome is to practice law with a law firm.\n\n### CHOOSING A LAW SCHOOL\n\nChoosing a law school is a big decision. Not only will the choice influence the next three years of your life, but it may also very well determine what type of law you practice, how you practice it, and how much you enjoy being a lawyer.\n\nNearly two hundred American Bar Association (ABA) accredited law schools exist in the United States. In many ways they are remarkably similar. Because of accreditation requirements, all law schools require at least two-and-a-half years of study. All teach many of the same courses, and almost all teach many of the basic courses in similar ways\u2014in large lecture halls using a case analysis method. All offer many of the same extracurricular opportunities, from law review to student bar associations to mock trial teams.\n\nWhen most people set about choosing a law school, they start by comparing law school rankings; weigh considerations such as the credentials of the faculty or the impressiveness of the building; and factor in geography, course offerings, and perhaps whether the school has a good football team. This is the wrong way to go about picking a law school.\n\n#### CONSIDER THE IMPORTANCE OF YOUR PEERS\n\nBy far the best indicator of a good law school match for you is how well you like, respect, and trust the students who will become your peers and whether you are stimulated by them. We think, when we decide to go to law school, that we are buying a legal education. In fact, it may be more relevant to say we are buying a peer group, choosing the people who will shape our values and approach to practicing law and who, in many cases, will be the people we deal with on both a professional and social basis in the decades to come.\n\nWe've already discussed how critical relationships are to happiness and how important it is to trust those people with whom you spend your days. But it is also increasingly evident that our peers determine our values and influence our behavior to a degree most people find surprising. In _Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives_ , Nicholas Christakis and James Fowler show how our peer groups influence everything from our weight to our sexual practices, not to mention our happiness. In fact, Christakis and Fowler come to the startling conclusion that not only is getting a $10,000 raise less likely to make you happy than having a happy friend\u2014it's also less likely to make you happy than having _a friend who has a friend_ who is happy! \"Network contagion,\" the phenomenon explored by Christakis and Fowler, is incredibly powerful and is rooted deeply in our evolutionary history.\n\nSo with that in mind, consider carefully the peer group you select. Do the students at a school you are considering have the ethical values you would like to have? Do they see the world as you see it, or would _like_ to see it? Are they, in general (there's a grouch or two everywhere) the type of people you would be proud to call your friends? When you go on your campus visit, remember to pay at least as much attention to the students as to the architecture. Interact as much as possible with students and ask yourself if they are the type of people you would like to be\u2014because they, in fact, will pull you in their direction for the next three years (at least). Their joys and sorrows will, to a larger extent than you ever imagine, be your joys and sorrows.\n\n#### CONSIDER VARIOUS TYPES OF RANKINGS AND SOURCES OF INFORMATION\n\nYou might be tempted, given the surface similarity of most law schools, to choose your school based on external ratings that emphasize the entering academic credentials of students, the financial resources and outside reputation of the school. You might decide to pick your law school based on the _U.S. News and World Report_ rankings of American law schools. For some of you, especially those of you interested in fast-tracking to judicial clerkships or high-salaried jobs at the nation's most prestigious law firms, that might well be a sound basis for your decision. No one can doubt that the ranking of a job applicant's school does influence hiring decisions for certain positions.\n\nYou may want to consider other factors as well in choosing your law school. It is worth noting, for example, that graduates of \"fourth-tier\" law schools (the lowest in the _U.S. News_ ranking system) actually report the highest level of job satisfaction. There are many possible reasons for this, including the satisfaction that comes from downward comparisons. Fourth-tier grads, like the bronze medalists at the Olympics, might be happy they made it through law school at all and thus appreciate their profession more than other law graduates. It might also be the case that the extremely competitive and intellectualized atmosphere of top-tier law schools, leading generally to highly paid and stressful jobs, is not conducive to a happy career path. The school that is most impressive on your resume may not be the one that will make you happiest in the long run. When you are thinking about attending law school, evaluate which law school will best launch you into a satisfying career.\n\nInstead of simply accepting admission to the highest ranked law school you can get into, consider carefully which law school seems to care about your future and can best guide you to a lifetime of happy lawyering. Contact law students and graduates and faculty at a prospective law school. What do they think about the place? Look at surveys such as the Law School Survey of Student Engagement, which evaluates law student happiness on many measures ignored in the typical national rankings of schools. Each year the _Princeton Review_ publishes its _Best Law Schools_ guide, which rates law schools based on more than 18,000 student responses in categories such as quality of classroom experiences, quality of life (including aesthetics of the law school and community ties), and professor accessibility and discusses the degree to which the school welcomes older students.\n\nTake a look at the school's catalog and see whether the school offers courses in areas of practice you especially care about. In short, kick the tires. Get a real sense of whether the school you're considering is truly a happy place likely to produce happy lawyers.\n\n#### VISIT THE SCHOOLS\n\nWhen you are comparing different law schools, visit them, ask upper-level students about their experiences, and go for a test drive\u2014sit in on several classes at each school you might want to attend. Is the law school one that makes efforts to equip its students to handle the stresses of practicing law? Is it one that encourages you to identify your own morals, beliefs, and standards? Are students encouraged to solve real-world problems? Is it a school where professors offer feedback and care about civility? Do the professors address the plights of people whose stories are the textbook cases? Do they seem to be trying to prepare students to have positive relationships with future clients? Supportive educational cultures promote student academic achievement, lead to success on the bar exam, and foster career motivation.\n\nThe best elixir is not money, but people. Search for a law school where professors understand and teach the importance of social bonds. Is your prospective law school one where students and faculty play together? Does it hold poker tournaments, faculty-student softball games, or annual skit shows? Do members of the law school community participate in Habitat for Humanity or other charitable projects? Ask current students if they have ever had a beer or a cup of coffee with a faculty member.\n\nIs the school one that prizes professional and personal mentoring? Historically, apprenticeships were the way that new lawyers were trained and socialized into the profession. Today, however, the interaction of professors and students at many schools begins and ends in the classroom. Check to see if your school offers a formal advising system by assigning each law student an individual faculty advisor. Formal advisement programs set the expectation that faculty are supposed to be more accessible to students. Several law schools have integrated members of the local bar into mentoring or inns-of-court programs to offer students both faculty advisement and attorney counseling.\n\nDoes the law school you are considering train for collaborative law practice? Does it encourage, or even permit, collaborative work among students? While students sometimes collaborate informally with peers on assignments, the \"vast majority\" of law students typically are not assigned to \"work together with other students on projects.\" Although numerous studies demonstrate the benefits of cooperative learning\u2014that it fosters engagement, depth, and critical thinking\u2014students at numerous schools still risk honor code prosecutions for cooperative behavior. Outside of some work in clinics and journals and a few team events such as moot court, most law schools \"make competition something of an art form to be practiced in relative isolation.\"\n\nA few schools are developing interdisciplinary and problem-solving courses, primarily in the clinical context, to encourage student teamwork in developing real-world solutions to problems of child and family services or healthcare. A very few schools are expanding these collaborations into the business curriculum\u2014for example, forming teams of business students and law students to teach entrepreneurial lawyering. Ask about the sorts of training your future school offers in collaborative law skills that will prepare you for the legal world you will enter. The practice of law is increasingly interdisciplinary, multidisciplinary, and collaborative. Trials are vanishing. In 2006, fewer than 1.5 percent of federal civil cases ended in a trial. The past quarter century has witnessed an exponential rise in the use of alternative dispute resolution mechanisms such as arbitration and voluntary mediation. Collaborative law practitioners report high rates of both client and lawyer satisfaction with the process and outcome. Not only is choosing a school that promotes collaboration likely to lead to a more satisfying career, it should also help prepare you for the changing nature of legal practice.\n\n#### CHOOSE A LAW SCHOOL WHERE PEOPLE MATTER\n\n_I'd rather learn from one bird how to sing than to teach 10,000 stars how not to dance._\n\n\u2014e. e. cummings\n\nLaw is a people profession. As some of the stories you will read in Chapter 7 illustrate, the vast majority of the best and worst experiences reported by lawyers we interviewed centered on the highs and lows of their interactions with other people. When you visit law schools, observe how faculty members treat their students. Try to find a school where professors respect students, their time, and their needs. An obvious form of respect, for example, is the learning and using of students' names. Students prefer to be called by their names rather than be identified as \"You in Back with the Green Shirt\" or have a professor's finger pointed in their direction. Other forms of demonstrating respect for students may be equally simple. See, for example, if classes start and end on time. Ask current students what's best and what's worst about their law school experiences.\n\nMore broadly, determine whether professors are adept at giving the subtle cues that people matter. It is easy to ask a student to give the opposing view to arguments made by another student; it is more difficult for a professor to communicate that she expects that student to look for and acknowledge the best in the argument made by the first student and then develop the other side. Were different points of view valued? Was explicit praise given when the students helped each other in class? Try to find out if the professors care about the students. As a student once told us, \"Good teachers care how students are doing in the classroom; great teachers care how students are doing in life.\"\n\nTeachers that care about people help counter declining civility in the legal profession and convey the idea that people and their feelings matter. They understand that in using the case method to push students to separate the \"relevant\" from the \"irrelevant\" this idea can be lost. Typically, if a fact or feature of a case is one that might affect the outcome of the case, it is \"relevant.\" If a fact or feature of the case, however much it might matter to the parties or even society as a whole, is unlikely to determine a legal outcome, the fact or feature is deemed \"irrelevant.\" Intense personal suffering, gross indignities, and abstract notions of justice all get thrown under the irrelevancy bus as the class discussion moves to its oh-so-logical conclusion. While this analysis might sharpen students' minds, professors risk signaling to students that all matters deemed legally irrelevant are, in the larger world of human experience, also irrelevant there\u2014when, in fact, they are anything but. There should come a time, in a discussion of a case, when we pause to measure the situation without our lawyer lenses. The practice of law is about more than winning or losing cases: it is about doing right for our clients while, consistent with that goal, doing as little damage as possible to everyone else.\n\nIn a school where people matter, professors will not overemphasize the combative or adversarial nature of the profession. Too often, the lesson drawn from professors' war stories is not the concept of intellectual engagement with ideas\u2014students begin to see a world filled with enemies:\n\nLaw students quickly learn to see other people as adversaries. Success means finding fault with the other side's, or the lower court's, position. If we focus on the positive, our words are deemed vacuous. This facet of our professional lives exacts a heavy toll on our ability to connect with other people in our lives, whether in professional or personal relationships.\n\nThe practice of dispassionately flipping legal arguments can cause lawyers-to-be to become insensitive to the people involved and their suffering. People behind the cases in textbooks can become dehumanized \"plaintiffs\" and \"defendants.\" Each case you will read was a series of real events\u2014perhaps life-altering\u2014for the people involved. The people in those cases lived with various social, economic, job, and familial pressures. None of them had, as we do now, the luxury of distance from the events in their lives. Christine Hurt, a torts professor at Marquette University Law School, reminds her students \"that every person in our textbook's cases, whether plaintiff or defendant, was a real person who had either been injured, or caused injury to others, and would therefore never be the same.\" How you think about the people behind the sterile appellate cases in the casebook lays a foundation for the ways you will think about clients in the future\u2014as demanding interlopers in an otherwise great relationship between the exchange of money and ideas or as complex people with frailties, needs, and hopes. If you learn to care about the people you represent, you will become a better professional and a more satisfied practicing lawyer.\n\n#### CHOOSE A SCHOOL THAT CARES IF YOU BECOME A HAPPY LAWYER\n\nThe traditional law school curriculum too often ignores career satisfaction. While more than one hundred colleges and universities offer classes in happiness research, we reviewed the available curriculum and course descriptions posted on the websites of almost two hundred ABA-approved law schools and found only six schools that offer specific classes in happy lawyering. Nine other schools address issues of career satisfaction as a small portion of much broader courses on law practice management, trial advocacy, or professional responsibility. Harvard Law School boasts more than four hundred seminars and courses but no courses on happiness or career satisfaction, although its undergraduate college offers one of the best-attended happiness courses in the country. The University of Michigan Law School lists the intriguingly named course \"Bloodfeuds,\" but again, no course on happiness. Perhaps law schools consider the topic the exclusive province of the career services office or one not worth academic inquiry\u2014or maybe the topic has simply not been considered at all.\n\nIdeas about job happiness, of course, may surface from time to time in other classes, such as legal ethics or professional responsibility. Discussion of these issues may help set expectations for careers and thus reduce dissonance between students' ideals and the later realities of legal practice. But the fact remains that more than 90 percent of all law schools in the country do not offer a single course that explicitly proclaims it will address issues of career satisfaction or lawyer happiness. How can law schools help students make good career choices if they offer no courses on career satisfaction?\n\n#### WHAT IF YOUR LAW SCHOOL DOESN'T?\n\nMaybe you are already in law school and as you read the above wish list, you realized that your law school was missing a few key features. Being the optimist that you are\u2014surely, there's a pony in here somewhere\u2014you might be wondering what you can do to make your law school more successful at promoting happy careers. You can always push for changes at your current school. Many, if not most, law schools make students participants in the schools' decision-making processes by giving them positions on law school committees affecting student life, including curriculum and appointment committees. Find the student organizations that host forums and invite speakers to your law school. Voice your desire to bring to your school speakers who can address lawyer happiness issues. You can request specific panels or presenters from your career services offices as well.\n\nWhy not approach a professor or a curriculum committee with the idea of creating a new course on satisfying legal careers? The course could introduce students to the general literature about the science of happiness and the different definitions of happiness. It could also provide comparative data on the satisfaction levels in various professions, identify characteristics of jobs with high and low levels of satisfaction, and consider dimensions of career satisfaction among lawyers in different types of practice. The class could include information from decision theory about how people respond to an overabundance of choices and what types of heuristic errors people make when predicting their reactions to events and situations. Course exercises could include projects that help students identify their own values or could require students to interview practicing lawyers. Finally, the course could weave in stories of lawyer heroes and heroines to remind students that they are embarking on a noble journey and to teach them, as philosopher Sidney Hook said in his 1943 book, the _Hero in History_ , the importance of not only doing their job well, but also of \"mak[ing] a unique contribution to the public good.\"\n\n### DEFINING SUCCESS IN LAW SCHOOL\n\nAll students applying to law schools write a personal statement telling why they want to be lawyers and what distinguishes them from the other 80,000 plus applicants. In encouraging students to present a sense of themselves, law schools echo Cyrano de Bergerac's Roxanne: \"Speak to me... be eloquent, be brilliant for me. Improvise, Rhapsodize... Please gather your dreams together into words.\" You probably wrote that you approached law school with the desire to make a difference. Perhaps you described your aspiration to fight for justice and make the world a better place. (We're generously assuming here, in these cynical times, you didn't just say whatever you thought would get you admitted.)\n\nSo what happens to your dreams?\n\nWhen you enter law school, it is easy to get on \"the institutional glide path.\" Law schools often define success for their students in terms of grades, class standing, and journal participation that lead to well-paying jobs at prestigious law firms. Law students report hearing about these prizes during law school orientation even before their first day of class. Consequently, students tend to internalize these prescribed measures of success and feel pressure to perform. As Harry Lewis, former dean of Harvard College, wrote in _Excellence Without a Soul_ , students try to obtain high grades as a form of credentialing rather than seeking understanding or mastery of the material. You can easily get sucked into believing that success and meaning can only be found in attaining a top ten percent grade point average, making Law Review, and having a full dance card for on-campus interviews. If you let these extrinsic markers of achievement define you, you are setting yourself up for unhappiness.\n\nOne of the first questions to ask yourself is whether other people are setting school and career expectations for you? Many students aspire to work in large law firms: some because of the exciting big impact cases those firms take, others in part because of the money or the prestige. Still others think they _should_ aim for big firm practice because law schools send the message that this is the pinnacle of success; at some law schools it is simply easier to find one of these jobs. Working at a big law firm and earning a high salary are not bad things. But three-fourths of law students in any given geographic area will not practice law with a large law firm. Ninety percent of law students will not graduate in the top ten percent of their class. (Go ahead, do the math.) And virtually all law students who embrace these traditional markers of success will be afflicted with the problem that these numeric measures are not good predictors for future career happiness. Buying into these metrics of success also has the unpleasant side effect of feeding a competitive law school culture and aggravating individual vulnerability, self-doubt, and isolation\u2014in other words, making your law school experience pressure-cooker miserable.\n\nTwo of the most important things the happiness literature reveals are that people who have a passion or purpose in their careers are happier and that those who define the meaning of their work for themselves will be happier doing it. Law schools tend to define success for people in narrow terms that begin and end with big numbers, judicial clerkships, and jobs at elite firms. That is a very limited vision of what is possible\u2014and it may have little or no relevance to what will make you happy.\n\nRemember your dreams. If you don't have dreams going in or just don't know what you want to do yet, explore widely, consider the kind of law _you_ want to practice, pay attention to the common errors you might stumble on as you try to make your future self happy, and consider whether you even want to practice law at all.\n\n### THE LAW SCHOOL EXPERIENCE\n\n_The Universal Fallacy: that the road to happiness runs through the top of the class._\n\nThe first year of law school is transformative. It is a time when you recognize that most legal problems do not have single, correct solutions\u2014that law is indeterminate. Recognizing that incoherence is embedded in law is unsettling, particularly for those of you who were drawn to law because you like rules. The reading assignments consist of a voluminous number of pages per night, much of it in what seems at first like a foreign language. Terms like _res ipsa loquitur_ and _mens rea_ may stump you even after you translate them from the Latin. In Contracts, \"consideration\" is not kindness. And \"horizontal privity\" in Property? Well, it doesn't mean what you think it means.\n\nYou must be prepared each day, if called on, to demonstrate your understanding of the material and perform in front of your classmates. The Socratic method, which is intended to probe students' understanding of the law and to show that factual differences can lead to different outcomes, can be wielded (highly improperly, we think) in ways that intimidate students and undermine self-esteem:\n\nMr. Jones, can you please tell us the facts in the case of...? Mr. Jones, what was the issue confronting the court in the case of...? Now, Mr. Jones, do you think the court was correct in finding that...? I see. Well, Mr. Jones, what would you have the court do instead? But Mr. Jones, now I am confused. You are telling us that.... Whereas, a few moments ago, you told us that.... Which is it, Mr. Jones...? What do you mean you're not sure, Mr. Jones? Not sure about what? Well then, let me change the facts a bit for you, Mr. Jones....\n\nSometimes the Socratic method seems like a cruel game of \"hide the ball.\" This can lead to a loss of trust. And, as we've seen, research in disciplines from social psychology to neuroeconomics shows that trust in relationships is crucial to happiness.\n\nCompetition and alienation from fellow classmates surface outside the classroom as well. Embedded in law school culture are other very public contests\u2014for positions as research or teaching assistants or on the moot court team or for clerkships, internships, and jobs. The architecture of law school is reminiscent of high school, with assigned classes, lockers, and relentless gossip. At some law schools, but certainly not all, students engage in more aggressive forms of bullying or hazing. They instant message each other during class to ridicule other students or play \"Gunner Bingo.\" Acts of aggression among law students may be the training ground for later incivilities among lawyers. This emphasizes the importance of selecting a school where you will like your peers.\n\nMost of you enter law school with high expectations. You were an academically successful undergrad or you wouldn't be here. Maybe you are also the sort of type-A personality that makes other type As look like B minuses. Grading systems are competitive; law school curves pit students against each other in class ranking battles. The nature of the rankings is that they create a small number of winners and a large number of losers. You may feel like a failure if your law school grades don't resemble your undergraduate grades, even though the curve is different, and your self-esteem may take a hit if you don't perform as well as your peers. First-year grades often determine who joins the law review and who wins prized summer clerkships, both of which pave the way for later jobs. What's worse, and makes you all the more anxious, is that all of those grades still typically ride on a single comprehensive test at the end of the semester. You will usually receive minimal individualized performance feedback before that single exam.\n\nIn response to these cumulative pressures, you may be seduced into changing both your life style and your attitudes: you will probably worry much more than you have before, give up regular exercise, and socialize much less with your family and friends. \"You'll have some significant changes in your personal habits: you'll lose your appetite or you'll want to pig out all the time; you'll want to sleep all the time or you won't be able to sleep; you'll want sex all the time or you'll never want sex again.\" You may suffer some manifestations of stress-induced illness: depression, headaches, viruses, allergies, or a host of other ailments\u2014and you may experience despair, difficulties concentrating, and anger at friends, family members and professors. You may try to anesthetize yourself with alcohol and drugs.\n\nYou might wonder whether law schools are admitting people destined to be unhappy. Law schools don't use a GPA-LSATMMPI admissions index, but studies show that law students start out with psychological profiles that are normal. Law students look very much like other students except for a couple of personality variables. They may be more competitive than the general undergraduate population, have a higher need for dominance or leadership, and have a tendency to be oriented to external measures of validation; but pre-law students have \"no greater incidence of psychological distress than the general population.\" Psychologist Kennon Sheldon and law professor Larry Krieger evaluated law students at two different schools during the course of their three years of legal education. They found that students entering law school were equally well- or better-adjusted than a comparable population of undergraduates. Then the students became increasingly unhappy in law school.\n\nAccompanying the intense competitive pressures to succeed is an insidious attitude transformation. Students are drawn away from intrinsic (inherently enjoyable) and idealistic motives such as helping others or curiosity and toward extrinsic motivations (activities that are \"a means to an end\"), such as \"impressing others, or gaining status and affluence.\" Success in law school is defined by these external measures and law students\u2014smart creatures\u2014internalize the goals of grades, law journal positions, clerkships, and high starting salaries.\n\nIn short, students buy into a system of superficial goals over which they may have little control, and they then suffer a loss of self-esteem from the competition. Krieger says simply that it is \"likely that your _attitudes_ (and anxieties) about the possible results, coupled with life style distortions, are wearing you out physically and emotionally.\" Even worse, many of these dysfunctional patterns established during law school\u2014the anxiety, depression, and drug and alcohol dependence\u2014carry over into practice. The competitive atmosphere of law school and the transition to practice can cement good or bad habits, especially for those for whom law school represents their first real experience with the full-time work world. Changing bad stress management habits once you have begun working is neither easy nor likely.\n\nIn the sections that follow, we consider the things you can do to avoid laying a poor foundation for career happiness. Before we do, though, let's examine an alternative outlook on law school. You often hear about the dark sides of the law school experience. What about the upsides?\n\n### A DIFFERENT VIEW OF THE LAW SCHOOL EXPERIENCE\n\nIs law school really _that_ bad? Does it inevitably set most law students up for disappointing careers? Just as many lawyers find the profession satisfying, many law students enjoy the ride. There is no one generic law school experience. Law schools are different. Law professors are different. Students will find professors at each end of the continuum measuring their concern for the well-being of their students.\n\nLaw schools across the country boast numerous graduates who have had a very different sort of experience than the stress laden competitive expedition described above. A 2007 survey of more than 27,000 law students showed that the vast majority of students were \"satisfied with their law school experience. Eighty-two percent of all students rated their law school experience 'good' or 'excellent.' Only 3% said their experience was 'poor.'\" Some students thrive in law school and emerge feeling excited about their upcoming careers. Some talk enthusiastically about the atmosphere of mutual support and sense of tight-knit esprit de corps they found in law school. An extension of that camaraderie is the network of fellow former students who will become your professional colleagues: \"Many of us look back to our law school days and see that as the time when we made some of the most enduring friendships of our entire life.\"\n\nThe wide array of clinics and externships offered at law schools gives students a window on career options that may be personally satisfying. Contrary to the bleak view of some commentators\u2014that law schools are creating disillusioned students who anticipate \"brutal hours, inadequate training and unfulfilling work\" at a large firm so \"they can hope to pay back their enormous debts\"\u2014a survey of third year students showed that 79 percent were either somewhat or very optimistic that they \"will have a satisfying career after law school.\"\n\nPerhaps one of the greatest services law schools can offer students in preparation for a satisfying career is to enable students' expectations to match the realities of practice. Law schools outfit students for fulfilling careers by preparing them for what is to come. The hard work that law school demands is appropriate training for a profession that has fairly grueling hours.\n\nYou may look back and realize that law school did equip you to handle many of the slings and arrows of practice: that you learned teamwork in sharing notes, that the Socratic dialogue prepared you to sharpen your thinking and articulate positions for your clients and to understand that there is no single right answer to \"most of the hard questions that real world practice poses,\" and that as a litigator\u2014and unlike Yogi Berra\u2014you need to be ready to \"tak[e] both forks in the road.\"\n\nStudents may have better or worse experiences in law school because of their own personality characteristics or the schools they choose to attend. The next section explores the types of attitudes that are associated with greater measures of satisfaction in law school and suggests some approaches you can take during law school to build a sound base for later satisfaction in practice.\n\n### DEFINING SUCCESS FOR YOURSELF IN LAW SCHOOL\n\n#### KEEPING YOUR OWN COMPASS\n\n_It is practically a \"given\" that great success\u2014top grades, high salary, or a prestigious job represent the fast track to happiness. This pervasive belief is false._\n\nSo much of learning how to actually practice law is on-the-job training. Yet there are a few keys to future career happiness that you can learn in law school. Surprisingly, they have little to do with pure academic successes. A recent study of more than 6,000 law students at fifty law schools showed that high LSAT scores are \"slightly negatively related to life satisfaction.\" The researchers concluded that \"one's ability level, as assessed by standardized admissions tests, does not predict life satisfaction; rather, rewards gained through hard work and engagement with the material predicts life satisfaction.\" In short, highly engaged law students are more likely to become happier lawyers.\n\nThose keys to life satisfaction have everything to do with learning to let internal motivations guide you. Students who define success in terms of their own values are better at avoiding the distress-depression-substance abuse triad. The individual values can include personal qualities that are important to you, such as integrity, responsibility, or openness to collaboration. Or they can shape intrinsically satisfying goals, such as representing clients effectively, telling a client's story, championing civil rights, or serving the public.\n\nHaving intrinsic focus does more than enhance self-esteem, build competence, and help avoid the competitive nature of a law school's system of extrinsic rewards. Research in learning theory shows that when students study toward the goal of mastering a topic and concentrate \"on acquiring the skills or knowledge that are the subject of study,\" they actually perform better academically than those students who set performance goals that focus \"on grades or other performances relative to... fellow students.\" In other words, if you are studying Property \"hard\" and agonizing over whether you are studying \"hard enough to get a B,\" you are likely to perform less well than if you study with the goal of being able to explain various rules about covenants that run with the land to a future client. If you learn to focus on your own objectives and consider what fulfills you personally, you will be more likely to pursue individually satisfying\u2014rather than socially-prescribed\u2014career directions.\n\nAnother key to success in law school that lays the foundation for future career satisfaction is to make friends. Your fellow students will be your professional peers for the rest of your career. Reach out, even if it makes you uncomfortable at first. Remoteness and a lack of sociability are not characteristics of either happy law students or happy lawyers. Studies show that those lawyers who have more close friends and a greater amount of social interaction are more likely to be successful (according to ratings of external evaluators) than lawyers who report they are isolated. Building connections to other people, and remaining close to your support network, takes work; but you will happily discover that much of what you will learn during law school and the best part of the overall experience takes place outside the classroom.\n\nDeveloping positive relationships with your professors is also important. Law students may perceive faculty as distant, particularly in first-year, large-section classes. Females and racial minorities in particular may be less comfortable than males and whites in seeking out professors for conversations about the course material or career advice. Developing social connections with faculty outside the classroom not only enhances learning, it \"makes the educational process more meaningful\" because professors can often help students reflect on their own values better in one-on-one meetings. To boot, professors may also have early notice of career opportunities.\n\nJust as you will ultimately want advisors outside of your own firm as well as inside once you start practicing, you should search for mentors outside the walls of the law school. Seek out mentoring. Learning to find mentors is part of your professional training. Networking and mentoring will be crucial to your career. Fortunately, you will probably discover that lawyers in the community are more than willing to be mentors. When asked to describe a peak experience he has had in the practice of law, Greg Castanias, a partner in the Washington, D.C., office of Jones Day, said that it was not any one event, \"Ultimately, it seems to come down to the mentoring I have received, and the mentoring I have been able to provide to others.\"\n\n#### AVOIDING DEBT-DRIVEN CAREER CHOICES AND SUBTLE STEERAGE\n\nThe literature on job satisfaction in particular law careers shows that on average those in government and public interest work are more satisfied with their work than those in private practice. However, many law students wonder whether they can afford to pursue their public interest passion and still repay their school loans. Studies show that when students enter law school, between 40 and 70 percent of them say they want to work in the public interest area. Yet, when exiting law school, only a fraction of those who expressed such a preference actually start work in public interest law. Looming law school debt provides an incomplete and perhaps partly misleading explanation for the crowds detouring away from public interest work. The other explanation is that the curriculum of most law schools is not geared toward training students in the administrative, practical, and human skills they will need to enter public interest work.\n\nMedian tuition at public law schools for in-state students was approximately $15,600 in 2008, $26,400 for out-of-state students at public schools, and $34,000 at private law schools. In 2009, the typical law school graduate entered practice with debts of approximately $100,000. Let's look at what that debt load means in relation to starting salaries.\n\nFor the class of 2008, the National Association for Law Placement (NALP) reports that the median starting salary among all law firms was $72,000, with a mean of $92,000. These figures can be misleading. Salaries are significantly stratified into a bimodal distribution: the mapping of them looks like a double-humped camel, with clumps of salaries at both the high and low ends. New lawyers are, upon graduation, salary-sorted into the \"haves\" and the \"have nots.\" Many law graduates received low starting salaries, a smaller number obtained large starting salaries, and there was not much middle ground. See figure 1.\n\nThe far right hump (really, more of a spike) of the camel shows that larger firms enter this salary competition believing their reputations are tied to their ability to compete with other big firms in terms of starting salaries. The other hump reveals that most other firms don't feel compelled to, or can't, enter the salary race\u2014and instead offer salaries that match their own markets.\n\nThe actual starting salaries vary significantly with the size and type of firm and the geographic area of the country. For instance, the national median starting salary in 2007 for firms with 2\u201325 lawyers was $68,000, for firms with 26\u201350 lawyers, it was $81,000, and for firms of 101\u2013250 lawyers, it was $105,000, with salaries starting at $130,000 for firms of more than 250 lawyers. First year prosecutors and judicial law clerks earned about two-thirds of the median salaries at the smallest firms ($46,000), while first year legal services lawyers earned only $38,000.\n\nFIGURE 1: Class of 2008\n\n_Source_ : NALP\n\nConsider how many of your current decisions are motivated by economics\u2014what you can afford to do, and whether you can repay your student loans. Fortunately, this debt may be more manageable than the looming large numbers first make it seem. Lawyers are not minimum wage earners and make a good salary over time. Tapping into recently enacted federal loan forgiveness programs\u2014programs that completely or partially relieve debt for lawyers working a certain number of years in particular legal services jobs\u2014is an option for some.\n\n##### _Loan Repayment Assistance Programs_\n\nSixteen states and the District of Columbia offer loan repayment assistance programs for lawyers entering public service jobs. These typically provide a yearly amount of from $2,500 to $12,000 of assistance to lawyers in specified public interest jobs whose annual income does not exceed $40,000-$65,000, depending on the state. Equal Justice Works lists 109 law schools that either have their own loan assistance repayment program or participate in one in their state or both.\n\nIn 2008, Congress passed the Higher Education Reauthorization and College Opportunity Act, which created four new loan forgiveness programs for public interest lawyers. Finally, the Department of Education has created a plan for Income Based Repayment (IBR) of federal student loans. Starting in July 2009, IBR creates a sliding scale based on income and family size for federal loans made to students (not parents), allowing them to qualify for reduced payment options. If you are interested in public interest work, examine these sources of assistance in debt repayment.\n\n##### _Income and Subjective Well-Being_\n\nAs you explore various career options, keep in mind the literature on the limited ability of money to make and keep you happy. Maybe money is the one thing you think will make you happy, but people overestimate the effect that money will have on their levels of happiness. Remarkably, even having higher income aspirations leads to reduced life satisfaction. People who look upward when they make comparisons and then evaluate their own situation in relation to their financial ambitions pay a price. Also, as our discussions of the \"hedonic treadmill\" revealed, the pleasure boost from a raise in income is transitory; then consumption levels increase, and you find yourself again in the position of wanting to have more than you do. What's more, particularly in certain types of law practice that emphasize billable hours, you will likely find that you do not have the time to enjoy the new toys your money provides.\n\n### IDENTIFYING A SATISFYING CAREER\n\n[F]or over twenty-five years, the National Association for Law Placement (NALP) has surveyed the nation's law schools and reported that over 90 percent of those seeking employment find it within approximately six months of graduation... . The fact that you will ultimately find a job does little to assuage the nagging fear in the minds of many law students that _they_ will be the ones left out in the cold, $100,000 in debt and panhandling outside the Grand Central Terminal just to stay alive... . Actually, the real danger is not that you will fail to find a job, but that you will find the wrong one.\n\nStudents attend law school for many different reasons and, thus, contemplate myriad career paths after graduation. When you start thinking about what job you want after law school, it's good to keep in mind that old law adage that there is no one right answer. Eduardo Punset cautions that because reality is too complex, we create ideals\u2014imaginary perfect models\u2014of lovers, of careers, and so on. There is no ideal job, but lots of jobs could be satisfying. There can also be jobs that are wrong for you.\n\n#### AVOIDING THE INSTITUTIONAL GLIDE PATH\n\nWhen you consider possible jobs, evaluate the sources of information and institutional pressures toward certain types of jobs. Law schools face a number of tensions that may undermine their steering students toward individually satisfying careers. Many of the subtle messages students receive about worthwhile occupations come from faculty members who, for the most part, worked for large law firms and did not have a large amount or a wide range of practice experiences. For law professors across the country the average time in practice prior to teaching is only 3.7 years. This is part of the glide path problem: professors may have a tendency to steer students toward the model with which they are most familiar\u2014a federal clerkship that rolls into a stint at a big firm.\n\nSchools also have a reputational interest in placing students in the biggest, best-paying jobs possible. This creates a prestige boost and produces more lucrative donors down the road.\n\nCareer services offices in a variety of ways cater to larger firms, which are their recurrent visitors during on-campus interviews (OCI) seasons and typically are the most frequent employers of graduates. The exposure of students \"to a significant level to a large firm presence... often leads to a perceived hierarchy of post-law school jobs, with employment at big firms being seen to be more prestigious than other non-large firm jobs.\" Another factor plays into the tilt toward \"Big Law.\" Career offices are judged in large part by their placement statistics. This may leave directors of career development in the position of spending buckets of time tracking students down for employment reporting purposes, a task low on anyone's list of favorite pastimes, and with much less time to do individual advising to assess career options that match interests and skills. Although law school personnel know that a mismatch between a given job and a student's interests and skill set will not result in a happy outcome, the traditional steerage toward the most prestigious job offered remains a well-embedded default position at most schools. Keep in mind that the conventional path may not be the one that will bring you the most long-term satisfaction. Although high status jobs and high salaries are \"seductive\u2014they create a nice picture of life... they are actually correlated with relative unhappiness.\"\n\nCareer service offices receive little encouragement to widely disseminate information that might put law schools into conflict positions with large firms in their areas, many of which are probably the law schools' principal supporters: the lavish donor firm that put the Jackson Pollock in the faculty lounge and the fountain in the student center. Although this may not be a conscious sentiment of those people giving career advice to students, there may be the vague concern that if law schools focus more on finding happiness-producing jobs for their graduates than high income-producing jobs, there may be an impact on law school finances over the long term.\n\nThe absence of information about which law jobs are more satisfying or firm-specific information about billable hours and work-life balance policies is compounded by firms having a marketing interest in the disclosure of less information rather than more. Remember that the lawyers who come to law schools for on-campus interviews are there, at least in part, to sell their individual firms.\n\nWhat all of these ingrained interests, default positions, veiled facts, and traditional ways of doing things imply is that you will need to be proactive in seeking out unfiltered information about measures of career satisfaction in various types of jobs. The Internet makes information about law firms (e.g., size, salaries, practice groups, services, and clients) instantly available. Rankings from various groups are splashed across its pages. Law Students Building a Better Legal Profession provides a website that provides an interactive \"report card\" on almost 350 firms in major markets across the country, based on factors such as the number of billable hours required. In 2008, _Working Mother_ magazine and the national consulting group Flex-Time Lawyers compiled its second annual list of the fifty best law firms for women. More information is available than ever before. Just as law professors have had to face the verdicts of their student reviewers on websites such as www.RateMyProfessors.com, law firms now are being reviewed on the Web by their associates and clients alike.\n\nMaybe your career services office is one that tries to ensure that the expectations of students match the realities of the firms or organizations they are joining. Perhaps it offers a chart of area firms' billable hour requirements, parenting leave policies, and approaches to work-life balance. Your career office might even keep data on alumni who go to different firms and whether they are happy there or whether they leave those firms to get a sense of personality matches between individuals and firms. If your career services office is one that doesn't do these things, consider asking why not?\n\n#### EXPLORING WIDELY VERSUS THE PARADOX OF CHOICE\u2014WHY THINKING LIKE A LAWYER MAY NOT HELP\n\nWhen you begin your search for jobs in law, the sheer number of possible paths can be daunting. You can have too many choices and be unhappy. You can also have too few choices and be unhappy. It's the law careers version of the Three Bears' dilemma.\n\nIf you take the path of least resistance, you wind up with only a narrow slice of possibilities. You participate in the on campus interviews, while moaning a little that the OCI favor the top 10 or 20 percent of students, and then pretty much limit your searches to the online lists of job postings on the career services website. This is a mistake.\n\nThere is often much more than OCI going on at a given school. Attend various career forums and panel discussions about career possibilities, meet-the-firms, and bar association networking events. Explore public interest job search guides, participate in mock interview programs, review binders with students' evaluations about their summer experiences, and attend educational programs about topics such as how to make yourself profitable to a firm or how to assess your values, personality, and career choices. Peruse the bar directory and write letters to firms in your interest areas. Seek out government internship or fellowship opportunities and inquire whether your career services office has a reciprocity arrangement with that of law schools in other cities where you might like to work. If you have a specific idea about your career path, pursue it by taking relevant courses, attending seminars or continuing legal education programs, and finding out how attorneys in the area landed their jobs. Explore widely, at least initially. But you can have too much of a good thing.\n\nRemember _The Paradox of Choice_? An overabundance of choices produces anxiety, undermines happiness, and leads to disappointment. Being able to select from among ninety different types of mutual funds and seventeen brands of toothpaste makes people feel overwhelmed. That same sprawling array of options\u2014coupled with the search for the perfect fit\u2014awaits you in the legal field. Too many options can paralyze you with indecision: \"Perhaps there comes a point at which opportunities become so numerous that we feel overwhelmed. Instead of feeling in control, we feel unable to cope.\" Even the possibility of too many choices can create inertia\u2014which is why most people stay with their telephone providers rather than shop calling plans.\n\nPeople feel most in control of their lives when they have choices\u2014but not too many choices. One way to deal with hyper-choice is to whittle down the multiplicity of options into a smaller number of clearly differentiated paths, a distinct variety of possibilities. The first step is distinguishing between \"choices that matter\" while \"unburdening yourself from the choices that don't.\" Don't spend all of your decisional time pondering the relative advantages of living in Pittsburgh versus Philadelphia\u2014and which city has the better baseball team. Think instead about the type of practice that interests you and best aligns with your values.\n\nChoosing an alternative that is good enough\u2014\"satisficing\"\u2014often leads to greater happiness than holding out for the perfect choice: \"With 'good enough' rather than 'best' as a criterion, the satisficer will be less inclined to experience regret if it turns out that an option better than the one chosen was available.\" Too often, we imagine that there is a perfect job and hold out for it. There are bad jobs and good jobs\u2014even some great jobs\u2014but there are no perfect jobs.\n\nMichael Melcher, a career coach and author of _The Creative Lawyer_ , says that many of the analytical strategies of \"thinking like a lawyer\" you learn in law school won't serve you well in thinking about your career path. Lawyers, he says, \"analyze rather than explore, focus on flaws and potential problems, look for clear precedent, require solutions of general applicability ('what would work for lawyers') rather than specific applicability ('what would work for me'), defer action in situations of uncertainty, [are] skeptical about possibilities, [and] avoid taking risks.\" It is not possible to imagine your whole career, see how it is going to play out, and know where you will be happiest. As Melcher concludes, \"the path from here to there is a zigzag, not a straight line, and... you are better off exploring and experiencing than assessing things through detailed analysis.\"\n\nSo, bearing in mind the paralysis that too many choices can cause, the worry that comes from too few choices, and your inability to reason your way out of this mess in a lawyerly fashion, what should you _do_? Is taking a job that passes the threshold of acceptability a good start, particularly given that your first job in law is highly unlikely to be your last? Or might that approach lead you down a path that locks you in to a particular type of work? We're going to give the typical law school answer: That depends. It depends on what your emotional and financial needs are, what jobs are available, and what ability you have to tolerate uncertainty. Too often, students seem desperate to Get A Job. Anxiety and fear tend to drive the job search. The best approach is to consider very specifically what would make you happy and what values, skills, and interests you have.\n\n#### REMEMBER YOUR PASSIONS\n\nThere are ways to both pursue your passion and distinguish yourself from the other 43,500 new law graduates who will flood the market each year. There may not be \"clear precedent\" for the type of law you think you might like to practice. If you have a combination of skills that permits you to offer a distinctive set of services to clients, you may be highly marketable and also very content. Consider the experiences of Kevin Travis, a lawyer who lives in the Midwest, but whose primary client is several states away in the South.\n\nI worked my way through both college and law school as a mechanic, and once I got my law license, I thought I'd never use my mechanical knowledge again. But that one client for whom I ended up spending all my time as a litigation associate at my first firm was a vehicle service contract administrator, and when they got sued over claims they had denied, they not only needed someone who knew how to defend a lawsuit, but also someone who could understand what had gone wrong with the car, and why, and could explain the mechanical aspects of the claim to the finder of the facts. Because I had the combination of legal and mechanical skills, I could do both, and suddenly, I'd found a specialty where I could stand out... not only in my town or my state, but in the whole country. I'm now doing the same type of work for a different client, but in the same industry, that is a thousand miles away. I get that work, rather than one of the many hungry local lawyers in the large town where my client is located, because of the unique combination of skills I can offer.\n\nOr take Julia Belian, now teaching in Michigan, who earned a master of divinity degree and then a law degree. With this set of degrees, she entered estate planning. She says she found a \"special joy\" in probate work because \"it really _matters_ , whether it is rich clients or not, because we're all mortal, and we all want to take care of our loved ones.... This isn't just _law_ we're discussing, it's life and death, quite literally.\" She is also particularly trained to help clients discuss their fears and make estate plans\u2014\"because to plan, you must first accept that you will die.\" Having these discussions with client, she says, is \"a ministry of the highest order.\"\n\nOr perhaps you liked horses when you were growing up. When Denise Farris was little, her family \"always had a horse in the back yard.\" Her \"horse hobby\" continued after law school. When she noticed the enactment of statutes creating liability shelters for equine professionals, she created a newsletter addressing those statutes and mailed the newsletter to stables in a tristate area as a public service. She was invited to speak on the topic by the Missouri and Kansas Horse Councils as well as the University of Kentucky Law School, and quickly became one of the nation's experts on equine law, which is now a substantial portion of her practice. She addresses numerous intriguing facets of equine law for clients: risk management contracts, stallion syndication agreements, \"who's your daddy\" testing following artificial insemination, zoning and environmental issues regarding manure and carcass disposal, and estate planning for owners wanting to ensure ongoing care for that special horse. Farris says it is richly rewarding to discover new practice areas can be created around life-long passions.\n\nBased on your interests, hobbies outside law, and special expertise, you may be able to create your own niche practice. But, if you don't have a pre-existing specialty area, then what?\n\n#### YOUR STRENGTHS, INTERESTS, AND VALUES\n\nEver since 1880, when John Venn introduced his famous diagrams, students have been taught to use overlapping circles to depict the intersection of information sets. Harvard University psychologist Tal Ben-Shahar proposes using Venn diagrams as a job finding tool. We believe this approach could benefit law students searching for that satisfying job.\n\nBen-Shahar urges job-seekers to use what he calls \"the Three Question Process.\" He identifies three inward-looking questions most helpful to the search: (1) What gives me meaning? (2) What gives me pleasure? (3) What are my strengths? Sources of meaning might include, for example, working with the elderly, solving problems, promoting religious values, or protecting nature. Sources of pleasure could include, for example, hiking, singing, solving crossword-puzzles, or reading books about history. Strengths might include problem solving, meticulousness, knowledge of history, and an ability to relate to a wide variety of people. Preparing lists of your own sources of meaning and pleasure and your strengths is an exercise that requires considerable reflection and time on your part. You might aim to identify, say, ten entries in each of the three categories. In identifying strengths, consider subject areas of expertise (for example, environmental law, intellectual property), your skill sets (for instance, writing, oral communication, analytical skills, organizational ability, ability to relate well to others), and your personality (are you compassionate, risk-tolerant, extroverted, patient?).\n\nTo complete the exercise, put your entries in circles and then, in Venn diagram fashion, examine the overlap between your circles to sharpen your thinking about what jobs might be the most satisfying. If you find meaning in working with the elderly (source of meaning), enjoy spending time with others or like discussing history (sources of pleasure), have expertise in the area of elder law or estate planning, and are patient and compassionate (strengths), then you might do well to consider positions with the Social Security Administration or an estate planning firm, to name a couple of obvious job possibilities. If you like helping underdogs, enjoy performing and strategizing, and relate well to people of all types\u2014well, we suppose you have already thought about becoming a plaintiff's personal injury lawyer.\n\nBen-Shahar warns that your own list is likely to be messy. Jobs might not come jumping off of your page of Venn diagrams. The real benefit of the Three Question Process for most of you is likely to be to sharpen your thinking about meaning, pleasure, and strengths and to gain an appreciation for how those concerns are central to your job satisfaction.\n\nAnother way of viewing your strengths comes from noted University of Michigan psychologist, Christopher Peterson, and Martin Seligman, former President of the American Psychological Association, as well as fifty-five other social scientists. They developed a classification of character strengths by examining concepts of positive characteristics of humans to identify six core virtues that religious and moral philosophers have, over history and across cultures, tied to human flourishing and achievement of the good life. The six virtues, according to these social scientists, are courage, humanity, justice, temperance, transcendence, and wisdom. These larger categories break down into twenty-four virtues or character strengths that are valued universally.\n\nPeterson, Seligman, and their group posit that people have a handful of these characteristics as their signature strengths. Indications of a signature strength are that people feel that the strength is an authentic reflection of who they are, have a fast learning curve for activities tied to that strength, and are invigorated and excited when using that strength. The idea is that people are better off trying to build on their individual signature strengths rather than trying to correct deficiencies. On his website, Seligman offers a questionnaire to help you identify your signature strengths. The key connection between these character strengths and happy lawyering is that \"[i]dentifying your strengths can help you find new work that is personally fulfilling and meaningful or alter your current work to become more personally fulfilling and meaningful.\" Peterson and Seligman suggest you consider a deep sort of happiness. Their focus is on ways to build on character strengths that may lead to longer term life fulfillment. Is what you are doing on a daily basis drawing on your strengths?\n\nSo you have answered Ben-Shahar's three questions and have taken Seligman's self-assessment to help identify your signature strengths. Where do you go from here? We suggest that you write not just a job description, but a sketch of the path you want your career to take.\n\n#### YOUR CAREER PATH\n\nUsing the questions below as a guide, draft a two-paragraph vision statement for your career. Paragraph one: What kinds of jobs would best match your personality, values, and skills? Paragraph two: Where do you want to be in ten years? Twenty?\n\n1. Do you have a general idea of what kind of career you want? Do you plan to practice law? What do you envision yourself doing in twenty years\u2014at the height of your career?\n\n2. What are your values? Do you value financial security? Time with your family and friends? Autonomy? Stability? Creativity? Do you want to have children? Do you like to take risks? Do you like variety? Do you like excitement and pressure or do you prefer life without stress? Do you desire leisure time to pursue various interests? Are your career plans consistent with your values?\n\n3. How do you want to spend your day? Do you like dealing with people? Do numerous meetings make you crazy, or do you welcome the time to noodle issues with colleagues? In what size groups do you work best: all alone, with one or two other people, with a larger group? Do you like spending large blocks of time researching and writing? Do you like detail work? Do you enjoy or hate business development? How do you feel about the prospects of litigation? Do you like to travel? Do you prefer new challenges or handling matters over which you have some mastery? Do you know what kinds of law do _not_ interest you?\n\n4. What kinds of hours do you envision yourself working as a lawyer: a mammoth number of hours, a 9 to 5 day, part-time? Are you someone who is able to create work life balance for yourself, or do you take on too much work?\n\n5. When you are considering a job, will the salary and benefits meet your basic minimum needs? Will some other factor be paramount in your job search (for instance, whether the work is meaningful to you or can keep your interest)?\n\n6. Are you interested in entering a type of law that might dictate that you live in a particular region (such as international law or water law)? Do you prefer cities? Large or small? Suburbs or rural areas? Domestic or international travel?\n\n7. Do you feel that other people (perhaps family members such as parents or partners) have specific expectations for what you will do with your law degree? Do they match yours? 8. What are your signature strengths as they relate to a legal career? Weaknesses?\n\n9. What are your interests or hobbies? Do they relate to your career plans? Can they?\n\n10. If you could not be a lawyer, what would you do? Do any of these alternate careers suggest what you could do with your law degree other than practice law?\n\nOnce you have a sense of the possible path you think you want to take, then you need to assess whether it is the right path. This is where you could encounter some cognitive stumbling blocks.\n\n#### THE FUTURE IS NOT EXACTLY CLEAR\n\nResearch in cognitive psychology shows that people systematically make inaccurate predictions about what will make them happy. In thinking about how to make your future self happy, you are likely to make two types of errors: prediction mistakes and, on the other hand, failures to follow those predictions that are accurate.\n\nLike other people, lawyers wrongly predict their emotional reactions to both positive and negative future events. They overestimate the joy that winning a multi-million dollar verdict will bring them, and they underestimate their ability to rebound from a denied promotion. In both circumstances, the context will moderate the effects of the event, and people will hedonically adapt and tend to return to their happiness set point. When you are considering that job offer from Parker, McBain & Chandler, LLC, you might assume that accepting an offer from one of the most prestigious firms in your city will bring you happiness. And it will\u2014for a while. Soon, however, the drudgery of day-to-day work will dampen the joy of the associational prestige.\n\nYou might also suffer from projection bias. People tend to assume that their future likes and dislikes will be similar to their current preferences. (This explains the tendency to overfill the grocery cart if you shop for food while you are hungry.) Daniel Gilbert observes, \"Because we naturally use our present feelings as a starting point when we attempt to predict our future feelings, we expect our future to feel a bit more like our present than it actually will.\" Your own behavior and many external factors will influence how your future self will react. If you are in your first year of law school, you may gravitate toward fast-paced jobs that promise the endorphin rush of trial work, such as being a prosecutor or public defender. You may, at present, have no other responsibilities so you can devote countless hours to trials. Three years from now, however, you may want a job with somewhat less excitement and more regular hours, so you can spend time with your nine month-old who isn't even a gleam in your eye at the moment. Gilbert suggests: \"if we want to predict how something will make us feel in the future, we _must_ consider the kind of comparison we will be making in the future and _not_ the kind of comparison we happen to be making in the present.\"\n\nEven when people make accurate predictions about the kinds of work that would make their future selves happy, they often fail to follow their predictions. Christopher Hsee and Reid Hastie, both professors of behavioral science at the University of Chicago, note that \"[a] major cause of sub-optimal decisions is impulsivity\u2014the choice of an immediately gratifying option at the cost of long-term happiness.\" You may have been a victim of your own impulsivity in the past if you have ever overindulged in food or drink. In the world of legal employment, at times people make impulsive decisions, accepting a job that has some immediate appeal\u2014it was the first one offered or a parent's good friend is a partner at the firm.\n\nAnother paradox of decision-making is that people are drawn to make rational or rule-based decisions, even if they predict that another type of experience would make them happier. For example, thinking they should follow a \"rule\" of seeking variety in experiences, people may choose a menu item other than their favorite when in fact they would be happier to repeatedly select the old standby item that they like the most. People also tend to assume that financial aspects of decisions should matter more than experiential aspects. Hsee and Hastie offer this example: \"when asked to choose between a 50\u00a2 small chocolate that looks like a heart and a $2 large chocolate that looks like a cockroach, most respondents opted for the larger cockroach-shaped chocolate, even though when asked to predict which they would enjoy more, most favored the smaller, hearted-shaped chocolate.\" If you are just embarking on your career, you may assume that the primary goal of the job search is to obtain the highest paying or most prestigious position possible. Instead, the evidence suggests that you should look for jobs that would give you the most experiential satisfaction or that would use your signature strengths.\n\nHow do you avoid the cognitive pitfalls? What if you think you would like a particular kind of legal job, but you just aren't sure? Or you were sure, at least until you read the last few paragraphs. We turn next to one of the best methods to steer clear of the shortcomings of our imaginations.\n\n#### INVESTIGATING POSSIBLE FUTURES\n\n_Before we set our hearts too much upon anything, let us first examine how happy those are who already possess it._\n\nThe best way to estimate your future happiness in a job is to ask people who currently hold that job whether they like their work\u2014look at surrogates for your future self. This is especially important for those of you just embarking on your law careers. Bill Davis, formerly a partner with a mid-sized law firm in North Carolina and then in-house counsel, suggests: \"look for 'success' stories in the environment you are considering and picture yourself relative to them, their skill sets and their personalities.\"\n\nYou need to get a picture of what, physically, you will be doing every day when you go in to the office. The actual day to day tasks and interactions can be as important as practicing in a particular substantive area. When you get up in the morning, you want to look forward to going to work. Every job has multiple dimensions, but two are key to happiness: what you will be doing on a daily basis, and with whom you will be doing it. \"Some days, one is more important than the other; but you need to have both,\" says Joan Heminway, a Tennessee law professor, who used to work in corporate mergers and acquisitions at Skadden Arps.\n\nPeople who practice the type of law that you think might be of interest can help you envision a day at the office. If you want more accurate information about whether you would like to try a capital case or write a patent application or conduct discovery in a behemoth class action, find people who are presently doing those things and ask them what they spend their days and night doing and particularly about the peaks and valleys in their work. One way to do this is to ask a lawyer in a particular specialty area for an \"informational interview.\" Don't just ask people whether they like their jobs\u2014most people say that they do. Ask them what the most satisfying parts of their jobs are, what aspects cause them the greatest stress or anxiety, what training they found most useful for their practice area, and whether they think their experiences are typical for that type of work.\n\nThe personal interview is not the only way to glean information about different careers from sources. Take advantage of public lectures. Approach professors who teach in the substantive areas that interest you. Most law schools also provide clinical opportunities and internships. According to a study done by Affiliates legal staffing service, \"90 percent of lawyers surveyed said internship experience was a key factor in selecting candidates for full-time positions.\" Internships or externships provide opportunities to assess your own reactions to certain types of practices. Summer associate experiences are also valuable, but courted and coddled \"summers\" would be well-advised to examine the lives of the junior associates they will become if they join the firm.\n\nThere is another source of potentially important information that could guide your job choice. After you get an offer and you are doing a final check on one or two firms, try to find out what has happened to associates from that firm from four or five years ago. Use Martindale-Hubbell (the books, the ones in the basement of the law library, not the up-to-date online version) to find the entering class of a firm from five years ago, compare that to the current associates list, see who left, and track several of them down to see why they moved to a new firm or position. By expanding your sample to include those who left and not just those who stayed, you can get a more accurate picture of the costs and benefits of your possible future job.\n\n#### INTERVIEWING FOR A MORE SATISFYING WORKPLACE\n\nMost career services offices offer a standard battery of books and handouts containing suggested interview guidelines. Most of these lists of sample questions are targeted toward impressing interviewers. Very few address ways to assess what a law firm's values are or the quality of life in that firm. We offer below a different sort of list, one that is attentive to both favorably impressing interviewers and helping you to probe more deeply for the kinds of things that will lead you toward a satisfying career.\n\n#### INTERVIEW STRATEGIES\n\nPresumably, before you have your first interview, you will have done your homework. You will know whether the firm's representative client list is big tobacco and axe murderers, or if it consists mainly of innocent victims of toxic mold and Bernie Madoff. You will know whether a senior partner has just been indicted or has just won the Congressional Medal of Honor. As you interview for a legal job, collect information about whether this is a place that fits with your values and interests, will appreciate your strengths, and will offer a satisfying work environment.\n\nThink about interviewing as roughly a two-stage process, corresponding in many instances to an initial round of oncampus interviews and a later in-house callback. An opening conversation with a firm or organization may not be the most opportune time to probe about the firm's work-life balance policy. During the initial interview you are trying to present yourself to law firms, rather than to assess all the benefits the firm can bestow upon you.\n\n##### _Initial Interviews_\n\nFor openers, most legal employers will typically ask a fairly standard series of questions, such as: Why do you want to practice law? Have you enjoyed law school and did you have a favorite course? What qualities do you have that you think will make you a good lawyer? Tell us about an obstacle you have overcome in life? Where do you see yourself in ten years? In these early conversations you can ask questions that reflect your interest in the firm's structure and your eagerness to work hard. Use your questions to glean information about the control the firm gives to associates, the feedback and training offered, and the interpersonal relations. May associates specialize in particular practice areas? Does the firm have any formal training, mentoring or business development programs for associates (e.g., are junior lawyers included in client pitches)? How does the firm evaluate associates' work, and are there any formal feedback systems? You should also ask what is the firm's governing structure (e.g., a single managing committee or several committee's on different issues), and what is the firm's overall growth plan? What is the typical length of the partnership track, and are attorneys expected to have an independent book of business for partnership? For litigation (or transactional work), what kinds of cases (or deals) would an associate be able to work on or exclusively handle?\n\nDepending on the available time and how the interview is going, you may also be able to ask additional questions about the kinds of people, qualities, and activities to which the firm attaches importance. Who are the heroes and heroines of the firm? Do you have a story about the way your firm has handled a case or some aspect of a case that is representative of your firm's values? You can learn a lot about whether a firm's values align with your own by hearing the victory stories (and war stories and horror stories) of a firm. The anecdotes people choose to tell you convey their own stories.\n\n##### _Later Interviews_\n\nDuring a second-round or in-house interview, you may be able to ask more probing questions designed to reveal how a firm or agency will match up with your happiness agenda. Below are some questions that you might want to ask of prospective law firms.\n\nThese questions do come with a warning label (forgive us, disclaimers are an occupational hazard): these aren't questions to ask as you would in a deposition or you will probably self-destruct during the interview. They are certainly not the only, probably not even the primary, questions you should ask during that second round. And they should be sprinkled in gently, over time, in between other questions, or you might want to figure the answers to them out for yourself without asking the questions directly.\n\n1. What are the firm's billable hour expectations? Do attorneys receive bonuses (or compensatory time off) if they exceed their targets or expectations?\n\n2. What kinds of responsibilities and decision latitudes are given to associates on cases? How about on personal matters? (e.g., Will I be able to arrange my own office, hang my own paintings, or bring in my own lamp?)\n\n3. Are attorneys able (or even encouraged) to set their own schedules? Are reduced hours, telecommuting, and flexible full-time schedules options for all lawyers (or only upon certain events, such as pregnancy or illness, or only after a certain number of years of service)? What percent of all associates and all partners opt for a reduced hours schedule? Are associates or partners who opt for reduced hours or flextime evaluated in the same manner as those working a traditional or set amount of hours? To what extent does the firm outfit home offices or provide technology reimbursement?\n\n4. How much vacation leave do attorneys get, but perhaps more importantly, do people really take it, do they feel pressured not to use it all, and are they tethered by e-mail and BlackBerrys during vacations?\n\n5. Is this a firm that plays to individual lawyers' strengths and interests? If lawyers want to pursue a new practice area interest as an associate or in midcareer, are there examples of the firm supporting lawyers to move in new directions?\n\n6. What amount of subsidized family (maternity, paternity, partner) leave does the firm provide? What percent of lawyers take the leave that is offered? Can lawyers negotiate unpaid sabbaticals apart from family leave?\n\n7. What type of personality does best in this firm? Are you looking for people who are driven or more laid back? How are conflicts (regarding cases or salaries or personnel) resolved?\n\n8. What efforts is the firm making to promote females and racial minorities into the top leadership positions at the firm? Does the firm have a director of diversity or a director of professional development? Do women\/racial minorities\/sexual minorities head practice groups? Sit on the firm's managing committee?\n\n9. What are the firm's turnover rates\u2014how many associates (and partners) leave each year?\n\n10. Apart from events to woo summer associates, in what kinds of events do members of the firm participate to play together (e.g., poker nights, sports, or a new parents group)? Do the lawyers in the firm go out to lunch together or get together regularly outside of work? What sorts of community involvement does the firm encourage and support?\n\nAfter the interviews are over, there is a question you need to ask yourself: do you think you will really like the people with whom you will be working?\n\n### FINAL THOUGHTS\n\nLet's end our conversation about your future on a fun and cheery note. Let's imagine your funeral.\n\nProfessor Larry Krieger, who is at the forefront of the humanizing legal education movement, offers a thought-provoking exercise: write your own eulogy.\n\nI ask participants to imagine a future time when they are retired and away from their current environment, perhaps traveling in a pleasant place. I have them imagine visiting a small, quiet gathering which then turns out to be a preview of their funeral. I then ask them to briefly write down the eulogies about themselves that, if they could attend their own funeral, they would like to hear from important others\u2014their life partner or best friend, a respected lawyer or judge that has known them in practice, a member of another community they valued during their life (church, neighborhood, service club, etc.), and if time allows, their child or another young person they had known. They may also be asked to write down the things that they would most like to be able to say about themselves\u2014the things about which they feel best when looking back on their life.\n\nThe results of this exercise are illuminating, because they show students and lawyers the kinds of things that matter most deeply to them. Participants are often surprised by the results. Almost invariably the qualities and values expressed in these eulogies are the most traditional human values and virtues: patience, decency, fairness, humility, courage, caring, integrity, willingness to work hard for worthwhile goals, helpfulness to others (family, friends, clients or community), and so forth. No one thus far in my experience has drafted a eulogy focused on a luxurious home, high grade point average, law review membership, or extraordinary income.\n\nWhile this exercise might seem to invite some of the predictive flaws of misimagining the future, it asks you to think now about how you want other people to think of you later. The point of it isn't to imagine an accurate future but to envision a future shaped by a life lived according to your personal values and intrinsic goals. It also brings home the point that these intrinsic virtuous qualities are unlimited and noncompetitive: \"In law school parlance, everyone can be in the Top Five Percent in kindness, patience, etc.\" If your focus in law school is on your own goals, and not those defined for you by the well-worn default paths, you are more likely to emerge from law school ready to embark on a career that will satisfy you.\n\n## **CHAPTER 6\n\nWhat Law Firms Can Do to Make Lawyers Happier**\n\n_We're a law firm. Time is billable. The client doesn't pay for small talk. Every minute you spend away from your desk is a minute the firm isn't making any money off your presence, even though you're still using the office supplies, eating the muffins, drinking the coffee, consuming the oxygen, and adding to the wear and tear on the carpets. You're overhead. And if you're not earning your keep, you shouldn't be here._\n\nJEREMY BLACHMAN'S SATIRICAL EXPOSE ON ELITE LAW firm life, _Anonymous Lawyer_ , a blog that became a book, is not the only bleak depiction of law firm life. William Keates kept a diary of his first year at a prestigious firm and published _Proceed With Caution_. Numerous other blogs, such as http:\/\/abovethelaw.com, catalog gossip, dubious achievements and horrors of law firm recruitment techniques, billable hour demands, associate abuse, office politics, and layoffs.\n\nIs it really that bad? Maybe not, but there is evidence that life, especially in larger firms, has taken a turn for the worse. The 2009 recession prompted salary freezes or cuts, delayed start dates for new associates and led to layoffs of both associates and partners. Beyond recession-induced turbulence, focus on the bottom line increasingly has made the comforts and relative stability of big firm life a thing of the past. From experienced attorneys the refrain is heard so often as to seem a mantra: \"Law is becoming just another business.\" Implicit in the complaint is the suggestion that law should be more than \"a business\"\u2014it should be a profession that places a high value on quality and service to the community. Older attorneys bemoan the emphasis today on meeting ever-higher billable hour requirements, the foregoing of face-to-face contact in favor of more efficient electronic communication, and the compromises in work product quality that are made to placate client cost concerns. They remember with fondness the clubbish colleagiality of their firm in days gone by, and they lament the loss of civility in their profession.\n\nThe good news is that, in response to these disturbing trends, some firms, corporations, and government agencies are reconsidering their personnel practices. A number of firms are implementing policies that allow their lawyers to find work-life balance or to reconnect with colleagues. If you are a partner at a law firm, the supervisor of in-house counsel, or the manager of a government or public interest organization (for shorthand reference, we will talk in terms of \"law firms\" for the most part), this chapter is for you. It addresses what you can do to make your firm a happier place and why you should do it.\n\n### FOUR REASONS LAW FIRMS SHOULD CARE ABOUT MAKING THEIR LAWYERS HAPPY\n\n#### **R EASON NUMBER 1:** HAPPY FIRMS KEEP GOOD LAWYERS\n\nThe primary resources of your law firm are your lawyers. If your firm is perceived as performing work of exceptional quality, you get the cases, the clients, and the fees. Your top talent can be lost if the lawyers are unhappy. The National Association of Law Placement conducted a survey of associate attrition at large firms in 2007 and found that within three years 37 percent of associates leave their firms, a figure that skyrockets to 77 percent within five years. Of course, there are a thousand reasons why a lawyer might leave a firm, but one of the most common is that the grass seems greener elsewhere\u2014in fact, the grass in the lawyer's former firm might have withered in the associate's eyes.\n\n#### **R EASON NUMBER 2:** KEEPING LAWYERS HAPPY SAVES MONEY\n\nEstimates of the amount it costs to replace a departing associate range from $200,000 to $500,000\u2014roughly one and one-half to two times the annual salary of that lawyer. If your firm loses five lawyers in a year, that is a one to two-and-a-half million dollar loss. The explicit costs of attorney turnover may include: severance pay and accrued vacation benefits, job advertisement for the now vacant position, recruiter's time and salary, interviewing expenses (including travel, meals, and hotel), reference checks, expenses for relocation, orientation and training of the new employee, and, possibly, signing bonuses. The behind-the-scenes costs\u2014such as lost productivity time, covering the work of the departing lawyer, client notification and perhaps mollification, as well as a morale drop among lawyers who remain\u2014can dwarf the explicit expenses.\n\n#### **R EASON NUMBER 3: HAPPY LAWYERS ARE** MORE PRODUCTIVE\n\nEvidence is mounting that happier workers are more productive and that happiness enhances performance. Happier people (who are more likely to have jobs in the first place) get better work evaluations and earn higher pay. They have less absenteeism and lower medical costs; they set higher goals and persist longer at cognitive tasks. In occupations outside of law, studies have shown that happiness is tied to numerous measures of better performance: people in a good mood are better negotiators than those in bad moods; they will make better managerial decisions, be more creative, and compliment the performance of those around them. It is reasonable to assume that positive expectations and happiness present the same correlations in law.\n\nThis is not to say your goal should be to keep lawyers in a constant state of euphoria. While happiness promotes creativity, it undermines precise focus because happier people are open to a variety of different kinds of information and lack the attention filter that negative emotions supply. Euphoria and detail work, the latter of which can be considerable in law firms, do not go hand in hand.\n\n#### **R EASON NUMBER 4: HAPPINESS IS ITS OWN REWARD**\n\nIn addition to saving valuable human resources and increasing productivity, creating a happier law firm will make you and the other lawyers in your firm happier. That, of course, is a reward in itself. As happiness begins to develop in a firm, it will spread of its own accord. You can \"catch\" happiness. If the lawyer next door is happy, if your secretary is happy, there's a good chance that their happiness will improve your mood as well.\n\nWe are social creatures, and our affective state is influenced by the emotions of people around us. People tend to spontaneously mimic the expressions and behaviors of other group members and to absorb, match, and mirror emotions of others through largely unconscious feedback mechanisms. Studies in organizational behavior show that group emotions\u2014such as morale, rapport, and cohesion\u2014and mood contagion in a workplace can be affected by words, nonverbal cues, policies, goals, and priorities. Negative moods also transfer between people and spread through a group. If you spend a significant amount of time around Debby Downer at your office, you may catch her negative mood. Disgruntled lawyers can spread their unhappiness like cold germs. The ability to infect other people works with cheerful as well as negative emotions. In short, both happiness and unhappiness are contagious.\n\n### OPTIMIZING, NOT MAXIMIZING, LAW FIRM HAPPINESS\n\nYour goal is not to maximize happiness but to optimize it. Ringing the party bell at 4:00 PM every day might boost firm happiness, but there are obvious costs to that as well. Your firm, after all, has clients to serve and income to earn.\n\nIf you want to just have a happy law firm, we'd suggest you hire a bunch of people with high happiness set points, typically optimistic extroverts. Hiring happy people, and only happy people, is the easiest way to improve firm happiness. But the serious business of law firms is better served by a mix of talented people with different happiness set points. Lawyers who have a tendency to be unhappy can still be excellent lawyers, better perhaps at worrying about details and negative outcomes. Remember Martin Seligman's observation about the positive dimensions of pessimism for lawyers. Happiness is a good thing, but it is possible to have too much of a good thing. For law firms today, however, too much happiness doesn't seem to be the problem. Too little happiness is.\n\n### ADDRESSING FIRM HAPPINESS: IT'S A WHOLE NEW GAME\n\nForces are driving law firms to revolutionize their traditional structures and ways of doing business. Push for change is coming from a variety of quarters\u2014not just from lawyers who would like time with their families but also from clients who prefer flat rate (per project) billing and who demand diversity in the law firms that do their work, from the American Bar Association (ABA) (which urged lawyers in 2002 to rethink the billable hour), and from students who will be the next generation of lawyers. Moving towards a happier law firm sometimes involves swimming with the current.\n\n#### UNDERSTANDING GENERATIONAL DIFFERENCES\n\n_Generation Y and women want a life balance.... The 50-year-olds don't understand that, but they're in a bind because there are not enough bodies to do the work. It's forcing law firms to think in different ways._\n\nJust when you were beginning to understand some of the generational differences of your firm's newest partners, the Gen X'ers, you realize that the next generation of your firm will be staffed by the Gen Y associates you recently hired. Seventy million strong, GenY'ers (or \"Millennials\") are the next wave of the American workforce. Lawyers from Generation X (born between 1965\u201379) and Generation Y (born between 1980\u20132000) often work in law firms where the senior partners are either Baby Boomers (born between 1946\u201364) or members of a Traditionalist generation (born before 1946).\n\nAs arbitrary as these labels might be for people born just a year apart (and odd that the \"generations\" vary from fourteen to twenty years in length), generational research can still offer some useful information. Obviously, these are group characteristics as well as sweeping generalizations. Some Gen Y'ers might have old-time values and some Baby Boomers have work attitudes indistinguishable from their younger co-workers.\n\nWork-life balance is more important to more contemporary generations than income. Ellen Freedman, a law practice management coordinator, says that Gen X'ers \"value flexibility in the work environment and expect significant recognition and rewards but are willing to take significantly reduced compensation in return for a more balanced lifestyle.\" Gen Y lawyers are sociable and want mentoring, teamwork, and feedback; they are also \" 'intolerant of drudgery.' \"\n\nGen X'ers and Gen Y'ers have much less firm loyalty or interest in a career commitment to a single firm. More than half (58 percent) of Traditionalists cited commitment to the company as most important, while less than a third (32 percent) of Gen Y'ers did so. For Traditionalists job changing carries a stigma; if you are a Boomer, you may see shifting jobs as creating a competitive disadvantage; for younger lawyers, changing jobs is necessary, expected, or routine. Remember that studies show that commitment is important to job satisfaction. If you are busy looking for your next job, it is hard to develop trust, commit to, and engage in your existing job. The reluctance of younger generations to commit to their present jobs might be an obstacle to happiness.\n\nDifferent generations come to the firm with different expectations. Boomers and Traditionalists are accustomed to face to face communications, including face time at the office, while younger generations who grew up with cell phones and computers are used to electronic communication and working from remote locations. Newer generations, raised by supportive, engaged parents who included them in decision making, expect transparency and the ability to participate in making decisions: \"They prefer egalitarian leadership, not hierarchies.\" Members of Gen Y have been treated as important and special, and they desire immediate and frequent positive feedback.\" 'Older workers think no news is good news. [Gen Y'ers] want pats on the back. Older people tend to be thankful for a job and younger workers really want some praise and recognition.' \" Members of this multitasking generation (known in some circles as \"the iGeneration\"), who text message at the speed of wifi and TiVo television shows, expect instant access to information and are accustomed to an unprecedented amount of control over their lives. While Boomers prize recognition in tangible forms, such as money or a corner office, Gen X'ers see freedom as the ultimate reward, while Gen Y'ers seek work that is meaningful to them.\n\nWhat does the current generational mix mean for sound law firm policies that promote overall happiness? It underscores the importance of recognizing that different things may make different cohorts more or less satisfied and appreciating that each generation brings helpful ideas to the table. The commitment espoused by Traditionalists is good for happiness and so is the feedback and mentoring promoted by Gen Y'ers. Gen X'ers want to work smarter rather than harder, and they tend to like flexibility. Gen Y'ers each have their individual playlists: they want to be judged based on their productivity and the quality of their work, not their seniority. As we discuss later in this chapter, some firms are moving to differential incentive systems, more family-friendly work arrangements, greater social interaction opportunities, and more explicit feedback systems. While there is no perfect recipe for law firm happiness, each generation offers promising ingredients.\n\nX'ers are looking for a team where they can make a meaningful contribution. Boomers did, and still do, want the same. X'ers want sufficient access to information. Doesn't everyone? X'ers want extrepreneurship, defined as room in the work to define problems, develop solutions at their own pace, and produce their own results. This is the very essence of a lawyer's work for her clients. X'ers want personal attention, mentoring, and loyalty. All successful lawyers feel the same.\n\n#### ADDRESSING CLIENT DEMANDS\n\nAcross the country, clients are calling for changes in traditional hourly billing methods\u2014the same thing that lawyers complain about from a different direction. Studies have demonstrated \"the potential conflict of interest between resolving disputes efficiently and billing clients by the hour.\" Financial incentives discourage efficiency and encourage padding of hours.\n\nIn some circles use of hourly billing may diminish due to pressures from clients. A number of large companies that outsource their legal work have begun to negotiate a fixed fee (one price for handling a series of matters) or a flat rate (a set cost per task) rather than billable-hour arrangements. Alternative fee arrangements still remain an alternative lifestyle for now: 83 percent of the general counsel responding to a 2008 Altman Weil survey said that, excluding discounted hourly rates, \"they used alternative fee arrangements less than 10 percent of the time.\" However, three-quarters of managing heads of Am Law 200 firms (the nation's two hundred highest grossing law firms) think that in the next decade \" 'many if not most' big firms will change their billing practices,\" and two-thirds said that some form of fixed-rate deals were being negotiated.\n\nSome of the country's largest corporations are also beginning to demand that the law firms to which they outsource their work not only hire appropriate numbers of female and minority attorneys but have those attorneys billing their time on the companies' work. In 2004, Rick Palmore, general counsel of Sara Lee, issued an open letter entitled A Call to Action: Diversity in the Legal Profession, urging corporations to insist that law firms performing work for them make a commitment to diversity. Since then, the commitment statement has acquired 136 corporate signatories, from such corporations as General Mills, General Motors, Hewlett Packard, Johnson & Johnson, Microsoft, Shell Oil, Starbucks, Target, and Walmart.\n\nThis corporate insistence on diversity has practical consequences for lawyers. To obtain the work of these corporate giants, law firms have responded by heeding the Call to Action and staffing their cases with lawyers who are women or racial minorities. Law firms are learning that diversity makes business sense. But does diversity lead to happiness?\n\n#### CONTEMPLATING WORKPLACE DIVERSITY\n\nFirms used to be good ol' boy networks, with no pressures to become more diverse along any dimension. Partners and associates were white, male, and upper middle class and played golf. Law firm demographics have changed enormously in the past several decades: law firms today are diverse in numerous ways. Although golf is still a constant, now it is as likely to be played with a Wii on a \"virtual golf course\" as it is at a country club.\n\nWhether a diverse law firm is, for that reason, a happy law firm, remains an open question. On the one hand, among the happiest nations in the world are those that are homogeneous, such as Scandinavian countries. The same can be said of homogenous neighborhoods, according to Robert Putnam, author of _Bowling Alone._ Putnam's study of diversity in communities found that ethnically diverse communities had substantially lower levels of \"neighborly trust,\" lower levels of voter participation, lower levels of confidence in government, a reduced rate of volunteerism, and \"less happiness and lower perceived quality of life\" than more homogeneous communities. At the level of personal interaction, considerable evidence suggests that people empathize more readily with others to whom they are similar in key ways. Much of this evidence (other country, community, and interpersonal relationship data) does not specifically apply to the workplace, and it is not clear what conclusions should be drawn for workplace happiness until more studies are conducted.\n\nOn the other hand, diversity unquestionably has educational and economic values, and differences enhance creativity. Lawyers who are members of Gen Y get satisfaction from being around others from different backgrounds. Social critic Malcolm Gladwell observes that \"in embracing the diversity of human beings, we will find a surer way to true happiness.\" Law firms are also seeing that attention to diversity\u2014training about micro-inequities, creating accommodations for attorneys having child or elder care responsibilities through part-time status or the ability to work from home, and developing \"parenting leave\" handbooks and mentoring programs\u2014improves satisfaction with working conditions and retention of women and racial minorities. The same individualized attention and flexible work policies are good for members of majority groups as well; firms that are family-friendly and sensitive to the individual needs of their lawyers tend to be happier places to work. And Putnam, despite the evidence he accumulated questioning whether diversity promotes happiness, ultimately concluded that diversity was a value worth pursuing. He noted that while diversity initially makes people uncomfortable, \"One great achievement of human civilization is our ability to draw more inclusive lines of social identity. The motto on the Great Seal of the United States (and on our dollar bill)\u2014e pluribus unum\u2014reflects precisely that objective\u2014namely, to create a novel 'one' out of a diverse 'many.' \" While the jury is still out on whether a more diverse law firm is a happier firm, we conclude that a more diverse firm is\u2014no small thing\u2014a better firm.\n\n### TEN STEPS THAT COULD MAKE YOUR LAW FIRM A HAPPIER WORKPLACE\n\nHappiness at a law firm has several aspects. The overall reputation of a firm\u2014is it a happy place to work or not\u2014is one dimension of employee satisfaction in the job domain. At the firm level, positive behavior, such as focusing on enhancing employees' attributes of loyalty, resilience or compassion (as opposed to criticism), correlates with employee commitment, retention, and productivity as well as various measures of corporate success. There is also \"facet-specific happiness,\" or attorney satisfaction with specific tasks while working at the firm, such as the degree of control over their own cases, clarity of directions, or feedback. Managers who are successful in raising attorney satisfaction will attend to generalized domain (or job) satisfaction as well as \"facet-specific\" happiness having to do with the attractiveness of different tasks. We will address some aspects of both of these dimensions throughout these steps.\n\n#### 1. PROMOTING LAWYER AUTONOMY\n\nStudies of attorney satisfaction show that if your job gives you significant autonomy, you are more likely to be happy. Conversely, satisfaction is lower if you are subjected to constraints on your decisions and work setting and circumstances. In 2001, the nonprofit research group Catalyst surveyed a sample of 1,400 lawyers who graduated from one of five elite law schools: University of California-Berkeley, Columbia, Harvard, Michigan, and Yale. Catalyst asked the participants to rank the top five sources of satisfaction for them in their jobs. Control over work was the leading source of satisfaction for men (68 percent) and the third highest source of satisfaction for women (57 percent). This result is consistent with other research on reported dimensions of job satisfaction.\n\nAn absence of control (low decision latitude), combined with high workload demands and high stakes outcomes, describes the work life of many associates. For firms that find ways to provide associates greater freedom to choose the cases they work on, or greater ability to control the cases they are assigned, the satisfaction, loyalty, and productivity payoffs could be immense. A step in the direction of greater lawyer autonomy, one that probably would not undermine other firm goals, would be to turn over smaller files to associates (combined with good mentoring of progress). If this were done at a significantly reduced hourly rate, more junior lawyers could gain valuable experience\u2014and clients would be more satisfied with the fees. Another approach might be to have younger lawyers, who are researching projects, explain their research results to the partner and then be the ones to brief the client. \"You learn how to practice law by practicing law\"\u2014and newer lawyers need meaningful responsibilities to acquire skills.\n\nAssociate happiness is also likely to be increased if you encourage development of each lawyer's specific strengths and skill sets. When you have a series of projects, rather than following the traditional route of individual partners tapping particular associates, send a memo to several associates with a brief squib about the projects and permit them to self-select work assignments. If you have a mentoring program, allow the junior lawyers who are being mentored to choose their mentors or express preferences for one of several possible mentors. This is essentially a no-cost proposition for firms, and it is a way of affording choices to newer attorneys. (If one partner is the perennial favorite and another is never selected as a mentor, perhaps that should indicate something about the appropriate compensation of those partners?)\n\nFinally, your firm can make its lawyers happier by letting employees make choices about their own work environments. Is it really necessary that every office have standard-issue furniture and the same antique white walls, or could employees be allowed to exercise some personal choice? Could that edict addressing the removal of personal pictures from office walls be rescinded? Could employees be given a greater say concerning what neighborhood in your sprawling four-stories of offices they are housed?\n\n#### 2. TAKING OFF THE BILLABLE HOURS STRAITJACKET\n\nWhen asked what they dislike most about the practice of law, many lawyers will answer \"billable hours\" or some variation on the theme: \"that my worth is tied to billing hours\"; \"unrealistic billable hour expectations\"; \"because of the hours I work, I have no outside life.\" In response to a 2005 National Association for Law Placement (NALP) Foundation survey asking attorneys the single most influential factor prompting them to change jobs, the factor checked by the largest percentage of respondents was \"reduction of work hours.\" Of the more than 2,300 respondents to an _ABA Journal_ survey, 84 percent said they would be willing to receive less income in exchange for a lower billable hours requirement. These are not just associates or younger attorneys. Almost half of the attorneys in management positions would gladly give up some portion of their salary in return for more family or free time. But, if lawyers have to ask for release time, flexibility, or alternative salary arrangements, rather than have the firm offer those as clearly available options, the lawyers who seek them may be seen as subpar or undedicated players. It will take structural changes to get around the perceptions that alternative time and salary arrangements are perversely different.\n\n#### 3. PROVIDING WORK-LIFE BALANCE\n\n_Nearly two-thirds of all lawyers report difficulty managing the demands of work and personal\/family life._\n\nLawyers want more work-life balance than they have now. The buzz-phrase is rampant; it's just that nobody seems to have much of the real thing. More than one-third of the male lawyers and almost one-half of female lawyers cited \"work-life balance as one of their top three reasons for selecting their current employer.\"\n\n##### _The Law Student Revolt_\n\nPressure for change is coming from law students at some of the nation's most elite schools. In the spring of 2007, a group of students at Stanford Law School and other law schools calling itself Law Students Building a Better Legal Profession (BBLP) sent letters to one hundred prestigious law firms urging them to reduce billable hours and implement work-life balance policies. The organization also produced rankings of firms based on how compatible firms' policies were with a balanced life for associates. BBLP urged firms to lower the number of billable hours expected per year, move toward flat rate or transactional billing, increase diversity, offer more time for pro bono work, and make work expectations transparent by publicizing the median hours that associates and partners work. Specifically, the group called for firms to implement \"balanced hours policies\" that would expressly approve of associates working \"80%, 70% or 60% of fulltime hours for proportional pay.\"\n\nProfessors Marc Galanter and William Henderson raise concerns about complications of staffing projects and question whether such a proposal is economically realistic: \"a lawyer who works 70% of the typical full-time load is unlikely to deliver a pro rata return on his or her time because fixed overhead for office space and support staff is not thereby proportionately reduced.\" Given this concern, compensation for less than full-time workers probably needs to reflect something less than the proportional hours worked\u2014say, 70 percent of full-time pay for working 80 percent of normal hours. Regardless of the specifics of the compensation scheme, firms need to make clear to the new generation of lawyers what work expectations are.\n\nCraig Segall, a Stanford Law 3L at the time of the group's founding, expressed the hope that law students at \"Tier 1\" law schools such as his own had \"the market power\" to make law firms more humane. Segall recognized the forces his group was up against, forces which have only been exacerbated by the recent recession. \"We are not so na\u00efve to think the situation will change over night,\" Segall wrote. \"Law students are hard to organize and this is a problem that's been developing for decades.\"\n\nOf course, not all law students and lawyers agree. In response to a _Wall Street Journal_ article about BBLP, some law students wrote of their willingness to work for long hours even if members of the new organization chose not to. One student, who indicated that he or she was \"dying to work hard,\" suggested to BBLP supporters: \"Can't take the heat? Stay out of the kitchen. Go work your 1600 hours in some other industry.\" Another unsympathetic lawyer wrote, \"Bottom line, they want the associational prestige of working at a brand name without having to put in the hours.\" Yet another lawyer suggested that the students consider working at smaller firms for \"sane hours\" but with less pay.\n\nThe student-promoted revolution may not be taking hold among associates\u2014particularly in tougher economic times. You are not likely to hear associates complaining as loudly as these law students about work-life balance issues, however unhappy they may be. As Segall noted, associates are poorly positioned to influence change: \"Associates have very little market power and, due to the lousy conditions they work in, many plan to leave after a few years.\"\n\nDoes this budding student revolt, centered at Tier 1 law schools, deserve a response from the nation's largest firms? Firms may conclude that the near constant availability of associates is critical to their clients, or that the economic efficiency that comes from assigning a single lawyer to work eighty hours instead of having two work forty hours each might seem irresistible. Firms could easily believe, and they might well be right, that their associational prestige will continue to attract a steady stream of topflight legal talent from law schools despite a generation's growing concern with work\u2013life balance. As one person commenting on the BBLP campaign saw it, the \"social capital\" of prestigious firms, with their \"cool buildings\" and \"cool addresses\" and \"cool art collections\" and especially high salaries, holds a huge appeal for graduates from national law schools.\n\nStanford Law School Dean Larry Kramer recently bemoaned the \"tendency of students from elite schools to go disproportionately to large firms\"\u2014a tendency which he said had grown \"more pronounced in recent years.\" Kramer worried that \"too many of our graduates have gone to these firms for the wrong reason\u2014and with predictable results, including boredom, frustration, and early lateral movement.\" The desire to \"add another prestigious line on a resume,\" says Kramer, proves irresistible to today's top students \"who have been taught from an early age to collect gold stars.\"\n\nEven if a steady stream of stellar recruits continue to come through their doors, we believe that BBLP raises issues that law firms should address. There are obvious economic and other efficiencies that derive from the current structure, but the willingness of new hires to work for less suggests that compromises can be made that won't seriously harm the bottom line. Whether the group can live up to the billing of its title is an open question, but the legal profession has never seen a movement quite like this one\u2014a grassroots organization, tapping the power of the labor market and using the capabilities of the internet to disseminate comparative data about law firms. More important, increasingly inhumane work environments have implications for how law is practiced, because harried, unhappy lawyers are undoubtedly more likely to be uncivil, cut corners, and engage in unethical practices. Moreover, weighing mightily for addressing some of BBLP's key concerns are all the advantages, laid out elsewhere in this book, that come to all members of a happy workplace.\n\nRichard Thaler and Cass Sunstein, in their important bestseller _Nudge: Improving Decisions About Health, Wealth, and Happiness_ , demonstrate how the architecture of choice shapes public and private decision-making. Their insights could be used by a firm seeking to maximize lawyer choice while moving towards a better work-life balance and overall level of happiness within a firm. Thaler and Sunstein note, for example, that people derive roughly twice as much unhappiness from losing a certain amount of money than they derive happiness from winning the same amount. A $50 loss causes the same value of negative feelings as a $100 win causes positive feelings. Most people, in other words, are \"loss averse.\" When people think about what they have lost, or what they didn't get that most people got, they are not happy campers. This would explain why most lawyers who are told \"You can give up 20 percent of your pay in return for working proportionally fewer hours\" might be unlikely to accept the offer, Instead, they will continue to grind away toward their unhappiness-producing 2,000 billable hours. However, if new hires are told, \"We expect you to put in 1600 billable hours for $X (an amount equal to, say, 20 percent less than what is paid those who produce 2,000), but you have the option of working 2,000 hours and receiving a _bonus_ (an amount that would bring pay up to the level currently paid for meeting the 2,000-hour expectation),\" a much higher number of recruits might opt for the reduced pay and reduced hours, thus improving the overall psychological health of the firm. Why? Because the reduced hours for reduced pay, having become the default option, no longer is associated in the new lawyer's mind with \"a loss.\"\n\nSusan C. Robinson, dean of career services at Stanford Law School, offers another idea for producing \"happy well-trained lawyers who not only stay in their firms, but in the profession.\" Robinson suggests that firms drastically cut starting salaries, say to the $80,000 range, and use the savings to create a formal apprenticeship period during which new lawyers would have reduced-hour expectations and gain the ability \"to cut their teeth on engaging legal work\"\u2014something they lost when clients began balking at paying \"high-priced billable hours for untrained associates.\" Several large law firms have already moved in the direction of apprenticeship-style programs. Howrey, a 600-plus lawyer firm, announced a pay cut for starting associates, coupled with training: \"from $160,000 to $100,000 in base pay plus a $25,000 bonus to pay down law school loans\u2014and they'll spend a good portion of their time attending classes with partners and shadowing them on client matters.\" The firm will bill out new associates at half of prior billing rates, and has low billable hour expectations for the associates: 700 hours the first year and 1,000 hours the second year. Although the program will cost Howrey in terms of lost billable hours and partner time for training, the firm expects costs savings in starting salaries and larger gains down the road in associate loyalty and efficiency, as well as client public relations. As Robinson sees it, an apprenticeship period will bring relief to the new associates \"burned out by the numbers of hours they are expected to clock and the tedium of the tasks they do.\"\n\n##### _Structural Changes in Firms_\n\nMany large firms are making structural changes that allow their lawyers to find better work-life balance. These range from providing greater parenting leave to allowing workers to adopt flex-time schedules. _Working Mother_ magazine and the national consulting group Flex-Time Lawyers compiled its second annual list of the fifty best law firms for women. Top ranking firms offer flexible work schedules and generous parenting leave. Among these top ranking firms, the average paid maternity leave is fourteen weeks; the average paid paternity leave is almost six weeks. Larger firms tend to have written policies, while smaller firms tend toward more variation, allowing leave that builds on sick leave and accrued vacation time. Of course, the more some firms extend maternity, paternity, and parenting leave, the more that other firms will need to do so to remain competitive.\n\nVarious types of flexible arrangements have been tested in law firms. These approaches include flex time (no standard working hours, and individual lawyers arrange their own schedules); a compressed work week (say, four ten-hour days instead of five eight-hour days); staggering work hours (for instance, working five nine-hour days one week, and four the next week), telecommuting with computer access to the law firm's files; part-time work (simply a reduced number of hours expected); and jobsharing. Some firms employ new position categories (such as \"of counsel\" or \"permanent associate\"), while others adopt the target hours approach. Under the target hours method, firms publish the average billable and nonbillable hours worked by lawyers in different practice areas, according to years in practice and status level. Each attorney and his or her supervisor agree on a target number of hours for the upcoming year based on the firm's needs and the attorney's individual circumstances. Deborah Epstein Henry, president of Flex-Time Lawyers, a national work-life balance consulting firm, points out that \"full-time\" and \"part-time\" are concepts that tend to stigmatize and do not accurately reflect work done in different areas of specialization: \"For associates at the same firm in the same year, a 'full-time' mergers and acquisitions lawyer may bill 2,300 hours and a 'full-time' trusts and estates lawyer may bill 1,800 hours, while a 'part-time' litigator may bill 1,800 hours and a 'part-time' family lawyer may bill 1,500 hours.\"\n\nOf Canadian lawyers who have used some type of flexible work arrangement, telecommuting (which eliminates commuting time and costs) was the most popular, with 77 percent of lawyers using it; flextime was chosen by 31 percent of lawyers, while a reduction in hours was used by 30 percent of lawyers. Telecommuting can boost happiness of telecommuters because it allows them to gain a huge amount of control over when and where their work is performed. However, because relationships matter, firms might not be creating happier workplaces by taking workers (except the real sourpusses) out of the office. We will see this tension repeated with other dimensions of workplace happiness: some solutions that may give a satisfaction boost in one direction undermine it in another way.\n\n##### _Underutilization of Existing Policies\u2014A Public Relations Problem?_\n\nAlthough 98 percent of the 1,500 mostly larger law offices responding to a NALP survey in 2008 said that they permit part-time work schedules, only 5.6 percent of all lawyers work part-time. Most of them are women. One feature limiting usage nationally is that more than one-half of firms offering part-time schedules precluded entry-level associates from using that option. Of those employees who are eligible, many may not be aware of part-time alternatives, and many more may be hesitant to pursue that career path. In one study, more than nine out often managing partners reported that their firms permitted attorneys to work less than full time, yet only six out often associates either knew of the policy or thought it was a realistic option.\n\nGender stereotypes of various types can keep participation in flexible work arrangements low. Female attorneys may assume that they have to work harder than male attorneys to avoid being perceived as less capable or committed to their careers. Male attorneys may even be told that a \"real man does not take parental leave.\" Even if your firm supports attorney freedom or family commitments, lawyers perceive that using parenting leave time or flexible work arrangements will undermine their status in firms and have a negative impact on their careers. A majority of lawyers responding to one survey said that they did not think a lawyer who opted for either a full- or part-time flexible work arrangement could ever make partner. Some research has shown that both men and women who use part-time or family-friendly policies in law firms are perceived as less competent than other full-time workers. If firms really want to promote work satisfaction, they need to make it possible for associates to work reduced schedules and still become partner.\n\nDespite the stigma and the problem of a part-time workload creeping into a full-time gig (while the attorney is still only paid for part-time work), lawyers who have these agreements are typically very satisfied with them. Consider the situation of Mary Jo Foster, then an associate at Streich Lang, who asked to work reduced hours when she adopted twin infants. She was expected to bill 80 percent of the usual billable hours target, and her compensation was reduced by 25 percent (the 5 percent difference went to overhead costs), but she kept full benefits. This arrangement eliminated most night and weekend work for her. She says, \"'Most people in the firm didn't even realize that I was working a reduced-hours schedule.' \" Attorneys with variable-hour arrangements who litigate confess that depositions, trials, and travel require juggling of their commitments but that they still prefer the freedom of time.\n\nOne of the best ways to help your attorneys create work-life balance is to set realistic billable hour targets or eliminate billable hour minimums and convey average expectations. Although most law firms in America do precisely the reverse, the ABA's Law Practice Management Section suggests limiting the rewards attached to billable hours exceeding your firm's expectations.\n\nAlthough the movement toward individual schedules is slow to take off, the demand for individualization is increasing, and new work-hour arrangements are emerging. Managers may worry that these changes could cause workers to make comparisons between their arrangements. However, comparisons in the workplace are inevitable. The focus of comparative judgments in the workplace today is based primarily on salary: full-time fourth year Associate X contrasts his salary to that of full-time fourth year Associate Y. With a more flexible approach, these comparisons will include both compensation and workload. Individuated hourly plans offer flexibility and respond well to employees' desires for control of their work flow. They also respond to the specific attention that newer generations of workers crave. Although individualization undoubtedly swallows a significant amount of management time, the benefits to law firms\u2014in attorney loyalty, firm morale, and economics\u2014can be enormous.\n\n##### _Flexibility and Law Firm Productivity_\n\nJames Sandman, the managing partner of Arnold & Porter, detailed the economic case for creating part-time programs. He observed that \"an accessible part-time program enables firms to compete both in the market for talent and for clients: recruits appreciate the option to attempt to balance work and personal needs and clients appreciate committed attorneys who are sometimes better able to focus on their cases because they are assigned fewer of them.\"\n\nAlthough limited data directly relates to law firms, experiences in other industries show that job flexibility reduces attrition, and there is no reason to suspect law would be different. One study of Washington, D.C., law firms showed that alternative scheduling arrangements were profitable for firms and helped them retain high performing attorneys. Michael Nannes of Dickstein, Shapiro, Morin & Oshinsky \"discovered a startling fact: 'We were losing lawyers not to other law firms, but to other schedules.' \"\n\nLaw firm managers tend to think in terms of comparing the productivity of lawyers working full time with those working reduced hours and conclude that a full-time worker would be more productive. However, if the analysis is based on \"productivity for the amount of time worked...lawyers working alternative work schedules are often more productive on a pro rata basis than their full-time colleagues. Because lawyers on alternative work schedules have a limited amount of time within which to complete their work, they are forced to be more organized and efficient.\" Studies confirm this. They show that attorneys who work part time are more efficient than their full-time counterparts. Anecdotal evidence indicates that attorneys who work part time typically work more than the agreed number of hours. One attorney who has worked in a job sharing arrangement for years comments, \"Hiring two attorneys to job share often leads to the productivity of 1.5 attorneys working full time.\"\n\n#### 4. ENCOURAGING A POSITIVE ATTITUDE\n\nAs a managing partner, you may use emotional contagion\u2014in the form of orientation lectures, pep talks, or training sessions\u2014to foster a \"can do\" atmosphere. Associates fresh from law school may need to hear this message early on, because many have been trained as problem finders rather than problem solvers. (It may also be in the associate's interest to spot the difficulties with architecting a deal or pursuing an avenue of discovery, because the associate will probably be the one doing the work, even while it is usually in the firm's interest to do the additional work.)\n\nAllen Rostron, now a law professor, tells a story about his first day as an associate at the prestigious New York firm of Cravath, Swaine & Moore. Much of the day was devoted to mundane orientation tasks, from filling out paperwork to being issued a coffee mug with the firm's name on it. The orientation also had a more serious side, as the new crop of nervous but eager associates gathered to hear several of the firm's partners talk about what the firm expected of its associates. The message was hardly a warm and fuzzy welcome. One of the partners, a very formal, somber lawyer from the corporate finance group, briefly welcomed everyone to the firm and then paused to look at each person around the table, as if to let anticipation build for a moment to underscore the importance of what he would say next. \"There is one thing that it's important for you to know from the very start,\" he finally continued. \"If anything goes wrong on any matter that you are working on, it is your fault. Nothing that happens is ever anybody else's fault. It's always your fault.\" \"For goodness sakes, it's my first day, I haven't even done anything yet, and it's already all my fault,\" Rostron remembers thinking. As the partner elaborated on what he meant by this, the reaction of his audience slowly but dramatically shifted. The partner explained that the firm expected and needed excellence from each of its attorneys, from the top down to the most junior associates. \"So when you get assigned to a deal, or a case,\" the partner said, \"you've got to realize that it's yours. You've got to have the attitude that you're responsible for it, and you're going to make sure everything works. You can't sit back and wait for somebody to give you a list of tasks to do. You've got to own these things, and tell yourself that you're going to be on top of it.\" That's why, the partner explained, you should feel like anything that goes wrong is your fault. \"You can't ever pass the buck,\" the partner said. \"If there's an error in the document, you can't say, 'well, it was the secretary's fault.' Because you should have given more precise instructions, double checked it, or done whatever was necessary to make sure it got done right.\" As the associates listened, they realized that the partner's message was not what it had seemed at first. It wasn't about getting yelled at and blamed or being afraid of getting in trouble. It wasn't about what a partner or anybody else was going to think about you or say to you. It was about how you were going to think and what you were going to say to yourself. Ultimately, and ironically, believing \"it's always my fault\" was a positive message about taking control.\n\nThe partner later offered another example of the \"can do\" attitude that the firm demanded of its lawyers, even the newly minted associates. \"Any time a client asks if the firm can do something,\" the partner said, \"the first word out of your mouth should always be 'yes.' Always. The answer is never 'no.' \" If the client calls one evening and asks if we can write a complex brief and file it the following morning, the answer should be \"yes.\" If the client asks if we can restructure the deal at the last minute, the answer should be \"yes.\" \"Now, of course,\" the partner went on, \"that 'yes' often will need to be followed by a lot more explanation. You may need to follow the 'yes' with a lot of serious warnings about the disadvantages and costs, the serious logistical difficulties that will have to be overcome, the alternative options, the corners that might have to be cut to file so quickly, the risks of making significant changes to what is already in place, and so on. But you always start with the 'yes.' \" Finally cracking a smile, the partner said, \"In fact, sometimes when a client asks if we can do something, you'll have to say 'yes\u2014absolutely, we can do that\u2014we can achieve your goal.... But of course we'll have to do it in a somewhat different manner, because what you just described is totally illegal.\" Again, the partner's point was that Cravath expected its associates to be problem solvers, not just problem spotters. \"There are a lot of lawyers out there who are good at telling clients what they can't do,\" a tax partner said later in the meeting. \"But that's not the sort of lawyer you want to be. You want to be the creative lawyer who figures out the way that it can be done.\"\n\nThe partners at Cravath spent time trying to shape a firm culture toward the positive. They made the points about the responsibility, control, and importance of each new associate\u2014qualities that the literature indicates are associated with happiness\u2014and emphasized that the new lawyers needed to be positive about achieving the goals the client wants. The danger of going too far in this direction is sometimes seen in business climates, where a \"can do\" attitude can become an Enron-like \"we can do whatever we want\" attitude, with excessive hubris and inappropriate attention to risks. But, as lawyers, we do not seem to have this dangerous unbounded optimism.\n\n#### 5. VALUING EMPLOYEES\n\nThe happiness literature does not dictate any single right way to structure law firm compensation systems. There is no one size-tickles-all model to recommend. Different systems will make various camps of people more or less satisfied. The one general statement than can be safely made is that lawyers are happiest when they think their work has been fairly valued.\n\n##### _Can't Buy Me Love: Partnership Structure, Firm Profits, and Satisfaction_\n\nHistorically, law firms had two categories of lawyers\u2014partners and associates\u2014with many larger firms using a seniority-based lock-step compensation system for their associates in which all associates ratchet ahead yearly with their class. Today your firm may have equity partners or shareholders, nonequity or income partners, associates, permanent associates, and a variety of \"of counsel\" positions, staff, and contract attorneys. Eighty percent of the country's two hundred largest firms have a two-tier partner structure. According to a study of Am Law 200 firms, whether your firm has a single-tier partnership structure or a modified partnership structure does not have a statistically significant effect on associate satisfaction. Associates' reactions to the structure of the partnership seem to depend more upon individual risk preferences and perceptions of how the structure might affect them: \"Some associates appreciate modified partnership structures' potential allowance for a stronger work-life balance. Others are angered by the additional structural barriers to election to equity partnership.\"\n\nA number of studies demonstrate an inverse correlation between law firm profits and the reported satisfaction levels of associates and partners. One study found negative correlations between both the profitability and prestige of law firms and lawyers' responses along various dimensions of satisfaction, such as family friendliness of the firm, desirability of working conditions, and firm openness regarding its finances. A study of University of Michigan law school graduates showed that the \"high-earning practitioners in large private practices report the lowest average job satisfaction.\"\n\nWhat accounts for this? One theory is that relentlessly profit-driven firms may not pay much attention to the personal satisfaction of their lawyers. Focus on the bottom line might lead to neglect of relationship building or other well-established sources of worker happiness. A second theory is that the cases that yield the highest levels of firm income provide less satisfying work than less lucrative cases. Whatever the explanation, an increase in firms' profits will not provide a trickle-down of happiness.\n\n##### _Associate Compensation_\n\nFirms today typically compensate lawyers in one of two basic ways: lockstep (a ladder system with a base salary for each rung of the ladder, tweaked according to various performance measures) or some variation of a merit system, usually with an \"eat what you kill\" component (rewards tied in some way to the business the individual lawyer brings in). Law firms often employ a bonus system, in which year-end bonuses are handed out to star performers (winners) without explicitly identifying less able performers (losers). If your law firm's primary objective is to make the _least_ number of associates _unhappy_ , you would do well to adopt lockstep compensation systems. However, the unhappiest associates under such a system might well be your hardest workers, and firms usually want to encourage productivity and reward excellent performance.\n\nCompensation systems, which may seem dissimilar on the surface, actually share common features. Almost all compensation models consider some measure of status within the firm. Even most lockstep systems are not really lockstep, because firms find ways to compensate exceptional work or use wage incentives to keep people from leaving: rainmakers and rock stars are rewarded. Once those negotiable components enter the system, the differences between systems tend to dissipate.\n\n##### _Partner Compensation_\n\nThe two primary factors in partner compensation are business origination and fees collected, although many firms have a formula or plan that also considers client and case responsibilities, contributions to management of the firm, pro bono work, and professional involvement enhancing the firm's reputation. Some formulas try to encompass ineffable qualities such as collegiality and cooperativeness. Other formulas incorporate measures of client satisfaction, skills, and contributions to the firm beyond simply the book of business and revenue production. Firms typically reserve some revenue for a bonus pool that can be divided pro rata or used to reward exceptional performance.\n\nThe recession has prompted some reconsideration of partner pay strategies. Because associates cost firms money, many of the larger firms are decreasing their associate numbers. Partners are hoarding work, and compensation compression is occurring, with incoming partners' compensation approaching that of more senior partners. These factors may reintroduce rigidity, bean (and hour) counting, and competitiveness in partner relations, all of which steer away from job satisfaction. Compression will probably prompt people to engage in more salary comparisons. A central finding of happiness research is that people care about their standing or status in a social group\u2014and law firms are no exception\u2014so relative income matters much more than absolute income to life satisfaction. Judge Patrick Schiltz tells the story of one lawyer who earned more than one million dollars a year who \"came within a whisker of quitting his firm\" when he learned that an income peer would received $10,000 more than he did as part of a year-end bonus.\n\n##### _Do Some Compensation Systems Make Lawyers Happier Than Other Systems?_\n\nNo study definitively proves one compensation system is superior to others in increasing lawyer satisfaction levels. Every compensation system will leave some lawyers in a firm unhappy. If relationships are key to life satisfaction, however, formulas that capture important contributions to firm life are more likely to promote a happier workplace. The ideal compensation system measures in some way an employee's overall contribution to the firm's well-being, satisfies the firm's key workers, is generally perceived to be fair, and is clearly explained up front to new members of the organization. The rub lies in the difficulty of measuring an employee's contribution to a firm's overall well-being. Based on happiness research, we know these contributions can come in many forms, including by supporting other employees, treating co-workers fairly, facilitating a transfer of more control to co-workers, and especially promoting trust within the office. Noted British economist John Helliwell's research indicates that \"among employed people, trust in management predicts life satisfaction more reliably than health, marital status, or income level.\"\n\n##### _Retention_\n\nFirms make a mistake when they place too much reliance on compensation to keep lawyers on board. Firms think that salaries matter so much to people that they can buy off a little unhappiness on the job by paying attorneys more. It works for a while. People overestimate how much happiness money will bring them. Over a relatively short time though, the happiness boost dissipates. \"[T]he effect of an income increase after four years is only about 42% of the effect after one year: the majority of the short-term effect of income vanishes over time.\" The hedonic treadmill begins to roll\u2014as one earns a greater salary, expectations rise and people adapt to the higher income level. Both associates and partners clamor for something more than money: they want a better lifestyle, more communications from their firm, autonomy, fairness, trust, and to feel valued.\n\nRaising salaries is not the best retention lure. In exit interviews, associates mention numerous reasons for leaving firms. Although partners often speculate that money is a primary reason, that is not borne out by surveys of associates. The most common three reasons associates cite for leaving firms are 1) an absence of mentoring, training and development, whether formal or informal; 2) a lack of or problematic relationships with partners; and 3) workload demands and a need for better work-life balance. Similarly, in a survey of more than one thousand lateral partners sponsored by _American Lawyer_ , respondents said that anticipated compensation with their new firms was one of the _least_ significant factors in their move. The most important reasons were their new firms' commitment to support or help expand their particular practice and the firms' ability to integrate them into the partnership. Understanding that money is not the answer to their retention problems, some larger firms are investing in programs that are intended to \"re-recruit\" their mid-level associates. In Washington, D.C., Howrey offers five academies for associate development each year. The firm has also created a mentorship program that gives each associate an individual partner-mentor and trains the partners in leadership and coaching. While rapport that evolves naturally might be more genuine, the firm is paying attention to key sources of satisfaction.\n\nThe main point here for managers is that, beyond certain base levels, relative income matters more to workers than actual income. A compensation system that promotes overall happiness should be perceived to be fair by as many of the valuable workers as possible. One way to create the fairness, even though it exposes the relative salary differentials, is by constructing a transparent compensation system.\n\n##### _Transparency_\n\nPeople's satisfaction with a job will be lower if they expect better rewards than they actually receive. If expectations are key in determining satisfaction with compensation, keeping a close match between expectations and results minimizes feelings of dissatisfaction.\n\nConsider the experiences of one large firm that revised its compensation and professional development practices. In 2001, the firm moved from a traditional lockstep system of pay and promotion to a level system of specifying competencies in seventeen areas, evaluating associates twice a year against these expectations, and adjusting compensation and associates' billing rates in accordance with those assessments. Compensation, including bonuses, became individualized and tied to growth in competency areas. The partner in charge of recruiting and career development for the firm says one of the most difficult implementation features was getting his partners to ignore time served: \"It is stunningly difficult for lawyers to let go of that because it is so deeply ingrained.\" Another feature of the plan is an upward evaluation of partners by associates in categories of supervision, availability, respect for associate workload, offering a broad perspective on cases, providing adequate lead time to complete projects, openness to questions, constructive feedback, and career guidance. The competency-based compensation system thus does something that firms increasingly see as important: it assesses the mentorship performance of partners and rewards partners who are good mentors and managers. Although attributing causation is impossible, the firm has noticed a marked drop in associate attrition under the competencies system. In 2001, the associate attrition rate was 26 percent (32 percent for female attorneys); by 2006, the associate attrition rate was 11 percent (10 percent for female attorneys).\n\nTransparency about the criteria used to evaluate performance and clear expectations seem to be two of the most important characteristics of a fair compensation plan. There is more than one good way to run a law firm. The important thing is to have a compensation scheme that is transparent so that people who are coming to the firm are those who would like whatever system the firm uses. Transparency lets people who have different objectives make a reasoned decision whether the firm fits their goals.\n\n#### 6. PROMOTING SOCIAL INTERACTION\n\n##### _Avoiding the Perils of Bigness_\n\n_The prevailing model of the large law firm is so shaky at this point that one of the things firms are going to have to do is rethink their basic organizational model and how they use their human resources.... We're in a moment where the large law firm model is under tremendous pressure and we're going to see whether people who organize the new models will take this into account and be more concerned about getting the maximum productivity and creativity out of their employees_.\n\nSomething important is lost as organizations grow past a certain size. In his book _The Tipping Point_ , Malcolm Gladwell makes the case that firms of over about 150 members lack the cohesiveness and collegiality to be a satisfying workplace\u2014and generally become significantly less efficient at accomplishing their goals.\n\nDrawing on the research of British anthropologist Robin Dunbar and others, Gladwell argues that once firms exceed about 150 members people cannot know their co-workers well enough that \"what they think of you matters\" and work becomes \"a very different kind of experience.\" Dunbar's research led him to conclude that the \" 'figure of 150 seems to represent the maximum number of individuals with whom we can have a genuinely social relationship, the kind of relationship that goes with knowing who they are and how they relate to us.' \" Up to a size of 150 or so, an organization can attain its goals informally, whereas larger firms are forced to rely on rules, regulations, and imposed hierarchies \"to command loyalty and cohesion.\" Firms of fewer than 150 members can have a sense of unity that comes from everyone knowing who to turn to for help in problem-solving. Each member knows what every other member brings to the table in terms of specialized knowledge\u2014and they can all feel comfortable approaching each other for assistance. Gladwell compares the well-run firm of fewer than 150 members to the family in its \"re-creation, on an organization-wide level, of the kind of intimacy and trust that exists in a family.\"\n\nWhat \"The Rule of 150\" suggests for law firms is that they should carefully weigh the social costs of expanding beyond that size. Most firms are highly unlikely to remake themselves into smaller firms, but Gladwell's point is to give consideration to other options once a firm approaches the 150 lawyer size: cap the firm size at 150, divide the firm into separate operations with distinct practice areas and distinct clients, or establish a satellite or branch office in a nearby city. While each of these options might, in certain cases, seem unrealistic, Gladwell suggests there will be significant costs to not taking one or the other step. Most large firms divide into separate practice groups for substantive areas. Being in a practice group helps, but doesn't really give you the _Cheers_ benefit of working in a place where everybody knows your name. Gladwell notes that strict adherence to the Rule of 150 has been the key to the much-admired success of Gore Associates, the firm that manufactures a wide variety of articles ranging from Gore-Tex fabric to Glide dental floss to hospital equipment. He quotes founder Wilbert Gore who reports, \"We found again and again that things get clumsy at a hundred and fifty.\"\n\nHaving warned that the road to happiness might be harder at a behemoth firm, we should observe that not all lawyers at giant firms are unhappy. Many enjoy the type of work most likely found at larger firms: the intellectual challenge of particular kinds of cases\u2014such as corporate restructurings, class actions, complex commercial litigation, or transnational work\u2014as well as the benefits of a variety of practice areas, the economic strengths of diversified practice areas, the competitive advantages of size and resources (cutting edge technology and the ability to deploy attorneys into public service projects), the caliber of clients, and the prestige of the firm.\n\nLarge firms themselves are changing form. One of the net effects of the recession of 2009 may be some dismantling\u2014or rearchitecting\u2014of giant firms. Douglas McCollam, a critic of large firm practice, offered this stark assessment:\n\nAt bottom, what's in question is the whole economic edifice of the modern American law firm. Like the pharaohs of old, big firms are enamored of constructing pyramids with an ever-widening base of associates and nonequity partners toiling on behalf of a narrowing band of equity partners at the top. Increasing a firm's \"leverage\"\u2014as expressed through the billable hour, one of the most pernicious creations in the annals of commerce\u2014has been the key metric driving profitability at big law firms over the last generation. Numerous studies have documented the deleterious impact this model has had upon the legal profession and clients. To date, nothing has been able to kill it. It would be ironic indeed if the economic downturn that has cost lawyers so much ended up being the very thing that saved the legal profession from its own excess.\n\n##### _Mentoring_\n\nHeading virtually any survey of associate dissatisfactions is a lack of mentoring. In one study, 43 percent of associates indicated that their supervising partners did not provide the mentorship the associates needed and wanted. One of the largest national studies on associate attrition found that the availability or lack of mentoring and feedback frequently affected associates' decisions to stay with their firms or to leave them. In a study of University Virginia law graduates, the aspect of the job setting with which men and women were the least satisfied was performance evaluation\u2014and that could encompass the absence of evaluation, the quality of the evaluation that was given, or that the evaluation itself was negative. People appreciate feedback that is phrased positively and gives a clear sense of expectations.\n\nMentoring can create profound social connections\u2014and in numerous studies of helping behaviors, both benefactors and recipients show elevated levels of happiness from acts of kindness, counseling, and other supportive activities. Professor Sonja Lyubomirsky says that this is more than just a \"helper's high\"; assisting other people \"can satisfy a basic human need for connecting with others, winning you smiles, thankfulness, and valued friendship.\"\n\nThose law firms rated among the nation's best are heralded for the training they provide. For example, the survey resulting in the fifty Best Law Firms for Women list found that \"68 percent offer mentoring for senior female associates and 62 percent offer management training for women attorneys.\"\n\n##### _Show Them the Love_\n\n_What people seek from others, more than anything else, is attention and appreciation._\n\nBeyond monetary valuation, in what ways can your law firm emphasize the importance of people as individuals? The various surveys of best companies and law firms\u2014from Catalyst, Vault, and _Working Mother_ \u2014show a number of common features in firms' treatment of their employees that lead to satisfaction. Favored firms promote their employees' dreams and encourage innovation and risk-taking; they value their employee's opinions, listen to their ideas, and implement their suggestions. They give feedback and promote the social network at work. It is unsurprising that employees who think their company values workers as individuals report greater job satisfaction and organizational commitment than those who don't. Similarly, employees whose values match those of their colleagues are also more satisfied and committed to stay. Successful law firms seem to have some sense of shared values or mission. Attorneys who are emotionally attached to their firms are the least likely to leave. On the other hand, \"[w]hen individual practices merely exist under the same roof, internal competition, hoarding of work, jealousy and suspicion develop.\"\n\nTo address a high level of associate attrition, Sullivan & Cromwell in 2006 held a training session for its partners on associate appreciation. This presentation encouraged partners to give associates feedback, to say \"thank you, and \"good job,\" and to [r]eturn associates' phone calls as quickly as you would a partner's or client's.\" The firm also arranged periodic associate lunches with the chairman of the firm and implemented a 360-degree review process\u2014to give associates feedback from subordinates and peers as well as supervisors. In 2007, the firm's attrition rate dropped from 30-plus percent to 22 percent.\n\nOf course, all of these activities cost money. We are not suggesting endless feedback or parades of perks, but a balance. One thing to weigh in that balance is the costs, both economic and human, of attorney attrition. Another concern to weigh in your balancing is that an unhappy attorney is unlikely to be a good ambassador for your firm.\n\n#### 7. GIVING THE PRO BONO BONUS\n\n_It is one of the most beautiful compensations of this life that no man can sincerely try to help another without helping himself._\n\n\u2014Ralph Waldo Emerson\n\nThe opportunity to do pro bono work is tied to greater job satisfaction for lawyers. Happily, the concept that people who have a sense of civic engagement are more satisfied dovetails with pro bono obligations in law. A recent ABA survey reported that only 46 percent of surveyed lawyers met the ABA's goal of offering \"50 hours of free pro bono services\" each year. Those who do provide pro bono legal services report a direct correlation between that form of giving back and their own satisfaction. Pro bono activities boost attorney morale.\n\nThe converse is also true: new lawyers identify the absence of pro bono opportunities as one of their largest areas of dissatisfaction with the practice of law. The ability to do pro bono work suffers from a lack of support from many firms and already-oppressive billable hour requirements.\n\nSome things are changing in the public interest world. Law firms are beginning to see pro bono opportunities as good business, a method to help recruit talented law school graduates, valuable training ground for associates, and a way to engage with communities. Esther F. Lardent, president of the Pro Bono Institute at Georgetown University Law Center, says, \"The nation's 150 largest law firms last year spent 4 million hours on pro bono, more than double what it was 10 years ago.\" Gradually, larger firms are appointing directors of pro bono activities, although many cut back on their pro bono activities during the recent recession.\n\nAttorneys gain a tremendous sense of personal meaning from representing people who are vulnerable and achieving some measure of justice for them\u2014for making a difference in people's lives.\n\nThe range of possible pro bono possibilities is vast: representing detainees at Guantanamo Bay, death row inmates, or hurricane victims denied financial assistance; helping veterans obtain disability benefits; assisting with the legal needs of low income immigrants; or filing adoptions. Conservative or libertarian public interest lawyers promote the constitutional rights of crime victims, campaign against eminent domain and affirmative action, and defend individual economic liberties and property rights.\n\nLawyers can pursue the intersection of their hobbies or interests and law. For instance, lawyers interested in animal protection and animal rights have given time to the Humane Society and the American Society for the Prevention of Cruelty to Animals. One lawyer has successfully challenged black bear hunts; another has saved condemned dogs from execution (when the \"client,\" a Rottweiler named Pookie, bit a child who stuck her foot through his fence). Call it \"pro bone\" representation.\n\n#### 8. CREATING A PLAYFUL WORKPLACE\n\nLaw firms can look for inspiration from trendsetters in other industries. Law firms are behind the curve when it comes to creating workplaces where a sense of play is encouraged. Consider the workplace strategies and incentives of Google, rated the single best place to work in this country in both 2007 and 2008, according to a _Fortune_ survey of more than 100,000 employees. While the compensation was top-notch (Google ranked 25th on the separate top-paying companies list in 2008), it was the attention to individuals, benefits, feel, and playfulness that rocketed Google to the number one spot on the best companies to work for list. Ninety-nine percent of employees receive stock options, and all of them can eat \"free breakfast, lunch, and dinner at 17 gourmet cafes, [and receive] a subsidy for buying a fuel-efficient car and free shuttle service to and from work.\" Part of the job satisfaction for Google employees may come from finding a workplace that prioritizes shared values, such as prizing tech wizardry, giving economic benefits for ecological acts, or sponsoring playfulness. Google's office in Zurich, for example, offers playrooms with pool tables and a huge slide from an upper floor into the lunchroom. Apart from all of the perks, employees gave Google the highest ratings for open communications, trust along various dimensions of respecting employees (such as caring for them as individuals and collaboration on relevant decisions), and camaraderie\u2014a sense of family or team spirit about the workplace.\n\nGoogle and other Silicon Valley firms are far ahead of anything most law firms have done to create a playful atmosphere at work. The exception may be boutique corporate defense firm Bartlit Beck in Chicago, which has a full-size mock courtroom in its offices and a basketball court in the middle of the firm\u2014with courtside offices for several of the partners. Its Denver office has a rock climbing wall in the lobby. Most law firms though have tended to be conservative, perhaps in part because of their image, but they are lagging dramatically in the workplace revolution. The new generation of workers in America places a much higher premium on play and creativity; allowing for these expressions can pay off in terms of revenue as well as morale.\n\nOf course, law firms are rarely all work and no play. Your firm may be one that knows the value of having partners, associates, and staff engage in recreation together. Some firms make use of firm-purchased luxury boxes at concerts and sporting events, offer wine tastings and cooking classes, or sponsor firm-wide trivia contests.\n\nPillsbury Winthrop Shaw Pittman, LLP, with fourteen offices and eight hundred lawyers, was one of only four law firms to earn a spot on _Working Mother's_ 100 Best Companies list. In addition to family-friendly flexible schedules and leave policies, the firm boasts an impressive roster of social events to help create personal relationships among its employees such as the following:\n\n\u2022 In May the firm's Washington, D.C., office arranged a private screening of Pirates of the Caribbean: At World's End for attorneys, staff, and their families. All enjoyed pre-show appetizers in Pillsbury's first-floor conference rooms, which had been turned into pirate-themed hangouts complete with servers in costume.\n\n\u2022 In July the firm's McLean, Virginia, office staged a \"Smooth Operator Challenge,\" during which attorneys made their favorite nonalcoholic smoothies as part of a friendly competition. Employees were encouraged to \"buy\" a full-size smoothie for $5, which benefited the Medical Care for Children Partnership. The firm matched every dollar raised for the charity.\n\n\u2022 Each office also holds an annual \"Family Fun Day\" at a popular amusement park or other fun venue.\n\nAt this point, the evidence is anecdotal and it sifts through mostly in stories about firm recreational events, but it seems that firms that play together stay together.\n\n#### 9. DESIGNING LAW OFFICES TO PROMOTE LAWYER HAPPINESS\n\n_Beauty is... the promise of happiness._\n\n\u2014Krister Stendahl\n\nChanging the physical design of a law office is a neglected tool in the search for greater lawyer happiness. Anyone who has ever spent much time in windowless cubicles or drab hotel rooms understands that interior spaces can affect moods. Some spaces rob us of optimism and a sense of purpose; other spaces make us happy and productive. Buildings can seem welcoming or forbidding, tranquil or exciting, harmonious or chaotic. A space that suits us just right at one time might not at another time.\n\nAlain De Botton, author of _The Architecture of Happiness_ , explains that buildings \"speak of visions of happiness.\" Not everyone shares the same vision of happiness, of course, so it is not surprising that two well-designed and happiness-promoting buildings might look very different from each other. One building might have an aristocratic feel while another seems more democratic. Anticipation of the future might be expressed in one building, while another communicates a fondness for the past. Nonetheless, there is growing agreement that certain design features tend to be associated with an increased sense of well-being in those who occupy the buildings fortunate enough to have them.\n\nDuring the last several decades, the architectural community has become increasingly focused on the ways people interact with architecture. Christopher Alexander and his colleagues at the Center for Environmental Structure published, in 1977, a groundbreaking book called _A Pattern Language_ , which identified and described 253 \"patterns,\" or design features, of towns, communities, and buildings that tend to promote individual well-being and smoothly functioning communities. A dozen or more of these 253 patterns are especially relevant to law office design and will be discussed below. The Academy of Neuroscience for Architecture, an organization founded in 2003, released data that supports our intuitions that natural light and natural air stimulate creativity at work. California architect Deborah Richmond agrees, arguing for \"a democratic distribution of natural light and air, which is proven to increase productivity and reduce sick days.\" The Academy of Neuroscience for Architecture has also published data showing that views of nature calm us and that low ceilings promote fine attention to detail while high ceilings facilitate more expansive thinking.\n\nSo what does the happy law office look like? To some extent, the answer depends on the personalities of the lawyers who occupy it (one person's thing of beauty is another person's bad and ugly), but architects and \"environmental psychologists\" are in general agreement that certain design features stand a good chance of making the smiles come easier.\n\nAmong the top recommendations is to rely as much as possible on natural light for illumination in every occupied office space. All the better if the natural light comes into the room from two sides. (There's a reason, other than prestige, why those corner offices are so sought after.) As Alexander notes, \"When they have a choice, people will always gravitate to those rooms which have light on two sides, and leave the rooms which are lit only from one side unused and empty.\" According to Alexander, \"this pattern, more than any other single pattern, determines the success or failure of a room.\" This insight is hardly new, as \"light on two sides\" was a tenet of the old Beaux Arts design tradition. Alexander argues that rooms lit on two sides create a desirable social atmosphere because they have less glare and less disturbing light-dark contrast around the windows. The light distribution in a room lit on two or more sides allows occupants to better read the facial reactions of others in the room. In short, Alexander claims, \"light on two sides allows people to understand each other\"\u2014and understanding other people makes us happy.\n\nThere are compelling reasons to avoid the traditional model of one enclosed office per attorney, spread along a stretch of hallway. Closed-off offices discourage interaction. On the other hand, offices that are too exposed feel uncomfortable. Ideally, offices should strike a balance between enclosure and exposure. This can be accomplished in a variety of ways, many of which have the added benefit of increasing flexibility. If work areas or offices are too small, claustrophobia can be a problem: studies show that spaces less than sixty square feet in area make people feel cramped.\n\nIf a law office includes, as most do, separate offices for each lawyer, then each office should have a view overlooking life. Alexander contends, \"Rooms without a view are prisons for people who have to stay in them.\" Worse yet, they might be bad for your health. A study of hospital patients showed that patients with rooms overlooking trees had shorter hospital stays and fewer post-stay medical complaints than those with rooms overlooking brick walls. The better the view, the freer one feels. A lawyer who can gaze out at a garden or pleasant street scene will be happier than one who looks out at a bulky air conditioner unit, the top of a parking garage, or an overgrown vacant lot. Obviously, not every law office can be located next to a botanical garden or an idyllic street, but in selecting a potential new office location, attention should be paid to the views it affords on all occupied sides. If staying put is the only viable option, well placed mirrors can sometimes be employed to create improved office views. You probably can't get the parking garage next door to move, but maybe you can peek around it.\n\nLaw firms, understandably, impose some limits on what individual lawyers can do with their own office space. _Playboy_ pin-ups, black velvet Elvises, or lava lamps send messages to clients that many firms would prefer not be sent. But allowing attorneys to fill their offices with meaningful things from their own lives (not items chosen just to impress visitors) gives them a greater sense of control, an important component of a happy life. Moreover, a room that reflects the history and tastes of the occupant facilitates connections with those who might come to visit. (\"Oh, I see you like to collect comic books\u2014I loved Superman as a kid too.\") Your office should tell your story. Unfortunately, as Alexander laments, too many are driven to \"the artificial scene-making of 'modern d\u00dfcor'\" that is \"totally bankrupt.\"\n\nA law office designed to promote happiness should include a large, comfortable space for lawyers to gather together and socialize. Major pathways in the office should be tangent to this area. This plan brings lawyers close to the socializing space on a regular basis and, as an added benefit, eliminates the long sterile corridors that dampen spirits. A well-designed gathering place would include a variety of welcoming things, such as a fireplace, coffee pot, a variety of comfortable chairs, and newspapers. Lamps rather than uniform illumination should be used, as studies show people seek out pools of light even when overall illumination is adequate elsewhere. Warm colored light, produced by wall paint, natural wood, or light sources, should permeate the space.\n\nEvery law office should provide a place for communal eating. Thomas Merton, author of _The Living Bread_ , writes, \"The mere act of eating together. . . is by its very nature a sign of friendship and 'communion.'\" Just as the table can become the center of family life, it can bring office colleagues together through shared food, drink, stories, and laughter. Communal eating spaces in a large firm can serve another function: they allow lawyers to meet and find colleagues who have mutual affinities that they might otherwise never get to know. In the well-designed eating space, lights directly over tables serve to focus attention on the communal activity and help bind groups.\n\nLarge undifferentiated spaces are not comfortable places for groups of lawyers to work. According to one large study of office workers, large spaces make workers \"feel unimportant\" and give the \"uncomfortable feeling of being watched all the time.\" For a legal team to do its best work\u2014and be happiest\u2014it needs a smaller work area or meeting room, preferably one that provides partial views out into a larger common area.\n\nOur affinity for certain features, such as natural plants and moving water, may trace back to our origins as a species. Plants of varying shapes and sizes should be scattered throughout the office. Moving water, most likely produced by a fountain, is associated with an increased sense of well-being and should be featured in a place where as many lawyers as possible might benefit from it.\n\nWhen it comes to law office design, there are indications that some firms are moving in the right direction. Firms are \"adding more communal spaces, from coffee bars to visitor lounges.\" San Francisco-based architect Barbara Gunn says law firms are \"looking for opportunities to encourage collegiality and chance encounters\" between lawyers. A New York lawyer overseeing the remodeling of his firm's offices agrees, saying that the goal is not \"to impress clients,\" but to create \"a work environment that really encourages and supports building collegiality and communication and teamwork.\" The new look in law firm architecture is a trend toward the building of \"teaming rooms where people from different practice groups can work together.\"\n\nIt is unrealistic to expect any office to offer all of the happiness-promoting design features described above, but the more that are part of the office landscape, the happier the lawyers who work (and sometimes play) in it are likely to be.\n\n#### 10. ASKING FOR FEEDBACK\n\nKeep in mind that mattering matters to people. People like feedback from their firm; they also like to be asked what they think about their working conditions.\n\nWhy not conduct an anonymous survey of your lawyers and staff to assess the dimensions of their job satisfaction? Using a survey questionnaire and scale of agreement (ranging from \"strongly agree\" to \"strongly disagree\"), you can ask whether your employees are satisfied with such measures as communication, feedback, training, mentoring, expected hours, ability to maintain a reasonable balance between work and personal life, collegiality, the physical structure of the workplace, and compensation. Assure anonymity, and add an open-ended area for comments on things that would enhance their satisfaction with the firm, the best aspects of working for the firm, and what bothers them the most. Merely asking for opinions provides a happiness boost.\n\n#### IS YOUR LAW FIRM A HAPPY PLACE?\n\nThis list is not intended as a one-size-fits-all checklist. The purpose of the enumerated questions is to translate those characteristics the happiness literature identifies as correlating with job satisfaction into pragmatic advice for law firm managers.\n\n1. In what ways do you tell valued players what they mean to the firm and that the firm wants them to be happy? Are all attorneys given rewards, recognition, emotional support, and encouragement to take risks and pursue dreams? What opportunities do you provide for career advancement? Do partners relay to associates good things clients say about them, and do they remember to give associates credit for the heavy lifting in front of clients? Are all attorneys trained in how to respect and work with staff?\n\n2. Does your work assignment system allow attorneys any flexibility to choose the projects or cases on which they would like to work? Are more junior attorneys given responsibility to manage cases independently? Are they given training in how to manage cases and develop business? Do partners frequently take associates with them to meet with clients? Do they routinely take associates out for social occasions?\n\n3. What proactive efforts does the firm take toward creating flexible schedules and compensation plans for attorneys and staff? Is your compensation structure transparent, and are new hires informed about it? What percent of your employees actually take advantage of flex- or part-time policies? Do you have policies offering generous maternity and paternity leave?\n\n4. Do you have a mentoring program for more junior associates that matches mentors and mentees based on the associates' developmental needs and allows participants to designate mentor characteristics they would prefer (such as practice area, gender, or family status) or select their own mentors? Does the program provide training for mentors that stresses understanding each associate's individual goals, creating a career roadmap, and promoting career development? Is an evaluation mechanism in place for associates to give confidential upward feedback on the quality of the mentoring they have been receiving? Is mentoring valued enough to factor into partner or senior associate compensation? Is the mentor also a guardian who can help resolve unreasonable conflicting demands on associates or run interference for them?\n\n5. Do attorney performance appraisals include benchmarks for skills and knowledge expected of associates? Are the criteria communicated so associates understand the expectations? Does the firm offer strategies for working on identified weaknesses? Does the firm provide regular and constructive feedback?\n\n6. To avoid surprises that make people feel insecure (\"Before you hear this from someone else . . .\"), does your firm have an established policy for the release of information about firm events and plans? Or do you rely on information leaking out and trickling down?\n\n7. Does your firm have regular opportunities to socialize (wine and cheese on the last Friday of the month, a co-ed softball game, or a group charitable project)? How often do your attorneys get together for lunch? What ties does your firm have to the community?\n\n8. Does the physical layout of your offices promote interaction and opportunities for the exchange of ideas? Does your firm offer places where attorneys can congregate and talk? Are you tolerant of water cooler time\u2014downtime where attorneys can share insights?\n\n9. Do you pay attention to indicators of attorney and staff satisfaction? What are your attrition rates for lawyers and staff at various levels? Are all attorneys, including those on alternate career paths, and staff asked about how firm policies affect their work? Are they asked for their ideas and input on innovations, efficiency, and firm practices?\n\n## **CHAPTER 7\n\nLawyers' Stories**\n\nTHE BEST WAY TO PREDICT WHAT EXPERIENCES WILL make us happy, or unhappy, is to look at how other people have reacted to those experiences. We think we are unique\u2014and, yes, we are\u2014but people are far more alike than they are different, and the reactions of others who have gone before us turn out to be the very best guide to our own possible futures. When we let our own imaginations, rather than the reports of others, predict our reactions to future experiences, we tend to miss the mark rather badly. There are many reasons why our imaginations fail us but chief among them are that we let our present attitudes and interests overly color our imagined futures, and we fail to appreciate how rapidly we adapt to both good and bad events.\n\nDaniel Gilbert says it is better that we rely on even \"a single randomly selected individual\" than our own imaginations when it comes to predicting how we will experience the future. Gilbert's suggestion is certainly open to a validity critique: \"Of course, relying on a sample of one is probably not going to tell you too much, especially when the sample is intrinsically biased. (Instead of talking to a yodeler, why not talk to someone who dropped out of yodeling class?)\" A better method of collecting information about career satisfaction in law\u2014better than relying on your own imagination or selecting the single yodeler\u2014is to talk to a variety of people in the profession.\n\nIn this chapter, we invite you to let other lawyers be your guide. In their stories they will warn you of missteps they have taken, identify the peaks and valleys of their experiences as lawyers, and point to various paths that lead in the direction of satisfying careers in law. We offer stories from happy lawyers, unhappy lawyers, and lawyers whose career satisfaction falls somewhere in between. Finally, we provide a few thoughts of our own on what these stories, and many others we haven't room to include, tell us about the pluses and minuses of practicing law and what you might expect as your career arcs from law school to retirement.\n\n### ABOUT BALANCING WORK AND THE REST OF YOUR LIFE\n\nWhen we asked hundreds of lawyers, in nearly every state and in a wide variety of practice areas, to identify what they've found in their careers to be the biggest obstacle to personal happiness, they most frequently cited their difficulty in achieving a satisfactory balance between the demands of their career and their other personal goals and responsibilities. \"The law is a jealous mistress and requires a long and constant courtship,\" noted U.S. Supreme Court Justice Joseph Story in 1829. Justice Story's observation seems as true today as it did nearly two centuries ago. Lawyers continue to make personal sacrifices to further their legal careers, but they are often not very happy about it.\n\nA typical complaint comes from an Atlanta attorney who says she \"felt a great deal of pressure to work longer rather than smarter\" and found that career satisfaction sank \"in an environment that rewarded people for the number of hours they spent away from their families.\" A Kansas City attorney says, \"It is very easy to get in a job and then find yourself looking around one day wondering where the last five years and all your friends went.\" A California attorney reports she is \"frustrated by the lack of time to develop my personal life\u2014to make new friends, to find a romantic partner, and to be my best self physically.\" \"There are only 24 hours in day,\" a hard-working New York tax attorney writes, \"so I can't do a lot of things that I would otherwise do, and that's tough.\" Another New York City lawyer notes that it is not just the long hours of practice that are tough but also the anxiety associated with always being on call: \"Having to stay in the office all night to review and comment on an unexpected series of documents sent from lawyers on the other side of a transaction was certainly not ideal, but the nights when I left at 5 or 6 and spent the evening worrying that I had left too early because something unexpected might arise were probably even worse.\" An Illinois attorney who is in-house counsel for a large corporation agrees, writing, \"It is difficult to completely separate myself from my job. Too many people and the success of my company depend upon me and my colleagues.\"\n\nThe large firm focus on billable hours is a source of much lawyer unhappiness, in part because it seemingly de-emphasizes work quality in favor of work quantity. A Missouri lawyer who \"takes pride in being efficient and cost-effective\" finds billable hours to be a source of frustration: \"I am being punished for being efficient, a perverse consequence of hourly billing,\" and, as a result, \"lose time with family and friends.\" Another attorney complains, \"At the end of the day, it seems my only value to the firm is based on billable hours.\" A California attorney finds that basing value on billable hours breeds resentment by associates against other associates who fail to work the same unreasonably long hours that they do. She quotes a colleague who once told her, \"All the galley slaves need to row at the same pace.\"\n\nThe long hours associated with law firm practice are especially tough on attorneys who have children. A marketing consultant for law firms puts it this way: \"'The law firm is probably the most family unfriendly place you can go... It's difficult to recruit women because they feel that somewhere down the line, they are going to have to choose between having a family and having a job, which is not a fair choice.'\" One attorney and mom tells this story about her difficulties in balancing work life and family life:\n\nIt was a Friday afternoon and my three-year-old son had a preschool Mother's Day program.... After the program, I checked my voicemail and was treated to a rant [from a partner for whom I worked] including curse words I hadn't heard before on how I was needed that afternoon, how my priorities were screwed up, and how I had left him in a lurch. I returned with my son to the office to find that some discovery had been hand-delivered and I wasn't there for him to give it to. Discovery! We had thirty days to respond to it! The real problem was that he was going on a canoe trip the next Monday for a week and wanted me to work on it before he left. As I left his office, I heard him yell at his secretary while I was in earshot that the firm needed to stop hiring mothers. The kicker is that this attorney had been appointed by the bar association to serve on a committee to improve the status of women in the profession.\n\nWhat can be done to restore a better balance between work and personal life? One approach is to change jobs. A lawyer who, as a litigator, found himself \"consistently at the mercy of deadlines\" decided he \"needed a change because of the demands of the job.\" His new position as a compliance officer for the athletics department at the University of Oklahoma has more regular work hours. Another lawyer (and mother) reports that her move from a large private firm to a three-day-per-week in-house counsel job restored the connections she needed to have with her young family. A lawyer who left private practice for a position with the federal public defender's office appreciates that as a federal employee, his number of vacation days increases each year and that he has a newfound flexibility for scheduling his dream trips.\n\nShort of changing jobs, lawyers offer other advice for achieving a better work-life balance. An attorney in a large private firm finds more hours for personal life this way:\n\nTrain clients not to call you at the last minute or on your cell phone unless it is an emergency. Don't schedule conference calls at lunch time. Don't schedule plane flights at 6:00 A.M. Take time to go to your kid's events, but take turns with your spouse. Go home in time to have dinner with your family regardless of the looks you get from your partners. Be happy with your compensation, regardless of others who are making more because they have different priorities than you do.\n\nAn attorney in a small town in the Midwest adds this: \"Know when to go home. And don't be a lawyer when you are with your family.\"\n\nSome lucky lawyers are able to find an appropriate work-play balance even within the confines of their law firm. One happy lawyer in a small workers' compensation firm enjoys his office foosball table, dartboard, and bar, where firm members gather regularly after work to toast the day and discuss sports, music, and other things on their minds. Because clients rarely meet at the law firm, he comes in jeans and regularly tosses a Frisbee around the office with his partner.\n\nAnother solo practitioner in Virginia echoes this concept of playfulness:\n\nI would tell students not to be afraid to try unusual places or ways in which to practice law. I would never have pictured myself as a solo practitioner in a poor area of rural Virginia, but overall I am extremely satisfied in life and practice here.... I can raise show dogs, go to my office in shorts, and still face exciting challenges and intellectual stimulation in my practice.\n\nSome firms help lawyers achieve a better work-life balance by adopting flexible work policies or by allowing lawyers to bring family members to the office. For example, a senior in-house counsel for a national association praises her organization's policy that offers a compressed schedule option. Attorneys in her company are able to take one paid day off for every two weeks so that they work eighty hours in nine days instead of ten. The lawyer is also grateful for the association's infants-in-the-workplace program that allows mothers and fathers to bring their infants with them to work until the children reach six months of age. The baby stays with the parent while the parent is working in the office, and there are two or three designated individuals who can stay with the baby when the parent needs to be away.\n\n### ABOUT WORKING WITH PEOPLE YOU LIKE AND TRUST\n\nWhen asked what about their job makes them most happy, lawyers often come up with the same answer: people. Lawyers who regularly interact with colleagues and clients that they like and trust tend to report high levels of career satisfaction. On the other hand, those who must frequently deal with ungrateful clients, unsympathetic supervisors, or unethical and unscrupulous opposing attorneys tend to be much less happy.\n\nA number of lawyers surveyed identify good relationships as their greatest source of happiness. \"Surround yourself with people you enjoy working with and make work fun,\" advises the managing partner of a small plaintiff's personal injury firm. This sentiment is echoed by the chair of a practice area at one of the country's largest law firms, who suggests, \"Learn to relate to your clients and colleagues as people\u2014you will not have much time for friends outside your practice and it is a blessing to have colleagues and clients who are friends.\"\n\nAn equally large number of lawyers see bad relationships as the cause of great career dissatisfaction. One managing partner who has practiced for more than twenty-five years complains about \"the lack of civility\" in the profession and the \"demoralizing\" willingness of some lawyers to compromise the truth in their \"incessant quest for money.\" Another lawyer from the Midwest believes, \"I have probably cut my life short by a matter of months stressing over unethical litigation tactics that certain adversarial attorneys have used against me and my clients.\" A Los Angeles attorney observes, \"I began to wake up every morning wondering what sort of dirty trick opposing counsel would pull that day.\" The director of the California Bar's Lawyer Assistance Project says, \"I've never seen such a lonely profession\u2014there's an inability to connect with other people at a deep level because there's so much of an adversarial relationship. The profession in a lot of ways makes it difficult to build trust.\" An unhappy big firm lawyer in the Southwest writes to complain about fellow lawyers who \"procrastinated and then shoved work onto others at the last minute,\" lawyers who \"took credit for other people's work,\" and about the \"general smugness\" that he found \"ubiquitous in large firms.\" A government lawyer complains about being frustrated by \"mid-level government bureaucrats who were shameless toadies to anyone above them and petty tyrants to those below them.\" Another attorney simply states his complaint about the practice: \"I have always said that the practice of law would be a happy profession if it wasn't for opposing counsel, clients and judges.\"\n\nAn East Coast lawyer recalls the morning he was taking a deposition in Connecticut when the receptionist mentioned that a plane had just crashed into the World Trade Center. As the morning went on, and the terrible fact of the twin towers' collapse became known, the deposition proceeded:\n\nWe didn't really know what else to do, and figured that it was going to be hard for people to get back to their homes anyway, so we just carried on with the deposition all that day and the next. And the striking thing to me was that everybody carried on pretty much as normal, bickering and squabbling over every little thing. And it just seemed very petty, given the events going on that day.\n\nJust as there are lawyers who report good and bad experiences with other lawyers, reports about client relationships are also a mixed bag. One attorney says \"rarely are any of my clients happy to see me, because if they need to see me, they are going through some sort of life crisis.\" A Kansas lawyer came to realize suddenly that clients were at times her \"greatest obstacle to happiness as a lawyer\":\n\nI was working late one night to get ready for a big and important trial where millions of dollars and jobs were at stake. In the middle of going over very technical, mind-numbing budget numbers, my client looked at me and said, \"You don't do any 'happy law,' do you?\" I was floored. Until then, despite the grueling hours, the lack of recognition, the abuse by clients, partners, and staff, the impact on my family, my health, and my sleep, I still loved my job as a bankruptcy lawyer. But I had never thought about it that way before. I realized that almost everything we do as lawyers involves unhappy people who are asking us to solve their unhappy problems, and many times there is no good solution that will make the client happy. And, I have found out, misery loves company and unhappy people want to make YOU unhappy.\n\nClients can be a source of unhappiness, a source of joy, and a source of the deepest anguish you can imagine in practicing law. Death penalty lawyer Sean O'Brien relates his worst experience practicing law:\n\nOn October 20, 1992, I had to break the news to Ricky Grubbs, a mentally retarded man, that the U.S. Supreme Court had declined to hear his case, and that he was going to be executed forthwith. I tried to break the news gently, \"The Court decided not to hear your case.... your appeals are over... Governor Ashcroft is not going to intervene.... there is nowhere else to go....\" He kept asking \"What's that mean?\" until I finally had to tell him that he was going to be executed right away, tonight. While I was talking to Ricky the guards came for him with the gurney, took him off the phone, strapped him down and killed him.\n\nThat case, O'Brien says, \"sent me to a dark, angry place for a long time.\"\n\nWhile some lawyers' darkest moments come from their clients, or what happens to their clients, other lawyers find them their greatest source of professional joy. A public defender from New Jersey says, \"My best experiences revolved around showing my clients that someone in the criminal justice system cared for them and their well-being. Some clients came to trust me after I was able to show them that I was completely on their side, no matter what they had done. That type of trust, which has to be first earned and then maintained, gave me the greatest satisfaction.\" An experienced attorney in Ohio, after having represented some of the same clients for over three decades, is of the same mind. He reports that several clients have \"become some of my closest friends\" and are the people he turns to when it is time to put away the briefcase and have fun.\n\nIf relationships are central to happiness, then it is hard to fault the advice of a South Carolina lawyer who writes, \"Try to avoid working with angry, nasty people. Try to find work with people who treat you with a basic level of dignity and respect. Try hard to treat the people around you, especially those down the food chain from you, the same way.\" Conflict, mistrust, and ungratefulness are, unfortunately, always going to be part of a lawyer's job. But how big a part they are will depend upon the job you choose.\n\n### ABOUT FINDING THE FLOW\n\nYou didn't need to buy this book to know that interesting work makes us happier than boring or anxiety-producing work. The challenge, of course, is identifying the work that interests you and then finding a way to get paid for it. The luckiest lawyers in the world are those who each morning can't wait to head into the office and find themselves asking, from time to time, \"How is it that I actually get paid for doing work that I love so much?\" Does that describe you? It probably does not. Law practice typically mixes interesting work with the not-so-interesting, includes fun cases and the not-fun cases, and offers both absorbing and tedious problems.\n\nThere is something appealing about the nature of law practice that got you to law school in the first place. Perhaps you always enjoyed puzzle-solving and recognized that law practice provides an opportunity to solve challenging real-world problems. What facts can I develop that will present my client's case most favorably? What case out there in Westlaw's or LexisNexis's cyberspace of cases best furthers my client's cause? What set of questions will best serve to weaken the power of an opposing lawyer's witness? Puzzles such as these, for the right person, can soak up time and attention.\n\nWe all know law practice is not just a bowl of cherries. There are humdrum cases, lengthy forms to fill out, footnotes to put in order, and billing documentation, in its six or ten or fifteen-minute increments, to complete. There are many times when law doesn't flow like a mountain stream in spring. Admit it: sometimes law practice is boring. Consider Catherine Kersh's description of a \"no flow\" experience when she worked as litigator at a big firm on the West Coast. Kersh, who now works for a nonprofit group administering scholarships, remembered spending a period of weeks in which she and many other associates were stuck in a room reviewing endless boxes of documents: \"Every day, for 12 hours, [we] fastened Post-it notes to legal briefs. 'You look around at the other associates, trying to remind ourselves, why did we go to law school?'\"\n\nThe goal is to improve the ratio of enjoyable work to tedious work. To some extent, this happens naturally over the course of a career, as your firm hires younger associates to do the more routine bottom-of-the-totem-pole work that almost every firm has. Also, in many cases, as your career progresses you will gravitate to the type of work you find most interesting.\n\nIn our survey, lawyers identify a number of tasks they find especially tedious or unpleasant. A New York tax associate says, \"A lot of what I do\u2014poring over contracts, in particular, can be pretty dry. But I knew this before I signed up, so fair warning.\" A real estate attorney complains bitterly about recording billable hours: \"Time keeping is the worst part of being a lawyer and it is a drudge. It's a pain and you have to plow through it and I hate every second of it.\"\n\nMost lawyers, however, find their work to be generally satisfying. An Indiana lawyer suggests that you \"remember that the best part of being a lawyer is the intellectual challenges that face you every day. You get paid to creatively solve problems\u2014and the vast majority of workers don't get that luxury.\" An Alabama lawyer finds satisfaction in \"looking at chaos and imposing order on it, either in the form of telling a coherent story about it for a statement of facts or in the form of synthesizing a set of rules from it (legal research).\" A lawyer from the middle of the United States says, \"I enjoy writing and working with words tremendously. I usually do about seven drafts of everything I write and I convince myself that each draft is better than the previous one.\" A Chicago employment lawyer finds flow \"the moment when I sit down to ask the first question in an important deposition,\" and on \"jury trial days.\" A happy Vermont lawyer describes how flow comes \"all the time\":\n\nI absolutely love writing a brief or legal memorandum or motion for a client; I enjoy the research and especially the challenge of putting all the pieces of the puzzle together in a way that is persuasive. I also love working on Excel spreadsheets and doing child support calculations and examining tax issues for a client, to see what the best financial solution might be for a client.\n\nA patent lawyer in New York agrees:\n\nOne of the things I love about practicing law is that it is so engaging that 'clock-watching' is rarely a problem. I often felt that way as a patent prosecutor: I loved the initial stages of patent prosecution: working with the inventors and writing a patent application. I could put in 17 hour days with barely a break, and go home happy, and start in early the next day again.\n\nA commercial lawyer from Nevada finds time passes quickly when he has the opportunity to work with colleagues on a meaningful case:\n\nI experienced flow regularly when working on team projects with two or more colleagues, putting together filings and strategizing. The solitary work of research and writing has never generated flow for me, but working in concert with others on a time deadline has almost been sublime, even when there was ostensible pressure and should have been anxiety.\n\nExperienced trial attorneys, unsurprisingly, report finding flow in trials. Noted criminal defense lawyer James Brosnahan says, \"Nothing compares to the electricity of an actual trial, and it is magnified when it is a jury trial.\" According to Brosnahan, there's great satisfaction in representing criminal defendants who feel the whole world is against them. He recalls his first meeting with John Lindh, the young California man who was the first American to face terrorism-related charges after 9\u201311. Lindh greeted him with the words, \"'Boy, am I glad to see you.'\" \"That's why I became a trial lawyer,\" says Brosnahan. Also finding flow in trials is an attorney who writes that the \"first few minutes in an opening statement is magic. The world belongs to me. Nothing matters but the words coming out of my mouth.\"\n\nWhen the flow is not there, consider changing jobs. A former New York tax attorney, who felt anxiety from having \"a lot of work dumped on me without explanation or guidance\" finds greater satisfaction in public interest litigation:\n\nI started to get the chance to feel like I knew what I was doing. I'd be the one who knew the most about the case, from reviewing all the documents and doing the legal research, and it's a good feeling to feel like you're on top of things and contributing. It's just like when you're a child\u2014it feels good to master things, and it's unpleasant when you don't know what you're doing and no one is helping you.\n\nA civil rights lawyer offers perhaps the best advice on finding happiness in the actual practice of law:\n\nEvery day, you're going to get up and do something (interview people, go through documents to find facts, negotiate, write persuasive arguments, draft documents, or something else), so make sure that it is something that you enjoy doing.\n\n### ABOUT ALIGNING WORK AND YOUR VALUES\n\nYou are happiest when you think your work matters. Like everyone else, you want to make a positive difference in the world. When work aligns with your values you feel good about what you're doing. It's a simple fact, an obvious fact, but one that often spells the difference between careers that are satisfying and those that are not.\n\nWhen a lawyer feels his or her efforts don't matter and aren't valued, even good pay and good colleagues are not enough to produce happiness. Sometimes unhappiness stems from projects that seem to be a waste of time. A Florida attorney writes:\n\nI did not enjoy working on matters that were not meaningful to me. Once I was working on a matter that required me to travel around a state to interview members within an industry. After toiling over a month on the interviews and the report, it landed in a pile of bureaucratic paper wastage and probably wasn't read. A nice income doesn't deaden that kind of suck from one's life-force.\n\nFor some attorneys, the problem is not that their work is of no account but rather that they sense they've joined the wrong side. They're working for the Empire, and no Force will ever be with them. A lawyer in Tennessee tells this story:\n\nMy firm represented a couple of slimy real estate developers. My managing partner assigned me to clean up all their dirty work. They were being sued by just about everyone with whom they had ever dealt. I was taking a deposition of a woman who was suing them over a landlord\/tenant dispute and she interrupted the deposition saying, \"You seem like a nice person. How do you sleep at night representing these scumbags?\" For one of the only times in my life, I was speechless! The truth hurts.\n\nIf some of lawyers' darkest moments come when their work is either unvalued or undermines their own sense of justice, it is also true that some of their brightest days come when they believe that their work made a positive difference in their clients' lives. Attorney Jim Husen, a family law lawyer in South Carolina, observes that \"people come to us when they are caught up in the vortexes of institutional power or face insurmountable obstacles or feel like they have been wronged.\" He says that to help clients, lawyers must \"draw on everything in our lives.\" In identifying so deeply with the causes of his clients, Husen finds satisfaction: \"I can think of almost no other job where that kind of commitment is called for.\"\n\nTime and time again, when asked to relate their peak experiences as lawyers, respondents in our survey point to cases in which they contributed to just outcomes that affected individual lives. Typical is the response of a Pennsylvania lawyer who says, \"When a client left court with the child custody she so desperately wanted, that made me happy.\" A Massachusetts attorney reports, \"I filed suit against a non-profit student study abroad organization for having placed my client in a home in Chile where she was repeatedly raped by her host brother. I settled the case with just filing the complaint. It's what I went to law school to do.\"\n\nA family law lawyer from North Carolina tells of \"a precious moment\" in her career that \"still brings tears to my eyes\":\n\nI was finishing a very contentious collaborative divorce. The husband and wife had been married almost 20 years and the marriage ended badly with a lot of discord but they shared a commitment to co-parenting their children. It was not the lovey-dovey case that most people imagine collaborative divorces to be, but they worked very hard and we came up with a deal that suited both of them. We gathered for the signing ceremony and were waiting for the notary to arrive. Someone suggested that we ought to sit and have a beer while we waited and our host happened to have some in the fridge. The husband and wife decided they didn't want a whole beer and decided to share one. It was an intimate moment that seemed to bring back memories of many times shared. The notary arrived, the papers were signed. As we stood up to leave, the husband and wife looked at each other and hugged. As they held each other, he tenderly petted her hair and each thanked the other for everything that had worked in their marriage and for working so hard to come to agreement.\n\nFor a Springfield, Missouri, lawyer a peak experience that came to mind was at the conclusion of a guardianship proceeding involving a female patient in the adult psychiatric unit in a hospital that she represented:\n\nHer partner was unable to care for her any longer, as he was very ill. Our patient was not safe to leave the hospital without a guardian, as she was severely mentally ill, and could not be relied upon to take her prescribed medications on her own. Thankfully, the court appointed the public administrator to serve as this patient's guardian. As I was leaving the courthouse that day, the patient's long-term partner approached me with tears in his eyes. He was frail and very thin, and was coughing up blood into his handkerchief. He told me that he was dying of AIDS and that he could no longer take care of his beloved. He had managed her medications for many years, but his doctor told him that he would die very soon, so he decided it was time to find someone else to take care of her. He took my hand, looked into my eyes, and thanked me for obtaining a guardian for his one true love. He explained that he could now die in peace because he knew she would be taken care of.\n\nSeveral estate lawyers tell us they find great satisfaction in their work. For example, an estate lawyer from Arizona says:\n\nI absolutely LOVE helping clients maneuver through two of the inevitable aspects of life: Death and Taxes. From the time I was a first year attorney, I knew I could provide great assistance to my clients by providing clear guidance to them which would ultimately make the transition of their assets after they die to the people and charities that mattered the most to them. I have always enjoyed teaching my clients about estate planning and dispelling the myths that others tell about probate or taxes. When a client dies and the estate plan \"matures,\" I feel it is an honor to be the person that the surviving loved ones turn to for help in handling the deceased client's affairs. It always gave me pride in being able to take the burden off those left behind so that I can focus on the paperwork, while they are able to focus on grieving as they need to and getting stronger emotionally day after day.\n\nAlignment of work with values comes naturally for lawyers who obtain positions with public interest firms that match their values and politics. For a New Jersey American Civil Liberties Union (ACLU) lawyer, satisfaction comes from working on a daily basis for a cause he strongly supports:\n\nEvery day that I go to work at the ACLU, I strive to advance values that make this country great: freedom, democracy, and equality. I work on a wide range of constitutional rights issues, including freedom of speech, freedom of assembly, freedom of the press, religious liberty, privacy, the right to be free from unwarranted police intrusion, the right to counsel, racial justice, gender equality, student rights, reproductive freedom, lesbian and gay rights, voting rights, and prisoner rights.... Those who work at the ACLU and other public interest organizations are not motivated by the desire to get rich, but rather by the desire to achieve the greater good. I believe that this difference helps to create a wonderful, collaborative, and supportive atmosphere in the office. Though we often work long hours, we do so because we want to and without resentment. Our work to preserve civil liberties and civil rights is a labor of love.\n\nSome lawyers find satisfaction in making things possible that tangibly improve their communities. For example, a large-firm real estate developer had this to say about what he found most satisfying in his practice:\n\nThe fun part of practicing revolves around feeling like you've accomplished something that's significant. Whether that's helping someone save a building and turn it into a hotel or helping the guys at a struggling shopping mall that's a liability to the city turn it into a useful asset.\n\nThe votes are in, and it is clear that aligning your work and your values contributes to lawyer happiness. But how do you do that? You have to pay the bills after all\u2014and sometimes the most satisfying cases are neither ones that pay well or do much to advance your career. A lawyer in our survey bemoans the day he chose prestige over principles:\n\nI went to law school intent on becoming a civil rights lawyer. My hero was Clarence Darrow because he was my mother's hero. Before I even knew what a lawyer was, she would tell me tales of this great man who took on unpopular causes, always championing the underdog. I ended up a victim of my own success. High grades and law review, followed by a federal judicial clerkship, opened doors to the highest-paying, most prestigious job opportunities. People I respected, my judge and my brother (a successful lawyer), told me no one in their right minds would turn down an offer from the [Biggest & Best] law firm. But it was never for me. I knew that at the interview. The partners were stuffy, arrogant, and conservative. It was not a good fit. But instead of following my own heart, I did what other people thought I should do. Bad mistake.\n\nMany lawyers do find themselves stuck in jobs they don't want. Don't wait too long to make your move, suggests a New Orleans environmental lawyer:\n\nI know so many people who feel stuck in jobs they don't want. In the beginning, they knew it was not for them, but said, I'll work here, advocating positions I don't like and working 2000+ hours a year for just a while. Once I make lots of money, I will move on. They get in it and find themselves vested and unable or unwilling to take the risk. Years of their lives will get by them with little reward except perhaps money, and that does not compensate them for having no life. I say, take a job that satisfies you and allows you to embrace your interests now, not later.... Working for people, and undertaking causes and cases, you believe in. I call that Tikkun Olam, leaving the world a better place.\n\nIf you get it wrong, don't be afraid to change course. You'll have company. Eighty-five percent of lawyers change jobs at least once during their careers. A lawyer who practiced in the state of Washington offers this advice on how to keep your values and work relatively aligned as your career progresses:\n\nWrite a letter to yourself explaining why you went to law school and what you hope to accomplish in your life with your law degree, for yourself, for your family, and for others. Put that letter in a safe place and look at it at least once a year. You can edit that letter from time to time. Circumstances and dreams change, after all. But if you change course, do it consciously and for good reasons. Be in charge.\n\n### ABOUT CAREER PATHS AND HAPPINESS\n\nHumans are happiness-seeking animals. When a choice, including a career choice, fails to bring the expected level of happiness and another opportunity arises, we usually leap to grab it. In the case of lawyers, the most striking trend over the course of a career is the number of lawyers who leave private practice for other law jobs. As the American Bar Association reports, \"As lawyers move deeper into their careers, fewer and fewer work in private practice.\" Nationally, eight out often lawyers depart from large law firms by their fifth year of practice. Unsurprisingly, most move to other law jobs, but a significant number of lawyers eventually find their way into nonlaw jobs. Among graduates of Stanford Law School, to take one example, 2 percent of the class of 2005 held nonlaw positions within three years of graduation from law school, but among graduates fifteen to twenty-five years out, more than 20 percent had shifted out of law practice entirely.\n\nMany who leave law practice have law to thank for their new careers. For example, one lawyer who worked at law firms representing musicians jumped to a nonlegal management position in a music company. For lawyers with the right credentials, this type of move is not unusual. Companies recognize that well-trained lawyers bring a lot to the corporate table. One entertainment industry executive observed, \"When you hire a lawyer, you're getting someone really smart and open to learning.\"\n\nOther former practitioners apply their knowledge of the law in new fields. Legal journalism is one example. Dahlia Lithwick is a former divorce attorney in Reno who became a popular commentator and reporter on cases at the U.S. Supreme Court for the online magazine _Slate_. She describes how she fell into her new job and found happiness. Working on divorces, Lithwick discovered, \"was not my thing.\" On a trip to Washington, D.C., to visit a friend, good fortune came from a chance phone call. _Slate_ called her friend, a professional writer, to see if she could cover a trial. After saying she couldn't, she handed the phone to Lithwick, who took the job. She found _Slate_ to be the \"perfect forum in which to write\u2014this instant, slightly quirky humor, kind of 'law as theater.'\" Lithwick says:\n\nI really do feel that I might be one of those lucky people who stumbled into the perfect job when she was 30 years old... I wake up every morning, and I'm so happy... I can't quite fathom what would be as wonderful as this.\n\nMost law school graduates do not leave the practice of law, but they still tend to get happier over time, even those who stay in practice. As time marches on, lawyers gain competence and new responsibilities and achieve a greater sense of control. One partner, reflecting back on his days as an associate, observes:\n\nWhen I was an associate, there were times when I would wake up and dread the day.... Having to answer to so many people was frustrating.... Associates don't normally get to see the project the whole way through which makes lack of control an issue for them.\n\nSome litigators conclude they would rather be transactional lawyers. The roller-coaster life of a litigator has great rewards, but it isn't for everyone. As one litigator tells us, \"The highs are higher and the lows are lower compared to the experiences of a typical transactional lawyer.\" Sometimes the ups and downs occur while working with the same case: a trial victory can quickly turn into an appellate loss, or vice versa.\n\nThe fact is you might be wrong about what kind of practice best suits your skill set and your personality. A federal district judge has these words of advice:\n\nMany new lawyers know exactly the law they would like to practice and they are right. Many think they know what kind of law they would like to practice and they are wrong. Others, like me, had no idea what type of practice will lead to the greatest job satisfaction. Of course, there is no way to know in advance what field of law will be most personally rewarding. All one can do is begin, discover and adjust.... I began my career representing banks, corporations and insurance companies. I concluded it representing people. I found that representing people was far more gratifying professionally. Coincidentally, it was more remunerative as well. If you think you know what you want to do, do it. If you do not know, engage in a general practice until you gravitate to that field.\n\nA sports law attorney regrets that legal education doesn't provide a systematic means for experimenting with career options:\n\nI wish the legal system had a 'residency requirement' similar to the practice of medicine. A young lawyer could work various 'rotations' to experience various practice areas to determine what they like\/don't like.\n\nThe type of practice you develop may not be one that you envision when you start out. When Bill Colby was a young lawyer and took on a pro bono probate matter, he never envisioned it would become the first right-to-die case to reach the U.S. Supreme Court. It was a guardianship for the parents of Nancy Cruzan, a young woman who had been in a persistent vegetative state for four years following a single-car accident. Her family was united in their belief that Nancy would not have wanted to live that way and they desired to remove her feeding tube. The Missouri attorney general opposed removal of medically supplied nutrition and hydration and the case wended its way to the U.S. Supreme Court.\n\nThe Court held that competent people have a constitutional due process right to refuse life-sustaining treatment. For an incompetent patient like Nancy, the Court ruled that the state of Missouri could require her parents to produce clear and convincing evidence of their daughter's wishes. Colby assembled that evidence when friends of Nancy came forward after hearing about the case in the firestorm of media coverage. A number of years after the Supreme Court decision in _Cruzan v. Director, Missouri Department of Health_ , Colby relinquished his partnership at the firm to write _Long Goodbye: The Deaths of Nancy Cruzan_. He later became a senior fellow with the Center for Practical Bioethics. In that capacity he was able to give lectures at medical schools, hospitals, and hospices about end of life issues. Part of his life's work has been dedicated to making sure other families will not have to suffer anguishing legal battles at the end of their loved ones' lives. Today, he is the general counsel at a large, inner city safety net hospital system.\n\nColby cautions that not every pro bono case will be like _Cruzan_ \u2014they are not all of national importance or life-altering for the lawyer. He tells law students \"that a lot of them get ungrateful clients, and witnesses that lie to you; and you don't feel that you're still helping to solve society's problems, which is what lawyers do.\" But many cases you take will be life-altering for your clients: \"I feel lucky that I got to know that family and had the chance to help them, and that I was involved in an issue that is important to society. But for me it was always about the family and getting to their end\u2014as bad as that end was.\"\n\nIn addition to not being afraid to take a new path, a large part of Colby's happiness may have to do with how he looks back at the path he has taken. If _you_ are fortunate\u2014and Colby's \"luck\" may relate to his general outlook on the world as much as the circumstances of that little pro bono probate matter he agreed to handle\u2014you may take away life lessons. The _Cruzan_ case, he says, \"was a great blessing for me, because it always gave me and still today gives me the ability (when I start to get the grump-its even a little bit) to say my life's pretty good\u2014which is nice to have.\"\n\nIf there is one consistent piece of career advice that experienced lawyers seem to give law graduates it is to follow their bliss. One former lawyer, now a teacher, has this to say:\n\nFollow your heart in choosing your career path. It's the most important advice I give my students each year. Such simple advice, yet so hard to follow. If you get it wrong the first time, which you are likely to do, don't be afraid to change course.\n\nHis sentiments are echoed by another attorney from the nation's capital:\n\nMy advice to a recent graduate is: find something that you love to do, something that will motivate you to get out of bed when you are tired, something that you will try to work on when you are sick, something that you would want to do even if you didn't have to earn a living, and then see if someone will pay you to do that work. And, understand that even when you're in the right job, doing the right thing, and happy, there will still be bad days, there will still be things that you don't like. There will be moments when you don't think you can do what needs to be done. Take a breath, take some chocolate, triage, prioritize, get something done, cross it off your list and move on to the next thing. Don't let the best be the enemy of the good.\n\n#### WHAT HAPPINESS RESEARCH PREDICTS ABOUT YOUR CAREER\n\n1. You are more likely to be happy with your career if you work in solo practice or a small firm than in a large firm.\n\n2. You are more likely to be happy if you work for the government than for a private firm.\n\n3. You are more likely to be happy if your compensation is above $70,000 per year than if it is below that amount.\n\n4. You are more likely to be happy if you have been a lawyer for ten years or more than if you recently graduated from law school.\n\n5. You are more likely to be happy if you've always wanted to be a lawyer since a young age.\n\n6. You are likely happy with the intellectual challenge that your law career affords.\n\n7. If you are a woman, you are more likely to be dissatisfied with your work-life balance than if you are a man.\n\n8. You are more likely to be happy if you like and trust the people you work with.\n\n9. You are somewhat less likely to be happy with your career if you graduated from a top national law school than if you graduated from a \"fourth-tier\" law school.\n\n10. You are more likely to be happy if you believe that your work aligns with your values.\n\n11. You are most likely to complain about lack of feedback, lack of work-life balance, unethical attorneys and ungrateful clients, and meaningless work.\n\n12. You are likely to change jobs three or more times before your career ends.\n\n13. You are likely, at the end of your career, not to regret your decision to become a lawyer (although there will have been days when you questioned your decision).\n\n## **CHAPTER 8\n\nSeeking Happier Ground**\n\n_To everything there is a season, and a time to every purpose under heaven... A time to weep, and a time to laugh; a time to mourn, and a time to dance..._\n\n\u2014 _Ecclessiates_ 3:1\u20133:4\n\n_How we spend our days is, of course, how we spend our lives._\n\n\u2014Annie Dillard, _The Writing Life_\n\nTHE BOOKS OF OUR CHILDHOOD TOLD OF LIVES LIVED happily ever after. As adults, we know that real lives are never like that. There is no kingdom of happiness that we cross into and then dwell within forever. Happiness is a shifting landscape on which no person ever becomes a permanent resident. The best we can hope for is to grab our share of visits. We also recognize that what brings us happiness at one time might not the next. If someone offers a surefire recipe for happiness, be skeptical. What happiness we experience is the result of a mix of genetics, circumstances, and our own thoughts and actions. The complexities of our emotions are beyond capture in a formula. We can, however, as this book has sought to demonstrate, increase the likelihood that we will experience more happiness in our legal careers. We do what we can, and we hope that it is enough. Sometimes, as songwriter Townes Van Zandt says, \"the blue wind blows our dreams away.\" But on other days, the gentle breeze comes, the clouds part, and the sun begins to shine.\n\nHow do we position ourselves to best capture that kind breeze when it comes along? That's really what this book has been about. But we need to understand that sometimes skies stay gray for longer stretches than we'd like; we need to appreciate our emotional lows as well as our highs.\n\n### MAKING THE BEST OF IT: THE USEFULNESS OF BOREDOM AND MELANCHOLY\n\nWe'll be the first to admit that you could act on every piece of advice offered in this book and still not be happy. Perfect happiness was not part of the plan. (Whether it's God's or Evolution's, we'll leave for you to decide.) Sometimes our genetic makeup, in the form of a low happiness set point or a predisposition to depression, keeps us tethered to unhappy ground. At other times, adverse circumstances\u2014ranging from marital or health problems to a bad slump for our favorite sports team\u2014block our path to happiness. Be patient. Those less-than-happy times of our lives are inevitable and it's important that we make the best of them.\n\nMartin Seligman, probably the most prominent voice in the positive psychology movement, admitted that he once believed that he could make a person happy just by ridding them of their sadness. He found, however, that he couldn't. When he simply focused on ridding a patient of his sadness what he got was not a happy person but an empty one.\n\nHappiness and sadness are not opposite sides of the same coin; they are sides of different coins. Both are necessary in any complete life. If you're feeling sad, sometimes you just have to accept that sadness is part of life. Would you value your happiness as much if you've never been sad? Could you say you really _knew_ happiness if you'd never met unhappiness? Carl Jung, founder of analytical psychiatry, noted that the \"word 'happy' would lose its meaning if it were not compensated by some sadness.\" Of all the many conversations Eric Weiner had during his worldwide exploration of \"the geography of bliss,\" none stuck with him more than the comment of a musician in the happy city of Reykjavik, Iceland: \"'I'm happy, but I cherish my melancholia.'\"\n\nNurture your melancholy and use it as an opportunity for personal growth or better self-understanding. Melancholy is critical to an understanding of what it means to be human. It leads, as Eric G. Wilson argued in _Against Happiness: In Praise of Melancholy_ , to an awareness of things passing, and from that awareness we gain a deeper sense of our own mortality and, it is hoped, a better grasp of the world's fragile beauty. Or, as Alan Wolfe recently contended, \"[A]n unhappy consciousness may take us further toward understanding than a Bovary-like contentment.\"\n\nThe notion that melancholy might have its beneficial aspects is not new. Aristotle contended that \"all men who have attained excellence in poetry, in art and politics, even Socrates and Plato, had a melancholic habitus.\" Charles Darwin, a rather melancholic soul himself, suspected that sadness serves an evolutionary purpose in that it is \"well adapted to make a creature guard against any great or sudden evil.\"\n\nNeuroscience research supports Aristotle's and Darwin's ideas that melancholy promotes creativity and helps focus the mind on problems. Negative moods increase activity in the prefrontal cortex and make \"it easier to pay continuous attention to a difficult dilemma.\" Persistence, a side-benefit of melancholy, is a trait associated with people who achieve success in creative fields; authors and artists often walk on the dark side. Also, research shows, sadness correlates with a \"more successful communication style,\" higher scores on intelligence tests, a better ability to judge the accuracy of rumors, and enhanced recall. In one study, shoppers remembered four times as many trinkets stacked near a check-out counter on rainy days than they did on sunny days. Jonah Lehrer, in a _New York Times Magazine_ article summarizing the recent research, concludes that sadness \"draws us toward our proplems, like a magnet to metal.\" When sadness enters your day at the office, tackle that complicated legal question that you've put on the back burner. There is, as Ecclesiates reminds us, \"a time to mourn and a time to dance.\" Accept what both times can offer you.\n\nBoredom is the pale hint of sadness. Just as into every life a little rain must fall, into every career a little boredom must creep. Not every task in a lawyer's day can produce flow. There will be repetitious tasks and tedious tasks. In a 1994 commencement address, poet Joseph Brodsky told Dartmouth College graduates, \"A substantial part of what lies ahead of you is going to be claimed by boredom.\" He warned that \"even unusually gifted individuals must endure boredom.\" Boredom compels us to invent means to escape its clutches. In the context of careers, your avenue of escape might entail looking for a more satisfying job. Although this search might keep you entertained for a while, Brodsky warned that \"before long this quest turns into a full-time occupation, with your need for an alternative coming to match a drug addict's daily fix.\" Instead, he urged that we learn from boredom one of life's most central lessons: Boredom \"is your window on time's infinity.... It puts your existence in proper perspective, the net result of which is precision and humility.... And the more finite a thing is, the more it is charged with life, emotions, joy, fears, compassion.\"\n\nDo not try to be happy 100 percent of the time. You will fail. More than that, you will not realize your potential. Comedian Conan O'Brien, being serious for a moment, ended his 2000 commencement address to Harvard University students with a wish that they experience both times of happiness and times of sadness: \"So that's what I wish for all of you. The bad as well as the good. Fall down, make a mess, break something occasionally, and remember the story is never over.\"\n\nDo not run from sadness and boredom. The path to happiness often runs through, not around, those often unappreciated emotional states.\n\n### THE WELL-LIVED LIFE\n\nWe've saved for this last chapter one of the most important messages we want to convey: happiness isn't everything. In fact, as odd as it may seem for the authors of a book called _The Happy Lawyer_ to say, happiness isn't even\u2014for everybody\u2014the most important thing.\n\nFor most of human history, happiness wasn't the issue, survival was. And when philosophers began their serious pondering, they by and large ignored the question of happiness and focused on other values such as virtue and honor and loyalty. The primary message of our central religious texts, the Bible, the Koran, and others, is not \"Be happy!\" In a world of war and woe, who could question the words of Thomas Jefferson: \"Perfect happiness... was never intended by the Deity to be the lot of his creatures.\"\n\nHistory shows us that obsession with happiness is a modern development made possible by the high standard of living now enjoyed by many occupants of the planet. Our expectations for achieving happiness\u2014at least in _this_ life\u2014outstrip our capacity to achieve it. Setting realistic goals about our own future happiness, including happiness in our careers, is the first step toward getting there. Your brain, unable to comprehend life's messy reality, has constructed an abstract and nearly perfect model of the ideal job. Your real job, in contrast to the ideal job lodged in your brain, will always fall short, given the sometimes triviality of day-to-day work in the office and a law career's inevitable disappointments.\n\nAs you stumble into dissatisfaction from time to time, remember that those other high values of old still matter. A single-minded striving for personal happiness can damage the fabric of society if it leads to neglect of family or community.\n\nFortunately, as we've suggested, the best path to happiness lies _in the direction_ of family and community. The point here is that the search for happiness, properly undertaken, can lead to a life well-lived. Being true to the noblest values of our profession will almost never hinder your quest for career satisfaction. If you turn out to be one of those persons constitutionally unable to find true happiness, find consolation in knowing that you have served your values and done well by others.\n\nBut, hey, you bought this book because you wanted a happier career as a lawyer, not because you want to be told that it's okay if that doesn't happen. And, of course, happier is better than not. We'll end this exploration of lawyer happiness by reminding ourselves of what we consider most central to satisfying careers in law.\n\n### GETTING READY FOR HAPPINESS\n\nYou move toward happier ground when you first open yourself up to the possibility of becoming a happier lawyer. Lawyers are purposeful people. You might be inclined to tell yourself, \"I'm going to become happier, damn it, and I'm going to become happier right now!\" In fact, such an attitude more likely will prolong your unhappiness than lead you out of it. As John Stuart Mill once noted, \"'Happiness should be approached sideways like a crab.'\"\n\nHappiness is experienced mainly in the present. Obsessing about your unhappiness, or overanalyzing your condition, takes you out of the present and into your past or future. To quote Mill again, \"Ask yourself if you are happy and you cease to be so.\" With that caveat in mind, what attitudes or steps get us heading toward a more satisfying career? Let's review what we consider to be the five factors most closely tied to lawyer happiness.\n\n#### 1. FIND WORK THAT INTERESTS YOU\n\nLet's consider the nature of your work. Does it interest you? Do you feel challenged by your work, yet not so much so as to be constantly anxious about your ability to meet the expectations of clients or supervisors? If the answer to either of these questions is \"no\"\u2014if you are bored by your work, or tired of filling out the same forms or making the same simple arguments\u2014it's time to explore whether you can change departments, clients, or the way you tackle your assignments. The field of law is not perfect, but it is, in many of its manifestations, challenging. If you don't feel challenged, look around, discover where the challenges are, and rise to meet them. A good first step to finding interesting work is to gain an understanding of your own competencies. Once you have identified your strengths, try to find a way to use them.\n\nGood work is challenging, but it also affords variety. As humans, we crave novelty. Performing the same task for the hundredth time will afford you little of the satisfaction that it once did. If you're lucky to find a job with not only content variety but also variety in work settings and colleagues so much the better. Novelty enhances the intensity of experience.\n\n#### 2. ALIGN YOUR WORK WITH YOUR VALUES\n\nThe closer your values align with your work, the happier you feel about your work. Even boring or anxiety-producing work has its benefits if it is consistent with your values. When asked to identify peak experiences, lawyer after lawyer in our study turned to cases where they believed that their work made a difference. There's a very important lesson here. Absorb it. Think about your deepest beliefs and, when there is a conflict between what your heart and what your head is telling you to do, listen to your heart.\n\nWhen you find work that you can love, commit to it. Commitment is a word associated most commonly with relationships, but it is no less central to a successful career than a successful marriage. Take that foot out of the door and you'll begin to build better workplace relationships and find greater career fulfillment.\n\n#### 3. BALANCE YOUR WORK AND THE REST OF YOUR LIFE\n\nAnother large theme in this book has been the importance of control. We've explained that control has many aspects, all of which are associated with increased career satisfaction. Still, for lawyers at least, one aspect stands above the others. If you feel good about the balance you have between your work obligations and your obligations to others (including family, friends, and self), you're likely to feel a lot better about your job.\n\nOf course, achieving a work-life balance is easier said than done. If your job makes such a balance impossible\u2014well, _do something_. If you are in a large firm, you might try to convince the powers-that-be that more concern about work-life balance issues is in the firm's long-term interest. If you are in solo practice or a small firm, perhaps you can learn to say \"no\" to some potential new clients, even if it means a loss of income.\n\nKids grow up quickly; your athletic prowess slips away; friendships die without attention. Don't keep postponing your happiness. You might just postpone it all the way to your graveyard.\n\n#### 4. DEEPEN WORKPLACE RELATIONSHIPS\n\n\"Social embeddedness\" is the single best predictor of happiness. Consider the people with whom your work hours are spent. If you are to become happier, in large part it will be because of them. If our exploration of happy lawyering has convinced us of anything, it is that relationships with other human beings are the greatest source of happiness. People who feel connected in meaningful ways with people they love and trust are best positioned to find happier ground. Trust: it's a small word with big meaning. The inability to trust will undermine happiness as surely as the capacity and willingness to trust will promote happiness\u2014in you and in those around you.\n\nPsychiatrist George Valliant devoted his lifetime to a seventy-year longitudinal study of Harvard University graduates of the late '30s and early '40s. When asked recently what the study taught him, he replied, \"That the only thing that really matters in life [is] your relationships to other people.\" Happiness is built above all on love, trust, and a sense of feeling valued. Should you not feel valued in your present workplace, should your present workplace not offer opportunities for deepening relationships, get out. There has to be some other job that would serve you better.\n\n#### 5. SAVOR THE SMALL PLEASURES\n\nBen Franklin urged people to find happiness in \"the little advantages\" of their daily lives. Almost all lawyers have some aspects of their jobs that give them pleasure. For you, it might be strategizing with colleagues about litigation, outlining an appellate brief, or just experiencing the sense of satisfaction that comes from clearing a few headaches out of your inbox. Whatever those moments are, recognize them. This requires some self-study: we are not all naturally attuned to what gives us genuine pleasure. The warm glow from achieving a good settlement, the buzz of excitement from brainstorming about an interesting case, the feeling of satisfaction from writing a well-turned phrase: Happiness is found in such moments. Savor them. Try to find ways to experience more of them. You might never experience the ecstasy of a winning a victory in the U.S. Supreme Court, but you will feel the pleasure of a client's compliment.\n\n### WHAT WILL BEING A HAPPY LAWYER DO FOR THE REST OF MY LIFE?\n\nAs nice as increased career satisfaction is, don't expect it to change your overall happiness level dramatically. As one career advisor pointed out, \"Asking a job to solve our unhappiness problem is asking too much of a job.\" She offered this analogy: It's \"like asking an exercycle to solve your weight problems without giving up doughnuts.\" Our overall happiness levels are more affected by our close personal relationships than they are by our career satisfaction. Developing a close set of nonwork friends, finding your true love, or making up with your estranged teenage daughter is the better bet for making life as a whole look sunny.\n\nStill, careers are important. In addition to meeting the innate need of humans to be competitive, a career adds to your \"self-complexity.\" Your career as a lawyer adds another dimension to your life. You may be, for example, a mother, a friend, a member of the P.T.A, a Presbyterian, and an Ohioan, but you are also a member of the legal profession. When things go sour in one or more of the other dimensions of your life, law can be a refuge or a source of pride. Your practice can be a buffer against setbacks in other domains of life. In general, the more self-complex we are, the more resilient we are\u2014and resilience in the face of difficulty is one of the best measures of a successful life.\n\nYou didn't need us to tell you that life is not one smile after another. You probably have realistic expectations as to how much more satisfying your law career can be. Even professional chocolate tasters have bad days at the office. We sincerely believe, however, that by taking some of the steps identified in this book you can have a happier career. The level of improvement is dependent upon such things as your happiness set point, what's going on in the rest of your life, the range of career options open to you, and how much you open yourself up to the possibility of greater career satisfaction.\n\n### A FINAL WORD\n\nIn the course of researching this book, we read a great many opinions about what makes us happy. Every author, it seems, uses the last page to identify the most important \"take away\" they want the reader to have. In our opinion, the best of these authors tended to circle around the same point: love makes us happier than anything. By this, we do not necessarily mean romantic love. Rather, it is the love inherent in _being attentive_ \u2014caring deeply about somebody or something. It could as well be love for justice or a cause or the life of the mind. British scholar Avner Offer puts the point concisely: \"Attention is the universal currency of well-being.\"\n\nWe generally think of love as it relates to our feelings for another human being. But it is also possible to love your job\u2014to pay attention to your work and to respect its traditions and noble goals. Like any marriage, your relationship with your work will have its tensions. There may be days when you wish you were doing something else. But as long as your work holds your attention, there's more than just hope for being a happy lawyer.\n\n## **Acknowledgments**\n\nWE ARE INDEBTED TO THE MORE THAN TWO HUNDRED lawyers across the country who took the (nonbillable) time to give thoughtful answers to our survey and share their stories.\n\nWe thank our colleagues and friends who brought so many different facets of their expertise to this project: Jasmine Abdel-khalik, David Achtenberg, Terri Beiner, Mark Berger, Bill Black, Paul Callister, Julie Cheslik, Bill Colby, Richard Delgado, Bob Downs, Alice Eakin, Jerald Enslein, Kathy Hall, Bob Hayman, Peter Huang, Mary Kay Kisthardt, Tony Luppino, Mira Mdivani, Lynda Moore, Sean O'Brien, Colin Picker, Judy Popper, Ortrie Smith, Barb Snell, Ellen Suni, Rob Verchick, Wanda Temm, Kevin Travis, and Dan Weddle. Special thanks to Lara Krigel Pabst, Andrew Schermerhorn, and Katie Woods for research assistance, to Elizabeth Johnson for administrative assistance, and to librarian Lawrence MacLachlan for research guidance. We also thank the University of Missouri-Kansas City Law Foundation for its research grant and the _Syracuse Law Review_ for permission to include a portion of our earlier article, _Happy Law Students, Happy Lawyers_ , 58 Syracuse L. Rev. 351 (2008), in this book. We are especially grateful to Naomi Cahn, June Carbone, Barb Glesner Fines, Aaron Geary, Andrew McClurg, and Allen Rostron for their thoughtful advice, constructive criticism, editing wizardry, playfulness, and unflagging support.\n\nOur greatest thanks are for our families, to whom this book is dedicated.\n\nOur gratitude is also an acknowledgement, that can't really be captured in words, of how important all of you are in our lives. You make us so happy.\n\n## **Notes**\n\n### _Foreword_\n\n. _Fax Poll: It Becomes a Miserable Profession_ , CAL. LAW., Mar. 1992, 3; Alex Williams, _The Falling Down Professions_ , N.Y. TIMES, Jan. 6, 2008, at 91.\n\n. Stephanie Francis Ward, _Pulse of the Legal Profession_ , A.B.A. J., Oct. 2007, 29, 32 (noting that only 44 percent of eight hundred survey respondents say \"I would recommend a legal career to a young person\").\n\n. Ashby Jones, _The Third Year Dilemma: Why Firms Lose Associates_ , WALL ST. J., Jan. 4,2006, _available at_ <http:\/\/208.144.:115-173\/salarydata\/law\/20060105-jones.html>.\n\n### _Chapter 1_\n\n. Robert Biswas-Diener et al., _Most People Are Pretty Happy, But There Is Cultural Variation: The Inughuit, the Amish, and the Maasai_ , 6 J. HAPPINESS STUD. 205 (Sept. 2005).\n\n. Sue M. Halpern, _Are You Happy?_ , 55 N.Y. REV. BOOKS 24 (Apr. 3, 2008), _available at_ <http:\/\/www.nybooks.com\/articles\/21197>.\n\n. John Monahan & Jeffrey Swanson, _Lawyers at Mid-Career: A 20-Year Longitudinal Study of Job and Life Satisfaction_ , 5 J. EMPIRICAL LEGAL STUD, I, 3 (forthcoming 2009).\n\n. RONIT DINOVITZER ET AL.,AFTER THE J.D.: FIRST RESULTS OF A NATIONAL STUDY OF LEGAL CAREERS 19 (2004); AMERICAN BAR ASSOCIATION, 2009 NATIONAL LAWYER POPULATION SURVEY, <http:\/\/www.abanet.org\/market research\/2009_NATL_LAWYER_by_State.pdf>\n\n. TOM W. SMITH, NATIONAL OPINION RESEARCH CENTER\/UNIVERSITY OF CHICAGO, JOB SATISFACTION IN THE UNITED STATES (Apr. 17, 2007) [hereinafter \"NORC Study\"]. The NORC Study surveyed more than 27,500 randomly selected people concerning their job satisfaction and happiness in 198 occupations. Clergy reported the highest levels of both job satisfaction (87.2 percent very satisfied) and general happiness (67.2 percent very happy). Roofers had the lowest mean score (only 25.3 percent very satisfied) for job satisfaction, and service station attendants the lowest mean score for general happiness (only 13.2 percent were very happy); 52.4 percent of lawyers said they were very satisfied with their jobs, and 43 percent said they were generally happy in life.\n\n. Peter H. Huang & Rick Swedloff, _Authentic Happiness & Meaning at Law Firms_, 58 SYRACUSE L. REV. 335, 343 (2008).\n\n. Stephanie Francis Ward, _Pulse of the Legal Profession_ , A.B.A. J., Oct. 2007, at 30, 32.\n\n. AMERICAN BAR ASSOCIATION YOUNG LAWYERS DIVISION, ABA YOUNG LAWYERS DIVISION SURVEY: CAREER SATISFACTION 17 (2001), <http:\/\/www.abanet.org\/yld\/satisfaction_8oo.doc>. The survey of 2,136 members of the Young Lawyers Division yielded 842 responses. \"Most young lawyers are at least somewhat satisfied both with their current job and with the practice of law generally.\" _Id._ at 1. These numbers match up with an American Bar Foundation study of Chicago lawyers in the mid-1990s, in which 84 percent reported being either satisfied or very satisfied with their work. John P. Heinz et al., _Content With Their Calling? Work Satisfaction in the Chicago Bar_ , 9 AM. B. FOUND. PUB. 1 (1998).\n\n. Susan Daicoff _Lawyer, Be Thyself: An Empirical Investigation of the Relationship Between the Ethic of Care, the Feeling Decisionmaking Preference, and Lawyer Wellbeing_ , 16 VA. J. SOC. POL'Y & L. 87 (2008).\n\n. AMERICAN BAR ASSOCIATION YOUNG LAWYERS DIVISION, _supra_ note 8, at 17.\n\n. For every report showing generally happy lawyers, fulfilled by their work, there's another with glum numbers suggesting the opposite. Is there an explanation for the contradictory results? One possibility\u2014and this is true for all the survey data\u2014may have to do with the different populations sampled. For instance, those who responded to the ABA Young Lawyers Division job satisfaction survey might disproportionately be people who like to belong\u2014joiners. Is it possible that ABA Young Lawyers are more gregarious and upbeat\u2014happier\u2014than the average lawyer? Sure, it is.\n\nAnother variable that can compromise the validity of studies is how they are phrased. If survey headings telegraph the results the researchers are seeking, respondents will be inclined to self-select either toward or away from participating and even give answers they think will please the researchers. As one example, \"the American Bar Association surveys conducted in 1984 and 1990 consisted of a mail-back questionnaire titled 'National Survey of Lawyer Satisfaction\/Dissatisfaction.' Although response rates for these surveys were fairly high, there is no way to know for sure whether disgruntled lawyers were disproportionately willing to participate.\" Kathleen E. Hull, _Cross-Examining the Myth of Lawyer Misery_ , 52 VAND. L. REV. 971, 972 (1999).\n\nSurvey results also depend heavily on the questions asked. If the question asked is \"Are you happy right now?,\" that will elicit a different set of responses than if the questions are about plans to leave a job within five years. Attorney satisfaction scores often stem from surveys that ask a domain-specific happiness question: \"Are you happy with your job?\" The reported domain-specific happiness scores can be affected by other life satisfaction variables or recent life events.\n\n. _Fax Poll: It Becomes a Miserable Profession_ , CAL. LAW., Mar. 1992, at 3; Alex Williams, _The Falling Down Professions_ , N.Y. TIMES, Jan. 6, 2008, at 91.\n\n. Heinz et al., _supra_ note 8, at 1.\n\n. Hull, _supra_ note 11, at 972. The _California Lawyer_ poll is discussed in the text _supra_ note 12.\n\n. Monahan & Swanson, _supra_ note 3, at 2.\n\n. Ronit Dinovitzer & Bryant G. Garth, _Lawyer Satisfaction in the Process of Structuring Legal Careers_ , 41 LAW & SOC'Y REV. 1 (2007).\n\n. Ward, _supra_ note 7, at 34.\n\n. Ashby Jones, _The Third Year Dilemma: Why Firms Lose Associates_ , WALL ST. J., Jan. 4, 2006, _available at_ <http:\/\/208.144.15-173\/salarydata\/law\/20060105-jones.html>.\n\n. Williams, _supra_ note 12, at 91.\n\n. DOUGLAS LITOWITZ, THE DESTRUCTION OF YOUNG LAWYERS: BEYOND ONE L 9 (2006).\n\n. Martin E.P. Seligman et al., _Why Lawyers Are Unhappy_ , 23 CARDOZO L. REV. 33,52 (2001).\n\n. Robert Kurson, _Who's Killing the Great Lawyers of Harvard?_ , ESQUIRE, Aug. 2000, at 84.\n\n. William W. Eaton et al., _Occupations and the Prevalence of Major Depressive Disorder_ , 32 J. OCCUPATIONAL MED. 1079, 1083 (1990); G. Andrew H. Benjamin et al., _The Prevalence of Depression, Alcohol Abuse, and Cocaine Abuse Among United States Lawyers_ , 13 INT'L J.L. & PSYCHIATRY 233, 240 (1990), _available at_ http:\/\/www.lawyerswithdepression.com\/uploads\/dep2.pdf. Other studies do not show that law is the occupational category with the very highest rate of suicide but still show that lawyers are in higher risk jobs. Steven Stack, _Occupation and Suicide_ , 82 SOC. SCI. Q. 384, 392 (June 2001).\n\n. Stack, _supra_ note 23, at 391.\n\n. In chapter 3 we explore further the question of whether the practice of law makes people unhappy or whether people prone to unhappiness are drawn to law.\n\n. _See_ Seligman et al., _supra_ note 21, at 45\u201449.\n\n. Hull, supra note 11, at 971.\n\n. _See, e.g._ , Monahan & Swanson, _supra_ note 3, at 2.\n\n. AMERICAN BAR ASSOCIATION, LAWYER DEMOGRAPHICS, 2008, <http:\/\/www abanet.org\/marketresearch\/Lawyer_Demographics_2008.pdf>\n\n. Dinovitzer & Garth, _supra_ note 16, at 7; Ward, _supra_ note 7 at 32.\n\n. _Study Reveals Conflicts in Attorneys' Personal Lives_ , Mo. LAW. WKLY. Feb. 20, 2006, _available at_ 2006 WLNR 9004939; Ward, _supra_ note 7, at 34.\n\n. David G. Blanchflower & Andrew Oswald, 7s _Weil-Being U-Shaped Over the Life Cycle?_ , J. SOC. SCI. & MED. (Aug. 14, 2006), _available at_ http:\/\/www.nd.edu\/~adutt\/activities\/documents\/BlanchOsUshapeCohorts14Aug2006.pdf (thirty-five-year longitudinal study); Mark Killian, _TV Advertising Dilutes Public Confidence in the Profession_ , 30 FLA. B. NEWS 1 (July 15, 2003).\n\n. Ward, _supra_ note 7, at 32.\n\n. BUREAU OF LABOR STATISTICS, U.S. DEP'T OF LABOR, OCCUPATIONAL OUTLOOK HANDBOOK 2008\u201309, LAWYERS, <http:\/\/www.bls.gov\/oco\/ocos053.htm> (last visited July 20, 2009). The median salary for associates who have at least three years of experience in Houston is $86,300; $89,200 in Philadelphia; $93,500 in Washington D.C; and $105,800 in San Francisco, with a national average of $79,500. For mid-career attorneys (ten to fifteen years of experience), salaries range from $128,600 in Houston to $134,700 in Washington, D.C., and $143,700 in San Francisco, with a national average of $120,500. Justin Rebello, _Pay Scale: Who Makes What Where?, in_ TWENTY THINGS LAWYERS NEED TO KNOW 20\u201322 (Nov. 2008). The median salaries of all lawyers nine months after graduation in 2005 was $60,000; for those who entered private practice, it was $85,000; for those entering government work, $46,158; for those in academics or judicial clerkships, $45,000. BUREAU OF LABOR STATISTICS, _supra._\n\n. _See_ David Leonhardt, _Money Doesn't Buy Happiness... Well, On Second Thought..._ , N.Y. TIMES, Apr. 16, 2008, at CI (\"In the United States, about 90 percent of people in households making at least $250,000 called themselves 'very happy' in a recent Gallup Poll. In households with income below $30,000, only 42 percent of people gave that answer.\").\n\n. Daniel Kahneman et al., _Would You Be Happier If You Were Richer? A Focusing Illusion_ , SCIENCE, June 30, 2006, at 1908, 1909.\n\n. Daniel Gilbert, _What You Don't Know Makes You Nervous_ , N.Y. TIMES, May 20, 2009, at A35.\n\n. Betsey Johnson & Justin Wolfers, _The Paradox of Declining Female Happiness_ , 1 AM. ECON. J: ECON POL'Y 190, 217\u2013224 (2009).\n\n. Nearly as many women as men enter the practice of law, although there is a decreasing entry rate into the profession for women. In 1980 slightly more than one-third of law students (34.2 percent) were women. This number peaked to almost half (49 percent) in 2002 and 2003, but it has dropped to just under 47 percent in the past five years. AMERICAN BAR ASSOCIATION, FIRST YEAR AND TOTAL J.D. ENROLLMENT BY GENDER, 1947\u20132008, http:\/\/www.abanet.org\/legaled\/statistics\/charts\/stats%20-%206.pdf (last visited Feb. 23, 2008).\n\n. AMERICAN BAR ASSOCIATION COMMISSION ON WOMEN, THE 2008 GOAL IX REPORT CARD\u2014AN ANNUAL REPORT ON WOMEN'S ADVANCEMENT INTO LEADERSHIP POSITIONS IN THE ABA, Feb. 2008, at 4.\n\n. DINOVITZER ET AT., _supra_ note 4, at 58.\n\n. MONA HARRINGTON & HELEN HSI, WOMEN LAWYERS AND OBSTACLES TO LEADERSHIP 12 (2007), <http:\/\/web.mit.edu\/workplacecenter\/docs\/law-report_4-07.pdf> _See also_ Theresa M. Beiner, _Not All Lawyers Are Equal: Difficulties That Plague Women and Women of Color_ , 58 SYRACUSE L. REV. 317, 326(2008).\n\n. Monahan & Swanson, _supra_ note 3, at 20, 24, 2.\n\n. Kenneth G. Dau-Schmidt et al., _Men and Women of the Bar: The Impact of Gender on Legal Careers_ , 16 MICH. J. GENDER & L. 49 (2009).\n\n. _Gender or Childcare? Study Sheds Light on Career Roadblocks_ , INDIANA L., Fall 2009, at 3.\n\n. _Id._ (quoting Ken Dau-Schmidt).\n\n. Dau-Schmidt et al., _supra_ note 44, at 72.\n\n. In 2009, 31 percent of all lawyers were women. AMERICAN BAR ASSOCIATION, 2009 NATIONAL LAWYER POPULATION SURVEY, _supra_ note 4. This compares to 30.2 percent in 2006\u20147, and 29.4 percent in 2005\u20146. These percentages were up from 1990 when women comprised 22 percent of the profession and from 2000 when women were 27 percent of lawyers. _See also_ Malaika Costello-Dougherty, _We're Outta Here: Why Women Are Leaving Big Firms_ , CAL. LAW., Feb. 2007, at 20.\n\n. Danielle M. Evans, Note, _Non-equity Partnership: A Flawed Solution to the Disproportionate Advancement of Women in Private Law Firms_ , 28 WOMEN'S RTS. L. REP. 93, 94 (2007). \"By 2000, 'only [sixty-one] percent of the female partners in the National Law Journal's 250 firms [had] equity status, as opposed to almost [seventy-five] percent of the male partners.'\" _Id._ at 94.\n\n. Laura T. Kessler, _Keeping Discrimination Theory Front and Center in the Discourse Over Work and Family Conflict_ , 34 PEPP. L. REV. 313, 316n.17 (2007).\n\n. _See_ AMERICAN BAR ASSOCIATION, VISIBLE INVISIBILITY: WOMEN OF COLOR IN LAW FIRMS 8\u201310 (2006); Jill Schachner Chanen, _Early Exits: Women of Color at Law Firms Tell ABA Researchers They Are Being Overlooked and Undervalued_ \u2014 _Maybe That's Why They Are Leaving in Droves_ , 92 A.B.A. J 32, 33, 35 (Aug. 2006).\n\n. Eric Swedlund, _Choose Law! Event Targets Would-be Lawyers_ , ARIZ. DAILY STAR, Mar. 3,2007, at B5 (noting that 3.9 percent are African American, 3.3 percent are Hispanic, 2.3 percent are Asian American, and 0.2 percent are Native American).\n\n. DINOVITZER ET AL., _supra_ note 4, at 64.\n\n. _Id._ at 64\u201365.\n\n. More than half of all minorities\u2014male and female\u2014leave within three years. See Wayne J. Lee, Brown v. Board, _Are We There Yet?_ , 51 LA. B.J. 404, 405 (Apr.\/May 2004).; Charles Toutant, _Women, Minorities Make Strides at Firms, But ABA Study Casts Pall_ , 185 N.J.L.J. 653, Aug. 21,2006, at 3; Debra Cassens Weiss, _Survey Reveals the Unhappiest Associates_ , A.B.A. J., Nov. 7,2007, <http:\/\/www.abajournal.com\/news\/survey_reveals_the_unhappiest_associates\/> (last visited Jan. 4, 2008).\n\n. Charles Toutant, _Women, Minorities Make Strides at Firms, But ABA Study Casts Pall_ , 185 N.J.L.J. 653, Aug. 21, 2006, at 3.\n\n. Dinovitzer & Garth, _supra_ note 16, at 12.\n\n### _Chapter 2_\n\n. Carlin Flora, _The Pursuit of Happiness_ , PSYCH. TODAY, Jan.\/Feb. 2009, at 62.\n\n. Ruut Veenhoven, World Database of Happiness, <http:\/\/worlddatabaseofhappiness.eur.nl\/> (last visited July 8, 2009).\n\n. Thinkers in numerous other disciplines, from economists and sociologists to epidemiologists and linguists, are drawing on and adding to these works in science and psychology. See, _e.g._ , BRUNO S. FREY & ALOIS STUTZER, HAPPINESS & ECONOMICS: HOW THE ECONOMY AND INSTITUTIONS AFFECT WELL-BEING (2001); Benedict Carey, _Does a Nation's Mood Lurk in Its Songs and Blogs_ , N.Y. TIMES, Aug. 3, 2009, _available at_ <http:\/\/www.nytimes.com\/2009\/08\/04\/health\/04mind.html>. Our focus is on the fields we think will be most instrumental in leading to happiness for lawyers.\n\n. David M. Buss, _The Evolution of Happiness_ , 55 AM. PSYCHOLOGIST, Jan. 2000, at 15.\n\n. EDUARDO PUNSET, THE HAPPINESS TRIP: A SCIENTIFIC JOURNEY 1(2007).\n\n. _Id._ at 16\u201317.\n\n. _Id._ at 16.\n\n. Susan Viebrock, _World-Renowned Scientist Delves Into Happiness Research_ , TELLURIDE DAILY PLANET, Mar. 27, 2007, <http:\/\/www.telluridegatewaycom\/articles\/2007\/03\/28\/news\/news03.txt>.\n\n. RICHARD LAYARD, HAPPINESS: LESSONS FROM A NEW SCIENCE 17\u201318 (2005).\n\n. JEROME KAGAN, WHAT IS EMOTION? (2007), quoted in Sue M. Halpern, _Are You Happy?_ , 55 N.Y. REV. BOOKS 24 (Apr. 3, 2008), _available at_ <http:\/\/www.nybooks.com\/articles\/21197>.\n\n. Viebrock, _supra_ note 8.\n\n. Natalie Angier, _A Molecule of Motivation, Dopamine Excels at Its Task_ , N.Y. TIMES, Oct. 27, 2009, at DI, D3.\n\n. LAYARD, _supra_ note 9, at 15.\n\n. Marnia Robinson, _Your Brain on Sex, Reuniting: Healing With Sexual Relationships_ , June 25, 2005, www.reuniting.info\/science\/sex_in_the_brain.\n\n. Angier, _supra_ note 12, at D3.\n\n. PUNSET, _supra_ note 5, at 126.\n\n. Natalie Angier, _The Biology Behind the Milk of Human Kindness_ , N.Y. TIMES, Nov. 24, 2009, at D2.\n\n. Viebrock, _supra_ note 8. Viebrock quotes researcher Richard Davidson:\n\nWe discovered that when expert practitioners meditated\u2014and our subjects had between 12,000\u201462,000 hours of meditation each over the course of their lives\u2014there were major, observable changes in the brain, some quite unusual. We saw the production of certain rhythms over extended periods of time, minutes, even hours. In normal individuals, these patterns occur very episodically and last only seconds. What we observed is the brain getting reorganized. We are using these findings to identify long-term end points achievable through intense practice.\n\n_Id._\n\n. Gretchen Reynolds, _Stress Relief: Why Exercise Makes You Less Anxious_ , N.Y. TIMES MAG., NOV. 22, 2009, at 16.\n\n. Of 178 nations ranked for happiness in 2006 by Adrian White, an analytic social psychologist at the University of Leicester in England, Japan came in as the 90th happiest nation, while Canada ranked 10th and the United States 23d. _Canada Scores High on World Happiness Map_ , TORONTO'S CITY NEWS, July 28, 2006, www.citynews.ca\/news\/news_2279.aspx .\n\n. ERICWEINER, THE GEOGRAPHY OF BLISS 310 (2008).\n\n. Sue M. Halpern, _Are You Happy?_ , 55 N.Y. REV. BOOKS, Apr. 3, 2008, _available at_ <http:\/\/www.nybooks.com\/articles\/21197>.\n\n. DANIEL NETTLE, HAPPINESS: THE SCIENCE BEHIND YOUR SMILE 49, 53 (2005).\n\n. Alan Wolfe, _Hedonic Man_ , NEW REPUBLIC, July 9, 2008, at 47.\n\n. David Lykken & Auke Tellegen, _Happiness Is a Stochastic Phenomenon_ , 7 PSYCHOL. SCI. 186 (May 1996), _available at_ <http:\/\/www.psych.umn.edu\/psylabs\/happness\/happyhtm> (\"From 44% to 53% of the variance in [wellbeing], however, is associated with genetic variation. Based on the retest of smaller samples of twins after intervals of 4.5 and 10 years, we estimate that the heritability of the stable component of subjective wellbeing approaches 80%.\").\n\n. _See_ Thomas J. Bouchard, Jr. et al., _Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart_ , 250 SCIENCE 223, 250 (Oct. 12, 1990); Auke Tellegen et al., _Personality Similarity in Twins Reared Apart and Together_ , 54 J. PERSONALITY & Soc. PSYCHOL. 1031, 1036 (1988).\n\n. SONJA LYUBOMIRSKY, THE HOW OF HAPPINESS: A NEW APPROACH TO GETTING THE LIFE YOU WANT 21 (2007).\n\n. MICHAEL F. MELCHER, THE CREATIVE LAWYER: A PRACTICAL GUIDE TO AUTHENTIC PROFESSIONAL SATISFACTION 78 (2007).\n\n. _Id._\n\n. Martin E.P. Seligman et al., _Why Lawyers Are Unhappy_ , 23 CARDOZO L. REV. 33,40 (2001).\n\n. WEINER, _supra_ note 21, at 14.\n\n. LYUBOMIRSKY, _supra_ note 27, at 21.\n\n. Jonathan Rottenberg, _State Happiness Rankings Reveal Americans' Happiness Insecurity_ , Psychology Today Blog: Charting the Depths, <http:\/\/www.psychologytoday.com\/blog\/charting-the-depths\/200912\/state-happiness-rankings-reveal-americans-happiness-insecurity> (last visited Dec. 29, 2009).\n\n. Lewis Diuguid, _Survey Says We Can Be in a Happier State, Like Utah_ , K.C. STAR, Dec. 28, 2009, atAi3.\n\n. Carl Bialik, _The Drag of Devising a State-by-State Mirth Meter_ , WALL ST. J., Dec. 23, 2009, at A8. See Andrew J. Oswald & Stephen Wu, _Objective Confirmation of Subjective Measures of Human Well-Being: Evidence from the U.S.A._ , SCIENCE ONLINE, Dec. 17, 2009. <http:\/\/www.sciencemag.org\/cgi\/content\/abstract\/sci;> Science. 118o6o6vi?maxtoshow=& HITS= IO& hits= 10& RESULTFORMAT=& fulltext=andrew+oswald& searchid=I& FIRSTINDEX= 0& resourcetype=HWCIT\n\n. Rottenberg, _supra_ note 33.\n\n. Forty-three percent of people who attend religious services weekly say they are \"very happy\" compared to 26 percent of people who never attend religious services. PEW RESEARCH CENTER PUBLICATIONS, ARE WE HAPPY YET?, Feb. 13, 2006, <http:\/\/pewresearch.org\/pubs\/301\/are-we-happy-yet>.\n\n. Ruut Veenhoven, _Hedonism and Happiness_ , 4 J. HAPPINESS STUD. 437, 452 (2003).\n\n. LAYARD, _supra_ note 9, at 15\u201317.\n\n. See James H. Fowler & Nicholas A. Christakis, _Dynamic Spread of Happiness in a Large Social Network: Longitudinal Analysis Over 20Years in the Framingham Heart Study_ , 337 BRITISH MED. J. 2338 (Dec. 4, 2008).\n\n. LYUBOMIRSKY, _supra_ note 27, at 44.\n\n. _Id._ at 17, quoting Daniel Gilbert.\n\n. LAYARD, _supra_ note 9, at 30.\n\n. Adrian White, _A Global Projection of Subjective Well-Being: The First Published Map of World Happiness_ , 2006, <http:\/\/news.bbc.c0.uk\/2\/shared\/bsp\/hi\/pdfs\/28_07_o6_happiness_map.pdf>. See _also Psychologist Produces the First-Ever \"World Map of Happiness_ , SCIENCE DAILY, NOV. 14, 2006, <http:\/\/www.sciencedaily.com\/releases\/2006\/11\/061113093726.htm>.\n\n. LAYARD, _supra_ note 9, at 69.\n\n. _Id._ at 64.\n\n. _Happiness. What Is It To You?_ , <http:\/\/www.esearch.com\/member\/quiz\/results\/happiness.shtml> (last visited July 12, 2009).\n\n. LYUBOMIRSKY, _supra_ note 27, at 6.\n\n. _Id._ at 20.\n\n. Lawrence S. Krieger, _Psychological Insights: Why Our Students and Graduates Suffer, and What We Might Do About It_ , 1 J. Assoc. LEGAL WRITING DIRECTORS 265 (2002).\n\n. BARRY SCHWARTZ, THE PARADOX OF CHOICE:WHY LESS IS MORE 88 (2004).\n\n. See Jane S. Schacter, _The Gay Civil Rights Debate in the States: Decoding the Discourse of Equivalents_ , 29 HARV. C.R.-C.L. L. REV. 283, 299 (1994).\n\n. DANIEL GILBERT, STUMBLING ON HAPPINESS 33 (2006).\n\n. LYUBOMIRSKY, _supra_ note 27, at 25.\n\n### _Chapter 3_\n\n. TOM W.SMITH, NATIONAL OPINION RESEARCH CENTER\/UNIVERSITY OF CHICAGO, JOB SATISFACTION IN THE UNITED STATES (Apr. 17, 2007).\n\n. Sonja Lyubomirsky et al., _The Benefits of Frequent Positive Affect: Does Happiness Lead to Success?_ , 131 PSYCHOL. BULL. 803, 826 (2005).\n\n. _See_ SONJA LYUBOMIRSKY, THE HOW OF HAPPINESS: A NEW APPROACH TO GETTING THE LIFE YOU WANT 128\u201332 (2007); AMERICAN BAR ASSOCIATION YOUNG LAWYERS DIVISION, ABA YOUNG LAWYERS DIVISION SURVEY: CAREER SATISFACTION 20 (2001), <http:\/\/www.abanet.org\/yld\/satisfaction_800.doc>.\n\n. ABA YOUNG LAWYERS DIVISION SURVEY, _supra_ note 3, at 17.\n\n. Stephanie Francis Ward, _Pulse of the Legal Profession_ , A.B.A. J., Oct. 2007, at 31\u201332.\n\n. ABA YOUNG LAWYERS DIVISION SURVEY, _supra_ note 3, at 31, Table 23.\n\n. Jonathan Foreman, _My Life as an Associate_ , CITY J., Winter 1997, _available at_ <http:\/\/www.city-journal.org\/html\/7_1_a2.html>.\n\n. The practice of law presents other pressures as well. In a \"satisfaction survey\" of Oregon bar members, 54 percent identified time and workload pressures as among the most frustrating aspects of their work, and 47 percent said they worried about making mistakes. Janine Robben, _Burnout_ , 69 OR. ST. B. BULL. 17,18\u201319 (Oct. 2008).\n\n. _See_ National Association for Law Placement, _How Much Do Associates Work?_ , Table 3. Billable Hours Requirements per Year by Firm Size, NALP BULL., Apr. 2009, <http:\/\/www.nalp.org\/may07billablehrs;> Julie A. Oseid, _When Big Brother Is Watching [Out For] You: Mentoring Lawyers, Choosing a Mentor, and Sharing Ten Virtues From My Mentor_ , 59 S.C. L. REV. 393,409 n.84 (2008). _See generally_ JEAN STEPANCIC & RICHARD DELGADO, HOW LAWYERS LOSE THELR WAY: A PROFESSION FAILS ITS CREATIVE MINDS 53 (2005).\n\n. RONIT DINOVITZER ET AL., AFTER THE J.D: FIRST RESULTS OF A NATIONAL STUDY OF LEGAL CAREERS 33 (2004). One study offered estimates of the average number of hours that lawyers spent at work, varying from a high of fifty-three hours per week at a large private law firm to forty-seven for a small private firm, forty-nine for lawyers working in business or finance, and forty-four for government lawyers. John Monahan & Jeffrey Swanson, _Lawyers at Mid-Career: A 20-Year Longitudinal Study of Job and Life Satisfaction_ , 5 J. EMPIRICAL LEGAL STUD, 1, 35 (forthcoming 2009).\n\n. National Association for Law Placement, _supra_ note 9.\n\n. DOUGLAS LITOWITZ, THE DESTRUCTION OF YOUNG LAWYERS: BEYOND ONE L 15 (2005).\n\n. National Association for Law Placement, _Salaries at Largest Firms Up Again_ , Aug. 21, 2008, <http:\/\/www.nalp.org\/salariesatlargestfirmsupagain>.\n\n. Susan Saab Fortney, _The Billable Hours Derby: Empirical Data on the Problems and Pressure Points_ , 33 FORDHAM URB. L.J. 171, 177 (2005).\n\n. William G. Ross, THE HONEST HOUR: THE ETHICS OF TIME-BASED BILLING BY ATTORNEYS 3\u20134 (1996) (quoting Paul Reidinger, _Confessions of The Rodent_ , A.B.A. J., Aug. 1995, at 82, 83).\n\n. Michael Asimow, _Embodiment of Evil: Law Firms in the Movies_ , 48 UCLA L. REV. 1339, 1377 (2001).\n\n. Amy Kolz, _Don't Call Them Slackers_ , AM. LAW., Oct. 3, 2005, _available at_ <http:\/\/www.law.com\/jsp.article.jsp?id=ii278983ii339> (last visited Jan. 3, 2008).\n\n. Patrick J. Schiltz, _On Being a Happy, Healthy and Ethical Member of an Unhappy, Unhealthy, and Unethical Profession_ , 52 VAND. L. REV. 872, 903 (1999).\n\n. _Retaining the Winners_ , NAT'L L.J. 18, Mar. 17, 2008, at 18.\n\n. NATIONAL ASSOCIATION FOR LAW PLACEMENT, KEEPING THE KEEPERS: STRATEGIES FOR ASSOCIATE RETENTION IN TIMES OF ATTRITION (1998), <http:\/\/www.nalpfoundation.org\/webmodules\/articles\/anmviewer.asp?a=6i>. For an elaboration of generational differences, see chapter 6 at notes 10\u201418.\n\n. In response to surveys, \"[j]udges and lawyers both agreed there has been 'a significant increase in instances of incivility\/unprofessionalism in civil litigation in the last decade.'\" Jeffrey A. Parness, _Civility Initiatives:The 2009 Allerton House Conference_ , 96 ILL. B.J. 636, 637 (Dec. 2008). Other evidence of growing incivility documents increased discovery abuses, competition for clients concerns, and disciplinary actions against lawyers. _See_ Melissa S. Hung, _A Non-Trivial Pursuit: The California Attorney Guidelines of Civility and Professionalism_ , 48 SANTA CLARA L. REV. 1127, 1133 (2008).\n\n. Kenneth A. Sprang, _Holistic Jurisprudence: Law Shaped by People of Faith_ , 74 ST. JOHN'S L. REV. 753, 758 n.30 (2000).\n\n. Lilia M. Cortina et al., _What's Gender Got To Do With It? Incivility in the Federal Courts_ , 27 LAW & Soc. INQUIRY 235, 235 (2002).\n\n. Christopher J. Piazzola, Comment, _Ethical Versus Procedural Approaches to Civility: Why Ethics 2000 Should Have Adopted a Civility Rule_ , 74 U. COLO. L. REV. 1197, H99n.i4 (2003);Ward, _supra_ note 5, at 31.\n\n. Thomas M. Reavley, _Rambo Litigators: Pitting Aggressive Tactics Against Legal Ethics_ , 17 PEPP. L. REV. 637, 638 (1990).\n\n. Joseph J. Ortego & Lindsay Maleson, _Under Attack: Professionalism in the Practice of Law_ , Mar. 20, 2003, _available at_ <http:\/\/www.nixonpeabody.com\/publications_detail3.asp?ID=303#refi9>.\n\n. <http:\/\/www.youtube.com\/watch?v=td-KKmcYtrM>.\n\n. Marc S. Galanter & Thomas M. Palay, _Large Law Firm Misery:It's the Tournament, Not the Money_ , 52 VAND. L. REV. 953, 960 (1999).\n\n. David Guenther, _To Be or Not Be a Lawyer, That Is the Question_ , 2001, <http:\/\/www.100megsfree3.com\/wordsmith\/2bornoti.html>.\n\n. Douglas R. Richmond, _Law Firm Partners as Their Brothers' Keepers_ , 96 KY. L.J. 231,263 (2007\u201308).\n\n. THE GALLUP POLL, HONESTY\/ETHICS IN PROFESSIONS (2007), _available at_ http:\/\/www.galluppoll.com\/content\/?ci=1654&pg=1.\n\n. AMERICAN BAR ASSOCIATION SECTION OF LITIGATION, PUBLIC PERCEPTIONS OF LAWYERS: CONSUMER RESEARCH FINDINGS 7\u20138 (2002), http:\/\/www.abanet.org\/litigation\/lawyers\/publicperceptions.pdf.\n\n. Michael Asimow, _Bad Lawyers in the Movies_ , 24 NOVA L. REV. 533, 537 (2000).\n\n. ANDREW MCCLURG, FIGHT CLUB: DOCTORS VS. LAWYERS\u2014THE RIVALRY BETWEEN AMERICA'S MOST REVERED, REVILED, AND MISUNDERSTOOD PROFESSIONS (forthcoming 2011, manuscript on file with authors).\n\n. Margaret Raymond, _On Legalistic Behavior, the Advocacy Privilege, and Why People Hate Lawyers_ , 55 BUFF. L. REV. 929, 930 (2007).\n\n. ANTHONY J. LUPPINO, CAN DO:TRAINING LAWYERS TO BE EFFECTIVE COUNSELORS TO ENTREPRENEURS, REPORT TO THE EWING MARION KAUFFMAN FOUNDATION, Jan. 30, 2008, at 6, <http:\/\/papers.ssrn.com\/s0l3\/papers.cfm?abstract_id=1157065>.\n\n. _Id._\n\n. Arthur Gross Schaefer & Leland Swenson, _Contrasting the Vision and the Reality: Core Ethical Values, Ethics Audit and Ethics Decision Models for Attorneys_ , 32 PEPP. L. REV. 459, 459 (2005).\n\n. Stephen D. Easton, _My Last Lecture: Unsolicited Advice for Future and Current Lawyers_ , 56 S.C. L. REV. 229, 244 (2004).\n\n. Charles Silver & Frank B. Cross, _What's Not to Like About Being a Lawyer?_ , 109 YALE L.J. 1443, 1476^77 (2000).\n\n. Frank B. Cross, _The First Thing We Do, Let's Kill All the Economists: An Empirical Evaluation of the Effect of Lawyers on the United States Economy and Political System_ , 70 TEX. L. REV. 645, 678 (1992).\n\n. Legal Underground, Feb. 14, 2005, <http:\/\/www.legalunderground.com\/2005\/02\/what_do_you_lik.html>.\n\n. Andrew Schepard & Theo Liebmann, _Law and Children_ , 235 N.Y.L.J., Jan. 18, 2006, at 3.\n\n. Guenther, _supra_ note 29.\n\n. Martin Luther King, _Where Do We Go From Here?_ , Aug. 16, 1967, <http:\/\/www.indiana.edu\/~ivieweb\/mlkwhere.html>.\n\n. Ward, _supra_ note 5, at 32.\n\n. Joshua Wolf Shenk, _What Makes Us Happy?_ , ATLANTIC, June 2009, _available at_ <http:\/\/www.theatlantic.com\/doc\/200906\/happiness>.\n\n. Monahan & Swanson, _supra_ note 10, at 2.\n\n. _Id._ at 22.\n\n. BARRY SCHWARTZ, THE PARADOX OF CHOICE: WHY LESS IS MORE 104, in (2004).\n\n. Pew Research Center, _Are We Happy Yet?_ , Feb. 13, 2006, <http:\/\/pewresearch.org\/pubs\/301\/are-we-happy-yet>.\n\n. DANIEL NETTLE, HAPPINESS: THE SCIENCE BEHIND YOUR SMILE 73\u201374 (2005).\n\n. Patrick Radden Keefe, _White Shoe, Black Hat: Michael Clayton's Devastating Critique of the Legal Profession_ , SLATE, Feb. 19, 2008, <http:\/\/www.slate.com\/id\/2184068\/>.\n\n. _See_ SUSAN SWAIM DAICOPP, LAWYER, KNOW THYSELF, A PSYCHOLOGICAL ANALYSIS OF PERSONALITY STRENGTHS AND WEAKNESSES 41 (2004).\n\n. _See, e.g._ , Don Peters & Martha M. Peters, _Maybe That's Why I Do That: Psychological Type Theory, the Myers-Briggs Type Indicator, and Learning Legal Interviewing_ , 35 N.Y.L. SCH. L. REV. 169, 169 (1990).\n\n. Susan Daicoff _Asking Leopards to Change Their Spots: Should Lawyers Change? A Critique of Solutions to Problems with Professionalism by Referenceto Empirically-Derived Attorney Personality Attributes_, 11 GEO. J. LEGAL ETHICS 547, 587\u201388, 595 (1998).\n\n. The \"lawyer personality\" is itself a category in the Myers-Briggs personality inventory. Based on the standard categorization of personality types, lawyers are much more inclined that the general public to be \"Thinkers\" (versus \"Feelers\"), \"Introverts\" (versus \"Extroverts\"), and \"Intuiting\" types (versus \"Sensing\" types). For example, 66 percent of all female lawyers are \"Thinking\" types compared to only 35 percent of women generally. (Male lawyers are even more likely to be \"Thinking\" types, with 81 percent getting that label, based on survey results.) On the Extroversion-Introversion scale, a full three-quarters of the general population turn out to be \"Extroverts\" compared to just 43 percent of lawyers. Finally, on the Sensing-Intuiting scale, 70 percent of lawyers score as \"Intuiting\" types, while in the general population, the percentages are exactly the reverse. (Intuiting types are considered to be past or future-oriented, impatient with the routine, doubt-ridden, and attracted more to the theoretical than the practical.) On only one of the standard personality scales, \"Perceiving\" types versus \"Judging\" types, do lawyers generally reflect percentages found in the population as a whole. MICHAEL MELCHER, THE CREATIVE LAWYER: A PRACTICAL GUIDE TO AUTHENTIC PROFESSIONAL SATISFACTION _76\u201379_ (American Bar Association, 2007).\n\n. Martin E.P. Seligman et al., _Why Lawyers Are Unhappy_ , 23 CARDOZO L. REV. 33, 34 (2001).\n\n. Catherine Gage O'Grady, _Cognitive Optimism and Professional Pessimism in the Large-Firm Practice of Law: The Optimistic Associate_ , 30 LAW & PSYCHOL. REV. 23, 37\u201338 (2006).\n\n. Seligman et al., _supra_ note 58, at 41.\n\n. Lawrence S. Krieger, _The Inseparability of Professionalism and Personal Satisfaction: Perspectives on Values, Integrity and Happiness_ , 11 CLINICAL L. REV. 425, 433,434 (2005).\n\n. Douglas O. Linder, _Who Is Clarence Darrow?_ , <http:\/\/www.law.umkc.edu\/faculty\/projects\/ftrials\/DARESY.htm> (last visited Aug. 11, 2009).\n\n. Alice Park, _A Primer for Pessimists_ , TIME, Apr. 6, 2009, at WI (quoting Martin Seligman).\n\n### _Chapter 4_\n\n. EDUARDO PUNSET, THE HAPPINESS TRIP: A SCIENTIFIC JOURNEY 55 (2007).\n\n. _Id._ at 56.\n\n. William W. Eaton et al., _Occupations and the Prevalence of Major Depressive Disorder_ , 32 J. OCCUPATIONAL MED. 1079,1083 (1990).\n\n. AMERICAN BAR ASSOCIATION YOUNG LAWYERS DIVISION, TABLE 23, ABA YOUNG LAWYERS DIVISION SURVEY: CAREER SATISFACTION 31 (2001), <http:\/\/www.abanet.org\/yld\/satisfaction_800.doc>.\n\n. _Id._ at 22.\n\n. Peter Warr, _Jobs and Happiness_ , SOCIETY FOR INDUSTRIAL & ORGANIZATIONAL PSYCHOLOGY, INC., Jan. 2007, <http:\/\/www.siop.org\/tip\/Current\/04warr.aspx>.\n\n. DANIEL GILBERT, STUMBLING ON HAPPINESS 23 (2006).\n\n. PUNSET, _supra_ note 1, at 78.\n\n. Stephanie Francis Ward, _Pulse of the Legal Profession: 800 Lawyers Reveal What They Think About Their Lives, Their Careers and the State of the Legal Profession_ , 93 A.B.A. J. 30, 34 (Oct. 2007).\n\n. Ross Gittins, _Happiness Is the Job You Like_ , SYDNEY MORNING HERALD, Mar. 10, 2004, _available at_ <http:\/\/www.smh.com.au\/articles\/2004\/03\/09\/1078594359806.html?from=storyrhs>.\n\n. INTERNATIONAL LABOUR ORGANIZATION, KEY INDICATORS OF THE LABOUR MARKET PROGRAMME 6, 10, <http:\/\/www.ilo.org\/public\/english\/employment\/strat\/kilm\/download\/kilmo6.pdf> (last visited July 9, 2009).\n\n. JEROME KAGAN, WHAT IS EMOTION? (2007), quoted in Sue M. Halpern, _Are You Happy?_ , 55 N.Y. REV. BOOKS 24 (Apr. 3, 2008), _available_ at <http:\/\/www.nybooks.com\/articles\/21197>.\n\n. DANIEL NETTLE, HAPPINESS:THE SCIENCE BEHIND YOUR SMILE 38 (2005).\n\n. American Public Media, _Bhutan's Falling Happiness Index_ , Nov. 14, 2007, <http:\/\/marketplace.publicradio.org\/display\/web\/2007\/11\/14\/consumed5_mmr_1\/>.\n\n. According to a recent study, 96 percent of people in the highest social class report being in control of their lives, compared to 81 percent in the lowest social class. NETTLE, _supra_ note 13, at 73\u201474.\n\n. ERICWEINER, THE GEOGRAPHY OF BLISS 251 (2008).\n\n. RICHARD LAYARD, HAPPINESS: LESSONS FROM A NEW SCIENCE 40 (2005). Richard Wilkinson and Kate Pickett amass substantial evidence that societies with greater economic and political equality have healthier and happier citizens. RICHARD WILKINSON & KATE PICKETT, THE SPIRIT LEVEL: WHY MORE EQUAL SOCIETIES ALMOST ALWAYS DO BETTER (2009).\n\n. SONJA LYUBOMIRSKY, THE HOW OF HAPPINESS: A NEW APPROACH TO GETTING THE LIFE YOU WANT 116 (2007).\n\n. WEINER, _supra_ note 16, at 114.\n\n. LAYARD, _supra_ note 17, at 16.\n\n. Warr, _supra_ note 6.\n\n. _Id._ at 15.\n\n. TOM W. SMITH, JOB SATISFACTION IN THE UNITED STATES (Apr. 17, 2007) (National Opinion Research Center, University of Chicago study).\n\n. Barbara Rose, _Money Can't Buy Happiness, Study Finds_ , CHI. TRIB., Apr. 17, 2007.\n\n. GILBERT, _supra_ note 7, at 166.\n\n. ROBERT PUTNAM, BOWLING ALONE: THE COLLAPSE AND REVIVAT OF AMERICAN COMMUNITY (2001).\n\n. LYUBOMIRSKY, _supra_ note 18, at 64, 89\u2013101.\n\n. Ward, _supra_ note 9, at 33.\n\n. Physorg.com, _First Ever World Map of Happiness Produced_ , <http:\/\/www.physorg.com\/news73321785.html>, July 28, 2006. Austria ranked third, followed by Iceland. The United States ranked 23d.\n\n. WEINER, _supra_ note 16, at 318.\n\n. Leslie A. Gordon, _Mid-Career Malaise: How to Find a New Path for Your 40s_ , A.B.A. J., Sept. 2008, at 38, 40.\n\n. _Id._\n\n. Debra Cassens Weiss, _Long Hours and Hard Work Took a Toll on Sotomayor's Relationships_ , A.B.A. J., July 10, 2009, _available at_ <http:\/\/abajournal.com\/news\/long_hours_and_hard_work_took_a_toll_on_sotomayors_relationships>.\n\n. TAL BEN SHAHAR, THE QUESTION OF HAPPINESS : ON FINDING MEANING, PLEASURE, AND THE ULTIMATE CURRENCY 46 (2002) (quoting Csikszentmihalyi).\n\n. MlHATYI CSIKSZENTMIHALYI, FINDING FLOW: THE PSYCHOTOGY OF ENGAGEMENT WITH EVERYDAY LIFE 29 (1997).\n\n. David Achtenberg, interview, Jan. 29, 2009.\n\n. _See generally_ CSIKSZENTMIHALYI, _supra_ note 35; MIHALYI CSIKSZENTMIHALYI, FLOW: THE PSYCHOLOGY OF OPTIMAL EXPERIENCE (1990).\n\n. GILBERT, _supra_ note 7, at 111^71, 212\u201333.\n\n. LAYARD, _supra_ note 17, at 48\u201349.\n\n. WEINER, _supra_ note 16, at 310.\n\n. Penelope Trunk, Brazen Careerist, <http:\/\/blog.penelopetrunk.com\/2006\/05\/15\/forget-the-soul-search-just-do-something> (May 15, 2006).\n\n. Daniel T. Gilbert et al., _The Surprising Power of Neighborly Advice_ , 323 SCIENCE 1617, 1618 (Mar. 20, 2009).\n\n. Trunk, _supra_ note 41, citing Daniel Gilbert.\n\n. _See_ Michael A. Cohn et al., _Happiness Unpacked: Positive Emotions Increase Life Satisfaction by Building Resilience_ , 9 J. EMOTION 361 (2009); _Smallest Joys Add Up to a Lot_ , K.C. STAR, July 12, 2009, at An.\n\n. LYUBOMIRSKY, _supra_ note 18, at 194.\n\n. BEN SHAHAR, _supra_ note 34, at 54\u201357.\n\n. _See_ MARTEN E.P. SELIGMAN, AUTHENTIC HAPPINESS: USING THE NEW POSITIVE PSYCHOLOGY TO REALIZEYOUR POTENTIAT FOR LASTENG FULFILLMENT (2002).\n\n. JEAN STEPANCIC & RICHARD DELGADO, HOW LAWYERS LOSE THEIR WAY: A PROFESSION FAILS ITS CREATIVE MINDS 14\u201315 (2005).\n\n. Deborah Rhode, _Foreword: Personal Satisfaction in Professional Practice_ , 58 SYRACUSE L. REV. 217, 224 (2008).\n\n. Ward, _supra_ note 9, at 33 (68 percent of public sector lawyers polled said they \"were satisfied with their professional lives.\").\n\n. PUNSET, _supra_ note 1, at 41.\n\n. Unfortunately, however, pro bono opportunities in most workplaces are very limited these days. One half of all attorneys report being dissatisfied with pro bono opportunities offered by their firms, and actual pro bono work averages less than thirty minutes per week. Rhode, _supra_ note 49, at 225. Only a quarter of lawyers say that their firms fully count pro bono work toward billable hours and about two-thirds believe that pro bono work will not help, or is more likely to hurt, their prospects for promotion or increased compensation. _Id._ at 226. Profits rule at most firms\u2014and the drive for ever higher profits (to pay ever higher salaries) has taken its toll on pro bono activity. The failure of firms to place a high priority on pro bono work is unfortunate.\n\n. _Id._ at 233.\n\n. _Id._\n\n. Penelope Trunk, Brazen Careerist, <http:\/\/blog.penelopetrunk.com\/2007\/01\/16\/the-connection-between-a-good-job-and-happiness-is-overrated> (fan. 16, 2007).\n\n. Erica Goode, _Exploring Life at the Top of the Happiness Scale_ , N.Y. TIMES, Jan. 29, 2002, at F6.\n\n### _Chapter 5_\n\n. _See, e.g._ , ANDREW MCCLURG, IL OF A RIDE: A WELL-TRAVELED PROFESSOR'S ROADMAP TO SUCCESS IN THE FIRST YEAR OF LAW SCHOOL (2008); HELENE SHAPO & MARSHALL SHAPO, LAW SCHOOL WITHOUT FEAR: STRATEGIES FOR SUCCESS (2d ed. 2002).\n\n. AMERICAN BAR ASSOCIATION, ENROLLMENT AND DEGREES AWARDED 1963\u20142008, <http:\/\/www.abanet.org\/legaled\/statistics\/charts\/statspercent20-percent201.pdf>. In that same time frame about 5,000 first year students left law school each year; approximately 1,400 left law school during their second year, and about 300 left during their third year. AMERICAN BAR ASSOCIATION, TOTAL J.D ATTRITION 1981\u20132006, <http:\/\/www.abanet.org\/legaled\/statistics\/charts\/stats%20-%2017.pdf>. These attrition rates include students who left voluntarily, flunked out, and transferred to another school. The numbers may say more about a school's admissions and retention practices than individuals' choices. In 2005 six law schools had attrition rates for first years above 30 percent. Debra Cassens Weiss, _lL Attrition Topped 30 Percent at Six haw Schools_ , A.B.A. J., Apr. 25, 2008, _available at_ <http:\/\/www.abajournal.com\/news\/1l_attrition_topped_30_percent_at_six_law_schools>.\n\n. Deborah Rhode, _Personal Satisfaction in Professional Practice_ , 58 SYRACUSE L. REV. 217, 223 (2008).\n\n. Allen K. Rostron, _Lawyers, haw & the Movies: The Hitchcock Cases_, 86 CAL. L. REV. 211,214 (1998).\n\n. People who engage with and care about other people report greater satisfaction over their life span. Nisha C. Gottfredson et al., _Identifying Predictors of Law Student Life Satisfaction_ , 58 J. LEGAL EDUC. 520 (Dec. 2008).\n\n. _Id._\n\n. Barbara Glesner Fines, interview, Apr. 21, 2009.\n\n. GARY A. MUNNEKE ET AL., NONLEGAL CAREERS FOR LAWYERS 3\u20134 (5th ed. 2006).\n\n. GARY A. MUNNEKE, THE LEGAL CAREER GUIDE: FROM LAW STUDENT TO LAWYER 5 (4th ed. 2002).\n\n. NICHOLAS A. CHRISTAKIS & JAMES H. FOWLER, CONNECTED: THE SURPRISING POWER OF OUR SOCIAL NETWORKS AND HOW THEY SHAPE OUR LIVES (2009).\n\n. Graduates of fourth-tier law schools are happier (43 percent are extremely satisfied) than graduates of elite law schools \"working in the most prestigious settings\"\u2014who are the least likely to report being extremely satisfied (24 percent) \"with their decision to become a lawyer.\" Ronit Dinovitzer & Bryant G. Garth, _Lawyer Satisfaction in the Process of Structuring Legal Careers_ , 41 LAW & SOC'Y REV. 1,25 (2007).\n\n. LAW SCHOOL SURVEY OF STUDENT ENGAGEMENT, STUDENT ENGAGEMENT IN LAW SCHOOL: PREPARING 2IST CENTURY LAWYERS (2008), <http:\/\/lssse.iub.edu\/2008_Annual_Report\/pdf\/J4u5h7e9\/LSSSE_2008_Annual_Report.pdf> [hereinafter LSSSE].\n\n. _Princeton Review's Best 174 Law Schools_ , Tax Prof Blog, <http:\/\/taxprof.type-pad.com\/taxprof_blog\/2008\/10\/princeton-revie.html> (Oct. 20, 2008).\n\n. _See_ LSSSE, _supra_ note 12, at 11\u201414.\n\n. _See_ Lawrence S. Krieger, _Human Nature as a New Guiding Philosophy for Legal Education and the Profession_ , 47 WASHBURN L.J. 247, 264 (2008).\n\n. Susan Sturm & Lani Guinier, _The Law School Matrix: Reforming Legal Education in a Culture of Competition and Conformity_ , 60 VAND. L. REV. 515, 532\u201433 (2007).\n\n. _See_ Patrick J. Schiltz, _Making Ethical Lawyers_ , 45 S. TEX. L. REV. 875, 879\u201385 (2004).\n\n. LAW SCHOOL SURVEY OF STUDENT ENGAGEMENT, ENGAGING LEGAL EDUCATION: MOVING BEYOND THE STATUS QUO 8, 13 (2006), <http:\/\/lssse.iub.edu\/2006_Annual_Report\/pdf\/LSSSE_2006_Annual_Report.pdf>.\n\n. Denise Riebe, _A Bar Review for Law Schools: Getting Students on Board to Pass Their Bar Exams_ , 45 BRANDEIS L.J. 269, 331 (2007).\n\n. _See_ Brigette LuAnn Willauer, Comment, _The Law School Honor Code and Collaborative Learning: Can They Coexist?_ , 73 UMKC L. REV. 513, 525\u201434 (2004).\n\n. Anthony J. Luppino, _Minding More Than Our Own Business: Educating Entrepreneurial Lawyers Through Law School-Business School Collaborations_ , 30 W. NEW ENG. L. REV. 151,166 (2007).\n\n. _See, e.g._ , Janet Weinstein & Linda Morton, _Interdisciplinary Problem Solving Courses as a Context for Nurturing Intrinsic Values_ , 13 CLINICAL L. REV. 839, 846 (2007).\n\n. Luppino, _supra_ note 21, at 178\u201486.\n\n. _The Endangered Trial Lawyer_ , 95 A.B.A. J. 63 (Mar. 2009).\n\n. PAULINE H. TESLER, COLLABORATIVE LAW: ACHIEVING EFFECTIVE RESOLUTION IN DIVORCE WITHOUT LITIGATION xx-xxi (2001).\n\n. Susan Grover, _Personal Integration and Outsider Status as Factors in Law Student Well-Being_ , 47 WASHBURN L.J. 419, 427 (2008).\n\n. Paula Lustbader, _You Are Not in Kansas Anymore: Orientation Programs Can Help Students Fly Over the Rainbow_ , 47 WASHBURN L.J. 327, 350 (2008).\n\n. Christine Hurt, _No Harm Intended_ , CHRON. HIGHER EDUC., May 25, 2005, _available at_ <http:\/\/chronicle.com\/jobs\/news\/2005\/05\/2005052501c.htm>.\n\n. _See_ Michael Hunter Schwartz, _Humanizing Legal Education: An Introduction to a Symposium Whose Time Has Come_ , 47 WASHBURN L.J. 235 (2008).\n\n. American Bar Association, _Alphabetical School List_ , <http:\/\/www.abanet.org\/legaled\/approvedlawschools\/alpha.html> (last visited Mar. 25, 2009). We are indebted to Lawrence MacLachlan for conducting this survey.\n\n. Lewis & Clark's \"Lawyering in Society,\" Temple's \"Law, Happiness, & Subjective Well-Being,\" the University of California at Berkeley's \"Effective and Sustainable Law Practice: the Meditative Perspective,\" the University of Missouri-Kansas City's \"Quest for a Satisfying Career in Law,\" the University of Virginia's \"Legal Careers and Life Satisfaction, and Yale's \"Happiness and Morality.\" _See_ Peter H. Huang & Rick Swedloff, _Authentic Happiness & Meaning at Law Firms_, 58 SYRACUSE L. REV. 335, 346\u201347 (2008).\n\n. _See, e.g._ , DANIEL GILBERT, STUMBLING ON HAPPINESS (2006); Daniel Kahneman et al., _Would You Be Happier If You Were Richer? A Focusing Illusion_ , 312 SCIENCE 1908 (June 30, 2006); Barry Schwartz & Andrew Ward et al., _Maximizing Versus Satisficing: Happiness Is a Matter of Choice_ , 83 J. PERSONALITY & Soc. PSYCHOL. 1178 (2002).\n\n. Doug Linder & Nancy Levit, _The Quest for a Satisfying Career in Law_ , Spring 2009, <http:\/\/www.law.umkc.edu\/faculty\/projects\/ftrials\/happylawyers\/Questions.html>.\n\n. Doug Linder, _Searching for Law's Heroes_ , 2001, <http:\/\/www.law.umkc.edu\/faculty\/projects\/ftrials\/trialheroes\/HEROSEARCH1.html>.\n\n. Doug Linder, _Does the Law Have Heroes?_ , 2001, <http:\/\/www.law.umkc.edu\/faculty\/projects\/ftrials\/trialheroes\/HEROSEARCH4.html>.\n\n. Boston College, _Law School Personal Statement_ , <http:\/\/www.bc.edu\/offices\/careers\/gradschool\/law\/lawstatement.html> (last visited Mar. 3, 2010).\n\n. WILLIAM H. COLBY, UNPLUGGED: RECLAIMING OUR RIGHT TO DIE IN AMERICA 109, 114 (2006). Lawyer Bill Colby created this term in another context, but it refers to a phenomenon of institutions creating structures that route people in prescribed directions.\n\n. Daisy Hurst Floyd, _Lost Opportunity: Legal Education and the Development of Professional Identity_ , 30 HAMLINE L. REV. 555, 562 (2007).\n\n. HARRY LEWIS, EXCELLENCE WITHOUT A SOUL: HOW A GREAT UNIVERSITY FORGOT EDUCATION 12\u201314, 140 (2005).\n\n. _See_ Luppino, _supra_ note 21, at 194.\n\n. LAWRENCE S. KRIEGER, THE HIDDEN SOURCES OF LAW SCHOOL STRESS: AVOIDING THE MISTAKES THAT CREATE UNHAPPY AND UNPROFESSIONAL LAWYERS 5 (2006), _available at_ <http:\/\/www.law.fsu.edu\/academic_programs\/humanizing_lawschool\/images\/E.pdf>\n\n. Pierre Schlag, _Hiding the Ball_ , 71 N.Y.U.L. REV. 1681, 1683 (1996).\n\n. Paul J. Zak & Ahlam Fakhar, _Neuroactive Hormones and Interpersonal Trust: International Evidence_ , 4 ECON.& HUM. BIOLOGY 412 (Dec. 2006).\n\n. Schwartz, _supra_ note 29, at 241 n.43 (This game \"involves assigning bingo squares to students who engage themselves in classroom discussions and then playing according to the normal bingo rules as those students participate in class.\").\n\n. Jess M. Krannich et al., _Beyond \"Thinking Like a Lawyer\" and the Traditional Legal Paradigm:Toward a Comprehensive View of Legal Education_ , 86 DENV. U.L. REV. 381, 385 (2009).\n\n. BARBARA GLESNER FINES, LAW SCHOOL AND STRESS (1999), <http:\/\/www.law.umkc.edu\/faculty\/profiles\/glesnerfines\/bgf-strs.htm>.\n\n. Gerald F. Hess, _Heads and Hearts: The Teaching and Learning Environment in Law School_ , 52 J. LEGAL EDUC. 75, 77 (2002).\n\n. A 1993 Association of American Law Schools survey of almost 3,400 students at nineteen law schools showed that 8.2 percent had used marijuana in the previous month, 8.8 percent had used some form of illicit drug, and 14 percent had drunk alcohol ten or more times in the previous month; during the previous year, 20.8 percent reported use of marijuana and 4.8 percent had used cocaine. _Report of theAALS Special Committee on Problems of Substance Abuse in the Law Schools_ , 44 J. LEGAL EDUC. 35, 41 (1994).\n\n. Intriguingly, while medical students suffered some similar symptoms of depression and anxiety, they did so at about half the rate of law students. Connie J.A. Beck et al., _Lawyer Distress: Alcohol-Related Problems and Other Psychological Concerns Among a Sample of Practicing Lawyers_ , 10 J.L. & HEALTH 1 (1995\u201396).\n\n. Susan Daicoff, _Lawyer, Know Thyself: A Review of Empirical Research on Attorney Attributes Bearing on Professionalism_ , 46 AM. U. L. REV. 1337, 1341\u201350 (1997).\n\n. Kennon M. Sheldon & Lawrence S. Krieger, _Does Legal Education Have Undermining Effects on Law Students? Evaluating Changes in Motivation, Values, and Well-Being_ , 22 BEHAV. SCI. L. 261, 264 (2004).\n\n. KRIEGER, _supra_ note 41, at 3 (emphasis in original).\n\n. INDIANA UNIVERSITY CENTER FOR POSTSECONDARY RESEARCH, LAW SCHOOL SURVEY OF STUDENT ENGAGEMENT: LAW SCHOOL REPORT 2007, OVERVIEW 7 (2007), _available at_ http:\/\/lssse.iub.edu\/pdf\/LSSSE%202007%20Overview_FINAL%20(PDF).pdf..pdf)\n\n. Peter F. Lake, _When Fear Knocks: The Myths and Realities of Law School_ , 29 STETSON L. REV. 1015, 1034 (2000).\n\n. James H. Backman, _Practical Examples for Establishing an Externship Program Available to Every Student_ , 14 CLINICAL L. REV. I, 4n.i6 (2007) (listing only eight law schools that do not have an in-house clinic).\n\n. Mitu Gulati et al., _The Happy Charade: An Empirical Examination of the Third Year of Law School_ , 51 J. LEGAL EDUC.235, 249 (2001).\n\n. Jonathan D. Rowe, _\"It Gets Late Early Out There\": Yogi Berra Tours the Law Schools_ , 77 MICH. B.J. 664, _666_ (1998).\n\n. KRIEGER, _supra_ note 41, at 4.\n\n. Nisha C. Gottfredson et al., _Identifying Predictors of Law Student Life Satisfaction_ , 58 J. LEGAL EDUC.520, 527 (2008). _See also_ Dinovitzer & Garth, _supra_ note 11, at 25.\n\n. _Id._\n\n. _See_ Sheldon & Krieger, _supra_ note 51.\n\n. _See_ Susan Daicoff, _Lawyer, Be Thyself: An Empirical Investigation of the Relationship Between the Ethic of Care, the Feeling Decisionmaking Preference, and Lawyer Wellbeing_ , 16 VA. J. Soc. POL'Y & L. 87, 133 (2008).\n\n. Michael Hunter Schwartz, _Teaching Law Students to Be Self-Regulated Learners_ , 2003 MICH. ST. DCL L. REV.447, 481\u201383.\n\n. _See, e.g._ , Edwin S. Shneidman, _Personality and \"Success\" Among a Selected Group of Lawyers_ , 48 J. PERSONALITY ASSESSMENT 609, 613\u201315 (1984).\n\n. Celestial S.D. Cassman & Lisa R. Pruitt, _A Kinder, Gentler Law School? Race, Ethnicity, Gender, and Legal Education at King Hall_ , 38 U.C. DAVIS L. REV.1209, 1263\u201364 (2005).\n\n. Gregory Bowman, _The Comparative and Absolute Advantages of Junior Law Faculty: Implications for Teaching and the Future of American Law Schools_ , 2008 B.Y.U. EDUC. & LJ. 171, 188.\n\n. Tan N. Nguyen, _An Affair to Forget: Law School's Deleterious Effect on Students' Public Interest Aspirations_ , 7 CONN. PUB. IN. LJ. 251, 252\u201352 (2008).\n\n. _Id._ at 257, 259\u201360.\n\n. AMERICAN BAR ASSOCIATION, LAW SCHOOL TUITION, http:\/\/www.abanet.org\/legaled\/statistics\/charts\/stats%20-%205.pdf (last visited Aug. 8, 2008).\n\n. Kathy Kristof, _The Great College Hoax_ , FORBES, Feb. 2, 2009, _available at_ <http:\/\/www.forbes.com\/magazines\/forbes\/2009\/0202\/060.html>\n\n. _The End of an Era: The Bi-Modal Distribution for the Class of 2008_ , Empirical Legal Studies, at <http:\/\/www.elsblog.org\/the_empirical_legal_studi\/2009\/06\/the-end-of-an-era-the-bimodal-distribution-for-the-class-of-2008.html> (June 29, 2009) (based on more than 22,300 recent graduates\u2014\"over half of all 2008 graduates\"\u2014who reported their starting salaries). This chart is copyrighted by the National Association for Law Placement (NALP) and reprinted with permission.\n\n. _Id._ You can easily find websites to research the starting salaries at individual firms in various locations, <http:\/\/www.infirmation.com\/shared\/search\/scored-search>.\n\n. National Association for Law Placement, _How Much Do Law Firms Pay New Associates? A 12-Year Retrospective as Reported by Firms_ , Oct. 2007, Table 1. Median Starting Salaries for First-Year Associates by Firm Size, Table 3, Median Starting Salaries for Selected Non-Firm Lawyer Jobs, <http:\/\/www.nalp.org\/2007octnewassocpay>\n\n. The states are: Arizona, Florida, Indiana, Kentucky, Maine, Maryland, Massachusetts, Minnesota, Missouri, Montana, New Hampshire, New Mexico, New York, North Carolina, Texas, and Washington.\n\n. Equal Justice Works, _State LRAPs_ , <http:\/\/www.equaljusticeworks.org\/node\/71>. Three states, Kentucky, Maine, and New Hampshire, have no income cap.\n\n. Equal Justice Works, _Law Schools With LRAPS_ , <http:\/\/www.equaljustice-works.org\/node\/66>\n\n. One provides state and local prosecutors and public defenders with $10,000 per year for a three year commitment. Another offers civil legal aid lawyers $6,000 for each year of a renewable three-year commitment (up to a maximum of $40,000). The third will give $2,000 per year (up to a maximum of $10,000) to public interest lawyers (including prosecutors, public defenders, and lawyers in low-income communities who work at nonprofit organizations). The fourth is a loan cancellation program which offers partial loan forgiveness for certain public service jobs. Equal Justice Works, _New Resource: Higher Education Reauthorization and College Opportunity Act 0\/2008_ , <http:\/\/www.equaljusticeworks.org\/node\/421>\n\n. IBRinfo, _What Are These New Programs?_ , <http:\/\/www.ibrinfo.org\/what.vp.html> (last visited Jan. 26, 2009).\n\n. DANIEL NETTLE, HAPPINESS: THE SCIENCE BEHIND YOUR SMILE 152 (2005).\n\n. MUNNEKE, _supra_ note 8, at 4.\n\n. EDUARDO PUNSET, THE HAPPINESS TRIP: A SCIENTIFIC JOURNEY 54 (2007).\n\n. Nguyen, _supra_ note 67, at 257\u201460.\n\n. David Hricik & Victoria S. Salzmann, _Why There Should Be Fewer Articles Like This One: Law Professors Should Write More for Legal Decision-Makers and Less for Themselves_ , 38 SUFFOLK U. L. REV.761, 769 (2005).\n\n. Joshua J.A. Henderson & Trevor C.W. Farrow, _The Ethical Development of Law Students: An Empirical Study_ , 72 SASK. L. REV.75, 98n.8o (2009).\n\n. Michael Sauder & Wendy Nelson Espeland, _Strength in Numbers? The Advantages of Multiple Rankings_ , 81 IND. L.J. 205, 211 (2006).\n\n. Dinovitzer & Garth, _supra_ note 11, at 4.\n\n. One survey of almost 19,000 associates asked them to rank firms along various dimensions, such as prestige, diversity, and quality of life (including measures such as hours and compensation, but also formal and informal training, and treatment by partners). Vault, _Top 100 Law Firms, 2009 Rankings_ , http:\/\/www.vault.com\/nr\/lawrankings.jsp?law2009=i&ch_id=242.\n\n. Judged, <http:\/\/www.judged.com\/jdfirmrankings.php> (last visited July 21, 2009).\n\n. _See_ BARRY NALEBUFF & IAN AYRES, WHY NOT? HOW TO USE EVERYDAY INGENUITY TO SOLVE PROBLEMS BIG AND SMALL (2003).\n\n. BARRY SCHWARTZ, THE PARADOX OF CHOICE: WHY LESS IS MORE 77\u201396 (2004).\n\n. _Id._ at 104.\n\n. _Id._ at 25.\n\n. _Id._ at 4.\n\n. _Id._ at 62; Barry Schwartz & Andrew Ward et al., _Maximizing Versus Satisficing: Happiness Is a Matter of Choice_ , 83 J. PERSONALITY & Soc. PSYCHOL.1178, 1179 (2002).\n\n. Michael Melcher, _Why Thinking Like a Lawyer Is Bad for Your Career_ , A.B.A. J., Apr. 15, 2009, <http:\/\/www.abajournal.com\/weekly\/why_thinking_like_a_lawyer_is_bad_for_your_career>.\n\n. _Id._\n\n. _Id._\n\n. TAL BEN-SHAHAR, THE QUESTION OF HAPPINESS: ON FINDING MEANING, PLEASURE, AND THE ULTIMATE CURRENCY 54 (2002).\n\n. CHRISTOPHER PETERSON & MARTIN E.R SELIGMAN, VALUES IN ACTION (VIA) CLASSIFICATION OF STRENGTHS, Jan. 4, 2003, <http:\/\/www.ppc.sas.upenn.edu\/viamanualintro.pdf>, at 4. CHARACTER STRENGTHS AND VIRTUES: A HANDBOOK AND CLASSIFICATION (Christopher Peterson & Martin E.R Seligman eds. 2004). The twenty-four strengths are the positive character traits that are components of the virtues. Courage encompasses bravery, industry (perseverance and diligence), honesty (authenticity, sincerity, and integrity), and vitality. Humanity includes strength in the areas of intimate attachment, kindness, and intelligence. Justice includes citizenship and teamwork, fairness, and leadership. Temperance consists of forgiveness and mercy, modesty and humility, prudence, and regulation of oneself. Transcendence is comprised of an appreciation of beauty and excellence (awe and wonder), gratitude, hope (optimism, future-mindedness, future orientation), playfulness and humor, and spirituality. Wisdom contains creativity (originality and ingenuity), curiosity, active open-mindedness, love of learning, and perspective.\n\n. Martin Seligman, _Authentic Happiness_ , <http:\/\/www.authentichappiness.sas.upenn.edu\/Default.aspx> (last visited July 4, 2009).\n\n. Peter H. Huang, _Authentic Happiness, Self-Knowledge and Legal Policy_ , 9 MINN. J.L. SCI. & TECH.755, 766 (2008).\n\n. Christopher K. Hsee & Reid Hastie, _Decision and Experience: Why Don't We Choose What Makes Us Happy?_ , http:\/\/papers.ssrn.com\/s0l3\/papers.cfm?abstract_id=9299i4&rec=1&srcabs=935470.\n\n. GILBERT, _supra_ note 32, at 137.\n\n. _Id._ at 143.\n\n. Hsee & Hastie, _supra_ note 102, at 3.\n\n. _Id._\n\n. Francois de La Rochefoucauld, quoted in Daniel T. Gilbert et al., _The Surprising Power of Neighborly Advice_ , 323 SCIENCE 1617 (Mar. 20, 2009).\n\n. GILBERT, _supra_ note 32, at 223.\n\n. Penelope Trunk, _The Connection Between a Good Job and Happiness Is Overrated_ , Brazen Careerist, <http:\/\/blog.penelopetrunk.com\/2007\/01\/16\/the-connection-between-a-good-job-and-happiness-is-overrated> (Jan. 16, 2007) (citing Daniel Gilbert).\n\n. Heather Brewer, _Snap Judgments_ , 9 Bus. L. TODAY 4 (Nov.\/Dec. 1999).\n\n. Lawrence S. Krieger, _The Inseparability of Professionalism and Personal Satisfaction: Perspectives on Values, Integrity, and Happiness_ , n CLINICAL L. REV.425, 435\u201336 (2005).\n\n. Id. at 436\u201337.\n\n### _Chapter 6_\n\n. JEREMY BLACHMAN, ANONYMOUS LAWYER 3 (2006).\n\n. Susan Deutschle, _Law Firm Retention Strategies Useful in Combating Attorney Turnover_ , COLUMBUS BUSINESS FIRST, Feb. 9, 2007, _available at_ <http:\/\/columbus.bizjournals.com\/columbus\/stories\/2007\/02\/12\/focus3.html?jst=s_cn_hl>.\n\n. Michael Renetzky, _The Smart Choice for Large Law Firms_ , <http:\/\/westle-galedcenter.com\/prm\/prmJSEjsf?id=50856i4> (last visited July 11, 2009).\n\n. ELLEN FREEDMAN, CALCULATING THE TRUE COST OF TURNOVER, 2005, <http:\/\/www.pa-lawfirmconsulting.com\/pdfs\/hr\/CALCULATING_THE_TRUE_COST_OF_TURNOVER.pdf>.\n\n. Peter Huang & Rick Swedloff, _Authentic Happiness & Meaning at Law Firms_, 58 SYRACUSE L. REV.335, 337 (2008) (citing studies).\n\n. _See_ Carol Graham, _Does Happiness Pay? An Exploration Based on Panel Data from Russia, Center on Social and Economic Dynamics Working Paper No. 28_ , May 2002, _available at_ <http:\/\/ssrn.com\/abstract=i0283i9>.\n\n. J.R. Minkel, _Happiness: Good for Creativity, Bad for Single-Minded Focus_ , SCI. AM., Dec. 18, 2006, _available at_ <http:\/\/www.scientificamerican.com\/article.cfm?id=happiness-good-for-creati>.\n\n. While numerous researchers have found effects in the populations they studied of co-workers spreading happiness, _see_ KIM S. CAMERON ET AL., POSITIVE ORGANIZATIONAL SCHOLARSHIP: FOUNDATIONS OF A NEW DISCIPLINE (2003); Sigal G. Barsade, _The Ripple Effect: Emotional Contagion in Groups_ , Oct. 2000, <http:\/\/papers.ssrn.com\/s0l3\/papers.cfmPabstract_id=250894>; Joanne H. Gavin & Richard O. Mason, _The Virtuous Organiza-tion:The Value of Happiness in the Workplace_ , 33 ORG. DYNAMICS 379 (2004); Marisa Salanova et al., _Flow at Work: Evidence for an Upward Spiral of Personal and Organizational Resources,']]._ HAPPINESS STUD, I (2006), other researchers have not found such workplace effects. See James H. Fowler & Nicholas A. Christakis, _Dynamic Spread of Happiness in a Large Social Network: Longitudinal Analysis Over 20Years in the Framingham Heart Study_ , 337 BRITISH MED. J. 2338 (Dec. 4, 2008).\n\n. See ELAINE HATPLELD ET AL., EMOTIONAL CONTAGION (1994).\n\n. Leslie A. Gordon, _Mid-Career Malaise, 94 A.B.A.j._ 38, 42 (Sept. 2008) (attorney career coach Debra Bruce).\n\n. _What Makes a Law Firm a Good Place to Work?, 26_ PA. LAW.14 (Dec. 2004).\n\n. Marc Galanter & William Henderson, _The Elastic Tournament: A Second Transformation of the Big Law Firm_ , 60 STAN. L. REV.1867, 1922n.235 (2008).\n\n. Diane Stafford, _Gen Y Reshaping the Workplace_ , K.C. STAR, July 6, 2008, at Ci.\n\n. PEGGY BLAKE GLEESON, MANAGING AND MOTIVATING THE GENERATIONS: IMPLICATIONS FOR THE STUDENT AND THE EMPLOYEE 7, Feb. 12\u201316, 2003, <http:\/\/www.uwsp.edu\/Education\/facets\/lmks_resources\/4413.pdf>.\n\n. NEIL HOWE & WILLIAM STRAUSS, _Characteristics of the Millennial Generation, in_ MlLLENNIALS GO TO COLLEGE (2003), _available at_ <http:\/\/wwwd.umn.edu\/advising\/MulennialTraits.doc>.\n\n. Shannon Henson, _Senior Partners Not LOLing at Gen Y's Perceptions_ , Oct. 9, 2008. <http:\/\/securities.law360.com\/articles\/71585>.\n\n. GLEESON, _supra_ note 14, at 7.\n\n. M. DIANE VOGT & LORI-ANN RICKARD, KEEPING GOOD LAWYERS: BEST PRACTICES TO CREATE CAREER SATISFACTION 84 (2000).\n\n. Lisa B. Bingham et al., _Exploring the Role of Representation in Employment Mediation at the USPS_ , 17 OHIO ST. J. ON DISP. RESOL.341, 350 (2002).\n\n. _See_ Lisa G. Lerman, _The Slippery Slope From Ambition to Greed to Dishonesty: Lawyers, Money and Professional Integrity_ , 30 HOFSTRA L. REV.879 (2002).\n\n. Gina Passarella, _Keeping a Legal Department Effective on a Smaller Budget_ , LEGAL INTELLIGENCER, Dec. 17, 2008, at 7.\n\n. Aric Press, _In-House at the American Lawyer_ , AM. LAW., Dec. 2008, at 11.\n\n. Call to Action: Diversity in the Legal Profession, _Corporate Signatories_ , <http:\/\/www.clocalltoaction.com\/> (last visited Dec. 1, 2008).\n\n. See Angela Brouse, Comment, _The Latest Call for Diversity in Law Firms: Is It Legal?_ , 75 UMKC L. REV.847 (2007).\n\n. _First Ever World Map of Happiness Produced_ , July 28, 2006, <http:\/\/www.physorg.com\/news73321785.html>.\n\n. Robert Putnam, _E Plurihus Unum: Diversity and Community in the Twenty-first Century_ , 30 SCANDINAVIAN POLITICAL STUD.137, 149\u201350 (2007).\n\n. MARTIN L. HOFFMAN, EMPATHY AND MORAL DEVELOPMENT: IMPLICATIONS FOR CARING AND JUSTICE 62 (2000).\n\n. See, _e.g._ , Michael E. Murphy, _The Nominating Process for Corporate Boards of Directors: A Decision-Making Analysis_ , 5 BERKELEY BUS. L.J. 131, 158 (2008) (\"While individuals value the strong relationships found in cohesive groups, they are most likely to find new ideas in social links that put them in contact with other social worlds or ways of thinking.\") (citing studies).\n\n. See _also_ Galanter & Henderson, _supra_ note 12, at 1922 (noting that Gen Y'ers \"demand a high level of racial and gender diversity within the firm's workforce\").\n\n. Malcolm Gladwell, <http:\/\/www.ted.com\/mdex.php\/talks\/lang\/eng\/malcolm_gladwell_on_spaghetti_sauce.html>.\n\n. See Muriel Goode-Trufant, _Beyond Diversity 2009: The Next Generation_ , 1722 PLI\/Corp 53, Feb. 25, 2009; David B. Wilkins, _From \"Separate Is InherentlyUnequal\" to \"Diversity Is Good for Business\": The Rise of Market-Based Diversity Arguments and the Fate of the Black Corporate Bar_, 117 HARV. L. REV.1548, 1557 (2004).\n\n. Putnam, _supra_ note 26, at 165.\n\n. Sunil J. Randall, _Enhancing Employee Performance Through Positive Organizational Behavior_ , 38 J. APP SOC. PSYCHOL.1580 (2008).\n\n. Peter Warr, _Jobs and Happiness_ , SOCIETY FOR INDUSTRIAL & ORGANIZATIONAL PSYCHOLOGY, INC., Jan. 2007, <http:\/\/www.siop.org\/tip\/Current\/04warr.aspx>.\n\n. CATALYST, WOMEN IN THE LAW: MAKING THE CASE 34 (2001), _available at_ <http:\/\/www.catalyst.org\/file\/165\/women_in_law_making_the_case.pdf>.\n\n. In the After the J.D. study, the measures on which lawyers expressed the highest degree of job satisfaction (on a scale of 1\u20147) were relationships with colleagues (5.7), level of responsibility (5.6), control over work methods (5.4), and intellectual challenge (5.4). Ronit Dinovitzer & Bryant G. Garth, _Lawyer Satisfaction in the Process of Structuring Legal Careers_ , 41 LAW & Soc'Y REV.1, 9 (2007).\n\n. Martin Seligman et al., _Why Lawyers Are Unhappy_ , 23 CARDOZO L. REV.33, 42 (2001).\n\n. Bruce A. Green, _Professional Challenges in Large Firm Practices_ , 33 FORDHAM URB. L.J. 7, 16\u201317 (2005).\n\n. _Id._ at 14.\n\n. SUSAN SAAB FORTNEY, IN PURSUIT OF ATTORNEY WORK-LIFE BALANCE: BEST PRACTICES IN MANAGEMENT 95\u201396 (Paula Patton ed. 2005).\n\n. Stephanie Ward, _The Ultimate Time-Money Trade-Off_ A.B. A. J., Feb. 2, 2007, at 2.\n\n. ABA COMMISSION ON BILLABLE HOURS, ABA COMMISSION ON BILLABLE HOURS REPORT ix n.iii (2002), <http:\/\/www.abanet.org\/careercounsel\/billable.html>; Francesca Jarosz, _Tipping Back the Scales_ , 16 Bus. L. TODAY 13, 18 (Mar.\/Apr. 2007).\n\n. CATALYST, BEYOND A REASONABLE DOUBT: LAWYERS STATE THEIR CASE ON JOB FLEXIBILITY 3 (Nov. 2006), <http:\/\/www.catalyst.org\/publication\/40\/beyond-a-reasonable-doubt-lawyers-state-their-case-on-job-flexibility>.\n\n. Joan C. Williams et al., _Law Firms as Defendants: Family Responsibilities Discrimination in Legal Workplaces_ , 34 PEPP. L. REV.393, 411 (2007).\n\n. Building a Better Legal Profession, <http:\/\/www.betterlegalprofession.org\/principles.php> (last visited July 5, 2009).\n\n. Galanter & Henderson, _supra_ note 12, at 1924.\n\n. Peter Lattman, _You Say You Want a Big-Law Revolution_ , WALL ST. J., Apr. 3, 2007, _available at_ <http:\/\/blogs.wsj.com\/law\/2007\/04\/03\/you-say-you-want-a-big-law-revolution>.\n\n. _Id._\n\n. _Id._\n\n. _Id._\n\n. Larry Kramer, _From the Dean_ , 81 STAN. LAW. 1 (Fall 2009).\n\n. _Id._\n\n53. _Id._\n\n. RICHARD H. THALER & CASS R. SUNSTEIN, NUDGE: IMPROVING DECISIONS ABOUT HEALTH, WEALTH, AND HAPPINESS 33\u201334 (2009).\n\n. Sharon Driscoll, _Law Firm Hiring: Time for a Change?_ , 81 STAN. LAW. 9, 11 (Fall 2009).\n\n. _Id._\n\n. Jeff Jeffrey, _Apprentice Programs Give First Years Extra Training_ , MIAMI DAILY Bus. REV., July 2, 2009, at A3.\n\n. Driscoll, _supra_ note 55, at 11.\n\n. Thomas Adcock, _10 N.Y. Firms Listed as Best for Women_ , N.Y.L.J., Aug. 15, 2008, at 24;Working Mother, _2008_ 50 _Best Law Firms for Women_ , 2008, <http:\/\/www.workingmother.com\/web?service=vpage\/2907>.\n\n. Deborah Epstein Henry, _Facing the FACTS: Introducing Work\/Life Choices for All Firm Lawyers Within the Billable Hours Model_ , <http:\/\/www.flextimelawyers.com\/pdf\/artio.pdf> (last visited Feb. 3, 2009).\n\n. CATALYST, _supra_ note 43, at 12.\n\n. For example, individualized work schedules may prompt lawyers to make relative comparisons.\n\n. Maria Vogel-Short, _Part-Time Lawyers Still a Rarity and Three-Quarters Are Women, Survey Says_ , N.J. L.J., Jan. 2, 2009, _available at_ <http:\/\/www.law.com\/jsp\/article.jsp?id=i202427i38453>.\n\n. Michael A. Scaperlanda, _Lawyering in the Little Way of St. Therese of Lisieux With Complete Abandonment and Love_ , 46 J. CATH. LEGAL STUD. 43, 48 (2007).\n\n. Joan Williams, _Our Economy of Mothers and Others: Women and Economics Revisited_ , 5 J. GENDER RACE & JUST. 411, 426 (2002).\n\n. CATALYST, _supra_ note 43, at 12.\n\n. Joan C. Williams & Stephanie Bornstein, _The Evolution of \"FRED\": Family Responsibilities Discrimination and Developments in the Law of Stereotyping and Implicit Bias_ , 59 HASTINGS L.J. 1311, 1329\u201330 (2008).\n\n. Ann A. Scott Timmer & Maureen Beyers, _Alternative Work Arrangements_ , 37 ARIZ. ATT'Y 40 (May 2001).\n\n. VOGT & RICKARD, _supra_ note 18, at 61. The idea is that firms should not pay bonuses for hours worked over a set amount. Instead, it is better to allow some carryover of hours into the next year or simply insist that your lawyers take the time off.\n\n. Audrey J. Lee, _Negotiating Part-Time Work: An Examination of How Attorneys Negotiate Part-Time Arrangements at Elite Law Firms_ , 6 PEPP. DISP. RESOL. L.J. 405, 414 (2006).\n\n. In 2004 IBM conducted a global work-life satisfaction survey of more than 42,000 of its employees in seventy-nine countries after implementation of flexible work options. It showed that 94 percent of all managers reported a strong link between those arrangements and the \"company's 'ability to retain talented professionals.' \" ARLENE JOHNSON ET AL., BUSINESS IMPACTS OF FLEXIBILITY: AN IMPERATIVE FOR EXPANSION 10 (2005), _available at_ www.cvworkingfamilies.org\/downloads\/BusinessImpac tsofFlexibility.pdf?CF ID=54713857& CFTOKEN=52379382. This experience echoed that of Deloitte in the 1990s when it undertook an initiative to reduce the high turnover rate for women. Surveys of its workers showed that flexibility was \"the factor most likely to improve retention of women,\" so \"the firm implemented flexibility as a key component of its women's initiative. Since then the turnover of women has dropped significantly, to the point that men's and women's turnover rates are now nearly equal.\" _Id._\n\n. Linda Bray Chanow, _The Business Case for Reduced Hours, Project for Attorney Retention_ , <http:\/\/www.pardc.org\/Publications\/business_case.shtml> (last visited June 20, 2009).\n\n. Law Society of Alberta, Alternative Work Schedules: Guidelines for Law Firms, <http:\/\/www.lawsocietyalberta.com\/resources\/modelEquityPolicies\/alternativeschedules.cfm> (last visited June 29, 2009).\n\n. DEBORAH L. RHODE, BALANCED LIVES: CHANGING THE CULTURE OF LEGAL PRACTICE 41 (2001).\n\n. Kira Dale Pfisterer, _When Three Fill Two: Part-Time Strategies for Full-Time Jobs_ , 51 ADVOCATE (Idaho) 15, 17 (Feb. 2008).\n\n. MILTON C. REGAN, JR., EAT WHAT YOU KILL: THE FALL OF A WALL STREET LAWYER 37 (2004).\n\n. William D. Henderson, An Empirical Study of Single-Tier Versus Two-Tier Partnerships in the Am Law 200, 84 N.C. L. REV. (2006), <http:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=871094>.\n\n. Ian J. Silverbrand, Note, _Modified Partnership Structures and Their Effects on Associate Satisfaction_ , 21 GEO. J. LEGAL ETHICS 165, 195 (2008).\n\n. William D. Henderson & David Zaring, _Young Associates in Trouble_ , 105 MICH. L. REV.1087, 1096 (2007).\n\n. Kenneth G. Dau-Schmidt & Kaushik Mukhopadhaya, _The Fruits of Our Labors: An Empirical Study of the Distribution of Income and Job Satisfaction Across the Legal Profession_ , 49 J. LEGAL EDUC.342, 346 (1999).\n\n. _See_ Silverbrand, _supra_ note 78. This might also be a function of the greater competition for partnership positions at the studied firms.\n\n. _Redefining How Your Law Firm Splits the Pie: What Works Now?_ , 06\u20148 LAW OFF. MGMT.& ADMIN. Rep. 2 (Aug. 2006) (\"56.1% of law firms with 100 or more attorneys use a lockstep system by class for associate compensation. The same is true for 45.4% of firms with 50 to 99 lawyers, but only 21.4% of smaller firms.\").\n\n. Joel A. Rose, _Firms Rethink Partners' Pay as Leverage Declines_ , 09\u201404 COMP.& BENEFITS FOR LAW OFFICES 1, 2\u20133 (Apr. 2009), _available at_ www.ioma.com\/law.\n\n. Maarten Vendrick & Geert Woltjer, _Happiness and Loss Aversion: When Social Participation Dominates Comparison_ , July 2006, <http:\/\/papers.ssrn.com\/s0l3\/papers.cfm?abstract_id=92io67>.\n\n. Patrick J. Schiltz, _On Being a Happy, Healthy, and Ethical Member of an Unhappy, Unhealthy, and Unethical Profession_ , 52 VAND. L. REV.871, 906 (1999)-\n\n. Stephen Overell, _A Working Recipe for the Quality of Life_ , FINANCIAL TIMES (London), Jan. 24, 2002, at 13.\n\n. Steve Crabtree, _The Economics of Happiness_ , GALLUP MGMT. J., Jan. 10, 2008, <http:\/\/grnj.gallup.com\/content\/103549\/Economics-Happiness.aspx>.\n\n. _Employers Urged to Focus on Training to Increase Job Satisfaction_ , Apr. 11, 2007, www.trainingfoundation.com\/page\/research\/2913 .html.\n\n. Andrew E. Clark et al., _Relative Income, Happiness and Utility: An Explanation for the Easterlin Paradox and Other Puzzles_ 19, June 2007, <http:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=998225>.\n\n. Elizabeth Goldberg, _Midlevel Blues_ , 28 AM. LAW.98 (Aug. 2006).\n\n. Kristin K. Stark & Blane Prescott, _Why Associates Leave: Research Shows That Attrition Has Very Little to Do With Money_ , LEGAL TIMES, May 7, 2007, at 45-\n\n. Jon Lindsey & Chuck Fanning, _After the Handshake_ , AM. LAW. Feb. 2007, _available at_ <http:\/\/www.law.com\/jsp\/tal\/PubArticleTAL.jsp?id=90000547278i>.\n\n. Drew Combs, _The Revolutionaries_ , 30 AM. LAW.102 (Aug. 2008).\n\n. Clark et al., _supra_ note 89, at 31.\n\n. PETER B. SLOAN, FROM CLASSES TO COMPETENCIES, LOCKSTEP TO LEVELS (Blackwell Sanders LLP 2007).\n\n. Faculty Colloquium, Oct. 10, 2008, UMKC School of Law.\n\n. _Gender or Childcare? Study Sheds Light on Career Roadblocks_ , INDIANA L., Fall 2009, at 3 (quoting Marc Galanter).\n\n. MALCOLM GLADWELL, THE TIPPING POINT: HOW LITTLE THINGS CAN MAKE A BIG DIFFERENCE 187 (2002).\n\n. _Id._ at 179.\n\n. _Id._ at 180.\n\n. _Id._ at 190.\n\n. _Id._ at 184. Other researchers, while agreeing with Dunbar that relationships in organizations break down once the group reaches a certain size, have come up with slightly different numbers for what that tipping point is. For example, anthropologists such as H. Russell Bernard have come up with a median estimate of 231 for the number of significant ties that a person living in the contemporary world can have. See H. RUSSELL BERNARD & CHRISTOPHER MCCARTY, THE NETWORK SCALE-UP METHOD: BACKGROUND AND THEORY 15 (Feb. 2009), _available at_ http:\/\/nersp.nerdc.ufl.edu\/~ufruss\/scale-up\/scale-up%20method%20theory%20and%2ohistory%20with%20 notes.pdf.\n\n. Douglas McCollam, _The End of Big Law_ , WALL ST. J., July 30, 2009, at A15.\n\n. Susan Saab Fortney _Soul for Sale: An Empirical Study of Associate Satisfaction, Firm Culture, and the Effects of Billable Hour Requirements_ , 69 UMKC L. REV. 239, 283 (2000).\n\n. NALP FOUNDATION FOR RESEARCH AND EDUCATION, KEEPING THE KEEPERS: STRATEGIES FOR ASSOCIATE RETENTION IN A TIME OF ATTRITION 14 (1998).\n\n. John Monahan & Jeffrey Swanson, _Lawyers at Mid-Career: A 20-Year Longitudinal Study of Job and Life Satisfaction_ , 5 J. EMPIRICAL LEGAL STUD, 1, 41 (forthcoming 2009).\n\n. SONJA LYUBOMIRSKY, THE HOW OF HAPPINESS: A NEW APPROACH TO GETTING THE LIFE YOU WANT 130 (2007).\n\n. Adcock, _supra_ note 59, at 24.\n\n. David Pollard, _Men Offer Appreciation; Woman Offer Attention_ , How To Save the World, <http:\/\/blogs.salon.eom\/0002007\/2006\/03\/26.html#a1478> (Mar. 26, 2006).\n\n. Elizabeth A. Amos & Bart L. Weathington, _An Analysis of the Relation Between Employee-Organization Value Congruence and Employee Attitudes_ , 142 J. PSYCHOL.615 (2008).\n\n. Ezra Tom Clark, Jr., _Characteristics of Successful Law Firms_ , 33 ARIZ. ATT'Y 16 (May 1997).\n\n. Peter Lattman, _Does \"Thank You\" Help Keep Associates?_ , WALL ST. J., Jan. 24, 2007, at B7.\n\n. THE AMERICAN BAR ASSOCIATION STANDING COMMITTEE ON PRO BONO AND PUBLIC SERVICE, SUPPORTING JUSTICE: A REPORT ON THE PRO BONO WORK OF AMERICA'S LAWYERS 5 (2005), _available at_ <http:\/\/www.abanet.org\/legalservices\/probono\/report.pdf>.\n\n. RONIT DINOVITZER ET AL., AFTER THE J.D.: FIRST RESULTS OF A NATIONAL STUDY OF LEGAL CAREERS 49 (2004); AMERICAN BAR ASSOCIATION, 2009 NATIONAL LAWYER POPULATION SURVEY, <http:\/\/www.abanet.org\/market-research\/2009_NATL_LAWYER_by_State.pdf>.\n\n. Linda Campillo, _A Dog's Best Friend_ , 57 OR. ST. B. BULL.27 (Jan. 1997).\n\n. _25 Top-Paying Companies_ , FORTUNE, Jan. 22, 2008, _available at_ <http:\/\/money.cnn.com\/galleries\/2008\/fortune\/0801\/gallery.bestcos_toppay.fortune\/25.html>.\n\n. Michelle Conlin, _Out of a Fishbowl_ , FORBES, Dec. 16, 1996, _available at_ <http:\/\/www.bartlit-beck.com\/articles\/detail.asp?whichid=1436407222004>.\n\n. Huang & Swedloff, _supra_ note 5, at 349.\n\n. Juliana B. Berry, _Motivating the Masses_ , LEGAL MGMT., 2007, _available at_ <http:\/\/www.alanet.org\/publications\/issue\/octnov07\/Motivation.pdf>\n\n. ALAIN DE BOTTON, THE ARCHITECTURE OF HAPPINESS 72 (2006).\n\n. CHRISTOPHER ALEXANDER ET AL., A PATTERN LANGUAGE: TOWNS, BUILDINGS, CONSTRUCTION (1977).\n\n. Jamie Friddle, _Finding Our Happy Place_ , COMMON GROUND, Jan. 2008, <http:\/\/commongroundmag.com\/2008\/01\/happyplace0801.html>.\n\n. ALEXANDER ET AL., _supra_ note 121, at 747.\n\n. _Id._\n\n. _Id._ at 890.\n\n. MARNI BARNES, HEALING GARDENS: THERAPEUTIC BENEFITS AND DESIGN RECOMMENDATIONS 59 (1999).\n\n. ALEXANDER ET AL., _supra_ note 121, at 1165.\n\n. THOMAS MERTON, THE LIVING BREAD 126 (1980).\n\n. ALEXANDER ET AL., _supra_ note 121, at 702.\n\n. Sheila Muto, _Law Firms Give Thought to Office-Design Issues_ , WALL ST. J., July 8, 2003, _available at_ <http:\/\/www.realestatejournal.com\/propertyreport\/office\/20030708-muto.html>.\n\n. _Id._\n\n. _Id._\n\n. EJ. ROETHLISBERGER & WlLLIAM J. DICKSON, MANAGEMENT AND THE WORKER 14\u201317 (1939) (describing the Hawthorne effect).\n\n### _Chapter 7_\n\n. DANIEL GILBERT, STUMBLING ON HAPPINESS 224 (2006).\n\n. Sue M. Halpern, _Are You Happy?_ , 55 N.Y. REV. BOOKS, Apr. 3, 2008, _available at_ <http:\/\/www.nybooks.com\/articles\/21197>.\n\n. The reports from lawyers in this chapter come from a variety of sources. Most stories come from about two hundred lawyers around the country who responded to a set of questions about career satisfaction that we sent by e-mail. In many cases respondents to the e-mail survey were contacted by phone and asked additional questions. Several other stories in this chapter come from student papers written for a short course we offered at the UMKC Law School. Students in the course were asked to report on interviews they conducted with experienced lawyers of their own choosing. A few additional stories, as indicated by footnotes, come from other publications.\n\n. Diane Curtis, _Billable Hours Intersect With the Profession's Woes_ , CAT. B.J., Jan. 2008, _available at_ http:\/\/www.calbar.ca.gov\/state\/calbar\/calbar_cbj.jsp?sCategoryPath=\/Home\/Attorney%2oResources\/California%20 Bar%20journal\/January2008&sCatHtmlPath=cbj\/2008-01_TH_o1_Billable-hours.html&sCatHtmlTitle=Top%20Headlines.\n\n. Melanie Lasoff Levs, _Best for the Business: \"Top Workplace\" Firms Garner Loyalty from Clients and Employees_ , 94 A.B.A. J. 34 (May 2008).\n\n. John Monahan & Jeffrey Swanson, _Lawyers at Mid-Career: A 20-Year Longitudinal Study of Job and Life Satisfaction_ , 5 J. EMPIRICAL LEGAL STUD, I, 26\u201427 (forthcoming 2009).\n\n. _Depression Among Lawyers: Chicken or Egg_ , <http:\/\/www.thedisgruntledlawyer.com\/law_school_advice\/> (last visited June 19, 2009).\n\n. Alex Williams, _The Falling-Down Professions_ , N.Y. TIMES, Jan. 6, 2008, at 91.\n\n. James Brosnahan, _\"Nothing Compares to the Electricity of an Actual Trial, and It Is Magnified When It Is a Jury Trial,\"_ 95 A.B.A. J. 50, 52, 63 (Mar. 2009).\n\n. Beyond the Underground, _What Do You Like Best About Being a Lawyer?_ , Feb. 14, 2005, <http:\/\/www.legalunderground.com\/2005\/02\/what_do_you_lik.html> (posted by NBT at 1:11 P.M.).\n\n. _Id._ (posted at 10:31 A.M.).\n\n. Monahan & Swanson, _supra_ note 6, at 2.\n\n. Edward A. Adams, _Survey : Young Lawyers Glad They're Attorneys_ , 95 A.B.A. J. 65, 65 (Mar. 2009).\n\n. Leslie A. Gordon, _Mid-Career Malaise_ , 94 A.B.A. J. 38, 39 (Sept. 2008).\n\n. Leslie Gordon, _Beyond the Law.JDs in All Walks of Life_ , STAN LAW.18, 21 (Spring 2008).\n\n. _Id._ at 22.\n\n. Dahlia Lithwick, _Legal Matters: The Importance of Being Irreverent_ , STAN. LAW.28, 30, 32 (Fall 2008).\n\n. _See_ chapter 1, text at note 36.\n\n. See Martha Neil, _Which Lawyers Love Their Jobs?, A.H.A.]._ Law News Now, Jan. 22, 2008, <http:\/\/www.abajournal.com\/news\/lawyers_who_love_the_law>.\n\n. _See_ chapter 5, text at note 11.\n\n### _Chapter 8_\n\n. Alice Park, _Wellness: A Primer for Pessimists_ , TIME, Apr. 6, 2009, at 2.\n\n. ERIC WEINER, THE GEOGRAPHY OF BLISS 182 (2008).\n\n. EDUARDO PUNSET, THE HAPPINESS TRIP: A SCIENTIFIC JOURNEY 53 (2007).\n\n. David Ian Miller, _Eric Weiner, Author of \"The Geography of Bliss,\" on What the Happiest Places on Earth Can Teach Us_ , S.F. GATE, Feb. 11, 2008, <http:\/\/www.sfgate.com\/cgi-bin\/article.cgi?f=\/g\/a\/2008\/02\/n\/findrelig>. DTL& type=printable, quoting Eric Weiner.\n\n. ERIC G. WILSON, AGAINST HAPPINESS: IN PRAISE OF MELANCHOLY (2008).\n\n. Alan Wolfe, _Hedonic Man_ , NEW REPUBLIC, July 9, 2008, _available at_ http:\/\/www.tnr.com\/booksarts\/story.html?id=3bcoe959\u20143b4e-44od-9b99\u201469078429b82c&p=2. Happiness has drawn other naysayers. Barbara Ehrenreich argues that positive thinking is a collective form of \"mass delusion.\" BARBARA EHRENREICH, BRIGHT-SIDED: HOW THE RELENTLESS PROMOTION OF POSITIVE THINKING HAS UNDERMINED AMERICA 13 (2009). A national ideology of excessive optimism, she says, has fueled things like the sub-prime mortgage crisis and the invasion of Iraq. _Id._ at 11. On the personal level, certain positive thinking techniques, such as \"self-hypnosis,\" promote \"magical thinking\" and require \"deliberate self-deception, including a constant effort to repress or block out unpleasant possibilities and 'negative' thoughts.\" _Id._ at 46\u201447, 5. We won't attempt to tackle generalizations about America's collective mental state (although we agree that unthinking indulgence in positive thinking certainly has its downsides), but chapter 2 is a response to her wholesale dismissal of the Science of happiness research.\n\n. Jonah Lehrer, _Depression's Upside_ , N.Y. TIMES MAG., Feb. 28, 2010, at 38\u201344.\n\n. Joseph Brodsky, _Listening to Boredom_ , HARPER'S MAG., Mar. 1995, at 11.\n\n. _Id._\n\n. _Id._\n\n. Conan O'Brien, Commencement Speech to the Harvard Class of 2000, Feb. 7, 2000, <http:\/\/www.february-7.com\/features\/c0nan.htm>.\n\n. Darrin M. McMahon, _The Pursuit of Happiness in Perspective_ , Apr. 8, 2007, <http:\/\/www.cato-unbound.org\/2007\/04\/08\/darrin-m-mcmahon\/the-pursuit-of-happiness-in-perspective\/> (quoting Thomas Jefferson's letter to John Page, July 15, 1763).\n\n. WEINER, _supra_ note 2, at 74, quotingjohn Stuart Mill.\n\n. _Id._\n\n. DANIEL NETTLE, HAPPINESS: THE SCIENCE BEHIND YOUR SMILE 87 (2005).\n\n. Joshua Wolf Shenk, _What Makes Us Happy?_ , ATLANTIC, June 2009, at 36, 46.\n\n. Penelope Trunk, _The Connection Between a Good Job and Happiness Is Overrated_ , Brazen Careerist, http:\/\/blog.penelopetrunk.com\/2007\/01\/16\/the-connection-between-a-good-job-and-happiness-is-overrated (Jan. 16, 2007).\n\n. _Id._ , citing professor of urban studies and creativity, Richard Florida.\n\n. NETTLE, _supra_ note 15, at 180.\n\n. _Id._\n\n. WEINER, _supra_ note 2, at 54.\n\n## **Index**\n\nadaptation, \u201342, , \u2013101,\n\naddiction, \u201327\n\nage factors and happiness, \u201310, \u201319, , , \u201373, \u201370,\n\nalcoholism, ,\n\naligning work and values, \u201351, , \u20138, , \u201317, \u201348, , \u201398, \u201326, \u201338\n\nanticipation and happiness,\n\narchitecture, , \u2013204,  n.120\n\nattention and happiness, \u201399, , , , , , , , , , , , , , , , ,  n.109\n\nAuthentic Happiness Center,\n\nauthenticity, \u201346,  n.99\n\nautonomy. _See_ control\n\nbillable hours, , \u201356, , \u201385, , , , , , \u201368, \u201374, \u201381, \u201313, , nn.9, ,  n.52,  n.42,  n.60,  n.4\n\nBen-Shahar, Tal, \u201347,  n.98\n\nboredom, \u201396, , , \u201318, \u201335, ,  n.8\n\nbrain chemistry. _See_ neuroscience\n\nBrodsky, Joseph, ,  n.8\n\nBuilding a Better Legal Profession, x, , \u201378, ,  n.45\n\ncareers\n\nchanging. _See_ jobs, changing\n\nchoosing a career (out of law school), , \u201357\n\nnon-law, , , \u201315, \u201349, , \u201328\n\npaths, , , \u201349, , \u201359, \u201331\n\nchecklists, \u201311, \u201349, \u20137,\n\nChristakis, Nicholas, \u201317,  n.10\n\ncircumstances (as a determinant of happiness), \u201315, , \u201343\n\ncivility, decline of, , \u201362, , \u201323, , , \u201314, nn.21,\n\nclients\n\nas a source of happiness, , , , , , , \u20138, , \u201323, , \u201345, , , , , , \u201330\n\nas a source of unhappiness, \u201354, \u201366, , , , , \u201368, \u201316\n\nColby, William, \u201330, ,  n.37\n\ncommitment to job, , \u201396, \u201310, , , , ,\n\ncompensation. _See_ income; lawyers, salaries of\n\ncompetence, , , , , , ,\n\ncontrol, \u201310, , , , \u201347, \u201353, , , , , , \u201386, , , , , , , \u201356, , \u201385, \u201389, , \u201313, , ,  n.15,  n.36\n\ncultural differences, \u201332\n\nCzikszentmihaly, Mihalyi, , nn.34, ,\n\nDavidson, Richard, , ,  n.18\n\ndepression, \u20137, , , , , \u201381, \u201330, \u201334,  n.23,  n.49,  n.7\n\ndownward comparisons, \u201390, ,\n\ndrugs and happiness, \u201328\n\nemotional contagion, , \u201317, \u201364, \u201383, nn.8,\n\neudaemonia, , \u20139, \u201336\n\n_See also_ happiness, long term (well-lived life)\n\nevolutionary psychology, \u201322\n\nexercise, \u201331, ,  n.19\n\nexpectations. _See also_ predictions\n\nlaw firms' work, , , , , , ,\n\nunrealistic expectations as a source of unhappiness, , , ,\n\nextroversion, \u201335, , \u201375, , , ,  n.57\n\nfeedback and happiness, , \u201395,\n\nflexible work arrangements, \u201386, \u201357, \u201374, \u201382, ,  n.68, nn.70\u201371\n\nflow experiences, \u2013100, , \u201320, nn.35, ,  n.8\n\nFowler, James, \u201317,  n.10\n\ngender, \u201314, , , , , , , , , , \u201371, \u201380, , ,  nn.39\u201340, , , \u201349,  nn.51, \u201356,  n.23,  n.57,  n.29,  n.35,  nn.59, ,\n\ngenerational differences, \u201370,  n.20\n\ngenetics, , \u201334, , ,\n\ngeography, , , \u201345, , , ,  n.21,  n.16,  nn.2,\n\nGilbert, Daniel, , , \u2013101, , , ,  n.37,  nn.42, ,  n.7,  nn.25, , , ,  n.32,  nn.103, , , ,  n.1\n\nGladwell, Malcolm, , \u201392,  nn.30,\n\nGoogle, ix, \u201398\n\nhappiness (and unhappiness)\n\nbenefits of happiness, , \u201364\n\ncircumstances and, , \u201333, \u201343\n\ncontagiousness. _See_ emotional contagion\n\ndefinitions of, \u201319,\n\ngenetics and. _See_ genetics\n\nlong term (well-lived life), , \u20139, \u201336\n\nmeasuring (and methodological concerns), \u20135, \u20138, \u201325, \u201332, , \u201350,\n\nsatisfaction. _See_ satisfaction\n\nset point, \u201335, , , , , , , ,\n\ntransient. _See_ pleasure\n\ntrust and. _See_ trust\n\nhedonic treadmill. _See_ adaptation\n\nHopkins, Michael, \u201331\n\nincivility. _See_ civility, decline of\n\nincome, \u201311, , \u201339, , , , \u201388, , \u201338, ,\n\nrelative comparisons, , , \u201390\n\n_See also_ money\n\ninterests, , , , , , , , \u201349, , , , , , ,\n\nInternet, ,\n\ninterviewing for a job, , \u201341, , \u201356,  n.55\n\nintroversion, \u201335, , ,  n.57\n\njobs\n\nchanging, , \u201373, , , \u201326,\n\nsatisfaction, x-xii, \u201316, , \u201373, , , , , \u20139, , , , , , , , , \u201380, , , , \u201397, \u20135, \u201328, , ,  n.3,  nn.5, ,  n.16,  n.28,  n.1,  nn.8, ,  n.23,  n.36\n\njob search. _See_ interviewing for a job\n\nKrieger, Larry, , \u201330, ,  n.50,  n.61,  n.15,  n.41,  nn.51, , , ,  n.111\n\nLSAT scores and happiness,\n\nlaw firms\n\nattrition, , , , \u201362, , \u201391, \u201395, ,  n.20,  n.2,  n.91,  n.105\n\nbenefit of happy lawyers, \u201364\n\ncompensation systems, , , , , , , , \u201381, \u201391, , \u20136, ,  n.52,  n.87,  n.29\n\ncompetition within firms, \u201361, ,\n\ndesign of offices, \u2013204\n\ndiversity, , , \u201370, ,  n.87,  nn.23, , , , , \u201370 n.31\n\nlarge firms, , \u20139, \u201315, \u201356, \u201363, \u201383, , , , , , , , , , \u201387, \u201393, , \u201314, ,  n.48,  n.10,  n.28,  n.59,  n.3,  n.38\n\nloyalty, , \u201367, \u201371, , ,\n\nmentoring in. _See_ mentoring\n\nmergers,\n\nplayfulness, \u201398, ,  n.99\n\npromotion, , , , , \u201391,  n.52\n\nsize of firm and happiness, \u201356, , , , \u201395, , \u201393\n\nsmall firms, , , , , , , , , , , , , , , , ,  n.10\n\ntraining, , , , , , , , , \u201396, \u20136,  n.36,  n.87,  n.88\n\ntransparency, , \u201391\n\nlaw practice\n\nalternate careers to, , , \u201315, \u201349, , \u201328\n\nchanging nature of, \u201386, \u201368, \u201374, \u201381\n\nethics, \u201359, , , \u201314,  nn.15, ,  nn.24, , ,  n.38,  n.85\n\ngovernment work, \u20139, \u201314, , , , , , ,  n.34,  n.10\n\nnobility of, \u201368\n\npublic interest work, , , \u201337, , , \u201397, , , \u201330,  n.52,  nn.67,\n\nsatisfaction with, changes over course of career, \u201372\n\nsources of satisfaction with, \u201370\n\ntechnology, impact of, , ,\n\ntypes of practice, , \u201315, , , \u201315, \u201320, , , , \u201346, \u201379, , , \u201331,  n.59\n\nlaw schools\n\nbias toward big firms, \u201341, \u201376\n\ncareer services, \u201324, \u201341, ,\n\nchoosing a, \u201324\n\ncourses on happy lawyering, \u201324\n\ncurriculum, , \u201325\n\ndebt (coping with), , \u201338\n\neffect of on happiness,\n\nexperiencing as a student, \u201334,\n\nhappiness as it relates to rank of, \u201315, \u201318\n\nimportance of peers in, \u201317\n\ntuition, \u201336,  n.69\n\nvisiting schools, \u201322\n\n_See also_ legal education\n\nlaw students\n\ncompetition, , \u201330,  n.16\n\nexperiencing law school, \u201334,\n\ngrades, , \u201330, \u201334,\n\nideals, , ,\n\ninterviewing. _See_ interviewing for a job\n\nintrinsic motivations, , , , \u201333,\n\npersonalities, , \u201329, , \u201347, ,  n.54\n\nsocial connections, , \u201334\n\nstatistics, , , , , , , \u201336,  n.39, \u201361 n.2, \u201364 n.48,  n.69\n\nstress, \u201333, ,  nn.41,\n\nsuccess in, , \u201376, \u201334, ,  n.1\n\nlawyers\n\nassociates, x, \u20136, \u201311, , \u201357, \u201362, , , , , \u201357, , , , \u201396, , \u20137, , \u201318, ,  n.3,  n.18,  n.34,  n.55,  n.9,  n.73,  n.87,  n.79,  n.91,  n.112\n\nfemale, \u201314, , , , , , , , , , \u201371, \u201380, , ,  nn.39\u201340, , , \u201349,  nn.51, \u201356,  n.23,  n.57,  n.29,  n.35,  nn.59, ,\n\nlaterals, , , ,\n\nlayoffs, ,\n\ngay, lesbian, bisexual, transgendered, , , , , ,  n.45,  n.52\n\npartners, , \u201313, \u201358, \u201362, \u201372, \u201357, , \u201367, \u201396, , \u20137, , , , ,  n.49,  n.30,  n.87,  n.16,  nn.77, ,  n.83\n\npersonality, \u201377, \u201357 n.56,  nn.57, ,  n.64\n\npublic opinion of, \u201368\n\npublic sector. _See_ law practice, government work\n\nracial minorities, \u201314, , , , ,  n.42,  nn.51, , ,  n.65,  n.29\n\nsalaries of, \u201314, , \u201363, , , , , , \u201336, , , , , , , , , , , \u201390,  n.34,  n.13,  n.52,  nn.71\u201373\n\nstatistics, x, \u20133, , , , , ,  nn.5, ,  n.34,  nn.39, \u201349,  n.52,  nn.8,\n\nunhappiness, causes of, xi, \u201317, , , , \u201367, , \u201378, \u201388, , \u201326, , \u201342, , \u201377, , , , , ,\n\nlawyers' stories, , \u201345, , , \u201331,  n.3\n\nlegal education\n\ncollaborative learning, \u201320,  n.20\n\ncurriculum. _See_ law schools, curriculum\n\nhumanizing, \u201323, ,  n.29\n\nlaw professors, , \u201323, \u201332, \u201340,  n.83\n\n_See also_ law schools\n\nloan repayment assistant programs, \u201337,  nn.74\u201378\n\nLykken, David, ,  n.25\n\nLyubomirsky, Sonja, , , , ,  nn.27, ,  nn.41, , , \u20133,  n.18,  nn.27, ,  n.107\n\nmarriage, ,\n\nmattering, , \u201384, , , ,\n\nmeasuring happiness. _See_ happiness, measuring\n\nmethodological concerns. _See_ happiness, measuring\n\nmeditation, , ,  n.18\n\nmelancholy, , \u201335,  n.5\n\nMelcher, Michael, , \u201343,  n.28,  n.57,  n.95\n\nmentoring, , , , , , , \u201367, \u201372, \u201391, , , ,  n.9\n\nmoney, , \u201311, , , \u201357, \u201363, , , , , , , \u201314, , , , , , , \u201377, , , , , ,  nn.35, ,  n.28,  n.24,  n.20,  n.41,  n.91,  n.116\n\n_See also_ income\n\nMyers-Briggs,  n.55,  n.57\n\nNettle, Daniel, ,  n.23,  n.52,  nn.13, ,  n.79,  nn.15,\n\nneuroscience, \u201331,\n\nNiebuhr, Reinhold,\n\nO'Brien, Conan, ,  n.11\n\noccupations other than law, , , \u20137, \u201352, , \u201365, , , , \u201393, , \u201315, , \u201328,  n.5\n\nOffer, Avner,\n\nparadox of choice, , \u201373, , \u201343,  n.51,  n.50,  n.90\n\npersonality as relates to happiness, , \u201335, , \u201377, \u201314, \u201330, , , \u201347, , ,  n.26,  n.54, \u201358 n.56,  n.57,  n.64\n\npessimism, , \u201377, ,  n.59\n\npleasure (short-term or transient happiness), \u201319, \u201330, , \u20135,\n\npoll data. _See_ surveys\n\npositive attitude, , \u201385,\n\npositive psychology, ix, xi, , , , ,  n.47\n\npredicting future happiness, \u201342, \u2013103, \u201352\n\npro bono work, , , , , , , \u201397, \u201330, ,  n.113\n\npublic interest work, \u201335, , \u201397, , ,  nn.67,\n\npublic perception of lawyers, \u201368,\n\nPunset, Eduardo, \u201324, \u201381, , , ,  n.5,  n.16,  n.1,  n.8,  n.51,  n.81,  n.3\n\nPutnam, Robert, , \u201370,  n.26,  n.26,  n.32\n\nrace. _See_ lawyers, racial minorities\n\nrecession, , \u201361, , \u201388, ,\n\nrelationships and happiness, , \u201338, , , , , , , , , , \u201396, \u201317, , , , , , \u201389, , , \u201316, \u201340,  n.28,  n.36,  n.102\n\nreligious beliefs and happiness, , \u201346,  n.37\n\nRhode, Deborah, ,  nn.49, ,  n.3,  n.74\n\nRostron, Allen, \u201384, ,  n.4\n\n\"Rule of ,\" \u201393\n\nsadness. _See_ melancholy\n\nsalaries. _See_ income; lawyers, salaries of\n\nsatisfaction\n\nkeys to, xi, \u201316, , \u201347, \u201352, \u201373, \u2013111, \u201318, , \u201324, \u201334, \u201347, , , \u2013209, \u201314, , \u201324, , , \u201341,  n.28,  n.3,  n.61,  n.44,  n.49,  nn.3, ,  n.31,  n.18,  n.36,  nn.71, , ,  n.88,  nn.104, ,  n.6\n\nSchwartz, Barry, , ,  n.51,  n.50,  n.32,  nn.90\u201394\n\nsecurity, , , \u201383, , ,  n.33\n\nself-complexity, \u201341\n\nself-esteem, , \u201328, \u201333\n\nSeligman, Martin, , , , , \u201347, , ,  n.21,  n.26,  n.30,  nn.58, , ,  n.47,  nn.99, ,  n.37\n\nset point. _See_ happiness, set point\n\nsexual activity and happiness, \u201322, \u201328, , , , ,  n.14\n\nsocial justice, , , ,\n\nsolo practice, , , , , ,\n\nStanford students. _See_ Building a Better Legal Profession\n\nstrengths, identifying, , , \u20138, , \u201347, , -238,  n.54,  n.99\n\nstress, \u201331, \u201356, , \u201381, ,\n\nsurrogation, \u2013103, , \u201353, \u20139\n\nsurveys, x, \u20137, , , , , , , , , , , , , , , , \u201385, , , , , , , , , \u201373, \u201380, , \u201395,  nn.1, ,  nn.5, , \u201347 n.11,  nn.12, ,  n.35,  nn. , ,  n.55,  n.3,  nn.6, ,  nn.21, ,  n.4,  nn.50, ,  n.87,  n.63,  n.71,  n.114, , n.13\n\nTellegen, Auke, ,  nn.25\u201326\n\n\"Three Question Process\" (for job searches), \u201347\n\ntools to build a happier practice, xi, , \u2013111, \u2013207\n\ntrial attorneys, \u201317, , , \u201320, \u201328\n\ntransactional work, \u201317, , , \u201328\n\ntrust, xi, , \u201340, , , \u201395, , , , , \u201389, \u201314, , ,  n.43\n\nunhappiness, work related, \u20137, \u201315, \u201335, \u201367, \u201378, \u201386, , \u20137, \u201364, \u201377, \u201395, \u201311, \u201318, \u201321,\n\n_See also_ happiness (and unhappiness); lawyers, unhappiness\n\nValliant, George, ,\n\nvalues, xii, , , , \u201353, , , , , \u20139, , \u201317, , \u201334, \u201342, \u201348, \u201355, \u201359, , , , , \u201321, , , , \u201337,  n.38,  n.61,  n.22,  n.51,  nn.99,\n\nVeenhoven, Ruut, ,  n.2,  n.38\n\nvirtues, six, \u201347\n\nWeiner, Eric, ,  n.21,  n.31,  n.16\n\nWilson, Eric G., ,  n.5\n\nwork-life balance, , , , , , , , , , , , , , , , \u201381, , , , , \u201313, , ,  n.40,  nn.71,\n\nWorld Database of Happiness, , ,  n.2,  n.2\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n#  \nCopyright \u00a9 2010, 2015 by Lamar Underwood and Nate Matthews\n\nAll rights reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018.\n\nSkyhorse Publishing books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.\n\nSkyhorse\u00ae and Skyhorse Publishing\u00ae are registered trademarks of Skyhorse Publishing, Inc.\u00ae, a Delaware corporation.\n\nVisit our website at www.skyhorsepublishing.com.\n\n10 9 8 7 6 5 4 3 2 1\n\nLibrary of Congress Cataloging-in-Publication Data\n\nUnderwood, Lamar.\n\n250 amazing hunting tips : the best tactics and techniques to get your game this season : deer,. bear,. waterfowl,. small game, and more! \/ Lamar Underwood and Nate Matthews ; illustrations by John Rice.\n\npages cm\n\nISBN 978-1-63220-303-8 (pbk. : alk. paper) \u2014 ISBN 978-1-63220-950-4 (ebook) 1. Hunting\u2014Handbooks, manuals, etc. I. Matthews, Nate. II. Title. III. Title: Two hundred and fifty amazing hunting tips.\n\nSK33.U527 2015\n\n639'.1\u2014dc23\n\n2015021854\n\nPrinted in the United States of America\n\n#  \nContents\n\nIntroduction\n\n1. White-tailed Deer Hunting\n\n2. Elk Hunting\n\n3. Mule Deer Hunting\n\n4. Bear Hunting\n\n5. Pig Hunting\n\n6. Moose Hunting\n\n7. Caribou Hunting\n\n8. Pronghorn Antelope Hunting\n\n9. Wild Sheep and Mountain Goat Hunting\n\n10. Turkey Hunting\n\n11. Duck Hunting\n\n12. Goose Hunting\n\n13. Grouse and Woodcock Hunting\n\n14. Pheasant Hunting\n\n15. Quail Hunting\n\n16. Squirrel Hunting\n\n17. Rabbit Hunting\n\nIntroduction\n\nThe other evening, after I had spent the day going over material for this book, my wife made a remark that brought doubt and fear crashing into my thoughts like lightning bolts. She simply said, \"The newspaper had a section today called '101 Grilling Tips.' I looked at them all and didn't find a single one to be useful.\"\n\nSuddenly, I felt almost ill. I thought about picking up the phone and calling Nate Matthews, my co-editor and writer on this book. Was a fate similar to \"101 Grilling Tips\" in store for our book\u2014multiplied two and a half times? Were our tips destined to be ignored, cast aside with shrugs and expressions of disappointment? It didn't seem possible, yet there it was: the possibility that we would not connect with our audience.\n\nInstead of allowing my anguish to continue unabated, I decided to look at \"101 Grilling Tips\" myself. I only liked a couple of the tips. The others were bland, without any juice to make me want to whip the cover off the Weber. My confidence in the work Nate and I were finishing began to rise again. Later, over the next few days as I gave our tips a final reading, that confidence level peaked out again. Today, ready to commit our tips to print, I guarantee our readers that they are going to find many, many useful tips. And I shall go even further by saying that many of the tips are so important that they mean the difference between success and failure on your hunt.\n\nYes, a single tip on hunting can carry that kind of importance behind it. If followed, it can mean you're literally bringing home the bacon. Ignored or forgotten, it can mean you're coming home only with that tired old expression, \"Sure was nice being in the woods today.\" Yea . . . sure . . . it was nice. There's no disputing that. But you have nothing new to put into the freezer.\n\nAllow me, if you will, to try to press my point about tips home with an example.\n\nMy personal deer hunting education was slow in evolving. I'm an Army brat, who grew up mostly in the south where my hunting was focused on smoothbore opportunities. Using dogs to drive deer to standers was still in vogue back then and never appealed to me, especially with the great hunting we had for quail, doves, ducks, squirrels, and rabbits.\n\nThere came a long-awaited morning, however, when as a young man I became a serious player in the great game known as whitetail deer hunting. I was settled against the trunk of an enormous white oak in the heart of classic whitetail country among the folded hills and ridges of Pennsylvania. Under a flashlight's glow, with my friend Larry's direction, I had swept aside leaves and sticks to make my nest as quiet as possible. I had a pack with a thermos and sandwiches, and in my hands rested a Winchester lever-action .30-30 with iron sights. The rifle triggered memories of the Daisy Red Ryder I had grown up using. When I was all set, Larry moved away to work another part of the ridge.\n\nAt first light, the horizon was cut from steel\u2014black lumps of white pines, the knife-like rim of a distant ridge looming above the forest. Without wind, the stillness seemed almost electric as I shivered in the cold, nervous to the raw edge at the thought of a buck coming my way.\n\nWhen the crows began to patrol, their raucous cries announcing that it was time for things to get moving, I detected the rustle of squirrels in the branches overhead. As the light came on, I began to hear occasional gunshots, echoing from the distant ridge.\n\nThe gunfire made me nervous. For some time, I had been hearing about reckless deer hunters, shooting at anything that moved. \"Oh, I had a couple of brush shots.\" That often-repeated comment wasn't funny to me.\n\nSuddenly, the sound of a limb breaking jerked my attention toward the trees to my left. I strained to see. Nothing! The sound came again, louder still, and followed this time by the dry rustle of leaves beings shuffled. I could only think of one thing that could make that much noise in the deer woods: another hunter! I was suddenly very, very afraid. I stood up, waving and shouting, \"Hi there!\"\n\nThere was a moment of total silence. Then, in an explosion of crashing brush, the buck of my dreams bounced into the full light in a clearing between the pines, white tail flagging, then was instantly gone before I could even think of raising my rifle.\n\nI stood there, stunned, hurt to the core. I was thinking terrible thoughts about myself. About being stupid, fumbling the ball, blowing my big chance. I was sure I would never have such an opportunity again.\n\nI felt like heading for our car, going home. But I sat back down instead, trying to get a grip on myself. I had no faith that another chance would come my way, but I knew the game would not be over until sundown. I would play it out, give Lady Luck a chance to shine on me again.\n\nGradually, the forest worked its therapy on my beleaguered spirit. Bird activity was plentiful\u2014 crows, blue jays, chickadees, woodpeckers, juncos were on the move and calling. As always when I'm on a stand, they furnished a interesting sideshow during the hours of waiting for game to show. Then, about noontime, I noticed that the squirrels were moving as industriously as they had been at dawn. Strange, I thought, there must be some dirty weather around. I can't see it coming, but they can feel it coming.\n\nI knew I was getting back into the groove then, focused on nature instead of myself. I dug into my pack for the sandwiches and coffee. They were very satisfying. I felt my confidence rising, and I realized now I owned something precious: the image of that buck in the clearing between the pines, the light full on what William Faulkner in a hunting tale called \". . . that rocking chair he carried.\"\n\nI did not get my \"rocking chair\" buck that day. That night, it snowed, and two days later I shot a fork-horn buck. For the first time in my life\u2014though certainly not the last\u2014I found myself saying to the other fellows, \"Well, boys, you can't eat horns.\"\n\nAs for the buck of my dreams . . . he would have to wait. And I could not help but think, rather bitterly: Why didn't Larry tell me: Bucks can make considerable noise when they're out for a morning stroll and haven't been disturbed. One single tip . . . that's all it would have taken. A tip I did not have.\n\nIf you've ever used the expression \"I'd rather be lucky than good,\" then you just might be a whitetail deer hunter. You're a card-carrying dreamer out to beat the odds, your heart set on a rack for the wall and a freezer where packets of venison take up most of the room. You're not discouraged by the possibility that this may not be your year to bag a buck, for the pull of the deer woods and deer camp are strong. You've just got to be out there, watching the weather, checking for sign, enjoying the company of good buddies.\n\nThat the task of bagging a whitetail buck\u2014any buck!\u2014requires skills is obvious to the brethren of Hunter Orange. While they are animals of fairly rigid habit, whitetail deer are interesting and complex creatures prone to a witch's brew\u2014for hunters anyway\u2014of contradictory behavioral patterns that are sometimes hard to understand and impossible to predict. When you think you have a lock on where they're feeding, they move to some other dining halls in another part of the forest or croplands. The white oak patch where you scouted a buck feeding a week before turns out to be empty on opening day\u2014except for its colony of squirrels. This list of failed game plans could go on and on.\n\nYou think you know what the deer will be doing. Perhaps you do. But all too often it seems like they're doing it somewhere else\u2014somewhere you don't know about! And don't think for a minute that your quarry doesn't know you're coming.\n\nOpening Day. The pre-dawn explodes with noise, like a bomb going off. A deer, especially a veteran \"survivor\" buck, would have to be deaf not to know what's going on. The cacophony includes the sounds of engines being revved up, wheels spinning in the mud, truck and SUV doors slamming, the snick-snick of gun actions being worked, and a myriad of shouts, \"Hey, Joe, you goin' to the Oak Tree stand?\" \"I'll meet you at noon by the apple orchard.\" \"Oh no, I forgot my shells!\"\n\nWell, perhaps we can get lucky. Happens all the time. Just drop by any check-in station, and you'll see people there who wouldn't know a scrape from a hog wallow. Nevertheless, they will have tags on bucks.\n\nThe goal Nate Matthews and I have set for ourselves is to give you something to cut the odds on your next hunt, for whatever game interests you\u2014whitetail deer, mallards, pheasants, moose, you name it.\n\nNate and I come your way from a background of working with some of the most highly respected magazines in the outdoor field\u2014Field & Stream, Sports Afield, and Outdoor Life. We do not present ourselves as experts in the various hunts we cover. But we know the people who are\u2014and you will find them in the pages ahead.\n\nIf you feel lucky, I'm sure you will carry that on your next hunt. As I have suggested, Lady Luck makes a fine companion. Nate and I hope, however, that you will be taking some of these tips along to supplement the Lady's efforts. You might be on a deer stand at dawn and hear something shuffling your way, breaking sticks, making noise. Sounds like another hunter, except . . . oh yes . . . there was this tip you heard somewhere. About bucks sometimes making noise instead of sneaking in. You keep watching . . . and then raise your gun. Yes!\n\nSometimes, just the right tip is all you need to have a great day.\n\nGood luck!\n\n\u2014Lamar Underwood\nCHAPTER 1\n\nWhite-tailed Deer Hunting\n\n1. Don't Shoot Bucks That Look Insecure\n\nWhen you first see a buck, take a moment to check its posture. Dominant bucks hold their heads high and walk loosely with their tails held straight out. A subordinate buck walks with stiff legs and a hunched back, and keeps its tail between its legs. If you see a good buck in a subordinate posture, consider holding your shot. It could mean there's a real monster in the area.\n\n2. Find Small Bucks Near Big Scrapes\n\nIf you're looking to shoot a buck quickly during the rut and aren't much concerned about the size of its antlers, look for a large scrape that's torn up, irregularly shaped, and looks like it's being used by more than one deer. Younger, more submissive bucks frequent such scrapes. Since these bucks are less wary and more numerous than trophy animals, you'll stand a good chance of filling your tag faster than you would when hunting scrapes made by solitary (and often bigger) deer.\n\n3. The Surefire Spot for Big Bucks\n\nDon't give up on a hunting spot when you learn a big whitetail has been taken there. If it was a dominant buck, a host of suitors for this vacated territory will soon move in. The sudden void may dramatically increase other bucks' activity. If you can hunt where another hunter has already bagged a big buck, do it!\n\n4. See More Deer by Scanning an Area Twice\n\nImmediately after stopping at a vantage point, allow your eyes to relax and move them slowly back and forth over the surrounding terrain without focusing on any specific feature. Relaxed eyes automatically focus on any movement within their field of vision. If no deer are moving in your immediate vicinity, shift to a tightly focused analysis of every piece of cover you can see. Peer into the shadows, looking for pieces of deer\u2014bits of antler, the curve of an ear, or the horizontal line of a back. Move to your next vantage point once you're satisfied that you've probed all the places a deer might be hiding.\n\n5. Don't Use Too Much Freshly Collected Scent\n\nIf you plan to use scent collected from the glands of a freshly-killed deer, make sure to use less of it than you would of the bottled stuff. The fresh gland scent will be much more potent than what you can buy commercially.\n\n6. Three Steps to Proper Still-hunting\n\nProper still-hunting can be described as a three-step process. Step one is to stand motionless behind an object that will break up your outline while searching the surrounding area thoroughly for any sign that deer are present. Step two is to remain still and use your eyes to pick out a way forward that lets you place your feet on the quietest ground cover possible, such as bare rock, moss, wet leaves, or soft snow. Step three is to scan the woods for deer one more time, then slowly and silently navigate the route you've picked out. Repeat steps one through three until you find your buck. Do not rush. A good still-hunter will sometimes take an hour to traverse 100 yards of heavy cover.\n\n7. Catch Wary Peak-season Bucks Off Guard during Lunch\n\nBecause of increased pressure during the rut, mature bucks will often change their patterns to avoid hunter activity. Many will become nocturnal, but a significant number instead spend more time searching for does during the middle of the day, when most hunters are back at camp taking naps and eating lunch. Try sitting your stand during the hours before and after noon to catch these deer off guard.\n\n8. Use Different Routes to Your Deer Stand at Sunrise and Sunset\n\nNever walk through a crop field in the early morning when approaching a deer stand set up near its edge. Deer are likely feeding in this field under cover of darkness\u2014you will startle them if you don't take a back route to your stand. The opposite is true when you're approaching the same stand during the afternoon or evening hours. Deer are likely bedded in the cover you used to hide your approach in the morning, waiting for the sun to go down before moving out to feed. You should approach your stand through the field at this time of the day.\n\n9. Guess a Deer's Sex by Analyzing Its Gait\n\nYou can tell buck tracks from doe tracks more easily when tracking deer through the snow. Does place their feet with precision; bucks sway from side to side while walking, a rolling gait that often leaves drag marks in powder. Longer drag marks may mean you've found the trail of an older or heavier buck.\n\n10. Hunt Near Food Sources When the Barometer Starts Dropping\n\nWhite-tailed deer feed heavily in the days and hours leading up to the arrival of violent low-pressure systems. Watch your barometer. When the pressure starts dropping, head to the edges of crop fields and alfalfa meadows, or to stands of mature oak where the ground is covered with acorns. As the front gets closer, start hunting from stands set up along trails that lead from these food sources to heavy cover where you know deer go to hunker down during nasty weather.\n\n11. Flush Big Bucks from beneath Downed Trees\n\nWhen still-hunting through mature forest during hot weather, keep your eyes peeled for large trees that have been uprooted recently by storms. The maze of shade and cover offered by the fallen limbs and branches provides one of the best hiding places in the forest, and big bucks will often bed down deep within their embrace to escape the heat of the day. Experienced animals feel very secure in such cover and may not flush unless you get extremely close to them. Approach every such tree you can find.\n\n12. Don't Get Too Hot to Sit Still in Cold Weather\n\nWhen hiking out to your tree stand in cold weather, do not wear all the layers you'll need to stay warm while sitting still. Doing so will cause you to sweat heavily on the way in, and this sweat will cool quickly once you stop moving, leaving you too chilled to remain quiet for long. Dress lightly and carry your outer layers in a backpack. Pull on warmer clothing only after you've climbed up into your stand and sat long enough for your heart rate to slow down.\n\n13. Read Rub Lines to Anticipate a Buck's Movements at Different Times of the Day\n\nRubs can show you to where, at what time of the day, and in what direction a buck normally travels. When you first find a rub, get down on your knees so that your field of vision is similar to that of the buck that made the rub. Scan for additional rubs in the area\u2014chances are good you'll find another from thirty to fifty yards away. Repeat this process until you've identified a series of rubs, called a rub line. Rub lines often mark the routes a buck uses to travel to and from his preferred feeding and bedding areas. Most rubs in a line will be made on the same side of each tree; this tells you the direction the buck travels when using the route. If the line leads from a feeding area to thick bedding cover, set up a stand nearby and hunt there in the morning. If the line leads away from bedding cover, hunt it in the evening.\n\n14. Hunt All Three Phases of the Rut\n\nA good deer hunter knows that there is not just one rut, but three. The first, called the pre-rut, occurs in early October, when mature, four- and five-year-old does first come into estrus. The second, known as the peak or primary rut, runs from late October to the last week of November, and is when the majority of female deer come into heat. The third, called the post-rut or late rut, takes place twenty-eight days after the end of the primary rut, as does that were not bred during October and November come back into estrus. These pre- and post-rut phases do not last long. Look for a sudden explosion of fresh buck sign, then hunt hard for several days using techniques, such as rattling, that take advantage of the increased aggression triggered by competition for a limited number of willing does.\n\n15. Gain Extra Seconds to Shoot When Rattling in a Buck\n\nBucks will often appear at the most inopportune moments, especially when you're rattling antlers to call them in. If you spot one while holding a rack in your hands, don't be afraid to put it down and pick up your gun. The buck you've called will be expecting to see some motion and will be less likely to startle immediately when he sees you, which gives you a few extra moments to shoot him. Using one smooth, unhurried motion, pick up your rifle, shoulder it, and fire. Make sure that you do not move too fast or jerk your arms, rifle, or body. Such sudden movements signal your excitement and may alarm the animal.\n\n16. Let Blood Color Tell You How Fast to Follow a Wounded Buck\n\nBlood trails don't just tell you the direction a wounded deer is moving. They can also provide good information about where on its body you shot the animal, and how quickly you should follow its trail. Bright red blood is full of oxygen and often means you've hit your deer in the lungs. Deer hit this way don't go far, so you can pursue them quickly. Dark red or purple blood may indicate a gut shot. If you find such blood, particularly in conjunction with bits of intestinal fat, and there's no precipitation forecast that could wash away or obscure the trail, consider giving the animal time to bed down and stiffen up before looking for a follow-up shot. Gut-shot deer often run long distances if they're chased immediately after being wounded.\n\n17. Tie Back Branches When Hanging Your Tree Stand\n\nWhile a good pair of pruning shears can be a bowhunter's best friend, there are many situations in which you should refrain from overclipping. For example, when setting up your tree stand, it is a good idea to try tying back any branches that obstruct your view, using a length of rope or wire. This method is quieter than clipping, will cause less damage to the tree you're using, and the branches can often be more easily positioned behind your stand in order to break up your silhouette.\n\n18. Find Deer in Transition Zones\n\nDuring the end of October and in early November, before the peak of the rut, deer switch from grazing in fields to browsing on twigs, branches, and buds. These foods are most easily found in wide transition zones of thick understory that grow up between mature forests and more open fields and meadows, where the shade cast by tall trees is not deep enough to inhibit the growth of younger saplings. The thick growth also serves as cover for the animals after leaves have fallen from the branches of more mature timber. Set your stand near deer trails close to rubs or scrapes in these transition zones, and you'll have a good chance of filling your tag.\n\n19. Still-Hunt Uphill in the Early Morning\n\nOn calm days when there's little wind, air will flow downhill as it cools in the evening, and uphill as it heats up during the day. Deer use these currents to keep track of their surroundings. To keep them off your scent, hunt your way uphill in the late evening and early morning hours, and down during the rest of the day.\n\n20. Hunt in Three Places at Once\n\nOne of the best locations to hunt is the intersection of three different types of vegetation. Look for a field corner bounded by timber on one side and a swamp, slough, or bottomland on the other, then hang your stand in a tree with a good view of any trails that lead from one to the other.\n\n21. Don't Hang Your Stand Too High in Steep Terrain\n\nWhile hanging your stand high in a tree will better hide your presence in level terrain, doing so in steep, hilly country may actually put you at eye level with deer working down the ridges you're hunting. Try lowering your stand to camouflage your silhouette in such conditions. A deer looking downhill will have a more difficult time spotting you against a backdrop of leaf litter than it will spotting you against the sky.\n\n22. Use Hunting Pressure on Public Land to Your Advantage\n\nIf you hunt public land that gets lots of pressure during the prime shotgun or rifle season, you're going to run into other hunters in the woods. Instead of letting them ruin your hunt, figure out how to use them to your advantage. Set your stand up on trails leading to thick cover near routes you know other hunters are using. Deer will flee to these areas when spooked by all the unusual sights, sounds, and smells in the woods, so you'll be in a good position to catch them as they sneak through.\n\n23. A Basic Rule for Knowing When to Move and When to Sit Still\n\nWhen you know deer are on the move, such as in the morning and evening hours, you should sit still in a good stand or other ambush point and wait for the animals to come to you. It is only during conditions in which deer stop moving that you should move to find them.\n\n24. Develop a Quiet Stride for Still-Hunting\n\nMaintaining proper balance is the key to walking quietly across the forest floor. A long stride combined with little forward momentum will often leave you tipping to one side or the other, which can force you to place your feet awkwardly as you catch your balance. To reduce the noise you make, learn to take smaller steps, and to place your feet heel or toe first. Shift your weight slowly onto your forward leg while rolling your foot from heel to toe (or toe to heel). When performed properly, this movement\u2014called the rolling compression step\u2014will allow you to feel any twigs, branches, or other objects that might make noise before you place your full weight on them. This lets you shift your weight to your back leg before the object snaps, then place your front foot in a new, less noisy spot.\n\n25. Analyze Stomach Contents to Pattern Feeding Behavior\n\nIt's a good idea to examine the stomach contents of a deer you've shot. Less-digested food is what the deer ate last; well-digested food was eaten earlier in the day. You can use this information to guess where the deer was feeding in the hours before you killed it, and then apply what you've learned toward filling any open tags you or your buddies still have.\n\n26. Easy Way to Check the Wind\n\nSave a few of the black neck feathers from your next ruffed grouse\u2014or the lightest feathers you can pick off doves, pheasants, or quail if they're your favorite birds. Hung from a bow limb with a piece of dental floss, a light feather is a great wind direction indicator.\n\n27. Don't Scare Big Bucks out of Bedding Sites\n\nIf you've located a good buck before the season, resist the urge to hunt him in his bedding site. This is almost certain to drive a wary trophy animal out of the area. Instead, hunt the travel zones between his bedding and feeding spots.\n\n28. Gauge How Well Your Stand Is Hidden Using Black-and-White Images\n\nA good way to tell if your stand or blind is well concealed is to photograph yourself sitting in it during the exact hours of the day you think you'll be hunting from it. Use a digital camera, and convert your images from color to black-and-white using the image-processing program on your computer. Deer are colorblind, so these black-and-white images will give you a good idea of the patterns, shapes, and tones that seem out of place. If you and your stand are easily recognizable, reconfigure its position and make sure that it is not too bright or too dark compared to its surroundings.\n\n29. Practice with Your Bow in Hunting Situations\n\nAlways practice shooting your bow under the same conditions in which you expect to shoot your deer. You may be able to stick five arrows into a circle the size of your fist at fifty yards when you're standing on flat ground and wearing a T-shirt, but that won't help you much if you don't know how to do the same thing while wearing a heavy jacket. Spend time during the preseason simulating live hunting situations. Use broadheads rather than field points, wear your hunting clothes, and practice shooting from awkward positions and elevated angles.\n\n30. Hunt Sleepy Bucks on Beds during a Full Moon\n\nWhen the moon is full and the sky is clear, white-tailed deer will feed heavily during the evening hours and move less often during the day than they will when the sky stays dark all night. Run drives through heavy cover or still-hunt other likely bedding areas after a well-lit night to increase your chances of filling your tag in such conditions.\n\n31. Watch a Feeding Deer's Tail\n\nFeeding deer always twitch their tails immediately before raising their heads to look around. If you immediately freeze when you see this motion, you'll be much less likely to alert the animal to your presence. Continue your stalk when the animal puts its head back down to feed.\n\n32. Look Downhill When Still-Hunting during Bad Weather\n\nThe best places to still-hunt during violent weather are found on elevated terrain. Benches crossing the sides of ridges make excellent routes to follow because they give you top-down looks into the kinds of thick cover where deer like to hole up. That extra field of view means you'll have more shots than you would if you were stalking through level ground, where the cover you're hunting will usually obstruct your line of sight.\n\n33. Don't Hang Your Stand Where You Find the Most Sign\n\nAreas chock-full of deer trails, droppings, rubs, and beds are not always the best places to hang your stand. The abundance of sign could mean that the area is being used as a sanctuary\u2014a place where deer congregate before heading out to feed, or where they bed down during the middle of the day. Since deer spend a great deal of time in such places, they become very familiar with them and will be sensitive to unusual sounds, smells, and sights. It can be extremely difficult to camouflage your presence under such conditions.\n\n34. Bagging a Lunchtime Buck\n\nSince most whitetail hunters are on their stands at first light, they tend to get restless from midmorning to noon. Many head back to camp, or their vehicles, seeking a sandwich and a chat with their buddies. That's when they inadvertently spook whatever deer are around, sending them sneaking away or bolting through the countryside. And that's exactly when hunters who have stayed on their stands, quiet and alert, reap the rewards of the biggest bucks.\n\n35. Use a Stick to Track Wounded Game over Sandy Soil\n\nSome soil types can absorb both tracks and blood, making trailing wounded deer difficult. If the trail you're following becomes indistinct, break off a straight stick that's the same length as the stride of the animal you're tracking. Place one end of the stick on the last clear track you can identify so that it's pointing in the direction you think the animal was traveling. Look for new tracks or flecks of blood at the other end.\n\n36. Sit Your Stand in the Morning When Hunting Hot Weather\n\nWhitetails are more active than normal during the night when the weather is unseasonably warm, and will stay bedded down in well-shaded cover that's close to a water source during the heat of the day. They may start moving again as the temperature starts dropping early in the evening, but when it's really hot the air won't begin to cool until well after darkness. The best time of the day to hunt deer during hot weather is during the first two hours of shooting light in the morning, when the air is coolest and you can catch your quarry moving from where they've been feeding to where they'll bed during daylight.\n\n37. Any Buck Is a Good Buck!\n\nIn my personal opinion, the concept of hunting only trophy deer seems misguided for most hunters. If you get a kick out of it, fine. Have at it. Most hunters, however, simply want to get out to the deer woods with the idea of bringing home a buck, any buck. The bigger the buck, the more thrills we might feel. But in the end, the old deer-hunting bromide is so, so true: You can't eat horns. My advice is to relax, enjoy your hunting, and stick to the premise that any buck is a good buck.\n\n38. Make Your Steps Sound like a Deer's to Spook Fewer Bucks\n\nThere are some situations in which it is impossible for you to walk silently through the woods. Dry leaves, for example, will crunch loudly no matter how carefully you place your feet. If you find it necessary to cover ground in such conditions (and there are no convenient game trails to follow that offer quieter places to walk), you will get closer to your quarry without spooking it if you learn how to pattern your steps so that they sound like the steps of a deer. Instead of a regular crunch, crunch, crunch, crunch, crunch (the standard human cadence), randomize the rhythm of your footsteps so that they form an erratic series of steps and pauses. Step, step, pause. Step. Pause. Wait. Step. Step. Step. Pause. And so on. You'll be much more likely to surprise bucks within shooting range if you use this pattern.\n\n39. Spot Bucks Down Low\n\nThe horizontal line of a whitetail's back is one of the best things to look for when still-hunting through thick brush. Most lines in the woods are vertical, and while you'll eyeball a great many fallen logs by keying in on the horizontal lines, you'll have a better chance of locating a hidden deer this way than you would by looking for a whole animal. Remember, though, that even mature bucks stand only three feet at the shoulder, so don't raise your eyes any higher than this when scanning the area ahead of you.\n\n40. Don't Face Your Stand at the Sun\n\nWhen placing your tree stand, avoid setting it up so that it faces a rising or falling sun. The rays beaming directly into your eyes will make it harder for you to see into shadows during critical low-light hours, and will glance off glasses, gun barrels, and reflective items, spooking deer that would otherwise be unlikely to spot you.\nCHAPTER 2\n\nElk Hunting\n\n41. How to Field-Judge a Bull Elk\n\nIf you're looking for a trophy elk, you need to be able to quickly judge the quality of its antlers before taking a shot at the animal. You can do this in three steps. First, try to count the points. A true trophy will have no fewer than six points on each side. Second, gauge the length of each antler's beam. A good bull should look as if it can tip back its head to scratch its rear end with the tips of his beams. Third, make sure that the bull's brow tines reach out over its muzzle, and that the other points have good length.\n\n42. Don't Scout for Elk like You Scout for Deer\n\nUnless you're hunting on private land, patterning an elk herd's activities before hunting season can be a waste of time. Elk react quickly to hunting pressure, and such pressure can be enormous on public lands, especially during the days leading up to the first of the season, when other hunters are out in force. Instead of scouting to identify a herd's normal, unstressed behaviors, focus your efforts on identifying where they go when spooked by the presence of other hunters. Look for heavy blowdown cover on steep slopes that are relatively close to meadows surrounded by thick timber. Elk feeding in these meadows will head toward such escape areas when startled. Identify good ambush sites along the likely routes they take, and you stand a good chance of shooting an animal pushed out by another hunter.\n\n43. Don't Let Elk Spot You Twice in the Same Spot\n\nIf you notice an elk in the herd you're hunting suddenly stop chewing its cud to stare at your position, the ideal response is to immediately stop all movement, no matter how awkward the position you're in, and remain motionless until it turns its head away. There will be times, however, when it will be impossible to stay still long enough for the animal that's spotted you to lose interest. If you must move to a more comfortable position, your best option is to lower yourself as slowly as possible to the ground. Lie there as long as necessary, then crawl to a new position before raising your head to take stock of the situation. Even if the elk did not spook, it will still be monitoring the spot where it saw you move.\n\n44. There's Only One Place to Shoot an Elk with a Bow\n\nIf you're hunting elk with a bow, the best shot to take is easy to remember, because there's only one ethical target to choose. Neck shots, shoulder shots, or shots at any other part of an elk's body are not recommended; to kill an elk quickly with a bow, you must shoot it directly behind its shoulder, piercing its heart and\/or lungs.\n\n45. Secure Your Elk's Carcass on Steep Slopes\n\nIf an elk you've shot falls on the side of a steep slope, your first action after making sure the animal is dead should always be to secure the carcass to a solid anchor, such as a tree trunk, using a stout length of rope. The last thing you want is for the animal's body to slide downhill, which could damage the meat and\/or put it at the bottom of a ravine where you'll have to work twice as hard to pack it out.\n\n46. Catch Elk Moving from Their Food to Their Beds\n\nThe worst time to still-hunt for elk is at midday, when they will be bedded down in thick cover that they've chosen because it is impossible to approach without them seeing or smelling you. You'll have more luck spotting them before they spot you if you still-hunt during the early morning and late afternoon, when elk are moving from the meadows and clearings where they feed to the heavy evergreen cover where they often prefer to bed, and vice versa.\n\n47. Bark like a Cow to Stop Startled Elk\n\nYou can briefly stop a spooked herd of elk by imitating the sound of a barking cow. A cow barks when she is alarmed. Other elk in the area will instinctively stop and look at her until she's identified the danger and run off, and then will follow her lead in exiting the area. If you're within range and have identified your target, use the moment when they stop and stare to take your shot.\n\n48. Spot Late Season Bulls at High Elevation\n\nLate in the season, you'll often find small groups of older bulls at higher elevations, where the snow is deep and the terrain is open. These bulls may be quite far from the rest of the herd, as much as a mile or more. Glass for them from below. When you spot such a group, try to pattern where the bulls feed and bed. Then set up a stalk to fit what you've observed. When planning your stalk, remember that if you spook the animals they will likely run to the nearest trees for shelter. If you're hunting in a group, post hunters along the tree line to intercept them.\n\n49. How to Recognize Elk Tracks\n\nElk tracks look like very large deer tracks, and a mature bull's prints will be much larger than those left by female or juvenile elk. Make sure that you do not confuse elk tracks with those left by moose or cows. Cow tracks are rounded and do not look much like those of a deer, and moose tracks are longer and narrower than those left by elk. Study a guidebook (or this illustration) before you hunt to avoid this problem.\n\nFrom left to right: bull elk, bull moose, cow elk, and juvenile elk tracks\n\n50. The Top Ten Elk States (by Elk Population)\n\nColorado: 200,000\n\nMontana: 150,000\n\nIdaho: 140,000\n\nOregon: 106,000\n\nWyoming: 85,000\n\nWashington: 60,000\n\nArizona: 55,000\n\nUtah: 50,000\n\nNew Mexico: 45,000\n\nCalifornia: 7,500\n\n51. Keep Predators Away from Your Kill with a Smoky Fire\n\nIf you are hunting elk on foot and shoot one far from the road, you will have to make more than one trip to pack out the meat, leaving the remaining meat alone in the backcountry. One way to keep bears, wolves, coyotes, and other predators from stealing your kill while you are away is to build a small, smoky fire from damp, pitchy wood near the quarters you had to leave behind. The smoke will help scare away predators, and you can use the plume to locate your kill quickly when you return to collect the rest of the animal. Make sure to do this only where it's both legal and safe, and never leave a fire untended in dry conditions where there is any possible risk that it might spread.\n\n52. Don't Bugle Too Often\n\nA bugling bull makes an impressive sound, and one that's not hard to reproduce using today's commercial calls. Because of this, and because many hunters have the unrealistic expectation that bulls often charge headlong toward the sound of a challenge, the bugle is the most overused call in an elk hunter's repertoire. Instead of relying on bugling to draw in a rutting bull, it's a better idea to use the call just to locate one, especially if you're hunting an area that gets lots of pressure. Once you've found a responsive bull, try to close with him by using other means. Call him in using cow calls, imitate the sounds of raking antlers, or even stalk silently downwind into shooting range.\n\n53. Change Vantage Points to Fully Scope an Area\n\nDon't dismiss an area as empty of game if you don't see any elk from a particular vantage point. You should always scope out an area from at least two positions so that you can view pockets and basins where elk may be feeding that are not easily identified from the beaten path. Changing vantage points takes work, and that's one reason why it can be such a successful tactic; most hunters won't be willing to make the effort and allow their desire for an easy hunt to let them dismiss an area before they've covered it fully. Use this to your advantage when hunting on public land.\n\n54. Approach Bedded Elk from Above\n\nElk like to bed in flat spots on side ridges where they can see well to the left, right, and downhill, and are high enough to feel comfortable that nothing will be approaching from behind. If you're following one's track and it suddenly turns upslope, it could mean the elk is ready to bed. Avoid following its tracks any longer, as the animal will be paying close attention to its back trail. Instead, circle uphill, staying downwind of the trail, and try to approach where you think the elk has lain down from above.\n\n55. Critical Advice to Elk Hunters\n\nCalling elk in areas of heavy hunting pressure may be counterproductive. Bugling will draw other hunters, and bulls that have been called in and spooked wise up quickly. It is better to simply spot the bulls, then stalk them.\n\n56. Look in Wet Meadows for Feeding Elk\n\nAlways pay close attention to wet meadows when looking for feeding elk. These small, moist clearings are rich in forbs and sedges, forage elk prefer. Wet meadows surrounded by thick timber are best; elk feeding in them will feel more secure when such cover is available, and will often feed later in the morning than they might in less accommodating terrain.\n\n57. Call Back Elk from a Busted Herd\n\nIf you flush a herd of elk while still-hunting and it splinters into multiple groups, you can use the animals' desire to herd back up to call them to you. If the animals did not smell you when first flushed, they will slow down and take stock of the situation from a few hundred yards away. Get out your cow call and blow it softly, imitating the sounds other cows make when they're regrouping. If a cow answers you, answer back, though wait for a few seconds first. If you do this right, the elk will think you're part of the group and will work back slowly in your direction.\n\n58. It Takes Two to Tag a Called-In Elk\n\nThe textbook method of calling a bull into bow range requires two hunters. One serves as the caller, the other as the shooter. The shooter sets up in the shadows of a tree or some other type of cover with a good view of the path the bull should take. The caller sets up twenty to thirty yards upwind of the shooter. Since most bulls will approach the caller by circling downwind, they should appear directly in front of the shooter.\n\n59. A Hot Weather Hot Spot for Daytime Elk\n\nWhen the weather is hot, look for elk taking shelter in forests of mature evergreens that have few low branches. The forest canopy protects the animals from direct sunlight, while the open understory lets in cool breezes.\n\n60. Find Elk in the Open after Bad Weather\n\nUnder most conditions, it is unusual to see elk feeding in the open during the middle of the day. The exception to this is if a period of strong winds or heavy rain or snow lets up at this time, or if the sun peaks through the clouds and warms up a slope that's been very cold. Elk will have been hunkered down, waiting out the bad weather, and will often feed heavily for a few hours once conditions improve. These are great times to glass food sources, trails, and bedding cover for movement.\n\nCHAPTER 3\n\nMule Deer Hunting\n\n61. Let Low Light Show You Hidden Deer\n\nThe best time to glass for mule deer is as the sun first appears early in the morning. The low-angle light causes the hair on their coats and light-colored rumps to glow, and will also gleam on the tines of a big buck's antlers. Low-angled evening light is also good for locating mule deer, but glassing in the late afternoon leaves you little time to stalk within range before the sun goes down.\n\n62. Glass Your Route before You Stalk\n\nSpotting mule deer is perhaps the easiest part of spot-and-stalk hunting. The stalk is just as important, and much more physically demanding. If you spot a bedded buck, don't immediately start moving into shooting range. First make sure that you're familiar with the route you're planning to take to get there. The best way to do this is to glass it thoroughly before packing up and moving out. Pick out landmarks you'll be able to identify (look for ones that can be recognized from multiple angles), and make sure to remember where they are in relation to where the animal is bedded down. Start your stalk once you've established a clear mental picture of the path you think will put you in a downwind location with a clear shot at the buck.\n\n63. Meandering Tracks Mean Beds Are Close\n\nIf a mule deer is traveling from a food source to bedding cover its trail will generally lead in a straight line (taking terrain into account). But as the animal approaches its bedding site it often starts to meander through the cover, nibbling on twigs and looking for a spot to lie down. If you're tracking a muley and see its trail start to wander, turn around and backtrack for a few hundred yards, then climb above where you think the animal has bedded down so that you can approach from a higher elevation. You'll get closer to the deer and will have a better shot at the animal if you spook it.\n\n64. Don't Shoot Deer You Can't Reach\n\nOne of the worst mistakes a mule deer hunter can make is shooting a buck on the other side of a ravine or canyon without knowing whether it's possible to retrieve the animal. Vertical cliffs, deep rivers, and maze-like canyons are difficult obstacles to negotiate, and some are impossible to cross. Make sure you know the terrain you're hunting before you take a shot like this.\n\n65. Mule Deer Prefer Sagebrush\n\nIf you're hunting in unfamiliar territory, one of the best places to look for muleys is in thick patches of sagebrush. Mule deer seem to prefer this plant throughout their range. Pay special attention to patches you find in places where sagebrush is relatively rare; the plant will act like a magnet for muleys in the area.\n\n66. Get in a Mule Deer's Zone\n\nAccording to Dennis Wintch, mule deer editor for Hunting Illustrated magazine, most mule deer habitat can be broken up into three distinct zones: a high zone (8,000 to 12,000 feet), a middle zone (5,000 to 8,000 feet), and a low zone (1,000 to 5,000 feet). Mule deer cycle through these zones depending on weather, hunting pressure, and food availability. The trick is to figure out which zone they're currently inhabiting. If you're hunting in a new area, look for fresh tracks and follow them until you're confident you know which level the deer are currently using.\n\n67. Getting the Range Right under the Big Sky\n\nThe most difficult adjustment easterners make when hunting the West for the first time is adjusting to the scale of thing. Deer tend to be significantly larger than what East Coasters are used to. Elk are five or six times better. Guesstimating ranges is a real crap shoot. Better depend on a rangefinder when hunting the wide-open spaces.\n\n68. Catch Bucks Seeking Shade in the Middle of the Day\n\nWhile the most productive times to glass for mule deer will be during first hours of morning and the last hours of the day, you can still spot deer moving during the middle of the day if you know where to look. Bucks will often change bedding sites during the middle of the day as shade shifts with the sun's movement, sometimes browsing for a few minutes before they lay down again. You'll be most likely to catch this movement if you're already glassing cover for bedded deer. Look under trees and around brush for anything that seems out of place with it surroundings. When a buck stands up you'll be focused in the right place.\n\n69. Tag Team a Spot-and-Stalk\n\nOne of the best ways to stalk within range of a bedded mule deer is to leave a hunting partner behind at the place you first spotted the animal. Your partner can then use hand signals to guide you as you sneak into shooting range. He'll need to far enough away from the bedded deer to not startle it when moving his hands, so bring a pair of binoculars and use them to check on him for instructions at regular intervals.\n\n70. Let the Wind Settle into One Direction\n\nWind currents can be fickle before the sun rises, so it's a good idea to wait until later in the morning before planning a stalk on a buck you've spotted. In most open-country habitat, the wind will usually settle as the air heats up, blowing in a consistent direction that makes staying downwind of your target much simpler.\n\n71. Don't Let Your Bow Keep You from Creeping on Your Stomach\n\nCreeping to within shooting range of an open-country mule deer is one of the most difficult and exciting challenges in all of bowhunting, especially when there's only an open stretch of tall grass between you and the buck you're stalking. To close with your target in such terrain you'll need to inch forward on your stomach, keeping your head down and using only your elbows to pull yourself forward. But it can be difficult to remain silent when you're carrying a bulky, awkward bow in your hands. You can keep your arms moving freely by placing the bow on your back. When you're close enough to shoot, simply slide the bow into your hands, knock an arrow, sit up on your knees, and shoot quickly.\n\n72. Glass for Bucks in Comfort\n\nIt is very important that you situate yourself comfortably when glassing for mule deer. Choose a place to sit that has a solid back support, such as a big rock or a stump, and bring foam pad, or, if you can spare the weight, a small, folding tripod stool to keep your butt off cold, hard ground. You should also invest in a tripod with adjustable legs for your binoculars and\/or spotting scope, which will reduce the strain on your arms. Never spot for deer with your rifle scope; you'll end up pointing the muzzle of your gun at targets you do not want to shoot.\n\n73. Stalk When the Wind Blows Strongly\n\nIf you spot a buck bedded in grassy, open terrain, wait to stalk it until the middle of the day, when the wind usually blows steadiest and strongest. The strong wind will wave grass, leaves, and branches back and forth, and you can use these natural movements to camouflage your progress into shooting position. Wear a ghillie suit for further protection from a mule deer's sharp eyes; the loose strips of fabric attached to the suit will move with the wind, making you look that much more like a part of the landscape.\n\n74. Use the Right Binocular\/Spotting Scope Combination\n\nLarge, 11x80 or 20x80 binoculars of the type designed for stargazers make excellent mule deer spotting tools because they gather a great deal of light, giving you an excellent picture during the twilight hours, which is when most muleys will be moving. But if you're hiking long distances to reach your hunting areas, you won't want to carry such heavy glass. A good compromise is to use a smaller pair of 10x50 binoculars and also bring a small, 20- or 25-power spotting scope. The combined weight of both these optics will be less than the weight of the larger binoculars, and you can break the weight up by storing your scope in your pack as you hike. Use the binoculars to spot movement at long range, then use the scope to get a closer look at whatever caught your attention.\n\n75. Be Prepared to Hike Long Distances\n\nAlways wear a daypack when you hunt mule deer. You often have to cover long distances to find them in the open country they live in, and you don't want to get stuck far from your truck without the proper equipment when the weather changes suddenly, as it often does in the mountains. Make sure your pack fits well and is roomy enough to carry a change of clothes, rain gear, a good knife, rope, binoculars, a compass, a survival kit, and extra food and water. Bring your pack even when you think you'll be hunting close to your camp or your vehicle; you never know when a fresh track might take you deep into the backcountry.\n\n76. Locate Clear-Cuts to Find Feeding Mule Deer\n\nOne of the best places to hunt mule deer in evergreen forest habitat is a recently logged clear-cut. Deer-friendly shrubs and plants grow rapidly in these clearings as their roots penetrate the disturbed soil and their leaves soak up sunlight normally blocked by large trees. You may have to walk a few miles to reach such clear-cuts, as most logging roads on public lands will be closed to public vehicle traffic, but with unpressured, top-notch feeding habitat as your reward, the hike will be worth the effort. Contact your local BLM or Forest Service office to get information on where these cuts have taken place (you want cuts that are 10 years old or younger), and plan your hunt accordingly.\n\n77. Use Thermal Currents When Stalking Bedded Bucks\n\nEarly morning air that has been cooling all night tends to flow downhill; later in the morning, as this air heats up in the light of the sun, it will reverse direction. These uphill\/downhill flows are called thermal currents, and they are important to remember when stalking hot-weather bucks. If you're glassing a clearing for feeding deer at first light, make sure you're positioned so that no downhill currents will carry your scent to the animals. Later, as the deer move uphill to bed, plan your stalk so that you approach them from above.\n\n78. Look for Bucks in Edge Cover on the North Sides of Ridges\n\nA great place to look when you're spotting for bedded mule deer is edge cover (thick brush lining clearings or other openings in which deer can remain concealed while still enjoying a good view of their surroundings). Look first along the north sides of ridges, which generally get more shade. Since deer here will already be bedded down, try to pick out pieces of the animals. Ear flicks, hind-leg scratches, and antler-glints may be the only clues you'll have to find the buck you're hunting.\n\n79. See and Be Seen\n\nWhen glassing for mule deer, keep in mind that if you're sitting in a place with a 360-degree view of the surrounding terrain, deer on all sides will have a good view of you, as well. Make sure to hide your silhouette by sitting with a rock, some brush, or a stump at your back.\n\n80. Follow the Farthest Track\n\nMule deer are very alert to the sounds and smells of animals following their back trails and will spook easily if they see you before you see them. To make sure you get the drop on an animal you're tracking, avoid looking too closely at its individual prints. Instead, keep track of the overall trail by picking out the furthest clear track you can see, then still-hunting up to it, keeping your eyes peeled for movement in the distance. Repeat until you find your target.\nCHAPTER 4\n\nBear Hunting\n\n81. How to Tell a Grizzly Bear from a Black Bear before Taking a Shot\n\nWhen hunting bears in the western states you must be very good at distinguishing between black bears, which are huntable, and grizzlies, which are protected under federal law. Color is not a determining characteristic; many black bears exhibit the same blonde coloration often associated with grizzly bears, and many grizzly bears come in darker colors. Instead, a hunter should look for two things. The first is the presence of a distinct hump on the shoulders; grizzlies have one, black bears do not. A hunter should also try to get a good look at the animal's face in profile. A black bear's nose slopes down from its forehead in a straight line. A grizzly bear's forehead dips inward from the forehead before pushing out into its nose, giving its face an indented, slightly concave appearance.\n\n82. A Bear of Many Colors\n\nNot all black bears are black. They come in many color phases, including blonde, cinnamon, chocolate, and even pure white or blue. Hunt the eastern states if you want a black phase black bear; other color phases are rare east of the Mississippi. Hunt the western states for a brown, cinnamon, or blonde-phase black bear. White phase bears (known as Kermode or spirit bears) and blue phase bears (known as Glacier bears) are found in British Columbia, but are protected from hunting by law in the province. Glacier bears, however, are also found in Alaska and can be hunted there.\n\n83. Look for Crop-Raiding Bears\n\nIf you live in a state that will not allow you to hunt bears over bait, you'll need to figure out where the animals are naturally feeding. One easy way to do this is to canvas farmers in the area you're hunting to see if any of them are having problems with bears raiding their corn fields, gardens, or other crops. You'll get a head start on the scouting process, and get the added bonus of expanding the amount of land on which you have permission to hunt.\n\n84. The Best Time for Bear Pelts\n\nThe earlier in the spring you can shoot a bear, the better quality its hide will be. Bears shed their winter coats as the weather warms up, often rubbing against trees, rocks, and fence posts to scrape off unwanted hair. If you shoot one later in the spring it may have a patchy, scruffy-looking hide. Catch them close to when they leave their dens you'll get a much better-looking pelt.\n\n85. When Black Bears Attack\n\n\"Indian hunters will tell you that a fighting black is more to be feared than either the grizzly or Kodiak, for the reason that the latter two species seem always to be in a frantic hurry about mauling a man, while a black will rip and tear at a victim as long as there is a spark of life remaining. This explanation of the Kodiak's tactics may account for the number of men who have lived to tell the tale after being mauled by the big brownies.\"\n\n\u2014Russell Annabel, \"Plenty of Bear,\" Field & Stream, 1937, reprinted in The Field & Stream Reader, Doubleday, 1946\n\n86. The Curious Eating Habits\n\n\"Black bear generally feed on berries, nuts, insects, carrion, and the like; but at times they take to killing very large animals. In fact, they are curiously irregular in their food. They will kill deer if they can get at them; but generally the deer are too quick. Sheep and hogs are their favorite prey, especially the latter, for bears seem to have a special relish for pork. Twice I have known a black bear to kill cattle.\"\n\n\u2014Theodore Roosevelt, The Wilderness Hunter, 1893\n\n87. Tell Black Bear Tracks from Grizzly Tracks\n\nYou can't tell a black bear's tracks from a grizzly's based on size alone. Large black bears will have tracks as big as medium-size grizzlies, and variations in sex and age make this an even more unreliable indicator. And while grizzlies usually have longer claws than black bears (translating into claw marks further from the tips of the toe imprints), not all surfaces pick up claw marks. The best way to tell which animal left a track you've found is this: find the imprint of the bear's front foot and draw a straight line across the base of the toes so that it's just touching the top of the front pad. If it's a grizzly track, most of the toes will be above this line. If it's a black bear the inside toe will be mostly below this line.\n\n88. The Differences between Boars and Sows\n\nIt can be very difficult to determine the sex of a black bear in the field. Male black bears, called boars, are generally much larger than females (sows) and have larger heads and longer bodies. Female bears rarely reach 350 pounds; boars commonly grow to 500 pounds or more. However, the only sure way to tell a younger male from a female is to look for a penis. This is easy in the spring and summer months, but can be difficult in the fall, when a bear's belly hair will be quite long.\n\n89. Why Late Season Is a Great Season\n\nFall bear hunting generally gets better later in the season. One reason is that bears move more often in cooler weather. Another is that in states or provinces where baiting is legal, hunters will often stop maintaining their bait piles after they fill their tags. Animals that were feeding on such bait will actively search for new food sources, making your own pile that much more attractive. Last, if you kill a bear in the late season, when the air is usually cold, you'll have an easier time of getting your meat out of the woods before it sours.\n\n90. Carry Tracking Line to Mark a Blood Trail\n\nAccording to Richard P. Smith, author of The Book of the Black Bear, one of the best ways to keep track of a bear's blood trail is to carry a couple of spools of Game Tracker line. This line is more commonly used by bowhunters (it attaches to their arrows and spools out after they shoot) but is also very helpful for hunters carrying guns. If you're trailing a wounded bear, simply allow the tracking line to pay out behind you. You'll get an easy to follow, continuous record of the trail you're following, and you won't have to break your concentration in order to tie bits of surveyors tape to branches or mark the trail some other way.\n\n91. Size a Bear by Looking at Its Tracks\n\nYou can get a good idea of how large a bear is by looking at its tracks. Average-sized black bears (150 to 200 pounds, dressed) will leave front tracks that are 3\u00bd to 4 inches wide, and rear tracks that are from five to six inches long. The tracks left by a trophy-sized animal will be much larger, with front pads five to six inches wide, and rear pads eight inches or longer.\n\n92. Talk Loudly to Avoid Startling Bears\n\nAccording to Richard P. Smith, author of The Book of the Black Bear, a hunter should rarely sneak when hiking in to hunt a bait pile. If a bear is already feeding there, the sudden appearance of a human may startle the animal, which could cause it to avoid the bait in the future, or to feed on it only at night. Instead, the hunter should warn any bears in the area of his approach well in advance of arriving at the bait. Whistle and talk in a normal, calm voice on the way in; this will notify the bear that you're on your way before you're close enough to startle the animal. Bears know that bait is left by humans and while they will move away off to avoid being seen when they hear people approaching, they will not be spooked as long as they are not taken unawares, and should return once they think you are no longer in the area.\n\n93. Give a Bear Time to Die\n\nBe careful when trailing a bear you've shot if you're only hunting with a bow. Pay attention to where your arrow hits the animal, and delay tracking it until you're confident that the bear has had time to die. Wait at least a half an hour if you hit it in the lungs, an hour if you hit it in the liver, and at least four hours if you shoot it in the gut. If possible (and legal), bring along a friend with a gun.\n\n94. What a Bear Trail Looks Like\n\nBecause bears will generally step in the exact same places when taking familiar routes, well-used bear trails often look like old, deep footprints worn into the forest floor rather than the smooth, groove-like paths normally associated with game trails.\n\n95. Trail a Drive to Bag a Trophy\n\nOne of the best ways to bag a big bear when putting on a drive through thick cover is to post a couple of shooters behind the drivers. Older bruins do not startle easily and will avoid leaving their security cover if at all possible. Instead of running out ahead of the drivers as a younger bear might do, a big bruin will often simply circle around them. Hunters following the drive stand a good chance of seeing bears that behave this way.\n\n96. Bowhunting Black Bears\n\nTime was when only the top experts even thought about taking a bear with a bow. Not today. Many of the same bowhunters who bag whitetail bucks with their bows are on the hunt for black bears\u2014mostly over bait in Canada. Opportunities to bag a bear with a bow abound in Canada. When Googling for information, make sure you type in the province you're interested in, and, of course, look for the guides and outfitters who have solid records in bowhunting.\n\n97. How to Measure a Trophy Bear\n\n\"Boone & Crockett recognizes four species of bear in North America: the Alaska brown bear, the grizzly bear, the black bear, and the polar bear. All are scored the same way, by measuring a dry skull's greatest width and adding it to its greatest length. The minimum scores required to make the all-time B&C books are:\n\nAlaska brown bear\u201428 inches\n\nPolar bear\u201427 inches\n\nGrizzly bear\u201424 inches\n\nBlack bear\u201421 inches.\"\n\n\u2014Tom McIntyre, www.fieldandstream.com\n\n98. Where to Shoot a Bear with a Gun\n\nThe best place to shoot a black bear with a rifle, shotgun, or handgun when the animal is broadside to you is directly in the center of its shoulder. If you're using the right caliber rifle (.270 or higher) this shot should break both of the animal's shoulders and penetrate its lungs. If the bear is facing you the best place to shoot it will be directly in the center of the chest. If it's facing away from you, shoot it in the center of its back, directly between the shoulder blades.\n\n99. Use Multiple Knives When Trimming Bear Fat\n\nBears spend the summer and fall months building up a thick layer of fat in preparation for their winter dormancy. If you shoot one in the fall, you'll need to trim off this fat before storing the animal's meat. The meat will keep longer and take up less space in your freezer, but the process takes time. Save some by having a few spare knives and a sharpening tool handy.\n\n100. Don't Plan a Spring Hunt Too Early\n\nWatch the weather when planning a spring bear hunt. A warm spring will get bears moving earlier in the season, but an unseasonably cold one will discourage them from leaving their dens. If you're traveling out west or to Canada to hunt spring bears, it's a good idea to build a bit of cushion into your schedule in case winter lingers longer than normal.\nCHAPTER 5\n\nPig Hunting\n\n101. Distinguish Hog Tracks from Deer Tracks\n\nAlthough hog tracks and deer tracks can be the same size, it is not difficult to tell them apart once you know what to look for. Hog tracks are blockier than deer tracks and have rounded rather than pointed tips. Deer tracks are teardrop shaped; hog tracks are square in both front and back, and have a more uniform width.\n\n102. Hunt Small Herds to Find Unpressured Pigs\n\nIf you know there are pigs in the area you're hunting but find only minimal amounts of sign, do not get discouraged. Smaller herds of pigs can be more predictable (and thus easier to hunt) because they are generally less pressured than larger groups, whose obvious trails and numerous wallows attract many more hunters.\n\n103. Wild vs. Feral\n\nVery few of the hogs in North America are truly wild\u2014most are descended from domestic pigs, and should be referred to as feral. A true Russian wild boar has a much longer nose and legs than a hog descended from domestic stock, it will have a pronounced ridge of hair running down the center of its back, and its tail will be straight.\n\n104. Shoot to Kill Quickly When Hunting Hogs\n\nA hog's vital organs are located lower in its body cavity than are vital organs in the body cavities of ungulates. To ensure a killing shot, always aim directly behind the shoulder as well as slightly lower than you would aim on a white-tailed deer. Be very sure of your shot before pulling the trigger or releasing your arrow. You do not want to have to follow one far after hitting it; hogs have a thick layer of fat beneath their skin that can quickly plug a wound, making blood trailing difficult, especially if you shoot the animal in a wet, swampy environment.\n\n105. Spot-and-Stalk Hogs in Open Country\n\nIn fairly open country, spotting wild pigs from a distance and then stalking them can be an effective method. Start out by trying to situate yourself you where you have a commanding view and the wind is coming toward you. If pigs are seen at a distance, walk slowly and quietly toward them, keeping the wind at your face and using available cover. Since wild pigs have poor eyesight, you may be able to get fairly close without being detected.\n\n\u2014www.jesseshunting.com\n\n106. Look for Thickets to Find Bedded Pigs\n\nWherever you hunt pigs, you can be confident that they'll spend their days holed up in the thickest vegetation to be found. Look for palmetto thickets in swampy bottomland, laurel tangles in the mountains, and grown-over clear cuts in forested country that are near a good source of food, such as an old orchard or grove of acorn-producing oaks. Set up your stand on trails that lead from their bedding cover to where they eat and make sure you're sitting in it long enough for the scent you left on the way in to disperse by the time they head out to feed in the evening.\n\n107. Here He Comes!\n\n\"When a wild boar means mischief, he makes his run with his head down. It is by a sudden thrust upward of his tusks that he does his deadly work. When he charges with his head high, he probably means that he just wants gangway.\"\n\n\u2014Archibald Rutledge, An American Hunter, Lippincott, 1937\n\n108. Look for Wallows When Scouting for Hogs\n\nWallows are muddy or dusty patches of ground where pigs roll to cool themselves off, remove parasites attached to their skin, and cover themselves in dirt to keep off biting insects. These are great places to look for when scouting, because you can use them to identify the sizes and numbers of animals in a herd. Tracks are easy to find in such places, and you can get an accurate read on a hog's size by measuring the imprints left by its body in the mud.\n\n109. Wild Boars: What You're Hunting\n\nWild boars in America are a mixture of feral (born wild) pigs from domesticated stock running loose in the woods for decades, even centuries, and of original European wild boars brought into this country and planted at different locations. One of the main plantings was by a man named George Moore who in 1912 put fourteen European wild boars on his 1600 acres of timbered land surrounding Harper's Bald, a mountain peak in the Snowbird Mountain Range of North Carolina. Moore thought of his land as a preserve, but, of course, the hogs roamed into the countryside and have been there for decades, plus spreading elsewhere in the Great Smokies. The pure European wild boars have also been imported in places ranging from New Hampshire to California to Georgia.\n\n110. Hunt Hogs near Old Homesteads\n\nOne of the best places to look for wild hogs is around an abandoned homestead. Pigs like these sites because they often contain abandoned orchards, overgrown gardens with wild-growing vegetables, and are located close to open meadows or overgrown pasture that offer a range of other food sources.\nCHAPTER 6\n\nMoose Hunting\n\n111. Don't Glass from the Same Spot You Call\n\nIt's always a good idea to change your location after you've finished an extended calling sequence so that you are glassing the area from a spot upwind of where you were calling. If there's a bull in the area, you can be sure he will have heard you and will remember the exact spot your calling came from. When he eventually comes in to investigate it's likely he'll first circle downwind of where he heard your calls. Switching spots lets you stay undetected if this happens.\n\n112. Be Patient When Calling in Moose\n\nMoose are curious creatures, but they can take their time satisfying that curiosity. Even during the peak of the rut it can take days for a moose that's heard your calling to decide to investigate. Be patient. There will be rare occasions when a nearby bull that's crazed with lust comes crashing into your setup right away, but most calling won't pay off until many hours later.\n\n113. Walk like a Moose\n\nMoose have sensitive ears and a keen sense of smell. It is very unlikely you'll be able to approach one on foot without it hearing you, even if you're working into the wind. Fortunately, you do not have to be completely silent. Instead, try to sound like a moose. First, make sure you're wearing no clothing that makes unnatural rustling sounds when brushed against branches or grass. Second, make sure any metal items you're carrying in your pockets, on your gun sling, or clipped to your jacket or pants are tightly secured so that they don't clink or clank while you're walking. Last, do not try to sneak. Predators sneak, tensing up their bodies in order to perform deliberate movements designed to minimize motion and reduce the sounds of their steps. Moose know what sneaking sounds like, and most humans are not capable of sneak quietly enough to fool one. You'll have more luck if you step as quietly as possible while still maintaining a loose, natural stride. Any sounds you make will likely be mistaken for the sounds of another moose, elk, or deer moving at a relaxed pace through the brush.\n\n114. How to Recognize a Pissed-Off Moose\n\nWhen a moose is irritated by your presence, it will use its body language to warn you that it's upset, pulling its ears back and flaring the long hair along its neck and hump, much the way a dog will when looking for a fight. It may even growl at you and lick its lips. When you see this, back away quickly and try to get a large obstruction between you and the animal.\n\n115. Russell Annabel on Alaska Moose Hunting\n\n\"Sheep hunting is great sport, bear hunting is packed with adventure and thrills, and there is a definite kick in risking your neck climbing the windy crags of the goat country\u2014but for downright fun, I'll take a moose hunt any old time. Like grouse shooting, it's a sport that goes with bright leaves tingling down through the branches of old trees, with quiet noonday watches on sun-drenched hillsides, with cautious sallies through the shadowy green-gold enchantment of deep forest aisles, and with campward horseback rides in the purple, star-shot dusk of mountain evenings. It is a sport for the man who appreciates the wilderness at its best, who has an eye for color and beauty\u2014and yet it also has its taut, pulse-quickening moments.\"\n\n\u2014Russell Annabel, \"In the Moon of the Painted Leaves,\" Field & Stream, 1936, reprinted in The Field & Stream Reader, Doubleday, 1946\n\n116. When to (and When Not to) Use a Big Scope\n\nOversize, 50mm objective lenses will always collect more light than standard-size rifle scopes, giving you a distinct advantage in low-light conditions\u2014brighter, sharper images of the animals you're aiming at. This does not mean, however, that you should always carry such a scope on your rifle. Oversize lenses are great choices if you're hunting from a stand or spotting and stalking in country where you expect to take long shots at unsuspecting moose. But if you plan to do much still-hunting, they can be a liability. There are two reasons for this. The first is that oversize scopes require higher mounts than standard scopes, which means it's more difficult to acquire a proper sight picture when you need to make a quick shot. The second is that oversized scopes are heavy! You'll be much happier if you mount standard-sized optics on the gun you use for still-hunting.\n\n117. Moose Hunting in a Big Wind\n\nUpland bird hunters hate days of the howling winds. Duck hunters love them. Moose hunters should love them. Working your way into position for a shot at moose is much easier in a big wind, if you plan your hunt to approach from downwind. First, the moose isn't going to catch your scent. Secondly, they can't hear much with the wind howling.\n\n118. Moose by Canoe\n\nMoose love water, and so do many moose hunters. This is because there are few ways to access unpressured moose habitat more quietly and with less effort than by paddling into it. And there are few ways of packing out moose meat more efficiently than by carrying its quarters in a canoe. Scan banks and shorelines for moose standing hidden in the brush, and pay close attention when paddling up or floating down rivers that connect ponds and small lakes\u2014moose travel along these streams because they often flow through flat terrain that's easier for them to traverse.\n\n119. Call like a Cow Using Only Your Voice\n\nCow calls are high-pitched groans that can be best described as a high, moaning \"eerrrrrrrr\" sound. These calls can be relatively short in duration, or can last for up to two minutes. You can imitate the sound using your voice alone. To do so, pinch your nose (a nasal sound does a better job of imitating a cow), cup your hands over your mouth, and start your \"err\" sound at a lower pitch, gradually raising pitch in the middle of the call, holding the \"r,\" and wavering your tone a bit before lowering the pitch as you taper off into silence.\n\n120. Stop a Startled Bull with a Cow Call\n\nIf you spook a bull, try making a long, loud cow call as he runs away. There's a good chance he'll stop to figure out where the sound is coming from, giving you an opportunity to make a quick shot.\nCHAPTER 7\n\nCaribou Hunting\n\n121. Caribou Migrate the Furthest\n\nSome caribou migrate more than 3,000 miles each year\u2014farther than any other land animal. They travel in herds every fall and spring from their wintering to their calving grounds, and arrive just in time to think about heading back.\n\n122. Multi-Purpose Hooves\n\nA caribou's hooves are wide, concave, and act like snowshoes, distributing the animal's weight on snow, ice, and melted muskeg. These hooves also work like paddles when caribou need to swim across fast-flowing rivers, or even large lakes. But they don't slow the animals down. Caribou have been recorded running faster than 50 miles per hour.\n\n123. Caribou Never Stop Moving\n\nCaribou are not the wiliest game animals a man can hunt, but that doesn't make them easy prey. First you have to find them. Then you have to decide whether or not to wait to shoot a trophy. You can't pattern a caribou because it never stays in one place. The herd you're stalking today might be miles and miles away the next. If you see a bull you like, pull the trigger, because you never know what tomorrow will bring.\n\n124. Bring the Right Optics\n\nUnless you're a bush pilot, hardest part of hunting caribou will be finding the herd. Bring high-power binoculars (at least 10 \u00d7 42) and carry a spotting scope. Make sure to use top-quality glass or you'll lose your ability to hunt during the morning and evening hours.\n\n125. Don't Spook the Herd\n\n\"When you're sneaking up on a bull, keep track of other caribou. They seem to float in from nowhere just when you want to move. Though caribou won't jet away like a whitetail buck when you surprise them at a distance, they will jog off and take other caribou with them. Then you have two options: Stay put and hope they stop so that you can stalk them again, or run after them. In my experience, spooked caribou seldom give you an easy second hunt.\"\n\n\u2014Wayne van Zwoll, \"Caribou: Noble and Mobile,\" www.fieldandstream.com\n\n126. How to Field-Judge a Caribou's Antlers\n\nWhen trying to guess the length of a caribou's antlers, use the animal's shoulder as a measuring stick. Most shoulders will be between 48 and 54 inches high. Look for antlers with curved main beams, which will generally be both longer and wider than straight ones (though they may look shorter from the side). A trophy animal's shovels will be broad, have multiple points, and extend far out over the muzzle. Kicker points, the spikes that grow off the back of a caribou's antlers, will add to the score, as will palmation and extra points at the tops.\n\n127. How to Read a Caribou's Body Language\n\n\"When caribou are not alarmed, they walk quite slowly, extending the head forward and downward. When alarmed, caribou perform a special behavior to warn other caribou of danger. They'll do this if a predator gets too close, but isn't about to catch them (or after they figure out that you're a person sitting on a rock). An alarmed caribou will trot with the head held high and parallel to the ground, and the short, normally floppy tail held up in the air.\"\n\n\u2014Alaska Department of Fish and Game\n\n128. A Good Gun for Caribou\n\nWhile a good shot won't need more than a .270 to take down a thin-skinned animal like a caribou, the animals are much larger-bodied than most whitetail deer. It can help to shoot a bigger gun when you're reaching out to knock one down at the long ranges you'll often find when hunting in the tundra. One great caribou cartridge is the .338 Win Mag. Loaded with a 200-grain bullet, the cartridge will hit three inches high at 100 yards. Elmer Keith loved this chambering for both caribou and elk, and you will, too.\n\n129. Winterize Your Caribou Gun\n\nAlways make sure to keep your gun clean, moisture-free, and either grease-free or treated with synthetic lube designed to function in extreme freezing temperatures. The last thing you want when firing your rifle at a caribou after a long, freezing, late season stalk is for the hammer, firing pin, or trigger to malfunction because your gun's oil congealed in the cold.\n\n130. Don't Put Wet Bullets into a Freezing-Cold Rifle\n\nAlways carefully clean any cartridges you've dropped on the ground if you're hunting in the far north during the late season. They make pick up moisture that causes them to freeze to the inside your gun's chamber, reducing your expensive rifle to a single-shot firearm.\nCHAPTER 8\n\nPronghorn Antelope Hunting\n\n131. Vision and Speed: The Pronghorn's Defense\n\n\"His eye is larger than that of cow or horse, nearly as large as that of an elephant; they give him somewhat the appearance of a huge beetle. He can see half or three-quarters of a mile away, with a range of vision keener than that of an 8-power glass. . . . I've seen a herd fairly fly across the plains up to the foothills and trees, then scorn the cover they have reached and circle back and back again, as if playing a game of tag with your bullets. The pronghorn is a real sportsman. He runs, but he never hides.\"\n\n\u2014Major Robert E. Treman, \"Wyoming Antelope,\" Field & Stream, 1937, reprinted in The Field & Stream Reader, Doubleday, 1946\n\n132. The Pronghorn Challenge\n\n\"Antelope shooting is the kind in which a man most needs skill in the use of the rifle at long ranges; they are harder to get near than any other game\u2014partly from their wariness and still more from the nature of the ground they inhabit . . . Even good hunters reckon on using six or seven cartridges for every prong-horn they kill; for antelope are continually offering standing shots at very long distances, which, nevertheless, it is a great temptation to try, on the chance of luck favoring the marksman.\"\n\n\u2014Theodore Roosevelt, Ranch Life and the Hunting-Trail, The Century Co., 1888\n\n133. Set Your Decoys with Stealth\n\nWhen hunting pronghorns with a bow, a decoy can make the difference between a fruitless stalk and a successful hunt. The trick is to creep close enough to set one up, and then to set it up without being spotted. Wait until the buck you're hunting is busy chasing a doe, or obscured from view by a clump of sage or other brush. If he sees your decoy rise up out of the grass he may grow suspicious of the unnatural motion and fail to come closer to investigate.\n\n134. Sit over Water\n\nYou can avoid strenuous stalks during midday heat if you switch to hunting a water hole. Pronghorns are creatures of the dry, high plains, but they need to drink periodically, and will often show up during the hottest hours to slake their thirst. If you know which water sources they've been using you stand a good chance of ambushing one.\n\n135. Give Your Blind Time\n\nIf you plan to hunt from a blind overlooking a water source, make sure the blind has been in place for a few days before you plan to sit in it. Pronghorns will be wary of this new addition to the landscape after it first appears; you want to hunt from it after they grow comfortable enough with its presence to wander within bow range.\n\n136. You Don't Need Long Shots\n\n\"The key to getting up on [pronghorns] lies in the seemingly flat land they inhabit, which is actually broken, cut, and intersected by coulees, ravines, gullies, washes, draws, ridges, hills, and divides. A smart antelope hunter can take advantage of this tortured topography to get close\u2014almost always less than 200 yards, and very often less than 100.\"\n\n\u2014David E. Petzal, \"Choosing the Right Rifle for Pronghorn Antelope,\" www.fieldandstream.com\n\n137. Glass from Your Truck\n\nMost pronghorn antelope have yet to associate vehicles with the hunters who drive them. This makes your truck an idea platform from which to glass for a trophy. Once you've spotted a buck you want to stalk, glass out the route you plan to stalk, then drive your truck far enough away that the animal won't see you getting out.\n\n138. The Pronghorn Antelope's Range\n\n\"True Americans, pronghorn are found only on the plains and grasslands of North America. Like bison, seemingly endless numbers once covered the west, stretching from Saskatchewan to just north of Mexico City. And like bison, they nearly became extinct. Populations declined from an estimated 30\u201360 million in the early 1800s to less than 15,000 by 1915. A moratorium on hunting lasting until the 1940s and a federal tax on firearms and sporting goods funding conservation efforts are credited with stopping the decline. Today there are almost 1 million pronghorn. Five subspecies are recognized: American\/common (found in most of range, Canada, and northern Arizona); Mexican\/Chihuahuan (found in New Mexico, Texas, formerly southeastern Arizona); Oregon (found in southeastern Oregon); peninsular (100\u2013250 animals, found in Baja, Mexico); and Sonoran (endangered, 500 animals found at Cabeza Prieta National Wildlife Refuge and Sonora, Mexico).\"\n\n\u2014Facts About Pronghorns, U.S. Fish & Wildlife Service, www.fws.gov\n\n139. Some Facts of Pronghorn Biology\n\nA male pronghorn weighs, on average, about 120 pounds. Females are slightly smaller, averaging 105 pounds. The animals are not large, standing approximately 3 feet at the shoulder. Both males and females have horns, though the male's are significantly longer, averaging 13 to 15 inches compared to the female's 3- to 5-inch horns. Pronghorns can live up to nine years in the wild.\n\n140. How to Field-Judge a Pronghorn\n\nA trophy antelope's horns will be longer than thirteen inches, which is the distance from the base of an average pronghorn's ear to the tip of its nose. Horns that appear to be twice as long or longer than the animal's ears are likely to break the Boone & Crockett record books, especially if they are curved, crooked, and look wider at their bases than the width of the animal's eye. Look for horns that split into prongs above the tips of an pronghorn's ears. You want an animal whose front prong extends at least 4 inches forward from the main horn.\nCHAPTER 9\n\nWild Sheep and Mountain Goat Hunting\n\n141. North America's Native Wild Sheep\n\nThere are two species of wild native sheep in North America, the bighorn sheep and the Dall sheep. There are three subspecies of bighorn: the Rocky Mountain bighorn (Ovis canadensis canadensis), the Sierra Nevada bighorn (Ovis canadensis sierrae), and the desert bighorn (Ovis canadensis nelsoni). There are two subspecies of Dall sheep: the Dall sheep proper and the Stone sheep.\n\n142. Glass in the Morning\n\nAlthough sheep move most during the morning and evening, it's generally a bad idea trying to find the animals late in the day. Sheep live in high, rough country. Unless you're prepared to spend the night, you don't want to get caught on the side of the mountain in the dark.\n\n143. Watch for Their Rumps\n\n\"Bedded sheep will almost always get up and move around a little . . . between noon to around 1 PM. They may feed for a few minutes, or move from one group of beds to another nearby. Or they may only get up, stretch, turn around and lie back down in the same bed. If you are watching the right spot at the right time, you'll see their white rumps and know exactly where they are.\"\n\n\u2014Bob Hagel, The Expert's Book of Big Game Hunting in North America\n\n144. The Colors of the Mountain\n\nBighorn sheep come in a variety of colors, from light tan to dark brown to a deep blue-gray. These colors vary from region to region (and sometimes within a single region). Because the animals tend to bed in spots that match the colors of their coats, it's a good idea to have a sense of the common shades found in the area you plan to hunt. Look for ground that matches these colors, then look for sheep hidden there.\n\n145. How to Field-Judge a Mountain Goat\n\nIdentifying a trophy mountain goat is not easy. The differences between a good billy's and a record-book animal's horns will often be less than an inch. The first step to finding a record is to make sure the animal you've spotted is, in fact, a male. Billies have high, humped shoulders, and shaggier coats than nannies. Once you've found an old male goat, straighten out his horns in your minds eye and compare that length to the length of the animal's head. If the stretched horns reach from the animal's nose to the bottom of its eye, they are less than nine inches long. If they reach from the nose to the base of the ear, they are at least nine inches long and will qualify as a true trophy.\n\n146. Creep across Crests\n\nWhen stalking sheep and goats, be extremely careful to never silhouette yourself against the skyline. If you must cross the crest of a ridge or a saddle, do so on your hands, knees, and belly, and move as slowly as you can.\n\n147. How to Field-Judge a Trophy Sheep\n\nWhen glassing for a record-book sheep, always look for a full, curling horn that's bottom extends below the line of the lower jaw. You want heavy, thick horns with broomed tips, which will score higher than unbroomed horns of the same length.\n\n148. Scan for a Silhouette\n\nSheep easily spot a skylined hunter's silhouette, especially when the hunter is moving. But the reverse is also true. When glassing for sheep, always keep a close eye on the tops of ridges, cliff edges, and the skyline over a saddle. Scanning for silhouettes is the easiest way to spot these animals.\n\n149. Don't Educate the Herd\n\nIf you're hunting with friends or as part of a guided group and are lucky enough to kill a sheep, don't immediately rush in to claim your kill after the shot. Stay hidden, watch to see where the animal falls, and wait to retrieve it until the rest of the herd has left the area. Avoid startling them and other hunters in camp will have a much easier time of stalking sheep in the area.\n\n150. Account for the Hump\n\nDon't let an old billy goat's hump throw off your aim. This massive growth of fat and hair covers a ridge of finlike vertebral spines and gives the animal a unique profile when compared with other big game. When preparing to pull the trigger, don't put your crosshairs roughly halfway up a mountain goat's body the way you would on an elk or a deer; the animal's vitals will be located in the lowest third of its body. Hold at the top of that lowest third to ensure a killing shot.\nCHAPTER 10\n\nTurkey Hunting\n\n151. Learn to Shoot Turkeys from Both Shoulders\n\nOne of the most difficult situations in turkey hunting is having a bird sneak up behind you when you're sitting at the base of a tree. It can be extremely difficult to twist your body around far enough to make an accurate shot, and it's nearly impossible if the bird is behind your right shoulder if you're a right-handed shooter (and vice versa if you shoot with your left hand forward). It's a good idea to practice shooting your turkey gun from your opposite shoulder before the season starts. If you're comfortable taking shots this way your chances of getting a bead on a turkey without spooking it will improve dramatically.\n\n152. Control the Volume of Your Box Call\n\nIf you're working a gobbler with a box call and he hangs up in the distance, you may be calling too strongly. Box calls are notoriously loud; the tom may think the hen you're imitating is closer to him than you want, and will often stop and wait, thinking that she will come to him. One way to get him moving is to reduce the volume of your call. Hold the call upside down, with the handle on the bottom, and slide your thumb up the sides to increase pressure on the call and gradually dampen the vibration. The gobbler will think the hen is moving away from him and may give chase.\n\n153. Five More Reasons Gobblers Are Easy to Miss\n\nNeed more reasons to miss a gobbler within 30 yards? Try these: (1) The bird is moving, and you panic slightly, raising your head from the gunstock just a bit; (2) You gun is new, unfamiliar, or one you haven't shot in weeks or months; (3) You're wearing gloves, deadening your touch, and you pull the trigger like it's a rusty nail; (4) Your guide or companion is doing the calling, and he whispers the command \"Shoot!\" to you. You instantly obey, even though you're not ready; and (5) You try for a head shot. Any of these reasons is enough to make a grown man cry.\n\n154. Use Decoys Late in the Season\n\nThe best time to use turkey decoys is after most hens are already sitting on their nests. A decoy is much less effective early in the breeding season when most toms will already be attended by hens.\n\n155. The Best Place to Set Up on a Roosted Gobbler\n\nIf you've done your scouting homework, you'll often know where a gobbler has roosted for the night. If you've done your extra credit, you'll know where he goes after he flies down. The best place to set up to call him in the morning will be between these two places, about 100 to 200 yards away from his tree (distance depending on how well leafed out the trees are). Get there well before first light, and sneak in as quietly as you can. Roosted birds are alert to unusual sounds and can pick up movements even in very dark conditions.\n\n156. Don't Let a Hot Gobbler Get Too Close\n\nThe next time a gobbling tom comes trotting into your setup, don't wait too long before taking the shot. Most turkey load patterns open up at around 20 to 30 yards; this is the distance at which you have the best chance of putting a pellet into his brain or spine. If you let him get to close your pellets may be packed so tightly together that a slight miscalculation will cause all of them to miss.\n\n157. Locating Roosting Gobblers\n\nWhen your calling or scouting has located a roosting area (and you've been careful not to spook the birds!), you'll hear them fly up into the trees\u2014big wings flopping, a great deal of noise. Be aware, however, that they don't pick the limb they wish to roost on from the ground, then fly up to it. It's after they are in the trees that they move around to a favored spot to spend the night.\n\n158. When Gobblers Get Lonely\n\nMany turkey hunters miss out on bagging their bird by not being alert to a hunting opportunity that takes place in the middle of the day. Sometimes around eleven o'clock in the morning, hens have left the gobbler to go to their nests. That's when the toms get lonely\u2014and start to gobble, betraying their location. You can get into position, set up, and call in your bird.\n\n159. Rake Leaves to Call in Hung-up Toms\n\nThe next time a gobbler hangs up in the distance, responding to your calls but refusing to approach, stop calling and start imitating the sound of a feeding hen by raking a hand through the leaf litter at your feet. If all goes according to plan, the gobbler will grow frustrated, wondering why the hen he can hear scratching for food won't respond to his calls, and will often come closer to investigate.\n\n160. Fake a Flock of Fall Turkeys\n\nMost hunters use turkey decoys to stimulate a gobbler's mating or competitive instincts during the spring season, but decoys also work well for fall turkeys. The trick is to use lots of them to simulate a small flock. If you can figure out where the turkeys you're hunting roost, and where they feed, set up your fake flock between them and use a couple of different calls to imitate the sound of a few feeding hens.\n\n161. Circle Gobbling Birds Hung up behind Obstacles\n\nGobbling turkeys hang up for lots of reasons, but one of the most common is that there's an obstacle between him and you. Streams, fences, and ravines will often keep a turkey from following up on the promise of a ready hen. In many cases you'll have the best luck killing your bird by crossing these barriers yourself. Using a crow call to keep him gobbling, circle around him until you're 180-degrees away from where you were set up before. He'll be more likely to return along a path he's already used than he will to work through less familiar territory, and you'll know there won't be any other obstacles on this trail that might obstruct his progress.\n\n162. Faking Out a Gobbler: A Desperation Tactic\n\nWhen a gobbler plays hard to get, and nothing else has worked try walking straight away from him, calling occasionally as you go. If he thinks his potential paramour is leaving him, he just might come running.\n\n163. Cover Ground to Find Mid-Day Gobblers\n\nTurkeys gobble more in the early morning than they do during the middle of the day. This makes them easier to find, but it does not mean they are easier to hunt. Most toms will gobble in the morning even if they're with hens, which means you stand a good chance of spending all morning talking with a bird that has no reason to come to your calls. If you find yourself working a bird that refuses to come to you, don't give up hope. Instead, go looking for a more accommodating tom. It is true that turkeys gobble less frequently when the sun is high, but the flip side of this behavior is that if you find one that does gobble, he is much less likely to be with a hen and will be far more willing to come in to your calls. Hike through your property and call every 100 yards or so until you get a response.\n\n164. Don't Get Mistaken for a Turkey\n\nNever wear any clothing or carry any accessories that contain the colors red, white, or blue. You should also keep your hands and head camouflaged when calling, and wear dark-colored socks and pants long enough to keep bare skin fully covered. These colors are found on the heads of wild turkeys, and you do not want to be mistaken for a gobbler by another hunter.\n\n165. Make a Gobbler Jealous\n\nYou can use a decoy to simulate a breeding hen by pushing a hen decoy's stake deeply enough into the soil so that the decoy's belly touches the ground. Hens take this position when they're ready to mate. Put a jake decoy behind her (a jake is an immature male turkey), as if he's about to breed her, and the tom you're hunting may become so upset that he approaches your setup with much less caution.\n\n166. Bust a Roosted Flock in the Fall\n\nThe most common tactic used by fall turkey hunters is to find and then scatter a flock of the birds and then sit down to call them back in. Turkeys will naturally want to regroup, and if you call well enough to imitate a lost bird, they will use you as a homing beacon. One good way to find a flock to bust is to identify where the birds roost. Scout the woods for large hardwood trees with lots of fresh droppings at their bases, and head out the evening before you plan to hunt to listen for the sounds the birds make as they fly up for the night. Creep into the woods before sunrise the next day, and then rush the flock as soon as it flies down.\n\n167. Coping with \"Shut-Mouth\" Gobblers\n\nIf pressure has forced spring gobblers into silence, try patterning a long beard like you would a deer. He'll have favorites route he takes to favored strutting areas and feeding spots. So glass open areas until you find where the birds are using, then set up an ambush.\n\n168. Don't Waste Time on Henned-Up Birds When You Can Hunt Somewhere Else\n\nIf you've got lots of land to hunt, don't waste time trying to bring in turkeys that aren't that interested in your calling. When a bird gobbles once in response to your calls but won't move any closer after 15 to 20 minutes, it's likely he's still with hens. Make a mental note of your location, then move on to search for another, lonelier bird who will respond with more enthusiasm. Later, however, if you still haven't filled your tag, return to the spot you were calling in when you first heard him gobble and try calling again. His hens may now be on their nests, and he'll be wondering what happened to the one that wouldn't come see him earlier in the morning.\n\n169. Why Gobblers Are Easy to Miss\n\nHow do hunters miss a big target like a wild turkey standing within 30 yards? My personal pet theory (and I've done it myself!) is that the shooter is so enthralled by the scene before him that he raises his head from the gunstock just slightly. Do that, and you'll miss every time.\n\n170. When Roosting Gobblers Fly Down\n\nWhen you hear hens fly down from the roost, while a gobbler lingers on his limb, still calling occasionally, your nerves will be as tight as they can get. But don't start thinking your bird is as good as in the oven. Next, a scenario can take place that virtually dooms your hunt. The hens may start walking away in a direction away from your setup. The gobbler flies down and joins them, oblivious to your calls.\nCHAPTER 11\n\nDuck Hunting\n\n171. Wind and Your Decoy Setups\n\nThe truth is that you can spread your duck decoys just about any way you wish, as long as you leave an open area for the birds to land into the wind. No matter which way they come from, or how much they circle, their final move down will be into the wind. No wind at all? It becomes a guessing game.\n\n172. Guide's Advice I Don't Want to Hear\n\nIn the duck blind, you'll often hear your guide urge you to, \"Stay down. Keep your head down. Don't watch the birds! I'll do the watching.\" Well, if you're not watching the birds, you're losing part of the joys of the hunt. Your blind should be good enough for you to peer through the stalks or brush just as the guide is doing. When the ducks are passing right overhead, neither one of you should be looking skyward. You'll spook the birds for sure.\n\n173. Pothole Sneak Attack\n\nIf you've scouted out a promising pothole or small pond and you're planning to jump-shoot the ducks that are resting there, try to sneak up on them with the wind at your back. When the ducks jump into the wind (which they most certainly will do), you might get a shot before they re-orient themselves and fly the other way.\n\n174. Local Birds: Use Small Decoy Spreads for Small Bunches\n\nMake a distinction between the resident ducks you hunt in early season and the large flocks that migrate in later on. You'll spot resident birds in pairs and small flocks, so decoy them accordingly and don't burn out any one place by hunting it too often. Save the big spreads for when the birds from up north show up.\n\n175. Mix 'Em up If You Want To\n\nSo you're thinking about adding some bluebills or canvasbacks (diving ducks) to your decoy setup of mallards, pintails, and gadwalls (puddle ducks) to give your spread more visibility. Go right ahead. It won't hurt your chances a bit.\n\n176. Jump-Shooting Joys\n\nJump-shooting ducks from a canoe or john-boat is a great way to hunt some creeks and small rivers. The best way is with a partner, one hunter with the gun at the ready, the other on the paddling. Stay quiet, anticipate the sharp bends where you may surprise a few mallards, blacks, or other puddle ducks. Listen carefully as you go. You just might hear the birds before you get to them.\n\n177. Too Hidden for a Good Shot\n\nWhen you're hunkered down in a blind so that you can't see the ducks you're working, when it comes time for someone to exclaim, \"Take 'em!\" you come up with your gun and have to find the birds before you get down to pointing and swinging the barrel. It won't be an easy shot.\n\n178. Gloves for Setting out Decoys\n\nGloves that stretch almost to your elbows and keep your hands dry are a must for setting out decoys. Shuck 'em off and wear your regular gloves when you get into the blind. See the \"Midwest PVC Decoy Glove\" at Mack's Prairie Wings, www.mackspw.com. Check other favorite waterfowl gear vendors for other options.\n\n179. Pond Shooting at Sunset: The Way It Used to Be\n\nWaiting for ducks at sunset beside ponds where the ducks would be coming to roost was once a mainstay of hunting tactics. Local wood ducks, mallards, and black ducks, puddle ducks of all sorts that had migrated into a particular area\u2014they all come hurtling into the ponds after sunset. Sometimes the shooting was so late, the birds had to be outlined against the western sky. Today, shooters who try this are easy marks for wardens waiting nearby to hear the sounds of gunshots after legal shooting hours. If you want to just watch the show (and you should!) leave your guns in the truck.\n\n180. It's All about Visibility, Visibility, Visibility\n\nUnless you're gunning a tiny creek-bottom or river location, surrounded by high trees, most of your duck-hunting locations will be in open areas where you hope passing birds can see your decoys and come on in. Anything you can do to increase the visibility of your spread will make a difference. Black decoys show up better from a distance. Magnum-size adds visibility. Canada geese decoys add visibility, whether you're hunting geese or not. Movement devices (the ones that are legal where you hunt) are critical if there's no wind blowing: spinners, battery-driven shakers, pull-cord movers\u2014whatever you've got.\n\n181. The \"Hole\" Is the Thing\n\nNo matter what shape of decoy spread you decide is right for your hunting location and conditions, it must contain a hole or two for the birds to land. If the water in front of the blind is solid with decoys, the birds will land on the outside of the spread, at long range or even out of range.\n\n182. The Outer Gun: The Key Position\n\nThe Outside shooter on the upwind side of permanent duck blinds or lay-out blind setups is in the key position and can absolutely ruin the shooting for everybody with him. It's happened to me more times than I can remember. The ducks, or geese, are coming into the spread against the wind, from his side. If he starts shooting too early, around the corner, the guns in the center and other side will get no shots, or shots at widely flaring birds only. Sometimes, to top off this little drama, the outside offender will turn to the other guys and say, \"Why didn't you guys shoot?\" Advice: Put an experienced shooter in that outside position, a shooter with the judgment and nerve to wait until the birds are into the spread enough so everybody can shoot.\n\n183. Where'd the Mallards Go?\n\nWhen you're on a marsh in the early morning where you reasonably expect a flight of mallards, don't be surprised if they don't show up until later in the morning. Your local birds, or even visitors from the north, may be feeding in the fields.\n\n184. Black Ducks\u2014Red Letter Day\n\nMy calling aspirations reached a sort of pinnacle years later. I was hunkered on an icy creek on the marshes of the Chesapeake Bay, near the famous Susquehanna Flats. A pair of black ducks flew down the creek, very high and in a big hurry, headed somewhere with express tickets. They clearly were not interested in my modest decoy spread, but when I hit them with my old Herter's call and the Highball, they turned like I had 'em wired. Interested then, they circled warily while I scrunched down. Now I started rattling off my Feeding Chuckle, and a few moments later they were cupped and committed. I could finally say that I knew how to call ducks.\n\n185. Where the Birds Want to Be\n\nPushing into a cove in the marsh or along a big river or lake, in the first pre-dawn light, you flush a big bunch of ducks or geese. Away they go, gabbling and honking. Never mind trying to follow them or heading for another spot. Set up right there. It's the place the birds want to be.\n\n186. \"Take 'Em!\"\n\nFew moments afield are as thrilling as those when a big flock of ducks sweeps into your decoys. You'll shoot a lot better when you are aware whether your birds are diving ducks\u2014like bluebills and canvasbacks\u2014or puddle ducks\u2014like mallards and pintails. Diving ducks will bore straight past when the shooting starts, while puddle ducks will bounce skyward as though launched from a trampoline.\n\n187. Tall-Timber Trick\n\nWhen gunning the hole in the tall timber with a few decoys out, give the water around your tree a good kick when birds are passing or circling to imitate splashing and feeding activity.\n\n188. Using the Wind with Your Decoy Spread\n\nDucks often want to land outside a spread of decoys\u2014even when the setup has left an inviting hole. That's why you want to set your decoys upwind\u2014not directly in front of the blind\u2014so that you'll still have a good shot at the birds coming in against the wind and trying to land on the outside of the decoy spread.\n\n189. Don't Be a \"Skybuster\"\n\nA \"skybuster\" is the most hated person on any marsh or field where there's duck or goose hunting. The Skybuster blazes away at birds that are clearly out of range, thereby frightening the birds away from the area and ruining chances others might have had on the incoming birds.\n\n190. Beating the Crowds in Public Hunting\n\nDucks quickly wise up to blinds on public hunting areas. You score more ducks if you seek out remote corners that see much less pressure. Use just a half dozen or so decoys and call only enough to get passing birds' interest.\nCHAPTER 12\n\nGoose Hunting\n\n191. The Most Effective Way to Set Out Goose Decoys\n\nVeteran Maryland call-maker Sean Mann guides early season duck and goose hunting in Alberta and is one of the most successful and experienced in the business. He told DU's Wade Bourne, in a tip for the DU Web site: \"To finish more geese when hunting over a field spread, set decoys 10 feet apart (three long steps), and face them in random directions. This set provides a natural, relaxed look, and it also offers incoming birds plenty of landing room inside the spread. By setting my decoys so far apart, I use half the number I used to. I can set up and tear down faster, and most of all, the geese work better. Our hunts are much more productive than when I set decoys closer together. Less really can be more.\"\n\n192. How to Change Your Luck with Snow Geese\n\nIn the October '08 Field & Stream, author Dave Hurteau, in an interview with veteran guide Tracy Northup, Up North Outdoors, www.huntupnorth.com, presents a deadly method for changing your luck with those tough, high-flying flocks of snow geese. In a tip called, \"Play the Wind,\" Northup says. \"Snow geese typically fly high and circle straight down, making it difficult to shoot them anywhere but right over a good spread. But a 30\u201340-mph wind keeps them flying nice and low.\" Northup recommends scouting out a location of snows where there are ditches or hedgerows a hundred yards or so from the fields where you can sneak into position to pick off the low-flying snows as they pass\u2014without spooking the main flock.\n\n193. Birds in Flight: Looks Are Deceiving\n\nBecause they are big, Canada Geese appear to be slow in flight, compared to ducks. And because of their long tails, pheasants appear to be slower than they really are. Swing your gun properly, lead the bird, and keep swinging as you pull the trigger. Or you'll be shaking your head, wondering how you missed.\n\n194. Snow Geese: Playing the Numbers Game\n\nThose large flocks of snow geese weaving across the horizon, clamoring constantly, are hard to pull into normal decoy spreads of just two- or three-dozen birds. The flying geese can see great distances, and they are looking for big groups of feeding birds. Savvy hunters have learned to cope with this by putting out decoys by the hundreds, if necessary, and to do this they'll use all the silhouettes they can haul to the site, plus whatever \"rag-type\" decoys they can fashion themselves from things like baby diapers and white garbage bags attached to a stake.\n\n195. Layout Blinds Take Getting Used To\n\nWhen using a layout blind, before the birds start flying take some time to try practicing the move it takes to rise into a shooting position. It takes some getting used to. If you don't practice it, you may not be in a good position with your face well down on the gun during the first critical seconds when it's time to, take 'em!\n\n196. When the Canadas Sleep Late\n\nIn below-zero weather, sleep in an extra hour or two. When it's that cold Canadas will stay roosted and fly out to feed only after the sun has come up and warmed things up a bit. It might be 10 AM before they leave the roost. The only thing you'll get by showing up at dawn is cold.\n\n197. Keep Those Silhouettes Visible\n\nWhen using silhouette decoys for geese, take care to position them so many of them appear broadside at every angle. When edge-on to the viewpoint of the flying birds, they become invisible.\n\n198. Layout Blinds: You're Part of the Action\n\nYou're lying in a field, totally hidden right among the decoys. No brushy blind, no boat, no pit blind, no elaborate box blind, no blind on stilts. Instead, you're tucked comfortably into a well-camouflaged layout blind, made further invisible by attaching brush to the blind's convenient straps and holders. You're wearing camo yourself, including a hat and mask. Even your gun is camouflaged. Unlike hunting from a brush blind where you have to keep your face down\u2014and thereby miss part of the spectacle of flying birds on the way in\u2014you're seeing the whole show, from the time birds appear in the distance, until they coming right into your face. There's nothing like it!\n\n199. Hiding Your Boat in Plain Sight\n\nThe john-boat or canoe you can put into the water and go wherever the ducks and geese are flying has gotten a lot easier to hide with the introduction of today's synthetic camo material. The material, imitating different shades and textures of marsh grass, comes in manageable mats you can attach to your boat, then roll up and put away after your hunt. Cabela's, www.cabelas.com, has a bunch of different patterns, including the excellent Avery, and there's a popular one called Fast-Grass that's available at the Knutson's waterfowling store and site, www.knutsondecoys.com.\n\n200. Don't Let Those Incoming Geese Fool You\n\n\"The approach of wild geese to a blind is one of the neatest optical illusions in nature. The geese just keep on coming. You think they are one hundred yards away, and they are two hundred. You think they are fifty yards away, and they are one hundred.\"\n\n\u2014Gordon MacQuarrie, \"Geese! Get Down!,\" Field & Stream, 1941, reprinted in The Field & Stream Reader, Doubleday, 1946\nCHAPTER 13\n\nGrouse and Woodcock Hunting\n\n201. Good Hearing Can Pay Off Big\n\nWhen the leaves in the woods are dry and crinkly, say on a perfect Indian Summer day, you can actually hear the few steps ruffed grouse take to launch into flight. The sound is a sort of dry, \"tick, tick.\" Once you've heard it a couple of times, followed by a flushing bird, you'll know what to listen for.\n\n202. Grouse Hunting Teamwork\n\nIt's fun to hunt grouse with a buddy, but you must know each other's location at all times, and even then you may not be able to take a shot. (That's the very reason most grouse hunters shoot better when they're hunting alone.) In keeping track of one another, use a simple call-out, like, \"Ho!\" or \"Over Here!\" instead of constantly shouting sentences like, \"I'm over here, Bob, on your right.\" The more grouse hear of such talk, the more likely they are to flush wild.\n\n203. Second Shots on Early Season Grouse Coveys\n\nWhen you flush a couple of early season grouse, and fire only one shot, don't break your gun to reload right away. You may be into an entire group of birds hatched that year, and one of the birds that's been sitting tight will jump late\u2014just when you break open your gun. On the other hand, if you've fired both shells, try to reload as quickly as possible.\n\n204. Keep Track of Where You Shot From\n\nWoodcocks are small birds and their feathers make for excellent camouflage on the forest floor. This can make finding one you've shot hard to retrieve, especially if you're hunting without a dog. After you knock one down, hang your hat on a branch or drop a spent shell on the ground where you were standing when you pulled the trigger. If you get confused about where you thought the bird landed you'll be able to return to the exact place you shot from to restart your search.\n\n205. Cock Bird or a Hen? How to Tell\n\nThe black band on the tail of all ruffed grouse tells you whether you've bagged a male or female. The cock bird always has a continuous band, while the hen's is broken. As in all things in nature, there are exceptions sometimes, mostly among young birds.\n\n206. Noise Flushes More Grouse\n\nYes, grouse can be a tight-holding bird, but when you're approaching their location with a lot of chatter and constant commands to your dog, you're almost guaranteeing you'll get a wild flush, out ahead of the point.\n\n207. Flushing Dogs as Grouse Dogs\n\nIt makes a lot of sense to use flushing\/retrieving dogs like Labs, Springers, and Goldens as grouse and woodcock dogs. (Pheasants too, by the way.) In addition to flushing the birds, they retrieve and make great family dogs. When trained properly to hunt close to the gun, these dogs can do a good job for you in grouse and woodcock covers. When not properly trained, running wide distances and out of control, they are worse than useless. You'd be better off walking up your birds alone.\n\n208. The Perfect Grouse and Woodcock Gun\n\nWhen it comes to grouse guns, arguments may rage over the bore\u201412 or 20 gauge\u2014but you'll find general agreement over these details: lightweight, short barrel, fast swinging, and with a stock that fits so perfectly that it instantly becomes part of you when your face touches the stock.\n\n209. Hitting More Grouse: The Big Secret\n\nHere it comes, the grouse-hunting technique that will put more birds in your coat than any other: When you hear the flush, don't stand there looking for the bird, then raise your gun. Your gun should be coming to your shoulder as you look toward the direction of the flush. At the first glimpse of the bird, the barrel should be coming onto what you're seeing, and you fire instantly. There's no tracking, no aiming. It's a \"throw\" shot, as I prefer to call it, instead of the oft-heard \"snap\" shot. Of course, this method is assuming a hunting buddy is behind you, out of harm's way.\n\n210. The Dead \"Giveaway\" on Woodcock\n\n\"Going into strange woodcock terrain 'cold' can be disappointing. Two coverts may look alike, yet only one many attract 'cock. Whitewash is the best clue, next to seeing actual birds. The white splashing disintegrate rapidly and when you see them you can almost count on woodcock being nearby.\"\n\n\u2014George Bird Evans, The Upland Shooting Life, Knopf, 1971\n\nCHAPTER 14\n\nPheasant Hunting\n\n211. When Ringnecks Sit Tight\n\nDespite their reputation for being track stars, running ahead of the dogs and hunters, pheasants are capable of making themselves invisible and sitting tight when they feel pressured enough. They can hide in the smallest clump of grass, letting you walk right past them.\n\n212. Don't Let That Tail Fool You\n\nCompared to gamebirds like quail and ruffed grouse, the takeoff of a pheasant, while exciting, is rather slow. However, in its initial, vaulting leap, the rooster pheasant's long tail and gaudy colors make the bird seem even larger than it really is, and the shooter fails to take a lead, but merely pokes the barrel at the body of the bird.\n\n213. How \"Cackles\" Betray Cockbirds\n\nWhen a pheasant \"cackles\" as it flushes, it's a cockbird. Every time. Hens do not cackle. This doesn't mean that cockbirds cackle every time they flush. They don't. Sometimes they fly away in silence. But when you hear a cackling bird, even if it's in the sun and you can't see its colors, you know it's a cockbird.\n\n214. Go Late for Western Ringnecks\n\nWhile the typical western ringneck hunt involves following short tails though long, golden grass in shirt-sleeve weather, some of the best pheasant hunting on prime habitats in the Dakotas, Nebraska, and Montana occurs after Thanksgiving. There is less pressure then, and the winter weather tends to push birds into flocks. Gaining access to great hunting spots is often easier then, with big game seasons ended. So consider a late season pheasant hunt in the West.\n\n215. The Key Maneuver\n\nWhen you work pheasant cover, always work toward a distinct end point, be it an irrigation ditch, road, creek, or open field. The birds will eventually figure they can't outrun you. As you and your dog approach that end point, the birds will flush.\n\n216. You're Walking Past the Birds\n\nPheasants tend to sit tighter in wet weather, so work cover more thoroughly than you would on a bluebird day.\n\n217. Coming up Empty!\n\nAlways reload quickly after shooting a flushing rooster; he may have compatriots with him. While you're patting yourself on the back for a great shot, other birds may flush as you stand there with an empty shotgun.\n\n218. They Need to Take a Drink\n\nLike four-legged critters, pheasants need a water source to thrive. In dry months in the fall, work cover near watering holes late in the afternoon to find birds.\n\n219. Pheasants in the Tracking Snow\n\nGot snow? Then go track up a pheasant. You don't need a dog, and you'll quickly learn the birds' favorite hiding spots and get a good feel of how many birds are in the area.\n\n220. After the Season Opens . . .\n\nHunting pheasants on opening day and then a week later will seem like the difference between night and day. The pheasant's \"disappearing act\" after opening day is one of the most remarkable in all upland bird hunting.\nCHAPTER 15\n\nQuail Hunting\n\n221. Shooting the Covey Rise\n\nA good covey rise\u2014say twelve to fifteen birds\u2014is one of the most exciting events in all wing-shooting. Alas, it's also the time when many shooters miss on their first shot, then hastily throw their second shot into thin air. From Day One, quail hunters are urged, \"Pick out one bird and shoot it.\" But they have a hard time doing it. The sight of all those birds, particularly wild birds, hurtling toward the trees\u2014or even through the trees\u2014keeps their face from getting down on the stock, which will result in a miss every time. Some hunters do well on coveys by telling themselves that they're going to shoot the first bird that flies\u2014just as though they were shooting singles. In my case, my scores on covey rises improved when I started really hunkering down on the gun, swinging toward one bird, and telling myself, \"I'm going to kill that bird.\" That's the kind of focus shooting covey rises requires, at least in my view.\"\n\n222. Leave That Safety on until the Shot!\n\nIn my formative gun-handling years, I learned to leave the safety on until the gun is moving toward my shoulder to make a shot. Many quail hunters take their safeties off when walking in to a point to flush the birds. The practice is dangerous. The birds may have moved, the dog will have to relocate, and you might have a lot more walking before the birds take flight (if indeed they ever do take flight). Walking around through briars and tangles with a loaded gun with the safety off is a prescription for disaster. Train yourself to leave the safety on until a bird is in the air and the gun is moving toward your shoulder.\n\n223. Marking down the Singles\n\nAfter a covey rise, many hunters make the mistake of diverting their attention to downed birds too quickly, instead of carefully watching the escape routes of the covey's survivors. Even after these escaping birds are seen to cup their wings and sail in a certain direction, they bear careful watching, for they can sail a long way and change direction quickly.\n\n224. Cold Front Moving in\u2014Great Hunting!\n\nThere's nothing quite like being in the field in an afternoon when low clouds are scudding about, it's starting to spit just a bit of snow or icy rain, and the temperature has been dropping sharply. Quail will be feeding like there's no tomorrow, leaving strong trails of scent your dogs can easily find. You can bag your limit on a day like this.\n\n225. Make Mine a Side-by-Side\n\nAll right, call me \"old-fashioned\" if you wish. But to me, a side-by-side 20-gauge double is the gun of choice for bobwhite quail. Barrels 26 inches, bored modified and improved cylinder. You can tramp a long time with a light double, enjoying its sweet feel of balance and sleekness, and when the gun comes to your shoulder, the broad sighting plane seems to flow onto the target. I like double triggers, and a straight stock, but again, I'm old-fashioned. For many hunters today, the over-under seems to be the favorite, especially when it's the same gun used for sporting clays. Makes all the sense in the world. But it's not for me in the quail woods.\n\n226. Leave Some for \"Seed\"\n\nShooting a covey of wild bobwhites down to two or three birds doesn't make any sense. From a covey of say twelve birds, set your personal take as five or six. No more. Now leave that covey alone for the rest of the season. There's always next year.\n\n227. Keep Your Head down on the Stock\n\nHunters sometimes wonder why they missed a seemingly easy shot\u2014a covey of quail bursting into flight, a flock of mallards right over the decoys, even a wild turkey strutting into plain view. Often the reason is simply because you were so excited and enthralled by what you are seeing, that you lift your head slightly from the gunstock. When you do, it's all over! You're going to miss the shot.\n\n228. Take It Easy, Get More Shots\n\nWith a good bird dog willing to \"hunt close\" under today's tough conditions, quail hunting is not the place to be in a hurry. Instead, just mosey along and take your time, working every nook and cranny along the edges of the fields thoroughly. You'll find lots more coveys than the hunter in a hurry.\n\n229. When Birds Are Running, Keep up with Your Dog\n\nWhen your dog is pointing for a few seconds, then moving ahead, then moving again and relocating as you come up, you've obviously got some running birds ahead. Try to keep up with the dog as he moves along. Chances are high that this covey is going to flush wild, well ahead.\n\n230. They're Closer Than You Think\n\n\"Most quail are killed within sixty feet of the gun. Before you say I am wrong, measure the next ten you kill. The bobwhite fades away so fast on the flush that many men won't shoot at a bird thirty to thirty-five yards away, believing that he is beyond good killing range.\"\n\n\u2014Ray P. Holland, Scattergunning, Knopf, 1951\nCHAPTER 16\n\nSquirrel Hunting\n\n231. Staying Put on Your Stand\n\nWhen you've selected a good spot to sit and wait for squirrels to show themselves, don't be in a hurry to walk over and pick up the first two or three you down\u2014with shotgun or rifle. Mark them carefully and keep sitting tight and watch for another target to show.\n\n232. The \"Trunk-Hugger\" Squirrel\n\nWhen a squirrel hugs the trunk, high in the tree and not moving, it usually is facing up the tree. By hugging the trunk yourself, you can outlast the squirrel into thinking the coast is clear. It will make a move, giving you a shot.\n\n233. A Squirrel Hunt Can Make Your Day Great\n\nMany hunters get so caught up in the pursuit of deer and \"glamour\" upland birds like grouse, quail, and pheasant that they forget the simple pleasures of a great day out squirrel hunting. Take a golden autumn day, a small pack with sandwiches and a thermos, a scope-sighted .22 rifle, and local knowledge of an area and you can walk for miles\u2014that is, walk when you wish to. A lot of your day will be spent sitting quietly at the basses of trees, watching the canopy overhead. One caveat: You can't do this hunt when and where firearm deer hunting is in progress.\n\n234. Playing \"Hide-and-Seek\" with Squirrels\n\nWhen a squirrel is on the opposite side of a tree, and keeps going around and around as you circle, trying to get a shot, try throwing something noisy over to the squirrel's side. A heavy fallen stick or something. You can often move him to your side this way.\n\n235. Those Wonderful Squirrel Dogs\n\nThey'll never have the glamour and appeal of labs and pointers and setters, but good squirrel dogs are worth their weight in gold. What kind are they? All kinds! Feists, curs, whatever. A good nose and desire to hunt are the credentials, and the best way to take advantage of them is to have your squirrel dog puppy hunt with an accomplished squirrel dog. The really good squirrel dogs hunt close to the gun and use their nose, eyes, and ears to find squirrels. Easing slowly through the squirrel woods with one of these dogs is a wonderful way to go hunting. In many areas, you'll have a bonus: You'll flush coveys of quail, and send rabbits bouncing away.\n\n236. Cutting-Edge Squirrel Loads\n\nIn a comprehensive article called \"Cutting-Edge Squirrel Loads\" on the Game & Fish magazine Web site, www.gameandfishmag.com, writer Mike Bleech gives today's squirrel hunters the latest details on shotgun and rifle loads that do the best job in bringing home a limit of bushytails. Bleech examines the important differences in magnum versus standard shot loads, with emphasis on the all-important velocity considerations. He brings the .22 hunter right up to date with test-firing comparing the venerable .22 Long Rifle against the scorching hot newcomer the .17 Hornady Magnum Rimfire and the newer, but slower, Hornady .17M2. The .17M2 gets Bleech's nod as the best way to go because, as he says, \"The light crack of the .17M2 is barely noticed by the squirrels.\"\n\n237. Mast Crop Low, yet Hunting Great\n\nWhen the crop of acorns on the white and red oaks is low, look for squirrels to be concentrated in areas where some trees have bucked the trend. Whatever good spots for acorns are available, the squirrels will find them\u2014particularly the white oak acorns. White-tailed deer do the same.\n\n238. The Pecan Tree Bonanza\n\nThe shady, fruit-bearing groves of pecan trees planted throughout the southeast and on into Texas are squirrel magnets. Forested edges of the groves will hold good populations of squirrels. With the landowners' permissions, you'll be set for good hunting.\n\n239. Midwinter Fox Squirrels\n\nWhen there's a crust of snow on the ground and the temperatures have dipped into the 20s, don't expect fox squirrels to be on the prowl in early morning. You'll find the best hunting doesn't get started until around ten o'clock. Although like gray squirrels, fox squirrels like acorns and other nuts, in midwinter they will always head for corn if it's nearby. Fox squirrels prefer belts of trees near croplands instead of dense forests, more so than grays.\n\n240. Try More Aggressive Squirrel Tactics\n\nSuccessful squirrel hunters today have come as many calls and calling techniques as duck hunters. Consider these two examples: On the Hunter Specialties Web site, www.hunterspec.com, pro Alex Rutledge shows how to use the H.S. Squirrel Call to do the \"Barking,\" \"Chattering,\" and the \"Young Squirrel Distress Call.\" All are effective, but \"Barking\" is one known to set off a chorus of squirrel answering calls and tip off their locations in woods you're hunting for the first time. Outdoor writer John E. Phillips, in his \"John's Journal,\" on his Night Hawk Publications site, www.nighthawkpublications.com, also likes \"Barking,\" along with some other calls he describes in his article, \"How to Hunt Squirrels Aggressively.\"\nCHAPTER 17\n\nRabbit Hunting\n\n241. The Hunter's \"Rabbit's Foot\" Luck\n\nRabbits are the Number One game for hunters everywhere\u2014and with good reason. They breed like crazy, which is great since few of them survive the predators they face for even one year. There are still plenty to go around for hunters, who prize delicious rabbit on their tables and like the way rabbits can be hunted with a variety of methods and a minimum of expensive gear.\n\n242. Watching for the Rabbit You've Jumped\n\nWhen your dog is running a rabbit, don't keep your eyes locked on the path the dog is taking. Keep looking in other directions, and you may spot rabbits that are trying to slip past all the action, or others that are just sitting tight waiting for you to go past.\n\n243. Top Tips for Late Season Rabbits\n\nLate season is a great time to hunt rabbits because deer hunters have ceased activities. Hunting can be tough, though. According to writer Ed Harp on the Indiana state site of Game & Fish magazine, www.indianagameandfish.com, the most important four considerations in finding late season rabbits are to find the clover, find blackberry and raspberry bushes, find the pine sapling stands where rabbits have chewed the bark in a circle, and find deer hunter food plots. Harp has six more top tips to help you get your limit in the article, \"Ten Tips for Taking Winter Rabbits.\"\n\n244. Use a Stick to Beat the Brush\n\nWhen hunting rabbits without a dog, use a brush stick, like a wading staff, to beat on the edges of thick cover where they may be hiding.\n\n245. Cold and Windy Mornings\n\nOn a windy morning, after a very cold night, look for rabbits to be on the sunny and lee sides of ridges, forests, and brush rows.\n\n246. Hunting the \"Slabfoot\" Rabbits\n\nHe's not called the \"Varying Hare\" for nothing, possessing two coats of fur to wear as needed. With the fall sun starts riding lower and lower in the sky, the days becoming shorter and shorter, the Snowshoe rabbit starts putting on its white coat whether needed or not by snowfall. Even with nary a flake on the ground, the big hares begin to change color\u2014a dangerous situation for them since they stand out in the forest so starkly. In Alaska, where I spent two teenage winters, Snowshoe rabbit hunting became a big part of my life. I loved hunting them and eating them.\n\n247. Snowshoes in Winter: The Going Gets Tough\n\nThe hunting in the hills not far from Fairbanks, where we lived, was easy until the snows came, then became more difficult with each passing day. The Snowshoe is mostly a nocturnal animal. During the day the rabbits hole up under the endless spruces and don't move until after dark. Kicking them out is tough work without a dog. When we were lucky enough to get one bounding away through the snow, they were remarkably fast and hard to hit in their great leaps over the snow. Old \"Slabfoot.\" I loved them.\n\n248. The Snowshoes' Survival Plan\n\nWhen winter grips the great North woods, the icy winds moaning through bare limbs without a scrap of vegetation, the snow piling on the endless spruce forests, the Snowshoe rabbit makes out just fine. Gone are the succulent plants of spring and summer, but the \"Varying Hares\" do just fine on a diet of willow, poplar, and other saplings of tender green bark. The Snowshoe has four big front teeth perfect for gnawing a bellyful of bark every night. Days are spent in cozy holes back under the spruces. No matter how cold, they make out just fine.\n\n249. Tactic for Snowshoes: Leave the Feeding Grounds Behind\n\nThe area where you find the great crisscrossing webs of Snowshoe tracks may not be the best place to make your hunt. They were there last night feeding; now they're dead certain to be in the thickest cover of the nearest swamps. That's where you want to make your hunt, moving slowly past the beaver dams, humps of brush, fallen trees, and limbs\u2014the places where Snowshoes spend their days, not their nights.\n\n250. Your Best Rabbit \"Scouts\"\n\nThe best rabbit scout you can find will be the farmer who owns the land you hunt on. Next will be the deer hunters who've been working the area. Many of them aren't interested in rabbit hunting, but they will have seen the spots where rabbits thrive.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nThe Last Chronicles of Thomas Covenant\n\nThe Runes of the Earth\n\nFatal Revenant\n\nAgainst All Things Ending\n\nThe Chronicles of Thomas Covenant the Unbeliever\n\nLord Foul's Bane\n\nThe Illearth War\n\nThe Power That Preserves\n\nThe Second Chronicles of Thomas Covenant\n\nThe Wounded Land\n\nThe One Tree\n\nWhite Gold Wielder\n\nG. P. PUTNAM'S SONS\n\n_Publishers Since 1838_\n\nPublished by the Penguin Group\n\nPenguin Group (USA) LLC\n\n375 Hudson Street\n\nNew York, New York 10014\n\nUSA \u2022 Canada \u2022 UK \u2022 Ireland \u2022 Australia \u2022 New Zealand \u2022 India \u2022 South Africa \u2022 China\n\npenguin.com\n\nA Penguin Random House Company\n\nCopyright \u00a9 2013 by Stephen R. Donaldson\n\nPenguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nDonaldson, Stephen R.\n\nThe last dark \/ Stephen R. Donaldson.\n\np. cm.\u2014(The last chronicles of Thomas Covenant ; Book four)\n\nISBN 978-1-101-63643-5\n\n1. Covenant, Thomas (Fictitious character)\u2014Fiction. I. Title.\n\nPS3554.O469L37 2013 2013016784\n\n813'.54\u2014dc23\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\nto Jennifer Dunstan,\n\nwho stood with me the whole time\n\nand to John Eccker,\n\nwho has put in more effort, and has been of more help, than I could ever have expected\n\nand to Robyn H. Butler:\n\n\"And they lived happily ever after.\"\n\n# Contents\n\nAlso in The Chronicles of Thomas Covenant\n\nTitle Page\n\nCopyright\n\nDedication\n\nMap\n\n What Has Gone Before\n\nPart One \"to bear what must be borne\"\n\n 1. _Betimes Some Wonder_\n\n 2. Nightfall\n\n 3. _Not Dead to Life and Use_\n\n 4. \"Try to Believe\"\n\n 5. Coming\n\n 6. Promises Old and New\n\n 7. Taking the Risk\n\n 8. The Right Materials\n\n 9. An Impoverished Temple\n\n 10. _But While I Can_\n\n 11. Back from the Brink\n\n 12. After Too Long\n\nPart Two \"the abyss and the peak\"\n\n 1. _A Tale Which Will Remain_\n\n 2. Toward Confrontation\n\n 3. Summoned to Oppose\n\n 4. Reluctances\n\n 5. \"No Prospect of Return\"\n\n 6. The Aid of the Feroce\n\n 7. At Last\n\n 8. Shamed Choices\n\n 9. Parting Company\n\n 10. All Lost Women\n\n 11. _Of My Deeper Purpose_\n\n 12. _You Are Mine_\n\n Epilogue: \"The soul in which the flower grows\"\n\n Glossary\n\n# What Has Gone Before\n\n\"The Chronicles of Thomas Covenant the Unbeliever\"\n\nAs a young man\u2014a novelist, happily married, with an infant son, Roger\u2014Thomas Covenant is stricken with leprosy. In a leprosarium, where the last two fingers of his right hand are amputated, he learns that leprosy is incurable. As it progresses, it produces numbness, often killing its victims by leaving them unaware of infections. Medications arrest its progress; but Covenant is taught that his only hope of survival lies in protecting himself obsessively from any form of damage.\n\nHorrified by his illness, he returns to his home on Haven Farm. But other blows to his emotional stability follow. His wife, Joan, abandons and divorces him to protect their son. Fearing the mysterious nature of his illness, the people around him cast him in the traditional role of the leper: a pariah, outcast and unclean. In addition, he becomes impotent\u2014and unable to write. Grimly he struggles to go on living; but as his despair mounts, he has episodes of prolonged unconsciousness, during which he seems to visit a magical realm known only as \"the Land.\"\n\nIn the Land, physical and emotional health are tangible forces, made palpable by an energy called Earthpower. Because vitality and beauty are concrete qualities, as plain to the senses as size and color, the well-being of the physical world has become the guiding precept of the Land's people. When Covenant first encounters them, in _Lord Foul's Bane_ , they greet him as the reincarnation of an ancient hero, Berek Halfhand, because he, too, has lost half of his hand. Also Covenant possesses a white gold ring\u2014his wedding band\u2014which they know to be a mighty talisman, able to wield \"the wild magic that destroys peace.\"\n\nShortly after he first appears in the Land, Covenant's leprosy and impotence disappear, cured by Earthpower; and this, he knows, is impossible. Indeed, the mere idea that he possesses some form of magical power threatens the stubborn disciplines on which his survival depends. Therefore he chooses to interpret the Land as a dream or hallucination. He responds to his new health with Unbelief: the dogged assertion that the Land is not real.\n\nBecause of his Unbelief, his initial reactions to the people and wonders of the Land are at best dismissive, at worst cruel. At one point, urged by sensations which he can neither accept nor control, and certain that his experiences are illusory, he rapes Lena, a young girl who has befriended him. However, her people decline to punish or reject him for his actions. As Berek Halfhand reborn, he is beyond judgment. And there is an ancient prophecy concerning the white gold wielder: \"With the one word of truth or treachery, \/ he will save or damn the Earth.\" Covenant's new companions know that they cannot make his choices for him. They can only hope that he will eventually follow Berek's example by saving the Land.\n\nAt first, such forbearance achieves little, although Covenant is moved by both the ineffable beauties of this world and the kindness of its people. During his travels, however\u2014first with Lena's mother, Atiaran, then with the Seareach Giant Saltheart Foamfollower, and finally with the Lords of Revelstone\u2014he learns enough of the Land's history to understand what is at stake.\n\nThe Land has an ancient enemy, Lord Foul the Despiser, who dreams of destroying the Arch of Time\u2014and with it not only the Land but the entire Earth\u2014in order to escape what he perceives to be a prison. Against this evil stands the Council of Lords, men and women who have dedicated their lives to nurturing the health of the Land, and to opposing Despite.\n\nUnfortunately these Lords possess only a small fraction of their predecessors' power. The Staff of Law, Berek's primary instrument of Earthpower, has been hidden from them. And the lore of Law and Earthpower seems inherently inadequate to defeat Lord Foul. Wild magic rather than Law is the crux of Time. Without it, the Arch cannot be destroyed; but neither can it be defended.\n\nHence both the Lords and the Despiser seek Thomas Covenant's allegiance. The Lords attempt to win his aid with courage and compassion: the Despiser, through manipulation. And in this contest Covenant's Unbelief appears to ally him with the Despiser.\n\nNevertheless Covenant cannot deny his reaction to the Land's apparent transcendence. And as he is granted more and more friendship by its people, he remembers his violence toward Lena with dismay. Thus he faces an insoluble conundrum: the Land cannot be real, yet it feels entirely real. His heart responds to its loveliness\u2014and that response has the potential to kill him by undermining his necessary caution and hopelessness.\n\nTrapped within this contradiction, he attempts to escape through a series of unspoken bargains. In _Lord Foul's Bane_ , he lends the Lords his passive support, hoping that this will enable him to avoid the possibilities\u2014the responsibilities\u2014of his white gold ring. And at first his hopes are realized. The Lords find the lost Staff of Law; their immediate enemy, one of Lord Foul's servants, is defeated; and Covenant is released from the Land.\n\nBack in his real world, however, he discovers that he has in fact gained nothing. Indeed, his plight has worsened. His experience of friendship and magic has weakened his ability to endure his outcast loneliness. When he is translated to the Land a second time, in _The Illearth War_ , he knows that he must devise a new bargain.\n\nDuring his absence, the Land's plight has worsened as well. Decades have passed there; and in that time Lord Foul has acquired the Illearth Stone, a bane of staggering power. With it, the Despiser has created an army that now marches against the Lords: a force which the Staff of Law cannot adequately oppose. The Lords need the strength of wild magic.\n\nOther developments also exacerbate Covenant's dilemma. The Council is now led by High Lord Elena, his daughter by his rape of Lena. With her, he begins to experience the real consequences of his crime: unlike the rest of the Council, he can see that she is not completely sane. In addition, the army of the Lords is led by a man named Hile Troy, who appears to come from Covenant's own world. Troy's presence radically erodes Covenant's self-protective Unbelief.\n\nNow more than ever Covenant needs to resolve his conundrum. Again he posits a private bargain. He will give the Lords his active support. Specifically, he will join Elena on a quest to discover the source of EarthBlood, the most concentrated form of Earthpower. But he will continue to deny that his ring has any magic. He will accept no responsibility for the Land's fate.\n\nThis time, however, the results of his bargain are disastrous. Using the Illearth Stone, Lord Foul slaughters the Giants of Seareach. Hile Troy is only able to defeat the Despiser's army by giving his soul to Caerroil Wildwood, the Forestal of Garroting Deep. And Covenant's help enables Elena to find the EarthBlood, which she uses to violate the Law of Death. She resurrects Kevin Landwaster, a long-dead High Lord, believing that he will have more power against Lord Foul than anyone living. But she is terribly wrong; and in the resulting catastrophe, both she and the Staff of Law are lost.\n\nCovenant returns to his real world knowing that his attempts to resolve his dilemma have served the Despiser.\n\nNearly broken by his failures, he visits the Land again in _The Power That Preserves_ , where he discovers the full cost of his actions. Dead, his daughter now serves Lord Foul, using the Staff of Law to wreak havoc. Her mother, Lena, has lost her mind. And the Lords are besieged by an army too powerful to be defeated.\n\nCovenant still has no solution to his conundrum: only wild magic can save the Land, yet he cannot afford to accept it. However, sickened at heart by Lena's madness, and by the imminent ruin of the Land, he resolves to confront the Despiser himself. He has no hope of victory, but he would rather sacrifice himself for the sake of an unreal yet magical place than preserve his outcast life in his real world.\n\nBefore he can reach the Despiser, however, he must first face dead Elena and the Staff of Law. Although he cannot oppose her, she defeats herself: her attack on him draws a fierce response from his ring\u2014a response which also destroys the Staff.\n\nAccompanied only by his old friend, the Giant Saltheart Foamfollower, Covenant finally confronts Lord Foul and the Illearth Stone. Facing the Despiser's savagery and malice, he at last finds the solution to his conundrum, \"the eye of the paradox\": the point of balance between accepting that the Land is real and insisting that it is not. On that basis, he uses the dire might of the Illearth Stone to trigger wild magic from his ring. With that power, he shatters both the Stone and Lord Foul's home, thereby ending the threat of the Despiser's evil.\n\nWhen he returns to his own world, he learns that his new-found balance benefits him there as well. He knows now that the reality or unreality of the Land is less important than his love for it; and this insight gives him the strength to face his life as a pariah without fear or bitterness.\n\n\"The Second Chronicles of Thomas Covenant\"\n\nFor ten years after the events of _The Power That Preserves_ , Covenant lives alone on Haven Farm, writing novels. He is still an outcast, but he has one friend, Dr. Julius Berenford. Then, however, two damaged women enter his life.\n\nHis ex-wife, Joan, returns to him, violently insane. Leaving Roger with her parents, she has spent years in a commune which has dedicated itself to Despite, and which has chosen Covenant to be its victim. Hoping to spare anyone else the hazards of involvement, Covenant attempts to care for Joan alone.\n\nWhen Covenant refuses aid, Dr. Berenford enlists Linden Avery, a young physician whom he has recently hired. Like Joan, she has been badly hurt, although in entirely different ways. As a young girl, she was locked in a room with her father while he committed suicide. And as a teenager, she killed her mother, an act of euthanasia to which she felt compelled by her mother's self-pity. Loathing death, Linden has become a doctor in a haunted attempt to erase her past.\n\nAt Dr. Berenford's urging, Linden intrudes on Covenant and Joan. When members of Joan's commune seek to sacrifice Covenant, Linden tries to intervene, but she is struck down. As a result, she accompanies him when he is returned to the Land.\n\nDuring Covenant's absence, several thousand years have passed, and the Despiser has regained his power. As before, he plots to use Covenant's wild magic in order to break the Arch of Time. In _The Wounded Land_ , however, Covenant and Linden learn that Lord Foul has altered his methods. Instead of relying on armies and warfare to goad Covenant, he has devised an attack on the natural Law which gives the Land its beauty and health.\n\nThe overt form of this attack is the Sunbane, a malefic corona around the sun which produces fertility, rain, drought, and pestilence in mad succession. So great is the Sunbane's destructiveness that it now dominates all life in the Land. And its virulence also serves to mask Lord Foul's deeper stratagems.\n\nHe has spent centuries corrupting the Council of Lords. That group now rules over the Land as the Clave; and it is led by a Raver, one of the Despiser's most vicious servants. The Clave extracts blood from the people of the Land to feed the Banefire, an enormous blaze which increases the Sunbane.\n\nHowever, the hidden purpose of the Clave and the Banefire is to inspire from Covenant an excessive exertion of wild magic. And toward that end, another Raver afflicts Covenant with a venom intended to cripple his self-control. When the venom has done its work, he will be unable to defend the Land without destroying the Arch.\n\nAs for Linden Avery, Lord Foul intends to use her loathing of death against her. She alone is gifted or cursed with the health-sense which once enabled the people of the Land to perceive physical and emotional health directly. For that reason, she is uniquely vulnerable to the malevolence of the Sunbane, as well as to the malice of the Ravers. Such evils threaten her to the core.\n\nAlthough her health-sense accentuates her potential as a healer, it also gives her the capacity to possess other people; to reach so deeply into them that she can control their actions. By this means, Lord Foul intends to cripple her morally: he seeks to make of her a woman who will possess Covenant, misuse his power. Thus she will give the Despiser what he wants even if Covenant does not.\n\nAnd should those ploys fail, Lord Foul has prepared other gambits.\n\nHorrified in their separate ways by what has been done to the Land, Covenant and Linden wish to confront the Clave; but on their own, they cannot survive the complex perils of the Sunbane. Fortunately they gain the companionship of two villagers, Sunder and Hollian, who help Covenant and Linden avoid ruin.\n\nBut Linden, Sunder, and Hollian are separated from Covenant near a region known as Andelain, captured by the Clave while he enters Andelain alone. It was once the most beautiful and Earthpowerful place in the Land; and he now discovers that it alone remains intact, defended from the Sunbane by the last Forestal, Caer-Caveral, who was once Hile Troy. There Covenant encounters his Dead, the spectres of his long-gone friends. They offer him advice and guidance for the struggle ahead. And they give him a gift: a strange ebony creature named Vain, an artificial being created for a hidden purpose by ur-viles, former servants of the Despiser.\n\nThereafter Covenant hastens toward Revelstone to rescue his friends. When he encounters the Clave, he learns the cruelest secret of the Sunbane: it was made possible by his destruction of the Staff of Law thousands of years ago. Desperate to undo the harm which he has unwittingly caused, he risks wild magic in order to free Linden, Sunder, and Hollian, as well as a number of _Haruchai_ , powerful warriors who at one time served the Lords.\n\nWith his friends, Vain, and a small group of _Haruchai_ , Covenant then sets out to locate the One Tree, the wood from which Berek originally fashioned the Staff of Law. Covenant hopes to devise a new Staff to oppose the Clave and the Sunbane.\n\nTraveling eastward, the companions encounter a party of Giants, seafaring beings from the homeland of the lost Giants of Seareach. One of them, Cable Seadreamer, has had a vision of a terrible threat to the Earth, and the Giants have sent out a Search to discover the danger.\n\nConvinced that this threat is the Sunbane, Covenant persuades the Search to help him find the One Tree; and in _The One Tree_ , Covenant, Linden, Vain, and several _Haruchai_ set sail aboard the Giantship Starfare's Gem, leaving Sunder and Hollian to rally the people of the Land against the Clave.\n\nThe quest for the One Tree takes Covenant and Linden first to the land of the _Elohim_ , cryptic beings of pure Earthpower who appear to understand and perhaps control the destiny of the Earth. The _Elohim_ agree to reveal the location of the One Tree; but first they paralyze Covenant's mind, purportedly to protect the Earth from his growing power. Led now by Linden, the Search sails for the Isle of the One Tree.\n\nUnexpectedly, however, they are joined by Findail, an _Elohim_ who has been Appointed to guard against the consequences of the quest's actions.\n\nLinden is unable to free Covenant's mind without possessing him, which she fears to do, knowing that she may unleash his power. When she and her companions are imprisoned in _Bhrathairealm_ , however, she takes the risk of entering Covenant, much to Findail's dismay. Covenant then fights and masters a Sandgorgon, a fierce monster of the Great Desert. The creature's rampage through _Bhrathairealm_ allows Covenant, Linden, and their companions to escape.\n\nAt last, Starfare's Gem reaches the Isle of the One Tree, where one of the _Haruchai_ , Brinn, becomes the Tree's Guardian. But when the companions approach their goal, they learn that they have been misled by the Despiser. Covenant's attempt to obtain wood for a new Staff of Law begins to rouse the Worm of the World's End. Once awakened, the Worm will accomplish Lord Foul's release from Time.\n\nAt the cost of his life, Seadreamer makes Linden aware of the true danger. She then forestalls Covenant. Nevertheless the Worm's restlessness forces the Search to flee as the Isle sinks into the sea, taking the One Tree beyond reach.\n\nDefeated, the Search returns to the Land in _White Gold Wielder_. Covenant now believes that he must confront the Clave directly, quench the Banefire, and then battle the Despiser; and Linden is determined to aid him, in part because she loves him, and in part because she fears his unchecked wild magic.\n\nRejoined by Sunder, Hollian, and several _Haruchai_ , Covenant, Linden, and a few Giants eventually reach Revelstone, where they challenge the Clave. After a fierce struggle, the companions corner the Raver commanding the Clave. There Seadreamer's brother, Grimmand Honninscrave\u2014with the help of a Sandgorgon\u2014sacrifices his life in order to \"rend\" the Raver. As a result, the Sandgorgon gains scraps of the Raver's sentience. Then Covenant flings himself into the Banefire, using its dark theurgy to transform the venom in his veins. When he is done, the Sunbane remains, but its evil no longer grows.\n\nAfterward, Covenant and Linden, Sunder and Hollian, Vain and Findail, and two Giants turn toward Mount Thunder, where the Despiser now resides. Along the way, Hollian dies. But in Andelain, Caer-Caveral expends his own life to resurrect her by violating the Law of Life.\n\nGradually Linden realizes that Covenant does not mean to fight Lord Foul. That contest, Covenant believes, will unleash enough force to destroy Time. Afraid that he will surrender his ring, Linden prepares herself to possess him, although she now understands that possession is a great evil.\n\nYet when she and Covenant finally face Lord Foul, she is possessed herself by a Raver; and her efforts to win free leave her unwilling to interfere with Covenant. As she has feared, he does surrender his ring. But when the Despiser turns wild magic against Covenant, slaying his body, the altered venom is burned out of Covenant's spirit, and he becomes a being of pure wild magic, able to sustain the Arch despite Lord Foul's fury. As a result, the Despiser effectively defeats himself; and Covenant's ring falls to Linden.\n\nMeanwhile, she has come to understand Vain's purpose\u2014and Findail's Appointed role. Vain is pure structure: Findail, pure fluidity. Using Covenant's ring, Linden melds the two beings into a new Staff of Law. Then she reaches out with the restored power of Law to erase the Sunbane and begin the healing of the Land.\n\nWhen she is done, Linden returns to her own world, where she finds that Covenant is indeed dead. Yet she now holds his wedding ring. And when Dr. Berenford comes looking for her, she discovers that her time with Covenant and her own victories have transformed her. She is now able to face her old life in an entirely new way.\n\n\"The Last Chronicles of Thomas Covenant\"\n\nIn Book One, _The Runes of the Earth_ , ten years have passed for Linden Avery; and in that time, her life has changed. She has adopted a son, Jeremiah, now fifteen, who was horribly damaged by the Despiser, losing half of his right hand. He displays a peculiar genius: he is able to build astonishing structures out of such toys as Tinkertoys and Legos. But in every other way, he is entirely unreactive, trapped in dissociation. Nonetheless Linden is devoted to him, giving him all of her frustrated love for Thomas Covenant and the Land.\n\nIn addition, she has become the Chief Medical Officer of a psychiatric hospital, where Covenant's ex-wife, Joan, is now a patient. For a time, Joan's condition resembles a vegetative catatonia. But then she starts to punish herself, punching her temple incessantly in an apparent effort to bring about her own death. Only the restoration of her white gold wedding band calms her, although it does not altogether prevent her violence.\n\nAs the story begins, Roger Covenant has reached twenty-one, and has come to claim custody of his mother: a claim which Linden denies. To this setback, Roger responds by taking his mother at gunpoint. And while Linden deals with the aftermath of that attack, Roger captures Jeremiah as well.\n\nSeparately, Linden and the police locate Roger, Joan, and Jeremiah. But when Linden confronts Roger, Joan is killed by lightning, and Roger opens fire on the police. In the ensuing fusillade, Linden, Roger, and Jeremiah are cut down; and Linden finds herself in the Land again, where she is informed that Lord Foul now has her son.\n\nAs before, several thousand years have passed in the Land, and everything that Linden knew has changed. The Land has been healed, restored to its former loveliness and potency. Now, however, it is ruled by Masters, _Haruchai_ dedicated to the suppression of all magical knowledge and power. And their task is simplified by an eerie smog called Kevin's Dirt, which blocks health-sense.\n\nYet the Land faces threats which the Masters cannot defeat. _Caesures_ \u2014disruptions of time\u2014wreak havoc, appearing randomly as Joan releases blasts of wild magic. In addition, an _Elohim_ has visited the Land, warning of dangers which include various monsters\u2014and an unnamed _halfhand_. And the new Staff of Law has been lost.\n\nDesperate to locate and rescue Jeremiah, Linden soon acquires companions, both willing and reluctant: Anele, an ancient, Earthpowerful, and blind madman who claims that he is \"the hope of the Land,\" and whose insanity varies with the surfaces\u2014stone, dirt, grass\u2014on which he stands; Liand, a na\u00efve young villager; Stave, a Master who distrusts Linden and wishes to imprison Anele; a group of ur-viles, artificial creatures that formerly served Lord Foul; and a band of Ramen, the human servants of the Ranyhyn, the Land's time-wise horses. Linden also meets Esmer, the tormented descendant of Kastenessen, a deranged _Elohim_.\n\nFrom Esmer, Linden learns the nature of the _caesures_. She is told that the ur-viles intend to protect her from betrayal by Esmer himself. And she finds that Anele knows where the Staff of Law was lost thousands of years ago.\n\nBecause she has no power except Covenant's ring, which she is only able to use with great difficulty\u2014because she has no idea where Lord Foul has taken Jeremiah\u2014and because she fears that she will not be able to search for him against the opposition of the Masters\u2014Linden risks entering a _caesure_. Accompanied by Anele, Liand, Stave, the ur-viles, and three Ramen\u2014the Manethrall Mahrtiir and his two Cords, Bhapa and Pahni\u2014Linden rides into the temporal chaos of Joan's power.\n\nThanks to the theurgy of the ur-viles, and to the guidance of the Ranyhyn, she and her companions emerge more than three thousand years in their past, where they find that the Staff has been hidden and protected by a group of Waynhim, relatives of the ur-viles. When she reclaims the Staff, however, Esmer brings a horde of Demondim out of the Land's deep past to assail her. The Demondim are monstrous beings, the makers of the ur-viles and Waynhim, and their power now threatens the existence of the Arch of Time.\n\nTo protect the Arch while she and her companions escape, Linden uses Covenant's ring to create a _caesure_ of her own. That disruption of time carries her company and the Demondim to her natural present. To her surprise, however, her _caesure_ deposits them before the gates of Revelstone, the seat of the Masters. While the Masters fight a doomed battle against the Demondim, she and her companions enter the ambiguous sanctuary of Lord's Keep.\n\nIn Revelstone, Linden meets Handir, the Voice of the Masters: their leader. And she encounters the Humbled, Galt, Branl, and Clyme: three _Haruchai_ who have been maimed to resemble Thomas Covenant in his honor. Cared for by a mysterious woman named the Mahdoubt, Linden tries to imagine how she can persuade the Masters to aid her search for Jeremiah. When she confronts them, however, all of her arguments are turned aside. Only Stave elects to stand with her: an act of defiance for which he is punished and spurned by his kinsmen.\n\nThe confrontation ends abruptly when news comes that riders are approaching Revelstone. From the battlements, Linden sees four Masters racing to reach Lord's Keep ahead of the Demondim. With them are Thomas Covenant and Jeremiah. And Jeremiah has emerged enthusiastically from his dissociation.\n\nIn _Fatal Revenant_ , the arrival of Covenant and Jeremiah brings turmoil. They are obviously real and powerful, yet they give no satisfactory account of their presence. And they refuse to let Linden touch them\u2014or to use the Staff of Law. Instead they insist on being sequestered until they are ready to talk to her.\n\nMeanwhile the Demondim mass at the gates, apparently preparing to destroy Revelstone; but they do not attack.\n\nShaken, Linden retreats to the plateau above Lord's Keep to await Covenant's summons. There she calls for Esmer, hoping that he will answer her questions. When he manifests himself, however, he surprises her by bringing more creatures out of the Land's distant past: a band of ur-viles and a smaller number of Waynhim, joined together to serve her. Cryptically he informs her that the creatures have prepared \"manacles.\" And he reveals that the Demondim are now working in concert with Kastenessen. But he avoids Linden's other questions. Instead he tells her that she \"must be the first to drink of the EarthBlood.\"\n\nWhen Covenant's summons comes, Linden meets with him and Jeremiah in their chambers. Covenant speaks primarily in non sequiturs, although he insists that he knows how to save the Land. At the same time, Jeremiah pleads with Linden to trust his companion. Feeling both rejected and suspicious, Linden refuses when Covenant asks for his white gold ring. In response, Covenant demands that she join him on the plateau, where he will show her how he intends to save the Land.\n\nLinden complies. She knows no other way to discover why and how her loved ones have changed. Instead of revealing their secrets, however, Covenant and Jeremiah create a portal which snatches her away from her present. Without transition, she finds herself with them ten millennia in the Land's past, during the time of Berek Halfhand's last wars.\n\nThey are near the dire forest of Garroting Deep\u2014and far from the place and time that Covenant and Jeremiah sought. Instead their purpose has been deflected by a man called the Theomach, who appears to have a mystical relationship with time. He is one of the Insequent, a race of humans who pursue arcane knowledge and power in complete isolation from each other: a race whose only shared trait, apparently, is a loathing for the _Elohim_. The Theomach interfered with Covenant and Jeremiah because he believed that their intentions were dangerous enough to attract the _Elohim_.\n\nThe result is that Linden, Covenant, and Jeremiah stand in the dead of winter many brutal leagues from _Melenkurion_ Skyweir, where Covenant and Jeremiah hope to use the EarthBlood and the Power of Command to defeat Lord Foul permanently. In desperation, Linden decides to approach Berek for help. She wins the future High Lord's trust by healing many of his injured\u2014and by introducing him to his own new-born health-sense.\n\nAfterward the Theomach accomplishes his own purpose by persuading Berek to accept him as a guide and teacher. To show his good faith, he speaks the Seven Words: a mighty invocation of Earthpower which Linden has never heard before.\n\nWith supplies and horses provided by Berek Halfhand, Linden, Covenant, and Jeremiah trek toward _Melenkurion_ Skyweir. But when the exhausted mounts start to die, Covenant and Jeremiah transport Linden to the Skyweir through a series of spatial portals. There Jeremiah reveals the magic of his talent for constructs. With the right materials, he is able to devise \"doors\": doors from one place to another; doors that bypass time; doors between realities. Building a door shaped like a large wooden box, he conveys himself, Covenant, and Linden deep into _Melenkurion_ Skyweir, to the hidden caves of the EarthBlood.\n\nCovenant is now ready to exert the Power of Command. But Linden drinks first, remembering Esmer's counsel. She then uses her Command to expose the secrets of her companions.\n\nAt once, a glamour is dispelled. Covenant shows his true form: he is Roger Covenant, not Thomas, and he despises all that his father loves. His right hand wields immense power: it is Kastenessen's, grafted onto him to wield Kastenessen's savage might. And on Jeremiah's back rides one of the _croyel_ , a succubus that both feeds from and strengthens its host. The sentience that Jeremiah has demonstrated is the _croyel_ 's, not his own. Gloating, Roger explains that he and the _croyel_ aspire to become gods. Bringing Linden into the past\u2014and bringing her here\u2014was an attempt to trick her into breaking the Arch of Time. So far, she has avoided that danger. But now she is trapped ten thousand years in the Land's past and cannot escape.\n\nA terrible battle follows, during which the Staff of Law turns black. Using her Staff, the Seven Words, and the EarthBlood, Linden opposes Roger and her possessed son. While an earthquake splits _Melenkurion_ Skyweir, however, Roger and Jeremiah escape, leaving Linden stranded.\n\nThe experience transforms her. She now believes that only Thomas Covenant can save the Land. At the same time, her determination to rescue Jeremiah becomes even stronger\u2014and more unscrupulous.\n\nAfter an encounter with Caerroil Wildwood, the Forestal of Garroting Deep, who engraves her Staff with runes of power, she is retrieved by the Mahdoubt. Here the Mahdoubt is revealed as one of the Insequent. When Linden is returned to Revelstone in her proper time, she learns that Liand has acquired a piece of _orcrest_ , a stone capable of channeling Earthpower in various ways. She also hears that a stranger has single-handedly eliminated the entire horde of the Demondim.\n\nHe is the Harrow, yet another Insequent. He covets both Linden's Staff and Covenant's ring, and he has the power to take them by emptying her mind. Fortunately the Mahdoubt intervenes. Violating the ethics which govern the Insequent, she defeats the Harrow, winning from him the promise that he will not wrest the Staff of Law and Covenant's ring from Linden by force: a victory which costs the Mahdoubt her own life. After assuring Linden that he will gain his desires by other means, the Harrow disappears.\n\nThen Linden, her friends, and the three Humbled summon Ranyhyn and ride away from Revelstone. Because she still has no idea where Jeremiah is hidden, her stated intention is to reach Andelain and consult with the Dead, as Covenant once did. For private reasons, she also hopes to recover High Lord Loric's _krill_ , an eldritch dagger forged to channel extremes of power too great for any unaided mortal.\n\nAlong the way, she and her companions come upon a village which has been destroyed by a _caesure_ : a _caesure_ which Esmer controls as a weapon against the Harrow. There she learns that the Harrow knows where Jeremiah has been hidden\u2014and that Esmer intends to prevent the Insequent from revealing his secret. At the same time, Roger Covenant attacks with a force of Cavewights. In the ensuing battle, Linden's company is soon overwhelmed. Frantic, she takes a wild gamble: she tries to summon a Sandgorgon. Six of them charge into the fight, routing Roger and the Cavewights, and allowing the Harrow to escape.\n\nLater Linden hears that a large number of Sandgorgons have come to the Land, driven by the rent remnants of a Raver's malign spirit. In Covenant's name, they answered Linden's call. But now they have repaid their debt to him. They seek a new outlet for their own savage hungers, and for the Raver's malice.\n\nWhen Linden and her companions have done what they can for the homeless villagers, they ride on to Salva Gildenbourne, a great forest which encircles Andelain. There they encounter a party of Giants, Swordmainnir, all women except for one deranged man, Longwrath: their prisoner. When the Giants and Linden's company reach comparative safety, they stop to exchange tales.\n\nThe leader of the Giants, Rime Coldspray, the Ironhand, explains that Longwrath is a Swordmain who has been possessed by a _geas_. With nine other Swordmainnir, the Ironhand has been seeking the cause or purpose of his compulsion. After acquiring an apparently powerful sword, he has led the Giants to the Land. Here it becomes clear that his _geas_ requires him to kill Linden.\n\nTo protect her, the Swordmainnir agree to accompany her to Andelain. But during the next day, they are assailed by the _skurj_ , fiery worm-like monsters that serve Kastenessen. Two of the Giants are killed. Yet Liand saves the company by using his _orcrest_ to summon a thunderstorm. The downpour forces the _skurj_ underground, and the surviving companions are able to flee.\n\nAt last, they reach Andelain. The sacred Hills are warded by the Wraiths, small candle-flame sprites that repulse evil by drawing power from the awakened _krill_. Thus protected, the companions hasten to find the _krill_.\n\nDuring the dark of the moon, however, the company meets the Harrow again. Indirectly he has offered Linden a bargain: if she surrenders the Staff of Law and Covenant's ring, he will take her to Jeremiah. But while he taunts Linden, Infelice, the monarch of the _Elohim_ , appears. She argues passionately against the Harrow\u2014and against everything that Linden intends to do. Yet Linden ignores them both as she approaches the _krill_.\n\nThere the Dead begin to arrive. While the four original High Lords observe, Caer-Caveral and High Lord Elena escort Thomas Covenant's spectre. Yet the Lords and the last Forestal and Covenant himself refuse to speak. None of them answer Linden.\n\nDriven to the last extremity, she raises all of her power from both her Staff and Covenant's ring, and commits those contradictory magicks to the _krill_. Doing so, she cuts through the Laws of Life and Death until she succeeds at resurrecting Covenant; drawing his spirit out of the Arch of Time; restoring his slain body.\n\nYet power on such a scale has vast consequences. Linden's actions also awaken the Worm of the World's End.\n\nIn addition, there are other problems. In _Against All Things Ending_ , she finds that Covenant's leprosy is active again. And the stress of his return to mortality has fractured his mind. As a result, he is often unable to control his thoughts, his memories, or even his attention.\n\nWhen the companions are informed that days will pass before the World's End is accomplished, Linden decides to accept the Harrow's bargain: one last attempt to rescue Jeremiah by surrendering her Staff and Covenant's ring. Then, however, another Insequent appears, the Ardent, who has come to ensure that the Harrow does not deal falsely with Linden. Under pressure, the Harrow\u2014with the Ardent's support\u2014transports Linden, Covenant, and all of their companions to the Lost Deep, elaborate caverns which were once the home of the Viles, creators of the Demondim. There Linden follows the Harrow to Jeremiah's hiding place.\n\nThe boy is still ruled by the _croyel_ , and he has concealed himself even from the _Elohim_ within one of his constructs. When Linden breaks the construct, Roger arrives to murder the Harrow, and to claim Covenant's ring. But Covenant opposes Roger with the _krill_ , and while they struggle, Esmer suddenly takes Roger to safety. Esmer then announces that the company's actions have awakened an ancient bane, a sentient and eternal being called She Who Must Not Be Named. Now She is rising to devour the company.\n\nDesperately Linden and her companions try to flee, bringing succubus-ridden Jeremiah and a helpless Covenant, whose awareness of his circumstances has been blocked by Esmer. But while the company, led by ur-viles and Waynhim, scrambles to escape, Linden falls prey to the bane indirectly: believing herself to be invaded by worms and maggots, she collapses into unconsciousness. However, Covenant's love for her enables him to overcome Esmer's influence. With the aid of the Dead\u2014who sacrifice the spectre of Elena, Covenant's daughter\u2014he forestalls the bane until the Ardent is able to transport the company away.\n\nOn the Lower Land a considerable distance from Mount Thunder, the companions try to recover. All are exhausted, and Linden is trapped in nightmares, unable to regain consciousness. Dismayed by her condition, Covenant takes a desperate risk: he holds her underwater, hoping that the sensations of drowning will bring her back to herself. Fortunately his gamble succeeds.\n\nThe Ardent informs the company that he has caused his own death by interfering with the Harrow. Before he passes away, however, he supplies the company with food. Later he also transports the Ramen Cords, Pahni and Bhapa, to Revelstone so that they can try to win the support of the Masters.\n\nWhen the companions have regained some of their strength, Linden attempts to free Jeremiah from the _croyel_ by entering his mind: a graveyard where all of his thoughts and desires are buried. She fails; and before anyone can try a different approach, the company is attacked by _caesures_. In the confusion, Liand tries to use his _orcrest_ stone against the _croyel_ ; but Anele\u2014suddenly possessed by Kastenessen\u2014kills the young villager.\n\nA different attack soon follows: Roger and a host of Cavewights are joined by Esmer. Although the Swordmainnir and the _Haruchai_ fight furiously, they are vastly outnumbered. But ur-viles and Waynhim respond by rendering Esmer helpless with their manacles. Now sane, Anele challenges the _croyel_ with the _orcrest_. And when the Humbled Galt unexpectedly sacrifices himself to preserve Anele, Anele is able to destroy the succubus. The effort kills the old man, but not before he transfers his legacy of Earthpower to Jeremiah.\n\nTo save the company, Linden uses wild magic, wreaking terrible carnage even though she is not a rightful white gold wielder. In the aftermath, Esmer begs her to kill him. But she cannot: she has done too much killing. However, Stave\u2014Galt's father\u2014spares her by using the _krill_ to end Esmer's life.\n\nLater Covenant leaves the company, taking only the remaining Humbled, Clyme and Branl, with him. He intends to confront Joan in an effort to end her torment and stop the _caesures_ ; and he is unwilling to risk Linden against a rightful white gold wielder. Also he believes that Jeremiah still needs her. The boy is no longer possessed, but he remains buried in his long dissociation.\n\nStricken by Covenant's departure, Linden and her companions decide to let the Ranyhyn choose their destination. After an encounter with the lurker of the Sarangrave, a wetland monster with terrifying appetites and strengths, the great horses run, taking Linden, Jeremiah, and Stave ahead of the exhausted Giants and Manethrall Mahrtiir. After many leagues, the three reach a crater full of the bones of ancient monsters. There Jeremiah begins to build one of his constructs while Linden defends him from _caesures_. But then Infelice arrives, intending to kill the boy because she believes that he will devise a prison for the _Elohim_ \u2014and because she knows that Lord Foul wants to use Jeremiah's talent to imprison the Creator. However, Stave's strength of will enables him to distract Infelice; and with Linden's help, Jeremiah completes his construct. When he enters it, he emerges with his mind restored.\n\nMeanwhile Covenant and the Humbled travel toward Ridjeck Thome, where Covenant first defeated Lord Foul. Along the way, they encounter the Feroce, worshippers of the lurker, who offer Covenant a bargain: they will help him overcome Joan's defenders if he will try to preserve the lurker from the Worm. Knowing that he cannot protect the lurker except by saving the Earth, Covenant agrees.\n\nTrue to their word, the Feroce sacrifice many lives; but their aid does not suffice. In order to reach Joan, Covenant and the Humbled enter a _caesure_ : a doomed gamble from which Joan rescues Covenant so that she can kill him herself. And Joan is possessed by _turiya_ Raver, who casts Covenant adrift in his memories. But before Joan can summon a killing blast, Covenant draws on her wild magic to heal his mind. When she is distracted by the Ranyhyn, he uses the _krill_ to end her life.\n\nA tsunami caused by the Worm follows. It nearly claims Covenant, Clyme, and Branl. And when it passes\u2014when a new day begins\u2014the sun no longer rises. The world has fallen into perpetual twilight: the onset of the last dark.\n\n## 1.\n\nBetimes Some Wonder\n\nLinden Avery's fate may indeed have been written in water. It was certainly writ in tears. They blurred everything; redefined the foundations of her life.\n\nStanding in Muirwin Delenoth, resting place of abhorrence, with Jeremiah clasped in her arms, she felt emotions as extreme as the dismay which had followed Thomas Covenant's resurrection and the rousing of the Worm of the World's End; as paralyzing and uncontainable as the knowledge that she had doomed all of her loves. But there, in Andelain, the scale of her distress had seemed too great to be called despair. Here, in the company of bones and old death, her glad shock at Jeremiah's restoration was too great and complex to be joy.\n\nStave of the _Haruchai_ stood waiting with his arms folded, impassive as a man who had done nothing, and had never lost a son. Three Ranyhyn waited near him, watching Linden and Jeremiah with glory in their eyes. In the distant west, the sun drifted down shrouded in the hues of ash and dust, casting shadows like innominate auguries from the stone blades and plates which rimmed the hollow. Heaved aside by the deflagration of Jeremiah's construct, the skeletons of _quellvisks_ sprawled against the far slope of Muirwin Delenoth as if they sought to disavow their role in his redemption\u2014or as if they had drawn back in reverence.\n\nSuch things were the whole world, and the whole world waited. But Linden took no notice. She was unaware that she had dropped her Staff, or that Covenant's ring still hung on its chain around her neck, holding in its small circle the forged fate of all things. She regarded only Jeremiah, felt only him; knew only that he responded to her embrace. A miracle so vast\u2014\n\n_I did it, Mom_. For the first time in his life, he had spoken to her. _I made a door for my mind, and it_ opened.\n\nJoy was too small a word for her emotions. Happiness and gratitude and relief and even astonishment were trivial by comparison. A staggering confluence of valor and trust had restored her son. At that moment, she believed that if the Worm came for her now, or She Who Must Not Be Named, or even Lord Foul the Despiser, her only regret would be that she did not get to know who her son had become during his absence.\n\nSomehow he had weathered his excruciating dissociation. In graves he had endured what the Despiser and Roger Covenant and the _croyel_ had done to him.\n\nShe was murmuring his name without realizing it, trying to absorb the knowledge of him; trying to imprint his hug and his tangible legacy of Earthpower and his unmistakable awareness onto every neuron of her being. He was her adopted son. Physically she had known every inch of him for most of his life. But she had never met the underlying _him_ until this moment: until he had arisen from his absence and looked at her and spoken.\n\nThe way in which she repeated his name was weeping; but that, too, she did not realize. She was no more aware of her tears than she was of Stave and the Ranyhyn and passing time and the ancient ruin of bones. Holding Jeremiah in her arms\u2014and being held by him\u2014was enough.\n\nShe had no better name for what she felt than exaltation.\n\nYet the exaltation was Jeremiah's, not hers. He had become transcendent, numinous: an icon of transfiguration. He seemed to glow with warmth and health in her arms as if he had become the Staff of Law: not _her_ Staff, runed and ebony, transformed to blackness by her sins and failures, but rather the Staff of Law as it should have been, pure and beneficent, the Staff that Berek Halfhand had first created to serve the beauty of the Land.\n\nThe gift that Anele had given Jeremiah elevated him in ways that Linden could not define. He had not simply become responsive and aware. He appeared to dismiss the past ten years of his life as if they had no power over him.\n\nSuch things could not be dismissed.\n\n\"Chosen,\" Stave said as if he sought to call her back from an abyss. \"Linden Avery.\" An uncharacteristic timbre of pleading or regret ached in his voice. \"Will you not harken to me?\"\n\nShe was not ready to hear him. She did not want to step back from Jeremiah. He vindicated everything that she had done and endured in his name. If she withdrew from exaltation, she would be forced to think\u2014\n\nAnd every thought led to fear and contradiction; to dilemmas for which she was unprepared. No one could endure what her son had suffered without emotional damage; without scars and scarification. Yet she could not discern damage. In her embrace, he felt more than physically well. He seemed entirely _whole_ , mentally and spiritually intact.\n\nThat Linden could not believe. She knew better.\n\n\"Mom.\" Like hers, Jeremiah's voice wept gladly. \"Mom, stop crying. You're getting me all wet.\"\n\nFor his sake, she tried.\n\nLong ago under _Melenkurion_ Skyweir, she had forgotten the sensations of being a healer. Although she had cared for her companions in various ways, she had responded to their injuries as if her own actions were those of a stranger. But she had not forgotten what she had learned during her years in Berenford Memorial, tending the wounded souls of the abused and broken.\n\nTraining and experience had taught her that an escape from unreactive passivity was a vital step, crucial to everything that it enabled\u2014but it was only the _first_ step. When a crippled spirit found the courage to emerge from its defenses, it then had to face the horrors which had originally driven it into hiding. Otherwise deeper forms of healing could not occur.\n\nShe realized now that she was expecting a rush of agony from Jeremiah: the remembered anguish of every cruelty which the Despiser and Roger and the _croyel_ had inflicted. That prospect appalled her.\n\nBut when she considered her son clinically, she recognized that the outbreak which she dreaded was unlikely. Immediate firestorms of memory were rare. More commonly, a new form of dissociation intervened to protect the harmed mind while its new awareness was still fragile. Full recall came later\u2014if it came at all. Jeremiah felt whole to her because his worst recollections had not arisen from their graves.\n\nFor all she knew, they might remain buried indefinitely.\n\nWhy, then, was she afraid? Why did she contemplate anything except her son's restoration? Why could she not be content with miracles, as any other mother might have been?\n\nShe could not because Lord Foul's prophecies might still prove true, if the Despiser contrived to recapture Jeremiah\u2014\n\n\u2014or if events triggered more memories than he could withstand.\n\nShe had failed to resurrect Covenant without his leprosy. Other restorations might go awry. With or without Lord Foul's connivance, predatory pain lurked inside Jeremiah: she could not believe otherwise. Suffering as calamitous as his possession by the _croyel_ might overtake him without presage.\n\nFor that reason, she needed to remain alert in spite of her gladness. But she did not know where to begin trying to identify the truths buried beneath her son's presence.\n\n\"Chosen,\" Stave repeated more sharply. \"Linden Avery. I comprehend the force of your son's awakening, and of your reunion with him. Who will do so, if I do not? I, who have lost a son, and may only yearn bootlessly for his return to life? Nevertheless we cannot remain here.\n\n\"It appears that the Falls have ceased. Yet should the Unbeliever fail in his quest, they will surely return. And the wider perils of the world will not await the culmination of your release from sorrow. The last crisis of the Earth gathers against us. Also the Ranyhyn are restive. I deem that they are eager to rejoin our companions, and that they discern a need for haste.\"\n\nLong before Linden was ready to release him, Jeremiah withdrew. For a moment, he gazed at her with gleaming in his eyes like the stars on the foreheads of the Ranyhyn. Then he turned toward Stave.\n\nLinden was too full of other emotions to be surprised when Jeremiah reached out and hugged the _Haruchai_.\n\nAlthough Stave did not respond, he suffered the boy's clasp until Jeremiah let him go. But when Jeremiah stepped back, the former Master lifted his eyebrow as if he were mildly perplexed.\n\n\"You are much altered,\" he remarked. \"Is your condition such that you are able to remember Galt, who kept the fangs of the _croyel_ from your neck?\"\n\nJeremiah nodded. \"I remember. He's your son. He let himself be killed so Anele could get that monster off my back. So Anele could give me all this power.\"\n\n\u2014the hope of the Land.\n\nLinden watched the boy with a kind of awe. Some part of him must have remained conscious throughout the long years of his dissociation. Other aspects must have been evoked or informed by the _croyel_ 's use of him. Otherwise he would not have been able to emerge so swiftly\u2014or to know so much.\n\n\"Then,\" Stave said flatly, \"I am content that you are indeed restored.\"\n\nAs if in confirmation, the Ranyhyn tossed their heads, and Hynyn trumpeted an imperious acknowledgment. From among them, Khelen came forward and nudged Jeremiah, apparently urging the boy to mount.\n\nJeremiah, Linden tried to say; but she had no voice. She did not know where to begin. Too many aspects of her relationship with her son had taken on new meanings.\n\nBriefly the boy stroked the young stallion's muzzle: a small gesture of affection. Then he turned back to his mother.\n\n\"Mom.\" There were tears in his voice again, if not in his eyes. His grin fell away. With his halfhand, he pointed at the bullet hole over her heart. \"I'm sorry. I never wanted you to get shot. But I'm glad, too. I needed you so bad\u2014\" For a moment, the color of his gaze darkened, hinting at black depths of pain. \"I needed you to come after me. I was worse than dead.\"\n\nHis pajamas remained torn and stained. The horses ramping across the tops were almost indecipherable. And Liand's blood still soiled the tattered bottoms, in spite of Linden's efforts to wash them. She could barely remember that the fabric had once been sky-blue. It would never come clean.\n\nBut before she could reply, Jeremiah shook his head hard; blinked until his expression cleared. Gesturing around him, he snorted, \" _Quellvisks_. They were good for something after all.\"\n\nSomething which Lord Foul had not foreseen. In a sense, the boy had reincarnated himself from the old bones of monsters.\n\nOh, my son. Linden needed to stop weeping. Really, she could not go on like this. When Stave said her name again, his tone had become more peremptory. And he was right. They could not linger here without food or water or their companions. The wonder of her son's emergence from his portal was a small detail compared to the threat of the Worm. The world's end would not pause for any instance of mere human exaltation and relief.\n\n\"Say something, Mom,\" Jeremiah prodded. His tone suggested a teenager's impatience. \"Say anything. Tell me you heard Stave. He's right, we need to go.\" His next thought made him grin again. \"And I want to see the Giants' faces when they see me. They are not going to believe it.\"\n\nLinden tried to refuse. She wanted nothing except to concentrate on her son. Her thirst for the sound of his voice was acute. There was so much that she yearned to know about him. About what he had endured\u2014and how he had endured it. It did not matter where she began, as long as she could search for the truth.\n\nI never wanted you to get shot.\n\nBut there was something else\u2014Something in Stave's tone nagged at the edges of her health-sense.\n\nShe absolutely had to stop crying.\n\nWhen she rubbed at her eyes, the emptiness of her hands reminded her that she no longer held the Staff of Law.\n\nShe felt strangely reluctant to retrieve it. It represented responsibilities which were too great for her. Nevertheless she was capable now of many things that would have surpassed her less than an hour ago. She was still the same Linden Avery who had raged and failed and despaired; yet somehow she had also been made new. And watching over Jeremiah was a task to which she could commit herself without hesitation.\n\nTo meet that challenge, she might well need every conceivable resource.\n\nUnsteadily she stooped to reclaim her Staff.\n\nAs her fingers closed on the engraved blackness of the wood, another faint pang touched her nerves: an evanescent breath of approaching _wrongness_. Frowning, she raised her head to scent the air, extend her health-sense.\n\nThe atmosphere had a brittle taste, as if it were compounded of a substance that might shatter. She knew that the season was spring; but that fact seemed to have no meaning on the Lower Land. Hideous theurgies and slaughter had made a wasteland of the entire region. Muirwin Delenoth was as desiccated as its bones: it had been shaped by death.\n\n\"Mom?\" Jeremiah asked; but still she did not speak.\n\nDrawing warmth and sensitivity from her Staff, Linden considered the slopes of the hollow, the ragged plates around the rim. Then she lifted her attention to the declining sun and the tainted hue of the sky. The pall of ash and dust overhead was wrong in its own fashion: it was unnatural, imposed by some force beyond the reach of her senses. But it was not malice; not evil or deliberate. The almost imperceptible frisson of _wrongness_ rose from some other source.\n\n\"Stave\u2014?\" She had to swallow hard to clear her throat. \"Do you feel it?\"\n\nThe former Master's silence was answer enough.\n\nSlowly she turned in a circle, pushing her percipience to its limits. She expected the disturbance to come from the vicinity of Foul's Creche; from Covenant's search for Joan. But she felt nothing there. When she faced northwest, however, she found what she sought.\n\nIt was faint, almost too subtle to be discerned. Yet it was thin with distance, not weakness. The fact that she could detect it at all across so many leagues bespoke tremendous power. As soon as she tuned her nerves to the pitch of this specific malevolence\u2014and to the direction from which it spread\u2014she knew what it was.\n\nIt was Kevin's Dirt, and it came from Mount Thunder.\n\nFor the first time, Kastenessen was extending his bale over the Lower Land.\n\nRepeatedly he had tried to prevent Jeremiah's rescue from the _croyel_. Now he was sending the fug of Kevin's Dirt to hamper Linden and the Staff of Law. When it spread far enough, his theurgy would numb her senses, and Mahrtiir's, and perhaps Jeremiah's. And it would aggravate Covenant's leprosy. If Joan did not kill him first. With forces drawn from She Who Must Not Be Named, the mad _Elohim_ strove to ensure that Linden and her companions would not survive.\n\nA shudder like a chill ran through her. Her fingers clenched the Staff until her knuckles ached. Reflexively she confirmed that she still had Covenant's ring. An old comfort, it had steadied her for years, until he had refused her.\n\n_\u2014the last crisis of the Earth._\n\n\"I understand,\" she told Stave abruptly. \"We should go. Kevin's Dirt is coming. And maybe the _skurj_.\" Or Kastenessen might decide to challenge her himself now that he had lost Esmer. \"We need to find the Giants and Mahrtiir. Then we'll have to decide what we're going to do.\"\n\nWithout Covenant\u2014\n\nShe meant to mount Hyn and ride at once. But when she looked at her son again, she faltered. He seemed eager: too eager. Did she detect an undercurrent of alarm? If so, she suspected that he chafed to flee from his memories before they could emerge from their coverts and ravage him. He needed movement.\n\nStave waited for her impassively. Almost pleading, Linden asked him, \"Do we have to ride hard? I need to talk to Jeremiah. There's so much\u2014\" Her son had become someone she did not know. \"If the Ranyhyn run, I won't be able to hear him.\"\n\nA quirk at the corner of Stave's mouth may have implied a smile. \"Chosen,\" he answered, \"the great horses have demonstrated that they are well acquainted with our straits. Mayhap they will moderate their haste for your sake, and for your son's.\"\n\n\"Then let's _go_ ,\" urged Jeremiah. \"I can't wait to see the Giants. And Infelice gave me an idea. I want to try it.\"\n\nHe startled Linden. An idea? What could he possibly have gleaned from the interference of the _Elohim_? And how? Who had he become? Was he simply trying to pack down the earth that shielded him from his immured hurts? Or had he somehow learned strengths which she could not imagine?\n\nIf his instincts prompted him to seek safety by outrunning his wounds, surely she should trust him?\n\nPushing herself into motion, Linden turned toward Hyn.\n\nAt once, Stave came to help her mount. And when she was seated astride the familiar security of Hyn's back, he did the same for Jeremiah, boosting the boy effortlessly onto Khelen. Then he sprang for Hynyn.\n\nHynyn whinnied a command to the other horses. Together the three Ranyhyn flowed into motion so smoothly that Linden felt no need to cling. Urged by Jeremiah's shout of celebration, they accelerated at the slope of the caldera, pounding upward, flinging clots and plumes of dry dirt from their hooves. But once they had crested the rim, passed between the sandstone sentinels, and started down the long slope northward, they eased their pace to a light-footed canter. Their strides raised a low drum-roll from the baked ground; yet when Linden settled herself to Hyn's rhythm, she found that she would not need to shout in order to make herself heard.\n\nAhead of her, Kevin's Dirt expanded its maleficence by slow increments. Fortunately its peril was not exacerbated by _caesures_. Their absence troubled her on Covenant's behalf\u2014they might now be aimed at him as he approached Ridjeck Thome\u2014but it also reassured her. For the moment, at least, she, Jeremiah, and Stave were relatively safe.\n\nRelying on the former Master and the Ranyhyn to warn her at need, she turned her attention entirely on her son.\n\n\"Jeremiah?\" She resisted an impulse to raise her voice over the rattle of hooves. \"Can you hear me all right?\"\n\nHe flashed a grin at her. \"Sure, Mom. I've been listening to you my whole life. I could probably hear you if you whispered half a mile away.\"\n\nThat simple answer was enough to stun her for a moment. Covenant had assured her, _None of the love you lavished on your son was wasted_. _That isn't even possible_. All those years of speaking her love to Jeremiah without any response\u2014and yet he had heard her. More amazing still, he had believed her in spite of what the Despiser and his natural mother had done to him.\n\n_Until we know more about what's happened to him, just trust yourself._\n\nA fresh rush of emotion made her awkward. \"Then you've probably already figured out most of the questions I want to ask.\"\n\n\"Maybe.\" He cocked his head to one side, considering. \"Let's see.\n\n\"That _croyel_ \"\u2014he made a spitting noise\u2014\"used me to say all kinds of things. You want to know how many of them are true.\"\n\nLinden nodded mutely. Everything about Jeremiah seemed to have the power to astound her.\n\n\"Well,\" he continued slowly, \"a lot of them were. True, I mean.\" His voice held a note of caution, as if there were details that he wanted to avoid. \"Mom, you tried hard to take care of me. I know that. It wasn't your fault you couldn't reach me. I just hurt too much. But giving me those racetrack pieces was like a miracle. I don't know how you came up with the idea, but it was perfect.\n\n\"Using those bones\"\u2014he gestured behind him\u2014\"was the second time I managed to make a\u2014I don't know what else to call it\u2014a door for my mind. That racetrack was the first. I couldn't do anything with my body except build. I wanted to. I just couldn't. But with my mind\u2014\n\n\"Most of what the _croyel_ said about that was true. When I went through my door, I was here. I mean, not _here_.\" He indicated the arid landscape. \"I mean in the Land. In this world. But I was still just a mind. I was just kind of floating around. In one time or another. One place or another. I couldn't touch anything, or talk to anybody.\n\n\"But there were people that noticed me anyway. Powers. Beings. And if they noticed me, they could talk to me. The Vizard was one, like the _croyel_ said. He wanted to use me. The Viles once, but they weren't interested. I think I met a Demimage, but he couldn't figure out what I was. A couple of Ravers. They _wanted_ me.\" Jeremiah shuddered. \"A few _Elohim_ , but mostly they tried to convince me to go away and not come back.\" With a snort of derision, he added, \"Like that was going to happen. It was the only escape I had. I couldn't give it up.\"\n\n\"And Covenant?\" Linden asked carefully. \"Did the _croyel_ tell the truth about him?\"\n\n\"As much as that monster could stand,\" Jeremiah replied without hesitation. She heard gratitude in his voice, saw affection in the brown warmth of his eyes. \"I mean about the real Covenant. Not about Roger. The real Covenant talked to me more than all the rest put together.\n\n\"He talked like he actually cared about me.\"\n\nTreading as cautiously as she could, Linden probed for more. \"What did he say?\"\n\nThe boy grinned at her again. \"He told me I could count on you. Like I didn't know that already. If I needed you, you would do anything to help me, even if it was impossible. He said you have no idea how strong you really are. He said it makes you wonderful.\"\n\nWonderful\u2014? That idea stunned Linden once more. It closed her throat; almost brought her back to tears. For long, terrible days, she had been tormented by the fear that her son secretly belonged to the Despiser; that he had acquiesced to the _croyel_ ; that he had been forever marked and marred by Lord Foul's bonfire, Lord Foul's malice. Yet Covenant had spent years of Jeremiah's childhood telling him that his mother was wonderful. And Jeremiah had believed the Unbeliever. Even in his dissociation, he had recognized something in Linden that she herself could not see\u2014\n\nWhile she tried to master her emotions, Jeremiah looked away. Frowning with concentration, he scanned the beaten terrain. \"And he talked about the _Elohim_. I didn't really understand, but I think he was trying to explain why they're important. They're like a metaphor?\" He sounded uncertain. \"A symbol? They represent the stars. Or maybe they _are_ the stars. Or maybe the stars and the _Elohim_ are like shadows of each other. The shadows of the Creator's children.\"\n\nHe shrugged, flexing easily with the beat of Khelen's strides. \"He wanted me to get it, but it didn't make much sense.\"\n\nLinden, too, did not understand. But she did not care about the _Elohim_. At the moment, she cared only about the ineffable fact that Jeremiah was speaking to her; that her son had found his voice when he had recovered his mind. And he had recovered his will as well: oh, yes, his will beyond question. His years of self-protective absence had taught him unexpected resources of determination.\n\nThey encouraged her to keep him talking.\n\nShe avoided the most crucial issue because he avoided it. Instead she inquired further about his encounters with Covenant's spirit.\n\n\"I probably shouldn't admit this,\" she offered tentatively, \"but I almost panicked when I saw Revelstone and Mount Thunder in the living room. I came close to taking you and running.\" She still believed that she should have done so. \"Then neither of us would have been shot.\"\n\n\"And we wouldn't be here to fight for the Land,\" Jeremiah put in at once.\n\nShe conceded his point. She did not want to discuss the cost of trying to carry burdens which were too heavy for human arms to lift. \"Of course,\" she continued, \"I didn't know then that your mind was coming here at night, when I thought that you were asleep. But what I'm trying to ask is, what inspired you to build those models?\" And to build them on the same day that Roger Covenant came to demand custody of his mother? \"Was that Covenant's idea? Did he tell you to do it?\"\n\nJeremiah thought for a moment. \"Not exactly. He never _told_ me to do anything. But he made sure I knew Revelstone and Mount Thunder were important. He said things could happen there that might frustrate Lord Foul.\" Suddenly vehement, he snapped, \"I _hate_ that bastard.\" Then, hunching his shoulders and knotting his fists, he calmed himself. \"So I wanted to warn you. Legos were the only language I had.\"\n\nThe only language\u2014Such things threatened Linden's composure. But Jeremiah had touched on his unspoken wounds, albeit obliquely. That demanded her full attention. Her own reactions could wait.\n\nIn the dirt ahead of her, she saw the marks of three Ranyhyn galloping toward Muirwin Delenoth: longer strides, deeper hoof-cuts in the ground, but the same track. Clearly Hyn, Hynyn, and Khelen were retracing their path away from the Swordmainnir and Manethrall Mahrtiir. They aimed to rejoin Linden's companions instead of pursuing some other purpose.\n\nInstead of taking her to Covenant.\n\nShe told herself that she was glad. She wanted to be reunited with her friends. Wanted them, in effect, to meet Jeremiah for the first time. In addition, she needed their support, their comfort, their ready courage. And she felt that she could not afford to be distracted from her son: certainly not by her yearning for the only man whom she had ever truly loved.\n\nAs though he had caught the scent of her thoughts, Jeremiah asked abruptly, \"Do you think he's dead? Covenant, I mean. When he left, he looked like he was going to die. Like he planned on dying.\"\n\nStartled, Linden countered, \"Why do you think that? What made you think he was going to die?\"\n\nThe boy studied her. \"Isn't that what _you_ think? I must have picked up the idea from you.\"\n\nLinden winced. She could easily believe that her reaction to Covenant's departure had conveyed the impression that she was bracing herself for his death.\n\nWhile her son faced her with concern darkening in his eyes, she sighed, \"No, Jeremiah. I don't think Covenant is dead. And I don't think he was planning to die. You've met him, but you haven't seen him in action. Practically everything he does is almost inconceivable, but he does it anyway. That's why the Land needs him. Why we need him.\" Her own needs were more complex. \"Maybe he really does have an inherent relationship with wild magic. Or maybe he's just _more_ than anyone else I've ever met. Either way, I don't believe that Joan can kill him. There isn't enough of her left, and that Raver can't make her into something she isn't.\"\n\nAfter a moment, Linden forced herself to be honest. \"But I do think something is dying. If it isn't already dead.\" Every word was bitter to her. It was gall on her tongue. She said it, and the next one, and the next, because she wanted to be worthy of her son. \"That must be what you saw in me when he left. He doesn't love me anymore. Or he's afraid of me. I love him, but ever since the Ardent brought us out of the Lost Deep, I've been watching what Covenant and I had together die.\"\n\nJeremiah listened with an air of impatience; but he waited for her to finish. Then he said as if he were certain, \"You're wrong, Mom. I've heard him. He still loves you. Whatever he's doing, it isn't about not loving you. That's what made me think he's planning to die. He left the way he did because he isn't sure he'll ever see you again.\"\n\nHer son meant well: Linden knew that. He might even be right. Nevertheless she doubted him. Her awareness of the many ways in which she had failed ran too deep. After all, what had she done to enable Jeremiah's escape from his prison? Sure, she had resisted Infelice as much as she could. And she had extinguished Joan's _caesures_. But in the end, her only real contribution had been trust: trust in the Ranyhyn\u2014and in Esmer's reasons for restoring Jeremiah's racecar.\n\nShe could not believe in Covenant's love because she did not know how to make peace with herself.\n\nIn self-defense, she reverted to her earlier questions. \"We were talking about your models. You explained Revelstone and Mount Thunder. What about your Tinkertoy castle?\" She had seen its original in the Lost Deep. \"Were you trying to tell me something there, too? Was that another warning?\"\n\nHad Covenant nudged Jeremiah to prepare her in some fashion? If so, the effort had been wasted. It was too cryptic. Knowing nothing of the Lost Deep, she could not have interpreted her son's faery edifice.\n\nThis time, Jeremiah shook his head. \"I was just practicing. I only visited the Lost Deep once. I mean, on my own.\" Without Roger and the _croyel_. \"But while I was there, I saw what the Viles could do. I fell in love with that castle. Then later, when I started to get the idea I needed to warn you somehow, I didn't want to make a mistake. So I tried to copy the castle.\n\n\"I hadn't done anything like that before. Everything else I built I just sort of found. Even the racetrack. I don't know how to explain it. I didn't start out with an idea. The shapes came from whatever I was using. They all just _came_. But if I wanted to warn you, I had to choose the shapes for myself.\n\n\"The castle was my first try.\" Linden saw satisfaction in his mien: satisfaction\u2014and a new surge of eagerness. \"It was easier than I thought. Until then, I didn't know I can choose anything I want. Now I do. I just need the right pieces.\"\n\nNow, Linden thought. While he was eager. While he felt sure of himself.\n\nIt was probably too soon. In her former life, she would have waited longer; perhaps much longer. But her son had so little time. The Earth had so little.\n\nHer heart seemed to crowd her throat as she asked, \"What was it like, having the _croyel_ on your back? What did it do to you? What did Lord Foul do?\"\n\nAt once, Jeremiah's manner changed as if he had slammed a door. He jerked his face away. \"You know what it was like. I don't want to talk about it. I want to forget it ever happened.\"\n\nThen he nudged Khelen away from Hyn. To Stave, he called, \"Can we go faster? I want to reach the Giants.\"\n\n\"Chosen?\" Stave inquired. His tone implied no opinion.\n\nCursing to herself, Linden muttered, \"All right. They're probably worried about us.\"\n\nThe Swordmainnir had been left behind because they were too weary after their long struggles to run with the Ranyhyn. And Mahrtiir had stayed with them so that Narunal could guide them across the wide wilderland of the Spoiled Plains to rejoin the other horses.\n\nStave nodded. Briefly he stroked the side of Hynyn's neck.\n\nWith a whicker of command to Khelen and Hyn, the roan stallion gathered speed so fluidly that Linden could not discern the precise moment when he began to quicken his gait. He galloped slightly ahead of them, but they did not lose ground in spite of their smaller stature. Indeed, Hyn matched his pace with apparent ease. As she had done before, the mare cast the hard ground behind her as if she could equal Hynyn's thundering haste for hours or days.\n\nStave rode effortlessly, like a man who had become one with his mount. In Khelen's care, Jeremiah waved his arms and shouted encouragement. But Linden gripped her Staff and prayed that she had not driven her son to bury his wounds more deeply.\n\nThe quality of the light in the stained air told her that the sun was setting beyond the barrier of Landsdrop. In the distance ahead, still scores of leagues away, she felt the advance of Kevin's Dirt more strongly. After their fashion, the Ranyhyn were trying to outrace a doom for which she had no answer.\n\nLinden's relief and joy at her son's restoration would have been greater if she had not been so afraid for him.\n\n_In your pre_ s _ent state, Chosen, Desecration lies ahead of you_. _It does not crowd at your back._\n\nIt was entirely impossible that he had not been maimed in some way by Lord Foul's malice and the _croyel_ 's cruelty.\n\n## 2.\n\nNightfall\n\nThe sun set, casting darkness across the Spoiled Plains; shrouding everything except the sensory glower of Kevin's Dirt. But Kastenessen's oblique assault on Earthpower and Law was increasingly vivid to Linden's percipience. Soon it would begin to hamper her. Even Jeremiah's inherited theurgy might be tainted. And the resources of the Staff would be diminished.\n\nIn addition, Covenant's leprosy would worsen. He might go blind, or lose the use of his hands altogether. He might find it difficult to keep his balance because his feet were numb.\n\n_I need to be numb_ , he had insisted in Andelain. _It doesn't just make me who I am. It makes me who I_ can _be._\n\nLinden did not understand that. The way in which he defined himself as a leper was like his relationship with wild magic, inherent, inexplicable\u2014and too ambiguous to be measured.\n\nCrossing terrain that made her feel numbed herself, Linden clung to the flowing reassurance of Hyn's back and prayed that some good would come of this long gallop through the threatened night.\n\nFortunately no _caesures_ appeared. Joan's attention was focused elsewhere; or _turiya_ Raver's was. Nevertheless Linden felt a growing disquiet across the region, an almost subliminal sense of disturbance that seemed separate from Kevin's Dirt. At first, she thought that she was tasting a nameless discomfiture in Hyn, a new anxiety that affected only the Ranyhyn. Yet when she pushed her percipience farther, she found a sensation of restiveness in the ground under Hyn's hooves. The foundations of the Lower Land appeared to be bracing themselves for an impact which they might not be able to withstand.\n\nAcross the leagues, Jeremiah's mood had changed. His eagerness had become impatience, frustration. He rode low over Khelen's neck, apparently urging the Ranyhyn to greater speed as if he fled from ghouls\u2014or as if he were filled to bursting with an unspoken sense of purpose.\n\nStars sprinkled the firmament overhead: the only light on the Lower Land. Surely the moon would rise soon? Even a slim crescent would do more than the lorn stars to soften the dark. But there was no moon. In its absence, the stars seemed strangely closer, at once more distinct and more vulnerable, as if they were drawing near to witness the outcome of their long yearning.\n\n_The shadows of the Creator's children_ , for good or ill: come boon or bane. They glistened like weeping in the absolute black of the heavens.\n\nWith growing urgency, Linden tried to recognize some specific feature of the terrain. But she had not attended to her surroundings during the ride to Muirwin Delenoth. She did not know where she was, and could only guess where she was going.\n\nHyn's unfaltering strides spoke eloquently of trust. Linden heard them well enough. She knew what they meant. Nevertheless her anxieties harried her through the night. Galled by them, she traversed an unreadable landscape in darkness like the onset of a nightmare from which there could be no awakening.\n\nHow much time had passed? An hour since sunset? Surely no more than two? Nonetheless the star-strewn dark seemed complete, as if it were the last night of the world.\n\nAbruptly Hynyn uttered a loud neigh like a blare of triumph in the face of oncoming evils. And a moment later, the stallion was answered. From the distance ahead came a welcoming whinny. Linden thought that she recognized Narunal's call.\n\n\"There, Chosen,\" Stave announced over the pounding of hooves. \"Our companions await us where we last found water.\"\n\nThe Ranyhyn were running between low hillocks like mounds inadequately cloaked in scraps of grass. Vaguely Linden smelled water. But her attention was fixed elsewhere, straining to discern the presence of the Swordmainnir and the Manethrall.\n\n\"At _last_!\" Jeremiah shouted. Then he began to halloo as if he expected everyone who could hear him to know his voice.\n\nIn moments, the Ranyhyn slowed their strides. Panting heavily, they dropped from a gallop to a canter, then to a jolting trot. Sure of their footing, they angled down into a gully where a small stream ran southward. As it muttered along its crooked path, it caught glints from the stars, a spangling of slight reflections which seemed to confirm that the lost lights were indeed becoming more distinct.\n\nSilhouetted vague and fireless against the faint glisten of the water stood ten shapes that Linden knew instantly: eight Giants, Manethrall Mahrtiir, and Narunal.\n\nAt once, Rime Coldspray and her comrades raised a loud huzzah that startled the night, shivering in the air like a challenge to calamity. Jeremiah replied gladly, and all of the Ranyhyn whickered their approval. Only Mahrtiir voiced neither pleasure nor exultation. His reactions were more complex.\n\nAs Hynyn, Hyn, and Khelen halted, Frostheart Grueburn and Stormpast Galesend surged forward to lift Linden and Jeremiah from their mounts. On Hyn's back, Linden almost felt equal to the exuberant relief of the Swordmainnir; but when Grueburn set her on her feet, the Giants towered over her, dwarfing her with their open hearts as much as with their size. She had more in common with Mahrtiir. While the Ironhand, Onyx Stonemage, and Cirrus Kindwind greeted Stave with claps on his back and shoulders that buffeted him in spite of his strength, Linden walked on legs stiff with riding toward the Manethrall. When she reached him, she dropped her Staff so that she could hug him with both arms.\n\nTaken aback by her display of affection, he resisted momentarily. But then he returned her clasp. \"Ringthane,\" he breathed softly. \"Linden Avery. Though I trust the Ranyhyn in all things, I must acknowledge that I have been sorely afraid. Also I am much vexed that I was not permitted to stand at your side. I am diminished in my own estimation. I must remember that I am Ramen and human. I must not judge myself by the majesty of the Ranyhyn.\"\n\nAs if she were answering him, Linden murmured privately, \"Jeremiah saved himself. Now I don't know how to help him.\"\n\nLike Mahrtiir, she would never be equal to miracles. She had to learn how to serve them, as he did.\n\nBut the Manethrall appeared not to understand her. \"Help him?\" he asked in a voice as low as hers. \"His alteration is plain. He is transformed beyond all expectation or conception. What manner of aid does he require?\"\n\nJeremiah was already talking to the Giants, practically babbling in his eagerness to tell his story. But _caesures_ and Stave and Infelice and Linden and the Ranyhyn and his racecar and Anele's legacy and a construct of bone all tried to find words at the same time: they tripped over each other and fell and bounced back up like tumblers performing some implausible feat of dexterity. Laughing at his own happy incoherence, he repeated his verbal pratfalls until he occasionally achieved a complete sentence. And the Giants laughed with him, rapt and delighted.\n\nOnly Stave stood apart. His native dispassion did not waver. If he took note of Linden's exchange with Mahrtiir, he feigned otherwise.\n\nWhispering so that she would not weep again, Linden told Mahrtiir, \"He doesn't want to remember what he's been through. I can't think about anything else. No one suffers like that without being damaged.\"\n\nThe Manethrall stepped back to regard her with his bandaged gaze. Still softly, he replied, \"That I comprehend, Ringthane. Who would if I do not, I who have lost eyes and use in a cause which exceeds my best strength? But I will speak once again of trust. Hear his vitality and joy. Hear him well. Far more than his wounds have been restored to him, and to you. If a lifetime of your love has not already wrought some healing, it will do so when its time is ripe.\"\n\nLinden had no response. She recognized his effort to reassure her, but she was not comforted. Jeremiah was not her only concern: other anxieties were tightening around her. His emergence required her to shift how she thought of herself.\n\nShe had no idea what had happened to Thomas Covenant. League by league, Kevin's Dirt swelled closer, expanding the ambit of Kastenessen's wrath and pain. Her awareness of a visceral alarm in the earth was growing stronger. And the Worm of the World's End was at work. Where its power was concerned, she doubted nothing that Infelice had told her; nothing that she had heard from Anele.\n\nThe company's circumstances, and the Land's, implied an imperative need for action. Now that she had rejoined her friends, she felt the pressure of events mounting. Instinctively she believed that she and her companions had to make decisions and act on them. Now, while they still could.\n\nYet she restrained herself for the sake of her son's rambling tale; and also for the sake of the Giants, so that they could gauge him for themselves. Raising both of her hands, she bowed her thanks and respect to Mahrtiir in the Ramen fashion. Then she retrieved the Staff of Law and went to the stream to quench her thirst. The Giants still carried some portion of the Ardent's largesse. Surely she could afford to eat a meal and rest before she imposed her tension on her friends?\n\nYes, she could afford that\u2014but she could not do it. When Jeremiah had given his audience a fairly complete description of what had occurred during his rescue or escape, she went in a gust of compulsion to join Rime Coldspray and Frostheart Grueburn and the rest of the Swordmainnir.\n\n\"Have you felt it?\" she asked without preamble. \"Kevin's Dirt is coming this way. Kastenessen knows where we are, and he intends to hurt us if he can. At this rate, Mahrtiir and I will start to lose our health-sense sometime around dawn. Even Jeremiah may be affected. And Kevin's Dirt is going to limit what I can do with my Staff. I won't be able to fight the _skurj_. I may not even be able to fight the Sandgorgons.\n\n\"Can you feel it?\"\n\nOne by one, the Giants turned toward her. She could not make out their expressions by starlight; but her nerves felt their enjoyment of Jeremiah subside, replaced by more somber emotions. The last of their laughter faded into the night. Standing with their Ironhand, the Swordmainnir regarded Linden gravely.\n\n\"Linden Giantfriend,\" Coldspray replied with an air of formality, \"we have felt it. But it will not assail us until dawn, as you have observed. For that reason among others, it is not our immediate consideration.\n\n\"You have ridden long and long without food or rest or sufficient water. And Giant that I am, I confess that my weariness clings to me, though we have bathed as well as we are able, and have conserved our endurance. Will you not partake of our remaining food? Will you not sleep for a time? The trials of the morrow will not be made less by effort in darkness, when we are scarce able to discern where we set our feet.\"\n\nLinden shook her head. Fears coerced her: she did not know how to relent.\n\n\"And there's some kind of distress in the ground,\" she countered. \"Can you feel that, too? It's like the rock under this whole part of the Lower Land is afraid. The Worm must be getting close. What else can it mean?\n\n\"I don't regret anything that we've done since we lost Liand and Anele.\" Anything except Covenant's departure\u2014and his desire to distance himself from her. \"But we're running out of time. We need to decide what we're going to do, and then we need to do it.\"\n\nThe Ironhand regarded Linden for a moment, apparently searching for some clue to the turmoil which goaded her. Then the leader of the Swordmainnir said more gently, \"You reveal a welcome alteration, Linden Giantfriend\u2014as welcome as your son's restoration in mind and power. Heretofore you have given your concern chiefly to him, heedless of the Earth's doom.\n\n\"I do not fault you in this,\" she hastened to add. \"We are Giants and adore children. Nonetheless other matters also weigh upon us. Your readiness now to challenge the foes of Land and life lifts our spirits.\"\n\nBefore Linden could find an appropriate response, Coldspray continued, \"Yet your need for food and rest remains. Though you did not choose to be so, you are the rock on which we have anchored our own purposes. Since our first encounter in Salva Gildenbourne, we have claimed a place in your company at every turn of the winds and currents. This we have done because we see more in you than you see in yourself, and also because we seek to make amends for the follies which led to Lostson Longwrath's _geas_. We will be guided by your heart.\n\n\"Still I must urge you to contain your apprehension for this one night. Much has transpired. Much has been asked of you\u2014and much given in return.\" She nodded toward Jeremiah. \"You would be more or less than mortal if you did not require time to absorb the gift of your son's restoration. And if you do not eat and rest now, you will be less able to withstand the coming storms.\n\n\"We will have need of you, Linden Giantfriend. You must grant to yourself some measure of kindness.\"\n\nThe Ironhand's consideration seemed to dissolve a barrier in Linden; to weaken or transform it. Her desire for decisions was as much an expression of incomprehension as it was of urgency. There were too many things that she did not understand. Covenant. Jeremiah. Lord Foul's plans for her son. And the _Elohim_ , who could have done so much differently.\n\nIn bafflement, she nodded to Coldspray. \"I'm sure you're right. Jeremiah must be hungry. And I could use a bath.\" The Ranyhyn had withdrawn into the night as if they had satisfied their own purposes; as if now they were content to wait until she determined hers. \"Let's all get some rest. Maybe we'll be able to see what to do more clearly in the morning.\"\n\nThe Giants replied with murmurs of approval; and Jeremiah yawned unexpectedly. \"I'm not just hungry,\" he announced. \"I'm _sleepy_. I thought I was too excited to sleep, but maybe I'm not.\"\n\nLinden nodded again. \"All right.\" Feeling suddenly drained, she turned to Stave. \"Will you guide me? I want to wash, but I'm not sure that I can find my way.\"\n\nWithout hesitation, the _Haruchai_ took her arm and steered her into the darkness away from the company. Trusting his friendship and his certainty, she accompanied him downstream.\n\nBut she wanted more than a bath. She wanted to understand. Questions about Jeremiah led her to _quellvisks_ , and to the _Elohim_. When she and Stave had gone beyond earshot of the Giants and her son, she asked him quietly, \"Why do you think they did it?\"\n\n\"Linden?\" the former Master inquired with as much gentleness as his dispassion allowed.\n\n\"Why did the _Elohim_ leave those bones where the Ranyhyn could find them? If they're so afraid of Jeremiah? They can move through time. The Theomach told me that. So did Esmer. They could have known that Jeremiah would need those bones. And they had the whole Earth to choose from. Why did they pick the Lower Land?\"\n\nWhy did they make possible a fate that they abhorred and then try to prevent it?\n\nStave shrugged. \"Mayhap they did not foresee him.\" Then he added, \"Their belief that they are equal to all things deludes them. They cannot perceive their own misapprehensions. How otherwise did they fail to foresee that you would permit ur-Lord Covenant to retain his white gold ring when you had become the Sun-Sage? Their fear of the Unbeliever's power and resurrection blinded them to other paths.\"\n\nBy slow increments, Linden began to relax. Stave's answer sounded reasonable. If nothing else, it implied that comprehension was attainable.\n\nAs far as she was concerned, the _Elohim_ had been wrong about her from the first.\n\nBefore long, the _Haruchai_ brought her to a small pool among the mounded hillocks. It was too shallow to let her immerse herself, had no virtue to assoil her sins; but it offered her enough water to scrub at the worst of her dirt and doubt. When Stave had assured her that he would stand watch somewhere out of sight, he faded soundlessly into the night, and she was alone.\n\nKneeling among the stones and sand at the pool's edge, she placed the Staff of Law beside her; lowered her face into the cold tang of the water. As long as she could hold her breath, she dragged her fingers through her hair and rubbed hard at her scalp. After that, she unbuttoned and dropped her shirt, removed her boots and socks, took off her grass-marked jeans.\n\nAlone with the stars, she did what she could to remove the stains of sweat and strain and dust and blood from her skin. With cold clean water, she tried to scour the soilure from her thoughts. Then she tossed her clothes into the pool and beat them like a woman who wanted to pound away every reminder that she was vulnerable to despair.\n\nhen she returned\u2014sodden, dripping, and chilled\u2014to her friends, she had not been made new. Her many taints had been ground too deeply into her to be simply washed away. Her runed Staff remained darkest black. If she raised fire from the wood, her flames of Earthpower and Law would be black as well, indistinguishable from the world's night. And there was an ache of apprehension in the ground that did not allow her to forget that her company and the Land and all of life were in peril. Nevertheless she had begun to feel the need for rest. And she knew that she was hungry.\n\n\"You look better,\" Jeremiah pronounced. \"I know how you like being clean.\" Then he snorted a soft laugh. \"I mean, I can guess. You sure gave me enough baths.\"\n\nLinden answered by wrapping him in a long, wet hug. She had no other way to express what she felt.\n\nIn her absence, the Giants had set out a meal for her: cheese, dried fruit, a bit of stale bread and some cured meat. Embracing Jeremiah, her nerves assured her that he had already eaten. Now she felt a tide of drowsiness rising in him. While she held him, he stifled a yawn.\n\n\"Mom. You're shivering.\"\n\nCold and over-wrought nerves had that effect, in spite of the heat clinging to the Spoiled Plains.\n\n\"You're right.\" Reluctantly she released him. \"Low blood sugar. I must be hungrier than I thought. Why don't you find a place to lie down while I eat something?\" Smiling crookedly, she added, \"If you're still awake when I'm done, you can tell me a bedtime story. I want to hear more about your visits to the Land.\" She particularly wished to hear more about Jeremiah's encounters with Covenant. \"They're bound to be more interesting than 'Bomba the Jungle Boy.'\"\n\nHe grinned, apparently remembering the books that she had read to him in another life. \"But I don't want to sleep.\" He made a sweeping gesture that included Stave and the Giants. \"This is too exciting.\"\n\n\"And it will still be exciting in the morning,\" Linden admonished him gently.\n\n\"Well\u2014\" He glanced around the floor of the gully. \"Maybe if I get comfortable somewhere.\"\n\n\"You do that.\" Inexplicably she wanted to weep again; but she swallowed the impulse. \"I really should eat.\" With a conscious effort, she turned to the meal that Frostheart Grueburn had left for her on a flat sheet of stone.\n\nNight covered Grueburn's face, and Rime Coldspray's. Linden could not see their expressions, but she felt them grinning. As Jeremiah moved away, looking for a clear stretch of sand and dirt, Cabledarm remarked quietly, \"Here Linden Giantfriend reveals yet another of her many selves. She is not merely the Sun-Sage, the Chosen, the indomitable seeker and guardian of her son. She is also the mother who provides care.\"\n\nLinden might have protested, if she could have done so with the same light-hearted kindliness that filled Cabledarm's voice. Instead she began eating; and after her first bites of hard cheese and stale bread, she was preoccupied with hunger.\n\nMahrtiir responded on her behalf. \"Are you taken aback, large ones?\" he said with a gruff attempt at humor. \"If so, I must chastise your lack of discernment. That she is a mother is plain.\"\n\nHaving spoken, however, he seemed disconcerted by the quiet laughter that greeted his gibe. Instead of laughing himself, he said more stiffly, \"Some have journeyed hard and long. Others have walked when they were weary and heart-sore. I have merely ridden and rested. I will stand watch with the Ranyhyn. And perhaps Stave will consent to join me. I have heard young Jeremiah's tale of great events. I would hear how those events are interpreted by the long memories and acute judgments of the _Haruchai_.\"\n\nStave glanced at Linden, then gave the Manethrall a barely perceptible nod. Together they walked away along the stream until they found an easy ascent out of the erosion-cut. A moment later, they were gone into the night.\n\nStill eating, Linden waited for the questions of the Giants.\n\nBut they did not question her. As if by common consent, they made themselves comfortable, some sitting against the walls of the gully, others half reclining beside the stream. Then in muted voices they began to tell old tales, stories which they all obviously knew well. None of their narratives went far: the Swordmainnir interrupted each constantly, sometimes with reminders of other tales, more often with good-natured jests. Nevertheless their interjections and ripostes had a soothing effect on Linden. That such strong warriors could be playful even now evoked an irrational sensation of safety. Indirectly they made light of their many perils and foes; and by doing so, they enabled Linden to relax further.\n\nSurely she could afford to rest while Mahrtiir, Stave, and the Ranyhyn watched over her and Jeremiah, and the Swordmainnir were content to amuse themselves with tales and gibes?\n\nWhen she had eaten everything that Grueburn had set out for her, she went to the stream for a long drink. Briefly she scanned the watercourse until her health-sense confirmed that Jeremiah was already asleep, sprawled unselfconsciously no more than a dozen steps away. Then she began to search for a place where she, too, could lie down.\n\nThe dampness and chill of her clothes were only vaguely unpleasant. She could have warmed them with her Staff, but she disliked the prospect of raising black fire here. It felt like a bad omen. And it might attract hazardous attention.\n\nRecumbent on the sand with only a few rocks to discomfit her, Linden rode the current of low Giantish voices as if it were a tide that lifted her into the worlds of dreams.\n\nThey were many and confusing, fraught with cryptic auguries and possible havoc. Muirwin Delenoth. An unleashed avalanche of water in the depths of Gravin Threndor. Resurrections. She Who Must Not Be Named. But one vision had more power over her than the others. In it, she and Jeremiah sat together in the living room that she would never see again, he on the floor surrounded by boxes of Legos, she in an armchair watching him. He was building an image of Mount Thunder in elaborate detail; and she loved watching him, as she had always done. The best part of the dream, however, was that he talked while he worked, happily explaining why he had chosen that image, what it meant to him, and how he had become so familiar with it, all in words which made perfect sense to her\u2014and which were forgotten as soon as they were uttered.\n\nOnce during the night, she was awakened by the visceral realization that a distant crisis had passed. Its aftershocks began to fade as soon as she became aware of them. Reassured by the knowledge that at least one cataclysm had kept its distance and run its course, she went back to sleep easily.\n\nShe yearned to return to Jeremiah and Legos, but that dream was gone. Instead, between one instant of consciousness and another, a hand touched her shoulder, and a low voice said her name. She recognized Stave before she knew that she was no longer asleep.\n\n\"Chosen,\" he said, still quietly, \"dawn draws nigh. Though the disturbance in the Earth has subsided, the Giants surmise that it is but the first of many. Indeed, they deem that some alteration has come to the Land. Having rested, they judge that it is now time to arise.\"\n\nIn an instant, Linden was fully awake. Jeremiah was stirring, roused by Stormpast Galesend. Like Stave, Manethrall Mahrtiir had returned. He conferred in whispers with the Ironhand, perhaps sharing any impressions that he had received from the Ranyhyn, while the other Swordmainnir secured their armor, checked their weapons, tied the scant remnant of their supplies into bundles.\n\nA low breeze drifted along the gully, touching Linden's nerves with an insidious sensation of change, not in the weather, but in something more fundamental, something in the nature of the air itself. The shift was not _wrongness_ or malice, yet it seemed to imply that it could be as destructive as evil.\n\nGripping Stave's arm and the Staff of Law, she climbed to her feet. \"Has anything happened? I mean, anything specific? Are the Ranyhyn worried?\"\n\nWith his usual detachment, Stave reported, \"The great horses appear restive. They snort at the air and toss their heads without any cause that I am able to discern. Nor do the Giants perceive any source of peril. Nonetheless\u2014\" He hesitated as if he were searching for contact with other _Haruchai_ minds; with memories which were beyond his reach. Then he continued, \"I share the apprehensions of the Swordmainnir. Some dire alteration approaches. We do well to meet it standing.\"\n\nA moment later, he added, \"It is in my heart that the Unbeliever has confronted his former mate, for good or ill.\" A hint of discomfort in his voice made him sound more formal. \"He has quelled her, or she has slain him. But the import of either outcome lies beyond my ken. Do such events conduce to the Earth's salvation or to its damnation? It is said that there is hope in contradiction, yet that insight surpasses me. I am _Haruchai_ , accustomed to clear sight or none.\n\n\"At your side, Chosen, I have made a study of uncertainty. Now I have learned that it is an abyss, no less unfathomable than the Lost Deep.\"\n\n\"Don't say that,\" Linden protested. She meant, Don't remind me that Covenant may be dead. We need him. _I_ need him. \"You understand more than you give yourself credit for.\"\n\nWithout uncertainty\u2014without hope in contradiction\u2014Stave would not have become her friend. He would not have stood with her against the united rejection of the Masters.\n\nStave appeared to raise an eyebrow. \"Where is the harm? Have I not made my allegiance plain? And did we not escape both the Lost Deep and the bane, though _skest_ and the _skurj_ also assailed us? Chosen, I do not fear to name uncertainty an abyss.\"\n\nLinden could have retorted, Sure, we escaped. After that bane nearly killed us. After we lost the Harrow, and the Ardent damned himself, and Covenant's hands were almost destroyed. After the Dead sacrificed Elena before I could ask her to forgive me. Don't you understand how _deep_ those wounds are? But she kept her bitterness to herself. All of her protests came to the same thing.\n\nShe had no hope for Covenant.\n\nInstead of responding, she left Stave and went to the stream. There she dropped her Staff, knelt, and plunged her face into the water, pulling her fingers through her hair while the cold stung her nerves.\n\nCovenant had asked or ordered her not to touch him. He had spoken as if he believed that she feared his leprosy\u2014or he feared it for her.\n\nThe Giants and now Mahrtiir conveyed the impression that they were waiting for her. When she glanced at the northwestern sky, she saw Kevin's Dirt glowering closer, riding the wind of Kastenessen's agony and virulence. In another hour at most, it would spread far enough to cover the company. Yet it remained hidden from mundane sight. It did not dull the stars. Indeed, it appeared to sharpen their brilliance and loss.\n\nLinden wiped water from her face, dragged her tangled hair back behind her ears, and rose to her feet. When she had retrieved her Staff, she moved to greet Jeremiah.\n\n\"Mom.\" She could not read his face except with her health-sense, but he sounded implausibly cheerful. \"Did you get some sleep? I sure did.\" He stretched his arms, rolled his head to loosen his neck. \"Now I feel like I can conquer the world.\"\n\nAs if he were performing a parlor trick, he snapped his fingers, and a quick spark appeared in the air above his hand; a brief instant of flame. In itself, it was a small thing, almost trivial. But it implied\u2014\n\nHe was already learning new uses for Anele's gift of Earthpower. Perhaps he was _becoming_ Earthpower.\n\nHis momentary display caught the attention of the Giants; but he ignored them to concentrate on Linden. \"What are we waiting for?\" he asked in a tone of rising excitement. \"We should go.\"\n\nInfelice had given him an idea\u2014\n\nHis manner troubled Linden. Instinctively she wanted to probe him again. She hungered to learn who he was in his new life. But she did not know what might happen if she interrupted his mood; his sense of purpose; his defenses. He might need such things more than he needed her understanding or sympathy.\n\nStave still stood nearby, a silent reminder of stoicism and rectitude. But he was more than that: he was also a reminder of trust. In the Hall of Gifts, she had confessed, _Roger said that Lord Foul has owned my son for a long time_. And Stave had replied, _I know naught of these matters_. _I do not know your son_. _Nor do I know all that he has suffered_. _But it is not so among the children of the_ Haruchai. _They are born to strength, and it is their birthright to remain who they are_.\n\n_Are you certain that the same may not be said of your son?_\n\nIf Linden asked him now, Stave might remark that Jeremiah had already proven himself in Muirwin Delenoth. The former Master might suggest that it would be better for her as well as for Jeremiah if she allowed him to discover his own path.\n\nShe was not ready for that. But the World's End would not wait for her to find enough courage. And when the Worm came, Jeremiah would share the Earth's fate no matter how hard she tried to save him.\n\nShe was responsible for the Worm's awakening. Now she needed to find better answers than the ones that had guided her here.\n\nSighing, Linden followed Jeremiah toward the Giants and the Manethrall. Sunrise would lift the darkness from the Lower Land. Perhaps it would shed some light into her as well.\n\nWhen she reached Mahrtiir, she said quietly, \"Kevin's Dirt is almost here. I hope that you'll let me know when it starts to blind you. I'll counteract it as much as I can. I don't like the way the air feels. We're going to need all the discernment we can get.\"\n\nThe Manethrall nodded. \"Ringthane, I hear you. I cannot evade the approach of Kastenessen's malevolence.\" Bitterness whetted the edges of his voice. \"It will make of me less than naught, a mere hindrance to my companions, as it did in the Lost Deep. Be assured that I will not scruple to seek your aid.\"\n\nThe promise appeared to cost him an effort of will or self-abnegation; but he spoke firmly, denying his pride.\n\nLinden rested her hand on his shoulder for a moment: a gesture of empathy to which he did not respond. Then she sighed, \"All right. We have a lot to talk about. Maybe it's time that we actually talked about it.\"\n\nBut she did not want to talk. She wanted to wait for the sun.\n\n\"Like you, Linden Giantfriend,\" Rime Coldspray offered, \"we mislike the touch of this air. It speaks of forces which lie beyond our ken. Perils draw nigh which have heretofore remained distant.\n\n\"Also the beings and powers which seek the World's End remain unopposed. I am the Ironhand of the Swordmainnir. I speak for my comrades when I say that we must now choose a new heading. And we must not dally in doing so, lest forces which we cannot oppose overtake us.\"\n\nLinden felt more than saw that night was ending. She smelled an easing of the dark. The first faint suggestion of daybreak drifted toward her from the east, riding the troubled breeze. But it did not dim the stars. Like the swift moil of Kevin's Dirt, the approach of dawn seemed to etch the profuse glitter overhead more precisely against the fathomless abyss of the heavens.\n\nStill she wanted to see the sun. With her Staff, she was capable of much. At need, the ready wood would answer her call with fire and heat and even healing. But she could no longer summon illumination. Jeremiah might be able to do so, if his mastery of his new magicks continued to grow. Covenant's ring would cast silver and peril in all directions if she forced herself to use it. But the stark ebony of her own access to Earthpower and Law precluded light.\n\nWhen the sun rose, the confused tangle of who she was and who she needed to be might begin to unravel like the recursive wards which had sealed the Lost Deep.\n\nStalling, she said uncertainly, \"We've been trusting the Ranyhyn. They've brought us this far. Maybe we should keep doing that.\"\n\nBut Manethrall Mahrtiir shook his head. \"Ringthane, they are Ranyhyn.\" She heard a note of finality or fatality in his voice. \"They wield neither ancient lore nor mighty theurgies. They have borne many of our burdens. Doubtless they will bear more. But they cannot determine the Earth's doom. The deeds required of us they cannot perform.\n\n\"Also,\" he added more sadly, \"I sense no clear purpose among them. They are restive, truly, and urgent to do what they may. But they neither command nor encourage us to ride. Rather they abide their discomfort, hoping\u2014or so I deem\u2014that we will soon determine our own intents.\"\n\nNow, Linden thought. Now the sun would show itself. Surely the east had begun to lighten? Certainly the funereal bindings of night had loosened their grip on the landscape. A kind of vagueness eroded the dark. In hints, the contours of the watercourse and the stream unveiled themselves. She could make out the Giants more clearly, starker shapes in the enshrouding gloom.\n\n\"That's all right, Mom,\" Jeremiah put in, impatient for his chance to speak. \"Like I told you, Infelice gave me an idea. I want to try it.\"\n\nLinden avoided his gaze. \"Can you wait a little longer, Jeremiah, honey? Just until sunrise?\"\n\n\"But\u2014\" he began, then stopped himself. Turning to the east, he frowned at the blurred outlines of the horizon. \"It should already be here. Why isn't it here?\"\n\nKevin's Dirt was less than a league away, a cruel seethe spurred southward by rage. Night continued to fade from the Lower Land, giving way to a preternatural dusk, an imposed twilight. Nevertheless there was no clear daybreak, no sign of the sun.\n\n\"This is wrong,\" Linden breathed. \"Something is wrong.\"\n\n\"Indeed,\" muttered Onyx Stonemage through her teeth. \"Something comes. I know not what it may presage, but my heart speaks to me of dread.\"\n\nThe stars shone like distant cries. Somehow Kevin's Dirt and even the swell of gloaming made them brighter, louder. A change had come to the firmament of the heavens, a change that threatened the isolate gleams. A change that caused them pain.\n\nNow? Linden thought. Now? Her sensitivity to organic truth assured her that the sun should appear _now_ ; that it should already have crested the crepuscular horizon. The absolute necessity of night and day required it, the life-giving sequence of rest and energy, relief and effort. The most fundamental implication of the Law of Time\u2014\n\nShe was wrong. There was no sun. There would be no sun.\n\nThe nature of existence had become unreliable.\n\nThe dusk softened until she could discern the faces around her indistinctly; until she could almost see the details of their grimaces and fears, their clenched expectations. But then the greying of the world seemed to stabilize as though it had found a point of equilibrium between night and day. After that, there was no increase of light.\n\nThe sun was not going to rise because it could not. Forces beyond Linden's comprehension held the Land in a gloom like the onset of the last dark.\n\nWhile Linden struggled to grasp the truth, several of the Giants gasped. Sharply Stave said, \"Attend, Chosen.\"\n\nShe flicked a glance around her, saw that all of her companions were staring upward.\n\nFor an instant or two, a few heartbeats, startlement confused her. The sky was too full of stars; of lights that glittered like wailing. She could not understand the panoply. She felt the leading edge of Kevin's Dirt, tasted the shock and horror of her companions, recognized a jolt of vehemence from Jeremiah; but she did not see what her companions saw.\n\nThen she did.\n\nOh, God\u2014\n\nStars were going out.\n\nOne. Then another. A pause while realities reeled. Two together as if they had been swallowed simultaneously.\n\nGod in Heaven! The sun was not the only casualty. And the Worm of the World's End had not yet reached the Land.\n\nThe stars were vast in number, of course they were: numberless beyond counting. By the measure of their profusion, their losses were small; almost trivial. But by the measure of brief human lives\u2014by any measure that included life and death\u2014the scale of the carnage surpassed conception.\n\nWhat kind of power could eat _stars_?\n\nWho could hope to stand against it?\n\n\"Mom!\" Jeremiah said urgently. \"You need to listen. I've been waiting long enough.\"\n\nShe could not hear him; could not drag her gaze down to meet his. She was transfixed by the incremental ruin of beauty. She had to watch it because there was no sun.\n\n\"Maybe it's a good thing I waited.\" Jeremiah's voice was taut with restraint. \"Maybe now you'll understand why my idea is important. Maybe now _I_ understand what Covenant was trying to tell me.\" But then he could not hold back a yell. \" _Mom!_ \"\n\nHis shout dragged at her attention. \"Jeremiah\u2014\" His name caught in her throat. Hoarse as a woman who had spent the night howling, she asked, \"What is it, honey? What's so important?\"\n\nDon't you see it? The stars are going out!\n\n\"You need to _listen_ ,\" he repeated. \"I know what to do!\"\n\nStave regarded the boy steadily. The former Master's gaze seemed full of the deaths of stars. Mahrtiir continued to peer blindly upward, but he appeared to be tracking the progress of Kevin's Dirt. Perhaps the stars were beyond the reach of his remaining senses.\n\nSlowly the Giants forced themselves to lower their heads. Blinking as though they had been appalled, they turned their eyes on Jeremiah. None of them spoke. Rigid as women who had become stone, they were too full of horror to express it.\n\nWithout stars, every sailor on the seas of the world would be lost. Every Giant aboard a ship, every seafarer from all the peoples of the Earth: trackless and doomed.\n\n\"All right.\" Jeremiah sounded incongruously satisfied and eager, as if the heavens held nothing fearsome. Nothing except an opportunity. \"I have an idea. I said that already. Infelice gave it to me. I mean, I got it from her. I'm sure she didn't mean what I heard.\"\n\nFortunately Kevin's Dirt had no immediate effect: it wrought its particular harm slowly. With her health-sense if not with her eyes, Linden watched her son. He no longer looked like a boy. He looked like a young man who did not need her.\n\nThe sight made her heart shiver as if she were feverish.\n\n\"You'll have to start from the beginning, Jeremiah. I don't know what you're talking about.\"\n\n\"You _do_ , Mom,\" he replied without hesitation. \"You were there. You just haven't thought about it enough.\n\n\"The stars going out.\" His assurance amazed Linden. It frightened her. \"That's the Worm. It's eating the _Elohim_.\"\n\nToo stricken to speak, everyone stared at Jeremiah. Beneath his familiar fierceness, Mahrtiir's visage betrayed an ashen dismay. The muscles of Rime Coldspray's jaws knotted and released like the hard beat of her heart. Latebirth had covered her eyes with her hands. Frostheart Grueburn gaped like a woman who had forgotten the meaning of her actions.\n\nEvery Giant\u2014\n\n\"So what are they afraid of?\" Jeremiah asked. \"I mean, the _Elohim_. I'm just a kid. Why are they scared of me? What do they think I can do that's worse than being _eaten_?\"\n\n_His purpose for us is an abomination, more so than our doom in the maw of the Worm._\n\n\"Infelice told us,\" he answered himself. \"She thinks I'm going to _trap_ them. And she knows I can do it. I can make a door they can't refuse. No matter how far they scatter, or how hard they try to hide. They can't refuse. That's part of who they are. They'll have to come if I make a door. I mean, the _right_ door. The right size and shape. The right materials. I can construct a doorway that _forces_ them. They'll have to pass through it.\n\n\"So of course she thinks I'll make a door they can't get out of.\" \u2014 _the Worm is mere extinction_. \"That's what the Vizard wanted. It's what she would do if she were me.\" _The prison which the boy will devise is eternal helplessness, fully cognizant and forever futile_. \"She thinks I'll trap the _Elohim_ forever.\"\n\nCaught in such a construct, Infelice and her people would _out-live the ending of suns and stars_.\n\nStave regarded Jeremiah without expression. Several of the Swordmainnir studied him as if he were changing in front of them, revealing unguessed aspects of horror or hope.\n\n\"But she doesn't know me, Mom.\" Jeremiah sounded almost smug. \"She doesn't know what I've been learning all these years.\n\n\"I'm not crazy like the Harrow. I know I can't build anything big or strong enough to hold the Worm. But I can make a door that sucks the _Elohim_ in. A door that takes them to a place where the Worm can't get at them. Only it won't be a prison because my door will let them leave whenever they want. I can keep them alive until they decide it's safe to come out.\n\n\"Then the stars will stop dying. And we'll have a better chance to stop the Worm.\"\n\nHe was moving too quickly for Linden. She scrambled to catch up with him; to untangle the significance of what he was saying. What had he told her about the _Elohim_? _They're like a metaphor?_ _A symbol?_ _They represent the stars_. _Or maybe they_ are _the stars_. _Or maybe the stars and the_ Elohim _are like shadows of each other_.\n\nThe idea made a weird kind of sense. Saving the _Elohim_ might actually stop\u2014or at least delay\u2014the destruction of the stars.\n\nStill Linden faltered. _His purpose for us is an abomination, more so than our doom in the maw of the Worm. But it is not the worst evil._\n\nInfelice believed that Lord Foul would eventually use Jeremiah to trap the Creator. Would that outcome be more or less likely if Linden's son contrived to preserve some of the _Elohim_?\n\nSuch questions were beyond her. She could not imagine their answers. She could hardly believe that they had answers.\n\nShe required an act of will to avoid looking up at the slow ravage of the heavens.\n\n\"I am exceeded,\" muttered Mahrtiir under his breath. \"Here even a youth of newborn mind surpasses a Manethrall of the Ramen. Serving only the Ranyhyn, my people are too small to comprehend or equal such powers.\"\n\nWhen no one else found a response, Linden asked tentatively, \"But Jeremiah, honey, what will that accomplish? We can't stop the Worm. We just can't. It's too much for us.\"\n\n\"But I can _slow it down_!\" Jeremiah crowed. \"If I can build my door before it eats too many _Elohim_ , I can buy us time!\" With exaggerated patience, he explained, \"The _Elohim_ are its natural food. If it doesn't get enough to eat, it'll be weaker. It'll move more slowly.\n\n\"Then who knows?\" He shrugged as though he knew nothing of uncertainty. \"Maybe we'll think of something. Or Covenant will. He's like that.\"\n\nIf Covenant were still alive. If he had survived his encounter with Joan and _turiya_ Raver. And if the Worm did not swallow Jeremiah's door whole. _By the measure of mountains, it is a small thing, no more than a range of hills_. It would dwarf anything that Jeremiah could build.\n\nAnd still the Worm would get all of the nourishment that it needed from the EarthBlood under _Melenkurion_ Skyweir. Anele had said as much. He had gleaned his knowledge from a stretch of veined malachite at the foot of the Hazard: stone lined with stains like Linden's jeans.\n\nThe prospect of acting on Jeremiah's desires scared her. She drew inferences from it that appalled her. If he did what he wanted to do, she would have to\u2014\n\nThat thought she could not complete. It led her toward places which were too extreme to be contemplated.\n\nThe construct which he envisioned would be vulnerable. It would need protection. She would have to\u2014\n\nAgainst the _Worm_? She had never had that kind of strength. No one with her did. Perhaps even Covenant did not.\n\nShe would have to\u2014\n\nHow could she make such choices? How could any mother put her son at risk and not stand ready to defend him?\n\nAnd yet\u2014\n\nHe was not the sum of her responsibilities. She had brought about the deaths of _Elohim_ and stars. Liand, Anele, and Galt. Even Esmer. All of Lord Foul's victims. She had awakened the Worm: she bore the burden of a world's ruin.\n\nHolding up a hand to ward off Jeremiah's eagerness, she said, \"I'm sorry, honey.\" She could not meet his hot gaze. \"I need to think about this. It puts a lot of pressure on you, and we can't be sure what the results will be.\" What materials would his construct require? And where in this blighted landscape could such things be found? \"I want to talk to Rime Coldspray.\" She already knew what Stave and Mahrtiir would say. \"Then I'll decide.\"\n\n\"Mom!\" he protested. But almost at once he bit down on his frustration. Sounding truculent, he muttered, \"Talk as much as you want. It won't change anything. I'm sure I'm right.\"\n\nLinden glanced at Stave, asking him with her eyes to watch over her son. Then she raised her head to the Ironhand. \"Do you mind if we talk alone?\"\n\nColdspray acquiesced with a shrug. Her jaws continued to bunch arrhythmically, chewing prayers or curses, as she walked away along the stream.\n\nConsumed by her own prayers, Linden followed.\n\nThey did not go far. Linden halted when Coldspray did, still within sight of their companions. Arms folded across her cataphract, the Ironhand stood rigid, waiting for Linden to speak.\n\nLinden understood her attitude: she read it in the lines of Coldspray's visage, the set of her shoulders. The Ironhand was not reluctant to talk to Linden. Instead she was shaken to the core by the sight of stars dying; by the sheer scale of what was being lost.\n\n\"Here's my problem,\" Linden began. Reluctance and doubt made her brusque. \"I don't know what to think of Jeremiah. He's my _son_. Seeing him like this is like seeing a new dawn. But I don't know what's happening to him\u2014or in him. After what he's been through, I don't understand how he can be so eager. It doesn't seem natural.\n\n\"Mahrtiir thinks that I should trust him.\" _Far more than his wounds have been restored to him, and to you_. \"That's hard for me. Where I come from, people who have been outrageously damaged don't suddenly become whole. I know that I haven't said much about my former life.\" She had been shot through the heart. Where she had been born\u2014where she belonged\u2014she had no life left. \"But back then, I was a doctor. A healer.\" Such assertions felt false to her now. She claimed them only so that Coldspray would understand her. \"I specialized in trying to help people with broken minds. And I never saw any of them recover completely without facing what happened to them. Not once.\n\n\"I'm afraid for him, Coldspray. I'm afraid of what might happen to him if he can do what he has in mind. I'm afraid of what might happen if he can't.\"\n\nEither outcome might enable Lord Foul to claim him.\n\nBrusque herself, Coldspray asked, \"Is your health-sense now dulled?\"\n\nLinden shook her head. \"Kevin's Dirt works slowly. It hasn't had time to affect me yet.\"\n\n\"Then I cannot counsel you as you wish to be counseled. Your son is closed to my discernment, as you are. Your perceptions exceed any that I am able to proffer.\"\n\nMore softly, the Ironhand admitted, \"Yet I am able to conceive of no course more worthy of our hearts and lives than his. What greater deed can we attempt, few as we are, and friendless in this gloom? For that reason alone, I would follow him wheresoever his eagerness leads. But there is more.\n\n\"Linden Giantfriend, my spirit is wracked by the deaths of stars. In their name, my counsel is young Jeremiah's. We must do what lies within our strength to preserve the _Elohim_.\"\n\nBefore Linden could respond, Coldspray continued, \"Nevertheless your son's purpose is perilous.\" Her tone tightened. \"Indeed, its hazards are extreme. Should he succeed in his intent, he will draw every surviving _Elohim_ to him. Doing so, he will also draw the Worm. They are its food. It will seek them out. Therefore his portal, his door, will require defense. It will require a defense greater than eight Swordmainnir, or eight score, or eight hundred can provide.\n\n\"For this reason, the choice must be yours. You alone among us wield true power.\" Sternly she concluded, \"Knowing the plight of the heavens, you will not turn aside.\"\n\nPerilous, Linden thought. Oh, Jeremiah! The same concern had occurred to her, although she had not gauged its implications so concretely. She dreaded what it might require of her.\n\nWithout realizing that she had lifted her eyes, she found herself staring skyward, transfixed by the calamity overhead. A gloom like bereavement covered the Lower Land. For all she knew, it covered the whole world. It would never be relieved.\n\nThen she realized that Rime Coldspray was right. She would not turn aside. She could not.\n\nNevertheless the Giants clearly did not grasp all that Jeremiah's desires entailed. They were dangerous, yes; but there was more. They meant that Linden would have to leave him. Abandon him to his peril. So that she could find a way to ward his construct when it was complete. In spite of her Staff and Covenant's ring, she was too weak. She would have to go looking for greater power.\n\nIf such power existed anywhere, and could be found.\n\nIf Covenant did not return\u2014\n\nShe saw no consolation in the gradual reaving of the stars. The heavens were an abyss of uncertainty. Stave did not fear such things. She did. She would have met a kinder fate in the maw of She Who Must Not Be Named.\n\nFinally she forced herself to meet Rime Coldspray's gaze.\n\nBecause she could not bear to say what she was thinking, she murmured, \"I would feel better about it if you were laughing. It's going to be hard.\" Earlier she had felt that the foundations of her life were shifting. Now they were being shattered. \"We don't just have to find whatever it is that Jeremiah needs to make his door. And we don't just have to protect him. Somehow we have to live through it.\"\n\nIn response, Coldspray managed a wan chuckle. \"Then I must concede that I have failed you. If joy is in the ears that hear, I have grown deaf. My hearing is whelmed by the clamor of an unrisen sun, and by the shrieking of slain stars.\"\n\n\"Don't worry about it,\" Linden answered as if she, too, were dying. \"That makes two of us. I'm so deaf, I keep forgetting to be glad that my son is alive and eager.\n\n\"Come on.\" She gestured toward the waiting company. \"Let's go find out what Jeremiah needs to save the _Elohim_.\"\n\nThe Ironhand nodded. \"Well said, Linden Giantfriend.\" Now she made no effort to force a laugh. \"Let us confront the challenge of these times together. While we do what we can, there is no fault in failure.\"\n\nConfront the challenge, Linden mused as she and Coldspray began walking. What choice did they have? But if they succeeded in any fashion, they would not do so together. Eventually she would have to face her fears. And she would have to face them alone.\n\nHer yearning for Covenant was so acute that it brought tears to her eyes.\n\nJeremiah seemed to swim through the blurring of her vision as he came to meet her. \"Well, Mom?\" he asked before she could say anything. \"What did you decide?\"\n\nInstead of replying, she wrapped her arms around him and hugged him hard, mutely pleading for his forgiveness. Then she took him with her to rejoin the rest of their companions.\n\nStave regarded her return impassively, as if his resolve sufficed for both of them. But the Giants and Mahrtiir were more troubled. Grueburn, Cirrus Kindwind, and the others studied Linden with doubt in their eyes. Perhaps they worried that her desire for Covenant ruled her; that she would insist on waiting for him. But the Manethrall's disturbance was of another kind. His sense of his own uselessness galled him like an unhealed wound. In the risk that Jeremiah wanted to take, Mahrtiir would be able to contribute nothing except his service to the Ranyhyn. He would have been better content if the loss of his eyes had killed him.\n\nLinden paused as though she wanted to be sure that all of her friends were paying attention. But in truth she was searching herself for courage, and trying to blink away her tears. She had always been vulnerable to the kind of paralysis that came from fear. From fear and despair.\n\n\"All right,\" she finally managed to say. \"I'm willing to do this your way, Jeremiah. What do you need to make your door?\"\n\nShe suspected that it could not be formed of bone. Bones implied mortality, and the _Elohim_ did not die. They could only be devoured. Or sacrificed.\n\nJeremiah's instant enthusiasm seemed to fill the gully from wall to wall. Indeed, it seemed to urge the stars closer so that they could hear him. Nevertheless his eagerness made him appear strangely fragile to his mother. What would happen to him if his intentions failed? Or if the Worm simply ate his door after he had gathered all of the _Elohim_ in one place? How would he bear it?\n\n\"Stone,\" he replied at once. \"A lot of it. In big chunks. I mean, really big. I won't be able to handle some of them, even with Earthpower.\" He flashed a glance around the Swordmainnir. \"I'm going to need all the help you can give me.\"\n\n\"Forsooth,\" Rime Coldspray responded in a noncommittal rumble. \"If aid you require, aid you shall have. But of stone the Earth is a vast storehouse. Even this parched wasteland is rich in forms and substances and textures and indeed purities of stone. Surely, young Jeremiah, the portal which you propose cannot be composed of random fragments. Even the theurgies of stonework practiced by Giants demand rock of particular natures and qualities. We must ask you to name the stone which you deem needful.\"\n\nAgain Jeremiah did not hesitate. Where his constructs were concerned, he seemed incapable of doubt. \"It's green. More like a deposit than actual rock. I don't know what it's called, but I saw some when you took me across the Hazard. Green like veins.\"\n\n\"Malachite,\" Onyx Stonemage pronounced; and Linden's stomach tightened as if the word were a prophecy.\n\nJeremiah nodded. \"That's it. But there it was just veins. I need plenty of it. It doesn't have to be pure. As long as there's malachite in the stone, I can use it.\" After a flicker of thought, he added, \"But if it isn't pure, I'll need more of it. I have to get the right amount. The less pure it is, the bigger the door has to be.\"\n\n\"Sadly,\" Cabledarm put in before Linden or the Ironhand could speak, \"we have seen no malachite since our escape from the Lost Deep. We are Giants, certain of stone. Our course in these last days has encountered no malachite.\"\n\nNow Jeremiah faltered. \"But you must\u2014\" he began, then stopped. After a moment, he admitted, \"I didn't see anything like it myself.\" His enthusiasm was crumbling. \"The _croyel_ controlled me, but it didn't control what I saw.\"\n\nCaught in his emotions, Linden tried to help him. \"Stave? The Masters scouted the whole Land. Did they find anything that resembled malachite around here?\"\n\nThe _Haruchai_ shook his head. \"We are not Giants. Seeking signs of peril, we observe in a different fashion.\"\n\nJeremiah's consternation dominated the dusk. It demanded answers.\n\nLinden faced him with disappointment in her eyes. \"Jeremiah, honey. I'm sorry. I don't know what else we\u2014\"\n\nHe cut her off. Ferocity flared in him as if he had suddenly become someone else: a creature of savagery and suspicion. His hands curled into claws. \"That's what you wanted to talk to Coldspray about,\" he snarled. \"You wanted to be sure I couldn't get what I need before you said yes.\"\n\nHis transformation shocked Linden. Suffering had done this to him, _this_. But she was not prepared for it. While she reeled inwardly, she could not respond.\n\nAround her, the Giants recoiled, as startled as she was, and full of disapproval. But Manethrall Mahrtiir's reaction was immediate anger. \"It is _not_ , boy,\" he snapped. \"There is no particle of her which does not desire your well-being\u2014aye, and the continuance of the Land. You speak now with the voice of the _croyel_ , and will _be silent_.\"\n\nSurprise stopped Jeremiah. For an instant, his vehemence faltered.\n\nAt once, Mahrtiir continued, \"Behold!\" With one arm, he flung a vehement gesture down the length of the watercourse.\n\nAs if by a flourish of magic, he dispelled Jeremiah's indignation. Instantaneously thrilled, Jeremiah wheeled to gaze where the Manethrall pointed.\n\nThe Ranyhyn were coming, four majestic horses bright with purpose. Prancing like pride made flesh, Hynyn led Hyn, Khelen, and Narunal along the stream toward the company.\n\n\"Their restiveness is answered,\" said the Manethrall. His tone was grim, but softer and more respectful, moderated by devotion. \"Their uncertainty was ours. We have now determined our need. Thus their path is made plain.\n\n\"Mount,\" he urged Linden and Stave. Jeremiah was already running toward Khelen, unable to contain his eagerness. \"Ride and hasten. The Ranyhyn have announced their will. Did they not discover bone when bone was needed? They will do as much for malachite. But we must not delay, lest the last _Elohim_ be consumed ere we are able to attempt their preservation.\"\n\n\"Aye,\" Rime Coldspray assented. She and her comrades made a visible effort to set aside their discomfiture. \"Make ready, Swordmainnir,\" she instructed. \"We cannot estimate the leagues which lie ahead of us, but we must traverse them swiftly.\"\n\n\"Yet again,\" grumbled Frostheart Grueburn. \"Must we run interminably?\" Nevertheless she did not dally as she tightened her armor and checked her sword.\n\n\"These great beasts,\" the Ironhand replied sternly, \"have given aid when we had no other. If they crave haste, they will learn that Giants comprehend its import.\"\n\nJeremiah had swarmed onto Khelen's back. Now he waved his arms like demands at the company. Hyn approached Linden, nudged her shoulder. For a moment, however, Linden did not react. Her heart was burning down to ash in her chest, and she did not know how to move.\n\nShe was sure now that Jeremiah's eagerness was his way of fleeing.\n\nWithout waiting for her consent, Stave boosted Linden astride the dappled mare. At the same time, Mahrtiir appeared to flow into his seat on Narunal. Mere heartbeats later, Stave mounted Hynyn; and the Giants announced their readiness.\n\nWith Khelen and Jeremiah in the lead, the company crossed out of the gully toward the northeast; toward the marge-land between the Shattered Hills and Sarangrave Flat.\n\nFollowing her son, and surrounded by Giants, Linden wept again. She had been given her first glimpse of Jeremiah's immured pain. She knew now that he needed her\u2014and that she was going to abandon him anyway.\n\nThat choice had been made for her. Acting on it would be worse.\n\n## 3.\n\nNot Dead to Life and Use\n\nBarely able to hold himself upright, Thomas Covenant stood on the cooled flow of Hotash Slay at the headland or boundary of the promontory where Foul's Creche had once ruled the southeast. Beyond him and against the cliffs on either side, wild seas thrashed in the aftermath of the tsunami. He heard their turmoil, a thunderous seethe and crash like the frantic labor of the ocean's heart. But through the surly dusk of a dawnless day, he could hardly see the eruption and spray and retreat of the lashed waves. There was no sun. Distinct as murders, the stars were going out.\n\nThis was a consequence of the Worm's rousing, as it was of his resurrection. It heralded the world's ruin. Now every death pierced him. Joan's end felt like a knife in his own chest. Killing her, he had wounded himself\u2014\n\nHe needed Linden. He did not know how to bear what he had become without her.\n\nBut he could not reach her. She was too far away\u2014and he was too badly injured. A shard of stone at the edge of the Shattered Hills had restored the old gash on his forehead: an accusation confirmed during his confrontation with Joan. Blood still oozed into the drying crust around his eyes and down his cheeks. Falling on rocks and coral had gashed his ribs badly. Some of them were cracked or broken. Splinters of pain gouged every breath. His jeans and T-shirt had been shredded. A lattice-work of torn flesh and more blood marked his arms and chest and legs.\n\nThe _krill_ 's heat must have burned his hands; his foreshortened fingers. But that damage, at least, he did not feel. Leprosy disguised his lesser hurts.\n\nBy comparison, the Humbled were almost whole. They, too, had been struck by scraps of flung rock. A cut marred the side of Branl's neck. Clyme's arms and tunic showed rents, contusions, small wounds. But they had not shared Covenant's floundering on the seabed, or felt Joan's blow. And they were _Haruchai_. They would be able to go on.\n\nNow they appeared to be watching for some sign that the doomed sun would rise, or that the incremental extinction of the stars would cease. But perhaps they were waiting for the Ranyhyn. If they permitted themselves anything as human as prayer, they may have been praying that Mhornym and Naybahn had survived the tsunami.\n\nWithout mounts, there was nothing further that Covenant or the maimed Masters could do to defend the Land. The Shattered Hills were an indurated barricade thronging with _skest_ , masterless and unpredictable. And the distance between him and Linden was impossible; scores of leagues\u2014\n\nHis need for her was just one more wound that could not be healed.\n\nThe gloom lightened until it resembled mid-evening or the last paling before sunrise. But it grew no brighter. All of the illumination seemed to descend from the precise and imperiled stars. It was their lament.\n\nThe Worm was coming\u2014and Covenant had no idea what to do. The light of the _krill_ 's gem had gone out. There was no wild magic left in him. Simply staying on his feet required every shred of his remaining strength. He bore Joan's ring in the name of an unattainable dream.\n\nOh, he needed Linden. He needed to make things right with her before the end.\n\nSuch yearnings were as doomed as the stars. The _Elohim_ had no hope of escaping the Worm's vast hunger.\n\nTime may have passed, but he did not notice it. He did not notice that he was still bleeding. The stab of abused ribs when he breathed insisted that he was alive; but he ignored it. He did not think about anything except Joan and stars and Linden.\n\nLong ago, he had promised that he would do no more killing. Now he was forsworn, as he had been in so many other ways.\n\nEventually Branl spoke. \"Ur-Lord, we cannot remain as we are.\" Faithful as a grave, he carried Loric's _krill_ clad in the remnants of Anele's apparel. \"We will forfeit our lives to no purpose. If the _skest_ do not assail us, privation and your wounds will bring death. We must delay no longer.\n\n\"If the Worm's advance may be measured by the fate of the stars, some few days will pass ere all time and life are extinguished. While they endure, a reunion with your companions\u2014and with the Staff of Law\u2014may yet be achieved. For that reason, we must abandon Naybahn and Mhornym. We must concede that they have perished. In their place, we must summon other Ranyhyn.\"\n\nAfter a pause\u2014a moment of hesitation?\u2014he added, \"And you must consent to ride. We cannot hope for your healing, except by the succor of the Staff.\"\n\nCovenant meant to say, No. He meant to say, Never. He could not break more promises. But those words eluded him. Instead his knees folded, and he sank to the stone. Some other part of him croaked, \"Here's another fine mess you've gotten me into.\"\n\nHe did not realize that he had spoken aloud until he tried to laugh. His chest hurt too much for laughter.\n\n\"Unbeliever?\" Undercurrents of anger fretted Clyme's tone. He and Branl had followed Covenant into a _caesure_. They had saved him when he was lost. \"Do you accuse us? These straits are not of our making.\"\n\nFor a while, Covenant could not imagine what Clyme was talking about. Then he managed to say, \"Oh, you.\" He dismissed the notion. \"I didn't mean you.\" Perhaps he should have laid the blame at the feet of the Creator; but he did not. \"I meant Foamfollower. This is all his fault.\n\n\"If he hadn't insisted on keeping me alive. Making impossible things possible. Laughing in the Despiser's face. He was always the Pure One, even if he didn't think so himself. None of us would be here without him.\"\n\nEven the Worm would not. Covenant would have died decades or millennia before Linden first met him.\n\nTime was a M\u00f6bius strip. Every implication looped back on itself. Every _if_ led to a _then_ which in turn redefined the _if_. But his human mind could not comprehend causality and sequence in such terms.\n\nThe Humbled regarded him as if he were babbling. Their faces kept secrets. _Try to believe that you are pure_. Who had said that to him? Like his heart, his mind was failing. He could not remember. Then he could. It was one of the _jheherrin_ ; one of the creatures who had aided him after he had denied their prayer for salvation.\n\n\"Ur-Lord,\" Branl said finally. \"Your hurts undermine your thoughts. Saltheart Foamfollower cannot be held to account for Corruption's deeds.\"\n\nBaffled by the simplification of such reasoning, Covenant tried to shake his head. Instead the twilight seemed to waver as if it were dissolving; as if reality itself were in flux. \"That's not the point.\" The point was that the _Haruchai_ had no sense of humor. \"The point is, I'm not going to ride the Ranyhyn.\" Foamfollower would not have known how to laugh if he had not been so open and honest in his grief. \"I made a promise.\" A vow. \"Promises are important. You know that at least as well as I do.\"\n\n\"We do,\" Clyme acknowledged. \"We are the Humbled, avowed to your service. We comprehend given oaths. Yet yours contradicts ours. If you do not ride, your death becomes certain. This we will not permit while choice remains to us.\"\n\nThey had entered a _caesure_ for Covenant's sake.\n\n\"Do you not comprehend the extremity of your straits? Weakened as you are, your oath cannot hold. Soon you will lapse from consciousness. Then we will summon the Ranyhyn and bear you away. This you can do naught to prevent. Where, then, is the harm in granting your consent?\n\n\"Did you not permit Mhornym and Naybahn to retrieve you from the path of the tsunami? Did their aid not violate your word?\"\n\nYou don't understand. Covenant was too weak for this argument. He could not explain himself to the Humbled. Clyme and Branl had carried him; the Ranyhyn had not. The horses had only helped the Masters help him.\n\nIn various ways, the Ranyhyn had always aided him\u2014but they did so because he did not ride.\n\nHe needed Linden. If nothing else, he had to ask her forgiveness. Express his love. Confess his sins. How else would he ever be able to put his ex-wife behind him? Nevertheless he could not face her like this. Not at the price of another broken promise.\n\nHolding out his halfhand, he murmured, \"Give me the _krill_.\"\n\nThe Humbled looked uncertain in the preternatural twilight. Branl may have lifted an eyebrow. Clyme may have frowned. But apparently they could think of no reason to refuse. After a moment, Branl placed Loric's dagger in Covenant's grasp.\n\nTrembling as though his burdens were too heavy for him, Covenant dropped the old cloth: Anele's last legacy. He did not need it now. The _krill_ was cold. Briefly he steadied the forged metal, peered at the inert gem. Then he reached up to pull the chain that bore Joan's ring over his head.\n\n\"You know why the light went out. Joan was the only rightful white gold wielder here. The only one with a ring that belonged to her. The _krill_ 's power died when she did.\n\n\"But I still have a claim on her ring. I married her with it.\"'Til death do us part. \"And I'm something more.\" He had become so in the inferno of the Banefire, and in the apotheosis of his death by wild magic at Lord Foul's hands. \"I'm white gold.\" How else had he been able to transmute Joan's power, using it to heal his mind\u2014and to refuse _turiya_ Raver's malice? \"Mhoram said so. Maybe I'm not the rightful wielder of _this_ ring, but I can still use it.\"\n\nShaking, he pushed Joan's ring on its chain onto the little finger of his left hand. It stuck at the remaining knuckle, but he did not try to force it. He did not intend to wear it long.\n\nWith as much care as he could muster, he closed both hands around the haft of the _krill_. Then, suddenly desperate, he stabbed the blade at the stone under him.\n\nThe dagger was only sharp when it was vivified by the possibilities of wild magic. Lightless, it was dull. It could not pierce cooled lava.\n\nBut it did. As he struck, the scale of his need and the fundamental strictures of his nature brought forth a familiar blaze from the gem: familiar and absolute, as necessary as breath and blood. It shone into his eyes like the nova of a distant star. The power-whetted blade cut inward as though the stone were damp mud.\n\nWhen he took his hands away, his fingers and palms felt no heat: the numbed skin of his cheeks felt none. Nevertheless he trusted the efficacy of wild magic; believed that the _krill_ was already growing hot.\n\nBlinking through dazzles, he squinted at Clyme and Branl. At first, they were bright with phosphorescence, as spectral as the Dead. Then they seemed to reacquire their mortality. But they were not diminished. Rather they looked as precise and cryptic as icons in the dagger's brilliance. Together they confronted Covenant's display of power as if they were prepared to decide the fate of worlds.\n\nAs distinctly as he could, Covenant said, \"I forbid you to put me on the back of a Ranyhyn. Find some other answer.\"\n\nThen he sagged. He thought that he had come to the end of himself. The Humbled were right: he could not hold out against his wounds. He had lost too much blood, and was in too much pain. If Branl and Clyme did not obey him, he would have to trust the great horses of Ra to forgive him.\n\nWhen he felt certain that he was done, however, he found that he was not. A distant sensation of power seemed to call him back from the collapse craved by his ravaged body. Involuntarily he straightened his spine, sat more upright. He imagined that he heard either Clyme or Branl say, This delay will prove fatal. Then he saw them recoil like men who had been slapped. He felt their surprise.\n\nDirectly in front of him, the figure of a man stepped into the light as though he had been made manifest by wild magic and the eldritch puissance of Loric's _krill_.\n\nThe newcomer seemed to emanate imponderable age. Indeed, he appeared to be fraying at the edges as he arrived, blurring as though he took in years and released vitality or substance with every breath. Nevertheless he looked taller than the Humbled\u2014taller and more real\u2014although he was not. His apparent stature was an effect of the light and Covenant's astonishment and his own magicks. He wore the ancient robes, tattered and colorless, of a guardian who had remained at his post, rooted by duty, for an epoch. Yet his features were familiar; so familiar that Covenant wondered why he could not identify them. A man like that\u2014\n\nAfter two heartbeats, or perhaps three, he noticed that Branl and Clyme were preparing to defend him. Or they were\u2014\n\nHellfire.\n\n\u2014bowing. _Bowing?_\n\nTogether they each dropped to one knee and lowered their heads as if they were in the presence of some august figure incarnated from the dreamstuff of _Haruchai_ legends.\n\nIn Covenant, memories reopened like wounds, and he recognized Brinn.\n\nThe _ak-Haru_. Brinn of the _Haruchai_ , who had outdone the Theomach in mortal combat to become the Guardian of the One Tree.\n\nHere.\n\nIf Covenant had ever doubted that the Worm was coming, he believed it now. There could be no surer sign than Brinn's arrival. Even the absence of the sun, and the slow havoc spreading among the stars, did not announce the Earth's last days more clearly.\n\nWhile Covenant stared, open-mouthed and helpless, the _ak-Haru_ approached until he was no more than two strides from the _krill_. There he stopped, ignoring the obeisance of the Humbled. His gaze was fixed on Covenant.\n\nIn a voice rheumy with isolation and too much time, he said, \"My old friend.\" Words seemed to scrape from his mouth as if they had grown jagged with disuse. The skin of his face had been seamed and lined until it resembled a mud-flat now baked and parched, webbed with cracks. \"I perceive that your plight is dire, as it has ever been. The fact that I have come is cause for sorrow. Yet it is cause for joy that my coming proves timely. Once again, I learn that there is hope in contradiction.\"\n\nIllumined by Loric's gem, Brinn's eyes shone among their wrinkles with a warmth of affection that Covenant had not seen in any other _Haruchai_ face.\n\n\"It is well,\" Brinn continued, \"that you have reawakened the Vilesilencer's _krill_.\" Strain complicated his tone, but not his gaze. \"Lacking some beacon to guide me across the wide seas, my search for you might have been delayed. However, you have done what must be done, as you have done from the first. For that reason among many others, I swallow my sorrow and greet you gladly, ur-Lord and Unbeliever, Thomas Covenant, friend.\"\n\nStill Covenant stared. Only the pervasive force of Brinn's acquired theurgy kept him from crumpling. Never in life had Brinn of the _Haruchai_ called him _friend_.\n\nSudden woe and rue and gratitude clogged his throat. He had to choke them down before he was able to inquire hoarsely, \"What are you doing here?\"\n\nAt the Isle of the One Tree, Brinn had told him, _That is the grace which has been given to you, to bear what must be borne_. Surely now Covenant had reached the limit of what he could be expected to endure?\n\nStill Brinn did not glance at either of the Humbled. His attention belonged to Covenant alone. Speaking more sternly, as if he were setting friendship aside, he replied, \"All things exist organically. This you know, Unbeliever. As one swells, another dwindles. As the Worm of death rises, the Tree of life declines.\" A lift of his hand referred to the heavens. \"After long ages of slumber, the Worm now draws nigh unto the Land, seeking its final sustenance. In natural consequence, the One Tree expires to its roots. Thus I am freed of my Guardianship.\n\n\"Alas, my powers diminish as the Tree fails. I am made less by the deaths of stars and _Elohim_. And it was never my task to preserve the Worm's sleep, except by protecting the One Tree. I have no virtue to oppose the World's End. Nor am I permitted to do so, regardless of the leanings of my heart. That burden is yours, Unbeliever, as it is the Chosen's as well, and also her son's. Together you must save or damn the Earth, as it was foretold in the time of the Old Lords.\"\n\nThen the _ak-Haru_ 's manner softened until it resembled his gaze. \"Yet I will not disregard the leanings of my heart. When I had achieved the stewardship of the One Tree, and you were thereby grieved, I assured you that good would come of it, when there was need. That promise I fain would honor. Therefore have I journeyed hither while some small portion of my strength endures, bringing both gifts and counsel. Mayhap thereafter I will also be able to perform a service or grant a boon, if my life does not fray and fall in the attempt.\"\n\nCovenant went on staring as though he had been made witless. Part of him heard hope in every word. Part of him had already fled toward Linden, thinking, Gifts? Counsel? A chance to make things right with her? And part of him remained stunned, too astonished to comprehend anything. Brinn had come like a figure in a dream. In another moment, he would depart in the same fashion, with the same effectlessness.\n\nBut the Guardian of the One Tree did not appear to take offense at Covenant's silence. His affection seemed to accept every facet of Covenant's condition. Nodding at what he saw, the _ak-Haru_ took one step back from the _krill_. Then at last he looked at Branl and Clyme, still half kneeling, still bowing their heads in homage.\n\nNow his mien darkened. Lines of anger tightened his visage.\n\n\"First, however,\" he pronounced severely, \"I will deliver myself of a reprimand which has long festered within me, tainting my regard for those whom I must name my people.\n\n\" _Haruchai_ , Masters, Humbled, I have come to reproach you.\"\n\nAt once, Clyme and Branl arose. The manner in which they surged to their feet and folded their arms conveyed surprise and indignation. In every line, their stances offered defiance.\n\nStolid as a graven image, Branl stated, \"You are the _ak-Haru_ who was once named _Kenaustin Ardenol_ , though you are now Brinn of the _Haruchai_. We do not lightly gainsay you. If you have cause to reproach us, however, you discern some fault which we do not find in ourselves.\n\n\"The weakness of uncertainty we acknowledge. Failure we likewise acknowledge. Against our given word, we have permitted Desecration, upon occasion because we were opposed by those whom we esteem, and upon occasion because the ur-Lord Thomas Covenant commanded it. Yet we have stood as Halfhands at his side. For his sake, we have dared the Lost Deep and She Who Must Not Be Named and Esmer _mere_ -son. We have confronted the _skurj_ and Cavewights and the Unbeliever's own misbegotten scion. We have entered into a Fall, hazarding endless banishment from time and life, and have there given aid to the ur-Lord when he could not aid himself.\n\n\"You are the _ak-Haru_. Would you have done otherwise in our place? Wherefore will you reproach us?\"\n\nBrinn dismissed Branl's protest with a soft snort. \"Your valor is beyond aspersion,\" he answered as if such things were trivial. Thunderclouds of ire seemed to gather about his head, contradicting the twilight and the clear stars. \"Set aside your pride and hear me.\n\n\"Doubtless others have spoken of arrogance. I do not. Rather the fault with which I charge you is _simony_.\" He spat that word. His eyes flashed dangerously, echoing the _krill_ 's radiance. \"You have grown ungenerous of spirit, demeaning what would else have been a proud heritage. You have withheld knowledge from the folk of the Land when knowledge might have nurtured strength. And you have withheld trust from Linden Avery the Chosen, setting yourselves in opposition to her efforts and sacrifices because you were unable to share her love and passion. These are the deeds of misers. They do not become you.\n\n\"Upon a time, the _Haruchai_ were not ungiving in this fashion. Had they not been ruled by open-handedness, they would have been less grievously stung by the Vizard's scorn. Yet open their hands were, and open they remained. The bonds among them were as vital as sun and snows, and as enduring as mountains. The wounds of scorn they sought to heal by open means, in direct challenge and honest combat. Thus it was that High Lord Kevin's generosity moved them to emulation. The Vow of the Bloodguard expressed an answering generosity, a desire to repay expansive welcome with expansive service until both welcome and service overflowed.\n\n\"Yet across the millennia of your Mastery you have allowed harsh times and cruel circumstances to bar the doors of your hearts. I will not cite your reasons for doing so, lest you deem yourselves thereby excused. Rather I say to you plainly that you have diminished yourselves until I am loath to acknowledge you as my people.\"\n\nInstinctively Covenant wanted to defend Clyme and Branl. Oh, he agreed with the Guardian. How could he not? Nevertheless the Humbled had stood by him like the _Haruchai_ of old. They had saved him again and again when he could not have saved himself.\n\nBut his companions did not turn to him for justification. They did not look at him at all. As if they were proud to be castigated, they faced Brinn squarely.\n\n\" _Ak-Haru_ ,\" Clyme replied, \"this accusation is unjust.\" Tautness marred his flat tone. \"We do not comprehend it. What deed of ours\u2014or of any Master\u2014has given rise to your wrath?\"\n\nAt once, the Guardian retorted, \"Are you truly so blind that you see no fault in naming yourselves 'the Masters of the Land'?\" His voice had become a distant rattle of thunder. In spite of his diminishment, his words had the power to summon storms. \"The Land is not a thing to be possessed as though it were a garment. It was not created for your use, that you might hazard it in a vain attempt to heal your ancient humiliation.\"\n\nUnmoved, Branl countered, \"Yet you yourself have done as we do. You are our exemplar. Our distrust of Linden Avery we learned first from you, who saw Corruption's hand at work in her, and who strove to preserve the Unbeliever from her errors.\"\n\nOmens of lightning glared from Brinn's eyes. \"I concede,\" he answered, \"that I trod your path when I forsook the Unbeliever's service. What of it? Did Cail not return to speak of the Chosen's salvific efforts at the Isle of the One Tree? And if you did not heed him, did you also fail to heed the First of the Search and Pitchwife when they described the forming of a new Staff of Law, and the unmaking of the Sunbane?\n\n\"No,\" he said harshly. \"Do not protest that you have endeavored to treat the Chosen with both restraint and respect. I am not swayed. Your restraint and your respect are as miserly as your deeds. Had you permitted them to do so, the Giants would have reminded you that open hands and open spirits were once valued among the _Haruchai_. Yet for many centuries you have offered the kindred of the Unhomed naught but unwelcome.\n\n\" _Unwelcome_ , forsooth!\" The _ak-Haru_ 's indignation was a thunderclap. \"For the _Giants_ , of all the peoples of the Earth. That is my reproach. Humbled, Masters, _Haruchai_ , I marvel that you are not shamed.\"\n\nNow even Covenant's numbed nerves and blunt health-sense felt tension rising in the Humbled. Brinn's objurgation stirred millennia of suppressed passions, of ire and resentment and denied helplessness, into living flames.\n\nSpeaking softly, ominously, Clyme asked, \"Do you seek to renew our humiliation? Is that the purpose which has brought you among us, the last purpose of your life?\"\n\n\"Paugh!\" The Guardian made a dismissive gesture with both hands. \"I am done with you. You do not hear, and so you cannot be redeemed. From this moment, I speak only to the Unbeliever. He will not disregard the remnants of my life, as you have done.\"\n\nHis gesture seemed to dispel the sensation of storms seething around him. He was definitely growing weaker, but he did not act weakened. Simply by turning away from the Humbled, he thwarted their outrage; cast them into shadow. Now they stood silent, like men whose mouths had been sealed. When Brinn faced Covenant again, he was smiling with a hint of remorse\u2014and also with an air of satisfaction.\n\nOn the far side of the _krill_ , he seated himself cross-legged in front of Covenant. His eyes in their nests of seams and wrinkles glittered with refreshed affection. He sat with his elbows braced on his thighs and his chin propped on his fists; held himself leaning forward to study Covenant more closely. When Brinn was comfortably settled, however, he said nothing. Instead he gazed at Covenant as if he, the _ak-Haru_ , had been made content by the sight of his old friend's face.\n\nCovenant wanted to lie down. His forehead throbbed, and broken bones gnawed like teeth in his chest, biting deeper with every slight movement. Brinn's obscure intentions and the dammed fury of the Humbled and his own wounds exceeded him. He ached to close his eyes and slump backward and let everything go.\n\nYet he did not. His heart had not forgotten its stubborn litany of loves and needs. And the Guardian had come because he wanted to help in some fashion. Covenant could not allow himself to lapse while so much remained unresolved.\n\nWith an effort that nearly made him sob, he muttered, \"You aren't exactly being fair. You know that, don't you?\"\n\nBrinn's smile grew warmer. \"It is for this that I esteem you, Thomas Covenant\u2014this among many other qualities. Regardless of your own plight, you do not neglect the hurts of your companions.\" Then his mien assumed more somber lines. \"But now we must take counsel together. Your wounds are grave, my friend. Some healing you must have. Yet with healing will come sleep. It must, for your need is extreme. Therefore we must converse before I expend my waning strength. If you have not chosen your course, these Humbled will determine it on your behalf\u2014and they will not determine wisely.\"\n\nCovenant groaned. \"You see me. You know what I've done. What's left? What can I possibly hope to accomplish?\"\n\nHe meant, Take me to Linden. If you have that kind of power, use it. Before I'm too far gone to tell her I'm sorry.\n\nThe Guardian nodded. \"Indeed, Unbeliever, I see you. Your desires are plain to me. You yearn to be reunited with Linden Avery the Chosen for the Land's sake, and for your own. Were these Humbled less parsimonious in their dealings, they would honor the passion which binds you to your loves. But I must urge you to reconsider the Land's peril.\n\n\"You have slain your former mate, a deed costly to you, and hurtful, yet nonetheless necessary. What then remains for you to attempt? Have you forgotten _turiya_ Herem? He who reveled in your former mate's agony and abasement? He is not slain. Of that I need not assure you. You are already certain of it.\"\n\nOh, hell, Covenant thought. _Turiya_? But he did not have enough life left to curse aloud. On the fall of a shuddering breath, he asked, \"You want me to go after _him_?\"\n\nBrinn's study did not waver. Instead of answering directly, he inquired, \"He has failed Corruption's chief intent for him. What will he now essay in restitution?\"\n\nHellfire. Covenant groaned again. He was in no shape to think, much less talk. Nevertheless he did what he could. Brinn had called him _friend_.\n\n\"He'll try to possess someone else. Or something else. He isn't good for much unless he's wearing a body.\"\n\nThe _ak-Haru_ leaned closer. \"Then whose flesh will he assume? Not yours, that is certain. He is not such a fool. Nor will he attempt the Humbled. Their intransigence has not waned. He cannot rule them. Among the _skest_ , he may perchance strive to attain your death. But they are little, and by nature timorous, readily cowed. Also I deem that _turiya_ Herem is too prideful to be contented by them.\"\n\nCovenant peered past the actinic brightness of the _krill_ as if he were going blind. \"So\u2014?\" His former companion faded in and out of focus. Give me a hint. I can't keep doing this.\n\nThe Raver had a long head start.\n\nBrinn watched as though his gaze could penetrate Covenant's soul. \"I ask again. Whose flesh will he assume? Of those that fear the Worm's coming, which is comparatively near? Which is driven by hungers apt for possession?\"\n\nCovenant flinched at an intuitive leap. \"What, the _lurker_?\" He stared through a blur of argent and failing consciousness. \"You want me to go after _turiya_ before he can possess the _lurker_?\"\n\nSo far, the monster had kept its word. True to the alliance, Horrim Carabal had sent the Feroce to rescue Covenant and the Humbled from the _skest_. But still\u2014The lurker of the Sarangrave had been a tale of horror for millennia. In some sense, it was the Despiser's creation. Directly or indirectly, Lord Foul had invoked an immense and sentient atrocity from the poisons leaking out of Mount Thunder.\n\nNow Brinn wanted Covenant to defend that\u2014that thing\u2014from _turiya_ Herem?\n\nThe Guardian replied with a grin as poignant as the deaths of stars. \"Name a better purpose, my friend, and I will honor it.\"\n\nCovenant meant to say, No. That's insane. But then he thought, So what? The Worm was coming. He had killed Joan. Everything was insane. The idea of trying to track down and stop a Raver\u2014in his condition\u2014was probably no crazier than his desire to see Linden again.\n\nOver the course of his life in the Land, he had caused or allowed terrible bloodshed. The Riders of the Clave whom he had killed personally were minor casualties compared to the uncounted villagers and _Haruchai_ that he had forsaken to slaughter while he searched for the One Tree. Saltheart Foamfollower had died helping him. Inadvertently he had killed Elena, his own daughter. Then he had brought about the sacrifice of her spirit to She Who Must Not Be Named.\n\nBut he had never struck a blow against the Despiser's most fatal servants. And the lurker possessed by a Raver would be an appalling foe. More insidiously dangerous than Roger and a whole host of Cavewights. Conceivably more powerful than _skurj_ and Sandgorgons. If that monster challenged Linden, she would have to face it without Covenant or love.\n\nThinking about her made his wounds burn. His damaged ribs were acid and remorse in his chest. He wanted\u2014Oh, he _wanted_. Nevertheless he understood Brinn.\n\nHe rubbed at the crust around his eyes, touched the fresh accusation on his forehead. Eventually he managed to mutter, \"Damnation, Brinn. I'm going to need a horse.\"\n\nThe _ak-Haru_ beamed at him like Loric's gem. \"And you will not ride the Ranyhyn. For this also I esteem you, ur-Lord. Yet a steed has been offered to you. You need only speak the beast's name.\"\n\nBrinn's voice invoked memories. As if from a great distance, Covenant heard the dying croak of the Ardent's last gift.\n\n\"Ah.\" In spite of his satisfaction, Brinn's sigh conveyed a tinge of regret. \"I see the recall in your gaze. My friend, you are indeed as I have remembered you. I am now content to provide those gifts which lie within my power.\"\n\nHis vigor seemed undimmed as he rose to his feet.\n\n\"Remain only a short while,\" he urged Covenant. \"Your healing will be my second gift. Here is my first.\"\n\nWhile Covenant watched, stupefied by too many hurts, Brinn raised a hand to his mouth and gave one sharp whistle as clear as a commandment.\n\nCovenant was losing his grip on consciousness. The only _Haruchai_ who had ever called him _friend_ had asked too much of him. He was no longer sure of what he saw or heard. The Guardian's call may have echoed through the maze of the Shattered Hills. The stars appeared to draw closer. They seemed to cry out. Perhaps their wailing was underscored by a clatter of hooves, irregular and indefinite.\n\nWhen the Ranyhyn arrived with their star-blazed foreheads shining like the emblems of _Elohim_ , Covenant thought that he saw four of them.\n\nTwo must have been Mhornym and Naybahn. They looked worse than Covenant felt. Ripped flesh hung in strips from their sides, exposing the damaged gleam of bones, especially along their ribs and on their knees. Blood oozed everywhere as if they were coated in ruin. They limped on legs that should not have supported them, and their eyes were dull with mute agony.\n\nBut they were still alive. They had heard Brinn's call. Somehow they had found the resolve to answer.\n\nProudly the _ak-Haru_ announced, \"Here are heroes. They have participated bravely and well in the defense of the Earth. Such battles are not won at a single stroke. They must be fought incrementally, by one selfless act of valor following another in its necessary sequence. Now Naybahn and Mhornym have completed their task. Their part is done. Though my strength wanes, I will preserve them. Then I will release them. While the Earth endures, no further service will be asked of them.\"\n\nThen he turned to the other horses, a palomino stallion and a black. \"And here are Rallyn and Hooryl. They have come to bear the Humbled on a quest which will require much of them, and of their riders. That they do so fearfully is no fault in them. They are Ranyhyn. Fear will not hinder their service.\"\n\nBriefly Covenant looked at Clyme and Branl. The sight of them made him wince. His senses were too blunt to discern anything except rigid indignation.\n\nBut Brinn ignored the Masters. Facing Covenant again, he said as if he were bidding farewell, \"Now, Unbeliever, Illender, Prover of Life, you must speak the name. Only its name will summon the steed and obtain its compliance.\"\n\nThe stars were too close. Covenant had never seen them look so near. Yet their proximity only accentuated the voids between them, the immeasurable gulfs of their isolation. Vaguely he wondered whether the _Elohim_ felt the same loneliness. Perhaps that explained their prideful self-absorption, their insistence that they were complete in themselves, _equal to all things_. Perhaps their surquedry was nothing more than compensation for prolonged sterility and sorrow.\n\nBut then the lamentation overhead and Brinn's kindness compelled him. Swallowing the taste of blood and woe, he did as the Guardian of the dying One Tree asked or commanded.\n\n\"Mishio Massima.\"\n\nBrinn's smile was a confluence of hope and regret as he stepped past the _krill_ to touch Covenant's blamed forehead lightly with one finger.\n\nAt the same time, he urged quietly, \"Recall that the _krill_ is capable of much. With use, it has become more than it was.\"\n\nHis touch seemed to light a star in Covenant's brain. Suddenly the dusk in all direction became a swirl of lights: the same swirl which had filled the Isle's cavern long ago when Covenant had tried to claim a branch of the One Tree. If Linden had not stopped him then, he might have brought about the world's end without realizing what he did.\n\nHe needed to make things right with her. He needed to tell her that he loved her\u2014and that he had killed Joan.\n\nBrinn had spoken of a service\u2014a boon\u2014but he had not revealed what it might be.\n\nThen the stars took Covenant, and he went to sleep as if he were falling into the heavens.\n\n## 4.\n\n\"Try to Believe\"\n\nSoreness and jostling finally roused Covenant. He had no idea where he was; but for a while, he did not care. If the flexing sensations of movement had not insisted on his attention, he would have tried to go back to sleep.\n\nHis whole body ached as though he had suffered a beating. A dull throb in his forehead matched the rhythm that carried him. But when he braced himself to draw a deeper breath, he found that the piercing hurt of broken ribs was gone. Bruises like groans had replaced the effects of sharp rocks and rending coral. His weakness felt more like convalescence than blood-loss.\n\nA week, he thought to the cadence of hooves, the flow of stubborn muscles. Just let me rest for a week. Then I'll open my eyes. I promise.\n\nHe did not have a week. He doubted that he could afford hours.\n\nVaguely he deduced that he was mounted. But not bareback: not on a Ranyhyn. The saddle under him reminded him of the Harrow's fallen destrier. And he was not held upright. No, he was sprawled resting along a long neck. The saddle horn dug into his abdomen. His legs dangled free of stirrups. The jolts were the beat of a hard canter.\n\nHe remembered Mishio Massima, the Ardent's mangy, shovel-headed horse. Clyme and Branl must have boosted him onto the steed while he slept. And they must have secured his arms\u2014perhaps with the reins\u2014so that he would not fall.\n\nMishio Massima's jarring gait punished his recent wounds. Nonetheless he was grateful. At Brinn's insistence, no doubt, the Humbled had honored Covenant's promise to the Ranyhyn.\n\nFor a time, he was content to rest as he was in spite of the prod of the saddle horn. The mystery of Brinn's aid remained with him; the miracle of Brinn's friendship. Covenant was less alone in the world than he had believed himself to be. Less alone than he felt with the rigid companionship of the Humbled. The dying Guardian of the One Tree had given him a profound gift\u2014\n\nBut it was not an unalloyed blessing. True, Brinn had mended the worst of his injuries. But the Guardian had also given him a task which he feared to contemplate.\n\nRemembering _turiya_ Raver, Covenant flinched. He needed to open his eyes. Hell, he needed to sit up. He had to know where he was. And where the Humbled were taking him. And how they had resolved their contention with their _ak-Haru_ \u2014if they had resolved it at all. And what the service or boon that Brinn had mentioned might be.\n\nThe possibility that _turiya_ Herem might take possession of the lurker of the Sarangrave frightened Covenant as much as the idea that he might never see Linden again.\n\nWith an effort, he lifted his head; lowered it again. Blinking, he tried to clear his sight. Then he made an attempt to free his arms.\n\n\"A moment, ur-Lord,\" Clyme said over the steady rumble of hooves. \"We will unbind you.\"\n\nNow Covenant realized that the hoof-beats of the horses were muffled. The ground where they ran was too yielding to be stone; too soft for bare dirt.\n\nPeering sideways through the gloom, he saw a shape veer toward him: a horse and rider. When Hooryl came near enough to brush his leg, Clyme bent down to undo the reins.\n\nBriefly Covenant fought the blur that marred his vision. It seemed worse than it should have been. He could still see stars overhead, but his companion's features were a twilit smear. He had to squint in order to discern that the horses were cantering on thick turf.\n\nHell and blood. He should have been able to see better than this. Brinn had healed him, and leprosy did not progress so swiftly.\n\nUnless\u2014\n\nStung by an intuitive apprehension, he pulled his awkward arms under him; pushed himself off his mount's neck. Then he clutched at the saddle horn to keep his balance.\n\nHe could not feel the horn at all, except with the nerves of his elbows and shoulders. His hands were numb.\n\n\"What\u2014?\" he panted. He seemed to need all of his strength to keep his seat. Insensate in their boots, his feet floundered for the stirrups and did not find them. \"What's going on?\" His voice was as vague as his vision. He had slept too long. \"What's happening to me? My eyes are going.\"\n\nAround him, the aegis of the gloaming was complete. It ruled everything. It was leaking into his head; into his mind. Only the stars as they died were vivid to him.\n\nClyme draped the untied reins over Covenant's forearms. Hooryl moved away from Mishio Massima, perhaps so that Covenant could move his leg freely while he groped for the stirrup.\n\n\"Kevin's Dirt has overtaken us.\" Clyme sounded angry. No, it was more than that. He sounded like a man who had given up pretending that he was not angry. \"It came upon us at midday. Clearly Kastenessen now directs his malice over the Lower Land, doubtless seeking to harm you, and also to hinder the Staff of Law. In this, he succeeds. To our sight, it is plain that Kevin's Dirt deepens your illness.\"\n\nCovenant had guessed as much. But he had not expected the effects of Kastenessen's brume to be so swift. Came upon us at midday? How much time had he lost?\n\nHe turned his head to confirm that Branl also rode beside him. The motion and his mount's strides made his head pound and his ribs throb. But those pains were more bearable than his earlier hurts; somehow more human. He could imagine that they would fade.\n\nBranl's visage wore a frown like a knot between his brows. It looked permanent, as if it had always been there; as if it had merely been masked by a learned and unnatural impassivity.\n\nSlowly the vagueness faded from Covenant's thoughts. After a moment, he was able to ask Branl, \"Where are we?\"\n\n\"Ur-Lord,\" the Humbled answered, \"the Ranyhyn are cunning. They eluded the snares of the _skest_ and escaped the maze of the Shattered Hills well before the onset of Kevin's Dirt. Now we return along the path of our approach to Kurash Qwellinir. The cliff above the Sunbirth Sea lies there.\" He gestured eastward. \"If your mount is able to sustain its pace, we will soon gain the region where we last found _aliantha_.\"\n\nCovenant sighed his relief. This was not the most direct route to the Sarangrave, but it was the shortest path to food. If Branl and Clyme had over-ruled their _ak-Haru_ 's counsel\u2014if they had decided to seek Linden and the Giants instead of pursuing _turiya_ \u2014they would have headed northwest from the Shattered Hills.\n\nCovenant looked around at the caliginous vista of the grass, the slope rising incrementally toward the east, the greying of the world. When he was ready, he announced, \"I want to stop for a while. I ache everywhere. I need to walk around some. I'm sure this nag\"\u2014he indicated Mishio Massima with his chin\u2014\"can use a break.\" In fact, the Ardent's beast seemed preternaturally hardy. Unlike the Harrow's charger, apparently, this horse had been bred for endurance. \"If nothing else, it probably wants grass. And we should talk.\"\n\nHe felt sure that the Humbled had much to tell him\u2014if they chose to do so.\n\nClyme and Branl consented promptly: a bad sign. Had they trusted Brinn's advice, they would have argued that Covenant required haste. But they slowed their mounts without a word. Mishio Massima eased to a bone-rattling trot, then jerked to a walk like a thing formed of tree-limbs rather than flesh and bone.\n\nBefore the beast halted, Covenant slid out of the saddle. At first, his legs refused to hold him, and he dropped to his knees. Fortunately the turf cushioned the impact. Then he forced himself to his feet. Stifling a groan, he began to stamp in a circle, trying vainly to drive some sensation back into his ankles and feet. Their numbness affected him like imminent vertigo: he needed to rediscover balance. As he moved, he twisted his trunk from side to side, testing the condition of his ribs. Briefly he rolled his head and swung his arms. When he had assured himself that he was substantially intact, he took a few deep breaths and braced himself to confront the Humbled.\n\nThey had dismounted. Now they stood facing him, Branl with his clenched frown, Clyme with his hands curled into fists. But the mounts were moving away, trotting westward. Covenant guessed that they had caught the scent of water.\n\nAlone with his companions, he rubbed at the crusted blood around his eyes; probed the new scar on his forehead with the nub-ends of his fingers. His fingers felt nothing, but the tenderness of the cut assured him that it needed more time to heal.\n\nThe Humbled had not endured their _ak-Haru_ 's reproach gently: that was obvious. Groping for a tone of respect, Covenant said, \"I'm not sure, of course. I was asleep. But I get the impression there are things you should tell me. Something happened while I was out\u2014and I'm not talking about Kevin's Dirt. Did Brinn say anything else? Did he\u2014?\"\n\nClyme interrupted him curtly. \"He did not. We were not heeded. No further speech was exchanged.\"\n\nCovenant stared. \"Are you sure? He said something about a boon. A service. He didn't tell you what it was?\"\n\nBrinn was _Haruchai_ : he could have spoken to the Humbled mind to mind more fluently and thoroughly than aloud.\n\n\"He did not,\" Clyme repeated, rigid as metal. \"He refused our mental communion, as only Stave has done heretofore. In his thoughts we found only silence.\"\n\nFrowning like Branl, Covenant wavered on his feet. Keeping his balance was as difficult as he had feared. Too much had happened. He needed the feedback of nerves which no longer communicated with the rest of his body.\n\nTo that extent, at least, he knew how the Humbled felt. The Guardian had undermined their foundations.\n\n\"What does that mean to you?\" he asked carefully. \"Has he given up on us?\"\n\nAfter a moment, Clyme appeared to relent. His shoulders released some of their tension. Less stiffly, he replied, \"When the _ak-Haru_ had extended his strength for your healing, he was much reduced. Indeed, he resembled a man drawing the last breaths of extreme age. We deem that he did not speak again of a boon because he had come to the end of himself. He could not do more.\"\n\nAh, hell, Covenant sighed. He hated to think that Brinn had simply passed away. After so much time and devotion\u2014He wanted to believe that his former companion would find some form of resolution or contentment; but Clyme gave him scant reason for hope.\n\nHowever, he could not afford to dwell on grief. Other issues were more compulsory.\n\n\"Then tell me what's changed for you.\" He strained his eyes to study the faces of the Humbled. When neither of them spoke, he made an attempt to sound gentle. \"Was being criticized by your _ak-Haru_ that bad?\"\n\nBoth men stiffened. Their anger made them vivid in the gloom. Branl's glower looked fierce enough to split his skull. Clyme knocked the knuckles of his fists together as if he were stifling an impulse to hit someone.\n\nLike the cut of a blade, Clyme stated, \"His words were hurtful to no purpose. He did not reproach what we have done. His reproach was that we are who we are. Is the wind to be faulted because it blows? Are the stones to be accused because they are not trees? We are _Haruchai_. We cannot be other than ourselves.\"\n\n\"Mayhap it was his right to speak as he did,\" Branl conceded. He was not less indignant than Clyme: he had merely assumed their shared burden of truthfulness. \"He is the _ak-Haru_ , Guardian of the One Tree. No other _Haruchai_ has equaled his attainments.\"\n\n\"Nevertheless,\" Clyme snapped. \"We care naught for his right to speak. Our true grievance, ur-Lord, is that he sought to counsel you, and his counsel was _false_.\"\n\nHe spat that word as if it were a curse.\n\n\"False?\" Covenant nearly choked. \"Hellfire! How do you get to a conclusion like that? You said it yourself. He's the _ak-Haru_ , for God's sake! How can you even _think_ a word like 'false,' never mind say it out loud?\"\n\nNow Clyme did not relent. His tone held an outrage so deep that it seemed to arise from the marrow of his bones.\n\n\"We do not charge him with malign intent, but rather with mistaken comprehension. As he has misesteemed us, so he has misjudged the Land's peril.\n\n\"The lurker's plight is of no consequence. That monstrous wight is an avatar of Corruption. A Raver's possession cannot increase its misbegotten appetites. It requires no urging to seek our ruin.\n\n\"Recall,\" he insisted as though Covenant had tried to interrupt him, \"that the Soulsease has found new depths among the roots of Gravin Threndor. The Defiles Course will not resume its accustomed flow until the immeasurable abyss of the Lost Deep has been filled. Thus the poisons which supply the lurker's most necessary sustenance have been much reduced. Already its hungers swell. They must. Having grown so vast, they must be vastly fed. Such a creature will not long remember that it fears your magicks, or Linden Avery's. Your alliance was a thing of the moment. It cannot endure.\n\n\"To abandon all other needs in the lurker's name is madness.\"\n\nMadness? Covenant wanted to protest. Is that what you think of Brinn? Is that what you think of _me_? But the Humbled were not done.\n\n\"That is reason enough to set aside the _ak-Haru_ 's counsel,\" put in Branl. \"Yet there are other reasons as well.\n\n\"Has not the Ardent cited the ravages of the _skurj_ and the Sandgorgons in concert? Has not Kevin's Dirt been sent to weaken us? And is not Kastenessen the source of both evils? There lies your true path, ur-Lord. You must join with Linden Avery to challenge the mad _Elohim_ 's malevolence. That task is paramount. An end to Kevin's Dirt must be accomplished.\n\n\"Doubtless Kastenessen is both spurred and guided by _moksha_ Jehannum. Certainly the Sandgorgons heed the Raver, seduced as they are by the remnants of _samadhi_ Sheol's spirit. Yet the power is Kastenessen's. There can be no true defense of the Land while he stands in opposition.\"\n\nFacing his companions, Covenant floundered. Anger he had expected. They were _Haruchai_ , Masters and Humbled; proud. Naturally they had taken umbrage at Brinn's judgments. But he had not expected them to express their indignation like this.\n\nShaken and dismayed, he felt a reflexive desire to argue. He could have pointed out that Kastenessen was almost certainly positioned somewhere among the secrets of Mount Thunder, and that the distance was insurmountable. No doubt Linden was closer; but finding her would not take Covenant nearer to Kastenessen.\n\nWhile he tried to assemble the necessary words, however, he realized that the distance was effectively irrelevant. _Turiya_ 's head start was already insurmountable. Under the circumstances, one impossible distance was much like another.\n\nIn any case, no rational argument would sway the Humbled. They were too angry. Behind their masks, their attitude was based on a passion that Covenant did not understand.\n\n_Something_ had stung a primal nerve in them: primal and intimate. They had been hurt in a place at once carefully hidden and exquisitely raw. The pain of that singular wound drove them to extremes of emotion which Covenant had not witnessed before in any _Haruchai_.\n\nUnsure of himself, he tried to be cautious. \"The Feroce saved us.\" Still he winced at his own bleakness, his tone of confrontation. In his way, he was as irate as the Humbled. \"Horrim Carabal held up his end. He didn't have to. He could have left us to the _skest_. After all, he hates wild magic. He hates the _krill_. But he kept his word anyway. We wouldn't be here talking about it if he hadn't honored his agreement. Maybe you can ignore that. I can't.\n\n\"First you wanted me to break my promise to the Ranyhyn. Now you want me to turn my back on an alliance. That doesn't sound like you. It doesn't sound like any _Haruchai_ I've ever met.\" He had to grit his teeth to keep from shouting. \"What's happened to you?\"\n\nDark as incarnations of wrath, Clyme and Branl glared at Covenant. For a long moment, they did not reply. They did not move. Perhaps deliberately, they gave him a chance to fear that they would turn away from him. The Masters had spurned Stave\u2014\n\nBut then, suddenly, Branl snatched the bundle of Loric's _krill_ from inside his tunic. With a flick of his wrists, he spun the blade free of Anele's tattered heritage. As the gem's argence blazed out, he stabbed the dagger into the grass.\n\nIn the _krill_ 's radiance, both Branl and Clyme looked hieratic, chthonic, as if they had already taken their places among the Dead. The reflections in their eyes gave them the authority of spirits unconstrained by the boundaries of life and time.\n\n\"Ur-Lord,\" Clyme announced, \"we are the Humbled in all sooth, the Humbled triumphant and maimed. Have you forgotten so much that you do not recognize the men whom we have chosen to become?\" His ire sounded more and more like lamentation. It sounded like fear. \"Do you not recall that it is our task to embody you among our people? You are the purpose and substance of our lives.\n\n\"If you do not return to Linden Avery, and do so swiftly, you will perish. We cannot stem the harm which Kevin's Dirt wreaks within you. Nor can the lurker of the Sarangrave succor you. Without the balm of the Staff of Law, your end is certain.\n\n\"Come good or ill, boon or bane, you must not heed the counsel of the _ak-Haru_.\"\n\nAs Clyme spoke, Covenant finally heard what lay behind the frustrated fury of the Humbled. As though the insight had come to him from the lost expanse of the Arch of Time, he understood; and he found himself trying to laugh, although he wanted to weep. Oh, Clyme. Oh, Branl. Have you come to this? After so much fidelity and striving, is this the best you can do?\n\nTheir beliefs were too small to vindicate the race of the _Haruchai_. At the same time, they were too much for Covenant.\n\nThat was their tragedy. They had attached an almost metaphysical significance to a lone and lonely man who could not bear the burden. He was unequal to the task of meaning, not because he was sick and weak\u2014although he was\u2014but because he was just one man, nothing more. Even if he transcended his own inadequacies indefinitely, he could not provide transcendence for anybody else. The _Haruchai_ needed to find it within themselves, not in him.\n\nNothing else would relieve the bereavement which had haunted them for millennia.\n\nBut they were not Giants: they would not respond to laughter; even to laughter as strained and loss-ridden as Covenant's. Their hearts spoke a different language.\n\nAs if he were translating alien precepts into pragmatic speech, he replied, \"Did I ever tell you that I respect you? I hope I did. I've said as many hurtful things as Brinn did, but none of it would have been worth saying if I didn't respect you absolutely. You're the standard I use to measure myself\u2014or you would be if I thought that highly of who I am. The idea that men like you care whether I live or die makes me want to prove you're right about me.\n\n\"But what's at stake here\u2014what we're talking about\u2014what we have to do\u2014isn't about whether or not I live through it. It's about the Land, and the Worm, and Lord Foul. We can't let the fact that I'm sick choose our commitments for us.\n\n\"I've made promises. Now I have to take the risk of keeping them. I have to be willing to pay whatever they cost.\"\n\nAnd his agreement with the lurker had been founded on a lie: the mistaken belief that he was the Pure One of _jheherrin_ legend. He needed to redeem that falsehood.\n\nClyme and Branl watched him without saying anything; without any expression that he could interpret. Clyme braced his fists on his hips. Branl folded his arms like barriers across his chest. If they grasped that import of his affirmation, they gave no sign.\n\nNevertheless Covenant went on as if he had won their consent to continue. \"But that cost\u2014It may not be what you think. Which is my fault,\" he added quickly, \"not yours.\n\n\"I don't say much about myself. I probably haven't told you or anybody that my disease\u2014that leprosy\u2014isn't fatal. Lepers can get worse for a long time without dying. Usually it's the things that happen to them _because_ they're lepers that kill them.\n\n\"Kastenessen can make me a whole lot sicker without stopping me. Kevin's Dirt is nasty stuff, but it won't save him. He only imagines it will because he's crazy and desperate.\n\n\"Meanwhile leprosy is like most of the things we struggle with. It's a curse, but sometimes it can also be a blessing.\"\n\nCast back by the _krill_ 's brilliance, the surrounding twilight seemed to deepen, drawing the stars ever closer to the world's doom. At the same time, the Humbled began to look both more substantial and more mundane; less like emblems from the realm of death. Unwillingly, perhaps, but irrefusably, they were being lured out of their moral reality into Covenant's.\n\nMore sure of himself now, the Unbeliever said, \"Look at it this way. Have you never wondered why none of the Ravers has ever tried to possess me? They've had me helpless often enough. So why am I still here? Sure, Foul told them not to take me. He didn't want them to get my ring. But why did they obey?\n\n\"Well, they've been his servants so long, you might think they're incapable of independent thought. That's one theory. But it can't be true. If it were, they wouldn't be much use. He would have to spend all his time telling them what to do. No, he has to be able to give them orders and then leave them alone while they figure out how to accomplish what he wants. They have to be able think for themselves.\n\n\"And they're by God _Ravers_. It's their _nature_ to be hungry for power and destruction.\" Just like Horrim Carabal. \"So why have they never, not once in all these millennia, ever tried to possess me? Why haven't they tried to take my ring?\"\n\nCovenant spread his hands, his foreshortened fingers, showing the Humbled that they were empty\u2014and that such appearances were as deceptive as the stoicism of the _Haruchai_.\n\n\"I think I know why. It's the same reason we can trust the lurker. And the same reason I have to do what I can to save him. Because they're afraid. They're all afraid. Horrim Carabal is afraid of the Worm. And the Ravers\u2014Well, of course they're afraid of Lord Foul. But I'm guessing they're also afraid of leprosy. They're afraid of what it might be like to possess a body and a mind as sick as mine. They're afraid of all this numbness, and going blind, and feeling crippled not to mention impotent even when they have wild magic to play with.\"\n\nHe shrugged as if he were susceptible to contradiction; yet with every word he felt stronger. \"Maybe being me would be too much like being the Despiser, trapped and helpless and full of despair even though he's too powerful and too damn eternal to be killed. Possessing other people, or other monsters, they can at least feel and hate and destroy. With me, they might not be able to do any of those things.\"\n\nHe was vaguely surprised to see Clyme and Branl blink in unison as if they were closing the shutters of their minds against illumination. But the moment was brief; no more than a flicker.\n\nAs if he were confessing an article of faith, Covenant concluded, \"That's why I might be able to save the lurker. It's why I have to be a leper. _Turiya_ won't even consider possessing me. Leprosy is my best defense. Even Lord Foul can't stop me if I'm numb enough.\"\n\nThen he held his breath. He could not read his companions: he saw only anger and blankness and inflexibility. Argent lit them against the backdrop of the sunless day, but did not reveal their hearts.\n\nThey were slow to respond. They may have been sifting through their imponderable storehouse of memories, testing Covenant's asseveration against their entire history with him.\n\nWhen Clyme finally answered, Covenant was not prepared for his response. Nothing in his manner, or in Branl's, hinted that the Humbled were capable of any reply except denial.\n\n\"How then,\" Clyme asked with the finality of a knell, \"shall we pursue the Raver? He is no longer hampered by the limitations of flesh. Even the Ranyhyn cannot equal his fleetness, and your mount is no Ranyhyn. How can the lurker be spared if we cannot overtake _turiya_ Herem?\"\n\nDimly through the dusk, Covenant saw Rallyn and Hooryl returning, bringing Mishio Massima with them. They seemed to know that the time had come to bear their riders again.\n\nHe exhaled hard; panted briefly for air. \"I have no idea,\" he admitted. \"I'll have to think of something.\"\n\nAt that moment, he believed that he would succeed. Like Brinn, Clyme and Branl had given him what he needed. While the Humbled stood with him, he could imagine that anything was possible.\n\nut he put off thinking until he and his companions had ridden far enough to find _aliantha_. He needed time to absorb Clyme's and Branl's acquiescence. And he felt thin with hunger. He had eaten nothing since he and his companions had left their covert in the cliff early the previous morning. The streams that the Ranyhyn discovered now eased him somewhat; but water was not nourishment\u2014and it was certainly not treasure-berries. He craved the rich benison of the Land's health and vitality. Without it, he could not reason clearly enough to untangle the riddle of _turiya_ Herem's head start.\n\nFortunately Branl and Clyme knew where they had last seen _aliantha_. And Covenant did not doubt that the Ranyhyn could have located the holly-like shrubs even without the guidance of the Humbled. The way seemed long to him, but Clyme pointed toward the first bush well before the unbroken twilight became midafternoon.\n\nThere Covenant dismounted. At once, Mishio Massima lowered its head to the grass as if nothing mattered except food. Carrying the _krill_ again, Branl remained with Covenant while Clyme rode ahead to gather more berries so that Covenant would not be required to waste time searching for a sufficient meal.\n\nAt the first tang of the fruit in his mouth, Covenant seemed to feel Brinn's hand reaching out to him across the leagues and hours; touching his sore forehead and damaged ribs and battered arms with renewal. In its own way, _aliantha_ was as much a gift as the _ak-Haru_ 's aid, and as precious. It answered questions which the Humbled had not asked.\n\nIt was for _this_ that Covenant had to find and stop _turiya_ , and then go on to the next battle, and the next. Not for the lurker. Not for the _Elohim_ , in spite of their slow, inexorable decimation. Not even for Linden, although his ache for her resembled weeping. No, it was for _aliantha_ that he had to fight: for treasure-berries, and for Wraiths; for hurtloam and Glimmermere and Salva Gildenbourne, Andelain and EarthBlood; for the Ranyhyn and their Ramen; for ur-viles and Waynhim; and for every mortal heart as valiant and treasurable as Liand's, or as Anele's. For their sake, he had to catch up with the Raver. He had to find a way.\n\nWhen he had eaten enough to take the edge off his hunger, he began to pace slowly, chewing fruit, scattering seeds, and talking. The numbness of his feet made him feel that he walked a friable surface tipping him toward vertigo. Nevertheless he persevered. He needed to hear his thoughts aloud in order to believe in them. And he needed movement to loosen the knots that bound him to his limitations.\n\nThe Worm was coming. Lord Foul's triumph drew closer with every hesitation, every delay. The Land could not be saved by anything less than extravagant efforts and hope.\n\nHope did not come easily to lepers. But Covenant had learned that there were better answers than grim survival and despair. He had been taught by more friends and loves than he could count.\n\nUnsteadily he ate, and marked out a circle on the giving ground with his steps, and talked.\n\n\"I keep thinking about Linden,\" he muttered as if he were speaking to Branl. With a wave of one hand, he dismissed a protest which his companion did not utter. \"I was watching her. I remember her life almost as well as mine.\n\n\"She should have died when she first arrived on Kevin's Watch. A _caesure_ broke the Watch right after she met Anele. All those tons of shattered granite collapsed like they fell from the sky. She should have been crushed. They both should have been reduced to pulp. But she kept them alive.\n\n\"I'm asking myself, how did she _do_ that?\"\n\nConcentrating on other things, he lost his balance as if he had tripped. He almost fell. The deadening of his nerves was becoming extreme. Still he was familiar with such dilemmas. The loss of sensation was like Unbelief. It could be managed. Sometimes it could be set aside. And under the right circumstances, it could become a form of strength.\n\nHow else had he twice defeated the Despiser?\n\n\"I was watching,\" he repeated as he resumed his tread. \"I saw what happened. I mean, what _literally_ happened. She slipped outside time. And she took Anele with her. Somehow she bypassed cause and effect and even ordinary gravity so that she and Anele came down on top of the rubble instead of under it. Hell, she didn't even break bones.\n\n\"But _how_? That was a neat trick. How did she manage it?\"\n\nPeripherally Covenant noticed Clyme's return. But the Unbeliever did not interrupt the awkward whirl, the vertigo in slow motion, of his paced circle.\n\n\"It's obvious, really. She did it with wild magic. She used my ring, even though she had no idea what she was doing, and she certainly never did anything like that before. It must have been pure reflex. Raw instinct. But that part doesn't matter. What matters is, she _did_ it. She proved it's possible.\n\n\"If wild magic is the keystone of the Arch of Time, it _participates_ somehow.\" Those words raised echoes for him. They implied memories which eluded recognition. \"You could say Linden did the opposite of what Joan was doing. Instead of shattering pieces of time, she found her way around them.\"\n\nThe Humbled studied him in silence. Their faces remained as blank as age-worn carvings.\n\n\"Well.\" Unaware of what he did, Covenant spread gestures in all directions as if he were flinging out his arms for balance; as if he sought to encompass the world. \"If she could do it, why can't we? After all, my poor son and that damned _croyel_ did it. They slipped through time to take her into the past. Which the Mahdoubt also knew how to do. And they slipped past distance to reach _Melenkurion_ Skyweir. Which both the Harrow and the Ardent knew how to do. So why don't we do the same thing?\"\n\nThere was something that he needed to remember, but he did not try to force it. Instead he let the past reach him in its own way.\n\nClyme slid down from Hooryl's back. Lifting the hem of his tunic, he showed Covenant that he carried a feast of treasure-berries. But Covenant did not pause. He could not stop talking now, even for the Land's largesse.\n\n\"Ignorance, I suppose. We don't know what Roger and the _croyel_ and at least some of the Insequent knew. If I ever understood how they did it, I sure as hell don't remember. And we probably haven't earned the knowledge. But when you can see a thing is possible, ignorance looks less irreducible. You can afford to try out theories or just plain guesswork because you know what you want to accomplish.\"\n\nAs if by an act of grace, the memory he sought came to him.\n\n_Time is the keystone of life, just as wild magic is the keystone of Time._ Among the Dead, the Theomach had said that. _It is Time which is endangered_. His counsel had inspired Covenant to risk a _caesure_ in order to confront Joan. _The path to its preservation lies through Time._\n\nThat was cryptic at best; hardly comprehensible. Nevertheless it sufficed.\n\nAbruptly Covenant stopped pacing, planted his legs for balance. His head continued its slow spin, but he faced the Humbled as squarely as he could.\n\n\"And Loric's _krill_ isn't our only instrument of power. We have white gold.\" He tapped his sternum where Joan's wedding band hung under his tattered T-shirt. \"If Linden can use my ring, I ought to be able to use Joan's.\"\n\n_You are the white gold._\n\n_Recall that the_ krill _is capable of much._\n\nWithout transition, he told Clyme, \"Give me some of that. I've got work to do, and I'm still hungry.\"\n\nHe had no real idea how to carry out his intentions. But he had found a place to start. And he could trust the Ranyhyn to help him.\n\nHe had suffered enough. Now he meant to surprise the hell out of _turiya_ Herem.\n\nefore long, he had satisfied his hunger. The bounty of _aliantha_ seemed to supply all of his immediate lacks. Each berry enriched his veins and muscles and even the fate written on his forehead until he was almost strong, almost steady. The threat of dizziness receded. His health-sense remained vague as a wisp, but he felt an unexpected tingle of renewed sensation in his ankles and wrists.\n\nWhen he was ready, he thanked Clyme. He urged the Master to save as many treasure-berries as he could. Then he asked Branl for the _krill_.\n\n\"I'm not sure what I'm doing,\" he admitted. \"But it's always helped me to have another source of power.\" The Staff of Law in Elena's hands. The Illearth Stone in Foul's Creche. Sunder's _orcrest_. Covenant had relied upon external catalysts or triggers until the Despiser's venom had eaten away his instinctive defenses, his visceral reluctance. \"And this ring is Joan's, not mine. Using it won't be easy.\"\n\nIn contrast, he had earned the privilege of wielding Loric's eldritch dagger. He had paid for it with bloodshed.\n\nBranl did not hesitate. Removing the wrapped blade from its place under his tunic, he delivered it to the Unbeliever.\n\nCovenant hefted the dagger, felt its weight and its implied power. \"Now what?\" he asked, thinking aloud again. In spite of his millennia within the Arch of Time, the prospect of theurgy still disturbed him. Magic suited Linden. Her health-sense guided her: she could control herself. Covenant was only a leper. Nevertheless he had come too far to start shirking hazards that scared him.\n\nHow often had he told Linden to trust herself?\n\n\"Well, let's see. I don't understand _how_ the Harrow and the Ardent did what they did. As far as I know, they just appeared and disappeared whenever they wanted. But Roger and the _croyel_ are another matter.\n\n\"They faced each other with Linden between them. They raised their arms to make an arch over her head. An arch like a door.\" Instinctively he began to pace again. \"A portal. But I can't do that. I can't stand in two places at once.\"\n\nCould Clyme or Branl assist him? He rejected that idea. _Haruchai_ did not wield magic. Whenever they could, they eschewed weapons of any kind. And Covenant had already required the Humbled to violate too many of their chosen prohibitions.\n\n\"Sounds like an impasse,\" he muttered. \"But it can't be.\" He lifted his burden with a shrug. \"So maybe I'm thinking about this the wrong way. Maybe Roger and the _croyel_ weren't making a door. Maybe it just looked like a door. Maybe it was really something else.\n\n\"Like what?\" For a moment, nothing occurred to him. Then he felt a surge of possibility. \"How about an _enclosure_? A way to keep everybody together while Roger and the _croyel_ combined their magicks?\n\n\" _That_ I can do.\"\n\n\"Ur-Lord?\" asked Clyme. Another man might have sounded baffled. The Humbled's tone expressed only polite disinterest. \"Your meaning is obscure to us. Speak more plainly.\"\n\nNourished by _aliantha_ , a sensation like eagerness throbbed in Covenant's veins. Perhaps lepers were capable of hope after all.\n\n\"Watch,\" he said as if he were sure of himself. \"Mount up.\" He took a few steps to increase his distance from the Humbled and the Ranyhyn. \"Keep my horse with you. I'll join you when I'm ready.\n\n\"And concentrate on _turiya_. Rallyn and Hooryl can find him if they know that's what we want.\"\n\nHis companions may have hesitated. If so, he did not see it. He had already turned his attention to his task.\n\nHis arms still ached. The _krill_ seemed too heavy to bear. In spite of Brinn's gifts, and the Land's, he remained weak. Nonetheless he used the stubs of his halfhand to pull out Joan's wedding band on its chain.\n\nAh, Joan\u2014Her ring had encircled a world of promises, but none of them were kept. If he got the chance, he intended to make better promises before the end.\n\nDrawing the chain over his head, he forced Joan's ring onto the truncated end of the little finger of his left hand. With the chain dangling, he unwrapped the remnants of Anele's raiment from the _krill_ , carefully keeping fabric between the ring and any part of the dagger.\n\nWhen the silver purity of the gem blazed out, defying the dusk in all directions, he paused. While his eyes adjusted to the shock of radiance, he tried to take stock of his condition; his fitness for what he meant to attempt.\n\nAt one time, he had feared white gold. He had been positively dependent on the idea that he was helpless; that he was capable of nothing, and that therefore nothing could be required of him. At another time, he had again feared wild magic, but for the opposite reason. Afflicted by Lord Foul's venom, he had raised fire from his ring too easily. He had become capable of appalling destruction and bloodshed at any provocation.\n\nNow he felt like an amalgam of those two Covenants, an alloy: the leper who feared the responsibility of any power, and the poisoned man whose violence threatened to defy constraint. He could imagine himself accomplishing _everything and nothing_ ,\n\n_hero and fool_\n\n_potent, helpless\u2014_\n\n_and with the one word of truth or treachery,_\n\n_he will save or damn the Earth_\n\n_because he is mad and sane,_\n\n_cold and passionate,_\n\n_lost and found._\n\nJust like lepers everywhere, he reminded himself so that he would not falter. Just like all of us. Everybody who still cares. We're all in the same mess.\n\n\"Well, hell,\" he drawled unsteadily. \"What's the point of dithering? Now's as good a time as any.\"\n\nDamned if you do, damned if you don't. The Despiser's favorite game.\n\nWincing as though he expected to be struck down, Covenant released his left hand from the blade's haft and slapped Joan's ring against the shining gem.\n\nIn that instant, his whole body became fire.\n\nHe was burning, but he was not burned: he blazed unconsumed. He felt as incandescent as the torrent of wild magic with which Lord Foul had once slain and freed him, yet he was not harmed. All around him, the twilight became darkness, impenetrable, impermeable. But within the ambit of his theurgy, silver reigned. It made every blade of grass along the sloping turf look sacred; distinct and ineffable. Argent lit Branl and Clyme on their Ranyhyn, holding Mishio Massima between them: it etched them against the sunless world as if it had incarnated them from the numinous substance of Covenant's imagination. Emblazonry shone on Rallyn's forehead, and on Hooryl's. Even the Ardent's horse resembled reified sorcery, ready to run between realities. Power surged in Covenant's veins until he did not know how to contain it.\n\n\"Ur-Lord!\" Branl called through the blare of light. \"Be wary! Such might is perilous!\"\n\nBut Covenant knew his limitations. He knew the difference between his puissance now and the immensely greater forces which he had wielded in his past life. In any case, he was still too frail to sustain so much power\u2014and this ring was not his. The _krill_ was probably burning his halfhand. For all he knew, Joan's ring was burning his finger. He simply could not feel the pain.\n\nDeliberately he dropped his left hand to his side, gripped the dagger with only his right. As he did so, the fire left him. He no longer spread brightness and flame in all directions; no longer poured out light as though his flesh were wild magic. But the _krill_ 's gem retained the radiance which he had summoned from it. Theurgy ran down the blade like water or blood.\n\nAt once, he stooped to touch the grass with the point of Loric's weapon. He let the blade's weight sink in as deeply as it wished, but he made no effort to drive the _krill_ deeper. Then he watched as the rough turf became lambent as if it had been touched with ecstasy.\n\nHe feared to see that the _krill_ 's touch had killed the grass, left it scorched and withered. But somehow he had invoked a form of power which was not destructive. Instead of dying, the turf continued to shine where he had cut through it.\n\nCrouched and stumbling, he began to drag the dagger in a line through the grass.\n\nHis heart strained as he moved on. He intended to draw a circle around the Ranyhyn and Mishio Massima; to enclose them in wild magic. But of course such precision was impossible for him. Rather than a circle, he was creating a ragged imitation of one. Nevertheless he persisted; and his silver clung to the grass.\n\nNow he could feel a throb of yearning from Joan's ring in his wrist and forearm. Her wedding band ached for more power. Perhaps it remembered the use which she had made of it, and craved ruin. But there was no wish for harm in Covenant's heart, and he was familiar with wild magic. His argent did no hurt.\n\nStaggering, he looked around to get his bearings. Then he went on, pulling Loric's dagger through the grass; inscribing his crude and hopeful mockery of a circle.\n\nHis heart beat harder. He tottered from step to step in a cripple's stoop that cramped his lungs, exhausted his muscles. He wanted to stop. Wanted rest. An end to all this striving and inadequacy.\n\nBut he wanted other things more.\n\nGradually he passed behind the horses. Over his shoulder, he could see the place where he had begun. It still shone as though it fed on streams of his life-blood.\n\nCome on, leper, he urged himself. Just take it one step at a time. One step. At a time.\n\nDrawing argent with him as if it were alive in the grass, he went on.\n\n\"Hold to your purpose, ur-Lord,\" Clyme urged. \"You near its completion.\"\n\nCovenant did not glance at the Humbled. His attention was fixed on the end of his leper's circle, his lurching enclosure. For no better reason than exhaustion, he was holding his breath. His muscles sobbed in protest. He nearly fell through the last few steps.\n\nThe argent would fade quickly if he did not continue to feed it. He did not have time to straighten his back, or breathe, or run to his mount.\n\nSomehow he had to do it.\n\nBut before he could decide to take the risk, Clyme snatched him from his feet. Cradled in Clyme's arms, Covenant was carried to his horse, tossed carefully into the saddle. At once, Branl caught Covenant's arm to steady him while Clyme sprang for Hooryl's back.\n\nThe world seemed to veer and yaw. There was not enough air, never enough air; or Covenant had forgotten how to inhale.\n\n\"Now, ur-Lord,\" Branl instructed him. \"It must be now.\"\n\nThe enclosure was already starting to flicker and go out.\n\nCovenant's companions raised his arms for him. They lifted Loric's _krill_ and Joan's ring high over his head. Together they helped him strike the dagger's gem with white gold a second time.\n\nJust for an instant, the Unbeliever became a conflagration again, a being of fire and theurgy. Then the Ranyhyn and Mishio Massima surged forward\u2014and the world vanished as though it had been erased from existence.\n\nhen his mount hit the ground at a full gallop, Covenant nearly lost his seat. His feet had not found the stirrups: he could not steady himself. And the after-flash of power filled his head. He flopped in the saddle like a loosely filled sack. Without the support of the Humbled, he would have fallen.\n\nHe had no idea where he was. The _krill_ 's brightness effaced his surroundings. It made black night where there may only have been twilight. Illumined by silver, the horses pounded the turf: he recognized nothing else. For all he knew, he and his companions had only traveled a dozen strides.\n\nBut then the Ranyhyn and Mishio Massima began to slow their gallop to an easy canter. Although images of wild magic still spun like vertigo in Covenant's mind, his body began to recover its center. His extremities were numb: the nerves of his torso and hips and thighs were not. They reacted reflexively.\n\nBy slow increments, he became aware that he was holding the dagger dangerously close to his mount. For Mishio Massima's protection as well as his own, he flipped the fabric that shielded his hands over the blade; covered the gem.\n\nAt once, darkness swept over him. It felt strangely like solace.\n\nHe almost said, Have mercy on me. Instead he managed to pant, \"What happened? Where are we?\"\n\n\"It appears, ur-Lord,\" Branl replied, \"that your efforts have succeeded.\" He took the _krill_ from Covenant, wrapped it more securely in Anele's raiment. \"We gauge that we have traversed some two score leagues, perhaps more. And our heading is to the northwest. The distance to Sarangrave Flat has been halved.\n\n\"In a sunless world, time is difficult to ascertain. Yet we are able to discern its passage. By our measure, an hour remains ere this gloom surrenders to true night. Our translation hither has not been altogether instant. Nevertheless we have been swift beyond comprehension.\n\n\"Ur-Lord\"\u2014for a moment, the Master appeared to hesitate\u2014\"if your strength suffices for a second exertion, we do not doubt that we will gain the marge of the Sarangrave. Mayhap we will do so ere _turiya_ Herem threatens the lurker.\"\n\nA second\u2014? Covenant groaned to himself. Hellfire! Ask me to bring back the sun while you're at it. The dusk seemed to wheel around him as if it arose from his dizziness; as if he were the source of the enshrouding twilight. His legs and back would not suffer the strain.\n\nIf he staggered just once\u2014if he pulled the _krill_ out of the grass for any reason\u2014he would have to start again from the beginning.\n\n\"Your weariness is plain,\" Clyme continued. \"But _aliantha_ will restore you.\" He showed Covenant his remaining treasure-berries. \"Then we will aid you.\"\n\n\"Aid me?\" Covenant asked. Mishio Massima cantered smoothly\u2014and yet he felt that he was seated on rolling logs or a canted boulder. \"How?\"\n\nClyme faced him through the dulled grey of the air. \"We will devise a means.\"\n\nCovenant stared. \"Well, damnation,\" he muttered after a few heartbeats. \"Since you put it that way\u2014\"\n\nWhen had any _Haruchai_ ever failed him?\n\nThe Ranyhyn appeared to understand. With Mishio Massima, they dropped from a canter to a trot and then a walk. In a moment, they halted.\n\nClutching the saddle horn with one hand for balance, Covenant reached to Clyme for food.\n\nHis hunger surprised him. He had eaten enough earlier; more than enough. But as soon as he bit into the first berry, he found that he craved the Land's nurturance. Convalescence was a harsh taskmaster; and his first expenditure of wild magic had depleted his stamina. Careless of future needs, he ate eagerly.\n\nIt was entirely conceivable that he had no future.\n\nStill the taste and efficacy of _aliantha_ gave him their blessing. After a few swallows, the whirling in his head subsided as new energy anointed him with possibilities. As if he were choosing his fate, he devoured Clyme's supply of treasure-berries. Formally, like an act of contrition, he thanked both of the Humbled. Then he announced that he was ready.\n\nHis mount seemed oblivious to everything except the chance to crop grass. But the eyes of the Ranyhyn rolled fretfully, and long tremors ran through their muscles. He did not believe that they were exhausted: they were the great horses of Ra; and they had not lacked for forage and water. Rather he guessed that they were afraid. They knew where they were going.\n\nSomething about the lurker\u2014Covenant had heard tales of their old trepidation, the only dread that they had never mastered. No doubt he had once known why they felt such fear. Now that memory was gone, lost when he had sealed the cracks in his flawed mind.\n\nThinking about the lurker, he felt a pang of his own. He had personal memories of Horrim Carabal; private reasons to be afraid. That the lurker of the Sarangrave feared white gold and the _krill_ was no comfort. If _turiya_ Raver managed to take possession of the monster, Horrim Carabal would resist Covenant with malice as well as terror.\n\nNevertheless he did not hesitate. \"What now?\" he asked his companions. \"How are we going to do this?\"\n\n\"I will bear you, ur-Lord,\" Clyme answered, \"in such a way that you need only press the _krill_ into the grass.\" He dropped from Hooryl's back, offered his arms to help Covenant dismount. \"Thus supported, you will complete the enclosure more swiftly.\"\n\nHe did not add that any circle he fashioned would be more symmetrical than Covenant's.\n\n\"Ah, hell,\" Covenant sighed. \"Why not?\" As he let Clyme lift him down, he muttered, \"But it's too bad you couldn't think of anything even less dignified. I should at least try to look as pitiful as I feel.\"\n\nThe Humbled gazed at him without expression. Neither of them replied. Calmly Branl surrendered Loric's blade.\n\nSwearing under his breath, Covenant accompanied Clyme away from the horses. He had never accomplished anything without help; and yet he still had not learned how to accept assistance gracefully. Being a leper had taught him to think and act and live alone. He ought to act on his decisions without hazarding anyone else.\n\nUnfortunately he could not pretend that he was strong enough for his task. When he and Clyme reached a safe distance, he said harshly, \"Let's do this. I'm not getting any younger.\"\n\nVexed at himself, he unwound cloth from the _krill_ 's gem. In the abrupt wash of radiance, he closed his left fist and punched the strange stone with Joan's ring.\n\nAgain he seemed to become argent delirancy. Power burned in his veins, flamed from his flesh, sprang toward the dying stars. In spite of his mortality, he felt that he had the resources of gods. The sensation was terrible and delicious, an exaltation of wild magic; capable of anything. But it was also brief. It forsook him as soon as he separated his hands.\n\nStill the effects of that moment clung to him, vivid as vision or prophecy. He hardly felt Clyme scoop him from the ground. He was scarcely aware that Clyme bent low, holding him within easy reach of the turf.\n\nAs if of its own volition, the dagger's blade sank until it pierced grass and cut soil, pulling Covenant's clasp with it. Then Clyme began to move so that the _krill_ sliced the earth with shining silver.\n\nSecured against vertigo by Clyme's unyielding arms, Covenant watched as the flow of power which sustained his line in the grass emanated from Joan's ring aching on his finger. Indirectly, therefore, it came from the secret recesses of his heart. That was why he had been left so depleted\u2014and so hungry. With wild magic, he expended his own spirit.\n\nHe wanted Clyme to hurry.\n\nClyme did not appear to make haste. Nevertheless he had already completed a perfect semicircle. From behind the horses, Covenant could see the spot where he had started. He would reach it in a score of heartbeats.\n\nPerhaps because Clyme moved with such alacrity in spite of his crouched, crab-wise steps, or perhaps because his circle was so exact, Covenant's power shone more brightly, promising translation across a greater distance. With the help of the Humbled, he might have been able to travel the Lower Land from border to border in mere hours.\n\nThe thought of such imponderable speed made him dizzy again. If he could stop _turiya_ \u2014and if he could do at least _some_ thing to help Horrim Carabal survive the Worm's arrival in the Land\u2014he might actually have time to rejoin Linden. Wherever her own exigencies had taken her, he might be able to find her.\n\nIf.\n\nThen the enclosure was done. It shone like the _krill_ , defining itself against the gloom. At once, Clyme surged upright. Sprinting, he carried Covenant toward their mounts. Before Covenant could regain his balance, he sat in Mishio Massima's saddle. Branl steadied him while Clyme mounted Hooryl.\n\nAs if he were pitching himself over a precipice, Covenant brought the ring and the gem together above his head.\n\nHe became an instant of wild magic; and reality vanished as the horses sprang into a gallop.\n\nHe could not perceive time. He had no opportunity to draw a breath. His heart did not beat, or he did not feel it measure out his life. The disappearance of the world was as sudden as a blink, complete as soon as it began. Yet time must have passed. When the world reappeared, the horses were running hard, pounding along uneven slopes at the full extent of Mishio Massima's strength. And the half-light, the gloaming\u2014\n\nThe dusk had deepened. The horses galloped in the core of the _krill_ 's illumination; but beyond it, the darkness looked solid as a wall. Covenant and the Humbled had ridden into a realm of shadows, or night had fallen.\n\nWhile he reeled, he tried to ask, Now where are we? But his throat was too tight to release words.\n\nAfter a moment, however, the horses began to slow; and Branl urged him to cover the _krill_. \"When you are no longer blinded by its light, you will perceive that the Sarangrave is nigh. It lies a stone's throw to the west.\"\n\n\"Here _turiya_ Herem's spoor is strong,\" added Clyme. His tone was sharper than Branl's, whetted by anger or anticipation. \"Nonetheless it appears that we are belated. The scent enters the wetlands ahead of us. Indeed\u2014\" The Master paused as if he were tasting the air. Then he stated, \"We discern struggle, a contest of powers. Frenzy lashes the waters at some distance. We deem that a battle has begun.\"\n\nBegun\u2014? Alarm ran like acid along Covenant's nerves. In an instant, he forgot dizziness, fatigue, depletion. \"Hellfire,\" he rasped. \"This is my fault. I took too long.\" Recovering. Thinking. \"Now I'm going to have to do this the hard way.\"\n\nInstead of veiling Loric's dagger, he held it over his head. A beacon\u2014\n\nSpectral against the coming night, tangled brush and gnarled trees became visible off to Covenant's left: limbs and twigs that resembled bleached bones in the silver light; clumps of reeds like thickets of spears; dark floating pads with nacreous flowers; noxious scum; troubled waters so black that they refused lumination. The tenebrous air was thick with stagnation and rot, the putrid remains of corpses. The fetor made knots in Covenant's guts. Instinctively he wanted to shy away.\n\nNevertheless the Ranyhyn and Mishio Massima cantered toward the area where _turiya_ Herem had entered Sarangrave Flat as if that were Covenant's truest desire.\n\nHell and _blood_. He was not ready for this. Not after everything that he had already endured.\n\nEven his blunt nerves sensed the inherited dread that gathered in Rallyn and Hooryl.\n\n\"Ur-Lord.\" Branl held out his hand, asking for the _krill_ as though he believed that he and Clyme could fight for the lurker in Covenant's stead.\n\nBut Covenant kept his only blade, his only light. He had no intention of risking his companions in the vile marshes of Horrim Carabal's demesne.\n\nFar away through the scrub and trees, the scrannel brush and marshgrass, he caught flickers of a diseased silver that reminded him of his one confrontation with the lurker many centuries ago. Instinctively he believed that the monster was exerting its malevolent theurgies against the Raver. If Horrim Carabal had welcomed _turiya_ 's possession, there would be no battle.\n\n\"Ur-Lord?\" Branl asked again.\n\nBloody damnation! Covenant had to act. He was already late. He chose to believe that the lurker was fighting hard; but as the Raver mastered more and more of Horrim Carabal's imponderable bulk, the monster's resistance would weaken. Soon the lurker might begin to submit.\n\nWhile the horses closed the distance, Covenant raised his voice. \"We need the Feroce! I won't ride in that marsh. Some of those waters can strip flesh off bones.\" This decision, at least, his companions would approve. \"And I don't know how else to communicate with the lurker!\"\n\n\"We are come too late,\" countered Branl. \"Already the Raver lays claim\u2014\"\n\n\"But he hasn't won yet,\" Covenant retorted. \"Horrim Carabal is _huge_. _Turiya_ can't overrun the whole lurker at once. Parts of that monster must be fighting back.\n\n\"I need to talk to it while it can still resist!\"\n\nIf Lord Foul's servant triumphed, Horrim Carabal would be a horrific foe.\n\nClyme's passion grew stronger, feeding on a private repudiation. \"We know not how to summon the lurker's acolytes.\"\n\n\"Then they'll just have to summon themselves,\" Covenant snapped. If they could discern his beacon. If their fear of white gold and Loric's _krill_ alerted them to his presence. \"If they don't, what good is an alliance?\"\n\nThe wetland was close: too close for Rallyn and Hooryl. Their fright showed in their flaring eyes; in the tremors which marred their strides.\n\n\"Stop!\" Covenant shouted to the horses. \"I want to stop here!\" Then he swung one leg over Mishio Massima's back; stood in the stirrup and braced himself to drop to the ground.\n\nHooryl and Rallyn complied. With the Ardent's mount between them, they slowed in sharp jerks, almost locking their knees. Within half a dozen strides, they halted, quivering as if they were feverish.\n\nAt once, Covenant let go and hit the grass, running toward the border of the Sarangrave, and waving the _krill_ : a signal to any being or creature capable of noticing him.\n\nClyme and Branl accompanied Covenant as if they had expected his unpremeditated rush. In the sweeping wash of argent, they looked as ghostly as the wide wetland; as vulnerable to banishment as the Dead. Still they were _Haruchai_ , as solid as their promises. Covenant did not doubt them.\n\nBut now he feared them. Their _ak-Haru_ had judged them severely\u2014and they bore an old grudge against Ravers. He shuddered to imagine how they would react when they learned that he meant to leave them behind.\n\n\"I'm here!\" he yelled as he hit soggy ground, stopped at the water's edge. \"We made an alliance! I want to keep it, but I can't if you _don't hear me_!\"\n\nHe needed to know how far into the marsh _turiya_ 's possession had spread. And he needed to get there; to the point of conflict, the heart of the struggle. Nothing that he tried would work if he did not first get ahead of the Raver.\n\nHe wanted the power to _forbid_ Lord Foul's servant, the ancient puissance of the Colossus; but that knowledge was lost.\n\nThrashed by distant fighting, the water at Covenant's feet heaved against its scum and muck. Gouts of tiny plant life rose into the air like miniature geysers, then slumped back into the slime. He thought that he heard screaming, inarticulate fury like far-off thunder; but he could not be sure through the slosh and slap of the disturbed wetland. He strained his eyes for hints of the Feroce, but the _krill_ 's radiance blinded him to everything beyond its reach. Again he yelled for attention\u2014and still there was no sign that he had been heard.\n\n\"God _damn_ it! What good is an alliance if you won't help me at least _try_ to honor it?\"\n\nNothing.\n\n\"Ur-Lord,\" Clyme offered, \"we will bear you. We discern the conflict, though it is distant. We will convey you to a place where you may strike with some hope of effect.\"\n\n\" _How_ distant?\" snarled Covenant. \"Is it leagues? Can you imagine what will happen to you if you try to carry me through _leagues_ of this stuff?\" He slapped a gesture at the marsh: bogs and quagmires; quicksand; depths and shallows; poisoned pools as harsh as vitriol. \"And _turiya_ is going to keep moving. What if he takes possession faster than you can travel? Our lives will be wasted.\"\n\nFacing the Sarangrave again, he howled, \" _I need the Feroce!_ \"\n\nHe had time to panic\u2014and time as well to admit that behind his alarm lay a secret relief at the possibility that he might be spared.\n\nThen Clyme nodded once. \"Ur-Lord, you are answered.\"\n\nHell and blood\u2014\"Where? I don't see anything.\"\n\nCovenant expected flickers of green like hints of the Illearth Stone, an approach of power the hue of sick and rotting chrysoprase. But though he searched until his temples ached, he found nothing except _krill_ -light and darkness.\n\n\"On other occasions,\" Branl answered, \"we beheld the Feroce bearing fires in their palms. Yet when the Masters observed them in centuries past, they moved within the Sarangrave without flames\u2014indeed, without any evident magicks. We surmise that they require theurgy only when they are parted from the wetland.\n\n\"Nevertheless we discern them. Two now approach.\"\n\nTwo? Covenant stared and saw nothing. Only two?\n\nWould two be enough?\n\nAt the limit of the light, he spotted a blur of movement. The creatures were stealthy, creeping behind clumps of scrub, stealing through pestilential grasses and mirkweed, crouching among trees that writhed as if they were in torment. He recalled the timidity of the lurker's acolytes during his earlier encounter with them. They had called him _the Pure One, wielder of metal and agony_ , and they had feared him. Without their High God's command, they would not have dared to enter his presence.\n\nBut he had no time for their craven courage. \"I'm _waiting_ , dammit!\" he shouted. \"I made a promise, and I intend to keep it! Your High God _needs_ me!\"\n\nFronds rustled some distance away. Passing bodies contradicted the sluggish distress of the waters. At unexpected moments, the large round eyes of the Feroce caught reflections of silver. They were hardly tall enough to reach Covenant's chest. And they were desperately afraid. Naked and hairless, clad only in the commandments that ruled their fright, they slipped between patches of cover or ducked under pads and rushes as if they believed that Covenant could extinguish them with a glance.\n\nBut at last they emerged. At the boundary of the marsh, they risked the _krill_ 's radiance.\n\nFlinching, the Feroce brought forth guttering emerald from the palms of their hands. Then they crept onto the mud that marked the border of the Sarangrave. There they stood before Covenant, cowering in supplication.\n\n\"Be merciful!\" they whimpered as if they shared one voice; one mind. \"You are the Pure One. You wield abhorrent metal and deliver agony. Such agony! Yet you accepted our High God's alliance. The Feroce surrendered many and many lives to complete his offered service. Take pity upon us now. Become the Pure One who redeems, as you have done before.\n\n\"Our High God cannot withstand the horror that assails him.\"\n\nTheir tone was piteous, but Covenant felt too much pressure to respond gently. \"I'm not the Pure One,\" he retorted. \"I've never been the Pure One. But I try to keep my promises.\"\n\nIn truth, he had not committed himself to fight for the lurker. Deliberately he had withheld that reassurance. As far as he was concerned, however, Horrim Carabal had exceeded the terms of their agreement. And he believed that the lurker had a role to play in the Land's defense, although he could not name it.\n\n\"Right now,\" he continued without pausing, \"I can't. I'm too far away. I'll fight for your High God, but first he has to help me. He has to take me where I'm needed.\"\n\n\"Not?\" quavered the Feroce as if they had heard only his denial. \"You are not the Pure One? We do not comprehend.\" Their protest sounded like the soughing of bogs, the suck of quicksand deprived of victims. \"You wield vicious metal. You bring excruciation. You have delivered such agony to our High God that he quails to hear you. You are required to be the Pure One. There is no other.\"\n\n\"Stop!\" Covenant demanded harshly. \"Call me whatever you want. We don't have time for this.\n\n\"Here!\"\n\nFrantic to show his good faith, he swept cloth around the _krill_ 's gem and blade. Instantly the light vanished. Night rushed over the region: it seemed to reel in its haste to fill the void left by covering the dagger. The fires of the Feroce revealed only themselves.\n\nUrgent and awkward, Covenant thrust the wrapped knife into the waist of his jeans, then jerked Joan's ring from his finger, looped the chain over his head, dropped the band under his shirt; made himself appear defenseless.\n\n\"I'll need metal to fight.\" Fear made him savage. \"And I'll have to hurt your High God. I'll have to hurt him _bad_. I need to cut off the infection,\" sever every portion possessed by _turiya_. \"I don't know another way.\" He had no idea how to kill a Raver.\n\n\"But I can't do anything if he doesn't _take me where I'm needed_!\"\n\nThe lurker was enormous. It could survive terrible damage.\n\nAs one, the creatures gave a quivering shriek as if he had appalled them to the core of their soft bodies. Their fires sprang high; dropped low. Flames dripped between their fingers like corroded flesh or spilth.\n\nCovenant swore in frustration. He should have gotten here sooner. If he were not so easily wounded, so damn mortal\u2014\n\n\"Ur-Lord,\" cautioned Branl. \"Ready yourself. Again you are answered.\"\n\nWhile Covenant strove to see, a dark shape arose from the waters.\n\nVisible only as a starker blackness in the dark, a tentacle rose and rose as if it were reaching for the heavens. It was thick as a cedar, tall as an elm. Its surface squirmed with desperation. In spite of Kevin's Dirt, Covenant felt the lurker's strength, its bitter hunger. Reaching high above him, its arm seemed to search with inhuman senses for the taste of its prey.\n\nCovenant had time to tell the tentacle or the Feroce, \"Leave my companions here. They can't help me. I'll need them later.\"\n\nThen the tentacle lashed down. Like a cracked whip, it snapped around him. Its fingers grasped every possible surface of his shirt, his jeans, his limbs. Coils clasped his arms hard to his sides. A heartbeat later, the tentacle sprang back; jerked him into the air with appalling ease.\n\nHe heard no response from the _Haruchai_. Only the voice of the Feroce scaled, frail and frantic, into the dark.\n\n\"Try to believe that you are the Pure One.\"\n\nIn a flicker as brief as a blink, he thought that he saw the Humbled take hold of Horrim Carabal's acolytes. Then the lurker snatched him through the sky as though the monster intended to hurl him into the heart of Sarangrave Flat.\n\nHellfire! He could not move his arms; could hardly breathe. Black trees and obscured streams rushed below him as if they were plunging into an abyss. If the lurker did not fling him to his death, it was going to squeeze out his life.\n\n_Your alliance was a thing of the moment._\n\nThe Feroce would have reacted differently if their High God had been mastered by _turiya_. The Humbled would have tried to ward Covenant. But he could not be sure that the lurker understood his intentions\u2014or knew how effortlessly he might be crushed.\n\nHe had no measure for direction or distance. The wetland seethed like a cataract below him. Night blinded every horizon. The roar of wind in his ears covered the stricken pound of his pulse. When he was thrown, he would soar for leagues before he hit and died.\n\nWithout warning, the coils wrenched him downward. Before he could even try to fill his lungs, Horrim Carabal slammed him into a pool, buried him in deep water acrid with poisons. His eyes would have been ruined in their sockets if he had not clenched them shut.\n\nBut the tentacle did not stop. It tore him through water and muck as easily as it had carried him above the marsh, as if he had no substance and did not need air.\n\nThe monster did not mean him harm. It had good reason to be terrified of white gold and Loric's _krill_. Good reason to fear wild magic. But it did not understand its own strength\u2014or Covenant's weakness. He was dying for air. The corrupt water stung him like a swarm of ants, biting and endless. Apart from suffocation and dread and pain, he felt only nascent fire, as if his mere presence sufficed to set the toxic waters ablaze.\n\nBut he was well acquainted with pain. It was human and inevitable: he could ignore it. And dread was akin to fury. _You are the white gold_. When his fear became a form of rage, he could burn his way free.\n\nSuffocation was altogether worse. Drowning was worse. He could more readily have endured the excoriation within a _caesure_. Drowning was desperation. It led only to unthinking frenzy.\n\nHe had to have air. _He had to have air_.\n\nOr he had to have peace: the silence of the last dark, voiceless and blissful: the surrender of every demand and desire.\n\nAir or peace: one or the other. He could not be given both.\n\nBut he wanted air.\n\nHe would never get it. He was already failing.\n\nStill his given body remembered its own exigencies, its own compulsory striving. It locked itself against the impulse to inhale death\u2014\n\n\u2014until the lurker suddenly ripped him upward.\n\nHe knew nothing; remembered nothing; could not interpret his changed rush through the fluid dark. But the flesh which Linden had fashioned for him was ruled by strictures that did not require conscious choices. As the tentacle heaved him out of the water and thrust him high, the pressure in his chest seemed to explode. Bursting, he found air.\n\nFor a time, nothing existed except wretched gasping and life. Blots like devoured stars swam across the void inside his eyelids, inside his head. Air and the wind of his blind movement exacerbated the sting of the waters until it felt feral, as fierce as wasps. Every breath was tumid and rank, difficult to take. The night tortured him with questions for which there were no answers.\n\n_Try to believe that you are pure._\n\nBecause he had to see, he slitted a glimpse outward and found ruptured dazzles there as well.\n\nHis eyes bled tears. Light smeared his vision. The shining was a noxious silver like and unlike the alloyed clarity of wild magic. And it was tainted by an underhue of emerald that resembled the virulence of the Illearth Stone. He did not understand it. The tentacle jerked him from side to side, asking its own febrile questions. The Sarangrave's fouled waters clung to his skin like scales. He felt blisters bubbling everywhere.\n\nBut tears washed away bitter minerals and evil. Blinking rapidly, he began to see.\n\nBelow him stretched a pool the size of a small lake. It veered one way and another as the tentacle squirmed. Its surface blazed with a nacreous lucence as dangerous as necrosis.\n\nFrom the depths of the water rose two more tentacles. They were thick as towers, supple as serpents, mighty as siege-engines. And they were locked in battle. One struck at the other while the other writhed to avoid blows that would have toppled oaks. The ferocity of their movement churned the pool to froth. Their struggle cast shadows like screams across the wetland, but did not quench the light.\n\nThe attacking arm feinted to distract the other. An instant later, the attacker flung itself like a noose around its foe near the water-line. It tightened and strained, apparently trying to rip the other arm in half.\n\nAt first, Covenant did not recognize what was happening. Then he did. The lurker seemed to be fighting itself, but it was not. It was resisting the Raver. Covenant felt _turiya_ 's loud malevolence in the caught tentacle. The Raver's mastery of the monster had reached this far along one arm. Now Horrim Carabal strove to tear off the possessed part of itself before _turiya_ could claim more.\n\nA doomed struggle: the lurker could not clench tightly enough, dismember itself swiftly enough. And it could not make the Raver flinch or shy because the Raver was not afraid. Moments after the monster grabbed its own arm, Covenant saw _turiya_ Herem's evil slip past the constriction and spread farther.\n\nThe lurker released that arm, tried for a new grip. What else could the monster do? But it could not preserve itself by that means. The truth was plain. The Raver's viciousness moved too easily. Even if the lurker contrived to stop _turiya_ in one place, Lord Foul's servant would simply shift his possession to another tentacle.\n\nA timid shriek thronged into the dark sky. Around the pool were gathered the lurker's worshippers, hundreds of them. Some stood to their waists in the water: others crowded the verge. From all of their hands shone green fires, bright desperation. Their wailing was a ululation of terror. But their hands and flames moved in unison, dropping low and then rising high as one, swaying from side to side like an invocation.\n\nIn the distance behind them crouched tormented growth and lurid streams, helpless in spite of innumerable toxins. Beyond the light lay beleaguered darkness.\n\nThe Feroce were trying to save their High God. Surely that was what they were doing? But Covenant had no idea what they sought to accomplish.\n\nThen he understood.\n\nTwo nights ago, in his cave above the Sunbirth Sea, the lurker's creatures had given him unexpected aid. Wielding their peculiar theurgy, they had caused the Harrow's prostrate destrier to recover its captious nature. _We have not given it strength_. _We cannot_. _But we have caused it to remember what it is_. That gift had enabled the beast to bear Covenant farther than he would have thought possible.\n\nNow the Feroce were fighting for the spirit of their High God with the only power they had: the power to impose memories. Frantically they struggled to help the lurker recall freedom.\n\nThat effort, too, was doomed. _Turiya_ Herem was stronger.\n\nNevertheless the effects of emerald worship and panic granted Covenant a little time to gather himself.\n\nHe could not help the lurker as he was, trapped in the tentacle's coils. But he had only one way to communicate with the monster, to explain his needs and intentions; and the turgid atmosphere resisted every breath. His gasping did not bring in enough good air to support a shout that the Feroce might hear.\n\nHe tried anyway.\n\n\"Listen,\" he croaked: a sound too small to pierce the forlorn shrieking; the savage slash and pound of tentacles; the turmoil of bright water. \"I want to fight, but I can't move my arms. I have to reach the _krill_. And your High God has to work with me. We have to fight together.\"\n\nHis flawed sight detected no sign that any of the creatures had heard him.\n\nStill the lurker of the Sarangrave feared possession more than pain. Doubtless the monster did not understand what Covenant had said. Yet it recognized that he had spoken. Perhaps it had felt his resistance as he squirmed against its coils.\n\nAbruptly a fourth arm reached out of the scourged pool. It snatched up a cluster of the Feroce. Wrapping them like Covenant, the massive appendage lifted them until he could look straight into their appalled eyes.\n\n\"Listen,\" he panted again. \"I need my arms. I have to reach my knife.\" It was likely that the Feroce did not know Loric's dagger by name. \"And your High God has to carry me to the right place. The place where I can cut off the horror, all of it.\n\n\"Make him understand. We have to do this _now_.\"\n\n_Turiya_ did not fear the lurker, but he would fear the _krill_. He would fear wild magic.\n\nRound eyes gaped at Covenant as if they had been blinded. The creatures had been crying out continuously. They did not stop. And there was no difference between the wailing in front of Covenant and the shrieks from below. All of the Feroce had one voice, the same voice. They uttered only anguish.\n\nYet the grip of the lurker's arm loosened. Its fingers shifted the coils lower on Covenant's chest.\n\nStill he could hardly breathe. The air was too damn thick\u2014\n\nWith all of his insignificant strength, he tried to grasp the _krill_.\n\nThe tentacle moved farther. After a moment that made dots of weakness dance across his sight, his halfhand found the dagger.\n\nNow, he thought. Hellfire! _Now_.\n\nHolding his weapon for his life, he drew it free. Dropped its covering. Raised it over his head in both hands.\n\n\"I'm ready,\" he gasped. \"Do it!\"\n\nWith actions as plain as language, Horrim Carabal chose agony. Any maiming was better than possession. In an instant, the monster stopped fighting itself. With a ponderous heave of its possessed tentacle, it extended the boundary between itself and _turiya_ Herem's mastery higher and then higher; away from the corrosive waters; closer to Covenant's elevation.\n\nAs if Covenant were an axe, the lurker swung him at a section of the massive arm which _turiya_ had not yet claimed.\n\nIn every limb, Horrim Carabal had the strength of half a dozen Giants. It struck with the force of frenzy. Covenant whipped forward like the crack of a flail. When his blade bit flesh, any ordinary weapon would have been ripped from his clutch. But wild magic whetted the edges of Loric's _krill_. Spitting flames, the dagger cut. Covenant hardly felt the impact.\n\nHis blow sliced partway through the tentacle. Vile blood fountained from the wound. It stank like distilled corruption. The whole of the Sarangrave seemed to erupt in an excruciated howl as if every leaf and stem and bog, every current, every swath of scum gave voice to the lurker's pain: a howl so vast that it effaced the thin shrieking of the Feroce.\n\nBut the tentacle was not severed. It was far too thick to be lopped off by a single slash. Through the gush of blood and the yowling, Covenant felt _turiya_ hesitate in alarm; draw back. In another moment, however, the Raver would surely control his fear. He would rush to pass beyond the cut deeper into Horrim Carabal.\n\n\"Again!\" Covenant rasped, although he could not hope to be heard. He could not hear himself. He needed the lurker to understand that if it did not ignore its hurt\u2014\n\nLike the Raver, the monster hesitated.\n\nThen it recovered its fury. Still howling like myriad ghouls, like the immeasurable torment of the damned, the lurker swung Covenant again.\n\nThe second slash cut through more of the appendage. Torrents of blood stained the waters, and were swallowed by shining. The lurker's roar seemed to batter Covenant's bones. Dazed by conflicting brightnesses, he could no longer see. The _krill_ 's heat ached in his wrists. Soon he would be too badly burned to hold on.\n\nBut now the monster did not hesitate. Savagely it swung yet again.\n\nFlopping like a doll in Horrim Carabal's coils, Covenant delivered a third cut.\n\nThe possessed arm was toppling. Still it had not been entirely severed. And while it fell, the Raver's lust to rule the lurker overcame his fear of Covenant's power. Vicious as a striking asp, _turiya_ Herem surged forward.\n\nAs if the monster's pain and rage had become his, Covenant thought, Over my dead body.\n\nIn a rush like delirium or exaltation, the Unbeliever and his ally hacked once more\u2014\n\n\u2014and the slain tentacle crashed down into the pool.\n\nStunned by howling and hot blood, Covenant struggled to retain his grip on the _krill_ ; his grip on himself. Cutting off the claimed limb was a temporary victory at best. The Raver had not been harmed. If Covenant did not strike again instantly\u2014if he did not force _turiya_ to defend against him\u2014Lord Foul's servant would escape. At need, the Raver could claim one or more of the Feroce. He might feel demeaned by their littleness, but he could conceal himself among them nonetheless. And if Covenant failed to locate him before he rallied his strength, he could make another attempt on the lurker.\n\nCovenant felt like a toy in the hands of an insane juggler, utterly disoriented, impotent with vertigo. Up and down had become the same thing. He could not distinguish any of his horizons. The violence of the monster's movements seemed to have dismembered him.\n\nStill he refused to accept a victory that might become defeat at any moment.\n\nThe lurker had done its part. The rest was Covenant's problem. He had to do something.\n\nNow or never.\n\nWith as much haste as Horrim Carabal's thrashing allowed, he tugged the chain holding Joan's ring over his head, clasped the hard circle in his left hand. Then he slapped the ring and the dagger's gem against each other.\n\nWithout transition, conflagration erupted in him as if his living flesh were tinder.\n\nSudden power anchored him. Disarticulated pieces of his surroundings were flung back into their natural relationships. But he did not care about his horizons, or his position in the air, or the lambent waters. The strange voice of the Feroce meant nothing to him. He needed\u2014\n\nThere, in the pummeled pool; in the corpse of the cut tentacle subsiding toward the depths: _turiya_ Herem. He felt the Raver's presence as if it were louder than the monster's roar.\n\n\u2014needed the lurker to drop him.\n\nHis fierce fire succeeded. It made the monster's coils flinch, loosen. Voluntarily or involuntarily, Horrim Carabal let go of him directly above his target; and he fell.\n\nFor an instant, he tumbled helplessly, out of control. But he was far from the waters when the lurker released him; and his fire made everything clear. Wild magic lit his nerves as if it were percipience. He had a sharp shard of time in which to master his limbs, twist his posture into a dive.\n\nStill holding white gold against Loric's gem, he struck the turbulence head first and plunged deep.\n\nAt the last moment, he remembered to shut his eyes. This water would blind him. It would scald his skin until it fell from his bones. But he was too frantic and furious to care. And here he did not need sight: there was nothing to see. He only had to sink faster than the tentacle. He had to reach it before _turiya_ could escape.\n\nHe sensed the Raver's terror. It filled the pool, as bright and bitter as the waters. But he also felt the slain appendage below him. It was close.\n\nAs he hit the still-squirming arm, he hammered the _krill_ into it and sent a blast of passion along its length, striving with his last breath, his last strength, to shred the Raver. If he accomplished nothing else with his life, he would at least give Linden the lurker of the Sarangrave as a potent ally rather than a lethal foe.\n\n_\u2014writ in water._\n\nDelirious and resolved, he poured out his heart until he felt _turiya_ Herem's spirit begin to fray.\n\nTorrents of wild magic ripped through the Raver. _Turiya_ was ancient and enduring, single-minded in his malevolence. He withstood more force than Covenant could have survived. But he was going to die.\n\nIf Covenant did not falter first.\n\nBadly burned, and dying for air, he grew weaker. He was human, after all, heir to every inadequacy that made life precious. No matter what his determination demanded of it, his body could not absorb unlimited quantities of damage. There were prices to be paid for the feats which he asked of himself. Collapse and unconsciousness were only the beginning.\n\nWithout the lore of forbidding\u2014\n\nBefore the end, however\u2014the Raver's end, or Covenant's\u2014the _krill_ was snatched away. Covenant almost dropped Joan's ring, but its chain was tangled in his fingers. Strong arms closed around him, bore him surging toward the surface. He had no time to remember that he was not done; that the Raver was still alive. His head was lifted into the air. Of its own volition, his defeated body fought for life.\n\nStentorian as a Giant, Branl shouted, \"These waters harm the Pure One! He must be relieved!\"\n\nThe Master supported Covenant with one arm. In his other hand, he clasped High Lord Loric's dagger.\n\nDeprived of theurgy, Covenant's head reeled. He could not understand what was happening, could not think, could hardly suffer the burns that ravaged his skin. Confused and thwarted, sick with vertigo, he did not recognize it when the tip of a tentacle slipped between him and Branl; when it curled around him and pulled him out of Branl's embrace. He only knew that now he hung in the air close to the seethe and lash of the pool. He did not know how or why.\n\nBelow him, Branl sculled as if he were waiting. In fragments of residual clarity, Covenant saw a fretwork of fine blisters on the Humbled's arms. The fabric of Branl's tunic appeared to be rotting on his shoulders. The gem of the _krill_ blazed with power that seemed purposeless, devoid of meaning.\n\nHellfire, Covenant groaned as his mind wandered among his defeats. Hell and blood. What have you done?\n\nThen he felt _turiya_ Herem rising. The viciousness of the Raver's aura pierced Covenant's bewilderment.\n\nWith the slow deliberation of a torturer, Clyme of the Humbled broke the surface in front of Branl. They were no more than two arm spans apart.\n\nAt the sight, Covenant's confusion became keening. That was not Clyme: it was _turiya_. The Raver's presence was too fierce to be mistaken for anything else.\n\nThe light of the acrid waters reflected in Clyme's eyes like the eagerness of depravity. The grin baring his teeth anticipated bloodshed and triumph.\n\nOh, hell. Hell and damnation. Clyme was possessed. _Turiya_ Herem had taken him.\n\nThat should have been impossible. Covenant had said as much. He knew it to be true. The _Haruchai_ could not be mastered by anything less than the concentrated evil of the Illearth Stone. They were too strong.\n\nNevertheless _turiya_ wore Clyme's body like a cloak. It was his to use or discard.\n\nCovenant's heart struck blows like knells inside his chest. His mind staggered, clutching at implications, inferences.\n\n_Turiya_ Herem could not have mastered Clyme. That was entirely impossible. It defied reality.\n\nTherefore\u2014\n\nGod in Heaven!\n\nTherefore\u2014\n\nCovenant wanted to wail.\n\nClyme must have _admitted_ the Raver. He _must_ have. No other explanation sufficed. Branl had interrupted Covenant, and Clyme had acquiesced to _turiya_ , so that the brother of _samadhi_ and _moksha_ would not perish.\n\nSo that Covenant would not sacrifice himself trying to destroy the Raver.\n\nSweet Christ! What have you _done_?\n\nIf Horrim Carabal had dropped Covenant again, he would have flung himself at Clyme in pure panic. But the lurker's coils held, and Covenant was too weak to break free.\n\nSwimming with his head and shoulders above the surface, Clyme glared delight at Covenant.\n\n\"Do you behold me, groveler?\" the Raver panted as if words were an unfamiliar exertion. \"You have attempted my end, yet you have not overcome me. Now your companion is mine.\n\n\"Will you slay him to assail me? I judge that you will not. Your heart is flawed. It cannot sustain such deeds.\"\n\nGroveler. That ancient epithet suited Covenant. He deserved it. He had become an avatar of abjection in the flagrant depths of the Sarangrave.\n\nBut _turiya_ Herem was not done. His malice demanded taunts which he spat out with extravagant glee. Only the effort, the obvious difficulty, with which he fought to speak hinted that his exultation was marred.\n\n\"The killing of your mate I acknowledge. There you surpassed my expectations. But her death bore the stench of mercy. That reek will not arise from the fate of Clyme _Haruchai_ , Master and Humbled. His execution at your hands will be purest murder.\" Clyme gnashed his teeth as if he were rending flesh. \"You lack the _belief_ necessary to the task.\"\n\nCovenant thought that he heard Branl say, \"Trust in us, ur-Lord.\" An echo of earlier promises. But he could not drag his attention away from _turiya_ 's scorn and Clyme's surrender. And the turmoil of the pool was loud. It masked every voice except the Raver's. Horrim Carabal's hurt and trepidation seemed to make no sound. Even the Feroce appeared to have fallen silent.\n\n\"Nonetheless,\" _turiya_ gasped savagely, \"I desire you to surpass yourself yet again. I crave the pleasure of your efforts to extinguish me. Should you discover within yourself the valor which you lack, you will learn its futility. Blithely I will disencumber myself of this mad _Haruchai_. While you expend your despair upon your companion's husk, I will possess myself of other lives\"\u2014he gestured around him\u2014\"which attend upon us in abundance. And if you seek me among these timorous wights, I will renew my mastery of their High God.\n\n\"Your death is thereby made certain. If I do not compel the lurker to slay you, the Sarangrave itself will do so. Already your passing has been too long delayed.\"\n\nClyme's grin stretched. He seemed to be screaming. Then he bit down on his lips until his teeth drew blood. The muscles at the corners of his jaw knotted like fists. For a moment, he squeezed the reflections out of his eyes. Anguish and resistance twisted across his visage like noisome creatures crawling under his skin. Blisters burst. They leaked dire fluids. His arms flailed.\n\nWhen he opened his eyes again, the light in them had changed. They caught the _krill_ 's radiance rather than the shining of the waters.\n\n\"This Raver lies.\" Clyme's voice was torment\u2014but it was _Clyme's_. \"He does not hold me. I hold him. I contain him as Grimmand Honninscrave once contained his brother. His mockery and struggles I disdain. He cannot flee. I will hold him while his ruin is achieved.\"\n\nAgain the Master's eyes were forced shut. In spite of _turiya_ 's opposition, however, he reopened them almost immediately. Ignoring the involuntary contortions which complicated his mien, he made his purpose clear.\n\n\"But his end must come swiftly. Though I am _Haruchai_ withal, and potent in my fashion, his malice undermines me.\n\n\"I will hold him.\" He looked, not at Covenant, but at Branl. \"The _krill_ must accomplish his death.\"\n\nAnd Branl did not hesitate. His people did not forgive. Because they did not mourn, they did not know mercy. Nor did they count the cost.\n\nOne swimming stroke took him close enough. Without a heartbeat's pause, he plunged the dagger into Clyme's chest.\n\n_Turiya_ 's shriek exceeded hearing. It scaled higher as though it had the power to make the whole of the Sarangrave tremble. The sound ripped along Covenant's nerves until they seemed to bleed.\n\nClyme's features looked like they were being torn apart. Still he retained the iron intransigence of the _Haruchai_. At the end of his life, he lifted his head to Covenant. While blood gushed from his mouth, he pronounced distinctly, \"Thus I answer the objurgations of the _ak-Haru_.\"\n\nBranl disagreed\u2014or his approval was so great that he could not contain it. Clyme's affirmation unleashed a kind of madness. Violence which had simmered beneath the impassivity of the _Haruchai_ for millennia exploded in the last of the Humbled.\n\nGripping Clyme's shoulder with one hand, Branl cut open the whole of Clyme's torso in a single slash. When the _krill_ found the friable barrier of bone between Clyme's hips, Branl dragged the blade in a circle through the Master's abdomen, disemboweling him. Then Covenant's companion drew back the dagger and began to hack\u2014\n\nCovenant tried to look away. In that attempt also, he failed.\n\nFlesh was soft to the _krill_ 's keenness. Bone meant nothing. In a convulsion of movements so swift that no part of Clyme had time to sink, Branl severed his comrade piece from piece until only gobbets and shards remained. Then finally they drifted away like stains upon the water; and the pool fed on them like a beast devouring tidbits. Moments after Branl ceased his butchery, they were gone, all of them.\n\nOh, Clyme! Is this what you think Brinn wanted?\n\nWhen he was done, Branl swam below Covenant, squinting upward with galled eyes.\n\n\"Are you content, ur-Lord?\" Grief clenched his visage. \" _Turiya_ Raver is unmade. Naught of him endures.\"\n\nBut the same was true of Clyme.\n\nCovenant had no answer. He wanted to weep; but he was in too much pain for tears. The Feroce called him the Pure One. They had asked him to _believe_. But he had not redeemed them, just as he not redeemed their distant ancestors, the _jheherrin_.\n\nThe Humbled had proven themselves. Nevertheless the difference between Saltheart Foamfollower's example and that of Branl and Clyme was more than Covenant knew how to bear.\n\n## 5.\n\nComing\n\nFury and thrashing were gone from the Sarangrave. By increments, the astringent light subsided from the pool. Breathing became marginally easier. With elaborate care, the lurker lifted Thomas Covenant high into the air. Another tentacle arose to bear Branl of the Humbled and High Lord Loric's _krill_.\n\nStately as a procession, at once celebratory and funereal, Horrim Carabal carried its saviors eastward above the writhen trees of the Flat.\n\nScores of the Feroce accompanied them. Scurrying around copses, sinking and then rising in quagmires, drifting like mist across streams and backwaters, the creatures went ahead of their High God's allies. And as they moved, they kept their emerald fires bright in their hands. Like the Wraiths of Andelain, the Feroce thronged forward in homage, escorting Covenant and Branl through the contortions and perils of the lurker's demesne.\n\nBut Covenant ignored the monster's acolytes. With the last of his scant endurance, he clung to Joan's ring and tried to stifle images of butchery. The manner in which Branl had destroyed Clyme burned in his mind. The sight seemed acid-etched inside his lids: whenever he closed his eyes, he saw it. The world had become a visceral dismay that refused utterance.\n\nAround him the fires of the Feroce shed little insight: a gleam of green across scum and mirkweed here, a brief flash on scrub and branches there. But the _krill_ still shone, casting its spectral light through the Sarangrave. The waters had destroyed the dagger's protective fabric covering. The haft's heat must have hurt Branl's hand; but if it did, he gave no sign. His true wounds ran deeper. His countenance was a fist which he could not unclench, and he did not glance at Covenant.\n\nIn the gem's echo of wild magic, tree limbs and marsh reeds as ghostly as spirits bobbed as if they were bowing. Harsh grasses swayed from side to side in consternation or awe.\n\nThen the tentacles paused above a small pond as clear and dark as the ravaged heavens: an eyot of starker blackness in the crowding mass of the Flat. There the damp voice of the Feroce rose. \"Our High God loathes the touch of such water,\" the creatures intoned. \"You will fall. But we have caused the water to recall its ancient purity. It will soothe you while we prepare a more worthy consolation.\"\n\nSoothe you, Covenant thought dully. That would be nice. His body was covered in blisters that stung like the tears which he could not shed. Anything cool\u2014anything that was not gall and bitter lamentation\u2014\n\nThe arms of the lurker sank close to the lightless pond. Briefly they hovered as if they were considering their options. Then they uncoiled.\n\nWith Branl beside him, Covenant dropped into cleanliness that resembled bliss.\n\nThe Feroce had spoken truly. Their magicks had made this water pure. He could drink from it, and drink, without any aftertaste of the seepages and rot which polluted the wetland. Nevertheless it did not heal. It was not Glimmermere. It did not wash away hurts or cleanse souls.\n\nHe needed something more. Untenable weeping filled his chest. He could not shut his mind's eyes against the brutal slash of the _krill_ in Branl's hand.\n\nThe _Haruchai_ swam at Covenant's back, supporting him. That was well. Covenant was too weak to move. And he did not want to look at Clyme's killer.\n\nPerhaps to ease his burned hands, Branl held Loric's knife underwater. That, too, was well. Darkness was another kind of balm. It eased Covenant's aggrieved nerves.\n\nAfter a time, he remembered to replace the chain of Joan's ring around his neck. Then he asked the dusk, \"Did you have to do that? Couldn't you just kill him and be done with it?\"\n\nHe had seen _Haruchai_ fight on any number of occasions, but he had never seen such an abandoned frenzy of violence.\n\nBranl's answer ached across the water. \"It was agreed between us. We remember Grimmand Honninscrave, and the Sandgorgon Nom, and _samadhi_ Sheol. By Grimmand Honninscrave's death, Nom rent the Raver. Yet shreds of that dark spirit endured within the Sandgorgon. They endure still, and cling to malice.\n\n\"We knew no other means by which _turiya_ Herem might be altogether unmade.\"\n\nCovenant nodded to himself. He accepted Branl's justification. What choice did he have? The Humbled had argued against pursuing _turiya_ , or considering the lurker's plight\u2014and still they had accomplished something that Covenant could not have achieved alone.\n\nLater he suggested like an offer of forgiveness, although he did not know how to forgive anything that had occurred, \"Then maybe you'd better explain how you did it. I told the Feroce to leave you behind.\"\n\nSilence held the gloaming for a while before Branl replied, \"It was not difficult to persuade the Feroce that you would have need of us.\" He sounded like the stars, forlorn and doomed. \"Our lives are memory. The creatures have no power to disturb or alter us. And their fear for their High God was extreme. Regardless of your command, they could not reject any form of aid. They summoned the arms of the lurker, that we might follow behind you.\n\n\"Thereafter Clyme and I determined our course together. I chose the task of your life, deeming that purpose paramount. Freely Clyme assumed the burden of the Raver.\n\n\"The _ak-Haru_ spoke of simony. We are\"\u2014his sudden pause had the force of a stab\u2014\"we were the Humbled. We could descry no other means by which we might correct our fault. How otherwise might we have become worthy of the Guardian, and of ourselves?\"\n\nIn a voice thick with woe, he concluded, \"I must believe that good may be gained by evil means.\"\n\nAnd now you're alone, Covenant sighed. This far from any of your people, you're cut off from everything that makes you who you are.\n\nAs isolated as a leper.\n\n_Simony_ , by hell! Covenant breathed faint curses to himself. Branl's people had never been as open-hearted as the Giants. But they had always been generous with their lives.\n\nEventually Covenant began to think that forgiveness might be possible after all.\n\nThen a shiver of anticipation or effort ran through the burning green around the pond. Sharing one voice, the Feroce announced, \"Consolation has been made ready. We are the Feroce. Our High God speaks in us. But you must remove yourselves from the water. To sustain its purity demands much of us, and our High God will not touch it.\"\n\nWith Covenant's consent, Branl swam toward the pond's edge. And when they were able to stand in the muck of the bottom, two tentacles snaked out of the surrounding marsh. As before, the lurker's arms closed carefully around Covenant and Branl, and lifted them high to avoid the trees.\n\nAgain the _krill_ shone silver in all directions, but its light revealed nothing that might be _consolation_.\n\nSide by side, Covenant and Branl arced upward, still moving eastward, into the accumulated dark of night in a sunless world.\n\nHorrim Carabal bore Covenant and his companion so lightly that he had no sense of duration or distance. He only knew that he was moving because the Sarangrave squirmed below him and the air felt like a rasp on his raw skin.\n\nSoon, however, the tentacles descended again. Then the lurker halted once more. This time, the monster held Covenant and Branl over a vaguely shimmering swath of dampness like a pit of quicksand eight or ten paces wide. Here, also, Feroce surrounded the lurker's destination. But now their numbers had become a multitude. Hundreds of the creatures waved their small fires, making the wetland garish, and chanted like worshippers in the presence of divinity.\n\nHorrim Carabal poised Covenant and Branl over the center of the quag, but did not drop them. The Feroce did not speak.\n\n\"Ur-Lord.\" Branl's tone changed. Surprise\u2014or something more than surprise\u2014penetrated his distress. \"Here is a great wonder. I would have avowed that such a\u2014I have no name for it\u2014that such astonishment could not exist in Sarangrave Flat. Surely it is precluded by the manifold illnesses and evils of the lurker's demesne. Yet it is unmistakable. It is\u2014\"\n\nHe stopped as though what he beheld had sealed his throat.\n\nCovenant peered downward, but he saw nothing that did not resemble quicksand or some other mire. He smelled only the cloying scents of rancid plants and putrefaction. The tumid exudation of the lurker's presence made breathing difficult.\n\n\"What is it?\" he murmured. \"What do you see?\"\n\nThe Humbled appeared to wrestle words past an obstruction. \"Ur-Lord, it is hurtloam. _Here_ , where no clean thing grows, and no health flourishes. It cannot be, yet it is.\"\n\n_Hurtloam_. The word sent conflicted squalls through Covenant in spite of his near-prostration and his complex pains. Hurtloam would heal his wounds; but it might also cure his leprosy. It had done so before. It could restore his crippled health-sense. It could make him potent and capable in ways which were denied to lepers.\n\nIt was life and ruin. It would rescue and damn him\u2014\n\n\u2014because his illness was essential to him. _I don't expect you to understand_ , he had told Linden's company in Andelain. _But I_ need _this_. _I need to be numb_. He had believed it then: he believed it now. _It doesn't just make me who I am_. _It makes me who I_ can _be_.\n\nHis leprosy was all that enabled him to hold the _krill_. In some sense, it was a defense against the Ravers. And he was not done. He had to remain as he was until the end.\n\nAnd yet he wanted to be healed. Oh, he _wanted_ it. He had become so much less than he needed to be. Wounds and weakness made him useless. He had nothing left to offer Linden. He would not be able to fight for the Land.\n\nInadvertently cruel, the Feroce and their High God proffered a gift which might also be a curse.\n\nAnd while hurtloam healed him, it would make him sleep. He would miss his chance to redefine his alliance with the lurker; perhaps his only chance. After everything that he and the Humbled had done to secure the terms of the bargain\u2014\n\nFearing the worst, he croaked, \"Wait!\" If the tentacle dropped him now\u2014\"Hellfire! Just wait!\"\n\nAt once, the Feroce stilled their chanting. Horrim Carabal did not let go.\n\nTogether the creatures spoke. \"Memory is a potent magic. We are the Feroce. We serve our High God. We have caused this small portion of his vast realm to remember what it was. The task has been arduous. We have expended much to complete it. But we are unworthy of the majesty which we worship. We have prepared this consolation because our High God has commanded it, and because we have failed in our service.\"\n\nImpassive now, Branl asked, \"How have you failed?\"\n\nA shudder passed through the throng. Emerald guttered in every hand. But the Feroce did not refuse to answer.\n\n\"We hazarded much, fearing the Pure One's wrath. Yet we are the Feroce. We serve our High God. For his life, we strove to awaken recall in the Pure One.\"\n\nWith those words, the small creatures drew Covenant's attention away from the conundrum of hurtloam.\n\n\"Our High God has not forgotten,\" they explained. \"He is vast in all things. He recalls a time when a strange force forbade the horror which you have slain from venturing beyond the great cliff in the west. We cannot conceive such might. But the Pure One knows forbidding. He has forgotten it.\n\n\"For our High God's sake, we sought to awaken memory. Forbidding would have served him better. It would have inflicted less agony. He would not have suffered abhorred metal and fire.\n\n\"Alas, the Pure One has sealed himself against recall. We could not elicit his knowledge. In that, we failed our High God. Our shame is great.\"\n\n\"Wait,\" Covenant demanded again. \"You mean you weren't fighting for your High God? You were trying to make me remember?\"\n\nThat accounted for his wasted regret that he had no lore to forbid _turiya_ Herem.\n\nThe creatures wailed. They cowered. \"Now you are wroth. Forgive, Pure One. Our High God is himself, great in wonder and sovereignty. He has no need of our small magicks. If you will not forgive our attempt, forgive our failure.\"\n\n\"Wait,\" Covenant insisted for the third time. \"You don't need my forgiveness. That's not important. But forbidding\u2014\"\n\nHe could not think. A fretwork of blisters covered his whole body. They seemed to cover his mind. Pain burst and bled wherever he turned. Whether or not he accepted hurtloam and healing, the Feroce were right: he had sealed himself against recall. For him, the strength of the Colossus was lost; irrecoverable.\n\nBut Linden\u2014\n\nShe was capable of surprises that appalled and delighted him. She might\u2014\n\nStruggling to articulate ideas as they formed, he said urgently, \"A message. I need you to carry a message for me. As fast as you can. To Lin\"\u2014he stumbled momentarily\u2014\"to the woman with the stick of power. The woman you tried to hurt. Tell her to remember forbidding.\n\n\"We're going to need it.\" _Without forbidding, there is too little time_. \"And she has resources we don't. If nothing else, she's met Caerroil Wildwood. He knows a thing or two about forbidding.\" In an ancient age, he had participated in the formation of the Colossus as an interdict against the Ravers. \"Why else did he give her those runes?\"\n\n_The end must be opposed by the truth of stone and wood,_ orcrest _and refusal._\n\n\" _Tell_ her,\" Covenant ordered; pleaded. \"Remember forbidding. Promise me you'll tell her.\"\n\nNow the Feroce appeared to grow stronger. They stood straighter. Their fires burned more brightly. \"It is done,\" they announced. \"Be assured, Pure One. Even now, your words hasten. We are little, but we are also many. We inhabit our High God's realm from verge to verge. Your command will be fulfilled.\"\n\nLike a sigh made flesh, Covenant sagged in Horrim Carabal's coils. He had done what he could. Now there was only one dilemma left to consider. One intolerable choice to make.\n\n_\u2014save or damn\u2014_\n\nHis frailty blurred such distinctions. The Lords had misremembered their prophecy about the wielder of white gold; or they had misunderstood it. The words should have been \"save _and_ damn.\" If he let himself die now, his end would be wasted. And if he let himself be healed, his life would be wasted later.\n\nTherefore he ought to choose life. While he lived, he could hope that something might change, for good or ill. _And betimes some wonder is wrought to redeem us_. Preferring death when life was offered was just despair by another name.\n\nBut, God, he was tired! He had already endured too much. In his present state, he imagined that the final darkness would be a kinder fate than hurtloam and more striving.\n\nAnd he was a leper. For a man like him, nothing undermined his foundations more than being cured. Because he was who he was, he did not know how to bear the moral contradiction of being spared.\n\nLike the _Haruchai_ \u2014\n\nBy that reasoning, he should have refused Brinn's succor.\n\nBut he had always been weak. Time and again, he had turned away from the strictures of his illness because he loved the Land. And Linden. In his own way, he also loved being human.\n\nAnd he had always needed help.\n\nUnder the right circumstances, weakness was a form of strength.\n\nWhile he wandered in his personal gyre, circling its edges like trapped flotsam, the Feroce renewed their thetic chant. The arms of the lurker held firm, waiting. But Branl grew restive. He, too, was in pain. The damage to his body he would doubtless survive. Certainly he would ignore it. The damage to his spirit was another matter.\n\n\"Ur-Lord,\" he said at last. \"Hurtloam awaits you. Will you not accept its benison? Alone, I cannot preserve your life. The lurker and the Feroce cannot. Kevin's Dirt will make corruption of your scalds until no recovery is possible.\n\n\"It was not for this that the _ak-Haru_ healed your earlier hurts. That he saw worth in the lurker's preservation does not entail that he desired your death.\"\n\nCovenant lifted his head, stared at the Humbled. With two words, Branl had shown him a way out of his confusion: Kevin's Dirt. Hurtloam would heal him as completely as his various maimings permitted. But it had never altered his essential nature. And some of its effects might be transient. His illness might thrive again under the bale of Kevin's Dirt.\n\nWas that not the underlying purpose of Kastenessen's curse? To thwart the deepest needs of those who loved the Land? In Linden's case, to limit her access to Earthpower? In Covenant's, to deny his fitness to be loved in return?\n\nSave and damn.\n\nFinally he faced the last of the Humbled. So that he would not be misunderstood, he told Branl, \"Only if you join me.\"\n\nOnce before, he had required the Master to accept healing. Now Branl needed it as much as he did, if for different reasons.\n\nHe had no idea what he would do if Branl refused. But the Humbled did not. Nodding once, Branl said, \"If that is your wish. I have traveled too far from myself to gainsay you.\"\n\nThen he announced to the Feroce, \"The Pure One has prepared himself. We accept your consolation, deeming it well-meant.\"\n\nIn response, the chant of the creatures became a shout. Green that shed too little light flared and danced on all sides. The arms of the lurker let go.\n\nWhen Covenant fell into the mud, his whole world became spangles of gold like the rising of little suns.\n\nater Branl drew him out of the hurtloam. Tentacles lifted Covenant and the Master again; carried them away. At the eastern edge of the Sarangrave, the lurker lowered them onto a swath of grass on a hillside unspoiled by ancient wars or poisons. Then the arms withdrew, leaving only a few of the Feroce to watch and wait.\n\nBut Covenant knew none of this. He was deeply asleep, resting as though he had received an act of grace.\n\nhen he awoke, he came from the depths of dreams which he did not know how to interpret. He had sojourned among the Dead: they had given him obscure counsel. But they had stood, not in Andelain, but on the friable span of the Hazard, speaking of doom while below them raved the many maws of She Who Must Not Be Named, as ruinous as the Worm. Behind them, Branl had slain Clyme again and again; but the Dead had paid no heed. With infinite relish, the bane had devoured Elena and Linden and the future of the Forestals, making them participants in an eternal scream.\n\nIn dreams, time blurred and ran, as chaotic and rife with death as the mingled perils of the Sarangrave.\n\nForbidding, the Dead had urged. Forgotten truths.\n\nThe Chosen's son.\n\nKastenessen.\n\nA-Jeroth of the Seven Hells, who desires all things unmade.\n\nRepeatedly Branl hacked at Clyme and _turiya_ until only gobbets and blood remained.\n\nBaffled and thwarted, Thomas Covenant opened his eyes to the grey murk of dawn in a world where the sun did not rise.\n\nBut his own condition seemed to repudiate Branl's ferocity and Clyme's death. He had slept deeply and long. God, he had _slept_. On this open grass, he had slept the sleep of renewed health, fathomless as the growing gaps between the stars. It was an anodyne that he had not expected, as salvific as hurtloam, and as necessary.\n\nNo doubt he had slept too long. Every hour counted against him. But he could not regret losing the night.\n\nWhen he opened his eyes and looked at the sky, he saw the stars clearly. Those that remained were as bright as gems of Time, and as disconsolate as condemned children. One after another, they went on dying.\n\nTheir slow plight grieved him. Yet it was countered by the sheer freshness of his physical sensations. Every burn and blister had been replaced by a tingling that resembled eagerness. His heart beat with a vigor which he did not recognize, as though it had been unshackled after a lifetime of imprisonment. His fingers flexed as if they had never known excruciation. Potential smiles twitched in the muscles of his face. And his feet\u2014By hell! He could feel his toes, actually _feel his toes_. They told him that his socks and boots were still sodden.\n\nHurtloam was a miracle: there was no other word for it.\n\nAnd like his body, his health-sense had become stronger. It assured him that his new life would be temporary. Kevin's Dirt shrouded the region, wreaking its incremental havoc; working against his restoration. Nevertheless he was grateful for any reprieve. The strange alchemy of hurtloam made even Clyme's death seem less bitter. At least for a little while, the future did not look as bleak as this day, the second without true sunlight. When numbness returned to his fingers and toes\u2014when his sight began to fail again\u2014he would be able to bear it.\n\nPropping his elbows on the thick grass of his bed, he raised his head and shoulders to gauge his circumstances.\n\nHe lay on a gradual slope that he did not remember, cushioned by turf like luxuriance. Therefore he was somewhere north of Lord Foul's many battlefields; somewhere in the long wedge of hale ground between Sarangrave Flat and the Sunbirth Sea. The lurker must have carried him here.\n\nShaking his head in surprise at such consideration, he regarded his companion standing like a sentinel a score of paces past his feet. Branl appeared to be keeping watch on the rank mass of the wetland. Or he may have been\u2014\n\nBeyond the _Haruchai_ , Covenant finally noticed a small cluster of emerald fires burning in the hands of four, no, five Feroce. They waited a few steps outside the border of their native waters. Branl may have been guarding Covenant against them; refusing them in some fashion.\n\nApparently their High God was not done with the Pure One.\n\nCovenant was reluctant to face them. He did not want to recall Horrim Carabal's peril, or to think about what the Humbled had sacrificed. But time was precious\u2014and the Feroce had blessed him with hurtloam. They had promised to speak to Linden for him. They had earned his attention.\n\nSighing at the ache of memories as cruel as Joan's suffering, Covenant pushed himself to stand.\n\nAround him, murk veiled every feature of the landscape, turned hills and grass and marsh and sky to an indeterminate, irredeemable smudge. Only the wavering fires of the Feroce contradicted the universal twilight; and they cast too little illumination.\n\nAwkwardly, as if he had forgotten how to walk, he went to join Branl.\n\nLike him, the Master still wore a second skin of mud. A trivial concern: it would flake away as it dried; and in the meantime, it provided a measure of protection against the increasing coolness of the air. But under the mud, Branl's tunic hung in tatters, eaten by the Flat's corrosive waters. Indeed, Covenant's own clothes were badly damaged. His jeans looked like they had been mauled, and his T-shirt was little more than scraps. Yet that, too, was trivial. Ruined attire suited the Unbeliever and his guardian.\n\nLooking more closely, Covenant was relieved to find that Branl also had been healed. In more ways than one\u2014A portion of the distress clenched and hidden behind his _Haruchai_ stoicism had been eased. He looked like a man who had finally come to terms with an amputation, or with some other old wound.\n\nResting his halfhand on the Master's shoulder, Covenant said, \"I'm sorry.\" Perhaps he would learn how to forgive Branl if he first asked forgiveness for himself. \"I can only guess what killing that Raver cost you. But I regret it. I wish I hadn't needed you to save me.\"\n\nAgain.\n\nBranl's gaze did not waver. \"You sought to spare us, ur-Lord,\" he replied as though every human tone had been hammered out of his voice. \"That you have ever done, though you have long known that no _Haruchai_ wishes to be spared. To be denied the outcome of our deeds implies a judgment of unworth. Yet you are the ur-Lord, the Unbeliever. As we are known to you, so you are known to us. By long travail, we have learned that your choices are indeed a judgment of unworth. But it is yourself that you judge, yourself and no other. Therefore we found no insult in your wish to confront _turiya_ Herem alone.\"\n\nInvoluntarily Covenant winced. The Humbled certainly knew him too well. But he did not like to think of his personal strictures in such terms.\n\nSighing again, he changed the subject. \"Do you still have the _krill_?\"\n\nBranl nodded. From the remains of his tunic, he drew out a bundle of broad leaves. \"Do you require its light, ur-Lord? I have covered it to appease the timidity of these Feroce.\" After a moment, he added, \"They crave speech with you yet again. For that reason, they have awaited your return from slumber.\"\n\nCovenant dropped his hand. \"Never mind. They're already scared enough. They've waited this long for me. I can wait a little longer to see where I'm going.\"\n\nHe had decisions to make, but he was not ready for them. He wanted Linden's forgiveness more than Branl's\u2014or his own.\n\nStanding at his companion's side as if he and the Humbled carried the same stigma, he addressed the Feroce.\n\n\"So far, you've honored your part of our agreement.\" That the lurker wanted something else from him made him brusque. \"I expect your High God to keep doing that. We've done more than I promised. You should do the same.\"\n\nThe Feroce flinched. Their flames guttered and spat. \"You are the Pure One,\" they answered, quavering, \"though you deny yourself. So it was at the time of the _jheherrin_. So it remains.\n\n\"You have exceeded the terms. This our High God acknowledges. The alliance is sealed.\"\n\nCovenant nodded; but he did not relax. \"And my message? Did you deliver it?\"\n\n\"We are the Feroce,\" the creatures replied. Their single voice sounded like mire forced to assume the shapes of language. \"We serve our High God in every pond and stream and quag of his glory. Your words have been conveyed. Their import we have striven to convey also.\"\n\nCovenant bowed his head in relief. Linden would understand. He had to believe that she would understand. And she would know what to do. _Something unexpected_. Something that he could not imagine.\n\nBut the Feroce were still speaking. \"If we have failed,\" they said, \"or if we are not heeded, our High God commands contrition. Our lives are forfeit. Should you wish to slay us, still the alliance is sealed. It will not be unsealed.\"\n\nThen the creatures stood and waited as if they were resisting an impulse to cower.\n\nTheir unrelenting fears troubled Covenant. \"Well, gosh,\" he drawled to disguise his dismay. \"That's magnanimous of him. Is everybody in this bloody mess trying to make amends for sins they haven't committed?\"\n\nThe fires of the Feroce quailed. Their large eyes reflected emerald alarm. They had tried to help him remember _forbidding_ \u2014they had given him hurtloam\u2014and still they expected to be punished.\n\nSwearing to himself, Covenant tried to soften the edges of his voice. \"You did what you could. If we exceeded the terms, so did you. What happens next isn't your fault.\"\n\nHe meant, You don't need to be afraid of me.\n\n\"So what does your High God want now?\" he continued. \"He's already sacrificed enough of you for my sake. I don't want more. What does _he_ want?\"\n\n\"He is our High God,\" the descendants of the soft ones replied. \"His greatness commands us. We do not refuse. We\u2014\"\n\nAbruptly they flinched like children at the first touch of a flail. Facing each other, they crowded closer together. Their flames seemed to gibber.\n\nFrom their circle of fire and fear, their voice arose like muffled wailing. \"Our High God commands. The alliance is sealed. It will not be unsealed. But he asks\u2014\"\n\nFor a moment, they appeared to lose control of themselves. Their green faded to flickers in their palms. Their voice became a thin cry like an echo of their earlier shrieking. Their bodies jerked as if they were appalled by what they had to say.\n\nBut then they mastered themselves\u2014or they were mastered. Their fires sprang back to life. The flames strained upward, striving toward the heavens. Garish emerald glared like malevolence on their weak features. Their wailing became words.\n\n\"Our High God craves a boon.\"\n\nCovenant stared at their chagrin. He required a moment to grasp that the Feroce were distressed by the notion that their High God had needs which could not be met by commands or alliances or raw power; that the lurker's tremendous size and strength could be reduced to pleading. In effect, Horrim Carabal had confessed an inadequacy that struck at the roots of their devotion.\n\nShaken on their behalf, Covenant said, \"You don't need to be afraid. There's no harm in asking. I'm not offended. Just say it. What does your High God want me to do?\"\n\nHe could not tell whether the Feroce understood him. They did not unclench their circle, or lower their fires, or cease their wounded cries. After a moment, however, their wailing became speech again.\n\n\"You are the Pure One. The Pure One redeems. Now havoc comes, a great and terrible hunger. It draws near. It is death. Utter death. Our High God cannot stand against it. He does not know what he must do. Will you heed him? Will you answer?\n\n\"Our High God must not perish!\"\n\nAh. Covenant nodded again. The lurker wanted to survive, and it did not know how.\n\nBut he was loath to suggest a course of action. \"That depends,\" he said carefully. He could not guess what the implications might be. \"I don't know exactly what you're asking. First tell me this. The havoc is coming. That's a fact. But _where_? Where is it coming?\"\n\nWould it head straight toward _Melenkurion_ Skyweir? Was it ready to end the world? Or did it want more food? More _Elohim_? Or something else\u2014?\n\nThe possibility that the Worm was hungry for _something else_ made Covenant's stomach twist.\n\n\"You are the Pure One,\" the Feroce replied in consternation. \"Do you not know that the havoc nears the heart of our High God's realm, the deepest waters? How is it that you do not know?\"\n\nThe deepest waters? Covenant frowned. That must mean Lifeswallower, the Great Swamp: the delta of the Defiles Course. He groaned at the idea. The ground on which he stood seemed to cant as if realities were shifting. Hellfire! The Worm was approaching Lifeswallower.\n\nBut it could have no interest in the lurker's demesne. It would find no sustenance in that polluted swampland. And certainly the Worm had no appetite for a monster like Horrim Carabal, a living corruption of Earthpower. Which meant\u2014\n\nCovenant dragged his hands through his hair, trying to steady his thoughts.\n\n\u2014that Lifeswallower was simply in the way. The Worm would merely pass through it. The instrument of the world's end had a different goal.\n\nPerhaps the Worm was coming from the north. Perhaps its path toward _Melenkurion_ Skyweir ran through the Great Swamp by chance.\n\nOr\u2014\n\nDamnation!\n\n\u2014it was going toward Mount Thunder.\n\nTo Kastenessen. Or to She Who Must Not Be Named.\n\nHell and _blood_!\n\nBoth explanations seemed plausible. Kastenessen was _Elohim_. He might be the nearest source of food. But he was tainted. He had merged a portion of himself with the _skurj_. Their sulfurous scent might make him unpalatable. In his own fashion, he was as corrupted as the lurker.\n\nShe Who Must Not Be Named was another matter. She was\u2014Covenant had no apt language for Her\u2014a gaoled god. She was not Earthpower. Nevertheless She was _power_. If the Worm sought to feed on Her\u2014\n\nThe battle between such beings would stagger the Arch of Time to its foundations. It might accomplish the purpose for which the Worm had been created.\n\nLord Foul had planned well. Oh, he had planned well! Here was another conceivable reason why Roger had hidden Jeremiah in the Lost Deep. To conceal the boy, of course. To preserve him for Roger's use\u2014and for the Despiser's. But also to arouse She Who Must Not Be Named if Linden discovered Jeremiah's covert.\n\n_It boots nothing to avoid his snares, for they are ever beset with other snares\u2014_\n\n\"You are the Pure One,\" repeated the Feroce in trepidation. \"Will you not answer?\"\n\nWith an effort, Covenant shook aside a whirl of sickening speculations. \"Oh, I'll answer.\" He did not know what he would say until he heard himself say it. \"But you still haven't told me what your High God wants. He can't believe _I'm_ going to stop the Worm. That havoc, as you call it, will swat me like I'm nothing. What does your High God think I _can_ do?\"\n\nStraining to respond, the voice of the Feroce scaled higher. The lurker's reply was naked supplication. \"Will you counsel?\" they asked as if they wanted to weep and had no tears. \"Will you reveal what must be done? For the alliance? For our High God's life?\"\n\n\" _Damn_ it,\" Covenant muttered to himself. His impulse to speculate was too strong. His mind wheeled. \"I can't.\" Even if the Worm hunted only Kastenessen, it was certain to encounter She Who Must Not Be Named. \"Not until I know where it's going.\"\n\nBefore the lurker's servants could muster more words, Covenant turned to Branl. \"What do you think? Maybe coming to Lifeswallower is an accident. Maybe the Worm is just passing through. But maybe it's aiming for Mount Thunder. Don't we have to know?\"\n\nThe gloom masked Branl's features; but the Humbled faced Covenant with a firmness that resembled certainty. \"Ur-Lord, hear me. You contemplate a journey to the last boundary of hills between Lifeswallower and the Sunbirth Sea. Such a quest will bear us many leagues farther from our companions, wherever they may be.\"\n\nCovenant braced himself to argue; but Branl was not done.\n\n\"Understand, ur-Lord, that I do not protest. Your task is mine. I am alone and have no path other than my chosen service. Yet I must observe that our need for an end to Kevin's Dirt is absolute. Your straits confirm this. Already your illness regains its force. The Worm in Mount Thunder may perchance bring about the cessation of Kevin's Dirt. Perchance it may not. Is it not therefore plain that our surest road to Kastenessen's defeat lies toward Linden Avery and her company? Your powers and hers together are more certain of success than any chance or mischance of the Worm.\"\n\nCovenant shook his head. Studying Branl while memories of Clyme's end scarified his thoughts, he said slowly, feeling his way, \"That makes sense, as far as it goes. But what if the Worm runs into She Who Must Not Be Named?\"\n\nRealizations seemed to swarm in Branl's gaze. Apparently he had not considered the bane. \"That outcome,\" he said slowly, \"must be prevented.\" Then he asked, \"Yet how can it be forestalled?\"\n\nCovenant grimaced. \"That's the problem. We have to know where the Worm is going. We might need the lurker against it.\"\n\nWhen his companion acquiesced, Covenant turned back to the Feroce.\n\nSwallowing a clot of apprehension, he said, \"Tell your High God this. I want him alive. I'll give him counsel, if I can think of anything. But not until I know more.\n\n\"I have to see this havoc for myself. Then we'll talk.\"\n\nThe idea that he would be moving farther from Linden made him ache; but he ignored that pang as well as he could.\n\nThe creatures fluttered their fires in alarm, but they did not protest. For a moment longer, they crowded together, mewling wordlessly while their theurgy pulsed in the twilight. Then they answered, \"You are the Pure One. The Feroce will await you. Our High God commands us. The alliance is sealed.\"\n\nAt once, they broke away from each other and hastened toward the wetland. As soon as their feet entered the waters of the Sarangrave, their flames went out. Covenant lost sight of them as if the marsh had swallowed them whole.\n\nHis mouth was suddenly dry, and his heart pumped dread. The enormity of what he meant to do seemed to thicken the murk. It made the air difficult to breathe. He had no real comprehension of the Worm's puissance. For all he knew, its power was too destructive to be gazed upon. The sight alone might scald his eyes in their sockets.\n\nFiercely he told himself, Or it might not. He would learn nothing if he did not take the risk.\n\nStop dithering. Just do it.\n\nThere was no other way to earn the necessary knowledge.\n\n\"We need the horses,\" he muttered to Branl. He would probably never see Hooryl again. He had to hope that Rallyn would be able to command Mishio Massima without help. \"And food. Water. From here on, everything is only going to get harder. I don't doubt that you can hang on indefinitely, but I have to keep up my strength.\"\n\nThe Humbled nodded. He did not speak of trust in the Ranyhyn, or in himself.\n\nThat was well. Memories of _turiya_ and butchery clung to Covenant. When the _Haruchai_ invoked _trust_ , the word meant too much. Long centuries ago, Covenant had asked the ancestors of the Humbled to preserve Revelstone. Clyme's death was only one of the results.\n\nut trust was still trust. It was earned, or it was not. As faithful as the _Haruchai_ , who remembered everything, Rallyn cantered out of the dusk in Naybahn's place, answering Branl's summons. And the palomino stallion brought the Ardent's mulish beast with him. When the Humbled had checked Mishio Massima's tack, he announced that the horses were ready.\n\nWith leaves to protect his hands, Covenant uncovered the _krill_. Then he removed Joan's ring from around his neck. As he had done before, he pushed the ring onto the stub-end of the last finger of his left hand; closed his fist around the chain to secure the band. As before, he struck the dagger's gem with the ring until his body blazed with wild magic. After that, he concentrated on pressing the point of the blade into the grass while Branl carried him around Rallyn and Mishio Massima.\n\nWhen Branl lifted him into his saddle, he nearly fell off the far side. A second Humbled should have been there to catch him. But he managed to steady himself on the saddle horn.\n\nWhile his line of silver lingered in the turf, the horses surged into motion, bearing him farther from his heart's desire.\n\nfter a blink of darkness which seemed to deny any possible passage, either through time or across distance, Covenant and Branl arrived galloping in a region that looked indistinguishable from the place which they had left. The hillside may have leaned at a slightly different angle. The slope ahead may have been less even. Conceivably Sarangrave Flat had receded to the west. But Covenant could not be sure. Beyond the _krill_ 's reach, the unnatural dusk masked details, and his vision was fading.\n\nBranl took Loric's dagger and covered it, giving Covenant's eyes a chance to adjust to the universal grey. The horses ran on as if they were determined to reach the edge of the world.\n\nBefore Covenant could swallow enough of his vertigo to frame a question, the Humbled pointed ahead. After a few moments, Covenant made out a deeper gloom like a clump of shadows in the rumpled ground: a small copse in a hollow. Soon he caught the faint glint of water. A stream purled over the contours of the hillside, hastening in the direction of the Sarangrave.\n\nAs the horses slowed, Branl stated with quiet satisfaction, \"The Land is provident\u2014as is Rallyn. Here we will find both water and sustenance. Corruption's wars did not extend into this region. Nor do the blights of Sarangrave Flat.\"\n\nCovenant did not doubt his companion, but he had other concerns. While he scrambled for balance, he asked, \"How far have we come?\"\n\n\"A score of leagues, ur-Lord. Perhaps somewhat more.\"\n\nCovenant winced. Only a score?\n\n\"Did we lose much time?\"\n\n\"No other mount could have borne us so swiftly,\" Branl replied with uncharacteristic asperity. He seemed to hear a complaint in Covenant's tone. But then he continued more flatly, \"Yet it is plain that our passages are not immediate. Though the sun no longer measures the day, I gauge that mid-morning is nigh.\"\n\nCovenant frowned, thinking hard. To some extent, at least, the distances that he and Branl could cover appeared to be controlled as much by Rallyn's instincts as by the size or even the precision of his argent enclosures. Nevertheless the abilities of the Ranyhyn clearly had limits. Otherwise they would not have needed two attempts to reach the Sarangrave the previous day.\n\nStill he was losing chunks of time. Where did the hours go? Where\u2014if anywhere\u2014did he and Branl and their horses exist during the interval?\n\nThe lag may have been inherent to his specific use of wild magic; or it may have been an outcome of his relationship with Joan's ring, a ring which was not his. After all, Linden had experienced something similar. When she had saved herself and Anele from the collapse of Kevin's Watch, she had done more than pass from one place to another. She had also moved through time: in effect, she had fallen more slowly than the broken remains of the Watch.\n\nAs soon as the horses halted near the stream, Mishio Massima jerked the reins away from Covenant and began cropping grass. Branl slid down from Rallyn's back; offered to help Covenant. But Covenant dismounted on his own. For a few moments, he braced himself against the Ardent's steed while the last sensations of vertigo faded, giving himself a chance to accept the returning numbness of his feet and the loss of sensation in his finger-tips. Kevin's damn Dirt\u2014Then he left the beast's side.\n\nWith Branl, he considered the nearby trees.\n\nThey were wattle, fast-growing and resilient. In sunlight, they would have been a verdant green, fresh and promising. Now they resembled shadows cast by a different version of reality, although they swayed in the tumble of a growing breeze. Certainly they appeared to offer nothing that Covenant could eat.\n\nNevertheless the Humbled seemed sure of his own perceptions. Firmly he beckoned Covenant to accompany him among the trees.\n\nThe copse was thick. Pushing his way between the trunks, Covenant soon tripped. When he looked down, he found that he had caught one of his boots on the thick stem of a vine.\n\nIn fact, vines twisted all over the ground among the trees. The whole stand was tangled with them.\n\n\"Do you recall this, ur-Lord?\" Branl sounded subtly amused. \"You were once familiar with it.\"\n\n\"Huh?\" Covenant had lost ages of memories, but he was sure that he had never heard one of the _Haruchai_ sound amused. \"When?\"\n\n\"During the time of the Sunbane,\" answered Branl, \"it provided nourishment when Corruption's evil spawned no edible growth, and _aliantha_ were scarce. It is _ussusimiel_.\"\n\nFor a moment, Covenant groped inwardly. Then he spotted the darker knob of a melon in the gloom; and he remembered. Long ago under a desert sun, Sunder had invoked vines and their fruit from parched, barren dirt. _At need it will sustain life_ \u2014\n\nIt did not taste as piquant as treasure-berries. And it lacked their extraordinary vitality. But it would be enough.\n\n\"Well, damn,\" Covenant muttered. \"If that isn't providence, I don't know what is.\" He felt unexpectedly cheered, as if an old friend had taken him by surprise. \"Hell, I don't even know what the word means.\"\n\n\"Then, ur-Lord\"\u2014Branl held up the wrapped _krill_ \u2014\"if you do not deem it an incondign use, I will harvest melons. While you break your fast, I will weave a net of smaller vines to carry a supply of the fruit.\"\n\nCovenant found that he was too hungry to argue. \"Do it. Somehow I'm sure Loric wouldn't object, even if he did spend damn decades sweating over that knife.\"\n\nBut he did not stay to watch Branl work. Instead he turned away, sparing his eyes the stab of the gem's shining. Lit by slashes of silver, he withdrew from the copse and went to the stream to drink.\n\nProvidence in all sooth. Even here, so many leagues away from the wonders of the Land that he had known in life, there were still gifts\u2014\n\nNow he prayed that food and water would sustain him well enough for what lay ahead.\n\nsecond self-contained violation of time or space took him and Branl nearly thirty leagues closer to their destination. As Rallyn and Mishio Massima galloped out of theurgy onto a long facet of exposed rock, Covenant clung frantically to his saddle horn, straining to contain a gyre of dizziness. But Branl rode as though he and Rallyn were more dependable than stone. Over one shoulder, the Humbled carried a net sack filled with enough melons to keep Covenant fed for a day or two.\n\nA wind out of the east buffeted the riders like the presage of a gale, but it was useless to Covenant. It did not stop the spin that sickened him, or lessen the blurring of his sight.\n\nAccording to Branl, one more passage of comparable length would convey them to the bluffs between the Sunbirth Sea and Lifeswallower, the headland which bordered the delta of the Great Swamp. From that vantage, they would be able to watch for the Worm without precluding contact with the Feroce.\n\nUnfortunately noon had already passed. Each translation by wild magic washed away time as well as balance. In some sense, the linear certainty of causality and sequence formed the ground on which Covenant's mind stood. His thoughts were moments; bits of bedrock. When he blinked from one location to the next, the change staggered him as if every nerve in his body had misfired.\n\nFor that reason, and because each exertion of Joan's ring drained him, he had to rest in spite of an accumulating sense of urgency. When the horses had slowed to a halt, he half fell out of Mishio Massima's saddle and lurched away like a wounded animal looking for a place to hide.\n\nHe yearned to be alone, at least for a little while; to soothe his vulnerability in isolation. But Branl followed him. After a silence, the Humbled pronounced, \"This frailty is an effect of Kevin's Dirt, ur-Lord.\"\n\nInstead of speaking, Covenant gritted his teeth and waited.\n\nInflexibly Branl added, \"The distress which results will fade more readily if I am permitted to hold High Lord Loric's _krill_.\"\n\nCovenant blinked at the knife bright in his grasp. Damnation. It's getting worse. Like the encroaching deadness of leprosy, vertigo was tightening its noose around him. In his confusion, the injured whirl of disorientation, he had not realized that he was still holding the dagger. He had not felt its heat\u2014\n\nWith a jerk of his arm, he surrendered the _krill_.\n\nAs Branl covered the gem, dusk flooded over the region. Under other circumstances, the sun's absence would have galled Covenant. Now, however, it felt like an act of kindness. Twilight was a kind of privacy. He needed it to recover his balance.\n\nThe lurker wanted counsel, but he had no idea what he could possibly say. If the Worm caught Kastenessen's scent, it would head toward Mount Thunder\u2014and toward She Who Must Not Be Named. Nothing would survive that encounter.\n\nTo prevent that outcome, Covenant might have to ask Horrim Carabal to sacrifice itself. But the monster would surely refuse. No alliance would persuade it to surrender its life voluntarily.\n\nHe had to hope that the Worm's approach to Lifeswallower was a coincidence; that it would ignore Mount Thunder. Otherwise he would have to think of a better answer for the lurker.\n\nHampered by Kevin's Dirt and vertigo, he could hardly think at all.\n\nortunately a third passage brought him to the headland. His mount hammered up a slope of saw-edged grass between bare juts of granite and basalt: a narrowing wedge of rising ground. To the north stood the bluffs which restricted the spread of Lifeswallower. In the east were the low cliffs bordering the Sunbirth Sea. Beyond the gap-toothed horizon ahead was nothing except grey sky and stars. They seemed to mark the edge of existence.\n\nThis time, the wind hit Covenant hard. Heavy as a torrent, it knocked him askew. When he tried to dismount, he toppled backward; landed on the grass with a jolt that stopped his breathing. The ground tilted from side to side, forward and back, in a sequence devoid of reason, as unpredictable and dangerous as dreaming. Gusts swept past him, sucking air out of his mouth. Blots marred his vision like the mottling of disease.\n\nBut then Branl took the _krill_. With a suddenness that resembled fainting, Covenant began to breathe again.\n\nWhile the stains faded from his sight, and the canting of the horizons eased, he was content to lie still and let the impact of his fall ebb. The troubled labor of his heart suggested that he had undergone an obscure ordeal. Nevertheless it reassured him. It confirmed that time endured, unbroken; that one thing led to another. The Law that constrained and enabled life held true.\n\nWhen he felt ready, he rolled onto one side, forced his arms and knees under him, pushed himself upright.\n\nGod, the wind\u2014He could barely stand against it; had to squint at the sting of tears. Without Branl's support, he might not have been able to move.\n\nBlinking, he scanned his surroundings. He had the visceral impression that he was standing on the highest peak of the world. But of course that was nonsense: this was not a mountain. Rather he had arrived downhill from the wedge-tip of the headland. To the east, the sea thrashed at the Land's last rock. He smelled salt on the blast. If he could find the vantage he sought, he would be able to see the surge of waves.\n\nAround him, the headland was a jumble of protruding stone, granite and basalt weathered smooth; gnawed across the millennia into shapes that resembled anguish and intransigence. Some of the rocks wore fringes of moss in the lee of the wind. Others had acquired threadbare cloaks of lichen.\n\nPeering behind him, he thought at first that the slope sank lower indefinitely. But when he squeezed the wind from his eyes and looked harder, he realized that the westward hillside was cut off by a line of darkness in the distance. There lay the Great Swamp, sweeping around the headland toward the sea. He could not smell Lifeswallower. The wind tore away the swampland's complex fetors. But below him the waters of the delta reflected a faint shimmer.\n\nAfter a moment, he spotted the horses. They were cantering down the slope, keeping their distance from the wetland as they descended. Apparently Rallyn believed that the riders had no immediate need of their mounts. And naturally both Rallyn and Mishio Massima wanted water as well as forage.\n\nThen Covenant noticed the emerald fires, small as dots, ascending slowly toward him.\n\nHe watched the creatures briefly. But they were still far away; and he had nothing to say to them. Turning back toward the tip of the promontory, he went upward with Branl's aid until he glimpsed the darker grey of the sea beyond the headland's rim. There he stopped.\n\nThe waves heaved frantically against their own weight, hacking across each other, rising into sudden breakers, erupting in spume. Some mighty pressure disrupted the normal scend and recession of tides. The seas were flung in frenzy at the cliffs, where they rebounded, smashed together, became chaos. The wind assailed Covenant's ears with their clamor as if the headland were under siege.\n\nGripping his companion's arm, he asked, \"Can you see anything?\"\n\nBranl studied the sea. \"I do not doubt that the Worm comes, as the Feroce have declared. In turmoil, the waves contradict themselves. Some cataclysm goads these waters. But its source is too distant for my discernment.\"\n\n\"How much time did we lose?\"\n\nA slight frown of concentration or surprise disturbed Branl's mien. After a moment, he replied, \"It appears that our final passage was prolonged. Mayhap the Worm's approach misleads my senses. Nonetheless I gauge that evening is nigh. Ere long, this dusk will turn toward true night.\"\n\nThe coming of night after a second sunless day felt like a bad omen. Covenant had no power against the World's End.\n\nNonetheless he had made promises\u2014\n\n\"In that case,\" he told Branl, \"I need to get out of this wind. Can you find a place where I can watch the sea and Lifeswallower? A place with some shelter?\"\n\nNodding, Branl drew him toward the stones which cluttered the corner of the headland. In the lee of a blunted fang as tall as Covenant, the Humbled urged him to sit and rest. Then Branl left. Still bearing his net of melons as well as Loric's _krill_ , he disappeared among the twisted shapes of basalt and granite, the motley of lichen and moss.\n\nCovenant sagged against the fang; rubbed his stiff cheeks with his insensate fingers; wiped away residual tears. Reflexively he confirmed that Joan's ring still hung under his T-shirt. The wind moaned miserably past the rocks, a raw sound like keening, but he tried to ignore it. Tried to think. Wind was only air in motion, he told himself. It merely reacted to forces beyond its control. If he heard lamentation in it, or auguries of havoc, he was misleading himself. The world did not _care_ : the natural order of things did not grieve or grow glad. Only the sentient beings who inhabited time wept and struggled and loved.\n\nThere was a kind of comfort in the notion that the Earth neither understood nor feared its own peril. Its life was not a reflection of himself. But such consolation was too abstract to touch him\u2014or his dying nerves did not feel it. Ultimately nothing ever mattered, except to the people who cared about it. To them, however, the import of the stakes was absolute.\n\nCovenant grimaced ruefully at his thoughts. Long ago, he had insisted that the Land did not exist, except as a form of self-contained delirium. In that sense, it _was_ a reflection of himself. And he was powerless in it because he could not change his own image in the mirror: it only showed him who he was. Therefore he could not be blamed for his actions; or for the Land's fate. Now he found himself arguing that the world was really nothing more than an impersonal mechanism inhabited by self-referential beings. Therefore no failure, here or anywhere, could be held against him.\n\nAfter so many years, he had changed very little. He was still looking for a way to forgive himself for being human and afraid.\n\nBut in fact he did not believe that the Land and its world were simply parts of a mechanism. They formed a living creation. And like all living things, they yearned for continuance. If he failed them, the world's woe would be as vast as the heavens.\n\nWhile it lasted.\n\nThere were hints of travail in the wind; suggestions of iniquity. But he did not know how to interpret them\u2014or he was not ready.\n\nHe was still wrestling with himself when Branl returned, no longer carrying his supply of _ussusimiel_.\n\n\"By good fortune, ur-Lord,\" the Humbled announced, \"there is a covert which I deem apt for your purpose. The wind is obstructed, yet views to the east and north are accessible. Will you accompany me?\"\n\nBriefly Covenant considered what he could see of his companion. Then he muttered, \"Well, hell. Why else are we here?\" Extending his arm, he asked for help.\n\nTrue to his commitments, the Master lifted Covenant upright. And he kept his hand on Covenant's arm for support and guidance. His grasp may have been meant as reassurance.\n\nCovenant glanced downhill to check on the progress of the Feroce. Their noxious fires shone more clearly now; but they were still no more than halfway up the slope. Trusting their uncanny ability to find him wherever he was, he turned away.\n\nAs Branl drew him among the stones, the Humbled asked, \"Ur-Lord, have you determined how you will counsel the lurker?\"\n\nBracing himself on contorted plinths and tall slabs, Covenant picked his way forward. \"It's like I said. I need to know where the Worm is headed. If it comes from the north, or the northeast, and doesn't turn, it's probably going straight for _Melenkurion_ Skyweir. In that case, the lurker isn't in danger. It doesn't need advice. But if the Worm comes from anywhere south of us, it's ignoring its direct line to the EarthBlood. That means it wants Kastenessen\u2014or She Who Must Not Be Named. Then I'll have to tell Horrim Carabal _some_ thing.\"\n\n\"To what purpose?\" countered Branl. \"That you desire to determine the Worm's immediate path, I comprehend. But what will any counsel avail? The lurker will not hazard its life at your word.\"\n\nCovenant stumbled to the left around one thrust of basalt, to the right past another. The cry of the wind was louder here. It pummeled him in forlorn gusts. But as he went farther among the stones, he was spared more and more of the wind's force.\n\n\"I'm still thinking,\" he answered through his teeth. \"There has to be something we can do.\" To accomplish what? Slow the Worm? _Stop_ it? He told himself not to be absurd. \"I just don't know what it is.\"\n\nThe Humbled may have shrugged. He did not argue.\n\nHis path twisted like a maze. It seemed long. But eventually Covenant came to a small patch of grass just wide enough to sit in. Branl's net of melons rested there in a notch between stones the size of Giants. Standing in the center of the grass, Covenant found that he had a clear line of sight northward. Through a gap in the jumble, he could see the rim of the bluffs perhaps ten paces away. And beyond the precipice\u2014\n\nThere the Sunbirth Sea assailed Lifeswallower with the mindless fury of a berserker.\n\nAt one time, perhaps only a few hours earlier, the waters of the Great Swamp had drained eastward in ramified channels like the branches of an immense tree. Among them had stood islands of unpalatable grass, tormented eyots of brush, clusters of hoary cypresses and other marsh-trees like sentinels watching over a sargasso. But such things were gone now. Indeed, every feature of the delta had been inundated or swept away. The mounting seas flailed in all directions, tearing apart or dragging under everything that defined this region of Horrim Carabal's realm. The portion of Lifeswallower that Covenant could see had become indistinguishable from the ocean's violence.\n\nThe sight made him shiver as if vertigo had already wrapped its cold fingers around his heart. Grinding his teeth, he turned to the east.\n\nAt first, he could not gain a view of the sea. Too many protruding rocks rose too high. But when he leaned to one side of his covert, he found an opening. There ages of wind and weather had scalloped the sides of several stones. And one slab of basalt had lost a substantial section of its center: it resembled a cripple hunching over a collapsed chest. The result was a window like an oriel, a gap that revealed an arc of the Sunbirth Sea.\n\nThrough the window came flicks and slaps of wind, occasional stings of spray; but Covenant was able to endure them for a few moments at a time.\n\nAt that distance, he could not discern any specific swell or cross-current. The whole ocean looked like a darker and more troubled iteration of the sunless sky. Even the horizon was no more than a smear of grey. If the Worm were coming from that direction, he saw no sign of it.\n\nBlinking hard, he moved back into shelter. With a gesture, he asked Branl to watch for him. Then he lowered himself to the grass and tried to believe that he had not come so far for nothing: that when the Worm arrived, he would know what to say.\n\nBranl scrutinized the east for a while; turned his attention briefly to the ruined delta in the north. Then he shook his head.\n\n\"Ur-Lord, I judge that the Worm is not imminent. I know nothing of its speed, but I will believe that a span of time remains to us. We are granted a respite.\" He removed the _krill_ from his tunic. \"Should you wish it, I will prepare _ussusimiel_.\"\n\nCovenant nodded. \"Sure. Why not?\" He needed strength. When the Worm came, he would have to flee, whatever happened. If he and Branl died here, their lives would be truly wasted.\n\nUncovering only the dagger's blade, the Humbled deftly took a melon, sliced it into sections, cut out the seeds. The pieces he handed to Covenant one at a time.\n\nCovenant ate until only rinds remained; but he did not notice the taste, or attend to what he was doing. He was listening to the unsteady ululation of the wind, trying to decipher its oblique message. Its salt tang and its keening were auguries that he did not know how to interpret.\n\nBranl offered to prepare another melon. Vaguely Covenant declined. He was not conscious of hunger; or he was not hungry for that kind of sustenance. He wanted the richer nourishment of an _answer_.\n\nAfter cleaning the blade, Branl put the _krill_ away and resumed his study of the east.\n\nWind and salt. The ravage of the delta. The Worm of the World's End. Kastenessen. She Who Must Not Be Named.\n\nAnd Linden, who was so far away that only Rallyn would know how to find her. The thought that he might not see her again before the end made Covenant's chest ache like a wound to the heart.\n\nBranl stepped back to gaze around the stones. After a moment, he said, \"Attend, ur-Lord. The Feroce approach.\"\n\nJerking up his head, Covenant spotted glints of emerald on the rocks. Fires guttered; flared more brightly; receded. Soon two of the creatures brought their flames and their timidity to the border of the grass. Two or three more Feroce followed behind them. Their eyes cast echoes of their theurgy into his shelter.\n\nIn their damp, squeezed voice, they asked, \"Pure One?\"\n\nCovenant faced them until he was sure that they did not mean to say more; that the two words of their question sufficed for them. Then he looked at Branl. \"What time is it?\"\n\nThe Humbled was a thicker shadow in the gathering murk. \"Evening becomes night,\" he answered. Responding to Covenant's underlying query, he added, \"I do not yet descry the Worm. Though its coming is plain, it remains beyond my discernment.\"\n\nAnd mine, Covenant sighed. Tightening his grip on himself, he turned back to the Feroce. \"Is the havoc close? The Worm? Do you know? Can your High God feel it?\"\n\nThe creatures replied with a thin wail, quickly cut off. Almost gibbering, they forced themselves to say, \"It is near. How do you not know that it is near? Our High God asks what he must do. He asks with desperation. His alarm is terrible.\"\n\nNear? Covenant muttered to himself. Hellfire!\n\n\"I'm sorry,\" he told the Feroce gruffly. \"You'll just have to wait. I won't know what to say until I see it.\" Almost at once, he went on, \"And I won't see anything until you get rid of those fires.\" They blinded him to everything else; cast a pall of memories over his mind. He remembered the Illearth Stone too well. \"If you can't survive without them outside the Sarangrave, hide them somewhere. I won't abandon you. I'll tell you as soon as I have something.\"\n\nThe creatures quailed. They moaned like the wind. But they did not protest. One by one, they retreated among the stones. For a while, their emerald lingered on rims of granite and basalt. Then Covenant lost sight of them.\n\n\"Branl?\" he asked anxiously. \"Anything?\"\n\n\"Perhaps,\" replied the Humbled. \"I am uncertain.\"\n\nCursing, Covenant surged to his feet. The wind seemed to blow darkness into his covert. Branl was little more than an outline against the rocks.\n\nIf the Master's acute senses were uncertain, Covenant would be effectively eyeless; but he had to look. Pressing himself against his companion, he stared through the eastward oriel until the strain of trying to see made his forehead throb as if he had bruised it. Still he found nothing.\n\nOr something.\n\nA hint of light at the boundary between sea and sky.\n\n\"There.\" He pointed. \"Did you see it?\"\n\nAt first, he thought that it was heat-lightning: a storm brewing. Almost immediately, however, he realized that he was wrong. The light did not flicker and glare. Instead it appeared to float on the distant turmoil of the seas.\n\nWind lashed at his eyes. It had become a gale.\n\n\"It resembles fog.\" The last of the Humbled sounded utterly dispassionate. \"A luminous fog, lit from within. Storms which arise nowhere else clash within it.\" After a moment, he remarked, \"The fog and its storms shroud an immense power. It brings havoc in all sooth, such havoc as no _Haruchai_ has ever witnessed. Yet the power does not harm the seas. It merely disturbs them.\"\n\nWaves hammered harder at the base of the cliffs. In spite of his numbness, Covenant felt the ground under his boots trembling.\n\nHell and blood. \"That's the Worm?\"\n\nComing from the east? Straight for the Great Swamp?\n\n\"I deem that it is. And it is swift. Yet the fog\u2014and indeed the storms\u2014run some distance ahead of their source.\" Branl turned to Covenant. \"Ur-Lord, I must speak of this. Time remains to us. If you wish it, we may flee in safety. Wild magic will enable us to traverse many leagues ere this peril achieves landfall.\"\n\nCovenant clenched his teeth until his jaws ached. \"Who do you think you're kidding? We can't leave now. Not until we see what that thing does.\"\n\nThe eerie glow expanded on the horizon. Already it was distinct even to his marred vision. He felt its force in the wind on his face. Its teeth seemed to gnash at his cheeks. The luminescence did indeed resemble fog, vapor filled with lightning. But the lightning did not waver or strike: it _endured_ , a convulsion of bolts without beginning and without end.\n\nAnd the fog did not flow toward the southwest. Rather it sent tendrils like arms ahead of the storms, questing over an area as wide as the delta. Soon, however, even the most distant streamers began curving inward, reaching for Lifeswallower.\n\nReaching as if they had found the spoor of the Worm's prey.\n\nOh, bloody hell!\n\nBands of fog drifted over the seas. They drew closer with every harsh thud of Covenant's heart. Wild winds hurt his eyes, but he could not look away. Now he saw that the actinic glare within the brume was not truly constant. Instead of jumping and crackling, it swelled and receded incrementally, a slow seethe which belied the speed of its advance; a gradual rhythm like the undulating heave of a tremendous body. And every surge flung the vehemence of the waves harder against the cliffs. Collisions and crashes sounded like thunder; like the blare of steerhorns announcing ruin.\n\n\"Ur-Lord,\" Branl stated, \"we must not delay. These forces threaten the headland. We cannot withstand them.\"\n\n_Damn_ it! The wind was trying to tell Covenant something. It urged him to _think_ \u2014\n\nThe inundation of Lifeswallower's delta. The bitter lash of salt.\n\nIf he judged only by smell, he would believe that the whole of the Great Swamp had already been ripped out of existence. Uncounted millennia of poisons no longer reeked; no longer spread their nauseating odors into the air. The fury of wind and water effaced every other perception.\n\nSurely that _meant_ something?\n\nStreamers full of fatal light swept closer, riding the blasts. One of them poured up the precipice in front of Covenant and Branl. Squirming like a serpent of moisture, it writhed among the stones. A ribbon as luminous as the enchanted stone of the Lost Deep brushed Covenant's cheek before he could jump back. For an instant like a heartbeat, it appeared to curl around Branl. Its touch was damp and gelid, bitterly cold, as fierce as the caress of a _caesure_. But the fog did not react to Covenant and his companion; to Joan's ring or Loric's _krill_. Oblivious to anything that was not food for the Worm, it ran on along the wind, gusting westward.\n\nNow Covenant saw a shape within the hermetic mass of the storms, a dark form limned by the heavy rise and fall of the lightning. Infelice had described the Worm as _no more than a range of hills_ in size. _An earthquake might swallow it_. But to him, it looked more like a chain of mountains breasting inexorably through the seas. Its power was staggering: he was barely able to keep his feet. Perhaps his appalled senses exaggerated the Worm's physical bulk; but nothing could measure its sheer _force_. He was too human to look at it for more than a moment at a time.\n\nBy comparison, the lurker was trivial in spite of its polluted mass. It could do nothing to thwart the Worm's passage. It could only die.\n\nAnd the World's End was definitely heading west. Toward Mount Thunder.\n\nHellfire! Hell and damnation! Covenant was thinking about the problem backward. The wind carried away the rancid effluviums of Lifeswallower and the Sarangrave. Of course it did. But considered from a different perspective, the gale _blocked_ the fetor.\n\nAnd how did the Worm find its prey? How did it locate the _Elohim_ in their myriad hiding places? By scent. It smelled them out. Not in any ordinary sense, no. They did not emit a mundane aroma. But their magicks, the mystical essence of who they were: _that_ the Worm could detect.\n\nIf those emanations could be detected, perhaps they could also be blocked. By a different kind of power. A force that was inherently _wrong_ for the Worm, antithetical to its appetites.\n\nMore urgently, Branl insisted, \"Ur-Lord.\"\n\nThe Worm's puissance had become explicit, even to Covenant's blunted nerves. Its might shone through the rigid rocks of the headland as if they were transparent.\n\nHe guessed that it was still two or three leagues out to sea. But at that speed\u2014He had no time to doubt himself. Practically reeling, he wheeled away from the oriel; away from the heedless band of fog.\n\nAnd as he moved, he yelled, \"Feroce! I need you!\"\n\nGlints of green showed in the jumble. They were too far away.\n\n\"I need your High God! I need him _now_!\"\n\nThe wind snatched words from his mouth. They disappeared among the stones, meaningless. Nevertheless the fires came closer. Gleams flashed from place to place, apparently running.\n\nAs the first creature emerged from the maze, the voice of the Feroce moaned urgently, \"Pure One? What must our High God do? He must not perish!\"\n\nStreamers searched the turmoil of the delta. Lightning pulsed with every heave of the Worm's bulk. Seas hurled chaos at the cliffs. The silent shout of storms constrained by the Worm's aura made the ground under Covenant lurch as if the foundations of the promontory were in spasm.\n\n_An earthquake might swallow it_. Under the right circumstances, Linden could trigger an earthquake. She and the Staff of Law had that kind of strength. Covenant did not: not with Joan's ring.\n\nHaste and frenzy gripped him. \"Listen fast.\" He was hardly coherent. \"Try to understand. I don't want your High God dead. He can't fight the Worm. But he has to _act_ like he's going to fight. He has to rear up. Make himself as big as he can. Right _there_.\" Covenant pointed at the drowned stretch of Lifeswallower to the north. \"I need him to block the way,\" confuse the Worm's instincts, fill the Worm's senses with corrupt emanations; mask the powers hidden in Mount Thunder.\n\n\" _Ur-Lord_ ,\" protested Branl.\n\n\"Pure One?\" The voice of the Feroce was a cry, a groan, a prayer. Their fires shuddered like the cliffs' bedrock. \"We are little. Our minds are small. We do not\u2014\"\n\nCovenant cut them off. \"Just _tell_ him!\" He wanted to tear his hair. \"I can't explain. I don't have time. I need him to _do it_. Rear up. Make himself _huge_. Pretend he's a barrier.\"\n\nIf the lurker did not panic\u2014if the monster kept its word\u2014\n\nFrantically Covenant strove to impose comprehension on Horrim Carabal's acolytes. \"The Worm doesn't want him. If he doesn't fight, it won't hurt him. But he has to look _big_ enough to fight.\n\n\" _Tell_ him! He can get out of the way if the Worm doesn't stop. But first he has to try to make it _pause_! He has to make it look somewhere else for food!\"\n\nWould that work? Of course not. Or not for long. But it might distract the Worm for a while. Slow it down. Buy a little time. Until the World's End found a different scent.\n\nThe Feroce could do what he asked of them. They could communicate swiftly enough. And the deeper waters of Lifeswallower were the lurker's true home. The core of the monster's mass and muscle lived there. If Horrim Carabal chose to do so, it could respond immediately.\n\nAlready the Worm had seethed a league closer.\n\nWind scattered the wailing of the Feroce among the stones. Their fires rose like screams. The gale did not touch their emerald theurgy, but the mounting convulsions beneath them did. The Worm's hunger made the flames flinch and bend.\n\nInstead of answering, they turned and fled.\n\n\"Ur-Lord!\" Branl demanded. He stood in the path of a glowing tendril, but it flowed around him as if he were nothing more than granite or basalt. \"We must depart!\"\n\nShaking his head, Covenant turned to peer down at the delta. \"I just need a minute! I have to see if this is going to work!\"\n\nPlease, God damn it! he begged the lurker. I almost killed myself against _turiya_. Clyme died for you. I know you're terrified. But you made a promise.\n\nWhy would Horrim Carabal comply? Covenant was asking the monster to dare its own extinction.\n\nThe lash of seas over Lifeswallower had become an undifferentiated flood. Incoming waters tried to withdraw and could not: the imponderable forces of the Worm's approach drove them farther into the Great Swamp. Night had overtaken the Lower Land, but it changed nothing. The fog shed its own light. Its radiance made the hard stone of the headland seem as insubstantial as dreams. Through obstructions of rock, Covenant felt every rise and dip of the Worm's heaving. The rhythm of its undulations was slow. It seemed almost casual. Or perhaps it was sluggish yet. Nevertheless its speed\u2014or its power\u2014filled him with dismay. His chest felt ready to burst.\n\nDesperately he stared past the rim of precipice, praying.\n\nBranl put a hand on his shoulder. \"Rallyn comes. We must ride.\"\n\nThe Humbled could have coerced Covenant; but Covenant ignored his companion. \"Look!\" Flailing one arm, he indicated the delta. \" _Look!_ Tell me what you see!\"\n\nInstead of pulling Covenant away, Branl moved to stand at the Unbeliever's side. _Your task is mine_. Leaning forward, he studied the thrash and clash of the flood. _I am alone and have no path other than my chosen service_. For a moment, he did not speak. Then he announced through the gale, \"Ur-Lord, you are answered.\"\n\nAnswered?\n\n\"The lurker gathers beneath the waters. Its bulk is immense. I cannot gauge its full extent. At present, it does not rise. It merely gathers. Yet I deem that it will heed your wishes. Its presence serves no purpose else.\"\n\n\"Tell me,\" Covenant panted. \"Tell me when it moves.\" The growing might of the Worm's aura snatched the air from his lungs. He struggled for every breath. \"I can't _see_.\"\n\nLuminescence shone through the stones, but it did not affect the Humbled. He seemed impervious to fog and catastrophe. He sounded more stolid than granite.\n\n\"Ur-Lord, there is more.\"\n\n\"More?\" Hellfire! \"Tell me!\"\n\nThe Worm was coming closer. In all the world, only a few moments remained; a handful of heartbeats. If the Worm passed the lurker toward Mount Thunder, nothing would stop it.\n\n\"The lurker begins its rise,\" reported Branl impassively. \"It is not alone.\"\n\nCovenant fought to see; fought to breathe. At first, he could only discern the tumultuous scourge and moil of seas, the accumulating pressure of the Worm's advance. But then he thought that he saw darkness swell near the boundary between the delta and the ocean. The waters there piled higher as if they were surmounting an obstacle.\n\n\"Do you descry them, ur-Lord? They cling to the lurker's sides.\"\n\nCovenant shook his head. He was sure of the monster now. In the center of his vista, it burst above the waves. Like a tectonic plate thrusting upward, the lurker jutted into the air. Breakers slammed against Horrim Carabal and were flung aside. Brandishing scores of tentacles like threats, it stretched higher, taller than any Giantship. Its central mass was a match for the Worm's. And it spread itself wide, wider than the coming catastrophe: a barricade against havoc. Clearly the monster understood its task.\n\nBut _them_? Clinging to the lurker anywhere? No. His eyes were too weak.\n\nThe lurker was too weak as well. In spite of its size and muscle, its emanations did not reach Covenant. He felt every surge of the Worm's approach; felt the harsh chill of the fog and the static charge of lightnings. But Horrim Carabal was nothing more than a shape in the distance, scarcely visible: too mortal to hinder the World's End.\n\nNevertheless the Worm slowed. Apparently it could sense the lurker's presence, although Covenant could not. A wall of malign toxins had arisen from the waters. The Worm slackened its haste as if it had become uncertain.\n\nThem?\n\nCovenant tried to plead for an explanation, but he had no air and no words.\n\nYet clearly Branl had not forgotten the effects of Kevin's Dirt on Covenant. The Humbled answered Covenant's soundless query. \"Ur-Lord, they are ur-viles. They are Waynhim.\"\n\nCovenant stared, and panted, and could not think. Ur-viles and Waynhim? Here?\n\nWhy?\n\nBranl pitched his voice to pierce the blast's lurid wail. \"I gauge that every surviving creation of the Demondim has come to oppose the Worm. Holding to the lurker's flesh, they wield their lore. Black theurgies with the appearance of corrosion spread from hand to hand among them. These magicks are not liquid. Rather they resemble strands of incantation. As they expand, they take the form of a web.\"\n\nCovenant cursed his inadequate sight. He ignored the shudders rising through the headland. Fervently he concentrated on Branl; listened as if he were counting every word.\n\n\"This web the creatures extend across the monster where it fronts the Worm. The sorcery of the web is fierce and bitter, rife with the unnatural fury of the Demondim, and of the Viles. I do not doubt that Linden Avery would name it _wrongness_. Yet the lurker takes no notice. Clearly the web does not pain the High God of the Feroce.\"\n\nCovenant groaned and swore because he could not _see_ it. He recognized only Horrim Carabal's bulk rising like midnight in the Worm's path. If the glow of the Worm's lurid aura glistened on the lurker's exposed flesh, or on the weird theurgy of the ur-viles and Waynhim, those sights lay beyond his reach.\n\nLike the world at the mercy of its own death, he was mostly helpless, yet not helpless enough to be spared the burden of bearing witness. And he was not blind to the Worm. Its power shone, vivid as etch-work, through every crouched or yearning menhir around him. It shone through the flesh of his arms and chest, lit every bone. He was as vague to himself as mist. Without Branl's solidity at his side, Branl's uncompromising substance, he might have been torn apart and scattered by the gale.\n\nIf he could not see the lurker distinctly\u2014and could not see the creatures or their lore at all\u2014he could still watch the approach of the World's End.\n\n\"It appears,\" Branl said, \"that your ploy may accomplish its intent. The lurker and the Demondim-spawn present a barricade of ill and evil, of ancient poisons and unnatural knowledge. It does not bar wind and storm and seas, though the lurker's form does so. Yet it disturbs perception. It would offend Linden Avery's percipience. It defies my efforts to name its essence.\"\n\nAnd it was working. Covenant felt that in every nerve of his disease-ridden body. It was _working_.\n\nLike the lurker itself, the strange theurgy of the web confused the Worm's senses. In spite of their fluid shapes and their arrogance, the _Elohim_ were beings of Law. They existed in accordance with the strictures of the Earth's creation. But Horrim Carabal was a perversion of Law. And the weird powers and comprehensions which the ur-viles and Waynhim had inherited or gleaned from their makers seemed to render Law meaningless. Together, the monster and the Demondim-spawn masked the scent of food.\n\nBaffled, the Worm slowed again. Gradually it heaved to a halt.\n\nA small tsunami pounded against the lurker, slashed at the web of sorcery. From border to border, the delta convulsed as if its foundations were vomiting. But Horrim Carabal withstood the assault. And the Demondim-spawn knew what they were doing. Their lore did not falter.\n\nThe Worm's storms and streamers searched to one side, explored the other. But the lurker had made itself _wide_. And the net of dark magicks covered Horrim Carabal from edge to edge. The webbing throbbed with acrid implications. The Worm's hunger hunted\u2014and did not find.\n\nThis eerie equipoise between ruin and darkness would not last: Covenant knew that. The Worm was too powerful to be stymied indefinitely. The lurker or the Demondim-spawn might flinch at any moment. They might all die. But they were holding _now_. If they could stand until the Worm detected the spoor of some other _Elohim_ \u2014or until its primal needs urged it toward _Melenkurion_ Skyweir\u2014\n\nA turn in the direction of the EarthBlood would bring the Worm straight at Covenant and Branl.\n\nClutching his companion, Covenant gasped, \"Let's go! While we still can!\"\n\nHis unlikely allies had achieved a tenuous pause. If the Land needed more time, Linden or some other power would have to provide it. Thomas Covenant had come to the end of what he could attempt as he was.\n\n## 6.\n\nPromises Old and New\n\nThe twilight did not change as Linden's company rode. A harsh grey held the landscape, a half-light without the softening of dawn or the soothing after sunset. It might have been the gloom before the onslaught of a storm, but there were no clouds. Despite the intrusion of Kevin's Dirt, the sky remained clear, fretted with doom, drawing the bright plaint of the stars closer, etching their deaths vividly against the fathomless abyss of their firmament. Linden could have believed that the Arch already trembled on the verge of collapse; but her health-sense insisted otherwise. The long strides of the Ranyhyn and the hoarse panting of the Giants insisted. Even in the absence of natural day, her pulse continued to measure out her life. And the blurred terrain continued to modulate around the company: a sign of movement that was also an affirmation that Time endured.\n\nRiding with the Staff of Law and Covenant's white gold ring into the last dusk of the world, Linden tried to think of the unrisen sun in terms that did not terrify her. After all, the sun was simply another star. The Worm's power to affect or even extinguish it made a kind of sense. And did not the gloom itself assert that the sun was not altogether destroyed? The final dark had not yet claimed the Earth. Even in this crepuscular blight, hope might be possible.\n\nKevin's Dirt asserted the contrary. Indeed, it seemed stronger here than it had on the Upper Land. Even now, no more than an hour or two after the failed dawn, the vile fug had begun leeching the sensitivity from Linden's nerves, blunting her ability to discern the conditions of her companions and even the nature of the terrain; promising failure.\n\nAccentuated by the dull light, the bloodstains that darkened the bottoms of Jeremiah's pajamas seemed to creep higher, opening like jaws to swallow him.\n\nBut the Ranyhyn ignored Kevin's Dirt. Running at a canter that accommodated the ragged endurance of the Giants, the horses had left behind the mounds surrounding the gully and the stream. Now they measured out the leagues across a hammered plain that appeared to stretch endlessly into an obscured future. Gloaming effaced the details of the landscape, rendered it effectively featureless in every direction. Still the eaten chart of the stars and Linden's tarnished health-sense confirmed that the horses had not altered their heading. They reached for the northeast with every stride, never hesitating.\n\nYet they did not neglect the needs of the Giants or their riders. In spite of Jeremiah's impatience, they paused at every clump of _aliantha_ , every thin rill and brackish pool. At such times, the boy refused to dismount. Instead he sat chafing until the company was ready to run again.\n\nBy mute agreement, Linden, Stave, and Mahrtiir drank little and ate none of the treasure-berries, leaving them for Coldspray and her comrades. Nonetheless it was clear that the Swordmainnir were suffering. Linden heard an ominous wheeze in Latebirth's respiration, and in Cirrus Kindwind's, and occasionally in Stormpast Galesend's. The others heaved for breath against the weight of their armor and weapons. Their faces were grey with exertion.\n\nAt a time that should have been mid-morning, Manethrall Mahrtiir brought Narunal to Hyn's side. \"Ringthane,\" he called over the clatter of hooves, \"we must consider what we do. If we do not soon gain our aim, the Giants will be too weary to aid your son. That they have come so far at such a pace bespeaks both great strength and great valor. Yet they are mortal withal. Ere long, even they must falter.\"\n\n\"What do you suggest?\" Linden could sustain the Swordmainnir with Earthpower for a while. But repeated infusions of imposed energy would exact a price. The women might well be left utterly prostrate when her assistance finally lost its efficacy. Earthpower and Law were only Earthpower and Law: they could not counteract the organic need for food and water and rest indefinitely. And Linden was reluctant for other reasons as well. Speed might be Jeremiah's only defense. \"Of course they need rest. We all do. But the Worm is coming. You said it yourself. We have to hurry.\"\n\nMahrtiir faced her with disgust in his mien, but it was not directed at her. \"For that reason, Ringthane, I deem that we must part again. While you accompany your son with Stave, I will remain to guide the Giants at a slower pace. Their aid may be much delayed, but they will rejoin you _capable_ of aid.\"\n\nAs if he expected Linden to demur, he added harshly, \"I serve no other purpose in this company. But I am able to ride brave Narunal, and to obey him\u2014aye, and also to comprehend his wishes. Therefore I await your consent.\"\n\nLinden saw that Jeremiah was listening; felt protests rise in him. She phrased her reply for his sake as much as for Mahrtiir's.\n\n\"That makes sense. Exhaustion won't help any of us.\" She forced a wry smile. \"And if anyone can convince Coldspray to be reasonable, you can. Maybe Stave and I can help Jeremiah make a start without you.\"\n\nJeremiah brandished a fist in approval.\n\nBut Mahrtiir hesitated. \"Then I crave a boon of you, Ringthane,\" he said after a moment. \"Restore my discernment to its fullest, that my use to the Swordmainnir may be prolonged. It will not endure. Of that I am aware. But I yearn to postpone the return of complete futility.\"\n\nIn spite of herself, Linden was loath to comply. She did not want to raise black fire in a lightless world. The prospect felt like a violation. Yet she could not refuse the Manethrall. Had her fears been his, he would have faced them at once, eager for struggle and combat.\n\nAdjusting her grip on the Staff of Law, she reached for Earthpower.\n\nAs she had expected or dreaded, her flames were barely visible. Their force was palpable enough, and to an extent comforting. But they were the hue of Jeremiah's fouled pajamas, the color of deepest night, and they seemed to thicken the gloom around them.\n\nNevertheless her magic was an expression of Law. Its inherent beneficence had not been altered. She had turned the wood to ebony in battle under _Melenkurion_ Skyweir. In the graveyard of Jeremiah's mind, she had become a form of blackness herself. If her power disturbed her now, it did so because it told the truth about her.\n\nAs if she were abasing herself, she covered first Mahrtiir and then herself in cleansing theurgy. And when her senses had recovered their acuity, she extended fire to the Giants, gifting them with all of the vitality that she could provide.\n\nThen Linden quenched her Staff. Slumping on Hyn's back, she told Mahrtiir weakly, \"Be safe. Catch up with us when you can. We'll need you.\"\n\nClarion as a whinny, the Manethrall replied, \"Fear nothing, Ringthane. We will come.\" Then he drew Narunal back from Linden's side so that he could speak to Rime Coldspray.\n\nHyn, Hynyn, and Khelen seemed to understand what had been decided without any word from Linden or Mahrtiir\u2014or indeed from Stave. Running like water on a smooth slope, they extended their strides into a full gallop. In the lead, Jeremiah yelled his excitement at the heavens. Then he settled himself along Khelen's neck as if he sought to increase the young stallion's speed.\n\nIn moments, the Giants were no longer visible behind Linden. For a short time, she continued to feel their presence. Then the Ranyhyn outran the range of her health-sense, and she was alone with Stave and Jeremiah once again.\n\nrom the plain, the riders entered a region of jagged stones piled against each other like the detritus of a mountain broken by earthquake or cataclysm. Some of them resembled the riven limbs and torsos of megalithic titans. Others were towers about to topple, or raw chunks of granite and obsidian the size of Giantships, or splinters as sharp as spears. Among them, the footing was treacherous, and the horses were compelled to pick their way at a gait little quicker than a trot. As if in compensation, however, springs and streams became more plentiful. Most were too thick with minerals and old rot to drink; but a fair number were merely brackish, and a few ran clear, gurgling untainted from some buried source. As before, Linden and her companions had left all of their supplies with the Giants and Mahrtiir; but they found more than enough good water to appease their thirst.\n\nPausing at a stream where the Ranyhyn drank as though they did not expect to discover more water for a long time, Linden asked Stave where they were. Sure of himself, he replied that they were approaching a region like an isthmus of the Spoiled Plains between Sarangrave Flat to the north and the Shattered Hills in the southeast. She had guessed as much; but she was relieved to hear that the marge-land was ten leagues wide or more, and that beyond it the Spoiled Plains expanded to fill the Lower Land between the Sarangrave and the Sunbirth Sea. If the horses kept to their present heading, they would have nothing to fear from the lurker.\n\n\"Come on,\" Jeremiah muttered. \"Come _on_.\" Then he sighed. \"I'm hungry. I hope we find _aliantha_ soon.\"\n\nSternly Stave remarked, \"In the ages of the Lords, there were no _aliantha_ on the Lower Land to the south of Lifeswallower. We are fortunate that they have taken root here during more recent millennia, sparse though they may be. But we cannot know how they were spread, or by whom. If we have ridden beyond their extent, we have no redress for their absence.\"\n\n\"That's easy for you to say,\" Jeremiah retorted; but he sounded impatient rather than irked. \"You're _Haruchai_. I'm not. If we don't find treasure-berries, I hope you can think of something else for us to eat.\"\n\nStave's only reply was a shrug.\n\nSoon the Ranyhyn were in motion again; and shortly after midday, they left that wrecked region behind. Now they ran, fleet as coursers, along a comparative flatland that lay at the foot of a long incline like the rim of a tectonic upheaval. There the running was easy, and the strides of the horses overcame distance as though the leagues were trivial.\n\nStill the stars died overhead. Like Jeremiah, Linden was hungry. But in addition, she was beginning to share his frustration. A part of her did not want to discover malachite. It did not. Her reluctance was a thin whimper in the background of every thought. More would be required of her than she could bear to contemplate. Nevertheless the plight of the stars\u2014and of the _Elohim_ \u2014infected her with urgency. The prospect of a lightless sky appalled her. Much as she disliked or even loathed the _Elohim_ , their peril seemed more important than her personal fears.\n\nThe grey gloom wore on her like an old sore, immedicable, weeping vital fluids. While Hyn's muscles flexed under her, and the mare's sweat soaked into her jeans, irritating her legs, Linden began to wonder whether night would ever come again\u2014and if it did, whether the Ranyhyn would allow themselves and their riders to rest. If Time remained essentially intact, surely some form of circadian cycle continued to rule the world? What would it mean if night did not come?\n\nHer private dread seemed to grow more petty with every surmounted league, every troubled thought. Now she wondered how anyone could refuse to take the innominate risks that lay ahead of her. How could she? If she ever hoped to hold up her head in front of Jeremiah and Covenant again?\n\nGradually the incline swung away, surrendering to erosion. Beyond it, Stave called Linden's attention to the fact that the Ranyhyn were adjusting their course. \"The northeast remains accessible,\" he informed her, \"yet now our path tends toward the Sarangrave.\"\n\n\"Do you know why?\" Memories of the marshland's fetor and the lurker's malevolence ached in her guts. She never wanted to approach the Flat again.\n\n\"I do not. _Haruchai_ cannot commune with Ranyhyn as the Ramen do. However, I surmise that the horses require fodder. Among the wetlands on the verges of Sarangrave Flat, they may find grasses to sustain them.\"\n\n\"Can't the lurker reach them if they do that?\"\n\n\"Indeed,\" Stave acknowledged. \"Yet sustenance they must have, and there is none in this region. Nearer to the coastline, the devastation of Corruption's wars and workings eases. There forage may be found. But the distance is too great, even for the endurance of the Ranyhyn. If they would continue to run as they do, they must dare their ancient foe.\"\n\nOh, good, Linden muttered to herself. Perfect. Just what we need. Another fight with that monster. But she could feel a new trembling in Hyn's muscles, hear hints of frenzy in Hyn's respiration. Stave was probably right.\n\n\"Then we'll have to protect them.\" She meant herself. Her companions could not oppose the lurker\u2014and the monster craved her Staff.\n\n\"Maybe we'll find _aliantha_ ,\" called Jeremiah. \"If the ground grows other plants, it can grow treasure-berries.\"\n\n\"Maybe,\" Linden conceded. To Stave, she added, \"If I get in trouble,\" if the Feroce cast their glamour over her mind again, \"take the Staff. I don't care if you have to hit me to get it. Just don't let that monster have it.\"\n\n\"I hear you, Chosen.\" The former Master sounded as passionless as marble.\n\nShe trusted him. Nevertheless he eased none of her trepidations.\n\ntill Khelen, Hynyn, and Hyn ran, defying their tangible exhaustion: the froth on their nostrils, the sweat on their coats, the ominous rattle in their mighty chests. At intervals, Linden refreshed them with brief blooms of Earthpower. But she did not use magic to extend her percipience. She did not want to know how near the Sarangrave might be.\n\nHeading more north than east, the riders rushed down into a wide lowland like an ancient caldera. There the Ranyhyn found a few patches of scrannel grass, only a few mouthfuls apiece, hardly enough to blunt the keenest edges of their need. Then they resumed their stubborn race against the reaving of stars. Laboring painfully, they pounded up the slight slope at the far rim of the lowland; and still they ran.\n\nIn this direction, they would certainly encounter the Sarangrave. Linden tried to tell herself that they might find what they sought at any time; that their ordeal might end beyond the next rise, or somewhere in the next shallow vale. But she did not believe it.\n\nAgain and again, she came back to _trust_. She had given the Ranyhyn the only gift that was hers to grant; but neither she nor they could afford to rely upon it. She would have to simply trust that they could accomplish what they had asked of themselves.\n\nA long time later, when her bestowed Earthpower had drained out of the horses entirely, the twilight began to thicken, become more viscid. A tumid dark crept out of the east to mask the contours of the landscape, deepen the bitter doom of the heavens. For a while, the dull light faded by minor increments, barely detectible: then it was gone altogether. Linden could not imagine how the Ranyhyn knew where to set their hooves. Nevertheless they did not falter. Perhaps they saw or felt the stars as clearly as she did. Perhaps they could hear the undefended lights pleading for redemption.\n\nAbsorbed by worries, she was slow to notice that she could smell water. It was dank and stagnant enough to be Sarangrave Flat, pervasive enough, fraught with implications of rot and dire corpses\u2014but it was water nonetheless. And where there was water, there might be provender for the Ranyhyn.\n\nAs if to answer her, Khelen whickered weakly; and Stave said, \"The Sarangrave is nigh, Chosen. It is shallow in this region. A fool who did not fear bogs and quags might wade for a league without encountering deeper streams. Yet I do not doubt that we are now within the ambit of the lurker's awareness.\"\n\nHe paused to let Linden respond. When she found nothing to say, he asked, \"Will you now surrender the Staff? I cannot wield it. Yet its absence from your hands may serve to ward you.\"\n\n\"Not yet.\" She was shivering at the cooler air as though she shared the extremity of the horses. Her memories of the Feroce and the lurker were too recent. And yet the Ranyhyn appeared to be on the verge of stumbling to their knees. They had to have food and rest. \"Not until we see the Feroce. They're the real danger.\" The theurgy of green fires cupped like instances of the Illearth Stone in their palms enabled them to enter her mind. They could erase the distinction between reality and memory. \"The lurker can't reach us if we don't get too close.\"\n\n\"I don't care about that,\" Jeremiah put in. His voice seemed to come from the bottom of an abyss. \"The Ranyhyn are desperate.\n\n\"I don't think I have the kind of power that's good for fighting. But I can be a distraction. I mean, since the lurker is so hungry for Earthpower. Maybe I can get its attention.\"\n\n\"Then stay back,\" Linden ordered hoarsely. \"If you're going to distract anything, do it from a safe distance. Let Stave and me protect you.\"\n\nAs she spoke, Hyn's strides began to slow. Just for an instant, Linden thought that the mare had come to the end of her endurance. But the smell of water was so thick that it hurt Linden's sinuses; and she recognized almost at once that Hyn was slackening her gait deliberately.\n\nStave responded by urging Linden to dismount. \"The littoral of the marsh is nigh. We must remain beyond the lurker's reach.\"\n\nWhen Linden nodded her consent, Hyn staggered to a halt. While Linden slid to the ground, Stave sprang down from Hynyn's back. Lurching, Khelen brought Jeremiah to Linden's side. In spite of his impatience, Jeremiah did not complain as he dismounted. Instead he patted Khelen's neck, muttering, \"Don't worry about me. I'll be all right.\"\n\nThe young stallion whickered thinly. Shambling into the darkness with Hyn and Hynyn, he headed toward water and forage.\n\nWhile Linden watched the horses, Stave spoke again. \"Await me, Chosen. I will attempt to discover _aliantha_. If I discern water which we may drink without harm, I will guide you to it.\"\n\nAt once, he followed the Ranyhyn. Like them, he disappeared as if he had been swallowed by the tenebrous air.\n\nHe may conceivably have wished to let Linden talk to Jeremiah alone. Beneath his _Haruchai_ dispassion lay a familiar capacity for solicitude.\n\nBut what could she say to her son? In certain respects, she understood him too well. Trapped deep within him, a terrible storm was brewing. He needed his defenses, his urgent focus on a vital task, to contain the violence of his refused memories. And he was altogether too young for his years. Lost in dissociation, he had not had time to learn how to live with himself.\n\nAs gently as she could, she murmured, \"You told Khelen not to worry, but I can't help it.\" Feeling him stiffen, she continued, \"Oh, I'm not worried about the waiting. You can do that when you have to. You've had plenty of practice.\n\n\"No, it's what you want to do for the _Elohim_ that scares me. A door like that\u2014You'll have to make it so _big_. It's going to take time. And when you're done, it's going to be vulnerable. If we can't protect it\u2014\"\n\nShe would need help. She could no longer ignore that truth. More help than any of her friends could supply. Covenant himself had said it. _We're too weak the way we are_. _We need power_. More power than Loric's _krill_ could summon, or the Staff of Law diminished by Kevin's Dirt, or a woman who was not a rightful white gold wielder.\n\nAt her side, Jeremiah relaxed a bit. \"I know,\" he admitted grimly. \"If we go through all that\u2014I mean, if we find enough malachite, and the Giants help me build what I want, and it pulls the _Elohim_ in, at least all the ones who're left\u2014and then the Worm just swallows my door\u2014\" He shuddered. \"That'll be worse than anything.\"\n\nHearing him reminded Linden of Kevin Landwaster and her own despair. Before she could respond, however, he said, \"But, Mom.\" He sounded as harsh as the night. \"I have to try. I don't know what else to do.\"\n\nThat, too, she understood. \"Then listen to me,\" she returned more sharply than she intended. \"Building your door\u2014That's your part. And it's enough. It's _enough_. The rest is up to us.\" It was up to her. \"We'll figure out a way to protect it. And if we can't, you'll just have to keep reminding yourself that _you did your part_. You aren't responsible for what happens after that.\"\n\n\"But it'll all be wasted!\" he protested. \"They'll all die.\"\n\nHigh Lord Kevin must have felt the same before the Ritual of Desecration. Nevertheless his ancestors among the Dead had forgiven him. And Linden had missed her chance to take pity on Elena. From She Who Must Not Be Named, Linden had learned how much her self-absorption had cost Covenant's daughter.\n\n\"No,\" she replied carefully, \"it won't be wasted. I can only imagine how bad it might feel to see something that you built destroyed. Especially something like _that_. But _listen_ to me. In a way, I've only known you for a day and a half, and already I'm so proud of you that I don't have any words big enough for it. Now I understand what parents mean when they talk about their hearts bursting.\"\n\nShe gathered passion as she spoke. Her own parents had never felt that way about her. Not once. The bitter legacies of her childhood filled her voice. Trying to sway her son, she was pleading for herself.\n\n\"For all of those years when I was taking care of you, do you know how many times I wondered if it was all wasted?\" If she had opened her heart and lavished her love for nothing? \"I'll tell you. I _never_ wondered. It was always worth doing, all of it.\n\n\"Of _course_ I cared about what might happen. Of _course_ I wanted you to find your way out. I wanted you with me. But I didn't adopt you and love you because of what _might_ happen. I did it because you were _always_ precious to me, every minute of every day. You were enough. I didn't need to know the future to know that you were worth everything.\"\n\nShe felt frustration from him, a rising denial; but she over-rode it.\n\n\"So maybe you won't be able to build your door after all. Maybe we won't be able to protect it after you make it. Maybe the _Elohim_ and the stars and all of us are doomed. So what? Right here, right now, you want to do everything you can to help, and that's wonderful. If the Worm eats your door, and you feel so hurt and angry and useless that you can't stand it, remember that I'm _proud_ of you.\"\n\n\"Stop it, Mom.\" He was crying, and trying not to show it. \" _Nobody's_ proud of a failure.\"\n\n\"That's nonsense.\" Instinctively she responded as if he were a normal boy, able to hear her. \"Failure isn't something you _are_. It's something you _do_.\" She needed to hear what she was saying. With every word, she pleaded for an answer to her mute dread. \"Having the courage to escape your prison is who you are. Wanting to help the _Elohim_ because the world needs them is who you are. My son is who you are. Everything else is just making mistakes, or not having the right materials or enough help, or not knowing enough, or trying to do something that's actually impossible. It just _happens_. It isn't who you are.\"\n\nWith her whole heart, she asked, \"How do you suppose Covenant managed to save the world _twice_? It isn't because he's stronger or smarter or greater than Lord Foul. He's just stronger and smarter and greater than Lord Foul _thinks_ he is. He's had the right kind of help. And he isn't afraid to take the chance that he's going to fail.\"\n\n\"Mom.\" Jeremiah was crying openly now. \"Mom, stop. Please. I need\u2014I need\u2014\"\n\nShe understood that as well. Who would, if she did not? Remembering Anele\u2014remembering _Must_ and _Cannot_ and the old man's last valor, an act of self-confrontation that humbled her\u2014she dropped her Staff and swept her son into her arms. Hugging him tightly, she murmured his name to him as if it confirmed his worth.\n\nLike a young boy, he sobbed hard for a moment\u2014and like a teenager, he suppressed his pain quickly. For a heartbeat or two, he held his mother as she held him. Then he let go of her, stepped back from her clasp. Snuffling loudly, he rubbed his face with both hands, wiped his nose on his forearm. In a congested voice, he asked, \"What's taking Stave so long? The Flat is right over there.\" He gestured uselessly in the darkness. \"I'm hungry. He should be back by now.\"\n\nWell, he was a fifteen-year-old boy, embarrassed by what he considered a show of weakness. For his sake, Linden smiled ruefully. Her sigh of regret she kept to herself.\n\n\"I'm sure\u2014\" she began. But before she completed the sentence, she felt the former Master's severe aura returning.\n\n\"It's about time,\" Jeremiah muttered. Then he called to Stave, \"Did you find _aliantha_? Are we that lucky?\"\n\nAlmost immediately, Stave arrived, a darker shape condensed from the raw stuff of night. His hands were full of damp plants. \"I did not,\" he answered. \"However, I have discovered tubers which I deem edible. They resemble the roots from which the Ramen prepare _rhee_. Cooked, they will provide sustenance.\"\n\nLinden smiled again. As warmly as she could, she thanked the former Master. Then she asked her son, \"What do you think? Can you use your Earthpower for cooking?\" Had he gained that much control over his inheritance? She hoped so. He needed a chance to recover his sense of competence. \"I can do it, but I'm more likely to attract attention that we don't want.\"\n\nShe did not doubt that the lurker would devour Jeremiah avidly. But she also felt sure that the monster would find her Staff better suited to its particular hunger.\n\nEager to put aside his distress, Jeremiah extended his halfhand, accepted a root. \"I'll give it a try.\"\n\nAs he did so, Linden retrieved the Staff, braced herself on its possibilities. Then she turned every dimension of her remaining discernment toward Sarangrave Flat, searching for some sign of the lurker\u2014or of the Feroce.\n\nAfter a moment, she located the Ranyhyn. They stood along the verge of a stagnant pool, cropping bitter grasses and vaguely pernicious shrubs with apparent unconcern. Clearly no hint of the lurker disturbed them. Nor did what they ate.\n\nThe wetland beyond them looked shallow. Its waters ran in sluggish streams or sat in rancid ponds interrupted by small eyots of grass or twisted brush; by occasional trees gnarled and stunted in putrefying mud; by brief swaths of reeds that nodded back and forth like conspirators in the currents and the breeze. Everything within the range of Linden's percipience reeked of age and decomposition and ancient malice. Darkness covered the Flat, as funereal as a grave-cloth. Nevertheless nothing suggested the presence of the lurker or its acolytes.\n\nAt her back, she felt a short burst of fire. At once, it winked out. Jeremiah snorted in quick disgust, but his concentration did not waver.\n\nA moment later, she sensed heat. It flickered, shrank, threatened to die out, then swelled more strongly. \"Ha!\" Jeremiah panted. \"So _that's_ how\u2014\"\n\nSoon he was able to hold his magic steady. The smells of cooking joined the thick odors of the Sarangrave.\n\nSomewhere in the depths of the wetland, a night bird cried: a wail of fright. Linden heard a sharp splash, a sucking sound. She may have heard the clamp of teeth. The cry was cut off. More distant birds squalled as they took flight. From other directions came the rustle of disturbed roosting; the squirm of thick bodies in mud; the plash of creatures that may have been fish. After its fashion, Sarangrave Flat was thick with life.\n\nStill nothing resembled the lurker. Nothing warned of the Feroce.\n\nBefore long, Jeremiah let his Earthpower dissipate. \"Ow!\" he muttered cheerfully. \"That's hot.\" Then he bit into the tuber. Through a mouthful of crunching, he announced, \"Tastes like dirt.\" But he did not stop eating.\n\nBy degrees, Linden began to relax.\n\nJeremiah took another root from Stave, summoned fresh theurgy. \"Your turn, Mom,\" he murmured as he worked. \"It's actually pretty good, if you pretend you can't taste it.\"\n\n\"Stave?\" Linden asked over her shoulder.\n\n\"I keep watch, Chosen.\" The _Haruchai_ 's tone hinted at reproof. She should have known that he was always alert. \"Doubtless the Ranyhyn also will give warning at need. Eat while you may.\" A beat later, he added, \"I have yet to discover clean water.\"\n\nLinden hesitated to lower her guard. She had encountered the lurker more than once\u2014and once was too often. But hunger overcame her uncertainty. With an effort, she turned her back on the Sarangrave.\n\nJeremiah had nearly finished cooking a second tuber. He held it in his halfhand with his left cupped over it. A faint glow of heat radiated between his palms. When he judged that root was ready to eat, he handed it to Linden.\n\n\"Just remember. Pretend you can't taste it.\"\n\nStave was right, of course: when Linden studied the steaming tuber, she saw that it was safe to eat. More than that, it would strengthen her if she ate enough of it. Swallowing hard to clear the discomfiture from her throat, she took a bite.\n\n\"Dirt,\" she answered Jeremiah's expectant gaze. \"Just like dirt.\" In fact, the crisp plant was bland at first; but it had a sour after-taste that made her yearn for the cleanliness of _aliantha_. Nevertheless she ate it while Jeremiah cooked another root for himself. She had no choice. She was facing a future which might never contain another meal.\n\nfter Jeremiah had finished preparing all of the roots, and he and Linden had eaten as much as their stomachs could tolerate, Stave left again to search farther for water.\n\nHe was gone for what seemed like a long time. While he was absent, the Ranyhyn withdrew from the edge of the wetland, putting a little distance between themselves and the disturbing seethe of the waters. But they did not go far. Linden felt them clearly enough, resting between her and the Sarangrave.\n\nWhen the former Master returned, he announced that he had located safe water in an eddy cast by the turbid seethe of the Flat. It was admittedly brackish and tainted, but not so foul that it would make Linden and Jeremiah ill. There they were able to quench their thirst before the impulse to gag became too strong to suppress.\n\nReturning to the place where they had eaten, Linden urged her son to get some sleep while he could. Then she searched out a relatively level patch of ground for herself. With the Staff clasped across her chest, and her eyes closed against the dying of the stars, she tried to take her own advice.\n\nBut her fears nagged at her. They seemed to crawl over her skin under her clothes. Soon, she knew, events might compel her to forsake her son. She had it in her to imagine a source of malachite, and the aid of the Giants, and a portal which would summon the _Elohim_. Those ideas only asked her to believe in the Ranyhyn and her friends and Jeremiah. But guarding the portal against the Worm would require a miracle, and she had none to offer. Therefore\u2014\n\nAh, God. Therefore she would have to go in search of a power great enough to accomplish what she could not. She would have to leave Jeremiah to the care of her friends. If she did not, everything that he hoped to accomplish would indeed be wasted.\n\nThe fact that she lacked the courage was no longer relevant. Like Jeremiah, she would have to try.\n\nOnly Covenant's return might spare her. She yearned for that. But she could not suppose that he would come. The task which he had undertaken was too dangerous, and he was too far away. No, the burden of preserving Jeremiah's construct was hers to bear in spite of her weakness. She could not hope to be spared. The Worm of the World's End was coming. Nothing that lived would be spared.\n\nGradually she found a kind of resignation. It felt like defeat, but it allowed her to drift into a sleep too stunned and shallow for dreams.\n\nynyn's shrill whinny awakened her with the suddenness of a knife. Even before Stave said her name, she began drawing black fire from her Staff.\n\nReflexively she glanced at the sky to gauge the time. Dawn was near, although it did not promise a sunrise. Nevertheless a certain amount of light was coming. Without it, the air would have been colder. Soon the darkness would become gloaming.\n\nThen she felt the Ranyhyn running. Urgently they fled from the vicinity of the Sarangrave.\n\nWhy did they not pause for their riders? They could have taken her and her companions to safety.\n\nBut she had no time to think about such things. At Stave's command, she surged to her feet.\n\nJeremiah was ahead of her. He stood squinting in the direction of the Sarangrave. Before she could speak, he pointed.\n\n\"The Feroce. They're coming this way.\" A heartbeat later, he added, \"I can practically smell the lurker.\"\n\n\"Indeed, Chosen.\" Stave sounded as calm as a clear day. \"Now you must release the Staff of Law to me. I will ward it.\"\n\nLike her son, Linden stared at the crouching malevolence of the wetland. At first, she discerned nothing except the movement of small bodies. As they bobbed past obstructions, they appeared to fade in and out of existence. But then they passed the last islets of trees and brush, and emerald flames the precise hue of the Illearth Stone opened in the darkness. At the same time, the air became thicker: more humid, rank with moisture.\n\n\"How many?\" She wanted confirmation. She counted six flames, therefore only three Feroce. But somewhere behind them she felt the bitter aura of the lurker. Surely the monster would not challenge her without more support?\n\n\"Three,\" Stave stated as if he could not be mistaken. \"Also I sense but one tentacle. More may come, but the one lingers a stone's throw behind its minions.\"\n\n\"Mom?\" Jeremiah asked anxiously. \"Shouldn't you give Stave the Staff? You said those things can mess with your mind.\"\n\nLinden ignored him. \"This doesn't make sense,\" she muttered. \"The last time, there were a lot more. And now I'm braced for them. What does that monster think three Feroce can do?\"\n\n\"Who can declare the lurker's thoughts?\" Stave responded. \"Yet the peril remains. And it will be directed at you, Chosen. The monster covets the Staff of Law.\"\n\n\"Then be ready.\" Linden tightened her grip on the ebon wood. \"If they're going to say anything, I want to hear it while I can still defend myself.\"\n\n\"Mom!\" Jeremiah protested. But Stave did not remonstrate.\n\nLeaving the marsh, the creatures approached with an air of hesitation or timidity. They were still some distance away, but her nerves read them clearly. They were hairless and naked, apparently frail. In their large round eyes, green glints reflected like threats. Only their flames implied any force. But their magicks were strange to her, nameless and unrecognizable. The capering fires could have been desecrated Wraiths, captured and cruelly transformed. Or\u2014\n\nHell, they could have been anything.\n\nWhy had the Ranyhyn fled without their riders?\n\nThe question suggested possibilities that nearly staggered her. Perhaps it was deliberate. Perhaps the horses had taken her close to Sarangrave Flat once before precisely so that the lurker's acolytes could draw her into the monster's reach\u2014\n\nThe Ranyhyn feared the lurker: she knew that. Mahrtiir had accounted for their terror clearly enough. But she could not believe that they had betrayed their own devotion. So maybe they had risked attracting their ancient foe\u2014then and now\u2014for a reason. A reason that had nothing to do with shelter or weariness.\n\n_What_ reason? she asked herself wildly. Did they _want_ her to lose her Staff? Did they want the lurker to have it?\n\nThey were the _Ranyhyn_. They would not forsake their riders without a compelling reason.\n\nThe Feroce were drawing near, still timorously, but still coming\u2014and Linden was out of her depth; foundering.\n\n\"Chosen,\" Stave said like the night. \"If you do not attend, I must claim the Staff without your consent.\"\n\n\"Give it to him, Mom!\" Jeremiah demanded. \"Do it _now_. You aren't paying attention!\"\n\nShe gripped the Staff as though her life depended on it. She was paying attention to too many things at once.\n\nThe Feroce stopped ten paces away. Instead of spreading out, they stood close together. \"We are the Feroce,\" they announced as if Linden had never faced them before. They all spoke, yet they seemed to share one voice: a voice as moist and malleable as mud. But they did not continue. In silence, they awaited a response.\n\n\"What is it this time?\" retorted Linden. \"You've already attacked us once. Isn't that enough? What do you want now?\"\n\nThe creatures flinched. Their voice quavered. \"Our High God commands. We must speak.\"\n\nAgain they fell silent.\n\nLinden trembled with remembered distress and pain. Covenant's farmhouse erupting in flame around her. Recursive memories looping back on themselves, blocking her escape. She Who Must Not Be Named. \"Then speak,\" she snapped. \"But don't think that you can hurt me again. I know you now. I won't leave any of you alive.\"\n\nThe Feroce recoiled a step. They needed a moment to rally their resolve. When they replied, their voice was faint, squeezed out of them by pressures which they could not refuse.\n\n\"We speak for our High God. We bear a message from the Pure One.\"\n\nThe Pure One? Where had Linden heard that term before?\n\n\"I'm listening.\"\n\nHer manner must have appalled the creatures. They quailed as though they might dissolve at any moment.\n\n\"Our High God has offered an alliance with the Pure One. It has been accepted. No harm will come to you that our High God or the Feroce can prevent. You will be given aid at need.\"\n\nLinden stared, reeling inwardly. She could hardly understand what she heard. Who would form an alliance with the lurker? Who was that crazy?\n\nGiven aid\u2014?\n\nPanting at the viscid air, she asked involuntarily, \"The Pure One?\"\n\nIn a tone like a sheet of basalt, Stave said, \"The _sur-jheherrin_ spoke of the ur-Lord as the Pure One. They esteemed him by that title, though he deemed the Pure One to be Saltheart Foamfollower.\"\n\n\"Covenant!\" Jeremiah crowed. \"He must have done something to that monster. It's afraid of him!\"\n\nNow Linden remembered. In the Sarangrave with Covenant, Sunder, and Hollian. A few _Haruchai_. Her first meeting with the Giants. The lurker's attack. Then the rescue by the _sur-jheherrin_.\n\nThe Pure One.\n\nCovenant was alive? Alive?\n\nThe Feroce reacted as though Jeremiah had offended them. Their posture stiffened. The fires in their hands grew brighter, shedding green light like malice. Their voice gained strength.\n\n\"Afraid? Our High God fears the cruel metal. He does not fear the Pure One. Nor does he fear the wielder of the stick of power. Blades and burning he withstands. Yet a havoc which he cannot withstand approaches. He must live. Failing to obtain the stick of power, he sought alliance with the Pure One. The terms were agreed.\"\n\nThen the creatures appeared to remember that they were little and frightened. They shrank within themselves. Their tone suggested awe or dismay. It may have held gratitude.\n\n\"The Pure One has exceeded the terms. This our High God acknowledges. The alliance is sealed.\"\n\n\"All right!\" Jeremiah exulted. \"All _right_!\"\n\nCovenant was _alive_. He had to be. Linden clung to that. She had only encountered the Feroce two nights ago, and Covenant had turned away from her two days before that, rushing to meet the crisis of _caesures_ and _turiya_ Raver and Joan. But he had so far to go\u2014He must have met with the lurker, or the Feroce, after Linden did, but before he found his ex-wife. Indirectly Infelice had confirmed it. In Muirwin Delenoth, she had said that the lurker's minions were aiding him. Yet when could he have fulfilled\u2014exceeded\u2014his promises to the lurker? He would not have allowed any agreement to distract him from Joan. Therefore he must have sealed his incomprehensible alliance _after_ that confrontation. He must have survived it.\n\nBut Stave's demeanor did not soften. \"Continue,\" he said, implacable as a force of nature. \"You bear a message from the Pure One. When was it given to you? Where was it given?\"\n\nThe Feroce made placating gestures. \"The Pure One named his wishes in the early hours of this same night. Our High God was lost. Then he was redeemed. Far to the east, the Pure One made his desires known.\"\n\nThis same night? Covenant had stopped Joan. He had lived through the ordeal. There was no other explanation. Linden wanted to fling herself at the creatures; hug them in gratitude. Covenant had _redeemed_ the lurker? But she could not move. The extremity of her relief held her.\n\n_This_ was what the Ranyhyn had done to her. For her. For the Earth. They had exposed her to their worst nightmares and fled so that she might inspire an alliance with the evil which had slain great _Kelenbhrabanal_ , Father of Horses.\n\nFrom the first, they had _trusted_ her\u2014\n\n\"Then deliver his message,\" Stave commanded.\n\nBobbing and cowering, the creatures complied. \"The Pure One has exceeded the terms,\" they repeated. \"Therefore our High God commands us to convey words from the Pure One. They are meant for the wielder of the stick of power. The words are these.\"\n\nWaved flames left emerald cuts across the darkness. \"Remember forbidding.\"\n\nLike a sovereign enchantment, that utterance altered the conditions of Linden's existence. Realities veered around her or within her, effacing the tangible world where she gripped her Staff; transforming the causes and sequences which ruled her known life. The night and the Feroce vanished. Stave and Jeremiah were gone. Every vestige of Sarangrave Flat passed away.\n\nFor one sickening instant, she understood that the creatures had done it to her again. They had imposed their glamour on her memories. Her belief that she was ready to resist was an illusion.\n\nThen that knowledge was swept away in a moil of altered revelations. It was forgotten as if it had no meaning.\n\nWithout transition, she stood on a fan of obsidian marked like her jeans with green stains; with streaks of malachite crooked as veins. The light of Liand's _orcrest_ defined the stone. Utter darkness filled the rest of the world. Imponderable leagues of stone stretched overhead, held in place by their preserved recollections. Other figures clustered nearby, but she could not see them. Before her, Anele lay prone on the fan with his arms splayed as if in crucifixion. Grief and enduring pain marked every line of his emaciated form, the mute woe of Mount Thunder's foundations.\n\n\"It is here.\" The words were etched in Linden's mind. \"The wood of the world has forgotten. It cannot reclaim itself. It requires aid. Yet this stone remembers. There must be forbidding.\" His voice sounded harsh as rock. In Salva Gildenbourne, he had referred to _the necessary forbidding of evils_. Now he insisted, \"If it is not forbidden, it will have Earthpower. If it is not opposed by the forgotten truths of stone and wood, _orcrest_ and refusal, it will have life.\n\n\"When the Worm of the World's End drinks the Blood of the Earth, its puissance will consume the Arch of Time.\"\n\nThe forgotten truths? Linden wanted to ask. What truths? But Anele's distress kept her silent.\n\nThen he lifted his head, looked directly at her with his blind eyes. As if he were speaking for someone else, he said precisely, \"Everybody concentrates on stone, but that's not the whole story. Wood is important, too.\"\n\nForgotten, she thought. Forbidding.\n\nIt requires aid.\n\nLike an affirmation or a denial, reality veered again. In silence that battered her like the clamor of mighty bells, she was driven deeper, farther. Anele and stone vanished. Mount Thunder's betrayed sorrow evaporated as though it had never existed.\n\nLinden feared the bitterness of killing her mother, the horror of watching her father's suicide. Instead she felt the barren dirt of Gallows Howe under her feet, bereft by the knowledge of endless slaughter, and crowded with wrath; avid to repay the cost of so much death. She sensed recrimination and the long butchery of trees. Music had brought her here, the fraught melody of Caerroil Wildwood's singing. Again she was not alone, but she could not see her companion. She saw only the Forestal.\n\nHe stood beside the dead trunks of his gibbet with song streaming from his robe as if the fabric were woven from threnodies and dirges. The silver vivid in his eyes hinted at wild magic, although he had no white gold. His beard had the luster of age and vigor and unending travail.\n\n\"While humans and monsters remain to murder trees,\" he mused, angry and doleful, \"there can be no hope for any Forestal. Each death lessens me.\"\n\nShowing more restraint than Linden had any right to expect, he sang, \"I have granted boons, and may do so again. But you have not requested that which you most require. Therefore I will exact no recompense. Rather I ask only that you accept the burden of a question for which you have no answer.\"\n\nHe enthralled and terrified her. Her own anger was fresh from her failure to rescue Jeremiah; from the carnage of stone under _Melenkurion_ Skyweir. Her heart was as hard as the mountain's, and as flawed.\n\n\"How may life endure in the Land,\" inquired Caerroil Wildwood, \"if the Forestals fail and perish, as they must, and naught remains to ward its most vulnerable treasures? Must it transpire that beauty and truth shall pass utterly when we are gone?\"\n\n\"I don't know.\" What else could she say?\n\nAnother voice, the voice of her companion, said, \"He does not require that which the lady cannot possess. He asks only that she seek out knowledge, for its lack torments him. The fear that no answer exists multiplies his long sorrow.\"\n\nAnd because she stood on Gallows Howe\u2014and because her spirit burned for Thomas Covenant after her failure to redeem her son\u2014and because Caerroil Wildwood could still wring her heart in spite of all that she had endured\u2014she made a promise that she did not know how to keep.\n\n\"I will.\"\n\nThen the Forestal took the Staff of Law, black as fuligin after her battle\u2014and she lay on her back on the hard ground with the night sky above her like the abyss that awaited all striving. A sensation of impact throbbed in her forehead, a shock too sudden to bring instant pain. The hurt would come later; soon. Her neck felt wrenched and torn. Dying stars filled her eyes, consumed one after another in slow sequence by the Worm's unappeasable hunger.\n\n\"Damn it, Stave!\" Jeremiah yelped. He plunged to his knees beside her. \"Did you have to hit her so hard?\"\n\nStave replied without inflection. \"She fell under the glamour of the Feroce. I could not scry what might transpire. And her grip on the Staff was urgent.\"\n\nLinden thought, You hit me?\n\nShe had told him to do so.\n\nShe could not look away from the ruin of the heavens; the inexorable depredations. Too many things had been made clear to her. The actions of the Ranyhyn were only the most immediate of her new insights\u2014and the least cruel.\n\n\"Mom!\" Jeremiah urged, tugging at her shoulders. \"Are you all right? Can you move? Stave is too strong.\"\n\nBecause he was her son, she dragged her gaze down from the sky. His face was a smear of darkness. Dawn was coming, but it did not ease the night. Without her health-sense, she would not have been able to trace the outlines of Jeremiah's alarm.\n\nWhat\u2014? she tried to ask; but she made no sound. Lingering comprehensions clogged her throat. And her neck was starting to ache. She might not be able to lift her head. The throb in her forehead became a rusty blade. Soon it would cut.\n\nSomehow she forced herself to ask aloud, \"What\u2014?\"\n\nQuick with concern, Jeremiah told her, \"The Feroce did\u2014whatever it was. You went blank. When you didn't come back, Stave hit you to take the Staff. I guess that broke you loose. Now they're leaving.\n\n\"But, Mom,\" he added, hurrying. \"The lurker\u2014You've got to see this. If you can stand. If Stave didn't break your neck.\"\n\nPercipience assured Linden that her neck was not broken. Raising her head would hurt; but it would only hurt.\n\n\"I,\" she breathed through a rising pulse of pain. \"Can. Try.\"\n\nIn one smooth motion, Stave scooped her from the ground. Cradling her neck, he held her in his arms for a moment. Then he lowered her legs gently. When her boots were settled on the dirt, he offered the Staff of Law to her weak grasp.\n\n\"It is yours, Chosen. I have no virtue to wield its healing, but you are able to relieve the harm of my blow.\"\n\nHer hands were numb. She could not feel the warm wood. Stave had hit her too hard. Nevertheless her Staff was there. She seemed to hear it murmuring to her, urging her to call on its benign strength.\n\nInstinctively she summoned black flames to lave her as if they were the waters of Glimmermere.\n\nHer heart seemed to stagger in its beat. Then Earthpower and Law took hold. Her shock and hurt began to ease. She recognized her surroundings more clearly, identified Jeremiah and Stave as if their substance had been affirmed by fire. In the distance, she discerned the Ranyhyn. They appeared to be waiting, watching; perhaps praying. When she turned her head hesitantly in the opposite direction, she saw Sarangrave Flat crouching along the horizon, a starker and more telic dark within the enshrouding night.\n\n\"Look, Mom,\" Jeremiah insisted. \"You won't believe it.\"\n\nNo proffered or sealed alliance could soften her fear of the lurker. But when she looked\u2014when she focused her health-sense as well as her inadequate sight\u2014she saw a single tentacle rising from the disturbed muck of the wetland. It stood taller than any Giant, far taller, and as erect as a sentinel. All of its many fingers, hundreds of them, were curled and clenched as though they feared an attack. Yet the lurker's arm did not flinch or waver.\n\nHorrim Carabal. The Ardent had told Linden the monster's name.\n\nShe gripped her Staff more tightly, absorbed more fire.\n\nWhen it was certain of her attention, the tentacle dipped as if in acknowledgment or homage.\n\n\"See that?\" Jeremiah breathed. \"Did you see that?\" He sounded proud. \"It bowed to you, Mom. The lurker _bowed_ to you.\"\n\nBefore Linden could reply, the tentacle slipped back into its fouled waters. At first, it appeared to leave no ripples in its domain. But then she saw the massive arm squirm away, a crooked seethe through the stagnant pools and mud. As it retreated, the air became easier to breathe. In moments, Horrim Carabal had withdrawn beyond the reach of her perceptions.\n\n\"Amazing,\" Jeremiah proclaimed more strongly. \"I don't know what Covenant did, but he sure got that thing's attention.\"\n\n\"Be wary, young Jeremiah,\" Stave advised. \"The lurker of the Sarangrave is malevolence incarnate. The actions of the Ranyhyn speak of this, if your own discernment does not.\"\n\nThen he turned to Linden. \"By your earlier account, Chosen, the glamour of the Feroce inspires you to relive events and perils belonging to your former world. We cannot interpret the lurker's intent until you speak of this new visitation.\"\n\nCarefully Linden leaned her head from side to side, tested the effects of Earthpower. Joints popped: stiffness lingered in her neck: pain still throbbed in her forehead. But she was essentially intact.\n\nAnd dawn was near. There was an unmistakable paling in the east.\n\n\"Well, I know one thing, anyway.\" Anele's prophecy and Gallows Howe left her hoarse. \"This is what the Ranyhyn were hoping for.\n\n\"The last time, they did it on purpose. They took us close enough to the Sarangrave for the Feroce to find us. They wanted us to meet the lurker and beat it so that it would know we were too strong for it. It already knew that it couldn't fight Covenant. Not when he has the _krill_.\" Clearly Horrim Carabal had not forgotten the agony of Covenant's power millennia ago. \"The Ranyhyn wanted it to understand that it can't fight us, either.\n\n\"After that, I'm guessing. But I think that the Feroce must have approached Covenant. Infelice said something\u2014\"\n\nStave would remember. Jeremiah might not. He had been absorbed in his construct.\n\n\"The Feroce must have talked to him about an alliance. Whatever he said, it must have satisfied them.\" And he had done more: that was obvious. She simply could not imagine what it might have been. \"In any case, the Ranyhyn brought us here tonight because\"\u2014she shrugged awkwardly\u2014\"well, because they needed fodder, of course. But they also wanted to know where the lurker stands now. Or they wanted us to know.\"\n\nAnd they had trusted both her companions to take care of her however the lurker reacted.\n\nWith a subtle air of satisfaction, Stave asserted, \"The Unbeliever has exceeded the terms, as is his wont. The alliance is sealed. It is my thought that the lurker fears the Worm. Therefore it craves power. And therefore it seeks allies.\n\n\"Yet, Chosen,\" he continued, \"you have said naught of your experience within this new glamour. Your interpretation of the Ranyhyn I accept. I have none better. Will you now speak of the visions imposed upon you?\"\n\nLinden did not want to reply. What she had learned or deduced was too great for her. She did not know where she would find the courage to bear it.\n\n_I will_. She had promised that she would seek out an answer for Caerroil Wildwood. An answer which she could not possibly possess. For that reason, and because her company did not suffice for its task, she would have to forsake her son.\n\n\"It wasn't like the last time,\" she said, striving for a steadiness that she could not feel. \"It wasn't terrifying. First I was down in the Lost Deep. Before we crossed the Hazard. I heard Anele reading that fan of obsidian and malachite. Then I was back on Gallows Howe with Caerroil Wildwood. The Feroce reminded me that I made a promise then. I told him that I would find out how the world could survive without Forestals.\"\n\nNow she felt certain that the world could not.\n\n\"But why?\" Jeremiah asked quickly. \"I mean, why did they want to remind you? It's not like you were ever going to forget things like that.\"\n\nLinden believed that she understood the point of Covenant's message. And she surmised that the Feroce had tried to ensure that she did not misinterpret it. But she did not say so. Instead she deflected Jeremiah's query.\n\n\"Maybe the lurker doesn't really understand alliances. It's used to having worshippers. Alliances are new. Sure, I was never going to forget. But the lurker can't know that. It's just choosing between powers that can hurt it. The Worm is going to destroy everything. The lurker is bargaining for its life.\"\n\nAfter a moment, Jeremiah conceded, \"That makes sense, I guess.\"\n\nStave regarded her with his customary lack of expression. Briefly he lifted his head as if he were scenting the air. Then he said, \"Dawn begins. It appears that the coming day will resemble the one past. And the Ranyhyn return. Doubtless they will feed again. Then we must hasten once more.\"\n\nLinden nodded to escape more inquiries. Unlike the horses, she was not hungry. The roots that she had eaten still lay in her stomach, a fibrous mass difficult to digest. At uncomfortable moments, its taste returned to the back of her throat. But water was a necessity.\n\n\"In that case,\" she told the _Haruchai_ , \"we should get something to drink while we still can.\"\n\nHe flicked a glance toward her, but did not demur. And Jeremiah agreed at once. Already he was eager again; impatient.\n\nIn the rising gloom of a new day\u2014the second since the sun had failed\u2014Linden and her son followed the former Master back to the eddy where they had risked the water the previous evening.\n\nThe possibility that they might not find the like again did not trouble her as much as the prospect of her own intentions. They were too much for her: one appalling risk piled on another until their sheer scale threatened to overwhelm her.\n\n## 7.\n\nTaking the Risk\n\nSoon Linden, Jeremiah, and Stave were mounted and running again, heading away from the Sarangrave directly into the northeast. Hyn's straining betrayed that the mare had not recovered her full strength. Both Hynyn and Khelen labored over the barren terrain. Still they had reserves of stamina. Linden understood their physical prowess no better than she comprehended their ability to find their way within _caesures_ , or their strange insight into the mind of Horrim Carabal. She knew only that Earthpower flowed richly in their veins. They seemed to draw their vitality from the Land itself, regardless of its blasted condition.\n\nKevin's Dirt loomed overhead, but she banished its effects almost reflexively. The threat of the lurker was behind her, and she no longer feared to exert her Staff.\n\nGradually the darker gloom of night became a kind of twilight over the region. Ahead of her, the ground undulated in slow dips and gradual rises toward its dulled horizons. Then the terrain became rougher\u2014the hollows deeper, the sides steeper\u2014until the horses appeared to traverse a protracted series of impact craters: the ancient outcome of fallen meteors, or of terrible bolts of theurgy. But the Ranyhyn were not daunted. Instead they seemed to gain fresh resolve from the difficulties, as if they were nearing their obscure destination.\n\nAnd eventually the terrain on Linden's right began rising. Along a line parallel to the path of the mounts, southwest to northeast, the stricken ground piled higher until it formed a ridge with a front as sheer as a cliff and a more gradual slope at its back. To her left were only more hollows or craters; but opposite them, the ridge jutted with its gutrock exposed as if a range of higher hills had been cleft.\n\nApproaching the highest point of the ridge, the Ranyhyn slowed. Rubble, boulders, and other detritus cluttered the base of the cliff, but did not extend far enough to obstruct the horses. Hyn, Hynyn, and Khelen had a different reason for easing their pace.\n\nIn the lead, Jeremiah rose on Khelen's back as if he were standing in stirrups. He punched his fists at the sky, defying the reaving of the stars. \"This is _it_!\" he shouted. \"Malachite! That cliff is _riddled_ with it!\"\n\nWhile Hyn jolted to a trot, Linden tried to spot what Jeremiah had seen.\n\nAt first, the profile of the ridge held her. From her perspective, it cut off perhaps a third of the heavens. Irrationally she hoped that she would not see more stars dying. But the slow carnage continued overhead. She could only spare herself that vista by lowering her gaze.\n\nEven then, she could not locate the source of Jeremiah's exultation.\n\nFortunately the Feroce had renewed her recollection of black rock elucidated by green veins. When she concentrated inward, tuned her senses to the hue and pitch and timbre of memory, and then studied the cliff-face once more, she began to discern flakes and small seams of the mineral she sought. They were difficult to detect, in part because they felt miniscule, too trivial for her son's needs, and in part because they were crowded among streaks of verdigris, knobs of blunt granite, porous patches of sandstone; masked by reflective facets of quartz, mica, feldspar, other crystalline stones. But there _was_ malachite.\n\nIt did not look like enough.\n\nYet Jeremiah's excitement was undaunted. As his mount halted, he vaulted to the ground; ran a few strides toward the ridge. \"There!\" he called as though he wanted the world to hear him. \"It isn't much. I mean, on the surface. But deeper\u2014! If we dig into the cliff far enough\"\u2014his hands sketched dimensions in the air\u2014\"some of it is practically pure!\"\n\nPointing, he indicated a section of the ridgefront a long stone's throw above his head.\n\nOh, God. Linden would have asked, How can we get at it? But a different problem had already occurred to her. Assuming that the cliff could be excavated, surely the stone above it would collapse? Anyone digging there would be crushed and buried.\n\nThe Ranyhyn had found what Jeremiah needed.\n\nIt was effectively inaccessible.\n\nHyn had stopped. Her breathing wheezed faintly as she waited for Linden to dismount. But Linden was too shaken to move.\n\nLike her, Stave remained mounted. His mien revealed nothing as he asked, \"Will not these boulders suffice, young Jeremiah?\" He nodded toward the debris at the foot of the cliff. \"They also contain portions of malachite.\"\n\nJeremiah turned to glare at the _Haruchai_. \"Sure,\" he snorted with the inadvertent disdain of a boy. \"If I wanted to build a door for mice. One that didn't go anywhere. But the _Elohim_ are bigger.\" He must have meant in personality and puissance. \"And the door has to take them someplace safe.\n\n\"No,\" he asserted. \"We need to get into that cliff.\"\n\n\"Then this is labor for Giants.\" Smoothly Stave slid down from Hynyn's back. \"While we await them, however, I will commence. Inform me when I have climbed to the place where you wish me to begin. I will discover what the strength of the _Haruchai_ can accomplish against such stone.\"\n\nIn response, Jeremiah laughed: delight, not derision. Flourishing his arms, he cast arcs of yellow flame across the gloom. \"I knew I could count on you. While you're doing that, I'll look at a few boulders. Maybe some of them have enough malachite. I'll need as much as I can find.\"\n\nStave nodded. Instead of approaching the ridge, however, he faced Linden. \"Chosen, you also must dismount. The Ranyhyn require rest. Indeed, they must depart in search of water and forage. And I will be unable to ward your son while I ascend the rock. That task falls to you.\" After a brief hesitation, he added, \"I do not dream that our foes have forgotten their craving for your son's gifts.\"\n\nRoger had an uncanny ability to appear out of nowhere. Distance was no obstacle to Kastenessen. And Lord Foul's powers\u2014even those of the Ravers\u2014were beyond estimation.\n\nFalls, Linden echoed; but she was not listening to Stave. Her mind followed other paths. The Masters called _caesures_ Falls. She could not conceive of any other way to keep her promises. But there were many kinds of falling. She could too easily imagine Stave crawling spider-like up the cliff\u2014until some hand- or toe-hold failed.\n\nWhat choice did he have? What choice did anyone have? Jeremiah needed malachite.\n\nShe shook her head, resisted an impulse to slap herself. She could not afford to sit on Hyn's back feeling stupid and defeated. Her son needed more than malachite. The whole Earth needed more.\n\n_When your deeds have come to doom_ \u2014\n\nShe had to think.\n\nStaring vacantly at the ridge, she told herself that the question was one of power. Surely it was a question of power? Even if Stave lived, what could he hope to achieve? And when the Giants came\u2014if they came\u2014they would be in as much danger as he, with as little chance of success.\n\nTherefore\u2014\n\nWell, obviously, the cliff would have to be broken open from a safe distance. What else? And that was a task for theurgy. Even if the stonewise Giants could devise an alternative, they were only eight\u2014and they were already weary. The work would take time. Not hours: days.\n\nPower was the only answer.\n\nBut what could she do? Fire she understood: black flame and burning. Yet merely scorching the face of the cliff would be a waste of effort. Heat alone would have no effect. In the Lost Deep as well as under _Melenkurion_ Skyweir, she had shattered stone; shaped it instinctively. If she could summon that form of strength or desperation again, she might be able to tear apart the ridgefront. But the malachite would be torn apart as well. Tons of mineral-seamed rock blasted to gravel would not serve Jeremiah's purposes.\n\nDoubtless the Staff of Law had other uses\u2014many of them\u2014but she was not lorewise enough to know what they were, or how they were done. And anything that she attempted with Covenant's ring would be worse. Wild magic resisted control. In that respect, it resembled the _caesures_ it created.\n\nHow could she open that ridge without risking lives?\n\n\"Chosen,\" said Stave more sharply. \"We cannot delay.\"\n\nBut then another possibility occurred to her. She had been given hints enough\u2014\n\n_Men commonly find their fates graven within the rock, but yours is written in water._\n\n_The lady's fate is writ in water._\n\n\"Wait.\" Scrambling to catch up her ideas, she slipped down from Hyn's back. \"Before you do anything rash. Do you know if there are any streams on the far side of this ridge? Any water at all?\"\n\n\"Mom,\" Jeremiah protested. \"We don't need water. We need to get started.\"\n\nShe and Stave ignored him. The _Haruchai_ met her gaze squarely. \"No, Chosen. This region is unknown to me. The Masters have found no cause to scout it. And I have discerned neither streams nor springs.\" After a flicker of thought, he said, \"Yet the Ranyhyn may discover what you desire. Doubtless their path lies toward water.\"\n\n\"Mom,\" Jeremiah objected again. \"What's so important about water?\"\n\nThe three mounts were already trotting back the way they had come. Eventually they would come to the place where the declining slope of the ridge met lower ground.\n\n\"I'm not sure yet,\" Linden answered. \"Maybe there isn't any. And if there is, I might not be able to use it.\"\n\nImpatiently Jeremiah came to join her and Stave. \"I don't get it. Sure, I'm thirsty, but it isn't bad yet. If you aren't going to drink it, what do you want it for?\"\n\nLife, Linden could have said. Hope. Fate. Doom. But she felt too uncertain to describe what she had in mind.\n\n\"Just wait,\" she urged her son. \"Watch the Ranyhyn. We'll know soon enough.\" To ease his frustration, she added, \"I don't want to risk Stave if we don't have to.\"\n\n\"But\u2014\" Jeremiah began, then clamped his mouth shut.\n\nDim as shadows, the Ranyhyn were only trotting. Nevertheless they appeared to cover distance rapidly. And as Linden watched, they began angling closer to the ridge.\n\nShe gripped the Staff hard; tried not to hold her breath.\n\nBefore long, the horses quickened their pace. Rushing at the slope, they ascended the dwindling silhouette of the ridge. For a moment, they labored upward. Then they gained the ridgeline and disappeared from sight.\n\nLinden sighed. She could assume that the Ranyhyn were seeking water; but that did not necessarily imply that it arose from a source within the ridge. The horses might have to search to the south or east beyond the thrust of the cliff.\n\nStill she could hope\u2014\n\n\"All right,\" she said finally. \"So maybe there's water. I won't know for sure until I find it.\"\n\nJeremiah had reached the end of his restraint. \"But _why_?\"\n\nImpelled by the pressure of yet another burden which she might not be able to carry, Linden started toward the ridge. \"The Lords,\" she replied over her shoulder, \"back when there were Lords\u2014They must have known how to do lots of things with a Staff of Law. But I can only guess what those things were. I don't know how to do any of them. I only know fire and healing.\" And brute force.\n\nWhile Jeremiah caught up with her, and Stave followed in silence, she continued, \"I can't heal anything here. But fire makes heat\u2014and heat makes water expand.\" Trapped water would be ideal, or water that could only rise to the surface in trickles. But buried springs and even pockets of moisture might conceivably suffice. \"Heat water fast enough and hard enough, and it explodes into steam. Maybe I can break part of the cliff.\"\n\nFor an instant, Jeremiah seemed stunned. Then he burst out, \"That's _brilliant_!\"\n\n\"It is a tenuous prospect,\" remarked Stave. \"The obstacles are many. I name only the site and quantity of water required, if indeed water exists within such a formation. Nonetheless the deed cannot succeed if it is not attempted.\"\n\nLinden was not listening. As she walked, she summoned Earthpower to sharpen her percipience, bathed her nerves in fire like condensed midnight. Then she began to explore the ridgefront. Concentrating on the section that Jeremiah had indicated, she felt her way inward, searching into and through multitudes of rock as if she were probing for wounds hidden deep within living flesh.\n\nAt the nearest obstruction, a boulder the size of a hut, she halted momentarily. But then she realized that she needed to be closer: close enough to study the face of the cliff with her hands. Cursing under her breath, she passed around the boulder and mounted a stretch of lesser rubble, the fallen residue of the cliff's severance. When she stumbled, she caught herself on the Staff and climbed higher.\n\nFinally she reached the main wall. From far above her, it loomed as if it were glowering in suppressed wrath. But she ignored its impending bulk, its ire, its enduring intractability. She needed nothing from it except water.\n\nIn one approximate location.\n\nIn sufficient quantity.\n\nAfter all, it was only a cliff. It was not the cunning subterfuge and malice of the Demondim, seething to mask the _caesure_ which gave them access to the Illearth Stone. Nor was it the recursive wards of the Viles, coiling themselves into a mad tangle to prevent intruders from entering the Lost Deep. It was only pieces and shards and spills and plates and torsos and veins and thews of the world's rock compressed by their own weight until they formed a front which had outlasted millennia. It had no defense against her health-sense.\n\nBut it was so _much_ rock. Of so many different kinds. In so many different shapes and structures. And it supported a mass which would have squeezed ironwood to pulp. Its secrets resisted discovery as if it had set its will against her.\n\nLeaning her Staff against her shoulder, Linden closed her eyes. Hesitantly at first, then more firmly, she placed her hands on the wall and began to insinuate her touch inward.\n\nStave had followed her as far as the rubble. There he kept watch. Jeremiah stood a bit behind her, but he did nothing to interrupt her concentration. At first, she felt his attention focused on her. Then she closed her mind. Deliberately she thought only about water.\n\nNow that she was not seeking them, she found streaks and facets of malachite everywhere. Crystalline deposits reflected her probing. Heavy granite ground against flows of basalt, reducing them to powder across the eons. Compacted dirt filled every crevice and crack. Schist blocked her search as though its memories and therefore its anger were more recent or more extreme than the rest of the rock.\n\nBut Jeremiah needed her. The _Elohim_ needed her. The Earth required its panoply of stars. And her friends would be at risk if she failed. They would have to hazard their lives if she could not open the cliff.\n\nWater, that was all she wanted: the most ordinary, necessary stuff of life. And it was everywhere in the created world. It rose from springs among the deepest roots of mountains. Beneath the desiccated purity of the Great Desert, it oozed and ran. The shores of every continent and island felt its surge and lash. From the sky it gave nourishment. And it could be violent. Oh, it could be violent! Linden had felt its force often enough to know what water could do with fury and turbulence.\n\nYet no Law required it to emerge where she could reach it.\n\nThen Jeremiah's halfhand clasped her shoulder; and for an instant, her concentration faltered. Almost immediately, however, she felt vitality flow into her from his touch. He was giving her Earthpower as the ur-viles had given her blood, so that she might be able to exceed herself.\n\nRiding the energy of his aid, she sensed a damp patch of dirt between a crumbling granite monolith and a writhen vein of sandstone.\n\nIt was small, little more than a suggestion of moisture; perhaps only a few drops. But it was water.\n\nGalvanized by hope and her son's support, she marked the dampness in her memory and pushed her senses farther.\n\nShe forgot hunger and thirst and weariness. Deeper in the ridgefront, higher, she found a second hint of water. By oblique implication, it led her to another pocket of moisture, and another. Another. There bits of damp marl and pumice were strung together like beads along a fissure between incompatible sheets of granite and obsidian. Linden marked them all, and followed them.\n\nThe detritus in the fissure became dense gravel. More water seeped in the gaps, fine droplets acrid with minerals. Carefully she extended her perceptions among them. The vein of gravel became a wedge, wetter and looser. Then it was plugged by schist. She stumbled within herself; leaned her forehead against the face of the cliff. That damn schist\u2014She did not understand how it obstructed her. But she could not spare the energy to study it. Insidiously, as if she sought to possess the rock without being noticed, she slipped her senses past the plug.\n\nBeyond it, she found what she sought.\n\nWater. A space like a bubble in the compressed flesh of the ridge. A cavity filled with water.\n\nIt was no larger than her head. And the water had not moved for an age of the Earth: it was cut off from its original source. But flaws packed with more gravel guided her to a pocket of water the size of her body. Farther in, she found a space big enough to hold a Giant's chest; then two more\u2014no, three\u2014each little more than a trapped fist; then, finally, a gap as large as the chamber where she and Anele had been imprisoned in Mithil Stonedown.\n\nAfter that, there was no more, or she had reached the limit of her reach, or her strength was failing.\n\nHad she located enough? Taken altogether, it was only a drop within the inland sea of the ridge. Nevertheless it would have to suffice. She would have to make it suffice.\n\nShe took moments or hours to ascertain that she could remember precisely where and what she had discovered. Then warily, as if she feared the cliff's animosity, she withdrew.\n\nGod, she could barely stand\u2014How had she become so tired?\n\nShe had no time for weakness. The marks in her memory would fray and fade. Reeling against Jeremiah, she took up the Staff again. His aid vanished, but she ignored its absence. Her eyes stared at nothing. She saw only the places that she needed to remember.\n\n\"Mom?\" he asked anxiously.\n\nShe staggered past him, nearly falling down the rubble toward Stave. When the _Haruchai_ caught her, she panted, \"Don't say anything. I have to concentrate. Just get me away.\"\n\nIf she succeeded, and they were too close\u2014\n\nStave seemed to understand. With an arm around her waist, he half carried her toward the hollows or craters in the northwest.\n\nHave mercy, she groaned as she stumbled along. I can't do this.\n\nShe had to do it.\n\nA long stone's throw from the ridge, Stave stopped; turned Linden to face the cliff. Jeremiah caught up with her there. He must have been able to see her fatigue. Standing behind her, he clasped her shoulders with both hands.\n\nFresh theurgy set fire to her blood. Flame ran in her veins. Her heartbeats were conflagration. Blackness bloomed from her Staff as if Jeremiah had invoked it without her volition. Fuligin etched everything that she saw and remembered against the tarnished grey daylight.\n\nShe told herself to start small. Begin with the tiniest bits of moisture. Try to force a few new cracks. Weaken the cliff.\n\nIf she could find them from this distance.\n\nA troubled wind out of the east tumbled over the ridgecrest, skirling in plumes and dust devils out across the wasteland of craters. It chilled the unnoticed sweat on her forehead, tugged loose a few strands of hair that were not matted to her cheeks and scalp. But it did not soothe her whetted senses; her urgency. In this season, the Land's prevailing winds were from the west.\n\nStraining, Linden Avery reached out to the cliff and tried to prove herself worthy.\n\nA damp patch between granite and sandstone. For a heartbeat or two, she focused her intentions there. Then she sent a dark burst of power to boil that small instance of moisture.\n\nShe almost felt the dampness swell; almost felt microfissures mar the surrounding rock. Almost. But she had expected nothing more. She was only trying to create a slight frailty.\n\nWith as much care as she could muster, she moved inward, upward.\n\nPieces of wet marl and pumice strung together like beads: a thin crack separating granite and obsidian: a more difficult challenge. If she failed to heat all of the beads at the same time, her efforts would lose some of their effect. Force would dissipate along the string.\n\nIt was too much for her.\n\nIt had to be done.\n\nGathering her resolve, she murmured the Seven Words. Her Staff became a scourge in her hands. Magic struck the string of moisture like a barbed flail. Black fire filled even the most miniscule hints of fluid with passion. For an instant, she thought that she heard the scream of over-heated water, the groan of stressed rock. Then the sound was gone.\n\nInvoluntarily she sagged as if she had been overcome. Blots swam and burst across her vision, stars and small suns, stains like abysms.\n\nBut Stave upheld her. Jeremiah gripped her shoulders, sharing the strength which Anele had concealed. Earthpower burned in her vessels and nerves, in the channels of her brain and the secret recesses of her heart.\n\nShe could not afford to fail.\n\nThe wind flicked grit into her eyes. She blinked rapidly, then shut them tight. Ordinary sight was a distraction. Looming huge and unmoved, the rock mocked her inadequacy. Only percipience would enable her to make her last attempt.\n\nShe had found six spaces filled with water. She meant to superheat all of them at once. Then larger cracks might join with minor flaws. They might trigger any inherent instability which the immense bulk of the cliff suppressed. They might cause seams and plates to slip\u2014\n\nIf one puny human being, exhausted and trembling, could induce something that size to shift.\n\nThe Staff shook in her hands as though it had become a burden too great for her to bear. Aching for puissance and accuracy, she invoked the Seven Words again. \" _Melenkurion abatha_.\" Her voice rose in desperation. \" _Duroc minas mill!_ \" The invocation became a wailing cry. \" _Harad KHABAAL!_ \"\n\nWhen she unleashed her fire, its blackness seemed to efface the world.\n\nIntense heat constricted by immutable mass created pressures which would have torn mere flesh to shreds. Pockets of water tried to expand. Granite and schist and a mountain's weight refused to move. The ridge had endured for millennia. Linden poured out power as if she were expending her soul. Given a voice, the stone would have laughed.\n\nThen it found a voice. Through the harsh beat of the Seven Words and her own gasping, she heard the cliff groan.\n\nA short sound, little more than a sigh; but it was enough to break her concentration. Unaware of herself, she dropped her Staff. Fighting a giddy swirl of phosphenes and oxygen deprivation, she opened her eyes; tried to see what was happening.\n\nIn two or three places across the ridgefront, dust puffed outward. Almost at once, wind dismissed the small exhalations as if they had never occurred.\n\nAfter that\u2014\n\n\u2014nothing. The cliff stood glowering in the gloom. It had not been touched, and did not care.\n\n\"Oh, Mom,\" Jeremiah moaned. \"No. That can't be right. I saw\u2014I felt\u2014\"\n\nLinden saw nothing. She felt nothing.\n\n\"Indeed,\" Stave pronounced. Abruptly he released Linden; left her to Jeremiah. Without explanation, the former Master strode toward the mountain.\n\nPerhaps he had decided to act on his original suggestion. Climb the ridge. Try to break loose pieces with his fingers.\n\nBut he stopped before he had crossed half the distance. From the ground, he picked up a rock. For a moment, he hefted it in his hand, tested its weight. Then, fluid as water, he flung it.\n\nIt struck the cliff-face above the places where Linden had seen puffs of dust. Three heartbeats passed. Four. Without her son's support, she would have collapsed.\n\nThen a grinding shriek appalled the air. The earth under her trembled. Tremors kicked up spouts of grit like gusts of pain everywhere between her and the ridge.\n\nWith the massive inevitability of a calving iceberg, a wide section of the wall shifted. For a moment, it seemed to hang on the edge of itself, clinging to its long stubbornness. But it could not hold against its own weight.\n\nWhen it fell in thunder, Linden fell with it. She had nothing left that might have enabled her to remain conscious.\n\nhe did not know how much time had passed when a glad halloo awakened her. Only moments, she thought at first. But her head felt too heavy to lift, burdened by sleep. And when she tried to gauge the condition of her surroundings, estimate the effects of plunging rock, she found that her reality had contracted. She recognized only the pressure of the hard ground against her body, the leaden weariness of her limbs, the ragged effort of her breathing, the parched ache of dust in her throat and lungs.\n\nEventually she realized that Kevin's Dirt had reclaimed her. Her health-sense was gone.\n\nNot moments, then. She must have slept for hours. Kevin's Dirt did not erode percipience so suddenly.\n\nWithout opening her eyes, she fumbled around her for the Staff of Law.\n\n\"It is here, Chosen,\" said Stave. The warm wood of the shaft was pressed into her hand. \"And now the Giants come. Manethrall Mahrtiir leads them. Soon the true labor of your son's purpose must begin.\"\n\nLinden hardly heard him. She had no attention to spare for anything except her Staff. Without her health-sense, she was less than useless.\n\nFortunately Liand\u2014lost Liand\u2014had taught her how to find the possibilities beneath the written surface of the wood, even when she had no enhanced discernment to guide her. He had given her more gifts than she could count. Pulling the Staff toward her, she held it close until its natural beneficence began to enlighten her nerves. After that, she was able to absorb Earthpower more quickly.\n\nStave had said something about the Giants\u2014and Mahrtiir\u2014\n\nSoftly through the dirt, she felt the tread of heavy feet: distant yet, but closing. Within that staggered beat, she detected the sharper impact of hooves. As her health-sense expanded, she identified Narunal.\n\nThen she located Jeremiah. He was closer than the Giants, but in a different direction. He must have been scrambling over the wreckage of the ridgefront; but now he stood waving his arms eagerly at the Swordmainnir.\n\nCoughing, Linden tasted the air. Between what should have been sunrise and sunset, the grey half-light remained uniform, undefined by any obvious passage. Nevertheless the flavor of the gloaming modulated incrementally, measuring time. Its faint savor told her that she had slept past midafternoon. A more natural twilight was only a few hours away.\n\nApparently Stave had kept watch over her for quite a while.\n\nNow the Giants and Mahrtiir had come. Soon she would have to face the fears which had harried her ever since Jeremiah had explained his intentions.\n\nShe did not need to raise her head to know that the stars were still going out one by one.\n\nPerhaps she should have been afraid; but she was too tired. She required more than mere sleep to restore her. She needed good food and drink, long rest\u2014and an easing of her ache for Thomas Covenant.\n\nInstead of thinking about what she meant to do, she turned to the question of keeping Jeremiah safe.\n\nIn spite of their shouted greetings, the Swordmainnir and the Manethrall did not hasten. Rime Coldspray and her comrades were profoundly weary. A little more time would pass before they came close enough to require Linden's attention.\n\nShe could at least try to talk to Stave.\n\nWith a muffled groan, she pulled her knees under her, pushed herself up with her arms. Her own fatigue felt as heavy as the ridge. She had to rest for a while before she shifted into a sitting position.\n\nMutely Stave extended his hand to help her rise.\n\nShe shook her head. She needed an entirely different form of aid from him\u2014and she had to talk to him about it alone. He deserved that.\n\n\"Stave,\" she said or coughed. Her throat was as dry as the wilderland. Deliberately she did not regard her son, or her approaching friends, or what she had done to the cliff. \"There's something that I want you to do for me.\"\n\nCruel days ago, the Mahdoubt had said of the former Master, _He has named his pain_. _By it he may be invoked_. That had been her last gift before she was lost to use and name and life. But Linden did not want to insist. She suspected that she would damage their friendship if she pressed him.\n\n\"Then speak of it, Chosen.\" His tone was uncharacteristically wry. \"Have you not learned that there need be no constraint between us?\"\n\nResponding to a question about Kevin Landwaster, he had once told her, _In your present state, Chosen, Desecration lies ahead of you_. _It does not crowd at your back_. She knew now that he was right. Nonetheless she hoped that he was also wrong.\n\n\"All right.\" She tried to clear her throat. Then she gave up. Coughing intermittently, she said, \"I have to go away, and you can't come with me. I want you to stay with Jeremiah.\"\n\nStave's silence seemed louder than curses. Was he not her friend? Had he not endured the spurning of the Masters for her? Had he not stood by her in every crisis? The ferocity with which he could have protested, and did not, made her flinch.\n\n\"Covenant said it,\" she explained hoarsely. \"It's all about power. I have to assume that Jeremiah has enough malachite. If he does, the Giants will find a way to help him. He'll be able to build his door. And it will work. The _Elohim_ will come. I have to assume all of that.\n\n\"So he's going to draw the Worm. I have to assume that, too. And when he does, he'll be in danger. He has too many enemies. I might be able to hold off Roger, but I can't fight Kastenessen. No matter how careful we are, a Raver might slip past us.\" If _moksha_ Jehannum took possession of Jeremiah\u2014\"I can't even imagine what Lord Foul is going to do. And we don't have a prayer of resisting the Worm.\n\n\"We need more power.\" She was pleading. \"I'm going to go look for it. But I can't bear to do that if you don't stay here for Jeremiah.\"\n\nStave's flat mien concealed his reactions. His aura seemed to assert that he had no emotions. Yet Linden had seen him grieve over Galt. And she knew his concern as well as his fidelity. Surely he had other human feelings as well, in spite of his stoicism and his vast memories? Surely he could understand her?\n\nHe sounded as ungiving as schist as he asked, \"Where will you go?\"\n\n\"I'll tell you.\" She was done coughing. \"Everyone has to know.\" She no longer flinched. \"But I'm not brave enough to say it more than once. This part is between us. It doesn't involve anyone else.\"\n\nAgain Stave was silent. Linden folded her arms over the Staff, held it against her heart, and tried to match him.\n\nAfter some consideration, he said, \"Do not mistake me, Chosen.\" His tone was like the dusk, unrelieved from horizon to horizon. \"I await only some mention of the Mahdoubt\u2014or perhaps of the Vizard. Were you not offered the means to command me?\"\n\nClinging to her weariness as if it were courage, Linden replied, \"I won't do that. I'm just asking. I'll beg if that's what you want. If Covenant were here, things would be different. But he isn't. I'm the one who has to go. I'm the only one who can. If I know that you'll protect my son.\"\n\nStave's manner conceded nothing. Nevertheless his response seemed to imply that a concession was possible. \"Yet some companion you must have.\"\n\nIn Muirwin Delenoth, he had argued that the participation of _the natural inhabitants of the Earth_ was a necessary condition for the world's survival, just as the presence of _beings from beyond Time_ was essential to Lord Foul's designs. And Linden knew that she needed help.\n\n\"I'll take Hyn,\" she answered weakly. \"And Mahrtiir, if he's willing.\"\n\nHe, too, could not assist Jeremiah. Nor could he fight Roger or Kastenessen or Ravers or\u2014\n\n\"Then, Linden,\" Stave said as if he were merely offering to help her stand, \"I will do as you request.\" A heartbeat later, he added, \"But do not doubt that my heart is torn within me. I will know neither certainty nor peace until your return.\"\n\nLinden's eyes were too dry for tears, but a sob twisted in her chest. \"All right.\" Bracing herself on the Staff of Law, she climbed to her feet. Then she dropped the wood so that she could wrap her arms around the former Master. \"Thank you.\"\n\nHe called her by her name so rarely\u2014\n\nShe would not have been surprised if he had stood rigidly passive in her clasp. But he answered her hug with his own. Almost gently, he murmured, \"You will not fail. Come good or ill, boon or bane, you are Linden Avery the Chosen. You will suffice.\"\n\nWhen he let her go and stepped back, he had done enough. As he had from the first, he had given her more than she had any right to expect.\n\nShe offered him a wrenched smile. \"If you say so.\"\n\nIn spite of her weariness, she stooped to retrieve her Staff. Then she turned to face Jeremiah and the Giants and Mahrtiir and the meaning of her life.\n\nWhile her attention had been fixed on Stave, Jeremiah had descended from the rubble which she had gouged out of the ridge. Now he was running toward the Ironhand, Mahrtiir, and the rest of their companions. Rime Coldspray quickened her pace slightly to meet him; and Linden thought that he would leap into the Ironhand's arms. But at the last moment, he restrained himself. Stopping suddenly, he braced his fists on his hips.\n\n\"What kept you?\" he demanded cheerfully. \"We've been waiting for _ages_.\"\n\n\"Alas,\" the Ironhand replied with a wan smile, \"we are Giants and perforce laggardly. Yet at last we have come.\" She kept on walking. With the boy trotting at her side to match her strides, she asked more soberly, \"What are your tidings, young Jeremiah? Malachite we see. And we see that it has been but recently torn from the thrust of yon ridge. A prodigious feat, and unexpected. Your tale must be equally prodigious.\"\n\nStooping under her burdens, Linden moved to intercept her friends. The exhaustion of the Giants was plain at any distance. To arrive so promptly, they must have marched through the night. Nevertheless her heart was drawn to Manethrall Mahrtiir.\n\nHis condition seemed as explicit as iconography. Uselessness and the loss of his health-sense had marked his mien until he looked haggard, too downtrodden to endure more: as deprived as he had been in the Lost Deep. But there he had been almost continuously active, and occasional gifts of Earthpower had eased his sense of futility. Here he had received no relief from the grinding depression wrought by Kevin's Dirt. Now his misery ached like an unhealed wound.\n\nTo ease his plight, Linden uncurled tendrils of flame from her Staff and stretched them over him. It was the least that she could do.\n\nFortunately this effort was within her strength. When her fire touched the Manethrall, he reacted as if he had been struck. For an instant, his misery seemed to spread its wings and become joy. Almost at once, however, he reassumed the glower which had become habitual during the past few days. But now his scowl had recovered its familiar combativeness.\n\nWhile Linden wielded Earthpower, some of the Giants paused to stare at her. Others cheered for her sake, or for Mahrtiir's. And Frostheart Grueburn called, \"Prodigious tidings, in all sooth! Here surely is a tale worthy to be told at length, and to be heard with laughter!\"\n\n\"It was all Mom,\" said Jeremiah proudly. \"She was brilliant! She found pockets of water in the cliff and made them explode. Now we have malachite.\" Then he made a visible effort to contain his eagerness. \"We can get started when you've had a chance to rest.\"\n\nThrough the twilight, Linden studied the faces of the Swordmainnir. Like Mahrtiir, they needed the Staff's gifts. Yet she could see that rest and refreshment were not their primary concerns. They were hungry for some reason to believe that they had not expended themselves in a futile cause.\n\n\"Alas,\" repeated Rime Coldspray, speaking to Linden rather to Jeremiah, \"our small store of viands we have consumed, lest we falter in our trek. Now I find that I am grieved by our failure of foresight. Your need for sustenance is clear.\"\n\nLinden started to say, Don't worry about it, but the Ironhand continued without pausing, \"In truth, we knew not how to measure your need against our own. And we did not imagine that we would encounter no _aliantha_ along our course.\" Then she grinned grimly. \"However, great Narunal is provident. We do not lack for water.\"\n\nWearily Grueburn, Latebirth, and Onyx Stonemage held up bulging waterskins.\n\nLinden wanted to express her thanks, but her throat was too dry for speech.\n\n\"Well, all _right_!\" Jeremiah answered for her. \"We haven't eaten since we cooked some roots near the Sarangrave. But it was like chewing mud. My stomach still isn't happy. And you have no idea how bad water tastes there. I could drink a gallon.\"\n\nThe Giants exchanged quick glances. Stonemage promptly lowered a waterskin, untied its neck, and held it for Jeremiah.\n\nAt the same time, Coldspray faced Linden with danger in her eyes. \"Beyond question, there are tidings here. What mischance or peril guided you within the lurker's reach? At no time did our own course approach the noisome banes of Sarangrave Flat.\"\n\nStruggling to moisten her throat, Linden admitted in an awkward rasp, \"That's a story. We have a lot to talk about.\" She looked around. \"Maybe we can find a place to sit down. I'm thirsty too,\" and so tired that her knees quivered. \"I can try to explain while we're resting.\"\n\nColdspray agreed with a nod. Pointing toward an area where several large chunks of the ridge formed an arc with a clear space among them, she said, \"There we may sit at our ease. When you have relieved your thirst, we will hear your tale.\"\n\nLinden nodded in turn. That place would be as good as any. It was far enough from the scar to be safe from late-falling stones and slides. And she wanted something to lean against: support for her back, if not for her raw heart.\n\nSighing, she accompanied Rime Coldspray.\n\nThe rest of the Giants followed with Stave and Jeremiah; but Mahrtiir brought Narunal to Linden's side. Dismounting, he bowed his homage to the stallion; watched briefly as the Ranyhyn cantered away in the direction taken by Hyn, Hynyn, and Khelen. Then he turned to Linden.\n\n\"Ringthane,\" he began gruffly, \"I have no speech adequate to my gratitude\u2014aye, or to my bitterness. It is my greatest wish to prove worthy of this company, and of the peril of these times, yet Kevin's Dirt renders me effectless. Having naught of merit to say on such matters, I will not speak of them again. Know only that I am avid for use\u2014and that my thoughts are clamorous with concern for the Swordmainnir. Giants they are in good sooth. Yet they have walked without respite for nigh unto two days and a night, and now an immense labor awaits them. Some succor they must have.\n\n\"Ringthane, I ask this of you. When you have rested, extend to the Ironhand and her comrades the same benison which you have bestowed upon me. They will have need of it.\"\n\nOh, Mahrtiir. Linden dragged her free hand through her hair, tugging to untangle emotions as complex and self-referential as the wards which had guarded the Lost Deep. \"As soon as I get some rest,\" she assured him. \"After what they've been through, I'm surprised that they're still on their feet.\"\n\n\"That is well,\" the Manethrall replied more quietly.\n\nWatching him sidelong, Linden saw that his spirit required more substantial nourishment. But she was not ready to speak of that.\n\nHe would get his chance to be of use.\n\nAs she and her friends reached the stones that Coldspray had indicated, several of the Swordmainnir groaned with relief. Cirrus Kindwind, Cabledarm, and Halewhole Bluntfist began loosening their armor. Latebirth, Stonemage, and Galesend handed their water skins to Grueburn. Then they, too, unclasped their cataphracts. When they had shrugged the shaped stone off their shoulders, they slumped to the ground.\n\nBefore seating herself, Frostheart Grueburn handed a waterskin to Linden, and another to Mahrtiir. Stave she did not neglect in spite of his ability to convey the impression that he had no physical needs. Then she joined her comrades, leaving only Rime Coldspray upright.\n\nJeremiah was too excited to sit. With his thirst satisfied, he began to pace as if he were already measuring out the dimensions of his construct. And Stave remained on his feet. But Linden sank gratefully to rest against a rough curve of rock. Fumbling, she untied the neck of her waterskin, lifted it to her mouth. For a long moment, she let the simple bliss of untainted water pour down her throat.\n\nAs she drank, fresh beads of sweat gathered on her forehead and were cooled by ragged winds. Tears stung her eyes; but on this occasion, she was glad to be a woman who could weep. Briefly she paused to let her flesh absorb the blessing which the Giants had brought. Then she swallowed more water.\n\nMahrtiir studied her closely. When he was satisfied with what he saw, he seated himself cross-legged near Latebirth: a position that allowed him to face Linden directly.\n\nWhile she could, Linden unfurled her power and spread it around the company as Mahrtiir had requested, sharing her only resource. But she did not heed the reactions of the Giants. Whatever she did for them would not be enough.\n\nGrateful for any relief, Coldspray removed her own armor, placed it near her feet so that her stone glaive was within easy reach. Rolling her shoulders and neck, she loosened her sore muscles. Then she folded her arms across her chest and waited for Linden's tale.\n\nBefore long, Jeremiah's impatience overcame him. \"Mom,\" he prompted. \"You said we have a lot to talk about. And I want to get started. The sooner we can build my door, the more _Elohim_ we can save.\"\n\nLinden sighed; set her half-empty waterskin beside her. \"I know. This isn't easy to talk about. We could discuss it for a long time. But I'm going to keep it short. If you want to argue, you'll just be wasting effort. I've already made up my mind.\"\n\nEventually she would have to call upon the strength of the Staff for herself. But not yet. Later she would need all of the energy that she could impose on herself.\n\nAround the arc, her friends and her son waited, watching her as though they heard omens in her tone; tocsins of dismay.\n\n\"Going to the Sarangrave wasn't our idea,\" she began. Wind twisted around her. She smelled dust from the wilderland of craters; dust and old death from the wound that she had torn in the cliff. In the sun's absence, the air had acquired a chill edge. \"The Ranyhyn made that decision.\"\n\nIn a few words, she described what had happened just before dawn. She recited the message that the Feroce had delivered. Even more tersely, she outlined the memories which the lurker's worshippers had invoked. While her companions considered her tidings, she drew her conclusions.\n\n\"So now I think that the Ranyhyn got what they wanted. Somehow they gave the lurker a reason to form an alliance with Covenant. An alliance with us.\" _No harm will come to you_ \u2014\"What that means, I don't know\u2014except that Covenant is alive. And he wanted me to remember _forbidding_. He wanted to remind me that it's necessary.\n\n\"I've thought about it, and as far as I'm concerned, we can only get what we need from wood. I don't mean the Staff of Law. I mean the forests. From the One Forest. From a time when the Forestals knew how to _forbid_ Ravers.\"\n\nShe felt Mahrtiir's reverent awe at the motives which she ascribed to the Ranyhyn. In the eyes of the Giants, she saw speculations and chagrin. Stave's mien revealed nothing; but Jeremiah stared at her as though he did not know whether to feel amazed or appalled.\n\nThe wind was growing stronger. Unexpected gusts brought tears to Linden's eyes again. She let them fall, careless of the streaks of dust with which they marked her cheeks.\n\nResisting an impulse to hurry, she drank again. Then she continued.\n\n\"We need power.\" Her tone was steady. Fatigue had the effect of calm. \"You all know that. We aren't enough. I'm not a rightful white gold wielder, and Kevin's Dirt limits what I can do with Earthpower. If we have to fight off Roger and Kastenessen and God knows who else\u2014if we want to protect Jeremiah's door\u2014if we want to save the _Elohim_ and the stars\u2014we aren't enough.\" Not without Covenant. \"We have to have more power.\n\n\"So\u2014\" Briefly the consternation mounting in Jeremiah's gaze undermined her, and she faltered. But she had prepared herself for this. And she had done as much as she could for him. The time had come to confront other concerns.\n\nEverything would have been different if she had known how to help him. But his needs were too deep for her to reach\u2014and she had too little time.\n\n\"So,\" she began again, \"I'm going to open a _caesure_ and force my way into the past. Hyn can take me where I need to go. She won't get lost. And I have Caerroil Wildwood's runes. They should be good for something more than bringing Covenant back to life. Maybe they can guide me.\n\n\"I'm not just too weak the way I am. I'm too ignorant. I don't have any lore. All I have is emotion,\" despair and love, joy and grief and dread, \"and it isn't enough. I want to find the Forestals and get them to teach me _forbidding_. There's no one else I can ask\u2014except the _Elohim_ , and they won't tell me.\" They considered Jeremiah's purpose abominable. \"If I want an answer, I have to get it from the Forestals. Then maybe I can use that kind of magic to stop our enemies. Maybe I can even use it to keep the Worm away from Jeremiah's door.\"\n\n\"Mom!\" Jeremiah protested hotly. \"You _can't_. _Caesures_ are _dangerous_.\" In a smaller voice, he said, \"And I need you. I need help.\"\n\nLinden avoided looking at him. The sight might break her. Leaving him felt like committing a crime; but she could not make any other choice.\n\n\"You'll have help. You've always had help. But you can do what you have to do. I'm not worried about that.\"\n\nShe wanted to be able to say the same of herself.\n\nShe had expected vehement objections from the Giants and even Mahrtiir; indignation and arguments; angry pleading. What she received was harder to bear. Her friends were shocked: that was obvious. But they did not react like people who believed that she had proposed a Desecration. Their emotions were vivid to her health-sense.\n\nWhat they felt after the initial jolt was hope.\n\nFor a long moment, none of the Giants looked at her. Stave appeared to regard some private vista which was visible to no one else. Only Mahrtiir and Jeremiah kept their attention fixed on Linden. The Manethrall watched her as if he were probing her defenses, looking for an opening. Jeremiah stared with dismay gathering like stormclouds in his darkened gaze.\n\nFrostheart Grueburn was the first to speak. As if to herself, she mused, \"Extreme straits require extreme responses. It cannot be otherwise.\"\n\n\"No!\" Jeremiah snapped immediately. \"Mom, you can't _do_ this!\" He seemed to keep himself from howling by an effort of will that made Linden's heart quake. \"Maybe you can go away. Maybe you can make a _caesure_ do what you want.\" When he clenched his fists, flames dripped between his fingers like blood. \"But you won't be able to _get back_!\"\n\nAt that moment, he sounded unutterably forlorn.\n\nDust bit at Linden's eyes. She blinked furiously to clear them. Don't say that, she wanted to plead. Don't make this harder than it already is. But she demanded a sterner reply from herself. The Land required more from her. Jeremiah himself required more.\n\nGod, she was tired\u2014\n\nMeeting her son's gaze with wind and dust and tears in her sight, she said, \"I made a promise to Caerroil Wildwood. I don't know how else to keep it. I don't know how else to save any of us. I'll find a way to get back.\"\n\nStave had turned his unyielding gaze toward Jeremiah. Manethrall Mahrtiir appeared to be suppressing a desire to speak. Tension mounted among the Giants, as restless as the wind. But the Ironhand still stood with her head bowed, studying the ground at her feet, saying nothing.\n\n\"And I'm talking about moving through time,\" Linden added before Jeremiah could respond. \"Remember that. How long it takes for me won't have any effect on you. If I can do anything that even remotely resembles what I have in mind, there's no reason to think that I won't get back before the Worm comes.\"\n\n\"But you won't _get_ back,\" Jeremiah insisted. His voice shook. Nevertheless he made a palpable effort to reason cogently. \"You can make a _caesure_ now because the Law is already weak. I mean Time and Life and Death. It's all been damaged. But back _then_ , when there are still Forestals, everything is intact. How can you make another _caesure_ that long ago? Just trying, you'll change the Land's history. Even if that doesn't break the Arch, you'll hurt it.\"\n\nDesperately he finished, \"We'll never see you again.\"\n\nHis wounds were so close to the surface that Linden could almost name them.\n\nAnd she understood his objection. It was apt in more ways than he appeared to recognize. If she reached the Forestals, her arrival would inevitably afflict them with knowledge\u2014or at least questions\u2014which they should not possess. That alone might do irreparable harm to the Arch.\n\nYet she had an answer. \"Then I won't go back to the oldest Forestals. I'll try to reach Caer-Caveral.\" Hile Troy. \"He was the last. Meeting me won't affect any of the others. And in his time, the Law of Death was already broken. He's about to break the Law of Life himself. I won't change his history.\"\n\nSurely Hyn could find her way through a _caesure_ to Caer-Caveral?\n\n\"In any case,\" she said, \"what else do you want me to do? I'm useless here. I'm useless to you. I don't understand your talent, and I can't carry boulders. My only alternative is to supply the Giants with strength until they work themselves to death\u2014and _that_ I can't bear.\n\n\"I know it's dangerous,\" she concluded. She was running out of words. \"But I'll get back somehow. Hyn will bring me.\"\n\nNeither her manner nor her appeal comforted her son: she saw that. He felt threatened, rejected. Forsaken when he finally had a chance to prove himself. He no longer looked at his mother. One finger at a time, he unclosed his fists. Then he spread his hands to reveal small gusts of fire cupped in his palms.\n\n\"You can say what you want.\" In the gloom, the stains on his pajama bottoms seemed to devour his legs. \"Talking won't help. I have more important things to do.\"\n\nLit by Earthpower, he turned away.\n\nThe sight twisted a knife in Linden's heart. She needed the kind of courage that Thomas Covenant had tried to teach her. But she did not have it, and he was not here.\n\nGrueburn and the other Swordmainnir squirmed. Rime Coldspray scowled thunderous disapproval at the dirt. The Manethrall's bandaged attention did not leave Linden's face.\n\nLeaning against her boulder, she waited for their reactions. She had chosen this crisis for herself. Come good or ill\u2014\n\nHow often had she heard those words?\n\nThey were better than despair.\n\nFinally the Ironhand raised her head. Gloaming veiled her mien, but it did not conceal the set of her jaw or the lines of her shoulders. Without preamble, she asked, \"Swordmainnir, will you gainsay me?\"\n\nHer tone was like the edge of her glaive.\n\nAs if they knew her mind, Latebirth, Onyx Stonemage, and Halewhole Bluntfist muttered, \"Nay.\" The others shook their heads. With both fists, Frostheart Grueburn punched lightly at the earth to emphasize her answer.\n\n\"Then,\" Coldspray announced harshly, \"I say to you, Linden Avery, Giantfriend, that you are a wonderment. I speak with respect\u2014aye, and with admiration as well, though my manner belies the fullness of my heart. That your intent is foolhardy beyond all reckoning cannot be doubted. Indeed, it appears to be as extreme as a leap into the abyss of She Who Must Not Be Named. Nonetheless you raise my spirits. In such times, all deeds must be extreme. The Earth's need requires it.\n\n\"Therefore my word to you is this. My comrades will give of their utmost to aid young Jeremiah, for his purpose is likewise admirable. Stave of the _Haruchai_ and I will accompany you, doing what we may in your service.\"\n\nThe other Giants nodded their approbation. Some of them started to applaud. But Stave cut them off. Peremptory as a challenge, he stated, \"I will not. My place is with the Chosen-son. And he will have need of your aid, Rime Coldspray, your labor and stonelore. You cannot be spared.\"\n\nQuick protests gathered in the Swordmainnir. Before any of them could speak, Stave declared, \"Yet some companion she must have. Should she attempt this quest alone, she will not return. In the absence of High Lord Loric's _krill_ , she cannot wield white gold while she holds the Staff of Law. The conflict of such theurgies must prove fatal.\"\n\nAt once, Manethrall Mahrtiir surged like a shout to his feet. \"Then this task is mine. It was foretold for me by the Timewarden himself while his spirit remained within the Arch.\"\n\nLinden had not forgotten. _You'll have to go a long way to find your heart's desire. Just be sure you come back._\n\n\"In Andelain,\" the Manethrall continued, gathering force as he spoke, \"Covenant Timewarden avowed, 'There is no doom so black or deep that courage and clear sight may not find another truth beyond it.' For that reason, and in the name of prophecy, and because I must, I will accompany Linden Avery, Chosen and Ringthane, Wildwielder.\n\n\"In the endeavor which young Jeremiah contemplates, I have no part. Yet I am Ramen, attuned to the Ranyhyn, and also acquainted with the perils of passages within Falls. Where I am weak, _amanibhavam_ will sustain me. I will not fail the Ringthane.\"\n\nJaw jutting, he averred, \"I speak for my people. We must become more than we have been, lest we prove unworthy of the Ranyhyn. The tale of the Ramen is too small to justify the service which defines us.\"\n\nColdspray and her people studied him with darkness in their faces. Some of them still wished to protest, especially Latebirth, who had often carried the Manethrall. Others showed resignation or grief, or waited uncertainly for their Ironhand's reply. But Linden bowed her head and let new gratitude flow through her. Although she wanted Mahrtiir with her, she had been loath to ask so much of him. His unrequested willingness eased her reluctance.\n\nAfter a long moment, Rime Coldspray raised her voice into the twilight. \"Manethrall of the Ramen, I am abashed.\" Her tone was gentler now, and more sorrowful. \"I confess it, Giant though I am. Eyeless, your sight is clear where mine is clouded. We must accede to your counsel.\"\n\n\"Then,\" Mahrtiir returned, \"I bid you farewell for a time. May our absence be brief. For my part, I am certain of you. When you have set your hearts to any purpose, you will accomplish it. So it was said of the Unhomed, and so it is with you. But where their tale has grown dim with age, yours will shine out, illuminating the last days of the Earth.\"\n\nHurrying as if he feared that Linden might object, the Manethrall turned to her. While she sat with her head lowered and a dull ache in her chest, he asked, \"Ring-thane, shall we depart?\" An eagle's eagerness sharpened his voice. \"That you are sorely weary is plain. Yet delay will not restore you. Doubtless you desire to be reconciled with your son. Yet delay will not comfort him. He spoke thoughtlessly, and will recant when he is calmer. I do not doubt that he will greet your return with joy.\"\n\n\"All right.\" Linden did not raise her head. \"All right.\" Carefully she took a last drink from her waterskin. Then she rested her hands on the blackness of her Staff. \"We should go while I'm too tired to be terrified.\"\n\nStill without looking at her friends, she said, \"Coldspray, Grueburn, all of you\u2014I'm not worried about you.\" Instead of facing anyone, she studied Caerroil Wildwood's runes as if they might suddenly reveal their meaning. \"You're _Giants_. If it can be done, you'll do it.\"\n\nHer weakness and dread were a sickness in the pit of her stomach, a foretaste of nausea and hornets and gelid emptiness as cruel as a chasm. They seemed bottomless.\n\n\"But, Stave\u2014\" she added unnecessarily. \"Be sharp.\" She could not meet his gaze. \"At some point, someone is going to try to stop Jeremiah. I hope that Mahrtiir and I can come back before that happens. If we don't, Jeremiah and the Giants will need everything that you have in you.\"\n\nThe former Master regarded her with no expression that her nerves could interpret. \"Linden Avery, I have said that uncertainty is an abyss.\" His flat voice contradicted the gust and swirl of the wind; the plumes of dust. \"Nevertheless I do not fear it. Only your self-doubt troubles me. You esteem yourself too slightly. For that reason, you are prone to darkness\u2014and for that reason alone. Forget such concerns. You are not Kevin Landwaster. Remember, rather, that you are loved by those who know you well.\n\n\"Go blessed by the goodwill of your companions here, and by the stalwart aid of the Manethrall, and by the prowess of the Ranyhyn. It may chance that you will accomplish something other than your intent. Yet good will come of it ere the end.\"\n\n\"All right,\" Linden repeated. What else could she say? But still she did not lift her head or rise to her feet. Her mortality was too heavy for her to carry.\n\nShe felt Frostheart Grueburn moving toward her; but she did not know why until Grueburn scooped her from the ground. Clasping her under her arms, Grueburn held her high, extending her into the grey light as if she were the standard around which all of the Swordmainnir rallied; and as Grueburn did so, the other Giants called Linden's name softly, celebrating her with murmurs. Then Grueburn set Linden on her feet.\n\nThere Mahrtiir took her arm. Baring his teeth like a hunter who had finally found the spoor of his prey, he said, \"Come, Ringthane. Lean upon me while you may. In a moment, Stave will summon the Ranyhyn. To spare our companions, we must gain a wary distance ere you attempt the creation of a Fall. We will walk while we await great Narunal and valiant Hyn.\"\n\nLinden accompanied him because he drew her with him. Her attention was contracting. Already the Giants were becoming dim. Stave had begun to fade. Jeremiah was little more than a will-o'-the-wisp bobbing among the boulders and shards. But she was not growing faint with fatigue and fear; not sinking back into the blankness which had overcome her in front of She Who Must Not Be Named. Rather she was concentrating inward, seeking the private door, secret and familiar, that opened on wild magic; the learned impulse which allowed her to invoke rampant argent.\n\n_Its imperfection is the very paradox of which the Earth is made, and with it a master may form perfect works and fear nothing._ Kasreyn of the Gyre had said that. But he may have been wrong. And she was not a master.\n\nStill she persisted. In recent days, she had surrendered any number of things. The time had come to surrender hesitation and doubt. Like a derelict, she limped over the cratered ground. Step by step, the stains mapped into her jeans and the runes which defined her Staff led her away from her son. Without the Manethrall's help, she could have fallen.\n\nVaguely she heard Stave whistling. Soon the Ranyhyn would come: yet another reason for gratitude. It impelled her to turn her mind outward once more.\n\nResting on Mahrtiir's support, she asked, \"You do understand, don't you? You can let Hyn and Narunal know what we want?\"\n\n\"Aye, Ringthane,\" Mahrtiir answered steadily. \"I comprehend. And that which I comprehend, our mounts will grasp as well. Are they not Ranyhyn, the great horses of Ra, Tail of the Sky, Mane of the World? Their devoir will both serve and preserve us.\"\n\nLinden nodded, but she was not listening. He had said enough. Now she needed wild magic, and it did not come naturally.\n\nPerhaps she managed a hundred paces. The scuff of her boots cast small plumes of dust into the swirling wind, the increasing chill. Then she heard or felt the approach of hooves.\n\nGratitude, she thought. Maybe that was the answer. Gratitude and trust. Jeremiah was alive and free. So was Covenant, in spite of Joan. And Covenant had urged Linden to take this risk. Hyn and Narunal would make it possible. Maybe if she remembered to be grateful and have faith, she would be able to avoid High Lord Kevin's tragic arrogance.\n\nWhen the mare and the stallion joined her, Mahrtiir spent only a moment in homage. Then he boosted Linden onto Hyn's capable back. A heartbeat later, he mounted Narunal. In the half-light, he looked to Linden like all of the Land's bounty incarnated in one mere human as frail and fallible as herself.\n\nPrompted by Narunal's imperious whinny, Linden passed the Staff of Law to the Manethrall. Covenant's ring she lifted from its hiding place under her shirt. Pressing the wedding band between both of her hands, she brought forth silver flame as if she had the courage to defy the Earth's doom.\n\nAs if she believed that good could be accomplished by Desecration.\n\nIn the distance, Jeremiah seemed to call her name. Overhead Kevin's Dirt appeared to catch fire and burn, lit by wild magic. But she paid no heed. Taking the risk, she created a disruption of Time and history that might destroy the world.\n\n## 8.\n\nThe Right Materials\n\nJeremiah was only a boy, but in some ways he knew too much. In others, he knew too little.\n\nDissociation had denied him the normal processes of growing up; the gradually acquired experience of passions and denials, of joys and disappointments. Even in the most practical matters, his development\u2014his acquisition of earned knowledge\u2014had been stunted. At the age of fifteen, he had never so much as changed his own clothes. Certainly he had never learned the most mundane social interactions. In that respect, he was younger than his years; unfamiliar with himself.\n\nYet he had learned other lessons too well. The flames of Lord Foul's bonfire had taught him that some pains were unendurable. And the moral rape of possession\u2014the manner in which he had been used by the _croyel_ to betray Linden's trust\u2014had shown him that hating what was done to him both aided and harmed him. It gave him the desire to fight back\u2014and yet it also convinced him that he would not have been so hurt if he did not deserve it. Hate cut both ways. If he had not been such a coward\u2014if he had not hidden himself away to escape his wounds\u2014Lord Foul and the _croyel_ would not have been able to possess him, use him. He had brought his worst suffering on himself.\n\nHe did not understand why that was true. Nevertheless he yearned to _pay back_ what had happened to him. At the same time, he hated what he felt. He hated himself for feeling it.\n\nBut there had been other forces at work in him as well. His mother's love and devotion had kept him alive. With Tinkertoys and Legos, Lincoln Logs and racetrack sections, he had constructed a sense of possibility and worth that might have eluded a less abused youth. And during his visits to the Land, Covenant's spirit in the Arch had offered him a one-sided friendship, compassionate and respectful.\n\nThe result was a conflicting moil of emotions which he did not know how to manage.\n\nAnd now Linden had abandoned him; actually _abandoned_ him in order to enter a _caesure_ with Mahrtiir. The fact that she had explained her actions did not ease him. It did not muffle the beat of indignation and fear in his veins. He had _counted_ on her. She had _taught_ him to count on her.\n\nAnd yet, strangely, he could hardly contain his excitement. Right here, right now, he had a chance to make his whole life worthwhile. If he succeeded, he would save some of the _Elohim_ , some of the stars. He would prove that Lord Foul and the _croyel_ and his natural mother were wrong about him. From head to foot, he trembled with eagerness to begin.\n\nThat contradiction was confusing enough; but he had more.\n\nHe had inherited Anele's legacy of Earthpower. It belonged to him now: the Land's living energy had become as much a part of him as the blood in his veins. He was inured against the vagaries of heat and cold, wind and wet. His bare feet endured sharp rocks and the ancient shards of weapons or armor without discomfort. His health-sense sloughed off Kevin's Dirt. He could fuse bones to make marrowmeld sculptures. He could even summon fire from his hands. And there might be more possibilities.\n\nFor him, Earthpower had become a piercing pleasure. It had enabled him to rescue himself from his prison.\n\nBut he had received other things from Anele as well. The old man had given him inarticulate scraps of knowledge, and horrific vulnerabilities, and an instinct for moral dread. Much as he treasured Anele's gifts, their implications appalled him.\n\nAnd because he had never learned how to manage among his emotions, he tried to ignore the worst of them. Nevertheless they clung to him. He was like his pajamas. His mother had dressed him in them and tucked him lovingly into bed. The horses rearing across their faded blue might have been Ranyhyn. Now they were torn and tattered; soiled with grime and dirt; defined by bullets. From the waist down, their innocence bore the stains of Liand's death. The _croyel_ 's gore marked the shirt.\n\nSo he had turned his back on Linden when she had insisted on throwing her life away in the Land's past. What else could he have done? He did not know who he was without her. He hardly seemed to exist. When her _caesure_ collapsed into itself and vanished, taking her and Mahrtiir and their Ranyhyn to a place and time from which they might never return, Jeremiah dissociated them in his mind, buried them away. Then he chose the excitement of building. It was his only escape.\n\n\"Come on!\" he called down to the Giants and Stave. \"Let's get started. The longer we wait, the more _Elohim_ we'll lose.\"\n\n_Elohim_ and stars.\n\nThat was why he was here, after all: to save things that could not save themselves. To delay the Worm's feeding, slow its progress toward the Blood of the Earth. To buy time until somebody came up with a better answer.\n\nBut the Giants ignored his shout. None of them glanced up at him. Even Stave did not. With the Swordmainnir, the former Master watched the place where Linden and Mahrtiir had disappeared as if he hoped or feared that she would return almost immediately. They were all acting like there was no need to hurry. Like Jeremiah did not need them\u2014or like the _Elohim_ and the stars and the whole world did not need _him_.\n\nWind skirled like travail around him, tugged at his pajamas. It carried dust from the gouged cliff, the fallen debris. Perhaps it would have stung his eyes if he had not been so full of Earthpower. Somewhere inside him was a small boy who wanted to cry because his mother had left him. But he refused to be that boy. The structure that he wanted to make both goaded and protected him.\n\nSomehow he swallowed the impulse to yell at the Giants in frustration. Here was another aspect of his confusion, his inability to resolve his own contradictions. The Giants were ignoring him\u2014but they were _Giants_ , and he had loved them ever since he had first seen them. When he and Linden and Stave had ridden to rejoin the Ironhand and her comrades, his response to the sheer size and wonder of who and what the Swordmainnir were had opened like a flower in his heart. They were Giants in every sense: he had no other word for them. And he had seen the delight in their eyes when they had gazed at him, the relief and welcome. They had made him feel that he was capable of putting his past behind him. Of cutting it off entirely. Under their influence, he had believed that he could accomplish something wonderful.\n\nIf they rebuffed him now\u2014\n\nAbruptly his frustration became chagrin. His health-sense was precise: he could see that he had offended the Swordmainnir. There was anxiety in the slump of their shoulders, worries aggravated by a great weight of weariness. And they carried griefs which Jeremiah did not recognize. But there was also anger. Their refusal to acknowledge his call was deliberate.\n\nHe had to talk to them\u2014and he was afraid of what they would say.\n\nHesitating, he took a moment to scan his surroundings. Above him hung the gouge which his mother had made in the ridgefront. It and its slope of rubble faced the north, or a bit west of north. At odd intervals, chunks of rock and clumps of dirt still fell from the upper surfaces of the gouge; but they clattered harmlessly to the sides. Buffets of wind scattered the dust before it could settle.\n\nThe ridge filled that side of the landscape. In every other direction, an almost featureless plain stretched out to the horizons, a beaten flat pocked with hollows like craters left behind by a barrage of huge stones or heavy iron, or of bolts of magic. In the cloying dusk, these hollows or craters gave the terrain a mottled appearance, as if it were stippled with shadows or omens.\n\nAs far as Jeremiah could see, nothing grew or moved. Nothing lived at all. And no springs or streams nourished the plain. In this region, the foundations of the Lower Land wore only a thin mantle of dirt, soil so barren that it refused even _aliantha_.\n\nAnd over it all lay the pall of the sunless murk, an augury of the last dark. As Jeremiah gazed around, he noticed that the afternoon was waning. Evening was not far off. Then would come full darkness, the second night since the sun had failed.\n\nEven now, the stars were visible, as bright as cries overhead. He could have watched them wink out of existence, had he been willing to face them. But at night\u2014\n\nAt night, the Giants would have more difficulty doing what he wanted from them.\n\nThe situation was urgent\u2014and still the Swordmainnir rested against their boulders. They had promised to help him. Now they acted like they had changed their minds.\n\nHe had to talk to them.\n\nHis private turmoil made him awkward as he began to descend from the rubble. Whenever he was working on one of his constructs, he was deft and graceful, full of confidence. But when he felt stymied, his muscles forgot what they were doing. He fumbled at the rocks, jerked downward, lost his balance and caught himself like a child half his age.\n\nHe hated being clumsy. He hated himself when he was clumsy.\n\nThe curve of boulders where the Giants sat faced away from him. Like Stave, they were not affected by Kevin's Dirt: they must have been aware of Jeremiah. Still they did not look in his direction. Earlier they had shed their armor and swords. Now they all rested against thrusts of stone. Only Stave remained on his feet, still watching the place where Linden and Mahrtiir had disappeared.\n\nBiting his lower lip, Jeremiah resisted a desire to start protesting before he reached his companions. Fortunately Rime Coldspray turned toward him while he was still a short distance away. Although her disapproval was obvious, her gaze steadied him. Clearly she did not intend to keep ignoring him.\n\nTroubled gusts stirred up dust, carried it away. Clad in twilight, the Giants resembled shadows or stones. Like shadows or stones, they looked deaf to persuasion. Still Jeremiah walked closer until he stood near Coldspray at the edge of the arc.\n\nNone of the Giants spoke. Stave did not. But they were all looking at him now.\n\nFor a moment, Jeremiah clamped his teeth down on his lip. Then he tried to say something that would not make his mother's friends angrier.\n\n\"I know you're tired.\" He was whining: he heard it in his tone. That, too, he hated. \"I know you need rest. But I can't tell how long this is going to take\"\u2014he gestured at the slope of rocks\u2014\"or how much time we have, or how many _Elohim_ we can save. And it'll be harder at night.\n\n\"I want to get started. Why is that wrong?\"\n\nHe felt the attention of the Swordmainnir. Nevertheless they conveyed the impression that they wanted him to go away.\n\nThe Ironhand shifted her shoulder so that she faced Jeremiah more squarely. Even seated, she was taller than he was. She seemed to glare down at him in the gloom.\n\n\"Young Jeremiah,\" she sighed, \"we are Giants. Children are more than our joy and our delight. They are our future\u2014if the notion of any future has meaning in these fraught times. We are endlessly indulgent.\"\n\nBefore Jeremiah could ask, Then why are you mad at me? she said more sternly, \"But by the measure of your kind, you are not a child. Much has been given to you. Therefore much is expected in return.\"\n\nWincing, Jeremiah retorted, \"I know that.\" The sound of his own truculence disgusted him. It sharpened the vexation of the Giants. But he did not know how to control it.\n\n\"Do you, forsooth?\" drawled Frostheart Grueburn. \"You conceal your wisdom well.\"\n\nLatebirth and Cabledarm offered their own ripostes; but the Ironhand gestured them to silence. On their behalf, she asked Jeremiah, \"Do you indeed comprehend what Linden Giantfriend has done for love of you?\" Her tone was a bared blade. \"Your manner suggests that you do not.\n\n\"I do not speak of her search for you across many centuries and uncounted leagues. Other mothers have done as much, if in differing times by different means. Nor do I speak of her surrender to the machinations of the Harrow, or of her perilous descent into the Lost Deep, or of her many efforts to relieve your absent mind. These things might other mothers have done as well. We ourselves have done much in Lostson Longwrath's name, and we are not his mothers.\n\n\"Now, however, Linden Giantfriend has exceeded our conceptions of love and fidelity.\" Rime Coldspray's voice cut. \"She has surpassed the hearts of Giants. Knowing that you have need of her, she yet prizes your worth so highly that she has hazarded more than her own extinction. She has dared the end of all Time and life. This she has done for the Land's sake, aye, but also for yours, that your endeavors here may accomplish their intended purpose.\n\n\"Does her attempt not express her devotion? Does it not merit your esteem?\"\n\n_Remember that I'm proud of you._\n\nJeremiah's immediate reaction was a flare of anger. \"She _left_ me.\" But then tears burned his eyes, and he wanted to weep. He understood what his mother was trying to do\u2014and yet he had treated her courage like a betrayal. Winds swirled around him like misery. Abruptly he sank to the ground; sat cross-legged with his elbows braced on his thighs and his head down.\n\nCome on, he commanded himself bitterly. Don't be a baby. If you start crying now, I'll never forgive you.\n\nIn a small voice, he asked the scoured dirt, \"What do you want me to do?\"\n\nGradually the Ironhand's aura lost its irate flavor. \"Attempt patience, young Jeremiah,\" she replied as if she had exhausted her reprimands. \"Grant to us an hour of rest. Linden Giantfriend's fire is a rare gift, but it cannot efface the cost of all that we have endured. When we have rested, two of us will commence the labor which your purpose requires. The others will sleep while they may. When the two must pause, they will awaken two others in turn. By twos, we will achieve what we can until all have slept. With the return of day, we will arise together to serve you.\"\n\nAfter a moment, she added, \"If need compels you, make use of the night. Doubtless there are preparations which will serve to hasten the morrow's labors.\"\n\nAttempt patience? That seemed impossible to Jeremiah. Patience was for people who were incapable of anything else. He had spent ten passive years exhausting his ability to _wait_. But when Coldspray suggested preparations, his heart veered. That he understood: identifying his materials; setting them out so that he would not have to search for them when the time came to put them in place. And he knew that he would have to spend a lot of time searching for the right sizes and shapes and quantities of malachite. While he did that, two Giants might be able to give him as much help as he could use.\n\nThinking hard, he grew calmer.\n\nA flurry of gusts out of the northeast slapped at the company. They tumbled against the ridgefront, scurried out across the plain. To Jeremiah's nerves, they felt like the leading edge of a gale. But the forces driving the wind were still distant. The full strength of the blast might not reach so far.\n\nA part of his mind was making calculations: measuring the mass of rocks against their hidden seams of malachite; estimating sizes and dimensions and positions. But that part of him was instinctive. It did not require his conscious attention. Instead of focusing on it, he tried to think of a way to make amends.\n\nHe did love Giants.\n\nGroping, he said tentatively, \"You've talked about Longwrath before. Lostson Longwrath. I heard you\"\u2014in spite of himself, he winced\u2014\"when the _croyel_ had me. But I don't know who he is.\n\n\"What happened to him? Where is he?\"\n\nAt once, Jeremiah felt a pang spread among the Giants, and he feared that he had made a stupid mistake. They looked at each other or turned away; shifted uncomfortably where they sat; touched their weapons. But then he saw that he had not irritated them again. Instead he had reminded them of a pain which they did not know how to relieve.\n\n\"Ah, young Jeremiah.\" The Ironhand sighed once more. \"You request a tale\u2014\"\n\nAbruptly Frostheart Grueburn heaved herself to her feet. Towering against the dimming sky and the lucid stars, she announced to her comrades, \"It is a tale which need not delay young Jeremiah's task. If Latebirth will consent to join me, we two will be the first to aid him. And while we do so, we will speak of Longwrath.\n\n\"I have borne Linden Giantfriend across many arduous leagues. In her name, I will bear this burden also.\"\n\n\"You are harsh, Grueburn,\" Latebirth retorted. \"You ask much. Scend Wavegift's death clings unkindly to me. Should Longwrath appear before us here, I would wish both to embrace him and to strike him down.\"\n\n\"As would we all,\" muttered Coldspray. \"Nonetheless Frostheart Grueburn's offer is a gift. Should you prefer to rest, Latebirth, I will join her.\"\n\n\"Nay, Ironhand.\" Groaning lugubriously, Latebirth pushed herself upright. \"I merely complain, as is my wont. Grueburn's thought is worthy of her\u2014\"\n\n\"A jest of two edges,\" remarked Onyx Stonemage. \"It both gives and takes.\"\n\n\"\u2014and I will endeavor to prove worthy as well,\" Latebirth finished without pausing.\n\nDucking his head, Jeremiah mustered the grace to say, \"Thanks. I know this is hard. But I really can't do it without your help.\"\n\nGrueburn swung her hand at his shoulder, a comradely clap that nearly knocked him off his feet. \"Waste no heed on us, young Jeremiah. We are Giants. We revel in bewailing our lot.\n\n\"Come.\" Followed by Latebirth, she steered him back toward the sloping rockfall. \"You will describe what is required, and we will speak of Lostson Longwrath while we attempt your desires.\"\n\n\"In that case\"\u2014with a nudge of his shoulder, Jeremiah redirected her toward a stretch of open ground at the foot of the rubble\u2014\"let's start there.\" Within three steps, his distress became excitement again. Wind slapped grit and portents at his face, but he ignored it. The preparations for his construct seemed to spring into focus of their own volition. \"I'll show you where I want to build.\"\n\nGrueburn nodded her approval; and Latebirth said, \"That is well thought, young Jeremiah. In the absence of plain commands, we would doubtless cause ourselves much unnecessary labor.\"\n\n\"And we would moan,\" Grueburn stated, feigning pride. \"Even among Giants, I am prized for the purity and pathos of my moans.\"\n\n\"I don't believe you,\" snorted Jeremiah. Carried on a rise of anticipation, he tried to emulate his companions. With gibes, the Swordmainnir refreshed their spirits: he saw that. Now he wanted to participate. \"You've probably never moaned in your whole life.\"\n\n\"Latebirth has not,\" Grueburn asserted while the other Swordmain chuckled. \"She is entirely dour. But I am capable, I do assure you, of the most extravagant and heart-rending moans.\"\n\n\"Enough, I implore you!\" pleaded Latebirth. \"Young Jeremiah's ears will bleed if you proceed to a demonstration.\" More soberly, she added, \"And we have consented to speak of Longwrath.\"\n\n\"Yet time remains to us,\" Frostheart Grueburn countered. \"When I regard the approach of the Worm, the hours appear as brief as heartbeats. But when I contemplate the exertions before us, mere moments are protracted to the horizons and beyond. If we lack time sufficient to speak at leisure, we also lack time for our task. Haste will gain naught.\"\n\nLatebirth grunted glum acquiescence. In silence, the two Giants accompanied Jeremiah to the span of ground where he proposed to build.\n\n\"Here,\" he announced at the edge of his goal. With a gesture, he asked Grueburn and Latebirth to halt. \"I'll mark out dimensions. If we don't pile rocks inside that space, they won't be in the way later.\"\n\nLatebirth scanned the area, muttered something that he did not hear. His attention had shifted. Images flared in his mind, becoming more explicit as he estimated shapes and masses, ratios of malachite, necessary boundaries. Stooping, he selected a fragment of basalt with a sharp point. For a moment longer, he studied the ground. When he was sure, he began gouging lines in the dirt.\n\nFour paces for a Giant straight toward the ridgefront. Five parallel to the spill of rubble. Four more to form the third side of a precise rectangle. And a line along the northwest to close the space. There he interrupted his marks to suggest a gap. Eventually that gap would become an entryway.\n\nWhile Jeremiah outlined his construct, Frostheart Grueburn began.\n\n\"Speaking of Lostson Longwrath is hurtful to us,\" she said gruffly. \"The fault of his plight lies with our forebears. From them, we inherit a shame which we do not bear lightly. For that reason, and because your kind is born to brevity, and because we must conserve our strength, I will be concise.\"\n\n\"Concise, forsooth,\" scoffed Latebirth. \"Already you falter in your intent.\"\n\nGrueburn ignored her comrade. \"Young Jeremiah,\" she went on, \"Longwrath's plight shares much with your former state.\"\n\nJeremiah flicked a startled glance toward her. But his task held him, and he did not pause.\n\n\"He is possessed,\" she explained. \"Forces which he did not choose and could not refuse have deprived him of himself. In the name of a foolish and unheeding bargain with the _Elohim_ , he is ruled by a _geas_ both cruel and minatory. Where he was once a Swordmain honored among us, he has become a madman bent on murder.\n\n\"And he is lost in another sense as well.\" Grueburn's tone was as personal as a plea. \"Though we were his guardians and caretakers, he was separated from us. Now we know not where he wanders, or indeed whether he yet lives. Nor do we know what form his _geas_ has taken. He failed in his first compulsion. Has he now been released? Is some new atrocity required of him? It is possible that Infelice might have answered us, had we inquired of her in Andelain. But we were consumed by our shame\u2014aye, and also by our wrath. We did not think to inquire.\n\n\"Whatever the burden he now bears may be, he was consigned to it by our thoughtlessness as much as by the _Elohim_.\"\n\nJeremiah tried not to listen. Grueburn raised too many echoes. They were as insistent as the erratic buffeting of the wind. But unlike the wind, they did not hurry past him. Instead they squirmed like crimes in the background of his mind.\n\nHe should not have asked about Longwrath.\n\nNevertheless he surprised himself by demanding when he meant to remain silent, \"What's your point?\"\n\nSome denied part of him wanted an answer.\n\nHis companions regarded him gravely. After a moment, Grueburn replied, \"My point, young Jeremiah, is that Longwrath's madness and pain do not foretell your doom. There is this difference between you. You were taken. He was bartered in a witless exchange.\"\n\nJeremiah flinched. Before he could stop himself, he retorted, \"It's the same thing.\" He did not want to say this. The words were compelled from him by pressures which he yearned to defy. \"My mother gave us away.\" He remembered it vividly. The _croyel_ had delighted in raising such spectres from their graves. \"I mean my natural mother, not Mom. She must have thought she was getting something. She sacrificed my sisters and me when she handed herself to Lord Foul.\" The bonfire had cost him two fingers. If he had not hidden from them, eyes as hungry as fangs would have claimed him. \"We were too young to know what she was doing.\"\n\nBut he had not been too young to be terrified\u2014\n\nGrueburn's shoulders slumped. \"Then I will grieve for you. And I will hold out hope for Lostson Longwrath, that he may evade his _geas_ as you have foiled your imprisonment.\"\n\nJeremiah poked at his leg with the tip of his rock, trying to suppress a residue of agony. Dust had already begun to fill his lines. In any case, they were shrouded in twilight, almost imperceptible. Resisting the unspoken appeal in Grueburn's voice, he asked roughly, \"Can you still see where I want to build?\"\n\n\"We are Giants,\" Latebirth replied as if she were certain of herself. \"We will not forget.\"\n\n\"Good for you,\" Jeremiah muttered under his breath: a sour whisper. Then he turned toward the rubble. \"Come on. We've wasted enough time.\"\n\nAlmost immediately, however, he regretted his tone. It sounded too much like petulance, the whining of a boy who did not want to grow up. As an apology, he clenched his hands into fists, then opened them with cornflower flames in his palms.\n\nLighting the way, he led Frostheart Grueburn and Latebirth onto the rockfall to search for malachite.\n\nome of the stones with their secret deposits of minerals and hope were small enough that he could manage them without help. Those he ignored temporarily. Instead he probed the rubble until he located two or three rocks that required Giants. These he indicated to Grueburn and Latebirth. When they assured him that they would be able to wrestle the stones from the slope without causing it to shift, he quenched his fires. In darkness softened only by the half-light of evening, he returned to the smaller pieces of granite and basalt, and began hurrying them downward.\n\nHe was going to need a lot of them. And dozens or scores of bigger chunks. The proportions of malachite were meager. With purer, richer deposits, he could have contrived a structure no taller than himself, its walls closer together: little more than a shrine. But with these rocks, his construct would have to be the size of an impoverished temple, crudely raised by people too poor to afford a better place of worship. And even then, he could not be sure that he would find enough malachite for his purpose.\n\nThe right materials in the right amounts with the right shapes. If he succeeded, the _Elohim_ would come. They would have no choice. But if he failed to locate enough malachite\u2014or to build his temple before every _Elohim_ perished, or before the Worm came\u2014then everything would be wasted. His own life would have no meaning. Mom would have saved him and then left him for nothing.\n\nWhile he fretted, however, other facets of who he was made their preparations with a confidence that seemed almost autonomic. Hardly thinking about his choices, he set the stones he carried where they would be readily accessible. As Grueburn and Latebirth struggled down the slope, supporting between them a massive boulder, and gasping stertorously, he estimated its shape in relation to its freight of malachite, then directed them to place it like a cornerstone where two of his lines met. When they dropped it where he indicated, and leaned on it to ease their trembling, he instructed them to turn it slightly. And as soon as they complied to his satisfaction, he followed flurries of wind back onto the rockfall to retrieve more fragments of his intent.\n\nIn moments, he found a chunk of a size that threatened to exceed him. But before he could pry the rock out of the slope, he felt Stave coming toward him.\n\n\"Permit me,\" the former Master offered. \"There is little else that I can do to aid you. I lack the stonelore of Giants. Nor, it appears, do my senses equal yours. Yet strength I have.\n\n\"Also I am not needed to stand guard. In the absence of such glamour as the Unbeliever's son has wielded, any force potent to endanger Swordmainnir will be perceived at some distance.\" With a gesture, he indicated the open plain. \"So far from the foes gathered in the region of Mount Thunder, I deem that we are in no imminent peril of attack. And I do not doubt that Hynyn and Khelen watch over us in their fashion.\n\n\"Therefore permit me, Chosen-son. Carry lesser stones as you have need of them. Provide guidance to the Giants. Permit me to make use of my strength.\"\n\nStave's voice conveyed an oblique impression of appeal. He seemed to want more than he asked. Apparently being separated from Linden was hard on him. He needed distractions while he waited for her return.\n\nNodding, Jeremiah stepped aside. When he had remembered to say, \"Thanks,\" he added, \"I'll show you more as soon as you're ready.\" Then he turned away to check on the Giants.\n\nLatebirth had a lump in her arms that she could barely lift alone. Tortuous with caution, she picked her way downward. At the same time, Grueburn strained to loosen a boulder which was too heavy for her\u2014and which might let the rockfall above her slip. Sure of himself in at least this one respect, Jeremiah told her to leave it. \"I'll need it\"\u2014it was veined with too much green to ignore\u2014\"but we can move it later. For now, we should look higher up.\"\n\nGrueburn gasped a sigh as she straightened her back. For a moment, she raised her face to the stars, groaned unfamiliar curses. \"Even among Giants,\" she admitted, \"I am proven foolish. Clearly movement here will weaken the slope. This I should have discerned without your counsel.\"\n\nJeremiah felt her weariness. It slapped at him like the wind. But he could think of nothing reassuring to say except, \"We still have plenty to choose from.\"\n\nUnsteady as an invalid, she accompanied him upward.\n\nHe studied her sidelong for a moment, remembering his mother and the Staff of Law. Then he slapped his hands together, lifted fire into the night. His flames were more than light and warmth. They were Earthpower. He wanted to believe that their uses were not limited to fusing marrowmeld structures and cooking sour tubers; but he had no one to teach him. He could only learn by trying.\n\n\"When Mom does this,\" he said more to himself than to the Swordmain, \"it helps.\" Reaching out, he grasped Grueburn's forearm.\n\nWhile he concentrated, trying to send his inherited magic into her, she watched him with a glint of hope in her eyes. After a few heartbeats, however, she murmured, \"A worthy attempt, young Jeremiah. Alas, it is not the Staff of Law. It warms and soothes. It does not restore.\"\n\nAs if he were flinching, he let her go. His failure was obvious. He did not need to hear it named.\n\n_Failure isn't something you_ are. His mother had told him that. _It's something you_ do. She had said it as if she believed it. But it did not feel like the truth. His inability to help Grueburn felt like just another demonstration that he was not _good enough_ to deserve success.\n\nWithout warning, he saw Lord Foul's eyes in the bonfire that had maimed him. Unbidden and compulsory, the memory cut him like the flick of a lash. It cut deep enough to draw blood.\n\nIn that instant, he wanted to hit back. He needed a lash of his own. He saw the _croyel_ 's neck gripped in his strangling hands; saw himself pounding the Despiser's head to pulp with a stone. His eagerness to _hurt them_ was so swift and unexpected that he was unable to control it. It snatched a snarl past his teeth before he could restrain himself.\n\nAt once, he slapped his halfhand over his mouth. But he was too late. Frostheart Grueburn had heard him.\n\nShe studied him anxiously. For a while, she seemed to flounder, uncertain of her course. But then she summoned her frayed strength. With elaborate care, softly, she said, \"Heed me, young Jeremiah. Linden Giantfriend fears for you. She fears that both the _croyel_ and the Despiser have wrought untold harm. Now I discern that she has good cause. But I do not perceive the form or substance of your distress.\n\n\"Will you not reveal yourself to me? There is much to be gained by the setting aside of such concealments. And I remind you that I am a Giant. The burden of joy is mine. It belongs to the ears that hear, not to the mouth that speaks.\"\n\nI don't believe you, Jeremiah retorted in silence. Hear joy? That's not even possible. People judge. The _croyel_ taught me that. _Mom_ taught me that. She judges herself all the time.\n\nBut his secrets were too dark for him. They implied too much vulnerability, too much helplessness. They would reduce him to a whimpering child. They might send him back to the safety of graves.\n\nSeething, he filled his hands with fire again. Then he scrubbed it across his rough cheeks, ran flames through his tangled hair. While light like dishonesty shone in his eyes, he avoided Grueburn's gaze.\n\n\"I don't know what you're talking about.\" Deliberately he tried to sound callous. \"Look for malachite higher up.\" He waved a dismissal. \"Find a piece you can lift. I see a few rocks I can manage.\"\n\nBitterly he turned away. He told himself that he was angry at Frostheart Grueburn because she had tainted the pure excitement of his talents and his task, but that was not the truth.\n\nLike the wind, the truth thronged with omens.\n\nt intervals, Grueburn and Latebirth asked for Jeremiah's opinion of one ponderous fragment or another. Most of the time, they labored without him. And before long, Latebirth limped away to rejoin her resting comrades. After rolling one more boulder down the slope, Grueburn followed her. Eventually the Ironhand and Halewhole Bluntfist came to toil in their turn.\n\nStave needed more of Jeremiah's guidance, but he gave no sign that he required any respite. He worked steadily, moving chunks and slabs that would have tested the thick muscles of Giants.\n\nAnd Jeremiah also did not tire. He considered that he had already spent ten years effectively asleep. That was enough. In addition, Anele's gift of Earthpower provided reserves that seemed boundless. Occasionally he paused to measure his growing collection of fragments against his temple's requirements. From time to time, he asked Rime Coldspray and Bluntfist\u2014or, later, Cabledarm and Onyx Stonemage\u2014to position their burdens on one of his structure's boundary-lines. But those interruptions were brief. Between them, he moved up and down the rockfall with the assurance of inspiration. Hidden deposits and delicate veins held his attention as if they made him complete.\n\nDisplayed against the fathomless velvet of the heavens, the stars continued their gradual dance of death. And with every loss, the lights that remained seemed to shine more brightly, like beacons pleading for rescue. Only their vast profusion, and the immeasurable distances between them, suggested that the Earth's demise was not imminent.\n\nFor a while, Cirrus Kindwind with one maimed arm and Stormpast Galesend relieved Cabledarm and Stonemage. But soon, too soon, they wore themselves to the fringes of prostration. Then only Jeremiah and Stave were left to carry on the task.\n\nMidnight had come. It had not passed. Dawn and more help seemed impossibly distant.\n\nThe wind gathered force through the darkness, rushing from nowhere to nowhere, and contradicting its own impulses incessantly. Dust and grit driven away in one direction flailed the ridge from another. Sudden blasts strong enough to stagger Jeremiah righted him in an instant. Nevertheless the gusts did him one service: they scoured away the dirt between the stones. As a result, he was able to locate portions of malachite more quickly. Chunks no bigger than his head and ragged menhirs the size of Giants revealed their secrets as if they were etched in possibilities.\n\nStill there was too little that he and Stave could accomplish alone. Long before dawn, they had gathered all of the lesser shards that the edifice required. Some pieces they put in place: others they could not. For the heavier labor, only Giants working together would suffice. And even when Jeremiah had identified every fragment that he would ask the Swordmainnir to move, he lacked one crucial element.\n\nEventually he would need a capstone, a culminating lump of malachite. Not a large one: nothing bigger than his two fists together, or perhaps his goaded heart. And its precise contours were not critical. Any approximation would serve his purpose. But it had to be pure\u2014\n\nWell, not absolutely pure. He could tolerate some slight admixture of other substances. But not much. Not much at all.\n\nWhere in or under this rubble was he going to find enough unadulterated green? So far, everything had been veins, tracery, small nodes; threads deposited in trickles across centuries or millennia. Otherwise he would have needed rocks of lesser bulk.\n\nWithout its capstone, his edifice would have no power over the _Elohim_.\n\nWhile the Giants rested, there was nothing more that Jeremiah could do to prepare or build. He could only search. And dawn was still three hours away.\n\nIncreasingly alarmed, he scrambled up and down the rockfall, moving with less assurance and more haste; gripped by a fever of trepidation. Over and over again, he told himself to slow down. He could not probe the slope deeply or accurately while he was hurrying. But he seemed to feel jaws snapping at his back, fangs wet with venom and malice, rabid agony. Memories\u2014At any moment, they might catch him.\n\nIf he failed now, he would not deserve anything that Linden had done for him.\n\nA slap of wind caught him rushing from one boulder to another. His foot missed its step as if the solidity of the world had faded. Without warning, the entire rockfall seemed to stand on its side. Then he plunged.\n\nIn an instant, realities transposed their definitions. Through the darkness, he saw as clearly as prescience that all of his conflicts and confusions would be resolved when his head smashed itself open on that looming jut of granite, _that_ one. He was falling too hard to twist aside. But now he understood that being overtaken by his fears was not the worst possible outcome. Even a retreat to his graves was not the worst. Anything could be destroyed, anything at all, by a senseless, childish accident.\n\nThen Stave caught his arm, swung him out of danger so suddenly that Jeremiah did not recognize Stave's grasp until Stave had settled him on a canting shelf of basalt. He did not feel the tight hurt of Stave's fingers until the first wildness of his heartbeat began to subside.\n\nHe was panting as though he had lost a race.\n\n\"Chosen-son,\" Stave said like a man who had seen nothing, done nothing, \"you appear troubled to my sight. Do not take it amiss that I say so. I am _Haruchai_. Your silence I deem condign. I gauge that you have concealed naught which may alter the choices of your companions. What purpose, then, is served by speech? Nonetheless you are mortal, as I am. And at the side of the Chosen your mother, I have learned that it is not shameful to request or receive aid. Therefore I will hear if you wish to speak.\"\n\nJeremiah was breathing too hard to think clearly. Mom wanted him to talk. Grueburn wanted him to talk. They wanted to probe horrible memories, expose parts of him that bore the marks of the _croyel_ and the Despiser. Of course he refused. But now he knew that there were worse things than failure.\n\nHe had in fact concealed something that might have affected Linden's choices. She did not understand the dark core of Anele's legacy.\n\nThe former Master had promised to watch over him. To keep him safe.\n\n\"Stave\u2014\" he began thickly. \"They don't know. I'm so afraid\u2014\"\n\nBut he could not continue. The words stuck in his throat.\n\nWhat purpose, then, is served\u2014? His mother was already gone.\n\nWhile Stave waited impassively, Jeremiah wrestled his demons into their familiar shapes.\n\n\"I'm afraid this is all wasted.\" He gestured awkwardly around him. \"There's a piece I need, and I can't find it. Without it, nothing else counts.\"\n\nStave lifted an eyebrow. \"What is it that you require, Chosen-son?\"\n\nJeremiah swallowed a groan. \"A lump of malachite. About this big.\" He put his fists together. \"And it pretty much has to be pure. But all I've got are traces. That whole ridge probably doesn't have any pure malachite big enough to save the _Elohim_.\"\n\nStave scanned the slope as though it did not interest him. \"Perchance it does not,\" he remarked. \"We cannot be certain until we have searched with greater care. Also it may be that the surface of the rockfall conceals its depths. I will accompany you until you are confident of your perceptions. If no hope is found, then mayhap we would do well to delve within the rubble.\n\n\"I see no cause for concern\"\u2014he may have meant despair\u2014\"until we have done our utmost. And even then, the lore of our companions may devise possibilities which elude us.\"\n\nJeremiah stifled a protest. He wanted to say, That isn't going to work. All of us together can't move this many rocks fast enough. But Stave's uninflected calm seemed to refuse objections.\n\nHow could he be right? He did not share Jeremiah's fears.\n\nHe was _Haruchai_. He had sacrificed his place among his people to stand with Linden. How could he be wrong?\n\nAfter a moment, Jeremiah nodded reluctantly. \"Sure. Why not? What else are we going to do?\"\n\nBracing himself on the former Master's dispassion, he filled his hands with fire. Earthpower might serve to sharpen his health-sense. And if it did not, it might comfort him anyway.\n\nTogether Jeremiah Chosen-son and Stave of the _Haruchai_ began the tedious task of scrutinizing the rockfall from every angle.\n\nor Jeremiah, time crept by in an ocean surge of frustration, inexorable as a tide, rising and falling from one moment to the next, but always climbing higher. An accumulating sense of futility lured his attention into darker places. His flames changed nothing, and he let them go; immersed himself once more in the world's darkness. Occasionally his heart rose at the glimpse of a deposit. When he saw that the amount of malachite was too small, his spirit sank again.\n\nBut Stave was always at his side, always calm\u2014and steadier than Jeremiah's pulse. Over and over again, Jeremiah swallowed his alarm and kept going for no better reason than because Stave was with him.\n\nStealthy as betrayal, dawn came closer; and still Jeremiah could not find what he sought. An hour before the moment when the sun should have risen, he and the _Haruchai_ completed the first stage of their search. They had looked everywhere. They had looked at everything. Now nothing remained except the imponderable labor of digging into the rockfall.\n\nHigh up on the slope, Jeremiah collapsed on a slab of granite with his elbows propped on his knees and his face hidden in his hands. He was tired now, worn out by defeat. Everything that felt like excitement or hope had drained out of him. No doubt Stave would go on searching. Jeremiah could not.\n\nThe _Haruchai_ remained standing nearby, glancing here and there with apparent unconcern. He may have been waiting for Jeremiah to recover. After a few moments, however, he said, \"Set aside discouragement, Chosen-son. Hope remains.\"\n\nThe flatness of his tone made him sound reproachful.\n\nJeremiah jerked up his head. As aggrieved as a child, he burst out, \"It does not! We've looked everywhere! And I don't care what you say about taking this rubble apart. Sure, we can look deeper that way. But we only have eight Giants\u2014and _they_ don't have any food. They'll have to shove rocks out of the way for _days_ while they starve. The world is going to end, and it'll break Mom's heart, and we'll still be here just digging!\"\n\n\"Softly, Chosen-son,\" Stave replied as if he were commenting on the condition of Jeremiah's pajamas. \"The time has not come to rouse the Swordmainnir. Doubtless they would answer your urging, but we have no cause to summon them. In one respect, you are mistaken. We have not extended our search to its boundaries.\"\n\nJeremiah stared. He wanted to shout something vicious, but Stave's manner stopped him. Briefly his mouth and throat worked without producing a sound. Then he asked hoarsely, \"What're you talking about?\"\n\n\"Chosen-son,\" Stave stated without hesitation, \"you have not turned your gaze upward.\"\n\nStill Jeremiah stared. What, upward? At the stars?\n\n\"Consider the ridge,\" explained his companion. \"Consider the wound which the Chosen has made. Your discernment exceeds my senses, but to my sight it appears that there is a source of malachite above us.\"\n\nJeremiah sprang to his feet as if he had been stung, flung his gaze at the source of the rockfall.\n\nAt first, he found nothing except blunt granite, blind basalt. Apparently every bit of green had already fallen.\n\nBut Stave was looking higher, studying the hollow near its ragged upper rim.\n\nA tall slab stood there, a monolith heavy enough to resist Linden's detonation. To a quick glance, the stone resembled granite or schist. But when Jeremiah looked harder, he saw that the slab was actually a flawed mix of igneous rock and more porous sandstone supported by rigid shafts of flint.\n\nAnd enclosed within the monolith were signs\u2014\n\n\"Really?\" he breathed. \"Are you sure?\"\n\nWas that his capstone? Exactly what his temple needed?\n\nIf so, it was inaccessible. Completely out of reach. Perhaps Linden could have used her Staff, caused the slab to topple somehow. Her son could not.\n\nWith enough rope\u2014\n\nThe Giants had no rope.\n\nScowling, Jeremiah clenched his fists until his fingers ached. \"I can't tell. It's too far away.\" Then he beat his knuckles against his thighs so that his frustration would not erupt into the night. The monolith appeared to lean as if it were taunting him; daring him to believe that it would topple. \"But even if it's enough, it's useless. We can't get at it.\"\n\n\"Chosen-son.\" Now Stave's tone was unmistakably a reprimand. He regarded Jeremiah as if the tugging of the fractured gale did not touch him. \"You judge in haste. Therefore you judge falsely. Have you come so far in Linden Avery's care and failed to learn that despair gives poor counsel? If the needed stone lies beyond your grasp, withdraw. Retreat to the foot of the rockfall. Acknowledge this truth, that you are not alone.\"\n\nJeremiah opened his mouth; closed it. A mordant voice inside him snarled, What're you going to do? Fly up there? I dare you. But that reaction arose from memories which he strained to suppress. He would have pulled down the ridge gladly to bury them. And Stave was impervious to Jeremiah's galled incredulity. Withdraw. Fighting himself, Jeremiah moved backward under the pressure of Stave's severe gaze. Retreat.\n\nMom! Where are you? I don't know what's going on.\n\nRetreat from _what_?\n\nAwkward as a youth who had never been sure of anything, Jeremiah went down the rubble as quickly as he could manage.\n\nWhen he reached bare dirt, he peered upward. Just for a moment, he could not locate Stave. But then a suggestion of movement snagged his attention. Squinting, he spotted a hard shape like a piece of condensed midnight untouched by starlight. Stave had already climbed beyond the top of the rockfall. Now he hung splayed against the ridgefront, searching with his fingers and toes for holds which would enable him to lift himself toward the immense hollow cut by Linden and Earthpower.\n\nHe must have been creeping: he hardly seemed to move at all. Jeremiah could not imagine how he found cracks and rims still solid enough to support him. Yet Stave did move. Sudden jerks conveyed the impression that a grip had failed, or a toehold. He appeared to swing from side to side, hanging by one hand; perhaps by one finger. Uncertain as hallucinations, bits of debris dropped away. But he did not fall.\n\nHe was _Haruchai_ , born to the crags and precipices and flensing winds of the Westron Mountains.\n\nIf he gained the gouge, he would be able to climb more easily, at least for a while. Its lower surface was not vertical. He would be halfway to the monolith.\n\nThe monolith itself was three times his height, many times heavier. It could have served as a monument for a Giant. He would not be able to dislodge it by simply throwing rocks at it. His only choice would be to work his way higher.\n\nBut toward the back of the hollow, the ascent would become steeper. Then the harmed stone above him would tilt outward. There the slab he strove to reach stood on a crude protrusion like a snout. That formation multiplied the hazards. He would have to climb beneath it, hanging precariously in the air\u2014\n\nJeremiah heard one of the Swordmainnir moving toward him, but he could not look away from the small flutter of darkness that represented Stave. Over and over again, he held his breath as if he believed that his own tension might protect the former Master. The whole night had come to this: the little increments, barely perceptible, of Stave's efforts.\n\nWrapped in winds, Rime Coldspray towered out of the night to stand beside Jeremiah. The Ironhand had left her armor and sword behind, but she moved as if she still carried them\u2014and had another Giant sitting on her shoulders. That she had slept was plain. But she needed more than rest. She needed sustenance. Above all, she needed relief. She and her comrades had known little except struggle and strife since they had first approached the Land.\n\nBriefly she regarded Jeremiah. Then she lifted her gaze toward the ridge and Stave.\n\nHe had almost reached the hollow. Holds broke in his hands; but he cast those shards away and hunted for better grips. Occasionally Jeremiah heard the clatter as rocks hit the slope. At other times, gusts carried the sounds away, and Stave seemed to climb in a preternatural silence, fraught as a clenched breath.\n\n\"Stone and Sea,\" murmured Coldspray. \"If this is not madness incarnate, it serves some purpose which I do not discern.\"\n\nJeremiah pointed. \"He's trying to reach that slab. It has malachite I need. But I don't think he can even get there. He won't be able to break it loose.\"\n\n\"Ah.\" The Ironhand released a sigh. \"Now I comprehend. The malachite itself is vague to my sight. But consider the stone within which it is concealed.\" She stared hard under her heavy brows. \"If distance and darkness do not mislead me, the stone stands somewhat apart. A cleft or flaw has detached it from the ridge.\n\n\"Stave Rockbrother will endeavor to dislodge it.\"\n\nJeremiah did not believe that Stave could do it.\n\nAs if to herself, Coldspray added, \"When it falls, he will also. Then he must perish. Though he is _Haruchai_ , his flesh is not iron. His bones are not. They will not withstand an impact from that height.\"\n\nWhile pressure mounted in Jeremiah's chest, Stave's unyielding shape crossed into the gouge. There he rose to his feet and paused, secure against the battering of the wind. For a few moments, he appeared to study the challenge ahead of him. Then Jeremiah saw the former Master wave one arm: a gesture of reassurance so unconvincing that it made Jeremiah wince.\n\nThis was impossible. It was all impossible. What Stave had done was already insane\u2014and there was worse ahead of him. When it falls, he will also. Jeremiah had not thought that far ahead.\n\nThen he must perish.\n\nAbruptly Jeremiah wheeled on Rime Coldspray, clutched at her arm. \"Do something,\" he panted. \"He's _Stave_. Mom will never forgive me if he dies.\" Because he was pleading with a Giant, the Ironhand of the Swordmainnir, he tried to tell the truth. \"I'll never forgive myself.\"\n\nWithout turning her head, Coldspray answered, \"This choice was not yours to make, young Jeremiah. It belonged to Stave Rockbrother. It remains his. He will suffer the cost because he chooses to do so.\n\n\"At present, his peril is diminished. Later it will become extreme. Should he fall within the hollow, we can do naught to aid him. We must trust his skill and agility to preserve him.\n\n\"The achievement of his purpose is another matter.\"\n\nStill watching Stave's wary ascent, she called, \"Ho, Swordmainnir! Bestir yourselves! You will wish to witness Stave Rockbrother's valor. And he will have need of you!\"\n\nAt first, there was no response.\n\n\"Frostheart Grueburn!\" shouted the Ironhand. \"Latebirth!\" She sounded more relaxed than Jeremiah felt; far more confident. \"Cabledarm! Onyx Stonemage! Hear me! Hear and come!\"\n\nAfter a moment, a bleary voice answered, \"We hear you.\" Grueburn. \"The very stars hear you.\"\n\nIf she said more, gusts carried the words away.\n\nFor a while, Stave moved more easily. But soon he reached the steeper recesses of the wound, where the stone had more cracks. He was forced to resume his earlier care, testing each handhold, each support for his feet, each small ledge and crack and bulge, before he committed his weight to it.\n\nYawning, Giants approached Jeremiah and Coldspray. He recognized them without glancing at them. Only Stormpast Galesend and Halewhole Bluntfist lagged behind\u2014or they were still asleep.\n\nWhile Stave crawled up the back of the hollow and began to creep toward the granite jut which supported the monolith, often hanging by his hands alone until he found places to anchor his feet, Coldspray explained his intentions to her comrades. Then she said, \"He is Stave Rockbrother, able and stalwart as the _Haruchai_ of old. He will not fail.\"\n\n\"When he succeeds,\" muttered Grueburn, \"he will fall. He must.\"\n\n\"And he will perish,\" Stonemage added grimly.\n\n\"Therefore,\" concluded the Ironhand, \"we must intervene.\"\n\nConsidering the problem, her comrades nodded.\n\nJeremiah wanted to ask, Intervene _how_? But Grueburn, Latebirth, Stonemage, and Cabledarm were already moving away. Apparently they did not need Coldspray's instructions. As they started up the rockfall, they separated. Grueburn and Latebirth on one side, Cabledarm and Stonemage on the other, they labored toward the ridgefront.\n\nAt first, Jeremiah could not imagine what they had in mind. Then he understood. They aimed to bracket the slab's likely path when it toppled. Clearly they meant to position themselves on either side of that path. If they could avoid being struck, they might have some chance of catching Stave.\n\nIf he did not fall first. If he managed to shift the monolith. If loosened rocks did not hit anybody. If just one of the Giants was quick enough to intercept his plunge. If his impact in her arms did not kill him as surely as the jagged rubble. If it did not break or kill her\u2014\n\nJeremiah was holding his breath again. He thought that he saw Stave's arms flailing. Dislodged debris spattered like rain into the hollow.\n\nBut Stave's indistinct form still clung to the rock. One grip at a time, he eased upward.\n\nCirrus Kindwind left the Ironhand's side, strode some distance up the rockfall. When she had climbed atop an especially tall boulder, she stopped to study the ridgefront. Then she raised her voice in a shout.\n\n\"Grueburn! Latebirth! Alter your heading!\" She waved her arms, directing her comrades to the left. \"You will be struck!\"\n\nThe two women did not respond to Kindwind's hail; but they must have heard her. They shifted their course.\n\nUnfortunately now they were no longer clambering up the rockfall's spine. Instead they were forced to straggle along the side of the slope. If Stave came down toward the crest of the rubble, they would not be able to reach him without sprinting upward\u2014and they were fatally weary.\n\nStill Stave made his way by undetectable increments. Only the erratic spatter of stones and the wind-torn fall of dirt showed that he was still moving. But he _was_ moving. One hand or foot or finger or toe at a time, he worked his way closer to the bulbous rock supporting the slab.\n\nJeremiah hardly dared to estimate the distance. Involuntarily he imagined Stave's fingers bleeding, his muscles trembling\u2014\n\nRime Coldspray rested a gentle hand on Jeremiah's shoulder. \"Remember that he is _Haruchai_ ,\" she murmured. \"He has performed wonders ere now. Mayhap he will surpass our fears yet again.\"\n\n\"But he's in trouble either way.\" If Stave failed to shift the slab, he would never be able to climb back down. \"Can they\"\u2014Jeremiah meant Grueburn and Latebirth, Stonemage and Cabledarm\u2014\"actually catch him?\"\n\n\"We are Giants.\" The Ironhand's reply was softer than the wind. \"Often we have been tested. Often we have prevailed.\"\n\nFor a time, Stave seemed to vanish. Hidden by the shape of the bulge, he had become indistinguishable from the stone.\n\nIn alarm, Jeremiah blurted, \"Where is he? What's happened?\"\n\n\"Gaze more closely,\" Coldspray advised. \"You will perceive that he is safe for a time. One arm he has wedged into the cleft between the monolith and the cliff. While he remains thus secured, his peril is diminished. Now the uncertain balance of the stone is the gravest threat. Should it tilt suddenly, catching him unprepared\u2014\" She allowed herself a sigh. \"In that event, opportunities to affect his fate will be slight. Far better for him if he must exert his full strength to shift the stone. Then his efforts will carry him outward, away from the precipice and ruin.\"\n\nAs she spoke, Jeremiah caught an image of Stave dropping like more rubble. Spinning out of control. Hitting the ridgefront over and over again until he was mangled beyond recognition.\n\nThe former Master had made his own choices\u2014but Jeremiah had inspired them. His whole body ached with a futile desire to keep Stave safe.\n\nStill resting her hand on Jeremiah's shoulder, Rime Coldspray continued, \"For the moment, I am primarily concerned by the width and depth of the cleft.\" She sounded deliberately casual. \"At this distance, I cannot gauge it. If the stone does not stand free of the cliff, it is unlikely to fall. And if the cleft will admit no more than Stave's arm, he will have scant leverage. Then the bulk of muscle which he will require might exceed even a Giant.\n\n\"No, we must hope that he will contrive to force his arms and chest\u2014indeed, his body entire\u2014into the cleft. For him as for us, that will be the most favorable circumstance.\"\n\nShe may have been trying to soothe Jeremiah by focusing his attention on practical details.\n\nTo an extent, she succeeded. As if involuntarily, he found himself imagining Stave squeezed behind the slab; Stave straining to shift the monolith. While Stave did such things\u2014if he did them\u2014he would not fall.\n\nWind stung Jeremiah's eyes. His pajamas fluttered around him in tatters. He ignored Stormpast Galesend and Halewhole Bluntfist as they drew near. Instead he watched Grueburn and Latebirth, Stonemage and Cabledarm. They had reached the places where they meant to wait for Stave. Now they stood motionless in the night. They were not immediately below the slab, but they were close enough to be struck by debris\u2014or by the slab itself if it bounced crookedly against the ridgefront. Still Jeremiah thought that they were too far from the cliff\u2014and too far from each other. He could not believe that they had a prayer of saving Stave.\n\nThe three Swordmainnir with Jeremiah studied the monolith and its pediment. From her boulder, Cirrus Kindwind did the same. Winds flailed like indignation in all directions, outraged by affronts too distant to be answered.\n\nWithout warning, Kindwind shouted, \"Ware, Frostheart Grueburn! The stone shifts toward you!\"\n\nIt might move in that direction if Stave could use only one arm. Or perhaps the rim of the bulge above Grueburn's position was simply weaker.\n\nFaintly through the tumult in his heart, Jeremiah heard Grueburn reply, \"I have seen it.\"\n\nShe did not step back. Nor did Latebirth.\n\nOnyx Stonemage and Cabledarm readied themselves to spring forward.\n\nUnder her breath, Coldspray murmured, \"Be swift, comrades. You must be watchful and wary, but above all you must be swift. We can ill endure any loss of life.\"\n\n\"Or indeed any injury,\" muttered Galesend, \"worn and weakened as we are.\"\n\nJeremiah beat bruises onto his thighs and strove to see.\n\n_There_. He had lost sight of Stave. The _Haruchai_ had thrust his way into the cleft; or the shape of the jut masked his presence. But the monolith had moved: Jeremiah was sure of it.\n\nIt did not move again.\n\nThen it did.\n\nAt first, it appeared to lean back toward the cliff as though it sought to crush the force which had disturbed it. For moments as long as heartbeats, as urgent as cries, it hung in place, scattering scree from its base.\n\nWith the suddenness of a calving glacier, the slab slid away.\n\nSilent as a cast-off leaf, it appeared to drift through the darkness until one end collided with the ridgefront. Instantly it broke apart, became half a dozen pieces. They rebounded from the impact, falling like a barrage toward the waiting Swordmainnir.\n\nGrueburn, Latebirth, Cabledarm, and Stonemage were all in danger; but the threat to Grueburn and Latebirth was greater.\n\n\"He is yours, Cabledarm!\" Grueburn yelled. Jeremiah saw her and then Latebirth leaping down the jagged slope of the rockfall.\n\nGranite thunder boomed. Heavy shards pounded the rubble where the two women had been standing.\n\nAt the same time, Cabledarm dodged a fragment which would have slain her. She surged upward. Unscathed, Onyx Stonemage braced herself; remained where she was.\n\nAbove them, Stave also struck the cliff. But he twisted as he dropped so that he hit with his feet. Somehow he planted himself long enough to flex his legs and spring away. His great strength transformed his plummet into an outward leap.\n\nArms spread like wings, he cast himself soaring into the mad roil of the winds.\n\nCabledarm was there when he came down.\n\nIn spite of his splayed posture, he was falling too hard, plunging like a chunk of the slab. Even a Giant could not hope to catch him safely. His weight and momentum would shatter bones, Cabledarm's as well as his.\n\nBut she did not try to catch him. She had other intentions. During the quick instants of his descent, she crouched low. Then she sprang to meet him, arching away as she did so; already pitching herself backward.\n\nHer huge hands found his hips. Her arms bent to absorb the collision. Then she gave him a prodigious heave.\n\nHis force and hers flung her, helpless, down the side of the slope. She tumbled like a piece of the ridge.\n\nBut she had redirected his fall. He was soaring again.\n\nToward Onyx Stonemage\u2014\n\n\u2014who caught him in both arms.\n\nLike Cabledarm, she did not try to hold him. Instead she swung him in an arc and released him so that she seemed to throw him in the direction of open ground beyond the rockfall.\n\nHe landed on his feet; dove and rolled to dissipate the last of his momentum. Then he rose to stand upright in the thick dusk.\n\nJeremiah began running before the _Haruchai_ came to a halt.\n\nThe monolith was broken. Its burden of malachite may have been shattered, made useless. Everything may have been wasted. Even Linden's ride into the chaos of a _caesure_ \u2014\n\nBut Jeremiah was not racing to locate the outcome of his only hope. He was running as if his heart might burst to find out if Stave and Cabledarm were all right.\n\nIn the east, a dull dawn announced the third sunless day.\n\n## 9.\n\nAn Impoverished Temple\n\nThe company gathered around Stave and Cabledarm. Jeremiah fought down an impulse to babble. I can't believe it! That was amazing! Are you all right? But he could hardly speak in any case. He was panting as though he had run an inconceivable distance, and had witnessed wonders.\n\nStave's arms and feet were latticed with scratches. His palms and fingers, his toes, the soles of his feet: all oozed blood. But those injuries were trivial. The effects of his impact with Cabledarm's hands were another matter. His whole body had struck and recoiled like a cracked whip. Now every joint looked torn; every muscle. His internal organs appeared to throb as if they had been beaten with clubs. Blood gathered at the corners of his mouth: he had bitten into his tongue. In spite of his _Haruchai_ stoicism, he was trembling.\n\nHe stood, but he seemed unable to speak. Like a man who had been blinded, he stared at nothing. If he felt the presence of his companions, he did not react to it.\n\nRime Coldspray studied him for a grim moment. Then she sent Cirrus Kindwind to retrieve a waterskin. She had nothing else to offer him.\n\nCabledarm's wounds were more obvious. They looked worse. Pitching Stave to Stonemage, she had flung herself down the raw edges and fanged splinters of the rubble. Like Stave, she had regained her feet beyond the slope. Unlike him, she stood hunched in pain, hugging her left arm against her chest. Giantish obscenities bubbled like froth past her lips. She was bleeding from half a dozen gashes, at least two of them deep enough to expose bone. Contusions covered her from shoulder to ankle. But her worst injury was to her left shoulder.\n\nThe force of Stave's plummet had ripped her arm out of his socket. It was dislocated so badly that Jeremiah could hardly bear to look at it.\n\n\"Only you, Cabledarm,\" the Ironhand muttered through her teeth. \"Only you could emerge so harmed from such a rescue.\"\n\n\"It is my gift,\" Cabledarm rasped. Then she groaned a curse. \"Stone and Sea! Am I not a Giant? And have I not vaunted myself the mightiest of the Swordmainnir? How am I thus humbled by mere falling?\"\n\n\"We need Mom,\" Jeremiah breathed miserably. \"We can't help her. And Stave looks like he's going to pass out.\"\n\nBut the Giants did not respond. Cabledarm's dislocation, at least, was hurt which they knew how to address. At a nod from Coldspray, Halewhole Bluntfist moved to stand behind Cabledarm. With one arm on Cabledarm's left shoulder near her neck, and the other across her chest under her right arm, Bluntfist grasped Cabledarm tightly enough to wring a moan from her comrade. Without a moment's consideration, Coldspray gripped Cabledarm's damaged limb and heaved; twisted.\n\nThe sound as the arm slipped back into place hit Jeremiah like a jab to the stomach.\n\nCabledarm roared. Briefly she wobbled as if she were losing consciousness. But Bluntfist held her until her faintness passed, and she began to curse again.\n\nGrimacing, Cabledarm moved the fingers of her left hand, managed a fist. When she was done swearing, she muttered, \"It is much and naught, Ironhand. It will hamper me, but it will mend. Only stanch some few of my rents, and I will name myself blessed. Stave Rockbrother lives\"\u2014she glanced quickly around\u2014\"does he not?\" Seeing the answer in the eyes of her comrades, she finished, \"Then will I name myself blessed in all sooth.\"\n\n\"For the present, however,\" the Ironhand commanded, \"you will conserve yourself, Cabledarm. Cirrus Kindwind brings water. While you drink and rest, we will contrive bindings for your wounds. You have earned the tales which we will tell of you. Now we will contrive to earn those which you will tell.\"\n\n\"Aye,\" Cabledarm assented: another groan. With Bluntfist's help, she lowered herself to the dirt. There she extended one gashed leg so that Bluntfist could try to stop the bleeding.\n\nJeremiah saw her injuries too clearly: the rich pulse of her blood and pain made him feel sick. About some things he knew too much. About this he knew too little. He could too easily believe that Cabledarm would bleed to death. That Stave was dying inside.\n\nFortunately Kindwind soon returned with several waterskins. Two she tossed to Bluntfist. A third she took directly to Stave.\n\nFleeing the sight of Cabledarm's torn flesh, Jeremiah joined Kindwind.\n\nStave did not react to their presence. He remained standing; continued to stare at nothing as though his whole world had become the abyss of the Lost Deep. Tremors ran through him like waves of fever. His hands shook. Even his lips quivered.\n\nJeremiah did not know what to say or do. Stave had promised to protect him. This was the result.\n\nFrowning, Cirrus Kindwind rested her hand on Stave's shoulder. \"You are not alone, Rockbrother,\" she assured him. \"Rest you need. So much is certain. But first you must drink. Your flesh has been much abused. It requires refreshment. And see?\" She unbound the neck of her waterskin, held it in front of him. \"Here is water.\"\n\nStave did not move. He did not appear to hear her. But when she touched his mouth with the lip of the waterskin, he raised his arms, accepted it from her. Trembling, he drank.\n\nJeremiah had never seen any _Haruchai_ do more than sip from the cup of one hand. Now Stave swallowed long gulps as though more lives than his depended on them; drank until he had emptied half of the waterskin. Then, slowly, he sank to his knees, settled back to sit on his heels. The waterskin he placed on the ground. His hands he rested, open palms upward, on his thighs. He seemed to nod.\n\nAfter that, he resumed gazing sightlessly at the twilight of the new day.\n\nBeckoning for Jeremiah to accompany her, Kindwind stepped away. When they had withdrawn a few paces, she said, \"We must trust, Chosen-son, that his folk restore themselves in this manner. It appears that his spirit has turned inward. But I will believe that a man who has performed his feats must soon heal himself and return to us.\"\n\nJeremiah swallowed against the dryness in his throat. \"I hope so. He doesn't deserve this.\"\n\n\"Ah, deserve,\" sighed Kindwind. \"The notion of deserved and undeserved is a fancy. Knowing both life and death, we endeavor to impose worth and meaning upon our deeds, and thereby to comfort our fear of impermanence. We choose to imagine that our lives merit continuance. Mayhap all sentience shares a similar fancy. Mayhap the Earth itself, being sentient in its fashion, shares it. Nonetheless it is a fancy. A wider gaze does not regard us in that wise. The stars do not. Perhaps the Creator does not. The larger truth is merely that all things end. By that measure, our fancies cannot be distinguished from dust.\n\n\"For this reason, Giants love tales. Our iteration of past deeds and desires and discoveries provides the only form of permanence to which mortal life can aspire. That such permanence is a chimera does not lessen its power to console. Joy is in the ears that hear.\"\n\nHer assertion startled Jeremiah. It seemed to question his foundations. If he closed his eyes, he could still see the extremity of Stave's fall. The hard throb of Cabledarm's bleeding and the excruciation of her shoulder cried out to his senses. Awkwardly he reached for Kindwind's last waterskin. When she released it, he drank as if his thirst\u2014his dismay\u2014had the force of a moral convulsion.\n\n\"So you're saying,\" he protested or pleaded, \"what Stave did is worthless? What Cabledarm did is worthless? It's all dust?\"\n\n\"Aye,\" Cirrus Kindwind assented, \"if that is how you choose to hear the tale.\" Her tone was mild. \"For myself, I will honor the effort and the intent. Doing so, I will be comforted.\"\n\nJeremiah wanted to shout. Instead he fumed, \"You sound like the _croyel_.\" Was joy in the ears that hear? Then so were agony and horror. So was despair. \"It was forever telling me everything Mom did was useless. Nothing matters. It's all dust. That's why Lord Foul laughs\u2014and Roger\u2014and those Ravers. They agree with you. In the end, they're the only ones who get what they want.\"\n\nKindwind looked at him sharply. Like the flick of a blade, she retorted, \"Then hear me, Chosen-son. Hear me well. There is another truth which you must grasp.\n\n\"Mortal lives are not stones. They are not seas. For impermanence to judge itself by the standards of permanence is folly. Or it is arrogance. Life merely is what it is, neither more nor less. To deem it less because it is not more is to heed the counsels of the Despiser.\n\n\"We do what we must so that we may find worth in ourselves. We do not hope vainly that we will put an end to pain, or to loss, or to death.\"\n\n_Failure isn't something you_ are. _It's something you_ do.\n\nWithout warning, Jeremiah found that he ached to share Kindwind's beliefs, and Linden's. Perhaps the monolith had never contained enough malachite. Perhaps the deposit had shattered. Perhaps Stave and even Cabledarm would die. Perhaps Mom would never come back. Perhaps futility was the only truth. Still Jeremiah would have to find a way to live with it.\n\nTo himself, he muttered, \"It's not that easy.\"\n\nCirrus Kindwind had never been possessed.\n\nHer response was a snort. \"We were not promised ease. The purpose of life\u2014if it may be said to have purpose\u2014is not ease. It is to choose, and to act upon the choice. In that task, we are not measured by outcomes. We are measured only by daring and effort and resolve.\"\n\nJeremiah wanted to insist, It's not that easy. It's _not_. But the words died in his mouth. Kindwind had already turned away. Several of the Giants around Cabledarm had turned away. They were gazing up at the spine of the rockfall.\n\nAt Frostheart Grueburn and Latebirth. As Jeremiah caught sight of them, they labored past the crest and began their descent. Between them, they carried a large chunk of stone.\n\nIt resembled a fragment of the monolith. He detected distinct signs of malachite.\n\nNot seams or veins, delicate trickles. A concentrated lode.\n\nIn an instant, he forgot everything else. Leaving Kindwind, he ran at the slope.\n\nThat was a piece of the slab: it had to be. And its mineral deposit was still intact.\n\nHow big was that sealed lump of green?\n\nThe two Giants came a little way to meet him. Then they set down their burden and straightened their backs, loosened their arms. Before he reached them, he felt their emotions.\n\nIn spite of the gloom, they were bright with vindication.\n\n\"By good fortune,\" Grueburn called to her comrades, \"the object of Stave Rockbrother's extravagance contained an admixture of sandstone. When it struck, it broke along its less durable seams. The malachite of its heart was preserved.\"\n\nJeremiah needed to see for himself. Filling his hands with Earthpower, he clapped them to the surface of the rock; probed inward with all of his senses.\n\nThen he wheeled away, flung his gaze down the slope toward Stave.\n\n\"You did it! Stave, you _did_ it!\"\n\nThe former Master knelt with his back to the rockfall. He did not lift his head or turn. He may have sunk so far down into himself that he did not hear.\n\nNevertheless he had succeeded.\n\nSome things were too easy. Accepting failure was one of them.\n\nor a time, Jeremiah was content to confirm the various locations of his materials, study their shapes, and plan. While he did so, the Giants finished tending Cabledarm's wounds. Then they rested.\n\nEventually Stave stirred. With an air of caution, as if he feared that he might break bones, he looked around at the cratered plain, the crepuscular day. Then he rose to his feet.\n\nThe relieved shouts of the Giants elicited no response. Jeremiah's gladness he acknowledged with no more than a nod. He gave the impression that he had forgotten speech, or gone beyond it. When he had surveyed the company and the rockfall, the beginnings of Jeremiah's construct, and perhaps the passage of time, he put a hand to his mouth and whistled.\n\nWhile Jeremiah and the Swordmainnir watched him, wondering, Stave waited for Hynyn.\n\nThe stallion came promptly. Although Jeremiah had seen no sign of the star-browed roan earlier, Hynyn appeared as if he had reincarnated himself from the substance of the gloaming. At Stave's side, he halted; stood patiently while Stave welcomed him by stroking his neck and shoulder. Then, together, they approached the Giants.\n\nAt once, Jeremiah hurried to join Rime Coldspray and her comrades.\n\nWavering on his feet, Stave stopped. He seemed to have achieved an unstable victory over his private wounds, one which might become defeat at any unexpected action, any unpremeditated word.\n\n\"You did it,\" Jeremiah said again, but hesitantly, unsure of himself in Stave's presence. \"You saved us.\"\n\nYou saved me.\n\nStave glanced at Jeremiah, then away. He did not meet Coldspray's gaze. With obvious difficulty, as if language required skills which he had forgotten or misplaced, he said, \"Hynyn will guide you to water. The way is long.\" His voice began to fade. \"But there is water.\"\n\nIn a husky whisper, he added, \"My thanks to Cabledarm. Also to Onyx Stonemage.\" He made an effort to gather himself. \"And to Cirrus Kindwind.\"\n\nStill cautiously, he turned his back. With the elaborate care of a man who feared falling, he walked out onto the plain until he was barely visible. There he knelt again, facing the northwest like a diminished sentinel.\n\nHynyn remained with the Giants. Clearly the great stallion understood the promise that Stave had made in his name. He waited for the women to act on it.\n\nAfter a brief consultation, Kindwind announced, \"With your consent, Ironhand, this task is mine. In the shifting of stones, I am hampered, but the bearing of waterskins will test only my dexterity.\"\n\nRime Coldspray nodded. \"Go with my thanks. Return as swiftly as you are able. Water we must have. The tasks remaining to us will be arduous.\"\n\nNodding to her comrades, Cirrus Kindwind left with Hynyn. The imperious arch of the stallion's neck seemed to assert that he could not be humbled by such mundane service.\n\nWhen they were gone, Rime Coldspray said, \"Now, Chosen-son. We have delayed too long. There is death in every lost moment. Instruct us, that we may begin.\"\n\nJeremiah's heart beat eagerly. At last\u2014\"I've found everything I need,\" he answered. \"But some of it still has to be moved. Then I'll need help putting the pieces in place.\"\n\n\"Indeed.\" Coldspray scanned her comrades. \"For the present, we are only six. But six are more than five, or three, or one. We must suffice.\n\n\"Instruct us,\" she said to Jeremiah again. \"Come good or ill, boon or bane, we will strive to do as you ask.\"\n\nUrged by relief and gratitude, Jeremiah tried to cheer. Then he turned to lead the Giants. With every step, he recovered more of his necessary excitement.\n\ny midday, the women had finished moving green-veined rocks to open ground. Before they were done, they were all trembling on the verge of exhaustion. But earlier Cirrus Kindwind had returned with every bulging waterskin that she could carry. The Swordmainnir had been able to continue working because they had enough to drink.\n\nNow they were sprawled in the dirt, resting as though they had been felled. The fraught rasp of their respiration sawed at Jeremiah's nerves until he felt as raw as their lungs; as desperate to be done. But they still had a lot to do.\n\nFor him, actually assembling his temple would be comparatively easy. It required no thought at all. His talents were certain, as instinctive as breath. He could have completed the structure without hesitation\u2014if he could have raised the heavier rocks alone.\n\nBut for his companions\u2014\n\nThe work ahead of them would demand more effort, not less. As the walls rose, massive chunks and boulders would have to be lifted higher. And the roof would be more difficult than the walls. The Giants would have to hold the stones in place at the height of their own shoulders until he could brace the construct with his last hunk of granite, his capstone of malachite. Only then would the temple stand without support.\n\nAt some point, Cabledarm had climbed upright. Walking stiffly, she had come to watch her comrades. But she was still too weak to stay on her feet. She had nothing to offer except the encouragement of her presence.\n\nStave had not moved. At some distance, he knelt facing the northwest as if he sought to ward off threats by nothing more than force of will. Or perhaps he was praying for Linden's return.\n\nStanding near the Ironhand, Jeremiah said uncomfortably, \"When you're ready.\" Erratic bursts of wind slapped at him. Grit stung his cheeks. Beyond his horizons, a fierce storm was brewing. The air was growing cooler. \"I know where everything goes. I can do this fast.\" _Elohim_ were dying. \"But you should take your time. We can't afford mistakes.\"\n\nInfelice had tried to prevent his escape from his graves. She should have known better. She should have trusted Linden.\n\n\"Yet it must be done,\" Coldspray replied in a low growl. \"Much depends upon it. When we are beset by storms as we sail the world's seas, we do not rest merely because we are weary. Rather we cling to our tasks, and to our lives.\" She seemed to be trying to convince herself. \"Matters do not stand otherwise now.\"\n\n\"Sooth,\" groaned Frostheart Grueburn. \"All that you say is sooth, Ironhand. We must\u2014yet I cannot. In the Lost Deep, I deemed that I had measured the depths of exhaustion. Now I learn that our flight from She Who Must Not Be Named was no more than a child's game by comparison.\"\n\n\"Nay, Grueburn,\" Stormpast Galesend countered like a pale imitation of herself. \"You misesteem us. Exertion alone does not justify our weariness. In addition, we lack viands. Do not discount that deprivation.\"\n\n\"Indeed!\" exclaimed Onyx Stonemage. \"I will give my oath that I am dwindling. Hunger diminishes me. My garments hang loosely, and my cataphract has become an encumbrance, and I fear that my sword has grown too long for easy use.\"\n\nFor a moment, the Giants were silent. Then Coldspray said like a sigh, \"You forget to whom you speak, Stonemage. All here know that in your care every sword grows too long for easy use.\"\n\nAnother silence followed while Jeremiah fretted. The Ironhand's comment may have been a jest. If so, he did not understand it.\n\nApparently the other Giants did. After a moment, they started laughing.\n\nAt first, their laughter was as weak as their limbs: a sound like moaning amid the confusion of the winds. But then Stonemage retorted, \"Mockery is ignorance. Occasions there have been in abundance, yet none have inspired complaint,\" and her comrades began to laugh harder. Soon they were laughing with such abandon that they could not lie still. Latebirth and Galesend tossed from side to side. Grueburn pulled her knees to her chest, hugged them. Even Cabledarm chuckled in spite of her wounds.\n\n\"I don't get it,\" Jeremiah protested; but the women went on laughing.\n\n_Joy is in the ears that hear_. Clearly the Swordmainnir lived by that creed. Jeremiah did not understand at all. They sounded hysterical. Yet when they subsided, they were stronger. Somehow laughing had restored them.\n\nThat was enough for him: he could accept it. When he was able to believe that the Giants were ready, he moved away toward the scant beginnings of his construct, beckoning as he went.\n\nThe rectangle that he had marked in the dirt was still vivid in his mind, although its visible lines had been erased. A few heavy stones had already been put in place for him. He had added a number of small rocks himself. But that was barely a start. Most of the building remained to be done.\n\nHowever, all of his materials were waiting for him. He could imagine their eventual positions precisely, as if they were lit by sunshine rather than masked by dusk. His part of the work that remained was simple.\n\nFollowed by the Giants, he thrust his way through the wind to select rocks in their proper sequence: a sequence that would allow him to prop each one securely before the next was lifted.\n\nThat the women were more willing than able was painfully obvious. Stones that one Swordmain had managed alone earlier now required the strength of two or three, or even four. Nevertheless their willingness did not waver. To spare themselves, they rolled rather than carried rocks to the edges of the nascent temple. Together they heaved the shards into position. Then Jeremiah scrambled to insert the chunks of granite and basalt that would brace the bigger pieces in place.\n\nThe Giants took turns, resting as much as they could. When their waterskins were empty, the Ironhand sent Kindwind to fill them again. And the women watched over each other. Whenever one of them faltered or stumbled, others moved to help.\n\nBy slow increments, the walls of the temple rose.\n\nEvery now and then, Jeremiah remembered to glance at Stave. Indistinct in the distance, the former Master still knelt with his back to the construct, motionless as a tombstone. He gave no sign that he was aware of his companions' efforts.\n\nThey could have used his help.\n\nBy the time that Kindwind returned, the walls were nearly complete. A crude slab had been set to form the lintel of the entrance. Without counting, Jeremiah knew that a dozen heavy rocks and twice that many smaller ones remained before the capstone could be wedged into place. He knew exactly where the pieces would go. But he did not know how his companions would be able to finish the work. They seemed entirely spent. He was not confident that their hearts would continue to beat much longer.\n\nWhile Kindwind handed around waterskins and her comrades rested, Jeremiah went to plead with Stave.\n\nBut when he reached the _Haruchai_ , he did not know what to say. He could see that Stave was healing. The former Master knew how to provide for his own recovery. Nevertheless his heart beat with palpable reluctance, the pulse in his veins was as thin as a thread, and his breathing barely lifted his chest. In spite of his native toughness, he looked like a man who might not stand again.\n\nJeremiah's appeal for help turned to dust in his mouth. Winds seemed to drive it back down his throat.\n\nStave did not turn his head, but his shoulders stiffened slightly at Jeremiah's approach. After a moment, he answered the supplication of Jeremiah's silence.\n\n\"Chosen-son.\" His voice was a wisp of its familiar inflexibility. \"Say what you must. I hear you.\"\n\n\"I don't know why you're still alive,\" Jeremiah blurted. \"But I don't know why the Giants are still alive either. They're way beyond exhausted. They can hardly lift their arms. And we still haven't done the hardest part.\"\n\nAbruptly he stopped. He had no idea how to continue.\n\n\"The hardest part?\" Stave inquired: a mere breath of sound.\n\n\"The roof. I'm making a temple. I mean, that's how I think of it. It has to have a roof. But it won't stay up until I brace it. That's what your lump of malachite is for. They'll have to lift the rocks and stand\u2014\" Simply thinking about such things hurt. \"They'll have to just stand there holding up the roof. And even if they can do that, I don't know how they're going to set the capstone. I can't imagine\u2014\n\n\"It has to be just right, or it won't work.\" He struggled to devise scenarios. They were all cruel. \"So even if four of them can hold up the roof, that only leaves two to lift the last piece because at least one of them has to climb up there,\" adding her weight to the rocks on the shoulders of the other Giants, \"and put that piece in place. I'll probably have to be there myself to make sure it's right.\"\n\nWithout warning, sobs crowded into his chest. If he let himself, he would wail like a child. He was tired to the bone, and all of his talents and excitements were useless now. He did not have the strength to complete his construct.\n\n\"It's terrible.\" He restrained himself by gritting his teeth. \"It's all terrible. I don't know how to make it better.\"\n\nStave did not react. For a time, he knelt motionless and said nothing, as if he had no interest in Jeremiah's distress. Eventually, however, he bowed his head in submission.\n\n\"Yet the attempt must be made.\" He spoke as if the wind tugged the words out of him. \"I will remain as I am for a time. Then I will come.\"\n\nWith that, Jeremiah had to be content.\n\nAs suddenly as it had arrived, his impulse to sob faded. He had reached the end of his emotions. Now he felt emptied. Matters were out of his hands. He had done what he could. Anele's gift of Earthpower did not make him mighty. It only made him vulnerable.\n\nSagging into himself, he left Stave and stumbled across the wind back toward the Giants.\n\nBut he did not go to them. He had nothing to tell them that they did not already know. Under kinder circumstances, they probably could have finished the task without him.\n\nInstead he made his way to his crude edifice. For a while, he studied the four walls and the northwest-facing entrance. Then he set to work.\n\nWith negligent, futile ease, he tossed small stones into their necessary positions along the tops of the walls. Doing so did not require thought: it required only certainty. But soon he had done what he could. Then he had to wait for the Giants.\n\nAround him, the day grew darker. That was wrong: his senses were sure. The time was early afternoon, no later. Yet the vague illumination was fading. He had become little more than a shadow to himself, a wraith in a distorted dream. His construct crouched in the gloom like the base of a tower broken by siege.\n\nCarried by baffled gusts and blasts, the darkness gathered from the east, or perhaps somewhat north of east. It advanced in tatters like the wind, moiling and routed, then surging ahead. And its source was still distant, scores of leagues away. Nonetheless the fading of the light was a warning.\n\n\"Ho, Swordmainnir.\" Rime Coldspray sounded improbably far away. \"Now or never. Behold! Night gathers against us prematurely. I know not how to interpret this augury, but I do not doubt that it promises ill. We must complete our purpose.\"\n\nA chorus of groans arose: protests and curses. Across the distance, Jeremiah felt the Giants climbing to their feet as if they were struggling out of an abyss. Even Cabledarm stood.\n\nLeaning against each other, the Ironhand and her women came to stand with Jeremiah.\n\nHe heard their exhaustion, their frailty. He seemed to taste it like charcoal on his tongue. He did not know how to bear it\u2014or how to ask them to bear it.\n\nBecause he was concentrating on them, a moment passed before he realized that Stave also had joined him.\n\nSeveral of the Giants greeted the _Haruchai_ , but he did not reply. Instead he regarded the walls of the construct. After a pause, he announced thinly, \"This is _suru-pa-maerl_. The folk of the Stonedowns formed such sculptures balancing and fitting stones to each other. In Muirwin Delenoth, Chosen-son, you devised a structure of marrowmeld. Now you have restored _suru-pa-maerl_ to the Land, or perhaps created it anew. Perhaps it gives cause for hope.\"\n\nThen he turned to Rime Coldspray. \"I have recovered strength enough for one effort. I will expend it here. Afterward I will pray that we have no more need of it.\n\n\"You must fashion the roof. When it lacks only its capstone, I will ascend. Receiving the stone from those below, I will place it as the Chosen-son instructs. That I will be able to do, that and no more.\"\n\nJeremiah winced. In her weariness, the Ironhand herself flinched. \"Will you?\" she asked, stern and anxious. \"Stave Rockbrother, the prospect troubles me. The monolith which you dislodged is broken. The portion containing malachite is small by comparison. Still it outweighs you.\n\n\"Your prowess is ever a cause for wonder. Nevertheless I fear that no _Haruchai_ could lift and settle that fragment.\"\n\nGloom masked Stave's visage. Even his lone eye was shrouded as if it had fallen into shadow. \"Yet the choice is mine,\" he answered. \"The strength is mine. The life is mine.\n\n\"If I am not needed, I will stand aside.\"\n\nColdspray rubbed her face like a woman disguising another flinch. First with one hand, then with the other, she slapped her cheeks. She seemed to dig deep into herself for a response.\n\n\"Certainly you are needed,\" she rasped.\n\n\"Thus in the end,\" one of her comrades muttered, \"even Giants may be reduced to brevity.\"\n\nStave nodded. \"Then have done with delay.\"\n\nJeremiah opened his mouth to argue; closed it again. How could he object? His construct was impotent without its capstone. Everything that he and the Giants and Stave had done here hung in the balance. If he wanted to spare the former Master, he would have to suggest an alternative; and he had none.\n\nSighing, the Ironhand said, \"Come, Swordmainnir. The task exceeds only our muscles and thews. It does not lie beyond our comprehension. We must believe that a feat which may be understood may also be achieved.\"\n\nIn response, Cabledarm lifted her head, flexed her arms. \"I will join you,\" she announced grimly. \"I am less than I was. What of it? I am able to stand. Therefore I will be able to stand under some weight of stone.\"\n\nColdspray nodded. \"That is well. You also are needed.\"\n\nLike a woman walking to an execution, she went to the nearest roof stone. There she told her comrades, \"Some will lift. Others will serve as pillars. The first pillar will be Kindwind. Cabledarm will be the last. When the roof is complete, Bluntfist and I will pass the final fragment to Stave. Thereafter we, too, will become pillars until the capstone is set.\"\n\nThe other Swordmainnir nodded their assent. When Cirrus Kindwind had entered the temple, Rime Coldspray and Stormpast Galesend rolled a chunk of granite inside. There they heaved it upward until Kindwind could crouch under it, accept its weight with her back and shoulders.\n\nAt the same time, Latebirth and Grueburn began shifting another stone. Onyx Stonemage joined Kindwind: a second support. Halewhole Bluntfist and Cabledarm readied themselves.\n\nJeremiah, too, was needed: he knew that. The sections of the roof had to be positioned exactly. Otherwise they would not remain in place when they were wedged by the capstone. Yet he did not move. He had lost every resource of excitement. Now he felt only a sickening apprehension.\n\nHow much more would his companions have to suffer because he had suggested building a sanctuary for the _Elohim_?\n\nor a while, he sank into a kind of paralysis. Matters of scale overwhelmed him: the extremity of the Giants; the consequences of failure. Possible deaths drained the volition from his limbs. But then his fears were thrust aside by a summons which he could not refuse.\n\nThe straining women did not call out to him. Stave did not. His construct did.\n\nIt was crude in every detail, and so tenuously balanced that a nudge might knock it down. At the same time, it was ineffable, capable of mysteries. Eloquent as a paean, it spoke the language of his talents, his deepest needs. He had to finish it.\n\nCompelled, he followed a Swordmain into the temple.\n\nNow he seemed calm to himself, although his voice shook and his hands trembled. Fervid and sure, he told the Giants, the pillars, where they had gone wrong; urged subtle corrections of tilt and fit; encouraged them to stand taller under their burdens. While darkness mounted across the plain, he guided the placement of his materials.\n\nSoon only Halewhole Bluntfist and Rime Coldspray remained to move the last stones. Cabledarm had already taken her place inside the temple. Blood seeped from her bound wounds, but she ignored it. With her comrades, she did what she could to keep the roof steady. But there were still two slabs to raise. One would have to rest entirely on the injured woman and the wall. The other she would be able to share with Cirrus Kindwind.\n\nThe gasping of the Giants sounded like anguish. They had to stand as rigid as foundations, but they could not stand straight. The finished walls around them were no higher than their shoulders. They had to lower their heads and bow their backs in order to balance the roof stones. That posture constricted their breathing. Their heavy muscles quivered on the verge of collapse. Any sudden shift might scatter them like dying leaves. Sweat streaming from their faces spattered the dirt, made marks like cries. Their staring eyes showed white like terror in the enclosed gloom.\n\nNevertheless Coldspray and Bluntfist forced the remaining stones upward. Somehow Cabledarm and Kindwind bore those added loads. Somehow they managed to turn and twist\u2014lowering one shoulder, raising another, shifting their feet incrementally\u2014so that the slabs fit where they had to be.\n\nJeremiah supervised all of this without thinking about it. He could not afford to regard the sufferings of the Giants, and nothing else required his consideration. As soon as Cabledarm and Kindwind achieved the right positions, he dashed out of the temple with the Ironhand and Halewhole Bluntfist at his back.\n\nStave waited there as if he were deaf to the desperation of the Giants. Shredded gales as fragmentary as the rocks of the construct gusted around him and away, but did not move him.\n\nFighting for breath, Coldspray and Bluntfist paused briefly; braced their trembling hands on their hips; straightened the cramps out of their backs and legs. Then Rime Coldspray nodded to Jeremiah and the _Haruchai_.\n\n\"Ready yourselves,\" she warned the other Swordmainnir as if she wanted to scream and did not have the strength. \"The end is near. One exertion remains, the last and the worst.\"\n\nAt once, she grasped Stave and wrenched him into the air. He landed on the roof as if he were as weightless as dust.\n\nThen it was Jeremiah's turn. He held his breath while Bluntfist lifted him; placed him beside Stave.\n\nWith his bare feet, he felt the ordeal of the Giants. The surface of the roof resembled strewn rubble. It shifted under him when he moved. The women were only moments from absolute exhaustion. The roof might yet cave inward. And there was one more rock\u2014\n\nIf Coldspray and Bluntfist could even raise that piece. If Stave could manage it alone in spite of his wounds.\n\nIf.\n\nEntire realities rested on one small word.\n\n\"Hang on,\" Jeremiah croaked. \"We're moving as fast as we can.\"\n\nHe was sure only of himself. The temple had been built correctly: it was exactly what it needed to be. When the capstone sat in its proper position, the whole edifice would become secure. Even rested Giants might not be able to knock it down.\n\nStone was not bone: he could not fuse it. Nevertheless there was power in shapes: the right shapes, the right materials, the right fit. The right words. The right talent. Even the right Earthpower. Such things could change the world.\n\nPraying, Jeremiah watched Stave at the edge of the roof. Coldspray and Bluntfist would have to do more than lift the last stone. They would have to hold it over their heads for the _Haruchai_. If he had to reach down for it\u2014if he could not crouch under it\u2014even his great strength would not suffice.\n\nGroaning like women whose hearts were about to burst, the Ironhand and Halewhole Bluntfist heaved. In their extremity, they half threw their burden at Stave.\n\nJeremiah did not understand how Stave caught it. He did not know why Stave's bones did not break; why Stave's muscles and heart did not rupture. The former Master was not breathing. He had no pulse. A convulsion seemed to stop his life.\n\nThe roof where he stood tilted. The stones on either side of him swayed fatally. Giants groaned in dismay.\n\nHe stayed upright, but he did not move. He looked like he could not. Every sudden thrash of wind threatened his balance.\n\nThen Coldspray and Bluntfist reentered the temple to help their comrades. Together they steadied the roof.\n\nSlowly, as if he thought that he could live forever without air or blood, Stave turned away from the edge. He took one abused step toward the hole in the center of the roof. Then he took another.\n\nAnd another, ascending the slope of the stones.\n\nStill his heart did not beat. He did not breathe.\n\nA sensation like terror gripped Jeremiah. He moved toward the _Haruchai_. He could not help Stave carry the stone, but he could guide it. As firmly as he dared, he placed his hands on the rock. By touch, he urged Stave to accommodate a subtle rotation: a shift of inches so that the rock would fit its intended seat.\n\nStave did not appear to look at his target. His eye seemed sightless. No part of him reacted to the pressure of Jeremiah's hands: no part except his feet. At his next step, he angled his failing stance slightly to match Jeremiah's wishes.\n\nWith the slowness of hindered time, one instant forced to pause for the next, he sagged to his knees. By rending increments, he extended his arms. Beyond the limits of his strength, he dropped his treasure of malachite into place.\n\nIn almost the same motion, he thrust himself away. From his knees, he fell onto his back. Soundless as a figure in a dream, he rolled down the slant of the roof, fell over the edge.\n\nThe jolt when he hit the ground restarted his heart. He began to breathe again. With a gasp that no one heard, he fought air into his lungs.\n\nJeremiah did not see him. Suddenly faint, the Chosen-son crumpled as if his own heart had stopped.\n\nut he was only unconscious for a moment. Then he jerked up his head like a swimmer who had been underwater too long.\n\nThe roof under him felt as solid as the cliff looming across the southeast. It looked like an accidental spill of stone too heavy to hang in the air; but it was not. It had become something more. Delicate strands and small deposits of malachite held the roof and the walls together as if they had become one with each other. The hidden green was now a mesh of theurgy able to withstand shocks which would have broken a house.\n\nAnd the whole edifice thrummed with power. It sent a thrill of summons along the winds, out into the twilight and the rising dark.\n\nThey had done it, the Giants and Stave and Jeremiah himself. Somehow they had vindicated Linden's faith in them.\n\nBut he did not know how many of his companions had survived.\n\nThen he did. As soon as he cast his health-sense farther, he located Stave. The _Haruchai_ lay prone in the dirt. Respiration barely lifted his chest. His heart straggled from beat to beat. Nevertheless he lived.\n\nApart from Cabledarm, the women were in no worse condition than Stave. Rime Coldspray, Cirrus Kindwind, and three others had managed to stagger out of the temple before they collapsed. Now they sprawled on the ground like invalids in the last stages of a wasting illness.\n\nFelled by their efforts, the remaining Swordmainnir lay like debris on the floor of the construct. Frostheart Grueburn and Onyx Stonemage were there. Their prostration resembled Coldspray's, and Kindwind's. Still Jeremiah could hope that they would recover. But Cabledarm's plight was more severe. She had lost too much blood. He had no idea how much longer her heart would be able to sustain its beat.\n\nYet she had succeeded. The whole company had succeeded. The construct was complete. It was exact. In some sense, it lived. That achievement counted. It may have been as costly as a defeat, but it was a victory nonetheless.\n\nJeremiah wanted to hear a song of praise. He should have sung it himself, but he did not know how.\n\nUnaware that he was hurrying, he gained his feet, went to the edge of the roof, dropped to the ground beside Stave. \"We did it,\" he told the _Haruchai_. \"You did it.\" Then he trotted around the corner to the front of the structure.\n\nThere he announced to the Swordmainnir, \"You did it. All of you _did_ it. You were _amazing_!\"\n\nThe Ironhand turned her head. She was too weak to lift it. Wan as a whisper, she asked, \"Do the _Elohim_ come?\"\n\nJeremiah looked up into a hard slap of wind, scanned his surroundings. Mottled by craters, the hardpan plain stretched away into the gloom. It looked as empty as a wasteland. Toward the east, darkness continued to swell, dimming the unnatural day, obscuring even the ravaged heavens. But full dark was still hours away.\n\n\"Don't worry about it,\" he answered Coldspray. \"They'll come. They have to.\"\n\nThey could not refuse without ceasing to be themselves.\n\n_His purpose for us is an abomination, more so than our doom in the maw of the Worm. But it is not the worst evil._\n\nInfelice believed that Lord Foul would use Jeremiah's gifts to form a prison for the Creator. _The eternal end of Creation is shadow enough to darken the heart of any being_. For that reason alone, her people had no choice. While any of them lived, they would make one last attempt to stop Jeremiah.\n\nBut the prospect did not scare him. He was looking forward to it. Infelice thought that she knew him. She was wrong.\n\n\"Then, Chosen-son,\" murmured Coldspray, \"I ask that you bring water. Cabledarm must drink. Water may ease her.\"\n\n\"Of course.\" Quickly Jeremiah looked around for the waterskins. All of the Giants needed to drink. Stave did. Jeremiah was thirsty himself. But Cabledarm\u2014\"I'll be right back.\"\n\nFortunately Kindwind's last trek to the distant spring or stream had delivered seven full waterskins: as many as she could manage. Several had been emptied, but Jeremiah found three that still bulged. With the strength of his inheritance, he carried two. One he left within Coldspray's reach. The other he took into the temple.\n\nHe did not want to look at Cabledarm. Her injuries still seeped blood, in spite of makeshift tourniquets and bandages. Her spirit had been reduced to embers. The idea that those sparks might fade twisted his heart.\n\nBut he could not both lift her head and hold the waterskin. She was too big for him, too heavy. After a moment's hesitation, he knelt beside Frostheart Grueburn, nudged her gently.\n\n\"I need you. Please. Cabledarm is dying. I've got water, but I'm not strong enough to help her drink.\"\n\nWith a strangled groan, Grueburn tried to raise her head. Her eyes opened, but at first she did not appear to see. Then her gaze focused on the waterskin. Groaning again, she flung out an arm. Her hand found the waterskin. She dragged it to her.\n\nWhile she drank, Jeremiah insisted, \"Cabledarm needs that. Did you hear me? She's dying.\"\n\nWearily Grueburn nodded. After a few swallows, she wedged her elbows under her, forced herself to rise to her knees. There she paused while she tried to remember strength or balance or at least determination.\n\n\"Chosen-son.\" Her voice was an exhausted rasp. \"Does your edifice stand?\"\n\nJeremiah was too anxious to answer. \"Cabledarm,\" he pleaded. \"Water.\" Grueburn would recognize the truth for herself when her mind cleared. \"I'll get another waterskin.\"\n\nIn a rush, he left the construct.\n\nOutside, he saw that Coldspray had managed to sit up and drink. In spite of her frailty, however, she was sparing with her own needs. Two swallows, or three: no more. Then she began to rouse her comrades.\n\nJeremiah allowed himself a quick drink from the third waterskin before he carried it into the temple.\n\nHe found Grueburn and Stonemage beside Cabledarm. Grueburn supported Cabledarm's head and shoulders while Stonemage held the waterskin to Cabledarm's mouth.\n\nGrueburn glanced up as he entered. \"Our thanks, Chosen-son,\" she said hoarsely. \"Cabledarm will perish, or she will not. In large part, the choice is hers. For the present, this must suffice.\" With a twitch of her head, Grueburn indicated the waterskin Jeremiah held. \"Succor to our comrades.\"\n\nGlad to be spared the sight of Cabledarm's peril, he turned away.\n\nIn the gloom beyond the entrance, Rime Coldspray was no longer the only Giant conscious. Halewhole Bluntfist sat nearby, rocking from side to side and holding her head. Latebirth had begun the arduous chore of prying herself out of the dirt. Stormpast Galesend was stirring. And Cirrus Kindwind was already on her feet. She had labored less than her comrades: she rallied with less difficulty. Now she was readying herself to go for more water.\n\nShe gave Jeremiah a grimace that almost became a grin. \"We live, Chosen-son. And we have accomplished our purpose. I have said that I honor effort and intent. Now I also honor their outcome. Few in life are given such gifts.\"\n\nThen she nodded in Stave's direction. \"How fares Stave Rockbrother?\"\n\nBefore Jeremiah could reply, he heard a sound in the wind.\n\nHe was expecting the chimes that announced the sovereign of the _Elohim_ , waiting for it: the crystalline clear ringing of small bells, lovely and delicate. Instead he heard a sharp clatter like the ruin of gongs; like a welter of huge iron crashing down. It was not loud. Indeed, it seemed imponderably distant, as if it had reached him from the far side of the world. Yet its tone and timbre were unmistakable. They spoke of shattering and calamity and irreparable loss.\n\nHe tried to call a warning to the Giants inside the temple, but the words stuck in his throat.\n\nInstinctively he believed that Infelice was coming to prevent _the worst evil_. To kill him before he could be reclaimed by the Despiser.\n\nIf so, none of his companions would be able to defend him. No Giant could stand against any one of Infelice's people. And the Swordmainnir were too weak to don their armor or swing their swords. Stave was not even conscious.\n\nBut Jeremiah did not flinch. He knew that Infelice was wrong about him. She would see the truth when she arrived.\n\nForgetting Stave and Kindwind and water, he went to stand at the entrance to his temple as if he had become its guardian.\n\nThe metallic clamor continued. It acquired intensity and ire. It was as sharp as knives forged to flense and flay. In spite of the distance, it cut. And it was coming closer. The Giants heard it now. The Ironhand and Bluntfist struggled upright, stood wavering with their fists clenched. Latebirth was at Galesend's side, rousing her comrade. Cirrus Kindwind moved to join her.\n\nAs Jeremiah reached the entryway, Grueburn and Stonemage emerged, supporting Cabledarm between them. They bore her a few paces to one side, lowered her carefully to the dirt. Then they stood over her as though they meant to fight for her.\n\nBut the wrath and repudiation of the _Elohim_ would not be directed at her, or at any of Jeremiah's companions.\n\nHe folded his arms across his chest; across the fouled blue and horses of his pajamas. He did not know how else to contain his trembling.\n\nRime Coldspray took a position on his left. Halewhole Bluntfist matched her on his right. Together they waited.\n\nHe expected to see forces gathering in the eastern darkness, anger as fierce as lightning, an army of eldritch beings. The shredded winds seemed to promise multitudes and violence. But when Infelice came, she came alone. And she did not arrive from any direction. Instead she incarnated herself in front of him like a star plucked out of the heavens. She was no more than five steps away.\n\nInvoluntarily he blinked. Her brightness stung his eyes. She was clad in light: an elegant profusion of gemstones\u2014emeralds and rubies, sapphires and garnets\u2014all shining with their own radiance, all arrayed like garments woven of glory. Only the iron clangor and desperation of her bells contradicted her deliberate loveliness, her stubborn will to believe that she was the crown of Creation.\n\nAcross the plain behind her, the wind fashioned illusions of movement in the hollows, illusions that made the ground look like it was squirming.\n\nHer vehemence seemed to appall the dusk. It buffeted Jeremiah's bones. Now he saw that her many jewels resembled tears, incandescent woe. Her wrath was weeping. The suzerain _Elohim_ 's form and raiment articulated fury indistinguishable from grief.\n\n\" _Abomination!_ \" she cried. \"Malign child! Thus you complete our despair! Better for us to be devoured by the Worm. Better had you never been given birth.\n\n\"I am able to decline entrance only because I am Infelice. I cannot continue to do so. My people have not come only because I prevent them. I cannot continue to do so. Soon we must accept eternal absence and futility, eternal continuance in a void in which we can do nothing, and from which we cannot return.\n\n\"This evil you have performed, though I have both striven and pleaded to avert it. In your heedlessness, you are a-Jeroth's servant, and all of your deeds conduce to his designs.\"\n\nColdspray and Bluntfist glowered uselessly. Farther away, Stormpast Galesend tottered to her feet between Latebirth and Kindwind. Grueburn and Stonemage knelt like shields on either side of Cabledarm.\n\nJeremiah should have been terrified. On some level, he was. Infelice had not given rise to the darkness mounting in the east. Her ire and lamentation had not caused the turmoil of winds. Something else was coming\u2014\n\nNevertheless his fears only made his hands tremble, only caused his heart to stutter. His crossed arms closed a door on that part of himself. Behind his fa\u00e7ade, memories of the _croyel_ barked in derision. Outwardly he faced Infelice as if he could not be daunted.\n\nIn spite of her supernal powers, she did not know him. He was exactly what she believed him to be. At the same time, he was something entirely different.\n\nHe raised his halfhand as if he expected her to respect it; to recognize that it did not resemble Covenant's. \"You're wrong,\" he said in a fevered voice. \"You don't know what you're talking about.\n\n\"Your people are dying. You need to get them here.\" Then he gestured behind him. \"But first you need to _look_.\" He wanted to shout in the _Elohim_ 's face. \"You've been wrong about me all along.\"\n\n\"Do you think to mislead me, boy?\" Infelice retorted imperiously. \"Do you believe that _I_ may be deceived?\"\n\nNonetheless she glanced past him.\n\nThen she stared. Confusion made chaos of her clangor and radiance. Her apparel thrashed around her like the storms of desire and misery which had haunted Esmer. Her visage modulated: it seemed to become scores of different faces in quick succession, as if all of her people were suddenly manifested in her. As if the entire meaning of their existence had been called into question.\n\nAn instant later, the clatter of falling metal ceased. Every wind dropped. Silence closed like a lid over the plain. The gems of Infelice's raiment corrected themselves, resumed their accustomed grace. When she spoke again, her voice was little more than a whisper.\n\n\"It is not a gaol. It is a fane.\"\n\nLike an antiphony, her bells chimed relief. They implied awe.\n\n\"That's right!\" Jeremiah crowed. Vindication rose in him. It felt like scorn for the ways in which the _Elohim_ had misjudged him. \"You have to go in, but you can come out whenever you want. If you want. If I were you, I would stay inside. Let the rest of us worry about the Worm. As long as you're in there, it can't reach you.\"\n\nFor a moment or two, Infelice looked so lovely that every aspect of her seemed to sing: every line of her face and form, every implication of her demeanor, every glad jewel. She was lucent with melody. But then she appeared to recall herself from a vision of hope. It had almost seduced her. Now she returned, unwilling, to the implications of her plight.\n\nFrowning, angry again, and strangely uncertain, she said as if she were asking a question, \"Yet the Worm will destroy the fane. Though we will not be consumed, we will be denied our place in life. That you cannot prevent.\n\n\"You have wrought a surpassing wonder. I acknowledge it. I acknowledge that we have misesteemed you. And your theurgy is\u2014\" Bells described her astonishment. \"Child, it is vast. My strengths are many, yet I cannot unmake what you have formed. Against any threat other than the Worm, this fane would stand.\n\n\"But you do not comprehend the Worm's power. It _transcends_. Sensing our presence, the Worm will devour the fane without thought or effort. Then it will continue its search for the EarthBlood and doom. Deprived of egress, we will be eternally lost.\"\n\n\"Mom is working on that,\" Jeremiah replied without hesitation. \"Sure, what we've done is vulnerable.\" Roger had smashed Jeremiah's Tinkertoy castle with the ease of contempt. \"And we don't have enough power to stop the Worm. But Mom went looking for somebody who can teach her how to do what we need.\n\n\"As long as she gets back\u2014\"\n\n\"Madness!\" Infelice cried at once. \"Utter madness!\" Apparently her fears had blinded her to other things. Preoccupied by carnage, she had focused on Jeremiah rather than Linden. Now she reached for arcane sources of knowledge. Revelations struck her like blows. \"The Wildwielder hazards the world's past. She seeks a Forestal forged from the substance of an _Elohim_. She seeks _forbidding_.\n\n\"It is madness.\" Infelice seemed to be speaking to herself. Arguing with her own instincts. \"Should she fail, she will destroy all Time and life ere the Worm achieves its culmination.\" But then her attention focused on Jeremiah again. Softer hues flowed through her raiment. \"Yet I see valor also in her, as we have from the first. Therefore we sought to forestall her darkest desires, and to serve her in defiance of her own wishes. Should she succeed\u2014\"\n\n\"That's right,\" Jeremiah said again. \"You'll still have a chance. You'll be safe, at least until the Worm gets to the EarthBlood. And it'll be slow. I mean, slower than if it ate you. We'll have more time.\"\n\nTime for Linden or even Covenant to come up with a better answer.\n\n\"A worthy effort,\" murmured the Ironhand, \"regardless of its hazards.\"\n\nThe other Giants remained silent.\n\nInfelice appeared to consider Jeremiah's assertion. Instead of contradicting or challenging him, she consulted the ineffable ramifications of her bells and Linden's daring and his construct.\n\nHe bit his lip; tried not to hold his breath. He had done what he could. If Infelice turned her thoughts now to what she had called _the worst evil_ , nothing that he had done\u2014nothing that he might say\u2014would satisfy her.\n\nAbrupt gusts broke free around the _Elohim_. Winds like the discarded scraps of a hurricane, tattered and imminent, gusted at the Giants, the fane, Jeremiah. Fretted with new grit, they rebounded from the ridge. The plain blurred and ran like a landscape in a mirage. Driven air did not touch Infelice, but it pulled like thorns at Jeremiah's pajamas, moaned in the gaps between the stones of his construct.\n\nIt was possible that Jeremiah had built hope for everyone else, and had left none for himself.\n\nFinally Infelice looked at him again. For the first time, he heard regret in her voice.\n\n\"You have exceeded our conceptions of you. This I confess freely, though it humbles me. Yet one threat remains unaddressed. Your companions have named you Chosen-son. I do so also. Yet you are chosen of a-Jeroth as you are of the Wildwielder. I have spoken of his desire to accomplish absolute evil. Chiefly for that reason, he has endeavored to possess you. He will do so again.\n\n\"You have completed your fane.\" The music of her bells became sharper. It cut against the winds. \"Your part in the world's doom is done. For the Earth's sake, and for Creation's, I must now slay you.\"\n\nHer words shocked the Giants. They hit Jeremiah hard even though he had expected them. He had no defense.\n\n\"I am loath to do so,\" admitted Infelice. \"Yet I cannot otherwise forestall a-Jeroth. The Worm will feed, or it will not. The Arch of Time will fall, or it will not. Still the Despiser will make use of your gifts. From your heart and passion and youth and weakness, he will devise imprisonment for the Creator. He will put an end to the very possibility of Creation. Only your death will prevent his eternal triumph.\"\n\nJeremiah stared at her; said nothing. Simply standing his ground required everything within him, his most intense love and his bitterest darkness.\n\nHe had inherited too much from Anele.\n\nBut Cirrus Kindwind rose to her feet. She spoke for him. With gems reflecting in her eyes, she said, \"You forget, _Elohim_ , though you are the highest of your kind. The Chosen-son is not alone.\"\n\n\"He is not,\" Rime Coldspray affirmed. She sounded as hard as a fist. \"Doubtless you discount his companions. And in this you are perchance correct. Our striving in your name has weakened us. We cannot oppose you.\" In spite of her weariness, her voice hit and tore as if its knuckles were studded with spurs. \"Nor do I name the Timewarden, whose deeds and purposes remain unknown to us. But having misesteemed young Jeremiah, will you now compound your error? Have you forgotten that Linden Avery, Giantfriend and Wildwielder, has proven herself capable of much? Have you forgotten that there is hope in contradiction?\n\n\"No. I will not credit it. You are _Elohim_. You do not forget. Yet one matter lies beyond your comprehension. Being who you are, you have no experience of it. Therefore I will say _this_ in the teeth of all who meditate ill toward the Chosen-son. He has _friends_. The Despiser may well attempt to possess him. If so, that evil will fail. No possession can hold one who does not stand alone.\"\n\nShe seemed to mean, One who is loved.\n\n\"Why otherwise,\" she concluded, punching home her avowal, \"is he now free of the monster which once ruled him? Doubtless foes who relied upon the _croyel_ were certain of their designs. Yet here he stands, relieved from mastery, and dedicated to the preservation of beings who abhorred him.\"\n\nConflicting responses appeared to twist Infelice's mien. Her raiment fluttered in disarray. At first, Jeremiah thought that she had taken offense; that she would react with wrath and violence. But then he saw her more clearly.\n\nThe sovereign _Elohim_ was diminished. Her assurance, her contentment in herself, had received a blow from which she did not know how to recover. The notion of _friends_ perplexed her; undermined her. Winds gyred around her like relief and dismay: a conundrum which she appeared unable to resolve.\n\nBut she did not hesitate long. Pressures that surpassed Jeremiah compelled her to a decision. Her voice wore discordant chiming like a funeral wreath. Though she was the highest of her kind, she had been wrong too often.\n\n\"I can delay no longer. I must acknowledge that I am answered, as the summons must be answered. You have spoken truly. We are _Elohim_. We have no knowledge of _friends_.\n\n\"This, then, is my word. Come what may, we who are great must now place our faith in you who are small.\"\n\nThen she found a brief severity. \"Be wary, Chosen-son. Your deeds bring perils which you do not foresee. We have given of our utmost, according to our W\u00fcrd. Now we can do naught. If your companions fail you, you are undone.\"\n\nTurning away, Infelice lifted a cry into the heavens: a resounding clang like a hammer-stroke on an immense gong.\n\nAt once, other _Elohim_ began to appear as if they had been brought by the winds; as if they had found their substance among the oneiric seethings that troubled the plain.\n\nOne after another, they flowed like liquid light toward the fane, so many of them that Jeremiah was astonished. He had seen stars dying: he had not considered the number that still lived. Perhaps the relationship between these beings and stars was more symbolic than literal. Nevertheless the heavens had not been entirely decimated. Those _Elohim_ that answered the call of Jeremiah's construct resembled a multitude.\n\nThe sight enchanted him. They were so beautiful\u2014! One and all, they were lovely beyond description. To his human eyes, they were men and women clad in elegance, and accustomed to glory: innocent of mortality; untainted by the dross of inadequacy and the burden of suffering; immune to the woes and protests that could only be stilled by death.\n\nThey were the _Elohim_ , eldritch and fey: as cryptic as prophecies in a foreign tongue, and as ineffable as the beauties of Andelain, or the melodies of Wraiths. An uncounted host of them had already perished: a throng remained, craving life.\n\nThey sanctified the unnatural twilight as if their coming were a sacrament.\n\nInstinctively Stormpast Galesend and Latebirth forced themselves to their feet. Even Cabledarm found the strength to stand. All of the Giants endeavored to square their shoulders, straighten their backs. In spite of their troubled history with the _Elohim_ , they set aside their exhaustion.\n\nGraceful as willows, stately as Gilden, each faery individual paused only to exchange a nod with Infelice, who stood aside for her people. Each glided into the fane and vanished from sight. And Jeremiah watched them stream past like a boy who had become magnificent in his own estimation, full of pride. He had caused this: _he_. He had justified Linden's highest hopes for him. Yet the swelling of his heart was not pride. At that moment, at least, it was gratitude. The success of his temple was not something that he had accomplished: it was a gift that he had been given. He did not waste himself on pride.\n\nFor that moment, while it lasted, he soared above his secrets as if he had been lifted into the heavens.\n\nExalted and transfixed, he could not brace himself against the convulsion that shook the ground like the onset of an earthquake. He had no answer for the blast of heat as fierce as an eruption of magma, or for the blare of savagery that seemed to repudiate the world. He did not understand the sudden cries of the _Elohim_ , or the haunted look that filled Infelice's eyes, or the frantic shouts of the Giants. He did not know what was happening until Kastenessen entered him, and all of his thoughts became anguish and slaughter.\n\nEcstatic agony. Rage so great that it could not be contained. Pain too extreme to be called insanity.\n\nThe mad _Elohim_ struck the plain like a fireball flung by a titan. At the impact, the very ground under his feet seemed to ripple and clench like water, liquefied by ferocity. He came roaring with triumph and lunacy and hate: a monster who no longer resembled the people who had imprisoned him; damned him. He was not lovely, not graceful. His visage was a contortion of suffering. Interminable pains gnarled his limbs. His vestments were fire. His eyes blazed like the fangs of the _skurj_. From his kraken teeth, slaver splashed the dirt and smoldered. And he dominated the horizon; cast back the gloom until even the darkness in the east appeared to wither and fade. He had made himself taller than a Giant, as tall as one of the avid worms which he had once restrained.\n\nHis right fist he held above his head, ready to hurl ruin at the fane.\n\nIt was not an _Elohim_ 's fist. It was Roger's, human and fatal. With it, Kastenessen could deliver devastations that no other being of his race might attempt or condone.\n\nBut he did not strike. He was not ready\u2014or he saw no need.\n\nHe had already taken Jeremiah, who stood on bare dirt. The boy had inherited this vulnerability from Anele.\n\nIn an instant, less than an instant, a particle of time infinitely prolonged, Jeremiah passed through the eager malice and sadism of the _croyel_ into pure fire, the catastrophic frenzy of bonfires. During that interminable flicker, his spirit was split. He seemed to become several separate selves, all simultaneous or superimposed, all cruelly distinct.\n\nNow he knew why Anele had chosen madness.\n\nOne Jeremiah realized that he had been possessed\u2014again!\u2014and tried to scream. One stood in the white core of a furnace, while another interpreted every form of pain as delight, as agony perfected to ecstasy. One watched the Giants, who should have scattered, saved themselves. But they did not. Doomed and determined, they placed themselves in the path of Kastenessen's savagery. And another Jeremiah relished the knowledge that he had become incarnate lava. The idea that his companions were about to die glorified him. It was for this that the Despiser had marked him. It was for this that he lived.\n\nSwift with glee, he moved to do his ruler's bidding.\n\nStill another self remembered every horror which the _croyel_ had inflicted upon him. He experienced again the misery of deluding Linden in Roger's company, cringed at what he had done under _Melenkurion_ Skyweir. Another aspect of his shredded identity fled for the safety of sepulchres. Another gibbered for the godhood of eternity. In that manifestation, he knew the keen pain of the _krill_ against his throat.\n\nAnd one\u2014\n\nOne of the many Jeremiahs _understood_.\n\nThis Jeremiah recognized the extremity of Kastenessen's need for ruin. He remembered the forbidden love, potent as delirium, and altogether delicious, which had drawn Kastenessen to mortal Emereau Vrai, daughter of kings. He felt Kastenessen's rage and dismay while he fought for his love against Infelice and others of the _Elohim_ , who should have valued him more highly. This Jeremiah knew intimately the unconscionable hurt of Kastenessen's Durance, his imprisonment against and among the _skurj_. This Jeremiah recalled in every detail the torment which had driven Kastenessen to begin merging himself with monsters.\n\nThis Jeremiah understood why Kastenessen cared only for the utter destruction of the _Elohim_. More, he knew why Kastenessen had not acted directly against Linden, or indeed against Jeremiah himself, until now; until all of his surviving people were gathered in one place. Although Kastenessen had used Esmer with remorseless brutality, he had not delivered his fury in person because any absence from the proximity of She Who Must Not Be Named would have put an end to Kevin's Dirt. His presence was required to channel and shape and direct the bane's fearsome energies. And he had believed, or _moksha_ Raver had persuaded him, that only the dire brume which hampered Earthpower and Law would make his revenge possible.\n\nNow Kastenessen had no more need for such stratagems. He had come in response to the fane's call, but he was not mastered by it. He was part _skurj_ and part human: he was in enough pain to refuse any coercion. No, he was here because he had achieved his desires. One of the Jeremiahs would carry out the last preparations.\n\nThat in turn was why Kastenessen raised Roger's fist, but did not strike. He had the power to shatter the fane, render it back to rubble. Nevertheless he withheld his blow, waiting for the certainty that every one of the _Elohim_ would be destroyed.\n\nNothing that happened in or to Jeremiah took any time at all. Part of him regretted that. He loved what he had become. He reveled in the purity of his given hate.\n\nIncandescent or incinerated in each of his separated selves, he flung himself at Infelice.\n\nIt was for this that he\u2014that Kastenessen\u2014had planned and waited and endured: so that the highest and mightiest and most dangerous of the _Elohim_ would be slain with the rest when he delivered his retribution.\n\nThree swift strides would be enough. Then Kastenessen in Jeremiah would wrap hate like molten stone around Infelice. He would hurl her through the fane's portal, the entryway to extinction. After that, only heartbeats would remain until the summons was complete; until every _Elohim_ was inside.\n\nUntil Kastenessen could unleash uncounted millennia of torment.\n\nJeremiah was sudden. He was quick.\n\nStave was faster.\n\nThe former Master was scarcely conscious. He could barely stand. Nevertheless he kept his promise to Linden. Lunging, he grasped Jeremiah's arm.\n\nHeat as fierce as brimstone savaged his hand, but he did not let go. Desperate and already failing, he delivered Jeremiah to the only protection that lay within his reach.\n\nAs Rime Coldspray had done to Stave himself earlier, the _Haruchai_ wrenched Jeremiah into the air. Off the bare dirt that exposed him to Kastenessen. Onto the stone roof of the temple.\n\nInto the direct line of Kastenessen's intended attack.\n\nThen Stave collapsed again. He did not rise.\n\nBut Infelice remained untouched outside the fane.\n\nKastenessen howled rage at the heavens, but Jeremiah no longer heeded him. As Jeremiah's feet left the ground, he crashed inwardly. His many selves seemed to smash against each other like projectiles, like bullets.\n\nThe force of their impact stunned him. It numbed his mind. He no longer thought or moved: he hardly breathed. Instead he lay still, wracked by revulsion; as weak as Stave. He could do nothing except watch and dread.\n\nKastenessen roared, but he did not strike. He wanted his full triumph. In moments, even Infelice would answer the fane's call. Then\u2014\n\nAlready the last of the _Elohim_ were passing inward. Their hope had become horror, and their features were written with dismay, but they had no power to reject their own natures. Two heartbeats, or perhaps three, no more than that, and Infelice would stand alone. Then she, too, would enter\u2014and Kastenessen would strike.\n\nNo, he would not. Not with Roger's hand. Never again.\n\nWhile Kastenessen readied his blast, a Giant surged out of a crater behind him. Jeremiah would not have known who the newcomer was if Frostheart Grueburn had not shouted, \"Longwrath!\"\n\nSwift as a bolt of lightning, the man reared high behind the deranged _Elohim_. In both fists, he gripped a long flamberge with a wicked blade. It edges gleamed against Kastenessen's lurid radiance as if starlight had been forged into its iron.\n\nOne stroke severed Roger's hand from Kastenessen's wrist.\n\nKastenessen screamed like an exploding sun. He staggered.\n\nLongwrath followed him to strike again.\n\nBut Kastenessen caught his balance. Blood pulsed from his wrist, the tainted ichor of Earthpower and lava. He did not heed it. Wheeling, he swung at his attacker with his good arm.\n\nPower erupted in Longwrath's chest. His armor had been damaged, torn apart at one shoulder: it could not withstand Kastenessen's virulence. The wrought stone sprang apart, spitting splinters as piercing as knives. But the shards evaporated or melted at the touch of Kastenessen's lava. Longwrath was flung backward, hurled away like a handful of scree. When he fell, he did not move again. Smoke gusted out of his chest as if his heart and lungs were on fire.\n\nRoaring once more, Kastenessen turned back to Infelice and the fane. Obscene heat mounted within him. He grew taller, blazed brighter. Acrid flames swirled higher, spinning about him like the birth-pangs of a cyclone. His sick brilliance stung Jeremiah's eyes, but the boy could not look away.\n\n\"Hear me, treacher!\" the mad _Elohim_ howled. \"I am more than you deem! Yon puerile fane cannot compel me! Still am I Kastenessen! Still my pain suffices to destroy you!\"\n\nRaving, he stoked his lethal energies, Earthpower and magma, _Elohim_ and _skurj_ , until they looked fierce enough to consume every life that had ever walked the plain. They were far more than he needed them to be. They would level Jeremiah's crude edifice as if it had no substance and no meaning.\n\nInfelice had been appalled earlier. Now, strangely, she was calm. She did not answer Kastenessen. Instead she remarked to Rime Coldspray, \"You think ill of us, Giant, and you have cause. But we are not as dark as you deem. For this also we laid our _geas_ upon your kinsman. For this also he acquired his blade. Failing one purpose, he has served another.\n\n\"He has not redeemed us, but he has weakened our lost brother. Now comes one who may achieve our salvation, however briefly. We cannot ask more of any who oppose the Worm.\n\n\"You will forgive your kinsman's passing,\" she added sadly. \"Alive, he would not lightly bear the recall of his deeds.\"\n\nThen the bedizened _Elohim_ faced Kastenessen across the gulf that separated their thoughts and desires, hers and his.\n\n\"I have heard you, doomed one.\" She did not raise her voice, yet it rang out, clarion and clear. \"Now you will hear me. Cease your striving. Enter among your people. Permit your hurt to be assuaged. We have dealt cruelly with you, but we are also kind. While life endures to us, we will provide a surcease from all that you have suffered.\"\n\nShe may have been telling the truth.\n\nNow comes one\u2014\n\nBut Kastenessen had spent long ages in his Durance. He had made choices which exacerbated his fury. Infelice's appeal could not reach him. For him, it may have been the final affront.\n\nHe gathered flames until they burst from his eyes and his mouth, from every limb and line of his towering form. He was becoming a holocaust, devastation personified: a bonfire high and hot enough to ravage the plain. His reply was one word:\n\n\" _Never!_ \"\n\nYet he was not given time to release his accumulated hate.\n\nFrom the northeast, a burst of extravagant argent opened the twilight. It cast back the darkness, dismissed the sunless gloom. It was as bright as Kastenessen, and as complex, but immeasurably cleaner. And it was brief, little more than a blink. Nevertheless it was long enough.\n\nOut of it came riding Thomas Covenant and Branl _Haruchai_ of the Humbled. Covenant held Loric's _krill_.\n\nThe shock of their arrival snatched Kastenessen away from his victims.\n\nCovenant rode a shovel-headed horse as ungainly and muscular as a mule. Branl was mounted on a Ranyhyn that Jeremiah had never seen before. And they were in a desperate hurry. Froth snorted from the nostrils of Covenant's horse, the muzzle of Branl's palomino stallion. Sweat reflected brimstone on their coats. They looked like they had galloped for leagues or days. Covenant lurched in his seat as if he were falling.\n\nAs soon as his mount's hooves struck the dirt, he pitched from his saddle. But he did not sprawl. Staggering like a holed ship in a storm, he managed to stay on his feet. Awkward and urgent, he confronted Kastenessen as if he had forgotten that the _Elohim_ could reduce his bones to ash.\n\nIn his maimed hands, the gem of the _krill_ shone like a kept promise in an abandoned world.\n\n\"You\u2014!\" Kastenessen began: a strangled howl. Rage clenched his throat, choked off his protest.\n\n\" _Try_ me,\" Covenant panted as if he were on the verge of prostration. \"Do your worst.\" He looked too weak to withstand a slap. Streaked by conflicting illuminations, his face had the pallor of a wasting disease. Still he was Thomas Covenant. He did not falter. \"See what happens.\n\n\"I killed my ex-wife. I helped destroy a Raver. And I've seen the Worm of the World's End. I am _done with restraint_!\" His teeth gnashed. \"I used to care how much you've suffered. I don't anymore. If you think you can beat me, go ahead. I'm _wild magic_ , you crazy bastard. I'll cut you apart where you stand.\"\n\nJeremiah stared and stared, and could not name his astonishment, when Kastenessen flinched\u2014\n\n\u2014and took an alarmed step backward.\n\nCovenant advanced, holding up the _krill_. It blazed like havoc, unmitigated and unanswerable. Its argent covered him with majesty. The silver of his hair resembled a crown.\n\nBranl came behind him, but did not intrude.\n\nKastenessen retreated another step, and another. Another. The passion in Covenant's eyes drove him. He must have realized that he was being forced toward Infelice and the fane; but he did not stop. Perhaps he could not. Perhaps he saw something in Covenant, or in Loric's numinous dagger, that cowed him.\n\nWith every step, he dwindled. Retreating, he became smaller. Lava seemed to leak out of him and fade, denatured like water by his own thwarted heat.\n\nCovenant stumbled and wavered, and kept coming. Kastenessen shrank away from him.\n\nGiants let him pass. They watched as if they were as stricken as Jeremiah; as transfixed.\n\nThen Infelice spoke Kastenessen's name like a command, and Kastenessen turned from Covenant to face her.\n\nTerror and loathing contorted his features. He conveyed the impression that he wanted to scream and could not because he feared that he might sob. Through his teeth, he spat words like fragments of torment.\n\n\"You have earned my abhorrence.\"\n\nInfelice's calm had become irrefusable. Placid as Glimmermere, she answered, \"We have. We will not ask you to set it aside. We ask only that you allow us to soothe your pain.\"\n\nHer response appeared to horrify him. \"It is what I am.\"\n\n\"It is not,\" she countered, undismayed. \"When it is gone, you will remember that you and you alone among the _Elohim_ have both loved and been loved.\"\n\nTo that assertion, he had no reply.\n\nShe did not repeat her invitation. Instead she reached out one hand to clasp his severed wrist. With chiming and mercy, she stanched his bleeding. If the pollution of the _skurj_ within him caused her any hurt, she accepted it.\n\nHis eyes bled anguish. He made no attempt to pull away.\n\nBriefly Infelice glanced at the Giants, at the Ironhand. \"Be warned,\" she told them. \" _Moksha_ Jehannum now rules the _skurj_. He will wield them with cunning and malice. And do not forget that the Chosen-son is precious to a-Jeroth.\"\n\nThen she surrendered at last to the imperative of Jeremiah's construct. Drawing Kastenessen with her, she entered the fane. In an instant, they were gone as if they had stepped out of the world altogether.\n\n\"Damnation,\" Covenant gasped. \"I wasn't sure I could do that.\"\n\nLowering his arms as if he had been beaten, he tried to approach the Swordmainnir. But his legs failed, and he dropped to his knees.\n\nOverhead Kevin's Dirt had already begun to dissipate. If more stars perished, they did so beyond the horizons. Jeremiah did not see them die.\n\n## 10.\n\nBut While I Can\n\nAs if they were each entirely alone, Linden Avery and Manethrall Mahrtiir rode through hell to save or damn the Earth.\n\nThey did not exist for each other. They were mounted on Ranyhyn that did not exist. Immersed in a cyclone of rent instants, they were consumed by the kind of hiving that drove men and women mad. Every nerve was stung beyond endurance, assailed by bitter particles of reality. At the same time, every perception had become white ice, gelid as the gulfs between the stars. Linden and her companion inhabited a frozen wilderland eternally unrelieved in all directions. They had entered a realm in which excruciation defined them. It was all they knew because it was all that they had ever known. It was all that they would ever know. One moment did not lead to the next, and so there was nothing to see or do or understand.\n\nIn that perfection of agony, Linden may once have imagined that she and Mahrtiir would be defended by experience. They had endured _caesures_ twice before, and had survived. Surely they would be sustained by the knowledge that what they were trying to do was possible? But she was wrong. Memory was meaningless in a place that contained all time and none simultaneously. One instant, _this_ instant, was the whole truth of who and what they were.\n\nYet it was not the whole truth of their plight. The _caesure_ imposed other dimensions of torment as well, other forms of futility. She had asked the Ranyhyn to take her and Mahrtiir backward in time, against the current of the Fall's wild rush; and that effort had consequences. While hornets burrowed into her flesh, and she occupied a bitter wasteland as if it were the summation of all her needs and desires, she also floated inside herself like a spectator, helpless amid the chaos, watching her own desecration as if she were dissociated from it.\n\nDays and days ago, she had once hung suspended like this inside Joan's mind, observing ruin through Joan's eyes because she had entered a _caesure_ of Joan's making. But now Linden was the cause of her own suffering. While other tortures failed to tear her apart only because their duration had no meaning, she also bore witness to herself.\n\nShe watched the Linden Avery who had always been inadequate to what her life required of her. The Linden who had allowed herself to be misled by Roger Covenant and the _croyel_. The Linden who had defied every Law by resurrecting Thomas Covenant, _compelled by rage_ \u2014and had nonetheless failed to resurrect him whole. The Linden who had been consumed by She Who Must Not Be Named, and had not sufficed to raise her precious son from his graves.\n\nThe Linden Avery who had roused the Worm of the World's End.\n\nBut there was more. Observing, she was able to recall things which the storm of time denied.\n\n_There is no doom so black or deep that c_ _ourage and clear sight may not find another truth beyond it_.\n\nCovenant had told her that. In the aspect of her anguish that resembled a shadow cast by her own flawed self, she yearned to believe him.\n\n_Trust yourself._\n\nOh, she ached for the ability to believe. But he had also said, _Don't touch me_ , as if he feared that her love would corrupt some essential part of him. She did not know how to trust herself. She was the daughter of her parents, a mother and father who had feared every hurt of living, and had raised her for death. That knowledge endured in her bones. A Raver had confirmed it. Unforgotten and unredeemed, it ruled her even now, in spite of Covenant and Jeremiah and the Land.\n\n_In your present state, Chosen, Desecration lies ahead of you. It does not cr_ _owd at your back_.\n\nIt was here. Was it not?\n\nBut because she was watching herself as if she were someone else, she was able to recognize that there were other ways to think. Her many friends had been trying to teach her that lesson ever since Liand had first introduced himself in Mithil Stonedown. By their devotion, they had assured her that she did not need to judge herself as if she were defined by her sins. In spite of her concealments and dishonesties, her fury _contemptuous of consequence_ , she was not alone.\n\nIf _courage and clear sight_ exceeded her, they did not surpass her companions. From the first, she had been supported by people whose hearts were bigger than hers; by loyalties more unselfish than hers. _Every essential step along the path_ , Stave had assured Infelice, _has been taken by the natural inhabitants of the Earth_. Linden's friends had urged _trust_ until even she had heard them.\n\nTrapped in the savagery of the _caesure_ , she found that desperation was indistinguishable from faith.\n\n_Attempts must be made_ \u2014\n\nHyn had carried her willingly into the Fall. Mahrtiir on Narunal had accompanied her willingly. She could believe in them.\n\n_\u2014even when there can be no hope._\n\nAnd she had done some things right. Witnessing herself with the detachment of a spectator, she could acknowledge those deeds. She had fought her way through the machinations of Roger and the _croyel_. She had provided for her son's rescue from the _croyel_ 's covert in the Lost Deep. And when every other action had been denied to her, she had given Jeremiah his racecar: the last piece of the portal which had enabled him to step out of his prison.\n\nIn those moments, no one else could have taken her place. To that extent, Anele had told the truth about her, as he had about so many things. _The world will not see her like again_.\n\nAnd there was more.\n\nNothing ameliorated the extravagant burrow and sting of dismembered moments. Nothing eased the cruelty of the frigid wasteland which would arise from Desecrations like hers. Nothing could. Nevertheless she still held Covenant's wedding band clasped in her hands. Silver fire still shone from the metal even though she was not a rightful wielder of white gold. It was as vivid to her as Covenant himself. It could be an anchor for her foundering spirit.\n\nThen she was no longer alone. She had always and never been alone. Manethrall Mahrtiir was at her side, holding the Staff of Law for her and looking ahead as if he had nothing to fear; as if he had finally identified the import of his life.\n\nAnd she was seated on Hyn's back, as she had always been. Narunal was at her side. The horses were not moving. Movement required causality: it depended on sequence. Yet they ran. Stride for stride, dappled Hyn matched Narunal's strength, Narunal's certainty, as the palomino stallion raced from nowhere to nowhere across the white wilderness.\n\nIn spite of the _caesure_ 's excoriation, Linden clung to Covenant's ring and endured.\n\nShe did not have to wait long. She had been waiting forever, and did not have to wait at all. This moment did not move on to the next because it could not, or because there was no _next_. Nevertheless the hard circle between her hands flared suddenly; and Hyn carried her out of chaos into sunshine under a summer sky.\n\nSunshine. A slow hillside clad in brittle grey-green grass as thick as bracken. A summer sky as lenitive as hurtloam.\n\nWithout transition, Linden was released.\n\nThe shock of change made her muscles spasm, made the world reel. Her stomach hurt as if she needed to spend hours puking. Blots of black confusion wheeled around her as though she were under assault by crows or vultures. The continuity of her personal world had been severed from itself. Unable to determine her position in time and space, she tumbled from Hyn's back, landed hard on the grass.\n\nFor a moment, she could not breathe; could not think. While her nerves floundered, she clung to the kind earth and wrestled with her impulse to vomit. She had arrived somewhere. Some when. Hyn had brought her here. She smelled summer in the air, felt an insistence on life in the stiff grass in spite of a prolonged paucity of rain. Straining to inhale, she caught a whiff of distant desiccation, as if she had arrived too close to a desert. The sky held too much dust. She had expected Andelain and lushness. She was unprepared for this baked hillside, this heat, this\u2014\n\nSomething had gone wrong.\n\n\"Ringthane,\" Mahrtiir croaked as if he were retching. \"Release the white gold. You must. Accept your Staff.\"\n\nShe heard him, but the words did not make sense. He sounded like an ur-vile, barking incomprehensibly. Something had gone wrong. The world was wrong: the grass, the sky, the sunshine. Only the writhing ruin of the Fall as it drifted away felt familiar. Narunal trumpeted a warning that she did not know how to interpret. Alarm fretted Hyn's answering whinny.\n\n\"Chosen!\" insisted the Manethrall. \"Linden Avery! Your _Staff_. You must quench the _caesure_! If it enters among the trees, it will wreak harm which no Forestal will pardon. We will not be heeded if you do not first spare the forest!\"\n\nLinden recognized a few sounds. The sigh of an arid breeze. The consternation of birds somewhere in the distance. A few words.\n\nWhen she remembered to let go of Covenant's ring, she began to breathe again.\n\nMahrtiir stumbled to her side. Roughly he rolled her onto her back. \" _Ringthane!_ \" Crouched against a glare of sunlight, he dropped the Staff of Law onto her chest. Then he fumbled at the dried remains of his garland, pinched off one of the last nubs of an _amanibhavam_ bloom. Scrubbing the nub between his palms to powder it, he slapped one hand to his nose, clamped the other over Linden's nose and mouth.\n\nToo many sensations. _Amanibhavam_ stung her sinuses as if she had inhaled acid. She had no time to notice that it dispelled her nausea. The sunshine wore a faint patina of dust. Shadows blurred Mahrtiir's visage.\n\nThen Earthpower flowed into her from the black shaft of the Staff; and she thought, Trees? A Forestal?\n\nOh, God.\n\nYou must quench the _caesure_!\n\n_Caesures_ destroyed stone. They would tear any forest to shreds. Even a forest defended by a Forestal\u2014\n\nWhere _was_ she?\n\nMahrtiir knew Andelain. Surely he would have called that woodland by name?\n\nReflexively she clutched the Staff. Then she heaved herself into a sitting position; staggered to her feet.\n\nThe Fall was already thirty paces away, forty. And it was big, as virulent as a tornado; a rip in the fabric of reality. Seething, it lurched toward a scatter of trees: Gilden, ash, sycamores, thirsty willows. They stood alone and in loose copses, punctuating the browning grass like the out-riders of an army in retreat. Like the grass, they looked parched, stricken by a persistent lack of rain, a dwindling watershed. She could not see past them to the forest itself, but she knew instantly that the forest was there. It seemed to glower in the distance, defying an inexorable drought.\n\nThe _caesure_ savaged the ground as it moved. It was going to plow a furrow of devastation into the heart of the woods.\n\n\" _Melenkurion abatha_ ,\" she gasped as if she were cursing. The burn of _amanibhavam_ sent flames like tendrils along the channels of her brain. \" _Duroc minas mill_.\" She felt as blighted as the trees, wan with thirst.\n\nWhere am I?\n\nWhat have I done?\n\n\" _Harad khabaal._ \"\n\nOne fire led to another; enabled another. As if she were turning her mind inside out, she drew ebon conflagration from the Staff and flung it like outrage into the core of the Fall.\n\nEarthpower and Law, the salvific antitheses of the time-storm. Her flames were as stark as fuligin, as black as the immedicable gulf of a night sky after every star had been devoured. But the darkness was hers: it was not inherent to the Staff's magicks. And here\u2014wherever _here_ might be\u2014she was not hampered by Kevin's Dirt. Riding the invocative force of the Seven Words, she hit the _caesure_ with a deluge of extinction as if she were pouring a lake onto an inferno.\n\nThe Fall could not withstand her. As she had done before, she caused the violent miasma to implode. With a sound like thunder, the _caesure_ swallowed itself as if it sought to suck her with it into nothingness. Then it was gone.\n\nIts passage had galled the earth\u2014a bitter wound\u2014but the nearest trees had not been touched.\n\n\"Mane and Tail, Ringthane!\" breathed Mahrtiir. Already he sounded steadier, stronger. Even withered, _amanibhavam_ retained the potency to restore him. \"That was well done. Another moment, and our quest would have failed. No tale of Forestals told among the Ramen makes mention of forbearance. They do not countenance the ravage of their woods.\"\n\nTrembling, Linden extinguished her flames. Well done? she wondered. Really? Mahrtiir was right, of course. She could not expect any Forestal to grant her desires after she had damaged his trees. But now she had no idea how she and her companion would return to their proper time. Too tired to think clearly, she had assumed that she would use the same _caesure_. An impossible idea. It had brought her to the brink of a terrible mistake. Yet the result was that she and Mahrtiir were trapped.\n\nShe could not imagine an escape that did not require another _caesure_ , another Desecration. And she could not guess what would happen if she violated Time in this era. Thousands of years separated her from Jeremiah, Stave, and the Giants; from any conceivable reconciliation with Thomas Covenant. The Law of Life had not yet been violated. A time-storm created here might consume every possible future.\n\nShe bit her lower lip in an effort to control herself; but she could not stop trembling. These trees did not belong to Andelain: she was sure now. The forest beyond them was too dark, too angry. And she knew of no time in which the heart of the Land had been gripped by a drought like this. Clearly Hyn and Narunal had ignored her desire to reach Caer-Caveral. So when _was_ she?\n\nWhy did the mood of the woodland seem familiar?\n\nNagged by the same concerns, Mahrtiir continued, \"Yet I confess that I am troubled. Was it not your intent to seek out the Forestal Caer-Caveral? That I conveyed to great Narunal. But this is not Andelain. Plainly it is not. Rather we have come to a place and time unknown to me.\" He muttered a Ramen curse. \"I cannot account for it. I am certain only that the gifts of the Ranyhyn are unerring. They have turned aside from your wishes for some good purpose.\"\n\nLinden nodded in bafflement. The presence of the great horses nearby offered an oblique solace. Still she could not keep the tremor from her voice as she asked, \"Do you recognize anything? Anything at all? Can you guess where we are?\"\n\nThe Manethrall scowled above his bandage. \"In these straits, Ringthane, blindness hinders me, though I am not constrained by Kevin's Dirt. I am able to assure you only that I have never stood in this region of the Land.\" After a moment of hesitation, he added, \"Among the Ramen, however, there are tales\u2014\"\n\nHis voice trailed away. Before Linden could prod him, however, he asked, \"The trees lie to the north, yes?\"\n\nShe nodded automatically, trusting his awareness of her.\n\n\"Are there hills in the east?\" he continued. \"Do they mount toward mountains?\"\n\nShe looked in that direction, summoned Earthpower to increase the range of her senses. \"If I'm not mistaken.\"\n\n\"And in the west? Do mountains also arise there?\"\n\nSquinting into the distance, she murmured, \"I think so.\"\n\nMahrtiir's manner became sharper. \"One question more, Ringthane. Does a waste extend at our backs? I perceive barrenness. Does it spread to the horizon and beyond?\"\n\n\"As far as I can tell. It looks like the edge of a desert.\"\n\nThe Manethrall stood taller, straightened his shoulders as if he had found himself in the presence of majesty. \"Then I must surmise,\" he announced so that the trees and even the wide sky might hear him, \"that we stand in the gap of Cravenhaw. Before us lies dire Garroting Deep. Narunal and Hyn have delivered us, not to Caer-Caveral, but to Caerroil Wildwood. If you would speak with him, Ringthane, we must dare his demesne.\"\n\nHe sounded almost eager.\n\nBut his words gave Linden a jolt. Details came together, formed connections.\n\nCaerroil Wildwood. Garroting Deep.\n\nNo wonder the darkness seemed familiar.\n\nShe had encountered Caerroil Wildwood when Roger and the _croyel_ had stranded her deep in the Land's past. At that point in his long life, the Forestal's puissance was undiminished. He had given her gifts: her life as well as runes for her Staff. In some sense, he had made possible Covenant's resurrection. And he had charged her with a question.\n\n_How may life endure in the Land, if the Forestals fail and perish, as they must, and naught remains to ward its most vulnerable treasures? We were formed to stand as guardians in the Creator's stead. Must it transpire that beauty and truth shall pass utterly when we are gone?_\n\nHe had understood that she had no reply. Nevertheless he had spared her. He had seen something in her, _the mark of fecundity and long grass_. _And the sigil of the Land's need has been placed upon her_. What sigil? For all she knew, then or now, he had referred to the bullet hole in her shirt. Or to the healed wound in her hand where the _croyel_ had stabbed her. Still she had felt compelled to promise an answer.\n\nBut long ago\u2014millennia later in the Land's life, a decade earlier in Linden's\u2014when she and Covenant had first met Caer-Caveral in Andelain\u2014when the guardian who had once been Hile Troy had sacrificed himself against the Law of Life\u2014he had been the last Forestal. By that time, Caerroil Wildwood had passed away. All of the Upper Land's ancient forests had been destroyed by the Sunbane.\n\n\"My God, Mahrtiir.\" Inferences linked themselves into language as rapidly as she could speak. \"This must be hundreds of years after I met Caerroil Wildwood,\" after the Mahdoubt had saved her. \"It must be before the Clave. Before the Sunbane.\"\n\nThe kingdom against which Berek Halfhand had waged his war was not a desert.\n\nHyn and Narunal had known what they were doing. Here she was in no danger of confronting the Forestal before her first encounter with him.\n\n\"That is well,\" averred Mahrtiir. \"Alas, to their shame the Ramen have no tales of events in the Land after the onset of the Sunbane. I must believe that Caerroil Wildwood perished striving against that abomination.\"\n\n\"No,\" Linden said at once. Covenant had learned the truth from the Clave. He had told her. \"He passed earlier. Before the Sunbane.\"\n\nThe Manethrall's surprise was plain. \"Then how was he brought to his end?\"\n\nShe bit her lip again. \"I'm not sure. It had something to do with the destruction of the first Staff of Law.\" Then she hurried on. \"But at least now we know _when_ we are\u2014approximately, anyway. Caerroil Wildwood is still alive. He may recognize me. If he doesn't, he'll recognize his runes. We have a chance.\"\n\nShe had intended to address her appeal to Caer-Caveral; but she saw now that Hyn and Narunal understood her needs, and the Land's, better than she did. What had she expected of Andelain's Forestal? Had she truly imagined that meeting her before her proper time would not affect his later decisions?\n\nIn _this_ time, here and now, she was in no danger of burdening Caer-Caveral with knowledge which he had not earned. The Ranyhyn had spared her a potentially catastrophic miscalculation.\n\n\"Thus,\" the Manethrall observed proudly, \"Hyn and Narunal vindicate their wisdom once again.\" Then he admitted, \"Yet queries remain. Are you able to summon the Forestal? To attract his notice is both perilous and necessary. And will he heed your desires? If the tales are sooth, Caerroil Wildwood will not grant a kindly hearing. Even in the days of dark Grimmerdhore, Garroting Deep was deemed the most wrathful of the forests. You have encountered this Forestal and lived\u2014aye, and were given ambiguous boons. Do you conclude therefore that he will bestow the knowledge you seek, though you wish to preserve a world in which he does not exist? Neither he nor any Forestal?\n\n\"Ringthane, if you are able to gain his heed, how will you sway him?\"\n\n_How may life_ _endure in the Land_ \u2014?\n\nLinden watched birds soar like questions among the trees. Beyond them, Garroting Deep brooded over its innumerable wounds and grievances, its savage hungers. Winds from the mountains which walled Cravenhaw on both sides did nothing to soften the heat swelling from the south. Already sweat gathered at the corners of her eyes. Dampness trickled down her spine.\n\n\"I don't know,\" she admitted. \"Caerroil Wildwood gave me his runes for a reason, and I don't think that it was just because I needed help. It had something to do with the question he asked me. He knew that I couldn't answer it, but he wanted an answer anyway. Maybe his runes were part of the question. Or he hoped that they might be part of the answer.\"\n\nMahrtiir considered her for a moment. Then he nodded with an air of renewed anticipation. \"For us, then, only the simpler query remains. How will you attract the Forestal's notice? He is said to be an imperious being, mighty and impatient withal, having good cause to loathe humankind. Also Garroting Deep is vast. He may be many leagues distant, unaware of our approach. Or he may be unwilling to acknowledge beings whose like have butchered trees beyond all counting.\"\n\nLinden frowned, shook her head. \"Hyn and Narunal got his attention for us. They brought us so close\u2014I almost let that _caesure_ hurt his forest. I don't care how far away he is. He must have felt that kind of violence. He'll come, even if all he wants to do is to kill us.\"\n\nThe Manethrall nodded again. \"Indeed, Ringthane. I cannot gainsay you. Thus my query becomes, how will you forestall his ire? Our tales assure us that the Forestals were mighty beyond comprehension. How otherwise did their puissance suffice to forbid the Ravers from the Upper Land?\"\n\nShe shrugged. \"I'm not exactly helpless myself.\" She had different concerns: fears that baffled her. _How may life endure in the Land\u2014?_ They were laden with doom. _Just be sure you come back_. \"But I won't fight him.\" She, too, loved trees. And she had not forgotten the lessons of Gallows Howe. \"I won't have to.\" Again she said, \"He'll recognize his runes.\"\n\nFor another moment, Mahrtiir scrutinized her as if he sought to gauge her resolve. Then he nodded once more. \"As you say, Ringthane. As ever, the deeds of the great horses conduce to hope. I grasp now that there is a fitness to your purpose. My own desires are thereby justified. Come good or ill, boon or bane, I will regret no moment of our quest.\"\n\n\"All right,\" Linden murmured. Gradually her attention shifted away from her companion. \"Then all we have to do right now is wait. And try not to go crazy.\"\n\nWhat choice did she have? She did not want to think about Jeremiah; about people and loves that she had left behind and might never see again. In one respect, her presence in a time where she did not belong was no different than any other crisis. In fact, it was no different than ordinary life. The only way out was forward. While Time endured, there was no going back.\n\nGripping her Staff for courage, she tried to put everything else out of her mind. At her side, Mahrtiir folded his arms across his chest like a man who knew how to contain his impatience. In the service of the Ranyhyn, he had learned the discipline of setting himself aside. He knew how to accommodate his frustration.\n\nHer former world had taught Linden similar skills. She had acquired a professional detachment in medical school and emergency rooms and Berenford Memorial. But she had lost that resource, or had left it behind with Jeremiah and the Giants, Stave and Covenant. She did not know how to stop fretting. Instead she gnawed on her fears as if she hungered for them; as if at the marrow she would find sustenance.\n\nShe needed her son and Covenant. She had to do what she could to keep them alive in spite of the intervening millennia.\n\n\"Oh, hell,\" she muttered abruptly. \"Who am I kidding? I can't just wait around. Let's at least get closer.\"\n\nHolding her Staff ready, she moved toward the nearest trees.\n\nTheir suffering without sufficient water was palpable. The willows in particular ached with distress, and the grass crackled under her boots. As the ground sloped down behind her toward the wasteland, moisture was wicked away from the woods. Apparently Caerroil Wildwood's music could no longer protect the outlying trees from the effects of the diminished watershed. Even if Garroting Deep faced no other perils, it was under assault by the perpetual drought in the south.\n\nTensely Linden crossed through stippled shade to bypass the forest's first fringes. Unaware that she was holding her breath, she approached the ragged edge of the Deep. At her shoulder, Mahrtiir matched her pace. A short distance away, Hyn whickered softly, and Narunal stamped his hooves; but the Ranyhyn did not follow.\n\nLinden passed a stunted copse, then a magisterial Gilden with leaves like scraps of clawed fabric, an oak mottled with brown stains like blights. The low rustle of breezes among their branches seemed to sound her name until she reached a stretch of open ground like a clearing. With enough water, it might have been a glade surrounded by verdure and consolation. Here it was simply earth that nourished little more than grass. Nevertheless the grass was healthier than it was beyond the trees.\n\nIn the center of the clearing, she stopped. Surely the Forestal was close? Surely he had felt her presence? But he was needed everywhere in Garroting Deep. He might turn away when he saw that the danger of the _caesure_ had passed. And she could only call out to him in one language: the speech of fire.\n\nUnsure of herself, she turned to Mahrtiir.\n\nThe Manethrall considered the forest for a moment. Then he offered gruffly, \"Here I am reminded of a tale concerning Lord Mhoram at a time when the forces of the Lords were threatened by an army commanded by a Raver. It is the same tale which relates the doom of Hile Troy. Risking much, the Lord approached Garroting Deep from Cravenhaw and raised fire in supplication. But he also dared to speak words of power, words which belonged to the Forestal. Therefore Caerroil Wildwood came.\n\n\"But those words were not repeated to the Ramen. The tale is known to us only because it was shared by Bannor and others after the _Haruchai_ had withdrawn from their service as the Bloodguard. They are a reticent people, as you know\"\u2014Mahrtiir sounded grimly amused\u2014\"and did not tell the tale fully.\"\n\n\"I wonder what they were,\" Linden mused absently. Her ears strained for hints of Caerroil Wildwood's singing: the poignant and feral melody of the Forestal's strength.\n\nA _caesure_ should have been inconceivable in this time. How could any lover of trees ignore such a threat?\n\nApparently Caerroil Wildwood could not. When Mahrtiir had been silent for a while, and Linden's trepidation seemed ready to burst out of her chest, she heard the first notes of a song that rent her heart.\n\nIt seemed to arch from tree to tree as if it were setting every leaf alight. Its power was unmistakable, a force as fraught as wild magic. But its potential ferocity was muted, held in abeyance: perhaps because its full might was not needed to rid the Deep of two mere humans; or perhaps because the Forestal was curious in spite of his unrelieved wrath; or perhaps because he recognized\u2014\n\nBreezes and tuned ire made the woods appear to waver like a mirage. Scraps of song more audible to percipience than to ordinary hearing spread like ripples. Boughs and roots added notes which should have been discordant, but which instead wove a dire counterpoint through the lamentation of leaves, the grief and objurgation of sap. Words that almost formed verses skirled through the grass at Linden's feet: _days before the Earth_ , and _its walk to doom_ , and _forbidding dusty waste_.\n\nWearing an aura of embattled music, Caerroil Wildwood appeared far back among the trees.\n\nHis steps were instances of a dirge, ancient and unreconciled; irreconcilable. Lucent melody rather than light cloaked him in lordship. A penumbra of sorrow etched with gall and despair surrounded him as he advanced; and he seemed to waft rather than walk, as if he were carried along by the chords of his puissance. He was as tall as a king; his flowing hair and beard and his long robe were white with antiquity; the silver authority of his eyes judged all things harshly. He commanded homage from the trees as he passed, but it was an obeisance of appreciation and reverence, not of servility. The service here was his: the forest did not serve him. In the crook of one elbow, he cradled a gnarled wooden scepter as if it were the symbol and manifestation of every trunk and shrub and seed-born wonder that he had ever loved. Loved and lost.\n\nAt her first glimpse of him, Linden bowed her head. Carefully she displayed the Staff of Law in front of her, lying like an offering across her hands.\n\nPlease, she breathed in silence. Just look. Don't decide anything until you look. I abandoned my son for this. If you don't help me, I've abandoned the whole world.\n\nBeside her, Manethrall Mahrtiir stood straighter. He held up his head as if he wanted to emphasize his bandage, the ruined sockets of his eyes; wanted the Forestal to see that he was unafraid in spite of his blindness.\n\nAt the edge of the clearing, Caerroil Wildwood stopped. He did not deign to come closer. He did not speak. His tune in its myriad voices spoke for him.\n\n_My leaves grow green and seedlings bloom._\n\n_I inhale all expiring breath,_\n\n_And breathe out life to bind and heal._\n\n_My hate knows neither rest nor weal._\n\nLinden ached to sing with him. If she could have replied with melody, he would have known that she had no wish for harm: not here; not in any forest. But she did not know the rites and cadences of his lore. Even her own magicks were mysteries to her. She could not address the Forestal in his natural tongue.\n\nStill she had to try. Singing, he seemed to become higher and mightier, exalted by the inadequacy of her silence.\n\n\"Great One\u2014\" she began. But then her littleness caught in her throat, and she faltered.\n\n\"Ringthane,\" Mahrtiir urged privately, \"you must. It is as you have said. He is aware of his doom. There lies the true heart of his torment. With every leaf and sprout of his realm, he cries out in bitterness and supplication. I hear now that he cannot decline to heed you. Some hope he must have. Is it not for this that he has clung to his devoir? Is it not for this that we have come, to proffer hope? Or, if not hope itself, then our striving in the name of hope?\"\n\nLinden feared the guardian of the Deep. Oh, she feared him! _From border to border, my demesne thirsts for the recompense of blood_. Now more than ever, that was true. He had known for millennia that he could not prevail over heedlessness and malice. His trees were too vulnerable\u2014\n\nVulnerable and precious.\n\nYet his insights when she had met him before had surpassed her comprehension. They might do so again.\n\nShe made another attempt. \"Great One. You know me.\"\n\nShe wanted to raise her voice even though she had no music to match Caerroil Wildwood's. But she could not be peremptory in his presence. She had to speak softly.\n\n\"You gave me a gift.\" She insisted on her Staff as if it were a pledge. \"And you asked me a question that I couldn't answer. I need your help.\"\n\nVexation spread through the trees. \"What is that to me?\" the Forestal countered: raw tatters of sound that seemed to arise from the woods at his back. \"In a bygone age, my heart was wrath. I was avid for bloodshed, and my ire suffused every leaf and twig and branch and trunk and root of my demesne. Yet now I recall that time as a halcyon era. Though I knew myself and all forests doomed, I remained capable of much, potent for both killing and nurturance in the name of trees and green. As you foretold, I feasted on the flesh of a Raver. But the years have become an age of the Earth, and the time of my power has passed. My strength withers in my veins. I cannot restore it.\n\n\"Do you ask my aid? I have none to give. My every effort is required to slow the ruin of all that I have held dear.\"\n\nHe fell silent, although his music went on weeping.\n\n\"You're right,\" Linden replied, forcing herself. Such honesty was difficult for her, but she had no other response. \"You've always been doomed. But soon it's going to get worse. Much worse.\" She meant the Sunbane. \"The Clave is going to create an evil like nothing that the Land has ever seen before, and it won't stop even when it has destroyed every last fragment of the One Forest. Eventually even Caer-Caveral will be gone. He'll reach the end of himself and let go.\n\n\"But long ago I told you that you would have a chance to make a Raver suffer. Now I tell you that the coming evil _will_ be stopped. White gold and Law and love will cast it out. A new forest will grow,\" Salva Gildenbourne, \"and it will be vast. The world will keep on turning, Great One.\n\n\"But that's not the end of what I know. Eventually there will be new evils. Worse evils. That's why I'm here. I can't offer you hope. I have to ask for it. I need _forbidding_. I need to know how to _forbid_. Otherwise my time will be as doomed as yours. Where I come from, the world won't keep on turning. The evil has gone too far. Nothing except forbidding can save it.\"\n\n\"What is that to me?\" the august figure asked again. \"The end of my days crowds close around me. I cannot forbid the waning of my own strength. My trees must perish. What will you forbid that I have not already failed to prevent? Soon or late, all things come to dust. I have no other purpose than sorrow.\"\n\n\"Now, Ringthane,\" Mahrtiir breathed like the breeze. \"You are acquainted with despair. Harken to his, and he will heed you. His song speaks to my heart. In this, he and I are one.\"\n\nLinden winced. She understood the Manethrall. She feared that she understood him too well. She had been given hints enough. But she could not afford to falter in her purpose.\n\n\"Great one, look at me,\" she implored. \"Look at my Staff.\" _This blackness is lamentable_ \u2014\"Look at your runes. You know what they mean. You gave them to me long ago, but even then you saw what was coming. You could already feel the hopelessness that eats at you now. You were so angry then because you were fighting your own futility. That's why you asked me a question.\" _How may life endure\u2014?_ \"Now I've come back. You've been waiting for millennia, and I'm finally here. Let me try to answer you.\"\n\nLet me tell you why I need you.\n\nThe Forestal did not reply at once. For a time that strained Linden's nerves, he sang to himself as if he were considering her plea, or her death; debating the many cruelties of his plight. Small winds carried plaints through the struggling grass.\n\nWhen he spoke at last, his melody sounded sharp enough to draw blood.\n\n\"Then come, human woman.\" He gestured imperiously with his scepter. \"Bring your companion if you must. If you would dare my scrutiny, you must stand upon Gallows Howe. In the presence of my doom and denunciation, you will speak. There you will live or be slain.\"\n\nBefore Linden could give her assent, Mahrtiir's voice rang out, pealing against the chime of the Wildwood's music. \"What of proud Narunal, Great One? What of Hyn, loyal and loving? They are Ranyhyn, as revered as trees. Without them, we are lost.\"\n\nFor the third time, Caerroil Wildwood demanded, \"What is that to me?\"\n\nThen Linden no longer stood on open ground. Ripples altered the surface of her senses, and she drifted among the trees where the Forestal had waited. The Forestal himself was gone: his song remained. It summoned her like a _geas_ from the depths of the dark Deep.\n\nWhile she staggered within herself, the fringes of the forest wavered like disturbed water. She stepped without transition onto a thin track like a path for deer wending crookedly among monarchs thick with age. She had no sensation of movement, but she had already come far. Aching branches swathed her in shadows defined by sunlight falling cleanly between the leaves. Caerroil Wildwood was drawing her toward the heart of his demesne. Doing so, he seemed to call her in the direction of her own past, and the Land's. With every stride, she crossed decades and leagues as if they were seamless, woven together by the fecund mutter of music from a thousand thousand voices.\n\nVaguely she was aware that Manethrall Mahrtiir walked at her side, passing farther and farther into Garroting Deep and time. He did not speak, and she did not. Like her, he appeared ensorcelled by the counterpoint of the Forestal's ubiquitous song.\n\nTogether she and her companion traveled among changes in the terrain: hills and streams; low stone buttresses grey with age and clad in moss; complex trails like a web of welcome for the animals that enriched the woodland. Variations among the trees themselves measured the progression of leagues: stands of new growth interrupted by the magisterial contemplation of ripe oaks; thickets crowded with orchids and _aliantha_ ; vibrant tracts of aspen and cottonwood on higher ground in the west, draped cypress and willow in lowlands and swales to the east. If the sun moved at all in the distant heavens, Linden did not notice it.\n\nHow far had they come? How far were Gallows Howe and the Black River from Cravenhaw? She did not know\u2014and could not care. While Caerroil Wildwood's trance carried her, compelled her, she only took one step after another, and filled her lungs with the woodland's wealth of scents, and marveled that so much largesse had withstood the depredations of centuries and humankind and malevolence.\n\nThe woods seemed as timeless as the chaos of a _caesure_ ; but Garroting Deep was not Desecration. In spite of its enduring bitterness, its galls of woe and ire, it had been formed for peace. The lost nature of the One Forest was irenic, as rapt as _Elohim_ , and as self-absorbed. Perhaps that similarity, that kinship, explained the willingness of the _Elohim_ to take action for the preservation of woodlands. The result was an anodyne for travail even when the trees were stiff with outrage. If Linden had ever been afraid, or desperate, or appalled, she had forgotten it; or the Forestal's music warded her from herself.\n\nFor her, the way was not long. Denser shade cut out more and more sunlight. Darker trees gathered gloom beneath their branches: their roots fed on shadows. Vines like hawsers tangled the underbrush on both sides of the trail where she and Mahrtiir walked. Swaths of leaves looked as black as dying blood except when brief glimpses of the sun revealed their true green. After unremarked moments or hours, she found herself approaching the barren slope of Gallows Howe.\n\nThe hill seemed higher than she remembered it: higher and more cruel, as if it had absorbed a terrible increase of savagery from the killing of a Raver's physical form, the destruction of a fragment of the Illearth Stone. The very dirt radiated hunger, thirst, desire, as though every clod and pebble craved the taste of blood; of slaughter enough to drench the soul of the forest. Here the Deep had no language for its bereavement except rage. Utterly dead, the Howe piled darkness upward as if it were impervious to sunshine; as if no light from the heavens could touch it. And near the crest arose the two dead trunks which supported the Forestal's gibbet.\n\nFrom the branches of the crossbar hung two nooses, ready and willing.\n\nBeside his gibbet, Caerroil Wildwood stood with his arms folded over his scepter as if he had been waiting for an age of the Earth. Around his neck he wore a garland braided from the stems and blooms of accusations. The song he spread around him had once been a dirge, but it had become as harsh and heavy as drumbeats announcing judgment.\n\nHis presence stopped Linden and Mahrtiir at the foot of the slope.\n\n\"Ringthane,\" the Manethrall breathed, suddenly aghast. \"This place\u2014Mane and Tail! Tales name it, but no Raman has beheld it. It is the Forestal's heart. It cannot be answered.\"\n\n\"I know,\" Linden said hoarsely. The labor of her pulse seemed to clog her throat. \"But he has a right to it. I felt like that, and all I lost was my son. What he's suffered is worse.\n\n\"The _Elohim_ made this possible.\" One of them had planted the seeds of power and lore which had germinated to become Forestals. \"But they don't die, so they don't grieve. They had no idea what his life would be like.\"\n\nA handful of Forestals had not sufficed to save the woods. As benign as Gilden and oaks, Caerroil Wildwood and his kind had been slow to recognize hate and heedlessness. They had taken too long to learn anger, too long to summon their strength. As a result, they had been forced to watch millions of living things in their care perish.\n\n\"But you said it yourself,\" Linden went on. \"He'll hear us anyway. He needs something to hope for.\"\n\nShe had to believe that.\n\nTouching Mahrtiir's shoulder, she urged him to join her as she began her ascent of Gallows Howe.\n\nHe may have faltered for a moment\u2014but only for a moment. Then he found his resolve, and his features seemed to become sharper. _You'll have to go a long way to find your heart's desire_. With his chin jutting, he moved upward at Linden's side.\n\nDeath accumulated under her boots at every step. The dirt heeded no appeal and would never be appeased: it had lost too much. On this hill in another era, she had found the granite rage which had carried or driven her from her battle with Roger and the _croyel_ to Thomas Covenant's resurrection. She understood the Howe's ire in the deepest channels of her heart.\n\nAs she climbed the hill's accreted hunger, however, she recognized other emotions as well. Listening with her nerves, her health-sense, she heard more. The passion of Gallows Howe was for revenge, retribution: the ground burned to repay its ancient pain. But that trenchant yearning arose from a foundation of unannealed bereavement. Trees beyond counting had been destroyed before the woods had awakened to anger. Grief came first. Without woe and protest, there would have been no wrath.\n\nThen inchoate perceptions which had tugged at the edges of her thoughts for days shifted toward clarity, and she heard still more. In spite of their avid bitterness, the songs sung by Caerroil Wildwood beside his gibbet were more complex than they appeared to be. First came grief. Yes. It led inexorably to rage. But it did so only because a different need had been denied. Between the underlying loss and the accumulated gall lay a yearning of another kind altogether: a vast, sorrowing, stymied desire, not for revenge, but for _restitution_. The forests, and the emblem of Gallows Howe, would not have grown so dark if they had not first failed to reclaim what they had lost. If the Forestals had not failed at restitution, they would not have succumbed to ire and viciousness.\n\nThat unbidden insight humbled Linden. It daunted her when she could not afford to flinch or turn away. It had too many personal implications.\n\nShe, too, had gone from loss to rage when her first efforts to find her son had failed. Nevertheless Jeremiah had been restored to her. Even though she had refused to forgive.\n\nShe could not ask Caerroil Wildwood to pardon the foes and forces which had ravaged his demesne. His vehemence was necessary. It was just. It was\u2014\n\nBut the Forestal did not wait for her to sort through her confusions. His music demanded more of her. While she and Mahrtiir neared his gibbet, the tall figure commanded, \"Speak, then.\" Melodies sawed across her hearing as if they sought to cut away subterfuge and falseness. \"I am done with forbearance.\"\n\nBecause she did not know what else to do, Linden lifted her Staff once more.\n\n\"Great One.\"\n\nEvery word required an effort of will. Inwardly she slogged through a quagmire. How could she ask Caerroil Wildwood for anything? He was doing what she would have done in his place; what she had already done. Nevertheless she made the attempt for Jeremiah's sake, and for the Land's, and for Covenant's\u2014and perhaps even for her own.\n\n\"A long time ago, you asked me a question. I think that I can answer it now. Or a piece of it, anyway.\n\n\"That isn't why I came. I've tried to imagine an answer for you ever since you spared my life, but I couldn't think of one. I wasn't even trying to reach you. But now that I am here, I see things differently.\n\n\"Great One, I need your help. If I'm right, that's your answer. You can help me.\" _Must it transpire that beauty and truth shall pass utterly when we are gone?_ \"And you're the only one who can. If you don't, beauty and truth will be just the first casualties.\" She meant the _Elohim_. \"Eventually the whole world is going to die.\"\n\nThe Forestal studied her. In a voice as low as a hum, and as piercing as an auger, he commanded her again. \"Set aside your blackness. I well recall the craving which inspired me to carve my will upon it. It has no virtue to preserve you.\"\n\nHe may have been asking for a show of good faith.\n\nAs if the written wood had burned her hands, Linden dropped her Staff.\n\nCaerroil Wildwood allowed the limbs of his trees a brief flourish of approval. But he did not dwell on it. Still stringently, he sang, \"You acknowledge that it was not your intent to seek me out. To that extent, I discern sooth. Now you will speak further. Do you ask me to credit that the desires of one human, or of two, or of a myriad myriad, suffice to determine the doom of the Earth? Justify your need, woman. Sway me or perish.\"\n\nLinden shook her head. \"You already know the truth.\" She had come too far to hold back. And she understood the peril of revealing things which might affect the Wildwood's role in the Land's history. \"You've known it for a long time. If you didn't, you wouldn't have given me your runes.\"\n\nHe would not have prevented Hile Troy from accepting Covenant's ring.\n\n\"You asked me how life can go on without Forestals. It can't. Thousands of years from now, it won't. Evil doesn't die. It doesn't stop. And where I come from, it's finally found a way to end everything. Unless you teach me how to _forbid_ it.\"\n\n_There must be forbidding._\n\n\"If I can do that, maybe I can save something.\" _Without forbidding, there is too little time_. \"I'll start with the _Elohim_ ,\" what was left of them. \"If that works, I'll do more. With my Staff and white gold and what you know, I'll stop as much evil as I can.\" _If it is not opposed by the forgotten truths of stone and wood_ \u2014\"But I can't do anything without the power to say _no_.\n\n\"You blocked the Ravers from the Upper Land. I need to learn how you did that. I need to be able to do the same thing. If you don't teach me, you might as well give up.\" From her perspective, his surrender had already happened. Something had driven Caerroil Wildwood to abandon his devoir long before she had first entered the Land with Covenant. \"There won't be any hope for any of us.\n\n\"Do you need a future for trees, Great One? This is your only chance. Without your help, I'm as lost as you are.\"\n\nSilver flared in the Forestal's eyes. It limned his gibbet and the surrounding trees, gave them a spectral cast as if they were etched with presentiments of ruin. The music of the woods became a threnody, forlorn and irredeemable. Leaves rose and sank like sobbing on their twigs. Song fell like tears on all sides.\n\n\"Then you are lost indeed. You speak words which you deem sooth. That I acknowledge. And you have striven to satisfy my query. That, also, I acknowledge. Therefore I will not require your heart's blood to repay the hurt inflicted by your plea.\n\n\"But I cannot grant your desire. You are human and ignorant, incognizant of deeper truths. You do not grasp that the forbidding which you seek is not lore. It is neither knowledge nor skill. It is essence. It is both my nature and my task. I cannot impart it.\"\n\nHis response was as simple as a sigh\u2014and as fatal as an earthquake. Linden staggered as if the Howe itself had shuddered under her; refused her. Cannot? She wanted to wail in chagrin. You _cannot_? After what she had done?\n\nBut he was Caerroil Wildwood, the Forestal of Garroting Deep. His grandeur and grief silenced her protests. Instead of yelling, she floundered for arguments.\n\n\"Then how did you make your Interdict against the Ravers?\" Her voice trembled on the verge of breaking. \"If you can't impart what you are, how did the Colossus of the Fall keep Ravers away from the Upper Land?\"\n\nAbruptly Mahrtiir took a step forward. Like the woods around the Howe, like the gallows, he looked sharp with intensity, whetted by the Forestal's shining. His eyeless visage seemed to yearn. His hands were ready for his garrote; for some demand that required death. But the Wildwood did not regard him.\n\n\"By transformation,\" the Forestal told Linden severely. \"By the alteration of essence. There is no other means. The _Elohim_ who became the Colossus of the Fall ceased to be who she was. She was made stone and could not unmake herself. Therefore her refusal endured. It did not wane until the forests dwindled, too grievously diminished to sustain her.\"\n\nWhile Linden cried out inwardly, unable to articulate her sudden despair, Mahrtiir took another step forward; upward. Then he stopped, holding himself poised as if for battle.\n\n\"Great One,\" he said, insisting on Caerroil Wildwood's attention. \"The world of our time requires forbidding\u2014and the forest of our time requires a Forestal to wield that stricture.\" His tone defied contradiction. \"You have it within you to create another of your kind, as you did with Caer-Caveral. Do so again with me. Make of me a Forestal for the woodland which will arise when its time has come, and for the preservation of the world. Permit me to carry on your labors. Share with me your mighty purpose, for I have none of my own, except to stand with those who shed their lives for the Land's sake.\"\n\nHe was certainly as blind as Hile Troy had been: as blind and as valiant. Like Troy, he had already chosen his doom.\n\nLinden tried to object; but the Manethrall's willingness and the Wildwood's singing closed her throat. _No_ , she pleaded, _no_ , but her voice made no sound, or no one heard her.\n\nThe guardian of the trees was going to refuse. Of course he was. His own fate was sealed, whatever happened. She had offered him only abstractions, vague predictions empty of substance. He had no reason to care about a world that did not exist for him.\n\nYet he seemed to stand taller, towering over the Manethrall. The multiplicity of songs around him acquired a new tune that vied against the woods' immedicable sorrow and ire. He raised his scepter. From its gnarled length sprouted notes woven to form harmonies which Linden had never heard before.\n\n\"That gift,\" he pronounced as if it were a sentence of death, \"is mine to grant.\"\n\nOh, Mahrtiir\u2014\n\nWere _all_ of her friends going to sacrifice themselves?\n\nIf she had snatched up her Staff, she might have been able to intervene. She could at least have made the effort. _And betimes some wonder is wrought to redeem us_. But she did not move. Perhaps she could not. Or perhaps she simply understood.\n\n_I seek a tale which will remain\u2014_\n\nStill she had to say something. \"Mahrtiir\u2014\"\n\n\"No, Ringthane,\" the Manethrall replied at once. \"You are the Chosen, but this choice is mine.\" He knew her too well. \"In this, Anele spoke wisely, as he did on other matters. To you, he said, 'All who live share the Land's plight. Its cost will be borne by all who live. This you cannot alter. In the attempt, you may achieve only ruin.'\" Then he gave her a fierce grin. \"It is done. The Forestal of Garroting Deep has heard me. His heart and his pain are great. He will not refuse.\"\n\n\"Indeed,\" Caerroil Wildwood hummed in harmony with his trees. \"I do not recant my gifts.\"\n\nA sharp skirl of music seemed to snatch her Staff from the dirt, carry it toward him. Holding his scepter in one hand, he caught the Staff in the other. The lines of his runes began to burn like his eyes, silver and severe.\n\n\"Yet I am grievously diminished. My strength falters. Therefore I will make use of your blackness to sustain me, as I have written that I must.\"\n\nWith the Staff and his scepter held high, he brandished gleaming like certainty over the Howe. \"Harken now. Hear my answer to your need.\"\n\nExtremes contradicted each other in Linden's heart, a turmoil of unexpected hope and dread. Possibilities that she had failed to foresee daunted and exalted her. In the dirt under her boots, complex emotions thrummed as if Gallows Howe had forgotten or surrendered none of its desires. Ahead of her, Mahrtiir stood with his hands open as if he were waiting for the weapon which would give them meaning at last: an import which no mere garrote could supply. Higher on the hill, and wreathed in compulsions which appeared to draw only purity from the Staff of Law, Caerroil Wildwood made his music louder, more encompassing, until it became a hymn chorused by the entire woodland. At the same time, he tuned his singing to a pitch that resembled language. Perhaps with her ears, or perhaps only with her health-sense, Linden listened to an arboreal melody more numinous than speech.\n\n\"It is my heart I give to you,\n\nMy blood and sap and bone and root,\n\nTo serve the woods with what we are\n\nWhile what we are endures to serve.\n\n\"I guard and grow the world's deep love.\n\nIts loveliness must justify\n\nThe sterner truths of rock and sea,\n\nFor they persist but do not grow\n\nAnd so their life is only Law:\n\nIt is not melody or joy.\n\nTheir substance, substanceless, is woe\n\nUnless it is redeemed by green,\n\nBy growth and verdure that relieve\n\nThe world from stone's commanding cold.\n\n\"If rock does not erode it does\n\nNot feed the trees that give it worth.\n\nIf sea does not give way to rain\n\nIt does not vindicate its surge.\n\nSuch passage is Creation's pulse:\n\nIts transformation brings forth love\n\nFrom Law's unending rest and flood,\n\nFor only life which passes on\n\nCan glorify remaining life.\n\n\"For loving's sake I guard the green:\n\nIts steward I became and am\u2014\n\nAnd you as well, for by my song\n\nIt is my soul I give to you\n\nTo serve the woods until we die.\"\n\nAnd while the Forestal's invocation swelled across tree and hill, Manethrall Mahrtiir of the Ramen began to change. Ineffable magicks wrapped him in their cocoon until he was barely visible. Swathed in Caerroil Wildwood's power, his bandage was burned away, and his raiment fell from him like dross. His lean form with its scars of struggle and its ropes of muscle was robed in samite that shone like incarnate cleanliness. An unalloyed argent too rare and refined to be wild magic transformed his visage. As if he had brought it forth from within him, a twig grew in his grasp until it became a sapling nearly as tall as himself: a child-tree crowned with new leaves, its roots clinging to a ball of rich loam, which he held with the ease of supernal strength.\n\nThe end of his human life had come upon him. When he emerged from the Forestal's theurgy, the man who had been steadfast in the face of every peril would be gone. Like the _Elohim_ of the Colossus, he would not be able to revoke his transubstantiation. Nevertheless his gladness aspired among the harmonies of Garroting Deep, and his eagerness for strife contributed a peal of joy.\n\nWatching him, Linden wanted to cry; but she had no tears for a friend who had found his heart's desire.\n\n## 11.\n\nBack from the Brink\n\nThomas Covenant could hardly stand. He felt like wreckage. Certainly he looked like a derelict, with his tattered jeans and T-shirt, and his silver hair wild. Only his boots had come intact through his immersions in Sarangrave Flat. If Rallyn had not led Mishio Massima through an arduous series of translations by wild magic, he would not have arrived anywhere. He and Branl would still be trudging along the edges of the lurker's wetland an impossible distance from where he was needed. Traveling through argent circles drawn on grass and stone and dirt with Loric's _krill_ , he had exceeded his image of himself.\n\nBut he had not done so without help. He was not as weak as he should have been, or as numb. Some of the effects of hurtloam lingered deep within him. He had drunk water made clean for him by the Feroce, and had eaten _ussusimiel_ melons. Aided beyond any reasonable expectation, he had been able to traverse the leagues.\n\nWith Kastenessen gone, Kevin's Dirt may have begun to dissipate; but if so, that was a victory which Covenant could neither confirm nor measure. Instead he was torn inside, frantic and grieving. Clyme's death remained as vivid as scars, as harsh as Joan's. The Worm was coming: it had already reached the Land. And Linden was not here.\n\nShe was not here.\n\nIn the instant of his arrival, he had seen things that deserved celebration. Jeremiah had escaped from his mental prison, or had been freed: that was obvious. Otherwise he would not have been able to design the crude structure at the foot of the rubble and the ridge. The Giants would not have known how to build it. And the edifice had succeeded. Infelice's presence at the portal, and Kastenessen's raging opposition, demonstrated that Jeremiah's efforts had achieved their strange purpose\u2014whatever that might be.\n\nBut the boy sprawled on the roof of the construct as if he had been felled. Stave lay motionless in the dirt near Infelice. None of the Giants wore their armor. They had no weapons. And there was no sign of Mahrtiir. Like Linden, the Manethrall had gone somewhere else\u2014or had been left behind\u2014or\u2014\n\nCovenant was stretched too thin to appreciate what the Land's defenders had accomplished.\n\nWithout any flicker of hesitation or pause for thought, he had flung himself toward Kastenessen armed only with Loric's eldritch dagger and his own extremity. _I killed my ex-wife_. Joan's ring seemed trivial against a being of Earthpower merged with brimstone and lava. _I helped destroy a Raver_. Yet Kastenessen had believed him. _And I've seen the Worm of the World's End_. Perhaps the _krill_ was capable of killing greater foes than Joan and _turiya_ Herem. _I am_ done with restraint _!_ Or perhaps Kastenessen had secretly wished to be swayed.\n\nBeyond the construct where Infelice and Emereau Vrai's lover had vanished, the gouged ridge rose like a barricade against the southeast; against memories of Joan. Covenant had sacrificed his own daughter. More than once. He had raped her mother. Ignored Triock's death. Permitted Clyme's. And he was Roger's father. He was responsible for that lost soul as well. In Morinmoss long ago, he may have killed the woman who had healed his mind. Hell, he had even ridden the Harrow's destrier to its death. He had committed wrongs enough to mark him as an acolyte of the Despiser.\n\nNow Linden was not _here_. He could not confess himself to her, or seek absolution.\n\nStave had said of her that she did not forgive. If that were true\u2014\n\nThe aftermath of Kastenessen's surrender left Covenant reeling with needs and ignorance. To defeat Joan, he had sealed the fracturing of his mind, but he had not rid himself of vertigo. Even the comparatively level plain felt like a precipice in the doom-clogged twilight. Slaps of wind raised dust on all sides as if every step altered the ground itself. He hardly noticed when Branl took the _krill_ to spare him at least that one burden.\n\nUnheeded or unneeded, Rallyn and Mishio Massima trotted away, presumably seeking water and forage.\n\n\"Unbeliever,\" Giants murmured or panted. \"Timewarden.\" Wan with exhaustion, Rime Coldspray called, \"You are timely come.\" And Frostheart Grueburn, \"Have you accomplished your purpose?\" And Cirrus Kindwind, \"Some ill end has befallen Clyme _Haruchai_.\" Other Swordmainnir repeated like groans, \"Longwrath,\" and, \"Lostson.\" They sighed the names of the _geas_ -damned man's parents, and ached for Moire Squareset and Scend Wavegift, both of whom were dead because of Longwrath.\n\nToo many questions. Too many contradictions to absorb. Covenant only knew that Jeremiah was still alive because the boy was looking at him; staring as if he were stunned, barely conscious. Stave lay like Longwrath. Like Longwrath's, Stave's flesh smoked as if Kastenessen had scoured his heart with scoria.\n\nIt was all too much. As if he were being ripped open, Covenant released a cry that seemed to come from the marrow of his bones.\n\n\" _What happened to Linden?_ \"\n\nThen he stood wavering as if he could not take another step without the woman whom he had loved for all of the Earth's ages.\n\nSwordmainnir hung their heads, too weary or overwhelmed to answer. But after a moment, Rime Coldspray summoned a vestige of resolve. She leaned into motion, came unsteadily toward Covenant. Tears that might have been relief or chagrin or sorrow\u2014that might have been anger\u2014made runnels through the dust on her face. When she was near enough to speak in a hoarse whisper and be heard, she stopped.\n\nLike Saltheart Foamfollower, the Ironhand towered over Covenant. The Giants had always been too much for him, more than he deserved. Trying to meet her gaze, he staggered until Branl steadied him.\n\n\"Timewarden,\" Coldspray breathed, \"our need is great. We have expended our last strength, Lostson Longwrath lies before us, slain for our salvation, and Stave Rockbrother is much harmed. For these ills, we have no anodyne. We must\u2014Ah, Stone and Sea. We must be more than we are.\n\n\"Yet I discern that your need is also great. Indeed, I fear that it exceeds comprehension. Therefore I will speak when I fain would hear.\"\n\nShe looked like she might collapse, but she did not. Even now, she met the challenge he represented.\n\n\"We deem that an alteration in Linden Giantfriend began when she was borne to the verge of the Sarangrave a second time.\"\n\nGloom held the plain. Beyond the stark brilliance of the _krill_ 's gem, the preternatural twilight seemed to defy every sentence. A second time? When was the first? Was that when she had hurt the lurker enough to inspire an alliance?\n\n\"There the Feroce conveyed a message, citing your command. They urged her to _Remember forbidding_. For that reason, she parted from us.\"\n\nThe Feroce had done Covenant's bidding. He had no one else to blame.\n\n\"Here we have fashioned a fane for the _Elohim_. It compelled them to come, and to enter, drawing Kastenessen with them. There its magicks will ward them from the Worm's feeding.\"\n\nCovenant stared up at her. Silver etched the lines of her visage, cut them into shapes that he feared to recognize. He believed the Ironhand, of course he did. Vast spans of time and knowledge were gone from him; but he remembered Jeremiah's importance, Jeremiah's talent. In some nameless sense, every future depended on Linden's son.\n\nChosen-son. The Giants had given the boy an epithet to call his own.\n\nExhaustion abraded the Ironhand's voice. \"All remaining _Elohim_ are now here. Therefore the Worm also must come. Preserved within the fane, they cannot be consumed. When the Worm destroys our edifice, however, they will have no egress. They will be eternally lost. For this reason, Linden Giantfriend determined that the Worm must be turned aside.\"\n\nThat, at least, Covenant understood. The Worm would come. It was coming. He wanted to ask, Then why gather all the _Elohim_ in one place? But he knew the answer: to give them a chance. They were helpless otherwise. And their presence here would not hasten the Earth's demise. Having reached the Land\u2014and having been blocked from Mount Thunder\u2014the Worm was bound to sense the location, the comparative proximity, of its final food. It would have come this way no matter what Jeremiah and the Giants did.\n\nBut it was not here now. The crazy turmoil of winds might mean anything. The clotted darkness on the northeastern horizon had more than one possible interpretation.\n\n\"Therefore Linden Giantfriend has invoked and entered a _caesure_. Accepting only Manethrall Mahrtiir as her companion, she seeks the deep past of the Land, where it is her hope that a Forestal will impart to her the forbidding which the world's peril demands.\"\n\nWithout Branl's support, Covenant might not have been able to stay on his feet. The deep past\u2014Oh, hell. Joan's death had not put an end to _caesures_. In spite of everything, Linden was still exceeding his expectations.\n\nBranl's expression was unreadable, as inarticulate as a mask of marble. However, his gaze was fixed, not on Rime Coldspray, but on Stave. Mind to mind, the Humbled may have been asking the former Master for confirmation.\n\nHow had Linden persuaded Stave to let her go without him?\n\nAt the end of her determination, the Ironhand said, \"If she succeeds\u2014and if the Arch does not fall\u2014and if she is able to return\u2014she will endeavor to refuse the Worm from this place.\"\n\nCovenant groaned aloud. Linden's absence was his doing. He had pushed her toward a risk so extreme that merely hearing it described made his pulse falter in his veins. He had been pushing her ever since she had returned to the Land, even though every stricture of Law and Time had screamed against such intervention. If she failed, the fault would be his.\n\nBut what else could he have done? He could not have acted differently without ceasing to be who he was.\n\nRime Coldspray was waiting. Her comrades were waiting. He had to say _some_ thing: something that was not more self-recrimination. He had done enough of that. It served no purpose.\n\nAnd Linden was not here to heal or curse him.\n\nGoading himself, he rasped, \"You already know some of my story.\" The Swordmainnir had heard his challenge to Kastenessen. \"Joan is dead. I rode the Harrow's horse until I killed it. We almost lost Mhornym and Naybahn. But they saved me. Then we went after _turiya_. Branl and me. And Clyme.\"\n\nNow that he had made a start, he meant to continue. But Coldspray held up her hand, asking him to pause. Other Giants were coming closer. Limping arduously, Frostheart Grueburn led the way with Latebirth and Cirrus Kindwind at her back. Onyx Stonemage and Stormpast Galesend moved like cripples, supporting Cabledarm between them. Wheezing, Halewhole Bluntfist labored after them.\n\nAs the women gathered beside the Ironhand, Covenant went on.\n\n\"Some things I have to guess,\" he admitted, \"but I gather Linden had a run-in with the lurker. You were probably there. Whatever happened, it got that monster's attention. Apparently Horrim Carabal can feel the Worm coming. It doesn't want to die. It needs more power. But it couldn't beat Linden\u2014or you and Linden. So the Feroce found me.\" They had accused him of being the Pure One. \"They offered us an alliance.\" _In pain and desperation_ \u2014\"Mutual help, safe passage, that sort of thing.\" _Already he suffers the presence of one who wanders lost within his realm_ \u2014\"That must be how Longwrath got here. The lurker let him through. And dozens of those little creatures died helping us get to Joan.\"\n\nHis listeners nodded again. Silver reflected like keening in their eyes. Longwrath's attack on Kastenessen must have astonished or appalled them. They had tried so hard to follow his _geas_ with him until he found peace. But they did not interrupt.\n\n\"After that,\" Covenant said, fierce and quavering, \"I wanted an alliance myself.\" He dreaded memories like this one. They were as piercing as images of Joan. \"It isn't hard to imagine we're going to need all the help we can get. And I was afraid _turiya_ would take the lurker.\" He did not mention the former Guardian of the One Tree. The Swordmainnir would recognize Brinn's name; but Covenant had no courage to spare for that explanation. \"Once I learned how to jump across leagues, we went after the Raver.\n\n\"We didn't catch up with him until he was in the Sarangrave. I tried to kill him, but I couldn't. Clyme and Branl did it.\" In spite of his private horror, he could not gloss over this detail. The Giants needed to hear it. They would understand. And the Humbled deserved at least that much homage. \"Clyme let _turiya_ possess him.\" Like Honninscrave in Revelstone. \"Then he held the Raver while Branl cut him to pieces.\" Covenant remembered hacked flesh, severed bones, blood. \" _Turiya_ wasn't just rent. He's gone. There isn't anything left.\"\n\nThe distress of the Giants showed in the way they looked at Branl; in glances teary with compassion and dismay. Perhaps more than any other living people, Coldspray and her comrades knew the cost of causing a Raver's end. Yet Branl's gaze gave them nothing. He was _Haruchai_ and did not accept grief.\n\nCovenant considered that rigidity a weakness, not a strength. He believed that forgiveness began with sorrow. But perhaps he was wrong. Perhaps a man who grieved would have spared Clyme. Then _turiya_ Herem would have lived. Eventually Horrim Carabal would have been lost\u2014and the Worm might have made its way, unresisted, to Mount Thunder.\n\nGrinding his teeth, Covenant went on.\n\n\"When Branl and I got out of the Sarangrave, we probably weren't all that far from where we are now. But the Feroce told us the Worm was getting close. That's when I sent my message. Then I wanted to see it for myself. We went to look.\n\n\"Now the Worm is here.\" He bit down on his voice to hold it steady. \"I can't describe it. I'm not going to try. But I can tell you this.\" Facing the clenched apprehension of his audience, he said, \"It wasn't headed _here_ ,\" toward Jeremiah's fane and the sealed _Elohim_. \"It was going west. Straight at Mount Thunder. At She Who Must Not Be Named.\n\n\"That scared me,\" he growled. \"Lord Foul likes convoluted plots. Every trap you face has another one hidden inside. If the Worm and that bane went after each other, we wouldn't have to worry about the _Elohim_ , or anything else for that matter, because we would already be dead.\"\n\nHe spread his hands; his truncated fingers. \"We're not, so I have to assume\u2014\"\n\nRime Coldspray looked like she wanted to interrupt then; but Covenant did not pause. As concisely as he could, he explained how the lurker and the Demondim-spawn had striven to deflect the Worm from Mount Thunder. Then he finished, \"I didn't wait around to see how long they could hold. Branl and I just ran. But they must have held long enough.\" Somehow. \"I can't imagine it would take this long for the Worm to get through Lifeswallower. Now I just have to hope we didn't lose them in the process.\"\n\nHe expected Coldspray or the others to question him. He had questions of his own. What had happened to Jeremiah and Stave? How had Jeremiah been retrieved from his dissociation? But before anyone spoke, Branl moved.\n\nWithout ceremony, as if the action did not require comment, the Humbled handed Loric's _krill_ to Rime Coldspray.\n\nShe accepted it reflexively, her eyes wide, as Branl strode past her toward Stave.\n\nWith Handir and the other Humbled, Branl had participated in punishing\u2014in excommunicating\u2014Stave for his defiance of his kinsmen; his devotion to Linden. The Masters had refused to acknowledge Stave's mental voice. Like Clyme and even Galt, Branl had treated Stave with disdain. Like them, Branl had challenged Stave more than once, tried to strike him down.\n\nNow the last of the Humbled approached Stave's prone form like a man who proposed to deliver judgment.\n\nCovenant should have stopped him; should have said something, anything. Clyme's death was only one example of the severity with which the _Haruchai_ judged themselves. But at that moment, Covenant was like the Giants. He had come to the end of what he could do.\n\nA darkening storm made omens in the northeast. Winds whipped Branl's legs, tugged at the tears in his tunic. But he ignored them. Implacable as a fanatic, he strode through the gusts.\n\nAt Stave's side, he stopped, braced his fists on his hips. For a moment, he bowed his head over Stave, apparently searching Stave's slack form for some sign of awareness. Stave's hand and forearm, his right, no longer smoked. Still he lay helpless, as if his mind or his heart had been as badly charred as his skin.\n\n\"I have named Stave 'Rockbrother,'\" Coldspray announced. She may have been warning Branl; but she seemed unable to raise her voice.\n\nAbruptly Branl stooped over the fallen _Haruchai_. With both hands, he lifted Stave upright.\n\nStave's head lolled to one side, then sagged until his chin rested on his chest. He dangled emptily in Branl's grasp.\n\nIn a blink of motion so swift that Covenant barely understood it, Branl released one hand and slapped the side of Stave's head. The blow made no sound that Covenant could hear through the wind.\n\nFrostheart Grueburn winced. Coldspray lurched toward the _Haruchai_ , going too late to Stave's defense.\n\nAs if the movement were full of pain, Stave slowly raised his head. His lone eye opened. Blinking, he fixed his gaze on Branl.\n\nBranl did not strike again. Unsure of herself, the Ironhand halted. Covenant watched with his pulse trapped like a cry in his throat.\n\nFor a few heartbeats, Branl met Stave's gaze. Then Stave's head sagged again; and the Humbled nodded once. Shifting his grip, he wrapped his arms around Stave.\n\nWith Stave hugged against his chest, Branl informed Covenant, \"The tale of the Giants is incomplete. We do not fault them. Their heed was consumed by Kastenessen and great weariness. It may be that they did not recognize the plight of Linden Avery's son. To Stave, however, it was evident that the boy received more than Earthpower at Anele's hands. The gift included Anele's openness to possession.\n\n\"Arriving in his fury, Kastenessen entered the boy. His apparent purpose was to drive Infelice into the fane, that he might destroy her with the other _Elohim_. But Stave intervened. By removing the boy from dirt underfoot, he severed the possession. Hence his burns, and his unconsciousness, and the boy's presence atop the fane, where he is warded by stone.\n\n\"There he might readily have been slain. Only Longwrath's coming, and yours, ur-Lord, preserved him. Yet more Stave could not have done. Without aspersion to the Swordmainnir, I assert that no other could have done as much.\"\n\nFresh vertigo sucked at Covenant. Realities shifted into new alignments: their implications veered like the world. Somehow Jeremiah had been rescued from his dissociation, or had saved himself. A mixed blessing: the Earthpower with which Anele had invested Jeremiah had made the boy vulnerable. No wonder Anele had hidden himself in madness; made himself blind. How else could he have concealed his true abilities, his secret purpose, from the Despiser? But Jeremiah did not have the old man's cunning. Lord Foul would be able to claim the boy whenever Jeremiah chanced to stand on the right surface, the right rough grass.\n\nThe ridge seemed to wobble from side to side, mocking Covenant. There were other inferences\u2014\n\nBranl had said _we_. He had addressed Stave in the fashion of the _Haruchai_ \u2014and he had listened to Stave's reply. Now he had reaffirmed his kinship with Linden's friend as if he felt pride in it. He spoke for Stave as well as himself.\n\nA profound change. If the Humbled had ever needed or desired Covenant's forgiveness, he earned it now.\n\nApparently the Ironhand also had heard and understood Branl's _we_. She raised the _krill_ so that its gem lit the _Haruchai_. Striving for formality, she replied, \"There is no aspersion, Branl Humbled. There is only praise, both for Stave and for you\u2014and for Clyme as well. At a better time, we will tell the full tale of Stave Rockbrother's deeds. We will honor your own. For the present, be assured that we esteem your courage and devoir.\"\n\nIndirectly she steadied Covenant. Breathing deeply to calm his private reel, he muttered, \"Then I guess Linden did the right thing when she made Stave stay behind.\"\n\n\"Indeed,\" assented Coldspray. And Branl said unexpectedly, \"In this, the Chosen has shown foresight. I am reminded of matters which Stave has not forgotten concerning her former service to the Land.\"\n\nAnother surprise. Covenant frowned through the silver light, and found that he had no response. For a moment, he almost wept.\n\nNone of the Humbled had ever called Linden by her title.\n\n\"And Stave Rockbrother himself?\" asked Cirrus Kindwind. \"How does he fare? He is closed to our discernment, as you are, Branl Humbled. We fear for him.\"\n\nBranl shrugged to indicate Stave. \"This state is not unknown among the _Haruchai_. More commonly, we have recourse to it when we are snared by storms among the high peaks of our homeland. When both passage and shelter cannot be attained, we withdraw as Stave has done to preserve the essence of our lives. Thus we endure the gales, emerging when they are spent. Upon occasion, however, we withdraw similarly to heal otherwise mortal wounds, or mayhap to weather such shocks and virulence as Stave has received. When he has restored himself, he will stand among us once more.\"\n\nCarefully he lowered Stave to the earth, then stepped back to resume his place with Covenant. He may have thought that he had said enough.\n\nBut Covenant had not entirely regained his balance. \"Wait a minute,\" he objected. \"There has to be more to it than that. Stave has touched Kastenessen before, and he wasn't hurt like this. Something is different now.\"\n\n\"It is, ur-Lord,\" admitted Branl. But he did not elaborate. Perhaps Stave had not remained conscious long enough to share those memories.\n\nRime Coldspray sighed heavily, a gust torn away by the moiling winds. \"I am too worn to bear the burden of Stave's tale. I will say only that he was gravely injured in the raising of young Jeremiah's fane. We will speak further when we have rested. I cannot remain upon my feet.\"\n\nIn fact, Covenant suspected that she was close to fainting. And the condition of her comrades was no better. Cabledarm's was worse. Even to his blunted sight, it was obvious that the injured Giant could not stand without support.\n\n\"Then don't worry about it,\" he said unsteadily. \"We'll have plenty of time for tales\"\u2014a grimace jerked across his face\u2014\"unless we don't, in which case it won't matter anyway.\n\n\"Is there any shelter around here? We should try to get out of this wind.\"\n\nGrueburn glanced at Jeremiah's construct. \"Can we not find calm within the fane?\" She sounded wistful. \"I am not so chary of _Elohim_ \u2014or indeed of Kastenessen\u2014that I would decline to ease my weariness in their presence.\"\n\nWhen Covenant followed her gaze, he saw Jeremiah lower himself to the ground. A moment later, the boy came running, waving his arms for attention. His ragged pajamas\u2014soiled with grime and stained by blood\u2014made him look destitute and desperate. Nevertheless he was recovering from Kastenessen's violation.\n\nTogether, the Giants turned to watch him. Rime Coldspray held up the _krill_.\n\nHe reached the company, jerked to a halt. \"You won't see them,\" he panted. Apparently he had heard Grueburn. \"It's just stones. The magic only works on them.\"\n\nA heartbeat later, he flung himself at Covenant, wrapped his arms around the Unbeliever. Suddenly he was crying\u2014and fighting to deny it.\n\n\"I'm sorry. I didn't have a chance. Mom left, and Stave was hurt, and Kastenessen\u2014he\u2014out of nowhere\u2014The _croyel_ was bad. He's worse. Much worse.\n\n\"I am _so_ glad to see you.\"\n\nCovenant returned Jeremiah's hug without hesitation. He ached for that himself; for any embrace if he could not have Linden's. And he, too, loved the boy.\n\nHe, too, feared for Linden's son.\n\nAs carefully as he could, he asked, \"What did Kastenessen do to you?\"\n\nJeremiah swallowed a sob; clung harder. \"He broke me.\"\n\nSudden compassion stung Covenant. Deliberately he eased his own clasp until Jeremiah did the same. Then he held the boy at arm's length, studied every detail of Jeremiah's mien: the rich brown of his eyes, the passionate mouth, the fine stubble on his cheeks; the lines cut by too much suffering. But he could not discern how deeply Jeremiah's hurts ran, or how badly he had been marred.\n\n\"How did he do that? What was it like?\"\n\nThe question seemed to transform Jeremiah. Ferocity darkened his eyes to the color of rot-laden silt. His mouth stretched, baring his teeth. The lines of his face assumed predatory angles. In an instant, he was no longer a boy confused by his wounds. He had become a young man crowded with bitterness.\n\nAlmost spitting, he snapped, \"I'm sorry Infelice let him inside. I'm sorry you didn't kill him. I want him _dead_. I hate being used, and _I don't want to talk about it_.\"\n\nHis vehemence shocked Covenant. A similar reaction twisted Coldspray's visage, and Grueburn's. Branl took a subtle step closer as if he sensed that Covenant was in danger.\n\nBut Covenant stood taller in front of Jeremiah. With a portion of his own ferocity, his rage for the damaged and the outcast, he retorted, \"Then hang on to feeling broken. Hang on to the pain. It can be useful. I should know.\"\n\nThen he lowered his voice. \"In any case, Kastenessen is different now. Without Kevin's Dirt, he's just another victim.\"\n\nJeremiah looked like he wanted to sink his teeth into Covenant's throat. \"I don't care.\"\n\n\"Chosen-son,\" the Ironhand murmured: a reprimand that lacked the strength to insist.\n\nTo himself, Covenant groaned, Oh, Linden. I'm so sorry. Nevertheless he held the boy's glare without flinching. Severe as a judge, he demanded, \"Then tell me something else. Is that temple a prison?\"\n\nHe already knew the truth, but he wanted to hear it from Jeremiah. He wanted Jeremiah to acknowledge it. It might help.\n\nAt once, Jeremiah's fury became chagrin. Without transition, he seemed altogether young and vulnerable. \"No!\" he cried as if Covenant had slapped him. \"I wouldn't do that. They can get out whenever they want.\"\n\nAh, hell. Covenant's relief was so swift that he sagged against Branl. Hell and blood. A host of fears drained out of him before he had managed to name them all. Sure, the boy was in pain. His rage revealed the depth of his wounds. But his distress now was as true as his desire for harm. And he had done what he could to forestall the world's end.\n\nAs soon as Covenant recovered his balance, he moved to hug Jeremiah again. \"Thanks,\" he breathed like a sigh at Jeremiah's ear. \"I knew that about you. I just needed to hear you say it.\n\n\"Linden will be so proud we won't know what to do with her.\"\n\nJeremiah sobbed again, a small sound like a plea. But he did not stiffen or pull away. Surrendering to Covenant's hold, he asked fearfully, \"Will she make it? Will she come back?\"\n\nHe seemed to say, I don't know who I am without her.\n\nCovenant knew how he felt. Striving for confidence, he countered, \"She's _your_ mother. Has she ever _not_ come back?\"\n\nCenturies ago among the Dead in Andelain, High Lord Elena had urged Covenant to take care of Linden, _that in the end she may heal us all_.\n\nBefore Jeremiah could reply, the Ironhand wavered on her feet: she nearly fell. Blinking as if she could no longer focus her eyes, she croaked, \"For mercy's sake, have done with contention. We must rest.\"\n\n\"Hellfire,\" Covenant growled. During his many experiences with Giants, he had seen them in every extreme of peril and pain; but he had never known them to be utterly exhausted. \"What're we waiting for? You don't just look tired. You must be half starved.\" Where could they have replenished the Ardent's supplies? \"Let's go lie down.\"\n\nInstead of responding, the Swordmainnir simply turned and trudged toward the fane like women who had come to the end of every desire except the wish for relief. Longwrath they left where he had fallen. After everything that they had done and endured, they were too weak to lament his end.\n\nnside the edifice, the company found better shelter than Covenant had expected. Though the walls were punctuated with gaps, and the ceiling looked precariously balanced, the winds outside were reduced to a confusion of mild breezes. In addition, the stones retained a suggestion of warmth: an aftereffect of theurgy. There Rime Coldspray returned the _krill_ to Branl, and the Giants stretched themselves out like dead women. But they did not sleep immediately. In low voices, wan and necessary, they began to talk, first Cirrus Kindwind, then Latebirth, then Onyx Stonemage. They told Covenant how Linden had broken open the ridge to uncover malachite. Passing the story from one to another, they described the building of Jeremiah's fane, and the extravagance of Stave's efforts, and the narrowness of his survival. And when their voices trailed away at last, Jeremiah gave Covenant a condensed version of his escape from his mental ensepulture.\n\nAmong his companions, Covenant sat with his knees hugged to his chest and tried not to rock from side to side like a child in need of solace. He wanted Linden, and had no way to reach her.\n\nCabledarm lay shivering as if she were feverish. Shock, Covenant thought. She had fallen hard and badly after deflecting Stave's plummet. Not for the first time, he felt bewildered by the abilities of the Giants. Saltheart Foamfollower had once walked through lava: a kind of _caamora_ , terrible pain and cleansing. Such deeds had come to appear almost normal for Foamfollower's people. Now Covenant wondered whether the Swordmainnir would be able to leave their present straits behind if they did not first find fire in which to release their sorrow.\n\nA comparable extravagance seemed normal for the _Haruchai_ as well. Covenant could hardly bear to contemplate what Stave had done to help complete Jeremiah's construct. And the sheer strength of will with which Stave had resisted Infelice for Jeremiah's sake, and for Linden's, left Covenant gaping inwardly.\n\nHow was it possible for any ordinary man\u2014or woman\u2014or boy\u2014to live up to the example set by the Land's other defenders, the natural inhabitants of this world?\n\nNonetheless Linden had taken Mahrtiir into the Land's past for a purpose as extreme as anything that the Giants and Stave had attempted. Covenant yearned to believe that she would succeed. And he had one tentative reason to think that she would not fail\u2014or had not failed yet. The Arch of Time still held. One moment led to the next. Covenant inhaled and then exhaled. He heard words arranged in comprehensible sequences. Therefore Law endured. Linden had not caused a fatal rupture\u2014or its ripples had not yet reached him.\n\nPerhaps that inference was an illusion. Perhaps he only experienced time chronologically because he was human, too mortal to perceive any other reality. Perhaps nothing truly existed outside the confines of his own perceptions.\n\nHe had considered such ideas before. At one time, he had trusted himself to them. Now he discarded them with a private shrug. They changed nothing. He was responsible for the meaning of his life, as he had always been: for his loves, and for his rejections. While he remained able to think and feel, he could not set such burdens down without betraying himself.\n\nNo doubt Linden believed something similar. How otherwise could she have hazarded a _caesure_? Without some kind of faith in the necessity of her commitments, how could she have ridden away from her son?\n\nNo wonder Covenant loved her.\n\nEventually the voices of his companions fell into silence. With the suddenness of a child, Jeremiah collapsed into slumber: an aftereffect of effort and Kastenessen's touch. Cabledarm stared wide-eyed at the ceiling, in too much pain to sleep, or to hear. But Bluntfist's eyelids closed, and she sank away. Onyx Stonemage resisted yawns\u2014briefly, briefly\u2014until they overcame her. Then Latebirth and Galesend slept as well. Soon only Coldspray, Grueburn, and Kindwind remained awake with Covenant and Branl.\n\nThe three women regarded Covenant, obviously waiting to hear more of his own tale.\n\nAs much for his own sake as for theirs, he began to speak. But he did not talk about what he had done and endured. He had no language adequate to Joan and the Ranyhyn, or to _turiya_ Herem and Clyme, or to the Worm of the World's End. Instead he spoke of Linden and Horrim Carabal.\n\n\"At least now we know what the Ardent meant when he said her fate is _writ in water_. Or part of it, anyway. She gave us an alliance with the actual lurker,\" who was nothing if not a creature born of water, made great by water. \"Hellfire! How improbable was _that_? But there's more. If she hadn't brought down that flood above the Lost Deep, none of us would have escaped. If she hadn't gone back to the Sarangrave, the Feroce might not have been able to give her my message.\"\n\nShe had used water to provide malachite for Jeremiah. And Covenant himself had broken her free from memories of She Who Must Not Be Named by holding her underwater. In retrospect, he trembled at his own daring.\n\n\"Long ago,\" he finished hoarsely, \"I told her to _Do something they don't expect_. If we ever find a way to stop Lord Foul, it'll be because she's taken him by surprise over and over again.\"\n\nAfter a long pause, Rime Coldspray mused, \"It indeed appears that many unforeseen outcomes were enabled by Linden Giantfriend's last effort among the caverns. But the same may be said of any deed. If she had not retrieved the Staff of Law. If she had not accompanied your false son and her possessed boy into the Land's past. If she had not dared all things to create a place for you among the living. Life is ever thus. One step enables another. For that reason, auguries are an ill guide. They tread perilously upon the borders of unearned knowledge.\n\n\"Still we are Giants. We crave an understanding of your own deeds. Will you not tell their tale more fully?\" She spread her hands in the light of the _krill_ as if she wanted to convince Covenant that they were empty and needed to be filled. \"Ignorance haunts us. It hinders rest.\"\n\nInstead of saying, No, or, Have mercy, or, I can't bear it, Covenant countered, \"I'm not sure that's true. I think it's Longwrath who haunts you.\" The Giants may have felt that they had failed him. \"You need a _caamora_ , and you don't know how to get one. It eats at you.\"\n\nThe Ironhand did not contradict him. Instead she asked, \"Are our hearts so plain to you?\"\n\nCovenant shook his head. \"I only think you need to grieve because I've known Giants for a long time. I can't _see_ you. And Kevin's Dirt just makes me blinder.\"\n\nNumbness was eroding his ability to _hold on_. When he could no longer grip, he would be effectively impotent.\n\n\"Then it will comfort you,\" interposed Frostheart Grueburn, \"to hear that Kastenessen's vile brume has faded from the heavens. His entry to the Chosen-son's fane has unbound his theurgies. Also the decimation of the stars has ceased. While the remaining _Elohim_ are preserved, it will not be resumed.\"\n\nFaded\u2014Covenant released his legs, sat up straighter. \"Well, damn.\" He had assumed that Kevin's Dirt was gone\u2014that its bale required Kastenessen's constant attention\u2014but he had not had an opportunity to ask for confirmation. \" _Thank_ you. If you hadn't built this place\"\u2014he gestured around him\u2014\"I might be useless by now.\"\n\nHe had always been useless without friends.\n\nBut the Giants were not deflected from their concerns. \"Nonetheless,\" Cirrus Kindwind remarked, still probing, \"your ailment has gained force. And we fear that you will refuse Linden Giantfriend's succor, as you once refused hurtloam in Andelain. It is your resolve that you must not be healed which most drives our desire to comprehend you.\"\n\nReflexively Covenant grimaced. In a quiet rasp, he said, \"I don't know how to explain it. Leprosy protects me somehow.\" If Lena had not given him hurtloam when he first came to the Land, he would not have been able to rape her. \"Sure, it costs me a lot. But it's also a kind of strength. It makes some things possible that I couldn't do without it.\"\n\nThen, to forestall more questions, he urged, \"You should get some sleep. We all need rest. Later I'll figure out some way to give poor Longwrath a _caamora_.\" Once before, he had done something similar. \"I have Joan's ring. And the _krill_. I should be able to manage a fire.\"\n\n\"Very well,\" the Ironhand murmured. She was already drifting. \"Though the burden of our woe is great, it is surpassed by weariness.\"\n\nIn another moment, her head sagged, and she was asleep.\n\nFrostheart Grueburn tried to swallow a cavernous yawn. Then she did what she could to make herself comfortable on the bare dirt.\n\nFor a while, Cirrus Kindwind continued to study Covenant through the _krill_ 's silver. But she did not prod him further. Nodding as if she were content, she said, \"Earlier I had occasion to remind Stave Rockbrother that he is not alone. I would proffer a similar assurance now. Whatever the substance of your fears or pains may be, you will not be required to confront it alone. We are merely weakened. We are not inclined to forsake you.\" She hesitated briefly, then added, \"And Linden Giantfriend has not forgotten her love for you.\"\n\nBefore Covenant could decide whether to weep or smile, Kindwind turned away and settled herself for sleep.\n\novenant dozed for a while himself. Most of his efforts since the struggle with _turiya_ Herem had been mental and emotional rather than physical, but they had drained him nonetheless. He did not mean to sleep while he hoped for Linden; but drowsiness overcame him, and he sank into a shallow slumber.\n\nLater some preterite instinct roused him, and he jerked up his head to look around. Squinting against the blur that marred his sight, he saw Stave enter the fane.\n\nThe former Master moved cautiously, as if he had become unsure of his balance. His right hand and forearm gave the impression that they ached. But his single eye as it caught the _krill_ 's shining was clear. It flashed argent at Covenant as if Stave had gained the ability to see into the Unbeliever's soul.\n\nPerhaps he had. He had allowed himself to grieve for Galt, his son. And he had let Linden convince him to remain with Jeremiah. To Covenant, those were astonishing changes. Of the _Haruchai_ whom he had known, only Cail had revealed a comparable willingness to go beyond rigid stoicism. Even men like Bannor and Brinn, Branl and Clyme, had measured themselves by standards which any other _Haruchai_ would have approved.\n\nCarefully Stave eased himself to the ground in front of Covenant. There he sat cross-legged, upright as a spear driven into the dirt, with the backs of his hands resting on his thighs. His eye seemed to transfix Covenant.\n\nWithout preamble, as if he were resuming a conversation, Stave said, \"I did not part willingly from the Chosen.\"\n\nHis manner rather than his tone suggested that he wanted to be understood.\n\n\"I know,\" Covenant answered quietly. \"But you let yourself be persuaded anyway. She asked, and you agreed.\"\n\n\"I did,\" the former Master admitted. \"I have found that I am no longer able to refuse her.\"\n\nCovenant's mouth twisted. \"I know the feeling.\"\n\nStave flexed the fingers of his right hand, testing them for residual damage. \" _Haruchai_ do not indulge in regret. Yet I am\"\u2014he appeared to search for a word\u2014\"unsettled. If she does not return, Timewarden, I will be unable to quench my sense of loss, or my remorse that I did not stand at her side.\"\n\nNow Covenant winced. \"I know that feeling, too.\" He had not simply turned away from Linden. He had told her not to touch him. More harshly than he intended, he said, \"But sometimes things like that have to be done anyway.\"\n\nStave nodded. \"Necessity demands. It does not countenance denial.\" Then, unexpectedly, he looked away, as if he rather than Covenant had cause to feel shame. \"Thus I am compelled to inquire of myself what purpose is served by regret\u2014or indeed by grief.\"\n\nWithout pausing to consider his reply, Covenant countered, \"How else do we know we're alive?\"\n\n\"By our deeds,\" Stave answered. \"By striving and service. By\u2014\"\n\nAbruptly he froze. His gaze sprang back to Covenant's. Nothing else moved.\n\nAfter a moment, he released a long breath. \"Ah.\" His regard did not waver, but his rigidity eased. \"Now I begin to grasp how it transpires that you and the Chosen have failed to comprehend the Masters\u2014and how the Masters have been misled in their apprehension of you. You and the Chosen\u2014those of your world\u2014The Chosen-son. Hile Troy. You judge by your hearts. It is by grief and regret that you know yourselves, rather than by deeds and effort and service.\"\n\nIn his turn, Covenant nodded. \"Well, yes.\" More than once, he had tried to explain himself to the _Haruchai_ ; but somehow he had failed to grasp the question implicit in their notions of service. \"Grief and regret. What else is there? Those are just other names for love. You can't feel bad about losing something if you don't love it first. And if you don't love, why else would you bother to _do_ anything at all?\"\n\nOf course, love was not so simple. He knew that as well as anyone; perhaps better than most. It spawned complications faster than it clarified them. It could be misguided or selfish. It could close its eyes. It could curdle until it became hate. And it implied rejection. Stepping in one direction required moving away from another. But at its core\u2014\n\nAt its core, love was the only answer that made sense to him.\n\n_There is hope in contradiction._\n\nFrom where Branl stood, the _krill_ left Stave's features in shadow. Covenant could barely discern the outlines of the former Master's mien. Only Stave's eye pierced the dusk.\n\nImpassive as any _Haruchai_ , he said, \"It is a terrible burden, Timewarden.\"\n\nCovenant shrugged. \"Look at Branl. Look at the Masters. Look at yourself.\" Briefly his old rage for the abused of the world rose up in him. \"Hellfire, Stave! Look at the _Elohim_.\" Then he subsided. Almost whispering, he asked, \"Is what you see any less terrible?\"\n\n\"It is not,\" Stave replied as if he were sure. \"It is more so.\"\n\nA moment later, something that may have been a smile tugged at the corner of his mouth. \"Were I inclined to the homage of mutilation\u2014which I am not\u2014I would now claim a place among the Humbled. Though they have aspired to emulation, they have not grasped the full import of their desires.\n\n\"Until now,\" he added in Branl's direction, acknowledging what Branl had done and endured.\n\nBranl lifted a shoulder slightly. \"Should the world endure,\" he promised, \"and the Masters with it, I will undertake to instruct our people.\"\n\nFinally Covenant bowed his head. The Humbled had made it surprisingly easy to forgive the manner of Clyme's death.\n\nradually the gloom within the temple became the more ominous twilight of late afternoon. By degrees, it thickened toward evening and full night. Branl remained standing, so motionless that he did not appear to breathe; holding Loric's dagger steady as a beacon. Stave still sat in front of Covenant, resting while his strength returned.\n\nIn the gathering darkness, the Giants began to wake.\n\nFrostheart Grueburn was the first. Muttering Giantish expletives, she rolled onto her side and struggled upright. Without a word, she left the fane. When she returned, she brought several waterskins. One she passed to Covenant.\n\nAs Covenant drank, Halewhole Bluntfist raised her head. After gazing blearily around her for a moment, she nudged Onyx Stonemage. Stonemage responded by ascribing a list of offenses to Bluntfist's parents; but she did not refuse to be roused.\n\nOne after another, the Swordmainnir arose. In the stark illumination of the _krill_ , they looked garish, like women who had become fiends while they slept\u2014or had been tormented by fiends.\n\nAmong them, Jeremiah woke up suddenly. His eyes seemed to give off glints of panic as he looked around for some sign of his mother. When he realized that she was still absent, he slumped back to the ground, covered his face with his hands. But then he practically flung himself to his feet. Ignoring his companions, he hurried out of the fane.\n\nThe Ironhand shrugged. No one said anything.\n\nBy turns, the Giants studied Cabledarm's condition, offered her what encouragement they could. Stormpast Galesend urged a little water into her mouth. They had no other help to give her.\n\nAll of them drank until they had emptied the waterskins. To no one in particular, Latebirth sighed, \"I would barter my sword\u2014aye, and my arm with it\u2014for a handful of _aliantha_ , and count myself fortunate in the exchange.\" Her comrades nodded mutely.\n\nWhile Covenant watched, Rime Coldspray stretched her arms and back, loosened her neck. Then she looked at him. \"Longwrath,\" she said curtly, reminding him of his promise.\n\nFire, he thought. Lamentation for the dead. The pain that consoles. In his own fashion, he understood how Giants grieved. Still he was reluctant to move. He had spent hours waiting for Linden: waiting and aching. Now he felt too heavy to stand, as if he were wrapped in iron chains. He would have preferred to go on waiting.\n\nBut he might not get another chance to keep his word. For all he knew, Cabledarm was dying.\n\n\"We need Linden,\" he muttered to no one in particular. \"I need her.\" Then he extended a hand to Branl, let the Humbled lift him upright.\n\nAfter sitting against the wall for so long, his muscles had stiffened. He felt like an assemblage of mismatched parts as he accepted the _krill_. But he was accustomed to that. And the gem of Loric's weapon shone steadily, answering the presence of white gold. With its magicks, he had already accomplished things which he had considered impossible. Why not more?\n\n_You are the white gold._\n\nHolding the dagger by its wrapped hilt, he led his companions from the shelter of the fane.\n\nOutside he found that night had almost claimed the plain. Beyond the reach of the _krill_ 's gem lay only blackness. Harsh buffets of wind seemed to hit him from every direction simultaneously. The chill pang of the air augured days of deeper cold. He had hoped for a moon; but it had not risen\u2014or it was left in darkness by the sun's absence from the world.\n\nHere even his blunted senses felt the violence glowering in the northeast: a crouched impression of storm as fierce as a predator, and as absolute as fuligin. He wanted to ask how far away it was, and how quickly it was moving, but the words caught in his throat.\n\n\"It is the Worm, ur-Lord,\" Branl stated like a man who could read minds. \"Yet it is many and many leagues distant. Also the fury of its coming outruns the Worm itself. It is not imminent.\"\n\nCovenant forced himself to breathe. After a moment, he managed to ask, \"How much time do we have?\"\n\nThe Humbled looked at Stave. Something silent passed between them. Then Branl said, \"If it does not increase its haste, it will not strike this region until the morrow, perhaps some hours after dawn.\"\n\nIn a low growl, the Ironhand confirmed Branl's estimate. \"Beyond question, the lurker and the Demondim-spawn have accomplished the wonders which were asked of them.\"\n\nThe force and confusion of the winds affected Covenant like vertigo. Lurching like a holed ship in an uneven gale, he moved toward Longwrath's corpse.\n\nThe two _Haruchai_ accompanied him, and behind them came the Swordmainnir. Cabledarm the Giants supported between them, although her mind wandered the borderland between consciousness and delirium. Maybe they hoped that fire would cauterize her internal bleeding.\n\nEventually Covenant spotted Jeremiah. The boy had climbed back onto the roof of the fane. Vague in the darkness, he stood there as if the crude edifice were a watchtower. Restlessly he scanned the plain from horizon to horizon, searching for some sign of his mother's return.\n\nCovenant felt a pang for the boy, but he did not allow himself to pause. Winds slapped at his face. They came at him from one direction and then another as if they were trying to nudge him aside from his purpose. The Worm was a condensed apocalypse: it pushed turmoil ahead of it like a bow-wave. He kept moving so that he would not relapse to waiting for Linden.\n\nLostson Longwrath lay where he had fallen, charred and lifeless: a darker shape like an omen outstretched on the benighted ground. Beside the _geas_ -doomed Swordmain, Covenant stopped. Too many lives had already been lost. No doubt the Worm had left tens or hundreds of thousands of deaths in its wake, perhaps millions\u2014and the carnage was just beginning. The _Elohim_ would not be the only casualties of Lord Foul's quest for freedom; his obsessive denial of his own despair. As it always did, Despite littered the world with victims.\n\nCovenant had to do what he could.\n\nWhile he secured his numbed grip on the _krill_ , however, Rime Coldspray said, \"A moment, Timewarden. One matter remains to be resolved. It concerns Longwrath's flamberge.\" She indicated the wave-bladed longsword where it lay near the man's burned fingers. \"He appeared to acquire it at the behest of his _geas_ , and therefore of the _Elohim_ , though we saw no clear purpose in it. Now, however, it appears in an altered light. The Harrow said of it that it was forged by theurgy to be potent against Sandgorgons. Its puissance has faded with disuse, he informed us. But those monsters have come to assail the Land, and we are too few and worn to oppose them. Therefore it is my hope that the blade's force has not altogether waned.\n\n\"If this weapon retains any virtue, one among us must wield it.\"\n\nA good point, Covenant thought. No matter what happened, the Land's defenders would need every conceivable weapon. But while the Giants looked at each other and frowned, weighing possibilities, Branl spoke.\n\n\"The ur-Lord will have need of the _krill_ , and I have no sword. A Giant's blade is an inconvenient length in my hands, but its weight does not exceed my strength, or my skill.\n\n\"The _Haruchai_ have ever eschewed weaponry. Nevertheless weapons we must have. If our people do not elect to reinterpret their service, they will render their lives effectless in the last crisis. Fists and feet suffice to oppose Cavewights, but they will not harm Sandgorgons or hinder _skurj_.\"\n\nStanding over the flamberge, he watched the women for their reactions.\n\nHesitating, the Ironhand asked, \"Timewarden?\" She seemed loath to let a man who was not a Giant take up Longwrath's only legacy. She faulted her ancestors for Longwrath's _geas_. It was the responsibility of the Swordmainnir to attempt amends.\n\nBut Covenant was sure. \"Why not?\" he returned. \"One way or another, we all have to reinterpret our notions of service. That's what Linden is doing. Stave did it a long time ago. Now Branl is doing the same. Why not let him keep going?\"\n\nColdspray gave her comrades a chance to object. Some of them scowled or looked away, shifted their feet uncomfortably; but no one contradicted Covenant.\n\nFinally their leader nodded. \"Let it be so. Branl of the _Haruchai_ , accept Lostson Longwrath's flamberge in the name of ancient friendship and faith. May you find worth in it\u2014aye, and give it worth as well\u2014to redeem the tale of a loved Giant thoughtlessly betrayed.\"\n\nBranl's reply was to take the longsword in both hands. Briefly he swung it around his head as if to demonstrate that he was equal to its heft. Then he stepped back from Longwrath's corpse, making room for Covenant.\n\nStave watched, expressionless as any of his kinsmen. Nevertheless he conveyed the impression that he approved.\n\nIt was time. Covenant had made a promise. Trembling, he drew out Joan's ring on its chain. With the ring's uncompromising circle in one hand, he raised High Lord Loric's _krill_ over the dead.\n\n\"Lostson Longwrath,\" he began. For the sake of the Giants, he strove to speak formally. \"Parents who cherished you named you Exalt Widenedworld, but they couldn't protect you from being hurt. Forgive one more wound. What you've suffered tears the hearts of your people, and I don't know another way to help them.\n\n\"A natural fire would be better for them, maybe even for you. I know that.\" Long ago, he had summoned the Dead of The Grieve into flames to find their release. \"But we don't have any wood. This is the best I can do.\n\n\"Whatever happens, remember that you saved the Giants who knew you best. None of them wanted this to happen. All of them are grieving.\"\n\nDeliberately Covenant put restraint out of his mind; pushed away his old fears of wild magic that refused control. Kneeling at Longwrath's side, he hammered the _krill_ through armor and raiment into Longwrath's chest.\n\nAt once, the gem blazed with such brightness that it seemed to erupt. Silver incandescence poured over the Giants and the _Haruchai_. It flooded the plain and the temple, drenched Jeremiah, ran up the slope of rubble to the ridge; denied the night. It was not fire, although Covenant thought of it as conflagration, saw it as flame. But it was capable of anything. In the hands of its rightful wielder\u2014sane or deranged, driven by love or _contemptuous of consequence_ \u2014it could shatter the foundations of Time. It made Longwrath's flesh and even his armor burn like kindling.\n\nAnd from the heart of the coruscation, Linden Avery came galloping on Hyn's back with the Staff of Law black as midnight in her hands.\n\n## 12.\n\nAfter Too Long\n\nThe flare of power from the gem resembled a scream. It struck Covenant as if it were tangible, a physical impact. He reeled backward. Quick as gusts, Branl and Stave dodged. The Giants scrambled aside from Linden's wild rush. They barely hauled Cabledarm out of the way in time.\n\nIn full career, Hyn pounded among them, past them, a prow of force cleaving a path for a second rider, a second Ranyhyn. They were halfway to the fane before Linden and her companion could bring their straining mounts to a halt.\n\nFrom atop the construct, Jeremiah flailed his arms with Earthpower blazing in each hand. If he yelled something, anything, Covenant did not hear it. A shock like vertigo seemed to unmoor his mind. He stood on a shattered world\u2014on fragments of comprehension\u2014and did not know what was happening.\n\nLinden\u2014? How\u2014?\n\nLinden hardly appeared to see her son; or she absorbed the sight of him in an instant, recognized that he was safe and whole. Wildly she wheeled as if she had arrived with furies and woe on her heels. The _krill_ 's shining glared like a crisis in her eyes.\n\nBut not alone. Covenant stared after her. Not alone?\n\nHe should have been able to identify that second Ranyhyn, that stallion. But he had no mind and could not think.\n\nThe stranger was singing\u2014or he emanated complex melodies like an aura. And he was not chasing Linden: he was her companion. Together they turned their mounts to confront the Giants and the _Haruchai_ and Covenant.\n\nThen Rime Coldspray called, \"Linden Giantfriend!\" and parts of Covenant's reality fell back into place. When she added, \"Manethrall Mahrtiir, most valiant of Ramen!\" Covenant began to regain his footing.\n\nMahrtiir? No. Impossible. _That_ was not\u2014\n\nOh, God. Blood and damnation.\n\nNarunal. The second horse was Narunal.\n\nNow Covenant recognized Mahrtiir's eyelessness, Mahrtiir's fierce visage. But the Manethrall was altogether changed; fraught with music and theurgy. His bandage was gone. In its place, the ravage which had cost him his eyes had become whole skin, seamless and new. He wore a robe of samite so white and pure that it might have been woven of starlight. Garlands of harmony draped his neck: a wreath of counterpoint adorned his head. And his mien\u2014His familiar combative frown had become radiance. It had become eagerness. Reflecting the _krill_ 's gem, he looked like wild magic cleansed of its extravagance and peril.\n\nIn his hand he carried a sapling\u2014a _sapling_ \u2014as if it were weightless in spite of its root-ball thick with loam and its wealth of new leaves like a gift of verdure to the barren plain.\n\nHis mere presence shed hymns like promises in all directions, and Narunal bore him as if the stallion had been exalted.\n\n\"No,\" Linden answered. She sounded hoarse and ragged, as if she had spent hours yelling\u2014or perhaps sobbing. \"Not Mahrtiir. Not anymore. This is Caerwood ur-Mahrtiir. If he ever gets the chance, he's going to be the Forestal of Andelain and Salva Gildenbourne.\"\n\nThe figure beside her nodded gravely. He may have been humming. Then he turned away as though Linden had introduced him to people who did not interest him. Guiding Narunal with his knees or his music, he rode, stately and ineffable, toward Jeremiah's temple.\n\nLinden remained where she was. Her eyes were full of frenzy. Too much had happened to her. Too much had happened while she was gone.\n\n\"You did\u2014\" She appeared to grope for words as if she had no names for what she saw and felt. \"Covenant, you\u2014\"\n\nYet Covenant faced her like the man who had chosen to forsake her days ago. He should have said something, _wanted_ to speak. Goading himself with curses, he strained to break the logjam of his emotions. But he was still stunned, still floundering.\n\nA _Forestal_? Of course. He had urged her to _Remember forbidding_. How else could she have done it? _Without forbidding, there is too little time_. The magicks of the ancient woodland guardians were not instruments which could be passed from one hand to another. They were inherent. So she had decided\u2014or Mahrtiir had decided for her. The Manethrall of the Ramen had been sacrificed.\n\nAnd Covenant had no idea where she had found the power to transform Mahrtiir; how far into the Land's past she had been forced to travel. Hellfire! It was no wonder that she looked wild, frantic. She had done and endured things which had shaken her heart to its foundations.\n\nHe wanted to ask, Who was it? Who did you find? He had the question ready; but he gritted his teeth against it. She needed something more from him. Something better.\n\nWhile Covenant stood paralyzed, silent and useless, Jeremiah came running from his construct. His hands burned like shouts as he sprinted toward Linden.\n\n\"Mom! You came back!\"\n\nLinden hardly glanced at him. Other concerns already gripped her. Her attention shifted from Covenant to the Giants. Her eyes widened in shock.\n\n\"God!\" she panted. \"What have you done?\"\n\nAbruptly she flung herself from Hyn's back. Unfurling fire as black as the distant storm, she strode toward the Swordmainnir. Toward Cabledarm.\n\nIn that instant, Covenant saw that the grass stains were gone from her jeans. She no longer needed them. And she looked clean, as if she had been refined by fire. Even her hair and clothes\u2014But the tatters of her shirt remained: the tearing of thorns, the bullet hole, the rent hem.\n\nShe ignored his scrutiny, his surprise. Focused on the dying Giant, she advanced as if she meant to hurl an attack.\n\nOnyx Stonemage and Stormpast Galesend flinched reflexively, then stood their ground, upholding Cabledarm between them. The other Giants stepped away to make room.\n\n\"God _damn_ it.\" Linden's voice was a raw mutter, barely audible, as if she did not expect to be heard. \"What happened to you? What have you done to yourself?\"\n\nThen she sent a torrent of flame at the damaged woman. Swift as empathy, she inundated Cabledarm with Earthpower.\n\nShe remained a healer, Covenant told himself, no matter how she judged herself. Wounds came first, pains and afflictions which she was able to treat. She had been through an ordeal: that was obvious. She must have been desperate to do _some_ thing that felt like restitution.\n\nHer effect on Cabledarm was not gentle. It was too urgent, too full of need. And perhaps she had not yet realized that the hindrance of Kevin's Dirt was gone. She seemed to scourge Cabledarm with healing.\n\nThe woman's head jerked back. Twisting against the grasp of her comrades, she gave a groan like a throttled scream. But she was not being harmed. Her pain was the hurt of internal organs violently mended, of bones roughly reset and sealed, of bleeding stanched as if it were being cauterized. When she fainted, her slackness\u2014and the new ease of her respiration\u2014suggested that she had already begun to recuperate.\n\nWatching, Covenant leaned on Branl as if he needed the comfort of the Humbled. He wanted to tell Linden that she was wonderful\u2014that he had been terrified for her\u2014that he was sorry\u2014that the world would _not see her like again_. But still he could not speak. He had no language for the extremity of his heart.\n\n\"That is well done, Linden Giantfriend,\" murmured Coldspray. \"Well done in all sooth. Now only Stave Rockbrother requires similar care.\"\n\nUnsteadily, as if she had assumed Cabledarm's fever, Linden looked around for Stave, who stood on the far side of Covenant's aborted fire. For a moment, she appeared to fix her senses and her bewilderment on the ashen remains of Longwrath's corpse. Her mouth opened for a cry of protest.\n\nThen she must have felt Jeremiah rushing toward her. She spun away from the fallen Giant and the wounded _Haruchai_ to catch her son in her arms.\n\n\"Jeremiah,\" she breathed. \"Oh, Jeremiah. I'm so sorry. I'm sorry that I had to leave you. I'm sorry that you had to do everything without me. You must have felt so abandoned\u2014\"\n\n\"Mom, stop.\" Jeremiah gripping her with flames. \" _I'm_ sorry. I acted like a kid. You did what you had to do, and I didn't even tell you I love you. I didn't tell you I understand.\"\n\nSome distance beyond the gathered company, Caerwood ur-Mahrtiir had paused in front of the fane, singing softly, rapt in contemplation.\n\n\"And we did it,\" Jeremiah added. Linden's return appeared to galvanize him. Abruptly he pulled away from her and gestured at the temple. \"We did it right. I mean, the Giants and Stave did it. They were amazing. And they came. The _Elohim_ came. They're inside. Even\u2014\"\n\nThere he faltered. His whole body seemed to clench at the memory of Kastenessen.\n\n\"I believe you,\" Linden assured him. \"I can't see them, but they left traces. They must have gone in to somewhere else, just like you said that they would. It must have been extraordinary.\"\n\nShe was making an effort to affirm her son. Nevertheless her tone was thick with tension.\n\n\"But are you all right? Did anything happen to you?\"\n\nShe must have been able to see farther into Jeremiah than Covenant could. _I'm sorry you didn't kill him_. _I want him_ dead.\n\nJeremiah ducked his head. \"There are worse things than being afraid, Mom. Being useless is worse.\" He indicated the fane again. \"The Giants did that. Stave did it. He and Cabledarm got hurt doing it. All I did was tell them what I wanted. Without them\u2014\"\n\nHelpless as a spectre, Covenant watched and listened. He loathed his silence, but did not know how to break it. Jeremiah did not mention Kastenessen or being possessed. And Covenant had no intention of telling the boy's story for him; betraying the boy's secrets. But he had so many other things to say.\n\n\"I know the feeling,\" Linden replied harshly. \"I've taken terrible chances because I felt that way. You've seen me. But it happens to all of us. We can't do everything alone. Or we can't do enough. Without help, we're all useless.\"\n\nShe was speaking to Jeremiah, but her words\u2014or her anger\u2014may have been directed at Covenant. How often had he said, _Don't touch me_? How badly had he hurt her by leaving her behind?\n\nHe, too, would have failed at everything without help.\n\nHe could almost see Jeremiah's needs aching in Linden's eyes as she turned away from her son. Innominate tensions and uncertainties ruled her. She seemed unable to stop moving.\n\nApparently she was trying to focus her attention on Stave.\n\nThe music emanating from the Forestal had changed. It had assumed a more telic mode, as if he were done with study. He had set his sapling upright directly before the fane's portal. Now he withdrew his hand\u2014and the sapling did not fall. He had already sung its roots into the hard dirt.\n\nStonemage and Galesend continued to support Cabledarm. The rest of the Giants gathered closer to Linden. Cirrus Kindwind rested her hand on Jeremiah's shoulder as if to soothe him until Linden could attend to him again.\n\nWhen Linden looked toward Stave, her gaze snagged on Branl. On the flamberge\u2014\n\nFor an instant, she froze. Fright flared in her gaze. Then her features knotted. From the Staff, she summoned a curling flame; sent fire licking up and down the rune-carved surface.\n\nBitter as bloodshed, she demanded, \"What're you doing with Longwrath's sword? Is that how you're going to kill me?\"\n\nWhat else could she think? Longwrath had tried more than once to cut her down. She had not seen him since the Wraiths had repulsed his desire for her death from Andelain. And the Humbled had distrusted her from the first, in spite of her history. They had threatened and opposed and judged her.\n\nNevertheless Covenant blurted, \"Linden, no.\" Distress broke through the blockade of his silence. Her reaction was too much for him. \"It's not like that.\"\n\n\"Why not?\" She did not glance away from Branl, or quench her power. \"He's wanted me dead ever since I resurrected you. What's changed?\"\n\nWhile she spoke, Covenant seemed to hear her crying, I woke up the Worm! Is no one _ever_ going to forgive me?\n\nYet Branl faced her without expression, without moving. He gave no sign that he regarded her as a threat.\n\n\"He killed Clyme,\" Covenant said in a frayed croak. \"Clyme let _turiya_ possess him. Then Branl killed Clyme. The Raver is gone. Everything's changed.\"\n\nAgain Linden froze. He could not read her.\n\nShe would not have forgotten Honninscrave's sacrifice against _samadhi_ Sheol, _turiya_ 's brother.\n\nNow Covenant felt driven to talk. He yearned to tell her about his alliance with the lurker. He wanted to convince her that she had made the lurker's efforts against the Worm possible. _Writ in water_. When she had eluded the snares of the Feroce, she had saved him and enabled Joan's end and given the Land precious days of life.\n\nBut he restrained himself. He needed to say such things\u2014but explanations of that kind were not what _she_ needed. She had been through too much: her nerves and her heart were too raw. An abstract alliance would not console her.\n\nNear the fane, Caerwood ur-Mahrtiir's sapling spread new branches and put forth fresh leaves and grew as if the Forestal had compressed years of rain and sun and rich soil into brief stanzas of hymnody.\n\nLinden seemed unable to move. Covenant's revelation must have shaken her conception of the Humbled. But he was not given time to continue. While he groped for better words, words that might ease her, Stave came to stand between Linden and Branl.\n\nImpassive as polished stone, he said, \"Chosen, I am here. I have done as you asked of me.\" Nothing in his gaze or his mien hinted at his intent. \"Now I am in need.\"\n\nDeliberately he showed her his savaged forearm and hand.\n\nOne of the _Haruchai_. Asking for help.\n\nAt the sight, something inside Linden snapped. Stave was her friend, one of the first. He had supported her against the combined repudiation of the Masters\u2014and had paid a cruel price. Her eyes filled with tears: she called up more fire as if her Staff's flames were sobs. But she did not reach out as she had to Bluntfist. Instead she cocooned herself in conflagration. Then she carried the dark blaze of her pain to Stave and wrapped her arms around him.\n\nAnd he returned her embrace as though he had grown accustomed to such familiarity. Accustomed to setting aside his native stoicism.\n\nA sigh of relief passed among the Giants. Jeremiah whispered, \"Mom. Mom,\" as if she made him proud.\n\nWhen she finally let Stave go, she was calmer. Quenching her power, she made the _krill_ 's illumination brighter in contrast. Still her eyes were full of darkness, as if her Staff's stain lingered in them. Trying to imagine how she had gained Mahrtiir's transformation\u2014and how she had managed her return\u2014Covenant shivered. He could only be sure that the cost had been high. But now at last she looked _present_ , as if she had been reclaimed by the time where she belonged.\n\nThat was well. The barriers inside him had broken. He could no longer remain silent.\n\nHe wanted to fall to his knees before her, abase himself somehow, plead with her. But self-recrimination was an expensive indulgence: he could not afford it. Controlling himself, he held her gaze until he was sure that he had her attention.\n\nCaerwood ur-Mahrtiir's sapling had become a young tree. Its leaves were spangled with melody as if the notes of his song were stars. And beneath the expanding spread of the branches, a furze of grass sprouted from the barren ground, punctuated by undefined clumps that might grow into shrubs. A liquid sound ran faintly through his singing like a promise of water.\n\nTo Linden, Covenant said, \"I killed her,\" as if the words burned his mouth, raised blisters on his tongue. \"I killed Joan. I promised myself I would give up killing. Now I hardly do anything else.\"\n\nLike the voice of the night, Branl asserted, \"It was not murder.\" Like an echo. \"It was mercy.\"\n\nStave nodded his assent.\n\nCovenant ignored the _Haruchai_. He concentrated on Linden's frown, and her eyes, and the tightness of her mouth.\n\n\"The Feroce cleared the way. Dozens of them died against the _skest_. Branl and Clyme helped me through a _caesure_ so I could reach her. She was going to finish me, but Mhornym and Naybahn distracted her. I killed her with the _krill_. I didn't know what else to do.\"\n\nLinden seemed to gather darkness where she stood, as if she were pulling the night around her, wrapping herself in shadows. \"Good!\" she snapped: a flare of vehemence like a reiteration of Jeremiah's rage at Kastenessen.\n\nAbrupt anger swelled in Covenant. \"There was a tsunami.\" Joan had suffered too much. \"It could have crushed us.\" Her weakness had not merited the use which Lord Foul had made of her. His voice rose, impelled higher by fury or supplication. \"Branl and Clyme and the Ranyhyn saved me.\n\n\"Hell _fire_ , Linden.\" His own heart was as raw as hers. \"Do you remember Brinn? Now that the Worm's awake, he doesn't have anything else to do. He showed up after the tsunami to tell us _turiya_ was going to possess the lurker. We went to try to stop that from happening.\"\n\nThen he forced himself to stop. In spite of his ire and numbness, he felt Linden withdrawing. She did not step back, but her frown became a scowl as her features closed against him.\n\n\"Why are you angry at me?\" Her voice shook. \"I haven't touched you. I wasn't even here.\"\n\nCovenant swore at his clumsiness, his difficult, stymied honesty. He bit down on his wrath hard enough to draw blood.\n\n\"I'm not angry at you. I'm ashamed. It's not the same thing.\"\n\nJeremiah may have tried to intervene. If so, the Giants kept him quiet.\n\n\"What are you ashamed of?\" Linden sounded impossibly distant, as if she had retreated to a redoubt where he could not hope to reach her. \"You put Joan out of her misery. She wasn't just in terrible pain. She was possessed. Death was the only way to give her any relief. And you stopped her _caesures_. Why is that something to be ashamed of?\"\n\n\"Because I failed!\" Covenant wanted to hit someone, anyone. If he could have felt what he was doing, he would have torn at his hair. Instead he knotted his insensate fingers together and twisted until his wrists ached. \"I wasn't strong enough to handle _turiya_ myself. That's why Branl had to kill Clyme. They both had to compensate for me.\n\n\"And because\u2014\"\n\nSuddenly awkward, he faltered. How could he say what was in his heart? To Linden? Like this? Beyond question, he was not strong enough. If he had ever been brave enough, he no longer remembered how that much courage felt.\n\nThe glittering among the leaves of the ur-Mahrtiir's tree had become a silver penumbra, purer that the brightness of Loric's _krill_ , and more melodious. The tree was a willow, graceful and arching. Soon it would be tall enough to spread its branches in a wide circle that included the fane. Its limbs drooped like weeping, though they grew like gladness. And under its shade, the thin grass was now turf, as lush as the greenswards of Andelain. Bushes grew like adornments under the dangling leaves around the verge of the grass. A delicate rill rippled argent past the Forestal's feet and chuckled away beyond the rubble, wending harmoniously toward the distant Sarangrave.\n\n\"Because?\" Linden prompted like a woman hiding behind a shield.\n\nHe was losing her. He did not know how to bear it.\n\n\"Because I hate the way I treated you! I hate the way I _left_ you. I had to go. I had to go alone. I couldn't risk you against Joan. And you had other things to do.\"\n\nFinally he managed to lower his voice. If he meant to tell the truth, he had to set aside the luxury of shouting; of judging himself.\n\n\"Linden, do you understand that _Kastenessen_ is in that temple? Have you realized yet that Kevin's Dirt is gone? If I hadn't left you behind, none of that would have happened.\"\n\nShe did not react. She had no attention to spare for victories.\n\nGroaning inwardly, Covenant confessed, \"But I shouldn't have treated you the way I did. I was just afraid. I was broken,\" maimed by fissured memories, \"and I didn't know how to live with it. I couldn't ask you to trust me,\" love me, \"because I didn't trust myself, or what I was becoming, or what I had to do. I wasn't sure I would have anything left when I was done. I couldn't say what I really meant.\"\n\nLoric's gem lit a subtle shift in Linden's gaze, a modulation in the darkness. Small black flames coiled like tendrils around her hand on the Staff. Covenant thought that he saw tremors in her shoulders.\n\n\"You told me not to touch you,\" she said as if the words were splinters of glass, sharp enough to pierce and rend. \"Isn't that what you meant?\"\n\n\" _No_.\" He gritted his teeth so that he would not cry out. \"It's what I needed. It's what I knew how to say. I'm a leper, for God's sake. It's how I cope with practically everything. But it is not the truth.\"\n\nNot the whole truth.\n\nShe appeared to be floundering: a drowning woman who nonetheless struggled against her desire to clutch at rescue. So softly that he barely heard her over the labor of his heart, she asked, \"Then what is the truth? What would you have said if you weren't broken or scared?\"\n\nObviously bewildered, Jeremiah watched his mother and his earliest friend. The _Haruchai_ betrayed no reaction; but the Giants gave the impression that they were holding their breath.\n\nDamn you, Covenant snarled at himself. Say it. _Do_ it. She can't read your mind.\n\nWhat did he gain by being a leper if numbness did not dull the edges of his fears?\n\nHis hands shook as he reached up to his neck. Fumbling, he grasped the chain that held Joan's ring under his T-shirt, pulled the chain over his head. For a panicked moment, his eyes failed him: he could not find the clasp. Then his fingers were too awkward to unclose it.\n\nBut he remembered who he was, and why he was here, and what was at stake; and a strange certainty came over him. The clasp seemed to open by itself, as if he had been given a blessing. _Attempts must be made_ \u2014How else could he believe in anything?\n\nHe dropped the chain. Holding the ring between the remnants of his thumb and forefinger, he extended it toward Linden.\n\n\"Linden Avery.\" His voice was hoarse, congested with emotions straining for release. \"I think I've earned the right to give this to anybody I want. But there's nobody else. I love _you_. That's all. I _love_ you. Will you marry me?\"\n\nShe flinched as if he had slapped her. For an instant, she recoiled, startled and uncomprehending.\n\nBut while she froze, caught in a maelstrom of surprise and consternation, disbelief and repudiation and self-doubt, the Forestal's song came clearly through the silence.\n\n\"It is my heart I give to you,\n\nMy blood and sap and bone and root.\"\n\nIn an instant, her turmoil was transfigured. Out of confusion and pain, she gathered herself. Her eyes reflected argent and recognition in patterns that spoke to Covenant. Without glancing at Stave, she tossed her Staff to the former Master. Its fire vanished before he caught it.\n\nHer gaze clung to Covenant's as she drew out his ring, freed it from its chain, discarded the strand as if it had become meaningless. For a few heartbeats, she closed the ring in her fist. Then she opened her hand, held the ring out on her palm.\n\n\" _Yes_.\" That one word seemed to contain her whole heart. \"Thomas Covenant, yes. I don't care what you've done, or what you're afraid of, or what you said days ago. I don't care how broken you were, or what's going to happen to us later. I only care about _now_. I love you.\"\n\nAs if she had summoned him past restrictions more personal than life and death, he started toward her. When he reached her, he took her left hand, lifted it to his lips, then slipped Joan's ring\u2014no, _Linden's_ wedding band\u2014onto her ring finger.\n\nWith this ring I thee wed.\n\n_And betimes some wonder is wrought\u2014_\n\nHe thought that she would offer his ring to the index finger of his halfhand, where he had worn it ever since he had grown gaunt. But instead she claimed his left. To his surprise, his ring finger accepted the band as if damage and scars had made him strong enough to wear white gold where it belonged.\n\n\"I'm yours,\" she murmured through a blur of tears. \"You're the only man I've ever really loved. You're the father Jeremiah should have had. As long as you wear this ring, I'm yours.\"\n\nHe knew what she meant. Long ago, he had surrendered his wedding band to the Despiser.\n\nHe was not going to do that again.\n\nWhen he took her in his arms and kissed her, he was trying to assure her that he would keep this promise.\n\nHer arms were around his neck. She returned his kiss as if she were opening her whole self.\n\nAnd slowly their embrace was transformed. It became a glow of wild magic. Alloyed argent expanded around them, wrapped them in light. Gentle as a caress, it swelled into the night, swirling warmly as it scaled higher and higher until they appeared to stand at the source of a gyre which might reach the stars. The gem of the _krill_ gave answer, as if High Lord Loric's ancient theurgy approved; but the effulgence of Thomas Covenant and Linden Avery out-shone it. Their power lit the battered plain to the horizons, reveled on the faces of the Giants and the _Haruchai_ and Jeremiah, emblazoned the outlines of the fane beneath the willow. Even Caerwood ur-Mahrtiir paused in his fertile labors to contribute a paean like a benediction.\n\nIf Covenant had been inclined to heed them, he would have heard the Giants cheering. He would have seen Jeremiah waving flares of Earthpower and grinning. He might have noticed Stave's brief, unconflicted smile.\n\nBut Covenant was kissing Linden. At that moment, nothing else mattered.\n\nhen he was finally able to look around, he saw that the Forestal had fashioned a bower.\n\nThe willow had grown as tall as a Gilden. Spangles of song lingered on its leaves, bedecked its branches with bright silver like the glimmering of unendangered stars. Illumination under the canopy of the boughs seemed to hold the memory of wild magic made tender by acquiescence. The tree stood directly before the fane's portal: its drooping arch almost concealed the construct. In the tree's shade, luxuriant grass cushioned the ground like a profusion of pillows.\n\nThe plashing runnel was now a grateful brook. It seemed to carry light and music with it as it chimed out across the plain. And near the edges of the circle, where the leaves trailed along the grass, Caerwood ur-Mahrtiir had invoked _aliantha_. A score or more of the holly-like shrubs with their viridian berries ripe surrounded the greensward, abundant as a feast.\n\nThe relative privacy of the bower suggested a form of sustenance that Covenant needed more than food. Perhaps that was the Forestal's intent. The heat in Linden's eyes affirmed that she felt as Covenant did. He was in a trembling hurry.\n\nBut the company had other needs: those took precedence. The privation of the Giants was extreme. They had given their last strength\u2014and then had given more. Covenant himself wanted more than the unsatisfying aliment of _ussusimiel_. Linden had probably gone longer without food. And Jeremiah was avid for treasure-berries.\n\nFor the sake of everyone with him, Covenant schooled himself to eat and drink and wait. When Linden smiled ruefully, he tried to match her.\n\nSpeaking for her comrades, the Ironhand gave thanks to the Forestal. They all bowed as if they declined to prostrate themselves only because they lacked the strength to rise again. Then they picked their fill of _aliantha_. The seeds they scattered around the plain and in the hollows like prayers for the Land's future. More boisterously, Jeremiah followed their example. As for the _Haruchai_ , Branl stood apart from the company as if all of his lacks had been satisfied by Longwrath's flamberge; but Stave ate without hesitation and offered the former Manethrall his gratitude.\n\nConsidering that they were Giants, inclined to relish the bounty of their own relief, Rime Coldspray, Frostheart Grueburn, and the others finished their meal quickly. They spent only a few moments thanking Caerwood ur-Mahrtiir. Then they passed beyond the thick willow-trunk to reenter the fane, taking Jeremiah with them so that Covenant and Linden would have some semblance of privacy.\n\nStave also went into the construct, bowing first to Caerwood ur-Mahrtiir, then to Covenant, finally and most deeply to Linden. However, Branl remained. \"Ur-Lord,\" the Humbled said with his usual absence of inflection, \"the return of the Chosen is a cause for gladness in itself, and is more so because she has restored a Forestal to the Land. Yet in one respect, it is misfortune. The Giants have been denied their _caamora_.\"\n\nTo the sudden inquiry of Linden's expression, he explained, \"The ur-Lord sought to relieve their sorrow by drawing flame from Longwrath's remains. Your arrival interrupted his efforts. Now Longwrath is naught but ash, and we have no wood.\"\n\nWhile Linden winced in regret, Branl addressed Covenant once more. \"Among Giants, denied lamentation is an enduring distress. Other tasks we have in abundance. And doubtless the Swordmainnir will be prompt to set aside their needs. Nonetheless I urge you to seek some blaze in which they may ease their loss.\n\n\"I am a Master of the Land,\" he said as if he were merely reciting a formula rather than acknowledging a profound change. \"I bear the taint of the unwelcome which the Giants have received at our hands. I would make amends, but have no means to do so.\"\n\n\"Oh, stop,\" Covenant protested. \"I forgot about that. We all had too much going on. But of course you're right. I\"\u2014he glanced at Linden\u2014\"we won't forget again.\"\n\n\"We won't,\" Linden affirmed. \"And I won't forget what you've done. I haven't been fair to you. I should have known better.\"\n\nInstead of nodding to her, as he had done so often in the past, Branl bowed. And when he had shown the same respect to both Covenant and Caerwood ur-Mahrtiir, he left the bower between the hanging branches to stand guard outside.\n\nAlone with the Forestal, Covenant and Linden faced each other as if they had lost the ability to look anywhere else; but they did not move.\n\nBriefly Caerwood ur-Mahrtiir sang words that Covenant recognized.\n\n\"I am the Land's Creator's hold:\n\nI inhale all expiring breath,\n\nAnd breathe out life to bind and heal.\"\n\nThen he faded into his music as if he had made himself one with the willow and the boughs, the leaves and the bedizening melody. In a moment, he was gone.\n\n\"Covenant\u2014\" Linden bit her lip, twisted the ring on her finger. \"I have too much to tell you. And there are so many things I\u2014\"\n\nHe interrupted her with a grin that felt like a grimace. \"Don't you think it's about time you stopped calling me 'Covenant'?\"\n\n\"Thomas, then,\" she offered. \"Thomas. Thomas of my heart.\"\n\nHe would have accepted anything, but he was grateful that she did not choose to call him _Tom_.\n\nWhen he opened his arms, she came to him like an act of grace.\n\nhen they were done, they lay relaxed on billows of grass, covered by the soft radiance of the bower. For a time, they talked casually, softly, reminding themselves of each other. But then they turned to more serious concerns.\n\nCovenant had his own questions, but Linden spoke first. Somber with doubt, she asked him what he thought about Jeremiah.\n\nHe sighed to himself. \"You mean, not counting the fact he's actually _with_ us? After what he's been through? It's amazing he can so much as speak, never mind design that sanctuary for the _Elohim_. He's already done a world of good. If you want more, you should talk to him.\"\n\nShe certainly needed to know how much her son had inherited from Anele. She needed to know about Kastenessen.\n\nShe cuffed him lightly. \"That's not an answer.\"\n\n\"I know. But I'm serious. He should tell his own story. He doesn't want to, but he should. Maybe you'll have better luck than I did.\"\n\nLinden gnawed at her lower lip for a moment. \"I'm not sure that I have the right to pry. He's already pushed me away more than once. I might do more harm than good.\"\n\nCovenant shrugged against her head on his shoulder. \"I'm not sure anybody has the right. Maybe prying does more harm than good. But look at it this way. He's too young for his years. He's had experiences that could cripple an adult, and he's never had a chance to grow into them. Parts of him are still a kid.\" And parts of him remembered the _croyel_.\n\n\"Sometimes kids need their parents to pry. Sometimes I think Roger wouldn't be such a mess if his mother ever took an interest in him.\"\n\nCovenant himself had never been given an opportunity with his son.\n\nLuminous in the warmth of Caerwood ur-Mahrtiir's music, Linden rolled over to rest her hands on Covenant's chest, prop her chin there and study his face.\n\n\"Thomas, what happened to you? What did you do after you left? How did you do it? What healed your mind? How did you change how Branl thinks?\"\n\nHe winced reflexively. But he did not refuse to answer. Eased by her love, he was able to describe the days that he had spent away from her.\n\nWhen he was done, she hugged him hard and wordlessly. For a time, she seemed to take his anguish and dread from him; and he thought about nothing except her.\n\nAfterward they rested. But neither of them slept.\n\nIn a more playful mood, she asked, \"So why aren't you growing a beard? You're human now. All the way human. As far as I can tell, the Arch of Time has lost its hold on you. Why isn't your beard growing?\"\n\n\"I don't know,\" he admitted. \"If I ever did, it's gone. But if I had to guess\u2014\"\n\nBriefly he rubbed at his cheeks, pushed his fingers through his transformed hair. \"You didn't have access to my physical self. That part of me died so long ago there was nothing left. And yet here I am. You must have created me out of my self-image.\" He spread his maimed hands. \"Apparently that includes leprosy, but it doesn't include whiskers.\"\n\nLong ago, shaving had been a form of self-abnegation for him, a punitive discipline. He was glad to be rid of the necessity.\n\nStroking her, he said, \"Now it's your turn. Linden, you're a mystery to me. And I don't just mean\u2014\" He gestured to indicate her adored body. \"I don't think I've ever been as surprised as I was when the Feroce offered me an alliance.\" Surprised and dismayed. \"Somehow you did that. Somehow you saved me.\" He would not have reached Joan, or survived his attempt on _turiya_ Herem, without the aid of Feroce. \"But you did more than that. You also rescued Jeremiah.\" When she shook her head, he amended, \"I mean, you gave him what he needed to rescue himself.\n\n\"That would have been enough for anybody else, but not for you.\" Not for a woman who thought so little of herself. \"After you brought Jeremiah here, you went to find the only possible source of forbidding.\" The only hope for the _Elohim_ , and perhaps for the Earth. \"Then you did something even more miraculous. You came back. Without using a _caesure_.\n\n\"Linden\"\u2014he kissed her eyelids, her nose, her mouth\u2014\"you amaze me. I want to know how you did it.\"\n\nHe saw her reluctance. It showed in the way she shifted to nestle against his shoulder so that he could not look into her eyes or watch her face. For a moment, he was afraid again. But then she began to answer, and his fear left him.\n\nBecause he knew the outcome, he listened calmly as she described how the Feroce had tried to lure her into the grasp of the lurker, and how Infelice had striven to prevent Jeremiah from freeing himself in Muirwin Delenoth. Jeremiah's desire to build a construct that might preserve the _Elohim_. The message of the Feroce. Her own decision to enter a _caesure_. Her arms tightened like grief around Covenant as she talked about her second meeting with Caerroil Wildwood, and about Manethrall Mahrtiir's transformation.\n\n\"But I still didn't know how to get back. After what Caerroil Wildwood did for us, the idea of making another Fall horrified me. I would have had to ruin an unconscionable amount of Garroting Deep. But I was desperate to return, and I couldn't wait until we left the forest. I didn't know what to do.\"\n\nCovenant heard the force of that emotional snare in her voice, the intolerable conundrum of being caught between mutually exclusive commitments. He recognized it.\n\n\"Mahrtiir\"\u2014she corrected herself\u2014\"no, Caerwood ur-Mahrtiir helped me. You should have seen him, Thomas. He stood here like a king, as if he had earned the right, and he sang things that I couldn't understand until Caerroil Wildwood nodded. Then Wildwood gave me another gift.\"\n\nLike suppressed weeping, she said, \"Oh, Thomas. Caerroil Wildwood said that he was tired of living. Tired of trying. Worn out by losing trees to people and wars. Law was getting weaker, and he knew that he was doomed. He'd faced too much evil. That was why he created Caer-Caveral, and why he made Caerwood ur-Mahrtiir. So that he could finally rest.\n\n\"He told me\"\u2014her voice broke\u2014\"that he still had no answer for the deaths of trees.\"\n\nThen she hurried to finish.\n\n\"Every leaf and branch all around the Howe sounded like it was sobbing, but he had made up his mind. He brought Hyn and Narunal to us. He gave us time to mount. 'By wild magic you came,' he said. 'Wild magic must guide your return.' When we were ready, he did something like what the Mahdoubt did for me. He didn't violate Time, he used everything he was to make an opening.\" Covenant felt her tears on the soft skin of his shoulder. \"Then he pushed us through so that Hyn and Narunal could find the way back.\n\n\"It killed him, just like it killed Caer-Caveral. All of his music and glory and anger and effort seemed to wail. The whole Howe was like a shriek. When we rode away, there was nothing left except screaming.\"\n\nTrying to comfort her, Covenant murmured, \"I wish I could remember.\" He did not care what he said: he only sought to acknowledge her distress. \"While I was still part of the Arch, I probably knew why Caerroil Wildwood decided to let go. Now that's gone. As far as I can tell, you found the only\u2014I don't know what to call it\u2014the only clean way to do what we need. The only safe way. The only way that doesn't change the Land's history.\"\n\nLinden wiped her eyes and nose. Under his caresses, her tension and remorse eased. \"I've been so scared. I didn't know what I was doing. Half of the time, I felt terrified. Otherwise I was just frantic. Jeremiah and the Land and even you needed more from me than I knew how to give. I only did what I did because I couldn't think of anything else.\"\n\n\"Hellfire, Linden,\" Covenant snorted. \"Don't sell yourself short. Miracles are becoming practically normal around here, and most of them are your doing.\"\n\nWhen she felt less troubled in his arms, he asked a different question. \"So how did you get rid of those stains on your jeans?\"\n\nShe lifted her head in surprise. After a moment, she sat up, snatched at her jeans, studied them. \"Oh my God. They're gone. I've had them for so long, I stopped seeing them. They must have faded when Caerroil Wildwood\u2014\"\n\nEyes wide, she faced Covenant. \"What does it mean?\"\n\nHe smiled crookedly. Still hungry for her, he said, \"Maybe Caerroil Wildwood took away those stains because you don't need them anymore. They were a map, and now you've found your way.\" She had found him\u2014or they had found each other. \"Maybe it just means we should try to take advantage of every minute we have left.\"\n\nFor a moment, she appeared to struggle against her uncertainty\u2014or against the particular intensity of his regard. But then she seemed to find that he had said enough. That his response sufficed. Dropping her clothes, she moved to put her arms around his neck.\n\nThat response sufficed for him as well.\n\nventually Linden asked a more difficult question. \"After Lord Foul killed you, you left your ring for me. You wanted me to have it, didn't you? So why haven't I been a 'rightful white gold wielder' all along?\"\n\n\"I'm not entirely sure,\" Covenant admitted. \"Sure, I wanted you to have my ring. But I didn't _give_ it to you. Lord Foul just dropped it. And I was in the same situation with Joan. I only got her ring\"\u2014he stifled a wince\u2014\"because she couldn't hold it any longer. That didn't make me a rightful wielder either.\"\n\nHe had experienced _rightfulness_. He knew what it meant.\n\n\"Now that's changed.\" With a gesture that felt effortless, he drew a brief streak of argent through the air, instantly ready, instantly quenched. \"So here's what I think. It isn't the getting that makes the difference. It's the giving. The choice. And the _kind_ of choice matters. Surrender is one kind. A vow is another. I didn't just give you a white gold ring. I gave you _me_. That's something the almighty Despiser is never going to understand. He's clever as all hell, but he's too self-obsessed or frustrated or maybe too damn miserable to figure out why he keeps losing.\"\n\nThen Covenant thought that he ought to warn Linden. \"But we still have to be careful. I don't have enough health-sense to feel the effects of what I'm doing. And you have the Staff of Law.\" It lay on the greensward beyond their clothes, its black shaft runed with auguries. \"I don't want to say wild magic and Law are antithetical. That's too simplistic. But the energies are incompatible. Wild magic refuses boundaries, and Law is all about boundaries. If you hadn't used the _krill_ when you resurrected me, you would have torn yourself apart. That's the _krill_ 's real power. It mediates contradictions.\"\n\nFor a moment, he thought that he heard the wind outside the bower blustering bitterly against the willow. But the blast did not trouble Caerwood ur-Mahrtiir's irenic singing, or ruffle his healing lumination.\n\nStill Covenant did not relax, in spite of his satiation. He sensed something unresolved in Linden\u2014or he knew that in her place he would not be at peace.\n\nAt last, she said, \"Thomas, I love you. I _love_ you. But I did a terrible thing when I forced you back to life. Waking up the Worm was bad enough. The Humbled were right about me. That was a Desecration. But I'm afraid that I did something worse at the same time. Do you remember what Berek said? I've made it impossible to stop Lord Foul.\"\n\nCovenant tightened his embrace as if he imagined that he could protect her. He remembered Berek's assertion perfectly. _He may be freed only by one who is compelled by rage, and contemptuous of consequence_. He recognized her fear.\n\n\"Now we can't save the world. We can't stop the Worm. We can only try to slow it down. Before long, Lord Foul will get his chance to escape.\n\n\"Thomas,\" she insisted, \"I _did_ that.\" In spite of all that she had done, she still found cause to accuse herself. \" _I_ did it.\" Then she admitted, \"But it didn't feel that way. Oh, I didn't care about the consequences. I can't deny that. But was I 'compelled by rage'? I don't think so. I was just desperate. Desperate for you. Desperate for Jeremiah. Desperate for _help_. I didn't know where else to get it.\n\n\"Is that all it takes to ruin everything? Is Lord Foul going to get free because of me? Is the whole Earth going to die because of me?\"\n\nAt that moment, Covenant would have given the remains of his fingers to reassure her. But he did not respond immediately. He had good reason to be cautious. During his early visits to the Land, he had justified himself falsely too often, and the cost of his obfuscations had been too high. And her needs were not his. Her desperation was not the same as his. It was more intimate, or more personal, or simply more consequential. He had only raped Lena and betrayed Elena and destroyed the first Staff of Law. He had not awakened the Worm. In an earlier age, Linden herself had prevented him\u2014\n\nNow he suspected that Jeremiah was more likely to be _compelled by rage_.\n\nHe wanted to say, Maybe you're right. Any one of us can destroy the whole world\u2014if it's _our_ world. All we have to do is destroy ourselves. But he demanded more of himself.\n\n\"Sometimes 'desperate,'\" he began, \"is just a convenient name for being so angry you can't stand it. After everything you went through\u2014after Roger and the _croyel_ and Esmer and Kastenessen and the Harrow and even Longwrath\u2014you finally got to Andelain\"\u2014he winced at the memory\u2014\"and I refused to talk to you. Hellfire, Linden! Only a dead woman wouldn't have been sick with fury.\"\n\nShe hid her face as if she were cowering; as if he had poured acid on her heart. \"Then I've done it. I've doomed\u2014\"\n\nIf she had pulled away from him, he might have cried out. He had hurt her enough to maim the bond which they had only begun to renew. But she still clung to him as if he were all that she had left. She still thought that he had a better answer\u2014or that he _was_ a better answer.\n\nAs gently as he could, he said, \"It's tempting to think that way. It lets us off the hook. If we've already made the only mistakes that matter\u2014or if somehow we just _are_ the only mistakes that matter\u2014we can't be expected to do anything else. But it's not that simple.\n\n\"For one thing, we aren't alone. We're all in this mess together. We're all making decisions and trying to justify the consequences. Whatever you've done, good or bad, you didn't do it in a vacuum. You've been reacting to people with their own agendas and situations you didn't cause. From the start, the Despiser has been pushing you where he wants you to go. And you've had help along the way.\n\n\"And for another\u2014\" Goaded by his own necessary passions, Covenant's voice rose. \"Linden, I just don't _believe_ it. I don't believe Lord Foul can't be stopped. I don't even believe the world can't be saved. Freeing Lord Foul wasn't the only thing Berek talked about. He also said there's _another truth_ on the far side of despair and doom. All we have to do is find it.\"\n\nShe did not react. He could not be sure that she was listening. He might have been speaking to the leaves and boughs, the harmony of gleams, rather than to the woman in his arms.\n\nNevertheless she continued to hold on to him.\n\n_You will not fail, however he may assail you. There is also love in the world._\n\nBecause she did not let go, he said more.\n\n\"And for another\u2014Oh, hell. I've written entire novels about this. 'Guilt is power. Only the damned can be saved.' Maybe that sounds cynical. Maybe it is. But who else _needs_ to be saved? Who else _can_ be? Not the innocent. They have their own problems.\" He was thinking of the Masters, who thought that rigid purity of service would relieve their ancient humiliation. \"They don't need anything as gracious or just plain kind as forgiveness.\n\n\"So maybe blaming ourselves is a waste of time. Maybe we should just admit that everybody goes wrong. Everybody does damage. That's what being human enough to make mistakes means. And if that's what being human means, then there's really only one question we have to answer. Is making mistakes _all_ it means?\n\n\"If it isn't, then _everything_ counts. Resurrecting me and waking up the Worm. Making love together and killing Cavewights. Hell and blood, Linden! I let my own daughter be sacrificed against She Who Must Not Be Named. And I didn't stop there. I went right up to the most pitiful woman I've ever known and stuck a knife in her chest. If you think I don't feel _bad_ about things like that, you haven't been paying attention. But if everything counts, then guilt is no reason to stop trying for something better.\"\n\nSomewhere among the music of his lights, Caerwood ur-Mahrtiir sang, \"It is my heart I give to you\u2014\"\n\nFinally Linden stirred. With small movements, she shifted the position of her arms, adjusted her head on Covenant's shoulder. For a time, she conveyed the impression that she was listening to the Forestal, or to the rebuffed thrash of the winds beyond the bower, or to the restless concern of Covenant's pulse. Then she brushed a delicate kiss across his chest.\n\n\"Here's the funny part,\" she murmured. \"I tried to say practically the same thing to Jeremiah. I used different words, but the point was the same. Maybe I should listen to myself every once in a while. You shouldn't have to make a speech whenever I think that I've done something wrong.\"\n\nSuddenly she yawned. \"If I weren't so sleepy, I would ask you to make love again.\"\n\nEntirely to himself, Covenant released a deep sigh of relief. There were any number of questions for which he had no answer; but for the time being, he was content with the one that she had given him.\n\n_You do not forgive._\n\nPerhaps she did.\n\n## 1.\n\nA Tale Which Will Remain\n\nWeary to the core, and yet eased in more ways than she knew how to name, Linden Avery dozed in Covenant's arms, _Thomas of my heart_. But she did not sleep deeply or long. After a time, a rustle among the willow branches plucked at her attention. She felt the pressure of hooves on the sumptuous grass, followed by the sounds of feeding. Casting a bleary glance over her shoulder, she found horses in the bower.\n\nHyn and Hynyn. Khelen. Rallyn. And the Ardent's mulish steed, Mishio Massima. In this lifeless region, their need for fodder had become imperative.\n\nLinden closed her eyes again, nestled against the anodyne of Covenant's shoulder. Her only true lover\u2014He had never stopped loving her: she believed that now. To some extent, she understood why he had seemed to spurn her days ago. And those aspects of his singular straits that still baffled her did not mar her gratitude. The sensation that he had vindicated her, body and soul, was more profound than her fatigue. It felt numinous and ineffable: a homecoming of the spirit. Every part of him had become as precious to her as a sunrise.\n\nThe ring on her finger resembled certainty. She could have spent days with her husband in the balm of Caerwood ur-Mahrtiir's bower, and done so gladly.\n\nBut eventually the snorts and snuffles of horses cropping grass prodded her to wonder how much time had passed. Motionless so she would not disturb Covenant, she extended her senses beyond the Forestal's bedizened canopy, and was surprised to discern that dawn was near: the feigned dawn of a sunless day. The fourth day\u2014was it really the fourth?\u2014since the sun had failed to rise.\n\nHer companions had left her alone with Covenant for most of the night. Even Jeremiah\u2014\n\nCurious now, Linden raised her head to look around.\n\nMelodies gemmed the leaves overhead as if they had been set in place to watch over her and Covenant; but of Caerwood ur-Mahrtiir there was no sign. He had hidden himself in the fecund intricacies of his hymns. Apart from the horses, she saw only the broad trunk of the tree, and beyond it the fane of the _Elohim_.\n\nGroaning softly, Covenant blinked his eyes open. When his gaze found Linden, he tried to smile: an awkward twist of his mouth. In the delicate light of the Forestal's music, the pale scar on his forehead seemed to glow. It might have been a nascent anadem, an old wound that was slowly becoming a crown. The stark silver of his hair promised flames.\n\nRemembering his ardor, she felt a delicious shiver like an intimation of the life that she wanted to have with him.\n\nAn impossible life while the Worm stalked the World's End, and Lord Foul plotted to reclaim Jeremiah.\n\nCovenant propped himself up on his elbows and looked her over with yearning in his eyes. He seemed to desire every contour. Then he frowned ruefully. Nodding toward the Ranyhyn and Mishio Massima, he muttered in mock-disgust, \"I probably shouldn't say this, Linden, but I don't really like horses.\"\n\nShe laughed softly. \"Neither do I.\" He made her name sound like a cherished endearment. \"But I'm very fond of Hyn,\" she added in case the mare understood her. \"And Khelen, of course.\"\n\nHow could she feel anything other than affection for them?\n\nAs if her response were a cue, Jeremiah called from within the fane, \"Mom? Can we come out? We're hungry. You have all the _aliantha_.\"\n\nShe was on the verge of saying, Sure, honey, when she remembered that she was naked.\n\nStifling a giggle, she answered, \"Give us a minute.\" She looked at Covenant, offered him a lop-sided smile, kissed him swiftly. Then she reached for her clothes.\n\n\"Hellfire,\" he growled under his breath. \"Bloody damnation.\"\n\nHe had not had enough of peace and privacy, or of her.\n\nShe pulled up her jeans, buttoned her shirt without regarding its tears and snags, its neat hole over her heart. Leaving her feet bare to enjoy the lush grass a little longer, she retrieved her Staff. Then she paused to study Covenant.\n\nHis leprosy had worsened in recent days. A slight haze occluded his vision. She suspected that he could not see clearly past twenty or thirty paces. And the numbness of his fingers stretched into his palms toward his wrists. His toes, and patches on the soles of his feet, had no sensation. Now the end of Kevin's Dirt had halted his deterioration. She found no indication that his symptoms were still spreading. Nevertheless he was farther from health than he had been when she had first resurrected him.\n\nHe fumbled into his jeans, worked his T-shirt over his head. While he tugged at the laces of his boots, she asked tentatively, \"Do you want any help, Thomas? I can heal\u2014\"\n\nHe hesitated for a moment, scowled, then shook his head. \"Thanks anyway. I can see well enough.\" He seemed to mean, Well enough for what I have to do. \"And I need my hands like this. The _krill_ gets hot. If I'm in too much pain, I won't be able to hold it.\"\n\nShe considered asking, Why is that important? How much do you know about what we have to do? But she rejected the idea. She did not want an answer: not really. She was in no hurry to think about the Despiser and the World's End.\n\nCovenant gave her a look full of hunger. Then he shrugged and nodded his readiness.\n\nHolding his gaze, she raised her voice. \"Come on out, Jeremiah. All of you. It's time.\"\n\nAt once, Jeremiah emerged from the temple. The sight of him both lifted and soured Linden's spirit. The emotions clenched inside him showed in his aura. He could smile because she had come back for him, and because she and Covenant were finally united\u2014and because he had been able to sleep. But the effects of Kastenessen's possession persisted: he did not know how to relieve them. And he had accomplished his one purpose. In the aftermath, he had lost the eagerness of his talent, the excitement which had driven and protected him. His ruined pajamas and his muddy gaze made him look haunted.\n\nBehind him loomed the Swordmainnir, grinning. Sleep and gladness had refreshed them, and their eyes as they regarded Linden and Covenant seemed to glow with warmth.\n\nRime Coldspray approached first, followed by Cirrus Kindwind, and then by Cabledarm brimming with restored wholeness. The other women carried their depleted waterskins. Among them, Stave walked like a man who had never been harmed.\n\nCovenant rose from the grass to greet them. With a mixture of pleasure and regret, he said gruffly, \"I should probably thank you. But I'm sure you can understand that one night just isn't enough.\" He touched Linden's shoulder briefly. \"I feel like I've been waiting for this my whole life, and now it's over\"\u2014he grimaced\u2014\"unless we can do things that are even more unlikely than what we've already done.\" Glowering like a man who did not know how to smile, he finished, \"Just once, I would like to face a challenge that turns out to be easy.\"\n\nLinden smiled for him. He had given her another gift to counterbalance the night's passing. Indirectly, perhaps, but unmistakably, he had already reassumed his rightful place as the leader of the Land's defenders.\n\n\"Yet betimes, Timewarden,\" replied the Ironhand, \"we are granted ease. To behold you and Linden Giantfriend as you are does not test my heart. It gives only joy.\"\n\nCovenant ducked his head. \"Maybe that's why I've always loved Giants. You remind me\u2014\" He spread his hands as if he had run out of words.\n\nLinden guessed that he was recalling Saltheart Foamfollower; or perhaps Pitchwife and the First of the Search.\n\nBut other matters quickly claimed the attention of the Swordmainnir. They were hungry, of course. And they knew as well as Linden did, or Covenant, that all of the company's deeds so far were only stopgaps. Branl outside the bower would have given warning of any imminent threat; but every peril was growing, and time was running out. With both pleasure and rue, the Ironhand and her comrades turned to Caerwood ur-Mahrtiir's abundance of _aliantha_ and clean water.\n\nBefore Jeremiah could join them, Linden stopped him with a hug. \"Can we talk, honey?\" she asked privately. \"I haven't had a chance to hear how you're handling what you've been through.\"\n\nHe avoided her gaze. \"There isn't much to tell, Mom. The Giants and Stave did everything. I mean, pretty much. All I did was organize the pieces and make sure they fit.\"\n\nShe recognized the deflection in his voice, but she did not question it. Instead she insisted mildly, \"I still want to hear about it. This may sound strange, but you probably know me better than I know you. You've been my son for years, but I feel like we've just met. I want to understand how you think. Just give me a minute to finish getting dressed.\"\n\nThe boy acceded with a glum nod.\n\nCovenant left her with Jeremiah, but he did not follow the example of the Giants. While she pulled on her socks and boots, he asked Stave abruptly, \"Is Branl saying anything?\"\n\nStave faced the Unbeliever with his customary lack of expression. \"Ur-Lord, the storm of the Worm's coming approaches. He gauges that an hour remains ere we must flee its ravages.\" The former Master glanced away briefly before adding, \"Should the Worm quicken its rush, we will receive warning.\"\n\n\"Well, damn,\" Covenant muttered. \"I should probably be glad. At least that thing isn't heading for Mount Thunder. But it's _hungry_. It's going to hit hard when it gets here.\"\n\nScowling, he went to the brook for water. Then he moved toward the nearest shrub and began to eat.\n\nLinden winced to herself. Covenant had seen the Worm before: she had not. But she imagined that it was huge and virulent\u2014and she had no idea whether the Forestal would be able to stand against it. The fact that the _Elohim_ were no longer physically present in this manifestation of reality might lessen the Worm's impulse to overwhelm Caerwood ur-Mahrtiir. Or deprivation might make the instrument of the World's End savage.\n\nMore savage than it was already.\n\nShe swallowed an urge to look outside the willow, confirm Branl's perceptions for herself. The Humbled was not likely to be mistaken. And her concern for her son was more immediate.\n\n_There are worse things than being afraid, Mom. Being useless is worse._\n\nAs far as she knew, a sense of purpose was all that had defended him against the cost of his emotional wounds. Now he had nothing to build\u2014and perhaps nothing to hope for.\n\nIf so, she knew the feeling. But she had her faith in Covenant to steady her. And long ago, she had been assured, _You will not fail\u2014_ She wanted to share those gifts with Jeremiah if she could. They were better than despair.\n\nPraying that she would be able to give him what he needed, she beckoned. \"Come on, Jeremiah, honey. Let's go into your temple. We can be alone there.\"\n\nHe flinched. He seemed to hide behind the silted hue of his eyes. His manner said, No, although he did not refuse aloud.\n\n\"I know that you don't want to talk,\" she offered patiently. \"I don't mean to make you uncomfortable. But I'm your mother. Worrying about their children is what mothers do.\n\n\"Come on,\" she repeated. \"If you help me understand, you might find that you feel less alone.\"\n\nJeremiah opened his mouth, closed it again. He looked around at the Giants, and then at Covenant, as if he hoped that one of them would intervene. But the women only nodded encouragement; and Covenant's attention was elsewhere.\n\nThe boy avoided Linden's gaze. Looking truculent and defensive, he joined her. When she turned past the willow trunk toward the fane's opening, he followed, scuffing his feet in protest.\n\nInside the construct, she found bare dirt between crooked walls supporting a ceiling that looked like it might fall on her at any moment. Gaps among the stones let patches of Caerwood ur-Mahrtiir's shining into the gloom, but that glow did not lift the shadows from Jeremiah's mien. He might have been little more than an emblem of the deeper night awaiting the Earth.\n\nFacing him, she put the Worm out of her mind; braced herself to concentrate on her son. He could not rid himself of his demons if he did not acknowledge them.\n\nHe began before she could choose a question. \"I don't know what you think we have to talk about. I already told you. The Giants and Stave did practically everything. After that\u2014\" A scowl concentrated his features. Its tightness reminded her of the twitch at the corner of his eye when Roger and the _croyel_ had lured her into the past. \"They must have said what happened. The _Elohim_ came. So did Kastenessen. Then Covenant showed up. Infelice took Kastenessen with her.\n\n\"That's _it_. That's all there is. The rest was just waiting for you and trying not to think you were dead.\" From his fists, small flames squeezed between his fingers. A caper of yellow light and shadows up and down his body made him look lurid. As if he were pleading, he added, \"Nothing else matters.\"\n\nLinden waited until he started to squirm under the pressure of her regard. Then she folded her arms over the Staff of Law, held it against her heart, and tried to be gentle.\n\n\"Jeremiah, honey. This isn't doing you any good. I'm your mother. I know that there's more. But there's something that you don't know about me.\"\n\nHer years at Berenford Memorial had taught her more than one way to probe the people who needed her.\n\n\"I'm more like you than you think. There were a lot of things that I refused to talk about. I kept them secret. That hurt me, of course, but I could live with it. The part that I didn't understand\"\u2014the part that she had been fatally slow to recognize\u2014\"was that I hurt my friends at the same time.\n\n\"Now I don't want any more secrets. I kept mine too long, and I finally learned something about them.\"\n\nWhile he stared at her, she told him the truth as if she were tearing away the scab from an unhealed wound.\n\n\"They feel like they protect us\u2014like we don't have to be ashamed of our secrets, or ashamed of ourselves, as long as no one knows about them. We tell ourselves that we're doing the right thing by keeping them. But that isn't true. Mostly we keep them because we don't trust the people who love us. And _that's_ just another way of saying that we don't trust ourselves. We really _are_ ashamed. We think that we're at fault and we're going to be condemned, or that we're weak when everyone else is strong, or that we actually deserve to be in pain and alone.\n\n\"My secrets were different than yours,\" she confessed. \"Of course they were. They're probably even more shameful. And they hurt everything and everyone that I love.\"\n\nEvery death caused by the Worm, every instance of destruction, was her doing: the loss of the sun; the reaving of the heavens. She was only able to live with that fact because Covenant loved her\u2014and because her son's mind had been restored\u2014and because she had friends. And because she did not know what else she could have done.\n\nIn spite of Jeremiah's defenses, she reached him. She felt his sudden uncertainty\u2014his alarm\u2014as if it were physically solid. In some ways, he was indeed younger than his years. Hearing his mother accuse herself made him feel threatened. For years, she had been his foundation. Now he could not be sure of her.\n\n\"Like what?\" he asked in a taut voice. \"What did you keep secret?\"\n\nFrom his perspective, there were too many possibilities. Most of them had the power to undermine him.\n\nLinden did not hesitate; but she could not keep the harshness out of her voice, the implied savagery.\n\n\"Resurrecting Thomas. I knew that I was going to break every Law the Earth absolutely needs to survive, but I kept what I had in mind to myself.\" \u2014 _compelled by rage, and contemptuous of consequence_. \"I made sure that no one had a chance to stop me. Now it isn't just the world that's doomed. As soon as the Worm gets to the EarthBlood, Lord Foul will be able to escape.\n\n\" _I_ did that, Jeremiah.\n\n\"But I didn't keep what I was going to do secret because I wanted those things to happen. I didn't think about the danger at all. I kept it secret because I was afraid that my friends would interfere. I didn't trust them enough to believe that they would understand, or that they would still be my friends if they knew the truth. And I felt that way because I was ashamed. I was ashamed of not protecting you from Roger in the first place. I was ashamed of letting him and the _croyel_ trick me.\n\n\"We're in this mess right now because I kept secrets.\"\n\nJeremiah nodded, but he seemed unaware of his own response. His eyes were full of dismay. He sounded small and inexpressibly forlorn as he admitted, \"I hate what's happened to me. I hate how _dirty_ the _croyel_ made me feel. I could hide from the pain. I knew how to do that.\" He had concealed himself for most of his life. \"But I couldn't hide from all that sneering.\n\n\"And I hated the way it made me hurt you. I couldn't prevent anything. I hated being too weak to stop it. I wanted to hurt myself, not you.\" Under _Melenkurion_ Skyweir, he had stabbed her hand\u2014\"But I couldn't. I just couldn't.\"\n\nFacing his unshielded need, Linden fought down her yearning to put her arms around him. He was both a child and a young man; but it was the young man who most needed her succor. The child understood too well how to bury himself away. The young man was the Jeremiah who would have to face what was coming. And that Jeremiah would not be consoled by hugs.\n\nBut he was not done. As if he were cutting himself, he said, \"Then Kastenessen took me, and I was helpless again. He reached out and _took_ me like I was nothing. Good for nothing. Useless. And I felt how he felt. He burned every nerve in my whole body until I thought I loved it. I thought it made sense.\n\n\"I'm ashamed of _that_. I _should_ be. I wanted him _dead_ \u2014I want Lord Foul _dead_ \u2014so I don't have to be ashamed anymore. And I don't want to talk about it because talking just makes it more real. It just tells everybody how useless I am.\"\n\nFor a moment, Linden could not respond. Kastenessen had taken him? She nearly cried out. Covenant had not told her. No one had warned her.\n\nHer son must have inherited more than Earthpower from Anele.\n\nI want Lord Foul _dead_. How else did she expect him to feel? She had once been possessed herself. The force of her own desire to see the Despiser's end made her tremble.\n\nNevertheless she had to offer Jeremiah something. She had to try.\n\nHoarse with empathy and suppressed outrage, she asked, \"Don't you think that maybe we all feel that way? He's the Despiser. He's spent eons doing as much harm as he can to the whole world. Don't you think that maybe everyone you know wishes he could be destroyed?\"\n\nQuick as a slash, Jeremiah retorted, \"But you aren't useless! Covenant isn't. The Giants are strong. Stave and Branl are strong. Covenant has his ring. You have a ring and the Staff of Law. I've already used up everything I know how to do. Now I'm just nothing.\"\n\nIt was too much. Without pausing to consider what she said, Linden snapped, \"That's how _I_ feel. _I've_ already used up everything I know how to do.\" Before he could protest or withdraw, she explained, \"Oh, I understand what you're saying. And you're right. Of course you are. There are probably all kinds of things I can do that you can't. But, Jeremiah, _I don't know what they are_. I've done everything I can think of. It doesn't matter how much power I have because I have no idea what to do with it.\" Her son also had power. \"Compared to the Worm\u2014hell, compared to the _Despiser_ \u2014I'm as useless as you feel.\" Deliberately she made her heart as naked as his. \"We have the same problem. What's happening is too big for us. It's just too big.\"\n\nJeremiah did not look at her. He stood half turned away like a boy who wanted to run and hide; a boy who already knew where he could go to feel safe. But he did not go. She felt his attention cling to her while his fears and his pain urged him to flee.\n\n\"Then how?\" he asked like a waif too lonely to wail. \"How do you go on?\"\n\nLinden did not hesitate. \"I've been here before.\" She had come too far to falter now. \"That's the advantage of being older. I've been here before. With Thomas. I've seen what he can do. Maybe _I've_ come to the end of what I can do, but _he_ hasn't. And he doesn't believe Lord Foul can't be stopped. He doesn't even believe the world can't be saved.\"\n\nThinking, Listen to me, Jeremiah. _Hear_ me, she finished, \"As long as that's true, I won't give up. I will not give up.\"\n\nAfter a long moment, she added, \"And I certainly won't give up on you.\"\n\nHis struggle was terrible to watch. He knew how to protect himself. His craving for the sanctuary of graves was visible in the way he stood, in the clench of his fists and the hunch of his shoulders. Sharing herself, Linden had not reassured him: she had precipitated a crisis which he had been fighting to avoid. But he also had reason to know that safety was a trap; that every sanctuary was also a prison. On some deep level, he had chosen to free himself from his long dissociation. More consciously, he had chosen to do what he could for the _Elohim_. He understood the choice that his mother wanted him to make now.\n\nIn the same tone\u2014forlorn and frail and alone\u2014he told her, \"I'll try.\"\n\nThen he let Linden hug him.\n\nWith that she had to be content. Perhaps it was enough.\n\nhen she and Jeremiah left the temple to rejoin their companions, Caerwood ur-Mahrtiir stood among them.\n\nAs before, he wore an aura of isolation, of harmonized and hermetic concentration, as if he were essentially alone. His eyeless visage did not regard the Giants or the horses. He appeared to ignore the _Haruchai_ and the Unbeliever. Nevertheless something in his stance or his singing conveyed the impression that he was aware of Linden. Melodies seemed to skirl around her like promises or compulsions.\n\nUnder the gemmed leaves and boughs of the willow, his music sounded like wrath.\n\nCovenant came to her at once, kissed her quickly, studied her with anxiety in his eyes. But she only returned his kiss and nodded: she did not answer his unspoken question. What he wanted to know would have to come from Jeremiah\u2014and at that moment, Jeremiah clearly did not mean to say anything. His face wore a sullen glower which masked his heart.\n\nThe Giants greeted her and Jeremiah with wry smiles and troubled frowns. Instead of asking questions, however, they busied themselves with necessary tasks. They had refilled most of their waterskins. Now they moved among the shrubs, gathering treasure-berries which they placed in the last two waterskins so that the company would not go hungry for a while.\n\nTo Linden, Stave bowed without any visible stiffness. After a moment's consideration\u2014or consultation\u2014he announced, \"Chosen, the storm of the Worm draws nigh. And its course lies directly toward us. We must depart.\"\n\nAh, God. Linden tightened her grip on the Staff until her hands ached. She was not ready\u2014and she had not eaten. Jeremiah had not.\n\nBut Hyn gave a soft whinny as if to confirm Stave's assertion. Facing Jeremiah, Khelen tossed his head and stamped one hoof. Restive and proud, Hynyn waited behind Stave.\n\nIn contrast, the Ardent's spavined horse, with its distinct ribs and slumped back, paid no heed to anything except grass. And Rallyn had already left the bower, presumably to join Branl.\n\nStudying Jeremiah, Covenant's expression settled into its familiar strictness, as exigent as a prophet's. \"I'm sorry, Linden,\" he said, muted and grim. \"We have to get out of here.\"\n\nBefore she could force herself to move, however, the Forestal spoke. He did not change his stance or gaze at anyone; but his song became words, as peremptory as commands. As if he were encouraging haste, he said, \"I have no staff.\"\n\nHe startled Linden; perplexed her. Fortunately Rime Coldspray seemed to understand him instinctively. Without hesitation, she replied, \"Great one, your lack is plain. If you will condone it, I will sever a branch to serve you, though I am loath to harm the loveliness and shelter which you have provided.\"\n\nCaerwood ur-Mahrtiir hummed to himself. After a brief pause, he answered, \"Do so. All of the world's woods know that boughs must fall like leaves\u2014aye, and the grandest of monarchs also\u2014when there is need.\"\n\nThe Ironhand bowed. Hurrying, she thrust her way between the hanging branches and lights to retrieve her stone glaive.\n\nWould a staff be enough? Would the ur-Mahrtiir himself suffice? Linden wanted to believe that. Long ago, the forbidding of the Forestals had blocked the Ravers along the whole length of Landsdrop. But the Worm was immeasurably greater than Lord Foul's most potent servants.\n\nHer hands on the Staff were suddenly damp. Sweat ran like spiders down her spine; like centipedes and maggots. Her flesh had not forgotten She Who Must Not Be Named. Nevertheless the Land's peril compelled her.\n\nHer voice shook as she asked the Forestal, \"Do you need any help?\" She had assured Jeremiah that she would not give up. \"Is there anything that I can do?\"\n\n\"There is.\" Caerwood ur-Mahrtiir's music gathered around her. \"The approaching puissance is vast. As I am, I cannot withstand it. I require your strength.\"\n\nInvoluntarily she quailed. Her old friend might need more from her than she knew how to give. But Caerwood ur-Mahrtiir wove the many strands of his music into a soothing counterpoint. He stood directly in front of her now. And as she regarded him, another face seemed to emerge within his, softening his unanswerable visage. Like shadows, blurred and tenuous, the former Manethrall's features joined those of the Forestal.\n\nHumming in a more human voice, he said, \"Yet I have not forgotten you, Linden Avery, Ringthane and Chosen. You bear dooms greater than the fate of the _Elohim_ , or indeed of the world's remaining trees. You must not perish in my aid. I ask only your blessing.\"\n\nMy blessing? She mouthed the words, but made no sound. Oh, Mahrtiir! My _blessing_?\n\nCaerwood ur-Mahrtiir unfurled ancient tunes around him, verse and refrain. \"This invoked bourne of verdure and health is small. By the measure of the world's end, it is little more than vainglory. But I will not have it so. I will not. Here stands the forgotten truth of wood, just as the fane which preserves the _Elohim_ expresses another truth also forgotten. While my bourne endures, it affirms that the Worm and death are not the sum of all things.\n\n\"Linden Avery, Ringthane, friend. Bless this beauty with your strength. Nourish it, that I may suffice in its defense.\"\n\nNow she understood. Relief and sorrow clogged her throat as if she had inherited them from Caerroil Wildwood and his gibbet. She could not speak. But she understood. At one time\u2014a time as forgotten as other truths\u2014she had been a healer. Behind the wrath of the olden Forestals, and the barrenness of Gallows Howe, lay passions of another kind altogether.\n\nWhile her companions waited, staring, Linden stepped back from the Forestal; cleared enough space to wield her Staff. Then she reached into herself, reached into the black shaft defined by runes between bands of High Lord Berek's iron lore, and brought forth Earthpower and Law for their intended purpose: not for battle and killing, but for sustenance and restoration.\n\nThis might be her last chance to use her Staff condignly. From this moment on, she foresaw only strife and carnage; possible Desecrations. With her whole heart, she sought to give her best to Caerwood ur-Mahrtiir's bower.\n\nHer health-sense guided her, first into recognition of the thetic nature of the Forestal's harmonies, then into awareness of their interplay, then into sensitivity to their tones and timbres. Her power was as black as the coming storm of the Worm, but it was made for this, God, it was _made for this_. Perhaps her magicks were flames. Perhaps she only imagined them as flames. Nevertheless they suited her purpose. When she had refined her fire to suit the chords and lines of the music which inspired the lush grass and the rushing brook, the willow with its limbs and leaves and glimmerings, the bedizened shade of the sanctuary, she poured out fuligin in the form of vitality.\n\nShe went deep into the dirt to fill it with Earthpower, feed every questing root. Baked and beaten earth she enriched until it became loam. From the soil, she brought Law and energy upward, encouraging sluggish sap, enhancing the hardiness of bark, suffusing boughs and twigs and leaves with anticipation. Among the branches, she added luster to the Forestal's gleams until they shone like refined stars.\n\nEverything that Caerwood ur-Mahrtiir had brought into being, she increased. The willow stretched taller, spread its shelter wider. Bursting from the ground, the brook became a stream gurgling with gladness. Grasses grew like dancing until they twined around the feet and ankles of the company. The faces poised before Linden were lit with spangles like epiphanies.\n\nIn response, the Giants bowed low, too entranced for speech. Covenant's eyes reflected the shining of leaves. Moved in spite of his mood, Jeremiah brought forth gentle flames the color of sunshine from his hands and forearms. Only Stave did not react. He stood with his arms folded as if the sole task required of him was to bear witness.\n\nAnd as Linden worked, the Forestal himself seemed to grow taller. His aura of exaltation and severity expanded until the nearest Giants and even Jeremiah backed away, giving themselves room for wonder. The promise of his mien became a cynosure, as compelling as a demand. Soon his fierce vigor filled the bower.\n\nHe needed only an instrument to wield his will against the Worm.\n\nThen the Ironhand returned, harried by winds, to give Caerwood ur-Mahrtiir what he lacked. While Linden withdrew her power and stepped aside, Rime Coldspray bowed deeply, showing her blade naked in her hands. When the Forestal nodded his consent, she moved to the edge of the bower, readied her glaive.\n\nWith one stroke, she lopped off a limb as tall as she was. As she did so, a sting of pain shot through the music, and the lights of the Forestal's theurgy glittered furiously. But the willow's distress soon passed, leaving a renewed tranquility under the canopy.\n\nLeaves and twigs and all, Coldspray brought the bough to Caerwood ur-Mahrtiir.\n\nThough it was twice his height, he accepted it easily; held it high as if it were the chorus of a hymn. For an instant, all of its leaves quivered. Then they began to glisten as if they were dewed with power.\n\n\"I am armed,\" he sang. \"Let every force and foe which disdains the glory of wood and green be warned. Though I have no forest to sustain me, I will not be thwarted while one tree stands at my back.\"\n\nThrough a quick blur of tears, Linden watched him as if he had been transformed again; as if he had surpassed his given exaltation.\n\n\"Linden,\" Covenant murmured as if he had no other language for what he felt. \"Linden. Hellfire.\"\n\n\"Nonetheless, ur-Lord,\" Stave put in brusquely, \"we must depart. If we do not attain a considerable distance, we will not survive the Worm.\"\n\nCovenant shook himself. He seemed to struggle for words. \"I know. We should go.\"\n\nHis tone said, _Now_.\n\n\"Aye, Timewarden,\" the Ironhand sighed. \"Doom crowds close upon us. We dare the Worm at our peril. We must trust that the Forestal who was once our friend and companion will not fail.\"\n\nMore firmly, she ordered her Swordmainnir to reclaim their armor and weapons. While Linden tried to break free of the spell which Caerwood ur-Mahrtiir had cast on her, the Giants gathered up their stores of water and _aliantha_. Then they ran from the bower.\n\nCovenant came to her with pride in his eyes. Wrapping his arms around her, he assured her softly, \"We can do this. Somehow we can do it. We just have to get started. As long as we're together\u2014\"\n\nHe steadied her. Somehow we can\u2014If the Forestal was strong enough.\n\nAfter a moment, she nodded.\n\nWith an air of regret, Covenant released her.\n\nA heartbeat later, Stave put his hands on Linden's waist and boosted her unceremoniously onto Hyn's back. As her muscles settled into their familiar places astride the mare, the former Master went to help Jeremiah mount. Covenant heaved himself into Mishio Massima's saddle. Stave sprang for Hynyn.\n\nBefore Linden was ready\u2014before she could possibly be ready\u2014the riders surged into motion.\n\nAn alteration in the Forestal's music parted the canopy to the northwest, opening a path out of the bower. Together Covenant and Linden, Jeremiah and Stave rode from shelter and solace into the bleak dawn of a sunless world.\n\neaving the protection of the willow and Caerwood ur-Mahrtiir was like passing from Andelain into the virulence of the Sunbane. Linden and her companions staggered to a halt. The Ranyhyn flinched; rolled their eyes. Mishio Massima shied and crabbed, nearly unhorsed Covenant. Caerwood ur-Mahrtiir's music had concealed the extent of the peril. Outside the bower, the storm's scale was unveiled.\n\nIt was enormous.\n\nDuring the night, the blast of presage had reconciled its confusion. Instead of writhing from one direction to another like a beast in agony, it had become a stiff assault from the northeast; a gale arising from the heart of the utter blackness that now loomed into the heavens like the front of an atmospheric tsunami. Eerie ululations like the anguish of ghouls sounded in the distance. Scourged gusts scooped groans from the craters that littered the ground; scaled into wailing on the ragged edges of the belabored ridge. If the Forestal's theurgy had not protected the willow, its leaves would have been torn away, scattered like debris. Boughs would have split like screams.\n\nThat was bad enough; but there was worse\u2014\n\nThe core of the storm was a blare of _might_ that defied perception: too loud to be heard, too dark for vision; too savage to register as anything except horror. But at the fringes of the Worm's approach, thunder crashed, a wild barrage like a convulsion that would never end. It seethed like the collapse of cliffs. Within it, armies of lightning stalked the plain, hammering the earth until the very dirt seemed to erupt and burn. Sudden and erratic, flashes lurid as bruises punctuated the blackness. On either side of the advance, desolations writhed like orgies, articulating the Worm's hunger.\n\nGod in Heaven! Linden had never\u2014\n\nInstinctively she snatched fire from her Staff. The sheer force of the blast threatened to extinguish her mind. But Earthpower sharpened her senses, made her more vulnerable. It seemed to _expose_ her, as if the magnitude of the storm served to measure her inadequacy.\n\nShe had unleashed this doom.\n\nHer strength left her. Her power became dust and ashes in her veins. Her heart lurched to a halt.\n\nCowering into herself, she did not feel the Giants running toward her. She hardly noticed them as they joined her. Their cataphracts would not protect them. Their swords were useless. She could not hear herself gasping, \"Oh, God. Oh, God.\"\n\nIn size, the Worm may have been _no more than a range of hills_ , but it had enough raw force to rive the world.\n\nHow close was it? Two leagues? Three?\n\nDistance meant nothing to such a creature. It was already too near. It would arrive more swiftly than any Ranyhyn.\n\nThen one of the Swordmainnir shouted, \"Behold the Forestal!\"\n\nLike a strike on an anvil, Linden's heart beat again. It began racing.\n\nMahrtiir!\n\nBehind her, Caerwood ur-Mahrtiir emerged from his bourne. Bearing his staff like an emblem of defiance, he strode to meet the gale.\n\nHe did not go far. Less than a stone's throw from the battered drape of the willow, he stopped; prepared to make his stand. He must have been singing, but the wind's thrash and groan and howl tore the sound away.\n\nLinden reached out for Jeremiah; caught his arm as if her mere grip had the ability to protect him. When he glanced at her, she saw a wasteland of shock in his eyes. Nothing in his life had prepared him for the magnitude of the Worm's violence.\n\nWhile she held her son, Stave held her. The Giants stared wildly, like women caught in the toils of the Soulbiter.\n\nCovenant struggled to keep his seat until Branl came to his side, helped him control the Ardent's horse. Then the Unbeliever panted hoarsely, \"Run! Hellfire! We have to _run_!\"\n\nTransfixed by Caerwood ur-Mahrtiir's daring, Linden could not drag herself away; but Hyn chose for her by surging into motion. Swordmainnir slapped themselves and each other, forced their limbs to move. Branl hauled on Mishio Massima's reins until the horse sprang forward. A stentorian peal from Hynyn seemed to take Khelen by the throat.\n\nThe company broke and ran as if it had been routed.\n\nOn some level, Linden recognized that she and her companions had to do more than simply evade the Worm itself. They had to get beyond the Worm's cloak of power. Those lightnings would sear the flesh from their bones. The winds would rip the riders from their mounts, knock even Giants to the ground. Yet she did not heed such things. Unregarded, her hand lost its hold on Jeremiah. She could not look away from the Forestal.\n\nSmall against the background of the bright willow, Caerwood ur-Mahrtiir stood before the blast. It wrenched at him, tried to shred his robe. Shafts of lightning marched closer with every heartbeat. Gales tore the branches of his staff. Still the leaves clung to their twigs: the glitter of song clung to the leaves. With music and wood, he opposed the dark as if he had within him the authority to deny annihilation.\n\nLinden could not believe that he was strong enough. He was a Forestal, transformed scion of a lineage potent against armies and Ravers. His puissance surpassed the Lords of old with all their lore. But the Worm exceeded every other living force. It dwarfed the exertion of wild magic and Law which had plucked the huge creature from its slumber. And Caerwood ur-Mahrtiir could not draw on the spirit of a spanning woodland, the will and energy of trees in their millions. He had only the willow at his back.\n\nThe willow\u2014and the fane with its treasure of _Elohim_.\n\nStill the company ran. Urgent and frantic, straining to their limits, the horses and the Giants ran. Branl warded Covenant. Stave brought Hynyn between Hyn and Khelen, watched over Linden and Jeremiah with his lone eye. Rime Coldspray and her comrades stretched their strides and raced for the horizon, running heavy as boulders, and yet fleet as driven seas.\n\nToo frightened to shout, Jeremiah flailed his arms, flinging streams of Earthpower in all directions as if he sought to haul his companions forward. The storm brushed his theurgy aside like dust.\n\nHolding her breath, Linden watched the Forestal.\n\nHe dwindled with distance, shrank in proportion to the Worm's vastness. As the storm towered over him, an ebon and unanswerable tsunami, his staff's gleaming and the willow's seemed smaller and smaller. They became puny things, ineffable and frail. At any moment, they would be extinguished. In its hunger, the Worm would swat them out of existence and take no notice.\n\nYet Caerwood ur-Mahrtiir stood. He sang, and refused to be silenced. The Worm's tumult was less than a league away, less than half a league; and still he stood. He was more than Caerwood ur-Mahrtiir. He was also Manethrall Mahrtiir, Raman, given to service. He refused as if his _No_ could sway even the unthinking appetite of the World's End.\n\nThunder shook the ground. When Linden risked a glance at the nearest lightnings, the boil of blackness, she saw that the company was too slow. The Giants and the horses were sprinting hard enough to burst the hearts of weaker beings, but they could not run fast enough. The storm was too wide: they had not begun their flight in time to avoid it.\n\nAnd yet the argent of Caerwood ur-Mahrtiir's forbidding endured.\n\n\"Linden Avery!\" Somehow Stave made himself heard through the chaos of running and winds, lightning and thunder. \"Chosen, attend! The Forestal succeeds! The Worm slows!\"\n\nImpossible! She stared in disbelief. The Forestal could not\u2014\n\nHe could. Caerroil Wildwood and Linden herself had given him enough.\n\nThe storm inundated her senses. Its might blotted out the heavens. Caerwood ur-Mahrtiir's light had become tiny in the face of the tremendous black. Nevertheless she saw the change of pace, not at the storm's core, but at its near edge.\n\nStave was right. The Worm was slowing down. It was actually _slowing down_.\n\nAnd slowing more and more as the Forestal's denial stiffened.\n\n\"Hell and blood!\" Covenant yelled. \"He's doing it! He's by God _doing it_!\"\n\nManethrall Mahrtiir, who had found his heart's desire\u2014and had come back.\n\nIt was not enough. Running as they were, Linden and her companions might escape the storm. If the Worm came to a complete halt\u2014if it paused to confront the Forestal, however briefly\u2014they might evade the lightning; the worst of the vehemence. But that alone would not save them. The World's End might then turn from Caerwood ur-Mahrtiir to follow the scent of EarthBlood. If it did, the storm would leave the bower and the fane intact. Instead it would swing in _this_ direction, away from the ridge. With one lunge, the Worm would send its ferocity raving toward the riders and the Giants. They would die like Joan in her former world, burned by blasts which no mortal flesh could withstand.\n\nStill the Worm _was_ halting. For this one moment, at least, the Forestal sufficed.\n\nWithout warning, Covenant also halted. Wrestling with the reins, he forced Mishio Massima to obey him. While the rest of the company wheeled in confusion, he swung out of the saddle, snatched at the bundled _krill_ , uncovered the blaze of the gem.\n\nWaving his arms, he shouted, \"Get together! As close as you can! I don't know how long Mahrtiir can forbid that thing! We have to get _out_ of here!\"\n\nLinden gaped at him. She felt snared by the Worm and the storm and Caerwood ur-Mahrtiir; unable to break free. But Hyn heeded Hynyn's whinny, or Rallyn's. In a rush, the mare crowded close to Hynyn and Khelen. Frantically the Ironhand and her comrades formed a tight cordon around Linden, Stave, and Jeremiah. Only Covenant and Branl, Mishio Massima and Rallyn stood apart.\n\nBranl had dismounted beside Covenant. With negligent ease, the Humbled tossed Longwrath's flamberge to the nearest Giant. At once, Covenant pitched himself into Branl's arms. As Branl crouched low, Covenant stabbed the _krill_ 's blade into the dirt.\n\nBright silver bloomed from the cut. Sustained by white gold and will, it clung to the ground as though it fed from a trough of oil.\n\nSwift as only the _Haruchai_ could be, Branl carried Covenant around the company while Covenant dragged the point of High Lord Loric's dagger through the earth. And as they moved, the _krill_ gouged a shining line in the earth, a curve becoming a circle.\n\nDirt was not tinder. It was not wood or oil. Nonetheless it held Covenant's power, undaunted by the gale, while the curve extended to enclose the company.\n\nMore quickly than Linden would have thought possible, Covenant and Branl completed their circle.\n\nImmediately the Humbled surged upright. Still carrying Covenant, he sprinted for the horses. Tossing Covenant deftly into Mishio Massima's saddle, Branl leapt for Rallyn's back.\n\nNow the line of light began to gutter and fade. But Covenant did not hesitate. With his left hand, he slapped his wedding band against the dagger's jewel.\n\nSudden incandescence surrounded the company. Without transition, the world vanished.\n\nLinden heard herself cry out for Mahrtiir, but there was nothing that she could do.\n\n## 2.\n\nToward Confrontation\n\nLinden Avery had passed through _caesures_. She had been taken out of her time by Roger Covenant and the _croyel_ , and had been returned by the compassionate lore of the Mahdoubt. The arcane abilities of the Harrow and the Ardent had conveyed her to and from the Lost Deep. Most recently, Caerroil Wildwood's last deed in life had restored her to her present.\n\nNevertheless she was not prepared for the sensations of being rushed out of time and space within a circle of wild magic.\n\nIf she could have stood apart from herself and watched, she might have noted the similarity between this translation and the reflexive evasion of linear time which had preserved her and Anele from the collapse of Kevin's Watch. She might have recognized that she and her companions occupied a void like a bubble in the blood of reality, an embolism that floated on its own currents, ignoring the natural pulse and flood of life. She might have realized that she herself was alight; that Covenant's use of the _krill_ and his wedding band drew a response from her own ring. She might have become aware that she was being reincarnated as much as translocated.\n\nBut she could not stand apart or think. Instead she simply went blank. And after an eternity or an instant, she returned to her mortality with a visceral crash while Hyn pounded beneath her, galloping back into the darkened world.\n\nShe felt blind, blinded, yet she saw everything at once; saw it limned in argent, distinct as a cut against the gloom, as if each detail had been etched in her brain.\n\nLed by Branl on Rallyn and Covenant on Mishio Massima, the company hammered the ground. They had been stationary: now they ran like panic. Fleet and certain, Hynyn kept his position between Hyn and Khelen, Linden and Jeremiah. Stave's flat visage showed no surprise. But Jeremiah reeled on his mount's back, caught off balance: only Khelen's care kept him from falling. Around him, the Giants staggered on the sudden surface. Their eyes rolled: they gasped and gaped. Yet they ran.\n\nTogether they followed the bottom of a wide depression which may once have been a swale, before it was baked dry. Patches of scrannel grass still clung to the dirt, rough-edged and stubborn. Between them, the ground was erratically cobbled with worn stones. Pummeled winds brought whiffs of dampness and rot from Linden's right: a direction which she instinctively knew was north. Ahead of the company, the terrain rose gradually toward a rumpled landscape a league or more distant.\n\nCovenant lurched in his saddle. He had dropped the reins to strike Loric's dagger with his ring. His boots had lost the stirrups. In another instant, he might fall. But then Branl caught Mishio Massima's halter to slow the beast. A heartbeat later, he snatched the _krill_ from Covenant. Covenant slumped forward, clutched at his mount's mane to keep his seat.\n\nThe Humbled had done such things before. He must have done them often.\n\nThe _krill_ 's brightness shrouded the heavens, made night of the twilit morning beyond its ambit. Everything outside its illumination had a look of fatality, of waiting, as if the unnatural dusk masked an ambush.\n\nAs Rallyn and Mishio Massima eased their pace, the other Ranyhyn shortened their strides. Around the riders, Rime Coldspray and her comrades relaxed their haste. Running became trotting; became walking. Covenant pushed himself upright, prodded his boots unsteadily into the stirrups.\n\nWinds boiled among the companions, tangled hair, flicked grit at faces. Here, however, the disturbance in the air was only a faint echo of the Worm's harsh turmoil. As Linden struggled to recover from the shock of translation, her first coherent thought was that the company must have crossed a considerable distance: far enough to pass beyond sight or sense of the Worm's storm. Whatever happened\u2014or had already happened\u2014to Caerwood ur-Mahrtiir and the fane, Linden and her companions had escaped.\n\nBut they had not done so instantaneously. She felt in her nerves that a portion of the morning was gone, perhaps an hour, perhaps more.\n\nHer ring still burned in response to Covenant's burst of wild magic, but its power was fading.\n\n\"Oh, wow!\" Jeremiah panted as if he rather than Khelen had galloped. \"How did he do that? Where are we?\"\n\nEager for solid ground, he leaned forward, swung one leg to slide off his mount.\n\nSilver-edged images lingered in Linden's mind, after-flashes of vision. Darkness clotted the surrounding twilight. Ahead of the company, the slope out of the depression or swale was still a stone's throw for a Giant away. Half buried stones staggered like the remnants of a broken road among stretches of rough grass. The grass blades were more grey than green, a hue like a memory\u2014\n\nLong ago, days or lifetimes in the past, Anele had stood on grass outside Mithil Stonedown. In a rancid voice, he had said, _There is more, but of my deeper purpose I will not speak_.\n\nOn grass that resembled this.\n\nAbrupt connections snapped into focus. Too late, Linden cried out, \"Jeremiah! _No!_ \"\n\nHe reached out and took me like I was nothing.\n\nBut Stave was faster. He seemed to know her thoughts; or he had his own fears. As she began to shout, he vaulted from Hynyn's back. Swift as thought, he caught Jeremiah before the boy's bare feet touched the ground and the grass. With a heave, Stave returned Jeremiah to Khelen.\n\n\"Mom?\" Jeremiah yelped. \"What\u2014?\"\n\nNow a different kind of shock reeled through Linden. Kastenessen was gone\u2014but he had not been Anele's only vulnerability. More than once, another being had possessed the old man.\n\n\"This grass,\" Stave stated flatly, \"is of another kind. That which cloaks the hills about Mithil Stonedown grows more thickly, and remains shorter.\"\n\nAnd nothing had harmed Anele among the lush verdure of the Verge of Wandering. Still\u2014\n\nLinden studied the grass, probed it with her health-sense. \"But it's similar. I'm not sure that it's safe.\"\n\n\" _Mom?_ \" Jeremiah insisted.\n\nThe Giants stared. Some of them gathered nearby. The others seemed content to stand and breathe. None of them interrupted Linden's concentration.\n\nCovenant turned his horse to face her. He watched her as though he knew what was in her heart.\n\n\"I'm sorry, Jeremiah,\" she said, thinking furiously; trying to calm herself. \"I didn't mean to startle you. But I don't know how much you've inherited from Anele. Kastenessen wasn't the only one who could use him. Lord Foul\u2014\" The memory of the Despiser's voice in Anele's mouth ached like a bruise too deep to heal. \"Whenever his feet touched a certain kind of grass, Lord Foul could take him.\"\n\nWhenever the Despiser had felt like taunting her.\n\nEven his aid had been manipulation. True, he had led her to hurtloam. Indirectly he had enabled her to avoid recapture by the Masters. But that ploy had served his purposes as much as hers. If the Masters had been able to prevent her from reaching the comparative sanctuary of the Ramen and the Verge of Wandering\u2014prevent Hyn from choosing her\u2014prevent her from retrieving the Staff of Law\u2014she would never have been able to find Loric's _krill_ and resurrect Covenant. But she also would not have awakened the Worm.\n\n\"I don't want that to happen to you,\" she told her son. \"It was agony for Anele, but at least he knew how to mask himself. There were parts of him that Lord Foul and Kastenessen didn't recognize or couldn't reach. If you have to defend yourself that way\u2014if you go back into hiding\u2014I'm afraid that you won't be able to get out again.\"\n\nHe had no bones with which he might devise another portal. His racecar was gone.\n\nThe troubled silt of Jeremiah's eyes held more than surprise; more than chagrin. Their sullen smolder looked like fury.\n\n\"That doesn't make sense,\" he protested. _I want Lord Foul dead_. \"I stood on grass when we went to the Sarangrave. When we drank at the edge of the marsh. Nothing happened.\"\n\n\"I know,\" Linden admitted. She had not known then that he was vulnerable. \"But maybe that was the wrong kind of grass. And Lord Foul isn't Kastenessen,\" _compelled by rage, and contemptuous of consequence_. \"He only shows himself when it suits him.\n\n\"We know that he wants you. At some point, he's going to try to take you.\"\n\nShe ached to protect her son; but her warning seemed to miss its target. His expression grew darker.\n\n\"Fine,\" he muttered. \"Let him try. I don't care. Kastenessen surprised me. Lord Foul won't.\"\n\nHis attitude stung her. For the first time in their lives together, she wanted to slap him; to get his attention somehow. But she held back. In spite of her alarm, she could see that he would not heed her: not on this subject, at this moment. His bitterness was too strong. And she knew how he felt. His manner reminded her of herself when she had been about his age, at her mother's abject bedside. If anyone then had told her not to end her mother's life, she would not have listened. Her own distress had ruled her, and she had already chosen her path.\n\n\"Then don't rush into it,\" she replied unsteadily. \"It's going to happen whether you're ready for it or not. And the Despiser is stronger than Kastenessen.\" A lot stronger. \"Give yourself as much time as you can.\"\n\nJeremiah glared at her for a moment. Then he turned his head away. \"Fine,\" he snorted again: a response that gave her nothing.\n\nLinden winced. She did not know what to say. She had been possessed once herself. More than once, she had fled within herself in terror and dismay. She knew at least that much about what he had endured\u2014and what lay ahead of him. But she could not simply _tell_ him what those experiences had taught her, or what they had cost. No description would suffice.\n\nAching for him, she sighed, \"All right. As long as you know what might happen.\"\n\nStill Jeremiah kept his head turned away as if he were thinking about something else; as if in his mind he had left her behind.\n\nKindwind opened her mouth to say something, then reconsidered and remained silent.\n\nNow Covenant came closer, steering Mishio Massima among the Giants. The comprehension in his eyes made Linden want to hide in his arms as if he had the power to spare her; as if his embrace might heal the wound of her son's straits. But when he reached her side, he said nothing about Jeremiah. Instead he announced, \"That was easier than I expected.\"\n\nHe may have been trying to deflect her from her fears.\n\n\"Those translations are draining. I can't even begin to tell you how tired I've been since I went after _turiya_. Branl had to carry me. By the time I got to Kastenessen, I was so exhausted I didn't think I could stay on my feet. But this time\u2014\n\n\"Hellfire, Linden. This time I had help. I felt it. With so many of us, it still should have been difficult, even for a rightful wielder. But you helped me.\"\n\nThen he changed the subject. Without transition, he asked, \"Can you see the stars? My eyes aren't that good anymore.\"\n\nThe stars\u2014?\n\nFor no apparent reason, the Swordmainnir began to relax. The Ironhand nodded. Frostheart Grueburn chuckled softly. Onyx Stonemage, Cirrus Kindwind, and the others looked bewildered for a moment. Then they smiled. They seemed to understand Covenant better than Linden did.\n\n\"The stars, Linden,\" he insisted patiently. \"Are they dying? Are they all dead?\"\n\nAfter an instant like another translocation, she caught up with him. The Giants were shaking their heads, but they let her answer Covenant.\n\nThe stars. When she looked at the sky, she saw that they were fewer than they had been mere days ago. The gaps between them were wider. Nevertheless no more of the forlorn lights were winking out. From horizon to horizon, they remained as bright as supplications in the black heavens.\n\n\"All right,\" she breathed as if she had forgotten to be afraid. \"All _right_. He did it. He's doing it. My God, he's _doing_ it.\"\n\nMahrtiir.\n\n\"No.\" Covenant spoke softly, but he sounded like he was crowing. \"You did that. You. You took Mahrtiir into a _caesure_ and brought back a Forestal. You made him strong enough to forbid the actual Worm of the World's End. Sure, _he's_ saving the _Elohim_. But _you_ made it possible.\"\n\nHe was looking, not at her, but at Jeremiah. He was trying to tell Jeremiah something\u2014\n\nBut he did not wait for some sign that the boy had heard him. Glancing around the cluster of Giants, he pointed at Linden with one foreshortened finger.\n\n\"My wife,\" he pronounced as if those two words were a celebration. \"Anele was right. The world won't see her like again.\"\n\nHe took her by surprise. For a moment, her eyes filled with tears. She could hardly remember being a woman who wept too easily.\n\nWiping her cheeks, she missed Jeremiah's immediate reaction. When she turned to him again, his shoulders were hunched, strangling emotions. \"Fine,\" he rasped yet again. He was talking to Covenant. \"I saved the _Elohim_. Stave did. The Giants did. You did. Mom did. Caerwood ur-Mahrtiir did. That's great.\n\n\"What do we do now?\"\n\nSeveral of the Giants grimaced at his tone. Scowling, Covenant showed his teeth as if he wanted to take a bite out of the boy. But it was Stave who answered.\n\n\"We have shared an abundance of _aliantha_ ,\" he remarked with particular dispassion. \"You and the Chosen have not.\"\n\nThe Ironhand nodded gravely. \"Indeed, Stave Rockbrother.\" At the same time, Cabledarm and Halewhole Bluntfist raised their waterskins.\n\n\"Fortuitously,\" Cabledarm proclaimed to Linden, \"we are Giants, and provident. In addition to water, we bear treasure-berries. They will feed us well enough for the present, and perhaps for the morrow as well.\"\n\nFor a moment, Linden simply stared while her emotions tried to go in too many directions at once. Then she murmured an inadequate thanks.\n\nYet she could not leave Jeremiah as he was, not without offering him some form of acknowledgment. In the act of reaching for Bluntfist's waterskin, she paused. \"But Jeremiah is right. What _are_ we going to do? Saving the _Elohim_ is just a delay. We have to do more.\" With a quick glance at Covenant, she suggested, \"Maybe we can talk about that while Jeremiah and I eat.\"\n\nJeremiah accepted treasure-berries from Cabledarm as if he were turning away. Nevertheless his gaze followed every shift and flicker of Covenant's reaction.\n\nFrowning, Covenant considered Linden's suggestion. After a moment, he muttered uncomfortably, \"We probably should.\"\n\nAt once, the Giants gathered around him. Stave remounted Hynyn, and Branl nudged Rallyn closer. In their disparate fashions, they all needed a sense of purpose as badly as Jeremiah did. A chance to live\u2014or to give the end of their days meaning.\n\nSitting Mishio Massima in the center of the company, Covenant asked Branl, \"How far have we come?\"\n\nThe last of the Humbled shifted his grip on Longwrath's flamberge. \"Not less than a score of leagues, ur-Lord, else we would be able to discern the storm of the Worm.\"\n\n\"Northwest, right?\"\n\n\"Our heading has been chiefly to the west. We stand some small distance from the southmost verge of the Sarangrave. We must pass the wetland to gain Landsdrop and the Upper Land.\"\n\nCovenant nodded. \"And how much time did we lose?\"\n\nBranl glanced at Stave. \"We gauge that our passage has consumed little more than an hour.\"\n\n\"Aye,\" Rime Coldspray assented. \"So it appears to me also.\"\n\nLinden did not bother to concur. While she listened, she concentrated on shaking out berries from the waterskin and tossing them into her mouth; relishing the abrupt tang of energy and health. The seeds she discarded as she ate.\n\nCovenant nodded again. \"Good. Since the Forestal has held on this long, we can at least hope he won't fail. The Worm will move on eventually, if it hasn't already. It's fast, and it'll go faster when it gets closer to _Melenkurion_ Skyweir. But we can be pretty fast ourselves now. Maybe we can get where we're going faster than Lord Foul expects.\"\n\nThe Giants watched him in silence, waiting for an explanation. Like Jeremiah\u2014like Linden\u2014they had come to the end of what they knew how to do. Now they sailed chartless seas and needed a heading.\n\n\"So.\" Covenant seemed to be reasoning aloud, speaking primarily to himself. \"Landsdrop. The Upper Land. Mount Thunder. That's where Lord Foul is. He has to be. He needs to be close enough to She Who Must Not Be Named to take advantage of whatever She does, but not close enough to be in danger himself. And he has to be able to organize his forces, all of which are somewhere inside or near Mount Thunder.\" Then he shook his head. \"Hell and blood. I think that's a mistake.\"\n\nLinden agreed. Of course it would be a mistake to approach Mount Thunder. She did not doubt Covenant, and Jeremiah had to be protected.\n\nBut the Giants passed puzzled glances back and forth; and Rime Coldspray held up her hands. \"A moment, Timewarden, I implore you. Your thoughts out-pace ours. Are you certain of our destination? Have you determined our purpose?\n\n\"Surely it would be folly to hazard Mount Thunder. You speak of She Who Must Not Be Named and other forces. I must also name your fell son, whose command of the Cavewights appears complete. I deem it unwise to trust that his puissance has been diminished by the severing of Kastenessen's human hand.\n\n\"Yet these are lesser concerns. Of greater import is the Worm, a threat to pale all other perils. I know not how we may oppose it, but we can accomplish naught if we do not make the attempt. Therefore surely we must follow, hoping to forestall the Worm from _Melenkurion_ Skyweir. How otherwise may the Land and the Earth and life be preserved?\"\n\nHer comrades murmured their agreement; but both Stave and Branl frowned as if they wanted to challenge the Ironhand's assessment. Jeremiah watched Covenant with an intensity that resembled nausea. His mouth shaped words that Linden could not interpret. They may have been protests.\n\n\"Well,\" Covenant said gruffly. For a moment, he appeared to wrestle with himself. Then he announced with an air of defiance, \"I disagree.\n\n\"I won't try to make your decisions for you. Even Linden and Jeremiah\u2014you all have to do what you think is right. But _I'm_ going to Mount Thunder. I have to try to stop Lord Foul. And I need you with me. I need you all.\n\n\"It's not just that I have no earthly idea what to do about the Worm. That thing is part of the created world. It's inherent to the way this world works. There isn't enough power anywhere to get in its way. But on top of that, I think the Despiser is more important. He's absolutely more important to _me_.\" Passion mounted in him. He did not raise his voice, but it thrummed with intensity nonetheless, with the authority of earned conviction. His whole body seemed to imply imminent wild magic. \"Ever since I first came here\u2014ever since he and the Creator picked me\u2014my life has been about Lord Foul. He scares me worse than any ordinary death, even if the people I love most are the ones who do the dying. I have to face that. I have to do something about it.\n\n\"Sure, if we could stop the Worm, Foul would be stuck in his prison. But we can't, and he won't. Think about that. Think about setting Despite loose in eternity, where it can pollute every new creation just like it's polluted this one. That's bad enough. Hellfire, _that's bad enough_! But it could become even worse. If he gets his hands on Jeremiah, he'll try to trade places with the Creator. He'll try to make a prison that will put an end to the very _possibility_ of creation. He'll wipe out everything that has ever lived, everything that ever might live, every conceivable world.\n\n\"If he can do that, _eternity_ will become the kind of wasteland we've only seen in _caesures_. Then there won't be anything anywhere ever again. Nothing except scorn until even Lord Foul's heart breaks.\"\n\nAmid the shocked silence of the company and Linden's dismay, Jeremiah asked like sneering, \"So your solution is to take me _closer_ to him?\"\n\nCovenant wheeled his mount to face the boy. \"Hell, Jeremiah, he can get you anywhere. All he needs is the right kind of grass and one mistake. Then you'll give him whatever he wants. That won't change if we're a hundred leagues from here fighting the Worm. And you'll never get a chance to find out what how you feel and what you can do are good for.\"\n\nThen he turned Mishio Massima toward Linden. His eyes blazed with need. \"Linden, I'm sorry. I have to do this. Eventually we all have to face the things that scare us most. And I'm not actually convinced that the Worm can't be stopped. I just don't think _we_ can stop it. There's more going on here than just the Worm and Lord Foul and Jeremiah and more enemies than we can count. I don't know what it is, but I don't believe\u2014I don't choose to believe\u2014that the way things look to us right now is the whole story. We have two white gold rings and the Staff of Law and Jeremiah's talent. We have friends who have never let us down. All of that has to be good for something.\"\n\nHe might as well have added, _And betimes some wonder is wrought to redeem us._\n\nBut he did not wait for a reply. The naked chagrin on Linden's face seemed to drive him away. Again he turned his mount.\n\nTo the Ironhand and her Swordmainnir, he said, \"So, yes. I do want to go to Mount Thunder. In spite of, or maybe because of, all the obstacles you mentioned. That's not the mistake I was talking about. The mistake would be to go there the way damn Foul expects.\"\n\nIn spite of their deliberate dispassion, the approval of the _Haruchai_ was plain.\n\nRime Coldspray held up her hands again. \"Enough, Timewarden.\" She and her comrades studied him with a mixture of rue and wonder. Cabledarm and Latebirth grinned openly. \"We cannot protest such passion. For us, any deed which can be attempted is preferable to one which cannot. If your purpose is clear to you, it will suffice for us. Unless,\" she added, \"Linden Giantfriend reasons against it. Then we will heed her as we have heeded you, and will await the outcome between you.\"\n\nLinden hardly noticed that everyone was looking at her now. She hardly recognized the confusion of dread and hope in Jeremiah's eyes. The light of the _krill_ and Covenant's extravagance filled her mind with gibbering.\n\nNo. Gibbering and carrion-eaters. Not She Who Must Not Be Named. I can't.\n\nAnd not Lord Foul. Not Jeremiah. _His worth to the Despiser is beyond measure_. I can't take that chance.\n\nNevertheless the bright gem of the dagger held her. Covenant's gaze held her. She had never been able to refuse him. From the moment of their first meeting on Haven Farm, he had compelled her by simply being who he was.\n\nFeeling bitter and beaten, she said slowly, \"Don't stop now. Tell us how you think we can get into Mount Thunder. Tell us how you think that's even possible.\"\n\nHe was still her husband.\n\nA sigh passed among the Giants.\n\n\"Mom,\" Jeremiah groaned: a low sound that did not distinguish between protest and relief.\n\nCovenant's eyes did not let Linden go. He spoke to the company as if he were answering only her.\n\n\"I've been inside Mount Thunder twice, and both times I went in by the front door. From the Upper Land along Treacher's Gorge to Warrenbridge, then into the catacombs. _That's_ the mistake. Foul is bound to be expecting us. We need another way in.\"\n\n\"Indeed, ur-Lord,\" remarked Branl. \"It is certain that other passages exist. One enabled the quest for the Staff of Law to evade Drool Rockworm. Another brought Cavewights and your son to assail us. But such paths are known only to the Cavewights. Also they are perilously small, ill-suited to Giants.\"\n\n\"Right.\" Covenant did not glance at the Humbled. All of his attention was fixed on Linden. \"We'll have to try a different approach.\n\n\"Forget the Upper Land. If the Sandgorgons and the _skurj_ were cutting into Salva Gildenbourne back when the Ardent brought us out of the Lost Deep, they'll be near Treacher's Gorge by now. Even if we get into the Wightwarrens ahead of them, they'll be right behind us.\n\n\"I think we should try climbing up from the Defiles Course.\"\n\nNo, Linden repeated. She could not stop herself\u2014and could not find her voice to tell him that he was wrong about her. _No_. What he said made sense. Nothing made sense. She Who Must Not Be Named was too strong for her.\n\n\"The waters are corrupt,\" objected Branl.\n\n\"Well, sure,\" Covenant countered. Every word was addressed to Linden. \"But they must have receded by now. The Soulsease has been pouring into the Lost Deep for days. Until all those chasms and caverns fill up, there won't be any water coming out. Or not much,\" he amended. \"There are probably other sources, but they're nothing like the Soulsease.\"\n\nBranl was not deflected. \"Also the path is unknown. Uncounted millennia of slime and filth and dire poisons will clog the channel. The inhalation of the vapors will cause sickness and death. The Giants will not be spared. The _Haruchai_ will not.\"\n\nAt last Covenant turned away as if Linden's silence and dismay had defeated him. He sounded sour and forlorn as he retorted, \"I'm not worried about the damn _vapors_. Linden has her Staff. We'll be fine. And we'll have another advantage. We'll be close to water.\n\n\"Hellfire!\" The scar on his forehead seemed to bleed silver. It squeezed out of his old wound like sweat. \"We'll be close to the lurker. If we need help, we'll get it. That monster has already staked its life on the alliance. We can do the same.\n\n\"Linden's fate is 'writ in water.' The Ardent told us that. What the hell else do you think he meant? The lurker can't reach the Upper Land, but the Defiles Course opens into Lifeswallower. That's where Horrim Carabal _thrives_.\"\n\nBut Branl did not relent. He and Stave had already shown their approval. Now the last of the Humbled seemed determined to judge Covenant's intentions accurately, as if he agreed with Linden.\n\n\"And also there is the matter of the Cords. They have been conveyed to Revelstone to seek the aid of the Masters. Should they succeed, that aid will not find us at the Defiles Course.\"\n\n\"Don't you think I know that?\" Covenant snapped. \"But they can't help us. If Bhapa and Pahni succeed, the Masters will head for Treacher's Gorge\u2014where they'll be slaughtered. They can't do anything against _skurj_ and Sandgorgons. For their sakes, we have to hope the Cords don't convince them.\n\n\"Whatever happens, we'll have to find the way by ourselves.\"\n\n_Writ in water_. Finally those words reached Linden. She remembered how Covenant had rescued her from her terror of She Who Must Not Be Named. He had gone to that extreme for her: her husband who loved her. How could she fault him for still being a man who went to extremes? When extremes were needed? And she knew that he was right about Jeremiah, although the truth appalled her. Lord Foul could reach him anywhere. The Despiser did not need proximity.\n\nWhile Covenant faced the company with his needs and his pain and his severe convictions, Linden found her voice. But she did not speak to him: she spoke to her son.\n\n\"What do you think, Jeremiah?\" Her voice shook. \"This has got to be harder for you than anyone else.\" He had said as much himself. He had no instrument of power. No weapon, no prowess, no great strength. \"Are you willing to go to Mount Thunder and take your chances?\"\n\nJeremiah's attention seemed to leap at her. \"Sure,\" he returned as if he had never questioned himself. \"Why not? Otherwise we're all just dead. If it's too much for me, I can always hide again. Lord Foul will still be able to use me, but I won't have to feel it. Not like I did with Kastenessen, and he only got me because I didn't expect it.\"\n\nHe gave the impression that he meant, Maybe I don't have to be useless. Covenant said he needs us. But Linden heard more. As if Jeremiah had spoken to her like the _Haruchai_ , mind to mind, she heard him say, I want Lord Foul _dead_.\n\nOh, my son\u2014\n\n\"Linden?\" Covenant asked. Now he sounded deliberately neutral, as if he thought that he had already put too much pressure on her. \"It's up to you.\"\n\nFrom him also, she heard more than he said.\n\nI know what I have to do.\n\nI can't do it without you.\n\nShe recognized the knots that defined his face, the lines like cuts, the clench of understanding and regret. How often had he regarded her like that? When he knew what the Land's need required, and regretted it for her sake rather than his own?\n\n_Eventually we all have to face the things that scare us most_.\n\nA flick of grit forced her to shut her eyes for a moment. She felt suddenly parched in spite of the lingering taste of treasure-berries; scorched by the heat of Covenant's gaze. She had ashes in her veins instead of blood. God, he was a cruel man sometimes. Cruel and terrible and irrefusable.\n\nBarely able to clear her throat, she said, \"You aren't just my husband,\" Thomas of my heart. \"You're Thomas Covenant the Unbeliever. And Jeremiah is willing. I'll go with you as far as I can.\"\n\nAt that moment, the sudden lift of relief and hope and even love in Covenant's gaze did not touch her. And she ignored the reactions of the Giants. Their Ironhand had already given her assent. Instead she remembered Berek Halfhand among the Dead.\n\n_He may be freed only by one who is compelled by rage, and contemptuous of consequence_.\n\nThe Lord-Fatherer's pronouncement made her want to weep. He may have been trying to warn Covenant rather than her. He may have been describing Jeremiah.\n\nOr he may have seen the Land's doom in all three of them.\n\nsecond circle of wild magic. A second rush of disorientation. A second reflexive response from Linden's wedding band. Then the horses and the Giants pounded as if they were deranged down the bottom of a ravine that Linden almost recognized.\n\nWeathered hills rose on either side. The cut between them was comparatively shallow, a crooked trough wide enough for the company. The sand and age-smoothed stones of the bottom provided an easy surface for the mounts and the Swordmainnir as they pounded along, slowing with every stride. And ahead of them\u2014\n\nBlack in the unnatural twilight of midday, a stream slid past a widening fan of sand punctuated by the jut of a few boulders. Complaining against rock on the far side, the water flowed down a small canyon that arced around the swath of sand.\n\nAs Hyn's gait eased, and Linden's nerves began to recover from the mad reel of translation, she realized that she did indeed know this place. Here the company had rested days ago. Here she had rejoined her companions after Covenant had retrieved her from nightmares of She Who Must Not Be Named. Here the Ardent had delivered a feast, and had lost his grip on name and use and life. And back there, behind her now, lay the ridge of fouled gypsum where Liand and then Galt had been slain, and Anele had perished; where Esmer had passed away: the crest crowned by cairns. In this low canyon, Covenant had ridden away with Branl and Clyme as if he did not want her love. It was a place of loss and struggle and butchery, a black omen.\n\nThe Ranyhyn must have chosen this destination. As far as she knew, Covenant did not have such control over his translations.\n\nFortunately the company had arrived in a region of calmer winds. The Worm seemed far away, as if it had lapsed back into abstraction.\n\nAs Mishio Massima slowed, Branl took the _krill_ from Covenant, held it up to light the way. Near the water's edge, the horses stamped to a halt. Heaving for air as if they had run for hours instead of moments, the Ironhand and her comrades stopped. Briefly silver glared like frenzy in their eyes. But within moments they began to breathe more easily. As they looked around, they nodded their recognition.\n\nAt the forefront of the company, Covenant practically fell out of his saddle, tottering like a man on the verge of prostration. But his unsteadiness was vertigo, not fatigue. He began to look stronger as he recovered his balance.\n\nStill mounted, Linden did not meet his gaze. She was not ready. She still felt stricken by his intentions and her own acquiescence\u2014and by her son's peril.\n\nAn awkward shrug clenched his shoulders. He left her to herself. Scanning the Giants, he drawled, \"Don't take this the wrong way, but you all look like you need a bath.\"\n\nColdspray gave him a lugubrious frown. \"We are clogged with grime, Timewarden, made filthy by long exertion. Indeed, we are altogether unlovely. How might your observation be interpreted wrongly?\"\n\nHe blinked at her as if he could not think of a response. Then he muttered in feigned disgust, \"Giants.\" More loudly, he remarked, \"God knows _I_ need one. Maybe my eyes are going, but I can still smell myself.\" To Jeremiah, he added, \"Come on. Let's at least try to get clean. Maybe we'll feel better.\"\n\nJeremiah had kept his seat on Khelen as if he were impatient to continue the journey. He avoided Linden's eyes as she avoided Covenant's. But he did not refuse. After only a moment's hesitation, he dropped to the sand. Together he and Covenant splashed into the stream.\n\nLinden held her breath until she saw that Covenant did not take Jeremiah beyond his depth. When could her son have learned how to swim? Then she looked away and made an effort to come to terms with her dismay.\n\nIt rose in her, a pressure that felt too strong to be contained. Covenant was taking Jeremiah to Mount Thunder. To Lord Foul. The hills crouched like threats on either side of the ravine, and on the far bank of the watercourse. The sunless stream looked more like vitriol than water. Beneath its vexed surface, it seemed to imply malice. Overhead the stars glittered as if they were trying to warn her.\n\nIf Jeremiah thought that anger and bitterness would preserve him, he was wrong.\n\nAround Linden, the Giants set aside their swords, then began loosening their cataphracts, shrugging the armor off their shoulders. Of no one in particular, Latebirth asked, \"Does the Timewarden mislike his odor? I cannot discern it. My own aroma precludes other scents.\"\n\n\"Aroma, forsooth,\" snorted Halewhole Bluntfist amid a chorus of muted chortling. \"If that is aroma, I am the suzerain of the _Elohim_. For my part, I do not scruple to name it 'reek.' \"\n\nWhile the other Swordmainnir jested, Frostheart Grueburn came to stand beside Linden. From Hyn's back, Linden only had to lift her head a little to regard Grueburn.\n\nIn contrast to her comrades, Grueburn looked grave, almost somber. Softly she said, \"Linden Giantfriend, perhaps you will consent to speak with me apart from these coistrels. A matter weighs upon my heart. You will do a kindness if you allow me to unburden it.\"\n\n\"All right.\" Linden's clothes were still clean, scrubbed by the benison of Caerroil Wildwood's power. Even her hair was clean. And she welcomed any distraction from herself. \"Let's talk.\"\n\nAs she slipped down from Hyn's back, Stave and Branl also dismounted. At once, the four Ranyhyn turned away from the stream and followed the ravine, taking Covenant's steed with them. No doubt they sought forage.\n\nFrostheart Grueburn loomed above Linden. With her back to the _krill_ , the Swordmain looked benighted, mired in shadows. A lift of her arm suggested the direction taken by the horses.\n\nLinden glanced at Stave. \"Keep an eye on Jeremiah?\"\n\nStave shook his head. \"Branl will do so.\"\n\nThe Humbled was headed toward the stream. There he stopped, watching Covenant and Jeremiah.\n\n\"All right,\" Linden said again. To Grueburn, she added, \"If you don't mind Stave's company.\"\n\n\"My concern is private,\" replied the woman. \"It is not secret. Stave Rockbrother's companionship is welcome at all times.\"\n\nLinden nodded. With Stave a few paces behind her, she accompanied Frostheart Grueburn up the ravine. At every step, she had to resist an impulse to stamp at the sand with her Staff. Did Covenant expect her to face the things that scared her most? She did not know how.\n\nPerhaps a dozen Giantish strides from her comrades, Grueburn halted. For several moments, she stood with her face raised to the sky as if she were studying the stars, or listening to them. When she lowered her head to look at Linden\u2014and past Linden at Stave\u2014her aura was troubled.\n\n\"Linden Giantfriend,\" she said quietly, \"my thoughts are awkward. I am uncertain how to speak of them.\"\n\n\"You're a Giant,\" Linden murmured. \"You'll find a way.\"\n\nGrueburn offered a strained smile. She seemed to shake herself. \"Toward you,\" she confessed, \"I feel more than friendship. Amid the perils of the Lost Deep, and at other times, I have cared for you, as you know. For that reason among many others, your place in my heart is great.\"\n\nWhen the woman paused, Linden said nothing. Grueburn was not waiting for a response. Rather she was hunting for a way to broach her concern.\n\nFinally Grueburn began. \"Some days past, while we traveled together after the Timewarden had parted from us, I chanced to stand with you while you and Stave Rockbrother spoke. Together you considered questions of Desecration.\"\n\nLike a slap of wind, Stave observed, \"Our words were intended for each other alone, Frostheart Grueburn.\"\n\n\"Yet I heard them. From that time to this, I have respected that they were not for me. Nevertheless my thoughts have turned often to matters of Desecration.\"\n\nLinden swallowed a groan. She did not want to talk about such things.\n\nTo Stave, Grueburn continued, \"Here I do not ask you to reveal what you have foreseen, or indeed what your insights may be. I do not seek to probe your heart. I wish to unveil my own.\"\n\nHer response seemed to satisfy Stave.\n\nFrostheart Grueburn returned her attention to Linden. Silver from the _krill_ caught the lines of the Giant's mien. With an edge in her voice, she said, \"You stand at the center of all that has transpired. I do not deem it unlikely that you will continue to do so. Your deeds are potent to cause some futures while ending others. And I say again that you are dear to me. Therefore my spirits were lifted to soaring by the outcome of your union with Covenant Timewarden. I saw gladness in you, the gladness and relief which dismiss Desecration. But now\u2014\n\n\"Ah, now, Linden Giantfriend, some new darkness hovers in you. For that reason, I am troubled. If you will consent to speak of your concerns, you will ease my own. Comprehension will open my ears so that I am again able to hear joy.\"\n\n_In your present state, Chosen, Desecration lies ahead of you. It does not crowd at your back._\n\nLinden bit down on her lip; steadied herself on that small pain. Then she countered, \"What are you afraid of?\"\n\nGrueburn sighed. \"Chiefly I fear that you sail a course which leads to the desecration of yourself. To my sight, it appears that you confront an impossible conundrum. You are a mother. You must preserve your son. Yet you cannot. You cannot ward him from the Despiser's malice. Nor can you ward him from the world's end. His doom\u2014if he is doomed\u2014lies beyond your intervention. His despair\u2014if he falls into despair\u2014is not yours to relieve. And in these straits, it may be that your distress is increased by your union with Covenant Timewarden, for how can a mother know gladness with her husband when her son is in peril? I fear the effect of this conundrum. Linden Giantfriend, I fear it acutely.\"\n\nWhile the woman spoke, Linden turned away. Beat after beat, she thudded one end of her Staff into the sand. She wanted to rebuff Grueburn. The Giant saw her too clearly. Perhaps they all did. But she had talked about _trust_ with Jeremiah; about the implications of withholding the truth. And the Swordmainnir were her friends. They were in as much danger, and had as much to lose.\n\nFacing the darkness, Linden replied, \"I don't know what to tell you. I don't know how to explain it, even to myself.\" Her horror at the idea of approaching Mount Thunder was too intimate to be named. \"But I can tell you this much. Thomas wants to walk right up to his worst fear and look it in the eye, but I'm not like that. Lord Foul isn't my worst fear,\" no matter how much she loved Jeremiah. \"And the Worm isn't. Even having to watch while everyone and everything I care about dies isn't. As long as Thomas is still alive, none of that is inevitable.\n\n\"My _worst_ fear\"\u2014this was as close as she could come to complete honesty\u2014\"is that there may actually be something I could do, and I won't be brave enough to do it.\"\n\nWhen her father had killed himself, she had been too young and little to stop him; but years later, when her mother had begged for death, Linden had done what her mother asked of her. Eventually she had learned to believe that there were worse things than Desecration. Letting the pain _go on_ was worse. It had to be healed. If it could not be healed, it had to be extirpated. And if it could not be healed or cut out, it had to be ended in some other way.\n\nThat \"some other way\" was her real conundrum. And her greatest fear was that she did not have it in her to resolve the contradiction.\n\nShe knew how Kevin Landwaster must have felt.\n\nFor a time, Frostheart Grueburn and Stave answered her with silence. What could they have said? She was who she was. Her fears were her own. But then Stave said like a man who had never known a moment's doubt, \"It is written in water, Linden. Deeds are not stones. Fears are not. And even stone may fail. No outcome is certain.\"\n\nBefore Linden could think of a response, Grueburn began to chuckle. \"Well said, Stave Rockbrother. As ever, Linden Giantfriend misesteems herself. She has restored joy to my ears, though she does not hear it.\"\n\nThen she added, \"Accept my thanks, Linden Giantfriend. You have comforted me. I regret only that you are not likewise comforted.\"\n\nAt once, the woman turned away. Perhaps she sensed that Linden wanted to be alone; that Linden needed time to accept what she had heard and said. Still chuckling, Grueburn went to rejoin her comrades. But Stave remained.\n\nHe said nothing further. For that, Linden was grateful. His presence was enough to remind her that she was not alone. No other answer would suffice unless she found it for herself.\n\nime brought her no clarity; but after a while, she felt steady enough to return to the company. While the stars were dying, they had called to her nerves like keening; like bright supplications. But now they were not vanishing from the heavens. Perhaps as a result, they looked less forlorn to her. They seemed to gaze down almost hopefully, as if they had found something to believe in.\n\nSighing, Linden rested a hand on Stave's shoulder to thank him. Then she began to make her way back to her companions.\n\nBefore she reached them, Covenant came to meet her, still dripping from his immersion. His face was full of shadows because the light of the _krill_ had shifted: Branl had taken the dagger to a nearby hillcrest upstream. Spectral as the elucidation of dreams, argent shone on Covenant's silver hair but left his features in darkness.\n\nAt once, the Swordmainnir withdrew. Some splashed into the water to bathe. Others moved away as if they were making room for Linden and her husband.\n\nWhile she wondered what she could say to him, he took her in his arms. Holding her close, he murmured, \"I'm sorry, Linden.\" His voice was little more than a husky rasp. \"I feel like I've hurt you, but I'm not sure how.\n\n\"I expected you to argue.\"\n\nShe let him hug her for a moment. Then she returned his clasp. \"I've been arguing with myself.\"\n\nHe stepped back enough to look into her eyes. \"What about?\"\n\nShe tried to meet his scrutiny, but her gaze slid away as if she were ashamed. \"I understand what you want to do,\" she told him with a rasp of her own. \"I don't have any better ideas. But you didn't explain what you want from me.\"\n\nOr from Jeremiah.\n\nCovenant's manner said, I don't want anything from you. I just want you. I just need you. But aloud he admitted, \"I know. I couldn't. I can't. It's all so _vague_.\" He rapped his forehead with his knuckles. \"I'm clear about what I have to do. What I have to try to do. But everything else is just impressions, instincts. It's not an accident that you and Jeremiah are here. It's not an accident that we're here together. Hell,\" he snorted, \"I wouldn't be here at all without you. But I have no idea what it means.\"\n\nHe hesitated for a moment. Then he squared his shoulders, shook Linden gently. \"The only thing I'm sure of is that _this_ \u2014the three of us together, with friends to help us\u2014is not what Lord Foul wants. We've already done things he couldn't have foreseen. Now I think we _are_ something he can't foresee.\"\n\nHe almost eased her. She believed in him: now she could almost believe him. But she was still afraid\u2014and she had not told him what she feared. She had not named it to herself.\n\n\"That's not enough,\" she said awkwardly.\n\nI'm not enough.\n\nHis voice hardened. \"Then I'll say something else.\" It set like cooling iron. \"If none of this works out\u2014if everything goes to hell no matter what we do\u2014if the worst turns out to be worse than we can imagine\u2014you might want to remember that we didn't _start_ it. It's the Despiser's doing. We couldn't have prevented anything. All we've ever done is react to what he does.\n\n\"Even if this whole world only exists in our minds\u2014even if damn Foul is just an expression of a part of ourselves we don't like\u2014we can't be blamed for it. We didn't _make_ ourselves. We were born into lives we didn't choose, parents we didn't choose, problems we didn't choose. We aren't responsible for that. We're only responsible for what we do about it.\n\n\"If what we do isn't enough, too bad. Let the Creator worry about what happens next. If he doesn't care, at least he can't accuse us of anything.\"\n\nGentle as a caress, Covenant cupped his palm to the side of Linden's neck and offered to kiss her.\n\nFor a moment, she resisted. He had not given her enough. Nothing would ever be enough. But he had given her what he had. And he was Thomas Covenant, her husband and lover. As much as possible, he was even her protector. And he would do what he could for Jeremiah. She could not refuse him. She did not want to refuse him.\n\nWhile she kissed him, she thought, Thomas of my heart. I can't do this.\n\nBut she imagined that perhaps she could. As long as he never let her go.\n\nong moments passed before she found the strength to step back. She was not done with Covenant. She needed his touch, his arms, his mouth. She could have held him, and been held, as long as time remained in the world. But she was also Jeremiah's mother. Her heart was divided.\n\nIn this, she knew, she was not alone. All hearts were divided, Covenant's as much as hers. She would not have been surprised to learn that his desire for her and his concern for the Land and his need to confront Lord Foul threatened to tear him apart whenever he faltered. But her divisions were more personal. And when she scanned the company\u2014the Giants washing in the stream and those waiting nearby\u2014she saw no sign of her son.\n\nHer stomach tightened reflexively. At once, she turned to Stave. \"Where's Jeremiah? You said that Branl would watch him.\"\n\n\"He does so.\" Stave nodded stolidly toward the shining on the rise beyond Linden. \"The Chosen-son parted from the company to wend upstream. Branl followed at a slight distance. He does not neglect his charge.\"\n\nLinden flung a glance at Covenant; but he shook his head. \"He didn't say anything. I tried to get him talking, but he had too much on his mind.\"\n\nIf something had happened to her son, she would have felt it. Surely she would have felt it?\n\n\"Beyond the hillside,\" Stave continued, \"the boy has discovered a stretch of grass among sheltered stones. It bears some resemblance to that which Anele had cause to fear. There he stands, offering demands and imprecations. Yet naught transpires. For that reason, Branl does not intervene.\n\n\"It appears that your son does not partake of the vulnerability or flaw which exposed Anele to Corruption. We conclude that the boy has inherited only Anele's openness to Kastenessen\u2014a peril which no longer threatens him. His wish to encounter evil is foolhardy, but it does not endanger him.\"\n\n\"Or,\" Linden countered over her shoulder, \"Lord Foul just hasn't taken advantage of it yet.\"\n\nShe was already running.\n\nBoulders like raised fists complicated her path. Possibilities reeled through her. Stave might be right. The gifts and curses which Jeremiah had received from Anele might have strict limits. She was not lorewise enough to know. But she could imagine other explanations.\n\nKastenessen could have used his ability to take Jeremiah at any time, whenever the boy stood on bare dirt. Yet the _Elohim_ had not done so, despite his driving pain and fury. Instead he had waited, bided his time until the opportunity he desired presented itself.\n\nIf he could exercise such restraint, Lord Foul could do so with ease. His malice was colder than Kastenessen's\u2014and far more calculating. The Despiser had allowed Anele to walk on rough grass unpossessed for a considerable distance during Linden's flight from Mithil Stonedown.\n\nYet naught transpires. Linden did not doubt Stave\u2014or Branl. Nevertheless the danger was real. It was always real.\n\nAnd Jeremiah did not understand it. He thought that he would be able to defend himself as long as he was not taken by surprise.\n\nThe slope ahead of her was not steep. And she was too frightened to feel tired. She should have been able to ascend easily, swiftly. Yet she grew weaker as she scrambled upward. Something profound within her had shifted. Her surrender to Covenant's intentions had diminished her. The strength drained from her limbs at every step. Her breathing was a hoarse gasp as she gained the crest.\n\nBranl stood there, gazing at her with only argent in his flat eyes. He might have said something if she had given him a chance, but she forced herself to hasten past him; downward.\n\nShe felt Stave only a few strides behind her. Covenant followed more slowly, accompanied by Rime Coldspray and Frostheart Grueburn. But Linden ignored them. Her attention was fixed on Jeremiah.\n\nHe stood in a hollow between hills too old and worn to glower down at him. And he had indeed found grass: a patch of saw-edged grey-green blades growing stubbornly where a cluster of rocks had collected soil from the erosion of the surrounding slopes. In the ghost-light of Loric's gem, those blades looked sharp enough to cut. Everything around Jeremiah was blades: the etched hillsides, the ragged edges of the rocks, the black lift and slice of the stream. To Linden, he resembled a child in the midst of shattered glass, heedless of the danger, about to take a step which would shred his soles.\n\nHe did not see her. She had come from the south, and he was facing the northward crease where the hills slumped to close the hollow. His head was bowed in concentration. Waves of tension made his shoulders twitch: the muscles of his back bunched and released. Between his teeth, he muttered words which did not reach Linden.\n\nShe forced herself to slow down. Yet naught transpires. Stave was right. Branl had seen no reason to take action because there was no reason. Jeremiah was only himself: taut with anger and dread, desperate to prove his worth, but untouched.\n\nShe stopped a few steps away. \"Jeremiah,\" she panted. \"Honey.\" God, she felt so _weak_ \u2014Unmade. As if her refusal to name her greatest fear had been her only source of strength. \"That's enough. You tried. You can stop now.\"\n\nStave arrived at her side. Covenant, the Ironhand, and Grueburn crossed the rise toward her. Coming to bear witness\u2014\n\nBranl trailed behind them, spreading the _krill_ 's light as far as possible.\n\nJeremiah lifted his head. Keeping his back to his mother, he made a scything gesture with his halfhand.\n\nWithout warning, a silent shock jolted the hills. For one small splinter of time, the world's Laws seemed to pause. Linden's heart did not beat. Her lungs did not stretch for air. The stream hesitated in its course, poised and motionless. Stave became one more stone in the hollow. Covenant hung between one downward step and the next. Coldspray and Grueburn froze.\n\nThen a second shock released the hollow. Linden's pulse hit like a blow on an anvil. Covenant lurched for balance. Stave readied himself.\n\nInstantly the air became attar, thick as the smoke of burning flesh, cloying as an inferno of incense. The heavens leaned down on Linden as if they had become lead. Even Stave flinched. Coldspray or Grueburn stumbled. One of them caught Covenant. Branl started downward with the _krill_ raised.\n\nIn Lord Foul's voice, Jeremiah announced like a grinding millstone, \"It may interest you to know, fools and servants, that your ploy has achieved its purpose. Your edifice stands, a worthy emblem of your wish to oppose me. Yet even there, your deeds work against you. Deprived of _Elohim_ , the Worm hastens onward. It _hastens_ , fools! The hour of my many triumphs approaches. You cannot thwart it.\"\n\n\"Branl!\" Covenant gasped. \"The _krill_. Give me the _krill_!\"\n\nThe Despiser and Jeremiah ignored him. They spoke only to Linden.\n\n\"Nonetheless,\" the crushing voice continued, \"this callow whelp thinks to challenge me. _Me!_ As guerdon for his puerile valor, I have given him a gift which will make him wise in the subtleties of despair. When I have need of him, I will claim him, and no endeavor of yours will suffice to redeem him.\"\n\n_If your son serves me, he will do so in your presence._\n\n\"And you, frail woman\u2014\" Lord Foul's mirth filled the vale. \"You have become the daughter of my heart. In you, I am well pleased. Ere the end, you also will serve me.\n\n\"Thus all things conduce to my desires.\"\n\nCovenant snatched the dagger from Branl. \"Is that what you think, Foul? Have you forgotten what we can do to you? Have you forgotten we're _coming_ for you?\"\n\n\"Forgotten, wretch?\" retorted the Despiser, bitter and gleeful. \"I rely upon it. I forget nothing. I am prepared for you. If you think to confront me, you will discover that your efforts harm only yourself.\"\n\nCovenant did not reply. With the _krill_ gripped in both fists, he advanced like an incarnation of wrath.\n\nInstinctively Linden barred his way. She had no idea what his intentions might be. If she had taken time to think, she would have realized that he would not hurt Jeremiah. He was bluffing again. But she did not think. Jeremiah was her _son_. And she was capable of responses which Covenant could not match.\n\n_Whatever you do to my son_ , she had promised the Despiser long ago, _I'm going to tear your heart out_. Now she knew that she would not. She was not the woman she had once been. Events since her arrival in the Land had taught her expensive lessons. Covenant was still teaching her.\n\nLike Gallows Howe, the world had more important needs than retribution.\n\nNevertheless she did not hesitate. She had made other promises as well, ones that she knew how to keep. With a sweep of her Staff, she unveiled Earthpower and Law.\n\nHer health-sense was precise. Her fire could be equally precise: as refined as a scalpel in spite of its blackness. Just for an instant, she sent it gyring skyward while she prepared it for her purpose; confirmed that it was exact. Then she swung it like the crack of a whip at her son.\n\nIt poured through Jeremiah without touching him. She had tuned her theurgy to the pitch and timbre of Lord Foul's malice rather than of Jeremiah's body, Jeremiah's appalled mind. Her dark flame struck only the Despiser.\n\nShe could do so because Lord Foul's mastery was of an entirely different kind than the _croyel_ 's. That monster had merely reached into Jeremiah; fed on him; used him: it had not existed within him. And his defenses\u2014his dissociation\u2014had protected him. But now he had arisen from his graves. He inhabited himself. That change enabled Linden to distinguish between his reclaimed self and the force which ruled him.\n\nShe may have been as frail as Lord Foul believed. She may indeed have become his daughter in despair. Still she was Linden Avery the Chosen, Jeremiah's mother and Covenant's wife.\n\nIn a burst of conflagration, she banished the Despiser. His malevolence burst and vanished like a punctured bubble. Intangible gales swept away the stench of attar. The laughter of broken rocks dissipated until it was entirely gone.\n\nLike a discarded puppet, Jeremiah collapsed to his hands and knees.\n\nLinden reached him a heartbeat later, dropped her Staff, flung her arms around him. Through his skin, she felt his warmth and dismay, his wholeness, his horror. He strained to breathe as if his lungs were clogged with the sweet, sick odor of a body arrayed for burial.\n\n\"Mom,\" he croaked. \"Oh, Mom. I can feel the Worm. I can _feel_ it. It's going up a cliff. A cliff! And it's going fast. Like the cliff was nothing.\"\n\nThe Despiser's gift.\n\nShivers that began in the marrow of Jeremiah's bones spread through him. Linden hugged him tightly, but could not still his trembling.\n\nLord Foul had taught her son to fear him.\n\n## 3.\n\nSummoned to Oppose\n\nAnother race through the interstices between instants and leagues brought the company to a twisted heave of hills that Linden had never seen before.\n\nShe had no idea how far Covenant's eldritch circle had carried the riders and the Giants. She could be sure only that she and her companions were still on the Lower Land. As Hyn slowed her wild gallop, following Rallyn's lead with Hynyn and Khelen, Linden saw Landsdrop massive on her left, thrusting its crooked rims thousands of feet above its foothills. And in the distance on her right, she caught troubled glimpses of water, grey and dim as tarnished silver: Sarangrave Flat between the barricade of the cliffs and deeper mire of Lifeswallower, the Great Swamp.\n\nOverhead the stars were no longer visible. Thunderheads like clenched fists battered each other back and forth between the cramped horizons, occluding the sky. The weather tumbled in confusion, affronted by the Worm's passing far away. With every sunless hour, the air grew cooler.\n\nAbruptly the company plunged down a steep hillside. Skidding on loose shale, the Giants floundered to keep their balance. Hyn locked her knees for a moment and slid. Then she lifted into a light-footed prance that carried Linden safely.\n\nJeremiah reeled on Khelen, but not because the young stallion had jostled him. Ever since he had regained his feet after his encounter with Lord Foul, the boy had wobbled as if some of his sinews had been cut. His eyes, always changeable, had acquired a nauseated hue. If they reflected his mind, his thoughts were a spew of vomit.\n\nCovenant also reeled. As Mishio Massima lurched to a halt, he pitched from the saddle. But that was an effect of vertigo. Every exertion of wild magic seemed to cast him into a whirlwind.\n\nOn more level ground at the foot of the slope, the company gathered among a few patches of scrub oak clinging to the thin soil between stubborn tufts of grass and weeds. A more gentle hillside lay ahead; yet no one proposed to hurry onward. Covenant had already returned the _krill_ to Branl. Now he folded cross-legged to the dirt, holding his head like a man trying to remember who he was. As before, the Swordmainnir labored for air as if they had been carrying monoliths on their backs.\n\nAlmost at once, heavy raindrops struck Linden's face. Spatters hit randomly around the area, raising small bursts of dust where they found dirt. Soon there would be more. Torrents were coming, a monsoon downpour entirely out of place in this season of the Lower Land.\n\nWincing in anticipation, she nudged Hyn closer to Khelen.\n\nAt the same time, Covenant heaved himself upright. Unsteady on his numb feet, he made his way among the Giants toward Linden and Jeremiah. Droplets ran down his cheeks like sweat.\n\nCarefully he asked, \"How are you doing, Jeremiah?\"\n\nThe boy glared past Covenant. He avoided Linden's concern. \"Stop worrying about me,\" he muttered. \"I'm fine. You can't do anything about it.\"\n\nHis hands still trembled as if he were feverish.\n\nCovenant looked questions at Linden.\n\nShe studied her son. Superficially he was undamaged: that was plain. The distress that appeared to disarticulate him was emotional, not physical. Only his spirit had been harmed.\n\nHe had spent too much of his life hidden: a powerful defense which had both shielded and hampered him. Crouching in his graves had preserved him in some ways, but had not taught him how to weather the Despiser's virulence. Possession and vicious scorn had withered his attempt at defiance.\n\nFrom Hyn's back, Linden reached out to touch Jeremiah's arm, get his attention. \"Is it that bad, honey? Can you talk about it?\"\n\nShe wanted to ask what had impelled him to risk exposing himself to the Despiser, but she suspected that she already knew the answer. He felt useless: he needed to do something that would help him believe in himself again. And Covenant had given him an oblique form of permission or encouragement.\n\n_Eventually we all have to face the things that scare us most._\n\nJeremiah glared at her for a moment, then turned his head away. To the coming storm, he muttered, \"You don't understand. You don't see it. I can't stop. All that power\u2014It isn't just terrible. It's more _real_ than we are. We're all going to die, and I get to watch.\"\n\nScattered raindrops struck at Linden like pebbles. Fiercely she wiped her face.\n\n\"You're right, Jeremiah. I _don't_ understand. But I still know how it feels. I'm not any braver than you are, or stronger, or better. My _God_ , Jeremiah. I let a crazy man stick a knife in Thomas because I couldn't make myself try to stop him. _Turiya_ Raver touched me, just _touched_ me, and I got so scared that I was gone for days. And _moksha_ actually possessed me. I know what that _feels_ like.\"\n\nHow much of her life had she spent ashamed? Despising herself?\n\n\"But I'm still here for the same reason you are. We aren't alone. We are not alone.\"\n\n\"Indeed,\" Cirrus Kindwind confirmed softly. \"We have spoken of this, Chosen-son. Giants affirm that joy is in the ears that hear because the telling of our tales binds us one to another. Speaking and hearing, we share our efforts to give our lives meaning.\"\n\nThe rain was falling harder. Soon it would be falling too hard to hear anything; say anything.\n\nThrough a slash of water, Jeremiah whispered, \"But you don't _see_ it. I don't mean anything.\"\n\nHis misery closed Linden's throat. She had no answer for him. She believed in Covenant, but she was afraid to believe in herself. Her greatest fear\u2014\n\nWhile she stumbled inwardly, Covenant put his hand on her thigh. \"We should get out of here.\" Slapping raindrops obscured the severity of his mien, the lines of his willing compassion. \"Maybe we can escape the worst of this storm.\"\n\nAs if he had triggered it, lightning shrieked overhead. Thunder made the air shudder.\n\n\"To my sight,\" Rime Coldspray remarked, pitching her voice to carry, \"the coming downpour does not appear extensive. Nonetheless it will be extreme. The Timewarden counsels wisely.\"\n\n\"Branl!\" Covenant barked over his shoulder. \"How far have we come?\"\n\nThe Humbled sat Rallyn with the _krill_ raised in one hand and Longwrath's sword leaning on his shoulder. \"Our translations have increased, ur-Lord,\" he replied. \"We have traversed nigh unto thirty leagues, and have lost but a portion of the afternoon.\"\n\nStave nodded in confirmation.\n\nLinden tried to remember how much ground the Ardent had covered when he had conveyed the company out of the Lost Deep. Another glare and shout among the clouds distracted her. The rain was becoming a deluge.\n\nCursing, Covenant started back toward the head of the company. Incipient torrents belabored his head and shoulders.\n\nAt once, Branl dismounted. The flamberge he handed to Onyx Stonemage. With Covenant and Loric's dagger, he strode beyond the company.\n\nRain hammered the ground. It beat the dirt to mud. Clotted rills squirmed past the feet of the Giants. Linden felt herself sinking under the weight of the downpour, hunching over her heart. Her son needed help, and she had nothing to give him.\n\nThe Giants braced themselves for a sprint which would have no perceptible beginning: it would simply come over them somewhere within the blank space created by wild magic and Loric's blade. Hyn tossed her head, repositioned her hooves. Khelen snorted a warning at Jeremiah. Lightning ripped through the gloom. Thunder roared against the cliff like the wrath of mountains.\n\nBranl moved swiftly, carrying Covenant. Covenant's line of fire defied the torrents as if dirt and rain were fuel. Flames danced like Wraiths on Linden's wedding band. Reflexively she held the Staff as far away from her ring as she could.\n\nWhen the world vanished, her heart plunged into darkness. She and her companions were taking Jeremiah to Mount Thunder.\n\nTo the Despiser.\n\nHe would relish her son.\n\nithout transition, the horses and the Giants were running as if their lungs would burst. They strained at a steep slope, labored forward against the obstructions of their mortality. Then they pitched down a hillside, plummeting like a landslide.\n\nThere was no rain. The dusk of late afternoon held the world under a sparse sprinkle of stars. Every breath sucked at a humid miasma of putrefaction and worse poisons.\n\nOn the left some distance away rose a high cliff sheer as a cut slab. And water lay _there_ , on the right: the rank wetland of the Sarangrave, brandishing its tortured trees and twisted scrub and fetor. Branches writhed like the beckoning limbs of demons. The companions hurtled toward the Flat as if they aimed to cast themselves headlong into its reek.\n\nThen Covenant heaved on Mishio Massima's reins, yelled at the beast to stop. Rallyn braced his legs against the descent: the Ironhand and her comrades dug in their heels. Stride by stride, the company slowed.\n\nA tree flashed past, and another. Ironwood? Hyn splashed through a stream. A glowering cluster of cypress reached out from the edge of the marsh. Following Rallyn and Covenant's mount, the companions veered away, angling closer to the cliff.\n\nAs Hyn mastered her momentum, Linden realized where she was. Although she had never seen the mountain from this perspective before, she recognized Mount Thunder. In profile, it resembled a titan kneeling against or within Landsdrop with its forearms and torso braced on the Upper Land, facing west. The nearby cliff was a side of the mountain rather than an extension of the great precipice which divided the Land. The hillside down which Hyn moved, trotting now, was one of the titan's calves. The other formed the far side of a valley leading from the base of the cliff into the Sarangrave.\n\nThe valley was wide enough to hold a large herd of Ranyhyn, long enough to accommodate several hundred Masters fighting Cavewights or _kresh_. On the lower slopes and in the bottom grew scattered ironwood trees nourished by streams of fresh water tumbling downward on both sides, north and south. Marsh grasses climbing out of the Flat wrestled for room to grow with bindweed and more noxious plants.\n\nBut the spine of the valley bottom was a riverbed that stank like a sewer.\n\nBlack water viscid as oil, putrid as excrement, ran from a gaping wound in the cliff between the mountain's knees. At one time, the river had thundered from that wound, flushing the bowels of Mount Thunder with the combined waters of the Upper Land, emptying the effluvium of banes and charnels, of disused Wightwarrens and discarded corpses and lakes of acid, into the welcome of the Great Swamp. But now the level had fallen far below the bed's rims. Even augmented by the streams, the Defiles Course barely carried enough water to cover the slimed rocks of its bottom. The wound in the cliff gaped like a waiting maw.\n\nInsects hummed with hunger past Linden's head. Some of them stung. Swarms of midges swirled here and there as if they fed on the odor of excreted toxins. As she rode downward, the cypresses appeared to rise up until they towered above her, avid and polluted. The ironwoods looked mighty, although they would surely have grown taller and broader in a kinder setting. Above them, the cliff extended itself to giddy heights.\n\nEven from lower ground, she could see that the exposed gutrock of the cliff was little wider than the valley between the mountain's calves. Slopes spread up at awkward angles from Mount Thunder's thighs into its back. There the mountainsides were rugged and threatening, riddling with clefts and flaws as if they had been hacked by gargantuan blades: they looked impassable. Nevertheless she suspected that Rime Coldspray and her comrades might be able to climb there, given time\u2014and perhaps rope. Stave and Branl could certainly do so. But Linden herself could not. For her, the higher sides of Mount Thunder were unattainable.\n\nHere she had no choice except to follow Covenant, unless she turned Hyn and fled, taking Jeremiah with her.\n\nIn the valley bottom near the trunk of an ironwood, Covenant finally halted. He handed the _krill_ to Branl; but he did not dismount. Instead he waited, peering into the gullet of the mountain, for the rest of the company to close around him. The tension in his shoulders, and the clutch of his hands on the saddle horn, told Linden that he was holding himself in his seat by force of will. His eyes bled tears as if the stink of the Defiles Course burned them; as if the fetor were remorse.\n\nWhile their respiration eased, the Swordmainnir scanned their surroundings anxiously, considering possible attacks or escapes. In contrast, the _Haruchai_ gauged the terrain with their characteristic dispassion. Alone among their companions, they remembered this place. No doubt their communal memories included recollections of the Defiles Course at its torrential height, when the flood in the lower end of the valley would have reached at least partway up the trunks of the cypresses. But Jeremiah did not lift his head or look around. Muttering to himself, he studied his hands and scowled as if their emptiness angered him.\n\nWhen he had regained his balance, Covenant announced, \"This is it. I guess that's obvious. We should rest while we can. I'm not sure we'll get another chance.\" Coming here had been his decision. Nevertheless his tone was thick with doubt. \"And we should send the horses away. They can't help us now.\n\n\"When we're ready, I'll try to get the attention of the Feroce. I'm hoping they can guide us at least part of the way.\"\n\nForestalling an objection which no one expressed, he added, \"Not that I think they've ever been _in_ Mount Thunder. But they're creatures of water. And not just any water.\" He gestured at the river. \"They thrive in this stuff. Plus they don't need light. Maybe they can lead us far enough to find the Wightwarrens.\n\n\"After that,\" he finished trenchantly, \"we won't need to know where we're going. We'll just have to fight. Eventually the way Foul defends himself will show us where he is.\"\n\nHe may have meant, Show us or herd us.\n\nWhile Linden tried to gather herself, Jeremiah glared at Covenant under his eyebrows. \"It's a waste of time,\" the boy rasped. \"I'm starting to recognize some of the landscape. The Worm is on the Upper Land. Beside a river. We'll still be groping around like we're lost when it reaches _Melenkurion_ Skyweir.\n\n\"And what makes you think you can trust the Feroce?\" He clenched his fists, apparently trying to muster flames. But his access to Earthpower eluded him. Perhaps visions of the Worm blocked it. \"You had an alliance while the lurker was scared. Now the Worm is moving away. Maybe the lurker thinks it doesn't need you anymore.\"\n\nCovenant shrugged. He faced Jeremiah squarely, but did not respond.\n\nFeeling helpless and dismal, Linden asked, \"Thomas, are you sure you want to do this?\"\n\n\"What else are we going to do?\" Leprosy blurred Covenant's eyes like pain or empathy. \"We're here now. If that's a mistake, it won't be my first. Most of what I do in life is just trying to make amends for things I've done wrong.\n\n\"Anyway you heard Jeremiah. We don't have time to try anything different.\"\n\nLinden did not respond. She had already lost this argument.\n\nBut Jeremiah was not done. \"But why bother? I thought I understood. I mean, before I could see the Worm. Now I don't. What's the point? We're all going to die anyway.\"\n\n_I have given him a gift,_ oh, Jeremiah, _which will make him wise in the subtleties of despair._\n\nLinden might have tried to reassure him. Covenant might have. But the Giants silenced them by the simple expedient of bursting into laughter.\n\nTheir loud mirth filled the valley. It seemed to startle the insects. Midges fled for the safety of the wetland's mire. Horseflies and mosquitoes skirled away, whining. Just for a moment, even the stinks of the Defiles Course and the Sarangrave became less daunting.\n\n\"Bravely said, young Jeremiah,\" Grueburn guffawed. \"A fine riposte.\"\n\nLatebirth and Halewhole Bluntfist doubled over, gasping for breath.\n\n\"'But why bother?'\" echoed Cirrus Kindwind. \"Why, indeed? You make sport of our fears, Chosen-son.\"\n\nStormpast Galesend slapped Cabledarm's back. Cabledarm aimed an elbow at Galesend's ribs.\n\nExpecting Jeremiah to take offense, Linden flinched. At the same time, however, she felt a rush of gratitude. Too much had happened since she had last heard laughter.\n\nWhile Jeremiah fumed, the Ironhand struggled for gravity. She scrubbed at her eyes until her humor receded to chuckling bursts. \"All paths lead to death,\" she said when she found her voice. \"This the Worm merely hastens. Nonetheless we must strive. How otherwise will we hold up our heads at the end of our days?\"\n\nLinden watched Jeremiah wrestle with himself. He must have felt mocked. Surely he felt that? But he also loved the Giants. And their mirth was too open-hearted to sound like ridicule. Briefly his mouth twisted: he almost smiled in spite of himself. Then he mustered a conflicted glower.\n\n\"Never mind. I wasn't serious. Have it your way.\"\n\nThat may have been as much grace as he could muster. If so, it was enough for Linden.\n\n\"Well, hell,\" Covenant drawled as the Giants subsided. \"Hellfire.\" Then he fell silent as if he had run out of words.\n\nAs if by mute agreement, Stave and Branl slipped down from their Ranyhyn. If the _Haruchai_ were capable of laughter, Linden had never heard it. Here, however, she caught a glint that looked like amusement from Stave's eye. Branl's manner as he leaned Longwrath's blade against the trunk of the ironwood hinted at the easing of subtle tensions.\n\nWhen Jeremiah dropped, dour and distant, to the ground, and Covenant dismounted, Linden joined them. The unfamiliarity of her wedding band or the aftereffects of wild magic made her finger itch. Holding the Staff in the crook of her elbow, she rubbed absently at the itch while she tried to think of a way to thank Rime Coldspray and her comrades.\n\nTwisting the kinks out of his back, Covenant made his way toward the nearest stream. The Ardent's steed cantered past him to thrust its muzzle into the water, blowing bubbles as it drank. The four Ranyhyn followed more sedately. Hynyn's wonted imperial air was subdued, and Hyn's head drooped as if she were weighted down with farewells. Khelen cast anxious looks at Jeremiah, but did not hang back.\n\nAfter the Swordmainnir had loosened their armor, Onyx Stonemage passed around the remaining waterskins of _aliantha_ : the last meal that Linden expected to eat. Covenant accepted treasure-berries. Even the _Haruchai_ did so. Then Kindwind and Grueburn carried the emptied sacks to another stream.\n\nThe valley's insects had forgotten their fright. A few flying things with stingers found Linden. One raised a welt on the back of her hand: another, on the side of her neck. Irritated by those pangs, and by the region's renewed fetor, she found herself remembering carrion. She remembered _being_ carrion; remembered the howling anguish and condemnation of She Who Must Not Be Named. Remembered Elena\u2014\n\nEntering the maw at the base of the cliff would resemble falling from the Hazard.\n\nSwearing to herself, she called Earthpower from her Staff to heal her little hurts, chase away the insects and the worst of the stenches; cleanse the recall of maggots and lice from her nerves. Then she extended the same small benison to her companions.\n\nJeremiah ignored her gift. Trapped in his own thoughts, he did not appear to feel any physical discomfort. Perhaps the same inheritance which protected him from cold and preserved his bare feet also warded him from stings. Whenever Linden thought that she should talk to him, she discovered that she was not ready. What could she have said? His ability to watch the Worm's progress was a wound for which she had no salve.\n\nLike her companions, she refreshed herself at the stream, ate her portion of the treasure-berries. Then she shared a hug with Covenant; leaned against the stubborn bones of his chest while his stubbed fingers ran awkward reassurance through her hair.\n\nAs the tenuous afternoon dwindled toward evening, vapors began to rise from the waters of the Sarangrave. At first, they were vague, visible only when they caught the light of the _krill_. But gradually they thickened into blots and tendrils of fog. By degrees, opaque arms and sheets found their way into the valley, traced the Defiles Course toward the sides of the mountain. Before long, the fog was a softly roiling wall that veiled the Flat. If it continued to expand, it would soon fill the valley.\n\nAs strands of fog coiled among the sparse ironwoods, the horses took their departure. Mishio Massima simply trotted away, tossing its head as if it had exhausted its patience for riders. The Ranyhyn were more formal. First they gathered around Covenant. As one, they reared, pawing the air as if he had won their approval. Then they separated, Hynyn toward Stave, Rallyn to Branl, Hyn and Khelen to Linden and Jeremiah. Hynyn nuzzled Stave while Stave stroked the stallion's nose. Khelen offered the same gesture of affection to Jeremiah. For a moment, Jeremiah appeared to rebuff the Ranyhyn. Abruptly, however, he flung his arms around Khelen's neck: a boy who did not want to be forsaken.\n\nBranl answered Rallyn's whinny with a salutation as old as the Lords. To Linden, Hyn lowered her head to the ground, bending one foreleg like a curtsey. \"No,\" Linden breathed as she hugged the mare, \"please. We're past that. I should be bowing to you.\"\n\nHyn replied with a soft nicker. The look in her gentle eyes implied sadness, pride, affection, regret, even an atavistic alarm. Nonetheless it seemed to aver that she had not lost faith.\n\nIn homage, the Giants drew their swords. Holding their blades high, they saluted the fidelity and service of the great horses.\n\nAs one, the Ranyhyn turned away. Together they followed Mishio Massima into the fog. If they neighed any last farewells, their calls were swallowed by the brume.\n\nThe stars were gone; masked. Damp vapors blurred the shape of the watercourse. The cliff lost its definition, its implacable rigidity. Around the companions, the _krill_ 's argent reflected back from the fog until they seemed to stand within a cynosure. A beacon. Beyond the light, the rest of the world was reduced to a slow seethe of blankness, moist and clinging.\n\nLinden regarded the fog with fresh apprehension. It seemed to imply perils which would strike without warning.\n\n\"What does it mean?\" demanded Jeremiah hoarsely.\n\n\"It means,\" Covenant replied, a low growl from the back of his throat, \"we've waited long enough.\"\n\nSuddenly brusque, he claimed the _krill_. With Branl at his side, he headed down the valley toward the Sarangrave. Loric's dagger thrust illumination ahead of him. At his back, fog crowded in to enclose the company. Where he stopped, the outermost twigs and boughs of cypresses were visible; but Linden could barely distinguish her companions.\n\n\"I'm here.\" Covenant appeared to shout, but the fog muffled some sounds while it accentuated others. The distant plash of water carried more distinctly than his voice. \"You called me the Pure One. We made an alliance. I've been keeping my part. Now we need to talk. I want the Feroce.\"\n\nWrapped in that fug, Linden found it impossible to believe that any of the Sarangrave's ears would hear him.\n\nDroplets beaded on her skin. The damp seeped through the flaws in her shirt. With her nerves rather than her eyes, she located her son's aura. His emanations conveyed the impression that he was crouching down inside himself; that he feared the touch of the vapors; that he wanted to flee.\n\n\"I discern no cause for alarm,\" Grueburn stated. \"Do our foes deem that mere fog will affright us? We have endured the toils of the Soulbiter, and have emerged scatheless. We are not so blithely overcome.\"\n\nShe may have been trying to comfort Jeremiah.\n\n\"Aye,\" answered the Ironhand. \"Yet fog occludes here as it does in the Soulbiter. Ready yourselves, Swordmainnir. Mayhap this brume is a natural exudation of the wetland. Or mayhap\u2014\"\n\nAround her, Giants tightened their cataphracts, loosened their arms and shoulders.\n\n\"Jeremiah?\" Linden felt an instinctive impulse to whisper. \"Listen to me. Are you listening?\"\n\nStave stood at her back, impassive and silent.\n\nCovenant may have been yelling at the Flat, but his words were lost. The _krill_ 's light did not penetrate the shroud over the Sarangrave.\n\n\"There's no stopping it, Mom,\" Jeremiah replied like a groan. \"You should see what it's doing to the plains.\"\n\nLinden grasped his arm. When he tried to pull away, she tightened her hold. \"I said, _listen_. Maybe there's a way out of what you're feeling. Maybe Foul gave you those visions to distract you. Maybe he doesn't want you thinking about other possibilities. Maybe your real problem is that you don't know how to defend yourself.\"\n\nJeremiah's tone changed. \"Mom?\"\n\n\"You have Earthpower,\" she explained, \"but it isn't a weapon. It's like _orcrest_.\" Or like Anele himself. \"It doesn't protect you. Maybe you wouldn't feel so hopeless if you had a way to fight.\"\n\n\"But I don't.\" In spite of her grip on his arm, he sounded as remote as Covenant. \"I'm useless.\" He may have meant broken. He had learned that his desire to repay the Despiser's malice was a foolish fantasy. \"All I can do is watch.\"\n\n\"No.\" Simply because her son's distress hurt her, Linden wanted to raise her voice. She had to force herself to speak quietly. \"Listen to me, honey. There's always something we can do, even if it's just changing the way we look at what's happening, or the way we look at ourselves.\n\n\"I think I know how you can defend yourself.\"\n\nWith her fingers, she felt his shock. \"How?\"\n\n\"Linden Giantfriend.\" The fog muted Rime Coldspray's tension. \"My heart misgives me. The Timewarden's hopes fail. The Feroce do not come. And this fog\u2014\" She made a spitting sound. \"Stone and Sea! I cannot persuade myself that it is natural. Some evil summons it.\"\n\nLinden closed her ears to the Ironhand. \"Try this.\" She pulled Jeremiah closer. \"Fill your hands with fire. You can do that. I know you can.\"\n\n\"Why?\" He tried to draw back. He had failed earlier. \"What good will that do? You just said\u2014\"\n\nShe cut him off. \"Just _do_ it. Then touch my Staff.\"\n\n\"Mom!\" he protested. \"I can't use your Staff!\"\n\n\"We don't know that yet.\" She strove to sound calm, but she trembled in spite of her efforts. \"We haven't tried it.\n\n\"First your Earthpower. Then my Staff. After that, I'll help you figure out what comes next.\"\n\nThrough her teeth, Coldspray muttered warnings which her comrades did not require.\n\n\"Hellfire,\" Covenant raged in the distance. The _krill_ 's shining throbbed ineffectually. \"I saved you from _turiya_ by God Raver. And I told you not to sacrifice yourself against the Worm. If you got hurt, it wasn't my doing. I kept my part of the deal. I've been keeping it. Now it's your turn.\"\n\nLinden felt his vehemence, but she did not hear an answer. Fog eddied around her head. She could barely make out Jeremiah's features.\n\nHe floundered in her grasp as if he wanted to resist and comply simultaneously. \"Mom\u2014?\" His distress came in bursts. \"I don't\u2014How can\u2014? Don't make me. I\u2014\"\n\nJust for a moment, she feared that she had pushed him too far. He was only a boy. And he had spent most of his life hiding. In effect, he had only known himself for a few days.\n\nBut then he stopped trying to pull away. Flames appeared in his palms as if his skin had caught fire.\n\nThey danced and fluttered, leaned raggedly from side to side like fires in a harsh wind. But they grew stronger as he gained confidence in them. By the measure of his needs, they were little things, no bigger than his hands. The sun-yellow of Earthpower did not push back the fog. Still these flames were _his_. They had been given freely.\n\nYes, Linden thought. If he could do that, he could do more. She would teach him somehow. His own health-sense would guide him if hers did not suffice.\n\n\"Giantfriend,\" the Ironhand insisted. \"Linden Avery.\"\n\n\"Now the Staff,\" Linden instructed Jeremiah, whispering again. \"It's full of possibilities.\" The runes. The iron heels as old as Berek Halfhand. The combined essences of Vain and Findail. Her own love. \"Try to feel them. Maybe they'll answer,\" Earthpower to Earthpower.\n\nShe had her wedding band. Covenant had made her a rightful white gold wielder. Surely she could fend for herself without the Staff of Law?\n\n\"It might not respond right away,\" she admitted. \"It isn't yours. I made it. I have a kind of symbiotic relationship with it. But if you keep trying, you should\u2014\"\n\n\" _Attend_ , Giantfriend!\"\n\nThe Ironhand's shout snatched at Linden. Involuntarily she wheeled away from her son's guttering hands.\n\nAt once, the distinctive reek of gangrene stung her nose. Impressions of necrosis seemed to hit all of her nerves, her whole body. She recognized that smell, those emanations; but for a confused instant, she could not identify them.\n\nThen she saw a lurid swelling of brimstone, a fierce gnash of lava. It was some distance away on the far side of the Defiles Course. Nevertheless it was hot enough to pierce the fog. She remembered roaring ferocity, fangs like scimitars in long rows, terrible jaws.\n\nOh, God\u2014\n\nBeyond her, the Swordmainnir strode down the slope to intercept the attack, spreading out so that they would each have room to strike and dodge. Stave stood a few paces in front of Linden as if he imagined that he could counter one of the _skurj_.\n\nCovenant may have been unaware of the threat behind him. He continued hurling his demands into the shrouded Sarangrave. The _krill_ slashed back and forth: cuts that had no effect. But now he was alone. Apparently Branl trusted that the lurker would not assail the Pure One, even if the monster had withdrawn its aid. With calm haste, the Humbled came back up the valley, gripping Longwrath's flamberge in both hands.\n\n\"Mom?\" Jeremiah called: a small sound like a whimper. \"Mom? What's happening?\"\n\nAbruptly a monster surged up from the eaten ground.\n\nNow Linden saw it clearly. The unthinking creature had devoured its way through the earth to emerge among the roots of an ironwood. Almost immediately, the tree exploded into flames. Bright as a bonfire, and hot as the ravaging of Covenant's home on Haven Farm, it heralded hunger and scoria.\n\nTall and thick as a Giant, the _skurj_ stood in conflagration with half of its full length braced underground. Roaring like an eruption, it twisted from side to side, apparently seeking the scent of its prey. Then it began its rush toward the company, drawing its whole body out of the dirt as it snaked into the valley bottom.\n\nUnder other circumstances, the river might have forced the monster to pause; perhaps to chew its way beneath the watercourse. But the Defiles Course was much diminished. The _skurj_ did not hesitate. Coiling its strength, it launched itself in a brimstone arc above the waters.\n\nIts fury dismissed the fog around it. Even at that distance, Linden felt waves of heat beat against her face.\n\nCovenant's shouting was hoarse and doomed. Still he persisted.\n\nLinden did not think. She had no time. Raising her Staff, she left Jeremiah's side. Black flames like the tails of a scourge pulled free of the wood and whipped around her as she hurried toward the Giants.\n\nStave accompanied her without question. He seemed to have no questions left.\n\n\"Don't move,\" she urged as she passed between Coldspray and Frostheart Grueburn. \"I can do this.\" She hardly heard herself. \"Take care of Jeremiah.\" In the back of her mind, she had already begun to pronounce the Seven Words. \"Lord Foul doesn't want him dead, but that monster probably doesn't care.\"\n\nThe _skurj_ was only one.\n\nIn Salva Gildenbourne, one alone had overwhelmed her in spite of her Staff. And during the company's flight toward Andelain, Kastenessen's monsters had been too strong for her. She could not have fought them in the Lost Deep.\n\nSince then, everything had changed. Kevin's Dirt was gone. Kastenessen's passing into the fane of the _Elohim_ had struck manacles from her wrists. While Covenant still believed that the lurker might heed him, she meant to guard his back.\n\n\" _Melenkurion abatha_ ,\" she promised softly while the _skurj_ arose from the riverbank and swept toward her. \" _Duroc minas mill_. _Harad khabaal_.\"\n\nDown the valley from her, Branl did not quicken his pace. He advanced with the remorseless inevitability of a breaking wave.\n\n\"Help her!\" Jeremiah panted at the Swordmainnir. \"That thing is going to _eat_ her!\"\n\nIf the Ironhand or any of her comrades replied, Linden did not hear them.\n\nNow! she told herself. Do it _now_.\n\nGet away from me, you overgrown slug. You cannot have my son! Or my friends. Or my _husband_.\n\nWith the Staff of Law alive and lurid in her grasp, she flung an ebon torrent of Earthpower and Law between the jaws of the _skurj_.\n\nThe creature's body radiated heat, but it did not emit light. All of the monster's radiance came from the cruelty of its fangs. They were lambent and infernal, curved for ripping: they blazed with havoc. Looking into that wide maw was like staring down the gullet of a living cremation.\n\nBut Linden was as ready as she would ever be. Her power was ready. And she was sick of frustration and fear, more profoundly infuriated that she had allowed herself to realize. She felt that she had not struck an effective blow since the day of horror when she had slaughtered uncounted Cavewights with wild magic: sentient creatures whose massacre at her hands still filled her with revulsion. She was by God _ready_ to oppose a monster which sought destruction merely to feed its own worst appetites\u2014and to satisfy a Raver's commands.\n\n_Moksha_ Jehannum had once possessed her. She remembered him vividly. Like Covenant, if only with her Staff, she was done with restraint.\n\nHer dark torrent tore a howl from the throat of the _skurj_. The monster reared back, balancing like a cobra on its length. For a moment or two, long enough for her to shout the Seven Words, it tried to swallow her power; gulp it down as if it were the natural drink of _skurj_. But it could not. Her power shredded its gullet, sent agony inward. Thrashing its head, it clamped shut its mouth, closed its jaws on its horrid lumination. Then it whirled away before she could inflict more pain.\n\nBranl intercepted the _skurj_ smoothly, as if he had foreseen the timing of his strike as soon as he had left Covenant's side. The flamberge was a streak of theurgy in his hands. One stroke cut halfway through the monster's neck.\n\nThen he sprang aside as the _skurj_ became a flailing fountain of blood as bitter as acid. Convulsions writhed through the monster: it seemed to snarl itself in knots. But it could not survive its wounds. While it gaped and snapped at the air, the light of its fangs faded into the fog.\n\nStill Linden did not stop. She had endured too much, and yearned to repay it. Branl had killed the _skurj_ for her: she turned her fire to quench the virulence of the monster's blood.\n\nEven when she had eradicated every spot and spatter from the dirt, she wanted to continue until she had reduced the corpse to ash. But she felt Stave's hand on her shoulder, heard him say, \"Enough, Linden. The monster is slain. Now you must conserve your magicks. Where one _skurj_ arrives, others will surely come.\"\n\n\"No,\" Jeremiah breathed, apparently to himself. \"Not more of those things. I can't stand it. How did it find us?\"\n\nIf he sought reassurance, no one offered it.\n\n\"Aye, Linden Giantfriend,\" rumbled the Ironhand. \"Your prowess raises a paean in our hearts. Yet Stave Rockbrother counsels wisely. In Kastenessen's absence, the _skurj_ are doubtless ruled by _moksha_ Raver. We must believe that a greater force follows this lone creature. We must spare our strength while we may.\"\n\nSomeone should have said that to Covenant. He was still trying to coerce a response from the Sarangrave, hacking at the fog with Loric's _krill_ , and yelling intermittently. The gem's argent spread out until the wetland smothered it. His voice made no sound that Linden could hear.\n\n\"In that case,\" she said as if she had only now begun to understand Rime Coldspray's warning, \"we need to _see_. We can't let whatever comes take us by surprise.\n\n\"Watch for me. I'm going to get rid of this damn fog.\"\n\nThe vapors baffled percipience. Like the Ironhand, Linden did not know whether they were natural or invoked. But she did not care. The fog itself was just suspended moisture. Earthpower and Law would dispel it.\n\n\"That would be a benison in all sooth,\" answered Coldspray. \"Make the attempt, Giantfriend. The Swordmainnir will ward your son.\"\n\nLinden nodded, but she had stopped listening. Again she prepared the Seven Words in her mind. Instinctively she moved away from her companions so that she would have space to work. With only Stave nearby, she tuned her senses to the pitch and timbre of mist. Then she lifted new flames from the Staff and sent them skirling upward.\n\nShe regretted the blackness of her fire. She would always regret it. But she had no idea how to relieve it. The fog was a simpler problem. And her stained theurgy was still Earthpower.\n\nWith her eyes closed, she summoned more and more of her Staff's potential. Her health-sense recognized and measured the vapors: their specific dampness on her skin; their distinctive currents and flavors. As if she were musing to herself, she murmured the Seven Words.\n\nThe only substantial obstacle to her intent was the extent of the fug. It arose continuously from the Flat, curled up into the valley without ceasing. To be rid of it, she had to dismiss it faster than it came.\n\n_Melenkurion abatha._\n\nObliquely she wondered whether it had been invoked by the lurker, perhaps so that the High God of the Feroce would have an excuse for ignoring Covenant's appeal. On a deeper level, she chewed the gristle of Jeremiah's question. She feared that she knew the answer.\n\n_Duroc minas mill._\n\nBut she had work to do and could not afford to distract herself. If more _skurj_ were coming\u2014\n\n_Harad khabaal._\n\nBehind her, the Giants muttered their approval. Stalwart as any of his kinsmen, Stave guarded her back.\n\nWhen she had cleared the air directly overhead, unveiled the stars and the onset of evening, she sent her fire toward the cliff above the Defiles Course; toward the steep slopes on either side of the exposed gutrock.\n\n\"How did it find us?\" Jeremiah repeated. He raised his voice, tried to make his question a demand. \"We can't get away if we don't know how it found us.\"\n\nThe _skurj_ were able to sense exertions of Earthpower; but Linden did not know how far their perceptions reached. Could they detect her power while they were ravaging in Salva Gildenbourne? Detect it past the bulk of Mount Thunder? And arrive so quickly? No: she did not believe it.\n\nShe no longer felt Covenant's irate, tattered summons; no longer sensed the _krill_ 's shining imprecation. Grimly she focused her attention on the Staff of Law and fog.\n\n\"There!\" one of the Swordmainnir barked softly.\n\nA quick pang of alarm disturbed Linden's flames. She bit her lip, resisted her impulse to falter.\n\n\"Where?\" asked the Ironhand. \"My sight has lost its youth. I do not descry\u2014\"\n\nCalm as mist, Stave said, \"Chosen. Direct your strength to the mountainsides beyond the Defiles Course.\"\n\nShe complied at once. _Moksha_ 's forces were more likely to round Mount Thunder from the north than the south.\n\nFresh tension spread among the Giants. Latebirth groaned. Stonemage and Grueburn cursed harshly.\n\n\"Chosen-son!\" snapped Cirrus Kindwind. \"Stand at my back. Move as I move. I will shield you.\"\n\nTo Branl, Coldspray rasped, \"You must defend the Timewarden. We cannot. If the Swordmainnir do not stand together, we will soon fall.\"\n\nLinden opened her eyes, but she did not need them to discern the Sandgorgons. She felt their eager ferocity in every nerve.\n\nThere were\u2014\n\nOh, God!\n\n\u2014at least a score of them. Two score? More?\n\nFatal as a landslide, they sped among lingering streamers of brume, hurtled down the mountainside toward the valley.\n\nOne led the way. It had pulled some distance ahead of the others. Behind it came three, no, four more Sandgorgons. Nimble on the pads of their feet, the strange backward flex of their legs, they cascaded over the rocks. The rest of the monsters followed, a pale rush angling across Mount Thunder's contorted slopes.\n\nFor an instant, Linden froze. How many Sandgorgons had left their home across the seas? More than this? Surely not _more_?\n\nThe company could not survive so many.\n\nWorse, Jeremiah would not be one of the victims. Lord Foul and _moksha_ Raver might not be able to control the _skurj_ ; keep them away from the Despiser's prize. The Sandgorgons were another matter. The shreds of _samadhi_ Sheol animated their minds. They would obey Lord Foul's wishes.\n\nAs if she had taken herself by the throat, Linden let out a black scream against the fog.\n\nThat was as much as she could do. She wanted to strike at the Sandgorgons before they reached the valley, do as much damage as she could from a distance. But she had already caught the reek of more gangrene.\n\nHigh above the Defiles Course, a second chancre had appeared, a second suppuration. The gutrock bled vile fluids like pus.\n\nGod in Heaven! We can't\u2014\n\nRime Coldspray adjusted the formation of the Swordmainnir. With Frostheart Grueburn, Latebirth, and Halewhole Bluntfist, she came to stand in front of Linden. The others positioned themselves to defend Jeremiah. He was trying to shout, but his voice broke into whimpers. Stave waited at Linden's side as if he were resting. In no apparent hurry, Branl returned along the valley bottom toward Covenant.\n\nRagged with strain, Covenant continued yelling at the Sarangrave.\n\n\"Linden Giantfriend.\" The Ironhand sounded almost nonchalant. The prospect of an impossible battle seemed to focus her combative nature. \"The _skurj_ we must entrust to you. If by kind fortune they approach singly, you may perhaps prevail. The Sandgorgons are mighty in all sooth, yet they wield only strength and ferocity. And we also are mighty. We are armed and armored. We will hope to stand against them. If they do not mass for a combined assault\"\u2014she shrugged to loosen her shoulders\u2014\"we will teach them to esteem us.\"\n\nThe pounding of Linden's pulse in her ears measured out Coldspray's words\u2014 _entrust to you._ After that, she recognized only one in three. Still she knew what was required of her.\n\nJeremiah had his defenders. Armed with a sword forged to fight Sandgorgons, Branl would guard Covenant. And Covenant was not helpless. If any residue of his victory over Nom lingered in the minds of the monsters, or in _samadhi_ 's, they might flinch from attacking him. That left the _skurj_.\n\nLinden believed that she could stop them\u2014\n\n\u2014if they came no more than one or two at a time.\n\nFierce and ruddy, a maw full of fangs burst from the granite high in the cliff. With grim satisfaction, Linden saw that the monster was directly above the Defiles Course. The riverbed held much less than its former torrents; but the remaining gush was still _water_ : polluted beyond estimation, yes, and stinking to the stars, but water nonetheless. Her fate was written in it.\n\nSwinging her Staff like the handle of a flail, and hissing the Seven Words past her teeth, she sent barbed fire at the _skurj_.\n\nThe leading Sandgorgon was already nearing the valley. The others did not gain ground, but they followed swiftly.\n\nThinking _Melenkurion_ and _minas_ and _khabaal_ , Linden found that the monster in the cliff had emerged near the limit of her range. She could not hit it hard enough to slay it. But she was fighting now: instinct and desperation guided her. She did not need to kill the monster directly. She could use the river. All she had to do was make the damn thing fall.\n\nDeliberately she harried the creature. She whipped fire at its jaws, made wounds in its gullet. Then she caused one of his fangs to rupture.\n\nRoaring in distress, the _skurj_ thrashed against the rims of its egress. The stone around it cracked and crumbled.\n\nIt was not a thinking creature. It did not observe and take care: it only hunted and fed\u2014and reacted to pain. After a moment, its own writhing broke loose a section of the cliff.\n\nAmid shards of gutrock as loud as thunder, the monster plunged down the face of the precipice.\n\nWhen the _skurj_ hit the Defiles Course, steam erupted from the impact. Fouled water sprayed upward, filled the valley bottom with a rain of poison and acid. But Linden had anticipated that. As the monster fell, she raised a curtain of black flame between her companions and the river. Earthpower burned ruin out of the air. Then, as the corrosive deluge subsided, she turned her fire against the _skurj_ again, burning to trap the monster in the river.\n\nInflicted hurts blocked the monster's escape. It shrieked like shattering as it swallowed spray and splashes, gulped down death. Then it collapsed, steaming furiously; stretched out its length in the current. A moment later, it was dead, and the Defiles Course flowed over it.\n\nLinden wanted a shout of celebration. She looked around for it. But sudden plague-spots dotted the far side of the valley; and more appeared on the near side, within a stone's throw of the company; and the first Sandgorgon raced off the mountainside onto lower ground, charging toward Branl and Covenant.\n\nThe _gaddhi_ of _Bhrathairealm_ had called the Sandgorgons _more fearsome than madness or nightmare_. Baked to an albino whiteness in the Great Desert, the creatures were destruction incarnate. They could pulverize granite with the prehensile stumps of their forearms. And their heads had been formed for battering, lacking eyes or other vulnerabilities. They breathed through slits like gills protected by tough hide on the sides of their heads.\n\nIf that Sandgorgon contrived to strike Covenant, it would snap every bone in his body.\n\nBut Linden could do nothing to defend him. Half a dozen _skurj_ had already thrust their heads and fangs out of the ground. More were close. Frantic and furious, she faced those threats, leaving her husband to Branl.\n\nShe had devised a new defense. Whipping flame from place to place, she concentrated Earthpower on the lambent fangs. From maw to maw, she caused eruptions like bursts of agony along the kraken jaws. Small hurts: the _skurj_ were huge, and their mouths held scores of scimitar-teeth. Nevertheless their pain was acute. It enraged the monsters\u2014but it also distracted them.\n\nIt slowed their emergence from the earth.\n\nGripping her glaive, the Ironhand breathed, \"Well done, Linden Giantfriend. I had not considered such a ploy.\"\n\nIt was no more than a delay, a transient interruption. But it might create openings for the Swordmainnir.\n\nWhile Linden lashed obsidian back and forth, accentuating her efforts with the Seven Words, Covenant and Branl finally turned to face the nearest Sandgorgon. As if they were sure of their strength, they strode to meet the charge. Branl held Longwrath's flamberge poised to slash. Covenant's halfhand gripped Loric's shining dagger by its wrapped hilt.\n\nBehind them came a cluster of Feroce, perhaps ten of the naked child-forms. They held out their hands like gestures of supplication or worship. Rank green flames twisted and flared in each of their palms.\n\nAt their backs, more fog piled out of the Sarangrave, obscuring the perils of the wetland.\n\nThe Sandgorgon gathered itself, sprang over the water. For the flicker of an instant, it vanished below the rim of the riverbank. Then another leap brought it out of the Defiles Course. Silent as the fog, as the boundary between life and death, it sped toward the Unbeliever and the Humbled. Between one stride and the next, it became a juggernaut.\n\nCovenant and Branl did not hesitate.\n\nInstead the creature faltered. Five of its strides from its targets, it jerked to a halt. Its head turned from side to side, scanning with its arcane senses. It seemed to remember Covenant. Its blunt forearms aimed confused blows at the air.\n\nBefore the Sandgorgon could recover\u2014before the thwarted scraps of _samadhi_ Sheol's sentience regained their mastery\u2014Branl delivered a cut that opened the monster's torso from its neck down through its chest to its opposite hip. Blood and strange guts spouted from the wound as the Sandgorgon toppled.\n\nBranl did not pause to regard the corpse. Four more creatures were only heartbeats away. One had already leapt the river. Another was leaping.\n\nBut Covenant turned to the Feroce in spite of his peril. \"That was impressive,\" he growled quietly. \"What did you do?\"\n\nThe Humbled continued his advance. His blade shed blood and strips of flesh as if its old magicks repelled the gore of the Sandgorgon.\n\nIn their one voice, moist and diffuse, the lurker's minions answered, \"We have caused it to remember that it is bestial, a creature of instinct, not of intent. We have caused it to remember that you are mighty. Alas, we are merely the Feroce. We are frail, unworthy to serve our High God. We cannot impose recall upon so many, or upon such savagery.\"\n\nAt the last instant, Branl stepped aside from the first creature, beyond the reach of its arms\u2014but not the length of his sword. The Sandgorgon had no defense as he slid the flamberge across its trunk below its ribs. Reflexively it clamped its forearms over the slash; but they were not enough to keep its life from spilling out.\n\nCovenant nodded to the Feroce. \"Do what you can,\" he said; demanded. \"And tell your High God I need more than just you. I need _him_. I need him _here_. This is what alliances are for. I have to have help.\"\n\nBranl spun into a horizontal cut that bit through obdurate bone, nearly severed the top half of a Sandgorgon's face and skull. But Longwrath's sword caught there, grinding between bones which could have smashed down a wall. The _Haruchai_ could not wrench his blade loose quickly enough to intercept the next creature.\n\nWailing, the Feroce brandished their fires as the third Sandgorgon swung a crushing blow at Branl.\n\nEven his preternatural strength was no match for the creature's. Yet he was _Haruchai_ , and swift. And he had not forgotten the ease with which a Sandgorgon had killed Hergrom, crippled Ceer. He evaded the blow by diving under the creature's arm. It did not touch him.\n\nHe landed on his feet, whirled back toward the creature. But now he was too far away to protect Covenant; and he had to retrieve his sword.\n\nAt the last instant, the theurgy of the Feroce took hold. The monster slowed its rush directly in front of Covenant.\n\nWincing and bitter, he raised the _krill_. The eldritch blade slipped as easily as murder into the Sandgorgon's heart.\n\nBlood sprayed from the creature's gills as it plowed into him. It was already dead. Still the impact sent him sprawling. He lost his grip on the dagger. It tumbled away across the dirt, sending dismembered flashes of argent through the new fog.\n\nFrom the ground, he glared wildly at the fourth Sandgorgon as though he imagined that he could defy it with nothing more than his gaze and his anger. Spangles like glints of frenzy gathered around his wedding band; but he had fallen too heavily to wield them.\n\nLeaping, Branl came down at that creature's back with the full force and magic of his flamberge.\n\nThe Sandgorgon staggered away in a welter of blood and bone. Its legs folded under it. It pounded its featureless face against the valley bottom while its muscles seized. Then it lay still.\n\nMore Sandgorgons were coming: too many. The first of them had reached the valley. In another moment, it would cross the Defiles Course.\n\nBranl appeared to shrug as he reached down to clasp Covenant's hand. In one effortless motion, he snatched the Unbeliever upright. A moment later, he retrieved the _krill_ , returned it to Covenant.\n\n\"Now or never,\" Covenant gasped at the Feroce. He could hardly breathe. Something in his chest felt broken. \"You said the alliance is sealed. We need help _now_.\"\n\nTogether he and Branl resumed their ascent along the valley. He lurched in pain. His companion looked as deadly as Longwrath's sword.\n\nThe Feroce followed at a slight distance. Their fires flared like mewling.\n\nLinden was not watching. She could not. While she harassed _skurj_ furiously, lashing Earthpower and Law at the bright lava of their fangs, another ironwood became instant conflagration. Burning sap burst from its trunk, its boughs, even its leaves. It was close: its heat slapped at her face as an open maw appeared, rabid and ravenous. Uprooted by the monster, the tree pitched down the slope as if it had been hurled aside.\n\nFrantic and off-balance on the cliff edge of her strength, Linden threw obsidian vehemence at the _skurj_.\n\nRime Coldspray stopped her. \"Withhold, Giantfriend! Assail more distant threats. We will oppose those that come near!\"\n\nWhile Coldspray shouted, Latebirth and Bluntfist rushed toward the residue of the blazing tree.\n\nLinden knew that the Ironhand was right. Still she lost herself in a moment of visceral terror. That monster was _close_. It could tear any Giant apart with one bite.\n\nJeremiah called out to her, but his voice seemed to come from the far side of the world. Roaring heat and viciousness muffled every human sound.\n\nYelling the Seven Words like curses, Linden flung the outrage of her heart at other _skurj_.\n\nAt least ten now howled beyond the river. Pustules in the dirt promised more. Joining the creature which Latebirth and Bluntfist faced, four had eaten their way underground to burst upward between the company and the watercourse. Linden started to hurt those four. Then she realized that they were not coming toward her. Instead they swarmed around the first monster which she had slain.\n\nThey were feeding. Eating their dead.\n\nJust for an instant, she believed that she and the Swordmainnir had been granted a respite. But she was wrong. These monsters reproduced by devouring their dead, absorbing the energies of the fallen. Then they split. With sufficient nourishment, one became two. Two might become four\u2014four might become eight\u2014if one dead _skurj_ supplied enough brimstone sorcery.\n\nHorror rattled in Linden's skull. It stung her whole body as if she had been caught in a rain of pebbles. While the monsters arrived faster than she and her companions could kill them, they could not be beaten.\n\nThey were arriving much faster.\n\nShe needed wild magic; needed a dozen staffs like hers; needed help.\n\nThere was no help.\n\nAnd she did not have time to drop her Staff so that she could invoke her ring. Nor did she have the _krill_ or any other catalyst which might ease her access to wild magic. Nor had she learned how to summon silver havoc instantly without aid. If she could have cleared her mind, concentrated her health-sense\u2014\n\nShe was already foundering. Any pause now might be her last mistake.\n\nLatebirth and Halewhole Bluntfist closed on the nearest creature from opposite sides. They endured the heat only because they were Giants. Latebirth lunged a thrust straight into the monster's side. But her thrust was a feint. As the jaws of the _skurj_ reached for her, she jumped back\u2014and Bluntfist rushed in. Raging like the monster, Bluntfist swung a two-handed cut at its neck with all of her mass and her prodigious strength.\n\nUnable to stop itself, the _skurj_ fell onto a second lunge from Latebirth. Her longsword drove between rows of tearing fangs into the back of the creature's throat; into the monster's brain. It collapsed in convulsions.\n\nLinden and Coldspray shouted a warning simultaneously, and Frostheart Grueburn charged; but they were too late.\n\nAnother _skurj_ erupted from the ground almost directly beneath Latebirth's feet. As if it had detected her scent while it chewed through the dirt, it knew exactly where to strike. She was hauling on her sword, pulling it free, when the monster emerged. In one fluid motion, it surged upward and _bit_ \u2014\n\nBluntfist sprang to Latebirth's aid. Grueburn was only three strides away. As if he had forgotten that he was helpless against such foes, Stave followed Grueburn.\n\nThe monster's jaws caught Latebirth below her arms, front and back. Fangs dug into her armor.\n\nLatebirth!\n\nThe hardened stone might have preserved her, at least for a moment; long enough for Bluntfist and Grueburn to arrive. But her cataphract was broken on one side, damaged in battle on the way to Andelain. The monster ripped through it as if it were sandstone; tore her open from chest to spine.\n\nOh, Latebirth\u2014\n\nHer killer was still swallowing blood and organs when Bluntfist and Grueburn hacked its throat to shreds. Fresh gangrene stained the earth around them.\n\n\"Ware, Swordmainnir!\" the Ironhand roared at the dismay of her comrades. \"The _skurj_ must not feed!\"\n\nThey were Giants, familiar with cruel storms and bitter fighting. They knew how to set their griefs and fears aside.\n\nLinden did not. Sick with distress, she sent a raving blast into the first creature that snagged her attention.\n\nIt had begun a leap over the Defiles Course. Half of its length was in the air as Linden's fire poured between its jaws, ran down its gullet. With Earthpower and fury, she ignited an explosion inside the long body.\n\nThen she had to hope that most of the monster would fall into the Defiles Course; that the river would prevent other _skurj_ from feeding. She did not have time to watch. More and more of the horrific serpents had reached the near side of the valley, or had appeared there. Grueburn and Bluntfist whirled away to face another creature. Onyx Stonemage and Stormpast Galesend left their places with Jeremiah, pounded down the slope to challenge a new foe. Stave picked up Latebirth's longsword. Wielding a weapon as tall as himself lightly, as if he had trained in its use for decades, he rejoined Rime Coldspray in front of Linden.\n\nBehind the _skurj_ , a torrent of Sandgorgons raced into the valley, speeding with the single-mindedness of spears toward Covenant and Branl.\n\nFaint amid the tumult, Jeremiah cried, \"Mom, _run_! _We have to run!_ \"\n\nAt every moment, more _skurj_ and Sandgorgons arrived, an inundating wave of monsters. Perhaps Linden would have fled\u2014perhaps the Giants would have\u2014if any of them had believed for an instant that they could outrun the monsters. If any of them had been willing to forsake Covenant and Branl.\n\nFrom the bottom of her heart, Linden brought up a howl of flame:\n\n\"Thomas!\"\n\nHe and Branl had come a third of the way up the valley. There they stopped. Apparently they had decided to stand their ground. Branl moved somewhat apart to accommodate the reach of his sword. Covenant held the _krill_ ready. \"Hellfire,\" he panted at the whimpering Feroce. \"Hellfire.\"\n\nDeliberately he shifted his left hand so that his ring touched Loric's gem. Then he uttered a shout of wild magic that halted the leading Sandgorgons as if he had _forbidden_ them. A dozen paces away, they paused to study him.\n\nOnce long ago he had fought Nom to a standstill. He had not tried to kill the creature; but he had defeated it, forced it to submit\u2014and to listen. He could do more. Yet his power then had not harmed Nom. It did not harm the Sandgorgons now. Their hides had some virtue against wild magic. They could withstand much of his ire. Against so many of them, he would have to unleash far more wrath, more than he could hope to control\u2014\n\nHe might shatter the cliff above the Defiles Course, sealing his only way into the mountain.\n\nWith argent bright in his eyes and silver burning on his scarred forehead as if his mind had become white fire, he ordered the monsters away. In Nom's name, and in his own, he commanded them to depart with their lives.\n\nThey did not acknowledge his authority. They were done with old respect and gratitude. Perhaps they now considered such emotions to be subservience. Instead they heeded _samadhi_ \u2014or _moksha_ Raver speaking to them through _samadhi_ 's remnants.\n\nWhile more monsters sped down the valley, those watching Covenant and Branl changed their tactics. Rather than obeying their instincts, trying to batter or crush any obstacle, they showed that they could think.\n\nFirst one of them crouched: then four more: then a score. One by one, they began hammering the ground with their forearms.\n\nOne was strong enough to cause vibrations that Covenant felt in spite of his numbness. Five made the earth under him shake, dislodged small stones, raised spouts from the dirt. A score\u2014\n\nHe staggered as if he had been taken by vertigo. Flailing to stay on his feet, he had to yank his left hand away from the _krill_. For one heartbeat, two, three, Branl seemed untroubled, as immovable as the roots of a mountain. Then he was compelled to shift his feet, correcting his stance against the tremors.\n\nAs more Sandgorgons arrived, they seemed to understand what the nearer creatures were doing. Without hesitation, they grasped their advantage. Hurtling forward, they struck like albino lightning at Covenant and Branl.\n\nOh, they could _think_ \u2014\n\nYet they remained bestial. _Samadhi_ 's mind was not natural to them, and it endured only in scraps of malevolence. Focusing on their foes, the Sandgorgons did not see a tentacle as thick as an ironwood unfurl itself from the Defiles Course; or they did not regard their peril.\n\nIn spite of his uncertain footing and Covenant's imbalance, Branl wrenched his companion out of the way as the tentacle swept like a scythe at the charging creatures.\n\nThe Sandgorgons were mighty. The lurker was mightier. It roared as if the tumult of fog had been given voice. With one heavy arm, Horrim Carabal blocked the charge. Swift as a spasm, the tentacle coiled around several of the monsters. Then it heaved them into the air.\n\nHowling and savage, the lurker snatched those Sandgorgons to the river and slammed them down; held them under the fouled water.\n\nAt the same time, a second tentacle stretched upward. Guided by the invocations of the Feroce, or by its own instincts, the lurker's arm crashed like a felled tree onto the crouching Sandgorgons.\n\nThat blow scattered the monsters. It stopped the tremors.\n\nIn an instant, Branl recovered. He righted Covenant. Then he rushed into the confusion of the Sandgorgons, delivering cuts like a whirlwind of blades. Some of the creatures lost arms, or forearms. One lost a leg. Two fell dead before the others rallied against the sorcery of Lostson Longwrath's flamberge.\n\nCovenant heard Linden's call then, but he had no chance to answer it. A screech from the lurker warned him. Turning, he saw the torn stump of the lurker's first tentacle writhe above the water, lashing the air with gore. He saw Sandgorgons spring, unharmed, out of the Defiles Course.\n\nAh, hell.\n\n\" _Don't stop!_ \" he yelled at the Feroce. \"I know he's hurt! Hurt is better than _dead_!\"\n\nSummoning himself, he wrapped both hands around the _krill_ once more. Then he moved toward the river. With every step, he mustered more of his power. In his grasp, the dagger seemed to grow longer, brighter, keener. The physical blade remained unchanged, but his wild magic became a longsword implied by Loric's theurgy.\n\nHe remembered the Seven Words. They were of no use to him. They bespoke Earthpower and Law. His force was of another kind altogether. He focused it with curses as familiar as leprosy.\n\nFacing a group of Sandgorgons, with more on the way, he did not hesitate.\n\nHe had slashed one and pierced another before they appeared to realize that he had become dangerous. Suddenly chary, they retreated from the cut of wild magic.\n\nCovenant's world contracted until it contained only Sandgorgons. Somewhere at the edges of his marred vision, innominate shapes swirled in and out of the fog; but he had no time to recognize them. Praying that they were some manifestation of the lurker's magicks\u2014that Branl had not fallen\u2014that Linden could contrive to preserve herself and Jeremiah and the Giants\u2014he anchored himself on his argent blade and assailed the creatures in front of him.\n\nHis wife had cried out to him, but he had not answered. He had only one answer left: one answer\u2014and no opportunity to try it. No way of knowing whether it would suffice.\n\nThe lurker's remaining arm pounded at the Sandgorgons again. Again. Some lurched, apparently hurt. One crumpled and did not rise again. Most withstood the blows as if they lived for such tests of their puissance.\n\nImpassive and lethal, Branl fought on. But his foes had changed their tactics again. He could no longer spin hacking and thrusting among them. Instead they backed away, gained a little distance. Then they spread out to surround him.\n\nAnd from the mountainsides still more Sandgorgons plunged downward. They seemed numberless: a horde of havoc.\n\nHigher up in the valley, Linden flung Earthpower like screams at the _skurj_. Her Staff sent out an unremitting barrage of flame, as black as death in the Lost Deep, and as extravagant as her struggle against Roger and the _croyel_ under _Melenkurion_ Skyweir. Theurgy that might have carved gutrock blasted monsters on all sides. Many she hurt, delayed, enraged. Some she slew. But they were creatures of lava, spawned in magma. They could shrug aside appalling quantities of her fire. And more came: so many that her every gasp filled her lungs with brimstone and putrescence. _Moksha_ Jehannum must have brought every living _skurj_ here from their former prison in the far north.\n\nHer horror was gone. She had sweated it out in heat and fury. Spots of anoxia danced across her vision like burgeoning infections. The wood of her Staff bucked and recoiled as though it might break into splinters at any moment. Her pulse had become an undifferentiated stutter in her veins, too ragged and urgent for individual beats. Even Jeremiah's sporadic shouts and warnings did not reach her. There was no room left in her for anything except Earthpower and _skurj_.\n\nShe was failing. For all her frenzy and desperation, her exertions did not suffice. The monsters far outnumbered her abilities. Even if she had been galvanized by the EarthBlood, as she had been under _Melenkurion_ Skyweir, she would have been no match for the host surging against her.\n\nAround her, her friends strove like demons against impossible odds. They fought in pairs, supporting each other: Stonemage and Galesend, Grueburn and Bluntfist. Cabledarm also had left Jeremiah. With Stave like a Giant at her side, she committed all of her strength to the fray. Exchanging feints and cuts, they wrought bloodshed among the monsters. Only Rime Coldspray stayed to ward Linden. Only maimed Cirrus Kindwind watched over Jeremiah.\n\nIn small bursts, momentary and localized, the Swordmainnir succeeded. They appeared to kill or cripple every creature they met. And Stave did as much as any titan\u2014until an untimely snap of jaws broke his longsword. After that, he had no choice except to throw the shard of his blade down the monster's throat, and to withdraw while he searched for some other weapon. His bare flesh could not survive any contact with the monsters.\n\nWithout him, Cabledarm fought alone.\n\nNevertheless the companions were doomed. The _skurj_ were simply too many to be overcome by frenetic Earthpower and a few Giants. And those creatures which did not attack fed. They multiplied. Much of the valley bottom had become a mad seethe of monsters as vicious as scoria, as fatal as the white core of a furnace. Many of the trees had gone up in flames, but their destruction made no difference.\n\nLinden no longer thought. In some sense, she no longer cared. She was too far gone to count her concerns. As far as she knew, her husband and Branl were already dead. She had only moments left. Jeremiah would survive only if Lord Foul or _moksha_ Jehannum turned the monsters aside.\n\nAbruptly Cabledarm went down. She did not rise again. Stormpast Galesend fell with a fountain of blood where her arm had been. One of the _skurj_ pounced on her before Linden could intervene.\n\nInstantly Stave dove into the struggle, claimed Cabledarm's sword. He joined Onyx Stonemage before she was overwhelmed.\n\nStrands of fog tumbled among the combatants, obscured details until crimson and obsidian fires burned holes in the streamers. Through the confusion, Linden saw too many Giants: half a dozen more than there should have been. Giddy with exhaustion and flame, she tried to count. Three Swordmainnir still fought. Coldspray and Kindwind made five. How could there be more?\n\n\"Welcome!\" the Ironhand shouted with a tantara in her voice. \"Well come in all sooth!\" Then she yelled, \"Assume my task, and Kindwind's, that we may give battle!\"\n\nThe others\u2014the _others_?\u2014were not Swordmainnir. Most of them were men. They wore canvas breeches and shirts rather than armor. And they carried no swords. Two had spears. Another appeared to drive an entire spar between the jaws of a _skurj_. Linden saw a collection of billhooks with whetted edges, belaying-pins longer than one of her arms, knouts studded with sharp stones, immense cleavers.\n\nSuch weapons should have been useless here; yet they wrought confusion among the nearest monsters. Billhooks tore open the hinges of jaws, left maws unable to close. Belaying-pins smashed teeth. Knouts distracted creatures while spears stabbed. Cleavers shed blood wherever they could. In spite of their bulk, the Giants moved with the agility of sailors trained to weather hurricanes.\n\nThey were a paltry force against the onslaught of _skurj_. Still they fought as if they were singing; as if they were glad to spend their lives in a hopeless cause.\n\nThe man who had fed his spar to a monster broke free of the battle, came toward Linden and Coldspray. \"Ironhand,\" he panted, grinning. Reflections of Earthpower and lava in his eyes resembled the exultation of hysteria or madness. \"I hear and obey. Stone and Sea! We are lost.\"\n\nRime Coldspray did not pause to acknowledge him. Roaring a Giantish battle-cry, she took her stone glaive into the heart of the turmoil.\n\nTo Linden, the man remarked, \"My name is unwieldy in such straits. For ease of use in peril, I am called Hurl.\"\n\nShe hardly heard him.\n\nA woman with the charred remains of a knout in one hand followed Hurl; hastened past Linden. As soon as the woman neared Jeremiah, Cirrus Kindwind ran to join the Ironhand. Swinging her longsword one-handed, Kindwind dealt furious cuts at every _skurj_ within reach. But she did not pursue her attack on any single creature. Her tactic was speed. Apparently her only objective was to cause pain; to weaken her foes with wounds.\n\nStave also relied on swiftness. Still he fought with the precision of a surgeon. He seemed inhumanly adept at slicing open the hearts of monsters. Somehow he avoided every slash of fangs, every scalding splash of blood, every brimstone touch.\n\nIt was all futile. One of the newly arrived Giants died directly below Linden. She could not save him. She had forgotten the Sandgorgons; forgotten Covenant and Branl. She had nothing left except a kind of autonomic ferocity. She had fought her way beyond the precipice of her strength and power. Now she could only flail and fall.\n\nYet the surprise of more Giants appeared to affect the _skurj_. It altered the focus of their rampage; or they received new commands from the evil which had replaced Kastenessen as their master. Their dim minds\u2014or _moksha_ Raver's\u2014recognized that Linden and the Giants were trivial: puny opponents easily eaten later. A greater enemy awaited their fangs, an antagonist whose power might provide a richer feast. Wild magic might slaughter every one of the monsters; or it might exalt them, if they were able to feed on it.\n\nIn a staggered cadence, as if some _skurj_ were more reluctant than others, they turned toward the lower end of the valley.\n\nThere near the Sarangrave and the Defiles Course, Covenant fought for time. He needed a respite, just a few moments for his only answer. His last gamble. He had to be able to stand back and _concentrate_ \u2014And even then, he might be too late.\n\nBut he could not take the chance while Sandgorgons forced him to struggle for every moment of life.\n\nHe saw Giants now. They seemed to come from nowhere, as if they were an incarnation of the fog. Five, no, six of them, two women, the rest men. Not Swordmainnir. They looked like sailors armed with implements from their ship. Their movements were swift and accurate, but they lacked the fluid poise of warriors.\n\nStill they were apt foes for Sandgorgons, more agile than Covenant, twice Branl's size. One against one, their sheer muscle matched the savagery of the monsters. Their skin was not hide bred in the extremes of the Great Desert and the brutal gyre of Sandgorgons Doom. They could not slough off crushing blows and ruinous waters. But their instincts and reflexes were not hampered by single-mindedness. They fought with intelligence as well as strength; with skills which they had earned in storms.\n\nAnd they did not fight alone. The lurker's tentacle continued its battery, pounding at as many creatures as it could reach. At the same time, Branl seemed to float through the contest as if he served his blade; as if he were a weapon wielded by the eldritch flamberge. If a Giant halted a Sandgorgon with a blow or a cut, the _Haruchai_ arrived bearing death.\n\nAlthough the newcomers were only six, they fought like furies. Their weapons soon failed them. Knives broke on the hides of Sandgorgons. Knouts had no effect. Spears only pierced when they struck perfectly. Still the Giants were Giants, powerful of fist and arm. Few as they were, they thwarted the onrush of monsters.\n\nSomehow the Giants and the lurker and Branl cleared a space around Covenant.\n\nThat should have been enough for him. He had been offered his chance. He only needed to gather himself.\n\nBut his damaged chest sucked air in wracked gasps; and vertigo filled his head as if he stood on an appalling height, peering down into the valley from the fatal slopes of Mount Thunder; and he had never unlearned his fear of unrestrained wild magic. He could too easily imagine shattering the high cliff above the river.\n\nThen he was given more than a momentary reprieve. A kind of convulsion seemed to grip the Sandgorgons as if an invisible hand had taken hold of their minds. They paused; scanned the valley as if they sought more satisfying opponents. An instant later, they wheeled away.\n\nSome of them delivered a last flurry of blows, but soon all of them were pounding back up the valley. Massed and eager, they formed a bleached river pouring irresistibly uphill. At the same time, the _skurj_ began to squirm downward, horrific numbers of the serpent-monsters. As the Sandgorgons ascended, they parted only to let scores of _skurj_ pass among them.\n\nThe attackers had traded targets. The Sandgorgons raced to assail Linden and Jeremiah and the surviving Swordmainnir. A tsunami of _skurj_ plunged toward Covenant and Branl and their unforeseen allies.\n\nCovenant's vision was too badly blurred: he could not tell how many Giants still stood with Linden. He recognized her only by the faltering fever of her Staff, her stained fire.\n\nDamnation. He did not know how she would fare against the Sandgorgons. Even aided by Branl and Giants, he would not be able to withstand the onslaught of _skurj_. Even if he ripped open the mountain\u2014\n\n\" _More!_ \" he cried at the Feroce. \"We need _more_!\"\n\nThe lurker's minions had withdrawn, flinching, toward the Flat. They may not have heard him. They or their High God may have chosen not to hear him.\n\nCursing himself for every lost instant, Covenant dismissed his wild magic longsword. Now or never. What good was leprosy if he could not trust its implications? If it did not enable him to bear what he required of himself?\n\nIn one quick motion, he pulled the blade of the _krill_ across his left palm, drew blood sluggish with dehydration. He had no staff, no instrument of Law. Like Berek Halfhand before him, he needed blood and desperation to accomplish what even wild magic could not. Clenching his cut hand, he slapped red drops against the dagger's gem. Then he flung his gaze upward, past Linden and the onset of Sandgorgons, past the outpouring of the Defiles Course, past the towering cliff to the highest slopes of the mountain. In his mind, he shouted the Seven Words: a prayer that had no voice.\n\nA prayer that meant, _Please_.\n\nAlmost immediately, he was answered.\n\nPower without shape or sound exploded in him, through him, around him. A detonation both silent and invisible shocked the valley from end to end. Theurgy as old as the world seemed to ripple across the fabric of reality. It jolted the Sandgorgons in their strides; bewildered the _skurj_ so badly that some of them turned on each other. Sights that should have been clear blurred and merged. The slopes on both sides of the river trembled.\n\nAfter the concussion came stillness: a quiet so profound that it appeared to stop time. Existence held its breath. The Sandgorgons began strides which they did not complete. _Skurj_ paused with their lurid jaws wide. Fangs or brains forgot themselves. Giants tried to flick glances at each other, or at Covenant, and found that they could not move. Only Branl\u2014\n\nLowering his blade, the Humbled bowed to Covenant as if he understood. As if he approved.\n\nA moment later, the entire sky became thunderheads, black as ur-viles, impenetrable as gutrock. The heavens poised themselves for a blast which would rattle Gravin Threndor to its roots.\n\nAs if on command, the lurker struck. From the Defiles Course, a tentacle lashed at the baffled _skurj_. It wrapped itself around one of the monsters.\n\nShrieking in pain, Horrim Carabal lifted the creature.\n\nThe tentacle caught fire: it burned like aged wood. Rabid flames streaked the air. The lurker's agony must have been extreme: worse than _turiya_ Raver's violation; worse than self-mutilation. Yet the sovereign of the Sarangrave did not let go. Instead it flung the _skurj_ eastward over the wetland.\n\nThat creature did not return.\n\nNor did the lurker. Its arm collapsed into the river, smothered flames in water and corruption. Sounds like the sobbing of marshes roiled through the fog. No other tentacles appeared.\n\nThrough Horrim Carabal's wailing came a deep concussion as unanswerable as a tectonic shift. Mount Thunder itself seemed to howl as gouts of sizzling rock swept downward. Storms boiled lower until they shrouded Gravin Threndor's high crown.\n\nAnd from the depths of the Flat, waters rose against the current of the Defiles Course as if they had been summoned by the mountain. Dark thrashing swelled between the riverbanks.\n\nCovenant hardly noticed the river. Dimly in the distance, he thought that he saw yellow fires break through the clouds. He thought that he saw discrete flames surge lower like the onset of an avalanche. They roared as if the very air had become conflagration.\n\n\"You are answered, ur-Lord,\" Branl announced distinctly. \"A worthy effort in all sooth. How the forces which you have unleashed may combat _skurj_ , who are themselves a form of fire, I cannot conceive. Nonetheless the summons is both valiant and unforeseen. I am proud that I am Humbled in your name.\"\n\nAt last, Covenant began to see the fires more clearly. They looked impossibly far away: too far away to reach the valley before the Sandgorgons and the _skurj_ remembered their savagery. But now he was sure that those flames were Fire-Lions. They embodied Earthpower and Mount Thunder's enduring spirit. They could be as swift as the theurgy which had called them forth.\n\nThe Sandgorgons rallied more quickly than the _skurj_. But the monsters of the Great Desert did not resume their charge toward Linden and her few companions. Their strange senses marked the rush of a new threat. And some deep part of them\u2014an instinct too atavistic to heed _samadhi_ Sheol\u2014responded with eagerness. They had been bred in scorching heat and flaying winds, and had been trapped for millennia within the scouring energies of Sandgorgons Doom. Their urge to prove themselves against any and every foe outweighed _samadhi_ 's urgings. It outweighed self-preservation.\n\nTogether they turned away from Linden, strode deliberately down into the bottom of the valley. There they stood like a wall, awaiting the landslide fury of the Fire-Lions.\n\nThey had already demonstrated that they had no cause to fear the rising waters.\n\nBeasts of flame became torrents on the mountainsides. They spread like wildfires toward the sheer drop above the river.\n\nMuttering mute curses like supplications, Covenant watched the cliff and the Sandgorgons. If _samadhi_ and _moksha_ did not regain control of those creatures\u2014if the uncertainty of the _skurj_ lasted just a little longer\u2014\n\nBehind Covenant, the Feroce gibbered for his attention. \"Pure One, hear us.\" Their pleading was a damp clamor, scarcely audible through the tumult of Fire-Lions, the scald and crash of ancient magicks. \"Our High God's flesh cannot endure the worms of fire. He must not hazard them. Yet the alliance has been sealed. Even in his anguish, our High God upholds it.\n\n\"You must seek higher ground. We have done what we have done. The Feroce can do no more.\"\n\nWhile Covenant stared, stricken witless, Branl called, \"Ur-Lord!\" He sounded uncharacteristically urgent. \"Heed the Feroce! The waters rise!\"\n\n\"Well said, _Haruchai_ ,\" muttered a Giant as he snatched Covenant into his arms. He had a seamed face, and skin toughened by wind and sun, yet he looked as slender as a sapling, or as incomplete, like a man whose body was decades younger than his visage. Nevertheless his muscles were hawsers. \"This fog masks a mounting flood. A tide gathers from the east. Even Giants cannot swim such waters.\"\n\nThe _skurj_ turned away from the cliff, away from the Sandgorgons. Those monsters which had bitten into other _skurj_ , seeking blood and sustenance, ceased their feeding. Rearing like serpents, they brandished their fangs at Covenant; at Branl and six unknown Giants.\n\nTogether the Giants scrambled out from under a breaking wave of reified lava. Covenant dangled, helpless in his rescuer's arms, trying to understand events which had become as sudden as vertigo. At the rear of the group, Branl fought alone, swinging Longwrath's flamberge in a blur of cuts. But he retreated as he slashed, moving quickly without giving the monsters his back. The thunder of the Fire-Lions sounded like ruin, the gutrock rumble of an earthquake powerful enough to tear Landsdrop apart. The tumult of water rising from the Sarangrave resembled the onrush of another tsunami.\n\nAt the full stretch of their long limbs, the Giants raced for the southern rim of the valley. A long stone's throw away, more Giants bore Linden and Jeremiah upward. Swinging a longsword, Stave accompanied them. Branl cut twice more at the nearest creatures, then turned to follow the Giants.\n\nWhen the Fire-Lions met the wall of Sandgorgons, and Horrim Carabal's flood found the _skurj_ , the result was cataclysm. It shook the foundations of the Lower Land for leagues in every direction. Struck by acrid eruptions of steam and fury, the thunderheads became a bludgeoning deluge that seemed to erase the valley from existence. Rain fell like the ultimate darkness.\n\nThen the Giants raised a huzzah, ragged and grateful. The monsters were dying, all of them. Dimly Covenant realized that most of his companions had survived. He had seen Linden's fire before the end. Lord Foul would not have permitted harm to Jeremiah.\n\nCarried by a Giant whom he had never met before, Thomas Covenant felt no relief. He had exhausted himself. Now he was too stunned to feel anything.\n\n## 4.\n\nReluctances\n\nThe downpour lasted until the Fire-Lions were done with the Sandgorgons; until all of the _skurj_ were dead, and the lurker's flood dwindled to the east; until _samadhi_ Sheol's sentience had faded entirely from existence. Then the thunderheads drifted apart as if they had forgotten their purpose. The chill of rain and darkness dismissed the fog. Glittering as if they trembled at what they beheld, stars pricked the night sky with loveliness.\n\nLinden did not see the Fire-Lions depart. For all she knew, they, too, had perished. But she did not think so. Gravin Threndor's ancient fire and glory were inherent to the world, as natural as the Worm. She doubted that they could be unmade.\n\nShe rested under the shelter of an ironwood high up on the side of the valley, as far as possible from the craters and carnage of the battle, the plague-spots like stigmata in the ground, the clinging reek of gangrene. Leaning against the hard trunk with the Staff of Law in her lap, she waited for some semblance of strength to return.\n\nShe was too tired to be afraid. Too drained even to stay on her feet after Hurl had delivered her here. Too depleted to regard Jeremiah, or Covenant, or the Giants. Instead she floated into the lucidity of exhaustion: that numb mind-set in which unbidden thoughts followed their own logic to conclusions that might not have made sense at any other time.\n\n_In your present state, Chosen\u2014_\n\nShe was done with fighting. That much had become clear to her.\n\n_\u2014Desecration lies ahead of you._\n\nGod, she had endured so much violence\u2014From her struggles against Roger Covenant and the _croyel_ to the horrors and killing beside the Defiles Course, she had fought and fought. With wild magic, she had shed the lives of scores or hundreds of misled Cavewights.\n\n_You have become the daughter of my heart_. It was enough. She was done. Ever since Jeremiah's escape from his graves, the foundations of her life had been shifting. They needed to shift further.\n\nShe did not mean that she had given up. Carried along by the syllogisms of prostration, she arrived at convictions which did not imply surrender. She had seen her husband find his way through an appalling conundrum of _skurj_ and Sandgorgons. She had seen Giants appear out of nowhere to hazard their lives; seen the lurker of the Sarangrave set aside its old malevolence and choose to endure terrible pain. Rime Coldspray and four of her Swordmainnir had given battle while three loved comrades were slain. Stave and Branl had fought as though they wielded the prowess of every living _Haruchai_. The fact that Linden and Covenant and Jeremiah were still alive meant many things. It did not entail or require surrender.\n\nBut she could not keep meeting peril with violence, striving to out-do the savagery of Lord Foul's servants and allies. She could not. She needed a different purpose, a better role in the Land's fate. She had passed through the wrath of Gallows Howe to the gibbet's deeper truths; to the vast bereavement which had inspired Garroting Deep's thirst for blood. The time had come to heed the lessons which her whole life had tried to teach her.\n\nIf she did not give up, and did not fight, what remained? She thought that she knew, although she trembled to contemplate it; or she would have trembled had she been less weary.\n\n_There is hope in contradiction._\n\nMaybe that was true. If she did not know how to forgive herself, she could begin by offering other forms of grace to people or beings who needed it more.\n\n_The daughter of my heart?_ she thought. Give me a chance. Let me show you what your daughter has in mind.\n\nShe was still the Chosen. She could make decisions and go in directions which the Despiser might not expect.\n\nAfter that, her helpless clarity looped back to its starting point. She was done with fighting; with violence and killing. One idea at a time, she followed the same logic to the same conclusions. Exhaustion was like that, she knew. Under the right circumstances, it shed a certain amount of light; but its own conditions prevented it from casting its illumination further.\n\nLater Hurl came to her with a satchel of dried fruit and cured mutton. He also offered her a flask of _diamondraught_ diluted with fresh water: enough of the Giantish liquor, he said, to restore her, but not so much that it would impose sleep. And when she had eaten a little and drunk more, she found that she felt strong enough to focus her eyes and look around.\n\nThe survivors were lit like reincarnations of themselves by the silver of the _krill_ in Branl's grasp. Jeremiah's distress called out to her. He sat huddled against the trunk of a tree nearby, but he did not look at her\u2014or at anything outside himself. With his arms wrapped around his knees and his face hidden against his thighs, he rocked back and forth like a child in too much pain. Stave and Cirrus Kindwind stood with him. The Giant murmured reassurances that Linden could not hear. Stave's stance suggested that he was keeping watch.\n\nHurl had joined most of the other newcomers a short distance away. From somewhere\u2014presumably among the fringes of the Sarangrave\u2014they had retrieved sacks bulging with supplies: food and more _diamondraught_ ; other things which they considered necessary, and which they must have carried for many leagues. As Stonemage, Grueburn, and Bluntfist gathered with them, the canvas-clad men and women handed out viands and refreshment.\n\nThe surviving Swordmainnir and several of the other Giants bore oozing scalds. Contact with the blood and entrails of the _skurj_ had burned them. But they were Giants, able to endure fulminating hurts. One and all, they were grieving over their fallen comrades. Yet that hurt, also, they were able to endure, at least for a while.\n\nDown the slope from Linden, Covenant stood with Branl, Rime Coldspray, and another Giant, an implausibly thin man who appeared to speak for the sailors. Like Stave, Branl was unscathed. The hunch of Covenant's shoulders told Linden that he had fallen hard, damaged his chest. Her nerves detected cracked ribs and some dislodged cartilage, but no broken bones. Nevertheless his manner resembled the ravaged hillsides.\n\n\"I swear to you,\" he was saying, \"I thought it made sense. This is what happens when I convince myself I know what I'm doing. Even after Lord Foul touched Jeremiah, I thought we could sneak in here. I'm still not sure we could get in any other way. But this was a disaster.\n\n\"Hellfire, Coldspray! I just about got us all killed.\" To the other Giant, he added, \"If you hadn't showed up\u2014\"\n\nOr if, Linden amended on his behalf, he had not feared his own power; if he had unleashed enough wild magic to cleanse the whole valley. If he had indeed been _done with restraint_. Yet she believed that he had done well to hold back. He had little health-sense, and wild magic tended always to resist control. He might have inadvertently killed his companions.\n\n\"Enough, Timewarden,\" the Ironhand replied, peremptory with fatigue and loss. \"It is bootless to fault yourself for an onslaught which you could not have foreseen. Our peril here was both extreme and bitter. Yet it has not exceeded the hazards of the more direct road. And here we have found aid as unforeseen as our foes.\"\n\nLinden nodded privately. Soon she would have to go to Covenant, if he did not approach her first. She needed his embrace to console her. And she wanted to explain herself as well as she could. She was done keeping secrets, especially from him.\n\nBut her son took precedence. She could only imagine what Lord Foul's visions and his own helplessness had done to him.\n\nShe allowed herself a bit more food, a few more swallows of _diamondraught_ -tinged water. Then she began the immense labor of rising to her feet.\n\nAt once, Stave came to help her. His hand on her arm lifted her, steadied her. His single eye studied her as if she were no longer closed to him. In silence, he supported her toward Jeremiah.\n\nAs Linden approached, Cirrus Kindwind moved away. Clearly she needed the solace of her own people.\n\nEvery step sharpened Linden's perception of her son's despair. Her nerves assured her that his mind was still present. Although he rocked back and forth like an abused child, he had not retreated to his graves. Nevertheless he looked lost in misery.\n\nFor a moment, she paused to think. But she was too tired and sure to reconsider anything. Lowering herself down the Staff of Law, she knelt facing Jeremiah. Then she set the Staff on the soaked ground between them.\n\n\"Jeremiah, honey. Can you hear me? Are you listening?\"\n\nHugging his face against his thighs, he rocked harder.\n\n\"Jeremiah, listen.\" Her voice was a sigh. \"I know it's hard.\" How many times had Thomas said that to her? \"But we're still alive.\" Others were not. \"This isn't the end. We can finish what we started.\"\n\nMuffled by his legs, Jeremiah whispered, \"You can. I can't.\"\n\nLinden searched herself for strength. \"What do you mean?\"\n\nSlowly his head came up as if he were summoning indignation; as if her question insulted him. Memories of Sandgorgons and _skurj_ capered like ghouls in his haunted gaze.\n\n\"Because I can't _do_ anything, that's why.\" He made a visible effort to sound angry, but his voice held only anguish. \"I wasn't even in danger. Foul wants me alive. But there were all those monsters, and I couldn't help you. I couldn't do anything except watch. And even when I did that, I could still see the Worm. Even when Latebirth and Galesend were dying, and it was horrible, and there was blood everywhere, and those fangs. I could still see the Worm. Every minute, it does more damage than all the _skurj_ in the world, and there's nothing I can do.\"\n\n_As guerdon for his puerile valor\u2014_\n\nAching for him, Linden summoned her courage. \"I know. It must have been terrible for you. That's why I want you to take my Staff.\"\n\nShe expected surprise, but he only looked away. \"Why? It won't make any difference. I can't use it. I don't know how. It isn't mine. You'll just have to take it back. You won't have any choice.\"\n\nShe was tempted to reach out and shake him; but she refrained. He was too full of dismay to appreciate what she was offering him. As calmly as she could, she admitted, \"We might have to take turns at first. The Giants and Thomas are hurt. They need me. But you can still get started. And I don't always have to hold it. I can use some of its power without touching it. That doesn't change anything. I still want you to have it. I want it to be yours.\"\n\n\"Why?\" Jeremiah repeated like a groan.\n\n\"Because you need to be able to defend yourself,\" he needed to believe in himself, \"and I don't need it anymore. I have white gold\u2014and I can't use both. No one can. Earthpower and wild magic together are too much. So now I want to learn how to handle my ring. I want you to learn how to use the Staff.\"\n\n\"I can't,\" he said again. \"I don't have any idea\u2014\"\n\n\"Jeremiah.\" She made his name sound like a reprimand. \"We talked about this. Of course you don't know how. But you can learn. You don't even need my help. You have your health-sense and your own power. You can teach yourself.\n\n\"And if you have something else to concentrate on, you might be able to stop seeing the Worm. Earthpower and Law can do all kinds of healing. Maybe they can cure those visions. Maybe they can even keep the Despiser from taking you again.\"\n\nTaking the risk, she finished, \"And maybe you can find a way to make the Staff clean again. I know that I can't. That blackness is too much a part of me.\"\n\nJeremiah stared at her. The bleak torment in his gaze became a muddy roil. Its ambiguous currents twisted in unfamiliar directions, disguising their own depths. For a moment, she feared that he would pull away completely; that she had asked too much of him. That he would choose despair and dissociation.\n\nBut then he reached for her Staff.\n\n\"I'll try. I can't stand the way I am.\"\n\nBlinking at an unexpected sting of tears, she said unsteadily, \"Just remember what I told you. Start with your own Earthpower. Use it to touch what the Staff can do. You should be able to feel it. Then you'll be able to do more. It won't be easy at first. But you'll get better.\"\n\nHe ignored her now. Already distracted, he stroked the written wood, familiarizing himself with its texture, exploring its arcane script. Briefly he considered its iron heels as though they held the secrets he needed. Then he surged to his feet, holding the Staff of Law as if he wanted to swing it around his head.\n\nAh, God. Feeling strangely naked, bereft, like a woman who had just said farewell to her son's childhood, Linden climbed upright. She was grateful for Stave's firm grip on her arm, reliable as a corner-stone; but she had no words to offer her friend. Before she could do or say anything else, she needed to stop weeping.\n\n\"I do not scry, Chosen,\" the former Master remarked without any discernible emotion. \"To my sight, the future holds only darkness. Yet I judge that you have acted wisely. The boy's need is great, and you have other strengths.\"\n\nFortunately Stave did not appear to expect a response. Without a sign from her, he guided her toward Covenant.\n\nAs she drew near, Covenant turned away from Rime Coldspray and the lean Giant. His gaze was feverish with pain, and the lines of his face had been cut deeper: he seemed to have aged years in the past few hours. Even without his memories of the Arch, he bore the burden of too much time. His damaged chest was the least of his hurts. At the core, he was defined by his rage for lepers; for the innocent victims of Despite. He hated the necessary fact that other people suffered so that he might oppose Lord Foul.\n\nWincing whenever his ribs shifted, he held out his arms to his wife.\n\nFearing that she had just sacrificed her son\u2014the first step toward sacrificing herself\u2014Linden stepped into Covenant's embrace as if she were falling.\n\nIt was a mercy that he did not speak. Words were demands. For a few moments, at least, she simply needed to be held. And no one else's arms felt like his. Even Jeremiah's hugs could not comfort her now.\n\nBut as she leaned on Covenant, she felt his injuries more keenly, his bodily hurts and his aggrieved spirit. He held himself responsible for too much. And she had done nothing to ease or heal him.\n\nWith her health-sense, she reached out for the Earthpower of the Staff. As she had done once to relieve a suffering Waynhim, she invoked healing from a distance.\n\nAt first, she focused her heart on the distress in Covenant's chest. But when she had restored the integrity of his ribs and cartilage, she turned the balm of Law on the scalds and exhaustion of the Giants.\n\n\"Thanks, love,\" Covenant murmured when she was done. \"That helps.\" His arms tightened around her.\n\nRime Coldspray and several of the other Giants stood straighter. In spite of their sadness, they smiled.\n\n\"Thomas.\" Linden held Covenant closer. She wished that she could talk to him privately. The things which she had to say were difficult enough: she did not want anyone else to hear them. But she had learned to distrust that impulse. \"I need to tell you something.\"\n\nWhile I still can.\n\nHe released a pent up breath. \"So tell me.\"\n\n\"I love you.\" There was no good way to say it. Words were inadequate. \"I want to help you. I want you to stop Lord Foul. I want the Land to be saved, and the Earth, and the stars, and the _Elohim_ ,\" although she could not imagine how any of those deeds might be accomplished. \"I want Jeremiah safe, and all of our friends, and everything that we've ever cared about.\n\n\"But I'm done fighting.\"\n\nCovenant stiffened as if she had frightened him. His voice was harsh with strain as he asked, \"And you think you have a choice?\"\n\nHe did not let her go.\n\nShe nodded against the thin fabric of his T-shirt.\n\n\"So tell me,\" he repeated through his teeth.\n\nTo make room for what she had to say, she eased away until she could touch his chest. Kissing the tips of her fingers, she slipped them through the old knife cut in his shirt. \"You said it yourself. We have to face the things that scare us the most. There's really no other way. Escape isn't worth what it costs.\n\n\"But the Despiser isn't what scares me the most. Even losing Jeremiah isn't. Or losing you. That might break me, but it isn't my worst fear. And the Worm\u2014\n\n\"Thomas, I've hardly seen the Land the way it was when you fell in love with it. That first time, when we came here together, it was all the Sunbane. And since then, we've lost too much, and I've been going crazy about Jeremiah.\n\n\"Oh, Andelain has changed my life. More than once.\" Glimmermere and _aliantha_ and percipience and the Ranyhyn had all changed her. \"But I simply haven't learned how to care about this world as much as you do. The Worm isn't my worst fear.\"\n\nBefore he could prompt her, she said, \"My worst fear is what I might become. Or what I've already become. I need to face that somehow.\"\n\n\"Then how\u2014?\" Covenant began. But he stopped himself. For a moment, he seemed to scramble like a man who felt the ground shifting under his feet. Then his head jerked up as if his chest had been pierced again; as if she had stabbed him. She felt the jolt of his intuitive leap. \"Oh. _That_ fear. Now I get it.\"\n\nLinden nodded again. Trying to be clear, she said, \"Days ago, you left me because you had to deal with Joan. If we live long enough, I'll have to leave you.\"\n\nAnd her son.\n\nGripping her shoulders, he stared like wild magic into her face. \"That's why you gave Jeremiah your Staff.\"\n\n\"One of the reasons,\" she conceded. Now that he understood, she found it comparatively easy to bear his gaze. \"Earthpower and Law can't help me. I have to use my ring.\"\n\nAt once, he pulled her close again, hugged her as though his heart refused to go on beating without her. \"Hellfire, Linden,\" he breathed. \"That's insane. It might be exactly what we need.\"\n\nShe matched his embrace. \"And I'm the only one who can even try. You said that, too. You have to face Lord Foul. And Jeremiah has to decide for himself. That leaves me.\"\n\n\"I remember,\" he said gruffly. \"I must have been out of my mind.\"\n\nThen he held her at arm's length again so that he could study the doubts and determinations following each other like ocean swells in her eyes.\n\n\"Well, why not?\" he growled. \"I didn't ask you and the First and Pitchwife to do my fighting for me when I decided to give up using power all those millennia ago, but you kept me alive anyway. Maybe I even expected you to do that. Why shouldn't it be your turn now? Sure, we have more enemies this time. But we also have more friends. And I think we're capable of things damn Foul has never seen before. Why shouldn't you get a chance to take your own risks?\"\n\nLinden smiled through a brief relapse of tears. \"I knew that you would understand.\" Then she added, \"But I haven't told Jeremiah. We aren't there yet. We might not live long enough to get there. And he has other things on his mind. I don't want to scare him until I'm sure that he needs to know.\"\n\nCovenant nodded; but abruptly he was distracted. \"I get it.\" He was no longer looking at her. \"But suddenly things aren't as simple as they were a minute ago.\"\n\nWhen she followed his gaze, her heart seemed to stop.\n\nHolding the Staff, Jeremiah had summoned his heritage of Earthpower. Small flames spread from his hands onto the shaft. They traced the cryptic lines of the runes, blossomed briefly on the iron heels, measured the wood.\n\nThey were his\u2014and they were stark black, as dark as ichor squeezed from the marrow of the world's bones.\n\n\"Jeremiah!\" When Linden's pulse resumed its labor, it pounded in her temples, in her ears, at the base of her throat. \"What are you doing?\" She had asked him to change the Staff. Instead her own darkness was changing him.\n\nHe did not glance at her. \"Don't bother me.\" His eyes echoed the hue of his flames. \"I'm trying to concentrate. This is temporary. I mean, I think it's temporary. I just don't know what to do about it yet.\"\n\nScowling to himself, he muttered, \"You're stronger than I thought. I can't figure out how you did it.\"\n\nLinden meant to intervene. She thronged with objections, warnings, supplications. But Covenant stopped her. With his hand on her cheek, he urged her to face him.\n\n\"Leave him alone for a while,\" he advised softly. \"He wants to try. Maybe this is how he has to learn. Maybe he has to go through you to get to himself.\"\n\nCovenant may have meant, Maybe he's starting to face his worst fear.\n\nLinden wanted to believe him, but she could not. Her father had kept her locked in the attic with him while he killed himself. Her mother had begged her to end her life. Linden had given her Staff to Jeremiah of her own free will; but she did not know how to distance herself from his peril.\n\nYet what else could she do? She had already decided to leave him when last came to last. When there was nothing left for her except the dark.\n\nInstead of stopping her son, she clung to her husband as if he were the only defense she had.\n\nut slowly food, diluted _diamondraught_ , and the aftereffects of Earthpower steadied Linden. By degrees, she regained a semblance of calm.\n\nThe same benefits wrought on her companions, the Giants if not the _Haruchai_ , until the frenzy and desperation of battle began to fade. And as the Giants recovered, their need for tales grew.\n\nClearly the newcomers and the Swordmainnir were well known to each other. But much had occurred since they had parted: both groups had much to tell, and to hear. In particular, the sailors wanted to understand the confluence of events which had brought about the crisis of the Defiles Course. Because they were Giants, they knew about Covenant and Linden; but everything that pertained to Jeremiah was a mystery to them. And the Swordmainnir were eager to hear how their people had contrived to arrive when they were most needed.\n\nWhen everyone had eaten, the sailors bundled up their supplies, leaving out a little food in case Linden or Covenant or Jeremiah wanted more. Then the Ironhand announced that the time had come, and her people gathered around her, aching and ready.\n\nLinden stood among them with Covenant behind her, his arms around her. Branl joined Coldspray so that the _krill_ would shed as much light as possible for the Giants. But Jeremiah seemed to have no interest in stories or woe. His immersion in his task was complete, as it always was when he worked on his constructs. His eyes watched flames while his hands made them dance and gambol on the Staff, or gave them shapes that suggested Ranyhyn, flickering portals, evanescent _Elohim_. Gradually he attuned himself. Nevertheless his every expression of magic remained as benighted as the world's doom.\n\nPerhaps to reassure Linden, Stave positioned himself near her son; but he did nothing to distract Jeremiah.\n\nThe Ironhand began by introducing the newly arrived Giants, seven men and four women. Hurl Linden had already met. Their leader was the Anchormaster of Dire's Vessel, the Giantship which had brought the Swordmainnir and Longwrath to the Land. His name was Bluff Stoutgirth, although he was lean to the point of emaciation; and his mien hinted that he was more inclined to hilarity than to command. Here, however, his manner was grave and grieving. His sailors and Rime Coldspray's Swordmainnir had endured much together during their voyage to the Land. They felt their losses keenly.\n\nFor Linden and Covenant, and for the _Haruchai_ , Bluff Stoutgirth named his comrades\u2014Etch Furledsail, Squallish Blustergale, Keenreef, Wiver Setrock, others\u2014but Linden doubted that she would remember them all. Still she was grateful for the knowledge that they had come from Dire's Vessel. That detail made the fact, if not the timeliness, of their arrival comprehensible.\n\nThe Anchormaster offered to tell his tale first. It was, he suspected, both shorter and kinder than that of the Swordmainnir, though perhaps no less unforeseen. With Rime Coldspray's assent, he began.\n\nAfter the departure of the Ironhand's company, Dire's Vessel had remained in the anchorage of ancient _Coercri_ , The Grieve of the Unhomed. For a number of days, the sailors busied themselves with the mundane tasks of repairing and maintaining their Giantship. Then they began to notice changes in the littoral's weather, disturbances in the sea. Storms lashed the coast and disappeared again without apparent cause. Downpours drenched Dire's Vessel out of clear skies. Currents ran awry, heaving the Giantship from side to side until anchors were set at every point of the compass. Still the Swordmainnir did not return. They had vanished among the uncertainties of their quest.\n\nFive mornings ago, however, the sun astonished the crew\u2014Stoutgirth said this with improbable good cheer\u2014by failing to rise. Stars began to disappear from the firmament of the heavens. Mighty swells from the southeast threatened Dire's Vessel's moorings. Such occurrences augured some immense and dolorous ill, but the sailors could not interpret the signs.\n\nYet on the following day a new astonishment appeared. Striding forth from tales many centuries old, a man made himself manifest upon the foredeck of Dire's Vessel.\n\nThat he was a man of immense age was plain. The lines upon his visage were such that they mapped a world. Indeed, his years had been so prolonged that they appeared to erode his substance where he stood. His raiment was ancient, an unkempt robe of indeterminate hue, and his limbs wore hatchments of scars. Yet he bore himself as one who could not be bowed, and his glances had the effect of lightnings.\n\nUnmistakably, Bluff Stoutgirth announced, the man was one of the _Haruchai_. Indeed, he was unmistakably Brinn, the companion of the ur-Lord Thomas Covenant and the Sun-Sage Linden Avery aboard Starfare's Gem: the _Haruchai_ who had become the Guardian of the One Tree.\n\nThe Guardian's tidings were dire in all sooth, Stoutgirth confessed. \"The Worm of the World's End is roused, seeking the ruin of all things. Therefore the One Tree withers. The life of the Earth nears its close.\" Yet when the Giants bewailed their lot, moaning the loss of love and wind and stone, of seas and joy and children, Brinn answered their lament.\n\n\"Yet good may come from loss as it does from gain. The decline of the One Tree has concluded my devoir. I am freed to remember the promises of an earlier age. And the Worm is not instant in its feeding. Life lingers yet in the world's heartwood. This gift is granted to me, that I may expend my waning strength in the Land's service.\n\n\"While I endure, I will guide you, for your aid will be sorely needed.\"\n\nNone aboard Dire's Vessel, the Anchormaster continued, could comprehend that need. Yet their hearts were lifted by the thought that they might yet be of use in the Earth's last peril. In a foreshortened Giantclave, the Master of Dire's Vessel, Vigilall Scudweather, determined that she and a half portion of the crew would remain to tend the Giantship, praying that events would allow them to serve some worthy purpose in their turn. Bluff Stoutgirth and the others prepared such supplies and weapons as they were able to carry swiftly. Then they followed the Guardian of the One Tree from The Grieve into Seareach, tending always to the southwest toward the toils of Sarangrave Flat and the renowned perils of the lurker.\n\nFor a wonder, they passed into and through the Sarangrave unthreatened. Indeed, their course was eased at every obstacle, though they had no understanding of the magicks which relieved their efforts. In another matter, however, fortune gazed less kindly upon them. The Guardian's diminishment was unremitting, and no succor of companionship or repast eased it. During the evening of the day now past, he frayed at last and faded, drifting away along the world's winds. Then the Giants feared that naught remained to thwart the Worm. Yet they persevered, for the Guardian had led them far enough to descry Mount Thunder. They knew their destination. Therefore they hastened onward, denying themselves all sorrow for Brinn _Haruchai_ , until they beheld turmoil upon the mountain. And at the last, fortune smiled once more. The Giants of Dire's Vessel did not come too late.\n\nSo Bluff Stoutgirth ended his tale.\n\n\"Joy is in the ears that hear,\" Rime Coldspray replied formally, \"not in the mouth that speaks. Upon occasion, however, both ears and mouth may know joy, for its causes are plain to all. When we foundered in strife and loss, your coming lifted our hearts. We are Giants and must grieve. Yet we are filled with gladness also. You are a brightness amid the world's dusk.\"\n\nThe other Swordmainnir offered their thanks and pleasure as well. But they fell silent when Covenant began to speak. Holding Linden tightly, he addressed the sailors with a familiar ache in his voice.\n\n\"Brinn talked about a service or boon. Even after he saved my life, he wasn't done. But he didn't tell me what he had in mind. Now I know. You're his last service. His boon. We weren't enough. We needed help. No matter what happens, we're going to need more.\"\n\nLinden nodded. Manethrall Mahrtiir had spoken truly. _And betimes some wonder is wrought to redeem us_. The Giants of Dire's Vessel had given Covenant time to summon the Fire-Lions.\n\nBut the Swordmainnir did not linger over their gratitude. Their weariness ran deep; and there was much that Stoutgirth and his crew needed to know.\n\n\"As you have surmised,\" the Ironhand began, sighing, \"our tale is both lengthy and unforeseen. It has cost us lives and blood and sorrow. The worth of our deeds is not ours to proclaim. Yet I will trust that worth resembles joy. It will be found in the ears that hear if the mouth that speaks cannot name it.\"\n\nThen Rime Coldspray gave the Giants of Dire's Vessel her story.\n\nAt first, Linden listened uncomfortably. The Ironhand described events and purposes in more forgiving terms than Linden could have managed, especially where Linden herself was concerned. She had to stifle an impulse to add her own stringent counterpoint to the arching cadences of Coldspray's narration. But gradually the Ironhand's tone filled her thoughts, lulling her until she drifted on the currents of Coldspray's voice.\n\nBeyond the reach of the _krill_ 's gem, darkness waited as though the whole truth of the world had become night. Overhead the watching stars seemed too disconsolate to value their hard-earned reprieve. Behind the episodes of Rime Coldspray's tale, the Sarangrave's lapping waters muttered reminders of venom and putrescence. Jeremiah's study of the Staff sent small flames skirling upward, but shed no light.\n\nYet within the ambit of the _krill_ 's argent, Bluff Stoutgirth and his comrades were transfixed. Where the Anchormaster and Hurl appeared to suppress jests at every turn of the tale, Keenreef and Squallish Blustergale looked dismayed to the heart. Etch Furledsail, Wiver Setrock, and one of the women\u2014had Stoutgirth called her Baf Scatterwit?\u2014stared at the Ironhand as if nothing made sense. Together Dire's Vessel's crew evinced every reaction except joy.\n\nNevertheless no one interrupted Rime Coldspray. Even Covenant did not, although he could no doubt have added his own interpretations. Instead he seemed distracted, as if he were thinking about something else.\n\nThen Coldspray was done. A long silence greeted her, until Stoutgirth announced brightly, \"A toothsome tale, Ironhand\u2014a veritable feast of clear peril and ambiguous vindication, strange beings and extravagant exertions. Doubtless we will gnaw upon it, seeking its marrow, while the world endures.\n\n\"Yet you have spoken of worth. For my part, Ironhand, I do not acknowledge it.\" He laughed happily. \"As matters stand, we resemble sailors snared in the ensorcelments of the Soulbiter. There can be no worth in the tale of those who fail and fall unwitnessed, for their doom is not redeemed by the telling of it. We must have boasting, Rime Coldspray! I will not name the deeds of this company worthy until the World's End has been forestalled. Only then may the tale be shared with those able to esteem it.\"\n\nLinden frowned, thinking that the Anchormaster had insulted her friends. But the Giants heard something different in Stoutgirth's assertion; or they heard it with different ears. Several of his sailors laughed, and both Grueburn and Kindwind chuckled.\n\n\"Then,\" Rime Coldspray replied, bemused and rueful, \"we must endeavor to win free of this Soulbiter, that we may thereafter brag of our survival.\"\n\nThe Anchormaster nodded. \"And toward that end, Ironhand, there is a matter which you have not addressed. How do you propose to sail these fatal seas? You have overcome the unwelcome of the _Haruchai_. And your companions are figures of legend, revered among us. Your purpose must be mighty indeed, to gather such a congeries of valor and puissance.\n\n\"Ironhand, what is your intent?\"\n\nColdspray opened her mouth to answer, then closed it. With a bow, she stepped aside, referring the question to Covenant; or perhaps to Covenant and Linden.\n\nCovenant's arms tightened momentarily. In Linden's ear, he whispered, \"This is the hard part of being a leper. I'm going to need your help.\"\n\nStartled, she turned to him with questions in her eyes; but his only response was a twisted smile as he stepped away from her. The sailors and the Swordmainnir towered over him, yet he faced them as though his stature equaled theirs.\n\n\"I hope you aren't expecting me to be sure of anything. We have too many enemies, and they have too much power. And all I really know about the Worm is that we can't stop it. But I don't want to just sit on my hands waiting to die. The Despiser started all this. _Him_ I think we might be able to stop. I want to put an end to his evil.\"\n\nHe pointed at the mouth of the Defiles Course. \"I want to get into Mount Thunder. Up into the Wightwarrens, if that's even possible. That's where Lord Foul is. I want to go find him.\"\n\nBriefly his shoulders hunched as if he were strangling his fears. \"But there's something else I want to do first.\"\n\nWhile the Giants studied him, he gestured Branl to his side. Taking the _krill_ , he held it up in his halfhand by its wrapped blade. Within its silver, he continued.\n\n\"The Ironhand told her story. The Swordmainnir have been through hell and blood ever since they left you. Fighting Longwrath, fighting for Longwrath, they lost Scend Wavegift. Against the _skurj_ , they lost Moire Squareset. And eventually Kastenessen killed Longwrath. All of that was bad enough. But now the toll is even higher.\" Although Coldspray and Stoutgirth had already acknowledged their dead, Covenant insisted on the names. \"Latebirth, Stormpast Galesend, and Cabledarm died for us, and Dire's Vessel lost a man I never even met. You called him Slumberhead, God knows why. He sure as hell wasn't dozing when he gave his life.\n\n\"It's too much. You're Giants, all of you. You can't ask yourselves to carry around that much grief indefinitely. You need a _caamora_. How else are you going to face what's ahead of us?\"\n\nThe Ironhand glanced at her surroundings. \"We have no fire,\" she said harshly, \"if we do not sacrifice yet another tree.\"\n\nAll of the ironwoods set ablaze by the _skurj_ had burned down to ash, or had been extinguished by rain. There was no flame in the valley apart from Jeremiah's experiments.\n\n\"And I won't ask you to do that,\" Covenant assured her. \"I promised you a _caamora_. I intend to keep that promise.\n\n\"When I made it, I thought I could use Longwrath's body. That seemed like a kind of acknowledgment. A way to make something good out of what he went through. But the Giants we've lost here have been mangled by the _skurj_. They already look desecrated. It seems disrespectful to use them.\n\n\"So I'm going to burn myself.\"\n\nTo the sudden alarm of his companions, he added quickly, \"I mean with wild magic. I'm going to light myself and hope that I can burn hot enough to console you.\n\n\"It's wild magic. It drains me. Hell, it even terrifies me. But it won't hurt me. The only danger is that I'll lose control. Too much might do more harm than too little.\"\n\nThen he turned back to Linden. \"That's why I need your help. Your health-sense. I want you to watch out for me. If I start to go too far, I want you to stop me.\"\n\nSeeing the raw need in his scowl, she felt a hammer pound in her chest. How could she stop him? Oh, she believed that he would not be harmed physically. His power was _him_. But the cost to his spirit might be extreme. His reluctance was necessary to him. It counterbalanced his extravagance: it was his way of managing his fear that he might commit havoc. If he damaged his friends\u2014if he damaged _anything_ \u2014he would not be able to forgive himself.\n\nHow could she stop him, except by possessing him?\n\nBut he did not give her a chance to protest, or to ready herself. He ignored the apprehension of the Giants, the doubts. Before they could say that they did not want him to take this risk, he touched his wedding band to Loric's cut gem.\n\nIn the space between instants, he became fire.\n\nShe could still see him. He stood incandescent in the core of a silver conflagration, a blaze like a bonfire barely contained, bound by force of will in the shape of a whirling pillar as tall as any Giant. As he burned, the _krill_ fell from his fingers: he no longer needed it. Flames seemed to burst from every inch of him. They looked pure enough to render his flesh from his bones. Yet he was not consumed. Instead his magicks appeared to exalt him. With wild magic, he could have brought life and time to an end without the aid of the Worm.\n\nNevertheless his power was also a howl. It tormented him. It was the contradiction which lay at the center of his plight in the Land, _the one word of truth or treachery_. Without wild magic, nothing could be redeemed. With it, everything might be damned.\n\nIn spite of her dismay, Linden understood. With wild magic, destruction came easily. _That_ she knew to be true. She had seen it in _caesures_ ; in the reaving of Cavewights. With fire, Covenant looked capable of ripping the stars out of the heavens. She did not know how to watch without weeping.\n\nFor a moment while Covenant blazed, Rime Coldspray and the other Giants hesitated. They did not know him as Linden did, but they could see how his attempt to both exert and restrain himself wracked him. At the same time, however, they recognized what he was offering. Even if they had not heard about the gift which he had once given to the Dead of The Grieve, they would have yearned to seize this opportunity.\n\nHe had chosen to risk himself. How could they refuse him?\n\nAbruptly the Ironhand reached into the whirl of fire, caught Covenant in her huge hands, and lifted him high. There she held him while his flames attacked her flesh as if they threatened to char her bones, reaching for her heart.\n\nHer grasp threatened his concentration; but he did not withdraw his power.\n\nHer pain was severe, as she needed it to be. She required such anguish to cauterize her bereavements. Without the cleansing of fire, her sorrow would have become bitterness. Eventually she would have lost her ability to hear joy.\n\nWhile Coldspray gripped him, Covenant fought to keep his balance between too much and not enough. But when she passed him to Frostheart Grueburn, his self-control faltered. Wild magic mounted higher.\n\nLinden watched him with her own agony. Cries that she could not utter closed her throat. Stave had come to stand at her back. He clasped her shoulders to steady her. Jeremiah had dropped the Staff of Law. He gaped at Covenant with consternation in his silted gaze. But she was aware of nothing except silver fire and Thomas Covenant.\n\nHow much could he endure? Three Swordmainnir remained after Grueburn. Stoutgirth and his crew numbered eleven. They, too, were eager for the healing hurt of a _caamora_. How could Covenant possibly\u2014?\n\nHow could she stop him?\n\nCirrus Kindwind received him from Grueburn, supported him awkwardly with her good hand and the stump of her maimed forearm. She kept him too long, and not long enough. Sensitive to his ordeal, she did not allow herself to anneal her whole lament. When she released him to Onyx Stonemage, she looked incompletely assuaged.\n\nLinden could not stop him. She could not help him. Not without possessing him. By imposing her choices on his. By using her health-sense to enter him as she had once entered Jeremiah; as she had done to Covenant himself several times long ago.\n\n_Good cannot be accomplished\u2014_\n\nHoarse gasps of strain burst between his teeth as Stonemage gave him to Bluntfist.\n\n\"Mom!\" Jeremiah yelled. \" _Do_ something!\"\n\nNear Linden's ear, Stave said sharply, \"Attend, Chosen. Your ring answers.\"\n\nAs soon as he said the words, she felt fire spitting from her wedding band.\n\nShe, too, was a rightful white gold wielder.\n\n_\u2014find another truth\u2014_\n\nIn the small gap of inspiration between heartbeats, she recovered her voice.\n\n\"Put him down.\" From her ring, she drew flames like streamers and wrapped them around her. She spoke fire. \"Put him _down_!\"\n\nThe Swordmainnir knew her too well. They could not resist her. Baffled and uncertain, Halewhole Bluntfist lowered Covenant to the ground.\n\nAt once, Linden rushed to him. Her arms and her love and her shining she flung around him. Then she gave herself to him\u2014or she made him hers. With percipience, she united their powers until she found a way to balance his extremity with her physician's caution.\n\nTogether they stood in conflagration while the Giants of Dire's Vessel crowded around them. Together Covenant and Linden burned as the sailors came two at a time to grip his shoulders or hers; to be flensed by pain and find release.\n\nA curtain of tears fell between Linden and her companions. For a moment, she was blind. She was almost deaf. But then the _caamora_ was done. When she felt the last of the Giants withdraw, she relaxed her fire, taking Covenant's with her. Her ring had answered his: now his answered hers. As if they had briefly become one, they let go of wild magic until they stood, unburned and unburning, in each other's arms.\n\nShe heard the Giants singing; but they seemed impossibly far away, and she did not listen to them. Instead she heeded only the need in her husband's embrace and the relieved beat of his heart.\n\n_There is also love in the world._\n\n## 5.\n\n\"No Prospect of Return\"\n\nAs if the _croyel_ still had the power to dredge up his buried past\u2014or as if Lord Foul had inherited that power\u2014Jeremiah remembered his sisters. Two of them, both barely toddling on their stick-thin legs. There was never enough to eat. Their names were\u2014? Their names were gone. He could not imagine their faces, except as pale smudges lit by the Despiser's bonfire. They had existed in a different world, on the far side of a wall of absence. He was not sure now that they had ever meant anything to him, except as squalling mouths that needed food worse than he did. And yet he remembered that they had been his sisters.\n\nLinden and Covenant did not know that about him. It was his last secret: he remembered his sisters.\n\nA scornful voice told him that he should have done something to protect them.\n\nHe _should_ have, even though he had gone first, he had put his right hand in the fire as soon as his mother finished screaming, and after that he was in too much pain to feel anything else. Even after he had learned how to conceal himself so that those terrible flames could not touch him again, the idea that he should have done something twisted his heart.\n\nWhy was he thinking about this now? It did not make sense. _Protect_ his sisters? _How?_ He was only five. His mother was always praying or crying. Just about the only thing he knew for sure was that he had to be good. He had to do what she told him. He had to obey Lord Foul's eyes in the bonfire. That was what kids did. It was how they stayed alive.\n\nYet they were your sisters, were they not?\n\nI don't even remember their names.\n\nYet you knew their peril, did you not?\n\nI was just a kid. I didn't know anything.\n\nYet you heard your mother's pain, did you not? You understood that fire burns, did you not?\n\nI was only _five_ , Jeremiah tried to protest. I had to obey.\n\nDid you? At such a cost?\n\nI couldn't do anything else! Everything hurt too much!\n\nYet they needed you, did they not? Had you refused the flames, would they not have done likewise? Are you not therefore the cause of their sufferings?\n\nI was just a kid.\n\nYet you are no longer a child.\n\nStop.\n\nAnd are you not as blameworthy now as you were then? For deeds and self-pity which imperil those whom you profess to love, are you not blameworthy still? Did you not reveal their heading and purpose by defying possession? You knew that peril also, did you not?\n\nStop. Yes. Stop.\n\nHow then do you now refuse blame?\n\nJeremiah had no answer for that voice. The sanctuary which he had designed for the _Elohim_ was not an answer. It was no excuse for standing on grass as if he thought that he could outface Lord Foul. He should have protected his sisters. He could not have protected them. He should have done it anyway. He deserved to watch the Worm while Linden and Covenant failed to save the world because of him. He had told Lord Foul where they were.\n\nSo now he concentrated obsessively on the Staff of Law: as obsessively as he had worked on any construct. As soon as he recovered from the surprise of the _caamora_ , the jolt of alarm, he picked up the Staff and resumed his study. Covenant was not hurt. Linden was not. Jeremiah could see that. They did not need him; and he had other things to do.\n\nWhen he held the strange black wood, he felt its possibilities. In a sense, it too was a construct. It was made of parts that he could identify. The living wood. The iron heels full of old magic. The language of the runes. The blackness, Linden's blackness: the deep ebony which had taken over his own Earthpower when he had tried to change it. How those parts interacted was a mystery, but that did not trouble him. How the parts of his own constructs interacted was a mystery. He did not need to think about it. Instead he tried to understand how the parts fit together. He wanted to see the _design_.\n\nIf he could do that, he would know how to use the Staff. He would have power. He would be able to _do_ things. Things that might make a difference. Things that might excuse him.\n\nThings that might silence the scorn in his mind, block visions of the Worm. Then he would have a chance\u2014\n\nLinden had given him that gift. His mother: the one who loved him, not the one who had put her own hand in Lord Foul's bonfire. Studying the Staff, he believed that he would cheerfully kill anybody or anything that tried to hurt her.\n\nBut the design\u2014the secret of Linden's gift\u2014eluded him. No matter how hard he tried, he could not _see_ it. He was beginning to sense some of the Staff's uses. A few of them might even be possible for him. And while he concentrated on those possibilities, Lord Foul's visions lost some of their harrowing vividness, their inevitability, their weight of ridicule. Still the design itself, the key that would unlock the gift, was beyond him. He could not alter the blackness of the flames.\n\nIn his heart, he was still only five.\n\nEventually his efforts to find his way felt less like his familiar obsession with building. They became a kind of fever, a ragged desperation that went nowhere. When Cirrus Kindwind offered him food, he ate. He accepted water. Vaguely he noticed the Swordmainnir and the sailors talking together, adding details and explanations to their stories, discussing the hazards ahead. He heard them decide to give their dead to the river, hoping that the lurker would convey the bodies to the cleaner waters of the sea. He saw Linden and Covenant wander away together\u2014not far, but far enough so they could at least pretend that they were alone. Without thinking about it, he knew that Stave and Branl watched over the whole company. But his real attention remained fixed on the Staff.\n\nIt should have been everything he wanted. Calling upon the resources of Earthpower and Law should have been as natural as reaching out his hand.\n\nIt was not. His ability to raise and shape flames like midnight blossoms mocked him with all that it was not. His fire did not extend his percipience or ease his fatigue. It was too insubstantial for healing. It had no force. And it was always black.\n\nThe laughter in his head derided him. Involuntary glimpses of the Worm made fun of him.\n\nAre you not therefore the cause of their sufferings? How then do you now refuse blame?\n\nThe Staff of Law required a Linden Avery\u2014or a Thomas Covenant\u2014and Jeremiah was just a kid.\n\nFinally he dropped it as if he were merely worn out. With both hands, he tried to scrub the bitterness off his face. Hiding behind a scowl, he gnawed on a dry sausage for a while, drank more water. Then he looked for a patch of level ground where he could stretch out.\n\nAlmost immediately, Linden called, \"Jeremiah, honey. Are you all right?\"\n\nHe wanted to retort, Leave me alone! I don't need you worrying about me. But of course if he said that everybody would know how he felt.\n\nInstead he muttered, \"Just tired, Mom. I need sleep.\"\n\n\"Rest as much as you can.\" Covenant sounded distant. He was thinking about something else. Probably about Linden. \"We're running out of time. I want to start before midnight.\"\n\nFine, Jeremiah thought. _You_ start. I'm going to lie down until somebody takes pity on me.\n\nBut he did not mean that. He meant, I'm lost. I need help. But you can't help me. You've already done everything. The rest is up to me, and I'm not enough.\n\ne expected to lie awake, chewing his misery while voices laughed and the Worm ravaged. But he was more tired than he realized. He surprised himself by falling out of the world.\n\nIn dreams, he watched the stars spin. At first, they wheeled slowly, as cautious and deliberate as if they were performing an unfamiliar dance. Later they moved faster. And as they swirled, they drew closer to each other, contracting their glitter, leaving the rest of the heavens drowned in blackness, as doomed as the Lost Deep. After a while, they began to collide and join. Yet the merging of one distinct gleam with others, and then still others, did not make their shining brighter. Instead their private lives seemed to extinguish each other. Soon hundreds or thousands of them had become one, and that one was scarcely visible: a dying ember in the fathomless ruin of the night.\n\nBut at the same time, that single dulled spark became heavier. Not bigger, no. Just more massive. And it leaned down on Jeremiah, pressed its intolerable weight against his heart. He did not breathe. There was no room in his chest for air. His heart no longer beat. It could not lift blood through his veins under so much pressure. He was becoming the sky, black and blank, infinitely desolate.\n\nHe awoke with an enfilade trapped between his ribs. Memories of bullets whined past him and into him, furious as hornets. Wildly he floundered to his feet, frantic for relief.\n\nHe nearly yelped when Stave grasped his arm.\n\n\"Still your alarm, Chosen-son,\" said the _Haruchai_ , almost whispering. \"There is no imminent peril. Dreams are not omens. They bespeak only your fears.\" Then he added, \"The Chosen slumbers yet, as do the Giants. Only the Ironhand and the Anchormaster stand watch with Branl. We do well to permit their rest.\"\n\nJeremiah resisted an impulse to cling to the former Master. There was no light: Branl must have covered the _krill_. Stave's grip felt like the only certainty in a reality which had lost its moorings. The boy half expected to see the stars continue their shrinking spiral, their fatal deflagration. But of course they remained where they were, clinging to their fate.\n\nThe air was thick with the complex reeks of the Defiles Course and Sarangrave Flat, of ironwood ash and drowned _skurj_ and the charred corpses of Sandgorgons. Around Jeremiah, darkness clotted like blood. It filled every span of ground and hidden niche. When he considered the movement of time, he found that midnight was near.\n\nAs quietly as Stave, he asked, \"Where's Covenant?\"\n\nThe _Haruchai_ pointed down the valley. \"There. He communes once again with the Feroce.\"\n\nJeremiah looked toward the marshy verge of the Sarangrave. At this distance, he could not descry Covenant. There was too much sensory clutter from the restless currents and predators. The lurker still complained over its pains, whimpering wetly. But near the wetland, Jeremiah spotted glints of emerald arrayed as if they had gathered to attend a potentate. Green flames fell and rose like sighs.\n\nBehind them, the Flat stretched eastward, growing darker with every league until its doom became the sky's.\n\nCloser to him lay the benighted shapes of Giants. A few of them slept against the boles of ironwoods near the crest of the slope. They snored and started fretfully, troubled by their dreams. Lower down, but still above the chancres and spilth of battle, the other sailors and Swordmainnir had found patches of ground where they could feign comfort.\n\nOverhead carrion-eaters flapped across the background of the stars. Slaps and splashes from the Flat sounded like feeding. The contorted carcasses of monsters littered the valley-floor like rubble. Bleached in the Great Desert for millennia, the dead Sandgorgons smelled only of sulfur and Fire-Lions. But the gangrene fetor of the _skurj_ clung wherever their blood had been spilled. If many of them had not been sucked into the marsh when the lurker's flood receded, the stench would have been worse.\n\nStanding with Stave in the last night of the Earth, Jeremiah pined for sunshine. He craved one more warm yellow wash of light. Trying to summon clean fire, he filled his palms with flame. But the blackness of his heritage persisted. Covered by darkness, his magicks were visible to ordinary sight only as deeper blots, stark as stigmata.\n\nStave still held his arm. \"Chosen-son.\" The former Master pitched his voice for Jeremiah alone. \"It may be that the task which the Chosen has offered is too extreme. She has asked of you an achievement which has surpassed her. If you will heed my counsel, therefore\u2014\"\n\nThe _Haruchai_ paused, apparently awaiting a response.\n\n\"Please.\" Jeremiah was tempted to snort, Don't bother. You can't help me. Contemptuous laughter echoed in his ears as if it had become a part of him, a cancer too insidious and personal to be cut out. More and more, the coming end seemed like an act of kindness. But he did not sneer at Stave. Any suggestion that did not make him feel smaller\u2014\"I've already tried everything I can think of.\"\n\n\"It is this,\" Stave replied. \"Set aside those tasks which daunt you. As your knowledge of the Staff grows, your strength will also. For the present, strive only to meet present needs. The lacks and requirements of this company are many. Choose among them one which lies within your compass.\"\n\n\"Like what?\" Jeremiah asked. Stave's manner seemed to banish scorn.\n\n\"Chosen-son,\" Stave returned, \"your senses are acute. And you will comprehend that our intended ascent into Gravin Threndor must present grave obstacles. Of these, the first is plain. The air is noisome. It discomfits us where we stand. It will become unendurable within the mountain.\n\n\"The Timewarden conceived that the Chosen would cleanse the air. However, the Staff of Law has now been entrusted to you.\" Stave stooped, retrieved the shaft, held it up. \"Therefore the task falls to you\u2014the task and the opportunity. An increase of strength comes from the use of strength.\"\n\nAs Stave spoke, bursts of surprise like little explosions ran through Jeremiah's veins. He clutched at the Staff. \"The _air_ ,\" he breathed. To his nerves, the atmosphere was as distinct as Earthpower. Its insidious taints were so clear that they were almost tangible. He had wasted so much time and effort. \"Why didn't I think of that?\"\n\nStave shrugged. Finally he released Jeremiah's arm. But Jeremiah hardly noticed. His mind raced. How had he let himself believe that he had to fail? Did the _croyel_ still have that much power over him? Did Lord Foul? Had he simply _assumed_ that the small flames which he could raise from the Staff were trivial? Ineffective because he did not know how to make them clean? Had he _tested_ them?\n\nHe had not. Instead he had let the Despiser and the Worm and even Linden's encouragement distract him. A stupid mistake, as stupid as breaking his own neck by not watching where he put his feet. And stupidity was worse than failure. It was worse than terror: it made him useless.\n\n_The purpose of life_ , Cirrus Kindwind had once assured him, _is to choose, and to act upon the choice_. If he could not do what Linden had asked him to do, he could do something else.\n\nHe could do _something_ that had to be done.\n\nefore long, Covenant started back up the valley, trailed by a cortege of Feroce with their nauseous emerald fluttering like banners. Along the way, Branl unveiled the _krill_. At the same time, Rime Coldspray, Bluff Stoutgirth, and the Humbled made their way down from the ridge of Mount Thunder's calf. Silver spread across the sleeping Giants as the Ironhand and the Anchormaster began to rouse them.\n\nFar back in Jeremiah's thoughts, images of the Worm squirmed. When they broke through his concentration, they stung his heart. Now he thought that he recognized the confluence of the Black River and the Mithil. If so, the Worm had crossed much of the South Plains. Furious as a perfect storm, the incarnate cataclysm flared and thundered ever closer to the hills which had once formed the boundary of Garroting Deep. And beyond the region of the lost forest stood _Melenkurion_ Skyweir. The companions did not have much time left. They had probably rested too long.\n\nBut now Jeremiah could push those nightmare visions away. The fangs that were Lord Foul's eyes, and the memories of the _croyel_ 's feeding, no longer consumed him. He had a job to do, a job he understood. In some ways, it resembled making one of his constructs: it involved pulling bits of good air toward him and rejecting poisons; forming a kind of breathable edifice. That may not have been how Linden cleaned the air, but he knew how to do it. The real challenge would be to _keep_ doing it. It would erode constantly: he would have to rebuild it constantly. And the erosion would get worse as the company moved. Still Stave's suggestion gave him hope. Watching Covenant's approach, Jeremiah felt almost ready.\n\nAbove and around him on the slope, the Swordmainnir shrugged their shoulders into the armor, examined their weapons. Without prompting, Wiver Setrock and the woman called Keenreef portioned out another meal, although their supplies were dwindling. Other sailors complained or jested. Of no one in particular, Baf Scatterwit asked where she was. Sounding sincerely confounded, she wanted to know where Dire's Vessel and her other friends had gone. But when Stoutgirth replied with instructions rather than answers, she complied as if she had forgotten her confusion.\n\n\"She is easily bewildered,\" one of the men\u2014Squallish Blustergale?\u2014remarked casually to Jeremiah, \"yet she is an adroit sailor, quick in every exigency. Aye, and doughty withal. None will outlast her on the sheets, or strive more fiercely when there is need. Also she is gentle in her bafflement. Therefore she is precious among us.\"\n\nFor her, Jeremiah felt a flush of sympathy. He knew too well that an absent mind fostered the illusion of safety\u2014and that the illusion was dangerous.\n\nMuttering to himself, he looked around for his mother.\n\nUntil Covenant had left to summon the Feroce, he and Linden had slept together on a stretch of churned earth thirty or forty paces closer to the high cliff which confronted the valley. She was awake now, brushing dirt from her clothes, combing her fingers through her hair. As she came toward Jeremiah, her right hand clung to her wedding band, turning it around and around her ring finger as if she feared that it would be taken from her.\n\n\"Jeremiah, honey,\" she asked when she drew near, \"were you able to sleep?\"\n\n\"Mom.\" He met her holding the Staff of Law in front of him like a promise\u2014or a defense. \"Don't worry about me. I'm making progress.\" He ducked his head to hide conflicting reactions: eagerness for what he might be able to accomplish; chagrin for what he could not. \"I mean, sort of.\"\n\nHer concern reached out to him. Argent reflections haunted her gaze like the residue of horrors. Wordless and worried, she hugged him tightly. Then she stepped back. \"Remember what I told you. There's no such thing as failure. _Sort of_ progress is better than nothing. Under the circumstances, it's probably impressive. We can only do what we can.\" The ruefulness of her smile twisted his heart. \"I need to remember that myself.\"\n\nBefore he could think of a response, she turned to meet her husband.\n\nCovenant came grimly up the side of the valley, walking like a man who had left behind anything that might have softened his severity, his personal commandments. The time had come to essay Mount Thunder; and Jeremiah could see that Covenant was as afraid as Linden. But for him, strangely, fear seemed to be a source of strength. In the illumination of the _krill_ , his silver hair shone like wild magic, the contained conflagration of his heart.\n\nHe returned Linden's embrace briefly; linked his arm with hers as he approached the Giants. Just for a moment, he looked like he might be on the verge of frenzy or tears. Then his expression hardened. The lines on his face resembled slashes.\n\n\"I talked to the Feroce,\" he announced unnecessarily. \"I guess that's obvious.\" The creatures stood a dozen steps behind him, as timorous as ever, and as compelled. \"They say they've never been inside the mountain. And they don't want to go. They call it a _Maker-place_. Lord Foul's home. It scares them.\n\n\"But the lurker didn't give them a choice. I didn't even have to argue. I only had to promise them that _that_ \"\u2014he pointed down at the gullet of the Defiles Course\u2014\"isn't a Maker-place. It's like the Shattered Hills. It defends Lord Foul, but he doesn't live there. He's somewhere up in the Wightwarrens, probably in Kiril Threndor. The Feroce can help us without going that far.\n\n\"They don't know what we'll find. They aren't sure they'll do any good. But they know water\u2014especially polluted water. They'll try to guide us. And\u2014\" Abruptly Covenant paused. For a moment, he covered his eyes as if he had been assailed by memories too painful to countenance. Then he controlled himself, shrugged stiffly. \"They'll try to make the water remember where it comes from. If they can do that, it might be as good as a map.\"\n\n\"What does he say?\" asked Baf Scatterwit. \"A map? Does he speak of a chart?\" She was becoming agitated.\n\nThe Anchormaster rested a lean hand on her shoulder, murmured a soft command which appeared to soothe her. She smiled at him, nodded, and did not speak again.\n\nIn a taut voice, Covenant finished, \"If what the Feroce can do doesn't take us into the Wightwarrens, we'll have to find our own way.\"\n\nThe Ironhand nodded sternly. \"Then, Timewarden, only two matters remain. You and Linden Giantfriend and the Chosen-son must eat to sustain your strength. And we must look to our survival within the mountain.\n\n\"We are Giants, lovers of stone. We do not fear to attempt the hidden passages. Also the Anchormaster and our comrades of Dire's Vessel will accompany us, for so they interpret the wishes of Brinn _Haruchai_ , the last Guardian of the One Tree.\"\n\nStoutgirth grinned as if he found her assertion risible; but he did not return a jest.\n\n\"Being sailors,\" Coldspray continued, \"they have borne with them a goodly quantity of rope. Such providence will surely serve us well.\"\n\nThe muscles at the corners of her jaw bunched. \"Yet we must breathe. It is certain that the airs within the water's channels will be foul beyond bearing. Ere long, respiration alone will prove fatal.\" Her tone was exposed gutrock. \"Therefore I am compelled to inquire. How can we dare Mount Thunder if we cannot breathe?\"\n\n\"Maybe the Feroce\u2014\" began Covenant darkly.\n\nJeremiah took a step forward. \"Wait.\" His hands itched with anticipation on the Staff. \"I've been working on this.\" He glanced quickly at Stave. \"I'm not sure, but I'm learning. Maybe I can\u2014\"\n\nAbruptly he closed his eyes; forgot words. Now or never. His mother had trusted him with her best instrument of power. If he proved her wrong, he would have to return it. Her hopes for him\u2014and his own\u2014would be gone.\n\nJust for a moment, malice pealed through the dark behind his eyelids. Prove her wrong, puppy? How can you not? You are naught but a tool, a means to an end. Your every deed serves my desires.\n\nBut Jeremiah refused to listen. The whole company was watching. And the Staff was _alive_. In small ways, it answered his Earthpower, his health-sense. He could believe that those responses would grow. And in the meantime\u2014Right here, right now, he could feel the air, taste it; almost touch its nature. He could distinguish between health and sickness.\n\nDeliberately he poured flames into the cups of his hands. Ignoring their taint, he wrapped them around the Staff. Then he asked the wood for more theurgy than his mere body contained. As hard as he could, he concentrated on breathing\u2014\n\n\u2014on pushing away poisons and corruption\u2014\n\n\u2014on rejecting putrescence and vilification\u2014\n\n\u2014and on drawing the cleanliness that remained toward him.\n\nAnd when he knew that he was inhaling and exhaling _life_ , he extended his edifice of good air toward his companions.\n\nSee? he told the mockery inside him. I can do this. I can _do_ it.\n\nThen he opened his eyes to see the effects of his efforts.\n\nLinden gasped as she took an unconflicted breath. \"Jeremiah,\" she murmured. \"My God\u2014\" Covenant filled his lungs and seemed to stand taller, as if the air had confirmed him. He gave Jeremiah a look like a shower of sparks from a whetstone. Rime Coldspray and Bluff Stoutgirth raised their heads, sampled the spread of vitality. Grins like promises showed their teeth. With gestures and relief, they exhorted their comrades to crowd closer.\n\nAs the whole company began to breathe more comfortably, the Ironhand announced, \"This is well done, Jeremiah Chosen-son. I confess that I did not foresee it. If you are able to sustain such exertions\u2014\"\n\nShe swallowed the rest of what she might have said; the questions she might have asked.\n\n\"It'll get easier,\" Jeremiah muttered self-consciously. \"I mean, I think it will. I'm not used to it yet. I just need practice.\"\n\nChuckling, Blustergale swung a clap at Jeremiah's back that would have felled him. But at the last instant, the Giant seemed to recall that Jeremiah was little. His hand patted Jeremiah gently and withdrew.\n\nStave bowed his approval. A tightening at the corner of his mouth hinted at a smile.\n\nBehind Covenant, the Feroce squalled in soft voices, as if they feared to be overheard; but Jeremiah did not know how to interpret their cries.\n\now that he had begun to prove himself, he was eager to try the uncertain ascent along the watercourse. But Rime Coldspray reminded him again that he needed food\u2014as did Covenant and Linden. Reluctantly Jeremiah let go of his magicks.\n\nWhile they ate and drank, the company discussed uncertainties and perils.\n\nThis approach to Mount Thunder's heart was Covenant's idea, but he did not know whether the path of the Defiles Course within the mountain would prove passable. In the past, he had only entered the Wightwarrens from the Upper Land. Certainly the Giants were skilled climbers and delvers. The _Haruchai_ were born to crags and cliffs. And they were adequately supplied for their immediate purpose\u2014or so the Anchormaster asserted. Nevertheless they could imagine obstacles which they would not be able to surmount. Water was water, after all. Under pressure, it could find its way through constrictions which would refuse Giants or _Haruchai_ or Feroce.\n\nIn addition, the Despiser clearly knew where to look for his enemies; and his servants were many. At any time, he might send Cavewights or stranger creatures to waylay the company. Long ago, horrors had formed a large portion of his forces. The companions could not assume that any stretch of their path would be uncontested.\n\nTo all of this, Jeremiah listened without paying much attention. For the moment, at least, he was content with food and the Staff of Law. Finally he knew what he had to do\u2014and how to do it. He had already shown that he could do it. The whole company trusted their lives to him. And Stave had assured him that he would get stronger. He might even learn how to do more than improve the air.\n\nIf Lord Foul tried to take him, sixteen Giants, two _Haruchai_ , and two white gold wielders might be able to protect him.\n\nSo he ate what he was given, and drank water lightly tinged with _diamondraught_ , and tried to mask his impatience while he waited for Mom and Covenant to finish this last meal.\n\nAt last, the company was ready. Keenreef and several other sailors shouldered packs of supplies. All of their quirts and spears had been destroyed, but most of Stoutgirth's crew still carried weapons: billhooks, longknives, belaying-pins. The Swordmainnir had their armor and their blades. And the _Haruchai_ had set aside the characteristic reluctance of their people to rely on weapons. Branl shouldered Longwrath's flamberge, while Stave bore Cabledarm's longsword.\n\nAmong such companions, Covenant and Linden looked small, vulnerable. But there was a dangerous promise in Covenant's eyes. And Linden looked withdrawn. She no longer seemed to care about details like difficult climbing and enemies. Only the way that she twisted her ring around her finger hinted that she was fretting.\n\nFormally the Ironhand drew her stone glaive. Holding it ready, she spoke in a voice of granite.\n\n\"Here we surrender every future which we have imagined for ourselves. We have no prospect of return. Indeed, we cannot trust that we will outlive another day. Our doom is this, that we enter Mount Thunder seeking to confront the most heinous of foes\u2014and yet the Worm hastens toward the World's End many scores of leagues distant, where no deed of ours can thwart it. Thus even the greatest triumphs within the mountain may come to naught, for no life will remain to heed the tale.\n\n\"Nonetheless I proclaim\"\u2014Coldspray swung her sword once around her head, then slapped it into its scabbard on her back\u2014\"that I am not daunted. _I am not daunted_. While hearts beat and lungs draw breath, we seek to affirm the import of our lives. The true worth of tales lies in this, that those of whom they speak do not regard how the telling of their trials will be received. When we must perish, my wish for us is that we will come to the end knowing that we have held fast to that which we deem precious.\"\n\nThen her tone eased. \"Doubtless this is folly. Yet when have our deeds been otherwise? Are we not Giants? And is not our folly the stone against which we have raised the sea of our laughter? What cause have we to feel dismay and hold back, when we have always known that no anchor is secure against the seas of mischance and wonder?\"\n\nPerhaps she would have continued; but the Anchormaster was already laughing. He tried to say something, but the words were lost in broad gusts of glee. For a moment, the other sailors were silent, dismayed by images of futility. But then Baf Scatterwit began to guffaw: the happy mirth of a woman who enjoyed laughing for its own sake. Her laughter broke the logjam of her comrades' fears. Carried along by her open-heartedness, the crew of Dire's Vessel roared as if they themselves were an exquisite jest.\n\nThe Swordmainnir were more restrained. They had lost too many of their comrades. But when Rime Coldspray started to chuckle, Frostheart Grueburn followed her example, and then Cirrus Kindwind. In their subdued fashion, the Ironhand and her warriors shared the delight of the sailors.\n\nPrivately Jeremiah thought that they had all lost their minds. Nevertheless he found himself grinning. He had heard too little genuine laughter in his life; and the mirth of Giants was especially infectious. At least temporarily, it made Lord Foul's scorn and the _croyel_ 's malice seem empty, like taunts from the bottom of an abandoned well.\n\nLong ago, Saltheart Foamfollower had enabled Covenant's victory over the Despiser by laughing.\n\nAs the Giants began to subside, Covenant muttered, \"Stone and Sea are deep in life.\" He seemed to be quoting. \"Two unalterable symbols of the world.\" Then he lifted his head to the dark heavens, the decimated stars. From his ring, a brief flash of silver challenged the night. \"I can't help it. I've always loved Giants. Any world that has _Haruchai_ and Ranyhyn and Ramen and Insequent and even _Elohim_ in it is precious. But there really is no substitute for Giants.\"\n\nJeremiah agreed with him.\n\nThe Ironhand answered Covenant's moment of power with a flash of her teeth. \"Then, Timewarden,\" she said, \"let us now vindicate your love.\"\n\nWith a sweep of her arm, she drew the Swordmainnir and Dire's Vessel's crew with her as she started down the side of the valley toward the throat of the Defiles Course.\n\nJeremiah followed them as if he, too, had been called. With the Staff and his own power, he drew clean air out of the ambient reeks.\n\nAfter a moment, Cirrus Kindwind came to his side. Frostheart Grueburn now accompanied Linden and Stave, and the Anchormaster had claimed a place with Covenant and Branl. Escorted by Giants and _Haruchai_ , Covenant, Linden, and Jeremiah picked their way between craters like maws and past rank corpses toward the cave where the Land's most ancient waters carried their burden of poisons and spilled evil into the embrace of the Sarangrave.\n\nApparently the Feroce had anticipated the company's movement. They already stood on the riverbank within an easy stone's throw of the cliff, a cluster of ten small creatures with emerald in their hands and naked fright in their eyes. They did not react as the first Giants approached them. Instead they stood in the stench of the Defiles Course, facing each other and quavering as if their deity had declared them expendable.\n\nBut when Covenant drew near, they turned away from their communion. Flinching, they spoke in their one voice: an eerie sound like squeezed mud, moist and attenuated.\n\n\"We are the Feroce,\" they said as if they were on the verge of weeping. \"We are only the Feroce. At our High God's command, we attempt aid. It exceeds us. We will not suffice.\"\n\nCovenant regarded them like a man who showed no mercy; but his words belied his manner. \"You don't have to suffice. You just have to try. When you can't do any more, you're free to go.\"\n\n\"Then,\" replied the creatures, \"we will begin. We have no wish to prolong our failure.\"\n\nTogether they faced the gaping mouth of the cliff. In a tight cluster, they started toward the deeper dark, a blackness that seemed to mock the _krill_ and the company, the night and the forlorn stars. Although no tangible power compelled them, they moved as if they were being scourged.\n\nCovenant watched them, but he did not follow. Instead he rasped to the Giants and the _Haruchai_ , \"Just remember. White gold is going to be mostly useless, at least for a while. I don't have much control. I'm more likely to cause a cave-in than accomplish anything useful. Plus I can't keep my balance worth a damn. And Linden hasn't had time to learn what she can do. We'll need all the help you can give us.\"\n\n\"This we have foreseen, Timewarden,\" the Ironhand answered calmly. \"If Giants are fools, they are also rock-wise, certain of foot on any stone. With your consent, we will bear you, and also Linden Giantfriend and Jeremiah Chosen-son. In our arms, you will be warded from many perils.\"\n\nNow Covenant looked back at his companions. \"Linden?\"\n\n\"I think it's a good idea.\" She made a palpable effort to sound confident; but Jeremiah heard the congested tension in her voice. \"Grueburn has carried me more times than I can count. I'm not worried about her. And I don't like the way that looks.\" She gestured at the river mouth. \"If nothing else, it's going to be slick.\" Her mouth twisted. \"I would rather be carried. If Grueburn doesn't mind.\"\n\nGrueburn's response was a snorted chortle.\n\nCovenant nodded. \"Jeremiah?\"\n\nJeremiah felt a touch of relief. \"Mom's right. I'm not as strong as I want to be. I mean with the Staff. If I don't have to do my own climbing, I can concentrate better.\"\n\nFor himself, Covenant did not hesitate. To Coldspray, he said brusquely, \"Thanks. I should have thought of that myself.\"\n\nThen he made a visible effort to relax as Bluff Stoutgirth lifted him from his feet.\n\nIn a moment, Jeremiah was sitting on Kindwind's forearm with his back against her breastplate. His lightless flames scurried up and down the length of the Staff. They were weaker than they needed to be, but they gathered enough purity to ease the company's breathing.\n\nFrom her position in Grueburn's clasp, Linden glanced at Jeremiah with an expression which he could not interpret. A warning? A prayer? Was she saying goodbye?\n\nShe had found her own sense of purpose, but he had no idea what it might be.\n\nOne after another, Rime Coldspray and all of the Giants followed the receding green of the Feroce. Holding the _krill_ above his head to extend its illumination, Branl walked close behind the Ironhand near Stoutgirth and Covenant. Stave took a position between Grueburn and Kindwind.\n\nStriding as if they were about to burst into song, the Swordmainnir and the sailors left the world they knew. Beside the Defiles Course, they entered Gravin Threndor and darkness.\n\n## 6.\n\nThe Aid of the Feroce\n\nAs Frostheart Grueburn carried her into the gutrock gullet of the Defiles Course, Linden lost her last glimpse of the heavens. It was cut off as if the whole of the world beyond the immediate channel, the immediate darkness, had vanished. As if the fate of every living thing, of life itself, had been reduced to this: impenetrable midnight; stone as slick as oil or black ice; Mount Thunder's imponderable tons, ominous and oppressive. As if she herself had become nothing more than a burden.\n\nThe decimation of the stars had been a constant reminder of the carnage which the Worm had already wrought. But what had been lost only made what remained more precious.\n\nYet she had set aside her responsibility for the world. She had chosen her task. It was necessary to her, the only choice that offered any hope of forgiveness. But it would not stop the Worm. It would not hinder Lord Foul, or save her friends, or spare her son.\n\nAt first, the watercourse became narrower, ascending in low stages like terraces or past obstructions like weirs. Beyond the Ironhand\u2014beyond Stoutgirth, Covenant, and Branl\u2014the Feroce clambered, elusive as eidolons, over a tumble of boulders barely wide enough to accommodate the Giants in single file. Long ages of poisons and leaking malice had pitted the stone, cut it into cruel shapes, left it brittle with corrosion. But the waters had also caked every surface with slime like scum. And wherever the tumult of the currents had left gaps, necrotic mosses clung, viscid as wax, treacherous as grease. Touching them would be like trailing fingers through pus.\n\nWhile the passage narrowed, however, its ceiling stretched higher. Here the Defiles Course ran down a fissure in Mount Thunder's substance. A few arm spans up the walls, the green of the Feroce gleamed sickly on moisture and moss: the residue of the river's former flow. Above that demarcation, the _krill_ 's argent faded into the dark.\n\nThe crevice was old: far older than Linden's knowledge of the Land. It had endured for eons, perhaps ever since the convulsion which had created Landsdrop. It might continue to do so. Nevertheless the gutrock overhead seemed fragile. The clutter of boulders where the Feroce led the companions demonstrated that stones did fall.\n\nBut the possibility that some tremor might release sheets of rock did not trouble her. She had more urgent concerns. More than the mountain or the darkness\u2014more than slick surfaces and vile moss\u2014she feared the air. It was not merely fetid and hurtful: it was thick with leached evils. Every breath brought dire scents from offal and corpses; from strange lakes of lava and ruin arising from the deep places of the Earth; from the detritus of horrid theurgies and delving. From time and rot and distillation.\n\nAnd from She Who Must Not Be Named. At intervals like the tightening of a rack, Linden tasted hints of the bane's distinctive anguish, terrible and bitter. She could only bear the miasma which she drew into her lungs because Jeremiah was ameliorating it with Earthpower.\n\nEarlier he had sweetened some of the air in the valley. He could not do as much here. The atmosphere was more concentrated. And the fact that his companions were forced to advance one at a time exacerbated his difficulties. He had to push the Staff's benefits too far. As a result, Rime Coldspray and the other Giants in the lead had begun to cough as if they were about to bring up blood. Between their stertorous gasps, Linden heard Covenant wheezing. Some of the Giants in the rear retched. The sounds of their distress rebounded from the walls; multiplied upward until they filled the crevice.\n\nThe air would continue to deteriorate as the company climbed. Leagues of unknown passages, dangerous footing, and pollution lay between the company and the more tolerable atmosphere of the Wightwarrens. And Jeremiah was already faltering.\n\nHe was not ready for this; not ready at all to have twenty-one lives depending on him for every breath.\n\nInstinctively she yearned to reach out for the Staff's resources; to wield them herself. Jeremiah was not far behind her: only Stave followed Frostheart Grueburn ahead of Cirrus Kindwind. Linden could siphon Earthpower and Law from the wood while he held it. Her chest _hurt_. She wanted good air.\n\nResisting her impulse to assume the work that she had given to her son was as painful as breathing.\n\nBut she had surrendered the Staff because Jeremiah needed it more than she did. Eventually he might need it absolutely. He had to become stronger. If she took back her trust prematurely\u2014if she made his challenge easier from the start\u2014she would undermine his efforts to believe in himself.\n\nYet the company was struggling. Sweat ran from Grueburn's face, although the stone and the water were cold as a crypt. Her distress ached through her lore-hardened armor. By degrees, frantic coughing spread among the Giants. In front of Grueburn, Baf Scatterwit was taken by a spasm so fierce that she slipped. She caught herself with both hands, avoided a plunge into the river, but not before her kneecap struck rock with an audible crack. Choking on Giantish obscenities, she hauled herself upright. Then, however, she was forced to halt, hunching over to massage her knee.\n\nFrom Coldspray or Covenant, ragged murmurs passed Linden's name back to her; but she did not need to hear it. She understood. Jeremiah had to do better.\n\n\"Jeremiah, honey.\" She was panting herself. \"You're trying too hard.\" He did not know himself well enough yet. \"It's easier than you think. It's the Staff of _Law_. It was made for this. You don't have to force it. You just have to encourage it. Guide it. Let it express how you feel.\"\n\n\"I can't.\" Jeremiah's protest was thick with dread. \"That doesn't make sense.\"\n\nLinden fought for patience. \"Try it this way. Close your eyes. Forget where you are. Forget what's happening. Forget the Staff, if you can. Concentrate on Earthpower and air, clean air, air that keeps you alive. It's like building one of your castles. You think about what you're making. You don't think about how you make it. The Staff is just a means.\n\n\"You can do this if you trust yourself.\"\n\nShe could almost hear his resolve breaking. \"That doesn't\u2014\" he began to insist. But then he stopped. \"All right,\" he said like a groan. \"I'll try building. That worked before. Just don't blame me if\u2014\"\n\nHe fell silent.\n\nFor a moment, the effects of his theurgy disappeared entirely. Linden drew air like shards of glass into her lungs. All of her muscles seemed to seize at once. Grueburn's gasps sounded like tearing flesh. Along the line, Giants stumbled to a halt, sank to their hands and knees. The _krill_ lit them like spectres, as if they had crossed over into the realm of the Dead.\n\nThe Feroce had caused some of Sarangrave Flat's mud to remember that it was once hurtloam. Covenant had said so. Surely they could do something similar to the air? If he asked them?\n\nThen Linden felt a stronger current of Earthpower emanate from Jeremiah and the Staff. It was tentative at first. It surged and receded. She found one healing breath, lost it again. Nevertheless her heart lifted. His access to the Staff's potential resembled the chamber hidden in her own mind, the room which could open on wild magic. Learning that the chamber existed had enabled her to locate it again. And each time, the search was more familiar. The door opened more easily. The same could be true for Jeremiah, if he refused to panic.\n\nHe was young and gifted. In some respects, his sense of himself was more flexible than hers, less conflicted by an awareness of his limitations. For a heartbeat or two, his power shrank; but it also became steadier. Then better air began to gust outward. Some of it escaped into the empty heights of the fissure. Most of it swept over the company.\n\nLinden snatched freshness into her lungs, fought for it. It was still tainted, but it became cleaner with every breath. Groans of relief spread among the Giants as Jeremiah expanded his efforts. Grueburn seemed to bite off great chunks of air, swallow them gratefully. A fierce grin bared her teeth. Still coughing, Baf Scatterwit started to laugh. One at a time, sailors and Swordmainnir joined her.\n\n\"Well done, Chosen-son!\" called the Ironhand. \"Well done in all sooth! It may be that our cause is doomed. It may be that we will soon perish. Yet miracles abound, and Jeremiah Chosen-son stands high among them.\"\n\nGradually Linden's companions stood straighter. They began to move again.\n\nThe Feroce had not paused. They may not have noticed the company's difficulties. Or they may not have cared. They had their own fears. Perhaps a stone's throw ahead of the Ironhand and Covenant, the troubled green passed from sight beyond a corner. Streaks of argent lit the rubble piled along the river as if the stones had tumbled there from Gravin Threndor's dreams.\n\nAs her respiration eased, Linden thought that she heard thunder.\n\nNo, not thunder. By degrees, the sound clarified itself. It was too wet, too complex, too constant to be atmospheric. It cast spray into the ambit of the _krill_ 's illumination. The company was approaching a waterfall.\n\nWhere the spray brushed her cheeks, it stung.\n\nShe could not gauge the height of the plunge by the timbre of its muffled roar; but she heard neither warnings nor chagrin from the Giants. The Ironhand did not hesitate as she bore Covenant out of sight, leaving Branl behind to light the way.\n\nIn moments, a few sailors and Onyx Stonemage scrambled to Branl's position, followed by Squallish Blustergale and more of the Anchormaster's crew. As Grueburn neared the turn, Linden became more confident that the water did not plummet from a great height. Still her anxiety did not relent until Grueburn carried her past the corner. Then she was able to see that the waterfall was no taller than one Giant standing on the shoulders of another.\n\nShe could not have climbed it. Perhaps Grueburn could not. But here the river's diminishment was obvious. A comparatively narrow gush of water pounded into the deep center of the channel. Beside the river on both sides, eons of a far heavier flow had left more gradual slopes. Broken rocks cloaked in mosses like shredded skin mounted upward in possible increments.\n\nA short way up the rise, Coldspray and Covenant waited for Branl and light. Above them, the Feroce scrambled for the rim as if they were in no danger of slipping. Their emerald glow wavered and gibbered on the walls as they scuttled out of sight. Then their fires faded as if the crevice had opened to accommodate a cavern.\n\nLinden looked back at Jeremiah. The radiance of Loric's gem revealed black tendrils of power like vines curling away from the Staff, making the air precious. As the boy worked, however, a scowl of strain clenched his features, and the wood trembled in his grasp. He was still trying too hard.\n\n\"Are you all right, honey?\" Linden asked over the shout of the water. \"Do you need rest? We should be able to survive for a few minutes.\"\n\n\"Don't bother me.\" He sounded distant, wrapped in concentration. She barely heard him. \"I'm fine.\"\n\n\"The Feroce act like they're in a hurry,\" Covenant offered, \"but I can ask them to wait\"\u2014he glanced at the waterfall\u2014\"once we catch up with them.\"\n\nWhen Jeremiah nodded, Rime Coldspray continued upward. Behind her, Bluff Stoutgirth gestured his crew forward. Moving as surely as the Giants, Branl passed Grueburn and Linden to rejoin the head of the line.\n\nAccompanied by argent, the Ironhand took Covenant past the lip of the fall, out of the harsh spray. At the rim, Branl waited again. Still in single file, Giants made the ascent. Ahead of Grueburn and Linden, Scatterwit limped over the treachery of the stones. She was obviously in pain, yet she chuckled in short bursts as if her damaged kneecap amused her.\n\nThen Grueburn crested the waterfall; and Linden stared in surprise. Ahead of her, _krill_ -light played across the black surface of a lake.\n\nIt may have been vast. The height of the cavern seemed to imply that it was; and the darkness beyond the _krill_ 's reach concealed the boundaries of the water. Liquid obsidian curved away to Linden's left, following the cavern wall out of sight. But ahead and to the right, the lake appeared to have no end\u2014or her senses were confused by intimations of power.\n\nIt was eerily motionless, as still as stone. Water dripped from mosses high on the walls, where until recently the cavern had been filled. Thin trickles fell here and there across the emptiness, perhaps dribbling from stalactites invisible in the dark. But there were no ripples: none at all. And no sounds. Drops struck the lake and were absorbed seamlessly. Water lay flat as glass against the rocks of the verge.\n\nThe Ironhand had halted with Covenant near the curve of the lakefront. One by one, the rest of the company reached them and stopped, peering into the blind depths or the veiled distance. Branl waved Loric's dagger for a moment, watched silver sweep across the immaculate ebony. Then he stepped back.\n\nA leaden silence ruled the cavern. From this vantage, even the waterfall appeared to make no sound. The Giants seemed unwilling or unable to speak. To Linden's eyes, the air over the lake looked as condensed and heavy as sweat.\n\nAgain her health-sense caught hints of She Who Must Not Be Named. Here they were stronger. The spilth of theurgies as old as the mountain\u2014as old as the Land\u2014stained the lake wherever she looked. Implied carrion-eaters tasted her skin.\n\nDistracted by noisome things, she was slow to notice that the Feroce were gone.\n\nGone?\n\n\"Thomas?\" The silence seemed to seal her throat. She had to swallow several times before she could say more than his name. \"The Feroce? Where did they go?\"\n\nRime Coldspray and her comrades scanned the cavern, the lake. Covenant gazed past or through Linden like a man who had lost his sight. \"Into the water.\" His voice sounded preternaturally distinct: precise and defiant. It should have raised echoes. Instead it fell stillborn. \"I don't know why. They didn't say anything.\"\n\n\"I am loath to believe,\" remarked Branl, \"that they have forsaken us.\"\n\n\"As am I,\" Stave agreed. \"They heed their High God.\"\n\nThe Ironhand coughed, cleared her throat. \"Without them\u2014\"\n\nAs if she had summoned them, delicate green flames appeared on the surface near the spot where the cavern's leftward sweep interrupted Linden's view. Untroubled by the waters, the creatures arose under their fires, lifted emerald from the lake. Their passing left no mark on the water's black sheen as they climbed the rocks.\n\n\"Why do you tarry?\" Their damp voice scaled into the heights. \"You must hasten. There is peril, much peril. Are you deaf to majesty? Blind to wonder? You must hasten.\"\n\nTensions ran among the Giants. They prepared to move. But Rime Coldspray stood where she was. From her clasp, Covenant called to the Feroce, \"What's going on? You picked a hell of a time for a swim. Did you wake something up?\"\n\nStudying the lake, Linden saw nothing, heard nothing. She felt only the noxious tickling of centipedes, tentative and eager.\n\n\"Will you not hasten?\" the creatures urged. \"We are merely the Feroce. There is no peril for us. Your lives are forfeit.\"\n\nCovenant appeared to freeze for an instant, startled into incomprehension. Then he snapped to the Ironhand, \"Go!\"\n\nAt once, Coldspray surged ahead over the hazardous stones. Behind her, Giants followed as swiftly as they could. But Baf Scatterwit's cracked knee slowed her. Grueburn waved her free arm, urged Halewhole Bluntfist and the trailing sailors to pass her. Then she drew her longsword; kept pace with Scatterwit. Cirrus Kindwind did the same, bracing Jeremiah with her maimed forearm. Holding Cabledarm's longsword, Stave positioned himself between them and the water.\n\nLinden did not glance at her son. The lake seemed to grip her. There were too many centipedes, more and more of them. Spiders. Maggots. Worms avid to feast on her sins. Only Jeremiah's tendrils of Earthpower and Law shielded her.\n\nA bulge appeared in the water.\n\nNo, not a bulge: a _body_. A stone's throw long. A double arm span wide. Lithe as a serpent, flowing up to bend the surface and then sliding back down endlessly. As dark as the lake, but rife with strength. If it had a head or a tail, Linden did not see them. No slight ripple or splash defined the immense monster's glide.\n\nFrom the rocks ahead of Covenant, the Feroce confessed abjectly, \"We sought to gaze upon our High God's god. We have done so. Its thoughts are broken. They lack glory. Only ruin remains. It will slay you all.\"\n\n\"Hellfire,\" Covenant rasped. \"I suppose even the lurker had to come from somewhere. If that's its mother, I've seen enough. I don't want more.\"\n\nBranl stood at the waterline, poised with the _krill_ and Longwrath's flamberge. Linden felt a throb of wild magic from Covenant's ring. Reflexively her wedding band answered his attempt to prepare himself. Imminent heat and argent chased away the things that scurried across her skin. The long arc of the monster's body continued to flow. If it felt the company's presence, its awareness was hidden in the depths.\n\nLinden tried to focus her attention on her ring, seeking to support Covenant; to dismiss the bane's touch. But she was too close to the Staff. Jeremiah's power seemed to block her, or she blocked herself. Wild magic and Law conflicted.\n\nColdspray and then the Anchormaster rounded the curve. Dire's Vessel's crew hurried after them, crowding close to the cavern wall. Weapons ready, both Stonemage and Bluntfist had taken positions like Branl's near the water's edge, guarding the rear of the company. Scatterwit whimpered as if she feared to be left behind.\n\nThe Feroce had disappeared again. Had they gone ahead? Linden did not know. She struggled to breathe. A moment passed before she realized that the company's alarm had pierced Jeremiah's concentration. He needed help.\n\nHere she could not call upon the Staff without touching it. The throb and itch of her ring interfered. Trepidation interfered. Leaning away from Grueburn's arms, she reached for Kindwind and Jeremiah; but she could not stretch far enough. Then Grueburn shifted closer to Kindwind, and Linden gripped the Staff by one iron heel.\n\nShe did not take it. Instead she added her will to her son's disturbed resolve, reinforced his intentions with her own.\n\nHe gave her a quick glance of thanks. Relieved, he settled back into himself. The pressure of poisons in her lungs eased. All of the Giants seemed to move more quickly. Even Scatterwit's pace improved.\n\n\"Attend!\" Branl called calmly. \"The water rises.\"\n\nLinden twisted her head to look.\n\nOh, shit. Branl was right. Still motionless, still silent, the midnight lake had begun to devour its borders, fed by some source beyond her discernment. It did not lap or splash against the rocks. It simply covered them.\n\nLed by the _Haruchai_ , Grueburn, Kindwind, and Scatterwit passed the curve at the rear of the company. At once, Branl ran ahead, carrying light. Now Linden saw that the cavern narrowed in this direction. The walls leaned closer to each other until they met in a sheet of running water. At first, the sheet appeared sheer, a straight waterfall thinned by its own width. It would be impossible to climb. And there were no slopes leading up to the tunnel that opened three or four Giant-heights above the lake. The Feroce stood facing the cul-de-sac as if they had been thwarted.\n\nBut then Linden saw that the pour of water reflected argent and emerald in a cascade of spangles. Under the waterfall, the stone was broken in scores or hundreds of places, pitted and interrupted wherever erosion and toxins had found flaws.\n\nSurely it would still be impossible to climb? The stone would be slick\u2014\n\nThe lake rose. The added water should have drained away as fast as it came, but it did not. Somehow the lurker's mad god heaved the entire surface higher. Grueburn, Kindwind, and Scatterwit were forced to pick their way closer to the wall.\n\nWithout explanation, Stave sprinted away. Inhumanly sure-footed, he caught up with the Humbled, moved among the Giants. He handed Cabledarm's sword to the Anchormaster. Linden heard him ask for rope.\n\nFavoring her knee, Baf Scatterwit stumbled into the edge of the lake. Her right foot went under. The impact of her weight had no effect on the water's massive lift.\n\nLinden had no idea what would happen then. The lake's power defied her senses. But Scatterwit scrambled out again. She tried to limp faster.\n\nLinden clung to the Staff's heel, struggled to help Jeremiah clean the air.\n\nFrom a sack of supplies, a sailor produced a coil of rope as thick as Stave's arm. He looped one end twice over his shoulder, secured it by tucking it under itself. Immediately he approached the swift sheet of the cul-de-sac. As if the difficulties were trivial, he began to ascend.\n\nWater pounded onto him. It splashed past him without affecting the eerie surface of the lake. He was drenched in ancient corrosives, distilled residues. But they did him no apparent harm. His flesh spurned the mountain's taints.\n\n\"Giantfriend,\" Grueburn rasped: a harsh scrape of sound. With her sword, she gestured at Scatterwit.\n\nLinden glanced in that direction, saw Scatterwit limping more heavily than before. Far more heavily. With every step, she lurched to the right, toward the lake, as if she had lost her balance. She seemed to recover by force of will.\n\nGod\u2014\n\nBaf Scatterwit's right foot had been cut away, severed at the ankle. A clean slice: clean and cauterized. There was no blood. She seemed unaware that her foot was gone. She moved as if only her damaged knee pained her.\n\nLinden started to shout a warning at the Giants; but Grueburn stopped her. Through her teeth, the Swordmain snarled, \"They know.\" Abruptly she slapped her sword back into its scabbard. With her free hand, she supported Scatterwit so that the woman could hurry without toppling.\n\nPanic and Grueburn's rush broke Linden's hold on the Staff. Earthpower and black flame faltered. The air dug a knife into Linden's chest. But Jeremiah tightened his grip a moment later, took up the slack. Complex stresses gleamed on his cheeks and forehead.\n\nBetween one urgent breath and the next, Linden saw Stave rise higher than Rime Coldspray's head. His fingers and toes gripped the damaged stone like claws. Another breath, and he had climbed more than halfway. Then he gained the lip of the tunnel and passed out of sight, trailing the rope behind him.\n\nNow, Linden thought. Now he has to secure it.\n\nThere was nothing that she could do for Scatterwit.\n\nStave did not have time. With Grueburn's help, Scatterwit joined the other Giants. Kindwind and Jeremiah came last. Bluff Stoutgirth gave Scatterwit a look of anguish, then jerked his head away. Other sailors chewed their silence as if they sought to break their teeth. They were all ready. Covenant now clung to Coldspray's back, leaving her arms unencumbered. But the lake still rose. In a few heartbeats, no more, it would threaten the nearest feet. It would sever\u2014\n\nStave's line jerked. At a word from the Anchormaster, Wiver Setrock grabbed it, tested it. Carrying more rope, Setrock swarmed upward, a sailor adept at ratlines and hawsers. Unlike Stave's, his feet slipped here and there; but those momentary skids hardly slowed him. If the corrupted water hurt him, he ignored the pain.\n\nHe reached the lip, vanished into the river's tunnel. Moments later, his line snaked down to his comrades. Then Keenreef and Hurl were climbing, each with new ropes.\n\nThe lake crept higher. The waiting Giants squeezed closer to the wall. Some of them stood in the waterfall, breathing with their mouths covered. The Feroce watched from a short distance. The water came to their ankles, then to their knees; but they did not fear it. Green and silver shone in their limpid eyes.\n\nLinden wanted to tell Coldspray and Kindwind to go next, take Covenant and Jeremiah to safety. But when she tried to speak, her voice failed. She could not imagine how Cirrus Kindwind would bear Jeremiah upward with only one hand.\n\nFrom the Ironhand's back, Covenant asked the lurker's creatures, \"What about you? We need you.\"\n\n\"The Feroce are the Feroce,\" they replied as if that answer sufficed. Sinking at every step, they began to back away. As they submerged, their fires flared briefly on the water, then went out.\n\n\"Hellfire,\" Covenant muttered. \"Bloody damnation.\"\n\nTo Coldspray, the Anchormaster said, \"In such straits, my will commands.\" His tone held an unexpected edge of authority. \"You and Frostheart Grueburn must ascend. Halewhole Bluntfist and Etch Furledsail will assist Cirrus Kindwind.\" He hesitated for an instant, then growled, \"Baf Scatterwit must hold the rear.\"\n\nScatterwit responded with a laugh like the croak of a raven.\n\nOf course, Linden thought bitterly. Scatterwit had been maimed. Therefore she was more expendable than her comrades.\n\nGroaning to herself, Linden worked her way around Grueburn until she reached Grueburn's back. The Swordmain would need both hands\u2014\n\nAbruptly Jeremiah's power evaporated. \"Sorry about this.\" Tension thrummed in his voice. \"I'll get back to it.\"\n\nLifted by Kindwind, he went over her shoulder to her back. As he shifted, he braced the Staff across her breastplate so that he could hold it with his arms on either side of her neck. There he hung, hugging the sides of her chest with his legs. Then he shut his eyes; began to exert Earthpower again. Black flame twisted upward in front of Kindwind's face.\n\nFour lines now dangled from the darkness of the tunnel. At once, the Ironhand, Onyx Stonemage, and two sailors hastened upward. As soon as they were clear, Grueburn started to ascend like a leap of fire. Through tainted torrents, Linden watched Bluntfist and Furledsail support Kindwind.\n\nSomehow Jeremiah continued to pour out Earthpower while water hammered down on him.\n\nThe lake still rose. It was no more than an arm span away from the rest of the Giants. Only Branl stood between them and the fatal surface.\n\nResting his flamberge on his shoulder, the Humbled crouched at the water's edge and prodded the tip of the _krill_ into a stone. There he waited, studying the lake as if he were daring it to touch High Lord Loric's blade.\n\nA moment later, Grueburn carried Linden up into the tunnel into the deeper darkness of the river's passage. At first, she saw nothing. Granite and black water filled her senses. But then Covenant's ring began to emit a soft glow. Strain knotted his forehead, bared his teeth, as he strove to elicit wild magic without losing control. Gradually his conflicted, tenuous light revealed the surroundings.\n\nBeyond its wide mouth above the cavern, the tunnel resembled a chute or flume angling sharply downward from somewhere far above. The diminished river filled its bottom, tumbling loudly over planes like shelves, gouged flaws, indurated obstructions. Covenant's silver bled along the splashing and spray. The Ironhand, Stave, and a few Giants had waded upward, forcing their way against the downrush to make room for their comrades.\n\nThere were no protrusions or stable boulders where Stave and the sailors could have secured their ropes. Instead Keenreef, Hurl, and two comrades anchored the lines by sitting in the river and bracing their feet in cracks and potholes. By plain strength, they supported the Giants ascending through the waterfall.\n\nEarlier Stave must have done the same\u2014\n\nGrueburn and Kindwind led Bluntfist and Furledsail upward. The river fumed against their knees, boiled to carry them away. But they were Giants: they kept their feet. All of the Swordmainnir were in the tunnel. More sailors swarmed up the ropes. By Linden's count, only Scatterwit, Squallish Blustergale, and Branl remained in immediate danger.\n\nThe river here was as corrupt as it had been around the lake. It reeked of the bane's exudations.\n\nAs Grueburn joined the Ironhand, Covenant gave Linden a look like a glare of fever. By its very nature, wild magic resisted restraint. It became more dangerous with repeated use. But Linden could not help him. There was too much Earthpower in the air. The chute constricted it. Reminders of She Who Must Not Be Named assailed her. Her wedding band no longer answered his.\n\n\"It's getting harder,\" Jeremiah groaned. He kept his eyes squeezed shut. \"The Worm\u2014I can see _Melenkurion_ Skyweir.\"\n\nGrueburn and Kindwind stood in the river shoulder to shoulder. Aching to relieve Jeremiah and Covenant\u2014to relieve herself\u2014Linden put her hand on the Staff again, added her determination to her son's.\n\n\"How far?\" she asked him. \"How far away is it?\"\n\n\"I don't know.\" Jeremiah was near his limits. \"Close enough.\" Then he added, \"But the Worm is in a river. It isn't moving as fast.\"\n\nLinden closed her eyes as well; listened to the tumid clamor of water. The Worm must have passed the boundary of the Last Hills. It was crossing the wilderland which had once been Garroting Deep. And along the way, it was appeasing its hunger by drinking from the Black River, which took its name from its burden of diluted EarthBlood.\n\nYet the Worm had traversed most of the Land with appalling speed. How much time remained before it forced its way into the depths of the Skyweir? A day? Less?\n\nA moment later, Covenant's wild magic faded. When Linden opened her eyes, she saw silver streaming from the _krill_ in Branl's grasp. It shone on the water frothing down the contorted length of the channel. At the same time, she heard shouts.\n\nHurl called, \"They are safe!\" And the Anchormaster crowed, \"Stone and Sea! We are Giants in all sooth! And the _Haruchai_ are Giants also, in their fashion. We live!\"\n\n\"The lake rises still,\" continued Hurl. \"Indeed, it swells more swiftly. Yet Scatterwit has suffered no further harm. And Blustergale has lost no more than two toes and a portion of a third. Had we been but a heartbeat sooner\u2014\"\n\nBlustergale interrupted him, roaring in feigned indignation. \"There is no pain! None, I say! Is this not an affront to fire the coldest heart? Am I not a Giant, as mortal as any, and as worthy of my hurts? Does the lurker's god think so little of me, or of Baf Scatterwit, or of all here, that it does not deign to cause _pain_?\"\n\nWhile Scatterwit chuckled, Bluff Stoutgirth commanded, \"Enough, Blustergale. Some among us deem toes needful. Demonstrate that you are able to ascend here, and I will suffer your umbrage. Should you slip or falter, however, I will regard you justly chastened.\"\n\n\"The lake rises still,\" Hurl repeated more urgently. \"Badinage and bravado will not slow it.\"\n\n\"Aye,\" the Anchormaster replied, \"and aye again.\" He had recovered his good humor. \"As you have seen fit to chide us, you will remain to mark the water's advance.\" Then he urged his sailors into motion.\n\nLed by Onyx Stonemage, the others thrashed ahead.\n\nBranl approached Coldspray, Grueburn, and Kindwind; Covenant, Linden, and Jeremiah. He held Loric's dagger so that its radiance did not shine into his eyes. Shadows obscured his mien as he announced, \"The lake did not heed the _krill_.\"\n\n\"The Feroce were right,\" Covenant grumbled. \"The lurker's god is crazy. That knife can cut anything.\" He peered into the darkness of the chute. \"Now I'm worried. We don't know what's up there. We're going to need those creatures.\"\n\n\"They've come this far,\" Linden sighed. In spite of their fears\u2014\"If they don't rejoin us, it's because they can't.\"\n\n\"I know.\" Tension throbbed in Covenant's voice. His arms were getting tired. He would not be able to cling to the Ironhand's back indefinitely. But Coldspray would need her hands to help her defy the weight of the river. Grueburn and Kindwind would need their hands. \"I just want to bitch for a while.\"\n\n\"Timewarden,\" Rime Coldspray replied like a reprimand, \"your tales are foreshortened beyond sufferance. They are ended ere I am able to hear joy in them. And you employ words strangely. 'Bitch,' forsooth. I will deem it a courtesy if you will refrain until we are better able to heed you.\"\n\nCovenant gave Linden a twisted smile, rolled his eyes. \"Have it your way. I'll do my complaining when we find the damn Despiser.\"\n\n\"And another,\" sighed the Ironhand. \"Is there no limit to your brevity?\"\n\nLinden wished that Covenant could laugh. She wanted to laugh herself. But she did not have it in her. The spray promised carrion. It implied horror. Even in the constriction of the flume, the sensations were oblique. Nevertheless they were getting stronger.\n\now the company did not tarry. A shout from Hurl announced that the lake was nearing the rim of the waterfall. Heaving against the pressure of the river, the Swordmainnir and the sailors fought their way upward. One of the Anchormaster's crew had tied a rope around Baf Scatterwit's waist. Giants ahead of her held the line. And Squallish Blustergale stayed with her, taking some of her weight. Together they struggled along behind their comrades.\n\nLinden's arms ached. Cramps threatened her thighs. Nevertheless riding Grueburn's back was easier than it might have been. All of the Giants moved hunching over, ready to catch themselves if they slipped on slick rocks or secreted moss. Grueburn's posture helped Linden to hang on.\n\nThe passage should have been impossible for the _Haruchai_. Water that reached the Giants' knees struck Stave and Branl above their waists. Nevertheless the two men forged ahead as if they were incapable of faltering. The _krill_ in Branl's grasp did not waver. He and Stave carried their cumbersome swords like men who had spent decades training with such weapons.\n\nBefore long, Hurl called to inform the company that the lake had reached the bottom of the chute.\n\nMuttering elaborate Giantish curses, the Swordmainnir and the sailors continued an ascent that seemed to have no end.\n\nEventually, however, Rime Coldspray came to a widening. There across the centuries the river had eaten deposits of sandstone and shale out of the walls. It had dug a pit in the underlying basalt. The result was a space in which all of the Giants could gather\u2014and a pool deep enough to swallow the _Haruchai_. Fortunately a few boulders clung to the sides. Here and there, stubborn granite ledges protruded from the walls.\n\nColdspray looked a question at Bluff Stoutgirth. When he nodded, she ordered a rest.\n\nGratefully Frostheart Grueburn sloughed Linden onto a boulder. Cirrus Kindwind put Jeremiah down beside Linden, stood straighter to ease the tension in her back. As Coldspray settled Covenant nearby, the Anchormaster arranged Stonemage, Bluntfist, and his crew, some standing to their chests in the pool, others leaning on boulders or propped against the walls. Then he asked for food and clean water.\n\nSailors unpacked chunks of cured beef and mutton, rinds of cheese, bread with the texture of hardpan, dried fruits, waterskins. As they did so, Linden accepted the Staff of Law from Jeremiah and assumed the whole task of purifying the air so that he could rest and eat. He had not questioned her assistance earlier: he was not loath to trust her now. Apparently he was learning to believe that she would not recant her gift.\n\nWhile she had the opportunity, she extended other forms of refreshment to the Giants; eased the trembling of Covenant's muscles; nourished Jeremiah's strength. As if to himself, the boy murmured, \"That's a neat trick. I want to learn it.\" But he did not reach for the Staff. Images of the Worm seemed to glide like ravens across the depths of his gaze.\n\nSome of the waterskins held diluted _diamondraught_. When Linden had swallowed enough to wash the taste of pollution out of her mouth and throat, she joined the Giants eating.\n\nShe had not heard Hurl's voice for a while. Surely he was able to stay above the lake? But if the Anchormaster felt any anxiety on Hurl's behalf, he concealed it with jests.\n\n\"Thomas?\" Linden asked. \"What do you think? How high can that monster lift so much water?\"\n\nHe opened his mouth to answer; closed it again. After a moment, he said, \"By damn.\" Surprise and relief. With the index finger of his halfhand, he pointed down the chute.\n\nIn the distance below the pool, unsteady emerald reflected wetly on the walls.\n\nThe fires of the Feroce were still some distance away, but they were coming closer. And before long, Linden made out Hurl's bulk looming behind them. In the green glow, he looked somehow ghoulish, like an avatar of the Illearth Stone. His grin resembled the grimace of a fiend. Nevertheless he was unharmed.\n\nThe condition of the Feroce was more difficult to gauge. Linden had never been able to sense the nature of their magicks. From her perspective, they seemed smaller, weaker, as if they had been reduced by their immersion in the lake. And when they finally waded into the pool, she saw that they had indeed shrunk. Although they floated effortlessly with their arms and flames above water, they appeared to have drawn into themselves as if their encounter with their High God's god had shamed them.\n\n\"They rose with the lake,\" Hurl proclaimed in a tone of wonder. \"I had surrendered all hope of them. Yet when the lake began to hint that it might recede, the Feroce emerged.\"\n\nThe creatures faced Covenant; but now they did not flinch or cower. Nor did they ask his pardon for their absence. \"We are merely the Feroce,\" they stated. \"We serve our High God. We do not question our worship. Commanded, we obey.\" The strangeness of their shared voice seemed to accentuate the corruption of the atmosphere, the taint of the river, the slick sheen of the walls.\n\n\"But we have beheld our High God's god. He is lessened. Perhaps he is lessened.\" They regarded only Covenant. Even their flames appeared to focus on him. \"Perhaps the Pure One is also lessened.\" Their emerald shone in his eyes. It gleamed like spray on his scarred forehead. \"You must hasten again. We do not question. Commanded, we obey. Yet doubt infects. It spreads. An end draws near. We fear it. It gladdens us.\n\n\"You must hasten.\"\n\n\"Or what?\" Covenant asked carefully.\n\nThe Feroce were no longer afraid\u2014or their fear had become a different form of apprehension. \"We are naught,\" they answered. \"Worship is all things. Or it also is naught.\"\n\n\"Mom?\" Jeremiah breathed. \"What's going on?\"\n\nLinden touched his shoulder to quiet him. She tightened her grip on the Staff.\n\n\"Then forget your High God,\" Covenant said almost calmly; almost mildly. \"Forget our alliance. Forget that Clyme died for it, and the Worm is going to destroy every god you can imagine.\" He did not raise his voice, but his tone became thicker, harder. \"Remember that the _jheherrin_ saved the Pure One. They were weaker than you are, and maybe more scared, but they helped him anyway. Then he set them free.\n\n\"Try remembering _that_. If doubt infects, so does courage.\"\n\nLinden held her breath. If the Feroce turned back now\u2014\n\nFor a long moment, they were silent. They did not move. Their large eyes remained fixed on Covenant. Nevertheless they conveyed the impression that they were conferring with each other.\n\nCovenant faced them steadily, waiting.\n\nFinally they sighed like slumping mud. \"We are the Feroce. We are ignorant of courage. We obey because we must.\"\n\nThey did not urge haste again. Instead they drifted away from Covenant, gathered in the center of the pool. There they faced each other, holding out their fires like questions for which they had no answers.\n\n\"Thomas?\" Linden asked.\n\nHe frowned at her, or at his own thoughts. \"I know. Not exactly reassuring.\" Then he grimaced. \"So what else is new?\n\n\"We should go,\" he told Rime Coldspray. \"We're running out of time.\"\n\nYet doubt infects.\n\nIt was contagious.\n\nNodding, the Ironhand addressed Bluff Stoutgirth. \"Anchormaster?\"\n\n\"Aye.\" Stoutgirth grinned. To his crew, he said as if he were jesting, \"Come, sluggards. Have done with feasting and sloth. While we dally, the world's doom grows fretful. Soon it may set its sails and depart unopposed.\"\n\nHis crew responded with snorts or groans, or with ripostes; yet they immediately began packing away their provisions. Soon they were ready.\n\nLinden hesitated, unsure of her son. But Jeremiah asked for the Staff without prompting. \"I feel better now,\" he assured her. \"I want to practice.\" He faced her squarely, held her gaze. \"But maybe you shouldn't help me anymore. You make it too easy. I don't have to push myself when you're doing half the work.\"\n\nShe winced. He was right, of course. He had to make himself stronger; had to earn his inheritance. But she already knew that she was going to abandon him again. She was even going to abandon Covenant. And when she did, she would leave without any hope that she might ever return.\n\nHer hands shook as she passed the Staff to her son. Unclosing her fingers required an act of will.\n\nHis attention shifted at once to the wood; but she continued to gaze at him, clinging. Carefully she said, \"I'm proud of you. Do you know that?\"\n\n\"Sure, Mom.\" His tone made it clear that he was not listening.\n\nThe theurgy which he summoned from his hands and his violated heart was as black as anything that she had ever done.\n\ned by the Feroce, the company struggled upward. Emerald oozed like infection down the river. The light of the _krill_ seemed to lurch from place to place as it struck irregular facets of stone. The channel felt interminable. Its twists and bends through Mount Thunder's gutrock blocked Linden's view ahead. She could not guess how far the company would have to climb.\n\nFortunately Jeremiah's use of Earthpower and Law was improving. The Giants were able to breathe more easily. And the hints of She Who Must Not Be Named which Linden had felt earlier were lessened by midnight fire in the confines of the flume.\n\nBlustergale continued to support Scatterwit. A few of her comrades took turns holding the rope tied around her. Like them, she labored ahead, striving toward an untenable future.\n\nSo suddenly that Linden only had time to flinch and grip, Grueburn slipped: she started to plunge. But Stave stopped her by anchoring her foot. She caught herself on her hands, regained her balance. Muttering rueful apologies, she bore Linden onward.\n\nOther Giants slipped as well. As their weariness grew, they lost their footing more often. Most of them recovered quickly, or were secured by their comrades. But one of the sailors fell hard enough to take Keenreef with him. Threshing their arms, they were swept downward. However, Wiver Setrock dropped to his knees below their rush, spread his arms, snagged his comrades before they collided with Grueburn and Kindwind. With another sailor and Onyx Stonemage at his back, Setrock helped the Giants find their feet.\n\nAnxiety and jests echoed down the chute. Coldspray and Stoutgirth shouted unnecessary warnings. Jeremiah looked around wildly for a moment: the only sign that he had noticed what was happening. Then he returned his attention to the Staff.\n\nDarkness. Green glaring dully. Flashes of argent. Loud water acrid with minerals and pollution. Treacherous rocks and mosses. More darkness. Covenant clung like a penitent to Coldspray's back. Jeremiah half knelt behind Kindwind, gripping the Staff across her cataphract. Linden listened to the effort of Grueburn's breathing, felt the strain in Grueburn's muscles, and could do nothing.\n\nShe had given up looking ahead when she heard the Anchormaster call, \"And not before time! Doubtless all things must have an end. After such an ascent, however, I would lief have gained a less ambiguous summit.\"\n\nLinden jerked up her head; saw that the fires of the Feroce no longer reflected on the walls. The _krill_ 's illumination seemed to imply an open space ahead. She tried to extend her discernment upward, but she could not. Her senses were blocked by Giants and fouled water, Earthpower and exertion. Even Loric's gem had the effect of obstructing her percipience.\n\nNone of the Giants spoke as they hurried to reach a place where they might be able to rest again.\n\nLike the Feroce, Coldspray and Stoutgirth had moved out of sight. Holding light for the sailors and the Swordmainnir, Branl stood at the edge of the channel-mouth. Now Linden was able to see that the river ran from another large cavity in the gutrock; but the scale of the space was still hidden from her.\n\nAs Grueburn labored upward, however, Linden heard more complex tones in the water's turbulence, new pitches and timbres. Another waterfall? No. The sounds lacked that deeper resonance. After a moment, she realized that she was listening to more than one torrent. From beyond the immediate rush and spray came the turmoil of other streams, two distinct sources of water, neither splashing from any considerable height.\n\nHalf a dozen sailors reached the Humbled. They passed him leftward, clambered out of sight. As Stonemage and Setrock gained the opening, they led Keenreef and more of Dire's Vessel's crew to the right. Together Grueburn and Kindwind carried Linden and Jeremiah to smooth stone at the rim of the tunnel.\n\nLinden peered out at a large cave like a bubble in Mount Thunder's igneous substance. By the measure of the lower cavern, its dimensions were modest. She could have hit the ceiling with a rock, or skipped a pebble halfway across the dark water in front of her: a diminished lake now little more than a pond marked by rancid strands and stains higher on the walls. At the water's former height, the rocks piled around the cave's bottom would have been covered, useless to the company. At the pond's present level, she could have scrambled anywhere in the cave.\n\nNevertheless the air was viscid, thick with omens. The hurtful tang of She Who Must Not Be Named was stronger here. Suggestions of ire and ruin felt like insects on Linden's skin, tangible and feeding. Without Jeremiah and Earthpower, she might have whimpered aloud.\n\nBut then Grueburn carried her aside, out of the way of the Giants behind them; and Linden noticed the water's inlets.\n\nThere were indeed two, one diagonally across from her on the left, the other opposite her and somewhat to the right. The stream on the left tumbled from a fissure in the wall, a crack barely wide enough to admit a Giant. The water frothing there conveyed the impression that it cascaded from somewhere far above the cave. In the _krill_ 's light, its spray shone silver.\n\nThe other stream boiled out of an opening beneath the lake's surface. Apparently it came from the base of a subtle flaw in the stone, a seam where distinct forms of rock had been reluctantly fused. Under the pressure of its own weight, water seethed into the pond.\n\nOnly the fissure on the left offered the company an egress. An ascent there would be difficult. If the crack narrowed, it might become impassable. But the water there was fresh.\n\nGod, it was _fresh_ \u2014It came from a clean spring, or from several. And the fissure was accessible. The company could reach it without enduring an immersion in the pond; without subjecting Linden to more of the bane's touch.\n\nThe Cavewights were entirely unlike the Feroce. Surely they required sources of clean water? Surely a source this abundant would lead toward the Wightwarrens?\n\nHer heart seemed to beat in her throat as she turned toward Covenant.\n\nRime Coldspray had set him down on the far side of the cave's outlet. Stave and all of the Giants had now emerged from the tunnel, and Cirrus Kindwind had already lowered Jeremiah to the rocks. He leaned on the Staff, resting, but he did not relax his efforts. Stark strands of power fluttered around the company, softening the atmosphere.\n\nHe was the only one not looking at Covenant. Stave, Branl, and the Giants watched the Unbeliever, the Timewarden, waiting for him to make a decision. As if there could possibly be any doubt\u2014\n\nBut he ignored them. Instead he faced the Feroce.\n\nWreathed in green, they clustered a few paces beyond him. Some of them stood with their feet in the pond, but they did not sink away. Again they appeared to commune with each other. Their fires danced like language in their hands.\n\nLinden winced at the sight. They were definitely smaller. Shrinking. Losing faith.\n\nCovenant's impatience showed in the clench of his shoulders, the rigidity of his back. He seemed to want some form of confirmation from the creatures, even though the company's path was obvious. After a tense pause, he demanded, \"Now what? We can't just hang around here. We don't have time.\"\n\nThe Feroce did not look at him. Their voice quavered as if they expected to be struck down.\n\n\"You will be wroth with us. You will not heed.\"\n\n\"What?\" Covenant's surprise echoed faintly around the walls. \"I'm going to get angry because you're trying to help? Why?\"\n\nTwo of the creatures pointed at the fissure. \"You must not enter there. It misleads.\" Two others indicated the rank moil of the second inlet. \"You must follow richer water.\" Then they crowded closer together. \"Now we perish. You will not suffer us.\"\n\n\"Thomas!\" Linden protested. She gestured urgently toward the crack. \"That water is _fresh_.\" It did not stink of threats.\n\nGiants nodded their assent.\n\n\"Oh, stop,\" Covenant growled at the Feroce. \"I'm not going to do anything to you. None of us are.\" He squinted over his shoulder at Linden, then addressed the creatures again. \"But we need an explanation. 'Richer' water? I assume you mean water with more crud in it. That doesn't make sense. Never mind that it's likely to poison us. Suppose you're right. Suppose it does run closer to the Wightwarrens. Even Giants can't swim against that kind of pressure. And we sure as hell can't hold our breath long enough to find air.\"\n\n\"Wait a minute,\" Jeremiah murmured. Ebony tendrils curled across the pond. They searched along the far wall. But he did not say more; and the alarm clamoring in Linden's ears prevented her from heeding him.\n\n\"So tell me,\" Covenant continued. \"Why _that_ water? Why is a trail we can't even follow better than one we can?\"\n\nThe creatures flinched. Their fires guttered. \"We are the Feroce. We obey, as we must. We cannot answer ire.\"\n\nCovenant swore softly.\n\nQuivering, the damp voice said, \"We do not know your goal. We do not know the mountain. But we taste the memories mingled here. Those waters do not hold the Maker's scorn. Other powers enrich them, yes. They urge false worship, abhorrent to us, seductive.\" The Feroce shuddered. \"Yet we are certain. Memory is certain.\"\n\n\"Wait a minute,\" Jeremiah said again.\n\nEveryone ignored him.\n\n\"The stream of mere water. The plain path. It misleads. It does not recall light. No light has shone upon it. No sun. No flame. No magicks.\"\n\nNo light\u2014?\n\n\"Stone and Sea!\" rasped the Ironhand. Other Giants muttered their chagrin.\n\nThe _Haruchai_ watched and listened as if they were drawing different conclusions.\n\n\"The richer waters,\" said the Feroce, \"remember much. They recall darkness and horrible strength. Strange theurgies. Time without measure. And light. Light! In a distant age, they have known the sun. They have not forgotten.\n\n\"The memory is there.\"\n\nAs one, the creatures pointed at the turbulence spewing from beneath the surface of the pond.\n\nOh, God. Floundering, Linden thought, Light\u2014The cascade of fresh water had never seen torches. It had never felt the ruddy glow of rocklight. Therefore its long plunge did not intersect the catacombs. Even Cavewights needed illumination.\n\nBut the other stream\u2014Ah, hell. That impassable gush came from the Soulsease. It had once traveled the Upper Land. It had known the warmth of the sun. And far to the west, the Soulsease entered the Wightwarrens. But only a few days ago, it had lost its way through the mountain. Now it plunged toward the Lost Deep. For that reason, it was fraught with the anguish and rage of She Who Must Not Be Named.\n\n\"Thomas.\" Linden's voice had fallen to a whisper. She was too frightened to raise it. \"I can't. There's no way\u2014\"\n\nShe could not submerge herself in water that reminded her of the bane. She would go mad.\n\nIn a taut rumble, the Anchormaster remarked, \"We know little of these Feroce. We of Dire's Vessel have heard your tale, but we have not lived it. Is it conceivable that they have turned aside from our purpose? If they are no longer ruled by their monstrous deity's will, it may be they who mislead, rather than the waters.\"\n\nCovenant shook his head. \"I doubt it. They had a perfect chance to abandon us back at that lake, but they didn't. The lurker still wants to live. They still want to live. We're the only hope they've got.\"\n\nLike him, Linden believed the Feroce. Nevertheless the company could not go where they suggested.\n\nShe should have been grateful. Instead she wanted to scream.\n\n\"I said, _wait_ a minute!\" Jeremiah demanded more strongly. \"You aren't paying attention.\"\n\nFierce as a blow, Covenant wheeled away from the Feroce. Bracing his fists on his hips, he glared past the spread of Giants and the mouth of the downward chute. \"Hellfire, Jeremiah! Paying attention to _what_?\"\n\nJeremiah faced Linden rather than Covenant. \"Look, Mom.\" Black fire played across the spout of fouled water, skirled up the seam of the wall. \" _Look_.\"\n\nLinden stared at him, thinking, Don't push me. I can't.\n\nBut he was her son. She could not refuse him. Trembling privately at the prospect of maggots, spiders, worms, she asked Frostheart Grueburn to put her down. When she stood beside Jeremiah, close enough to borrow some of the Staff's Earthpower, she turned her senses toward the fused stone. Alarm hampered her, but she forced it aside. Unsteadily she directed her percipience into the water; into the wall.\n\nThere.\n\nInstinctively she recoiled; closed her throat against a moan.\n\nThe rock along the seam was thin. It looked thin enough to break. And beyond it\u2014\n\nShe bit her lip until she drew blood.\n\n\u2014stretched a different fissure, a wedge with its tip at the seam. It was narrow, but it widened into the distance until it passed beyond her discernment. And it was full of water.\n\nNo, she realized a heartbeat later, not full. Everywhere under Mount Thunder, the Soulsease had shrunk to a fraction of its former flow. Before that, it had been a mighty torrent. That hidden fissure had indeed been full. And the cave itself had been full as well: a fact which probably explained why the weight of water had not broken through the seam ages ago. The cave had served to equalize the pressure. But now\u2014\n\nAh, now the level behind the wall had dropped. The fissure had emptied itself until the water stood, waiting to drain, little more than the height of a Giant above the pond. If the rock broke, the issuing flood would be fierce. Still the Giants might be able to withstand it. Stave and Branl might. When the river found a new level, a new equilibrium, the company might be able to ascend against it.\n\n_Writ in water_. God help me.\n\nLinden was not ready. She would never be ready.\n\n\"Linden?\" Covenant called in frustration. \"Jeremiah? What is it? Damn it, I can't _see_.\"\n\nIn a voice so small that she hardly heard it herself, Linden answered, \"That wall is thin. There's a crevice behind it. I can't tell how high the crack is, or how far it goes. But if we break the wall\u2014\"\n\nShe did not have the courage to say more.\n\n\"That's it,\" Jeremiah confirmed more loudly. \"That's the way. We can go there.\"\n\nThe Swordmainnir peered across the cave in wonder. At a nod from the Anchormaster, Hurl and Wiver Setrock began to work along the wall to the right. \"We are wise in the lore of stone,\" Stoutgirth explained unnecessarily. \"We will ascertain whether our strength may suffice to open the way.\"\n\nWhile he waited for Hurl and Setrock, he sent four of his crew leftward to refill their waterskins from the clean stream.\n\n\"The stone,\" intoned the Feroce, \"remembers endurance. It will not surrender to fists or blades.\"\n\n\"That's not the problem,\" Covenant muttered over the clamor of waters. With Coldspray's help, he crossed the slick outlet, then scrambled toward Linden and Jeremiah. \"The problem is control. Too much is easy. Just enough is hard.\"\n\nLinden turned to him as if she were falling. When he reached her, she put her arms around his neck, leaned against him.\n\n\"Oh, Thomas,\" she whispered to him alone. \"I can't do this. I can feel She Who Must Not Be Named.\"\n\n\"What, _here_?\" he breathed. Alarm tightened his grip. \"Is She close?\"\n\nLinden shook her head. \"I didn't mean that. I don't know where She is. But I can feel Her power. It's leaking into the water. Just smelling it is bad enough. If I touch it\u2014\"\n\nHorror crouched in the pit of her stomach, in all of her nerves. The bane was death to her. It hungered for her soul.\n\nJust for a moment, Covenant held her like a man who wanted to howl. Then he mastered himself. \"I understand,\" he said stiffly. To retrieve her from carrion, he had been forced to hold her underwater; threaten her with drowning. \"I'll spare you if I can. If the Giants aren't strong enough, I'll try\u2014\"\n\nShe felt him grimace as if he had bared his teeth. \"Hellfire, Linden. I might bring down the roof.\"\n\nShe knew what he meant. He had too little health-sense\u2014and he feared himself too much. His passions were too extreme for restraint.\n\nHurl and Setrock reached the seam. While Hurl pressed an ear to the stone, listening, Setrock thumped the flaw with the heel of one palm, gently at first, then harder. Harder. Then Hurl stepped back. To the Anchormaster, he called, \"Water lies beyond this stone. Linden Giantfriend and the Chosen-son gauge acutely.\" His voice carried like streaks of argent across the surface of the pond. \"Yet the Feroce do also. The stone has suffered much across the ages\u2014aye, and absorbed much to harden it. It will not yield to blows or iron.\"\n\nBranl raised Longwrath's sword as if it were a question.\n\nHurl shook his head. \"The theurgies of that blade are obscure. I cannot conceive that they will suffice here.\"\n\n\"The _krill_?\" asked Branl.\n\nThe idea wrung a flinch from Linden. \"No,\" she told Covenant. \"Not the _krill_. It might cut deep enough. But whoever holds it will be too close.\" Tons of water and rock would crash outward\u2014She had no choice. \"I have to do it.\"\n\nHe pulled back his head, peered into her eyes. \"Are you sure?\"\n\nShe could not hold his gaze. Leaning her forehead against his chest, she sighed, \"I can try. I need to start using my ring. It might even help.\"\n\nWhen he had invoked wild magic in the lower cavern, his wedding band had summoned a response from hers: a silver throb which had muted the sensations of pincers and scurrying. Perhaps her own power would shield her from the bane's tormented, tormenting seepage.\n\nShe felt Covenant gather his resolve. Briefly he tightened his hug again. Then he wheeled away.\n\nShouting so that everyone would hear him, he demanded, \"Listen! Linden is going to break through that wall for us, and she's going to have to do it from here.\" Opposite the seam. \"But when she does, she'll release a hell of a lot of water. I don't want it to touch her! I don't care how you do it. _Think_ of something. Just keep her out of the water!\"\n\nJeremiah gaped at him. In surprise, the boy lost his grasp on Earthpower. The air failed in Linden's lungs. She started to gag. But then Jeremiah recovered. Renewed flames spread outward.\n\n\"Mom?\" he asked anxiously. \"Mom?\"\n\nPanting, she urged him, \"Don't worry about me. Your job is hard enough. If I can do this, there's going to be a flood\u2014but we'll still need air. The Giants will take care of me. I'll be fine as long as we can breathe.\"\n\nThe Ironhand and the Anchormaster exchanged a few quick words. Then Rime Coldspray announced, \"Timewarden, it will be done. The water here is vile. It will become more so. Yet we are hardy against such affronts. With your consent, I will entrust Linden Giantfriend to Bluff Stoutgirth and those in his command. They are not hampered by armor and swords. Frostheart Grueburn will stand ready to receive the Giantfriend when our passage inward has been opened.\"\n\nCovenant did not object. Linden could not.\n\nAt once, Stoutgirth called a few orders of his own. Almost immediately, everyone except Linden was in motion.\n\nThe sailors refilling the waterskins helped each other past the fissure, moving directly toward the seam. Blustergale supported Baf Scatterwit in spite of her insistence that she did not need aid. Flashing a smile of encouragement at Linden, Grueburn strode away, followed by Cirrus Kindwind carrying Jeremiah. With a boost from Stave, Covenant climbed onto the Ironhand's back. A look of nausea filled his eyes as if he had been overtaken by vertigo.\n\nFor an instant as Branl passed her, Linden considered asking him to wait with her. She could use the _krill_ 's gem to trigger wild magic: she had done so once before. But the outcome then had appalled her. She remembered too well the charred remains of Cavewights, scores or hundreds of them. Her spirit still wore the stains of slaughter. She nodded to the Humbled, but did not speak.\n\nStave came to her side. He gave her a grave bow, regarded her with his single eye. \"In such straits, Chosen,\" he remarked, \"it may be that Giants are better able to ward you than one _Haruchai_. Nonetheless I will not be parted from you. I have accepted once an absence from your side. I will not do so again.\"\n\nOf the friends who had first joined Linden after her return to the Land, Stave was the last. The Ramen were gone. Liand and Anele were dead. And in some ways, Stave had endured more than any of them. She had no words for her gratitude.\n\nTrailing behind Stonemage, Bluntfist, and the last of the sailors, Etch Furledsail stopped with Linden and Stave. Even among Giants, she was tall: a graceful and comely woman no longer young, with grey scattered through her hair, a gleam in her eyes, and a weathered face. \"It may appear to you,\" she offered, \"that our intent for your protection entails needless hazard. I assure you that it does not. I dare not attempt true haste over the hazards of these rocks. Therefore we will bear you across the water.\n\n\"Fear nothing,\" she added. With a wave, she indicated Setrock and Hurl beside the seam. \"Where one Giant may fail, three will succeed. And we are adept in water. Here it is noxious in all sooth.\" She frowned at the pond. \"Still it will not harm us.\n\n\"Giantfriend, I ask only that you do not resist when the wall has been opened. To evade such torrents, we must move swiftly.\"\n\nLinden said nothing. She had stopped listening. Her gaze followed her companions as they gathered on both sides of the flaw where she meant to strike, but she was not watching. Her attention had turned inward. While Furledsail's voice passed over her, she searched for the door hidden within her, the specific intersection of intention and emotion and openness, of need and willed desperation, which gave her access to wild magic.\n\nFurledsail raised an eyebrow at Linden's silence. Stave replied with a slight shrug.\n\nAt first, Linden could not find her way. Too many things could go wrong. If she ruptured the wall, a tremendous amount of water would crash straight toward her. It would hit hard enough to make pulp of the Feroce, who still stood on the other side of the cave's outlet. Or the Giants poised beside the seam might be struck by shards, caught in the cascade, torn away. Jeremiah's concentration might falter again. Then the bane's insidious fetors might overwhelm Linden. And she could not be sure that the company would be able to force a passage along the crevice behind the wall. If that crack held more water than the cave could release\u2014\n\nFurledsail intended to carry her into the pond; into memories of horror and anguish\u2014\n\nBut then Covenant called her name. Jeremiah shouted, \"Mom!\"\n\nSteady as gutrock, Stave said, \"You are Linden Avery the Chosen, named in honor Ringthane, Giantfriend, and Wildwielder. Much is asked of you, but much has also been given. The time for doubt has passed. Only deeds or death remain. On other occasions, you have dared Desecration. You need not fear it now.\"\n\nAnchored by the voices of people who were dear to her, Linden closed her mind to the clamor of too much trepidation, too many possible disasters. She was not alone. Her husband and her son loved her. Her friends had faith in her. She could trust them.\n\nBut they could not reach into her secret places for her. That she had to do for herself. And she knew how. She had done it before. She only had to retrace her mental steps.\n\nFollowing her health-sense inward, she found the intimate chamber of her power. It was masked on all sides by fears and sins, unforgiven, but it was a part of her nonetheless. She had a right to it.\n\nNow or never. How often had she said that to herself?\n\nWhen argent stark as lightning began to blaze from her ring, she did not hesitate. And she did not hold back. She was not Covenant, fraught with ungovernable potential. Causing _caesures_ had required precision, supreme delicacy: attacking granite and basalt demanded only _force_.\n\nShe delivered _force_ as if she had suddenly become mighty.\n\nAs the stone cracked along the seam, the whole cave seemed to shriek. Rubble and water burst from the wall like screams.\n\nBefore Linden could snatch another breath, everything became chaos.\n\nAn avalanche of water slammed into the pond, but she hardly saw it; hardly saw the Giants gripping each other so that they would not be swept away; hardly saw Jeremiah flail black fire in all directions, wild and useless. She had already been lifted into Furledsail's arms. At once certain and cautious, Furledsail moved into the pond. But she did not confront its upheaval directly. Instead she angled away to the right, past the immediate thrash and spray.\n\nAt the same instant, Hurl and Setrock dove. Hurl stretched out, long and shallow, crossing as much distance as he could. Setrock went deeper, shorter. As Furledsail sank to her waist, Hurl broke the surface beyond her.\n\nAbove the water, the lash and rebound of waves, Furledsail tossed her burden upright to Hurl.\n\nLinden caught a frantic glimpse of Stave swimming. Then Hurl's hands caught her.\n\nHe did not hold her. Treading water, he pitched her back over his head. A blind throw\u2014\n\nBlind and unerring. Deftly Wiver Setrock snagged her out of the air. In the same motion, he, too, flung her behind him.\n\nA heartbeat later, Linden lay in Grueburn's clasp at the water's edge. With wary haste, Grueburn retreated up the rocks to press her back against the wall among the other Giants. Her grin as she regarded Linden was feral with glee.\n\nLinden's mind had gone blank. She stared up at Grueburn as if she did not recognize the Swordmain.\n\nBut somewhere deep inside her, a voice was crowing.\n\nYou did it. You _did_ it.\n\nDid you call me your _daughter_? she shouted at Lord Foul. Watch and learn, you smug bastard!\n\nShe could cheer and threaten because the Despiser was not her greatest fear. He was Covenant's problem. She had chosen a different path to the World's End.\n\nWild magic was a necessary step.\n\n## 7.\n\nAt Last\n\nTumults crashed inward. They threatened to fill the cave, drown the entire company. The Feroce vanished in roaring waves. The air that came with the flood stank of minerals and trapped hate. It surpassed Jeremiah; surpassed the Staff of Law.\n\nBut the constrained volume of the river was less than it had been scant days ago, much less; and the cave's outlet swallowed the immediate brunt of the inrush. On either side, waves slammed like heavy seas against the walls, fell back onto each other. The pond became a boiling cauldron, a contained squall. Surges tore at the Giants' ankles, knees, thighs. Fluid blows hammered Stave and Branl. Yet moment by moment the flail and rebound of the waters ran down the mountain's throat.\n\nGradually the flood seemed to find its balance. Its force receded as the cleft drained. Turmoil slapped at the walls and the company, but did not claim them. Smaller waves sank to the level of knees and then ankles. Soon the water only splashed the feet of the Giants. Its thunderous howl faded.\n\nAt the same time, the air tumbling from the opened crevice lost some of its virulence. It had been blocked for ages or eons, and its contagions had congealed until they were thick as mire. But now it ran out like the river; and as it emptied the cleft, it drew air from some cleaner source. Gasping, Linden tasted hints of something that resembled life. When Jeremiah regained his grip on Earthpower, the whole company began to breathe more easily.\n\nHis efforts confirmed that he was unharmed.\n\nBut the Feroce were indeed gone. If they had survived the torrents, they had allowed themselves to be swept away: back to the cavern and the black lake, to the Defiles Course and the Sarangrave. Linden wanted to think that they were still alive. They had done what they could. Perhaps their High God would forgive their doubts.\n\nA shout from the Ironhand announced that the gap into the crevice had become passable. Branl carried the _krill_ closer to light the way as the Anchormaster and half a dozen of his crew dropped down into the water's former channel, then began scrambling upward. The river frothed against their legs, but they labored higher until they were out of sight.\n\nThrough the raw clamor of the current, Covenant told Branl to go ahead. With Coldspray's assent, the Humbled took Loric's dagger into the crevice. For a moment, the gem left slashes of argent on the pond's turmoil. Then the Ironhand followed, bearing Covenant with her, and her size blocked most of the light. The remaining streaks and gleams made the cave and its water look ghostly, transient, as if the whole place were dissolving; losing its place in the reality of time.\n\nHalewhole Bluntfist went next with Setrock and Furledsail. Cirrus Kindwind carried Jeremiah after them. Then it was Frostheart Grueburn's turn. As Linden scrambled onto Grueburn's back, she saw a rope trailing from the crevice: a lifeline. Onyx Stonemage gripped the end while someone\u2014Bluff Stoutgirth or one of his sailors\u2014pulled it taut. Muttering her approval, Grueburn held the line to steady her as she bore Linden into the crevice with Stave behind her. Squallish Blustergale supported Scatterwit. Stonemage brought up the rear.\n\nThe lifeline was necessary. Somewhere beneath Grueburn's feet, there was stone: there had to be. But long turbulent millennia had deposited thick layers of silt as cloying as quicksand. The water pounding against Grueburn's thighs was not the greatest obstacle to her ascent. The silt was worse. She sank to her calves and higher in muck that dragged at every step. While she hauled one foot out of the mire, her weight drove the other deeper. She needed the rope.\n\nFor that reason, any Giant above her who happened to find secure footing paused to anchor the line. The result was progress in arduous surges as sailors and Swordmainnir pulled themselves or each other from one patch of solid ground to another.\n\nHow the _Haruchai_ managed to ascend, Linden could not imagine. Glancing behind her, she sensed an uncharacteristic frown of vexation on Stave's visage. The strain in his muscles was as palpable as Grueburn's. At intervals, he clutched at the lifeline, obviously reluctant to require its aid.\n\nHow long could he continue? How long could the Giants? Linden had often been amazed by their endurance, but still\u2014The crevice was too narrow for the companions to assist each other side by side, and the silt was _deep_. Each new step seemed to demand more effort than the one before.\n\nA call from above warned the company that Stoutgirth had floundered into a pit where the mire seemed bottomless. His sailors dragged him back; but then everyone else was forced to wait while the Giants in the lead probed for a way past the pit.\n\nLinden felt a flutter of panic. The walls seemed to be leaning in. Surely the crevice was becoming narrower? The current boiling past Grueburn's legs carried glints of She Who Must Not Be Named like flakes of shed malice: lightless, invisible, yet distinct to Linden's nerves.\n\nIf Frostheart Grueburn lost her balance\u2014If Linden plunged into the water\u2014\n\nApparently Stoutgirth's fall and rescue had released gases trapped in the pit. Heavy as fog, sulfur and putrefaction rode the stream. They burned Linden's eyes, stung her nose, bit into her chest, until the tug of running water took them away.\n\nShe could hear Covenant swearing at his helplessness. Jeremiah jerked his head from side to side, flung black fire along the river. Spray stood like sweat on his skin.\n\nThen the Anchormaster reported success. The line began to lurch forward again.\n\nIn their turn, Kindwind and Grueburn reached the pit. Now Linden understood Stoutgirth's mistake. Her health-sense could not measure the varying depths of the silt. It was all so old, so laden with refuse and minerals, so full of the aftereffects of dire theurgies, that it refused percipience.\n\nHelped by Bluntfist and Furledsail, Cirrus Kindwind bore Jeremiah around the rim of the pit: a narrow path. Linden shifted until she hung from Grueburn's shoulder; dangled over the pit as Grueburn forced her way around it. Stave crossed by floating on his back and pulling himself along the rope. Grueburn and Kindwind waited while Blustergale ensured Scatterwit's safety. Then Blustergale sent Scatterwit ahead. He stayed behind to assist Onyx Stonemage.\n\nIn heaves and sags, the company struggled upward. Aching for Grueburn, and for Jeremiah, Linden concentrated on clinging to Grueburn's armor\u2014and on holding still so that she would not disturb Grueburn's balance.\n\nHere the air was definitely better. It became cleaner, demanded less from Jeremiah, as the river dragged its atmosphere with it. Hints of the bane persisted, but they were diminished.\n\nOn into darkness, interminably. The fissure became wider. It narrowed again. At intervals, indurated juts of stone interrupted the silt. For long stretches, the muck seemed deeper. The Giants fought for breath to feed their straining muscles, their accumulating exhaustion. Their gasps filled the crevice above the rush of water. Linden could not remember when they had last rested.\n\nThen the rope was drawn tighter. Grueburn gripped it with both hands. She began to move a bit more easily. Behind her, Scatterwit chortled, a sound as forlorn as a groan. The light of the _krill_ reached farther down the cleft. It touched Kindwind's head, flared like fire in Jeremiah's hair. The wall on the left had begun to lean away from the river. The darkness overhead felt more open.\n\nThe leading Giants must have found a place where they could stand; where they could gather on firm rock and brace their feet.\n\n\"Soon, Giantfriend,\" Grueburn panted. \"Soon.\"\n\n\"It better be.\" Jeremiah coughed the words. \"I can't hold on much longer.\"\n\nLinden watched the silver on the walls grow brighter as more and more of the company moved past the _krill_. In moments, she caught sight of Branl. Where he stood, the left wall appeared to fall away. But then she saw that the fissure simply became wider. Beyond a rough edge like a doorpost, that wall curved back, continuing the crevice. The river ran there, tumbling more slowly between sheer sides now farther apart. Past the turning, rough stone formed a floor like a platform above the water, vaguely level, and perhaps ten or fifteen paces across.\n\nBluff Stoutgirth and his immediate companions waited there, chests heaving. Coldspray had put Covenant on his feet. He stood squinting past the glare of Loric's gem, impatient for Jeremiah and Linden. With Setrock and Furledsail, Bluntfist had taken the rope. Together they hauled as if they hoped to raise their comrades from a crypt. Silt caked their legs, but they ignored that discomfort.\n\nOn the platform, some of the sailors began unpacking waterskins and bundles of food.\n\nEager to slip down from Grueburn's back\u2014eager to put her arms around Covenant\u2014Linden did not look around. Her legs stung as she dropped to the stone. Moving toward Covenant, she stumbled, had to catch herself. Then he was holding her tight. The urgency of his hug matched hers.\n\n\"Hellfire, Linden,\" he murmured near her ear. \"I thought that was never going to end.\"\n\nIt was not ended now. The companions had merely found a respite.\n\nFrom the downward fissure, Stonemage herded Blustergale and Scatterwit out of the river: the last of the Giants. As similar as brothers, Stave and Branl came toward Linden and Covenant.\n\nLinden felt Jeremiah quench the power of the Staff. Instinctively she flinched. But the atmosphere here was kinder to her lungs. Although it was thick with dust and disuse, stale, acrid, the river carried most of its wastes and poisons with it. She could breathe without choking.\n\nWhen she had held Covenant long enough to ease her heart, she turned to her son.\n\nJeremiah was sitting on the stone, hugging his knees against his chest in an effort to control the tremors in his limbs. He had dropped the Staff beside him. Dully he stared across the water, a gaze as expressionless as the far wall. Saliva collected on his drooping lower lip: a sight which Linden had not seen since he had emerged from his dissociation.\n\nShe knelt at his side, put her arm over his shoulders. \"Jeremiah, honey? Are you all right? It's no wonder you're tired. You've been keeping us all alive.\"\n\nHis eyes did not shift. He hardly seemed to blink or swallow. His voice was a low rasp, a scraping like the sound of a creature crawling on its belly.\n\n\"It isn't fair, Mom. It's not. I'm so tired. I can't go on. I can't. But I have to have Earthpower. Without it\u2014\" Abruptly he released his legs, slapped at his face as if his weariness revolted him. \"It protects me.\n\n\"You don't know what it's like. That mountain is _huge_. And the Worm is in the river. It's drinking every bit of Earthpower it can find, but it wants more. It wants it _all_.\"\n\nOh, Jeremiah\u2014\n\nUselessly Linden told her son, \"You'll get stronger. You're already stronger. We'll eat something, rest for a while. You'll feel better. Then we'll need you again. We'll have to go back into the river. You'll be able to protect yourself.\"\n\nLeaden with depletion or despair, his head turned toward her. \"What are you talking about?\" He peered at her as if he were going blind. \"The river? Why?\" With one hand, he pointed up the wall. \"That's the way. It has to be. The air's better there. You won't need me anymore.\"\n\nShe frowned, momentarily confused. Then she stood to look around.\n\nGiants cast grotesque shadows, shapes that appeared to caper across the walls. Between them, however, the _krill_ lit this section of the crevice clearly.\n\nUnder tremendous pressure long ago, layers of stone on this side had shifted. Diagonally beginning half a dozen paces beyond the Giants and angling erratically into the darkness overhead, ancient forces had pulled the higher reaches of the wall back from the lower. The result was a crude ledge or shelf: a natural formation that lurched upward, lying level in some places, jutting like a titan's stairs in others; obstructed here and there by piles of rubble. For short distances, it looked wide enough to accommodate horses. Other stretches were too narrow to let more than one Giant pass at a time.\n\nIt ascended beyond the _krill_ 's illumination, beyond the range of Linden's senses, climbing into the secrets of the crevice. She had no way of knowing where it led. But the air drifting down was unmistakably cleaner.\n\nSurely even stone-dwelling Cavewights required unfouled air?\n\nIn any case, the ledge went higher. It might go far enough to reach the catacombs.\n\n\"You're right,\" she murmured to Jeremiah. \"We have to go up.\" Then she added quickly, \"But that doesn't mean we don't need you. It just means that you can stop wearing yourself out for a while. Maybe you can learn other ways to use the Staff.\"\n\n\"Like what?\" he asked as if she had suggested something unimaginable. He had already failed to affect the hue of the wood. He could not undo its effect on him.\n\nInstead of giving him a direct answer, Linden said, \"You're here for a reason, honey. It's no accident.\" For his sake, she spun a web of inferences that made her tremble. \"Of course, you're here because Roger took you. He wanted a way to make me give him Thomas' ring. And Lord Foul wants revenge. He thinks that you can help him trap the Creator. He's trying to fill your head with despair so that you won't fight him.\n\n\"But it isn't that simple. Lord Foul isn't the only one who chooses who comes to the Land. He picks us because he thinks that he can manipulate us, or because he thinks that we're already his. But the Creator chooses us, too. They both picked us.\" Covenant had taught her this. Now she pushed it further. \"The only difference is, the Creator doesn't manipulate us. He lets us make our own decisions.\"\n\nIgnoring the rest of the company, Linden hurried to make her point before her courage failed.\n\n\"The Creator sees hope in you, honey. He sees things that you might choose for yourself, things that might make a difference. That's why\u2014\" Oh, God. Did she have to say this? Did she have to face it? \"That's why he didn't warn me before Roger got to you. If he had given me any hint that you might be in danger, I would have stopped Roger somehow. I would have taken you away so that he couldn't find you.\"\n\nShe had almost done so when she had seen images of Revelstone and Mount Thunder in her living room.\n\n\"The Creator didn't warn me because he needs you.\"\n\nHer claim seemed to strike a spark into the tinder of Jeremiah's aggrieved spirit. Unsteadily he stood to face her. The murk of his gaze clung to her.\n\n\"Needs me _how_? What am I supposed to do?\"\n\nFor that, Linden had no answer.\n\n\"What you've always done,\" Covenant put in roughly. He had come to stand behind Linden. She felt the tension in his muscles, heard the clench in his voice. \"Something damn Foul doesn't expect.\"\n\nJeremiah's head snapped toward Covenant. His mouth hung open.\n\n\"Maybe,\" Covenant went on, \"you think he marked you. Maybe you think being a halfhand means he has some kind of claim on you, some kind of special power over you. But that's backward. _He_ didn't cut off those two fingers. Your _mother_ did. And she did it so she could save the rest of your hand. Being a halfhand doesn't make you a victim. It makes you free.\n\n\"The Despiser doesn't know you as well as he thinks he does. He can't. Filling your head with visions is just a trick to keep you off-balance. He doesn't want you to see the truth. You're only his if you choose him.\"\n\nJeremiah gaped at Covenant. Linden watched turmoil seethe like Lifeswallower's mire in her son's eyes. The whole company seemed to pause while he struggled to understand: even the river seemed to hold its breath. The _krill_ cast light and shadows in all directions.\n\nAs if he were choking, Jeremiah protested, \"But what I see is _real_. The Worm is _real_.\"\n\nHe may have meant, We're all going to die.\n\n\"Well, sure.\" Covenant's tone conveyed a shrug. The Despiser did not lie. \"But that's not the point. The Worm isn't more real than _you_ are. It's just more dramatic.\"\n\n\"I don't get it,\" Jeremiah groaned. \"I can't\u2014Lord Foul is too strong.\"\n\nHis confusion and need twisted Linden's heart; but Covenant did not relent. \"Then let him be too strong. You don't need to beat him. Just do _something_ he doesn't expect. Be yourself.\"\n\nA young man with the Staff of Law and his own Earthpower: a young man with a talent for _making_. Even the Despiser in his fury and frustration could not satisfy all of his desires without the ability to create. Linden understood what Covenant was saying. She knew why Lord Foul needed her son.\n\nBut she could see as clearly as if she had entered him with her health-sense that Jeremiah did not understand. He was too young to know how much he did not know about himself. When he ducked his head to mutter as if he were ashamed, \"Maybe Roger had the right idea. Maybe we should all try to become gods,\" she seemed to hear the _croyel_ in him: the legacy of being possessed.\n\nYet she did not hear scorn. Bitterness, yes. Fear. Self-pity. But not contempt. He had other birthrights as well.\n\nSurely she could try to believe that they would come to his aid when he needed them? Surely she should trust him, no matter how much his distress hurt her, or how much she feared for him? She would not be there for him when his plight came to its crisis. Trusting him now might be the last gift that she would ever be able to give him.\n\nhen the companions had eaten another meal, shared their waterskins, and refreshed themselves as much as they could on the better air drifting into the crevice, they started upward. Once again, the Ironhand and the Anchormaster took the lead; but this time Covenant walked behind them with Branl and Halewhole Bluntfist. After Hurl, Keenreef, and several other sailors, Linden and Jeremiah essayed the terraced ledge accompanied by Stave, Frostheart Grueburn, and Cirrus Kindwind. Onyx Stonemage and more of Stoutgirth's crew came next. As before, Blustergale and Baf Scatterwit brought up the rear.\n\nIn places, the surface they trod resembled sheets of slate, and there the going was easy. Some of the stairs where the rock had crumbled were minor obstacles. But occasionally the sheared steps reached to Linden's waist. A few were taller than she was: they cast shadows as threatening as chasms. Like Covenant and Jeremiah, she had to be lifted to the next level.\n\nThe walls leaned toward and away from each other, tracing the variations of Mount Thunder's flaws and stubbornness. By increments, the river fell below the reach of the company's illumination. The rush of water became distant, as if it were fading out of the world; and with it the spilth or detritus of She Who Must Not Be Named also receded. In gusts and eddies, the air improved.\n\nLike the crevice, the width of the ledge undulated. At intervals, Linden was able to walk at Jeremiah's shoulder as if she could still shield him. More often, the company was forced to go in single file. When the ledge became dangerously narrow, Cirrus Kindwind kept her hand on Jeremiah's shoulder, and Frostheart Grueburn did the same for Linden.\n\nAfter some distance, Rime Coldspray and Bluff Stoutgirth came to a break in the ledge. Linden could not see how they crossed it. Giants blocked her view. Her every step was obscured by shadows. But when she and Jeremiah reached the gap, she found that the sailors had stretched a rope over it, held taut by Hurl on one end and Wiver Setrock on the other. Using the line for support, Kindwind and Grueburn helped Jeremiah and Linden to the far side.\n\nWhen the last of the Giants were safe, the company continued to climb.\n\nLinden lost her sense of duration. Nothing in the mountain's perpetual midnight marked the passage of time. Gradually the river passed out of hearing. After that, there were no sounds apart from the efforts and breathing of the companions. The _krill_ 's light shifted as Branl moved, but it revealed only rock and more rock, enduring and unrelieved. Beyond it, darkness crowded thick as obsidian or basalt.\n\nStill the river pulled air downward with it: a guttering breeze on Linden's face. For a while, she derived a sense of progress from the declining pressure of taints in her lungs. Soon, however, the changes became too subtle to be distinguished. Then weariness and strain became her only measure for the meaning of her steps.\n\nAt intervals, Jeremiah extended tentative flicks of theurgy from the Staff, but their purpose eluded Linden.\n\nIn the distance ahead, the crevice bent sharply to the left. Beyond a blind corner, another high step or shelf interrupted the ledge. This one reached the chests of the Giants. Some of the sailors were able to gain the next level unassisted; but the Swordmainnir were more heavily burdened, and their weariness was more profound: like their smaller companions, they needed help.\n\nWhen Grueburn had lifted her past the shelf, Linden paused to scan her surroundings.\n\nWithin the ambit of the _krill_ 's illumination, the ledge looked wide as a road, comparatively level. But the crevice was narrowing. After its sweep to the left, it curved gradually back to the right; and as it did so, the opposite side restricted her view ahead. Overhead the walls leaned together: she supposed that they met somewhere in the darkness, closing the fissure. Above her at the farthest extent of the light, a line across the near wall suggested the possibility of another ledge.\n\nThe far wall was pocked with holes like the mouths of tunnels, open maws where the gem's radiance did not penetrate. They looked big enough for Giants. A few were level with the company's path, but most were scattered higher around the curve.\n\nLinden peered at those holes, frowning, until she felt Covenant's tension. It poured from him like the heat of a fever. He was glaring along the ledge ahead with his fists clenched and his shoulders tight, as if he were expecting a blow.\n\nWhen she followed his gaze, she saw bones.\n\nThey littered the ledge as far as she could see: thighs and ribs, arms, hands and feet, skulls. Small heaps like crushed children. Whole skeletons piled atop each other. Femurs and ulnas randomly discarded. Smashed skulls grinning at their own ruin. Hundreds, no, thousands of them. Most of them suggested Cavewights, but some made Linden think of ur-viles\u2014or stranger monsters.\n\n\"I don't like this,\" Covenant muttered. \"It's probably good news. Somebody tossed them here. We must be getting close to the Wightwarrens. But hellfire! I think we're in trouble.\"\n\nIn the _krill_ 's silver, the bones looked desiccated, bleached: they seemed to ache with age. But when Linden studied them more closely, she saw that only some of them were old. Others still wore gobbets of flesh, shrouds of blood. The breeze drifting past her held a tang of new rot\u2014\n\n\u2014and another odor, one which she did not want to recognize. She remembered it too well.\n\nThe fresher piles seethed with rats. They cleaned the bones fearlessly, creatures that had never been threatened. Occasionally a dark eye glittered at Linden. Whiskers twitched. Plump bodies fought for every shred of meat.\n\nLong ago aboard Starfare's Gem, she had seen them swarm at Covenant, possessed by a Raver and eager for his blood.\n\n\"Thomas,\" she whispered: a dry croak.\n\nHe reached out to her. \"What is it?\" When she took his hand, he gripped her hard. \"Do you sense something?\"\n\n\"I can\u2014\" Linden tried to say; but her throat closed. She had to force out words. \"Oh, Thomas. I can smell _moksha_.\"\n\nThe precise evil of Ravers was imprinted on her nerves. Her memories of _turiya_ were bad enough. What _moksha_ had done to her was worse.\n\nCovenant stared at her. \"Damnation.\" Darkness and light warred in the background of his gaze. Then he wheeled away.\n\n\"Branl!\" he barked. \"Coldspray! We're going to be attacked!\"\n\nThe Ironhand called a question; but her comrades reacted before he could answer. Bluntfist, Kindwind, and Grueburn urged Covenant, Jeremiah, and Linden farther along the ledge, closer to the wall. Between them and the plunge of the crevice, Stonemage drew her sword. Branl thrust the _krill_ into Hurl's hands, flourished Longwrath's flamberge. He and Stave flanked Stonemage.\n\nBaffled, the sailors heaved Baf Scatterwit above the edge. As she scrambled away, they stretched their arms for Squallish Blustergale.\n\n\"Are you sure?\" Covenant panted to Linden.\n\n\"Of course she's sure.\" Jeremiah made a palpable effort to sound fierce, but his voice came out in a yelp. \"We always get attacked.\n\n\"I can't see!\" He shoved at Kindwind's back. \"I can't do anything if I can't see.\"\n\nGripping her longsword, Cirrus Kindwind shifted to cover him more completely. Bluntfist and Grueburn readied their blades.\n\nHerding Scatterwit and Blustergale ahead of them, Coldspray and Stoutgirth strode closer. \"Setrock!\" the Anchormaster commanded. \"Keenreef. Furledsail. Lead us! Clear bones from our path. If we are assailed, we must have sure footing.\"\n\nThe three sailors surged forward. Scatterwit started after them, hopping. Two of her comrades caught her arms, dragged her aside. Blustergale and another Giant followed Setrock, Keenreef, and Furledsail to help sweep debris from the ledge.\n\nMany of the bones crumbled when they were kicked aside. They released a fume of age.\n\nInstinctively Linden siphoned Earthpower from the Staff, sent her health-sense farther. The holes in the far wall looked deep. They felt empty: tunnels leading nowhere. The rats had a musty fetor, the smell of carrion and ancient dust. And the Raver\u2014\n\nImplications of _moksha_ Jehannum burned her nerves, but she could not locate their source, any source.\n\nHurl held the _krill_ above his head to extend its light. Wiver Setrock and his companions brushed through the bones. The sailors behind them pushed more into the crevice. Rats scurried away, chittering angrily. _Moksha_ remained hidden.\n\nAbruptly Coldspray announced, \"There is no gain in waiting.\"\n\n\"Aye,\" assented the Anchormaster. \"If we are to be assailed, our foes must approach along our path. They cannot surprise us.\"\n\nAt a word from the Ironhand, Stonemage and Bluntfist started after Blustergale. With Bluff Stoutgirth, Hurl, and the rest of the sailors, Coldspray followed. Kindwind and Grueburn drew Covenant, Linden, and Jeremiah away from the wall. With Branl and Stave, they trailed behind the rest of the company.\n\nWithout a cordon of defenders around her, Linden felt exposed. The holes in the stone across from her seemed to watch like eyes, black and malicious. But Jeremiah was visibly relieved: now he could see. He loosened his arms, swung the Staff from side to side as if he were testing its reach. Determination clenched his features. And Covenant strode after Setrock, Keenreef, and Furledsail as if he feared for them more than for himself.\n\nBranl kept pace with Covenant. Stave stayed with Linden.\n\nFrostheart Grueburn rested a hand on Linden's shoulder, kept Linden between her and the wall. Calmly she assured Linden, \"Stoutgirth Anchormaster speaks sooth. Our foes cannot surprise us here. They will seek some advantage of position or concealment.\"\n\nLinden was not comforted. She could feel _moksha_ Raver crouching somewhere near. Hurl's grasp on the _krill_ was not as steady as Branl's. As he moved, shadows reeled across the walls, along the ledge. Threats seemed to lurk in all directions.\n\nUrged by Coldspray, the company advanced more quickly. At the same time, Stoutgirth called to his sailors in the lead, ordered them to wait for Bluntfist and Onyx Stonemage.\n\nSpread out along the ledge, the companions rounded the curve. Linden searched past the Giants for a glimpse ahead. Mutely she prayed for haste. The holes scattered across the far wall disturbed her. For no reason that she could name, she wanted to get out from under their black glower.\n\nA moment later, Setrock shouted in frustration. When a gap opened between the Giants, Linden saw that the ledge ran straight for a short distance beyond the curve. Then it was blocked by a pile of large boulders. The last of the rats vanished among them.\n\nBlustergale and his companions were still kicking away bones. Other Giants studied the boulders, testing their bulk, looking for a way past them.\n\nAs the Ironhand and the Anchormaster strode closer, Furledsail turned toward them. \"The balance of this obstruction appears precarious,\" she reported. \"We may be able to shift the stones.\" She hesitated, glanced at her comrades, then added, \"Yet the formation is not natural. Moreover, it is recent. I deem that it was placed to thwart us.\"\n\n\"Oh, hell,\" Covenant muttered. \"Hell and blood.\"\n\nHe sounded tense enough, anxious enough, to tear the barricade aside with wild magic. He knew what _moksha_ had done to Linden.\n\nHis power might shatter the ledge.\n\nBiting her lip, Linden pushed her senses among the boulders. She wanted to discern what lay beyond them. But before she could extend her percipience far enough, a sharp cracking sound distracted her. From high above her came a granite impact, one rock massive as a menhir colliding with another\u2014or bouncing off the wall.\n\nShe jerked up her head. Saw nothing.\n\nAn instant later, a boulder the size of a Giant struck somewhere far overhead. It rebounded in a spray of shards. Splinters as keen as knives hissed past the ledge. The remaining mass arced away, hammered the far wall below the holes, burst into rubble. She did not hear the fragments hit water. The fissure was too deep.\n\nGiants yelled. Linden, Jeremiah, and Covenant were shoved against the wall again. Kindwind and Grueburn crouched over them, shielded them with lore-hardened armor.\n\nApparently unconcerned at the edge of the drop, Stave pointed at the line or ledge a long way up the fissure. \"There,\" he announced. \"The stone fell from that height.\"\n\n\"Don't _stand_ there!\" snapped Covenant. \"If it was supposed to hit us, there's going to be more!\"\n\nStave glanced at Covenant. \"Indeed, Timewarden. From this vantage, I will have forewarning. The wall provides a measure of shelter, yet it also obscures sight.\"\n\nPeering upward, he said, \"I discern no\u2014\" Then he spun toward the Giants near the blockade. \"Beware!\"\n\nToo late, Linden felt the swift hurtle of another boulder.\n\nThis one did not strike the walls. It came straight down, hard as a meteor.\n\nSailors thrust Scatterwit aside as the second rock struck within a stride of where she had been. It tore off a chunk half the width of the ledge as it bounced away, squalling with debris.\n\nMore than half the width. Only an arm span remained.\n\nLinden, Covenant, Jeremiah, and their immediate defenders would have to pass that break in order to follow their companions.\n\nCovenant's vertigo\u2014\n\n\"Giants!\" roared the Anchormaster. \"Shift the barricade! We must pass onward!\"\n\nWith Keenreef and Furledsail, Setrock began straining at the pile. Others of Dire's Vessel rushed to add their strength. Hurl moved to give them more light. Scatterwit lurched after him.\n\nOne long stride took Stoutgirth past the break. Coldspray crossed behind him, then looked back to verify that the rear of the company was safe. Linden, Covenant, and Jeremiah. Branl and Stave. Kindwind and Grueburn.\n\nA third boulder seemed to detonate against the far wall. A granite fusillade ripped across the ledge.\n\nBlustergale went down with blood spurting from his shoulder. A shard had pierced an artery. Fragments whined off Bluntfist's armor, staggered Stonemage. A sailor whose name Linden did not recall was torn apart. For an instant, his whole body spasmed. Blood and fluids sprayed from half a dozen wounds. Trying to regain his balance, he pitched off the ledge.\n\nA scream that she could not utter choked Linden. Heedless of the danger, Covenant ran toward the break. Jeremiah looked around wildly.\n\nAs if from nowhere, a stone spear appeared in the center of Hurl's chest. He sprawled backward, crashed against the wall. The impact knocked the _krill_ from his hand. It hit the ledge, skittered away\u2014\n\nShadows pounced from all directions.\n\nFaster than Linden's fear, Branl leaped the break, dove headlong. Sliding in Blustergale's blood, he snagged the dagger at the lip of the drop.\n\nSomehow he kept his longsword.\n\nMore spears crossed the crevice, a volley of stone shafts. Setrock and his comrades were driven back from the blockade. Now Linden saw a Cavewight standing in each of the tunnels in the opposite wall. The holes spat spears. Then those Cavewights moved aside. More creatures with spears strode into view, stepped into the force of their throws.\n\nRime Coldspray shouted orders louder than Bluff Stoutgirth's. At the same time, she returned over the breach to intercept Covenant. Ignoring his curses, she hauled him off his feet, swung him onto her back so that she could shield him.\n\nGrueburn and Kindwind guarded Linden and Jeremiah with their armor. Grueburn's blade batted a spear aside. Stave knocked another out of the air.\n\nBluntfist sprang close to the edge, protected as many sailors as she could. Limping, Stonemage joined her. Bluntfist let one spear splinter against her cataphract while she chopped at another. Stonemage deflected two shafts. The Giants behind her dodged.\n\nSurging upright half cloaked in blood, Branl raised the _krill_. One-handed, he swung his flamberge. A spear shattered. Pieces fell into the crevice. Bright silver spread over the ledge, along the fissure. Shadows capered, jeering.\n\nMore spears came in continuing waves.\n\nYears among shrouds and ratlines had made the sailors agile. They twisted and ducked; shoved each other out of the way; blocked spears with belaying-pins and fists. When they could, they armed themselves with the Cavewights' weapons.\n\nSobbing, Scatterwit clamped her hands to Blustergale's shoulder. \"Ward yourself!\" he gasped. \"The wound is mine. I will stanch it.\" But she ignored his protest.\n\nNear Linden, Earthpower burgeoned. Abruptly Jeremiah pushed past Kindwind's protection. Yelling words which he had heard Linden use, he found an open space, aimed the Staff of Law like a lance. \" _Melenkurion abatha!_ \" From the wood's iron-shod end, black flame blared. \" _Duroc minas mill!_ \" Magic lashed like lightning across the fissure, scoured its way into one of the tunnels. \" _Harad khabaal!_ \" The Cavewights there caught fire, blazed in agony.\n\n\"Take _that_ , you bastards!\" Like Scatterwit, he was sobbing. \"I'm learning! I'll kill you all!\"\n\nTo Linden, Cavewights implied Roger Covenant. She shouted Jeremiah's name, fearing an eruption of Roger's laval fury. But she hardly heard herself over the roars and efforts of the Giants, the sharp strike of spears.\n\nAs the boy readied another blast, Stave reached him. Turning his back on the Cavewights, on the spears, Stave stepped in front of Jeremiah, forced Jeremiah to look at him. Calm as a breeze amid the turmoil, the former Master said, \"Wield such strength with care, Chosen-son. It is new to you. Therefore it is uncertain.\"\n\n\"Rockbrother!\" called Frostheart Grueburn.\n\nStave did not glance at the Swordmain. \"Also,\" he told Jeremiah, \"the ur-Lord's maimed son may join the assault at any moment. You must prepare to oppose him.\"\n\nCursing, Grueburn left Linden, leaped to stop a spear aimed at Stave. The frantic sweep of her sword missed: she took the shaft's point on her breastplate. It glanced away, clattered on the ledge.\n\n\"Roger?\" cried Jeremiah. \"You want me to fight _Roger_? How am I supposed to do that?\"\n\n\"With care,\" Stave replied evenly. \"With passion, certainly, but also with care.\" Step by step, he urged Jeremiah back into the shelter of Cirrus Kindwind's bulk and armor.\n\nFrantic and afraid, Linden searched the confusion with her senses; but she found no sign of Roger.\n\nAbruptly the barrage of spears stopped. Responding to a signal that Linden did not hear or feel, all of the Cavewights withdrew from the gaping tunnels.\n\nA moment later, the barricade beyond the company collapsed as if a keystone had been removed. Huge rocks rolled over the edge, dropped soundless into the dark. In an instant, most of the barrier was gone as though a door had been kicked open.\n\nAlong the ledge charged a throng of Cavewights, howling.\n\nThey were armed with spears and falchions, cudgels and axes. Plates of stone hanging from their shoulders served as armor. The hot red of their eyes scorned the _krill_ 's wavering radiance.\n\nTheir suddenness caught the Giants off balance. A cudgel like a battering-ram struck the side of a sailor's head, knocked him into the fissure. Cries followed him down. Furledsail fell back with gore spilling from a slash below her ribs. She, too, might have gone over the edge; but Setrock sprang after her, snatched her into his arms. Her attacker Keenreef stopped with one hard punch to the center of the forehead. Then he had to wheel away from the vicious stroke of an axe.\n\n\"Withdraw!\" The Anchormaster's command echoed up the wall. He sounded preternaturally unperturbed; accustomed to gales. Perhaps he was also accustomed to loss. \"Withdraw, Dire's Vessel! This is work for Swordmainnir!\"\n\nBut he did not retreat himself. Jerking the spear out of Hurl's chest, he advanced to meet the Cavewights. He seemed to be laughing.\n\nOnyx Stonemage and Halewhole Bluntfist were already running to counter the charge. Rime Coldspray shouted for Branl; consigned Covenant to the Humbled. With her glaive in her fist, she raced after Stonemage and Bluntfist.\n\nThe ledge was too narrow for a massed assault. No more than four Cavewights led the attack; and even then, they hampered each other. Stonemage and Bluntfist let the remaining sailors scramble between them. Then the two Swordmainnir faced the creatures.\n\nColdspray stopped Stoutgirth three paces behind her comrades. She and the Anchormaster braced themselves for flung spears; prepared to cut down any Cavewight that fought past Stonemage and Bluntfist.\n\n\" _Damn_ it, Branl!\" Covenant demanded. \" _Do_ something! I can take care of myself!\"\n\nBranl studied Covenant for a moment; shrugged delicately. Then he handed Loric's dagger to the Unbeliever. Springing away over the break in the ledge, he went to join the Ironhand.\n\nCovenant, Linden, and Jeremiah were left behind. Stave, Grueburn, and Kindwind. But soon sailors came to them dragging Blustergale, carrying Furledsail. Harried along by Setrock, Scatterwit retreated from the fray.\n\nLinden watched Giants and Cavewights fight in darkness relieved only by the _krill_ in Covenant's grasp, and by the crimson glow of eyes. At first, Bluntfist and Stonemage seemed implausibly effective. They were skilled and mighty. They had room enough between them to swing their blades. And they could afford to let creatures lunge past them: Coldspray and Stoutgirth protected their backs. Rabid thrusts and slashes were beaten aside. Bodies toppled from the ledge in welters of blood.\n\nBut the Cavewights were mighty as well, born with the strength to delve in gutrock by hand. They were nearly as tall as Giants. Their arms were longer. And they were many, more than Linden could count. Eventually their sheer numbers would overwhelm the Giants. Already Bluntfist and Stonemage were driven backward. The Ironhand and the Anchormaster were forced to retreat as well.\n\nBranl strode between the commanders. He passed Bluntfist and Stonemage, drifted like a shadow among the Cavewights. With the rippled edges of his longsword, he seemed to reap creatures all around him. Howls became shrieks. Bodies fell. In the press of Cavewights, his shorter stature was an advantage. Creatures fighting at the height of Giants could not block his flurry of cuts, his swift dance. For a moment, he stopped the advance. Linden almost believed that he would be able to turn the battle.\n\nStill the Cavewights were too many. And they were not mindless. Quickly they adjusted their tactics. Those in the lead sprang aside, cleared a space which allowed other creatures to level their weapons and their strength at the Humbled.\n\nBranl dodged a spear, cut off the arms of its wielder. As if in a single motion, he blocked a cudgel on one side, countered a sword on the other. He slashed at thighs, knees, ankles.\n\nBut more Cavewights came. In spite of his prowess, he was beaten backward.\n\nSoon Covenant would have no choice. Jeremiah would have none.\n\nAt the edge of her vision, Linden thought that she saw another boulder plummet into the depths. She saw or imagined a Cavewight sprawling through the air after it.\n\nShe shook her head to dispel the image. She could not help Branl and the struggling Swordmainnir. She had promised herself\u2014\n\nBut she could meet other needs. Snatching Earthpower from the Staff, from Jeremiah, she aimed fire at Blustergale's shoulder and Furledsail's side.\n\nShe worked fiercely. She had no time for kindness. The battle was coming closer. Like an act of violence, she stopped Blustergale's bleeding, mended the bones, closed the wound; poured energy into his veins: healing as brutal as abuse. When she was sure that he would live, she treated Furledsail in the same fashion.\n\nAll of the Giants had cared for her. Some had given their lives. This was how she rewarded the living.\n\nThen Stave called her name. She jerked up her head, flung her gaze at the fighting.\n\nHe still stood near her. Nevertheless another _Haruchai_ had joined Branl. The newcomer had acquired a falchion from a fallen creature. Together he and Branl struggled to slow the Cavewights so that the Swordmainnir would not be overrun.\n\nAnother\u2014?\n\nLinden did not recognize him.\n\nA heartbeat later, a second unfamiliar _Haruchai_ landed lightly on the ledge. He must have been working his way down the wall. Even one of his people could not plummet onto stone with such ease from any great height.\n\nHe had the grizzled hair of a veteran: his face was a lattice of old scars. He paused to glance around at Stave and the sailors. Grueburn and Kindwind. Linden and Jeremiah. Then he stared at Covenant.\n\nFor the first time, Linden saw open astonishment on the impassive face of a _Haruchai_.\n\n## 8.\n\nShamed Choices\n\nWild magic swelled in Covenant. It yearned for release. His wedding band ached on his ring finger. The ambush had already killed three of the Giants. Hurl's body lay against the wall, transformed by a stone spear into fresh feasting for rats. Two of the men from Dire's Vessel had plunged into the crevice; into the distant embrace of the Defiles Course. Still an uncountable number of Cavewights pressed forward: a storm of red eyes and ferocity squalling like ghouls. Covenant did not know how much longer the company's defenders could withstand the attack.\n\nStonemage and Bluntfist. Coldspray and Stoutgirth. Branl with Longwrath's flamberge. The eldritch blade's magicks were meaningless here: its edges were not. Together the Humbled and the Giants were more effective than Covenant could comprehend.\n\nThe Anchormaster's remaining sailors had armed themselves with spears. Even Blustergale and Furledsail had regained their feet after Linden's violent healing; had claimed weapons. But they could not enter the fray. The ledge was too narrow.\n\nCovenant loathed killing, but his abhorrence for the suffering and loss of those who stood with him was greater. To save them, he would have incinerated every Cavewight on the ledge. And he would have borne the cost; added those deaths to the stains on his soul. His ring seemed to plead for use.\n\nYet he suppressed its fire, swallowed his ambiguous power. He did not have enough control to strike at the Cavewights without harming the people whom he longed to save. He could not protect Linden and Jeremiah. He had never been able to spare anyone who chose to fight for the Land.\n\nAnd he did not understand why the unfamiliar Master stared at him in such amazement.\n\nSurely the _Haruchai_ had come for this? Summoned by Bhapa and Pahni, they must have rushed to Mount Thunder to join the Land's last defense. Why else were they here? How else had they arrived when they were needed?\n\n\"Thomas,\" Linden panted. \"Thomas.\"\n\nCovenant barely heard her. He gripped Loric's _krill_ as if he had forgotten it.\n\nWhy was this Master surprised?\n\nFortunately he had not come alone. Armed with a Cavewight's falchion, the other warrior now supported Branl. In perfect harmony, they appeared to flow and eddy among the creatures, delivering bloodshed and death with the grace of dancers: a cut here, a thrust there, a spinning feint, on and on, all too swift for Covenant to grasp. Maimed and dying Cavewights were flung like sleet into the fissure. And those that fought past the two Masters were met by the hard iron of the Swordmainnir, or by Stoutgirth's spear.\n\nThe strength and skill of the defenders slowed the charge. They halted it.\n\n\"Ur-Lord.\" Stave pitched his voice to pierce the clamor and rage, the screams of pain, the raw gasping. \"Here is Canrik of the Masters. His comrade is Dast. Above us, Ulman and Ard await the outcome here.\" Stave's tone had a sardonic tinge, trenchant and vindicated. \"They were unaware of your return to life.\"\n\nUnaware\u2014? The idea staggered Covenant. Realities shifted. The ledge tilted to one side. It began to turn. He stood on impossible stone, could not keep his balance. The crevice called his name, a chiaroscuro of alternating seductions and commands. The _krill_ fell from his numb fingers.\n\nWhat had Bhapa and Pahni told the Masters? Had the Cords even reached Revelstone? Had the Ardent failed in his dying gift? Then why were the Masters here?\n\nCovenant wanted to howl silver until the ledge stopped; until everything stopped.\n\nThrough the whirl, Linden's arms found him. \"Thomas!\" He thought that he saw Stave holding the _krill_ nearby; but he heard only his wife. \"The Masters came. Stave says that two hundred of them came!\" With every word, she tried to summon him back. \"But they didn't know where to look for us. There are too many tunnels. They had to spread out. Four of them found a place where the Wightwarrens open on this crevice. Somewhere up there.\" She seemed to be pointing. \"Two _hundred_ , Thomas! We'll have more help as soon as the others learn where we are.\n\n\"Hang on, Thomas! You have to hang on.\"\n\nReeling, he struggled to focus on her. His hands fumbled their way to the sides of her face. He held her directly in front of him, almost nose to nose, so that she would wheel with him\u2014or so that the truth of her would remain stationary. She was not spinning. The ledge was not. Even the world was not. It was all in his mind.\n\nHe should have been accustomed to such things. He had been dizzy often enough\u2014\n\n\"Mom?\" Jeremiah asked. He seemed to be pleading. \"Are they going to save us?\"\n\nCanrik spoke. \"Ur-Lord.\" His voice was hard. His amazement had become anger. \"There are questions which must be answered.\" Then he seemed to relent. \"They cannot be answered here.\n\n\"Giants!\" he called. \"Do you possess rope? We would do well to gain the ledge above. Ulman and Ard will aid us. We have no other path.\"\n\nA cudgel caught the side of Stonemage's leg. She went down. Lunging, Stoutgirth spitted her assailant. Blood gushed from the Cavewight's mouth, splashed the Anchormaster's face. But Stoutgirth was not done. In spite of his leanness, he was strong enough to heave the Cavewight into the air on the end of his spear. Furiously he pitched the creature over the edge.\n\nStonemage's pain made Linden flinch. She pulled away from Covenant. \"Help them!\" she yelled at Canrik. \"Stonemage can't stand!\"\n\nThe Master faced her, glaring. \"There is no need. The attack fails. A rout begins.\"\n\nHarried by fears like furies, Covenant forced his inward whirl aside. The whole crevice continued turning, but he ignored it. Standing with his legs wide, he looked along the ledge.\n\nThrough a blur of failing sight and vertigo, he saw that Canrik was right. Only seven or eight Cavewights still fought. The other Master, Dast, pursued creatures trying to retreat. Branl spun to help Bluntfist and Coldspray with their opponents. His blade spilled entrails, flung red spray. With every slash, Coldspray drew bright gore. Bluff Stoutgirth threw his spear: a final strike that gouged chips from a plate of armor. Then he stooped to Stonemage, pulled her arm over his shoulder, hauled her upright. Together they staggered toward the rest of the company.\n\nBlustergale did not wait for instructions. From one of the company's sacks, he produced a heavy coil of rope. Baf Scatterwit tried to take it from him: his healed shoulder was still weak. He refused her; gave the rope to Wiver Setrock. To console her, he said, \"Stand ready. You will have other tasks.\"\n\nShe hooted a laugh. \"I am ready. Am I not ready always?\"\n\n\"Anchormaster!\" shouted Setrock. \"The Master counsels an ascent! Other Masters wait to assist us.\"\n\nManic in his mask of blood, Stoutgirth grinned, rolled his eyes. \"Sluggard! Why do you delay? If we do not accept aid when it is offered, we are not merely fools. We are witless as well.\"\n\nAt once, Setrock moved to the rim of the ledge, peered upward. For a moment, he gauged the distance, hefted his coil of rope. Then he nodded. Crouching to gain force, he threw his coil at the ledge high above the company.\n\nIt disappeared in the darkness for a moment. Then one end of the rope came snaking down.\n\nCovenant drew a steadier breath, watched his surroundings settle back into their necessary positions.\n\nStoutgirth lowered Stonemage to the ledge, settled her leaning against the wall. \"Another line,\" he commanded Scatterwit cheerfully; too cheerfully. Anguish in his gaze belied his tone. \"Rig three cradles. I will not entrust the Timewarden or Linden Giantfriend or Jeremiah Chosen-son to the strength of their arms.\"\n\nHe did not add that Cirrus Kindwind had only one hand, or that Blustergale and Furledsail were still recovering, or that Onyx Stonemage was hurt, or that Scatterwit herself had lost a foot.\n\nCovenant approved. He did not believe that he would be able to hold on when fresh vertigo urged him to fall.\n\nQuestions which must be answered?\n\nCanrik was glaring at Linden again as if she were a viper. As if he felt betrayed\u2014\n\nUnder the force of his gaze, she seemed to shrink inside her clothes. She had endured too much distrust from the Masters; too many judgments. Her history with them hung on her shoulders like a millstone. But she did not reply to Canrik's plain ire. Instead she turned to Jeremiah. Like a woman who wished to demonstrate something, she said distinctly, \"I need Earthpower, Jeremiah. For Stonemage. Do you mind?\"\n\nApparently she wanted Canrik to understand that the Staff of Law now belonged to the boy.\n\nJeremiah frowned. \"She's hurt.\" He looked baffled. \"You don't have to ask. She needs you.\"\n\nHe seemed to mean, I don't know how to help her.\n\nThe idea that the Masters still saw harm in Linden made Covenant want to hit Canrik in the face. Trembling at the intensity of his own wrath, he watched her walk toward Onyx Stonemage.\n\nThe injured Swordmain sat on the far side of the place where a boulder had broken the ledge. She kept her hands clamped around her thigh to block the sensations from her knee, prevent the pain from breaking through her self-command. Linden did not try to cross the gap unaided\u2014and did not wait for help from the Giants. Instead she halted near the breach and bowed her head, concentrating her senses on Stonemage's injured leg. As if of its own accord, fire unfurled from the Staff of Law, an ebon tracery stark in the _krill_ 's shining. It spun whorls like intaglio as it reached toward Stonemage.\n\nBeyond them, Rime Coldspray kicked a Cavewight off the ledge: apparently the last of the creatures. Breathing hard, she and Halewhole Bluntfist studied the distance for a moment, where Dast harried the remnants of the attack. Then they raised their swords to salute Branl.\n\nHe replied with a _Haruchai_ bow. His expression acknowledged neither pride nor satisfaction. If he had gleaned anything from Dast's thoughts, he did not reveal it.\n\nBriefly Covenant faced Canrik. He wanted to demand, How dare you? How _dare_ you? After everything she's been through while you were sitting on your damn hands? But he restrained himself. There was too much here that he did not understand. Too much that the Master did not.\n\nDeliberately he shifted his attention to Jeremiah. Harsh as a rasp, he asked, \"Where is the Worm now?\"\n\nLike Linden, he intended a demonstration.\n\nJeremiah winced. He studied his hands twisting on the Staff. \"It's still in the river.\" His voice shook with bitterness. \"Still above ground. But it's getting close. I can't see _Melenkurion_ Skyweir anymore. There's just a huge cliff with a crack where the river comes out.\"\n\nOver his shoulder, Covenant looked at Canrik again. Did you hear that, you self-righteous bastard? You think you've got questions? You have no idea.\n\nEverything that Linden had done for her son's sake since Covenant's return to life was justified.\n\nThen he told Jeremiah unsteadily, \"Don't worry about it.\" The boy was ignorant of Linden's fraught history with Canrik's people. When would she have explained it? Why would she? Galt had saved Jeremiah's life. \"I know what's happening to you is cruel. I can only imagine how much it hurts. But you'll get your chance to do something about it. And the Masters will help us.\"\n\nAt least until their questions were answered.\n\nAs if in response, Canrik said, \"The Masters have been given lies. Stave conceals his thoughts. Branl of the Humbled must reveal truth.\"\n\nThe openness of Branl's mind did not trouble Covenant. Of course Branl would tell the truth. He had promised to instruct his people. Covenant trusted that he would tell the whole truth.\n\nBut lies? Who had lied to the _Haruchai_? Who had taken that risk? And how had the discernment of the Masters been foiled?\n\nStave regarded Canrik with a flatness that seemed to imply disapproval; but he did not reproach the Master.\n\nAround Covenant, the Giants hurried through their preparations to leave the ledge. A sailor called Spume Frothbreeze braced his feet on the wall. With a second coil of rope over his shoulder, he pulled himself upward hand over hand. Scatterwit's line had been knotted around his waist so that he could drag it behind him.\n\nTo Covenant's blurred sight, the height of the next ledge seemed unattainable. If Ard and Ulman were there, he could not distinguish them. Within moments, Frothbreeze faded into obscurity.\n\nBut the Giants did not hesitate. At once, a woman followed with the company's last rope: Far Horizoneyes. Like Frothbreeze, she climbed with the ease of long experience.\n\nKeenreef and Setrock took the remaining supplies, hastened upward. Covenant scowled at the cradles knotted into Scatterwit's line: three of them tied in sequence so that he, Linden, and Jeremiah could sit in them and simply hold on while Giants and _Haruchai_ raised them. He did not want to do this. He would lose his balance again. And all three of them would be vulnerable. If the Cavewights renewed their attack, threw more spears\u2014\n\n\" _No_.\" The Ironhand's voice snatched him out of his fretting. Although she spoke quietly, her vehemence shocked him.\n\nTurning, he saw Bluff Stoutgirth rise to his feet with Hurl's body across his shoulders.\n\n\"No,\" repeated Coldspray, furious or grieving. \"Anchormaster, no.\"\n\n\"One I lost to the _skurj_ ,\" Stoutgirth replied like a lament. \"For him, I have been granted a _caamora_. But three were slain here, and two fell beyond the reach of sorrow.\" He bared his teeth through his veil of blood. \"All were in my command, and their guerdon was death. I will not forsake Hurl to the feeding of rats.\"\n\n\"You _will_ ,\" countered the Ironhand. \"I do not gainsay your bereavement. Nonetheless you are the Anchormaster of Dire's Vessel, and you have not been relieved of command. Storms do not abate when a Giant falls from the rigging. Nor is our peril eased by the loss of comrades precious among us.\n\n\"The world's ending will be _caamora_ enough for any woe. You will not hazard your life for a corpse.\"\n\n\"Will I not?\" Stoutgirth did not meet her gaze. \"Is this your word, Rime Coldspray? Do you speak thus, you who have lost five of your Swordmainnir, and have seen the purpose of your striving across the seas fail? Ironhand, your heart is stone. Mine is water.\"\n\nColdspray clenched her fists: anger glared in her eyes. Before she could retort, however, he jerked up his head, laughed like a loon. Two strides took him to the edge of the chasm. There he crouched, braced his arms under Hurl's body, and heaved it into the depths.\n\nLaughing or crying, he said, \"Hurl I give to the river. May it bear my heart to the surcease of seas, as it does him.\"\n\nHis wracked mirth rose until it seemed to fill the crevice: a broken man's threnody for the world's fallen. But he did not permit his rue to hold the company back from the ropes.\n\nhen Covenant reached the higher ledge, he had to sit down. Freed from his knotted cradle, he collapsed against one wall of the crude tunnel leading away from the crevice; drew his knees to his chest and hugged them urgently; hid his face. He felt unmanned by vertigo, by impossible demands and contradictions. He had barely known Hurl. He could not even remember the names of the other slain sailors, Giants who had lost their lives without striking a blow in their own defense. And his decisions had led them to ruin. It was his responsibility to make their deaths worthwhile.\n\nIt could not be done. Nothing that he ever did would assuage Lord Foul's countless victims. Nothing would suffice to honor the valor of those who still struggled for the Earth.\n\nStill Covenant had to try. He had to close his ears to the siren song of dizziness and futility. He had to believe\u2014\n\n_There is no doom so black or deep that courage and clear sight may not find another truth beyond it._\n\nHe was a leper. Surely he could believe whatever he chose? As long as he was willing to pay the price?\n\nFortunately he was not alone. In the Land, he had seldom been alone; but this time he had been given more than companionship and aid. Linden was coming toward him. He did not need health-sense to recognize the love in her eyes, the raw concern. Jeremiah followed behind her, clutching the Staff of Law as if his sanity depended on it. Stave brought the light of Loric's _krill_ into the tunnel. Branl had gone to extremes that still appalled Covenant. Two Masters\u2014Ard and Ulman?\u2014stood on the ledge, helping with the ropes. And there were still Giants.\n\nGod, Giants\u2014Five of Rime Coldspray's comrades: four of the Anchormaster's sailors: all gone, as lost to the world as Lostson Longwrath. Nevertheless those who remained outnumbered the dead.\n\nAnd two hundred Masters had come to the Wightwarrens. Two _hundred_ \u2014\n\nIf Covenant's ability to choose what he would and would not believe was one side of being a leper, this was the other: he did not know how to bear such abundance. He had spent decades in one world and millennia in another learning how to stand alone.\n\nYet he could not pretend that he was not grateful. When Linden sat down beside him and slipped her arm over his shoulders, he found that he was able to meet her gaze.\n\n\"It isn't all bad,\" he said roughly. \"At least we're still together. Some of us made it.\"\n\nHe meant, I love you, Linden Avery.\n\nHer hug seemed to say that she understood.\n\nBlinking uselessly, he looked around. \"How are we doing?\" Shadows and stark silver confused the shapes gathering around him. \"We can't stay here.\"\n\n\"We'll be ready soon,\" she told him. \"Some of the Giants need help.\" Cirrus Kindwind and Onyx Stonemage. Baf Scatterwit. Squallish Blustergale. Etch Furledsail. \"They're being hauled up now. That only leaves Canrik and Dast.\"\n\nOf course, Covenant thought. Naturally the Masters would insist on coming last.\n\nTwo hundred of them were in the Wightwarrens somewhere. Against how many Cavewights? He had no idea. Roger had had plenty of time to summon every living creature in Mount Thunder. And _moksha_ Raver remained a threat. He might still be able to command any number of Lord Foul's servants. Covenant was not sure that two hundred _Haruchai_ would be enough.\n\nAnd in spite of what he had said to Jeremiah, he was not confident that he could count on their help. _The Masters have been given lies_. He did not know what those lies were. Therefore he could not guess how the Masters would react to the truth.\n\nTogether, Kindwind and Stonemage were heaved onto the ledge. When the last sailors and _Haruchai_ had been pulled upward, Canrik strode among the Giants toward Rime Coldspray.\n\n\"Ironhand,\" he said at once, \"we must not tarry in this passage.\"\n\nColdspray looked down at him. \"Aye. Our foes are certain of our presence. They will surely come against us. And here we cannot retreat. We will perish\u2014we must\u2014if we do not discover a choice of headings. Are you able to guide us?\"\n\nCanrik nodded. \"Our older knowledge of the Wightwarrens is slight, but we have not forgotten our path hither. And as we rejoin with our kinsmen, our knowledge will increase.\n\n\"What do you seek? Where do you hope to discover Kastenessen\"\u2014he cast a caustic glance at Linden\u2014\"if it remains your intent to confront one deranged _Elohim_ while the Land and the Earth are unmade?\"\n\nWithout pausing for thought, Covenant surged to his feet. \"Kastenessen?\" he snapped. Lies? \"Where did you get that idea? Didn't you feel it when Kevin's Dirt faded? What did you think that meant? Kastenessen gave up days ago.\"\n\nClearly the Masters knew that the Worm of the World's End had been roused\u2014\n\n\"We are not blind, ur-Lord,\" retorted Canrik. \"We are aware that Kevin's Dirt has ended. But we were misled, Stave does not speak to us, and Branl is\"\u2014the Master appeared to search for words\u2014\"strangely reluctant. We cannot divine your purpose.\"\n\nCovenant made an effort to swallow his anger. The Masters were not his enemies. He was simply outraged that they thought ill of Linden.\n\n\"I have to get to Kiril Threndor,\" he rasped. \"If that's not too much to ask. I want to find the Despiser. And Cavewights aren't our only problem. My son is here somewhere. He's scared enough to try anything. Plus there's _moksha_ Jehannum. He's probably mad as hell.\n\n\"I don't know what's bothering you, but it's trivial. We don't have time for it.\"\n\nFor a moment, Canrik stood as if he had been silenced. Slowly a frown settled onto his forehead. Then he stated, \"Our questions must be answered.\"\n\nWithout waiting for a reply, he strode down the tunnel.\n\nColdspray glanced sharply at Covenant; but she did not delay. Hailing her Swordmainnir, she sent Halewhole Bluntfist and Frostheart Grueburn after Canrik. Then she followed him herself, taking Cirrus Kindwind with her, Ard and Ulman; leaving Onyx Stonemage with Covenant, Linden, and Jeremiah.\n\nA subdued Anchormaster marshaled his crew. His uninjured sailors\u2014Wiver Setrock, Spume Frothbreeze, Keenreef, Far Horizoneyes\u2014he sent ahead. With Scatterwit, Blustergale, Furledsail, and Dast, he trailed the rest of the company.\n\nInstinctively Covenant took Linden's hand, rested his halfhand on Jeremiah's shoulder. Accompanied by Stave and Branl, they started along the passage.\n\nJeremiah did not resist, but he walked with his head down, paid no attention to where he put his feet. His hands tightened and relaxed on the Staff, urgent as heartbeats. At intervals, he jerked up his head and glared around him. But he did not speak; did not appear to notice Covenant or Linden.\n\n_Maybe Roger had the right idea. Maybe we should all try to become gods._\n\nThe notion made Covenant's stomach burn as if he had swallowed acid. He refused to believe\u2014\n\nLinden studied her son for a moment. Then her eyes flinched away. She looked at Covenant, pleading like a woman who had no language for her needs. Almost at once, however, she turned her attention to Stave.\n\nIn a low voice, she asked, \"What's bothering the Masters? Did Pahni and Bhapa reach them?\"\n\nShe might have asked, Do they think that Pahni and Bhapa lied? They can't believe that. If they do, why did they come?\n\n\"Chosen,\" replied Stave, \"I must accord to our people the respect which I will require of them.\" His tone suggested that he was keeping his distance. \"They will speak of their doubts and indignations when we have evaded immediate pursuit. It is their right to be who they are, and to determine what they will become.\n\n\"Yet I am free to acknowledge that the Masters have heard and questioned the Cords. True to his service, the Ardent delivered Bhapa and Pahni to the vicinity of Revelstone. Their words gave the Masters cause to come in search of you, Chosen.\" He gave a subtle emphasis to Linden's title. \"Now the Cords accompany the Masters. If our foes and our fate permit it, you will be reunited with them.\n\n\"More than that I will not disclose.\"\n\n\"Damn it, Stave,\" Linden muttered. \"That's not enough. How could they not know that Covenant is alive? Didn't Bhapa or Pahni tell them?\"\n\nHow had the Cords goaded the Masters into action at last, if not by insisting that the ur-Lord needed them, the Unbeliever, the man who had twice defeated Corruption?\n\n\"We'll hear about it soon enough,\" Covenant put in. He did not have the heart to challenge Stave's scruples. Instead he tightened his grip on Linden's hand, trying to reassure her. \"Or we'll spend what's left of our lives fighting, and we won't hear anything at all. Either way, it doesn't matter. They aren't just Masters. They're _Haruchai_. Eventually they'll help us, even if they think we did something terrible behind their backs. They have to. They're too ashamed to do anything else. They've already passed up two chances to face Lord Foul with me, not to mention once with Kevin. They don't know how to live with it.\"\n\nStave nodded like a shrug. Branl did not offer his opinion.\n\nFor a long moment, Linden studied the ungiving stone ahead of her. When she finally spoke, her voice was so soft that Covenant barely heard her.\n\n\"Don't let them get in my way. This is my last chance. We can't stop the Worm. It's my fault, but I can't do anything about it. That's why I have to\u2014\"\n\nAbruptly she stopped.\n\n\"I know,\" Covenant sighed. \"We're all in the same boat. The only thing that might be worse than facing our fears is not facing them.\"\n\nLinden did not reply; did not lift her head. She clung to his hand as if she were drowning.\n\nCovenant knew the feeling. He believed that she would find the courage she needed. A woman who could do what she had done would be able to do more. But he was not at all sure how he would bear losing her.\n\nThe sheer scale of his anger at the Despiser was becoming a liability. Often it had kept him going when he should have failed. But now he needed a better answer\u2014and his anger threatened to blind him.\n\nThat was the paradox of his leprosy. In order to confront Lord Foul, he positively required numbness. He had to be untouchable: immune to every affront; impervious to the extremes of wild magic. Unaffected by the implicit betrayal of Roger's allegiance. Yet numbness might also leave him impotent. It had done so before.\n\nWhen Linden left, she would take his heart with her. If he allowed fury to fill that great hole in his chest, he was sure to fail.\n\nven in this unfamiliar passage, Covenant recognized the Wightwarrens. He knew them by the crudity of the Cavewights' delving\u2014the careless walls and ragged ceiling, the irregular protrusions of stone where the creatures had neglected to finish what they started\u2014and by the instinctive cunning with which the tunnel followed veins and lodes within the gutrock. From here, anyone who knew the catacombs well would be able to find Kiril Threndor, Heart of Thunder, where Covenant had once surrendered to the Despiser.\n\nBut he had no idea how far he still had to go. And he felt sure that the company would be attacked again before he reached his goal.\n\nAs if to prove him right, a warning shout came from the darkness ahead: the Ironhand's voice. He heard yells and effort, the clash of weapons. At once, Onyx Stonemage gestured for a halt. She went three paces farther, then stopped, waiting with her longsword in her fists.\n\n\"Mom?\" Jeremiah asked uselessly. \"Mom?\"\n\n\"Cavewights bar the passage,\" Branl announced, \"a small force. I surmise that they did not anticipate our ascent from the crevice. They were not prepared against us. Yet the constricted space aids them. They suffice to\u2014\"\n\nStave shook his head. Briefly Branl narrowed his gaze. Then the Humbled said, \"They do not suffice. Four Masters assail the creatures from the rear. Openings are created for the blades of the Swordmainnir, and for Canrik and Dast. Three Cavewights have fallen. Five. Now eight.\" After a moment's silence, Branl stated, \"The passage has been cleared.\"\n\n\"Is anyone hurt?\" Linden asked.\n\nBranl appeared to hesitate before saying, \"Skill and armor shielded the Swordmainnir. _Haruchai_ do not regard their hurts.\"\n\n\"In other words,\" she snapped, \"they don't want me to insult them by offering to treat them.\"\n\nCovenant ground his teeth. She was right, of course.\n\nStave shrugged. \"There is much which the Masters do not comprehend.\"\n\n\"Mom,\" Jeremiah breathed thickly. \"I smell blood.\"\n\nLinden glanced past Covenant at the boy. \"I know, honey. I'm sick of all this killing. But we can't stop. If we don't fight, they'll kill us.\"\n\nAs if to herself, she muttered, \"It galls the hell out of me that the Cavewights would probably be on our side if they knew how Foul is using them. They can _think_ , for God's sake. They just don't think clearly enough.\"\n\nAnd they probably love their children, Covenant added for her. They probably hate us for what we're doing. But he kept that thought to himself.\n\nJeremiah murmured something that Covenant did not hear. Stonemage was beckoning them into motion.\n\nStill holding Linden's hand, still resting his palm on Jeremiah's shoulder, Covenant started forward again.\n\nSoon he, too, could smell blood: blood and more bitter fluids. In the distance ahead, the _krill_ 's illumination caught glints of crimson on the floor and walls. It looked dark as ichor. The Giants and Masters leading the company had moved past the site of the fray, leaving hacked and gutted corpses behind them. Blood lay in thick pools around bodies and spilled guts. Stonemage strode through the carnage as if she could not afford to acknowledge it. Stave and Branl stepped, heedless, in swaths of red, trod with apparent unconcern over dripping corpses. But Covenant had to let go of Linden and Jeremiah so that he could pick his nauseated way among the dead.\n\n_God_ , it was hard not to hate the Despiser. Rage felt like the only sane response.\n\nAs the Giants bringing up the rear passed the slain Cavewights, Branl told Covenant, \"The Swordmainnir have gained an intersection of passages. The path familiar to Canrik and his companions lies to the right, but there the air is fraught with peril. Samil, Vortin, and other Masters approach from the left. They report that their search did not tend toward Kiril Threndor. Therefore the Ironhand wishes to continue ahead. She awaits only your consent, ur-Lord.\"\n\nCovenant hesitated momentarily, trying to guess the consequences of every choice. Then he rasped, \"Tell her to trust herself. More Masters will find us. Eventually some of them will know how to reach Kiril Threndor.\"\n\nBranl and Stave nodded. Branl's manner hinted at increased concentration as he conveyed Covenant's reply.\n\nCovenant looked to Linden for her approval; but her attention was fixed on Jeremiah. The boy stood staring straight ahead as if he had gone blind. His hands shifted up and down the Staff as if he were wrestling with the Worm.\n\nGroaning to himself, Covenant trailed after Onyx Stonemage.\n\nWhen he and his companions reached the intersection, they found Frostheart Grueburn, Halewhole Bluntfist, and Dast waiting for them. Blood dripped from a cut the length of Grueburn's left forearm. A spear had gashed Bluntfist's right cheek. But their wounds seemed superficial. In the _krill_ 's argent, their grins looked garish as grimaces.\n\nThey gestured Covenant and the others onward. \"The Ironhand deems,\" Grueburn explained, \"that we are no longer required in the forefront. Therefore we will ward the rear.\" As Grueburn added, \"Though we are Giants, we counsel haste,\" Bluntfist chuckled. \"Yon tunnel\"\u2014she indicated the one on Covenant's right\u2014\"is rife with odors. It augurs unpleasantness.\"\n\n\"Be careful,\" Covenant warned them unnecessarily. \"We can't lose you.\"\n\nHe had to stifle an impulse to start running.\n\nThis tunnel climbed steeply; dipped down; rose again. It turned at odd angles. After a while, the clang of iron echoed after Covenant. Muffled snarls, thudding blows. Branl reported that Cavewights assailed Grueburn, Bluntfist, and Dast. But now the confines of the passage aided the Swordmainnir and the Master. They could afford to retreat as they fought, following the company. And soon they were able to beat back the creatures. The sounds of struggle faded.\n\nBranl continued to relay information from his kinsmen. In the distance ahead, the vanguard reached a branching. Four more Masters were there. These _Haruchai_ reported that they had found a cave, a space like a small cavern with a shallow basin for a floor and openings into other passages: a place where the companions could be questioned.\n\n\"I don't like it,\" Covenant complained to Branl. \"We're running out of time.\" And he did not want to hear accusations from the Masters.\n\n\"In this, they speak with one voice,\" replied Branl. His tone concealed his personal reaction. \"They require an account of our deeds and purposes.\"\n\n\"They will be answered,\" Stave returned. \"Yet I also mislike the prospect of delay. We can have no effect upon the outcome of the world if we do not achieve our ends before the Worm drinks of the EarthBlood.\"\n\nThe Humbled shrugged. \"If the Masters are denied, they may respond with denial.\"\n\n\"Oh, God,\" Linden sighed. \"Just what we need.\"\n\nCovenant swore to himself. Whatever else Linden had done, she had not lied to the Masters. But they might not be able to see past the fact that she had set in motion the Earth's ruin.\n\nAloud, he demanded, \"Can't you convince them, Branl? It doesn't matter why they're here. Hellfire! It doesn't even matter if Bhapa and Pahni lied to them. We need help. Holding us back now is just surrender. We might as well kill ourselves.\"\n\nThe Humbled held Covenant's glare. \"I cannot sway them, ur-Lord. I am not as I was. My thoughts no longer accord with theirs. They deem that they would not have acted as I have done. In their minds, they would have forestalled the Worm's awakening. This belief justifies their wrath.\"\n\nJeremiah was squirming. \"That's stupid,\" he snorted as soon as Branl finished: scorn thick as venom. \"Covenant wouldn't be here without it. And I wouldn't be _here_. I would already be helping Roger and that _croyel_ become _eternal_.\n\n\"Did you tell the Masters _that_?\"\n\n\"To what purpose, Chosen-son?\" countered Branl. \"They would reply that Corruption could not threaten creation while he was imprisoned within the Arch. And while he was imprisoned, much might have been attempted to thwart him. Only the Worm's awakening assures his triumph.\"\n\nBefore Covenant could think of a response that was not rage, Linden spoke. \"If it's up to me,\" she told the Humbled, \"I'll answer anything. I don't know how much time we have. I don't know if we can afford to stand around arguing. But the Masters are important. I'll do what I can.\"\n\n\"Chosen.\" Branl's visage revealed nothing. Yet when he bowed, he gave her his full respect. Then he turned away, bearing the company's only light down the tunnel.\n\nWell, damn, Covenant thought. My wife\u2014\n\nBaring his teeth, he tried to grin. When that failed, he concentrated on catching up with the Humbled.\n\nI am not as I was.\n\nAnd Linden was facing the most immediate of her fears.\n\nefore long, Covenant and his immediate companions reached the place where the tunnel forked. There four Masters awaited him. He recognized Ard and Ulman. The other two were Vortin and Samil.\n\nThe _krill_ lit momentary wonder in Vortin's eyes, and in Samil's, as they bowed to Covenant. It exposed their ire when they regarded Linden. But they did not linger. While Branl explained that they would help Grueburn, Bluntfist, and Dast guard the rear, the four men moved into the blackness of the passages.\n\nCovenant heard weapons behind him again. Giantish oaths echoed like gasps along the tunnel. Bluff Stoutgirth's voice harried Scatterwit and Blustergale.\n\n\"They are swift enough,\" a Swordmain responded to the Anchormaster. Grueburn? Bluntfist? \"Expostulation will not speed them.\"\n\nGritting his teeth, Covenant followed Stonemage with Linden, Jeremiah, and Stave. Among them, Branl strode along like a man whose uncertainties had become faith.\n\nThis passage also ascended and dipped as it wandered; but now each rise took the company higher into the mountain. Covenant had no idea where he was in relation to the ancient Heart of Thunder. His human memories of the catacombs were confused by the dangers which he and his companions had faced then. Surrounded by this darkness, this weight of stone, he could not imagine how far he still had to go.\n\nTired as he was, the erratic climb felt long. The vagaries of the rough corridor blocked his view in both directions: he could not see beyond the _krill_. Like the Giants ahead of him, those behind seemed insubstantial, as if they had faded from the world. Only Linden and Jeremiah were real. Branl, Stave, and Onyx Stonemage.\n\nBut then the tunnel angled downward so sharply that Covenant had to lock his knees to keep his balance. When he was free to look up again, he saw the end of the passage, an opening into a wider space. From his perspective, that space resembled a pit, black and bottomless. But the figures walking into and through it demonstrated that it was not deep.\n\nAs he followed Branl into the cave, he found his companions gathering near its center. The Ironhand and Cirrus Kindwind. Four of Stoutgirth's sailors. And Masters\u2014\n\nTheir number confused him until he realized that more _Haruchai_ had joined the company. They bowed to the Giants, gazed with closed faces at Covenant and Linden. Then they spread out around the space, leaving only scarred Canrik with Rime Coldspray.\n\nThe cave was vaguely circular, with walls that looked natural rather than hewn, and a knuckled ceiling like an array of clenched fists. The floor was a complex jumble of fallen rocks on sunken patches where the underlying granite had contracted or cooled, sending a fretwork of narrow cracks through the surface. Four more tunnels opened like throats at irregular intervals around the walls: gullets choked with darkness where the light did not penetrate. In pairs, the Masters moved to stand guard at each entrance.\n\n\"Thomas?\" Linden asked softly. \"Where are Pahni and Bhapa?\"\n\n\"They'll come.\" Covenant tried to sound sure. The Wightwarrens were vast; but when the mental communion of the _Haruchai_ reached enough of their people\u2014\n\nWhen or if.\n\nShe frowned. \"We need them. The Masters seem to think that I can answer their questions, but I probably can't.\" She fell silent briefly. Then she added like a sigh, \"I want to see Pahni and Bhapa again.\"\n\n\"Me, too,\" he muttered. He had already lost too many friends\u2014and he was going to lose more. He did not know how to avoid it.\n\nHe and Linden greeted Coldspray and Kindwind, acknowledged Wiver Setrock and the other sailors. Covenant scowled at Canrik, thinking, Be careful what you say. Be very careful. But he did not warn the Master aloud. Instead he turned to watch the Anchormaster and the remaining sailors enter the cave.\n\nBoth Furledsail and Blustergale were helping Baf Scatterwit now. She had torn open the stump of her ankle trying to walk as if she had not lost her foot, and every step left smears of blood. Nevertheless she grinned hugely as she rejoined her comrades.\n\nA few moments later, Grueburn, Bluntfist, and five Masters entered the small cavern. They arrived at a trot, but they slowed when they saw the rest of the company. Carmine streaks stained their limbs, their tunics and cataphracts; but little of the blood was theirs. They did not move like people with injuries. The two Swordmainnir approached their Ironhand. Dast, Ard, and Ulman joined the _Haruchai_ keeping watch at the entrances. Vortin and Samil took places with Canrik.\n\nRime Coldspray's jaws worked as if she were chewing curses. \"Here the Masters require answers,\" she grated, \"though every delay serves our foes.\"\n\n\"We do,\" said Canrik, impervious to her indignation. \"We comprehend exigency. Nonetheless we will await the coming of the Voice of the Masters, and of the Ramen Cords. Our thoughts have reached out to other Masters, and thence to more distant kinsmen. Handir and those with him now hasten toward us. They will stand in the presence of the ur-Lord before the end. They will demand sooth from Linden Avery, who has brought the Worm upon us, and has given rise to falsehoods.\"\n\nCovenant beat his fists together, punching the hard circle of his wedding band to control his ire. Linden was right. He repeated that to himself again and again. She was right. He could do nothing against Lord Foul if the Masters refused him. Five Swordmainnir, eight Giants from Dire's Vessel, Stave, and Branl were not enough to oppose thousands of Cavewights, never mind Roger and _moksha_ Jehannum and any other force that the Despiser summoned.\n\nDistinctly Jeremiah said, \"You don't know Mom.\" His eyes looked blank, as if he were thinking about something else. From his hands, black power ran like oil through the Staff's runes. \"Why would she lie? She isn't afraid of you.\"\n\nJust for an instant, Linden's features crumpled. Then she covered her face with her hands. When she lowered them again, her expression had hardened, and the gaze that she fixed on Canrik was bleak.\n\nAs if he were choking, Covenant asked the Master, \"How long do you expect us to just stand here?\"\n\nCanrik regarded him gravely. \"The pursuing Cavewights have been slain. No others gather within reach of our discernment. We conclude that they do not know where to seek for us. No assault is imminent.\" He glanced around at the Giants. \"And your companions must welcome any respite.\"\n\nHe appeared to believe that his reply would content Covenant.\n\nCovenant said nothing. In spite of his weak sight, he could see that the attitudes of the Masters had changed. They had recovered from their initial surprise. Now they conveyed more anger. They appeared to feel betrayed, not by Covenant himself, but by the fact of his presence. And they blamed Linden\u2014\n\nDays ago, they had been misled by an image of Covenant. When Roger had ridden disguised by glamour into Revelstone with Jeremiah and the _croyel_ , the Masters had failed to discern the truth. They had reason for doubt.\n\nNevertheless Covenant wanted to yell at them. Linden had already endured too much from Handir and the other Masters. She did not deserve more.\n\nUncharacteristically brusque, Stoutgirth Anchormaster told his crew to distribute food and water. Before they could comply, however, more Masters began to arrive. In groups of four, they entered the cave from various passages: a score of _Haruchai_ ; then two score. To the Giants, they bowed impassively. To Linden and Covenant, they gave flat stares as fierce as castigations. Jeremiah they seemed to ignore. Then they spread out to form a cordon around the company; but whether they did so to defend Covenant, Linden, and Jeremiah, or to defend against them, Covenant could not tell.\n\nHandir had once threatened to wrest Linden's implements of power from her.\n\nI won't stand for it, Covenant told himself. I can't.\n\nBut he could not make the Masters' decisions for them. _The necessity of freedom_ belonged to them, as it did to everyone else.\n\nStill pressures rose like water within him. Soon he would have to start raging at somebody, anybody, for no better reason than because he needed an outlet.\n\nHe bit down on his tongue to stifle a shout when Handir finally strode into the cave.\n\nThe silver of Handir's hair, and the scars which seamed his visage and forearms like emblems, testified to his years and stature. He was the Voice of the Masters, accustomed to authority.\n\nThree of his people accompanied him, but they were not alone. Among them, they escorted Manethrall Mahrtiir's Cords, Bhapa and Pahni.\n\nAt the sight, Covenant's anger fell away like a wave from a cliff. He could see that the Cords had changed. The Pahni whom he had known might have forgiven Linden for refusing to attempt Liand's resurrection. That girl might have run to hug Linden; might have shed tears of gratitude and relief. And brave, diffident Bhapa would have stood back only because he did not consider himself important enough to demand attention.\n\nNot now. Somehow both of the Ramen had inherited Mahrtiir's spirit. Pahni swept forward like a striking raptor, and her eyes were bright with vindication, keen as whetted iron. Bhapa approached more slowly, but not because he was reluctant or daunted. Rather he walked with the firm tread of a man who had been flensed of his weaknesses.\n\nThe two of them gave the impression that they had brought the Masters from Revelstone by force of will.\n\nPahni offered the Ironhand a Ramen bow. The Cord's gaze flicked among the Swordmainnir, counted their losses. Then she bowed again more deeply, acknowledging their fallen. But she did not greet Covenant. Although her eyes widened when she saw the Staff in Jeremiah's hands, her attention did not linger on him.\n\nThe look that she fixed on Linden was simultaneously proud and defiant. She seemed to dare Linden to tell her that she had done wrong.\n\nLinden started toward the girl, then stopped herself, biting her lip. Her eyes were bruises.\n\nBhapa's manner was more reserved. He honored Coldspray and her comrades formally. To Covenant, he bowed as well, saying only, \"Timewarden.\" His brows lifted as he regarded Jeremiah; but he, too, did not pause for wonder. In a voice as tight as a rope, he said, \"Some tidings we have received from Handir. We have been assured that Manethrall Mahrtiir has not fallen. For that we are grateful. But the tale of his transformation we must hear at another time.\" Then the Cord came to stand in front of Linden.\n\nHer arms lifted to him, but he did not grant her an embrace. Instead he sank to the stone; prostrated himself before her as if she had become his suzerain, as honored as a Ranyhyn.\n\n\"Bhapa\u2014\" Linden's voice broke. \"Oh, Bhapa.\" Tears ran down her cheeks. \"What are you doing? What's happened to you?\"\n\nSeverely the Cord rose to his knees. After studying her face for a moment, he stood. Dampness softened his gaze, but his manner did not relent. He spoke as if he were offering a pledge.\n\n\"Linden Avery, Ringthane and Chosen, I cry your pardon. I will account for my deeds. But I must first assure you that Cord Pahni is innocent of fault. She spoke as she did at my command. The blame of the outcome is mine and no other's.\"\n\nClarion as a whinny, Pahni announced, \"He has become my Manethrall. He has honored my life with service. Where he leads, I follow gladly.\"\n\nGod in Heaven, Covenant thought. What have you done?\n\nLinden bit her lip again, struggling to contain a torrent of emotions.\n\nJust for an instant, Jeremiah looked like he wanted to put his arms around her. But he caught himself, stepped back. He had known Bhapa and Pahni only for a short time\u2014and only through the veil of the _croyel_ 's derision. He did not know how to interpret what was happening.\n\nFrowning, Handir had contained himself while the Cords preceded him. Now he spoke.\n\n\"I am Handir,\" he said in an astringent tone, \"by right of years and attainment the Voice of the Masters. Much lies between us. It must be answered.\n\n\"We have learned to our cost that our discernment cannot pierce the glamour of Corruption's servants. Here we behold one who appears to be the ur-Lord, yet we have seen his like before. We require some assurance that he is indeed the ur-Lord rather than a new display of glamour.\"\n\nLinden flinched. Disapproval spread among the Swordmainnir. Branl allowed himself a scowl. Stave's mouth tightened.\n\nBut Covenant responded first. Handir's challenge brought back his anger in a rush.\n\n\"Oh, stop,\" he snarled. \"Branl must have told you who I am. Are you so sick with suspicion that you can't even trust one of the Humbled? The _last_ of the Humbled?\n\n\"Here, I'll show you.\"\n\nFiercely he stamped toward Jeremiah. To Jeremiah's surprise, Covenant slapped his halfhand onto the black wood, gripped it for a moment. Then he wheeled to face Handir again.\n\n\"Do you remember the test of truth? I thought you remembered everything. When Roger was pretending to be me, he didn't let Linden use the Staff. Hell, he didn't even let her touch him. Now she's my wife. My _wife_ , do you understand?\"\n\nThe _Haruchai_ were passionate about their mates\u2014\n\nThrough his teeth, Covenant gritted, \"I don't know what's bothering you people. I don't really care. You want answers? If you keep this up, you'll all answer to _me_.\"\n\nThen he stopped. Handir's evident satisfaction silenced him. Now he realized that Handir's demand had served an oblique purpose. Inadvertently Covenant had just confirmed Handir's authority; his right to judge.\n\nIt was possible that Handir had not doubted Covenant's identity\u2014\n\nThe idea made Covenant reel. What had Bhapa and Pahni told the Masters?\n\nWith a defiance of his own, Stave said impassively, \"Be at ease, Timewarden. The Masters crave stone, yet they stand upon quicksand. They are indeed misled. Uncertain of his devoir, Handir masks his deeper apprehensions.\"\n\nThe Voice of the Masters did not react to Stave's assertion.\n\n\"Then get to it,\" Covenant told Handir. \"At least one of us will by God answer your questions.\"\n\nDeliberately Vortin and Samil moved to stand beside their leader. They appeared to ask or expect Branl to join them; but the Humbled remained with Covenant and Linden, Jeremiah and Stave.\n\nSensitive to the tensions in the cave, Bluff Stoutgirth took his crew aside; out of the way. Glowering, Rime Coldspray did the same with her comrades, although the Swordmainnir plainly wanted to defend their friends. The cordon of the Masters tightened around them all. Within it, Pahni remained nearby, apparently waiting. But Bhapa stood with his back to Handir, facing Covenant and Linden. Again Covenant felt the force of the former Cord's new demeanor, his earned severity.\n\n\"Cord,\" Handir said harshly: a reprimand or a warning.\n\nBhapa ignored him.\n\n\"Ringthane and Timewarden,\" the older Raman began, precise as a garrote, \"the wrath of the Masters is mine to endure. You had no part in their misapprehension. Their umbrage rests, not upon a falsehood, but upon a withheld truth. For this, I again cry your pardon, Ringthane. The choice was mine. Cord Pahni spoke as she did at my command.\"\n\n\"And would do so again,\" declared Pahni. \"I serve my Manethrall as I do the Ranyhyn.\"\n\nBut Bhapa did not pause for her. \"The dilemma of the Masters is this. They did not know of your return, Timewarden, because I did not permit them to know. They were informed of the rousing of the Worm. They were told of the Worm's hunger for the Blood of the Earth. But naught was said concerning your resurrection, Timewarden. Your name was not spoken. I did not concede the knowledge that you were restored to us by the selfsame deed which awakened the Worm.\"\n\nIn spite of their stoicism, Handir, Canrik, and the other Masters betrayed their indignation. Branl must have explained Covenant's return. Nevertheless they were unprepared for Bhapa's confession. Millennia ago, the Bloodguard had trusted the Ramen\u2014\n\n\"Rather, Ringthane,\" the older Cord continued to Linden, \"I encouraged them in their belief that the blame for the world's doom is yours.\" His tone was a strangle-hold. \"Speaking as I had instructed her, Pahni gave them cause to imagine that your sole purpose from first to last has been the restoration of your son, that you have given no heed to the havoc which you have unleashed. Therefore the Masters have come seeking retribution for the final crime of the Earth.\"\n\nCovenant listened with his mouth open, wordless and appalled. Linden stared as if Bhapa had betrayed her: a man who had sworn himself to her. The Giants cursed softly, gripping their weapons. Only Jeremiah did not react. Apparently images of the Worm had reclaimed him.\n\nThen Handir barked, \"Enough! Am I a child, that a Raman must assume my place?\"\n\nSwift as threats, Samil and Vortin approached Bhapa. They grasped him roughly, dragged him aside.\n\nLinden looked like she might wail. To restrain her, distract her, Covenant said reflexively, \"This is my fault.\" Her distress was worse than his. She already blamed herself\u2014\"I should have told Bhapa and Pahni what to say. I didn't because I thought the Masters deserved a chance to make their own decisions. It never occurred to me\u2014\"\n\nWhat had possessed the Cords?\n\nLinden did not look at him. Her whole face seemed to plead with Bhapa.\n\nIgnoring Handir's indignation, Bhapa told her, \"Faithful to his word, the Ardent delivered us to the vicinity of Revelstone. There we were able to speak privately ere the Masters greeted us. The burden of my wishes I gave to Cord Pahni because her need was plain. I prayed that her passion would prevail where my own ire might undermine me.\"\n\nHe tried to say more; but Samil silenced him with a hand on his throat: a choke which nearly lifted him from his feet.\n\nWithout transition, Covenant's wedding band burned. Sudden fire crowded his mind, straining for release.\n\nThe threat to the Cord was too much for the Swordmainnir. In an instant, Rime Coldspray reached Samil and Bhapa, her glaive in her hands. Frostheart Grueburn followed a step behind her.\n\nAround them, the cordon of Masters closed like a noose. Giants brandished their weapons: swords and spears. Pahni's garrote appeared in her fists.\n\n\"Handir!\" Covenant snapped. \" _Handir!_ \"\n\nHandir's jaws bunched. He nodded once.\n\nSamil released Bhapa. Samil and Vortin stepped back.\n\nCovenant took a deep breath, made an effort to quench his heart's fire.\n\nIn a small voice like a cry, Linden asked, \"Why, Bhapa? Why did you do that?\"\n\nHandir spoke over her. To the Ironhand, he said, \"Withhold your blows. You cannot stand against us. For that reason, we will not strike. We scorn unequal combat. Samil sought only to impose silence upon the Cord.\"\n\nReluctantly Coldspray sheathed her sword. Grueburn and the other Giants lowered their weapons. As they did so, the Masters relaxed their ring around the company.\n\nTheir Voice faced Linden. \"We share your query, Linden Avery. We will hear it answered. But first we must have some confirmation of what has occurred.\"\n\nBefore she\u2014or Bhapa\u2014could protest, Handir turned, not to Covenant, but to Branl.\n\n\"Are your thoughts sooth?\" he demanded in the full light of High Lord Loric's _krill_. \"Stave has learned concealment. Therefore he is suspect. Concealment enables falsehood. Are you now likewise capable of falsehood?\n\n\"Is _turiya_ Herem truly slain? Has Linden Avery indeed restored a Forestal to the Land? Has her fated boy provided for the preservation of the _Elohim_ , and for an end to Kevin's Dirt? Have you defeated Sandgorgons and _skurj_? Does the ur-Lord now seek to challenge Corruption in Kiril Threndor?\"\n\nBranl lifted an eyebrow. Then he shrugged like a man who did not deign to take offense. \"I am _Haruchai_ ,\" he said. \"More, I am Humbled. I do not sully my mind with lies.\n\n\"Nor,\" he added more sharply, \"will I condone aspersion to the Ramen. As do you, Handir, Voice of the Masters, I require an account of their deeds. Yet they have been at all times steadfast and valiant companions. They have given of themselves utterly while the Masters remained effectless in Revelstone. I will endure no denunciation of them.\"\n\nHandir studied Branl. He appeared to search Branl's mind.\n\n\"We are not effectless now,\" the older man retorted. \"Two hundred Masters have entered the Wightwarrens, seeking Linden Avery and Kastenessen as we were urged. Two hundred more strive toward _Melenkurion_ Skyweir, where they, too, will give of themselves utterly against the Worm, if their arrival is not belated.\"\n\nAt once, Pahni countered, fierce and proud, \"Did the Ranyhyn consent to bear you?\"\n\nHandir glanced at her. \"You know the truth of this, Cord Pahni. Do not aggravate your fault with insolence. You will be judged when you have justified your deeds.\"\n\nThen he said to Covenant as much as to Branl, \"Ranyhyn bore us hither. Without their aid, we could not have come so swiftly. But the Masters who ride to _Melenkurion_ Skyweir do so on lesser beasts. The great horses declined to be ridden there.\"\n\nThe shining of Pahni's eyes resembled exultation. \"Thus the Ranyhyn approve Manethrall Bhapa's purpose.\"\n\nThe Voice of the Masters permitted himself a vexed frown. \"I do not hear you,\" he told the Cord. \"It becomes evident, however, that I must heed the last of the Humbled. By him, as by the ur-Lord's presence, the lies of the Ramen are exposed. Now Linden Avery's query must be answered.\n\n\"Bhapa of the Ramen, it is not in the nature of your people to scheme and mislead. Why have you betrayed their legacy? Why have you concealed necessary truths?\"\n\nCovenant was holding his breath. He forced himself to let it out. The idea that two hundred Masters intended to oppose the Worm directly appalled him. He shook his head to dispel images of pointless slaughter.\n\nWary and unrelieved, Rime Coldspray and her Swordmainnir studied Bhapa, measuring the man in front of them against their memories of him. The Giants of Dire's Vessel did not know the Cords, but they remained poised to support the Ironhand. Only Baf Scatterwit did not seem tense. She was chuckling to herself as if everyone in the cave amused her.\n\nJeremiah muttered something that Covenant could not hear. The boy scowled darkly, as if he were contemplating murder. The absence in his eyes suggested that he was watching the Worm burrow into _Melenkurion_ Skyweir.\n\nBhapa rolled his head to loosen his bruised throat. He came closer to Linden and Covenant. In the open center of the gathering, he stopped: a man who needed room for the fire of his emotions. His eyes were white flames in the surrounding gloom.\n\n\" _It was for this_ ,\" he told Handir in a tone of throttled fury. \"That you might here encounter the truth of the Ringthane, the Chosen, Linden Avery\u2014encounter it and _know shame_.\"\n\nThen he turned his back on the clenched repudiation of the Masters.\n\n\"Ringthane\"\u2014he addressed his appeal directly to Linden\u2014\"you are dear to me. My esteem you won by your care of Sahah, who is both Pahni's cousin and half my sister. No succor known to the Ramen could have brought her back from death, yet you contrived to do so.\n\n\"My heart you won in the aftermath of First Woodhelven, when you redeemed Manethrall Mahrtiir's life\u2014aye, and preserved also his place as my Manethrall. At that time, I could not have met the peril of these times without his guidance. Sparing him, you spared me also.\"\n\nLinden listened with tears spilling from her eyes, but she made no sound.\n\nThe older Cord's voice rose as he continued. Anger grated like thunder in the background of every word.\n\n\"And since those great deeds, I have been stunned to the soul by your devotion to your son, by your valor in the greatest extremity, and by your enduring love for the Timewarden. I know nothing of _turiya_ Herem, or of Forestals, or of _Elohim_. Yet I know with a certainty which surpasses utterance that the awakening of the Worm was the outcome of Fangthane's cunning, not of any desire for Desecration in you. You acted only upon your love for the Timewarden, and upon your love for your son.\n\n\"Linden Avery, Chosen, Ringthane, I am _offended to the marrow of my bones_ that these sleepless ones have dared to think ill of you. They have named themselves the Masters of the Land, but they do not _serve_. True service submits itself to the cause which it serves, deeming that cause holy. This the Ramen comprehend. True service does not judge the deeds which are asked of it. It does not consent to _this_ and refuse _that_ , according to the dictates of its own pride. It gives of itself because the cause which it serves is worthy.\n\n\"The self-will of these Masters _offends_ me. It is an offense to every good which they have sworn to preserve.\"\n\nAs if he were unaware of the lifting of Covenant's heart, unaware of the bright approval in the eyes of the Swordmainnir, unaware even of Linden's weeping, Bhapa said more softly, \"That is my justification. I did not mislead the Masters for the Land's hurt, or for their own. I merely\"\u2014he spat the word\u2014\" _encouraged_ them in their judgments and pride, praying that they would ride forth in wrath to confront Desecration. Thereby I hoped to impose upon them a confrontation with their own folly.\n\n\"If I must say more, I will add only that I did not invoke the Timewarden's name because I feared that the Masters would not heed it. When have they ever stood with him in his last need? I feared that their notions of service would compel inaction.\"\n\nThen the Cord was finished. Briefly he slumped as if his passion had drained from him. But after a moment, he squared his shoulders and lifted his head, bracing himself to accept the consequences of what he had done.\n\nLinden's only answer was to say his name like a sob as she went to him. To his look of surprise, she replied by putting her arms around him and holding him tight.\n\nCovenant wanted to weep himself. He wanted to laugh, and to shout out his joy in the Cords, and to rail at the Masters. But he contained his turmoil, set his own emotions aside in order to concentrate on Handir.\n\nFates of every description stood on the lip of a precipice. One misstep now might be fatal. Covenant should have felt dizzy; but he found that his faith was equal to this moment. Bhapa had brought the Masters to a crisis of rectitude, a challenge which would search their definition of themselves to its core. And here they had the power to save or damn Covenant's intentions. Nevertheless he was content to await the outcome. He called himself the Unbeliever, but he believed in Bhapa, whose name meant \"father.\" In Pahni, whose name was \"water.\"\n\nAnd he had always trusted the _Haruchai_.\n\nThe Voice of the Masters did not speak. His mien revealed nothing. No doubt he was engaged in a vehement discussion with his kinsmen; but they masked their thoughts.\n\nWhen Linden had satisfied her gratitude, she released Bhapa. Blinking to clear her eyes, she gave him a crooked smile. Then she turned to Pahni.\n\nClearly she was unsure of herself with the young woman. Pahni had not spoken a word to her since Linden had refused to attempt Liand's resurrection. Instead of offering to hug the Cord, Linden asked with an ache of yearning in her voice, \"My God, Pahni. How did you do it?\"\n\nHow had a woman who had been little more than a girl when she found her first love in Liand discovered the strength to face down the assembled Masters in Revelstone?\n\nIn spite of her slight stature, Pahni met Linden's question with an imperious air. She looked whetted, as if she had spent days applying her heart to a grindstone. Without hesitation, she replied, \"I made of my grief a form of rage. I spoke to excoriate, goading the Masters to bestir themselves. We are the life which remains. They could not stand idle while a mere Cord faulted them for permitting the world's Desecration. They had no answer for the charge which I brought against them.\"\n\nThey did not grieve. Therefore their bereavements ruled them.\n\nHarsh as the call of a hawk, Pahni added, \" _I_ do not cry your pardon, Ringthane. I am a Cord of the Ramen. I will not regret that I have abided by the command of my Manethrall.\" But then her manner softened somewhat. \"And I also am offended in your name. I, too, crave the shaming of the Masters.\"\n\nAt that, Linden covered her face with her hands.\n\nRelieved and grateful, Covenant went to Bhapa. When the older Cord met his gaze, he said without rancor, \"You took a hell of a risk. What were you going to do if it didn't work?\"\n\nBhapa's mouth twisted. He almost smiled. With a hint of his former diffidence, he said, \"Timewarden, I would have spoken of you. Your need outweighs my wrath. Had the Ringthane's name failed, yours might have prevailed\u2014though,\" he admitted ruefully, \"in that event the burden of shame would have become mine to bear.\"\n\nCovenant nodded. Under his breath, he murmured, \"You're a brave man. I'm glad you're here. But maybe you should have trusted them with the truth. This\"\u2014a twitch of his head indicated the Masters\u2014\"isn't settled.\"\n\nStill Handir and his people said nothing, revealed nothing. They guarded the cave and the company, motionless as graven images while they carried on their mental debate.\n\nImpatient for a decision, the Giants fretted among themselves. While Grueburn and Stonemage spoke in low voices to Bluff Stoutgirth's sailors, telling them more about Bhapa and Pahni, Rime Coldspray approached the Cords. She greeted them kindly in spite of her obvious exasperation, praised their courage, thanked them for their fidelity to Linden. Then, however, she reached the end of her endurance. Striding past the Ramen, she confronted Handir and Canrik, Samil and Vortin.\n\n\"Enough of this!\" she called so that every Master could hear her. \"While you query yourselves, our foes rally against us. Such uncertainty ill becomes you. If you will not stand with us, stand aside. We must attain Kiril Threndor.\"\n\n\"Must we then countenance shame?\" snapped Canrik. \"Is that your counsel, Giant? You who know nothing of the strictures which form and inform the _Haruchai_?\"\n\nThe Ironhand started to retort; but Handir gestured abruptly for silence. Ignoring Coldspray, he faced Covenant across the shining of the _krill_.\n\n\"Nonetheless this also is folly.\" He spoke with his accustomed rigidity\u2014and yet his tone conveyed a cry of protest. \"Doubtless Linden Avery has become a rightful white gold wielder. And your endeavors against Corruption have twice exceeded every expectation. Yet when the Worm feeds, wild magic cannot counter it. Only Law can withstand the Earth's destruction, but the Staff is held by a boy who has not mastered it. Why do you wish to expend our lives where no good outcome can be achieved?\n\n\"If we must be shamed, we will bear it. We are _Haruchai_. Yet it is cruel\u2014is it not?\u2014to insist upon our service in the name of folly. In the name of futility, ur-Lord. In the name of _waste_.\"\n\nCovenant grinned at him fiercely. \"You tell me. Which would you rather do? Die here fighting Cavewights? Take the chance that something good might happen? Or be swept out of existence while you stand around complaining about waste?\"\n\nThe Voice of the Masters paused for only a moment. Then he said without inflection, \"We will fight.\"\n\nCovenant clenched his fists; stifled an impulse to punch the air. \"Then get me to Kiril Threndor. Protect Linden as long as you can. Keep Jeremiah safe. And brace yourselves. We've already surprised the hell out of Lord Foul. Maybe we'll surprise you, too.\"\n\nAfter that, he could no longer contain himself. Turning away from Handir, he shouted at the ceiling, \"Did you hear that, you tormented bastard? The _Haruchai_ are going to _fight_!\"\n\nThe Ardent's last service had accomplished its purpose.\n\n## 9.\n\nParting Company\n\nCovenant wanted to talk to Linden, remind her that he loved her, do what he could to reassure her. In addition, he meant to check on Jeremiah. The boy's elsewhere gaze was changing: his whole face seemed to be changing. The silted hue of his eyes had acquired a crimson tinge, as if his irises were bleeding. And his visage looked leaner, deprived of its youthfulness by dismay and nascent horror. His hands no longer gripped the Staff tightly, no longer spilled the black flames of his transformed legacy. He may have forgotten that he held it.\n\n_As guerdon for his puerile_ _valor_ \u2014\n\nHe was losing his ability to ward himself from visions of the Worm.\n\nCovenant wanted to say something, ask questions, understand; give comfort if he could. But he had no time. While the echoes of his defiance lingered in the cave, the cordon of Masters surged into motion.\n\nResponding to the mental shouts of the sentries, _Haruchai_ sprinted toward the chamber's openings. Around the company and the Cords, a few Masters formed a protective circle: Handir and Canrik, Samil and Vortin, Dast and Ulman. Stave held the _krill_ high in one hand, hefted Cabledarm's longsword in the other. Branl readied Longwrath's flamberge.\n\n\"Cavewights,\" the Voice of the Masters announced, passionless as stone. \"They have massed their forces. Now they advance.\"\n\nCovenant spun, scanned the entrances. \"Where?\"\n\n\"On all sides, ur-Lord,\" Branl replied.\n\nNodding to the Anchormaster, Rime Coldspray and her comrades joined Handir's defensive formation. The sailors arranged themselves to support the Swordmainnir.\n\n\"Hellfire!\" Covenant's ring itched for use. He felt an irrational desire to fling wild magic at the knuckled ceiling. \"Then pick one! Which one goes toward Kiril Threndor?\"\n\nLinden's face was pallid with fright as she grasped Jeremiah's arm, prepared herself to pull him into motion.\n\nHe threw her off. \"Again?\" he protested petulantly. Then his voice darkened. \"Of course. We're always attacked.\" He sounded like a different person, someone older, inured to abuse. \"Somebody should tell them they're as doomed as we are.\"\n\n\"Jeremiah!\" cried Linden softly. \"Honey? What's happening to you?\"\n\nFor an instant, the boy's eyes rolled back in his head. Then he bared his teeth. His gaze came into focus.\n\n\"I'm getting it, Mom.\" Again he sounded different, as if this time he had arisen from some other grave. \"I don't care what Stave says. I'll show you.\"\n\n\"We do not know the way,\" Handir told Covenant. \"None here have trod familiar passages. We must estimate our road. We are certain only that Kiril Threndor lies in that direction.\" He pointed above and behind Covenant. \"We will endeavor to clear a path there\"\u2014he indicated the tunnel closest to Kiril Threndor's heading\u2014\"hoping to encounter other Masters. Their knowledge may extend farther.\"\n\n\"Sure,\" Jeremiah muttered. \"Why not?\"\n\nBhapa and Pahni stood with Stave beside Linden and Jeremiah. The Cords held their garrotes in their fists.\n\nCovenant heard a noise like the sizzle of rain on hot stone: running feet. It swept closer. Before he could respond to Handir, Cavewights charged into the cave on all sides. In an instant, they filled the space with chaos and howling.\n\nThey came brandishing spears and truncheons, falchions heavy as spars, axes shaped to behead Giants. They burst into the cave from every entrance in such numbers that they could have inundated their foes, left no one standing.\n\nBut they did not come so far. Three strides into the chamber, they crashed like breakers against a seawall of Masters.\n\nHardly able to understand what he saw, Covenant watched the warriors meet the attack with a fanged front. At each entrance, tight wedges of three or four men bit like teeth into the brunt of the charge. Even as they fell in spurts and gushes of blood, the _Haruchai_ drove confusion among the first creatures; forced them to veer away on both sides. Some of the Cavewights tripped over bodies, did not rise again. Others spilled past the formations and scattered their lives against a bulwark of Masters.\n\nThe wedges did not hold. They could not. There were too many Cavewights. But the _Haruchai_ were at their most devastating when they fought singly. As their front failed, they spun among their assailants, fighting as though carnage exalted them. They leapt and ducked, avoided and struck. Punches snapped arms, broke necks. Kicks dislocated knees, smashed feet. And many of the Masters snatched up weapons. They cut like scythes through the Cavewights, reaping entrails, brains, gore.\n\nNevertheless the creatures were many; and they had spent millennia nurturing their hatred and savagery, their resentment of peoples who had repeatedly foiled their singular dreams. They fought with the ferocity of beasts. Slaughtered themselves, they delivered slaughter in return. Covenant watched dozens of Masters go down amid scores of Cavewights. Wherever he looked, he seemed to see _Haruchai_ killing or crippling creatures\u2014and yet at every moment the Masters were driven back. Axes took heads, ripped torsos. Spears, bludgeons, brutal swords: all wrought havoc. Even the armed warriors died, cut down from behind while they slew the foes in front of them.\n\nCovenant could have stopped this\u2014but only by killing everyone in the cave, rendering every living thing to ash. His thwarted heart burned, accomplishing nothing.\n\nStill more Cavewights surged inward, striding long-legged over the mounting rubble of corpses. Their weapons flung red ruin. Step by step, the fighting closed around the company. Handir prepared his defense. The Swordmainnir waited with their blades poised.\n\nBehind them, Linden and Jeremiah faced each other, apparently arguing. Alarm stretched her features. He gnashed his teeth as if he were biting off hunks of desperation. She may have been shouting\u2014they both may have been shouting\u2014but Covenant could not hear them. Howls and screams deafened him, the sickening sounds of torn flesh, the hard smack of blows, the crack of breaking bones.\n\nAs if he were answering his mother, Jeremiah raised the Staff of Law. He held it over his head like a quarterstaff, braced to hammer down fire. The look in his eyes was agony.\n\nAbruptly Branl gripped Covenant's arm, turned him toward the tunnel where Handir had proposed to leave the cave. At the same time, the _Haruchai_ between the company and that exit changed their tactics.\n\nImponderably graceful amid the viciousness and turmoil, those Masters drew back, leaving an open line for the Cavewights, an aisle straight toward the clenched center of the defense.\n\nCovenant thought that he heard Linden yell, \" _Now_ , Jeremiah!\"\n\nRoaring triumph, the creatures rushed forward\u2014\n\nNow or never.\n\n\u2014and Jeremiah swung the Staff.\n\nBlack lightning raged from the shaft. Earthpower struck at the Cavewights, fire hot as an inferno. It set them ablaze as if their bones were kindling. Their roars became shrieks. Lit like torches, they blundered away, trying to escape.\n\nMore creatures charged. More creatures caught fire. Jeremiah screamed as if his efforts were claws tearing at his heart. His eyes wept anguish. Nevertheless he poured out power in a convulsion of killing.\n\nFor a moment\u2014if only for a moment\u2014he cleared a path.\n\n\"Now!\" Linden cried again. \" _Run!_ \"\n\nThis time, she was shouting at the Giants.\n\nThe company obeyed. Shielded by Masters and Swordmainnir, and then by the Giants of Dire's Vessel, Branl hauled Covenant forward. With Bhapa and Pahni, Stave herded Linden and Jeremiah. While the surviving _Haruchai_ gathered to ward the rear, the Land's defenders dashed along Jeremiah's path.\n\nA moment later, the boy's power failed. He crumpled as if his tendons had been cut. He dropped the Staff: he may have fainted. But Far Horizoneyes snatched him off the floor, cradled him without missing a step. Furledsail grabbed the Staff and kept running.\n\nCavewights crowded the passage ahead. They had only paused, shocked or startled by screaming. But while they were in the tunnel, their movements were constricted. With Canrik and Samil\u2014with Vortin, Ulman, and Dast\u2014Handir tore into the creatures, broke them like boughs in a rending wind. And those Cavewights that withstood the force of the _Haruchai_ fell to the blades of the Swordmainnir.\n\nTrampling bodies, the company gained their exit.\n\nBut now the Masters also were hampered. Their speed and agility became less effective. Dodging a spear, Ulman stepped into the stroke of a falchion. The blade opened his side, cut deep enough to reach his spine. He fell, fountaining crimson. The other warriors in the lead survived only because they were supported by the swift skill of the Swordmainnir, the lick and thrust of longswords.\n\nThe _Haruchai_ holding the rear did so without the aid of battle-trained Giants. The Anchormaster and Frothbreeze gave what aid they could: still the losses among the Masters were grievous. While they struggled against swords and axes, massive clubs, they also had to contend with spears hurled over their heads to strike at Stoutgirth's crew. Leaping to intercept some of those shafts left the Masters defenseless. They were cut down or spitted.\n\nBehind the warriors, Keenreef and Setrock swung their sacks of supplies, blocked spears with bundled waterskins and food.\n\nAs the Masters died, the Cavewights drove closer. How many _Haruchai_ remained in the rear? Ten? Less?\n\nCovenant heard Scatterwit laughing amid the clamor: a horrific sound, shrill and urgent, feverish as hysteria. It jerked him around to watch as Scatterwit thrust her way among the Masters. Stoutgirth's shout, and Blustergale's, carried after her, but she ignored them.\n\nLurching on the stump of her ankle, she rushed the Cavewights with her arms spread wide as if she wanted to embrace every creature within reach.\n\nIn an instant, the point of a spear jutted from her back. A truncheon crashed onto her left shoulder. An axe bit between her ribs on the right. Her laughing was cut off; but she did not falter. Four, no, five Cavewights she hugged to her chest. Using them as a shield, she drove her great strength and weight against the pursuing creatures.\n\nFor a moment, she was impossibly successful. Somehow she cleared a space between her comrades and their foes. Five paces. Seven. Ten. When the blade of an axe came down on her head, spilling brains and ruined bone, she sagged. Still her legs thrust her forward. Supporting herself on the creatures in her arms, she kept fighting.\n\nThen she was done. Strength and life drained out of her: her legs failed: she dropped to her knees. Propped upright by corpses, she knelt there until her foes hacked her to pieces.\n\nScreaming, the Anchormaster tried to follow her. Frothbreeze and Blustergale caught his arms, held him back.\n\nRage filled Covenant's throat. He could hardly breathe. \"The _krill_ ,\" he gasped. \"I need the _krill_!\"\n\nScatterwit had opened a gap. If he could reach the rear before the Cavewights resumed their advance\u2014\n\nStave and Branl must have understood him. Without hesitation, Stave slapped the bright _krill_ into Covenant's hands. At the same time, Branl moved past Covenant. With one arm, the Humbled parted the sailors so that Covenant could pass.\n\nWhile Linden cried his name, Covenant brought up a rush of wild magic.\n\nBut he did not unleash its raw force. Instead he shaped silver fire along the blade of the _krill_. As he had done against the Sandgorgons, he fashioned an argent sword fierce as the white core of a furnace.\n\nWith Branl, he went to meet the Cavewights.\n\nBehind the two men, the rest of the company fled, following Handir's embattled cadre and the striking Swordmainnir. Supported only by the last of the rearguard, Covenant and Branl carried bloodshed among their attackers.\n\nCovenant made no attempt to defend himself. He had no skill, and was burning too hotly to care. He left his own protection to Branl's flamberge, to the fleet prowess of the few Masters. Wielding his chosen theurgy, Covenant became incarnate killing.\n\nWith every slash and thrust, every frantic swing, he appalled himself. He had to goad himself with curses like groans in order to keep moving. Otherwise he would have plunged to his knees, crippled by abhorrence. The Cavewights were only simple in their thinking: they were not unintelligent. And they had a long history. On their own terms, they had a civilization. They had never deserved the use which Lord Foul had made of them. They did not deserve what Covenant did to them now.\n\nHe promised himself that the Despiser would pay for this; but no promise sufficed to condone such slaughter.\n\nBranl and the Masters exacted their own toll. They were as precise as surgeons, as fluid as wind. But where they cut and blocked, punched and fended, Covenant ravaged.\n\nThe Cavewights seemed endless. Those still alive after the struggle in the cave were joined by more issuing from the other passages, entire hordes of creatures mad with blood-lust and ancient resentments. Yet even they could not withstand a blade forged of wild magic that shone like condensed stars. Nor could they match the skill of the _Haruchai_. Their screams and shrieks raced back down the tunnel, pierced the hearts of the Cavewights behind them. Their rage became fear. It became terror and panic. Fighting the press of their fellows, they tried to flee.\n\nAt first, they failed. The creatures advancing from the cave were not yet afraid. They resisted the impulse to retreat. But loud desperation filled the passage. It flooded through the Cavewights, carried away their fury. They turned to run, leaving their piled dead to guard their backs.\n\nThere Covenant flinched to a halt. His eldritch longsword frayed and faded: the _krill_ dangled in his numb clasp. Hellfire, he tried to say. Hell and damnation. But he could not catch his breath. There was no air anywhere. There was only blood.\n\nBlood and bodies, some still writhing in their last throes.\n\nIf he had been able to speak, he would have asked Branl and the Masters to forgive him. Of the _Haruchai_ guarding the rear, only seven remained; and most of them bore wounds. How many of them had already given their lives? Covenant could not bear to guess.\n\nSurely he had the right to defend himself? To fight for the people he loved, and for their world? Surely the Despiser was responsible for all of this blood?\n\nOf course, Covenant told himself. But the fact of his antagonist's malevolence did not relieve him of culpability. He had done so much of the actual killing\u2014\n\nThere was a price for such deeds. He intended to pay it\u2014as soon as he could breathe again. As soon as he found his way to Kiril Threndor.\n\nWithout a word, Branl took his arm, urged him into motion. Beyond the _krill_ 's reach, the rest of the company had vanished around a bend in the tunnel. But he could still hear fighting. Muffled by distance, blows and yells echoed out of the darkness. Clearly Handir's comrades and the Swordmainnir were able to beat back the Cavewights blocking their path. But the creatures had not given up. They contested every step.\n\nThey were not _Haruchai_. They had no way of knowing what Covenant had done\u2014and could do again.\n\nPulled into a trot, Covenant ran after his wife and his friends, stumbling on his numb feet like a man who had never drawn a clean breath.\n\nPast the bend, he nearly fell when the _krill_ 's light revealed the body of a Swordmain among the strewn corpses of Cavewights.\n\nCirrus Kindwind sprawled against the wall, propped at an awkward angle by a spear driven through one eye and out of the back of her skull. Her longsword lay a few paces away, as if she had tried to throw it with her last strength. Her features had closed around the spear: they held it in place like an act of defiance.\n\nShe had been fighting in darkness. Covenant carried the only light.\n\nBlinded by intolerable tears, he ran again, trusting Branl to guide him.\n\nAbruptly the sounds of fighting ahead ceased.\n\nQuiet as the dark, Branl said, \"Other Masters have come to assail the Cavewights. The way has been cleared.\" After a moment's pause, he added, \"It will not remain so.\"\n\nCovenant tried to clear his vision, but he saw no sign of his companions. He found only bodies and spilled fluids rank as offal.\n\nThe tunnel turned again. It rose steeply. At the top of the incline, he had to clamber over terrible mounds of the dead. He feared to look at them; feared to see some of Handir's people, another Swordmain, the Cords. Linden or Jeremiah. His friends had been fighting an uphill battle when they were rescued.\n\nBeyond heaps of Cavewights, he caught up with the company.\n\nAt first, he could not see past Bluff Stoutgirth and his crew. They had spread out in a wider section of the passage: their tall forms blocked his view. But then the sailors stepped aside, and the _krill_ 's silver fell on other survivors.\n\nIn the vanguard, the Voice of the Masters stood with Canrik and Dast, Vortin and Samil. They had been joined by nine or ten of their kinsmen. A quick glance showed Covenant a multitude of wounds and stains. Nevertheless all of the _Haruchai_ bore themselves as if their hurts were superficial; as if they had not lost scores of their people, and had never known sorrow. Closer to Covenant, still heaving to control their breathing, Frostheart Grueburn, Onyx Stonemage, and Halewhole Bluntfist waited with the Ironhand. Gore streaked their cataphracts: their longswords trembled in hands made weak by weariness. But their injuries looked shallow. Only the darkness in their eyes betrayed the loss of Kindwind.\n\nStoutgirth's dismay was more overt. His jaws worked as he tried to summon some sound from his throat, some shout or cry which might relieve his pain. Yet he remained mute: a man for whom all laughter had gone out of the world. At his side, Squallish Blustergale wept openly. The other sailors hung their heads in shock and fatigue.\n\nBhapa and Pahni stood apart from the rest of the company as if they had no place in it. They had not fought. Nor had they known any of the fallen except Cirrus Kindwind. And they were Ramen, lost without open skies to unfetter their spirits.\n\nAmong the Giants, Covenant found Linden and Jeremiah with Stave.\n\nThe boy was conscious now; on his feet. He had reclaimed the Staff of Law. Holding it upright, he scowled at his hands as they moved over the shaft, tracing the runes as if he were searching the written wood for the answers to questions which he did not know how to ask. He did not glance up when Covenant arrived. His concentration excluded everyone.\n\nBut Linden's gaze leapt at once to her husband. Her mouth shaped his name.\n\nThe sight of her made Covenant feel like weeping again. He recognized the complex consternation in her eyes: fear for her son and her friends, and more particularly for him, combined with a flagrant dread which had not yet become resolve. And something else, a kind of horror\u2014\n\nUntil he saw her expression, he did not realize that he was drenched in blood.\n\nHe went to her at once. But he did not touch her; foul her. He did not dare. His hands made truncated gestures, then fell back to his sides. The _krill_ in his grasp cast cavorting shadows that seemed to mock the faces around him.\n\nLinden's mouth repeated his name. Thomas. And again, Thomas.\n\nHandir moved among the Giants toward him. \"Ur-Lord,\" said the Voice of the Masters, \"we must not delay. Two paths to Kiril Threndor are now known.\" He must have acquired them from the minds of the newcomers. \"One is the more direct. It is also the more perilous. If we must, we will attempt it. We await only your word.\"\n\nJeremiah stamped the Staff on the stone. His voice cracked. \"We don't have time. Don't you understand? The whole _mountain_ is coming down.\" He did not look up from his hands. \"The Worm doesn't even feel it.\"\n\nCovenant groaned. _Melenkurion_ Skyweir was falling like Kevin's Watch. Hellfire\u2014\n\nLinden studied her son. Her face twisted. Then an obstacle within her seemed to break; or perhaps she pushed it aside. She went to Covenant, threw her arms around his neck, pressed all of herself against his soaked T-shirt and jeans as if she ached to embrace his sins, his accused soul.\n\n\"Thomas,\" she breathed in his ear. \"Oh, Thomas.\"\n\n\"Ur-Lord,\" Handir repeated more loudly.\n\nCovenant dropped the _krill_ so that he could wrap his remaining strength around his wife. What else could he do? He had no words for his distress; no language that might soothe his clawed heart. He was going to lose her. The Worm was making his choices for him.\n\n\"Are you sure about this?\" Linden asked in an aching whisper. \"I mean about Kiril Threndor?\" She may have meant, About everything? \"Are you sure that Lord Foul is there?\"\n\nAre you sure that you want to face him?\n\n\"Of course he is.\" Covenant clung to her acceptance. \"Or he will be when I get there. Where else would he be? Sure, he wants us all dead.\" All except Jeremiah. The Despiser had probably laid a _geas_ on the Cavewights so that Jeremiah would be spared. \"But if that doesn't work, he wants me to find him. He wants the pleasure of finishing me.\"\n\nSo softly that Covenant barely heard her, Linden murmured, \"Then help me. I can't do this.\"\n\nHe wanted to tell her, You can. You're the only one who can. But he did not. She had heard his professions of faith often enough.\n\n\"Ur-Lord!\" insisted Handir; but Covenant was not listening. He was already covered in blood. It was too late to count the cost. Maybe someday he would be forgiven.\n\nHe released Linden. When she loosened her arms, he stepped away from her to confront Jeremiah. Deliberately he placed himself in front of the boy, braced his empty fists on his hips.\n\n\"Can you hear me?\" he demanded. \"I need you. You have to hear me. I need your help.\"\n\nLinden might rally if he could show her that her son was not as lost as he looked.\n\nJeremiah did not glance up from the Staff. Shadows seemed to redefine his face. In a caustic tone, as if he were speaking for the _croyel_ , he snarled, \"Then you might as well give up. I can't even _see_ you. I can't see anything. The Worm is under that mountain. That's all there is.\"\n\nThinking, Forgive me, Covenant barked, \" _Jeremiah!_ Snap out of it! You think this is bad? It's going to get worse. Have you forgotten? Foul wants to _use_ you. He's going to do you more damage than you can imagine.\"\n\nThe boy flinched as if Covenant had struck him. Darkness writhed across his visage.\n\n\"Thomas!\" Linden objected.\n\nCovenant ignored her.\n\n\"Right now, he's just softening you up. Soon he'll get serious. He'll try to tear you apart, turn you inside out, hurt you so much you'll be _eager_ to do what he wants. If you don't help me, he wins.\"\n\nLinden tried to come between Covenant and Jeremiah. Stave held her back. The spurned _Haruchai_ seemed to understand\u2014\n\nJeremiah looked like he wanted to weep. In a different voice, abused and abject, he whimpered, \"I can't\u2014\"\n\nAs if he had lost patience, Covenant retorted, \"You _can_. You have that right. You were _born_ with it. All you have to do is choose,\" _must_ or _cannot_. He pushed his fingers through his hair, tried to harden his heart. Deliberately harsh, he rasped, \"Otherwise you might as well go back into hiding. You'll be useless.\"\n\nSlowly Jeremiah's silted gaze settled into focus on Covenant. He seemed to return from some other dimension of reality; some private hell. When it came, his answer was distinct.\n\n\"I don't want to go back there.\"\n\nCovenant felt like cheering. Grimly he stifled the impulse. \"Then trust yourself. Trust the Staff. There's a way to fight back. You just have to find it.\n\n\"And remember I need you. You might do something better than surprising the Despiser. You might surprise yourself.\"\n\n\"Ur-Lord,\" Handir demanded, peremptory as a cudgel. \"Do you not hear me? Every delay is fatal. You must select a path.\"\n\nStill Covenant ignored the Voice of the Masters. He had to face Linden.\n\nShe was glaring at him, furious and bitter. Her hands clenched as if she wanted to hit him. He had hurt her son.\n\nBefore she could speak, he said harshly, \"Maybe I'm wrong.\" With the fingers of his halfhand, he massaged the scar on his forehead. \"Maybe I'm not. Look at him. What do you see?\"\n\nFor a moment longer, her indignation raked Covenant; but she could not refuse him.\n\nWhen she focused her senses on Jeremiah, her eyes went wide. Realizations scudded across the background of her gaze. In a startled tone, she breathed, \"You brought him back.\"\n\nCovenant nodded. He felt suddenly drained, weak in every limb, as if he had passed a test which might have broken him.\n\nTo Handir, he said in a wan voice, \"The direct road. Jeremiah is right. We don't have time for anything else.\"\n\nHe knew what _direct_ meant. It would require more killing.\n\nRime Coldspray stood over the old _Haruchai_. \"If the path is perilous,\" she asked, \"what form do its hazards take?\"\n\nHandir frowned up at her. \"For a portion of its length, Ironhand, we will be exposed to assault on all sides.\"\n\nShe snarled a curse. Then she gave Covenant a look full of reflected argent. \"Aye, Timewarden. If we must kill and die, then let us do so swiftly and be done.\"\n\nAt once, she turned to the sailors. Sure of herself now, she told them to help the Masters guard the rear of the company.\n\nBranl had retrieved the _krill_. As he restored it to Covenant, he said, \"Be wary, ur-Lord. Your son has not yet opposed us. _Moksha_ Raver remains. And we do not doubt that Corruption has other servants.\"\n\nWith a mental command, Handir sent the newly arrived Masters to support the sailors. Joined by Canrik and Dast, Samil and Vortin, he started along the passage. The Swordmainnir followed at his back. Gesturing for the Cords, Covenant accompanied Branl. Stave urged Linden and Jeremiah forward.\n\nThrough the thick midnight of the Wightwarrens, Covenant bore the only light. He tried to hold it steady, but his arm wavered like his thoughts. Be wary. Roger and Cavewights and _moksha_ Jehannum. Cirrus Kindwind. Baf Scatterwit. Scores of slain _Haruchai_. And for what? Not for him. Not even for Linden. Lord Foul was not afraid of them. He believed that he had already triumphed. Nothing that they did could stop the Worm.\n\nNo, it was all for Jeremiah: all the threats and bloodshed, all the striving and woe. So that the Despiser would be able to take him.\n\nCovenant could only pray that Jeremiah would eventually find a way to resist.\n\nefore long, the company's progress became a running battle, frantic and almost continuous. The tunnel branched more frequently, intersected other passages; and at almost every junction, massed Cavewights waited, or small bands of Masters, or both.\n\nWith their acquired weapons\u2014heavy falchions, spears nearly as tall as Giants, axes that Covenant could not have lifted\u2014Handir, Canrik, and their comrades led the way. Deceptively swift, they slipped among their foes, slashing or stabbing at exposed limbs, throats, groins. Together they disrupted one attack after another.\n\nAnd behind them came the Swordmainnir. Rime Coldspray and her women fought in a kind of fury, pitiless and brutal. Their blades flung blood. Crimson stained the air, streaked the walls, glazed the floor. They wore it as if it nauseated them, but they did not falter.\n\nCavewights went down, screaming or already dead. _Haruchai_ fell as well. New warriors joined the company. Together they hastened from one struggle to the next.\n\nFor the time being, at least, Lord Foul's forces did not attack from the rear. Masters reported that Cavewights crowded the tunnel behind them; but the creatures appeared content to follow at a distance. They feared Covenant's wild blade\u2014or they desired a surer chance to strike.\n\nGuiding by newly acquired memories, the Voice of the Masters turned left at one branching, passed straight through two intersections, angled sharply to the right at a third. The passage lurched upward in stages like terraces. The cries of the dying trailed like spectres behind the company.\n\nThirty or more _Haruchai_ had now joined the company. The losses of the Cavewights were far greater. But there were thousands of Cavewights. Tens of thousands. At present, the tunnels themselves were the company's best defense. Covenant and his companions survived primarily because Handir's route avoided another open killing field like the cave.\n\nAt Covenant's side, Linden hurried as if she were hunkering down inside herself, trying to make herself too small for her fears to find her. Nevertheless she remained Linden Avery. With Jeremiah's consent, she borrowed Earthpower from the Staff at intervals and spread it among the Giants, fed vitality to Covenant's dwindling reserves. To Bhapa and Pahni, she offered the same gift; but they declined it. They were Ramen. They could have run through these passages indefinitely, fleet as horses in spite of their loathing for enclosed spaces.\n\nTo the Masters, she gave nothing. She knew better.\n\nAlong the way, Jeremiah made his own use of Staff-fire. Instead of extending flames outward, however, he appeared to draw them into himself. They ran up his arms as if he intended to broil his own skin, excoriate himself. Then they faded into his chest. And as he absorbed the Staff's magicks, his eyes darkened until they seemed to refuse light. They glittered at Covenant's silver like chunks of obsidian.\n\nCovenant had no idea what the boy was trying to accomplish, but he did not question it. He had provoked this reaction. Now he had to trust it.\n\nThe gem's shining restricted his view ahead, but he thought that he saw\u2014\n\nAbruptly he lowered the dagger, shaded it with his free hand. \"Branl?\" He was breathing too hard to articulate a question. \"Branl?\"\n\n\"Indeed, ur-Lord,\" the Humbled replied as though he understood.\n\nThere: in the distance above Covenant, beyond the dark shapes of the Swordmainnir hastening upward temporarily unopposed: a faint glow. Reddish, but not crimson; warmer and more yellow than the laval eyes of the Cavewights. It seemed to flicker as Giants interrupted it, but Covenant could guess what it was.\n\nThen Linden grabbed his arm, breathed his name; and he was sure.\n\nRocklight. The company was approaching one of the lit regions of the Wightwarrens.\n\nThe glow grew stronger. Summoned by Handir, Masters from the rear ran past Covenant and Linden, Jeremiah and the Cords. Fresh rage and iron rang along the passage. Hard impacts. A rabid stutter of screams, howls, frenzy.\n\n\"We will ward you,\" Stave said suddenly, \"but you must also defend yourselves. A multitude awaits us.\" He touched Jeremiah's shoulder. \"Do you hear, Chosen-son? You must turn your thoughts to our peril. It may be that our lives will require your aid.\"\n\n\"What do you want from me?\" Jeremiah panted. \"More killing? That's not what Law is for. I can't forget the Worm. I'm not strong enough.\"\n\nLinden regarded him with desperation in her eyes.\n\n\"Then don't worry about it,\" said Covenant between sickened breaths. \"You're getting ready for a different kind of fight.\" As was Linden. \"Leave this one to the rest of us.\"\n\nTo Giants and _Haruchai_. And to Covenant himself, who had already shed enough blood to drown him.\n\nHe gave himself no other choice. In a former life, he had turned his back on power. Now he demanded it of himself.\n\nRocklight washed over him. Rime Coldspray and her comrades passed an opening, spread out to both sides. Blows and shouts pounded down the tunnel, but the sounds were strangely muffled. A gulf seemed to swallow their force.\n\nStraining for air, Covenant went a step or two ahead of Linden and Jeremiah; ahead of Stave. The _krill_ he held at his side so that it would not blind him. With Branl, Bhapa, and Pahni, he drove his weakness out of the tunnel onto a ledge as wide as an avenue.\n\nThere he found himself facing a rocklit chasm.\n\nIt was not a fault or flaw in the gutrock, although it resembled a crevice: long and high, but not wide, little more than a stone's throw from wall to wall. Rather it had been fashioned, dug out over centuries or millennia. The ruddy light everywhere testified to the effort and theurgy which had formed the space. Overhead, and to left and right, it stretched beyond the reach of Covenant's dimmed sight. But when he moved closer to the rim of the ledge, he could see the bottom of the excavation: a crude trough crowded with debris, as full of refuse as a midden.\n\nIn spite of Stave's warning, he stopped and stared, momentarily unable to do anything except look. For a few heartbeats, he forgot fighting; forgot his peril entirely. He needed time to comprehend what he saw.\n\nA ledge opposite him resembled the one where he stood. It was the lowest of five, six, no seven levels like communal passages, each carved into the wall two or three Giant-heights above the next. And at the back of each horizontal cut, each shaped road, were openings like doorways. They measured out the chasm in both directions at intervals of perhaps twenty paces. Stone doors closed some of them. Others stood open, revealing lit chambers.\n\nHabitations. Covenant could hardly think. He struggled for air as if he were inhaling dismay. Dwellings. Homes.\n\nHomes implied families. Families implied children.\n\nThere were hundreds of doorways near enough for his failing vision; and the chasm was long. If the wall where the company had emerged mirrored the one across from it, the space held thousands.\n\nThousands of homes. The Cavewightish version of a city.\n\nAh, hell. Covenant had brought bloodshed to a place where the creatures were vulnerable, where their mates and children could be killed. A place which they would defend for reasons better than obedience to the Despiser.\n\nEverywhere he looked, he saw Cavewights mustering. On every level, armed bands gathered and ran, converging\u2014\n\nAny uncontrolled wild magic here would incinerate children.\n\n\u2014on bridges that spanned the chasm.\n\nHellfire! There were dozens of the damn things, wrought granite roads as wide as the ledges. A few stretched straight across, level to level; but most of them arced, connecting the walls at differing heights. On Covenant's left, the nearest bridge reached to the third level opposite it: another farther away on his right extended to the fourth. An elaborate and apparently random network of spans crisscrossed the space, giving every ledge access\u2014direct or indirect\u2014to every other.\n\nAnd on every bridge, Cavewights raced across the air, rushing to give battle.\n\n_\u2014exposed to assault on all sides._ Bloody damnation!\n\nCovenant wheeled on Branl. \"We have to get out of here! These are their homes! We can't start killing their _children_!\"\n\nThe Humbled shrugged. \"We do what we must. Foes now throng the passage at our backs. We have sacrificed the choice of retreat.\n\n\"Our path lies there.\" He pointed to the nearest bridge. \"From the third level opposite, we must cross to the fifth above us. At that height, a passage leads toward Kiril Threndor. Its constriction will defend us once again.\"\n\n\"Then _run_!\" Covenant yelled. \"Before they can stop us!\"\n\nHe could not unleash wild magic here. Even to save the Earth, he could not.\n\n\"Thomas!\" Linden clutched his arm, tugged at him. \"Look!\"\n\nFor an instant, his mind reeled. Then he dragged his attention away from possibilities which horrified him.\n\nOn both sides, his companions were already fighting.\n\nTo the left, the Ironhand and Frostheart Grueburn slashed like furies through the press of Cavewights. Among them, Handir and half a dozen Masters dodged and struck. Onyx Stonemage and Halewhole Bluntfist had gone to the right. With more _Haruchai_ , they held their ground against three times as many creatures. The cacophony of battle was terrible. It seemed more terrible because it dissipated in the high chasm as if it were meaningless.\n\nThe Giants of Dire's Vessel had arrived behind Covenant. The last six or seven Masters prepared to block the tunnel, protect the rear of the company.\n\nNow Covenant spotted more Masters on the levels above him: groups of four widely scattered. They were too few to save his companions; too far away.\n\n\" _Coldspray!_ \" he cried as if he were falling.\n\nThe Ironhand and Handir exchanged shouts. Coldspray bellowed commands at Bluff Stoutgirth. The Anchormaster answered with curses. His glare held madness.\n\nClutching their unfamiliar weapons, the sailors charged to the right. With strength and mass, if not with skill, they rushed to support Stonemage and Bluntfist.\n\nBhapa and Pahni hesitated for a moment, spoke to each other. Then they followed the Anchormaster.\n\nTogether _Haruchai_ , Giants, and Ramen began to force the Cavewights backward.\n\nAt the same time, Coldspray and Grueburn appeared to redouble their efforts. They chopped down creatures, tore through flesh and bone, flung bodies off the ledge. Handir and Canrik fought as one, striking high and low simultaneously. Samil and Dast knocked Cavewights off their feet. Vortin and his comrades broke necks, cracked skulls, disabled limbs.\n\nFor a moment, Covenant did not understand. The Ironhand had divided the company. Surely she had made it weaker? But then he realized that she had also divided the Cavewights. They were fighting for their homes now, not for Lord Foul. They rushed to oppose two threats instead of one.\n\nNear Covenant, Coldspray's tactics seemed to accomplish nothing. Only savage fighting pushed the creatures back.\n\nNevertheless fewer foes gathered on the bridges which the company had to cross.\n\nColdspray and Grueburn gained the foot of the nearest span, the shallow arc to the third level. Handir and his warriors fought to secure the Ironhand's position.\n\nCavewights tried to burst from the tunnel at Covenant's back. Masters repulsed them.\n\nBranl hauled Covenant after Coldspray. Covenant caught Linden's hand, pulled her with him. Stave brought Jeremiah.\n\n\"Mom!\" Covenant could barely hear the boy. \"What do you want me to do?\"\n\n\"Stay with Stave!\" she called back. \"He'll tell you!\"\n\nColdspray and Grueburn started onto the bridge. Ahead of them, a fresh onslaught of Cavewights came howling down the span. Branl and Stave followed the two Swordmainnir with Covenant, Linden, and Jeremiah. Somehow Handir and his comrades finished their immediate foes.\n\nSwift as swords, Handir and Canrik ran to join the Ironhand. Dast and Samil. Vortin and a few _Haruchai_ guarded the rear.\n\nAt the mouth of the tunnel, Masters died one by one. Numberless Cavewights gained the ledge. Some sped after the group escorting Covenant. Others pursued the sailors.\n\nWith Bhapa and Pahni, warriors reached the crossing to the fourth level. Stoutgirth and his crew fought as if they were caught in a hurricane. Stumbling on a slashed leg, Far Horizoneyes fell from the ledge. Blustergale scattered creatures with every swipe of his spear. Stoutgirth's shouts sounded like hysteria as he herded his crew onto the bridge.\n\nColdspray and Grueburn surged upward. Confusion spun through Covenant, lethal as vertigo, fatal as blades. The chasm gaped below him. It breathed his name. If Branl and Linden had not held him\u2014\n\nDizzy and wandering, he followed the Swordmainnir.\n\nThey were still a dozen strides away from a collision with charging Cavewights when other creatures began to fling spears from the upper levels.\n\nPartially protected by their cataphracts, the Ironhand and Frostheart Grueburn did not pause. Cursing fiercely, Coldspray hastened to meet her foes. Grueburn slapped shafts aside with her longsword or her open hand.\n\nHandir and his warriors formed a shield around Covenant, Linden, and Jeremiah. The _Haruchai_ countered a barrage of throws. Handir caught one spear, blocked another. A third pierced his chest, cast him silent as a stone into the chasm.\n\nSnagging shafts from the air, Canrik and the other Masters advanced as though their leader's death changed nothing. When one of Vortin's comrades mistimed a catch and was gutted, none of the warriors flinched.\n\nCovenant felt the shock as Rime Coldspray crashed against the torrent of Cavewights. She should have fallen: the impact would have split a slab of marble. Yet she stood. At her back, Grueburn braced her with one hand\u2014and Canrik, Dast, and Samil attacked as if they were born to the use of weapons\u2014\n\n\u2014and the enfilade of spears stopped. The creatures thronging along both walls could not throw now without hitting their own kind.\n\nCavewights plunged like detritus from the bridge as Coldspray and Grueburn powered ahead.\n\nOut of the heights, a boulder struck the span where the sailors and the Cords raced upward. Bouncing away, the stone took two Masters with it.\n\nAfter that, Covenant lost sight of Stoutgirth and the others. He hardly knew where he was. His boots skidded in blood: he could not imagine how Branl and Linden kept him on his feet. His mind was whirling madness. He seemed to rise borne on a gyre of carnage.\n\nThen he was gasping on the flat shelf of the third level, and Linden was shouting his name, urgent as fever, and the bridge back across the gulf to the fifth level was only a dozen paces away. Cavewights came from both directions, but he had no time for them. He caught his balance on the sight of the span he had just crossed. Up the curve slick with slaughter, more Cavewights rose like executioners; like deserved death. They poured from the passage where the company had entered this habitation, gushed upward in a flood released by the dying of Masters.\n\nThey were too many. That was all: they were just too many. The Swordmainnir and the _Haruchai_ were already fighting desperately, drenched in blood. Trusting Covenant, Linden, and Jeremiah to Stave, Branl sprinted to support his kinsmen. Jeremiah trembled on the verge of panic, ready to hurl black devastation in all directions. Linden stood with him, but she looked lost, unable to help him: appalled or paralyzed. A deranged part of Covenant wanted to demolish the whole place, children and families and everything living. He and his companions could not survive _more_ Cavewights.\n\nSuddenly calm, almost at peace with his dizziness, he went to face the creatures rising in rage up the bridge. Once again, he shaped wild magic along the blade of the _krill_ , formed a longsword of fierce argent. With it, he began hacking great hunks of granite out of the span.\n\nWhen the Cavewights there saw what he was doing, they froze.\n\nThree blows cut halfway through the indurated substance of the bridge. The fourth sent shivers down its length. The stone screamed at its own weight.\n\nShrieking, the creatures turned to flee. Most of them reached the lower ledge before the bridge fell in thunder. The rest plummeted.\n\nStill swinging, Covenant nearly followed the wreckage into the depths. Stave dragged him back.\n\nCovenant did not pause. Every thought was gone from his head: every notion or awareness except a compulsory desire to get his people out of here. He would never rid himself of the taste of blood. Brandishing slaughter, he ran to help his companions reach the next bridge.\n\ne and those with him were only able to gain the fifth level because new groups of Masters entering the habitation converged where they were needed. Fresh and unbloodied, they threw their lives into the mass of Cavewights. They were _Haruchai_. In a distant region of the Land, two hundred of them rode to oppose the Worm of the World's End with their bare hands. Fighting and dying like men who had never known fear and did not count the cost, they helped Rime Coldspray and Frostheart Grueburn clear the top of the span.\n\nOf the Masters ascending with the Swordmainnir, only Canrik and Samil remained. Branl alone guarded the rear, contesting every step with Longwrath's flamberge. Somehow Stave kept spears away from Covenant, Linden, and Jeremiah.\n\nFortunately the tunnel toward Kiril Threndor was near. And the Cavewights blocking the way had been scattered by unexpected Masters. From the opposite wall, more creatures came, loud as thunder, vehement as lightning; but most of them were not close enough to strike.\n\nStill they were too many, as they had been from the first. They would follow the company into the passage ahead. Eventually they would kill everyone.\n\nAt Canrik's urging, Coldspray and Grueburn led their companions into the blind dark of the tunnel. He and Samil joined Branl and Stave guarding the rear. The surviving Masters arrayed themselves at the opening, braced to die so that the Cavewights could not pursue.\n\n\"No,\" Covenant panted at them. \"Come with us.\"\n\nHe had seen too many _Haruchai_ killed.\n\nBranl silenced him. \"Will you seal the passage, ur-Lord?\"\n\nCovenant struggled to breathe. \"Yes.\"\n\nHe could not have done so in the earlier tunnels. The company might have needed to retreat. Now he had gained a path to the Despiser. There was no going back.\n\n\"Then,\" said Branl flatly, \"these Masters will aid the other Giants and the Cords.\"\n\nCovenant tried to move; tried to lift the _krill_. Are you serious? You want me to leave them out there? His arms refused to obey him until the warriors outside the tunnel met his frantic gaze and nodded their approval.\n\nEven here, they made their own choices. He could not gainsay them.\n\nGroaning curses, he forged fire along the blade of Loric's dagger for the last time. Unsteady as a man who had forgotten the use of his limbs, he slashed silver at the walls and then the ceiling. With wild magic, he cut down great chunks of stone until the passage was sealed.\n\nAfter that, he collapsed inwardly. He could still walk, still go where he was guided; but he could not think or speak. Images of slaughter filled his head. Wounds gaped at him like the grins of ghouls. The tumult of falling stone volleyed against the boundaries of his mind. So much killing. So many dead. And he had lost the sailors. He had lost everyone with them.\n\nHe had brought carnage into the dwelling-place of the Cavewights: just one more item on the long list of his crimes.\n\nWhat was it all for? Covenant knew his own reasons, but Lord Foul's daunted him. The Worm could not be stopped. At last, the Despiser could be sure of his long-sought freedom. Then why had he been so profligate with the lives of his servants? Did he simply _enjoy_ sacrificing them? Or did he secretly fear that Covenant might yet find a way to thwart him?\n\nNo. The Despiser knew Covenant too well.\n\nBut Lord Foul did not know Linden and Jeremiah: not with the same intimacy. The fane which had preserved the _Elohim_ and delayed the Worm demonstrated that he had underestimated Covenant's wife and her adopted son. Without their efforts, their opposition, he might already have escaped the Arch of Time.\n\nMaybe that explained the brutality of his defenses.\n\nThe tunnel rose. Dragging the weight of his sins behind him, Covenant trudged upward.\n\nAt his side, Linden stared ahead, wide-eyed as a woman who saw a holocaust waiting for her. Jeremiah wrung the Staff as though he wanted to twist it apart. His every step was a flinch. Leading their few companions, Coldspray and Grueburn slumped like derelicts. Only Stave and Branl, Canrik and Samil paced the ascent like men who could not be appalled by any sacrifice.\n\nA rift cut across the tunnel. It split the floor as though it had been made by an axe sharp enough to wound mountains. It yawned at Covenant, too black to be relieved by the _krill_ 's shining. But it was thin: a fracture no wider than his thigh. Pretending to ignore it, he stepped across.\n\nMore fissures appeared. They were little more than cracks, yet they served to remind him of the times when violence had torn through Kiril Threndor, Heart of Thunder.\n\nHe was getting close\u2014\n\nWhen the Giants halted, he nearly walked into them. Blinking and stupefied, he looked around.\n\nThey had entered a chamber like an exaggerated vesicle, a natural formation left behind by some accident of volcanism. The passage continued, but Coldspray and Grueburn stood wavering as if they had come to the end of themselves: they looked like they wanted to lie down. The cavity was more than large enough to accommodate them prone. It could have held a dozen sleeping Giants.\n\nTo one side rested a pair of large boulders. They seemed strangely out of place. Covenant could not imagine how they had come to be here. But plenty of room remained, and the floor was approximately level. When he found himself swaying on his feet, he realized that he was tired enough to stretch out and rest in spite of the Earth's peril.\n\nAnd yet his weariness was a drop in the ocean of Coldspray's and Grueburn's exhaustion. Even the _Haruchai_ were probably worn down, although they concealed it.\n\nGrueburn's longsword dangled from her fingers. \"Is it conceivable,\" she asked, plaintive as the cry of a distant tern, \"that we are done with combat? I cannot raise my arms.\"\n\n\"'The mightiest of the Swordmainnir,'\" muttered Coldspray dully. \"So I have vaunted myself, and so I am. Behold.\" She lifted her glaive. \"My hand is firm.\" It shook like a dying leaf. \"My eye is keen.\" Fatigue glazed her gaze. \"Beyond question, I am\u2014\" Abruptly she dropped her sword. Her shoulders slumped. \"Stone and Sea! I am undone by woe and killing. I cannot spit out the taste of blood. It will fill my mouth to the end of my days.\"\n\nSighing, Covenant roused himself enough to respond, \"Join the club.\"\n\nJeremiah said nothing. He appeared to have lost interest in everything except his ambiguous struggle with the Staff of Law. Folding his legs, he settled himself against one wall, sat cross-legged with the black wood resting across his thighs. His head he kept bowed as if he did not want anyone to see the darkness deepening in his eyes.\n\nLinden studied him for a moment, then turned away. She had spent too long clenched inside herself; too long crowded with needs and fears which she had not allowed herself to express. She was a rightful white gold wielder: for hours now, she could have struck her own blows. Yet she had contained herself, passive as dust amid the winds of battle. Somehow she had withheld\u2014\n\n_But I'm done fighting._\n\nIn spite of endless provocations, she had kept faith with her decision. The cost of so much restraint must have been severe. Now she seemed ready to explode.\n\nNevertheless her voice stayed clenched as she asked the Ironhand, \"What about the others? We left them to die.\"\n\nHer bitterness resembled the edges of Longwrath's sword.\n\nColdspray shook her head. \"They will not perish while they are able to fight and flee.\" She spoke as if she sought to reassure herself. \"Having lost us, they will retreat for their lives. My commands were plain. And Halewhole Bluntfist and Onyx Stonemage are Swordmainnir. They comprehend that they must not sacrifice the Anchormaster's crew and the Masters of the Land\u2014and assuredly not the Ramen Cords\u2014to no purpose. Rather they will seek an egress from the habitation.\"\n\nThen her tone frayed. It seemed to tear. \"Now we have played our part. Ask no more of us. We can go no farther.\"\n\nOnce before, Covenant had seen despair in the eyes of a Giant, when Saltheart Foamfollower had tasted the ecstasy of killing Cavewights\u2014and had found that he wanted to kill more. That despair had kept Foamfollower alive when all of his people were murdered. Coldspray's surrender, and Grueburn's, made Covenant want to weep.\n\nHe drew a shuddering breath. Well, then, he told himself. This is as good a place as any.\n\nHell and blood.\n\nTo the Ironhand, he said, \"Don't worry about it. You've brought us far enough. Nobody could have done more.\"\n\nThen, wincing inwardly, he told Linden, \"If you're going to do it, now's the time. You won't get another chance.\"\n\nOn the walls, silver made dark streaks like the ichor of mountains.\n\nAlarm flared across her face as she turned to him; but she did not protest. Instead she tightened her grip on herself, increased the pressure until it threatened to break her. \"Already?\" she asked without hope or humor. \"Are you sure? I still want to live.\"\n\nHer gaze said, I still want to live with you.\n\n\"Kiril Threndor isn't far.\" Covenant choked for a moment. He had to swallow a rush of grief. \"You can't go there with me. Neither can Jeremiah. This is it.\"\n\nAs if he were asking for forgiveness, he added, \"I'll take Branl. Jeremiah will have Stave and Canrik and Samil.\"\n\nShe looked away. Her eyes avoided Coldspray and Grueburn as if she felt shamed by the prices which they had paid for her. Instead she regarded her son again.\n\nTo no one in particular, she said, \"All right. I chose this. Some of those poor Masters might still be alive if I had made a different choice.\" She seemed to choke momentarily. \"Or Baf Scatterwit. Cirrus Kindwind. God, I loved her\u2014\n\n\"Losing them will be wasted if I change my mind now.\"\n\nCovenant's vision blurred. He squeezed his eyes shut to clear them. Taunting her, Lord Foul had called Linden his _daughter_. He was wrong.\n\nFrom the floor, Jeremiah asked suddenly, \"What're you talking about?\"\n\nLinden did not let herself look away. \"Jeremiah, honey\u2014\" Her voice was breaking. \"I have to go.\"\n\nIn one motion, Jeremiah surged to his feet, lifted his gaze into the light of the _krill_. His eyes were as black as the Staff. Even the whites had become midnight.\n\n\"Go where?\"\n\n\"I can't put it off any longer.\" She sounded tight enough to snap. \"I need to face the only thing that scares me worse than losing you. You and Thomas.\"\n\nHis face twisted. Protests clawed at his features. \"But you'll come back,\" he said as if that were not a question. \"That's what you do. You come back.\"\n\nShe flinched\u2014but she did not falter. \"I don't think so, honey. Not this time.\"\n\nJeremiah stared horror at her. \"You're going to leave me? You're going to let Lord Foul have me?\"\n\n\"No, Jeremiah.\" Her tone sharpened. \"I'm not going to _let_ him anything. But I can't fight him for you. Even if I took back the Staff and stood right in front of you, I couldn't help you.\" More gently, she said, \"I wish that I could spare you, but I can't. If you don't want him to take you, you have to stop him yourself.\n\n\"I know it's hard\u2014\"\n\nHer son cut her off. Vicious as a denunciation, he sneered, \"'I know it's hard.' You keep saying that. You don't know anything. I've already tried to fight. I'm not strong enough. The _croyel_ thought I was easy. How am I supposed to stop the Despiser?\"\n\nLinden shook her head. Her distress made Covenant ache. \"I don't know. But I believe in you. You can do it.\"\n\n\"I _can't_!\" His shout was like the tearing of flesh, full of pain and awash with blood. \"I'll have to watch the Worm destroy _everything_!\"\n\nCovenant's balance shifted. Only grief kept him from dropping to his knees. Only a whetted empathy kept him from raging at Jeremiah. But grief and empathy were enough. He braced himself on them when everything else spun away.\n\n\"You can always decide to give up,\" he said as if he were steady and sure; as if he had strength to spare. \"You have that right. If it's what you really want.\" Or the boy could join Lord Foul. \"But I need you. I'm going to need you absolutely. And Linden can't help me. Nobody else can. There's only you.\n\n\"But first we have to let Linden go.\"\n\nJeremiah flung a look black enough to kill at Covenant.\n\nA heartbeat later, the boy turned his back on his mother.\n\n\"Then go.\" He sounded as lightless and fatal as the path toward Kiril Threndor. \"You never loved me anyway. I was just an excuse. You don't want to have to blame yourself for letting me put my hand in that bonfire.\"\n\n\"Jeremiah\u2014\" Linden was weeping now. \"Honey\u2014\"\n\nAh, hell, Covenant thought. Visions of the Worm had raised all of Jeremiah's demons. He had spent days suppressing them. They ruled him now. Deliberately he sat down again, put his back to his mother; to Covenant and their companions. His hands wrestled ebon flames along the wood of the Staff as if he wanted to rewrite Caerroil Wildwood's runes.\n\n_Maybe we should all try to become gods._\n\nThe Giants watched blank-eyed, caked in drying blood, mute as cenotaphs. Branl studied Jeremiah with a speculative frown, as if he were considering where to cut the boy.\n\nCovenant gave the _krill_ to Stave. Then he took Linden's arm and pulled her away. While she stifled sobs against his chest, he held her tight.\n\nWith as much tenderness as he could manage, he promised quietly, \"I'll talk to him, love. He doesn't know it yet, but he's just proving your point. You can't do his fighting for him. No matter what happens to him, he's the only one who can do anything about it.\"\n\n\"Oh, Thomas.\" Distress shuddered through her, harsh as spasms. \"I'm so scared. What if he gets it wrong?\"\n\nFor a moment, she could not go on. She slumped against Covenant as if she had lost the will to stand on her own.\n\nHe hugged her in silence. He had no words\u2014\n\nBut gradually she responded to his embrace; drew a steadier breath. Freeing one arm, she wiped her face, smeared tears and blood across her cheeks. \"And I swore that I would love you as long as you never let me go. Now I'm the one who's leaving. I have to let both of you go.\"\n\nCovenant held her as hard as he could. \"I understand. You can't get rid of me this easy.\" Then he said more seriously, \"In any case, I'm like you. I believe in Jeremiah. He has to feel this way. If he doesn't, he won't ever get past it.\"\n\nAt one time, Covenant himself had embraced despair\u2014\n\n\"Also,\" Stave put in like a man who had been biding his time and was done with patience, \"you will not depart alone.\" The _krill_ shone full on his face; on the scar of his lost eye. \"Linden Avery, I have said that I will not be parted from you again. The Chosen-son I entrust to Canrik and Samil, and to the Swordmainnir. You I will accompany.\"\n\nSurprise seemed to loosen some of Linden's tension: surprise or relief. She ignored the former Master long enough to kiss Covenant quickly, wipe her face again. Then she turned to Stave.\n\n\"Do you know where I'm going?\"\n\n\"Mayhap.\" Stave may have smiled. \"Or mayhap I am mistaken. I care naught. At one time, I declared that Desecration lies ahead of you. Now I am persuaded that there is no Desecration in you. I will not stand at your side to ward against you. I will do so because I have not learned humility, though you have endeavored to teach me. I crave further instruction.\"\n\nHis assertion sounded like an example of _Haruchai_ humor.\n\nLinden tried to say his name. Apparently she could not. Instead she went to him, put her arms around his neck.\n\nPast her hair, Stave met Covenant's gaze. \"You have wed well, Timewarden,\" he said as if his characteristic stoicism had become a form of jesting. \"I will strive to ensure her return.\"\n\nCovenant nodded. What could he have said? There were no words in all the world for his gratitude.\n\nWhen Linden released the former Master, he returned Loric's dagger to the Unbeliever.\n\nCovenant took it; gripped it. His throat was as tight as his grasp on the _krill_. He had to force himself to ask Linden, \"Are you ready?\"\n\nThe corner of her mouth twisted: a failed smile that nearly broke his heart. \"I'm never ready. I've given up waiting for it.\"\n\nHe rubbed his scar roughly, tried to compose himself. \"Then remember I love you. I _love_ you.\n\n\"And don't worry about Jeremiah. You did your part. I refuse to believe anything you did for him is wasted. The rest is up to us.\"\n\nHer mouth said, \"I'll try.\" Her eyes said, Thomas of my heart.\n\nThe Giants offered her no farewell. Frostheart Grueburn set her teeth on her lower lip: a woman stifling protests. Tears streamed openly down her cheeks. Rime Coldspray hung her head as if she could not bear her weariness\u2014or her dismay.\n\nJeremiah did not look at any of them.\n\nTogether Linden and Stave moved to a clear space a few steps from the walls and the Swordmainnir. There they waited like contradictions or counterweights. His poised relaxation balanced her trembling tension. After a moment's consideration, he tossed Cabledarm's sword to Canrik. No weapon would serve him now.\n\nGrim as a deliverer of damnation, Covenant stood beyond Linden's reach. He could not afford to hesitate now. He had no time; and his resolve might fail at any delay. He knew where she was going. He was more afraid for her than he was for himself.\n\nAs if he had begun preparing for this days ago, he gave fresh wild magic to the dagger's gem and thrust the blade into the stone between his boots. When the hard surface caught silver, he dragged the _krill_ to the side, cutting granite like damp clay. Step by step, he sliced a circle around Linden and Stave.\n\nAlong the line of his cut, his power shone as if rock were the fuel for which it yearned.\n\nHe did not need a large circle to enclose his wife and the former Master. In spite of his awkwardness and grieving, he returned to his starting point quickly. Then he forced himself upright. Wild magic reached for the ceiling. Through its brightness, he met Linden's gaze.\n\nFor his sake, she kissed a promise onto her wedding band, held it up with her hand clenched.\n\nWeeping like Grueburn, Covenant slapped his ring against the _krill_ 's gem.\n\n_The world will not see her like again._\n\n_Care for her, beloved, so that in the end she may heal us all._\n\nToo late, Jeremiah cried out, \"Mom!\" Linden and Stave were gone.\n\nCovenant turned away as if he were falling.\n\nElena, he thought obliquely, I'm so sorry. I'm doing what I can. Somebody else has to care for you.\n\ntill he had no time. He could not afford his own weakness, or the wailing of his wrenched heart. He had to keep moving. He would find some form of peace soon enough.\n\nAh, God.\n\nJeremiah was standing now, showing Covenant a face fretted with ruin. \"I keep doing that,\" he said in such misery that Covenant wanted to turn away. \"It's like I don't even remember her until it's too late.\" His head hung as if he were talking to the floor. \"By the time I understand what she's doing, she's already gone. I don't even say goodbye.\"\n\n_I'm never ready._\n\nCovenant knew the feeling.\n\nHe allowed himself to postpone speaking to Linden's son for a moment. While he tried to gather up the shreds of his courage, he asked Branl, \"How much farther?\"\n\nThe Humbled glanced at the tunnel ahead. \"Kiril Threndor is near, ur-Lord.\" Then he frowned. Tension in the lines of his face betrayed anxieties which his tone concealed. \"Yet my heart misgives me. I cannot credit that Corruption has no other defenses close about him.\" Briefly he appeared to consult with Samil and Canrik. \"Also, ur-Lord, I do not discern Corruption's presence. His malice is particular. It cannot be mistaken. That some great evil awaits us is plain. Yet it is not Corruption. He is absent\"\u2014Branl cocked an eyebrow at a sudden thought\u2014\"or veiled by glamour.\"\n\nCovenant swore privately, but he could not pretend that he was surprised. Lord Foul knew that he was coming\u2014and the Despiser was cunning.\n\nRubbing numbly at the scar on his forehead\u2014the mark of his sins\u2014Covenant turned to Jeremiah.\n\n\"It's probably a good thing you can see the Worm.\" He did not try to be gentle. \"You'll know when it's time.\"\n\nJeremiah jerked up his head. \"Stop that.\" His doom was stark in his eyes. Tattered and soiled, stained with old blood, his thin pajamas made him look as unloved as an empty house. \"Stop saying things you know I can't understand. You keep saying you need me, but you won't tell me how or why. You act like you think I'm important, but I don't know what you're talking about.\n\n\"Why can't I come with you?\"\n\nCovenant grinned without humor or kindness. \"It's fun, isn't it. You're like all the rest of us. Nobody ever hands you an answer. The only thing you can do is guess. Then you have to take your chances.\"\n\nAt once, however, he reached out, wrapped the fingers of his halfhand around the Staff of Law. Another test of truth: he wanted the boy to believe him.\n\nTo his touch, the wood felt dead; almost brittle. Ripe for consummation.\n\nStartled, Jeremiah quenched his flames. But he did not look away. His gaze clung to Covenant's. For a moment, his eyes resembled the Harrow's, deep as voids, hungry for some life that was not his own. But slowly they became harder, flatter: the black of obsidian and anger.\n\nDistinctly Covenant said, \"You can't come with me because I don't want you that close to Lord Foul until I can distract him. But I do want you to come. I think you'll know when. You'll be able to sense it.\" He glanced at the Masters. \"Or Canrik and Samil will. Or watching the Worm will tell you.\"\n\nJeremiah stared.\n\nHolding the Staff and the _krill_ so hard that his forearms ached, Covenant tried to explain.\n\n\"I need you because I don't think I can beat Lord Foul by myself. You aren't strong enough? Neither am I. He's too much a part of me.\n\n\"When the Worm drinks the EarthBlood, the Arch of Time will start to crumble. That's when Foul can escape. More than anything else, he wants _freedom_. If he has to, he'll even give up trying to trap the Creator. Being stuck here\u2014\" Covenant let go of the Staff. He shoved his fingers into his hair and pulled, trying to drag his thoughts into language. \"There's no word big enough for that kind of despair.\"\n\nIf Jeremiah understood nothing else, he would understand _that_.\n\nAgain Covenant found himself swaying, unsure of his balance. His intentions became impossible as soon as he articulated them. He wanted to fall down; just hit the floor and lie there while he could still choose the moment of his last collapse.\n\nBut he had made promises to Linden. Hell, he had made promises to practically everybody, one way or another. And he could not turn his back on Jeremiah's distress.\n\n\"I need your help to keep him busy. If we can, I want to make him miss his chance. As long as he's stuck here with us, he'll be vulnerable. Then I might be able to find an answer of my own.\"\n\nIs that plain enough for you? Hellfire, Jeremiah! It's all I've got.\n\nThe boy glared blackness. His breath came in ragged chunks, as if the labor of his heart did not leave room for his lungs. He swallowed as if his mouth and throat were full of blood.\n\n\"I _can't_. Don't you understand? He's the _Despiser_. He can take me whenever he wants. I won't able to do _anything_.\"\n\n\"Oh, stop,\" Covenant snapped. He might have yelled, We're out of time! \"There's always something you can do. You have talents. You have the Staff. And you know what _possession_ is like.\" _He broke me_. _I hate being used_. \"If nothing else, you can just hide. You can hide as long as you want.\"\n\nJeremiah had freed himself from years of dissociation. Maybe he would be able to find his way out of Lord Foul's grip.\n\nThe boy bared his teeth as if he wanted to take a bite out of Covenant; but Covenant was done with him. Intuitively, if not with any of his truncated senses, he felt the end of Time approaching. He had to go.\n\n\"Help me,\" he finished. \"Don't help me. It's up to you. I'm out of time.\"\n\nLike a man who had recovered his balance, he turned his back on Jeremiah's stained struggle; on the lost boy's naked need. At one time, Covenant had risked the Land's ruin for the sake of a snake-bitten child. More than once, he had approved when Linden had made similar choices. This was different. No matter what Jeremiah believed about himself, he was not helpless. He was _not_.\n\nAnd Lord Foul did not understand him. After all of this time, the Despiser still had no real idea what he was up against.\n\nAs Covenant left Linden's son, Rime Coldspray spoke. In the _krill_ 's light, she looked like a closed door. Her voice was rusted iron, a blade gnawed by neglect. Yet her gaze was sure in its mask of blood.\n\n\"Do not fear, Timewarden. While we live, we will stand with the Chosen-son. If we cannot guide him, mayhap Canrik and Samil will do so. They have shown their worth. They will not fail in Stave's stead, or in Branl's.\"\n\nMute as an unmarked grave, Frostheart Grueburn nodded her assent.\n\nWith that hope, Covenant followed Branl out of the chamber.\n\nhe Humbled held Longwrath's flamberge ready. He walked lightly, silent as a breeze. _That some great evil awaits us is plain_. Behind him, Covenant stepped over cracks in the wracked stone, carrying the illumination of Loric's courage and lore into darkness. The tunnel twisted from side to side as if it were writhing. _I cannot credit that Corruption has no other defenses close about him_. Here and there, flecks of mica or quartz in the walls caught silver and glittered like eyes.\n\nMore fractures flawed the gutrock. The forces unleashed here must have been appalling: High Lord Prothall's struggle with Drool Rockworm for the old Staff of Law; Lord Foul's fierce and increasingly frantic efforts to destroy Covenant's spirit. Clutching the _krill_ , Covenant rushed past thin splits that called out to him, urging vertigo and surrender. He had surrendered once. Not again. Not now. Linden had gone to meet her worst fear. He intended to do the same.\n\nThen argent caught the edges of an opening ahead. Covenant smelled sulfur, the dire reek of brimstone. He felt distant heat like the withering touch of Hotash Slay long ago. And attar.\n\n\"Ur-Lord,\" Branl said sharply. \"Be warned. There is might and evil. Though I cannot name their source, they vow death.\"\n\nAttar, Covenant thought. The sweet sick stink of funerals; of preserved corpses. Lord Foul.\n\nThe _Haruchai_ as a people did not know that smell. They had never confronted the Despiser.\n\nHardly aware that he was struggling for breath\u2014that sweat ran like tears down the galls of his visage\u2014that his hands shook as if he had fallen into fever and delirium\u2014Thomas Covenant accompanied the last of the Humbled into Kiril Threndor.\n\nHe knew this place. He would have recognized it in any nightmare. Here he had been killed with his own power, his own ring. Here he had ascended in agony to participate in the Arch of Time, to defend with his soul the most necessary of the Laws which made life possible.\n\nThe space was a chamber like an abscess in the deep chest of Mount Thunder, Gravin Threndor: round and high, large enough to hold scores of Cavewights worshipping, and acute with patches of rocklight like plague-spots. Random illumination oozed like pain from the walls. The walls themselves looked like they had been shaped by a brutal blade, cut angrily into facets that cast radiance in all directions. From the ceiling, the light was thrown back like a spray of shattered glass by stalactites that resembled burnished metal: reflections so bright and broken that they seemed to swirl on the verge of madness. Some of the stalactites, too, had shattered, leaving gaps like gouges overhead, scattering their debris across the floor. Around the cave's borders, tunnels opened like unuttered screams. Among them were scattered a few boulders that resembled the stones where Covenant had left Jeremiah, displaced by violence or theurgy from where they belonged.\n\nHere was the source of the gutrock's fracturing, here in Kiril Threndor. Those cracks were memories of terrible battles, recollections expressed in the language of wounds. Within the chamber, more splits spread insanity across the floor. From their depths, darkness swirled into the air. In places, the surface had buckled, tilting slabs at tormented angles.\n\nBut the fissures did not touch the time-worn dais in the center of the chamber. Flaws avoided that stone as if they had been denied; as if no form of harm could alter the fundamental substance and meaning of the low platform.\n\nTwo steps into Kiril Threndor, Covenant halted. He no longer noticed the stench of attar. He did not regard the allure of cracks in the floor, or the entrances from which Cavewights might pour forth at any moment. He was transfixed where he stood by the figure on the dais.\n\nThe sight was as _wrong_ as a knife to the heart; as hurtful as the piercing which had twice ended his life, once in the woods behind Haven Farm, once here at the Despiser's hands.\n\nRoger Covenant.\n\nObviously waiting, Roger faced his father. A grin like a rictus stretched his fleshy cheeks. The slouch of his shoulders and the heaviness of his torso betrayed his disregard for his mortal flesh; his disdain. On his shirt and pants, he bore the scorch-marks of his battles with Linden. The puckered skin of healed burns showed through holes and tears in the fabric. For his deeds, he had paid a price in pain\u2014\n\nHis hands were empty of weapons, of any instruments of power. But his right was Kastenessen's, hot and ruddy as lava, flagrant with power. It blazed like the jaws of the _skurj_. It, too, must have cost him excruciation.\n\nHe gnashed his teeth at Covenant. \"Well, hi, Dad.\" His mouth sneered; but his voice was a tortured thing, twisted on a rack of unappeasable desires until its joints opened and its sinews tore. \"You took your own sweet time getting here.\"\n\nHis eyes were Lord Foul's, carious as rotting fangs.\n\n## 10.\n\nAll Lost Women\n\nLinden had chosen this. It was not a reaction to the Despiser's manipulations: it was her own doing. She had stepped off the path of his desires. If she served him now, she could not pretend that she had been misled or tricked.\n\nHer choice. Her doing, for good or ill.\n\nAnd she had promised herself that she would remember; that she would allow no effect of shame or pain, horror or failure, to confuse the fact that she had acted of her own free will. She would not blame Lord Foul, or fault Thomas for failing to spare her, or think less of Jeremiah because he had been weak.\n\nShe had made that promise to herself. Nevertheless she forgot it in the first instant of translation. She forgot who she was, and why she was here, and what she had intended to do. Such things were washed out of her by a scend of enchantment. Her world had become magic and majesty, and nothing was required of her except wonder. Something more than translation had occurred. She had entered a realm of transubstantiation where delight was the only possible response. Here she found contentment in awe and tranquility, the ineffable mansuetude of the redeemed.\n\nThe rich rug luxurious under her feet was distilled solace. It overlapped others as hieratic as arrases depicting scenes of worship, humility, sanctification: tableaux in which the devout ached with joy. She could have gleaned comfort endlessly from each of them; but her eyes and her heart were enticed by rapture on all sides. Somehow the richness of the rugs was both complete and transparent, solid and evanescent. They lay on a lucent floor pristine as aspiration, enduring as marble. Enhanced by the intervening substance of the rugs, the stone seemed polished to the point of incandescence. She was only able to bear its marmoreal radiance because she had been exalted to the tone and timbre of her surroundings.\n\nGazing around her, rapt and delirant, she saw a space like the ballroom of a grandiloquent palace; saw beauties in such profusion that she could not hope to appreciate them all. Loveliness effloresced in every direction. Near the walls, braziers of burnished gold offered flames redolent with incense and purity. Among the rugs, delicate filigree shafts like spun glass clean as crystal stretched upward to form arms that supported chandeliers as bright as the splendor of worlds. Beyond them, wide staircases graceful as wings swept toward higher levels and finer glories. Yet their treads and their immaculate banisters did not call her to rise and explore. She was satisfied where she stood, more than satisfied; already so dazzled and enraptured that any ascension\u2014any movement\u2014would diminish her perfect peace.\n\nHigh above her, mosaics sang like choirs: a reverent hymnody audible only as praise. They displayed constellations and firmaments like burgeoning creations, like galaxies and stars and worlds always new.\n\nYet more delicious to her senses than any other munificence was the fountain. A geyser in the center of the floor, it reached high, flawless and faceted as a single diamond, until it spread its arching waters wide: a feathered spray of droplets as precise as wrought gems. There no small jewel fell. Each clinquant bead hung in abeyance, suspended, motionless. Static and lovely as ice, the fountain displayed its own splendor: an icon of transcended time, sealed against change as though its perfection had been made eternal\u2014and eternally numinous.\n\nBespelled, she gazed about her like a figure in a dream, forgetting life and love and peril for the sake of an ecstasy that surpassed comprehension.\n\nBut Stave stood in front of her. She did not know him; or she did not see him; or he had no significance capable of distracting her from wonder. The scar of his lost eye dragged a frown across his visage. His hands gripped her shoulders and conveyed nothing.\n\n\"Chosen,\" he said as if he spoke from the far side of the world. \"Linden Avery. Will you not hear me?\"\n\nShe gazed past him or through him as though he were only a figment, too tenuous to require notice. He may have been no more than a blur in her vision. Soon her sight would clear, and he would be gone.\n\n\"This place is known to me.\" Every word vanished as soon as he uttered it, absorbed by astonishment. \"I have learned to set aside its power.\" For no apparent reason, he studied her closely. \"It is known to you as well, Linden. We stand where we have stood before, among the mazements of the Lost Deep. Then, however, Earthpower and the Staff of Law enabled you to turn aside from enchantment. Now you must reclaim yourself by other means.\"\n\nIn a small voice, Linden asked, \"Why am I here?\" But she was not talking to Stave. She simply did not understand how she had come to be blessed by so much beauty.\n\nHis frown deepened. \"A query with many replies, Linden. One is that I have guided you hither, knowing no better place for your purpose.\" He hesitated; gave a slight _Haruchai_ shrug. \"I have no apt language for such matters. It is my belief that translations by wild magic are directed by clarity of intent. Heretofore our courses and destinations were determined by the Ranyhyn. Matters obscure to us were plain to them. Now we have found our way unaided.\"\n\nHis hands tightened on her shoulders. \"Here, however, you did not choose our course. The burden of clarity was mine. As I once conveyed you to Revelstone without your consent, so also I have brought you to the Lost Deep. If I have erred, the fault is mine.\"\n\nHe was fading. Linden could hardly see him. By slow increments, an exquisite pleasure erased him from her sight. Soon her eyes would be clear, as untrammeled as the palace, and as precious. She wanted nothing in life except to see and hear and touch and smell\u2014\n\n\"Why have we come?\" he continued as if he did not know that he was almost undone. \"Another reply is that the bane rises. Though the distance is great, Her emanations are distinct. Seeking your son, Linden, we roused She Who Must Not Be Named. Thereafter it was conceivable that She would relapse to somnolence. She had been deprived of her prey by the Timewarden. Doubtless Her wrath was great. Yet She had also fed upon the soul of High Lord Elena. At another time, She might have resumed Her ancient sleep.\n\n\"Yet now I perceive that She could not. The flood which was released against the _skurj_ has filled the abyss of Her slumber. Indeed, those waters are withheld from the Lost Deep only by the lingering theurgies of the Viles. Such an inundation cannot harm a being such as She Who Must Not Be Named. Nonetheless it vexes Her. Therefore She rises.\"\n\nThe man's tone became more urgent, although he existed only as vagueness. \"She _rises_ , Linden. And I fear\u2014\" His fading hands shook her. \"Linden, hear me. I am _Haruchai_. I fear nothing, yet I tremble. I fear that the bane will ascend to Kiril Threndor. I fear that She will discern the scent of Corruption and the puissance of the Timewarden. I fear that She will fall upon them in fury and lay the Timewarden waste.\n\n\"For that reason I have guided you here. It is my hope that you will call out to Her with wild magic. It is my prayer, Linden, that you will draw Her to us before She nears Kiril Threndor.\"\n\nHe must have wanted something from her. Why else did he mar the palace with his voice, his hands, his insistence? But each word evaporated as soon as his mouth shaped it. He might as well have made no sound. He persisted in her sight as nothing more than a dwindling imperfection among the meretricious entrancements of the ballroom.\n\n\"Linden.\" Although he sounded as calm as snow-clad peaks in clear sunshine, he conveyed a subtle desperation. \"You must hear me. All of life tilts on the edge of a blade, and I am afraid. My hand remains able to strike you, and to strike again, until I am heeded. Alas, my heart will not suffer it. You must hear me.\"\n\nShe did not. She had forgotten him. She had almost forgotten that language had meaning. His words slipped past her. Then they were gone. Only ensorcelments remained.\n\nBut Stave was not alone. At his back, an array of creatures crouched in the act of rising to their hind legs. Black things, no more than a dozen. And grey ones, smaller, half that number. Above the cruel slits of their mouths, they had no eyes. Wet nostrils dominated their faces. Pointed ears twitched on their skulls. Their heads and bodies were hairless.\n\nStave turned and bowed to them as if they had earned homage.\n\nThey made chittering sounds like his, language without meaning. One of them taller than the others held a jerrid of black iron, a scepter like a short javelin. A fuming liquid as dire as poison dripped from the iron. The tall creature snuffled at Linden, then turned away. With low growls and snarls, it used the point of its jerrid to sketch incomprehensible symbols in the air. Acid drops scattered here and there; but they evaporated before they touched the floor.\n\nFor a moment without measure or duration, nothing changed. Linden remembered nothing. Only the ballroom endured. Like Stave, the creatures faded, the black ones and the grey. Like him, they were almost gone.\n\nThen a subtle tremor ran through the fountain, a vibration so brief and untenable that it defied sight. She could not believe that she had seen it. She hardly recognized her own fright.\n\nSlow and horrid as a plunge from a nightmare precipice, a single jewel of water high in the fountain began to fall.\n\nLight shone all around the small bead. It looked like an epiphany; like the essence of the Earth's gems; like the last gleam of the ravaged heavens. It fell and fell forever, infinite and fatal; and while Linden watched it, her heart did not beat, her lungs did not draw breath. When at last it reached the floor, the largesse of the rugs, it made a tiny splash: the first faint quiver of a world about to shatter.\n\nSomewhere in the distance, hundreds of leagues away, the Worm\u2014\n\nLinden blinked. A small frown knotted her forehead. Her heart offered up a weak beat.\n\nThe creatures continued their guttural invocation\u2014and another bead of water began its interminable demise\u2014and Stave stood in front of her again, clutching her shoulders.\n\nWhen he repeated her name, she wanted to weep.\n\nA second little splash. A few ripples. In rugs? In marble?\n\nA third rare jewel of water, and a fourth, dropping from perfection into time and ruin. When they struck, they made a pattering sound, delicate and awful.\n\nOh, God, she thought. Stave. The Worm. The bane.\n\nUr-viles and Waynhim. Once again, they had come to her rescue when she did not know how to save herself.\n\nIn this place, rescue was an atrocity. It destroyed a supernal achievement, the triumph of lore which had preserved the palace through the ages. And the effect on Linden was no less cruel. Raindrops brought back memories like devastations. Thoughts were carnage and cataclysm.\n\nJeremiah. Thomas!\n\nSomehow she reached out to the _Haruchai_. Her voice was softer than the accumulating drip of the fountain. Her eyes should have been full of tears.\n\n\"What did you say? About the bane?\"\n\nHis back straightened. His chin rose proudly. His eye shone.\n\n\"She rises, Linden. If you do not call out to Her, She will assail the Timewarden. She will consume your son.\"\n\n_Damn_ it! Linden wished that Stave had hit her. She wanted to pummel herself. She had chosen to face her worst fears. Then she had forgotten all about them. And while she had lost herself among marvels, the Earth's peril had increased beyond bearing.\n\nShe Who Must Not Be Named might take Thomas and Jeremiah.\n\nIn a different life, a bullet had struck Linden. A scar over her heart matched the perfect circle in her shirt. There was no going back. Choices made could not be recanted.\n\nThomas had unforeseeable strengths. He might survive. But mere Law and Earthpower would not suffice to ward Jeremiah.\n\nFor Linden's sake, or for the Earth's, the ur-viles and Waynhim had disrupted the prolonged theurgies of the Viles; sacrificed their own heritage of splendor. Around the ballroom, a light drizzle fell. The fountain cast a fine mist that gathered into droplets. Drips leaked from the music of the ceiling. Ripples ran down the stairways. Gradually the chandeliers released their lights. Spots of water stippled the woven rugs, the immemorial floor.\n\nNo going back. Now or never.\n\nGod help me.\n\nLinden delayed only long enough to say to the ur-viles and Waynhim, to the eyeless features of the loremaster, \"You keep helping me, no matter how much it costs, and I still don't know how to repay you.\" Then she wheeled away.\n\nClenching her fists, she raised her face to the leagues of blind stone above the Lost Deep. Rain spattered her cheeks and forehead. Its sheer age stung her eyes. In her mouth, the drops tasted like dust.\n\nAs if she had always known what she could do, she invoked her wedding band. She had no more use for despair and recrimination; inadequacy. Only power would serve. Like a woman screaming, she flung a roar of wild magic into Mount Thunder's gutrock.\n\n\" _I'm here!_ You lost me once! Come get me now!\"\n\nHer theurgy could have torn vast stone to powder; could have brought the weight of the mountain crashing into the caverns of the Lost Deep. But her health-sense was precise. She did not hurl silver against the rock: she tuned it to pass through Mount Thunder's substance, sharpened it to a pitch that only the bane would be able to hear.\n\n\"Come and _get_ me! I can save you!\"\n\n_Melenkurion_ Skyweir was already falling. She felt its massive collapse like atmospheric pressure on her skin, heard it like the grumble of impossible thunder. At any moment, the Worm would begin to drink EarthBlood from the world's heart.\n\nSlowly the drizzle became rain. Details among the mosaics blurred and ran as their melodies dwindled to liquid. The staircases slumped, shrugging thin streams from their sides. The shafts of the chandeliers bowed as if they had lost faith in themselves. Rills curled around Linden's boots, flowing nowhere. Argent made raindrops as bright as exploding stars.\n\n\"I can tell you how to save yourself!\"\n\nShe felt Stave's hand on her shoulder. His touch seemed almost diffident as he asked for her attention. But she did not acknowledge him until her power and her shouting failed; until she could no longer sustain her summons.\n\nSilver stains danced like little suns across her vision as she turned back to her friend.\n\nThrough a veil of rainfall, Stave told her, \"It is enough. If your call is not heard, no other will suffice.\"\n\nThe ur-viles and Waynhim barked to each other like dogs, excited or fearful. The loremaster gestured resignation or encouragement with its jerrid. Water glistened on the skin of the creatures as if the fluid were dying, giving up its last magic.\n\n\"Therefore I must speak,\" continued the former Master. \"I will not be vouchsafed another occasion to do so.\"\n\nLinden glared and squinted, trying to clear the spots from her eyes. Wet hair straggled across her cheeks.\n\n\"I must state plainly, Linden, that you have become wondrous in my sight. Here my life is forfeit. It may be that the bane will heed you. Me She will not suffer. In Her sight, all men are betrayers. I will be devoured.\"\n\nWater streamed on Linden's face, scattered from the lines of her jaw. Drops snapped against her skin. Here my life is forfeit. How had she failed to consider this? For hours, she had imagined her intentions as though they threatened only her. But of course Stave was right. He could not withstand the bane. She Who Must Not Be Named would not tolerate him.\n\n\"As farewell,\" her final companion told her, \"I must say aloud that I regret nothing. My fears are gone. You risk much, as you have ever done. Whatever now ensues, know that I am made proud by my place at your side.\"\n\nShe Who Must Not Be Named only slew men; only killed and ate them. She had no other use for them. Women She consumed in an entirely different fashion. She craved the torment of their living spirits when their bodies were destroyed. Her hunger was for the anguish of their souls, undying and endlessly tormented. It resembled or confirmed or justified Her own agony.\n\nIn some sense, literally or metaphorically, the bane was here because Lord Foul had betrayed Her; seduced and ruined Her with lies; gaoled Her within Time. Now She could only suffer\u2014and feed on the sufferings of any woman who came within Her grasp. _Diassomer with fear and dread_ \u2014Unforgiven Elena, Covenant's daughter by rape. Emereau Vrai, Kastenessen's mortal lover. An Insequent whom the Ardent had called the Auriference. Hundreds or thousands of women across the ages of the Earth. As far as She was concerned, all women and every love had been betrayed.\n\nIf She had not forgotten Her true name\u2014Her real scope and power\u2014She would have brought everything to an end long ago.\n\nLinden peered through splashes and rivulets at Stave. The rain was becoming torrential as millennia of lore failed, unloosed by these few ur-viles and Waynhim according to the arcane dictates of their Weird. Lashing drops and spray fraught with residues stung like acid. She tried to find her voice; swallowed bitterness so that she might shout refusals at her friend. If he would not ask the Demondim-spawn for protection, she meant to plead on his behalf.\n\nBut she did not. She was already overwhelmed.\n\nI am made proud by my place at your side.\n\nIn the small space between instants, the rainwater running over her body became vermin. It became centipedes as long as her hand, feasting maggots, spiders with hundreds of pincers, lice that scuttled and squirmed, worms burrowing. Noisome things crawled and clawed and pecked everywhere, intimate as lovers, avid as eaters of death. Desperate to quash the feeding, she thrashed like a madwoman, hit herself frantically, dug at her scalp until she drew blood.\n\nStave may have shouted her name. If he did, the rain slapped his voice from the air.\n\nCascades filled her mouth with biting insects. They laid their eggs in her eyes, breeding. When she tried to breathe, she gasped abhorrence into her lungs and retched. Beetles and centipedes scuttled down her throat.\n\n\u2014 _written in water_. The Despiser had named her fate. Water was horror. It was eager excruciation. It transformed her to carrion and shrieking.\n\nNow Lord Foul laughed at her from an insurmountable distance. _You have become the daughter of my heart_. Laughed as he must have laughed at She Who Must Not Be Named. Soon Time would begin to crumble, and he would be free. Linden had brought this on herself. She had given it to the world as if it were the sum and consummation of her life. It would never stop. Across every inch of her flesh, it drove her mad. She could not bear it. If she had been given a knife, she would not have hesitated to flay the skin from her bones.\n\nSuch desecration should have finished her. But it was endless. It could always get worse.\n\nAnd while Linden flailed in torrents, the bane shouldered Her way into the cavern.\n\nHer power was immense. No doubt She could have shaped Herself to slip through the passages of the Lost Deep. Yet She did not. Damage suited Her: She liked wreckage in Her wake. As She entered, the rolling bulk of Her fury made a ruin of the stone. With every shrug, Her advance flung rubble at the walls. Her many faces were etched in fire. Mute screams stretched their mouths. Torment gouged their eyes.\n\nWithout knowing what she did, Linden stopped thrashing. The scale of the bane's extremity and rage demanded her absolute attention. Suddenly worms and maggots were no longer sensations. They became insights.\n\nWhen She Who Must Not Be Named spoke, the impact of Her voice seemed to stop Linden's heart. The ferocity of the sound changed the rain to steam and scalding.\n\n\"Do you speak to me?\" The roar crushed Linden's hearing. \"Do you speak to me of _save_? _Do you dare?_ My pain cannot be redeemed. It can only feed and _grow_.\n\n\"You are mine. I will relish you. I require only a moment to chew the marrow from the bones of the man who has betrayed you to me. Endure your suffering. It will be brief. Then I will consume you, and you will know the ecstasy of eternal woe and regret\"\u2014She gathered Herself to cry like a beast\u2014\" _and agony_!\"\n\nLinden could not protest. The bane's intent was just. Linden deserved centipedes and spiders. Horror was her true heritage: the legacy of her pitiful, self-pitying parents. By audacity and blind carelessness and insufficiency, she had awakened both the Worm of the World's End and her own worst nightmares. She had brought this doom upon herself.\n\nNevertheless it was intolerable. The bane would kill _Stave_ , her friend when she had no other. The knowledge that he was about to die for her sins was more than she could bear.\n\nDays ago, the foundations of her life had begun to shift. Now they settled into new alignments. Like a woman rising from her own grave, she changed. In a rush, her whole reality was transformed. Faster than the febrile stutter of her heart, maggots and squirming and misery became a wail of wild magic.\n\nShe had no power to equal the bane's. She Who Must Not Be Named transcended everything mortal. Nevertheless Linden was Thomas Covenant's wife. He had wed her in love and joy. In passion and courage, he had made of her a rightful white gold wielder. She was not helpless.\n\nSwift as her pulse in her veins, she spun silver puissance around her treasured friend, caught him in a fist of bright flame: a fist or a circle. She had no _krill_ to enable a translation, but she had other resources. She had the unthinking reflexes which had allowed her to step outside the sequences of time during the collapse of Kevin's Watch. She had the whetted senses with which she had created _caesures_ without stumbling into Joan's madness. And she was not hampered by her husband's necessary reluctance.\n\nWhile the bane surged forward, Linden grasped Stave and _threw_ him. Away from this moment. Away from this place, this stone, this fate. Trusting his instincts\u2014his _clarity of intent_ \u2014to choose his destination, she spared him the cost of her choices.\n\nPerhaps he would forgive her.\n\nWhen he was gone, she wrapped herself in wild argent a heartbeat before the bane pounced on her, shrieking.\n\nThe ur-viles and Waynhim did not try to help her now. Shrouded in rain, they stood apart like witnesses: creatures condemned to watch the extinction of their obscure hopes.\n\nThe bane's rage took Linden, snatched her into incandescence and infernal torment. But the bane did not _have_ her. Vermin and pestilence did not have her. She was cloaked in her own fire, cocooned heart and soul. Within the bane's appalling body, she was not devoured. Instead she left the sensations of horror and eaten death behind as if they had become irrelevant.\n\nAccording to Kasreyn of the Gyre, white gold was an imperfect tool able to fashion perfection in a flawed world. But she did not seek perfection. She wanted only to preserve herself until she could at least try to keep her promise.\n\nShe thought of herself as an embolism, a tiny clot or bubble in the flagrant bloodstream of She Who Must Not Be Named. Untouched because she was trivial. Wild magic warded her against time, against mortality. She controlled nothing. She could not harm the bane. But she could remain herself. She could think and strive. The vast being roared in frustration and bafflement, thwarted hunger; but Linden ignored Her.\n\nLinden Avery had chosen this fate. She knew why she had done so. She knew that she was lost. She would die as soon as her resolve and her fire failed. Nevertheless she did not falter. While she could, she pursued salvation.\n\nThrough the tremendous roil of wracked souls, the seething turmoil of the bane's victims, Linden searched for Elena.\n\nElena Lena-daughter, child of rape, prey of Despite. Seeking to oppose Lord Foul, she had broken the Law of Death to raise Kevin Landwaster's spectre\u2014and by that crime, she had become the Despiser's servant. When Linden had seen her among the Dead in Andelain, Elena had still borne the galls and wounds of her self-Desecration. Yet Linden had given her no pity, no kindness. Of Elena's later sacrifice to the bane, Linden knew only what she had been told. But she remembered too well what she herself had done to the first Law-Breaker. Now she considered it the least forgivable of her sins.\n\nAs if she had the right to judge\u2014 _she_ , who had set the world's last crisis in motion\u2014she had denied to Elena the understanding and consolation which Berek and Damelon and Loric had given Kevin. Instead of mercy, she had offered Elena only demands: the selfish expostulations of her own guilt.\n\n_Stop feeling sorry for yourself. It doesn't accomplish anything._\n\nThat memory still made Linden cringe. It had brought her here. Because of it\u2014and because the implications of carrion required this\u2014she had forsaken her husband and her son and the imminent destruction of the Earth.\n\nThroughout the bane's clamoring chaos, she drifted, searching for Covenant's daughter.\n\nScores of faces wailed in front of her and fled. Hundreds. Thousands. She believed that she knew how to save them all. Or perhaps she only hoped. But she had to start with Elena, who had been four times betrayed: by the circumstances of her birth, by her own actions in the cave of the EarthBlood, by Linden, and by being cast into the inferno of the bane's mad agony.\n\nElena was here. Finding her was only a matter of time\u2014and Linden was immune to time while her strength lasted.\n\nWhen the aghast ravage of Elena's face appeared, Linden clutched it with silver before the spectre could be swept away.\n\nElena did not struggle, yet Linden could barely hold her. The bane's wrath lashed the eidolon in every direction. She Who Must Not Be Named pressed down on Linden's shield with all of Her accumulated mass: the weight of ages. Only Linden's cocoon preserved her. Only wild magic kept Elena with her, face to face.\n\n\"Oh, Elena.\" She spoke in flames. She had no other voice for her remorse and shame. Her words were the lament of her wedding band: the grief of an absolute promise broken absolutely. \"I'm so sorry. You didn't deserve what I did to you. You and Caer-Caveral brought Thomas to me. I should have been grateful. But I couldn't think about anything except how much I hurt. I treated you like it was your fault. I wanted you to be stronger than I was. I wanted you to forgive me, but I couldn't say that. I can't forgive myself.\n\n\"I'm like Kevin. I chose my own Desecration. You just made a mistake. You don't deserve\u2014\"\n\nElena's cries made no sound that Linden could hear. Nevertheless the High Lord's protests silenced Linden. They appeared like avatars in her mind; like reifications of every injury which had ever flensed Elena's heart.\n\nWhy have you come? My suffering is enough. I do not desire the sufferings of others. I did not call you to this doom.\n\nTerrible pressures distorted Elena's features: stretched them until they tore; compressed them into granite knots. Her eyes were wounds.\n\nDo you conceive that I was compelled to eternal horror? The Dead are not so cruel. I acceded to the pleas of Sunder and Hollian out of love for my father, and because you are his beloved, and because you must be preserved.\n\nLinden Avery, you multiply my torment. You have damned yourself. I must go mad, as She is mad. Why have you come?\n\nAt an unconceivable distance, the thunder of _Melenkurion_ Skyweir's destruction boomed. Surely the Worm had begun its feeding at the wellspring of the Earth's Blood? Surely the world's remaining life could be measured in heartbeats?\n\nLinden did not care. She had been trained as a physician, a surgeon, a healer. She knew in her bones that her first and only responsibility was to find an answer for the need in front of her.\n\n\"To free you,\" she answered in conflagration. \"I'll free as many of you as I can. I'll tell the bane how to free Herself. But I have to start with you. You're the one I hurt.\"\n\nThomas Covenant's daughter, as precious as her own son.\n\nElena's wailing was inaudible. Still Linden heard her. Her voice seemed to burst from her eyes, from the veins throbbing in her temples.\n\nYou cannot. Do you hear? You cannot. We are souls. Her anguish binds us. As we are, we cannot be divided from Her. We must live again to be free of Her. We must have flesh. We must be truly separate, spirit from spirit, thought from thought. Pain from pain. To release us, you must unmake our deaths.\n\nWe cannot be freed!\n\nThat cry rent Linden's heart. It nearly snapped her resolve. For a moment, she could only gape at Elena. _Unmake_ your death? _How?_ Elena was not Thomas: she was not imprinted on Linden's nerves, Linden's needs, Linden's love. Her body was gone beyond comprehension. And Linden did not have either the _krill_ or the Staff. She had only her ring.\n\nDon't tell me that I have to leave you like this!\n\nBut then her preconceptions shifted. She had spent her life making promises that she did not know how to keep. She had never sufficed to keep them. And yet she had accomplished more than she could have imagined. But not because _she_ was more\u2014or not only because she was more. No, she had been able to do so much because she was not alone. From Liand and Anele and Stave to the Ranyhyn and the Swordmainnir and Thomas himself, she had been aided in every deed. She had been given gifts\u2014\n\nThey had taught her truths which should have been obvious, but which had nonetheless eluded her. Berek Halfhand had seen Gallows Howe in her, a mound of ruin made barren by bitterness and slaughter. In Garroting Deep, however, she had discovered a deeper truth beneath the drenched dirt.\n\nMore than bloodshed and revenge, the olden forest had yearned for restitution. The trees would have turned their backs on killing entirely if they could have recovered their ravaged expanse and majesty.\n\nShe understood that now. She recognized, if the bane did not, that healing was both more arduous and more worthy than retribution. And sometimes healing required measures as extreme as the patient's plight. Surgeons amputated or extirpated. They performed sacrifices. They transplanted. They did not judge the cost. They only did what they could.\n\nAnd even here, in the Lost Deep at the onset of the World's End, Linden was not alone.\n\nIn a blaze of wild magic, she reeled against the current of the bane's savagery, dragging Elena with her.\n\nThe bane's resistance was brutal and blind, undirected. She Who Must Not Be Named could shatter entire landscapes, but She did not know how to fight within Herself. She had never needed to do so before. Linden seemed to struggle endlessly\u2014and to find what she sought in an instant.\n\nThrough the flame and hunger and abhorrence of the bane's boundaries, she saw the Demondim-spawn.\n\nUnder a deluge of collapsing theurgies, the ur-viles and the Waynhim stood together as if they had finally become kin, united by a common interpretation of their Weird. As one, they studied the bane with senses other than sight; or they studied Linden.\n\nTime and again, they had helped her when she had not known that she needed their gifts. Like Thomas. Like the Land itself.\n\nPeering at the creatures, she understood at last that they had not unbound the ancient magicks of the Viles merely so that she would be able to remember and act. They had cast down their purest heritage for reasons greater than her needs and desires. Their Weird demanded more. They had undone the wonders of the Lost Deep for the same reason that they had aided her and the Land repeatedly: so that they would be vulnerable now. So that they would be accessible\u2014\n\n_If you can ever figure out a way to let me know what you need or want from me\u2014_\n\nWhat had the ur-viles and the Waynhim ever wanted, except to escape their loathing for their own forms?\n\nLinden waited until the loremaster met her gaze; until the tall creature nodded its assent. Then she did what she could for Elena.\n\nRisking the shroud of wild magic which protected her, Linden flung Elena out of the bane; tossed her like a wisp of hope or a kept promise into the waiting embrace of the loremaster.\n\nThe creature appeared to swallow. The spectre of Elena seemed to vanish. Linden could not be sure. She Who Must Not Be Named was roaring: a howl that stunned Linden's chest, rattled her mind in its chamber of bone, stopped her ears and eyes and mouth and lungs. She hardly knew who or what she was.\n\nTime was fraying at the edges around her, starting to unravel. Soon its deterioration would unweave the world. Reality would lose its shape. Existence would cease.\n\nStill wild magic shielded her. Her own needs shielded her; her own loves. She was not done.\n\nShe grasped the first spectre shrieking past her: a woman who could have been anyone, Diassomer Mininderain, Sara Clint, Joan herself, anyone at all. As she had with Elena, she gave the savaged soul to the loremaster, or to all of the Demondim-spawn. Then she reached for another victim.\n\nBefore Linden could do more, the bane found a defense. Her ferocity seemed to have no beginning and no end as She began to compress Herself, condensing Her might and bulk around Linden, making Herself more solid. Linden no longer drifted on currents of fire and fury. Pressures great enough to rive mountains clamped down on her. Forces which dwarfed her threatened to rupture her eardrums, burst vessels in her lungs, squeeze blood from her eyes. Lost women were held motionless in their unutterable screams.\n\nBut Linden did not need ears or eyes or air to hear She Who Must Not Be Named.\n\n\"You diminish me! You dare to diminish me! You will not! You speak of save, but your purpose is _betrayal_. I will not permit you!\"\n\nLinden had no voice. It had been crushed out of her. She could speak only with wild magic: the blazing paradox, _save or damn_ , which formed the keystone of life.\n\n\"This isn't betrayal. It's kindness. I can save all of these poor women. I can tell you how to save yourself.\"\n\n\" _How?_ \" The bane's roar was a sneer, contemptuous as vitriol. \"You are nothing! What do you offer that I have not attempted endlessly?\"\n\nLinden could not move. She was effectively dead. The bane's power was too much for her. Nevertheless she answered.\n\n\"The Arch of Time is breaking. If you don't believe me, look for yourself. You can see it. The Worm of the World's End is drinking the EarthBlood. Everything is going to be destroyed. Your prison is starting to fall apart.\n\n\"While it falls, you can slip out. You'll be free. But you have to go now. Otherwise I don't know what might happen. You belong to eternity. If you don't leave\u2014if you stay inside Time\u2014you might be extinguished along with everything else.\"\n\nPerhaps She Who Must Not Be Named craved extinction, an eternal end to Her suffering. If so, Linden had failed. But at least her own anguish would end as well.\n\n\"All I want,\" she insisted in fire, \"is to release your women. You don't need them anymore. Not now. They're part of this world.\" They were dross, imperfections. \"If you take them with you, they'll only hinder you. They may even prevent you. You won't be truly free.\"\n\nThe bane contracted around her. Terrible strength made pulp of her flesh and organs, her bones, her mind. Nothing existed for her except the raw rage of She Who Must Not Be Named.\n\n\"Fool! Madwoman! Treacher! Do you conceive that I desire _freedom_? You do not know me. Freedom is _agony_. It is _abhorrence_. It is not _redemption_. I am anguish because I have forgotten who I am.\n\n\"The destruction of this world is nothing to me. I cannot die. _I must have my true name!_ \"\n\nConvulsions shook Mount Thunder to its roots. Shocks distorted the definitions of existence. Slabs fell from the ceiling and were pulverized. Granite sifted like ash onto the heads and shoulders of the ur-viles and Waynhim. Stone lurched under their feet. Yet they stood as if they were rooted by pride: legs straight, backs regal, arms open to welcome released souls. The loremaster's eerie visage shone with an inward exaltation.\n\nLinden felt ripples like imminent _caesures_ trembling toward her, confusing the structure of instants. There were no risks left except this one.\n\n\"Then give me Emereau Vrai.\" Kastenessen's lover: the only woman who had ever been loved by an _Elohim_. He had given her some of his magicks. How else had she been able to create the _merewives_? Perhaps he had also revealed secrets which no one mortal\u2014which none of the bane's other victims\u2014could have known. Why else had his people considered his crime so heinous that he deserved his Durance? Linden had heard long ago that he had been punished for harming an ordinary woman with his love; but she did not trust that explanation. When had the _Elohim_ ever been so protective of individual lives? Emereau Vrai might know\u2014And if she did, the Demondim-spawn might be lorewise enough to understand her. \"Let me show you that I'm telling the truth.\"\n\nI'm a woman, damn it! I don't want to seduce you.\n\nThe bane contracted in fury. Her vehemence increased. It was unbearable, unanswerable. Though Linden clung to wild magic\u2014to her wedding band\u2014to the promise of Thomas Covenant\u2014she was little more than a spark, a fading ember within the virulence of She Who Must Not Be Named. Hundreds or thousands of women shrieked their pain and despair, but they made no sound.\n\nThen the pressure eased. Yowling to Herself, the bane receded slightly. Linden remembered to breathe. She blinked at the blood in her eyes.\n\nAn excoriated face appeared in front of her. A voice that registered only in Linden's mind said, I am Emereau Vrai. Does Kastenessen love me still? I am betrayed to this doom, but not by him. It was his kindred who made of me a plaything for damnation. All that I have done, I did because he was taken from me.\n\nYou have spoken my name. Know that I forgive nothing. Alone among this host, I approve my fate. She Who Must Not Be Named is my god. My anguish is worship.\n\nLinden might have said, Of course he still loves you. In his heart, he never let you go. He made himself insane for you. But she did not have the strength. Her life and her will were almost gone. She needed the last of herself to clasp Emereau Vrai and send Kastenessen's lover into the arms of the Demondim-spawn.\n\nThey accepted her gladly, barking their homage amid the devastation of the Lost Deep.\n\nThen Linden was done. Wild magic drained out of her, and she was swept unshielded into the excoriation of the bane. As far as she knew, she only remained alive because she had slipped into a fracture between instants. When the currents of the bane's fury carried her back into the sequences of time, she would die.\n\nYet that fracture\u2014or some other pause\u2014held her. She did not die, or move, or think. Entire realms of pain slid past her as if she had become untouchable.\n\nAs if she had finally become worthy of her husband.\n\nWith senses other than vision or hearing or touch, she recognized the Demondim-spawn. They stood like kings in the wreckage of their eldritch legacy. Every visage among them now shone like the loremaster's. The proportions of their bodies were changing, as if they were becoming human; sharing the loremaster's transfigured spirit. They seemed taller.\n\nIn unison, they chanted at the bane: a paean or invocation as alien as their guttural speech, and as incomprehensible. With every rise and fall, every beat, their hymn appeared to accrue peril, as if they were hazarding more than their own destruction; as if the accumulation of their words threatened the pediments of reality. And yet their eagerness was plain on their eyeless faces. Somehow they had arrived at a crisis of extermination or apotheosis toward which they had striven for millennia.\n\nThey may have been extolling the bane\u2014or forbidding Her.\n\nHer response was a cry that sent spasms through the gutrock for leagues in all directions:\n\n\"I AM MYSELF!\"\n\nWhen Linden's heart beat again, she was no longer inside the bane. Instead she had the sensation that she was being carried; cradled with the tenderness of a lover. Powers that surpassed understanding protected her from the ruination of the Lost Deep.\n\nShe was given a moment to watch the bane release souls into the waiting arms and mouths and bodies of the ur-viles and the Waynhim: a torrent of long anguish so suddenly relieved that she could not name what became of it. Then the bane began to rise like music, intangible as mist, and potent as divinity, through Mount Thunder's stubborn foundations; and Linden was lifted with Her, passing among the mountain's complex rocks and cavities as if she were as transient as a wraith.\n\n## 11.\n\nOf My Deeper Purpose\n\nFor a moment that felt like a protracted sob, Jeremiah watched Covenant and Branl recede along the passage toward Kiril Threndor; watched the silver of the _krill_ fade like the last light in the world. Then he folded back down to the floor. Sitting with the Staff of Law gripped across his thighs and images of the Worm chewing at the edges of his mind, he stared into absolute blackness and tried to believe that he was not out of time. That the subtle trembling of the stone did not announce the collapse of the Arch. That Covenant would come back to him, since Linden had said that she would not. That he would be spared.\n\nHis mother had not even bothered to explain where she was going; or why.\n\nHe was angry; too angry to speak or grieve. Linden and Covenant had left him with an impossible burden, as if he were somehow responsible for saving the Earth. As if he were not still the same boy who had been too small to rescue his sisters from Lord Foul's bonfire.\n\nOn some level, he knew that he was also angry at himself. Angry because he hated his own childishness. Because he felt useless and stupid. Because he had not tried to get an explanation from Linden, or to change her mind, or to say goodbye. Angry because Covenant expected too much from him. But that anger belonged to some other Jeremiah\u2014to a piece of who he had become when Kastenessen had broken him\u2014not to the boy who had been left by his mother and his first friend.\n\nSure, he understood Covenant's reasons for walking away. _I don't want you that close to Lord Foul until I can distract him._ The words sounded like they made sense. _But I do want you to come. I need your help to keep him busy._ That was simple enough.\n\nBut it was not simple at all. Covenant had also said, _You aren't strong enough?_ _Neither am I_.\n\nAnd _Then let him be too strong. You don't need to beat_ _him_. And _Just do_ something _he doesn't expect._\n\nSo what was _that_ supposed to mean?\n\nAnd what would it accomplish? Nothing that Jeremiah or Linden or Covenant himself ever did would stop the Worm. It was already drinking EarthBlood: Jeremiah could feel or see or hear it. The whole world did not contain enough power to prevent its own death.\n\nWhat good would it do to make Lord Foul _miss his chance_?\n\n_As guerdon for his puerile valor\u2014_\n\nJeremiah was angry, all right. Of course he was angry.\n\nIn some ways, sitting there in Mount Thunder's stark midnight hurt more than being possessed by the _croyel_. That bitter creature had made him truly helpless, as unable as a corpse to affect how he was used or what he became. But he was not helpless _now_ : not literally. He had the Staff of Law and his own Earthpower. He could kill Cavewights. Eventually he could maybe teach himself how to help Coldspray and Grueburn recover. If nothing else, he should have been able to fill this cave with light and warmth. But the Staff's possibilities only taunted him. They emphasized all of the things that he could not do.\n\nCovenant and Linden might as well have asked Jeremiah to remake the world.\n\nGnawing his futility, he ignored the exhausted rasp of the Giants' breathing, the useless stoicism of the _Haruchai_ , the slow drip of blood from too many wounds. He had nothing to say to his companions. They could not help him.\n\n_Maybe Roger had the right idea. Maybe we should all try to become gods._\n\nThe idea was a cruel joke.\n\nHe should never have listened to Linden. He should never have accepted her Staff. He should have stayed in his graves, hidden. He would have been better off. Nobody would have expected him to produce miracles.\n\n\"Chosen-son?\" asked Rime Coldspray: an abraded whisper. \"Jeremiah? Do you hear me?\"\n\nHe wanted to snarl at her. The floor trembled under him. A fever gripped Mount Thunder's gutrock. In the distance, the implied roar and clatter as _Melenkurion_ Skyweir collapsed shook the world. He could feel it. Towering plumes of dust and ruin cast a pall across the Land's last dusk. He could see it.\n\nCovenant was wasting his time. Linden had thrown her life away.\n\nBut naturally the Ironhand and Grueburn did not hear what Jeremiah heard. He was alone.\n\n\"I'm busy,\" he muttered. \"What do you want?\"\n\n\"Chosen-son.\" Rime Coldspray made a palpable effort to speak clearly. \"I am loath to burden you further. We are not altogether sightless in such dark. And I do not doubt that the vision of the _Haruchai_ exceeds ours. Nonetheless some small flame would comfort our spirits.\n\n\"I do not ask a _caamora_ ,\" she added as if she feared that he would misunderstand. \"I am undone by weariness, and have no heart for lamentation. Yet fire and light would be a kindness.\" She sighed. \"Mayhap they would enable me to remain upright until we are summoned by the Timewarden's need.\"\n\n\"Aye,\" breathed Grueburn. She sounded too weak to say more.\n\n\"Then you should sit down.\" Jeremiah remembered seeing a couple of large boulders against the walls. They were invisible now, blank to his health-sense, indistinguishable from the surrounding stone; but the Giants could rest on them. \"Don't you feel it? The floor is starting to shake. The Worm is sending out ripples. The more it drinks, the bigger they'll get. Soon you won't be able to stand. You'll last longer if you don't try.\"\n\n\"Stone and Sea!\" the Ironhand panted. \"Does the world end? Does time remain for the Timewarden to accomplish his purpose? Have we come so far at such cost, and arrived too late?\"\n\n\"How should I know?\" countered Jeremiah sourly. \"I've never watched a world die before.\" Then he rasped, \"Of course we're too late. That's what all those Cavewights were for. Lord Foul sent them to slow us down.\"\n\nWe were doomed, he added to himself, as soon as Mom and Covenant started thinking I could hold up my end.\n\nBut Canrik said like a reprimand, \"He is the ur-Lord, the Unbeliever. Twice he has wrested life from the clutch of Corruption, for the Land if not for himself. We are Masters and have doubted much. Now we are done with uncertainty. While Branl remains able to speak to us, we will fear nothing.\"\n\nJeremiah grimaced. \"Fine. You do that. Fear nothing as long as you want. Just don't say I didn't warn you when this place starts to shake so hard you fall down.\"\n\nThe darkness of the cave and the darkness inside him mirrored each other. He could not distinguish between them.\n\n\"Ah, Chosen-son.\" Coldspray's voice seemed to scrape the floor. It sounded as unsteady as the stone. \"Your straits are indeed bitter. I know not how you may be consoled.\n\n\"Yet surely you also would find comfort in light.\"\n\n\"Don't you think I'm trying?\" Jeremiah retorted, caustic as lye. \"I've been trying ever since Mom\"\u2014he raised the Staff, slammed it back onto his thighs\u2014\"gave me this thing. But I can't change what I am. It's all just black.\"\n\nThe Staff had turned against him soon after he had begun trying to use its stained resources. Before that, his power had been the warm yellow of sunshine. He could have provided at least a taste of kindness for Coldspray and Grueburn. But his efforts with the wood had not changed it. Instead it had stripped away his denials, his defenses.\n\nIt had exposed the truth\u2014\n\nThe Ironhand sighed again. \"Ah, then.\" She may have shrugged. \"Lacking other illumination, I will emulate the certainty of the _Haruchai_. I will trust that Linden Giantfriend and Covenant Timewarden will exceed every expectation, as they have done from the first. And also\u2014\" She groaned softly. \"Also I will heed your counsel, Chosen-son. Until we are summoned to Kiril Threndor, I will rest.\"\n\nJeremiah heard the creak of her joints as she forced herself to move. He felt the mute crying of her muscles, the catch and strain of her respiration, the lurch of her pulse. With Grueburn, she went to the wall opposite him. The weight of their armor and swords seemed to make their shoulders moan as they lowered themselves to lean or sit, apparently on the boulders.\n\n\"The ur-Lord has begun,\" Canrik announced. \"He confronts two great evils. Branl now discerns that Corruption has taken possession of the ur-Lord's son. They stand as one.\" A moment later, he added, \"In such a conflict, Branl is of little use.\" His tone had a grim tinge. \"His flesh cannot withstand the fire and fury of the _skurj_. Therefore he cannot ward the ur-Lord.\"\n\nTaken possession, Jeremiah thought. Oh, joy. In spite of his own despair, he felt a reflexive pang for Covenant's son. When Roger lost his partnership with the _croyel_ , he must have decided that Lord Foul was his only path to godhood; his only way to survive the shattering of the Arch. But he should never have trusted the Despiser. He must have been so desperate\u2014\n\nThen Jeremiah forgot about Roger. The ur-Lord has begun. Time was running out\u2014and Jeremiah was still as helpless as a kid.\n\nMore than anything else at that moment, he wished that he had refused the Staff of Law. How could he have believed that _he_ would be able to make a difference?\n\nThe Worm appeared to drink slowly: it looked ecstatic. Nevertheless shockwaves multiplied among the Land's bones, ran through the gaps between instants. Far to the southwest, time was beginning to twist and flow. Mountains which had once leaned against _Melenkurion_ Skyweir slumped as if they were melting. Confusion distorted the foothills. Trees which had died thousands of years ago in Garroting Deep flashed into existence and blurred away.\n\n_Melenkurion_. The Seven Words. Abruptly Jeremiah decided to try them. He could not imagine what they might do, but he had to try _some_ thing. Anything would be better than simply waiting to die.\n\n_Melenkurion abatha. Duroc\u2014_\n\nHe blinked; scowled into the darkness. There was light in the cave. How had he not noticed it before? It was faint, yes. But still\u2014\n\nIt had to be new. It must have appeared while he was distracted by the Worm.\n\nFaint but distinct: a disturbing actinic blue, eerie as necromancy. Except where it was blocked by the Giants, it limned the boulders as if they had begun to bleed magic. And yet it conveyed nothing to Jeremiah's nerves. His health-sense insisted that the light did not exist.\n\nIn the strange glow, he saw the _Haruchai_. Vague as ghosts or will-o'-the-wisps, they faced Kiril Threndor with their backs to him and the Giants and the stones. He could feel their tension, their desire to aid Covenant.\n\nHe blinked again and again. What was causing that acrid blue? And why was it only visible to ordinary sight?\n\nHe tried to say Coldspray's name, or Grueburn's. He struggled to speak the Seven Words aloud. But his mouth and throat were suddenly too dry.\n\nHe and his companions were not alone in the cave.\n\nWith a ponderous ease that made him flinch, the boulders began to expand.\n\nThey unfolded like crouching behemoths: monsters of living rock that had concealed themselves by curling down until they resembled balls. Now they stood, pitching the Swordmainnir headlong. Jeremiah saw lumpen heads without necks, actinic eyes, massive arms and legs outlined like sketches in phosphorescent blue.\n\nSoundless as figments, voiceless as hallucinations, the creatures moved.\n\nColdspray and Grueburn crashed to the floor. At the sound, Canrik and Samil wheeled. As if they did not need time to gauge their peril, they sprang at the monsters.\n\nBurning eyes flared. Jeremiah watched in horror as one of the creatures moved to meet the Masters. A swinging arm hit Samil like a bludgeon, threw him against the wall. Jeremiah heard the horrid smack of smashed bone when Samil's skull split. The _Haruchai_ collapsed in a mess of blood and brains, sprawled lifeless as a doll.\n\nCanrik evaded a killing blow. He delivered a kick to the monster's shin, a strike that nearly broke his leg. Then he was swatted away like a clod of dirt. Only a frantic twist in the air kept him alive when he collided with the wall.\n\nAt the same time, the other stone-thing approached the Giants. Lifting one heavy foot, the creature stamped at Coldspray's back, tried to crush her spine.\n\nShe struggled to roll aside; failed. But her armor protected her. The stomp drove the air from her lungs. Her backplate cracked from neck to waist. Nevertheless she was not broken.\n\nThen Frostheart Grueburn heaved herself to her knees, swung her longsword in a wild cut at the monster. The iron bounced away, ringing like a shattered bell: it almost tore itself from her grasp. The stone-thing appeared unharmed. But her blow forced it to step back while it recovered its balance.\n\nPanting curses, the Ironhand wrenched herself upright, gripping her lore-hardened glaive in both fists.\n\nCanrik came to attack again. He moved as quickly as he could; but even his great strength could not mask his limping, or his unsteadiness.\n\n\"No!\" Coldspray gasped. \"Await an opening! We must combine our efforts!\"\n\nHe staggered to a halt.\n\nAt once, she raised her blade as if she meant to chop at the monster's head. Then she surged forward, committed all of her bulk to a straight kick at the creature's chest.\n\nJeremiah thought that he heard the thews of her knee tear, but she did not cry out. The stone-thing was driven two steps backward, three\u2014\n\n\u2014and Canrik leapt onto the creature's back, clamped his hands over its eyes\u2014\n\n\u2014and Grueburn rushed the other monster. Discarding her longsword, she tackled the creature, wrapped her arms around it, forced it away from Jeremiah. By plain force and desperation, she strove to pitch it into a fall\u2014\n\n\u2014and _moksha_ Jehannum slipped into Jeremiah as easily as an indrawn breath.\n\nAfter that, Jeremiah only knew what was happening to his companions because the Raver cast glances outward. Everything that he might have chosen for himself was taken away.\n\nhe first jolt of possession was cruel as the heat of a wildfire. It burned through Jeremiah leaving nothing but ash. Yet the scalding emotional violation passed in an instant. It was gone before he could even try to scream.\n\nIn its wake, it left an utter and unutterable peace.\n\nThe tranquility of complete helplessness dismissed his fears, his bitterness, his frantic floundering. Sudden as a crisis of the heart, every responsibility and desire and need was lifted from him. Nothing more could be asked of him\u2014he could ask nothing more of himself\u2014because there were no choices left. He was free at last of anything that resembled humanity.\n\nOh, he was conscious of _moksha_ Jehannum's presence and power, aware in every nerve and fiber. He knew that he had been claimed. He felt the Raver's vast glee, a sensation of triumph like ecstasy or delirium. He recognized the Raver's insatiable hunger for havoc. He knew that he had finally become _moksha_ 's tool, and Lord Foul's: a thing that only lived to serve the Despiser.\n\nYet the effect was not hurtful. It was pure relief, a soothing that mimicked bliss. This act of possession was a gift, a benison, a benediction. It eased him like an act of grace. He had finally become the boy he was meant to be; the boy he should have been ever since he had thrust his hand into Lord Foul's bonfire ten years ago. He had come home to himself.\n\nDo you now discern truth? asked the Raver kindly, eagerly. Long have you striven to evade our intent, long and at great cost. Long have you concealed yourself from suffering, though your wounds festered with every avoided day. Do you now grasp that there can be no surcease or anodyne for an implement, except in its condign use? Do you comprehend that there is both freedom and exaltation in the acceptance of service?\n\nThis all true believers know. They submit every desire and gift to the will of beings greater than themselves, and by their surrender they gain redemption. Self-will accrues only fear. It achieves only pain. The highest glory is reached solely by the abdication of self.\n\nDo you understand? Do you acknowledge at last that you are the Despiser's beloved son, in whom he is well pleased?\n\nThere the Raver paused. He appeared to be waiting for a response from Jeremiah; a sign of acquiescence. But Jeremiah did not reply. He had forgotten himself and did not remember what was at stake. He was simply at peace. The only part of him that seemed to have an independent existence was the part that regarded the Worm. Yet that sight conveyed neither dread nor anticipation. It had no personal implications. It merely _was_ : a fact as real as possession, and as inevitable.\n\n_Moksha_ did not prod him. Patient as the ages, the last of Lord Foul's Ravers waited as if together he and Jeremiah could take all the time in the world. When moments or hours or years had passed, and still Jeremiah had not stirred from his relief, _moksha_ Jehannum looked away as if he were mildly interested in the fate of Coldspray and Grueburn and Canrik.\n\nIn spite of their exhaustion, Jeremiah's companions fought. With a shout that seemed to rend her heart, Frostheart Grueburn succeeded at toppling her foe. But the stone-thing twisted as it fell, pulled her beneath it. When it landed on her, the impact broke her cataphract as if it were dried clay, tenuous and brittle. Air burst from her lungs.\n\nNevertheless she rolled away as the monster shifted to strike her. Its blow shook the floor; or the Worm's feeding did. A fretwork of cracks marred the rough surface. Gasping frantically, and shedding shards of armor, she regained her feet.\n\nThe other creature flailed blindly, trying to fling Canrik from its back. But its arms could not reach him. Somehow he kept his hands over its eyes. It could not see Coldspray. Through _moksha_ , Jeremiah heard or felt the wail of pain from Coldspray's damaged knee. Still she was the Ironhand. She did not relent. She kicked the stone-thing in the chest again; growled through clenched teeth; kicked again. At the same time, Canrik exerted all of his strength to drag the creature's head back. Off balance, the creature stumbled toward the wall.\n\nWhen it hit, Canrik would be crushed.\n\nThey were Jeremiah's friends. Even Canrik\u2014\n\nSamil was already dead.\n\nA vague unease drifted through the boy's tranquility. He felt himself or the Raver frowning.\n\nTo _moksha_ , Jeremiah admitted, I don't know how.\n\nHow? asked the Raver. He sounded bright as new coinage: shining gold stamped with Lord Foul's feral eyes.\n\nI don't know how to be a tool. He hardly heard himself. I don't know enough. I'm like a knife that's too dull. I haven't been sharpened. I'm not ready.\n\nWell said. _Moksha_ Jehannum's approval had a salacious tinge; a hint of slaver. All implements must be refined to their purpose. The Despiser's intent is glorious beyond utterance. No mortal born is apt to his hand. You must become greater than the greatest of your former aspirations. You must transcend every demand placed upon you by those lesser beings who sought the profit of your gifts, misnaming their desires love. By submission, you will attain the stature of eternity and awe. The Raver chuckled: a sound like the jaws of a trap closing. As will I in the perfection of my service. Then his attention became more acute. For that reason, I am within you.\n\nCruel blue silhouetted the fighting beyond Jeremiah. The monster with Canrik on its back appeared to recognize its opportunity. It heaved its granite mass at the wall. But at the last instant, Canrik sprang away. He uncovered the creature's eyes just in time to let the stone-thing see Rime Coldspray thrust her glaive at its throat.\n\nBlue glared like delight. Her blade's point splintered: her sword skidded aside. Fragments as keen as poniards scattered to the floor. Weakness and her own force dropped Coldspray to her knees. Despair gripped her features like nausea.\n\nFrostheart Grueburn did not hazard another clinch with her foe. Evading heavy blows, she retreated, circled. As soon as she could, she dove to retrieve her longsword, rolled back to her feet. An instant of consternation twisted her mien when she saw the notch that her first blow had left in the iron. But she had no other weapon. Parrying frantically while the metal shivered and shrilled, she retreated again.\n\nReason? asked Jeremiah.\n\nIndeed, the Raver answered. Take no offense when I observe that you are sadly ignorant. There is no fault in you. The _croyel_ was sent to teach rather than to torment you. Alas, it was a petty being, seduced by its own desires. It did not prepare you. Therefore I have intervened.\n\nMy task is to whet the dull blade. Yet you are not mere iron. Neither force nor fire will refine you. You require knowledge.\n\nThat knowledge I will grant. Behold!\n\n_Moksha_ Jehannum gestured in Jeremiah's mind, and the Staff of Law appeared there as though it had been translated out of his clasp. His hands still held it: his fingers curled like claws on the black wood; like an atavistic denial. Nonetheless he saw its image, precise and tangible, with the vision of thought.\n\nThis instrument, said _moksha_ , I will not touch. It is loathly and vile, fashioned to thwart me. In your grasp, however, it is mighty, capable of wonders. When it is made to serve your gifts\u2014and when those gifts in turn serve the Despiser\u2014it is potent to affect eternity, shaping order out of shapelessness.\n\nI will guide you to the lore of its proper wielding.\n\nOh, Jeremiah breathed. Order out of shapelessness. The idea pleased him. Constructs. Building. His one joy. To his granted peace was added an unforeseen happiness, a sense of possibilities.\n\n_We do what we must so that we may find worth in ourselves._\n\nHe was beginning to understand that there was more than one path to godhood.\n\nBeyond the Staff in his mind, the Staff in his hands, the Giants and Canrik still struggled. Though their strength was waning\u2014though every step and effort drained the life from their muscles\u2014they circled and evaded, apparently trying to maneuver the monsters away from Jeremiah. But the stone-things were no threat to him. They protected him. They had been sent to keep his companions away from the Staff of Law.\n\nYes, Jeremiah said. Yes.\n\n_Moksha_ 's approval seemed to make reality bend and ripple. His voice seemed to be the Worm's.\n\nThen observe closely. That noisome wight, the hated Forestal of abhorred Garroting Deep, has written his will and power upon your instrument. He is among the most despised of our foes, yet even he must serve our lord and master. Such is the Despiser's majesty and cunning. Harken well while I read the runes.\n\nTheir import will distress you. This saddens me. The Raver did not sound saddened. I desire only your exaltation. Alas, all knowledge is hurtful. Yet it is also needful. And your discomfort will be brief. You will swiftly return to joy.\n\nJeremiah nodded his consent. Masked within himself, within the private quietude of graves, he began to ask questions which the Raver did not hear. His time as the _croyel_ 's host and victim had taught him that possession was torture. He had only been able to endure it because he had no choice. Why, then, had _moksha_ entered him bearing only relief and ease? Why did the Raver trouble to lull him with peace or pleasure?\n\nHe suspected that he knew the answer. He had heard too many people talk about _the necessity of freedom_.\n\nAnd Kastenessen had broken him; but that violation had not destroyed him. Now he realized that the experience had taught him something useful. He knew how to be more than one Jeremiah at a time, each distinct from the others. He could think his own thoughts as well as the Raver's.\n\nWhat Lord Foul wanted from him, he told himself secretly, was not something that could be compelled. Like wild magic, his talent could not be coerced beyond the small uses which the _croyel_ had made of it. No matter how much he was whetted, he would not be able to exceed anything unless he agreed to it. At some point, the Despiser would need Jeremiah to serve him by choice. To submit. The tranquility which _moksha_ gave or imposed was a lure.\n\nThe idea did not disturb Jeremiah. The Raver's mastery did not allow resistance, or the emotions of resistance. It banished distress. Nevertheless there was more than one Jeremiah\u2014and some of them could be concealed or dissociated in ways which did not attract _moksha_ Jehannum's attention.\n\nBubbling with glee, _moksha_ read the Staff. His magicks lit the abstruse symbols, not with fire or shining, but with a deeper black that scorned human notions of darkness. His disembodied finger traced the script as he interpreted it. Yet he did not explain it in words. Instead he gave Jeremiah images.\n\nWhile the runes came to life, Jeremiah found himself standing on the ruined dirt of Gallows Howe surrounded by the ire of trees.\n\nHis presence there was only a vision. He had not passed through time to an age when Caerroil Wildwood's outrage ruled Garroting Deep. His body still sat on the floor of a cave in Mount Thunder, holding the Staff of Law across his thighs, feeling tremors rise through the gutrock; apparently watching his companions fight with their last strength. But his mind\u2014\n\nHis mind had followed the Forestal's symbols into the recesses of _moksha_ Jehannum's memories.\n\nEverything that Jeremiah beheld, _moksha_ viewed with hate, with savagery and revulsion. The dirt under his feet had drunk the deaths of Ravers. Their assumed bodies had dangled from the gibbet of the Howe while their spirits had shrieked in agony. Anywhere else in the Land, anywhere at all, _moksha_ or _samadhi_ or _turiya_ could have simply slipped away when their flesh was taken, sparing themselves the horror of being slain. But in Caerroil Wildwood's demesne, they had been denied that luxury. The Forestal had _forbidden_ them. They could not escape.\n\nThe recollection made _moksha_ Jehannum froth with fury and frustration. Nonetheless what the Raver sought was here, in the innate lore of forbidding; in Caerroil Wildwood's ability to draw power or sentience or resolve or rage from every leaf and branch, every twig and trunk and root, throughout his loathed realm\u2014and then to express that force in ways which _moksha_ and his brothers could not withstand.\n\nFor the Raver, Gallows Howe summed up everything that he abhorred about forests. But his hatred was more than that. It was wide as well as deep. It included every tree of every variety everywhere: young and old, graceful and gnarled, upright and outstretched. Alone they were each as vulnerable as kindling. Together they were as mighty as mountains. Therefore _moksha_ hated them with a vehemence that trembled in every particle of his being. They were everything that he was not: stately, grand, generous, welcoming, austere, fecund. Their existence justified every stretch of ground where they flourished\u2014and the Raver hungered for their extinction.\n\nJeremiah saw all of this as _moksha_ Jehannum saw it. He felt the Raver's fulminating outrage so keenly that he appeared to share it. And he knew that _moksha_ wished him to share it. But he also saw the Howe and the Deep with his hidden eyes. He knew the wrath and grief of the innumerable trees. He understood how those passions formed the essence of the Forestal's power. More, he recognized that the forest's vast appetite for bloodshed was not inherent. It was a response to a terrible crime.\n\nThe force which lay behind it was not rage, but rather a bereft adoration for the green and living world in all of its fragile guises. The substance and sorrow of everything that Caerroil Wildwood had been and done was his love.\n\nAnd Garroting Deep was an emblem of the Land. _Moksha_ 's hatred of trees was only one manifestation of a more encompassing evil: the fury and despair that despised or feared every aspect of the Land's rich beauty.\n\nThis, too, did not trouble Jeremiah. He felt no indignation, no desire to protest. Instead he considered it among his private selves. He resisted nothing, and so nothing was taken from him. Passive as a victim, he kept his thoughts to himself, as he had done for most of his life.\n\nFrostheart Grueburn still circled on unsteady legs, flailing with her blunted longsword. Rime Coldspray hacked and hacked at her foe until her glaive was shattered to the hilt. Canrik twisted between the stone-thing's legs, trying to trip or topple the monster. But that tactic failed him. The creature was too strong, too heavy.\n\nStill the _Haruchai_ struggled. And he had resources of stamina which exceeded even the Swordmainnir: he could still think. When he realized that he was too weak to bring down the monster, he slipped away. Snatching up a long sliver of the Ironhand's sword, he sprang again onto the creature's back. His ragged dirk he pounded into one of its eyes.\n\nThe force of his blow sliced open his hand. Blood spurted between his fingers. But the sliver penetrated. Actinic blue blazed for an instant. Then the eye went dark.\n\nThe stone-thing had no voice. It could not scream. Nevertheless the reflexive slap of its hands at its face was as wounded as a shriek. One hand swept the shard from its eye. The other caught Canrik's wrist. A fierce swing flung him away.\n\nEntirely by chance, the monster threw him into the tunnel toward Kiril Threndor. He vanished from the cave.\n\nJeremiah did not see what became of him. He did not know how the Giants stayed on their feet. Yet this sight also did not distress him. He watched his friends impassively, as if he had already succumbed.\n\nHe understood forbidding now: the how of it, the why, the necessary power. He had absorbed it without the hindrances of language because _moksha_ and the Despiser needed him to understand it. It was essential to Lord Foul's deeper purpose. But Jeremiah's epiphanies went further. On Gallows Howe, with Garroting Deep unfurled like a banner around him, he realized that forbidding was essential to other purposes as well, to desires which were not the Despiser's.\n\nForbidding was Earthpower, of course; but it was Earthpower transformed by trees and their Forestal into an entirely different form of magic.\n\nTo _moksha_ , Jeremiah said, I need more.\n\nIf forbidding alone had been enough, the Forestals could have defeated Lord Foul themselves.\n\nIndeed. _Moksha_ Jehannum's approval was incandescent. Abhorrence is but one refinement. Other whetstones are needed to perfect the blade.\n\nWhile Jeremiah watched, helpless and unmoved, the Raver took him on a coruscating plunge through other memories, other expressions of recalled lore.\n\nHis passage was a whirlwind, a giddy chiaroscuro, a torrent of glimpses and insights. He did not try to grasp them: he hardly looked at them. Instead he simply accepted them; allowed them to be imprinted on his nerves, written into his brain. Some were millennia old: a jeweled casket sunk deep into the mire of the Great Swamp, a tapestry sealed in a cavern lost among the snows of the Northron Climbs, a periapt as crowded with knowledge as a tome. Others were immeasurably ancient: the creation of Forestals from the substance of an _Elohim_ , the complex theurgies which had fashioned the Colossus of the Fall, the invocation of Fire-Lions. He did not need to make sense of them because they were already his, ready for his submission and use.\n\nBut among the swift confusion of those recollections, Jeremiah found one memory that filled _moksha_ Jehannum with a particular delight. It was the Raver's recall of that horrific, wonderful moment when _moksha_ had taken possession of Linden.\n\nPerhaps her straits should have appalled Jeremiah; yet they did not. He was intimately familiar with the excruciation which the Raver had inflicted on her, the relish for her torment. He had survived such things himself. And he knew that she had somehow expelled _moksha_ Jehannum for Covenant's sake, or for the Land's. She was Linden Avery. _Moksha_ 's cruelty could not define her.\n\nHowever, some of her own memories lived among the Raver's; and _those_ wrung Jeremiah's heart. They erased his calmness, dismissed his given relief as if it were nothing more than a mirage. For the first time, he learned what his mother had suffered when she, too, had been just a kid.\n\nRemembered by _moksha_ , Jeremiah stood in the attic with her, watching her father bleed out of his cut wrists, and helpless to force the blood back into his veins. Already gashed and dying, that aggrieved man had locked her in with him so that she would not be able to go for help. In effect, he had compelled her to witness his surrender to self-pity: her father.\n\nShe had been only eight.\n\nMom. Jeremiah wanted to wail. _Mom_. But the Raver was not done.\n\nCrowing, _moksha_ remembered Linden's mother. At about Jeremiah's present age, she had been at her mother's bedside while her mother had prayed for death. According to _moksha_ , the woman's illness may not have been terminal. But Linden had heeded her mother's pleading. Her mother had blamed her, Linden, for causing her husband's death; for making her life unsupportable. And Linden had been left alone to provide care. Wipe away sweat. Mop up dribbling mucus. Tend bedpans. So when Linden had exhausted her own misery, she had\u2014\n\nJeremiah did not know how to bear it.\n\n\u2014taken wads of tissues and forced them down her mother's throat; forced more and more of them down until her mother would never blame anyone else again.\n\nThe Raver reveled in those events. _Moksha_ wanted Jeremiah to understand that his mother had always been a victim and a killer. The woman who had claimed to love him was as pitiful and weak as his natural mother. Linden's parents had made her who she was. She would never be anything more. Because of her\u2014 _moksha_ Jehannum insisted on this as if the truth were beyond question\u2014Jeremiah had always belonged to Lord Foul. From the first, he had been raised to serve Despite by women who had earned their own victimization.\n\nThe gift that Lord Foul offered now was more than mere peace, more than simple relief: it was transcendence. Jeremiah's submission would be rewarded with a place in eternity, a form of godhood in which his wounds and struggles would have no meaning. He would be free at last of his inherited unworth.\n\n_Moksha_ urged this vision of Jeremiah's future as if it were perfected delight. And Jeremiah heard the Raver. He recognized what the Raver wanted from him. But he was no longer listening. Within his secret silence, he cried out for the woman who had chosen to be his mother when no power in life could have required her to claim him.\n\nYes, he told Lord Foul's servant. Yes.\n\nEntirely dissociated from his real circumstances\u2014entirely concealed from his possessor\u2014he meant, Watch your back, you piece of shit. I'm coming for you.\n\n_Just do_ something _he doesn't expect._\n\nSpasms shook the cave. Forerunners of temporal rupture broke chunks of rock from the ceiling, scattered debris across the floor. Grueburn staggered from side to side gasping for breath, barely able to stand. Canrik lurched back into the cave. He kept his fist clenched to stanch the bleeding of his hand. Desperation twisted his features as he searched for a way to aid the Giants.\n\nIneffective as a cripple, Coldspray stood directly in front of Jeremiah. The one-eyed monster advanced on her, ready to strike. She waited for it as if she had come to the end of herself and could no longer raise her arms.\n\nBut when it reached out to wrap her in a crushing embrace, she lifted the remains of her glaive and hammered the pommel into the creature's good eye.\n\nAs the light of that eye died, the blinded stone-thing lashed out. In mute pain, it tossed the Ironhand aside as if she had become trivial.\n\nNow, however, the monster could not see. Confused by its hurts, it seemed unable to locate Coldspray. Instead of pursuing her, it continued its advance. Swinging its massive arms, it came toward Jeremiah.\n\nOne inadvertent impact would be enough. He would not survive even a glancing blow. Lord Foul's plans for him\u2014\n\nInside Jeremiah, _moksha_ Jehannum snarled an obscenity. Distracted, he snatched Jeremiah's halfhand off the Staff of Law, drew a swift symbol in the air.\n\nThe creature began another step. Halfway through the motion, it suddenly collapsed into dust: a pile of remains stirred only by the tremors rising through the floor.\n\nDuring that brief instant, Jeremiah took his chance.\n\nHe had absorbed astonishing kinds and quantities of lore from the Raver, more knowledge than he could have named. Forbidding was a part of it. An expression of Earthpower called a Word of Warning was a part. The wood-magicks of the _lillianrill_ were a part, as were the elaborate healings which the Lords had once wrought in Trothgard, and the music with which Caerwood ur-Mahrtiir had invoked a bower among the wastes of the Lower Land. He knew how the great tree-city of Revelwood in the Valley of Two Rivers had been fashioned.\n\nBut that was not all: he had learned more. If he had been released, he could have devised a prison which would have snared _moksha_ Jehannum until Time was extinguished. Given a few uninterrupted days, he could have repaired the damage that ancient violence had done to Mount Thunder's heart. With a few years and a Forestal's aid, he could have made a garden of the Lower Land.\n\nBut the Raver had not released him, and he had only an instant. When his opportunity came, he did not hesitate.\n\nOne small sip of Earthpower from the Staff restored his inherited theurgy. Then he rose up from helplessness to trade places with his possessor.\n\nIn the space of a single heartbeat, he trapped _moksha_ Jehannum inside himself.\n\nThe Raver struggled, screaming. Of course he struggled. He knew everything that Jeremiah did. He had long ages of experience to guide him. He had frenzy and ripe terror. And Jeremiah was only mortal. He lacked the intransigent metal of a _Haruchai_. He did not have the great spirit of a Giant. He had no inborn capacity to defy possession.\n\nBut he had resources which Lord Foul's servant could not match. Linden had blessed him with long years of care and tenderness. Anele had given him power. He had learned how to walk away from the helplessness with which he had protected himself. And he was not afraid to grasp the Staff of Law.\n\n_Moksha_ howled horror at the ceiling. He thrashed and writhed, raked frantic claws across the barriers which Jeremiah raised against him, sank sharp teeth into the flesh of Jeremiah's resolve. Wild and despairing, the Raver fought.\n\nYet Jeremiah refused the fight. He did not need to measure his strength against his foe. Instead he relied on knowledge which _moksha_ did not share. Retracing his own past, he _dissociated_ the Raver; committed Lord Foul's servant to the graveyard where he himself had once lain, hidden and lost. Almost effortlessly, he dropped the Raver into the waiting earth.\n\nWith Earthpower and newly acquired lore, he clamped down on _moksha_ Jehannum until he could no longer hear the Raver's screams. He piled dirt over the malign spirit, stamped the grave flat. Then he turned away.\n\nAt one side of the cave, Rime Coldspray tried to regain her feet, but she could not. Trying to evade the second monster, Frostheart Grueburn had crumpled to her knees. Canrik had found another splinter of Coldspray's glaive. Now he looked for an opening, a chance to sacrifice his other hand.\n\nGritting his teeth, Jeremiah rose up in power. A detonation like a thunderclap from one heel of the Staff tore the stone-thing apart. Rendered to powder, it fell.\n\nThe floor heaved. The ceiling shed more rocks. Cracks yawned open, grated shut. Here and there, wounds split the walls. Patches of gutrock oozed and ran as if their essences were being squeezed out of them.\n\n\"I'm sorry,\" Jeremiah panted: a faint echo of his friends' gasping. \"I mean, I'm sorry that took so long. First I didn't know how to do it. Then I had to wait for a chance.\"\n\nA chance which the Swordmainnir and Canrik had given him.\n\n\"Do not heed us,\" the Ironhand managed to say between broken breaths. \"The Timewarden\u2014The Worm\u2014\"\n\nJeremiah did not have time to think. Covenant needed him. Canrik was already waiting for him at the tunnel toward Kiril Threndor.\n\nHe took the time. \"You're joking.\" His tone hinted at _moksha_ 's glee. He had enjoyed immuring the Raver. \"I can't leave you like this. You don't look strong enough to stand.\n\n\"This is Mom's Staff. It doesn't really belong to me. But I know how to use it now.\"\n\nThen he released a second blast of Earthpower.\n\nThis detonation was as fierce as the force which had destroyed the stone-thing; but it was an entirely different kind of theurgy, a more natural magic. It hurt Coldspray and Grueburn, but it did not damage them. Instead it delivered violent healing, a ferocity of repair. He had learned too much too quickly: he was not capable of gentleness. And the Worm was feeding. Concussions spread through the substance of the world. Disruptions of Time mounted toward the last crisis of the Earth. He had to reach Kiril Threndor and Covenant.\n\nIn a moment, he was done. He stamped the Staff on the floor once because he had no words for what he felt. Then he gathered himself to follow Canrik.\n\nUntil he saw Rime Coldspray climb to her feet and test her limbs, trembling as if she were feverish\u2014until he felt Frostheart Grueburn standing near him, and Canrik watching with open surprise\u2014Jeremiah did not notice that the cave was full of warm light. He had taken it for granted\u2014\n\nThe Staff felt like recognition in his hands. It sent out broad swaths of flame as kindly and soothing as sunshine. Its shaft shone with the cleanliness of healthy heartwood. Along its surface, Caerroil Wildwood's runes remained, distinct as promises, but their meaning was no longer obscure. They were an offering and an appeal: they enabled and prayed.\n\nTo Jeremiah Chosen-son, the descendant of Sunder and Hollian in spirit if not in body, the Forestal's script pleaded for restoration.\n\n## 12.\n\nYou Are Mine\n\nAt the edge of Kiril Threndor's high chamber, Thomas Covenant stood motionless, held by shock and fury while he scrambled to absorb what he saw.\n\nAnger was not what he needed here: he knew that. If he had failed to see the truth for himself, he could have heeded High Lord Berek among the Dead. _He may be freed only by one who is compelled by rage_ \u2014Ire would mislead him when he absolutely had to be the master of himself.\n\nBut he could not control what he felt.\n\n_Well, hi, Dad_. That was his son. His _son_ , wracked like a plague victim by power and malice. _You took your own sweet time getting here_. His son with Lord Foul's putrescent eyes.\n\nThe Despiser had claimed Covenant's lost boy at last. Lord Foul had taken possession\u2014\n\nThe sight set a spark to the driest tinder in Covenant's soul. Between one breath and the next, he became conflagration; incandescent wrath. Wild fire flushed across his skin in waves like the urgent knot and release of his heart. Flames spat from his eyes, lashed out from his arms and chest. His vehemence cast argent through the diseased chiaroscuro of rocklight. Bright killing gathered like a blade in the scar on his forehead.\n\nBerek had warned Linden. He had warned Covenant. But he had said nothing about the means by which Lord Foul might gain freedom.\n\n\"What's the matter, Dad?\" Roger glared as though his whole being had been consumed by scorn; as though he had been torn apart and put back together wrong. Denied anguish contorted his visage. At every moment, he looked more like a maimed thing, twisted beyond recognition. His right hand was sick lava, fuming and rotten. \"Aren't you glad to see me?\"\n\nHis plight demanded pity. For Covenant, pity was rage.\n\nA step ahead of Covenant, Branl regarded the figure on the dais. He held Longwrath's flamberge negligently, as if he had no further use for it. \"Ur-Lord,\" he remarked as though he had been studying a particularly uninteresting icon, \"I now comprehend why I was unable to discern the presence of Corruption. His aura was both blurred by Kastenessen's _skurj_ -born theurgy and disguised by his human vassal. Here his evil is plain. Corruption has taken your son, or your son has given himself. We must oppose both or neither. We cannot harm the spirit while the flesh shields it.\"\n\nHell and blood. Covenant had no answer for the Humbled. He had none for Roger. Wreathed in flame, he tightened his grip on the _krill_ and started forward. Fissures marred the floor in front of him like the outcome of his anger; but he ignored them. Dizzying reflections and stalactites and tortured slabs of granite meant nothing to him. With every stride, he raised Loric's dagger higher. The radiance of its gem filled his voice.\n\n\"Let him go,\" he snarled at the Despiser. \"This is between you and me. _Leave him out of it_.\"\n\n\"Dad!\" Roger feigned surprise. He feigned dismay. \"You still don't get it.\" He lifted his inhuman hand to match the _krill_. A brimstone stench covered the reek of attar. The redder heat of magma daunted the rocklight. \"None of this would have happened if you and that damn woman hadn't interfered. All I wanted was the _croyel_ \u2014the _croyel_ and Jeremiah\u2014but you wouldn't let me have them. If you had stayed out of my way, I wouldn't be here.\n\n\"This is _your_ doing, Dad. It's the only choice I had left.\"\n\n\"I don't care,\" Covenant retorted. \"You did this to yourself. Nobody forced you. All you had to do was take pity on your mother,\" on poor, deranged Joan, who had no defense, \"and none of this would have happened.\"\n\n\"Really?\" drawled Roger. His grimace mimicked a sneer. \"You actually think that? You should care. I'll tell you why. Since you seem oblivious to what's been going on, I'll explain it.\n\n\"My _mother_ \"\u2014he spat the word\u2014\"was useless. She couldn't help me. She was just a distraction to keep you away from me. The _croyel_ and Jeremiah were my way out. While I had them, I didn't have to _serve_ anybody. _I_ didn't have to care. But you took that away, you and that damn woman. You slammed the door on me, _Dad_. This is what I have left.\n\n\"I'm not going to die no matter what you do, and do you know why?\" Pressures within Roger clawed terrible shapes across his face. Lurid fires filled his eyes. Threats dunted from his halfhand. \"Lord Foul is going to take me with him. That's the deal. I gave myself to him, and he's going to give me eternity. We're just waiting until the Arch crumbles enough to let us out. Then we'll be gone. It'll be like you and this whole disgusting place never existed.\n\n\"I'm letting him do what he wants because _he's going to save me_!\"\n\nHalfway to the flawless dais, Covenant halted; froze on the verge of howling his fury. The pain in Roger's voice stopped him. He could almost hear the hollowness of his son's soul.\n\nBranl was right. Of course he was. Covenant could not strike at Lord Foul without hitting Roger first. He would have to kill his son in order to hurt the Despiser\u2014and he had already killed his son's mother.\n\nHe needed a better answer. Somehow he had to set anger aside, swallow horror. Roger's sarcasm and arrogance masked the truth. The young man was appalled by what he had done to himself.\n\n\"No,\" Covenant snapped, wrestling for composure. \"He won't take you with him. Whatever he offered you won't be what you think it is.\" He had cloaked himself in fire and outrage as if they were a shield, but he could shrug them off his shoulders if he dared. If he could find the courage. \"You're scared, Roger. You're too scared to think. You aren't using your brain.\n\n\"You're physical. Don't you understand that? You're mortal. Time is all you've got. It's the only thing that makes life possible. Without it, you're nothing. You're just\u2014\"\n\nThe floor heaved, shaken by Roger's impatience or the Worm's feeding. Cracks groaned in the walls. The stalactites scattered rocklight and silver in pieces sharp as shards. Covenant lost his balance, staggered until Branl caught him.\n\n\"Well, duh,\" Roger snorted. \"Of course I'm physical. That's why he needs me. That's why I can trust him. He needs me to get rid of you.\n\n\"I'll give you this, Dad. Lord Foul is afraid of you. You've surprised him too often when he thought he had you beat. But that won't happen this time. That's what _I'm_ for. That's why he made a deal with me. I'm going to make sure you don't surprise him again.\"\n\nResisting a rush of frenzy, Covenant shouted, \"No! He's just _using_ you. He doesn't _need_ you. He can be as physical as he wants whenever he wants.\" Covenant had not forgotten the tangible impact of the Despiser's contempt when Covenant had faced the Illearth Stone in Foul's Creche. \"But you can't be as eternal as you want. You're _dross_ to him. You're more than a hindrance, you're a prison. He can't escape the collapse of Time while he's inside you. If he tries that, he'll die when you do. He won't get out unless he leaves you behind.\n\n\"And when he does, you won't be able to follow him, and you sure as hell won't be able to accompany him, because you're just _you_. You aren't made for eternity. You're just a frightened man who can't stand being afraid. Giving yourself to Foul isn't hope, it's _panic_.\"\n\nRoger was roaring like his hand, poised to strike; but Covenant did not pause. \"You're going to die like the rest of us,\" he insisted. \"No deal can save you. Foul can't make you a god. He knows that as well as I do. If you can convince yourself otherwise, you've been serving him longer than you think.\"\n\n\" _No_ , Dad.\" Tremors like hysteria shook Roger's voice. Pain wrenched at the corners of his mouth. \"You've got it all wrong. Lord Foul doesn't lie. He promised I would stop being afraid. He promised what's happening now is temporary. He promised I would never be in pain again.\"\n\nSure, Covenant wanted to reply. It's all true. You _won't_ be afraid. You'll be dead.\n\nBut a sudden surge of power from the dais closed his throat. Abruptly Roger's voice changed. \" _Enough_.\" It became the sound of crushed boulders, falling mountains. It had the depth and resonance of a tectonic upheaval. \"That promise I will honor. I will put an end to your fawning. Now you will be silent. I will speak to this doomed wight who deems himself my foe.\"\n\nInvoluntarily Roger bit down on his tongue. Blood leaked from his mouth. His eyes bled venom.\n\nAt the same time, behind or within or through Roger, Covenant saw another figure, a towering shape taller than Giants, mightier than the spectres of High Lords. Authority and rocklight limned the form; but within its outlines was nothing but absence, an emptiness like the chasm of the Lost Deep. The figure's sole feature was its fanged eyes. They resembled Roger's, yellow and bitter. But the ferocity in them, or the despair, was fiercer than Roger's denied terror.\n\n\"Ur-Lord,\" Branl warned unnecessarily. \"Corruption manifests. Yet he also retains possession of your son.\"\n\n\"Oh, good,\" Covenant snarled at the Despiser. \"I'm glad. Now I can talk to you directly.\n\n\"You really ought to be ashamed of yourself. You don't need surrogates. You should have the decency to let Roger go. Or if you can't manage decency, you should at least have the dignity. Using him just makes you look craven.\"\n\nLord Foul expanded. He made himself too big to be confined by Kiril Threndor. Yet his lambent silhouette remained visible, as if he had superimposed himself on the rock.\n\n\"We are well met, Timewarden.\" He did not shout, yet every word was a blast of ruin. \"In times past, I have named you groveler, anile and foolish, but I now perceive that you have become worthy of me. Your death has been made certain. No exertion is required of me to assure it. Nevertheless I acknowledge that at last you merit extinction at my hand.\"\n\n\"Sure.\" Covenant readied the _krill_. As much as he could, he ignored Lord Foul's fierce shape, concentrated on Roger. \"Try it. I'm not going to surrender again. And I am done with restraint.\"\n\nThe Despiser laughed like grinding stones. \"Yet you have not forgotten folly. That pleases me.\" His eyes and Roger's bit at the air. \"I find delight in your misbelief that you are potent to oppose me.\n\n\"Have you forgotten, Timewarden? Does mortal recall fail within you even now? I have assured you that you are mine. You have been my servant always, though you have twice refused submission. Each and all of your efforts to thwart me have conduced to my present triumph. Because you have dared to oppose me, I will be made free.\"\n\nCovenant shook his head. \"Maybe _you're_ the one who's forgotten. We've talked about this before. It goes both ways. If I'm yours, you're also mine. Maybe I've always been yours, but I made you mine when I let you kill me.\n\n\"And apparently you've forgotten Linden. You tried to tell her the same thing. According to you, everything she does guarantees your escape. But she's still here. She's still doing things you didn't expect and couldn't imagine. She may even find a way to keep you here when reality falls apart.\"\n\nThe Despiser swelled. He appeared to gather vehemence. But Covenant did not flinch.\n\n\"And haven't you forgotten Jeremiah? Don't you need him? Isn't he essential to your _deeper purpose_? How can you even hope to use him when he has the Staff of Law?\"\n\nLord Foul's laughter was savage. It felt unanswerable.\n\n\"Indeed, the boy holds the Staff of Law. But my servant _moksha_ has taken possession of him. Even now, he awaits my will. Through him, Law itself promotes my intent.\"\n\nOh, hell! In spite of his fire, Covenant faltered. _Moksha_ had Jeremiah? The walls of the chamber seemed to contract around him. Futures for which he had prayed faded like hallucinations. He had gambled on the boy: gambled and lost.\n\nHow would Linden bear it when she learned that her son served the Despiser?\n\nAt that moment, Roger struck. His halfhand hurled a bolt of incineration at his father.\n\nReflexively Covenant caught the blast with Loric's _krill_ ; blocked it with the gem's radiance and an outpouring of wild magic. Argent against laval crimson, flame against the savagery of molten stone, he fought to save himself.\n\nBut he hardly knew what he was doing. He lost track of Branl. The dagger bucked in his grasp: Roger's force tried to tear it from his numb fingers. The coruscation of powers blinded him. Briefly Kiril Threndor inverted itself. He depended from the floor, felt himself falling toward the ceiling. Then the whole chamber reeled, giddy as vertigo.\n\nHe clung to the _krill_ instinctively, sent his heart's need like lightning through the blade's cut jewel; floundered to survive.\n\nHis son's might appalled him. Roger was stronger now. The severing of his human hand from Kastenessen had not weakened him. Nor had Kastenessen's passing into the fane of the _Elohim_. Roger's given fist retained the ravaging force of the _skurj_. And Lord Foul stood behind him or within him, supporting him.\n\nSoon the _krill_ would start to melt. It had to. Nothing mortal-made could endure Roger's virulence, or Covenant's wild response.\n\nUpright beyond the ceiling and the stalactites, the breaking gutrock, Lord Foul watched. His eyes gnashed approval.\n\nBlasts like magma knocked Covenant's weapon from side to side. Feral heat chewed into his hands, gnawed at his arms. And his dead nerves betrayed him. They spared him from the worst of the pain, but they also weakened his grip. The hilt twisted. The skin of his fingers seemed slick as spilth. He could not hold.\n\nHe had to hold. The moment of his last crisis was upon him. Catastrophes burned in the bones of his forehead. Everything that he required of himself while life remained in his body depended on his ability to grip and hold.\n\nSomehow he withstood Roger's assault. He had more than the _krill_ : he had wild magic. In some sense, he _was_ white gold. The power possible for him was limited only by his humanity, his flesh and sinew and passion. Loric's dagger was not melting. Even Covenant's hands were not. They were preserved by the theurgies that saved and damned; by the contradiction of renewal and ruin that formed the keystone of the Arch of Time. As long as he did not let go\u2014\n\nBut he could not do more; could not advance to threaten Roger or the Despiser. Together they were too strong. Roger's savagery demanded his utmost, and his utmost was not enough.\n\nAnd while he fought to withstand lava and malice, he gave no heed when the boulders against the walls opened themselves and became monsters.\n\nTwo of them. Three.\n\nApparently the Despiser was not satisfied. He desired Covenant's death too much to let Roger fail.\n\nThe stone-things were vacancies. Despite their actinic auras, they were only visible to ordinary sight. Branl did not sense them. His attention was fixed on Covenant's struggle. One step at a time, he circled obliquely closer to the dais. But he was looking for an opening, a chance to attack while Covenant distracted Roger. He was not watching for other threats.\n\nAs massive as monoliths, and as silent, two of the creatures lumbered toward the Humbled from opposite sides. The third advanced on Covenant.\n\nCovenant saw nothing except white fire and ruddy brimstone; felt nothing except the tearing heat of Roger's theurgy. Roger had called him _oblivious_. He was oblivious now. There was no room in his heart or his mind for anything beyond the extremity of his need to hold on.\n\nBut Branl was _Haruchai_. He may have been as transfixed as Covenant; may have felt as desperate. Nevertheless he was a warrior to the bone, defined by combat. A heartbeat before the nearer stone-thing drew close enough to hit him, he saw it.\n\nWhatever he thought or felt at that moment, he did not hesitate. Spinning away from the dais, he swung a two-handed cut at the side of the creature's neck.\n\nThe clang of iron shivered among the stalactites. The flamberge bounced back, singing with stress.\n\nThe monster lurched to a halt. A third of its throat had been sliced open.\n\nBranl needed an instant to regain control of his blade. Then he swung again.\n\nThis time, the creature folded to its knees. Slow as a sigh, it collapsed on its face and became dust.\n\nFebrile with pain and hate, Roger fed the mounting holocaust. Through the glare, Covenant descried Roger's features. Their agonized contortion seemed to cry out, wailing of needs and fears that surpassed sound, exceeded the firestorm of powers. Roger's mouth shaped words which Covenant could not hear.\n\nDad, Covenant's son seemed to be saying, help me.\n\nAbruptly his own dread and hurt fell away. The burning of his hands lapsed into numbness. His grip steadied the _krill_ against Roger's onslaught. Wild magic rose to a pitch too acute for perception. _Moksha_ Jehannum had taken Jeremiah. Covenant did not know what had become of Linden, but he knew that She Who Must Not Be Named was too strong to be defeated. And the Worm of the World's End was feeding. Forces mightier than Covenant's struggle shook Mount Thunder to its roots. He was losing everything that he had ever striven to preserve. Nevertheless he was not daunted. He still had something to fight for.\n\nHis son was possessed. Roger bore the immedicable wound of Kastenessen's hand. He had been a fool\u2014a fool and a coward\u2014but that changed nothing. He had not chosen his parents; had not caused his mother's weakness or his father's absence. Now the extravagance of his distress made Covenant's voluntary hurts seem trivial.\n\nA different kind of anger dismissed Covenant's pain; his earlier wrath. This new ire resembled his old, familiar rage for lepers. It was a passion colder, calmer, and more complete than his desire to hurt the Despiser: a sympathy so furious that it felt like exultation.\n\nClenching Loric's dagger, he concentrated his outpouring of fire through the gem. Then he began to force his way toward the dais. One step at a time, he advanced against torrential magma and malevolence.\n\n\"No!\" the Despiser shouted. \"I will not permit it!\"\n\nWhile Branl stood over the fallen stone-thing, the second creature came at his back. One sweep of its granite arm smashed his shoulder, flung him at the wall. Noiseless amid the cacophony of magicks, the flamberge clattered to the floor. He struggled to rise, but his legs failed him.\n\nIn that instant, Stave appeared in Kiril Threndor as though he had dropped from the ceiling. Somehow Linden had translated him here. He would not have left her side willingly.\n\nNonetheless he was _Haruchai_ : he did not need time to gauge what was happening around him. As his feet touched the floor, he dove for Branl's longsword. A roll brought him upright with the flamberge in his fists. His momentum carried him into a straight lunge at the creature which had struck the Humbled.\n\nIn spite of its antiquity, the blade retained some vestige of Kasreyn's lore. It drove deep into the monster's chest. When Stave wrenched out the longsword, the stone-thing toppled to one side. Dying, it turned to powder and drifted away.\n\nReflections of brimstone and wild magic flashed in Stave's eye as he hastened to stand between Covenant and the third monster. His mien was a taut mask of outrage and grief.\n\nLinden, Covenant thought. Oh, God. What have you done?\n\nBut he did not stop fighting.\n\n\"No!\" Lord Foul roared again. \"I will not _permit_ it!\"\n\nScourged by his possessor, Roger shifted his aim. Fierce as a scream, he turned his power away from Covenant.\n\nA mistake\u2014In the space between instants, Covenant thought that the Despiser had misjudged his foes\u2014or had simply been overcome by his own fury. The _Haruchai_ could not oppose him. Covenant was the real danger.\n\nThen, however, Covenant saw the frenzy in Roger's eyes\u2014saw the Despiser's bitterness dulled by a more human anguish\u2014saw Roger hurl coerced scoria, not at Stave, who shielded Covenant, but at Branl, who could not.\n\nThe Humbled lay gasping against the wall. One shoulder had been shattered. Other bones were broken. His legs refused to hold him. Still he managed to wrench himself aside.\n\nRoger's blast did not destroy him. Instead it made a smoking ruin of his wrecked arm, stripped the flesh from his ribs. Even that lesser damage might have killed him; but Roger's attack cauterized as it burned. Branl was stricken unconscious: he did not bleed. His chest still heaved for air.\n\nRoger had done that: _Roger_. It was as close to an act of mercy as he could manage. In spite of Lord Foul's mastery, Roger had left Stave alive to protect Covenant.\n\nAnd Covenant\u2014\n\nCovenant recognized his chance.\n\nIn a stumbling rush, he ran at Roger, gained the dais. Faster than he could think, he slashed with the _krill_.\n\nOne swift stroke severed Kastenessen's hand.\n\nThe hand exploded; or Lord Foul's presence in Roger did. The concussion tossed Covenant away. He hit hard enough to crack his skull. A whirlwind of little suns wheeled across his mind. He lost the dagger somewhere. Blood started from his eyes. It ran from his ears. He could not feel his arms, his legs. A gyre of disconnected instants sucked at the verges of reality.\n\n\" _You_ ,\" raged the Despiser, \"will not _prevail_!\"\n\nA clutch of theurgy yanked Covenant from the stone, threw him farther. He skidded like scattered bones over slabs and fissures.\n\nHe had no strength, no weapon. He might as well have had no limbs. Another throw would finish him.\n\nSightless and desperate, he answered with wild magic. His mind became white fire. Violent flames poured from every part of him that still had living nerves and could feel pain.\n\n\"You bastard.\" Roger seemed to be shrieking at Lord Foul, but Covenant heard only whispers. \"You lied to me.\"\n\n\"And do you now take offense, little man?\" snorted the Despiser. \"I do not regard your umbrage. I do not speak lies. If you heard falsehood, it was of your own making. Now you will suffer the outcome of your folly. Take comfort in the knowledge that your abjection will be brief.\"\n\nRadiating fire like waves of fever, Covenant tried to blink the blood out of his eyes; struggled to see.\n\nHe lay on a canted sheet of basalt. Vaguely past its rim, he glimpsed the unharmed dais, the broken clutter of stalactites. The furious shape of Lord Foul still dominated the chamber, too immense to be opposed or endured.\n\nBranl lay where he had been struck. Stave had vanished or fallen. Had he confronted another monster? Covenant had no idea how many stone-things still moved in Kiril Threndor.\n\nBut over there, to the left of the dais, stood Roger, unpossessed and human. Fountains of blood had streaked his clothes, stained his face. Facing the Despiser, he huddled over his pain with his gushing wrist clamped under his arm to slow the bleeding. He glanced at Covenant; at Covenant's undifferentiated, useless flail of power. Then he turned back to Lord Foul.\n\nTremors ran through the floor. They staggered Roger, rocked Covenant mercilessly. The Despiser and the dais they did not affect.\n\nLord Foul's biting eyes loomed over Covenant. \"As for you,\" he sneered, \"beaten Unbeliever, impotent Timewarden, I have reconsidered your doom. Though I hunger for your death, I also crave your despair. Therefore I have asked of myself which end will wound your spirit more grievously, a death in agony at my hands, or an occasion to witness the final devastation of all that you hold dear. Remain as you are, and you may observe my return to majesty. Continue to oppose me, and I will snuff your frail life as you would a lantern.\"\n\nSquinting, Covenant located the _krill_. It was too far away.\n\nGrip and hold.\n\nTry it, he panted, although he could not speak. See what happens. He could hardly move. You haven't won yet.\n\nNevertheless his shining faltered. He let his power fall away.\n\nThen he found himself rising to his feet. Stave lifted him from behind, supported him when he could not stand alone.\n\nThe last of Lord Foul's stone defenders was gone.\n\nThe chamber juddered as if it had been struck by the leading edge of a tsunami. Covenant's guts and chest knotted, threatening to retch blood. But Stave's arms sustained him.\n\nSoftly Stave breathed, \" _Moksha_ Jehannum has taken the Chosen-son.\" He had dropped the flamberge. He had no more use for it. \"Canrik cannot succor him. The Ironhand and Frostheart Grueburn cannot. Samil has been slain.\"\n\n\"Linden?\" Covenant coughed: an effort that seemed to grind the broken ends of ribs against each other.\n\n\"I know not.\" Stave did not disguise his bitterness. \"She cast me from her ere she was claimed by the bane. I desire to hope that she lives, yet I cannot.\"\n\nA moment later, the former Master whispered, \"I do not comprehend, Timewarden. Time comes unbound. Soon it will unravel entirely. Why does Corruption remain?\"\n\nThrough a mouthful of blood, Covenant panted, \"He's enjoying himself too much.\" After uncounted millennia of imprisonment\u2014\"He knows he's already won. He's just waiting for Jeremiah.\"\n\nAnd while Lord Foul waited\u2014\n\nCovenant wanted to strike. He ached for the strength to stop the Despiser. But he was too weak. Too badly hurt. Sick with grief for Linden and Jeremiah. He had nothing left except waiting.\n\nRoger deserved a better father.\n\nRoger was crying. He may have wanted words, but he could only manage sobs. A young man who had dreamed of eternity\u2014\n\n\"Timewarden,\" Stave demanded, uncharacteristically urgent, \"some deed we must attempt. We cannot condone this doom.\"\n\nI know, Covenant thought dimly. I just need a chance to breathe.\n\nHe needed something to believe in. Something to hope for.\n\nWhat kind of idiot thinks he can save the world by himself?\n\nHe had forgotten how seductive despair could be.\n\n\"Hear me, Timewarden,\" ordered Stave. \"I will endeavor to retrieve the _krill_. Should I succeed, you must wield it. You must\u2014\"\n\nCovenant gripped Stave's arm weakly; tried to restrain the _Haruchai_ , although of course he could not. Spitting blood, he croaked, \"Wait. He wants Jeremiah. We still have time.\"\n\nToo much wild magic would only hasten the fall of the Arch. It would ease the Despiser's departure.\n\nStave did not move. He may have trusted Covenant. He may have simply hesitated.\n\nLord Foul's gaze had turned away. He appeared to peer through rock toward the cave where Covenant had left Jeremiah. His eyes dripped eagerness. He was as vulnerable as he would ever be.\n\nWe still have time.\n\nCovenant had abandoned Linden's son to _moksha_ Raver.\n\nSuddenly the Despiser's eyes flared. They blazed like torches. His outrage stunned Covenant's ears. Kiril Threndor lurched in the mountain's chest as though Mount Thunder had suffered a fatal crisis.\n\nStave said something. He may have been shouting, but Covenant could not hear him.\n\nRoger was moving.\n\nBroken as a derelict, as the wreckage of his dreams, Roger stumbled toward the dais. He crouched. When he rose again, he clutched High Lord Loric's dagger.\n\nAs he raised his arm, fresh blood pumped from his severed stump. Red splashed across the stone like an accusation.\n\nHis screaming seemed soundless as he hammered the blade into Lord Foul's impalpable shape.\n\nA puny attack, too low and frail to accomplish anything. And the Despiser was mighty: he was scarcely physical. Nevertheless wild magic coruscated in the dagger's gem. Loric had forged his blade to mediate between irreconcilable possibilities. It was the highest achievement of his vast lore. Somehow it _hurt_ \u2014\n\nIn spite of Lord Foul's vast power, the _krill_ appeared to nail him where he stood; fix him in one place. He gathered his fury into a fist. With a single punch, he crushed Roger to wet pulp. But he did not leave the dais. Did not slip past the restrictions of time.\n\nRoger\u2014\n\nNow Covenant heard Stave yelling, \"The Chosen-son has freed himself!\"\n\nAt last. Now or never.\n\nCovenant was battered and deadened, too weak to support his own weight, broken in ways which he was too fraught to name. But he was still a white gold wielder, a by God _rightful_ white gold wielder. And he had made promises. _I am done with restraint_. He hit Lord Foul with fire as fierce as a bayamo.\n\nThe Despiser thrashed, howling. As if the effort were insignificant, he expelled the _krill_. Then he turned on Covenant. Enraged and savage, he countered with so much force that Covenant's bones should have been pulverized.\n\nStones heaved. Igneous slabs were tossed like dried leaves. Repercussions ripped down the remaining stalactites, filled the air with whirling debris.\n\nBut Covenant withstood the blast. Wild magic withstood it. He had surrendered once. Never again.\n\nJeremiah had found a way to defeat _moksha_ Jehannum. Help was coming. All Covenant had to do was survive. And keep hurting Lord Foul. Prevent his escape. The Despiser must have believed that he would still be able to claim Jeremiah before Time collapsed in on itself. Covenant had no intention of letting that happen.\n\nPowers mounted in Kiril Threndor. Incinerating silver and Lord Foul's sledge-hammer blows staggered the chamber. Covenant only knew that Stave still lived because he, Covenant, had not fallen to his knees. He no longer saw anything, heard anything. Yet he _felt_ everything as if his nerves were white gold, as if his senses were wild magic. He recognized every concatenation of Lord Foul's malevolence. He could have named each of his own responses.\n\nHis millennia within the Arch of Time had not been wasted on him. His heart and his mind and even his leper's body understood wild magic. He was half translated out of reality himself, refined by fire and determination until he hardly needed his own physical existence.\n\nHe could not keep the Despiser here: he knew that. Instants were fraying. Moments bled into each other. Causes and sequences were becoming confused. Lord Foul might outlive such uncertainties: Covenant could not. He fought only to distract his foe, to engage the Despiser's endless hatred. To make the Despiser _miss his chance_.\n\nThen the chance came, Lord Foul's or Covenant's.\n\nWith flame and effort rather than sight, Covenant saw Jeremiah enter the chamber; saw Jeremiah running wreathed in Earthpower as clean and necessary as sunlight. The heartwood Staff in his hands blazed with a purity that pierced rocklight and argent, defied Lord Foul's savagery.\n\nBehind him came Coldspray, Grueburn, and Canrik, but this contest was not for them. Like Stave and Branl, they had done more than Covenant could have asked or imagined. Their part in the Land's fate was finished. Only Jeremiah had the power to alter the terms of Covenant's struggle.\n\nAnd Jeremiah knew what was needed. While Covenant fought to block Lord Foul, preclude Jeremiah's possession, Jeremiah fashioned his magicks\u2014\n\nThe Despiser's instant reaction was glee, triumph, exultation. He reached for Jeremiah as if he were pouncing. But wild magic tore through the hands of Lord Foul's power, shredded his grasp. Covenant ripped apart the Despiser's clutch while Jeremiah wrought Earthpower.\n\nIn the guts of Mount Thunder, the consequences of the Worm's feeding expanded. Shock after shock, they mounted toward their final outcome. Waves ran up and down the walls as if the rock had become water. Granite pain dripped from facets of rocklight. Unnatural heat and cold gusted at Covenant's face like gasping, like strained exhalations of time.\n\nIn a moment or an hour\u2014in no time at all\u2014Lord Foul appeared to realize what was happening. He appeared to recognize that he had to flee. If he wanted freedom, he had to abandon his _deeper purpose_ against the Creator. He would be trapped otherwise. He would cease to exist.\n\nShrieking like the deaths of stars, he turned away.\n\nBut he was already too late. Because Jeremiah\u2014\n\nOh, God, Jeremiah!\n\n\u2014had learned how to _forbid_.\n\nWith Earthpower and extravagance\u2014the whetted extremity of a boy who had been hurt too much and was finally done with helplessness\u2014Jeremiah forbade Lord Foul's escape.\n\nIn horror, the Despiser wheeled to face his foes again.\n\nCovenant he ignored. Wild magic ripped through his fleshless form, sent fiery harm careering everywhere along his disembodied nerves; but he was not dissuaded. He knew pain too well: he had spent eons wrapped in his own agony. Damage and diminishment could be repaired. His chance for freedom would never come again.\n\nEvery force at his command, Lord Foul focused on Jeremiah. But now he did not strive to take possession. Instead he sought to destroy.\n\nHe knew more about forbidding than Jeremiah did. He was stronger than the boy would ever be. When Covenant wounded him, he could call on long ages of despair to secure his concentration.\n\nAt first, Jeremiah wielded the Staff with an exalted certainty. He had freed himself from _moksha_ Raver: he had earned his power. And he had spent too much of his life immured in dissociation. His need to repudiate Despite defined him. Nevertheless he was only himself; only human. Lord Foul was the Despiser, eternal and insatiable. Although Covenant fought as hard as he could, flailed desperately and did ferocious damage, Jeremiah began to falter.\n\nThe Staff trembled in his grasp. His arms shook. His eyes were cries of dismay. He gave his utmost\u2014and it was not enough. Bit by bit, his forbidding began to crumble.\n\n\" _Jeremiah!_ \" Covenant yelled: a shout of conflagration. \"Hold on! _I'm coming!_ \"\n\nWith Stave's help, he floundered toward the dais, flaying his foe as he approached. But he already knew that he would fail. He could have torn open Mount Thunder's entire torso\u2014he felt destructiveness on that scale within him\u2014but he could not block Lord Foul's flight. Wild magic was the wrong kind of power. Like the Despiser, white gold aspired to freedom; and any forbidding required the structures and commandments of Law.\n\nJeremiah dropped to one knee. Blood burst from his mouth. Earthpower pouring from the Staff began to gutter. In another moment\u2014\n\nJeremiah! Oh, God!\n\nWithout warning, an overwhelming thunder swept through Kiril Threndor. It staggered the whole mountain. For an instant, Covenant thought that the Worm had drunk its fill; that the World's End had come. Then he saw more clearly.\n\nA hand like the fist of a god struck down the Despiser. Strength that threatened to crack Covenant's mind left Lord Foul crumpled on the dais, almost corporeal, almost whimpering. A transcendent touch secured Jeremiah's forbidding. As if as an afterthought, something supernal deposited Linden at Jeremiah's side.\n\nA heartbeat later, the thunder passed on, leaving the Earth to its own ruin. In the power's absence, the rising convulsions of the Worm's feeding felt like a reprieve.\n\nLinden clasped Jeremiah, helped him stand again. Her return renewed his resolve, his strength. Fresh Earthpower crowded the chamber. Refusals tightened around the Despiser.\n\nCovenant believed that he was deaf as well as blind. Wild magic was all that kept him alive. Nonetheless he heard Linden say, \"She Who Must Not Be Named is gone. I gave Her what She needed. This must be what She calls gratitude.\n\n\"I love you, Thomas.\"\n\nIt's enough, Covenant thought. Thank you. It's enough.\n\nBut he could not afford to pause. Reality was coming undone around him, and he had not confronted his worst fears.\n\nHe could do that now. Linden had come. She was whole and here. The emblem and summation of all betrayed women had given Covenant that gift.\n\nMustering his own gratitude, he urged Stave to support him until he gained the dais.\n\nThe Despiser was smaller now, beaten down or reduced by the bane's retribution. He was almost Covenant's size. He hunched into himself as though he sought to hide. As though he wanted to be smaller still.\n\nWith wild magic and leprosy, Covenant reached out to him. With pity and terror, Covenant lifted Lord Foul upright.\n\nThis was his last crisis. There could be no more.\n\n\"Do you understand?\" he asked like a man bidding farewell. \"If I'm yours, you're mine. We're part of each other. We're too much alike. We want each other dead. But you're finished. You can't escape now. And I'm too weak to save myself. If we want to live, we have to do it together.\"\n\nThe Despiser met Covenant's gaze. \"You will not.\" The voice of the world's iniquity sounded hollow as a forsaken tomb. His eyes were not fangs. They were wounds, gnashed and raw. \"You fear me. You will not suffer me to live.\"\n\n\"Yes,\" Covenant answered, \"I will.\"\n\nHe was blinded now, not by fires and fury, but by tears as he closed his arms around his foe. Opening his heart, he accepted Lord Foul the Despiser into himself.\n\nhen it was done, Thomas Covenant turned to the people who had redeemed him. If he could have looked at himself, he would have seen the scar on his forehead gleaming.\n\n\"Thomas,\" Linden breathed. Earthpower and argent shone like wonder in her gaze. \"Oh, Thomas. I don't understand. I don't know what it means. I'm just glad that I got to see it.\"\n\nStave nodded his acknowledgment. His assent.\n\nCanrik's face was hidden. Squatting beside Branl, he did what he could for the Humbled. Rime Coldspray and Frostheart Grueburn simply stared, too exhausted to recognize their relief.\n\nKiril Threndor stumbled as if Mount Thunder itself had flinched. Chunks of the ceiling broke loose. Fissures clenched the walls, unclenched. In the distance, the mountain's shoulders shrugged avalanches. Covenant felt the Earth's foundations failing. But Jeremiah's forbidding protected everyone in the chamber. He hardly seemed to notice his own prowess.\n\n\"So am I,\" the boy admitted. More sourly, he said, \"Too bad we won't get to enjoy it.\"\n\nCovenant tried to smile. \"What are you talking about?\" He spoke to Jeremiah, but he poured out his heart to Linden. \"This is our chance. We can't stop what's happening, but that doesn't mean we can't try to save the Earth. I know that sounds impossible, but maybe it isn't. We don't have to create an entire reality from scratch. We just have to put the pieces of this one back together.\n\n\"If we follow the Worm\u2014and if we pick up the pieces fast enough\u2014and if we know where they belong\u2014\"\n\nPerhaps the Arch and the world could be rebuilt from the fragments of their destruction.\n\n\"We have everything we need,\" he assured Jeremiah. \"Two white gold wielders. The Staff of Law. Linden's health-sense. Your talent. Hell, we still have the _krill_. And I think\u2014\" His face twisted with pain and chagrin and hope. \"I'm not sure, but I think I know everything Lord Foul knows.\"\n\nThe Despiser had striven for eons to escape his prison. His knowledge of the created world was both vast and intricate.\n\nJeremiah stood straighter. His hands tightened eagerly on the Staff. \"I've learned a few things myself.\"\n\n\"And I've seen She Who Must Not Be Named without all of that agony and bitterness,\" offered Linden. \"I know what She means.\"\n\nIn spite of its galls and strain, hers was the most beautiful face that Covenant had ever seen.\n\n\"We can do this,\" he said as if he were sure. \"We can do it together.\"\n\n_There is no doom so black or deep\u2014_\n\nLinden looked at Jeremiah. \"Then you had better get rid of that Raver. He's holding you back.\"\n\n_Moksha_ had probably exacerbated Jeremiah's faltering earlier.\n\nJeremiah nodded. He closed his eyes. For a moment, he grimaced. He may have feared losing what he had gained from the Raver; feared losing a part of himself. But then he became a brief flare of Earthpower and forbidding.\n\nDarkness billowed out of him. _Moksha_ writhed uselessly, seeking a body that could sustain him. But the Giants were too weary to be used, Branl was too severely injured, and Stave and Canrik were too obdurate. Howling, the Raver fled.\n\nBraced on Stave's shoulder, Covenant left the dais. When he had reclaimed Loric's dagger, he stabbed it into the stone where Lord Foul had stood. It had held the Despiser there briefly. Perhaps it would do something similar for Mount Thunder's heart.\n\nIn the light of the gem, Covenant went to stand with Linden and Jeremiah.\n\nTheir faces were starting to blur. Bits of them seemed to fade in and out of solidity. The ichor of the mountain streamed from the walls, spattered from the ceiling. The dust of pulverized gutrock rose like spume from the cracking floor. For an instant, Branl appeared to be whole again. For another, he resembled a desiccated corpse. Canrik's wounds and those of the Swordmainnir wavered between past and future.\n\n\"If it will be done,\" Stave said, or had said, or would say, \"it must be done now. Do not fear for us. We are at peace. Our deeds here would content the heart of any _Haruchai_.\"\n\n\"And of any Giant,\" Rime Coldspray managed faintly.\n\nCovenant took the time to embrace Linden; to give her the best kiss that he had in him. He delayed long enough to ruffle Jeremiah's hair. Then he said simply, \"Now.\"\n\nWith his halfhand, he clasped Linden's left. Sharing his burdens, he raised both arms, held high his bright wedding band and hers. After an instant's hesitation, Linden reached out to grip the cleansed Staff between Jeremiah's hands, trusting the influence of the _krill_ , or the accelerating collapse of Law and Time, or her own rightful use of wild magic to protect her from incompatible theurgies. She smiled at her son. He was concentrating too hard to smile back.\n\nA final convulsion tore through Kiril Threndor. Wracked beyond endurance, the whole chamber became rubble.\n\nLifted by fire, Covenant, Linden, and Jeremiah stepped into the wake of the World's End and rose like glory.\n\n## Epilogue\n\n\"The soul in which the flower grows\"\n\nTogether in deep night, Thomas Covenant, Linden Avery, and Jeremiah walked west from the slopes of Gravin Threndor through the enduring woodland of Andelain.\n\nAt first, they could hear the distant turmoil of the Soulsease as it rushed between the walls of Treacher's Gorge: a plaint like a lament, compelled and swift. But gradually the sound faded among the rich hush of the trees. Stately Gilden and high oaks comforted the heavens. Broad-boughed sycamores and gnarled cottonwoods spread their limbs in welcome. Occasional rills chuckled through the dark, and lush greenswards cushioned walking. Amused breezes wafted their small jests here and there, caressing the Andelainian largesse with tranquility as pellucid as Glimmermere. Along the hillsides, _aliantha_ and flowering forsythia gathered like guides or guardians, confirming a path through the night.\n\nThe three carried no light, although Covenant and Linden could have etched the trees with argent, and Jeremiah bore the restored Staff of Law as well as his legacy of Earthpower. They preferred to make their way among the monarchs and nobles of the Hills without other illumination because they themselves had become light. The three of them glowed gentle silver as though they lived half in the realm of the Dead; as though they were in transition, passing into or leaving a dimension of refined spirit. And the scar on Covenant's forehead held a more concentrated lucence both oneiric and definitive. He wore it like an implied coronet, the crown of all that he had loved and done.\n\nThe ambiguous auguries of their marred clothes were gone. Instead of ruined red flannel or a cut T-shirt or blood-soaked pajamas, instead of jeans and boots, they were clad in robes of fine sendaline supple as woven ghost-silk, soothing to their hard-used skin, and their feet were bare. In their passage beyond Kiril Threndor, they had been made clean.\n\nLifted by the verdant luxury of the grass, they walked easily, and the crisp air was an elixir in their lungs. On some other night, an atmosphere which had not known the sun's touch for days might have left them shivering. On this night, the chill was refreshment, balm: an anodyne for iniquity and travail.\n\nThe three figures luminous as spectres did not feel distance. They did not notice time. They had done what they could to answer their own questions, and were free of impatience. Certainly Covenant and Linden could have walked for hours in silence, content with Andelain, and with the communion of their clasped hands. But Jeremiah was young. He spoke first.\n\n\"We did it.\"\n\nLinden smiled at him. \"We did.\"\n\nAfter a while, Jeremiah asked, \"Did we do it right?\"\n\n\"I think so,\" Covenant said. Old and present pains complicated his tone. He did not share himself with his essential enemy without cost. \"It's hard to be sure.\" Too much had been lost.\n\nThen he gestured ahead. There a glade bedecked with wildflowers opened among the trees. \"But we did that part right.\"\n\nPast the boughs, the reaching twigs, the abundance of leaves, a vast multitude of stars emblazoned the heavens, distinct and glittering and inspired, complete in their loveliness. Their myriads made magnificence of the sky's black void.\n\nThe three stopped in the heart of the glade. For a time, they simply gazed upward, rapt and reveling.\n\n\"Of course,\" Covenant added, \"we had help.\"\n\nFrom an innominate distance, Infelice came to stand with them. Sumptuous in her gems and beauty, the suzerain of the _Elohim_ was herself an incarnation of stars. \"Indeed, Timewarden,\" she said like the chiming of faraway bells. \"We who were preserved from the Worm have given our aid, though our diminishment has been grievous. Chiefly we have concerned ourselves with guiding the Worm's return to its proper slumber. Doing so, we have assisted in the restoration of the One Tree to its full leaf and bloom. Yet these were lesser tasks gladly undertaken. The greatest deeds were yours, Timewarden, and yours, Wildwielder, and also yours, Chosen-son. Your achievements transcend us.\n\n\"You have made the world new.\"\n\nJeremiah nodded, grinning.\n\n\"But all those people,\" Linden said sadly. \"Millions of them. Tens of millions. All that devastation. I did that. I have to live with so much death\u2014\" She did not continue.\n\nCovenant tightened his grip on her hand.\n\nInfelice shook her head. \"Yet had you not roused the Worm,\" she replied, \"he whom you name the Despiser would have wrought graver harm by some other means. Damning the Earth, you enabled its redemption. Therefore do not fault yourself, Wildwielder. Though it shames me to confess it, your folly has surpassed the wisdom of the _Elohim_. We erred in our opposition, erred cruelly. Now we accept the outcome without regret.\"\n\n\"'Beings from beyond Time,'\" murmured Linden.\n\n\"Indeed,\" the _Elohim_ said again. \"For that reason, if for no other, there can be no fault in you. You were chosen for your task. You did not seek it out. Nevertheless you have found it within yourself to prevail.\"\n\nThen she faced Covenant. \"For your sake, Timewarden, I am grieved. You have elected to bear the lasting burden of this restoration. You have given the living Earth a gift which exacts anguish. The Despiser is not defeated. He strives within you. While you live, he must be defeated continuously. I have come to proffer my obeisant gratitude\u2014and also to inquire how you contrive to endure your triumph. Your willingness defies my comprehension. I could more readily grasp the surrender of your spirit to the Arch of Time. Your acceptance now surpasses me.\"\n\nCovenant grimaced. He almost smiled. \"It's easier than it looks. Or it's harder. Or maybe it's just worth the effort.\" He ran his halfhand through his hair. \"I don't know how else to explain it. Lord Foul makes us strong.\"\n\n\"Strong?\" Jeremiah objected. \"The Despiser? He would have slaughtered the whole world and laughed about it.\"\n\n\"Well, sure.\" Covenant shrugged. \"But ask yourself why he's like that. Berek said it. 'Only the great of heart may despair greatly.' All that malice and contempt is just love and hope and eagerness gone rancid. He's the Creator's curdled shadow. He\u2014\" He grimaced again. \"I'm not saying this right.\n\n\"He gives us the chance to do better.\"\n\nJeremiah and Infelice studied him, frowning.\n\n\"In any case,\" Covenant added, \"taking a stand against him is what makes us who we are.\" He looked more sharply at the _Elohim_. \"When we don't, we aren't anything. We're just empty.\"\n\nUncharacteristically gracious, Infelice bowed. \"A just charge, Timewarden. I perceive now that it is condign. I am content to acknowledge it.\n\n\"Contemplating the paradox of your folly and wisdom, I bid you joy.\"\n\nRiding a delicate loft of bells, she took herself away.\n\nLinden watched her husband's face and smiled like a new day.\n\nAs if he were answering her, Covenant said, \"I can feel my fingers. They seem to have nerves again, what's left of them. And the soles of my feet\u2014They used to be numb. Now I know I'm standing on grass. I can almost feel individual blades.\n\n\"I've always thought you were beautiful, but I had no idea you're _so_ beautiful.\"\n\nShe kissed him for a time while Jeremiah rolled his eyes. Then the companions walked again.\n\nThe Hills displayed themselves like treasures. Leagues may have passed, unmeasured by Andelain's kind ease. In the east, Mount Thunder's dark bulk showed against the paling sky. Intimations of morning lifted birds into the air. Chirps and twitters began like introits, the preliminaries of worship. Every in-drawn breath was a sacrament. Every exhalation released care.\n\nAnd from out of the fading night came Wraiths to do homage.\n\nFleet as candle-flames, and glad as an aubade, throngs of living fires danced among the trees, two or three at first, then scores, then innumerable hundreds. Sharing warmth and brightness like wealth, they gathered in the air. Harmoniously they measured the sequences of a stately gavotte around Covenant, Linden, and Jeremiah. One at a time, they wafted closer to kiss blessings onto Linden's forehead, and onto Jeremiah's. But in front of Covenant they appeared to falter as though they were abashed or frightened, dismayed by awe. Eschewing his forehead, they touched lightly on his wedding ring, then scampered away, relieved and eager, piquant as trills.\n\nWhen the Wraiths had bestowed their approval and were done, the companions resumed their effortless travel.\n\nLater, on a rise crowned with larch and plane, they heard a snatch of song. There they paused to listen.\n\nSwelling around them, melodies arched and ached among the boughs. A counterpoint as deep as roots joined the music, and leaves offered a fluttering descant: the strophes of an ode to spring and fertile burgeoning, to anticipations of ripe summer. Soon the whole woodland seemed on the verge of full-throated song. But then the chorus shrank or condensed until it became Caerwood ur-Mahrtiir striding upward with the earned remains of his staff cradled in one arm.\n\nCalling his name, Linden started toward her dear friend. After a few steps, however, she halted at the sight of the crowd ascending behind the former Manethrall.\n\nTall figures followed the Forestal, creatures sculpted and kingly in their perfection. A few were grey, the rest as black as the departing night. Like Caerwood ur-Mahrtiir, they were eyeless. But he had lost his orbs in battle: they had been fashioned without the need for ordinary sight. Where gaping nostrils had once dominated their faces, they now had more human noses which they appeared to bear proudly; and their mouths could smile. The straight strength of their limbs matched the symmetry of their forms and the sovereignty of their carriage.\n\nThe tallest of the creatures accompanied the Forestal a step behind his right shoulder: the loremaster. The rest of the transformed ur-viles and Waynhim stopped a few strides away. The loremaster carried its fearsome iron jerrid in one fist, but the other creatures had exchanged their eldritch knives for wands like twigs with which they appeared to shape the verdant music.\n\n\"Mahrtiir?\" Linden began. \"Caerwood ur-Mahrtiir? You have no idea\u2014Are these\u2014?\" Unable to complete a question, she said through her tears, \"I am so glad to see you!\"\n\n\"We are well met,\" mused the Forestal, \"well met in all sooth, Ringthane, Linden Avery, friend. And well met also, Covenant Timewarden and young Jeremiah. Among unforeseen wonders, you are a particular delight. Though I sang against the Worm with every aspect of my given strength, I did not prevent the world's death. Nor could I evade it. Yet I am here. Indeed, all who clung to life at the moment of the Earth's extinction live still. While the restored Arch endures, you will be remembered and honored among all of the wide world's forests.\"\n\n\"But how did you get here?\" asked Linden. \"We left you\u2014I don't even know how far we've come.\"\n\n\"Andelain is here,\" answered the ur-Mahrtiir. \"Salva Gildenbourne is nigh. When the Worm had turned aside from my service to the fane, I wished to meet my passing among the trees and richness and innocence which I love. Therefore I sang to these woodlands, and was conveyed hither.\"\n\nAt once, he continued, \"I will not linger. The sight of you suffices for me, Linden Ringthane. A task immense and needful awaits, and I am avid to begin while my powers freshen within me. Much of lands and peoples, of wood and mountains, has been laid waste, much that cannot be restored. Yet much remains. And there can be no true healing that does not commence with trees.\n\n\"I am become the Earth's Forestal.\"\n\n\"Alone?\" Linden inquired like a plea. \"Alone, Mahrtiir?\"\n\nCaerwood ur-Mahrtiir sang mirth. \"Assuredly not, Linden Avery, friend. With me are these ur-viles and Waynhim, the last of their kind. Aye, they are Demondim-spawn, given life by lore rather than by natural birth. But they are also High Lord Elena redeemed from torment. They are the Auriference and Emereau Vrai and Diassomer Mininderain and many other women. They are the dark yearning of _merewives_ and the sunlit absorption of _Elohim_. And now they are also Forestals.\n\n\"Encountering each other here, and filled with wonder that we had been spared, we spoke at length, these regal creatures and I. I proposed to them a new interpretation of their Weird, one suited to their perfected forms and exalted spirits\u2014and they adjudged the meter and harmony and timbre of my music worthy. I will not labor for the Earth's renewal alone.\n\n\"In sooth,\" the ur-Mahrtiir admitted, \"our task is too great for us. But we are not daunted. We will grow, Linden Avery.\" His singing rose until it shivered every leaf, flourished along every bough; and every creature sang with him. \"We will _grow_.\"\n\n\"Guardians,\" Linden murmured as the Forestals carried their melodies through the Hills toward other, more distant forests. \"In the Creator's stead.\" _How may life endure in the Land, if the Forestals fail and perish, as they must, and naught remains to ward its most vulnerable treasures?_ \"I would never have guessed that Demondim-spawn were the answer to Caerroil Wildwood's question.\"\n\n\"They weren't,\" said Covenant. \"You were. You and Mahrtiir. You kept that promise, just like you kept your promise to the ur-viles and Waynhim.\n\n\"And you saved my daughter. Here I was, planning to punish myself eternally for what happened to her, and you\u2014\"\n\nJeremiah scowled, feigning disgust. \" _Please_ don't start kissing again. It's gross.\"\n\nLinden laughed until her son laughed with her. Then the three of them resumed their walk into the west.\n\nBehind them, Gravin Threndor\u2014mighty and long misused\u2014grew distinct as the sun ascended from its imposed ensepulture. Across the heavens, the stars appeared to withdraw, making way for daylight. The greying sky became pearlescent with promise. Winged flights graceful as birdsong articulated the air and the treetops like runes in motion, a script constantly modulating toward new interpretations. Implied flames touched the tips of the highest sequoias.\n\n\"Amazing,\" Linden breathed. \"Something as simple as sunrise. I didn't think that I would ever see it again.\"\n\nCovenant grinned. \"You call that amazing? _I_ didn't think I would ever see well enough to know the difference.\"\n\n\"I can't wait,\" Jeremiah said. But whether he felt impatient for the sun, or for some other wonder, he did not explain.\n\nGradually light came to the heights of Andelain. Bright day spread down branches and boles as though Mount Thunder had granted it passage. The mountain wrapped its cloak of shadow closer about itself. Sunshine enlivened the leaves with memories of music.\n\nAnd in a wide hollow defined by stands of mimosa, by wide-spread jacaranda and flowering rhododendron, with a giddy brook running past an abundance of _aliantha_ , Covenant, Linden, and Jeremiah found the friends and companions with whom they had shared so much weariness and strife.\n\nRime Coldspray and Frostheart Grueburn were there, Onyx Stonemage and Halewhole Bluntfist. The Giants of Dire's Vessel, those who had survived their many battles: Bluff Stoutgirth, Squallish Blustergale, and their few comrades. Canrik and perhaps two score other Masters, all that remained of two hundred. Manethrall Bhapa. Cord Pahni. Branl, the last of the Humbled, who had killed Clyme and sacrificed an arm and become certain. And Stave, the former Master.\n\nThey had been healed and refreshed, all of them, and their raiment restored by the re-creation of the Earth. They lacked only the silver glow and sendaline of _beings from beyond Time_. They had come together in the hollow to feast on treasure-berries, drink pristine water, and share their astonishment.\n\nThey must have been conveyed here while the fraying strands of Time were rewoven.\n\nThey did not immediately notice Covenant, Linden, and Jeremiah. But then the three were announced. Among the trees at the edge of the hollow, Ranyhyn whinnied a proud welcome: Hyn and Hynyn, Rallyn and Khelen, Rohnhyn and Naharahn. And as their call carried over the Hills, full sunlight struck the horses, burning away the last vestiges of dusk from their glossy coats. Among them, the Ardent's mount cropped grass as though it had no use for mere relief and wonder.\n\nBut the star-browed Ranyhyn did not remain to receive greetings or gratitude from Linden and Jeremiah, as they must already have done from Stave and Branl, Bhapa and Pahni. They were eager to rejoin their herds and their Ramen. They galloped away, taking Mishio Massima with them, and trumpeting praise to the new day.\n\nGiants and _Haruchai_ lifted their heads. Bhapa and Pahni looked around.\n\nA moment later, jubilation and awe filled the air. Linden wept for gladness, and Jeremiah wavered between shouts and tears. Covenant spread his arms like a man who yearned to embrace everyone simultaneously, and his scarred forehead shone like incarnated starlight.\n\nThen there were shouts and much laughter among the Giants, hugs and clasps and affectionate congratulations. As one, Stave, Branl, and the Masters did more than bow: they sank to one knee and lowered their heads in homage. Unable to contain himself, Manethrall Bhapa put his hands on Linden's waist and lifted her high until she begged him to put her down. With more restraint and sadness, Pahni offered her hopes for Linden's happiness, and for Covenant's.\n\nJeremiah joined the mirth and effusion of the Giants. Linden took the Ramen away from the others to make her peace with Pahni's bereavement, to speak of Caerwood ur-Mahrtiir, and to share her heart with friends who had been as faithful as Liand. For his part, Covenant spoke first with the Ironhand and Stoutgirth Anchormaster, while Stave, Branl, and Canrik attended him.\n\nHis efforts to find words for his gratitude, the Giants brushed aside. \"The thanks are ours to give,\" Rime Coldspray proclaimed. \"We are wont to avow that joy is in the ears that hear. Upon such occasions, however, it is also in the mouth that speaks. Though our hearts are galled by loss, they also overflow with gladness. Wherever Giants remain in the Earth, the names of Covenant Timewarden and Linden Giantfriend and Jeremiah Chosen-son will be uttered in celebration and reverence.\"\n\nBluff Stoutgirth nodded his approval. But he smiled with difficulty, and his need for a _caamora_ was plain. He was a sailor, not a warrior: his losses bore a different emotional weight than Coldspray's. Nevertheless he accepted them with a spirit slowly lifting.\n\nCovenant had only one question for them: what now?\n\nThe Anchormaster answered without hesitation. \"With the Giants at my command, I will return to Dire's Vessel. It is my hope that we will sail at once for our homeland. I pine for the harborage of Home. I ache to learn the fate of our kindred. And I yearn for new ears to soften my sorrow with their joy.\"\n\nCovenant understood. He had his own sorrows to assuage. \"And you?\" he asked of Coldspray.\n\nBefore she could reply, Stave spoke.\n\n\"With the Ironhand's consent, we will welcome her and her Swordmainnir to Revelstone. We have much for which we wish to atone. First among our faults, doubtless, is the ignorance which we have inflicted upon the folk of the Land. Yet more immediate to us here is the manner in which we have rebuffed the friendship and valor of the Giants. We hunger to make amends.\"\n\nCovenant cocked an eyebrow at the outcast Master's use of _we_. But he did not interrupt.\n\n\"Also,\" Stave went on, \"I would seek a boon of the Ironhand, and perhaps of her comrades also, a boon which pertains to Revelstone, and which Revelstone may sway her to grant.\"\n\nNow both Coldspray and Stoutgirth stared at him, as surprised as Covenant.\n\nStave faced them with a smile: another surprise. \"You crave explanations.\" Amusement sparkled in his eye. \"Know, then, that I am Stave, by right of years and attainment the Voice of the Masters. I speak for these _Haruchai_ assembled here, and also for those who have retained the benison of their lives elsewhere.\"\n\nMore gravely, he said, \"Your example, Covenant Timewarden, and also that of Linden Avery the Chosen, and indeed of Jeremiah Chosen-son, have turned our thoughts to new paths. We have concluded that the Land has no need of Masters. Rather it will be better served by Lords. Therefore we wish to claim a different purpose. If you do not gainsay us, ur-Lord, we will form a new Council, emulating with our best strength the service begun by Berek Lord-Fatherer.\n\n\"And the boon which we will ask of the Ironhand is this, that she and her Swordmainnir join with us in that Council. By their kindness and merriment, we hope\"\u2014he smiled again\u2014\"to avoid the snares of our long past and severe judgments until the time when the folk of the Land discover a desire to stand among us.\"\n\nJeremiah had wandered closer while Stave spoke. Now the boy said, \"I can tell you where to find Kevin's Wards.\"\n\n\"And we will welcome that knowledge, Chosen-son, when our need for it is ripe.\"\n\nCovenant shook his head, but not in disapproval. \"I don't know what to say. It sounds practically ideal. But you'll have to give up your rejection of Earthpower. Or lore. You'll have to start from scratch.\"\n\n\"As we should, ur-Lord,\" Stave replied. \"The Earth has been vouchsafed a new beginning. The _Haruchai_ also must begin anew.\"\n\nAfter a moment's thought, Covenant observed, \"You'll need a High Lord. You, Stave?\"\n\n\"I?\" Stave countered. He seemed to hear a jest in Covenant's question. \"No. I do not stand so high in my own estimation. And I do not doubt that the day will come when the Voice of the Masters must speak for the _Haruchai_ rather than for the Land. The Council of Lords and the High Lord must regard wider concerns.\n\n\"I have named Canrik to lead the first Council. He is newly acquainted with uncertainty, and will gain much from an immersion in the necessary doubts of the Lords.\"\n\nCanrik nodded, expressionless as any Master or Bloodguard.\n\n\"But Branl\u2014?\" Covenant asked. \"Surely he's earned it?\"\n\n\"I will not shoulder that burden,\" the _Haruchai_ halfhand stated flatly. \"Clyme's death mars my heart. I desire a different atonement. I will return to Gravin Threndor, seeking High Lord Loric's _krill_.\"\n\nHe held up his remaining hand to forestall objections. \"Certainly the Cavewights will greet me with enmity. However, Corruption no longer goads them to madness. And they, too, must feel awe at their continuation in life. It is my hope, therefore, that soft words and a refusal to do harm will dissuade them from bloodshed. They are not mindless, ur-Lord. And I am not helpless in my own defense, though I will cause no more hurt. Mayhap I will elude death until they perceive that we are no longer foes.\n\n\"Should I succeed, I will bear the _krill_ to Revelstone. And should I fail\u2014\" Branl shrugged delicately. \"I will die content in myself. I will not perish grieving.\"\n\nCovenant thought of Cail, who had been rejected by his people, and had gone to find his fate alone. Branl was rejected only by himself. Still he would have to find peace on his own terms.\n\nFinally Rime Coldspray said to Stave, \"The boon you seek is too great to be granted readily, Rockbrother. My comrades and I must speak of it at length. Indeed, many Giantclaves await us, and we will spend whole seasons in delight and sorrow and hope. But first we will gladly accompany you to Revelstone. How can we refuse? We are Giants.\"\n\nTogether, Stave, Canrik, and Branl bowed their thanks.\n\nAfter a while, Linden came to join Covenant and Jeremiah. Resting one hand on her son's shoulder, she pointed into the west. \"Who do you suppose that is?\"\n\nLooking there, Covenant saw a lone figure standing in sunlight at the rim of the hollow. A woman, he thought, although he could not be sure. The figure's head was wrapped in cerements like the Theomach's. Ribbands as garish as the Ardent's ornamented the figure's upper body, while from its waist hung a motley skirt as haphazard and arcane as the Mahdoubt's.\n\nTo Covenant's gaze, and Linden's, and Jeremiah's, the figure replied with a beckoning gesture.\n\nAt first, Covenant smiled. \"It looks to me,\" he said wryly, \"like the Insequent are finally giving credence of the idea of acolytes.\" He almost chuckled. \"In fact, if I had to guess, I might say that's _the_ Acolyte.\"\n\nBut then his eyes darkened, and for a moment he resembled a man who had never recovered from his oldest wounds.\n\n\"It's time. We have to go.\"\n\nAs he spoke, the figure drifted out of sight.\n\n\"Go?\" Jeremiah protested at once. \"Why? We just got here.\"\n\nLinden studied her husband quizzically, but she did not contradict him.\n\n\"The Chosen-son speaks for me as well,\" began Rime Coldspray.\n\n\"And for me,\" put in Bluff Stoutgirth.\n\n\"We have sung no songs to honor you,\" Coldspray added. \"We have not truly begun to voice our wonder and gratitude, our esteem deep as seas. We have not told you of our love. And we have heard neither Linden Giantfriend's tale nor Jeremiah Chosen-son's. In sooth, we are scarcely able to estimate your own.\n\n\"What compulsion requires you to depart, Timewarden?\"\n\nCovenant rubbed his glowing scar to disguise a clench of woe and regret. \"Unearned knowledge,\" he answered brusquely. \"Right now, we're too dangerous. Jeremiah and me. Maybe even Linden. Jeremiah needs time to figure out what he's going to do with everything he got from _moksha_. He has to learn what it all means and decide how he wants to use it. Linden freed She Who Must Not Be Named. She freed Elena\"\u2014his voice caught for a moment\u2014\"and who knows how many other lost souls. That must have been shattering. She hasn't had a chance to recover. And I'm carrying the Despiser around inside me. What he knows isn't a problem for me. I used to be part of the Arch of Time. But he's _Lord Foul_. If I let him, he might spit in your faces. Or he might find a way to use my ring. I hope I can persuade him to relax. Maybe I can even convince him to think of me as something more or better or at least kinder than his worst enemy.\n\n\"We all need time.\"\n\nAnd possibly a teacher, he mused ruefully. If so, one of the Insequent might serve. The Theomach had certainly guided Berek Halfhand well enough.\n\nSoftly Rime Coldspray said, \"Though you conceal it, your hurt is evident, Covenant Timewarden. None here would choose to deny you. Do not take it amiss when I confess that your departure will sadden us.\"\n\nWith an effort, Covenant set aside his aching. He reached out for Linden's hand, smiled at Jeremiah. \"It isn't permanent,\" he said more cheerfully. \"It can't be. Our old lives are finished.\" By degrees, his distress receded. \"There's no going back. You can't get rid of us this easy.\"\n\nThen a new mood came over him, one that he had not felt for a very long time; and he found himself laughing as if he were a man for whom laughter came naturally.\n\nTake _that_ , he told his inner Despiser. And all this time, you thought I hated you.\n\nWhen he subsided, he said to his friends, \"I can't tell you how good it feels to know we can see you again whenever we want.\" Still chuckling, he added, \"But we won't until we're ready.\"\n\nLinden gave him a smile that sang in his heart; and Jeremiah nodded awkwardly, discomfited by recognitions for which he had not prepared himself. Together they walked away in the direction taken by the Insequent: the Unbeliever and his new wife and his obliquely adopted son.\n\nAnd as they walked, spring rainclouds gathered to the southwest. In the distance, sudden showers streaked the air, falling like chrism to the reborn ground. Struck by sunlight, the showers returned a rainbow to the heavens: one bright instance of the world's inherent splendor.\n\nWhen it faded, Covenant, Linden, and Jeremiah appeared to fade with it. But their silver lingered for a time, until the day moved on.\n\nHere ends\n\nThe Last Dark\n\nand\n\n\"The Last Chronicles of Thomas Covenant\"\n\n# Combined Glossary for The Chronicles of Thomas Covenant\n\n**_Abatha_ :** one of the Seven Words\n\n**Acence:** a Stonedownor, sister of Atiaran\n\n**_Ahamkara_ :** Hoerkin, \"the Door\"\n\n**Ahanna:** painter, daughter of Hanna\n\n**Ahnryn:** a Ranyhyn; mount of Tull\n\n**Aimil:** daughter of Anest, wife of Sunder\n\n**Aisle of Approach:** passage to Earthrootstair under _Melenkurion_ Skyweir\n\n**a-Jeroth of the Seven Hells:** Lord of wickedness; Clave-name for Lord Foul the Despiser\n\n**_ak-Haru_ : **a supreme _Haruchai_ honorific; paragon and measure of all _Haruchai_ virtues\n\n**Akkasri:** a member of the Clave; one of the na-Mhoram-cro\n\n**_aliantha_ :** treasure-berries\n\n**Alif, the Lady:** a woman Favored by the _gaddhi_\n\n**_amanibhavam_ :** horse-healing grass, dangerous to humans\n\n**Amatin:** a Lord, daughter of Matin\n\n**Amith:** a woman of Crystal Stonedown\n\n**Amok:** mysterious guide to ancient Lore\n\n**Amorine:** First Haft, later Hiltmark\n\n**Anchormaster:** second-in-command aboard a Giantship\n\n**Andelain, the Hills of Andelain, the Andelainian Hills:** a region of the Land which embodies health and beauty\n\n**Andelainscion:** a region in the Center Plains\n\n**Anele:** deranged old man; son of Sunder and Hollian\n\n**Anest:** a woman of Mithil Stonedown, sister of Kalina\n\n**Annoy:** a Courser\n\n**_anundivian yaj\u00f1a_ :** lost Ramen craft of bone-sculpting\n\n**Appointed, the:** an _Elohim_ chosen to bear a particular burden; Findail\n\n**Arch of Time, the:** symbol of the existence and structure of time; conditions which make the existence of time possible\n\n**Ard:** a _Haruchai_ ; a Master of the Land\n\n**Ardent, the:** one of the Insequent\n\n**_arghule\/arghuleh_ :** ferocious ice-beasts\n\n**Asuraka:** Staff-Elder of the Loresraat\n\n**Atiaran:** a Stonedownor, daughter of Tiaran, wife of Trell, mother of Lena\n\n**Audience Hall of Earthroot:** maze under _Melenkurion_ Skyweir to conceal and protect the Blood of the Earth\n\n**Aumbrie of the Clave, the:** storeroom for former Lore\n\n**Auriference, the:** one of the Insequent, long dead\n\n**Auspice, the:** throne of the _gaddhi_\n\n**_aussat Befylam_ :** child-form of the _jheherrin_\n\n**Baf Scatterwit:** a Giant (woman); crew aboard Dire's Vessel\n\n**Bahgoon the Unbearable:** character in a Giantish tale\n\n**_Banas Nimoram_ :** the Celebration of Spring\n\n**Bandsoil Bounds:** region north of Soulsease River\n\n**Banefire, the:** fire by which the Clave affects the Sunbane\n\n**Bann:** a Bloodguard, assigned to Lord Trevor\n\n**Bannor:** a Bloodguard, assigned to Thomas Covenant\n\n**Baradakas:** a Hirebrand of Soaring Woodhelven\n\n**Bargas Slit:** a gap through the Last Hills from the Center Plains to Garroting Deep\n\n**Bareisle:** an island off the coast of _Elemesnedene_\n\n**Basila:** a scout in Berek Halfhand's army\n\n**Benj, the Lady:** a woman Favored by the ****_gaddhi_\n\n**Berek Halfhand:** Heartthew, Lord-Fatherer; first of the Old Lords\n\n**Bern:** _Haruchai_ slain by the Clave\n\n**Bhanoryl:** a Ranyhyn; mount of Galt\n\n**Bhapa:** a Cord of the Ramen, Sahah's half-brother; companion of Linden Avery\n\n**_Bhrathair_ :** a people met by the wandering Giants, residents of _Bhrathairealm_ on the verge of the Great Desert\n\n**_Bhrathairain_ :** the city of the _Bhrathair_\n\n**_Bhrathairain_ Harbor:** the port of the _Bhrathair_\n\n**_Bhrathairealm_ :** the land of the _Bhrathair_\n\n**Birinair:** a Hirebrand, Hearthrall of Lord's Keep\n\n**Bloodguard, the:** _Haruchai_ , a people living in the Westron Mountains; the defenders of the Lords\n\n**Bluff Stoutgirth:** a Giant; Anchormaster of Dire's Vessel\n\n**bone-sculpting:** ancient Ramen craft, marrowmeld\n\n**Borillar:** a Hirebrand and Hearthrall of Lord's Keep\n\n**Bornin:** a _Haruchai_ ; a Master of the Land\n\n**Brabha:** a Ranyhyn; mount of Korik\n\n**Branl:** a _Haruchai_ ; a Master of the Land; one of the Humbled\n\n**Brannil:** man of Stonemight Woodhelven\n\n**Brinn:** a leader of the _Haruchai_ ; protector of Thomas Covenant; later Guardian of the One Tree\n\n**Brow Gnarlfist:** a Giant, father of the First of the Search\n\n**_caamora_ :** Giantish ordeal of grief by fire\n\n**Cable Seadreamer:** a Giant, brother of Honninscrave; member of the Search; possessed of the Earth-Sight\n\n**Cabledarm:** a Giant; one of the Swordmainnir\n\n**Caer-Caveral:** Forestal of Andelain; formerly Hile Troy\n\n**Caerroil Wildwood:** Forestal of Garroting Deep\n\n**Caerwood ur-Mahrtiir:** a Forestal; formerly Manethrall Mahrtiir\n\n**_caesure_ :** a rent in the fabric of time; a Fall\n\n**Cail:** a _Haruchai_ ; protector of Linden Avery\n\n**Caitiffin:** a captain of the armed forces of _Bhrathairealm_\n\n**Callindrill:** a Lord, husband of Faer\n\n**Callowwail, the River:** stream arising from _Elemesnedene_\n\n**Canrik:** a _Haruchai_ ; a Master of the Land\n\n**Cavewights:** evil creatures existing under Mount Thunder\n\n**Cav-Morin Fernhold:** former Forestal of Morinmoss\n\n**Ceer:** one of the _Haruchai_\n\n**Celebration of Spring, the:** the Dance of the Wraiths of Andelain on the dark of the moon in the middle of spring\n\n**Center Plains, the:** a region of the Land\n\n**Centerpith Barrens:** a region in the Center Plains\n\n**Cerrin:** a Bloodguard, assigned to Lord Shetra\n\n**Chant:** one of the _Elohim_\n\n**Char:** a Cord of the Ramen, Sahah's brother\n\n**Chatelaine, the:** courtiers of the _gaddhi_\n\n**Chosen, the:** title given to Linden Avery\n\n**Chosen-son:** Giantish name for Jeremiah\n\n**Circle of Elders:** Stonedown leaders\n\n**Cirrus Kindwind:** a Giant; one of the Swordmainnir\n\n**_clachan_ , the:** demesne of the _Elohim_\n\n**Clang:** a Courser\n\n**Clangor:** a Courser\n\n**Clash:** a Courser\n\n**Clave, the:** group which wields the Sunbane and rules the Land\n\n**_clingor_ :** adhesive leather\n\n**Close, the:** the Council-chamber of Lord's Keep\n\n**Clyme:** a _Haruchai_ ; a Master of the Land; one of the Humbled\n\n**_Coercri_ :** The Grieve; former home of the Giants in Seareach\n\n**Colossus of the Fall, the:** ancient stone figure guarding the Upper Land\n\n**Consecear Redoin:** a region north of the Soulsease River\n\n**Cord:** Ramen second rank\n\n**Cording:** Ramen ceremony of becoming a Cord\n\n**Corimini:** Eldest of the Loresraat\n\n**Corrupt, the:** _jheherrin_ name for themselves; also the soft ones\n\n**Corruption:** Bloodguard\/ _Haruchai_ name for Lord Foul\n\n**Council of Lords, the:** protectors of the Land\n\n**Courser:** a beast made by the Clave using the Sunbane\n\n**Cravenhaw:** a region between Garroting Deep and the Southron Waste\n\n**Creator, the:** maker of the Earth\n\n**Croft:** Graveler of Crystal Stonedown\n\n**Crowl:** a Bloodguard\n\n**_croyel_ , the:** mysterious creatures which grant power through bargains, living off their hosts\n\n**Crystal Stonedown:** home of Hollian\n\n**Currier:** a Ramen rank\n\n**Damelon Giantfriend:** son of Berek Halfhand, second High Lord of the Old Lords\n\n**Damelon's Door:** door of lore which when opened permits passage through the Audience Hall of Earthroot under _Melenkurion_ Skyweir\n\n**Dance of the Wraiths, the:** the Celebration of Spring\n\n**Dancers of the Sea, the:** _merewives_ ; suspected to be the offspring of the _Elohim_ Kastenessen and his mortal lover\n\n**Daphin:** one of the _Elohim_\n\n**Dast:** a _Haruchai_ ; a Master of the Land\n\n**Dawngreeter:** highest sail on the foremast of a Giantship\n\n**Dead, the:** spectres of those who have died\n\n**Deaththane:** title given to High Lord Elena by the Ramen\n\n**Defiles Course, the:** river in the Lower Land\n\n**Demimage:** a sorcerer of Vidik Amar\n\n**Demondim, the:** creatures created by Viles; creators of ur-viles and Waynhim\n\n**Demondim-spawn:** another name for ur-viles and Waynhim; also another name for Vain\n\n**Desolation, the:** era of ruin in the Land after the Ritual of Desecration\n\n**Despiser, the:** Lord Foul\n\n**Despite:** evil; name given to the Despiser's nature and effects\n\n**_dharmakshetra_ :** \"to brave the enemy,\" a Waynhim\n\n**Dhorehold of the Dark:** Forestal of Grimmerdhore\n\n**_dhraga_ :** a Waynhim\n\n**_dhubha_ :** a Waynhim\n\n**_dhurng_ :** a Waynhim\n\n**_diamondraught_ : **Giantish liquor\n\n**Diassomer Mininderain:** in myth, a woman betrayed by Lord Foul and imprisoned in the Earth as punishment\n\n**Din:** a Courser\n\n**Dire's Vessel:** Giantship used by the Swordmainnir to convey Longwrath\n\n**Doar:** a Bloodguard\n\n**Dohn:** a Manethrall of the Ramen\n\n**Dolewind, the:** wind blowing to the Soulbiter\n\n**Doom's Retreat:** a gap in the Southron Range between the South Plains and Doriendor Corishev\n\n**Doriendor Corishev:** an ancient city; seat of the King against whom Berek Halfhand rebelled\n\n**_drhami_ :** a Waynhim\n\n**Drinishok:** Sword-Elder of the Loresraat\n\n**Drinny:** a Ranyhyn, foal of Hynaril; mount of Lord Mhoram\n\n**_dromond_ :** a Giantship\n\n**Drool Rockworm:** a Cavewight, leader of the Cavewights; finder of the Illearth Stone\n\n**_dukkha_ :** \"victim,\" Waynhim name\n\n**Dura Fairflank:** a mustang, Thomas Covenant's mount\n\n**Durance, the:** a barrier Appointed by the _Elohim_ ; a prison for both Kastenessen and the _skurj_\n\n**_durhisitar_ :** a Waynhim\n\n**During Stonedown:** village destroyed by the _Grim_ ; home of Hamako\n\n**_Duroc_ :** one of the Seven Words\n\n**Durris:** a _Haruchai_\n\n**EarthBlood:** concentrated fluid Earthpower, only known to exist under _Melenkurion_ Skyweir; source of the Power of Command\n\n**Earthfriend:** title first given to Berek Halfhand\n\n**Earthpower:** natural power of all life; the source of all organic power in the Land\n\n**Earthroot:** lake under _Melenkurion_ Skyweir\n\n**Earthrootstair:** stairway down to the lake of Earthroot under _Melenkurion_ Skyweir\n\n**Earth-Sight:** Giantish power to perceive distant dangers and needs\n\n**eftmound:** gathering place for the _Elohim_\n\n**eh-Brand:** one who can use wood to read the Sunbane\n\n**_Elemesnedene_ :** home of the _Elohim_\n\n**Elena:** daughter of Lena and Thomas Covenant; later High Lord\n\n**_Elohim_** **,** **the:** a mystic people encountered by the wandering Giants\n\n**_Elohimfest_ :** a gathering of the _Elohim_\n\n**Emacrimma's Maw:** a region in the Center Plains\n\n**Emereau Vrai:** Kastenessen's mortal lover, now victim to She Who Must Not Be Named\n\n**Enemy:** Lord Foul's term of reference for the Creator\n\n**Eoman:** a unit of the Warward of Lord's Keep, twenty warriors and a Warhaft\n\n**Eoward:** twenty Eoman plus a Haft\n\n**Epemin:** a soldier in Berek Halfhand's army, tenth Eoman, second Eoward\n\n**Esmer:** tormented son of Cail and the Dancers of the Sea\n\n**Etch Furledsail:** a Giant (woman); crew aboard Dire's Vessel\n\n**Exalt Widenedworld:** a Giant; youngest son of Soar Gladbirth and Sablehair Foamheart; later called Lostson and Longwrath\n\n**_fael Befylam_ :** serpent-form of the _jheherrin_\n\n**Faer:** wife of Lord Callindrill\n\n**Fall:** _Haruchai_ name for a _caesure_\n\n**Fangs:** the Teeth of the Render; Ramen name for the Demondim\n\n**Fangthane the Render:** Ramen name for Lord Foul\n\n**Far Horizoneyes:** a Giant (woman); crew aboard Dire's Vessel\n\n**Far Woodhelven:** a village of the Land\n\n**Father of Horses, the:** _Kelenbhrabanal_ , legendary sire of the Ranyhyn\n\n**Favored, the:** courtesans of the _gaddhi_\n\n**Feroce, the:** denizens of Sarangrave Flat, worshippers of the lurker, descended from the _jheherrin_\n\n**Filigree:** a Giant; another name for Sablehair Foamheart\n\n**Findail:** one of the _Elohim_ ; the Appointed\n\n**Fields of Richloam:** a region in the Center Plains\n\n**Fire-Lions:** living fire-flow of Mount Thunder\n\n**fire-stones:** graveling\n\n**First Betrayer:** Clave-name for Berek Halfhand\n\n**First Circinate:** first level of the Sandhold\n\n**First Haft:** third-in-command of the Warward\n\n**First Mark:** Bloodguard commander\n\n**First of the Search, the:** leader of the Giants who follow the Earth-Sight; Gossamer Glowlimn\n\n**First Ward of Kevin's Lore:** primary cache of knowledge left by High Lord Kevin\n\n**First Woodhelven:** banyan tree village between Revelstone and Andelain; first Woodhelven created by Sunder and Hollian\n\n**Fleshharrower:** a Giant-Raver, Jehannum, _moksha_\n\n**Foamkite:** _tyrscull_ belonging to Honninscrave and Seadreamer\n\n**Fole:** a _Haruchai_\n\n**Foodfendhall:** eating-hall and galley aboard a Giantship\n\n**Forbidding:** a wall of power\n\n**Forestal:** a protector of the remnants of the One Forest\n\n**Fostil:** a man of Mithil Stonedown; father of Liand\n\n**Foul's Creche:** the Despiser's home; Ridjeck Thome\n\n**Frostheart Grueburn:** a Giant; one of the Swordmainnir\n\n**Furl Falls:** waterfall at Revelstone\n\n**Furl's Fire:** warning fire at Revelstone\n\n**_gaddhi_ , the:** sovereign of _Bhrathairealm_\n\n**Gallows Howe:** a place of execution in Garroting Deep\n\n**Galt:** a _Haruchai_ ; a Master of the Land; one of the Humbled\n\n**Garroting Deep:** a forest of the Land\n\n**Garth:** Warmark of the Warward of Lord's Keep\n\n**Gay:** a Winhome of the Ramen\n\n**_ghohritsar_ :** a Waynhim\n\n**_ghramin_ :** a Waynhim\n\n**Giantclave:** Giantish conference\n\n**Giantfriend:** title given first to Damelon, later to Thomas Covenant and then Linden Avery\n\n**Giants:** the Unhomed, ancient friends of the Lords; a seafaring people of the Earth\n\n**Giantship:** a stone sailing vessel made by Giants; _dromond_\n\n**Giantway:** path made by Giants\n\n**Giant Woods:** a forest of the Land\n\n**Gibbon:** the na-Mhoram; leader of the Clave\n\n**Gilden:** a maple-like tree with golden leaves\n\n**Gildenlode:** a power-wood formed from the Gilden trees\n\n**Gleam Stonedown:** a village decimated by _kresh_\n\n**Glimmermere:** a lake on the plateau above Revelstone\n\n**Gorak Krembal:** Hotash Slay, a defense around Foul's Creche\n\n**Gossamer Glowlimn:** a Giant; the First of the Search\n\n**Grace:** a Cord of the Ramen\n\n**Graveler:** one who uses stone to wield the Sunbane\n\n**graveling:** fire-stones, made to glow and emit heat by stone-lore\n\n**Gravelingas:** a master of _rhadhamaerl_ stone-lore\n\n**Gravin Threndor:** Mount Thunder\n\n**Great Desert, the:** a region of the Earth; home of the _Bhrathair_ and the Sandgorgons\n\n**Great One:** title given to Caerroil Wildwood by the Mahdoubt\n\n**Great Swamp, the:** Lifeswallower; a region of the Land\n\n**Grey Desert, the:** a region south of the Land\n\n**Grey River, the:** a river of the Land\n\n**Grey Slayer:** plains name for Lord Foul\n\n**Greywightswath:** a region north of the Soulsease River\n\n**Greshas Slant:** a region in the Center Plains\n\n**_griffin_ :** lion-like beast with wings\n\n**_Grim_ , the:** (also the na-Mhoram's _Grim_ ) a destructive storm sent by the Clave\n\n**Grimmand Honninscrave:** a Giant; Master of Starfare's Gem; brother of Seadreamer\n\n**Grimmerdhore:** a forest of the Land\n\n**Guard, the:** _hustin_ ; soldiers serving the _gaddhi_\n\n**Guardian of the One Tree, the:** mystic figure warding the approach to the One Tree; formerly _ak-Haru Kenaustin Ardenol_ ; now Brinn of the _Haruchai_\n\n**Haft:** commander of an Eoward\n\n**Halewhole Bluntfist:** a Giant; one of the Swordmainnir\n\n**Halfhand:** title given to Thomas Covenant and to Berek\n\n**Hall of Gifts, the:** large chamber in Revelstone devoted to the artworks of the Land\n\n**Hamako:** sole survivor of the destruction of During Stonedown\n\n**Hami:** a Manethrall of the Ramen\n\n**Hand:** a rank in Berek Halfhand's army; aide to Berek\n\n**Handir:** a _Haruchai_ leader; the Voice of the Masters\n\n**_Harad_ :** one of the Seven Words\n\n**Harbor Captain:** chief official of the port of _Bhrathairealm_\n\n**Harn:** one of the _Haruchai_ ; protector of Hollian\n\n**Harrow, the:** one of the Insequent\n\n**_Haruchai_ :** a warrior people from the Westron Mountains\n\n**Healer:** a physician\n\n**Heart of Thunder:** Kiril Threndor, a cave of power in Mount Thunder\n\n**Hearthcoal:** a Giant; cook of Starfare's Gem; wife of Seasauce\n\n**Hearthrall of Lord's Keep:** a steward responsible for light, warmth, and hospitality\n\n**Heartthew:** a title given to Berek Halfhand\n\n**heartwood chamber:** meeting-place of a Woodhelven, within a tree\n\n**Heer:** leader of a Woodhelven\n\n**Heft Galewrath:** a Giant; Storesmaster of Starfare's Gem\n\n**Herem:** a Raver, Kinslaughterer, _turiya_\n\n**Hergrom:** one of the _Haruchai_\n\n**High God:** title given to the lurker of the Sarangrave by the Feroce\n\n**High Lord:** leader of the Council of Lords\n\n**High Lord's Furl:** banner of the High Lord\n\n**High Wood:** _lomillialor_ ; offspring of the One Tree\n\n**Hile Troy:** a man formerly from Covenant's world; Warmark of High Lord Elena's Warward\n\n**Hiltmark:** second-in-command of the Warward\n\n**Hirebrand:** a master of _lillianrill_ wood-lore\n\n**Hoerkin:** a Warhaft\n\n**Hollian:** daughter of Amith; eh-Brand of Crystal Stonedown; companion of Thomas Covenant and Linden Avery\n\n**Home:** original homeland of the Giants\n\n**Hooryl:** a Ranyhyn; Clyme's later mount\n\n**Horizonscan:** lookout atop the midmast of a Giantship\n\n**Horrim Carabal:** name given to the lurker of the Sarangrave\n\n**Horse, the:** human soldiery of the _gaddhi_\n\n**horserite:** a gathering of Ranyhyn in which they drink mind-blending waters in order to share visions, prophecies, and purpose\n\n**Hotash Slay:** Gorak Krembal, a flow of lava protecting Foul's Creche\n\n**Hower:** a Bloodguard, assigned to Lord Loerya\n\n**Hrama:** a Ranyhyn stallion; mount of Anele\n\n**Humbled, the:** three _Haruchai_ maimed to resemble Thomas Covenant in order to remind the Masters of their limitations\n\n**Hurl:** a Giant (man); crew aboard Dire's Vessel\n\n**Hurn:** a Cord of the Ramen\n\n**hurtloam:** a healing mud\n\n**Huryn:** a Ranyhyn; mount of Terrel\n\n**_husta\/hustin_ :** partly human soldiers bred by Kasreyn to be the _gaddhi_ 's Guard\n\n**Hyn:** a Ranyhyn mare; mount of Linden Avery\n\n**Hynaril:** a Ranyhyn; mount of Tamarantha and then Mhoram\n\n**Hynyn:** a Ranyhyn stallion; mount of Stave\n\n**Hyrim:** a Lord, son of Hoole\n\n**Illearth Stone, the:** powerful bane long buried under Mount Thunder\n\n**Illender:** title given to Thomas Covenant\n\n**Imoiran Tomal-mate:** a Stonedownor\n\n**Inbull:** a Warhaft in Berek Halfhand's army; commander of the tenth Eoman, second Eoward\n\n**Infelice:** reigning leader of the _Elohim_\n\n**Insequent, the:** a mysterious people living far to the west of the Land\n\n**Interdict, the:** reference to the power of the Colossus of the Fall to prevent Ravers from entering the Upper Land\n\n**Irin:** a warrior of the Third Eoman of the Warward\n\n**Ironhand, the:** title given to the leader of the Swordmainnir\n\n**Isle of the One Tree, the:** location of the One Tree\n\n**Jain:** a Manethrall of the Ramen\n\n**Jass:** a _Haruchai_ ; a Master of the Land\n\n**Jehannum:** a Raver, Fleshharrower, _moksha_\n\n**Jerrick:** a Demimage of Vidik Amar, in part responsible for the creation of _quellvisks_\n\n**Jevin:** a healer in Berek Halfhand's army\n\n**_jheherrin_ :** soft ones, misshapen by-products of Lord Foul's making\n\n**Jous:** a man of Mithil Stonedown, son of Prassan, father of Nassic; inheritor of an Unfettered One's mission to remember the Halfhand\n\n**Kalina:** a woman of Mithil Stonedown; wife of Nassic, mother of Sunder\n\n**Kam:** a Manethrall of the Ramen\n\n**Karnis:** a Heer of First Woodhelven\n\n**Kasreyn of the Gyre:** a thaumaturge; the _gaddhi_ 's Kemper (advisor) in _Bhrathairealm_\n\n**Kastenessen:** one of the _Elohim_ ; former Appointed\n\n**Keenreef:** a Giant (woman); crew aboard Dire's Vessel\n\n**Keep of the na-Mhoram, the:** Revelstone\n\n**Keeper:** a Ramen rank, one of those unsuited to the rigors of being a Cord or a Manethrall\n\n**_Kelenbhrabanal_ :** Father of Horses in Ranyhyn legends\n\n**Kemper, the:** chief advisor of the _gaddhi_ ; Kasreyn\n\n**Kemper's Pitch:** highest level of the Sandhold\n\n**_Kenaustin Ardenol_ : **a figure of _Haruchai_ legend; former Guardian of the One Tree; true name of the Theomach\n\n**Kevin Landwaster:** son of Loric Vilesilencer; last High Lord of the Old Lords\n\n**Kevin's Dirt:** smog-like pall covering the Upper Land; it blocks health-sense, making itself invisible from below\n\n**Kevin's Lore:** knowledge of power left hidden by Kevin in the Seven Wards\n\n**Kevin's Watch:** mountain lookout near Mithil Stonedown\n\n**_Khabaal_ :** one of the Seven Words\n\n**Khelen:** a Ranyhyn stallion; mount of Jeremiah\n\n**Kinslaughterer:** a Giant-Raver, Herem, _turiya_\n\n**Kiril Threndor:** chamber of power deep under Mount Thunder; Heart of Thunder\n\n**Koral:** a Bloodguard, assigned to Lord Amatin\n\n**Korik:** a Bloodguard\n\n**Krenwill:** a scout in Berek Halfhand's army\n\n**_kresh_ :** savage giant yellow wolves\n\n**_krill_ , the:** knife of power forged by High Lord Loric; awakened to power by Thomas Covenant\n\n**Kurash Plenethor:** region of the Land formally named Stricken Stone, now called Trothgard\n\n**Kurash Qwellinir:** the Shattered Hills, region of the Lower Land protecting Foul's Creche\n\n**Lake Pelluce:** a lake in Andelainscion\n\n**Lal:** a Cord of the Ramen\n\n**Landsdrop:** great cliff separating the Upper and Lower Lands\n\n**Land, the:** generally, area found on the map; a focal region of the Earth where Earthpower is uniquely accessible\n\n**Landsverge Stonedown:** a village of the Land\n\n**Landwaster:** title given to High Lord Kevin\n\n**Latebirth:** a Giant; one of the Swordmainnir\n\n**Law, the:** the natural order\n\n**Law of Death, the:** the natural order which separates the living from the dead\n\n**Law of Life, the:** the natural order which separates the dead from the living\n\n**Law-Breaker:** title given to both High Lord Elena and Caer-Caveral\n\n**Lax Blunderfoot:** a name chosen by Latebirth in self-castigation\n\n**Lena:** a Stonedownor, daughter of Atiaran, mother of Elena\n\n**_lianar_ :** wood of power used by an eh-Brand\n\n**Liand:** a man of Mithil Stonedown, son of Fostil; companion of Linden Avery\n\n**Lifeswallower:** the Great Swamp\n\n**_lillianrill_ :** wood-lore; masters of wood-lore\n\n**Lithe:** a Manethrall of the Ramen\n\n**Llaura:** a Heer of Soaring Woodhelven\n\n**Loerya:** a Lord, wife of Trevor\n\n**_lomillialor_ :** High Wood; a wood of power\n\n**Longwrath:** a Giant; Swordmainnir name for Exalt Widenedworld\n\n**Lord:** one who has mastered both the Sword and the Staff aspects of Kevin's Lore\n\n**Lord-Fatherer:** title given to Berek Halfhand\n\n**Lord Foul:** the enemy of the Land; the Despiser\n\n**\"** **Lord Mhoram's Victory** **\"** **:** a painting by Ahanna\n\n**Lord of Wickedness:** a-Jeroth\n\n**Lord's-fire:** staff-fire used by the Lords\n\n**Lord's Keep:** Revelstone\n\n**Lords, the:** the primary protectors of the Land\n\n**loremaster:** a leader of ur-viles\n\n**Loresraat:** Trothgard school at Revelwood where Kevin's Lore is studied\n\n**Lorewarden:** a teacher in the Loresraat\n\n**loreworks:** Demondim power-laboratory\n\n**Loric Vilesilencer:** a High Lord; son of Damelon Giantfriend\n\n**_lor-liarill_ :** Gildenlode\n\n**Lost, the:** Giantish name for the Unhomed\n\n**Lost Deep, the:** a loreworks; breeding pit\/laboratory under Mount Thunder where Demondim, Waynhim, and ur-viles were created\n\n**Lostson:** a Giant; later name for Exalt Widenedworld\n\n**Lower Land, the:** region of the Land east of Landsdrop\n\n**lucubrium:** laboratory of a thaumaturge\n\n**lurker of the Sarangrave, the:** monster inhabiting the Great Swamp\n\n**Magister, the:** former Forestal of Andelain\n\n**Mahdoubt, the:** a servant of Revelstone; one of the Insequent\n\n**Mahrtiir:** a Manethrall of the Ramen; companion of Linden Avery\n\n**_maidan_ :** open land around _Elemesnedene_\n\n**Maker, the:** _jheherrin_ name for Lord Foul\n\n**Maker-place:** _jheherrin_ name for Foul's Creche\n\n**Malliner:** Woodhelvennin Heer, son of Veinnin\n\n**Mane:** Ramen reference to a Ranyhyn\n\n**Maneing:** Ramen ceremony of becoming a Manethrall\n\n**Manethrall:** highest Ramen rank\n\n**Manhome:** main dwelling place of the Ramen in the Plains of Ra\n\n**Marid:** a man of Mithil Stonedown; Sunbane victim\n\n**Marny:** a Ranyhyn; mount of Tuvor\n\n**marrowmeld:** bone-sculpting; _anundivian yaj\u00f1a_\n\n**Master:** commander of a Giantship\n\n**Master, the:** Clave-name for Lord Foul\n\n**_master-rukh_ , the:** iron triangle at Revelstone which feeds and reads other _rukhs_\n\n**Masters of the Land, the:** _Haruchai_ who have claimed responsibility for protecting the Land from Corruption\n\n**Mehryl:** a Ranyhyn; mount of Hile Troy\n\n**_Melenkurion_ :** one of the Seven Words\n\n**_Melenkurion_ Skyweir: **a cleft peak in the Westron Mountains\n\n**Memla:** a Rider of the Clave; one of the na-Mhoram-in\n\n**_mere_ -son:** name or title given to Esmer\n\n**_merewives_ :** the Dancers of the Sea\n\n**_metheglin_ :** a beverage; mead\n\n**Mhoram:** a Lord, later high Lord; son of Variol\n\n**Mhornym:** a Ranyhyn stallion; mount of Clyme\n\n**_Mill_ :** one of the Seven Words\n\n**_Minas_ : **one of the Seven Words\n\n**_mirkfruit_ :** papaya-like fruit with narcoleptic pulp\n\n**Mishio Massima:** the Ardent's mount\n\n**Mistweave:** a Giant\n\n**Mithil River:** a river of the Land\n\n**Mithil Stonedown:** a village in the South Plains\n\n**Mithil's Plunge, the:** waterfall at the head of the Mithil valley\n\n**Moire Squareset:** a Giant; one of the Swordmainnir; killed in battle by the _skurj_\n\n**_moksha_ :** a Raver, Jehannum, Fleshharrower\n\n**Morin:** First Mark of the Bloodguard; commander in original _Haruchai_ army\n\n**Morinmoss:** a forest of the Land\n\n**Morninglight:** one of the _Elohim_\n\n**Morril:** a Bloodguard, assigned to Lord Callindrill\n\n**Mount Thunder:** a peak at the center of Landsdrop\n\n**Muirwin Delenoth:** resting place of abhorrence; graveyard of _quellvisks_\n\n**Murrin:** a Stonedownor, mate of Odona\n\n**Myrha:** a Ranyhyn; mount of High Lord Elena\n\n**na-Mhoram, the:** leader of the Clave\n\n**na-Mhoram-cro:** lowest rank of the Clave\n\n**na-Mhoram-in:** highest rank of the Clave below the na-Mhoram\n\n**na-Mhoram-wist:** middle rank of the Clave\n\n**Naharahn:** a Ranyhyn mare; mount of Pahni\n\n**Narunal:** a Ranyhyn stallion; mount of Mahrtiir\n\n**Nassic:** father of Sunder, son of Jous; inheritor of an Unfettered One's mission to remember the Halfhand\n\n**Naybahn:** a Ranyhyn; mount of Branl\n\n**Nelbrin:** son of Sunder, \"heart's child\"\n\n**_Nicor_ , the:** great sea-monster; said to be offspring of the Worm of the World's End\n\n**Nom:** a Sandgorgon\n\n**North Plains, the:** a region of the Land\n\n**Northron Climbs, the:** a region of the Land\n\n**Oath of Peace, the:** oath by the people of the Land against needless violence\n\n**Odona:** a Stonedownor, mate of Murrin\n\n**Offin:** a former na-Mhoram\n\n**Old Lords, the:** Lords prior to the Ritual of Desecration\n\n**Omournil:** Woodhelvennin Heer, daughter of Mournil\n\n**One Forest, the:** ancient forest covering most of the Land\n\n**One Tree, the:** mystic tree from which the Staff of Law was made\n\n**Onyx Stonemage:** a Giant; one of the Swordmainnir\n\n**_orcrest_ : **a stone of power; Sunstone\n\n**Osondrea:** a Lord, daughter of Sondrea; later high Lord\n\n**Padrias:** Woodhelvennin Heer, son of Mill\n\n**Pahni:** a Cord of the Ramen, cousin of Sahah; companion of Linden Avery\n\n**Palla:** a healer in Berek Halfhand's army\n\n**Peak of the Fire-Lions, the:** Mount Thunder, Gravin Threndor\n\n**Pietten:** Woodhelvennin child damaged by Lord Foul's minions, son of Soranal\n\n**_pitchbrew_ :** a beverage combining _diamondraught_ and _vitrim_ , conceived by Pitchwife\n\n**Pitchwife:** a Giant; member of the Search; husband of the First of the Search\n\n**Plains of Ra, the:** a region of the Land\n\n**Porib:** a Bloodguard\n\n**Power of Command, the:** Seventh Ward of Kevin's Lore\n\n**Pren:** a Bloodguard\n\n**Prothall:** High Lord, son of Dwillian\n\n**Prover of Life:** title given to Thomas Covenant\n\n**Puhl:** a Cord of the Ramen\n\n**Pure One, the:** redemptive figure of _jheherrin_ legend\n\n**Quaan:** Warhaft of the Third Eoman of the Warward; later Hiltmark, then Warmark\n\n**_quellvisk_ :** a kind of monster, now apparently extinct\n\n**_Quern Ehstrel_ :** true name of the Mahdoubt\n\n**Quest for the Staff of Law, the:** quest to recover the Staff of Law from Drool Rockworm\n\n**_Questsimoon_ , the:** the Roveheartswind; a steady, favorable wind, perhaps seasonal\n\n**Quilla:** a Heer of First Woodhelven\n\n**Quirrel:** a Stonedownor, companion of Triock\n\n**Rallyn:** a Ranyhyn; Branl's later mount\n\n**Ramen:** a people who serve the Ranyhyn\n\n**Rant Absolain:** the _gaddhi_\n\n**Ranyhyn:** the great horses of the Plains of Ra\n\n**Ravers:** Lord Foul's three ancient servants\n\n**_Raw_ , the: **fjord into the demesne of the _Elohim_\n\n**Rawedge Rim, the:** mountains around _Elemesnedene_\n\n**Reader:** a member of the Clave who tends and uses the _master-rukh_\n\n**Rede, the:** knowledge of history and survival promulgated by the Clave\n\n**Revelstone:** Lord's Keep; mountain city formed by Giants\n\n**Revelwood:** seat of the Loresraat; tree city grown by Lords\n\n**_rhee_ :** a Ramen food, a thick mush\n\n**_rhadhamaerl_ :** stone-lore; masters of stone-lore\n\n**Rhohm:** a Ranyhyn stallion; mount of Liand\n\n**_rhysh_ :** a community of Waynhim; \"stead\"\n\n**_rhyshyshim_ :** a gathering of _rhysh_ ; a place in which such gathering occurs\n\n**Riddenstretch:** a region north of the Soulsease River\n\n**Rider:** a member of the Clave\n\n**Ridjeck Thome:** Foul's Creche, the Despiser's home\n\n**_rillinlure_ :** healing wood dust\n\n**Rime Coldspray:** a Giant; the Ironhand of the Swordmainnir\n\n**Ringthane:** Ramen name for Thomas Covenant, then Linden Avery\n\n**ring-wielder:** _Elohim_ term of reference for Thomas Covenant\n\n**Rire Grist:** a Caitiffin of the _gaddhi_ 's Horse\n\n**Rites of Unfettering:** the ceremony of becoming Unfettered\n\n**Ritual of Desecration, the:** act of despair by which High Lord Kevin destroyed the Old Lords and ruined most of the Land\n\n**Rivenrock:** deep cleft splitting _Melenkurion_ Skyweir and its plateau; there the Black River enters Garroting Deep\n\n**River Landrider, the:** a river of the Land, partial border of the Plains of Ra\n\n**Riversward:** a region north of the Soulsease River\n\n**Rockbrother:** Swordmainnir name for Stave\n\n**Rockbrother, Rocksister:** terms of affection between humans and Giants\n\n**rocklight:** light emitted by glowing stone\n\n**_roge Befylam_ :** Cavewight-form of the _jheherrin_\n\n**Rohnhyn:** a Ranyhyn; mount of Bhapa after Whrany's death\n\n**Roveheartswind, the:** the _Questsimoon_\n\n**Rue:** a Manethrall of the Ramen, formerly named Gay\n\n**Ruel:** a Bloodguard, assigned to Hile Troy\n\n**Ruinwash, the:** name of the River Landrider on the Lower Land\n\n**_rukh_ :** iron talisman by which a Rider wields the power of the Sunbane\n\n**Runnik:** a Bloodguard\n\n**Rustah:** a Cord of the Ramen\n\n**Sablehair Foamheart:** a Giant, also called Filigree; mate of Soar Gladbirth; mother of Exalt Widenedworld\n\n**sacred enclosure:** Vespers-hall at Revelstone; later the site of the Banefire\n\n**Sahah:** a Cord of the Ramen\n\n**Saltheart Foamfollower:** a Giant, friend of Thomas Covenant\n\n**Saltroamrest:** bunk hold for the crew in a Giantship\n\n**Salttooth:** jutting rock in the harbor of the Giants' Home\n\n**Salva Gildenbourne:** forest surrounding the Hills of Andelain; begun by Sunder and Hollian\n\n**_samadhi_ :** a Raver, Sheol, Satansfist\n\n**Samil:** a _Haruchai_ ; a Master of the Land\n\n**Sandgorgons:** monsters of the Great Desert of _Bhrathairealm_\n\n**Sandgorgons Doom:** imprisoning storm created by Kasreyn to trap the Sandgorgons\n\n**Sandhold, the:** the _gaddhi_ 's castle in _Bhrathairealm_\n\n**Sandwall, the:** the great wall defending _Bhrathairain_\n\n**Santonin:** a Rider of the Clave, one of the na-Mhoram-in\n\n**Sarangrave Flat:** a region of the Lower Land encompassing the Great Swamp\n\n**Satansfist:** a Giant-Raver, Sheol, _samadhi_\n\n**Satansheart:** Giantish name for Lord Foul\n\n**Scend Wavegift:** a Giant; one of the Swordmainnir; killed by Longwrath\n\n**Search, the:** quest of the Giants for the wound in the Earth; later the quest for the Isle of the One Tree\n\n**Seareach:** region of the Land occupied by the Unhomed\n\n**Seasauce:** a Giant; husband of Hearthcoal; cook of Starfare's Gem\n\n**Seatheme:** dead wife of Sevinhand\n\n**Second Circinate:** second level of the Sandhold\n\n**Second Ward:** second unit of Kevin's hidden knowledge\n\n**setrock:** a type of stone used with pitch to repair stone\n\n**Seven Hells, the:** a-Jeroth's demesne: desert, rain, pestilence, fertility, war, savagery, and darkness\n\n**Seven Wards, the:** collection of knowledge hidden by High Lord Kevin\n\n**Seven Words, the:** words of power from Kevin's Lore\n\n**Sevinhand:** a Giant, Anchormaster of Starfare's Gem\n\n**Shattered Hills, the:** a region of the Land near Foul's Creche\n\n**She Who Must Not Be Named:** an ancient bane slumbering under Mount Thunder, now composed of many lost women\n\n**Sheol:** a Raver, Satansfist, _samadhi_\n\n**Shetra:** a Lord, wife of Verement\n\n**Shipsheartthew:** the wheel of a Giantship\n\n**_shola_ :** a small wooded glen where a stream runs between unwooded hills\n\n**Shull:** a Bloodguard\n\n**Sill:** a Bloodguard, assigned to Lord Hyrim\n\n**Sivit:** a Rider of the Clave, one of the na-Mhoram-wist\n\n**_skest_ :** acid-creatures descended from the _jheherrin_\n\n**_skurj_ :** laval monsters that devour earth and vegetation; long ago, the _Elohim_ Kastenessen was Appointed (the Durance) to prevent them from wreaking terrible havoc\n\n**Slen:** a Stonedownor, mate of Terass\n\n**Slumberhead:** a Giant (man); crew aboard Dire's Vessel\n\n**Snared One, the:** ur-vile name for Lord Foul the Despiser\n\n**Soar Gladbirth:** a Giant; youngest son of Pitchwife and Gossamer Glowlimn\n\n**Soaring Woodhelven:** a tree-village\n\n**soft ones, the:** the _jheherrin_\n\n**Somo:** pinto taken by Liand from Mithil Stonedown\n\n**soothreader:** a seer\n\n**soothtell:** ritual of revelation practiced by the Clave\n\n**Soranal:** a Woodhelvennin Heer, son of Thiller\n\n**Soulbiter, the:** a dangerous ocean of Giantish legend\n\n**Soulbiter's Teeth:** reefs in the Soulbiter\n\n**Soulcrusher:** Giantish name for Lord Foul\n\n**South Plains, the:** a region of the Land\n\n**Sparlimb Keelsetter:** a Giant, father of triplets\n\n**Spikes, the:** guard-towers at the mouth of _Bhrathairain_ Harbor\n\n**Spoiled Plains, the:** a region of the Lower Land\n\n**Spume Frothbreeze:** a Giant (man); crew aboard Dire's Vessel\n\n**Spray Frothsurge:** a Giant; mother of the First of the Search\n\n**springwine:** a mild, refreshing liquor\n\n**Squallish Blustergale:** a Giant (man); crew aboard Dire's Vessel\n\n**Staff, the:** a branch of the study of Kevin's Lore\n\n**Staff of Law, the:** a tool of Earthpower; the first Staff was formed by Berek from the One Tree and later destroyed by Thomas Covenant; the second was formed by Linden Avery by using wild magic to merge Vain and Findail\n\n**Stallion of the First Herd, the:** _Kelenbhrabanal_\n\n**Starfare's Gem:** Giantship used by the Search\n\n**Starkin:** one of the _Elohim_\n\n**Stave:** a _Haruchai_ ; a Master of the Land; companion of Linden Avery\n\n**Stell:** one of the _Haruchai_ , protector of Sunder\n\n**Stonedown:** a stone-village\n\n**Stonedownor:** one who lives in a stone-village\n\n**Stonemight, the:** a fragment of the Illearth Stone\n\n**Stonemight Woodhelven:** a village in the South Plains\n\n**Storesmaster:** third-in-command aboard a Giantship\n\n**Stormpast Galesend:** a Giant; one of the Swordmainnir\n\n**Stricken Stone:** region of the Land, later called Trothgard\n\n**Sunbane, the:** a power arising from the corruption of nature by Lord Foul\n\n**Sunbirth Sea, the:** ocean east of the Land\n\n**Sunder:** son of Nassic; Graveler of Mithil Stonedown; companion of Thomas Covenant and Linden Avery\n\n**Sun-Sage:** one who can affect the Sunbane\n\n**Sunstone:** _orcrest_\n\n**_sur-jheherrin_ :** descendants of the _jheherrin_ ; inhabitants of Sarangrave Flat\n\n**_suru-pa-maerl_ :** an art using stone\n\n**Swarte:** a Rider of the Clave\n\n**Swordmain\/Swordmainnir:** Giant(s) trained as warrior(s)\n\n**Sword, the:** a branch of the study of Kevin's Lore\n\n**Sword-Elder:** chief Lorewarden of the Sword at the Loresraat\n\n**Syr Embattled:** former Forestal of Giant Woods\n\n**Tamarantha:** a Lord, daughter of Enesta, wife of Variol\n\n**Teeth of the Render, the:** Ramen name for the Demondim; Fangs\n\n**Terass:** a Stonedownor, daughter of Annoria, wife of Slen\n\n**Terrel:** a Bloodguard, assigned to Lord Mhoram; a commander of the original _Haruchai_ army\n\n**test of silence, the:** test of integrity used by the people of the Land\n\n**test of truth, the:** test of veracity by _lomillialor_ or _orcrest_\n\n**The Grieve:** _Coercri_ ; home of the lost Giants in Seareach\n\n**Thelma Twofist:** character in a Giantish tale\n\n**The Majesty:** throne room of the _gaddhi_ ; fourth level of the Sandhold\n\n**Theomach, the:** one of the Insequent\n\n**Thew** : a Cord of the Ramen\n\n**Third Ward:** third unit of Kevin's hidden knowledge\n\n**Thomin:** a Bloodguard, assigned to Lord Verement\n\n**Three Corners of Truth, the:** basic formulation of beliefs taught and enforced by the Clave\n\n**thronehall, the:** the Despiser's seat in Foul's Creche\n\n**Tier of Riches, the:** showroom of the _gaddhi_ 's wealth; third level of the Sandhold\n\n**Timewarden:** _Elohim_ title for Thomas Covenant after his death\n\n**Tohrm:** a Gravelingas; Hearthrall of Lord's Keep\n\n**Tomal:** a Stonedownor craftmaster\n\n**Toril:** _Haruchai_ slain by the Clave\n\n**Treacher's Gorge:** ravine opening into Mount Thunder\n\n**treasure-berries:** _aliantha_ , nourishing fruit found throughout the Land in all seasons\n\n**Trell:** Gravelingas of Mithil Stonedown; husband of Atiaran, father of Lena\n\n**Trevor:** a Lord, husband of Loerya\n\n**Triock:** a Stonedownor, son of Thuler; loved Lena\n\n**Trothgard:** a region of the Land, formerly Stricken Stone\n\n**Tull:** a Bloodguard\n\n**_turiya_ : **a Raver, Herem, Kinslaughterer\n\n**Tuvor:** First Mark of the Bloodguard; a commander of the original _Haruchai_ army\n\n**_tyrscull_ :** a Giantish training vessel for apprentice sailors\n\n**Ulman:** a _Haruchai_ ; a Master of the Land\n\n**Unbeliever, the:** title claimed by Thomas Covenant\n\n**Unfettered, the:** lore-students freed from conventional responsibilities to seek individual knowledge and service\n\n**Unfettered One, the:** founder of a line of men waiting to greet Thomas Covenant's return to the Land\n\n**Unhomed, the:** the lost Giants living in Seareach\n\n**un-Maker-made, the:** in _jheherrin_ legend, living beings not created by the Maker\n\n**upland:** plateau above Revelstone\n\n**Upper Land, the:** region of **** the Land west of Landsdrop\n\n**ur-Lord:** title given to Thomas Covenant\n\n**ur-viles:** Demondim-spawn, evil creatures\n\n**_ussusimiel_ : **nourishing melon grown by the people of the Land\n\n**Vailant:** former High Lord before Prothall\n\n**Vain:** Demondim-spawn; bred by ur-viles for a secret purpose\n\n**Vale:** a Bloodguard\n\n**Valley of Two Rivers, the:** site of Revelwood in Trothgard\n\n**Variol Tamarantha-mate:** a Lord, later High Lord; son of Pentil, father of Mhoram\n\n**Verement:** a Lord, husband of Shetra\n\n**Verge of Wandering, the:** valley in the Southron Range southeast of Mithil Stonedown; gathering place of the nomadic Ramen\n\n**Vernigil:** a _Haruchai_ ; a Master of the Land guarding First Woodhelven\n\n**Vertorn:** a healer in Berek Halfhand's army\n\n**Vespers:** self-consecration rituals of the Lords\n\n**Vettalor:** Warmark of the army opposing Berek Halfhand\n\n**_viancome_ :** meeting place at Revelwood\n\n**Victuallin Tayne:** a region in the Center Plains\n\n**Vidik Amar:** a region of the Earth\n\n**Vigilall Scudweather:** a Giant (woman); Master of Dire's Vessel\n\n**Viles:** monstrous beings which created the Demondim\n\n**_vitrim_ :** nourishing fluid created by the Waynhim\n\n**Vizard, the:** one of the Insequent\n\n**Voice of the Masters, the:** a _Haruchai_ leader; spokesman for the Masters as a group\n\n**Vortin:** a _Haruchai_ ; a Master of the Land\n\n**_voure_ :** a plant-sap which wards off insects\n\n**Vow, the:** _Haruchai_ oath of service which formed the Bloodguard\n\n**_vraith_ :** a Waynhim\n\n**Ward:** a unit of Kevin's lore\n\n**Warhaft:** commander of an Eoman\n\n**Warlore:** Sword knowledge in Kevin's Lore\n\n**Warmark:** commander of the Warward\n\n**Warrenbridge:** entrance to the catacombs under Mount Thunder\n\n**Warward, the:** army of Lord's Keep\n\n**Wavedancer:** Giantship commanded by Brow Gnarlfist\n\n**Wavenhair Haleall:** a Giant, wife of Sparlimb Keelsetter, mother of triplets\n\n**Waymeet:** resting place for travelers maintained by Waynhim\n\n**Waynhim:** tenders of the Waymeets; rejected Demondim-spawn, opponents and relatives of ur-viles\n\n**Weird of the Waynhim, the:** Waynhim concept of doom, destiny, or duty\n\n**_were-menhir(s)_ :** Giantish name for the _skurj_\n\n**Whane:** a Cord of the Ramen\n\n**white gold:** a metal of power not found in the Land\n\n**white gold wielder:** title given to Thomas Covenant\n\n**White River, the:** a river of the land\n\n**Whrany:** a Ranyhyn stallion; mount of Bhapa\n\n**Wightburrow, the:** cairn under which Drool Rockworm is buried\n\n**Wightwarrens:** home of the Cavewights under Mount Thunder; catacombs\n\n**wild magic:** the power of white gold; considered the keystone of the Arch of Time\n\n**Wildwielder:** white gold wielder; title given to Linden Avery by Esmer and the _Elohim_\n\n**Windscour:** region in the Center Plains\n\n**Windshorn Stonedown:** a village in the South Plains\n\n**Winhome:** Ramen lowest rank\n\n**Wiver Setrock:** a Giant (man); crew aboard Dire's Vessel\n\n**Woodenwold:** region of trees surrounding the _maidan_ of _Elemesnedene_\n\n**Woodhelven:** wood-village\n\n**Woodhelvennin:** inhabitants of wood-village\n\n**Word of Warning:** a powerful, destructive forbidding\n\n**Worm of the World's End, the:** creature believed by the _Elohim_ to have formed the foundation of the Earth\n\n**Wraiths of Andelain, the:** creatures of living light that perform the Dance at the Celebration of Spring\n\n**W\u00fcrd of the Earth, the:** term used by the _Elohim_ to describe their own nature, destiny, or purpose; could be read as Word, Worm, or Weird\n\n**Yellinin:** a soldier in Berek Halfhand's army; third-in-command of the tenth Eoman, second Eoward\n\n**Yeurquin:** a Stonedownor, companion of Triock\n\n**Yolenid:** daughter of Loerya\n\n**Zaynor:** a _Haruchai_ from a time long before the _Haruchai_ first came to the Land\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## Dedication\n\n_In memory of George Nicholson, who taught me so much_\n\n#  \n\n## Contents\n\n  1. Dedication\n  2. Part 1: Violet\n    1. Chapter 1\n    2. Chapter 2\n    3. Chapter 3\n    4. Chapter 4\n    5. Chapter 5\n    6. Chapter 6\n    7. Chapter 7\n    8. Chapter 8\n    9. Chapter 9\n    10. Chapter 10\n    11. Chapter 11\n    12. Chapter 12\n  3. Part 2: Celine\n    1. Chapter 1\n    2. Chapter 2\n    3. Chapter 3\n    4. Chapter 4\n    5. Chapter 5\n    6. Chapter 6\n    7. Chapter 7\n    8. Chapter 8\n    9. Chapter 9\n    10. Chapter 10\n    11. Chapter 11\n    12. Chapter 12\n    13. Chapter 13\n    14. Chapter 14\n    15. Chapter 15\n    16. Chapter 16\n    17. Chapter 17\n    18. Chapter 18\n    19. Chapter 19\n    20. Chapter 20\n    21. Chapter 21\n  4. Part 3: Goose\n    1. Chapter 1\n    2. Chapter 2\n    3. Chapter 3\n    4. Chapter 4\n    5. Chapter 5\n    6. Chapter 6\n    7. Chapter 7\n    8. Chapter 8\n    9. Chapter 9\n    10. Chapter 10\n    11. Chapter 11\n    12. Chapter 12\n    13. Chapter 13\n  5. Part 4\n    1. Chapter 1\n    2. Chapter 2\n    3. Chapter 3\n    4. Chapter 4\n  6. Excerpt from _Beheld_\n  7. Prologue\n  8. Chapter 1\n  9. Back Ads\n  10. About the Author\n  11. Books by Alex Flinn\n  12. Credits\n  13. Copyright\n  14. About the Publisher\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Chapter 1\n\n  1. vi\n  2. v\n  3. \n  4. \n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. iv\n\n# PART 1\n\n# _Violet_\n\n#\n\n#\n\n# _1982_\n\nI was a strange child. Strange looking, for certain, with buckteeth, red hair (and matching invisible eyelashes), a hooked nose, and barely the hint of a chin. My classmates at Coral Ridge Elementary teased me about these defects as if it was their God-given right. Maybe it was. After all, if I wanted to fit in, wouldn't I just act more normal?\n\nBut even if I could change my hair, my nose, my chin, I couldn't perform plastic surgery on my soul. Some secret part of me remained stubbornly different\u2014brought egg salad when everyone else knew it was gross or used vocabulary words like _whimsical_ , words no one else understood.\n\nAnd shortly after my tenth birthday, I learned I was different in another way.\n\nA special way.\n\nThat day, a rare, cold January day in Miami, I was walking alone on the playground. I had a jump rope. But when I'd asked the two girls who sat behind me, Jennifer Sadler and Gennifer Garcia, if they wanted to jump with me, they'd turned to each other with perplexed looks.\n\n\"Jump rope?\" Jennifer's blond ponytail swung from side to side as she shook her head. \"Who jumps rope anymore?\"\n\n\"We do in PE.\" It was the only sport at which I excelled.\n\nJennifer rolled her enormous blue eyes. \"Yeah, for a grade.\"\n\n\"We could do something else then,\" I said. \"Kickball?\" The Jennifers were part of a kickball game that included most of our class. I was never invited.\n\n\"Sorry, I think the teams are set.\" Gennifer Garcia wrinkled her perfect nose.\n\nAs I trudged away, trailing my red-and-white-striped rope, I heard Jennifer say loudly, \"You are too nice to that girl. It just makes her think she can talk to us.\"\n\nGennifer laughed, then said softer, but still loud enough for me to hear clearly, \"You can't just tell people they're freaks.\"\n\n\"Why not? It would help her to know.\"\n\nDid being beautiful make you cruel? Were they really being cruel if I was so defective? And, if I were suddenly pretty, would I be mean too? Not that that would ever happen.\n\nSo I sat alone, shivering on a swing, looking at the toes of my navy Keds against the lighter blue clouds, when I spied a cluster of kids gathered around something. Ordinarily, I avoided unsupervised groups of my peers, but this particular group included, at its outskirts, one Gregory Columbo, a sensitive boy with coal-black hair and the warmest brown eyes I'd ever seen. Though he wasn't exactly nice to me, he wasn't mean. He was quiet, more the type to read a book than to play baseball. Now, he wiped at his eyes with his gray sweatshirt sleeve. And something else made me gravitate toward the group, something I didn't quite understand.\n\nI jumped off the swing and went before I could chicken out.\n\n\"What's everyone looking at?\" I wanted to ask Greg why he was crying, but probably, he was pretending not to.\n\nGreg turned away, touching the side of his face with his fingers.\n\n\"Greg?\"\n\n\"Leave us alone, ugly.\" Jennifer smirked. \"It's just a stupid, dead crow.\"\n\nGreg turned and I caught sight of his face. His cheeks and eyes were mottled red, like blotchy, red glasses, and I had a distant memory of him and a woman\u2014his mother\u2014standing outside his house, filling a blue bird feeder. Greg's mother had died the year before.\n\nThe crowd was dispersing except for two boys, Nick and Nathan, who stood over the bird, poking it with a stick. I hated them. Hated. I could see the crow now, shiny and black as Greg's hair against the dappled greenish-brown grass, one black bead of an eye staring at me. I stooped beside it.\n\n\"Eww!\" Gennifer said. \"Are you going to touch it?\"\n\nI glanced back to see if Greg was still there. Nick hovered over me with his stick. I fixed him with a stare.\n\n\"Don't touch that. It's disrespectful of the dead.\" I remembered my mother saying something like that. Nick laughed and began to respond, but I said, \"Go away.\" I heard violence in my voice.\n\nHe backed away. \"It's dumb anyway.\" Nathan and the Jennifers followed him off.\n\nI reached down and touched the bird. It was not cold, not yet at least. But it lay as still as the airless day. Its licorice-black wings caught the light, reflecting green and purple. I felt the ground move with the thumping of my enemies' retreating sneakers. I shivered, then slid my hand underneath the bird. It felt like leaves. I turned my back to Greg. Then, I placed my other hand over it and picked it up. It was larger than both my hands, but it weighed less than air. I remembered someone, Greg maybe, saying in class that birds' bones were hollow so they could fly.\n\nWithout knowing why, really, I raised the bird to my face, opened my hands, and blew.\n\nThe bird blinked.\n\nNo, it hadn't. It was dead.\n\nI stroked its smooth feathers. Behind me, Greg stood silent, but I could hear his breath. I found a tiny wound under the bird's wing. When I touched it with my finger, it seemed to disappear.\n\nThe bird blinked again. This time, I was certain it had.\n\n\"Are you okay, then?\" I barely whispered.\n\nIt cocked its head toward me. Behind me, I heard a gasp. Greg. I did not, could not look at him. I whispered, \"Are you ready to go now?\"\n\nAgain, the bird closed then opened its black bead eye, and in my head, I thought I heard it say, \"Do you want me to?\"\n\n\"Whatever you like. Your family is probably worried, though.\"\n\n\"It was a boy,\" the bird said, \"the one with the stick. He threw a rock. It hurt. It _killed me._ \"\n\n\"It didn't kill you. You're alive.\"\n\nThe bird twitched its head. \"It killed me. I was dead.\"\n\nI couldn't understand this. \"Hmm, maybe you should get away from him.\"\n\n\"I guess so!\" said the bird. \"Can you help?\"\n\n\"How?\" I drew in a deep, shaky breath.\n\n\"Just . . . raise me up.\"\n\nI nodded, then stood, still holding the crow in both hands. I lifted it high above my head.\n\nIt rose, shifting first to one clawed foot, then the other. It spread its black wings toward the graying sky. They gleamed purple and green in the strained sunlight. It fluttered, then lifted into the air. I watched it until it was merely a small, black speck against the clouds. I turned toward school.\n\n\"Hey,\" a voice said behind me. \"That bird was dead.\"\n\nGreg. I looked up to where the bird had been. \"No, it wasn't. It flew away.\"\n\n\"But it was _dead._ \"\n\n\"No.\" It seemed, now, very important for me to impress this upon him. After all, I could barely believe what I'd done myself. It couldn't have happened, but it had. \"No. It was only scared. Nick hit it with a stick. It's okay now.\"\n\nGreg glanced at the ground. I did too and saw an irregular puddle of blood, mixing with the dusty dirt. \"Okay,\" he said finally.\n\n\"We're going to get in trouble,\" I said.\n\n\"I'll tell Ms. Gayton what happened, that you helped the bird.\" His eyes were just nice looking, the way his eyebrows came down in the center, all concerned.\n\nI shook my head. \"She won't believe you.\"\n\n\"She will. Teachers like me. They feel sorry for me because of\u2014\"\n\n\"Your mom.\" I was thinking they liked him because he was so cute, those big eyes.\n\nHe nodded. \"I can get away with anything.\"\n\n\"They hate me.\" It was true. Even though I aced every test, teachers had no smiles for me. They liked the prettier girls, the ones who didn't care about learning but only copied one another's homework.\n\n\"I'll tell her you were helping me then,\" he said.\n\nI thought about it. \"Okay.\"\n\nWe began to walk. As we did, my hand brushed Greg's. I wanted to grab it, but I didn't.\n\nHe said, \"That was pretty cool.\"\n\n\"I didn't do anything.\"\n\n\"You did.\"\n\n\"No.\" I said it too loudly, too emphatically, then regretted it. I sounded like I was lying. \"I'm sorry, but the bird was probably just shaken up or something. I mean, it flew away like nothing.\"\n\nI remembered the wound under its wing, the wound that had closed. _How_ had it?\n\nGreg shrugged. \"Okay.\" He started up the stairs to the school building, taking them two at a time, keeping ahead of me. He thought I was weird, though I'd tried so hard to prevent it, though I wanted him to like me so much. I'd have done anything to make him like me. Anything.\n\nI didn't speed my step. What would be the point? As I trudged along, I felt something, like a pair of eyes, peering at me through the trees. I looked back and saw a shadowy figure dressed in green like the leaves. Or maybe it was the leaves. I shivered, remembering what my mother had said about perverts who lurked on playgrounds, waiting for children alone.\n\n\"Hello?\" I said.\n\nNo answer except a riffle of wind across the grass and my own footsteps.\n\nI stopped, glanced back. No one there. A caw, maybe the same crow, echoed from the treetops. I felt each individual hair on my arms stand on end. My feet planted on the ground.\n\nFinally, I ran after Greg to the school building.\n\nNothing unusual happened for a year or so. Maybe I made nothing happen. But then, one day something did. Something that changed everything.\n\n#\n\n#\n\n# _1985_\n\nFrom that day on, Greg and I became friends. Sort of. We didn't really talk much at school, but if a teacher assigned a group project, we chose each other as partners. We'd work at my house or usually Greg's, making brochures to entice people to move to the Colony of Delaware, or a travel poster for Mars. I loved going to Greg's house. His father was always building something like a flower box or a huge tree house, and he let Greg and me help. Once, when we were in seventh grade, it was a birdhouse.\n\n\"Violet loves birds,\" Greg told his dad.\n\n\"Do you?\" His father's eyes were brown like Greg's, and he was tall, but not skinny. He had a belly, which seemed very dad-like. \"So you're a bird expert?\"\n\n\"Not an expert. But they're pretty.\"\n\n\"Right, she's not an expert,\" Greg said, laughing. \"The first time I spoke to Vi, she was holding a crow in the palm of her hand. Since then, I've seen them follow her.\"\n\nI stared at him, startled. I didn't know he'd noticed, but it was true. Birds seemed to gravitate toward me, animals too; even wild ones like possums and raccoons didn't scatter as they did when others approached. I kept that quiet, though. It didn't exactly make me seem normal.\n\n\"I think they even talk to her sometimes.\" Greg smiled. He had a dimple, just one, on the left side of his mouth.\n\n\"Right, Greg. Sure.\"\n\nGreg's dad laughed along. \"Well, if you could talk to the Carolina wrens, maybe ask them what's the proper opening size for their birdhouse. Or you could just look it up.\"\n\n\"Is that really important? My mom just bought the birdhouse she thought was pretty.\"\n\n\"Do you see many birds in it?\" Mr. Columbo asked.\n\n\"No,\" I admitted. \"Never.\"\n\n\"If you look it up for me, I'll build a birdhouse they'll use\u2014and I'll make another one for your house.\"\n\n\"I think I'd rather come over and look at yours.\" I wanted to be with Greg, always. The Columbos felt more like a real family than mine, even though it was just two of them. At home, it was just me and my mother, who spent more time on her nails than talking to me.\n\nStill, Greg and I went to the school library during lunch the next day. We looked up the information in a book, finding that the Carolina wren needed a hole between one and a quarter and one and a half inches. Mr. Columbo let me drill the hole myself and even, against my protests, made a second one for me. \"You can still come over, Violet.\" His eyes crinkled around the edges. \"You're welcome any time.\"\n\nAfter that, Greg invited me over without needing a magazine project as an excuse. We walked home together, at least as far as his house, every afternoon and checked out the birdhouse. One day, we saw a small brown bird moving in. \"Wow,\" I said, \"your dad's going to be so excited.\" There had been birds in my own birdhouse since the first day, but I didn't tell Greg that.\n\nGreg grabbed my hand and squeezed it. His own hand was cool, dry, and I felt a jolt of electricity run up my arm. \"We've never had a bird in our birdhouse before, no matter what he said. They came because of you.\"\n\n\"Right.\" I could barely speak from concentrating on the moment, the dizzying tingle of his hand in mine.\n\n\"You should stay, at least until Dad gets home.\"\n\nI nodded and walked into the house. I thought I could hear one of the birds say _Thank you_ , __ but when I looked over at Greg, he hadn't heard anything. We spent the whole afternoon sitting side by side at the window, watching the wren build her nest, until Mr. Columbo called us in for dinner. Greg smelled of pencil shavings and Irish Spring soap, and even though he let go of my hand, I could still feel it against mine, far, far into the night.\n\nAfter that, I went to Greg's house every day after school. We watched the wrens to see if there were eggs or babies, but we also sat on the sofa, eating brownies we made from a mix and watching _Family Ties_ and _The Cosby Show,_ shows about big, happy families like ours weren't. People at school called us geeks in love, and while I pretended to cringe, I secretly enjoyed that people thought it possible. Greg wasn't my boyfriend\u2014not yet\u2014but I wanted him to be.\n\nBut the summer before eighth grade, Greg went to sleepaway camp for two whole months. \"It's like this really outdoorsy camp my dad went to when he was my age,\" he told me. \"I hate it. I think it's supposed to make me a man.\"\n\n\"Sounds fun.\" I rolled my eyes, thinking he was perfect as he was. In fact, I was hoping he'd invite me to the Halloween dance at school in the fall. Halloween was my favorite holiday because it let me be someone else, someone better. Or maybe someone worse. I was hoping maybe, costumed as a witch or vampire, I would have the courage to lean in, to will him to kiss me. But that was a long time away. \"Can I write you?\"\n\n\"Sure. That'd be cool.\" He smiled his cute, one-dimpled smile. \"I know my dad won't write that much, not like a mom would.\"\n\n\"I'll write.\"\n\nOnce he left, the long, lonely summer stretched ahead of me, and I wanted to write every day. I didn't, though. Some dim part of me knew that would look crazy, like I had no life, which I didn't. I actually did things like an Everglades bird tour, just so I'd have something to write about. I worked on the letters every day but sent them once a week. Greg wrote back twice, then stopped. I told myself it was because he was busy, then because camp was almost over. I'd see him soon. But when he got back, he was always busy. In rushed phone calls, he said he had family in town one day, shopping for school another. So I didn't see him until the first day of eighth grade.\n\nI searched for Greg in the crowds outside. Usually, he wasn't hard to spot. He was tall and stayed on the outskirts. But I didn't see him.\n\nThen, I did. He wasn't on the outskirts, but in the middle of the crowd.\n\nGreg had always been tall, but now he was taller. Not skinny anymore, though. Suddenly his shoulders were broader, his face more manly. He was standing with some people, people like Nick and Nathan, Jennifer and Gennifer, people who'd always picked on me and ignored him. Popular people. Greg was laughing, his black hair shining in the sun like a crow's wings, his smile like the sun itself. I made myself walk past him, and even though he seemed to look right at me, he said nothing.\n\nAll week, I tried to catch Greg's eye, to find a way to talk to him, and all week, nothing. It wasn't like he was being actively mean. It was worse. It was like I was a stranger. He was just this boy, this suddenly popular, handsome boy at my school, and I was nobody he knew. It was like we'd never done all the things we'd done together, like he'd never been my friend, like I was some stupid girl with a crush on a stranger.\n\nThursday, I finally got up the courage to call him.\n\nHis father answered. \"Violet! Long time, no see. We have a new woodpecker.\"\n\n\"Oh, that's great!\" I smiled on purpose, hoping he could hear it in my voice. Maybe it would all be okay. \"Is Greg there?\"\n\nBut, when Greg got on the phone, I knew it wouldn't be okay. He sounded different, awkward, like someone wearing too-tight shoes but trying not to show it. \"So, um, what do you want, Violet?\"\n\nSuddenly I didn't know. I didn't know what I wanted. I wanted everything to be the same as it had been the year before. Or not the same. Better. There had been the promise of something more, and now, it was gone, and I wanted to change that. I wanted to change . . . time.\n\n\"Are you going to . . . ?\" _To what? Invite me over? Say hello in class? Do anything? Be normal?_ \"I haven't talked to you since you got back.\"\n\nA pause. In the living room, my mom was flipping through television channels. I heard the _Family Ties_ theme start. I bet we've been together for a million years; And I bet we'll be together for a million more.\n\nMom changed the channel, and _Entertainment Tonight_ came on instead.\n\n\"Yeah, about that,\" Greg said. \"The thing is, my friends don't like you.\"\n\n\"Your friends? What friends?\"\n\n\"Jennifer Sadler and them. They say . . . you're bossy and mean to them.\"\n\nI sucked my breath in. Unreal. Popular people always had some reason why you deserved for them to be mean to you. \"And you believe that? That I could boss Jennifer around, that I bully _her_? __ All you have to do is look at the two of us to know . . .\"\n\nI stopped. That was the problem. He had looked at us.\n\n\"I don't know what to believe. I just want to have friends. I don't want to be alone.\"\n\n\"You had a friend. You weren't alone.\"\n\n\"You know what I mean. I've always had a thing for Jennifer.\"\n\nI didn't know that. Why was he doing this to me? Why did Jennifer even want to be friends with Greg? But I knew. Because he was beautiful. And beautiful guys were catnip for Jennifer. A beautiful guy like Greg didn't belong with an ugly like me. The second she showed interest in him, he knew. I felt so stupid for thinking anyone would ever really like me, especially someone as great as Greg.\n\n\"I'm tired of people thinking I'm weird,\" he said.\n\n\"I thought you were weird, in a good way. I liked you the way you were.\"\n\n\"I don't want that anymore.\"\n\nI stood, clutching the phone. I didn't want to put it back in its cradle. The click it would make would change everything. I felt like, if I just held on, I could hold on to my life. But Greg said he had to do homework. I hung up before he could hear me crying.\n\nThe next day, in civics class, Mrs. Davis assigned a group project, an ad for a mock presidential candidate. I tried to make eye contact with Greg, my usual partner, but he turned toward Jennifer Sadler.\n\n\"Does anyone not have a group?\" Mrs. Davis asked.\n\nI raised my hand, barely flipping up the fingertips, glancing around to see if anyone else raised theirs. No one. So humiliating. Mrs. Davis asked if anyone only had two in their group.\n\n\"Yeah, us.\" Nick gestured to himself and Nathan.\n\n\"Okay, you can join them,\" Mrs. Davis told me.\n\n\"Great,\" Nathan muttered when I started over to them\u2014as if I was going to be the liability in their group when everyone knew I was the smartest person in class and they were just about the dumbest.\n\n\"I can just do the whole thing,\" I said. \"It's easier.\"\n\n\"Will you say we helped?\" Nathan asked.\n\nI looked over at Greg, who sat deep in conversation with StupidGennifer and StupiderJennifer.\n\n\"Depends. Will you refrain from being complete jerks for the duration of this project?\"\n\nBlank stares. I tried to figure out which word they hadn't understood.\n\nI revised. \"Will you be nice?\" All single syllables.\n\nThey both nodded.\n\n\"Fine then.\"\n\nI glanced at Greg, but he wasn't looking. Beside him, Jennifer mouthed, _Ugly._\n\nI realized what I had known, probably all along, what ugly girls since the beginning of time had been trying to deny: Beauty was all that mattered. I might tell myself that if people _really_ knew me, they'd look past my weak chin and non-eyelashes, would see into my soul and like me despite it all. But, watching Greg giggle with Jennifer and Gennifer, I knew that was not the case. Greg Columbo had looked into my soul\u2014but he still couldn't see past my nose. And, if he couldn't, for sure no one else could. I was disgusted at myself for liking him.\n\nBut I still did.\n\nFor the next week, Nick, Nathan, and I worked on our project. Or rather, I worked on our project while Nick and Nathan read comic books under their desks.\n\nTuesday, I asked my mother to take me to the drugstore for supplies. \"I need a poster board, Sharpies, stencils, and one of those scissors with the cool-looking edge.\"\n\n\"Isn't this a group project? Can't someone else buy this stuff?\" She squinted at herself in the mirror, looking for age spots. It would be hard to pull her away.\n\n\"They're sort of worthless. You know how it is.\"\n\n\"Where's Greg been lately? He was always a nice kid.\"\n\n\"He changed over the summer. Can we go to Eckerd's now? You can look at makeup while I find this stuff.\"\n\n\"Like I'd buy makeup from the drugstore.\" Still, she started toward her purse, since it must have been obvious I wouldn't change my mind. \"Changed how?\"\n\n\"What?\" This was more interest than she usually took in my life.\n\n\"How did Greg change?\"\n\n\"Oh, I don't know. Got too handsome to hang with me.\" I faked a laugh.\n\nMy mother, of course, was beautiful. Not beautiful the way every kid thinks her mother is beautiful, but actually beautiful. I'd barely known my father. He died when I was little, leaving Mom with enough money that she never had to work, never had to remarry \"another old, rich guy,\" as she said. Mom had no photos of him she'd admit to, but he must've been really ugly because, for sure, I didn't get my looks from her side. She was tall, with the build of a dancer, blond hair the color of starlight, and eyes the exact shade of the Mediterranean Sea in photos of Greece. Her brows arched high, making her appear wide-eyed and innocent. Her lips were dark and pouty, the type I imagined boys wanted to kiss. No wonder she didn't know about some people having to do all the work on projects. I bet guys were falling all over themselves trying to do her homework for her when she was in school.\n\nAnother mom would have said something about looks not being important or that I'd get pretty one day when I was older. That she said neither proved that she didn't believe those things. Instead, she said, \"Oh, I guess that happens. Come on. Let's go.\"\n\nAs we started toward the door, I looked back into her mirror. She spent so much time in front of it, I half expected it to talk to her.\n\nOn Friday, I brought in my\/our project. It was perfect, better than a professional graphic designer would have done. I set it up in the front of the room, noting the peeling tape on Greg's group's poster, the shaky handwriting on another. Mine\u2014I mean, ours\u2014was the best in the class. I took my seat, imagining that even my rivals were stunned by its beauty.\n\nA lot of teachers, when we did group projects, handed out an evaluation form so students could grade themselves and their peers. The idea, I guess, was that if one person did all the work, he could rat out his partners. Like that would ever happen.\n\nI sometimes gave bad grades, though. I had nothing to lose socially since everyone already hated me. Now, I picked up the worksheet and contemplated it. The first part was easy: evaluating myself. I'd been a joy to work with, of course. Cooperativeness: A; completed assignments: A; creativity: A.\n\nI moved to the second section, where I was supposed to grade my partners. Of course, they deserved an F in every category. They'd done nothing. No, they'd done something. They'd left me alone. About that, they'd been completely cooperative.\n\nI penciled in their names and wrote, _Cooperativeness: A_.\n\nI'd promised, after all. I didn't like to lie, but they'd met every deadline because I hadn't given them any. And, as far as creativity went, I guessed they'd creatively managed to avoid work. I penciled in As for that too.\n\nWhen Mrs. Davis collected the papers, I handed in mine with a clear conscience. Group projects were stupid. Teachers said they were supposed to teach us how it worked in the real world. I already knew. The real world sucked.\n\nI saw Nick and Nathan hand in their papers, sort of smiling at each other. Of course, they were thrilled to have gotten away with doing nothing and still getting multiple As.\n\nWalking home that day, I saw Greg walking with Jennifer. I'd seen him walk with her before, but I'd told myself it was for the project. Now, the project was over, and he was still with her. Were they going to his house to look at the wrens?\n\nI couldn't breathe.\n\n#\n\n#\n\nMonday, Mrs. Davis handed out grades for the projects.\n\nOf course, we'd gotten As for neatness, artistry, and accuracy. But when I got to the peer-graded portion, I saw that Nick and Nathan had each given me Fs for the three categories of cooperativeness, completed assignments, and creativity. At the bottom of the page, Mrs. Davis had written: _Overall project grade: C._\n\nI felt my heart actually hammering against my ribs. This was not possible. I'd done all the work, even lied to give them As. That's what they'd been snickering about in class.\n\nI went up to Mrs. Davis's desk.\n\n\"Stay in your seat, Violet. We're supposed to be working on our chapter outlines.\"\n\n\"I did that last night. I have to talk to you about this.\" I thrust the grade report at her and pointed to the Fs with a shaking hand.\n\nShe didn't take it. \"Well, if you don't do the work on a group project\u2014\"\n\n\"I did the work. I did all the work.\" My voice sounded shrill, even in a whisper, and I felt like my face might crack. Everyone was looking up from their papers.\n\nMrs. Davis glanced around. \"Violet, there's really nothing\u2014\"\n\n\"They did nothing!\" I burst out. \"I did it all.\"\n\nShe sighed. \"What grade did you give them?\"\n\n\"I gave them As,\" I admitted, \"but that was because . . .\" I took a breath.\n\n\"Because what?\"\n\nI looked at Mrs. Davis. She was almost six feet, big and broad-shouldered, with curly red hair. I bet she'd been picked on in school.\n\n\"Because I promised them I would if they left me alone.\"\n\n\"Left you alone?\"\n\n\"I hate group projects,\" I whispered, knowing every eye was on me. \"People are mean to me because of . . . my looks. Nick and Nathan didn't want me in their group. They bullied me into doing everything.\" I knew if I said I was bullied (which was true), she'd take it seriously. The school worried about bullying. Or, rather, they worried about fights. Never mind if people slowly died inside, year after year.\n\n\"They said I had to do the whole thing,\" I continued. \"If you look at the handwriting, you can see it's all mine. And the project. I brought it in and set it up.\"\n\nShe took the paper from my hand. \"Okay, I'll look into it. Sit down now.\"\n\nA minute later, I saw Mrs. Davis looking at the project. Then she called Nick and Nathan up. After class, she called me back to her desk.\n\n\"I changed your grade and theirs. When I confronted the young men with the evidence that yours was the only handwriting on the poster\u2014not to mention that they'd fooled around during all the class time in which you worked on it\u2014they admitted that you had done all the work.\"\n\n\"Wow, thanks.\" I was sort of amazed she couldn't have figured that out without my telling her. \"So what happened to Nathan and Nick?\"\n\n\"I'm sorry.\" She looked down. \"I'm not allowed to discuss another student's discipline with you.\"\n\nWhich was how I knew they'd really gotten reamed.\n\nSure enough, when I got to language arts class, I heard one of the Jennifers saying she'd seen Nick and Nathan going into I.S.S. I smiled.\n\nThat afternoon, I was walking home, smiling at the knowledge that my grades were again perfect. I walked alone, trying not to notice Greg taking off in the opposite direction with Jennifer, probably heading to her house. It was near Halloween, and the air had gone from summer-hot to chilly. A gust of wind swept up the empty street, and I shivered.\n\nThen, suddenly, I heard footsteps behind me.\n\nAt first, they were distant. I resisted looking back, though I wanted to see if it was maybe Greg. It wasn't Greg. It wouldn't matter if it was Greg. There was no Greg for me.\n\nThe pounding steps got closer. And harder. I could tell now there were two pairs of feet. Boys' feet. Another gust practically knocked me over, sending leaves and dirt into my face. Usually, no one else walked this way, toward the outskirts of town. I sped up. As soon as I passed Salem Court, I knew they'd part from me. They had to be going there. I matched my step to the rhythm of theirs. Yet they grew closer. My backpack was heavy, digging into my shoulders, slowing me down, and my sneakers cut into my heels.\n\nI, then they, passed Salem Court. They didn't turn. They were following me. That was the only explanation. No one was outside, no one to help me. There was one house, dark and lonely, with peeling, once-white paint. They said an old lady lived there, an old lady or a witch, but I'd never seen her. No kids, though.\n\nSomeone yelled, \"Hey, ugly!\"\n\nI turned to see who had shouted. Nick and Nathan. They broke out laughing. \"Look at that!\" Nathan yelled. \"She answers to ugly.\"\n\nThey were following me. And they were angry. I broke into a run. My sneakers were like blades, slicing into my heels, but I ran. I ran!\n\nAnd, behind me, I heard them running too. Something hit the side of my head, hard. A rock. It stung, and I dropped my backpack to run faster, dropped it even though I knew they'd take it, knew they'd steal my books and scatter my papers to the wind. I ran as fast as I could.\n\nAnother rock hit me. \"Stone the ugly witch!\" And they were on me, pushing me to the pitted pavement, slamming my head to the hard ground. Their fists rained on me, on my face, into my stomach. I couldn't breathe. I couldn't fight them. The world should have gone black, almost did go black with the blows to my face, but instead, I stared upward at the blue sky. A bird, a black crow or maybe a grackle, sat on one bare tree branch. _Help me,_ I thought.\n\nStrangely, I remembered that day in grade school, the day I'd first spoken to Greg. _Help me._ Then, I was floating, no longer inside my body, but above it. I was the bird, perched high in the tree branches, waiting. I opened my beak and gave a mighty caw, spreading my wings and showing my black feathers to the sun. With my beaded eyes, I looked down at the girl, the ugly girl on the pavement, being beaten by two big boys. She looked tiny, shriveled. I cried out again.\n\nSuddenly I wasn't alone anymore. And the sky was no longer blue. It was black with the wings of dozens, no hundreds of birds, blackbirds, grackles, crows, ravens, even larger birds, birds I'd never seen before, lunging and diving below, pecking at my attackers, at their faces, their eyes, not stopping even as the boys ceased beating me and began to beat at the birds. Their beaks pecked the boys' hands, their arms, drawing blood, and I watched from my tree branch, spreading my wings in joy.\n\nFinally, the boys stumbled up and ran, the birds pursuing them down the street. Only one remained, a single crow, glossy wings reflecting the light in purple and green.\n\nI watched from above. _I_ was a bird. Then, I was a girl again, a small girl. In my body, on the ground. I gathered myself up. I felt no pain. I stood and walked over to get my forgotten backpack. The crow stood, unmoving, as if it had something to say to me in some secret crow language. Still, I walked around it, gingerly, carefully. I picked up my backpack. The street was again deserted. Nick and Nathan were truly gone. I wondered if I looked like I'd been beaten. I ran my fingers through my hair. Even though I was ugly, I hated to be messy. Why make it worse than I already was? My mother had taught me better. Finishing that, I trudged toward home.\n\n\"Hello?\" A voice came out of nowhere.\n\n#\n\n#\n\nI started. It was a woman, standing as if she'd always been there. Yet, I hadn't seen her before. The streets had been quiet, empty.\n\n\"Hello?\" she repeated. \"Are you all right? Did they hurt you?\"\n\nSo she had seen? But how? There had been no one there. No one! The boys would never have thrown rocks at me with witnesses. They were dumb but smarter than that. Bullies always knew how to hide it.\n\nFinally, I spoke. \"I'm . . . I'm fine.\"\n\n\"I see that.\" She stepped closer. I noticed the crow was gone. It had been exactly where the woman now stood. \"You don't have a scratch on you. How is that . . . Violet?\"\n\nA chill wind rippled through the trees. \"How did you know my name?\"\n\nShe shrugged. \"Lucky guess, I suppose. Was I right?\"\n\n\"You know you were.\"\n\nShe smiled. She had long, black hair and wore a dress of sheer, iridescent material, first black, then purple, now green, flowing around her. Her hair caught the strained sun and seemed to do the same. I couldn't determine her age. She was beautiful. \"You look like a Violet, I suppose.\"\n\n\"No, I don't.\" First off, no one was named Violet. If you wanted to guess the name of a girl at my school, you might choose right with Jennifer, Kathy, Lisa, or Michelle. But I was the only Violet. \"Violets are pretty, with their little faces turned to the sun, hopeful. I'm not pretty. I'm not hopeful either.\"\n\nShe walked closer. Her black hair blew around her face. \"You could be anything you want to be.\"\n\nI laughed. That sounded like something a mom would say. Anyone's mom but mine. \"I can only be what I am.\"\n\n\"Sometimes, what you are is more than enough. How did you get those birds to come?\"\n\n\"They just showed up.\"\n\n\"Pretty convenient, wouldn't you say? Ever hear of birds attacking anyone like that?\"\n\n\"In a movie once.\"\n\n\"You won't hear about this time either. The boys will consider telling their parents, but, eventually, will decide it makes them sound guilty. Or crazy.\"\n\nThere were no cars anywhere. We were alone. Her eyes were a strange bright green, like a Sprite bottle.\n\n\"How about you?\" she asked. \"Have you had any other experiences with birds?\"\n\n\"Who are you?\"\n\n\"Oh, I'm sorry. I'm Kendra. I live here.\" She pointed at the house on the hill, the one I was sure was abandoned. \"I'd ask you in, but, of course, your mother would disapprove of your coming inside a stranger's house.\"\n\nNow her eyes seemed brown.\n\n\"Actually, I doubt she'd care.\" I knew the second I said it that it was the wrong thing to say. What if she was a kidnapper or something? But it just popped out. Besides, I'd never heard of a woman kidnapper.\n\n\"Ah, so she knows.\"\n\n\"Knows what?\"\n\n\"That you can take care of yourself. And, of course, she's right. You can. Self-sufficiency is one of the few benefits of being lonely.\"\n\n\"How did you know . . . ?\" I stopped. I was going to ask how she knew I was lonely, but of course, I knew: Anyone as ugly as I was would be lonely.\n\nI started to walk away, then turned back.\n\n\"Once, I saved a bird.\"\n\nI expected her to react with surprise or, at least, interest. Social interaction wasn't a huge thing with me, but I thought the normal thing to say was, \"You did?\" or \"How did you save it?\" Instead, she just nodded, as if unsurprised.\n\nMaybe I should have walked away. Yet I always felt I owed people an explanation, so I told her the story. \"I guess I didn't really save it,\" I concluded. \"It just sort of felt like\u2014\"\n\n\"No, you did save it,\" she said.\n\n\"How do you know?\"\n\n\"I was there. You were ten years old.\"\n\nThe hairs on my arms stood on end. How was that possible? I remembered the bird, the one I'd seen watching me. Then, the birds today, attacking Nathan and Nick. Were they related? Was it possible that the bird that day somehow remembered me, had told the other birds?\n\n___Crazy._\n\nBut the woman\u2014Kendra\u2014repeated, \"I was there.\" Then she waved her hand in the air and disappeared. Where had she gone? A crow cawed from somewhere. I looked down. It was right where Kendra had been standing. Then, in a heartbeat, the crow was gone, and Kendra was back.\n\n\"How did you do that?\" I felt breathless.\n\nI knew the answer, though. Magic.\n\n\"There are people in this world who have powers, Violet.\" The sun was already beginning to set, streaking the sky behind her in strange shades of purple and orange. \"I am one of those people. And so are you.\"\n\n\"That's crazy. If I have powers, why can't I . . . ?\" I stopped, unsure, for once, how to express the thought: Why did everyone hate me, if I was so powerful?\n\n\"Think of what you know of witches. Does anything you have heard or read lead you to believe they are universally beloved?\"\n\n_Witches._ I turned the word over and over in my head, not understanding at first, as if it was a foreign language.\n\nI thought about witches in books, the old woman in the gingerbread house, the green-faced crone in _Wizard of Oz._ But they were evil. The gingerbread witch had tried to bake Hansel and Gretel. The Witch of the West had captured Dorothy with an army of flying monkeys.\n\nKendra said, \"I was ten when I first started noticing my own powers. But nothing big happened until I was thirteen. How old are you now, Violet?\"\n\n_Thirteen._ But I didn't, couldn't say it. I stepped back. If this woman was a witch, would she try to kidnap me? Bake me? Hold me hostage? I wanted to turn and flee. Yet my feet felt suctioned to the ground.\n\nAnd I wanted to stay. I needed to hear what else she had to say. Was she saying that she was a witch? Or that I was?\n\nKendra chuckled, not a high witch cackle, but a low sound from the bottom of her throat. \"You think all witches are evil, yes?\"\n\nI didn't know what to say. If she was a witch, I didn't want to insult her\u2014especially if she _was_ evil. I noticed the street I'd just passed, Salem Court, named for a place where women were hanged as witches. In school, we'd learned they weren't really witches. Now, I wondered.\n\n\"I've only read about witches in books. Are they real?\" I said to the woman who'd changed from a human to a bird. \"Are they evil?\"\n\n\"Witches are all different, just like everyone else. Some are nice, some not. What we all are, however, is lonely.\"\n\n_Lonely._ The word washed over me like summer rain.\n\n\"But occasionally,\" Kendra continued, \"I meet a kindred spirit, and when I do, I keep my eye on that person.\"\n\n\"Keep your eye on me? Have you been spying on me?\"\n\nI expected her to deny that, but instead, she nodded. \"Since that day at the playground, I've watched you.\"\n\n\"As a bird?\"\n\n\"Or with this.\" From the air, she produced a shining object, a mirror, surrounded by silver curlicues. She held it out. I cringed at my ugly face.\n\nBut she said, \"This is a magical mirror. With it, you can see anything, anyone.\"\n\n\"How?\" I reached for the mirror.\n\n\"Just ask.\"\n\n\"Ask.\" I had a hard enough time talking to people. What would I say to a mirror? They'd always been my enemies.\n\n\"Think of someone, anyone in the world you want to see, and the mirror will show you.\"\n\nWithout hesitating, I said, \"Show me Greg.\"\n\nMy hideous face faded from view. The image changed to a room, a house I didn't know. Greg sat with Jennifer, books spread out before them, studying. Or, at least, Greg was trying to study. Jennifer was babbling on. I searched Greg's face for signs of annoyance. Greg took studying seriously, like I did. He'd once threatened not to study with me when all I'd done was ask if he wanted a glass of water!\n\nBut now, Greg smiled, then laughed\u2014laughed!\u2014at something Jennifer had said. He pointed at the math book just as Jennifer was trying to turn a page. Their hands touched. Greg turned away, blushing. He always blushed. I knew why he didn't mind Jennifer's chatter. Jennifer was beautiful, unlike me. Jennifer was everything I wasn't. The light gleamed off her blond, straight hair. I could feel my own frizz curling on my neck. Jennifer turned her fair, unblemished cheek toward Greg, and I could feel the hurt of the zits on my own cheek. Greg leaned toward her and then . . .\n\nHe kissed her! Greg actually kissed Jennifer!\n\nThe mirror fell from my hand and clattered to the ground. It shattered like ice against the black pavement. I jumped when a shard cut my ankle. \"Oh!\"\n\nI knelt down in the splintered glass, not caring if the bits and pieces embedded themselves in my hands and knees. \"I'm sorry. I'm so sorry. Oh, but I hate mirrors. I didn't do this on purpose, though.\" I picked up one of the larger fragments. It caught the waning sun, reflecting it into my eyes. I saw my own face in it, red, blotchy, a tear dripping from my too-light blue eyes.\n\nAnd then, the fragment moved. I almost dropped it a second time, but I reached over with my other hand and caught it. The moving fragment sliced my palm. I wanted to cry out, but I didn't, couldn't speak, for at that moment, I saw what was happening.\n\nThe fragment, _all_ the fragments, moved in the air like shimmering leaves, catching the light. As they fluttered together, my slice of glass slid from my hands. It joined the others, forming a silver oval. Then it flew into Kendra's hand.\n\nI felt my mouth hanging open. I closed it. \"How . . . ?\" My finger had a heartbeat.\n\nKendra laid her hand upon mine. \"Think about it. I'm sure you can work it out.\"\n\nI flinched under her touch, but somehow, I couldn't remove my hand. When she finally pulled hers away, the cut had disappeared.\n\n\"What did you do?\"\n\n\"Shh. People use too many words nowadays, always talking on the phone, in person. There is a place in the world, I believe, for thought.\"\n\nI started to say something else, but I couldn't. It was almost as if someone was covering my mouth.\n\n\"Silent thought. One minute. Begin now.\"\n\nI didn't want to think. I wanted to run. And yet, I couldn't because, more than that, I wanted to know. The mirror. The cut. The birds. Kendra was telling the truth about being a witch. And if she was telling the truth about herself, was it true about me too?\n\nWhat did it mean?\n\nI stared at Kendra. Her eyes looked green again. Finally, she said, \"Are you willing to speak now?\"\n\nI tried to put it into words. \"If I'm a witch, can I make things . . . ?\" What was the word I wanted?\n\n\"Better? Maybe. Different? Yes.\"\n\n\"Happen. Can I make things happen?\"\n\n\"Depends what you're asking for. World peace? End to hunger? Because, no, you can't do those things. No one has enough power for that.\"\n\nBefore I could stop myself, I blurted, \"I want to be beautiful. Can that happen?\"\n\nIt sounded so bare, out in the open like that, out in the empty street. Yet, it was the only thing I wanted, had ever wanted. Well, that and Greg. I knew beauty was nothing. But it was also everything.\n\n\"Yes, yes, I can do that. Or, rather, you can.\"\n\n\"How?\"\n\nShe stared into the distance. The street was silent, no cars, no people, only that wind that picked up the dead leaves and whirled them around, finally sending them skittering away like so many winged insects.\n\nEventually, she said, \"Not all at once. Changing things too quickly is how one gets discovered. But slow changes are fine. I've found that most people are stupid and unobservant.\"\n\n\"So how\u2014?\"\n\n\"Come here tomorrow.\" She pointed at the boarded-up house. Again, I remembered my mother\u2014or probably someone else's mother\u2014saying not to go with strangers. And yet I knew my mother wouldn't mind. If this woman could make me beautiful, she'd think it was a risk worth taking.\n\nI said, \"Can I see the mirror again?\"\n\nShe drew it back out from the folds of her gown. Her eyes were brown now. \"Don't break it.\"\n\n\"I won't.\"\n\nI grasped it and brought it up to my face. I studied myself, crooked nose, freckles, frizzy hair, everything.\n\n\"Can I . . . can you change _one_ thing now?\"\n\nShe smiled. \"Something small?\"\n\n\"Something big. My nose. Can you make it smaller or, at least, not have a bump on it?\"\n\nShe laughed. \"Funny how society stereotypes witches as having long noses. In fact, it's the first thing most witches would change.\"\n\nI noticed her nose. It was adorable, tiny, and turned up.\n\n\"Very well,\" she said. \"Close your eyes. It will only be a moment.\"\n\nI closed them. Around me, I heard the wind pick up, felt the dirt and rocks pelting my ankles. I wondered what she would do to my nose and, for the first time, I wondered what she _could_ do. Make me even uglier? How could I trust someone I'd never seen before?\n\nAnd yet, I knew I had nothing to lose. Still, I held my arms around myself, shivering in anticipation and maybe fear.\n\nA moment later, she said, \"Okay.\"\n\nI opened my eyes. She was holding the mirror toward me. I stared at it.\n\nIt was my face, still my face, ugly, pale, blotchy, chinless. I still had no eyelashes and horrible hair. No one would notice the difference.\n\nBut there was a difference. The bump on my nose was gone.\n\n\"Oh.\" I turned sideways to admire it. \"Oh, thank you. _Thank_ you.\"\n\n\"Now do you trust me enough to come back tomorrow?\"\n\nI nodded. I still held the mirror in my hand, not wanting to stop looking at it. Finally, I handed it back to Kendra.\n\nShe smiled. \"Power can be a wonderful thing, Violet, a wonderful, terrible thing.\"\n\nI was still thinking about the mirror, about me, my face. I wanted to ask her how it could be terrible. But, when I looked up, she was gone.\n\nI thought about power. A chill ripped through me.\n\n#\n\n#\n\nWhen I found Mom, she was plucking her eyebrows. \"Hello, Mommy.\"\n\nShe barely looked up. \"Hello.\"\n\nI just stood there. I wanted to see if she'd notice a difference. But she still didn't look. She plucked one hair, then searched for her next victim. Just when I thought she'd forgotten I was there, she said, \"Violet?\"\n\n\"Yes?\"\n\nShe plucked another hair, still not looking up.\n\n\"You'd tell me if I had hairs on my chin, wouldn't you?\"\n\nI had to smile. My mother would never, in a million years, have a hair on her chin. A chin hair would be a flaw, and my mother was flawless. She didn't even have freckles. Not one.\n\nI must have taken too long to answer because she said, \"Oh, my GOD! I already have them, don't I? You've noticed, and you haven't told me. Violet, what is the point of even having a daughter if she doesn't tell you about your chin hairs?\"\n\nShe abandoned her eyebrows and started searching her chin, positioning it in front of the mirror and rolling her eyes down to try and look. Well, that just summed up our relationship. I wanted to let her suffer, but I also wanted her to look at me, instead of her chin, so I said, \"No, Mom, of course you don't have chin hairs . . . that I've noticed anyway.\"\n\nI had to add that last.\n\n\"Are you sure? Can you look? Because I was talking to Marge Holcomb today, and since she's so tall, I was looking right at her chin, and you'll never guess what I saw there.\"\n\n\"A chin hair?\"\n\n\"Haha. No. _Three_ chin hairs. Three! That was all I could look at. I felt terrible for her.\" She laughed, the sound of a breaking mirror. \"I would die if anyone looked at me and saw chin hairs.\" She laughed again.\n\n\"You're lucky you're not that tall.\"\n\n\"I am . . . not.\" Still, she searched her chin. \"That's not the point, Violet. The point is, I shouldn't have any. Do you know how hard it is to see your chin? I hate getting old.\"\n\nShe still hadn't looked at my face. Unbelievable. \"I don't think you have any. Want me to look?\"\n\n\"Would you?\" She broke into a smile, and like when I was a kid, it made me happy, so happy. My mother was smiling at me.\n\nWhen I was little, I used to watch her get dressed to go on dates. She had the most beautiful clothes, nothing like other mothers. Silk blouses in jewel colors and strapless gowns like Vanna White on _Wheel of Fortune._ After she left, when the babysitter thought I was in bed, I'd sneak into my mother's closet and try everything on, clothes, shoes, jewelry, makeup, always arranging it back very carefully as if it were a booby trap she'd set to catch me in the crime of pretending to be her.\n\nSometimes, when she was home, I'd ask to try the things she used, the powders and creams that widened her eyes, blackened her lashes, and made her so pretty. I thought if I could look like her, she would love me, and maybe she would have\u2014if I'd looked like her.\n\nBut I never did.\n\nOne thing she loved to do was watch pageants, Miss America, Miss USA, Miss Universe, Miss World. The girls in those pageants competed for scholarships, but their sparkly dresses and big hair probably cost more than they'd ever win. The contestants danced sexily onstage and then talked about ending world hunger. Some of the girls probably weren't even pretty before they applied all the hair spray, false eyelashes, and goo, and they did stupid talents like ventriloquism and hula dancing.\n\nAnd yet, every time they crowned a new queen, the camera panned the audience, found their families. I knew those moms loved their daughters, loved them enough to take out a second mortgage to buy a case of hair spray. Which was more than my mother loved me.\n\nSo I was cherishing this mother-daughter moment even if it did involve hunting for chin hairs. \"Move into the light where I can see you better. Maybe by the window.\"\n\n_Where you can see me._\n\n\"Oh, okay. Let me put on my contacts first. I always take them off when I pluck, to help me see close. But then I can't see anything. I can't even see your face.\"\n\n\"Sure.\" I handed her the case. I would have stuffed the lenses into her eyes myself had it meant it would happen faster. Could she tell?\n\nFinally, they were in. She blinked at me. \"Okay, then, where did you want me?\"\n\n___In my corner._ She still didn't seem to notice any difference. Was I so ugly that a little improvement didn't help? No, she probably hadn't adjusted to the contacts. \"Come here.\"\n\nThe sun was close to setting, and the western-exposed window filled the room with strained light. I stood close. \"Tilt your chin up.\" She'd be looking right at my nose in that position.\n\nShe obeyed. I held the tweezers, searching for hairs. None there. My perfect mother with a hair on her chinny-chin-chin? Impossible. But I pretended to search, waving the tweezers near her face, wondering if I should pretend to see something.\n\n\"Anything?\" Her voice sounded breathless.\n\n\"I'm . . . not . . . sure.\" _Look at me._\n\n\"Violet, you must be very certain. I have a date tomorrow, someone new, someone rich. I can't have . . . a hair.\"\n\nNo reaction to my new and improved nose. But, of course, she was too fixated on herself, her wonderful self, as she always was. Maybe if I found a hair, she'd look at me.\n\n\"Oh, here it is!\" I inched the tweezers up to one downy, blond, regular peach-fuzz hair. I grabbed it. \"Got it!\"\n\nA moan escaped her lips, and with that sound, I gave the skinny hair a mighty yank. My mother gasped.\n\n\"There!\" I held the empty tweezers in triumph.\n\n\"You're sure that's all?\" she asked.\n\n\"Positive.\" _Look at me. Look at me!_\n\nBut she turned away. I wanted to say something, anything else. Ask for help with my homework. Ask her to make me cookies? Hardly. Ask if she thought I was pretty. Ha. I knew the answer to that one without asking. I never would be.\n\nUnless . . .\n\n\"Okay, Mommy, I have lots of homework.\"\n\nI did have homework, but for once, I let it slide. I could do it at school. One B was hardly going to wreck my average.\n\nHad I just thought that? I'd gone ballistic about the group project, about anything less than an A.\n\nBut the difference was, then, I'd thought it mattered. Now, I knew it didn't. Middle school grades didn't matter. Grades didn't matter. Or how smart you were. Or what college you got into. Nothing mattered if you didn't have the one thing that did: beauty. I was going to get it, and I wasn't going to wait.\n\nI ran to my room and looked for a mirror. There was none over my vanity. I hadn't wanted one. Now, I did. I needed the truth of my ugliness, laid out before me, to see my work. I had no compacts, no powders or blushers, nothing with a mirror in it. Finally, I spotted the little jewelry box my grandfather had given me. It played music when opened, a plastic ballerina whirling round and round, the mirror behind her reflecting her every move. I broke the ballerina off and threw her aside, leaning on the spring that had connected her to the box. It kept spinning even after the dancer was removed. I tried to examine my face.\n\nI couldn't see much in the tiny mirror, just my nose and my nonexistent lashes. How I hated my lashes! I tried to think back on the conversation with the witch, the conversation and what had happened before it. The boys. They'd attacked me, and suddenly, magical help had come. I remembered the other time the magic had come. The broken bird. What did those two experiences have in common? In both cases, I'd really wanted something to happen. Was that it? Was it enough? No. If merely wanting something to happen was enough, I'd be beautiful already.\n\nAnd my mother would have a beard like Santa Claus!\n\nNo, it had to be more than wanting. There had to be some kick in the butt, something to jump-start the magic like cables on a car battery.\n\nThe anger. That would explain it. Maybe longing and need too. Ordinarily, I was what people in books called mild-mannered, accepting as an ugly girl needed to be. Pretty girls could have fits of pique, but girls who looked at me should be nice. Usually, I was.\n\nBut today, I hadn't been. How could I have? Nor that day with the bird. I loved birds, and I hated the kids who would harm one. Someone who hurt an innocent little bird would hurt me, or anyone.\n\n_Hate._\n\nI stared into the mirror as best I could, patted the spring that had now stopped wavering in the tiny gold frame. I hated my face, my eyelashes especially.\n\nThey weren't hard to hate, sparse and pathetic, almost invisible. I'd told myself that my ugliness made me stronger, a survivor. Now I knew I'd lied. Ugliness wasn't power. Beauty was. Those pageant girls knew it. So did the network execs who only hired beautiful airheads to do the news. They were making us stupid as a society. And Greg, my poor Greg, forgetting his best friends because of the beautiful, evil Jennifer. Beauty was power, and I hated it, hated Jennifer and Nick and Nathan, hated all of them, everyone. I wanted the power to hurt them.\n\nI felt a strange tingling in my face. I expected to see that my eyelashes had grown instantaneously, but they hadn't. They were still short, still whitish. I squeezed my eyes shut against their hideousness, and I felt the room begin to whirl. The music box started playing _Swan Lake,_ the story of a lovely swan. I remembered _The Ugly Duckling._ Me. I wanted to be the swan. The room spun around me, and when I opened my eyes, the colors kept swirling. I couldn't see my face. I felt like I couldn't breathe. I dropped the jewelry box. It clattered to the floor, still playing, spilling its contents. I felt myself falling too. Then, everything went black.\n\nI woke, sprawled on the mint-green Berber carpet. I pushed myself up.\n\nI started to stand when I noticed the music box lying open on its side, on the ground. It was just out of my reach, but I crawled toward it, expecting nothing when I looked in its mirror. Sitting up, I held the box close to my face, peered into the mirror.\n\nThe eyes that met mine were unrecognizable. Or rather, I did recognize them. They were my mother's eyes, deep blue, fringed with black. Whose eyes were these?\n\nI struggled up. The clock said 7:30. Light streamed through the window. Time for school. I ran to the bathroom and peered into the sparkling, silver mirror above the sink.\n\nIt was still me. At least, most of it was me. Weak chin\u2014check. Thin, fuzzy hair\u2014still awful. But the eyelashes were the Maybelline-commercial lashes I'd wished for. The eyes themselves weren't really that different. The lashes just made them look that way. But, it occurred to me, I could change them too.\n\nI could change them. I had the power to change anything.\n\nAnything.\n\nI could be the most beautiful woman in the world.\n\n#\n\n#\n\nWhat I learned the next day in school was this: No one looked at me. Maybe they looked when I was younger, long enough to realize I was beneath their notice. But then, they stopped. So they didn't notice my eyelashes or even my nose.\n\nKnowing this would've been a big time-saver, had I realized it earlier. I thought of all the hours I'd spent washing, brushing, and blow-drying my hair, covering zits in tinted Clearasil, or choosing outfits that wouldn't get made fun of. Had I known all along that I was invisible, I could have taken up a hobby. Or discovered a cure for cancer.\n\nEven Greg didn't notice the change in me. In fact, he didn't look at me at all. Why did I care so much? Why did I still want him?\n\nAt least _I_ was enthralled by my new look. I excused myself to look in the girls' room during each class, and I went in between too, seeing the same people, the druggies, cutters, and truants each time. The third time, I saw this girl, Molly, a brunette I'd known since kindergarten. She squinted at me through what I was pretty sure was a pot-induced haze and said, \"Don't I know you?\"\n\n\"Yeah, I'm Violet. We went to Coral Ridge together.\"\n\nHer eyes flickered with recognition. \"Ohhhh, I remember. You were the freaky chick who picked up the dead bird.\"\n\n\"It wasn't dead. People just thought\u2014\"\n\n\"Noooo, it was dead. You made it come back to life. Poof!\" She made a sort of _abracadabra_ gesture with her hands. \"But when I told people that, they said I was on drugs.\"\n\n\"You probably were. How could I do that. I'd have to be\u2014\"\n\n\"A witch. I told them you were, but they didn't believe me. They said just because someone has a bad nose doesn't mean they're a witch.\"\n\n_Out of the mouths of potheads . . ._\n\n\"Hmm, I see.\" I turned toward the mirror. Even though she was acting suspicious, I couldn't resist looking at myself.\n\n\"By the way,\" Molly said, \"you look really good. The first time you came in, I almost didn't recognize you. You get a nose job or something?\"\n\nI nodded, watching myself in the mirror. I realized that, up until now, I'd wondered if the whole thing was my imagination. \"Yeah. Yeah, I did. Thank you. Gotta get back to class.\"\n\n_It was real! It wasn't wishful thinking! It was real!_\n\nAfter school, I pushed through the crowds, past Nick and Nathan, who avoided me after yesterday. Once I was out of sight, I broke into a run. In minutes, I was walking up a driveway overgrown with roots and weeds breaking through the asphalt. The house's steps were covered in weeds too. Did Kendra really live here? I peered through the grimy window. Inside was dark, abandoned. Had it all been my imagination?\n\nI remembered Molly's comments, the boys avoiding me. No. I hadn't imagined it.\n\nI reached for the cobweb-crusted door, avoiding the dead leaves and insect skeletons. I knocked softly. The sound seemed to reverberate around the empty porch. No footsteps, so I was startled by the door opening.\n\nThe girl\u2014because that's what she was\u2014looked about my age with short hair that was actually purple. She wore a long, black gown that looked a hundred years old, and bizarre makeup with black sparkles highlighting dark eyes. She looked like a stunning vampire or something, a creature of the night. Was this even Kendra?\n\nBut she said, \"Come in, Violet.\"\n\n\"Is it . . . Kendra?\"\n\n\"Oh, I forgot.\" With a wave of her hand, she transformed into the older, dark-haired woman from yesterday, then back. \"Everyone likes a change sometimes. Yesterday, I took a form that would make you comfortable, a middle-aged woman. But really, I can be anything I want.\"\n\nI stepped inside. It was like the Palace of Versailles, marble floor and glowing chandeliers, all brand-new. She led me to a little sofa covered in tapestry fabric. How had she done it all? Witchcraft, of course. Could I do it too?\n\nMy heart whispered that I could.\n\n\"I see you've been experimenting.\" She focused on my eyes.\n\n\"What?\" I tried to look confused. \"No, I haven't.\"\n\nShe stared at me. \"Of course not. I must be mistaken. Your eyes have always looked like the second coming of Elizabeth Taylor.\"\n\n\"Who's that?\"\n\n\"Sorry, I date myself. She was an actress, lovely, violet-colored eyes, long, lustrous lashes. Some say she was the most beautiful woman on earth. Or the fairest one of all.\"\n\n\"The fairest . . . huh?\"\n\n\"Sorry. It's something we used to say when I was a girl. No one says that anymore.\"\n\n_The fairest one of all._ Man, would that be something. I'd have been happy just not being ugly, but the _fairest._\n\n\"Okay,\" I admitted. \"I guess I experimented with my lashes a little. I figured no one would notice, and no one did.\"\n\n\"No one?\"\n\n\"This one druggie girl. But my own mother didn't.\"\n\nKendra nodded. \"How did you do it?\"\n\nI shrugged. \"Don't you know?\" She'd probably been watching me in her mirror.\n\n\"I want you to tell me.\"\n\n\"Okay. I channeled my feelings, I guess.\"\n\n\"What do you mean?\"\n\n\"Once, when I was in fourth grade, two boys got into a fight in PE. Instead of getting mad at them, the coach told them to channel that anger into the softball game we were playing. Choke up on the bat, and hit real hard. And it actually helped. At least, one of them got a home run on his next at bat.\"\n\n\"And this has what to do with you?\"\n\n\"Just, that's what I did\u2014channeled all the anger, the rage over the way people treat me, at girls like Jennifer and Gennifer, who wouldn't even let me play kickball with them in grade school, at Greg, for not noticing how awful they are, or not caring, anyway, for ditching me. I put all that in my head and concentrated on . . . my eyelashes.\"\n\nShe laughed.\n\n\"It seems trivial, I guess.\" I knew it wasn't.\n\n\"No, eyelashes are very important.\" She wasn't kidding.\n\n\"Is that how it always works? Does magic come from rage and hatred? It did with Nick and Nathan too.\"\n\nBut not with the bird. How had that happened?\n\n\"Maybe hatred and rage, maybe fear, maybe even love, though I've never tried that one myself. It comes from a strong emotion that overwhelms you like an ocean's waves. For me, it began with desperation. When I was your age, a terrible plague swept through my town. My father died, mother too. Then, one by one, each brother and sister was taken from me until, finally, I had only one left, my youngest brother, Charlie. And Charlie lay dying in his bed.\"\n\n\"That must have been terrible. No one could help?\" I didn't have siblings, only my mother. But the thought of losing her was too terrible to bear.\n\nKendra had a faraway look in her eyes. \"No, no one could help. I was thirteen and alone. Half the town lay sick and dying. The rest mourned as I did. Every day, Mr. Howe, the gravedigger, brought his wheelbarrow down our street, asking if we had any dead to bring out, and one by one, my family left me.\"\n\nI shuddered. \"What did you do?\"\n\n\"I went to a woman, a healer in town. Her name was Lucinda, and she had been my friend, had told me that someday, I might be a healer like her. But she was gone too.\"\n\n\"Was she dead?\"\n\nKendra shook her head. \"She'd just disappeared. At that moment, I felt more emotion than ever before, emotions crowding inside me, crawling over one another, clamoring to get out like Pandora's box, anger, grief, desperation, loneliness, and they poured out of me and onto Charlie.\"\n\nI remembered the night before, the room spinning, my vision going purple.\n\n\"So the emotions were what triggered your magic?\"\n\n\"I didn't know at the time, but yes. When I woke the next day, Charlie was awake, alive. He was cured of his sickness as if nothing had been wrong. This had happened to no one else. Everyone who had sickened had died. There was only one reason this could have happened: me. I had cured him.\"\n\n\"Wow.\" It was incredible to think that such powers existed, that I could have them too. \"Wait. When was this?\"\n\nKendra hugged herself, her slim hands crushing down the black fabric of her dress. \"The year the plague struck England. 1666.\"\n\n\" _Sixteen_ sixty-six?\" It was impossible. \"So you're . . .\"\n\n\"Immortal, yes. All witches are.\" With a wave of her hand, she transformed again, this time into a young girl from another era, blond braids streaming down her back. She wore a long, blue dress with full sleeves and a red apron. \"It is a blessing, but a curse as well. One gets lonely. There are so few of us.\"\n\nFrom another room, I heard a clock ticking. \"So nothing can kill you . . . us?\"\n\n\"Nothing but the flame. I have managed to avoid it these three-hundred-odd years, sometimes just barely.\" She waved her hand and was herself again, at least, the self I'd seen before. \"Come now, let's work on making some magic.\"\n\nI wanted to. I especially hoped to be able to work magic without passing out. Kendra made it look so easy.\n\n\"So, my friend, what should we try next? Something small.\"\n\n\"Does it have to be small? People at school wouldn't even notice if I showed up six inches taller. I'm invisible to them.\"\n\n\"You'd be surprised what people notice.\"\n\nI thought of Molly from the bathroom. I nodded.\n\n\"And once people notice, they look for ways to use your magic against you. That's how people like my friend Lucinda disappear.\"\n\n\"But you said she was immortal, unless . . .\" I shuddered, picturing someone being burned at the stake, the wood piled high around her, flames lapping at her feet. Would the fact that she couldn't die except by burning mean she couldn't asphyxiate, that her heart couldn't burst, but rather, she would have to be burned alive, watching the skin peel off as painful first- and second-degree burns gave way to the blessed relief of death?\n\n\"No,\" Kendra said, \"she was not burned. She merely disappeared. Witches do. Now, let's find something suitably small. Your complexion, perhaps?\"\n\nI had a few blemishes, nothing terrible but, of course, I was self-conscious about them. I nodded.\n\n\"Hate drains the energy,\" she said, stroking my skin gently, like my mother never had. \"It's not a safe emotion. That's why you passed out last night. Is there another strong emotion you can tap into?\"\n\nMy first thought was love, my love for Greg. But that emotion was all tied up with other emotions, my hatred for Jennifer, my anger at Greg himself for ditching me when he suddenly became hot. I tried to think of happier memories, but they all failed me. I thought how I felt every day, how I felt alternately invisible, ignored, ridiculed. How I felt alone, even in my own house, with my own mother.\n\nI remembered what Kendra had said about witches being lonely.\n\n\"Would loneliness work?\"\n\nKendra smiled, barely turning up her lips. \"Yes, dear, I know it will. It is an emotion I use quite a bit myself.\" She reached toward me. Her hand was small and white, as if it had never seen sun. \"Close your eyes.\"\n\nShe passed her fingers down my forehead and across my face. I closed my eyes gladly. The bright light against white walls was suddenly tiring.\n\n\"Now remember . . .\" Her voice was soft, soothing. \"Remember the loneliest you've ever felt.\"\n\nSo many memories to choose from. The time my school had Lunch with Your Child Day, and my mom was the only one who didn't come. She hadn't had time, she said, but she sure had time for dumb things like hair appointments. No, it was because I wasn't presentable, wasn't pretty. She was ashamed to be seen with me.\n\nI'd thrown my sandwich in the garbage, feeling like the ugliest person in a world full of beauty.\n\nBut perhaps this memory edged too close to hate, to anger. I remembered other days, mundane things, walking home from school alone, remembered not having a partner on field trips and having to sit by someone else's mother who was chaperoning, party invitations handed out to everyone but me, no one talking to me except to ask me to please switch seats so their friend could sit there. I remembered . . .\n\n\"It worked.\" Kendra's voice interrupted my thoughts.\n\n\"What did?\" I realized I was weeping, hot, salty tears seeping out from under my eyelids.\n\n\"Look.\"\n\nI did, mopping at them as I looked. The mirror, Kendra's lovely mirror, was before me. I gazed into it. My skin was clear, unblemished, smooth, pink, like Mom's. And still, the tears kept coming, coming out of me.\n\nKendra's arms tightened around me. \"There, my darling. Someday, you will be able to do this without crying.\"\n\n\"Will I?\"\n\nShe stroked my back. \"I promise.\"\n\n\"Can everyone do it then? I mean, does everyone have the power to channel their emotions?\"\n\nI hoped not.\n\n\"Oh, no. Not everyone. It is a rare thing indeed. No, my darling. You are special.\"\n\n_Special._ No one had ever called me that before, not even teachers at school.\n\n\"Come, my darling. You have worked hard. Let me get you some gingerbread.\"\n\n\"Gingerbread?\"\n\nShe shrugged. \"I am sentimental. The witch who taught me, she made gingerbread.\"\n\nI remembered the story of _Hansel and Gretel_ , the children made into gingerbread. What if Kendra turned out to be like that witch, a cannibal bent on murder? Would I have the strength to run away from the one person who praised me and thought I was special? I stroked the smooth skin of my cheek, feeling Kendra's arm around me. I wasn't sure. I wanted Kendra to teach me. Desperately.\n\nBut the gingerbread she brought me wasn't shaped like children. In fact, it wasn't even a cookie, but a cake in a square pan. Kendra cut a still-steaming chunk for each of us and served it with glasses of cold milk. The hot gingerbread warmed my mouth and soon my tears were forgotten.\n\n\"What happened to the witch who taught you?\" I asked.\n\nKendra brushed some crumbs that had fallen onto the lace tablecloth. They vaporized instantly. \"Alas, she was burned.\" She looked down.\n\nI waited for her to elaborate, to explain, but she didn't. There was only the sound of our forks on china. \"I'm sorry,\" I said.\n\nShe shook her head and still didn't speak.\n\nFinally, she said, \"You should go home. Your mother will miss you.\"\n\nI doubted that, but I said, \"Can I come back tomorrow?\"\n\n\"Best to wait a little. Thursday, perhaps, so as not to excite suspicion. I will see you then, my dear.\"\n\nAnd suddenly, she wasn't there. The air felt chilly as, one by one, the objects in the room disappeared too, and I was all alone in the old, abandoned house.\n\nI touched my cheek.\n\nI wondered what else I could do.\n\n#\n\n#\n\nI stepped outside. The door slammed shut. The sound echoed down the silent street. I trudged through the weedy yard. The sun had been shining when I'd entered Kendra's yard. Now the clouds blocked any sign of it.\n\nOther than the brief, magical time when I'd had Greg, I had always been lonely. Yet the realization that Kendra was the only person _voluntarily_ to speak to me in months chilled me. What was wrong with me? It couldn't just be that I was ugly. Yet it had to be. What else could it be? It had to. If it wasn't about my appearance, then changing it wouldn't change anything. And I wanted to change everything.\n\nEverything.\n\nDown the block, I saw a lone, white cat playing by the roadside. I remembered hearing that most white cats were blind. Or was it deaf? The cat was scrawny, maybe a stray, and, suddenly, I wanted to pick it up, take it home with me. I'd never asked for a pet. Could my mother really say no? I walked faster, suddenly wanting the cat, hoping it didn't have a collar.\n\nSuddenly I heard a rumbling behind me. A car! I jumped, then ran under a tree, feeling the whoosh of air as the car sped by.\n\nMy heart was pounding. I screwed my eyes closed. Then, I heard a dull thud. My eyelids flew open. The cat! The cat, crushed under the wheels of some Mustang.\n\nI waited for the car to stop, but it roared on as if the driver hadn't noticed. Or just didn't care.\n\nThen, I was screaming, \"Stop! No!\" But the words were lost in the motor's roar, and the pounding of my footsteps on black pavement.\n\nThere was surprisingly little blood, only a bit coming from the kitty's mouth. Black tire marks marred its white coat. I held my hand to its chest, feeling for a heartbeat. There was one, but only faint. I knew it wouldn't last long.\n\nI gathered the cat in my arms, hating the driver. How could people be so uncaring? He didn't even stop.\n\nSomething, a jagged, broken bone, penetrated the cat's coat. The loneliness and sadness rose up in my throat like bile, then came spilling out of my mouth in words like vomit, words I didn't understand. I just sat there in the road, rocking the cat back and forth, saying I didn't know what, and suddenly, the pointy bone retreated inside its body. The cat's heartbeat quickened, and then its whole being began to vibrate.\n\nIt was purring! Purring and rubbing up against me! I knew I had fixed it, my magic and I had. If my magic only did one worthwhile thing, saving the cat was enough. More than enough.\n\nI picked up the cat and carried it home. To my mother's questioning look, I said, \"I found a cat. I'm going to keep it.\"\n\n\"Were you going to ask me?\"\n\n\"No. I'll take care of it.\"\n\nI stared at her, and maybe there was something in my eyes\u2014or my new eyelashes\u2014that made her say, \"Okay. We'll have to get cat food tomorrow, but I have some old tuna tonight.\"\n\nI fed the cat\u2014whom I named Grimalkin after a witch's cat in a book\u2014and took her to my room. She curled up on my bed and purred while I did my homework. She loved me already.\n\n\"The cat will have to stay outside while you're at school,\" my mother said the next morning. \"We don't have any litter. It will pee all over the place.\"\n\n\" _She_ will run away if I leave her outside.\" The cat had slept on my bed all night, between my legs, her purring lulling me to sleep. I'd awakened more rested than I'd ever been and had spent most of the morning admiring the cat's blue eyes and white whiskers. I had to keep her.\n\n\"If you feed it, it will stay.\"\n\n\"We don't have any cat food. Can't I just leave her in the bathroom?\"\n\n\"Violet, do you know what cat pee smells like?\"\n\nI didn't answer, assuming the question was rhetorical. In fact, I did know, and I hoped my mother wouldn't notice the smell on a pair of jeans that had been crumpled on my desk chair. I'd stuck them into the washing machine and planned to wash them after school.\n\n\"The cat can come in once we get litter, but for now, it will have to stay outside.\"\n\nI drew the cat into my arms. Most cats\u2014I knew from painful experience\u2014didn't like being picked up, but this one began to purr and rub her head against mine. \"I can't just throw Grimalkin outside!\"\n\n\"Uck, why give it such an ugly name? Call it something pretty like Tiffany or Courtney.\" My mother turned back toward her room. \"It has to go out.\"\n\nI went back to my own room. I considered skipping school, but there was a test in social studies, and I knew Mom would never write a note for me. Too much trouble. I shut the door, thinking perhaps I could hide Grimalkin inside. But she started to meow.\n\nFinally, I waited until Mom was engaged in the delicate contour drawing that was her morning beauty ritual, then I took the cat outside.\n\n\"Stay there.\" I placed her on the step.\n\nThe cat looked at me as if she understood. She blinked once, then again in the morning sun.\n\n\"Stay there,\" I repeated.\n\nShe lay down on the step, curling herself into a ball.\n\n\"Perfect,\" I said, even though I knew the cat would probably be gone when I got home, and the loneliness was like a paper cut on my heart. Why couldn't Mom just get some litter? Be a human for once? But I knew she wouldn't, so I started toward school.\n\nThe cat stood and followed me.\n\n\"No,\" I said. \"Stay.\"\n\nBut the cat kept following me, like Mary's lamb.\n\nI remembered what I'd learned about channeling my emotions. The cat could not leave. She couldn't. It was completely unreasonable for me to get all involved with a cat, but I had. My loneliness made me unreasonable.\n\nI picked the cat up and held her close, willing her to stay there, wanting her to. I was facing east, and the sun was already high and at the perfect angle to get in my eyes. I shut them, still feeling the warmth on my eyelids, on my cheeks, the purry warmness of the cat, who was not struggling in my arms. She nuzzled me, and I knew I couldn't go to school and leave her. I held the warm, fluffy, vibrating ball, rocking her like an infant. Behind my eyelids, I could see the sky changing colors, blue to red to purple and a burst of bright pink like fireworks. Stay. _Stay._\n\nThen, suddenly, I felt the cat's back feet digging into my stomach. She wanted to leave. You can't hold a cat who wants to go, and this cat did. I dropped her and opened my eyes.\n\nThe sun had gone behind a cloud, a cloud that hadn't been there before. I'd stood there longer than I'd realized.\n\nI looked at the cat. She was chasing a squirrel. It went up a tree that overhung our neighbor's property. Grimalkin followed it. I knew she would be lost. The squirrel would go further and further, and she'd go with it.\n\n\"Stay,\" I said.\n\nShe did not, of course, look back at me. She followed the squirrel across the branch, toward the neighbor's yard.\n\nBut when the squirrel entered the neighbor's yard, Grimalkin didn't follow. Instead, she stopped, as if realizing she could go no further. She looked insulted, as cats do. Then she turned and ran down the tree trunk and stood at my feet. She began to lick her right front paw.\n\nI glanced at my watch. Five minutes until school started. I had to leave, and I had to run. I gathered my books and lunch box, dropped in my trance. I patted Grimalkin's head, then started toward school.\n\nWhen I looked back, the cat was still sitting under the tree, unmoving.\n\nWhen I returned that afternoon, she was still there, waiting for me.\n\n#\n\n#\n\nI visited Kendra after school the following day and the day after that and every day for the next two weeks. She taught me things, magic things. While she cautioned me not to alter my appearance too drastically, I did alter it some, straightening my teeth so much that my orthodontist declared, with a shocked expression, that I no longer needed braces. I also gave my hair a wavy, just-out-of-the-salon perfection. Finally, people noticed. At least, my mother complimented me on finally taking an interest in my appearance. Mom was great at making a compliment sound like an insult.\n\nBut Greg didn't notice.\n\nOr, more likely, he didn't care.\n\nKendra taught me some party tricks, moving stuff with my mind like witches did on television shows like _Bewitched._ She didn't say anything about casting spells on other people, and I didn't ask. I knew she'd disapprove since she'd told me several times that my magic could backfire.\n\n\"Backfire how?\" We were at her house, which was now decorated in some sort of British colonial, dark furniture and stuff with elephants on it.\n\nKendra adjusted her linen blouse. \"Oh, sometimes the unexpected will happen, or you'll realize the magic you thought you wanted, you didn't want at all.\"\n\n\"All I want is to be pretty. And have friends.\"\n\n\"All?\" Kendra asked.\n\nWell, no. Not all. I wanted Greg to love me. And I wanted Jennifer to contract a bad case of leprosy, to help that along. But I didn't say that. Across the room, a brown spider crawled up a dining room table leg that looked like a lion's paw. I liked spiders. I never killed them. Unlike most people, I knew they weren't harmful, not usually. In fact, they killed mosquitoes and flies, bugs that spread disease.\n\nThe spider lifted first one leg, then another. Was it a brown recluse? Those could be harmful, resulting in skin death and terrible scarring. But even they didn't usually bite. People who got bitten by spiders brought it upon themselves. They weren't careful.\n\n\"Can you give me an example of backfiring?\" I wondered if I could get the spider to come toward me. Kendra had taught me many tricks, but since the day of the cat, I hadn't used my magic on another living creature. I wanted to. It would be cool to be able to manipulate others, like maybe make Jennifer scratch herself like a gorilla. But, of course, I'd have to be able to do it so no one knew it was me.\n\n\"So hard to think of just one example,\" Kendra said. \"I have been alive hundreds of years, and I seldom get the opportunity to tell my stories.\"\n\nI laughed. That was obvious. For each bit of magic I learned, there was at least an hour of talk. But Kendra's stories were fascinating. She looked my age, so she was like a friend. Yet she'd lived hundreds of years. Someday, I'd be as experienced and smart as she was. \"I'm sure it will be a great story.\"\n\nKendra leaned on an umbrella stand shaped like an elephant's foot and stared out the window into the waning light. \"I once knew a tsar who had twelve beautiful daughters.\"\n\n\"A tsar? So you lived in Russia?\"\n\n\"I've lived everywhere. But yes, this particular tsar was in Russia, and he had twelve daughters, each with so many suitors that the tsar could not decide\u2014which was a high-class problem to have. He planned to have a month of balls and events and invited them all to a huge house party\u2014or, rather, castle party. I was employed as a maid, so I knew of all the preparations.\"\n\n\"A maid? Why would someone with your powers want to be a maid?\"\n\n\"A maid is an easy job for someone with my powers. A blink of my eye, I can clean the silver, fluff a hundred comforters, and try on all the jewels in the house. But a maid is where the action is. As a maid, I traveled on the finest ships, including the great _Titanic_ , lived in palaces the world over. And people say things when the maid is in the room, secret things.\"\n\n\"So why aren't you a maid now?\"\n\n\"Alas, there are few palaces and even fewer kings. The world has become more democratic, but also more boring. What happened in the tsar's palace could not happen today.\"\n\n\"What happened?\" Across the room, the spider still crawled.\n\nKendra continued. \"Many great preparations were made. Every stick of furniture was dusted to gleaming. Every sheet was washed in lavender to stimulate restful sleep. I was a lady's maid to Manya, one of the middle daughters. It was my job to make certain her clothes were in order and to do her hair. She had lovely titian hair.\"\n\n\"What is titian?\"\n\n\"Oh, I am sorry. I forgot that people of your generation know so little. Titian is a dark red shade favored by an artist also named Titian who painted many red-haired women.\"\n\nI nodded. Titian sounded so much prettier than my own carroty red hair. Perhaps I would change my hair to titian sometime.\n\n\"Anyway, Manya had red hair, so she liked dresses in blue and green with satin slippers to match. Since the party would last a month, I made certain she had thirty gowns (it would not do for a princess to repeat) and six pairs of dancing slippers, three each in green and blue. I took great pride in the hairstyles that I\u2014or, rather, my magic\u2014could accomplish. But one morning, the princess had a terrible illness.\"\n\n\"What was wrong with her?\"\n\n\"I had no idea. It seemed like a sleeping sickness I'd seen on my travels, but there was no fever or cough. The princess simply couldn't stay awake. And worse, when I left her chamber to get help, I found that all her sisters were similarly stricken. They did not want to leave their beds, and when the governesses forced them to, they dragged around the floor as if half dead.\"\n\nI wondered how all the princesses would have been so close in age. Were they sextuplets? Octuplets? But I figured Kendra wouldn't appreciate the interruption.\n\nShe continued. \"We all ministered to the princesses, but none got any better. On the third day, though, I noticed a startling change. A pair of Manya's dancing slippers was missing.\"\n\n\"How strange.\" A missing pair of slippers wasn't my idea of high intrigue. My eyes wandered again to the spider. It had moved to the underside of the table and was starting to make a web. As Kendra talked on, I concentrated on the spider, the bit of thread emanating from its spinnerets. I wanted it to come over. Could I make it do so by just wanting it? To do magic without Kendra noticing was my goal.\n\n\"I looked everywhere for the lost shoes,\" Kendra said. \"The princess had gone nowhere, so nothing should have moved. Finally, I asked one of the other ladies' maids if she had seen them. To my surprise, she reported that a pair of her princess's shoes was missing also!\"\n\nShe paused when she said this, as if expecting some reaction. I had none, so I said, \"Hmm.\" I was still staring at the spider, willing it to come toward me, concentrating all my love and hate, joy and longing into that one task.\n\n\"After polling the other maids,\" Kendra continued, \"we found that, indeed, each young lady was missing shoes. A search ensued, and finally, we found a pile of slippers down the rubbish chute. Each pair had been danced into rags!\"\n\nI saw the spider's thread disengage itself from the table to which the spider had attached it. I tried to contain my excitement, to concentrate.\n\n\"It was so strange,\" Kendra said, \"because Manya and all her sisters had done nothing but sleep. And, a few days later, another pair of shoes disappeared from each closet.\"\n\nThe spider raised a tentative leg and started the long journey down the table.\n\n\"The tsar was so concerned about his daughters' illness that he offered a reward: Any man who found the cause of his daughters' malaise could choose a princess to marry. Many young men came to take the test, but none could solve the mystery. Those who failed were brutally dispatched. Meanwhile, another twelve pairs of slippers were danced to rags.\"\n\nThe spider reached the ground and began to traverse the gleaming marble floor.\n\n\"One day,\" Kendra said, \"I was out in the garden gathering flowers when a young man approached me. 'You are the maid to one of the princesses, I believe?' he said.\n\n\"I nodded, for I recognized him as a boy I saw sometimes in town, a boy who worked in the blacksmith's shop. He said, 'Then, perhaps, you can help me. It's about the princesses. The challenge the tsar has made. I want to try.'\n\n\"'You shouldn't,' I told him. 'It's a good way to lose your life.'\n\n\"But he told me he was very much in love with Princess Svetlana. He knew this could be his only chance to marry her. I asked him how he could be in love with her. He would barely have seen her. Svetlana was the oldest daughter and thought to be the most beautiful.\n\n\"But he told me I was wrong. Svetlana was an avid horsewoman. When her horse needed new shoes, she came with the groom. In fact, she came when any of her sisters' horses needed shoes also. 'She is so kind,' said the blacksmith's boy, 'so modest, not a haughty princess at all. She even has smiles for the lowly groom whom she accompanies. She treats him as well as a lord. Perhaps if I could win her, she could love a poor peasant too. This is my only chance.'\"\n\nThe spider lifted first one leg, then the other, walking toward me, getting closer as Kendra continued with her story.\n\n\"I told him I didn't understand how he expected me to help. After all, if I knew what was making the princesses sick, I would already have solved the mystery. But he said, 'I think you have other skills that would help me.'\n\n\"I knew,\" Kendra said, \"what skills he must mean, my witch skills. But how had he found out? I had done everything in my power to hide my abilities, for if I was discovered, I would be at best run out of town, at worst, tried as a witch. 'I do not know what you mean,' I said.\n\n\"But he told me his aunt was a witch. He knew how they functioned. He had seen how things changed when I was about, the way the crows followed me and the horses reacted around me. And he said, 'I saw how Ivan Vangeloff's horse went lame when Ivan angered you.'\n\n\"There he had me. Ivan had been my beau, my ex-beau, a shopkeeper's son. When he had glimpsed pretty Katrina in the town square, he called upon me no more. I had seen his horse in the blacksmith's shop, and, forgetting myself, I had bewitched it. Now I was paying the price. I would have to help the boy.\"\n\nThe spider came closer and closer. If I could perform spells upon animals unnoticed, I could probably do the same on people. A very interesting possibility. _Come to me_ , I told the spider with my thoughts. _I have no wish to harm you. Not you_.\n\n\"So it was settled,\" Kendra said. \"I would use my magic to give the blacksmith's boy\u2014his name was Alexei\u2014the power of invisibility.\"\n\nI gasped. \"You can do that?\" It was all I could do to hold my gaze on the spider, for this seemed the most wonderful power of all. To go anywhere, spy on people, even.\n\n\"Of course,\" Kendra said. \"It is just another way of changing shape. If one can shift one's own shape, it is nothing to change the shape of another, even to nothing.\"\n\n\"Wow.\" I sighed in amazement. The spider walked still closer, and I wondered if I could change its shape. If I made its legs longer, altered the color a bit, would it be a harmless daddy longlegs instead of a poisonous brown recluse? Could I transform its nature by shifting its appearance?\n\nAnd, if so, could I change who I was, become an outgoing, happy girl, a winner instead of a loser? I'd have to ask Kendra sometime, maybe later, after I'd listened to her boring story about how having magic powers was somehow a _bad_ thing.\n\n\"That night,\" she continued, \"Alexei arrived, announcing that he would solve the mystery of what was happening to the princesses. The staff snickered, for the boy was small and pathetic. He was put up in a guest room, but as soon as the castle had gone to bed, I snuck him into Manya's room. Then, I made him invisible. The next morning, when the castle woke\u2014except for the princesses\u2014Alexei declared that he had the answer.\"\n\nKendra paused, and I knew I was supposed to say something.\n\n\"What did he tell you?\" The spider was close enough now that I could make out details. Its legs were short and wide, bent like a crab's. I remembered a daddy longlegs I'd seen. Its legs were slender, arched like the supports on a bridge. I didn't know how to change one to the other, but I'd try.\n\n\"The blacksmith's boy said he had followed the princesses through a secret trapdoor hidden beneath one of their beds. He did not say how he'd been able to do so without being seen, and of course, no one knew.\n\n\"Under that door, he said, was a staircase, and down that staircase was a canal. The princesses, dressed in their best gowns and dancing slippers, traipsed down the staircase and entered the gondola. The gondolier took the princesses (and, unbeknownst to him, the blacksmith's boy too) down the canal to a secret dock, where they were met by twelve young commoners who escorted them to a secret ballroom where they danced the night away. At dawn, the princesses boarded the gondola again and went back to their rooms to spend the day in sleep.\n\n\"'Impossible,' said the tsar. 'Why would my daughters sneak out at night to dance with commoners?'\n\n\"But that night, we stayed up to watch the princesses. On my watch, one of the princesses got up, as if looking for something. But when she saw me watching, she fell back into fitful sleep.\"\n\nKendra's words got lost as I concentrated on the spider, then on the magic words I had learned and the ways I could use them. My vision blurred, but I struggled to remain focused. Just as I was about to give up, one of the spider's legs began to stretch, then another. The spider wobbled on its path, but finally, all eight legs matched. It skittered toward me, its body elongating as it went, the violin marking leaving its back. Success!\n\n\"The next morning,\" Kendra said, \"we reported what we had seen. The tsar had the room checked and found the hidden door. The canal. 'It seems you are right, young man,' he said. The young man chose to marry the princess, Svetlana.\"\n\nI tore my eyes away from the spider. \"What's wrong with that? You said your magic backfired, but that's a happy story. A poor blacksmith's boy marrying a princess\u2014it's like Cinderella.\"\n\n\"Ah, but Cinderella's prince wanted to marry her. That's why he searched the kingdom. Svetlana did not wish to marry the blacksmith's boy, and when she heard her father's declaration, she ran sobbing from the room. Indeed, all her sisters did. They wept all day, but, that night, the first of the royal guests were to arrive, and their father commanded the princesses to dry their eyes and come down to greet them. When I was fixing Manya's hair, I asked what was wrong.\"\n\n\"'Don't you see?' she asked. 'We don't want to marry the blacksmith's boy or anyone of Father's choosing. We do not wish to be auctioned like cattle. Svetlana was\u2014is\u2014in love with the groom. That is why she went with him to the blacksmith's shop. In fact, we all have secret loves, commoners we visit at night. But now that we are discovered, we will be married to princes. We shall go far away and never see our darlings again.' She sighed. 'If only I could dance with Viktor just one more time.' And, again, she began to weep.\"\n\nThe spider was within inches of me now. I reached out, and it crawled onto my hand. I stared at it, marveling at its coffee-bean-shaped body, its sunburst of legs. I had done that, changed one thing to another. I turned away so that Kendra would not see my smile.\n\n\"And so it came to pass,\" she said. \"Svetlana married the blacksmith's boy, and each of her sisters also married a man she did not love. The end.\"\n\nMesmerized, I reached out to touch the spider.\n\n\"Ouch!\" It bit me on the knuckle.\n\n\"What is it?\" Kendra asked.\n\n\"This spider. It bit me.\" Already, I could feel the venom seeping into my system. \"Help me, please.\"\n\nKendra looked at it. \"Oh, don't be silly. It's a daddy longlegs. Humans aren't affected by their bites.\"\n\n\"But it's not a daddy longlegs. It's a brown recluse. I . . . changed it. Unless you're saying it's turned into a real daddy longlegs. Then it won't hurt me.\"\n\nKendra reached out toward the spider, and then, with a tiny tap, changed it back to its true shape. She placed it on the ground and shooed it away. Then she glanced at my hand.\n\nA small, white blister began to form, but with the touch of her finger, that too was gone. \"No. It is still a brown recluse. You can change a thing's appearance, but not its nature. Perhaps that, along with my story, is enough learning for today.\" She nodded at me. \"Run along.\"\n\nAnd I did what she said. What I had learned was valuable indeed: Any harm I did to myself, I could undo.\n\nAs I walked toward the door, I made certain to stomp upon the spider until it was just a brown blot on the floor.\n\n#\n\n#\n\nThat night, I decided Kendra was crazy. Oh, sure, she'd recognized what I was, taught me magic, and made me the powerful witch I was becoming. But that didn't mean she wasn't also crazy. Or senile. How could she say magic wasn't a positive thing? Even in the story she'd told, magic had ended the carnage.\n\nOf course, Kendra was still my best, my only friend. I needed her. And I needed her to teach me too.\n\nI thought about what I'd done with the spider. I'd been able to change its shape, if not its nature. I could do that to people. Changing their looks would be enough. That way, I could keep Gennifer and Jennifer the same shallow girls they were\u2014trapped in fat, bacne-ridden bodies.\n\n_If you can't change their nature,_ something mean inside me whispered, _you can't make someone fall in love with you. Greg never will; you will find out the hard way._\n\n_Stupid._ I told myself I didn't need to change anything. Greg had loved me for myself all along, all the time we were friends. He just got sidetracked. He was shallow. Boys were. He wanted a prettier girlfriend. Since I was going to be the most beautiful girl in the world, he'd have to love me. I'd just have to get beautiful Jennifer out of the picture, to be sure.\n\nI knew I'd have to change her appearance as I did my own, slowly, gradually, so it wouldn't be obvious, even to her.\n\nAs I drifted off to sleep, I made my plan. I would put it into action in language arts, the one class I had with Jennifer and not Greg.\n\nThe next morning, I stood by my locker, brushing my hair. I'd recently installed a stick-on mirror, so I could admire myself between classes. My classmates still didn't notice the difference in me, but I did. I straightened up. Greg was passing by, alone for once. I met his eyes with confidence, then looked away on my own terms.\n\nMaybe that's what I needed, to be on my own terms. Probably Greg would eventually get tired of Jennifer on his own without any help from me. She was really stupid. It was silly of me to want to hurt her. After all, if Greg didn't like me for me, what good was he? Maybe I should\u2014\n\n\"Hey, watch it, ugly!\" Someone slammed into me.\n\n\"Sorry.\"\n\n\"Your face is sorry.\" It was Jennifer, Jennifer with brand-new highlighting and a face full of makeup. \"You just exist to get in my way, don't you?\" She shoved past me toward Greg, who'd stopped to wait for her.\n\n_Okay. Game on._\n\nIn class, we were reading _Animal Farm_ aloud, painful because Mr. Cameron had students take turns reading and, apparently, some still couldn't. I read ahead, pages ahead, but then he'd stop to discuss it, and I wouldn't remember where we were. So, instead, I just zoned out as Colby Buckner read, \"Man is the only real enemy we have. Remove Man from the scene, and the root cause of hunger and overwork is ab . . . ol . . . ished forever.\" I thought about pigs. The pigs in the story were supposed to be like the people in power. People were like pigs. If only the people who were like pigs could _look_ like pigs.\n\nI contemplated Jennifer, who sat two rows to the side of me. She wasn't even pretending to read. Her eyes fluttered closed, then open. Her forehead drooped forward. Her nose was adorable, slim, and turned up. What if it turned up a little bit more . . . ?\n\nShe saw me looking at her and mouthed, _Pig_.\n\nI thought of everything Jennifer had ever done to me, the insults, taking Greg, the snickering, taking Greg. Then I thought of everything anyone had ever done to me. All my life, I'd been an outcast, a pariah, and why? Because I wasn't pretty enough? Because I was too smart to matter but not smart enough to play stupid? I closed my eyes, but I could see Jennifer's face, Jennifer's perfect, blue-eyed, laughing face, her enviable nose. Then, in my mind, it morphed into a pig nose. She squealed in horror, just like a pig, and held her lively, nail-polished fingers up to hide it. She squealed again, then started to cry. I smirked in satisfaction.\n\nIn the room, Colby was still reading. I put my head down, looking at Jennifer so she couldn't see me.\n\nShe was still whispering to Gennifer. Her nose was still perfect.\n\nWhy hadn't it worked? Guess I hadn't done it right. Maybe it was harder to work magic on others. But hadn't Kendra cured her brother the first time out? Hadn't I gotten the birds to fight off Nick and Nathan? I leaned my hand on my face and looked up.\n\nHuh. My face felt different. My nose felt . . . piggy.\n\nWhat? How could that be? I'd seen it so clearly in my mind, Jennifer's face changing, not mine. Not mine!\n\nEven as I held my hand up, I felt my nose hardening, my nostrils spreading even more. My head was heavy, and I remembered reading that a pig's snout weighed about a pound. I cradled my head in my hands like I had a headache, leaning to cover myself with my hair.\n\n\"Violet, will you read next?\" Mr. Cameron asked.\n\nI began to cough, still holding my hands over my face. I managed to gasp out, \"Bathroom!\" Around me, everyone was laughing. Without waiting for Mr. Cameron's response, I bolted to the girls' room, still coughing. I looked in the mirror and saw . . . my snout.\n\nIt was pinkish-white with black spots and stood at least three inches from my face!\n\nWhat . . . the . . . ?\n\nI ducked into a stall, shaking, and tried to bring up the magic. My thoughts were racing. How would I get out of here? What would Mr. Cameron say? Could I put my face back? I realized that, while anger had been an awesome motivator, fear was a terrible one.\n\n_Breathe. Breathe. Stop thinking about how you can't leave the building like this. Forget how you left your backpack in Cameron's class. Breathe._\n\n_Breathe!_\n\nI remembered the spider. How I'd changed its shape, slowly. If I could do that, I could do anything. Anything. Anything except give Jennifer a pig nose. I'd ask Kendra about that. Clearly, I still had a lot to learn about magic. Thank God for Kendra.\n\nFinally, I felt my heart rate slow. And my breathing. It was hard to breathe through the snout. I sat on the toilet, breathing. Breathing. Breathing. Imagining my nose\u2014not my nose, but Michelle Pfeiffer's nose, Diane Lane's nose, or model Brooke Shields's adorable, famous nose. Yes, that was it. Perfectly upturned with not too much nostril. I'd once read that it was nearly impossible to achieve this surgically. But magic surgery had to be better. The breathing, the heavy pig snout, the concentration made me feel weak, almost light-headed, and the metal stall walls began to blur around me. I held my hands up to my face and felt the snout shrink beneath them. Relief! I straightened my neck, held my head up, opened the door.\n\nEven though I knew I was late, I couldn't resist a glance in the mirror.\n\nIt was perfect, almost too perfect. No, there was no such thing as too perfect. I would be perfect, all of me!\n\nAfter school, I ran to Kendra's house.\n\n\"I tried to use magic, and it backfired on me.\"\n\n\"Backfired?\" Kendra looked bemused\u2014and maybe a little amused too, arranging herself on a red bench that looked like it belonged in the Museum of Modern Art. \"How could it backfire?\"\n\n\"Um, I don't know, turned my nose into a pig's snout. That's all.\"\n\nKendra chuckled. She was wearing a black lace ball gown that had barely made it through the door. \"So you were trying to change your nose into this stunning creation\u2014Diane Lane's nose, I believe\u2014and it turned into a snout instead? Is that exactly what happened?\"\n\n\"Not exactly.\"\n\n\"I didn't think so. Perhaps you were trying to give someone else a pig's snout?\"\n\n\"How did you know?\" Kendra seemed to know a lot of things\u2014even knowing I'd copied Diane Lane's nose. Could she read my mind? Or was she spying on me with that mirror?\n\n\"It's about discretion.\" She pulled me by the arm to sit beside her.\n\n\"Discretion?\"\n\n\"'Like a gold ring in a pig's snout, is a beautiful woman who lacks discretion.' That's from the Bible.\"\n\n\"You didn't strike me as the religious type,\" I said.\n\n\"I have nothing against religion. Since I'm never going to die, I don't have to worry about impressing God for the afterlife, but that doesn't mean I don't want to use my powers for good. Also, if you go around turning your enemies' noses into pigs' snouts, you'll get caught. That's why it's impossible to do.\"\n\n\"To do what?\"\n\nShe waved her hand and produced two plates. \"Gingerbread?\"\n\n\"No, thank you. What's impossible?\"\n\nWith another wave of her hand, the plates were gone. \"Rules of magic. It is impossible to change someone without their knowing.\"\n\n\"You changed me the first day. You changed my nose.\" _Not as well as I changed it._\n\n\"Ah, but you knew about it at the time. It was your choice. Once, I turned a proud, cruel boy into a beast, but he knew about it. To work magic on someone else, you must reveal yourself. It keeps others from being blamed. But had you changed your classmate's nose, she wouldn't have known how it happened. That's not allowed by the rules of witchcraft.\"\n\n\"So I can't even give her . . . zits? Diarrhea? A bad SAT score when she's a junior?\"\n\nKendra shook her head. \"Not without also giving those to yourself. It keeps you from abusing your power\u2014and from making everyone suspect you. But you can do wonderful things for yourself, travel the world, give yourself incredible talents, never pay for cute new clothes.\" With a wave of her hand, she changed her dress to fuchsia. \"You should hear me sing opera\u2014I'm like a mermaid.\"\n\n\"All I want is for Greg to love me again. I don't care about that other stuff.\" I sort of wanted the gingerbread back. It was comfort food.\n\n\"Then you will have to win him back with your own looks and abilities\u2014not by harming Jennifer. But it will be a difficult task.\"\n\n\"Why's that?\"\n\n\"Because, my dear, he never really loved you in the first place. He was friends with you because he was lonely. With Jennifer, he has a whole circle of friends.\"\n\nI remembered Greg and me, walking to his house after school, doing crazy science experiments like putting Mentos in Diet Coke so it would explode, checking our birdhouses daily. I'd had my first wren of the year last week, and I'd so wanted to tell Greg. But I knew he wouldn't care anymore. I guessed he'd never cared. I nodded, knowing Kendra was right.\n\n\"It's so unfair. Why do they hate me, Kendra? I always thought it was because I was ugly. But now, I'm not that ugly, and they still hate me. What's wrong with me?\"\n\nKendra frowned. \"I think you chose to like the wrong girl's boyfriend.\"\n\n\"But that's not fair. I saw him first.\"\n\n\"Since when are bullies fair?\" Kendra asked. \"Do you think they issue some sort of Bully Code of Conduct\u2014only pick on people who deserve it?\"\n\nIn truth, I guess I felt I had deserved it. Why would they pick on me if I didn't? I'd deserved it because I was an ugly freak. But now I knew they'd picked on me because they could, and maybe because I cared. Some part of me had once longed to be friends with Jennifer, to sit at her lunch table and go shopping at Dadeland after school. I couldn't explain _why_ I wanted that. She was horrible. But part of me wanted to deserve them, the beautiful girls.\n\n\"I don't know why no one likes me. I thought it was my looks, but now, I don't know.\"\n\n\"I like you.\" Kendra put her arms around me. The dress was taffeta, a stiff fabric that felt like hundreds of Pringles chips when I hugged her. But I sunk into her embrace. She was the absolute coolest friend in the world.\n\n\"There, there,\" she said, \"I love you. And, someday, others will too. You're going to be an incredible woman. You'll see.\"\n\n\"I don't want to be an incredible woman,\" I sobbed. I knew I was no better than anyone else. I'd only been happy to be smart because I was ugly. Really, I wanted to be beautiful and be loved. \"I only want Greg!\"\n\n#\n\n#\n\n# _1989_\n\nOver the next few years, I changed everything about myself. Everything I could, at least. My hair. The color of my eyes. My height. By seventeen, I was beautiful, tall with the body of a _Sports Illustrated_ swimsuit model, strawberry blond hair that never got messy, and eyes the color of lavender dish soap\u2014or, well, violets. I hadn't had a zit in four years, and even little details like the amount of space between my eye and my eyebrow (ideally the size of another eye) or my philtrum (the area between lips and nose\u2014ideally about fifteen millimeters) didn't escape my notice. I was Pygmalion to my own Galatea, every part sculpted perfectly.\n\nNo one cared.\n\nOr noticed.\n\nI also developed talents, as Kendra had suggested. I could sing like Whitney Houston, dance like Paula Abdul. I used these abilities in school musicals and on the Cougarettes dance team. I was so talented they couldn't reject me even if they hated me.\n\nWhich they did.\n\nIt wasn't as overt now. I wasn't ridiculed, not usually. But even though strange men, and even women, approached me on the street to beg me to visit their modeling agency and new boys at school asked me on dates, I had no one to love me, no one I wanted. And, at the end of Cougarettes practice, when the popular girls got into their cars to go to the mall together, have dinner together, do homework together, I was the only one stuffing my pom-poms into my backpack alone. And watching Jennifer leave with Greg.\n\nYes, after all this time, they were still together. It was the one high school relationship that lasted longer than a rock star's marriage. They were voted Cutest Couple in middle school, and they'd be in this year's yearbook too. Greg was the star wide receiver, recruited by colleges. Jennifer was on dance team, at his side after every game. They belonged together.\n\nAnd, of course, Jennifer got everyone else to hate me, just like she always had. Maybe now that I was beautiful, she saw me as a threat. Greg didn't care what a bitch she was. He didn't even know I was alive.\n\nAnd yet, I still wanted to change Greg's mind, engineering ways to run into him without her. I tried different routes to my classes at school until I found ways to cross his path, just so I could say hello or look at him. I should have moved on, but moving on wasn't a thing with me.\n\nOnce, I passed him walking home from school. Greg had no car, but Jennifer had one, so she drove him most days, after her dance team practice and his football. But Wednesdays, she had Student Council (yes, I'd memorized their schedules), so Greg walked.\n\nI had a car, a little blue Mazda Miata convertible I'd gotten for my sixteenth birthday. My mother was generous now that she was proud of me. Every Wednesday, I drove slowly from school, stalking Greg, fantasizing about offering him a ride. But, usually, someone else gave him one. After all, he was popular.\n\nBut that day, it happened.\n\nAs a bonus, it was raining, a sun shower that promised to get harder. Leaving the school parking lot, I saw Greg jogging toward home.\n\nI pulled alongside him, onto the grass. \"Want a ride?\"\n\nHe wore black athletic shorts and a green tank top that showed every wet muscle. I longed to run my hands over him, to touch the hardness of his perfect body, the softness of that crow-black hair I'd loved since I was ten.\n\nAt first, he didn't seem to recognize me. He approached the car, squinting in the sparkly sun-rain. I smiled, showing my kissably puffed lips and that philtrum. I shook my hair a little. \"Hey.\"\n\nHe backed away. \"Oh . . . Violet. It's you. I probably shouldn't.\"\n\nI feigned confusion. \"Why not? It's awfully wet out.\"\n\n\"Yeah, I know.\" He was getting soaked.\n\n\"It's dry in here.\"\n\nHe was thinking about it. His hand approached the door handle. I wished I could control his mind.\n\n\"Why don't you sit in the car while you're working out the deep, philosophical problem of whether to accept a one-mile ride from an old friend.\"\n\nHe pulled his hand back a little. Way to go, Violet. But he looked at me, stared at me, actually, like he was seeing me for the first time. I parted my lips, knowing I finally had a tool, a tool more powerful than magic I could use on him. I stared back at him, lowering my eyelids. \"We used to be such good friends, Greg.\" Showing my straight, white teeth. I was harmless. Beauty was always trustworthy. \"Don't you remember?\"\n\n\"Oh, yeah.\" A gush of air. He opened the door, letting the rain in, and himself. \"I remember. You're right. It's stupid.\"\n\nI tried not to grin. \"You're getting the seat wet. Let me . . .\"\n\nI pulled a towel from my dance bag and began wiping at the seat, just the seat at first. \"Let me get the other side too.\" I leaned across him, trying to touch him but make it look accidental. Like me, he'd become more beautiful each day. His arms were sculpted bronze ropes, and after I dried the visible expanse of seat, I began using the towel to stroke each muscle. \"Sorry. My mom will kill me if anything happens to the leather.\"\n\n\"It's a great car.\" His voice sounded strained, as if he was struggling to keep it even. \"You're really lucky.\"\n\n\"Yeah, my mom and I have been getting along better. Let me get behind you.\" I got real close to his ear.\n\n\"What?\"\n\n\"Your back.\" I caressed it, moving him forward. \"Let me dry it.\"\n\n\"Sorry.\" He obliged, and I dried behind him. We were close, closer than I'd been in so long. I felt a little breathless, but I tried not to let him know that my pulse was racing. I could have kissed him if I'd wanted. But I knew I shouldn't. Get him to trust me first. I dried the seat and his back, leaning close enough that my long hair brushed his chest as I did. I didn't dare touch his chest, his legs, on purpose. I wasn't that confident. But I dried his shoulders, his neck. He didn't try to take the towel from me. Was it because he trusted me or because, as I hoped, he didn't mind my touching him. My arm brushed his, and I felt his muscles stiffen.\n\n\"Guess that's good enough.\" I handed him the towel. \"Let's go.\"\n\nI pulled out. The car was tiny, and it had a manual transmission, which meant every time I used the stick, my hand was practically in Greg's lap. His legs were long, so there was barely room to move.\n\n\"That was a great kickoff return touchdown you made the other day against Bradford,\" I said.\n\n\"Oh, you saw that?\" But he looked pleased.\n\n\"Well, of course. I was there on the dance team. Remember?\"\n\n\"Yeah. It doesn't always seem like those girls are watching the game.\"\n\nHe meant Jennifer. I grinned. \"I watch you all the time, Greg. Of course I do. You're, like, the whole team.\" We approached a stop sign, and I downshifted, my hand brushing his leg as I did. \"It's beautiful to watch you play, the way you got around those guys. It was . . . poetry.\"\n\n\"Wow. Thanks. That's really flattering.\"\n\n\"Doesn't Jennifer watch you?\" Was I pushing my luck?\n\nBut he said, \"Yeah. I mean, of course she does. She's my girlfriend. She just doesn't always seem to understand it. I guess football's complicated.\"\n\nWhich was why thousands of drunks could understand it. \"Well, it's nice that she tries.\"\n\n\"Sure.\" He actually looked a little unsure.\n\n_What do you even like about her?_ But I didn't ask. \"Well, I thought you were wonderful, a hero. And I heard there was a scout at the game.\"\n\nHe looked down, embarrassed. \"Well, yeah, it's not really big colleges. Division Two.\"\n\n\"Are you kidding? Do you know how many guys would kill to be scouted by anyone? Your dad must be so proud.\" I patted his arm, loving the warmth of him under my hand.\n\nHe grinned. \"Yeah. Yeah, he is.\"\n\nWe approached his house now. The ride was ending soon. I searched for something else to say. \"How is your dad? Remember when I used to go to your house after school?\"\n\n\"Yeah. We were just kids then.\"\n\n\"I know. But we had so much fun, didn't we? Like when we built the birdhouses?\"\n\n\"Sure.\"\n\n\"And then, the wrens came.\"\n\nHe nodded. \"I remember.\"\n\nI wanted to ask him if he ever did anything like that with Jennifer, but I realized I might not want to know the answer.\n\nHe squirmed in his seat. \"Look, Violet, I . . .\"\n\n\"You once told me we couldn't be friends because I was too weird.\"\n\n\"I didn't say that.\"\n\n\"You did. I remember it like the answers to last week's vocabulary quiz. And, since then, I have worked really hard not to be weird, to be worthy of you. I mean, worthy of your friendship.\"\n\n\"Violet, I didn't ask you to do that.\"\n\n\"I know you didn't. I wanted to. I wanted to . . . be friends again. I missed you . . . that so much.\" We were dangerously close to his house. I was driving slowly. I couldn't drive any slower, but I couldn't let him out of the car without some kind of affirmation that we'd at least talk again.\n\n\"Friends?\" He looked doubtful.\n\n\"Sure. Friends. Friends like we used to be.\" I wanted so much more, but there was time.\n\n\"I guess we can be friends,\" he said finally.\n\n\"That's so great!\" I had to stop myself from doing a little seat dance. \"So maybe I could drive you home every Wednesday.\"\n\nHe looked like he'd sat on a half-eaten Slurpee. He drew away.\n\n\"I don't think Jennifer would like that.\"\n\n_And, of course, you're not allowed to have independent thought. Does she really need to know?_\n\n\"Well, then, maybe I could call you after school sometimes.\"\n\n\"I guess. Maybe. Well, maybe I should call you.\" We were right by his house, and he had his hand on the door handle like he might want to jump and run. \"Look, I should go. Could you just\u2014?\"\n\n\"Sure.\" I pulled into his driveway, and he got out. I'd blown it.\n\n\"Thanks, Violet. We'll talk soon.\"\n\nThen he got out and sprinted into the house.\n\nI sat there a long time, staring at the closing door, then the seat which still, despite my towel drying, retained the damp imprint of his body. I curled over and lay against it until the seat got cold, and I couldn't feel him anymore.\n\nI knew he'd never call me.\n\n#\n\n#\n\nI got to see Jennifer three afternoons a week at Cougarettes practice. We wore cougar ears and drawn-on whiskers (okay, it may have been slightly\u2014just slightly\u2014dumb). We danced at football games and in competition. The juniors and seniors took turns choreographing, and this year was my first turn. I loved dance and didn't just rely on magic for my talents. It wouldn't be as satisfying. I spent hours working on my extensions and practicing jet\u00e9s in the long hallway between the bedroom and kitchen of our house, trying not to kick my mother or the cat. I could leap higher than anyone. I'd also spent a long time on the routine, forgoing homework to make up the choreography. Kendra and I had gone to an Indian movie at the local art cinema, and I loved the dance routines. I wanted to do something like that. It would be different. Special! Like nothing any dance team had ever done before!\n\nWhich would probably mean people would think it was weird.\n\nStill, I had to try. If worrying about people thinking I was weird stopped me, I'd never leave the house. It wasn't like the girls in my grade were going to like anything I did. But there were eight new girls on the squad, three freshmen, five sophomores, none of whom knew I was weird. They weren't in any of my classes, so they didn't think I was a loser. I could impress them with my completely special routine.\n\nThat day, a Friday, I stepped in front of the group, trying to be proud, trying to at least _look_ confident. I was, after all, six inches taller than I'd been in middle school with perfect 34 Cs and thighs like a Barbie doll's, that never squooshed together. Genetics didn't make bodies like mine, no matter what the magazines tried to tell us. But still, I stood before the group of fifteen girls in the otherwise empty gym, unsure how to start. I wondered if orchestra conductors felt like this, not completely confident that the violinists, oboists, or the guy playing the rainstick would obey his commands. Probably not. Probably everyone felt confident but me. Witchcraft didn't change the nature of a thing, so witchcraft couldn't make me confident.\n\nBut I tried to change my own nature. As I took my place before the group, I straightened my shoulders, shook my perfect ponytail, smiled, and tried to channel the other girls who'd stood in my spot before\u2014the Nicoles and Julies and Merediths who'd stood confidently on the foul line with _Cougars_ painted in green, and said:\n\n\"Okay, this is going to be awesome!\" I clapped my hands, astounded by the awesomeness of it all. \"I thought we'd do something different, a Bollywood-style routine!\"\n\nA few of the new girls nodded, like they knew what that meant, and one of them even said, \"Like those Indian movies? That sounds so cool.\"\n\nEncouraged, I went on. \"So, for the first eight beats, this is what we'll do. Start with one arm out to the side, like this, in front of your chest.\" I demonstrated, picturing myself as the Indian actress in the movie I'd seen. \"Then, shoulders up-down, up-down.\" Most of the girls copied me, but not all. I decided to keep going.\n\n\"Now, switch your arms the other way, and shoulders up-down, up-down.\"\n\nJennifer, who hadn't been following me, cleared her throat. Or, rather, she had a coughing fit, and most of the returning girls stopped dancing. They put their arms down.\n\n\"Come on, guys.\" I tried to act like everything was normal. \"I don't want to go on until we have the first eight. Up-down, up-down, then switch.\"\n\nI was screaming, but my voice got lost in the big gym.\n\nNow, none of the returning girls were following me, but the new girls were, and the routine was pretty easy. So I decided to pretend nothing was wrong. \"Freshmen, you're doing great! Show those seniors. Okay, one more time, then we'll move on. Up-down, up-down, then switch.\" Some of the new girls had dropped out too.\n\n\"Come on, guys!\" My voice cracked a little. I looked at the fluorescent lights to keep from tearing up.\n\nSuddenly Jennifer sat on the floor. Her cohorts, Gennifer and Meighan, followed. Then, the other junior girls, and the seniors.\n\nJennifer spoke. \"What is this from, some kids' show? It's dumb even for you, Violet.\"\n\nOne of the new girls, the one who'd said it was fun, started to open her mouth. Then, she shut it.\n\n\"It's Bollywood-style, like movies from India. The Indian film industry\u2014\"\n\n\"Does anyone care about this?\" Jennifer's hands were on her hips. \"We're gonna look stupid. Why don't you think of something else, and we'll try next week.\"\n\nI looked around. The adviser, Miss Levin, was supposed to be, um, advising, but she'd retreated into her office to call her boyfriend. And really, what was I going to do? Run to her and cry? Jennifer was the cocaptain. \"Fine,\" I said.\n\n\"Great.\" Jennifer stood up. \"I'm sure you'll be able to come up with something better if you . . . think about it more.\"\n\nI stared at the light. I knew I couldn't come up with something better. What I'd done was perfect. Besides, even if I did, they'd hate it. I couldn't get a break with Jennifer. She had Greg, had everything I'd ever wanted. Why did she have to be such a bitch?\n\n\"Meighan, show us your routine,\" she said.\n\n\"Gladly.\" Meighan smiled and walked up front. For the next twenty minutes, she taught a routine based upon Madonna's \"Papa Don't Preach,\" __ which was 1) borderline obscene; 2) copied from the music video; and 3) such an old song that all the teams had already done it. Everyone followed her like she was an innovator. Un. Freaking. Real.\n\nI thought about how freeing it would be to throw caution to the winds and announce that I was, in fact, the Wicked Witch of the West and make all their heads explode. The news coverage: \"High School Dance Team's Heads Explode.\" The T-shirts: \"Guns don't kill people: Pissed-off witches kill people.\" There'd be protests by religious groups, complaining that taking prayer out of schools had caused my breakdown. It would be worth being ostracized, though. I was already ostracized. I didn't think they burned anyone at the stake anymore. And I'd be safe because I did it openly. That was the stupid rule, right?\n\nBut, of course, I was too nice. Scratch that\u2014too much of a wimp\u2014to do it.\n\nWhen practice finally ended, I headed for the locker room, ahead of the others. It stunk of the sweat of thousands of freshmen. I got my gym bag, which I'd left on a bench. I wanted to change quick and get out.\n\nSomething was sticky.\n\nSomething was crawling on my clothes! Then, up my arms. Ants. Hundreds of ants.\n\nFrom the doorway, Jennifer giggled. \"Look!\" She was pointing at me.\n\nShe'd covered my clothes in something. Syrup? No, honey. Jennifer had poured honey into my bag! On my clothes!\n\n\"Something wrong, Violet?\" She came to where I was standing. A few other girls lagged behind her like the followers they were.\n\nI stared down, remembering all those days in grade school when I'd eaten alone, all those times they'd made fun of me\u2014 _she'd_ made fun of me. I felt the skin on my forehead tighten.\n\n_Don't stinking cry. Calm down._ This was easy. It was a few ants. I could control higher orders of animal than ants.\n\nI drew my clothes out of the bag. The honey, the ants, were all gone. I tried to keep my voice even. \"Wrong? What could be wrong, Jennifer?\"\n\nShe gasped, and her mouth quivered like Grimalkin's when she watched birds through the front window, knowing she could never get them, but longing. Sort of like I did with Greg.\n\nI fixed her with a stare under the long lashes of my lovely violet eyes. \"I know you would never do anything to my backpack, would you? You're so sweet and nice, and everyone loves you, and besides . . .\" I fluffed my T-shirt in the air to show her how not covered in ants it was. \"It's a bad idea to mess with me.\"\n\nShe laughed. \"And why's that?\"\n\nI had a headache, and my eyes felt hot. Only crying would relieve it.\n\nBut I wouldn't cry. I'd rather have a headache forever. \"It just is. You can figure it out.\"\n\nI took off my leotard and stood there in bra and panties, not even trying to be modest, letting her and the others see every inch of my perfect body, more perfect than hers, more perfect than anything nature could make. My brain, I had no confidence in, but my body was excellent. I clenched my back teeth to keep from sobbing, but I smiled. \"I know it would be too much to ask you to be . . . decent. So I'm just telling you to leave me alone.\"\n\nShe laughed. \"Decent? Omigod, you are so crazy.\"\n\nI pulled my T-shirt over my head, slowly, aware most of the girls were watching. I shrugged. \"Who knows how crazy I am?\"\n\nI slid my perfect, tanned legs into my little white shorts, then slipped on my sandals, admiring the pedicure that never chipped. Jennifer had stopped laughing. Maybe she'd realized nothing was funny. Maybe she was scared.\n\nThey all remained silent as I left the room.\n\nI stood outside, contemplating the blue and white afternoon. I lifted my head and squinted at the sun. A warm breeze set a nearby wind chime in motion. Across the street, a dog barked.\n\nI allowed my face to break for a moment, feeling the hot tears running down my cheeks, the blessed relief of the dam breaking against the tension in my head.\n\nI looked at the dog.\n\nI had seen the dog before, a Doberman, wiry as a welterweight boxer and miserable about being fenced in. It saw me too and began to bark and fling itself against the chain-link fence. I felt its pain. I was fenced like that dog, fenced by people's expectations.\n\nI lowered my eyes to meet the dog's. It was across the street, but it saw me. It calmed. It lowered its rump to the ground, sat, then curled into a ball. I held eye contact. _Good girl._\n\nSo I couldn't do anything to Jennifer without bringing the same on myself. But could I have the dog do something to Jennifer?\n\nI had nothing to lose by trying. Whatever I messed up, I could easily fix.\n\nThe dog was still staring at me. Its slender, black tail bounced up and down, as if it'd heard me.\n\n\"Good girl,\" I said aloud.\n\nI examined the fence that contained it. Chain link already bowed down from previous escape attempts. It was almost low enough for it to scale in a single jump. Almost.\n\nI visualized a weight, like an anvil in a cartoon, crushing down the fence. A wind whipped through the parking lot and across the street. The fence bowed. The dog sat, obedient, wagging its tail. This time, I pictured a steamroller, crushing it. The wind whistled through the trees. The fence bowed further, tapping the resting dog on the buttocks. It started, then woke.\n\n\"Stay,\" I whispered.\n\nIt stayed, but on the alert now, ready to spring.\n\nThe fence lay low on the ground, no protection at all.\n\nBehind me, I heard laughter coming closer, then a voice from inside. \". . . so funny! Did you see her face when no one would do it?\"\n\n\"Her routine wasn't that bad,\" another voice said.\n\n\"Oh, of course it was.\" I recognized Jennifer's high-pitched whine. \"Anything she does is total poop.\"\n\n\" _She's_ total poop,\" Meighan agreed.\n\n\"I can't believe she's even on the team,\" Jennifer said. \"Where are the standards?\"\n\nThe voices grew closer. I stared at the dog. _Good doggie. Stay one moment longer._\n\n\"She's so ugly,\" Jennifer just kept going. \"Disgusting, really.\" I heard the thunk-squeak of someone pushing the bar that opened the door. I moved to the side, out of the way, focusing on the dog, communicating. _Good doggie. Nice doggie. I'm your friend. And you're going_ __ _to hurt the mean girl who hurts me\u2014one bitch to another._\n\nCould I do this?\n\nSuddenly my life started flashing before my eyes, but it was all Jennifer, like a collage of all the mean things she and her friends had done, said to me, the turned backs and mocking faces, Jennifer's voice in my ear, always, saying, \"These seats are saved\" or \"Omigod, what happened to your hair? Did you brush it with a spaghetti server?\" The pick-pick-picking on my skin, my hair, my nails, my body. \"Are those new shoes, Violet? Where'd you get them\u2014the orthopedic store? Can you believe what a suck-up she is, getting an A on the test when everyone else failed? Suck-up, asking questions in class. If I had that nose, I'd hide my face. If I had that hair . . . those eyelashes . . . that body . . . little bitch.\"\n\nThe dog was up on its haunches, backing up, then clambering, springing over the fence. The rage and hatred in its eyes matched my heart. The dog accelerated, dodging a passing car, intent on its target. Too late, Jennifer saw it charging toward her. \"Oh, no!\" she screamed. Her friends, true to form, ditched her, edging back into the building. Jennifer, poor, sweet Jennifer, was left all alone as the black beast's savage jaws came closer, closer.\n\nAnd then, the dog was atop her, teeth ripping at her clothes, her arms. She was screaming, but her shrieks were drowned out by my own cries. \"Bad dog! Bad dog! Not her arm!\"\n\nThe dog sunk its teeth into Jennifer's cheek and pulled.\n\nIt was enough. The dog had done its damage. Jennifer's cries were still in my ears. I came from my hiding place. Jennifer's friends were in the door, cowering. It was just me, the dog, Jennifer, her face bloodied, with the teeth marks. I ran for the dog, breathless, as if I'd been running the whole time. I laid hands on the dog. \"Stop!\" I felt a jolt of electricity running from my body to the dog's. I pulled the dog off Jennifer. It stopped, motionless, as if Tased. I stroked the soft fur. It was, after all, a good dog. It backed away, then trotted to its own yard. I went with it, as if I were taking it there, but partway back, I let it break away.\n\n\"Hide,\" I whispered.\n\nI glanced at the fence. It sprung back into place.\n\nI ran and knelt over Jennifer. Her face was bloody, flesh torn, and she was sobbing, holding her hand to her cheek. \"Are you okay?\" She whimpered in response. I pulled a damp towel from my dance bag and ripped it in half. I handed it to her. \"Here. Apply pressure.\" I helped her do it. I was so nice.\n\nNow that the danger was over, Jennifer's friends came back, all concerned, all, \"Are you okay? What happened?\" pushing me aside. Someone ran to dial 911.\n\nThen, Greg was there. He must have been at practice. He knelt over Jennifer, beside me. \"What happened?\"\n\n\"It was a dog,\" I said, a little breathless just from being near him. \"It came out of nowhere and attacked her. I pulled it off her.\"\n\n\"Where did it go?\" Meighan said.\n\nJennifer was still screaming, sobbing, incoherent. But then, she formed words, painful words. \"It was her.\" With her good hand, she pointed at me. \"She did it.\"\n\n\"It was a dog,\" Gennifer said. \"A pit bull.\"\n\n\"A white one!\" said Meighan.\n\n\"It was her. It was her.\" Jennifer's voice rose to a violin's pitch. \"She made it attack me.\"\n\nGreg's eyes sought mine, and I made mine wide, confused. \"Jennifer, the dog attacked you. I got it to stop. I helped you.\"\n\n\"Noooo! You didn't. You made it attack me. You made it because of what I did to your stuff.\" She was cringing in pain but still strong enough to accuse me.\n\n\"Poor Jennifer.\" I laid my hand on her arm. \"You're in such pain from that bad dog. But of course, I can't control the dog. And besides . . .\" I moved my hand to her bloodied face. The wound was jagged, a lightning bolt from eye to mouth, streaks across her once-perfect nose. \"Why would I want to hurt you? We're teammates, right? Friends.\" I stroked her hair until her eyes closed. \"That's good. Probably better to sleep.\"\n\n\"You're a hero, Violet,\" someone said.\n\nA siren sounded in the distance, distracting everyone. So I was probably the only one who heard Jennifer's voice, softer than the breeze in the grass, whispering, \"You're a . . . witch.\"\n\nI turned to Greg. \"The paramedics are here. It sounds like she doesn't want me around, so I'll go.\"\n\n\"Thank you for helping her, Violet,\" he said.\n\nI shrugged. \"I was just so scared when it came after her. It was all, like, adrenaline when I pulled the dog off her. I mean, it could've attacked me. Maybe call me later and let me know how she's doing.\"\n\n\"Of course. You're a good friend.\"\n\nI looked at Jennifer. My spell had her sacked out on the lawn, three Cougarettes over her. \"Well, that's exactly what I want to be, a good friend.\" I smiled more at the sight of bleeding bitch Jennifer than at that thought. If I couldn't have Greg, I could at least have revenge.\n\nBut I meant to have Greg.\n\n#\n\n#\n\nGreg called that night, and we talked about Jennifer. Jennifer. Jennifer\u2014the pain she was in, the surgery she'd require, the way they didn't let him ride in the emergency vehicle and how mad that made him, the way they'd looked all over the neighborhood for the pit bull but no one had found it. Jennifer. Jennifer. Her name was like the bells, bells, bells in the Edgar Allan Poe poem, driving me mad.\n\n\"Did you see the dog?\" Greg said, \"the dog that attacked Jennifer? I thought it might have been that black Dobie across the street, but the fence was totally intact.\"\n\n\"No, it was definitely a pit bull,\" I said, not wanting to implicate the Doberman. \"Big, white one. It ran away after I got it off Jennifer.\"\n\n\"You're really brave. Okay, I thought maybe Jen was confused about the breed. She's been kind of loopy from the drugs they gave her, confused in the head. She kept saying _you_ attacked her.\"\n\nI laughed. \"I must have really sharp teeth.\"\n\n\"I know. I told her that was crazy, that you were always sweet and gentle. I still remember how you helped that bird when we were kids.\"\n\nI smiled. \"Of course I remember.\" I wondered what had happened to that sweet girl, the one who loved all living creatures. Kendra said you couldn't change the nature of a thing. The ability to strike out at my tormenters must always have been there. Now I was using it.\n\nI didn't mind. I'd had enough.\n\n\"Thanks for keeping me posted. Will Jen be back at school soon?\" I still didn't know how bad it was.\n\n\"Not for weeks. She has to have surgery for the scarring. The thing ripped off half her nose.\"\n\n\"Oh, that's terrible.\" It was. Plastic surgery might make her even prettier. I wanted Jennifer to know what it meant to be ugly. \"Let me know if there's anything else I can do.\"\n\n\"I will. Everyone says you saved her life.\"\n\nI winced. \"Well, anything for a teammate.\"\n\n\"It's actually good to hear your voice, Violet.\"\n\n\"Is it? Then, maybe you'd want to\u2014I don't know\u2014meet at school to talk some more tomorrow.\"\n\n\"That would be great. I usually sit with Jennifer at lunch.\"\n\n\"I know.\" How I knew.\n\n\"We could sit together until she gets back.\"\n\n\"And talk about birds. Get your mind off things.\"\n\nI hung up and went to bed happy, Grimalkin purring at my feet. Jennifer would be gone for weeks. Weeks! I had Greg all to myself. And now, I wasn't the frizzy-haired, hook-nosed loser I'd been. I was beautiful, talented, confident\u2014at least on the outside. And, on the inside, I was the girl he'd liked all along, the smart girl. At least, smarter than Jennifer.\n\nThe next few weeks were the happiest of my life. Greg and I ate lunch together, studied together, I drove him to school. Even Jennifer's friends were nice to me. They'd seen me pull the dog off her. At first, Greg and I mostly talked about Jennifer, how amazing she was, but soon, we branched off into less-annoying topics, subjects like life, college (we both wanted to be environmental lawyers), current events, subjects an idiot like Jennifer couldn't possibly talk to him about.\n\nWe planned to see the movie _Dead Poets Society,_ which was playing at the mall. \"It wouldn't be a date,\" I told Greg. \"We could just go as friends.\"\n\n\"I guess it's okay. Jennifer didn't want to go to that movie anyway. She said it looked stupid.\"\n\n_Jennifer's stupid._ But I didn't say it. I didn't want Greg mad at me. I wished there was some magic I could work to make Greg see how awful Jennifer was, how we were meant to be together. I couldn't use magic to make bad things happen to people, at least not directly. But what about making someone fall in love with me?\n\nI stopped by Kendra's house the day of our movie non-date. School was out now, and Greg was visiting Jennifer. I put the question to Kendra.\n\n\"Tell me again why I can't cast a spell to make Greg love me?\"\n\nKendra winced. She hated the word _spell_. Said it smacked of mystical books and the silly TV series _Bewitched_ , about a housewife who made magic by wiggling her nose. Magic, she said, came from deep within. It was a matter of harnessing it, rather than learning it.\n\n\"You can't make someone fall in love with you. Love comes from within too.\" She reached out her hand, an awkward gesture for her, and touched my shoulder. \"Unfortunately, you had several opportunities to work your magic, so to speak, with Greg. It hasn't worked.\"\n\n\"It's so unfair.\" I bit my cuticle. \"I have magic powers. I should be able to have anything I want. But this is the only thing I ever wanted.\"\n\n\"Is it really? The only thing?\"\n\nI thought about it, shredding my cuticle as I did. Of course it wasn't the _only_ thing. But Greg was the main thing. Had I gotten him easily, maybe I'd be satisfied, but, as it was, I wanted more. I wanted to be beautiful, more beautiful than Jennifer, than everybody. I was. And powerful. I was that too. But that wasn't enough, or hadn't been. I was so sick of people making fun of me that now I wanted to be better than everyone, at everything. And obliterate my enemies. \"Okay, maybe he's not _all_ I want, but he's the main thing. That and world domination.\" I waited for Kendra to laugh, but she didn't. \"But without Greg, I'll never be happy.\"\n\n\"Oh, dear.\" Kendra pressed her finger to her brow.\n\n\"What?\"\n\n\"I'm just worried you'll never be happy.\"\n\nI took another bite of my cuticle. This time, it bled, but I immediately stopped it.\n\nGreat. I could save money on Band-Aids. What an astounding ability I had.\n\nGreg and I did go to the movie, though, which was about this boarding school teacher who encourages his repressed students to pursue their dreams, write poetry, seize the day. Then, one boy kills himself because he wants to be an actor, but his parents don't see it that way. Of course, the teacher gets fired.\n\n\"So that was a downer,\" I said to Greg as we walked through the mall afterward. \"The message is that if you write poetry and think for yourself, you end up either dead or so beaten down you won't dare have an original thought again.\"\n\nGreg laughed. \"You're right. But most of us don't have parents as bad as that guy's.\"\n\n\"Speak for yourself,\" I muttered.\n\n\"I think the real theme was _carpe diem,_ \" Greg said.\n\nIt was late, and the mall was nearly empty. I could hear, even feel our footsteps on the white marble floor. We were completely in step, just like when we were kids. I wanted to grab Greg's hand.\n\n_Carpe diem. Seize the day._\n\nI didn't grab it. I was too scared he wouldn't take it. Still, our arms brushed as we walked.\n\n\"Do you believe that?\" I asked. \"Seize the day? Do what you want because each day might be your last.\"\n\n\"I do. Don't you?\" He stopped walking to look at me. I could see us, reflected in the mall doors, him, tall and dark, me smaller, my hair flowing down my back. We belonged together.\n\nI shook my head. \"Guess I'm more cautious.\" Still I wondered how it would be to seize the day, do what I wanted. If I grabbed Greg now and kissed him, would he kiss me back? I looked up at him. His eyes met mine. It would be so easy to do that.\n\n\"You're so different than you used to be,\" he said. \"It kind of shows how things can change. One day, you were this ugly duckling. Sorry. But look at you now\u2014a swan.\"\n\n\"Thanks.\" He would kiss me back. He would.\n\n\"Or what happened to Jennifer.\"\n\n\"What do you mean?\" I wanted to go back to talking about me, about how I was a swan.\n\n\"That dog could have killed her. And then, she'd have been gone, and she'd never have known how I felt about her. That accident inspired me. You saved her for me.\"\n\n___Saved her? For you? You've got to be kidding!_\n\n\"Can I show you something? I've been carrying it around because I'm afraid I'll lose it.\"\n\nAnd then before I could say no, no, I don't want to see anything that has to do with Jennifer, he pulled something from his pocket, a red velvet box, and held it near my face.\n\nI backed away. \"What is it?\"\n\n\"It's a promise ring.\" He opened it. There was a silver ring with a heart and the tiniest diamond chip known to mankind in it.\n\n\"Promise? Promise what?\" I felt like my legs might buckle under me, like I'd be on the floor.\n\n\"That we'll get married someday.\"\n\n\"Married? You and Jennifer?\" I felt the bile coming up in my throat. I couldn't speak anymore.\n\nGreg was nodding, grinning like an evil doll in a horror movie. \"I'm giving it to her when she comes back to school. Unless you think I should seize the day and give it to her sooner.\"\n\n\"I . . . I . . .\" I tried to speak but could only choke. I caught sight of myself again in the mall doors. I was so beautiful, so beautiful, so . . . so what? What good was it? It was everything I'd dreamed of, and it wasn't enough.\n\nSuddenly the glass door shattered, blowing out like a bomb had hit it.\n\n\"Whoa!\" Greg jumped back. \"What happened? It didn't hit you, did it?\"\n\nBut it wasn't like he threw his body over it to save me.\n\n\"N-no.\" I saw myself in another door. Then that shattered too. My magic was ungoverned, out of control, getting away from me, and I knew I had to stop before Greg realized it was me, then realized I'd hurt Jennifer. I veered toward the one open door, looking down, half closing my eyes so as not to see myself, and I ran. Beside me, Greg was shouting, \"Whoa! Whoa, watch out! What is that?\" He put his arm around me, shielding me, and I wished I could pretend he was doing it out of love. But I knew much better. We ran through the parking lot. I dared look up, only to see myself in a car's windshield. It, too, shattered, but I hoped Greg didn't notice. I had to get out, get home, get away.\n\nFinally, closing my eyes entirely, I got to my car. I told Greg I thought there was glass in my eye, so he drove us home, me seeing nothing. Thankfully, talk about flying glass at the mall drowned out any thought about talking about Jennifer. Jennifer. Jennifer. I excused my looking down by crying. From the shock of it, I told Greg. I didn't have to pretend. My tears were real.\n\nI should have let her die. I had every chance, but I'd stopped it. Why? Why? What had Jennifer ever done but torment me? Why was I such an idiot?\n\nI wouldn't make that mistake again.\n\nI got home, finally, and groped my way to my bedroom, trying to avoid my mirrored closet door, the giant frame mirror over my dresser. I didn't wash my face or brush my teeth, fearing to enter the bathroom.\n\nBut when I stumbled to bed, something cold and hard touched my flesh.\n\nA mirror. The mirror. Kendra's same mirror she had given me the first day. How was it here?\n\nI stared into it. It didn't shatter. \"Kendra?\"\n\nThus summoned, Kendra appeared. She could see me too, for she stared at me. \"My darling, what's wrong?\"\n\n\"It's useless! He'll never love me! He _loves_ Jennifer! He wants to marry her. Marry! They're seventeen! He's supposed to go to college and be a football star. And my magic is all out of whack. I broke stuff.\"\n\nIn the mirror, Kendra nodded, but her eyes narrowed. \"Were you at Cutler Ridge Mall?\"\n\n\"Yes. How did you know?\"\n\n\"The news reported a bombing, broken glass everywhere. But they couldn't find the bomb.\"\n\n\"That was me.\" I was shaking, thinking about it. \"I couldn't stop it. I don't know if I can stop it now. I'm afraid to look at anything.\"\n\n\"My darling.\" Kendra's face held all the sympathy I never got from anyone else. \"When witches are unhappy, they generally make their displeasure known. It's unfortunate you were in such a public place, but you must calm yourself now.\"\n\n\"So what do I do?\"\n\n\"Try to think about someone else. Try to be happy. Concentrate on other things. Your studies. You're such a smart, wonderful girl. Find someone else. Forget him. Let him go.\"\n\nShe was saying all the things my mother should have said, if she hadn't sucked. Yet it seemed like she was speaking a foreign language. \"Forget Greg? I can't. I love him, only him.\"\n\n\"He doesn't love you. People get over failed romance all the time.\"\n\n\"No, never. Don't you understand? Everyone hates me. _Everyone._ They won't even do a dance routine if I choreograph it. They hate me, all of them. I even hate myself. I'm worthless. Worthless. Greg is the only one who ever cared for me, ever saw anything in me but ugliness.\"\n\n\"Shh.\" Kendra put a finger to her lips. \"That's not true. I love you, Violet. You are my true daughter. And you'll find someone else, someone better. Someone who will love you.\"\n\nI nodded, but I knew I wouldn't, couldn't. I would try to be happy, but the only way I could be was by getting Greg, by plotting Jennifer's utter destruction. I'd missed one opportunity. I wouldn't miss it again. Not just a dog bite, something worse.\n\nI'd have Greg\u2014someday\u2014if it took everything I had.\n\n\"Kendra?\" I asked. \"Am I the fairest one of all?\"\n\n\"Aw, honey, of course you are. Please get some sleep. It will be better. You'll meet someone else, someone who appreciates you as he never did.\"\n\nI nodded. \"Okay.\"\n\nAnd then, she disappeared, and I was looking at my own face.\n\nI was so beautiful.\n\n# PART 2\n\n# _Celine_\n\n# _(The Present)_\n\n#\n\n#\n\nWhen I was eight, my Girl Scout troop went to a sleepover at the zoo. My mom, our leader, had agreed to do it even though animals terrified her\u2014as long as the troop sold enough cookies to cover the cost. I think Mom thought it was a safe bet. We were rich, suburban kids who didn't have to work for much. But the thing is, everyone wants cookies. We figured if we each sold 150 boxes, we'd have enough for the sleepover _and_ matching leopard-print bandannas. Then we nagged our moms to take us door to door or hold booths in front of Publix supermarket so we could make our goal.\n\nMom seemed way less excited than scout leaders are supposed to be when she announced we'd made it. I didn't blame her. She was really scared of animals, and with good reason. When she was in high school, she was attacked by a dog that had bitten her face and arms so badly she needed plastic surgery. After that, she freaked at every barking dog. More than one had come after her. So, of course, we'd had no pets. It wasn't just dogs that stalked her. A few times, birds had attacked her, and once, when we were bike riding, a squirrel jumped right into her basket, ran up her arm, and grabbed her face. She'd crashed, and I crashed into her. She shook for an hour. So my mother's fears weren't unreasonable. But I really wanted to do the sleepover, so she agreed. Besides, the zoo animals were in cages.\n\nThat night, we got to the zoo at six. After a dinner of hot dogs, Cool Ranch Doritos, and pink lemonade (I remember all the details because of what happened after), we went on a special after-hours tour. One of the docents, a strikingly beautiful woman with bright auburn hair, took us first to the reptile house. Mom clutched my hand as we watched the savannah monitor show its pointed teeth. We walked on. Quickly. Next were the wild animals who'd been brought in for the night. The lion roared. Beside me, I felt Mom stiffen. I hugged her in solidarity.\n\n\"It's okay, hon,\" said our tour guide, assuming I was the baby who was scared. \"The cage is made of steel and has two separate doors. He's not getting out.\"\n\nMy mother laughed. \"I know. I'm being silly.\"\n\n\"Not at all. Lions are powerful animals. Better to fear too much than too little.\"\n\n\"Thank you.\" Mom squinted at the woman. \"You . . . you look so familiar. I wonder if we went to high school together.\" She touched the scar on her arm. \"But no, you'd be too young. You can't be more than twenty-five.\"\n\nThe docent nodded. \"I get that a lot. I seem to look like a lot of people.\"\n\nWe started to walk back to our sleeping quarters. The zoo at night was an eerie place, full of strange cries and creeping shadows. Above us, the canopied trees seemed to move, even though there was no breeze. Then, I saw a shape.\n\n\"Look! A monkey!\"\n\nIt wasn't in one of the trees contained in a cage. Instead, it was in a ficus tree growing near the path.\n\nAll the girls looked up, picking it out from the shadows. \"Aww, how cute,\" my friend Laurel said.\n\n\"Is it supposed to be out here?\" Mom said. \"It's not in a cage. Did it get out?\"\n\n\"Oh, Mom, it's just a little monkey.\" I heard her heavy breathing.\n\nAnd just as I said, \"little monkey,\" the creature let out a high-pitched shriek, swung down from the lower branch, then launched itself right at my mother. I stepped in front of it too late. It was on her.\n\nFrom a distance, in the dark, it had seemed like a tiny monkey. Close up, it was way bigger, the size of a cocker spaniel, with long, powerful arms. My mother crumpled to the ground, and the monkey was grabbing her hair, pulling it over and over so her head was slamming, slamming, slamming on the pavement. The monkey sunk its teeth into her cheek, her chin. I screamed. I screamed and was kicking at it, over and over, get it off my mother, away from her! My mother must have screamed too, but all I could hear were my own shrieks, see the dark blur of my fellow troop members running for cover, the docent calling for help on her radio, my mother shielding her face, the monkey biting her arms, her legs. I fell backward from kicking, but the monkey was still on my mother, biting her.\n\nFinally, someone came. They shot the monkey with a dart gun and pulled it off her. It was too late.\n\nMy mother was airlifted to the hospital, but the damage was too great. After a few days in ICU, they took her off life support.\n\nMy mother was dead.\n\n\"It was all my fault,\" I kept saying at her wake. Everyone was there, saying things I couldn't hear, didn't want to hear. My mother was dead. My mother was gone.\n\nI couldn't even see her. It was a wake, not a viewing. The funeral home hadn't been able to fix her up enough to show. Despite her scars, my mother had been a beautiful woman. She'd taken great pride in her looks. She wouldn't have wanted anyone to see her looking less than her best.\n\n\"Of course it wasn't your fault,\" a woman's voice said above me.\n\nI looked up to see an angel's face with bright blue, almost violet, eyes; fringed, long black lashes; all framed by wavy, auburn hair. I recognized her. The docent from the zoo.\n\n\"If anything,\" she said, \"it's mine.\" She turned to my father. \"I've been over and over it in my head, so many times, reliving it\u2014how the monkey got out, how I should have seen it first, taken you girls to shelter. Warned her.\" As she spoke, a tear coursed down her face. She wiped it away, all the while staring at Dad.\n\n\"Do I know you?\" he asked. \"You look so familiar.\"\n\nMy mother had said the same thing.\n\n\"I volunteer at the zoo,\" the docent said. \"I took the girls around . . . that night.\"\n\nDad shook his head. \"It was the fault of the zoo management or whoever should have locked that beast in a cage. If volunteers were responsible for safety, that is a sad state of affairs. Poor Jennifer was terrified of animals, and now . . .\"\n\n\"I am so sorry,\" said the docent.\n\nDad's eyes glistened. He looked at the woman again. \"Really, though. You look so familiar.\"\n\nThe docent wiped another tear. \"Don't you remember me, Greg? We went to school together. Violet Appel.\"\n\n\"Violet?\" My father stepped closer, staring at her. \"Violet, it's been so long.\"\n\nI could barely keep my eyes off her myself. I hadn't seen her that well that night at the zoo.\n\nMy mother had been beautiful, and people said I was too. But this woman, Violet, was different. She was stunning. Like, I actually felt stunned to look at her, like electric shock.\n\nMy father took her hand and sobbed. \"Violet!\"\n\n\"There, there, it will be okay, Greg,\" she said.\n\nThey stood there a long time, Violet holding his hand, my father crying, until finally, she said, \"I should let you greet the others.\"\n\nDad nodded. \"I'm so glad to see you, Violet. Maybe we can talk later.\"\n\n\"Of course, Greg.\" Violet moved away. \"I've missed you.\"\n\nDad said to me, \"Violet was my best friend in elementary school. Such a nice girl. I haven't seen her in years.\"\n\nAt the time, I didn't think about it, but now I realize that Violet had told my mother she didn't know her. She'd obviously lied.\n\nAfter the wake, she was back. She seemed to really want to talk to Dad. \"After what happened, I quit volunteering at the zoo. I couldn't . . . I couldn't go back.\" She shuddered a little. \"But I've loved animals since we made those birdhouses with your father.\" She touched Dad's hand.\n\nDad smiled. \"I remember that. Dad was asking about you last time I saw him. He remembers all that old stuff.\"\n\n\"Grandpa made birdhouses?\" Grandpa didn't remember my name, usually.\n\nViolet knelt beside me. She smelled like roses. \"Sweet child. Has your father never made a birdhouse with you?\"\n\nDad said, \"Jennifer was terrified of animals since the dog attack in high school\u2014you remember. She avoided them, and yet, they couldn't seem to stay away from her. Cats peed on her shoes. A raccoon lived on her car for a month and wouldn't leave when shooed. We had to sell the car. She had a job at an ad agency downtown, but she was forced to quit when pigeons ganged up on her on the street. Now this . . .\" His face broke, and he began to sob. \"I should have taken the troop myself! I should never have let her near the zoo.\"\n\nViolet embraced him, rocking him back and forth while he sobbed in her arms. When I think about it now, it was weird, like he was a little kid. But at the time, it seemed so perfectly comforting that I hugged both of them. She said, \"Don't blame yourself, Greg. There's no way you could have known.\"\n\nWe stood there, crying, the three of us, and somehow, Violet's presence made it better, made it almost right. It was like magic.\n\nTwo days after the funeral, Violet arrived with an animal carrier. From it, she drew a small ball of fur, orange and white. \"Someone left her in my yard. My cat was my fondest friend when I was a lonely teenager, so I thought a kitten might help. A cat always helps.\"\n\nI couldn't believe anyone as beautiful as her had ever been lonely.\n\nShe placed the kitten on the ground. I reached out, but she started to run away from me. Violet touched its back, and suddenly it switched direction and gamboled toward me.\n\n\"I asked your dad if you could have it,\" she said.\n\n\"Really?\" The kitten pawed at my leg. I scooped it up, and it began to purr, like the small motorboats we used to rent on vacation at the lake.\n\n\"Really,\" my father said from behind me. \"A child should have a pet.\"\n\nThat made me feel bad. The reason we hadn't had a pet was because of my mother, her fears. Now, I could have a cat. I'd rather have had my mother. But the kitten curled itself into a ball on my lap. It felt so warm.\n\n\"She likes you,\" Violet said, \"but who wouldn't like a sweet, pretty girl like you?\"\n\n\"Oh, I'm not pretty,\" I said, though I'd been hearing it all my life. In fact, I thought I was strange-looking, with jet-black hair and white skin like my father's, though I had my mother's blue eyes. I hated when people said I was beautiful because it was always something like what Violet had said, something that implied I should have no problems because of my looks.\n\nObviously, that wasn't true. If anything, my looks made people like me less. The first day of kindergarten, the kids had just stood around, staring at me. But then no one asked me over to their houses or to sit together at lunch. Mom had always said they were scared of me. That was why she'd volunteered to be the scout leader, so she could help me make friends. Once people got to know me, they forgot about my looks\u2014sometimes.\n\nBut I wanted Violet to like me, so I didn't say any of this. The kitten nuzzled my face. Violet stood real close to Dad, smiling up at him.\n\n\"Your modesty makes you all the more beautiful,\" she said. \"Maybe one day, you can come over and see my cat. I'll make you dinner too. I'm a good cook.\"\n\n\"Okay.\" I hugged the little cat.\n\n\"We'll make a date.\" Dad touched Violet's shoulder. \"Thank you.\"\n\nI named the kitten Sapphire for her blue eyes, and the next week, we had dinner at Violet's house. She made what she called \"gourmet\" mac and cheese, with some kind of weird smoked cheese in it and told stories about working as a lawyer for the US Attorney's office, putting criminals away in jail. I played with her cat, whose name was Grimalkin.\n\nIn the next few months, we spent more and more time with Violet until we were seeing her every day. I didn't mind. Violet had two passions, animals and her beauty routine, so some weekends, we went on nature walks, like in the Everglades, seeing alligators and birds and once, a panther. Other times, she did my hair so easily it seemed like magic, or we went for mani-pedis, stuff I'd done with Mom. Of course, Violet wasn't my mom, but having her around made it a little easier. And Dad was a lot less sad. I noticed the photos of my mother went from the living room to his bedroom. Then, one day, they disappeared. I found a big photo of the three of us in his closet. I took it out and put it under my bed, but I didn't let either Dad or Violet see it.\n\nA year later, they got married. I wanted to be a flower girl in the wedding, but Dad said it wouldn't be formal, and it had to be small. \"I'm a widower. It's only been a year since your mom died. We don't want a public wedding.\"\n\n\"Yes, no fancy dress for me, but I've got my man.\" Violet leaned over and kissed my father on the mouth. I looked down. They kissed a lot. \"Besides, none of your dad's school friends liked me much.\"\n\n\"Why wouldn't they? Were they jealous because you were so beautiful?\"\n\nViolet looked at Dad and laughed. Then, she hugged me. \"Oh, my sweet girl, you crack me up.\"\n\nI didn't know what the joke was, but I laughed along anyway.\n\nI wanted Violet to love me. She was the only mother I had, after all.\n\n#\n\n#\n\n\"Can you guys stop making out?\" Violet was doing my hair for dance. At least, she was until my father came in. Then, she launched at him like a pumpkin in a catapult.\n\n\"We weren't making out. I'm just kissing my wife hello.\"\n\nI rolled my eyes. They were always kissing, and since Violet had my hair in her hand, it was getting tugged. \"I'm going to be late.\"\n\n\"Okay.\" Violet tore herself away from Dad and went back to brushing. \"Your hair's so pretty. Black as a crow.\"\n\n\"A crow?\" I wrinkled my nose. \"Crows are ugly.\"\n\n\"Have you ever really looked at a crow, Celine?\" Violet continued to brush my hair. I was eleven, old enough to do my own bun, but it was hard to tell Violet that. \"I mean, up close? They aren't completely black. In the light, you can see purple and green. They almost shimmer. And each feather is lined up perfectly with every other one. So beautiful.\"\n\n\"Why do you love crows so much?\"\n\n\"I love all birds. When I'm old, I mean to travel the world and see as many as I can. But crows and ravens will always be my favorite. It was because of a crow I met your father.\" She looked up at Dad. \"Remember?\"\n\n\"Of course. We rescued it on the playground. Some boys were throwing rocks at it, and Violet saved it.\"\n\n\"After that, we were always friends,\" Violet said.\n\nI detected a false note in her words. She and my father hadn't \"always\" been friends. She'd just showed up after my mother died. But there was no point in saying that, particularly when she had my hair in her hands. Besides, I loved Violet, and I liked birds too. Sometimes, we went for walks and counted how many different kinds we saw. \"Well, if you like crows, I'll like them too.\"\n\nShe hugged me for that, though her eyes were on Dad. Then, she put the last pins in and sprayed my bun. I always had the best hair in class.\n\nBut then, everything changed.\n\nI was thirteen when I had my first period. I had it at school, of course, because that's just how things go with me. And, when I went down to the clinic to get a pad, one of my friend's moms was there. \"Do you want me to call your dad?\" she asked.\n\n\"What? No! He'll get all weird. Call Violet if you have to call anyone.\"\n\nLaurel's mom got kind of an odd look on her face. \"You and Violet actually get along, huh?\"\n\n\"Yeah, she's nice. We're going bird-watching in Texas for spring break.\"\n\nShe nodded. \"Okay. I wish your mom were here. This is a day a mother and daughter should share. I can't believe it's been five years.\" She gazed at me. She and my mother had been best friends when they were kids. \"You look so much like your mom.\"\n\n\"You think so?\" I could barely remember her face. When I tried to picture her, I could only see the photo I still kept under my bed, her in the same pink dress, the same frozen smile. I couldn't remember the look and feel of her at all anymore. It was like the photograph had painted over my real memories of her. \"I have black hair like my dad.\"\n\n\"Yeah, but those big blue eyes, they're Jennifer's eyes.\"\n\nShe looked about to cry, so I said, \"It's all right.\" Because what else, really, was there to say? I just wanted out of there.\n\nFinally, she called Violet and shared the news. \"You can go back to class now,\" she said after.\n\nI got out of there as fast as I could.\n\nBut half an hour later, Violet was there, carrying dark jeans and checking me out of school.\n\n\"You didn't have to,\" I said. \"We're doing a lab in science. I sort of wanted to\u2014\"\n\n\"Science can wait. This is a special occasion for us girls.\"\n\nI shrugged. It was sort of weird that she and Dad hadn't had other kids, but it was better for me. So I guessed if I had to act like her daughter, it was worth it.\n\nShe took me to Marble Slab Creamery. We both got chocolate with Oreos mixed in, and as we paid, the college boy at the register looked me up and down. I turned away, blushing like I always did when guys stared at me. Later, when Dad got home, Violet made my favorite dinner (still the mac and cheese) and filled him in. \"Your little girl's becoming a woman.\"\n\nI squirmed when she said that, squirmed more when he said, \"A beautiful woman.\" With his hand, he turned my face toward him. \"Like your mother.\"\n\n\"She doesn't look like her.\" Violet touched my dad's face. \"She looks like you.\"\n\n\"More like Jenny,\" he said. \"My lord, Vi, she's even as beautiful as you.\"\n\nIt was only for a second, so I thought I imagined it. But, in that instant, I saw Violet's eyes turn from lavender to black as a crow's. Then they went back to normal.\n\nBut that couldn't have happened. She wasn't a witch or anything.\n\n\"Do you think so, Greg?\" Violet asked. \"Yes, I suppose she is, just as beautiful.\"\n\nShe, too, gazed at me, and I saw hatred behind her smile. From that day on, she never looked at me the same way again.\n\nAnd when we went to Texas over break, a red-footed booby swooped down and attacked my face.\n\n#\n\n#\n\n# _Sophomore Year_\n\nWhen I get on the bus, wearing my usual hoodie, army boots, and jeans a size too large, Whitney Jacobs stage-whispers that I look like a bag lady, but Alex Abercrombie does a Parisian pass, brushing his hand against my ass to get by me. I ignore them both. _Eyes on the prize._ Laurel's holding a seat for me. She's near the back, where we live, waving and generally looking more excited than anyone has the right to look at six-forty in the morning.\n\nI know why. And sure enough:\n\n\"Is your mom letting you go to the concert?\" she asks the instant I sit.\n\nThe concert! I let down my guard and do a little seat dance to let her know I'm excited too, then say, \"My wicked _step_ mother, if that's who you mean, isn't in on the decision. But my dad says yes as long as your mom's going.\"\n\n\"Well, of course. What did he think, a couple of fifteen-year-olds are going to hitch to Orlando?\"\n\n\"For Jonah Prince, I completely would.\"\n\n\"Can I get an 'amen'?\"\n\nAnd we squee in unison.\n\nWhitney and her mean girlfriends look back at us and roll their eyes. I smile.\n\n\"Jelly?\" Laurel says.\n\n\"Not of anyone who still says 'jelly.'\" Whitney turns away.\n\n\"Is 'jelly' not a thing anymore?\" Laurel whispers to me.\n\n\"Don't worry about it. She is such a hater.\" I pat Laurel's shoulder.\n\nJonah Prince, as everyone knows, is an incredibly gorgeous and gifted singer-songwriter. Previously the front man for the Boyz Band, he went out on his own when he noticed he was the only one with talent. And beautiful green eyes. And a hot British accent. I'm in love. Laurel and I host unofficial Jonah Prince pages on every social networking site I can think of, even Facebook, which only parents use.\n\nOkay, so I have no life. I study, hang with Laurel, and listen to Jonah. But I know that, when we go to his concert in Orlando (where we'll get front row floor seats because we're in his fan club and plan to spend the entire night before tickets go on sale on the Ticketmaster site, entering CAPTCHA codes so we can get tickets the very first second the presale begins\u2014and if that doesn't work, I'm spending a year's worth of babysitting money to buy them on StubHub), Jonah will pick me out from the crowd, his eyes attracted by the extraordinarily beautiful and artistic sign I'll make, and my total memorization of all his songs. He'll lead me onstage. I'll have the perfect outfit, of course, something that's actually figure flattering. Once there, he'll sing every song to only me, ignoring the boos of the other poor girls. I'll ask him to bring Laurel onstage to meet his drummer. After the concert, we'll talk for hours about the charity work he did in Haiti last year or the new songs he's composing. He'll ask me to come along on his tour, so I can get permanently away from my stepmother, who hates me.\n\nYeah, so it's a bit of a stretch. But it _has_ to happen. There has to be something really good coming to me, to make up for Violet.\n\nIn some cultures, like the Greeks and Turkish, they believe in the Evil Eye. It's the idea that, if someone envies you, bad things will happen. Since that fateful day when my father said I was as beautiful as Violet, I've become more beautiful (don't hate me\u2014I can't help it) and Violet's eyes have been seriously evil. Curling irons attack me, leaving scars. Tweezers rip out my eyelashes. Cleansers turn toxic and give me rashes. I avoid Violet\u2014and beauty products\u2014as much as possible. I wear no makeup, no hair spray. I don't get fake nails or even nice clothes. But people still stare at me. And Violet notices. And hates me for it.\n\n\"That's really trite,\" Laurel says, pointing to the chem notebook I'm studying for today's test. It has _J.P. 4-ever_ written on it in pink highlighter.\n\n\"Nothing about Jonah Prince is trite.\"\n\n\"Do you even know what _trite_ means?\"\n\n\"Yes, I do.\" The bus hits a bump, and I steady my notebook, covering the writing with my palm. \"But it probably makes us sound like nerds who use words like\"\u2014I lower my voice\u2014\"trite.\"\n\n\"Okay,\" she whispers, then giggles.\n\n\"Jonah is special. He writes his own songs and plays four instruments.\"\n\n\"Preaching to the choir. But my mom says he's just like Justin Bieber. Or someone named Bruce Springsteen, who was apparently famous when she was a teenager. Or the Beatles. Or Elvis. She says the names change, but the pathetic-ness of a teenage girl, scribbling initials of a guy she's never met, _will_ never meet, is always the same.\"\n\n\"Laurel, I love your mom. She's the closest thing I have to a mom, and she lets me call her Gennifer. Plus, she's letting us use her credit card to buy tickets.\"\n\n\"And driving us to Orlando,\" Laurel reminds me.\n\n\"And driving us to Orlando,\" I agree. \"Your mom is completely legit.\"\n\nLaurel laughs because she knows that's something her mom would say, trying to be cool. Laurel and I have been best friends since we were born a month apart fifteen years ago, to moms who were best friends. I'm told that, usually, when people's mothers want them to be friends, they inevitably end up hating each other. Laurel and I are the exception. At this point, she's kind of the only person I hang with, but Dad says it's better to have one good friend than ten bad ones. Violet sort of rolls her eyes when he says that, but it's true. Also, it's Laurel's house I hide in almost every weekend. Laurel's mom knows how weird Violet is, so she lets me sleep over. Without Laurel, I'd spend my weekends watching my dad and Violet suck face.\n\nStill, I say, \"But your mom's old, Laurel. She doesn't remember what it's like to be our age. When Jonah sings 'Beautiful but Deadly,' it's like we've already met, like he's looking into my soul.\"\n\nLaurel rolls her eyes. \"Yeah, I know. Being the most beautiful girl on the planet is hard for you.\"\n\nI look back at my chem notes. She knows I hate when people say I'm beautiful.\n\n\"Celine?\" Laurel tries to look over my notebook. She's super-cute with wavy, dark hair. She thinks she's fat, but she's not. She's curvy.\n\n\"I'm sorry,\" I say. \"I'm on a chemistry grind. I want to get an A so I can take AP chem next year. And this chapter is hard.\"\n\n\"All work and no play . . . ,\" she singsongs. \"You know, you could go for a normal guy. Bryce Richardson is into you, and he sort of looks like Jonah.\"\n\n\"Bryce Richardson isn't into me.\" Bryce is the hottest guy in our class\u2014and so knows it. I've seen him make fun of smaller guys, heavy girls, people with acne. I heard he wouldn't go out with one girl because her eyebrows were too close to her eyes. The guys with huge egos always like me. The right ones never do. I just want a smart, funny guy, but it's like guys my age are scared of me.\n\n\"I've seen Bryce looking at you,\" Laurel insists.\n\n\"I don't think so. You should go for him if you like him. I'm a little too sapiosexual for him.\"\n\n\"Sapiosexual? What's that? Sounds dirty.\"\n\n\"It means attracted to smart people. That kind of lets out Bryce Richardson.\"\n\nShe sighs. \"I'm giving up now. Next order of business: _Oliver!_ auditions are after school today.\"\n\nI sigh and close the notebook. \"Guess I'm done studying.\"\n\n\"You know you'll ace it.\"\n\nI don't want to talk about this any more than I want to talk about why I don't like Bryce Richardson. But Laurel's totally obsessed with theater. \"Why are we trying out for a musical again?\"\n\n\"Because I need to pay my dues in the chorus\u2014or, hopefully, a small but pivotal role\u2014so I can get a lead next year.\"\n\n\"I'm sorry. I wasn't clear. Why am _I_ trying out? I have stage fright. And no talent.\"\n\n\"Because you're my best friend. And because, if I'm in the play and going to rehearsals every day, you can't hang out at my house to avoid Lady Violet\u2014unless you want my mom to teach you quilting. She'd completely love to do that.\"\n\nLaurel knows how important Violet avoidance is to me. At least, she gave me the heads-up on the play. She made me watch the movie version a few weeks ago, and it was pretty good. I guess being in the chorus wouldn't be horrible, as long as I can be in back where no one sees me.\n\n\"Okay,\" I say, \"so what do I have to do? Hopefully, not dance. Because I'm bad at that.\"\n\n\"Just a song. The drama teacher said 'Happy Birthday' __ was okay for chorus. They just want to see if you can sing on pitch.\"\n\n\"I think I remember all the words to 'Happy Birthday,'\" I tell Laurel, \"but we can go over them at lunch, just in case.\"\n\nLaurel rolls her eyes.\n\nWhen we get off the bus, Pierre Duval, a senior I don't know, is rolling by in his BMW convertible. \"Nice jeans, Celine,\" he yells. \"I'd love to get in them.\"\n\nI move to the other side of Laurel and try to ignore him.\n\n\"I am not trying for the role of Oliver,\" the guy onstage tells the drama teacher, Mrs. Connors. \"I may be little, but I'm not a kid. And I'm not cute.\"\n\nI've noticed this guy before, even though he's a year ahead of me. He was in my bio class last year. His name is Goose Guzman. He has dark, wavy hair, olive skin, a wicked sense of humor . . . and he's maybe four and a half feet tall. Is it wrong to say I want to meet him _because_ he's a little person? I figure he knows how it feels to be stared at for something beyond your control. Plus, he's funny\u2014as he's demonstrating at the moment.\n\n\"You're definitely not cute, Goose,\" Mrs. Connors says.\n\n\"Thank you.\" He half smiles, lifting only one side of his mouth. \"I'm also not a soprano.\" He makes his voice real deep when he says that.\n\n\"What part did you want to try for?\" Connors asks.\n\nGoose shrugs. \"Bill Sikes. Maybe Fagin.\"\n\nShe sighs. \"Bill Sikes is supposed to be a big, scary guy, Goose. I don't think\u2014\"\n\n\"Oh, I see how it is. You're being heightist.\" Goose stands up very straight. \"Someday, when I'm the next Peter Dinklage or Warwick Davis, I'll tell people my high school drama teacher wouldn't cast me because I was too short.\"\n\nI chuckle. The guy is awesome.\n\nConnors rolls her eyes. \"I'm not refusing to cast you, just refusing to cast you as Bill Sikes. I won't consider you for Widow Corney either, if that's okay. Maybe you can be\u2014I mean try for\u2014the Artful Dodger.\"\n\nI remember from the movie that the Artful Dodger was a clever teenage pickpocket. Good call. Goose seems okay with that idea. At least, both sides of his mouth are now up. \"Fine. Either that or Fagin.\"\n\nMrs. Connors shrugs. \"Try for both. We'll see what happens. But who am I going to cast as Oliver? Most of the boys here are huge.\"\n\n\"Cast a girl,\" Goose says, \"a little, cute girl. Cast her.\"\n\nAnd he points right at me. I shrink down in my seat and try to think of some clever retort about how he doesn't seem to mind looks-based casting when he's not the one being stereotyped. But I realize it's probably not the same. Mrs. Connors looks at me and claps. \"Splendid idea. Anyone under five foot three\u2014except Goose\u2014will read for Oliver.\" She walks over to me. \"Can you sing 'Where Is Love?'\"\n\nI remember that song from the movie too. It perfectly described my life, a lonely child yearning for a lost mother.\n\nBut I don't want to be Oliver. I hate being the center of attention. This reminds me of second grade, when we did _Cinderella_ , and I really wanted to be a mouse. I got Cinderella. Not only did I have to memorize a ton of lines, but all the mouse girls hated me. I glare at Laurel, who's five eight, then say to Mrs. Connors, \"I only wanted chorus. I was going to sing 'Happy Birthday _._ '\"\n\nShe sighs audibly. \"Maybe just try?\"\n\nAnd, like the people pleaser I am, I say, \"I guess if you have the sheet music.\"\n\nTwenty minutes later, I'm standing onstage singing \"Where Is Love?\" Three other girls\u2014girls who actually wanted to be Oliver\u2014have already tried, and I could barely hear them. I could do that. No one's heard me sing before. I even avoid karaoke at birthday parties. I could waver off pitch on the high notes or sing so softly that no one could hear me. But when I get onstage, I start thinking about Mom, about how we used to bake cutout cookies at Christmas, and the Hannah Montana costume she made me for Halloween when I was five and thought Hannah Montana was cool. I remember all the crafts we did for Girl Scouts before that fateful day. And I start to sing about poor miserable Oliver, searching for his mother.\n\nHalfway through, I see Mrs. Connors wipe away a tear.\n\nWhen I finish the song, there's silence except a few sniffles. Sniffles!\n\nSomeone\u2014Goose\u2014starts to applaud and whistle. Thanks, dude. Then, everyone else applauds too.\n\nI go back to my seat, sort of hoping no one else tries out. Maybe I want the part after all. I wonder if this is how Jonah feels onstage. That would be something else we could bond about.\n\nNo one else tries out. Mrs. Connors starts with the girls who want to play Nancy. Which is all of them, except me. So after Laurel and a few others go, I head for the ladies' room.\n\n\"She thinks she's all that,\" I hear one of the other girls whisper as I walk by.\n\nSo unfair! I want to round on her and ask what, exactly, I did to make her say that. Of course I don't.\n\nWhen I come out, angrily, I pass Goose by the water fountain. \"Hey,\" he says.\n\n\"Hey,\" I reply.\n\nHe doesn't quite reach my shoulder. He has these chocolate brown puppy-dog eyes. \"You really moved me in there. My parents, they take in foster kids, and when you sang that song, it made me think of them, like, think of their _struggle._ Also, those are cool boots.\"\n\n\"Thanks. My mother died when I was little. The song reminded me of her.\" Why did I say that? I never talk about my mother. To anyone, let alone complete strangers. I hug my notebook to my chest. \"Um, I'm Celine.\"\n\nHe nods. \"I know. We had bio together last year. Goose.\"\n\n\"Oh, yeah. I didn't think you remembered me. I was just a freshman.\" He was always at the center of a group of theater kids, making people laugh, while I sat off by myself and acted studious because I didn't know anyone in the class. All my friends took Earth-Space, the normal ninth grade class, but I love science, so I was ahead.\n\n\"You were the one who stood up for the sub that day,\" he says.\n\n\"Oh.\" Ugh. I look down, embarrassed. But since he's shorter than I am, down is right at his face. \"You remember that?\"\n\n\"It was memorable.\" He waggles his eyebrows, smiling that half smile again.\n\nYeah. One day in class, we had a sub. A new one, probably fresh out of college, and he stuttered. Badly. Since the class didn't have assigned seating, he had to call roll. Which was painful. It took maybe ten minutes to call thirty names, but it seemed way longer because people were being so rude. He searched for C-C-Columbo and G-G-Guzman with everyone laughing at him. Well, almost everyone. I actually specifically noticed that Goose wasn't. But he wasn't telling his friends to shut up either.\n\nWhen the sub reached the last name, Torres, his stutter made it come out Tit-tit-Torres, and everyone lost it, including the sub, who looked about to cry. It really bugs me when people make fun of someone who can't help it, and I didn't have any friends in that class anyway.\n\nSo I stood up. I had nothing to lose. I turned, faced the class, and just glared at everyone. And people actually stopped laughing, at least enough to hear me when I said, \"What are you, four years old? I'm sure it's soooo funny.\"\n\nWhich is the bravest thing I've done since trying to kick the monkey off Mom.\n\nAnd, for whatever reason, that shut everyone up. I knew people thought I was a bitch, but I sort of thought the same thing about them.\n\n\"It wasn't a big deal,\" I say. \"I just wanted to get started with biology. I'm going to study nursing, so I need good grades in science.\"\n\nGoose shakes his head like my dad does when he knows I'm lying. \"Don't downplay how brave you were, standing up to a roomful of people. You weren't the only person who was annoyed, just the only one who said anything. I didn't. You were like a warrior.\"\n\nI shrug and look away so he can't see my smile. Warrior. I like that.\n\n\"Serious badass. And you were brave just now, onstage.\"\n\nGlad to change the subject, I say, \" _That's_ true. I was terrified. I'm probably the one person here who doesn't want a lead. Hopefully, I won't get it.\" Now that it's over, I'm backing off the fact that I really sort of want it.\n\n\"Oh, you'll get it.\" He points to my notebook. \"So who's J.P.? Your boyfriend?\"\n\nHe's looking at the writing on my notebook.\n\nSo. Embarrassing. I try to sneak my fingers on top of the writing. \"Um, actually, he's a singer. Jonah Prince.\" I realize this does not make me sound like a warrior-type. \"It's dumb. My friend Laurel and I are getting tickets for his concert in Orlando this summer, so I was, you know, being a fangirl.\" I try not to look like someone who wrote three Tumblr posts about Jonah in the past week.\n\n\"Jonah Prince. That's the guy with the dance moves, right?\" Goose executes a Jonah Prince\u2013like spin. \"I've got some moves of my own.\"\n\nI stifle a giggle. He says, \"It's okay to laugh. I wasn't being serious.\"\n\n\"I guess we should go back in,\" I say. \"I think there's a dance audition.\" Which Laurel totally lied about.\n\n\"Wouldn't want to miss that. As you can tell, I'm a brilliant dancer.\"\n\nThat time, I do laugh. We head for the auditorium. Goose holds the door for me, then takes a seat with one of the better Nancy candidates. Another girl is onstage, singing \"As Long as He Needs Me\" off-key. I sit by Laurel.\n\nThe next day, when the cast list goes up, I'm Oliver. Goose Guzman is listed as the Artful Dodger, and he's hugging the girl he was sitting with, who got Nancy.\n\n\"I'm someone named Charlotte, and chorus,\" Laurel says. \"We'll never rehearse together.\" But I can tell she's thrilled she got a speaking part.\n\n\"Congratulations. Yes, we will. Oliver's in almost every scene.\" _Le sigh._ \"I'll be at your rehearsals and a bunch more. At least, it gets me away from Violet. And squee! Charlotte is a great part!\"\n\n\"The beginning of a brilliant theater career,\" Laurel agrees. And we jump up and down.\n\n#\n\n#\n\nThat night, at dinner, I tell Dad and Violet about the play.\n\nDespite a full-time lawyer job, Violet is an incredible cook. She scours food blogs and cookbooks and makes recipes that look like they should take hours, but still gets dinner on the table by seven. I think she stays up all night, chopping stuff. So, while normal kids are eating frozen breaded chicken breasts or even ramen, my family gets shrimp \u00e9touff\u00e9e with grits or, tonight, beef Wellington.\n\nGod, how I wish we could eat frozen chicken breasts sometimes. I love frozen chicken breasts with barbecue sauce on them. It's such a mom thing to make.\n\nI miss my mom.\n\nBut I eat everything in hopes that it will make Violet hate me less. It doesn't work.\n\n\"What's this in the middle?\" I ask, picking at my steak, which has been rolled in a crust like a pie and has something gray that looks like pureed mouse in the center.\n\n\"It's really good.\" Dad smiles at Violet and rubs her back. \"You're such a great cook. I don't know how you have time for this.\"\n\n\"We make time for those we love,\" she says.\n\nAnd then, he kisses her. On the lips. With tongue. Long pause while I throw up in my mouth.\n\n\"But what is it?\" I start to scrape it off, figuring they won't notice since they're so busy sucking face.\n\n\"Darling, don't do that. It's p\u00e2t\u00e9, okay? It's delicious. It's . . .\"\n\n_Liver! Do you know they force-feed the geese through a tube to make their livers more \"buttery\"?_ But I don't say it. I don't use any unnecessary words with Violet. I try to stuff the liver into my mouth to keep the words from coming out.\n\n\"Mmm. You're a great cook, Violet, um, Mom.\" In the beginning, she'd asked me to call her Mom, but it had seemed wrong. But, as years went by and everyone else had a mother, I changed my mind. It wasn't like Mom was going to know about it. And Violet was nice. Then.\n\n_Was nice._\n\nShe'd seemed happy about it. Dad said Violet had never had a good relationship with her own mother, a beautiful woman who'd joined us for Thanksgiving exactly once in the years Violet and Dad had been together, and who'd criticized the dry turkey (It was perfect, like everything Violet cooked) and left before dessert. I try to remember that when I get mad at Violet, but it's not easy.\n\n\"So I'm going to be in the school musical,\" I say, mostly to Dad. \" _Oliver!_ \"\n\n\"Oh, that's wonderful,\" Violet says. \"It's always so much fun to be in the chorus.\"\n\n\"Absolutely,\" Dad says. \"Violet did musicals in high school, and she was on the dance team.\" He's so pleased we have something in common.\n\n\"That's so cool,\" I say. \"I suck at dance. What were you in?\"\n\nViolet is actually momentarily happy, talking about herself. \"Senior year, we did _My Fair Lady_. I played the lead, Eliza. And junior year, I was Irene Molloy in _Hello, Dolly!_ \"\n\n\"She was breathtaking,\" Dad says, and takes a big bite of his liver-y steak. \"Just like this meal. I don't know how one woman can be so good at everything.\"\n\nAlmost like she's a witch.\n\n\"So breathtaking you never gave me a second look,\" Violet says. \"You were busy looking elsewhere.\"\n\nDad looks down at his plate. \"Mmm.\"\n\nI know I have to tell them. \"Actually, I'm not in the chorus. It's kind of a funny story. They were looking for a short girl to play Oliver.\"\n\n\"So you have the lead?\" Dad says. \"That's wonderful.\" Because he is just that clueless to how Violet will react.\n\nShe's already frowning. I say, \"It's not really the lead. All the girls wanted Nancy.\"\n\n\"Don't be an idiot,\" Violet snaps. \"The play's called _Oliver!_ not _Nancy!_ \"\n\n\"But I'm playing a boy. I'll have to wear my hair in a cap and look ugly.\" She should love that.\n\n\"Maybe you should cut it off, immerse yourself in the role of a London street urchin. When I played Eliza, I talked in a British accent twenty-four seven.\"\n\n_So weird._\n\n\"I don't think that's necessary,\" Dad jokes. \"Everyone can't be as dedicated as you, Violet.\"\n\nI can tell he's trying to help me, but it's still annoying that he always takes her side. I chomp on my beef Wellington. \"This is really incredible. You're a great cook, Violet.\" I chew real quick, like a rabbit, then take another bite. Dad and Violet are chewing too, which is better than tonguing each other.\n\n\"Anyway,\" I say when I've eaten enough to leave, \"I'll probably have rehearsal most days. Laurel's mom can drive me when Laurel's rehearsing too, but otherwise, maybe you can pick me up on your way home?\"\n\nI'm looking at Dad, but Violet says, \"Of course.\"\n\n\"Thanks.\" I take one last bite. \"Mmm.\" Chewing.\n\n\"You must have a very beautiful voice,\" Violet says.\n\nI can hear the jealousy tinging her own voice. But why would she be jealous? She's beautiful and talented herself. Why hate me so much?\n\n_Because she's a total wack job._\n\n\"I think it was more that no one wanted to be Oliver. They tried to talk this one short boy into it, but he refused. And the few girls who tried couldn't sing the notes right.\" I take another bite even though I don't want it, then stand.\n\n\"I'm sure you're being modest.\" Violet's stopped eating, watching me.\n\nI sit back down, finish my bite, take another. When I've finished the whole gross, goose-torture experiment, I ask to be excused. \"I have a lot of homework.\"\n\nLater, when I go to wash my face, the soap feels hot on my cheek. I throw it aside and rinse off the water.\n\nAn ugly red welt spreads across my face. When I go to bed an hour later, it's even worse. I finally fall asleep with an ice pack pressed against my cheek. I put on Jonah's music to take my mind off it.\n\nThat night, I dream that Jonah and I are trapped in a burning building. He rescues me and takes me away, away from everything.\n\nThe next morning, my face is normal, like nothing happened. Before I catch the bus, I look for Violet. Dad's already left, and I want to remind her I'm sleeping over Laurel's. Violet's room is quiet. Maybe she left too.\n\nThe bedroom is carpeted, so my feet make no sound. A light shines from the bathroom. Violet's there. I can see her sitting at her vanity. In her hand, she holds a mirror, a big, round, silver one with a long handle. It's beautiful. There's a mirror over the sink, of course. Maybe she's trying to see the back of her hair. But no. As I stand there, staring, she speaks.\n\nTo the mirror.\n\n\"Mirror, mirror, in my hand, who is the fairest in the land?\"\n\n_Who's the most batshit crazy?_\n\nI bolt from the room, but not before I think I hear the mirror say something back.\n\n_Now who's crazy?_\n\n#\n\n#\n\nWe have a read-through after school that day. The second Laurel and I reach the auditorium, Goose runs to meet us.\n\n\"Please, suh, can I have some maw?\" he says in a British accent.\n\nI recognize Oliver's famous line and smile.\n\n\"Knew you'd be perfect for it,\" he says. \"I practically discovered you.\"\n\n\"And how did you know that?\" I try to give him some attitude\u2014I'm a drama geek now.\n\n\"All right, you caught me. I threw you under the bus so I didn't have to play a little kid. I hate playing kids.\"\n\nI nod. I get it. The same reason I hate playing princesses\u2014typecasting.\n\n\"Anyway,\" he says, \"enough chitchat. When Connors calls roll, say 'chop' instead of 'here,' okay? And pass it on.\"\n\n\"Why?\" Laurel says.\n\n\"Everyone's going to do it.\"\n\n\"That sounds like those bad antidrug videos they make us watch in homeroom.\" Laurel puts on a stoner voice. \"Hey, man, everyone's doing it.\"\n\n\"I'm sure this isn't going to end with one of us OD-ing on pot brownies.\" I look at Goose. \"Right?\"\n\n\"Right. So you're in?\" When we nod, he runs off to tell other people.\n\nWhen Mrs. Connors calls the first name, the guy playing Fagin answers, \"Chop.\" Mrs. Connors looks a little uncomfortable but calls the next name, Willow something. The girl playing Nancy answers, \"Chop.\" Then me, Bill Sikes, Mr. Bumble, all the way down to the chorus.\n\nAs the last name is called, all the older drama students yell, \"Timber!\" and pretend to fall on the floor like trees. They pull us uninitiated people down too. Mrs. Connors rolls her eyes and looks right at Goose. He grins at her, and she starts laughing too.\n\nAfter the read-through (during which my Cockney accent is roundly ridiculed in case I was getting too full of myself), Goose comes up to me again. \"How's J.P.?\"\n\n\"Still unaware of my existence,\" I say. _And my identity as his future bride._\n\n\"I heard one of his songs on the radio last night.\"\n\n\"Oh, really?\" I hold my breath. Guys just looove to say that Jonah sings like a girl. Obviously, they're jealous, but it's still annoying.\n\nBut Goose says, \"That song, 'Beautiful but Deadly,' it's pretty scathing social commentary about society's attitudes about appearance. I wasn't expecting that.\"\n\n\"Yes! Exactly! Most people don't get that. At least, most guys.\" Is he making fun of me?\n\n\"Guys like to hate on rock stars because they get all the girls. Me, I'm more confident in myself.\"\n\n\"I can see that.\" I smile. \"You should go to his concert. Lots of girls to choose from.\"\n\n\"I'd probably get trampled by hundreds of screaming women. Not a bad way to go, though.\"\n\nLaurel interrupts. \"My mom's here. You ready?\"\n\n\"Sure.\" To Goose, I say, \"Concert tickets go on sale tomorrow. You'd probably be the only guy there. Big advantage.\"\n\n\"Good tip. By the way, if you need help with your accent\u2014not that it's bad or anything\u2014you could come over after school. My mom likes this British soap called _EastEnders_ , and sometimes, we all speak in Cockney at dinner, just for fun.\"\n\n\"That _does_ sound fun.\" A lot more fun than watching my parents make out. \"I may take you up on that.\" Even though he obviously thinks my accent is bad.\n\n\"He's really nice,\" I tell Laurel as we leave.\n\n\"OMG,\" Laurel says. \"You didn't actually buy that stuff he said about J.P. Guys do not listen to Jonah Prince. He's either making fun of you, or he's a total horndog and using Jonah to flirt. Like he'd be your type.\"\n\nI guess she means because I'm pretty and he's short. Because, apparently, being beautiful automatically makes a person shallow. Like Violet.\n\n\"I don't know that I have a type.\" I don't like what Laurel is implying. Plus, it seems like she's judging Goose by his looks as much as people judge me by mine. I don't like to think that's true so I say, \"Besides, I think he's dating Willow.\"\n\nGoose has stopped to wait for her. I go back to him. \"Say, Goose, what was your favorite part of the song you listened to?\"\n\nHe thinks a second. \"There were lots of parts I liked.\"\n\nLaurel's nodding like, _sure._\n\n\"Well, what was one part?\" I ask, testing him, hoping Laurel's not right.\n\nHe thinks another second, then starts to sing:\n\nIf I could see you through your eyes;\n\nAnd you could see me through mine;\n\nThe world would change for the better.\n\nOnly then could we feel love divine.\n\nHis voice is strong and mellow. He looks away as he sings, like he's suddenly shy about it. For a second, it's like Jonah's there, singing to me.\n\nWhen he finishes, we just stand there, silent. Then he coughs. \"I don't think I remember any more of the words\u2014but I remember I liked it!\"\n\nI laughed. \"It's okay. That's my favorite part too.\"\n\nWillow\/Nancy comes up to us. \"I didn't know you were such a Jonah Prince fan.\"\n\n\"I'm full of surprises,\" Goose says. \"You ready?\"\n\n\"I am,\" she says. \"My mom says it's fine if I come over on a school night, since I told her it was to watch _EastEnders_ for the play.\"\n\n\"Great.\" He offers her his hand. \"Your chariot awaits.\"\n\nWillow leans in and kisses him. Then they walk off together. I give Laurel a look, like, _See?_ because Goose and Willow are obviously a couple. She's also tall, at least six inches taller than I am, so they make an interesting one.\n\n\"See you guys Monday,\" I say.\n\n\"Have fun buying tickets,\" Goose says.\n\nWhen I get to Laurel's mom's car, I text my dad to remind him I'm staying over. He doesn't reply. He doesn't care, of course, since it will give him more time to stay with his beloved, completely insane Violet.\n\n#\n\n#\n\nOur fan club membership, plus incredible persistence on four different devices, gets us floor seats! After that, we get every calendar we can find (wall, cell phone, the agenda books they give us at school\u2014the ones with inspirational quotes like, \"If you reach for the moon, even if you miss, you'll land among the stars\"\u2014which isn't even accurate) and write countdowns on them. We start a series for Tumblr and Instagram too. One hundred thirteen days exactly. We plan to cross off each day together, to share the anticipation.\n\nWe sit at the kitchen table, and while Laurel's mom makes us eggs, plan the entire day out. \"What should we wear?\" I ask.\n\n\"We could get T-shirts.\"\n\n\"But then we'll look like everyone else.\"\n\n\"I know! I know!\" Laurel starts jumping up and down in her seat. She's so cute. \"We could _make_ T-shirts. That would really get his attention.\"\n\nI try not to sigh. My fantasy of meeting my future husband does not involve looking like a screaming fangirl in a shirt that says _Waiting for My Handsome Prince_ in glitter. I read once that Elvis Presley fell in love with his wife because she didn't think he was that big a deal. Her parents even threatened not to let her see him anymore when she broke curfew. Talk about hard to get. It's bad enough my only shot at meeting Jonah is at a concert. I have no idea how I'll get him to notice me, short of a miracle where his eyes meet mine across a crowded basketball arena, and he just _knows._ Still, I plan on leaving in his private limo. It's destiny. I believe in destiny, so why not?\n\n\"T-shirts are unflattering,\" I tell Laurel. \"We should be different, rock some really fabulous outfits. We have months to plan and budget.\"\n\n\"Oh, okay.\" She looks sort of surprised. I don't usually show much interest in fashion. \"So you're going to buy clothes that fit and stuff?\"\n\n\"Yes. Okay. Hey, we could make signs. You're great at that artsy stuff.\" Signs are part of my master plan to get him to notice me. I had this idea about putting a line from the poem \"The Love Song of J. Alfred Prufrock,\" which I read is Jonah's favorite poem. How cool is it that he has a favorite poem? I thought about putting \"Dare to eat a peach,\" which is about taking chances. Or \"Dare disturb the universe,\" which is about not being afraid to be controversial. No one else would know what it meant, just Jonah. He'd get that I'm more than a fangirl\u2014I'm his soul mate. It would be like secret code. I decided not to tell Laurel that yet in case\u2014you know\u2014she thinks I'm crazy. Maybe I'm a little crazy, but you can't win if you don't try.\n\nOver breakfast, while Mrs. Mendez reads the newspaper, we discuss colors. \"Jonah's favorite color is purple,\" Laurel says, \"so we should use purple poster board.\"\n\n\"Won't everyone use purple because it's his favorite color?\"\n\n\"Pink and purple?\" Laurel amends.\n\nI guess peach would be a bit much. I'm not even sure if they make peach poster board. Orange isn't the same. \"Dare to eat a peach\" is probably too weird anyway.\n\nWe decide to make two signs.\n\n\"What will yours say?\" Laurel asks.\n\nSuddenly embarrassed, I say, \"I'm not sure yet. Maybe we shouldn't make it right away. It might get ruined.\"\n\nLaurel nods. \"The glitter will fall off.\"\n\n\"Last time I used glitter for a school project, Violet's lame-o cat, Grimalkin, rubbed against it, then licked herself. She yakked up glitter all over my bed, and all Violet cared about was whether the cat was okay.\"\n\n\"Grimalkin?\" Laurel's mom looks up from the paper. \"She still has that cat?\"\n\n\"As long as I've known her, I guess.\"\n\n\"She used to talk about that cat in high school,\" Laurel's mom says. \"I remember the weird name.\"\n\nI forgot that Laurel's mom had known Violet in school. But that was a long time ago. I remember Dad had his twenty-year reunion a few years ago. So unless Violet got the cat on the last day of high school, it would be even older than that. \"It must be a different cat. Cats don't live that long, do they?\"\n\n\"Probably not. It was a white cat, I remember, solid white. She had a picture of it in her locker. We thought that was so . . .\" Her voice trails off.\n\n\"You can say it. So weird.\" Violet's cat, Grimalkin, is white. It doesn't seem that old at all, though. \"Maybe she just gives all her cats the same name, like Lisa on _The Simpsons._ \" Violet is exactly that weird.\n\n\"Let's make the posters here,\" Laurel suggests. \"That way, Grimalkin Five won't eat the glitter.\"\n\nI giggle. I know mine will be purple and gold, colors of royalty. But I'll wear peach to stand out from the throngs of girls wearing purple.\n\nSunday afternoon, Dad shows up at Laurel's house, unannounced. I gather my belongings . . . slowly, and come to the door when Mrs. Mendez calls my name for the third time.\n\n\"Hey,\" he says when I'm in the car. \"We missed you.\"\n\n\"Really? I thought you and Violet liked having alone time. It gives you a chance to make out. Constantly.\" Since Violet stopped liking me, it seems like my dad and I never do anything together, so this was one of the few chances I had to talk to him alone.\n\n\"What's that supposed to mean?\"\n\n\"It means no one's parents do that. It's super-weird and makes me uncomfortable.\"\n\n\"I don't think that's true.\" His arm sort of flexes, holding the wheel.\n\n\"Did you see how mad she was that I got the lead in the play? She practically burst into flames when I told her. She's never happy when anything good happens to me.\"\n\n\"Hey, wait, what's bringing this on?\"\n\n\"She hates me. She's jealous. Or I remind her of Mom.\"\n\n\"That's crazy. You look much more like me.\"\n\nHis answer is so immediate that I know he thought about it before. I decide to ask him the question I've been wondering about.\n\n\"Why didn't you marry her in the first place? Why did you choose Mom?\"\n\nHe brakes to avoid a squirrel that runs in front of our car. I pitch forward. Silence. Or as silent as it can get in our neighborhood with a lawn mower going, a small dog yapping its head off, all the tranquil joys of suburbia.\n\nFinally, he says, \"Believe me, that's a sore spot for Violet.\"\n\n\"I bet it is.\" It's like I thought. She hated my mother for taking Dad. And that's why she hates me.\n\n\"I'm afraid I wasn't very nice to her. We were friends, always together, and then, one day, I just . . . stopped. I knew Violet had a crush on me, but she was sort of homely and I just wanted Jennifer. There was never a time when I didn't like your mother. Jen was so pretty, so much more outgoing and confident than I could ever be. And, since she hated Violet's guts, I had to choose.\"\n\nWait. I want to rewind to the part where he said Violet was homely.\n\n\"You're saying you dumped Violet because she was ugly? Ugly?\"\n\n\"Well, not dumped her. We weren't dating.\"\n\n\"But Violet was _ugly_?\"\n\nHe shrugs. \"I guess she was a late bloomer.\"\n\n\"So she had acne? Or was overweight?\"\n\n\"No, not that. A bad nose and stuff. Not much of a chin. I don't know, she was homely. Everyone made fun of her. Stop asking me about it.\"\n\nI feel a twinge of pity for poor Violet, ugly and with one friend, my dad. Then he ditched her for the pretty cheerleader. I push back the feeling that my mom was a mean girl. But I remember what Violet said about the cat being her best friend when she was a teen, what Laurel's mom said about her even having pictures of the cat in her locker, probably to get her through the day when everyone picked on her. It was sad.\n\nBut Violet isn't sad and pathetic anymore.\n\n\"No. What happened? Did she get plastic surgery?\" I always knew Violet had work done.\n\nDad looks surprised. \"No. I mean, I don't think so. She was just pretty in high school. It was sort of weird. Like one day I looked at her, and she wasn't ugly anymore. I don't know.\"\n\nI shake my head. \"You get that Violet is, like, the most beautiful woman I've ever seen?\"\n\n\"I know. Some people are just late bloomers, I guess.\"\n\n\"You said that before.\"\n\nWe're in our driveway. Dad says, \"Want to go out with us for pizza?\"\n\nI throw open the car door. \"Not really. I ate at Laurel's. And I have homework.\" Both are lies. I start toward the house.\n\nDad follows me. \"Hey, I think we should eat one meal together the whole weekend.\"\n\nI keep walking. Violet intercepts him, kissing him. I start upstairs.\n\nWhen I reach the steps, stupid Grimalkin throws herself at my legs, claws out, and rakes them down my calf.\n\n\"Ouch!\" Reflexively, I kick at the cat to get her off me.\n\n\"Don't do that!\" Violet shrieks. \"She's old!\"\n\n\"She attacked me out of the blue!\" I scream back. Like when that monkey went after my mother.\n\nThe cat ran\u2014ran like a kitten\u2014toward Violet and rubbed against her legs.\n\nAfter they leave for dinner, I go to Dad's closet, to a box where he keeps old yearbooks and stuff. I used to love looking at the photos of Mom and Dad, power couple. Mom was so beautiful in her dance team uniform and her homecoming dress. I never looked for pictures of Violet before.\n\nFirst, I check the high school ones. Violet's there, tall and beautiful as expected, star of the school play in a big black-and-white picture, and on the dance team. The pictures could have been taken yesterday. Nothing has changed except her poufy 1980s hairstyle.\n\nBut when I look further back, to eighth grade, the name Violet Appel reveals an awkward, hunched girl with a crooked nose, an overbite, and no eyelashes. I only recognize her from the name.\n\nIn a group photo, with the middle school choir, she stands in front, one of the shortest.\n\nIn the high school dance team photos, tenth grade Violet is in the center of the back row, statuesque and beautiful.\n\nViolet could have gotten a nose job.\n\nShe could have gotten a boob job, her eyelashes dyed, braces, makeup.\n\nBut you can't get surgery to grow six inches in two years, which is what she did.\n\nThe change is like magic.\n\n#\n\n#\n\nI have _Oliver!_ rehearsal every day after school. It's a great Violet-avoidance ploy, but also, I'm admitting to myself that I'm not completely miserable about being Oliver. In fact, I'm glad Laurel got me to try out. I've always thought I hated being the center of attention. Now I realize I hate being the center of attention _for my looks._ I'm sick of everyone fawning over my hair, my eyes, liking me because of how I look\u2014or hating me for the same reason. But, as Oliver, with a cap covering my hair, I'm getting attention for my singing and acting. Not that I want to be an actress or anything, but it's fun for now.\n\nYesterday, we practiced a scene where Oliver's locked in the funeral parlor alone, overnight. At tryouts, people sniffled when I sang \"Where Is Love?\" __ Now they flat out bawled. It feels great that people care so much.\n\nToday, we're practicing a scene Laurel's character is in. It takes place in the funeral home too. In it, a bully named Noah is picking on Oliver. He's a lot bigger, so Oliver just takes it. But then, Noah insults Oliver's mother, and Oliver goes nuts on him, pushing Noah into a coffin and kicking him. Charlotte, Noah's girlfriend, runs in screaming her head off. Laurel had to scream at the audition, and that's why she got the part. She's a great screamer. Once, she saw a marine toad eating dog food on her patio, and she screamed so loud the neighbors called 911.\n\nI can so relate to Oliver getting mad at someone insulting his mother. Violet always makes little comments about mine when Dad isn't around to hear. Like last year, the lady next door commented that I was growing up pretty, just like my mom. Violet said yeah, Dad had been so taken with Mom's beauty (yes, she'd actually said, \"taken with\"\u2014she talks like that) that he hadn't noticed how dumb she was. Mrs. Hernandez acted like maybe she'd heard Violet wrong. I felt like one of those cartoon characters, when they have smoke coming out of their ears. But, of course, blowing Violet up was out of the question.\n\nNow, when Tedder Strasky, the guy who plays Noah, says to Oliver (me), \"Your mother was a real bad 'un,\" I just picture Violet's face on Tedder's body, and I launch myself at him as hard as I can. I figure I can't really hurt him because Tedder's at least a foot taller than I am and outweighs me by a hundred pounds. But, to my surprise, he gives a yowl that doesn't sound like acting and falls on his butt, then slides to the edge of the stage. Goose and Willow are sitting in front, but they clear out real quick when Tedder goes barreling toward them. Tedder just sits there, winded, so I can't push him into the coffin.\n\nWhen Mrs. Connors yells, \"Cut!\" Tedder turns to me.\n\n\"Man, you're strong, girl. I did not see that coming.\" He looks annoyed, even though he's trying to laugh.\n\n\"Oh, sorry.\" I look down. \"Guess I really got into the part.\"\n\n\"No, it was good. It was kinda hot. I just wasn't prepared.\"\n\nI ignore the \"kinda hot,\" but say, \"Thanks.\"\n\n\"Try not to maim the other actors, Celine,\" Mrs. Connors calls.\n\n\"Will do.\" I turn away, blushing, but not before I see Goose take his seat again and give me a thumbs-up. He mouths a word: _warrior._ Yeah, that's me. I roll my eyes.\n\nWe try it again, and now Tedder's ready for me. Connors says, \"Remember what it was like when she took you by surprise. That was really good.\"\n\nI'm hoping we're going to finish up with the funeral home scene so Laurel's mom can drive me home, and I don't have to bother Dad or, worse, Violet. But after I punch Tedder in the gut (gently) for the fourth time, Mrs. Connors says, \"Good work. I think we can move on.\" She calls Goose onstage for the scene where Oliver and the Artful Dodger first meet.\n\n\"Thank God,\" Tedder says. \"Do I get padding for the actual show?\"\n\nEveryone laughs, but I apologize again. Even though I think he's being a big baby at this point.\n\n\"Don't worry about it, little girl. I can handle you.\" I don't think he's totally kidding.\n\n\"You've got some real anger management problems, huh?\" Goose says when he comes onstage.\n\nWell, I do, but I say, \"I feel really bad.\"\n\n\"Don't. Strasky's exactly like the character he plays. He was a huge bully in middle school. He actually did steal people's lunch money. It's not just a clich\u00e9. It would almost be worth playing Oliver to get a chance to beat him up.\"\n\n\"You could take him,\" I say. \"He obviously has problems dealing with someone with a low center of gravity.\"\n\n\"I think he had trouble dealing with a girl\u2014he couldn't just break your face. But I like that: low center of gravity. I'm not short. I just have a low center of gravity.\"\n\n\"Go out for wrestling,\" I suggest.\n\n\"Can't mess up my pretty face for showbiz.\" He poses like one of those guys in the Calvin Klein ads, except with a shirt.\n\nMrs. Connors calls for us to start. She blocks the scene up to the song and runs through it a few times. Then, she glances at her watch. \"We'll start 'Consider Yourself' __ tomorrow.\"\n\nI look at my watch too. Five-thirty. Too early for Dad to pick me up, perfect for Violet. I start down the steps.\n\n\"What?\" Goose was walking toward Willow, who stayed to wait for him. But he stops and looks over his shoulder at me.\n\n\"Hmm?\" I turn back.\n\n\"You sighed.\"\n\n\"Oh, nothing. It's just too early for my dad to pick me up, and Laurel just left. It's okay, I can walk.\"\n\n\"I was just about to offer you a ride.\" He looks at Willow. \"You don't mind, do you?\"\n\n\"No, no,\" Willow says in a fake Cockney accent. \"We can give 'er a lift. You 'av to talk Cockney, though.\"\n\n\"What? Oh . . . um, I only live a wee bite awie from he-ah.\" It sounds more Irish than Cockney, but Willow nods. I remember what Violet said about speaking in a British accent during _My Fair Lady_ rehearsals.\n\n\"We're gonna stop at Tawget first,\" Goose says. \"Ye mind?\"\n\n\"No, that's fine.\" I have to outline a chapter for chem, but I'd still rather get home closer to when Dad does.\n\n\"Why are we goin' to Tawget awl of a su'in?\" Willow asks as we walk out.\n\n\"'Member wot I tole you?\" Goose says.\n\nWillow grins. \"Oh, it's a prahnk.\"\n\n\"Not a prahnk,\" Goose says. \"A sociological experiment.\"\n\n\"You 'av way too much time on your hands,\" Willow says.\n\nI laugh. Goose does seem to have a day with a few more hours in it than everyone else's. In addition to the timber prank, the other day he got someone to hide a walkie-talkie in the dropped ceiling in the dressing room, then talked into the other one, yelling, \"Let me out! Let me out!\" Half the cast thought he was stuck up there.\n\n\"I'm still cracking up about the ceiling thing,\" I say.\n\n\"See there,\" he says to Willow. \"And she is a h'independent observer.\"\n\nWillow musses his hair. \"I 'eard about this school where these kids released three pigs, labeled one, two, an' faw. The principal spent the 'ole day, looking fer number three. You should do that.\"\n\n\"As soon as I find someone to give me three pigs. Do you 'av any pigs, Celine?\"\n\nI shake my head.\n\n\"Maybe for the experiment, we shouldn't do the accents,\" Goose says. \"It kind of makes us seem . . . weird.\"\n\n\"You think the accents make us seem weird?\" Willow says. \"Not, say, your personality?\"\n\n\"I think that's a good idea,\" I agree, relieved.\n\nWe're in the school parking lot. Goose holds up his keys, and a blue Civic beeps. I'd assumed Willow would be driving, which was stupid. Still, I wonder how he reaches the pedals.\n\n\"Special pedals,\" he says, reading my mind.\n\n\"What?\" I say.\n\n\"My car,\" he says. \"I call her Nelly. She has extended pedals. People always wonder how I can reach. So I'm saving you the trouble of asking or rubbernecking from the backseat.\"\n\n\"That's cool,\" I say, embarrassed at being so obvious. And average.\n\nWe get in the car, a typically messy boy's car with crumpled assignments and McDonald's fry wrappers. \"Sorry it's such a mess,\" Goose says. \"I drive my brothers and sister around sometimes.\"\n\n\"Sure, and they're always leaving their chem homework.\" Willow uncrumples a paper she almost sat on. \"You'd get better grades if you handed this stuff in sometimes. And fasten your seat belt.\"\n\n\"God, stop being such a harassenger,\" Goose whines, but he does the seat belt. \"Mrs. McKinney said she'd take the homework late. She loves me. Everyone loves me.\"\n\n\"Don't know how you talked her into that,\" Willow says. \"But remind me to staple it to your forehead tomorrow.\" She mimes stapling it.\n\n\"So what's the sociological experiment?\" I ask.\n\n\"Yes,\" Willow says, \"what is this grand idea that's keeping me from studying for my gov test tomorrow?\"\n\n\"Selfies with strangers!\" Goose says. He's turned toward me, backing up the car, so I guess he sees the blank look on my face as well as I see the grin on his. \"You go to a public place, walk up to people, and take a selfie with them. Someone else films their reactions.\"\n\n\"And you don't get beat up?\" This is so not something I can do.\n\n\"Not so far. I only did it once before. A few people got weirded out, but others thought it was funny.\"\n\n\"I'll get the person with a can of pepper spray.\" Still, I wish I were like him. I've spent my whole life hating when people looked at me. Goose seems to love being the center of attention. It must be cool to be so comfortable in his skin. \"Can I be the one filming it?\" I know I'm a geek, but I suspect Goose wants an audience more than anything else anyway.\n\n\"We take turns,\" Willow says. \"It wouldn't be fair otherwise.\"\n\n\"Besides, you ladies are so hot, people'd probably rather have their picture taken with you than me.\"\n\nWillow slaps Goose's shoulder. \"Just for that, you're going first. That way, if we get thrown out for harassing people, there won't be any photographic evidence of me being there.\" But she's laughing.\n\n\"Gladly,\" Goose says.\n\nWe're near Target now, and I feel a little nervous, but also, sort of anticipating. Goose pulls into the parking lot and parks near where an old lady is loading bags into her car. \"How about her?\" Willow asks him.\n\n\"The old lady?\" Goose asks. \"Easy.\"\n\nBefore I can even process what he's doing, Goose is out of the car and running up to the old lady. \"Smile!\" he yells.\n\n\"What?\" the old lady says as he stands next to her. She's not much taller than he is, and at first she looks a little freaked out. \"What are you doing?\"\n\n\"Just a picture?\" he asks. \"Please? And I'll help you with your bags.\"\n\nFinally, the old lady smiles, and Goose snaps the picture with his phone. Willow's filming the whole thing.\n\n\"Why do you want my picture?\" the old lady says.\n\n\"No reason,\" Goose says. \"You look beautiful. Can I help you with those?\" He grabs one of her bags by the handles.\n\nThe old lady looks a little confused, but when Goose puts the groceries into her car, she smiles again. \"Aren't you sweet?\"\n\n\"Yes. Yes, I am. Can we all get a picture with you?\"\n\nShe notices me and Willow, who is still filming. \"Oh, aren't you pretty girls? Of course. You made my day.\"\n\nWe gather around. Goose takes a selfie of all of us, and we finish helping the lady with her bags. She tries to give Goose a dollar, but he says, \"Free service.\"\n\n\"Well, bless you,\" she says.\n\nWe walk away, Willow saying, \"It's easy if you're going to offer to load their bags.\"\n\n\"What can I tell you? I'm just a wonderful person,\" Goose says. \"Now, I'm blessed too.\"\n\n\"I don't know,\" I say. \"It could have gone another way, and she could have kicked him in the face when he touched her groceries.\"\n\n\"She did seem a little like a ninja,\" Willow says.\n\n\"I'm nonthreatening,\" Goose says. \"We shouldn't approach anyone with kids, though. They'll think we're pervs.\"\n\n\"Good thought,\" I say.\n\nWillow goes next, accosting a nerdy-looking guy our age in the entrance. He's happy when she runs up next to him to take a picture.\n\n\"He was looking at your boobs,\" Goose says when she comes back. \"That's easy too.\"\n\n\"He was _not_ looking at my boobs. Why can't you acknowledge my gifts?\"\n\n\"I see your gifts. And so did he. Show me the picture then.\"\n\nWillow gets out her phone and glances at it. \"Okay, so he was looking at my boobs. At least I didn't have to do chores for him.\"\n\nNext, we go to accessories and try on knitted hats and scarves. \"Ooh, put on that red hat,\" Goose says. I put on a red hat and a striped scarf that looks like _Where's Waldo?_ It's silly, but Goose takes my picture. \"Model it, darling!\"\n\nWillow takes another photo with a different rando who also looks at her chest, but when Goose tries to take one with a girl our age, her boyfriend shows up and gets mad. We take more pictures of each other too, and a selfie of all three of us. Then, we head for the grocery section.\n\nGoose says, \"Your turn, Celine.\"\n\n\"Hey, I'm just along for the ride.\"\n\n\"No such thing as a free ride,\" Goose says. \"You need to come out of your shell. You're in theatah now.\"\n\n\"You'd better do it,\" Willow says. \"Otherwise, I'll never get home to study for my test.\" Willow looks around for a target. \"How about her?\"\n\nShe points to a woman. I can't tell how old she is, except she's older than us. She's in the frozen food section. She's wearing sort of a crazy outfit, a long, green, velvet skirt, black boots, and a black turtleneck. Her dark hair comes to her waist and has one purple streak. I'd actually love to get a picture of her to show Laurel. She'd like her style. And she looks nice.\n\n\"Okay,\" I tell Goose. \"I'm going in.\"\n\nI ready my phone, then run up to the woman, trying to be as confident as Goose and Willow. \"Selfies with strangers! Smile!\" And I snap her photo.\n\n\"Oh, look at you,\" she says. \"So pretty. Is this some kind of school project?\"\n\n\"Yes. Sort of.\"\n\n\"Okay. Well, glad I can help. What's your name?\"\n\n\"Celine.\" I am sooo embarrassed. \"Celine Columbo.\"\n\n\"Columbo.\" Her eyes narrow in recognition. \"Any relation to Greg Columbo?\"\n\n\"He's my father.\" Great. Now, she'll report back to Dad, and he'll think I'm on drugs. \"How do you know him?\" Hoping she'll say she hasn't seen him in years.\n\n\"Oh, I haven't seen him in years. I've known him since he was a boy, though.\"\n\nWhich is weird, because she doesn't look as old as Dad, nor does she have that Botoxed look of the fake-young. But then, I could say the same about Violet.\n\n\"Listen, we were just messing around,\" I say.\n\n\"I know. I'm Kendra, by the way.\" She backs up a few steps, staring at me, a weird expression on her face. \"My, you are lovely, Celine. I'd heard you were, but I never imagined.\" She reaches up and touches my face, just staring at me. \"So sweet and innocent.\"\n\n_Weird._ I back away. \"Listen, I have to go. You won't tell my dad you saw me, will you? I can delete the picture if you want. We were just kidding around.\"\n\nGoose, who has been uncharacteristically silent during this encounter, comes up to me now. He says, \"Hey, Celine, we should get going. Sorry, ma'am.\"\n\n\"Not a problem,\" Kendra says. \"I won't say a thing to your dad. I doubt I'll see him anytime soon.\" She turns and goes back to choosing a bag of peas. We hightail it around the corner.\n\n\"Oh, God, kill me now,\" I say. \"That was crazy embarrassing.\"\n\n\"What? I really liked hearing about how lovely you are,\" Goose says.\n\n\"Yeah, so sweet and innocent,\" Willow says. \"So she's crazy? There's lots of crazy people. We've all done it, now let's get out of here.\"\n\nShe seems a little annoyed so I say, \"Yeah, we should go.\"\n\nWe do, but Goose teases me the whole way home. I don't mind.\n\nGoose drops Willow off first because she keeps griping about studying. My house isn't much farther. Now that we're done talking about the prank, I can't think of much to say. I always get this way, tongue-tied around new people. It actually takes longer than I thought because Goose makes a wrong turn. Fortunately, he talks for a while, about the play, about how they're having the cast party at his house, and it's going to be awesome. But then, there's silence. I say, \"Thanks for driving me home. I hate having to ask my stepmother. And thanks for taking me with you. It was so much fun.\"\n\n\"Did you really think so?\" He stops at a stop sign and looks at me. Sitting, we're almost the same height. It's mostly his legs that are short. \"You seemed nervous.\"\n\n\"I was. I'm not good at talking to strangers. But I think it's important to face your fears.\" I think of my mom, how facing her fears had killed her. \"I mean, sometimes, like being afraid of talking to people. Or stage fright.\"\n\n\"You're afraid of _me_?\" He shakes his head. \"No one's afraid of me.\"\n\nI realize this sounds crazy. \"Not afraid, exactly. But sometimes, with this group, I feel like I can't keep up. Everyone's so outgoing.\"\n\n\"So you're shy?\" He nods like he's processing this new idea. \"That's why you always sit with your friend and only talk to her?\"\n\n_Do I do that?_ \"I guess. Why? Did you think I was a snob?\"\n\n\"Yeah, sort of. But maybe I just assumed that because you're so . . .\" He stops.\n\n\"What?\"\n\n\"Nothing.\"\n\n\"So what? What am I?\" Though I know.\n\nHe shakes his head. \"Nothing. I was going to say I'd see you in the hall at school, even before the play, walking by yourself, and I figured you were a snob because you're so beautiful. Okay? Usually, girls who are complete tens don't have a reason to be shy, so you assume they're snobby if they don't talk. But that was idiotic to think, so I wasn't going to say it.\"\n\n_Et tu, Goose?_ I wonder if other people think that, people who aren't as honest as he is.\n\nI say, \"When you pointed to me that day and said I should be Oliver, I could have killed you. I only tried out for the play at all because Laurel wanted me to. She's my best friend. I wanted to be in the chorus and dance in back, not be a star. I hate when people look at me. I hate when people think they know what I'm like.\"\n\n\"Like I just did. Got it.\"\n\n\"You're not the only one.\"\n\nA car comes up behind us and honks. Goose starts driving again.\n\n\"Sorry,\" he says. \"I shouldn't have judged you by your looks. You might not realize this, but people judge me by my looks all the time.\"\n\nI smile. \"No. It was a good thing. An opportunity. Once I got the part, I enjoyed it.\"\n\n\"You're awesome in the play. I tear up when you sing, 'Where Is Love?' It's . . . lovely . . . so sweet and innocent . . .\" He laughs.\n\nI laugh too, realizing the conversation had gotten too serious for him. I roll my eyes. \"Anyway, I guess I have you to thank for me getting the part, since you didn't want to play it.\"\n\n\"Are you kidding? People would have come up to me, going, 'Please, suh, can I have some maw?' for the next year.\"\n\n\"And they won't do that to me?\"\n\n\"It's cute when you're a girl, right?\" He looks at my eye roll. \"Guess not.\"\n\n\"I'm sorry you didn't get to be Bill Sikes like you wanted.\"\n\nHe laughs. \"I didn't want to be Bill Sikes. I'm not stupid. I know I can't be Bill Sikes with a Nancy who could kick my ass. I wanted to be the Dodger. I figured by pretending I wanted to be Bill, I could guilt her into giving me Dodger, instead of making me be Oliver.\"\n\n\"Genius.\"\n\n\"That's me.\"\n\nWe're in my driveway now. I wish I could freeze time, freeze this afternoon and hang with him and Willow, instead of having to go home to Violet. Fun and laughter seem to follow Goose around, and I wish I could too. I like him, and suddenly it's really important that he like me. \"So you don't think I'm a snob anymore, do you?\"\n\n\"Nah, you went along with my stupidity, so you must be okay.\"\n\n\"I enjoyed it,\" I say. \"Thanks.\"\n\n\"Any time you need a ride, let me know.\"\n\n\"Oh, you don't have\u2014\"\n\n\"Any time, Celine.\"\n\n\"Sure.\" I get out of the car and walk toward the house. Halfway there, I notice Violet gazing out the window at me. When she sees me seeing her, she closes the curtain.\n\nI glance after Goose. He waited while I walked to the door, but now he's pulled to the bottom of our circular driveway. He looks back at me and grins. I wave.\n\nI walk inside, yelling, \"Hello!\" Even though Violet was there two seconds ago, there's no answer. Sapphire is on the staircase. I bend to pet her. A claw flashes, and my arm is bleeding. I pull back. \"Sapphire, you hurt me!\"\n\nAs if realizing her mistake, she nuzzles my arm, then licks the spot she scratched.\n\n\"Oh, are you sorry? You should be. I don't know what got into you.\"\n\nI remember Violet's face at the window, that odd, envious expression she's had on her face all the time lately. Could she have turned Sapphire against me?\n\n_Don't be crazy._\n\nAlmost as soon as I get to my room, Violet calls me for dinner. I think about saying I don't feel well to avoid it. But I'm actually hungry.\n\n\"Just a minute,\" I say.\n\nI take out my phone and go through the pictures we took at Target; Goose, Willow, and me with the confused old lady. Then, all of us, trying on accessories. I'm looking for the one I took of the strange woman, Kendra her name was. I want to send it to Laurel. But weirdly, it's not there. I remember telling Kendra I could delete it, but I didn't think I had. Had someone else taken the picture, not me? No. It was _selfies_ with strangers. I go through every picture on my camera roll, but it's like it was magically removed.\n\nFinally, I go down to dinner.\n\nDad and Violet have already started. Violet stares me up and down, like she's trying to figure out who I am.\n\n\"Mmm, this looks good,\" I say, just to say something.\n\n\"What happened to your arm?\" Dad asks.\n\n\"Sapphire scratched me. Can you believe it?\"\n\n\"That's not like her,\" Dad says.\n\n\"I know,\" I say. \"She doesn't even seem like she remembers she has claws.\"\n\n\"Sometimes, animals just turn on people,\" Violet says. \"Like someone will have a pit bull for years, and then one day, it attacks someone. Or what happened to your mother.\"\n\nDad and I both stare at her. Outside, someone's mowing their lawn. Otherwise, it's silent.\n\nI say, \"I must have scared Sapphire. She acted sorry afterward. She licked my arm.\"\n\n\"Well, be careful,\" Dad says.\n\nI glance at Violet. \"I will.\"\n\nDad goes to the gym after dinner. Violet never goes. She's one of those lucky people who has a perfect body, seemingly without exercising. I go up to my room to do homework.\n\nBut later, when I go down to get a glass of water, I pass their room. Violet is sitting on the bed, staring into the mirror. It looks like she's talking to it again, a whole conversation.\n\n_What a weirdo._\n\nI hear the words, _so beautiful._ But no, Violet's lips aren't moving.\n\nOkay, so I'm crazy. But no crazier than Violet.\n\nWhen Violet sees me, she turns the mirror facedown on the bed.\n\nI walk away.\n\n#\n\n#\n\nSo now, I stay away from cats, straight irons, beauty products, hair dryers, anything sharp, hot, caustic, or with the potential to become sharp, hot, or caustic.\n\nAnd Violet. I stay away from her as much as possible. I go to Laurel's house most nights for dinner and all weekend, every weekend. Dad doesn't even seem to notice. When Laurel's mom picks us up for the fifth time in one week, she comments, \"We're seeing a lot of you lately, Celine.\"\n\n\"I'm sorry. I sort of hate being home lately.\"\n\n\"You're always welcome, Celine. You do more chores than my own children.\" She glares at Laurel, who glares at me. It's true. I've been nauseatingly helpful at their house, doing all the dishes and volunteering for other chores so they won't get sick of seeing me so much.\n\n\"What's wrong at home?\" Mrs. Mendez asks. \"Does her name begin with a _V_?\"\n\n\"Oh, it's not . . . Violet just doesn't like me that much. Guess I remind her of my mother.\"\n\n\"Yeah, they hated each other. Everyone else loved your mom, everyone but Violet.\"\n\n\"And now, she hates me. Only three more years until I leave for college.\"\n\n\"You can come over whenever you want, Celine. You can even just stay here if your dad says it's okay.\"\n\n\"Really?\"\n\n\"Absolutely. I'd do anything for Jennifer Sadler's daughter.\"\n\n\"And my best friend,\" Laurel reminds her.\n\n\"Well, Jennifer was my best friend,\" Mrs. Mendez says. Then she stares out the window, like she's trying not to cry, or maybe like she's crying.\n\nAfter a while I say, \"So, what was Violet like when you were kids?\"\n\nMrs. Mendez wrinkles her nose. \"I hate to say it, but Violet was a total freak. We were a little mean to her, but honestly, it was hard to resist. Violet went out of her way to do weird stuff. Like, once, in fourth grade, she picked up a dead bird on the playground.\"\n\n\"A dead bird?\"\n\n\"Well, it turned out not to be dead. But it looked dead, and it probably had some kind of bird disease. We were all grossed out, but Violet just walked over and picked it up with her bare hands. Like it was nothing. It was such a typical Violet thing to do.\"\n\n\"Violet loves birds,\" I say.\n\n\"Violet loved being a weirdo. For a while, we referred to anything incredibly weird as 'Violetish.' Oh, we were terrible. She was just . . . awkward. Poor thing.\" She sounds a little like the mean girls at school when they know they're being mean, but they just enjoy it.\n\n\"But you said the bird wasn't dead. Maybe she was trying to help it.\" I remember, now, Violet told me this story, a long time ago. She said they'd rescued the crow.\n\nMrs. Mendez shrugs. \"I didn't see it, but Greg said it flew away.\"\n\n\"After she held it?\"\n\n\"After she held it.\"\n\nI think of something else. \"My father said she was really ugly when she was a kid.\"\n\n\"Oh, yes, sooooo ugly. Poor thing. Crooked nose and no eyelashes. She looked like a parrot with no feathers.\" She smiles a little.\n\n\"And then, she got prettier?\"\n\n\"It was like a miracle, really.\"\n\n\"And she didn't have surgery or anything?\" I figured my dad, being a man, wouldn't be smart about stuff like that. A woman would know the truth. And maybe she was wearing five-inch heels in the dance photo.\n\n\"Not that anyone knew about. It wasn't like someone who got a nose job over summer vacation. I did that. With Violet, it just . . . happened. Everything in maybe a month or two. And she just kept getting more beautiful. More power to her, though. We were awfully mean to her.\"\n\nI noticed she had gone from \"a little mean\" to \"awfully mean.\"\n\n\"Including my mom?\" I asked.\n\nShe stares out again before speaking, then says, \"Not to speak ill of the dead, but especially your mom. She was brutal, even more so after she started dating Greg. Violet _always_ liked him, and Jen couldn't resist rubbing it in her face. I think part of her attraction to Greg in the first place was that it made Violet so mad.\"\n\nWhich makes me feel a little sorry for Violet. I hate that my mom was so mean to her, and kind of my dad too. My parents were obviously like the Bryce Richardson and Whitney Jacobs of their class. I hate that.\n\nAt least, until I get home and the mockingbirds in our oak tree dive-bomb me.\n\nAnd I see Violet watching me out the window, smiling.\n\nAnd I remember that she's decided to be mean to me, to make up for it.\n\n#\n\n#\n\nThere's a song in _Oliver!_ called \"Who Will Buy?\" In it, Oliver, who has survived the workhouse, being sold to a funeral home operator, and living in a den of pickpockets, has been taken in by the kindly Mr. Brownlow. He looks out the window at the beautiful morning\u2014a beauty he knows can't last, not for him. He wishes someone would buy the wonderful morning and the feeling in his heart so he can remember it forever. He knows he can't.\n\nIt's my favorite song in the show, and it's how I feel about being in the play too. After that day with Goose and Willow, Goose gets everyone to include me when they do stuff after rehearsal or on weekends. I bring Laurel along, so suddenly, we're part of this cool, weird group of friends instead of sitting home every night with Violet and her disgruntled cat. Everyone's also raving about my voice. Mrs. Connors says I should take drama and try for thespian conference next year. Laurel and I are both going to. This is the happiest I've been since I was a kid in Brownies, and my mom was alive.\n\nAnd, like Oliver, I know the skies won't stay blue forever.\n\nThe play is this weekend. There are two more days before dress rehearsal, so we're staying late every night. I'm barely home, which is good because the cats are plotting to murder me. Grimalkin waited by my bed last night and attacked my leg when I walked to the bathroom. When I fell (hard!), Sapphire went for my face. It felt like an ambush. Appliances are still turning against me too. I've started taking baths because the shower never stays the temperature I choose. Dad says he'll call the plumber, but he keeps forgetting until he hears my screams.\n\nBut the show's songs are in my head all the time. Today, in the bathtub, I start singing, Who will buy this wonderful feeling? I'm so high, I swear I could fly.\n\nWhen I walk downstairs to go to the bus, Dad and Violet are there, sharing coffee. Like, they're literally sharing it\u2014she's sipping out of a cup that he's holding. _Kill me._ I brush past Dad to get to the refrigerator for my lunch.\n\n\"Hey, you sounded incredible up there,\" Dad says.\n\n\"What?\" Violet stops drinking coffee. \"Who sounded incredible?\"\n\nDad brushes Violet's butt with his hand, a gesture that always makes me wince. He says, \"I heard Celine practicing in the bathroom. She has a beautiful voice, just like her mother . . . oh, and you too, Violet. But, of course, she inherited it from Jennifer. I sure can't sing.\"\n\nViolet looks like someone who found a pube in her ice cream. Then, she smiles. \"How wonderful. The play must be soon.\"\n\n\"Yes, when is it?\" Dad asks.\n\n\"Um, it's Friday and Saturday. But you don't both have to go if you don't want.\" I want Dad to go, not Violet.\n\n\"Of course we're going. What else would we do? Friday night? I'll put it on my calendar.\" Dad takes out his phone. \"Is there preferred seating for parents of the star?\"\n\n\"I'll check.\"\n\nViolet's sitting there with a weird, strangled expression on her face. I know she'll ruin it somehow. \"We're both looking forward to it,\" she says. But her eyes don't match her words.\n\nWhen I wake Friday morning, I have no voice. None at all. I take a decongestant, but I don't have a cold, so it doesn't help.\n\n_No!_ I want to scream it, but nothing comes out, not even a shriveled whisper. I am literally speechless.\n\nThis is impossible. I've never lost my voice in my life. Now, on the first day of a play I have the lead in, it's gone.\n\nI know Violet's behind it.\n\nThere have been so many clues, so many reasons for believing it's true, and only one reason for not believing it:\n\n_There's no such thing as witches._\n\nBut how can we know that's true, really true?\n\nThroughout time, we've believed in magic, not cute, Disney magic that makes stuff fly around the room, but black magic, magic that caused plagues and brought down churches. That's why they burned witches at the stake. That was what they feared when they hanged women as witches in Salem. But the women in Salem weren't really witches. Not the women who died anyway. Real witches would have been better at avoiding detection. Real witches wouldn't have been captured. Real witches wouldn't have died. Real witches would have taken revenge on their enemies.\n\nLike Violet had.\n\nViolet was an ugly child. Dad says so. Mrs. Mendez too. The middle school yearbook confirms it. But now, Violet is beautiful without surgery or anything.\n\nAnd my mother was Violet's enemy.\n\nTwice, in Violet's presence, my mother was attacked by animals. The second time, an animal killed her.\n\nViolet made them attack my mother. Violet killed her.\n\nI am my mother's daughter.\n\nSo Violet hates me.\n\nNow, objects and animals are attacking me.\n\nYet I know no one will believe me. I know my father won't. Why?\n\nBecause there's no such thing as witches.\n\nI don't know what I can tell Mrs. Connors or all my friends in the cast. I'm letting them all down.\n\nI knock on Dad's door. I want to go to a doctor.\n\n\"Come in.\"\n\nWhen I walk in, he and Violet are still in bed, watching the morning news, wrapped up in each other as usual.\n\n_Nauseating._\n\nDad sees me. \"Hey, it's your big day.\"\n\nViolet smiles, happier than I've seen her in a long time. She waves but says nothing.\n\nDad says, \"Would you believe it? Violet lost her voice. Can't say a word.\"\n\nI point to my throat, gesture and nod, like, _Me too_.\n\nThe doctor can't do anything. He sees nothing wrong with me. \"No cold. Throat isn't red. Nothing wrong with her tonsils,\" Dr. Alvarez tells Dad. \"Seems like selective mutism.\"\n\nI shake my head so hard my hair hurts my eyes, then I write a note. _I did not select to be this way_.\n\n\"I didn't mean you're faking. Perhaps a better term is hysterical mutism. Is anything upsetting you?\"\n\n_Other than having no voice and a stepmother trying to kill me? No._\n\n\"She has a play tonight,\" Dad says, \"a play she's been practicing for weeks. She has the lead.\"\n\n\"Maybe she's nervous,\" Dr. Alvarez says. \"Are you nervous, Celine?\"\n\nI shake my head. On my pad, I write, _I want to be in this play_. My eyes are full of tears.\n\nDr. Alvarez nods.\n\n\"But the strange thing, Doctor,\" Dad says, \"is my wife is having the same symptoms. They couldn't both be hysterically mute at the same time, could they? It would be an awfully big coincidence.\"\n\n_It would be except that Violet's faking. She cast some sort of spell on me, then pretended to lose her voice too so it would seem like an illness. That was obvious._\n\nBut I can't say it. I mean, aside from the obvious fact I can't speak, insulting Violet seems dangerous, considering she killed my mother. I'm just realizing that's what I'm saying about her. I believe that's true. I believe she murdered my mother so she could claim my father for her own.\n\nAnd, if that's the case, I don't want to get her any angrier at me than she already is.\n\nIf she killed my mother, she could kill me next.\n\nI get Dad to drop me off at school after the doctor. I have to tell Connors.\n\nShe sort of flips out when I do. It takes me a while to persuade her that I can't talk at all. She says maybe I'll be better for tomorrow's performance. I know I won't.\n\nI attend both performances, in solidarity with the cast. I watch my understudy screw up my part. I cry softly in the audience because softly is the only way I can cry. I tell Goose I can't go to the cast party. The cast party is for people who were in the cast. He begs me to come anyway, but I'd feel too awkward.\n\nWhen I wake Sunday morning, my voice is back. Like I knew it would be.\n\nSo is Violet's. Like I knew it would be.\n\n\"It's like magic,\" she says.\n\n_Exactly._\n\n#\n\n#\n\nI spend as little time at home as possible. I study with Laurel, stay at her house most nights, and only go home to refill on clothes or money or if I need something signed for school. My drama friends haven't ditched me (most pity me), so I hang with them too. When I do go home, I try to leave for the bus as early as possible.\n\nSunday night, two weeks after the play, Dad picks me up at Laurel's. I hear him in the kitchen, talking.\n\n\"So have you adopted my daughter?\"\n\n\"She and Laurel love spending time together,\" Mrs. Mendez says. \"It's so nice that they're friends. She said she asked you if she could stay over those days. Didn't she?\"\n\n\"Yeah, she texts me at nine at night and tells, not asks.\"\n\n\"Is that true?\" Mrs. Mendez notices I've entered the room.\n\n\"What's the big deal?\" I ask Dad. \"Violet doesn't want me around. I'm just granting her wish.\"\n\n\"That's not true,\" Dad says.\n\n\"It isn't? So if Violet had a choice between living with both of us or just you, she'd choose both?\"\n\nMrs. Mendez takes the pot she's stirring off the stove, then starts for the door. I don't blame her. \"I'll just let you two talk.\"\n\n\"We're leaving. Thanks for having her.\" Dad takes my overnight bag from me. \"Violet loves you.\"\n\n\"Yeah, I can see the adoration in her eyes. She practically busted a gut from pride when I got the lead in the play. She wasn't crazy-jealous at all.\"\n\n\"We'll discuss this in the car.\" He opens the door and waits for me to go out.\n\nIn the car, he puts on his serious Dad voice and matching expression that he must copy from reruns of _Full House_. \"Celine, Violet is a successful lawyer. An adult. Why would she be jealous of a teenage girl?\"\n\n\"You'd have to ask her that. I just know she is.\"\n\n\"Violet tries. She just doesn't know about kids. That's why we didn't have any.\"\n\nShe has him so brainwashed. I wonder if _that's_ witchcraft. \"Give me a break. I'm not some baby crying. She was fine when I was a kid. It's now, now that I'm old enough to be a threat.\"\n\n\"Why would you be\u2014?\"\n\n\"Don't tell me you haven't seen the stuff that's happening.\"\n\nHe stares at the road, taking the Dad-eyes off me. \"What stuff?\"\n\n_Well, first off, my mother being killed by a freaking MONKEY._\n\n\"You ignore it,\" I say. \"You ignore it all. You're so hot for her you don't see what she is.\"\n\n\"And what's that?\"\n\n\"A witch.\"\n\nStunned silence. All I can hear is the car's air-conditioning, blowing too cold on my leg.\n\nThen, Dad laughs. \"Good one, Celine. I thought you were going to at least tell me she was a criminal, something that could be real.\"\n\n\" _This_ is real. You know it. She reanimated a bird when you were kids. Then, she made that dog attack my mother in high school. Then, she made a monkey kill her. You were there when all those things happened. You saw it!\"\n\n\"You're insane.\" My father is practically shaking. \"You need help. I should\u2014\"\n\nI stop talking. Violet would be completely happy to have him put me in some facility for troubled teens. I may be troubled, but I'm not crazy. I've just finally\u2014finally\u2014realized the truth.\n\nBut Dad's never going to see it. He'll never believe she's a witch even with all the evidence. No one would, I guess. But he's not even going to admit she hates me, that she hurts me on purpose.\n\nThe other, harder, truth is that my father loves Violet more than he loves me.\n\n\"Fine,\" I say, \"maybe it's my imagination. But I'm not imagining that she doesn't like me. You know she doesn't want me around, even if you won't say it. So why can't I just stay at Laurel's?\"\n\n\"You can go there after school sometimes, but you can't live there. You're my daughter.\"\n\nWe're pulling into our driveway now. I say, \"Fine,\" knowing I'm going to do whatever I want. I jump from the car as soon as it stops moving.\n\n\"Violet has a good heart,\" he calls after me.\n\nI don't answer. Anything I say would be worse than saying nothing.\n\nI don't speak to Dad for the next few days. It's not obvious because I just go to school early and come home late. I sleep at home and avoid the cats like they're murderers\u2014which they may be.\n\nFriday, I'm standing, waiting for the bus with Laurel, when Goose comes running up to me. \"There's been an accident! It's your dad.\"\n\n\"What? What kind of accident?\" I flash back to the day with my mom, and it feels like a fist squeezing my heart. Would Violet hurt Dad?\n\n\"A woman came up to me and told me to get you. Your father's been in a car accident. He was airlifted to Jackson. That's all she said. I can take you there.\"\n\n\"Who? What woman? Did she have red hair?\"\n\n\"No. She looked a little familiar, but I'm not sure from where. Come on.\"\n\nI follow him and Willow in silence. What is this? Why would Violet hurt Dad? And who was the woman Goose talked to?\n\nPlease let Dad be okay.\n\nWe pull up in front of the emergency room. Now that we're here, it seems so much more real. Airlifted to Jackson. What if he's really hurt or . . . worse? \"I don't know what to do,\" I tell Goose.\n\n\"I'll go in with you. Let me figure out where to park.\"\n\nThen, I see Violet running toward me.\n\n\"That's my stepmother,\" I say to Goose. I start out of the car.\n\n\"I can still come with you. Just follow her, and I'll catch up.\"\n\nI really want Goose to stay. Really. But I know he probably wants to leave. And I'm sure Willow does. It would be too weird to have them stay at the hospital.\n\n\"It's okay. Thank you. I'll go with Violet.\"\n\n\"All right. Let me know what's happening. Text me, day or night.\"\n\n\"Okay.\" I run after Violet.\n\nViolet's hair is half up, half down. She's taken off her five-inch heels to go faster and holds her shoes in her hand. When I catch up to her, I can see that her face is tearstained. It's the one time I've seen her not perfect. She runs to the front desk. \"I'm looking for my husband, Gregory Columbo. He was airlifted here. He was in a car accident.\" A huge sob rips from her throat.\n\n\"You're Mrs. Columbo?\" a nurse asks.\n\n\"Yes.\" Violet's still sobbing. \"Where is he? You have to take me to Greg!\"\n\n\"Just wait a moment. I'll get the doctor.\"\n\n\"Please! You have to take me\u2014\" But the nurse walks away.\n\nViolet follows her. \"Maybe you shouldn't follow her in there,\" I say.\n\nShe looks at me like she's just realizing I'm there. \"I have to. Time is ticking.\" I can hear her breath, shallow, like a panting dog's, and that's how I know she suspects what I do: He's dead. My father is dead. I go to put my arm around her, but she shoves me aside and runs after the disappearing nurse, through the emergency room doors. \"Greg! I have to see him! Greg!\"\n\nShe's intercepted by a female doctor in a white coat. \"May I help you? You shouldn't be back here.\"\n\n\"It's my husband, Gregory Columbo. He was brought here. I need to see him.\"\n\n\"Yes, I was on my way out to you. I'm Dr. Martinez.\" The doctor is about Violet's age, short with blond hair in a messy bun. She looks like a mom, and her voice is soft. \"I'm so sorry. We did everything we could, but he didn't make it.\"\n\n\"Noooooo!\" It's a shriek. I feel tears spring to my eyes, and I wish Goose had stayed. I wish Goose had stayed. But Violet's reaction is much, much more. The sound is inhuman. \"I have to see him! You have to take me to him!\"\n\nThe doctor tries to get in front of Violet, who shoves against her. \"He was badly injured. It may be upsetting. In a little while, you can identify the remains.\" She places her hand on Violet's shoulder.\n\n\"No! Not a little while!\" Violet's voice fills the room, and the doctor jolts back as if shocked. \"You have to take me to him now! Right! Now!\"\n\nI hear glass shatter. The window on one of the doors has broken, but no one touched it.\n\n\"Very well. Just calm down,\" the doctor says. \"You might want to have the girl stay here. She shouldn't see him like this.\"\n\n\"She can stay.\" Violet wipes the tears from her face, but they just keep coming. \"I have to see Greg! Greg!\" It comes out a wail.\n\n\"Follow me.\" I notice the doctor keeps her distance from Violet.\n\nViolet follows, still breathing like a mastiff, practically running and almost overtaking the doctor. Despite the doctor's instructions, I follow, but when I get to the door she holds open, I stop.\n\nMy father\u2014his body\u2014lies motionless on a bed, a sheet covering most of him. Still, I can see that his head is bashed and bleeding. His face is almost unrecognizable. I stop. He isn't my father anymore. The doctor is right. I don't want to see him like that. But Violet elbows past me into the room.\n\n\"Greg!\" She throws herself onto his body, embracing him, like she's trying to touch as much of him as possible, give her life to his, and she begins to make weird noises. It sounds like she's praying or speaking in tongues, but not exactly. Not exactly words, either. She isn't crying anymore, but her voice rises to a wail in the quiet room, and her whole body vibrates.\n\nI remember what Laurel's mom said about the bird that was dead, then wasn't. Is Violet trying to bring my father back? Can she?\n\nI would be willing to deal with everything about Violet if she could do that, anything to have him back, even for a minute, anything to make things right.\n\nThe doctor comes up behind me. \"I'm so sorry. I shouldn't have let her . . . I'll get a nurse for your mother, with a sedative.\"\n\nI nod, biting the back of my lips, then get out, \"My stepmother. She adored him.\" I know it's true. If I ever doubted Violet's love, seeing her writhing, covered in my father's blood, changes that. Violet is insane with grief. I feel my own, like a weight on my chest. I have no one I can turn to. I have no one but Violet.\n\n\"I'm sorry for your loss,\" the doctor says.\n\nViolet's on the floor now, still wailing. But now, I understand words. \"Greg! Come back to me! I can't lose you so soon! I can't . . . I can't! I caaaaan't!\"\n\nI run to her, trying to avoid looking at Dad's body.\n\n\"Violet, let's get out of here. Looking at him can't help.\"\n\n\"He can't be gone!\" she screams. \"What good is it to be able to fix birds and cats when the only one I want is dead on a table?\"\n\n_Birds and cats._ But I lead her out into the hallway. I try to embrace her, but she falls to her knees. She throws her head back and shrieks so loudly that the floor seems to shake like the ground is opening. The doors rattle, and again, a glass window breaks. Then another. And another. How is this happening? Is her grief so huge that she could make this happen? Is there a tornado outside? Did Violet cause it? Is she really a witch?\n\nI know the answer. If I didn't before, I know it now.\n\n\"Violet, stop!\" I yell. \"You can't do this. It won't\u2014\"\n\n\"Violet!\" A woman is walking toward us. She's about Violet's age, with a beautiful face and dark hair streaming down her back. The woman from Target. Kendra.\n\nGoose said the woman who spoke to him looked familiar. It was her.\n\nShe wears a black dress that looks like it's from another era. An orderly tries to stop her, but she stares at him, and he backs off. \"Violet, I'm here.\"\n\n\"Kendra.\" Violet collapses in a ball onto the floor. \"How did you know I was here?\"\n\nThe woman, Kendra, kneels by her, embracing her. \"I knew, my darling. I am always there for you, Violet. It's all right, my sweet.\"\n\n\"No!\" she sobs. \"No! He was my life! Now, he's gone. He's dead, and it's all for nothing. Nothing! It's all worthless. _I'm_ worthless without Greg!\"\n\nThe orderly who tried to stop Kendra is with Dad now, covering his body, covering his face so I don't have to see it again.\n\n\"This is a punishment!\" Violet wails. \"A punishment for what I've done, for what I am!\"\n\n\"There, there, Violet.\" Kendra rocks her, like a mother with a small child. \"There, there.\"\n\nBetween sobs, Violet says, \"But Greg was the only one who ever loved me!\"\n\n\"No, dear,\" Kendra says. \"I love you. I love you, and I will be with you forever. Forever and ever and ever when everyone alive today is gone. I love you.\" She holds Violet for a long time, letting Violet's sobs shake them both.\n\nFinally, she looks up at me. I know I'm staring, and I feel that my mouth is open. I shut it.\n\n\"You're Celine. We've met before,\" she says. When I nod, she adds, \"Poor child, both parents gone. You and Violet are all each other has.\"\n\nHorrible thought. My throat feels full at that thought, like I might never swallow again, like I might choke and die and be with my parents sooner, and be happy. But I nod. I breathe through my nose until I can speak. \"But who are you?\"\n\n\"I'm Kendra,\" she says. \"I am Violet's sister.\"\n\n#\n\n#\n\nKendra, at least, turns out to be normal, though I'm still not sure about the sister thing. Wouldn't we have known if Violet had a sister? She talks to the hospital and police about my father's body. Together, she and I visit the funeral home and make the plans. Violet stays in bed the whole time.\n\n\"Celine?\" Kendra taps my leg. I look up from my phone. I'd been thinking about sending a text to Dad. It seems like I should be able to. After all, it's only been a day. How can everything change in a day?\n\n\"I'm sorry, dear,\" Kendra's saying, \"do you prefer dark wood like mahogany or cherry? Or we could get something light like poplar.\"\n\n\"What?\" I put in Dad's name, text _I love you._\n\n\"For the casket. I'm so sorry.\"\n\n\"They're all nice,\" I whisper. Now that I've sent the text, I can see the whole line of others, all from me, unanswered, saying I'm sleeping at Laurel's. I slip the phone back in my purse and jab my finger at a reddish-brown one. \"That one.\"\n\nKendra nods. She plans everything and doesn't ask me anything else.\n\nThe day before the funeral, I go to school. It seems better than staying home with Violet, better than thinking about Dad being gone.\n\nBut as soon as I get off the bus, I realize it's a huge mistake. I stand there, not sure what to do, staring down at the sidewalk where the class of 2014 had painted lots of happy, green cougar paw prints, walking toward the school. All I can see are everyone's feet, yellow Converse, blue Converse, plaid Converse, flowered fake combat boots, white Vans, all walking with and against the paw prints, turning into an impressionistic painting as my eyes fill with tears.\n\n\"Hey, hey, why are you here? You shouldn't be here.\"\n\nI'm still looking down, but I can see the top of Goose's head, one dark curl going into another. He looks at me, his eyes meeting mine.\n\n\"I just figured that out,\" I say, and then, I start to sob.\n\nGoose begins to reach up to put his arm around me, then stops and tugs my hand instead. \"Come on,\" he whispers.\n\n\"Come on where?\"\n\n\"Shh. Quick. Get in my car. It's still early.\"\n\n\"What about you?\"\n\n\"Shh. Be quiet. Have to be casual in case someone sees us. School doesn't start for twenty minutes.\" He tugs on my hand.\n\n\"Won't Willow wonder where you are?\"\n\nHe looks away. \"We broke up. I don't want to discuss it. Or anything. Be quiet.\"\n\nHe pulls me along, through the Converses, Vans, Keds, and I follow. I don't want him to take me home.\n\n\"Don't worry. I won't take you home,\" Goose whispers, reading my thoughts. Then, real loud, so everyone around can hear, \"We'll just go get that paper you forgot, then come back.\" He tugs my arm. He is brilliant at navigating through the legs of taller kids, and he just pulls me along.\n\nI've stopped crying, at least. \"Okay, let's hurry then.\"\n\nFinally, we reach his car. The parking lot isn't crowded yet. The bus always gets there so early. I think I hold my breath the whole time we're buckling our seat belts and pulling out against traffic, but in a minute, we're free. Goose drives a block, then another, not looking at me. When we reach the park, he pulls into the parking lot.\n\n\"Are you okay?\" he says.\n\nWhen I collapse in tears against his shoulder, he says, \"Okay, dumb question. Dumb question. Aw, Celine.\" He puts his arms around me. \"I'm so sorry.\"\n\n\"I have nobody,\" I say. \"There's nobody left.\"\n\n\"There's nothing I can say. I wish I could make it better, but . . . I'm going to shut up, like I never do.\" He holds me harder. His arms are surprisingly strong, and I sink against him and sob. My head, my neck, my jaw hurt. Everything aches with the emptiness that comes from wanting to talk to Dad and realizing I never will, never again. Goose just holds me.\n\nFinally, when it's almost eight, I say, \"I don't want to go home. I came to school because it feels so empty there.\"\n\n\"I understand. We can go to my house and watch TV or something. My mom's home. I mean, I'm not trying to lure you there for immoral purposes.\" He's trying to make me laugh.\n\n\"Your mom's home? Won't you get in trouble?\"\n\n\"She works from home. She's an artist. And nah, she'll write me a note. She knows people need personal days sometimes. I needed a ton of them in seventh grade, when I took PE.\"\n\n\"That must've been tough.\"\n\n\"I'm not into sports . . . especially when people try to use me as the ball.\"\n\nI finally laugh. \"You're the best.\"\n\n\"I know. I'm awesome. We covered that. Come on. I hate missing Hoda and Kathie Lee. I hear they're doing makeovers.\"\n\nI really don't want to go to school or anywhere else. \"Okay. I love a good makeover.\"\n\n\"Girl, there is nothing to make over about you.\" He turns the key in the ignition.\n\n\"Yeah, just my life,\" I say. \"I don't know what I'm going to do now.\"\n\nWe pull out of the parking lot, driving right past a police car. Goose nods at the guy. \"Seriously, if you ever need a place to go, you can stay with us.\"\n\n\"Right.\"\n\n\"No, really. My mom is big on taking in strays. Stray dogs, feral cats, foster kids. She has a kind heart. She probably wouldn't even notice you're there.\"\n\n\"Great. I'm a stray.\"\n\n\"I didn't mean it that way. I meant you're a friend, and if you were in trouble, we'd help you.\"\n\n\"Okay.\"\n\nHe stops the car again. \"Look at me.\"\n\nSince there's not much choice, I do. The sun streams through the windows, and I blink at him.\n\nHe says, \"I'd do anything for you.\"\n\nI'm stunned for a second. It's a strange thing to say. Then, I recognize it. He's quoting the song \"I'd Do Anything\" __ from _Oliver!_ I laugh. \"Anything?\" I parrot the song.\n\n\"Anything.\" He says it seriously, not a joke.\n\n\"Okay. Thanks. Can we go now, before we get picked up for truancy?\"\n\nHe nods. \"Anything.\" He starts the car again.\n\nWe pull up to a big yellow house with marigolds planted in the front flower boxes. \"After you, milady,\" Goose says in a Cockney accent. A spotted cat jumps out of our way as we approach a black door that's a little wider than usual. \"The woman who owned the house before us was in a wheelchair, so they had lower counters and stuff,\" Goose explains. \"That's why the door's so wide, which we didn't need, but the other stuff was perfect for us.\"\n\nI notice the cat doesn't try to murder me. I like that in a cat.\n\nHe walks in and calls, \"Mom!\"\n\nThe house smells of coffee, a homey smell. A blond woman, shorter than Goose and wearing light blue sweatpants, comes out of the kitchen. She's carrying an African American baby in one arm and a bottle in the other hand. \"School's out a little early, isn't it?\" She sees me. \"Oh, I didn't know we had company.\"\n\n\"Celine, this is my mom, Stacey. Mom, this is Celine. She's having a rough day.\"\n\n\"Aren't we all?\" She looks at Goose. \"Don't you have a test in history today? You can't just blow off school for no reason.\" She has a southern accent. Over her head, I can see three kids, two boys and a girl, eating cereal at the island in their downsized kitchen. An orange cat is walking around. It rubs against my legs, but it doesn't try to attack me or anything.\n\n\"It's more than a normal bad day, Mom,\" Goose says. \"Celine's father died Friday.\"\n\n\"Oh, my God.\" Stacey rearranges the baby in her arms so she can give him the bottle. \"Oh, sweetie. I'm so sorry. You didn't want to stay home with your mother?\"\n\nMy throat feels tight. \"I don't have one of those either. She died when I was eight. I just have my stepmother now, and she's flipping out. I had to get out of there, but then, I couldn't handle school. Goose was really sweet and brought me here. I hope he's not in trouble because he was just being nice.\"\n\n\"Yes, my sweet boy. You poor thing.\" The baby turns away from her, but she coaxes him to take the bottle. He sucks for about a second, then starts fussing again. \"I have to get them ready for school, but you stay. Of course, stay as long as you want.\" She adjusts the baby again.\n\n\"Thank you. Um, can I hold him for you?\" I suddenly really want to be useful.\n\n\"You don't have to do that.\"\n\n\"I want to.\"\n\n\"Well, that would be nice, thank you. Do you want to sit on the sofa and hold him?\"\n\nI can tell she's worried I'll drop him, so I sit down on the squashy white sofa. She settles him into my lap. \"His name is Jeron. After he drinks the bottle, you rub his back until he burps.\"\n\n\"Okay.\" The baby feels warm and a little sweaty, but when he is in my arms, he starts to suck on his bottle contentedly. \"Good boy, Jeron.\"\n\n\"He likes you.\" Stacey disappears into the kitchen, where I hear her telling the kids to hurry. Goose sits next to me and starts flipping through the channels, which are mostly morning news shows and cartoons. He settles on _SpongeBob_ for a minute, then changes his mind. Stacey walks through the room with the three kids behind her, wearing blue and white school uniforms. The two boys look about the same age, maybe eight, and they're dwarves or little people (I don't know which is the right term, and I'm afraid to ask). One has dark hair and looks a lot like Goose. The other has red hair and freckles. The little blond girl is about five, an average-sized five-year-old.\n\n\"Are they all your brothers and sisters?\" I ask.\n\n\"Yes,\" Goose says. \"But if you mean did my parents have all of them, no. Tyler, the one with red hair, is adopted. His parents abandoned him. He's the same age as Tony, my other brother. Department of Children and Families contacted my mother because he was a person of short stature, and they thought she'd deal with him better than his parents had. She was fostering before that.\"\n\nI file away _person of short stature_ in my mental bank. I wonder if Tyler's parents ditched him because he was one. It seems impossible, but clearly, some parents suck.\n\n\"And Jeron is a foster kid, but we might get him too. We've had other babies that got sent back to their parents. My mom really likes babies. She had one that died, right after my sister, Isabella, was born. There's this genetic thing. So that was sort of the end with her, and she started fostering instead.\"\n\n\"So there are seven of you here?\" I say. \"Must get crazy in the morning.\"\n\nHe shrugs. \"I'm usually out of here before they wake up.\" He flips through the channels some more and settles on a movie. \"Oh, man, I love this movie. I love this! Have you ever seen it?\"\n\n\"What is it?\" Onscreen, a red-haired girl in crazy clothes and a hat is talking to a guy in even weirder clothes. All the extras look like corporate lawyers. Like, they're high school students who don't own jeans.\n\n\" _Pretty in Pink._ It's a John Hughes movie. He was, like, the god of teen flicks in the 1980s.\"\n\n\"Yeah, my dad made me watch _Breakfast Club_ together last year. That was him, right? I loved it.\" My stomach drops at the realization that, now, I will never watch another movie with Dad. I feel my eyes starting to fill, but I inhale and try not to cry.\n\n\" _Breakfast Club_ 's great too.\"\n\n\"Let's watch this,\" I say. \"Did it just start?\"\n\n\"Yeah, it's pretty close to the beginning. It's about this girl, Andie, from the wrong side of the tracks, and she likes this rich guy, Blane.\"\n\nSo I get totally engrossed in the story of red-haired, offbeat-dressing Andie and her best friend, Duckie, a short, weird guy who has a crush on her. People at school make fun of them because they're poor and also sort of weird. By the time Stacey comes back with the kids trailing behind her, Jeron is asleep in my arms.\n\n\"Oh, my gosh,\" Stacey whispers. \"You got him to sleep? That kid never sleeps.\"\n\n\"I forgot to burp him,\" I whisper back.\n\nStacey says, \"Hey, if he went to sleep, that's better. Do you want to try to put him in his crib?\" She says it like she doesn't think it's a great idea.\n\n\"Nah, he might wake up. I can keep holding him if you want. I'm not going anywhere.\"\n\n\"We're watching _Pretty in Pink,_ Mom.\" Goose has paused the TV.\n\n\"That's a great movie. Good music.\"\n\n\"If you like music from the eighties,\" Goose teases.\n\n\"Hey, _I'm_ from the eighties,\" Stacey says.\n\nThe little girl is starting to sit down with us, but Stacey gestures to her to come on. \"We'll be late, Isabella.\" To me, she says, \"I'll take Jeron off your hands when I come back.\"\n\n\"It's no problem. It's sort of . . . life affirming.\" The same cat from before rubs against my legs again. I sort of jump. \"Sorry, cats hate me.\"\n\n\"Oh, these cats are harmless.\" Stacey takes Isabella and Tyler by the hands and says, \"Come on\" to the other boy. He follows her out.\n\n\"She seems like a really great mom,\" I say. \"You're lucky.\"\n\n\"Guess I am.\" Goose unpauses the movie. \"This is my favorite part.\" Onscreen, Andie's friend Duckie is wearing a yellow jacket and doing a crazy dance to an old song, \"Try a Little Tenderness,\" __ trying to impress her and obviously failing miserably. I giggle.\n\n\"That's love,\" Goose says, \"when you're willing to make a total ass of yourself for a girl.\"\n\n\"Have you ever done that?\"\n\n\"Not so far. Not on purpose anyway. But . . .\"\n\n\"But what?\"\n\n\"Nothing. I talk too much. Shh. Let's watch this.\"\n\nI wonder what he means. Stacey doesn't come back for a while, and in the movie, Andie accepts a date with Blane, breaking Duckie's heart. Then, Blane breaks her heart by asking her to the prom, then canceling because his rich friends don't approve. My arm is starting to ache, and I ease Jeron onto the sofa, planting myself on the floor so I can keep him from sliding off.\n\n\"I don't see what Andie likes about Blane so much,\" I say to Goose. \"He's got no spine. He doesn't want to admit to his friends that he likes her.\"\n\n\"I guess he's supposed to be hot,\" Goose says.\n\n\"He's not that hot. And hotness only goes so far.\" After seeing my dad spend seven years with beautiful but crazy, I know that for a fact. \"After you've known someone a while, you stop looking, I think. Character is more important.\"\n\nIn the end, Andie wears a pink dress and goes to the prom by herself. But Blane apologizes, so she walks off into the bright prom lights with Blane.\n\nAnd Duckie tells her it's fine!\n\n_Totally lame._\n\n\"That was a crazy-stupid ending,\" I say to Goose. \"The girl had no pride.\"\n\n\"Right?\" he agrees. \"Did you know that in the original version of this movie, Andie wound up with Duckie, but test audiences didn't like it. So they went back and changed it. They re-filmed the whole ending to make it that bad.\"\n\n\"Really? Test audiences have no souls. She should totally have ended up with Duckie.\"\n\n\"Absolutely. You get it. I knew you'd get it.\"\n\n\"It was so stupid. He would never have said that.\" I imitate the actor who played Duckie. \"'He's not like the others. Go to him.' It must have hurt the actor's soul to even have to say that.\"\n\nGoose nods. \"I agree.\" He switches the TV back to _SpongeBob_ , which is still on. Or on again. Patrick is obsessing over getting nachos. \"So you'd go with Duckie instead of hot Blane?\"\n\n\"I told you, hotness only goes so far. My stepmother is obsessed with appearances, and it makes her crazy. Besides, I sort of think Duckie is cute. In his own way.\"\n\n\"Really? He didn't strike me as your type. You're still marrying Jonah Prince, right?\"\n\nI laugh. Does he think I'm silly for obsessing over a rock star? \"Oh, absolutely.\" I turn over my hand so he can see the faded writing on my fingers, where I've written _J.P. 4-ever._ \"We talked last night. He promised to take me to homecoming next year. On his private jet. And then we'll go to Paris and buy a horse\u2014two horses, one for each of us.\"\n\n\"Get me a horse too.\"\n\n\"Oh, yeah,\" I agree. \"He totally said three horses. Wanted to know what color you wanted. I said palomino.\"\n\n\"Good call.\"\n\nWe fall back into silence, watching _SpongeBob._\n\n\"I'll miss school again tomorrow for the funeral,\" I say after a while.\n\n\"I was thinking about going to that too.\"\n\n\"Really?\" I know Laurel's going, and her mom. But having more people as a buffer between me and Violet would be good. The thought of sitting there all alone with Violet is just freaky, almost as freaky as my father's body in a box in the ground.\n\n\"Would you want me to go?\" Goose asks.\n\n\"Yes. Yes, I would so want you to go. But . . . you'd do that?\"\n\n\"Like I said, I'd do anything.\"\n\n\"Okay,\" I say. \"Can we just sit here and watch _SpongeBob_ for the rest of the day?\"\n\n\"Absolutely. Sometimes, watching _SpongeBob_ is the best thing. _SpongeBob_ never disappoints.\"\n\nSo we keep watching. My eyes feel about to close. In the background, I can hear Stacey talking to Jeron and making him another bottle, the next-door neighbor's lawnmower, and the squirrels outside, at war with the blue jays over a bird feeder. We watch maybe three more _SpongeBobs,_ maybe more than that. I'm not sure because, at some point, I give in and I fall asleep.\n\n#\n\n#\n\nAfter the funeral, Kendra moves in with us. She says it's to help Violet. I'm still not sure I believe she's Violet's sister, but it's good to have her there because Violet is definitely several pins short of a bowling alley. She doesn't go to work. She doesn't eat. She doesn't even cry.\n\nInstead, she spends day and night sitting on her bed, staring into her hand mirror.\n\nShe doesn't talk to it anymore, just stares. I wonder if it still answers back.\n\nA week after my father's death, I hear Kendra talking to her as I'm passing by on the way downstairs.\n\n\"Don't you think you should go to work?\" she asks.\n\nViolet draws a brush through her hair. She holds it up so the auburn strands catch the light. \"I have no reason to go to work.\"\n\n\"Of course you have. Your job is important. You help people.\"\n\n\"I don't care about helping people. When have I ever?\"\n\n\"As a girl you did.\"\n\n\"I'm not that girl anymore . . . thank God.\"\n\n\"I loved that girl,\" Kendra says.\n\nGrimalkin brushes by me. She takes a swipe at me with one paw. I should go. I start toward the stairs again.\n\n\"Your beauty, then,\" Kendra says, and I stop again. \"It used to be so important to you to be beautiful. Perhaps you could do some modeling, commercials.\"\n\nI see a circle of light moving on the wall, as if someone was flipping a mirror over and over. The cat is stalking back toward me. I can't bring myself to kick her down the stairs, even though I hate her.\n\n\"I am beautiful, aren't I?\" Violet's voice says.\n\n\"Of course you are, my darling,\" comes Kendra's reply. \"But your hair is messy.\"\n\n\"Am I the most beautiful woman in the whole world?\" Violet asks.\n\nKendra hesitates. \"You are one of them.\"\n\n\"I mean, if I brushed my hair?\" Violet asks.\n\nStill, Kendra doesn't just say yes. _Just agree with her! Lie to her!_\n\nBesides, who is more beautiful?\n\n\"My dear, you are very beautiful. Any man would be happy to have a woman of your beauty.\"\n\n\"I don't want any man. I want Greg! I've only ever wanted Greg!\"\n\nFrom the bedroom, I hear glass break. I run downstairs before they can know I was eavesdropping.\n\nThat day, I forget my book for English class. I ask the teacher for a pass to go to my locker. When I get there, I find a mirror inside.\n\nViolet's mirror.\n\nWhich is weird in and of itself because I didn't put it there and I didn't leave my locker open.\n\nThe hallway is silent. The white floors, unoccupied by hundreds of feet, gleam like ice. I don't want to touch the mirror. It might be poisoned or attack me. Violet's things tend to do that.\n\nYet, something compels me to reach for the handle. I turn it over.\n\nSecond weird thing: It's not broken, though I heard it shatter this morning.\n\nI lift it and search for my face inside.\n\nBut my face isn't there.\n\nKendra's is.\n\n\"I need to speak with you,\" she says.\n\nThe mirror slips from my hand, shattering against the terrazzo. I look around to see if anyone has noticed, if anyone heard.\n\nNo one there. I debate running, leaving the splintered glass there, hoping someone will clean it up before fifth period. I have to. I grab _The Book Thief_ from my locker and slam the door. It bounces open, so I hold it closed. I start to replace the lock. My hands are trembling.\n\nAs I do, before my eyes, the pieces of the mirror rise from the floor, hundreds of slices of silver, shining in the fluorescent light.\n\nThey dance before my eyes, then form into an oval mirror, neatly replacing themselves in the frame.\n\nIt flies up into my hand.\n\n\"Careful this time.\" Kendra's voice is a whisper. \"I need to speak with you after school. Can you come to my house?\"\n\n\"What? How? How are you doing this?\"\n\n\"Isn't it obvious?\" She smiles. \"I'm a witch.\"\n\n\"There's no such thing as\u2014\"\n\n\"I know you believe in witches, Celine. You've been living with one for years.\"\n\nValidation. Someone confirming all my crazy suspicions. I wish I could tell Dad. Now, the victory seems hollow. A thousand times a day, I remember he's gone, and it's like he has died again.\n\nI say, \"You know . . . about Violet?\"\n\n\"Of course. And I know you're not safe with her. But you have to get back to class. Come to my house after school. It's the big, gray one on the corner of Salem Court, the one that looks abandoned. Get your friend to drive you, the boy from Target.\"\n\n\"Goose.\"\n\n\"He'll drive you.\"\n\nI know he would, if I asked. And I know the house. I didn't know anyone lived there. \"How do I know I can trust you?\"\n\n\"You don't. But you know you can't trust Violet. Also, if I wanted to kill you, you'd already be dead. I want to help you.\"\n\n\"Violet hasn't killed me . . . yet.\" My hand is shaking, rattling the mirror back and forth.\n\n\"Yet. Because of your father. Now, your father is gone.\"\n\n\"So you're saying . . . ?\" She's saying Violet is going to kill me.\n\n\"My house. After school. With the boy.\"\n\nHer face disappears from the mirror.\n\nAnd then, the mirror disappears from my hand.\n\n#\n\n#\n\nTwo hours later, I'm in front of the abandoned house with Goose. I have no idea why, but Goose, at least, had been excited about it. \"Really?\" he said. \"Someone lives in that house? She invited you? Cool. We always thought it was haunted. On Halloween, we dared people to touch it.\"\n\nI nodded. \"So did we. Laurel and I did it last year.\" I've explained as little about this as possible. It seems better to just let it unfold.\n\n\"So I get to go inside. Cool.\"\n\nI laugh. \"You said that.\"\n\nBut now that we're here, he seems less sure. The driveway is almost impassable, with weeds and branches scraping the sides of the car. At least two windows are broken, and the paint that used to be white is dirty gray, where there even is paint.\n\n\"You're sure someone's not playing a joke on you?\" Goose says as he negotiates the driveway.\n\n_Considering she came to me inside a mirror, no._\n\n\"I don't think so.\" A flock of crows lands in the yard, settling on the bushes and in the trees. I wonder if they're friends of Violet's.\n\nGoose puts the car in park. \"You think those crows are going to shit on my car?\" When I shrug, he says, \"Maybe wait here while I look around, in case something's weird?\"\n\n\"That's okay. I'm glad you're here, though.\" I open my door and step out. A branch scratches my arm. One crow caws. Then, they all begin to until the yard is filled with their cries.\n\nGoose swears under his breath, but follows me to the doorstep. The house doesn't look any less abandoned up close. Leaf-covered cobwebs drape the door, as if it hasn't been opened in years. Grass and flotsam collect in the porch corners, and the doormat is almost entirely worn away. I knock. My hand chips away a layer of dust and dead bugs.\n\n\"Why?\" Goose says.\n\n\"Why what?\"\n\n\"Why are you glad I'm here? You won't let me be the big hero and go in ahead of you while you wait at a safe distance. What if someone comes after you with a knife?\" He tries to stand in front of me.\n\n\"No one's going to come after me with a knife. And, if they did, I wouldn't sit in the car while you bleed to death on the steps anyway.\" It sounds funny, except when you think about it. \"What I really need is someone to be here with me, experience it with me, so I'll know I'm not crazy.\"\n\nAt that, the door opens with a creak.\n\n\"Ah, I see you're both here,\" Kendra says.\n\nShe's holding the mirror.\n\nKendra steps aside so we can enter. My foot goes from dusty slate steps to . . .\n\n. . . shiny black-and-white-checked marble floor.\n\nThe inside of Kendra's house is nothing like the outside. Everything inside is brand-new, and it's all black, white, or hot pink. It looks like a set for the musical _Hairspray_ (which we're supposed to be doing at school next year), with pink walls, a pink jukebox, pink Lava Lamps, and a soda fountain with black tables and hot pink chairs. Kendra, wearing a fluffy, pink and black dress that could go to the prom, points to a spiral staircase that leads to a loft above the room.\n\n\"Follow me.\"\n\nWe do. The staircase is narrow, and, through the steps, I can see the room swimming pinkly below. I notice Goose is looking up at the ceiling.\n\n\"Are you okay?\" I ask him.\n\n\"I have a thing about heights.\" He looks a little green, which clashes with the pink, and takes the last steps very deliberately. Soon, we're at the top in another black-and-white-tiled room with hot pink sofas. Kendra gestures that we should sit. I notice Goose takes the seat farthest from the stairs.\n\n\"Good to see you,\" Kendra says. \"I wish it could be under better circumstances.\"\n\n\"You saw me this morning,\" I remind her. \"At my house.\"\n\nShe nods. \"Indeed, I did. But I will be seeing little of you from now on, I'm afraid.\"\n\n\"What? Are you going someplace?\" Weird though she is, I sort of like having Kendra around, as a buffer between Violet and me. I can't imagine staying with Violet forever. Yet where else is there? Three of my grandparents are dead. My father's father lives in a nursing home. He probably doesn't even know Dad died.\n\n\"No. You are.\"\n\n\"What?\" The pink fluorescent lights are disorienting. I look down, but the checkerboard floor seems to be moving. \"What do you mean?\"\n\n\"You're in danger. You're in danger, and you must leave that house.\"\n\nGoose says, \"What kind of danger?\"\n\nI look at my hand. It glows in the flashing pink jukebox lights. My head is light, spinning. I wonder if this is how it feels to be on drugs.\n\nI say, \"My stepmother is a witch.\"\n\nGoose laughs, but an uncomfortable laugh. \"Yeah, my mom can be a witch sometimes too.\"\n\n\"You are using the term _witch_ in a derogatory way?\" Kendra straightens her shoulders.\n\n\"Is there another way?\" Goose looks confused.\n\n\"A literal way,\" Kendra says. \"For someone who has magic powers and can use them for either good or evil.\"\n\n\"Oh. Right.\" He laughs again, unbelieving.\n\nI'm saying nothing because my head is hot. I put my hand to my forehead.\n\n\"Besides, your mother isn't a witch in either sense. She has no powers, and she is a kind woman who takes in foster children.\"\n\n\"How did you know that?\" Goose asks.\n\n\"I know everything.\"\n\n\"I guess Celine told you.\" He looks to me for confirmation.\n\nKendra shakes her head. \"Celine told me nothing.\" She looks at me. \"Are you all right?\"\n\n\"I . . . the lights . . . all the pink. It's very pretty, but I think it's giving me a headache.\"\n\n\"I think life is giving you a headache. Understandable. How's this?\"\n\nShe waves her hand, and in a second, the room changes to a French provincial d\u00e9cor, yellow flowered sofas and dark wood. Much calmer.\n\n\"Better. Thanks.\"\n\nHuh. She just waved her hand and changed everything. This does not surprise me as much as it should.\n\nGoose is looking at his hands like he expects them to melt. \"Whoa. What the\u2014\"\n\n\"There are witches.\" I look around, trying to figure out a way of explaining what I don't completely understand myself. Even after the mirror, I can't quite believe this. \"There are witches, and my stepmother is one of them. She used to be an ugly girl, and then, she made herself beautiful.\"\n\n\"She can do that?\" Goose asks.\n\n\"Yes. And I'm afraid Violet is a bad witch.\" Kendra sighs. \"She wasn't always so. When I met her, she was a little younger than you, a sweet girl, a victim. But, perhaps that's what made her what she is. She always wanted revenge on her enemies, and she got it.\"\n\nGoose is up. He starts to pace, then sees the stairs and sits back down. I say, \"Got it how?\" Though I think I know.\n\n\"When she killed your mother.\"\n\nEven though I thought I knew this, it still shocks me. Confirmation of my worst suspicions.\n\n\"You . . . you knew she did that? Why didn't you say anything?\"\n\n\"Who would have believed me?\" Kendra asks. \"A woman is attacked by a monkey. Should I tell the police that Violet manipulated it? Do you think they'd believe someone can communicate with animals? Your father didn't, and he saw her do it twice.\"\n\n_The dog._\n\n\"And I couldn't risk being detected. Witches belong to no age, no time. I have no driver's license, no birth certificate but a family Bible dated 1652, no passport. Even my fingerprints have worn away. That happens in three hundred years.\"\n\n\"You've been alive three hundred years?\" Goose says.\n\nI say, \"But you should have told before she hurt someone.\"\n\n\"That's what I'm doing. I'm telling _you_ before she hurts _you._ \"\n\n\"Hurts me? You mean kills me?\"\n\n\"Yes.\"\n\nThe air leaves the room. Silence. Goose has stopped, staring around the room like he can't believe it. Why would he? I barely believe it, and I've been living with a witch for years. I now realize there were so many signs I should have seen, little things like Violet's ability to whip up a recipe with dozens of ingredients in the half hour after she got home from work, like the fact that she still looked twenty-five years old, more than twenty years after she graduated high school. I was stupid. Dad was stupid. Still, I would love to go back to a time when I could not believe it.\n\nAnd she wants me dead. Dead.\n\nGoose looks at Kendra. \"So you're saying her stepmother is an actual witch, she killed her mother, and now she's trying to kill Celine?\"\n\n\"Yes,\" Kendra says. She says it like it's obvious.\n\n\"Did she kill Celine's father?\"\n\n\"No. She loved my father.\" I look at Kendra. She nods that it's true.\n\n\"That she did. She loved him too much. It made her crazy. And now that he's gone, she has nothing and is even more insane. She's consumed by loneliness and her jealousy of you.\"\n\n\"Why is she so jealous of me? I know she is, but I've never understood it. I don't have anything she would want.\"\n\n_She wants me dead. I still can't believe it._\n\nGoose laughs. \"Even I know you're the most beautiful girl I've ever seen. Who wouldn't be jealous of that?\"\n\nI stare at him, stunned. He knows how much I hate to talk about my looks. At least, I thought he knew.\n\n\"I'm sorry, Celine, but it's the truth. You're beautiful, so beautiful it's almost unreal. And people hate you for it, even beautiful people. You can't ignore that anymore.\"\n\nI wonder if he resents me. I hate this. My beauty has never done me a bit of good. It makes people hate me, makes idiots want to be my friend. I wish I could just be average. But I say, \"Fine. But Violet is beautiful too. Obviously, she made herself as beautiful as she could with her magic.\"\n\n\"Yes,\" Kendra says. \"As beautiful as she _could_. But there are limits to magic. Your innocence, your goodness, are part of what make you beautiful, a part Violet does not have. She lost it when she began to play with the darkness, and now there is nothing she can do to be as beautiful as you. As long as you live, she will always be second, just as she was second to your mother.\"\n\n\"Second? What about Keira Knightley? Amanda Seyfried? Tyra Banks?\" I name the most beautiful women I can think of. \"Should they be watching their backs too?\"\n\nKendra grimaces. \"Perhaps. But they don't live in her backyard, so perhaps she'll forget them. We'll keep her away from movie theaters, just in case,\" she joked.\n\n\"We'll have to keep her away from Celine, you mean,\" Goose says. He's standing again, and he comes to stand in front of me, all protective. \"If Violet can't make herself more beautiful, why can't she just make Celine ugly, give her zits or something? Wouldn't that be enough for her? She doesn't have to kill her.\"\n\nI mouth _Gee, thanks_ at him.\n\nKendra says, \"She could, but only if she reveals herself. She can change someone else's appearance only if they know about it. Otherwise, any spell she casts reverberates back to her.\"\n\nI understand now. \"Is that why she lost her voice when she made me lose mine?\"\n\n\"Yes. But there are other ways she can hurt you, as she found a way to hurt your mother.\"\n\n\"You mean kill my mother.\"\n\n\"Yes.\" She sits closer to me. \"You must go into hiding.\"\n\n\"Hiding? But how? I have school.\"\n\n\"School will wait. It has to.\" She places her hand on my shoulder. \"Violet has confided her plan to murder you. I've offered to help.\"\n\n\"Help?\" Goose says. \"What kind of sick\u2014?\"\n\n\"I lied,\" Kendra says. \"I told her she needs to withdraw you from school, say you're moving in with your aunt in another town. Then, I will take you away and murder you. I told her I'd help because she's been such a delight to me. And she was. She was like my own daughter.\" She looks down. \"Where did I go wrong?\"\n\n\"I'll tell you where,\" Goose says. \"Where you didn't tell the world about her years ago, when you knew she had something to do with what happened to Celine's mother.\"\n\nSuddenly Kendra is gone. The space where she was is just empty. Seconds later, she materializes downstairs.\n\n\"There's not a jail cell that can hold her,\" she says. She disappears again.\n\nThen, she's back with us. \"Besides, I didn't think it would get this bad. I loved Violet like my daughter. She was all I had. It's like how parents of killers always say they never suspected.\" She turns to me. \"I will take you away, but to a safe location. Is there someplace you can go, to hide?\"\n\nI stare at Kendra. She looks different than the first day I saw her, younger. I have a feeling she can change everything about her looks, if she wants. She could kill me and disappear, and no one would know. I have to trust her because there is no point in not trusting her. It's like she said before: If she wanted to kill me, I'd already be dead.\n\n\"My friend Laurel. I always stay with them. Her mother was my mom's best friend. She hated Violet. If she thought I was in danger\u2014\"\n\n\"Ah, yes, but you do often stay there, which is precisely why it wouldn't work. Violet would immediately look there to see if you were hiding. No, it needs to be someone else, someone she doesn't know, a distant relative, perhaps. Or a friend.\"\n\nShe's looking at Goose, who has paced closer again. He picks up on the cue.\n\n\"She could stay with me, with my family.\" He holds out his hand. \"I'd do anything for Celine.\"\n\n\"Yes,\" Kendra says. \"I know.\"\n\nAnd I know too.\n\n#\n\n#\n\nGoose drives Kendra and me to his house. The whole time, my head is throbbing from the conversation with Kendra. Could it be true? Could Violet actually want to have me killed? But, of course. It makes perfect sense. Violet has never loved me, maybe not even back when she acted like she did. Did I think that, without Dad, the one thing that bound us together, she would suddenly want to be my mommy? Of course not. Her loathing for me would only increase. Without her darling Greg there, I was only a reminder of my mother, a girl she'd hated.\n\nMy stomach, my heart, my entire body feel empty: _There is no one now in the entire world who loves me._\n\nIt's hard to remember a time when there was.\n\nWe're at a stoplight. Goose reaches over and touches my arm. \"It'll be okay, Celine.\"\n\nI shake my head. \"How do you know that?\"\n\n\"Because it has to be.\"\n\nThe light turns green, and we drive on.\n\nIt seems so drastic, to abandon school, abandon my whole life on the word of one woman, one witch. Yet what choice do I have? I've seen this all happen. I know it's real. Goose says he can talk his parents into letting me stay. He's so nice. Why is this guy I barely know so nice while the person who is supposed to care about me is so horrible?\n\nKendra insisted on coming along to explain it all to Stacey. \"They need to know what they're getting into. It's not just letting a friend of yours stay a few days. It's not even the same as a foster child. They would be harboring a runaway. Your life is in danger. They'd need to keep you safe and tell no one.\"\n\nTerrific. Why would they put themselves in that kind of danger for me?\n\n\"Would it ever end?\" I ask Kendra.\n\n\"At some point,\" Kendra says, \"Violet will have to move on. Witches always do. Then, you could start a normal life again, maybe go to college.\"\n\nIt doesn't seem like much of a plan, but there's nothing else to do. For some reason, I keep thinking about my AP exams. I won't even get to take them.\n\nWhen we get to Goose's house, Stacey is browning meat on the stove. Jeron is hanging in a baby swing. The other three kids are spread out at the kitchen table, doing three different homework assignments.\n\n\"Oh, you brought company.\" Stacey wipes her hands on her apron. She gives Kendra's pink prom dress a quick up-and-down look. \"My son always finds such pretty girls.\"\n\n\"Should I have worn something else?\" Kendra whispers to me. \"Is this too much? Should I change now?\"\n\n\"No!\" Goose and I both say, real quick.\n\n\"It's fine,\" Goose adds. \"Mom, this is Kendra, and you remember Celine.\"\n\n\"Of course.\" Stacey moves the pan off the stove. \"How have you been, dear?\"\n\n\"Fine,\" I say automatically. \"Well, except . . .\"\n\n\"That's why we're here, Mrs. Guzman,\" Kendra says. \"Celine has a big problem, and we're hoping you can help.\"\n\n\"I live to solve problems,\" Stacey says.\n\nI look to see if she's joking, but the funny thing is, she doesn't seem to be. Her eyes are full of concern.\n\n\"Mom, what's an adjective?\" Tyler asks.\n\n\"Stupid,\" Tony says. \" _Stupid_ is an adjective. Don't you ever do Mad Libs? It's a word that describes someone, like _fat_ or _freckly_.\"\n\nStacey gives Tony a look. \"Antonio, we don't call anyone stupid around here.\"\n\n\"I didn't call him stupid. I just said stupid is an adjective.\"\n\n\"Don't you need help sometimes?\" Stacey demands.\n\n\"I need help _now,_ \" Tony says, \"with math.\"\n\n\"Oh, gosh.\" Stacey looks at Kendra and me. \"Can you just wait one second?\"\n\nQuickly, she dumps the browned meat into a colander to drain, then she heads over to the two boys.\n\nIn the swing, Jeron laughs. Goose is so lucky to get to live with such a happy family.\n\nMaybe I will be too.\n\n\"Can I help him with his math, maybe?\" I ask.\n\n\"Oh, could you?\" Stacey says. \"But I thought you needed to talk to me.\"\n\n\"Goose can explain. I like math.\"\n\n\"Oh, good, because it's not my favorite.\" She walks back to Goose and Kendra. \"What can I do for you?\"\n\nI don't hear most of their explanation because I'm reading about the builder who needs 244 bricks to build a house and has to buy them in boxes of six. Isabella's crayons keep rolling off the table. I almost think she's doing it on purpose, to get my attention. Still, I keep retrieving them.\n\nSo the first thing I hear is Stacey saying, \"I'm sorry, but you expect me to buy that?\" She elbows Goose. \"I'm sort of amazed you believe that.\"\n\n\"I know it sounds crazy.\" He looks up at Kendra. \"Isn't there some way you can make my mother understand, like what you did at your house, with the decorations?\"\n\nKendra lifts her hand to her chin. \"Another parlor trick, so to speak? Oh, okay. But perhaps we should go into the actual parlor\u2014meaning the living room\u2014so as not to freak out the children.\"\n\n\"Good idea,\" Goose says. They walk around the corner.\n\nTony's doing better with his math, and I don't think I'm supposed to do it for him anyway. I sort of want to know what's going to happen. So I stand. I walk over to where the ground beef is draining and pour it back into the pan. I never get to cook at home, but Mrs. Mendez sometimes makes spaghetti with meat sauce, which I guess is what Stacey's making. There's a jar of sauce and a box of pasta on the counter. I don't know if Stacey wants to add anything else to the sauce, but I find a pot and put on some water to boil.\n\nSuddenly, from the corner of my eye, I see a commotion in the living room. Lights are flashing purple and green, sparkling like falling fireworks. Small objects fly by. I run to see what's happening in time for everything to stop.\n\nKendra and Stacey are standing there with a very tall man. He's definitely at least six four, and he definitely wasn't there before.\n\n\"What the\u2014?\" the man says.\n\nI do a double take, look at his face.\n\nIt's Goose, a foot taller than Kendra, where before, he'd been a foot shorter.\n\n\"You . . . did this?\" Stacey is backing away, screaming, \"Put him back!\"\n\n\"How do I look?\" Goose peers into the mirror on the wall, but since it's hung low, he has to bend down to see. \"I can't tell because my face looks the same\u2014all studly and hot. But look at you down there\u2014so cute!\"\n\n\"I liked you better before,\" I say. \"Kendra, you have to put him back. What will people at school say?\"\n\n\"They'll make me the star of their basketball team.\" Goose mimes a slam dunk.\n\n\"Put him back. Please,\" Stacey says.\n\n\"Fine.\" Kendra is pouting. \"People say they want a demonstration. Then they get all mad when you demonstrate.\"\n\nShe looks up at Goose, then goes into a sort of trance. It all repeats, the room, the lights. At the end of it, Goose is back to normal. He stares up at me.\n\n\"You could make me an inch taller, couldn't you?\" he says to Kendra. \"Or maybe just give me better hair?\"\n\nShe shakes her head. \"That's how Violet got started. Little things.\"\n\n\"Celine, what's an adverb?\" Tyler asks from the kitchen.\n\n\"A word that modifies a verb. Sort of like an adjective. They mostly have 'ly' on the end. Like, 'He grew shockingly.' _Shockingly_ is an adverb.\"\n\n\"Thanks,\" Tyler calls.\n\n\"I put on some water to boil,\" I tell Stacey, who looks like she's about to fall over. \"Do you need a glass of water?\"\n\nShe nods, shakily, and I go to get it. She backs up to lean on the wall.\n\nWhen I come back, she's saying to Kendra, \"So you're saying that Celine's stepmother has . . . powers like that, and she's going to use them to hurt Celine?\"\n\nKendra nods. \"To kill Celine.\"\n\n\"Oh, sweetie.\" She takes the water from me. \"I'll have to discuss it with my husband, of course.\"\n\n\"That's just a formality,\" Goose says. \"He never says no to you.\"\n\n\"He's right,\" Stacey admits. \"I still have to ask, though. But he'll be happy to let you stay. How could we say no?\"\n\n\"Thank you. I'll help around the house. I'll tutor the kids. I'm really good at math. I'll\u2014\"\n\nStacey puts her arms around me. They only come up to about my waist, but the hug is warm and strong, like a mom's. I lean into her. She says, \"It's okay. I know you will. You've been through so much. I want you to be safe. Every child deserves to be safe.\"\n\nThat may be the first time anyone has ever said anything like that to me.\n\nWhen Mr. Guzman comes home he is _not_ real happy about the idea. Kendra demonstrates her powers yet again (this time, by cleaning up the kitchen using only her mind\u2014which I can tell Stacey appreciates). Then, they retreat to a bedroom to argue. Goose and I listen through the door.\n\n\"She's a minor,\" Goose's father says. \"We could be accused of kidnapping.\"\n\n\"We're not kidnap\u2014\"\n\n\"Doesn't matter. You take someone else's kid, you're kidnapping. It's the law. We could try to get a legal guardianship.\"\n\n\"And then, everyone would know where she is,\" Stacey says. \"This woman already killed her mother.\"\n\n\"And you believe that?\" Mr. Guzman says.\n\n\"I do. And if you're wrong, and the girl dies, how will you feel? You fight for people's rights for a living.\"\n\n_If the girl dies._ Me. I'm the girl. My whole life depends on the outcome of this conversation. The outcome of this conversation and Kendra, which is a scary thought.\n\nNext to me, Goose whispers, \"He's a lawyer, unfortunately. My mom can talk him into anything, though.\"\n\nMr. Guzman says, \"I know I do. I'm trying to protect my family here. You want to protect the whole world.\"\n\n\"Yes, I do!\" Stacey yells. \"It takes a village, Jorge.\"\n\n\"Why do we always have to be the elders of the village?\"\n\n\"Someone has to be!\"\n\n\"We don't even know this girl,\" he says.\n\n\"Maybe we should put the sauce on,\" Goose whispers. \"I'm thinking we can probably still hear them from the kitchen.\"\n\n\"Good idea.\" I'm not sure I want to hear them.\n\n\"Your son knows her,\" Stacey says. \"Are you saying you don't trust his judgment?\"\n\n\"He's seventeen. He sees a pretty girl and\u2014\"\n\n\"Come on!\" Goose takes my hand and hustles me to the kitchen before I can hear what he does when he sees a pretty girl. Kendra is sitting at the table helping with homework, her pink tulle skirt puffing out over the tabletop. The boys are finished with what they were doing and are on to a history assignment, which is the same for both of them. Kendra is holding crayons while Isabella colors.\n\n\"You're pretty smart,\" Goose says.\n\n\"I've done high school thirty-seven times,\" Kendra says. \"I enroll whenever I get bored. If I went to your high school, I'd be the valedictorian without even using magic.\"\n\nGoose laughs. \"I may need some help with chemistry then.\"\n\n\"We can study together,\" I say before I remember that maybe I won't be studying, won't even be going to school.\n\n\"Your dress is so pretty.\" Isabella pets Kendra's poufy skirt.\n\n\"If you do a good job, I'll make one just like it for you,\" Kendra says.\n\n\"I think she usually adds an onion to the spaghetti sauce,\" Goose says.\n\nI find an onion and a knife to start chopping it up. Tears fill my eyes, and I pretend it's the onion. I want to stay in this warm, safe house with Goose's warm, safe family. But I wish I didn't need to. I wish I could go back in time to when I was eight years old and everything didn't suck.\n\nKendra comes up behind me and whispers, \"It's okay. If they say no, we'll think of something else. You're safe. I'll make sure of it.\"\n\nBut that's when Stacey comes into the kitchen and says, \"You can stay with us.\"\n\n\"You talked Dad into it?\" Goose asks.\n\n\"I didn't have to talk him into it,\" Stacey says. \"He knew it was the right thing to do. Right, Jorge?\"\n\nHe walks in behind her. \"I can't ignore someone in trouble. Welcome to the family, Celine.\" He sticks out his hand. \"Don't get us sent to jail.\"\n\nI wipe the onion juice off on my jeans before offering my own hand. \"I won't.\"\n\nThat night, Kendra invites me to visit her house after school tomorrow. She does it loudly, in front of Violet. \"In fact,\" she says, \"I can drive you to school in the morning too.\"\n\nShe tells me to pack any items I really want in my backpack. \"Nothing Violet would notice missing. You're supposed to be dead. I might be able to bring you some other things later, but it's better to be safe. Violet is a powerful witch, so I'm not sure what I can slip past her. Take your favorite things.\"\n\nIt's amazing in a situation like that, how few things matter. I take photographs of my mom and dad, a bracelet my father bought me made of tiny golden seashells, some clothes, underwear, and toiletries. I take all my saved-up birthday money in case I need anything else, or for an emergency.\n\nLast, I go to the computer and print photos off my phone: Laurel and me, wearing matching _I_ \u2665 _Jonah_ T-shirts, me and Goose at Target. I put the latest Jonah Prince album on a disc. I delete the photos of Goose from the hard drive so Violet won't know about him.\n\nI hand my phone to Kendra.\n\n\"I guess I can't use this anymore.\"\n\nKendra shakes her head. \"Stacey knows you're coming tomorrow, late morning. She's ready for you.\"\n\nI've already told Laurel I'm moving in with my aunt in Tennessee. I told her I'll be living in the mountains with little internet access. \"I was thinking maybe you should make a sign that says, 'Dare to eat a peach,'\" I told her.\n\n\"I'm not going,\" she said.\n\n\"You should. Take someone else. Just, you know, think of me when you're there.\"\n\nShe sighed. \"This sucks.\"\n\nI can't believe I'm not going to see Jonah.\n\nNow I ask Kendra, \"Can I write to Laurel ever?\"\n\n\"Not for now,\" Kendra says. \"It's too risky. What if her mother ran into Violet and said they'd heard from you?\"\n\n\"That's unlikely. Her mother hates Violet.\"\n\n\"It's possible,\" Kendra says. \"I'll help Laurel to . . . forget you.\"\n\n\"Just what I want, my best friend to forget me.\"\n\n\"I know, dear. I'm sorry.\" She's said it a hundred times. I know she blames herself. I can't say I don't blame her for parts of it. But she's the only person who can help me now. At least, I hope she can.\n\n\"How are you going to withdraw me from school? You're not a relative.\"\n\nKendra looks down a second like she's deep in thought. Her dark hair cascades over her eyes. When she looks back up again, the eyes that meet mine are Violet's. Her whole face is. And her hair transforms from brown to auburn. \"They won't know that.\"\n\nIt's sort of terrifying. I want to go back to that time in my life when I knew nothing about magic, when I didn't suspect it existed.\n\nI look around the room. This will be my last night going to sleep here. I've lived here my whole life. I lived here with my mother, back when I was happy. I wonder if I'll ever be happy again.\n\nI guess it's up to me now. I have Goose's family to protect me. But I need to protect them too\u2014from Violet.\n\n#\n\n#\n\nThe next morning, Kendra, disguised as Violet, takes me to school. I don't say good-bye to Violet, who is crying in her room. I amble down the stairs and to the front door, taking in each wall, each rug, each stick of furniture. I know I'll never be here again. When I get outside, I pause. The yellow birdhouse I built with Dad when I was ten hangs in our Hong Kong orchid tree, uninhabited. We had wrens last year, but now they're gone. I walk over to the tree. I glance up at Violet's window to make sure she isn't watching me. Nothing. I pull down the birdhouse from the tree.\n\n\"I'm ready,\" I say when I get in the car.\n\nKendra is wearing one of Violet's favorite work suits, black skirt with a bright purple jacket. She doesn't drive the car. It sort of drives itself. She doesn't even put her hands on the wheel.\n\nI watch as the car stops for a school bus with no input from Kendra. \"So Violet thinks you're just going to take me to your house after school to kill me?\"\n\n\"Pretty much.\"\n\nI feel a chill go up my arms. I remember Violet, when I was a kid, taking me to the park to feed the ibises, how I used to laugh as they pecked at the crumbs with their slender beaks, and Violet told me they were the first birds to come out after a hurricane. She didn't hate me then. How could she hate me so much now? What would be happening had Kendra not intervened? What if Violet finds out?\n\n\"Thank you,\" I say. \"I know this is hard for you.\"\n\nKendra strokes my arm where the little hairs are standing on end. \"I wish I had done more, sooner. I loved Violet so much I didn't see it, didn't see what she'd become. Love can blind us.\"\n\nWe're at school now. The car steers itself into the parking lot, and we get out.\n\n\"Act like you like me, so no one gets suspicious.\" Violet-Kendra takes my hand.\n\nI pull away. \"If we held hands, that would be suspicious. No girl my age would hold hands with her stepmother.\"\n\nI walk ahead of her to the office.\n\nWhen she finally gets someone to help her, she says, \"I'm Violet Columbo. I'm withdrawing my daughter from school.\"\n\n\"Withdrawing from school? Why?\" The woman's eyes immediately go to my waist. Oh, God, she thinks I'm pregnant.\n\n\"I mean she's transferring to another school. Here.\" Kendra reaches into her purse and pulls out Violet's driver's license along with a sheet of paper.\n\nThe lady looks at the paper. \"Tennessee. Far away. We're sorry to see you go.\"\n\nI look sad. It's not hard. I am sad. My whole life is over, school, taking drama next year, seeing my friends. It's almost like Violet _is_ killing me, killing everything about me. I feel my face get hot around the eyes.\n\nBut, soon enough, it's over and I'm back in the car with a Kendra who looks like Kendra again. \"What happens when my info goes to that other school?\"\n\nKendra wrinkles her nose. \"What other school?\"\n\n\"The one in . . .\" I realize what she means. \"Oh, I see.\"\n\nWhen we pull in to the Guzmans' driveway, Stacey has just pulled in too. She must be coming back from the kids' school. She's taking Jeron out of his car seat.\n\n_This situation is so . . . awkward. I don't know this woman, but I'm moving in with her._\n\nBut as soon as Stacey sees me, she turns around. \"Welcome to our home.\"\n\n\"Oh, thank you.\" I want to hug her, but her hands are full, so full without me, fuller now. \"You have no idea\u2014\"\n\n\"It's fine, sweetie. You're safe here. Come on inside.\"\n\nI turn to say good-bye to Kendra, but she and the car are both gone. How does she do that? And will I ever even see her again?\n\nI follow Stacey inside. I offer to take Jeron, but she says she's got him. She hands me her keys. \"Lock the door behind us,\" she says.\n\nI notice the door has three locks, the regular one and two dead bolts. She sees me looking at them. \"I've fostered kids who were removed from their parents' custody. Drug dealers, really bad people. It seemed like a good idea to have some extra barriers.\"\n\nI nod. I know that, if Violet finds me, no door will keep her away. And yet, here with Stacey, I do feel safe, safer than I've ever felt since I was little.\n\nI lock all three locks. Each makes a satisfying clunk.\n\nWith Jeron on her hip, Stacey takes me on a tour of the house. \"Your room's in the back, where it's quieter. You have to share with Izzie, I'm afraid.\"\n\n\"That's okay. She's sweet. I've always wanted a sister.\" The room has bunk beds. The bottom is made up with _Little Mermaid_ sheets. The top is more sedate, but still pink, waiting for me. The walls are Pepto-Bismol pink with giant snails and starfish painted all over them. Stacey must have painted these, and I love it all.\n\n\"See if you think so in a week. We had to clean for a couple of hours to get it presentable for you. But Isabella helped. She's excited about having a big sister.\"\n\nI smile, feeling good for the first time in a week. \"I'll help her clean up. I'll help with all the housework.\"\n\nStacey doesn't protest or say I'm a guest. She just thanks me. I'm glad. I have no intention of sitting on my butt all day, sponging off them. It strikes me that I never saw Violet clean the house. She must have used magic. Still, I'm guessing I can figure it out.\n\nStacey shows me around the rest of the house. It's big and pretty and most of the walls are bright yellow, cheerful enough to maybe let me forget that, even though I'm with these great people, I'm a prisoner here. There's a piano in the living room, and I ask Stacey who plays.\n\n\"I used to, when I had more time. Now, the boys do.\"\n\n\"Goose plays the piano?\"\n\nShe laughs. \"Can't picture him sitting still that long, huh? I made him learn when he was younger. He complained so much, said the teacher was mean, but now, he plays great. I'll make him play for you.\"\n\nI say, \"I'm so grateful to you.\"\n\n\"Stop being grateful. This family, we help kids who need it. You need it. Besides, my spoiled son can talk me into anything.\"\n\n\"He's a talker.\" I grin.\n\n\"He wanted to stay home today.\" She steers into the kitchen and puts Jeron in his swing. She starts taking out stuff for a bottle.\n\n\"Can you show me how to make that? I want to help.\"\n\nShe hands me a canister of powdered formula and the bottle. She shows me how much water to add. \"I only got him to go today by saying it might be suspicious if he missed too much. He's a little lazy.\"\n\n\"He gets good grades, though. I saw him at the honor roll assembly.\"\n\n\"Because I stand on his neck.\" She laughs. \"Not literally. He doesn't like homework, though.\"\n\n\"I can help you with that too. I'm good at homework.\"\n\n\"Don't worry so much, Celine.\"\n\nBut I do. Still, I nod and offer to give Jeron a bottle.\n\nAfter that, Stacey puts him down for his nap. \"I have to get some work done now.\"\n\n\"What do you do?\"\n\n\"I illustrate kids' books. I can show you my studio later. We built out the garage. But right now\u2014\"\n\n\"I know. You've got to work while you can.\"\n\n\"Make yourself at home. You must be exhausted from all the stress.\"\n\nI don't think I am, but she probably needs to work, so I go to my\u2014our\u2014room and climb the ladder to the top bunk. I survey the room. The sea creatures watch me. I guess Stacey painted them. It's something my mom would have liked. When I was little, my room had a border of cows jumping over moons. I kept it long after I was too old, but eventually, Violet redecorated. From above, I can see that Isabella didn't do that thorough a job of cleaning. Clothes and papers are stuffed behind the dresser and desk. I'll offer to help her, or maybe just clean up myself.\n\nBut first, I want to shut my eyes.\n\nI crawl under the crisp, pink sheets that smell of laundry detergent and bury my head in the pillows. I wish I could have brought my own pillow from home\u2014I've had it for years, and it's smooshed just the right way\u2014but it was too big. Also, my music, which was on the phone. I have the CD I made of Jonah, but no CD player. I'll have to ask Goose if he has one. Still, the bed is comfortable, and I close my eyes.\n\nIn the dream, I am little. Or, at least, everyone else is bigger. I'm someplace cooler than Florida, and it is fall, apple-picking time. I'm at an orchard with Dad. Then, I realize my mother is there too. She's so young and beautiful, her blond hair flying behind her. We are so happy, running, stumbling, falling through the piled red and gold leaves to trees heavy with fruit. Mom holds my hand as my father hoists me onto his shoulders. I reach for the sky, grasping a crimson apple instead. I start to bite into it.\n\nIt falls apart in my hand.\n\nSomething is crawling out of it. Insects. Maggots. And then, they are all over me, down my hand, up my arm, devouring me, my mother, my father.\n\n\"No!\"\n\n#\n\n#\n\n\"Hey!\"\n\nSomeone is shaking the bed, making the world vibrate.\n\nI wake, staring at my hand. It is still there, still the same. Gradually, objects take focus behind it. I stare into the eyes of a blue stingray, then a red crab. Where am I?\n\nI look down to where the shaking came from. Goose is there, by my bed.\n\n\"Are you okay? I wasn't going to come in, but you were . . . you seemed . . .\"\n\n\"It was just a dream, just a nightmare.\" I smile. \"I'm glad you're here.\"\n\nHe looks up. \"Heights. My favorite.\"\n\n\"I'll come down.\" I gesture for him to move over, then throw my legs over the side of the bed and slide down.\n\n\"How was your first day?\" he asks.\n\n_You mean, other than the fact that my parents are dead and I'm living with strangers because a crazy witch is stalking me?_\n\n\"Great,\" I say.\n\n\"I doubt that.\"\n\n\"I miss my house, and my school, and everyone\u2014and especially, my dad. But I'm glad you're here now. It's a lot less lonely.\" I reach for his hand. \"You're so sweet.\"\n\nHe grins. \"When my brothers and sister get home, you're going to wish for lonely.\"\n\n\"I doubt that. I've never had siblings. Laurel has a brother, though, and he messes with her, but they're always there for each other. I always sort of wanted a big brother.\"\n\nHe looks down. \"I guess I can be your brother, then.\"\n\n\"I didn't know you played piano.\"\n\n\"One of the many things I'm awesome at. Want to hear me?\"\n\nI remember my promise to Stacey. \"Don't you have homework?\"\n\n\"What are you, in cahoots with my mother?\"\n\n\"Maybe a little. Your mom is awesome.\"\n\n\"That's because you don't know her.\"\n\n\"I know she let me come live here. She's the best.\" I put my hand on his arm. \"Come on. I can sit with you while you do it. I was taking chem this year too. I could do it with you. That way, when I retake it, I'll be all set.\"\n\nI stop. Will I ever take chem again? Will I ever get to walk down the street again? Or will I be hiding forever? Goose is right. I barely know his family. I don't even know him that well. Did Kendra put some kind of spell on them to make them have me? Or did she just know they were the kind of people who would?\n\nGoose says, \"Okay, deal. But first, let's make smoothies. Our neighbor, Mrs. Ozanich, has a bunch of mango trees. She gives us tons of mangoes every year, and my mom freezes them. We still have frozen mangoes left from last year, and it's almost time for the new ones to ripen. So Mom says we have to make smoothies every day.\"\n\n\"I've never had a mango,\" I say.\n\n\"Wow, how can you live here and not have mangoes?\"\n\nWe go to the kitchen and take out the frozen mangoes, yogurt, and juice for smoothies. We make enough for Goose's siblings too. \"To keep them quiet,\" he says. Then, we start on chemistry. Mrs. McKinney is one of those teachers for whom copying from the book is a sort of religion. She's assigned them to copy fifty-plus definitions. I offer to read them aloud while Goose writes them.\n\n\"I'd rather do it the other way around, I read and you do the copying,\" he says, \"but I guess they'd notice if I suddenly developed pretty, girly handwriting.\"\n\n\"How do you know my handwriting is girly?\"\n\n\"Hmm, one, you're a girl, and two, I saw how you wrote 'C.C. loves J.P.,' which, three, is a girly thing to do.\"\n\n\"Oh, it is?\" Of course I know it is.\n\n\"Yup. A guy wouldn't write that on a notebook even about his actual girlfriend.\"\n\n\"So guys don't express their emotions. Got it.\" I gesture for him to get out some paper.\n\n\"I didn't say that. Guys express their emotions by doing things.\"\n\n\"Like what?\" I say,\n\n\"I don't know, taking care of a girl, fighting off bears, slaying dragons, repairing plumbing, going to work sixty hours a week to feed their families, like my dad does.\"\n\nI think, again, of my dad, about how mean I was the last time we spoke. He thought Violet and I loved each other when he married her. I did love Violet then. How did it all go so horribly wrong?\n\nGoose waves his hand in front of my eyes. \"You there?\"\n\nI snap to. \"Women do things for people they love too.\"\n\n\"Like what? Bake cookies?\"\n\n\"I guess. Yeah, we make things nice, and cookies might be involved. What's wrong with cookies?\" There was a sign up at Offerdahl's, the sandwich place near my house, that said, _Cookies Are Love_.\n\nHe laughs. \"Nothing. I love cookies. What else?\"\n\nI make a mental note to make cookies for his family, maybe tomorrow. I'll ask Stacey if she has chocolate chips. Otherwise, there should at least be stuff for sugar cookies. \"I don't know. I've never been in love\u2014other than with rock stars who don't know I exist. Have you?\"\n\nHe thinks about it a second before shaking his head. \"No, not really.\" He gets paper from his backpack. \"We should do this. I want to dazzle you with my musical ability before I get stuck helping my brothers with their homework. What's the first one?\"\n\n\"Acid-base titration. That's t-i-t . . .\"\n\n\"Ha! Made you spell a bad word.\"\n\n\"Idiot. R-a-t-i-o-n.\" I wait for him to finish writing. \"It means a procedure that is used to determine the concentration of an acid or base.\"\n\n\"Got it.\"\n\nThe definitions take over an hour, and then he has pre-calc, which I can't help him with except by keeping his siblings away from him. Isabella has developed a little girl-crush on me, so I get her to help me clean up behind the dresser while Stacey helps the younger boys with homework. Then, Isabella and I set the table.\n\nIt's after dinner before we go to the piano, but Goose actually does dazzle me, playing the _Moonlight_ Sonata __ from memory. Then, he takes out a book of vocal selections from _Oliver!_ and makes me sing along.\n\n\"You're incredible,\" I say.\n\n\"I wouldn't say incredible. You mean because of my size?\"\n\n\"No. I didn't say that. I would never\u2014\"\n\n\"No, you didn't. I'm sorry. People tend to underestimate me. When my mom first took me for lessons, the guy didn't want to teach me, said I wouldn't be able to play because my fingers would be too short or something. I believed him, but my mom found a different teacher, and she told me about this French dude, Michel Petrucciani. He had a genetic disease that not only made him small, but also made his arms ache when he played. He died young. But he became a really famous jazz pianist. They said he was a dwarf but played like a giant. So, after that, I knew anyone could play.\"\n\n\"I'd like to hear him sometime. But I just meant you play really well for anyone. I wish I'd learned. I was begging my parents for lessons when . . .\"\n\nI don't want to say it. It feels like I've been thinking about Mom a lot, like I'm always complaining.\n\n\"You didn't because your mom died?\" he finishes.\n\n\"I love singing, though.\"\n\n\"I could teach you to play,\" he says. \"You could practice during the day, as long as you don't wake Jeron. My mom will kill you if you wake Jeron.\"\n\n\"Really? You'd teach me?\"\n\n\"Sure, why not? I think they sell beginner books for older students in the school bookstore. But we have my brother's books for now. It's a little babyish.\" He pulls one out from behind his book. There are cartoons of birds sitting on a musical staff. \"Okay, it's a lot babyish, but it shows all the notes. You start with middle C.\" He gestures for me to sit down. I do, next to him, hip to hip. He picks up my hand and places it so the thumb is on a white key that's next to another white one. \"Each finger has a number. The thumb is one, forefinger is two, and so on. That's how you know which finger to use, so you don't tie yourself in knots. CDEFG.\"\n\nHe guides my hand, playing each one. My blue nail polish is chipped. I'll have to ask Stacey for nail polish remover.\n\n\"What's that?\" Goose points to a burn scar on my finger, the one from the curling iron.\n\n\"Oh, it's weird. It's a burn. Violet could make appliances . . . turn on me. I know it sounds crazy.\"\n\n\"Nah.\" He strokes it with his finger. \"Poor little finger.\"\n\n\"It didn't hurt that much.\"\n\n\"Mom wants you to take out the garbage.\" Tony comes up behind us. \"And I'm supposed to practice.\"\n\n\"You could go a week without practicing. You only want to practice now because I am.\"\n\nTony grins. \"Recital's coming up.\"\n\nI stand. \"I'll help you with the garbage.\"\n\n\"No. You can't go outside,\" Goose says. \"It's not safe.\"\n\n\"I just feel like I should help more. Your family is helping me so much\u2014\"\n\n\"You helped me do a ton of chem busywork. It would've taken twice as long without you. I hate that crap. I'll be back in a few minutes.\"\n\n\"You're not going to watch me play, are you?\" Tony says.\n\n\"Only if you want me to,\" I tease.\n\n\"I don't.\"\n\nSo I go into the family room and watch a reality singing show __ with Stacey and Isabella. A few minutes later, Goose comes back in. He stands in the doorway, motioning for me to come over.\n\n\"Goose wants to talk to his girlfriend.\" Isabella giggles.\n\n\"Shh,\" Stacey says.\n\nI head over to him. \"What?\"\n\nHe pulls me away. I can hear Tony practicing an unrecognizable song for the tenth time.\n\n\"There was someone out there, watching me,\" he whispers. \"Don't go near the windows.\"\n\n\"Someone? Like who?\" I'm whispering too, so Stacey won't hear. Or someone outside.\n\n\"I don't know. I heard rustling. I saw . . . a shape. Our neighbor, she's outside gardening a lot, but not at night.\"\n\nHe's worried. So am I. If Violet's watching Goose, she's as good as found me. How would she, so quickly?\n\n\"I'll check,\" Goose whispers. \"I just want you to stay away from the windows, okay?\"\n\nThe doorbell rings. We both jump.\n\n\"I'll get it!\" Tony starts to get up.\n\n\"No! You keep practicing. You have a recital coming up\u2014and it's not sounding real good at this point.\" Goose walks to the door. \"And remember, no one's supposed to know about Celine being here.\"\n\n\"I know, I know,\" Tony says. \"It's your dear little sister that's going to be the weak link there.\"\n\nGoose waves him off. At the door, he says, \"Who is it?\"\n\n\"It's me,\" says Kendra's voice.\n\nGoose starts to open the door.\n\n\"Wait. How do we know it's Kendra?\" I ask.\n\n\"Only Kendra would know you know Kendra,\" she says from outside.\n\nGood point. Still, Goose opens the door warily.\n\nA woman who looks nothing like Kendra stands there, a blond housewife in yoga pants. She steps inside. \"Close the door.\"\n\nI do, and as I do, she melts into Kendra again, dark hair, purple-streaked, wearing a veiled black hat and high-heeled boots that make her look from the Victorian era.\n\n\"It's done.\" She looks at me. \"You're dead.\"\n\nI draw in a breath. \"You mean\u2014\"\n\n\"I mean Violet believes you're dead. I brought her the proof she sought.\"\n\n\"What proof?\"\n\nShe opens her hand. I step back. In her palm is an object, a finger. _My_ ring finger, wearing a turquoise ring my dad brought back from a trip to Arizona. The blue nail polish is chipped, and on the knuckle is a burn scar. I look at the finger that is still on my hand. They're identical, other than the blood.\n\nAnd suddenly, it all becomes real, too real. I feel a tightness in my throat. My vision looks black, gray, like someone is fiddling with the lights, and my head feels tight.\n\nAnd then, I'm on the floor. Someone is pressing a wet rag to my forehead. Someone else is holding me.\n\n\"Celine?\" It's Stacey who's cradling me in her lap. \"Are you okay?\"\n\n\"Yes, it was just . . .\" I search for Kendra. Gradually, she comes into focus.\n\n\"I shouldn't have shown it to you. Sometimes, I don't think.\"\n\n\"Yeah,\" I say. _Yeah, seeing my severed finger was a little jarring._ \"Will it work? Will she believe I'm dead because of that?\" Stacey's stroking my hair like a mom.\n\nKendra says, \"She knows I'm a witch, so she knows I can make things up. But she trusts me. If she stops trusting me, she'll get suspicious.\"\n\n\"So that means\u2014\"\n\n\"It means I have to pretend I adore her when she completely disgusts me. Ah, well, if anyone can do it, I can. I've had centuries of practice.\"\n\nI take Kendra's hand\u2014the one that isn't holding my finger\u2014and squeeze it. I know I'm not the only one Violet has hurt.\n\n#\n\n#\n\nThat night, when I go to sleep, I dream of the orchard again. But this time, when I reach for the fruit, I fall from my father's shoulders to the ground, into a pile of apples, some ripe, some rotten. Down, down, I sink deep into them, and they multiply like cancer cells, massing around me, eating me alive. I look up, searching for my father's hand, but he is doing nothing to save me.\n\nI wake, sweating, sobbing. I reach out, searching for the nightstand, the clock. There is only air. Gradually, I remember where I am. Goose's house. Safe. For now. Objects take shape in the darkness. The buzzing of a thousand bees becomes a ceiling fan. A monster in the corner becomes Isabella's giant stuffed bear, the kind people win at the fair. I try to sleep, but now, I am awake. I can't listen to music because I left my phone with Kendra. I don't want to disturb the family, especially Isabella, but the more I lie here, the more terrified I become, thinking of Violet coming for me. Not only for me, but for Goose and his parents, his siblings, even little Jeron. I'm putting them all in danger. They are all at risk for me, and there's no end in sight. Kendra has no plan. Maybe I should run. Or just give myself up to Violet.\n\nIn the darkness, I climb down the ladder. My backpack is still there, packed. I take out jeans and a T-shirt, put them on. Isabella stirs in her sleep. I stand still until she settles back in. I pick up the backpack and my shoes.\n\nIn the family room, I sit on the sofa to put on my sneakers. I leave the lights off. I don't know where I'm going. I have my birthday money, over a hundred dollars. Maybe I could take a bus somewhere, somewhere far from Violet, then look for a shelter. I start to tie my shoelaces.\n\nThe lights go on, blinding me. I blink, then look toward the door. It's Mr. Guzman.\n\n\"Night owl, huh?\" he says.\n\n\"I had a nightmare. Then, I couldn't sleep, and I feel . . .\" I stop.\n\n\"What?\"\n\n\"Nothing. Just insomnia.\" I know if I tell him I feel guilty about being here, he'll just try to reassure me, tell me how much they want me here. I am already taking enough from these kind people, who are risking so much for me. I don't need to burden them with my guilt too.\n\nHe says, \"Are you okay? What was the dream about?\"\n\nI notice he's dressed for the office already, carrying a briefcase. I remember what Goose said about how hard he works.\n\n\"I don't want to make you late,\" I say.\n\n\"I'm not late, I'm early. I like to read the paper when it's quiet. You're . . .\" He hesitates, looking me up and down. \"You sleep with your jeans on?\"\n\nI nod.\n\n\"Shoes too?\"\n\n\"I was just putting them on.\"\n\n\"I can see that. My question is why? I know you can't be thinking of taking a walk.\"\n\n\"I was thinking about leaving.\" It just pops out.\n\nHe doesn't reply, only nods. I add, \"I guess I just . . . I feel bad, making you hide me.\"\n\nHe shakes his head. \"You didn't make us do anything. We volunteered.\"\n\n\"I know, but\u2014\"\n\n\"But nothing. We're responsible for you. You think if you left, we'd just forget about you, like a cat that shows up on our doorstep for a while, then leaves?\"\n\n\"I guess not.\" I really thought he didn't want me to stay. I know Goose and Stacey pressured him to keep me.\n\n\"My wife and son didn't talk me into anything,\" he says, reading my thoughts. \"I'll admit I was . . . hesitant to take you in, but now that we committed, we're in this. If you left, we'd have to look for you. We couldn't just leave you out there on your own, in danger. The police would be involved. Do you want that?\"\n\nI shake my head.\n\n\"Me neither. So why don't you go back to bed for a while? It's not even six.\"\n\n\"Maybe I will.\" I stand and start to go.\n\nWhen I reach the doorway, he says, \"Celine?\"\n\nI turn back.\n\n\"I have a big trial today. I've been preparing for months, and my client stands to lose big if I do a bad job.\" He pauses, staring at my feet, still in sneakers. \"I don't have to worry about whether you're going to run away, do I?\"\n\n\"No, sir. You don't need to worry. Thank you.\"\n\nI go back to bed until Isabella wakes me, wanting to know if I can braid her hair. Goose has already left for school.\n\nI practice the piano all day, except when Jeron is asleep. That's when I make oatmeal cookies from the recipe on the box of oatmeal. I learn Tony's whole piano book. That day, after school, Goose brings me an _Adult All-in-One Piano Course_ book and a book of Jonah Prince's greatest hits. \"I stopped at the music store in Suniland. So you have something to write 'I heart Jonah' in,\" he says.\n\nI'd wondered where he went. I flip through the pages. It has \"Beautiful but Deadly,\" my favorite song, and all these pictures of Jonah.\n\n\"I heart your family,\" I say.\n\n\"You should've known we'd be awesome, having met me.\"\n\n\"I did. I just didn't know _how_ awesome. You're so lucky.\"\n\n\"We don't want anything to happen to you,\" he adds.\n\nI wonder if Goose's dad told him about my escape attempt. I hope not. I don't want him to know. And I don't want anything to happen to them either.\n\nLater, in my room, I cut one of Jonah's pictures out very carefully, with scissors, and tape it up next to my bed. A souvenir of the life I used to have, when I cared about things like rock stars.\n\nThat night, I don't have any nightmares. Instead, I dream I'm with Jonah. It's the same kind of weird dream I used to have, where he rescues me. This time, I'm a princess, like Sleeping Beauty, comatose in my golden bed. Jonah comes through the window and kisses me. I wake and gaze into his eyes. A handsome prince! He rescues me, taking me into his strong arms. We're in love, will be in love forever.\n\nOf course, I know it's not real. Still, I wonder if the dream means I'm safe.\n\n#\n\n#\n\nWhen you're little, you think it would be cool not to have school, to just sit home and watch TV all day, every day. When my mom was alive, and sometimes even in the early Violet years, I used to love staying home, eating soup, and watching _Fairly OddParents._\n\nLet me tell you: It gets old real fast. There's a definite limit to how many game shows, courtroom dramas, and soap operas a person can watch\u2014especially if that person isn't eighty years old. When you find yourself _really_ rooting for the uniformed soldier to win the hot tub in the showcase, __ it is time to turn off the TV.\n\nI do. I study for classes I'm not taking at school. I'm way ahead in Spanish. I research different ways to French braid Isabella's hair. I practice on her dolls. I bet she has the best hair at school. I listen to the Jonah Prince CD I made, and Goose loans me an old iPod he has. I make cookies almost every day, which makes me Tony and Tyler's favorite person. I also get good at the piano real fast. By the end of the first week, I can play Minuet in G __ by heart _._ But, when I tell Goose I want to learn _F\u00fcr Elise_ , he says I need to learn to play scales first.\n\n\"Okay, that should take a day,\" I say, feeling jazzed.\n\n\"You need to play them well,\" he says, \"two octaves, twelve different scales, and that's just the major ones.\"\n\nI roll my eyes. \"Fine. Teach me.\"\n\n\"You're welcome.\" He sits beside me on the piano bench. He's wearing some cologne that smells citrusy. Then, he starts in on C, showing me how to sneak my thumb under the first two fingers to reach the F. His hands are small, but he makes up for it with speed and skill, then he tells me to try. I do.\n\n\"Arch your hand more.\" He reaches for it, lifting my palm.\n\n\"Next, you're going to hit my knuckles with a ruler.\" I stare at my fingers, remembering the one Kendra held in her hand.\n\n\"Only if you don't practice the right way. Again.\"\n\n\"Such a power trip,\" I say. \"You've just been waiting for someone to boss around.\"\n\n\"Nah, I've had siblings for years.\"\n\nI've noticed that, since I've been here, Goose comes right home after school. I know he didn't before. He hung out with friends, went to their houses, played pranks at Target. Even Stacey commented on how much he's around. Thank God for Goose. I'd explode from loneliness if he didn't spend the hours after school with me. I don't care if I have to help him with his homework. I'd _do_ his homework for him if he wanted. He is my best, my only friend now.\n\nToday, I ask what I've been wondering about for weeks. \"So . . . why'd you and Willow break up?\"\n\n\"The F major is a little different than the first few,\" he says. \"You use the first four fingers, then roll the thumb onto the C.\"\n\n\"Goose?\" I say.\n\n\"I'm ignoring you,\" he singsongs.\n\n\"Fine. Don't tell me.\"\n\n\"It was mutual,\" he says. \"Now, you try.\"\n\nI start the scale, but I don't remember where I'm supposed to use my pinky. So I stop.\n\n\"So, okay, mutual's good,\" I say.\n\n\"Yeah, we _mutually_ decided that since she's a senior and I'm only a junior, and we're not in love or anything, she should date someone who'd look better in her prom pictures.\"\n\nHe plays the scale for me again, so I can see where my fingers go. \"You actually don't use your pinky at all,\" he says.\n\n\"You're kidding,\" I say. \"I mean about Willow, not the pinky. She said that to you?\"\n\n\"Not in those words.\" He points to my fingers, and I try again. I get it sort of right.\n\n\"What words, then?\" I cross my arms in front of my chest. I'm done playing until he talks.\n\n\"Well, for context, you have to know that when I took her to homecoming, her mom made me stand on a step stool for photos. And I was still a head shorter than her.\"\n\n\"That's because she's super-tall.\" When he gives me a look, I say, \"Well, partly.\"\n\n\"Oh, completely. It was definitely her freakish _tallness_ that was the issue.\" He stands up. Have I overstepped? But he says, \"So then, when I asked her about prom, she said she thought maybe she should go with someone else. Not anyone in particular, just 'someone else,' like anyone but me. Then, she said\u2014actual words here\u2014she didn't want me to have to find a tux because she knew it was hard for me to rent in my size.\"\n\n\"Seriously?\" I say. \"Well, it's her loss.\"\n\nHe shrugs like he did before, but doesn't sit down. \"I wasn't in love with her. She wasn't 'the one.' It just sucks to get dumped.\"\n\nI nod. \"She sucks. You'll meet someone better.\"\n\nHe sits back down. \"I guess.\" He gestures at me to try again.\n\nI do. This time, I get it perfect, but he yells at me to arch my hand. \"You're hitting extra notes with your palm.\"\n\n\"Okay, okay.\" I start again, arching my hand sarcastically high. Goose seems to think it's perfect.\n\n\"Good. Again,\" he says. When I start to play, he says, \"Sometimes, I get tired of always being the court jester.\"\n\nI'm not sure I heard him right, over the music. I stop playing. \"What?\"\n\n\"Nothing.\"\n\n\"I think it was something.\" I cross my arms again.\n\nHe gestures for me to play again, but I don't, waiting for him to elaborate. He says, \"People only like me because I'm funny.\"\n\n\"That's not true.\"\n\n\"It is true. Like, last year, in English class, we had to write a poem and read it aloud. I don't know if you know this about me, but I write poetry.\"\n\n\"How would I know that?\" I start to play the scale again but also sneak glances at him out of the corner of my eye. He has strangely long eyelashes, and, in the lamplight, they make a shadow on his face like a moth flying.\n\n\"I guess you wouldn't,\" he says. \"But I do. This should have been an easy assignment. But I knew that, if I read some teen angst poem, or even something about a tree, people would've busted a gut laughing. It would have challenged their image of me in a way they wouldn't want it challenged. So I wrote a limerick and got a C. But at least when they laughed, they weren't laughing at me.\"\n\n\"Are limericks that easy to write?\" I finish the scale. \"I thought they were hard.\"\n\n\"They're easy if you write about a girl from Nantucket,\" he says.\n\n\"I'm surprised you even got a C.\" I laugh, but then say, \"Okay, so here's what I think about that. First off, you're too hard on yourself. People love you. You're super-popular. You could be class president if you wanted.\"\n\n\"Yeah, right. If they love me, they love me because I'm funny.\"\n\nI shake my head and go on. \"They like you because you're _fun._ You're the most fun person I know _,_ which is different than just being funny. Secondly, you don't know if they'd have laughed because you didn't give them a chance. You were the one who didn't want to challenge their image of you. You chickened out.\"\n\n\"I know what would've happened.\"\n\n\"Maybe. Maybe not. But third, you're probably right. You probably do have to change who you are some to be popular. Everyone does. When I was in middle school, Whitney and the mean girls wanted me to be in their clan. But they only liked me because of my looks, and I didn't want to be a mean girl. So I decided just to stop trying, not have a group. That's why I'm not popular.\"\n\nHe thinks about it, then says, \"Okay. But I wasn't _normal_ not popular before I changed. You could be popular if you wanted. Your looks are enough. My looks are another story.\"\n\nI nod, acknowledging that's probably true. I know what it's like to look different, but I can't pretend that being beautiful is the same\u2014at least outside of Violet's house.\n\nI say, \"Tell me about it.\" Because I know he wants to, even if he won't admit it.\n\n\"I'd rather teach you the G major scale,\" he says.\n\n\"Let me guess. One sharp, so it goes like this.\" I play exactly the same fingering as C major, but starting on G with the F a half step higher.\n\n\"That was right,\" he says, looking sort of stunned.\n\n\"I'm gifted,\" I tease. Actually, the G major scale was in my piano book, right before Minuet in G, and I worked on it. \"Now, tell me your sad, sad story, and I'll tell you mine. Bet I win.\"\n\nHe laughs. \"Challenge accepted. Okay. So, when I was a kid, I had no friends. Zero. I was smaller than everyone, and no one wants to hang out with the weird-looking kid. Occasionally, people would be _nice_ to me, like take obvious pains to include me, to show they weren't assholes. Or because their parents told them to. But I was never the kid who just got invited to hang out after school.\"\n\n\"That sucks,\" I say.\n\n\"It did. And there's more. Once, when I was nine, this kid, Coleman, invited me to his birthday party. I was excited because, usually, I didn't get asked. People would hand out invitations at school, and every boy would get one except me and this kid, Ricky, who picked his nose until it bled. But Coleman invited me. So I showed Stacey the invitation, and the next day, I told him I could go.\"\n\n\"Okay.\" I finger the G major scale again, to distract him from the fact that he's actually expressing an emotion for once, instead of being a total _guy_ like he usually is. \"So then what?\"\n\n\"He said, 'Oh, I only invited you because my mom said I had to ask all the boys.' I felt like someone kicked me in the stomach.\"\n\n\"Ouch,\" I say, wanting to reach back through time and hold that little nine-year-old boy's hand. But I play the scale again. I hate Coleman. I don't even know him, but I hate him.\n\n\"Yeah. Anyway, I wasn't really invited, so I didn't want to go. Obviously.\"\n\n\"Obviously.\"\n\n\"But I'd already told my mom about it, and I didn't want to tell her why I'd been invited either. She was so happy. She bought this big gift, a Lego _Star Wars_ starfighter set. So that Saturday, I faked a stomachache to get out of it. Stacey was so upset. She kept asking me if I felt better, telling me it was okay if I went late. She was worried I wouldn't get invited again if I blew it off. She was even a little mad at me. Finally, I told her the truth. Man, did that reek.\"\n\n\"What happened?\" Though I can imagine.\n\n\"She cried. She flat out bawled. She kept saying how cruel kids were, threatening to call Coleman's mom and tell her. I begged her not to. It wouldn't make things better. Finally, I told her to please stop talking about it. So then, she gave me Coleman's gift. I spent that day making a starfighter with my mommy because I had no one else to make it with.\"\n\n\"Stacey's the best, though.\" His story is so sad I sort of want to cry myself, but I suppress it. \"Do you want to show me A major?\"\n\n\"Nah, that's enough new stuff for one day. Work on those. Try with both hands.\"\n\n\"Okay.\" I start playing C major, but softly.\n\n\"Anyway,\" he continues, \"one day right after that, I told a joke in class, just by accident, said something funny. I don't even remember what. I'd been funny to my family all along, but never in school. I was too shy. But everyone laughed. Actual positive attention from my peers. I liked it. So, I decided I was going to be funny all the time. That weekend, I went to the library and got all these joke books and books of insults, and when I got to school, I started telling them. The first kid who insulted me, I said, 'You're so dumb, you're flunking recess.'\"\n\nI laugh. \"That's funny.\"\n\nHe rolls his eyes. \"Yeah. If you're nine, it's hilarious. Everyone laughed and started making fun of him, instead of me. So the next time someone insulted me, I said, 'Is that your nose, or did the _Millennium Falcon_ park on your face?' I thought of that one myself. I was a huge _Star Wars_ freak, and this guy had an enormous nose, so it was perfect. It was easy to come up with jokes when I did it ahead of time, not on the spot. And people laughed at that too. The next week, someone invited me over to his house for real, not just to be nice. It was the same guy I'd told the nose joke about. Turned out he liked _Star Wars_ too. And he respected me now.\"\n\n\"Or feared you,\" I say.\n\n\"Nah, we were friends. We still are. It was Tristan Hernandez.\"\n\nI gape. That was the guy who played Bill Sikes. Tristan's huge (as is his nose), and they're best friends. \"You're kidding.\"\n\n\"Nope. So ever since then, I've spent every spare moment thinking of funny things to say, coming up with pranks and stuff. It's hard work. But no one makes fun of me anymore. They want to be my friend. I'm cool. But they don't know the real me.\"\n\n\"Do I know the real you?\" I hold my breath, awaiting his answer.\n\nHe shrugs. \"This is him. You like him?\"\n\n\"He's a good piano teacher,\" I say.\n\nHe smiles, showing a dimple on one cheek, but not the other. \"Thanks.\"\n\nSilence. I want to say something else, tell him he's _not_ that ugly kid he obviously thinks he is, that he's funny and charming, but he's also handsome, especially when he smiles. Sure, the first thing I noticed about him was his size, but his beautiful, brown eyes were a close second. But that would be awkward, so instead, I just sit there with a dumb grin on my face, playing F major.\n\nThen, we both speak at the same time.\n\n\"Is the real you _ever_ funny?\" I ask.\n\n\"It's not that I'm never funny,\" he says, then laughs when he realizes what I've said. \"Of course I'm funny. I'm hysterical, obviously. Just not all the time. I have deep thoughts too.\"\n\n\"Got it. Deep thoughts. I'd hate to think it was all a lie. Can I read your poetry sometime?\"\n\nHe looks away. \"I shouldn't have told you about it.\"\n\n\"Why can't I?\" I stop playing and make my lips an exaggerated pout.\n\n\"'Cause it's embarrassing. What if you hate it? What if you think it's stupid?\"\n\n\"I wouldn't. I'm your friend.\"\n\n\"Right. Friends.\" He nudges me over and starts playing \"Clair de Lune,\" a piece he says he plays to relax _._ He's still not making eye contact. The music is soft and gentle, moonlight over a river. \"Okay, how about this? Someday, I might leave a poem lying somewhere, where you can find it. Just don't ever tell me you read it, okay?\"\n\nI roll my eyes. \"Guys are such idiots. Don't want anyone to know they have souls.\"\n\n\"Soul? What's that? I sold mine to the devil in exchange for piano-playing ability.\" He keeps playing, showing off, not looking at me. Now, the music sounds like falling water. \"Hey, this is huge for me. You're the only one I've ever said this to. You're the only one who knows the real me, the me that gets pissed off at the world sometimes, the me that thinks it's not fair.\"\n\n\"Life's not fair,\" I say. \"I know a lot about getting pissed off at the world.\"\n\nHe nods. \"I bet you do.\" He keeps playing, his arm brushing mine with each arpeggio, stronger, then softer, like a river, flowing toward the ocean, then crashing into rocks. For a minute, it's only the sound of the piano. The piano and our breathing.\n\nThen, at the highest point, he says, \"Okay, your turn. How old were you when your mom died?\" and I wonder if he's timed his question, as I timed mine, so I'll think he's not listening, concentrating on playing.\n\n\"Eight,\" I answer. \"So you were not getting invited to Coleman's party, and my mom was planning the Zoo Sleepover of Death. They said it was a freak accident\u2014I mean, she got attacked by a freaking monkey. One day, she was there. Then, she's gone, just like my dad.\" I suck in a shaky breath, closing my eyes and letting the music flow over me. It's so beautiful and a little sad. I still can't believe my dad is gone forever. I barely had time to process it before I had to deal with this, with Violet. \"It feels like I'm sleeping over a friend's house, and when I get home, he'll be there. I regret every time I stayed at Laurel's and wasn't with him. Who knew it was my last chance? Violet took it from me.\"\n\n\"You've been through a lot.\" Goose stops playing, giving me his full attention now that it's not about him. The silence seems louder where the music used to be.\n\nI say, \"When my mom died, people were really nice, at first. They brought so much lasagna our freezer was full for months. They offered to watch me after school. I had tons of invites. But then, it sort of stopped. It was like everyone forgot us. They'd do things with their families, and wouldn't ask us, even if they used to when my mom was there. The only one who was still friends with me was Laurel.\"\n\n_Laurel._ I swallow hard. Yesterday, Goose left his laptop on the kitchen table when he went to school, and I saw Instagram photos of Laurel and this girl, Britney, wearing matching pink shirts that said _Waiting for My Handsome Prince._ I bet they're going to the concert together too. Britney's willing to wear the lame-looking T-shirts, and she'll be sitting in my seat. _My_ floor seat with _my_ best friend seeing _my_ Jonah Prince. The concert's in two weeks.\n\nStill, I say, \"Laurel was my best friend, and her mom was my mother's best friend since they were kids. One day, I was complaining about how no one ever wanted to hang with me anymore. She was playing at our friend Cassie's house, but Cassie hadn't asked me.\" I start playing C major again, just with my right hand.\n\nI say, \"Laurel, she got this sort of weird look on her face. She said that Cassie had told her that her mom didn't want her to play with me anymore. Her mom said that when girls didn't have mothers, they went wild. She thought I'd be on drugs or something. I was eight, and her mom already had me pegged as a future crack addict. Laurel said she wasn't going to be friends with Cassie anymore either.\"\n\n\"So your mother was dead, and your friend ditched you too?\" Goose said. \"That's harsh.\"\n\n\"Friends, plural. All __ my friends ditched me. Maybe they weren't all as mean about it as Cassie, but they totally forgot about me. The only one who stayed with me was Laurel. I think maybe the others never really liked me in the first place. Their moms were friends with my mom. She was the leader of the Girl Scout troop. Once she was gone, they didn't care about me, and neither did their kids.\"\n\n\"How could they not like you?\" he says. \"You're so . . .\" He stops.\n\n\"Beautiful?\" I roll my eyes.\n\nHe shakes his head. \"That wasn't what I was going to say. I was looking for the right word.\"\n\n\"What word is that?\"\n\nHe thinks about it. \"Fierce. That's a word. You're fierce. People think you're like this fragile flower because you're so little and pretty. But you're really strong. You've been through all this, and it hasn't broken you. And it hasn't made you less sweet.\"\n\nWow. That's maybe the greatest thing anyone's ever said to me.\n\nI say, \"I guess they don't see that about me. And I don't even know if it's true. I feel so alone. My parents . . . my dad. I have nobody. Nobody.\" My eyes fill up with tears. They just do that randomly now, but usually not in front of anyone. Just out of the blue, I'll be wallowing in self-pity. I hate it, but sometimes I can't help it, with all that's happened. The piano keys swim before me so I can't see which is which.\n\nGoose sees. He puts his arms around me and pulls me toward him. \"You have me . . . us. I know it's not the same.\"\n\nI sniffle, then sob. His hold on me tightens. His arms feel so warm, and he smells of lemony cologne.\n\n\"I'm an orphan,\" I say. \"I wish I could just take care of myself.\"\n\n\"I'll take care of you. I'll get my parents to. They took in Tyler and Jeron. They can take you in too. It doesn't have to be temporary. You can stay here forever.\"\n\n\"That's crazy.\" But I remember what his dad said, about not treating me like a cat. Maybe it isn't crazy.\n\n\"It's not crazy,\" he says. \"It's okay. You'll never have to go back to her or anywhere you don't want. We'll take care of you, Celine. I know it's not the same as having your parents, but I . . . we'll love you, my family. You'll never be alone.\"\n\n\"But . . .\"\n\n\"Shh. Just stop . . . stop thinking I'm wrong, okay? I'm telling you the way it is. While I am around, you will never be alone. I know I'm not a big guy, but I'm big enough. We all are.\"\n\nHe takes me by the shoulders and backs off, making me look at him. His brown eyes hold conviction. He believes what he's saying. It's incredible. I've only known him a few months. Who knew there were people out there who were so kind? I'm not used to it.\n\n\"You're so . . . why are you so sweet to me?\"\n\nHe laughs. \"That's me, Mr. Sweet.\" He lifts a hair out of my eyes, wiping a tear. \"You win.\"\n\nI say, \"Win what?\"\n\n\"The pity prize. You said you were going to win, and you did.\"\n\n\"I don't want to win. I don't want pity.\"\n\n\"I know.\"\n\nI bury my face in his shoulder and sob with abandon.\n\nHe holds me tight, and he lets me.\n\n#\n\n#\n\nThe days go by. Slowly. Stacey and Jorge attend spring concerts, and Isabella comes home with a dance recital costume that makes her look like a duckling. So cute! I help her sew the straps to be the right size. Tony's piano recital has happened. The school year is almost over. When Kendra had arranged for me to live with the Guzmans, we had agreed it would be temporary, a few months, a year at most. Then, she said, Violet would get past it, move on. Once I was off Violet's radar, Kendra would help me create a new identity, move me to a new town, get me a new family, sort of a Witness Protection Program with witchcraft.\n\nThe part I had pushed back in my mind, the part I'd avoided acknowledging, the part I'd definitely never told Goose's parents was this: Violet never \"got over\" anything. She hadn't gotten over her love for my father, whom she'd met at age ten. And she'd killed my mother years after she'd last seen her in high school. She'd stayed, like an alligator waiting, mouth open, for its unsuspecting prey, until my mother had felt safe. Only then did she strike.\n\nOf course, she thinks I'm dead. Maybe that will help.\n\nStill, it won't be as easy as Kendra thinks. I had secretly hoped to be able to go to the Jonah Prince concert with Laurel in June. But it's late May, and I haven't seen Kendra since the day I moved here. Will I never go to school again? Never have friends, other than Goose? Never have a boyfriend, marry, or have children, just stay here like one of those kidnapped girls who lives her whole life in someone's walls\u2014albeit with really nice people? Would that be enough for me? And what about the Guzmans? Always sheltering me, swearing the kids to secrecy about me, that had to be hard on them.\n\nOne day, after practicing _F\u00fcr Elise_ for two hours, I retreat to my\u2014our\u2014bedroom to listen to my Jonah CD. Goose is doing math homework, so I don't want to disturb him. I bring the book Goose got me, to look at the pictures. I wish I could get a copy of _J-14_ or _Tiger Beat_ that would be sure to have pinups of Jonah. But it's not like I can ask someone to buy me something so silly\u2014even with my own money. So all I have is the songbook. That, and my dreams.\n\nAt night, I listen to Jonah on my headphones, and I try to conjure up the dream I had about him again. I know it's just a fantasy, but my reality is so crazy-awful. But instead, I dream of apples, rotten, exploding apples. Lately, I've dreamed of being rescued, but the guy who saved me wasn't Jonah. At least, I don't think he was. I couldn't see his face. Still, I can feel my rescuer's arms around me, his lips on mine. I seek out his eyes and know that I am safe!\n\nOnly to be awakened by Isabella's snoring. They should take her to be allergy tested! Actually, something in the air is giving me allergies too. It hangs heavy, filling my lungs, making me feel like I'm half asleep all the time.\n\nMy fantasies are taking over my life, and I will go insane.\n\nCould work. If I'm locked in a mental institution, Violet won't find me. I could make friends, insane friends. I can grow old and fat, pretending I'm in Jonah's arms.\n\n_Le sigh._\n\nToday, I'm listening to \"When I See Your Face\" when Isabella comes in.\n\nIsabella is the one person who keeps me firmly grounded in reality. It's pretty hard to live in a fantasy world when a five-year-old keeps begging you to play with My Little Ponies. Fortunately, I really _love_ playing with My Little Ponies. Isabella insists on being Pinkie Pie, but she lets me be Rainbow Dash sometimes. I think it's important for her to learn to share. I gave Goose money to buy her Pinkie Pie's helicopter, from me, for her birthday.\n\n\"I like this song.\" She does a little dance around the room.\n\n\"Do kids your age like it?\" I ask, hoping they don't. If kindergarteners like the same music you do, that's bad.\n\n\"I don't think they've heard it.\" She points to the songbook in my lap. \"Who's that?\"\n\n\"That's Jonah Prince, the guy who sings it.\" I turn over the book so she can see the color photo on the cover.\n\n\"You have his picture up by your bed too.\"\n\n\"How do you know that?\" I ask, wondering if she had climbed up onto my bed. I knew she could. Last week, when she couldn't find her tap shoe, it had turned up under my pillow.\n\n\"I can see it.\" She points to the picture and, since I'm sitting on the floor, I can see it's clearly visible from her perspective. Which means Goose sees it too. At worst, he knows I ripped out a page of his gift. At best, he thinks I'm a lame fangirl who keeps a rock star's picture over her bed. Of course, he already knows I wrote Jonah's name on my notebook.\n\nOkay, so I _am_ a lame fangirl.\n\nThey said on _Entertainment Tonight_ (which I watch because I am now the only teen on the planet not on social media) that Jonah's dating this Teenz Channel star, Allegra Kendall. They also say he drinks and parties a lot. Not exactly my type, but it's his music I love. Besides, I know I'm not really going to marry him. I just want to go to his concert like a normal girl, and maybe fall in love with a normal boy someday. I'm not actually insane . . . yet.\n\nNow, Goose comes in, so he must finally be finished with his pre-calculus. \"Yay, you're done!\" I stand and try to block his view of the photo.\n\nHe looks at Isabella, who's still dancing. \"Did you ask her about the mangoes?\"\n\n\"What about the mangoes?\" I love the smoothies we make, but I thought we were out of frozen mangoes.\n\n\"Good news\u2014Mrs. Ozanich brought us new mangoes. They're ripe again. My mom wanted us to cut them up and peel them, to put in the freezer.\"\n\n\"Sounds like fun.\" If Goose said this, it would be sarcasm, but I actually like helping. Feels like earning my keep.\n\n\"Yeah, fun.\" Still, we head to the kitchen where the counter is stacked with a dozen mangoes, red, orange, and gold, and oozing sap. I think of my dreams, of the apples. Could these too all turn to muck and maggots? But I haven't had the dream in a few days. I feel safe. The mangoes are beautiful, the colors of sunshine.\n\nI pick one up. There are two peelers, and I hand one to Isabella. \"Do it over the sink,\" I tell her. Since the kitchen is fitted for smaller people, she can do this easily. It's harder for me. When I first moved here, I kept leaning on counters that weren't quite where I expected, but now I'm used to it. Still, I'll let Goose cut them up.\n\n\" _Ferris Bueller_ _'s_ on tonight,\" Goose says.\n\n\"What's a Ferris Bueller?\" I ask.\n\n\"Celine, didn't you ever watch TV?\"\n\n\"Maybe not as much as you did.\" Laurel and I watched a ton, but it was all stuff like _Bridezillas_ and _Say Yes to the Dress_ , __ stuff you wouldn't share with a guy friend.\n\n\"Cultural illiterate. Well, it's a John Hughes movie, the same guy who wrote _Pretty in Pink. Ferris Bueller's Day Off_ is my favorite. You have to watch it.\"\n\n\"Okay.\" I try to slide the peeler under the mango skin. It's not as easy to peel as an apple, but at least it stays whole in my hand. Suddenly there's a knock on the window.\n\nI look out. It's someone I don't recognize, an older woman, dressed up with a hat, like a Jehovah's Witness. But they usually come to the front door.\n\nShe mouths _Kendra_ so I understand.\n\nGoose sees too and starts toward the door. I glance at Isabella. Goose says, \"Hey, isn't your dance show on, Isabella?\"\n\n\"Mom says I have to do this, right?\"\n\n\"Nah, we got it,\" he says. \"Why don't you go watch?\"\n\nIsabella goes to the family room and, again, Goose starts for the door. But, then, Kendra is in the room.\n\n\"Oh!\" She puts her hand down onto a pile of mango peels. \"What a mess.\" In an instant, the mangoes are peeled and chopped, flying into the Ziploc bags we had for them, and the kitchen is cleaner than before.\n\n\"That's better.\" She turns to face us. \"I have good news. I think Violet has decided to move on.\"\n\n\"Really?\" That so does not sound like her.\n\nKendra nods. \"She was talking about taking a job transfer to another office, another city. She says no one in this town has ever liked her, so she should go someplace new. I encouraged her. I told her she could travel, as I had. We could even travel together. She agreed that she could forget Greg better if she didn't live in his house. Wasn't that emotionally healthy of her?\"\n\n\"Very.\" _It sounds fictional._ \"Other than the fact that she only got there by thinking I was dead.\"\n\n\"We take what we can get,\" Kendra says. \"She's applied for a job transfer. Once she leaves, I really think you can move on with your life.\"\n\n\"Move on?\" Goose asks. \"Like she can go back to school? And get a foster family?\"\n\n\"Well, she has to move, of course. Violet thinks she's dead. She can't just show back up at school. Violet would still be her guardian. She'd find out. I'll take Celine someplace far away and exotic, like France or Italy, or maybe Ohio.\" She touches my cheek. \"I've decided it would be safer if you live with me, like my daughter.\"\n\n\"Wow. France?\" Goose says.\n\nI look at Goose. I know he's thinking the same thing I am: We'll never see each another again. I'll miss him.\n\n\"It will probably be several weeks still,\" Kendra says. \"Don't get too excited yet.\"\n\nGoose looks down. \"No, not too excited.\" He turns to Kendra. \"Hey, you want a smoothie? We have all these mangoes.\"\n\nShe shakes her head. \"Can't. Allergies. But thanks.\"\n\nAnd, before I can say _Witches have allergies?_ she's gone.\n\nGoose looks up at me. \"I guess you can go to summer school now, to catch up.\"\n\nI nod and try to smile. \"Yup. In France.\"\n\n\"Maybe you could talk her into Ohio. At least they speak English.\"\n\n\"It's still really far from Florida.\"\n\nHe turns away and starts feeding mangoes into the blender. \"Yeah. Really far.\"\n\nWe make smoothies. They don't taste the same, though. Mangoes are usually sweet. These were so pretty, but they taste like turpentine. Then, we try to go into the family room like nothing is wrong. After all, nothing is. This is what I wanted, to be away from Violet forever, to be free. It's not like Stacey and Jorge were planning on keeping me forever. It's not like they need another kid to add to the five they already have. It's not like they'll miss me the same way I'll miss them. They have each other. It's not like\u2014\n\n\"Sit by me!\" Isabella shoves her mother over to make room.\n\nStacey and Isabella are watching a reality show _._ Jorge is on his laptop, and the boys have disappeared into their rooms. Since Isabella is taking ballet-tap, we've all gotten completely addicted to this show about dancers and their crazy stage moms. Now, one of the moms is threatening to quit because she doesn't like her daughter's costume.\n\n\"She quits every week,\" Stacey says.\n\nTwo grown women are screaming and stomping their feet. It's like a traffic accident. You can't stop watching. But I say, \"This is so nice.\"\n\n\"What is?\" Goose says. \" _Dance Moms_?\" The dance teacher is screaming at all of them.\n\n\"Just . . . this. Watching TV together. My family never did that. You're so lucky.\" Where will I be a year from now? With Kendra in Ohio? Or France? But I don't want to watch _Dance Moms_ with Kendra. I want to stay here and help Isabella do her hair every morning, help Goose with chemistry, learn to play something a little harder on the piano.\n\nStacey laughs. \"Well, you've come to the right place. That's all we do together around here, watch TV. Can't get anyone out for some exercise.\"\n\n\"You know, they've got power yoga on every day at six. I was thinking we could do it together.\" Maybe if I pretend I'm not leaving, I won't have to. Yeah, that'll work.\n\n\"That'd be cool,\" Stacey says. \"Maybe DVR it for Jeron's naptime.\"\n\nAfter _Dance Moms,_ the rest of the family goes to bed. Goose says, \" _Bueller_ time.\"\n\n\"You should go to bed too, Goose,\" Stacey says. \"Don't you have a test tomorrow in chem?\"\n\n\"Day after tomorrow,\" he says.\n\n\"Still, you need to sleep. Your grades are important to you. You want to get into a good college.\"\n\n\"We'll just watch half,\" he says. \"Please. Celine's alone here all day.\"\n\n\"Alone?\" Stacey laughs.\n\n\"You know what I mean. I just want to spend some time with my friend.\"\n\nStacey frowns. \"Okay. But just half the movie.\"\n\n\"Absolutely.\" Goose waits until she turns the corner into the hallway before he whispers, \"Nah, we're watching the whole thing.\"\n\n\"She's right, you know. You shouldn't ruin your GPA. Junior year is the most important for college.\"\n\nHe rolls his eyes. \"What you're not understanding is that I'm a genius. I get As without studying.\"\n\nI remember what Willow said about him not handing stuff in. \"Can you get them without sleep?\"\n\n\"Sleep's for babies. Sit down, and stop acting like my mother.\" He sits and pats the seat beside him. \"You're going to love this.\"\n\nI do. It's a funny movie about this cool guy, Ferris, who skips school with his girlfriend and his very reluctant best friend, Cameron. They outsmart his parents and the principal and go to a fancy restaurant, a baseball game, and an art museum.\n\nGoose says, \"It's true what Ferris said in the movie. Life moves pretty fast. If you don't stop and look around once in a while, you could miss it.\"\n\nI notice he's wearing that cologne again, the lemony one. Probably, whenever I smell lemons for the rest of my life, I'll think of him.\n\nI want to stay with him.\n\nI say, \"That is true. I probably haven't looked around much lately. Maybe once I stop hiding out in your house I could do that. Except I'll miss you.\"\n\nHe looks sad, then suddenly smiles like he's trying hard to. \"We should run away for the summer, go backpacking or something.\"\n\n\"My stepmother . . .\"\n\n\"We'll go someplace she'd never look, like the Grand Canyon.\"\n\n\"I'd like that. It's supposed to be beautiful.\"\n\nGoose glances at the TV. \"Oh, wait\u2014this is my favorite part. Ferris is in a parade.\"\n\nAnd then, in a second, he jumps up onto the coffee table, imitating Ferris. \"'And I'd like to dedicate it to a young man who doesn't think he's seen anything good today,'\" he lip-synchs with Ferris. \"Give me Izzy's hairbrush.\"\n\nI give him a look like WTF, but hand him the hairbrush. So when Ferris starts singing \"Danke Schoen,\" __ Goose does too. Danke schoen, darling, danke schoen. Thank you for all the joy and pain.\n\nI can see the actor, Matthew Broderick, on the television, and Goose doing a perfect imitation of his every expression, every gesture.\n\n\"You've obviously seen this movie a lot,\" I say, laughing.\n\nHe ignores me, still singing, \" _Danke schoen_ . . .\" and dancing. Then, he holds out his hand. I take it, and I dance with him. It's fun, and for a while, I forget that my life is a mess, that my family is gone, and I'm never going to go to school again. Goose is good at making me forget the bad stuff.\n\nHow will I live without him?\n\nBut that night, when I go to bed, I have another nightmare. Again, I'm picking apples with my parents. This time, when I pick the ripe, green apple, it turns black, then melts to molasses. The molasses spreads up my hand and turns my arm black. Then, my whole body. It consumes me until I am gone.\n\nI wake, remembering something: Violet's maiden name. Appel.\n\nViolet is the apples. They destroyed my family, and they will destroy me.\n\nThe dreams mean I will never be safe. She'll find me sooner or later.\n\n#\n\n#\n\nEven though I can sleep all day, I never get enough rest because my dreams keep me up all night, worrying, not just worrying about Violet and Kendra but worrying about the Guzmans, what they're risking, keeping me. Despite Jorge's assurances that they want me, I can't help but think they'd be better off if I just disappeared.\n\nOf course, if what Kendra says is true, I'll be disappearing soon anyway, and that worries me even more. Where will I go? Who will I live with? Will I ever have any friends? And what if Violet finds me anyway? I can put these questions out of my head in the day, but at night, they dance in my head like sugarplums on acid, keeping me awake for hours.\n\nBut, the night after _Ferris Bueller_ , __ I decide to turn in early, to read until my eyes shut without my help. I know Goose has that chem test the next day, and I want him to study. But when I get to our room, Isabella is still coloring.\n\n\"You should put that away,\" I say. \"Your mom's going to tell you to soon.\"\n\nIsabella has been very cutely obedient. Doing everything I say. Goose says, jokingly, it's because she looks up to me. I think it's just because I'm new.\n\nBut, apparently, the novelty has worn off because she says, \"I'll wait until Mommy tells me.\"\n\n\"And then, about ten minutes more, I bet. And then, she'll be mad and take away TV tomorrow. Don't you want to watch _Liv and Maddie_ together? It's our favorite show. Come on, I'll help you clean up.\" I start gathering her crayons. It's a big box with a hundred twenty, and I know she likes when I put them in rainbow order, so I start with the reds. \"You get the oranges.\"\n\nShe does. It takes forever because she's just a little kid and some crayons are in the middle between red and orange, like mango tango. Finally, she hands me the ten oranges, and I fit them next to the reds. \"Now, start on yellows, and I'll do greens.\"\n\nShe picks up a lemon-yellow crayon and holds it aloft.\n\n\"Do you love my brother?\" she asks.\n\nThe question startles me for a second, and the magic mint crayon slips from my fingers. Once I catch it, I consider her question. I know any answer I give will be trumpeted not only to her brothers but to the neighbors and everyone at her school. I also know that five-year-olds only define love one way: a boy and a girl K-I-S-S-I-N-G. Neither yes nor no will yield a good result.\n\nI think of Goose at the piano, playing music like falling water.\n\nFinally, I say, \"Of course I love all your brothers. And your mommy and daddy and you too.\" I pick up the screamin' green.\n\n\"I meant my brother Goose. Are you his girlfriend?\" she says.\n\n\"Silly!\" I laugh. \"We've never even been on a date. We're friends.\"\n\n_Blue violet, red violet, violet blue . . ._\n\nIsabella rolls her eyes. \"I mean\u2014\"\n\n\"Izzy! Time for bed!\" Stacey's voice mercifully interrupts us.\n\n\"In a minute!\" Isabella says.\n\n\"Now. Brush your teeth. I think I saw something crawling around between them earlier. I'm checking your toothbrush.\"\n\nIsabella thrusts the various yellows into my hand and walks out, huffing.\n\nAfter she leaves, I quickly gather the blues and violets and purples, then the neutrals. I'm in bed with my eyes closed and the pillow over my face by the time she comes back, safe from questions I can't safely answer.\n\nFortunately, she doesn't revisit the subject the next day. I feign sleep in the morning until she's gone. In the afternoon, I'm listening to Jonah when she gets home. She shares my Jonah obsession to a major degree now.\n\n\"He's soooo cute!\" she says.\n\nI laugh. \"You're, like, five years old.\"\n\n\"Six. You know I had my birthday last week.\"\n\n\"Okay, six. Sorry. You're not supposed to be obsessing over rock stars.\"\n\n\"Why not? You do.\" She turns away, singing \"Yes, Baby, Yes,\" and shakes her hair to indicate she wants me to braid it again. I oblige. I saw a style with a braid across the forehead, and I've been wanting to try it. I get her hairbrush and start brushing out her golden waves.\n\n\"Well, I'm older than you,\" I say. \"I've been obsessed with Jonah for, like, a year, and I'm getting less obsessed.\" It's true. Without Laurel's influence, it's less fun. \"Do you know I was supposed to go see him in concert?\"\n\n\"You were? Like, see him in person? Wow.\"\n\n\"I know. My friend Laurel and I got tickets the first day they went on sale. We had floor seats right near the stage where he could see us if he looked down, and we were going to make posters so he'd notice us.\"\n\n\"What were the posters going to say?\"\n\nI can't tell her about _Dare to eat a peach._ Not only will she not get it, but it also sounds completely stupid when I say it out loud. So I say, \"But now, Laurel's going with this other girl, Britney. It makes me so mad. I really wanted to go.\" I want to cry, not about Jonah. That was a complete fantasy. But about Laurel, being trapped inside, basically losing my life.\n\n\"That's what you're upset about? A stupid concert?\"\n\nIt's Goose. He was late coming home from school today, and I've been waiting for him. I finally mastered _F\u00fcr Elise,_ and I've waited to play it for him, my only audience. Now, he's standing in the doorway, staring at me with something like disdain.\n\n\"Jonah Prince, really?\" he says. \"You're stuck here all the time. You've had to quit school. My parents are taking risks having you here. I never go anywhere anymore, just so I can entertain you, and you're upset because you can't go see Jonah Stinking Prince and his diaper pants?\"\n\nHis words are like a bee sting, or a hundred. I turn on him.\n\n\"It's not the only thing I'm upset about, and you know it.\" Even more, I want to cry. Why is he being such a douche? Can't he see that Jonah is a _symbol_ of all those other things? Like having to go to Ohio or _France_ with Kendra. Like maybe never seeing him again? \"You don't have to stay home for me. No one told you to. Go out with your other friends if they're so great.\"\n\nHe looks at me, sucking in a breath. \"Maybe I will. And maybe if you'd stop playing his insipid music night and day, I could actually think straight and study.\"\n\nThe chorus of \"Yes, Baby, Yes\" is playing. Those are pretty much the only lyrics. It is insipid. Still, I say, \"You said you liked it. Were you just playing me?\"\n\n\"Yes. I mean, no.\" He rolls his eyes. \"I was being nice.\"\n\n\"Yeah, well, maybe you should keep _trying_ to be nice.\"\n\nHe stalks over and pulls the plug on the speaker. Isabella starts screaming that he's in her room. Finally, he leaves. I scramble up the ladder to my bed to cry, but not before I rip down the photo. I don't want to throw it out, though. It's from the book Goose got me. Even though Goose hates me, it has meaning. Instead, I hide it under my pillow. Isabella turns the sound back up as soon as Goose leaves, louder than before, so Jonah is screaming, \"Yes, baby, yes!\" but when I close my eyes, I can't picture Jonah's face, only Goose's face, disappointed and angry at me.\n\nStacey calls us for dinner, but I say I feel sick. I'm mostly embarrassed. When Isabella comes back to the room, I pretend I've gone to bed early, even though it's only seven and still light out. Eventually, she leaves.\n\nA little later, there's a knock on the door. I ignore it, even when I hear Goose's voice, saying, \"Celine? Come on, Celine. I didn't mean it. Come play the piano with me. Or just talk to me. Anything.\"\n\nI bury my head deeper under the pillow, ignoring him, even though I know I'm being a brat.\n\n\"Celine?\"\n\nI don't answer, and finally, he goes away.\n\nBut I don't sleep. I can't. I stay awake, listening to the muffled noise of the television, the whirr of the blender making more smoothies, the boys fighting and flushing toilets. Goose knocks two more times, and I want to talk to him, but now, I've pretended so long that I can't. Of course he was totally right. It's dumb to fantasize about a rock star I'll never meet. I know that. Eventually, Isabella goes to bed and all the noise gives way to the wind chimes on the patio and the stop-start of the air conditioner. I'm lying there, wide-awake.\n\nHours later, I hear a knock on the window.\n\n\"Celine?\" The voice seems to be coming from inside my head. Kendra.\n\nI slide down to the floor and pad toward the window in my bare feet. When I get there, I have to look down. She is disguised as Stacey. But her voice is still Kendra's.\n\n\"Celine? May I come in?\"\n\n\"Come in, then,\" I whisper. And immediately, she is there beside me.\n\n\"It's, like, midnight. What is it?\"\n\nShe melts into her own face and grows about a foot. \"I'm sorry, Celine. I think I was wrong last week. About Violet. She's not leaving.\"\n\n_What else is new?_\n\n\"And I think she might be onto me. That's why she lied. She may know I helped you.\"\n\nThis is worse news. \"So she knows I'm alive?\"\n\n\"I'm not sure. But you should definitely lay low.\"\n\n\"How much lower can I lay? I'm in total hiding.\" I have another horrible thought. \"Does she know I'm here? Are the Guzmans in danger?\"\n\nShe shakes her head. \"I don't think so.\"\n\n\"You don't _think so_?\" I glance at Isabella, sweet Isabella asleep in her bed, her French braids still intact. \"How about you _know_ instead of thinking? I can't put them at risk.\"\n\n\"I'll find somewhere else for you to stay, another place, another town. Just don't talk to anyone right now. Don't even trust me if I come to your window.\"\n\nI'm thinking maybe I shouldn't have trusted her in the first place, but I don't say it. She's trying to help. If it weren't for her, I wouldn't even know about Violet's intentions. I'd probably be dead. My head is awhirl with jumbled feelings, but mostly regret, regret that I have to leave here, leave the people I love, people who have protected me. Goose, whom I adore despite his recent meanness. Yet I don't see any other way.\n\n\"How will I hear from you?\" I ask.\n\n\"Through the mirror, only the mirror,\" Kendra says.\n\nAnd then, she's gone.\n\n#\n\n#\n\nI stumble back to bed and find the mirror under my pillow, as I knew it would be. But now I can't sleep at all. I have to go away. I have to leave to protect the Guzmans. It's not fair to them. I should rest up, then sneak away in the morning. It's hours before I sleep again, and then, I am awakened too soon, by a hand in the darkness.\n\n\"Celine?\"\n\nGoose. He's climbed up the ladder even though he's afraid of heights. He nudges my shoulder. I smell that citrus cologne he's been wearing.\n\nI have never been good at giving people the silent treatment. When Laurel and I would fight, we'd say we were never speaking again. That would last an hour. I can never handle someone I love being mad at me. One of the most heartbreaking things about my father's death is that I was so mean to him. I would have broken\u2014if he hadn't died. I don't want to leave on bad terms with Goose. I may never see him again.\n\nSo I say, \"What, Goose?\" not even sounding mad.\n\n\"I'm so sorry. I . . . it wasn't you. I had a crap day. I got a D on a pre-calc quiz.\"\n\n\"A D?\" He's good at math, usually. At least, he understands the chem math a lot better than I do. \"Ask if you can do extra credit, maybe.\"\n\n\"Yeah, maybe. That's not the point. The point is, I was a douche, and I'm sorry. I'm upset that you're leaving, mainly.\" He doesn't know how soon I'm leaving, and I decide not to tell him. He'll just try to get me to stay.\n\n\"Okay,\" I say. \"But listen. You can do stuff with your other friends, you know. Just leave me home playing ponies with Isabella. I like it, and I'm not going anywhere.\" A total lie.\n\nHe tries to brace himself on the rail of the bed. \"I know. I want to hang with you. Can't you tell that I . . . ?\" He stops, then drops down to the floor.\n\n\"Tell what?\"\n\n\"Nothing. I don't remember what I was going to say. But we have fun together, don't we?\"\n\n\"So much fun I ruined your grades?\" I know the D is because he was hanging out with me too much.\n\n\"That's my fault, not yours,\" he says. \"I know you're lonely, no matter what you say. I'm sorry you can't go to the concert. I know it was important to you.\"\n\n\"It's not that big a deal. I know the Jonah thing is stupid.\" He must think I'm such an idiot. \"I was more upset about Laurel. Laurel's going with someone else. It feels like she forgot all about me.\"\n\n\"No, she misses you.\" He kicks the floor. \"Shit. I forgot to tell you. I saw her the other day, and she said she misses you so much. She doesn't understand why she hasn't heard from you.\"\n\n\"Oh, wow.\" Now, I feel worse because she probably thinks I forgot all about her, and I can never tell her otherwise, tell her I miss her.\n\nFrom below me, Isabella yells, \"Would you guys be quiet! I'm trying to sleep. I'm going to tell Mom you were in my room.\"\n\n\"Okay, I'm sorry.\" Big talk from a kid whose hair I do three times a day. To Goose, I say, \"I'll come down.\"\n\nWe go to the kitchen, and over Cheerios, Goose says, \"Maybe they'll do one of those 3D concert movies next year, and I can take you to it.\"\n\nI smile. \"You're so sweet.\" Even though a 3D movie isn't the same as actually being in the room with Jonah. But I think of what Kendra said. By this time next year, I will be in some other place, far away from everyone I know.\n\nOr I could be dead.\n\n\"Listen, you'll get out. You'll come back here, or I'll go to Ohio. Or even France. We'll do stuff together. You heard what Kendra said. Violet will move on. No one's that crazy.\"\n\n_You don't know Violet._ \"You're probably right.\" I don't want him to worry any more than he already has.\n\n\"Of course I am. What are you doing today?\"\n\n_Leaving._ Suddenly, I just know it. I have to leave, have to protect them from Violet. That's more important than anything else, even than seeing him again. \"I don't know. I guess I'll practice the piano a lot. Your mom has a PTA meeting, and then, she's taking Jeron for a checkup, so I'll be all alone.\"\n\n\"I wish I could stay home with you, so you won't be lonely.\"\n\n\"I'm fine. Really. You should go suck up to your math teacher.\" _Forget me._\n\n\"I will.\"\n\nI watch him trying to catch a stray Cheerio that's floating away like a little life ring. This is probably the last time I'll ever see him. I stare at him, memorizing his eyes, his dimple, everything about him. I don't want to go. Yet, what choice do I have? A lock of hair falls into his face, and I reach to move it. He looks up at me, raising an eyebrow.\n\n\"Violet always told me people treat you better if your hair is neat,\" I say. \"Besides, it covers your beautiful eyes.\"\n\nI remember what Dorothy said to the Scarecrow: _I think I'll miss you most of all._\n\nHe smiles halfway. Our eyes meet, and for a second, we just stare at each other. He smiles big. \"Yeah. My hair should be more of a priority, I guess. I should get going.\"\n\nBut he stands there, still staring. Suddenly, I want to tell him lots of things, that I don't really care about Jonah, for one. That no one in my entire life has ever been as nice as he is, for another. It's weird that you can just meet someone, and right away, they mean so much to you. But if I say that to Goose, he'll know something's up, that I'm running away, and I don't want him to know. I don't want him to stop me. I don't want to leave, but I have no choice.\n\nI wish I could stay here. With him. Forever.\n\nSo I wait until he's picked up the cereal bowls and is on his way out before I say, \"I really appreciate everything you and your family have done.\"\n\nHe shoulders his backpack. \"Please stop thanking me. Anyone would do this, anyone decent.\"\n\n\"Guess I don't know many decent people.\"\n\n\"Why don't you get some more sleep?\" he says. \"I'll see you later.\"\n\nIt's just barely light out. I say, \"Okay.\" I want to hug him, feel his arms around me one last time. But, instead, I wait until he closes the door, then go back to my room.\n\nI watch him out the window, feeling like my bones may crumble, as he runs to his car in the morning rain. Then, I watch the car's taillights get smaller and smaller until they disappear entirely.\n\nI lie in bed, tears running down both cheeks, my fist in my mouth so as not to wake Isabella with my sobs. I will never see him again.\n\nAnd he is the only one who ever rescued me, truly rescued me when I really needed it.\n\nI sleep, fitfully, dreaming of apples, exploding like the bomb at Hiroshima, taking me up in a mushroom cloud to the top of the world. The dreams make me sure that leaving is the only way. I'm not safe. Nothing will make me safe. At least, if I leave, I won't endanger others. I can't put the family I've come to love in danger.\n\nWhen I wake, Isabella is gone, Stacey and Jeron and the boys too. I'm sorry I can't say good-bye to them, but I can't. I stuff my few possessions back into my backpack. I take the songbook Goose bought me, not as a souvenir of Jonah, but as a souvenir of Goose, of our time here.\n\nI go to the bathroom to brush my teeth.\n\nOn the counter, I find a sheet of paper.\n\nIt's a poem, the poem Goose said he'd leave for me someday. I see the title, \"Going to Target With Her.\" I smile and feel like I'm about to cry at the same time. It's about that day at Target, with Willow. I'm a little surprised that he would write a poem about Willow, or leave it for me, when he said he wasn't in love with her, when she dumped him. Then, I read it.\n\nGoing to Target With Her\n\nGoing to Target with her\n\nDriving the ass-backward long way to Target to spend more time with her\n\nRather than cramming for the test I need to ace\n\nTime I couldn't have gotten otherwise\n\nAt Target with her\n\nTrying on cheap, stupid, beautiful red and purple Target hats,\n\nTrying to make her laugh and pose for pictures I take\n\nTelling her we're there for a different reason\n\nLying that we're there for a different reason\n\nWhen really, I just want to be with her\n\nAt Target with her\n\nHoping her tiny, white butterfly hand will brush against mine across the displays of socks and gloves,\n\nAnd she'll see me differently.\n\nDifferently than everybody else.\n\nBut she doesn't.\n\nTaking pictures of her trying on stupid hats\n\nTaking selfies, but really, training my phone on only her\n\nAt Target with her\n\nGoofing around in frozen foods\n\nAt Target with her\n\nHer and another girl, the girl I was supposed to be at Target with\n\nThe girl I used to think I maybe liked until the first day I met her, the first time I heard her voice, the first time I talked to her\n\nAnd I knew\n\nThat she was the arrow that hit the target that was my heart.\n\nThe paper smells like his cologne, just a little. Underneath the poem is a photo, a selfie Goose took that day. He'd taken it of all three of us, but in the photo, you can only see part of Willow's arm.\n\n_Training my phone on only her._\n\nI stare at it a second, realizing that I, not Willow, am the \"her\" of the poem. The poem is about me, about that day at Target. _Driving the ass-backward long way._ I remember how long it took to get there and, especially, back, how Goose made a wrong turn on the way from Willow's house to mine. Stupid. He asked me to go because he liked me.\n\nMaybe loved me.\n\n_Loves me._\n\nI remember what he said about showing me his poetry: \"Okay, how about this? Someday, I might leave a poem lying somewhere, where you can find it. Just don't ever tell me you read it, okay?\"\n\nBut how could he expect me to say nothing?\n\nI sink to the floor, reading it over and over to see if I could be wrong, yet I know I'm not.\n\nI wanted to leave today, right now, to protect him, his family, from Violet. I love them and need to protect them. This makes it worse. He endangered himself because of how he feels about me. Yet how can I leave without talking to Goose? If I leave, he'll think I ran away because of the poem. But that's not it at all. Not at all.\n\nThe doorbell rings. My pulse quickens. Is it him? I know he skips sometimes, or takes \"personal days.\" But he wouldn't ring the doorbell, unless he forgot his key. I want to see him again, so much. Maybe he did forget his key. I'll check. I start for the door, then go into the bathroom to put the poem back where I found it, facedown on the pink tiled counter. Maybe he changed his mind about showing me. Maybe he came home because he wants to get it back. I sprint for the door.\n\nBut when I bend to look out the peephole, I see only an old woman holding a Publix grocery bag of mangoes, standing in the rain.\n\nKendra.\n\nI have to get her to help me leave tonight, after I talk to Goose. I open the door.\n\nShe looks confused. \"Who are you?\"\n\nNot Kendra.\n\n_Through the mirror, only the mirror._ That's what Kendra said. But this isn't Kendra. She must be the neighbor, the nice lady with the mangoes. I make up a name. \"I'm, um, Mary, the Guzmans' cousin. I'm staying with them awhile.\"\n\n\"Pretty girl.\" With her free hand, she reaches out to touch me. She is old, with wrinkles atop her wrinkles, white hair piled on her head. I back away. Stupid. She's just an old woman, a _nice_ old woman. I step forward, letting her touch me. The rain is falling, but she doesn't seem to care.\n\n\"Thank you, ma'am. Stacey's not home.\"\n\n\"I came to bring her these.\" With great effort, she lifts the bag of mangoes. \"Do you like mangoes?\"\n\n\"I do.\" They smell overripe, rotten. \"We've been making smoothies from them.\" I start to take the bag from her, but she pulls it away.\n\n\"Have one plain.\" She reaches into the bag and takes out one that is mostly scarlet, about the size of her clawed hand, perfectly firm with no brown spots. \"Nothing like a mango, fresh from the tree, juicy . . . succulent.\"\n\n\"Thanks. I'll have it later.\"\n\n\"Oh. Okay, if you don't like them.\" But she keeps holding it out in her veiny, spotted-brown hand. \"I always save the nicest ones for my neighbors, and this is the best one of all. Isn't it pretty?\"\n\nI nod. It is beautiful, so red it almost glows. The most beautiful mango I've ever seen.\n\nThe old woman is looking down. She drops the hand holding the mango, slowly. \"It's just, they don't last very long.\"\n\nIt's so ruby red.\n\n\"I . . . okay. I'll try it.\" I hate to hurt her feelings. She seems like a nice old lady.\n\n\"Just one bite. You've never had one fresh picked, I'll bet. Why, when I was a little girl, we used to put a straw right into them and drink the nectar. You should try that.\"\n\n\"I don't think we have a straw, but another time.\" I reach for the mango. It is hot, probably from the sun, though it's morning, and the light isn't too bright yet. I examine its surface, looking for the perfect place to bite. It is smooth as porcelain, red and yellow as flame, no green, every part as perfect as the next. \"Looks yummy.\"\n\n\"Try it then. It won't make you fat, if that's what you're worried about.\" She laughs.\n\nHer lips are so wrinkled, and there are hairs sprouting atop them. Suddenly, I don't want to bite it, but what else can I do? Throw it back into an old woman's face? Just a small bite. I choose a red part and sink my teeth into the thick flesh. The bite mark shows a crescent of yellow, like the sun.\n\nOnly after I've bitten it does the old woman look up at me. I notice her eyes. They're not old at all, and they're familiar, so familiar.\n\nIt's the color. They're not blue.\n\nThey're violet.\n\nThe bite of mango catches in my throat. I choke as I fall to the ground.\n\nI hear her laughter.\n\n# PART 3\n\n# _Goose_\n\n#\n\n#\n\nI decided yesterday in chem that I was going to stop being in love with Celine.\n\nOkay, that sounds a little arbitrary, but life is arbitrary. It was arbitrary life that put her in my path, that made her audition for that play, that let me fall in love with someone so completely unattainable.\n\nAnd I need to butch up. My grades are sucking wind. I'm getting a C in pre-calc because I can't stay awake in class. Every day, I sleepwalk through school, just waiting for the time when I can go home, when I can see her, talk to her, stay up all night watching John Freaking Hughes movies with her, hoping our hands will meet across a bowl of cut-up mangoes or teaching her piano just so I can sit next to her on the bench.\n\nAnd every night, I crawl into bed an hour before I have to be up for school, wondering why I haven't told her how I feel.\n\nBut really, I know.\n\nI'm afraid. Who wouldn't be? What guy wouldn't be scared to declare his love to the most beautiful girl on the planet\u2014a girl who could have anybody, even a rock star? Why would she want _me_?\n\nBut maybe she could. Maybe she appreciates my intelligence, my talent, my sense of humor, my low center of gravity. Maybe she's special enough to see who I really am.\n\nSo, yesterday, I made my plan. I would walk in there, find her, and say, \"Celine, I love you.\" No. Too bold. Maybe I'd say, \"Celine, I was thinking I'd like to make you dinner. No, not mac and cheese, but something special, a special dinner for the two of us, like spaghetti with crumbled-up hamburger in the sauce, like my mom makes. Okay, maybe my mom will just _make_ it for us, but then, she'll leave us alone so we can have dinner together. Like a date? Yes, like a date. Actually, a date.\" I could say that, couldn't I?\n\nMaybe.\n\nYes. Yes, I could ask her out. Or in. I've asked girls out before. That wasn't scary. And they've accepted. I didn't have to declare my love for her and ask her to marry me. In fact, that would be creepy since I'm only seventeen, and it's not 1940. I could just ask her on a date. What was there to lose? I was, after all, saving her life by hiding her from her wack job stepmother. That had to be a point in my favor, a way out of the friend zone and into the end zone. And, if I asked her out, and she laughed at me, I could just pretend I was kidding. I'm a super-big kidder.\n\nI decided that's what I would do.\n\nThe experiment about acids and bases, the pre-calc class, and the reading of _The Kite Runner_ all sort of ran together after that because I was contemplating the possibility, the impossibility that she might actually love me. It wasn't so crazy, was it? I was a great guy. And she had said herself that she thought that Andie should have ended up with Duckie. I was confident that I was at least as good-looking as Duckie.\n\nI mean, if Duckie was four foot five inches tall.\n\nBut, except for that, I was actually better looking than Duckie. I didn't have that 1980s bouffant hairdo that you could tell had been weird even then because the cool guys in the movie didn't have it. And hadn't Celine herself said I had beautiful eyes? I do have beautiful eyes.\n\nA year after _Pretty in Pink_ , John Hughes did another movie, _Some Kind of Wonderful,_ about a guy who falls in love with the popular girl, ignoring his tomboyish female friend, who adores him (and who was actually hotter than the popular girl, but I digress . . .). But in the end, he realizes it's the nerdy friend he wants. _Some Kind of Wonderful_ ends the way _Pretty in Pink_ should have, with the hero, Keith, running after Watts and giving her \"a kiss that kills.\"\n\nThat's how I wanted my story to end too.\n\nI was actually semi-confident, but when I went in to talk to her, she was babbling about how much she loved Jonah Prince, how she wanted to go to the concert so she could meet him. I pictured his eyes (under one of his stupid backward baseball caps), meeting hers across a crowded stadium, and I remembered one universal truth:\n\nGirls like hot guys.\n\nAnd a second truth:\n\nI'm not one.\n\nSo I sort of freaked out on her, which was stupid because it wasn't her fault. But we made up this morning, and everything is fine. But then, I had to leave that stupid poem in the bathroom. I thought it would be easier than telling her how I felt, but now, I realize it's worse. Because, if she doesn't feel the same way, there's no way I can back out, no way I can laugh it off and say I didn't mean it. It's there, in black-and-white. I wrote a _poem._ About her. And if she thinks I'm a pathetic loser, I can never go back.\n\nSo the question is, what kind of drugs was I on when I thought leaving that poem was a good idea? Like, did Stacey spike my cereal milk with absinthe? What other explanation could there be?\n\nSo, even though I _need_ to go to school to keep from actually failing math, I turn around and head back home. I consider calling my mom. I could ask her to find the poem, to rip it up before Celine sees it. But I don't want her to read it either, don't want her to know about it and think, _Isn't he cute_. So I wait at McDonald's until it's time for her to take my brothers and sister to school, and then, I head home. Hopefully, I can pick up the poem before Celine wakes up.\n\nAnd then, I'll tell her in person or maybe just walk up to her and kiss her, which somehow works for guys on sitcoms.\n\nSure, maybe. Ten years from now.\n\nBut, when I get to the house, something is different. The front door is open. It's raining and the rain has come in. There are mangoes, red and gold, scattered down the steps.\n\nThen, I see Celine.\n\nShe's spread out across the entranceway, her feet against one wall, her body slumped against the other, like she's just asleep. But no one sleeps getting rained on.\n\n\"Celine!\" I run toward her through the driving Miami rain. \"Celine!\"\n\nShe doesn't move.\n\nI grab her wrist and, at the same time, put my head against her chest, hearing her heartbeat, feeling her pulse. She isn't dead. She isn't dead!\n\nYet, she isn't moving.\n\n\"Celine!\" I start to shake her hand, furiously. I get closer to her ear. \"Celine, wake up! Wake up!\"\n\nShe doesn't move. I have to call my mother, or maybe 911. I take out my phone. My hands are shaking as I dial it. I look around as it rings.\n\nA shiny red-gold mango lies by Celine's bare feet. It has one crescent-shaped yellow bite missing, just one. How did it even get there? She must have opened the door to someone. Why did she open the door? She knew not to!\n\n\"You have to help me,\" I tell the 911 lady. \"I think she's been poisoned!\"\n\n\"Who's been poisoned?\"\n\n\"My . . . friend. She was lying on the floor when I got home. Send someone quick.\" I can barely remember my own address. It's like everything that's happened before this moment has been a dream.\n\n\"Okay. Is she breathing?\"\n\n\"Yes! Yes, but she's not conscious. She's lying on the floor. Send someone quick!\"\n\n\"Someone is coming, sir. Did she hit her head?\"\n\n\"I told you. I think she's been poisoned. Hurry!\"\n\n\"What do you think she ingested?\"\n\nI look at the mango. It's crazy to say I think she had eaten a poisoned mango, but what else could it have been?\n\n\"This is going to sound crazy, but there's a mango here, and I think it might have been poisoned.\"\n\n\"A mango?\" the lady says, like it's the most normal thing in the world. \"Does she have allergies?\"\n\nI hadn't thought of that, but I say, \"She eats them.\"\n\n\"Is it stuck in her throat? Did you attempt to remove the obstruction?\"\n\n\"No. I mean, I don't know if there's an obstruction.\"\n\nShe tells me to sweep my finger through Celine's throat to see if there's something stuck in there. I slide Celine's head onto my lap, then ease my index finger between her lips and into her throat. _Shit, please don't die._ She gurgles a little, but there's nothing there. \"Please hurry!\"\n\n\"They're coming, sir. They're on their way. Don't hang up.\"\n\n\"No, there's nothing there. I'm sorry.\" I hold Celine tight.\n\n\"It's okay. They're coming. She's still breathing?\"\n\n\"Yes. Yes.\"\n\nShe asks me some other stupid questions, who Celine is, who I am. Finally, I hear the siren in the driveway. I watch as they take over, then take her away.\n\n#\n\n#\n\nAccidental overdose. That's what the paramedics and, later, the doctors at the hospital, think it was, and that's how they treat it. I know it was neither, but they won't listen to a kid. They do all the stuff I already did. They check her vital signs and hook her up to a bunch of tubes, then take her to the hospital. They pump her stomach, give her charcoal. The whole time, I feel like it's _me_ they're hooking up to all these machines, me who can't speak or move, who's clinging to life. They ask me questions I can't answer. They want to know what I think she OD'd on. Nothing, I tell them. I tell them to test the mango. I brought it with me. They act like that's crazy.\n\nThey also want to know where her parents are. \"She doesn't have any,\" I tell a nurse for the fourth time. He's a burly Cuban guy twice my size. They've wheeled Celine out, away from me, into some curtained-off area where I can't see her anymore. I wonder what they're doing to her. \"Her parents are dead. She lived with her stepmother, but she was abusive. She took her out of school. If she came to my house, she was probably running away.\"\n\n\"Do you know where she lives?\" the nurse asks.\n\n\"No, um . . . I was at her house once, but I don't really remember.\" Not that I drove by there twenty or a hundred times or anything. But I'm buying time. If Violet comes, maybe she'll finish her off. She could tell them to unhook her oxygen. Or unhook it herself. Or put something in Celine's tube. Or use magic to kill her and make it look like an accident. I'm hoping maybe if I say she was abusive, they won't let her in. And it's not a lie anyway.\n\nThe nurse turns over the clipboard he had all ready to write the information. \"Okay, let me know if you remember. The police will probably want to talk to you too.\"\n\nThe police? Maybe they are taking this seriously. \"Then I should probably wait until my parents get here.\"\n\nMy mom shows up soon after, then Dad. Dad says they can try to file papers to become Celine's guardians. He says I did good, even though I was skipping school and what's that about. He also says we probably have to tell them about Violet.\n\n\"But she'll kill her, Dad.\" I picture Celine, before they took her away, so white and still. \"She already almost died.\"\n\n\"We won't let her,\" he promises.\n\nI give them Violet's name and address. I actually don't know if they have a landline, or any phone number. Celine would know, but she's suspended in some still, white place with too-little air. \"She's a really bad person, though. Celine was afraid of her.\"\n\nThat's completely true. I remember something Celine showed me once, that first day we played the piano.\n\n\"Violet burned her. Celine showed me the scar. It's on her ring finger.\"\n\nAnd that's when Violet walks into the emergency room.\n\nAnd starts crying.\n\nAnd begs to see \"my darling, darling Celine.\"\n\nI try to follow her when they take her out, but the nurse stops me. \"Family only,\" he says.\n\nBut Violet says, \"It's all right. He saved her, didn't he? He was like a brother to her.\"\n\nI stare at her. She knows why, and she stares back. As she does, her eyes turn from blue to brown, then back again.\n\nKendra.\n\nKendra with Violet's ID and insurance cards . . . and her face.\n\nI breathe in, a shaky breath, and follow her into the curtained area where Celine lies. \"We're going to admit her to the ICU,\" the nurse says. \"We don't know what she ingested.\"\n\n\"Try everything,\" Violet\/Kendra says. \"It might be something you don't suspect. May I speak to a doctor?\"\n\n\"Yes, ma'am. Just a moment.\" He excuses himself and is gone.\n\nI stare at Celine. Her face is white as her pillowcase, and she doesn't look sick, only sleeping. She is so beautiful, her dark hair fanned out beneath her, a frame for her doll-like face. Celine hates being called beautiful. She says people judge her for it as much as they'd judge someone else for ugliness. But her beauty is so undeniably . . . there. Of course, it's the first thing you see. But, gradually, as I've gotten to know her better, I've forgotten it, forgotten how frightening that beauty is. She's become just Celine, a girl I know, the girl I love, Celine in motion, talking, singing, playing with my sister, lusting after Jonah Prince.\n\nNow, a motionless shell, the outer beauty is all that's left. It crashes over me like a wave, taking me down into it. I'm choking, drowning.\n\nCeline's beauty isn't what I love about her. It's what I hate about her because it's what makes her so unattainable. I love the girl inside, but I'll never have her.\n\nNow, it's more than her beauty that makes her unattainable. What if she never wakes up?\n\nWhen I first noticed her, it wasn't because of her looks. Well, not _just_ because. It was because she was really brave, really fierce. She stood up to a roomful of sophomores who were picking on this stuttering substitute. She shamed us all, including me. I should have done what she did, but I didn't want to stand up to my friends. From that day on, I wanted to know Celine Columbo. It took a year, but when I finally did, she was everything I'd imagined. She was beautiful on the inside, like those ABC Family shows say you're supposed to be.\n\nAnd now, she's dying. I reach out my finger and touch a little peach-colored freckle on her arm right above where the tube goes in.\n\n\"Ahem.\" Violet\/Kendra interrupts my drooling reverie. I look up at her.\n\n\"Why is she like this? I thought Violet wanted to kill her. I mean, not that I want her dead, but this . . .\" I jut my hand toward Celine.\n\nI hate that Kendra looks like Violet.\n\n\"Violet did try to kill her. She didn't die because I used enchantments to protect her as best I could. But I couldn't defend her completely, and I can't wake her.\"\n\n\"What can?\"\n\n\"Violet, of course, but that isn't going to happen.\"\n\nNo, it's not. \"Anyone else?\"\n\nViolet\/Kendra shakes her head. \"I don't know. Usually, with spells, there are hoops to jump through. The feather of a golden bird, the scale of a dragon, but until someone tells you . . .\"\n\n\"You don't know.\" I look back at Celine. Her chest rises and falls with her breathing. She's still breathing, but for how long? \"Is there a time limit? How long will she stay like this?\"\n\n\"Indefinitely, as long as she's nourished and hydrated. I assume they'll feed her intravenously.\"\n\n\"So someone could just tell them to disconnect it.\"\n\n\"Someone could.\"\n\nWe exchange a look. We both know who that someone is.\n\nEarlier this week, I was playing the piano with Celine, sitting beside her on the bench, our hips touching, her hand so close to mine. Why didn't I take her hand? Why didn't I beg her to run away with me, someplace where no one would find us?\n\n_Because I'm just a kid. I can't do anything._\n\n_I have to do something._\n\n\"I have to talk to her,\" I say.\n\n\"Talk to . . . ?\"\n\n\"Violet. The real Violet.\"\n\nViolet\/Kendra shakes her head. For a moment, I see a glimpse of her real self. \"You're getting overconfident, talking to me disguised as Violet. The real Violet won't speak to you, much less do what you ask.\"\n\n\"Maybe she will.\"\n\n\"Maybe all the world's children will join hands and sing 'Kumbaya,' but it's unlikely,\" Kendra says. \"I've been trying to talk to her for weeks. You see where that's gotten me. And she likes me.\"\n\n\"What do you think I should do then? Just leave Celine here to die? Or stay in a coma for the next fifty years?\"\n\nI look at Celine again. I remember that day she came over, right after her father died. She fell asleep almost in my arms. I'd watched her sleep for over an hour, and it gave me hope. She felt that comfortable around me. Maybe I'll never have what I want, but just knowing that Celine is alive on the planet makes the world better. At least my world.\n\nKendra's smile is sad. \"No, I guess not. You love her very much, don't you?\"\n\n\"Is it that obvious?\"\n\n\"I saw it that first day at Target. That's how I knew you would help her.\"\n\n\"I'd do anything for her. I mean, I'd prefer not to die for her, but if that's the risk in talking to Violet, I'll take the risk.\"\n\n\"Don't take unnecessary risks, though,\" Kendra says. \"Talk to her in a public place, her office building or the train station.\"\n\nI hate the Metrorail station. It's an elevated train with views of the city, and I can't stand on the platform without holding on to a wall the whole time. Even then, it's terrifying. \"Where does she work?\"\n\n\"I'll show you. And take this with you.\"\n\nFrom nowhere, she produces a silver mirror with a handle. It's about a foot long, and when she hands it to me, it weighs maybe ten pounds.\n\n\"What's this for?\" I ask.\n\n\"If you look into it, you can see whoever you want. We can stay in touch that way.\"\n\n\"You don't have, I don't know, a phone?\" I turn the crazy thing over. It looks like Marie Antoinette might have owned it, all curlicued and fancy. And I'm supposed to talk into it?\n\n\"Sorry. I can't make that kind of technology. Mirrors are easier. If you get in trouble, let me know. Otherwise, try not to let Violet see it.\"\n\n\"Why? What will happen if she does?\"\n\n\"Bad things.\"\n\nShe doesn't elaborate any more than that. I probably don't want to know.\n\nI take Celine's hand. It's tiny and white, with little pink nails like shells. And it's cold. I clutch at it, warming it up. Finally, the orderlies come to wheel her to the ICU.\n\n\"What's your plan?\" Kendra says as she leaves.\n\nI laugh. \"I wish I had one.\"\n\nBut it's really not funny because my whole plan is just to talk to Violet. What else can I do?\n\nAnd yeah, I'm terrified. But I'm more terrified of what will happen if I do nothing.\n\n#\n\n#\n\nViolet's office building turns out to be a skyscraper with balconies, way worse than the train station. Until I change my mind and go to the train station. Then, that's worse. The escalator going up is three stories high with soaring views of the surrounding tall buildings. And by \"soaring,\" I mean sickening. My stomach feels like I swallowed a wad of chewing gum. A quarter of the way up, I picture myself plummeting from the height, tumbling over and over to the ground, then farther, down to hell. Halfway up, I picture myself jumping. I close my eyes, remembering a news story I saw once about a guy who fell twenty-three stories through what was supposed to be a stable glass floor at a hotel.\n\nAbout me and heights: I avoid them. Florida is a flat place, so it's not that hard, but there have been moments. My earliest memory of being afraid of heights was being about five years old and going to Tom Sawyer Island at Disney World. Tom Sawyer Island is one of the lamest attractions in the parks. Only little kids go on it. To reach the island, you have to cross this skinny, rickety bridge, supported by barrels floating in the water. I got halfway across, and I couldn't go on. I was sure I was going to die, and I stood in the middle of the swaying bridge, screaming like a girl, while the people bottlenecked behind me yelling in a very un-\"Happiest Place on Earth\" manner for me to get out of the way. Finally, my dad and a Disney employee dressed as Huck Finn picked me up and carried me off, to the cheers of the assembled guests.\n\nThis turned me off to theme parks. Obviously. I tell people it's because I'm not tall enough for the rides, but I actually can go on all the rides at Disney. You only have to be three foot eight to go on Space Mountain. I just don't want to. Other things I avoid include rock-climbing walls, rope courses, skyscrapers, hotels with high balconies, and\u2014oh, yeah\u2014the Metrorail.\n\n\"Are you okay?\" Kendra has decided to accompany me, at least this far, probably in hopes of talking me out of it. Which she never will. I wanted her to stay with Celine. My mother is at the hospital, but Kendra would be better protection. We rehearsed things for me to say to Violet, but now, they're out of my head.\n\n\"You're turning green,\" she says.\n\n\"It's not easy being green,\" I try to joke. I feel the hospital cafeteria tacos in my throat.\n\n\"I didn't realize you were this big a chicken.\"\n\n\"I'm no chicken.\" But I still don't open my eyes, preferring to stare at the red and black shadows behind them. \"I have a problem with heights, but I'm here. I'm in this. I'm brave. Brave\u2014whoot! A chicken would say, 'Oh, well,' and go home.\"\n\n\"Yes, he would.\" Kendra touches my arm. \"You're going to want to open your eyes now and step forward, brave boy. You're at the top. You've arrived.\"\n\nI open them just as my left foot hits the metal at the top of the escalator. I made it. The first half of my journey seemed to take as long as a drive to Fort Lauderdale with my brothers fighting. The second, with the distraction of my annoyance with Kendra, took seconds. For an instant, the floor swims below me, and I stumble. But Kendra takes my arm and points to the turnstiles. \"Wait for Violet inside. No scary heights, and she comes through here most days around 5:10. And be careful.\"\n\nMy watch says 5:07. We cut it close. When I turn to thank Kendra, she is gone. I hurry to buy a ticket. Then, I notice I have one in my hand. It wasn't there before. Thanks, Kendra.\n\nHer \"be careful\" rings in my head.\n\nI go inside, and wait where Kendra told me. I don't know what I'll say when Violet gets here, but I have to try.\n\nIf I feel like the climb was the scary part, that all changes when Violet shows up.\n\nI've only seen the real Violet twice, once for a brief second as she rushed into the hospital, the other time at the funeral. Both times, she was frantic and messed up, miserable about her husband's accident. She still looked inhumanly beautiful.\n\nNow, her beauty is something else.\n\nKendra spoke of Celine's beauty as being from within, a light of kindness and innocence. Celine and I joked about it, because, who talks like that. But it was true. Celine has the face of an Old Master's _Madonna_.\n\nViolet is a comic book super-villainess. Catwoman? Harley Quinn? Elektra? No, she's Poison Ivy, Batman's nemesis, live and in person in the Metrorail station with long, bright auburn hair and a body you can't help noticing, hugged by a bright blue suit that shows off her curves\u2014and her eyes.\n\nThose eyes. They're blue, almost purple, and huge. And evil. I can see them even through the mobs of commuters. In high-heeled boots, she sticks out above the crowd and way over me. Every guy in the place pivots to stare at her, and more than a few women.\n\nShe is the most intimidating person I have ever beheld. I practically expect to see vines sprout from her hands. A girl in front of me smacks her boyfriend, who's staring.\n\nAnd I have to confront her. Now. Preferably before she goes out to the scary trains.\n\nI pursue her. \"Violet!\"\n\nShe looks around, hearing her name but, at first, not seeing me. I shove through all the legs. Some guy calls me an asshole. It's rude, but this is life-and-death. No time to apologize.\n\n\"Violet, wait!\"\n\nShe turns and stares. She looks at me like I'm a mangy dog or a leper. Maybe I am.\n\n\"May I help you?\"\n\n\"Violet.\" I'm panting. \"I'm Goose, Goose Guzman. I'm\u2014\"\n\n\"I know who you are. You're my stepdaughter's little friend.\" She emphasizes _little_. Her eyes bore into me, and I see something like revulsion at my appearance. \"You brought her home once, and to the hospital, when Greg . . . but Celine is gone now. She's living with her aunt in . . . Tennessee.\"\n\nPeople are stopping to stare, and I realize we must look like characters in some fantasy movie. Their eyes give me courage. Violet can't do anything to me in front of all these witnesses.\n\nNothing except _not_ help Celine.\n\n\"You know that's a lie,\" I say. \"You didn't send her to Tennessee. You sent Kendra to kill her. But Kendra didn't. You know that too, because you saw Celine this morning.\"\n\nViolet looks away. \"I don't have to listen to your babbling, dwarf.\" She turns on her heel and walks toward the glass-brick escalator, the one that will carry her\u2014and me\u2014even higher to the train platform.\n\n\"Wait!\" I run after her, trying to get in front of her, or grab her hand. But she eludes me, striding through the crowds so I have no choice but to pursue her up a second scary escalator. I want to take the elevator, but I can't risk losing her.\n\nI'm shaking when my foot hits the bottom step. By the top, I can't breathe, much less speak. I stumble after her.\n\n\"Why?\" I gasp out. Everyone's staring at me, but I don't care. I follow Violet. The train station is laid out with a platform in the middle, north and southbound tracks on either side. She stands on the southbound side, too near the edge, away from the crowds of people.\n\nThe train station overlooks the city. How great if it was underground, like the New York subway, where the only drop is onto the tracks themselves. Instead, I can see for miles and miles, practically to my house. I hang back, or try to, but Violet seems to realize my fear for she steps closer to the edge, her beauty silhouetted against the blue and silver skyline, teetering on four-inch heels. I could push her over. It wouldn't help Celine, though.\n\n\"Why do you hate her so much?\" I'm shivering, though the day is hot and windless, a June day in Miami.\n\nShe smiles, then laughs. \"Come closer, and I'll tell you.\" With her hand, she beckons me. Yep, she definitely knows I'm afraid.\n\nI edge closer. When I'm near a high place, I always picture myself plunging, flying at first, then diving down, dropping like a . . . well, dropping like a guy falling off a train platform. Must concentrate. I look into Violet's eyes, but they're almost as scary as the height.\n\nI will myself forward. When I'm five feet away, far enough that, if I fell, I wouldn't go over, I stop. I try again.\n\n\"Celine loved you,\" I say. \"When she first met you, she did. She told me how wonderful you were. You got her a kitten. You did her hair, and then . . .\"\n\n\"And then, she turned into her mother. I hated that bitch.\" Violet's voice is a knife in the noisy station. \"She was cruel and heartless. She never loved Greg. She only took him because I wanted him. You can't imagine how she bullied me. And Celine is the same.\"\n\nI hear the wind down the tracks. \"I think I can imagine being bullied. But Celine isn't like that. She's not a bully. I've seen her stand up for people, not bully them.\"\n\n\"Please. She's exactly like her mother.\"\n\n\"I wouldn't be friends with her if she was like that.\"\n\nViolet smiles and gets a little gleam in her eye. \"Is that what you are, dwarf? Friends? Because that's not what I see.\"\n\n\"What do you see?\" But I think I know. And I guess she's just going to keep calling me \"dwarf\" like she's some evil sorceress.\n\nWhich, I guess, she sort of is.\n\nShe throws back her head so her orange hair streams down her back. \"I see a beautiful girl, using some loser who's in love with her. I see Jennifer all over again.\"\n\nI shake my head. \"She's not like that.\" But I wonder. Violet is half right. I am in love with Celine. Is she _all_ right? Am I just deluding myself about Celine?\n\nNo. That's not true. Celine was nice before she knew me, definitely before she needed me. If anything, I was using her, pretending to be her friend in the hope of being something more.\n\n\"Believe what you want.\" Violet takes a step backward. \"I've seen the look on your face when you're beside her on the piano bench, holding her hand, trying to edge closer. Or when you watch TV together at all hours of the night. It's the same look I used to have with Greg, sad, longing. Pathetic.\"\n\n\"How did you . . . ?\" But I know. Magic. All those things happened in the past few weeks, after Celine had moved out. Violet had watched us the whole time. She'd known Celine was alive, waited until she could harm her. I take a half step toward her, saying, \"What did Celine do to you? What did she do that was so awful that you want her dead?\"\n\nViolet thinks for a moment. \"She is a constant reminder that her father preferred someone else to me. Even when we were married, he never really loved me. He was enthralled, but his love was all for Jennifer.\"\n\n\"But her father's gone. Either way, he's gone. Couldn't you, in his memory, just be the bigger person? Just let his daughter live?\" In the distance, I see the train coming. What if I fell onto the track, if I was swallowed up in it? I feel dizzy, like my head is an escaped balloon, floating in the fluorescent lights above me.\n\nViolet chuckles. \"You're really afraid, aren't you?\" When I don't answer, she says, \"With a blink of my eye, I could send you hurtling over that track, then crash! Down onto the sidewalk. They wouldn't even recognize you.\"\n\nI feel her words as if she did it. But I say, \"Why don't you?\" Trying to keep my voice steady, though I hear it shaking.\n\nViolet smiles. \"I know that, whatever I can do, Celine can hurt you more, hurt you the way Greg hurt me. That's better, my little friend.\" The train is pulling into the station now, brakes squealing, covering the treacherous view. \"My train is here. I have to go. I'll do you a favor, dwarf. I won't kill your darling.\"\n\nI feel my knees start to give way, but I catch myself. \"Really?\" Sensing a trick.\n\n\"Really. But I won't revive her either. She will sleep forever. As long as you both shall live, you can fantasize that she might have loved you, if only Evil Violet hadn't taken her away. You can worship her shell of a body, wrap your little tiny arms around her, and dream that she is yours instead of knowing the cruel reality, as I did with Greg.\"\n\nHer eyes are mesmerizing, like Christmas lights.\n\n\"And what reality is that?\" I barely whisper it.\n\n\"That someone beautiful like her could never love a little freak like you.\"\n\nHer words snap me out of my trance. I stumble forward, wanting to hurt her. The train screeches to a stop, but I barely notice it, the people, the height. My heart is banging like crazy, and I advance toward her, remembering every kid who ever made fun of me in elementary school, every fear I had about Celine, every doubt I had about myself, sinking that anger into Violet.\n\n\"You know, Violet, I may be a freak, but I have a family that loves me, unlike you. I have tons of friends, unlike you.\" _I'm saying this to a witch?_ But I don't care. I can't stop myself. The words just come out, like vomit all over the yellow, rubber barrier that separates the train from the station. \"You're a beautiful woman, and it hasn't helped you. Your beauty is nothing, worthless.\"\n\nI take a shaky breath. I look up at Violet, and she's smiling.\n\nShe laughs. \"Anything else?\"\n\n\"Yeah. You're evil, and you've paid the price. You have no one. Greg is gone, and everyone else hates you.\"\n\nI look up to see if I've hurt her. I want to hurt her. But her face is immobile, unreal.\n\nI add, \"So, sometime, you may want to ask yourself which one of us is really the freak.\"\n\nAnd, before she can change me into a toad, I turn and walk toward the elevator. When I look back at her, the train doors are just opening. Yet, she is gone.\n\nI'm sweating when the elevator comes. I take it down, then sit in the train station for half an hour, trying not to hyperventilate, wanting my mother. Finally, I summon Kendra with the mirror. She walks with me back to the hospital.\n\n\"I'm sorry, but not surprised. If a good talking to was all it took, I would have done it. It is, alas, her nature to be cruel.\" She touches my shoulder. \"I am sorry.\"\n\n\"I'll find another way.\"\n\nCeline is still in the ICU, beautiful and dead-looking. I think of what Violet said about her being a shell. It's not true. I want to take her into my arms, give some of my own life to her. I could give her ten years, twenty, half the time I have left. I would if I could. Was Violet right? Would I rather Celine be like this forever than face her rejection?\n\nNo. I love Celine, and part of loving someone is wanting the best for them, even if the best for them isn't being with you. I'm not Violet. Even if Celine will only ever be my friend, I want her alive.\n\nBut how can I make that happen?\n\nI gaze at Celine's pale white face, and I feel the weight of utter hopelessness upon me.\n\n#\n\n#\n\nIt's Sunday now. Celine is still in the ICU, stable, but no different. The doctor assured Violet\/Kendra that Celine has a good chance for recovery. Of course, the doctor doesn't know about the magic.\n\nFriday and Saturday, day and night, I sat in a hard, metal chair by Celine's bed, holding her limp hand, squeezing it, willing her to squeeze back. The nurses didn't even try to kick me out. Kendra said it was okay, and I guess they felt sorry for me. Maybe they realize I can barely move. I feel petrified, not like when people say they're petrified meaning scared, but actually petrified like wood, like parts of me have turned to stone, and if Celine doesn't come out of this, maybe I'll just turn to stone beside her. Of course, I can't help but think of what Violet said about worshiping Celine's shell of a body. How long can I go on, waiting for someone who may never awaken?\n\n___Forever._\n\nViolet has been true to her word. She hasn't been here. Still, I won't leave Celine alone.\n\nWhen I went home to get my clothes Friday afternoon, the poem I left for Celine was still on the bathroom counter, facedown, exactly where I'd left it. She never saw it. And that was when I realized, I wanted her to. Even if I was going to be totally crushed by her reaction. When you love someone, you have to tell her.\n\nI lean close and whisper in her ear, \"Please wake up.\"\n\nBut she doesn't. I squeeze her hand. I have my American history exam tomorrow, and I've been reading aloud to her from the book. In between World War One and World War Two, I tell her all the things we can do together if she wakes up. After Korea, I beg her to wake up. Vietnam, the Cold War, \"I miss you so much.\" Now, I've studied all I can, so I just watch her.\n\nStacey's texted me about a dozen time, telling me I have to sleep at home tonight, long texts about how she understands I'm upset, but I have to finish out the year. Finally, I text back that I'll be home by nine.\n\nIt's a little before eight when Kendra shows up. She's disguised as Violet, and I shudder. As soon as she closes the curtain, she changes into herself, red hair melting to purple-streaked black, her clothing turning to a long, black dress.\n\nShe looks excited. \"I've figured out what will help Celine.\"\n\n\"Really?\" I clutch Celine's hand, hopeful.\n\nUntil the next words out of Kendra's mouth.\n\n\"Yes. A handsome prince! I remember once, there was a girl who slept for over a hundred years. It was a curse like this one, except a fairy placed it. The girl pricked her finger on a spindle and fell down as if dead. A century passed\u2014they didn't even have feeding tubes then\u2014and everyone forgot her until, one day, a prince came riding by on his horse. He kissed the girl. She woke and they lived happily ever after.\"\n\nI try to keep my voice even. \"So what year was this?\"\n\nKendra thinks about it. \"The kiss happened in 1675, but the princess went to sleep a century before.\"\n\n\"And this was in Europe? England?\" I feel suddenly tired, like I could just put my head down on Celine's bed and go to sleep.\n\n\"Or Germany, maybe. One of those countries.\"\n\n\"One of those countries with princes riding around on horses?\"\n\n\"Yeah.\" She finally gets my meaning. \"Oh.\"\n\n\"This is America, Kendra. No princes here.\"\n\n\"There could be. It probably doesn't have to be from a reigning family. One of the Romanovs was mayor of Palm Beach for a while\u2014the great-grandson of the tsar of Russia.\"\n\nPalm Beach is about two hours' drive from here. \"Do you know him?\" I ask.\n\n\"He died in 2004.\"\n\nNot helpful. I shake my head. \"Could there be another way?\" But, even as I say it, I remember her saying the other princess slept for a hundred years. A hundred years before they found a way to wake her. Celine might live a hundred years in a state of suspended animation.\n\nBut I won't.\n\nI take a long look at Celine, the shell of Celine. Is this all I'll have of her, ever? I want to shake her, slap her, do anything to wake her, but it won't work. No, that's not what I want. I want to take her in my arms and kiss her. But I'm no prince, just some poor slob who loves her. She's not mine to kiss.\n\nSuddenly I have to go. I can't look at her anymore today.\n\n\"You'll stay?\" I ask Kendra.\n\nKendra smiles sadly. \"Don't worry. I won't leave her.\"\n\nI turn toward the door.\n\n\"You know, Goose, the prince was also her true love.\"\n\nI shrug. No help there.\n\nWhen I get home, I can't sleep. Even though I've already spent the whole day studying, I lie in bed, reading the flash cards I made, because it gets my mind off Celine. After a while, I guess I drift off.\n\nAt three, I wake up. The lights are on. There are three-by-five cards all around me and only one thought in my head.\n\n_Jonah Prince._\n\n#\n\n#\n\n\"He is, technically, a prince,\" I tell Kendra. \"And Celine's not-so-technically completely hot for him, so maybe he's her true love. There's no royalty in Florida, unless you count Disney World. I know it's a crazy idea. . . .\"\n\n\"No,\" Kendra says. \"It might work.\"\n\n\"Really?\" I'm impressed with myself. We're in Celine's hospital room, where Kendra guards her motionless body. I try not to look at Celine. It makes me sad that she looks like a beautiful doll that should be displayed in a plastic bubble.\n\n\"Sure,\" Kendra says. \"I've seen people get off on technicalities before. But how are you going to get Jonah Prince here? Isn't he, like, a rock star?\"\n\n\"I figure that's where you come in. You could use your magic powers to just . . . zap him here so he can kiss her.\"\n\nKendra rubs her forehead, the way my mom does to smooth out wrinkles, except Kendra doesn't have any wrinkles. \"I can't do that with people. Even if I could, I wouldn't.\"\n\n\"Why the hell not? What kind of lame powers do you have?\"\n\nKendra huffs a bit at that. \"What if Jonah Prince was onstage, or just sitting down for dinner with his family, and he suddenly disappeared. People would freak. It would be in all the papers . . . and everyone would know that witches are real.\"\n\n\"And that would be bad?\"\n\nKendra nods. \"You've heard of Salem, witch burnings in Europe?\"\n\n\"That was a long time ago.\"\n\n\"Women are being burned as witches in New Guinea to this day. I can't take the chance. And it isn't the right way. He must come of his own free will.\"\n\n\"Wow.\" I reach over and run my hand across Celine's. It feels cold, but I warm it. \"Okay, but you can still help me, right? If I get him to come?\"\n\nKendra nods. \"I can help you.\"\n\nUnder my hand, I can feel Celine's pulse. Touching her makes me feel better, knowing she's still alive, warm, real. I turn toward her. In the elevated bed, her face is close to mine, so close I could kiss her, close enough to hear her breathing.\n\nInstead, I lean my head against her shoulder and close my eyes.\n\nI remember something, the argument Celine and I had about Jonah. I raise my head.\n\n\"There's a concert coming up. He's going to be in Florida.\"\n\nAn hour later, between Kendra's mirror and my phone, I know everything I need to know about that concert. It's next Tuesday night in Orlando. I can't tell where he's staying, but Kendra promises to spy on him with her mirror and let me know as soon as he checks in.\n\nKendra is pacing back and forth in Celine's room. \"Okay! So all you have to do is go to Orlando, locate Jonah Prince, and ask him to come back here and kiss Celine.\" She counts these things off on her fingers like they're done. \"Easy-peasy!\"\n\nI can't believe she just said _easy-peasy._ I mean, aside from how annoying that phrase is, it won't be. \"You get that he'll have tons of bodyguards, right? Big bodyguards, possibly with guns.\"\n\n\"I have confidence in your ability.\"\n\n_Glad one of us does._\n\n\"And you get that he's a serious asshole?\" I say. My research about Jonah also revealed an uncomfortable number of accounts of him urinating in public, sideswiping bicyclists with his Maserati, spitting on people, wearing diaper pants to meet the president\u2014the list of douchery goes on and on. What did Celine see in this guy? \"He's not necessarily going to be helpful.\"\n\nShe shrugs. \"If you don't want to do it, I'm sure I can keep Celine comfortable here.\"\n\n\"It's not that I don't want to.\" I look at Celine, noticing for the hundredth time the two little freckles on one side of her nose. I have always been a person who pushed myself, someone who tried to ignore limits and strived not to give in if someone said I couldn't do something, whether it was playing the piano or climbing the jungle gym or being the Artful Dodger. But this sounds . . . really hard. And if it does work, I'll lose her to Jonah.\n\nAnd yet, if I don't do it, Celine could die. Who's to say Violet\u2014the real Violet\u2014won't sneak in and put some real-world poison in Celine's feeding tube. At best, she might live in a vegetative state. I have to try, at least.\n\n\"No,\" I say. \"I will do anything for Celine. And that means overcoming any obstacle.\"\n\nThe main obstacle turns out to be my mommy.\n\n\"No. Are you crazy?\" She's tenderizing meat with a mallet. With it in hand, she is four feet of scary. \"Of course you can't do that.\" _Whap!_ \"You can't drive to Orlando on your own.\" _Whap!_ \"You can't sneak into a rock star's hotel room.\" _Whap!_ \"What if you were arrested?\" _Whap! Whap! Whap!_ \"You'd never get into college.\" _Whap!_ \"You could get shot.\" _Whap!_ \"I won't let you put yourself at risk in that way. Let Kendra do it.\"\n\nFunny how you never get over fearing the wrath of Mom.\n\nI tell her, \"Kendra needs to stay here with Celine. What if Violet comes back?\"\n\nAnother series of _whaps!_ I jump at each one. She keeps pounding as she says, \"What if Violet does come back? What if she comes after you? Or our family? No!\" _Whap! Whap! Whap!_\n\n\"But I love her!\"\n\nIt's the first time I've said it to anyone. I've only just started admitting it to myself.\n\nI feel naked.\n\nAt least Stacey stops pounding.\n\n\"I love her,\" I repeat.\n\nStacey sucks in a deep breath. \"I know you do. We love her too. And we've done a lot for her. She's a sweet girl. But, don't you see? We have to think of you first.\" Stacey puts down the mallet and holds out her arms like she's about to hug me. But, when she walks toward me, I dodge her.\n\n\"If she dies, I will never forgive you.\"\n\nI turn and leave. There is silence. Then, the pounding starts again, louder.\n\nIn the dwindling days of classes, I contact Jonah Prince's agent, publicist, producer, the Florida Citrus Bowl, where the concert is being held, and the local news, all with the same sad story. My friend is dying. Maybe a visit from Jonah Prince will help. No one bites. The guy's a total douche, and everyone who works for him knows it.\n\nI give Stacey begging looks every time I pass her. When that doesn't work, I stop looking at her completely.\n\nOn Sunday morning, two days before the concert, I make the decision I always knew I'd make.\n\nI'm going to Orlando with or without my parents' permission. Stacey hid my car keys. I hate her for that. Hate. But I will find another way.\n\nI have never wanted anything the way I want this.\n\nHow can my parents not understand that it's like my heart is trapped inside her?\n\n#\n\n#\n\nA few years ago, I heard about the show _Game of Thrones,_ starring Peter Dinklage, a guy about my size (and my personal hero), as Tyrion Lannister, the sometimes heroic, sometimes not, son of a lord. I begged my mom to let me watch it. She protested that the show was TV-MA, with tons of nudity and violence in it. She suggested _Little People, Big World_ as a substitute.\n\nBecause what any normal, red-blooded teenage guy really wants to see is a show about a bunch of little people, operating a wedding farm.\n\nBut I gave it a shot. Also, I read the book, _A Game of Thrones,_ which is over eight hundred pages long. Then, I started on _A Clash of Kings._ And I begged my mom again. I pointed out that, if I wanted to see boobs, I could find them on my phone, just by Googling _boobs._ It wasn't about boobs.\n\nIt was _mostly_ not about boobs.\n\nThat time, she let me. She'd watched the show herself by then, and she said okay\u2014if I promised not to be influenced by the characters' drinking and whoring. I swore I wouldn't hire any prostitutes without asking her first.\n\nThen, I marathon-watched the whole series in a single weekend.\n\nBest. Show. Ever. For a lot of reasons. But, especially because of Dinklage. That's not just my opinion. If you look at any poll about people's favorite characters, Tyrion is the hands-down winner, besting even the hot blond dragon chick who gets naked a lot (okay, I like her too). Whether that's because people love to champion the underdog or because of the character's quick wit, I don't know.\n\nBut I sort of lied to Stacey. Tyrion's character had a huge influence on me. Here was a guy who looked like me (only old), using military tactics, wearing armor, and marching into battle with an ax. A hero . . . even if he usually got injured. And the legions of _GoT_ fans found that believable. It made me see that a hero had nothing to do with size, made me want to be a hero too, realize I could be.\n\nBut how many battles do you get to fight in the twenty-first century? How many fair maidens are there to rescue from the cruel king?\n\nCeline is my true love, even if I'll never be hers.\n\nIt's time to be a hero. And a hero must have a quest.\n\nI have one.\n\nAnd I'm starting on the freaking Metrorail.\n\nThe train to Orlando leaves at 8:10 a.m. I've never taken a real train, the kind that goes to other cities, instead of the lame commuter trains. The pictures on the Amtrak website make their trains look like the ones that go to Hogwarts.\n\nAnd the best thing is, they stay firmly on the ground.\n\nThe worst thing? I have to take the Metrorail to get to the Amtrak train.\n\nHave I mentioned I hate the Metrorail?\n\nOh, and lying to my parents. That's another bad thing. But I'm so pissed at them, so pissed that they won't let me go, that I don't care that much.\n\nWhen I was a kid, I was a really bad liar. My mom said she trusted me completely because I was so bad at covering my tracks. Like once, when I was eleven, I took the leftover Fourth of July fireworks from the garage (in November). I was going to set them off with my friends, but I left such a trail of matchbooks and wrappers that I got caught red-handed before the first crackle. Dad said he was concerned that I lacked the logic skills to be a more proficient liar (yeah, he talks like that; he's a lawyer). Since then, I've improved a little, but I still don't lie much. It sounds nerdy to say, but I'm close to my parents, or I was. When I lie, it weighs on me. Besides, everything always comes out.\n\nThis definitely will. Sneaking out and training it to Orlando to stalk a rock star\u2014hard to hide. It might even make the paper, and not in that good way parents like, like when you earn your Eagle Scout or sing at a charity concert or have perfect attendance for thirteen straight years. No, this would be more like the stories you see on the \"Florida Man\" Twitter feed, the guy who collects all the stupidest, craziest things done by folks in the Sunshine State: There was \"Florida Man tries to remove face tattoos with welding grinder,\" and \"Florida Man caught with sushi sampler stuffed down pants,\" not to mention the classic, \"High school graduation canceled after Florida Man etches massive penis on football field.\" I can be, \"Florida Man arrested in former boy band star's dressing room.\" The humiliation. It has never been my ambition to be a Florida Man.\n\nBut it's worth it if I can save Celine.\n\nThat's what I try to remember when I go to my dad\u2014to _lie_ to my dad\u2014before he leaves for work.\n\n\"Hey, can you drive me to the train station? I want to go to Jackson to see Celine.\"\n\nHe looks at his watch. \"It's awfully early.\"\n\nIt is. It's 6:30. Dad usually leaves at 6:45. I'm counting on that to be able to make the 8:10 train.\n\nI say, \"I got up early so I could get a ride. So I don't have to bother Mom since she took away my car keys.\" I swallow. Angrily.\n\nHe shifts from foot to foot. \"You have to understand\u2014\"\n\n\"I will _never_ understand. Never.\" He frowns, but doesn't say anything. I add, \"Look, all I want is a ride to the train. It's on your way. But if you can't help me with this one little thing, I guess I'll figure out the bus.\"\n\nI can't. I would never make it on time if I took the bus. But I try to sound casual anyway.\n\nHe shakes his head. \"It's okay. I can take you. Be ready in ten minutes.\"\n\nI don't take much with me. I don't want to arouse suspicion. With any luck, I'll be talking to Jonah before people notice I'm gone.\n\nI put Kendra's mirror, a book, and my sweatshirt in my backpack from school, enough to look normal, not enough to look like I'm running away. I take all the money I saved from my jobs tutoring kids on their monologues for drama auditions, close to five hundred dollars. I cashed out my bank account.\n\nI turn off my phone and leave it on my desk. If anyone tries to use it to find me, they'll get nowhere.\n\nI wish I didn't have to lie. But nothing else worked.\n\nAt 6:42, I'm in the car with Dad.\n\n\"How's she doing?\" He's trying to make conversation.\n\n\"No difference.\"\n\n\"She could still recover,\" he says.\n\nI know he doesn't believe it. Still, I say, \"I know.\"\n\nWe drive in silence. The streets are pretty empty without people going to school. Even the joggers and dog walkers slept late. It's cloudy, and I hope I can get to Amtrak before it starts to pour.\n\nWe reach the Metrorail's \"kiss-and-ride\" lot. There's a huge escalator leading up, just like at the downtown station I went to with Kendra. Dad eyes it dubiously. \"You're going up that? I remember Tom Sawyer Island.\"\n\nYeah, thanks for reminding me. But I say, \"I love her. If the only way I can see her is this train, I'll do it. I don't give in to fear like some people.\"\n\nHe starts to say something, then stops. I hope he's not going to offer to drive me to Jackson, but he just gives me sort of a pitying look and rubs his forehead with two fingers. \"Let me know if you need a ride home.\"\n\n\"Thanks.\" I get out of the car.\n\nI'd asked Kendra if she could transport me magically to Orlando, but again, she'd been unhelpful. \"The only thing I could do is turn you into a crow to fly there.\"\n\nI declined, but now, as I head upstairs once again, I wonder if it would have been easier. A bird is in control of itself, its wings, its destiny. As it is, I'm at the mercy of stairs and tracks I didn't build, putting myself at risk.\n\nBut, as I told my father, I can't give in to those fears.\n\nSo, as the escalator bears me up, instead of closing my eyes, I concentrate on the sky.\n\n\"Where's your mom?\" the lady at the Amtrak ticket counter asks me.\n\n\"My dad dropped me off. I'm seventeen.\"\n\nIt's one of those high counters I can barely see over. The woman stares down at me. \"I can't sell you a ticket without your parents.\"\n\n\"Actually, you can. I . . . I mean, my parents and I looked this all up on your website. Minors who are sixteen or seventeen can travel without restriction.\" I look at my watch. Seven forty-five. The train leaves soon. \"I'm seventeen. Look, here's my driver's license. I'm visiting my grandma in Orlando. If I don't get off that train, she'll worry about me.\"\n\nI've experienced this before, the disconnect between my size and my age. Adults don't like to sell tickets to an R-rated movie to someone who's as tall as their eleven-year-old. I don't mind getting in places cheap, but they'll be handing me kids' menus when I'm fifty. Maybe I'll grow a mustache.\n\n\"I'll have to get my supervisor.\"\n\n\"No. You don't.\" I'm trying hard not to lose it because yelling will make me seem younger. I tap my driver's license on the counter, which is above my head. \"What you need to do is look at my license and check my age. Then, sell me a ticket so I can go see my grandma. Please.\"\n\nShe doesn't answer.\n\nI say, \"Please. She's an old lady, and she misses me. The train leaves in twenty minutes, and the next one isn't until twelve.\"\n\nFinally, the woman relents and takes the license from me. She compares my face to the photo, checks the birth date, counts on her fingers. \"Oh, you are seventeen. I'm sorry. You look much younger.\"\n\n\"What can I say? I have a great moisturizer.\"\n\n\"I could use some of that.\"\n\nHuh. She doesn't get that I'm kidding. How cute. Just print the ticket.\n\n\"Okay, will that be one way or round trip?\"\n\n\"One way.\" Part of me says I should buy round trip to avoid this problem when I come back, but I'm really hoping to come back in Jonah's private jet. Why be pessimistic? \"I may stay the whole summer.\"\n\nI get to my seat right before the door closes. There's a bunch of Boy Scouts in the same car, probably going to Disney. They're running around like someone gave them too much sugar. In fact, someone did give them too much sugar. There's a box of donuts open on one seat and a mom is offering around another. Wish I could take one. I forgot breakfast.\n\n\"Would you like a donut?\"\n\n\"Uh . . .\" I look around, realizing the Boy Scouts are all wearing matching navy blue T-shirts and matching baseball caps. I'm wearing a navy blue polo. I could make this work.\n\n\"Yes, please.\" I take a jelly, not looking up. I know from my siblings that little kids hate jelly. It would've been left anyway.\n\n\"Aren't you polite!\" The lady musses my hair.\n\nAs soon as she walks away, I scarf down the donut and head for the bathroom. I take out Kendra's mirror. \"Show me Kendra.\"\n\nShe appears. I can tell she's at the hospital. \"How's Celine?\"\n\n\"The same.\"\n\nI want to ask to see her, but first I say, \"Any updates on where Jonah's staying?\"\n\n\"He's still on the plane. You could keep an eye on him in the mirror too.\"\n\n\"I will. Can I see Celine first?\"\n\n\"It won't make you happy.\" But Kendra turns the mirror away anyway. She's wrong. Just seeing Celine's peaceful face, knowing I'm helping her, at least trying to, helps. Maybe it won't work. In that case, I'll have plenty of time to be miserable. But, for now, I'm hopeful.\n\n\"Thanks,\" I tell Kendra.\n\nBefore I leave the bathroom, I ask the mirror, \"Show me Jonah Prince.\"\n\nIt does, as if it's a television. Dude's in an airplane seat, first class. No\u2014a private jet. He takes his gum out of his mouth and sticks it underneath the shiny walnut trim on the wall beside him. Gross. Even though he can't hear me, I say, \"Please, guy, please be everything she thinks you are, everything she wants you to be. It's crazy, but you're her only hope.\"\n\nI take my seat, which is next to one of the moms from the scout troop.\n\n\"I think you have the wrong seat,\" she says. \"Where's your mom?\"\n\nI look up at her. \"Back in Miami.\"\n\n\"Oh.\" She starts a bit. \"Sorry. I thought you were from my troop. I had an empty seat next to me.\" There's an edge to her voice.\n\n\"I bought my ticket at the last minute. I'll be quiet, though, at least quieter than this group.\"\n\n\"You'd kind of have to be.\" She eyes a group of boys who are playing soccer with a squashed Bavarian cream donut.\n\n\"Where are you headed?\"\n\n\"Happiest place on earth. I need a drink.\"\n\nIt's eight-thirty in the morning.\n\n\"Kidding.\" She must see my rising eyebrows.\n\nI can tell she wasn't. \"Sure. Where are you staying? I mean . . .\" I realize this sounds stalkerish. \"Are you camping? Because you'd really need a drink then.\"\n\n\"No, thank God. We're staying at the All-Star.\"\n\nThe All-Stars are the cheaper Disney hotels, which are still pretty expensive, but nowhere Jonah would stay.\n\nI nod. \"So you have a bus picking you up?\" I'm forming a plan in my head.\n\n\"I certainly hope so.\" The donut sails past us at eye level. \"Guess I should do something about that, huh?\"\n\nMy mom wouldn't put up with that crap for one second, but I say, \"They'll get tired soon.\"\n\n\"They never get tired.\"\n\nFinally, one of the other moms takes the boys to the lounge car. The mom by me follows her, probably to get that drink. I take out the mirror and set it up between me and the window.\n\n\"Show me Jonah Prince.\"\n\nThey're off the airplane now. Jonah's surrounded by people, bodyguards, an older, balding guy I think is his agent, and a girl I recognize from TV, Allegra Kendall. She used to be on this show Isabella likes, _What a Girl Needs_ , and she looks, basically, like every other teen star, the Demi Lovato model, with long, dark, wavy hair, brown eyes, and a ton of lipstick _._ She teeters on high-heeled boots and waves at a group of people inside the terminal as they walk across the tarmac.\n\n\"Can you be a little polite?\" she says to Jonah. \"Like, look up at them.\"\n\n\"I'm dead tired,\" he says. __\n\n\"It's that pill you took on the plane. Live in the now, Jonah. They're your fans.\" She says it through her teeth, still smiling and waving.\n\n\"I have a concert tonight. I shouldn't get stressed out.\" He tugs at his pants, which are so low in the crotch he looks like he shit his pants. \"You don't know what it's like to have all these people demanding things of you.\"\n\n\"I don't know?\" She blinks at him and puts her hand on her hip. \"Of course I know. But it's really important to them. They camped out waiting for you. Can't you at least look up?\"\n\nHe rolls his eyes and looks up. Through the windows, I can see girls jumping up and down and screaming. Allegra gives him a kiss on the cheek.\n\n\"Quit it,\" he whispers. He tugs at his pants again. I remember reading about this rock star who actually lost his pants onstage because they were so loose.\n\n\"Don't be such a grump.\" Allegra tries to smile as they walk inside the airport. Fans scream.\n\nJonah puts on a big, fake grin. \"I'm not being a grump. You're being annoying.\"\n\nThis conversation is painful. I'm just listening because I'm waiting for Jonah or someone to say the name of their hotel. I know his concert's at seven, and then, he'll go back to the hotel. I just need to sneak into his room before then and beg him to go with me.\n\n_Just._\n\nJust need to sneak into a rock star's room.\n\nJust need to get this jerkwad with pants down to his knees to do something out of the goodness of his heart.\n\nThey're in the airport now. Girls are thronging to meet Jonah. His bodyguards and the airport police are trying to hold them back. I wonder if they purchased tickets just to get through security, or if this many random fans are just wherever he is. Allegra is trying to interact with them, but most want nothing of her, only him. And he's ignoring them, talking to the balding guy, one hand firmly on his pants.\n\n_Please just say where you're staying before Drunk Mom comes back and I have to put away the mirror._\n\n\"Will the hotel be like this too, Harry? 'Cause I'm planning on breaking up with . . .\" He nudges his head toward Allegra, who is hugging a little girl. \". . . so it would be nice to have some privacy in case she screams her head off.\"\n\nThe guy\u2014Harry\u2014shrugs. \"I don't think there are any leaks, but these girls tend to find out.\"\n\n\"They'd better not,\" Jonah says. \"Get extra security.\"\n\n\"I'll alert the Cornwallis.\"\n\nBingo. The Cornwallis is a fancy non-Disney hotel on Disney property.\n\n\"The Cornwallis?\" Jonah shoves past the fans. \"Nice British name. Will it be private?\"\n\n\"We have a floor all to ourselves, you, Allegra, your staff.\"\n\n_What floor? What floor?_\n\n\"What floor?\" Jonah asks.\n\n\"The top floor,\" Harry says. \"The floor below is a health club. People make all sorts of noise. She can scream as much as she likes.\"\n\n\"Good. I need to be rid of her. The stress is affecting my voice.\"\n\n\"Wouldn't want that,\" Harry says just as a fan breaks through and touches Jonah. His bodyguard pushes her away.\n\nA few of the kids who didn't go to the lounge run past me. A blond boy is holding a handheld game, and a redhead tries to get it back. Just as they get to me, the redhead tackles the blond, barreling into my legs and almost knocking the mirror from my hands.\n\n\"Hey, watch it,\" I say.\n\n\"Sorry.\" The redhead stares at me. \"Hey, how old are you?\"\n\n\"Seventeen?\" I say. \"You?\"\n\nHe ignores my question. \"So you're, like . . . ?\"\n\n\"A smaller-sized adult? Yes.\"\n\n\"Cool. I thought maybe you were from one of the other troops.\"\n\n\"How many troops are there?\"\n\n\"Three, I think.\"\n\n\"Come on,\" the blond kid says.\n\n\"Okay.\" And they run off.\n\nI notice a blue baseball cap lying under my feet. I pick it up and start to call after the redhead who lost it. But they're gone.\n\nI have an idea.\n\nIt's one when the train pulls into the Orlando station ahead of schedule. Thirty little boys in thirty blue T-shirts and twenty-nine blue hats are herded as neatly as you'd expect out of the car. I follow them to their bus and put on the thirtieth hat. I make sure to move around, not stand by anyone in particular. I pull the hat low. I'm guessing no mom here knows all thirty kids.\n\n\"Hey, you!\" One of the moms grabs my shoulder.\n\nI freeze.\n\n\"Stay with the group.\" She pushes me forward into the line.\n\nI follow the others onto the bus and sit in back. I keep my head down.\n\nIt's not a short ride, but I know if I make it, I'll be on Disney property. I can use their transportation to get to the Cornwallis.\n\nOf course, getting to Jonah himself will be another story.\n\nWhen we reach the All-Star Sports Resort, I find the red-haired kid and hand him his hat. \"Hey, you dropped this.\"\n\nThen, I run in the opposite direction.\n\n#\n\n#\n\nAn hour later, through a maze of transportation, I have gone from the All-Star to the Magic Kingdom, the Magic Kingdom to the Cornwallis Hotel's own shuttle. I let myself close my eyes on this one. I picture Celine, the way she looked that day we went to Target, smiling at me. I want her to smile again, even if it's not at me.\n\n\"That boy fell asleep,\" the kid next to me says.\n\n\"He must have had a fun day,\" his mother replies.\n\n_And it's not over._ I open my eyes, and we're pulling into the Cornwallis Hotel. Parking attendants dressed like Cinderella's footmen, with powdered wigs and purple hats with giant plumes greet the shuttle. I step out, and they bow with a flourish.\n\nI glance at my watch. Two-thirty.\n\nSo far this was easy, too easy.\n\nI'm sure it will get harder.\n\nI take the elevator to the twentieth floor. There's no lock, no restriction. But when I get off, a guy who looks like an extra on _Game of Thrones\u2014_ two feet taller than I am with a scar on his cheek that looks like it was made by a sword\u2014comes at me.\n\n\"What are you doing here?\" he growls.\n\n\"Just heading for my room. Twenty-fifteen.\" That's Harry's room. I heard them discussing it one of the ten times I checked the mirror while on the bus.\n\n\"Wrong!\" the guy growls. \"No one on this floor but Jonah Prince's people.\"\n\n\"I know. I'm with him. Harry, his manager's my dad.\" I start forward. One good thing about being my size. You can get away with stuff because you look nonthreatening.\n\n\"Hey, Otto,\" the guy holding me says. \"You remember Harry having a kid who looks like this?\"\n\nHe grabs my waistband and drags me over to another, even bigger guy. The disrespect is mind-boggling. He lifts me up. I almost have a clear shot at kicking the guy in the nose, but I'm thinking that might not be a wise choice. Instead, I yell, \"Put me down!\"\n\n\"Pretty sure I'd remember him,\" the bruiser says.\n\nOf course, the bad thing about being my size is, I'm also very memorable. I had one shot to walk off the elevator, and I blew it.\n\nThe guy turns around and shoves me into the elevator. \"Bye-bye, little guy.\"\n\nSighing, I press the button to go down to the eighteenth floor.\n\nTime for another plan. Or a plan at all. Since I didn't know which hotel he'd be at, I hadn't been able to research the layout or anything. I'd thought maybe luck would be on my side. Like I could go up with a housekeeper. But even the minute I spent on Jonah's floor let me know there was no housekeeper there. They probably cleaned his floor first, so he could check in early.\n\nThe housekeeper on this floor is still working. A door stands open. I hear vacuuming. I wonder if I can talk her into sneaking me upstairs. Doubtful. I try the stairs, but the door to the twentieth floor is locked, so I come back down. The hallway is empty.\n\nI pull out the mirror.\n\n\"Show me Jonah,\" I say.\n\nThe mirror does. He's in a fancy room, talking on his phone, alone.\n\n\"I'm seventeen, and I have a hundred million dollars. I don't need you telling me what to do.\" He holds the phone away from his ear to avoid what I'm sure are her shrieks. \"Yes, I suppose I do want to break up. After tonight. Let's just get this tour over with. You're still coming to the concert, right? What? What?\"\n\nHe paces back and forth, tugging at his pants even though no one's there to see him. \"Is that a threat?\" He nods. \"All right, then. Call my mum. She's in New York, and she won't care, but go ahead and call her. Oh, stop crying. You know you never really liked me. It was all a publicity stunt. What?\" Again, he pulls the phone away. \"Just come to the concert tonight. We leave at six-thirty.\"\n\nHe hangs up. Then, he walks to the window and stares out, shaking his head. \"Call a guy's mum, why don't you?\" He must notice a crowd outside because he backs away and flops down onto the bed.\n\n\"Show me Allegra,\" I tell the mirror.\n\nIt does, switching to a similar-looking room. Allegra is collapsed on the bed, phone in hand, crying, saying, \"He broke up with me. He's so out of control! He's not the guy I used to know. I had to stop him from giving these little girls the finger.\" She stops talking and sobs. \"I know. You're right. Thanks.\"\n\nShe puts down the phone and sobs some more. I feel bad, watching something so private. Still, I don't know what else to do. Also, I feel like I know this girl, having watched a gazillion hours of _What a Girl Needs_ with Isabella.\n\n\"You're better off without him, sweetheart,\" I say aloud but, of course, she doesn't hear me. A minute later, she sits up and looks at the phone on her nightstand. She crawls across the bed and looks at it, reading the instructions, then picks it up and dials.\n\n\"Hey.\" She sniffles. \"How do I get room service?\"\n\nA few seconds later, she says, \"Hello, this is, um, Mrs. Kendall, room 2016. Can you send up, um, a bottle of wine . . . white wine . . . what _kind_? Um, I don't know. Whatever you recommend. Oh, and a hot fudge sundae.\"\n\nThis gives me an idea. I tell the mirror, \"Show me the room service person she's talking to.\"\n\nAs the mirror melds into the person she's talking to, I start toward the elevator.\n\n\"Yes.\" It's a girl with dyed-black hair and a white apron over black shirt and pants. She rolls her eyes. \"I understand it's an emergency. I'll send up the 2008 Didier Dagueneau Silex right away . . . and the sundae . . . my name. It's Kasey. With a K. Yes, I understand. It will be there in twenty minutes.\"\n\nTwenty minutes. I have twenty minutes to work if I'm going to be part of Allegra's room service order.\n\n\"Isn't that a clich\u00e9,\" Kasey says when she gets off the phone. \"Little teenybopper star wants to get drunk. And stupid room service is going to buy that it's her 'mother' calling.\"\n\nThe second the elevator hits the ground floor, I'm off and running toward the restaurant. I figure the kitchen's probably behind there. It's a dead time of day, after lunch, but way before dinner. When I stroll in, there's only one woman manning the ma\u00eetre d' station. \"May I help you?\"\n\n\"I'm looking for my sister. She works here. Kasey? The kitchen's through there?\"\n\nShe nods and goes back to the game she was playing on her phone.\n\nI keep my back against the wall as I enter the kitchen. The room has an island with high, stainless steel counters. I hide under one near a wheeled room service trolley that is already topped with a bottle of white wine. Kasey's standing by it. A tall red-haired girl is bringing over wineglasses. \"How many?\"\n\n\"Just one, I suppose.\" Kasey makes a tsk-ing sound with her tongue. \"Poor little Allegra, drinking alone.\"\n\n\"And ice cream,\" says the second girl. \"She got dumped for sure.\"\n\nI peek over the counter. Kasey snaps a photo of the wine bottle. \"Got dumped, and now, she'll get drunk.\" She takes a business card from her apron pocket. \"This guy, he's a paparazzi . . . paparazzo . . . photographer; he told me to call with any tips. Said he'd make it worth my while. He'll love hearing about Allegra getting faced.\"\n\nI feel a twinge for this girl I don't know. Well, like I said, Isabella loves her. I notice the trolley's about the width of the TV in my parents' bedroom, with a tablecloth over it. I could fit under it if I could get there. Then, if they pushed me up to Allegra's room, I'd at least be on Jonah's floor. Maybe they even have adjoining rooms. But right now, both girls are standing by the trolley. If only one would move.\n\n\"That's really mean,\" the tall one says. \"Kick her when she's down.\"\n\n\"You have such a soft heart, Caitlin. How 'down' can she be, spoiled little starlet.\"\n\n\"My kid sister loves her. She'll be so disappointed. Why do they always turn out to be crazy, drunks, or sluts?\"\n\nKasey shrugs. \"Not my fault. Hey, make that sundae. Her majesty said she needed it quick.\"\n\nCaitlin moves away. Kasey takes out her phone, starts to call someone. I make my move, sliding under the counter, which requires some knee bending, even for me. I get into the trolley, which requires more. It's tight. I'm short, but I have shoulders, and I'm not a contortionist who can dislocate them. At one point, the trolley starts to move. I hear the wineglass rattle against the cooler. Kasey doesn't notice. She's too busy talking.\n\n\". . . leaving in a couple hours. She should be stumbling drunk by then.\"\n\nFinally, I stuff myself in, sitting on my feet, shoulders gathered around my ears. My left side aches bad, and I wonder if I dislocated my shoulder after all. In any case, I can't move except to give the tablecloth a tug. I hope Caitlin will hurry.\n\n\"There we go.\" I feel the thunk of a huge sundae above my head. \"With extra whipped cream. I'm sort of excited to see her. Wonder if she'd let me take a picture.\"\n\n\"They'll kill you if you ask. Have some dignity.\"\n\nCaitlin pushes the trolley. It barely budges. \"Wow, I need to start working out.\" She gives it a big shove, and we're on our way.\n\nShe drives me through the kitchen, where Caitlin manages to ram into every possible counter, and into the dining room, which is carpeted, so the thing will barely roll with my weight on it. Each bump and thump makes my shoulder ache worse, and I grit my teeth to keep from crying out. Finally, finally, we hit a marble floor.\n\nIt's smooth skating for about twenty feet. Then, we reach the elevator, which is another death struggle as we get over the gap.\n\nAnd off to the top floor. So I have a solid minute to sit and think about how much my shoulder hurts, how both feet are already asleep, how I'd love to clear my throat, scratch my head, burp, fart, or all four.\n\nI try to remember that tomorrow, this will all be over. I'll have succeeded or failed, but hopefully succeeded.\n\nAnd, if I succeed, Jonah will wake Celine. Jonah Prince is as close to a prince as we have, and she loves him.\n\nSo they can fall in love, and she can live happily ever after.\n\nWith that asshole.\n\nI really need to scratch my nose. Why am I doing this again? Oh, yeah. Love. True love. Unselfish love.\n\nAnd a fascinating topic for college essays about an obstacle I've overcome.\n\nIf I don't get arrested and can still apply to college.\n\nFinally, finally, the elevator reaches the top floor.\n\nI breathe out, though I have no idea why I'm relieved. It's not like I can get out and walk around now. Rather, I'm going to have to wait until Allegra passes out or leaves or, at least gets drunk enough not to notice a guy walking around her room. Depending on her drinking habits, that could take awhile.\n\nI could come out of this experience a hunchback. Another good college essay topic.\n\nI really want to crack my neck.\n\nCaitlin wheels the trolley down the hall. I hear her explaining to the bodyguards who she is. They let her through. Finally, she knocks on the door.\n\n\"Room service!\"\n\nThe door opens. \"Thank God you're here.\" Allegra's voice.\n\nSilence.\n\n\"Well, aren't you going to bring it in?\" Allegra asks.\n\n\"Oh! Sorry.\" Caitlin pushes against the trolley. \"I was just sort of . . .\"\n\n\"Slow?\" Allegra says.\n\n\"Awestruck. My little sister watches your show all the time. She won't believe I met you.\"\n\nAllegra seems to gather herself. \"I'm sorry. That's great! Can you bring the tray in now?\"\n\n\"Oh!\" Caitlin gives the trolley a shove, and I have to grip the bottom to keep from falling out. \"Of course. I'm sorry.\"\n\n\"It's okay. Maybe set it by the bed.\"\n\nBy the bed. That's good because maybe I could sneak out and get under the bed.\n\nIn an hour or so.\n\nCaitlin pushes the trolley across the marble floor for a long time, turning to get through a door or something. Finally, it crashes into the bed.\n\n\"Really, I'm sorry,\" Allegra says. \"That's sweet that your little sister likes the show so much. Would you like a picture so she'll believe you met me?\"\n\n\"Omigod! Omigod! Omigod! Really? That would be so awesome. I can get a selfie of us together. Thank you sooo much!\"\n\n\"No problem.\" Allegra sounds happier. \"It's great to meet a fan.\"\n\n\"You have so many fans,\" Caitlin gushes.\n\n\"Obviously not enough for Jonah,\" Allegra mutters.\n\n\"What?\"\n\n\"Oh, nothing,\" Allegra says. \"Do you have a phone with you? Okay.\"\n\nI wish Caitlin would leave so there'd be one less person to notice if I tumble out of this thing. My legs ache, and so does my stomach. I had a burger on the train. Obviously, a mistake. What if I . . . experience gastric distress? Just thinking about it makes my stomach hurt worse. Actually, there's no part of my body that doesn't ache. My arms, shoulders, even my head are pulsing, pulsing, pulsing.\n\nI think of Celine, comatose, feeling nothing. I have to hold it together.\n\nLast night, I printed out some photos of Celine, the one I took at Target, one from the house, and one of her in the hospital. I did it since I wouldn't have my phone, so I could prove her existence to Jonah. They're still in my backpack, flattened against my back. I wish I could look at them, see her face. Maybe it would inspire me. I close my eyes and visualize her. I don't need the photo. Her blue eyes stare into my soul, and I see every detail, every eyelash, every freckle, every blush. Every time she smiled at me.\n\nIt works. I relax and can sit there on my sleeping foot while Caitlin snaps a dozen selfies and Allegra coos about Caitlin's sister, asking how old she is and if she wants an autographed eight by ten. It takes at least ten minutes before Caitlin says, \"Omigod! Your ice cream's melting! I'm sooo sorry. Should I bring you a new one?\"\n\n\"That's okay.\" Allegra's feeling better, it seems. \"But I guess I should eat it now.\"\n\n\"Of course. Sorry. Thank you. I'll get going. You're the best!\" She stumbles toward the door, and Allegra is left alone.\n\nAlmost alone.\n\nI hear her taking the cover off her ice cream, then clinking the spoon. The trolley shakes as she digs into her sundae. Then, she bursts into tears.\n\n\"I hate you!\" she screams. \"I hate you, Jonah Prince! I hate you!\"\n\nWith my unique view of the floor, I see her remove first one strappy sandal, then the other. I hear her hurl them against the wall.\n\nI try not to move, to be very Zen-like, meditative, enlightened, planning my course of action. I'm in Allegra's room. After Allegra either leaves or passes out, I'll take out the mirror and contact Kendra. Hopefully, she'll have figured out the relation between Allegra's room and Jonah's. Hopefully. Hopefully, they're adjoining with a connecting door. If so, I can simply walk into Jonah's room and talk to him when he gets back.\n\nAnd hopefully, he won't be a complete douche.\n\nMy neck hurts so much. So much. I try to adjust it.\n\nAnd tumble to the floor.\n\nFor a second, time stops. I lie there, dazed, thinking\u2014I don't know\u2014thinking maybe if I don't move at all, Allegra won't notice a guy lying on her floor.\n\nBut no, I can see her eyes. She knows I'm here.\n\nAnd she's winding up for a good scream.\n\n#\n\n#\n\n\"No, please!\" I stumble to my feet. \"Please don't be scared!\"\n\nIt's hard to stand with both feet asleep, and for a second, I think I'll fall. Allegra gapes at me, mouth open but no words coming from it.\n\nI say, \"I am your _biggest_ fan. I just had to see you in person. I'd never hurt you.\" I stare at her like I'm starstruck. \"Wow, you're so beautiful!\"\n\nShe really is. I see her decide not to scream. She closes her mouth, then opens it right back up again.\n\n\"You're . . . little.\"\n\n_You're . . . observant._\n\nBut I say, \"Yeah.\"\n\n\"Sorry. My brother's about your size, my older brother. He's away at college.\"\n\nSomewhere in the far recesses of my mind, I think I knew this, the way you just know things about famous people without actually knowing where you know it from. Probably Isabella. Kid is a storehouse of completely worthless information.\n\nOr maybe not that worthless.\n\nI say, \"Do you miss him?\"\n\nAnd her face sort of breaks. She starts to cry, dark hair falling over her face. \"I miss him so much. Caleb's his name. He goes to school in Louisiana, and he's always saying I should just . . . chuck this and come home. I miss him and my mom and dad. I miss everyone. I want to go home!\"\n\nOn shaky legs, I approach her. What is it about me that makes women burst into tears in my presence? Still, better than screaming. Being from Miami, I'm a hugger. But that's out of the question. I keep my distance as much as possible, holding out my hand.\n\nShe reaches for it. \"What's your name?\"\n\n\"People call me Goose.\"\n\nShe shakes her hair away from her face. Her eyelashes are moist, with a bit of mascara under her eyes. \"I'm Allegra. I guess you knew that. I'm so lonely I kept that room service girl here until my ice cream melted. I have no one to talk to, no one but Jonah, and he h-hates me.\"\n\n\"I'll talk to you.\" She still hasn't let go of my hand, so I squeeze hers. \"Talking's my superpower.\"\n\nShe smiles. \"My brother's like that too.\" Finally, she drops my hand. \"You're positively not going to kill me, right?\"\n\nI shake my head. \"If I was, wouldn't I have done it already?\"\n\n\"I guess.\" She inhales a huge load of snot. \"I could use a tissue.\"\n\nI wonder if she usually has someone wipe her nose for her. \"Let me get one. Which way's the bathroom?\" Her room is huge, and it connects to another room, a living room, which is why it took Caitlin so long to push the trolley through. There's a balcony on one end of the room, but no connecting door. I glance into the living room. There's another balcony, an entry door, and yes, a closed door\u2014maybe it connects to Jonah's room. I head into the bathroom and grab a wad of tissues. I hand them to Allegra. \"Cry away, milady,\" I say in my Cockney accent. \"Sorry, I was in the school play, _Oliver!_ , and we talked like that.\"\n\nAllegra sniffles into a second and third tissue. Her face is red and puffy. \"I'd love to be in a school play. It sounds so . . . normal.\"\n\n\"Most people in my school play would love to be on a TV series.\"\n\n\"That's because they don't know what it's like to have your career be over at seventeen.\"\n\n\"You mean just because your show went off the air? You'll get another.\"\n\n\"No, I won't. People hate me.\" She picks up tissue number four.\n\n\"The girl who delivered your room service didn't hate you. She said she was a fan.\"\n\n\"No, she said her little sister's a fan. People watched my show when they were younger, but then they outgrew it, so they hate me. Do you know there are 'I hate Allegra Kendall' Facebooks, Twitters, Instagrams, and Tumblrs? One of the Twitters is called DieAllegraDie. People post about what a whore I am, or how I can't act. If I wear a bikini, someone takes a picture and says I'm a slut. If I wear a big sweatshirt, they say I'm getting fat. I optioned this book I liked, because I wanted to make it into a movie, and people said I was doing it because I had to pay to get roles. They hate that I date Jonah, so it should make them happy that he dumped me.\"\n\n\"He's kind of a jerk, isn't he?\"\n\n\"Yes!\" Her eyes widen, and she nods. \"He's a total jerk. Cute only gets you so far. He's rude. He loogeyed on his fans once. More than once. But they love him and hate me. They're all just waiting for me to get drunk and do a sex tape. My agent thinks that wouldn't be a bad thing, prove I'm not a baby anymore.\"\n\nI eye the bottle of wine which, I notice, is open but still full. \"Is that what you think too?\"\n\n\"Yes. I mean, no. I mean, I don't know why I ordered it. I guess because my boyfriend broke up with me, and you're supposed to get drunk. But I don't really like drinking. I like being in control of myself.\" She looks at me, like she wants my opinion.\n\n\"I think that's smart.\" This goes against my plan to wait until she passed out. But I guess that went out the window when she saw me anyway. Besides, I feel bad for her. I remember the room service girl, Kasey. If Allegra gets drunk to go out with Jonah, the paparazzi will take pictures and make her look bad. I can sneak into Jonah's room after Allegra leaves\u2014if she trusts me.\n\nAnd I'm trustworthy. I say, \"If you get drunk and the press gets photos, won't it hurt your chances of getting into college?\" She stares at me like I'm crazy. \"Sorry. My mom's always worried about college.\"\n\nShe laughs. \"Yeah, college. What a joke. I could never go to college.\" She takes the last tissue.\n\n\"Why not? Lots of kid stars go to college. Jodie Foster was a kid star, and she went to Yale. Natalie Portman went to Harvard. Emma Watson started at Brown, but she quit because she was working too much.\" More stuff I know without knowing _how_ I know.\n\n\"High-class problems.\" She sniffles deeply.\n\n\"Let me get you more tissues.\" I go back to the bathroom and, this time, remove the whole Kleenex box from its holder and hand it to her.\n\n\"Thanks. You are so nice.\" She blows her nose. \"I could never get into a college like that. I'm not smart enough.\" She starts picking up the pile of used tissues.\n\n\"I bet you are. Or just go to a normal college. Where's your brother go?\"\n\n\"Loyola.\" She sighs and tries to stuff the tissues into her pocket. \"Back home. God, I'd love to go to school with Caleb. I miss him so much.\"\n\n\"You should go. I bet you have a ton of money saved up. Then, if you still wanted to act, you could come back afterward, when you don't have to get drunk or pose nude to prove you're a grown-up.\"\n\n\"I would never pose nude,\" she says. \"But you're right.\"\n\nWe talk like that for maybe an hour, maybe more. I tell her about the classes I take in school and how it's my ambition to play Boq in _Wicked_ on Broadway. She tells me about her family, the TV show, and how everyone pushed her to date Jonah, to be part of Jollegra. But then, the teen girls hated her even more because she \"took him\" from them. \"I really barely know him. We never once talked the way I'm talking to you right now. I have no friends. This is the longest conversation I've had in a year, and you're some guy who snuck into my hotel room.\"\n\nI shrug. \"Yeah, sorry about that.\"\n\n\"No, no. Don't be. I'm glad you did. You kept me from doing something really stupid.\" She glances at the clock. \"Oh, gosh, I have to get out of here soon. Command performance with his highness. I bet I'm a mess.\"\n\n\"No, you're beautiful. Your eye shadow's just smeared. Here, let me help you.\"\n\nI get the makeup I saw in the bathroom and help her clean up. Then, we tackle what to do with the wine, and with me. She's worried that, if someone knows about the wine, they'll think she drank it, and if they know she has a guy in her room, they'll say she's a slut. \"I haven't done one slutty thing ever,\" she says. \"They don't care.\"\n\n\"Don't worry about me. I'll sneak out after you leave.\"\n\n\"Are you sure?\"\n\n\"I snuck in. And maybe you can give the wine to the maid, like a tip.\"\n\nSo we write a note to the maid and put it on the wine. Allegra takes a picture of the bottle with her phone, to prove it.\n\n\"Do you want a picture of us?\" she asks me.\n\n\"Huh?\" Then, I remember, I'm supposed to be her biggest fan. Also, if my parents don't murder me, I could show it to Isabella. \"I forgot my phone. Can you take one with yours and send it to me?\"\n\n\"Good idea. Then, I'd have your phone number, and you'd have mine. We could talk sometimes. You could tell me about high school.\" She crouches down beside me to snap the photo.\n\n\"And you could tell _me_ about college, when you go back to Louisiana.\"\n\nShe squeezes my shoulder and takes another shot. \"I think I might actually do that. I mean, yeah, the press will assume I'm pregnant, but I'll know the truth.\"\n\n\"So will I.\"\n\nShe looks at the photos. \"It would be so cool to be friends with you.\"\n\n\"That's me, every girl's BFF.\"\n\n_And no one's true love._\n\nShe sends the photo to the number I give her. Just as she does, there's a knock on the door.\n\n\"That's Jonah. Gotta go.\" She looks at me. \"Are you sure you're okay?\"\n\nI nod. \"I'll think of a way out.\" I already have.\n\nShe leans and kisses me on the cheek. \"You're an incredible person, Goose. You've really opened my eyes.\"\n\n\"I'm glad.\"\n\nI hide behind the curtains as she opens the door. I hear Jonah's voice. \"So, I suppose we're doing this,\" he says in his British accent.\n\n\"I suppose we are,\" Allegra says. \"Look, I'm sorry about before.\"\n\n\"I don't want to get back together.\"\n\n\"No, neither do I.\" Her voice is calm, even happy. \"I just want to get along and stop fighting and everything.\"\n\nA pause. \"Sounds good. Shall we?\"\n\n\"We shall.\"\n\nI hear the door close behind Allegra. I wait a minute, then two, before I move from my spot, in case anyone comes back. No one does, so I check out the living room.\n\nThe connecting door is locked from the other side. If Jonah's room is on either side of Allegra's, there's only one way to get to it.\n\nClimb over the balcony.\n\n#\n\n#\n\nDid I mention I'm afraid of heights?\n\nAfter Allegra leaves, I contact Kendra through the mirror.\n\n\"You're still in Allegra's room?\" She's wearing a black lace mantilla, but otherwise, she looks like Violet.\n\n\"Yeah, how did you know?\"\n\n\"Hmm, magic. That's how I know you're in room 2016, and Jonah was next door in 2014, so you only have to get one room over.\"\n\n\"Yeah. Only.\" I look out the glass doors. It's so high, I can see Cinderella's castle in the Magic Kingdom. \"And you still can't magically zap me there?\"\n\n\"Still, no.\"\n\n\"Really? What's the good of having magic powers?\"\n\nKendra grimaces. \"I often wonder that myself.\"\n\nI hang up or sign off or whatever you do when you talk to someone in a magic mirror. I ask the mirror to show me Jonah.\n\nHe's in a limo, sitting with Allegra, but pretty far apart. She's trying to talk to him. \"I was thinking about going back to Louisiana.\"\n\nHe says, \"That's nice.\" He smiles at his phone, takes a selfie, then stares at it. He picks his teeth with his pinky.\n\nTerrific. They are well and truly broken up. That means when he meets Celine, if she does wake up, they can fall in love. He'll be available. And he's such a douche. I'm going through all this so a girl I like can ride into the sunset with a douche.\n\nMaybe he won't like her.\n\nAny guy with eyes would like her.\n\nI visualize Celine's smile. Of course he'll like her.\n\nI decide to wait in Allegra's room for a while. The concert's at seven, so Jonah won't be back until at least nine. Maybe if I climb over the balcony after dark, I won't be able to see the ground below.\n\nI look in the mirror. \"Show me Celine.\"\n\nIn the mirror, I see her, dark hair fanned out over the crisp, white pillow, her heart-shaped mouth curled into a tiny smile. Is she dreaming? Does she know I'm gone?\n\nI remember what Violet said about letting her live forever, comatose. Would I rather not know if she could never love me than lose her?\n\nNo. I want her to be happy. She's been through so much, losing her mother, her dad. She deserves to be happy, even if it's not with me.\n\n\"I love you,\" I tell the mirror.\n\nCeline doesn't respond. Of course, she couldn't hear me even if she was awake.\n\nI stare at her another minute.\n\nI decide to check out\u2014just check out\u2014the situation on the balcony. I pick up my backpack, open the sliding glass door, and head outside.\n\nThe balcony is a large one, spanning both the living and bedroom areas of Allegra's suite. At its front is a white aluminum railing. I don't know much about construction, but it seems pretty flimsy, with vertical rails about six inches apart and some ornamental scrollwork on top. I edge out, remembering the part in season one of _Game of Thrones_ where Tyrion's imprisoned in the sky cells, these dungeons on the side of a sheer cliff with no wall on one side and an abrupt thousand-foot drop. At least I have a railing. At least I'll only fall into the pool area and have the remote possibility of just being a quadriplegic.\n\nExcept\u2014oh, right\u2014Tyrion was _fictional._\n\nStill, I have to look. I walk to the side of the balcony closest to Jonah's room. It's not a balcony that bumps out. I won't have to scale a wall. Allegra's balcony and Jonah's are only inches apart.\n\nFor an average-height guy, it would be a no-brainer, just lift himself over the railing (which reaches my chest but would reach someone else's hip at least), swing his leg over the other railing, and drop down.\n\nDown.\n\nI picture myself, swan-diving off the railing, splattering on the pavement.\n\n_Florida Man plummets from hotel balcony._\n\nMy parents will hear about it on the news.\n\nMy mother will know she was right.\n\nAnd Celine will stay in a coma.\n\nI look across again, aware of my breathing, which is crazy-hard, hard enough that it sounds like a car with a busted muffler, and my teeth feel like they're buzzing.\n\nOkay, I need to go back inside, just for a while, a second. Slowly, carefully, like Tyrion in the sky cells, I edge toward the door.\n\nShit, there's someone in Allegra's room. The maid.\n\nI rush to the corner of the balcony, sit down, and hide.\n\nThe good thing about being my size is it's easy to hide.\n\nThe bad thing is, I feel like I could fit through the railings.\n\nI can't. They made it so kids can't fit through. Isabella couldn't. I'm bigger than a kid.\n\nThat doesn't change my racing thoughts and heart. If I could just be reasonable, I wouldn't have this fear. But that's not how it works.\n\nI close my eyes and picture myself, over and over, splattering to the ground. Even the solid stucco wall behind me doesn't help.\n\nI want my mother.\n\nNo. No. I don't. I'm here. I want to do this. Put on your big boy pants, Guzman. You can do this.\n\nOnce again, I take out the mirror.\n\n\"Show me Celine.\"\n\nThe mirror shows me her room again, her face. Kendra, disguised as Violet, sits in the visitor's chair while an elderly nurse checks Celine's chart. I watch until the nurse walks out.\n\nDesperate _not_ to look down, I ask the mirror to show me Stacey.\n\nShe's at home, in our kitchen. God, I wish I was in our kitchen. She's cleaning up dinner dishes. She hasn't noticed I'm gone yet. At least, she hasn't started worrying. I check my watch. Seven-thirty. Half an hour into the concert.\n\n\"Show me Jonah.\"\n\nHe's in his dressing room. I can hear the opening band in the background. He looks stoned, and he's meeting some fans who are posing for cell phone pictures.\n\n\"Show me Violet.\"\n\nThe mirror shows me the same elderly nurse who was in Celine's room.\n\n\"No, show me Violet.\"\n\nStill, the nurse.\n\nI get it. The nurse is Violet. But Violet promised to leave Celine alone.\n\nViolet lied.\n\nOf course Violet lied. Violet is at the hospital, disguised as a nurse. With access to drugs, access to needles.\n\nAnd access to Celine.\n\nI look at her name tag as she walks out. It says, _Lavinia Barnes, RN_.\n\n\"Show me Kendra.\"\n\n\"I still can't zap you into Jonah's room.\" Kendra still looks like Violet. This is really confusing.\n\n\"Listen! Violet's in the hospital.\" I tell her about Nurse Barnes, what I saw. \"Please don't let her hurt Celine, and maybe . . .\"\n\n\"What?\"\n\n\"Maybe get my parents, my dad. See if there's something they can do to protect her.\"\n\nHer face, Violet's beautiful, horrible face, shows confusion. \"But if I go to your parents, I can't stay with Celine.\"\n\n\"I was thinking maybe you could use the phone.\"\n\nShe rolls her eyes. \"Ohhh, the phone. Of course.\" She turns and looks at the old-fashioned beige phone on the bedside table. \"How does it work?\"\n\nGeez. I walk her through it. Several times. Finally, she gets it. She says she'll call.\n\nI've been trying not to look around too much, not to look _down._ Instead, I look out. It's after eight now. The sun is sinking into the orange pool of the sky. The mirror shows me Jonah, singing onstage. It's almost time.\n\nI stare through the glass door. The maid is still in Allegra's room. She's drinking the wine Allegra left. What if she sees me? I walk to the edge of the balcony. Even though it's June and hot, I suddenly feel chilled to the bone. My legs, my arms, my hands are shaking, teeth chattering.\n\nAs I reach the very edge of the balcony, my hands shake harder. The mirror slips from my grip.\n\nI hear it shatter to the ground, twenty floors below.\n\nShit.\n\n#\n\n#\n\nShit. Shit. Shit.\n\nI dive to the floor in case someone below saw the mirror (how could they not?), guessed where it came from.\n\nI look through the bars, down, twenty floors below. The ground swims up to my eyes, and I feel like a cartoon coyote.\n\nOn the floor as I am, I'm not worried about falling. That won't happen. I won't fall off the balcony. Even I know I can't fall from the floor.\n\nI'm worried about the entire balcony falling, detaching from the building and crashing to the ground\u2014but only after it hits every single balcony in between. How is it even attached to the building? Screws? Concrete? Who put it up there? Were they drunk? Disgruntled? Insane? I was able to push these thoughts from my head when I could look at the mirror or talk to Kendra. Now, with nothing to drown out the noise in my head, all I can think of is the screws that hold the railing up and when was the last time they were inspected?\n\nProbably never.\n\nI have no idea whether Celine is safe from Violet or when Jonah will be back. I know nothing.\n\nThe good news is, I hear nothing from below. Hopefully, they'll think some kid at the pool broke the mirror.\n\nBecause kids are always running around with antique sterling silver mirrors.\n\nThe bad news is, it's almost dark.\n\nOr maybe that's the good news.\n\nI reach for my backpack with a shaking hand. I fumble for the one distraction I have left. Celine's picture. I take it out and stare at it in the waning light.\n\nThe day it was taken was the first day I'd told myself, screw it, I'm never going to _not_ love this girl, the day I knew it would never work with Willow or any other girl because I would never stop thinking about Celine.\n\nShe'd been so beautiful that day, in a sweater the color of iceberg lettuce that perfectly set off her pale skin and dark hair. They'd been practicing the scene before Oliver and Dodger meet, the scene where Oliver is bullied by Noah Claypool. This guy, Tedder Strasky, was playing Noah, which was perfect casting because Tedder's a serious bully, like the kind that puts guys' heads into toilets (not mine, but still . . .). Celine had been playing Oliver halfheartedly. Acting wasn't really her thing. But when Strasky said his line about Oliver's mother being \"a real bad 'un,\" everything changed.\n\nOliver's supposed to attack Noah, and considering Strasky is about twice Celine's size it should have taken an impressive amount of stage combat to make it work. But as soon as Strasky said his line, Celine stood, launched herself at him, and practically pushed him off the stage. For a second, it was so real. I knew.\n\nI knew, whatever happened, I wanted that girl on my side. She was a fighter.\n\nJust like that day in biology class.\n\nNow, she's in bed, maybe dying.\n\nI know if our positions were reversed and it was Celine in a situation where she could save me, she would not be cowering on the floor. She would not be worrying about how the balcony was screwed in.\n\nShe'd be fighting her way into Jonah's room.\n\nAnd that's what I'm going to do.\n\nWhat I am doing.\n\nI pull myself up on the railing. I feel it wobble a little, hear it creak. I drop back down.\n\nNo, it didn't wobble. It's solid.\n\nI'm solid.\n\nI take a deep breath. Okay, I take five. I'm doing this. I don't look down. I can barely see.\n\nThe distance between the two balconies is less than a foot. All I have to do is climb up on one, then down the other. All I have to do.\n\nAnd if it falls off and crashes to the ground, I will just die. That's all.\n\nI read a book once, about auditioning. It said that you could combat nerves by imagining the worst-case scenario. Like, you don't get the part, so you have no money so you starve and die. Death is the worst-case scenario. Some comfort.\n\nBut the worst-case scenario right now is that I _don't_ do it and Celine stays in that cold, gray place for the next fifty years.\n\nWorse than death.\n\nDeath, I'll risk.\n\nI stuff the photo into my backpack, zip it, and throw it onto Jonah's balcony.\n\nI take one last breath and hold it.\n\nI pull myself up on the railing and over.\n\nMaybe it's adrenaline rush that lets mothers lift cars off their infants. I pull up first one leg, then the other. I'm on top of the railing like I'm Spider-Man.\n\nOkay, not exactly like Spider-Man, but pretty good. It holds. It's not crashing to the ground. But the whole thing is like slow motion, like I've been here for an hour, and just as my foot is searching for the other balcony, the sky lights up with an explosion of red and gold.\n\nFireworks. Disney fireworks, which means there are a lot of them. Fifteen minutes at least.\n\nI freeze. The balcony, the building, the entire city is shaking with explosions, first from one side, then another.\n\n_Think of Celine. Be brave like Celine. For Celine._\n\nMy foot finds the other rail. I don't want to move. I want to stay, hug the railing forever.\n\nNo. I want to land on the other side, save Celine.\n\nThe fireworks explode like bombs bursting in air all around me. I wonder if this is how it feels to be in a war, like in the 1940s in London or Berlin.\n\nThe balcony's trembling, and I'm trembling with it. I picture Celine on the other side, arms outstretched, beckoning to me, telling me that loving her isn't a crazy idea.\n\nMy feet hit the ground.\n\nI crumple to the floor.\n\n\"Hey, what are you doing here?\"\n\n#\n\n#\n\nOkay, so I know I'm small. And being small sort of skews your perspective. So, possibly, you might see someone who's just a little tall and think they look like Fezzik, the giant in _The Princess Bride._ Especially if you were already completely freaked out from hanging from a balcony two hundred feet up, during a fireworks display.\n\nBut really, I think this guy is literally eight feet tall.\n\nAnd there are two of him.\n\nWait, the second guy is the one I talked to at the elevator, the one with the cut. He's only seven feet tall. My bad. And I get a good view of the cut on his face because the first guy is lifting me by the shirt collar. It's a festering, weirdly swollen, open wound.\n\n\"You really should get that looked at,\" I gasp as Fezzik strangles me with my own shirt because, even in stress situations, I can never just shut up. My mom says, no matter what happens, I always have my mouth to keep me company.\n\nThese guys aren't talking much, though, unless you count cursing. Fezzik carries me into the room and swings me like a pendulum against the wall, then drops me. He starts to pick me up again.\n\n\"Stop!\" I yell, because that seems like a reasonable thing to say.\n\nAnd, weirdly, they both do stop. They stop and stare at me like they think I'm going to say something brilliant. I try.\n\n\"I'm not here to hurt Jonah. I mean, how could I? Look at me.\" I stare at the guy with the cut. \"Does that thing feel hot when you touch it?\"\n\nThe guy holds his monster-hand up to his face. \"Yeah, really hot. Is that bad?\"\n\n\"I'm not sure.\" I'm glad no one's picking me up, and I'm trying to prolong that. As I said to Allegra, talking is my superpower, my only superpower. \"It gets a little warm just because it's healing. But if it's really hot, it might be infected. I knew a guy with a wound like that, and he was seriously ill.\"\n\nActually, I don't \"know a guy.\" It was a character from a movie _._ And he died. But I keep that information on a need-to-know basis. \"Anyway, you should have it looked at.\"\n\nThe behemoth puts his hand to his face. \"Thanks, man.\" He turns to his friend. \"Otto, do you think it looks infected?\"\n\nOtto squints at it. \"Could be. He's right. You should get it looked at.\" He turns back to me. \"We need to get him out of here. He'll be back soon.\" He starts to pick me up again.\n\n\"Wait!\" I scream. \"Wait! Wait!\"\n\nHe drops me again. Hard. My head is ringing. \"What?\"\n\n\"Please,\" I say, channeling Westley from the same movie. I'm on my knees, more because I'm already down there than because I'm begging. But partly because of begging. \"I have to talk to Jonah. I came all the way from Miami and climbed over a balcony.\"\n\n\"What are you, in love with him? 'Cause he likes girls, lots of girls.\"\n\n\"I know. That's what I need to talk to him about, a girl. She's my friend. And she's dying.\"\n\n\"Haven't heard that one in a week.\" The scarred guy is still touching his cut.\n\n\"No, it's true. If you let me get my backpack, I could show you pictures of her. You could see.\" I wish I had the mirror. With that, I could prove lots of things, including the existence of magic. I could check on Celine too. I push aside my worries about Violet. Kendra's taking care of it.\n\nBut the mirror's gone. I have, as usual, nothing but my big mouth.\n\nSo I start talking, telling them the whole story, about how beautiful Celine is, and how nice, all the things that happened to her, her parents dying and everything. At some point, the scarred guy (whose name, I find out, is Sherman) does get my backpack. He takes out the photos and shows them to Otto. I've got one of Celine, lying still on the bed. When he gets to that one, I'm practically crying. And so are Otto and Sherman.\n\n\"So you see,\" I say, \"it's really important that I find Jonah. Only he can help.\"\n\nOtto looks at Sherman and shakes his head. \"That little prick's never going to help. He'll probably fire us for letting him stay.\"\n\n\"Tell him it would look bad in the papers if Jonah's bodyguards beat up a shorter statured individual like myself.\" They look at me, confused. \"A little person, a dwarf. If a big guy beats up someone smaller, that's . . . frowned upon.\" This has gotten me out of many a fight (when my mouth has gotten me into one), questioning the guy's manhood for hitting someone smaller.\n\n\"How would the press find out?\" Otto asks.\n\n\"I'd tell them, of course. It would be right there on the cover of the _Enquirer_. I'm an actor. I love publicity. And then, it would be all your fault.\"\n\n\"He makes an interesting point,\" Sherman says.\n\n\"And, meanwhile,\" I say, \"it would be really _good_ publicity if he visited a sick girl in the hospital. And couldn't he use some of that after the thing with the bicycle last week? After his breakup with Allegra.\"\n\n\"He didn't break up with Allegra,\" Sherman says.\n\n\"Oh, I think you'll find he did. He was a total jerk and broke her heart. Even if my friend doesn't wake up, he could take pictures, prove Jonah visited. We could call _Extra_. It would be a lot better than another story about him getting drunk and doing something stupid.\"\n\n\"He's right,\" Sherman says. \"We could ask Harry. That's his manager. And you seem like a really nice guy, um . . .\"\n\n\"Goose,\" I say.\n\n\"Goose. We should try to help Goose, Otto.\"\n\nBut, just at that moment, the door opens. And Otto picks me up and drops me again.\n\n#\n\n#\n\nYou know in old cartoons, when someone gets beat up and they see stars going around their head? That's how it is. I literally see stars. Then, I realize it's the last of the Disney fireworks out the window. Still, my head really hurts.\n\nOtto starts to pick me up again. Why did they turn on me?\n\n\"Man, stop!\" I scream. \"Dwarf tossing's been illegal in Florida since 1989!\"\n\nAn aside: This is actually true. In 1989, the Florida legislature voted to ban the \"bar sport\" of throwing little people against mattresses. This may be one of the best and least-stupid laws Florida has ever passed. Makes me proud to be a Floridian. It's still legal other places. But there are actually people campaigning to bring it back, to \"create jobs.\" If that's ever my job, just kill me.\n\nSherman's yelling at Otto to stop too. \"Don't go crazy, man! Dude's tiny. God, this is gonna look really bad if the papers get wind of it.\"\n\nAnd then, I understand.\n\nWhen Otto picks me up the second time, he whispers, \"When I drop you, stay down and act hurt.\"\n\n\"Shouldn't be hard.\" It comes out a grunt.\n\nHe drops me, and every bone in my body aches. But I get it now. It's a show for Jonah.\n\n\"I just wanted him to visit my sick friend!\" I yell.\n\n\"What's going on?\" A British accent. Jonah. Wearing purple diaper pants and a backward baseball cap. \"You're beating up a . . . a midget in my room?\"\n\n\"Actually,\" Sherman says, \"that word is considered offensive. I believe the preferred term is 'little person.'\"\n\n\"Right,\" I grunt.\n\n\"But you're beating one up. In my room. That's my point, really.\"\n\n\"Otto is,\" Sherman says. \"I tried to stop him. I told him it would be absolutely horrific publicity if this guy went to the press.\"\n\n_Horrific?_\n\n\"He snuck into your room,\" Otto says. \"On the balcony.\"\n\n\"Ouch!\" I yell, partly for show but partly because it really does hurt.\n\n\"Stop it,\" Jonah says.\n\nThe guys back off. Behind Jonah's back, Otto winks at me.\n\n\"Who are you, and why are you in my room?\" Jonah demands.\n\n\"My name is Goose. Goose Guzman, and I want you to visit the most beautiful girl in the world in the hospital.\"\n\n\"Oh, brother.\" Jonah sighs.\n\nI pick up Celine's picture, which has fallen on the floor. \"That's her. She's in Miami. She's in a coma. And I think meeting you might be the only thing that will wake her up.\"\n\nAnd then, I tell him everything else about Celine, how great she is. \"I know you have a ton of fans, but this girl is special. And, what's more, she's an orphan.\"\n\n\"An orphan?\" Jonah smirks. \"Like Oliver Twist?\"\n\n\"Exactly like Oliver Twist,\" I say, glomming on to the fact that he's British and has actually heard of Oliver Twist. \"In fact, that's how we met, in a school play, _Oliver!_ She sang, 'Where Is Love?' and that was how I knew.\"\n\n\"Knew?\"\n\n\"What a great person she was. Like no one else. And you have the opportunity to help someone like that. And, frankly, it would help you too.\"\n\n\"What? How would it help me? Why do I need help?\"\n\nIs this guy for real? Does he have no idea that everyone thinks he's a complete turd?\n\n\"Well, after the pictures of you peeing on your neighbors' lawn last week and practically running over that kid in your Maserati, not to mention your big breakup with . . .\" I stop, realizing I'm probably not supposed to know about the breakup. \"Don't you think it would be nice to have some good publicity?\"\n\n\"Now listen here,\" Jonah says. \"It's not my fault the press follows me about and reports on my every move\u2014normal teenage stuff like\u2014\"\n\n\"Like mooning a group of Catholic schoolchildren out the window of your limo?\" I ask.\n\n\"Nuns are so funny!\" He giggles. \"They look like penguins!\"\n\n\"Or when your monkey bit that waitress?\"\n\n\"She shouldn't have tried to pet it.\"\n\n\"So you don't care that you're perceived as the biggest douche in the universe?\"\n\n\"Girls don't think so.\" He glances in the mirror and adjusts his hat.\n\nIt's true. Celine liked him a lot. I always figured it was a blind spot on her part, that she somehow didn't see what he was like. It's like on _The Simpsons,_ how Lisa reads _Non-Threatening Boys Magazine,_ that girls like unattainable, girlish-looking boys because they're scared of real ones.\n\nMy head hurts. In fact, my whole body hurts. My brain hurts, and I just want a ride to the train station so I can go home\u2014after calling my mom to assure her I'm fine and apologize.\n\n\"Fine,\" I say. \"I'm sorry I thought you'd want to help this girl, who's a really big fan\u2014and, incidentally, a really cool human being. My bad.\" I turn and start for the door.\n\n\"Actually, I think it's a fabulous idea.\"\n\nBefore me stands the biggest, blondest woman I've ever seen. She's wearing a pink suit and a matching hat with huge white roses all over.\n\nI turn to stare. Jonah turns too. \"Mum, what are you doing here?\"\n\n\"Allegra phoned. I've come to check on the mess you've made of your life.\" She looks him up and down. \"Pull up your trousers.\"\n\n\"Mess? I'm an international sensation.\" But he does pull up his pants, which fall back down as soon as he lets go.\n\n\"An international sensation with no soul.\" She grabs his baseball cap. \"Take that off. It's disrespectful.\"\n\nJonah tugs at his pants and the cap at the same time. \"Mum, what have I done?\"\n\n\"You spit at that crowd of fans last week. You wore those horrible trousers to sing the national anthem at a ball game. The whole country saw your crack, the whole world, maybe.\" She gives them another tug.\n\n\"Mum, quit it.\"\n\n\"When you visited the Washington Monument, you said you were sure Washington would have told a lie to get to one of your concerts.\" The roses on her hat tremble with each word.\n\n\"He might've.\"\n\n\"And you haven't been inside a church in a year. This is not how I raised you, Joshbekesha!\"\n\n\"Josh\u2014what?\" I say.\n\n\"That's not my name!\" Jonah snaps. \"I'm having it legally changed when I'm eighteen.\"\n\n\"But you're seventeen now, and you do as I say. And I say you should do one nice thing for every ten rotten things you do.\"\n\nJonah nods. \"Yes, Mum.\" He looks at me. \"Perhaps I can help you out after all.\"\n\nJonah's Amazonian mother smiles at me. \"Now, this looks like a nice boy who listens to his mum.\"\n\nI feel my ears get hot, which literally has never happened. Between my olive complexion and my high threshold for embarrassment, I'm not a blusher. But I assume that's what's happening now. I say, \"Usually, you're right. I am a great son with a great mum\u2014uh, mom. But I'm afraid today has been an exception. If you'd let me use the phone, I could make it up to her, though.\"\n\nTwo hours later, I am\u2014as hoped\u2014on Jonah's private plane. We had to drive through what looked like a cornfield of teenage girls to get out of the hotel. I don't know how they knew he was leaving, or maybe they just live in the parking lot. I called my mother on Josh's mom's cell phone, and she only slightly freaked out. I guess she figured out where I was and was happy I wasn't dead or arrested.\n\n\"Can you check with Kendra?\" I ask. I'm trying to figure out what to say not to get her more worried. But I'm worried, so it's hard. \"Can you just . . . make sure Celine's okay?\" I don't want to tell her about the evil nurse.\n\n\"Okay,\" Stacey says. \"Just get home safe.\"\n\nJonah's sitting in the seat across from mine. He has on khaki pants and a blue button-down his mother brought him and sort of looks like a waiter at TGI Fridays. He's saying, \"Yes, mum\" a lot.\n\n\"Yes, Mum, I did notice how Goose called his mum so she wouldn't worry,\" he says.\n\n\"You're right, Mum. It probably wouldn't kill me to volunteer at a soup kitchen.\"\n\n\"Yes, I'll get a haircut. It would look nicer.\"\n\n\"Of course I'm not on drugs.\"\n\nHis speaking voice, like his music, is sort of . . . soothing. It's after 1:00 a.m., which means it's been almost a full day since I've slept. And, even then, I barely did because I was so worried about Celine.\n\nI feel like closing my eyes.\n\nMaybe I will.\n\nYes, maybe I will . . .\n\nI will . . .\n\nI feel a bump beneath me. I start awake. It takes me a moment to realize where I am. Across from me, someone is saying, \"Of course, Mum. Of course I realize it should be about the music.\"\n\nJonah. Jonah's plane. I've actually succeeded. He's going to go to the hospital and kiss Celine.\n\n_Kiss Celine._\n\nI push back all the feelings that causes. I can't think about how much it's going to suck to see him kiss her right now. Or ever. Celine is my friend, and I should want what's best for her. And if this . . .\n\n\"Really, Mum, how was I supposed to know I shouldn't text at a funeral?\"\n\n. . . if this idiot is what's best for her, then that's what I should want. At least he has a nice mom.\n\nI look out the window. The night outside is black. At least, since it's 2:30 a.m., there shouldn't be too many girls waiting in the terminal.\n\nOkay, I spoke too soon. As soon as we leave the secure area, there are _hundreds_ of girls, crushing together, craning to see Jonah.\n\n\"Is that him?\" one yells.\n\n\"Couldn't be, in that nerdy outfit.\"\n\n\"It's a disguise! It's a disguise!\"\n\n\"Omigod! That's his mom!\"\n\n\"I love his mum!\"\n\n\"Who's the little guy?\"\n\n\"Are you famous too?\"\n\n\"Not yet,\" I can't resist telling them, \"but I'm going to be.\"\n\nOtto and Sherman and a bunch of other bodyguards I don't know fight against the surging mob. How did they even know he was going to be here? Don't they have mothers to tell them not to go to the airport at two in the morning?\n\nOh, yeah. They probably blew off their mothers like I did.\n\nFinally, we make it to Jonah's limo, one of three limos that peel off in separate directions. Ours goes to the hospital.\n\nWhen we get there, it's blissfully quiet. It never occurred to anyone that Jonah would go to a hospital instead of a club or a South Beach restaurant where you eat dinner in bed. We head for the entrance.\n\n\"Where's the photog?\" Jonah's manager, Harry, is griping. \"The photog was supposed to meet us outside. Damn, there's always paparazzi around when he's pissing on a monument, but never when he's doing something nice.\"\n\n\"Perhaps it's because he's so seldom doing anything nice,\" Jonah's mom says. \"Go on, love.\"\n\n\"Maybe he could go up now and the photographer can come when he gets there, when Celine wakes up.\"\n\n_Oh, please, let Celine wake up._\n\n\"I think we should wait for the photographer,\" Jonah says. \"After all, it's the whole reason we're\u2014\" He's interrupted by the mother of all nudges from his mum. \"I mean, of course I'd love to go up and meet the gi . . . young lady right now.\"\n\nHe looks to his mother for approval, and she pats his shoulder.\n\n\"Come on, then.\" I gesture for Jonah to follow me to the elevator. \"It's probably better if it's just the two of us.\"\n\nThe elevator is one of those big ones that can accommodate a gurney. We stand far apart and don't talk. Jonah's probably tired from the tongue-lashing, and me, I don't have anything to say. I don't have anything to think. At least, nothing I _want_ to think. If I was thinking\u2014which I'm really trying not to do\u2014I'd be thinking this is _it._ End of the line. If Jonah's kiss doesn't wake Celine up, maybe nothing will. Maybe Celine is really and truly gone forever.\n\nThe hospital is so silent, which is bad because it allows me to be alone with my thoughts but good because it's quick. Only one nurse gets on the elevator. She doesn't seem to notice Jonah, and she's going to the same floor we are, twelve.\n\nI watch the numbers. I don't want to talk to Jonah. Celine thinks he's so profound, but really, he's an idiot. She'll be disappointed.\n\nShe'll be disappointed _if_ she wakes up.\n\n_Five._\n\nThis has to work.\n\n_Six._\n\nIt will work.\n\n_Seven._\n\nWhat if it doesn't?\n\n_Eight._\n\nNo point thinking about it.\n\n___Nine._\n\nBut what if it doesn't?\n\n_Ten._\n\nStop it. Stop it!\n\n_Eleven._\n\nI'll know in five minutes. Two if we run. Almost there. At least the wondering will be over.\n\nThe elevator jolts to a stop.\n\n\"I'm afraid I can't let you boys off,\" the nurse says.\n\n#\n\n#\n\n\"What the\u2014?\" Jonah yells.\n\nBut I know. Of course it's Violet. She looks different than the nurse I saw before, but I know her by the expression on her face. And, um, the fact that she's not letting us move.\n\nThe first thought that flashes through my head is that I must be right about Jonah. I must be close. Violet hasn't bothered Celine until now. If she's suddenly trying to stop me, she must know that Jonah is the handsome prince who can wake Celine.\n\nI lunge for the alarm button. It starts ringing. Jonah's screaming, \"Help! Help! I'm being kidnapped! I'm a rock star!\" But then, just as suddenly, my arm, my whole body freezes. Jonah's screams stop. I can see that he, too, is frozen, stone-like, like Medusa's victims. I can only move my eyes, and with them, I see Violet push the button for the roof.\n\n\"I'll drag you up and throw you off. Falling from a great height is your destiny, dwarf.\"\n\nI feel the elevator start up again. I can't do anything about it.\n\nAnd then, there's another person in the elevator. She looks like Violet. My eyes take in blazing red hair and high-heeled boots. \"What are you doing here?\" the nurse-Violet screams.\n\n\"Saving them!\" the other Violet screams, so I know it's Kendra. \"It's too late to save you. Violet, you disappoint me.\"\n\nThe elevator again jolts to a stop.\n\n\"Disappoint you? I always disappoint you,\" Nurse Violet mocks. \"I disappoint everyone.\"\n\n\"That's not true. I thought you were the daughter I never had. It breaks my heart to have to stop you, to have to use tough love.\"\n\n\"Then don't!\" Nurse Violet screams.\n\nAnd suddenly, a ball of fire flies right at Kendra and me. I can move, and I duck to avoid it. Kendra somehow quashes the flame, but there is another, and another. Jonah is shrieking. The doors open, and Kendra screams at us to run, even as she uses a fireball of her own to hold Nurse Violet at bay.\n\n\"How?\" I look out the door. The elevator is several feet above where it's supposed to be, hovering above the floor. The white linoleum floor looks slick and hard as ice.\n\n\"Just jump!\" Kendra says.\n\nAnd, amazingly, tugging Jonah behind me, I do.\n\nI fall hard, but I don't die. Jonah lands neatly, and I yell, \"Come on!\" I don't look back. I hear the door close. I think, hope, we're on the right floor, Celine's floor. I check the numbers on the doors, 1201, 1202. Yes! We skid around a corner and almost hit an oncoming nurse.\n\n\"Slow down!\" she yells.\n\nAt least it's not Violet. We slow. Jonah's been making frightened, incredulous sounds, combined with lots of cursing. Once we pass the nurse, he says, \"What the hell was that?\"\n\n\"A witch.\" I don't look at him. \"I didn't tell you because you'd have thought I was lying. Or crazy. But now you know. A witch put a spell on Celine to make her go to sleep. I want you to kiss her, so you can wake her up.\" I keep walking fast, not looking at him. Eyes on the prize. Celine.\n\nPresumably because of what he's seen, Jonah doesn't seem to think I'm crazy. \"You think I can break the spell?\"\n\n\"You're a handsome prince, aren't you? Or as close as we have.\" We reach a corner. I grab the wall to stop myself, then check around it.\n\n\"Oh, yeah.\" He's panting, but he grins. \"Guess I am.\"\n\nWe round the corner at a fast walk, me working hard to keep up with Jonah's longer legs. I say, \"So you'll do it?\"\n\n\"Sure. Why not?\"\n\n_Why not? Because you might be putting yourself in a witch's path._ But I don't say it. Why would I? If he's too dumb to realize it, I'm not going to enlighten him. I just need him to wake Celine. He can leave right after. In fact, I'd prefer it.\n\nWe fast-walk around a last corner, then to Celine's door.\n\nI open it.\n\nShe is so beautiful. It's been a day since I've seen her, and I am stunned by her like it's the first time. She lies there, so pale against the white sheets. Her black hair is fanned out behind her on the pillow, and her full, red lips are exactly the ones I've always wanted to kiss.\n\n_Please let this work. Please come back to me._\n\nI jut my hand toward her, in case Jonah doesn't get _which_ comatose girl exactly I meant. \"That's her.\" It's hard to form words. \"Celine.\" Something's wedged in my throat, making it hard to talk. The idea that, if this doesn't work, maybe nothing will. She might never awaken, she might die, and with her, the possibility\u2014however slight\u2014that I keep with me every night as I drift off to sleep, the possibility that she could someday love me.\n\nOf course, if it works, if Jonah's kiss wakes her, that possibility will be gone anyway. He doesn't know her, but once he sees how pretty and sweet and funny she is, he'll fall in love with her. Even a douche like him would know she's special. They'll walk off into the night together, like Andie and Blane, and I'll be left all alone\u2014well, alone with a houseful of people, but without her. So, alone.\n\nStill, I have to try it. I love her. She needs to be alive on this planet, even if it's not with me. That's what love is, after all, wanting the best for the other person, not yourself. I learned that the hard way.\n\nJonah looks at her, and smiles. \"She's lovely.\" His admiration is genuine, of course. With his accent, it comes out all _loff-lee_ , which is probably why girls think he's so hot. Maybe someday, I can move to another town where nobody knows me, pretend to be a Brit, and get all sorts of girls, short girls, tall girls, lots of girls. Just not Celine.\n\n\"She is,\" I say. \"Loff-lee. She's nice too, and funny, and talented and . . . good with kids. She's like no one I ever met before, which is why I need you to help her.\"\n\n\"And you think my kissing her . . . ?\"\n\n\"I hope so.\" _Do I hope so? I do._ \"I can be straight with you now that you've seen the witches, seen what they can do. This is her only chance.\"\n\nHe shrugs. \"Guess we can try.\" And then, without another word, he leans down toward her and . . .\n\nI can't look. I turn away. This is what I wanted, dammit. This is what I wanted, the reason I traveled so far, lied to my parents, hid in the room service cart, and hung from a balcony. I want this. I just want her to wake up, no matter what.\n\nThe room is silent. God, are they still kissing, all this time? I want to look, but I don't want to see if she's, you know, enjoying it too much.\n\nFinally, Jonah's voice says, \"I don't think it worked, man.\"\n\nMy chest is a deflating balloon. My eyes ache like I just came out of salt water. I squeeze them together. I wanted it to work. I did. Now what?\n\nI turn back, opening them. Jonah's staring down at Celine. \"I was really hoping it would, and not just because I wanted the reputation of having my magic lips raise someone from the dead. I knew if you were willing to go to all that trouble for her, she must have been pretty special.\"\n\nIt's a really coherent thing for him to say. Still, the past tense about kills me. My eyes are damp, but I'm not going to wipe them, not in front of him. \"She was. She is.\"\n\nHe tosses his hair a little girlishly. \"So this spell, the spell the witch put on her, it said _I_ had to kiss her?\"\n\nI shrug. \"I don't know. In all the fairy tales, the girl gets awakened by a handsome prince. I was going with that idea.\"\n\nHe chuckles. \"Perhaps not handsome or princely enough.\" He looks down, thinking. \"My mum used to read me those storybooks. It was nice. I was a shit to my mum, wasn't I?\"\n\n\"Yeah, sort of.\" I was a shit to mine too, and it didn't even help.\n\n\"You know,\" Jonah says, \"most of my books talked about true love as well. Perhaps that's a factor. Perhaps that's what's missing, the love part. Beautiful as she is, I don't love her.\"\n\nThat must be it. Still, I say, \"She loves you, though. She listens to all your songs, has posters of you all over the place, writes your initials on her notebook in pink highlighter . . .\"\n\nHe throws back his head then and laughs. \"But that describes half the teenaged girls in the world, these idiots who camp out in the airport. Do you think I can resuscitate all of them too?\"\n\n\"They probably don't all need it.\" Celine's not an idiot, but he does have a point.\n\n\"Still, I think your definition of love may be a little thin.\"\n\n\"My definition of love isn't thin at all.\" I take Celine's hand and squeeze it in the silent room. It's so soft, and I remember teaching her to play the piano, one finger over the other. I love her fingers.\n\nJonah sees me and nods. \"Surely there must be someone who actually loves her, who _knows_ and loves her. She's quite pretty.\" He looks back at her, and now, I hate him looking at her, since it didn't work. He turns back toward me.\n\n\"She's very pretty,\" I say. \"Nice too, and talented and fun. But she doesn't have a boyfriend, if that's what you mean.\" When you think about it, it's crazy that someone as cool as Celine doesn't have hundreds of guys in love with her. She once told me she didn't like guys who think they're hot. That describes most guys at our school.\n\nJonah tugs at his pants, then seems to realize they aren't falling down, since they're the geeky pants his mom got him. \"Well, maybe not a boyfriend, but it strikes me that someone who went to all this trouble for her\u2014I mean, someone who could have been arrested dozens of times. Someone who did get beaten up by my bodyguards\u2014that, perhaps, that person may in fact be her true love.\"\n\nOh. Duh. He means me. Am I that easy to see through? Is he smarter than I thought? \"Yeah, well, of course I love her. But that doesn't mean she loves me back. I mean, look at her, and then look at me.\"\n\n\"Of course,\" he says, and I sort of want to hit him for agreeing so quickly, but then, he says, \"Ahem. I mean, of course, we don't know what she thinks, and she isn't awake to tell us. She could love you. You have rather a charming personality.\"\n\n\"Gee, thanks.\" This guy's getting less charming by the minute.\n\n\"But even assuming you're right, does it have to be mutual? If you are truly in love with her, might that not be enough love?\"\n\n\"I don't know. I hadn't thought of that.\" I hadn't.\n\n\"Should you not, perhaps, try?\"\n\nI think about it. I've wanted to kiss her since forever, or at least a few months. But the thing that stopped me was her reaction, what she'd think of me. If I didn't kiss her, we could be friends. I could be with her all the time, like buddies, see her every day. But if I kissed her, it would get all awkward if she didn't love me back. I didn't want to upset things.\n\nNow, she's in a coma. She wouldn't need to know I kissed her. In fact, I'll take it to the grave. Is that pervy? Maybe. Perhaps, as Jonah would say. But does that matter? The fact is, she holds my heart in her body, and if she doesn't awaken, I may die.\n\nI nod. I step up to Celine. Her lips are so full in her heart-shaped face. I wish she'd open her eyes so I could see those too, so I don't feel like I'm taking advantage. But, of course, that defeats the purpose of kissing her. I touch a lock of the shiny, black hair on the white pillowcase. It feels soft like the satiny ribbons my mother uses on packages at Christmas or on my sister's hair. Celine used to do Isabella's hair in ribbons. I can hear her breathing, smell the sweetness of her breath. I imagine for a moment that she loves me. It's not so impossible, is it? I'm a great guy. I picture us sitting at the piano that one night, me playing \"Clair de Lune,\" trying to impress her _._ She could have loved me then. I reach forward and adjust her face so it's leaning toward me. I feel like there's no air in the room.\n\nIt's not like I've never kissed a girl before. Just not this girl, the one that matters. And there's the part about her being asleep. I inhale through my nose. Then, my lips meet hers.\n\n_God._\n\nI mean to give her a small kiss, a polite kiss, not be like one of those guys who waits until a girl passes out, then mauls her. I love this girl. I love this girl, but I don't want her like that, not by fraud. In my fantasies, she wants me too. And yet, when our lips meet, I feel a flash of something\u2014call it electricity, call it magnetism, call it magic\u2014binding us together, and I can't let go, I can't let go, and I'm kissing her like I've always imagined.\n\nFinally, I back off. I more than back off. I pull my lips off her like a plunger getting yanked out of a toilet. I run behind Jonah, then out the door.\n\nIt didn't work. I knew it wouldn't. A regular guy like me couldn't possibly be the true love of the most beautiful girl in maybe the whole world. I'm not Blane. Heck, I'm not even Duckie. Still, I hoped it would work. I hoped it would because now, I'm out of ideas, and Celine's still in a coma and I am there with her. Maybe she'll die or just stay there, suspended, forever, and I will never have anyone to teach piano to or watch John Hughes movies with, no one to tell me I don't have to be funny for people to like me.\n\nAnd, at that moment, with no one there to see except the nurses (who are probably used to it), I give way to the tears that had been threatening to seep out of my eyes for the past week. I bury my face in my hands and sob. _Celine._\n\n\"You're here. You?\"\n\n_What?_ A voice from inside the room. Not Jonah's voice. A girl's voice. Celine's voice! But how?\n\n\"I am here,\" Jonah's voice says back. \"Was it . . . Celine?\"\n\n\"Yes. I had a dream about you. You were in it. In my dream, you kissed me. I thought it was only a dream because it sounded so crazy. I mean, why would Jonah Prince be here with me? It's so incredible.\"\n\n_She's talking to Jonah. She's so happy to see him._\n\n\"Why indeed,\" he says. \"A little friend of yours came to see me, to tell me about you.\"\n\n_No. No. Don't tell her I kissed her. It will ruin everything._\n\n\"A little . . . oh, you must mean\u2014Goose! In my dream, you weren't the only one who kissed me. In my dream, Goose kissed me, and that's what woke me up.\"\n\n_What?_\n\n\"Goose!\" She's calling me. \"Where is he?\" Her voice holds a note of panic I can't help but imagine is from missing me. Could it be? I wipe away the embarrassing tears. My face hurts.\n\n\"Celine?\" I step from behind the door frame before I have time to chicken out. \"I'm . . . I'm here.\" Is she mad? Will she laugh at me?\n\nShe's half sitting up on the bed, leaning on her hands, her blue eyes wide open. When she sees me, her face breaks into this huge smile that wasn't there before, not even for meeting Jonah.\n\n\"Goose! You are here! You went away, and I missed you so much!\" She adjusts herself on the bed, then reaches out her hand.\n\nI run to take it. God. It's not like I haven't touched her hand before, but this is so . . . intentional. Almost like kissing her, but almost kind of better than kissing her because she knows I'm doing it. Is it possible? Could my kiss have awakened her?\n\n\"You . . . missed me?\" I ask. \"You knew I was gone?\"\n\n\"Of course. Everyone thinks that people in comas don't hear anything, don't know anything. But we do hear or at least _I_ did, and we have a lot of time to think, too, about . . . everything. I knew you were here all that time. And then, I knew you left. Why did you leave? I missed you, even when I was asleep.\"\n\n_She missed me._ I get more daring. I squeeze her hand. I see the little burn scar I noticed that day. It's really her hand I'm holding. \"I left to find Jonah, to get him to kiss you.\"\n\n\"Why did you want him to kiss me?\" She squeezes back, only instead of just a little squeeze, she clings to my hand.\n\n\"Kendra said a handsome prince might break the spell. I thought . . . Jonah Prince. Prince. I thought he was your handsome prince, your true love.\"\n\nShe laughs, shaking her head. The light from overhead makes her hair sparkle like those black stones goth girls have in jewelry. \"He's not my true love. I just like his music.\" She looks at Jonah. \"No offense.\"\n\n\"None taken,\" he says. \"You seem quite a pleasant girl, but I don't love you either.\"\n\n\"I do love your music,\" she says politely.\n\n\"But you woke up,\" I say. \"How?\" Because, even though the thought has been forming in my head, I want her to say it. After all, maybe it's just that I love her. That's what Jonah said. Maybe me loving her is enough.\n\n\"You silly goose! How could you have awakened me without knowing the answer?\" I guess I'm still staring at her blankly because she says, \"True love, right?\"\n\n\"True love?\" Obsidian. That's what those black stones are called. They're supposed to be magic. Her hair is like obsidian. \"What? Who?\"\n\nShe's still clutching my hand. With the other, I gesture to myself. \"Me?\"\n\n\"You.\" She loosens her grip. \"I mean, if you feel the same way. Maybe you don't.\"\n\n\"If I feel . . . ?\" And suddenly, my mouth is stretching so far, my face smiling so hard it hurts. \"You mean you and me? You don't mean you love me as a friend, or . . . any of those other things girls say?\"\n\nShe's just staring at me weird, and she says, \"Nuh-uh. None of that. That first day at auditions, I was like, 'This guy is awesome.' You were so bold. And I agreed to be Oliver partly so I could know you better. And then, when I did, you were sweet and funny and smart. And brave. You protected me. You _saved_ me, like a hero.\" It's like in my dreams, every dream I'd had. Her voice, saying she wants me, and it's finally dawning on me, what she's saying. \"You're the one who woke me up after all.\"\n\n\"I did, but . . .\" I step closer, wanting to take her in my arms now. \"You really . . . ?\"\n\n\"Perhaps I'll go get a nurse,\" Jonah says, \"let her immortalize this moment on film to tweet to my fans and make my agent happy.\"\n\nI guess he leaves. I'm not really paying attention.\n\n\"I read the poem,\" she says, \"the one you wrote.\"\n\n\"You did? It's really . . . embarrassing.\" I'm still not completely wrapping my head around the idea that this is happening.\n\nShe shakes her head. \"I loved it.\" Celine holds her other hand out, wiggling her fingers until I come closer. Then, she touches my face. Her hand is so soft. \"It let me know you felt the same way I did. I hadn't admitted it to myself before then, even when Izzy flat out asked me.\"\n\n\"Really?\" I wonder how long she felt this way, how long I was wondering when I didn't have to.\n\nShe nods. \"God, you're so adorable.\"\n\n\"Really? Adorable? That's the adjective you're choosing? Like I'm a kid or a teddy bear?\" But I'm thinking, _She thinks I'm adorable_.\n\n\"Oh, don't be stupid. People call big guys cute all the time. Adorable as in, I _adore_ you. I adore you G . . . what's your first name? It's strange to love a guy called Goose. When I was sleeping, I tried to remember if I'd seen it in the program for _Oliver!_ , but I couldn't envision it.\"\n\nI laugh not because it's really funny but just because I'm happy. \"Nope. It wasn't there. They listed me as Goose Guzman. That's what I told Connors to do.\" Now, I want to stop talking and kiss her again.\n\n\"But that's not what's on your birth certificate. At least, I hope it's not. I mean, when you graduate, your diploma won't say Goose Guzman, will it?\"\n\nI laugh again, all stupid-happy like our neighbor's shih tzu, who practically turns himself inside out from ecstasy when you pet him. \"Nah. My father wanted me to have a big name since I was a little guy. His name's only one syllable, Jorge; two if you pronounce it the Spanish way, Hor-hey. They gave all of us big names, Antonio, Isabella, and me, Mauricio. It's a dumb name.\"\n\n_I'm talking too much. Less talking, more kissing._\n\nShe smiles. \"Mauricio. I like it.\" She rolls closer to the edge of the bed. \"Aren't you going to kiss me again, Mauricio?\"\n\nI do. I do, and the sparks and the magic and the fireworks are all there just like before.\n\nSuddenly there are people in the room, and there's music, a guitar, and a voice, singing.\n\nSometimes when I see your face,\n\nIt takes me to a better place.\n\nWhen I look into your eyes,\n\nWalls fall down and curtains rise . . .\n\nAnd I'm kissing Celine, holding, crushing her to me like I always wanted to and never dreamed possible. She loves me and I love her, and she's alive and safe, and we're together.\n\nFinally, though, we break apart, and I say, \"What is that music?\"\n\n\"Oh, sorry.\" It's Jonah. \"I was feeling a bit of the third wheel here, so I thought that, if this was a movie, there would be music. I do love a happy ending. Harry brought up the guitar, but perhaps it is a bit\u2014\"\n\n\"No.\" I grin. \"There would be music.\"\n\n\"There definitely would be,\" Celine agrees.\n\nThe other people in the room are nurses, and then a doctor shows up too. And Kendra, who _looks_ like Kendra and a little worse for wear too. I know I'll hear what happened later. They're all pretty shocked to see that Celine is, in fact, not dead, but I resist the urge to gloat. What are doctors supposed to know about magic spells? So we let them think they cured her. After some debate, they unhook Celine's feeding tube and everything, and after about six hours of tests and my dad (who filed an emergency petition to become her legal guardian, after Jonah and I explain about Violet attacking us in the elevator) filling out a ream of paperwork, and photographers taking tons of photos to prove Jonah was here, the hospital lets Celine go home. With us. With me.\n\n\"So what do you want to do when we get there?\" I ask Celine on the way downstairs.\n\nShe squeezes my hand. \"I was thinking we could make some smoothies and watch _Some Kind of Wonderful._ \"\n\n# PART 4\n#\n\n#\n\n# _Kendra_\n\nThe elevator door slams shut behind them, and a ball of fire comes at me.\n\n\"Hey!\" I yell. \"If you don't stop, you're going to kill us both!\"\n\n\"Does it really matter?\" Violet asks.\n\n\"It does to me. I want to see how this turns out, whether he wakes her.\" For I have realized that Violet wouldn't be here if she didn't think Celine would wake. And I've realized something else. I've figured out who can wake her. It's not Jonah Prince.\n\nI see Violet gearing up for another attack. Her magic is no greater than mine. It may be less. But she's more ruthless. She loves no one and has nothing to lose. I once felt that way, but now . . . well, I want to check on Celine. I can hold Violet at bay long enough for Goose and Jonah to reach her, I hope, but I can't stop Violet forever.\n\nI've frozen the elevator on the fourteenth floor. Another explosion rocks it.\n\nI dodge the fireball, putting it out with a neat blast of water.\n\n\"Play with fire,\" I say, \"and we'll both get burned.\"\n\nViolet smiles. \"The difference between us is I don't care if I die. I don't care about anything.\"\n\n\"Then why hurt that sweet child?\"\n\n\"Because that child is evil, the spawn of evil.\" She walks away as much as possible in an elevator. I know why she's doing it, to make me think she's pacing, make me let down my guard. But I never let down my guard with Violet. I did it early on, and I regret it.\n\n\"You think she's like her mother?\" I say, eyes firmly on her.\n\n\"Her mother, or her father. Greg was no better. I see that now. He wanted Jennifer as a trophy, no more. That's why he preferred her even when we were both beautiful. She had more status.\"\n\n\"Maybe he loved Jennifer, and not you.\" I can't resist twisting the knife. What difference does it make? Being careful around Violet didn't help.\n\nViolet ignores me. \"He wanted her as a trophy. And then, when she was gone, he wanted me for the same reason. He never loved either of us, not really.\" She shakes her head sadly.\n\n\"Celine loved you,\" I say. \"Before she knew you killed her mother, she loved you.\"\n\nViolet looks over her shoulder at me, her face like a crumpled gardenia. \"No one has ever loved me, not really.\"\n\n\"You know that isn't true.\" I walk over to her and lay a hand on her shoulder. \"I have loved you like a daughter since the first day we met.\"\n\nAnd, like a real daughter, she fights against me, pushing me away with hands suddenly burning hot. I cringe, and she says, \"You don't love me.\"\n\n\"I do, though I did you a disservice, allowing you to change yourself so much.\"\n\nI remember how she looked that first day, so small and pale, beaten down by those horrible boys. I'd had my own experiences with boys like that, and I'd have done _anything_ to help her. I had done anything. I had given her my knowledge, my instruction, my magic, my heart. At what point did I give too much, do too much? And could I have stopped her if I'd tried?\n\n\"A disservice?\" Violet murmurs, as if she hasn't quite heard. \"I was miserable. You . . . tried to help me.\"\n\n\"But did I help you? Or did I make things worse?\"\n\nShe shakes her head, still not looking at me. \"I don't know.\"\n\nA voice comes through the elevator's intercom. \"Are you all right in there? We're sending help.\"\n\n\"It's fine.\" But, obviously, time is running out. I can't keep Violet in this elevator forever. She could leave if she chose. I only hope she'll stay. I must persuade her to make peace with Celine or, at least, leave her alone.\n\n\"Celine isn't who you think,\" I tell her.\n\n\"Of course not,\" Violet snaps. \"I'm always wrong about everything.\"\n\n\"Maybe not everything, but this. You told Goose that Celine could never care about him. You were wrong about that too.\"\n\nShe shakes her head. \"I wasn't. Little bitch would only want the captain of the football team. She'd never appreciate that kid, even after all he went through for her.\"\n\nI pull a mirror from my voluminous skirt. Violet knows what it is because she has one just like it. I gave it to her so many years ago, and we have spoken through it almost every day. I say, \"Show me Celine.\"\n\nThe scene in the mirror shifts to Celine's hospital room. Celine is awake, looking around the room. She says, \"True love, right?\" My heart feels tight in my chest. It worked!\n\nViolet pushes the mirror back toward me. \"That proves nothing. The pop star, he woke her up.\"\n\nI angle the mirror toward her. Now, Celine is holding Goose's hand, gazing into his eyes.\n\n\"She loves him, always has. So you were wrong about that. What else were you wrong about, Violet?\"\n\nShe stares at the mirror like someone in a fog. \"I don't know.\" She takes it from me, gazing at the happy couple. \"This is all I ever wanted. This. Love. But when Greg died, there was no chance left for me.\"\n\n\"No chance? You have every chance. You're immortal, magical. There is always another chance.\" It's a conversation we've had before, unsuccessfully. She doesn't seem to comprehend how long her life will be. I, with hundreds of years behind me, know that life stretches before her like a patchwork quilt with many experiences, some beautiful, some heartbreaking. \"Go someplace where no one knows you, and start again.\"\n\n\"Someplace else.\" The mirror catches the ceiling lights. \"Yes, someplace else. Will you take care of her then, of Celine?\"\n\n_So you can always know where to find her?_ But I don't say it. She is staring so oddly that I wonder what she has in mind. \"Of course Celine will be taken care of. But what about you, my darling?\"\n\n\"Yes.\" Her voice is a shredded whisper. \"What about me?\"\n\nAnd suddenly the elevator begins moving, down this time. It doesn't stop at twelve or anywhere but goes all the way to the bottom. When the door opens, Violet steps forward, then out. \"Good-bye, Kendra. And thank you. I know what I have to do.\"\n\nShe presses the twelve button, gives a tiny wave of her fingers. They are, as usual, perfectly manicured. Everything about her is lovely, luminous. If you didn't know her, you'd think she was perfect.\n\nShe smiles as the door closes, and I know I will never see her again.\n\nOn the twelfth floor is celebration. Celine, Goose, Goose's father, Jonah, all celebrate Celine's revival. A photographer snaps pictures. It is hours before we leave, and when we do, I go in the car with Goose, Celine, and his father. Goose and Celine sit in the backseat, holding hands.\n\n\"I think you should stay with me,\" I tell Celine. \"I can protect you in case . . .\" I'm not actually sure Violet intends to do anything, but I realize I want Celine with me. \"You don't need protection anymore, probably. But I'm alone and you're alone. It was meant to be.\" She can be my daughter, and I can do a better job this time.\n\n\"That might be good,\" Goose's father says. \"If Goose and Celine are . . . together, it wouldn't be right for Celine to live with us.\" I see him raise his eyes in the rearview, but then he smiles.\n\n\"Hey,\" Goose says. \"I thought you said you couldn't just zap people someplace. Looked like you kind of zapped into that elevator.\"\n\nI shrug. \"Every rule has an exception.\"\n\nCeline says, \"I'd like to live with you, Kendra. But you don't think Violet will try to harm me again?\"\n\nI start to say I don't know what Violet will do, but that I will try to protect her.\n\nThen, something catches my eye.\n\nOff in the distance, a plume of smoke, a brush fire maybe. But it's not in the right direction for a brush fire, not to the west. Rather, it's in the direction of\u2014\n\nI nudge Goose's father and point. \"Drive that way.\"\n\n\"What? Why?\"\n\nBut then, he too sees it, an orange blur, a flame, just for a second.\n\nCeline, noticing, shrieks, \"Oh, no! No! Do you think\u2014?\"\n\nI shake my head. \"I'm not sure.\"\n\nWe follow the smoke until we are on a familiar street, Violet's street. Celine's street. Celine's house.\n\nThe house is in flames.\n\nI hear glass breaking, and a crow flies overhead.\n\n#\n\n#\n\n# _Violet_\n\nSomething is burning. It's my house. It's burning down. I myself struck the match, a wooden match from a restaurant matchbox, someplace Greg and I used to go. _Greg! Did you ever love me? Or was I just a poor substitute for someone else? Did you ever even love her?_ A sob escapes my throat, or perhaps I'm choking. I am lying in the bed I used to share with my husband, waiting. If I rest my nose on his pillow, I can still smell him, barely. Except it's hard to breathe. I stare into the silver mirror in my lap, expecting to see the girl I was, the ugly girl. I'm still beautiful, but all I want is to die. What the dwarf said was right. Everyone hates me. It isn't my face, not anymore, but me. And yet, as the flames lap closer and closer to the bed, the mirror in one hand, my wedding photo of Greg and me in the other, I can't help but wonder if that could change. Maybe there is another way, another place. I could do as Kendra said, go somewhere else where no one knows me, start over as many times as I need to. Change my appearance and fly like a crow to faraway places.\n\nThe room is hot. A window breaks, and I am sweating, blinking my eyes against the gray smoke. A mortal would have succumbed to it long ago, but I am no mortal and can only die from the pain of the flame. I dread it, coward that I am. I squint at the silver mirror. \"Show me Celine,\" I tell it.\n\nThere she is, black hair and white skin, a beautiful girl, a girl who once loved me. A girl I loved. She's with the dwarf, sitting in a car holding his hand. Her eyes widen, and she leans to embrace him, gazing at him as if he is the most beautiful man she has ever seen. He turns and smiles, and suddenly, he is beautiful, dark brown eyes shining from a handsome face. I see his beauty as I wished others would see mine. I know I was wrong about Celine. I was wrong about so many things. Was I wrong about myself too? Can there be hope for me?\n\nI feel a spark on my shoulder. The bedsheet has caught fire, and soon, I will be consumed by it. I am not tied to this bed, though. I can still flee. I make my decision. I take one final look at the photo, at Greg. Greg, who never loved me at all, not really. I feed it to the flames. I watch it burn.\n\nThere is nothing left of Greg but Celine. There is nothing at all left of Violet. Violet is dead. Quickly, I manufacture something, a dummy version of the girl I was, the ugly girl. I remember reading _The Picture of Dorian Gray._ In the end, when the beautiful main character died, he became the hideous old man in the picture. That was how they found him. They could only recognize him by the rings on his hand. That is how they will find me\u2014or think they did\u2014my charred remains lying on the bed. But I will be gone, far, far away from all of them.\n\nThen, I make my escape, flying on jet-black wings out the window and away, into the warm summer night.\n\nI will begin again . . . somewhere!\n\n#\n\n#\n\n# _Celine_\n\n# _February, the next year_\n\n\"You know we don't actually need to go shopping,\" Kendra says as we pull into the Target parking lot where it all began. I'm driving the red VW Bug Jorge helped me buy with money from the trust he set up for me out of my father's estate. Goose taught me to drive because Kendra definitely didn't know how. \"That's one of the great things about being a witch, no money needed.\"\n\n\"I know,\" I tell her. \"But you know you like shopping. It gives you ideas.\"\n\nBoy, does it give Kendra ideas. Since I've started taking her on weekly shopping trips, our entire house has gone from French provincial to Early College Dormitory with every kind of thing Target sells, all in pastel polka dots. Kendra buys none of it. It all just appears. \"You know you love the dollar section.\"\n\n\"That's true. You think they have that mint foot rub?\" She exits the car, fluffing her purple tulle bustle. \"Maybe I should get a job there.\"\n\nGoose and I exchange a look. Kendra, work? Kendra, wear a uniform? Kendra, deal with the public?\n\n\"What?\" She looks from one of us to the other. \"It would be easy for me and give me something to do when you go away to college.\"\n\nWe start toward the shopping carts. There are tons of black birds, crows, or grackles on the lights overhead, and they're cawing and chirping so loud it makes my head hurt. I'm freaked out by birds, have been ever since we had to read \"The Birds\" __ in English class this fall. Goose loves that and loves to mess with me. \"They're gathering, Celine,\" he says in a creepy voice. \"They're making plans.\"\n\n\"Quit it!\" I slap his shoulder. He recoils like I've hurt him, but I know he's messing with me. Still, I give his shoulder a pat.\n\nKendra has also been staring at the birds, but now, she says, \"Come along, children.\" She grabs a cart and starts booking it to the entrance.\n\n\"Maybe you should take Kendra to some higher-end places,\" Goose says, yelling over the cawing. \"Get ideas for a prom dress.\"\n\n\"Splendid idea,\" Kendra says, still walking extra fast.\n\nI love Kendra, but since I've been living with her, she has this great idea I should dress like her. I'm more of a prep, but I'll occasionally let her design a dress for me. Just not for prom.\n\nI try to change the subject. \"Is that your way of inviting me to prom?\" I ask Goose.\n\n\"I sort of thought it was a given we were going together, since I'm the love of your life.\"\n\n\"It is, and you are. But it's still nice to be asked.\" I was sort of expecting an elaborate \"prom-posal\" out of him. He's theatrical, after all. He's left roses in my locker twice, and once planned an elaborate scavenger hunt, involving teachers, students, even the football coach, all to give me my birthday present, a bracelet with charms representing both our families, and us. So I was expecting at least a song with the lyrics changed to include my name, sung at a pep rally. Which would be super-embarrassing, actually.\n\nWe've reached the entrance. I'm ready to go in, to get away from the birds. But Goose takes my hand and gets down on one knee. \"Celine, my darling, will you accompany me to the prom?\"\n\nI laugh. \"Of course I will.\" And part of me is thinking, _Get up._ But the other part of me knows he's perfect, that I need someone just like him, someone who doesn't mind being stared at, who helps me get out of myself. Who loves me for me. Finally. So I wiggle my fingers. \"Now, kiss my hand.\"\n\nHe does. The birds are screeching, cawing. I tug at his hand to help him up. \"Let's go in. The birds are freaking me out.\"\n\nJust as I say that, one bird swoops down from the rest. It's a big one, and flying sideways, it looks like a black kite. It flaps its wings right in my face. I run behind Kendra and Goose, remembering my mother, her fear of animals. But the bird doesn't peck or attack me. Instead, it flutters down and rests on the shopping cart handle, right by Kendra's hand. It stands there, staring at me. I grab Goose's hand at the same time Kendra grabs my other one. So we form a weird human chain, me and the two people I love best. No one can hurt me, not with love and magic on my side.\n\nThe bird cocks its head to one side, watching us.\n\nIt blinks, then flies away.\n\nI stand, holding Goose's and Kendra's hands, and watch it disappear into the sky.\n\n#\n\n#\n\n_Goose_\n\nAnd we live happily ever after.\n\nReally. That's all.\n\nThe End\n\n## Excerpt from _Beheld_\n\n## Prologue\n\nI know that children don't read fairy tales anymore. Oh, they see the movies\u2014animated, sweet ones with helpful birds and talking raccoons, problematic ones where passive young women simply sleep and wait for their princes to come. But those are made-up stories. The real stories, stories that have recurred time and time again, are far more brutal. Stepmothers ordering their daughters' hearts brought to them to eat raw. Young women cutting off their toes to fit an idealized vision of female beauty. And those are just the romances!\n\nI know, for I have been alive for much of this time. Not all, of course. These tales date back to the ancient Greeks, and I'm not _that_ ancient. Still, I have lived as a witch since my birth in 1652, and as a teenager since I was one, over three hundred years ago.\n\nIn that time, I have sought love. Once, I found the man I thought would be mine forever. But I have lost him time and time again. This story is about how I found love and lost it. I don't know how it will end.\n\nBut it started in Salem, Massachusetts. I may, in previous accounts, have fibbed a bit when I said I wasn't there. It is _such_ a clich\u00e9 to claim one was in Salem. But I was. Most of those accused as witches there weren't actually witches, but a few of us were.\n\nOr, at least, two.\n\n##\n\n_Witches and Wolves_\n\n_Salem, Massachusetts_\n\n_January 1692_\n\nI might not have stayed in Salem had it not been for James. I might have been safer. But I have never been one to court safety above all, and I wasn't in 1692.\n\nIt was in 1692 that I fell in love with James.\n\nThen I had been alive close to two score years, but like most magical beings, I did not look it. Nay, I did not feel it either. This was convenient, as few things in my life were, for appearing mature carries with it certain expectations\u2014that the person will marry, have children, _be_ mature. I wanted none of that, for few people were like I was. They would age. They would die, as my family had.\n\nI would not, as long as I stayed clear of fire. Fire was the only thing that could kill a witch. Still is.\n\nI knew not to play with fire.\n\nI knew, also, not to play at love. Love would only lead to painful loss.\n\nBut then I met James.\n\nIt happened one morning, early, so early that my breath was a silver cloud on night black as my cloak. I was out chopping wood for the family's needs. I was a servant, but the Harwoods were not wealthy, so I was rather a maid-of-all-trades\u2014chop the wood, darn the socks, watch the babes. It reminded me of life with my own family, back when I had one.\n\nThat morning, the spring breezes had not yet chased away the winter cold, but I was warm, for I was working. Goody Harwood kept a close watch on me, so I could not use magic. Not all the time, anyway.\n\nIf you think I was working like the mature woman I should have been, you do not know me well. I was slim, as I still am. Every swing of the ax was a herculean effort. I had been out close to an hour and had only two bone-thin logs to show for it. I knew that soon, she would be there, spying for me, accusing me (not incorrectly) of malingering. I had to move quickly.\n\nI picked up the ax.\n\nJust as I did, a black shape crossed my vision. Bird!\n\nThis was enough to make me stop again. The birds had left for winter and, thus far, had not returned. And this was no robin redbreast, but a crow.\n\nI had a history with crows.\n\nI examined the bird. It was a large one with a yellow bill. It flew around me just above my head and, finally, settled on the very log I had been about to split.\n\nI laid down my ax, sighing as it sank into a snowdrift. My hands were bare and would surely freeze when I reached in.\n\nI shooed the bird.\n\nIt did not move. Nor the second nor the third time, either. It merely stared with its black bead eyes, as if it intended to speak.\n\nFinally, I reached for the ax. The blade was freezing. I meant to swing it just once.\n\nWhen I rose, the bird had disappeared.\n\nNot entirely pleased at the end of my excuse for idleness, I returned to my chopping.\n\n\"Mistress!\"\n\nA voice interrupted me, startled me.\n\nI whirled to see where it came from, for I had been sure I was quite alone.\n\n\"Your humble servant,\" someone said, and he bowed.\n\nHe wore black, at least what I could see, from the toes of his shoes to his hat. With his face thus obscured, he might have been any man I had seen before, any man in Salem, farmers beaten down by the winter's struggle, old before their time.\n\nBut when he rose, I knew I had never seen him before.\n\nI would have remembered.\n\nThe man staring back at me was beautiful in an unearthly way, with hair the color of fallen pine needles, skin that had never known harsh sun or harsher winter cold, and eyes a shade bluer than the bluest ocean. He was perhaps two years older than I\u2014meaning two years older than I appeared, so still in the bloom of youth, tall and strong.\n\nI hesitated. I wanted his help as much to keep him there as to get out of my work. But neither motive was proper for a girl my age, a girl any age in Salem. I glanced around. No signs of life anywhere except for the trickle of smoke from the chimney. I had built a fire when I'd risen. With any luck, the Harwoods would gather by it and Goody Harwood would not come looking for me when she needn't.\n\nI nodded, trying to pull my gaze down like a proper young lady.\n\n\"If you please,\" I said.\n\nHe moved closer and, at first, I started at his nearness. Then I realized he meant to take the ax from me. I held it out to him, trying to lower my eyes.\n\nI saw him notice, and his gaze upon me made me look down all the more. Yet I so wished to stare at him. I held my arms around my body, pretending only to be affected by the cold.\n\nI knew it was more than that.\n\nHe took the ax, brushing each of my gloveless hands with his own. They were so warm, and I sank a bit when the weight was removed.\n\nFinally, I glanced up, for he was very tall, and when I did, I saw him smile.\n\nI pulled my eyes away, but his smile remained in my memory. He was _so_ handsome.\n\n\"There now.\" He spoke with a bit of an accent, from Scotland. \"You are too young and too lovely for such hard work.\"\n\nI looked down harder.\n\n\"I am not as young as you might believe.\" I backed away.\n\n\"Nor am I.\" He made no move to chop the wood. \"And I know things. Have you heard about what is happening in Salem?\"\n\nI had. At least, I thought I had. There were rumors of children bewitched by demons. But I did not want to admit that it concerned me. If I did, he might suspect how much it did. And why.\n\nSo I said, \"I know little. I spend my days and nights just as you see me and my Sundays in worship.\"\n\nThe left corner of his mouth came up as if to call me on this lie. \"Like any God-fearing young maiden.\"\n\nI nodded. \"Of course.\"\n\nHe nodded, half gravely. \"Then I should tell you. It happened in town, at Reverend Parris's house. His daughter, Betty, and niece, Abigail, have been behaving . . . bizarrely.\"\n\nI had heard it. Young girls barking like dogs, writhing and crying out as if in pain. I had not done it. Nor were there any other witches in these parts. Perhaps there was a fungus in their flour. Perhaps they just wished for attention. But I knew better than to say that.\n\n\"I see.\" I managed a nod.\n\n\"But did you know that in Boston four years ago, a young woman was stricken with similar symptoms?\"\n\nAye. I had heard something of that.\n\nI shook my head.\n\n\"She was, and a woman named Ann Glover was hanged as a witch based upon the suspicion that she had enchanted the girl.\"\n\n\"What has this to do with me?\" I asked. \"Why are you telling me this?\"\n\nI had stood out too long with too little work, and now my body was cold, so cold it felt as if the bones might snap.\n\nHis words did nothing to warm me. \"Because it concerns you, Kendra.\"\n\n\"Why?\" How did he know my name?\n\nBut then I heard the creak of the opening door. I whirled to make my excuses to Goody Harwood, but she smiled.\n\n\"Oh! I thought to hurry you along. The fire is waning, and you must make the breakfast still. But I see you have been harder at work than I suspected. I suppose I couldn't hear the thuds for the gusts.\"\n\nAs if to answer, the wind whipped through me, ruffling my hair. I turned away.\n\nGoody Harwood had not mentioned the man who was there, and when I turned, I saw why. He was gone, gone as if he had never existed. But in his place was a cord of neatly stacked logs. A crow set atop them.\n\nI took a shaky breath. I felt about to choke. \"I will be but a moment longer.\"\n\nAnother gust shook the branches, and she shut the door against it.\n\nWhen I turned back, the wood was still there, and the man. I had not imagined it, any of it.\n\n\"How did you . . . ?\" A thousand questions leaped to mind, but I completed the one I had started. \"How did you know?\" _My name? That I was a witch?_\n\n\"I knew because I knew. James Brandon, at your service.\"\n\n\"I have to go inside, sir.\"\n\n\"Nay.\" His blue eyes were intense now. \"You should leave Salem, and quickly. This place is not safe. For you. For any of us. But I will stay and see it out, to protect innocents. You should protect yourself.\"\n\nDid he mean to say that he was a witch\u2014a wizard\u2014himself? I wanted to know, and yet my need to flee him was stronger. \"I must go inside, sir. The family will wonder about me. I have to make the breakfast.\"\n\nHe gathered some of the wood and brought it to me. As he did, he met my eyes, and for a moment, the wind ceased and the air became first warm, then hot around me, until I felt like I might burn through the drifted snow and not be unearthed until springtime.\n\n\"Then I will see you soon, Kendra,\" he said. \"I will see you every day until you agree to leave. Now go inside.\"\n\nI could not turn away from him easily, but I forced myself. I had great experience in taking leave of people. I reached for the doorknob.\n\n\"One other thing.\" His voice interrupted me. \"Beware of wolves.\"\n\nI turned back, but when I did, he wasn't there. In his place was the black crow, staring at me with bright eyes.\n\nIt flew away.\n\nThe wind began to howl again and did not stop until I was inside the house.\n\n## Back Ads\n\nDISCOVER\n\nyour next favorite read\n\nMEET\n\nnew authors to love\n\nWIN\n\nfree books\n\nSHARE\n\ninfographics, playlists, quizzes, and more\n\nWATCH\n\nthe latest videos\n\nTUNE IN\n\nto Tea Time with Team Epic Reads\n\n## About the Author\n\nPhoto credit Gene Flinn\n\n**ALEX FLINN** loves fairy tales and is the author of the #1 _New York Times_ bestselling _Beastly_ , a spin on _Beauty and the Beast_ that was named a _VOYA_ Editor's Choice and an ALA Quick Pick for Reluctant Young Adult Readers. _Beastly_ is now a major motion picture starring Vanessa Hudgens. Alex also wrote _A Kiss in Time_ , a modern retelling of _Sleeping Beauty; Cloaked_ , a humorous fairy-tale mash-up; _Bewitching_ , a reimagining of fairy-tale favorites, including _Hansel and Gretel, Cinderella, The Princess and the Pea,_ and _The Little Mermaid_ , all told by Kendra, the witch from _Beastly_ ; and _Towering_ , a darkly romantic take on _Rapunzel_. Her other books for teens include _Breathing Underwater, Breaking Point, Nothing to Lose, Fade to Black_ , and _Diva_. She lives in Miami with her family. Visit her online at www.alexflinn.com.\n\nDiscover great authors, exclusive offers, and more at hc.com.\n\n## Books by Alex Flinn\n\nBEASTLY\n\nBEASTLY: LINDY'S DIARY\n\nBEWITCHING\n\nMIRRORED\n\nTHREE BEASTLY KENDRA CHRONICLES\n\nFOUR BEASTLY KENDRA CHRONICLES\n\nBEHELD\n\nCLOAKED\n\nA KISS IN TIME\n\nTOWERING\n\nA MAGICAL TRIO\n\nBREATHING UNDERWATER\n\nDIVA\n\nFADE TO BLACK\n\nNOTHING TO LOSE\n\nBREAKING POINT\n\n## Credits\n\nCover art \u00a9 2015 by Howard Huang\n\nCover design by Heather Daugherty\n\n## Copyright\n\nHarperTeen is an imprint of HarperCollins Publishers.\n\nMIRRORED. Copyright \u00a9 2015 by Alexandra Flinn. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.\n\nwww.epicreads.com\n\n* * *\n\nLibrary of Congress Cataloging-in-Publication Data\n\nFlinn, Alex.\n\nMirrored \/ Alex Flinn. \u2014 First edition.\n\npages cm\n\nSummary: A modern, multigenerational tale of Kendra, the witch from \"Snow White,\" who trains Violet, an ugly, lonely, and heartbroken girl in the 1980s who transforms herself into \"the fairest one of all\" but still cannot win Greg's heart, and Celine, Greg's daughter with Violet's high school rival, Jennifer.\n\nISBN 978-0-06-213451-6 (hardcover)\n\nEPub Edition \u00a9 August 2015 ISBN 9780062134547\n\nVersion 11282016\n\n[1. Beauty, Personal\u2014Fiction. 2. Witches\u2014Fiction. 3. Popularity\u2014Fiction. 4. Schools\u2014Fiction.] I. Title.\n\nPZ7.F6395Mir 2015 |  2014041196  \n---|---  \n[Fic]\u2014dc23 |  CIP  \n|  AC\n\n* * *\n\n15 16 17 18 19 PC\/RRDH 10 9 8 7 6 5 4 3 2 1\n\nFIRST EDITION\n\n## About the Publisher\n\n**Australia**\n\nHarperCollins Publishers Australia Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\nwww.harpercollins.com.au\n\n**Canada**\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON M4W 1A8, Canada\n\nwww.harpercollins.ca\n\n**New Zealand**\n\nHarperCollins Publishers New Zealand\n\nUnit D1, 63 Apollo Drive\n\nRosedale 0632\n\nAuckland, New Zealand\n\nwww.harpercollins.co.nz\n\n**United Kingdom**\n\nHarperCollins Publishers Ltd.\n\n1 London Bridge Street\n\nLondon SE1 9GF, UK\n\nwww.harpercollins.co.uk\n\n**United States**\n\nHarperCollins Publishers Inc.\n\n195 Broadway\n\nNew York, NY 10007\n\nwww.harpercollins.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## Trisha Ashley\n\n## The Magic of Christmas\n\n## Dedication\n\nFor my son, Robin Ashley,  \nwith love.\n\n## Contents\n\nTitle page\n\nDedication\n\nPrologue: December 2005, Winter of Discontent\n\nChapter 1: Old Prune\n\nChapter 2: All Fudge\n\nChapter 3: Bittersweet\n\nChapter 4: Mushrooming\n\nChapter 5: Sweet Mysteries\n\nChapter 6: Driven Off\n\nChapter 7: Loose Nuts\n\nChapter 8: Well Braced\n\nChapter 9: Soul Food\n\nChapter 10: Cornish Mist\n\nChapter 11: Popped Corks\n\nChapter 12: Just Desserts\n\nChapter 13: Raspberries\n\nChapter 14: Slightly Curdled\n\nChapter 15: Drink Me\n\nChapter 16: Unrehearsed Entrances\n\nChapter 17: Tart\n\nChapter 18: Simmering Gently\n\nChapter 19: Stirring\n\nChapter 20: Freshly Minted\n\nChapter 21: Slightly Stewed\n\nChapter 22: Given the Bird\n\nChapter 23: Put Out\n\nChapter 24: Flamb\u00e9\n\nChapter 25: Cr\u00e8me de Coeur\n\nChapter 26: Crackers\n\nChapter 27: Charmed\n\nChapter 28: Cold Snap\n\nChapter 29: Clueless\n\nChapter 30: Unscheduled Appearances\n\nChapter 31: Middlemoss Marchpane\n\nChapter 32: Hoar Frost\n\nChapter 33: Well Stirred\n\nForget the Jimmy Choos, Chocolate Shoes And Wedding Blues Is the Only Accessory You Need For Spring 2012...\n\nTwelve Days of Christmas\n\nAbout the Author\n\nOther Books by the Same Author\n\nCopyright\n\nAbout the Publisher\nThe Magic of Christmas is loosely based on one of my earlier novels, Sweet Nothings, with the addition of a lot of new material. I felt there was so much more to say about the village of Middlemoss and all the characters who live there, especially Lizzy and her friends in the Christmas Pudding Circle, the annual Boxing Day Mystery Play and the vanishing squirrels!\n\n## Prologue: December 2005, Winter of Discontent\n\nThe venue for the last Middlemoss Christmas Pudding Circle meeting of the year (which was usually more of an excuse for a party) had been switched to Perseverance Cottage because Lizzy's thirteen-year-old son had come down with what she'd thought was flu and she wanted to keep an eye on him.\n\nLater, looking back on the events of that day, it seemed to Lizzy that one minute she'd been sitting at the big pine table in her kitchen, wearing a paper hat and happily debating the rival merits of fondant icing over royal with the other four members of the CPC, and the next she was frantically snatching at the card listing the symptoms of meningitis, which she kept pinned to her notice board, and shouting to Annie, her best friend, to ring for an ambulance.\n\nAt the hospital, Jasper changed frighteningly fast from a big, gruff teenager to a pale, sick child, and Lizzy tried urgently to contact her husband, Tom, who was away on one of his alleged business trips. But as usual he didn't answer his mobile and was nowhere to be found, so all she could do was leave messages in the usual places... and several unusual ones.\n\nThe hospital radio was softly warbling on about decking the halls with boughs of holly, but Lizzy, filled with a volatile mixture of desperate maternal fear and anger, wanted to deck her selfish, unreliable husband.\n\nIt was just as well that Annie was such a tower of strength in an emergency! During that first long day while Lizzy anxiously waited for the antibiotics to kick in, her friend popped in and out between jobs for the pet-sitting agency she ran, visited Perseverance Cottage to feed the poultry and let out Lizzy's dog, and reassured Tom's elderly relatives up at the Hall that she would keep them updated with every change in Jasper's condition.\n\nThen in the evening she returned to the hospital and she and Lizzy spent the long night watches sitting together while Jasper slept, reminiscing in hushed voices about when they first met and became best friends at boarding school. Lizzy had begun spending the holidays with Annie's family in the vicarage at Middlemoss, where she was quickly absorbed into the Vane household, much to the relief of the elderly bachelor uncle who was her guardian \u2013 and it was also in Middlemoss that she'd met Tom and Nick Pharamond, cousins who were often farmed out with relatives up at the Hall in the school holidays.\n\nNick was the eldest: quiet, serious and appearing to prefer the company of the cook at Pharamond Hall to anyone else's. Tom, who was really only nominally a Pharamond, his mother having married into the family, was the opposite: mercurial, charming and gregarious, though he'd had a quick temper and a sharp tongue, even then...\n\nNick was the first to fly the nest. Having inherited the Pharamond cooking gene in spades, it wasn't a huge surprise to anyone except his staid stockbroker father when he took off around the world at eighteen, tastebuds and recipe notebook at the ready. Now he was chief cookery writer for a leading Sunday newspaper and author of numerous books and articles, while Tom, in contrast, had dropped out of university and gravitated down to the part of Cornwall where many of his more useless friends had also ended up.\n\nWhen he set eyes on Lizzy again after a long interval, it was across a buffet table at a large party in London, where he was a guest, and where she and Annie, who'd done a French cookery course after school, were helping with the catering. He fell suddenly in love with her, a passion that also embraced her rose-tinted dreams of a self-sufficient existence in the country.\n\nSomehow she'd forgotten about his dark good looks, his overwhelming charm and his quirky sense of humour... Before she'd had time to think \u2013 or to remember his quick temper, occasional sarcasms and how short-lived his enthusiasms had been in the past \u2013 he'd swept her off her feet, into a registry office and down to the isolated hovel he was renting in Cornwall.\n\n'Marry in haste, repent at leisure,' she said to Annie, as Jasper stirred restlessly in his hospital bed. 'You tried your best to warn me not to rush into it.'\n\n'You fell in love and so did Tom: there was no stopping you,' Annie said. 'Besides, you were addicted to all those books about living in Cornish cottages, with donkeys and daffodils and stuff.'\n\n'True,' Lizzy agreed wryly, 'and it was blissful that first summer \u2013 until the reality of living in a dank, dilapidated cottage in winter with a newborn baby set in, especially after Tom started vanishing for days on end without telling me when and where he was going.'\n\n'He was worse after Jasper was born, wasn't he? I think he resented not being the centre of attention,' Annie said.\n\n'He still does, though how you can be jealous of your own son, goodness knows! Anyway, it was like living with a handsome but unreliable tomcat... and nothing much has changed, has it?' Lizzy asked bitterly.\n\n'Perhaps not, but at least two good things came out of your marriage,' Annie pointed out, being a resolutely glass-half-full person: 'Jasper and your books about life in Perseverance Cottage.'\n\n'True, and it was thanks to your telling Roly how cold and damp the cottage was, after you visited us, that he offered us a house on the estate rent free, so that actually makes three good things.'\n\n'Oh, yes \u2013 and it was marvellous when you came back to Middlemoss to live,' Annie agreed fervently. 'I'd missed you so much!'\n\nHer voice had risen slightly and Jasper woke up and grumpily demanded why they were muttering over him like two witches. Then he complained that the dim light hurt his eyes, and a nurse appeared and firmly ushered them out of the room for a while.\n\nThe following morning it was clear that the antibiotics were working. Great-uncle Roly visited Jasper in the afternoon and by evening he was so obviously on the mend that Lizzy managed to persuade Annie, who'd brought sandwiches and a flask of soup ready to share a second night's vigil with her, to go home instead and get some sleep.\n\nLizzy herself intended spending a second night there, of course: by Jasper's bedside when allowed, or in the stark waiting room, with its grey plastic-covered chairs and stained brown cord carpet.\n\nIt was in the latter room that Tom's cousin Nick Pharamond found her, having driven non-stop halfway across Europe since Roly had given him the news about Jasper. His brow was furrowed with added frown lines from tiredness, and the dark stubble and rumpled black hair didn't do much to lighten his usual taciturn expression. Lizzy always imagined that Jane Eyre's Mr Rochester would have been exactly like Nick, but she was still both delighted and relieved to see him because, unlike Tom, you could always rely on him to turn up in an emergency.\n\nAlthough she wasn't normally a weepy sort of person, she instantly burst into tears all over his broad chest, while he patted her back in a strangely soothing way. Then he made her drink the hot soup Annie had left and eat a sandwich she didn't want: he was forceful as well as reliable.\n\nThe only downside to his presence during the rest of that long night was that Lizzy became so spaced out with shock and exhaustion that something unstoppable took over her mouth. She could hear her own voice droning on and on for hours, telling Nick a whole lot of really personal stuff about the last few years that she'd only previously confided to Annie, like how bad relations had become between her and Tom, especially since she found out about his latest affair.\n\n'I don't know who this one is, but she's been having a really bad influence on him. He's played away before, of course, but it was never serious. He says it's my fault anyway, for being so wrapped up in the cottage, the garden and Jasper \u2013 and perhaps it is.'\n\n'That's totally ridiculous, Lizzy: of course it isn't your fault!' Nick said. 'He should grow up!'\n\nFilled with gratitude at his understanding, she'd fished out a petrol receipt from the bottom of her handbag and on the back of it feverishly scribbled down her cherished recipe for mashed potato fudge, a creation she'd first invented while trying to cook up some comfort from limited ingredients down in Cornwall (and which was much later to be christened Spudge by Jasper).\n\nIn return Nick, who was normally pretty tight-lipped on anything personal, divulged that Leila (his wife) refused all his suggestions that they both cut down their working hours to spend more time together, so they seemed to be seeing less and less of each other. This was really letting his guard down, so the night-watch effect must have been getting to him, too.\n\n'Do you think everything will be all right with me and Tom once Jasper's off to university in a few years and I'm not so tied to Middlemoss and the school run?' she asked Nick, optimistically. 'I could even go with him on some of his business trips to Cornwall.'\n\n'I honestly don't know, Lizzy, but it won't be your fault if it isn't,' Nick said, and gave her a big, wonderfully comforting hug.\n\nThen something made her look up and over his shoulder she caught sight of Tom standing in the doorway staring at them.\n\n'Oh, Tom, where have you been?' she cried, releasing herself from Nick's arms. 'Still, never mind \u2013 you're here now, that's the main thing.'\n\nTom ignored her, instead demanding suspiciously of Nick, 'What are you doing here, that's what I want to know?'\n\nHe was still looking from one to the other of them as if he'd had an extremely odd idea, which it emerged later he had \u2013 one that would finally turn what had already become a very sour-sweet cocktail of a marriage into a poisoned chalice.\n\nBut at the time, all Lizzy registered was that his first words were not an urgent enquiry about his only child and, in one split second, not only did the last vestiges of her love for Tom entirely vanish, but they took even the exasperated tolerance of the previous years with them, so there was absolutely no hope of resuscitating their marriage.\n\nIf Tom had ever possessed the core of feckless sweetness she'd believed in, then some wicked Snow Queen had blown on his heart and frozen it to solid ice.\n\n## Chapter 1: Old Prune\n\nHere in Middlemoss Christmas preparations start very early \u2013 in mid-August, in fact, when the five members of the Christmas Pudding Circle bulk-order the ingredients for mincemeat and cakes from a nearby wholefood cooperative. Once that has arrived and been divided up between us, things slowly start to rev up again. It always reminds me of a bobsleigh race: one minute we're all pushing ideas to and fro to loosen the runners and then the next we've jumped on board and are hurtling, faster and faster, towards Christmas!\n\nThe Perseverance Chronicles: A Life in Recipes\n\nThe members of the Christmas Pudding Circle were sitting round my long, scrubbed-pine kitchen table for the first meeting of the year. It was a hot, mid-August morning, so the door was open onto the sunlit cobbled courtyard in order to let some cooling air (and the occasional brazen hen) into the room.\n\nI poured iced home-made lemonade into tumblers, then passed round the dish of macaroons, thinking how lovely it was to have all my friends together again. Apart from my very best friend Annie Vane, there was Marian Potter who ran the Middlemoss Post Office, Faye Sykes from Old Barn Farm and Miss Pym, the infants' schoolteacher. The latter is a tall, upright woman with iron-grey hair in a neat chignon, who commands such respect that she's never addressed by her Christian name of Geraldine, even by her friends.\n\n'Oh, I do miss our CPC meetings after Christmas each year,' Annie said, beaming, her round freckled face framed in an unbecoming pudding-bowl bob of coppery hair. 'I know we see each other all the time, but it isn't the same.'\n\n'I was just thinking the same thing,' I agreed. 'And it doesn't matter that it's midsummer either, because I still get a tingle down my spine at the thought that we've started counting down to Christmas.'\n\n'I suppose we are in a way, but it's more advance planning, isn't it?' Faye said.\n\n'Yes, and we'd better get on with it,' Marian said, flicking open a notebook and writing in the date, for she organises the CPC just as she, together with her husband Clive, run most of the events around Middlemoss. As usual, she was bristling with energy right down to the roots of her spiky silver hair. 'First up, are there any changes to the list of ingredients for Miss Pym to order?'\n\n'I still have last year's list on my computer, so it will be easy to tweak it before I email it off,' Miss Pym said, helping herself to more lemonade. An ice-cube cracked with a noise like a miniature iceberg calving from a glacier.\n\nBut there was not much to tweak, for of course we mostly make the same things every year: mince pies, Christmas cakes and puddings. We need large quantities too, for as well as baking for our own families, we also make lots of small cakes for the local Senior Citizens Christmas Hampers, which are annually distributed by Marian and the rest of the Mosses Women's Institute.\n\n'Who has got the six small cake tins for the hamper Christmas cakes?' asked Annie.\n\n'Me,' I said.\n\n'I'll put you down to bake the first batch then,' Marian said, scribbling that down, then she handed out the CPC meetings rota. We're supposed to take it in turns to host it in our homes but I don't know why she bothers, because after the first one it always goes completely haywire for one reason or another.\n\nThe important business of the meeting concluded, I got out some coffee granita I'd made. It never tastes quite as perfect as I hope it will, but they were all very kind about it. Then the conversation turned to frozen desserts in general and we discussed the possibility of concocting a brandy butter ice cream to go with Christmas pudding. I think Faye started that one: she makes a lot of ice cream for her farm shop.\n\nWriting the CPC meeting up later for the Chronicles, I added a note to include the recipe for the brandy butter ice cream to that chapter if one of us came up with something good, and then laid my pen down on the kitchen table with a sigh, thinking that it was just as well I had the Christmas Pudding Circle to write about.\n\nAlthough my readers loved the mix of domestic disaster, horticultural endeavour and recipes in my Perseverance Chronicle books, I could hardly include bulletins on the way the last, frayed knots of my failed marriage were so speedily unravelling, which was the subject most on my mind of late. I had become not so much a wife, as landlady to a surly, sarcastic and antisocial lodger.\n\nThe first Perseverance Chronicle was written in a desperate bid to make some money soon after we were married, influenced by all the old cosy, self-sufficiency-in-a-Cornish-cottage books that I had loved before the reality set in. Mine were a little darker, including such unromantic elements as the joys of outside toilets when heavily pregnant in winter and having an Inconstant Gardener for a husband.\n\nIt was accepted by a publisher and when we moved back to Lancashire I simply renamed the new cottage after the old and carried on \u2013 and so, luckily, did those readers who had bought the first book.\n\nMy self-imposed quota of four daily handwritten pages completed (which Jasper would type up later on the laptop computer Unks bought him, for extra pocket money), I closed the fat A4 writing pad and turned to my postcard album, as to an old friend. This was an impressively weighty tome containing all the cards sent to me over the years by Nick stuck in picture-side down, since interesting recipes were scribbled onto every bit of space on the back in tiny, spiky handwriting.\n\nHe still sent them, though I hadn't seen very much of him in person, other than the occasional Sunday lunch up at Pharamond Hall, since the time Jasper was ill in hospital. And actually I was profoundly grateful about that, what with having poured my heart out to him in that embarrassing way, not to mention Tom suddenly getting the wrong idea when he arrived and found Nick comforting me...\n\nAnd speak of the devil, just as I found the card I wanted, a dark shape suddenly blocked the open doorway to the yard and Tom's voice said, 'Reading your love letters?'\n\nHe was quite mad \u2013 that or the demon weed and too much alcohol had pickled his brain over the years! The album was always on the kitchen bookshelf for anyone to read, so he knew there was nothing personal about the cards \u2013 unless he thought that addressing them to 'The Queen of Puddings' was lover-like, rather than a sarcastic reference to one of my major preoccupations.\n\nMind you, Tom was not much of a reader, though luckily that meant he had never, to my knowledge, even opened one of my Perseverance Chronicles.\n\n'No, Tom, I'm looking for a particular marzipan petit four recipe for the Christmas Pudding Circle to try,' I said patiently. 'The only love letters I've got are a couple of short notes from you, and they're so old the ink's faded.'\n\n'So you say, but I don't find you poring over them all the time, like you do over Nick's precious postcards,' he said, going to wash his hands at the kitchen sink.\n\nI dished out some of the casserole that was simmering gently on the stove and put it on a tray, together with a chunk of home-made bread, since he now preferred to take all his meals alone in the sitting room in front of his giant TV. Jasper and I had the old set in the kitchen and tended to leave him in sole possession.\n\nHe picked up the bowl of stew now and stared into it like a sibylline oracle, but the only message he was likely to read was 'Eat this or go hungry.'\n\n'What are these black things, decayed sheep's eyeballs?'\n\n'Prunes. It's Moroccan lamb tagine.'\n\nFrom his expression you would have thought I'd offered him a dish of lightly seasoned bat entrails.\n\n'And I suppose Nick gave you the recipe. What else has he given you lately?' he said, with a wealth of unpleasant innuendo. 'Don't think I haven't noticed that your son looks more like him every day!'\n\n'Oh, for God's sake, don't start on that again!' I snapped, adding recklessly, 'You know very well why Jasper looks like Nick, just as you look like Great-uncle Roly: your mother must have been having an affair with Leo Pharamond while she was still married to her first husband! Why don't you ask her?'\n\nIt was certainly obvious to everyone else, since those slaty purple-grey eyes and raven-black hair marked out all the Pharamonds instantly. But Tom went livid and hissed like a Mafia villain in a bad film, 'Never ever malign my mother's name again like that \u2013 do you hear me?'\n\nThen he followed this up by hurling the plate of hot casserole at the wall with enormous force, shattering it and sending fragments of bowl and spatters of food everywhere. He'd never been physically violent (I wouldn't have stood for it for one second) so I don't think he was particularly aiming at me, but a substantial chunk of green-glazed Denby pottery hit my cheekbone and fell at my feet.\n\nIt was a shock, though, and I stood there transfixed and staring at him, one hand to my face, in a silence broken only by the occasional slither and plop of a descending prune. Suddenly finding myself released from thrall, I turned and walked out of the door, dabbing lamb tagine off my face with the hem of my pale green T-shirt as I went, then headed towards the village.\n\nI must have looked a mess, but luckily it was early evening and few people were about, for the Pied Piper of TV dinners had called them all away, using the theme tune of the popular soap series Cotton Common as lure.\n\nI didn't have far to go for refuge. Annie's father used to be the vicar here, but now that he and his wife are alleviating the boredom of retirement by doing VSO work in Africa, Annie has a tiny Victorian red-brick terraced cottage in the main street of Middlemoss.\n\n'Lizzy!' she exclaimed, looking horrified at discovering me stained and spattered on her doorstep. 'Is that dried blood on your face and T-shirt? What on earth has happened?'\n\n'I think it's only prune juice and gravy, actually,' I reassured her, touching my cheek cautiously. 'A bit of plate did hit me, but it must have had a round edge.'\n\n'Plate?' she repeated blankly, drawing me in and closing the front door.\n\n'Yes, one of those lovely green Denby soup bowls we had as a wedding present from your parents.'\n\n'Look, come into the kitchen and I'll clean you up with warm water and lint while you tell me all about it,' she said soothingly.\n\nThe lint sounded very Gone With the Wind \u2013 but then, she has all the Girl Guide badges and I don't suppose the First Aid one has changed for years. So I followed her in and sank down on the nearest rush-bottomed chair, my legs suddenly going wobbly. Trinity (Trinny, for short), Annie's three-legged mutt, regarded me lambently from her basket, tail thumping.\n\n'There's nothing much to tell, really,' I said. 'Tom flew into one of his rages and lobbed his dinner at the wall.'\n\n'Oh, Lizzy!'\n\n'I said something that made him angry and he just totally lost it this time. I don't think he was actually aiming at me, though it's hard to tell since he's such a rotten shot and \u2013 ouch!' I added, as she dabbed my face with the warm, damp lint.\n\n'The skin isn't cut, but I think you might get a bruise on your cheek,' she said, wringing the cloth out. 'I could put some arnica ointment on it.'\n\n'I don't think I could live with that smell so close to my nose, Annie,' I said dubiously, but her next suggestion, that we break out the bottle of Remy Martin, which she keeps in stock because her father always swore by it in times of crisis, met with a better reception.\n\n'I think you really ought to leave Tom right away, Lizzy,' Annie suggested worriedly. 'He's been so increasingly horrible to you that it's practically verbal abuse \u2013 and now this!'\n\n'I'm just glad Jasper wasn't there,' I said, topping my glass up and feeling much better. 'He's gone straight from the archaeological dig to a friend's house, and won't be back till about ten.'\n\n'His exam results should be here any time now, shouldn't they?'\n\n'Yes, only a couple more days.' I sipped my brandy and sighed. 'Even though I'll miss him, it'll be such a relief to have him safely off to university in October, because I live in dread that Tom will suddenly tell him to his face that he doesn't think he's really his son. That would be even more hurtful than ignoring him, the way he's been doing the last couple of years.'\n\n'I don't know what's got into Tom,' Annie said sadly. 'He always had so much charm... as long as he got his own way.'\n\n'He still does charm everyone else. I'm sure no one would believe me if I told them what he's really like at home.'\n\n'True, but he's so used to me being around, he's let the mask slip sometimes, so I've seen it for myself,' Annie said. 'He was all right with Jasper for the first few years, though, wasn't he?'\n\n'Well, he didn't take a lot of notice of him, but he was OK. But he started to turn colder towards me even before he got this strange idea that I had a fling with Nick, so I think whoever he's been having an affair with since then has had a really bad effect on his character.'\n\n'You did have a fling with Nick,' Annie pointed out fairly.\n\n'Oh, come on, Annie! I was way too young and anyway, it only lasted about a fortnight before he told me he was going abroad for a year because he wasn't changing his life-plans for my sake. I didn't see him after that until the day I got married to Tom and he turned up then with Leila in tow \u2013 do you remember?'\n\n'Gosh, yes. She was so scarily chic, in a Parisian sort of way, that she made me feel like a country bumpkin \u2013 she still does! But I thought it was nice of Nick to make the effort, even though he and Tom had grown apart over the years. They never had a lot in common, did they?'\n\n'I think the main problem was that Tom always felt jealous of Nick, since Nick was a real Pharamond and Roly's grandson, whereas he was just a Pharamond because his mother had married one. Allegedly,' I added darkly.\n\n'It's odd how things turn out,' mused Annie, putting away the bowl of water and tossing the lint into the kitchen bin. 'You always had much more in common with Nick than with Tom.'\n\n'How on earth can you say that, when we argue all the time?' I demanded incredulously. 'The only thing Nick and I have ever had in common is a love of food, even if mine is much less cordon bleu.'\n\nThough of course it is true that food has played an important part in both our families. The search for a good meal in the wrong part of a foreign city was the downfall of my diplomat parents and would be the downfall of my figure, too, were I ever to stop moving long enough for the fat to settle.\n\nAs to the Pharamonds, the gene for cooking was introduced into the family by a Victorian heir who married the plebeian but wealthy heiress Bessie Martin, only to die of a surfeit of home-cooked love some forty years later, with a fond smile on his lips and a biscuit empire to hand on to his offspring.\n\n'You and Nick have both got short tempers and you love Middlemoss more than anywhere else on earth,' Annie said. 'And of course I know that Jasper is Tom's son, but it's unfortunate that he's looking more and more like Nick with every passing year.'\n\n'Well, yes, that's what Tom said earlier, so I reminded him about the rumours that his mother had an affair with Leo Pharamond before her first husband was killed, and that's what started the argument off! He always flies into a complete rage if I say anything against his sainted mother.'\n\n'It's quite a coincidence that Leo Pharamond and her first husband were both not only racing drivers but killed in car crashes,' Annie said, 'though there did seem to be a lot of fatal crashes in the early days.'\n\n'Someone told me they called her the Black Widow after Leo died, so it's not surprising her third husband gave it up and whisked her off back to Argentina,' I said.\n\nTom's mother had started a whole new life out there, but her firstborn was packed off to boarding school and farmed out at Pharamond Hall in the holidays. That made us both orphans in a way, which had once seemed to make a bond...\n\nAnnie said, 'Tom's hardly seen his mother over the years, has he?'\n\n'No, or his half-siblings. He blames it all on his stepfather, of course, and won't hear a word against her. Come to that, I've only met her a couple of times and we can't be said to have bonded.'\n\n'You'd think she'd at least be interested in her grandson \u2013 Jasper's such a lovely boy,' Annie said fondly.\n\n'I used to send her his school photos, but since I never got any response, I gave up. In fact, with all this rejection, it's wonderful that poor Jasper isn't bitter and twisted, too!'\n\n'Oh, he's much too sensible and he knows we all love him: me, Roly, even Mimi.'\n\nI considered Unks' unmarried sister, Mimi, who is not at all maternal and whose passions are reserved for the walled garden she tends behind the Hall. 'You're right, she does seem to like him, despite his not being any form of plant life.'\n\n'And Nick is fond of him \u2013 Jasper and he get on well.'\n\n'He only really sees him during our occasional Sunday lunch up at the Hall, when we're all on our best behaviour for Roly's sake, because Tom's made it abundantly clear he isn't welcome at Perseverance Cottage.'\n\n'How difficult it all is!' Annie sighed, which was the under-statement of the year. 'I always agreed with Mum and Dad that marriage should be for ever, but once Tom started having affairs and being really nasty to you and Jasper, I changed my mind. He's not at all the man you married.'\n\n'Oh, I don't know,' I reflected. 'I think perhaps he is, it's just that his true nature was hidden underneath all that charm. His sarcastic tongue has suddenly become a lot more vicious, though, which I expect is because he really wants me out of the cottage now, but I mean to try and stick to my original plan and hang on until I've got Jasper settled at university. It doesn't do a lot for my self-confidence when Tom's constantly belittling me and telling me how useless I am, though.'\n\n'You're not useless,' she said, 'you've been practically self-sufficient for years in fruit, vegetables and eggs, made a lovely home for him and Jasper, and written all those wonderful books.'\n\n'I don't actually get paid very much for the Chronicles \u2013 they're a bit of a niche market \u2013 and I'm running late with the next, what with one thing and another.'\n\n'I suppose it's hard to think up funny anecdotes to go between the recipes and gardening stuff, what with all the worry about Tom. But if you want to leave him right now, you know you and Jasper can move in here any time you like, and stay as long as you want,' she offered generously.\n\n'I do know, and it's very kind of you,' I said gratefully, not pointing out that her cottage isn't much bigger than a doll's house: two tiny rooms up and down, crammed so full of bric-a-brac you can hardly expand your lungs to full capacity without nudging something over. Jasper, when he visits, tends to stand in the corner with his arms folded so as not to damage anything.\n\n'Once Jasper is at university I might have to take you up on that offer, but very temporarily. I'll still need to make a home for him to come back to. I'll have to get a job stacking supermarket shelves, so I can rent somewhere. I'm not really qualified to do anything else.'\n\n'Then what about Posh Pet-sitters? Business is expanding hugely since I added general pet-feeding and care to the dog-walking, and I could do with an assistant.'\n\nAnnie set up Posh Pet-sitters several years ago with a loan from her parents, and business seemed to be building up nicely, due to the patronage of several of the actors from the long-running drama Cotton Common, set in a turn-of-the-century Lancashire factory town, who have suddenly 'discovered' the three villages that comprise the Mosses.\n\nWhere they led, other minor celebrities followed, since although off the beaten track, we're within commuting distance of Manchester, Leeds, Liverpool and the M6, and in pretty countryside just where the last beacon-topped hills slowly subside into the fertile farmland that runs west to the coast.\n\nSome of the actors live in the new walled and gated estate of swish detached houses in Mossrow, but others have snapped up whatever has appeared on the market, from flats in the former Pharamond's Butterflake Biscuit factory, to old cottages and farms.\n\n'Did you go and see Ritch Rainford yesterday?' I asked, suddenly remembering how excited Annie had been at getting a call from the singer-turned-actor who plays Seth Steele, the ruggedly handsome mill owner in Cotton Common. (All that alliteration must have been too much for the producers of the series to resist!)\n\nHe's bought the old vicarage where Annie's family used to live, a large and rambling Victorian building with a brick-walled garden, in severe need of TLC and loads of cash. (The new vicar is now housed in an unpretentious bungalow next to the church.)\n\nAnnie's pleasantly homely face, framed in a glossy pudding-bowl bob of copper hair, took on an unusually rapt \u2013 almost holy \u2013 expression and her blue-grey eyes went misty. 'Oh, yes! He's...' She stopped, apparently lost for superlatives.\n\n'Sexy as dark chocolate?' I suggested. 'Toothsomely rum truffle?'\n\n'Just \u2013 wonderful,' she said simply. 'He has such charisma, it was as though a... a golden light was shining all around him.'\n\n'Bloody hell! That sounds more like finding all the silver charms in your slice of Christmas pudding at once!' I stared at her, but she was lost in a trance.\n\n'Lizzy, he's so kind, too! When I explained that I used to live at the vicarage, he took me around and showed me all the improvements he's made, and told me what else he was going to do. Then he just handed me a set of keys to the house so he could call me up any time to go and exercise or feed his dog.'\n\n'Well, if your clients didn't do that, you wouldn't be able to get in,' I said drily. 'What sort of dog does he have?'\n\n'A white bull terrier bitch called Flo \u2013 very good-natured, though I might have to be careful around other dogs.'\n\n'And what's the new vicar like?' I asked, but she hadn't noticed, being full of Ritch Rainford to the point where her bedazzled eyes couldn't really take in another man. However, a crush on a handsome actor was not likely to get her anywhere.\n\nAnnie was once engaged, but was jilted with her feet practically on the carpeted church aisle. Since then she had safely confined her affections to unsuitable \u2013 and unattainable \u2013 actors.\n\n'I've heard he's single and has red hair,' I said encouragingly since, despite her own copper locks, she has a weakness for redheaded men.\n\n'He hasn't got red hair, he's blond!' she protested indignantly, and I saw that she was still thinking of Ritch Rainford. Perhaps I ought to watch Cotton Common to see what all the excitement was about.\n\nEventually Annie ran me home, since I wanted to be there when Jasper returned. I was by then attired in one of her voluminous cardigans \u2013 a bilious green, with loosely attached knitted pink roses \u2013 to hide the dried but dubious-looking stains on my T-shirt.\n\nShe said she was going to come in with me and give Tom a piece of her mind, which would not have gone down well, but luckily Tom, his van and some of his clothes had vanished. He'd also locked me out; but not only did Annie have our key on her ring, I kept one hidden under a flower pot, so that wasn't a problem.\n\n'Looks like he's gone away again,' I said gratefully. 'Thank goodness for that.'\n\nOf course he hadn't thought to feed the hens, who had put themselves to bed in disgust, or the quail, so Annie helped me to shut everything up safely for the night.\n\nAs we walked back to the cottage Uncle Roly Pharamond's gamekeeper, Caz Naylor, sidled out of a small outbuilding and, with a brief salute, flitted away through the shadows towards the woods behind the cottage.\n\nHe's a foxy-looking young man, with dark auburn hair, evasive amber eyes and a tendency to address me, on the rare occasions when he speaks, as 'our Lizzy', thus acknowledging a distant relationship that all the Naylors in the area seemed to know about from the minute I set foot in the place for the first time at the age of eleven.\n\nAnnie looked startled: 'Wasn't that Caz? What's he doing here?'\n\n'I let him have the use of the old chest freezer in there. Since I cut down on the amount of stuff I grow, I don't need it,' I said, for I'd been slowly running things down ready for the moment that I knew was fast approaching, when I must leave Perseverance Cottage. 'He comes and goes as he pleases.'\n\nShe shook her head. 'All the Naylors are strange...'\n\n'But some are stranger than others? My mother was a Naylor too, don't forget! Descendant of some distant ancestor who made good in Liverpool, in the cargo shipping line \u2013 which at least explains why I'm such a daughter of the soil and feel so firmly rooted here.'\n\nShe smiled. 'I expect Roly told him to keep an eye on things after that animal rights group started targeting you.'\n\n'More likely he's keeping an eye on his freezer,' I said, though it was true that the only evidence of ARG (as they are known locally) I'd spotted around the place lately were the occasional bits of gaffer tape where a banner had been ripped off my car or the barn. 'Perhaps they just aren't bothering with me that much. I mean, I can see why they might target Unks and Caz, especially since no one knows what Caz does with all those grey squirrels he traps, but why me? I'm not battery farming anything.'\n\nAll my fowl lived long, happy and mainly useless lives, except for an excess of male quail and the occasional unwanted cockerel, which Caz dispatched for me with expert efficiency.\n\n'I expect they just include you in with the Pharamond estate, since your cottage is part of it,' she agreed. 'It's not personal.'\n\nWe cleaned up the mess in the kitchen as well as we could and then Annie left, since it was clear enough that Tom wasn't coming back that night, at least \u2013 and I thanked heaven for small mercies.\n\n'What happened to your face, Mum?' Jasper asked, getting his first good look at me in the light of the kitchen, when a friend dropped him home later. 'That looks like a bruise coming up. And why are you wearing one of Auntie Annie's horrible cardigans?'\n\n'Your father dropped a plate and a piece hit me,' I explained. 'Annie loaned me the cardigan to cover up the gravy stains on my T-shirt and I forgot to give it back when she went home.'\n\nHe looked at the dent and new marks on the plastered kitchen wall and said, 'He dropped a plate horizontally?' in that smart-lipped way teenage boys have.\n\n'Yes, he was practising discus throwing,' I said, and he gave me a look but let the subject drop.\n\nHe didn't ask where his father was. But then, at that time, he never did.\n\n## Chapter 2: All Fudge\n\nWe are in the middle of a hot spell and the air is fragrant with sweet peas and roses and full of the dull, drowsy drone of bees drunk on nectar. Yesterday I divided up the bigger clumps of chives and began drying herbs for winter, crumbling them up as soon as they were cool and storing them in cork-topped containers, though the bay leaves have simply been left in bunches hanging from the wooden rack in the kitchen. But soon they, too, will be packed in jars and put away in the cupboard until needed.\n\nAs I used up the final jar of last year's mincemeat for brownies, I wondered if mincemeat would also work as an ingredient in fudge \u2013 maybe even in Spudge, the mashed potato fudge I invented while we were living in Cornwall...\n\nThe Perseverance Chronicles: A Life in Recipes\n\nTom had been gone a couple of days when Jasper pointedly enquired after dinner one night if there was anything I wanted to discuss, but I just said we would have a little chat before he went to university and he gave me one of his looks.\n\nI knew he was now an adult, and at some point I'd have to explain to him that I was going to leave his father and the cottage as soon as he'd gone off to university, but at that moment he was so happy that he'd got the exam grades he needed for his first choice, I didn't want to rain on his parade.\n\nNext day, when I let out the hens, I found it was one of those delicious late summer mornings that reminded me of the early honeymoon weeks of our marriage in Cornwall: dreamy swirls of mist with the warm sun tinting the edges golden, like pale yellow candyfloss wisps. You could easily imagine King Arthur and Queen Guinevere riding out of it in glorious Technicolor, all jingling bridles and hooded hawks, though if they had they would probably have been surprised to find themselves transported from the land of legend into a Lancashire backwater like Middlemoss.\n\nThe last remaining acres of darkly watchful ancient woodland that crowded up to the back of Perseverance Cottage would have looked normal enough to them, I suppose \u2013 apart from Caz Naylor, who as usual was camouflaged from headband to boots, Rambo-style. I spotted him flitting in and out of the trees only by the white glint of his eyeballs and the sweat glistening between the green and brown streaks on his naked chest. A blink and he was gone, back to wage war on the dangerous alien life form known to the uninitiated as the grey squirrel.\n\nStill, even in Arthurian times they would probably have had some kind of shamanistic Green Man and so would be used to such goings-on, and the duckpond, chickens and vegetable patch out front would look reassuringly normal to them. But what would they have made of the huge, tumble-down old greenhouse, the remains of a previous tenant's abortive attempt at market gardening? Or my battered, once-white Citro\u00ebn 2CV? A 2CV that, I now noticed, had its hood down, so the seats would be soaked with dew and very likely lightly spattered with hen crap. Or even, which was much, much worse, duck gloop.\n\nIt was also listing drunkenly on one seriously flat tyre.\n\nTossing the last of the feed to the hens, I stuck my head inside the cottage door.\n\n'Jasper?' I called loudly up the steep stairs, expecting him to be still asleep. By nature, teenagers are intended to be nocturnal, so it felt cruel to have to drag him out of his lair under the eaves each morning.\n\nInstead, he loomed out of the doorway next to me, making me jump. 'I'm here, Mum. What's up?'\n\n'Flat tyre. You have your breakfast and get ready while I change it. I hope it's a mendable puncture \u2013 the spare's not that brilliant and if I have to buy a new one it'll be worth more than the rest of the car put together.'\n\nOne of the Leghorns had followed me into the flagged hallway (a Myrtle: all the white hens are called that; and the browns, Honey) and I shooed it out again. There's something terribly cement-like about hen droppings when they set hard.\n\n'I'll change it,' he offered. 'Or I can cycle over.'\n\n'No, I'll have it done by the time you've had breakfast, and you'll be late otherwise.'\n\nThe medieval dig he was working at was only a few miles away, but the lanes between the site and us were narrow and twisty, so I worried about his safety. Annie calls it 'mother hen with one chick' syndrome, but she is just as dotty about Trinity, her rescued dog. And if I hadn't been an anxious mother, then maybe I wouldn't have demanded the right treatment for Jasper's meningitis that time he was rushed into hospital, even before the tests came back positive... It didn't bear thinking about.\n\nJasper wandered out again a few minutes later holding a piece of toast at least an inch thick, not counting the bramble jelly and butter, removed the wheel brace from my hand (giving me the toast to hold in exchange), and unscrewed the last nut.\n\n'Thanks, that was stiff. You'd think if I'd tightened it up in the first place, I'd be able to undo it easily, wouldn't you?'\n\n'Dad not back yet?' Jasper asked, glancing across at the large, ramshackle wooden shed Tom used as his workshop, with the 'Board Rigid: Customised Surfboards' sign over it.\n\n'No.'\n\n'Well, remember that time you asked him to go and buy a couple of pints of milk, and you didn't hear from him for a week?' he said, clearly with the intention of comforting me should I need it. But actually, I was sure he shared my feeling that his father's increasing number of absences were a blessing, even though I was usually the one on the receiving end of Tom's viciously sarcastic outbursts.\n\nHe couldn't help but have noticed the way Tom had estranged himself from both of us, behaving more like a lodger than a husband and father.\n\nJust let me get him safely off to university in October, then I can sort my life out \u2013 somehow, I prayed silently.\n\nJasper said nothing more, but retrieved his toast and went back into the house.\n\nThe first golden glow of the morning was fading, much as my love for Tom had quickly vanished once I'd grasped what kind of man I'd married: the mercurial type, an erratic moon orbiting my Mother Earth solidity. For years I'd thought that deep down he loved and needed me, and he'd always managed to sweet-talk me into forgiving him for anything and everything, although my exasperation levels had slowly risen as my son matured and my husband remained as irresponsible as ever. Have you ever imagined what it would be like to be married to Peter Pan once the novelty wore off? A Peter Pan with a dark side he kept just for me... like a sweet chocolate souffl\u00e9 with something hard at its centre on which you could break your teeth \u2013 or your heart.\n\nHis cousin Nick, whose Mercedes sports car was slowly bumping down the rutted track towards me, scattering hens, wasn't any kind of souffl\u00e9 \u2013 more like one of his own devilishly hot curried dishes. He does cook like an angel, though, and he's an expert on all aspects of food and cooking, writes books and articles and has a page in a Sunday newspaper colour supplement.\n\nThe Pharamonds didn't seem to do marriage terribly well and he'd had a volatile, semidetached relationship with Leila for years. She's another chef, which was at least one too many cooks on the home front, by my reckoning. I was glad to see she wasn't with him that day, because Leila is a lemon tart. Or maybe, since she's French, that should be tarte au citron?\n\nMiaou.\n\nI resolved not to be catty about her, even if every time we met she contrived to make me feel like a lumbering great carthorse. She's an immaculately chic, petite, blue-eyed blonde, while I am tall and broad-shouldered, with green eyes flecked with hazel, fine light brown hair in a permanent tangle, and the sort of manicure you get from digging vegetable beds without gloves on.\n\nUnks \u2013 Great-uncle Roly \u2013 didn't like her either. He said if it weren't for her refusing to stop working all hours in her restaurant in London and settle down, there would have been lots of little Pharamond heirs by then. But he couldn't have thought this through properly, because if they were a combination of the scarier bits of Nick and Leila, that would be quite alarming indeed.\n\nLeila was married before and was fiercely independent, with her own swish apartment above her restaurant; while Nick had a small flat in Camden. And considering he spent at least half his time at Pharamond Hall, which Leila rarely visited, you'd wonder when they ever saw each other.\n\nI certainly hadn't seen Nick for ages. He always phoned up for any eggs, fruit or vegetables he needed when staying at the Hall and working on recipes, but I just dropped them off with Unks' cook, Mrs Gumball.\n\nYet here he was, deigning to pay me a visit. As his Mercedes pulled up I removed the jack and then slung the punctured tyre in the back of the car, where Jasper's bike already reposed. You can get anything in a 2CV, if you don't mind being exposed to the weather.\n\nNick got out. He was wearing dark trousers and an open-necked soft white shirt with the sleeves rolled up, the glossy, thick black plumage of his hair spikily feathering his head. His strong face, with its impressively bumpy nose, can look very attractive when he smiles, though the last time he'd wasted any of his charm on me was in the hospital when Jasper had meningitis. And after the way I'd bared my soul to him in the night hours, I could only feel profoundly grateful that I hadn't seen much of him since then.\n\nI distinctly remember telling him how I hoped that once Jasper was at university, things would get better between me and Tom \u2013 and instead, from that very moment they'd rapidly got worse and worse...\n\nI became aware that Nick was waving his hands slowly in front of my face, like a baffled stage hypnotist.\n\n'Planet Earth to Lizzy: are you receiving me?'\n\n'Oh, hi, Nick \u2013 long time, no recipe,' I said, wiping my filthy hands up the sides of my jeans \u2013 they were work ones, so it wasn't going to make a lot of difference. I only hoped I hadn't run them through my hair first, though since I didn't remember brushing it this morning, a bit of grease would at least hold the tangles down.\n\nHe frowned down at me. 'I sent you a card from Jamaica.'\n\n'That was ages ago, and a recipe for conch fritters isn't exactly the most useful thing to have in the middle of Lancashire \u2013 the fishmongers don't stock them. Anyway, what are you doing here at this time of the morning? Have you driven straight up from London?'\n\n'Yes, I'm looking for Tom,' he said shortly, checking me over with eyes the dark grey-purple of wet Welsh slate, as though he wasn't sure quite what species I was, or what sauce to serve me with. 'What have you done to your face?'\n\nI flushed and touched the bruise on my cheek with the tips of my fingers. 'This? Oh, a plate got dropped and one of the pieces bounced up and hit me,' I said lamely; it was almost the truth.\n\nHis brows knitted into a thick, black bar as he tried to imagine a plate that explosive.\n\n'It looks worse than it is, now it's gone all blue and yellow \u2013 it'll have vanished in a day or two. And Tom's away,' I added. Thank goodness!\n\nFrom the way Nick was looking at me I thought I'd said that aloud for a minute, but finally he asked, 'Oh? Any idea when he'll be back?'\n\n'No, but he's been gone since Monday, so I'll be surprised if he doesn't turn up today.'\n\nHe raised one dark eyebrow. 'And do you know where he's gone?'\n\n'He didn't say and there is no point in ringing his mobile because he never answers or gets back to me.' I shrugged, casually. 'You know what he's like. He might be off delivering a surfboard. I'm pretty sure he's not doing a gig with the Mummers, they don't usually go that far from home.'\n\n'A gig \u2013 with the what?'\n\n'The Mummers of Invention: you know, that sort of folk-rock group he started with three local friends?'\n\n'No,' he said shortly. 'I'm glad to say I don't.'\n\n'You must do because one of them's that drippy female Unks rents an estate cottage to \u2013 she sells handmade smocks at historical re-enactment fairs. And if you ever came up for the Mystery Play any more, you would have seen them \u2013 they provide the musical interludes. Tom played Lazarus as well, last year. He stepped in at the last minute and the parish magazine review said he brought a whole new meaning to the role.'\n\n'I can imagine \u2013 and I do intend being here for the next performance.'\n\n'I thought Leila couldn't leave her restaurant over Christmas?'\n\n'She can't; I can,' he snapped, and I wondered if their marriage was finally dragging its sorry carcass to the parting of the ways, like mine. 'So, you've no idea where Tom is, or when he'll be back?'\n\n'Probably Cornwall, that's where he mostly ends up, and if so, he's likely to be staying with that friend of his Tom Collinge, the weird one who runs a wife and harem in one cottage.'\n\n'I suppose he may be there by now, but he was in London on Monday night, Lizzy. I ran into him at Leila's restaurant, but he left in a hurry \u2013 without paying the bill.'\n\n'He did?' I frowned. 'That's odd. I wonder what he was doing in London?'\n\n'Well, it evidently wasn't me he'd gone to see, since he bolted as soon as I arrived.' He looked at me intently, as though he'd asked me a question.\n\n'Oh?' I said slowly, trying to remember whether Tom had actually ever said which of his friends he stayed with when he was in London.\n\n'Still, you know Tom,' I tried to laugh. 'He probably just found himself near the restaurant and dropped in.'\n\n'Then just took it into his head to shoot off without paying when I turned up unexpectedly? Leila said she didn't want to charge him for the meal anyway, since he's a sort of relative.'\n\n'That's kind of her,' I said, amazed, because it wouldn't surprise me if she gives even Nick a bill when he eats there!\n\n'Yes, wasn't it just?' he said drily. 'And one of the staff let slip that he'd stayed in her apartment the previous night, too \u2013 the staff seemed to know him pretty well. But I told Leila, business is business and she'd never let sentiment of any kind come before making money before, so I would just drop the bill in on my way up to the Hall. Here it is.'\n\nI looked at his closed, dark face again and suddenly wondered if he suspected that Tom and Leila had something going on. Surely not. It would be totally ridiculous! I knew that Tom had been having a serious affair for the last few years, of course, but not who it was with, although I assumed it was someone down in Cornwall where he spent so much time. It couldn't be Leila... could it?\n\nMy mind working furiously, I took the offered bill and glanced down at it, then gasped, distracted by the staggering sum. 'You must be absolutely rolling in it, charging these prices!'\n\n'Not me \u2013 Leila. And the prices aren't anything out of the ordinary for a restaurant of that standard. She's just got a Michelin star.'\n\n'Congratulations,' I said absently, staring at the bill, the total of which would have fed the average family of four for about a year. More, if they grew most of their food themselves, like I do. 'But I'm sorry, I don't have that kind of money on the proceeds of my produce sales \u2013 and in case you haven't noticed, I've scaled that side of things down drastically in the last eighteen months.'\n\n'Come on, you must get good advances for your \"how I tried to be self-sufficient and failed dismally\" books. You can't plead poverty,' he looked distastefully down at the mess he was standing in, 'whatever it looks like here!'\n\n'You should have looked before you got out of the car,' I said coldly. 'The ducks have been up. And one small book every two or three years doesn't exactly rake in the cash. I only get a couple of thousand for them. I'm lucky to still have a publisher! My agent says it's only because my faithful band of readers can't wait to see what else goes pear-shaped every time. And they like all the recipes.'\n\n'Ah yes, the Queen of Puddings!' He wrinkled his nose slightly.\n\n'What?' I said indignantly. 'Just because it's wholesome, everyday stuff, it doesn't mean it isn't good food! At least my recipes don't need ninety-six exotic ingredients, four servile minions and a catering-sized oven to produce.'\n\nHe grinned, as though glad to have got a rise out of me, and I began to remember why our boy-girl romance never got off the ground: an interest in food is the only thing we've ever had in common, whatever Annie says, and he never tires of reminding me that mine is not gourmet, and it's largely focused on sweets and desserts.\n\n'And this is not my bill, so you'll have to come back and speak to Tom about it later,' I added, sincerely hoping that that was all he wanted to talk to Tom about. Clearly he was harbouring suspicions... But no, whoever Tom was having an affair with, it couldn't be Leila, his own cousin's brittle little acid drop of a wife, however strange the circumstances might look!\n\n'If I can catch him,' Nick said, the grin vanishing. He abruptly changed the subject. 'Jasper had his results yet?'\n\n'Oh, yes!' I said, happily diverted. 'Yesterday and they were just what he needed for Liverpool University, to read Archaeology and History. He's having breakfast at the moment \u2013 why don't you come in and talk to him? He hasn't thanked you for that Roman cookery book you sent him, yet.'\n\nJasper's keenly interested in food and drink too, but only from a purely historical perspective. Delving about in medieval cesspits and middens, which was what he seemed to be spending his days doing at the dig, suited him down to the ground.\n\nNick looked at his watch. 'I haven't time today, so congratulate him for me, won't you? I'd better be off. I'm doing some articles on eating out in the North-West \u2013 out-of-the-way restaurants and hotels \u2013 so I need to drop my stuff off up at the Hall and get on with it. Breakfast awaits, then lunch and dinner...'\n\n'Lucky you,' I said politely, though sitting in restaurants isn't my favourite thing. I'd rather pig out at home than eat prettily arranged tiny portions consisting of a splat, a dribble and a leaf, in public.\n\nHe was frowning down at me again. 'You know, Lizzy, two thousand is peanuts compared to what I get for my books. No wonder you're living in a hovel \u2013 especially with Tom spending his earnings as fast as he makes them.' He gestured at the giant satellite dish, incongruously attached to the side of the cottage.\n\n'We don't need a huge amount of money and Perseverance Cottage is not a hovel,' I began crossly. 'Uncle Roly had all the mod cons installed before we moved in, and it's exactly how I like it. I've got everything I want.'\n\n'Have you? Or perhaps you've got more than you bargained for,' he said drily, his eyes again resting speculatively on my bruised cheek.\n\nI hoped he didn't think Tom had taken to physical violence \u2013 or that I would have stayed to be a punchbag if he had! I was just about to disabuse his mind of any suspicions in that quarter when he turned round to survey my domain and remarked suavely: 'I wouldn't say the family have come a long way from the heady days of Pharamond's Butterflake Biscuits, but they have certainly diverged in their interests.'\n\nThen, before I could point out that he at least was still vaguely in the bakery line, he got back into his car and reversed away in a cloud of dust. A lot of gritted chickens shot out from under it.\n\n'Wasn't that Uncle Nick?' Jasper asked, coming out ready for the off.\n\n'Yes, but he couldn't stay. He had an urgent appointment with breakfast, though he did send you his congratulations on the exam results. Get in. I'll just wash my hands and we'll go.'\n\n'Can I drive?' he asked hopefully. He'd recently passed his test, lessons courtesy of a lucky win on the gees at Haydock by Great-uncle Roly.\n\n'OK. Turn it round while I get ready.'\n\nHe'd left the cottage door open, and one of the hens had made a small deposit on the rag rug.\n\n## Chapter 3: Bittersweet\n\nWe are more than halfway through August, the time of year for eating fruits and salads as they come into season; but all too soon we will be bottling, brewing, jamming and preserving as if our lives depended on it and famine was sure to follow glut. And the minute the Christmas Pudding Circle receive their bulk order of dried fruits, peel, nuts and other ingredients, we will all be making our mincemeat too, for we use a marvellous Delia Smith recipe that keeps for ever.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nAll the way to the dig, while the loud music chosen by Jasper drowned out even the possibility of conversation, I wondered whether it could possibly be Leila that Tom had been having an affair with for the last couple of years \u2013 or the main one, because I'm sure he still scattered his favours pretty widely.\n\nWas Nick really hinting that he suspected that, or had I imagined it? But things certainly didn't sound too friendly between him and Leila, even by their semidetached, sweet-and-sour standards!\n\nAnd what would I say to Tom when he returned? While saying nothing would probably be the most sensible option until my plans to leave were in place, I couldn't let what he'd done pass, even if I didn't really think he was trying to hurt me physically.\n\nMaybe I should have left before, even if it did mean disrupting Jasper's schooling? The situation had certainly been affecting him \u2013 he seemed practically to have given up going out with his friends in the evening when Tom was home. Instead, he lurked in his room with the laptop Unks bought him, only suddenly looming silently up between us whenever voices were raised.\n\nSo now was probably the moment to clear the air and tell Tom straight that I was not prepared to put up with his behaviour any more, so I was leaving him. I was convinced this was what he'd been angling for, so he could play the hurt innocent party to everyone and, perhaps, install someone else here in my place...\n\nI found that a particularly horrid thought, but Perseverance Cottage belonged to his uncle Roly, so obviously if anyone were moving out it would have to be me. And I simply wouldn't ask Roly to help me, for not only did I not want to disillusion him about Tom, whom he had treated like another grandson, but he'd already been so kind and generous to us all these years by letting us have the cottage rent free.\n\nI expected I could find new homes for the hens and quail, but finding a new home for me would be the major problem. While the recent influx of newcomers into the area (especially the Cotton Common crowd) might mean that Annie's Posh Pet-sitters could expand enough to employ me part-time, on the downside, it also meant property rentals had soared out of my reach.\n\nIt was all depressingly difficult! Oh, why couldn't Tom just vanish into thin air, never to be seen again, like those mysterious disappearances you read about in the newspapers?\n\nIn need of comfort, I stopped off at Annie's cottage on the way home from dropping Jasper at the dig. It was still early, but she'd already made a chicken casserole and popped it in the slow cooker for later.\n\nShe seemed to have learned a lot more practical stuff than I ever did on that French cookery course we did in London after we left school, where volatile Madame Fresnet screamed at us all day long in French, the language in which we were supposed to learn to cook, thus killing two birds with one stone. At the end of the six months we all emerged with shattered eardrums, shattered French and the ability to whip up tartelettes au fromage at the drop of a whisk.\n\nTrinity skipped up to greet me, and Susannah, Annie's deaf white cat, regarded me with self-satisfied disinterest from the top of the Rayburn.\n\n'All right?' Annie asked anxiously, scrutinising my face.\n\n'Fine. Tom's not back yet and Jasper's at the dig \u2013 I just dropped him.'\n\n'It's great he got his first choice university, isn't it?' she said, getting down another mug from the rack and pouring me some coffee. 'Do you want a chocolate croissant? They're hot from the oven and I don't think I can eat the last one, I've had two already.'\n\n'Your eyes are bigger than your belly,' I said vulgarly, accepting the plate, and sat down at the kitchen table, keeping my eyes firmly away from Trinny's pleading dark ones, because the last thing a dog with three legs needs is to be overweight.\n\n'I saw Nick this morning,' I told her, dunking the croissant into my coffee so the bittersweet dark chocolate began to melt into it. This makes a change, because I usually do it the other way round and dip my food into melted chocolate, especially strawberries. It's amazing what you can coat in chocolate \u2013 and I'm not talking about that revolting body paint, because I prefer to keep the two greatest pleasures life can hold completely and unmessily separate... or at any rate, I did. I think I have forgotten how to do one of them.\n\n'That's really what I came to tell you about, Annie. He called in early on his way up to the Hall, and he said Tom was in London on Monday.'\n\nI described my conversation with Nick. 'Don't you think that sounds like he suspects Tom and Leila might be having an affair?'\n\n'Oh, no, surely not? Not with his own cousin's wife?' she exclaimed, looking horrified. Annie is just too nice for her own good, but I suppose being a vicar's daughter didn't exactly help to squash her natural inclination to think the best of everybody if she possibly could.\n\n'I don't know, but I certainly hope not. I can't really see him and Leila getting it together, can you? She's quite scary, in a beady-eyed and elegantly chic way. And I always thought it must be someone local or down in Cornwall, so perhaps Nick has got the wrong end of the stick.'\n\n'I'm sure he must have,' she agreed, and then her eye fell on the kitchen clock. 'Look at the time! I promised I'd put in a couple of hours at the RSPCA kennels. The flu's hit the staff and volunteers hard. There are no pet-sitting jobs that I can't handle myself this afternoon, but tomorrow will be busier.'\n\nShe looked slightly self-conscious: 'Ritch Rainford has asked me to go in at lunchtime and walk Flo, because he'll be at the studios in Manchester all day.'\n\n'You'd better get off, then, if you're sure there's nothing you want me to do. I'll see you at the Mystery Play committee meeting later, when I'll finally get to meet the new vicar.'\n\n'Oh, yes, he's... he seems nice,' she said vaguely, but I could see that her mind was still too taken up with the delights of Ritch Rainford to bother with lesser mortals.\n\n'Oh, before you go, can I borrow that candyfloss maker you bought for the last Cubs and Brownies' bazaar? Some lusciously lemon morning mist has given me ideas.'\n\n'Of course. Now, where did I put it?' She vanished into the pantry and came back with a large cardboard box. 'The instructions and everything are still in there. Do you need anything else? Sugar?'\n\n'No, I'm OK for sugar,' I assured her.\n\nI left her putting Trinny in the back of her car, and then drove down to the other end of the village to drop the punctured tyre off with Dave Naylor at the local garage, Deals on Wheels. (And I know it seems confusing at first that most of the indigent Mosses population who are not Pharamonds are either Naylors or Gumballs, but you quickly get used to it.)\n\nThen I headed for home, passing the entire contingent of the Mosses Senior Citizens' Circle waiting to board a coach for the annual trip to Southport Flower Show... including Unks' alarmingly spry octogenarian sister, Mimi Pharamond. I slowed down, staring, and she waved at me gaily, the rainbow-coloured Rastafarian knitted hat Nick brought her back from Jamaica flapping over one eye.\n\nSince Juno Carter, her long-suffering companion, was currently laid up after an accident, letting Mimi loose alone on the flower show seemed a recipe for disaster. I only hoped someone had been delegated to keep an eye on her. And a firm grip.\n\nThere was still no sign of Tom's van outside the cottage and you couldn't miss it because it had 'BOARD RIGID' in big fluorescent orange letters up the side and the logo of a stickman surfing. The workshop door was closed too, but in the bedroom I found his dirty clothes scattered on the rug as though washed up there by a high tide, so he'd either gone out again, or come back without his van.\n\nStill, clearly he had returned from wherever it was he'd been. I gathered up Tom's clothes, added some of Jasper's and mine, and then went down to stuff them in the machine. There was no beating them on a rock for me, even in the first flush of self-sufficiency in Cornwall, though before I bought the washing machine out of my first Perseverance Chronicle sale, I used to do the laundry by trampling up and down on it in the bath. Then I would pass it through an old mangle in the yard, which was not fun in winter.\n\nIt hadn't taken me long to realise that most books on self-sufficiency were written by men in warm, comfortable rooms, while their wives were out there dealing with the raw realities of life. Or that Tom, while initially enthusiastic, soon lost interest and succumbed to the burgeoning surfing culture instead. Once you added a tiny baby into the equation, the offer of a cottage on the Pharamond estate up in Lancashire was one I was determined we wouldn't refuse.\n\nAs I pointed out to Tom at the time, if you have the contacts, you can customise surfboards anywhere, and besides, Middlemoss was as close to a home as I had ever got, and I longed to return there.\n\nTom's jeans crackled when I picked them up to stuff into the washer, but then I always had to empty his pockets of a strange assortment of objects, from board wax to fluffy sherbet lemons.\n\nThis time the haul was a dark blue paper napkin tastefully printed in the corner with the word 'Leila's' in gold, a teaspoon that probably came from the same place, since it was definitely classier than any of our mismatched assortment, a stub of billiard chalk, a red jelly baby with the head bitten off and a piece of pink paper folded tightly into the shape of a very small rose.\n\nTom had doodled in origami roses as long as I'd known him, which could be very irritating when it was my shopping list or the top page of a stack of book manuscript; but equally, it used to be rather endearing when it was an apology for forgetting to tell me he was going off somewhere. At least, it was until the novelty wore off, along with my patience.\n\nI flattened this one out and found it was the last part of a letter, abjuring my husband to 'Tell old Charlie Dimmock you've found someone else and give her the push', and promising, if he did, to tie him up \u2013 and maybe even down if he really begged her to. It was signed 'Your Dark Heart'.\n\nWell, that didn't sound like Leila, did it? I could imagine she'd give anyone a good basting, but would she have time in her busy schedule for bondage?\n\nA horrible image flashed before my eyes of a naked Tom, trussed and oven-ready, and I found I was sitting on the quarry tiles feeling sick and recalling the last time we made love, which was just before Jasper was taken ill.\n\nI'd accused Tom of having yet another affair, but this time his attempt to sweet-talk me round hadn't entirely worked and he'd said I was so unresponsive he felt like he was practising necrophilia. And then I'd said that I felt much the same, since he might look like the man I married, but the part of him I'd loved seemed to be quite dead.\n\nThis threw him into the first of his really frighteningly violent rages during which he said that living with a cold bitch of a wife who thought food could cure anything was enough to send any man off the rails, and stormed out.\n\nAfter this I began sleeping in the small boxroom off our bedroom, and things between us went downhill rapidly. He made no attempt to conceal his affairs, though this note was the first hard evidence that any of them were serious...\n\nFeeling suddenly dizzy, I put my head on my knees and closed my eyes, wishing our old lurcher, Harriet, was still around to snuffle sympathetically in my ear.\n\nWhen the feeling passed off I resolutely got up and washed my face in the kitchen sink with cold water, ate an entire packet of those chocolate mini-flake cake decorations, then went out to the workshop, from where faint strains of Metallica now wafted through the Judas door.\n\n## Chapter 4: Mushrooming\n\nI know a lot of people dry mushrooms, or freeze the button ones, but I either eat them freshly gathered from the nearby fields, or not at all.\n\nBut I do make marzipan mushrooms sometimes and give them as gifts in the sort of little paper-strip baskets we used to weave at infants' school for Easter eggs. The mushrooms are very easy: you simply make the cap from marzipan and place it upside down. Then press a disc of more marzipan onto it, coloured brown with a little cocoa powder, and make a ribbed effect with a fork. Add a marzipan stalk, and hey presto! Realer than real.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nTom had his back to me when I went in, spray-stencilling some intricate, hand-cut Celtic design onto a surfboard. He was wearing a mask and baggy dungarees over his T-shirt and jeans, and his dark hair curled onto the nape of his neck in a familiar ducktail.\n\nWhere his cousin Nick was built on a large and rugged scale, Tom was a slight, wiry man and every slender bone of his body was beautiful. But despite (allegedly) not being a Pharamond other than in name, he did have the unmistakable look of one, so I was convinced that all the rumours about his mother were true.\n\nI stood there for a minute, thrown by that familiar curl of hair, shaken by the stirring of a tenderness I had thought long dead. Then he must have felt my presence, for he turned cold grey stranger's eyes on me, pushing down the mask. The CD came to an end and there was silence.\n\nHis eyes flicked to the fading bruise on my cheek and away again. 'You're still here, then? Thought you might have cleared off.'\n\n'Like where?' I demanded. 'And Jasper? The animals? Did you think I had an ark ready and waiting somewhere?'\n\n'Ah, yes, I forgot: my great-uncle by marriage, my cottage. Poor little orphan Lizzy has nowhere to go, has she?'\n\n'Don't think I intend staying with you any longer than I have to,' I told him coldly. 'The minute Jasper's off to university, that's it. And if you're interested, his results came and he got into Liverpool.'\n\n'It's always Jasper, isn't it?' he said pettishly.\n\n'You should be pleased because he's your son too, whatever mad ideas you've got in your head. But I'm not playing your games any more, Tom \u2013 you can believe what you like.'\n\n'Oh, come on, Jasper's the spitting image of Nick, my dear old no-blood-relation cousin \u2013 and don't forget I caught you in each other's arms at the hospital when Jasper was ill.'\n\n'I've told you repeatedly that he was just comforting me \u2013 and you could have been doing that, if I'd been able to get hold of you! But I conceived Jasper practically as soon as we'd got married and I never even looked at Nick in that way \u2013 or any other man! No, there's another obvious reason why both you and Jasper look like Pharamonds, only you'd rather believe ill of me than your mother!'\n\n'We'll leave my mother out of this,' he said, that ugly look in his eyes. 'But the sooner you clear out, the better.' Turning back towards his board he said dismissively, 'Fetch me a beer out will you? There's some in the fridge.'\n\n'Fetch it yourself. I didn't come out here to wait on you. Oh, and here's a restaurant bill from Leila. I only hope the meal was worth it!'\n\n'What?' He swung round and snatched it from me, glanced at it and then looked up suspiciously. 'Where did you get this?'\n\n'Nick called by early this morning. You left Leila's without paying the bill, and she wants her money.'\n\n'Oh, I don't think this is Leila's idea,' he said, crumpling the bill into a ball and tossing it into a corner. 'I've already paid her \u2013 in kind. Bed and board. So now you know, and presumably Nick also knows.'\n\n'Suspects, perhaps... but... Leila can't possibly be \"Dark Heart\"!' I blurted.\n\nHe took a menacing step towards me. 'What do you know about Dark Heart?'\n\n'I found a bit of a note in your pocket when I was sorting the washing, but it didn't sound like Leila,' I said, standing my ground.\n\n'It isn't,' he said shortly. 'It's someone else... someone more conveniently local, who's prepared to please me in ways you wouldn't have, even if I'd asked, dearest wife.'\n\n'Is it someone I know, Tom? And Leila \u2013 was that a one-off? She isn't the woman you've been having an affair with since before Jasper was ill, is she?'\n\nHe didn't reply, just smiled rather unpleasantly. I hoped he hadn't been running two of them in tandem even then. But someone local... who could it be?\n\nOh God, he hadn't got drunk and started an affair with that drippy girl who played the electric violin and sang in the Mummers, had he? I'd noticed she hadn't been able to look me in the eye for months, but thought she'd maybe been one of his one-night flings. Evidently, he wasn't going to tell me anyway.\n\nI thought of something else. 'Where's your van?'\n\n'It broke down in a lay-by about twenty miles away. I had to get the garage to bring it in \u2013 think the gearbox's had it. Now, any more questions? Only I need to finish this board because I'm off down to Cornwall at the weekend to deliver it, assuming the van's fixed by then.'\n\nI stared at him, thinking how normal a monster could look.\n\n'If you aren't leaving immediately, you could make yourself useful and fetch that beer,' he suggested.\n\n'Fetch it yourself! I'm going for a walk in the woods to think all this over, and then later I've got a Mystery Play Committee meeting, the first of the year,' I said, and saw a flash of anger in his eyes.\n\nAs I left I heard the music restart, and the hissing of the spray.\n\nOutside I practically fell over Polly Darke, our local purveyor of stirring Regency romances \u2013 and I use the term 'Regency' very loosely, since she never let historical facts come between her and the story. She gave me one of them once and I noticed the words 'feisty' and 'lusty' appeared on practically every page to describe the heroine and hero.\n\nAnd now I came to think of it, she never let facts come between her and a modern story either, since she was always snooping about under one pretext or another, and twisting things she saw and heard into malicious gossip. Divorced, she had lived in her hacienda-style bungalow between Middlemoss and Mossedge for several years, and I'm sure was convinced that she was accepted everywhere as a local.\n\nWhile I didn't suppose she could have heard anything much through a wooden door, that wouldn't prevent her from spreading lurid rumours about me and Tom around the three villages by sundown.\n\nShe was looking her usual strange self, in a severely truncated purple Regency-style dress, and with her hair cropped and dyed a dense, dead black. She clutched a small blue plastic basket of field mushrooms to her artificially inflated bosom, which might or might not be a fashion statement \u2013 are plastic baskets currently a must-have accessory?\n\nApart from the kohl-edged eyes and puffy, fuchsia-pink lips (which reminded me, strikingly, of a baboon's bottom), her face was pale as death. Paler.\n\n'Oh, Polly, are you all right?' I asked. 'You haven't been eating your own home-bottled tomatoes or anything like that, have you?'\n\nFrom time to time she fancied herself as the Earth Mother type and tried her hand at jams, chutneys and bottled goods, which she then gave to all and sundry, in my case together with a generous dose of botulism or something equally foul. Just my luck to get that one!\n\n'Oh, no, I haven't had time for any of that, Lizzy \u2013 I've got a book to finish, you know.'\n\n'Yes, Senga does like you to keep them coming, doesn't she?'\n\nHaving fallen out with two agents and three publishers, Polly had been taken on by my own agent, Senga McDonald \u2013 and may the best woman win.\n\nHer dark eyes slid curiously to the closed workshop door and back to my face. 'I thought I heard raised voices \u2013 is everything OK with you and Tom? Only sometimes lately you haven't seemed entirely happy, and you know you can always depend on me if you need a shoulder to cry on.'\n\nOh, yes, but only if I kneel down first, I thought, as she smiled at me in a horribly pseudo-sympathetic sort of way.\n\n'I'm fine,' I said shortly. 'We were just discussing business. Were you looking for me?'\n\nShe gave a start. 'Oh, yes. I picked loads of mushrooms in the paddock this morning early and I thought you might like to swap them for some quail eggs? But if it's inconvenient, it doesn't matter.'\n\n'No, not at all. I'm just off for a walk, but you know where they are in the small barn? Help yourself and leave the mushrooms there,' I told her, and walked off, not caring whether she thought me rude or not. When she first moved to Middlemoss she went all out to be my best friend, but we had absolutely nothing in common (apart from Senga). Anyway, I already have a best friend in Annie.\n\nNor, it occurred to me, was she the type to skip about the fields at dawn gathering mushrooms, which in any case looked suspiciously like shop-bought ones, small, clean and perfectly formed. My marzipan mushrooms looked earthier than those!\n\nI headed for the woods, for I found their dark, cool depths wonderfully soothing, especially on a hot day. They restored a sense of my unimportance in the great scale of things, shrinking my problems down to a more manageable, acorn size.\n\nLuckily I was wearing a pinky-red T-shirt, so Caz would spot me if I strayed onto the smaller paths he stalked so relentlessly. But if he was out there with his gun, he didn't make himself known. He's not much of a talker in any case; but then, most of his dealings are with squirrels, so he doesn't need to be.\n\nAfter a while I found my thoughts turning away from more painful subjects onto the comforting one of food, wondering which member of the Christmas Pudding Circle would come up with the best recipe for brandy butter ice cream.\n\nMore than likely it would be Faye, since she's a farmer's wife who has diversified by opening a farm shop and caf\u00e9, where she sells her own home-made organic ice cream. She was already perfecting a Christmas-pudding-flavoured one.\n\nEventually, as the shadows lengthened, I reluctantly had to turn for home, even though I dreaded seeing Tom again. But there was no need: he wasn't there and, more to the point, neither was my car.\n\nCome to that, even the punnet of mushrooms Polly Darke had presumably left had vanished into thin air, though possibly Caz had been around and fancied them. He knows he can help himself to anything edible he can find, though it seemed a bit greedy to take them all. (He keeps the freezer I gave him locked, so goodness knows what's in there. Better not to know, perhaps?)\n\nI searched for a note saying where Tom and my car had gone to, but there was nothing. Unless he came back by the time I returned from the Mystery Play Committee meeting, Jasper was going to have to cycle home that evening, and I would be extremely annoyed.\n\nI fed, watered and generally cared for everything that needed my attention, then changed and set off for the village hall \u2013 on foot.\n\n## Chapter 5: Sweet Mysteries\n\nThe Mystery Play Committee will reconvene on the 19th of August with rehearsals to start in September as usual. If any member of last year's cast cannot for any reason continue in their role, would they please inform Marian and Clive Potter at the Middlemoss Post Office.\n\nMosses Messenger\n\nThe members of the Middlemoss Mystery Play Committee were gathered around a trestle table in the village hall, which exhibited reminders of its many functions: the playgroup's brightly coloured toys poked out from behind a curtained alcove and their finger-painting decorated one wall, while the other bore posters of footprints illustrating the various new steps the Senior Citizens' Tuesday Tea Dance Club were trying to master.\n\nPersonally, I thought salsa might give one or two of them a bit of trouble, but I was sure they would all give it a go. Their line dancing ensemble at the last Christmas concert had been a big hit, and Mrs Gumball, the cook up at Pharamond Hall, had got so excited she fell off the end of the stage. But fortunately foam playmats were always stacked there after an incident a few years back, when one of Santa's little elves fell over, causing a domino effect along the line until the last one dropped off and broke a leg.\n\n'I think we might as well start, Clive,' I suggested to the verger, opening the plastic box of Choconut Consolations I'd brought with me and setting it in the middle, so everyone could help themselves. 'I don't know where Annie's got to, but Uncle Roly's gone to the races. He said after all these years he could do the Voice of God in his sleep, so you could sort it all out without him.'\n\nThis year's committee was formed of the usual suspects; some of them also CPC members. There was Dr Patel, our semi-retired GP, Miss Pym the infants' schoolteacher, the new vicar \u2013 untried and untested and looking more than a little nervous \u2013 and Clive and Marian Potter, who between them ran the post office, the Mosses Messenger parish magazine and also pretty well everything else that happened round Middlemoss, including directing the annual Mysteries. Then there was my humble self, for Clive liked to have a token Pharamond on tap, since Uncle Roly was inclined to give his duties the go-by if something more interesting came along. Annie was presumably held up somewhere.\n\n'Very well. I've convened this meeting earlier than usual for two reasons,' announced Clive, who is like a busy little ant, always running to and fro. Marian is the same, and I have a theory that they never sleep, just hang by their heels for the odd ten minutes to refresh themselves, like bats. Come to that, they're so in tune with one another they have probably leaped up the next rung of the evolutionary ladder and communicate in high-pitched squeaks us mere bog-standard humans can't hear.\n\n'First off, I thought the vicar might need a bit more time to get to grips with the Mysteries, it all coming as a bit of a surprise to him, like.'\n\nThe vicar, a carrot-haired, blue-eyed man with a naturally startled expression, nodded earnestly: 'But I'm delighted, of course \u2013 absolutely delighted.'\n\nI wondered if anyone had warned him that the last vicar was currently having a genteel nervous breakdown in a church nursing home near Morecambe. An elderly man, he'd been hoping for a quiet country living, I feared, where he could jog along towards his retirement, not the whirl of activity that is the Mosses parish. But at least the new one was younger and unmarried. I observed with interest the way he suddenly went the same shade as his hair when Annie, breathless and dishevelled, rushed into the room.\n\n'Sorry I'm late,' she said, subsiding into the seat next to me. 'One of the dogs slipped its lead and was practically in Mossrow before I caught him.'\n\nShe smiled apologetically around at everyone and, apart from the vicar, who was still looking poleaxed, we smiled back, since Annie is Goodwill to all Mankind personified. Even though I'm her best friend, I have to admit that she is a plump, billowy person the approximate shape of a cottage loaf and, although her hair is a beautiful coppery colour, that pudding-bowl bob does not do her amiable round face any favours. She certainly doesn't normally cause men to go red and all self-conscious...\n\n'We were only just starting,' I assured her. 'Clive's called the meeting to familiarise the vicar\u2014'\n\n'Do all call me Gareth,' he interrupted eagerly, finding his voice again, but I expect most of us will just carry on addressing him as 'Vicar' because we are nothing if not traditionalists in Middlemoss.\n\n'And you must call me KP,' said Dr Patel agreeably, 'like the nuts.'\n\n'And I'm Lizzy,' I put in hastily, seeing Gareth's puzzled expression at KP's old joke. 'You've already met Annie, haven't you?'\n\n'Oh, yes.' He swallowed, his Adam's apple bobbing. 'At church.'\n\nHe was just Annie's type and clearly smitten, but she didn't seem to notice!\n\n'Perhaps we'd better get on?' suggested Clive. 'Only the Youth Club will be in here tonight for snooker, and I'll need to set the tables up. First, could you all please read this quote from a recently published book.'\n\nHe passed round a bundle of photocopies.\n\nAlthough called the Middlemoss Mysteries, this surviving vestige of a medieval mystery play, annually performed in an obscure Lancashire village, is in reality a much debased form. At some point in its history it was reduced to a mere series of tableaux illustrating several key Biblical scenes, such as the Fall of Lucifer, Adam and Eve and the Nativity. Then, early last century what little dialogue remained was rendered into near-impenetrable ancient local dialect by Joe Wheelright, the Weaver Poet, and this is constantly reinterpreted by each generation of actors. The head of the leading local family, the Pharamonds, traditionally speaks the Voice of God.\n\nWe all read it in silence.\n\nThen Annie said, 'Well, it's not so bad, is it, Clive? We can't hope to keep the Mysteries a total secret, so we always do get some strangers coming along, especially since the Mosses have suddenly become so terribly trendy to live in.'\n\n'No, it's the folksy visitors who would want to take over and fix the whole thing like a fly in amber that we want to discourage,' I agreed. 'The Middlemoss Mystery Play is just for the locals, something we've always done, like that Twelfth Night celebration they have up at Little Mumming.'\n\n'That's hardly comparable with our play, dear, since I'm told it's only a morris dance and a small miracle scene of George and the Dragon,' Marian said.\n\n'That's right,' agreed Clive. 'But they keep it quiet: I've even heard that they block the road into the village with tractors on the day, to deter strangers.'\n\n'I think the best thing about our Mysteries is the way each new generation of actors adds a little something to their parts, even if we do now stick more or less to the Wheelright version,' I said, though actually, while the acting itself is taken very seriously, I often suspect the Weaver Poet of having had a somewhat unholy sense of humour.\n\n'I don't think \"debased\" is a very polite description,' Marian said, looking down at her photocopy again and bristling to the ends of her short, spiky silver hair. 'And what does he mean, \"impenetrable dialect\"? If the audience doesn't know the bible stories before they see it, then they should, so they'd know what was going on!'\n\n'Er... yes,' said the vicar, with a gingerly glance at Dr Patel, who was sitting with his hands clasped over his immaculately suited round stomach, listening benignly.\n\n'Oh, don't mind me,' the doctor said, catching his eye. 'I went to infants' school right here \u2013 my father was the senior partner at the practice \u2013 so I know all the bible stories. So did all the Lees from the Mysteries of the East Chinese takeaway in Mossedge, and there's usually at least one of that family taking part in the play.'\n\n'We have many mysteries,' I said helpfully. 'Even the pub is called the New Mystery.'\n\n'Little Ethan Lee made such a sweet baby Jesus last year,' Miss Pym said sentimentally. 'He simply couldn't take his eyes off the angels' haloes.'\n\n'None of us could,' Annie said. 'We'd never had ones that lit up before.'\n\n'Oh?' said Gareth, clearly groping to make sense of all this. 'Well, Clive has kindly loaned me the videos of last year's performance, which I've watched with... with interest.' He cleared his throat. 'While I've seen the Chester Mystery Plays and, er... although the format of scenes from the Old and New Testaments have similarities to that, otherwise they don't seem much alike...'\n\n'They're not, Vicar,' Clive said. 'They might have been at one time \u2013 you'd have to ask Mr Roly Pharamond, he's got all the records. But when the Puritans took over and tried to ban it, the squire \u2013 another Roland, he was \u2013 he told the players to cut it right down, so it could be performed in one day up at the Hall, instead of here on the green.'\n\n'Yes,' agreed Marian, 'and on Boxing Day instead of Midsummer Day, because fewer strangers would be travelling about then. Then, when it was safe to perform the Mysteries in public again \u2013 well, we'd got used to doing things our way.'\n\n'So it's still performed up at the Hall on Boxing Day?' Gareth asked.\n\nMiss Pym nodded. 'In the coach house. The doors are opened wide and the audience stands in the courtyard, with lots of braziers about to keep them warm. The stables on either side are used as dressing rooms. It lasts about five hours, with breaks for refreshments, of course, and musical interludes.'\n\n'Musical interludes? Indeed?' Gareth brightened. 'Hymns, perhaps? I'm hoping to breathe a little life back into the church choir.'\n\n'No, actually a local group perform \u2013 the Mummers of Invention,' I told him. 'My husband sings with them and they're quite good. Sort of electric folk style.'\n\n'Mummers of Invention?' he murmured, looking bemused.\n\n'The last vicar had the strange idea that the play was blasphemous in some way,' Clive said, 'but you could see yourself from the video that it's the exact opposite, couldn't you? It's all bible stories, and the entire parish is involved right down to the infants' school. The children always play the procession of animals into the ark.'\n\n'And they helped me to make the Virgin's bower last year with wire and tissue paper flowers,' Miss Pym said, 'though since it kept falling on Annie's head (your fifth and last appearance as Virgin, wasn't it, dear?) it could not have been called an unqualified success.'\n\nAnnie caught the vicar's eye, went pink, and looked hastily away \u2013 but at least now she had noticed him.\n\n'And you run the Mysteries committee, Clive, and direct the play?' Gareth asked.\n\n'Yes, that's right. In September we start giving out the parts and rehearsing. No one can play the same role for more than five years except God, so things change, and different people come forward or drop out.'\n\n'Some of the new actors who've moved into the area lately have volunteered,' I said.\n\n'Yes, like Ritch Rainford,' Annie murmured dreamily, and I gave her a look. I hope she's not going to get a serious crush on the man, since it's unlikely to lead anywhere.\n\n'But most of them don't live here all the time, Annie, and you need people who do, especially when there are more rehearsals just before Christmas.'\n\n'Yes, so the parts are usually played by local people and someone always volunteers if there's an emergency, like last year when Lazarus broke both ankles falling off his tractor,' Dr Patel said. 'He could have lain down, but there was no way short of a real miracle he was ever going to rise up and walk. So Lizzy's husband, Tom, stepped in at the last minute.'\n\n'He made a very good Lazarus: I gave him four stars in the parish magazine review,' Clive broke in.\n\nGareth turned to me. 'So, your husband is Tom Pharamond, and he also plays in a band called the Mummers? I don't think I've met him yet, have I?'\n\n'I shouldn't think so, he's not much of a churchgoer. And he said he wouldn't take part in the play again this year, it was a one-off, Clive \u2013 sorry. You'll need a new Lazarus.'\n\n'Pity,' Clive said regretfully. He coughed and shuffled his papers together. 'So, we'll ask for nominations for the parts and rehearsals will start in the middle of September in two groups, one on Tuesdays and the other, Thursdays. As usual, I'll need a director's assistant for each scene. Lizzy, will you take on the Fall of Lucifer, the Creation, and Adam and Eve? You are still doing Eve this year, I hope?'\n\n'Yes, my fifth and final go too, thank goodness \u2013 even a knitted bodystocking is perishingly cold in December. I had to keep warming myself over a chestnut brazier last year and a couple of my fig leaves got singed.'\n\n'You could try thermal underwear?' suggested Miss Pym. 'Those thin silk ones for under ski suits.'\n\n'That's an idea! Not so bulky.'\n\n'Miss Pym will do Noah's Flood, of course, and Marian will oversee Moses. One of the tablets broke last time; someone will need to make a new one...' Clive made a note, and ticked off Moses.\n\n'Vicar, if you could be in charge of the Nativity \u2013 Annunciation, Magi, Birth of Christ, Flight into Egypt?'\n\n'Yes, of course,' Gareth agreed, though rather numbly, I thought. But unlike the last vicar, at least he hadn't started gibbering and lightly foaming at the mouth by this stage.\n\n'Dr Patel has offered to do the Temptation of Christ, the Curing of the Lame Man, the Blind Man, and the Raising of Lazarus: all short scenes.'\n\n'Seems appropriate,' agreed the doctor, adding generously, 'and the Water into Wine and Feeding of the Five Thousand too, if you like.'\n\n'I'll see to the Last Supper, Judas, the Trial and Crucifixion myself this year \u2013 the Crucifixion's always tricky, but you might want to take that on next year, Vicar \u2013 and then that leaves just the Resurrection, Ascension and Last Judgement.'\n\n'I'll do those again,' offered Annie.\n\n'We do the final dress rehearsals for the whole thing up at the Hall in a couple of sessions before Christmas,' Marian helpfully explained to the vicar. 'In random order, or it would be unlucky. But since at least two-thirds of the players will have done their parts before, it's just a question of making sure the new ones know their lines and where to stand, that's all.'\n\n'Oh, good,' said poor Gareth weakly. He looked at his watch. 'I'd better get back \u2013 I've got a funeral to prepare.'\n\n'Yes, our Moses \u2013 such a sad loss,' Miss Pym said. 'We will have to recast that part, too.'\n\nClive stuffed his papers and clipboard into a scuffed leather briefcase and then he and Marian started transforming the hall into a snooker parlour for the Youth Club, turning down my offer of help.\n\nWhen I went out the vicar was already halfway across the green with Annie, heading in the direction of the church. I bet they were only talking about something totally mundane like Sunday school, though, and she hadn't noticed at all that he fancied her.\n\nMiss Pym climbed into her little red Smart car and vanished with a vroom, and Dr Patel wished me good night and got into his BMW.\n\nI wended my way home to Perseverance Cottage, where I did not find my husband or, more importantly, my car, but did find a telephone message on the machine from Unks, asking me to ring him back. When I did, he told me that Mimi, his elderly sister who lived at the Hall with her long-suffering companion Juno, had been arrested by the police at the Southport Flower Show, having temporarily got away from Mrs Gumball, who'd volunteered to keep an eye on her. You can't blame her, though, since Mimi is very spry for an octogenarian while Mrs Gumball is the human equivalent of a mastodon, so moves slowly and majestically.\n\nUnfortunately, Mimi is a plant kleptomaniac: no one's garden is safe from her little knife and plastic bags, and she really just can't understand why anyone should take exception to her habits. Still, the police had merely cautioned and released her this time and, since the coach had by then set out on the return journey, drove her and Mrs Gumball home in a police car.\n\nRoly said she was under the impression they had done it to give her a treat, and was hoping next year's flower show would be as much fun.\n\nThen he added, rather puzzlingly, 'And I hope Tom told you that you can stop worrying about ever losing Perseverance Cottage, my dear, because after I'm gone, it's yours and Tom's. I would have said before, if I'd known it was on your mind.'\n\n'But I wasn't worried, Unks! In fact, the thought never even entered my head,' I assured him. Since I would have to leave soon, it was immaterial to me, but Tom had evidently used me as an excuse to find out how things had been left. How Machiavellian he's becoming!\n\nAfter this, I unpacked Annie's candyfloss machine to distract myself from worrying until Jasper arrived safely home. The instructions absolutely forbade me to use any natural essences or colourings other than special granulated ones designed for the purpose, which was disappointing from the point of view of making Cornish Mist, until I discovered one of the tubs in the box was lemon.\n\nFascinating how the floss forms inside the bowl like ectoplasm, and you have to wind the near invisible threads onto wooden sticks. Fine, sugary filaments drifted everywhere, and the kitchen took on the hot, sweet, nostalgic smell of funfairs.\n\nIt was really messy but fun, which Jasper said was a good description of me, too, when he got home and saw what I'd been up to, though by then I was sitting among the debris, writing it all up for the Chronicle.\n\nMaybe I'll have 'messy, but fun' as my epitaph.\n\n## Chapter 6: Driven Off\n\nI wonder if plastic bags of fluffy white candyfloss labelled 'edible Santa beards' would go down well with children at Christmas? I expect they would try them on and get terribly sticky, though.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nThere was still no sign of my car next morning and, in a furious temper, I rang all of Tom's friends that I knew about, or who I had mobile numbers for although trying to contact his surfing buddies down in Cornwall was always like waking the dead, and I got little sense out of them even when they did answer the phone.\n\nThe first time or two he went missing for a few days I also rang the local hospitals and the police, but after that I learned my lesson.\n\nI woke Jasper early and saw him off by bike to the dig, then I called Annie to tell her I was without transport; but luckily she only wanted me to exercise the two Pekes and a Shitzu belonging to one of the more elderly members of the Cotton Common cast, Delphine Lake. She'd bought one of the expensive flats in part of the former Pharamond's Butterflake Biscuit factory in the village and I'd walked her dogs several times before.\n\nUncle Roly sold the Pharamond brand name out to a big conglomerate years ago for cash, shares and a seat on the board, which was both a smart and lucrative deal; so now the factory has been converted to apartments, a caf\u00e9-bar called Butterflakes, and a museum of Mosses history.\n\nDelphine's dogs may be little, but they loved their walks, so it was late morning before I got back to the cottage and found a female police officer awaiting me on the doorstep. An adolescent colleague sat biting his fingernails behind the wheel of a panda car.\n\nI immediately thought the worst, as you do. 'Jasper?' I cried. 'Has something happened to Jasper?'\n\n'Mrs Elizabeth Pharamond?' she queried solemnly.\n\n'Yes!'\n\n'I'm Constable Perkins and I'm afraid I have some very bad news for you.'\n\nShe paused, and I was just about to take her by the throat and shake her when she added,\n\n'About your husband.'\n\n'Oh \u2013 thank God!' I gasped devoutly, then burst into tears of relief.\n\nWresting the keys from my nerveless fingers, she ushered me into my own home, where she broke the news that Tom had had a fatal accident. He'd driven off the road into a disused quarry, which was odd in itself, since there's only one place within a radius of about fifty miles where he could have managed to perform that feat, and it's up a little-used back lane.\n\nWhile her colleague made me tea, she spoke to me with skilful sympathy, though my reactions clearly puzzled her. But all I was feeling was an overpowering sense of relief that it wasn't Jasper.\n\nAnd then I got to thinking that this was all so blatantly unreal anyway, that it couldn't be true: it must be just some dreadful nightmare!\n\nThis was a very calming idea, since I knew I'd wake up sometime, so I agreed quite readily to go and identify Tom's body. My head seemed to be this helium-filled thing bobbing about on a string \u2013 or that's what it felt like, anyway \u2013 but there's no accounting for dreams.\n\nAnd Tom, apart from his thin, handsome face being a whiter shade of pale, looked absolutely fine. He was always one to land butter-side up...\n\n'Is this your husband?' the policewoman asked formally.\n\n'Yes \u2013 Thomas Pharamond. Are you sure he's dead? Only he looks just like he did when he was playing Lazarus.'\n\nShe gave me a strange look, but assured me that Tom had broken his neck in a very final manner. Then she offered me yet another cup of tea, which I didn't want, and took me home again, sitting beside me in the back seat while the adolescent did the driving. He feasted on his fingernails at every red light and I don't know why, but it suddenly reminded me of the stewed apple with little sharp crescents of core snippings that they used to give us at school for pudding.\n\nThe policewoman whiled away the journey by telling me that they thought the car (my car, which was now a write-off) had been at the bottom of the quarry for a few hours before it was found, and he must have died instantly, but I expect they say that every time. There would have to be a post-mortem examination, and probably an inquest. I think she said there would be an inquest. I wasn't taking it all in, because of course it wasn't real.\n\nWhen we got to Perseverance Cottage, she asked if there was someone who could stay with me.\n\n'Oh, yes \u2013 I'll phone the family right now,' I assured her, suddenly desperate to get rid of her. 'Thank you for... for \u2013 well, thank you, Officer. I'll be fine.'\n\nShe looked a bit dubious, but drove off leaving me to it, and I thankfully closed the front door and leaned against it: that seemed solid enough. So did the cold quarry tiles beneath my feet when I kicked my sandals off...\n\nIt began slowly to dawn on me that this really was happening and Tom was actually dead! In which case, I could only be glad that Jasper was at his dig, since I'm sure he would have insisted on coming with me to identify Tom, though actually his face had looked peaceful enough, if vaguely surprised by the turn of events. I felt a sudden pang of guilt, remembering how glad I had been that it was Tom who had died and not Jasper.\n\nBut now I'd have to break the news to him about his father... and to Unks and Mimi and Tom's mother out in Argentina...\n\nStiffening my trembling legs I tottered into the sitting room and dialled the Hall, getting Uncle Roly.\n\nI don't think I was the mistress of either tact or coherence by this stage, but he took the news well, if quietly, and offered to phone Tom's mother and stepfather in Argentina himself, which was a huge relief. Then he said he would also try and contact Nick, still off touring the eateries of the rural North-West.\n\n'And Jasper?' he asked. 'I take it he is at the dig, and doesn't know?'\n\n'Yes, and I think I'll just wait for him to come home before I tell him,' I decided, for why rush to give him the bad news? 'Anyway, Tom was driving my car \u2013 his van broke down \u2013 so I haven't got any transport.'\n\nWhen I phoned Annie she was out and the message I left was probably unintelligible.\n\nRoly thoughtfully called in later in the Daimler to say Joe Gumball was driving him over to the dig to collect Jasper and he could break the news to him on the way home, if I wanted.\n\n'Oh, Unks, you are kind!' I said, gratefully. 'But it must be just as hard for you. You don't have to do it.'\n\n'My dear, having lived through the war, I'm inured to breaking bad news.'\n\nI offered him some of the damson gin I'd been drinking to try to dispel that feeling of being underwater with my eardrums straining, but which had just seemed to make everything more unbelievably bizarre, and said anxiously, 'I can't believe Tom isn't going to walk back in through that door at any moment, the way he always turned up after he'd been missing for a few days.'\n\nHe patted my hand. 'There, there, my dear. Leave everything to me. I'll be back with Jasper in no time.'\n\nMimi phoned me up just after he'd left, but halfway through offering me her condolences in a graciously formal manner, she completely lost the thread and said she was too busy to talk to me just now. Then she put the phone down.\n\nBut at least her call had jarred me into remembering to feed the poultry. It was a bit late, but when I called, 'Myrtle, Myrtle, Myrtle \u2013 Honey, Honey, Honey!' they all came running.\n\nRound the side of the big greenhouse I came unexpectedly nose to bare (except for the camouflage paint) chest of Caz Naylor, who indicated with a nod of his head and a raised eyebrow that he would like to know what was happening.\n\n'Tom's driven off the quarry road,' I said. 'In my car.'\n\n'Dead?'\n\n'So they say.'\n\n'Car?'\n\n'That's a write-off, too.'\n\nHe grunted non-committally, then handed me a small blue plastic basket containing one slightly decayed mushroom. 'Poison,' he said, prodding it with a slightly grimy finger.\n\n'I know,' I began, recognising it, but he turned and flitted off back through the shadows until he'd completely vanished into the woods.\n\nThat was the longest conversation I'd had with him for ages... and what was the significance of the poisonous fungi in a punnet that looked suspiciously like the one Polly Darke had brought me full of field mushrooms... was that only yesterday? Perhaps she'd inadvertently picked a poisonous one? After my previous experience of Polly's way with foodstuffs, I should have been more cautious in accepting them anyway!\n\nOr perhaps Caz had simply taken to giving brief nature lessons in his spare time.\n\nJasper was very quiet and pale when he came in, and though we shared a long hug, said he'd like to be alone for a bit and vanished up to his room. I thought it best to leave him to talk in his own time.\n\nHe did reappear when Annie arrived and seemed pretty composed by then, though he being the quiet stoical type it's hard to tell, even for me.\n\nI thought I was quite composed too, but as soon as Annie walked through the door I burst into tears, as though her arrival was some kind of absolute proof that it really wasn't all a ghastly nightmare. I left a full set of grubby fingerprints up the back of her lavender Liberty cotton shirt.\n\nShe hugged Jasper too, something which he would normally go out of his way to avoid, even though he is fond of her. Then we all just sat around in a fuzzy cloud of disbelief and damson gin.\n\nIt was the sheer unreality: Tom had gone missing so many times, it was hard to believe he wouldn't just walk through that door at any minute with the TV remote control in his hand (he secreted it away somewhere in his workshop when away), and sit watching endless films on Sky, which he'd had installed soon after he got the giant TV.\n\nHe'd always been supremely selfish. Even the Tom I fell in love with, charming though he'd been, really only thought about himself for at least ninety-five per cent of the time, which is why he always did exactly what he wanted and apologised afterwards.\n\n'Yes, I know,' Annie agreed when I shared this gem with her, together with the rest of the bottle of gin, after Jasper had gone up to bed (or at least, back up to the Batcave). 'But when he was around he seemed to cast a spell of charm, so people didn't realise it until later. Or if they did, they didn't mind, because they thought he wasn't doing it intentionally to hurt anyone, it was just how he was.'\n\nThe gin might not have been such a good idea after all, for my past life seemed to take on a darkly ominous pattern. 'Why?' I demanded. 'What have I done to deserve this? Why do I have to lose everyone? I know I didn't love Tom any more, but I didn't want any harm to come to him either!'\n\n'We all have to die,' Annie pointed out soothingly, passing me the plate of ginger parkin she'd found in the fridge while looking for something to blot up the alcohol. I must have sliced and buttered it earlier, on automatic pilot.\n\n'Yes, but why don't my loved ones die naturally of old age? Look at my parents! OK, Daddy was a diplomat, but of all the British Consulates in all the world, why did they have to be sent to that one? And having got there, why did they have to immediately sit in the wrong restaurant and get blown up? Couldn't they have settled for baked beans on bagels at home, and then lived nice, peaceful lives and been more than a few faded snapshots and some stored furniture to their only daughter?'\n\n'But you had nothing to do with it \u2013 you'd just arrived for your first term at St Mattie's,' she pointed out. 'You weren't even in the same country. Stop imagining you're some kind of Angel of Death! What would Daddy say if he could hear you?'\n\nFrom past experience I could confidently predict that Annie's father would go wandering off into a scholarly monologue on angels of death, the existence and symbolism of, which would be soothing, but not precisely helpful.\n\nAnnie gave me a hug. 'It's not your fault that Tom was killed and you did your best to save your marriage. I know what it's been like the last few years, and you're a saint to have stayed with him.'\n\n'I'm not a saint. I stayed for Jasper, really, and because we both loved living here.' A tear rolled down my cheek and landed onto the half-eaten slice of parkin I was holding, though I didn't remember taking a piece.\n\n'I'm sure for the first few years Tom did love and need you, Lizzy. He wandered off, but he always came back again.'\n\n'Perhaps, but there were always other women. I tried to shut my eyes to it, but it hurt, Annie.' I swallowed hard. 'But I think I'm grieving for the Tom I married, even if the man I thought he was never existed. And I still feel guilty for being so relieved that it was Tom, rather than Jasper.'\n\nAnnie comforted me as well as she could, and I have a vague recollection of her helping me up to bed, where I must have passed out.\n\nWhen I staggered down next morning, feeling like Lady Lazarus, everything had been cleared and tidied and washed up.\n\nThere's probably a Girl Guide badge for coping with a friend's bereavement too, together with the Advanced Award for staying in control of your faculties while under the influence of damson gin.\n\n## Chapter 7: Loose Nuts\n\nCandied citrus peel makes a good gift and although the traditional process is messy and time-consuming, there is a quick method, which I have used with some success. When candied, the pieces can be dipped in good dark chocolate for a tasty treat.\n\nThe Perseverance Chronicles: A Life in Recipes\n\n'Oh, my husband was really selfish,' I said to PC Perkins, when she came back again later that day for what she called 'a little background detail'. This, oddly enough, seemed to consist of asking me what Tom had been like, but I expect she'd been on some kind of Dealing with the Victims of Bereavement course, or something.\n\nI'd finished quick-candying the orange peel left from yesterday and today's breakfast juice, and was just writing the recipe up for the latest Perseverance Chronicle, so even the sitting room, when I led the way into it, still smelled enticingly of citrus and hot sugar.\n\nI seemed to be going through the motions of normal life, but most of the time my brain was entirely absent, so I must have been doing it on automatic pilot.\n\nJasper, who had phoned up the dig earlier to explain his absence, followed us in and loomed about protectively. After the previous night's hair-down, damson-gin-fuelled wake with Annie, I had given up trying to hide things from him. I don't think it worked in the first place.\n\n'Oh, really?' she said encouragingly, seating herself on the armchair Tom had favoured for his telly watching. I made a mental note to do something about that giant blank screen, which was like having a dead eye in the room...\n\nI shuddered and she eyed me speculatively.\n\n'You don't make your husband sound terribly attractive, Mrs Pharamond!'\n\n'Actually, he could be very charming, and when I fell in love with him I thought the way he used to vanish for days without a word was endearingly absent-minded and eccentric. But really, he was just too wrapped up in himself to bother doing anything he didn't want to, a bit like a cat.'\n\n'But you can still love a cat,' Jasper pointed out. 'Most cat owners seem to think their cats love them back, too.'\n\n'He did seem fond of me, in his way, until the last few years \u2013 and of you, too, Jasper, when you were small,' I assured him, wiping a runny tear away. 'Some men just aren't good with children.'\n\n'I expect we'd have got on better if I'd surfed, or was interested in weird folk-rock music and stuff \u2013 fitted into his interests,' Jasper agreed. 'History and archaeology bored him.'\n\n'Yes, and he wasn't even interested in food, was he, except from the eating it point of view?'\n\nThe police officer, who'd been listening in a sort of fascinated silence, now broke in, notebook at the ready. She seemed to have an agenda of her own. 'Just a couple of questions, Mrs Pharamond \u2013 and I'm sure you have a few you would like to ask me.'\n\nShe gave me a reassuring smile, though it contained no warmth. Yesterday she'd seemed so kind and sympathetic, so maybe she could switch a fa\u00e7ade on and off at will, like Tom. She also had coral-pink lipstick on her front teeth and it was so not her colour.\n\n'Perhaps your son \u2013 Jasper, isn't it? \u2013 could make some tea,' she suggested.\n\n'I think I'll stay here,' Jasper said thoughtfully, settling down on the sofa next to me.\n\n'Can you tell me what time your husband left here on the Wednesday? You said you last saw him then, didn't you?'\n\n'I don't know when he left, because I went for a walk in the late morning \u2013 a long walk in the woods \u2013 and when I got back my car had gone.'\n\n'Did he often borrow your car?'\n\n'No, practically never, because I usually made sure he couldn't find the keys. His van had broken down, that's why he took mine.'\n\n'So you were surprised to find your car gone?'\n\n'Yes, and annoyed when he didn't come back in time for me to go and collect Jasper from the dig... or at all. I needed my car.'\n\n'He would probably have come back in good time if the accident hadn't happened, Mum. His mobile was in the workshop and I expect he'd have taken it with him if he hadn't just popped out for something,' Jasper said. 'Wonder where he was going. I checked it for messages, but he'd wiped them, so that was no help.'\n\n'I don't know,' I said dubiously. 'He probably just forgot his phone.'\n\n'Where do you think he might have been going, Mrs Pharamond?'\n\n'I've no idea. But he told me earlier he had to finish a surfboard to deliver this weekend, so I was surprised when he didn't come back.'\n\n'Finish a surfboard?'\n\n'He customised surfboards for a living. You know \u2013 spray-painted designs on them? He was a keen surfer, too...' I stopped, having a sudden vision of Tom freewheeling into space off the quarry road and wondering if he found the sensation exhilarating? I wouldn't put it past him, and of course he'd never expect anything he did, however dangerous, to actually kill him.\n\n'And you were here all evening?'\n\n'Yes. After I got back from the Mystery Play Committee meeting in the village hall I was experimenting with candyfloss, so I was pretty busy.'\n\nShe gave me a strange look but didn't follow that one up. Instead she turned her attention to Jasper.\n\n'And you were at this archaeological site all that day?'\n\nHe nodded. 'Occasionally I cycle there in the mornings, but Mum usually picks me up in the evening. The narrow roads round the site have become a bit of a rat run since everyone got satnav and she thinks I'll get knocked off the bike,' he said tolerantly. 'When I got home she'd been making lemon candyfloss. Yummy.'\n\n'Right,' she said, scribbling away. I nearly asked her if she would like me to whip her up some Cornish Mist, but I could see she had no sense of humour.\n\n'So, Mrs Pharamond, you must have been angry about your husband taking the car?'\n\n'I was, and even more so when he didn't come back. But I knew if I didn't turn up at the dig, Jasper would cycle back, he really didn't mind.'\n\nI was starting to feel strangely worried, despite knowing I had nothing on my conscience other than guilt for that profound moment of relief I'd felt on hearing that it was Tom who'd had the accident and not Jasper.\n\n'Jasper, perhaps tea would be a good idea? Or coffee. Would you mind?'\n\nHe gave me a look, but rose to a gangling six foot and, stooping under the low beam, went to the kitchen, though he left the door ajar. This is not a cottage where you can have private conversations... or indeed, private much of anything.\n\n'Can you tell me how the accident happened yet? I thought he must have had a seizure, perhaps, or a heart attack, even though he seemed a bit young for that? Or perhaps the brakes failed, or something?'\n\n'Actually, it looks as though one of the Citro\u00ebn's wheels came off.' Her eyes were fixed on my face to gauge the full effect of this pronouncement.\n\n'A wheel came off? But would that have caused him to veer off the road?'\n\n'Not necessarily. It's usually possible to drive on three wheels to a safe halt.'\n\nA sudden, rather nasty, thought struck me. 'Do you know which wheel came off?'\n\n'The front driver's side.' She looked at me intently again, and I realised I must've turned pale. 'Why?'\n\n'I had a flat tyre... it must have been that same morning, so I changed the wheel for the spare and took it in to be mended. Jasper undid the last nut \u2013 it was stiff \u2013 but I changed the wheel and put the nuts on again,' I said firmly. 'Jasper had gone back into the house by then. And what's more, it was absolutely fine on the drive to the dig and back!'\n\n'Mrs Pharamond, I'm not accusing you of anything!'\n\nWasn't she? It began to sound amazingly like it!\n\n'Isn't it just possible you didn't tighten them up quite enough, so they slowly worked loose? Accidents do happen.'\n\n'You mean I might have accidentally killed my husband?'\n\nNow I saw which way she was heading with this, I thanked God it was me who had tightened the nuts and not Jasper!\n\n'If they were a bit loose, then the tight bends of the quarry road could have completed the job,' she said. 'It's a possibility. We haven't found any of them yet.'\n\n'But I'm sure they were tight, because I used a wheel br\u2014' I stopped as Jasper came back in carrying a battered tin tray of mugs and an open carton of milk.\n\n'Yes, they were,' he said, putting the tray down on the coffee table with a thump that slopped some coffee over the rims. 'I could hear what you were saying from the kitchen and Mum put the wheel back on and tightened the nuts. And then when she went in to wash her hands, I tightened them up even more.'\n\nWe gazed at him, though presumably not with the same mixed feelings of affection and exasperation.\n\n'Oh, Jasper,' I said, 'I'm not being accused of anything except carelessness, so you really don't have to try and protect me!'\n\n'I'm not, Mum, it's quite true. I left you putting the wheel back on, but I checked it was tight enough later, when you weren't about.'\n\nI wondered how often he'd felt he needed to check up on me, and from my expression he deduced that he ought to add something. 'It was fine \u2013 I thought it would be.'\n\n'Of course it was! Any idiot can change a wheel,' I said indignantly.\n\nPC Perkins had lost interest in the ins and outs of our dispute, and turned to Jasper, notebook at the ready. 'So you are quite sure that the wheel was in a safe condition?'\n\n'Absolutely. And I often checked them and the tyre pressure since I passed my test, for the practice.'\n\n'So, how do you account for the same wheel coming off?'\n\n'I don't \u2013 that's your job, isn't it? But we don't know how long he'd been out, so he could have left the car standing about, and loosening the wheel nuts might have been someone's idea of a joke.' He shrugged. 'Mum's car was ancient, so who knows? Maybe the threads had gone or something, even?'\n\nI stared at him, thinking that he certainly didn't get his coolness and sang-froid from me or Tom \u2013 but, of course, my father was in the diplomatic service.\n\nShe closed her notebook with a snap. 'Once the post-mortem has been completed, if everything is in order, an inquest will be opened and adjourned and an interim death certificate issued,' she said briskly, by which I presumed she meant unless they found I'd been feeding him Cyanide Chutney for months. (Or Polly Darke's poisonous tomatoes. Pity I hadn't thought of that one!)\n\n'The funeral can then take place, and the inquest proper will open at a later date.'\n\n'Must there be another inquest?'\n\n'Yes, it's standard procedure in cases of this kind.'\n\n'Which kind?' I demanded, when I heard the kitchen door suddenly burst open and crash back against the wall, rattling all the china on the dresser. Then Polly Darke stumbled over the sitting-room threshold like a dishevelled, shrink-wrapped Bacchae, all billowing green chiffon sleeves, stick-thin legs and enormous boobs.\n\n'Well, stay me with flagons,' I said, surprised (damson gin for preference), for even Polly wasn't usually this avid to garner news.\n\nHer slightly prominent eyes passed over the policewoman and fixed on me. 'Is it true?' she demanded thrillingly. 'Is Tom really dead? They're saying he had an accident \u2013 in your car!'\n\nPresumably this was rhetorical, for with an anguished cry of, 'Tom! Tom!' she threw herself into the nearest chair and burst into hysterical sobs.\n\nJasper and I exchanged glances. Attention-seeking taken to extremes, combined with a raging desire to know what was happening was, I'm sure, our first thought.\n\n'This is Polly Darke, Officer,' I explained resignedly. 'She's a novelist and lives near Mossrow.'\n\nPolly looked up, her face like a drowned flower (a slightly withered pansy). 'I can't believe it. Only the night before last Tom was with me, and now he's gone. Gone!'\n\n'Why was he with you?' asked Jasper, puzzled. 'I thought he'd finally finished those Celtic murals you asked him to do ages ago?'\n\n'Because he loved me!' she exclaimed tragically and began to sob gustily again.\n\n'He was with you the night before last?' I stared at her, my mind whirling faster than a tumble dryer. 'Good heavens, don't tell me that you, of all people, are Dark Heart? No, it can't possibly be you!'\n\n'Yes it is! Why not?' she demanded belligerently, straightening from her pose of utter despondency. 'I could give him what he needed\u2014'\n\n'Tie him up, tie him down?' I suggested a bit numbly. You know, I'd never even considered her as a possible suspect, because to me she was a rather pathetic and ludicrous creature, though perhaps men might see her differently? But not young men, apparently, for Jasper looked even more incredulous than I was.\n\n'Dark Heart?' he queried.\n\n'Yes, your father was having an affair with someone, but though I found a note in his pocket on the morning of the day he vanished, it was only signed \"Dark Heart\", so I didn't know who it was.'\n\n'You mean, Dad was having an affair with her?'\n\n'Evidently, but I certainly thought it would be someone younger.'\n\nI'm quite sure Polly is much older than I am \u2013 well the other side of forty \u2013 even if she does try to hold back the years with every ancient and modern art at her disposal.\n\n'What do you mean?' she demanded indignantly, glaring at me. 'I'm only thirty-five!'\n\n'And the rest,' Jasper said drily.\n\nI'd entirely forgotten the policewoman was there until she interjected into the sudden lull in the proceedings, 'So you knew your husband was having an affair, Mrs Pharamond?'\n\nHer notebook was open again, I saw, pen poised.\n\nI glanced uneasily at Jasper. 'He... well, he had had lapses occasionally in the past, but they didn't mean anything. Then I found out about a more serious affair about five years ago, when my son was ill \u2013 and I'm so sorry, Jasper: I didn't want you to find out about your father's affairs, especially like this.'\n\n'Oh, I knew all about the women, Mum,' he said calmly. 'I even caught him at it with that girl out of the Mummers once, when I walked in on them in the workshop.'\n\n'You did?'\n\n'That's a lie!' Polly yelled furiously, but Jasper just glanced coolly at her, one eyebrow raised, as though she were a failed souffl\u00e9. He looked terribly like Nick. I don't think Polly is any kind of souffl\u00e9, though, more of a synthetic Black Forest gateau with poisonous cherries.\n\n'So you were not on good terms with your husband,' the policewoman suggested to me, 'although he'd had affairs in the past to which you hadn't objected?'\n\n'Of course I objected!' I exclaimed. 'What do you take me for? And they were usually more in the nature of one-night stands than anything serious. For a long time I used to believe him when he said he loved me and they meant nothing.'\n\n'Yes, but that was the old Dad, not the nastier model we've had to live with lately,' Jasper pointed out. 'Even I've overheard him, taunting you about some woman he's been seeing \u2013 and he's not coming across as a very admirable-sounding character, is he?'\n\nThe police officer said patiently, 'So this time he was having a serious affair, Mrs Pharamond? He would have left you?'\n\n'No, it had to be the other way round, because this cottage belongs to his great-uncle by marriage, Roly Pharamond. So I intended leaving, once Jasper was at university and I'd found new homes for the livestock and sorted out somewhere to go, some sort of job...' I trailed off.\n\n'That's so not true! I heard you arguing in his workshop that very morning and when I questioned him about it later, he told me he'd asked you to leave and you'd refused!' Polly cried. 'He was afraid Roly Pharamond would take your side and he'd lose the cottage and everything he'd worked for.'\n\n'Obviously you didn't hear much, Polly!' I said, surprised. 'What I actually told him was that I'd had enough and was going to leave him as soon as I could. And if anyone worked around here and stood to lose everything, it was me!' I added incautiously, and the policewoman's pen skidded quickly across the page.\n\n'Well, at least you don't have to do that now, Mum,' Jasper remarked, and a small silence ensued.\n\nI sighed. 'We might still have to move, Jasper. It depends on Uncle Roly.'\n\n'Unks won't put you out, Ma. He's really fond of you.'\n\n'So,' said the officer to Polly, 'you overheard an argument, and what then?'\n\n'She came out,' Polly said, with a venomous look at me. 'So I said I'd brought her some field mushrooms to exchange for eggs, and she said, \"Help yourself, I'm going for a walk.\" She was really odd \u2013 she looked furious. When she'd gone I spoke to Tom briefly and he said he'd come over later, after he'd finished the board he was painting \u2013 which he did. And that's the last time I saw him, because when I woke up early next morning he'd gone. He parks around the back of the house, out of sight, so I'd no idea he hadn't come in his own van,' she added. 'I just assumed he had.'\n\n'No, it was still at the garage,' I told her. 'But if he hadn't taken my car, when he knew very well I wanted it later, it might have been me and Jasper who had the accident.'\n\n'It should have been you!' she said venomously. Her reddened eyes and sharp nose made her look like a particularly unsavoury rodent.\n\nJasper stood up slowly and said in a tone of menace I'd never heard from him before, 'I think you've said \u2013 and done \u2013 quite enough. Why don't you clear off?'\n\nShe floundered hastily and inelegantly out of the chair and backed towards the door. PC Perkins jumped up and stood between them.\n\n'If I could have your name and address, Ms Darke? I'll follow you over and ask you a few more questions in your own home, if I may?' She turned to me with a thin smile: 'Thank you for your assistance, Mrs Pharamond.'\n\nI had a horrible feeling she suspected me of loosening the wheelnuts on purpose, then leaving the keys out where Tom was sure to find them. And goodness knew what Polly would tell her!\n\n'Jasper,' I said when they'd gone, 'you were wonderful!'\n\n'Don't worry, Mum, that cop may have a suspicious mind, but we know there's nothing to find, so they can't pin anything on you.'\n\n'Thank you, darling,' I said weakly, then had a thought. 'I wonder if Tom had anything to eat at Polly's? Only if he had an attack of food poisoning, that might account for why he lost control of the car when the wheel came off.'\n\n'I don't think he went there to eat, Mum,' Jasper said, before vanishing back up to his Batcave.\n\nIn the kitchen I discovered that half the candied peel had vanished, presumably eaten by Jasper while waiting for the kettle to boil, but then, it's very moreish. But it didn't matter, I was only going to dip it in dark chocolate as a treat for later.\n\nMeanwhile, there was a whole row of bolting lettuces (I'd planted too many, as usual) to toss to the hens, and fruit to pick: a fresh strawberry Pavlova would be wonderfully comforting.\n\n## Chapter 8: Well Braced\n\nOnce our bulk order for dried fruit, peel and all the other ingredients has arrived and been divided up among the five members of the Christmas Pudding Circle, you can tell where we all live by the rich aroma of cooking mincemeat wafting from the doors and windows. We tend to make it early and of course it's useful all year round, for making mincemeat brownies, stuffing baked apples and a host of other things \u2013 not least the famous Middlemoss Marchpane tart.\n\nI'm going to make a bumper quantity this time, before I really get going on all the bottling, preserving and freezing of garden fruits and vegetables that starts to build up momentum at this time of year: the making of chutneys, jams, curds and relishes...\n\nThe Perseverance Chronicles: A Life in Recipes\n\nOn the Monday Jasper returned to his dig (by bike) and I went to the Christmas Pudding Circle meeting. I was glad of any distraction from the turmoil of mixed emotions caused by Tom's death and Polly's revelations, though it would have been impossible to describe what I felt. It wasn't even as if Tom had played a major part of our lives for the last few years, except in a negative, passing storm-cloud sort of way, but still, grief of some kind was an element and Jasper, I was sure, felt much the same. And also, I was increasingly uneasy at the direction the police enquiries seemed to be taking...\n\nMarian's carefully drawn-up CPC meetings rota had already gone to pot. This one was held at Faye's farm instead of Annie's cottage, because our ingredients had arrived and she had more room in her kitchen than anyone else for dividing everything up.\n\nAnnie picked me up and ran me there and was concerned to know how Jasper was doing.\n\n'Still confused, poor boy,' I said. 'Tom had been so horrible to him lately \u2013 but he was still his dad, after all.'\n\n'I expect he feels all angry and sad and muddled up,' she agreed. 'And you, too!'\n\nWhen we got to Faye's, the others were already there and expressed their condolences, before we got down to the business of the meeting: dividing up our purchases of flour, dried fruit and peel, flaked almonds and all the rest of it. Faye's cavernous farmhouse kitchen slowly became redolent with the spicy fruity smell of Christmas and I found that strangely comforting.\n\nSo too was the tea Faye laid on afterwards, with strawberry jam and clotted cream to spread on the freshly baked scones. It was no wonder her little tearoom was perpetually packed out, so that she had to take on extra staff!\n\nI felt so much better after spending an hour or two in the undemanding company of my friends. And then the making and bottling of my mincemeat over the following days, along with producing some jam and chutney from a basket of ripe apricots given to me by Marian, proved a pretty good distraction.\n\nDue to Unks ringing an old number instead of that of Nick's BlackBerry, it was Wednesday before he tracked him down to give him the bad tidings, and by then I'd already received a postcard of Morecambe Bay he'd posted a couple of days ago. It bore a scribbled recipe for spiced potted shrimps, which I found immensely comforting.\n\nUnks said Nick sent his love and would call me when he got back, because, of course, being a true professional, he will complete his assignment and send in his copy first. This was more than I seemed able to manage, for the end of August deadline for sending in my newest Perseverance Chronicle was fast approaching.\n\nI rarely mentioned Tom in the books \u2013 though when I did I referred to him only as 'the Inconstant Gardener' \u2013 but I couldn't entirely ignore what had happened to him, so bringing the latest one to a close on any kind of upbeat note would be impossible. Unless, that was, I ended it just before Tom's demise. Then I could include it in a foreword at the start of the book after that, which would come out when a decent interval had passed, the misery blunted by time.\n\nI could even end my current Chronicle with an apocryphal near-death by mushrooms instead. It's the sort of thing my readers seemed to enjoy and I often embroidered the truth to make a good story. Saved by Caz Naylor in the nick of time... assuming he had found the poisonous fungi in the basket of mushrooms Polly left me, which I don't think was ever clearly established. He could have been giving me a hint about Polly and Tom. I'm sure Caz must often have been flitting about the place in the evenings like a shade, so may well have seen and heard some of what had been going on.\n\nBut I should have known better than to accept anything edible from Polly's hands, even if the mushrooms had looked suspiciously like supermarket ones, brought as an excuse to snoop around Perseverance Cottage \u2013 or maybe, now I know about her affair with Tom, in order to see him without my suspecting anything.\n\nI had a sudden horrible thought. If I'd cooked the mushrooms without spotting the poisonous one, Jasper might have been made ill too! (But not Tom, who didn't like them.) That put me right off mushrooms, whereas before I loved them.\n\nA lovely letter of condolence came from the Vanes this morning. They must have posted it practically the minute Annie told them about Tom. Of course, they only really knew the old, charming Tom and not the monster I'd been living with lately, but it was very kind and comforting all the same.\n\nThe latest issue of the Mosses Messenger was in the letter box too, and carried a Mystery Play notice, which I found a bit poignant.\n\nIMPORTANT NOTICE!\n\nEveryone wanting to take part in the next Mystery Play should put their name forward immediately to Clive and Marian Potter at the Middlemoss Post Office, including any of last year's performers intending to reprise their roles. (Applicants must be residents of Middlemoss, Mossedge or Mossrow and able to devote one night a week to rehearsals.) Sadly, due to bereavement, the important roles of Moses and Lazarus need to be recast and, as always, we need a new baby Jesus. Rehearsals will start in mid-September.\n\nI'm sure I will never be able to watch the Raising of Lazarus scene again without thinking of Tom, though actually when Lazarus's mother says to Jesus, 'Our Lazarus hath popped his clogs, and he were my only child,' my eyes tend to well up anyway.\n\nOf course, Jesus immediately tells Lazarus to stop larking about, because his mother is in a proper state about him, and up he jumps with a cheery, 'Hello, our ma. Is summat up?'\n\nBut there will be no miracle this time for Tom, who hath well and truly popped his clogs.\n\n'Hi, Lizzy, I hear fate caught up with Tom before I did,' was Nick's bluntly uncompromising opening gambit later that day when he did finally phone. 'Roly told me what happened.'\n\nI gripped the receiver tightly. 'Oh, Nick! Did Unks tell you it was my car? The wheel came off, and it was the one I'd just changed the tyre on, so I'm sure the police think I did it on purpose!'\n\n'Don't be so bloody stupid, of course they don't think that! Why would you tamper with the wheel on your own frigging car?' he snapped, and I stopped wilting over the receiver and glared at it, as though Nick could see me.\n\n'That's true, I wouldn't \u2013 but his van was at the garage and for once I'd left my keys out somewhere where he could find them. And we'd just had an argument, which Polly Darke overheard \u2013 she told the police. Nick, she was the one Tom was having the affair with, but I don't think anyone except the police and Jasper know about that yet.'\n\n'Well, I certainly didn't, but it doesn't surprise me, because apart from you, he always did have crap taste in women.'\n\nThat was as close to a compliment as I'd ever got from Nick \u2013 and he can't include his own wife in that statement either, can he? So he mustn't really think Tom and Leila were having an affair, after all. The idea of Leila as Tom's lover is even more ludicrous than Polly, so I expect Tom was simply after free bed and board in London and lied to me to spur me into finally leaving him.\n\n'I had to go and identify Tom,' I said abruptly, my thoughts taking a sudden, darker turn. 'It was all right until we actually went in, because I thought it was a nightmare, so it didn't matter. But then there he was and... I kept expecting his eyes to open. I couldn't believe he was really dead, even though they said his neck was broken. Now I keep thinking about him, so white and... gone.'\n\n'Don't think about it, then,' he said sensibly. 'It's a pity I wasn't home to identify him for you. Anyway, it wasn't your fault and I think he's given you a rough time the last year or two. I've kept my nose out, but I've heard things.'\n\nTears pricked behind my eyes. 'Well, it's over now and they're releasing the body. They've done the post-mortem and adjourned the inquest,' I said, shivering. 'The inquest will reopen again later, but not for ages probably, so the funeral can go ahead on Tuesday.'\n\n'Then they don't suspect you of anything, you stupid bat,' he said, with what sounded suspiciously like relief. Surely Nick didn't secretly think I'd bumped Tom off, too? 'I'm heading down to London tomorrow,' he added, 'but I'll be back in a couple of days, if you need any help with the arrangements.'\n\n'Thanks, but I've got Annie \u2013 and the rest of my friends in the CPC have volunteered to help. The new vicar seems very pleasant and helpful too. In fact, everyone has been so kind. Roly offered to have the bunfight after the funeral up at the Hall, but I wanted it here. I'm going to do everything just the way the Tom I married would have wanted it \u2013 the fun one with a sense of humour.'\n\n'Let's hope it's a nice day then, because you won't get more than six people standing up in your sitting room.'\n\n'I'm going to use the old greenhouse: that's certainly big enough. I was going to get rid of it, so it's fortunate I haven't got round to it yet.'\n\n'It'll still have to be a fine day, because it leaks like a sieve,' he objected.\n\n'Caz Naylor asked me if he could do anything, and he's out there now mending the cracked panes with gaffer tape.'\n\nSaying Caz had volunteered to help was a slight exaggeration, since his precise words had been a questioning, 'Do owt, our Lizzy?'\n\n'You're honoured,' Nick said drily. 'Whenever Roly or I ask him to do anything, he vanishes. How's Jasper taking all this?'\n\n'Stoical and quiet, but much the same as me: in some ways it's a relief to know Tom's not coming back, but then we feel guilty for even thinking that, and sad at the same time and...' I broke off, my voice wobbling.\n\n'Do you want me to come straight back now?' he asked abruptly.\n\n'No, of course not! What could you do?' I said, braced by his tone.\n\n'Nothing, I suppose. Where's Jasper now?'\n\n'He's been back at the dig since Monday. He thought he might as well, rather than mope around the house.'\n\n'Very sensible. Has he sorted out his university accommodation yet? Or is he going to live at home?'\n\n'I wanted him to live in the hall of residence for the first year at least, because that's what university is all about, isn't it \u2013 getting away from your parents and making your own life? Only now he says he might share a house with one of his friends and some other students instead.'\n\n'He's very level-headed for his age. I'd let him do whatever he wants.'\n\n'Yes, I suppose so, and at least Liverpool is close enough for him to come home for the weekend, or for me to drive over, if he gets homesick or anything.'\n\n'I expect he'll quickly have other distractions.'\n\n'I should think the course will be distraction enough. I'm so proud of him, getting onto it. And he should get the full student loan now too, I think, because it'll be based on my income and I've hardly got any, because I don't suppose barter counts. Unks told Jasper yesterday that he was going to make him an allowance. He's so kind, and really, we have no claim on him.'\n\n'Roly thinks of you as family. And speaking of family, have you heard from Tom's mother and stepfather? You'd think they would offer to help!'\n\n'Oh, they have, and it was dreadful! He phoned and said Tom's mother was too upset to speak to me and probably wouldn't be well enough to travel all this way for the funeral, but he would pay for it all! I told him I didn't want his money and I haven't heard anything since.'\n\n'Well, burn all your boats at once, why don't you?' he said sarkily.\n\n'You're such a comfort to me!' I snapped, but beginning to feel much more like the real Lizzy Pharamond under all this bracing common sense.\n\n'Fellow feeling, darling. Leila and I are going to split, though I hope not in quite so final a manner as you and Tom.'\n\n'You are? I'm so sorry!'\n\n'Are you? Then don't be! Things haven't been good between us for a long time, though she's always refused to discuss divorce.'\n\n'Well, she's Catholic,' I pointed out. 'That's probably it.'\n\n'Only nominally, and she's going to have to get used to the idea, so the sooner the better. That's why I'm going straight down there now, to tell her.'\n\n'You mean she doesn't know yet?'\n\n'She knows how I feel: our marriage is dead in the water, and it's time to call it quits. And I want to spend much more time in Middlemoss: I feel more creative there.'\n\n'Lawrence Durrell's \"spirit of place\",' I agreed. 'Middlemoss gets your creative juices flowing. Mine, too \u2013 this is my real home and I'm not sure I could write anywhere else.'\n\n'If you can call your stream-of-consciousness burblings writing, any more than you can describe your recipes as cookery,' he said. 'We can only be grateful you do your ghastly Chronicles under your maiden name and disguise anything that might give away the location!'\n\n'At least anyone can make my recipes, you don't need a thousand pounds worth of equipment and three underlings to help you!' I shot back rather unfairly, since I know very well he whips up his recipes in his own kitchen, or the one at the Hall: personally tried and tested before being unleashed in his Sunday newspaper cookery page, or in his books. They're mostly straightforward recipes too, not nouvelle cuisine or anything, though some are still a little fancy for my taste. I like to keep things simple.\n\n'Plebeian', he once called me, when he found me devising a recipe for strawberries and custard bread-and-butter pudding. But then, as I recall, he ended up eating two plates of it... and now I come to think of it, I haven't made that for quite a while and it's yummy... real comfort food.\n\n'I think we're digressing,' he said, sounding pleased as always to have got a rise out of me. 'I'm going to London to tell Leila that I'm instructing my solicitor to start divorce proceedings, whether she wants it or not. After that, I'll be back at the Hall, so I'll be around if you want me. Don't tell anyone about the divorce yet. I'll break the news to Roly later. I'm not sure how he'll take it, and I don't want to give him any more shocks.'\n\nActually, I thought Roly would be pleased rather than shocked, but I didn't say so. 'No, I won't mention it to anyone and\u2014' I broke off as a thought struck me. 'Nick, I don't think Leila knows about Tom yet! How awful, I forgot to tell her!'\n\n'I'll tell her and I'll be back before the funeral,' he said, and put the phone down.\n\nIf he'd been here in person, I wondered if he would have given me a big, comforting hug like he had in hospital that time, when Jasper was ill... and looked at me with that same startled expression in his slate-coloured eyes, as though surprised to find himself doing it?\n\nHe does have a softer side and, although he can be a bit taciturn, he's all bark and no bite.\n\nIn the event, Nick wasn't back before the funeral, calling from London to explain briefly that Leila insisted on being present at it and was refusing to discuss anything about the divorce until afterwards, so he'd be driving her up on that morning.\n\nThere wasn't anything for him to do, anyway \u2013 I was pretty well organised, Annie having taken over the finer details. When you've been Leader of the Pack (Brownies) for years, these things come naturally to you.\n\nThe funeral being on a Tuesday, the preceding Monday's CPC meeting had been cancelled and instead my friends all brought to Perseverance Cottage food for the buffet and then stayed to help get everything ready. Annie must have spent half the night making little sausage rolls, Faye had baked both sweet and savoury scones, Marian brought the makings of three different kinds of sandwich and Miss Pym had assembled two huge platters of cold meats.\n\nSo by late Monday afternoon the preparations for the Feeding of the Five Thousand \u2013 or however many turned up to be fed and watered after the ceremony \u2013 was complete. Every surface in the kitchen and larder groaned under the weight of plates and bowls and platters. The fridge door kept trying to spring open, and plastic bags of yellow candyfloss swung from the rack above the kitchen table. The very air could have been sliced up and served with whipped cream, it was so loaded with mingled aromas.\n\nAnnie came back later, and she and I sat in the tiny sitting room, drowning our sorrows in elderberry wine and eating some of the mincemeat brownies intended for tomorrow, while Jasper was out the back, immolating Tom's favourite surfboard on the garden bonfire. He was accompanied by the strange, small dog (rather like a hairy haggis with legs) which Annie had brought with her, along with Trinny, and it had immediately attached itself to Jasper.\n\nHe came in from his bonfire with the creature under one arm and vanished up to his Batcave in the attic to bludgeon his emotions with loud music.\n\n'Annie, that dog\u2014' I began.\n\n'Jasper's agreed to foster it until he goes to university,' she interrupted brightly. 'The kennel was full, and no one seems to want to adopt it.'\n\n'You surprise me,' I said tartly. 'It nipped my ankles when it came in and it sheds so much hair it leaves a trail behind it across the carpet.'\n\n'I expect Jasper will give it a good brushing. You've still got all Harriet's stuff, haven't you?'\n\n'Yes, but I don't want another dog at the moment, and what if Jasper gets attached to it? He can't take it to university with him. You'll have to take it away with you right now!'\n\nPutting my glass down I went upstairs, determined to oust the creature before things went too far. Jasper's door was open just a crack and through it I saw him sitting on his bed, his face buried in the hairy haggis and his shoulders shaking.\n\nSilently I backed away and tiptoed downstairs.\n\n'It can stay for a couple of weeks,' I conceded to Annie, 'but that's it. You'll have to keep looking for a permanent home for it.'\n\n'OK,' she agreed. 'Unless you find you want to keep her, after all.'\n\n'I doubt it. I've got puncture marks in my ankle.'\n\nAnnie went home soon after that. She was going to come here straight after the church service the next day, instead of attending the interment, and organise the Women's Institute volunteers who are manning the buffet at the funeral feast in the greenhouse.\n\nAround three in the morning, entirely unable to sleep, I went downstairs and whipped up a batch of strawberries and custard bread-and-butter pudding \u00e0 la Lizzy Pharamond, and then Mimi wandered in out of the night, dressed in wellies and with a man's Burberry overcoat over her nightie.\n\n'Hello, dear, I've come for tea,' she said brightly, sitting down at the kitchen table. 'And to tell you that Tom's dead.'\n\n'I know,' I said, handing her a portion of bread-and-butter pudding and the cup of cocoa I was just about to drink myself.\n\nShe seemed very taken with the words, and was still repeating softly: 'Tom's dead, Tom's dead!' all the time I was walking her back up the dark drive to the Hall later, which was a little trying.\n\nWhen we got there, Juno had just discovered her absence and was frothing gently at the mouth. But there was no harm done, though the sooner her leg is healed so she can keep tabs on Mimi again, the better.\n\n## Chapter 9: Soul Food\n\nI had one of those confused moments standing at the edge of the grave, where I couldn't remember where I was \u2013 or even who I was \u2013 let alone who was six feet below me, tastefully attired in sustainable Norwegian pine. The coffin was crowned with a home-made wreath of dried hops (Tom had been a great devotee of real ale), bearing the handwritten epitaph: 'For the Tom we loved, from Lizzy and Jasper'. We had refrained from adding 'if he ever existed'.\n\nThe circle of eyes fringing the grave reminded me of a stargazy pie, except that they were not blank and dead, but expectant \u2013 and fixed on me. What could they want?\n\nThere was Nick's tall, broad-shouldered figure, his purple-grey eyes dark and brooding, possibly because his chic French wife was hanging tightly on to his arm, as she did to all her possessions.\n\nNext to him was his father, Nigel, in whom the strong Pharamond genes had surprisingly been subjugated by the more nondescript ones of his mother, his expensive suiting trying to turn a sow's ear into a silk purse.\n\nDr Patel, Marian and Clive Potter and Faye, wearing a borrowed-looking black hat jammed over her dark curls and with her squarely-built, rosy-cheeked husband in tow. Miss Pym, nodding encouragingly at me, as if I were a recalcitrant four-year-old. A ragbag of Tom's old surfing chums, looking shifty. Polly Darke, wearing a short and inappropriate black chiffon garment cut low over the twin pink Zeppelins of her bosom, hovering uninvited and unwanted on the fringes of the crowd. Gareth, the new vicar, with his pale, interestingly knobbly face, bright red hair blowing in the slight breeze like a fiery halo...\n\nJasper nudged me with a bony elbow. 'Mum?'\n\nAs though his action had opened the sluice gate, a scummy dark tide of realisation rushed into my head: I was Lizzy Pharamond, widow, mother of the willowy youth next to me, and now expected to toss earth onto my late husband's remains like a cat tidying up after itself.\n\nThe husband who had once had a quirky sense of humour, until something dark, angry and increasingly nasty had slipped in to inhabit that space instead. I'd been mourning the loss of the old Tom for a long time, but now these last rites seemed to form an epilogue to our life together and a full stop.\n\n'Mum?' Jasper said again, more questioningly, draping a sinewy arm across my shoulders. For a teenage boy this was touchingly demonstrative and, for the first time that day, I felt painful tears at the back of my eyes, though earlier I'd struggled to suppress grossly unbecoming giggles during the vicar's eulogy, when he tried to reconcile wildly conflicting descriptions of Tom's character by using surfing as a metaphor for his journey through life and on into the great ocean that was Death.\n\nI remembered what was expected of me. Slowly I reached into my large, gaily embroidered shoulder bag and took out Tom's mobile phone and the TV remote control, then tossed them with a clatter into the open grave on top of the coffin. Grave goods: the things most dear to him \u2013 apart from his favourite surfboard, immolated by Jasper of course. But even that was here in spirit, for as I turned and left amid stunned silence, I stumbled over its effigy worked in wired flowers, with a card attached reading, 'Yo, dude! Catch a big one.'\n\nFrom behind me came the light patter of earth as the mourners hastened to cover up the evidence of my eccentricity, though I fear Tom will be gone but not entirely forgotten until the battery on his mobile runs out. He was always popular with his drinking companions.\n\nAt the end of the gravelled path stood Tom's white van, which had done duty today as his hearse, and I suddenly recalled how the six mismatched surfers and Mummers of Invention had earlier tried to shoulder the coffin before carrying it into the church.\n\nA hysterical bubble of laughter attempted to force its way up my throat, though I managed to stop it escaping by clamping my lips together. But two painful tears squeezed out and ran down my cheeks, compounded of laughter and sorrow, inextricably mixed together with an over-heavy seasoning of the guilt that seems to be an inescapable accompaniment to death.\n\n'Ow-do, missis,' Dave Naylor said. The proprietor of Deals on Wheels had driven Tom's van to the funeral and was now leaning against it, rolling a cigarette between scrubbed but darkly cracked fingers.\n\nAnother Naylor, you note \u2013 and also, on less official days, likely to address me as 'our Lizzy'. I really must do a bit of family research some time!\n\n'Bear up, lass. It's all sorted now and a great send-off it were, too. Them Mummers singing \"Amazing Grace\"?' He shook his head in slow wonderment. 'By heck, we'll never hear the likes of that again.'\n\n'Oh, I do hope not,' I agreed fervently. 'And thanks for driving the van, Dave. You are coming back to the cottage, aren't you?'\n\n'Aye, but I'll let the fancy cars go first. I'll take the van back to the garage with me afterwards and drop your new car off in the morning.'\n\n'That's fine \u2013 see you later,' I said gratefully, and carried on to where Roly Pharamond awaited me in his long black Daimler on the main pathway, having sensibly eschewed the interment in favour of a sit-down and a swift nip of brandy. His sister, Mimi, seemed to have eschewed the funeral altogether.\n\nJoe Gumball, husband of Roly's cook and jack of all trades up at the Hall, got out of the driver's seat and opened the door for me. He was wearing the hat and jacket of a chauffeur over faded blue dungarees and wellington boots.\n\nJasper, who was silently following me, got in the front and I slid onto the leather back seat, where Roly patted my hand with his thin, dry one and said gently, 'All done and dusted, my dear?'\n\n'All done and soon to be dust,' I agreed numbly. 'It seems so surreal \u2013 and the way everything keeps undulating slightly isn't helping,' I added. This underwater rippling feeling had been going on ever since I got the news of the accident, and nothing, not even the best elderberry wine, could entirely make it go away.\n\nHe shook his head sadly. 'I never thought to outlive Tom \u2013 but there, anyone can have an accident. Well, better get the bun fight over with, I suppose. Mimi should be along later with Juno.'\n\nJoe pulled out and headed for the cottage where, with the help of Annie and two ladies from the WI, the big, ramshackle greenhouse would have been turned by now into a venue for the funeral baked meats. Trestle tables and folding chairs had arrived this morning, borrowed from the village hall, along with tea and coffee urns.\n\nWe'd moved what plants there were towards the far end, but an aroma of tomatoes and moist earth scented the air. Still, I'd judged that better than holding it in Tom's wooden workshop, with its stale smell of dope, and the spray paint he used to customise the surfboards, several of which were propped in various stages of completion around the walls.\n\n'You know, I still expect to open his workshop door and find him there,' I said, following this train of thought. 'Just like all the other times when he vanished for a few days and turned up as though he'd never been away.'\n\nJasper turned around and looked anxiously at me, and I summoned a smile from somewhere. Luckily he couldn't hear us, because the sliding glass partition was shut and Joe was playing muted country-and-western music.\n\nWe drove over the hump-backed bridge crossing the stream, scattering the gaggle of five vicious geese, which had taken up residence there among all the innocently stupid ducks.\n\n'Must get something done about those creatures,' Roly said absently. 'The children are all too frightened to go to the playground, and I'm told you can't feed the ducks without being attacked.'\n\n'That local animal rights group, ARG, might have something to say about that,' I said. 'But the geese are getting more and more aggressive, and they leave such a mess behind them, too. Someone is bound to skid on it eventually and then there will be hell to pay, though I don't know who you can sue if no one owns them?'\n\n'Perhaps, since I own the green and the stream, I own them, too \u2013 or at any rate, the right to deal with them,' Roly suggested. 'I'll ask my solicitor \u2013 Smithers will know. Or perhaps I'll just get Caz Naylor to quietly round them up one night and move them somewhere else.'\n\n'How's he doing with the squirrels?'\n\n'Very well. Constantly patrols the exclusion zone, of course, but that's what you have to do, to keep the grey buggers out. Only way. Reds, that's what we have at Pharamond Hall. Always have, always will. On the coat of arms, even.'\n\nI thought this showed a touchingly Canute-like optimism, since the tide of grey squirrels seemed to have swept over most of Britain. But then, the reds had got Caz on their side.\n\n'I think the signs Caz has put up on the main pathways might have caused some talk,' I suggested. '\"Red or Dead!\" is a bit ambiguous and that new one just inside the gate that says, \"Warning! Keep to Path!! Trespassers May Be Unexpectedly Terminated!!!\" is a bit over the top.'\n\n'Only to outsiders \u2013 and what are they doing wandering all over my estate, that's what I want to know? Locals \u2013 yes. They know the score: keep to the public footpaths, don't wear grey.'\n\n'Are you still being targeted by ARG?' I asked. 'I don't seem to be bothered by them so much now, but I suspect that's because Caz's keeping an eye on the place.'\n\n'Well, family, aren't you?' Roly said vaguely. 'And they've eased up on the estate a bit since I put that piece in the parish magazine saying Caz uses humane live traps to catch the grey squirrels. Ingenious things: the reds can get out again, but the grey's too big.'\n\n'Mmm,' I said, because of course the question not to ask is: what does Caz do with the grey squirrels after he's caught them?\n\nWe passed between the impressively pineapple-finialed gateposts of Pharamond Hall, then turned sharp right onto the track that led down to Perseverance Cottage, which is just inside the estate boundary wall. I'd stuck a sign up earlier deflecting the mourners away from the Hall, but the ravening and curious horde would be hard on our heels, probably expecting an abundance of finger food and alcohol in a suitably sombre setting. Instead, they would find themselves in a huge glasshouse, eating home-made scones spread with jam and cream, strawberries and custard bread-and-butter pudding and other, even less usual, comestibles (I got a bit carried away yesterday), all washed down with tea or coffee.\n\nI'd noted the police presence at the funeral (PC Perkins and Little Boy Blue), but somehow their car had managed to arrive at the cottage first. As I got out, so did they, and Perkins came over and said they'd come to offer their condolences. But there was an underlying implication that she thought I was a merrier widow than I let on, and she'd been expecting me to cast myself onto the coffin with a last-minute confession. But perhaps I was becoming paranoid.\n\n'Do stay for refreshments in the greenhouse,' I said politely and, after a small, uncertain pause, she said they would follow us over. At least her colleague would have something to eat other than his fingernails.\n\n'How the wheel came off is destined to be one of life's great mysteries,' I mused aloud, as we walked across the cobbled yard. 'And why he didn't stop the car from going over the edge. Still, at least I know who Dark Heart is now, so that's one puzzle solved.'\n\n'Dark Heart?' Roly said. I'd quite forgotten he didn't know about Tom's affair.\n\n'Dark Heart's how the woman Dad was having an affair with signed herself,' Jasper explained helpfully. Roly was leaning on his arm as he picked his way over the smooth, slightly slippery stones. 'It's that novelist woman \u2013 Polly Darke. Would you believe it? She's got to be even older than Mum, and a complete hag.'\n\n'Thank you, darling,' I said. 'I think she is fashionably haggard, rather than a hag, really. And I'm sorry, Roly, I didn't mean to tell you about her. It just slipped out.'\n\n'Boy must have been mad!' Unks exclaimed, patting my arm. 'Wondered why she turned up to the funeral. Wearing a black see-through nightdress, too. Woman's got more bones than a picked chicken carcass!'\n\n'Was she in church? I didn't actually take in most of who was and who wasn't.'\n\n'She was there, Mum, but I thought Uncle Nick was going to throw her out when she tried to sit in one of the front pews. And it was a black chiffon dress, Unks \u2013 but she's, like, fifty years too old to wear it.'\n\nI think that might have been a slight exaggeration, but I didn't feel I was a winner in the sartorial stakes either. I didn't have any black, so was wearing a floaty, dark, paisley-patterned Indian dress over a long pink crinkle-cotton skirt, both borrowed from Annie. I'd belted it severely in around my waist, but I still looked like Widow Twankey.\n\n'Woman's got a nerve turning up at all, if she was carrying on with Tom! Well, well, it just goes to show you that the boy wasn't himself the last few years.' Roly shook his head.\n\n'His character had changed...' I began cautiously, and then broke off as the nose of Tom's white van appeared, followed by a cort\u00e8ge of assorted vehicles. 'Here they come! Dave must have got tired of waiting and decided to lead the way instead.'\n\n'Let's have a drink before battle commences, shall we?' Roly suggested.\n\n'It's only tea or coffee, Unks, unless you want Jasper to fetch you a glass of elderberry wine? I've hidden a few bottles under the table in the corner, just in case. I didn't want this to turn into some kind of drunken revel and go on for hours.'\n\n'Quite right, but I've got some brandy in my cane,' he said. 'Ah, Annie, my dear...!'\n\nWe established him in the greenhouse, on one of the chairs grouped before a veritable thicket of tomato plants and rampant bell peppers, and he unscrewed the top of his cane and poured a generous slug of brandy into his tea while I went and peeped out of the entrance.\n\nAs the cars arrived, people milled about in front of the cottage, so Jasper went out to usher them in the right direction and I retreated back to Unks. I expect I should have stood in the doorway and accepted their condolences as they came in, but I couldn't face it.\n\nThe mourners massed in the entrance like worried sheep, nearly balked and broke away, then came slowly in a surge towards me. At the last minute most of them sheered off and spread out to range up and down the trestle tables, probably looking for alcohol.\n\nI accepted a slug of brandy in my teacup from Roly and sat down.\n\n'That's it: let them come to you and do the polite,' Unks said. 'The ones with manners, anyway. Half of this lot weren't at the funeral \u2013 probably just after food and drink. Freeloaders!'\n\n'Weren't they? The church did seem to be full, though, and I thought the new vicar made a brave job of it.'\n\n'Not bad. Bit much to expect him to do a complete stranger's funeral two seconds after he arrives. Let's hope he lasts longer than the last vicar. Ah, here's Nigel.'\n\nRoly's portly retired stockbroker son was indeed forging towards us, though there was no sign of his wife. Still, I hadn't expected even Nigel, who didn't seem to care much for the North \u2013 or indeed Tom, for that matter, whom he considered a cuckoo in the nest \u2013 to make the effort to come today.\n\n'My son and heir,' Roly said drily in my ear. 'And unless Nick and that French woman get a move on, nearly the last in the line.' He nudged me with a sharp elbow. 'What do you think, Lizzy \u2013 is she past it?'\n\nLeila, chic in a vintage Chanel suit, bold red lipstick and with her hair drawn sleekly back like a bleached Paloma Picasso lookalike, stood poised in the doorway looking capable of anything, though the curl of her lip said exactly what she thought of her current location. Just wait till she saw the catering!\n\nNick, wearing his best sardonic Mr Rochester expression, loomed behind her.\n\n'I wouldn't get your hopes up, Unks,' I said.\n\n## Chapter 10: Cornish Mist\n\nI found myself mentally writing the recipe for a funeral feast, as though it was something I could put into one of the Chronicles.\n\nRecipe for a Fine Funeral Feast\n\nStep 1: In a large greenhouse mix together approximately sixty assorted mourners, half of them uninvited, several wearing wildly inappropriate surfer garments, and some that you're certain you've never met before in your entire life hovering furtively around the edges.\n\nStep 2: Add two police officers steeped in dark suspicions, a mad historical novelist dressed like the porno version of a Poldark widow, and a depleted folk\/rock group all wearing identical black 'Gaia Rocks!' T-shirts.\n\nStep 3: Gently fold in your best friend, trying valiantly to suppress her natural expression of cheerfulness; a mourning, lightly intoxicated nonagenarian great-uncle by marriage; a tall, dark and glowering chef and his acidulated and tangy French wife (soon to be ex and who might, or might not, have been having a fling with your late husband), and the man from the garage who'd volunteered to drive Tom to the funeral in his own white van with the 'Board Rigid' logo up the side.\n\nStep 4: Throw in a pompous stockbroker, the entire Mystery Play committee, including a nervous but well-meaning vicar and a furtive-looking youngish man in a strangely mossy green suit, who has forgotten to remove his Rambo-style headband.\n\nStep 5: Sprinkle with borrowed WI tea and coffee urns and crockery, garnish with triangular sandwiches, halved scones spread with clotted cream and home-made blackcurrant jam, trays of cubed chocolate Spudge on cocktail sticks, wedges of bread-and-butter pudding and bowls of Cornish Mist.\n\nStep 6: Stir well before it coagulates into clumps or \u2013 even worse \u2013 curdles.\n\nStep 7: Garnish with assorted hens, a three-legged whippet, and one yapping haggis-sized hairball that someone, probably Jasper, has let out of the cottage.\n\nStep 8: Stand back, since once warm the mixture could go up like a rocket \u2013 and down like the stick...\n\nBut now the Pharamond family (and one or two of the bolder hens) began slowly to converge on me from all directions. Nigel reached me first, but once he'd kissed my cheek and said he was very sorry he wandered off again, probably in search of non-existent sherry.\n\nGreat-aunt Mimi was pushing Juno in a wheelchair, though she'd told me last night she was hoping Dr Patel (not KP, but his daughter) was going to let her start walking again by the end of the week. The expression on her face was grimly stoical, at odds with the gaily frivolous arch of tissue-paper flowers Mimi'd attached to the back of the chair.\n\n'Isn't that the bower from the Adam and Eve temptation scene that the Infants' School made last year \u2013 the one that kept falling on the snake?' Annie asked.\n\n'Yes. Thought it would brighten things up,' Mimi agreed. 'Damned gloomy things, funerals,' she added. 'Better if he'd gone ages ago, because he was nowhere near as much fun, the last few years.'\n\n'Mimi!' protested Juno. 'What a thing to say!'\n\nI focused slightly (for the shot of brandy had been generous, on an empty stomach; I hadn't been able to eat anything before the funeral). 'Oh, did you notice that too, Mimi?'\n\nShe leaned over Juno's chair towards me. 'Possessed!' she hissed. 'The devil was looking out of his eyes!'\n\nJuno twisted her head upwards. 'Mimi, you haven't taken your pill today, have you? Where have you hidden it?'\n\nMimi backed off, looking guilty and slightly agitated. 'I didn't need it \u2013 why do I need it? Lizzy doesn't need one, and she's seen the devil looking out, too!'\n\n'She's quite right, I did,' I assured Juno. 'And Mimi seems fine to me. I'll get her a glass of elderberry wine, that'll do her good. I've hidden it under the end table, so the surfers and Mummers don't get drunk and silly. Sillier,' I added, for they seemed to have joined forces behind the tomato plants, together with some flagons and bottles I certainly hadn't supplied.\n\nAs well as the wine I collected a plate of food, since I was suddenly ravenous. One or two people spoke to me (including one of the WI ladies, who complimented me on the garnish of golden quail eggs, an effect I achieved by boiling them wrapped in onion skins), but I was glad I'd decided to dispense with the moving-about-graciously-thanking-everyone-for-coming thing, because I didn't feel at all thankful to most of them.\n\nThe Mummers of Invention \u2013 or what was left of them \u2013 now struck up a lively tune at one end of the greenhouse. The drippy girl (and I can only assume Tom was desperate when he slept with her) stuck one finger in her ear and started to drone a song. She'd have to do her own harmonising without Tom, but since she talks to herself worriedly all the time just like Alice's white rabbit, that shouldn't cause her any great problem.\n\nI gave Mimi her glass of wine and then sat next to Unks, eating steadily, as is my usual wont when stressed, unhappy, worried... or happy, cheerful and optimistic. Let's face it, I eat. That morning's fast was simply an aberration.\n\nLeila seemed to have the same thought. 'If you didn't do all that digging and outdoor work, you would be as fat as a pig,' she commented, coming to a halt in her stilettos in front of us, Nick right behind her like an attendant thundercloud. Then she kissed the air about an inch from Unks' cheek and said, 'Well Roland, how are you?'\n\n'Fine, Leila,' he said cautiously. 'Nick, you haven't heard how the two thirty at Haydock went, have you? Only I had a sure thing running. Happy Wave out of Surfer's Paradise \u2013 couldn't go wrong with that pedigree, could it?'\n\n'On a day like today, you can have no interest in horse racing!' Leila exclaimed, scandalised, and Roly looked slightly abashed.\n\n'Joe's probably listening to it on the car radio, Unks,' I said. 'I'll go and ask him in a minute. In fact,' I added slightly worriedly, 'I ought to go and see where Jasper is.'\n\n'It's all right, I saw him sitting on the Daimler running board talking to Joe when I came in,' Nick said. 'That's a weird-looking dog he's got! I thought you'd go for another lurcher, like Harriet.'\n\nI pulled a face. 'It's one of Annie's strays. We're only supposed to be fostering it, so it wasn't put down, but she and Jasper have taken a shine to each other. I'm going to end up looking after it when he goes to university, I can see.'\n\nOne more responsibility, when it came to leaving Perseverance Cottage and forging a new life on my own, too...\n\nLeila seemed to be thinking along the same lines, for after looking me over in her usual critical way, as though she found me wanting in all aspects (which she probably did), she said in her strongly Parisian-accented English, 'So, Lizzy, I expect you will be moving on, now? Those books are your only real source of income, are they not? And those you can write anywhere.'\n\n'But they are the Perseverance Chronicles,' Juno pointed out, 'about her life here in Perseverance Cottage \u2013 that's the whole point! All Lizzy's daily struggles \u2013 that's what her readers like.'\n\n'They'll certainly love the next volume, then, even though I'm ending on a happier note, before the tragedy. After that,' I shrugged, 'who knows?'\n\n'But you must keep writing them and including the wonderful recipes,' Juno enthused, her square, rather weather-beaten face animated. I hadn't realised she was a fan.\n\n'Oh God,' groaned Nick, temporarily distracted from silent thundercloud mode, 'don't remind me! Why anyone should find Lizzy's sugary, fatty recipes for nursery puddings and inedible bakes of any interest at all entirely beats me. And what's more, wasn't that bowl of yellow, lemon-flavoured stuff someone just offered me candyfloss?'\n\n'Cornish Mist,' I said automatically, and Leila snorted.\n\nRoly, who'd been sitting looking quietly baffled, suddenly put in, 'What did you mean, Leila, that Lizzy would be moving on? Moving on where?'\n\n'I suppose Leila meant that I would be looking for somewhere else to live, Unks.' I smiled at him affectionately. 'It's been wonderfully kind of you to let us have Perseverance Cottage all these years, when you might have rented it out for a decent amount, but I realise you'll have other plans for it now.'\n\n'That's right,' agreed Nigel belligerently. He'd suddenly reappeared behind his father and was swaying slightly, like a pot-bellied palm tree in a breeze. I couldn't imagine him hobnobbing with the Mummers and surfers, so he must have found the elderberry wine the way pigs find truffles. 'Told you it was stupid, not charging them rent all this time, Father \u2013 and Tom not even a real Pharamond! Besides, the place is too big for Lizzy on her own. I'd do it up and sell it \u2013 get a fortune for it, the way house prices are rising round here.'\n\n'Would you?' Unks said coldly. 'Perhaps you'd sell off the rest of the estate for housing too, while you were at it?'\n\n'Probably wouldn't get planning permission. Sodding red squirrels and trees are more important than people,' he slurred. 'But you could divide the Hall up into apartments \u2013 executive ones \u2013 and keep the best for yourself. Make a huge profit!' His eyes lit up with the fire of greed.\n\n'So, that's what you would do to the Pharamond estate, Nigel, is it?' Unks said. 'If you inherited it, of course.'\n\n'But he is the heir, is he not?' Leila said. 'And his ideas, they are eminently practical!'\n\n'Nigel shouldn't count his chickens before they are hatched. The entail on the estate was broken long ago, don't forget, so it is mine to leave to whoever I wish.'\n\nRoly turned his head and smiled at me: 'I told Tom that I was making Perseverance Cottage over to him and Lizzy, but now it will be Lizzy's home for as long as she needs it \u2013 and Jasper's, of course.'\n\n'Oh, thank you, Unks! That is so sweet of you!' I said, tears coming to my eyes, and kissed the top of his head.\n\n'It's been a pleasure having you living so close, my dear. And the boy's a Pharamond \u2013 he should grow up here and regard it as his home.'\n\n'But he isn't really a Pharamond, Father,' Nigel pointed out, swiftly sobered by the possibility of disinheritance.\n\n'Course he is,' Mimi said, leaning on the wheelchair and waving her empty glass emphatically, and Juno twisted her head round and gave her another anxious look. She would be getting a crick in it at this rate. 'Only have to look at him to know that. The boy's the spit of Nick.'\n\nThere was a small silence, which Unks broke by saying drily, 'Before anyone lets their imagination run riot, when Tom's mother asked me to take Tom on, she confessed to me that she was having an affair with Leo before her first husband died, and she was pretty sure Tom was his. I promised not to say anything, but I feel current circumstances absolve me from that now.'\n\n'His mother said that? Oh, I wish she'd told him!' I said indignantly. 'He always had a chip on his shoulder about not really being a Pharamond, but he wouldn't hear a word against her. Instead, he pretty well accused me of\u2014'\n\nI broke off hastily, meeting Nick's eyes, and went pink. 'Well, I just wish he'd known, that's all.'\n\n'I think you, Roland, are just saying this to cover up the truth! Jasper is Nick's child \u2013 Tom told me this himself,' Leila exclaimed unguardedly. 'They had an affair when they were very young, and then they picked it up again after she married Tom.'\n\n'Rubbish!' Unks said firmly. 'Boy-and-girl affair one summer and fought like cat and dog \u2013 and then Nick went off on his travels and that was that. Dare say they hardly even saw each other for years, and Lizzy was mad about Tom when they married, anyone could see that!'\n\n'Of course it isn't true,' I said hotly. 'Jasper is Tom's child. But I'm glad to have the truth I suspected come out at last, even if it is a bit late in the day.'\n\nNick was glaring at his wife. 'You mean, Tom told you that and you believed it? When did he tell you?'\n\n'Oh, years ago.'\n\n'It can't have been that many years ago. He only got that stupid idea after Jasper had meningitis,' I said.\n\n'I think we all know you've had a difficult time with him lately,' Roly remarked, to my surprise. 'I hear things \u2013 know more than you think.'\n\n'Yes, but I'm afraid it had got to the stage where I was going to leave him, Unks,' I confessed. 'I couldn't take any more. I was just waiting for Jasper to be settled at university.'\n\n'Well, now you don't have to,' Mimi said brightly. 'You can stay here and dear Jasper can come home in the holidays. Such a clever boy!' She took a bowl of Cornish Mist from a passing tray and dug in her pastry fork. 'Mmm!'\n\nThen, yellow-moustached, she looked up again at our rather silently thoughtful tableau. 'Where's Tom? He's missing all this lovely food and he is so fond of his stodgy puds.'\n\n'He's gone, Mimi,' Juno said shortly. 'I'll explain later.'\n\nMimi's brain had clearly done one of its little loops and shunted Tom's death into a siding. I wished mine would.\n\n'Oh, will you? Good,' she said, her eyes wandering around the very motley crowd. 'Look, that woman's wearing a black baby-doll nightdress! Should I have come in fancy dress too, Lizzy?'\n\n'Good grief, it's that Polly Darke woman! As if it wasn't bad enough her coming to the funeral, without turning up in your home, Lizzy!' Annie exclaimed, scandalised, appearing beside me suddenly with her bob of coppery hair attractively ruffled and her cheeks flushed \u2013 but then, it was getting slightly steamy in the greenhouse, in more ways than one.\n\nPolly had staggered backwards through the concealing tomato plants, where she'd presumably been drowning her sorrows with the surfers and Mummers. She regained her balance with an effort, turned, and then fixed her dark-ringed, sunken eyes on me. 'Lizzy! Not so much the grieving widow, I see! Maybe you're even celebrating, because it wouldn't surprise me if you loosened that wheel on purpose, and then told Tom to borrow your car!'\n\n'Polly, that's entirely stupid. And you watched me walk off into the woods that morning, don't forget! When would I have done it?'\n\n'You could have come back. You were probably hiding, waiting for me to leave.'\n\n'Do you all know Polly Darke?' I said resignedly. 'You can probably spot without my telling you that she's a novelist of the raunchier kind, but she was also Tom's mistress.'\n\n'Good grief!' Juno exclaimed incredulously. 'Was she really?'\n\n'More, much more, than that, Tom loved me! He had to get Lizzy to move out, so the split looked like her fault, and then Roly would leave him the cottage; but after that we were going to get married,' Polly cried, looking about her wildly, as though expecting a sympathy vote and not getting it.\n\n'You are quite mad \u2013 a fantasist!' Leila said, looking her up and down disgustedly. 'He would not have even a tiny affair with a woman such as you.'\n\n'I'm afraid he did have a mistress, but I've only just found out that it was Polly,' I told her. 'It was a shock to me, too!'\n\n'And me,' Unks said drily. 'Revelations, indeed!'\n\nLeila narrowed her eyes at Polly, like a snake. 'What could you possibly have to offer him?'\n\n'You'd be surprised,' I said darkly. 'I certainly was.'\n\n'This is fun, isn't it?' Mimi said brightly, but no one took any notice.\n\nLeila drew herself up. 'She is lying. It is true that Tom had a lover, yes \u2013 but it was me!'\n\nNick's hand closed in a vice-like grip on her arm. 'But you swore you'd never been more than friends with him! And you wouldn't agree to a divorce.'\n\nShe shrugged. 'He planned to move in with me eventually, but it is as this woman says: he wanted to keep in with his great-uncle, who is fond of Lizzy, and inherit the cottage, so he was forcing her out. She deserved it; she was still having an affair with you! Tom said even in hospital when Jasper was so ill, you couldn't keep your hands off each other.'\n\n'That is so not true,' I said hotly.\n\n'No, Lizzy isn't like that in the least,' Annie agreed loyally, 'and neither is Nick.'\n\nPolly, who'd been quiet, rallied again: 'Oh, I knew he had an old mistress who wouldn't accept that their affair was over. But it was me he loved \u2013 and me he spent his last night with!'\n\n'I don't believe you. He was with me only a few days before he died, and he swore he loved me and soon we would be together always,' Leila declared.\n\n'Presumably after I had obligingly died, and he had inherited the cottage?' queried Roly mildly.\n\n'You are ninety-two, Roland,' Leila said defensively. 'In France we are more practical about these things.'\n\n'Not practical enough, my dear. I was leaving the cottage to him for life only, then to Jasper after him, not outright.'\n\n'And you were going to tell me all this when, exactly?' Nick demanded of his wife in a voice that reverberated through the greenhouse like thunder. 'And why refuse me a divorce?'\n\n'I knew that you would try and take half my restaurant that I've worked for so hard \u2013 half my money. Why should I give you what is mine? I thought once Lizzy was free, you might be more reasonable about it, and I would wait. And I will fight you through the courts for every penny!'\n\n'I don't want your money \u2013 I don't want anything of yours, just my freedom!' Nick said furiously. 'I'll sign a statement to that effect any time you choose \u2013 would have done before, if I'd known what was worrying your mercenary little heart. You and Tom seem to have been made for each other!'\n\n'No they weren't. She's a lying cow and it's me he loved and wanted to spend his life with!' Polly said, quivering with rage. Then she entirely lost it and lobbed her plate of food at her rival. Half a scone daubed with cream and red jam clung to the side of Leila's face like some exotic wart, before sliding slowly down her cheek and dropping off, smearing her expensive suit on the way.\n\nMimi giggled, but Leila gave a scream of rage and lunged at Polly with her long, sharp red nails. I was just thinking that I'd put my money on Leila, when Nick seized her arms from behind and two of the surfers, who must have followed Polly through the tomato plants, leaped forward and grabbed her too.\n\n'Put them out!' Unks snapped. 'You!' he said to Polly. 'Whatever the truth of the matter, you should have had the decency to stay away. If you don't go now, I'll have you removed.'\n\nPolly went limp and started sobbing, and the two surfers let go of her arms. She stumbled towards the door on her extremely high heels and, as the crowd parted to let her through, I realised we had unwittingly been providing entertainment for everyone within ear-and eyeshot \u2013 which, in a greenhouse, is pretty well everyone who could cram in.\n\nThe only good thing was that the police and the vicar seemed to have left before the floorshow.\n\nNick, still grasping Leila's arms, snapped in my direction, 'Excuse us! Back later, Roly \u2013 hope your horse won.' And he marched her off. I would have liked to have been a fly in the car on the way back to London: Nick's invective can be quite inventive when he's in a rage.\n\nUnks looked pleased. 'Well, she was always a bitch,' he remarked happily. 'Don't know why he took up with her, except she was beautiful, I suppose, and they had the cooking stuff in common. Once he's over her, he can start again \u2013 with a sensible Lancashire lass this time, perhaps?'\n\n'They're both lying old bags!' quavered the small, pathetic voice of the drippy girl from the Mummers right next to me, and it was quite lucky from her viewpoint that neither of them was there to hear that description, because it would have been tantamount to staking a kid to attract tigresses. 'He loved me \u2013 and what's more, I'm having his baby!'\n\nWhite as a sheet and naturally rather pop-eyed, Ophelia Locke seemed an unlikely candidate for Tom's attentions, so I was probably the only person who believed her. Still, it was another reason to be glad the other two had gone. There was clearly a general feeling among the onlookers that this was a scene too far and we were into the farce; but then, before anyone could rally enough to say so, Ophelia fainted backwards.\n\nCaz Naylor stepped forward and caught her neatly, then slung her over his shoulder, where she dangled limp as a shot rabbit. 'Not right in't head, Mr Pharamond,' he said tersely.\n\n'Evidently, poor girl,' agreed Unks, looking rather taken aback.\n\nHanging upside down must have sent the blood rushing to Ophelia's head, for she revived enough to beat weakly on Caz's back and whimper, 'Put me down, you big bully!'\n\nCaz ignored her and carried her off, the crowd parting to let them through. There was a spontaneous spatter of applause.\n\nUnks unscrewed the top of his cane. 'Anybody want a shot of brandy? I certainly do! And wasn't that the girl I let an estate cottage to \u2013 makes handicrafts, or some such stuff?'\n\n'Barbola work?' suggested Mimi. 'And is Caz walking out with her?'\n\n'Smocks,' I said. 'Yes, Unks, that's the girl.'\n\n'Nobody does barbola work these days,' Juno said. 'And I shouldn't think she's Caz's type.'\n\n'Oh? Well, that was all just like a play!' Mimi said, still clapping her hands. 'Was it a play, Juno? Is that the end?'\n\n'Yes, time for us to go home,' she agreed. 'Come on, that gardening programme you like will be on by the time we get there.'\n\nMimi whirled the chair about with no more ado. 'Lovely party \u2013 thank you for having me!' she called politely over her shoulder. The remains of Polly's jam scone squidged under the wheels.\n\nThe excitement clearly over, some of the remaining guests followed them out, though the last two Mummers were still obliviously droning on in the background and I could see one of Tom's surfing chums stretched out under the trestle tables, snoring.\n\n'Jasper,' I said, as he finally came in, closely followed, nose to heel, by the little dog, 'Unks says we can live at Perseverance Cottage for as long as we want to. Isn't that kind?'\n\n'Really? Thanks, Unks. I was worrying what would happen to Mum when I went off to university, but now I won't need to any more.'\n\n'You don't have to worry about me at all!' I said indignantly. 'I'm perfectly capable of taking care of myself.'\n\nJasper eyed me uncertainly. 'Are you all right, Mum? Only you look a bit strange.'\n\n'Strange? Why on earth should I look strange?' I demanded, though I could feel hysteria trying to tweak my mouth into an idiot's grin.\n\n'I think we're all tired and a bit overwrought,' Annie said quickly. 'What an exhausting day! All these scenes and revelations.'\n\n'What scenes and revelations?' asked Jasper, looking from one to the other of us.\n\n'I'll tell you later,' I promised, which I would have to, even if only the edited lowlights. 'I wonder how we can get rid of the last of the guests.'\n\n'I'll send Joe back to turf out the stragglers,' promised Unks, hoisting himself to his feet with Jasper's help. 'But most of them will go when they see me leaving.'\n\nHe advised me, before Joe drove him away, not to dwell on recent events, but instead remember Tom as he once was. He looked frail and tired, so I hoped it all hadn't taken too much out of him.\n\nI was fine \u2013 too numb and full of elderberry wine and brandy to feel anything except a desire for oblivion.\n\nIt was early evening before Joe Gumball loaded the last drunken surfer into the back of the taxi called to take them to their B&B in Mossedge, and finally persuaded the two Mummers to go away. They had been too drunk to notice the scene with Ophelia, or even question where she had vanished to. (And what had Caz done with her?)\n\nThe urns and crockery had long been efficiently removed by the WI ladies, with my grateful thanks, and Marian and Clive Potter had supervised the local Cubs and Brownies in carrying the trestles and chairs back to the village hall, so that was that.\n\nAnnie saw to the poultry, and then cooked us a meal we none of us really felt like, except the dogs, before going home.\n\nFinally alone with Jasper I felt almost too exhausted for the effort of explanation, but when I told him that he really was Unks' great-great-nephew he just said, 'Yes, I know. Unks told me ages ago, when I asked him why I looked more like Uncle Nick than Dad.'\n\n'You did?' I stared at him. Then I sighed tiredly, and decided to tell him about the other pretenders to the throne of love and get it all over with at once. 'Did you also know that your father was having an affair with Leila as well as Polly? They nearly came to blows this afternoon.'\n\n'Oh, so that's what she and Nick were arguing about when they left! He pushed her into his car and roared off, and then Polly came out and drove away too, though she didn't look fit to be behind the wheel. And another thing,' he added thoughtfully, 'why was Caz giving that Mummer girl a fireman's lift? She didn't seem to appreciate it.'\n\n'She fainted \u2013 right after declaring that it was really her your father loved, not the other two. And, Jasper, she says she's pregnant!'\n\nHis eyebrows rose. 'She does?'\n\n'I hope it isn't true, because goodness knows, things are complicated enough without that.'\n\n'Well, look on the bright side,' Jasper said, with the breezy insouciance of youth. 'At least there's no chance the baby's mine!'\n\n## Chapter 11: Popped Corks\n\nGinger Beer\n\nLet me give you a few words of wisdom culled from many years of making the stuff. (And you will find the recipe for making a ginger beer culture in Book 1 of The Perseverance Chronicles.)\n\n1. Ignore the Quatermass-experiment effect once the yeast starts working \u2013 whatever it may look like, the ginger culture will not take over the world... yet. And isn't it amazing that water and a bit of yeast and ginger scum can turn into something so delicious?\n\n2. Ask everyone to save their screw-top plastic pop and water bottles for you, because if you go the traditional route of glass bottles and corks, expect your house to explode at frequent intervals.\n\n3. Even with screw-top bottles, it is not a good idea to transport large quantities of ginger beer about, especially in a car. Nor should you be tempted to celebrate any special event by shaking the bottle before opening.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI was woken early next morning by a thirteen-gun salute, which proved to be half my remaining stock of ginger beer exploding.\n\nAfter cleaning the sticky mess up and loosening the rest of the caps so I wouldn't lose all of it (Jasper and I are very partial to ginger beer), I went out to see to the poultry.\n\nThere appeared to be fewer quail. Presumably Caz had taken the opportunity the previous day to cull the male ones again, though I wasn't sure how he found the time, unless he'd popped back after taking Ophelia to... well, where had he been heading? So long as it wasn't to his giant freezer, I didn't suppose it really mattered. Perhaps he had just put her in a Mummer's car and left her to it. Or carried her home to her estate cottage, which is not far from his, protesting all the way? (Though I didn't think most local girls would protest if he wanted to carry them off, the foxy sheikh of the western Lancashire world.)\n\nI hadn't really thought of him in the knight-errant role before, even if he appeared to have been protecting me against the ARG activists. Perhaps he'd got something going with Ophelia. But then, that did seem a bit unlikely too, since he was definitely carnivore and I remembered Tom telling me once that all the other Mummers were vegan and wouldn't even wear leather shoes.\n\nThe quail, in their little pens, all made identical cheeping sounds. I'd never managed to tell one from the other, which was lucky since I didn't actually get attached to them, like I did to all the Honeys and Myrtles, and even the ducks. But sometimes even they were so nasty and vicious to each other that I stared to feel maybe their real destiny was to be on a plate with stuffing and gravy, or orange sauce. Nature is red in beak and feather, as well as tooth and claw \u2013 someone should tell ARG that.\n\nBut after wrapping a couple of dozen quail eggs in onion skins during the sleepless night watches before the funeral, I really didn't care if I never ate another one, let alone a quail itself... so I thought I'd get rid of them; find them another home.\n\nThe duck population was here before me and was pretty self-sufficient, but the hens could just naturally reduce as they died of old age \u2013 so long as I found the eggs before they hatched. After all, it was just going to be me here at Perseverance Cottage most of the time, and if I didn't produce much more food than I could eat, with the bit extra for barter, then I'd have time to throw myself into helping Annie expand Posh Pet-sitters, and earn some money. Barter worked well up to a point, but not with electricity bills and the Inland Revenue.\n\nYes, even the enormous, aluminium-framed glasshouse \u2013 relic of a doomed attempt at market gardening by a previous tenant \u2013 could go. There was a small one behind the cottage that did well enough for my needs. I decided to get Jasper to put it on Freecycle, where you can advertise anything you want to get rid of \u2013 but also get things you need, for free. I expect someone will want it. But what I was to do with Tom's workshop contents I couldn't imagine...\n\nBack at the cottage I was surprised to find Jasper up and getting ready to go to the dig, which on the whole I thought a good idea (though possibly not in the Diesel jeans that had cost me a fortune). I think we both had that spaced-out, anti climactic feeling, and the alternative was sitting about thinking unproductive thoughts.\n\n'Do you know how many eggs I had to sell to buy those jeans?' I demanded, but he took the question as rhetorical and carried on cutting multilayered doorstep sandwiches.\n\nI couldn't drive him there, since, even if the van hadn't been down at the garage, now it had done duty as a hearse neither of us fancied ever getting in it again. But Dave Naylor was coming that morning with the ancient but, he assured me, very practical Land Rover I had in a rash moment agreed to swap it for.\n\nSo Jasper cycled off to the dig with the furball (now called Ginny, short for Guinevere, though anything less like a Guinevere I never saw) in the carrier, together with a bowl and a big bottle of water. He said there was plenty of shade to tie her up in and he could walk her at lunchtime, though from what I could make out of her shape beneath the fuzz, rolling her about a bit would probably do just as well.\n\nDave brought my new old car about an hour later, after I'd drafted the advert for the quail. His son Gary, who played Jesus in the Mystery Play for the first time last year, followed him down the track in Tom's van, to save him the quarter-mile trek back. That damned van with the horribly apposite 'Board Rigid' logo seemed to be a recurring motif, and I'd be glad when it was resprayed and sold on, preferably away from the area. I'd managed to find the paperwork for the van, which I expected would soon reappear for sale on the garage forecourt after a quick makeover.\n\nThe Land Rover had obviously had a hard life, being battered, dented and with high mileage, but Dave assured me they lasted for ever and, as soon as I had mastered the rather unyielding clutch pedal, I would be fine.\n\nJasper thought the vehicle swap was a good idea and was looking forward to driving the Land Rover. It was fortunate I have a son who prefers antiquities. I tootled it around the yard and up the track, and it certainly felt more solid than my 2CV ever did. Later I steered it cautiously down to the post office and handed Marian the quail advert for the parish magazine. She said I was just in time for the next issue, which would be out at the end of the week, but that was sheer luck, since their publication dates are erratic and entirely depend on how Clive feels about it.\n\nIn the late morning one of Tom's surfer friends, who'd been a bearer at the funeral, phoned up and asked if he and his mate Jimbo could drop in.\n\n'That's nice of you, Freddie,' I said, assuming they wanted to say goodbye before setting off back down to Cornwall. It was amazingly thoughtful of them and, I would have previously said, totally out of character.\n\n'Well, we wanted to say goodbye, Lizzy, but we've also got a proposition.'\n\n'A proposition?'\n\n'Yes, we thought we might take on Tom's business and wanted to discuss it with you. If you've no other plans for it, of course.'\n\n'No, I haven't really had time to think of it. But of course you can come down and we can talk about it,' I said, for it would be a relief. I had no idea what to do with all his equipment and half-finished boards and stuff, and it would be rather nice if his friends carried on with Board Rigid.\n\nSo far as I could gather from male-bonding rituals, Freddie and Jimbo had been two of Tom's best friends. They'd all been at Rugby School together, dropped out of university and then washed up in Cornwall, bumming around on family money. But I suppose even that and the patience of your family is finite, and when a man is heading rapidly for his forties it's time to stop being the playboy of the western UK, and earn your own bread and wetsuits.\n\nI went into the workshop and opened the big front door to the sunshine for the first time since the accident, instead of the little Judas door.\n\nThings looked dusty already and there was that familiar smell of paint, varnish and dope (both kinds). Tom's wetsuit swung from a hook like the spent chrysalis of some strange creature: which I suppose it was, when I came to think about it.\n\nOne or two boards that were obviously commissions were almost finished, and there were several of the ones he bought in and painted to sell on through shops. I had a rough idea what they were worth and the people who had ordered boards had their names and phone numbers taped to the back: Tom's idea of paperwork. Mind you, the system seemed to work, which is more than could be said for his tax returns: I only hoped the Inland Revenue was not now going to fall on me like a ton of bricks.\n\nTom's friends turned up so quickly they must have phoned from a mobile up the lane and were very kind, kissing my cheek and hugging me as if they had always liked me, which they didn't. I was the wrong sort of girl: Tom should have married someone sea-sporty, acquiescent, well connected and, above all, rich.\n\nJimbo has a long body, stumpy legs and a big nose, but he didn't get his nickname from his appearance: his name is James Bow. Freddie is tall and skinny, with grey-blond hair fuzzing over a head like a bleached coconut. They must have to get their wetsuits made specially.\n\nAfter looking over the workshop as though it was a dubious garage sale, they um-ed and ah-ed a bit, then said they would take the worry of it off my hands for three hundred pounds, what did I say?\n\n'What, for the whole lot?' I gasped, thinking I hadn't heard right.\n\n'Well, let's say four hundred, to be fair. There's not much here, and we'd have to finish off the old orders and deliver them, of course,' Freddie pointed out, as if this would be a big favour.\n\n'But Tom's almost finished them, and he hasn't been paid yet. And the other boards that belong to him are worth more than that alone!' I protested, stunned.\n\nThey exchanged quick looks and I realised they hadn't expected me to know anything about the business or the value of it, and be too upset to think straight. I'd fallen among thieves.\n\n'Oh, no, Lizzy, they're not top-quality boards, you see,' Jimbo said quickly, 'and several are half-finished \u2013 you'd never sell them like that. Besides, you don't have the contacts, do you? But we do.'\n\nI looked at them: these were supposedly Tom's closest friends. He'd been to school with them, hung out with them over the years, and they were well off by my standards, even if they did look like bums most of the time. And now they wanted to make a quick profit by cheating his widow!\n\nThey were also cunning beach bums, for Freddie now produced a neatly printed agreement. I read it through a couple of times, noting the words 'sale to include everything pertaining to the business of Board Rigid', but my brain wasn't really up to coping with possible pitfalls.\n\n'Trust us \u2013 we're doing you a favour, Lizzy,' Jimbo said persuasively. 'I mean, who else would be interested?'\n\n'Take it all off your hands \u2013 one less thing to worry about,' agreed Freddie, shiftily avoiding my eyes.\n\nBut when I looked around the workshop again I discovered I didn't have any fight left in me. I really didn't care that much \u2013 and Tom wouldn't have seen it as anything other than a smart move by his friends.\n\nSo, reluctantly, I signed. They paid me cash from a wad of notes: I said they were rich. 'Do you want a pint or two of my blood as well?' I asked bitterly. 'Or a kidney, perhaps?'\n\n'Poor Lizzy,' Jimbo said sadly, 'I can see Tom's death was a huge shock to you. You're doing the best thing, putting all this behind you.'\n\n'Yes...' Freddie agreed, looking around again with more of a proprietorial air, 'once it's gone, you'll feel much happier, you'll see.'\n\n'Might as well load it up and take it all now,' Jimbo suggested. 'Where's the van parked, Lizzy?'\n\n'The van?' I repeated blankly.\n\n'Tom's van \u2013 the one we've just bought,' Jimbo said patiently.\n\n'But you didn't mention any van!'\n\n'Well, not as such, perhaps, but you did agree to sell us everything pertaining to the business, lock, stock and barrel.'\n\n'Two smoking barrels!' I said, woken out of apathy into indignation. 'Look, you've already screwed me over this deal, you can't possibly have expected to get Tom's van for four times that much!'\n\n'Lizzy, you don't know how big a favour we're doing you. Of course we only offered you that much money because of the van. It's worth more than the stuff in the shed,' Freddie began.\n\n'I know, because I bought it myself, out of the advance for the third of The Perseverance Chronicles! And anyway, you're too late: I've already disposed of it.'\n\nThey stared at me, aghast. 'Disposed!' they exclaimed as one.\n\n'If you're interested, it's down at Deals on Wheels.'\n\n'Deals on Wheels?' echoed Freddie.\n\n'The garage in the village. I swapped it for a Land Rover this morning, so the van is now the property of Dave Naylor. I suggest you go and talk to him, if you're still interested in it.'\n\nThey were not happy, but the van was legally my property and they couldn't do anything about my having already disposed of it, which certainly made me feel better. Eventually they stopped trying to browbeat me into getting it back and giving it to them, and went off to talk to Dave at the garage, saying they would be back later to empty the workshop.\n\n'Fine.'\n\n'Perhaps you could give us the key, so we don't have to disturb you again?'\n\n'Not until I've removed his personal things. The CD player and stuff like that.'\n\nBut when it came to it, there wasn't actually much in there that I cared to take, so when they came back again in Tom's van, which I'd hoped never to see again, I left them to it. I bet Dave struck a hard bargain, and serve them right, too.\n\nLater, I went to collect Jasper and Ginny in the Land Rover and when we got back the workshop doors swung open, and it was empty of life except for some curious hens, one of whom had laid an egg on the old easy chair among the burst stuffing.\n\nJasper and I have decided to sell Tom's insanely huge TV and buy a small one that he can take to university with him, along with his laptop \u2013 and I am going to put the money I got for Tom's workshop contents towards a computer and printer of my own.\n\n'Then you can bring me up to speed on using the internet and emailing before you go,' I suggested.\n\n'Good idea,' he agreed, 'your skills do need honing a bit. It's a pity we still only have dial-up connection, though, because it's really slow. Middlemoss must be the last area in Lancashire that hasn't got broadband yet.'\n\n'Miss Pym says we will get it by the New Year and anyway, slow suits me fine to start with,' I said, reflecting how much I had changed from the first days of my marriage when I hadn't even wanted a TV, to now carrying a mobile phone everywhere (even if I forgot to switch it on half the time) and accepting that a computer was going to have to be part of my everyday life.\n\nStill, adapt and survive...\n\nWhen it began to get dark I went out to shut everything up for the night and while coming back noticed that the light was on in the empty workshop.\n\nMy heart stopped dead for a moment, until reality set in and I thought it might be Freddie and Jimbo, returned in a fit of pique to make a thorough job of it. There were still the tattered old sofa and chairs, the ancient upright piano and the kettle, for instance. No resurrected Board Rigid van stood outside to load anything into, it was true, but I wouldn't have put it past them to have parked it on the road and sneaked in through the side gate.\n\nThen again, it could be Caz, curiosity stirred by the unlocked door; or ARG, setting dynamite charges. Even Mimi on the loose...\n\nI crept up to the open door and peeped in to find, to my astonishment, that the workshop was infested with Mummers. They weren't doing anything, just hanging about with an air of aimless expectancy, but when they saw me they drew together into a defensive, sheep-like huddle.\n\nOphelia was mumbling to herself as usual and I thought her pale froggy eyes were going to pop out altogether. 'Oh God! Oh God! It's her \u2013 she'll kill me \u2013 she'll kill me! Oh God!'\n\nI looked dispassionately at her. She was as limp and wet as seaweed and I was still finding it hard to square what Jasper had said about finding her and Tom in flagrante, with the reality of what she actually looked like. In fact, she has all the allure of a white blancmange, so it must have been simply availability allied with drink, or the demon weed. After the Leila\/Polly revelations it didn't seem of much moment anyway, unless she really was pregnant by him? Hard to tell, when she dresses in her own smocked garments most of the time.\n\n'Ophelia didn't mean what she said, yesterday,' Mick said hastily, his fingers fiddling with the blue feather in his hat. The three of them edged even closer together.\n\n'No, no-no-no!' whimpered Ophelia, while nodding rapidly.\n\n'She's not pregnant?'\n\n'No, she meant that bit.'\n\n'Tom loved me!' Ophelia said, but rather uncertainly.\n\n'Nah, we keep telling you \u2013 he just wanted a quick shag,' Jojo said brutally.\n\n'So the baby could be Tom's?' I asked him.\n\n'It could be anyone's \u2013 Caz Naylor's even.' They stared accusingly at her. 'Sleeping with the enemy!'\n\n'The enemy? Caz's your enemy?' I frowned over that one, but I suppose gamekeepers and rabid vegans are oil and water and shouldn't mix, though clearly at some point two of them had.\n\n'Well, come to that, it could be mine \u2013 or even yours,' Mick pointed out fair-mindedly to his friend.\n\n'Ooh!' moaned Ophelia, chewing her lip frantically like a mad albino rabbit, huge pink eyelids fluttering.\n\n'You've got around a bit,' I said to her, though feeling a bit sorry for her now Mick and Jojo were being so horrible. But Ophelia's having slept around was quite a good thing in a way from my point of view, because it seriously lessened the chances of the baby being Tom's. I supposed I would just have to await the outcome, and so would she.\n\n'Mr Pharamond's bound to put her out of her cottage, after what she said at the funeral,' Jojo suggested helpfully.\n\nOphelia wrung her hands and stared at me with the eyes of a mad martyr embracing her doom. 'Yes, yes, it's all my own fault and I deserve to be punished!'\n\nOh, good heavens, she wasn't another kinky one, was she? But then I sighed resignedly, for if there was any chance she was pregnant with Tom's child, I couldn't let her be turned out.\n\n'You two stop trying to stir things up,' I said severely to the men. 'Of course Roly won't give you notice, Ophelia \u2013 or at least not until after the baby's born. I'll speak to him about it, but I'm sure Mick and Jojo are quite wrong.'\n\n'Don't think he could do it, anyway,' Mick said belligerently. 'She's got her rights!'\n\n'Possibly not, but I don't think the question will arise. Anyway, what are you all doing here?'\n\nThey shifted uneasily again, looking around the near-empty workshop as though expecting something \u2013 or someone \u2013 to materialise out of the dark shadows.\n\n'He's not coming back, if that's what you're all waiting for,' I said evenly. 'Tom's played his last gig and you'll have to get a new singer: not that I thought he was much good, anyway.'\n\n'His voice harmonised with mine very well,' Ophelia blurted, then blushed as she caught my eye, like she usually did \u2013 as well she might. 'Oh God!'\n\n'And he wrote most of the lyrics to my tunes,' said Jojo, slowly turning the gold hoop in his ear as though tuning what remained of his brain.\n\n'No, actually, that was me,' I said incautiously and they stared. 'He used to hammer them out on the old piano and I'd try to fit words to them \u2013 just give you a base to work it up from, you know? I mean, they weren't really mine when you'd finished with them, because they evolved into something else \u2013 something better, usually.'\n\nFor at their best (after a pint or two of Mossbrown Ale), the Mummers sometimes acquired a near-Pentangle unity that was quite hypnotic. 'But the last couple of years, he didn't ask me to help him with them any more.'\n\n'Thought they'd gone off,' Mick, the one who looked like an escapee from The Clan of the Cave Bear, said. 'Can you sing, too?' he asked hopefully.\n\n'No.'\n\n'But\u2014'\n\n'Absolutely not,' I said firmly. 'Look, I don't mind if you want to come and keep using the workshop to practise in, as long as you don't bother me. But you'll have to find a new Mummer.' A thought struck me: 'Annie \u2013 you know Annie Vane, don't you?'\n\nThey nodded.\n\n'She pet-sits for an ex-pop singer who bought the old vicarage. Ritch Rainford, he's called and he's an actor now, playing a Victorian mill owner in that Cotton Common soap.'\n\n'Not Ritch Rainford from Climaxxx?' Miss Drippy said breathlessly. 'I thought that was just a story, that he'd bought the place. He's famous... but old,' she added belatedly.\n\n'Nah, he can't be much more than forty-five, at most,' Jojo said, giving her a dirty look and adjusting his bandanna over his bald spot to the point where it almost became a headscarf. It'd have to be the pirate look next, low down on the forehead. 'And he might want to keep his hand in, do a couple of gigs with us \u2013 worth asking... Good to talk to him, you know?'\n\n'You do that,' I agreed.\n\n'But would the Mysteries Committee let us play for them, if one of us wasn't from the Mosses?' said the girl. 'You know how stuffy they are about second-homers. I shouldn't think he lives here all the time.'\n\n'I don't know, Ophelia, but you could ask. It's not like you're performing in the plays and have to go to all the rehearsals, is it? Just incidental music and filling in between scenes.'\n\n'Olivia,' she corrected me.\n\n'What?'\n\n'My name's Olivia, not Ophelia.'\n\n'Oh?' That was a surprise, but I fear due to her water-dipped appearance she will forever remain Ophelia in my mind. It seemed to strike a chord with the other two as well.\n\n'Suits you,' Jojo said, and Mick agreed.\n\n'Why not change your name \u2013 new name, new start?'\n\n'Yeah! Ophelia Locke \u2013 cool,' she agreed, brightening slightly. 'I'll do it! Ophelia... Ophelia... Ophelia.'\n\nLoopy Locke, more like.\n\n'Looks different in here, somehow, without Tom,' Jojo remarked intelligently.\n\n'That's because I've sold the surfboard business and all the stuff's gone,' I said patiently. 'Look, Jojo, here's a key to the workshop \u2013 I've got a spare. It'll save you asking for it if it's locked, and you can leave equipment here safely if you want to.'\n\n'Thanks,' he said and, giving my arm an earnest squeeze, added, 'And, you know, anything you want \u2013 do anything we can...' He trailed off, made earnest eye contact and let me go.\n\n'Ophelia Locke...' whispered Miss Drippy in an ecstatic undertone. There were brown rabbits stencilled on the pockets of her smock.\n\nWatership Down has a lot to answer for.\n\n## Chapter 12: Just Desserts\n\nAs you will see in the preface, life took a sad turn here at Perseverance Cottage with the sudden loss of the Inconstant Gardener. However, my friends are all rallying round to divert my mind from unhappy thoughts, especially my fellow members of the Christmas Pudding Circle.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nLess than a week had passed since the funeral, yet with disconcerting rapidity summer had slid into September and what passed for normal life resumed. Even when Tom had been home he'd never played much part in the family rounds, so his absence was not really missed, insofar as you would miss a ticking time bomb.\n\nJasper seems to be feeling much the same, though it didn't help that half the time Mimi forgot what had happened and we had to explain it to her all over again.\n\nWe made an expedition to buy the laptop and printer Jasper thought I should have and then he set me up a little workstation in the window of the sitting room, which provided at least a temporary distraction for his thoughts.\n\nStill, at least Jasper could escape to the dig every day and I had way too much to do to brood, for the garden had taken advantage of my lack of attention to burgeon forth into a burst of flowerings and fruitings like a butterfly dancing along the edge of winter. I was harvesting and bartering the excess, bottling tomato chutney and pickling shallots.\n\nThis morning the members of the CPC all went to Faye's farm again for the meeting, since she wanted us to taste the Christmas ice creams she was developing \u2013 and it turned out she had also managed to produce the perfect brandy-butter one, too! I think it will be a lovely change with the Christmas pudding and we have all ordered a tub each.\n\n'Thank you all for helping me with the buffet for the funeral too,' I said, when we had settled down to the coffee and gossip part of things. 'I don't know how I would have managed without you.'\n\n'That's all right \u2013 what are friends for, if not to help each other?' Marian said, and the others murmured agreement.\n\n'There were certainly a few eye-opening revelations about Tom, weren't there?' Miss Pym said forthrightly. 'Neither that French wife of Nick Pharamond's nor Polly Darke appear to have any moral code whatsoever!'\n\n'Or Ophelia Locke,' Faye pointed out.\n\n'I don't think you can entirely blame Ophelia \u2013 she's obviously a sandwich short of a picnic,' Annie said generously.\n\n'No, and I do feel a bit sorry for her, even if she is exasperatingly silly,' I agreed. 'I've persuaded Unks not to evict her from her estate cottage.'\n\n'It's a pity it's not in his power to evict Polly Darke from her house though, isn't it?' Marian suggested. 'She's been seen everywhere, dressed in weirdest widow's black, playing for the sympathy vote.'\n\n'Well, she won't get it from any of us,' Annie said. 'In fact, she's not at all liked locally.'\n\n'She asks me every year if she can take part in the Mystery Play,' Marian said, 'and I turn her down. She's just attention-seeking.'\n\n'I'm glad my part is small,' Faye said, who was currently playing Mary Magdalen. 'I hate getting up in front of all those people.'\n\n'Me too,' I agreed. 'I saw Gary Naylor the other day and he said he was going to be Jesus again.'\n\n'Oh yes,' Marian said. 'He made quite a good job of it last year, once we'd persuaded him out of wearing black during his scenes, especially those big boots with all the metal studs.'\n\n'Jasper says he's an Emo,' I explained.\n\n'What's an Emo?' asked Faye.\n\n'Sort of a gloomier Goth, I think.'\n\n'Only a year older than Jasper, isn't he?' Miss Pym said. 'Expect he will grow out of it soon. He was a good boy at school... and speaking of which, term starts again next week, so I will soon be rehearsing the little animals for the Noah scene.'\n\nMiss Pym must be long past official retirement age and they are too afraid to tell her, but though she now only works part-time, she is still very much in control of the small infants' school and, I suspect, always will be.\n\nJasper and I agreed that there was nothing to beat a supper of globe artichokes with a little pot of melted butter for dipping, and fresh bread and cheese, with blackberry fluff and cream to follow.\n\nOne afternoon I was out in the garden waging a Canute-like attempt to assert my authority over Nature, when the vicar visited me.\n\nWhile Gareth ostensibly came to see how I was going on and offer comfort and a shoulder to cry on, should I need it, I quickly discerned that he really wanted to talk about Annie. So I told him all about our long friendship, dating back to our schooldays at St Mattie's, where she was hockey captain and my best subject was Nature Studies, and the French cookery course we did afterwards in London.\n\n'Neither of us was academic, you see. After the course we worked for a party catering firm for a couple of months, until I married Tom and she came home to live in Middlemoss.'\n\n'And now she has her own pet-minding business?'\n\n'Yes, and it's very successful,' I said, and told him what a lovely, trusting person Annie was, even after being jilted practically at the altar several years ago, when her fianc\u00e9 ran off with one of the prospective bridesmaids. Then I pointed out her many activities within the parish.\n\n'Of course,' I added casually, tossing a handful of weeds into the wheelbarrow, 'the way to Annie's heart is through her love of dogs. She even puts in several hours a week as a voluntary helper at the RSPCA kennels.'\n\nGareth left carrying bags of salad vegetables and runner beans, and looking thoughtful, so I hoped I'd planted some idea of how to win her affections: now the ball was in his court.\n\nHe was very nice in a serious and terribly good way, so I thought they would be very well suited. I couldn't give my best friend in marriage to just any old eligible bachelor, he had to be Mr Right. Or, in this case, Mr Bright.\n\nThis could be just the right moment for him to make his move, too, for I'd got the impression lately that Annie was finding Ritch Rainford disconcerting, now her initial bedazzlement was wearing off. She'd never had one of her fantasy men become flesh before.\n\nIf Gareth played his cards right, she could very well rebound quite happily into his arms.\n\nAnnie had managed all the Posh Pet-sitting stuff herself since the funeral, but I told her I could cope now if she needed help. So the following morning I walked Delphine Lake's three little dogs, who were very glad to see me, because Delphine's idea of a walk is from the car into the house.\n\nThen I went to collect a cat from the vet's surgery and returned it to the owner \u2013 or rather, the owner's au pair, who didn't seem very pleased to have it back. But I expect once it had got over its indignant rage at being confined in a carrying box it would soon calm down. How weirdly vocal Siamese cats are!\n\nOn the way home I popped in at Annie's little terraced cottage again to see if anything else needed doing, and found her making a chart of her Posh Pet-sitting for the next fortnight, with different coloured stars for the regular customers and fluorescent spots for the one-offs \u2013 very organised. Even the keys for houses where she lets herself in were starred and spotted to match.\n\n'I see Ritch Rainford's bagged all the gold stars,' I commented, having made us each a cup of coffee.\n\n'Well, he is our major celebrity so far,' she said defensively. 'And he seems to be turning into a regular customer, though he doesn't give me much notice. He's terribly casual \u2013 handed out the keys and the code for the burglar alarm to me and his new cleaning lady before he really knew us at all.'\n\n'Who has he got cleaning for him?'\n\n'Dora Tombs. She's a Naylor \u2013 niece or great-niece of Ted, the gardener up at the Hall.'\n\n'We Naylors get everywhere, like Mile-a-Minute.'\n\n'I think you're more of a rambling rose than a Russian Vine,' she said kindly.\n\nA brazen strand of hair fell into her eyes, and she pushed it back and clamped it down with a white Scottie dog hairslide. She has no taste: even the smock she was wearing over her cord trousers and T-shirt had dogs stencilled around the bottom and made her look like a pregnant bun loaf.\n\n'What on earth are you wearing?' I demanded. 'Isn't that one of Ophelia Locke's little creations?'\n\n'Yes. The big pockets are really useful and it wasn't expensive. She sells most of them at those historical re-enactment fairs, but she printed one with dogs as a special order for me.'\n\n'It does nothing for your figure,' I told her frankly.\n\n'I haven't got a figure.'\n\n'Yes, you have, an hourglass one with a very small waist. But that thing doesn't go in in the middle at all. You look entirely globular. Take it from me, it's a mistake.'\n\n'It's very practical, which is why I wanted it,' she said defensively. 'Anyway, no one is interested in my figure.' Then she blushed underneath all the little freckles.\n\n'Come clean, obviously someone's interested! Tell Auntie Lizzy,' I said encouragingly. Had Gareth actually made his move already?\n\n'They're \u2013 he \u2013 he's not really, it's just that he can't seem to take his eyes off my bust when I'm wearing a T-shirt,' she confessed. 'So I feel happier covered up.'\n\n'What, the vicar?' I exclaimed.\n\nShe looked at me as if I'd run mad. 'The vicar? Of course not, Lizzy! No, I meant Ritch Rainford. I thought he was lovely at first, so charming and amazingly handsome. Only there's something in his eyes when he's talking to me and everything he says seems to have some kind of innuendo in it... and... well, I'm simply not used to that kind of thing. It makes me feel very gauche and uncomfortable, though I'm sure he doesn't mean anything by his manner, it's just his way.'\n\n'Oh? So he's turned out to be mad, bad and dangerous to know?'\n\n'It's just me being silly and not knowing what to say back, I expect. For instance, when I was bending over patting Flo the other day he walked into the kitchen and stared at my chest, then said, \"You don't get many of those to the pound!\" I simply didn't know where to put myself.'\n\nI choked. 'Oh dear! That was very rude and un-PC of him.'\n\n'Yes, but you would have known what to say to him, wouldn't you?'\n\n'Probably. Or socked him one.'\n\n'I find I just \u2013 just don't want to go there any more in case he's in, although Flo is a very nice dog.'\n\n'Then don't go! I'll do his pet-sitting instead. He knows you've got an assistant, doesn't he?'\n\n'Oh, yes. But, Lizzy, he might be even worse with you, because you're so pretty!'\n\n'I'm not pretty at all, you daft lump,' I said, surprised. 'My hair is a really boring light brown colour and I'm way too tall! So don't worry, I can deal with him, no problem. Some of Tom's friends were rather oddball too, don't forget. So hand over his keys and pass on any requests for pet-sitting \u2013 I'll sort him.'\n\n'Well, actually, Lizzy, he wants me to go in to walk and feed Flo tonight, because he's out at some party until late. But then Gareth \u2013 the vicar \u2013 suddenly asked me to dinner and I said yes without thinking, so now I don't know what to do.'\n\n'He has? There, I knew he fancied you. I could tell at the Mystery Play Committee meeting, and he made a beeline for you at the funeral feast!'\n\n'No, of course he doesn't fancy me,' she protested, blushing again. 'I can't imagine how you got that idea!'\n\n'So, why has he invited you to dinner, then?'\n\n'We just sort of got chatting earlier. He was admiring Trinny and said he would love another dog \u2013 his last one died of old age just before he moved here. Isn't that sad? Only he's out such a lot he doesn't feel it would be fair just at the moment. So I told him about the kennels and the rescue dogs and how they always needed people to walk them, and he said he would come with me whenever he has time. So then he asked me if I would have dinner with him and help him understand what he's supposed to be doing with the Mystery Play and village things like that. There's a lot for him to take in, coming into somewhere like Middlemoss.'\n\n'There certainly is,' I agreed, 'especially pitchforked straight into a stranger's funeral, poor man. But he's quite right; who better than the former vicar's daughter to help him make sense of it? But don't wear the smock.'\n\n'Of course not!' she protested, then added thoughtfully, 'He's very good-looking, isn't he?'\n\nWhile I was glad to see she'd stopped mooning over Ritch Rainford and transferred her interest to Gareth, I couldn't help but feel that calling him good-looking was pushing it a bit, unless you particularly fancied knobbly, flame-haired, blue-eyed, lanky men who didn't seem to be fully in control of their limbs.\n\nIf something comes of this, their children will all be ginger-nuts (though perhaps they will raise a family of rescue dogs instead).\n\n'He's delightful,' I agreed, hastily banishing my mental picture of their possible progeny. 'And don't worry about Mr Rainford: I'm not afraid of the big bad wolf.'\n\nWhen I got home the latest Mosses Messenger with my advert in it had been pushed through the door, and there were two messages on the answering machine. I seemed very popular, suddenly.\n\nThe first was from Nick, saying he was finally returning from London, though why he thought I would be interested in his movements I don't know. Or what he thought I would do with the postcard of Camden Lock, with 'Sorry!' and a recipe for jellied eels scrawled on the back, which arrived the other day. I couldn't possibly eat eels \u2013 they're too snaky. And what was he sorry about? Leila and the permanently absent Tom are the ones who should be sorry!\n\n'Hi, Lizzy, I'm on my way back,' Nick's deep voice said. 'Things took longer than I expected because I couldn't persuade Leila I really didn't want any of her precious assets, even after her solicitor drew up an agreement for me to sign, until I stuck his ebony paperknife in my thumb and signed in blood. A bit melodramatic, perhaps, but it seemed to do the trick.'\n\nThere was a pause, then his voice resumed with just a hint of rueful laughter in it, 'OK, very melodramatic. And the solicitor didn't seem to want the knife back \u2013 said I could keep it as a souvenir. Anyway, the divorce is on its way, a clean split, and no claims on each other's property or earnings. We never shared anything anyway, so that makes it easier. Oh, and it's all given me an idea for a recipe. Got any raspberries, or am I too late?' The message clicked off.\n\nRaspberries?\n\nThe second message, from my agent, Senga McDonald, was short and to the point. After drumming her fingers and humming a brief snatch of 'Will ye no' come back again?' she said, 'Lizzy? Can you send me the new Chronicle, pronto? Only Crange and Snicket want it right now, and your sales figures aren't so good that you can afford to miss your deadlines. You did say you'd finished it and it just needed a polish, so slap it into the post right this minute!'\n\nWell, don't stop the carnival on my account, even if I have only been widowed for five minutes! I thought.\n\nStill, it was as finished as it would ever be. I only needed to read through it and make final corrections before it went off. But I was starting to wonder if I would ever finish another Chronicle, because I was not exactly hitting my target of four pages a day any more and said as much to Senga when she rang me back later to make sure I'd got her message, and was obeying orders.\n\n'Oh, don't worry your wee head about that one just now. I've had a brainwave and sold Crange and Snicket on the idea of a collected book of your recipes and hints, with the odd anecdote thrown in. They think it's a great idea \u2013 they're going to call it Just Desserts. Shouldn't take you too long to do, should it? Toss in some new recipes to liven it up.'\n\nI stared at the receiver as if it had bitten me, while her voice rolled inexorably on.\n\n'What? When? I mean, when do they want it?' I broke in urgently, when she stopped for breath.\n\n'Oh, not until early next year \u2013 January, say. Loads of time. Now, have you put that manuscript in the post?'\n\n'Tomorrow,' I promised. 'I'm just making final corrections and I might have to reprint a few pages later.'\n\n'See you do \u2013 I'll be expecting it. Send a copy directly to Crange and Snicket, too.'\n\n'I've got my own computer now and I'm getting going with it, so I'll be able to email the next book to you, instead of posting it,' I told her, because she'd made it pretty plain that she and the publishers would prefer my books that way, rather than printed out and posted. I'd simply have to move with the times if I want to stay published.\n\n'Well, welcome to the twenty-first century at last!' she said sarcastically and rang off.\n\nIt was just as well I would be able to print my own pages off, too, because I'd just remembered that Jasper wasn't going to be home until late. He was going straight to Liverpool with a friend after he finished work at the dig.\n\nNot, of course, that he'd ever objected to my using his laptop and printer; it had just seemed like a personal intrusion to use them when he wasn't there.\n\nI wandered rather aimlessly round the kitchen for a few minutes, then dolloped clotted cream and raspberry jam onto some meringue halves I made yesterday with leftover egg whites.\n\nThey were so yummy I ate six, and I think I'll put them in the latest Chronicle and Just Desserts as an alternative to scone cream teas.\n\nI did try to read through the manuscript, but I just couldn't concentrate and found myself staring blankly at the same page. Eventually I gave up temporarily and went to do a bit of gardening instead. I picked loads of strawberries and even a few raspberries \u2013 everything was still burgeoning forth like nobody's business. I wished someone would tell my garden it was time to start winding down into autumn and taking it easy.\n\nAs I worked I thought about Ophelia Locke, also burgeoning forth with a baby that might just possibly be Tom's. But whoever the father proved to be, the poor little thing was trying to grow on a vegan diet and I wasn't sure Ophelia was capable of seeing it got enough of the right nutrients.\n\nWith a resigned sigh I fetched a big wicker basket from the outbuilding, and began to pack it with fresh fruit and salad vegetables, a bunch of baby carrots, eggs and ripe tomatoes. Then I set off up through the woods to Ophelia's estate cottage.\n\nLike me, she lives right next to the boundary wall of the estate, so it would have been quicker to walk up the road, but not as pretty as the woodland paths, or as cool. The sky was a brazen blue and it was Indian summer hot.\n\nOphelia had attached a nameplate to her front door that said 'Whitesmocks', but whether she meant that as a name or a description of her way of making a living, was unclear. There was unlikely to be any passing trade interested in purchasing antique-style clothing down here anyway, since her only next-door neighbour was the old and very deaf gardener who was Mimi's sparring partner and occasional accomplice in plant larceny, and I couldn't see Ted taking to smocks. Caz's cottage was quite nearby, only set further back in the woods in isolation, on the other side of a small stream bridged by mossy, ancient slabs of stone.\n\n'Oh God!' Ophelia said predictably, opening the door and staring at me, 'Oh, no, oh God!' and fell to chewing her lower lip.\n\nOh, my ears and whiskers!\n\n'Hello, Ophelia,' I said bracingly. 'Can I come in? Only I've brought you some spare fruit, vegetables and salad stuff, which will do you good. And eggs, though I wasn't sure if you ate those.'\n\nShe fell back rather reluctantly and I stepped straight down the one worn step into the tiny living room. It smelled of unbleached calico and herbal tea. A sewing machine was set up by the window, and white material was festooned everywhere. A clothes rack on castors crammed with the finished product swayed slightly in the breeze from the open door.\n\n'That's kind \u2013 that's so kind!' she said, and for a horrible moment I thought she was going to burst into tears or embrace me, or something equally embarrassing. Instead, she wrung her hands and stared at me despairingly. 'But Ted, the old gardener next door, he says that Mr Pharamond will give me notice to quit the cottage because I was... well, he won't want me here.'\n\n'No, didn't you get my message? I've spoken to him and he's no intention of throwing you out, so you don't need to worry about that. We'll see what happens after the baby arrives \u2013 which will be when, do you think?'\n\n'I don't know really \u2013 January, maybe?' she said vaguely. 'I didn't realise I was pregnant for ages and then I thought I'd just sort of wait and see...'\n\nI frowned. 'Wait and see what? Have you visited your doctor?'\n\n'Oh, no! I believe nature should take its course. One of my friends will come for the birthing.'\n\n'The birthing? Come on, Ophelia, this isn't the Dark Ages! You need to see a doctor and have the baby in hospital. Nature's way may turn out to be pre-eclampsia, or something like that.'\n\nShe stopped chewing her lower lip and looked stubborn.\n\n'I hope you're eating well, anyway? Couldn't you give up the vegan stuff for the duration?'\n\n'Vegan is healthy. There are lots of vegan mothers.'\n\n'Then make sure you vary your diet as much as you can. I'll keep leaving you fresh fruit and vegetables on your doorstep every week, a bit like one of those organic delivery services.'\n\n'You don't have to,' she began, blinking nervously. Her big pink eyelids reminded me of those festooned blinds.\n\n'No, but I want to. Whatever you did isn't the baby's fault, is it?'\n\n'Noooo...' she muttered, walking backwards until she was half-enveloped among the hanging smocks. 'No, no, no!'\n\n'There you are then, that's settled,' I said soothingly, following her and thrusting the basket into her arms.\n\n'I'll \u2013 I'll unload this, so you can have it back!' she gasped and, fighting her way out of the folds of material, escaped into the lean-to kitchen.\n\nI looked in from the doorway. Although basic, it was at least clean and tidy, though yet more eternal rabbits had been stencilled everywhere. On the little gate-leg table lay a roll of familiar-looking recycled yellow paper, a big black felt-tip pen and a coil of silvery gaffer tape \u2013 and something clicked in my head.\n\n'Ophelia, when Jojo and Mick said you'd been sleeping with the enemy, meaning Caz Naylor, did they mean because you three are all members of ARG?'\n\nShe dropped the bunch of baby carrots. 'No, no no... not me. Not us. No, I\u2014'\n\n'You're a terrible liar,' I said dispassionately. 'I recognise the paper on the table from the posters you put on my barn and my car \u2013 and even over my front door!'\n\n'No, no, no!' she jabbered, backing away, her prominent eyes starting. It is a pity she didn't say 'no' more often when men were around; it would have saved her a lot of trouble.\n\n'Yes, yes, yes,' I said firmly, quite certain now. 'So, Ophelia, let's get this straight: you're living here on the Pharamond estate and, as a member of ARG, targeting not only the owner of that estate but also his gamekeeper, with whom you've been having an affair?'\n\n'Oh, no! He \u2013 we didn't! Or only once, when Caz caught me putting up a banner on the Hall gates and... and I don't know what came over me! But I said I wouldn't go with him again, because he was evil and persecuted the poor woodland creatures!' she whimpered.\n\n'Once?'\n\n'Or... maybe twice... three times,' she conceded, which reminded me of the joke about only being unfaithful once, with the Household Cavalry. 'And Caz's not really murdering the grey squirrels, he catches them and takes them away!' she said earnestly. 'I still have to put the posters up, though, because the Pharamonds are on the ARG hit list, but Caz takes them straight back down again, like he was doing with the ones at your place.'\n\n'I suppose you might have some kind of case for targeting Caz and the estate, though it's a pretty shaky one to my way of thinking, but none at all for me. I mean, there are battery hen farmers and goodness knows what else in the area. Why?'\n\n'S-someone said that you were cruelly exploiting your hens and the poor little quail, but when I said I wouldn't do it unless ARG told me to, this person made me do it...'\n\nI frowned, while she stood there wringing the feathery stems of the carrots between her hands. I'm not entirely sure I'd trust her with a small baby when she's overwrought. 'So someone made you target me? Who? And how? Do they know something about you?'\n\n'No \u2013 yes, oh God!' Ophelia went white. 'If ARG find out I've been targeting someone not on the hit list, I'll be expelled from the organisation!'\n\n'Jolly good idea, especially in your condition. Who exactly is this person with a spite against me?'\n\nShe pressed her lips together firmly and said nothing.\n\n'You know,' I said to her severely, 'there are better and perfectly legal ways of campaigning for animal rights, aimed at the people who really do abuse animals...'\n\nThen I remembered something I'd read in the local paper. 'Ophelia, were you involved in that big raid at the end of last year, on the lab rat farm over near Skem?'\n\nIf possible, she went even paler. 'Nooo...' she moaned. 'I wasn't there \u2013 she didn't see me \u2013 there wasn't any proof!'\n\n'Ah, I see. So it's a she, and she knows you were involved in that raid?'\n\nShe moaned again. 'I was the last into the back of the van and her headlights caught me.'\n\n'Don't you all wear masks, or something?'\n\n'I forgot my balaclava and the scarf slipped,' she said rather sulkily. 'But now Caz says I don't have to do anything she says any more, because he's got something on her, so she can't make me \u2013 and now I know the truth about her, I wouldn't anyway!'\n\n'So Caz knows all about it, and who this person threatening you is?' I said, trying to disentangle her sometimes cryptic and convoluted utterances. I had a growing suspicion that I knew who it was now, too.\n\n'He already knew most of it, he'd been watching me... following me! He made me tell him everything,' she said, blushing.\n\nI didn't ask how. 'At least Caz seems willing to take some responsibility for you.'\n\nShe went pink and looked away. 'He's not so bad really \u2013 not now I know what he does with the squirrels.'\n\n'What does he do with the squirrels?' I asked, then added quickly: 'No, don't tell me, I don't really want to know!'\n\nShe snivelled. 'I thought Tom loved me, I really did, but I see now it's just like Jojo and Mick said: I was a handy bonk.'\n\n'At least you seem to have grasped the realities of the situation,' I agreed. 'I suppose it's none of my business, but since Caz rescued you the other day after the funeral, he must care about you.'\n\n'He's furious with me for making a scene and saying that about... about Tom,' she whispered, twisting a strand of dishwater-blond hair around her fingers. 'Only I was so angry, hearing those two going on as if they meant something to him, when I was the one who was pregnant!'\n\n'And did Tom really say he loved you?' I asked curiously.\n\n'He said I knew how he felt about me, and so I thought he meant he loved me. But he wasn't any different to me afterwards \u2013 and sometimes he could be very cruel, even though he didn't really mean it!'\n\n'Yes, I know.' I wondered how many times Jojo, Mick and Caz had... No, even I couldn't ask that, though it seemed that the odds on it being Tom's baby were at least three to one. We would just have to await the birth and guess. Or do DNA tests, or something. If it were Caz's, I suspect the entire Naylor clan would instantly know it anyway, in a very Midwich cuckoo way, just like they recognised me the second they saw me.\n\nI walked home, thoughtfully swinging the basket. Someone had been angry or jealous enough of me to force Ophelia to include Perseverance Cottage in the ARG campaign, which was an act of petty spitefulness. And when it came down to it, I could only think of one likely person.\n\nI'd love to know what Caz has got on her, but I don't suppose he would tell me even if I asked.\n\n## Chapter 13: Raspberries\n\nSeptember means the last of the late-fruiting raspberries and picking blackberries for jams, crumbles and wines, and the making of damson gin by Miss Pym, one of the CPC members, to a secret family recipe. It is so delicious that the barter rate is very high!\n\nThe Perseverance Chronicles: A Life in Recipes\n\nWhen I got home Nick's car was there and he was just walking out of one of the outbuildings, brushing light-coloured feathers off his sleeve. When he saw me he stopped dead and looked slightly taken aback, though goodness knows why: it was my house, after all.\n\n'Oh, hi, Lizzy! I was just talking to Caz.'\n\nI expect he meant that literally, since he was unlikely to get a complete sentence in exchange, let alone a conversation. Caz often makes me feel like a Shakespearean actor embarking on a long monologue, although of course he does nod, shake his head or scratch his nose from time to time.\n\n'It was kind of you to let him have that big freezer to store stuff in,' he added, carefully closing the door behind him.\n\n'Well, why not? I wasn't using it any more now I don't grow so much. The other one in the larder is quite enough.' I turned to lead the way to the cottage, since I was hot and thirsty after my walk. 'God knows what he puts in it, but I've a horrible feeling it might be all those grey squirrels from the humane traps.'\n\n'Oh, no, not squirrels,' he assured me, so I assume Caz had let him see into it, which was quite a relief, really. I expect it is just rabbits and stuff, for the pot. (And I thought again what a strange mismatch it was that vegan, animal rights Ophelia and carnivore Caz should ever have come together!)\n\n'I've been downsizing the fruit and vegetable production for nearly two years, ever since I realised I was going to have to leave Perseverance Cottage. Only now, of course, I don't have to. Unks is so kind.'\n\nHe followed me into the kitchen and leaned against the fridge, arms folded and glowering darkly, like Mr Rochester in a strop. 'I hadn't realised things were quite so bad between you. You should have told me. Caz says when he's been down here in the evenings he's heard Tom being really abusive.'\n\nThat accounts for the feeling I often got that our arguments had an audience \u2013 and also why Unks was not surprised that I had intended to leave Tom, because Caz must have given him a hint. I really should draw the downstairs curtains more, though you tend to forget when other houses don't overlook you. But who knows when Caz might be flitting past on his nocturnal activities?\n\n'I was suspicious when you had that bruise on your face and were so evasive about how you got it,' Nick said.\n\n'That wasn't intentional! He was never physically violent \u2013 what sort of doormat do you take me for? I'd have left immediately with Jasper if he'd tried anything like that,' I said indignantly.\n\n'No, sorry, I expect you would. But I also thought you might have told me what was going on.'\n\n'It was none of your business. What could you have done?'\n\n'Probably made things worse,' he admitted ruefully. 'It was pretty clear that Tom didn't want me around here the last few years, so I thought you both might patch things up if I gave you a wide berth.'\n\n'No, things just steadily got worse. Do you want some ginger beer?' I offered, as a peace-making gesture. The top came off and it fizzed gently into glasses. I simply love the sensation when the bubbles get up the back of your nose and ginger explodes in your head.\n\n'I still think you could have told me things were so bad that you were going to leave him, Lizzy.'\n\n'Like when?' I demanded. 'I could hardly have bellowed out the information while we were all sitting round the dining table up at the Hall over a Sunday lunch, with Tom turning on the charm for Roly's benefit, and you in and out of the kitchen worrying over the roast beef and whether the horseradish sauce was just a trifle too \"piquant\". Maybe I should have sent you a postcard?'\n\nHe ran a hand through his black hair so that it stood up like a cockatoo's crest and said more reasonably, 'After what you said at the hospital, I thought it was best to distance myself a bit \u2013 especially after Tom accused me of having had an affair with you.'\n\n'He did? I hadn't realised... But did you also know he suspected Jasper was yours?'\n\n'Yes, but I told him he was mad, so I hoped he'd quickly come to his senses.'\n\n'Well, you might have told me that you knew.'\n\n'I wasn't sure how much he'd said to you, so I thought it better not. And, of course, I didn't know then that my wife was having an affair with him,' Nick added. 'But when I found him in Leila's restaurant that night it all sort of clicked into place, even though she denied it.' He looked sombrely at me. 'Perhaps he only started the affair in the first place to get revenge on me for something he imagined I'd done \u2013 and I'm sorry she made that scene at the funeral, Lizzy.'\n\n'Well, it wasn't your fault, was it?'\n\n'No, but I brought her, and I knew she was still lying when she insisted there was nothing between them, although I didn't know then why she wouldn't agree to the divorce.'\n\n'And then it turns out she was sharing Tom with Polly Darke, and Ophelia Locke was the slightly wilted side salad! Oh \u2013 you did hear about Ophelia's revelations \u2013 the pregnancy?'\n\nHe nodded. 'Roly says she's deranged and invented it.'\n\n'No, but she deluded herself into thinking a quickie meant something deeper. She seems very naive for her age. There's a good chance the baby isn't Tom's, though. We'll have to wait and see.'\n\n'She's spread her favours a bit?'\n\n'I don't think she's capable of saying no. But at least Ophelia is just credulous and silly, while with Polly and Leila it all boils down to greed and self-interest, doesn't it?' I said sadly. 'And the ironic thing is that Tom thought Unks was going to leave him the cottage outright when he died, and even I could have told him that he would never have split the estate up like that!'\n\n'No, and I'm afraid my poor father scuppered his chances of inheriting by giving us his plans for turning Pharamond Hall into apartments,' Nick said ruefully.\n\n'I expect Unks will leave it to you, because you wouldn't do that to it, would you?'\n\n'No, of course not! I love it the way it is. And actually, Roly says he is leaving it to me, though keep that to yourself. Which means that I do have every right to worry about your welfare, if only because you live on the estate,' he added. 'And that, I suppose, puts Jasper in line as next heir, by rights, since it's not entailed.'\n\nI stared at him. 'Don't you dare put that idea into his head! You'll remarry, probably to some young, skinny model type like that photographer who came to the Hall last year for that feature on you \u2013 Lydia, was it? \u2013 and have a multitude of children.'\n\n'I don't think so,' he said shortly.\n\n'Famous last words. Well, I'm certainly not getting married again. It's the single life for me from now on. Pretty much as before, really, only without the threat of intermittent verbal abuse hanging over me.'\n\n'And I married a woman who was so territorially defensive she insisted on keeping her own flat on and pencilled me into her schedule if I wanted to see her! But now Leila and I are divorcing, I'll be spending most of my time up here in Middlemoss. I might even sell my London flat, because Roly never uses it, he prefers his club. I send most of my work in by email these days, so I can base myself anywhere.'\n\nNick picked up his glass and absently drank his ginger beer down in one go, which made his eyes water because it's good, strong, peppery stuff. 'My God, you are just so Enid Blyton at times, Lizzy! Aren't you going to offer me a sticky bun and an adventure, too?'\n\n'No, but there are Choconut Consolations in that tin, if you want one?'\n\n'No, thanks, you know I don't like sugary stuff much.' He finally sat down, uninvited, on a spindly old kitchen chair that groaned slightly. 'Lizzy, didn't it occur to you that Unks would have been sympathetic if you'd told him what was going on? He would certainly have helped you and Jasper to find somewhere else to live, for instance.'\n\n'Since Tom was always his old, charming self to me whenever Unks or anyone else was around, I wasn't sure he would believe me! And I didn't want to tell him about the affair, especially since I had no idea who it was... and \u2013 Oh, Nick, I'm so sorry!' I exclaimed. 'I keep forgetting it was Leila.'\n\n'Among others. And I don't really care any more, except on your account. I've been trying to get her to agree to a divorce for ages, so one good thing has come out of all this.'\n\n'The last couple of years with Tom have been fairly hellish, but all's well that ends well... though I suppose I shouldn't say that when it is Tom's death that's made everything come right. It makes me feel so mixed up \u2013 guilty and relieved and sad, all at once...' I sniffed and took a gulp of ginger beer.\n\nBeing Nick, he didn't rush to comfort me, but instead said bracingly, 'Time to move on, for both of us. And since I'm going to be around a whole lot more from now on, if you want any help with anything, you only have to ask.'\n\n'Thank you, that's very kind. But I do most of the work here myself and, as I said, I've scaled things down. In fact, Jasper's put the details of the big greenhouse up on the Freecycle website to see if anyone will dismantle it and take it away, and I've put an advert in the parish magazine about the quail. I'm selling that enormous TV of Tom's, too.'\n\n'Didn't you throw the TV controller into Tom's grave?'\n\n'Yes, but Jasper had one of those universal ones all the time, so he could watch the History Channel on the big screen when Tom was away.'\n\nI took another sip of ginger beer and reached for the biscuit tin. 'At least I got rid of the goats years ago, when they learned how to climb trees.'\n\nHe gave me a scathing look. 'Goats can't climb trees, Lizzy!'\n\n'You obviously don't know much about goats. And you wouldn't believe how strong they are! If a goat sets its mind on going somewhere, there's not a lot you can do to stop it. Anyway, I never did get used to goat-flavoured milk and yoghurt and I thought the cheese tasted like brown soap.'\n\n'What are you going to do with Tom's business?'\n\n'That's already sorted,' I said, taking a crunchy bite of Choconut Consolation. 'The day after the funeral two of Tom's friends, Jimbo and Freddie \u2013 do you remember them? \u2013 came and made me an offer for everything, and I accepted it. I could have got much more, but I wanted to... well, I wanted to just get rid of it all! They got a really good deal, but the funny thing is that they assumed Tom's van was included, only I'd already swapped it with Dave Naylor for that Land Rover in the yard, so they had to go and buy it back!'\n\n'Really, Lizzy, you should have left all that to me. I'd have got you a good price for everything!' he said, not seeming at all amused.\n\n'It's none of your business,' I said tartly as a few faint, plangent notes wafted across the courtyard and through the open kitchen door.\n\n'What the hell's that?' he demanded, startled. 'It seems to be coming from the workshop!'\n\n'It's just the Mummers of Invention, Nick. I let them carry on using the workshop to practise in. I wasn't using it.'\n\nHe gazed blankly at me. 'But Ophelia Locke is\u2014'\n\n'Pregnant with Tom's child? Chances are it isn't his, but I think she was more sinned against than sinning because she's so very easy and persuadable. Gullible, even. I've just been up to her cottage to take her some spare fruit and veg to build her up a bit, but if I'd known she was coming down here tonight she could have taken it back with her.'\n\n'You're crackers!'\n\nThis didn't seem to be the moment to tell him she was also one of the members of ARG who'd been targeting my cottage and the estate. Anyway, she'd said she'd stopped now and I expect Caz would make sure she did.\n\n'Strangely enough, I suspect Ophelia's half in love with Caz, only she doesn't want to admit it,' I said, following that train of thought.\n\n'Is she? Well, they do say that opposites attract! Maybe we could have made a go of it all those years ago, if you hadn't suddenly decided to marry Tom on the rebound after we split up.'\n\n'I did not marry Tom on the rebound. We fell in love months afterwards!' I said hotly. 'And we only split up because you decided to go off on a world recipe-finding mission for a year, don't forget.'\n\n'You should have understood \u2013 and waited. I wrote to you.'\n\n'No you didn't, you only sent me recipes on postcards!'\n\n'That's the same thing.'\n\nHis eyes, the purple-grey of wet Welsh slate, were baffled.\n\n'Well, whatever,' I said. 'It's pointless having post-mortems at this stage, isn't it? We married other people and moved on.'\n\nThere was a tap on the door and a shadow darkened the threshold. 'Hello!' called a deep and attractive male voice. 'Mrs Pharamond?'\n\nThe man, who was tall with curling blond hair and a ruggedly attractive face, stopped halfway through the door. 'Oh, sorry if I'm disturbing you,' he said, with a charmingly apologetic smile. 'I'm Ritch, you know \u2013 Ritch Rainford?'\n\nAs my eyes met his incredibly blue ones, it struck me that Annie's description of his charms had been wildly understated. A force field could not have held me faster at that moment and I fear I might even have been drooling \u2013 but then, he did make me think of slabs of golden-brown Honeycomb Crunch...\n\nHis gaze released mine and he looked enquiringly at Nick, who was sitting there with his arms crossed like a terribly gloomy wooden Indian.\n\n'You're not interrupting anything,' I said, managing to get my voice back. 'This is my late husband's cousin, Nick Pharamond.'\n\n'Hi,' Ritch said in a friendly manner, but the two men seemed to me to be eyeing each other in a very sizing-up-for-battle way. It reminded me of a film I once saw of bull elephants fighting, probably a territorial thing. They were both big, fit men, so I wouldn't know which one to put my money on.\n\n'I just wanted to say how kind it was of you to let the group continue practising in the workshop, Mrs Pharamond, and to let you know I'll be up here jamming with them sometimes. So if you see a stranger around the place, it's me.'\n\n'Do call me Lizzy,' I said. 'And actually, I'll be the stranger round at your place tonight, if you still want Posh Pet-sitters to come, because I'm Annie's assistant and I'll be seeing to Flo.'\n\nHe gave me another warm \u2013 very warm \u2013 smile, his blue eyes crinkling at the corners, and I could see what Annie meant, because there was just something about his expression that made you feel hot under the collar (or the smock, in her case).\n\n'Great, I can see she'll be in good hands. Well, better get back to the gang, I suppose. Not my sort of music, really, but it's good to keep my hand in!'\n\nAnother one-hundred-and-fifty-watt smile and he was gone. I sighed involuntarily, watching his retreating, lithe figure until it vanished into the workshop. Ophelia might think he was old, but I bet that was before she saw him in the flesh.\n\n'Well, you do seem to be managing everything very well without my help,' Nick said, abruptly getting up and banging his head on the ceiling light, which swayed alarmingly. 'Bloody hell!'\n\n'Jasper's started doing that too, now he's over six foot. I must shorten the chain, or something. Thanks for coming though, Nick. Jasper will be sorry to have missed you.'\n\n'He's at the dig?'\n\n'Yes, and then his friend was going to pick him up and they were going over to talk to the owner of the student house his friend's brother rents with a couple of others. Jasper's taking that dog that Annie dumped on us, and thinks he can sweet-talk the landlady into letting him keep it in the house, even though it's supposed to be no pets. He'll be home late. But it's probably just as well, because I need to read through my new book tonight \u2013 if I can bring myself to concentrate on it for long enough to spot any mistakes!'\n\nHe gestured at the dog-eared heap at the end of the table. 'Is that what all this stuff is? Yet another glorious Perseverance Chronicle?'\n\n'Yes, and possibly the last. I'm not sure I'll be able to finish the next, because I'm only managing to write about a paragraph most days instead of four pages. And I tried to go through the manuscript earlier, but my brain got stuck and I read the same page over and over,' I said despairingly. 'My agent will kill me if it isn't in the post tomorrow.'\n\n'I'll read it for you,' he offered, to my surprise.\n\n'What \u2013 you?'\n\n'Why not? I'm literate and I've nothing in particular to do for the rest of the day. It's not that big a manuscript, so it won't take me long. I'll red-pen any mistakes and drop it back later.'\n\nIt was amazingly kind of Nick, but perhaps he just wanted to do something to help, so I pushed it all into a manila folder and handed it to him thankfully. 'That is such a weight off my mind! I'll be out around seven to see to Mr Rainford's dog, but you know where the key is if I'm out, don't you?'\n\n'Raspberries,' he said.\n\n'What?'\n\n'I left you a message earlier. Do you have any?'\n\n'Oh yes, there's a September-fruiting bush behind the greenhouse, but there aren't an awful lot of them. I put them aside for you in the small barn.'\n\n'That's OK. I bought some supermarket ones too, just in case, though they don't have the same flavour. I can mix them together.'\n\n'What are you making?'\n\n'A variation on liquorice ice cream, with a raspberry coulis.'\n\nI looked at him doubtfully, but he seemed to be serious. Then he stunned me by stooping and swiftly kissing me on the mouth, which he absolutely never does, and left with my manuscript under his arm.\n\nMy lips tingled. I supposed we were kissing cousins, if only by marriage... but I wouldn't have described that as an affectionate peck on the cheek! Also, since he was the sort of man who gets a five o'clock shadow five minutes after he'd shaved, I got a free exfoliation into the bargain.\n\nStaring after his car as he drove off (and it was very nearly pressed duck for dinner), I realised what I hadn't admitted to myself before: that in recent years I'd really, really missed our invigorating exchanges of opinion. Every life, especially one so literally down to earth as mine, needs just a little vinegar in the mix.\n\nBut Ritch Rainford, now \u2013 he was more of a sweet treat...\n\nGoing back in, I slightly loosened all the caps on the ginger beer bottles, just in case, then looked out the Honeycomb Crunch recipe. It is much the same as cinder toffee, really \u2013 sugar, white vinegar, water and bicarbonate of soda, only with added butter and golden syrup. It's the mixture of bicarb and vinegar that makes them go all bubbly.\n\nI added that thought to the current chapter of the next Chronicle, before the inspiration bubbles went flat again.\n\n## Chapter 14: Slightly Curdled\n\nAfter making Honeycomb Crunch, it occurred to me that if you crumbled it up, it would make a wonderful topping for home-made vanilla ice cream. You could wrap a chunk in a plastic bag and hit it with a rolling pin \u2013 that would do the trick.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nJasper phoned to tell me his new landlady had agreed that Ginny could take up residence in the rented house with him and his friends at the end of the month, so that was sorted. I would have preferred him to live in a student hall of residence for the first year, but the whole point of your children going away to university is so they can live their lives, not yours, so I would just have to go along with his decision.\n\nThe thought of my little boy exposed to all the big city temptations of drugs, unprotected sex and being knifed in the street... well, I had to swallow hard before I could say brightly, 'Oh good, I'm so glad, darling. That will be great, living with your friends and having Ginny with you. What time will you be back tonight?'\n\n'Chris's mum is going to drop me off, probably about eleven. She doesn't like him driving late at night \u2013 she's nearly as bad as you are for fussing.'\n\n'Jasper, I don't fuss! How can you say I fuss?' I demanded indignantly. But then, despite my best intentions, asked, after a moment's pause, 'What are you doing tonight?'\n\n'Sex, drugs, tattooing our arms with old syringes off the street, that kind of thing,' he said good-naturedly.\n\n'Jasper!'\n\n'Watching DVDs, making popcorn, drinking beer,' he amended.\n\n'I'm going into the village in a minute, to pet-sit an actor's dog \u2013 Ritch Rainford. He used to be some kind of pop star in the eighties.'\n\n'Never heard of him,' he said, unimpressed. Had it been a bosomy model from one of the boys' magazines I expected he would have been much keener, but the only creature with artificially inflated breasts in the Mosses is Polly Darke, and I knew he classed her with the pensioners.\n\nHow can silicone be sexy? Isn't it peculiar that many men find artificial breasts just as much of a turn-on as real ones? Another one of life's strange mysteries to ponder when you are examining your marrows.\n\nI ate a generous portion of my own version of Lancashire hotpot \u2013 good and peppery, with rich gravy and a shortcrust topping \u2013 and read the rest of the latest issue of the Mosses Messenger, before going out.\n\nAccording to 'The Verger's Village Round-up', Caz Naylor had 'kindly volunteered to resolve the goose situation in Middlemoss, because children and the elderly were being terrorised, and the mess they left was proving a danger to life and limb. Caz has therefore now caught and rehomed them to somewhere more suitable, a solution we know will be acceptable to all interested parties.'\n\nThere was also a notice that rehearsals for the Mystery Play would be starting in the village hall on the fourteenth, Tuesdays (generally acts 1\u20139) and Thursdays (acts 10\u201322). Luckily, we've filled the vacancies for Moses and Lazarus without any trouble, and Miss Pym, who is a dab hand with papier-m\u00e2ch\u00e9 after a lifetime spent teaching infants, is making a new Thou Shalt Not Commit Adultery commandment tablet.\n\nI set off for the old vicarage just before seven, since it's only a short walk, but first I changed into a pair of decent, clean jeans and a pale green T-shirt with pretty old buttons sewn in a border all round the neck, an idea I got from one of Annie's magazines. I sewed them onto my best Indian leather toe-post sandals too, though they kept getting ripped off.\n\nWhen I rigorously brushed all the knots out of my hair it went into a ripply light-brown mass round my head like something Rossetti would have painted, though thankfully sans the sulky Pre-Raphaelite trout-pout.\n\nThe T-shirt brought out the green in my eyes, and my face looked glowingly healthy for a recent widow. I wondered about applying a bit of make-up (I often think about make-up, but rarely bother to do anything about it), but then suddenly thought, what the hell am I doing, getting all duded up to walk Ritch Rainford's dog? Am I crackers? Do I think Flo is going to give her master my marks out of ten for effort and appearance when he staggers home from his party?\n\nBut it was too late to change, so I set off as I was. As I passed the vicarage bungalow, I wondered how Annie and Gareth were getting on. It seemed very daring of him to invite a single lady to dine with him alone. Well, I assumed they were alone (apart from Trinny), unless he's invited half the village round for support?\n\nThe old vicarage now sported the new name of Vicar's End. I unlocked the front door with the key Annie had given me and stepped inside. It was always unlocked when Annie's father was vicar, so that seemed odd in itself. And somehow, I still expected the cool, tiled hall to smell of lavender, floor polish and beeswax, just like it used to, rather than of some exotic artificial household fragrance mixed with slightly acrid cigarette smoke.\n\nThe old hallstand had been replaced by a glass-topped console table bearing an indecent bronze sculpture and a severely tailored arrangement of decayed-looking black orchids among spiky foliage. I was just touching them to see if they were real (they were), when, with a clatter of claws, Flo hurtled down the hall to meet me, velvet coat rippling and tail thrashing about. Annie had said she would be delighted to see me, even though I was a total stranger, and she was quite right. In fact, had I been a burglar, I expect she would have been equally pleased.\n\n'Good girl, Flo!' I said, patting her. 'Good girl!'\n\nMy instructions were to let her into the garden and feed her, then hand her a chewy rawhide bone and shut her in the kitchen on departing. I thought I would feed her first, since she didn't seem particularly interested in going out.\n\nWhat happy, smiley faces white bull terriers have! 'Come on then, Flo, din-dins,' I said, and had just started towards the door at the back of the hall that led into the kitchen, when a deep, instantly recognisable masculine voice from above called out, 'Tobe, is that you? I'm on my way!'\n\nHe was, too: leaping athletically down the stairs two steps at a time, Ritch Rainford landed in the hall almost at my feet, though an advance wave of expensively intrusive aftershave just beat him to it.\n\n'Yark!' I squawked inelegantly.\n\nHe looked equally surprised for a moment, then smiled. 'Sorry, thought you were my lift! Did I startle you, Lizzy?'\n\n'Well, yes,' I said, swallowing. 'I thought you'd have been long gone.' Despite myself I was answering that effulgent smile, drowning dizzily in the depths of his cerulean-blue eyes...\n\n'I'm so glad I wasn't,' he said, to which I couldn't think of a thing to say.\n\nWas he flirting? My flirting abilities were not great, even in my youth, and by now had atrophied to the point of no return. I looked at him doubtfully, but decided it was just his usual manner and got a grip on myself. I even started breathing again: in, out \u2013 it was quite easy now I'd remembered how to do it.\n\n'No... well, since you are still here, you won't need me to see to Flo, will you? You can do it before you go,' I suggested.\n\nA horn sounded: 'No time \u2013 that's Toby. You know Toby Little, plays Rufus Grace in Cotton Common?'\n\n'No, I'm afraid not, I haven't seen it,' I confessed. 'I don't watch much TV. I quite like gardening and cookery programmes, but anything else just doesn't seem to hold my concentration, probably because it isn't real. Life's much more interesting, isn't it?'\n\n'Is it?' he said, looking at me curiously. The horn hooted again, impatiently. 'Look, Lizzy, why don't I tell Tobe to hang on a couple of minutes while we sort Flo, and then you could come with me, meet some of the cast, come on to the party?' he suggested.\n\n'What \u2013 me? Oh, no, thanks, I couldn't! Jasper \u2013 my son \u2013 is coming home later...' I began, automatically stammering out excuses, though secretly a little bit of me was rather tempted by the idea in a fascinated-by-a-snake kind of way.\n\n'Isn't he grown up? I don't suppose he needs his mum on the doorstep, waiting for him,' Ritch said, with another dangerously beguiling smile.\n\nHow did he know that? Did he know all about me?\n\nHe turned serious again, blue eyes concerned and sincere. 'But I'm sorry \u2013 you were very recently widowed \u2013 what was I thinking of? Of course you don't want to go to parties yet!'\n\n'No,' I agreed, having entirely forgotten about Tom until that point. 'And his cousin, Nick \u2013 you met him earlier \u2013 is coming back this evening too. He kindly offered to correct the manuscript of my next book, which needs to be posted off tomorrow, so it would be very rude of me not to be there. Sorry.'\n\n'Your book? Are you another novelist, like Polly Darke?'\n\n'Ah, yes, our Trollope of the North,' I said sourly. 'No, I write sort of autobiographical books with recipes. Do you know Polly?'\n\n'Met her around a few times,' he said vaguely. 'She had a fling with a friend of mine. Apparently she's pretty fit \u2013 does some weird yoga stuff. Muscles like knots on string, he said.'\n\n'Really? I've never seen her arms, she always wears long sleeves. I thought she might be on drugs or something.'\n\nHe shrugged, then his eyes flicked over me and he gave me a slow, sexy smile. 'I prefer natural, curvy women to skinny ones with monster boob jobs, every time.'\n\n'That must make you fairly unique,' I said tartly, and he grinned.\n\nThe front door was thrust open and a voice bellowed: 'Ritch, what the hell are you doing, you bastard? Are you coming or what?'\n\n'On my way out!' Ritch called back. Then he turned to me, 'Well, see you later, then. Be good, Flo.' And off he went.\n\nWas he actually flirting with me, Lizzy Pharamond, mature (even overripe) Middlemoss tomboy, or was he like that with all women? I could quite see why he'd disconcerted Annie, though, because I felt slightly and interestingly singed around the edges. The words 'moth' and 'flame' came vividly to mind.\n\nSizzle, sizzle.\n\nIf he comes across on the TV like that, no wonder Cotton Common's ratings have risen drastically since he joined the cast \u2013 he's trouble at t'mill!\n\nI spent almost an hour with Flo, who is a delightful dog and will fetch a thrown rubber ball indefinitely, though she seemed quite sanguine about swapping my company for a chewy bone when I left.\n\nHalfway home, while sauntering past the eerily quiet, goose-free green, I suddenly remembered about Nick and broke into a guilty run.\n\nHe'd obviously been at the cottage for some time, since he'd let himself in and there was an empty coffee cup on the kitchen table.\n\n'Oh, sorry, Nick \u2013 have you been here long?' I said, panting slightly. 'I was playing with the dog and didn't notice the time.'\n\nHe rose to his feet, heavy brows practically meeting across his impressive nose, and snapped, 'You look pretty smart for dog-sitting. Going somewhere?'\n\n'No,' I snapped right back, flushing. 'I don't spend all day, every day, in gardening clothes, you know!'\n\nHe took me in with his slaty, sardonic eyes, from gold sandals to waving, if now dishevelled, hair. 'You always did scrub up well. Hope your client appreciated it.'\n\n'How did you know Ritch was still there?' I gasped, startled, then felt myself going pink again.\n\n'You just told me!'\n\n'Well, he was, though on his way out. And don't call him a client in that tone of voice, like I was a hooker!'\n\n'Sorry!' he said, but didn't sound it. 'I'll be on my way. Didn't find too much wrong with your manuscript, except a bit of tailoring of the truth.'\n\n'I have to. Nobody would believe the real things that happen. They're much too incredible, and anyway, it would be too depressing. My misfortunes are supposed to be funny.'\n\n'Yes, your formula for success with your readers does seem to be a series of pratfalls linked with nursery-pudding suggestions,' he said unkindly.\n\n'You offered to read through it, I didn't make you!' I said indignantly.\n\n'I wanted to help.' He ran a hand through his black hair, which stood up on end. 'Look, seeing you're all gussied up, why don't you come out somewhere quiet for a drink with me? I'll even give you some pudding ideas for your new recipe book.'\n\nThe poet Wendy Cope puts it so well about men being like buses: there isn't one for ages and then two come along at once, flashing their signals. (Not that that necessarily means they're going to stop.) And I didn't even want to catch one!\n\n'That's kind of you, Nick, but I really don't feel like going out tonight and I need to make those alterations to the manuscript and pack it up. Anyway, we'd only argue like we usually do.'\n\n'Not necessarily, but please yourself,' he said, and walked out.\n\nI stared after him, which was well worthwhile, because his rear view was just as good, if not better, than Ritch's.\n\nI thought he was just trying to be kind to me in his way, but Nick's kindness moves in mysterious ways, its wonders to perform.\n\nIt was still just about light when Clive Potter cycled up for some tomatoes and to tell me that Adam (as played by a local farmer) had given himself a hernia while lifting bales, and had to drop out of the Mystery Play.\n\n'We'll have to audition for a new one, I'm afraid, Lizzy, unless anyone comes forward.'\n\nI had a sudden mental vision of playing my Eve to Ritch Rainford's Adam, but firmly suppressed it: an innocent in the Garden of Eden he certainly was not. More like the snake.\n\nHe displayed even more snake-like tendencies later, when he phoned from somewhere noisy to invite me to his house-warming party at Vicar's End on Friday night!\n\n'Just a few people \u2013 you could come to that, couldn't you?' he said persuasively, and I won't say I wasn't tempted for a minute, before common sense reasserted itself and I politely declined.\n\nIt got me thinking about Honeycomb Crunch all over again, though, and I decided to try using it in a variation of Eton Mess. I called it Cinder Cream and I thought I was on to a winner.\n\nWhile I worked away putting in the corrections to the manuscript that Nick had marked, I thought how odd it seemed in the cottage without Jasper there... but I supposed it was a foretaste of what was to come and I'd just have to get used to it.\n\n## Chapter 15: Drink Me\n\nI was up early, picking sound, firm apples for storing. I don't know the names of the varieties of all the old trees that were already here when we came, but by now do know by trial and error which will keep and which are best eaten or cooked straight away. I thought I might do baked apples filled with mincemeat and drizzled with cream for dessert that night, always a favourite.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nNext day Annie relayed a request from Ritch that I take Flo to the Mossedge canine beautician for nail clipping and a bit of pampering.\n\nShe'd been quite right about the Posh Pet-sitting taking off, because I was already down to look after two cats in a converted barn over at Mossrow for two days, plus a couple more one-off jobs. I'd be so busy that weeds would soon outnumber vegetables in my garden and the finished crops would remain uncleared.\n\nI managed to fit in a visit to the cats while Flo was being done, and then returned her to Vicar's End, where Ritch was flirting with his cleaning lady, Dora Tombs, whom he called 'Dorable'.\n\n'Get away with you!' was her standard response to each sally.\n\n'Morning, our Lizzy!' she said, as I went in (being a Naylor, and so distantly related). 'Keep that dog off my clean floor until it's dried \u2013 and the same goes for you,' she added, jabbing at Ritch's feet with her mop.\n\nRitch took a step back and gave me a lazy, glinting smile that took me in from top to bottom, lingering thoughtfully on the way.\n\n'Flo's clean as a whistle, Mrs T,' I assured her. Flo skittered and slid over to her bowl and started wolfing biscuits as though she was famished.\n\n'They said at Doggy Heaven that she was good as gold,' I told Ritch, adding severely, 'and you could have taken her yourself, if you're not going to work, and saved some money!'\n\n'Ah, but then I wouldn't have had two beautiful women at my beck and call, would I?'\n\n'Get away with you, you daft bugger!' Dora said. 'You're all mouth and trousers, you!'\n\n'Don't you ever work?' I asked, unimpressed.\n\n'Actually yes, and I'm on my way. Just waiting for the car.'\n\n'You don't drive?'\n\n'Lost my licence and I've got another twelve months before I get it back again,' he said ruefully, but I didn't feel sympathetic because I expect it was drink driving, which in my opinion is a criminally stupid thing to do.\n\n'Tough luck,' I said, but when he smiled at me I found myself smiling back. He's clearly as self-centred as most of the male race, besides being unable to resist flirting with any female who comes within range, but I must admit he is extremely attractive.\n\n'I could use a part-time chauffeur?' he suggested, raising a questioning eyebrow.\n\n'I've got enough to do. My garden will be an impenetrable jungle if I don't spend more time at home.'\n\nA horn beeped outside. 'Pity \u2013 and there's my car. See you later, girls!'\n\nAfter he'd gone, seeming to take the sun with him, Mrs T put the kettle on and we had tea, toast and gossip.\n\nShe's also Polly Darke's cleaner, which was fascinating: apparently she had a whole room devoted to some weird kind of yoga, and worked out in there twice a day.\n\n'Fit as a flea and strong as an ox,' Mrs T averred, crunching toast. 'Wouldn't think it to look at her, would you? And I'm that sorry about your troubles,' she added, which is as close as anyone's got to mentioning the goings-on at the funeral feast. 'I could tell sometimes she'd had a man in the house, but if I'd known who it was, I'd have told you.'\n\n'Oh, thanks,' I said. 'But it's all water under the bridge now, and I'm trying to move on and put it all behind me.'\n\n'That's right \u2013 and I'm sure Mr Nick will help you sort everything out. He's a proper man.'\n\n'A proper man as compared to what?' I asked curiously.\n\nShe gave me a Mona Lisa smile. 'Eva Gumball says he's divorcing that foreign woman and going to be living up there at the Hall most of the time, now. His granddad's that made up about it!'\n\nInformation in the Mosses travels as fast as thought. 'I'm glad for Uncle Roly's sake that Nick will be spending more time in Middlemoss, but I certainly don't need anyone to help me sort things out,' I said firmly.\n\n'That Polly Darke's turned a whole bedroom into a walk-in wardrobe, too,' Mrs T said, changing the subject back.\n\nOn Friday night I felt restless, especially after I'd popped into Delphine Lake's earlier to walk her dogs, and she'd said she was going to Ritch's party.\n\n'For cocktails at eight, dear. But us old ones will clear off early and then I expect it will go on until the small hours!'\n\nI don't actually like parties, except family ones, so I can't imagine why I felt left out... All right, perhaps I did, because my mind kept presenting me with scenarios involving Ritch that were quite unbecoming to a widow of such recent date.\n\nI had an unsettled night and then, when I went downstairs early next morning, I found Mimi fast asleep on the sofa in the sitting room.\n\nWhy do I even bother locking my front door when absolutely everyone seems to know where I hide the spare key?\n\nJuno, who was now allowed on her feet again, arrived in search almost immediately, limping gamely. 'I wish you'd stay in your bed at nights!' she scolded Mimi, who simply gazed blandly at her like a comfortable cat.\n\n'Stay to breakfast?' I invited. 'Jasper's getting up \u2013 he's going to the dig.' Thuds and yapping from above were evidence that Ginny was doing her best to help. The ceiling light swayed gently and small flakes of plaster drifted down, like the grey-white feathers Nick had been brushing off his sleeve when he came out of the small barn the other day...\n\nIf Caz hadn't fitted a padlock to his freezer, I'd have had a quick snoop by now!\n\n'No, thanks, we must get back. Mrs Gumball always cooks enough for twelve, and think of the waste!' Juno said, propelling the reluctant Mimi away.\n\nI didn't think Mimi would be terribly hungry anyway, because when I opened the fridge to get the milk, I discovered that half a bowl of experimental Cinder Cream had been eaten, and it's surprisingly filling.\n\n'Come up to the Hall later \u2013 around eleven!' Mimi said, clinging to the doorframe with both hands and smiling at me. 'Nick's invited us all to try out some ice cream he's making \u2013 yummy!'\n\n'He invited me, too?' I asked doubtfully.\n\n'Especially,' Mimi confirmed, still beaming but losing her grip on the gloss paint, and then was borne away until her cracked soprano singing, 'Hokey pokey, a penny a lump!' faded into the distance.\n\n'You've just missed Mimi and Juno,' I told Jasper when he finally came down. Ginny shot past my ankles and scattered the chickens in the yard, but unintentionally, I think. She probably couldn't see them for all the hair in her eyes.\n\n'I know, I heard. Mimi sounded happy.'\n\n'She mostly does. Oh, there's the phone.'\n\nI should have said, rather, 'where's' the phone, since I couldn't find it until I traced the long flex from the kitchen into the sitting room. Mimi seemed to have built a nest for it with all the cushions.\n\nBy the time I got to it, it had stopped ringing, but the caller had left a message: Ritch, sounding very gin-and-cigarettes gravelly. 'Lizzy? If you get this, come round and sort Flo out right away, will you? I'm feeling a bit rough this morning and she keeps yapping... I don't think Dora's coming until this afternoon... just let yourself in.'\n\nI could hear faint barking, and then Ritch groaned (rather sexily, it has to be said) and put the phone down.\n\nWell, he might at least have let the poor dog out, even if he did have a hangover! It would be nearly an hour until I could get there, since I wanted to drop Jasper off at the dig first, so by that time he would probably have given in and done it himself. And didn't he have to go to work every day? I know nothing about these things; perhaps they record the shows in batches or something? Or not on Saturdays?\n\nThe phone rang again while I was carrying it back into the kitchen, but it was just a man who had spotted the greenhouse last night on Freecycle and asked me for my phone number, wanting to arrange to come and look at it.\n\n'You're very popular this morning, Mum,' Jasper commented. 'And a bit pink,' he added, but I ignored that. I'd already let the hens out and fed them, collected the eggs, watered the garden and greenhouse, put a load of washing in the machine, made an especially nice packed lunch for Jasper and cooked bacon and eggs. Who wouldn't look flushed?\n\nWhen I cautiously let myself into Vicar's End, there was no sign of life other than a muffled barking from the kitchen.\n\nPoor Flo had been unable to keep all four legs crossed and left a puddle by the door, about which she seemed to feel apologetic, though it was not her fault, as I told her while I let her out before finding the mop and disinfectant and cleaning it up.\n\nThen I filled her bowl with fresh water and put a few crunchy dog biscuits down to keep her going for a while. I didn't know what Ritch wanted me to do, but I was quite sure he could afford the Posh Pet-sitter prices, so after that I took Flo for a nice long walk. It had rained in the night, so she wasn't such a clean, white and glossy creature on our return, though she was a very much happier one.\n\nI hadn't even started out clean and glossy, being back to gardening jeans and old T-shirts, Nick's remarks having rankled slightly.\n\nWhile I was still rubbing Flo with a tartan towel helpfully inscribed 'DOG' that I found hanging in the scullery next to her lead, Ritch wandered into the kitchen, obviously fresh from the shower, in gilt-edged designer stubble and a very short white towelling robe. Clearly he's a natural blond, because the hair on his legs was golden right up to the hem. He was carrying a glass beaker of straw-coloured liquid, which he set down on the counter.\n\n'Morning. I could do with a rub down too,' he said with a wicked if rueful smile. Then, opening the fridge, he bent over and rummaged around. I looked away hastily.\n\n'Thanks for coming,' he said, emerging with an opened carton of milk. 'Don't know what we were drinking last night \u2013 that's the trouble with cocktails, and after a couple you don't care any more \u2013 but today I feel like hell.' He picked up the glass beaker again. 'I'll just finish this, and then make some coffee: want some?'\n\n'What is it?' I asked cautiously.\n\nHe grinned. 'I meant, do you want some coffee! I don't think you'd want any of this, though I could be wrong \u2013 it's pee.'\n\nHe drained the last drops and put the glass in the dishwasher. Did he say pee? Eeeugh!\n\n'Er, no,' I said, backing away slightly. 'Did you say you were drinking...?'\n\n'My own urine? Yeah, every morning \u2013 everyone's doing it. It's good for you.'\n\nAfter last night I should think his pee was at least forty per cent proof. 'I... hadn't heard about that,' I said, wondering if he was quite mad. 'How interesting!'\n\nHe gave me a wicked smile, but it wasn't working any more. 'It cures anything. That and frequent sex are all a man needs to keep healthy.'\n\n'Really?' I felt as if some miraculously attractive bubble had burst and taken all the rainbows with it, but managed with an effort to gather my wits together: 'I've taken Flo for a walk and changed her water, so she's OK. I'd better go now \u2013 I've got things to do.'\n\n'Sure you can't stay awhile?' He switched on one of those espresso machines that look as if you need a whole generator and a degree in engineering to make them work.\n\n'No, really.' I wasn't drinking any more coffee out of any of his cups, now I knew about his habits.\n\n'Shall I settle up with you now, or do you want to send me a bill?'\n\n'Oh, Annie will send you one at the end of the month, if that's OK? I put it all down on her chart and she does the bookwork. Bye, Flo.'\n\nI bent to fondle her smooth, velvety head and, when I rose to go, Ritch followed close behind me up the hall and reached out a long arm to open the front door, brushing casually against me as he did so.\n\n'Oh \u2013 thanks,' I said, unnerved by the proximity of all that naked male flesh, and shuffled past into the sunlight just as Annie, towed by four large hounds, was passing the end of the drive. Unable to wave, she began to smile, then caught sight of Ritch lounging in the doorway in his mini-robe. The smile wavered, she went pink and hurried on.\n\nI dashed after her, calling out: 'Hey, Annie, wait for me!'\n\nShe turned reluctantly. 'Lizzy!'\n\n'That was not what it looked like,' I said severely. 'Honestly! You should know me better by now! He phoned me this morning and asked me to go and sort Flo out, because he had a hangover.'\n\nActually, in that bathrobe it was almost a hang under, so it's just as well it was only Annie who spotted us, because she's probably the only one who would have believed me. I took two of the dog leads.\n\n'He simply couldn't be bothered to let Flo out this morning, which is terribly selfish, and has put me right off him.'\n\n'I should think so, too,' she said indignantly. 'The dog must have been desperate!'\n\n'She'd made a puddle, but as close to the back door as she could, poor thing. But speaking of pee, Annie, you'll never believe what Ritch does with his!'\n\nAnd I was right: she didn't believe me and insisted he must have been joking, though I'm sure he wasn't. It's not a fad likely to catch on in Middlemoss.\n\n'So, how did you and Gareth get on the other night?' I asked, changing the subject. 'You never really said.'\n\nShe blushed under her freckles. 'Fine... he is so nice. But he can't cook, so it was just reheated ready-meals. I told him about the French cookery course we did after we left school and I'm going to get him a slow cooker like mine and show him how to use it.'\n\n'You should invite him back to dinner at your place, only early enough so he can help you cook it,' I suggested. Cooking together is, I think, a very intimate thing to do.\n\n'That's a good idea. Something simple but nice, like that chicken in white wine thing you do, or a risotto.'\n\n'And a stodgy pud. Bet he likes those \u2013 most men do.' Even Nick, though he pretends he doesn't, just to wind me up.\n\n'Yes, he'd bought a chocolate gateau,' she agreed, 'and he ate quite a bit of it. Oh, well, must go and take these dogs back. I'll put the extra Ritch pet-sitting on the chart for you when I get home... and Lizzy,' she added anxiously, 'you aren't falling for him, are you?'\n\n'No, though he's very attractive \u2013 or was, until I got grossed out by his habits! But even were I looking for another man, one who thinks pee and hot, casual sex will cure anything is obviously operating on a different wavelength from mine.'\n\n'Gosh, yes!' she agreed, innocent blue-grey eyes open wide. Ritch is not the only one in Middlemoss operating on a different wavelength from me, but I love her anyway.\n\n'I've served my time in the prison of love, though I might get another dog later, once Jasper's taken Ginny off to university with him.'\n\n'Tell me when, and I'll find you a nice stray,' she promised, beaming. 'Is Ginny going to university with Jasper?'\n\n'Yes, he's persuaded the landlady of a student house to let him keep her with him, but don't ask me how.'\n\n'Oh, he can be just as charming as Tom was, when he wants to be,' she said. 'Only of course, he is much more solid, reliable and kind.'\n\nThe church clock struck eleven, galvanising my memory. 'Oh, must fly, Annie! I'm supposed to be up at the Hall tasting some ice cream Nick's made, and Mimi said he invited me especially, so he'll be cross if I don't go.'\n\nThrusting the dog leads back into her hands, I rushed off.\n\nIn fact, Nick seemed totally surprised, but not displeased, to see me. He was wearing a blue-striped apron, a smudge of sugar and a streak of raspberry, and looked good enough to eat... if you liked that kind of thing, of course.\n\nI looked at Mimi suspiciously and she waved her spoon at me and called gaily, 'Just in time!'\n\n'Hello, my dear,' Unks said. 'I didn't know you were coming. This is going to be a treat, isn't it?'\n\nHe, Mimi, Juno, Mrs Gumball and even Caz Naylor, half-concealed by the shadow of the inglenook, were all sitting round the kitchen table, spoons poised.\n\n'Here, have mine, Lizzy, and I'll get some more,' Nick said, and I took the proffered bowl and sat down, looking at it dubiously. The ice cream was sort of grey, like town snow turned to slush, and the blood-red raspberry coulis swirling over it contrasted strangely.\n\n'It tastes better than it looks,' Mimi remarked. 'I love liquorice! Yum!'\n\nShe was right. Nick sat down again next to me, long legs brushing mine. 'What do you think?'\n\n'It looks horrible in a sophisticated sort of way, but tastes great.' I turned to see what Caz was making of it, but he'd quietly stolen away, leaving only an empty dish behind, which was tribute enough, I suppose. 'It's the opposite of coffee granita, which I always expect to be delicious, but never quite comes up to expectations.'\n\n'Oh? I'll have to see what I can do about that.' His eyes gleamed.\n\n'Nothing I haven't already tried!' I snapped.\n\n'You want to take a bet on that?'\n\nI might have been tempted to rub ice cream into that superior smile, if I hadn't already eaten it all.\n\nMrs Gumball was still daintily spooning hers in. 'What that boy will think of next!' she said, shaking her head so that all the silvery-grey curls, tied up on top of her head in a skittish whale spout effect, quivered.\n\n'Great,' Juno said, laying down her spoon. 'Mimi, don't lick the bowl!'\n\n'Why not, when we're just family?' she demanded indignantly.\n\n'It's still not polite.'\n\n'Roly eats roast duck with his fingers and then licks them.'\n\n'Would you like to go for a drive?' Juno asked, in an attempted diversion. 'I think my leg's up to it now, if we don't go too far.'\n\nMimi clapped her hands. 'Martin Mere to feed the ducks!'\n\n'Oh good, good,' Unks said. 'Bit of fresh air will do you both good.' He got up. 'Must go and study the form a bit \u2013 got a horse racing on Saturday. Snowy Sunday.'\n\n'Not in September, surely?' I said, puzzled.\n\n'Name of the horse. Snowy Sunday out of Weekend Blizzard.'\n\nUnks has shares in three racehorses, but they usually seem to fall over, or go backwards, or do something that doesn't involve getting past the post.\n\n'Cold lunch in the dining room at one,' Mrs Gumball said, heaving herself to her feet. 'I'm just off to see to my Joe's dinner. Mind my kitchen's clean and tidy again when I get back, Nick Pharamond!'\n\n'Don't I always clear up after my cooking?' he demanded indignantly.\n\n'I'll load the dishwasher myself,' I promised her.\n\nOne by one they went, and Nick and I quickly sorted out the kitchen in fairly amicable silence.\n\n'That's that,' I said finally, looking round to see if we'd missed anything. Nick is a very messy cook and it was surprising how many pots and pans he had used just to produce ice cream and sauce. 'I'll have to go, I've got someone coming to look at the greenhouse and he said around lunchtime.'\n\n'You have? Someone you know?'\n\n'No, a stranger who saw the ad on the Freecycle website.'\n\nHe frowned. 'I'd better come and deal with him for you. You should get me to do this sort of thing. Anyone might turn up on your doorstep.'\n\n'I can handle it! I mean, I'm only selling an old greenhouse, not the crown jewels,' I said firmly. I'm used to doing everything myself, so why would I suddenly need a man to do it for me?\n\n'I'll walk down with you anyway, just make sure\u2014' he began to insist, as though I were some frail little flower; then his BlackBerry went off and while he answered it I slid quietly away home.\n\nThe man was waiting for me in the yard outside the cottage, leaning against an old pick-up truck and smoking a roll-up, and I instantly rather regretted refusing Nick's offer so hastily, because there was just something about him I didn't like, even apart from the aroma of stale alcohol.\n\n'Had a bit of trouble finding you,' he said, straightening with a leer and running bloodshot eyes over me as though I were a dubious filly. 'Lonely down here, isn't it?'\n\n'Not really, people go past on the main road all the time and my family live up the drive,' I said briskly. 'So, Mr...'\n\n'Roach,' he slurred.\n\n'So, you're interested in having the greenhouse?'\n\n'Well I was, but I took the liberty of having a look at it while I was waiting, and it's in pretty poor condition.'\n\n'The supports are fine and it should dismantle easily. Anyway, what were you expecting for nothing?'\n\n'It'll cost you a fortune to pay someone to take it away for you,' he said. 'But I don't suppose a lady like you would know about that. Recent widow, aren't you? Bet you miss having a man about the place.'\n\nHe came a bit closer, flicking his cigarette onto the cobbles.\n\n'Look, are you interested or not?' I demanded, ignoring the innuendo but backing off slightly. The yard brush was leaning against the wall behind me and I reckoned that if desperate, I could always beat him into submission with it: it was a good, sturdy one.\n\n'Perhaps I might be,' he said, with what was definitely a leer. 'Maybe you'd like to discuss it somewhere more comfortable? I wouldn't say no if you invited me in.'\n\n'Oh, Nick!' I exclaimed thankfully, as his tall, broad-shouldered figure appeared round the end of the barn. He looked from one to the other of us from under dark brows and I took his arm and squeezed it meaningfully. 'This is Mr Roach. He came to look at the greenhouse.'\n\n'Loach,' the man corrected me, backing off warily. 'But I've had second thoughts. It's not quite what I wanted. Doubt it's what anyone wants,' he added. 'Well, see you!'\n\nHe climbed back into the cab of his pick-up and drove off with a bit of unnecessarily macho revving and tyre screeching.\n\nNick looked at me with one raised eyebrow, but nobly refrained from saying, 'I told you so.'\n\nRealising I was still clinging to his arm, I hastily let go and moved away.\n\n'Do you want to come in?' I asked. 'Or go back for lunch? I'm going to have ham and split-pea soup and home-made bread.'\n\n'I'll stay, but only if you promise to let me handle anything else you want to get rid of. I'm sure Roly will back me on this one \u2013 and the cottage is part of the estate.'\n\n'Oh, all right!' I agreed. 'But don't think I couldn't have handled that horrible man without you, because I could! I've already had much nastier people chasing me up for money they say Tom owed them, when they don't seem to have the least bit of proof.'\n\n'You haven't paid them, have you?'\n\n'Despite what you think, I'm not entirely stupid,' I said with dignity. 'I asked Unks' advice, and he told me to pass them all on to his solicitor, Smithers, and he would deal with them for me and let me know which were the genuine ones that had to be settled.'\n\n'I don't think you're stupid,' he said, following me into the cottage. 'You just take independence a little too far sometimes, that's all.'\n\n'Did you come down just to lecture me, or did you want something?' I asked pointedly.\n\nHe smiled innocently and said, 'I just wanted to know what recipe you'd already tried for the coffee granita?'\n\n'One you sent me from Italy on a postcard of the Leaning Tower of Pisa,' I said with satisfaction. I hauled out the postcard album to look for it and he seemed amazed that I'd not only kept them all, but also carefully put them into a book. However, show me anyone interested in cooking who wouldn't have hoarded the recipes.\n\nWe ended up looking through the album for almost a whole hour without arguing, which must be a record. Maybe he was mellowing.\n\n'I'll see you at the first Mystery Play rehearsal on Tuesday night,' were his surprise parting words before striding off whistling, the sun glinting on his blue-black hair.\n\nWhat could he mean? Had he volunteered to help Clive?\n\nRitch phoned and invited me for a quiet drink at the caf\u00e9-bar in the former Pharamond's Biscuit factory on Tuesday night, so he could 'get to know me better'! He said he was tied up until then, which gave me a bad moment because I remembered that note from Polly Darke I found in Tom's pocket just before he died...\n\nI told him I couldn't go, since I had a Mystery Play meeting and the cast tended to go to the village pub afterwards, but he said that was OK, he would meet me and come along too, and I simply couldn't think of a way of telling him not to on the spot.\n\nI looked long and hard in the mirror and was at a loss to understand why, unless he was currently desperately short of another woman for his frequent, healthy sex rota.\n\nHe probably wouldn't bother after all, but if he did, I'd just have to hide behind the crowd and hope avid Cotton Common fans mobbed him, which seemed quite likely.\n\nI tried out two varieties of coffee granita on Jasper, later, but although he said they were delicious I still felt they lacked that little extra something...\n\nOn the other hand, some mincemeat fudge turned out really well.\n\n## Chapter 16: Unrehearsed Entrances\n\nI tried out my mincemeat fudge on the members of the Christmas Pudding Circle yesterday and it went down a treat, as did Marian's variation on gingerbread biscuits for hanging on the Christmas tree. She had cut out an inner star and placed a crushed boiled sweet in the middle, which melted during the baking process to form a stained-glass effect. I used to do something similar for Jasper when he was little \u2013 traffic light biscuits. We are all busy cooking for the annual Autumn F\u00eate next Saturday too, where there will be much friendly rivalry for the various prizes. Annie has asked me to make some bags of candyfloss and will collect them on Friday...\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI was a little late setting off for the first Mystery Play rehearsal on the Tuesday evening, because I'd spent the afternoon making a start on my Christmas cake \u2013 and not just my own, but the six small ones that each member of the CPC makes for the local Senior Citizens Christmas Hampers, to be distributed by Marian and the rest of the Mosses Women's Institute.\n\nI like to soak the fruit in dark rum for about five days \u2013 no faffing around pouring alcohol into holes in the base for me! \u2013 so, after an afternoon of chopping and mixing, I had three enormous covered bowls sitting in the larder gently marinating, and an aching arm.\n\nWhen I got to the village hall the actors were standing about in groups, chatting. Acts 1 to 9 of the Mystery Play rehearsal would be supervised by me, the vicar, Miss Pym and Marian, whose husband, Clive, was, as usual, overall director. Roly, Voice of God in perpetuity, never put in an appearance until the final performance, since having played the role for so many years he could do it in his sleep. On the actual day he sits in a corner of the courtyard in a little striped canvas pavilion like something from a jousting field, well wrapped up and with a warm brazier, and speaks his lines into a mike connected to the speaker just inside the barn door.\n\nClive was putting cardboard signs up in the parts of the room where the different acts were to gather and I found Marian filling the boiler in the kitchen area behind the hatch, ready for its long, slow journey towards the tea break.\n\nI was just handing her my offering of treacle flapjacks, to go with what was left of her experimental gingerbread biscuits, when Nick loomed up beside me.\n\n'Hi, Lizzy: should I have brought something to eat, too?'\n\nI nearly dropped the box: he moved disconcertingly quietly for such a big man. 'Oh, hello Nick!'\n\n'No, that's all right. These of Lizzy's are just extras, because I always bring a tin of Teatime Assortment with me,' Marian said, 'though the current one's down to them pink wafer things, which no one seems to go for. But nice of you to offer, Nick \u2013 and that lamb recipe of yours in the Sunday supplement was a right cracker! I'd never have thought of doing that with olives.'\n\n'Oh, thanks, Mrs Potter,' Nick said modestly with one of his most charming smiles, and she blushed. I was quite sure he knew the devastating effect these had, like silvery sunshine breaking out from behind lowering, purplish storm clouds.\n\nClive, clipboard in hand, bustled up. 'Ah, Nick, you've told Lizzy you've volunteered?' he beamed.\n\n'To help out?' I asked. 'Continuity man? Props?'\n\n'To be Adam to your Eve,' Nick told me blandly. 'Delving while you spin. Giving in to your temptations. Eating the apple from your hand.' He looked down at me quite seriously, though one eyebrow was quizzically raised. 'Passing you the fig leaves.'\n\n'You?'\n\n'I'm sure we're all delighted,' Clive said. 'I hadn't thought to ask him before, because he's been here so infrequently in recent years, but now he'll be able to make most of the rehearsals, so that's all right. And it's only one fairly short scene, isn't it?'\n\n'Er \u2013 yes,' I agreed, still taken aback.\n\n'If I do miss any rehearsals, Lizzy can put me through my paces at home,' Nick suggested. 'I'll be based up at the Hall from now on.'\n\n'I'm sure we're all very glad to hear that,' Marian said warmly.\n\n'Oh, there's Annie,' I exclaimed, spotting her walking in. 'I thought she'd be going to the other rehearsals on Thursdays?'\n\n'She's very kindly going to assist the vicar, too, since he hasn't done this before,' Clive said, then clapped his hands and announced into the resultant silence, 'To your places, please!'\n\nEveryone separated to his or her group. Mine was small, since I was in charge of only the Fall of Satan, the Creation and my own scene in the Garden of Eden.\n\nWe began with the Fall of Satan, which is basically just the Voice of God (which I read, in Roly's absence) and Lucifer, plus nine entirely silent angels.\n\nAs always, God has the last word: 'I am reet disappointed in thee, Lucifer, thou art too sharp for thine own good. I loved thee like a son, yet thou art naught but a foul fiend, fit only for t'pit of damnation. Get thee gone!'\n\nAfter this, Lucifer vanishes in a puff of yellow smoke, signifying the sulphurous fumes of Hell, with a receding wail \u2013 or in some years a shriek, depending on the interpretation of the role. At any rate, he vanishes. We ran through it a couple of times, then I released the angels to go over to the vicar's corner, where they were required during the Nativity to join with the shepherds and Three Wise Men in a stirring rendition of 'Silent Neet, 'Oly Neet' around the manger.\n\nThe Creation is a monologue by the Voice of God, with sound effects off, and after I'd read through that I handed over the directing to Lucifer while I put Nick through his paces as Adam. He refused a script and had evidently been studying the video of last year's Mystery, because he was word perfect and you couldn't fault his Lancashire accent.\n\nIt went smoothly right up to the point where he asked, 'What are thou eating, Eve?'\n\nI replied: ''Tis a fruit the snake told me were reet tasty \u2013 and see, yonder birds peck at it and come to no ill. Dost thou want a bite, flower?'\n\nI offered him an imaginary apple and he said warmly, 'From you, darling, anything!'\n\n'That's not in the script,' Lucifer objected.\n\n'I thought we could change the script if we liked?' Nick said innocently.\n\n'Only if it's an improvement \u2013 this is serious,' I said severely. 'Stop messing about.'\n\n'I was serious,' he protested, and Lucifer grinned. I was just glad I'd sent the angels away, reducing the audience by nine in one stroke.\n\n'And don't think me coy, but on the night, how do we preserve our modesty, Lizzy? I've forgotten.'\n\n'In my case, with a very long wig and a bodystocking. The last Adam wore a pair of beige swimming trunks and carried a strategically placed leather bucket. Apparently, in the old days Adam and Eve used to speak their lines standing behind boards painted to look like bushes.'\n\n'I'll see what I can do,' he said gravely. 'Is it a big bucket?'\n\nI gave him a stern look: we seemed to be rapidly reverting to the innuendo and sparring of our teenage years, and this was a Nick I'd pretty well forgotten ever existed. Perhaps the euphoria of pending divorce had brought it out in him again?\n\n'Thou hath tasted t'fruit of knowledge that wor forbidden thee and found it sweet \u2013 yet shall it be bitter henceforth on thy tongue!' read Lucifer, in his temporary role as Voice of God, though still grinning. We stopped for refreshments after that, before I ran everyone through their lines again: I would work on the movements and check the props and costumes during later rehearsals.\n\n'That'll do for tonight,' I said finally, and went to see how Annie and Gareth were getting on with the Nativity cycle.\n\nDave Naylor from the garage, as Joseph, was wheeling Mary to Bethlehem on the back of an old-fashioned butcher's boy bike. On the night of the performance a star lantern is hauled slowly across the stage on a wire, a very pretty effect.\n\n'Not far to go now, luv. Bethlehem's on t'horizon, ower yonder.'\n\nMary, who was inspecting her fuchsia-painted fingernails, replied absently, 'The sooner the better, chuck, for my time's close \u2013 but will we find a place t'lay our heads?'\n\n'I'll find thee a roof ower thy head this night, flower,' Joseph promised, 'no matter what, don't thee worry thi'sen.'\n\n'So,' the vicar said to Annie, 'the bike represents the donkey. But there is a real donkey on the night, is there?'\n\n'Not any more. We stick to the bike for the performance, too.'\n\nGareth looked baffled.\n\n'We used to have a donkey,' I chipped in, 'but it was more trouble than it was worth, and when it died of old age someone suggested the bike. The rack on the front is really handy for carrying baby Jesus on the flight into Egypt later, too.'\n\n'Er... yes,' he agreed. 'I suppose it would be.'\n\nI looked critically at Mary, who worked at the hairdresser's in Mossedge. 'I hope Kylie's going to lose the false fingernails and not chew gum on the actual night.'\n\n'Yes, she's really taking it very seriously,' Annie assured me. 'She said she was going to put that cushion up her T-shirt tonight even though it's not a costume rehearsal, so she'd get the right posture for sitting on the bike.'\n\n'That's the spirit,' Nick said, having followed me over.\n\nClive clapped his hands again, thanked everyone for coming and said he would see them every Tuesday until Christmas.\n\n'Now we all go and unwind in the pub,' Annie informed Gareth.\n\n'Oh, do we?' He was looking a little dazed, as well he might, but his eyes when they rested on Annie were almost dog-like in their devotion, which was bound to appeal to her.\n\n'Where's Jasper tonight?' Nick asked me as we left.\n\n'At home. One of his friends is staying over and I left them pizza ready to heat up, and some strawberries dipped in chocolate as a treat.'\n\n'You're a good \u2013 if weird \u2013 mother,' he commented, as we all trooped out of the door onto the green, where Ritch Rainford stood leaning against a tree, smoking.\n\nMarlboro man.\n\nUntil that moment I'd entirely forgotten what he'd said about turning up \u2013 not that I'd really thought he'd meant it in the first place. I stopped dead and Nick practically fell over me.\n\nRitch ground the cigarette out under his heel and walked over rather beautifully, as if the cameras were on him. 'Hi Lizzy, good rehearsal?'\n\nHe glanced around at the others, smiling generally, like warm sunshine. 'Lizzy said she thought you wouldn't mind if I tagged on tonight to the pub?'\n\n'No, I didn't!' I muttered half under my breath, though clearly Nick thought it was an assignation because he gave me a dirty look. I couldn't see what it'd got to do with him anyway, though, even if I had been making assignations with Ritch, except, I suppose, that it was a bit early for his cousin's widow to be involved with another man.\n\nBut I wasn't involved, and I was sure Ritch couldn't be seriously interested in me, so perhaps he just wanted to get out and meet the locals.\n\n'I'm Ritch Rainford, you know,' he told everyone. 'I've just moved into the village.'\n\nFortunately, most of them did know who he was and swept him along with us. Some of the locals were certainly glad to meet him: Kylie was hanging on to his every word, and she's terribly pretty despite two nose rings and shocking pink hair, so I might well be worrying needlessly.\n\nAt the pub, which was rather full, I managed to slide onto a bench seat next to Annie. She had Gareth on her other side and he was looking about him as if he'd never been in such a place before, which for all I know he hadn't, an innocent abroad. But he was in safe hands.\n\nRitch's arrival had caused a minor sensation and, even if he'd wanted to sit by me, which I dare say he didn't, by the time he'd disentangled himself from his admirers Nick had got there first.\n\nHe rang the bell on the wall behind my head. 'This has got to be one of the few pubs left where you can ring for service and have your drinks brought over,' he said with satisfaction. 'That way, you don't lose your seat when you get up.'\n\nRitch pulled up a chair opposite and, leaning over, said in his warm, intimate, liquid-chocolate voice, 'So... rehearsals go well, Lizzy? What part are you playing?'\n\n'Eve, and yes, we made a good start.'\n\n'And Nick is playing Adam, Annie tells me, now the original actor has had to drop out,' Gareth said. 'Keeping it in the Pharamond family!' He smiled around genially.\n\n'Really? I wish I'd known about it because I'd have loved to have played Adam to Lizzy's Eve!' Ritch said. 'She could tempt me to anything!'\n\n'I'm sure you'd have been brilliant, but the actors need to be local people, because of the commitment,' Annie explained diplomatically. 'I expect you're much too busy to give up all that time!'\n\nWhen our drinks came, the barmaid also apologetically handed me a folded paper. 'Landlord says he's sorry to bother you, Mrs Pharamond, but if you could see your way to settling up this bill, he'd be very grateful.'\n\nI unfolded it and glanced at the total, which was pretty huge. Nick leaned over and took it from my hand, though I tried to snatch it back. 'Tom's bar tab? I'll take it back with me to add to the rest, for Roly to settle.'\n\n'I wish you'd give it back! I don't see why Roly should have to settle all Tom's bills \u2013 and for goodness' sake, how many more are there?' I added despairingly.\n\n'Oh, I should think that's about the last,' he said, tucking it into his jeans pocket, from where I certainly wasn't going to try to retrieve it.\n\nI glowered at him, but he smiled blandly and turned to talk to Marian Potter. Kylie had cornered Ritch's attention again \u2013 he was either giving her his autograph or his phone number, or possibly both. On my other side, Annie and Gareth were totally engrossed in their own conversation.\n\nThe noise level in the packed bar was now so high it was hard to hear what anyone who wasn't right next to you was saying, though I nodded and smiled at friends who seemed to be mouthing in my direction. After a while, feeling I'd had enough, I nudged Nick in the ribs with my elbow.\n\nHe grunted and turned.\n\n'Do you think you could let me out? I'd like to go home.'\n\nRitch caught my eye and, leaning forward until his golden head practically touched mine, said, 'Are you going, Lizzy? I'll walk you home.'\n\n'No need,' Nick said, draining his pint and slamming down the empty tankard, 'I'm going the same way.'\n\n'But I'd like to,' Ritch said stubbornly, half-rising to his feet.\n\n'Do stay, Ritch!' I said hastily. 'Nick has to walk right past my cottage to get to the Hall anyway, so we might as well go together.'\n\nBetter the devil you know, after all.\n\nRitch looked at me, eyes bluer than cornflowers and, I'm sure, as sincere as a quagmire. 'Right... well, I'll ring you, then.'\n\n'Ring you about what?' Nick demanded as soon as we were outside in the blessedly cool, quiet evening.\n\n'Pet-sitting Flo, his bull terrier \u2013 and I'm so sorry to drag you away from your long and clearly engrossing conversation with Marian.'\n\n'I wanted the secret ingredient in her Lancashire hotpot recipe,' he said simply.\n\n'Oh.'\n\nThere was a silence as we walked up the quiet village street and along the lane. Then, as we turned through the stone gateposts of the Hall, I said, following my somewhat tortuous train of thought, 'What do you think about drinking your own pee?'\n\nHe stopped dead, though it was a bit too dark to make out his expression. 'What? You do say the damnedest things! Do you mean like on the Bounty, when they ran out of water and there was nothing else? At least, I think it was the Bounty.'\n\n'No, it's some health thing \u2013 supposed to be good for you.'\n\n'I do vaguely remember reading about it, now I come to think about it,' he said slowly, 'but I don't think it's likely to take off, Lizzy! You aren't thinking of trying it, are you?'\n\n'God no!' I shuddered. 'Horrible thought.'\n\n'Why are we talking about it, then?' he asked reasonably.\n\n'I know someone who does it and I just wondered. It seems very odd to me.'\n\n'It seems very odd to me, too,' he agreed. 'Lizzy\u2014'\n\n'What?'\n\n'Nothing!' he snapped abruptly and, abandoning me outside the cottage, turned and strode off into the darkness, back towards the drive.\n\n'Hi, Mum, nice night?' Jasper asked as I staggered wearily in. He was collecting Cokes and an indigestible assortment of snacks out of the fridge.\n\n'Don't ask!'\n\nHe grinned. 'We're playing computer games in my room. OK if we drop Stu home on the way to the dig in the morning? It's not far out of the way.'\n\n'If you can get him up that early,' I agreed. 'And me. I'm off to bed, I'm shattered!'\n\nI did look for the chocolate strawberries first, though, but they'd all gone. I ate a slice of cold cheese and onion pizza instead, followed by a frozen banana dipped in maple syrup and chopped nuts.\n\nIt's no wonder I had bad dreams.\n\n## Chapter 17: Tart\n\nMost villages have a summer f\u00eate but in Middlemoss, for reasons long forgotten, we have ours in mid-September. Despite this, the weather is always Indian summer good, which is generally thought to be a reward from on high for performing our local Mystery Play so faithfully every year.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nAnnie and the vicar were also helping with the second Mystery Play rehearsal and I popped down under the pretext of dropping off the bags of candyfloss she'd asked me to make for the fete, though really I was simply curious to see how the other scenes were going.\n\nI arrived just as Satan was tempting Jesus. 'Si'thee, Jesus, tha's been biding here in t'wilderness some time now,' he said. 'Art though not famished? Would thee not like a bite of Hovis and a sup of brew?'\n\n'Nay, gi'ower \u2013 there's nowt thou can tempt me wi', for thy beautiful face hides a foul heart,' responded Dave Naylor's son Gary, bringing an interesting touch of Goth gloom to the part. 'Man can't live b'bread alone and I want none of thy offerings \u2013 get thee gone!'\n\n'I don't remember anyone mentioning Hovis before,' I commented to Annie, handing her the bags of candyfloss.\n\n'No, I think that's a touch of Gary's own,' she agreed. 'I only hope he doesn't turn the water into Alcopop during the Miracles!'\n\nThen the actors were called for the Resurrection and I left before Marian could rope me in for anything.\n\nMost of the indigent population spent the week before the f\u00eate polishing their marrows, arranging flowers, baking, or labelling jars of preserves and pickles, for the various prizes were always strongly contested. Annie and I were no exceptions: her cheese scones always swept the board and her fruit tea bread was legendary.\n\nI enter only a few categories (but usually win them), among them Best Plate Apple Pie and Best Middlemoss Marchpane, the latter being a local delicacy consisting of an open mincemeat tart in an almond pastry case, the top water-iced and criss-crossed with marzipan strips. Yummy.\n\nAs expected, on Saturday morning the warm sun shone down on the small field next to the village hall where two marquees had been erected. They were the same ones used to keep the local sheep dry during lambing and they are also hired out for wedding receptions, so we've all come to associate the smell of damp wool with celebrations.\n\nThe f\u00eate is always run with military precision by Marian and Clive, who I'm sure would both spontaneously combust from sheer, pent-up excess energy if they did not have the entire management of every activity that goes on in the Mosses.\n\nOne or two of the minor celebrities now living amongst us had probably half-expected to be asked to open the f\u00eate, but Mimi always does it, having assumed the role as of right on the death of Unks' wife years ago. It would certainly be more than the Potters' lives were worth to attempt to change something so fixed in her head.\n\nWearing a drab cotton safari skirt suit sporting many pockets, slightly muddy Gertrude Jekyll boots and incongruously lacy cotton gloves, Mimi briskly admonished the gathered throng to spend lots of money, because the church roof had seen better days and, with no more ado, declared the f\u00eate open.\n\nThen she leaped from the podium with surprising agility and trotted over to the plant stall, which she appeared to think was some kind of free lucky dip for gardeners, snatching up anything that took her fancy and leaving poor Juno to pay for (and carry) it all, like a royal lady-in-waiting.\n\nTed, the old gardener who helps her up at the Hall, hovered at her elbow, offering unwanted advice in an agonised bleat: 'You don't want that, Miss Mimi! That won't do well, Miss Mimi, not in our soil it won't \u2013 and who's going t'plant all of 'em, that's wor I'd like t'know?'\n\nHe gloomed away unheeded until Juno suggested he help carry everything to the Daimler, at which point he switched to the subject of his bad back and melted away into the crowd, so I gave her a hand instead.\n\nI thought I might as well, since I felt at rather a loose end: Jasper was at his dig, which meant that this was the first year I'd come here alone, and it felt odd... but it was something I was quickly going to have to get used to. In fact, he'd probably just been good-naturedly humouring me the last year or two by coming with me!\n\nThe sudden realisation that life wasn't ever going to be the Lizzy and Jasper Show again was almost unbearably poignant. How thin and near-transparent the folds of time are! I could almost step through them into another dimension, where the child Jasper would put his hot, sticky small hand in mine, dragging me towards the swingboats, or to ride his favourite giraffe on the little roundabout... I could see him excitedly dipping into the bran tub, or clutching some cheap, hideous, furry toy I'd spent pounds winning for him.\n\nI'm a sentimental idiot: even the smell of the toffee apples made the tears come to my eyes.\n\nWhen I'd helped load Mimi's haul into the Daimler I did a stint behind the counter of the hoopla stall, where the bags of candyfloss I'd made were hung up as prizes. (I'd given the borrowed candyfloss maker back to Annie, because it was such fun I'd splashed out on one of my own.)\n\nAnnie usually got roped into organising the children's races, where she was very popular, since she could always be guaranteed to have pockets bulging with little prizes for any disconsolate losers \u2013 sherbet dabs, jelly worms and flying saucers.\n\nI had quite a good view of the goings-on from the hoopla stall, but there was no sign of Nick, who is so tall that his dark head can usually be spotted above any crowd. And Ritch must have been working, for once, for otherwise I was sure he wouldn't have been able to resist treating the peasantry to the sight of his golden magnificence.\n\nBut Polly, brazen as ever, was there with a group of rowdy friends, though I noticed she avoided me \u2013 and Caz, when he emerged from the shadows of the beer tent to perform his part, with silent and ferocious concentration, among the morris dancers.\n\nThe Mosses morris team eschew the traditional white clothes, hats and streamers in favour of all-black outfits, including leather waistcoats, and you really wouldn't want to try to tie a bell on any of them.\n\nAfter this, Caz applied his skills to the coconut shy, awarding the resultant pink teddy bear to Ophelia, who seemed to be constantly near him while looking as if she didn't know quite why. She also suddenly appeared very pregnant, even under the bunny smock.\n\nI noticed the way Annie and the vicar gravitated together between their various duties like those magnetic ladybirds \u2013 so it must be love, love, love! But if so, it's a strange, old-fashioned, Jane Austen-ish love, with no declarations or physical contact whatsoever.\n\nWhile I'm sure they think their passion is a big secret (and it certainly seems to be a secret from each other), I expect the whole parish is indulgently watching the progress of their romance with almost the same avidity they confer on the twice-weekly episodes of Cotton Common.\n\nAnnie's cookery lessons now seemed to take place most evenings when Gareth wasn't otherwise engaged, and they'd been seen together walking rescue dogs up near the RSPCA kennels. But they had not been observed holding hands, or engaging in any other lover-like activities.\n\nIn the absence of her parents, I could see I'd soon have to ask him if his intentions were honourable, or this state of affairs could go on indefinitely. And if they married, that would be yet another change: for though of course Annie and I would always be best friends, since she had fallen in love with Gareth I already saw much less of her than I used to.\n\nEver felt totally isolated in a crowd? I was just wallowing in a murky trough of self-pity when I was distracted by spotting something that made me doubt the evidence of my own eyes. So when Jojo and Mick came to try their hands at the hoopla (they were useless, but I gave them a bag of candyfloss each anyway), I said curiously, 'I didn't just see Caz and Ophelia sharing a hot dog, did I?'\n\n'Caz told her there wasn't any meat in hot dogs,' Jojo said, 'and the stupid bat believed him, so God knows what she thinks they make them out of. ARG's thrown her out. I shopped her \u2013 living with a gamekeeper on the hit list!' He looked around furtively to see if anyone could overhear.\n\n'Your secrets are safe with me, Jojo,' I assured him. 'I'm glad ARG's thrown Ophelia out and I hope you two aren't going to do any more silly things. Can't you just join the Green Party, and Friends of the Earth and that kind of thing, and lobby peacefully for what you want?'\n\nMick gave me a pitying look but didn't deign to answer this question. 'I expect it's just the pregnancy that's made her go weird,' he suggested. 'She might be all right afterwards.'\n\n'Doesn't that depend on how you define \"weird\"?'\n\n'Nah,' Jojo said, 'she'll be shacked up with him by then. He's already got her twisted round his little finger. He even hangs around your cottage on the nights when we're practising, so he can take her home.'\n\nI could see there was a bit of jealousy going on, but you could hardly blame Caz for keeping tabs on a girl who so notoriously found it difficult to say no to other men. Somebody needed to.\n\n'He hangs around my cottage anyway,' I said. 'He has the use of the freezer in one of the outbuildings, and also it's part of the estate, so he keeps an eye on things.'\n\n'He's certainly keeping an eye on Ophelia,' Mick said.\n\n'Are they really going to move in together?'\n\n'They pretty nearly are already \u2013 matter of time,' said Jojo disgustedly.\n\n'Mum's the word about ARG,' Mick said, tapping his nose as they moved away.\n\n'Mummer's the word,' I amended, watching them shamble off. At least, unlike Tom's avaricious surfer friends, they were mostly harmless. I didn't feel any need to shop them to anyone, including Nick, because I was sure they'd given up targeting the estate for the moment, and targeting me had been all Ophelia's doing.\n\n'We're going home,' Juno said, suddenly bobbing up next to me out of the throng, Mimi in tow carrying bags of popcorn, candyfloss and a half-eaten hot dog. 'Forgot to mention, Nick said to tell you he was coming, but he'd be late.'\n\n'If he doesn't come soon it'll all be over,' I said. 'And I can't see why he thought it mattered to me whether he was coming or not. I don't care.'\n\n'Don't care was made to care,' Mimi chanted. 'My governess used to say that.'\n\n'Well, come along \u2013 you're overexcited,' Juno said firmly. 'Tears before bedtime!'\n\n'She used to say that, too,' Mimi said, being borne away in the direction of the car. I would have said 'sick before bedtime' was more likely than tears.\n\nThe flaps of the marquees had been firmly closed while the vicar and two other local worthies judged the exhibits, but were now thrown back. I felt fairly complacent about the outcome. After several years of walking off with golds in two or three categories, I was already counting my book tokens and debating whether to give them to Jasper, or blow them on myself for a change.\n\nSo, when I was relieved of duty on the stall, I strolled over there, prepared to be gracious and modest as I collected my prizes. And I had won Best Fruit Chutney and Best Middlemoss Marchpane \u2013 but Nick, a surprise late entry, had beaten me into second place for Best Plate Apple Pie.\n\nI stared at the gilt-edged card with his name on, feeling surprisingly infuriated, as though he had performed a mean and underhand trick. Several local ladies were looking secretly pleased to see me pipped for gold this time, and there was some nudging and whispering as Nick, doing his silently materialising trick, reached past me and collected his prize.\n\n'Congratulations,' I said through gritted teeth, while fanning myself slightly ostentatiously with my own handful of cards. 'Was that Mrs Gumball's recipe?'\n\n'No, a variation of my own,' he said, smiling modestly, then was engulfed in a tide of congratulatory and admiring women. I stalked out of the tent.\n\nNext year he wouldn't find it so easy, as I told him when he walked home with me. Not that I wanted him to walk home with me: I pretended not to hear him when he called out, 'Wait for me, Lizzy!'\n\nCatching up, he demanded, 'Are you sulking?'\n\n'Why on earth should I sulk? And anyway, I never sulk!'\n\n'Oh, no? Isn't this the cold shoulder because I won the pie prize? Do you have to be the queen of all the puddings?'\n\n'Don't be silly,' I said, striding briskly off again.\n\n'You show me yours and I'll show you mine?' he said suggestively.\n\n'What?' I exclaimed, coming to a stop and turning to stare at him.\n\n'Apple pie recipe,' he explained innocently.\n\n'No way!' I snapped.\n\nApart from popping out to do a couple of pet-sitting jobs for Annie, I spent most of that Sunday making Christmas cakes: one big one for us and the little ones for the Senior Citizens' hampers. I knew I would need to pass on the small cake tins to someone else at the CPC next day, since we were all to make six each.\n\nMy arm certainly ached after all that stirring, but the cottage smelled totally delicious, and Jasper scraped the mixing bowls out with a teaspoon, just the way he'd always done.\n\n'What gift is the WI giving the Senior Citizens along with the hampers this year?' I asked Marian at the CPC meeting.\n\n'It's some kind of fleecy blanket with arms to snuggle into while watching the telly, called a Slanket.'\n\n'I like the sound of that, I wouldn't mind one myself,' Annie said.\n\n'I never seem to sit down long enough to make it worthwhile,' I said.\n\n'Me neither,' agreed Faye. 'Coming to the CPC meetings is the only time off I ever seem to get.'\n\n'If I feel at all chilly when I sit down with my bedtime tot of whisky to watch the news at ten, then I wrap myself in my pashmina,' Miss Pym said. 'It is light, but warm.'\n\nI certainly couldn't imagine her wrapped in anything going by such an inelegant name as a Slanket!\n\nI handed over the little cake tins to Faye, who was next on the rota, and then we discussed what extra treats to make for the hampers \u2013 rum truffles, peppermint creams, petits fours and things like that. I already make most of those anyway, for ourselves and for gifts, so it would be no trouble to make extra.\n\nAfterwards, when everyone had gone, I rang Unks to remind him to get his solicitor, Smithers, to tell me how much Tom's debts came to once he had wound up his affairs, but he said it would be a while yet, and not to worry about it.\n\nTom really didn't have any affairs to wind up, so I suspected Roly was going to take care of it and not tell me at all, but he was vague about it and hard to pin down.\n\nStill, after many delays, the insurance company had disbursed a paltry sum to compensate me for my lost 2CV \u2013 barely enough to buy a decent bicycle, let alone another car. Just as well I had had Tom's van to swap for the Land Rover.\n\nBut I used some of the money to order knitted silk long johns and a vest at off-season prices \u2013 quite a bargain. They were white and so I would have to tint them flesh pink when they arrived, so they wouldn't look too obvious from a distance under my Eve bodystocking, wig and figleaves. This year, I would be a very well-padded Eve.\n\nThey'd be useful afterwards in winter for gardening too, though not sexy, but since I wasn't intending to show my underwear to anyone, sexy was entirely irrelevant. Anyway, I wouldn't have the time or energy, since suddenly we were into the peak season of mellow fruitfulness so for the foreseeable future I'd be jamming, freezing, salting, chutneying, bottling and cordial making as though Famine were just outside waiting to knock on the door. Dried apple rings were already hanging in festoons above the stove; wine bubbled in every corner and bunches of onions, lavender and bay leaves dangled from the wooden rack above the kitchen table.\n\nI always felt much happier \u2013 sort of safer \u2013 once the cottage was full to bursting with a huge store of food and drink. I must naturally have a siege mentality.\n\nI dashed out between jobs to see to Ritch's dog. He wasn't there, but I had seen him once or twice on the evenings when he'd turned up to jam with the Mummers in Tom's old workshop. He'd taken to calling into the cottage kitchen afterwards, where I plied him with food and drink while I worked, just like everyone else who dropped by.\n\nI rather liked the company. Jasper was out much more in the evenings, another foretaste of my solitary existence soon to come \u2013 and I didn't mind the flirting now I knew it was just his manner, nothing personal. Besides, he was decorative to have around and he said it would be impossible to make a better apple pie than mine, it was perfect.\n\nWhile I was seeing to Flo, Dora Tombs told me Kylie had been spotted sneaking out of Ritch's house at the crack of dawn, and said he'd better watch out when her soldier fianc\u00e9 comes home on leave.\n\n'And she's not the only one!' she added darkly. 'A regular harem, he seems to be running. He's a charmer \u2013 like honey to humming birds, he is.'\n\n'That's terribly poetic, Mrs T!'\n\n'Saw them when I went over to Canada, to see our Sara,' she explained. 'Vancouver Island \u2013 wings whizzing round like bike wheels.'\n\n'Ritch reminded me of Honeycomb Crunch the very first time I saw him. You know, a bit like cinder toffee?'\n\nShe gave me a sharp look. 'And what does Mr Nick remind you of, then?'\n\n'Nick?' I said, surprised. 'Oh, hot spicy curry every time!'\n\n'Better for you than toffee,' she remarked cryptically. 'Honey or not.'\n\nClive and Marian Potter had got to hear the rumour about Kylie, and confided to me at the second Mystery Play rehearsal that they thought it conduct unbecoming, considering her role.\n\n'Just look at her over there, playing the Virgin Mary,' Marian said, scandalised. 'Butter wouldn't melt!'\n\n'Mary, thou art chosen as the mother of the Son of God, so think thisen lucky,' the Angel Gabriel was telling her.\n\n'By heck, then, thee'd better have words with my Joseph smartish and explain t'matter,' she riposted forthrightly, 'or t'wedding's off!'\n\n'It's probably just gossip,' I told Marian and Clive, but when Ritch, who was waiting for us outside again, swooped down and kissed me a smacker right on the lips, they exchanged meaningful glances as if to say, 'What, you too, Eve?'\n\nActually, I thought the way he and Kylie avoided each other in public was much more obvious than if they'd been entwined for the whole evening. I was sure they had an assignation set up for later, because although Ritch sat next to me and flirted outrageously, when I got up to go he didn't offer to walk me home this time \u2013 I went alone.\n\nNick had already left, much earlier. Maybe he'd got the Lancashire hotpot recipe out of Marian? Mission accomplished.\n\nJasper was home when I got there, eating cheese on toast with Ginny snuffling hopefully round his feet for crumbs. He said Nick had phoned to say he'd found a taker for the big greenhouse, and it was to be dismantled tomorrow morning and removed on a lorry.\n\n'I don't see why he didn't tell me that earlier!'\n\n'He said even with your clothes on he found you so distracting as Eve that he forgot, and goodness knows what he would be like at the actual performance.'\n\n'He did? What kind of thing is that to say to my son?' I demanded, scandalised.\n\nJasper grinned.\n\n'And he doesn't mean it either,' I said snappily. 'He's just being sarcastic and horrible.'\n\n'No, he's not: he likes you, Mum.'\n\n'I think I know your uncle Nick by now and all his little ways,' I said firmly.\n\nAfter dropping Jasper and Ginny off bright and early at the dig next morning, I went home and changed into my oldest jeans, ready to help dismantle the greenhouse.\n\nBut the new owner proved to be one of those men who doesn't recognise any woman's existence in a business deal and so addressed himself totally to Nick, who was hanging around looking taciturn. I returned to the kitchen and my jam making.\n\nWhen they had gone and I went out, I found my little domain totally changed, with the last plants that had been inside it pathetically huddling together in the open, as if for warmth.\n\nIt was another thing sorted out, though \u2013 and the huge TV has already gone, after I put a card in the post office window. On the proceeds, Jasper had chosen a small one with an integral DVD player to take to university with him. I only hoped he was going to work and not spend his entire student loan on films, drink and stuff.\n\nI'd begun to notice that if Jasper was home when Ritch called by, he seemed to be in and out of the kitchen all the time, and having six foot of sardonic teenage youth critically observing him rather cramped Ritch's flirting. I didn't know why Jasper disliked him, unless he had joined the ranks of those trying to pair me up with Nick (I was not blind to all the hints various people, including Juno and Mimi, had been dropping), though I could tell them right then that this was only wishful thinking and not a horse that was ever going to run.\n\nBut clearly everyone saw Ritch as some kind of threat, to either my heart or my virtue (or both), for when Jasper wasn't there, Caz seemed to be hanging about the yard until Ritch left, instead.\n\nAnd I was forever finding Nick wandering about the place, as if he owned it... which I supposed he sort of did, come to think of it, though he didn't own me.\n\nIn fact, my cottage seemed suddenly to have become one of the most popular spots in the Mosses.\n\nI invented a recipe for potato and nut biscuits, which came out so well I tried making chocolate flapjacks with mash, too.\n\nHowever, Jasper said they were more like 'mudflaps', so a little more experimentation was clearly in order.\n\n## Chapter 18: Simmering Gently\n\nI've been potting hyacinth bulbs and putting them away in a dark cupboard today \u2013 pink, blue and white. It seems indulgent to take the time, when there's so much to do in the garden and the blackberries still cluster thickly on the brambles, begging to be picked and turned into wine, jelly and jam. But when they flower, they'll be like a breathy promise of spring to come.\n\nAnd we might need it, for a heavy crop of berries on all the bushes means a hard winter, according to old country lore.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nThe skiing underwear I'd ordered was sitting in a neat brown parcel on my doorstep when I got back from dropping Jasper at the dig one morning, so I went straight upstairs to try it on.\n\nIt fitted tightly, but I had quite a job getting the old, stewed-tea-coloured bodystocking on over it (one of us was losing our stretch with age \u2013 or maybe both of us).\n\nWhen I looked in the mirror I saw that I presented a strangely padded and seamed appearance, like an unsuccessful home-made Cabbage Patch doll, especially once I got the totally unrealistic flaxen wig out of its storage box and completed the ensemble.\n\nSexy it was not, and when I heard someone walk into the hall downstairs I nearly died: my heart certainly stopped and I went still as a mouse. There was no way I was going to let anyone see me like this!\n\n'Lizzy?' called a familiar voice.\n\n'Oh, thank God!' I muttered devoutly. I'd quite forgotten I'd arranged for Annie to come over and help me sort out Tom's clothes and personal stuff, a task I'd been cravenly putting off.\n\n'Come up \u2013 I'm in the bedroom,' I called.\n\nShe clumped upstairs and I turned to face the door, striking a soulful pose and twiddling one long gold ringlet round my fingers.\n\n'Have you already started?' she was saying as she walked in. 'I'm a bit late, I\u2014'\n\nHer jaw dropped and her stunned expression sent me into near hysteria. After a dumbstruck minute she started to giggle too and soon we were both entirely incoherent.\n\n'So,' I said finally, trying to wipe the tears from my face with a long tress and finding that nylon wasn't very absorbent, 'you don't think I should play the Eve part for laughs, then?'\n\n'Oh, Lizzy, can you imagine Nick's face if he walked on and saw you like that?' she said, sitting down on the bed, limp with laughter.\n\n'I could \u2013 but I'd rather freeze to death first!'\n\n'Well, you can't possibly, anyway. But perhaps if you bought a new Spandex bodystocking it might be warmer than that old one?'\n\n'It hardly seems worth the expense when this is the last year I'll play Eve.'\n\n'Never mind, you looked stunning, even in the old outfit, last year,' she said loyally, 'though quite indecent from a distance!'\n\n'Speaking of indecent, I don't know what Nick intends wearing, if anything. Do you?'\n\n'I asked him, in case he hadn't given it any thought,' Annie said innocently. 'He said he'd ordered footless tights from a ballet-clothing place and would wear his cricket box under them to protect his modesty.'\n\nThat conjured up quite a vision... which I hastily dispelled, though not without some difficulty. 'He'll freeze,' I said with conviction. 'We both will!'\n\n'I expect he'll be all right, because it's only a few minutes, the Garden scene, isn't it? Then you can rush into one of the loose boxes and put your warm clothes back on.'\n\n'Still, that does it \u2013 if he's going to be half-naked, then I'm not going out there padded up like Michelin woman! A new Spandex outfit it is. I've still got some car insurance money.'\n\nI thought I might even go completely mad and get myself a pretty frock, too, for Christmas dinner, which we always had up at Pharamond Hall. I couldn't remember when I'd last bought myself something new to wear that wasn't vital, like jeans.\n\nI changed back into ordinary clothes and then we got down to sorting out Tom's stuff, which I'd collected up and pushed into his wardrobe or drawers out of sight.\n\n'Jasper's taken a couple of things \u2013 cufflinks, mostly \u2013 but the rest can go to the charity shop, or in the recycling bin.'\n\n'I'll take it all down to the Animal Shelter shop,' she offered. 'I brought the car up rather than walk, in case.'\n\nTom didn't have a huge wardrobe of clothes, so it didn't take long. I was so glad I wasn't doing it alone, though, because memories, mostly painful, tended to tumble out of every open door and drawer.\n\n'How are you and Gareth getting on, Annie? You're seen almost everywhere together, like Siamese twins,' I teased, once we'd loaded her car and retired to the kitchen for a well-earned cup of coffee and a restorative plate of chocolate slab cake. 'And the cookery lessons too! Is he teaching you anything in return?'\n\n'No,' she said, her face clouding over. 'Lizzy, I enjoy being with him, but I think he's just being friendly. I'm that type of girl, aren't I? Men don't think of me romantically at all, so I expect I'm exactly like a sister to him!'\n\n'You daft bat!' I said, regarding her incredulously. 'He's absolutely dotty about you! When he looks at you he has that soppy sheep expression in his eyes, and every time he speaks to you he goes red as a beetroot. And he told me he thought you were very pretty!'\n\n'He didn't!' Annie went pink with pleasure.\n\n'He did.'\n\nShe looked at me doubtfully. 'Then why...? I mean, you must be wrong, Lizzy!'\n\n'Oh, I'm sure he's just shy. Encourage him a bit.'\n\n'I couldn't possibly! What if he only wants to be friends? Think how embarrassing it would be if I'd made a fool of myself and we had to go on meeting as if nothing had happened...'\n\n'But you do fancy him, don't you, Annie? I mean, this is love's young dream and all that?'\n\n'More love's not-so-young dream,' she said ruefully.\n\n'Rubbish, we're still thirty-somethings, and that's a very good age for love.'\n\n'It might be, but I daren't risk destroying my friendship with Gareth to find out.'\n\n'You won't. You wait and see.'\n\n'You aren't going to do anything, are you?' Her soft, blue-grey eyes looked at me anxiously.\n\n'No, of course not,' I reassured her quite untruthfully. 'I'll await events to prove me right and then I expect to be matron of honour at the wedding, in a mid-calf-length puce taffeta dress with those puffed shoulders that make you look five feet wide.'\n\n'Not puce,' she said seriously. 'The church carpet and hassocks are scarlet, so it would clash.' Then she sighed, her eyes refocusing, as though abandoning a beautiful dream. 'Anyway, enough of my boring affairs, Lizzy \u2013 what about you? The whole village is talking about the way Ritch Rainford flirts with you!'\n\n'Oh, come on, you must have realised I'm just a smokescreen for the women he is having affairs with, Kylie among others. But I do like him, and I enjoy flirting with him. At least he makes me feel I'm still attractive.'\n\n'Perhaps, but it's making Nick jealous, haven't you noticed?'\n\nI stared at her. 'Well, yes, but not jealous of me personally \u2013 he simply doesn't like Ritch paying attentions to his cousin's widow. Come to that, he just doesn't seem to like Ritch. But I'm sure his attitude's mainly a territorial thing.'\n\n'I think you're wrong, and he's fond of you,' she said earnestly.\n\n'Annie! It's bad enough Juno and Mimi \u2013 and now even Jasper \u2013 trying to matchmake, without you joining in!'\n\n'Do they? I hadn't realised. But there, you see \u2013 even the family think it would be a good thing if you got together!'\n\n'Annie, it's not going to happen \u2013 and it would certainly be a marriage made in hell, not heaven.'\n\n'I don't think Nick would agree with you,' she persisted stubbornly.\n\nI considered it seriously for a minute, remembering the way he'd kissed me once or twice in a most uncousinly manner, and how he'd referred back to our short-lived romance as if he couldn't understand why it hadn't worked out... But apart from that, there wasn't anything to suggest he was harbouring an undying passion for me.\n\n'No,' I said firmly, 'we bicker more than we agree, and drive each other mad: too many cooks again. He might still fancy me \u2013 I don't know \u2013 but he isn't in love with me.'\n\n'So you're just friends \u2013 like Gareth and me?'\n\n'Well, not quite. More sparring partners. The family \u2013 and probably the whole village, going by the hints Dora Tombs has been letting fall \u2013 just wants a neat and tidy ending: you in the parsonage, me in the Hall, all's well that ends well. Only life isn't like that.'\n\n'I suppose not.'\n\nShe sighed sadly, but I was determined that at least her romance would turn out right. All Gareth needed was a bit of encouragement.\n\nThat evening Ophelia came round, driven by Caz in his ancient Land Rover (even more ancient than mine and painted with camouflage, just like him sometimes), and to my astonishment asked me if she could buy the quail!\n\n'You haven't anywhere to keep them,' I pointed out, 'and anyway, what would you do with them? You won't eat them or the eggs, so you'll be overrun with male quail in no time.'\n\n'No she won't, then,' Caz drawled, leaning against the bonnet, his khaki hat tipped over his nose.\n\nReally, he's getting almost loquacious! It must be love. Anyone would think it was spring, the way Cupid's fiery darts are flying in all directions.\n\n'There's the old piggery behind my cottage \u2013 I can keep them in that for now,' Ophelia said, rabbited at her lower lip a bit and added, 'I've decided to become vegetarian while I'm pregnant, and eat fish and eggs.'\n\n'Good idea!'\n\nShe gave Caz a half-defiant look: 'But not flesh of any kind!'\n\nWhat does she think fish are made of?\n\n'Right, I'll start including more eggs in your basket of fruit and vegetables. I wasn't sure if you were eating them or not.'\n\n'No, no, no, you shouldn't! It's too kind and... and I don't see why you should be kind,' she muttered, her bulging eyes taking on that sainted martyr look. 'I don't deserve it.'\n\n'You may not, but we have to think of the baby. It needs good wholesome food to grow properly.'\n\n'But you carry it all the way up to my cottage and it must be really heavy!' She wrung her hands in an anguished sort of way. Let's hope she is never holding a quail \u2013 or, indeed, the baby \u2013 when she's in one of these states. 'Don't \u2013 please don't do it any more. Caz says he'll fetch it.'\n\n'OK \u2013 that will be good. I'll leave the basket near the eggs in the outbuilding on Monday mornings, how about that?'\n\nI couldn't really see Caz in the Little Red Riding Hood role with a basket, but that was his problem. Maybe he'd bring a backpack.\n\nShe nodded like a car mascot and then added, after another of her lip-chewing ruminations, 'Thank you for the bottled tomatoes.'\n\n'I didn't give you any bottled tomatoes!'\n\nShe blinked slowly. 'Yes, you did. They were on the doorstep last night, with that yellow checked material tied over the lid, like all your jars of stuff.'\n\n'Gingham? I bought a whole roll of it years ago, and never got to the end of it. But I didn't leave any jars of anything on your doorstep.'\n\n'But... it must be you. I don't know anyone else who bottles tomatoes.'\n\nI had a sudden horrible thought. 'I know someone who tried to,' I said grimly. 'Polly Darke! And they gave me botulism or something equally ghastly.'\n\n'But she's not... she isn't... she wouldn't...' Ophelia trailed off, and then wrung her skinny hands together again distractedly, her eyelids frantically fluttering.\n\n'Look, I guessed she was the one who made you do the ARG stuff to me, out of sheer spiteful jealousy. There was no one else it could have been.'\n\n'But Caz made her stop, so perhaps she's sorry now, and this is a present?' Ophelia suggested. 'But I don't want her peace offering!'\n\n'If it is a peace offering. We know she's spiteful enough to do something nasty. I'd throw it away, just in case, if I were you.'\n\n'I'll do that,' Caz said, the stony expression on his face boding ill for Polly \u2013 though would she have tried something that might have harmed Ophelia's baby? I thought back to what I knew of her, which, due to my avoiding her as much as possible, was not a lot.\n\n'I could be quite wrong about her, but it is odd that I was the only one who got the dodgy jar of tomatoes when she was handing them out to half the village that time,' I said slowly. 'And another thing: although I've only been to her house once for a book launch party, I was horribly sick afterwards, though I didn't hear of anyone else being taken ill.'\n\n'Toadstool,' Caz said meaningfully.\n\n'Toadstool?' For a minute I thought he'd run mad, then I remembered: 'You mean that poisonous one you showed me \u2013 was it in the basket of field mushrooms Polly brought me to swap for eggs?'\n\nHe nodded grimly.\n\n'And wasn't it the kind you only get in woods, not open fields?'\n\nHe nodded again.\n\n'Could she be that jealous and vindictive? It's so downright nasty!'\n\nCaz shrugged.\n\n'Well, if it is true, let's hope she doesn't present any more little gifts to other people disguised as my offerings! I'd better tell Marian Potter tomorrow that someone is maliciously leaving tainted jars of food that look just like mine on doorsteps and she will spread the word. And I'll stop covering the jars with anything except Cellophane from now on, even if they don't look as pretty!'\n\nOphelia had lost interest by now and wandered off to commune with the quail by means of little cheeping noises. She seemed to be frighteningly at one with them mentally, which didn't bode well for the intellect of her future offspring.\n\nI followed over. 'So, what are you going to do with the quail?'\n\n'Give them a happy life and I can sell the eggs, too. I think that's all right,' she said earnestly.\n\n'OK, they're yours,' I agreed.\n\nShe was totally impractical, but I expected Caz would just quietly go in and do what had to be done with the birds when she wasn't looking.\n\nThey seemed to be settling down into a pair, though an odd couple they made. Maybe knowing secrets about each other forms a bond, for he was aware she was in ARG, and he'd told her what he did with the grey squirrels he caught. And Ophelia seemed very malleable, apart from a bit of occasional stubbornness, so I expected he'd slowly bend her into the shape he wanted over time.\n\nWhy did that make me think of brandy snaps?\n\nCaz found some cardboard boxes, which he punched holes into, and loaded the quail up then and there, dismantling the pens and taking those, too, since I would have no further use for them.\n\nWhat with the big bare space where the greenhouse used to be, and the lack of cheeping, moving feathers in the barn, things were looking quite deserted, apart from the hens and ducks. They'd all sheered off while Caz and Ophelia were there, but now came back looking for any pickings.\n\nWith a sigh, I went back into the cottage and looked out my recipe for brandy snaps, even though trying to wind them round the handle of a wooden spoon is such a pain that I couldn't usually be bothered.\n\nNext day when I was on my way back from a spot of pet-sitting duty, I spotted Caz and Nick in the woods near the drive. Caz seemed to be talking, or at least, replying, which was amazing! Then they shook hands...\n\nWhat was that all about? Had some kind of deal been done?\n\nOn Sunday we were invited up to lunch at the Hall. Actually, we had an open invitation, but sometimes I was too busy, or wanted to go out with Jasper for the day while I'd still got him.\n\nNick was cooking, as he often did because he said it gave Mrs Gumball a rest. Though she protested at having her kitchen taken over, I think she was quite pleased really: she was no spring chicken any more, after all.\n\nHe was in and out of the kitchen when I got there and spurned my offer of help, though he accepted Jasper's even though he only knows the theory of cooking and not the practicalities. So I left him to stew in his own jus and sat down at the dining table with the others.\n\nUnks smiled at me fondly, then went back to reading the sports section of the Sunday paper.\n\n'We're going on a garden tour by coach,' Mimi informed me chattily.\n\nI stared at Juno. 'Is that a good idea?'\n\n'Don't see why not \u2013 I'm fully fit again. I'll frisk her for knives, scissors and plastic bags before we set off, and I won't take my eyes off her for a minute within fifty paces of anything green. It's a late-booking bargain.'\n\nMimi smiled innocently.\n\n'I don't think you will get a lot out of it, Juno,' I said. 'You could do with a more restful holiday, after your accident.'\n\n'There are entertainments in the evenings at the various hotels,' Mimi said. 'It'll be fun. But Juno and me have got to share a room. People will think we're an odd couple.' She giggled.\n\n'Pity, I was hoping to pick up a toyboy,' Juno said, 'preferably a rich one who could whisk me away from this madhouse.'\n\n'She doesn't mean it,' Mimi confided to me. 'Her heart belongs to Sean Connery.'\n\n'I've never been one of those romantics,' agreed Juno, 'but I like a man to be a man.'\n\n'Do you think Nick is a macho man, Lizzy?' Mimi asked, with one of her disconcertingly clear looks, just as the man himself strode into the room carrying the soup tureen, with Jasper following up behind with a silver basket of bread.\n\n'If macho is big, male and overbearingly bossy, yes,' I said sweetly.\n\nNick gave me one of his slaty looks, the purple-edged sort.\n\nIt was good soup, followed by roast beef and perfect Yorkshire puddings, and he'd made one of his apple pies for dessert, which I have to admit was delicious, though I certainly wasn't going to tell him so. Instead, I said the pastry was a little dry and asked for more cream.\n\nJasper and Nick were talking quietly in the kitchen when I carried through the dessert plates, and I hoped it was serious male stuff, because my attempt to discuss Safe Sex and STDs with Jasper certainly hadn't gone down too well.\n\nNick seems to be Confidant of the Moment \u2013 but not mine. Been there, done that, and once was enough!\n\nI was stuffed to bursting point after the coffee, but then Nick practically dragged me back into the kitchen and made me taste three different coffee granitas before I left. When I said I didn't think any of them had that extra something, he went very sulky, even though (unlike what I said about the apple pie) it was quite true.\n\nOn the way home Jasper said I'd hurt Nick's feelings and I should have pretended one of the granitas was great, and I said, astonished, 'Why should I, when he's always so rude about my cooking?'\n\n'But you know he doesn't mean it. He's just joking \u2013 it's affectionate.'\n\n'I'm not so sure about that,' I said darkly.\n\nAnyway, I'm never again going to tell a man something is wonderful when it's not. It's my un-New Year resolution.\n\n## Chapter 19: Stirring\n\nWe're heading towards October and I'm catching up with the garden: clearing away the finished crops and storing layers of carrots in boxes for the winter. At the Christmas Pudding Circle the small cake tins changed hands yet again and soon we will have enough for all the Senior Citizens' hampers. When Marian first suggested we bake the cakes we also offered to make individual Christmas puddings; but it turned out that when the WI asked them for likes and dislikes, they all preferred bought microwavable puddings and cartons of ready-made custard.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI passed on the warning about gifts of possibly tainted bottled goods being left on doorsteps by the simple method of mentioning it one morning at the Christmas Pudding Circle meeting. Marian especially is permanently plugged into the local grapevine via the post office, so by afternoon everyone within a five-mile radius would know, like dropping a pebble into a pond and watching the ripples spread.\n\nAfter that I hadn't intended to give the matter much further thought. In fact, I was half inclined to think Ophelia's jar of tomatoes had been a gift from some well-meaning villager, who'd simply reused a gingham circle from something of mine. Goodness knows, I've supplied enough preserves and pickles to village fairs, f\u00eates and bazaars over the years!\n\nBut then Leila phoned me out of the blue in mid-afternoon while I was making a carrot cake and, to my complete astonishment, apologised for what she'd said at the funeral.\n\n'Of course, much of it was true, but it was not the time or place for such matters. I had come simply to pay my last respects.'\n\n'Quite... and... thank you,' I replied cautiously, though not sure quite what I was thanking her for.\n\n'I see clearly now I was deceived by Tom and also, perhaps, by Nick. But that is life, so now I am resolved to stay single. Nick says he will still review my restaurant in his articles; it will not make a difference, our divorce,' she added, sounding surprised and slightly scornful of his magnanimity. 'So, we should all bury the hatchet and move on, yes?'\n\n'Er, yes... and it's nice of you to phone me,' I said doubtfully, wondering if there was a catch, for example, exactly where she meant to bury the hatchet.\n\n'I could do no less, after you sent me the peace offering, though a pot of blackberry jam, that is not sensible to put in the post, even packed so well.'\n\n'Jam?'\n\n'It says \"Blackberry Jam: Middlemoss Autumn F\u00eate\" on the label, so I knew it must come from you.'\n\nI'd provided some to be raffled off for charity, so it must be one of those. 'You haven't eaten any, have you?' I demanded urgently.\n\n'No. I do not eat jam, it is not in my diet regime.'\n\n'Then don't! I didn't send it, someone else did \u2013 and I'm afraid it might be... tainted.'\n\n'Tainted? You mean poisoned? Someone is trying to kill me? But that is ridiculous!' she said witheringly.\n\n'No, I'm sure she doesn't intend to kill you, just make you sick and pin the blame onto me. She's tried the same thing with Ophelia Locke, but I wasn't sure\u2014' I broke off. 'Oh, you don't know Ophelia, do you?'\n\n'I know of her. I have been told she claims to be carrying Tom's child, but she sounds a type most hysterical and neurotic.'\n\n'I suppose she is a bit,' I agreed, wondering who Leila's spy in the village was. 'But she is pregnant and there's a possibility it could be Tom's \u2013 about one in four, if you want the odds. But anyway, she found some bottled tomatoes on her doorstep the other day and assumed they were from me, and they weren't. And then I remembered once having a bad experience with bottled tomatoes someone gave me, and I got suspicious. Only it seemed so incredible that then I thought I must be imagining it.'\n\n'But you are confusing me with all this talk of bottled tomatoes! Who \u2013 and why? And...' There was a pause. 'It is Tom's other woman doing this, that weird person, Polly something?'\n\n'I think so,' I admitted. 'I can't imagine who else it could be, and there are a few too many coincidences. She's been blackmailing the local animal rights campaigners into targeting me, too, so I wouldn't put it past her.'\n\n'I will set the police on to her!'\n\n'There isn't any proof, so I don't think they could do anything, but I'm going to let her know that she's found out, so she'll think twice before trying anything else!'\n\nLeila still maintained that the police should be involved, but I was wary: what if they thought I'd done it myself, as a sort of double-blind?\n\nBut in the end she agreed she would do nothing for the present, and rang off, after adding that if I was ever in the vicinity of her restaurant there would always be a table free for me, an offer I thanked her for but was unlikely to take up.\n\nWhen I next popped into Annie's cottage on the way to see to Flo, I told her exactly who I'd meant when I'd warned them at the CPC meeting that someone was leaving poisoned preserves on doorsteps. Then I retailed my conversation with Leila.\n\n'Poor, poor woman!' she said sadly.\n\n'There's nothing poor about Leila!'\n\n'No, I meant Polly, to be so consumed with spite and jealousy that she could do such awful things!'\n\n'Well, that's one way of looking at it,' I said. 'Trust you to feel sorry for her! And it's all very well playing these pranks on me and Leila, but Ophelia's pregnant and it might have made her really ill or harmed the baby! Goodness knows what she put in those tomatoes.'\n\n'That's true, and she might do something else! Perhaps I ought to tell Gareth so he could go and reason with her?' she suggested doubtfully.\n\n'Absolutely not! There's no way she's going to cast herself upon his bosom and weep tears of repentance, and it would be like sending Daniel into the lioness's den.'\n\n'What are we going to do, then?'\n\n'I'm going to speak to Polly, preferably in a public spot with other people about. I'll tell her I know about her tricks, and if she does anything else I'll report her to the police. That should stop her.'\n\n'Oh, I hope so,' Annie said earnestly. 'Perhaps it will shock her into realising how badly she's been behaving, so she can move on.'\n\n'I wish she would move away,' I said, getting up. 'Well, must go and see to Flo on my way home. Ritch isn't going to be back until late and she's probably got all four paws crossed by now.'\n\n'You really aren't falling for him, are you, Lizzy?' she asked anxiously.\n\n'No, of course not, though if celibate widowhood ever palls on me, it's nice to know my options for random sex are still open.'\n\n'Lizzy! You wouldn't!'\n\n'Probably not \u2013 especially if he continues with all his dubious habits.'\n\nIn the afternoon Nick rang and demanded I go up to the Hall and taste his newest version of coffee granita, but I declined, since I was in the middle of a huge quince jelly-making operation by then and could hardly down tools at his bidding.\n\nHe slammed the phone down, but strode into my kitchen not ten minutes later, carrying an insulated box cradled in his arms like a baby.\n\n'This one's perfect!' he said, the light of battle in his eyes and his dark hair sticking up in an angry crest. 'I defy you to find fault with it!'\n\n'Look, I'm up to the elbows in this, I can't sit down and eat,' I protested, so he followed after me around the kitchen, feeding me teaspoons of granita as if I were a stubborn toddler. Much though I would have liked to find fault, however, I couldn't. The colour, taste and texture were all pure perfection.\n\nWhen I said as much he tossed the spoon into the Belfast sink with a clatter, grabbed me and planted a triumphantly emphatic kiss on my lips before I could fend him off with the ladle.\n\n'Mmm... you taste of the perfect coffee granita!' he murmured, half-closing his eyes.\n\n'I don't know what else you'd expect, when you've been force-feeding me the stuff for the last ten minutes,' I snapped, taking a step back. We unpeeled rather stickily.\n\n'You're a very messy jam maker,' he said severely.\n\n'And you are a very messy cook, full stop. I've never seen anyone use so much equipment to make even the simplest dish.'\n\n'Like my apple pie?' he said, a gleam in his eyes. 'Come on, Lizzy, you know there was nothing wrong with it last Sunday.'\n\n'It wasn't bad,' I conceded, then I smiled at him innocently and asked, 'That granita... just what did you add to make it taste like that?'\n\n'Wouldn't you like to know!' he said tantalisingly. 'Well, see you at the play rehearsal later, Eve!' Then, picking up his cold box, he walked out.\n\nHe was still being exasperating at the rehearsal \u2013 and so were the Nine Angels, who kept gurning at each other when they thought I wasn't looking.\n\nI didn't go to the pub afterwards, because I simply wasn't in the mood, but sneaked out of the side door and dashed home. Anyway, the kitchen was still a sticky mess and I wanted to clear that up and then, while still in quince mode, make some wine.\n\nFriday was Jasper's last day at the dig and he returned smelling of real ale and with some kind of excavation certificate, with which he was highly pleased. I knew he'd been feeding Ginny pork scratchings at the pub after the dig, because she threw most of them up behind the kitchen door.\n\nJasper went straight off after dinner to stay for a couple of days with his friend Stu, whose family lived in Ormskirk, and although I wanted to spend every precious moment left with him before the start of his first university term, I didn't try to persuade him not to go. Instead, I drove him there and dropped him off myself, hoping Stu's mum was expecting Ginny, too \u2013 and thank heaven she came ready-house-trained.\n\nI also hoped Jasper would behave himself, though frankly there are not many dens of iniquity in the lovely old market town of Ormskirk. But before he left home Unks gave him a huge amount of spending money. He must have had a win on the horses, to have so much cash about him.\n\n'You'll be sensible, won't you, Jasper?' I said, hugging him before he got out of the Land Rover, which he suffered me to do in a resigned sort of way.\n\n'It's not me who needs to be sensible, when that Rainford man's forever dropping in and hanging out with you, or chatting you up in pubs,' he protested. 'Not to mention phoning you up and asking you to go round to his house all the time!'\n\n'He's just being friendly, Jasper, and I only go round to his house to look after the dog.'\n\n'I don't see why he can't look after his own dog.'\n\n'Well, neither do I, really \u2013 or get one of his many girlfriends to do it.'\n\n'So you do know about all the girls he takes back there, then?' He sounded relieved.\n\n'Honestly, Jasper! I'd have to be deaf, dumb and blind not to, even if Dora Tombs didn't tell me. The whole village is talking about him, which at least means they've moved on from going over what your dad got up to with Leila and Polly, and taking bets on who the father of Ophelia's baby is,' I said tartly. 'Now, you stop worrying about me, because anyone would think I was some naive innocent out of a Victorian melodrama, about to be taken advantage of by the villain of the piece!'\n\n'He does do that all the time in Cotton Common.'\n\n'But not in real life,' I said firmly. 'And I'm not interested in Ritch that way \u2013 or any other man. I just want to be left alone with my hens, my garden and my recipes.'\n\nEspecially the search for the perfect apple pie and coffee granita...\n\n'Try not to fall out with Uncle Nick while I'm away,' was his final admonishment as he removed his holdall and slammed the door. He must have been reading my mind \u2013 and bossiness seems to run in the Pharamond blood, just like cooking.\n\nI didn't get a chance to fall out with Nick, since it turned out that he had left for London and was then going on to Cornwall in search of fish recipes for some forthcoming article.\n\nI felt a bit... piqued, I think is the word. I'd sort of got used to having him around again, annoying though he is, because he's someone to bounce food ideas off and argue with. Annie's interested in food, but I wouldn't call her a creative cook, and she's so even-tempered I couldn't pick a quarrel with her even if I tried.\n\nMind you, since every second word she utters these days is 'Gareth', exasperation might eventually lead me to smother her to death, probably with a hassock.\n\nRitch was off in London too, shooting some cameo role for a film, so Dora Tombs and I were taking care of Flo between us. On Sunday I joined the depleted party up at the Hall for lunch and Mimi and Juno were full of talk of their holiday and deep in planning the installation of a water feature in the walled garden, while Unks was happily doing his own thing, as usual, mainly involving studying racing form.\n\nI was still extremely busy myself, perpetually preserving, storing, gardening and pet-sitting, but it gave me a foretaste of what it was going to be like once Jasper went off to university...\n\nI suspected Nick wouldn't give up his flat in London after all, but go back to dividing his time between there and the Hall. Eventually, once the divorce was finalised, he'd marry someone young and beautiful and start a family. I was quite convinced this would happen, because even I had to admit he was wildly attractive, except when he was annoying me... so it was just as well he annoyed me most of the time, wasn't it? And clearly he had no great interest in me, whatever my misguided relatives thought, since he couldn't even be bothered sending me postcards any more!\n\nAnyway, I liked being on my own, and Jasper wouldn't be a million miles away, so I couldn't imagine why I was feeling suddenly so depressed.\n\nBut of course I wasn't totally alone, for I still had my friends in the Christmas Pudding Circle, and at the next Monday meeting Miss Pym gave us all a surprise gift of candied angelica. Marian had brought me her giant round two-part metal Christmas pudding mould, too, so the following afternoon I abandoned everything else and made a huge spiced fruit cannonball instead. The smell of the ingredients took me right back to happy times at the vicarage with Annie's family, and was very comforting.\n\nStirring, I made a wish.\n\n## Chapter 20: Freshly Minted\n\nI have been preserving apples in wine and making green tomato chutney, before clearing away the tomato plants. I'm not sure what I'll do with the bare patch where the huge glasshouse used to stand, but I'm very tempted to turn it into a little apple orchard, with several unusual old kinds.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nThe most dreadful thing! Gareth called to tell me that Tom's mother and stepfather had just arrived in the UK and were on their way down to pay their respects at Tom's grave!\n\n'What? But why didn't they call me?' I asked, stunned. 'I haven't heard a word since the funeral \u2013 and it's years since they saw Jasper, too, and he's away at the moment. If only they'd let me know they were coming!'\n\n'I don't know, but I thought I'd check to see if you knew about it, because they didn't mention you and you ought to be there. I gave Mr Barillos directions to the graveyard. They've hired a car and were only about an hour away when he rang.'\n\nI ran a distracted hand through my tangled hair. 'Well, thanks for telling me, Gareth. They're such odd people that I suppose them behaving like this shouldn't surprise me, but you're right and I'd better meet you at the graveyard.'\n\nI hoped they wouldn't be too disappointed that the stone has not yet been erected. It's ordered, but these things do take time.\n\nQuickly I changed into something clean, though equally unsuitable for the occasion, and set off, collecting Annie on the way for moral support.\n\nGareth was standing by the grave, which did indeed look forlorn, especially on such a grey, cold October day as this one: a grassy mound in the Pharamond corner. I intended planting spring bulbs there, once it had its stone, and possibly a bit of lavender. I don't much like cut flowers left dying on graves, or in those little stone urns.\n\nI wasn't sure I would have recognised the Barilloses if I'd met them in the street, but on a gravel path in an old country graveyard, they stood out like sore thumbs. For a start, there was something glossily expensive about their clothes and, although it was not a sunny day, they both wore huge, wraparound reflective sunglasses.\n\nThe skin visible on Tom's mother's face looked stretched, smooth and peachily tinted, while her skittish curls were a rich brassy blond. Her husband looked positively withered and prune-like in comparison, apart from having a head of hair like black Astroturf.\n\nGareth stepped forward and clasped their hands in turn, murmuring a few earnest words. Then, since they were pointedly ignoring me, he gestured and said, 'And here's your daughter-in-law, Lizzy, come to meet you and her friend, Annie Vane, whom I don't think you've met.'\n\n'Well, Elizabeth, I didn't expect to see you here,' Jacqueline Barillos said coolly.\n\n'My wife did not wish to see her \u2013 you should not have told her we were coming,' argued her husband.\n\n'But she's Tom's widow,' began poor Gareth, baffled. 'Who better to offer comfort and\u2014'\n\n'But we know she did not care about him and was a bad wife,' interrupted Jaime Barillos. 'We have had many beautiful letters from the woman he did love, whom he wanted to marry. We know how grieved he was when he found out about his wife's affair and realised that even his own son was fathered by his cousin!'\n\n'Now, just a minute!' I broke in angrily. 'I think I can guess who's been telling you this pack of lies, but it most definitely is not true!'\n\n'Certainly not!' Annie defended me stoutly. 'It was Tom who was the unfaithful one, not Lizzy, and Jasper is Tom's son.'\n\n'And you, of all people, must know why my son looks so like a Pharamond!' I added pointedly to Mrs Barillos.\n\nShe gave me a dirty look, then threw a dramatic hand towards the grassy mound and cried, 'Here's the proof \u2013 does this look like the tomb of a loved husband?' Clearly she'd missed her calling and should have been on the stage.\n\n'The stone is ordered. These things take time,' I explained.\n\n'Not even any flowers...' she sobbed, turning to her husband, who put his arm around her and glared at me.\n\n'Please,' began Gareth, 'please don't distress yourself, Mrs Barillos! Look, why don't we all go back to the vicarage and talk this through? I fear you're letting the natural grief of a mother lead you to unwarranted conclusions\u2014'\n\n'No!' she declared, lifting her head and turning her dark lenses in my direction like an inimical ant. 'I would like you all to go away so I can pay my respects to my son \u2013 alone.'\n\n'Then afterwards, perhaps...' suggested Gareth tentatively.\n\n'No. Leave us in peace,' she said implacably.\n\nI turned and walked off before I could say something I would regret, and Annie followed me, though Gareth paused to speak to them before catching us up.\n\n'This has been a bit of a shock, Lizzy. Why don't you come back to the vicarage for a cup of tea anyway?' he suggested kindly, but I insisted I was fine and, despite their protests, set off for home. I didn't even want Annie's company just then.\n\nThat ugly little scene had seemed too melodramatic to be true at the time, especially with the Barilloses resembling nothing so much as a pair of Thunderbirds puppets, but now, suddenly, my legs began to feel trembly and I realised it had affected me more than I'd thought.\n\nSo when a sleek dark red sports car slid to a purring stop next to me and Ritch offered to run me home, it was a relief to get in. I didn't have to talk, either, because he was full of what he'd been doing.\n\nIn fact, it's sometimes pleasantly relaxing being with a man who notices nothing much other than himself, though it was kind of him to take me home when he'd just driven all the way up from London.\n\nWhen we got to Perseverance Cottage I pulled myself together and thanked him for the lift, then added firmly that I knew he wouldn't mind if I didn't ask him in, since I had lots to do before the Mystery Play rehearsal.\n\n'That's OK. I'll see you later, after it,' he said, and I managed to smile at him before climbing out of the car and waving him off.\n\nTurning, I spotted Caz through the open door of the barn, doing chin-ups on a crossbeam, like a very strange clockwork toy. Hadn't he got a beam of his own to swing from?\n\nWithout Nick, that night's rehearsals were a bit... flat, I suppose is the only word to describe them.\n\nThe only highlight was when I overheard the new Moses, in answer to God telling him that he'd written down Ten Commandments on tablets of stone, reply testily, 'Could thee not find something lighter? I'm no spring chicken, tha knows! Just as well I hadn't t'carry 'em up t'mountain as well as down!'\n\nI felt hugely tired and unusually down, which was probably reaction from that horrible scene in the graveyard, so I might have just sneaked straight home again rather than on to the pub with the others, except that home was empty without Jasper.\n\nThere wasn't any sign of Ritch after the rehearsals, but he was already in the pub, the centre of an admiring circle. I expect he was telling them all about his cameo film role, too. I sat quietly in the corner with Gareth and Annie for a while, then left early and fairly abruptly when I spotted Polly coming in, wearing a wrapover dress that made her breasts look like a giant pair of loosely packaged white puddings.\n\nI pounced on her near the door, grabbing her sinewy arm. 'I want to talk to you! I know what you've done, Polly.'\n\nShe went the colour of clotted cream and stared at me through a spidery inch of clogged mascara. 'I don't know what you mean!'\n\n'Those tricks you've been playing \u2013 the jars of jam and tomatoes made to look like mine, the ARG harassment, the poisonous fungi in the mushrooms, even the lies you've been telling the Barilloses! And I'm warning you, if there's any more of it, I'll go to the police.'\n\nHer colour came back in a rush. 'Tell them, then, and see if they believe you!' she hissed, then wrenched her arm away with surprising strength and shoved her way through the crowd towards the bar.\n\nThere was no point in following her, but Ritch caught me up outside.\n\n'Wait for me! I said I'd run you home tonight \u2013 you look exhausted.'\n\n'It's been quite a day,' I agreed wearily, though it just goes to show that you shouldn't misjudge people: he might have seemed totally self-absorbed in the pub, but he'd still noticed I wasn't exactly a sparkling little star in the firmament tonight.\n\nSo when we pulled up outside Perseverance Cottage I didn't resist when he put his arm around me and kissed me: apart from suddenly feeling too exhausted to move, I needed comfort.\n\nHe tasted only of minty mouthwash. Now, that was a relief.\n\nAfter a couple of minutes he must have detected a certain lack of enthusiastic cooperation because he sat back again. 'OK, I know when I'm flogging a dead horse. You don't really fancy me, do you?'\n\n'I wouldn't exactly say that,' I replied honestly, 'but...well, to be frank, I find the thought of your morning pee-drinking sessions a bit off-putting!'\n\n'Really? You know, you're the second woman to say that \u2013 though it's perfectly natural, you know \u2013 everyone is doing it.'\n\n'Not round here they're not!'\n\n'Well anyway, I've given it up, now. I get crates of Elyxr delivered instead.'\n\n'What kind of elixir?' I asked curiously.\n\n'Oh, Elyxr is the name they've given to a very special and expensive ionised mineral water. It comes from a secret source high in the Himalayas, where everyone's over a hundred and the local men all father children into their nineties and beyond. What do you think?'\n\n'I think you've got more money than sense. What's ionised water?'\n\n'I've no idea,' he confessed. 'But anyway, now you know I've given the other thing up, does that make a difference?' he asked hopefully.\n\n'Not really, because it's pointless \u2013 I'm simply not harem material.'\n\n'You can be chief concubine,' he offered, cheekily.\n\n'Thanks, but no thanks. And you'd better watch your step with Kylie, too. Did you know she has a very tough boyfriend, in the army?'\n\nHe grinned unrepentantly. 'In the army and in another country, though. Anyway, she's not serious and neither am I; it's just a bit of fun. But you and I could be serious...'\n\n'No we couldn't, don't be daft! There isn't a serious bone in your body,' I said severely, fending him off, then gasped as a spectrally pale face appeared at the window.\n\n'What the hell...?' began Ritch explosively, letting me go just as the door on my side was pulled open.\n\n'Hens,' Caz Naylor said succinctly, with a jerk of his head.\n\n'Oh God!' I scrambled hastily out. 'I entirely forgot to lock them up for the night before I left, and there's been a fox about. Good night, Ritch, thanks for the lift!'\n\n'But, Lizzy\u2014' he began to protest, though when I ignored him and started across the yard towards the henhouse he gave up and drove off.\n\n'I've done 'em,' Caz said from behind me, stopping me in my tracks, and then he turned and loped silently off into the darkness.\n\nMen.\n\n## Chapter 21: Slightly Stewed\n\nThere is already an autumnal feel to the air, along with the nostalgic hint of dead leaves and wood smoke. I put grease bands around my cherry and apple trees, and cleaned out the small greenhouse behind the cottage, before lining it with bubble wrap to conserve warmth through the winter to come \u2013 and perhaps it will be another hard one, in which case I will soon be cooking up my home-made version of fat balls to help the birds get through.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nNick was still away, but very early one morning a postcard of Penzance arrived, with a recipe scrawled on the back for a dessert consisting mostly of Cornish clotted cream. It appeared to have travelled the length of the country before coming home to roost, possibly because the front was tacky so it had stuck to other mail. I had to sponge and dry it before adding it to the album. I expect he wrote it in a restaurant over a lush dinner, probably in the company of some equally lush Poldarkian beauty.\n\nI was about to embark on my daily larder-filling (Lizzy the human squirrel), gardening and pet-sitting activities, though guiltily feeling that I should instead make a proper start on the Just Desserts book, when \u2013 speak of the devil \u2013 Senga rang me.\n\n'I thought you might like to know that Polly Darke and I have parted company,' she told me crisply. 'She rang me up hysterically demanding I drop you as my client, or she would leave me. So I told her to take her business elsewhere.'\n\nI nearly dropped the telephone. 'But, Senga, she earns much more than I do!'\n\n'Perhaps, but she's ten times the trouble and I'm not having one of my authors telling me who else I can or can't represent. Anyway, if she carries on like she's doing, pestering her publishers and doing a prima donna act all over the place, they will drop her, too.'\n\nWhile I was grateful that Senga decided to keep me and ditch Polly, it does give Polly one more thing to hate me for. But now she knows that I know she did all those spiteful things, surely she wouldn't be stupid enough to try anything else?\n\n'Crange and Snicket want to know how Just Desserts is coming along,' Senga said. 'And so do I.'\n\n'I'm collecting recipes,' I assured her hastily.\n\n'Don't forget that you can use old stuff from all the Chronicles, though you need at least fifty per cent new material or your readers will feel cheated.'\n\n'I will, and I'm picking Nick Pharamond's brains, too, only his recipes tend to be pretty sophisticated and I have to dumb them down to my level.'\n\n'I'd forgotten he was some kind of relation of your husband's \u2013 that's lucky. And he's really attractive as well, isn't he?'\n\n'Lots of women seem to think so. He's in the middle of getting divorced \u2013 shall I put in a good word for you?'\n\n'God, yes!' she said enthusiastically.\n\nIn the afternoon Jasper returned and I needn't have worried about what he was going to spend Unks' holiday money on, because he came back laden with stuff to take to university with him: his own kettle, mugs and crockery, plus tons of archaeology books because he'd struck a rich vein in a second-hand bookshop.\n\nHe also sported a strange haircut and lots of new clothes, including some oddly worded T-shirts that probably meant something a mother shouldn't know about: I didn't ask for a translation.\n\nGinny sported a new collar. Jasper tried to persuade me that she was pleased to see me on her return, citing the fact that she hadn't yet nipped my ankles as evidence, but I wasn't convinced. I think she is a one-man bitch.\n\nI went out early next morning to clean out the cage of a rather vicious African Grey parrot, leaving Jasper getting his stuff ready to go to university the following day.\n\nWhen I got back he said Nick had phoned. 'He wanted to wish me good luck for university.'\n\n'That was kind. Where is he?'\n\n'Back in London, but he's coming home soon. He has to do something about the divorce first \u2013 go to the solicitor's maybe, and sign something? He said he and Leila were getting on better now they were divorcing than they ever had while they were married,' he added.\n\n'How lovely. I'm so happy for them.'\n\nJasper grinned. 'He said he was sorry to miss you, and had you managed to make a decent apple pie yet.'\n\n'Ha, ha!' I said sourly. My shortcrust pastry is so light it practically floats off the plate, so I still can't see why the judges at the f\u00eate gave Nick the gold prize!\n\nAfter a quick lunch we set off to the museum and botanical gardens near Southport, which was one of Jasper's favourite trips out as a small boy. He seemed to enjoy the outing as much as I did, though I expect he was just humouring his old mum again. I did let him drive the Land Rover, though, so that might have had something to do with it.\n\nWhen we got home he brought down all his boxes and bags of stuff, and stacked them in the hall, while I cooked his favourite dinner of roast chicken with crinkly, thick-cut chips, followed by little pots of rich, dark chocolate mousse.\n\nIt had been a lovely day and, even if I was sad to see Jasper leave home, I was also happy for him too, because this was how it should be.\n\nSo there I was next day, about to leave my beloved offspring, bag and baggage, marooned in a strange place to start a part of his life that I would only peripherally be involved in.\n\nNot that Jasper was entirely among strangers, of course, since not only was Ginny present, but also his friend Stu and another friend's elder brother were among the students sharing the terraced house.\n\nJasper's bedroom was on the ground floor in what had once been the morning room, so at least if there was a fire he could get out fast... And even after half an hour I was still sitting quivering on the bed from the effect of having driven through the Liverpool traffic system, trying to find the place.\n\nJasper had somehow managed to fit all his stuff in the Land Rover, but now it had exploded to four times its original bulk, like popcorn in a microwave.\n\n'When I went to London to do my cookery course with Annie,' I remarked, looking at it all with amazement, 'we had\u2014'\n\n'Just one rucksack and a sleeping bag each,' he finished for me, ripping the tape off a cardboard box and delving inside for coffee and mugs. He had his own little sink in the corner of the room, which was handy. 'I know, Mum, you've told me.'\n\n'And I had a guitar.'\n\nHe frowned. 'I don't think you ever mentioned the guitar. And you can't play a guitar.'\n\n'No, so I swapped it with someone in the first week for that glass pig with the three little pigs inside it.'\n\n'Oh, yeah, cannibal pig.' He plugged in his brand-new kettle and switched it on. I was ready for a cup of coffee by then.\n\n'You don't have to stay any longer, you know, Mum,' he said kindly, looking up. 'It'll be dark before you get home if you don't get off soon. Besides, no one's going to come in here while my mother's hanging about. I can hear them talking in the kitchen, so I'll go and take my food and stuff through in a minute when you've gone.'\n\n'It'll be dark anyway by the time I get home,' I pointed out, feeling slightly hurt. 'But perhaps I had better go and leave you to get on with it. I only hope I can find my way back out of Liverpool again.'\n\n'It'll be easier finding your way home than getting here, because you'll know where you are once you're out of the city. Come on, I'll see you off.'\n\n'I'll phone you tonight, just to make sure everything's all right, shall I?'\n\n'Well, I might be out somewhere too noisy to hear it,' he said dubiously, 'but you could leave a message.'\n\n'No, that's OK \u2013 you call me when you feel like a chat,' I suggested, with a brightness I certainly didn't feel. 'I'd love to know how you're getting on.'\n\nOut on the pavement I gave him a hug, which he suffered with saintly resignation, then got into my now empty Land Rover.\n\nHe leaned in at the open passenger door and said, 'Now, Mum, remember what I said, and don't try changing any plugs!'\n\n'I only melted one once!' I replied indignantly.\n\n'And switch the light off before you change a light bulb. Don't mess about with the timer on the boiler \u2013 and if the flame goes out again, ask Unks to send Joe down to do it, or Uncle Nick.'\n\n'Now look here, Jasper, I'm not completely helpless, you know! I may not have an affinity with anything electric but\u2014'\n\n'Actually, you're the kiss of death to anything electric,' he interrupted firmly. 'Anyway, I expect I'll come home for the odd weekend before Christmas, so you can save anything that wants doing until then, if you like.'\n\nHold on, I thought, shouldn't this be me giving out the instructions? And not about electricity either, but drugs, safe sex and eating properly (though the eating bit wasn't so pressing since I'd packed enough food and drink to last him for about ten years).\n\n'I expect I'll survive,' I said, then looked at him \u2013 tall, skinny, his dark hair whipping about in the brisk breeze \u2013 and swallowed hard.\n\n'Goodbye then, darling. Hope you settle in quickly,' I said slightly huskily, though I did manage a smile, before starting the engine and heading in the direction I hoped would take me home.\n\nThe sun was sinking in a clear sky, but just like the song, it was raining, raining in my heart.\n\nAnnie had kindly been to the cottage to shut up all the Myrtles and Honeys, and left me a pineapple upside-down cake with a nice note telling me to ring her if I wanted to, though ten to one Gareth was there, or she was out doing something godly and good with him.\n\nThe cottage looked desolate and empty, which was exactly how I felt. Jasper had been the centre of my universe for over eighteen years: what was I going to revolve around now?\n\nThen Unks called to ask me how Jasper had settled in and I told him how he'd turned the tables on me in the good advice stakes, which made him laugh.\n\nAfter that I turned to food and drink (last year's apple wine and flapjacks) for comfort until bedtime, when I fell into a state of comatose indigestion.\n\nI wasn't feeling much happier in the morning, especially since I also had a hangover. It was lucky I didn't have any pet-sitting jobs to do, for I kept bursting into tears, which was quite unlike me.\n\nI stripped Jasper's bed so that it would be made up all nice and fresh if he was homesick and popped back for a night or two. But I didn't linger in there, with the bare surfaces where his computer had been and the gaps on the bookshelves. It was too poignant.\n\nWhile the bedding was going through the wash and dry cycle, I gave the cottage the sort of thorough cleaning it only gets when I'm trying to distract myself from something. It didn't quite work, though, because I kept finding things of Jasper's that set me off again.\n\nIn fact, Nick walked in on me just before lunch and discovered me slumped in a sobbing heap in the old wicker basket chair in the kitchen.\n\n'Lizzy? What on earth's the matter?' he demanded, coming to a sudden startled stop and staring at me. 'I don't think I've ever seen you cry, except that once at the hospital!'\n\nI snuffled back the tears and held out the large black Snoopy sock with a hole in the heel that I'd found down the back of the radiator. 'It's J-jasper's!' I wailed.\n\nLooking relieved, he bent down and hauled me to my feet, then gave me a little shake. 'Oh, is that it? Come on, Lizzy, stop wallowing in pathos! He's only a few miles away at university, not the other side of the world. You can go and see him any time you want.'\n\nI sniffled and tried to pull away, but he didn't let go. 'No I can't! He's got to make his own life there, so I couldn't possibly keep popping in and fussing. But it's such a long time until Christmas!'\n\n'I expect he'll come back for a weekend before then. Look, there's no point in sitting here moping like a wet weekday,' he added, 'so get your jacket and let's go.'\n\n'Go? Go where?' I said indignantly, though at least I'd stopped feeling weepy.\n\n'I'll take you out for lunch. You look quite decent \u2013 no need to change. Come on.'\n\nConsidering I was dressed in my oldest jeans and a washed-out sweatshirt, he had to be joking.\n\n'I don't think I want to go out, thank you, Nick,' I began, but he wasn't taking no for an answer, so in the end it was easier to give in, though I insisted on changing, bathing my face in cold water and brushing my hair first.\n\nLunch was fish and chips, well laced with salt and vinegar, eaten out of newspaper on the seafront at Southport, with the expanse of beach stretching away under a cold blue sky scudding with clouds.\n\nBut that was after he'd made me walk for miles, so by then I was starving and, I admit, feeling much more optimistic.\n\nAfter he dropped me back at the empty nest, Annie came around with someone else's Maltese terrier, so I didn't have time to start moping again and, anyway, all that fresh air and exercise seemed to have numbed the pain a little.\n\nI even resisted the urge to call Jasper and see what he was doing, but instead got hot and sweaty stuffing his duvet back into its clean cover, a task somewhat like giving birth in reverse, but without the excruciating agony followed by a stranger trying out their embroidery skills on your private parts.\n\nThe good thing about being pregnant (possibly the only good thing apart from the eventual baby) is that at least you know where your child is and have a pretty good idea what it's doing. Now we were only joined by the frail umbilical cord of Jasper's new mobile phone.\n\nMimi and Juno's Glorious Autumn Garden Colour luxury coach tour set off just after seven in the morning and, since I'm always up early, I volunteered to drive them to the pick-up point outside the New Mystery pub.\n\nMimi was highly excited, but I was sure Juno had thoroughly searched both her luggage and her person for any sharp implements with which she might attempt to steal cuttings from the various stately homes they were to visit, so provided she kept her eye on her I was hopeful they would have an enjoyable holiday.\n\nWhile Juno was seeing to the luggage being stowed away, Mimi dashed into the corner newsagent's shop to buy cough candy and Uncle Joe's Mint Balls for the journey, and came back carrying several little paper bags, her cheeks bulging like a hamster's.\n\nAs the bus pulled away I looked up at Mimi, who was seated in the window, a sweetly angelic little old lady in a pink velvet Alice band and matching pearly beads. She waved benignly at me and I could have sworn something glinted in her hand...\n\nBut I must have imagined it, or she was wearing a ring... though I'd never known her to wear one, since they got in the way of her gardening.\n\nThe coach vanished down the high street and I turned the car for home, calling at Annie's cottage on the way as much from the hope of hot croissants as to see if any more pet-sitting jobs had come up.\n\nIt was a Danish pastry day, which was nearly as good, and she was bubbling with suppressed gossip.\n\n'Gosh, Lizzy,' she said excitedly, 'there were ructions last night in the village! Kylie's boyfriend got leave from the army and came home a couple of days ago, and, of course, someone told him all about what Kylie's been up to with Ritch. So he went round to Ritch's and there were loud voices, then they had a fight! Or Kylie's boyfriend hit Ritch, at any rate, and the police got called.'\n\n'Well, I did warn Ritch about the boyfriend, so it's entirely his own fault.'\n\n'And Kylie's,' Annie pointed out fairly.\n\n'That's true, but I bet she put all the blame onto Ritch.'\n\n'Ritch's just phoned me up. He tried you first, but there was no answer. He wants one of us to go round and take Flo out, because he isn't feeling well.'\n\n'Meaning battered and bruised? I'd better go on my way home and I'll ring you later and tell you what he says \u2013 if I can catch you,' I added. 'You're always out these days.'\n\nMainly with Gareth, whom I still hadn't had that quiet word with. What was holding him back? Perhaps I ought to give him the birds-and-bees talk I gave Jasper before he went to university?\n\nExcept that that wasn't such a great success, come to think of it: Jasper said he was sure he already knew much more about it than I did.\n\nRitch took his sunglasses off to show me his black eye, but he seemed to have escaped relatively unscathed otherwise.\n\nTigger certainly hadn't lost any of his bounce. 'Kiss it better?' he suggested hopefully.\n\n'No chance!' I told him severely. 'And I did warn you.'\n\nHe shrugged. 'It was just a bit of fun on both sides, nothing serious. I don't know what Kylie told him, but he swung a punch at me as soon as I opened the door and took me by surprise, then we had a bit of a scuffle until the police came and broke us up. I didn't press charges.'\n\n'Magnanimous of you!'\n\n'It is really, because Make-up are going to have their work cut out disguising this shiner for a couple of weeks, unless they can work it into the storyline... That's an idea,' he added thoughtfully, heading for the phone.\n\nAs you can imagine, the fight was the main topic of conversation at the Christmas Pudding Circle on Monday, when we were all crammed into Annie's tiny cottage. But once we'd exhaustively thrashed that out, we got down to deciding which sweets we would make for the WI hampers and then divided up the big packet of Cellophane circles we'd ordered, from which we would make little cone-shaped bags, to tie up with shiny ribbon. Rum truffles fell to my lot this year, but I also volunteered to make sugar-free Fruit and Nut Munchies for our one diabetic Senior Citizen.\n\nKylie was much subdued at the Mystery rehearsal, though at the same time strangely triumphant: I expect that's the glow from having two men fighting over your charms. She didn't look a bit embarrassed during the Annunciation scene, but I'd have curled up and died.\n\nAfterwards her fianc\u00e9, a stocky and pugnacious-looking young man, collected her and whisked her away, though he didn't have to worry about Ritch tonight, because he wasn't anywhere to be seen.\n\nNick, who seemed to be in a foul mood, said sourly, 'Looking for Casanova? I hear Kylie's boyfriend's sorted him out big-time.'\n\n'Actually, there's hardly a mark on him except a black eye, and I think he only got that because he was taken by surprise,' I said coldly.\n\n'Oh? And did you kiss it better?' he asked sarcastically, so clearly his mind ran along similar lines to Ritch's. Must be a man thing.\n\n'No,' I said shortly, going slightly pink. 'We're not on kissing terms.'\n\nHe raised an eyebrow. 'That's not what Caz says.'\n\n'Caz? You've been discussing me with Caz? And I haven't\u2014' Then I broke off, recalling that actually I had shared a kiss with Ritch.\n\n'Slipped your memory?'\n\n'Not that it's any of your business, but Ritch did give me a kiss \u2013 for comfort. Tom's parents showed up while you were away. They didn't tell me they were coming, but Gareth let me know and I went to meet them at the churchyard. Only Polly's been writing lies to them and turned them against me, so there was a horrible scene. It really upset me. Ritch picked me up on the way home and he was very kind.'\n\n'I bet he was,' Nick said nastily.\n\nUp to this point I'd been automatically ambling in the direction of the pub, but I stopped dead and demanded, 'What is the matter with you tonight? You were really nice to me the other day too, when I was upset about Jasper leaving home, and now you're being horrible!'\n\n'I hadn't had a chance to speak to Caz then.'\n\nI glared at him: 'Have you told Caz to spy on me?'\n\n'To watch over you, after he told me about the mysterious jar of tomatoes and the mushrooms \u2013 which, by the way, you might have mentioned to me yourself!'\n\n'I dealt with it,' I said shortly. 'And it did sound a bit unlikely \u2013 The Case of the Poisonous Mushrooms \u2013 so I thought you might not believe me anyway.'\n\n'True, I can't say it really convinced me until Leila told me about getting that pot of jam in the post. She had a friend run some tests on it.'\n\n'She did? And was it poisoned?'\n\n'It contained a strong emetic.'\n\n'Well, I didn't send it.'\n\n'I didn't think you had, but there doesn't seem to be any real proof that Polly did either, does there? Still, I wasn't taking any chances, so I asked Caz to watch out for you while I was away.'\n\n'There's no need, I can look after myself. And what's more, whatever I get up to is my own private business!'\n\n'Pardon me for caring!' he snapped.\n\n'Look, Nick, I don't know why you're needling me like this tonight, but I've had enough, so good night!' I said, and walked off. Though I felt his eyes boring between my shoulder blades, he didn't follow me, so I expect he had other fish to fry.\n\nIn the morning I found an old envelope on the doormat, with Marian Potter's secret Lancashire hotpot recipe scribbled on it in Nick's spiky handwriting, so I suppose it was an apology of sorts for his foul mood.\n\nThen when I opened the front door a parcel fell on my feet: The Perseverance Chronicles copy-edits had arrived with amazing speed. There was also a rough proof of the book cover, which was too twee for words: why did they always turn my solid northern sandstone cottage into something thatched, gabled and timbered to within an inch of its Anne Hathaway life?\n\n## Chapter 22: Given the Bird\n\nThis morning I made some sugar-free sweets, suitable for diabetics: a sort of sugar plum, without the sugar! I tried various blends of chopped dried fruit and nuts: apricot, almond, brazils, raisins, with a little orange juice to bind the mixture together. I rolled some into little log shapes and others into balls, and left them on a sheet of baking parchment to dry out for a couple of hours. I think they would keep very well in the fridge, but they were so delicious they didn't last long.\n\nThe Perseverance Chronicles: A Life in Recipes\n\n'Gareth found three large frozen geese on the vicarage doorstep this morning,' Annie said.\n\n'Frozen geese?' I questioned, putting a mug of tea and a plate containing three small wedges of apple pie in front of her, each marked with a different coloured cocktail stick flag. 'I know we've had some night frost, but it hasn't been that cold yet!'\n\n'Plucked, cleaned, oven-ready, with an anonymous note saying they were for the Senior Citizens' Christmas dinner. Gareth called me right away, to ask my advice.'\n\n'That was a pretty generous donation,' I said thoughtfully. 'Frozen...?'\n\n'Yes, they were in those big, see-through roasting bags, but there were no supermarket labels on them or anything like that.' She looked down at her plate. 'Three pieces of pie?'\n\n'Little pieces. I want you to taste them and tell me which one you think is best. Here, have some cream.'\n\n'All your apple pies taste wonderful.'\n\n'But Nick's obviously taste better, or he wouldn't have won the gold at the f\u00eate.'\n\n'Yes, but you won practically everything else, including Best Middlemoss Marchpane, Lizzy, so you might let him have that one small triumph.'\n\n'Not if I can help it.' I watched her drizzle cream over the pie and asked, 'So, what did you and Gareth do with the frozen geese?'\n\n'We took them up to the Hall. I thought we might as well, since Mrs Gumball cooks the Senior Citizens' dinner anyway \u2013 with your help, of course \u2013 and there are huge freezers in the cellar she can store them in.'\n\nShe tried another forkful of pie and chewed thoughtfully. 'Nick was there, and he said this year he was going to help cook the dinner.'\n\n'He is? Then Mrs Gumball won't need me as well, will she?'\n\n'I think she will, because I overheard Nick telling Gareth that you would both be helping her,' Annie said. 'And when we ran into Clive Potter on the way back, Gareth told him about the geese and how Nick and you were going to help Mrs Gumball to cook them for the Senior Citizens, so I expect it'll be in the next Mosses Messenger.'\n\n'Then it can come straight back out again,' I said sourly. 'I have no intention of being the Demon Chef's whipping boy. He can find another one.'\n\n'Whipping girl \u2013 and you're very crabby today,' she observed, then looked down at her empty plate as if surprised that she'd cleared it down to the last pastry flake. The coloured sticks lay round the edge like the hours on a clock.\n\n'Which slice did you like best?'\n\n'To be honest, I couldn't really taste much difference,' she confessed. 'They were all delicious!'\n\nI sighed. 'Maybe he uses Dark Arts and says a spell over his cooking pots?'\n\n'You'll be able to watch him cooking the Christmas dinner and find out,' she teased. 'Has he done something to annoy you?'\n\n'When does he not do something to annoy me? But yes, he has excelled himself, because he set Caz on to watch me while he was away!'\n\n'But Caz already does watch you. I mean, he keeps an eye on the place because it's part of the estate \u2013 and probably also because he thinks of you as family, however remote the connection.'\n\n'Yes, I know that, but Ritch kissed me the other night when he drove me home and Caz not only interrupted us, but he reported it to Nick!' And I repeated to her what Nick had said, about not being sure it was Polly playing the nasty tricks and even thinking I'd imagined them at first.\n\n'Of course, I haven't told him about Ophelia being in ARG, and I don't suppose Caz has either. I'm not sure how he would take it and he might feel he has to tell Unks.'\n\n'He'd probably find it funny, Lizzy!'\n\n'I wouldn't put it past him. He seems to find most things I do funny \u2013 except the bits involving Ritch. And that was a perfectly innocent kiss, even if it was any of his business who I kiss, which it isn't.'\n\n'It was?'\n\nI grinned. 'Well, it was a perfectly enjoyable one, at any rate, but just between friends. Ritch knows I don't want a relationship with him \u2013 or anyone else,' I stated firmly. 'I'm getting quite tired of Unks, Mimi, Juno and now even Jasper trying to throw Nick and me together, when it must be clear it's a complete non-starter.'\n\n'But, Lizzy, it's very evident even to me that he's concerned about you, and jealous!'\n\n'Do you really think so?' I considered the matter carefully, then dismissed it. 'I've noticed he's a bit jealous, but I'm sure it's not of me personally, but some dog-in-the-mangerish male thing to do with property. I'm part of his family and living on his land, as it were... except it's still Unks' land, of course.'\n\n'No, you're wrong, Lizzy: you think I don't notice these things, but I've seen the way he looks at you and I'm sure he's in love with you.'\n\n'No way,' I said positively. 'Just because you're in love, you think everyone else should be!'\n\nShe went slightly pink under the freckles, but carried on doggedly, 'And you missed him when he was away, so I don't think it's all one-sided, either.'\n\n'No I didn't,' I lied, 'which is just as well, because that's what he always does, isn't it? He goes away. We split up the first time because he wasn't going to let a little thing like our romance come between him and his globetrotting, recipe-collecting expedition. I'm only surprised he's stayed around Middlemoss so long now, because he's obviously getting itchy feet.'\n\n'Now, Lizzy, you know if it weren't for Tom, he'd have spent much more time here in the last few years. He's quite sincere about loving the place.'\n\n'He may love the place, but he's only passionate about his cooking.'\n\n'Well, he's using it to try and impress you,' she insisted stubbornly. 'He knows the way to your heart is through food.'\n\n'He's certainly not going to do that by snatching my gold prizes away. Though if he has got any designs on my virtue, I'd probably do almost anything he asked, for that coffee granita recipe.'\n\n'Lizzy!' she exclaimed, shocked.\n\n'Just joking,' I said hastily. 'But you've got it all wrong, because I'm not interested and he's not interested. After all, if he cared about me that way, what's to stop him from telling me?'\n\n'But you were only widowed in August and his divorce isn't through yet, so that might be holding him back, don't you think?'\n\n'Yes... I suppose it might if he felt about me like that, which he doesn't. And I don't really feel newly widowed, now I'm over the shock, since Tom and I were so estranged.'\n\n'If Nick was a dog, he'd be a big, dark, Irish wolfhound,' she said inconsequentially. 'Ritch is a bit of a tomcat.'\n\n'I don't like the way your mind's working,' I said severely. 'Gareth has been a demoralising influence on you. I'm afraid poor Juno's feeling a bit demoralised too, because she rang Unks and told him she had to explain precisely what Mimi was doing to someone's stately garden with nail scissors. You know, I thought I saw her holding something metal up when the coach pulled away the morning they left. She must have got them in the corner shop when she was buying sweets.'\n\nAfter she'd gone I got out the copy-edits of my book and went through them for errors for the second time, before parcelling them up and taking them down to the post office. The cover I pinned to the kitchen notice board in the hope it would grow on me, but it hasn't yet. Nor do all the flowers depicted in the cottage garden bloom at the same time anywhere other than in the artist's demented imagination \u2013 or not in this hemisphere, anyway.\n\nJasper seemed to be settling in well, and I sent off a box of chocolate-coated candied peel. I hoped he was eating properly and not pickling his liver with spirits, like many students do.\n\nI'd decided Annie was right about my apple pies and you couldn't improve on perfection, so Nick winning first prize must have been a fluke. The final one I tried that night, though, using dark Barbados sugar to sweeten the apples, had an interesting slight toffee-apple flavour, which made a nice change.\n\nA policeman who sounded like that boy who favoured finger food called to say that one of the 2CV's wheel nuts had been handed in by a metal-detecting member of the public and added, in a seemingly casual aside, that it appeared to be in a perfectly good state of repair, the thread undamaged. Then he informed me the inquest was set for the end of January, and rang off.\n\nThe light was fading fast. I put on a warm coat and went to lock up the hens and, as I did so, Caz emerged slowly from the shadow of the barn. Since my slight contretemps with Nick I had entirely ignored Caz when he was around, which hadn't seemed to bother him in the least \u2013 if he'd even noticed.\n\nNow I ran my hand distractedly through my tangled hair and said, 'Oh, Caz, the police have found one of the missing wheel nuts \u2013 or rather, I think someone found it and handed it in \u2013 and the thread on it looks fine. I'm sure they still think I loosened them on purpose and encouraged Tom to take my car!'\n\nCaz glanced at me in his usual obliquely wary yet not unfriendly way, then said, 'Don't you fret, our Lizzy, it'll all come out in t'wash,' and loped off towards the woods, his gun under his arm.\n\nI stared after him: if he continued getting so garrulous, we might soon be able to hold an entire conversation.\n\nHe'd probably be back later, too, because the Mummers would be coming round to the workshop to practise. I hoped Ritch came too and popped in as usual for coffee and chat afterwards. I felt like some company.\n\nRitch not only stopped by, he took me out to the caf\u00e9-bar in the former Pharamond's Butterflake Biscuit factory, which was very pleasant now I had firmly established my unavailability for his healthy sex rota.\n\nOr at least, I think I have...\n\nOf course this didn't stop him flirting with me, but actually I found that quite enjoyable now that Nick had suddenly gone even more morose and distant than usual. And also, of course, now I knew Ritch had stopped his more dubious habits he had regained a little more of his previous attraction!\n\n'You look great tonight \u2013 that top really shows off your curves,' he said, leaning forwards towards me across the table. 'I'll butter your pie dish any time you give me the word!'\n\nI grinned. 'Hanging round my kitchen has given you a whole new vocabulary!'\n\n'And a whole new taste for real home-baking,' he agreed.\n\nAfterwards Ritch dropped me off at the cottage and I let him have a good-night kiss, though I didn't want to make a habit of it... well, not if Caz was watching.\n\nThere was no sign of him, but that didn't mean he wasn't still lurking about somewhere.\n\nMimi and Juno got back from their coach tour late on the Saturday, then walked down next afternoon to give me my present of a pair of flower-patterned gardening gauntlets, and for Juno to show me the photographs she had taken on her digital camera.\n\nMimi was, as usual, not only unrepentant about her bit of petty plant-pilfering, but entirely failed to understand what the fuss was about. 'I don't suppose they will take, because it is the wrong time of year, but anyway, gardeners should share things,' she said.\n\n'It might have been politer to ask first,' Juno said patiently.\n\n'Oh, they didn't really mind,' Mimi said. 'I promised them some cuttings from my rare lavender when it's the right time to take them \u2013 and then perhaps Tom could drop them off next time he is heading in that direction.'\n\n'Tom's dead, Mimi,' I said gently.\n\n'Oh, is he? Then perhaps if I wrap them in wet kitchen towel and a plastic bag and post them, they will be all right. Not that the postal service is all it used to be,' she added.\n\nWe had to have the next CPC meeting at Marian's, because she had boxes and boxes of big green apples a friend in the WI had given her and she was desperate to get rid of them.\n\n'Maggie said they were eaters, but they cook well too, and it was such a bumper crop this year that she didn't know what to do with them,' she explained. 'So I said I was sure we could divide them up among our group.'\n\n'There are an awful lot of them,' Fay said, 'and I've got more than enough of my own to be going on with.'\n\n'Me too,' I agreed, but since there were loads left even after the others had all taken some, I ended up with the lion's share, simply because I couldn't bear to see them go to waste. I couldn't imagine what I was going to do with them.\n\nI spent a large part of that Tuesday filling the freezer with apple pies, pur\u00e9e and crumble, yet had hardly made a dent in the apple mountain...\n\nAnd Nick was not at the Mystery Play rehearsal, because he'd gone off on his travels yet again, according to Mimi, who'd come down to the village hall with Juno, being in one of her restless phases. Then she added meaningfully that he was catching up on things he should have done before, only he hadn't wanted to leave Middlemoss, so I expect he'd finally just got bored and restless.\n\n'Oh, it's the nativity, my favourite bit!' Juno said, as the vicar called, 'Shepherds and angels to the crib, please!'\n\nAs they gathered round, I noticed for the first time that the Nine Angels of the Annunciation all had bright white patches on their wings where they had repaired them with fresh sturdy new feathers.\n\n'See yonder breet light shining on t'owd stable?' said the First Shepherd, adjusting his tea-towel headdress.\n\n'Aye, I do that, and a right bobby dazzler it is an' all!' replied the Second Shepherd, then nudged his friend as the Three Wise Men appeared. 'Hey up, we've got company.'\n\n'Why wasn't there a Wise Woman too?' Mimi asked in a penetrating whisper.\n\n'I don't know \u2013 perhaps they didn't fancy riding a camel all that way?' suggested Juno.\n\nBy the end of a very hectic week, the larder shelves were groaning with apple-based jams and jellies, apple sauce, apple chutney, apples in wine and spiced apple... you name it, and I'd made it.\n\nOf course I'd had to keep up with the pet-sitting and gardening too, but apart from escaping to Butterflake's for an hour or so again one evening with Ritch, the days passed in a sticky haze.\n\nThen suddenly I was down to the last few apples in the bottom of a box, and made a batch of fritters with them, which I ate sifted with brown sugar, drizzled with honey and blobbed with thick cream. After that I felt completely appled out and never wanted to see another one again, unless, of course, it was as apple wine.\n\nI ceded the apple pie prize to Nick in perpetuity. And speaking of Nick, I'd heard nothing from him for days and then suddenly had a spate of postcards all at once \u2013 every single one bearing some kind of tart recipe! Perhaps that was what his next article would be about?\n\nI gave all my CPC friends a jar of apple chutney at the next meeting, but I had a feeling I'd still be trying to offload the remainder next autumn.\n\nNick was back next day, just in time for the Mystery Play rehearsal but, like Ritch, appeared to have lost interest in going to the pub afterwards. I wasn't finding it very tempting either, because I could only take playing gooseberry to Gareth and Annie for so long, no matter how kindly they went out of their way to include me in their conversation.\n\nAs we came out of the village hall, I thanked Nick for all the postcards, then asked curiously, 'But why are all the recipes for tarts?'\n\n'I thought you deserved them,' he said shortly and then strode off into the night, looking distinctly Mr Rochester again.\n\nI don't know what was biting him, unless he'd heard about my occasional friendly drink at Butterflake's Bar with Ritch, and misconstrued it?\n\nI spent the rest of the evening catching up with the next Perseverance Chronicle, having not had time to keep up with it while so preoccupied with apples \u2013 and apples formed the basis of most of what I wrote! I also made a few more notes for Just Desserts, because I needed to get going with that soon, before Senga started snapping at my ankles.\n\nIf Nick hadn't been sulking I could have asked him for some ideas, but at least he'd provided the inspiration for a whole chapter devoted to tarts...\n\n## Chapter 23: Put Out\n\nThe vicar is delighted to announce that due to an anonym ous benefactor, this year's Senior Citizens' Christmas dinner (to take place on 1 December) will be roast goose with all the trimmings! As usual it will be cooked by Mrs Eva Gumball up at the Hall, most kindly assisted by Mrs Lizzy Pharamond and also, adding that touch of cordon bleu, Mr Nick Pharamond! It will be delivered piping hot to the village hall, courtesy of our friendly local Meals on Wheels volunteers.\n\nMosses Messenger\n\nThe dankness of early November set in and the children had been collecting firewood for Friday's Bonfire Night for the last week or so. I thought Jasper might come home for that, but he said he was too busy, though busy with what, he didn't inform me, and it was probably better not to ask.\n\nStill, he seemed to have settled down very happily at university and was enjoying his lectures. He told me what had been discovered about the Vikings' dietary habits, from excavating their cesspits at York, at more length than I really wanted to know. It's quite amazing what passes through the digestive tract more or less whole, isn't it?\n\nMarian brought the latest issue of the Mosses Messenger and pointed out the announcement about my involvement in cooking the Senior Citizens' Christmas dinner. It was a fait accompli, because once it had been proclaimed in the parish magazine, there was no getting out of it... unless they were to forget about my helping by then? It was almost a month away.\n\nSaying I never wanted to see another apple again was obviously tempting fate, for Marian then asked me to make toffee apples for Bonfire Night \u2013 she runs a little refreshment table with the proceeds going to charity. As usual, Miss Pym would provide a tray of treacle toffee, Annie gingerbread pigs for the children and Faye would bake parkin. There was usually someone roasting chestnuts too and I absolutely adored those.\n\nMarian had yet more apples in her car to give me, but I didn't mind really, especially after I thought up a variant, Treacle Toffee Apples, and added it to my Just Desserts collection. Then it occurred to me that the Bonfire Night celebrations in Middlemoss would make a whole chapter of the next Chronicles, if I included a few other interesting snippets of information, like the fact that we always burned an effigy of Oliver Cromwell, warts and all, and not Guy Fawkes like everyone else.\n\nIt was not so much that the villagers were all staunch Royalists in the Mosses, just that they knew how to enjoy themselves and deeply resented the Puritans, or anyone else, trying to put a damper on their fun, and especially the Mystery Play.\n\nAnd speaking of the Mystery Play, the Tuesday rehearsal went very well... and in my experience, if the November rehearsals go well, then the final dress rehearsal is a total disaster! But the performance itself on Boxing Day would go down wonderfully, whatever happens, because everyone would be well oiled with mulled wine and marinated in anticipation by then.\n\nBut on Tuesday even Nick had cut out the innuendo from his interpretation of Adam and played it straight. Sombrely, even. He appeared to be still sulking, though I wasn't sure what about, and he went off again straight afterwards. There was no Ritch in the pub, either, so I played gooseberry with Annie and Gareth for a while, then went back home, where I ate two toffee apples. Just as well I'd made a lot.\n\nI sent Jasper some of the treacle toffee left over from the apples, which I'd moulded into a square and then broken up, plus a rawhide bone for Ginny \u2013 might as well blunt her teeth before he brought her back for Christmas. I was missing him so much. It seemed to hurt more as time went on, though part of me was also, of course, happy that he was having a good time at university.\n\nThe toffee and treacle-toffee apples were all wrapped in circles of Cellophane and piled back into the empty apple box by Thursday, when Marian collected them after the second Mystery Play rehearsal of the week. Apparently that one had also gone swimmingly, so we were both now convinced that the dress rehearsals were doomed to some kind of disaster.\n\nBonfire Night, the following day, was likely to be freezing, and I felt increasingly sure my guess about another cold, snowy winter would be right: the amazing number of berries on everything was a dead giveaway.\n\nAnnie never went out on Bonfire Night, staying in to comfort Trinny, who clearly associated loud firework bangs with some unimaginable terror from her past and became a shivering heap. She said Gareth would show his face at the event, before joining her to roast chestnuts over the fire and watch the home videos Annie's parents had sent her of their VSO work in Africa.\n\nI expect there will be at least a chaste foot of sofa between them.\n\nUnks was away and Juno wouldn't bring Mimi down for the bonfire, since she got much too excited, but instead would treat her to a short private display of Emerald Cascades and Glittering Fountains in the walled garden before cocoa and an early night.\n\nWhat Nick was doing I had no idea, and nor was I even remotely interested, so I set out on my own at seven, torch in hand, my innards warmed by a strong slug \u2013 or maybe two \u2013 of Miss Pym's rightly famed damson gin: last year's had been an excellent vintage, but I was down to the last couple of bottles.\n\nThe fire was well alight when I got there and the first of the fireworks were going off, under the direction of Clive Potter. There was quite a crowd about and the refreshment table was doing a roaring trade. I didn't buy one of my own toffee apples, but I did purchase a plastic cup of mulled wine.\n\nLooking round the faces brightly lit by the fire I spotted many familiar ones, though some, like Polly Darke and her friends, were not so welcome. On the other side of the bonfire I could see Jojo, Mick, and almost the entire Mysteries cast. In fact, most of the Mosses residents had turned out as always, though I expect the event wasn't sophisticated enough for Ritch and his crowd.\n\nFrom time to time, one or two of the more rebellious teenagers sneaked off into the darkness outside the firelight to set off explosive fireworks, and were yelled at for their trouble: it was all much as usual.\n\nI sat on a log and peeled hot chestnuts out of a paper cup, then got another tumbler of punch and began to feel a lot happier. 'This is fun, isn't it?' I said, finding myself standing next to Ophelia, who was swathed head to foot in a Tolkien-style woollen cloak with a tasselled hood. It looked pretty weird by firelight, but probably not as odd as the full-length knitted coat that I was wearing, a labour of love presented to me by Annie last Christmas. It had lots of little hanging daggy bits like a raddled sheep, and the strident colour combination meant I could only wear it in the dark.\n\nOphelia's white face was upturned and rapt, watching a sunburst of stars. 'Oh... it's sooo beautiful!' she sighed rapturously. 'Beautiful, beautiful stars. Stars...'\n\nThen as the firework flickered and went out she turned to me and said excitedly, 'Star! Of course! I'll call the baby Star!'\n\n'Star Locke?' I said doubtfully, though of course it might by then be a Star Naylor. 'If it's a boy, it might sound a bit odd.'\n\n'No, no... beautiful!' she murmured, and another firework shot up into the sky and exploded into a galaxy of pinprick lights. 'Better than Rambo...'\n\n'That's very true,' I agreed, beginning to feel a bit muzzy and wondering if my earlier shots of damson gin hadn't been such a good idea. Or perhaps the punch was stronger than usual. Whichever it was, I had the feeling the chestnuts were sloshing about in an awful lot of liquid, and it was probably about time to call it a day and go home... especially since Nick had suddenly materialised out of the shadows nearby like the Prince of Darkness.\n\nHe was looking at me with what appeared to be acute disapproval: so nothing new there, then.\n\n'Must find Caz and tell him about stars,' Ophelia said, looking around her vaguely, though you'd need ESP to find our chameleon of Middlemoss if he didn't want to be found.\n\nShe wandered off and I too turned to go, but had only taken a step or two away from the firelight when something landed with a thud just where we'd been standing and immediately exploded with a horrendous bang and a shower of bright sparks.\n\nI put my hands over my ears and staggered, almost falling \u2013 and then was suddenly knocked flat by someone large and heavy. He landed on top of me and rolled me over and over and even winded, shocked and with my face pressed into icy mud, I somehow knew it was Nick. After what seemed like ages his weight was removed and urgent hands ripped my woolly coat off.\n\nThere was a smell of singed wool, and also, possibly, singed me.\n\nI turned over slowly, dazed and winded, then sat up in time to watch him jumping up and down on my coat. I knew it was ghastly, but it didn't quite merit that treatment.\n\n'Lizzy, are you all right?' Marian cried, running over and trying to haul me to my feet, only my knees seemed to have given up and I was a dead weight.\n\n'I'm fine,' I gasped, reinflating my lungs and trying to wipe the mud and grass from my face.\n\nClive appeared out of the darkness and declared vengefully, 'I don't know who threw that firework, but if I find him, he'll wish he hadn't!'\n\n'No one would be stupid enough to throw it in this direction on purpose. It must have been an accident, Clive,' Marian said. 'Those boys just wouldn't be told!'\n\nI looked around suddenly. 'Ophelia? Is she all right? Only we were talking together just before the firework went off.'\n\n'Don't you worry about her, she was well out of range and that Caz's with her,' Marian said soothingly. 'You were closest: did it burn you anywhere?'\n\nNick picked up my mangled coat and examined the limp and ruined remains with satisfaction. 'There, that's out. Only just caught it, though.' Then he bent down and hauled me effortlessly to my feet, though he had to keep one arm around me to stop me falling over again.\n\nWhen he realised I was trembling violently from a mixture of shock and cold, he shrugged out of his leather jacket and wrapped it around me, the silk lining warm and slithery.\n\n'I think Lizzy may have singed the back of her legs a bit, Nick,' Marian pointed out worriedly. 'Her jeans are charred in a couple of places.'\n\n'Yes, and I can't seem to stand up,' I said weakly.\n\n'Shock,' Marian said. 'Stand back, everyone, and let her get some air!'\n\nUntil that moment I hadn't even realised that the ring of spectators was pressing close, watching avidly, including Polly Darke, a half-smile on her lips like a slightly warped Mona Lisa. Then her eyes shifted sideways to Nick and she slowly took first one step back, then another, until she vanished into the darkness.\n\nI blinked. Maybe I'd imagined her...\n\n'Drink's more likely than shock, the way she was knocking the punch back,' Nick was saying unsympathetically. 'I don't think there's much harm done, but I'll take her home.'\n\n'Perhaps you should bring her to the post office first and Marian can see if she's burned?' suggested Clive. 'It might be bad enough for Accident and Emergency.'\n\n'I don't think so,' Nick said, 'but if it looks worse than I think it is when I've got her home, I'll phone the doctor.'\n\n'You do that,' Marian agreed.\n\n'Your voices sound strange,' I commented, and so did my voice, too \u2013 frail and far away. And then everything seemed to be shifting dizzyingly...\n\n'I expect the blast deafened you a bit,' Clive suggested.\n\n'No, I think I'm going to\u2014' I began, and then the darkness closed over my head like water.\n\nI woke in Perseverance Cottage lying on my own sofa in front of the glowing fire, with Nick wiping the mud from my face with a wet flannel. A cold wet flannel: I expect that's what brought me round.\n\nHis face, concerned and intent, was very close to mine. 'At last!' he said with relief when he saw my eyes open muzzily. 'I was starting to get worried.'\n\n'What... happened?'\n\n'You fainted.'\n\n'I never faint!'\n\n'Then maybe my first guess was right, and you passed out from all that punch you were knocking back, then,' he said.\n\n'I didn't have that much, and there's usually very little alcohol in it,' I said, attempting to sit up and feeling strangely disconnected.\n\n'How do you feel now?'\n\n'All right \u2013 a bit shaky.'\n\n'I expect that'll go off. There are two small burns on your leg. I've put some antiseptic and dressings on them, but I don't think they're much to worry about.'\n\nActually, I was more worried by the sudden realisation that he'd removed my jeans! Under a concealing blanket, all I was wearing on my lower half were my sensible cotton pants.\n\nMy face burned and I sat up straighter and primly tucked the blanket around my legs. 'I think I ought to thank you for \u2013 well, for putting me out. That's why you threw yourself on top of me, wasn't it?'\n\n'Yes, and I'm sorry about that, but I could see your coat was catching and it was the quickest way of smothering the flames.' He got up and came back holding the sad remains of my coat. 'I'm afraid I've made a bit of a mess of it.'\n\n'You certainly have \u2013 and Annie knitted it for me. Now I expect she'll make me another even more hideous one, because I told her I loved it.'\n\nThen I had an evil thought: perhaps I should tell her he jumped on it because he was jealous, and then she might knit him one, too? She whips them up in no time, on giant needles.\n\n'You ought to go to bed. Do you want me to carry you up?' he offered.\n\n'No, I don't,' I said firmly, shivering again. 'But I'd like you to fetch the bottle of damson gin from the kitchen and then lock the door behind you when you go.'\n\n'I don't think you should drink any more alcohol! You're in shock and would be better trying to go to sleep, and you don't have to be nervous, because I'll stay here tonight on the sofa. Go to bed and I'll make you some cocoa.'\n\n'I'm not nervous, I don't need you to stay here with me, and I don't want cocoa \u2013 I want gin. And if you aren't going to get it for me, then I'll get it myself,' I said, attempting to rise from a tangle of blanket on slightly wobbly legs.\n\nNick sighed and got up. 'OK, but don't blame me if you feel terrible in the morning.'\n\nMy hand trembled so much that the glass rattled against my teeth, so he had to sit down with his arm around me and hold it. But it did the trick and I soon began to stop shaking and calm down \u2013 or maybe 'go comatose' is a better description. The warmth of the fire and the soft pink light from the table lamp were very soothing...\n\n'I think Polly might have thrown the firework,' I said drowsily, relaxing against his broad chest, which was invitingly close. Anyway, it was that or fall over sideways.\n\nHe'd put the empty glass down, but hadn't removed his arm from around my shoulders and now he rested his chin on top of my head. 'I was looking at you so I didn't see where it came from. Polly was there, but the chances are she wouldn't do something that stupid. It was just boys messing about, and you were unlucky.'\n\n'Perhaps you're right, but she looked so... so pleased afterwards...' I yawned hugely.\n\n'Come on \u2013 you're all in, so I'll carry you up to bed.' He gathered me up as though I was a loose-limbed doll, but before he could rise to his feet, some compulsion made me slide my arms around his neck.\n\nHe went quite still and our eyes met and held, his like unfathomably deep, dark pools in the lamplight. Then he gave a resigned sort of sigh, tightened his grip and kissed me.\n\nHis lips tasted of inevitability: there was never anything of the minty mouthwash about Nick Pharamond.\n\n## Chapter 24: Flamb\u00e9\n\nI don't know why, but whenever I need a little comfort I find myself mixing up a batch of the quick and easy confection I call Choconut Consolations. They couldn't be easier to make: simply melt some good-quality chocolate (milk or plain, according to your preference) and stir in unsalted peanuts until it is a thick, lumpy mixture. (Those nuts that have been roasted in their shells give the best flavour, I've found \u2013 but remove the shells and then rub the red skins off before using, of course!) Spoon into petits fours cases, or onto a tray covered in baking parchment and leave to go hard in a cool place, though not in the fridge.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nNext morning I awoke slowly, with that languorous, totally sated and exquisitely guilty feeling you get after a really bad chocolate binge \u2013 blissed out.\n\nBut when I opened my eyes to find I was not lying in my bed but on the sofa, I instantly remembered it wasn't chocolate I'd pigged out on last night. In fact, I recollected every single moment only too clearly, right the way from Nick knocking me flat and battering me into the mud, to our kiss and more than make up... though I suppose that at least had the advantage of not involving icy wet earth.\n\nFor a woman whose memory span was normally similar to that of a goldfish, this was quite something, though the action replay going on in my head could have done with some soft-focused editing around the edges to hide all that urgent hunger \u2013 which surely hadn't been all on my side, even if I'd started it, had it?\n\nThe curtains were still drawn and the lights were off, though the fire was burning brightly enough behind the brass firescreen for me to see that I was alone. But that was no surprise, for I'd instantly sensed on waking that the cottage was empty apart from me \u2013 long empty. Slowly I heaved myself to my feet and, clutching my blanket, tottered into the kitchen on my singed legs, wincing at every step.\n\nPropped against the kettle was a brief note in Nick's distinctive handwriting:\n\nLizzy, it's six and I'm supposed to be in London at ten for the shortlist photoshoot for Cookery Writer of the Year. I'll phone you later. Mud brown suits you, by the way \u2013 you should always wear it.\n\nNick\n\nAnd that was it! I read through it twice, as though some hidden message might reveal itself, then crumpled it into a ball and threw it with some force at the wall opposite. It bounced off and fell behind the fridge.\n\nThen I slumped down on the chair, feeling humiliated and angry. This was worse \u2013 much worse \u2013 than when I confided in him at the hospital, because this time I gave him more than my secrets and Spudge recipe \u2013 and all he could think about was some stupid cookery award!\n\nBut so be it, I resolved: from now on, let him eat cake. I know what I'll be eating \u2013 Humble (or should that be Humiliation?) Pie. Here's one I prepared earlier:\n\nMix just enough alcohol with a bad shock and a dash of unadulterated essence of lust.\n\nPut in a warm, dark place.\n\nRemove any inhibitions and stir a little.\n\nThe leftovers can taste bitter if eaten cold next day.\n\nI've changed my mind about Nick being like spicy curry. Now I think he's more like that rich, dark chocolate that's been spiked with extra-hot red chillies, and one chunk is definitely enough.\n\nAnnie, receiving news from the milkman at the crack of dawn about those parts of my sizzling evening that were common knowledge, hotfooted it round the second she'd finished the first dog-walking session.\n\nShe found me slumped in the kitchen in my dressing gown over a plate of Choconut Consolations, though I'd roused myself enough earlier to stagger out into the painful daylight and let out the disgruntled hens, before showering off the last traces of mud and Nick's subtly intrusive aftershave, while singing 'I'm Gonna Wash That Man Right Outta My Hair' through gritted teeth.\n\nWhile she was applying some of her Girl Guide first-aid skills to re-dressing my singed leg, I confessed to her that the most sizzling part of the evening hadn't been the firework-throwing incident.\n\nShe stopped heartily slapping on the Savlon, which was a relief, and stared up at me, blue-grey eyes round and startled. 'You don't mean you and Nick...?'\n\n'Yes, me and Nick!' I confirmed gloomily. 'I can't imagine what got into me, apart from a little too much of Miss Pym's damson gin. Perhaps the shock of nearly being blown up sent me temporarily insane?'\n\n'There you are,' she beamed, ignoring this suggestion, 'I knew you were in love with each other all the time!'\n\n'Love had nothing to do with it,' I said tartly. 'I don't know what it was \u2013 shock, gin, propinquity, comfort, hormones, a substitute for chocolate... whatever.'\n\n'Oh, no, Lizzy!' she protested. 'I'm sure Nick\u2014'\n\n'Nick was gone long before I woke up, so I don't know what his excuse was, but he kindly left me a note making it plain some trashy award is far more important than I am. Read this!'\n\nShe finished pressing a huge Elastoplast into place and I handed her Nick's terse little note, now crumpled and looking slightly the worse for wear.\n\n'Why's it got cobwebs on it?'\n\n'Because it's been behind the fridge. Read it and tell me if it sounds even remotely lover-like to you.'\n\nShe did, lips silently moving, then looked up uncertainly. 'Well, I suppose he had to go to the photoshoot if he's been shortlisted for Cookery Writer of the Year, Lizzy.'\n\n'Big deal,' I said sourly. 'But never mind, at least he makes it clear that food is still much more important to him than I am, just in case I was harbouring any illusions.'\n\n'Yes, but food is pretty important to you, too.'\n\n'Maybe, but I still put relationships first.'\n\nShe sighed. 'Then perhaps men see things differently and he thought you'd understand.'\n\n'He was wrong, then, wasn't he?'\n\nShe pored over the note again. 'It's very Nick, isn't it? You couldn't describe it as romantic.'\n\n'Not by any stretch of the imagination, and it's short to the point of being terse,' I agreed.\n\nAnnie was still trying to find excuses for him. 'I expect he was in a rush, but you'll be able to see him at the award ceremony on the telly on Monday.'\n\n'No I won't, because I've sold Tom's and the one in here is on the blink.'\n\n'You can come and watch mine, then.'\n\n'Thanks, but I think I'll stay home for a couple of days. My leg is very sore and I'm covered in bruises from Nick throwing himself on top of me. I had to hobble out in my dressing gown to let the hens out and I'm going stiffer by the minute.'\n\nShe went pink. 'Lizzy! Too much information!'\n\n'When he was putting the flames out,' I explained patiently. 'He rolled me in the mud.'\n\n'Oh, how quick-witted and brave of him! He's a hero!'\n\n'Don't start going all dewy-eyed and romantic again: it's pointless. I only wish I never had to see him again, because it'll be even worse than when I babbled my entire life history to him at the hospital, while Jasper was ill.'\n\n'You'll feel differently after he's talked to you,' she suggested, ever the optimist. 'And he will phone you up \u2013 look, he says here in the note that he's going to \u2013 and then you'll see he really cares about you.'\n\n'He'll find that difficult, since I don't intend answering the phone. I'll let the machine take the messages.'\n\n'Come on, you know you won't be able to resist answering, in case it's Jasper.'\n\nShe's quite right, I do tend to snatch it up at the first ring \u2013 and it rang right then. We both froze and stared at it.\n\nAt the sixth ring she gave in and lunged for the kitchen extension that hung on the wall by the fridge. 'Hello? Oh, Nick, it's you! Yes, Annie... No, I've just put a fresh dressing on it. It's not too bad, but it'll be sore for a couple of days... I'll ask her.' She covered the phone and held it out towards me enquiringly.\n\n'Tell him I've got much more important things to do than talk to him,' I said loudly, and started hobbling round the kitchen, opening the cupboard doors and slamming things about.\n\n'I'm afraid she can't come to the phone at the moment... Oh, you heard?' She looked up. 'He says, what's more important than talking to him?'\n\n'Food, of course \u2013 he should understand that,' I said pointedly. 'I'm making some giant rum truffles to send to Jasper. They're one of his favourites.'\n\nAfter a moment she put the phone down. 'He says he's sorry he had to dash off, but he'll come and see you when he gets back, and to be careful. Careful of what?'\n\n'I suppose he means careful in case the thrown firework wasn't some stupid adolescent prank last night, but Polly stepping up her campaign.'\n\n'Oh, no, I'm sure even Polly wouldn't do anything so dangerous.'\n\n'No... perhaps not. She's only done petty, spiteful things so far.'\n\n'I still find it hard to believe anyone could be so nasty. Couldn't it all just be coincidence, after all?'\n\n'The ARG stuff was certainly her idea and, besides, when I told her I knew what she was up to, she didn't deny it.'\n\n'Then I expect she's stopped now and the firework was an accident,' Annie said.\n\n'Speaking of accidents, I'm afraid I was wearing that lovely coat you knitted for me last night, and by the time Nick had finished trampling it into the mud, it was beyond repair.'\n\n'Never mind the coat, at least you're OK, that's the main thing. I can always knit you another.'\n\n'That would be lovely,' I agreed, then added, lying through my teeth, 'Nick said it was such a shame it was spoiled because it was wonderful, and he wished he had one just like it.'\n\n'Did he? Then I'll knit him one, too,' she said kindly. 'Well, I'd better be off \u2013 take it easy for a day or two, won't you? I can manage all the pet-sitting until you're fit again.'\n\n'I'm just a bit stiff really, there's nothing wrong with me.' To prove it I got up again to see her out.\n\n'When I arrived, Caz was in the barn doing exercises and Ophelia was sitting on a bale of straw watching him,' she said, pausing on the doorstep to look across the courtyard. 'But it looks like they've gone now, doesn't it?'\n\n'It's a pity he wasn't here early enough to let the hens out. You know, I'm beginning to think I might as well convert all the outbuilding into accommodation, so everyone can just move in with me,' I said a little sourly.\n\nDespite what I'd said earlier, I walked down to the village later, thinking the exercise might help loosen me up a bit. I still felt as though I'd gone three rounds with a gorilla.\n\nI went into the post office to mail Jasper the box of giant rum truffles I'd made that morning, and an Advent calendar with a chocolate behind every window. I only hope he doesn't get zits. The post office was busy and everyone in the queue was still talking about my near-roasting, though the news of Nick's TV appearance had also got out and was causing much excitement. I said I expected it would all come down to a brief glimpse of him among the also-rans, then realised how sour grapes that sounded and shut up.\n\nOf course I wanted him to win it, since clearly it meant so much to him. Of course I did...\n\nOn the way back home I noticed that Gareth's car was parked outside the vicarage and, on impulse, paid him a visit.\n\nHe gave me tea and I got right down to brass tacks.\n\n'Look, Gareth, I hope you don't mind my speaking frankly, but Annie is my oldest friend and all this dithering about is making her miserable. So I want to know whether your intentions towards her are honourable. Is Barkis willing?'\n\nHe choked on his arrowroot biscuit, but when I could get any sense out of him it was just as I thought: they were both pussy-footing around, each thinking the other one only wanted to be friends.\n\n'She loves you, you dimwit, she told me so,' I said plainly, but he was so modest it took a while to convince him. When it finally did, he stared at me with dawning hope in his blue eyes.\n\n'She'll be at home now, having lunch,' I said casually. 'I know she's got a busy afternoon, because she's covering my pet-sitting jobs as well as her own today, but I'll be fit to work again tomorrow. I'm...'\n\nBut I was talking to myself, because he'd gone without so much as grabbing his coat or saying goodbye. As I let myself out, I only hoped he had a key. We didn't want our vicar arrested for breaking into his own home, did we?\n\nAnnie phoned me up between pet-sitting jobs, almost incoherent with happiness, to announce that Gareth had proposed and they were now engaged. They're trying to get through to her parents to give them the glad tidings, but communication with that remote area of Africa is a little difficult at present.\n\nBut I'm sure when they do hear they'll be very happy and, if anyone deserves wedded bliss, I'm sure Annie does.\n\nAs for me, I remembered a recipe in one of my books for rabbit with chilli-chocolate sauce \u2013 and it was definitely different.\n\n## Chapter 25: Cr\u00e8me de Coeur\n\nSunday started bright, crisp and frosty, so I really threw myself into tidying up the garden, raking up dead leaves for compost and clearing the annual herb beds, accompanied by a lot of hopeful hens.\n\nIn the afternoon, I covered my own Christmas cake and the six individual ones I'd made for the WI hampers with marzipan. Then, having some left, I made petits fours by sandwiching marzipan between two walnut halves and put them in paper cases.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI found it hard to sleep on Saturday night, despite having spent the entire evening making yet more rum truffles \u2013 this time little ones for the WI hamper goody bags.\n\nEvery time I closed my eyes, lowlights of the night before kept running on a loop through my head and I became conscious of all the aches and pains of my poor bruised, singed and battered body.\n\nThere was little work to do in the garden now that winter had arrived, but on Sunday I still found enough both there and in the kitchen to keep myself occupied. That night I did finally fall asleep, exhausted and still smelling of the rum truffles I'd been putting into little Cellophane bags tied up with silver ribbon.\n\nMonday's post brought me a card depicting the Tower of London with a Cr\u00e8me de Coeur recipe scribbled on the back, but I decided Nick could keep his heart to himself \u2013 if he'd got one.\n\nAt the CPC meeting I handed over all the little bags of truffles to Marian, another task done. I'd gone braced for lots of discussion of my Bonfire Night mishap, but luckily the news of Annie's engagement and Nick's forthcoming TV appearance were much more exciting.\n\nWe toasted the bride-to-be with the bottle of elderflower champagne I'd taken with me for the purpose, though of course she hadn't yet got a ring to show off.\n\n'But I gave Gareth my little silver dolphin ring for size, and he was going to buy one today,' she said, looking rosy-cheeked and very pretty, though I think the latter partly attributable to her having finally listened to my advice about growing out her fringe and abandoning the pudding-bowl haircut.\n\n'Do you trust him to get something you'll have to wear for the rest of your life?' asked Faye.\n\n'Oh, yes,' she said simply. 'I'm sure it will be lovely.'\n\nWe discussed the wedding pretty exhaustively and also the impossibility of Gareth marrying himself, though of course if Annie's parents got back in time her father could perform the ceremony.\n\n'I'd really love that,' she said wistfully. 'To be married by daddy in his old parish \u2013 that would be so special!'\n\n'Well, I don't see why not, if that's what you want,' Miss Pym said.\n\nWe had a second elderflower champagne toast and then, since we were once again at Faye's, we had another ice-cream tasting. The rum and raisin had the edge on the toffee apple, but they were both good.\n\nFaye said my champagne had given her an idea and next year she might try concocting an elderflower ice cream, which she thought would have a delicate but interesting flavour.\n\nI hadn't been home for long when Unks rang up and absolutely insisted I go up to the Hall to watch the awards ceremony on TV with him, Mimi and Juno that evening so, rather than upset him, in the end I agreed.\n\nWe all crowded round the big set in his den, which is decorated with a mixture of old racing prints and early Pharamond's Butterflake Biscuit posters, reflecting the varied strands that make up the family character.\n\nThe Cookery Writer of the Year award was just one among many, so we had to sit through Sport, Fashion and goodness knows what, before we got to it. It was just as well I'd taken up some of the walnut petits fours I'd made on Sunday evening and a big slab of very gingery parkin.\n\nThe cameras kept panning around the room and I caught a brief glimpse of Nick at one of the tables. I'd seen him in a dinner jacket before, of course, but never at a distance. It was like looking at a stranger... and I had a better chance to examine the effect when he went up to collect his award. Yes, he won the thing.\n\n'I knew he would,' Mimi said complacently, through a mouthful of parkin.\n\n'You can't have known,' Juno said. 'He's not the only good cookery writer around.'\n\n'He's in a league of his own,' she said loyally. 'And don't you think he looks handsome in his dinner jacket, Lizzy? He's the best-looking man in the room.'\n\nHe was certainly the tallest and though with those strongly marked features you couldn't in all fairness call him handsome, he was possibly the most attractive-looking man there. I realised I was sitting forward and leaned back again, casually.\n\n'He scrubs up well,' I agreed grudgingly.\n\n'He's coming home in a couple of days and bringing some people with him,' Unks told me. 'He asked me if I would mind if they decorated the hall, kitchen and dining room up as if it was already Christmas, so they could photograph it for an article.'\n\n'Isn't it a bit late for that? I thought magazines did everything well in advance.'\n\n'It's for the Sunday magazine he writes for and they were ready to feature some footballer or other; only now he's involved in a big sex scandal and his wife and children have left him, so they asked Nick instead.'\n\n'It will be such fun, like having two Christmases,' Mimi said rapturously. 'Will we have presents too, Roly?'\n\n'No, it's all fake. Don't get your hopes up. They'll bring everything they need to decorate the place with them, including a pretend Christmas dinner. Lizzy, Nick said to ask you if you would come, because they need extra guests for the photographs. And perhaps Annie and the vicar, too?'\n\n'Kind of him,' I said drily.\n\n'We'll all be in the magazine,' Mimi said, 'pulling crackers and opening parcels.'\n\nI seemed to have already pulled a cracker and I hadn't recovered from the big bang yet.\n\nUnks may have detected a certain lack of enthusiasm, because he asked anxiously, 'You will come, won't you, Lizzy? And Jasper too, of course, if he's home.'\n\n'I shouldn't think he will be, though he'll be back for the real Christmas, of course.'\n\n'I'm going to wear my blue lace dress,' Mimi announced.\n\n'You'll have to: it's the only decent dinner dress you've got left,' Juno pointed out. 'I keep telling you not to garden in them.'\n\n'Well, you've only got that black thing. Unless we buy new ones? What about you, Lizzy?'\n\n'I don't have anything suitable at all, so I'd better not come,' I said quickly. 'I'm sure Nick won't want me there, anyway.'\n\n'He said to ask you specially,' Unks said and they all seemed to be looking at me meaningfully... though, of course, they knew about the accident, for news travels fast round here.\n\n'He was so brave at the bonfire, wasn't he, Lizzy? If he hadn't been so quick-witted you might have been badly burned,' Mimi said, reading my thoughts.\n\n'Then he carried you back to your cottage after you fainted,' Juno sighed. She clearly has a much more romantic streak than her bluff exterior would lead you to believe.\n\n'Yes, he's a real hero. He even spent the night on the sofa afterwards, in case I was suffering from shock and needed anything,' I added pointedly. This was quite true, though of course I didn't mention that I had spent the night there too and, unfortunately, had needed something... or someone.\n\nMimi looked very thoughtful, but before she could say anything else I said quickly, 'Did you all know about Annie's engagement to the vicar?'\n\nThis proved distraction enough to keep them going until I went home.\n\nI'd put the phone down on Nick twice since he'd left. When I heard his voice I couldn't think of a thing I wanted to say to him \u2013 or nothing polite, anyway. He got the message eventually and stopped phoning. Good.\n\nHe missed yet another Mystery Play rehearsal, though we were all word-and position-perfect, and into sorting out the costumes and props, ready for our final dress rehearsal before Christmas. This we always do up at the Hall, in two sessions in random order, as ordained by Clive and Marian, and Adam and Eve are sometimes excused from wearing their skimpy outfits on this occasion, because of the cold. (I was so hoping it would be cold! The less exposure in my new Spandex outfit, the better.)\n\nMimi and Juno must have been at a loose end, because they came down to watch the rehearsals again and Juno kindly read Nick's part in her big, deep voice.\n\n'Might as well come along to the pub with you afterwards for a quick snifter,' she said heartily when we'd all finished.\n\n'Actually, I'm going to the Butterflakes caf\u00e9-bar with Ritch, tonight,' I said, slightly self-consciously, though there's nothing particularly secret about our occasional friendly drinks. And I certainly didn't want Nick thinking that there weren't lots of other men interested in me, even if he wasn't...\n\n'Oh, that sounds such fun!' Mimi said. 'I've never been to the Butterflakes bar. Why don't we go too, Juno?'\n\n'Oh, we couldn't possibly intrude on Lizzy's evening,' Juno said, and Mimi's face fell.\n\nRitch was awaiting me outside, but when I turned round to introduce Mimi and Juno they'd vanished. Somehow I wasn't completely surprised to find them already ensconced at a corner table at the caf\u00e9-bar when we went in. They smiled and waved.\n\n'Friends of yours?' asked Ritch.\n\n'Roly Pharamond's sister, Mimi, is the one with silver curly hair. Juno is her companion \u2013 and I'm starting to suspect they're here tonight to keep an eye on me.'\n\n'Do you need keeping an eye on?' he asked, brightening. 'Got anything interesting planned for later tonight?'\n\n'No, just a drink, a chat and then home \u2013 alone,' I said pointedly.\n\n'Oh, well, worth a go. Dora told me Nick Pharamond was the hero of the hour at the bonfire and you and he are, as she put it, only waiting for your six months' mourning to elapse before naming the day.'\n\n'Dora said that?' I gasped, because if so, then the whole of Middlemoss probably thought the same! How on earth did these rumours get about?\n\n'Yes. Sounds like something straight out of Cotton Common, doesn't it, though you should be wearing widow's weeds to look the part.' He glanced at me curiously. 'So, are you and the Young Master going to get hitched, then?'\n\n'No!' I said forcefully.\n\n'Well, you needn't bite my head off, I didn't suggest it! Though come to think of it, he does give me jealous looks whenever he sees me talking to you!'\n\n'That's just a sort of general disapproval,' I explained. 'He thinks of me as part of the estate's goods and chattels.'\n\nWhen we left, Mimi and Juno followed us out, though they could hardly hitch a lift in Ritch's sports car.\n\nJuno stooped and said to me through the window, 'Might just call in on the way home, Lizzy. Roly wanted some more of that lemon marmalade and it slipped my mind earlier.'\n\nSo I was definitely being chaperoned! But I was sure it was entirely their own idea, and they couldn't keep it up twenty-four hours a day. It did the trick tonight, though, because Ritch dropped me off and left immediately. I think he found Juno rather alarming.\n\nWhen the terrible twosome turned up ten minutes later, Mimi had a miniature paper umbrella behind one ear and was full of exotic cocktails, giggly and overexcited, but I thought she'd go out like a light once her head hit the pillow.\n\nI should be so lucky.\n\nThere didn't seem any way of getting out of the photoshoot, especially since Annie and Gareth were very excited about it, but I had no idea what to wear and neither did Annie.\n\nNone of our clothes looked smart enough, especially if Juno and Mimi were getting dressed up in their best. In the end we thought we'd better buy something new, and I drove us both over to Southport in the Land Rover.\n\nAnnie wanted to look in bridal shops anyway, because she and Gareth hoped to get married in January, if Annie's parents could get back for it, and that didn't leave a lot of time for the preparations. She wanted me to be chief matron of honour and the rest of the CPC to form her bridal retinue, preferably dressed in pink, like a posy of slightly pass\u00e9 flowers. But her thrifty little soul was shocked by the high prices so we gave up after a while and searched out our dinner outfits instead.\n\nAnnie chose a midnight-blue chiffon tunic top with matching palazzo trousers, cinched in around her narrow waist (her shape is a very curvy hourglass) with a gold chain belt. Mine was a dark, clingy dress in a holly-leaf colour that made my eyes look very green. It had interestingly draped bits and looked like nothing on the hanger, but it certainly made the most of what assets I possessed when it was on.\n\nIn fact, it was dead sexy, and not at all the sort of thing I would normally wear, but in it I felt armoured for any eventuality.\n\nWhen Nick came back very early on the Thursday morning, he instantly threw the Hall into a flurry of preparation for the Christmas photoshoot, though they didn't have to put any decorations or a tree up, because everything was supplied and the house was to be professionally 'dressed'.\n\nMrs Gumball had sent Joe down for some eggs, which is how I knew Nick was back, and after a few hours he finally managed to tear himself away and walked into my house without a by-your-leave.\n\nI gave him one glance, as he lounged in the doorway in dark thundercloud mode, and then concentrated on beating my fruitcake mix to death: when he was wearing his Mr Rochester expression it was never a good sign.\n\n'Why did you keep putting the phone down on me, Lizzy? Don't we have something to talk about?'\n\n'We have nothing to talk about, Nick,' I said firmly.\n\n'Yes we have! The other night\u2014'\n\n'Shouldn't have happened, and as far as I'm concerned it never happened,' I interrupted firmly. My arm was starting to ache by now, so I stopped beating and began buttering the cake tin instead.\n\n'But, Lizzy\u2014'\n\n'Look, I don't want to think about it, let alone talk about it!' I snapped, crashing the tin back down on the tiled work-surface and glaring at him.\n\n'Why? What was so wrong about\u2014'\n\n'I'm not discussing it,' I said. 'Just forget it, OK?'\n\nHe gazed at me, black eyebrows drawn together in a ferocious frown. 'Forget it? Come on, Lizzy, it must have meant something to you!'\n\n'Shock makes people do the strangest things, Nick. But if you like, you can put it down to an excess of gratitude that you saved me from serious burns,' I suggested.\n\n'I don't want your damn gratitude,' he snarled, and then slammed out, making everything on the dresser rattle.\n\nAfter putting the cake in the oven I sat down and scraped the mixing bowl clean. It tasted salty \u2013 but that was probably all the angry tears dripping into it.\n\n## Chapter 26: Crackers\n\nIt's best to leave a few days between covering your cake in marzipan and icing it, but it works perfectly well even if you don't. Mince pies freeze very well and defrost quickly, so I usually start to bake batches of them around mid-November. I've heard some people put sugar in their shortcrust pastry, but that sounds too sickly for words: the sweetness should come from the mincemeat filling. Nor do I dredge the tops of mine in yet more sugar... The whole world seems to have gone sugar-crazy!\n\nThe Perseverance Chronicles: A Life in Recipes\n\nIn the morning I had just begun icing the Christmas cakes when Mimi and Juno popped round to describe how the family had been banished to the kitchen and small morning room, while the hallway, staircase, drawing room, dining room and family silver were all being buffed up to a high polish.\n\n'Only there isn't much family silver,' Mimi said, 'unless we drink out of Roly's racing trophies.'\n\nShe was more than a little overexcited, so I suspected that Juno's main reason in bringing her was to get her away from the Hall for a little while.\n\n'You've never seen anything like it!' Juno said. 'Mrs Gumball is in high dudgeon and says if people wanted to see their faces in the furniture, they should have let her know years ago.'\n\n'Yes, and Juno slipped and nearly fell in the hall, because they had polished under the rug,' Mimi said, helping herself to a scrap of fondant icing. 'She might have broken her leg all over again!'\n\nShe didn't sound terribly regretful about this, but I think she'd relished the unusual amount of freedom she'd enjoyed while Juno had been laid up.\n\n'Stupid thing to do,' Juno agreed. 'But it's all nearly ready for tomorrow now.' Then she added, casually, 'Nick's been very bad-tempered today.'\n\n'Isn't he always?' I asked.\n\n'Oh, no, he only looks gloomy most of the time, he isn't really,' Mimi said. 'But since he said he was going to visit you yesterday, he's been really ratty. Did you fall out?'\n\n'None of your business, Mimi!' Juno said reprovingly.\n\n'We may have had... just a little misunderstanding,' I confessed. 'But we know where we both stand, now.'\n\n'Where's that?' Mimi asked irrepressibly, but was frowned down by Juno.\n\n'Are all these little cakes the ones for the WI hampers?' she said, tactfully changing the subject.\n\n'Yes, we've done six each but we decorate them all the same, so no one knows who baked which.' I opened the old biscuit tin in which I kept the cake decorations and laid out half a dozen plastic sprigs of holly, robins sitting on logs and gold Merry Christmas plaques. 'Perhaps you could put one each of those on the tops for me, Mimi, while I look out the cake bands?'\n\nI also got out the fruitcake I made yesterday, and neither of them complained about the taste, so perhaps I only imagined the mix tasted salty.\n\nWhen they'd gone I put the small cakes away in the larder, ready to deliver to Marian at some point, then turned to our own, decorating it exactly the same way I do every year, with Santa emerging from a little forest with his reindeer sleigh. The bristly fir trees were firmly stuck together in the tin like an early form of Velcro, and had to be prised apart.\n\nAnnie and Gareth collected me in her car next day and we all agreed that it seemed very odd to be getting glammed up for a smart dinner before it was even lunchtime. Mind you, I dress up smartly so infrequently that it would have felt odd at any time.\n\nThey were looking forward to it, but I would have cried off, even at this late stage, if I could have done it without upsetting Unks. I mean, it wasn't even as if we could eat the damned food, since it was all going to be either fake and glazed with something to make it photograph prettily, or sit under the lights for so long it would be rife with three strains of salmonella.\n\nMimi had already rung me to describe how Christmas had arrived at Pharamond Hall very early that morning in a large van, along with a miscellaneous assortment of photographers, food technicians and the like, plus a snootily elegant grey-haired woman, whose job was to 'dress' the rooms they were to use: deck the Hall with boughs of holly.\n\nMrs Gumball let us in at the kitchen door and said that they'd already photographed in there and the hallway with its garlanded oak banister, until she was fit to scream, and she'd be glad when they were done. Then she took our coats and sent us through into the dining room, which looked strangely unfamiliar.\n\nAlthough it was barely midday, the crimson curtains were shut and the only light came from thousands of candles glittering off the polished dark panelling. Very realistic swags of festive foliage studded with gilded baubles were draped everywhere, in a colour scheme of crimson, gold and an ecclesiastical deep purple that must have made Gareth feel quite at home.\n\nThere were a lot of strangers milling about with cameras and lights and things near the dining table, on which gleamed an unfamiliar silver candelabra and a lot of sparkling cut glass, but the family were all gathered round the fire next to a large fake Christmas tree, among a litter of discarded festive giftwrap.\n\nNick, looking darkly morose and Mr Rochester in an immaculate dinner jacket, leaned on the mantelpiece with his foot on the fender, gazing into the flames and barely acknowledging our entrance.\n\nUnks made up for this, however, by saying jovially, 'Come in! What excellent timing, because they're almost ready for us to do the Christmas dinner scene. Annie, my dear, you look lovely,' he added, kissing her. 'Being engaged suits you! You're a very lucky man indeed, Gareth.'\n\n'I certainly am,' the vicar agreed, looking devotedly at Annie, and she blushed.\n\n'Lizzy looks pretty too,' Mimi commented brightly, 'don't you think so, Nick?'\n\n'She certainly looks different,' he said, actually looking at me properly for the first time and taking in the figure-enhancing effect of my new dress with a raised eyebrow. I suddenly wished I hadn't bothered dressing up, but come in the dungarees I wear when I whitewash the henhouse.\n\n'We've been opening presents,' Mimi said. 'They were all empty, but we are having real crackers.'\n\n'And real wine,' Unks added. 'Need something to keep us going!'\n\n'The Christmas tree pops up, decorations and all, like magic,' Mimi confided to us. 'Roly, can we have one of those?'\n\n'No, I like the real thing, smelling of pine,' Roly said. 'And you like decorating it, don't you, m'dear?'\n\n'Oh, yes, I hadn't thought of that,' she agreed.\n\n'Can you take your places at the table, please?' someone called, and we went where we were directed, which in my case was between Juno and Nick.\n\nPeople darted in to tweak, dab and twitch everything to perfection as we posed, slightly self-consciously, as directed. I was already aware that my new green dress fitted where it touched \u2013 and it touched almost everywhere \u2013 but when the photographer kept zooming in on my cleavage I began to wonder if it might be a bit over the top in more ways than one. Then Nick glared at him and he backed off a bit.\n\n'Can we pull the crackers now?' Mimi asked plaintively. 'Haven't they finished yet?'\n\n'OK... go ahead,' a man's voice said from the dark shadows.\n\nThey were certainly big, expensive-looking crackers, with equally pricey-looking novelties inside. Mine, which I pulled with Nick, contained a gold-plated pen, a gilt cardboard crown and a tightly rolled piece of paper.\n\n'Does your motto make sense?' Juno asked, puzzling over hers. 'I think mine is supposed to be a joke, but I'm not sure. I mean, how could you cross an elephant with a mouse? That's not physically possible!'\n\n'Read yours aloud, Lizzy!' ordered Mimi gaily. She was becoming flushed and excited.\n\nI unrolled the long, thin strip of paper and found it entirely covered in Nick's instantly familiar spiky handwriting. 'Mine doesn't make sense either,' I said quickly, crumpling it into my hand. 'Do you want my pen to go with your little photo frame and gold dice, Mimi? They seem to match, don't they?'\n\n'Oh, yes, please!' she said, but just as I was handing it over, there was the sound of a loud altercation outside the door and a bit of scuffling.\n\nThen Mrs Gumball lumbered in, with a small, rotund and apoplectic man hard on her heels. She jerked a thumb over her shoulder in his direction. 'It's that little twerp Lionel Cripchet, from over Rivington way.'\n\n'Sir Lionel,' he snapped, bobbing up in front of her and glaring generally round, but though we must have presented a very Night Watch sort of tableau, the strangeness of it escaped him under the urgency of his anger: 'I'm here for an explanation!'\n\n'Are you?' Unks said mildly, taking another sip of wine. 'Well, now you are here, I wouldn't mind an explanation myself about that supposedly sound horse you sold me a couple of years back. Remember? The one that mysteriously went permanently lame the day after I bought it?'\n\n'I'm not here to talk about horses, but squirrels! Yes, that's taken you by surprise, hasn't it? I suppose you thought I wouldn't find out!'\n\n'Is the man mad?' Juno asked. 'Why is he blethering on about squirrels?'\n\n'Yes, spit it out, Cripchet,' Roly said amiably. 'Why are you blethering on about squirrels?'\n\n'You know very well,' he exclaimed slightly wildly, looking at the vicar and Annie as if he expected them to come out in support, despite their baffled expressions. 'I've been overrun with the little grey bastards these last two years and now \u2013 last night \u2013 I finally caught him in the act!'\n\n'Who?' asked Nick, then added, after a moment's thought, 'And what?'\n\n'Your gamekeeper, Caz Naylor. His Land Rover was parked up a track next to my estate at one this morning! Now, what do you say about that?' Sir Lionel demanded triumphantly.\n\n'Is Caz still around?' Roly asked Mrs Gumball.\n\n'In the kitchen. Shall I send him in?'\n\n'Do,' he agreed. 'I am quite sure he has a perfectly innocent explanation.'\n\n'Ha!' said Sir Lionel, moustache bristling.\n\nCaz slid silently into the room a moment or two later, but no further than the dark shadows just beyond the reach of the candlelight.\n\n'Ah, Caz, Sir Lionel wants to know what your Land Rover was doing parked up a track next to his estate in the early hours of the morning,' Roly said. 'Were you indeed there?'\n\nCaz nodded, almost imperceptibly.\n\n'And I'm sure you had a very good reason?' prompted Unks.\n\n'Of course he had a damn good reason!' yelled Cripchet, going puce and practically dancing up and down on the spot. 'He was releasing hordes of flaming grey squirrels onto my land, that's what he was doing! There's standing room only and they're fighting for territory. It's like World War Three out there!'\n\n'What do you say to that, Caz? What were you doing?'\n\n'Courtin',' he said laconically.\n\n'Courting?' demanded the infuriated baronet. 'Courting? You can't expect me to believe that, Caz Naylor!'\n\n'Actually,' the vicar interjected quietly, 'Caz and his fianc\u00e9e, Ophelia Locke, have just asked me to put up the banns, so I see no reason to doubt him.'\n\n'That's right,' agreed Caz, and then, clearly feeling that enough had been said, sidled back out of the door again.\n\n'But the squirrels...' began Sir Lionel, baffled and furious.\n\n'You know, Caz said he'd found a lot of the traps sprung but empty lately,' Unks said, with an air of sudden illumination. 'And that animal rights group, ARG, are very active around here, so I dare say they've been releasing them onto your land, that's what it is.'\n\nCripchet's lips worked silently and his skin went an even more ominous shade of puce.\n\n'A drink before you go?' suggested Nick hospitably.\n\nSir Lionel looked from one to the other of us and said slowly, 'It's a damned conspiracy! You're all in league together!' and then he slammed out.\n\nThe magazine crew, who'd been watching with silent appreciation, broke into a spatter of polite applause. Roly bowed.\n\n'There, all's well that ends well, isn't it?' he said happily. 'And if you have finished with us, too, ladies and gentlemen, then I suggest we adjourn to the kitchen for something real to eat and leave you to pack everything up.'\n\nWhen I got to my feet I discovered I still had the curl of paper from the cracker clenched in my hand and, for want of a handbag, shoved it down the front of my dress when no one was looking.\n\nMrs Gumball had hot soup and sandwiches ready, and by the time we'd finished those, Christmas had been dismantled, packed away and driven off again. Evidently it took much less time to do that, than set it all up.\n\nAnnie and Gareth were giving me a lift home and as we left, Nick called out to me, 'Lizzy, I'll have to go away tomorrow, but I'll be back by the end of the month, so I'll see you up here bright and early on the first of December.'\n\nI stopped dead. 'You will?'\n\n'Yes, it's the Senior Citizens' Christmas dinner, remember? Mrs Gumball is expecting us to both help cook it.'\n\n'That's right,' she agreed.\n\n'But surely, if Nick's helping, you won't want me under your feet, too?' I suggested hopefully.\n\n'Many hands make light work,' she said firmly. 'And I've three geese to cook!'\n\nThe note in the cracker was Nick's recipe for prize-winning apple pie and I instantly saw that there was no startling difference between his and my own. So if his really was better, then it must mean that he had a lighter hand with the pastry, which was even more unforgivable.\n\nIf he wanted a motto for his cracker, 'I shot myself in the foot' would do admirably.\n\n## Chapter 27: Charmed\n\nWe had the first snowfall of winter last night and I awoke to find everything fuzzily flocked in white and looking Christmas-card pretty. This is when the birds are glad of the bright-berried bushes like pyracantha, holly and viburnum \u2013 but it's still only mid-November, so let's hope there are still lots of holly berries left for the Christmas decorations!\n\nThe Perseverance Chronicles: A Life in Recipes\n\nJasper and Ginny arrived unexpectedly next afternoon, dropped off for a couple of hours by his friend Stu, who had bought a car.\n\nI wished he could have given me warning, so I could have cooked him something he loved, but it was wonderful to see him. I know we've talked over the phone a lot, but it wasn't the same as actually being able to put my arms around him and give him a hug. He looked sort of subtly grown-up too... and even Ginny refrained from trying to bite my ankles, so perhaps absence made the heart grow fonder. I told him about the photoshoot and what a hollow mockery of a real Christmas it had been, then after a while, he popped up to the Hall to see the family and came back later with the news that Nick was off travelling abroad for the next couple of weeks, which is something he does do from time to time, filing his copy from wherever he is. But yesterday he didn't say a word about where he was going!\n\nMind you, he hardly said a word to me at all...\n\nMrs Gumball had given Jasper half a cold roast chicken and an apple turnover to take back to university with him, Unks a fifty-pound note (I didn't even remember seeing one of those before) and Mimi, not to be outdone, presented him with the gold crown from yesterday's cracker. I'd already packed up a moveable feast as my contribution to the student larder, so they would none of them starve before the end of term.\n\nWhen Stu came to pick Jasper and Ginny up again I heroically refrained from quizzing him about how good his car brakes were, or telling him to drive carefully. I think I deserved a medal for that.\n\nAt the CPC meeting I gave Marian the mini Christmas cakes and so did one or two of the others. 'Oh great,' she said, 'the hampers are coming along really well, I just need to bulk buy the boxes of mince pies nearer the delivery date.'\n\nOf course we would have happily made those, too, but when surveyed the majority of the Senior Citizens preferred shop ones, though I've no idea why.\n\nWe were all crammed into Annie's tiny cottage and once she had poured the coffee and passed the Fondant Fancies, she tipped a big bag of pine cones onto the newspaper she had spread over the table and showed us how to turn them into Christmas tree decorations.\n\n'I got the idea from a magazine and tried it with the Brownies last year, and they looked lovely,' she said, 'though I didn't let them loose with the Superglue, of course.'\n\nShe demonstrated how to glue on a ribbon loop to the top, then dabbed a little gold paint around the cone, sprinkled it with glitter, and shook off the excess, before placing it in an old egg box to finish drying.\n\nI'd wondered why she'd asked us to bring old egg boxes with us! And come to think of it, they would be perfect for storing any breakable tree ornaments too.\n\nThat evening the Mummers were in the workshop, practising the songs they would play in the intervals of the Mystery Play, and afterwards Ritch came over to the cottage and said he thought Ophelia was imminently going to give birth to a chest of drawers.\n\n'She has suddenly grown a big bump, all out at the front,' I agreed. 'Though I suppose those smocks she wears have been hiding it for months and so it's only just become noticeable. She's no idea when it's due. You'd think she was living in the Middle Ages, the way she avoids modern medicine. Did you know she and Caz are going to get married?'\n\n'Are they?' He looked at me over a table spread with home-made goodies (I'd half-expected him to come tonight) and unleashed his glowingly attractive smile. He appeared so blondly wholesome that I admit my heart gave a bit of a thump... Then I reminded myself that even if he had given up one of his dubious habits there was still the philandering and the healthy sex rota.\n\n'Yes, they've put the banns up already. And Annie's parents are delighted about her engagement, too \u2013 they're flying back in the New Year on leave for the wedding.'\n\nAnnie, now sporting a modest sapphire ring, was going about in a permanent glow of happiness.\n\n'She and the vicar seem perfectly matched,' Ritch said. 'Maybe I should try it?'\n\n'What, marriage?' I said, startled.\n\n'Why not?' He gestured at the table. 'A woman who can cook like this is worth hanging on to!'\n\n'Don't be daft,' I said, though rather flattered. 'Monogamy isn't in you!'\n\nThe Cotton Common cast were kept hard at it this week, so Annie and I were also kept busy dogwalking: I was becoming very fond of Flo, and also of Delphine Lake's little dogs.\n\nNick was not at the Mystery Play rehearsal that evening, of course, and it was all a bit unexciting. But then, it was mainly costume adjustments and props, for there were only a couple more rehearsals before the final dress one up at the Hall, so that was only to be expected.\n\nMarian and Clive were also very involved in the annual Mosses Christmas Show, which was to take place in early December, so naturally they liked to have the Mystery Play well in hand in order to concentrate on that at the end of November instead.\n\nI did go to the pub for a bit with the others, where Annie and Gareth revealed to me that they've decided that spending a fortune on a big wedding was immoral, so instead they were going to have a thrifty one and make a large donation to charity.\n\n'Luckily I'm the same size as Mummy was when she got married, so she suggested I wear her lovely wedding dress,' Annie said on the Thursday, when I called in after taking Flo for a walk. 'I took it out of storage this morning, with the veil and everything, and it was perfect. It's hung up in my bedroom now. And dear Miss Pym says she'll try and find four bridesmaids' outfits in shades of pink on eBay, which she can alter to fit. She makes almost all her own clothes.'\n\n'I'd never have guessed,' I said untruthfully. 'But what about a reception? It's a pity I got rid of the big glasshouse really, though I suppose it would have been too cold in January.'\n\n'Oh, we'll have it in the village hall, with a simple buffet: perhaps all the guests could bring a contribution.'\n\n'I'll make the wedding cake, that can be mine,' I offered. 'And do you know, I think getting married this way is going to be much more fun!'\n\n'Yes, that's what I think, too,' Annie said, her eyes shining. 'And it will make our special day even more wonderful, knowing that we'll be helping others.'\n\n'Spread the love,' I agreed, giving her a hug. 'Now, show me this beautiful wedding dress!'\n\nIt was Stir-Up Sunday, and the day when the church service traditionally included the prayer beginning with the words, 'Stir up, O Lord, the wills of your faithful people', which always used to be the signal for the Christmas puddings to be made.\n\nMy own huge, round one was long since made, but I spent most of that day cooking the ones for the Senior Citizens' Christmas lunch. Mrs Gumball does all the rest of it, but I rather like making the puddings.\n\nWe'd stopped putting charms in them after a minor disaster when one of the Senior Citizens broke his dentures on a Bachelor's Button: in any case, they always refused to give them back, and it got expensive buying new ones every year.\n\nI felt the first twinge or two of excitement that Christmas always gave me: dim but happy memories of those spent with my parents and more recent ones with the Vanes. And whatever difficulties I'd had with Tom, I'd always tried to make sure that Jasper, too, would have a hoard of joyous treasured memories of Christmas.\n\nIt had been ages since Nick had gone abroad, but not a single postcard, with or without tart recipe, had arrived. Meanwhile the CPC had become more of a Wedding Circle, since we spent almost the whole time discussing Annie and Gareth's big day!\n\nMiss Pym put in low eBay bids on bridesmaids' dresses and had already secured two, which were on the way.\n\n'And there does not seem to be much interest in bidding on the others, so I will know by this evening whether we have those as well,' she said. 'They are all in shades of pink, so though we won't match, we will have a theme.'\n\n'Lovely,' I said resignedly, though pink is definitely not my colour and, to be honest, it's not going to do a lot for Faye's ruddy complexion, either. 'Though let's hope it doesn't snow, or we will freeze to death!'\n\n'What about if we all get an ivory-coloured pashmina or wrap?' suggested Marian.\n\n'Good idea,' I said, for at least a pashmina is likely to be useful later.\n\n'The meeting had better be at my house again next week, so we can have the first fitting,' Miss Pym suggested, but by then we had completely lost the thread of our rota, as usual, so it might have been her turn anyway.\n\nAlthough Ritch still often called in at the cottage after the Mummers sessions, we hadn't been to Butterflake's for a drink for ages, probably because he has another woman \u2013 or even two or three \u2013 on the go. Caz was still hanging around the cottage just as much as usual, though, but since he'd staked his claim on Ophelia, I supposed he would want to keep an even closer eye on her.\n\nI finally got a postcard from Nick, with a Turkish delight recipe on the back. I wondered if that was an improvement on tarts...\n\nWe all had a mince pie tasting at the CPC over at Miss Pym's neat bungalow in Mossedge. We had to wait until we'd had our bridesmaids' dresses fitted first, though, so they didn't get marked.\n\nThe four of them varied from baby pink to a deep rose (mine) and are all the traditional tight-bodiced, full-skirted type, with big, puffed sleeves. There was much pinning and tacking, then we had our mince pies and a modest sherry, since most of us were driving.\n\nMiss Pym had also found some cheap pashminas on the internet and we gave her the go-ahead to buy them.\n\n'And Roly is providing all the flowers, including decorating the church and my bouquet,' Annie said gratefully. 'It is so kind of him. In fact, everyone is being wonderful.'\n\n'That's because we all love you, dear,' Marian said. 'You will have a splendid day, just you wait and see!'\n\nWhat with organising the Christmas Show, Senior Citizens' Christmas hampers and lunch, and directing the Mystery Play (among other things too numerous to list), Marian was, by the end of November, starting to look even thinner, her huge dark eyes sunken and her cropped silver hair bristling with electricity. But she and Clive always insisted they loved to keep busy: and they must have done or they wouldn't have volunteered for everything!\n\nLuckily for their peace of mind, Nick returned just in time for the next Mystery Play rehearsal, albeit bleary-eyed, unshaven and very, very grumpy. He snapped out his lines with barely a look at me, which boded well for the next day's early morning start helping Mrs Gumball to cook the Senior Citizens' Christmas Lunch.\n\nI can't describe to you how much I wasn't looking forward to that.\n\n## Chapter 28: Cold Snap\n\nI make my own version of those fat balls for wild birds that you can buy, mixing birdseed, dried fruits, nuts, bacon rinds and crumbs with some melted dripping or lard. You can either put blocks of it on the bird table or refill those coconut shells that are pierced for hanging up. The cold weather seemed to be set to continue into December and though the child in me found pleasure in the idea of a White Christmas, it would be hard on the birds and other small creatures.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI was up at the Hall before dawn, carrying two baskets containing the big Christmas puddings, along with some brandy butter I'd whipped up the night before. It's not far, especially if you take the shortcut through the woods and the walled garden, but by halfway I'd begun to wish I'd taken the Land Rover.\n\nMrs Gumball and Nick were already hard at work by the time I arrived, and had divided the cooking between them, leaving me the role of skivvy. I quickly discovered that Nick is hell on wheels in a kitchen, too, and takes no prisoners. Had it not been for such a good cause I wouldn't have stood it for a second \u2013 but never again! The moment when it was all packed into the Meals on Wheels van and trundled down the drive was wonderful \u2013 as was the stiff drink and long soak in the bath I had as soon as I got home, despite it being only lunchtime. Nick had offered to drive me back, but by then I wasn't speaking to him \u2013 if I had been in the first place, which was a moot point.\n\nAnnie, who had also risen early that morning to help put the Christmas decorations up in the village hall and then stayed to serve dinner, popped in to Perseverance Cottage later to report that it had all been a great success: the geese were delicious, and we'd all got a vote of thanks for our labours at the end.\n\nClive was going to write it all up for the Mosses Messenger, with photos... and come to think of it, I did vaguely recall that he'd been up in the Hall kitchen earlier and a flashbulb had gone off right in my eyes...\n\nMarian and Clive rushed out the first December issue of the Mosses Messenger at record speed, and it was as I feared: there was a photograph of me looking hot, cross, shiny and dishevelled in the Hall kitchen, flanked by Nick, in gleaming chef's whites and Mrs Gumball, wearing a crisp, frill-edged pinny and with not a hair on her head out of place.\n\nHowever, there was a lovely picture of the Senior Citizens toasting Annie and Gareth's engagement in dandelion and burdock, sherry, beer or Pinot Grigio, according to their tastes.\n\nOnce I'd recovered from that, I threw myself into giving the cottage its annual big Christmas clean, from the attic downwards. Unfortunately, when I went up to the attic I found I hadn't fully secured the bottom section of the loft ladder, so that it slid up when I was near the top. I clung to it, swinging to and fro over the stairwell like Tarzana of the Apes but, luckily, finally dropped off when over the landing. I lay there on my back for a few minutes, winded and giggling slightly hysterically, but after that I double-checked the ladder before trusting my weight to it.\n\nI didn't do much up there anyway, other than sweep away the cobwebs and collect the boxes of Christmas decorations... especially after I discovered a few more forgotten odds and ends of Tom's. And I don't know why, but they made me burst into tears. I didn't miss him \u2013 in fact, there was a sense of relief that he wasn't ever going to be coming home again \u2013 but I think that made me feel even guiltier.\n\nPerhaps there was added guilt, too, about what happened with Nick on Bonfire Night \u2013 but of course that was just a combination of alcohol, shock and a need for comfort, not love. Nick may be attractive (even when he's at his worst, barking orders at me in the Hall kitchen), but he's also exasperating, and that's twice he's dropped me like a hot potato and gone off doing something food-related and therefore far more important.\n\nIn between all my cleaning, I baked some Christmas tree gingerbread shapes for the next CPC, though wedding mania was still holding sway. Miss Pym intended making more adjustments to our dresses, since Marian was losing weight, while I was putting it on. I blame it on being unable to do much in the garden, which lately is either frozen hard, covered in snow, or both.\n\nI couldn't believe that already it was the last Mystery Play rehearsal in the village hall, and really we didn't need it, we were all word-perfect. So after a quick run-through, we all turned to helping Marian and Clive set the hall up ready for the village Christmas Show the following evening. I was looking forward to that, since I never got involved, so all I was expected to do was buy a ticket and go to watch it.\n\nIt was late and bitterly cold when we went out, so most of us headed straight for home. Nick silently fell into step beside me, but instead of seeing me to my door he strode off at the turn from the drive up to the Hall with a brusque 'Good night!', leaving me to it.\n\nEveryone goes to the Christmas Show, including Roly, Mimi and Juno. Even Nick came this year, but when Juno offered to change places so I could sit by him, I said quickly that I was quite happy next to Roly, and Nick glowered at me.\n\nThe evening followed its usual pattern: Ted the gardener gloomily produced rabbits out of a battered top hat and silk scarves out of the ears of members of the audience. He was followed by the infants singing carols, which always reduced most of the audience to tears, and Dave Naylor singing 'O Sole Mio', which didn't.\n\nThe Senior Citizens' Tea Dance Club's display of salsa dancing was particularly memorable. Some of the others may have been more technically perfect, but the fire and liveliness of Mrs Gumball's performance more than made up for any little mistakes.\n\nOn the Friday I went to Liverpool to fetch Jasper, dog and baggage home for the Christmas holidays, though I took a wrong turning and circled one of the two cathedrals twice, before charging off in what luckily turned out to be the right direction.\n\nIt was lovely to see him again, but Ginny was still about as attractive as a hairball, and gave an experimental nip or two at my ankles as I hugged Jasper.\n\nHis belongings seemed to have doubled since I left him there in October, and we had a job getting them into the Land Rover. I treated that like a sort of three-dimensional jigsaw puzzle, which is something most women are good at since life is a three-dimensional jigsaw puzzle containing several trick two-sided pieces. (I'm sure Nick is one of those, from an entirely different puzzle.)\n\nAll the way home Jasper was silently texting messages on his phone and when I asked who to, he said his girlfriend! He didn't expand on this interesting remark but I expect he'll reveal all eventually.\n\nThe day after Jasper came home Nick slammed in through the kitchen door like a whirlwind and demanded, 'Why didn't you tell me Ophelia Locke was the ARG supporter who was targeting you \u2013 and at Polly Darke's instigation?'\n\n'How did you find that out?' I blurted, taken off guard.\n\n'Caz just told me, among several things he suddenly decided I ought to know \u2013 and I might have taken the other incidents more seriously if I'd known about it.'\n\nJasper, who'd been sitting at the table finishing off a late, late breakfast, looked up. 'Ophelia was? What, with those animal rights people?'\n\n'You mean, you didn't know about it either?' Nick said in a quieter voice, seeming slightly mollified.\n\n'I didn't tell him \u2013 or about the other incidents,' I said, 'because I didn't want to worry him.'\n\n'Which other incidents?' asked Jasper.\n\nNick gave him a quick r\u00e9sum\u00e9 of what had been happening and then added, 'And there was a firework thrown at her at the bonfire, did she tell you about that?'\n\n'We don't know that was Polly,' I said, going pink as usual when anyone mentioned Bonfire Night.\n\n'Actually, we do, because Caz spotted her doing it.'\n\n'He did? Then why on earth didn't he say so?'\n\n'You know how he feels about the police. It took him long enough to tell me.'\n\n'You won't tell Unks about Ophelia being in ARG, will you?' I asked anxiously. 'Only they've thrown her out now, and since she and Caz are getting married it would be a pity to spoil everything.'\n\n'You are the strangest woman!' Nick exclaimed, looking exasperated.\n\n'She certainly is,' Jasper traitorously agreed. 'Do you know, I found her crying over her postcard album when I came downstairs earlier and when I asked her why, she said there was something terribly sad about Cr\u00e8me de Coeur!'\n\nNick seemed strangely cheered by the thought of my misery. 'She did? Well, well!'\n\n'Shouldn't we do something about this woman, if she's playing nasty tricks on Mum?' suggested Jasper.\n\n'Something is going to be done,' Nick assured him. 'Leave it to me.'\n\n'Oh, right,' Jasper said, looking relieved. 'Well, come on, Ginny. Mum, can I borrow the car, if you don't need it today?'\n\n'Why, where are you going?' I asked automatically.\n\n'Meeting Stu and some other friends, and maybe going to see a film and have a pizza, but I won't be late. And I won't drink and drive,' he added patiently.\n\nI handed him the keys to the Land Rover. 'Are you meeting your girlfriend?'\n\nJasper tapped the side of his nose infuriatingly, which was all the reply I got. Nick followed him out and I saw them talking together before I closed the door against the icy wind.\n\nWhen I looked out again, the yard was deserted and the hens had retired to huddle somewhere warmer. The very last Honey, her thick brown feather bloomers blown up like an inside-out umbrella, was running up the ramp into the henhouse.\n\n## Chapter 29: Clueless\n\nToday's meeting of the CPC was our Christmas party, because instead of the next one we were all going to help pack and distribute the WI Senior Citizens hampers. Everyone came to Perseverance Cottage bearing food \u2013 little triangular sandwiches, quiche, individual cream-topped sherry trifles decorated with green diamonds of angelica and, of course, Christmas cake. We ate our slices at the end with a chunk of crumbly Lancashire cheese on the side.\n\nWe had a lovely time, but after they'd gone and I was clearing the table, I couldn't help remembering back to when my son was taken ill on the same occasion, five years earlier...\n\nThe Perseverance Chronicles: A Life in Recipes\n\nI scribbled a heartfelt 'but thank goodness he pulled through!' to end the paragraph, thinking how lovely it was to have him home again, even if he did seem to be out of the house most of the time. There was certainly nothing wrong with his appetite: food vanished from the fridge and cake tins overnight, and I was making mincemeat flapjacks on a daily basis.\n\nHis Christmas present wish list seemed to consist almost entirely of books and CDs, although I'd already collected a few bits and pieces, including a spectacular Swiss army knife with millions of gadgets, which I rather coveted myself. I was sure it would come in handy.\n\nI had an awful lot of handwritten pages of notes for my next Chronicle and the Just Desserts book to type up, which would keep me occupied between all the Christmas stuff. But then, I'd already made the Christmas cake and pudding, and I didn't need to think about Christmas dinner itself, because we always had it up at the Hall with the family. It will be yet another goose... but then, it usually was.\n\nThe first Mystery Play dress rehearsal (for which I wasn't needed) took place up at the Hall, and apparently went quite well, with only one or two minor mishaps. Clive and Marian randomly mix up the various acts of the play for the two dress rehearsals because there's a feeling that it would be unlucky to do the complete thing right through before the actual performance. I could only hope that the snow had thawed and it was not quite so bitterly cold when I came to rehearse in my Eve costume the following Tuesday.\n\nWhile I was out on pet-sitting duty, Caz dropped a freshly cut Christmas tree off at the cottage, and by the time I returned Jasper had set it up in its stand in the sitting room and was opening the boxes of decorations we'd collected over the years, along with some old family ones I could remember my mother hanging up. Out came the fragile glass violins, trumpets and bells; the bright birds with purple and pink feather tails and the gaudy strings of slightly balding tinsel.\n\nWe don't have lights because I'm convinced they will set the house on fire. I don't know why, though perhaps distant memories of the way the bulbs used to pop when my father turned them on might have had something to do with it. I expect that's where I get my uselessness with electricity from.\n\nLater, while Jasper finished the decorating, including hanging a stocking for Ginny from the mantelpiece, along with his own, I baked thin, crispy star-shaped spice biscuits to hang on the tree with ribbon, the finishing touch.\n\nWhile I did this, the sound of carols on the CD player, the mingled smell of spices and pine... the memory of the cold, crisp air outside \u2013 all these seasonal elements combined until the magic of Christmas, as always, had me in its thrall.\n\nUnfortunately, next morning I found PC Perkins standing on my doorstep, her dark uniform lightly frosted with snowflakes like a rather odd Christmas card. She very politely suggested that we go and look in the outbuilding where I kept my gardening tools, because she'd received an anonymous tip-off.\n\nShe didn't say a tip-off about what, but I said she was welcome to go and look, and I would follow her over once I'd put my wellies and anorak on.\n\nWhen I got there, having waded through an audience of interested hens, she was standing staring up at the wall rack where my tools hung fairly neatly \u2013 and there, hooked among them, my blue steel cross-shaped wheel brace.\n\n'Is that the one you used to change your tyre, on the day your husband took your car?' she enquired.\n\n'It certainly looks like it,' I began, reaching up for it, but she put her hand on my arm to stop me.\n\n'Please don't touch it, Mrs Pharamond.'\n\nI let my hand fall to my side. 'But... I'm sure it wasn't there before! I'd have noticed it when I was hanging up the tools, because it doesn't live there. I always kept it in the car.'\n\n'So when did you last see it?'\n\nI frowned, trying to remember, though the events of the summer seemed an awfully long time ago now. 'I'm pretty sure that when I'd finished changing the wheel, I slung it in the footwell behind the driver's seat,' I said slowly. 'Didn't I already give you a statement about that? But of course Jasper checked the wheel too, while I was in the cottage, and I can't recall what he said he did with it. He's gone up to the Hall, but I'll ask him when he gets home, shall I?'\n\n'If you don't mind,' PC Perkins said, unfolding a large plastic envelope and inserting the wheel brace into it. 'And I'll just take this and check it for fingerprints, if you have no objection?'\n\n'Not at all,' I said politely, 'but you'll only find mine and Jasper's, won't you?'\n\n'Just routine. We like to tie up all the loose ends,' she said, giving me that 'I'll get you yet, you murderess' smile. After such a long silence, I'd convinced myself that I'd only imagined the police were suspicious of me, but clearly I'd been quite right all along!\n\nWhen he came in, Jasper said he thought he might just have propped the wheel brace up against the barn wall when he'd finished tightening the nuts, but he couldn't be sure. He could equally well have tossed it into the back of the car, where it usually lived.\n\n'But whichever way, someone must have put it with the gardening tools recently and then told the police,' I said, puzzled, 'because I'd definitely have noticed it if it had been there all this time, since I'm constantly taking tools out and putting them back \u2013 and it was hung on top of my favourite spade. But what's the point, when finding it won't tell the police anything they didn't already know?'\n\n'I wouldn't worry about it, Mum. I expect she really meant it, about tying up loose ends. And you are vague sometimes, so you might have moved it to get at the spade, and not noticed it was there.'\n\n'I'm not that vague. And who tipped them off about it, and why?'\n\n'It's a mystery, but not one that's important. I'd forget it,' he advised. 'Or you could tell Uncle Nick about it and see what he thinks.'\n\n'No, thanks,' I said crisply. 'He'd probably just accuse me of losing my marbles, like you.'\n\nJasper had now become even more antagonistic towards poor Ritch, if that were possible, and warned me that if he became his stepfather he would leave home! I assured him that even if I had been tempted to remarry, which was the last thing on my mind, I would certainly not replace one chronic philanderer with another.\n\nI expected it was all because he overheard Ritch jokingly asking me to marry him again, when he caught us having a Christmas kiss under the mistletoe I'd suspended from the drying rack. (I hang mistletoe up every year, but that was the first time I'd struck lucky.)\n\nRitch and I had already exchanged presents. He gave me a delightful little sparkly crystal snowman brooch and I gave him a box of home-made Turkish Delight (from the postcard recipe sent to me by Nick) and a large rawhide bone for poor old Flo, about to be immured in kennels while her master flew off to stay with friends in the Caribbean over Christmas.\n\nI might have felt compelled to offer to have Flo myself if it hadn't been for Ginny: one snap of Flo's powerful jaws and Ginny would be only a lingering memory. However, Flo was booked into the local luxury Dogtel, with heated beds and her own run, so I didn't suppose she'd find it too traumatic.\n\nJasper didn't, however, extend his antagonism towards all my male visitors, even on one occasion helpfully pointing out the mistletoe to Nick before he went out, though luckily I don't think he heard him.\n\nBut at least Nick now seemed to have finally accepted that I simply wanted to forget what happened on the night of the bonfire and continue as we were before, so we were back on our old, slightly argumentative but fairly amicable terms, and he was helping me with recipes for Just Desserts.\n\nHe began bringing down bundles of his old notebooks for me to copy things out of, though his idea of what was suitable and what wasn't didn't exactly coincide with mine.\n\nIt was no wonder I was putting on weight, because I'd adjusted my chocolate intake to compensate for... well, I didn't really know what for, but it was very comforting. Have you ever tried hot chocolate custard?\n\nJasper and I went up to the Hall for Sunday lunch and Roly had the newspaper magazine with the Christmas photoshoot article in it. And actually, it all looked really magical, lush and quite swish, in a slightly medieval sort of way, not fake at all.\n\nMimi said she thought Juno looked just like Edith Sitwell, but luckily Juno mixed her up with Edith Cavell and was vaguely flattered, saying she knew she was a heroine putting up with Mimi but that might be going a bit too far.\n\nI'd taken up a box of the spice biscuits, all ready threaded with ribbon for hanging, and we decorated their big Christmas tree in the hallway after lunch, with Nick leaning over the banister to place a porcelain-faced angel on the very top.\n\nI was convinced that their Christmas tree lights were made of Bakelite! I only hoped they'd had an electrician check them in the last fifty years.\n\nI was so glad I'd got a Land Rover, because I used to be very nervous about driving on snowy roads and now I wasn't in the least. We had another light snowfall on top of the last lot, which had half-thawed on the roads and then refrozen, making it pretty treacherous, but I made it easily down to the village hall to help with the hampers.\n\nEach recipient had ticked boxes on a form giving their likes, dislikes and preferences (Marian was nothing if not organised) so we just had to select from the list and assemble each box, which were the cardboard sort printed with a green holly pattern, with pop-up handles.\n\nSeveral of the WI members had four-wheel-drive vehicles, so were going to deliver the hampers that afternoon, when the roads had been gritted.\n\n'Another job done,' Marian announced with satisfaction as the last of them drove away. 'There's just the final Mystery Play dress rehearsal tomorrow, and then we can all relax and just enjoy ourselves over Christmas.'\n\n'Except we actually have to do the play on Boxing Day,' I pointed out, and the thought of shivering in the snow in my new Eve outfit was not an enticing prospect.\n\n'But that's the fun bit,' Annie said, then sighed. 'I will miss our CPC meetings until we start again in summer, though.'\n\n'This year I think we need to start again right after Christmas,' I said, 'only as Wedding Organisers instead!'\n\n## Chapter 30: Unscheduled Appearances\n\nThe snow lingers and, though the local farmers have kept the roads around the village open, more is forecast. It won't stop the Mystery Play, though \u2013 nothing has ever done that, not even Cromwell!\n\nThe Perseverance Chronicles: A Life in Recipes\n\nOn Tuesday afternoon I walked up to the Hall after lunch for the second of the Mystery Play dress rehearsals. I left Jasper typing up some of my latest Chronicle onto my new laptop. He was much faster than me, so that was a big help.\n\nHe said Unks wanted him to go up to the house later in the afternoon, so he would see me there.\n\nThe cobbled courtyard of Pharamond Hall where the audience stand to watch is bound on one side by the kitchen wing and on the others by stables and outbuildings, making it very sheltered. The entrance is through a large arched gateway with, directly facing it, a second arched doorway to the coach house, which forms the stage for the performance.\n\nMarian, Clive and most of the cast for the rehearsal scenes were already there, milling about, while the Mummers of Invention (minus Ritch, of course, who was on his way to the Caribbean) stood in one corner, running through the song for the first interval. Ophelia was wearing a knitted poncho in three shades of mud brown and it was stretched to the limit over her now enormous baby bump. Various bits of scenery and old props had been dragged out of storage and the loose boxes on either side set up as changing rooms. I knew Joe Gumball had already hung up the stiff, heavy canvas curtains in the entrance to the coach house, because Jasper and Nick had helped him, and now he was checking that the star lantern slid easily across the wire behind it.\n\nThere was a chilly wind blowing, and since the courtyard was not warmed by braziers and a massed audience, as it would be on the night, we ran through our scenes pretty briskly. Clive was reading the Voice of God today and started with Lucifer being cast out of Heaven. The silent angels, with their freshly flighted wings, trooped on and off on cue, but when Moses did his scene he interjected more than a little acerbity into his lines: his rheumatism was clearly still playing him up.\n\nI was on next, but luckily, due to the extreme cold, Clive kindly excused Adam and Eve from having to change into costume, which was a relief. I didn't know about Nick, but I was having serious doubts about the decency of my new Spandex outfit. Still, at least we were back on reasonably good terms again and from the tone of our voices you would have thought we were discussing the price of fish, not contemplating any kind of temptation.\n\nAfter that, Miss Pym and some of the parents brought the infants up from school in an orderly but excited crocodile, carrying their animal masks, to practise the Ark scene.\n\n'And all the animals came into t'ark out of the rain, and, by heck, it were pouring down,' Noah said, standing next to Mrs Noah, who was seated on a bucket. The children started to march past two by two, growling, roaring, hissing and generally sounding like a zoo at feeding time. Last of all came a solitary unicorn.\n\n'There's two of every darn thing \u2013 except t'unicorn. Yon's not going to breed on its own, Wife.'\n\n'Well,' said Mrs Noah, reluctantly looking up from her knitting, which was presumably a late Christmas present she was keen to finish, 'there's no more of 'em. Reckon that's the end o' the line for t'poor little beast. I never did see much use for it, though it's proper bonny.'\n\n'It attracts virgins, so they say,' said Noah.\n\n'Well, it's just thee and me now, chuck, so I reckon them have died out an' all,' said Mrs Noah. 'Knit one, purl two!'\n\nAfter the Ark scene we always have a break before the Nativity, so all the little animals can see Father Christmas before going home. By now Annie and Gareth had arrived together and helped Miss Pym shepherd the excited children through the arched gateway to be lined up again, sans masks, outside the front door of Pharamond Hall.\n\nThe rest of us went through the kitchens the back way to the cavernous hallway, where a log fire roared and the fairy lights flickered on the huge tree like so many weak fireflies (and I am sure they are not supposed to do that). Roly was sitting in an ancient carved chair next to it, dressed in the red, fur-edged hooded suit and black boots traditional on these occasions, and puffing at a cheroot, which was not. Over the years his wig and beard had yellowed with nicotine, so that I'm sure the scent of tobacco would forever remind successive generations of local children of Christmas.\n\nIn the shadows just behind the chair lurked Caz Naylor, the largest elf you ever saw, wearing pointed Spock ears and with his hat jammed down hard over his eyebrows, waiting to hand the presents to Father Christmas. What always surprised me was the way he could move so silently when his outfit was entirely covered in little bells. Perhaps he'd stuffed them with something?\n\nThe fire glowed in the huge hearth, and the candle bulbs in the cartwheel of evergreen foliage that was suspended from the ceiling were dimmed. The house smelled of cinnamon and burning fir cones, hot mince pies and spiced punch from the bowl on the trestle table laid out ready.\n\nFrom beyond the great oak front doors came the sound of a lot of reedy young voices belting out 'Good King Wenceslas' at the top of their lungs: distillation of pure Christmas magic, again.\n\n'Here we go,' Unks said, regretfully removing the stub of cheroot from his mouth and tossing it accurately into the fire. 'Let the little blighters in.'\n\nNick swung the door open and a tide of children rushed forward, only to be halted in their tracks by Miss Pym, who has a presence and voice that could command armies.\n\n'Stop!' she commanded.\n\n'Ho, ho, ho,' Unks said benevolently. 'Come in, one and all!'\n\nJoe Gumball activated the CD player and 'White Christmas' began to chirrup merrily in the background.\n\nThere was a gift for every child, and while Miss Pym orchestrated the queue, the adults fell on the food and drink. A few older children appeared as parents began to turn up to collect their offspring, but there was a bag of extra gifts for this contingency, so no one went away empty-handed.\n\nBy now Jasper had arrived too, and was talking very seriously to the vicar in the corner. At a rough guess, I'd say they were discussing the eating habits of Biblical folk or something like that, unless Gareth had a personal hobby horse and a stronger will than Jasper's. Annie had gravitated across to join them, and Trinny, wearing a collar of tinsel, was circling Ginny in a vaguely menacing manner, probably trying to decide which end was which.\n\nJasper picked Ginny up and Trinny immediately lost interest and wandered off under the table, where there were rich pickings in crumbs and discarded pastry. Mrs Gumball's idea of children's party food ran to miniature pork pies, tiny triangular sandwiches, and little jellies in paper cases with a blob of cream and a diamond of angelica on top of each. The hot mince pies and punch were strictly for the adults.\n\nEventually Santa announced that the reindeer were getting restless and he had to leave, which was the cue for the last of the tired but happy children to be taken home. As the final car vanished down the drive, Clive Potter firmly rounded the cast up for the final part of today's rehearsal: the Nativity, and another song or two from the Mummers.\n\nJasper had already gone home and Nick was helping Roly out of his robe, boots and wig, but the rest of us trooped out again into the growing dusk, warm, full and a bit reluctant. Marian and Kylie made for the loose box changing room to adjust her Mary costume, while our Joseph, Dave Naylor, leaned against a wall in his striped robe, smoking a cigarette.\n\n'Where's Ophelia?' Jojo asked, looking around him vaguely. 'We're supposed to be playing \"While Shepherds Watched\" before the next scene starts.'\n\n'Dunno. Haven't seen her for ages,' Mick said, then cupped his hands round his mouth and bellowed, 'Ophelia!'\n\n'That's funny,' I said. 'Now I come to think of it, I don't remember seeing her in the Hall, either. Has she gone home? I hope she's feeling all right.'\n\nCaz appeared, without his elf ears.\n\n'Caz,' I called, 'do you know where Ophelia is? Only no one seems to have see\u2014'\n\nI faltered as a piercing howl of anguish echoed from the stables we used as dressing rooms. Then Marian Potter's cropped, silvery head appeared over the half-door and she cried wildly, 'Help! Is there a doctor in the house?' before bobbing down again, more Punch and Judy than Mystery Play.\n\nUnfortunately Dr Patel had rehearsed his scenes the previous week so he wasn't present. There was a second's breathless hush, then we all rushed across the yard. Caz beat Annie and me to it, but it was a close-run thing and the others crowded up behind.\n\nInside the dimly lit stable, with no more ado than a couple of pangs and an animal urge to be alone, Ophelia had chosen to give birth, if not in the manger, certainly right next to it.\n\nShe lay pale, spent and panting slightly on the straw, her big eyelids closed, while Kylie, clearly revolted, was holding a messy and screaming baby at arm's length.\n\nIt was amazing! I'd never seen an infant that so closely resembled a fox cub, and there could be absolutely no doubt that it was Caz's.\n\n'Something to wrap him in,' Marian ordered distractedly, but Joseph was already passing his voluminous striped towelling headdress over. Caz slipped through the door, removed his child from Kylie's uncertain grasp and enfolded it in the warm material. Then he sat down on an upturned bucket. The baby, as if by magic, stopped bawling and stared up at him.\n\nGareth, who was still leaning over the half-door between Annie and me, now said slightly uncertainly, 'Bless you, my child!' like an aged bishop. Still, I don't expect this is a situation he's ever had to contend with before.\n\nClive, efficient as ever, had already trotted back to the house to tell Unks and call an ambulance. He said the horse might have bolted, but a check-up of mother and baby was clearly indicated.\n\nThere was a feeling of anticlimax about the rest of the rehearsals once Ophelia, the infant and Caz had been whisked away to hospital. Ophelia hadn't wanted to go, and Caz had almost balked at the sight of the ambulance's brightly lit and clinical interior, but Nick had reappeared by then and firmly shoved him in and closed the door.\n\n'I wonder if she's actually got anything ready in her cottage for the baby's arrival,' I mused.\n\n'Oh, yes. Dave says all the Naylors have rallied round with baby clothes and equipment,' Annie assured me.\n\n'I think their wedding had better be postponed until after the christening,' Gareth remarked thoughtfully. 'Or perhaps we can do both on the same occasion? I'll have to consult the bishop.'\n\n'If they can decide on a name,' I said. 'Star and Rambo seemed to be frontrunners last time I talked to Ophelia.'\n\nGareth gave me a doubtful smile: I expect he thought I was joking.\n\nWe rushed through the Annunciation, Nativity and Flight into Egypt at breakneck speed. Kylie was distinctly huffy, and clearly felt she had been upstaged, though Joseph, bare-headed, performed his part with perfect sang-froid.\n\nAfterwards, most of the cast set off for the New Mystery, and I bagged a lift with Gareth and Annie. Nick followed us down in the estate pick-up, with nine angels crammed in the back and Lucifer sitting beside him.\n\nWhen we got there I took our usual corner seat with Gareth and Annie, but it was only when we sat down that I realised Nick hadn't followed us but was smiling and talking with Polly Darke over near the bar. He's so tall he must have had a bird's-eye view down her cleavage: her twin prows were jutting out like the front of a catamaran.\n\nShe noticed I was watching and flashed a triumphant look in my direction as he steered her away to a darker corner, one hand under her elbow and his glossy dark head bent towards hers.\n\nMy mouth must have been hanging open, because Annie nudged me with her elbow and asked anxiously, 'Are you all right, Lizzy? You look a bit odd.'\n\n'I'm fine,' I said with an effort, 'just a bit tired, suddenly.'\n\n'Yes, me too. Where's Nick? I thought he followed us in.'\n\n'He decided to go for a bit of a tramp,' I explained.\n\n'I expect he needed some fresh air,' Gareth said vaguely, as if we hadn't already spent most of the day out in the cold, freezing our socks off.\n\n'If you don't mind, perhaps I'll just get off home, after all,' I said. 'I feel a bit tired, and there's such a crowd it'll take her ages to come for our orders anyway.'\n\n'Don't you want to wait and we'll drive you back?' asked Annie. 'We won't be long, because Trinny's in the car, and she'll get cold.'\n\n'No, that's all right,' I said, getting up. 'It's only five minutes away and Jasper will be there. I'll let you know if I hear any more about how Ophelia and the baby are doing.'\n\nOn my way out I sneaked a glance at the corner where Nick and Polly were still sitting, their heads close together.\n\n'Come on, Mum, obviously he's doing it for a reason,' Jasper said, when I told him about Nick's betrayal \u2013 which I did about five seconds after arriving home. It was that or burst.\n\n'Oh, yes, I could see that,' I said shortly. Ginny, not liking the tone of my voice, ran her teeth thoughtfully up and down my ankle.\n\n'No, Mum, I meant it must be part of some plan he has, because he said he would deal with her, don't you remember?'\n\n'Then he's going about it in a strange way! And if you're right, why didn't he tell me what he was going to do?'\n\n'You kept everything a secret from him, didn't you? He only found out from Caz and Leila what was going on. And I expect if he'd told you, your reaction when you saw them together wouldn't have looked half as authentic. Now she'll think she's putting one over on you.'\n\n'Maybe she is: he wouldn't be the first man unable to see past a pair of pneumatic boobs.'\n\n'Not Uncle Nick,' he said stoutly. 'Really, Mum, you can't possibly believe that \u2013 she's a complete dog.'\n\n'Bitch,' I said, absently, because I was wondering if he could be right. Then I realised what he'd said. 'That was a bit rude, Jasper!'\n\n'I suppose it was \u2013 but I only meant that she's no competition, so you don't need to worry about Uncle Nick falling for her.'\n\n'I'm not worried in the least, he can fall for anyone he likes,' I assured him, then rather spoiled the effect by adding, 'but while we're speaking of bitches, Jasper, do you think you could teach yours not to nip my ankles?'\n\n'She's just being friendly,' he said fondly, bending down and giving her a pat. 'By the way, Unks rang and told me about the nativity at the Nativity \u2013 sorry I missed it!'\n\n'Just don't expect a repeat performance on Boxing Day,' I warned him. 'I think we'd all better stick to the script from now on.'\n\n## Chapter 31: Middlemoss Marchpane\n\nI just made a chocolate, fruit and nut Christmas wreath, by packing melted chocolate mixed with puffed rice breakfast cereal into a ring mould, then studding the surface with whole nuts of various kinds, crystallised cherries and other candied fruits, glued on by half-dipping them in more melted chocolate. I'm going to have it as our table centrepiece on Christmas Eve, with a red candle in the middle.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nWord had it that Ophelia discharged herself from hospital almost as soon as she had been checked over, but the Naylor clan were rallying round.\n\nI spent the next couple of days doing Christmas baking, including the fine ham that Roly had sent down for me (he does this every year), and making a big trifle and a Middlemoss Marchpane.\n\nI put the recipe for the latter, with one or two small adjustments, into Just Desserts. I was going to save it to take up to the Hall with us on Christmas Eve, for we always went to listen to the carol singers, whose first call it traditionally always was. But then Jasper's friend Stu came over to stay the night and they demolished it, so I had to set to and make another.\n\nI drove up to the Hall, since as well as the Marchpane I had my contribution to tomorrow's Christmas dinner with me: a vat of mulligatawny soup and the giant round Christmas pudding. There was also a box of presents to put under the tree, most of them home-made and edible.\n\nWe've always had Christmas dinner up at the Hall: Mrs Gumball would go in early to cook breakfast and put the goose into the oven, then I would finish the cooking and serve it. But this year Nick was here, so apart from my soup and pudding contributions (and some brandy butter ice cream I'd got from Faye), he was doing it solo. He'd have to, because after my previous experience as chef's skivvy, I'd no intention of ever letting myself in for that again.\n\nIn fact, I was now trying to avoid him altogether, since every time I looked at him a nasty picture of his and Polly's heads, flirtatiously close together, slid into my mind. I might have agreed with Jasper that it was all just a cunning ploy to get information out of her, but I wasn't a hundred per cent convinced...\n\nJoining in with the carol singers round a roaring fire up at the Hall always seemed a significant moment and by the time they'd all trooped off again, full of sherry and mince pies, I felt as full of anticipation as a child.\n\nBack home once more, we had our usual Christmas Eve supper of thick slices of the Christmas ham with egg and chips, followed by first go at the big sherry trifle I'd made. Then we watched an old film in the sitting room on Jasper's little TV, which he'd brought back with him, along with all his other stuff. He'd fixed the kitchen one, too, by the simple expedient of changing the plug.\n\nJasper's stocking, which had been knitted for him by Annie when he was a toddler, hung next to Ginny's at one end of the mantelpiece. Mother Claus would fill it and hook it over the handle of his bedroom door later, as she always did... and I suspect she'd better hang Ginny's there too, or there would be trouble.\n\nThere was quite a heap of gifts under our tree. I couldn't resist fingering the ones from the family we had brought back with us, but of course I couldn't open them until next day, or it would spoil the surprise...\n\nWe had an orgy of unwrapping next morning while Ginny chewed noisily on a rawhide version of a candy cane and, although I'm sure we both thought of Tom while dividing up the presents into two piles, rather than three, neither of us mentioned his name. He'd hardly been around much for the last four or five years anyway, spending as little time in our company as possible, so the spirit of Christmas past didn't really haunt us, even if we were briefly saddened by the ghost of what might have been.\n\nJasper gave me a pen, the kind with liquid inside that you tilted so an Egyptian sarcophagus lid slid open to reveal a mummy's mask. There was a mummy-shaped biscuit tin too, so he'd obviously found a good museum shop somewhere. I had soaps, bath oils and gardener's handcream from Mimi and Juno, an antique-looking ring from Roly \u2013 and a new postcard album, bound in soft blue leather, from Nick. He must have noticed my old one was full up to overflowing...and he must also intend sending me a lot more, too, so I expected I was right about him soon tiring of staying in one place and he'd soon be off on his travels again.\n\nJasper retired to his room, wearing the long Dr Who scarf Annie had knitted for him, to have a private conversation on his mobile phone with his girlfriend \u2013 about whom I still know practically nothing, except that her name is Kelly \u2013 while I tidied up the discarded wrapping paper and ribbons.\n\nThen I put on my new slinky green dress (why should all the honours go to Polly?) and we went up to the Hall for Christmas dinner.\n\nAnnie and Gareth had been invited too, and it all felt a bit like d\u00e9j\u00e0 vu after the photoshoot one, except we actually got to eat the food, and Lionel Cripchet didn't burst in and start going on about squirrels. And Jasper was there too... and Ginny, who was sick behind the door from Mimi feeding her too many titbits, so that was different from last time.\n\nI was wearing the old and valuable-looking ring that had been Unks' gift to me, but with the large, oval emerald turned inwards so it didn't catch the light. I felt increasingly sure it was a family heirloom, in which case I really had no right to it. So, as soon as I got the chance for a quiet word, while we were going into the drawing room for coffee, I asked him if he was sure I should have it.\n\n'Yes, my dear, it's quite fitting,' he assured me. 'Don't you like it? Would you have preferred a modern one?'\n\n'Oh, no, I love it! Only I'm sure I've seen it in one of the portraits in the gallery, so it must be a family piece.'\n\n'It's the betroth\u2014' began Mimi, who'd caught up with us, spotting it for the first time, but a glance from Unks silenced her and she wandered off again with a giggle.\n\n'I want you to have it,' Roly said firmly. 'Humour an old man, m'dear?'\n\nI thanked him, but thought that the first opportunity I got I'd check out the portraits in the gallery and see if I could spot it, because if I was right, Nick might not be so happy about having part of his inheritance given away. Meanwhile, I'd have to remember not to wear it when gardening, or it would go the way of my wedding ring, back into the earth, never to be seen again.\n\n## Chapter 32: Hoar Frost\n\nThe annual Middlemoss Mystery Play on Boxing Day marks the end of the old year and the start of the new and I expect the Mystery Play replaced some much older, pagan ritual that would have taken place at about the same time.\n\nThe Perseverance Chronicles: A Life in Recipes\n\nThey say the sun shines on the godly, and certainly just as I arrived there on the morning of the Mystery Play with Jasper a weak, golden light began to spread over the courtyard of Pharamond Hall.\n\nThe farmers had cleared the local roads of the last fall of snow and, though icy in places, they were passable with care. In any case, many of the audience preferred to walk there.\n\nThe Mosses Women's Institute was setting up the refreshment stand near the kitchen door (the money raised goes to local charities), and I handed over my contribution of ginger parkin, fruitcake and bags of vanilla candyfloss. I managed to restrain myself from suggesting they make themselves Santa beards out of it, because this is quite a serious occasion, really.\n\nJasper went off to help Caz and Joe Gumball with the myriad last-minute jobs: lighting the charcoal braziers that were set around the courtyard, testing the microphone in Unks' little striped tent, from where he would speak as Voice of God, and moving scenery. I stored my Eve costume, wig and figleaves (which are threaded onto elastic, so they are quick and easy to put on for the Expulsion) in one of the loose boxes used for changing rooms: men to the left of the coach house, women to the right.\n\nWhen I came out again the audience had started to arrive, bearing picnics, folding chairs and rugs, and Jojo and Mick were warming their hands at one of the charcoal braziers. I hadn't thought how depleted the Mummers would be, since Ritch was still away basking in the Caribbean, and Ophelia, of course, had just given birth; but when I spoke to them they told me that actually Ophelia and the baby were in the kitchen with Mrs Gumball, who would mind the infant while she popped out and performed as usual.\n\n'Is that a good idea, so soon?' I asked doubtfully.\n\n'Yeah, she's fine, she wants to do it,' Jojo assured me, but I imagine the poor girl's performance will be even limper than usual.\n\n'Have they decided on a name for the baby yet?'\n\n'Sylvester Star, according to Ophelia,' Mick said, 'but I heard Caz Naylor calling it Sly.'\n\n'That's got to be better than Rambo, though,' I said, and they agreed.\n\nAlthough there was an old outside toilet behind the stables (Victorian vintage, with shiny mahogany seating), in recent years Roly has also arranged for a portable toilet block to be set up next to it, which saves much queuing during the breaks. I sensibly repaired there before putting on my Eve costume under my clothes: it would certainly be impossible to go again in that outfit. Makes you wonder how Spiderman and other superheroes manage, doesn't it?\n\nThe Spandex felt odd under my jeans, but quite warm. I left my wig hanging on the post outside the loosebox, together with my figleaves, and went outside again. The courtyard was now quite full and noisy, and the WI ladies were doing a roaring trade in hot drinks. The air was cold and smelled of spices and roasting chestnuts \u2013 or, if you suddenly and unexpectedly found yourself in the vicinity of Polly Darke and her little circle of friends, as I did, civet cats.\n\n'You've been making candyfloss again, I see,' Nick said, doing his silently materialising act right next to me.\n\n'I call it Hoar Frost and I'm dedicating the recipe to Polly,' I said tartly. 'What is she doing here? And why hasn't anyone run her off the premises?'\n\n'She's here because I invited her specially and told her it just wouldn't be the same without her,' he said, with an enigmatic smile. 'I suppose you feel much the same about Ritch Rainford. Poor Lizzy \u2013 didn't he invite you to go to the Caribbean with him?'\n\nI felt myself blush, because actually Ritch had, though I knew he was only flirting, as usual.\n\n'Yes,' I said shortly and ambiguously. 'It's a pity he isn't here for the play,' I added, fingering my sparkling little snowman brooch rather ostentatiously. 'Several of the other Cotton Common cast members are, though I don't know if they'll have the stamina to stay for the whole thing.'\n\nNick's hand captured mine and he stared at the ring on my finger. 'I didn't get a good look at that last night,' he said thoughtfully. The flat green stone gleamed with restrained opulence in its heavy, antique gold setting.\n\n'I hope you don't mind Unks giving it to me? I suspect it's a family heirloom, but I did ask him if he was sure he wanted me to have it.'\n\n'Well, then, I suppose you could say he's given you the family seal of approval,' he said blandly. 'And look, he's arrived, so we must be about to start. Who's that with him?'\n\n'Delphine Lake, one of the actresses in Cotton Common.'\n\nPretty as a picture from silver curls to tiny, pointed blue shoes, Delphine had somehow managed to insinuate herself into Roly's royal pavilion, but then, he always did have an eye for an attractive woman. There was just enough room for another folding canvas chair, and their heads were close together in earnest conversation.\n\nClive Potter came out and stood in front of the canvas curtains, holding up his hands for silence, and then bid everyone welcome to the Middlemoss Mysteries.\n\n'Now let our play begin!' he said dramatically, bowed and walked off.\n\nA small silence ensued, then there was a squeak as Nick leaned in and switched on Unks' microphone before his voice could be heard, confiding to Delphine, '... and then blow me if it didn't pick itself up at the fifth, overtake the field and gallop home by a head!'\n\n'Voice of God!' Nick whispered urgently.\n\n'Ah, yes \u2013 excuse me, my dear...' There was a rustling noise, as of paper being picked up. 'I AM GOD, THE ALL-POWERFUL, ALL-KNOWING,' he declaimed loudly, then lowering his voice to a more normal level, continued, 'Listen to my words \u2013 take heed of the mysteries that will unfold before your dazzled eyes.'\n\nThe curtain was pulled back to reveal Lucifer and nine angels against a gilded cloudy backdrop and the Mysteries were well and truly up and running. (Or bicycling, as would be the case during Mary and Joseph's journey to Bethlehem and subsequent flight into Egypt.)\n\nVarious interesting noise effects accompanied God's description of the Creation, which I could hear as I shrugged off my clothes in the changing room and concealed as much of myself as possible with the long, blond wig.\n\nThen we were on.\n\nIf you've ever tried to remember your lines while inches away from a tall and attractive man dressed in little more than ballet tights, you'll understand why I found it hard to keep my eyes on the apple. He was carrying a small sheaf of hay, which may have preserved his modesty from the audience, but was not much help to me. I expect ballerinas quickly get blas\u00e9 about this kind of thing.\n\nOf course, it might have helped if he'd stuck to the text when I offered him the apple, like he's done at all the recent rehearsals, instead of soulfully telling the audience in the most hammy way that I'd already had his heart and he didn't think a piece of fruit was much of an exchange.\n\nThey loved it, but I was tempted to elope with the snake.\n\nThen he took a bite, tossed it over his shoulder into the wings and led me offstage to cover my modesty with figleaves.\n\n'Nice costume,' he said, casting away his sheaf of hay and adjusting his figleaves like a hula skirt. The effect was interesting. 'Need any help with yours?'\n\n'No, thanks,' I said primly. Whoever plays Eve next year will need new elastic: the twang had quite gone out of mine.\n\nWe quickly took our places behind the painted bushes and the Voice of God demanded why we had eaten the forbidden fruit? I only wished I knew.\n\n'The woman tempted me,' Nick said, passing the buck, just as men have done from time immemorial, and we were expelled from Eden.\n\nThe curtain came down and the Mummers began to play something lilting while the scene was changed for Noah's Ark.\n\nNick dashed off for his changing room and I headed for my own warm outfit, shivering. I quickly dressed and then went out into the courtyard through the back doorway, avoiding the scurrying animal-headed infants and a harassed-looking Miss Pym.\n\nNick was already in the courtyard, talking to Polly. I elected to watch Noah's Flood from the other side, with Annie and Gareth.\n\n'What's Polly doing here?' Annie whispered to me worriedly, when Gareth had kindly gone to get me a hot drink (I was still freezing). 'And why is Nick chatting to her like that, and laughing and... well, flirting?'\n\n'Search me! He said he'd invited her, so perhaps he's fallen prey to her fatal beauty.'\n\n'No, I'm sure he hasn't, because he was flirting with you in the Adam and Eve scene, Lizzy, and he couldn't take his eyes off you! He must have an ulterior motive for making up to Polly.'\n\n'That's what Jasper says,' I agreed grudgingly, 'but I think he seems to be enjoying himself too much.'\n\nThe curtains closed on Noah's Ark and the animals, and then reopened revealing a tetchy-looking Moses seated on a mountain.\n\n'Here are my commandments, writ on tablets of stone,' said the Voice of God.\n\n'Could thee not find something lighter? I'm no spring chicken, that knows!' grumbled Moses.\n\n'There are ten of them \u2013 see thee obey the rules,' ordered God, while Moses hobbled about collecting them up in his teatowel headscarf.\n\n'I'll give it me best shot, Lord, and I can't say fairer than that.'\n\nIgnoring this sally, God ran briskly down the list then demanded finally, 'Dost thou understand?'\n\n'Yea, Lord,' Moses said obediently, though with an evil look in his rheumy blue eyes. 'I'm not deaf, tha knows! I'm going back down t'mountain as fast as me legs can carry me, and I'll be straight on t'case. Idol worshipping and other ungodly goings-on will be reet out t'window.'\n\n'Good, good \u2013 for I see everything, you know, I am omnipresent,' God added conversationally.\n\nThen the mike squeaked and his voice suddenly boomed, 'IN FACT, POLLY DARKE, I KNOW WHAT YOU DID LAST SUMMER! I KNOW IT WAS YOU WHO LOOSENED THE WHEEL NUTS ON LIZZY PHARAMOND'S CAR, CAUSING THE DEATH OF HER HUSBAND.'\n\nEveryone, including me and a flummoxed Moses, turned to stare at Polly Darke. Nick let go of her arm and stepped away, but I could see from his face that this was no surprise to him: God's accusation had been prearranged.\n\nShe found herself the centre of a staring, whispering circle of shocked faces: even her friends were wide-eyed.\n\n'Polly did?' I exclaimed. 'But\u2014'\n\n'No, no, I didn't!' Polly yelped, looking from face to face for some sympathy. 'Why on earth would I do that? I loved him!'\n\n'Because you expected Lizzy to drive the car, not Tom,' Nick said clearly and coldly. 'It was just one of a series of little spiteful accidents you arranged for her, because you were eaten up with jealousy.'\n\n'No! No, I didn't! I haven't\u2014'\n\n'Good heavens! Surely she wouldn't do something so evil?' gasped Annie, shocked to her soft-centred core, and Gareth put his arm around her consolingly. I wished someone would put their arm around me: I was shaking even more now, and not from the cold.\n\nI didn't notice PC Perkins and her youthful associate until she was actually putting handcuffs on Polly, just like in a film, and saying clear enough for everyone to hear, 'Polly Darke, you are under arrest...' and proceeding to give her the official caution.\n\nPolly stared around like a hunted animal, but there was no escape: I could see the flashing lights of police cars beyond the archway, and other officers. Then her eyes fixed on me.\n\n'It was her \u2013 her!' she cried. 'I've said so all along... you've no proof!'\n\n'It was not!' boomed God into his microphone. 'And there was a witness to your wrongdoing.'\n\n'That's right,' Caz agreed loudly from the shadows. 'I seen her doing it.'\n\n'EVIL WOMAN, BEGONE!' God added, with finality. I think the excitement of the moment had quite rushed to his head.\n\nThere was a buzz of excitement as she was escorted out and we all listened until the scrunch of gravel under tyres vanished into the distance.\n\nI was struggling to take it all in, but when I saw Nick talking to Caz, I suddenly realised who must have hidden the wheel brace with Polly's fingerprints on it among my gardening tools, and then tipped off the police!\n\nWould Polly have had the strength, once Jasper had tightened the nuts, I wondered \u2013 then remembered what Ritch and Dora Tombs had said about her working out. And really, anyone can change a wheel with one of those cross-brace things, it's not that difficult.\n\nBut it had been my car, me she wanted to hurt, not Tom. Not kill me \u2013 none of her little tricks had been intended to go quite that far; though I expect she would have looked on my death as a bonus.\n\nAnd in the end Caz must have told Nick everything he knew, and so they had set this very public accounting up \u2013 as revenge? I didn't suppose Polly could be charged with anything terribly serious.\n\nThe courtyard was still buzzing, but then Moses suddenly awoke as if from a trance, and banged his shepherd's crook on the floor a couple of times to regain the audience's attention.\n\nSlowly they quietened and turned back to the stage.\n\n'If that's all, Lord, I'll be getting off, then,' Moses said, back to the script.\n\n'Aye, go with my blessing upon you,' God said, sounding exhausted, and invisible hands began to draw the canvas curtains across the front of the arched doorway.\n\n'A hot rum toddy, that's what I need,' Roly added, forgetting to switch off the microphone. 'Delphine, my dear, you'll join me, won't you? There's a short break before the next acts for refreshments, and I'm sure we all need them.'\n\nJojo and Mick picked up their instruments and began to play, and Ophelia, looking harassed and frightened, ran out of the house, fiddle in hand.\n\nNick forged his way through the crowd and handed me a plastic tumbler of hot toddy, which I took automatically and drained in one: I needed it.\n\n'Well,' he said thoughtfully, 'I don't know how we'll be able to follow that next year.'\n\nI turned on him accusingly. 'You knew that was going to happen \u2013 you, Caz and Uncle Roly set that up. How long have you known it was Polly who sabotaged the car?'\n\n'Not long at all. I thought that, at least, was an accident, until Caz told me what he'd seen. He'd been watching from the woods that day and saw Polly come out of Tom's workshop and look at the car, then pick up the wheel brace (which Jasper must have left leaning against the wall, by the way) and start unscrewing the nuts \u2013 by sheer coincidence on the same wheel you'd changed earlier. Then she put the wheel brace back where she found it and left. Caz was going to go down and see what she'd been up to when the coast was clear, but Tom drove off in your car before he had the chance. So Caz wrapped the wheel brace in sacking and put it behind the freezer you let him use.'\n\n'Why? And why didn't he say anything?'\n\nHe shrugged. 'Well, you know Caz. He said he thought at first it was something to do with ARG and he didn't want to get Ophelia into trouble. It was before he knew that Polly wasn't a member of the group, just forcing Ophelia to target you.'\n\n'He's not keen on the police anyway. But he confided in you.'\n\n'Yes, he finally told me the whole thing, because he was so angry that Polly was prepared to harm Ophelia, even when she was pregnant. I negotiated with the police and they're going to forget that Ophelia was ever a member of ARG in return for Caz's statement.'\n\nHe seemed to feel this was worthy of praise, for he paused expectantly.\n\n'Oh, well done, Nick!' Annie, who had been listening admiringly, exclaimed. 'You are clever!'\n\nI gave her a withering look. 'I think you might have let me in on what was happening, Nick!'\n\n'Why? You didn't tell me anything! I found it all out for myself.'\n\n'Yes, but Lizzy was upset when you were flirting with Polly, Nick,' Annie said traitorously.\n\n'No I wasn't!' I exclaimed indignantly. 'I\u2014'\n\n'Ssh... afterwards,' Nick said, a gleam in his slaty dark eyes, 'they're starting again.'\n\nI gave him a glare and moved away, avoiding him for the rest of the entertainment, which isn't easy when you're enclosed in a small courtyard. I can't say my mind was completely on the play either \u2013 or even on the refreshments, which just goes to show how churned-up and confused I felt.\n\nBut eventually I began to be caught up in the Mysteries again, just as I was every year.\n\nKylie was a subdued and modest Mary, with only one or two wisps of violently pink hair escaping from her hooded robe, and her fingernails unpainted. The huge rock that sparkled in the muted light on her engagement finger was not quite in role, though: Kylie had clearly got her man.\n\nThere were the usual moments of light relief during the Miracles: it didn't matter that the audience had heard the lines before.\n\n'Get up, thou great lazy lummock,' Jesus told the Lame Man forthrightly. 'Pick up thy pallet and walk.'\n\n'I'll be reet glad to, lad, 'tis no life for a man, this. What did tha say thy name wor?' asked the Lame Man, getting up.\n\n'Jesus of Nazareth.'\n\n'Is that ower near Burnley?'\n\n'Nay,' said Jesus, moving on to the next supplicant. 'What's t'matter wi' him?' he asked one of the disciples.\n\n'He can't see owt, master.'\n\n'That's reet,' agreed the Blind Man. 'But I believe thee can cure me and so my friends hath brought me here.'\n\n'I'll touch thy eyes, and if thee believe, then thee will see. How many fingers am I holding up?'\n\n'All of 'em, Lord.'\n\nThe audience cheered, then sobered for the final darker scenes before the second interval. But once the curtain was drawn across the crucifixion scene (excellently performed by Gary Naylor) the holiday spirit returned and everyone headed for the refreshments to fortify themselves for the resurrection and the grand finale.\n\n## Chapter 33: Well Stirred\n\n'We're happy, Lord, to see thee again,' Faye said stolidly, in her role as Mary Magdalen. 'Thee said that thee would come back and thou were right. Wilt thou stay awhile?'\n\n'Nay, I must get home to my Heavenly Father.'\n\n'Well, I reckon he'll be reet glad t'see thee, and thou art done thy bit for mankind.'\n\n'My father hath many mansions, Mary, and all who believe in him will be welcome in t'Kingdom of Heaven.'\n\n'That'll be proper champion, that will,' Mary said gratefully.\n\n'I'll be off then,' Jesus said, suiting the action to the words, and Mary followed him behind the drawn curtain.\n\nAn angel appeared, the new white goose-feather patches on his wings glistening, and stood with one hand cupped to his ear, as if listening intently.\n\n'Here is my judgement, and the pure of heart need fear nowt,' said Roly as Voice of God, refreshed and speeding up considerably now the finishing post was in sight.\n\n'What is thy wish, Lord?' asked the angel.\n\n'That retribution shall visit the wrongdoers.'\n\n'Lord, it shall be done.'\n\n'Let it be so, for as the old year dies, another, Lazarus-like, rises anew. Our play is played out, our Mysteries unfolded,' said God.\n\nThe angel, who'd been gazing vaguely up into the rafters, now turned to look directly at the audience and said weightily, 'Look into t'mirror of thy heart and, if thou like not what thou see, then freshly start again, fer Christ died fer thee.'\n\nGod, as always, got the last word. 'Heed my commandments. Keep thy conscience clear. Remember, I'll see thee agin, this time next year!'\n\nGoing by the wild applause it was certainly another Middlemoss Mystery success, but more than one mystery had been enacted, revealed and resolved today. It had been a cathartic and exhausting experience, and the audience was subdued as they slowly began to leave, while I felt like a well-wrung-out dishrag.\n\n'Everyone involved in organising the play has been invited to the house for a hot toddy before we go home,' Annie said, taking my arm and giving it a squeeze. 'You're coming too, aren't you, Lizzy? Look, there's Jasper going in. And I want to know all the details about Polly, too \u2013 did you really not know any of that was going to happen?'\n\n'No, of course I didn't!' I snapped, finding myself being swept through the kitchen and along the passage to the Great Hall, where the steaming silver punch bowl and a tray of sandwiches were laid out before a blazing log fire. 'I'd have told you.'\n\nRoly beckoned me across to where he was sitting with Delphine. 'Well, my dear,' he said, 'that seems to have worked out for the best, doesn't it? Justice for poor Tom has been served, and everything is sorted out satisfactorily.'\n\n'Is it?' I said, slightly sourly.\n\n'You were very good as Eve,' Delphine said kindly. 'Quite beautiful in that costume.'\n\n'Yes, you're much better with Nick as Adam,' agreed Roly. 'But if he isn't playing it next year, he can take over as Voice of God.'\n\n'That was Lizzy's last turn in the role, wasn't it?' Nick said, having come up behind me unobserved. 'I'm not playing Adam to anyone else's Eve.'\n\nRoly looked from one to the other of us and, beaming, took our hands and clasped them together in his. (Theatricality also runs in the Pharamond bloodline.) 'Let it be a New Year, a new beginning for both of you!' he said sentimentally.\n\n'I don't know what you mean, Unks,' I said, trying and failing to loosen my hand from Nick's strong grip. 'And I'm afraid I'll have to be going home now. Jasper?'\n\n'I'm going out again, to help clear up,' Jasper said quickly. 'I'll see you later.'\n\n'I'll walk you home, Lizzy,' Nick said, 'but first there's something I want to show you.'\n\nI couldn't imagine what he'd got that I hadn't already seen. But I let him lead me upstairs to the long gallery, switching on the wall lights as we went. He came to a stop in front of the portrait of an eighteenth-century Pharamond bride, who posed with one slender hand resting on a book \u2013 and on her finger, my ring. I just knew it was an old family piece.\n\n'There \u2013 you see?' he said.\n\n'Nick, I can't possibly keep a family heirloom, whatever Unks says. Please take it back!' I protested, tugging it off my finger and handing it to him. He accepted it, then calmly took hold of my other hand and shoved it over the knuckle of my ring finger instead.\n\n'What on earth are you doing?' I said, trying to pull away.\n\n'It's the betrothal ring of the Pharamonds.'\n\n'I dare say it is, but we're not betrothed\u2014'\n\n'I think we are, and Unks thinks we are \u2013 so you're outnumbered. Just as well he would never let Leila have the ring, because I'd never have got it back.'\n\nI glared at him. 'This isn't the Middle Ages, so I do have a say in all this, Nick Pharamond \u2013 and I'm not engaged to you! You are an underhand, devious\u2014'\n\n'Yes, I know,' he said soothingly, pulling me close, 'but I do love you. I think, deep down, I always did.'\n\n'You have a damned strange way of showing it!'\n\n'There wasn't much point, when we were both married to other people... but the postcards showed I was always thinking of you. I never wanted quite to let go of you. And you kept them all.'\n\n'Only for the recipes,' I said quickly, fighting a rear-guard action, for close proximity was scrambling my brain cells and weakening my knees, just as it so disastrously had on Bonfire Night. 'Besides, we argue all the time and you despise my cooking!'\n\n'No I don't, I just like to wind you up. You should know that by now.'\n\n'You think your cooking is more important than I am!' I accused him.\n\nHe grinned. 'No, I think it's a pretty even match, actually. I don't see why I can't have my cake and eat it.'\n\n'I do. And anyway, we're just too different \u2013 it'd never work,' I said firmly, then ruined the effect by smiling back at him.\n\n'If mayonnaise works, I don't see why mixing the two of us together shouldn't \u2013 if we do it slowly and very carefully.' His lips moved over my face and then lingered on my mouth before I could point out that curdled mayonnaise was a lot easier to rescue than a curdled marriage.\n\nOh, hot chilli chocolate sauce! I thought, but more in resignation than revolt. He was the one who broke that clinch: I couldn't have, even if you'd waved a giant Mars bar in front of me.\n\n'And I've had a great idea! Once you've finished Just Desserts, we'll collaborate on a joint recipe book of all the postcards. We'll call it A Feast of Romance,' he added soulfully.\n\nI laughed. Senga was going to absolutely love it \u2013 and him. 'That is a totally corny idea! And what's more, I'm not cooking anything with you, because I always end up doing all the donkey wo\u2014'\n\nI stopped dead as I spotted Caz silently slinking out of the dark shadows at the end of the gallery. Downstairs someone had turned up the music so that the heavenly sound of a choir singing Silent Night drifted down the long, dark gallery, while over our heads a mistletoe ball rotated in the slight draught from the open door. Caz jerked his head back in the direction of the stairs. 'Mr Roly says t'champagne's open. You two done, yet?'\n\n'Rising nicely,' Nick said, glancing up at the mistletoe ball thoughtfully, then reaching for me again.\n\n'Half-baked!' I amended, giving him a quelling look. 'Caz, tell him we'll be down in a minute \u2013 and perhaps you ought to cork up the champagne again, because there are just a few rules of engagement I need to thrash out first, before I even consider this insane idea.'\n\n'Like what?' Nick asked suspiciously.\n\n'Separate kitchens,' I said, smiling sweetly. 'And that's just for starters!'\nLoved the seasonal treats in this book?\n\nThen why not try and make them yourself?\n1) Mincemeat Flapjacks\n\nThese are very easy to make!\n\nIngredients:\n\n4 oz butter\n\n2 tablespoons of golden syrup\n\n2 oz Demerara sugar (or a soft, dark brown sugar, if you want a slightly 'treacly' taste)\n\n5 heaped tablespoons of mincemeat, either bought or home-made\n\n5 oz rolled oats\n\nMethod:\n\nPreheat oven to gas mark 3, 160\u00b0C, 325\u00b0F and grease a seven-inch baking tin. If using a cake tin instead, then I would line the base with baking paper, too.\n\nMelt together the sugar and syrup in a pan over a low heat, then stir in the mincemeat and, once warmed through, the oats.\n\nRemove from heat and mix well, then spoon into the baking tin and spread it out, flattening the top.\n\nPut into the oven for about half an hour: it should be slightly golden brown. Remove and leave to cool for fifteen minutes before marking into squares or slices.\n\nWhen cool, store in an airtight container.\n\n2) Christmas Mincemeat Spudge  \n(Mashed potato fudge)\n\nIngredients:\n\n5 oz of mashed potato\n\n1 oz butter\n\n1 lb icing sugar (and some extra, in case it is needed)\n\n4 tablespoons mincemeat, either bought or home-made\n\nA few drops of almond essence\n\nMethod:\n\nGrease a small baking tray or pie dish.\n\nMash the potato with the butter and, while still warm, stir in the icing sugar. When smooth, mix in the mincemeat and the almond essence to taste.\n\nDepending on the runniness of the mincemeat used, you may need to add extra icing sugar \u2013 you are aiming for a very stiff consistency that has to be spread into the tray. (The first time I tried this variation I didn't put quite enough sugar in, so it didn't set hard, but it made a lovely fudge topping for vanilla ice cream!)\n\nWhen cold it will be firm and can be cut into pieces and stored in an airtight box in the fridge.\n\nVariations:\n\nYou can make plain Spudge by omitting the mincemeat. Instead try adding vanilla essence and\/or two tablespoons of de siccated coconut. When cold, cover with a layer of melted chocolate.\n\n## Forget the Jimmy Choos, Chocolate Shoes and Wedding Blues is the only accessory you need for spring 2012...\n\nWhen eccentric Tansy Poole suddenly finds herself the newest proprietor of Bright's Shoes, a run-down shoe shop tucked away in the idyllic Lancashire village of Sticklepond, she is inspired to take the business in a new direction. And soon, Cinderella's Slippers opens to the public-promising to provide the footwear to make any fairytale wedding come true.\n\nStocking everything a bride would want to walk down the aisle in, from exquisite satin pumps to quirky vertiginous heels, Tansy's shop soon expands to carry shoe-shaped wedding favours, bridesmaid gifts and even sumptuous artisan chocolate shoes. It's the dream destination for anyone who loves shoes!\n\nIf only everything in her personal life could be as fulfilling \u2013 but with a fianc\u00e9 unwilling to name the day until she fits into a size 8 wedding dress, not to mention disturbing family revelations having recently come to light, Tansy takes refuge in making Cinderella's Slippers a success. Her hard work pays off when word of mouth attracts customers from far and wide.\n\nBut one man isn't thrilled by the stream of eager shoe lovers thronging to Cinderella's Slippers... Shakespearian actor Ivo Hawksley, resident of the cottage next to the shop, is troubled by a dark secret in his past and has come to Sticklepond to nurse his own broken heart, not have his peace shattered by the trill of 'Here comes the bride' every time the shop door opens!\n\nHowever, Ivo is shaken by the realisation that he and Tansy have a link in their past and soon, they both find out how secrets shared can make a very strong bond indeed...\n\nDon't miss the brand new book from The Queen of Yummy Lit, guaranteed to put a spring in your step.\n\nChocolate Shoes and Wedding Blues \u2013 due to be published by Avon spring 2012\n\nISBN \u2013 978-1-84756-277-7\n\n## TWELVE DAYS OF CHRISTMAS\n\n## Trisha Ashley\n\nChristmas is coming...\n\nChristmas has always been a sad time for young widow Holly Brown, so when she's asked to look after a remote house on the Lancashire moors, the opportunity to hide herself away is irresistible \u2013 the perfect excuse to forget about the festivities.\n\nSculptor, Jude Martland, is determined that this year there will be no Christmas after his brother runs off with his fianc\u00e9e and he is keen to avoid the family home. However, he will have to return by the twelfth night of the festivities, when the hamlet of Little Mumming hold their historic festivities and all of his family are required to attend.\n\nMeanwhile, Holly is finding that if she wants to avoid Christmas, she has come to the wrong place. When Jude un expectedly returns on Christmas Eve he is far from delighted to discover that Holly seems to be holding the very family party he had hoped to avoid.\n\nSuddenly, the blizzards come out of nowhere and the whole village is snowed in. With no escape, Holly and Jude get much more than they bargained for \u2013 it looks like the twelve days of Christmas are going to be very interesting indeed!\n\nISBN: 978-1-84756-115-2\n\nOut now.\n\n## About the Author\n\nTrisha lives in beautiful North Wales, together with the neurotic Border Collie recently foisted onto her by her student son and an equally neurotic but also vain, bad-tempered and chancy Muse. Her previous book, Twelve Days of Christmas, was a Sunday Times bestseller in 2010.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins authors.\n\nPlease visit Trisha's Facebook fan page for exclusive recipes, quizzes and regular updates from Trishaworld!\n\n## Other Books by the same author:\n\nSowing Secrets\n\nA Winter's Tale\n\nWedding Tiers\n\nChocolate Wishes\n\nTwelve Days of Christmas\n\n## Copyright\n\nThis novel is entirely a work of fiction.  \nThe names, characters and incidents portrayed in it are the work of the author's imagination. Any resemblance to actual persons, living or dead, events or localities is entirely coincidental.\n\nAVON\n\nA division of HarperCollinsPublishers  \n77\u201385 Fulham Palace Road,  \nLondon W6 8JB\n\nwww.harpercollins.co.uk\n\nFirst published in Great Britain by HarperCollinsPublishers 2011\n\nTHE MAGIC OF CHRISTMAS. Copyright \u00a9 Trisha Ashley 2011. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nTrisha Ashley asserts the moral right to be identified as the author of this work\n\nA catalogue record for this book is available from the British Library\n\nISBN-13: 978-1-84756-116-9\n\nEPub Edition \u00a9 AUGUST 2011 ISBN: 978-1-84756-301-9\nAbout the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\n<http:\/\/www.harpercollins.com.au>\n\nCanada\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON, M4W, 1A8, Canada\n\nhttp:\/\/www.harpercollins.ca\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\nhttp:\/\/www.harpercollins.co.nz\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\nhttp:\/\/www.harpercollins.co.uk\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\nhttp:\/\/www.harpercollins.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nBegin Reading\n\nTable of Contents\n\nAbout the Author\n\nPhotos\n\nCopyright Page\n\nThank you for buying this\n\nFarrar, Straus and Giroux ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nThe most beautiful thing we can experience is the mysterious. It is the source of all true art and science.\n\n\u2014Albert Einstein\n\nAutumn Fog\n\nPrivate First Class Vonnegut prepared to die.\n\nAt the bottom of a snowy hollow, he fixed his bayonet and waited, huddled in a group of roughly fifty soldiers. Their unit, the 423rd, had been at battle for three days, since December 16. They'd been lost for most of it. They must be somewhere in Luxembourg, someone said. Now they were surrounded, herded into a small depression in the unfamiliar land. Kurt hunched into his coat\u2014he had a tall man's habit of hunching\u2014but he couldn't get warm. That December\u20141944\u2014was one of the coldest and wettest ever recorded in Europe.\n\nThe Germans were shouting at them. Kurt and the other soldiers couldn't see them, but they could hear accented voices telling the Americans to give up. They were surrounded, the Nazis said. It was useless to resist. The men bunched together, pointing their bayonets out the way soldiers do in the movies. Time slowed down. Kurt had always liked being part of a clan, and here, at the end of the line, the soldiers became almost one being, a big porcupine bristling with steel quills. For a few minutes, it was kind of nice.\n\nHe had waded ashore to the European theater less than a month earlier and ground to the front in a truck buffeted by sleet. He was still somewhat in shock. His mother had died of a drug overdose\u2014was it a suicide? an accident?\u2014just before he shipped out. The sadness hung thick over his departure, complicating and deepening his fear. He longed for the feeling that someone loved him, followed his every move with boundless devotion. He hadn't realized how much he needed that until his mother was gone.\n\nStill, for the first time in his life he felt beyond reproach. No longer a flunking chemistry student or a college dropout, he was where he was supposed to be, a soldier putting his life on the line. He was now the sort of person honored in the grandiose Soldiers' and Sailors' Monument at the center of his hometown, Indianapolis. Not even his big brother, Bernard, could say that much. Bernie, the A student, the brilliant scientist, the MIT man like their father. The one who launched the chain of events that landed Kurt here, a pacifist about to be swallowed by war.\n\nThe Germans fired on the trees above the soldiers' heads. Branches and splintered steel rained down. A couple of guys were hit. They might be dead. Twenty-five years later, Kurt would introduce a character named Edgar Derby to the world and describe his experience of this very battle. He would call what rained down on him \"the incredible artificial weather that Earthlings sometimes create for other Earthlings when they don't want those other Earthlings to inhabit Earth any more.\"\n\nCome out, the Nazis ordered again. The Americans came out. When he saw the Germans, Kurt couldn't help but note their white snowsuits. That made so much more sense, he thought, than his absurd army drab. The Americans were always in olive, as if wars were never fought in white places, in white weather.\n\n* * *\n\nThe battle had started three days earlier at 5:30 a.m., before dawn crept over the frozen landscape of Schnee Eifel, or Snow Mountain. They were manning a lightly defended spot along the old Westwall, a reinforcement ridge the Allies called the Siegfried Line. It was quiet: no one expected much to happen on this front. Then out of the predawn darkness came the attack, and it sounded like the sky falling in. For eighty miles along the Westwall, Allied soldiers woke up to German artillery raining down on them: fourteen-inch shells from railroad guns, the hacking cough and plunk of mortars, the high-pitched whistles of what the Americans called \"screaming meemies\" but the Germans called Nebelwerfer: fog throwers. The forest along the front was leveled as the Germans fired on trees. Even the Nazis were impressed by their own artillery storm. \"A hurricane of iron and fire,\" one German major called it.\n\nFor nearly an hour, the onslaught pounded the American troops. Then, suddenly, there was an eerie silence. As the soldiers tried to get their bearings, they heard a clank, and then they were lit like stage actors from the east. Through the dense fog, the Germans were flooding them with searchlights\u2014a new intimidation tactic they called \"artificial moonlight.\" To the battered Allies, it felt as if the Nazis had commandeered control of nature. Finally, out of the white came snow monsters: German infantry.\n\nSomething big was happening, but no one knew what. Telephone lines were blasted; whole divisions lost contact with command. Strategists got garbled and patchy reports; some thought the barrage was merely a \"spoiling\" attack, a futile lashing out by an enemy who knew he was defeated. After all, the war was meant to be winding down. General Eisenhower had a standing bet with the British field marshal Montgomery that Europe would be won by Christmas. It took days before the Allied generals realized that a major offensive was under way and diverted troops to stop it.\n\nIn retrospect, it was clearly a mistake to man twenty-five miles of front with one Allied division. But the Germans were outnumbered and out-armed. Why would they launch an offensive? They had only one thing on their side: the Allies called it \"Hitler's weather.\"\n\nIn war, soldiers fight more than the enemy. They fight topography. They fight time. Most of all, they fight weather. Kublai Khan might have overrun Japan, but a typhoon destroyed half his ships. The Spanish Armada fell to Britain because of storms on the North Sea. Napoleon was especially unlucky in weather: he lost Waterloo because of a rainstorm, and his march on Russia was beaten back not by war craft but by winter.\n\nIn World War II, weather mattered more than ever. It was the first war in which airpower would be decisive, and the U.S. Army Air Forces were especially vulnerable to bad weather: cloud cover disrupted bombing runs; snow scrambled radio signals; icing forced planes to land. But weather could waylay the Navy too: the day Kurt woke up to the German attack on the western front, the Third Fleet faced a typhoon in the Pacific. The storm sank 3 destroyers, wrecked 146 aircraft on carriers, and killed 778 troops, racking up a higher death toll than any Japanese attack. And weather tormented the infantry: rain and snow slowed tanks, troops, and supply lines. Fog could conceal enemy movements.\n\nWeather forecasting had been part of most militaries since the early nineteenth century, but when World War II broke out, the generals realized they needed more meteorologists than ever. Colleges were enlisted to train thousands of weather officers for the new Air Weather Service. MIT, America's leading meteorology school, established a special program, bumping its enrollment from thirty students to around five hundred. The department head, Sverre Petterssen, left MIT to join General Eisenhower's meteorology team. He played a key role in the war's most famous weather suspense story.\n\nDuring World War I, a new physics-based school of meteorology had arisen in Norway called air mass analysis. Petterssen was Norwegian and a proponent of this scientifically rigorous new approach. When Ike was poised to invade Normandy, Petterssen told him to wait. The skies looked clear, but the upper air situation was unstable. Petterssen told Eisenhower's team that the weather was likely to turn bad and scuttle the invasion. He based this forecast not just on what he could observe but on the idea that large wind patterns were battling each other high in the sky, atmospheric echoes of the clash of armies below. Such wind patterns are common knowledge today, but in the 1940s not everyone believed that weather was shaped by the huge, invisible air masses that meteorologists had given a warlike name: fronts.\n\nThe American meteorologist Irving Krick, who was also on the team, scoffed at Petterssen's approach. Krick, using the classic forecasting technique\u2014looking at weather maps from the past to determine how the future might shape up\u2014declared that the offensive should go on as planned.\n\nThe largest invasion in history hung in the balance as the weathermen argued about the winds. Finally, the chief meteorological officer made the call. D-Day was postponed for twenty-four hours, and indeed the weather turned stormy. The next day looked no better, but Petterssen pointed to the changing barometric pressure as a sign that the weather would improve. There would be a window of opportunity before the next bad day. Gambling on Petterssen's prediction, General Eisenhower launched the attack.\n\nNow it was the Germans who were using the weather to their advantage. The attack that would come to be called the Battle of the Bulge was planned for December with good reason. Under cover of heavy cold fog, the Germans had amassed 410,000 troops, 1,400 tanks, and 2,600 artillery weapons for the predawn offensive. The weather had slowed the delivery of Allied troops and supplies to the front, and the bitter cold ensured that the Allied infantry, huddled in foxholes and trenches, was distracted by the need to keep warm. Best of all, the heavy cloud cover would keep British and American planes grounded, depriving the Allies of air support. General Alfred Jodl had foreseen it in the detailed operational plan he drew up for the Reichsf\u00fchrer. He called it Herbstnebel: Autumn Fog.\n\n* * *\n\nKurt's regiment, the 423rd, got the worst of Autumn Fog. Its troops were far enough forward to be on German ground, and before they even realized what had happened, they were cut off. The men of the 423rd, like the entire 106th Division, were green; they had never seen action. Many, like Kurt, had been pulled out of the Army Specialized Training Program (ASTP)\u2014a combined college and military program that was meant to lead to a degree and an officer's commission. It was canceled in 1944 because the Army didn't need more officers; it needed riflemen. Hitler was going to be defeated not by strategists or engineers but by numbers. College boys like Kurt were plucked from classes on thermodynamics, calculus, and mechanical engineering, given a few months' hasty training in combat skills, and shipped to Europe. Kurt tried to get assigned to public relations, but his efforts failed. His unit, the 106th, was the last American infantry division to be mobilized in World War II. Two-thirds of its troops were single men under the age of twenty-three.\n\nThe Schnee Eifel was where these young men were supposed to get \"blooded\"\u2014to practice their battlefield skills before facing actual combat. Now they were thrust in the middle of the very bloody real thing, and they didn't know what to do. Kurt's regiment commander, Colonel Charles Cavender, was told to dig in and hold the Germans back as best he could. He was promised an airdrop of ammunition and supplies. As the Germans moved inexorably forward, the men of the 423rd split into small groups and huddled together like sheep. By nightfall, the sheep were surrounded. For the next two days, they fought as best they could, in small groups or larger ones, while the Germans streamed around them and toward St. Vith.\n\nFor three long days, the 423rd and its sister regiment, the 424th, tried to hold their ground, like ants clinging to a boulder in a rising flood. By the morning of December 19, hundreds of men from the 106th were dead or wounded. The promised airdrops weren't coming, and there was no sign of reinforcements either. Colonel Cavender sent six of his men out to reconnoiter. One of them was Kurt. They weren't looking for the enemy; they were looking for their own artillery. Wandering the snowy hills, the six men found about fifty more Americans. And then the Germans found them.\n\nThe Americans surrendered as they were taught: they dismantled their weapons and threw them into the snow. Coming out of their gully, they said things like \"take it easy\" and \"don't shoot.\" They wanted to go on living if they possibly could. Kurt knew a little German; his German American parents spoke it, and he'd had two years of it in high school. He tried out a few words. The Germans asked him if he was of German ancestry. He gave them his last name, Vonnegut.\n\n\"Why are you making war on your brothers?\" they asked him. The question made little sense. He was a Hoosier, not a Kraut. But they weren't completely off base. When he said his own name, he said it, as his father did, in the German way: Kooort.\n\nThe Germans pointed their guns at the Americans. They told Kurt Vonnegut Jr. to march.\n\n* * *\n\nTwo years earlier, he had been a relatively carefree undergraduate, writing columns called \"Innocents Abroad\" and \"Well All Right\" for The Cornell Daily Sun, buying Old Grand-Dad bourbon for a dance he hoped his sweetheart, Jane, would attend. The war meant this to him: Cokes were being rationed on campus. The university was banning house parties and out-of-town dates. Fraternities were going to be strapped for cash. Kurt devoted a whole column to the looming frat-house financial crunch. \"It's a nasty picture no matter how you look at it,\" he wrote in May 1941. \"From an abstract point of view it will be interesting to watch, just like bombing.\"\n\nHe'd been sixteen when the war began. He and two friends were wrapping up a summer road trip by spending a few days as the guests of Frank Phillips of Phillips Petroleum. Phillips had a resort in Bartlesville, Oklahoma, called Woolaroc Ranch. Woolaroc was a teenage boy's fantasy, where Kurt and his pals spent their days riding horses, fishing, and swimming, their evenings drinking beer and smoking twenty-five-cent cigars. They goofed around with the lodge's player piano and ransacked their host's library for spicy crime novels and anthropological accounts of the sex lives of primitive tribes while the radio tallied the mounting threat in Europe. It all seemed impossibly far away. The day they drove home, Hitler invaded Poland. As they pounded the road for seven hundred miles, Kurt had looked back on the whole adventure and wondered if he'd ever be that happy again.\n\nHis family was pacifist; the Vonneguts had always been freethinkers. Kurt clung to his antiwar conviction long after most others had succumbed to patriotic warmongering. In one column for the Sun, he defended the unpopular isolationist sentiments of Charles Lindbergh. In another, he criticized the extreme anti-German bias of the American media. Later, he blasted Wendell Willkie, \"political yo-yo from the Hoosier state,\" for advocating the opening of a second front. It wasn't that he was pro-German. He was just antiwar. He came from a long line of Germans, yes; his grandfather had designed the gorgeous Indianapolis social center formerly known as Das Deutsche Haus. After World War I caused a wave of anti-German sentiment, Das Deutsche Haus was renamed the Athenaeum. But in Kurt's family, ethnicity was less important than ethics, intellect, and wit.\n\nHe had learned early that the best way for a third child to be heard at the dinner table was to crack a joke. Being funny was the only way he got them to stop interrupting and listen to him. Besides, his brother was brilliant, and his sister was artistic and beautiful. He couldn't compete on brains or talent or glamour. So he nurtured his penchant for humor, and this served him well at the Sun. His fellow students at Cornell didn't always agree with his isolationist sentiments, but they liked his snappy writing. In March 1942, he was appointed assistant managing editor of the paper.\n\nHe bragged about that to Jane Cox. He was always trying to impress her. They'd known each other since they were small children. In a way, she was his best friend. He recognized things in her\u2014imagination, ambition, idealism\u2014that he saw in himself. They were going to get married and live a blissful life together, full of books and music and smart conversation and ultimately kids\u2014seven of them. He knew all of this, felt it somewhere deep inside him, even if Jane, busy acing her classes at Swarthmore and acting in plays and going on dates with a roster of eligible young men, hadn't come around to it yet. Kurt wrote it over and over in his letters to her. She was alternately encouraging and distant. She was hell to get along with, Jane, but he loved her, and he always would. Nineteen forty-five was the year he had picked for their wedding. He wrote it in one of his columns.\n\nBefore he left for Europe, they became lovers.\n\nAt Cornell, he had spent all his time working on the newspaper, to the detriment of everything else. His grades in his major\u2014chemistry\u2014suffered. He was supposed to be earning an officer's commission in the ROTC, but he got kicked out after writing an irreverent column: \"We Impress Life Magazine with Our Efficient Role in National Defense.\" In it, he claimed that he and the other ROTC boys had little idea what they were doing, but when a Life photographer visited, they gamely ran around and disassembled a rifle while shouting things like \"Flathatcher! Biffleblock!\" to seem like crack militiamen. The ROTC was not amused. It wasn't the first time Kurt had mocked the warlike exertions of the college boys. An earlier column was cast as a letter to the military department from the school's zoologists, who claimed to be just as ready for service as the chemical engineers and advance drill squads.\n\n\"Up in the front lines our commanding officer will say 'Vontegal... what the hell kind of butterfly is that,' and we'll be the only man in the trench that can tell him. That's the sort of thing that wins wars!\"\n\nIt was, on some level, a sly crack at his brother, Bernard, who was doing war work. He'd been asked to leave his peacetime job and go back to MIT to work in the Army's Chemical Warfare Service laboratory there. He was exempted from the draft because of it. He couldn't tell his family what he was working on, but the family was proud of him; they were always proud of Bernard. How could Kurt resist mocking the notion that scientists would win the war? Besides, that was pretty much the only way he ever got a jump on Bernie: he made fun of him.\n\nBy his sophomore May, Kurt's grades were so bad he was put on academic probation. He made light of his woes in a column titled \"The Lost Battalion Undergoes a Severe Shelling.\" He was the lost battalion.\n\nWhy didn't he just switch his major to English or journalism? He was a newspaperman at heart. In Ithaca, his happiest moments were when he was editing the Sun, just as in Indianapolis they had been when he was editing the Shortridge Echo, the nation's first high school daily. His experience there convinced him that not only did he like newspaper work, he was good at it. Toward the end of high school, he'd even managed to land a job offer from The Indianapolis Times. He wanted to take it. But becoming a newspaperman wasn't what his father and Bernie had in mind for him.\n\nSure, Kurt senior and Bernie agreed, young Kurt could write, and he was funny\u2014the family clown, the class clown\u2014but when he graduated from high school, it was time to get serious. At one point, Kurt thought he might like to become an architect, like his father and his grandfather Bernard, who had designed the Athenaeum. The opulent building was still the heart of the Indianapolis German American community, and Kurt had spent many a night there as a kid, admiring the elaborate woodwork and leaded glass windows as the adults talked or danced or listened to music. It must be nice to make something so beautiful. But that was before the Great Depression had ruined his father. The disheartened Kurt senior wouldn't hear of Kurt following in his footsteps. Be anything, he said bitterly, but an architect.\n\nWhen he was a young man, Kurt's dreams\u2014shared with Jane\u2014were all about writing. They both fantasized about being news correspondents in Europe. Sometimes, when Jane was playing along, they envisioned the house they might share: a courtyard with an oak tree at its center and a studio out back where they would sit side by side and type out masterpieces. But even Kurt had a hard time imagining writing for a living. He would have to do something else to support those seven kids.\n\nBernie knew what Kurt should do; he should be a scientist, like him. So Bernard and Kurt senior decided that Kurt should study chemistry. That was a useful, practical field. Kurt didn't necessarily disagree. He believed, as they did, in science. It had more answers to the questions of life, he told Jane, than fields like psychology or philosophy. Science was going to make the world a better place. To be part of the utopian future, he should do as his brother said.\n\nThe older men didn't think Kurt junior was MIT material, so they settled on Cornell. When it looked as if Cornell might not take him, Bernard drove Kurt to Harvard, where he was given a provisional acceptance. But then Cornell came through, and Bernard thought he would have a better time there. That, Kurt said later, \"was his idea of me, sort of third rate.\"\n\nSo in the fall of 1940, Kurt went off to Cornell to study chemistry. But he wasn't a born scientist like Bernie. When it came to the actual class work, it just didn't grab him. Not the way writing did. So he ignored his classes and did what made him happiest\u2014keeping late hours in the offices of the Sun. Even the warning in his sophomore spring didn't set him straight. By Christmas break of his junior year, he was flunking out. He came down with pneumonia at home and decided not to go back. But his draft number was coming up. In March 1943, he enlisted in the U.S. Army.\n\nSo now the whole chain of events boiled down to this: Private Kurt Vonnegut Jr. was a prisoner of war.\n\n* * *\n\n\"The war is over for you,\" the Germans told the Americans. Kurt joined the long line of Yank soldiers marching east, toward Germany. German cameramen stood filming as the prisoners limped along. This might be the Nazi propagandists' last chance to convince the weary folk at home that victory was still within reach. Kurt saw them pointing their lenses at the broken men. Twenty-five years later, in Slaughterhouse-Five, he would describe them as having run out of film\u2014a perfect symbol for the empty pointlessness of it all: the propaganda, the offensive, the Nazi war machine, the whole goddamn war. He picked up his tired feet and marched, past dead soldiers unfurling from tanks, past men frozen in snowy fields, arms stretched toward the sky in fruitless supplication. The Germans did have film. In it, the Americans looked dirty, disheartened, and exhausted. Some supported wounded comrades or lugged makeshift pallets. The rest trudged miserably along. They had survived the German offensive, but many would not survive what lay ahead.\n\nWhen they came to the top of a hill, the captured men could see a long line of prisoners, as far forward as the eye could see. Seven thousand Allied soldiers had been bagged by the Germans. It would have been different if the Allies had been able to get their planes in the air. Air support could have nipped Herbstnebel in the bud. But in all that fog, the airplanes failed them. The weather had fought for the other side.\n\nAs they marched into Germany, some of the prisoners must have known the Germans were wrong about one thing: the war wasn't over for them. Kurt and the other captured American troops were marching into a strange gray area, a place where they were neither soldiers nor civilians, neither at peace nor at war. The autumn fog swallowed them whole.\n\n* * *\n\nBernard Vonnegut's life had always gone according to plan.\n\nAs a kid doing science experiments in the basement, he had planned to go to college and become a scientist. The Depression might have prevented that, but his family didn't let it, keeping him in his elite private high school, Park School. (They economized by enrolling Kurt in public school.) After graduating, Bernard studied chemistry at MIT, publishing his undergraduate chemistry thesis on X-ray analysis of crystalline bromine, then stayed at MIT for grad school, with the help of scholarships and a teaching fellowship. He planned to join his father's fraternity, Kappa Sigma, and by grad school Barney\u2014as his MIT friends had rechristened him\u2014was earning room and board by tutoring Kappa Sigma freshmen.\n\nHe hadn't planned for the swim team to lose every meet in his senior year, but he swam gamely anyway. \"Vonnegut should be commended for his excellent performance in the 200-yard breast stroke event,\" declared the yearbook, Technique. He swam on a 300-yard medley relay team that set an MIT record. He was a good team player. He never needed to be the star.\n\nAfter grad school, his plan was to get a good job as an industrial research scientist and start publishing papers, and that's exactly what he was doing as a chemist at Hartford-Empire Company, working on the physical and chemical properties of glass, when war came. And then his plans started to unravel.\n\nNow here he was, stuck on one of his frequent trips to Minneapolis, where he lived at the Hotel King Cole (Royal Guest Rooms! Food to the King's Taste!), away from his Cambridge apartment and Bow. He had married Lois Bowler just over a year earlier, on Christmas Day. They had been set up by friends, and she fit his plans too. His nickname suited her: with her long brown hair and delicately sculpted features, she was extraordinarily beautiful\u2014even Kurt had to admit it\u2014and also somehow fragile, as if she might come untied. Bow didn't like to be alone; she usually went to her parents' home in Elizabethtown, New York, when Bernie traveled.\n\nNow, instead of working at the glass factory, he was traveling back and forth between MIT and the Ice Research Base in Minneapolis, where he spent his days folding his lanky form into the tiny heated compartment he and his team had fashioned for the B-24 they named \"the Flying Icing Wind Tunnel.\" An Army Air Forces pilot would guide the bomber plane into any available freezing clouds, and a sergeant would open the bomb bay doors and lower from the fuselage an ingenious rotating cylinder device Bernie had designed to measure water content and drop size. As the device slowly iced over, bitter wind would whip through the plane. The sergeant would sit casually by the hole in the airplane floor, reading a comic book.\n\nBut things were about to get back on track. The war was winding down, meaning his job in deicing would soon end. And Bow had recently told him she was pregnant. So in the quiet of his royal guest room, Bernie took a piece of hotel stationery and wrote the date: February 15, 1945.\n\nOne day nearer to victory.\n\nThat was the cheery banner message at the bottom of the page in The Indianapolis Times that listed his brother, Kurt, as missing in action, unaccounted for after the awful, late-stage offensive, the Battle of the Bulge. The family had found out in January. The paper printed a note that read like an obituary: \"He attended Cornell University and after entering the service received ASTP training at the Carnegie Institute of Technology and the University of Tennessee. When the program was closed, Pvt. Vonnegut was transferred to the infantry and assigned to the 106th.\" That was it, Kurt's twenty-two years summed up in two sentences. The family was not without hope, exactly, but it seemed possible that the war had brought a second tragedy\u2014first their mother, now Kurt\u2014to the Vonnegut family.\n\nHis brother's last letter, written on December 15, had been to Bernard.\n\nBernard was an optimist by nature, not given to brooding or melancholy. The best thing, as he saw it, was to look to the future. He addressed his letter to Chauncey Guy Suits\u2014Guy to his friends\u2014the new director of the General Electric Research Laboratory, the nation's oldest and most renowned industrial research lab. He opened by apologizing for not having written sooner, but he'd been on the go nonstop since leaving Schenectady, New York, ten days earlier.\n\n\"I am very favorably impressed with the facilities of the laboratory, its policy and attitude towards research, and the opportunities there for the sort of work that interests me,\" he wrote. \"As you know from our talk, I am not yet in a position to reach a final decision, however... I am most interested in working for General Electric.\"\n\nHe couldn't say exactly when he'd be free from his war obligations, but they all knew the war was likely to end soon. He stated that he expected a salary of $5,000 to $6,000 a year.\n\nHe was going to have a family to support, after all.\n\nHe had been in Schenectady ten days earlier to talk to Irving Langmuir about ice. It had been a heady experience. The brilliant chemist\u2014the first industrial scientist to win a Nobel Prize\u2014crackled with ideas. Some people's eyes glazed over after only a few minutes of speaking with Irving, but Bernie loved it. For the last few months, he had been corresponding with Langmuir and his assistant, Vincent Schaefer, about aircraft deicing, and even though Bernie was director of the MIT meteorology department's deicing project, he was the one doing most of the learning. Langmuir was a chemist, like Bernie, but he had also done significant work in physics and crossed over into biology, mathematics, and even psychology when it suited him. The scientific disciplines were just beginning to divide into a series of silos, their boundaries patrolled by ever-more-focused specialists, but Langmuir was an old-school generalist. Like Bernie, he had never expected to become fascinated by the problem of deicing.\n\nIcing has been a problem since the dawn of aviation. Ice clogs carburetors, adds weight, and interferes with the wings' aerodynamics. An airplane wing's shape provides its lift: add enough ice and it can simply fall out of the sky. With helicopters, it's even worse.\n\nIn early days, pilots simply didn't fly in freezing conditions. But as aircraft grew more robust, they were expected to fly in any weather. Methods of deicing were invented and tried. There were coatings, heat applications, inflatable rubber boots that would expand and knock the ice off the wing. Sometimes they worked; sometimes they didn't.\n\nBernard had never thought much about aircraft deicing, but he had thought a lot about ice. For his graduate thesis, \"A Freezing Point Apparatus,\" he designed a device to measure the exact point at which water with other substances dissolved in it will freeze. Bernie was proud of it. He liked devising tools and gadgets: his first patented invention was an easy-to-clean pipe he designed jointly with his father. The aircraft program required lots of apparatus, because the scientists needed to understand how ice formed. So Bernie once again found himself doing something fun: conducting experiments and coming up with gadgets to get at data in new ways.\n\nAs a scientist, Bernie was both old-fashioned and newfangled. He was not unlike MIT in that way. When Bernie's father had attended Tech\u2014as its students called it\u2014the college was a trade school that trained students to design electrical systems or engineer wastewater plants. America's top scientific minds had always gone off to Germany for grad school. But as war with Germany approached for the second time, Americans felt a new urgency about training their own scientists. In the 1930s, MIT's president, Karl Compton, and vice president, Vannevar Bush, began shifting the school's focus from practical skills, such as drafting, to pure science.\n\nBy the time Bernie had arrived at MIT, even the undergrad curriculum was centered on pure science. His classes were mostly chemistry, math, and physics, with a smattering of biology and humanities thrown in. He'd had one class in drawing and descriptive geometry, the one required of all first-year chemistry majors. Still, he never shied away from drawing his own illustrations or designing his own instruments. Both imaginative and pragmatic, he actually liked doing paper-clip-and-string kinds of experiments, what he called \"Victorian science.\" When he was first called back to MIT to become a research associate in the Chemical Warfare Service, he was handed a German gas mask and told to reverse engineer it. Bernie got Manhattan District clearance so he could use radioactive tracers to measure smoke penetrations in the mask's filters, then designed an optical apparatus to measure the penetrations quickly. He even applied for a patent on it.\n\nBernie was a tinkerer. For him, tinkering was what science was about. You played around with something until you understood it. He'd been playing around with aircraft deicing since 1942, when a friend invited him to come work on a meteorology project at MIT. He wasn't trained as a meteorologist. He sometimes joked that he'd never really noticed the atmosphere until someone pointed it out. Meteorology had always been a poor stepsister to hard science disciplines like physics and chemistry. But he recognized right away that the field's puzzles were problems of basic chemistry and physics, manifested in everyday life. Rain, snow, clouds, storms: all were deeply familiar yet fundamentally strange. For a scientist driven, as Bernie was, by curiosity, it was an ideal subject. To study the weather was to ask science's most basic question: Why does the world work the way it does?\n\nOnce, when he was in his early teens and the Vonneguts were vacationing in Chatham, on Cape Cod, Bernie wandered down to the beach while the rest of the family slept. He had become enthralled by the dark and the sound of the waves. He walked and walked, passing the lighthouse, winking its beacon into the blackness. When he came to a jetty, he walked out to the end and stood there, surrounded by fathomless dark, hearing only the suck and roar of the ocean battering the breakwater. It was terrifying and thrilling, being so small, so alone in boundless ocean and air.\n\nHe never really lost the feeling. The earth's systems\u2014water and weather and atmosphere\u2014were huge and complicated, full of mystery and delight. Bernie had no idea yet that weather would occupy the rest of his life. Or that his interest in ice and his new deicing colleagues would shape not only his future work but also his brother's. He just knew he wanted to keep conducting the kinds of scientific explorations that fired up his brain. GE seemed like a good way to do that.\n\nAwash in government contracts, GE had expanded enormously during the war, especially the Research Laboratory, and the company planned to continue growing. President Charlie Wilson, known as Electric Charlie, directed Guy Suits and other managers to go on a hiring spree. He planned for GE to become a postwar powerhouse and frequently shocked even fellow executives by citing \"two billion\" as the number he expected net sales to reach before long. But that was going to require new ideas and eager new men. So Suits invited Bernie, among other young men, to apply for a job.\n\nWith war winding down and Bow pregnant, the timing of Suits's letter was perfect. Bernie had responded right away and had gone for an interview in early February. There he met some of GE's scientific luminaries: tall, grave physicist Albert Hull, a pioneer of the vacuum tube; plump and outgoing Saul Dushman, one of the earliest experts in quantum mechanics; Gorton Fonda, whose team had famously developed an American version of the radar screen in just fourteen days. Suits had sent Bernie back to Minneapolis with an application form, and he filled it out in his hotel room. Under \"available for employment,\" he wrote, \"Probably at termination of war.\" For racial extraction, he wrote, \"German.\" Under education, he approximated his class standing, with typical modesty, at \"lower part top half.\" On the line for draft status, he wrote, \"2B\": deferred because of work in a war-related industry.\n\nHe had been spared his brother's fate because he was doing critical work: helping get Allied airplanes safely in and out of their bombing runs.\n\nBernard addressed the envelope and went to bed. There were more test flights to be made on the B-24, more huddling over instruments in the Flying Icing Wind Tunnel. But the war would be over soon; so everyone said. The Russians were moving in from the east, and German cities were being pounded by Allied bombers. The Minneapolis Star-Journal had just run the headline \"2,250 Planes Hit Dresden.\" The German city, one of Europe's most beautiful, was already burning after a night assault by the RAF; now American bomber planes were adding fuel to the fire.\n\nAn Atlantis, Kurt would call Dresden later, sinking beneath waves of flame.\n\nDid Bernie envision firestorms as he drifted off to sleep in his royal bed at the Hotel King Cole? It seems unlikely. There, in the blank white of Minnesota winter, it would have been quite a leap to imagine the storm of flames his work there was meant to enable. A firestorm is the most complex form of artificial weather warring earthlings can conjure. First there's the search for a target city blanketed in clouds. Then the first wave of planes dropping small explosives, to buckle weaker buildings and blow out windows and doors. The people running for safety. The second wave of incendiary bombs falling with a gentle rain-like patter, then bursting to life. The fires, fed by oxygen circulating through the perforated city, building into currents of wind. Hot gales sweeping through the streets; the city melting, escape routes blocked by a greedy whirlwind of death. The aftermath: blackened husks that were once human beings, and basement shelters turned to tombs.\n\nOne thing it is certain Bernard did not imagine: that as he wrote his letter to Guy Suits and laid the plans for his future, his brother was huddled underneath Dresden as the city was wiped from the map. Bernard, his mind fixed on clouds and water and ice, could hardly have imagined Kurt hiding in a slaughterhouse basement, expecting at any moment to die by fire.\n\nIt would be many years before the brothers would learn the code name for the destruction of Dresden: Operation Thunderclap.\n\nPrecipitating Events\n\nSomeone was knocking on the door of Dean Langmuir's apartment on East Forty-Eighth Street. It startled his younger brother Irving. Irving had arrived in New York City the day before, and Dean had put up the usual cot. Hotels had been crowded enough before VE Day. Now with troops on their way home or reassigning to the Pacific, getting a room in New York was downright impossible. Especially on short notice\u2014and short notice was how Irving Langmuir did things.\n\nDean had already gone to his Wall Street job, so Irving opened the door. In the hallway stood two Army officers, and they were looking for him, not Dean. Once inside, they told him they had come to ask him to change his plans. They would prefer that he decline the invitation to Russia.\n\nLess than a month had passed since Germany's surrender. On May 8, 1945, Langmuir wrote \"VE Day\" in his GE notebook and underlined it. Then he started a list of meteorology papers he wanted to read. For the last three years, he had been doing war-related research, taking up whatever question the government needed answered. He had designed smoke generators, investigated airplane deicing, and studied precipitation static\u2014the tendency of airplanes flying in snowstorms to pick up electrical charge that disrupts radio communications. Somehow all the projects he had been handed had converged in the fascinating field of weather study. That was what he was going to work on next.\n\nThe GE Research Lab encouraged its scientists' fixations the way parents encourage a child's passion for dinosaurs or ants. When Langmuir first started there, in 1909, Dr. Willis Whitney\u2014nicknamed Doc\u2014told him to look around and find something fun to work on. That's what all new scientists were told: Doc Whitney was convinced that was the best way to ensure they produced quality work. He was known for making daily rounds in the lab and asking the scientists, \"Are you having fun today?\"\n\nAfter a few weeks of playing around with whatever piqued his interest, Langmuir told Doc Whitney that he was enjoying himself, but he didn't see how this was doing much for the company. Whitney told him not to worry about that; usefulness was Doc's problem, not his. And sure enough, before long, Langmuir's playful experiments had led to a complete redesign of the lightbulb\u2014meaning longer life for customers and a whole new set of patents for GE. Soon after, his goofing around with lightbulbs led to the invention of atomic hydrogen welding. Langmuir had been following his own fancy ever since. And his fancy had led him to weather.\n\nNow the Russians\u2014and who knew more about snow?\u2014were offering him a chance to meet their top scientists. The invitation had come just days earlier, addressed to \"Prof. Irving Langmuir, Shenektady, N.Y.\" Irving and another twenty-five prominent American scientists had been invited on a monthlong, all-expenses-paid visit to Russia to celebrate the 220th anniversary of the Soviet Academy of Sciences. Never mind that there had been no Soviet Union 220 years earlier. Irving packed his bag and phoned his brother. The Russians, he told Dean, were picking him up in two days.\n\nWhen Irving got to New York, Dean raised a few concerns. The whole thing seemed haphazard. The invite said very little about the program or schedule. Many of the scientists' names were even misspelled. And did Irving know anything about Russian festivities? He would be expected to drink toast after toast of vodka. To refuse would insult Mother Russia. But to comply... well, one might find oneself in a compromised position. And what were the Soviets up to, less than a month after VE Day, inviting some of America's most prominent scientists\u2014many of whom had been doing war work\u2014on a junket? Sure, the Russians were allies, but after the inaugural United Nations Conference in San Francisco, where their petulance had nearly scuttled passage of a UN charter, American suspicions about Soviet motivations were on the rise.\n\nIrving had dismissed all of this. That was politics, and politics did not concern him. He thought the trip would be scientifically interesting, so he packed his bags. That's how Irving was. It was fortunate he had even remembered to pack: a few years back, he showed up for a weeklong sailing trip in a blue serge suit and dress shoes, without so much as a windbreaker. Irving's obliviousness to real-world concerns was legendary. He could walk right by a colleague without so much as nodding. When a woman fell down on the stairs in front of him, he famously stepped over her and continued on. Once, he stepped in a can of paint, pulled his foot out without pausing, and kept on walking, leaving a trail of safety-yellow footprints in his wake.\n\nSo Dean thought it reasonable to make sure that his brother knew what he was getting into. Irving had assured Dean that it was all cleared with the State Department, so Dean had shrugged and gone to work. And then the Army officers appeared.\n\nIrving listened with his customary air of unruffled intensity, but beneath his patrician reserve he was livid. Born in 1881, Irving Langmuir had grown up in an urbane and academically inclined family, graduating from Columbia's School of Mines and the University of G\u00f6ttingen. At GE, he had gone from triumph to triumph. His initial lightbulb successes were just the start. He was one of the first scientists to conduct experiments on ionized gases with strange electrical and magnetic qualities, which he named \"plasmas,\" inventing the field of plasma physics. He improved sonar detection and advanced understanding of the atom's structure, writing a famous paper describing his \"concentric theory.\" He intuited the relationship between winds and ocean circulation. In 1932, he became America's first industrial scientist to win a Nobel Prize, for his work on monolayers\u2014surface films only one molecule thick. The work launched the field of surface chemistry, with applications in mining, aviation, medicine, and water resource management, as well as in understanding the fundamental structure of matter.\n\nAs GE's celebrity scientist, his salary rivaled those of top executives. He worked on whatever he wanted and traveled as he saw fit, frequently giving papers at academic conferences and before professional societies. His opinion was sought by radio journalists and newspapermen. People asked for his autograph. He was not accustomed to being bossed around, not even by the Department of Defense.\n\nThe officers explained that the Army would prefer that someone with Irving's level of security clearance not visit the Soviet Union at this time. Irving asked if they were ordering him not to go. No, nothing like that, they said. They were requesting that he not go. They told him they had made the same request of the physicist Edward Condon and he wasn't going. They neglected to mention that the State Department had helpfully revoked Condon's passport. Later it would be clear why: Condon had worked on the Manhattan Project.\n\nIrving didn't make trouble. But as soon as they left, he phoned GE. He really wanted to go on this trip. A chance to see Soviet scientific labs and talk to Soviet scientists was a rare and thrilling opportunity. Besides, he didn't know half as much as the Army seemed to think he knew about military secrets. None of the war work he had done was highly classified. The smoke screens he had developed were now being openly used in the Pacific to protect ships from kamikaze attacks. It seemed to Irving that the Army was letting a misplaced concern about secrecy get in the way of scientific learning.\n\nExecutives made a few calls. As one of the government's largest war contractors, GE had sway with the military. The Army rescinded the request, and Langmuir was free to go to Russia.\n\nOn June 10, 1945, one of President Truman's own C-54 transport planes left New York's Air Transport Command airfield. On board were sixteen of America's most prominent scientists: chemists, mathematicians, a hydrologist, and an anthropologist. Conspicuously absent were any physicists. Nevertheless, everyone aboard agreed: once things in the Pacific wrapped up, a new era of peaceful international cooperation would surely begin. And scientists, united by the apolitical pursuit of knowledge, were going to lead the way.\n\n* * *\n\n\"Now, no emotions, please!\" Kurt cried. His adored sister, Alice, ran and hugged him anyway, crying, and he too wiped away some tears. His father hugged him after Alice, and his uncle Alex\u2014another favorite\u2014happily pumped his hand. It's unlikely that any of the soldiers eating ice cream, playing the jukebox, or dancing with WACs at Officer Club 1 paid much attention. This was a familiar scene at Camp Atterbury, with its rows of hastily constructed clapboard barracks lined up like hay bales in the Indiana fields. Now that the war in Europe was over, families\u2014the lucky ones, anyway\u2014were turning up daily to pick up their boys.\n\nIt was the Fourth of July, and Kurt junior\u2014Kay to his family\u2014was home. His family had only found out he was alive when his letter of May 29 reached Indiana the previous month.\n\n\"Dear People,\" it began. He was alive\u2014not only alive, but still very much Kay, simultaneously mordant and offhand as he reeled off the inventory of horrors he had survived in the last six months. The letter\u2014ten succinct paragraphs\u2014was a masterpiece of concision. It was his first attempt to write about his wartime experience, and it would remain his best for a quarter century. Like one of those seeds that lies dormant until the conditions for germination are perfect, it would wait until the Vietnam era to sprout into Slaughterhouse-Five.\n\n\"I've been a prisoner of war since December 19, 1944,\" he wrote, since \"seven fanatical panzer divisions\" cut his division off from the others. After that, \"the supermen marched us, without food, water or sleep,\" for about sixty miles, then packed them into boxcars and shipped them across Germany. The prisoner train was strafed by the British on Christmas Day; a carload of officers was killed. South of Berlin, the men were unloaded and deloused. \"Many men died from shock in the showers after ten days of starvation, thirst and exposure,\" he wrote. \"But I didn't.\"\n\nHe was shipped off to a work camp in Dresden, where conditions were dreadful. One boy starved, and two were shot dead for stealing food. \"On about February 14th,\" he wrote, \"the Americans came over, followed by the R.A.F. Their combined labors killed 250,000 people in twenty-four hours and destroyed all of Dresden\u2014possibly the world's most beautiful city. But not me.\"\n\nThe rest of the story was crammed into short paragraphs\u2014his forced labor carrying corpses for incineration, the evacuation as the Russians approached, the Nazis' abandonment of the prisoners, a bizarre wagon ride across Germany, his final repatriation to a Red Cross camp in Le Havre. He hoped to be home, he wrote, in a month.\n\n\"I've too damned much to say,\" he concluded. \"The rest will have to wait.\"\n\nWait it would. The letter, with its repetitive short phrase, \"But not me,\" was a preview of Slaughterhouse-Five with its ironic catchphrase, \"So it goes.\" Its theme of grim good luck\u2014the meaninglessness of both death and survival\u2014would anchor Kurt's lyrical antiwar masterpiece. The letter's evocation of war was brilliant. Even Bernard could see that. He and Lois gave a copy to Lois's hometown newspaper. The Adirondack Record\u2014Elizabethtown Post printed Kurt's letter on its front page, under the title \"Vet Describes His Experiences as POW of Germany.\" But Bernard didn't tell Kurt about it. In 2008, after Kurt's death, the letter would be included\u2014incorrectly\u2014in a collection of previously unpublished works.\n\nEven before Kurt wrote to his family, though, he had written to Jane. He took a chummy tone, pointing out that they'd parted on pretty friendly terms. He was alluding to the fact that they were lovers and that he'd written to her several times to make sure she wasn't pregnant. That was his typical mode\u2014jokey understatement\u2014when it came to writing about matters of sex.\n\nJane was by then working in Washington at the OSS\u2014the predecessor to the CIA. She had graduated from Swarthmore Phi Beta Kappa in history, even though her thesis was somewhat controversial. She had argued that history teaches us nothing except that history is meaningless. The historians didn't appreciate that much, but Kurt did. He loved the way her mind worked. Throughout boot camp and Army training, he had continued to write her love letters. As his time in the military wore on, he felt as if all his dreams were dying except one: his dream of a life with Jane.\n\nJane, for her part, was as busy as ever. She had other suitors, including a younger man still at Swarthmore, but Kurt was much on her mind. Then her letters of November 27, December 16, and December 27 were all returned to her, with the ominous stamp \"Missing.\" Like his family, she hadn't known if he was dead or alive until she got his letter in late May.\n\nHe told her he was still alive, though he looked kind of starved. He was in possession of some money and a furlough and was hoping she hadn't gone and gotten married.\n\nShe hadn't.\n\nSo Kurt had stopped off in Washington to see Jane on his way to Indiana's Camp Atterbury, and soon she would be back in Indianapolis too. He was planning to ask her once more to marry him, and she was expecting him to ask.\n\nHis family had to be shocked at his appearance. He was forty-five pounds lighter, and his skin had been ulcerated by vitamin deficiency. But he was still the same old Kay. He insisted on taking the wheel, and as he drove the forty miles from Camp Atterbury to Indianapolis, he talked nonstop. The family was transfixed as he unfurled a tale of suffering, near starvation, abuse, and dumb luck. He lamented the destruction of Dresden and railed against the cruelty of the SS. When he got to the part where he and three others were forced to dig a grave for Michael Palaia, a fellow soldier executed for stealing a can of pickled string beans, he burst into tears.\n\n\"The sons of bitches!\" he cried, still driving.\n\nThey all listened intently. No one interrupted, as they would when he was little. No one could correct him or gainsay him, because he was the one who was there. Uncle Alex couldn't get over how articulate the young man had become. Kurt's account of the fall of Germany was terrifyingly observant. And his thoughts about the future were bleak.\n\n\"I know what's going to happen in Europe,\" he said. \"Now the trouble really starts. The French hate the Americans; the Poles and Russians hate the Germans; the Poles hate the Russians.\" Uncle Alex told Kurt he'd have to be patient with \"civilians,\" who would have so little idea of what he'd been through.\n\n\"Oh, hell,\" Kurt replied. \"I want to be a civilian myself. I'm sick and tired of being in the infantry... I've had enough of it. And I'm goddamned sick and tired of the whole damnfool bloody mess.\"\n\nHe was not alone. Many of the citizen soldiers who came back from the war were sick of it. They weren't professional soldiers. They were college boys and local kids, yanked out of normal lives and hurled into a maelstrom. Most who survived just wanted to put the whole thing behind them. Like so many of his fellow vets, Kurt had come home determined to throw himself vigorously into peacetime life. That meant finishing school, getting married, starting a family, and getting a job. The war played no part in any of that. It was time to take up where he had left off, as if the previous two years had never happened.\n\nSo Jane Marie Cox came to Indianapolis to see her mother, and when she went back to her job in Washington, she was wearing an engagement band made from Kurt's mother's ring. The original ring, an artifact of the family's affluent pre-Depression life, had boasted two large diamonds. Bernard and Kurt each got one to give to their wives.\n\nKurt couldn't wait to get on with married life. He was full of dreams and plans for their future, which would begin the minute he got his discharge. The only reason he stayed behind in Indianapolis instead of spending his precious leave in Washington, near his future wife, is that he was stuck at home waiting for Bernie.\n\nBernard had not been there to greet Kurt; he and Lois were in Elizabethtown, New York, awaiting the birth of their first child. They were still living in Cambridge while Bernie wrapped up his deicing work, but Bow wanted to have the baby at the Community House hospital near her parents. Three days after Kurt's homecoming, Peter Vonnegut was born. Bernie sent photographs of the baby, and Kurt and Alice were amazed to see how much like Bernie he looked. It was hard to believe, Kurt told Jane in a letter, that there were now two such creatures on earth.\n\nHe wrote to her every day. After such a long separation, it was nearly unbearable to be apart once more. But Bernie kept delaying his trip. And he was hoping that after his trip to Indy, Kurt would come back to Cambridge with him, to visit and meet his son. Kurt agreed, planning to spend a few days with Bernie and Bow before heading to Washington and Jane. He wanted to see his brother, but he was eager to start his new life. He consoled himself with the thought that maybe the know-it-all Bernie would pass on some explicit marital tips. If so, he promised to relay them to Jane pronto.\n\nWhile awaiting Bernie's arrival in Indianapolis, Kurt went downtown to the offices of The Indianapolis News and asked to see issues from the second week of February. He wanted to find out how the bombing of Dresden had played in America. Later he would describe finding a column half an inch long, a tiny paragraph merely noting that Dresden had been bombed and two American planes had been lost. He had experienced a life-altering cataclysm, and Americans hadn't even heard about it.\n\nIn fact, the paper's reporting on Dresden was much more thorough than he claimed. Strategic bombing was the front-page story, under a banner headline: \"Great US and RAF Air Raids Drive on Reich.\" The paper reported that 2,250 American bombers had attacked industrial centers in Germany, including delivering \"one of the biggest blows of the war\" to Dresden, which was \"already burning from a night assault by heavy R.A.F. bombers.\" The next day the paper reported that Dresden lay in ruins.\n\nIt's not that Kurt lied or even misremembered. The newspaper's story was simply not enough. Surrounded by movie listings, sports scores, bond prices, and ads for hats, given equal billing with Indy getting a new airport and the manpower shortage in trash collection, how could it suffice? All that heavenly life had been going on back home while he was huddled in hell. And the paper, like all American wartime newspapers, took the standard patriotic tone. The morality of firebombing civilians was not brought up. Kurt had watched Atlantis disappear beneath the waves, and the newspaper had merely reported a fruitful rain. Schoolgirls had boiled to death in water tanks; zoo animals had charred to blackened husks. Kurt saw the giraffe and wished he hadn't. To cause that kind of suffering without feeling shame could only be considered a sin.\n\n* * *\n\nBernie arrived in Indianapolis on August 7, 1945. He had repeatedly delayed the trip because Bow was having terrible headaches after Peter's birth and he didn't think he should leave her alone with the baby. Finally, on Monday, August 6, he left, heading to Indianapolis by rail. The trip took the better part of two days. On Tuesday afternoon, the train stopped in Dayton. He called the family to tell them he'd be in Indy at 4:40.\n\nSomewhere in there, he must have seen the newspapers. A single bomb had been dropped on Hiroshima, and banner headlines were declaring that the world had changed. \"Atomic Bomb May Spell Annihilation for Japs,\" proclaimed that morning's Indianapolis Star. A drawing just underneath the headline showed a uranium atom being exploded into two new atoms.\n\nThe article called atomic energy \"the most terrible destructive force ever harnessed by man.\" And it was now in U.S. hands. The president was insisting on immediate surrender, or the Japanese could expect \"a rain of ruin from the air, the like of which never has been seen on this earth.\"\n\nBernard knew they weren't posturing. He understood atomic fission; any scientist working at his level did. He had used radioactive tracers in his work with gas masks at MIT. So he knew why the Star, like papers across the nation, had very little to report about what had actually happened to Hiroshima. No trains were going in or out, and the city was under an impenetrable cloud of dust and smoke. The Japanese were reporting that Hiroshima lay in ruins, but many Americans thought they must be exaggerating. Scientists like Bernie knew better. A city had been wiped from the map, and with that the atomic era had begun, born not in the war rooms of Hitler or Hirohito but in America's halls of science. Born in places like the one where Bernie was about to start work. This wasn't Victorian science: this was something completely new and appalling. The usually unflappable Bernie was sick at heart.\n\nHiroshima cast a pall over the brothers' reunion. Bernie took in Kurt's emaciated appearance, and Kurt took in Bernard's horror at the day's news. Kurt hadn't realized, until he saw his brother's face, how dramatically the world had changed. Just that morning, he had written a cheerful letter to Jane with a 7 inscribed in pencil behind the typed words. Seven for the date, for the seven kids they planned to have. When he saw Bernie, his mood changed.\n\nLike everyone else in the nation, Bernie followed the newspapers' race to piece together the story in the days that followed. The United States, in secret from its citizens and Congress, had entered \"the battle of the laboratories.\" It was a two-and-a-half-year effort, employing more than 125,000 people, costing $2 billion, and requiring the construction of three top secret cities\u2014Oak Ridge, Hanford, Los Alamos\u2014but in the end America had won the war of the labs. The White House called it \"the greatest achievement of organized science in history.\"\n\nTwo days later, on Thursday, organized science annihilated Nagasaki.\n\nThe war was over: so the president and the military were saying. Families surged with new hope of seeing their sons. Many Americans, worn down by months of harrowing reports from the front lines of Iwo Jima or the liberation of the Japanese prisoner of war camps, felt a grim satisfaction: The Japs had gotten what they deserved. And it had been American know-how that gave it to them. The Indianapolis Star ran a breathless story under the headline \"Scientists' Dream Comes True\u2014the Atom Is Split!\"\n\nIt was a dream, but it was a nightmare too. In the days following the two atomic attacks, the nation experienced a kind of cognitive dissonance. The newspapers and radio could talk of nothing but nuclear fission, but the conversation ricocheted between hope and dread. On the one hand, there were stories about atomic airplanes and atomic cars, about atomic medicine that would cure cancer and atomic energy that would power the nation for pennies. The National Press Club in Washington introduced an \"atomic cocktail,\" and Hollywood publicists christened the voluptuous new starlet Linda Christian the \"anatomic bomb.\" On the other hand, there was fear. What would happen when some other nation got its hands on the terrible new weapon? That was inevitable, Bernie told Kurt. There was no keeping this secret. War had grown more frightening than ever: it had commandeered the power to destroy the earth. When the Japanese agreed to surrender, revered radio personality Edward R. Murrow summed up the national mood. \"Seldom, if ever,\" he intoned, \"has a war ended leaving the victors with such a sense of uncertainty and fear, with such a realization that the future is obscure and that survival is not assured.\"\n\nBernie's soon-to-be employer was mentioned in many stories about the bomb; GE's contributions had been critical. And lest anyone think that the scientists could now turn to peacetime pursuits, GE was running ads in papers nationwide pointing out that \"military and naval power drove this enemy to defeat down a road built by research.\" Democracy owed its survival to scientific know-how. That work must not cease with the end of war. \"Scientific progress and productive efficiency,\" declared the ads, \"are the most wonderful weapons of all time because they do not have to be laid aside when the fighting ends. They must not be laid aside.\"\n\nBernie returned to Cambridge earlier than he had planned and without his brother. With the end of the war weeks if not days away, he had to get busy wrapping up his government contract. D. E. Chambers, assistant director of the GE Research Lab, had already written to him asking when he intended to report to his new job. Scientific progress, as the ads declared, must remain on the march.\n\n* * *\n\nKurt and Jane took the small, leaky rowboat out onto Lake Maxinkuckee. It was called the Beralikur\u2014for Bernard, Alice, and Kurt\u2014and it had been the boat the kids used every summer, when the family spent weeks at their cabin on the lake and the three kids spent their days running around with a gaggle of cousins and friends. Those were some of the happiest days of Kurt's life.\n\nThe honeymoon trip had a valedictory feel: this boat ride would be the last in the Beralikur. The cottage, a final vestige of the family's pre-Depression prosperity, had been sold; Kurt and Jane would be the last Vonneguts to stay there.\n\nThe papers were still full of atomic horror as the newlyweds rowed out into the lake. They were adrift in a world gone mad, but at least they had each other.\n\n\"I swam all the way across this lake when I was eleven years old,\" Kurt told his new wife.\n\n\"You told me,\" she said.\n\n\"I don't think you believe I could really do a thing like that,\" he insisted. \"But you ask my brother and sister if it isn't true.\" Bernard and Alice had been there, egging him on. He was tall and gangly, a kid once given a Charles Atlas bodybuilding set by a sadistic high school coach. But in the water, he felt beautiful. As he swam across the lake, he'd felt buoyed up not just by water but by his siblings. For once he'd earned their admiration. He wanted Jane's admiration too, but she wasn't about to hand it over for something as banal as swimming across a lake a decade earlier. She had higher hopes for him, the kinds of hopes he barely dared hold for himself.\n\nSometimes he could hardly believe they were married. But it had been real, all of it, the silver vases full of white gladioli on the Cox backyard terrace, the shimmery chords of a harp, modest words spoken by a minister from First Friends Church. The wedding had been moved from September 14 to September 1 after Kurt received orders to report to the Miami Beach Redistribution Center. There was not enough time for Bernard to get there, so Kurt's high school friend Ben Hitz took his place as best man.\n\nIn the newspaper announcements of their wedding, Jane's academic accomplishments\u2014Tudor Hall, Swarthmore, Phi Beta Kappa\u2014outshone Kurt's. He had only \"attended Cornell.\" But there was also a write-up on the Indianapolis News society page where the columnist Filomena Gould had gushed over the young private's \"fresh and cogent humor.\" He even showed her some of his writing about the war, which she declared \"surpasses any firsthand account of an American soldier's existence in enemy hands.\"\n\nJane liked his writing too. She wanted him to keep doing it, and he was; in his free time, he worked on short stories and humorous essays and fired them off to The New Yorker or The Saturday Evening Post, reporting on his submissions\u2014and rejections\u2014to Jane. He assured her, however, that he had no illusions about trying to write full-time, as a career. He'd get a good job to provide for their seven kids.\n\nA loon popped out of the lake and gave its long, mournful cry, and the newlyweds were silent.\n\nIt was all an ocean. That was one of Jane's favorite sayings, from a book Kurt hadn't read: The Brothers Karamazov. The priest Father Zossima says it to the youngest Karamazov brother, Alyosha. The elderly monk is explaining why someone would ask forgiveness of the birds. \"It's all an ocean,\" he says, meaning the birds, the sky, the clouds, even himself; it's all part of one big surging life force, and the name of that force is God.\n\n\"It's all an ocean!\" Jane would cry when she was struck by how everything was interconnected. It made her happy to believe, fervently, in things that lent mysterious magic to the world. She thought Dostoyevsky's treatise on the human need for that magic was the most brilliant book ever written. She suggested Kurt start reading it on their honeymoon.\n\nThe lake, the loon, the forgiveness of birds, the rain-like patter of incendiary bombs, the exploding sun over Hiroshima: It was all an ocean, but how did it fit together? Kurt knew there were currents connecting the ruin rained down on Japan with the firestorm that haunted his dreams. The link between Hiroshima and Dresden was indiscriminate bombing, what the Allies called total war. But it was deeper than that; it was something about man's inhumanity to man, about technocrats who valued demonstrations of know-how so much they lost sight of the fact that they were killing people: schoolgirls and geezers and mothers pushing babies in prams.\n\nDresden, Hiroshima, Nagasaki: the world was spinning off into a series of atrocities, each one worse than the last. It was enough to make a man lose his faith. Kurt's buddy Bernard V. O'Hare had told him on the ship home that the war had killed his faith in God. Kurt didn't believe in God, but he still thought that was too much to lose. But now he too was losing his own faith, faith in knowledge and technology. As a child, Kurt had drawn pictures of futuristic cars and planes and houses. As a young man, he had acquiesced in his brother's and father's desire that he dedicate his life to science. Now it seemed that the urge to discover, to invent, was not noble at all, but evidence of a sickness in the human soul. \"Scientists' Dream Comes True.\" What kind of dream was destruction and death in industrial quantities? This could only be the dream of a species that hated the earth and hated itself too.\n\nJane said they could build a life of poetry and art and beauty. They could go back to school, leaving science to others, and throw themselves into the study of the human soul. They could have friends and raise kids and build a home filled with music and books, rich conversation, and a well-stocked bar. They could make paintings and write stories and make those things their life, a life that, in its own small way, would help change the world. She showed him another of her treasures, a slip of paper given to her by a professor who copied a quotation for each student at semester's end and tucked it into a walnut shell. Jane's quotation was \"Some good sacred memory, preserved from childhood, is perhaps the best education.\"\n\nRight there on their honeymoon, Kurt painted his first painting. For his subject, he chose a chair\u2014something solid, and homey, and redolent of possibility. And he started reading The Brothers Karamazov.\n\n\"My brother asked the birds to forgive him,\" Father Zossima told Alyosha. \"That sounds senseless, but it was right for all is like an ocean, all is flowing and blending; a touch in one place sets up movement at the other end of the earth.\"\n\n* * *\n\nIrving Langmuir took his seat in a room full of people who had just changed the world. It was September 19, 1945, and every famous physicist in the nation\u2014along with eminences from other sciences and humanities as well\u2014seemed to be here at the University of Chicago. Langmuir had seen some of them at a dinner the previous month for the American-Soviet Science Society. Now they had all been summoned to Chicago by the university's dynamic young chancellor, Robert Hutchins, for a secret conference on the scientific and social ramifications of the atomic bomb.\n\nChicago had played a critical role in the Manhattan Project: Enrico Fermi and Leo Szilard had achieved the first self-sustaining nuclear chain reaction there in 1942. Now many of the nation's most prominent physicists were headed for Chicago to join its new Institute for Nuclear Studies. But it was crucial that the bomb be contemplated not just as a technical problem but as a social one. The atomic era had begun, and no one knew what came next.\n\nThe general mood was grim. A terrible thought was troubling many scientific minds: not every scientific advance was necessarily good for humanity. Leo Szilard set the conference's tone with a speech full of foreboding. The bombs that had fallen in Japan, he said, had a one-mile radius of destruction. Atomic bombs in ten years would have a destruction radius ten times greater. The Soviets would inevitably get the bomb, probably in two and a half years, and in six years they would have enough of them to destroy every major city in America. The United States by that time would have built enough atomic bombs to destroy every major city on the planet. The only hope was international control of all atomic energy, which required something drastic: world government. This would probably only come about through World War III: he estimated the chance of instituting world government without war at only 10 percent.\n\nSome in the audience objected to such a dire forecast, arguing that world peace could be possible without world government. Others argued that Szilard was probably wrong about the Soviets getting the bomb so soon. Irving Langmuir spoke up then. He had been to Russia just a few months earlier, he said, and he agreed with Szilard. The Soviets would catch up in less than five years, probably more like two or three. Their science would surpass ours in ten. The critical question was how much information the United States should share with them. In his opinion, the United States should share its science freely.\n\nMany scientists agreed. Science abhors secrecy. It was said that the Army was going to seek legislation giving it total control of all atomic bomb research so it could continue to keep the details classified. But to the scientists, the idea of keeping the atomic secret was absurd. Other nations would get atomic capability; the only question was when.\n\nWhat the scientists hoped for was international control, including an inspection system that would keep nations from building more bombs. Langmuir pointed out that it wasn't such a bad thing that the Russians would have the bomb soon: once other nations had nuclear weapons, it would be less likely that the United States would use them. The military strategist Bernard Brodie was there, and he agreed with Langmuir. The following year, he would publish \"War in the Atomic Age\" in a collection called The Absolute Weapon. The essay would be a founding text of what came to be known as the theory of nuclear deterrence.\n\nBut many in the room recoiled from the idea of peace through a balance of power based on fear. To most of those present, seeing atomic war as a kind of game was morally repugnant. Worse still would be a world in which nations raced to out-arm one another, building bigger and more fearsome weapons. Already the Manhattan Project physicist Edward Teller was advocating building a \"Super bomb,\" a hydrogen atomic bomb that would make the first A-bombs look puny. But not many in Chicago agreed with him. Most of the scientists gathered there were convinced that an atomic arms race would be nothing less than the end of the world.\n\nIn fact, many seemed to feel guilty about their role in having created the first superweapon. This seemed illogical to Irving. The atomic bomb had ended the war and unquestionably saved many lives, Japanese as well as American. Yet many of the Chicago scientists seemed to feel it shouldn't have been dropped or even that they shouldn't have made it. \"The physicists have known sin,\" the head of Los Alamos, J. Robert Oppenheimer, would later declare, \"and this is a knowledge they cannot lose.\"\n\nMany scientists had read H. G. Wells's 1914 novel The World Set Free, a story that takes place in a future where scientists have harnessed atomic energy. Rather than ushering in a new era of ease and enlightenment, atomic power causes massive social upheaval and world war. The world is finally \"set free\" when humans realize that a technological advance as momentous as atomic power calls for an end to nationalism and a new era of world government in which science becomes \"the new king of the world.\"\n\nSzilard was so taken by the book that he ordered a copy of it for the Chicago scientists. Along with most of the conference participants, he agreed with Wells's main thesis: once scientific knowledge reaches a certain level, separate sovereign states are no longer possible. In the atomic era, nations would go the way of coal: they would be replaced by something better.\n\nAt the conference, the scientists hammered out a consensus based on three simple points: there was no secret that could be kept, there was no defense against the bomb, and there must be international control of atomic energy. They decided to form a group, the Federation of Atomic Scientists (later changed to the Federation of American Scientists), dedicated to educating the public\u2014and the government\u2014about this new technology. It was, the anthropologist Robert Redfield declared, their moral duty. With that, what was known as the Scientists' Movement was born. It made its public debut with a two-page statement published in Life magazine. The statement hewed to the three main points: no secret, no defense, international control.\n\nAs one of America's most famous scientists, Irving Langmuir was in a position to take their message to the people. He stuck to the script, at least at first. In October, he testified before a Senate subcommittee, predicting that Russian science would trump America's in ten to twenty years. In November, he addressed a joint meeting of the American Philosophical Society and the National Academy of Sciences, insisting that an atomic war could make the earth uninhabitable and that \"world control of all atomic energy seems the only alternative.\" But in late November, back before the Senate to argue for an international agreement, he urged that \"the Governments of the United States, Britain and Canada make immediate contacts with the Russian government to secure, if possible, their tentative agreement instead of relying solely on the more cumbersome machinery of the United Nations.\"\n\nAt heart, Irving was a pragmatist. Sure, he'd had his youthful flirtation with socialism, but like so many men of his age and status he had settled into a low-key laissez-faire conservatism. Theoretically, he had no problem with the concept of world government. But instituting it was likely to be difficult. Science would progress most smoothly if the Western powers simply stuck together and did their best to pacify the Russians.\n\nAlso unlike many of his colleagues, Irving knew and accepted that science would never return to its prewar openness. He was not an academic but an industrial researcher at one of the nation's biggest defense contractors. Government security clearance had always been an informal requirement at the GE Research Lab. Soon it would become mandatory for employment. This didn't worry Irving. He didn't see how it could possibly become a problem.\n\n* * *\n\nKurt stood in the shower, thinking of Jane. His wife, Woofie. That was her old nickname. \"Woofie on the dance floor or Woofie in a seminar or Woofie in a bull session or Jane Marie out for tea,\" declared the 1944 Swarthmore Halcyon, \"is the same piquant treat.\" Jane was putting a stop to that now, though. Sometimes he called her Wifey. He was trying out other nicknames too: Lovey, Sweety, Dear Heart, Darling, Lambykins. They were jokey, but his love was real.\n\nWater was drenching him in a shower stall in Kansas, but he was seeing Jane standing in the Florida ocean. Her dress was hiked up around her hips\u2014oh, how he loved those hips\u2014and her ankles were being licked by the waves. She had come down to Miami Beach while he was there awaiting his post-combat reassignment to Fort Riley. They had stayed at the Roney Plaza Hotel: palm trees, cabanas, pool, ocean beach, and their own bed; it was bliss.\n\nWhat other shores, she had wondered there in the surf, had the water molecules that were touching her touched before? She was always saying things like that, brilliant things that emerged from the part of her he thought of as her fourth dimension, the part full of wisdom and inspiration and poetry. There was Jane, the smart, charming woman the world knew, and there was Jane Marie, his Jane Marie, lyrical and spiritual and capable of depths that other people could only imagine.\n\nAt Fort Riley, he had been assigned the job of clerk-typist. In between typing up reports of correspondence received, he was writing to anyone he could think of to speed up his discharge. At least he had finished reading The Brothers Karamazov. Here, in the miserable flat Kansas landscape, where he cycled between apathy, depression, and joyful memories of Jane, the book was like a beacon of sanity. Dostoevsky was saying something he had always suspected: that if there wasn't a God, humans would have to invent him. They needed that order, that illusion that things made sense. But the real salvation, the true holiness of the world, came only from the world itself.\n\nNext up, he was going to read War and Peace. Also a life of Beethoven and a textbook on calculus. He had to be worthy of Jane\u2014and of the University of Chicago.\n\nThe day the acceptance letter arrived was the happiest in his life. He was going to finish his undergraduate degree on the GI Bill, and this time it wouldn't be in science. He would study anthropology\u2014which at Chicago was a program that led to a master's degree, not a bachelor's. And Jane had been given a fellowship for grad school in Slavic languages and literature. They would enroll\u2014if his damn discharge papers would finally come through\u2014in the winter semester. They would finally start living the beautiful, intellectual life they had always wanted.\n\nChicago had been an easy choice. Kurt had thought about going to its law school even before going to war. His cousin Walter, a close friend, was studying philosophy there now, also on the GI Bill. Walter came from the \"artsy\" side of the Vonnegut family: his parents were actors. And he and Kurt had another thing in common: Walter had been a navigator on B-17s during the war, and after being shot down, he'd been a German POW for two years.\n\nKurt couldn't wait to be done with the Army. Fort Riley was dreary, and he had little to do. The only good part was he had enough free time to write short stories. One of the very first he attempted was \"Atrocity Story,\" a thinly disguised account of his fellow POW Michael Palaia's execution. Kurt changed his name to Steve Malotti and set the story in the Red Cross camp at Le Havre, where the narrator and a couple of others try unsuccessfully to get the bureaucrats in the War Crimes Commission tent interested in Malotti's case. The story ended\u2014as the real event had\u2014in a frustrating lack of closure. But Palaia's story haunted Kurt, and it would until he got it right.\n\nPaper was scarce, so he typed his stories on the backs of the meteorological briefs that rolled off the office Teletype: \"Scattered thundershowers occurred early this morning in Minnesota, northern Wisconsin and western Upper Michigan, and are occurring again this afternoon in Northern Minnesota.\" On the flip side of the weather report, he typed another attempt at his war material: \"Brighten Up!\" This time, he aimed for a lighter tone than in \"Atrocity Story,\" focusing on the black marketeering of a \"dissipated little weasel\" named Louis Gigliano, who manages to use his time as a POW to rack up massive profits. It was an improvement, story-wise, funnier and not so pedantic. Friends of his at the base who read it said he could write.\n\nWhen he finished his stories, he sent them to Jane. It was her critical eye he really cared about. She had been an editor on Swarthmore's literary magazine, The Dodo, and could write herself. He trusted her. She typed up his stories, editing them in the process. She picked out a selection to send to the \"writer's consultant\" Scammon Lockwood to get some advice. Kurt waited anxiously, fearful of what she might think of him if Lockwood said he was a hack. Fortunately, Lockwood thought he had promise and for another twenty dollars, would help edit his work. They declined, because they didn't have the cash. But Lockwood's praise convinced Jane more than ever that Kurt could write for a living one day. He protested: How could he support seven kids on a writer's income? He thought he should get a job in a newsroom or a school. He knew from his misery at Fort Riley that an office job wasn't for him. But something steady and salaried, that's what he wanted. Jane was wonderful, but she was impractical. He told her she didn't understand finances. She said that she just knew it would all work out.\n\nSometimes it scared him, Jane's conviction that he would become a great writer. She told him he was a modern-day Chekhov, that if he'd lived in the Elizabethan age, he'd have been Shakespeare. She believed his works would help create the literature of the postwar era. He thought that was pushing it. It was easy to believe he was a genius when he and Jane were daydreaming, but now, as he toiled away as a clerk-typist in Kansas, it seemed like mere fantasy. They were just normal people, he told her; they should not aim for greatness. Bernie was the great one in the family, the one who was going to make a contribution.\n\nBut one day in November, reading the foreign affairs section of Newsweek, he had a strange sensation. He knew about these things. In fact, he knew more than the writer did. He knew what Europe looked like, how it tasted. He knew what artillery sounded like coming in, what it felt like to be crammed into a boxcar with sixty smelly men, what it sounded like to be bombed, what it felt like to begin to starve. He knew how to make iron ration soup, how a man who had given up hope stared into space with blank despair. He knew all this because he was there. His experience had taught him things that were important. Writing about them wouldn't be just for him. People should hear about what he knew if they were going to make better decisions in the future. He told Jane in a letter that he would take his writing seriously. She had given him the courage at least to do that.\n\nWhen his discharge finally came, he was ready. He was going to go back to school and to write. From this point on, he was going to live his own life.\n\nHead in the Clouds\n\nSchenectady, New York, was divided into three parts. South of downtown lay neat rows of modest homes, where almost all of the workers lived. In the east, just beyond the leafy campus of Union College, was the GE Realty Plot, where the managers and executives had stately homes. And in the center, just south of the Mohawk River, belching and clanking at the end of Erie Boulevard, were the machines.\n\nThe sprawling Schenectady Works, as the industrial complex was known, was GE's world headquarters and one of its largest factories. A brick and steel compound ringed by a fence, the hulking, humming city within a city contained forty thousand employees in more than two hundred buildings. The Works had a hospital, a fire station, a power plant, and a foundry. It had clubhouses, restaurants, employee stores, its own sound studios, and a radio station, WGY. Blinking down on all of it through the smokestack haze was the sign atop Building 37: yellow letters ten feet tall spelling out \"General Electric,\" crowned by the giant red GE insignia. \"The initials of a friend,\" the company called its logo. Employees called it \"the meatball.\"\n\nFor the last three years, the Works had turned out the tools of war in quantities never before imagined. Now its three daily shifts cranked out the fruits of peace: the steam turbines, generators, motors, electrical equipment, and control systems that would help build the nation's postwar prosperity. Americans and foreigners alike came to gape at the wonder of it. Tour groups clutching maps followed guides from building to building, wide-eyed at this mecca of American industry.\n\nBernard wanted to be close to work, but he didn't want to live in the noisy, bustling factory town. Bow had grown up in a sylvan village in the Adirondacks. They had a dog that needed room to romp. He ran ads in the Schenectady Gazette for a farm or a home in the country. Postwar housing was tight, so when something came on the market that was almost in the country, he grabbed it. The house was in Alplaus, a tiny village right across the Mohawk River from Schenectady. It sat on the village's main street but was set far back on a big lot, and one whole side hung out over a tangle of trees and vines at the edge of Alplaus Creek. It was a peaceful spot, just a short walk to the block-long village center, with its firehouse, grocery store, and post office at the back of a bike shop. Most of Alplaus's residents worked for GE. It was a short drive to the Works, and if Bow needed the car, Bernie could take one of the daily buses that ran from Alplaus to the GE gate.\n\nThe buildings at the Works were numbered so people could navigate the sprawling complex, but Bernie was lucky: he only had to get himself to Building 5, right near the gate. The GE Research Lab's cozy connection with management was reflected in its location right next door to Building 37, the main administration building; the two brick office blocks were connected by an arched sky bridge. In the midst of the sprawling Works, the Research Lab stood apart, a cluster of workrooms where brainy Ph.D.'s cooked up experiments with little regard for their practical value. It was a temple of science lodged in a city of trade. Yet the Research Lab was the branch of GE most essential to proving the company motto: \"Progress is our most important product.\"\n\nThe General Electric Research Laboratory was the brainchild of GE's first celebrity scientist: Charles Proteus Steinmetz. A four-foot-tall, hunchbacked, cigar-puffing socialist, the eccentric Steinmetz was a mathematical and electrical genius who dominated GE in the early twentieth century. GE's generators wouldn't have been the same without him. One of the most famous stories told about Steinmetz is probably apocryphal, but it conveys the reverence in which he was held. Mid-career, the story goes, Steinmetz was called in to look at a broken generator in one of GE's plants. Steinmetz asked for the generator's mechanical drawings and spent a couple of days poring over them. Then he approached the machine and put his ear to it. After a few moments, he took out a piece of chalk and marked an X on the side. Open the generator there, he told the engineers, and remove so many turns of wire from its turbine. Asked the fee for his services, he said $1,000. Shocked by the amount, the plant engineers insisted on an itemized invoice. Steinmetz sent them a bill. It said,\n\nMarking chalk \"X\" on side of generator: $1.\n\nKnowing where to mark chalk \"X\": $999.\n\nThat was Steinmetz's particular genius: knowing where to mark the X. And yet the company was initially reluctant to embrace his plan for a research lab\u2014until 1900, when GE patents began to expire. Facing the loss of market dominance in lightbulbs, management suddenly saw Steinmetz's point: keeping a group of research scientists on the payroll meant anything they invented or discovered would be the property of GE.\n\nOther industrial labs were narrowly focused on churning out new or improved products. GE's was different. In a radical new approach to industrial research, the scientists worked only part of the time on practical tasks like coming up with new patentable types of lamp filaments. The rest of the time they were allowed to conduct pure research.\n\nIt was a fantastic success. Given the opportunity to explore freely, the scientists came up with all kinds of new inventions. By World War II, the GE Research Lab could claim responsibility for the electric range, transoceanic radio, the portable X-ray machine, the turbo-supercharged jet engine, and autopilot. Its contributions to both world war efforts further boosted its importance to a company whose growth was increasingly driven by government contracts. GE dubbed it \"the House of Magic.\" The scientists hated the name\u2014they were doing science, not magic tricks\u2014but the boys in PR loved it.\n\nBernard fit the place temperamentally. The Research Lab was sometimes described as having a university atmosphere. But in its early days, it was more like a salon, a mecca of what Bernie called Victorian science: the eccentric but brilliant researchers followed their curiosity wherever it took them, moving easily between physical, material, and even biological sciences. Having a Ph.D. in physics didn't mean you couldn't work in neurology if you liked; having a Nobel Prize in chemistry didn't mean you left statistics to the mathematicians. The scientists discussed their work with each other in weekly cross-disciplinary presentations and frequently had far-flung hobbies outside the lab as well: Steinmetz had kept a greenhouse full of strange plants and a menagerie of deadly animals, including alligators, black widow spiders, and rattlesnakes. The lab's director, Doc Whitney, was famous for his obsession with turtles. Albert Hull taught Greek before becoming a physicist and contributed the many classical coinages for the lab's new inventions\u2014thyratron, magnetron, kenotron, dynatron\u2014that one inventor later dubbed \"Graeco-Schenectady.\" Irving Langmuir spent much of his spare time studying the hydrodynamics of the Adirondacks' Lake George, where he had an island house, and his assistant Vincent Schaefer collected arrowheads, archaeological artifacts, and natural history specimens.\n\n\"Research is appreciation,\" Doc Whitney liked to say. He often told the story of how, as a young professor at MIT, he had given students an assignment in which they combined the elements of sulfur and iron in a glass tube and heated them over a Bunsen burner. As the iron and sulfur combined into iron sulfide, the vessel was suffused with a bright glow. Students writing up the experiment almost inevitably reported that the elements had combined. When they did, Whitney would send them back to the lab with the instructions \"Repeat and note the glow.\"\n\nBernard never had to be told to note the glow. Like Whitney, he considered research an expression of appreciation for the natural world and its principles, and he couldn't wait to get down to doing it. But almost as soon as he arrived in Schenectady, he was denied the opportunity. GE was on strike.\n\nAlmost before the VJ Day celebrations wound down, the nation was swept by a wave of strikes. Throughout the war, laborers had accepted wage controls and strike bans, even while upping production, and profits had soared. Workers expected corrective wage increases once the war was won. Instead, after VJ Day, employers, fearing the loss of government contracts, began laying off workers and cutting hours. So in the fall of 1945, union members across the nation\u2014oilmen, coal miners, autoworkers, truckers, meatpackers, newspaper printers, phone and telegraph operators\u2014began walking off their jobs.\n\nBernie probably hadn't expected to see it happen at GE, any more than President Charlie Wilson had. The company was supposed to be a liberal, employee-friendly place: it was widely known that in 1937, while other corporations were fighting unionization, GE had willingly hammered out a contract with the United Electrical Workers. Employees called it the \"Generous Electric Company\"; management didn't think its workers would strike over a smaller-than-requested pay raise. Yet in early January, a hundred thousand of them did.\n\nThe strike hobbled the entire company. Dozens of factories, including the Schenectady Works, were effectively shut down as picketing electrical workers denied access to everyone, including non-union, white-collar employees. Bernie, along with all the Research Lab scientists, was told to work from home.\n\nPublic opinion favored the striking workers. Desperate to avert a public relations disaster, GE settled with the union on a wage increase of eighteen and a half cents an hour. Now at last Bernie could start his new job\u2014by looking around for a good problem to work on. If he didn't find something quickly, someone would assign him something, and he preferred to choose a research project himself. So, after years of wartime interruption, he finally got his life back on track.\n\n* * *\n\nChicago was a dream come true. Kurt and Jane's apartment on Ellis Avenue was within walking distance of the university and from Walter and his wife, Helen, who lived on Lake Park Avenue. The cousins were frequently at each other's homes or strolling down to Lake Michigan together. Kurt and Jane loved the lake. They loved having Walt and Helen in town, and they loved the university. Chicago was a heady place to be. The school had always been a magnet for leftists, skeptics, bohemians, and civil rights activists; now the issues raised by the bomb gave its intellectual life a new urgency. But Chicago's thinkers weren't wallowing in atomic dread. They were stepping up to help shape this new atomic age. Hyde Park buzzed with a new prospect: that, just as in H. G. Wells's novel The World Set Free, humanity had learned its lesson at last.\n\nKurt and Jane embraced the idea. The horror of Hiroshima could at least bring this new hope: that human beings might finally change, might finally move beyond the outmoded nation-state and form a world government. If so, there might be hope for the elimination of war. Some scientists were even acting on their conviction that they must lead the way in renouncing violence. In the spring of 1946, as the government prepared to test atomic bombs by dropping them on a fleet of captured enemy ships at Bikini Atoll, some prominent scientists, including J. Robert Oppenheimer, refused to take part.\n\nIt was gratifying that Chicago was at the center of this movement. Chicago chancellor Robert Hutchins and physicist Leo Szilard were leading spokesmen for the world government cause. The Chicago-based Federation of American Scientists published the first issue of its journal, the Bulletin of the Atomic Scientists, the month after Kurt and Jane arrived. Soon after that, it put out the bestseller One World or None: A Report to the Public on the Full Meaning of the Atomic Bomb, a collection of passionate essays advocating international control of atomic energy. Contributors included Albert Einstein, J. Robert Oppenheimer, Leo Szilard, Edward Condon, and Irving Langmuir.\n\nKurt's own anthropology professor Robert Redfield had helped form the University Office of Inquiry into the Social Aspects of Atomic Energy and had joined Chicago's Committee to Draft a World Constitution. He believed that the social sciences\u2014especially anthropology\u2014were going to be critical in helping humanity adjust its institutions for the new atomic age.\n\nKurt admired Redfield, whose work he began to follow when he switched his specialization to cultural anthropology. He had started off studying physical anthropology, which had charts and maps and measurements and all kinds of complicated theories, making it comforting to someone who valued scientific thinking. But there were also tedious scientific tasks, like measuring the size of early human brains by pouring grains of rice into their skulls, then measuring the rice. Disappointed, Kurt had gone to his adviser and confessed that the science bored him; he would rather study poetry. The adviser had recommended cultural anthropology, \"poetry which pretends to be scientific.\" Kurt liked the idea, and he threw himself into the new field.\n\nHe filled his notebooks with cramped notes: phases of different cultures; notes on cultivation, ceramics, human sacrifices, and metallurgy; ideas about literacy, agriculture, the development of moral orders. He learned about a vast array of cultures: Finns, Kazakhs, South Pacific Islanders, Mayans, Incas, ancient Chinese, Turks, Native Americans. He devoured journal articles, marking them up with red pencil. He studied Turkish linguistics. It was still rigorous and precise. But now all that precision was being directed toward something he actually cared about: not molecules or cells or atoms, but human beings and the cultures they made.\n\nHe was especially taken with Professor Redfield's ideas about the changes and dislocations that result when small isolated communities\u2014Redfield called them folk societies\u2014evolve into sprawling, heterogeneous cultures. Culture, Professor Redfield declared, was in a constant state of flux; even the moral order was always forming, dissolving, re-forming. Culture wasn't something that simply happened to humans; it was something they could tinker with. You could see it happening as the atomic age washed over Chicago and the university's best minds worked on reshaping the culture to accommodate it.\n\nKurt could play his part in that by writing. So he attempted, once more, to write about his Dresden experience in an essay called \"Wailing Shall Be in All Streets.\" It was, like his letter home, pointed and concise. It began with an account of basic training and how the infantry were urged to \"kill, kill, kill,\" without being told very clearly why they were fighting. \"A lot of people relished the idea of total war,\" he wrote; \"it had a modern ring to it, in keeping with our spectacular technology.\" But the war, he wrote, had left him \"sick at heart.\" The reason was simple: \"In February, 1945, Dresden, Germany, was destroyed, and with it over one hundred thousand human beings. I was there.\"\n\nI was there. It was something he would say again and again.\n\nHe went on to tell of his experience helping to find and burn bodies after the firestorm. But then the essay devolved into several pages of outright moralizing.\n\n\"There can be no doubt that the Allies fought on the side of right and the Germans and Japanese on the side of wrong. World War II was fought for near-Holy motives. But I stand convinced that the brand of justice in which we dealt, wholesale bombings of civilian populations, was blasphemous.\"\n\nHe sent \"Wailing\" to The American Mercury, which rejected it, but editor Charles Angoff suggested Kurt send the magazine something else.\n\nHe tried a different tack in an essay called \"I Shall Not Want,\" where he tried to convey what it felt like to starve. Here he tried to keep the tone light, thinking maybe he could sell it to Gourmet. Some parts worked, such as his account of how the prisoners kept journals full of fantastical descriptions of meals they would eat once free, arguing about whose imagined menus were best. But other parts were over-formal, tendentious even. Jane went through it with a pen, striking out stilted constructions and wordiness. It helped, but not enough. Kurt sent it off to The New Republic's college essay contest. The prize was a summer internship at the magazine. He didn't win.\n\nIn July, Kurt and Jane vacationed at Lake Maxinkuckee with Walt, Helen, and their baby son, Kit; Kurt and Jane were his godparents. The two couples played cards and stayed up late talking about politics. That month, Congress passed an act creating the Atomic Energy Commission. It felt hopeful: like the first step on the road away from madness and toward sanity through world government.\n\n* * *\n\nNot long after his arrival at GE, Bernie dropped in on the lab of his old deicing acquaintances. Irving Langmuir was away for most of the spring, giving the prestigious Hitchcock Lectures at Berkeley, but Vincent Schaefer was in Schenectady, moving forward with their new investigations. Bernie told Vince that he had grown interested in the process of supercooling: lowering the temperature of a liquid or gas without converting it to a solid. Vince said he and Irving were working on the same thing. Specifically, they were trying to figure out if they could manufacture snow.\n\nSnow had brought Vincent Schaefer and Irving Langmuir together. Vince was president of the Schenectady Wintersports Club, and Irving, who also adored skiing, was a member. In the early 1930s, Vince got the idea of sponsoring a snow train for skiers out of Schenectady. Irving, an avid pilot, offered to take him up to look for likely routes in his open-cockpit Waco plane. Before long, the startled Vince found himself flying the plane\u2014banking, turning, diving, even practicing stalling and recovering, all under the tutelage of the brilliant scientist. They flew over the Catskills and scouted out ski hills. By the time they landed, Vince was groggy with incipient hypothermia. Langmuir took the younger man to his own house, and Marion, Irving's wife, fed him hot tea and cookies until he felt better.\n\nTo some people, Vince seemed an unlikely assistant for Irving Langmuir: he had never even finished high school. His father was sickly, and Vince had to leave school at sixteen in order to help support the family. On the advice of an uncle, he joined the GE apprentice program and trained as a machinist, landing a job as a drill press operator. Eventually graduating to model maker for the Research Lab, he quickly became known as the person who could put together any kind of apparatus the scientists needed. Intelligent, ambitious, and with a burning desire to be a \"real\" scientist, he was soon not simply making lab machinery but helping in the design of experiments.\n\nIn 1932, when Langmuir's old assistant retired, Vincent became his right-hand man. Vince called Irving \"the Boss,\" but they were really scientific partners. Irving put little stock in degrees or credentials: he cared about whether a person was curious and thoughtful. Bernie was the same way: it never seemed odd to him that Vince had made himself indispensable.\n\nLike Bernie, Irving and Vince had grown intrigued by supercooling during the war. In their work on precipitation static, they had attempted to conduct experiments at the research station on New Hampshire's Mount Washington, home to some of the worst weather in the world. One day, as they were hiking up\u2014they preferred to hike up and ski out when they had work to do at the summit\u2014Langmuir stopped and indicated some clouds that hovered on the mountain's peak. They were heavy and ominous, but only one lone snowflake drifted to the ground.\n\n\"Look, Vince,\" he said. \"With all these clouds everywhere, there's only a flake here and a flake there. Why? I think we ought to do some more studies on that.\"\n\nThey knew that the clouds on the mountain peak were often supercooled. Objects at the summit research station could amass ice three feet thick without a drop of rain or snow falling. Irving intuited that this must happen because the clouds passing over the peak contained large numbers of water droplets that were colder than 32 degrees Fahrenheit. Those droplets didn't freeze until something disturbed them, such as sudden contact with a metal surface. It was the same process that caused airplanes to accumulate ice when flying through supercooled clouds. But were there ways to force the supercooled water droplets in a cloud to freeze spontaneously, after which they would fall as snow or rain?\n\nPeople have dreamed of inducing precipitation for as long as they have been thirsty. But efforts to make rain were usually mystical, not scientific. Military men had long claimed that rain tended to fall after big battles, leading to the idea that firing cannons into the sky might \"bust\" the clouds. In the nineteenth century, some scientists argued that filling the air with particulate matter was likely to bring down rain; James Pollard Espy, the nation's first government meteorologist, proposed burning large swaths of the nation's forests to make rain for the arid West. But most purported rainmakers were charlatans and con men. No one really knew how to make rain because no one understood how rain happened.\n\nSnow and rain seem like some of the most basic phenomena on the planet, yet they were still largely mysterious. Meteorologists knew that clouds form when moist air cools to its dew point, converting its water vapor to cloud droplets. At some point, some of those droplets would convert to ice crystals. The ice crystals would collect water and grow until they were heavy enough to fall. But what made those first droplets turn to ice?\n\nIn the 1930s, the meteorologist Walter Findeisen proposed that they required a nucleus\u2014a small atmospheric particle of dust or salt spray for the water to cling to. Findeisen suggested that if he was right, it should be possible to introduce artificial nuclei into clouds to stimulate rain. Langmuir and Schaefer had not yet read Findeisen's foundational work; they preferred to start their work with experiments, rather than the literature. But they had guessed correctly that the cloud droplets require a nucleus to convert to ice. So, with the war over, they had decided to test different chemical substances for their ability to do just that.\n\nBecause Vince was already conducting experiments on supercooled water, Bernie decided that he would focus on supercooled metals. But he kept in touch, dropping in on Vince periodically to see how he was progressing. In the early summer, Vince had an idea that was startlingly simple. He requisitioned a GE chest freezer. When it arrived, he lined it with black velvet and aimed a light at its innards. He chilled it to 10 degrees below freezing and breathed into it, and voil\u00e0\u2014his breath made a cloud! Because cold air is heavier than warm, the cloud stayed there in the freezer, even with the lid open. Now, with a laboratory cloud, he could begin trying to make laboratory snow.\n\nFor several weeks, Vince assailed his cloud with substances that might function as ice nuclei: sulfur, magnesium oxide, volcanic dust, talc, diatomaceous earth. As the summer heated up, his lab partner Katharine Blodgett\u2014the first woman to receive a Ph.D. in physics from Cambridge University\u2014noted that his experiment was cleverly timed: it required him to spend sweltering days in the lab hanging the upper half of his body into a freezer. But one day in July, it got so hot the freezer couldn't keep up. To get his freezer back down to the right temperature quickly, Vincent got a large block of dry ice, which the Research Lab always had on hand, and threw it in. To his surprise, the entire cloud inside instantly converted into shimmering crystals of snow. After all the substances he'd tried, dry ice\u2014nothing more than solid carbon dioxide\u2014had nucleated the cloud.\n\nVincent wrote up the experiment in his lab notebook: \"I have just finished a set of experiments in the laboratory which I believe points out the mechanism for the production of myriads of ice crystals.\" When he wandered into the adjoining lab, he was less circumspect. To the astonished scientists he announced, \"Now I know how to make it snow.\"\n\nWhen Langmuir returned to Schenectady in late July, Vincent showed him the freezer experiment. The Boss saw the point at once. In his GE lab notebook he wrote, \"Control of Weather.\"\n\nThey both knew this was their next big project. Irving had recently heard that his name was on the list of prominent figures President Truman was considering to head up the Atomic Energy Commission. But now nothing could be further from his mind than atomic bombs and nuclear power. He was onto something even more important. In his small notebook, he recorded his plans. They would fly to the tops of clouds and seed them with pellets of dry ice or a stream of liquid carbon dioxide. They would make flights through various types of clouds measuring wind speed, temperatures, vertical currents, and the distribution of water droplets. \"Develop theory of growth of rain drops,\" he wrote. \"Can we cause cloud of uniform droplets to give rain?\"\n\nHe spoke to Guy Suits about getting a plane. But he would also need instruments to record all the data they would collect. Then he had another thought.\n\n\"What is Vonnegut doing?\"\n\n* * *\n\nOn the last day of August 1946, The New Yorker dedicated its entire issue to John Hersey's thirty-one-thousand-word article \"Hiroshima.\" Rather than report on the event in the traditional way, Hersey followed six characters in detail from the moments just before the bomb fell, documenting their activities in the horrific days that followed. Written in a flat, almost clinical tone that avoids overt moralizing or rhetorical flourish, the piece piles detail upon detail until the cumulative result is far more harrowing than any attempt to sermonize. The issue sold out at newsstands in a few hours. Rarely has a magazine piece\u2014which soon became a book\u2014caused such powerful soul-searching. Albert Einstein sent a thousand copies to fellow scientists, urging them to consider its implications.\n\nIt was the sort of response Kurt would have liked to provoke with \"Wailing Shall Be in All Streets.\" But his little piece on Dresden was not gaining traction. After The American Mercury rejected it, he sent it to Harper's, The Atlantic, Time, and The Yale Review. Edward Weeks at The Atlantic was positive enough that Kurt revised it and sent it again. But in the end, no one wanted it. He threw himself back into his anthropology studies at Chicago.\n\nAnd then, before the fall semester could even get going, Jane told him she was pregnant. It had happened: an atom of Kurt and an atom of Jane had smashed together and split into two new cells. Two to four to eight to sixteen to thirty-two and on and on\u2014a chain reaction bringing life, not death.\n\nShe would drop out of school. That's what women did. Together they went looking for her adviser, a gloomy Russian who had fled Stalin and washed up on the shore of Lake Michigan to bestow the pearls of Russian culture on frat boys, ex-GIs, and coeds. They found him in the library. Jane told him she was going to resign her scholarship and drop out of grad school. And then\u2014she couldn't help it\u2014she broke down and wept.\n\n\"Mrs. Vonnegut, pregnancy is the beginning of life, not the end of it,\" he sniffed.\n\nYes, it was a beginning, the beginning of a life they had imagined together, their married life with seven kids. But it felt like the end of something too. They shared so many dreams: to write, to learn, to travel, to help cure an ailing world. Having seven kids was among their dreams, but now it was cutting to the head of the line, especially for Jane.\n\n* * *\n\nWhen he heard the whine of an airplane over the village of Alplaus, Bernie rushed outside. He had come home for lunch; his house was just a mile from the airport. Vince and Irving were still at the airstrip, experimenting. In fact, that might be Vince overhead. Bernie peered at the airplane intently. Sure enough, a long unnatural trail of something that looked like ice crystals was streaming out behind it. He ran for his camera.\n\nBernie was now officially part of Langmuir and Schaefer's team. They had invited him to join them soon after Schaefer's freezer experiment. For months now, Vince had been conducting more freezer tests while Bernie scoured the crystallographic tables, trying to find other substances that might nucleate ice crystals. Meanwhile, Langmuir was filling his notebook with theoretical calculations of the number and size of snow crystals likely to be produced in different circumstances.\n\nThey had all been longing to try out dry ice nucleation on a real cloud. But the fall had been frustratingly mild: sunny, warm days, with barely a cloud to speak of in the heavens. The bright blue Schenectady skies deepened to a rich royal tone at dusk, and the stars blinked the promise of another clear day. Every morning, Vincent was up at dawn, scanning hopefully for clouds. He told the others he was so eager to test his theory he couldn't sleep.\n\nBy November 12, unable to bear the suspense, they had decided to make a practice flight. They rented a single-engine Fairchild plane and got Curtis Talbot of the GE test flight division to fly it. Using a motorized dispenser he had designed, Vince released crushed dry ice into cumulus clouds at three to five thousand feet. But the cloud temperature was nowhere near cold enough to produce snow. There were some more promising stratus clouds at around twelve thousand feet, but Talbot told them the single-engine plane could go no higher than ten thousand feet.\n\nA day later, the weather had finally given them what they wanted. The temperature was around freezing, and the sky to the east and north of Schenectady was filled with parallel bands of thick stratus clouds. An excited Vince had called Talbot as soon as he stepped outside, then driven to the local dairy and bought six pounds of crushed dry ice. Now he was up there, dispensing it.\n\nBernie retrieved his camera and went back outside. He fumbled with the aperture setting, then searched the sky again for Curtis and Vince in the Fairchild. Langmuir had had a two-way radio installed on the plane after the previous day's flight, so he was probably talking to them now from the tower.\n\nThe plane Bernie was watching flew in an odd pattern, the trail of snow behind it tracing its path. The path was growing more elaborate. It almost looked like a P. Then it looked like an E. Suddenly his heart sank.\n\nP... E... P... S... I...\n\nIt was not Vince and Curtis at all. Bernie couldn't help but laugh at himself. Ruefully, he went inside and put his camera away.\n\nBut when he got back to the lab, everything was in an uproar. Vincent had not done any skywriting that morning. He had done something even better: he had made snow.\n\nIrving and Vince recounted the story over and over. The stratus clouds were high up in the sky, so after takeoff Curtis had begun climbing, taking forty minutes to reach ten thousand feet. Vince spotted a promising cloud in the vicinity of Mount Greylock, over the Massachusetts border. Its base was well above them at around thirteen thousand feet.\n\n\"Can we get to it?\" he asked Talbot. Curtis, getting into the spirit, nodded and urged the little plane upward.\n\nThe plane's single propeller churned away. It seemed to take forever. The last four thousand feet took half an hour to climb. Looking into the cloud, Vincent saw shimmering iridescent ice crystals around its edges\u2014a sign that the inside was probably supercooled. He checked the thermometer. It read \u221217.5 degrees, and its bulb was beginning to ice over. Curtis swung the airplane around, and they flew into the cloud. Vincent dispensed three pounds of dry ice. Then his ice dispenser jammed. He was breathing heavily. His head swam, and his heart pounded\u2014the effects of altitude in the unpressurized plane. He picked up the cardboard box and opened the Fairchild's window. Wind sucked the remaining dry ice into the white.\n\nAfter that, they made a big loop and flew back through the cloud again. This time, they were surrounded by glinting crystals of snow.\n\n\"We did it!\" Vince cried over the roar of the engines. He and Curtis shook hands on their triumph.\n\nIn the tower, Langmuir had his field glasses glued to his eyes. Shortly after the plane disappeared into it, the cloud almost seemed to explode. Then it began to split horizontally, dividing into two parallel clouds. Falling from the space between them were long streamers of snow.\n\n\"It works,\" Irving wrote in his notebook. When the plane landed, he rushed across the tarmac to greet Vince.\n\n\"This is history,\" he said.\n\nBack at the lab, the normally reserved Langmuir could hardly contain himself, glowing with excitement as he described the snow they had made.\n\n\"I could see it forty miles away,\" he marveled.\n\nGuy Suits pointed out that they would soon need a better plane to continue the experiments. He had worked closely with the military during the war, heading up GE's war research division, and had many close contacts there. He proposed bringing in General Curtis LeMay. Surely the Air Force would be interested in a technique for dissipating fog and clouds.\n\nBefore long, the phones were ringing. GE's publicity department, the News Bureau, was staffed with real journalists, and like reporters everywhere they had a nose for breaking news. Before long, two men from the News Bureau had arrived at Building 5 with portable typewriters and were tapping away as fast as Langmuir could talk. The PR men returned to the News Bureau that evening, bursting with reports of man-made snow. An assistant editor at The GE Monogram\u2014the company's in-house magazine for managers\u2014was there to hear the tale.\n\n\"Well, Schaefer made it snow this afternoon over Pittsfield!\" he reported to his colleagues back at the Monogram. \"Next week he walks on water.\"\n\nThe GE press release went to the papers the very next day: \"Scientists of the General Electric Company, flying an airplane over Greylock Mountain in western Massachusetts yesterday, conducted experiments with a cloud three miles long, and were successful in transforming the cloud into snow.\"\n\nThe release quoted Langmuir's estimate that a single pellet of dry ice the size of a pea \"might produce enough ice nuclei to develop several tons of snow.\" He went on to give a primer in cloud physics. When a cloud was seeded and its water droplets froze, he said, latent heat was released. Ironically, the freezing process generated heat in the cloud, which would produce turbulence, the kind that causes cumulus clouds in thunderstorms to billow upward. The turbulence, he said, \"enables the process to spread as a type of chain reaction and draws more moisture into the active region.\"\n\nIt was typical of Langmuir's brilliance that, although not trained as a meteorologist, he had intuited something fundamental about clouds. They are not, as most people think, reservoirs of water hanging out in the sky. They are hives of activity, constantly taking on water vapor from the atmosphere, the ground and surface water, and sometimes converting that water to precipitation. Langmuir had figured out that seeding did more than force clouds to release stored water. It could literally make clouds more efficient. He and Schaefer had discovered a method not of \"milking the skies,\" as it was sometimes described. They had found a way to build better clouds.\n\nThe newsmen were less interested in cloud physics than in what this all meant for the man on the street. They packed the press release with thrilling claims. Scientists could now fill reservoirs, deliver more water to hydroelectric dams, clear dangerous clouds that iced airplanes, make snow over ski resorts, and divert storms from urban areas. A new era of managed climate was dawning.\n\nIt was headline news. \"Snowstorm Manufactured,\" announced The Boston Globe in inch-high banner type. \"Three-Mile Cloud Made into Snow by Dry Ice Dropped from Plane,\" said The New York Times. \"Man Does Something About Weather\u2014Makes It Snow,\" declared the New York Post, and Newsweek quipped, \"Deliver One Blizzard.\" Time described the event as if the dry ice had been an atomic bomb and the cloud had succumbed to radiation sickness: \"Almost at once the cloud, which had been drifting along peacefully, began to writhe in torment. White pustules rose from its surface. In five minutes the whole cloud melted away, leaving a thin wraith of snow.\"\n\nFile clerks struggled to keep pace as more than ten thousand clippings descended upon GE. Delighted newsmen took publicity photographs of the News Bureau snowed under with news.\n\nFor Bernie, Vince, and Irving, the next few weeks were a whirlwind. Letters and telegrams poured into the Research Lab. It seemed almost everyone had ideas about how to use this fabulous new tool. Airline meteorologists wanted to dispel icing clouds and dissipate fogs. Water managers and irrigation districts wanted to produce more rain. A Chilean government agency requested a team of GE scientists to draw up a plan for fixing Chile's arid areas. A U.S. Marine Air Corps navigator offered to fly the GE scientists to Seattle to dispel fog that was slowing rescue efforts for a plane that had crashed near Mount Rainier. Ski clubs and ski resorts offered their slopes to further the work. A classroom of California schoolkids sent postcards requesting a snowstorm. A film crew wanted snow for a shoot in Buffalo. Buffalo itself wanted snow sent back over Lake Erie.\n\nThere were naysayers, but not many. The Boston Herald editorialized that \"bringing more snow to snowy New England is very much like carrying coals to Newcastle... Can the General Electric create a snow-repellant as well as a snow-producer?\" A columnist in the New York Sun had more philosophical objections to monkeying with nature, asking, \"Who wants to see a child look out the window at the crystals from fairyland on a winter morning and exclaim, 'Oh, mumsy! Look what General Electric is doing'?\" Vincent, riding high on his new fame, sent the columnist a sardonic apology, claiming that he was \"repenting inside the igloo doghouse\" to which he'd been consigned and assuring the columnist that \"the day will never come when each and every snowflake carries a G-E monogram.\"\n\nBut most of the world seemed giddy at the prospect of GE snow. And why not? Only a year earlier, scientists had harnessed the most fundamental power in the universe and used it to end a seemingly endless war. Why shouldn't they move on from mastery of atoms to atmosphere? This was just the next step in man's taking control of nature.\n\nOne letter sounded a different theme, one of particular interest to Guy Suits. Simon Goldstein, an insurance broker from New York City, wrote to warn GE about the many injurious effects that a manufactured snowfall might cause: car accidents, falls, floods, property damage, even the expense of snow removal. \"This is likely to produce lawsuits against your Company,\" he wrote. \"It would therefore seem dangerous to leave yourselves unprotected in these circumstances. May I hear from you?\"\n\n* * *\n\n\"Time to go home, Barney,\" Katharine Blodgett called out as she gathered her things. It was 5:30, and everyone else had already left. But Barney\u2014Bernard's undergraduate nickname had followed him to GE\u2014was huddled over Vincent's freezer. Approaching him, Katharine realized he wasn't likely to be heading home anytime soon. He was on the trail of something interesting and was determined to see it through.\n\nThat was the quality in Bernie that made him such a good scientist\u2014and also at times such a frustrating husband, father, or brother. Once he got some new idea in his teeth, no force on earth could tear him away from it. Now his obsessive nature was zeroing in on a new method of cloud seeding: silver iodide.\n\nHe had paged through the entire crystallographic handbook, looking for things with crystal structures similar to water. He had found three promising substances and tried them in Vince's freezer. Only one seemed to have any effect. So he had been trying it in different forms, including smoke. When he vaporized silver iodide, it worked like a charm. Not only that, but the ice nuclei created with silver iodide lasted for half an hour or more. They lasted even after more moist air was added to the freezer. The effect was as striking as what happened with dry ice, and it was likely to be even more durable.\n\nHe was trying, he told Katharine, to figure out if silver alone would do the trick, or iodine alone, or if it had to be silver iodide. He had hours of experimenting to do, and he wasn't about to stop now. Bernie would stay in the lab for as long as it took, because he was onto something big. Dry ice seeding was causing a national sensation. But Bernie might have found something even better.\n\nThe next time Vincent went up in the Fairchild, Bernie went with him. They conducted two days' worth of inconclusive tests with dry ice. On the second day, Vincent thought perhaps they had over-seeded the cloud, producing so many ice nuclei that they were too small and light to fall. But Bernie couldn't help but notice that even when their seedings produced immediate and dramatic results, they did not propagate further snow. Dry ice worked, but only for a short time.\n\nBack in the lab, he built a small smoke generator based on the ones Langmuir and Schaefer had made during the war. He put silver iodide in it and ran it for fifteen seconds, and the lab filled with nuclei. When he blew a puff of air into the freezer, it filled with ice crystals. The effect persisted for an hour. Bernie put the device on a windowsill and let it blow silver iodide smoke out into the atmosphere. But it was late in the day and dark, so there was no telling if the chemical had any effect on the clouds.\n\nIn early December, Bernie, Irving, and Vince met with the Army. After a formal lunch downtown at the Hotel Van Curler, they gave Dr. Michael Ference of the Signal Corps a tour of the lab. Vincent created an ice crystal fog in the freezer. Bernard brought a balloon filled with silver iodide smoke he had produced in the boiler of Building 37, and they created another round of ice crystals with that. Ference was impressed. He thought they could work out a contract for at least a year, including the long-term loan of a bomber plane and its crew.\n\nOn December 20, Vincent and Curtis went up in the Fairchild again. Bernie and Irving were in the tower, but the airplane radio failed, so they had to wait for the plane's return to hear how the seeding went. After landing, Vince reported that they had made four runs dispensing dry ice and liquid carbon dioxide. On their way back to the airport, they saw a new cloud full of fine crystalline snow hanging just below the cloud deck. They figured it had been made by seeding.\n\nThe first snowflakes fell shortly afterward. Snow fell all afternoon and into the evening. By the time it stopped, around 11:00 p.m., Schenectady was buried under nearly ten inches of snow.\n\n\"A very interesting storm,\" Langmuir wrote in his notebook.\n\nFor the next couple of days, they researched it. Bernie collected data on snowfall times and measurements for all Weather Bureau stations within two hundred miles of Schenectady. They made maps showing the storm's development. Irving woke up before dawn with equations filling his head, waiting until 9:00 a.m. before calling Vincent or Bernie to talk it through. Once they felt certain, Irving called Guy Suits and told him what they had concluded: their seeding had caused the unusually heavy snowfall in the Schenectady region and beyond.\n\nSuits told him not to tell anyone.\n\nBut on the day after Christmas, Irving wrote to C. N. Touart in the Air Weather Service of the Army Air Forces. \"Schaefer made some seeding runs Friday morning, Dec. 20,\" he wrote, \"which look as though they may have produced wide-spread effects upon the development of the snow storm that swept over parts of New York state and New England.\" If Suits didn't want to go public, the best way to ensure continued research was to keep the military interested.\n\n* * *\n\nThe meteorologist Harry Wexler, chief of scientific services for the U.S. Weather Bureau, respectably liveried in flannel suit and wire-framed glasses, sat up in his conference chair and gaped. He had come to Boston to learn what could be learned for the good of his governmental agency, but he hadn't expected anything as dramatic as this. At the front of the room, Vincent Schaefer was showing slides of man-made snow while Irving Langmuir predicted GE's imminent victory over the climate.\n\nWexler was at the joint meeting of the American Association for the Advancement of Science and the American Meteorological Society (AMS). The whole conference was abuzz with the GE scientists' bombshell. The night before, GE had hosted a cocktail reception in the Georgian Room of the posh Statler Hotel. Vincent had brought the laboratory freezer he had used to make his early experiments, and as the assembled reporters and scientists drew near, he had breathed into the box to create a cloud. Ice cubes rattled in glasses as the gods of this miniature world waved their hands and filled its sky with shimmering snow.\n\nNow Langmuir was talking in sweeping terms about their experiments outdoors and what would come next: deserts would bloom, storms would be quelled, snow would fall where people wanted it and not where they didn't. After the official paper was over, he spoke even more broadly. He told people about the flight of December 20 and the unusual snowfall that had followed. Cloud seeding, he was telling people, had created a spectacular snowstorm.\n\nHarry Wexler had studied meteorology at MIT, under the supervision of pioneering Swedish meteorologist Carl-Gustaf Rossby, and gone on to become an editor at the Journal of Meteorology and a member of the council of the AMS. He had joined the Weather Bureau in 1934, and he was now on a mission to bring the new, mathematically rigorous meteorology of the Scandinavians to the United States. He had an ally in the Weather Bureau chief, Francis Reichelderfer. Long a bastion of old-school weather mapping and forecasting, the government's weather service was undergoing a makeover under Chief Reichelderfer. He was going to drag it\u2014kicking and screaming, if necessary\u2014into the new scientific age.\n\nAs Harry and Chief Reichelderfer saw it, the facts were simple. The laws of physics underlay all weather, from the tropical hurricane to the sudden gust of wind. The planet spun, the sun shone, the tides rose and fell, the winds blew, all of it following basic equations of hydrodynamics that had been around for two hundred years. With enough study, it must be possible to come up with a mathematical model of the atmosphere that would enable accurate forecasts of its future behavior. This, after all, was the scientific revolution's great insight: that the universe could be described\u2014all of it\u2014in purely numerical terms. In other words, it was knowable, and if it was knowable, it was predictable.\n\nScientists had a name for this insight: the clockwork universe. In 1814, the French mathematician Pierre-Simon Laplace had imagined a creature intelligent enough to \"comprehend all the forces by which nature is animated... an intelligence sufficiently vast to submit these data to analysis.\" For that creature, \"nothing would be uncertain and the future, as the past, would be present to its eyes.\" Scientists had made up a name for this mystical being with perfect knowledge. They called it \"Laplace's demon.\"\n\nHarry Wexler had dreamed of being able to modify the weather. What meteorologist hadn't? But one of his favorite quotations was from Ben Franklin: \"He who would master nature must obey her laws. He must learn her laws and then obey them.\" If weather control ever became real, Wexler knew it would be the result of what he had dedicated his life to: learning the physical dynamics of the atmosphere. It would happen when humans had learned enough to become like Laplace's demon. That's what the atomic scientists had done: they mastered nature's laws to master nature. But here, at the AMS meeting, scientists who knew nothing about the mathematics of atmospheric circulation were announcing they could make snow\u2014when they didn't even know how nature made snow! It was as if the nuclear physicists of 1942 had been approached by a couple of chemists who claimed to have gone outside and smashed some atoms.\n\nThe GE scientists' revelations were the \"outstanding contribution\" of the meeting, Harry Wexler reported to Chief Reichelderfer. But they were meteorologically naive, and their approach was downright primitive. The real way to take charge of the climate was through something more rigorous\u2014like the ten nonlinear equations on punch cards Harry had right there in his briefcase. He had brought them to Boston to see if he could get some time on Harvard's high-speed calculating machine. Once the equations were solved, he would take them back to Princeton, New Jersey, where the Weather Bureau was taking part in an all-out mathematical attack on the weather. Because as Harry Wexler saw it, the future was not in experiments but in equations\u2014equations so complex they had to be worked out by an artificial brain. Laplace's demon might finally be coming to life, not at GE, or in any human mind, but in the tiny, electrically ignited glow of a vacuum tube.\n\n* * *\n\n\"I promise to scrub the bathroom and kitchen floors once a week,\" Kurt typed, \"on a day and hour of my own choosing. Not only that, but I will do a good and thorough job, and by that she means that I will get under the bathtub, behind the toilet, under the sink, under the icebox, into the corners; and I will pick up and put in some other location whatever moveable objects happen to be on said floors.\"\n\nThe baby wasn't even born yet, but things would never be the same. Jane missed going to class, and she had terrible morning sickness. She pestered Kurt so much about the household chores that in January he finally typed up a contract in which he promised, in addition to scrubbing the kitchen and bathroom floors weekly, to take out the garbage promptly, hang up his clothes, and refrain from putting out his cigarettes or dumping his ashes in wastebaskets. If he failed in these duties, Jane was \"to feel free to nag, heckle or otherwise disturb me until I am driven to scrub the floors anyway\u2014no matter how busy I am.\"\n\nAnd he was busy. With the baby coming, Kurt had taken on a job in addition to his schoolwork. He was moonlighting for the City News Bureau, the team of crack reporters who provided much of the copy for Chicago's papers. He wasn't experienced enough to be taken on as a reporter, so he was toiling away as a copy boy on the night shift. He'd come home for a few hours' sleep before going off to classes.\n\nHe was happy to be back in a newsroom again. But the move was driven more by necessity than desire. He was a family breadwinner now.\n\nIt was exhausting, but he still loved Chicago. It all seemed so relevant. A foolish fight might be raging in Congress over whether David Lilienthal was sufficiently anticommunist to head up the new Atomic Energy Commission, but Chicago thinkers were already past such ridiculousness and hashing out a draft of a world constitution. In the anthropology department too, Kurt felt as if he had found a sort of extended family. It was a small community working together to figure out big questions about people and their cultures. He wanted to be part of it, to be one of them.\n\nHe finished up his contract with Jane by declaring it effective until the baby was born, \"when my wife will once again be in full possession of all her faculties, and able to undertake more arduous pursuits than are now advisable.\" He was looking forward to that.\n\nSomehow Kurt managed to paint: he exhibited three paintings in an open student exhibition on campus. None of them sold. No matter. He bragged about the exhibition in a letter to Walt and Helen. The point of doing art was doing art. He was his own man now, finally freed from the hyperrational world of science to focus on what he really cared about: human beings.\n\nBolt of Lightning\n\nPeople said John von Neumann was a demigod. He had made a good study of Homo sapiens, declared his colleagues in Princeton and Washington and Los Alamos, so good he could impersonate our species with nearly perfect panache. Every so often, though, when he cut quickly to the heart of a mathematical quandary or began reciting from memory a book he'd read decades earlier\u2014in ancient Greek\u2014or when one totted up the sheer number of appointments and directorships and consultancies the man held, then the facade would slip. The dapper suits, the glittering parties, and the urbane aplomb that won him the nickname Gentleman Johnnie\u2014sometimes all of it just couldn't disguise the obvious: this creature could not be human.\n\nHarry Wexler liked him a lot.\n\nThey were alike, Harry and Johnnie. Both were Jewish mathematicians, Wexler the son of prosperous Russian immigrants, von Neumann from Hungary's Jewish upper class. Harry had taught at the University of Chicago and Aviation Cadet School during the war; von Neumann had taught in Berlin before joining the exodus of professors fleeing Germany in the 1930s. Harry found his home at the U.S. Weather Bureau in 1934, one year after Johnnie found his at the Institute for Advanced Study in Princeton, New Jersey.\n\nKnown as a \"paradise for scholars,\" the Institute for Advanced Study was founded in 1930. It's not a part of Princeton University but a private research institution located in the same town. Von Neumann became one of the initial five members, along with Albert Einstein and the mathematicians Oswald Veblen, Hermann Weyl, and James Alexander. Johnnie was the youngest of these luminaries, but he had already made vital contributions in the mathematical fields of game theory, measure theory, fluid dynamics, quantum mechanics, and ergodic theory\u2014the study of long-term average behavior in dynamic systems like traffic jams and weather. Like Harry, he had a reputation for asking searching questions and efficiently getting to the heart of a matter.\n\nBoth Harry and Johnnie felt comfortable working with the military. Von Neumann had done the math confirming the implosion method used in the Trinity test and in the plutonium bomb dropped on Nagasaki, but in contrast to many scientists he had no moral qualms about his work on the bomb. He had been present for the Trinity explosion and the nuclear tests on Bikini Atoll. Like his friend Edward Teller, he believed the United States should get on with building the \"Super\"\u2014a thermonuclear, or hydrogen, bomb. In fact, he was an advocate of preventive nuclear strikes: it was what game theory said a rational actor should do. But now, he told people, he was thinking about something much more important than bombs. He was thinking about computers.\n\nIn 1946, \"computer\" was generally understood to mean a human being\u2014usually a woman\u2014who sat with a slide rule or an adding machine and did equations. But although Johnnie was known to have an eye for pretty women, in this case he was thinking about something else: high-speed calculating machines\u2014the mechanical computers that would soon replace the human ones. Von Neumann had served as a consultant on the design of the world's first electronic general-purpose computer, ENIAC, but even before the Army received it in 1946, he was consulting on the design of an improved computer, EDVAC. Now he was refining the architecture for an even better one he planned to build at the Institute for Advanced Study. And that's where Harry Wexler came in.\n\nVon Neumann needed a problem he could give his new device publicly to demonstrate its chops. The military-funded machine would mainly be used to run calculations for the hydrogen bomb, but those calculations would be secret. Johnnie needed something he could talk up for reporters, and weather modeling would be perfect. He was an expert in hydrodynamics and was friends with the eminent meteorologist Carl-Gustaf Rossby. Past attempts to model the weather with numerical methods had failed for lack of computing power. John von Neumann's new machine would solve that problem.\n\nVon Neumann brought in a friend from RCA Labs, the engineer Vladimir Zworykin. Zworykin was a radio pioneer whose work would lay the foundation for television, but he also had an interest in weather. In October 1945, Zworykin issued \"Outline of Weather Proposal,\" a report for which von Neumann wrote an enthusiastic letter of support. The outline declared that advances in computing technology would now \"permit the prediction of [air] mass movements for perhaps several days in advance with results obtained within a few minutes.\" Once the weather prediction problem was solved, he wrote, attention could be turned to weather control.\n\nZworykin and von Neumann went to the Weather Bureau in January 1946, where they painted a dramatic picture of the climate tamed: rain made by seeding clouds with dust or chemicals, storms redirected with flamethrowers or atomic bombs, water temperatures raised by spreading oil over water, and air temperatures increased by painting the earth with carbon or aluminum to increase absorption or reflectivity. The ultimate goal, as the outline put it, was to \"channel the world's weather, as far as possible, in such a way as to minimize the damage from catastrophic disturbances, and otherwise to benefit the world to the greatest extent by improved climactic conditions where possible.\"\n\nThe first step was to develop a whole new approach to weather prediction. In the past, meteorologists had made forecasts by comparing present weather maps with past ones, assuming the atmosphere would behave much as it had before. Numerical weather prediction would be different. Von Neumann planned to divide the atmosphere up into a grid and collect as many data about each point in the grid as possible. He would then apply mathematical equations from thermodynamics to predict the likely future behavior of each point given the behavior of the points near it. It was a wholehearted embrace of the clockwork universe: turn the weather into an equation and solve for X.\n\nThe Institute for Advanced Study's Meteorology Project, funded by the Office of Naval Research in May 1946, was designed to take the \"first steps towards influencing the weather by rational, human intervention.\" By the time GE announced the manufacture of snow over Mount Greylock, Johnnie was assembling a meteorology team in Princeton to restate meteorological problems as equations in fluid dynamics, then feed them into his artificial brain. At the same time, he was gathering a team to build that brain. It was a method about as far from the GE scientists' approach as could be imagined. Instead of staring at the sky, the institute team would be staring at their calculations. Instead of conducting experiments in the natural atmosphere, they would program a machine to do math. But the goal was the same: figure out how the weather worked so that they could bend it to their will.\n\nThe Weather Bureau's man on the project was Harry Wexler.\n\n* * *\n\nBernie lowered his popgun into Vince's freezer and squeezed the trigger. He had bought the gun for seventy-five cents in the toy department of H. S. Harney, downtown. It went off with a pop that sounded like a firecracker or a fart.\n\nBeans, beans, the magical fruit:\n\nThe more you eat the more you toot...\n\nBernie loved that rhyme. He wasn't as big a fan of slapstick as Kurt and Alice, but like them he had an earthy sense of humor. Farting was funny. Poop was funny. Playing jokes was funny too, and the popgun was a kind of joke on nature. When he popped it, the cloud chamber filled with ice nuclei. It was just the kind of experiment he liked: a trick played on the natural world to make it yield up its secrets. It worked because the air in the popgun was under pressure. Compressing gases increases their temperature; decompressing them cools them. When released, the air in the popgun expanded rapidly, cooling the air around it, and the cold nucleated the ice. The same thing happened, Bernie found, when he opened a bottle of soda pop in the cooler or popped a bubble in a sheet of bubble wrap. He did it over and over, grinning every time, until Katharine Blodgett, a few feet away at her desk, thought she might go mad. But the lab's constant stream of visitors loved it.\n\nAt the meeting of the American Physical Society at Columbia University that January, Bernie, Irving, and Vince delivered a joint paper and demonstrated seeding with the freezer, using dry ice pellets, the popgun, a bursting balloon, and a hot wire carrying silver iodide. The New York Times ran a story about their paper the next day, pointing out that \"work has not progressed far enough to permit predicting actual uses.\" GE was now insisting that they be much more circumspect in explaining what they were up to.\n\nSince the first media frenzy, Research Lab director Guy Suits had been worrying about legal entanglements. Now every time something strange happened with the weather, GE was suspected of interfering. So many clippings credited Vincent Schaefer with strange weather phenomena that when the director of the Berkshire Museum was injured slipping on a patch of ice, someone sent Vince the clipping, noting that for once he wasn't blamed.\n\nBut Suits had a plan for limiting liabilities. In February, he sent Bernie, Vince, and Irving copies of their new contract with the government. GE would no longer be engaging in airborne experiments to modify the weather. Instead, flight experiments would be \"conducted by the government, using exclusively government personnel and equipment, and shall be under the exclusive direction and control of such Government personnel.\" The lead agencies would be the U.S. Army Signal Corps and the Office of Naval Research. GE employees working on the project were merely advisers and were to \"refrain from asserting any control or direction over the flight program. The GE Research Laboratory responsibility is confined strictly to laboratory work and reports.\"\n\nTheir project was being handed off to the military. But no one was terribly upset. The experiment was the heart of the matter, and for it to go forward, the lawyers needed to be appeased. The idea that cloud seeding would be used as a weapon didn't seem real, the military men around them just an irritating condition of work. Besides, there were advantages. The Signal Corps had smart scientists and even better airplanes than GE. And they were all given raises when the government contract went into effect. For Bernie, it was his second raise that month. They discussed a code name for the program. Vincent suggested \"Project Cirrus.\"\n\nThey all applied for security clearance. President Truman had just announced a new loyalty program for federal workers mandating FBI investigations of all government employees and making it possible to dismiss any federal worker who could be shown to have \"sympathetic associations\" with organizations deemed communist, fascist, or subversive. Just to be safe, Irving resigned his sponsorship of the National Council of American-Soviet Friendship. Membership in such organizations could get one investigated by the House Un-American Activities Committee, as was happening now to the physicist Edward Condon, whom Truman had nominated to head the National Bureau of Standards. The former Manhattan Project physicist was a follower of the suspiciously \"revolutionary\" theory of quantum mechanics, and had come out strongly in favor of civilian control of atomic energy. Now he was undergoing a direct personal attack by Representative J. Parnell Thomas, the chairman of HUAC.\n\nThat wasn't going to happen to Irving. His resignation letter struck all the right notes, disavowing his previous positive attitude toward Russia, equating \"appeasement\" with \"national suicide,\" and noting that unless the United States got tough, the Russians would build atomic bombs. He did not mention that he himself had told Congress that Russia would inevitably have atomic bombs within five years. For all his absentmindedness, Irving was no political novice.\n\nBernie didn't have to worry about such things. He had his political views, but he kept them to himself. He was more of a behind-the-scenes innovator, not someone who put himself forward, who got asked to put his name on the letterhead of organizations like the National Council of American-Soviet Friendship.\n\nGE and the Army Signal Corps jointly issued a press release announcing the creation of Project Cirrus. In keeping with their new publicity aims, the announcement focused on more modest effects than the ones Irving liked to tout: it mentioned dispelling fogs and clouds over airports, noting that only after much more research might the work lead to \"manipulation of gigantic natural forces for the benefit of mankind everywhere.\" GE made sure to note that its scientists would \"provide advice and instruction, but [would] not take part in the flight program.\"\n\nAnd then things returned to what passed for normal at the House of Magic: Vincent continued making flights, Langmuir continued working out equations, and Bernie built a generator on the roof of Building 5 to dispense silver iodide into the Schenectady sky.\n\n* * *\n\nAs the baby's arrival approached, Kurt amped up his efforts to find a \"real\" job. He and Jane might fantasize about writing for a living, but with a baby he needed a paycheck and security. He'd applied for a job writing catalog copy for Sears & Roebuck\u2014getting as far as being given a tour of its Chicago offices\u2014but no luck. In the solicitation letter he sent to a range of possible employers, he described himself as about to receive a master's degree from Chicago, \"twenty-four, married, and the father of a very young child.\" He mentioned his service in the war and noted that his anthropology work had been \"extremely satisfactory from a personal standpoint.\" His grasp of human relations, he said, should make him valuable as a \"personnel or labor relations man.\"\n\nHe got back some positive replies. He'd been offered a couple of copywriting jobs at ad agencies. He could continue on as a reporter. He had an offer of a teaching post in a private school and another for an editorial spot at an educational publisher. All he had to do was finish up the degree. So, as Jane's due date approached, he hammered out a thesis plan. Called \"A Comparison of Elements of Ghost Dance Mythology with That Mythology of a More Tranquil Period,\" it proposed comparing late-nineteenth-century Native American uprisings with the work of Cubist painters in Paris in the early twentieth century. To his mind, it was bold and original, the kind of work other people in the department did: drawing connections among cultures without suggesting one was better than another, striving to find the elements that made human beings in one time and place much like those in another.\n\nOn May 11, their son was born. Mark, named after Mark Twain, came squalling into the world right around the time the University of Chicago anthropology department rejected his father's thesis proposal.\n\n* * *\n\nOnce again, Bernie had to get out his popgun. The demonstration always amused him, but lately it sometimes felt as if they spent more time glad-handing visiting dignitaries than doing science. On this bright August day, a team of high-ranking Army and Navy men had come to Schenectady for the official launch of Project Cirrus. They got the tour and the cold-box demonstrations that were beginning to become a little like a dog-and-pony show. But the military brass loved it. Weather was almost like a new toy for them. As Rear Admiral Luis de Florez had written in The American Magazine the previous September: \"We must consider the possibility that man-controlled weather can become a terrifying weapon. We must reckon with rivers and lakes as potentially terrible enemies. Sea disturbances\u2014call them man-made tidal waves\u2014may well be a factor in the next war... America must expand the scope of her thinking if she is to retain her position among nations in the world of tomorrow.\" The article was titled \"Weather\u2014the New Super Weapon.\"\n\nThat wasn't the kind of application Bernie hoped to see their work producing. He would rather see it used to save lives. That's what he'd tried to do earlier that month, when a telephone call came into the lab asking Project Cirrus to help put out a forest fire in California.\n\nThe fire had started in early August, in the tinder-dry brush north of Pasadena. Within a day, warm southwest winds had swept the flames through Big Tujunga Canyon, where they consumed more than three thousand acres. On August 6, a forest service employee and a volunteer were killed as they frantically worked to make a firebreak. Residents were evacuated, highways were closed, and the Navy dispatched 280 seamen to help. Still the supervisor of the Angeles National Forest went on the radio pleading for more volunteers. As many as 800 people were fighting what was called the worst wildfire in a quarter century when someone thought to call GE.\n\nFrank Backus of the Los Angeles GE office was on the line when Bernie got to the phone. The Los Angeles Times had lined up a DC-3 and four hundred pounds of dry ice. What should they do now?\n\nBernie had tried to explain how to seed a cloud. But even as he talked, he knew it was futile. The problem was the clouds were low in the sky. The plane was above them at fourteen thousand feet, meaning that in California's climate they were probably too warm to contain supercooled water. Still, the flight crew wanted to try. They made five passes over the cloud deck, dispensing 60 to 120 pounds of dry ice each time, as per Bernie's instructions. Unfortunately, the cloud, reported the Los Angeles Times, \"failed to cooperate.\"\n\nThe effort was unsuccessful, but it had illuminated the high stakes of getting this right. Once Bernie got silver iodide to work in the natural atmosphere, it would be easier than ever to bring water and life, instead of fire and death, down from the clouds. At least that's what Bernie was hoping for. The Signal Corps colonel and the Army brigadier general visiting the lab might have had something else in mind. Perhaps they were thinking of what General George Kenney, of the Strategic Air Command, had recently told the graduates and alumni of MIT: \"The nation that first learns to plot the paths of air masses... will dominate the globe.\"\n\nBernie lowered his popgun into the freezer and seeded the miniature cloud. This time, it was for the News Bureau photographer too. The PR men never missed an opportunity to show GE in the nation's service, and the colonel and the general looked impressive, their lapels paved with rows of colorful ribbons. In the photograph sent to newspapers, Bernie would be smirking as he popped the toy gun. The officers would be smiling slightly too as they leaned forward to look into the freezer. The bursting flashbulbs would catch something slightly acquisitive in the military men's expressions, perhaps an eagerness for a time when the freezer would be replaced by the sky and the gun they aimed would be real.\n\n* * *\n\n\"To Walt and Helen.\" Kurt and Jane clinked martinis. Walter had graduated, and he and his family had moved to the San Juan Islands off the coast of Washington. They were going to be homesteaders, building their own cabin, growing their own food, probably raising sheep and knitting sweaters from their wool. It was an admirable escape from everything\u2014money pressure, school pressure, job pressure. Kurt and Jane missed them, and they envied them a little too.\n\nNext they raised their glasses to the UN Atomic Energy Commission, which had been spending August hammering out plans for international control of atomic energy at the UN's temporary home in Lake Success, New York. Two days earlier, the United States, Great Britain, China, and France had given the plan their support. Kurt and Jane let themselves enjoy a moment of optimism about a peaceful future, free from fear of the bomb. It was August 31, the eve of their anniversary, and they were celebrating at the restaurant Jacques. It was an overpriced French joint, with a mural of the Eiffel Tower and waiters sporting fake French accents, but they needed a treat. They sat outdoors on the famous flagstone terrace and contemplated possible futures.\n\nMark was now almost four months old. But the joy of a new baby was somewhat undercut by the financial anxiety that arrived as predictably as night feedings and dirty diapers. Kurt's job as a copy boy at the City News Bureau paid next to nothing. And because the anthropology professors hadn't gone for his comparison of Native American uprisings and Cubist painters, he'd had to spend the summer cooking up a whole new thesis project\u2014with money from the GI Bill running out.\n\nThe department's unanimous rejection of his thesis proposal had been galling. It was as if they were suggesting he was no better at this than he had been at science. As if they considered him third-rate.\n\nHe'd gone back to his writing. While working on a new thesis outline, he had fired off stories and essays to magazines. Finally, he'd finished the new proposal and in late summer had submitted it to the department. In response to their earlier comments, it was much more modest. His new project took as its starting point the philosopher Georges Sorel's claim that periods of social change cause the rise of new mythologies. Looking at Native American religious and spiritual movements in the late nineteenth century, he was going to analyze what effect the rapid assimilation caused by white colonist expansion had on the Indian myths. It was less ambitious, less creative, less out there than his previous effort.\n\nThe department didn't hate it. Some of the professors even said encouraging things. With some more effort, the message was, he would get this right. But in August, he was offered the opportunity to become a reporter\u2014full-time\u2014at the City News Bureau. He took it. He'd had enough of the eggheads humiliating him. Pounding the streets looking for good copy\u2014anything involving a dead body was best\u2014that was something he felt confident he could do. He could be a reporter and keep working on his own essays and stories. Somehow he would also find time to work on his thesis.\n\nBut August brought a relentless flood of rejections. \"Wailing Shall Be in All Streets\" had been rejected six times. \"Brighten Up!\" was rejected by Glamour on August 11. He sent it off to Charles Angoff at The American Mercury, reminding him of his encouraging note a year earlier. Angoff returned it a week later, just a couple of days after \"I Shall Not Want\" was rejected by Coronet.\n\nHis writing was getting nowhere. And the City News Bureau job was grueling, all wrong for a family man. He was exhausted, and he and Jane hadn't had a Sunday together in weeks. It was time to start considering other options. So over martinis and pricey French food, they talked through the possibilities.\n\nA job offer from The Dayton Daily News was tempting, because Kurt loved being in a newsroom. But there was also an offer from the educational publisher Bobbs-Merrill in Indianapolis. There was a lot of appeal in the idea of going back to Indy. Their kids could grow up near their grandparents and with the kids of their old friends. But the couple would have to face the stifling social life of a city where family connections had carved out roles for them already. Besides, a new option had recently come into the picture: Schenectady and GE.\n\nEarlier that week, Kurt had gotten a phone call from George Griffin at the GE News Bureau. He was responding to a letter Kurt had sent at Bernie's suggestion. Bernie had already told George his younger brother might be a great hire for the News Bureau: he had a science background and real newspaper experience. The GE News Bureau was a publicity department, but it aimed to be as much like a real newspaper as possible\u2014minus the objectivity, of course. In exchange for giving up journalistic credibility, one got the security and perks of a corporate position.\n\nThe GE job paid better than the others. It was in Schenectady, not too far from New York City, the center of the publishing world. And it was also near Bernard. As much as Kurt chafed under his brother's influence, he also longed to be surrounded by family. Their kids could play together. Their wives would have each other for support. It was a new town and a new start, but with Bernard there it wouldn't be lonesome. For the first time, they wouldn't have to worry about money. They could buy a house. And without having to work nights and weekends, maybe Kurt could get some real writing done.\n\nJane agreed it was a good opportunity. The GE recruiter dropped by their house the very next night, as they were still basking in the mellow martini haze of their anniversary dinner. It was a great job with a real future, he assured Kurt. Of course, GE only hired college graduates. Kurt assured the recruiter that wouldn't be a problem. He would soon be in possession of a master's degree from Chicago. He was still planning to take his exams and write his thesis. He'd just have to do it while working.\n\nTo take the job at GE would be once more following his brother's plan for his own life. But Bernie was only acting in Kurt's best interests. Why resist?\n\nHe told the GE man yes.\n\nEye of the Storm\n\nBernard was treating Kurt like a prince. First he had helped him find a nice house\u2014two bedrooms and an alcove where he could have an office\u2014right around the corner from Bernie and Bow in Alplaus. Now he was taking Kurt appliance shopping. GE employees got discounts on the best new models\u2014another reason the company was known as Generous Electric.\n\n\"I own a home now,\" Kurt wrote proudly to his father. \"Albeit humble, it's ours, and we'll love it I'm sure.\" He was in an expansive mood, because by cashing in some bonds, he'd been able to handle the $7,000 price tag without taking on debt. He felt manly enough to sign his telegram to Jane, as he did whenever he felt plucky, \"Tarzan.\" Jane was still in Chicago packing up their furniture, which they would store in Bernard's barn until they could move into their own house.\n\nAt the employee store of the Generous Electric Company, Kurt ordered a refrigerator, a stove, and an automatic washer that he was convinced would make Jane's life easier. In his letter to his father, he called it a \"Bendix.\" He hadn't yet realized that in Schenectady, you didn't call a GE refrigerator a Frigidaire or a GE washer a Bendix. GE expected every one of its employees to serve as brand ambassadors for the company.\n\nAnd what a company! GE was bringing in more money than ever before. In 1947, the company's net sales topped $1.3 billion, more than double the sales of the peak prewar year, 1940, and rapidly rising toward Charlie Wilson's magical number of $2 billion. World War II had been good to GE. The company had produced more for the war than anyone else, cranking out $4 billion worth of what America needed to win. Aircraft turbochargers! Bazookas! Howitzers! Gun turrets! Turbines! Three-quarters of the Navy's propulsive power! Radar detection equipment, gun directors, generators, and electrically heated flight suits for pilots! Searchlights so strong a man eight miles away could read the newspaper by them! The company had met the war's more quotidian needs too: fuses, wires, and, of course, the start of it all, lightbulbs. A B-29 Superfortress bomber carried 170 electric motors, 26 motor-generator sets, and 15,000 feet of electric wiring, all of it made by GE.\n\nThe company put out a book celebrating its war effort in 1947. Men and Volts at War described the Schenectady Works as \"the nerve center of one of the world's biggest and most complex war machines.\" Postwar, the company was still a war machine, a paragon of the new and powerful alliance between industry and the military. When Kurt arrived, GE was building the Knolls Atomic Power Laboratory, a nuclear research facility it was going to manage for the government, next door to its new state-of-the-art campus for the GE Research Lab. It was already operating Hanford, the government nuclear production complex in Washington.\n\nGE's non-war-related industries were surging ahead too. The company was building a huge turbine construction facility in Schenectady, to keep pace with the nation's increasing demand for electricity. It had just completed three new chemical plants and a top-of-the-line electronics facility in Syracuse and broke ground on a new induction motor plant in San Jose, California, and an artificial-lightning research facility in Pittsfield, Massachusetts. But consumer products were where the real growth was. A GE stove and refrigerator in every kitchen! A GE washing machine in every house! A GE toaster on every counter and of course GE lightbulbs in every socket, from Park Avenue to Skid Row and everywhere in between, lightbulbs issuing forth from its five new lamp plants, lightbulbs that lasted longer and shone brighter and grew cheaper all the time, because as the company motto said, progress was its most important product. Progress! Ever onward GE marched. It had armed the nation for war, and now it was equipping America for peace, for the electrically lit and atomically powered prosperity that know-how would deliver next.\n\nAs a \"junior writer,\" Kurt would, for the first time in his life, be making enough money to support his wife and son in true American style: a house, a yard, a pension plan, a kitchen full of GE appliances, the monogram on each one shining like a promise of the best of everything. And at least his job would be writing. He knew how to craft news stories, to dig up information, and to shape it into a magazine feature or science story. And that was exactly the skill the News Bureau wanted.\n\nFounded in 1919, the GE News Bureau was a novel approach to corporate publicity. Instead of issuing standard press releases\u2014thinly disguised ads, accompanied by unappealing illustrations\u2014GE hired professional reporters to pitch newsworthy stories, including layouts with illustrations and photographs. A News Bureau story wasn't just press agent puffery. It was a ready-made piece of journalism, with a hook, supporting information, quotations from experts, and a sense of the big picture.\n\nBy the time Kurt arrived, the News Bureau's light blue envelope had become a welcome sight on many editors' desks. The department had originally been located downtown, to emphasize its separateness from the advertising arm, but not long after Kurt arrived, the bureau moved into the Schenectady Works. There, it became part of the company's increasingly impressive corporate communications wing, the largest corporate publisher in the nation. GE produced internal publications like the Schenectady Works News (for blue-collar workers), the General Office News (for white-collar employees), and the Monogram (for managers). There were also internal publications that were distributed beyond the company: the GE News Graphic (for appliance salesmen), the GE Review (for engineers), and the GE Digest (for overseas personnel). It produced a magazine for young people and well-packaged scientific reports. GE didn't just manufacture goods: it mass-produced ideas.\n\nHis new bosses gave him a physical. They gave him a desk. They gave him a typewriter of his very own! They gave him a beat: the Lighting section, the Service Engineering section, and the Schenectady Works Research Lab\u2014a lab less prestigious than the GE Research Lab, where Bernie worked. Get out there and find stories that would get the GE name in front of America's eyeballs; that was his mission. Stories about things like a set of mechanical hands for working on radioactive materials, or the GE sales force refrigerator named Junior that walked and talked\u2014and even once got arrested in Cleveland for whistling at pretty girls. It wasn't like covering thirteen police stations, two Coast Guard stations, and a fire department, and he was glad of it. The work would be easier. He could handle it and get his thesis done too. Maybe he'd even be able to spend some more time doing his own writing.\n\nSo Kurt joined the human tide of workers arriving at the plant every weekday, flowing in four lanes of bus and car traffic down Erie Boulevard to the Works, a sea of fedoras and overcoats surging through the huge plant gates. When he walked through the entrance, Building 5, where Bernard worked, was on the right. Directly across the street, Kurt turned left, in to Building 6.\n\nThe press room of the News Bureau looked like the newsroom at any paper. Desks were shoved up against each other and strewn with papers, ashtrays, pencils, beer cans, Remington manuals, and rotary phones. A table along one wall held a row of portable typewriters, to be grabbed for jobs on the go. A pin board above that was pasted with eight-by-ten publicity photographs. The fact was not lost on Kurt that many of them featured the impish grin and distinctively cumuliform pompadour of his older brother.\n\n* * *\n\n\"A hurricane is a complicated thing,\" Vincent told the reporters, \"not the simple whirl of an artist's conception.\"\n\nVince was grasping at straws. He was exhausted. He'd flown to Florida and back in the last four days, a twenty-six-hour round-trip. He'd barely landed in Schenectady before GE hauled him to a press conference and reporters started pelting him with questions. What they all wanted to know was, what just happened?\n\nThe problem was, Vince couldn't tell them. It wasn't that he didn't know. He wasn't allowed to tell them. There'd been too much talk already; GE was in damage-repair mode.\n\nFrom the very start, Irving, Bernie, and Vince had dreamed of cloud seeding hurricanes. Once Project Cirrus was formed and their experiments were officially attributable to the government, it began to seem possible. GE was insulated from liability concerns by a solid wall of military brass. The team had access to military airplanes and pilots who wouldn't balk at flying into huge storms. They had a steering committee that included meteorologists from the Navy and the Army Signal Corps and bosses eager to try out this new tool. Finally, they were in a position to attack a real hurricane and see if they could change its course.\n\nHurricanes begin when warm water evaporates from the ocean surface. As the water condenses in the air, it releases latent heat. This heat builds up, and the warm air rises, creating a low-pressure area. Air rushes into this depression, creating winds. Because of the Coriolis effect\u2014a slight deflection of moving objects caused by the earth's rotation\u2014the winds begin to spiral inward toward the center of the low-pressure region, clockwise in the southern hemisphere, counterclockwise in the northern. As the storm spirals, it pulls in more water vapor, which condenses and creates more latent heat. Eventually, bands of showers and thunderstorms begin to form. The most intense band, the eye wall, encircles the calm eye at the center. Hurricanes spiral their way with the prevailing winds until they reach colder water or land where there is no more warm water vapor to serve as \"fuel.\"\n\nHurricanes are the most vicious storms on earth, causing billions of dollars' worth of destruction, along with injury and death. An average one contains hundreds of times more energy than an atomic bomb, a fact Irving Langmuir liked to repeat. The military had upped its hurricane research program during the war, beginning a regular program of flying airplanes into hurricanes to collect temperatures and wind speeds as an aid to forecasting. But the Project Cirrus scientists were convinced they could do better than studying the storms. They wanted to alter them.\n\nEveryone agreed that \"busting\" a hurricane with dry ice seemed unlikely, given the enormous amounts of energy involved. But diverting one\u2014changing its course just enough to send it harmlessly out to sea\u2014that could be possible. Especially if they cloud seeded it when the hurricane was young, before it had built up an elaborate storm system. After all, a giant boulder rolling down a mountain could change its path after hitting the tiniest pebble. Like Steinmetz marking his chalk X on the faulty generator, all they had to do was figure out where to put the pebble. The steering committee drew up an official plan for the experiment, stating its goal as determining whether seeding could \"modify the normal growth and development of tropical storms.\" And then the scientists waited for the right hurricane to come along.\n\nSeveral promising hurricanes hit in September, but the team hadn't lined up all its planes. Finally, in October, a good-sized storm headed up through the Caribbean toward Florida. The Weather Bureau didn't yet have a protocol for naming hurricanes, but the Project Cirrus team had been christening the storms in alphabetical order. In October, a team flew to Florida to cloud seed Hurricane King.\n\n\"We accomplished our purpose,\" Vince told the reporters in Schenectady, \"to obtain scientific data on the hurricane.\" It was what he had been told to say. \"We never had any plans or hopes to break up the storm.\"\n\nThat's not what The New York Times had reported just days before. As Hurricane King spiraled toward Florida, the paper reported that Project Cirrus was planning to spray a thousand pounds of dry ice into the storm to see what effect it would have. \"The result of this will lead us to further accomplishments in efforts to break up a hurricane,\" Navy meteorologist Daniel Rex of Project Cirrus told the Times as the Project Cirrus team was en route to Florida.\n\nFor more than a month, in fact, newspapers had been full of reports that Project Cirrus was going to attempt to bust a hurricane. \"The next great tropical storm to whirl through the Atlantic,\" reported the Associated Press, \"will be dusted with carbon dioxide (dry ice), silver iodide or some other cooling crystal which\u2014perhaps\u2014will condense its water vapors, cause a record-smashing rainfall at sea, and dissipate the swirling air mass.\" The giddy reporting caused some anxiety at Project Cirrus. It sounded great, but what if they didn't deliver? Or worse, what if something bad happened? The steering committee composed a press release insisting that \"contrary to earlier, unofficial reports, plans do not call for any 'hurricane-busting' attempts.\"\n\n\"We are in far too preliminary a stage to think of stopping a hurricane,\" Langmuir was quoted saying in the release. \"At this point, we are only interested in seeing and recording any effects the dry-ice technique will have.\"\n\nNevertheless, when the Project Cirrus team stayed overnight in Mobile, Alabama, en route, Vince awoke to the local newspaper headline \"Hurricane-Busting Plane Lands in Mobile as Storm Approaches Miami and Florida East Coast.\"\n\nBy the time the Navy plane arrived at MacDill Air Base, near Tampa, Hurricane King, traveling in a northeasterly direction, had already crossed the tip of Florida, leaving a trail of devastating floods across the Everglades and the southeast coast. According to the Weather Bureau, it was heading out to sea in the Atlantic Ocean and weakening quickly. Thinking it might be too small to bother with, the team debated whether to return to New York. But hurricane season was almost over, so they decided to carry on with the experiment.\n\nThere were three planes. Kiah Maynard, a junior team member recently recruited by Bernie, was on board the seeding plane, a B-17 flying just above the cloud deck. The second plane, flying higher up, carried military photographers. Vince was on the third plane, a B-29, which stayed eight thousand feet up and fifteen to twenty miles back, putting him in the best position to see larger patterns and changes.\n\nWhen they caught up with the hurricane, about 350 miles northeast of Jacksonville, they realized immediately that the Weather Bureau had been wrong. The storm was not weakening but maintaining its fierce spiral, a large, intense squall line at its outer edge. It was chugging steadily out to sea on an east-by-northeast course.\n\nThe seeding plane dropped eighty pounds of dry ice along a 110-mile-long track parallel to the squall line and another hundred pounds into the top of a large cumulus cloud boiling up from the storm. At first, it looked as if this produced little or no effect. But as the planes circled to take more photographs, observers saw a new line of rain falling parallel to where they had seeded. It looked like a trench shoveled out of the hurricane's spiral of clouds. Commander Rex estimated that about 300 square miles of storm had been affected. They took 250 photographs and flew back to Florida.\n\nThe next morning, Vince settled into his seat on the plane home, notebook in hand. He had thirteen hours in the air and was planning to use the time to record his observations. But not long after the B-17 was airborne, it began to buck and rattle. The turbulence increased until it was the most intense buffeting any of the passengers had ever experienced in the air. Vince's handwriting grew steadily worse, until writing was out of the question.\n\nLater, they found out the reason for the rough skies. Immediately after they had seeded it, the hurricane made a dramatic dogleg turn. Inscribing a huge 7 over the Atlantic Ocean, it turned westward and plowed back to the mainland. It was battering Savannah with eighty- to one-hundred-mile-an-hour winds when they flew over. As the unknowing scientists clutched their armrests in the sky above, a thousand people were taking shelter in Savannah's city hall. Windows were smashed, boats were tossed ashore, sugarcane fields were flattened, a downtown bank lost its roof, and several parked planes were destroyed. Charleston took a hit too, with waves topping its seawall, parts of the city flooded, and a lumberjack killed by a falling tree. All in all, damage from the storm's second landfall was estimated at $23 million.\n\nBy the time the team landed in Schenectady, a storm of outrage was following in the hurricane's wake. A letter writer in the St. Petersburg Times declared that locals blamed GE and were \"pretty sore at the army and navy for fooling around with the hurricane.\" A Miami weatherman called the diversion a \"low Yankee trick.\" The sheriff of Savannah declared that if anyone from Project Cirrus showed up in his town, he'd throw him in jail.\n\nWhich is why GE had hurried Vince into a press conference: the company needed him to convince reporters that Project Cirrus was not responsible for the storm's change of course.\n\n\"Could you see any effects?\" reporters demanded. Vince told them he was not allowed to say; the Army had classified all the reports. They asked him if he thought that the team's experiment had caused the storm to alter its course. He said no. Reporters were not convinced. Hurricanes just didn't behave like this one had, heading one way, then suddenly making a hairpin turn. Even the fiercely loyal Schenectady Gazette said the question had to be asked: \"Did science, for the first time in history, 'bust' a hurricane on October 13?\"\n\nIrving Langmuir was certain that it had. And he was elated. The redirection of the hurricane proved that he really was on the verge of something huge. His dreams of widespread weather control were not pipe dreams at all. He couldn't wait to tell the world\u2014even if GE was insisting he keep it quiet.\n\nHe seemed oblivious, however, to another implication of the hurricane experiment, one that would come to haunt Bernie more and more: the possible busting of Hurricane King was the first suggestion that you didn't necessarily have to be planning to make a weapon in order to make one.\n\n* * *\n\n\"Perhaps you saw the news reports that mentioned Bernard as having been aboard the hurricane-busting B-17,\" Kurt wrote to their father. \"This was a newsman's mistake. Bernard got into the farewell picture but not into the plane.\"\n\nHe enjoyed being able to report on his brother's activities. But that wasn't the main reason he had set up an office for himself at home. His plan was to make progress on his master's thesis. He had hoped he might even take some trips to New York City, where he could work at Columbia's Butler Library. It weighed on him that he had told the recruiter he would have his diploma before starting, as GE required for white-collar jobs. Without a college degree, he didn't deserve his position or his paycheck. What would happen if somebody in Schenectady found out he lacked credentials? This wasn't the kind of town that laughed things like that off.\n\nSo he set up a desk where he could work on his thesis. And then he sat down at it and began writing stories.\n\nIt was another secret, this desire to write, his ambition as confidential as his un-degreed status. GE was better off not knowing that one of its junior writers had aspirations beyond making GE look good. Nor did his new colleagues need to be made aware that he was different. Methodically, he went to work and found GE stories and typed up GE press releases; he bantered with his fellow writers and took part in Monday meetings. But at home, he was just as methodical in figuring out how to break into the magazine world as a writer of fiction.\n\nHe knew he had to write what editors wanted. Clearly, they didn't want his war stories. So he turned his mind toward figuring out what they did want. As always, Jane helped. They studied the magazines together, trying to determine where he had the best shot. Like any aspiring author, Kurt longed to see his byline in The New Yorker. To get in there was to join the pantheon of Writers Who Mattered. But there were other periodicals that also bestowed literary prestige: Story, The Yale Review. Then there were the popular magazines that were still highbrow: The American Mercury, The Atlantic, Harper's. Beneath those were the \"slicks,\" the general-interest glossies that published scads of short fiction: The Saturday Evening Post, Tomorrow, Collier's, Life, Esquire. They liked their stories punchy and quick, with surprising plot twists or unexpected revelations right at the end. In the 1940s, all the magazines did, even The New Yorker. Before television, fiction was popular entertainment.\n\nAt the bottom of the ladder were the \"genre\" magazines, pitched at readers who liked a specific kind of story: detective tales, westerns, science fiction. Magazines like Spicy Detective, which Kurt read as a kid, and those he had never read, like the women's magazines: Redbook, Mademoiselle, Cosmopolitan, Glamour, McCall's. To appeal to editors there, stories had to feature things women cared about: relationships, family, love. He and Jane bought copies of those too and read them, analyzing the fiction they published.\n\nKurt was not above shaping his stories to an audience, even an audience of women. He wrote stories that were classic women's magazine fodder: one called \"Ruth\" about a young war widow coming to terms with the possessive grief of her mother-in-law, another called \"City\" that narrated the thoughts of a man and a woman who cross paths at a bus stop and long to speak to each other. He wrote a novella called \"Basic Training\" about a young orphan who goes to live with his cousins in the country and falls in love with one of them, disrupting the harsh militaristic rule of their father. He submitted them all to women's magazines, but they rejected them, like everyone else.\n\nEven as the negative responses piled up, Jane's confidence never wavered. Kurt tried to let her faith in his genius buoy him as it did her. They put candles in three miniature wine bottles, one for \"keep,\" one for \"on,\" one for \"trying.\" It had become their unofficial motto. He felt bolstered when the rejections were at least encouraging. Sometimes an editor wrote a brief note at the bottom: \"Sorry, not for us, but try again!\"\n\nJane had always said he was really a writer. He had insisted that he was just a regular Joe. Now, as he pretended to be a regular Joe, he was beginning to realize it wasn't what he wanted to be at all.\n\n* * *\n\nBernie left Building 5, merging into the mass of people flowing out Gate 85. He went to the parking lot and joined the glacier of cars inching out of the Works and down the blazing \"white way\" of Erie Boulevard, the enormous artificial moon of the GE sign behind them. Schenectady was a bustling town, especially with the holidays approaching. Shoppers were rifling through Harney's, Sears, and Carl's, revelers were spilling out of bars and restaurants, the four downtown movie palaces were ablaze, and a man dressed as Mr. Peanut was coaxing hungry shoppers to stop for a snack. Bernie barely noticed any of it. He was looking forward to his latest nighttime pursuit.\n\nAs he walked into his house one early winter night, it had occurred to Bernie that the moisture in the cold, clear air must be supercooled, just like water in a cloud. He went inside, got an electric heater, doused it with silver iodide, and turned it on. When he went outside and waved it around, he could see a fog of ice crystals materializing around it. It was more proof of how effective silver iodide was.\n\nHe couldn't get far enough with the electric heater, so he put some silver iodide on a newspaper and stuffed it into an oil burner. Then he went for a walk. It was beautiful. The fog crystallized around him like smoke, drifting from his hand and down through the dark village streets. Now, whenever it was cold enough, he experimented outside. Often their new junior teammate Kiah Maynard would come over, and the two of them would make fog, then get in the car and drive around to see where the wind took it. Sometimes they found it as much as a mile away.\n\nTonight, however, he was alone. Everything was quiet: The grocery store was closed, the post office at the back of the bike shop shuttered. Even the firehouse was dark. The comfortable houses of Alplaus gave off the soft glow of families at home having dinner, couples listening to the radio, kids taking their baths and going to bed. Outside, Bernie walked alone, fog streaming behind him like a wake.\n\nShortly after Bernie got home, his phone rang. It was his neighbor John Fisher, a fellow GE employee and a friend; John's wife, Jo Ann, was in the GE Research Lab Newcomers' Club with Bow.\n\n\"You fooling around?\" John asked.\n\n\"Yes,\" Bernie said. \"Why?\" John explained that he had looked out his window and noticed that even though it was a clear night, the house next door to his was shrouded in a thick miasma.\n\nYes, Bernie told John. He had made that fog.\n\n* * *\n\nKurt rolled a piece of paper into one of the News Bureau typewriters and went to work. He was answering a request for a photograph. The photo was of Bernie.\n\n\"The photograph of General Electric's Dr. Bernard Vonnegut originated from our office,\" he typed. However, he informed the requester, there were no more prints, and the negative belonged to the Army Signal Corps. \"Moreover,\" he typed, \"we have a lot more to do than piddle with penny-ante requests like yours.\"\n\nHe'd only been at GE for about a month when the letter had landed on his desk. It was from his own uncle Alex, sent originally to the Schenectady Gazette. Alex had seen a photograph of Bernard in his hometown paper, with a photo credit from the Gazette. He wrote asking for a copy, explaining that he was \"a wee bit proud\" of his famous scientist nephew. He enclosed a dollar for the newspaper's trouble. The Gazette passed the request on to the GE News Bureau, source of the photograph, and George Griffin\u2014out of courtesy or mischief, who can say\u2014gave the job of responding to Kurt.\n\nHe must have snickered to himself as he typed. \"We do have some other photographs of the poor man's Steinmetz,\" he wrote, \"and I may send them to you in my own sweet time. But do not rush me. 'Wee bit proud,' indeed! Ha! Vonnegut! Ha! This office made your nephew, and we can break him in a minute\u2014like an egg shell.\"\n\nIt was true too; Kurt could see that now. The News Bureau was HQ for the corporate boosterism that poured down on Schenectady as steadily as the Works soot. The scientists were celebrities because the News Bureau promoted them as such. In other words, it was Kurt's office that made people like Bernie into stars.\n\nSurely, he thought, Uncle Alex would realize it was all a joke\u2014especially when he got to the final lines, where Kurt closed by returning the dollar, pointing out that \"one dollar to the General Electric Company is as the proverbial fart in a wind storm.\" He signed off \"Guy Fawkes: General News Bureau.\"\n\nThere actually was a guy named Fawkes in the News Bureau, but Guy Fawkes\u2014surely Uncle Alex wouldn't think someone would actually be named after a notorious English traitor. He sent the letter off, thinking it was a hilarious family prank. Later, he would find out that Uncle Alex had been outraged. He even consulted a lawyer, asking how he might demand compensation for such shabby treatment. Poor earnest Uncle Alex stewed over the insult until someone pointed out to him that Guy Fawkes was clearly a joke name and reminded him that his nephew Kurt worked in the GE News Bureau. Surely Alex was the victim of nothing more than a jesting sibling rivalry. Who but Kurt could so deftly manage to praise and insult his brother simultaneously by calling him the \"poor man's Steinmetz\"?\n\nLater, Kurt speculated that his favorite uncle must have been furious with him. But Alex said nothing at the time. He simply held on to the offending letter. And at some point, he gave it to Bernie.\n\n* * *\n\nHarry Wexler grew increasingly outraged as he listened to Irving Langmuir talk about hurricanes. Langmuir was giving the keynote address before the National Academy of Sciences, and even though it was clear GE's legal team had eliminated any verbiage that might implicate their company in property damage or worse, Irving's excitement was palpable. He clearly believed Project Cirrus had accomplished something by seeding a hurricane with a couple of hundred pounds of dry ice. He seemed to think, in fact, that they had changed its course. At the end of his heavily vetted account of Hurricane King, he declared that he planned, the following year, to study hurricanes some more \"to see if we cannot, by seeding them, in some way modify them or shift their positions.\"\n\n\"I think the chances are excellent,\" he declared, \"that with increased knowledge... we should be able to abolish all of the evil effects of these hurricanes.\"\n\nHurricanes! Harry Wexler knew hurricanes! He was the first meteorologist to fly directly into a hurricane for the purpose of collecting data back in 1944. He and two Army Air Forces pilots took an A-20 Havoc right into the eye of the storm, riding its updrafts as coolly as businessmen taking an elevator to the fortieth floor. The New York Times had even written it up! Harry probably knew more about hurricanes than anyone else alive. Certainly he knew more than Irving Langmuir, who had just declared\u2014incorrectly\u2014that it was atypical for hurricanes to have squall lines. Harry Wexler had published a paper about squall lines just last year! Project Cirrus was wading into the meteorologists' territory, but its scientists weren't bothering to educate themselves about the meteorology field.\n\nAfter the talk, Langmuir was even more unrestrained, declaring that seeding by Project Cirrus had almost certainly altered the track of Hurricane King. He also talked about a new theory he was developing, that raindrops could be formed inside clouds not just by nucleation but by a kind of chain reaction. A single ice crystal or droplet of water might, under the right conditions, trigger an increasingly large event. One snowflake would lead to two, two would lead to four, four to eight, eight to sixteen, and so on, until the entire cloud would precipitate. The theory accounted for rain in places where clouds did not always reach the freezing point before precipitating. It also meant, he said, that it could be possible to induce rain simply by seeding clouds with water. A bucketful could cause the chain reaction, or even a single drop\u2014just as smashing a single atom could wipe Hiroshima or Moscow or even Schenectady right off the map.\n\nChief Reichelderfer was at the meeting too, and he was as appalled as Harry. Project Cirrus's fantastic claims were increasingly causing headaches for the real weather experts at the bureau.\n\n\"Army and GE Join to 'Make Weather.'\" \"Rain to Order.\" \"Snow Made in a New Way.\" \"Scientists Get Ready to Do Something About the Weather.\" It seemed as if a week didn't go by without some outlandish headline announcing a story about weather control, thanks to the experimentalists of GE. The company was cranking out press releases with an avidness that suggested America's weather was soon to be issuing forth from Schenectady as reliably as America's washing machines.\n\nWorse still, the public wanted the government to get on board. Letters poured in to the Weather Bureau insisting that the agency make rain, eliminate hail, put out forest fires, and bust hurricanes, and with each one Reichelderfer and Wexler seethed a bit more. Langmuir and his team didn't even understand the mathematics of the atmosphere. And yet their reckless claims were building up the public's expectations to the point where, every time the Weather Bureau failed to forecast a freak storm, someone started making noise about cutting the bureau's budget, because the military, or even private industry, could do its job better.\n\nWhich is why the bureau had gone on the offensive. Harry was following up on every Project Cirrus report, going over the raw data in an attempt to find errors. Chief Reichelderfer convened an advisory committee on cloud physics to reproduce the experiments and refute the claims of Project Cirrus. And he assigned a full-time Weather Bureau observer to Project Cirrus: William Lewis. Lewis was a young bureau meteorologist whom Harry had sounded out on the topic of cloud seeding. He was sufficiently skeptical. Assigning Lewis to the Project Cirrus team would make it look as if the bureau were taking the project seriously, while providing Reichelderfer and Wexler with a mole to report back on the goings-on in Schenectady and help them to prove once and for all that the GE scientists were not rainmakers but charlatans.\n\n* * *\n\n\"Some 15 different types of finely divided soil, mostly from desert or arid regions of the country,\" Kurt's story began, \"have been found to be capable of producing snow in the laboratory, Vincent J. Schaefer... revealed recently in Chicago.\"\n\nIt was an odd situation. The GE Research Lab was not on Kurt's beat: manager Roger Hammond had the plum assignment of writing almost all of the Project Cirrus publicity. But now, in April, the News Bureau had sent Kurt to Chicago with Bernard, Vince, and Irving for the annual meeting of the American Chemical Society. To the bureau, it must have made sense: Kurt was a former chemistry major. He had worked in the Chicago newspaper world, so he had the contacts to place his stories there. And he was a graduate of the University of Chicago\u2014or so they thought.\n\n\"Mr. Schaefer, Dr. Irving Langmuir and Dr. Bernard Vonnegut, G-E snowmaking scientists, spoke recently before the 113th national meeting of the American Chemical Society,\" Kurt typed. That was GE-speak, always hypercorrect about everyone's titles and credentials.\n\nKurt didn't often get a chance to hear Bernard publicly presenting his work. Privately, of course, Bernie talked of almost nothing else. Clouds and nuclei and ice and precipitation\u2014he could go on about theories and experiments for hours, blissfully unaware if he was boring his listeners. That was Bernie. He didn't love playing in the snow the way Vince and Irving did\u2014two months earlier he had fractured an ankle when Vince tried to teach him to ski\u2014but he had endless stamina for talking about the stuff.\n\nGE cultivated this obsessiveness, this focus on science and science alone, by making the Research Lab into a kind of playground. The Research Lab scientists didn't have to punch a time clock. They didn't have to do whatever tasks their bosses assigned them. They didn't have to get approval for a lousy two-page press release from the News Bureau boss, the contracting agency, Engineering, Patents, Legal, the Works office, and the Atomic Energy Commission. They were in the business of seeking truth\u2014not, as some folks considered publicity, buggering truth for money. And truth meant whatever nifty scientific puzzle was at hand\u2014not the truth of the real world, where things were very likely going to hell. The GE scientists could remain completely withdrawn from the outside world if they wanted to. Did Irving Langmuir even know that the planet was seemingly on the brink of World War III? The Soviets had just ordered Allied military personnel out of East Germany and were putting increasing pressure on supply lines to West Berlin. Kurt read the newspaper avidly and worried about things like that. But as far as he could tell, Irving paid no attention to anything outside his own research.\n\nWorse than that, he didn't worry a whit about what use might be made of his inventions. He just fiddled away on the strings of his brain as if it were a toy violin and his whole purpose on earth was to play it. During the last AMS meeting, Irving had some time to kill, so he went to the American Museum of Natural History. He got so engrossed in pondering the exhibits that he failed to hear the closing announcements or notice when everyone else left. He stayed lost in thought until the lights went out, and he had to grope his way along dark corridors for half an hour before finding a surprised security guard to let him out.\n\nBernie could be like that too. He wasn't quite as absentminded as Irving Langmuir, but he was often lost in his own head, unaware that others might not share his enthusiasm for its contents. Kurt could get annoyed at his brother's endless talk about his own work. But often he was interested in what Bernie had to say. Sometimes he even solicited scientific explanations\u2014as he had when he read a Fortune magazine piece, \"Weather Under Control,\" that February. The magazine reported that Irving had answered a fundamental question: why supercooled water droplets nucleate into ice crystals at \u221239 degrees, even though theoretically they should stay liquid forever. The answer had to do with ice-2.\n\nIf subjected to enough pressure while at around \u221236 degrees, supercooled water forms an ice crystal that looks like a tetragon, rather than its usual hexagonal shape. The scientist who discovered this, Percy Bridgman, named the crystalline variant ice-2. Irving realized that cloud droplets are so tiny that their small surface area squeezes their contents, increasing their internal pressure. If small enough, a droplet's high pressure causes it to convert to ice-2. After crystallizing, it grows again and reconverts to regular, six-sided ice.\n\nKurt had not realized that there were different kinds of ice crystals. Of course, Bernie was happy to explain. He told Kurt to imagine cannonballs stacked on a courthouse lawn. Just as the balls could be piled up into different shapes, ice crystals could stack up in different configurations. In fact, Bridgman had described a whole series of ice phase variants, ice-1 through ice-6. And who knew\u2014there might be more to come.\n\nKurt socked away that idea and the image of cannonballs on a courthouse lawn. Bernie might be pedantic sometimes, but his explanations were often damn good.\n\nIt was regular old ice that Bernie was talking about at the Chicago conference. He was describing a spray-nozzle smoke generator that would burn a combination of hydrogen and silver iodide, vaporizing the silver iodide into tiny particles perfect for precipitating snow. He described lab experiments that showed that the vaporized silver iodide particles continued to exist in a supercooled cloud long after being introduced, causing snow crystals to form for as much as an hour afterward.\n\n\"He contrasted this to seeding with dry ice,\" Kurt typed, \"which generates large numbers of minute ice crystals immediately upon contact with the supercooled cloud.\"\n\nIt was the clearest statement yet of why Bernie was convinced silver iodide was the real future of rainmaking. Sometimes Kurt's explanations were damn good too.\n\n* * *\n\nAs the boat from Henderson Harbor neared Association Island, at the eastern end of Lake Ontario, the men on board could see the funnel of a tuba rising over a throng of straw boaters. The band was playing \"Marching Along Together,\" the Camp GE theme song for 1948.\n\nSwinging along with GE.\n\nWorking along in stride\n\nWithout a question\u2014without a doubt\n\nWe're GE's fighting crew\n\nOh rum ti-did-dle di let's all shout\n\nTwo billion we will do.\n\nVince filed off the boat with the others toward the banner reading, \"Welcome: Camp General Electric.\" Here was the ultimate proof that the company valued him: he had been invited to Camp GE with 270 other men, managers from every branch and byway of GE. Every second summer, up-and-comers were invited to a three-day festival of networking, team building, and leadership training. Happily, he walked down the gangplank and got in line, awaiting his turn to be handed his GE monogram T-shirt, his camp boater, and his songbook. His tent number was 47. His team was Blue.\n\nHe'd heard about Camp GE for years. Irving and Doc Whitney were frequently at the island, representing the Research Lab. Guy Suits had gone too. Now it was Vince's turn. This honor followed close on the heels of another one: that June, the University of Notre Dame had granted him an honorary doctorate. He was deeply gratified; unlike Bernie and Irving, Vince actually cared about such formalities.\n\nVince had worked as Langmuir's right-hand man for decades. He had contributed immensely to Langmuir's research and published papers in respected scientific journals. He was a scientist, no doubt about it. But he had never been \"Dr. Schaefer.\" It had always been \"Mr. Schaefer.\" Now, even though it isn't customary for recipients of honorary degrees to use the title \"Doctor,\" Irving Langmuir had begun referring to Vince as Dr. Schaefer, and it stuck.\n\nSo the newly minted Dr. Schaefer was ready to stand tall during the flag-raising ceremony, to play pinball or Ping-Pong in the Tom Catte room, to gather under the Old Elfun Elm during the inspirational \"charge to the rookies,\" and to do his best for the Blue. Divided into four teams, Blue, Green, Gold, and Red, the men competed in a variety of sports. Volleyball, shuffleboard, trapshooting, and horseshoes were all possibilities, but the big event was the softball game. Competition was fierce, with friendly rivalry amplified by signs planted in the public gathering spaces saying things like \"The green team welcomes you to Camp General Electric!\" At the end of the three days, whichever team had the most points won the coveted title of Camp Sports Champions. Vince's camp songbook had the words to the Blue team's war cry, sung to the tune of \"Yankee Doodle\":\n\nBlue's the best team on the Island\n\nThe others should have stood in bed\n\nWe have no trouble with the gold and green\n\nAnd say the hell with the red.\n\nThe songbook also had the words to the marching song, the president's song, \"GE Will Shine\" and \"Underneath the Elm,\" and, of course, all the classics: \"Roll Out the Barrel,\" \"Auld Lang Syne,\" \"My Wild Irish Rose,\" \"Pack Up Your Troubles,\" and \"I Want a Girl.\"\n\nEach morning, there were business meetings\u2014lectures about marketing and manufacturing, research and public relations, as well as skits based on the year's leadership theme: \"Our Heritage! Our Responsibility! Our Destiny?\" The skits were written by Lemuel Boulware, manager of employee relations. Boulware had been appointed to his position after the strike of 1946, when plants under his management had stayed on the job. A zealous free marketeer, Boulware was on a mission to reeducate GE employees: his skits were ideological pageants about the evils of unions and the rewards of unfettered commerce. The company's president, Charlie Wilson, would be at Camp GE as well. He usually arrived late, and a song would be sung in his honor to the tune of \"Auld Lang Syne\":\n\nStrong hearts and hands across the land\n\nThe loyal men and true,\n\nFrom Eastern coast to Western strand,\n\nCharlie, they're all for you.\n\nThe afternoon was dedicated to sports and to impromptu amusements like the insurance game Police Protection. In the late afternoon, cocktail hour began, followed by dinner and entertainment, and the famous brotherhood ceremonies led by an actor dressed as an Indian.\n\nVincent joined in with his usual vigor and team spirit. He carefully saved his program, his tent assignment, and his songbook. But he was also eager to get back to work. He had an even more exciting trip to take in the fall: Project Cirrus was heading out west. They were planning to start their program of large-scale weather control with a particularly useful undertaking: making rain in the desert.\n\n* * *\n\nKurt was doing his best for GE. But it wasn't enough to applaud every new gadget or machine the company cooked up as if it would change the world. It wasn't enough to obey your GE boss and play softball on a GE team and buy your appliances at the GE employee store. The company wanted to tell you how to think too.\n\nEvery week or so, a new poster went up, Lemuel Boulware's florid signature at the bottom. \"Why must we SAVE more\u2014as well as PRODUCE more?\" \"Should pay be equal everywhere?\" \"What is Communism? What is Capitalism? What is the Difference to You?\" You could be sure Mr. Boulware\u2014a.k.a. Mr. Bullwhip\u2014would tell you the answers. He had all the answers, Mr. Bullwhip did. Mr. Bullshit was more like it, at least as Kurt saw it. Boulware's messages to the employees were unabashedly pro-America and antilabor. Higher wages? Bad for workers because they increased inflation. Equal pay? A fallacy. Huge corporate profits? A clear sign of virtue. Communism? Nothing less than slavery. Redistributing wealth? The end of America.\n\nKurt's boss, George Griffin, was not much better. He had been a colonel in World War II, and he never seemed to forget it; now he was a company man through and through. George taped cartoons to his office door, like the one showing two executives looking at a chart of crashing sales. \"It can't be our product's quality,\" one was saying to the other. \"We make the finest buggy whips in the world!\" George saw it as a crack at other companies. GE wouldn't be caught dead making buggy whips. Its most important product was progress.\n\nKurt tried to fit in, but he had brought some of his Chicago bohemianism to GE. He wore sneakers to work, shed his jacket as often as possible, and sometimes completely forgot his tie. He took up a collection at the office for the United World Federalists, an organization dedicated to the cause of world government. He got all his fellow junior writers to give a dollar each. Colonel Griffin did not appreciate these efforts. In Monday morning staff sessions, Kurt frequently found himself on the listening end of a lecture.\n\nAt least he had companions. Kurt's friend Ollie Lyon was a hell-raiser too, a fellow infantry veteran who had been on the same troopship to Europe as Kurt, though they didn't meet then. At GE, they became friends. Kurt also got to be friends with Bob Pace, another junior writer and a fellow graduate of Shortridge High. Bob had been two years ahead of Kurt there, and now they remet at GE. It was all an ocean, as Jane said.\n\nOn slow afternoons in the pressroom, the junior writers would repair to Walker's Pharmacy on State Street to eat pickled hard-boiled eggs washed down with beer. They talked politics, and there was plenty that summer to talk about. The Allied airlift had subverted the Soviet blockade of Berlin. American and British pilots, flying around the clock, were averting World War III, at least for now. But at home, Red-baiting was becoming the norm. Attacks on scientists were growing frequent, and it had taken prolonged exertions from the scientific community to stop the long and brutal smear campaign against the physicist Edward Condon, whom the Atomic Energy Commission had finally cleared to continue as director of the Bureau of Standards. But the HUAC chairman, J. Parnell Thomas, was not backing down. He was claiming that the Oak Ridge atomic laboratory was \"heavily infested\" with Reds. Government security clearance was now necessary to work in the GE Research Lab, and those who were denied it had no recourse. Frequently, they didn't even know the cause.\n\nMost Americans felt it was reasonable for the government to keep track of communists; it followed that it was only right for GE to be careful about whom it hired. If you didn't have anything to hide, people figured, you didn't have anything to fear. They didn't know that the FBI was compiling dossiers on the GE scientists, that even scrupulously correct Irving Langmuir had an FBI file listing luncheons he had attended, speeches he had made, phone calls he had received, positive statements about him printed in The Daily Worker. The government is reasonable, people thought. It wouldn't ruin a man's career without good cause. And GE... most people trusted GE even more than they trusted the government. Kurt saw that as a kind of sickness.\n\nKurt and Bob Pace often talked about how to avoid catching \"GE Disease.\" The main symptom was an overidentification with General Electric\u2014or \"the Company,\" as such people called it. The affliction wasn't uncommon. In fact, GE encouraged it, especially among the PYMs, the Promising Young Men. The PYMs joined company discussion groups about protecting the free market and angled for invites to leadership conferences and Camp GE. They took memory-building classes at the Schenectady YMCA or attended GE colloquiums, all in the interest of becoming better organization men.\n\nTucked away at Walker's Pharmacy, out of sight of the electric boosterism of the company sign, one could gripe about GE. It was a relief to spend time not putting on an act. Quietly, Kurt was also keeping his eyes open in case a better job turned up. At the end of his first summer at GE, he submitted an essay to The Reader's Digest on the pleasure of working with one's hands. In his cover letter, he casually inquired about job openings there. Editor DeWitt Wallace wrote back to say they had none but would keep Kurt's qualifications in mind if any arose.\n\nIn the fall, an opportunity came up to better his position at GE. The News Bureau needed another writer in the magazine division, and Kurt lobbied hard to get the position. Being responsible for pitching stories to magazines like The Saturday Evening Post and Life meant reading those magazines would be part of his job. Meeting editors would be a cinch. Perhaps he could parlay it into selling some of his own work too.\n\nKurt got the promotion. It was a step up at GE. But it was also one step closer to what Jane had always said he should be doing, writing for a living. More and more, he knew that she was right: he was biding his time at GE, but secretly he was planning his escape.\n\n* * *\n\nBernie was frustrated. He wanted to try out his silver iodide burner, but he couldn't get his hands on any hydrogen to fuel it.\n\nIt was October, and Project Cirrus was in New Mexico on an exploratory mission. Langmuir had decided that they should try to modify storms not in the East, where they simply traverse the land, but in the desert, where storms actually start. And Vincent had lined up an invaluable contact in the region: Everly John Workman, known as Jack.\n\nBernie liked Jack Workman; they all did. Acting president of the tiny New Mexico School of Mines, Jack was busily remaking a sleepy trade school for mining engineers into a serious institute of technology with a specialty in his passion: atmospheric physics. Many of his initiatives\u2014such as requiring calculus for all students and disbanding the school's basketball team\u2014were wildly unpopular. But he was energetic and pugnacious, and he always got his way. Best of all, he was a paper-clip-and-string scientist, like Bernie. He had cobbled together a passionate team and a ragtag weather research caravan: vans and old school buses and former Army trucks all kitted out with jerry-built instruments aimed at the sky. Barreling through the desert on the heels of a thunderstorm like a steampunk motorcade, Workman's group was Victorian science at its best. Jack's personal storm chaser was an open Packard to which he'd welded a lightning-proof steel roof mounted with electric field meters. It seemed perfectly safe, except for the part where he would floor the accelerator and then dive beneath the dashboard to read his instruments while the car careened down desert roads. The first time he ever did this with Project Cirrus team members in the car, it was raining, and the car careened wildly in the muddy road, nearly sliding off one side, then the other. Everyone was terrified except Irving, who calmly took out a notebook and scribbled calculations before announcing that the car's cyclic maneuver had a frequency of 176 per minute.\n\nThey were all excited about studying clouds and rain in New Mexico. The Southwest's weather is as different from the East's as its traffic. Both skies and highways are congested in the East, clogged with huge flows from countless sources. But out in the open desert, especially in the summer, there are few big weather fronts. Instead, the warm air rising off the sun-heated ground forms small, isolated clouds that typically move only a few miles before dissipating in the dry New Mexico air or boiling up into dramatic, short-lived thunderstorms.\n\nWorkman's team had agreed to help Project Cirrus with radar tracking and photography. And the Project Cirrus team only had three days there, so they got right to work. On October 14, they seeded a cumulus cloud with dry ice. The results were spectacular. The cloud billowed into an angry thunderhead and promptly produced reams of precipitation. Doing the calculations later, Langmuir concluded that the storm had produced rain over a forty-thousand-square-mile area, about a quarter of the state of New Mexico. He attributed it all to the seeding. \"The odds in favor of this conclusion,\" he wrote, \"as compared to the assumption that the rain was due to natural causes are many millions to one.\"\n\nMeanwhile, Bernie was determined to get his silver iodide generator up and running. But he spent all three days trying to lay hands on some hydrogen out there in the middle of nowhere. By the time he finally acquired the fuel, the Project Cirrus group was preparing to leave. Fortunately, Workman's team was eager to continue the work, so Bernie left a spray-nozzle silver iodide generator behind. The School of Mines researchers agreed to run it sometimes, as he requested, after the Project Cirrus scientists left.\n\nIt had been a frustrating trip, but he went back to Schenectady with high hopes. The hot, open desert sky seemed as if it might provide what the humid, busy air of the East was refusing him: definitive proof, in the real atmosphere, of what silver iodide could do.\n\nWatersheds\n\nBernie's work space was a disaster area. Bottles, beakers, Bunsen burners, microscopes, thermometers, cameras, tubes, and every other conceivable piece of lab equipment commingled on his workbench with the flotsam and jetsam of work life: pens and notebooks and coffee cups and ashtrays brimming with butts. Vincent even noted Bernie's mess in his work review but added that it wasn't really a problem. No one complained. Doc Whitney had once done a statistical study of workplace tidiness, showing that the most productive scientists were the ones with the messiest desks, so official Research Lab policy was to leave the slobs alone. Occasionally, however, an incredulous co-worker couldn't help commenting on the chaos.\n\n\"If you think that's bad,\" Bernie once replied, tapping his head, \"you should see what's up here.\"\n\nNow Bernie and his mess were moving to \"the Penthouse,\" GE's name for the huge top-floor laboratory designated for Project Cirrus in the company's new facility. The Knolls Research Laboratory had not been officially christened yet, but by late 1948 most of the scientists had moved in. The $10.5 million building sat high on a bluff over the Mohawk River in Niskayuna, a sleepy little town about five miles east of downtown Schenectady and the Works; a bus constantly ran the \"GE Loop\" between them. The suburban locale gave the place a self-contained, campus feel. The new building housed a library, a cafeteria, an auditorium, and an employee store. The grounds had once been a private estate. Its mansion was restored and occupied by the Research Lab employees' exclusive Whitney Club, and its tree-lined drive and formal gardens lent it a patrician grandeur.\n\nThe sleek modernist building was designed to be a researcher's dream. Walls, piping, and wiring were modular, so they could be easily reconfigured for evolving projects. Compressed air, steam, hydrogen, and illuminating gas were piped in along with hot and cold water. There were even vacuum pipes. You turned on the tap, one scientist told the Schenectady Gazette, and \"nothing\" came out. That might have been Bernie; it was his kind of quip. In the Penthouse, he now shared a huge lab\u2014a combination of rooms 507 to 513\u2014with Vincent and Katharine Blodgett; Irving had a private office nearby. On the roof was a complete meteorological station that captured a constant stream of weather data recorded on instruments in the offices below.\n\nThis place was designed not for Victorian science but for the specialized, industrialized discipline that was coming to be called Big Science. The electron microscope, vacuum furnace, and low-temperature lab were up and running; the future would bring a chemical pilot plant and a radiation building with betatrons, synchrotrons, and high-voltage X-ray equipment. And next door, the Knolls Atomic Power Laboratory (KAPL)\u2014a government facility operated by GE\u2014was nearing completion. KAPL would be focused on military projects like designing nuclear submarines, but the alliance between it and the similarly named Knolls Research Lab was unmistakable. The white-collar world of the scientists, allied with the government and the military, was edging away from the blue-collar world of the Works and into its rarefied technocratic sphere. One News Bureau liaison was given an office in the Knolls. The rest of the News Bureau\u2014including Kurt\u2014stayed behind at the Works.\n\nIrving was not there for the move; he was home, recovering from cataract surgery. But he'd be back in full swing soon. Langmuir was approaching seventy, but he had no intention of letting age run him out of science. Once, Dr. Michael Ference, the team's Signal Corps representative, made the mistake of asking Langmuir how his retirement might affect Project Cirrus.\n\n\"You'll know when I'm retired,\" Irving snapped. \"I'll be dead.\"\n\nWhy would he retire when he was just commencing the most significant work of his career? Project Cirrus was now working like a well-oiled machine. With two B-17s fitted with special instruments for recording atmospheric data, as well as photo panels to document results, the team had carried out more than fifty experimental flights, seeding clouds over Cape Cod, Vermont, New Hampshire, and New York, over the Catskills, the Berkshires, and the Adirondacks. They had attacked stratus clouds and cumulus clouds with dry ice. They had even tried seeding clear skies.\n\nWith their new planes, they were finally getting some good documentation. In November near Schenectady, they had dispensed dry ice in the pattern of a Greek gamma twenty-four miles long, then photographed it from above. The gamma shape was entirely clear. Within half an hour, they could see the ground through its opening. When Vincent drove north to Rotterdam to photograph it from below, the seeded opening stretched from horizon to horizon, and curtains of snow were still falling from its edges. Later the same day, they attacked a cloud bank that was sitting over Rome, New York, this time flying a racetrack pattern. Below, controllers at the Rome airport gaped in wonder as two parallel bands of blue suddenly opened up in their otherwise overcast sky.\n\nProject Cirrus had seeded clouds using Bernie's silver iodide technique only three times. On all three, observers on the planes agreed that the silver iodide had affected the clouds. But while the dry ice seeding runs were producing dramatic photographs, the silver iodide runs yielded only blurry fields of gray that looked as much like abstract paintings as they did clouds. Bernie's project remained a low priority.\n\nScience Newsletter ran a story about his seeding method, putting his picture on the cover above the words \"Fire for Rain.\" He stood behind his ground-based hydrogen burner igniting an aerosolized solution of silver iodide. The streaking lines of fire cast his tall form and curly pompadour in shadow so that he looked like a movie mad scientist. It was a nifty photograph, but it was all smoke and mirrors until he could prove that silver iodide worked in the real world, not just in the lab.\n\n* * *\n\nShop Electric threw its holiday bash at the Rotterdam Democratic Club. Wire and Cable joined forces at the Town Tavern. High Voltage, Nucleonics, and Tool Design all had theirs at the popular Hans Grell's. Production took over Lloyd's with piano playing, singing, and dancing; one eager employee even attempted an Irish jig on the tabletop. By late November, reports of such antics at the holiday parties were crowding the gossipy pages of the Schenectady Works News.\n\nKurt, like the nation, was upbeat. The threat of nuclear annihilation seemed to recede as time passed. No other country had yet succeeded in getting the bomb, and there was still hope for averting a nuclear arms race and instituting some kind of world government. The United Nations was getting a chance to prove its effectiveness, working on brokering an armistice between Egypt and Israel. The Berlin Airlift had now been supplying West Berlin for five months, proving it was possible to stand up to Soviet aggression without provoking war. Best of all, the incumbent, Harry Truman, defeated the Republican challenger, Thomas Dewey, in the presidential election, upending the experts' predictions. The close race had led to the conservative Chicago Tribune's famously mistaken headline: \"Dewey Defeats Truman.\" Kurt undoubtedly laughed about that; he despised the Trib. And the election was close not because national sentiment was moving to the right. It was close because the six left candidates on a ticket of eight had divided the progressive majority. Kurt considered himself a socialist, but he'd take Truman over Dewey. Things, it seemed, were moving in the right direction, even if they weren't yet perfect.\n\nAs Kurt and Jane prepared to go out on the last Friday in November, Kurt was in an ebullient mood. They were headed for a festive black-tie affair in the ballroom of the Mohawk Golf Club\u2014the first dance of the season for the Schenectady Junior League, where Jane was a provisional member. Their friends Ollie and Lavina Lyon were babysitting, and when the Lyons arrived, Kurt came downstairs in his tuxedo. On his feet were sneakers. It was something his sister, Alice, used to do: go to school dances in a party dress with sneakers on her feet. She was so beautiful and exotic that somehow she could pull it off.\n\nJane told him to go upstairs and change his shoes. She liked wackiness too: it was her nail polish on the nails of their claw-foot tub after all. But they were still carving out their place among the swells of Schenectady. They had made friends with other GE couples and attended cocktail parties and bridge games. Jane had joined the PTA and the American Association of University Women. She volunteered one night a week at the local hospital. Kurt had signed on as an Alplaus volunteer fireman. They were fitting in, or at least pretending to.\n\nOn weekdays in Building 6, Kurt performed the role of the dedicated junior writer, buckling down to pitch compelling stories about GE to editors, trying to flood them with the conviction that whatever new gadget GE was about to unveil would top the second coming of Christ. In his new position at the magazine division, he was churning out peppy overviews of subjects as unpromising as a leak detector shaped like a football or the new lab for low-temperature physics. He had spent enough time studying magazines to predict what editors wanted, and he seriously hoped to land GE an article in one of the national glossies. It would make things better for him at work. But he knew that he lacked what the company men had: some fundamental organ or special circuitry in the brain that would have enabled him to form an emotional attachment to the company, or even just to give a damn.\n\nSo in the evenings and on weekends, he parked himself at the tiny desk he and Jane had set up at the end of the upstairs hallway and wrote. From wherever she was in the house, Jane could hear the sound of typing. She was getting to recognize its rhythms. A smooth, steady clacking meant things were going well. When the typing stopped, she worried. If the writing was going badly, Kurt was hell to live with.\n\nOnly a few trusted friends knew about Kurt's aspirations. Ollie was one. The Lyons hesitated to drop by unannounced on evenings or weekends. They knew they'd find Kurt at his desk, smoking Pall Malls and typing, and they didn't want to bother him. Plenty of guys had literary dreams, but Kurt also had discipline.\n\nBehind the facade of the hardworking junior writer, the good team player, the loyal employee, the tiny flickering hope would not be extinguished: that one day he might quit and write for real. Jane remained convinced that her husband was a genius and that the world would figure it out eventually. Her faith didn't waver as the rejection slips piled up: she simply decoupaged a wastepaper basket with them. Kurt used them as scrap paper. When Jane gave him a grocery list, asking him to pick up corn, six wieners, Duff's devil's food, Junket, and instant icing, he jotted the items down on the back of a rejection slip from Reader's Scope.\n\nBut all that was unknown to most of his colleagues. Good company men dreamed of nothing more than steadily ascending the company ladder. And Schenectady was a company town full of company men. They rode on company buses wearing company suits and company fedoras. They had immaculate company wives and adorable company kids, and they lived in leafy company neighborhoods. That's not to say the company owned their homes, but in a community where two out of three schoolkids had a dad who worked for GE, where the baseball teams, bowling leagues, and ski trains were sponsored by GE, where people's social lives centered on GE clubs and GE lectures and GE parties, and where the local newspaper could pretty much be counted on for a GE story on the front page every day, it was hard not to feel that the company owned your soul.\n\nThat wasn't him: he was holding his soul apart. Still, when Jane asked, Kurt went upstairs and changed his shoes. He understood the need to fit in, and he was doing his best. But it made him nervous. He had a sneaking suspicion that if you pretend to be something for long enough, you might actually become it.\n\n* * *\n\nA few days before Christmas, Bernie oversaw Project Cirrus's fourth airborne attempt to seed clouds with silver iodide. It was flight fifty-seven for Project Cirrus. He had devised a new method for getting the chemical into the sky: impregnating small chunks of charcoal with silver iodide, then loading them into a burner that projected from the tail of the B-17. The scientists could light and unlight the burner as the plane flew.\n\nThe bomber plane took off from the Schenectady airport. It was a gray day, the sky slabbed with the concrete clouds typical of winter in upstate New York. The B-17 easily reached seventy-two hundred feet and began flying a gamma-shaped pattern over a solid bank of gray stratus clouds. Dry ice was dispensed first, for about a minute. Then Bernie's silver iodide burner was turned on for fifteen seconds, burning three or four pounds of silver-iodide-infused charcoal. Then another round of dry ice was released, to bracket the seeded spot. Kiah Maynard, the designated GE observer aboard the plane, scrupulously noted everything in his GE notepad. Bernie listened in from the tower.\n\nWhen the plane circled back over the seeded area, everyone on board could see the results. Like the dry ice, the silver iodide had attracted water vapor from the cloud's supercooled droplets. Three long furrows had opened up, each filled with sparkling ice crystals. Man-made snow was falling. The Signal Corps photographer shot forty-four pictures.\n\nBack at the lab, the photographs were developed. When he saw them, Bernie was elated. Flight fifty-seven had produced the first photographic evidence that silver iodide seeding worked on actual clouds. Excitedly, he calculated the probable number of snowflakes his seeding had produced. Five hundred per square centimeter of cloud, he figured. That was equivalent to dry ice seeding.\n\nThe implications were huge. Dry ice seeding was limited to local effects. Silver iodide's persistence offered the chance to change the weather for real\u2014and not just one cloud at a time. If Bernie's silver iodide smoke could be introduced to the right air current, the results could be seen across entire weather systems. He just had to convince the rest of the team.\n\n* * *\n\nKurt was convinced he had invented a winner: Alfred Moorhead, a classic company man. Alfred was saddled with the usual accoutrements: a dull job, a numskull boss, a nagging wife. Would it be so bad if he figured out a way to take harmless revenge on all of them? And so, right there in Building 6, his first successful GE story began to take shape.\n\nAlfred is an office functionary in a huge corporation: the United Manufacturing Company. He's nothing special until he takes a company seminar in mnemonics. At United Manufacturing, as at GE, a good memory is considered a business asset. So when Alfred becomes a memory whiz, his career takes off.\n\nHis old boss and his nagging wife hardly know what to think of this new and improved Alfred. Magically, he's been transformed from a loser to a winner. What no one realizes is that Alfred's new memory power comes from attaching violent images to anything he has to recall. When he has to remember the date he started work at the company, he thinks of a crashed airplane with soldiers marching past it carrying a flag emblazoned \"17.\" This brings to mind the date March 17, 1929\u2014the year of the \"crash.\" To recall his boss's extension, 717, Alfred imagines the boss with two 7-shaped hatchets in his body and a 1-shaped dagger in his throat. When his wife calls and asks him to pick up carrots, molasses, Dreft detergent, and liverwurst, he doesn't need to write it down. He just imagines sticking carrots in her ears, dumping molasses and soap flakes on her, and stuffing a liverwurst in her mouth. For the last item on the list\u2014matches\u2014he adds a backdrop for his sadistic tableau: the United Manufacturing Company in flames.\n\n\"The More Vivid, the Better\" was a nasty little story, an office revenge fantasy with a dollop of marital resentment. But it worked; Kurt was pleased with it. He redrafted it a couple of times, changing its title to the catchier \"Mnemonics.\" He figured it was just the kind of thing the magazines were looking for. It had a \"hook\" and humor and was set in a milieu that people would recognize.\n\nIt was also in keeping with the times. Like so much of the era's fiction, \"Mnemonics\" openly expressed anxiety about the conformist corporate culture taking shape in the nation. But Kurt still couldn't separate that anxiety from the war. The downed airplane with its (B-)\"17,\" the marching soldiers, the factory in flames: on some level, the Bulge and Dresden were still haunting his imagination. Like Salinger's \"Perfect Day for Bananafish\" or Sloan Wilson's Man in the Gray Flannel Suit, the early version of \"Mnemonics\" hints that the corporate blandness of the company man might be concealing war trauma. There was logic to this, coming so soon after the liberation of the Nazi death camps. Organization men were followers, and Dachau, Auschwitz, and Bergen-Belsen had introduced the world to the destruction blind followers can wreak.\n\nKurt sent the story off to The New Yorker. It was rejected right away. He told himself to keep on trying, but he felt disheartened: he knew it was his best effort yet. He put \"Mnemonics\" away instead of sending it somewhere else. Maybe he wasn't cut out for the writing life after all.\n\n* * *\n\n\"There is no use worrying about a need of one hundred pounds of dry ice,\" Irving told the audience at the January 1949 meeting of the American Meteorological Society in New York City. \"One gram is enough to do a good job.\"\n\nThey still weren't presenting their experiments on silver iodide; they didn't have enough data. But Bernie was in the audience as Irving and Vincent outlined their latest thinking about the correct amounts of dry ice to use in seeding clouds. The team was coming to think that sometimes less was more. Vince showed photographs of the trenches and L-shaped channels they had carved in the clouds. At the end of his talk, Irving gave an account of an October flight experiment in New Mexico, on a day when the Weather Bureau had not predicted any rain for the state. But after Project Cirrus seeded a large cumulus cloud near Albuquerque, the cloud became a thunderhead and precipitated. More clouds developed in the wake of that one. Irving showed slide after slide of the seeded cloud growing and developing into a squall line. He estimated that a hundred million tons of water had originated in that single seeding.\n\nWhen Irving was done talking, Ross Gunn stood up to comment. Director of physical research for the Weather Bureau, he was head of the Cloud Physics Project, designed to check the claims of Project Cirrus. Gunn pointed out that he had often seen cloud troughs such as those in the Project Cirrus pictures while flying and doubted they were the result of seeding. Project Cirrus could show all the photographs they wanted, but they didn't prove anything. And it was simply outrageous to claim that mile-wide holes could be produced in a stratus cloud deck by a mere pellet of dry ice.\n\nThen came the Cloud Physics Project's report. Recounting their experiments to date, the Weather Bureau scientists announced their conclusion: clouds could be somewhat modified in appearance by seeding, but there was no evidence that these modifications could induce self-propagating storms.\n\n\"The experiments showed that the artificial modification of cumuliform clouds is of doubtful economic importance for the production of rain,\" the report declared. Nor was the attack over: immediately after the Cloud Physics Project report, Weather Bureau chief Reichelderfer gave the conference lunchtime address, and in it he proposed that money be spent attempting to predict the weather accurately, rather than on attempting to control it. He didn't say it outright, but Reichelderfer, like Harry Wexler, was pretty sure he knew how better weather prediction was going to be achieved: with John von Neumann's computer. Once they had that up and running, it was going to be easier than ever to dismiss the work of Project Cirrus.\n\nAfter the talk, Irving took Reichelderfer aside. Why was he saying such starkly negative things when the evidence was still coming in? Reichelderfer had a blunt reply: to counteract Irving's exaggerations.\n\n\"Which of my statements do you consider the worst?\" asked Langmuir, with his usual scientific detachment.\n\nThat was easy. Reichelderfer said it was Irving's claims that the seeding of hurricanes should be studied with an eye to learning how to steer them. Particularly deplorable was Irving's National Academy of Sciences speech, where he had announced that \"with increased knowledge we should be able to abolish all of the evil effects of these hurricanes.\" It was a gross exaggeration, Reichelderfer said.\n\nIrving was unruffled. The larger the storm, he said, the more energy in it, and the more energy in it, the easier it should be to have a big effect on it. A hurricane could probably be modified by just a single pellet of dry ice. You just had to know where to put the pellet. To think otherwise, he told the astonished Reichelderfer, was like claiming that a massive forest fire could never be set by a single match.\n\n\"Rain-Making Held of No Importance,\" declared The New York Times the next day. \"Weather Men See Little Value in Scientists' Efforts to Alter Natural Patterns.\" GE dispatched Bernie to do the rounds of the radio networks: he was a good interview subject, able to explain things clearly. Patiently, soberly, he described how cloud seeding worked and explained why they believed the experiments should keep going forward. But the battle lines had been drawn. The GE scientists led one side, and the Weather Bureau headed up the other. Maybe, Vincent told Bernie and Irving later, they should start seeding clouds in the pattern of the GE logo, imprinting the \"meatball\" on the sky instead of racetracks or gammas. Then people might believe they had done something.\n\nIrving had a better idea. As soon as he got back to Schenectady, he began arranging for the Norwegian meteorologist and D-Day forecaster Sverre Petterssen to visit GE and go over the Project Cirrus findings. Petterssen was highly respected and scientifically rigorous. But he was also now the director of scientific service for the Air Force Air Weather Service, and while the Weather Bureau might be staffed with naysayers, the military could be counted on for enthusiasm.\n\nThat was becoming evident to Bernie as well. The Navy and the Signal Corps liked his silver iodide generator so much, they were refusing to declassify information about it. And not long after the AMS conference, he received a letter from the Office of Naval Research informing him that a group of high-ranking officers was coming to Schenectady to discuss the cloud nuclei counter he was developing. It would be sometime next month: they would get back to him with the exact date. They simply assumed Bernie would be there to tell them what they wanted to know.\n\n* * *\n\nKurt was thinking about what might happen if a scientist refused to tell the military men what they wanted to know. A scientist had recently done just that: Norbert Wiener.\n\nProfessor Norbert Wiener of MIT was one of the world's leading mathematicians. He was good friends with John von Neumann, but the two men could hardly have been more different. In January 1947, Wiener had made national news by canceling a talk at an MIT symposium on high-speed calculating machines because the conference was funded by the military. Just before the conference, The Atlantic had published Wiener's open letter to an aircraft company researcher who requested a copy of a paper on controlled missiles Wiener had written during the war. Wiener refused to give it to him. It was simple, Wiener wrote. He had done that work under government contract because he thought he should assist the war effort. But then he had seen the results.\n\n\"The policy of the government itself during and after the war, say in the bombing of Hiroshima and Nagasaki,\" he wrote, \"has made it clear that to provide scientific information is not a necessarily innocent act, and may entail the gravest consequences. One therefore cannot escape reconsidering the established custom of the scientist to give information to every person who may inquire of him.\" Scientists had now become arbiters of life and death, he declared, and he was censoring himself because \"to disseminate information about a weapon in the present state of our civilization is to make it practically certain that that weapon will be used.\"\n\nWiener's letter had made a big impression in Schenectady. The scientists Kurt knew at GE often discussed issues like those he raised. Had it been morally right to drop the atomic bomb on Hiroshima\u2014and on Nagasaki? If morally wrong, were the scientists who built the bomb as guilty as the generals who decided to use it? Wiener's vow of noncompliance was admired by many in the Scientists' Movement and beyond, but it was disturbing too. Should scientists censor themselves? Didn't knowledge belong to the world at large?\n\nTwo years of debate had not settled the question. In November 1948, the Bulletin of the Atomic Scientists had published Wiener's follow-up essay, \"A Rebellious Scientist After Two Years.\" Again, Wiener pulled no punches. \"The degradation of the position of the scientist as an independent worker and thinker to that of a morally irresponsible stooge in a science-factory has proceeded even more rapidly and devastatingly than I had expected,\" he wrote. \"In view of this, I still see no reason to turn over to any person, whether he be an army officer or the kept scientist of a great corporation, any results which I obtain if I think they are not going to be used for the best interests of science and of humanity.\"\n\nA morally irresponsible stooge in a science factory! Some people might look at the GE Research Lab and see exactly that. GE made no effort to hide its cozy relationship with the military; on the contrary, it boasted of it. But what did this mean for scientists like Bernie? Were they really kept scientists, ethically deficient worker ants mindlessly contributing to a proliferating war machine? Kurt knew his brother was at heart a pacifist; their parents had raised them that way, and the war had only deepened the conviction for them both. But as soon as cloud seeding was made public, its military uses were being discussed. There was General Kenney telling the graduates of MIT, \"The nation that first learns to plot the paths of air masses accurately and learns to control the time and place of precipitation will dominate the globe.\" That made Bernard uncomfortable. But because of his discovery, he was now answering to the generals. If a scientist like Bernard wanted to stop his work from being used for violent ends, what might he have to do?\n\nKurt's new story was different from anything else he had written. There was no grieving widow or office romance, none of the crowd-pleasing claptrap he had been churning out, hoping to please the slicks. Nor was there mention of World War II. Instead, right there at his desk in the Schenectady Works, from his vantage point in GE's science factory, he began to imagine an antiwar scientist named Professor Barnhouse.\n\nProfessor Barnhouse discovers something shocking: he has the ability to control things previously thought to be uncontrollable. But his dream of using his power for humanity's benefit soon crumbles as he realizes that the military men only see its value as a superweapon.\n\nHe called the story \"Wishing Will Make It So: A Comprehensive Report on the Barnhouse Effect.\" On one of the first full drafts, he crossed out \"Barnhouse.\" Was it too close to his brother's nickname, Barney? He made the professor Brenhaltz instead. Lots of scientists had German names. Before long, he restored it to Barnhouse.\n\nThe story is narrated by Barnhouse's student because the professor himself has disappeared. Barnhouse's student has been tasked with explaining the history of the \"Barnhouse effect.\" It began, he explains, when Barnhouse, an artillery private in the Army, rolled sevens in his first barracks dice game\u2014ten times in a row. Like Langmuir with his New Mexico storm, Barnhouse went off and excitedly calculated the odds of that happening by chance: they were one in sixty million. He tried rolling dice again and realized he could produce the effect: he had telekinetic powers. He began consciously cultivating this power, \"dynamopsychism,\" until eventually he could destroy entire buildings from miles away.\n\nBarnhouse keeps his power secret at first, annoying the student-narrator with what seem like irrelevant questions. His favorites are \"Think we should have dropped the atomic bomb on Hiroshima?\" and \"Think every new piece of scientific information is a good thing for humanity?\" He reveals his power to his student-narrator when he decides to declare it publicly, in a letter to the secretary of state.\n\n\"I have discovered a new force which costs nothing to use,\" he writes, \"and which is probably more important than atomic energy.\"\n\nIt was exactly the same claim Langmuir was making about weather modification. And like Langmuir, Professor Barnhouse quickly finds himself taken up by the military. A Senator Warren Foust\u2014sounding a lot like General Kenney on the topic of weather control\u2014declares, \"He who rules the Barnhouse Effect rules the world!\" The military starts a dynamopsychism program, naming it \"Project Wishing Well.\" It secludes the professor and his student in a safe house under the protection of guards on loan from the Atomic Energy Commission and plans a secret test of the Barnhouse effect called \"Operation Brainstorm.\" It succeeds brilliantly. Barnhouse, sitting on a couch, uses his dynamopsychic powers to destroy ten V-2 rockets fired in New Mexico, bring down fifty radio-controlled bombers over the Aleutians, and completely disarm 120 target ships headed for the Caroline Islands. The generals are so elated by the operation's success they don't notice when Barnhouse slips away. It's left to the student to read aloud the manifesto the professor has left behind.\n\nThe manifesto opens with the kind of pun Vonnegut could never resist. \"Gentlemen,\" the professor writes, \"As the first superweapon with a conscience, I am removing myself from your national defense stockpile. Setting a new precedent in the behavior of ordnance, I have humane reasons for going off.\"\n\nThe manifesto goes on for another page and a half. The tone is Norbert Wiener's, but the politics are even more overt. In fact, the manifesto could have come directly from a United World Federalists position paper. Barnhouse points out the fallacy of trying to forge peace by building weapons and declares that the world cannot afford a nuclear war. He chastises the generals for failing to put their faith in government\u2014in particular, the downtrodden United Nations. He explains that henceforth he will be making sure UN recommendations are carried out. And he demands that there be no more vetoes\u2014just as the Scientists' Movement had insisted a few years earlier. The story ends with the professor noting that if the world leaders don't like it, they can lump it; he's in control now.\n\nEven as it addressed the vexing ethical quandaries raised by science, \"Report on the Barnhouse Effect\" was an optimistic story. The professor is a good person, an idealist with the power to enforce his principles. He sets an example for scientists much as Norbert Wiener did: he refuses to cooperate with the war machine. If enough scientists like Barnhouse would just step forward, the madness of an arms race could be averted.\n\nProfessor Barnhouse was the first in a long line of fictional scientists Kurt Vonnegut would write into being. He was also the most unambiguously noble, born at one of the last moments when Kurt thought that politics might still turn around, that the world might come to its senses and find its way to a new, enlightened era of prosperity and peace.\n\n* * *\n\nIrving was delighted with the man the Weather Bureau had sent to destroy Project Cirrus. He loved that William Lewis seemed determined to find every flaw in their data, every weak spot in their equations. Skepticism was how science progressed. He was convinced that he and Bernie and Vince would win Lewis and the Weather Bureau over in the end.\n\nLewis spent a long time formulating his position. In June 1949, he handed the Project Cirrus team a memo about the New Mexico storm they had reported on in January. He outlined his objections to the Project Cirrus interpretation of events in five single-spaced pages with three hand-drawn charts. The main problem was simple: How could the team know for sure that they had actually caused the rain they observed? Here was the heart of the matter, the scientific flaw that was really an epistemological problem. \"It is not possible,\" Lewis wrote,\n\nin any particular instance, to decide with any degree of certainty whether the rainfall observed in an area presumably affected by seeding, was in fact due to the seeding, or would have occurred anyway. You may surmise that the rain resulted from the seeding while I surmise that the seeding had very little to do with it, but neither of us knows. There is no evidence for a positive decision either way.\n\nIn order to produce proof that cloud seeding worked, Lewis said, \"it will be necessary to design the experiments in a way that will permit the use of statistical evidence in the verification of the results.\"\n\nStatistics! It was a brilliant idea. Given the particular set of conditions, what are the odds of rain? Even Irving Langmuir hadn't thought of that. But he was absolutely thrilled. Statistical analysis was tremendously popular now that high-speed calculating machines could be used to sort data. Scientific American had even performed an analysis of the news coverage of the House Un-American Activities Committee investigation of the physicist Edward Condon by reducing assertions about Condon in New York City's nine daily newspapers to holes on punch cards and running the cards through an IBM sorting machine. It had proved, objectively, that the liberal newspapers favored Condon and the conservative newspapers maligned him.\n\nNumbers didn't lie. They couldn't be accused of bias, and they were oblivious to the prejudices of their observers. No one could refuse to believe something the numbers revealed. Irving embraced Lewis's idea at once. It might be impossible to prove what would have happened in the past. But one could prove the statistical likelihood of what did happen. And if what happened was statistically unlikely, over and over, the evidence would add up until the conclusion was irrefutable. Numbers would come to the rescue of his research. Now he just had to do one thing: teach himself statistics.\n\n* * *\n\nHere was a change: Kurt's boss was delighted with him. The microchemistry story he had written up in December was picked up by Life. The magazine even sent a Life photographer to Schenectady to shoot photographs of employees holding tiny lab instruments in their hands. It was Kurt's biggest score yet for the News Bureau. His triumph even merited a mention in the Schenectady Works News.\n\nBut privately, Kurt was far more interested in his own writing. He had finished \"Barnhouse,\" and he felt good about it. It revived him from the letdown of the \"Mnemonics\" rejection. With the new story in hand, he figured he had enough material to seek representation. Aiming high, he sent seven stories off to Russell & Volkening Inc. One of New York's most respected literary agencies, Russell & Volkening represented top-notch writers like Eudora Welty, Henry Miller, and Saul Bellow, and it had a reputation for liking offbeat fiction. But Diarmuid Russell wrote back disappointingly soon, declining to take Kurt on. The stories were brisk, he said, but he didn't think editors would like them. Still, he suggested Kurt send the Barnhouse story to Collier's and the Post.\n\nKurt wasn't one to turn down free advice. So in March, as \"his\" microchemistry story (absent his name) came out in Life, he mailed off \"Report on the Barnhouse Effect\" to The Saturday Evening Post. It was rejected almost at once.\n\nStill, he had faith in \"Barnhouse.\" Not only was it a good yarn; it was in keeping with the tenor of the times. That month, three thousand delegates were gathering at the Waldorf-Astoria in New York for a peace conference. Organizers included Albert Einstein, Arthur Miller, Lillian Hellman, and Norman Mailer, who made a sensational speech about capitalism being the ultimate cause of wars. Norman Cousins of the United World Federalists spoke as well. The State Department declared the conference to be communist propaganda, HUAC insisted it was all part of a subversive Red \"peace offensive,\" and thousands of protesters from veterans groups, religious organizations, and Eastern European immigrant groups picketed outside the Waldorf, singing patriotic songs, reciting prayers, and shouting anti-Russian and anticommunist slogans. But people were talking\u2014shouting even\u2014about peace and internationalism. His story was not just sharp; it was relevant.\n\nKurt sent \"Barnhouse\" to Collier's in April. It came back with a form rejection. But at the bottom, in an almost unreadable scrawl, was a short note: \"This is a little sententious for us. You're not the Kurt Vonnegut who worked on The Cornell Sun in 1942, are you?\" He didn't recognize the name, and he wasn't sure he wanted to own up to being his feckless undergraduate former self, so he filed the rejection away and sent \"Barnhouse\" off to Story.\n\nIt was languishing there when Jane told him she was pregnant again.\n\nTwo kids! It was a thunderbolt. If it was hard to quit a corporate job with one kid, it would be even harder with two. The noose was tightening around his neck. He had to get his writing career off the ground, or he'd be like one of the countless corporate drones who longed to write but who couldn't bring themselves to leave the security of the paycheck, the life insurance, the paid vacation, the employee discount on appliances. After twenty-five years, GE employees got thanked with a reception and a commemorative pin. The Quarter Century Club, they called it. Every year the quarter centurions had a clambake. Clams, cigars, cocktails\u2014the best of everything to honor the GE lifers. He was headed for his own commemorative clambake if he didn't get out soon.\n\nDesperation made him do what he had scrupulously avoided until now: he told one of his work colleagues about his writing. George Burns was a photographer who sometimes worked with Kurt on News Bureau stories. He had been on staff at GE once, but now he was freelance. He and his brother Jimmy had a photography studio on Schenectady's State Street. George was a fun guy, an adventurer. When GE built its new radio tower, George scaled it to get a photograph from the top. Like Kurt, he was a veteran; he'd served as an enlisted man in the Pacific. George shot the famous flag raising at Iwo Jima for Yank magazine; his was not the version that became famous, but he was easygoing enough not to care. He was standing by when Prime Minister Tojo tried to kill himself, and he was one of the first photographers to document the results of the atomic bomb. After flying over Nagasaki in a B-25, he wrote that the city looked as if giants had stomped through it, grinding every building into the ground.\n\nGeorge had never read any of Kurt's fiction, but he could relate to creative aspirations. He immediately recommended that Kurt contact a war buddy of his from Yank, a straight-up guy who had written an impressive story on the firebombing of Tokyo. The guy was an editor at Collier's now, George said; he might be able to help. His name was Knox Burger.\n\nThe name rang a bell. Kurt went home and pulled the Collier's rejection letter for \"Report on the Barnhouse Effect\" from his file. Suddenly the scribbled letters took shape: Knox Burger.\n\nKnox had been a friend of Kurt's in college, the editor of a campus humor magazine called The Cornell Widow. The Sun and the Widow were rivals, frequently taking potshots at each other in print, but Knox and Kurt had liked each other. Once, Knox had even taken Kurt's suggestion about toning down some apparently anti-Semitic caricatures of Sun editors that his own magazine had produced.\n\nKurt sat down and wrote to Knox right away. It was already June, and Jane was due in December. His letter was alternately full of self-deprecation and braggadocio. \"Sorry you didn't care for my story,\" he wrote. \"I got a typewritten letter back on it from the Post. Story has now had it for a month. You're right. It was a dog.\" But he had other things in the works, including a completed novella and another that would be done soon. He closed by saying he would be in New York for two days of the following week. Would Knox like to have lunch? Knox replied by telegram three days later, telling Kurt to call him at his office when he arrived in town. It seemed, to Kurt, as if his breakthrough might finally be at hand.\n\n* * *\n\nOn a cool July morning in Socorro, New Mexico, Bernie turned an impish yet calculating gaze to the sky. He'd done enough lab work, enough backyard experiments. Now he was determined to make a decisive demonstration of what his silver iodide generator could accomplish. He would see if he could do anything about those puffy, cumulus clouds dotting the blue.\n\nIn its second interim report, the Weather Bureau's Cloud Physics Project had not only dismissed the idea that dry ice might produce rain of economic importance but also claimed that no significant rainfall resulted from the seeding of cumulus clouds with \"persistent nuclei.\" Bernie had published a short rejoinder in May. The bureau team, he noted, had described experiments using nuclei of lead oxide and potassium iodide. His lab tests showed that silver iodide vaporized for one second was ten thousand times more effective than other chemicals in producing nuclei for the formation of ice crystals. If the bureau scientists were using other compounds, it wasn't surprising they had gotten poor results.\n\nBernie was ready to prove his case. He had perfected the method, and now, with his little homemade smoke generator, he would make an effective demonstration\u2014something that William Lewis, who had come to Socorro with them, would have to acknowledge. The problem was that Vincent and Irving had arrived in Socorro a week early to plan the summer's research, and they hadn't included any experiments with silver iodide in the program.\n\nSo, on July 21, as the others prepared for the day's dry ice seeding flights, Bernie lugged his generator to a spot out in the middle of the dry desert scrub. He fired it up at around 6:00 a.m., right after the sun came up. He took notes on the wind, the temperature, and the location of clouds in the distant sky. Then he went to breakfast. When he came back, the wind had picked up, and the burner was still humming along.\n\nNew Mexico was a stratiform world: first brown and yellow desert, then a layer of mountains on the horizon, and then the clouds\u2014low cumulus, billowing up like a second mountain range, and above them the white smear of stratus. There was so much sky here, a blank blue page on which the sun's work was written every day, then erased by night's cool dark. Every morning it started all over. The air began to warm around breakfast. By mid-afternoon it was so hot the men would strip to the waist.\n\nAt around 8:15, Bernie saw cumulus clouds forming near the Sandia Mountains, the range of peaks east of Albuquerque, and also over the Manzanos, farther south. The clouds looked like the region's typical orographic cumulus: small puffs of white formed by heat funneled up the side of the mountains. Most orographic clouds just cling to their hills like hats, growing a little but not precipitating. But as Bernie ran his generator and watched the sky, the clouds began to build. By 9:30, the cloud over the Manzanos looked heavy with ice. An hour later, it was developing into the classic anvil shape of a brewing storm.\n\nPeriodically, Bernie released a balloon to show him how the winds were blowing so he could gauge where his silver iodide smoke was headed. The balloons drifted right to the Manzanos and from there to the Sandias. That meant the winds were carrying his silver iodide smoke to exactly the places where the clouds were whooping it up.\n\nAt lunch, he told Langmuir he'd been running his silver iodide generator and that he thought it might be affecting the clouds. Langmuir, focused on the afternoon's dry ice seeding flight, had no response. He barely seemed to register Bernard's remark.\n\nBernie went back to his burner around noon. He changed its hydrogen tank and kept it running. And then he heard thunder. He looked up and saw flashes of lightning. The air beneath the Manzano clouds was dark with virga: wisps of rain that fall and evaporate before reaching the ground. It was working! Before long, he could see a line of cumulus clouds building up to the northwest. The entire area where his silver iodide was going was a mass of roiling storm clouds. He was so excited he kept forgetting to use military time in his notes. He turned his burner off at 1:30. He didn't want to interfere with the team's dry ice seeding from the air, which was scheduled to start over the Sandias at around 3:00.\n\nSchaefer, aboard the B-17 that afternoon, was surprised to find that the cumulus clouds they planned to seed were already a seething mass of thunderheads when they arrived, pouring down rain as they boiled their way northeast. Even more surprising would be the data from Albuquerque's nearby radar station showing a radar echo of precipitation appearing in the silver iodide\u2013seeded cloud at around 10:00 a.m.\u2014exactly when Bernie's calculations said the silver iodide should be reaching it. The precipitation area began by covering about one square mile at 20,500 feet\u2014atypically low for New Mexico. It then expanded with mind-boggling speed, increasing to seven square miles in four minutes and rising to 34,000 feet in another two. It was one of the most dramatic storms they had ever\u2014possibly\u2014made.\n\nThat evening, Bernie told Langmuir again that he had been running his generator that morning. All at once, the information seemed to break through Langmuir's interior monologue. The scientist stared at Bernie.\n\n\"You were running the generator today?\" he asked.\n\n\"Yes,\" Bernie said.\n\nWith that, Langmuir's attention was finally caught. He wanted to know everything. How often had the generator been running? Had he been running it before? Had the October dry ice experiments possibly been affected by silver iodide from the ground? All of their data needed to be reassessed in the light of this new fact: silver iodide worked. It had been working all along. More silver iodide experiments needed to be done. The summer's program had to be rewritten. That night, Irving stayed up late, dictating eight thousand words of notes. \"Taking all in all,\" he concluded, \"I feel that the evidence is now practically overwhelming. The evidence for the production of rain, and even thunderstorms, by silver iodide seeding is now about as certain as the opening up of clouds by stratus cloud seeding.\" In his diary, he was more succinct: \"Vonnegut made a thunderstorm.\"\n\nThe next day, Langmuir flew to Los Alamos, elated. He'd started doing the calculations on Bernie's thunderstorm, and they looked tremendous. The silver iodide seeding, he figured, had brought millions of gallons of water down on New Mexico.\n\nOften when he flew, Irving offered to take the controls. The FAA had ruined flying for him: when the regulations decreed that logbooks had to be a particular color, he'd sold his plane in disgust. No one was going to tell Irving Langmuir what color his logbook should be. But he still liked to practice when he had a chance. Today, however, he was more interested in looking out the window. Eagerly, he scanned the New Mexico desert unfurling beneath them. Near Santo Domingo, he saw what he was looking for: a swollen, roiling river entering the Rio Grande.\n\n\"What river is that?\" he shouted to the pilot, pointing it out.\n\nThe pilot peered out.\n\n\"Galisteo Creek,\" he yelled back. He said he was surprised to see so much water in it: its streambed was usually dry. Langmuir noted this with satisfaction. He would get stream flow data. He'd get rain gauge data. The numbers would bear out what he was seeing beneath the plane: Vonnegut's thunderstorm still filling the desert's dry wash. Here, finally, was something on the scale they'd been imagining all along.\n\nWhen he arrived at Los Alamos, Irving excitedly described Vonnegut's thunderstorm to the Manhattan Project physicist Edward Teller. Teller had little patience for Langmuir's effusions: he was once again deeply embroiled in debates over his thermonuclear bomb, or \"Super.\" He'd wanted to build the Super since the start of the Manhattan Project, and the objections of the peace-minded Scientists' Movement had not changed his mind. Teller was born in Hungary, and the implosion of Europe had cast a pall on his spirit, as it had for many expatriates. He had little patience for talk of UN control or international disarmament. To him, the only option was to stay a step ahead of the Reds by developing the Super at once.\n\nJ. Robert Oppenheimer disagreed, as did David Lilienthal, head of the Atomic Energy Commission, and many of the scientists who had rallied around the cause of peacetime arms control. Many in the public, too, still held hopes for peace and world government. But since 1946, a different consensus had been growing in Washington, one that foresaw the world not uniting but dividing into two armed camps. Events in Berlin only solidified this conviction. The journalist Walter Lippmann had just given the situation a name in his new book: The Cold War. That was helpful to Teller. He felt certain that the political climate was going to swing his way in the end. Even Oppenheimer, he was convinced, would support the Super if the problems with the physics were solved.\n\nAs Langmuir enthused about his storm's destructiveness, Teller grumpily wondered if the GE chemist saw rainmaking as \"competition\" for the atomic bomb. It was a surprisingly insightful thought: Langmuir did see weather control as trumping atomic fission. As he liked to point out, a storm system like a hurricane contained hundreds of times as much energy as an atomic bomb, even a hydrogen bomb. Controlling that force would be an achievement as great as\u2014if not greater than\u2014harnessing the atom.\n\nAnd it excited people. For all his absentmindedness, his ability to lose himself so deeply in his thoughts that he seemed barely aware of other people, Langmuir liked to communicate. He was a scientific missionary, surprisingly adept at explaining complex scientific ideas to laymen. He liked elucidating things for children, for lab visitors, for anyone who would listen. His proselytizing for science never stopped. When he held toddlers, he bounced them on his knees, first with both knees in sync, then with his knees alternating. \"In phase,\" he would chant, \"out of phase!\" Kurt Vonnegut once overheard him saying that anyone who couldn't explain his work to a fourteen-year-old had to be a charlatan.\n\nHis zeal for science had led Langmuir to invent many things in his career\u2014the gas-filled incandescent lamp, the mercury-condensation vacuum pump, the atomic-hydrogen welding torch, the thoriated tungsten filament. But not one of those achievements had ever captured the public imagination\u2014until now. Even winning the Nobel Prize paled in comparison with taking charge of the climate. Langmuir believed it would be his legacy to humanity.\n\nAnd if it topped the atomic bomb in its peacetime value, it only stood to reason it would be valuable in war as well.\n\n* * *\n\nIt was awkward at first, the lunch between Knox Burger and Kurt Vonnegut\u2014a meeting of two men who had known each other in another life, on another planet: Planet Cornell. Planet pre\u2013World War II.\n\nKnox Burger was a man's man with a sharp tongue. Some people found him brusque. He had a craggy face and balding pate and an appreciation for good booze and pretty girls. He spent so much time fly-fishing in cold creeks he said his feet never got warm. After work, he could frequently be found at the Algonquin, holding forth for a crowd of publishing types. You could locate him by his roaring laugh.\n\nThe two men liked each other. Like Kurt, Knox had enlisted, and in the Army he had become a staff correspondent at Yank, the magazine by and for enlisted men. He had flown with a B-29 bomber squadron, then gone to Yank's Saipan bureau. After the Japanese surrender, he moved to Tokyo to cover the occupation, writing smartly reported, man-on-the-street stories about Americans in Japan, delving into the cultural meanings of things like tea and geishas. But he also wrote a hard-hitting account of the firebombing of Tokyo. At the time, his estimated death toll of 100,000 struck many as extreme, but later studies validated it.\n\nSince the war, Burger had gone to grad school, gotten married, and written stories and articles. He had come to Collier's in 1947 and became its fiction editor the following year.\n\nCollier's was a general-interest magazine, not unlike The Saturday Evening Post. It had around 2.5 million readers who enjoyed its mix of fiction, nonfiction, cartoons, and photo essays. Its spunky history of muckraking appealed to Kurt. Postwar, though, there was less of that. The Collier's creed, published monthly on the editorial page, was \"to keep always before its readers a high, sane and cheerful ideal of American citizenship.\"\n\nKnox gave Kurt a tour of the magazine's editorial offices, treating him like a serious writer, not like a guy with little more to his literary credit than a stack of rejection slips. At lunch, Kurt told Knox about his novella and his desire to write slick short fiction. Knox suggested Kurt try his hand at short-shorts\u2014stories under about two thousand words. Collier's published one short-short every week, usually with three regular-length stories and two installments of serialized novellas.\n\nKurt wasn't about to waste time. He bought that week's Collier's and read it on the train home. There were a couple of adventure stories, a romance, an adventure-romance, and a lighthearted short-short by James Kirch called \"Morning After,\" in which a man mollifies his angry wife after a party by insulting all the other women there. The story Kurt liked best was a sentimental yarn by Robert McLeod about a neglected little girl whose father is attending college on the GI Bill. The girl believes\u2014with misplaced optimism\u2014that her inattentive parents are planning a surprise birthday party for her and invites everyone in Vets Village. The narrator, also a vet, swaps cynical commentary with his wife about the whole sad spectacle, but in the end each plans a birthday surprise for the little girl.\n\nIn concept, the stories weren't far off from what Kurt had been writing, but their execution was better. Back at home, Kurt started working on a short-short right away. Two days later, he sent Knox his novella \"Basic Training,\" complimenting the Robert McLeod story in his letter. The very next day, he finished the short-short \"Bonanza\" and sent that too, as well as \"Mnemonics,\" the office story about sadistic fantasies, which The New Yorker had rejected. Then, four days later, even though he had yet to hear back on the three things he'd already submitted, he sent another brand-new short-short, \"The Case of the Phantom Roadhouse.\"\n\nTwo days later, \"Phantom Roadhouse\" came back, with a note saying it didn't come together correctly at the end. Then Knox's assistant returned \"Bonanza,\" calling it \"flat and inconsequential.\" The next day, Knox returned \"Mnemonics,\" declaring Kurt's pen name, Mark Harvey, the most pompous he had ever come across. In spite of the mockery, it wasn't all bad news. He thought \"Mnemonics\" had potential, and he might be able to use it if Kurt did some more work\u2014and then he went on to give three paragraphs of detailed instructions for fixing it. The mnemonics course needed more description. Perhaps Kurt could have Alfred try boosting his memory by imagining a pleasant scene, then switch to something violent. That would help readers in the hinterlands understand that Alfred was actually enjoying his sadistic reveries. Then the last fantasy needed more work. He should link it to the office. The wife should provoke not just hostility but something pleasant. Unless he wrote in a secretary and made the pleasant thoughts about her.\n\nIt was a barrage of criticism. But Knox was encouraging him to rewrite, and Kurt wasn't about to waste a chance like that. He went through Knox's letter and typed out his suggestions in a numbered list. Then he went down the list, revising. Four days later, he sent the story back.\n\n\"I seem to be sending MNEMONICS back for further revision,\" Knox replied two days later. He liked Kurt's changes, but more were now needed\u2014a full single-spaced page of them. The hero was not likable. His boss and his wife should be more annoying, to justify their savaging in Alfred's daydreams. And perhaps there could be a grace note at the end.\n\nKnox's letter was detailed and patient, explaining his changes and why he wanted them. He had a good sense of story structure, of how the reader should be carried along. But he was a magazine editor who needed to acquire and edit six stories a week. He didn't have time to teach Kurt Vonnegut to write. So at the end, he mentioned that he had given Kurt's name to Kenneth Littauer, of the literary agency Littauer & Wilkinson. \"I told him I thought you might turn out to be a skillful and prolific writer,\" he said.\n\nKurt knew he was at a crossroads. Again he typed out the suggestions in the form of a list and made his way down it, checking each item off as he made the corresponding change. Then he sat down to type a cover letter to Knox. He chose his words more carefully than usual. First he thanked him for being extraordinarily patient with him. Then he crossed that out with a row of x's. He wrote that he felt like a pilot who had been talked down from a foggy sky by air traffic control. But he crossed that out too. He finally settled for thanking Knox for giving him reasonable instructions that were nearly as long as the story itself. He hoped he had managed to satisfy them. And he thanked Knox for mentioning him to Kenneth Littauer. He would definitely like an agent.\n\nAfter all that, Collier's rejected \"Mnemonics.\"\n\nThe head honchos, Knox reported, thought \"the net result lacks conviction. It just leaves a bad taste in your mouth.\" He also rejected \"Basic Training.\" Kurt had sent another short-short in the meantime called \"Robot Cop,\" pointing out in his cover letter that GE really was developing robots, so it wasn't far-fetched. Knox returned it almost immediately.\n\nKurt refused to get discouraged. Two days later, he sent Knox \"City.\" Knox apologized for throwing up so much smoke without making a fire\u2014and rejected that too.\n\nKurt, meanwhile, had gotten in touch with Kenneth Littauer, and Littauer agreed to take him on as a client. \"City,\" Kurt told Knox with some satisfaction, should be the last piece of his to come direct. After this, Knox would be hearing from his agent.\n\nBut first, Kurt started hearing from his agent.\n\nKenneth Littauer was old-school, a sweet talker when it came to wooing clients or charming editors. But when he rolled up his sleeves and got down to work, he didn't pull his punches. In August, Kurt sent him a story called \"Enterprise,\" along with \"Mnemonics\" and \"Report on the Barnhouse Effect.\" \"Barnhouse,\" Kurt pointed out, had received encouraging rejections from both the Post and Collier's. He figured it had the best chance of being bought, if only because it expressed an enthusiasm for world government that was presently in the air.\n\nHe was referring to enthusiasm for the United Nations, which had finally brokered an armistice between Egypt and Israel. Collier's itself had recently run a feature article by the New York Times reporter A. M. Rosenthal defending the UN. And although an unpleasant civil war was brewing in Korea, hope for peace through world government had not yet died. State legislatures across the nation were passing pro-world-government resolutions; in September, the Schenectady County board of supervisors would vote unanimously to do the same.\n\nKenneth Littauer didn't give a damn if the entire world voted to abolish nation-states and sing peace anthems under the UN olive branch. Stories had to be good on their own terms, and Kurt's weren't. Littauer didn't much like \"Mnemonics.\" \"Enterprise\" was revolting. As for \"Barnhouse,\" it was better, but not good enough. Maybe it could be fixed. After all, it started out fine. But it failed to resolve itself like a short story and instead, with its wordy manifesto, flamed out in an onslaught of political speechifying. No editor would stand for such a violation of the basic rules of storytelling\u2014not even if he agreed wholeheartedly with the sentiment behind it.\n\nThe difference between a writer who makes it and one who doesn't is the ability to take criticism. Not just to take it, but to learn from it. Kurt could have taken umbrage at his new agent's dismissive tone toward his most promising work. But he didn't. He wrote back thanking Kenneth for letting him know how the story really ended. He agreed that \"Enterprise\" was revolting. A dog, in fact. He softened the blow by partly blaming Jane. The two of them had looked at some issues of Redbook and concluded that was what the women's magazines wanted. He promised not to try anything like that again.\n\nAnd then he went to work on Professor Barnhouse.\n\n* * *\n\nBernie was finally making progress. The Project Cirrus team had come back from New Mexico more excited than ever. The program of research they were planning gave a new prominence to seeding experiments with silver iodide.\n\nIn August, the Albany Times-Union ran a multipage spread in its Pictorial Review section about the coming era of weather control. GE's work, the article said, had brought the world to the brink of a new age in which rainfall and snowfall would be precisely determined in advance, great storms would be countered by \"anticyclones,\" damaging hailstorms would be bombed out of the sky, artificial hills would draw rain to arid deserts, and cold climates would be warmed up by deliberately melting the polar ice caps and diverting the Gulf Stream. A futuristic drawing superimposed on the Manhattan skyline showed how entire cities would reside under climate-controlling \"roofs\" in which scientists equipped with telescopes, rockets, and astronomical observatories would dole out the sunlight, manage humidity, and carefully calibrate the rain.\n\n\"Only the descendants of the men and women who believed that airplanes would never fly, that atomic fission was impossible and that men would die if they ever traveled at 60 miles an hour, doubt that one day man will achieve considerable control over local weather, to the extent of being able to decide where, when, and how much it shall rain or snow,\" the paper declared. This imminent state of affairs meant that international weather disputes were likely, unless an overriding authority, a \"Meteorological United Nations,\" took charge.\n\nThe article was illustrated with photographs from the GE News Bureau: Vincent holding a cloud meter, Bernie igniting fireworks from his silver iodide burner, Irving discussing photographs of snowflakes with a lab visitor. And, of course, there was a photograph of military officers gazing into Vincent's freezer as if it were a crystal ball.\n\n\"'Command of the air,' in the weather sense, over enemy territory will become a decisive weapon,\" declared the piece, \"and it is no secret that, because of this, some nations have given the highest priority to research into weather control.\"\n\nThat month, the Soviets exploded their first atomic bomb.\n\nRainmakers\n\nMarion Mersereau Langmuir had always been game. Her courtship with Irving, consisting of grueling mountain hikes, frigid ski trips, and scrabbles through dirty caves, had won the scientist's affection away from her prettier, but less intrepid, sister. As Langmuir's wife, she was the ideal helpmeet, joining him on his travels or staying home with their children, Kenneth and Barbara, while he visited laboratories and attended symposia. She indulged her husband's obsessions, even hitching up her skirts to climb into a canoe and paddle around Lake George with a payload of boulders to armor shorelines he thought were eroding. She didn't fuss when, after a long time away, he absentmindedly left her a tip on the breakfast table, and she didn't even put her foot down when Irving tried to cure Kenneth of a cold by immersing him in hot baths to induce a fever. But when Irving went outside and began setting all their toilet paper on fire\u2014on a weekend when their vacation house was full of guests\u2014she might have thought he had gone too far.\n\nEver since the Project Cirrus team had returned from New Mexico, Bernie had been coming up with new methods for employing silver iodide. On July 28, he impregnated some tissue paper with silver iodide and burned it in the lab. A couple of hours later, Katharine Blodgett pointed out to him and Irving that enormous banks of cumulus were rolling into Schenectady from the northwest, west, southwest, and southeast. A delighted Langmuir stood at the window watching for half an hour. He was absolutely convinced: Bernie had called up those clouds.\n\nIrving asked Bernie to make him a small bottle of the silver iodide solution, to take along to his house on Crown Island in Lake George. There, on Saturday, he proceeded to impregnate his home's entire supply of toilet paper with the chemical, laying each sheet in the sun to dry. He used single-thickness and double-thickness paper. The silver iodide, he noted, gave the toilet paper a distinctly yellow color. Once the sheets were dry, he held them one at a time in kitchen tongs and set them alight.\n\nHe started burning toilet paper at about 12:20. Less than an hour later, a large cumulus cloud developed over French Mountain, at the south end of Lake George. From his vantage point in the middle of the lake, he watched the small thunderstorm develop, travel southwest, then reverse itself to head northeast. Rain fell on the eastern shore. By 3:20 it was all over.\n\nEagerly, he burned another ten sheets of toilet paper. Once again, fifty-five minutes later, he saw a cloud forming over French Mountain. It started raining, and this time there was more thunder. Irving went out onto the lake in his motorboat, Penguin. The storm was unlike any he'd ever seen in the thirty years he'd had his house on Crown Island. The wind was blowing hard from the east, and rain was pounding the east side of the lake, but the west side was perfectly dry.\n\n\"I believe that silver iodide has done the trick,\" he wrote in his notebook. If silver iodide seeding could be done so easily, he noted, there was little hope for government regulation of weather control. After all, anyone with a little chemical know-how and a few rolls of toilet paper could call up a local deluge.\n\nIn late August, Irving sat down with Guy Suits and told him what he was thinking. Convinced by William Lewis that a statistical approach would be the only way to prove his results, he had ordered himself a statistics textbook and gone to work proving that Bernie's New Mexico thunderstorm had produced the heavy rains that followed. Combining their data with all the rainfall and stream gauge figures he could get his hands on, he had come up with a startling conclusion: Bernie had not only triggered a rainstorm back in July; he had produced all the rain that fell on the entire state for twenty-four hours after his generator ran. Irving proposed doing massive experiments with silver iodide, increasing the amounts until they got definite weather modification results over huge areas\u2014say forty thousand square miles. They had found the key to large-scale weather control.\n\nSuits was alarmed. Langmuir's claims were sweeping, and he was proposing that GE scientists do exactly what Suits and the GE lawyers had ordered them not to do: attempt to effect big changes in the weather. The lab director went to Bernie and Vincent separately and asked them if they agreed with Irving's conclusions. Had their experiments really caused such huge effects?\n\nBernie had been there when his generator called up the storm. He had seen it happen. But he was, at bottom, a scientist, and while the results were good, they were not yet definitive. So, like Vince, he gave a guarded reply. Nothing had been scientifically proved, he told Suits. The experiments simply suggested that they should keep exploring cloud seeding with silver iodide.\n\nBut Irving, now armed with a basic knowledge of statistics, was irrepressible. He divided the entire state of New Mexico into octants radiating outward from Albuquerque and began collecting the Weather Bureau rainfall data for all of them. He needed to establish the statistical proof that their cloud seeding had modified the weather. Once he had done that, he could go on to tell the world the new idea that was brewing in his head: he was becoming more and more convinced that the weather was going to be easier to control than to predict. That's because it was a perfect example of a divergent phenomenon.\n\nFor decades, Langmuir had been working out a theory of two types of phenomena in the world: convergent and divergent. He defined convergent phenomena as those \"in which the behavior of the system can be determined from the average behavior of its component parts\"\u2014in other words, phenomena in which the normal laws of cause and effect apply. If you do X, you can be pretty sure the result will be Y. Convergent phenomena are things like the laws of classical physics and the orbits of the planets. Divergent phenomena are \"those in which a single discontinuous event (which may depend upon a single quantum change) becomes magnified in its effect so that the behavior of the whole aggregate does depend upon something that started from a small beginning.\" In other words, the Xs can be so minute that the Ys become impossible to predict. Divergent phenomena are things like history, economics, and weather.\n\nJohn von Neumann used the words \"stable\" and \"unstable\" to talk about something similar. Stable systems were deterministic and resisted disturbance by small things. They were like a game of checkers: once a being\u2014or a machine\u2014with enough calculating power learned all their laws, that player could not lose. Unstable phenomena were upset by small disturbances, making them hard to predict. But sufficient calculating power would allow these processes to be controlled. Once scientists determined where exactly the instability began, they could intervene, introducing the disturbances that produced the desired outcome instead of the undesired one. The weather, Johnnie liked to say, contained both stable and unstable systems: ultimately, the stable systems would be predicted, and the unstable ones would be controlled. It was simply a matter of knowing when and where to chalk the X.\n\nUnlike von Neumann, Langmuir saw all weather as fundamentally unstable. It was always too big and unwieldy a system to be captured by equations. Divergent phenomena flew in the face of the clockwork universe. No matter how much information you had, you could never accurately predict their behavior.\n\nNorbert Wiener had been telling von Neumann something similar. Wiener thought that Johnnie misunderstood weather: he failed to take into account that his model would never have complete information about the atmosphere. The atmosphere was always in flux, always flowing and blending; a touch in one part could affect the next and then the next until a cascade of changes set up movements on the other side of the earth. How, even with the fastest computers, could you ever account for all of it? How could you even see all of it? Even with a network of meteorological sensors blanketing the earth, the accuracy of your measurements would never be perfect. Your data were thus always on some level approximate, which meant your forecasts, too, were approximations. And the instabilities in those approximations multiplied rapidly. There would never be an absolute solution, as there could be with a zero-sum game, only a series of statistical likelihoods.\n\nAll three men understood something that would take physicists another couple of decades to develop into a theory: the idea that tiny and ultimately unpredictable events could alter the course of huge systems and that this made the behavior of such systems difficult to predict. But Langmuir\u2014like von Neumann\u2014still hoped that control was within his grasp. In fact, he believed he had found the place to chalk his X on the atmosphere; he just needed to prove to others that it was there.\n\nWhat he needed was a signal, something to break through the noise and show that his experiment was having an effect. The idea he came up with was brilliantly simple: have Bernie's generator pump out silver iodide on a regular basis\u2014say on three specific days each week\u2014and see if that introduced a periodicity into the weather. A regularity that matched his seedings in the vast, unpredictable atmosphere would be evidence of the human hand as unmistakable as signing the GE monogram in the sky.\n\nSuits had a different idea. He called the Project Cirrus representatives from the Signal Corps and the Navy into his office and reemphasized the GE position: company employees must never engage directly or indirectly in seeding experiments that might lead to large-scale weather modification. They were advisers only. All action taken in Project Cirrus was taken by the government alone.\n\nThen he dictated a letter to Irving, Bernie, and Vince that laid out his position in no uncertain terms. Any extensive seeding experiments, he reminded them, \"should be carried out entirely by the Government and its employees.\" GE's role\u2014as it had been from the beginning\u2014was only \"to appraise [sic] them of the facts and on the basis of these facts to let them make their own decisions in this field.\" He appended a copy of the letter he had sent the team in early 1947 stating that they were merely advisers. And he gave them an ultimatum. If, he said, \"the program develops in such a direction as to subject the Company to serious hazards from a liability standpoint, it may very well become impossible for us to continue with this work.\" In other words, toe the line or GE would kill Project Cirrus.\n\nThen, to leave no doubt that he was serious, he declared that the company would not approve for publication any papers or reports that made it sound as if GE employees were engaging in widespread weather modification. Their job from now on was to keep quiet about what they knew.\n\n* * *\n\nKurt came in the front door after work, and there it was, propped up on the piano. An envelope\u2014a check envelope\u2014addressed to him from Collier's. Jane had put it on the upright so he'd see it right away.\n\nIt had finally happened.\n\nExcept for his marriage to Jane, Kurt had never had a more productive working relationship than the one with Kenneth Littauer. The agent had gone to work on Kurt's stories as if they were race cars and he were a pit mechanic at the Indy 500. They started with \"Barnhouse.\" Kurt revised the story as Littauer directed, and Littauer sent it back to Knox Burger. Knox had lots of changes and thought it was too long. Nor did he like Kurt's latest pseudonym: David Harris.\n\n\"Why doesn't he use his real name on the story?\" he wrote to Kenneth. \"Afraid GE will fire him for a Red?\"\n\nIt wasn't an idle concern. Truman's announcement of the Soviet nuclear test in September had jolted Americans out of a comfortable feeling of supremacy, and GE, like the nation, was swinging rightward. The shock was not mitigated by the fact that the scientists had been predicting this development for years. Irving Langmuir had even foreseen exactly how much time it would take.\n\n\"Ever since atomic energy was first released by man,\" the president declared, \"the eventual development of this new force by other nations was to be expected.\"\n\nOnce people had a technology, it seemed impossible not to use it. There was no Professor Barnhouse in the real world; the suicidal course of history could not be changed. Now the nation was winding itself up with paranoia over Soviet aggression and communist infiltration. GE printed two fretful essays as a special supplement in the September issue of Monogram: \"Should We Fear a Welfare State?\" (the answer was yes) and \"Free Men Forge Their Own Chains!\" The word from the corporate higher-ups was clear: in the face of this new and frightening world, America must get tougher, not nicer. The hope of peace through internationalism was fading, as was the notion that winners should give losers a helping hand.\n\nWith the specter of nuclear war once again haunting the public sphere, Kurt had gone back to work on \"Barnhouse.\" In October, he fired a revised version off to his agent. The manifesto was gone: all that remained was the punning opening paragraph. And instead of fizzling out in a political tantrum, the story ended with what happened next: Barnhouse destroys the world's arms stockpiles from his hiding place and sends the student-narrator a letter tipping him off to the secret of the Barnhouse effect so he can carry on the work when Barnhouse is gone. On page 1, Kurt crossed out \"David Harris.\" He wrote \"Kurt Vonnegut, Jr.\" at the top.\n\nKurt knew in his heart it was good. Jane did too. But, he had told Littauer, he would await confirmation of its worth in a more interesting form: money.\n\nNow in late October, at long last, it had arrived: a check for $750 (minus Littauer's well-earned 10 percent). Nothing, not even gin, is as good as a check for soothing a writer's anguished soul\u2014especially a check that adds up to two months' labor at your mind-numbing corporate job. Of course they threw a party, spending a good part of the money on food and booze. But as Jane giddily told a neighbor, they'd just eat cereal until the next story sold.\n\n\"I think I'm on my way,\" Kurt wrote in a triumphant letter to his father. It would only take four more such sales to have the equivalent of a year's pay at GE. And then, he declared, he would \"quit this goddamn nightmare job, and never take another one so long as I live, so help me God.\"\n\nA God in whom he did not believe. Still, his father shellacked the whole boast to a piece of Masonite with a quotation from The Merchant of Venice: \"An oath, an oath, I have an oath in Heaven: \/ Shall I lay perjury on my soul?\"\n\nKurt was happier than he had been since he was first accepted at Chicago. It had all been worth it: the writing on nights and weekends, the rejections, the revisions that went on and on. He hunkered down to write the four more stories that would add up to a year's salary and issue him a ticket to a better life\u2014a life that would finally be plotted by him.\n\n* * *\n\nBernie was listening to a talk in a room at the enormous Hotel Jefferson in St. Louis. Harry Wexler was showing slides and describing a startling atmospheric event that he actually did believe was man-made: the Donora smog.\n\nIt happened on October 27, 1948. A thick fog had settled on Donora, an industrial town on the Monongahela River outside Pittsburgh. Persistent fogs are often due to temperature inversions, where cold air settles near the ground and a layer of warmer air lies above, instead of the more common reverse. Without the usual mixing mechanism\u2014warm air rising, cold air falling down\u2014the atmosphere becomes stable and stagnant. What made that disastrous in Donora was the presence of two belching factories: the American Steel and Wire plant and the Donora Zinc Works. The effluent pouring out of their stacks was bad on a normal day, peeling paint from buildings and depositing a layer of grime on local homes. But on this day, and the ones following, the smokestacks' pollution had nowhere to go, and Donora began to fill with acrid smog. Before long, the town had to keep the streetlights on all day because the smog was so thick residents could barely find their way around.\n\nOn October 30, people began to die. Thousands more poured into the local hospitals, complaining that they couldn't breathe. The fire department went from house to house, giving oxygen to those in distress. They had a hard time finding their way. People tried to evacuate, but thick smog and heavy traffic made getting away nearly impossible. Twenty people died in fourteen hours, in a town where thirty usually died in a year.\n\nThe nightmare ended when a cold front arrived and rain and wind cleared the air. But the story had grabbed the nation's attention. Air pollution was something people were just beginning to talk about seriously. The Donora smog showed it could be deadly.\n\nBernie was compelled for another reason. This was the first time the AMS meeting had included an all-day session on pollution. His paper came after Harry Wexler's, and he talked about using the condensation nuclei counter he had developed for Project Cirrus to measure smog. For Bernie, this was an exciting new development. The work he was doing for Project Cirrus could be used for even more beneficial purposes than rainmaking, such as investigating air pollution.\n\nHe wasn't declining to work on weapons. But he was finding other applications for his research.\n\nThe media continued to see Project Cirrus as a weapons program. A month earlier, Harper's had run a long article about rainmaking, emphasizing its military significance. Had Project Cirrus been around in World War II, it declared, it might have been able to clear the cold fog shrouding the western front in 1944, preventing the Battle of the Bulge.\n\n* * *\n\nSomething had sent Kurt back to his war material. Only this time, he was trying to make it funny.\n\nEvery spare moment was spent at his desk, filling ashtrays and pounding keys. He knew what Collier's wanted now; he just had to crank it out. Page after page rolled off his typewriter, the kinds of pages he thought would sell. The problem was that they weren't selling. Knox Burger had encouraged him, but now he wasn't satisfied with anything Kurt wrote.\n\nOnce you got an agent, it was supposed to be easy. Once you cracked the formula and sold your first story, it was supposed to be easy. Once you made friends with an editor, it was supposed to be really easy. It didn't take Kurt long to realize the central truth of being a writer: it's never easy.\n\nKnox had suggested he submit short-shorts, and Kurt liked that idea, because he could write one in a single sitting. But Knox could reject them as fast as Kurt could turn them out. So Kurt revised \"City\" again and sent it to Kenneth. Kenneth thought it was a decided improvement over the two earlier versions, though he still didn't love the story. \"I don't guarantee it,\" the agent wrote to Knox, \"but neither do I despair.\"\n\n\"Not quite good enough,\" Knox fired back.\n\nIt was Kenneth collecting the rejections now, usually with a line or two of explanation: \"slight,\" \"a trifle stilted and precious,\" \"would be better if he did not attempt to pander to popular tastes,\" \"over-obvious and didactic.\" He softened the blows in writing back to Kurt.\n\nIn December, as the birth of his second child approached, Kurt had gone back to a story he'd drafted a few times, \"Das Ganz Arm Dolmetscher.\" Starting with the title\u2014a purposeful butchering of the German for \"the very poor interpreter\"\u2014it was a strange little tale, a comic piece about an infantryman on the western front who gets appointed chief interpreter for his battalion even though he can't speak German. It was a return to Kurt's war material but with a lighter, more ironic tone than he had managed in the past. There was even a hilarious playlet in the middle in which the narrator imagines seducing the Belgian burgomaster's daughter with the only lines of German he knows\u2014the first stanza of Heinrich Heine's \"Die Lorelei\" and some commands from the Army phrase book:\n\nDOLMETSCHER (to BURGOMASTER'S DAUGHTER): I don't know what will become of me, I am so sad. (Embraces her.)\n\nBURGOMASTER'S DAUGHTER (with yielding shyness): The air is cool, and it's getting dark, and the Rhine is flowing quietly.\n\n(DOLMETSCHER seizes BURGOMASTER'S DAUGHTER, carries her bodily into his room.)\n\nDOLMETSCHER (softly): Surrender.\n\nBURGOMASTER (brandishing Luger): Ach! Hands up!\n\nDOLMETSCHER and BURGOMASTER'S DAUGHTER: Don't shoot!\n\nKurt thought it was the funniest thing he had ever written. In fact he'd spent too many evenings when he should have been writing a new story giggling over the pages of this one.\n\nUnfortunately, Knox Burger didn't agree. He called the playlet hackneyed and suggested Kurt get rid of it. He also didn't like the opening, and he insisted that Kurt get the thing down to five pages. Kurt sat on his pride and managed to redraft the story and send it to the typist before December 29, when Edith was born and the Vonneguts became a family of four.\n\n* * *\n\nBernie was at New York's Hotel Astor giving a paper on another nifty gadget he'd designed: a vortex thermometer to measure true air temperatures from an airplane. Irving Langmuir was there to give a paper too, and Bernie was probably the only person present who knew the stupendous bombshell that was going to be dropped in it.\n\nNineteen fifty was shaping up to be an annus mirabilis. Early in the New Year, the Schenectady weather, as if mimicking the increasing chilliness of world affairs, had turned viciously cold. An ice storm grounded the Project Cirrus planes. The Schenectady Railway Company's B29 bus hit a patch of ice on Alplaus Avenue, slid for half a mile, and crashed through the guardrail of the bridge near Bernie's house. It was only prevented from tumbling into the creek ravine below by one of Bernie's trees. For years, Bow had been urging him to cut the tree down, and now it had probably saved the lives of the four people aboard. Newspapers published the dramatic photograph of the bus, its front wheels hanging off the bridge, its flat nose pressed into the tree trunk.\n\nEven that was not as surprising as the January 7 announcement of Irving's retirement. Irving had always scorned the very idea. But his purpose in \"retiring\" was not to cease work. It was to redouble his efforts on weather control. He would keep his office at GE and continue to work as a consultant for the company. But his energies would be dedicated to Project Cirrus.\n\nLangmuir was still convinced that he was doing the most important work of his career, and at sixty-eight he might not have many years left in which to do it. His two older brothers had already died; his younger brother, Dean, collapsed and died of a heart attack just four days before Irving's retirement luncheon. He himself had had an operation for colon cancer before the war and had undergone his second cataract operation in December. His recovery time had been longer the second time around. He was slowing down, and he wanted to conserve his energy for what mattered. In announcing his retirement, he had told reporters that he fully expected to be diverting large snowstorms and rainstorms from population centers within three years. It was just the sort of thing GE wouldn't have allowed him to say.\n\nAnd now, at the annual AMS meeting, Irving was going to deliver what he considered the most important paper of his career. He was going to describe Bernie's New Mexico thunderstorm for the assembled meteorologists, and in it he would announce that Bernie's silver iodide\u2013induced storm had produced 320 billion gallons of rain on one day\u2014more than enough to fill all of New York City's reservoirs. This was bound to be of great interest in a city that was running out of water.\n\nIn 1950, New York City had nearly 7.9 million people. While the city's expansion was not as explosive as it had been in the nineteenth and early twentieth centuries, the population was still growing steadily, increasing water demand. Individual water usage was up too. But the water supply infrastructure was out of date: the last new reservoir had gone into service in 1926. It was the second of two in the Catskill Mountains, a hundred miles north of the city. Construction on a planned four more in the Catskills watershed had been halted, first by interstate wrangling with New Jersey and Pennsylvania and then by the war. By 1947, three of those reservoirs were under way. The city was racing to get them built when the drought hit.\n\nIt started with a dry winter in 1949, followed by a dry summer and fall. By early 1950, New York had been in a state of water crisis for nearly a year. Car washes, sprinklers, and swimming pools were banned. Residents were urged to put barrels and rubber rafts on their rooftops to collect rainwater for dishwashing and toilets. Wearing \"Save Water\" armbands, volunteer \"conservation commanders\" went door-to-door ferreting out leaky plumbing. The city instituted \"Dry Fridays\" and \"Thirsty Thursdays,\" where people were meant to avoid bathing and even drinking water. The Central Park Zoo rationed its hippo's water, and Tiffany used gin for the pool in its window display. The mayor had stopped shaving. But none of it seemed to be working. The New York Times had instituted a regular feature called \"The Water Situation\" that tallied rainfall and reservoir levels and noted how many days were left before water pressure failed. When Bernie and Irving arrived in New York, the Times had given the city sixty-six days before its taps went dry.\n\nIrving began his talk with a survey of natural precipitation but quickly moved on to describing the artificial seeding process. Cloud seeding, he pointed out, could produce too many ice nuclei, actually preventing rain. In some cases, a single pellet of dry ice might do a better job than too much of it. Heavy rain, he declared, can often be brought down \"by using a single pellet of dry ice shot into the side of a cloud\" from a flare gun. He discussed where to aim the pellet, the altitude at which the clouds were best seeded, and the rate at which dry ice should be dispersed into the tops of clouds. These specifics, he explained, were why the U.S. Weather Bureau had thus far failed to produce \"rainfall of economic importance\" in its Cloud Physics Project. Cloud seeding was an art, he implied, and the Weather Bureau seeders were essentially hacks.\n\nHe then gave an outline of probability theory before launching into a description of their work in New Mexico. He gave detailed reports of two days during which the team had operated Vonnegut's silver iodide generator. On both days, the Weather Bureau had predicted no rain, but on both, heavy rain and thunderstorms were observed.\n\nUsing river flow and rain gauge data, Langmuir calculated the similarity of rainfall distribution on the two heavily seeded days. They had a correlation coefficient of +0.78. The chance of getting such a high correlation randomly, he declared, was one in ten million.\n\n\"We must conclude,\" he declared, \"that nearly all of the rainfall that occurred on October 14, 1948[,] and July 21, 1949[,] was the result of seeding... Silver iodide seeding produced practically all of the rain in the state of New Mexico on these two days.\" Furthermore, after the July experiment, a band of heavy rainfall progressed from New Mexico across southern Colorado and Kansas, bringing three to five inches of rain to those states. All that precipitation, too, might have been caused by Bernie Vonnegut's generator.\n\nThe experiment, Langmuir said, proved that the entire United States could double its rainfall for a couple of hundred dollars. Thirty milligrams of silver iodide released into a cumulus cloud six miles in diameter would liberate as much heat\u2014the latent heat of condensation\u2014as the explosion of an atomic bomb. The time was ripe, he concluded, for turning to the study of hurricanes. Silver iodide generators at sea, he was convinced, could modify hurricanes and even prevent them from reaching land.\n\nThere was a buzz in the room when Irving finished. He had thrown down a gauntlet for the Weather Bureau\u2014at the very session chaired by Chief Reichelderfer. He had even tossed back in the chief's face the \"exaggeration\" the chief so detested about steering hurricanes. Reichelderfer remained professional, cordially thanking Irving for his \"very, very interesting paper\" and noting there were only three or four minutes for discussion. A flurry of comments ensued, starting with the objections of the Weather Bureau's observer William Lewis, who thought that the conditions in New Mexico might have led to rain anyway.\n\nBut the reporters in the room were uninterested in counterarguments. When the session was over, they mobbed Irving. Was he saying that New York City could make rain to relieve its drought?\n\nIt very well might, Langmuir replied. There were no guarantees, but when the cost of an attempt was so low, why not try?\n\nThe next day, the newspapers were filled with the story. The city's water commissioner, citing Weather Bureau skepticism, had dismissed cloud seeding just as quickly as he had dismissed a group of Indian chiefs who offered to come to town and perform a rain dance. Now Irving Langmuir was saying that rainmaking might relieve the city's water woes. Langmuir was, The New York Times editorialized, \"no rain-making crank, but a Nobel Prize winner who ought to be consulted in the present extremity.\"\n\nNot long afterward, the Project Cirrus team received a phone call from the New York City water commissioner, Stephen Carney. Might he and Edward Clark, the chief engineer for the water department, visit Schenectady? A meeting was arranged for the following week.\n\nLangmuir triumphantly took William Lewis out to lunch.\n\n\"Do you still believe that no seeding experiment has increased rainfall by even as much as 10 percent?\" he asked. Lewis, showing significant pluck, replied that he did.\n\n\"Did any single point that I brought out in my January 25th paper appear to you to have any significance?\" Irving pressed.\n\n\"I don't know of any,\" responded Lewis, refusing to be cowed by the eminent scientist.\n\n\"So you think my paper was mostly bunk?\"\n\n\"I personally wouldn't use that word,\" Lewis said, \"but it does describe my opinion reasonably well.\"\n\nIrving laughed. He loved nothing better than a fight, and he couldn't help but admire the guy for standing up to him.\n\n\"Why don't you undertake something constructive?\" he said, and gave Lewis an assignment looking for seven-day periodicities in any weather trait in the northwest states.\n\nHe had his reasons for this. In November, Workman's team in Socorro had started running Bernie's silver iodide generator every Tuesday, Thursday, and Saturday. Langmuir was hoping to introduce a regular pattern\u2014a periodicity\u2014into the weather to prove that he was making rain. But in January, something strange happened. Bizarrely heavy rains were reported in the Ohio and Wabash River basin. It was a long way from the generator, but it made some sense. The air over New Mexico didn't carry much moisture. The silver iodide, encountering little water vapor to nucleate there, might just be floating east with the prevailing winds. As it entered the Mississippi River basin, it would collide with moisture-laden winds sweeping up from the Gulf of Mexico. The winds would carry the chemical north to the Ohio River basin, where it would finally cause rain.\n\nThe members of the Project Cirrus steering committee, concerned that they might be causing damaging floods, met in late January and decided to reduce the number of days on which they seeded to two per week. As soon as they did, the rainfall in the Ohio River basin let up. It was all looking more than coincidental.\n\nThe steering committee was eager to tell the military. At the end of January, President Truman had announced that he had directed the U.S. Atomic Energy Commission\u2014the organization originally meant to help deter a worldwide arms race\u2014to build more atomic bombs. He specifically included the Super, the hydrogen bomb Teller and others had been advocating. But Project Cirrus might be providing them with a superweapon that was even better.\n\n* * *\n\nKurt Vonnegut's first published story, \"Report on the Barnhouse Effect,\" hit the newsstands on February 11, 1950, in Collier's. The Schenectady Gazette ran an article announcing his triumph. Typically for the credential-loving company town, the paper announced that he was \"a graduate of the University of Chicago.\"\n\nHe was trying to repeat his success, but it was hard. \"Mnemonics,\" \"City,\" and \"Das Ganz Arm Dolmetscher\" were being turned down all over town. Knox Burger was rejecting everything they sent him. He was, Kurt grumbled, turning to Yaddo again, going back to buying work from the well-known writers who went to fancy writers' colonies and won prizes and hung out drinking cocktails in New York. Kurt was cranking out reams of prose to no avail.\n\nTwo weeks after \"Barnhouse\" was published, the News Bureau got a call from a librarian at the Works Library in Building 2. She had looked up from her desk, and there, looking right back at her through the window, was a deer. Somehow it had gotten into the Works, and now it was trapped inside the walls of the industrial city. The librarian had called security to come get it out, but she thought the News Bureau would want to know about it.\n\nA photographer was dispatched at once. The deer, a young doe, was backed into a corner, her head ducked and her spindly legs splayed. Two members of the GE patrol approached, and as the photographer snapped away, they grabbed the doe around the middle. She was so small that they easily lifted her up on her hind legs to truss her front ones, then wrestled her to the ground and trussed her back legs too. They heaved her into the backseat of a GE patrol vehicle, and a GE security officer got in with her, wrapping a stocky arm around the doe's neck as if they were on a date.\n\nThe General Office News ran a page of photographs of the whole drama, with droll captions giving the deer's point of view: \"What a way to treat a Republican!\" and \"All I wanted to do was sneak out of work at 4:30.\" The News Bureau issued a press release slyly commenting that the deer had forgotten to wear her ID badge. GE was always hassling the employees about leaving their badges at home.\n\nIt was all supposed to be cute and funny, but Kurt knew how the deer felt. Trapped. Cornered. Baffled by this strange, ugly place and the bizarre rituals of the creatures who inhabited it. He knew what it felt like to want out.\n\nFive years later, \"Deer in the Works\" would be the first story Kurt sold to Esquire. In it, a newspaper writer takes a job in publicity at a big corporation because he's worried about supporting his growing family. On his first day, a deer gets into the Works, and the new guy is assigned to get the story. As the writer wanders through the labyrinthine plant encountering terrifying assembly lines, alcohol-soaked sales meetings, and labs with mysterious names, he realizes he has made a terrible decision in leaving his small-town newspaper for the security of being a company man. He cannot thrive in this place. He cannot even breathe the soot-filled air. Finally, he wanders hopelessly into a ball field and the deer appears, pursued by Works policemen with pistols drawn. The writer does the only thing he can: he opens the gate and lets the deer escape. And then he follows her out. The story ends simply, \"He didn't look back.\"\n\nThat's what Kurt was living for now, his own escape. Four more salable stories were all he needed. The irony was, it was GE that was handing him the material he needed to break the company's bonds.\n\nOne day not long after the deer affair, he was researching a News Bureau story when he learned about the stratosphere. It was a great word. There were other great words to describe the strange zones between the atmosphere and space. He wrote them down on the back of an envelope: \"stratosphere,\" \"troposphere.\" In particular, he was drawn to the notion of the ionosphere, a layer of ionized air one hundred to four hundred miles above the earth's surface, the boundary between the earth's atmosphere and the empty vacuum of space.\n\nWhat might be up there? Weather, of course\u2014that he heard about all the time. And for a few years after the war, scientists had puzzled over strange, erratic blips that showed up on radar screens, dots that flitted about with apparent disregard for the laws of physical motion. First noticed by a scientist at Bell Labs, the blips captured the imagination of reporters, who called them \"ghosts\" or \"angels.\" Then, in early 1949, newspapers announced that the mystery was solved: they were insects.\n\nBut what if there really were ghosts or angels up there, drifting around in those atmospheric zones?\n\nKurt made his main character a scientist with a Germanic last name. Dr. Groszinger is the kind of man who loves science for its own sake, who finds comfort in the \"dependability of the physical world.\" He is working on a top secret experiment funded by the military, but he isn't really focused on that. \"The threat of war was an incident,\" Kurt wrote, \"the military men about him an irritating condition of work\u2014the experiment was the heart of the matter.\"\n\nDr. Groszinger's first name is Bernard.\n\nBernard Groszinger is supervising Project Cyclops, an experiment to send a controlled spaceship into orbit above the earth. There is a man on board whose job is to report the weather conditions over enemy territory and the accuracy of guided atomic missiles should war break out. As the story opens, the spaceship has been launched, and Bernard is chain-smoking while he awaits the first weather report from its passenger, Major Rice. He is relieved when contact is made. But soon it becomes apparent that all is not well. Major Rice is hearing voices in space\u2014voices that seem to be coming from the dead. And the dead have plenty to say.\n\nBernard Groszinger is convinced that the man in orbit is insane. But when the general in charge of Project Cyclops has the messages from the dead investigated, they seem to check out. Dr. Groszinger then thinks it must be an elaborate hoax\u2014until the major reports that a woman speaking with a German accent has asked for Dr. Groszinger. When the major quotes his mother's favorite lines from Goethe, Bernard realizes the guy is not making it up.\n\nThe general doesn't give a damn about the voices\u2014as long as they are kept secret, so as not to disclose that his nation has a spaceship in orbit. But for Groszinger, the voices create a moral quandary. Doesn't the public have a right to know that the dead can be contacted? Wearily, he thinks perhaps no one would even be surprised. \"Science had given humanity forces enough to destroy the earth, and politics had given humanity a fair assurance that the forces would be used,\" he thinks. \"There could be no cause for awe to top that one. But proof of a spirit world might at least equal it. Maybe that was the shock the world needed, maybe word from the spirits could change the suicidal course of history.\"\n\nWhen amateur radio operators stumble onto the secret frequency and overhear the major's transmissions, the general is forced to jam the frequency and bring the spaceship down, killing the orbiting major. Dr. Groszinger is sworn to silence about the whole affair.\n\nThe story concludes with Dr. Groszinger denying everything to a crush of reporters. He tells them that the unidentified object seen crashing into the Atlantic was most likely a meteor. When a reporter asks what's out beyond the stratosphere, he says it's just dead space. He gives it a name: the thanasphere.\n\n\"Has a nice scientific ring to it, don't you think?\" he asks. The reporters press him to tell them when the nation will have a rocket in space.\n\n\"You people read too many comic books,\" he tells them.\n\n\"Thanasphere\" was a good story: tight, sharp, and just thoughtful enough for the slicks. In a light way, it addressed the things Kurt saw going wrong with the nation: militarism, technocracy, secrecy. The quest for bigger and better weapons was shunting aside the things that made human beings human. The story's general turns a blind eye to the desperate voices of the dead\u2014and the entire existence of the spirit world\u2014in order to maintain secrecy and efficiency. It's a dark vision of military values. But it didn't find much to admire in scientific values either. After all, it's the scientist, Bernard Groszinger, who makes the whole operation possible. Like his namesake, he has his moral qualms. But in the end, he does what the general wants.\n\n* * *\n\nIn mid-February, New York water commissioner Stephen Carney and engineer Edward Clark visited Schenectady. They went first to Langmuir's home, where Irving gave them an overview of Project Cirrus. Then, because his own driveway was blocked by seventeen inches of snow that had fallen the night before, Irving rode with them to the Knolls. There, Carney and Clark were treated to the usual song and dance: Vince seeded the freezer with dry ice, and Bernie deployed his popgun with a puckish smile. As always, the News Bureau team was there to get photographs, and though Carney had a long, mournful face and Clark wore a skeptical frown, at the end of the day, the two men met with a large group of assembled reporters at the Hotel Van Curler and made a startling announcement. Cloud seeding, Carney said, was worth a try. He was going to request that the city's mayor hire a consulting meteorologist to put a program in place. Langmuir added that the city should be sure to find \"a competent meteorologist who has nothing to sell and who will study and plan the project with a conservative approach.\"\n\nThis was a meaningful comment. Bernie, Vince, and Irving were beginning to realize they had created a cottage industry. Dozens of private \"rainmakers\" were hawking cloud-seeding programs all over the nation. One of them, Irving Krick, had already been trying to sell his services to New York City. The GE men knew Krick. Lately, they had begun to see him as a problem.\n\nIrving Krick was a confident cowboy of a weatherman from California. He had been on Eisenhower's meteorology team, where his old-fashioned forecast had come close to causing D-Day to fail. But after the war, he had spun the story to cast himself as the hero. A charismatic self-promoter with a wavy pompadour and a Laurence Olivier mustache, Krick had a weather consulting company specializing in long-range forecasts that the Weather Bureau considered hokum. Now he had taken up the cause of cloud seeding with gusto. A year earlier, he had visited GE and shared data with the Project Cirrus team. But since then, the GE scientists watched with growing alarm as Krick bounced all over the country selling rainmaking programs that they considered scientifically sloppy. Worse still, Krick was hinting that he had some kind of affiliation with GE. News Bureau manager Roger Hammond told the Cirrus team that Krick was \"using GE and the names Langmuir, Schaefer and Vonnegut to his commercial advantage.\" And lately, Krick had started telling reporters that he had a \"new way\" to make rain for New York: seeding with silver iodide.\n\nThe next time Krick visited GE, he found his reception significantly chillier than it had been a year earlier. The scientists suggested that he confine himself to dry ice seeding, leaving the silver iodide to the experts. Then Langmuir took him aside and told him, in no uncertain terms, to stop taking credit for Bernie's work. It was something Bernie would never have done himself. He was never particularly concerned about getting credit, even when he deserved it; for him, the science was its own reward.\n\nNot long after Clark and Carney returned to New York\u2014their trip was delayed by the weather\u2014New York City's mayor, William O'Dwyer, announced that he had hired a scientist to begin formulating a plan: Wallace Howell. A longtime colleague and friend of the Project Cirrus team, Howell was a consulting meteorologist at Harvard's Blue Hill Observatory. Vince had suggested him, and the Project Cirrus team was pleased he was hired: Howell was a real meteorologist, not a charlatan. And he would consult with them regularly.\n\nProject Cirrus was riding high. Their work was finally going to prove its worth to the world! The Industrial Bank of Commerce ran an ad with a picture of Irving, Bernie, and Vincent standing by their freezer under the words \"Salute to the Future.\" There was a quotation from Guy Suits declaring that the weather work \"may prove in the future to be as important to human welfare as the advent of atomic energy.\"\n\n\"Here,\" declared the ad, \"is another example of industry at work for humanity. It represents progress for all mankind, and its beneficial effects on our civilization are truly immeasurable.\"\n\nThe week New York City hired Howell, Vincent made one of his frequent appearances on the WGY radio program Science Forum. His topic was the \"sunny side of rainmaking.\" Outside, a hard rain was washing the winter snow into a slushy brew as Vincent speculated for listeners at home about the coming day when blizzards and thunderstorms would be averted, aircraft icing forestalled, forest fires prevented, and fog simply swept away. It was all nearly within their grasp.\n\n* * *\n\nDr. John Herbert Hollomon invited Kurt and George Burns into his study. Herb was one of GE's most promising young men, a wunderkind who had come to GE Labs to head up the new physical metallurgy division in 1946. Bernie had worked with Hollomon before moving to Project Cirrus. Herb was a friend of Bernie's and, like many of Bernie's friends, had also become a friend of Kurt's.\n\nAlso, he liked model railroading.\n\nGeorge Burns set up his camera to get a shot of Herb with a model train. They were there to do a profile, and this was just the sort of humanizing detail GE loved to give out about its scientists. Kurt had just written a press release about a speech urging scientists and engineers to engage in activities unrelated to their work. Now here was Herb Hollomon, hoisting a tiny locomotive, the perfect example of a man with a wholesome hobby. An exemplary hobby. Unless of course the man who loved model trains was to get so obsessed with them that his family life suffered... like, for instance, Earl \"Hotbox\" Harrison, a model-train maniac oblivious to his wife's distress, in the story \"With His Hand on the Throttle.\"\n\nKurt worked on short story ideas constantly. General Electric contained a rich vein of characters, settings, and concepts he could mine. It was a microcosm, a corporate culture that reflected many aspects of the national culture Kurt found disturbing. The same month he did the News Bureau story on Herb Hollomon, he wrote a GE press release about a robot tool dolly.\n\n\"A one-armed robot on wheels, which can close doors, turn valves, take apart and reassemble complex machinery, and perform virtually every task the human hand can perform,\" he wrote, \"has been developed by General Electric engineers for work in radioactive areas, it was revealed here today.\"\n\nKurt could easily summon the gee-whiz attitude GE liked to use for unveiling inventions, but his own feelings about the things he saw were complex. Was every scientific advance necessarily good for humanity? Professor Barnhouse's question would hang over everything Kurt Vonnegut ever wrote. The tool dolly, for instance, made radioactive contamination into a neat problem, something that could be solved with enough know-how and engineering spunk. Nothing like a whiz-bang gadget to take the sting out of a nuclear holocaust!\n\nSomewhere in there, Kurt was taken to see another nifty invention: the GE automated milling machine. A cathedral of steel with motors strung from electrical cords, it was designed to cut the rotors for jet engines and gas turbines. GE had just received a patent on the machine's tracer control system. Thanks to that control system, every rotor could now be cut to exact, identical specifications, every contour decreed not by the skill of the machinist but by the dictates of the little clicking computer attached to the blade. The machinists who worked milling machines were some of the top-paid laborers at GE. Or at least they had been. At the behest of GE, they had lent their movements to the tape-driven control system, and now the clicking box and its robot cutter would replace them.\n\nKurt couldn't blame the inventors. It wasn't necessarily vicious or antisocial to make such a machine. But it was definitely going to be a bad thing for a lot of people who had been proud of their skills. No one at GE seemed to be thinking about that.\n\nAnd automation of mechanical tasks was only the beginning. GE was now getting into building high-speed calculating machines. The company was the first in the world to own a differential analyzer, an analog computer of the type first designed by Vannevar Bush. A shockingly noisy contraption made up of thousands of gears, it occupied an entire floor of the old athletic building. Langmuir and Katharine Blodgett had used it during the war to calculate the liquid water content of clouds and particle size distribution inside them. Now GE had built a new device, the Ordinal Memory Inspecting Binary Automatic Calculator: OMIBAC for short. With its thirty-three hundred electron tubes and circuits, OMIBAC was five thousand times faster than a human computer at doing math. It could solve a problem in days that would take months or even years for a mere person. This was obviously a good thing. Or was it? Increasingly, some people were beginning to wonder if such devices might ultimately diminish human beings. Even the Monogram article about OMIBAC was accompanied by a cartoon of a man cowering in a box, with the caption \"Are machines smarter than ME?\"\n\nKurt had read Norbert Wiener's book Cybernetics, which came out in 1948. Practically everyone in Schenectady would have. The Saturday Review of Literature said it was \"impossible for anyone seriously interested in our civilization to ignore this book.\" It was so widely admired that Wiener was persuaded to write a \"popular\" version without the mathematics; it was published in 1950 under the title The Human Use of Human Beings. Together, the books provided the nation with an introduction to an entirely new way of thinking. Wiener defined cybernetics as \"the science of control and communication in the animal and the machine.\" Machines, Wiener pointed out, are very much like human beings, or even human cultures: They operate by sending and receiving messages in an attempt to control their environment. Such feedback allows them to learn.\n\nWiener's idea\u2014that the brain is a kind of computer and the computer a kind of brain\u2014has become so much a part of our thinking now that it's hard to imagine how revolutionary it was at the time. And Wiener was also already thinking about the human implications of this breakthrough. Alone among computing pioneers, he was asking what the human downside to the computing revolution might be. At present, a computer's output was numbers. But eventually it would produce other things: information, pictures, songs. Cybernetics explained that the computer was a kind of mechanical slave destined to usurp not only physical labor but mental labor too. The first industrial revolution had proved that human handwork could be done by machines; the second would prove that human thinking could too.\n\nThere was a short story in that. Kurt named his story\u2014and his computer\u2014\"EPICAC.\" The narrator is a mathematician who works the night shift programming EPICAC. The military men who control EPICAC, and the scientist who designed it, Dr. Ormand von Kleigstadt\u2014another Germanic name\u2014want the computer to \"plot the course of a rocket from anywhere on earth to the second button from the bottom on Joe Stalin's overcoat.\" As it turns out, EPICAC can do much more.\n\nThe narrator is a programmer in love with his colleague, a \"crackerjack mathematician\" named Pat, but she repeatedly declines his marriage proposals because they aren't poetic enough. One night, sitting at his keyboard, \"trying to think of something poetic, not coming up with anything that didn't belong in The Journal of the American Physical Society,\" the narrator types his troubles into EPICAC: \"My girl doesn't love me.\"\n\n\"What's love? What's girl?\" asks EPICAC. After the narrator defines the terms and then defines poetry, the computer writes him a poem to give to Pat. She loves it, so the narrator gets EPICAC to produce another one. Slowly, passing off EPICAC's creations as his own, the narrator wins Pat's affection. He programs EPICAC to produce the perfect marriage proposal. Of course, the computer thinks it's proposing for itself. The narrator tells it that women can't love machines; they can only love human beings.\n\nPat accepts the narrator's proposal on the condition that the narrator write her a poem every year for their anniversary. That night, EPICAC kills itself by overloading its own circuits. The narrator finds its suicide note: \"I don't want to be a machine, and I don't want to think about war... But fate has made me a machine. That is the only problem I cannot solve.\" EPICAC has left the narrator a wedding gift: five hundred more poems to give his wife.\n\n\"De mortuis nil nisi bonum,\" the story concludes. \"Say nothing but good of the dead.\"\n\nKurt finished \"EPICAC\" quickly and sent it to Kenneth, who liked it enough to send it on to Knox Burger. Knox returned it with notes, commenting that the title sounded like something you took when constipated. But in response to Kurt's complaint that he wasn't managing to sell anything, he offered a glimmer of hope. His stories, Knox wrote, were much more readable than most science fiction. It was perhaps the first time Kurt heard his work called that.\n\nOn April 4, Knox had good news. Collier's was buying \"Thanasphere.\" Elated, Kurt sat down to work on a story about a scientist who invents a new way of making ice that turns out to be horribly dangerous. He started out calling it \"The Crystal.\" But before long, it had the title \"Ice-9.\"\n\n* * *\n\nThe first flight of New York's rainmakers was delayed by rain. Kept on the ground at Floyd Bennett Field until the squall passed, the police department's Grumman Goose\u2014a comical amphibious plane that looks like a cartoon whale balancing a surfboard on its back\u2014arrived in the Catskills too late. Any clouds that could have been seeded had already moved on.\n\nThe city's newspaper writers were crestfallen. The entire metropolis had been eagerly awaiting its first municipal rain. The New York Times Magazine had run a detailed article explaining the different types of cloud seeding and when they would be deployed. Bernie had provided drawings for it.\n\nUpstate residents, meanwhile, awaited the event with more trepidation than glee. Before the first rainmaking attempt even happened, lawsuits were threatened. Albany's mayor, Erastus Corning, tried unsuccessfully to get the state Water Power and Control Commission to stop New York City from \"intercepting\" rain meant for Albany's watershed. Then a group of property owners and civic associations in the Catskills had issued a court summons in Ulster County to block the rainmakers. The city's lawyers advised the mayor to ignore it.\n\nOn April 13, Wallace Howell and a police pilot took off from La Guardia Airport in the Grumman Goose. They flew north, up the Hudson, and circled the Catskills watershed area, scouting out likely clouds. Howell sprinkled a hundred pounds of dry ice into clouds near the Ashokan Reservoir. The plane returned to Floyd Bennett Field at 6:42 p.m.\n\nSome observers on the ground insisted that there was no snow until the plane flew back and forth overhead, at which point it began snowing for the next hour and a half. Others said there was snow off and on all afternoon, but it increased in intensity after the plane passed over. The state police at Phoenicia insisted that there were light squalls all day, with no observable change in the afternoon.\n\nThe next morning, New Yorkers awoke to an unseasonable snowstorm. Slush coated the sidewalks, snow blanketed Easter lilies in the city parks, and city workers had to dig shovels out of spring storage. The snow was oddly patchy, sometimes falling on one block but not the next one over. City hall's wires overflowed with complaints from irate drivers shocked at the icy conditions on bridges and roads. City dwellers called it \"Howell's snow.\"\n\nEven as he fired up two generators mounted on station wagons to cruise the watershed and blow silver iodide into the clouds, Howell insisted there was no way to tell if the snow had resulted from seeding.\n\n\"It would be completely impossible for anybody to say whether or not we increased the yield of today's snow flurries over the watershed area,\" he told reporters. \"On any individual flight, how can anybody tell?\" After at least ten flights, he might have a better idea. But the media were not put off so easily. The New York Times ran a photograph of Howell standing near a snowbank at city hall under the title \"Is It His or Nature's?\"\n\nAll throughout the spring, Howell seeded clouds with silver iodide from the air and from ground generators mounted on trucks. And all that spring, rain fell on the watershed. The local newspapers breathlessly reported on each increase as the city's water supply inched upward. News reports took on a martial tone: the city was mounting \"double-barreled attacks\" on the clouds, \"bombarding\" them from the ground and conducting \"aerial assaults.\" On April 20, torrential rains in the Catskills caused the Schoharie Reservoir to spill over, ending seeding operations for a few days. With the reservoirs up and New Yorkers still drastically reducing usage on Thirsty Thursdays and Dry Fridays, Commissioner Carney said that the bans on sprinklers and swimming pools might be lifted if the trend continued through summer.\n\nThe first court case landed in the state supreme court in May. Ben Slutsky, owner of the Nevele Country Club, demanded an injunction against the city's rainmaking because it could damage his business. Why would people take vacations in the Catskills, he asked, if they figured it was going to rain the whole time? The supreme court, unmoved, found for the defendant, declaring the plaintiff had \"no vested property rights in the clouds or the moisture therein.\" Furthermore, the court must weigh Slutsky's \"remote possibility of inconvenience\" against the city's need to maintain \"an adequate supply of pure and wholesome water\" for ten million people. The city's interest won out. The court did not even question the assumption that cloud seeding worked.\n\nOn May 1, a sixteen-hour seeding operation was followed by three solid hours of rain, adding nearly two billion gallons to the reservoirs. Howell remained reserved, refusing to take credit. It might be his. It might be nature's. But rain was falling. Because of Bernie's invention, New York City was saved.\n\n* * *\n\nDr. George Hoenikker held a vial containing a milky white shard. His invention: Ice-9. His wife didn't understand it. His kids didn't understand it. No one saw its significance, no one but him. But he was about to become famous. Maybe he'd win the Nobel Prize. But one thing was for sure: he wasn't going to work at General Forge and Foundry anymore. He wasn't going to help them build the Blue Fairy Rocket. He had fled to Cape Cod and was thinking of taking up woodworking.\n\nBut then things went wrong. He got mixed up with a blowsy married woman. He got in a car accident. The story careened off in crazy directions. It had no arc.\n\nSo Kurt killed Dr. George Hoenikker. He threw the whole \"Ice-9\" story away and started over. He made several outlines. But they all started in the same place: with a scientist whose conviction that his work carries no social responsibility allows him to invent Ice-9.\n\nIce-9 was Irving Langmuir's idea. Someone had told Kurt about it at a cocktail party. GE parties often featured funny stories about Irving. There was the time his secretary was out sick and Irving dealt with the temp for a whole day without noticing the substitution. There was the time he went to pick up Marion and got so lost in thought that when he neared his wife and she waved, he simply waved back and drove past. There was the time he went out with his daughter, Barbara, got caught up in a conversation, and came home without the little girl. But this story was different. It didn't concern Langmuir's absentmindedness. It concerned his idea for a novel.\n\nYears earlier, the partygoer told Kurt, H. G. Wells had been invited to tour the Schenectady Works. Wells was quite famous at the time, so he was given GE's most famous scientist as a guide: Langmuir. Irving took the esteemed writer around the plant, but he was a little unsure of how to talk to a novelist. Because H. G. Wells wrote fiction, Irving thought he would try to entertain him by offering up an idea for a novel. Why not write a book, he suggested, about a scientist who invents a form of ice that's stable at room temperature? The ice somehow gets into the wrong hands and contaminates the water supply. All of the earth's water freezes instantly, and life on the planet is doomed.\n\nH. G. Wells had not been interested in Irving's idea, but Kurt was. How strange that his brother's older colleague should invent such a perfect parable for the dangers of what he himself did\u2014inventing things with no regard for the human consequences! And the stable ice\u2014he had come to think of it as Ice-9\u2014was an intriguing concept, particularly given what Bernie had told Kurt about different types of ice crystals. At another GE party, Kurt had tried the concept out on a GE crystallographer. The scientist seemed intrigued. He went over to a chair, sat down, and stared blankly into space. After half an hour, he came back to Kurt.\n\n\"No,\" he had said. \"It's not possible.\"\n\nScientifically impossible, maybe, but it was a damn good story idea. Kurt kept thinking about it. Wells didn't want it and Irving had no use for it, so he figured, finders, keepers. And now, after a number of false starts with \"Ice-9,\" he thought he had finally found the right way to tell it. He would set up the ethical conflict by using two characters\u2014two men, a scientist and a sociologist\u2014with different attitudes about the moral duties of scientists.\n\nHe turned Dr. George Hoenikker into Dr. Arnold Macon. Macon has invented Ice-9 while living in an island banana republic where he is pet scientist for the island's repressive dictator, General Monzano. Monzano\u2014whose name recalls the Manzano Mountains where Bernie had his big breakthrough\u2014is faced with a civil uprising. As rebel forces threaten the dictator's stronghold, Dr. Macon is visited by a sociologist friend, Franklin Dale. Macon can't resist telling Dale about Ice-9.\n\nDale immediately recognizes the potential for disaster. He and Macon quarrel about the scientist's ethical obligations. Macon insists he has none: he just does research and leaves the uses of his discoveries to the engineers. Dale insists he shouldn't work on inventions that have the power to do great harm. He is proved correct when General Monzano gets his hands on the Ice-9 and holds the world hostage, threatening total annihilation of life on earth if the rebels don't back down.\n\nThe rest of the novella plays out in rousing confrontations and chase scenes as Macon tries, first, to convince the world that his invention is real and capable of doing great harm and, second, to retrieve the Ice-9 from Monzano.\n\nIn the final scene, Macon finds General Monzano apparently dead, and the Ice-9 dangling over the sea on a chain attached to the hands of a clock. At the stroke of twelve, it will drop into the water and destroy the world. Macon hesitates, and suddenly he sees that he has been the real villain all along. Secretly, he loved his invention's destructive power. As he holds the Ice-9 and contemplates ending the world himself, Monzano\u2014alive after all!\u2014shoves him into the sea. Macon pulls the general with him, but the Ice-9 catches on a tree branch, suspended above the high-tide line. The world is saved\u2014for now\u2014through sheer dumb luck.\n\nKurt figured the novella was perfect for magazine serialization. The only thing that worried him was that, like much of his work, it lacked female characters. Still, he thought it was the best thing he'd ever written.\n\nThe month he was finishing the draft of \"Ice-9,\" GE's president, Charlie Wilson, addressed the ethical obligations of scientists in Schenectady's annual Steinmetz Memorial Lecture. There was no use criticizing scientists, Wilson declared, simply because the truths they happen upon can be used for evil ends. \"There is nothing moral or religious or hostile or inspirational in science unless man puts it there,\" Wilson declared, sounding a lot like Dr. Macon.\n\nKurt thought otherwise. It wasn't necessarily evil to invent Ice-9 or a machine that replaced a human being, like OMIBAC, EPICAC, or the \"contour-following system\" that made skilled workers obsolete. But it was unethical to do these things without thinking about their implications for humanity. Increasingly, that's what his stories were saying.\n\nHe sent \"Ice-9\" to the typist with renewed hopes of making it to his goal of selling five stories. Kenneth had sold \"Das Ganz Arm Dolmetscher\" to The Atlantic: that made three. He was only getting $125 for it, but getting published in The Atlantic was a coup for his reputation as a writer. Before long, he might be able to quit, and when he did, it was becoming increasingly clear to him what he would do. He would write a novel about science and progress and ethical quagmires\u2014a novel, in short, about GE.\n\n* * *\n\nBy late May, New York City newspapers were announcing water gains on every \"front.\" By mid-June, the tone was a little less gleeful. The reservoirs were near full, but constant rain was ruining the vacation season upstate. Resort owners grumbled about lost tourist income; farmers in neighboring Orange and Sullivan Counties complained that their vegetable crops were being washed out; sports fans were outraged when Giants and Dodgers games were canceled. Ironically, even the annual picnic of the Department of Water Supply, Gas, and Electricity was rained out.\n\nOn June 15, with reservoir levels surging, prohibitions on watering lawns and filling swimming pools were lifted. In July, residents were allowed to wash their cars. In August, as he prepared a report on his first six months as the city's consulting meteorologist, Wallace Howell finally said what the press wanted him to say.\n\n\"I have made rain,\" he told city reporters.\n\nGE was trying to get its rainmaker to stop saying the same thing.\n\nIrving had submitted his American Meteorological Society paper\u2014the one that had sparked New York City's experiment\u2014to an editor at Science. The GE lawyers, fearing that the paper could be used as evidence in a civil suit against the company, had the News Bureau head, George Griffin, call up Science and withdraw it. No one informed Langmuir.\n\nWhen Irving found out, he was furious. GE was demanding he go along with the official position that GE scientists never caused rain. But his future work and reputation depended on convincing people that he had made rain. In the face of his anger, the GE lawyers backed down. Langmuir might be controversial, but he was still their most famous scientist, their only Nobel laureate. The paper appeared in Science in July. By then, Irving, Bernie, and Vince were all back in New Mexico, flying regular seeding runs to see if they could introduce definitive changes in the weather\u2014not just in that state, but across the entire nation.\n\nOut of the Blue\n\nPrime Minister Liaquat Ali Khan, generous of smile, rotund of body, arrived at GE sporting his traditional fez. He was trailed by his brocaded, bejeweled wife and flanked by nine handlers, two from the Department of State. The News Bureau had diligently organized the photo ops. At the Works, the prime minister was photographed smiling while a manager explained the engine assembly line. Then he was taken to the turbine department. Gamely, he stood by as a worker drove gleaming rivets into an enormous flywheel. Flashbulbs detonated like lightning.\n\nThe prime minister had left Pakistan to learn what was to be learned in the most powerful nation on earth for the good of his spanking-new country. His state visit, hastily arranged after Stalin invited him to the Soviet Union, was designed to cement young Pakistan's alignment with Western democracy and free enterprise.\n\nThe Ali Khans received a full state tour of Washington, including visits to see President Truman, the Naval Academy, Mount Vernon, and the Tomb of the Unknown Soldier. Then they went to New York, where the prime minister received an honorary degree from Columbia University president Dwight D. Eisenhower. After that, they were hustled around the country\u2014Chicago, San Francisco, Houston, New Orleans\u2014inspecting everything from hospitals and factories to crop dusters and prize cattle. In Kansas City, they visited the home of an average American family. The average American husband and wife stood nervously in their Sunday best while Liaquat Ali Khan sat stiffly in their average American living room with their average American son on his lap.\n\nWhen he and his wife arrived in Schenectady on May 24, 1950, Prime Minister Ali Khan had exactly 510 days to live. He would be assassinated on October 16, 1951, by an Afghani Pashtun ultranationalist. Conspiracy theories alleging U.S. or Soviet involvement would never be proved or disproved. But no one knew this on May 24. All anyone knew that day in Schenectady was that a brand-new nation's head of state must be dazzled by the wonders of American industry.\n\nKurt didn't write the press release for the prime minister's visit, but he couldn't have missed the fuss. GE loved nothing more than treating visiting dignitaries to a tour of the company's showcase factory. But what was it like for someone like Liaquat Ali Khan to see the Schenectady Works? Did any of the guests treated to this spectacle of technological prowess and entrepreneurial efficiency ever question the value of all those toasters and refrigerators and washing machines and jet engines and tanks? Did anyone ever dare ask what it was all for?\n\nAt home, Kurt put a fresh piece of paper in his typewriter and banged out the words \"Outline for a Science Fiction Novel.\"\n\nHe figured that's what he'd been writing: science fiction. Not crazy exotic tales of alien races and outer space and time travel; that wasn't what he meant. He had never read the genre magazines full of monsters and Martians, intergalactic wars and improbable future worlds. He meant fiction about science and the probable future that science was spawning in the here and now. Because he'd been living at the leading edge of that future for a couple of years, and he had some issues with what he saw.\n\nProgress: the one true church. The smartest people understood that it had a downside, people like Norbert Wiener, but no one was listening to them. Take, for instance, David Lilienthal's recent speech. Lilienthal had recently retired from the Atomic Energy Commission, and a group of scientists had quickly arranged for him to come to Schenectady and speak. They thought he would address the morally urgent issues behind atomic energy, now that he could say what he really thought. Instead, Lilienthal blathered on about nuclear medicine and nuclear power and how the main uses of the atom would be for peace. He dismissed the \"cult of horror\" of certain scientists and compared the discovery of fission to the discovery of fire.\n\nAs if fire had never harmed a fly.\n\nDuring the question-and-answer period, someone in the audience asked how many atomic bombs it would take to destroy Schenectady.\n\n\"I haven't given it a thought,\" Lilienthal said.\n\nKurt was giving it a thought. His novel would be about progress and the dark side of it no one wanted to discuss.\n\nIt was Kenneth Littauer who had suggested writing a novel. An editor friend at Doubleday thought Kurt should expand \"Report on the Barnhouse Effect\" to novel length. It was not impossible, Kenneth said, that the publisher would offer Kurt an advance of $2,500 if he could provide a detailed proposal. Half a year's salary!\n\nSo now Kurt was writing an outline. He had an idea about the novel he wanted to write, and it wasn't a longer version of \"Barnhouse.\" He wanted to write a novel about GE. It would be science fiction in the way that most of his stories were science fiction. Science fiction like Aldous Huxley's Brave New World or George Orwell's 1984, which had been so acclaimed the previous year. Kurt's novel would cover many of the same issues\u2014freedom, totalitarianism, warmongering\u2014that those well-respected novels did. But the totalitarian world of his novel would be based not on socialist England but on Schenectady. He wasn't going to imagine some fantastic future world with babies in bottles or spy screens in every home; he was going to play out the implications of what was already happening right here, in the heart of the free market, in the capital of industrial know-how. He was going to bite the hand that fed him.\n\nHe'd never written a novel outline before. In his pr\u00e9cis for Kenneth Littauer, he reverted to his anthropology-student habits. He started off by claiming broad cultural relevance: machines were taking over human labor, he declared, and this was a more significant cultural development than atomic energy. As proof, he inserted a long quotation from Norbert Wiener's Cybernetics:\n\nThe first industrial revolution... was the devaluation of the human arm by the competition of machinery... The modern industrial revolution is similarly bound to devalue the human brain at least in its simpler and more routine decisions. Of course, just as the skilled carpenter, the skilled mechanic, the skilled dressmaker have in some degree survived the first industrial revolution, so the skilled scientist and the skilled administrator may survive the second. However, taking the second revolution as accomplished, the average human being of mediocre attainments or less has nothing to sell that it is worth anyone's money to buy.\n\nThat was exactly the world in which his novel would take place: a world sharply divided into an elite class of people intelligent enough to remain necessary and a vast, unhappy underclass whose usefulness has been usurped by machines. It was sort of like the physical split that had already occurred at GE, where the scientists and the managers moved to the Knolls and the laborers stayed behind at the Works, watching as machines took over their purpose. The central character would be an engineer, intelligent enough to be part of the elite but not really comfortable in it. He would be discontented with the whole situation without really understanding why. Eventually, there would have to be a rupture of some sort.\n\nThe novel outline hinted at a deeper animus behind Kurt's resolve to quit GE. The job was making him crazy. And he was making Jane crazy. But it wasn't just boring company politics and the soul-sapping task of public relations that were wearing him down. For him, GE was a brave new world, and he didn't like it.\n\nIt wasn't just machines or technology. It was the use of those things to divide the world into winners and losers. When Kurt was young, the Depression made losers of most of the nation, so the shame of losing had faded. But after the war, the nation staged an orgy of victory: VE Day, VJ Day. Suddenly Americans were ahead, and it felt good. It felt right. Even the arms race was exciting\u2014a game the nation could win. The old socialist ideal of equalizing things was out of favor. In fact, people were starting to be afraid of being labeled socialists\u2014or worse, communists\u2014for advocating things that used to be uncontroversial, like trade unionism. People like Joseph McCarthy, the first-term senator from Wisconsin who claimed to have a list of 205 known communists working in the State Department, were whipping up a frenzy of Red-baiting. A Senate subcommittee was now investigating a number of individuals who were suspiciously friendly toward losers.\n\nIt was as if some people forgot the most basic truths of being human: we are frail, imperfect, vulnerable creatures always in need of other humans for support. Technology was evil if it was used to make some people fabulously comfortable and toss others out with the trash. It was evil if it made that cruelty seem rational. It was evil if it removed individuals from their humanity, if it suppressed the fundamental insight that we're all in this together.\n\nKurt knew that he could shape these themes into an important novel. He gave his two-page outline to Kenneth and went back to writing short stories. The possibility of $2,500 was just that\u2014a possibility. You couldn't buy Junket or wieners with it. Besides, a popular writer could make $2,500 on two stories. Writing a novel was something you did to gain literary credibility. Kurt wanted that\u2014fervently\u2014but he also wanted to quit his job and make a living by writing. Short stories were the quicker route to prosperity.\n\nOr would be, if someone would only buy them. As summer arrived, Kurt continued to crank out stories, and Knox continued to reject them. \"Keep on trying\" wasn't working. Kurt groused to friends that Knox acted as if he were lining up prizewinners for the O. Henry Awards, not picking entertainment for a slick magazine. He couldn't believe Collier's rejected \"EPICAC.\" Even Kenneth liked that one; he submitted it to The New Yorker. \"It's about love in the technological age. I thought you might think it funny,\" he wrote to the editors there. They didn't.\n\nBy now, all of Kurt's free time was consumed sitting at his desk. He started and restarted, wrote and rewrote, as the waves of life broke over him. Mark had his third birthday party; kids brought donations for the Schenectady Cerebral Palsy Fund in place of gifts. Jane waitressed at the Schenectady Symphony's Junior League pops concert. Bernie's son Peter won a prize for his flower arrangement at the Alplaus Methodist Church flower show, and Bow was pregnant again. PTA meetings, volunteer nights, dinner parties, school picnics: all of it eddied and churned around the beachhead Kurt built at the end of the hall upstairs. Ashtrays filled, wastebaskets overflowed, and the typewriter clacked away like the engine of a ship headed for a distant shore.\n\n* * *\n\nBernie peered out the window of the B-17. His cloud\u2014a clump of billowing cumulus over the mountains west of Socorro, New Mexico\u2014was rearing up from the cloud bank like a gray sea monster. It was his cloud because he had seeded it with silver iodide half an hour earlier. Now it was growing ominously heavy and turning the same dark gray as wet concrete. But no rain was falling from it. As he watched, the cloud thrust farther upward, taking on the anvil shape of a thunderhead. Still no rain.\n\nIt was over-seeded, he figured. They had long suspected that over-seeding could cause clouds to grow while creating droplets too small to fall, and here was further proof, a cloud pregnant but unable to give birth. Fifty grams of silver iodide had been too much. Tell that to the private operators who were going haywire all over the state, seeding clouds with a hundred grams or more of silver iodide, on the mistaken theory that more was always better.\n\nNew Mexico was parched by drought. Before Project Cirrus arrived, the state's Economic Development Commission had held a meeting on the topic of rainmaking. Scientists, Weather Bureau meteorologists, and private rainmakers were invited to brainstorm. Irving went out to New Mexico early to attend. Although Bernie had been expressing concern about the burgeoning of rainmaking for profit, Irving had not quite realized how many people had decided to get into the business of weather control. At the meeting, one commercial cloud seeder after another spoke, making it clear that the whole state was blanketed in rainmakers, dispensing dry ice from airplanes, burning silver iodide on the ground, shooting silver iodide from guns: all of it paid for by farmers desperate to save wilting crops and dwindling herds. The amounts of material the cloud seeders were using were ridiculous, and no one was keeping track. Irving finally saw Bernie's point: Project Cirrus's entire program was being jeopardized by careless amateurs.\n\nIrving had stood up at the conference and made an impassioned speech urging the establishment of a central repository for data about the cloud-seeding operations across the state. He didn't like regulation, but he suggested a voluntary agreement among rainmakers to submit detailed reports of any activities to the New Mexico School of Mines. Jack Workman's group there would assemble and collate all the data and make them available to researchers. He spoke with conviction, and his usual charisma worked its magic. Everyone at the meeting agreed to the plan. But afterward, when one of the conference's private rainmakers started a new operation and the Economic Development Commission wrote asking for its report, the reply was a flat refusal from the group's lawyer stating that all data were \"the property of the client.\"\n\nNevertheless, the Project Cirrus team had come back to New Mexico for more test flights; the pilots stayed at the base, while the scientists packed into Poverty Row, the School of Mines' war-era bunkhouse, with its paper-thin walls and spartan living quarters. Vince and Irving brought their wives, but the trip was too much for the pregnant Bow. Bernie brought his nephew Albert Lieber, whose family lived in Scottsdale, Arizona. A high school senior headed for Caltech, Albert was delighted to spend his summer tagging along after Irving, recording data, and hanging on the great scientist's every word.\n\nBernie's plane banked sharply and flew back around the seeded cumulus cloud to get more photographs before turning back toward Socorro and the radar site. At Socorro, they headed northwest, toward another slowly forming cumulus system. This one they seeded with dry ice, two pounds per mile. Almost as soon as they had dropped it, radar echoes began to show precipitation. The clouds ballooned upward, their action captured in time-lapse photographs.\n\nWithin an hour, rain was pounding New Mexico. The Rio Salado and the area's dry arroyos quickly became engorged with rushing water. A truck and a car that were in the dip where Route 85 crossed the normally dry riverbed were swept downstream, halting all traffic on the road. Vince later estimated that the rivers had carried a thousand cubic feet of water per second to the Rio Grande. The floodwaters raged all that afternoon and continued through the next day as well.\n\nThe team was delighted. The photographs were excellent, and with the radar echoes they offered powerful evidence that their seeding was capable of dramatic effects. It had been even more powerful to observe it in real time from the air. Bernie felt in his heart that they had made that storm: He saw it. He was there. Unfortunately, William Lewis of the Weather Bureau was not. He had chosen to sleep in that morning and had missed the whole operation.\n\nThat night, they read in the news that the UN Security Council had passed a resolution committing itself to military action in Korea. Two weeks earlier, seventy-five thousand soldiers from Korea's communist North had crossed the 38th parallel and invaded the democratic South. In response to this clear breach of the peace, the UN Security Council had recommended that member nations begin making their military forces available to the American unified commander. For the first time, the young UN was taking up its role as planetary peacekeeper.\n\nIt was a UN military action, but it still meant that America was once again at war. Men would be drafted, ships and aircraft would be sent to the front, and military personnel on noncombat assignments would be called into active duty.\n\nIn other words, Project Cirrus was going to lose its planes and its pilots.\n\nAt least Bernie's work could carry on. Workman's team in New Mexico would continue dispensing silver iodide on their regular schedule, and the GE team would continue to analyze the data, searching for proof that Bernie's generators were changing the nation's weather patterns.\n\n* * *\n\nKurt looked out over the oily harbor in Gloucester, Massachusetts, and felt glum. He loved being near the ocean, even if, as a Great Lakes person, it made him feel as if he were swimming in chicken soup. He and Jane had been excited to bring Mark and little Edie to New England for their August vacation. They hadn't counted on the world going to hell.\n\nThe newspapers were full of it. War again. The first UN observer\u2014Colonel Unni Nayar of India\u2014had just been killed by a land mine, along with the two British journalists riding in his jeep. The UN had ejected a pacifist protester who tried to hand out leaflets advocating mediation instead of war. President Truman was rattling his saber. It all felt so familiar.\n\nBut it was new too, a civil war that felt like the opening skirmish in a global battle of ideologies. Some of the American GIs who came back wounded were insisting they had seen Russians driving the tanks that attacked them. America's UN delegate, Warren Austin, called the North Koreans a \"Russian zombie.\" The armed camps were summoning their ranks; the Cold War was turning hot.\n\nLooking out over the ocean, Kurt couldn't help but picture it: seven thousand miles across that expanse of dark water, it was all happening again. Young men\u2014as young as he had been\u2014were being sent into another maelstrom. They thought they were soldiers, but they were pawns really, pawns in a nasty game of brinksmanship between the world's two superpowers. The Korean War had nothing to do with them, little even to do with the Korean peninsula, and everything to do with the growing enmity between communism and capitalism, between what were called East and West but were really the Soviet Union and the United States.\n\nRight there in Gloucester, he started a new story: \"King's Knight to Queen Five.\" In the story, later retitled \"White King,\" the soldiers are literally pawns. Twelve soldiers, their commander, Colonel Kelly, and Kelly's family\u2014on their way to a military attach\u00e9 post in India\u2014have been blown off course by a sudden storm over China and crash-landed in territory held by a communist guerrilla chief, Pi Ying. In the presence of a Russian \"observer,\" Major Barzov, Pi Ying offers to let them all go\u2014but only if Colonel Kelly will play chess for their lives, using his troops and his family as chessmen. Whenever he loses a piece, that person will be shot. If Colonel Kelly refuses to play, they will all be killed.\n\nThe Americans take their places, and the game commences. As pieces are lost and soldiers dragged off to their deaths, Colonel Kelly realizes that Pi Ying has a sadistic fascination with what the American will do. Major Barzov seems cold and distant, Pi Ying's mistress is blank-faced, but the communist guerrilla chief is watching the game with the avid fascination of a young boy watching a colony of ants being drowned by a flood.\n\nKurt drew chessboards and sketched out the game, because the story would hinge on an actual move. Colonel Kelly realizes that he can beat Pi Ying by luring him into a trap where his sadism will blind him to the ramifications of his move. He has to make a sacrifice that will trick Pi Ying into a checkmate. As he contemplates his options, he realizes that only one sacrifice will suffice: he must lose one of his sons.\n\nThe cold resolve deserted Kelly for an instant, and he saw the utter pathos of his position\u2014a dilemma as old as mankind, as new as the struggle between East and West. When human beings are attacked, x, multiplied by hundreds or thousands, must die\u2014sent to death by those who love them most. Kelly's profession was the choosing of x.\n\nFour soldiers have already been killed when Kelly moves his son into a vulnerable position. He pretends to realize the sacrifice too late and begs to take the move back. Pi Ying refuses. As the boy is about to be hauled off and shot, Pi Ying's mistress intervenes, murdering Pi Ying and killing herself. Major Barzov continues the game, but he quickly realizes the cleverness of Kelly's move. He allows the survivors safe passage out of the region and doesn't insist on killing the boy. The story ends with banter between Barzov and Kelly about a possible rematch\u2014at the time and place of the Soviet Union's choosing.\n\nIt was a dark story, bleaker than much of what Kurt had been writing. He made a sketch of himself on the back of one of his drafts, a lanky figure hunched feverishly over his typewriter. He drew quickly, the lines electric with urgency. All his hopes for world government, for peace and disarmament, were being dashed by the action in Korea. The world powers were once again playing games with the lives of men. And he was becoming more and more convinced that the root of it was the ability to see human beings not as individuals but as pawns. The danger of the technocratic worldview was that it made human conflicts, human drama, even war, into a kind of game.\n\nWhile they were in Gloucester, he made a painting of the face of a clown and put it up for sale in a local gallery. It didn't sell, so they brought it home with them. Kurt's clown had a bulbous red nose and straw-like blond hair and a big white greasepaint smile, but his blue almond-shaped eyes\u2014eyes much like his own\u2014were unmistakably sad.\n\n* * *\n\nBernie looked out over the audience of high school teachers in Connecticut. Often at events like this one, a meeting of the New England Association of Chemistry Teachers, he gave a general overview of rainmaking and its results to date. But tonight he was doing something different. He was talking about the ethical implications of science.\n\n\"Many farmers, ranchers and civic-minded people in many parts of the country are now engaged in cloud seeding,\" he declared. He was careful to point out that the amateurs were not bad people. But they \"can unwittingly cause large scale, and perhaps adverse, modifications of weather many miles from the scene of their operations.\" Not only that, but their efforts could be polluting the data of serious scientists trying to conduct rigorous experiments.\n\nThe media were still enamored of cloud seeding. An issue of Time magazine had just hit newsstands naming Langmuir the man of the week, his likeness on the cover holding an umbrella that doubled as a test tube. The writer noted Irving's intense curiosity: \"He has been known to sit for half an hour beside a rock surrounded by rising water just to see what a dozen ants will do when their refuge is submerged.\" But though the article featured a long list of cloud-seeding projects under way across the nation, it gave no indication that this deluge of rainmaking might be problematic. That, it seemed, was going to be Bernie's job.\n\nLike Professor Barnhouse, like Dr. Macon in \"Ice-9,\" Bernie was beginning to fear that his invention was getting out of hand. But, also like his fictional counterparts, he couldn't prevent people from using it recklessly until he had proved it was real.\n\n\"Legislators in the future may well face many problems connected with cloud seeding,\" he told the chemistry teachers. \"Laws may be necessary to prohibit and police seeding operations which are contrary to the best interests of the public. Licensing of seeding operations and permits for seeding may be desirable.\"\n\nIt was a risky thing to be suggesting at a time when the future of Project Cirrus was hardly guaranteed. The number of flights had plummeted as Project Cirrus pilots were reassigned to Korea. GE's president Charlie Wilson had recently resigned his post to head up the Office of Defense Mobilization, and his successor, Ralph Cordiner, was a meticulous and humorless leader with an eye glued to the bottom line\u2014a trait that earned him the nickname Razor Ralph. Razor Ralph was unlikely to have much enthusiasm for a project that did nothing for profits, especially as it exposed the company to all kinds of risks.\n\nFor an example of those risks, one only had to look to the Catskills, where residents of New York City's watershed were still trying to get the city to stop cloud seeding. At a meeting of the local Farm Bureau Federation, hay farmers and apple growers bitterly complained about damage to crops, and the general secretary for New York demanded government regulation. The Boards of Supervisors in Orange and Sullivan Counties had passed resolutions demanding the city cease all rainmaking. The Palisades Amusement Park had even offered to pay the consulting meteorologist Wallace Howell twice his city salary if he would quit. \"Why Don't You Let Us Alone, Rainman,\" griped a headline in The Kingston Daily Freeman, summarizing the general feeling. But even though the reservoirs were replenished, Howell and the New York City Board of Estimate carried on. Their data would be more valuable if they seeded for four consecutive seasons.\n\nThe New York Times's editorial board commended Bernie's endorsement of regulation, noting that his position differed from Langmuir's. Irving too was frustrated with the amateurs who might be polluting his data, but he believed regulation was impracticable, suggesting only a voluntary agreement among rainmakers. That was in keeping with the times, where \"planning\" and \"regulation\" were increasingly unpopular words. Bernie was swimming against the tide of unbridled capitalism. It was not a role he had ever expected to assume. But someone had to make sure that scientists paid attention to the damage they and their inventions might unwittingly do.\n\n* * *\n\n\"Ice-9,\" Kurt's story about a scientist whose invention nearly destroyed the world, was languishing. After Argosy rejected it, hard on the heels of The Saturday Evening Post, Kurt glumly told Kenneth they should probably hawk it to Astounding Stories for $300. That was highway robbery for something he thought was the best thing he'd ever written, but it was better than having the whole novella end up in the garbage can.\n\nBut he was hard at work on a new story, a political allegory, which was perhaps not the smartest choice. Already an editor at Collier's had complained that his chess story, \"White King,\" savored of knee-jerk anticommunism. Kurt objected to Knox.\n\n\"I am a registered Democrat, pro\u2013Fair and New Deal, distressed by the new anti-subversive laws, hate McCarthy, enflamed by Communist smears on liberals\u2014etc., etc.,\" he wrote. \"But, dammit, Knox, I don't like Communist Russia any more than I did Nazi Germany. There are some 3,000 Americans dead in Korea, killed in a chess game with Russia looking on.\" People who considered his depiction of Pi Ying sadistic, he declared, were perhaps \"too insensitive to a casualty list as long as King Kong's arm.\" He signed the letter \"George Sokolsky\"\u2014the name of a newspaper columnist known for loving free markets and hating Reds.\n\nNow, unrepentant, Kurt was writing another anticommunist story, using petrified ants to tell a morality tale about totalitarianism. He made it even more direct by setting it in the Soviet Union. The problem was, he couldn't figure out how it ended.\n\nKurt had been looking at William Morton Wheeler's book Ants: Their Structure, Development, and Behavior. According to Wheeler, ant societies had progressed through stages of social behavior analogous to those of human cultures. Modern ants lived in mindless, often militaristic societies in which the hereditary hierarchies of caste and function were evident as physical traits: soldier ants were big, with pincers for fighting; worker ants were small, with enlarged mandibles for carrying things. There was no individualism, no escaping one's physical destiny. What, Kurt wondered, if human beings were headed in the same direction?\n\nLike \"Ice-9,\" the new story centered on a conflict between two men, one doctrinaire, the other doubtful. But in this story, the men are brothers, Josef and Peter. Both are myrmecologists\u2014scientists who study ants\u2014but they have different approaches to their discipline. The elder brother, \"Josef the rock, the dependable, the ideologically impeccable,\" plays along with the Communist Party, while the younger brother, Peter, chafes under party orthodoxy. Peter has a blot on his reputation, a paper he wrote about slave-raiding ants that was declared ideologically incorrect, requiring him to issue an apology. Now Josef, who ghostwrote the retraction for him, is trying to help him steer clear of trouble as they investigate some newly discovered fossil ants.\n\nThe story is set in the Erzgebirge, or Ore Mountains, the rolling mountain range dividing eastern Germany from Czechoslovakia. The occupying Russians, while searching for uranium there, have discovered a cache of fossilized ants. When the brothers see them, they are amazed. The pre-Mesozoic ants are large and pincerless and have smaller mandibles than modern ants. And petrified along with them is evidence of a prehistoric ant culture: houses, art, musical instruments, and books.\n\n\"Josef,\" Peter says, \"do you realize that we have made the most sensational discovery in history? Ants once had a culture as rich and brilliant as ours. Music! Painting! Literature! Think of it!\"\n\nGoing through the layers of rock, the brothers see how the \"magnificent ant civilization\" evolved into \"the dismal, instinctive ant way of life of the present.\" Artistic, intellectual, individualized ants have disappeared, leaving only warlike, cultureless drones. The brothers slowly realize they will never be allowed to publish their find: the parallels with their own totalitarian culture are far too obvious.\n\nThe political allegory was heavy-handed. But what gave Kurt even more trouble was the relationship between the two brothers. The story is told from Peter's perspective. But Kurt struggled with the character of the older brother, Josef. In early versions, he made Josef a Communist Party dupe. He wrote a scene in which Josef and Peter go out in a thunderstorm and argue about Peter's desire to speak the truth. Josef tells Peter that he mustn't speak \"some kinds of truth\" and urges him to \"overlook certain things.\" Peter suddenly sees his brother for the sorry character he is: a \"frail figure in a whirlpool, clinging desperately to a raft of compromises.\"\n\nThe scene needed to be fixed. It didn't convey Peter's respect for his older brother. Kurt rewrote it several times, once even having the brothers get into a physical fight, but he couldn't get it right, so he took it out. He was also struggling with the ending. In one version, Josef falsifies their scientific report while Peter escapes to the West. In another, the brothers end up in Siberia, near a huge atomic bomb factory that blows up. Yet another concludes with the brothers rewriting their findings to adhere to the party line but still getting banished to Siberia, where they bitterly reflect on the similarity between men and ants.\n\nHe drafted more than ten versions of the story. His struggles to figure out Peter's fate mirrored his struggle to determine his own. He was trapped at GE just as Peter was trapped by the Communist Party. Would he continue living a lie or take the risk of escaping?\n\nBut he was also wrestling with his view of his brother. Bernard was the rock, the dependable scientist. But he wasn't like Josef, a frail man clinging to a false ideology out of fear and conformity. Living in Schenectady, being part of Bernard's work, Kurt could see that Bernie was no kept scientist, toeing the science factory's company line. He too was trying to stay true to his ideals in a culture that didn't share them. The simplistic characterization of the older brother as a thoughtless organization man wasn't working anymore.\n\nOn a blank page, while working on his ant story Kurt typed up a to-do list. One thing on it was his income tax. Another was to get back to writing his novel. For the first time he gave it a title: Player Piano.\n\n* * *\n\nBernie's phone rang at the lab. It was Bow, with a shopping list of things to pick up on the way home. Bernie grabbed the nearest thing at hand, a letter from a Park Avenue matron thanking him for giving a talk about cloud seeding to her women's club, and scribbled as Bow dictated: books, pen, cigarettes, matches, newspaper...\n\nBow was at home, recovering from the birth of their twin boys a few weeks earlier. She'd suffered headaches and gloominess after Peter's birth five years earlier, so Bernie was trying to help out as much as he could. But it was hard to tear himself away from the lab. Project Cirrus was getting closer to proving that their periodic seedings had caused widespread changes in the weather.\n\nThe day after the twins were born, GE formally dedicated the Knolls by hosting the annual meeting of the National Academy of Sciences. Irving gave a lecture about their periodicity experiments in New Mexico. He saw it as a chance to change the minds of the meteorology professors who had written a report for the Department of Defense branding his claims of widespread weather modifications \"extraordinarily extravagant.\" When the report was declassified, something like a brawl had ensued at the AMS. Harry Wexler wanted to publish the report in the Bulletin. Vincent, who had been elected to the AMS council, insisted that Langmuir be given a chance to respond. Chief Reichelderfer pointed out that Langmuir never gave anyone else a chance to respond to his GE reports. \"It seems to me it is time we began to play up the authentic meteorological opinion in this matter of rainmaking,\" he wrote to the council heads, \"in view of the complete abandon with which the proponents of artificial rainmaking on a large scale have expressed themselves in public over the past two or three years.\"\n\nFor the National Academy, Langmuir had marshaled their statistics and their best photographic evidence. He had Weather Bureau charts showing rainfall at stations throughout the Midwest enlarged to enormous size so the seven-day pattern was stunningly obvious, even from the back of the room. Chief Reichelderfer was unable to attend, but he dispatched two Weather Bureau meteorologists to Schenectady soon afterward. Bernie was back at work by then, and Langmuir enlisted him to assist in lecturing the Weather Bureau men, laying out their charts and data over the course of two days.\n\nIrving was growing frustrated with the meteorological community's refusal to grant scientific credibility to his life's most important work. He was particularly annoyed by the Weather Bureau's claim that no one could say cloud seeding worked until they understood the mechanism precisely. After all, meteorologists might never know all there was to know about the atmosphere. But they could still change it. It was like the germ theory of disease, he reasoned. For decades, doctors knew that washing their hands led to lower mortality among patients. The fact that they didn't know why was not sufficient reason for refusing to lather up.\n\nBernie, meanwhile, was determined to track down the truth about what the commercial rainmakers were doing. He began writing to chemical supply companies like Braun, Dow, Merck, and Pfizer to ask whether their sales of silver iodide had gone up. \"It is hoped that it will be possible on the basis of the information received to estimate the rate at which silver iodide has been and is being introduced into our atmosphere,\" he wrote. \"If this quantity is sufficiently large, an attempt will be made to evaluate the effects which may be produced by this seeding.\" Many companies reported no increase in sales, but a few had seen large jumps. Elmer & Amend, a division of Fisher Scientific, reported that its 1950 sales had gone from three pounds to twenty-seven. Eight pounds of that was sold to New York City for use in the watershed.\n\n* * *\n\nIn late November 1950, a winter storm hit the Eastern Seaboard. It started in North Carolina and eventually swept across twenty-two states. All-time record low temperatures were set across the Southeast, historic snowfalls pounded the Appalachians and the Midwest, hurricane-force winds hit New York, and tidal flooding surged northward up the coast of New England. The Big Ten championship game between Ohio State and Michigan was played in a near whiteout, earning it the nickname the Blizzard Bowl. A million people lost power, and 353 people lost their lives.\n\nThe Catskills were hit especially hard. But New York's rainmaker Wallace Howell wanted a full year of data. So, as heavy winds and hard rain lashed the region, a city water department crew ran a silver iodide generator attached to a trailer in Fahnestock State Park, near Cold Spring, a location chosen so that the prevailing winds would carry the silver iodide over the watershed. At the same time, a city airplane seeded clouds closer to the reservoirs.\n\nThe storm intensified that night. The northern part of Ulster County got a downpour of eight inches, and ten of the region's twenty-four rain gauges recorded more rain than in the famous hurricane of 1938. A month's worth of water was added to the city coffers in three days.\n\nThe damages were epic. Dozens of bridges and highways were washed out. Farms and villages were flooded. Parts of the New York Central's Delaware and Ulster branch railroad bed were washed away, leaving tracks hanging from hillsides like strings of Christmas lights. Three small dams burst in the village of Pine Hill, draining the town's recreational lake. In Arkville, a seventy-two-ton fuel tank broke loose and barreled a quarter mile down State Route 28, and a covered bridge was swept a thousand feet downstream. Many local residents declared it the worst flood they had ever seen. The Weather Bureau would later call the storm one of the most destructive ever recorded on the Eastern Seaboard. And its meteorologists had failed to predict any of it.\n\nIt was a freak storm, the Weather Bureau said. No one could have predicted it. Irving Langmuir didn't necessarily disagree. He doubted anyone could have predicted it. But he had a sneaking suspicion that Bernie's cloud seeding had caused it.\n\n* * *\n\nBernie and Bow didn't have far to go to get to 18 Hill Street; a short walk around the corner, and they were at Kurt and Jane's front door. Tonight the house was brightly lit and buzzing with voices. It was a couple of days after Christmas, the night of the Junior League holiday ball. Kurt and Jane were throwing a dinner party before the dance. Bernie and Bow were joined by the Fishers, the Hollomons, the McCartys, the Yarboroughs, and the Metcalfs\u2014all GE employees and their wives.\n\nThere was a lot to celebrate. After Langmuir's recent paper presenting his evidence for widespread periodicities in the Bulletin of the American Meteorological Society, the meteorologists were taking a more conciliatory tone: Harry Wexler had even gone to Socorro earlier that month to let Vince and Jack Workman present the Project Cirrus case. They disagreed on whether cloud seeding was sufficiently proved but had all gained more respect for the other side's position. It seemed as if it were only a matter of time before the meteorological world at large realized the significance of the Cirrus work.\n\nBut if Bernie had reason for optimism, Kurt was in an even better mood\u2014jubilant, in fact. The story logjam had finally broken when Knox Burger bought \"EPICAC.\" Knox had never liked the story much, and he sarcastically blamed the purchase on the higher-ups. He told Kurt his rewrite was shoddy, asking him to take into account the fact that thinking machines had become common knowledge: there was one at Harvard and maybe even one at MIT. He also suggested Kurt try to make the scientist in the story sound a little less puerile.\n\nBut the rough handling was a sign that Kurt was finally a real magazine writer. In November, Knox had bought \"White King,\" upping Kurt's fee from $900 to $1,250. Kurt wrote \"SOLD\" in big letters with a red crayon on the story's GE News Bureau folder. It was a turning point, the turning point, at long last. He had sold \"Thanasphere,\" \"Das Ganz Arm Dolmetscher,\" \"EPICAC,\" and now \"White King\" in the year since selling \"Report on the Barnhouse Effect.\" Five stories total: the number sufficient to quit his job. And he had a bunch more in the pipeline, along with his idea for a novel.\n\nSo this party was a farewell of sorts. In mid-December, Kurt had given notice at GE. He was quitting as of the start of the New Year. New year, new life. Kurt Vonnegut Jr. was no longer a PR flunky. He was finally going to be what he'd always wanted to be: a writer. And he knew what he was going to write about too: science in the House of Magic.\n\nCold Fronts\n\nKurt doodled on the back of his manuscript. Then he got out the Scotch tape. He cut parts out and taped in replacements. Once again, he was reworking \"Ice-9.\" He had given himself ten days.\n\nNow that he was a real writer, he had to be serious about producing work quickly and efficiently. This was no longer something he did in his free time; it was his job. So he had typed up a schedule for himself, planning out every writing day for two months. He allowed three days to revise a story, ten or so to write a new one. Ten days for the \"Ice-9\" revision, and then there was his novel, Player Piano. He allotted three weeks for chapter 1 and a couple of days for chapter 2, but there he stalled. He didn't really know how long a novel took to write.\n\nHe'd finished a draft of the first story on his schedule, \"Between Timid and Timbuktu,\" but he didn't like it. It focused on a recently widowed painter, David, who misses his dead wife so much that he becomes obsessed with finding a way to be with her again. Maybe, David speculates, there are creatures in the universe who can hop back and forth in time to visit previous parts of their lives. Then a local man has an accident, and his heart stops. He is quickly revived by the town doctor and upon waking reports that he saw his whole life pass before him. Inspired, David hatches a secret plot to kill himself and have the doctor revive him so he can go back in time and see his wife again. But the doctor gets another call and fails to show up at the appointed hour. David, having given himself a fatal injection, dies.\n\nIt was Kurt's first attempt to do something with the idea of time travel. Norbert Wiener's Cybernetics included a chapter called \"Newtonian and Bergsonian Time,\" in which he contrasted the time of physics, which is reversible, with the time of biology, which travels in a straight line. The time of biology, or Bergsonian time, moves relentlessly in one direction\u2014birth, growth, decline, and death succeeding one another in an inexorable forward march. But in Newtonian time, the time of physics, everything has already happened: nothing new can be created, and nothing can ever really die. An interesting thought experiment, Wiener noted, would be to imagine an intelligent being whose time ran in the opposite direction of our own.\n\n\"Between Timid and Timbuktu\" was a somewhat clumsy first attempt at a topic that would play a big role in Kurt's work. Variations on Newtonian time would recur frequently in his fiction as plot devices, always echoing the simple human longing of that early story: the desire to cheat death and hold on to the past. Most famously, he would invent the alien Tralfamadorians in Slaughterhouse-Five, who live in Newtonian time, where everything has always existed and will always continue to exist. They thus do not stand in awe of death, as we do, merely saying, when someone dies, \"So it goes.\"\n\nAt some point, critics would label Kurt's use of time travel a device for demonstrating the \"absurdity\" of the world. To do so disregards the fact that Bergsonian time is also always operating in the novels. Human beings live out their lives and die; they mourn their lost loves and dead friends. The existence of the inhuman time of quantum physics only highlights the poignancy of time as we experience it, advancing ever onward to our graves. Who wouldn't want to escape from one world to another, to follow the allure of science into a world where we never die?\n\nBut unlike later iterations, \"Between Timid and Timbuktu\" is sentimental, even maudlin at the end, as the dying David gazes wistfully at a painting of his wife. Kurt gave it three days and set it aside. It was a dog. Eventually, he would give up on it and swipe its title for a chapter head in his second novel, The Sirens of Titan. There, too, time would be a theme, with the chain of events set in motion when a character\u2014Winston Niles Rumfoord\u2014slips into a time warp.\n\nAfter setting \"Timid\" aside, Kurt had worked some more on the ant story, which was also refusing to behave. Now he was back to \"Ice-9.\"\n\nFrom his typewriter, he could look out the window. He had a real office now: he had rented a bedroom from a woman whose daughter was off at college. It was right on the main street of Alplaus, and from his desk in the coed's room he could watch the buses heading off to GE with their cargoes of company men. Poor bastards. Even as he struggled with his stories, he knew he was well out of that.\n\nBut if he had escaped, his mind hadn't: the stories he wanted to write were GE stories. He didn't call it GE. He called it General Forge and Foundry, or General Household Appliance Company, or Federal Apparatus Corporation. But the company discernible behind all of them was the one he knew so well: the scientists, the PR hacks, the visitors on Works tours, the imperious managers, and the girl pool where typists Dictaphoned the days away, awaiting rescue by diamond ring. His stories were a portrait of the GE Works itself: the numbered buildings, the departments with absurd names (\"Wire and Cable,\" \"Nucleonics\"), the offices and gates and fences and smokestacks belching their acrid benediction over the Electric City. But most of all he wrote about company attitudes: the worship of the company man; the mindless team spirit; the faith in progress, technology, and science; the enthusiasm for anything that reeked of the future\u2014the machine-made, computer-coded, semi-conducted, radioactive future.\n\nThat was certainly what he was writing about in \"Ice-9\" too. He believed in the story, but it had been rejected now by more magazines than he cared to tally. He thought maybe he knew what was wrong. It was the lack of a woman. No one wanted to read a story without a little romantic frisson.\n\nSo he changed the sociologist Professor Dale into a gorgeous redhead named Marion, a former lover of the mad scientist Dr. Macon. Once again, Macon invents Ice-9 and brags that it could destroy the planet's water. When Marion doesn't understand, Macon, sounding just like Irving Langmuir, explains that crystallization happens by chain reaction. It's like a single match starting a forest fire.\n\nAfter that, things unfold much as they did in the earlier version, with Macon insisting that he has no moral duties as a researcher and Marion insisting that he does. Once again, the evil dictator Monzano gets his hands on the Ice-9, and Macon has to try to get it back. But Kurt had to rewrite the conclusion: Professor Dale is murdered in the earlier version, and Marion couldn't be dead if there was going to be a romantic ending. Instead, Monzano jumps into a cave pool with the Ice-9, but Dr. Macon manages to get to it and melt it before the tide rises and floods the cave. The world is saved, and, of course, boy gets girl.\n\nChanging Dale to Marion helped, but it didn't fix the fundamental problem with \"Ice-9.\" Kurt couldn't quite put his finger on what was wrong in the time he had allotted, so he set it aside too and moved on. By late January, he'd completed and sent to Kenneth Littauer a serviceable story called \"Bockman's Euphoria,\" about a scientist who invents a new kind of radio wave that has a drug-like euphoric effect on anyone who hears it. People put their lives on hold just to lie around listening to it. That was a defter and less personal take on the same topic: a nifty invention that gets put to use before anyone figures out the kind of damage it can do.\n\n* * *\n\nIn Kentucky they called it the Great Appalachian Storm of 1950. In Ohio they called it the Thanksgiving Storm. Many places called it the Storm of the Century, and in Pittsburgh it was simply known as the Big Snow. But in the Catskills, where it did so much damage, they were calling it the Rainmakers' Flood. And they were furious.\n\nThe first 13 claims against New York City were filed on February 16, 1951. Within a week, there were 30 more. Eventually, more than 130 upstate clients would file legal claims against New York City for the destructive November rainstorm that they believed Wallace Howell had made. Many were individuals whose homes were damaged, but businesses filed claims too: the Pine Hill Country Club, G. W. Merritt Lumber Company, the Funcrest Hotel, Levy's Liquor Store. The town of Shandaken demanded $167,500 to repair forty-eight roads and bridges. The township of Middletown asked for $60,000 and the village of Margaretville, $23,645.91. The avalanche of claims totaled more than $1 million.\n\nThat month, a detailed meteorological account of the whole storm appeared in the journal Weatherwise. Irving read it with approval and shared it with his team. There were many anomalies that he thought supported his hypothesis that cloud seeding had caused the storm. For starters, it traveled northwest, the exact opposite of the usual weather pattern. The cold front that precipitated it had appeared out of nowhere and deepened with dizzying speed. That was why it had taken the Weather Bureau\u2014and everyone\u2014by surprise. But as Irving saw it, the 1950 storm had been caused not by Wallace Howell but by Project Cirrus and its fellow rainmakers out West.\n\nIn meteorology, large, circulating low-pressure systems are known as cyclones. Cyclones can be single storms or continent-wide circulation patterns that shape the weather across huge areas. Irving had begun a statistical analysis of the year's weather anomalies nationwide with the Weather Bureau's long-range forecasting expert, and to his eyes, the signal of the periodicities was coming through loud and clear. But weather oddities were cropping up all over the place. He believed this was because cloud seeding was affecting cyclonic development. If that was the case, silver iodide in the West might be changing the whole nation's climate.\n\nBernie was less certain than Langmuir that they had caused every large-scale abnormality in the North American weather. But he did agree that silver iodide seeding could create dramatic effects. His own paper about the cloud-seeding experiment of July 21, 1949, declared that his generator had dispensed enough particles that day to bring down 320 billion gallons of rain. He stopped short of saying that meant he had caused the resulting storm.\n\nBut Irving was unabashed about making such claims. He was even asserting, in talks about the seeding of Hurricane King, that their efforts had likely caused the storm's divergence from its course. Since that experiment, he had been conducting an intensive study of hurricane paths. In the entire recorded history of Atlantic hurricanes, only one other storm had made a hairpin turn like the one made by Hurricane King. And the other hurricane had only done so after it had slowed down and nearly stalled. Hurricane King was near its maximum velocity when they seeded it. The storm made its turn shortly thereafter. What were the odds that the two events were unrelated? Langmuir had an answer: 1 in 170.\n\nHe was growing impatient with the objections. When a young meteorologist at an MIT symposium objected to Langmuir's conclusions about Hurricane King, Irving dismissed him with a few undiplomatic words, saying he was so stupid he didn't deserve an explanation. Bernie would never have been so rough on the young man. But he understood Irving's frustration. They were collecting data as fast as they could, and everything they learned pointed in the same direction. If the Weather Bureau and the mainstream meteorology community had a better explanation, the GE group was willing to hear it. But it was almost as if the meteorologists refused to listen. The AMS Bulletin had just published a paper summarizing the research to date and stating that there was no proof that cloud seeding could produce large-scale effects. Most meteorologists seemed inclined to leave it at that: with no irrefutable proof, there was no need to investigate further. Which left the GE team alone in trying to figure out if the strange storms and droughts that seemed to be appearing all over the country had anything to do with the gold rush of rainmaking that was happening in every state west of the Mississippi.\n\n* * *\n\nHarry Wexler spent Valentine's Day in Princeton, plotting the next phase in his campaign against Project Cirrus. The previous April, with John von Neumann's computer still under construction, the Meteorology Project team had gone to the Aberdeen Proving Ground in Maryland to try out their forecasting equations on the ENIAC. The Army computer was a beast, with eighteen thousand vacuum tubes and six thousand switches. It was so big you literally worked inside it, sticking electrical wires into the plugboards that stored its programs. The team that trekked down to Maryland had worked with ENIAC around the clock for five weeks, managing to produce six retroactive forecasts\u2014predictions of weather that had already happened. They weren't perfect. Forecast accuracy was erratic. Actual computer time to produce a twenty-four-hour forecast was twenty-four hours. Nonetheless, the results looked enough like the real weather that the programmers returned to the Institute for Advanced Study triumphant. Weather prediction by equation was going to work.\n\nNow, at the Valentine's Day meeting, the Princeton meteorologists wanted to discuss something different: Irving Langmuir's alleged seven-day periodicity. Harry had brought a Weather Bureau statistician to Princeton to explain that the seven-day pattern Langmuir thought Project Cirrus had created was not the only such periodicity that had ever happened. In the last fifty years, there had been two other prolonged periods\u2014four to five months\u2014in which the weather showed a remarkable seven-day regularity. Neither was as striking as the one that had occurred in 1949\u201350, however, while Bernie's silver iodide generator was running on a seven-day schedule. But did that mean Project Cirrus was necessarily the cause? Von Neumann, his meteorologists, and a couple of Princeton University statisticians all agreed that the answer to that was still unclear. They thought Langmuir's experiment should continue for a few more years.\n\nThis had to be disappointing to Harry Wexler. For him, the ultrarational institute project was a counterbalance to Langmuir's baseless conviction. It was a way of proving that the atmosphere behaved according to scientific precepts too powerful to be overcome by the puny interventions of one human\u2014especially one who didn't even understand them. He who would master nature must learn her laws and then obey them! But it was easy for Harry to forget that John von Neumann was just as interested as Irving Langmuir was in weather control.\n\nHarry wanted more than ever to prove the fraudulence of Project Cirrus and its claims. So he had a new idea. The Meteorology Project's next round of tests was going to try out new equations developed to predict the weather in an unstable atmosphere, one with cold fronts and warm fronts battling it out. It's these unstable systems that give rise to cyclones. No numerical model to date had successfully predicted their genesis.\n\nOnce the equations were ready, the team had to choose a past occurrence of cyclogenesis on which to try them out. And Harry Wexler knew just which historic storm they should try to forecast: the Thanksgiving Storm of 1950, the so-called Rainmakers' Flood. If their equations could travel back through time and accurately predict that past event, that would definitely prove that the storm had been produced by nature and not by the hand of man.\n\n* * *\n\nIt was the future, Knox Burger told Kurt over cocktails. Nineteen sixty-one to be exact.\n\nThe last ten years had been tough. The Russians invaded Yugoslavia in 1952, launching World War III. The Soviets had A-bombed cities in Europe and America. The UN had retaliated by A-bombing Moscow. A suicide mission by American paratroopers in the Urals destroyed the Russian stockpile of atomic weapons in 1953, at which point the Soviet Union began coming apart at the seams. Dissidents rose up in the satellite states, then in Russia itself. By 1955, many key generals had defected to the UN. As the U.S.S.R. slid into chaos, the UN sent in occupying troops. Six years later, Collier's was reporting on the collapse of communism and the rebuilding of the world.\n\nThat was the premise of a special issue he and the other Collier's editors were planning. Kurt and Jane listened eagerly as Knox described it. It was spring, a beautiful mild week, and they were in New York City, staying at the fabled Algonquin Hotel. They had friends to meet and a party to attend, and Jane was finally getting to meet Kurt's adored (when he was buying) and reviled (when he was not) editor. The three of them met for cocktails. Kurt had to be gratified that Jane was there for this: the editor pitching him, trying to get him on board for an important special issue. Jane had to be gratified that her convictions about Kurt's success were turning out to be true.\n\nThings were going well. Kurt had stories in Collier's in January and February. He had rewritten \"Mnemonics,\" his story about the company man with the great memory, and finally he got it right. The breakthrough came when he got rid of Alfred's wife and introduced his secretary as a secret love interest. Then he changed all of the mental pictures Alfred Moorhead uses to boost his memory from images of violence and war to images of Hollywood starlets holding, wearing, or straddling mnemonic devices. When, for instance, he has to remember to check that Davenport Spot-Welding and Davenport Wire and Cable have not been confused in his invoices, he imagines Lana Turner in a leopard-print sheath and Jane Russell in a telegram-sarong lying on opposing davenports. At the end, Kurt added a parade of beauties that Alfred thinks up when he has to remember a really long to-do list: when he has checked off the list and one image remains, he grabs her lustily, thinking she's in his head.\n\n\"Now, baby,\" he says to her, \"what's on your mind?\" The woman turns out to be his secretary, who murmurs, \"Well, praise be, you finally remembered me.\"\n\nWith the elimination of all its sadism\u2014the part Knox said left a bad taste in one's mouth\u2014the story changed from an outburst of rage at cubicle conformity to a slightly silly office sexcapade. But what the hell? For the equivalent of a third of a year's salary at GE, Collier's could have all the libidinous diversions its editors wanted. He also sent Kenneth a slight but effective romance called \"Little Drops of Water.\" It was the sort of thing the agent\u2014and the market\u2014liked: a story about a commitment-averse bachelor who loves his predictable single life, and the clever girl who gets him to propose by slowly making herself an indispensable\u2014if annoying\u2014part of his daily routine. It had a spirited, light tone and a happy\u2014but not sappy\u2014ending.\n\nThen he wrote \"Happy Birthday, 1951\" from the part of him that brooded about war, that couldn't help but feel glum every time he read news of Korea or the Cold War or America trouncing the godless communists. Set in an almost mythic world of endless war, it described how an old man who has raised an abandoned baby through seven years of fighting comes to realize that the child he nurtured knows nothing of peacetime life. It was fast-paced and efficient and had a peach of an ending, a sharp, startling exchange on the boy's birthday that summed up, in a few words, the point: if we build a militaristic world, war is all that kids will know or care to know. Kenneth would say it was gloomy, but that was okay. One of these days, he was going to be able to write exactly what he wanted to write.\n\nIn fact now, over drinks with Knox, he was being invited to join a project that was just the sort of thing he wanted to do: a special issue of Collier's dedicated to demonstrating the madness and futility of war. It was going to make a big splash, Knox said, with top-notch writers already on board: Edward R. Murrow, Arthur Koestler, Robert Sherwood, maybe even John Steinbeck or William Faulkner. And Knox wanted to place Kurt among them.\n\nThe Collier's editors hadn't decided what to call it yet, and the whole thing was shrouded in secrecy. In office memos and conversations with potential writers, they referred to it by code name: Operation Eggnog.\n\nOn the spot, Kurt pitched Knox on an idea for a story about a cabinetmaker in occupied Czechoslovakia. He is making a booby-trapped desk for the Russian commandant in charge of his town, but when the Americans take over, the cabinetmaker continues making the same deadly desk for the new captain: Russian, American\u2014it doesn't matter to him. One occupier is as bad as another. Knox thought it sounded promising.\n\nKurt Vonnegut Jr., as Knox had predicted to Kenneth Littauer, was turning out to be quite a skillful craftsman of salable short stories. Knox had finally bought \"Mnemonics\" and \"Bockman's Euphoria,\" now titled \"The Euphio Question,\" and another story, \"The Foster Portfolio,\" was already in his hands. Knox had even offered to talk to book editors about Kurt's novel. His career was finally taking off.\n\nWhen Kurt and Jane arrived back in Alplaus, they realized the next step was clear. They were free to go wherever they wanted, to find a community better suited to the artistic life they craved. Why should they stay on in GE's orbit, shuffling around to Junior League balls and corporate clambakes? They could cut themselves loose from Schenectady altogether and drift in an ocean of possibility.\n\nThey put their house up for sale.\n\nFired up by the trip and the prospect of moving, Kurt was awash in inspiration. He wrote to Knox about another idea he had for Operation Eggnog. He could write a short-short about Dresden. Not just about Dresden, but about the morality\u2014or lack of morality\u2014in saturation bombing of civilians. Knox wrote back to say that it sounded a little risky: \"Mr. A would shy away from the notion.\" This was a light way of pointing out that Edward Anthony, the publisher of Collier's, was more conservative than Kurt realized. Collier's was not, as Kurt seemed to think, putting together another \"Hiroshima.\" The high, sane, and cheerful ideal of American citizenship was still where it set its sights, even when it came to nuclear war.\n\nJust focus on the carpenter story, Knox advised. Maybe Kurt could slip something about saturation bombing in there.\n\nJust over a week later, Kurt sent a completed draft, titled \"The Commandant's Desk,\" to Knox. He also told Knox the good news: he and Jane were leaving Schenectady. They were going to move to somewhere on the Atlantic coast. They hadn't decided where yet, but they both wanted to be near the ocean. As a trial run, they were going to rent a house in Provincetown, Massachusetts, for the whole summer. Kurt invited Knox to come visit them there.\n\nHe was on a roll. Putting his house on the market gave him such a kick he sat down and cranked out a marketable story in just one day. Called \"Build Thee More Stately Mansions,\" it was about a woman who lives in squalor but obsessively clips ideas from home decor magazines and redecorates her house in her head. When she is hospitalized, her husband redecorates to surprise her, following her clippings exactly. But upon returning, she doesn't even notice what he's done. To her, the house looks just the way she's always seen it.\n\nKenneth liked the story and sent it off to Knox right away. Before long, Kurt wrote to Knox again, this time with news that his house had sold, at a profit. Knox congratulated him, then urged him to get down to editing \"The Commandant's Desk.\" The characters were not complex enough\u2014the Czech too noble, the American major too villainous. In other words, the story was too simplistic, too blunt an antiwar instrument. The purpose of Operation Eggnog was not simply to warn the world about the dangers of nuclear war. Collier's intended to convey that nuclear war should be avoided, but if one occurred, the United States was going to win.\n\nKnox didn't explicitly say what Kurt should have known: times were changing. The nation was at war again. It was possible the United States would use nuclear weapons in Korea. It was getting increasingly important to come down hard on communism and to squelch any doubt about American virtue, even when it came to the use of superweapons. The Atomic Energy Commission was ramping up its research programs, the United States was constructing an atomic proving ground in Nevada, and the military had just begun a new round of atomic tests in the Pacific that many speculated were going to demonstrate the feasibility of the hydrogen bomb. The time for asking whether America should have dropped the atomic bomb was over. The time for talking up America's atomic supremacy had begun.\n\n* * *\n\n\"You say here that 'We are of the opinion that the Weather Bureau should continue to be responsible for leadership in this field,'\" Senator Clinton Anderson of New Mexico queried the witness. Willard McDonald, assistant chief of the Weather Bureau, sat on the witness stand looking cranky. His prepared statement had been interrupted.\n\n\"How much has been done in the knowledge of rainmaking by the Weather Bureau?\" Anderson prodded. \"I am just trying to find out whether you have leadership or whether the private rainmakers in General Electric have leadership.\"\n\n\"They have leadership in advertising,\" McDonald shot back.\n\nThat was surprising. Bernie was sitting right there in the room, with Vincent and Guy Suits, listening to McDonald take potshots at Project Cirrus\u2014but advertising? Even for the Weather Bureau, that was a low blow. But McDonald was agitated. Senator Anderson, committee chairman, was treating him as a hostile witness.\n\nThe Senate hearings pertained to three bills introduced in Congress. The main questions behind all three were the same. Should the federal government conduct research in weather modification? Should it regulate the weather modification work of private enterprises? And if the government did get involved, which agency should be in charge\u2014the Department of the Interior? Agriculture? Or Commerce\u2014meaning the Weather Bureau?\n\nNew Mexico was suffering a debilitating drought, and Senator Anderson was very much in favor of anything that would forward the cause of weather modification and help him get his constituents more rain. Or at least redirect some research dollars so he could tell his constituents he was getting them more rain. Anderson's sidekick was Senator Francis Case of South Dakota. That state's western half was also parched, so Senator Case, too, was a fan of cloud seeding. He had even taken it upon himself to fill a cottage cheese container with dry ice, hire a pilot, and fly around the state seeding clouds from the airplane window with a teacup. Whenever he saw rain, he had his pilot land at the nearest ranch, where he explained that he was responsible.\n\nNeither senator was going to let Assistant Chief McDonald get away with scoffing at the work of GE. Senator Anderson demanded to know if the Weather Bureau had come up with the idea of using silver iodide to make rain. McDonald had to admit it had not.\n\n\"Who has leadership, that is what I am trying to find out,\" Anderson insisted. \"Who does have leadership in the field?\"\n\n\"Are we talking about leadership in advertising or leadership in science?\" the assistant chief stubbornly replied.\n\nThis was turning out to be far more interesting than Bernie could have expected when he and the others arrived in room 224 of the Senate Building. But they weren't surprised by Senator Anderson's tone. The politician had been briefed. In November, he had visited Schenectady, where he had gone over the proposed bills with Guy Suits and spent an hour with Irving, Vince, and Bernie discussing their work in New Mexico. Like so many others who encountered the charismatic Langmuir, Anderson had come away a believer. Now he was openly scoffing at the idea of putting the Weather Bureau in charge of weather control experiments.\n\n\"It would be like turning over the development of the atomic bomb to some group that says 'We will try it but we know it cannot be built,'\" he declared.\n\n\"Mr. Chairman, you certainly are putting words in our mouths,\" protested McDonald.\n\nSenator Anderson then turned to his fellow committee member Senator Case to ask whether he had received any Weather Bureau assistance for his own \"rainmaking\" program\u2014his tempests dispensed from a teacup\u2014in South Dakota. Needless to say, Senator Case had not.\n\n\"Did the Weather Bureau evaluate those results?\" demanded Anderson of the assistant chief.\n\n\"I presume not,\" replied a dour McDonald. Did they really expect the bureau to take seriously some barnstorming senator heaving dry ice from a Cessna? \"I do not know whether that particular experiment came to our attention or not.\"\n\nAnderson began talking about the amount of silver iodide being sprayed into the air, accusing the bureau of being uninterested in finding out whether the activity was having any effect on the nation's climate. McDonald objected to the presumption that silver iodide was doing anything. This allowed Anderson to accuse the bureau of obstructionism.\n\n\"I do not claim that spraying silver iodide into the air produces rainfall,\" the senator said at one point, \"but I say if you spray it on Friday and it rains on Saturday, and you spray it on Friday and it rains on Saturday and you do that for 20 straight weeks, I begin to say to myself there might be some connection. The Weather Bureau says you can't prove it and therefore there is not any.\"\n\n\"Senator,\" McDonald shot back, \"I think that line of connection is just as reasonable as to say because the washerwomen hang out their clothes generally on Monday and it rains on Wednesday, that there is a connection between those two things.\"\n\n\"If that were true, you could,\" interjected another senator.\n\n\"That is exactly why I question the advisability of turning over any money to the Weather Bureau,\" declared Anderson triumphantly. \"You see where the trouble is.\"\n\nMcDonald tried to object that he was not suggesting there was absolutely no connection between the silver iodide and rainfall.\n\n\"You just got through saying that it was like hanging out the wash on Monday and it raining on Wednesday,\" Anderson shot back.\n\n\"As far as connection is concerned, it might be as remote as that,\" McDonald replied.\n\n\"Then, that is an absolute statement on your part, that the silver iodide has nothing to do with the rainfall.\"\n\n\"No, sir,\" replied the beleaguered assistant chief, \"that must not be considered so.\"\n\nMcDonald tried to return the discussion to the bureau's efforts to get at the truth of the matter, but once he had invited the washerwomen into the room, they wouldn't leave.\n\n\"I was just wondering,\" put in Senator Case, \"if the women in a certain community all hung their wash on Monday for 20 consecutive weeks and if then on 20 consecutive Wednesdays it did rain, would the Weather Bureau feel that there was no relationship between the two?\"\n\n\"You know where we would stand seriously on a question like that?\" McDonald said, stoically forging ahead as laughter rippled through the chamber. \"We would want to try that for more than 20 weeks, because we know in weather\u2014it is not as funny as it may sound\u2014we know that in weather strange repetitions, strange sequences are so common, and there are repetitions back in history of unaffected weather, weather prior to any of these things being introduced, and we will find weather patterns which are exactly like these patterns which occur after these things are begun. I think we are justified in a degree of conservative agnosticism, if you please. It is not skepticism. It is agnosticism. We do not know.\"\n\n\"The question is,\" put in Senator Guy Cordon from Oregon, another state where rainmakers were feverishly attacking the clouds on behalf of farmers, \"are you interested in finding out?\"\n\nMcDonald talked about statistics and evidence, insisting that there still wasn't enough proof to presume that cloud seeding worked.\n\n\"We would not want to draw a hasty conclusion at all,\" he declared. \"It is not our way of doing business.\"\n\n\"I am sure of that,\" responded Anderson. After pushing McDonald to admit how little money the bureau had spent researching the topic, the senator dismissed the Weather Bureau representative. McDonald had become so upset at one point he had pounded the table with his fist. That was simply not done at Senate hearings. It was a PR disaster for the Weather Bureau.\n\nThe General Electric scientists now took their seats at the witness table, and the committee's tone immediately changed from combative to fawning. Senator Anderson seemed eager to get certain facts into the record right off. He asked Guy Suits to verify that Bernie had originated the silver iodide method of cloud seeding.\n\n\"Does he have in his statement how many products he had to check before he got down to finding one?\" the senator asked. Suits said he didn't think Bernie did.\n\n\"How many did you check, Dr. Vonnegut?\" Anderson queried.\n\nBernie demurred. \"I personally have checked very few products,\" he said. \"Schaefer has checked a great number and so have other investigators.\" But the senator was not having any of Bernie's modesty.\n\n\"Did you not go through about 1,300 before you came down to one?\" he insisted.\n\n\"What I did was to look in the handbook for a compound having a particular property and there were 1,900 listed,\" Bernie replied. \"I just looked through the list until I found one that looked right. I did not try the 1,900. I just looked at them.\"\n\n\"That is pretty good evidence,\" declared the senator, determined to get Bernie's achievement into the record, \"that you did not waste time.\"\n\nGuy Suits read his statement first. He compared weather modification to the atomic bomb in its importance to the nation. He showed photographs, calling them \"irrefutable\" visual evidence of cloud-seeding results. He listed Langmuir's credentials and declared that Irving was having \"the time of his life\" in the scientific dispute over rainmaking. If he lost this argument, it would be a first. \"I place my bets on Langmuir,\" Suits declared. Avoiding the subject of regulation, he expressed support for Senate Bill 222, the only bill under consideration that would limit the legal liability of government contractors doing weather modification research.\n\nAfter Suits, Vince read his prepared statement. He went a little further than Suits had in declaring the desirability of establishing a federal commission to launch a research program in rainmaking. He pointed out the importance of knowing what kinds of weather modification activities were taking place and suggested that it would be better to license rainmakers than to try to control the field.\n\nWhen he was finished, Senator George Smathers of Florida waded into dangerous territory.\n\n\"Mr. Chairman, have not experiments been conducted in attempting to dissipate hurricanes?\" he asked. \"It seems to me I recall down off Puerto Rico, about a year or so ago, the Air Forces from Orlando, in conjunction with scientists, attempted to break up a hurricane heading toward Florida, and, I might add, without success.\"\n\n\"First of all,\" Vince quickly said, \"I would like to correct a misconception here. I was on that flight, and the main reason we made the flight was to see what a hurricane was made up of on top. With their supercooled clouds, how many supercooled clouds were there; could those clouds be changed? We did not try to break up the hurricane.\"\n\nThe senators bantered among themselves about the vaguely remembered hurricane and whether it was headed toward Florida or away from it. No one seemed to recall Hurricane King's dramatic change of direction, and Vince did not remind them of it. He told them that the team had waited until the hurricane was far out to sea and had then conducted a localized seeding. Asked what the results were, he stuck to the company script and said he didn't know. Guy Suits interjected to point out that no one would conduct hurricane research without the liability provisions proposed in Bill 222.\n\n\"Dr. Suits,\" asked Senator Anderson, \"was it not true that the hurricane, if it subsequently turned back toward the United States and it had destroyed property, it might have been a very expensive procedure for General Electric?\"\n\nGuy Suits did not point out that the hurricane had done just that.\n\n\"That is right,\" he said.\n\nDoubling down on the rewriting of history, Vincent jumped in again.\n\n\"It so happens that we did not conduct the experiments,\" he said, \"but nevertheless I concur in the need for much further studies of this very important problem.\"\n\nVince then ceded the hot seat to Bernie.\n\nBernie had written his statement carefully. He was going further than either Guy Suits or Vince had gone. He got right to the point. Cloud-seeding techniques, he declared, were going to make it possible to extend considerable control over the weather for the nation's good. But in order to make sure this power was used for good and not evil, regulation was necessary.\n\n\"Despite a strong personal dislike of restrictions and regulations,\" he declared, \"I am convinced that in order to achieve these benefits, cloud seeding must be placed under strict Federal regulation. The problems of weather control are so large and of such nationwide importance that only Federal legislation can ensure that this powerful new tool will result in the greatest good for the largest number of people.\"\n\nIt was a tricky thing, asking for federal regulation. It made you look like someone who wanted a planned economy. That was the sort of thing that got you branded a Red. He needed to make it clear that he had good reasons.\n\n\"In the absence of this legislation, I believe that the development of the benefits to be derived from cloud seeding may be greatly retarded or prevented and that possibly much harm can result from storms, droughts, or floods produced by uncontrolled seeding.\"\n\nHe proposed that the federal government provide funds for research and that the research not be classified but be freely shared with other scientists. That was even more dangerous territory, speaking out, if subtly, against military control of the science. Part of what got Edward Condon and other scientists investigated by HUAC was outspoken support for civilian control of atomic energy and for the creation of a national science foundation to move research out of the military sphere and back into civilian hands. Condon had won the battle with HUAC, but at great personal cost. Others were not so lucky.\n\nBernie stepped into this minefield with care. He was a supporter of regulation, in fact, but he couched his claim in a \"strong personal dislike,\" making himself sound more conservative than he actually was. He made a point of exempting dry ice seeding from federal control, though he gently recommended licensing it. He patiently explained to the puzzled senators that the effects of dry ice were localized, while the effects of silver iodide could persist for hundreds or even thousands of miles.\n\n\"The potentialities, both for good and bad, which attend silver iodide seeding are so large that the development and use of this technique must be placed in the hands of the Federal Government,\" he said.\n\nA senator asked him about clouds. Whom did they belong to\u2014the state, the nation, the people? To Bernie it must have seemed like a dumb question: clouds weren't things that could be construed as property. But he couldn't insult the senator. Nor could he say that clouds belonged to the nation or the people without sounding as if he were advocating nationalization.\n\n\"I think I am a poor one to ask a question like that,\" Bernie demurred, \"because my training is primarily scientific and I would not venture a legal opinion.\"\n\nSenator Lester Hunt, who had asked the question, was from Wyoming, whose legislature had just declared sovereignty over its atmosphere.\n\n\"Last year, South Dakota had some very profitable experiments in rain making,\" Senator Hunt noted. \"We think they stole some of our water, maybe.\"\n\n\"I might say that the prevailing winds, though, are generally from the northwest,\" put in Senator Case of South Dakota, implying that if his state stole anyone's water, it was Iowa's.\n\nThis was just the sort of problem that was going to become common if the government didn't regulate rainmaking. But wisely, Bernie stayed out of the rain-rustling dispute. Completing his statement, he pointed out the difficulty of regulating cloud-seeding operations, suggesting that perhaps the best method would be simply \"to make it clear to the public that operations of this sort are contrary to their own best interests and to the country as a whole.\"\n\nBernie must have been relieved when he headed back to Schenectady that day. He had managed to tread lightly on the delicate issues of civilian control and federal regulation while conveying his concern for how his invention might be used if it fell into the wrong hands.\n\nThe hearings went on for two more days, continuing to be less a debate about the need for regulation than a forum on the efficacy of cloud seeding. The senators brought in a slew of rainmakers and treated them with great courtesy. Irving Krick was given a whole day for his testimony, during which he expounded on his own successes. Wallace Howell told the subcommittee his rainmaking had probably increased rainfall in New York's watershed by 14 percent. Weather modification skeptics, on the other hand, were grilled aggressively. The young professor Irving had insulted at the MIT symposium, Charles Hosler, tried to argue that cloud-seeding results should be proved and verified in the lab before any outdoor experiments were made.\n\n\"How did you feel about the atomic bomb?\" he was asked. Things got even worse when he suggested that seeding was \"more or less like praying. It seems to be, at this point, a matter of faith.\" When the senators pointed out that many esteemed scientists believed they had seen it happen, Hosler made a grave error.\n\n\"I saw a man pull a rabbit out of a hat once, and he did it by saying two funny words, and I cannot prove otherwise,\" he snapped. \"Therefore, I think we should have legislation to produce rabbits and send the meat to India.\"\n\nSenators do not take kindly to being mocked. Hosler was not allowed to speak again after that.\n\nAt the end of day three, Vincent returned to show time-lapse photographs of clouds being seeded, then expanding into thunderheads and pouring down rain. For many of the senators, that clinched it. Rainmaking had to be real: they were seeing it happen. Guy Suits invited them all to come to Schenectady for more demonstrations. GE would provide a plane.\n\nBut after the hearings ended, the subcommittee received a letter from a witness with a different position: the Department of Defense. Its letter commenting on all three bills was far from skeptical about the possibility of weather control. But it nevertheless came out firmly against regulating weather modification in any way. The implications were clear: the military services were planning to conquer the atmosphere. And it would be easier to do that in the absence of legal constraints.\n\n* * *\n\nIlium fuit; Troja est.\n\nIt was the motto of Troy, New York, the manufacturing town near Schenectady: \"Ilium was; Troy is.\"\n\nKurt was not concerned with what was or what is. He was writing about what might be.\n\n\"Ilium, New York, is divided into three parts,\" he wrote, invoking the famous opening line of Julius Caesar's Conquest of Gaul. This was going to be a serious novel.\n\n\"In the northwest are the managers and engineers and civil servants and a few professional people; in the northeast are the machines; and in the south, across the Iroquois River, is the area known locally as Homestead, where almost all of the people live.\"\n\nHe saw it as an American version of Arthur Koestler's Darkness at Noon, another portrait of a man imprisoned for treason by a state he had helped create.\n\nThe protagonist of Player Piano is the plant manager of the Ilium Works, a \"triangle of steel and masonry buildings\" where \"machines hummed and whirred and clicked, and made parts for baby carriages and bottle caps, motorcycles and refrigerators, television sets and tricycles\u2014the fruits of peace.\" The Ilium Works was once a private factory but now, like all factories, is under control of the National Manufacturing Council, administered from the central planning office, its every baby carriage and bottle cap built according to specifications and quotas set by EPICAC, the computer in charge of the entire economy\u2014and really the entire government.\n\nAldous Huxley's Brave New World was about a horrific future world where genetically identical human beings are manufactured in test tubes and brainwashed to conform to their castes in a strict social hierarchy. George Orwell's 1984 was about a horrific future world divided into party members and downtrodden proles, with submission to groupthink enforced by a sadistic surveillance state and endless war. Kurt Vonnegut's Player Piano would be about a horrific future world where world peace has been achieved, hunger and privation banished, and nothing more awful has happened to people than that machines are now doing their jobs.\n\nKurt named his main character Paul Proteus. Doctor Paul Proteus. The name echoed that of the early genius of Schenectady, the man who knew where to chalk the X on a broken generator: Charles Proteus Steinmetz. Dr. Paul Proteus is a poor man's Steinmetz: he's smart, but not nearly as smart as his illustrious father, Dr. George Proteus, the \"first National Industrial, Commercial, Communications, Foodstuffs, and Resources Director, a position approached in importance only by the presidency of the United States.\" But Paul is well respected and affluent, part of the technological elite, so he should be happy. The war that had racked the world had been won with the help of the machines: \"Democracy owed its life to know-how.\" But the start of the novel finds him feeling vaguely at odds with the world. Kurt wrote and rewrote several openings. In the one he settled on, Paul has found a cat in the Works and is trying to adopt it\u2014ostensibly to catch mice among the machines. His desire for the cat, emblematic of his longing for life and connection in the midst of his sterile technological domain, comes to a bad end when the cat flees a machine and dies on the plant's electric fence.\n\nPaul is preparing a speech for the anniversary of the plant's takeover by the National Manufacturing Council. His ambitious wife, Anita, is hoping his speech will help forward his promotion to the position of manager of the Pittsburgh plant. But for Paul, the speech is a chance to air some of the doubts that have been creeping into his head about the world he and his fellow engineers have made. He puts a quotation into his speech: Norbert Wiener's claim that the second industrial revolution would devalue mental work.\n\nHis amorphous sense of dissatisfaction draws Paul to Homestead, the side of town where those not smart enough to obtain graduate degrees live. Unable to compete with the machines economically, people of merely average intelligence are given a choice: join the Army or work in the Reconstruction and Reclamation Corps\u2014known as Reeks and Wrecks. There's little interaction between them and the elite managers. But Paul keeps finding himself returning to Homestead's shoddy saloon, with its rinky-dink player piano. There, encountering the people's longing for purpose, he nurses the nagging feeling that something about the system is just not right.\n\nPaul's pretense to normalcy begins to unravel when his old pal Ed Finnerty shows up. Paul idolizes Finnerty: his brilliant friend \"might have been an architect or physician or writer.\" Lately, Ed has been working for the National Industrial Planning Board in Washington, a prestigious job, but he has a reputation as a rebel\u2014a man insufficiently invested in the brave new technological world, not to mention too given to women, cars, and booze. And sure enough, soon after Finnerty arrives in Ilium, he announces that he's quit his job.\n\n\"Sick of it,\" he tells Paul. \"I looked around me and found out I couldn't face anything about the system any more. I walked out, and here I am.\"\n\nThese two central characters have personalities reflecting two parts of Kurt himself: Paul, the guy who's doing his best to be the company man, and Ed, the rebel who uses messiness and irreverence as a rebuke to a society he can't respect.\n\nIn setting up his protagonist's dilemma\u2014stay and work for a system he feels is somehow bad for humanity, or revolt and become a nonentity in the techno-utopian world order\u2014Kurt recorded the details of life at GE that had irked him as vaguely yet insistently as Dr. Paul Proteus is irked. Like Schenectady, Ilium is a world divided into those with Ph.D.'s and those without them. It is a world of intense awareness of social rank and fervent adherence to convention and petty social rituals, such as competitiveness between color teams at the company's \"orgy of morale building,\" an annual camp on an island called the Meadows to which the most promising young men are invited every year. The successful in this world possess, as Paul does not, as Kurt did not, \"the ability to be moved emotionally, almost like a lover, by the great omnipresent and omniscient spook, the corporate personality.\" Above all, it is a world in which machines are established as more efficient than humans and are therefore assumed superior.\n\nRecalling his feelings while looking at GE's motorized milling machine, Kurt wrote a scene where Paul fixes a group of automated lathe machines controlled by a magnetic tape loop on which \"were recorded the movements of a master machinist turning out a shaft for a fractional horsepower motor.\" Paul thinks back to when he and Finnerty were sent to make a recording of the best machinist for the tape loop, and he remembers the man's name\u2014Rudy Hertz. \"This was the essence of Rudy as far as his machine was concerned,\" Paul thinks.\n\nAs a counterpoint to GE's techno-utopianism, Kurt imported many ideas from Norbert Wiener's Cybernetics. Speculating in his book about the inevitable chess-playing computer, Wiener admitted that it would play an optimum game according to the logic of von Neumann's game theory. But, he wondered, would it \"offer interesting opposition to a player at one of the many levels at which human chess players find themselves\"? Kurt played out that scenario in a scene where a group of young executives in training challenge Paul to a game of checkers. As Ilium's undisputed checkers champion, he accepts with pleasure\u2014until he realizes that he is being pitted against a machine called Checker Charley. Angrily, Paul tries to concede.\n\n\"I can't win against the damn thing. It can't make a mistake,\" he tells Anita when she tries to intercede. He eventually agrees to play and wins, but only because Checker Charley has a loose connection and catches fire.\n\nIn Cybernetics, Wiener describes a machine that performs a human function as a kind of mechanical slave. \"Any labor that accepts the conditions of competition with slave labor,\" he declares, \"accepts the conditions of slave labor, and is essentially slave labor.\"\n\n\"What have you got against machines?\" one character asks another in Player Piano.\n\n\"They're slaves.\"\n\n\"What the heck... I mean, they aren't people. They don't suffer. They don't mind working.\"\n\n\"No, but they compete with people.\"\n\n\"But that's a pretty good thing, isn't it\u2014considering what a sloppy job most people do of anything?\"\n\n\"Anybody that competes with slaves becomes a slave,\" the first man replies.\n\nAs Kurt saw it, Wiener's questions should be asked more in Schenectady. So he created a fictional Schenectady\u2014Ilium\u2014and put a Wienerian protagonist in it. Paul had helped to create something wonderful\u2014a society without inefficiency or privation. But he had failed to foresee its potential for damage, because he, and his entire culture, had been careless about upholding human values. They had been too spellbound by their shiny, mechanical toys, too enthralled by their exciting new science to consider its effect on real human lives.\n\n* * *\n\nBernie had drawn a map of the United States. He was shading in vast portions of the West. Those were the areas where silver iodide was filling the air.\n\nVince had been writing to the commercial rainmakers, trying to find out how many acres they had under contract. Weather Modification Incorporated. North American Weather Consultants. Water Resources Development Corporation. The Range Development Company. Snow Incorporated. The explosion of private, for-profit rainmaking operations was startling. Around 10 percent of the nation was now under commercial cloud-seeding contracts, almost all in the arid West. Pretty soon, every cloud from the Rockies to the West Coast would be sprayed with silver iodide or shot full of dry ice.\n\nIt wasn't surprising really. Precipitation for profit was an obvious outgrowth of the Project Cirrus research. It came from the same entrepreneurial spirit that aimed to put a GE refrigerator in every kitchen, a GE lightbulb in every socket. Consumers wanted GE rain on every field, GE snow on every ski slope, and until GE was willing to provide it, others would. So the world raced headlong to embrace every new thing under the sun that might lead to material gain.\n\nProject Cirrus had been formed with the idea of benefiting humanity: ranchers could improve their rangeland, utilities could make cheaper power, farmers could divert the hailstorms that caused $15 million worth of crop damage every year. But it was clear to Bernie that there had to be some oversight. Otherwise, they were never going to know what was happening up in the skies. How could they even trust the results of their own experiments? For all they knew, out-of-control cloud seeding might mean human beings were changing the whole climate.\n\nCase in point: massive floods in the heartland. Kansas and Missouri had been hit by unprecedented rains in May, June, and July. The Kansas, Osage, Neosho, Verdigris, and Missouri Rivers all jumped their banks, causing damages of around $1 billion. Both state capitals\u2014Topeka and Jefferson City\u2014were devastated. Manhattan, Kansas, was drowning in eight feet of water. In Kansas City, levees were topped, and factories, warehouses, and stockyards were washed away. Ten thousand farms lost their topsoil. Twenty-eight people were dead.\n\nSenator James Kem of Missouri visited the GE offices in July looking for answers. Irving, typically unconcerned with the ramifications, swiftly declared that the heavy rains had probably been caused by cloud seeding. Vince and Bernie were more circumspect. They felt it was impossible to establish direct causality yet but agreed that the floods should be investigated by competent meteorologists. Senator Kem asked them to draft a statement about regulation for him to use in the next round of congressional debates over rainmaking, the ones that would be closed to the public. Meanwhile, the governor of New Mexico summoned Jack Workman to Santa Fe and told him to turn off the Project Cirrus generators or lose state funds for the School of Mines: New Mexico could not risk being liable for drowning two other states. Irving thought this was an excellent idea: if the periodicity ended when the seeding did, it would further prove his results. But the Army Signal Corps did not want to stop the experiment.\n\nAs the possibility of legislation loomed, the military was getting increasingly involved. A meeting was called of the Research and Development Board\u2014part of the new Department of Defense\u2014to discuss military applications of weather control. Guy Suits telegrammed Langmuir, who was in Hawaii at the invitation of the Pineapple Research Institute, and asked him to come home early to attend. But when Irving got to Schenectady, Suits told him that the board didn't want him there after all. Vincent went to the meeting instead. Langmuir decided he would independently prepare a report for the military on the periodicities.\n\nThis was not the life of research Bernie had envisioned. Leaving graduate school and taking a job at Hartford-Empire, he'd figured he would spend the rest of his life playing around with glass, helping to make better beer bottles or fog-free mirrors. Then, during the war, someone had shown him the atmosphere. It seemed like a neat problem, figuring out what made the natural world work the way it did. But now his life was wrapped up in hearings and controversies and contracts and behind-the-scenes intrigue, even as he spent his days trying to figure out if he had invented something that was causing major harm. His research had been intended to bring the benefits of water down from the sky, to create a kind of anti-Dresden, an anti-Hiroshima. An explosive showering of life, not death, from the clouds. Now the undersea warfare branch was negotiating with GE to have Bernie come work with it.\n\nBernie didn't want to spend the rest of his life making someone's dreams of undersea warfare come true.\n\nWhy shouldn't he just do something straightforward like make toys? He was an inventor; he always had been. His vortex whistle for measuring fluid flow and true air speed would make a good toy. He wrote to Guy Suits, asking for a release of patent on the vortex whistle just for musical instrument or toy applications. Suits wrote back denying the request. GE couldn't just go around releasing patents, he said, especially when it was part of a much larger patent application. Besides, getting involved with trying to develop and market some kind of toy would distract Bernie from what he should be doing. He should focus on the work at hand\u2014by which Suits meant, do what GE wanted him to do.\n\n* * *\n\nDon't come to Provincetown, Jane implored her mother. Please don't come visit your daughter and grandkids.\n\nCape Cod had seemed like such a good idea. They would shed the tedium of the company town in a place that was its exact opposite, with wild scenery, wilder artists, cheap, shabbily gorgeous houses, and of course the ocean: the glassy bay on one side, the pounding surf on the other, and the little spit of bohemian paradise known as Provincetown in between. And it had worked. It had been an exhilarating summer. They met Norman Mailer at the beach and had cocktails with the young writer whose war novel The Naked and the Dead had made him a literary sensation. They reveled in books and paintings and jazz and freedom from everything Schenectady stood for: convention, the Junior League, the Mohawk Golf Club, the Monogram, the omnipresent, omnivorous company.\n\nThey hosted a family reunion\u2014Bernie and Bow came with Peter and the twins, Scott and Terry. And Alice and Jim came toting the Adams gang: Jimbo, Steve, and Kurt. It was nice, as it always was when the three siblings got together\u2014lots of talk, raucous laughter, beer, cigarettes, and that warm feeling of being a tribe. Kurt was finally living the life he wanted\u2014a life chosen by him and not for him\u2014and it was going well. Stories were selling steadily. Knox bought \"More Stately Mansions\" in June and \"The Commandant's Desk\" for Operation Eggnog in July, the same month that Scribner officially optioned Player Piano, the novel that was going to make his name. They lined all the kids up outside the house and took a picture, seven boys and one baby girl, their sun-streaked hair incandescent in the Provincetown sun. They were literally aglow with possibility.\n\nWhat Kurt and Jane hadn't counted on was becoming a hotel. A house near the beach on Cape Cod, especially a charming, shake-shingled cottage on Commercial Street, steps from galleries, clubs, theaters, and beaches and blessed with four breezy bedrooms upstairs, was an irresistible enticement to scads of friends and acquaintances who suddenly just had to catch up with their old pals the Vonneguts.\n\nKurt and Jane had totted it up: they had ten days to themselves all summer. The rest of the time they played hosts, which meant getting good at a game they called \"Get 'em plastered faster.\" The idea being, of course, that the sooner your friends were blotto, the sooner your husband could get back to writing the novel his publisher (oh, that did sound nice) was breathing down his neck to finish. The problem was, by the time one made all those martinis, one had sampled a few oneself, which meant no more writing that night, and the next morning would require a sluggish approach to the breakfast table, let alone the typewriter, through a martini haze. While the guests, of course, slept off their hangovers, oblivious to the fact that they were impeding the Progress of Literature.\n\nFinally September came; their last guests were headed home, and Jane had written to everyone who threatened to come that month and told them to stay away, even her mother. She wanted to see her mother, naturally, but enough was enough! The last guests left just in time for Kurt and Jane to remember that they had less than a month to figure out where they were going to live, because their rental ended October 1.\n\nThey had expected to spend several weeks traversing the coast like vagabonds in search of a place to settle. But their wonderful summer narrowed it down for them: they wanted to stay on Cape Cod. The artistic, intellectual lifestyle, the natural beauty, the quaint villages, and the ever-present ocean\u2014all of it fit the dreams they had hatched together years earlier. Provincetown, however, was a little remote, isolated on the very tip of the Cape, sixty miles from the mainland bridge. Kurt would probably need to fly to New York pretty frequently, to meet with editors and his agent, so they decided to start their search by looking for a place near Hyannis, which had an airport. And then, amazingly, they found a house in their first week of looking.\n\nIt wasn't their dream house. Jane had always dreamed of finding a wreck of a beach house or, better yet, a run-down old barn and turning it into a home. What they found instead was a spanking-new red ranch house in the town of Osterville, with a fenced-in backyard and a studio where Kurt could write. Aesthetically, it was not their style. Jane didn't even like writing the words \"ranch house\"\u2014so dull, so cookie-cutter! But given that they had two small kids and a novel to finish posthaste, and the house\u2014a model house, no less\u2014needed no work, they decided it would suit them. Osterville seemed quaint, there was a good school in Hyannis, and best of all they could move in at the end of September. So with the same impulsiveness with which they had sold their Alplaus home, they bought the Osterville ranch house. Kurt could get down to work at last.\n\nActually, he had been getting more done than Jane's letter suggested. Kurt had managed to make all the revisions that Knox Burger requested on his Operation Eggnog story, \"The Commandant's Desk,\" toning down the story's bitter antiwar tone. At Knox's suggestion, he had even rewritten the ending to make a point about the decency and sensitivity of the American occupiers. The galleys had arrived in August.\n\nMore important, Kurt had made real progress on Player Piano. Paul's story was moving inevitably toward the point where he would make his escape; he has already bought an old farm and is attempting, in secret, to learn how to live off the land. His increasingly antisocial activities have raised suspicions at the company, but his boss, Kroner, has suggested they tell the central office that Paul is working as a company mole, because there is a revolution brewing among the masses and the company wants to crush it.\n\nPlayer Piano's shadowy underground movement is known as the Ghost Shirt Society\u2014referencing the Native American religious uprising the Ghost Dance Society that Kurt had planned to write a thesis on at Chicago. The Works managers don't know much about it, but they suspect its headquarters are in Ilium. Once Paul is known to be a discontented manager, they hope to get him inside the Ghost Shirts as a spy. Paul goes along with the plan, although he has no intention of actually giving the company any information. Or does he? Is he a mole pretending to be a revolutionary or a revolutionary pretending to be a mole? Dr. Paul Proteus was developing into an interestingly complex character.\n\nKurt had an excellent new idea too: he would write a series of inter-chapter vignettes showing the lives of common Ilium people under the present technological regime. He thought they would give the novel a depth that one man's story alone might lack. The only problem was figuring out how to fit those sketches in. He could just insert them, maybe put them in italics. But it would be better to weave them in somehow so they didn't feel like an interruption. At some point during the summer, he had hit upon the perfect device. He would create a character, a luminary of sorts, and send him on a grand tour like that given the prime minister of Pakistan, Liaquat Ali Khan, in 1950. Just as the prime minister visited farms and schools and factories and of course Schenectady, Kurt's luminary would visit homes and factories and of course the Ilium Works. Maybe he'd even get to see EPICAC.\n\n\"The Shah of Bratpuhr, spiritual leader of 6,000,000 members of the Kolhouri sect,\" he began, \"wizened and wise and dark as cocoa, encrusted with gold brocade and constellations of twinkling gems, sank deep into the royal-blue cushions of the limousine\u2014like a priceless brooch in its gift box.\"\n\nHe had fun making up words for the shah's native language, drawing on his Chicago studies of linguistics. Khabu meant \"where?\" Brahouna meant \"Live!\" And takaru meant \"slave\": the shah is constantly mistaking American citizens for takaru. Kurt had the shah visit the same kinds of places the prime minister had\u2014Army installations, a barbershop, even an average American home. He wrote humorous set pieces\u2014the Army private's resentment, the family man's sad adulteries, the barber's interminable analysis of current affairs. He had particular fun with the character of the shah's tour guide from the State Department, Dr. Ewing J. Halyard, graduate of Cornell, as sartorially florid as he was occupationally stultified. But all that would change. Kurt had plans for Dr. Halyard. He might find out that his boring job is not as secure as he thinks.\n\nAll in all, Player Piano was coming along nicely, guests or no guests. Which was good, because in September, Kurt received some bad news from Knox Burger. Although it had already paid for the story, Collier's killed \"The Commandant's Desk.\" He was kicked out of Operation Eggnog. Kurt figured he knew why: the magazine's conservative publisher, Edward Anthony, wasn't willing to risk running anything that could be construed as critical of the American army. Kurt had changed a story about the unrelieved awfulness of all wars and all occupations into a story about Americans as reluctant, but fair, occupiers. But his changes weren't enough. The tenor of the times\u2014at least as Collier's saw it\u2014had veered away from complexity and embraced dogmatism. So Knox had found something to replace \"The Commandant's Desk,\" undoubtedly something anodyne and blandly patriotic. Kurt's political sentiments had lost him the most significant publication of his literary career to date.\n\nHe redoubled his work on Player Piano. It was growing difficult to ask the questions he thought needed asking in short stories for the slicks. The novel would put him on the map as a writer of serious literature.\n\nShifting Winds\n\nBernard entered the East Ballroom of the giant Curtis Hotel in downtown Minneapolis, his name tag stuck to his left lapel, and took his seat on the dais. The panel on cloud seeding at the October AMS meeting was packed. Members of the public as well as meteorologists crowded the room: Minnesota's farmers wanted to know how to make rain.\n\nJoining Bernie to give papers that afternoon were Irving Krick, Wallace Howell, Robert Elliott, and Paul MacCready\u2014commercial rainmakers, all of them. The only other serious researcher listed on the docket was Sol Resnick, a professor at Colorado A&M. But there was a last-minute addition to the roster\u2014Herbert C. S. Thom, statistical expert from the Weather Bureau, added through bureau machinations.\n\nBernie was going to outline the construction of his spray-nozzle smoke generator for dispensing silver iodide. The commercial rainmakers were giving papers much more ambitious in scope. Wallace Howell was surveying methods of evaluating artificial rainmaking programs. Paul MacCready was reporting on fantastically successful cloud-seeding operations over ten thousand square miles of Arizona. Robert Elliott was showing time-lapse movies demonstrating the effects of using differing amounts of silver iodide. And Krick, in typically enterprising fashion, was holding forth on the topic most likely to interest the locals: the economic benefits of rainmaking, including dollar-for-dollar returns in crop yields and better grazing conditions that western farmers could expect if they hired someone like him.\n\nAfter all that, Professor Resnick was going to report on a study showing that rainmaking didn't work, and Herbert Thom would denounce the methods of statistical analysis that suggested it did.\n\nThis was the absurd position Bernie found himself in. He wasn't one of the boosters, the salesmen who drove the Weather Bureau mad with their unsubstantiated claims. He wasn't even as big an advocate as Langmuir, who was now declaring that cloud seeding was probably causing most of the nation's weather glitches by altering cyclonic development. Langmuir had finally succeeded in getting the Project Cirrus generators in Socorro turned off and was expecting the periodicity to vanish, finally proving irrefutably that silver iodide was its cause.\n\nBut if Bernie wasn't an unthinking promoter, he wasn't a naysayer either, one of the people Irving called \"wet blankets\" who made it their mission to prove that rainmaking was hokum. Bernie believed that it probably worked and that they should quit bickering and get down to investigating possible unintended results. Because ever since the Great Plains floods, Bernie had been growing more alarmed.\n\nEarlier that month, Bernie had voluntarily attended the Project Cirrus steering committee meeting. Afterward, he wrote a letter to the Signal Corps' Dr. Michael Ference, head of the committee. He had attended the meeting, he said, because he was concerned about the possible relationship between cloud seeding and floods. Langmuir was convinced there was a connection, and the Weather Bureau was convinced there wasn't. Bernie didn't quite know what to think, but he thought the floods that summer had been tragic. If they were preventable, it was even more tragic.\n\nBernie politely suggested that Project Cirrus release a measured statement. He thought it could be worded so that it simply pointed out potential dangers without provoking controversy. It was important that people racing to use this new technology understand that it might have unintended human consequences.\n\n\"Project Cirrus is supported by the public, and it seems to me that it is very clearly our responsibility to inform the public to the best of our ability,\" he wrote. \"If we fail to do this now, and new floods occur which can possibly be attributed to the cloud seeding, I think we will rightfully share in the blame and deserve the harshest criticism.\"\n\nThe kind of criticism, that is, that he was reading in almost every story his brother wrote.\n\n\"I have taken the liberty of expressing my views to you despite the fact that I am not a member of the Steering Committee,\" Bernie wrote in closing, \"because I feel I have a share in the responsibility involved.\"\n\nHe was sticking his neck out because he wanted to do what was right. He wouldn't let love of his invention blind him to his duty to humanity.\n\nBernie still loved the science. But here in Minneapolis, it was more obvious than ever that the very topic of rainmaking caused people to behave in completely unscientific ways. They either became fanatical crusaders or closed-minded skeptics. They either hoped to profit from it before the science was complete or refused to believe any science that didn't fit with their preconceived ideas. Where were the calm voices of reason that might keep real harm from being done?\n\nIt was not unlike what was going on in the nation, in the world in fact. Russia was evil, and America was good. Or, if you were Russian, the Soviet Union was good, and America was evil. Politics was not even a battleground anymore; it had become a kind of puppet theater where caricatures of goodness browbeat caricatures of evil, and no one wanted to hear the more complex story told.\n\nScience\u2014real science\u2014was always a complex story. It was rare for things to be black-and-white, for evidence to be irrefutable and results to be obvious to all. Science was a conversation; it was the back-and-forth dialogue that mattered. When the dialogue degenerated into fanaticism and politics, it no longer functioned as science. Finding the truth and mapping the best course of action became impossible.\n\nBernie gave his paper on the spray-nozzle generator to the assembled meteorologists and farmers. He would publish it, giving co-authorship to Kiah Maynard. But it would be his last published paper on making rain.\n\n* * *\n\nCollier's special issue, \"Preview of the War We Do Not Want,\" hit newsstands in late October. The cover depicted a sorrowful but grimly determined American MP standing in front of a colorful map of the Soviet Union. The word \"occupied\" was stamped on the Ukraine, and a UN flag was planted on Moscow. The list of writers underneath the picture was impressive: Robert Sherwood, Arthur Koestler, Walter Winchell, Edward R. Murrow, J. B. Priestley, Philip Wylie. It would have been nice to see Kurt Vonnegut Jr. among them. But in the place of \"The Commandant's Desk\" was a limp story by John Savage called \"Trouble at Tuaviti,\" in which a brave American missionary and his loyal Pacific Islanders foil a Soviet sneak attack.\n\nKurt read the issue with growing distaste. The whole magazine treated atomic war as something that could be won\u2014that would be won, naturally, by the United States. And then everything would come out okay in the end, because for America everything always did. The issue was lavishly illustrated: Washington, D.C., after a nuclear bomb assault; parachutists dropping into the Ural Mountains; Grand Central Terminal after a bombing; women packing a Moscow stadium for the first post-Soviet fashion show\u2014because of course after atomic war has devastated the planet, the first thing a woman's mind turns to is hemlines and hats. Collier's must be raking in the profits, because the issue was packed with advertising: General Motors, Body by Fisher, Pall Mall, Pabst Blue Ribbon, Firestone, Frigidaire, and of course GE, using the prospect of World War III to sell washing machines.\n\nKurt particularly disliked Philip Wylie's story, \"Philadelphia Phase,\" a sentimental love triangle between an American officer, his blue-blooded girlfriend, and a Russian immigrant, set in a nuclear-war-ravaged Philly. It turns out the Russian girl has been made sterile by radiation, so she kills herself, leaving the hero to go back to his Main Line sweetheart. Boy gets girl, even amid the inconvenience of atomic holocaust.\n\n\"The Commandant's Desk\" had been true to the grating inhumanity of war and the moral hazards of occupation. It had treated soldiers as complicated human beings and war as a threat to our better selves. That, Kurt knew, was why his name wasn't flying off the newsstands with the rest of them.\n\nBut he couldn't spend time worrying about it. He had more important work to do. Scribner's Harry Brague, who was editing Player Piano, had written to say that he loved the new inserts about the shah of Bratpuhr. He assured Kurt that this was going to be a good novel, a novel Scribner would be proud to publish. And while not pressuring him exactly\u2014the publisher did want him to write the best book he could\u2014he informed Kurt that in order to make the spring list, the book needed to be in Scribner's hands by November 15. That was less than a month away, and there was still quite a lot to go. Paul Proteus had come to his crossroads, the place where he would have to decide whether he was a company man or a rebel. He would either inform on the revolution or join it.\n\nWhen he started the novel, Kurt had not known what Paul would do. He had made several outlines. Ed Finnerty would become a revolutionary, but Paul would side with the machines. Or Finnerty would rebel, and Paul would reluctantly testify against him. Or Finnerty would persuade Paul to join the revolution, and Anita would testify against Ed to get Paul back.\n\nBy October, he had made up his mind. He was going further than any of his previous endings. Dr. Paul Proteus, having quit his job, would accept the role of leader of the revolution and would refuse to inform on his fellow rebels, even though he would lose everything for it\u2014his wife, career, friends, status.\n\nIt was his first scientist character since Professor Barnhouse who would refuse to let love of his invention blind him to his duty to humanity.\n\nHe brought it all to a head at the Meadows. Here was a chance to expose the absurdity of Camp General Electric, that craven corporate pep rally on Association Island, and he used all the ammunition he had. The men arrive at the island by boat, just as in real life. They line up for flag-raising ceremonies under the old oak\u2014counterpart to GE's treasured elm. They are divided into four color teams and attend a play that seems to be directly based on Lemuel Boulware's skit for 1948, the summer Vince Schaefer and the News Bureau's Roger Hammond attended. The loudspeaker at the Meadows blasts songs from the Association Island songbook.\n\nAs he wrote the Meadows section, Kurt wove in two new storylines that were also rooted in his GE experience. In one, Dr. Ewing Halyard of the State Department gets a letter informing him that he never completed his physical education requirements at Cornell, invalidating his undergraduate degree. With no bachelor's degree, \"he had never been entitled to his Ph.D., his classification numbers, or, more to the point, to his pay check.\" Halyard is put on probation until he can make up the missing credits. He goes to Cornell, only to discover that the head of the athletic department is still angry about a letter Halyard wrote to the alumni magazine complaining about the football team's rowdy postgame behavior at an establishment called Club Cybernetics. The coach takes his opportunity for payback, flunking Halyard on his PE exam. Halyard will never get his degree, and he will lose his job because of it.\n\nIn another section, the shah of Bratpuhr experiences a bout of lust. He begins shouting untranslated but obviously indecent suggestions to women from the window of his limousine: \"Pitty fit-fit, sibi Takaru? Niki fit-fit. Akka sahn nibo fit-fit, simi Takaru?\" A reluctant Halyard prepares to play pimp; he's done it before. But then a woman hearing the shah's catcalls agrees to get in the car. She seems like a normal American housewife, and Halyard tries to tell her that she has made a mistake. The woman says she knows exactly what she's doing. \"He was asking for something, wasn't he?\" Halyard says yes. \"There's been no misunderstanding,\" she tells them. She has agreed to prostitute herself because her husband has been fired. Four days earlier he had the classification number W-441, or \"fiction novice.\" He turned in a book that was beautifully written, but twenty-seven pages too long, almost ten points above the acceptable \"readability quotient,\" and with an anti-machine theme.\n\n\"So he was ordered into public-relations duty,\" she concludes.\n\n\"So the story has a happy ending after all,\" Halyard says.\n\n\"Hardly. He refused,\" the wife tells them. Halyard is shocked that a man would rather have his wife prostitute herself than go into public relations. \"I'm proud to say,\" the woman replies, \"that he's one of the few men on earth with a little self-respect left.\"\n\nThat was him, of course, the novel bending back on itself to suggest that Player Piano might be the very book\u2014twenty-seven pages too long, ten points beyond readability, with an anti-machine theme\u2014written by a writer husband who despises PR. It was the first of Vonnegut's many surprise cameos\u2014Hitchcock-like\u2014in his own novels. Together the two tangential stories\u2014Halyard's and the writer's\u2014reflected the anxieties that had colored his time at GE. Kurt lacked the degree that made him worthy of his rank and pay scale, and he had been improperly consigned to public relations, a job he ultimately refused. His novel thus raised and vanquished the twin demons that had haunted him at GE: the fear that he didn't fit in, and the fear that he did.\n\nAnd he did it at the same time that he was resolving his feelings about Paul Proteus, the scientist inventor, the golden boy of the Ilium Works who must choose between an easy life as a company man and what he knows in his heart is right. When he is captured in a raid on a Ghost Shirt meeting, Paul is given a stark choice. He can claim he was acting as a spy and name the leaders of the revolution. Or he can confess to having joined the revolution, forever destroying his chance of returning to his comfortable place among the elite. Paul recognizes the mythic quality of his decision:\n\nHere it was again, the most ancient of roadforks, one that Paul had glimpsed before, in Kroner's study, months ago. The choice of one course or the other had nothing to do with machines, hierarchies, economics, love, age. It was a purely internal matter. Every child older than six knew the fork, and knew what the good guys did here, and what the bad guys did here...\n\nBad guys turned informer. Good guys didn't\u2014no matter when, no matter what.\n\nPaul Proteus would end the novel as a good guy. He would claim his position as leader of the Ghost Shirt Society and happily accept the manifesto written in his name by one of the revolution's leaders, the political science professor Ludwig von Neumann.\n\nLudwig von Neumann is so unlike John von Neumann it seems likely his name was selected for irony. When the revolution backfires, it's he who bemoans the fact that the revolutionaries failed to destroy EPICAC. His manifesto ends with an incantatory list of affirmations of what is human in human beings:\n\nI hold, and the members of the Ghost Shirt Society hold:\n\nThat there must be virtue in imperfection, for Man is imperfect, and Man is a creation of God.\n\nThat there must be virtue in frailty, for Man is frail, and Man is a creation of God.\n\nThat there must be virtue in inefficiency, for Man is inefficient, and Man is a creation of God.\n\nThat there must be virtue in brilliance followed by stupidity, for Man is alternately brilliant and stupid, and Man is a creation of God.\n\nYou perhaps disagree with the antique and vain notion of Man's being a creation of God.\n\nBut I find it a far more defensible belief than the one implicit in intemperate faith in lawless technological progress\u2014namely, that man is on earth to create more durable and efficient images of himself, and, hence, to eliminate any justification at all for his own continued existence.\n\nImperfection, frailty, inefficiency, stupidity: these were precisely the qualities John von Neumann had dedicated his life to eradicating not just in meteorology but in every human endeavor.\n\nKurt mailed the final manuscript to his editor Harry Brague with a stipulation. If Scribner liked the novel and decided to publish, it must never publicize his relationship with General Electric. The company, he explained, was holding a hostage: Bernie. He did not want his brother's career put at risk because of something he had written.\n\nFor the first time in his life, Kurt found himself in the position of looking out for his older brother.\n\n* * *\n\nMore than four hundred people crammed into New York's Roosevelt Hotel for the January 1952 AMS meeting. But to those who were paying attention, the tide had turned. The Weather Bureau campaign was beginning to vanquish belief in rainmaking.\n\nThe Weather Bureau scientists were taking a more diplomatic tone; ever since the disaster of the May Senate hearings, McDonald's pounding fist had been thudding in Chief Reichelderfer's head. Congress controlled the bureau's budget: the negative press after McDonald's outburst had cost it real money! The campaign was now being conducted mostly behind the scenes. Weather Bureau analysts were double-checking all the Project Cirrus data, and Harry Wexler was ever present in the background, quietly making sure anyone who was exposed to Project Cirrus heard the Weather Bureau's side of things too. Most important, Weather Bureau statisticians were working to disprove Langmuir's claims.\n\nLangmuir was unconcerned. He didn't mind that his AMS meeting talk on the seven-day periodicities was scheduled for a panel with two Weather Bureau statisticians. He actually thought their work would help him to prove that his seeding had caused the weather periodicities. In fact, it was doing the opposite. For all his brilliance, Langmuir had failed to see the irremediable flaw in his method pinpointed by the bureau statisticians. Langmuir's calculations of probabilities for rainfall in seeded areas assumed that rain in one area could be considered independently of rain in the next\u2014that in looking for patterns, he could treat rainfall levels as if they were random numbers. But rain in one place is probabilistically related to rain in a place nearby; a touch in one place eddies and flows, setting up a touch in the next, and so on, until you have reached the other side of the earth.\n\nIt was the statisticians who would, in the end, cast the Project Cirrus research into doubt and consign Irving Langmuir, once the nation's most famous chemist, to a marginal place in the history of science. Irving did not see it coming. He did not see that his whole way of doing science\u2014his generalist, do-it-yourself, paper-clip-and-string mode of Victorian science\u2014had become a liability. Science was being sorted into silos, and interaction between them was strictly regulated. As a chemist who had strayed into meteorology and then statistics, his failure to stick to his specialty branded him an outlier, someone who could safely be ignored.\n\nOnly one group of people considered him relevant. Recently, two officers from the Joint Chiefs of Staff had approached Irving and said they'd like to meet him in Schenectady to discuss military applications of weather control. Happily, he gave them his contact information. He had already renewed his security clearance, expecting he would soon be working directly with the military. Word was getting around that GE planned to discontinue Project Cirrus as of July, the end of the current contract. Under the reign of \"Razor Ralph\" Cordiner, the company was being reorganized into business units\u2014even the Research Lab. Lab managers were given orders to organize teams of researchers to carry out prescribed projects\u2014projects that were likely to lead to monetary results for GE. Doc Whitney's question, \"Are you having fun today?\" was no longer the watchword in the House of Magic.\n\nAfter the AMS meeting, Irving returned to his island on Lake George, where he was now conducting most of his work. Visitors and colleagues who wanted to talk to him had to write in advance, because he had no telephone on the island. Then they would go to the Adirondack town of Bolton Landing, where arrangements would be made for Langmuir's Chris-Craft, Wendy, to bring them across Lake George to the laureate's leafy Neverland.\n\n* * *\n\nWhat are people for? Kurt was pondering that on Christmas Eve. The line was the very heart of the novel, the most succinct statement of his theme, and his editor Harry Brague had cut it out.\n\nEditors!\n\nPlayer Piano wound down with a failed revolution. After refusing to rat out the insurgents, Paul is tried for treason. During his trial, the Ghost Shirt Society starts an uprising, freeing Paul and aiming to take the world back for humanity. But the attack on the machines rapidly turns into a free-for-all. Drunk on destruction\u2014and on liberated booze\u2014the people smash everything from the Ilium Works assembly lines to bakeries and sewage disposal plants. Weakly, Paul, Finnerty, and the other leaders try to stop them from destroying the useful machines, but it's hopeless. There is no middle ground between fanatical love of the system and fanatical hate. Technology is either an absolute good or an absolute evil: the more complex story Paul wanted to tell is lost in the joyous upheaval of the downtrodden taking charge of their destinies.\n\nHomestead is isolated by government troops, who refuse to come in and help clean up the mess until the revolution's leaders are handed over. Paul and the others briefly hope to build a truly human existence: to chop wood and grow food and build shelters. But, in a final irony, as soon as the frenzy of ruin has given way to the cold light of day, people miss the machines. The revolution's leaders come upon a group gathered around a smashed vending machine that once dispensed a soda called Orange-O. Everyone has always hated Orange-O, but they are cheering on a comrade as he repairs the machine. The people, it seems, are doomed to reassemble the very world that had oppressed them.\n\nRemarkably prescient, Player Piano foresaw a world divided between well-educated whiz-kid executives who believe technology is the answer to every human problem and alienated service workers showered with shiny new techno-gadgets in place of real roles as citizens. At the center of this world is the computer, deified by the paternalistic, paranoid culture of the modern corporation. Previous dystopian novels taught readers to look for hope in the success of the revolution\u2014or despair in its failure. Winston Smith's revolution fails in 1984, leading to his brutal torture and \"re-education.\" Bernard Marx's attempt to escape the system in Brave New World leads to his exile. The dark irony of Player Piano is that no torture, no exile, is required. Before the revolution's smoke has even cleared, the rebels are at work rebuilding the very technology they revolted against, because technology tells far too seductive a lie. It tells us we can transcend our banal physical limitations; we can travel at the speed of sound, think at the speed of light, live forever in a shiny digital Eden. We humans, the novel implies, will always crave Orange-O machines and computers and video games and iPhones and self-driving cars, even if we suspect that these false gods are robbing us of our humanity.\n\nDisconsolately, Paul and the other leaders turn themselves over to the state.\n\nThe original draft of Player Piano concluded with a scene between Ewing J. Halyard and the shah of Bratpuhr. Halyard decides to escape demotion by immigrating to Bratpuhr with the shah. But the shah explains that he will be a slave there if he can't rotate his navel. Halyard rages at the ridiculousness of assigning status based on something so pointless, but the shah is implacable. So Halyard performs his last duty as a member of the State Department, offering the shah and Bratpuhr an American \"modernization\" package that will endow them with all the machines and computers necessary to re-create the American economy. The shah refuses, instead giving Halyard one more question for EPICAC: \"What are people for?\"\n\nThat question was the one Kurt wanted to leave ringing in the reader's head. But early in November, Kenneth Littauer had called to say that Harry Brague didn't like Kurt's ending at all. The whole book contract hung in the balance. Kurt flew down to New York to meet with Harry in person and figure out a better conclusion. Together, the writer and the editor decided that it was wrong for the book to end on Halyard. The protagonist was Paul; the story should end with him.\n\nKurt wrote a new conclusion to follow the shah's departure, an epilogue in which, seventeen years after the failed revolution, Paul Proteus is released from prison and comes home to his farmhouse to find his former boss Kroner waiting to welcome him. Paul has spent his incarcerated years as the prison librarian, and his embrace of classic literature has given him a new, more spiritual view of life. He recognizes that the world will always change, and he just hopes humans can hold on to their spiritual values despite the relentless forward march of technology.\n\nHarry Brague didn't hate the new conclusion, and at long last he issued Kurt a contract for the book. But after reading it a few times, the editor wondered if the epilogue was necessary. In fact, he thought Kurt could dispense with the last two chapters, ending the novel on the willing surrender of Paul and other leaders of the revolution. It would be a starker ending, and more serious. Kenneth Littauer thought it would make the novel feel pessimistic, but Kurt liked the idea. It might leave people wondering, arguing: Was the ending happy or sad? Was the writer an optimist or a pessimist? Was there hope for humanity? He wanted his novel, like Orwell's or Huxley's or Koestler's, to start conversations, arguments even.\n\nBut he was unwilling to lose the shah's question for EPICAC: What are people for?\n\nThe question is reminiscent of the penultimate chapter of Norbert Wiener's Human Use of Human Beings. \"Our papers have been making a great deal of American 'know-how' ever since we had the misfortune to discover the atomic bomb,\" Wiener wrote. \"There is one quality more important than know-how and we cannot accuse the United States of any undue amount of it. This is 'know-what': by which we determine not only how to accomplish our purposes, but what our purposes are to be.\" In order to explain the difference between \"know-how\" and \"know-what,\" Wiener gave the example of a \"prominent American engineer\" who bought an expensive player piano. But the engineer was not interested in the music; he was interested in the piano's mechanism. \"For this gentleman,\" Wiener wrote, \"the player piano was not a means of producing music, but a means of giving some inventor the chance of showing how skillful he was at overcoming certain difficulties in the production of music.\"\n\nOn perhaps the last innocently joyous night of his life, just before he heard that Hitler had invaded Poland, Kurt had spent an evening with his buddies at Woolaroc Ranch in Oklahoma, smoking cigars and loading rolls of music into the player piano. They hadn't been interested in the piano's mechanism, in the know-how of it. They had been interested in hearing the tunes while they hung out, a tribe of three, enjoying one another's company. Like his brother, Kurt loved music: jazz and classical and the Beatles\u2014it all partakes of the know-what, the higher truth that gives beauty and purpose to human existence. It's the music, not the mechanism, that people are for.\n\nSo, on Christmas Eve, Kurt found a place earlier in the book where the shah's question \"What are people for?\" could be reinserted, and he typed up an insert and mailed it to Harry Brague. And with that, his first novel was complete.\n\nKurt was so excited for the book to come out he could hardly stand it. He was eager for the royalties to start rolling in. He needed the money. Harry told him to sit tight and put the book out of his head. Get going on the next one, he advised. The only thing left to do now was to write a dedication, if he wanted one. Kurt sent one back right away: \"For Jane\u2014God bless her.\"\n\nWhen the galleys of his first book arrived in March, he thought they were the most beautiful thing he had ever seen. He wrote to Harry suggesting a few people to whom Scribner should send advance copies of the book, hoping they might help spread the word. One of them was Norbert Wiener.\n\n* * *\n\nJohn von Neumann's computer was a thing of beauty. There it sat, eight feet long, six feet high, and two feet wide\u2014a Corvette in a world that until now had only seen 18-wheelers. At a mere thousand pounds, it was smaller and sleeker than Harry Wexler had ever imagined a computer could be. It had twenty-three hundred vacuum tubes and forty raked cylinders along its sides and would be faster than ENIAC by far, doing 2,000 multiplications or 100,000 additions per second. It would whip through a twenty-four-hour forecast in a mere three hours.\n\nIn other words, it had roughly as much computing power as the kind of handheld calculator you might find in a cereal box today. And it was going to revolutionize computing.\n\nIt was May 1952, and von Neumann and his engineers were still working out the kinks, but the computer was finally operating. And unlike ENIAC, it didn't break down all the time. The summer before, it had run smoothly for sixty straight days doing a large thermonuclear calculation. Soon it was going to start in on the weather. The Meteorology Project's mathematical model had been growing increasingly complex, but the team believed it was sophisticated enough to predict cyclogenesis. As Harry Wexler had suggested, they were going to test the new equations by attempting to predict the formation of the 1950 Thanksgiving Storm.\n\nA year from now, the computer would successfully forecast the cyclogenesis that led to the Thanksgiving Storm of 1950, proving that the so-called Rainmakers' Flood was a predictable act of nature. But even now, before that triumph, when Harry looked at the computer, he felt the satisfaction of having reached a peak he had only vaguely discerned during the long ascent. The days of predicting the weather using history and human intuition were about to end. Imperfection, frailty, and inefficiency were not part of Johnnie's new machine. The atmosphere could be understood deterministically. It could be reduced to equations and solved. And after the atmosphere, what next? The computer would continue to grow speedier and smarter, would solve more and more human problems. Harry Wexler was looking at the future not just of weather but of technology. The future of science. Of humanity. He was seeing a world just like that of Player Piano, but unlike Kurt, he had no problem with it.\n\nOne of the project engineers later summed up the magnitude of that moment in history. \"A tidal wave of computational power was about to break and inundate everything in science and much elsewhere,\" he wrote, \"and things would never be the same.\"\n\n* * *\n\nGuy Suits had asked Bernie to come see him. Bernie figured it was about his future at GE.\n\nProject Cirrus was officially shut down. GE was getting out of weather modification; the liabilities were too big and the profit potential too small. Vincent and Bernie were unhappy about it, but there was nothing they could do. Under the new regime of Razor Ralph, they had to work on what they were told to work on.\n\nThe most fun Bernie had had at work lately was trying to \"play\" a germanium crystal on the Project Cirrus facsimile machine. It hadn't worked. The etched surface of germanium had a complicated structure, like a record, but the facsimile machine was not sensitive enough to pick up its signal. It had been worth a try. Purified germanium was a semiconductor, and its structure had been of great interest to the entire world since 1948, when scientists at Bell Labs had introduced the world to a new device that used a germanium semiconductor: the transistor.\n\nTransistors! These tiny, pea-sized devices were all anyone could talk about. Smaller, faster, more durable, and more efficient than vacuum tubes, transistors were going to revolutionize electronics. It was the dawning of a new age for that industry, which experts predicted would soon grow to rival the chemical industry. Forget about better living through chemistry: better living through electronics was the future now. Power transfer, communications, lighting\u2014even thinking was going to be done by transistors one day. There would be televisions half the current size! Radios that could fit in your pocket! Computers that could sit on a desk! They were all just around the corner because of Bell Labs' new invention.\n\nEveryone wanted in on the promising new product, and GE was no exception. Research Lab scientists were being redeployed to the semiconductor section, Bernie among them. Vincent, too, was told that he should consider taking up semiconductors now that GE was divesting itself of weather.\n\nBernie wrote up his research in Research Lab Report RL-723: \"Variations in the Contact Resistance of a Copper Point Moving over Etched Surface of Germanium Crystal.\" Diodes were interesting, but his heart wasn't in semiconductors and transistors. And he didn't want to be told what to research.\n\nHe had begun to look around for other jobs. He was corresponding with Bill Hubert, formerly with Project Cirrus, who was now at the Institute of Meteorology in Stockholm. The institute wanted a good cloud physics man to come to Stockholm for a few months. Bill had suggested Bernie, and Bernie was tempted. But Sweden was a big trip for the family, and he had also received a job offer from Arthur D. Little, a private research company in Cambridge, Massachusetts. Arthur D. Little conducted research in many different fields, and its scientists had more control of what they worked on than scientists at the GE Research Lab did now.\n\nGuy Suits had clearly gotten wind of the fact that Bernie was thinking of leaving GE. When Bernie got to his office, Suits had a speech prepared. It was a big step Bernie was thinking of taking, he said. He hoped Bernie wouldn't take the decision to leave GE lightly. After all, there were a lot of factors to weigh.\n\n\"One of the things you must consider,\" he declared, \"is your equity in the pension plan.\"\n\nHere it was again, that same fork in the road that had confronted his brother. Of course, the choice at hand had nothing to do with pension plans, Quarter Century Clubs, appliances, or clambakes. It was purely an internal matter. Every kid past sixteen knew this fork, what the good guys did here, and the bad guys. Good guys stayed true to their love of science, their pursuit of knowledge for the good of humanity. Bad guys were venal. They made choices based on money.\n\nBernie knew at that moment, listening to Guy Suits, that he had made his decision and that it was the right one. He would leave GE, which in 1952 was no longer the company that had hired him in 1945. It had changed, radically. But then, so too had the nation.\n\nAnd so, in fact, had he.\n\n* * *\n\nKurt was crushed: Norbert Wiener hated Player Piano.\n\nIt was the latest in a series of small crises in the run-up to the publication that had left Kurt a nervous wreck. First he spent a couple of weeks worrying that a soda company somewhere might actually make something called Orange-O and that it would sue him for saying no one liked it. He had persuaded Harry Brague to have the Scribner lawyers look into it. The lawyers could find no evidence of a real Orange-O.\n\nThen he had somehow got it in his head that Scribner was holding off publishing the book until fall, nearly a year after he'd completed it, and he freaked out, fearing he wouldn't get any royalties until he and his family had starved. Harry assured him that the book was slated for late summer. Somewhat reassured, Kurt retired to his study to write promotional taglines and sketch ideas for the cover. He sent these to Harry, who ignored them, along with Kurt's concepts for marketing campaigns and avowals that his next book would be better.\n\nBut this latest blow was the worst of all. Norbert Wiener had written Scribner a scathing letter. Kurt thought Wiener would love the novel! After all, Wiener's ideas were threaded throughout the whole book. Kurt even had Paul Proteus credit him with the idea of the second industrial revolution. He'd asked Scribner to send Wiener the book because he thought the mathematician would take it as a kind of tribute. Instead, Wiener accused Kurt of setting the novel in a dystopian future in order to avoid indicting what was actually happening in science today.\n\nIt was a problem Kurt would bump into over and over in his career. People would read his work as some sort of futuristic science fantasy, persisting in seeing his books as comic space operas and druggy head trips, when he thought he was writing pointed social satire.\n\nIn 1973, David Standish would ask him in Playboy why he had turned to science fiction for his first novel. Kurt told him about working at General Electric and seeing things like the automated milling machine.\n\n\"So science fiction seemed like the best way to write about your thoughts on the subject,\" Standish pressed.\n\n\"There was no avoiding it,\" Kurt said, \"since the General Electric Company was science fiction.\"\n\nIt's unclear what set Wiener off, though he was notoriously touchy about his ideas being co-opted, often declaring himself \"not a Wienerian.\" Yet Kurt had captured exactly the moral questions raised in Cybernetics and The Human Use of Human Beings, questions Wiener would continue to explore in later works like God and Golem Inc. It may be that some of Wiener's hostility resulted from jealousy. He had literary aspirations himself\u2014he published science fiction stories in the MIT magazine Tech Engineering News and would write a novel in 1959\u2014and Kurt had conveyed many of Wiener's ideas in a form more accessible and enjoyable than the mathematician's own.\n\nWiener seemed most upset by the fact that Kurt had used his friend John von Neumann's name for one of the novel's revolutionaries. In his letter, he told the folks at Scribner to tell Vonnegut \"he cannot with impunity... play fast and loose with the names of living people.\" This of course is balderdash: writers play fast and loose with the names of living people all the time. What really upset Wiener might have been that Professor Ludwig von Neumann\u2014described as \"a slight, disorderly old man who had taught political science at Union College in Schenectady\"\u2014might better have been called Ludwig Wiener. Ludwig von Neumann sounds absolutely Wienerian when he declares in his Ghost Shirt manifesto, \"Without regard for the wishes of men, any machines or techniques or forms of organization that can economically replace men do replace men. Replacement is not necessarily bad, but to do it without regard for the wishes of men is lawlessness.\"\n\nAfter Scribner forwarded Wiener's letter, Kurt and Jane spent a couple of days fantasizing about suitably mean retorts. Then Kurt wrote Wiener a chilly note thanking him for troubling himself with Player Piano and stating that he felt his indictment of contemporary science should be clear. He apologized for having innocently given offense, claiming he had picked the name von Neumann at random. He might have thought he did, but Kurt never picked names at random. He pulled them from his personal store of words with private significance. Senator Warren Foust in \"Barnhouse\" was likely suggested by Commander Elwood Faust, one of Project Cirrus's key pilots. Winston Niles Rumfoord of The Sirens of Titan and Bertram Copeland Rumfoord of Slaughterhouse-Five were almost surely born of Benjamin Thompson, Count Rumford, who endowed the Rumford Prize won by Irving Langmuir. George M. Helmholtz, a recurring character in Vonnegut's stories, invokes the mathematician Hermann von Helmholtz, whose equation was utilized in much of the Project Cirrus work. GE names, like GE ideas, would surface persistently and evocatively in everything Kurt ever wrote.\n\nIn mid-August, Player Piano landed in bookstores across the country\u2014except for those in Schenectady. Not one bookstore in GE's company town would agree to stock it; every bookseller made an improbable excuse. Bernie had received his copy, signed with love from Kurt. He had probably already read it when the review in The New York Times Book Review came out. On August 17, 1952, the eminent critic Granville Hicks praised the book's humor, calling Kurt a \"sharp-eyed satirist.\"\n\n\"It is a little like Brave New World,\" Hicks wrote, \"except that Mr. Vonnegut keeps his future closer to the present than Aldous Huxley succeeded in doing, and his satire therefore focuses more sharply on the contemporary situation. The machines he is talking about are not gadgets he has dreamed up; they are in existence, as he is careful to point out.\"\n\nThe next day was Bernie's last day at the GE Research Lab. Between the final edits and the Orange-O freak-out, the Wiener letter and The New York Times, somewhere in there Bernard Vonnegut had walked into the office of Guy Suits and handed in his official resignation.\n\nLater, he would say that his career at GE ended when he went on vacation and someone cleaned off his desk. Whether or not the story was true, it was an elegant way of expressing the choice he had made at his own crossroads, a choice to trust his own messy, imperfect spirit over the order and efficiency of GE. Confident he was doing the right thing, Bernie turned in his keys, his all-hours pass, his patent notebook, his lab supplies, his meters and instruments, and his K parking permit, and he walked out of the Knolls Research Lab. He didn't look back.\n\nThe secretary carefully filled out his termination sheet, noting his new address: 704 Country Way, North Scituate, Massachusetts. His new home was close to the ocean and close to Kurt and Jane.\n\nFor the first time in his life, Bernie was following in his brother's footsteps, instead of the other way around.\n\n* * *\n\nIn June 1953, a mile-wide tornado plowed through Worcester, Massachusetts. Tornadoes in Massachusetts are rare, and the storm cell that spawned this one was so huge that people on Cape Cod could see it as it churned northeast, killing ninety-four people and leaving ten thousand more homeless in the eighty-four minutes it took to get to the coast. Later it was called the Worcester Twister and given a rare F4 rating on the Fujita scale.\n\nAs news of the storm broke, Bernie went down to the beach in Scituate with his camera to watch the giant stomp across the Boston area. The storm was about 160 kilometers away, but he could see its electrical discharges filling the sky. Blinking his eyes as fast as he could, he couldn't get a glimpse of the storm not lit up by lightning. The release of all that energy must be heating the air and creating updrafts, which could explain the storm's size. He estimated the storm's top to be about twenty kilometers up, in the stratosphere, which would mean its vertical winds were traveling upward at a hundred meters a second. The lightning was still going like gangbusters when the storm reached the shore and headed out to sea.\n\nHe was as awed as he had been as a kid, when he snuck out of the house in Chatham and went down to the ocean. He was witnessing a firestorm made by nature. Scientists didn't even know if the winds caused the lightning or the lightning caused the winds. But Bernie had an intuition that electrical charge might play a larger role than people thought.\n\nThe original impulse behind Project Cirrus was a simple mystery: What makes clouds give rain? But the query about the natural world had nearly been lost in a whirlwind of controversy and competing desires. The military men wanted weapons, the commercial rainmakers wanted profits, GE wanted free publicity, and the meteorologists wanted to protect their turf.\n\nIn 1953, Congress appointed the new Advisory Committee on Weather Control to investigate, once more, whether the government should conduct research on the topic. Bernie was asked to write a report for the committee. Five years later, when congressional hearings were held, he asked to testify. Of his own volition, he wrote up a statement on House Bill 86. The bill, he said, was based on a false premise: that anyone could know what kind of research would lead to weather control. In fact, he told them, Vince's original experiment had been driven by the desire to understand how nature works, not by a desire to control it.\n\n\"I believe the best way to achieve weather control,\" he wrote, \"will be to sponsor basic research in the physical sciences necessary to the understanding of weather.\"\n\nThat same year, Commander William Kotsch of the U.S. Navy wrote to Bernie asking for a copy of his report on the uses of weather as a weapon. Bernie sent him a copy of the report he wrote for the Advisory Committee on Weather Control. But the commander was bound to be disappointed, he said. He dealt \"very sketchily\" with the use of weather as a weapon.\n\nIn fact, he didn't mention it at all.\n\nAfter leaving GE for Arthur D. Little, Bernie would sometimes accept funding from the military. But never again would he work on something that had obvious military applications. Instead, he began developing a new theory about thunderstorms, proposing that updrafts and downdrafts caused electrical charge in thunderclouds and that electrical charge led to rain, instead of vice versa. One of his cleverest experiments concerned the fact that thunderstorms almost always have a positive charge at the top and a negative charge at the bottom. One summer in New Mexico, Bernard and his colleague Charlie Moore strung a two-kilometer wire between two mountains to release negative charge. When thunderstorms came through and encountered the negative charge, they reversed their polarity. Over the course of two weeks, Bernie and Charlie Moore created three storms that discharged positive charge to the ground\u2014electrical storms turned electrically upside down. It was perhaps his best joke on nature. Bernie spent much of the rest of his life marshaling evidence for his theory about electrical charge in thunderstorms, but he once admitted he wasn't really driven by the need to prove it.\n\n\"I'm [just] trying to find out,\" he said, \"what's going on.\"\n\nHe no longer wanted to control. He wanted to know. And the purpose of knowing was not, in the end, to banish mystery. It was to appreciate it. To note how it glows.\n\n* * *\n\nIn May 1954, Kurt tried and failed to get a job at Time Inc. He was broke. Player Piano had received a few polite notices but had not sold well and had quickly faded from view. It had not made his fortune or his name. Worse, Collier's had stopped buying his stories. The short story market was shrinking. And Jane was pregnant with their third child. Desperate for income, he sent Harry Brague at Scribner all he had, six chapters of a book he was now calling Cat's Cradle.\n\nThat month, Collier's published a sensational cover story called \"Weather Made to Order?\" Written by Howard Orville, chair of the Advisory Committee on Weather Control, the piece trotted out all of the most dramatic claims about the coming era of man-made weather: hurricanes, tornadoes, and thunderstorms would be quelled; deserts and dust bowls would bloom; forest fires and floods would be prevented. Weather would be used as a weapon, deluging enemies with rain or causing them to starve by preventing it. According to Collier's, this was all laudable, American enterprise at its finest.\n\nKnox Burger was not at Collier's anymore; he had left to become a book editor at Dell. Kurt's last two stories for Collier's, \"Poor Little Rich Town\" and \"With His Hand on the Throttle,\" were both GE based, the first about a GE-type efficiency expert and his attempt to streamline life in a small village and the second about a model railroad fanatic inspired by Herb Hollomon. But the magazine market for fiction was rapidly drying up as popular entertainment was increasingly delivered by television. Collier's itself would cease publication in 1957.\n\nKurt could see that clearly enough, and he had turned to writing plays. He had even purchased a television and was starting to write teleplays in the hope of breaking into the new medium. But they weren't selling. And he couldn't seem to get very far with the new novel. Ever since the disappointing debut of Player Piano, Kurt had been racked with writer's block. Harry Brague had encouraged him, trying to convince him that the novel's reception had actually been good and that he should sit down and write a new one before the public forgot his name. He had even advanced Kurt $500 on the unspecified next novel when Kurt was desperate for cash. It didn't help.\n\nHarry Brague didn't quite know what to make of the six chapters Kurt sent of Cat's Cradle. In fact, it would take ten years and two more books before the former \"Ice-9\" would morph into Kurt Vonnegut's breakthrough book. Kurt would fix its problems by stepping back and turning the straightforward adventure story into a story about stories, an adventure stumbled into by a writer-narrator called Jonah who is trying to research a book about the day the first atomic bomb was dropped on Hiroshima. He wants to call it The Day the World Ended.\n\nIn the final version, Dr. George Hoenikker becomes Dr. Felix Hoenikker, \"father of the atomic bomb\" and inventor of the even more dangerous ice-nine. Dr. Hoenikker is already dead as the book opens, but Jonah researches his book by tracking down Hoenikker's three children and traveling to Ilium, New York, home to the Research Lab of General Forge and Foundry, where Felix Hoenikker worked. The Research Lab is described as a kind of playground, where \"men are paid to increase knowledge... the most valuable commodity on earth.\" Felix Hoenikker is its star scientist, viewed by the Research Lab's main client, the Pentagon, as \"a sort of magician who could make America invincible with a wave of his wand.\"\n\nAs Jonah explores Ilium, he collects stories about Dr. Hoenikker, many of which are versions of stories about Irving Langmuir. Dr. Hoenikker played cat's cradle with his son Newt\u2014one of Langmuir's favorite games to play with children. Hoenikker once left a tip for his wife, as Irving had for Marion; he was known to declare, as Kurt had heard Irving say, that any scientist who couldn't explain his work to a child was a charlatan. A secretary at General Forge and Foundry tells Jonah that Dr. Hoenikker once challenged her to tell him something that was absolutely true, to which she replied, \"God is love.\" This is what Clare Boothe Luce told Irving Langmuir when they were chatting before appearing on the radio together and he challenged her to think of a statement that was true but not provable. But Langmuir's comment on the exchange was \"She sure had me there!\" Kurt has Felix Hoenikker reply, \"What is God? What is love?\"\n\nThe connections between Felix Hoenikker and Irving Langmuir made it clear that, as Kurt would frequently declare in years to come, the absentminded scientist was based on his brother's former boss. But Felix Hoenikker's last gift to humanity\u2014ice-nine, \"a new way for the atoms of water to stack and lock, to freeze\"\u2014that invention was Bernie's.\n\n\"Suppose,\" explains Dr. Asa Breed, vice president in charge of the Research Laboratory of General Forge and Foundry,\n\nthat the sort of ice we skate upon and put into highballs\u2014what we might call ice-one\u2014is only one of several types of ice... And suppose... that there were one form, which we will call ice-nine\u2014a crystal as hard as this desk\u2014with a melting point of, let us say, one-hundred degrees Fahrenheit, or, better still, a melting point of one-hundred-and-thirty degrees.\n\nDr. Breed tells Jonah to imagine the many ways oranges could be stacked in a crate, or cannonballs piled on a courthouse lawn. It was exactly how Bernie had explained cloud nucleation to Kurt.\n\nOne of the things critics rarely understand about Cat's Cradle is that it is not about the misuse of science. It's about the failure of people to understand the significance of a scientific advance. The threat in the book is not that scientists will produce something dangerous but that the community won't recognize it when they do. Felix Hoenikker is a bad guy not because he invented ice-nine but because he failed to warn anyone about it. Only creators can really understand the hazards of what they create: it's their moral obligation to make sure the rest of the world understands them too.\n\nAt GE, Kurt had witnessed his brother trying to do just that.\n\nIn putting the story in the mouth of the anthropologist-like Jonah, Kurt made the major breakthrough of shifting the emphasis from Dr. Felix Hoenikker to his three children: Franklin, the genius scientist with \"a wiry pompadour... that arose to an incredible height\"; Angela, the tall sister who plays the clarinet brilliantly; and Newt, the glum midget who paints pictures. It's not hard to see the similarity between the names Hoenikker and Vonnegut, or the way in which the Hoenikkers are versions of the three Vonnegut siblings: Bernard, the genius scientist; Alice, the ethereal artist; and Kurt, the glum wordsmith painting clowns as the world goes to hell.\n\nHis other main breakthrough was to bring in the outlawed religion of Bokononism. In short, evocatively titled chapters\u2014each about the length of a press release\u2014Kurt gets Jonah and the Hoenikker siblings, each carrying a shard of ice-nine, to the island of San Lorenzo. The banana republic is run, as it was in early drafts, by the dictator Papa Monzano, but it has a new element: Bokonon and his made-up faith.\n\nThe conclusion of Cat's Cradle echoes the earliest drafts of \"Ice-9\": the novel ends with apocalypse. Papa Monzano has gotten hold of some of Felix's ice-nine, and he takes it to kill himself. When the San Lorenzo air force bombs his palace, his body slips into the sea, and the world's water supply is converted to ice-nine by chain reaction. An epic cyclone reduces most of the planet to rubble. Jonah and his love object, Mona, survive by hiding in a bomb shelter constructed in the palace dungeon. When they emerge, most of the population is dead; for the sad remnant left behind, it's only a matter of time until they accidentally ingest some ice-nine and their own water molecules seize up. Angela Hoenikker dies when, unconcerned, she picks up a clarinet and plays. Just as in the Vonnegut family by then\u2014Alice died of breast cancer in 1958\u2014the Hoenikker brothers are all that remains of the family. As human history draws to its close, Franklin, the scientist, builds an ant farm and passes his days marveling at ant behavior. When provoked, he issues a \"peevish lecture on all the things that people could learn from ants.\" It's a scene reminiscent of the end of \"The Petrified Ants,\" a story never published in Kurt's lifetime, where the exiled brothers Josef and Peter marvel at the adaptability of the insects.\n\n\"Men could learn a lot from ants, Peter my boy,\" Josef tells his younger brother.\n\n\"They have, Josef, they have,\" Peter replies. \"More than they know.\"\n\nIn Cat's Cradle, as in \"The Petrified Ants,\" Kurt used a pair of brothers to address the moral duties of scientists. But in the earlier story, the siblings were meant to be contrasted with each other: the older one morally compromised by conformity, the younger one an individualist working to maintain his integrity. In Cat's Cradle, the brothers are equally compromised\u2014they both carry shards of ice-nine\u2014and equally ineffective at saving the world. Newt, the younger brother, occupies himself much as Franklin does in the apocalypse. He too is just farting around waiting to die, making paintings with scavenged paint. There's no wisdom to be gleaned from the end of the world, from destruction on a global scale.\n\nThe real hero of Cat's Cradle is Bokonon, the prophet who admits that his wisdom is based on lies but whose invented religion gives human beings comfort. Like the Ghost Shirt Society in Player Piano and the Church of God the Utterly Indifferent in The Sirens of Titan, Bokononism provides an alternative to the sterile technological belief in truth personified by Felix Hoenikker. It uses false myths and harmless lies\u2014foma\u2014to encourage humans to love one another, to acknowledge their connectedness, and to spend their short time on earth with grace and compassion.\n\nIn crafting the creed of Bokononism\u2014for which he would finally, years later, be granted his anthropology degree from the University of Chicago\u2014Kurt found perhaps his best embodiment of the truth he had learned all those years ago in The Brothers Karamazov: If God did not exist, human beings would have to invent him. And if that's the case, why not invent a kind and loving God, a God who encourages us to find the sacred in nothing more, and nothing less, than our own human selves?\n\n\"Let us remember how good it was once here, when we were all together,\" Alyosha says at the end of Dostoevsky's novel, \"united by a good and kind feeling which made us, for the time we were loving that poor boy, better perhaps than we are.\" He is appealing to the boys to embrace their best natures in his absence because he is leaving, about to accompany his brother Dmitri to Siberia.\n\n\"You must know that there is nothing higher and stronger and more wholesome and good for life in the future than some good memory, especially a memory of childhood, of home,\" he says. \"People talk to you a great deal about your education, but some good sacred memory, preserved from childhood, is perhaps the best education.\"\n\nThat was the line that Jane's Swarthmore professor had shared with her and that she shared with Kurt on their honeymoon. Kurt always kept that piece of paper. It reminded him of the sacredness of memory. He had his own sacred memory preserved from childhood, the memory of his childhood swim across Lake Maxinkuckee. Thanks to the way the human mind lets us travel in time, he could revisit that lake anytime he wanted. He could dive in and swim across it just as he had done as a child, his arms and legs churning like an engine, the waves sliding off his body, a body that always looked best, he thought, in the water. In the water, he was beautiful. In the freshwater lake, he was at home. The water would buoy him up, and he would swim, happy to be swimming in what to him was an ocean, the second-largest lake in Indiana, with his sister and his brother at his side.\nEpilogue: Rainbow's End\n\nKurt sat in his office around 9:30 in the evening. In one hand, he had a cigarette, and in the other a piece of paper. Bernie had given it to him earlier that day.\n\n\"We do have some other photographs of the poor man's Steinmetz, and I may send them to you in my own sweet time.\"\n\nIt was his letter to Uncle Alex, written in 1947, fifty years earlier, when he was still a young man. He was an old man now. Bernie was even older, of course. Old and sick. He had handed the letter to Kurt in Albany, where Kurt went every week by train to see his older brother as he faded away from lung cancer.\n\nBy now, Kurt was famous, very famous. But success had not come easily. After Player Piano, he was blocked for years. Scribner had sold the paperback rights to Bantam for $4,000, and the book that was going to earn him some literary respect was retitled Utopia 14 and released in a lurid mass-market edition, complete with a futuristic city and writhing half-dressed people on the cover. \"Man's revolt against a glittering, mechanized tomorrow,\" it read, next to the cover price of thirty-five cents. Thus began the process of consigning Kurt Vonnegut novels to the shelf marked \"science fiction,\" a designation he would alternately embrace and bemoan.\n\nFor years, all he could produce were short stories to pay the mortgage, and even those were not plentiful enough. He took copywriting jobs, teaching jobs; he opened one of the nation's first Saab dealerships. It failed. Cape Cod wasn't quite ready for Saabs.\n\nWhen he finally broke through his writer's block and wrote The Sirens of Titan\u2014published as a paperback original\u2014he was still writing about the war. War remained the backdrop for his books for another fifteen years: battles, mortar attacks, artillery shellings, air assaults, ships firing cannons, bombs dropping on jungles and castles and cities, hordes of nameless refugees, legions of dead. He even designed and tried to market a military strategy board game called GHQ, with counters representing infantry, artillery, and paratroop units arrayed on a checkerboard, as if he were turning his story \"All the King's Horses\" into an actual game, as if he were refighting the Battle of the Bulge.\n\nHe started and restarted his Dresden book, five times, ten times, more times than he cared to guess. He wrote it as a teleplay, a short story, a nonfiction essay. He was working on it when Cat's Cradle was published in 1963, and ice-nine\u2014along with the terms karass, foma, and wampeter\u2014entered the language. He was working on it when he published Mother Night and God Bless You, Mr. Rosewater, each admired by a few critics and contributing to his underground reputation. He was working on it when his sister, Alice, died of cancer, two days after her husband, Jim Adams, was killed in a freak train wreck on New Jersey Transit. He was working on it when he and Jane took in her four orphaned boys and suddenly had seven kids, as they always knew they would.\n\nFinally, in the late 1960s, with the nation cycling through gloom and guilt and rage over events in Vietnam, Kurt was offered a teaching position at the Iowa Writers' Workshop. The mood in the nation plus the literary stimulation of Iowa and the escape from his own routine broke the logjam that had lasted for a quarter century. He began to rewrite what he had always called Slaughterhouse-Five, wrestling the material that had haunted him since 1945 into his own unique novel form. He was finally able to use Norbert Wiener's idea of \"Newtonian time\"\u2014reversible time in which nothing new can ever happen\u2014to full effect. When Billy Pilgrim begins to experience time as the Tralfamadorians do, it is not just a far-out concept of quantum physics but also the time we all live in, where at any moment we might be transported back to a former moment in our lives.\n\nOne of the last typescript drafts before Slaughterhouse-Five began with the line \"Billy Pilgrim has come unstuck in time.\" Above it, Kurt wrote one word by hand: \"Listen.\"\n\nGrowing up as the youngest child, he always yearned to say this to his parents and siblings. As the younger brother who wanted to be a writer, not a chemist, he had longed to say it to Bernard. As an adult, he finally learned to say it, learned to work his desire to say it into the style we now recognize as uniquely his.\n\nAt once an antiwar treatise and a psychological depiction of post-traumatic stress, Slaughterhouse-Five electrified readers and made Kurt a literary icon. His time in Iowa salvaged his reputation, assured his family's financial future, and gave him what he had always wanted: a place in the literary pantheon. It was also where he launched an affair that would begin the process of ending his marriage to Jane. So it goes.\n\nRecently, as Bernie lay dying in Albany, he had told Kurt that he didn't think scientists made very good husbands. They were too focused on the fascinating things going on in their own heads.\n\nYou could say the same for writers, of course.\n\nBut Jane's confidence in Kurt had shaped him, and her voice was there in the work. In Slaughterhouse-Five, he concluded one of the most compelling descriptions of a massacre ever written with the voice of a bird: the last word in the novel is \"Po-tweet?\" It summed up Jane's undergraduate history thesis at Swarthmore: no sense can be made of history; no meaning can be gleaned from a massacre. The only thing to do is to listen to the birds.\n\nAnd, like Dostoevsky's Father Zossima, to ask forgiveness of them.\n\nTwo years after Kurt published Slaughterhouse-Five, newspapers broke a fantastic story: the CIA had been using cloud seeding as a weapon of war. Since 1966, U.S. planes had flown more than twenty-six hundred cloud-seeding missions over Indochina, spraying the clouds of Vietnam and Laos with aerosolized silver iodide. The intent was to make rain: rain to wash out river crossings and muddy up the Ho Chi Minh Trail, rain to slow supply lines and troop movements in North Vietnam. Like so many weather modification projects, the program had a sci-fi-like name: Operation Popeye.\n\nBernard was horrified to see his invention used in actual combat. But news of the missions led to an international outcry and, eventually, something he had always desired, a UN treaty banning the use of weather modification as a weapon of war.\n\nBy the time Bernie heard about Operation Popeye, he had left industrial science and become an academic. His career at Arthur D. Little in Cambridge had been spent investigating electrification in tornadoes and thunderstorms. In the summers, he continued to chase clouds and lightning in Socorro, based at what became the Langmuir Laboratory for Atmospheric Research. Then, in 1967, he accepted a faculty position at the Atmospheric Sciences Research Center, founded six years earlier by Vincent Schaefer at the State University of New York at Albany. Along with several other former colleagues from the Project Cirrus team, Bernie became a professor and continued his research while teaching, nurturing young scientists with his passion for weather.\n\nFrom Schenectady to Scituate to Albany, through childbirth and schooling, through children growing up and succeeding or struggling, through his wife's death, and his son's illness: through all those years, Bernie had kept Kurt's letter to Uncle Alex. He must have found it funny. He liked a good joke as much as any of them. But maybe he found the letter touching too, a reminder of the younger sibling unable to resist a chance at a potshot, the jealous kid brother who lives on in kid brothers everywhere.\n\nIn 1985, MIT invited Kurt to give a speech. He must have been gratified to be invited to hold forth at his brother's and father's and grandfather's alma mater. To make the scientists at one of the biggest science factories of all listen to him. He felt as if he might actually be able to do some good. So he told the graduating class about his brother, Bernard, a fellow graduate of MIT.\n\n\"My brother knew early on that he would be a research scientist, and so could not be self-employed,\" he told them. To make a living, his brother was going to have to work for somebody else, to make someone else's technological dreams come true. Bernie, he told them, got his doctorate on the eve of World War II:\n\nIf he had gone to work in Germany after that, he would have been helping to make Hitler's dreams come true. If he had gone to work in Italy, he would have been helping to make Mussolini's dreams come true. If he had gone to work in Japan, he would have been helping to make Tojo's dreams come true. If he had gone to work in the Soviet Union, he would have been helping to make Stalin's dreams come true. He went to work for a bottle manufacturer in Butler, Pennsylvania, instead. It can make quite a difference not just to you but to humanity: the sort of boss you choose, whose dreams you help come true.\n\nIt was a reprise, in some ways, of an address Kurt gave to the American Association of Physics Teachers in 1969 called \"The Virtuous Physicist.\" Scientists at several major research universities were protesting the misuse of science as a tool of war. After his talk, Kurt was asked by reporters what defined a virtuous physicist. He told them it was simple: \"one who declines to work on weapons.\"\n\nSomeone should write an oath for them, Kurt told the scientists at MIT nearly twenty years later. One modeled on the Hippocratic oath, something like \"The regimen I adopt shall be for the benefit of all life on this planet, according to my own ability and judgment, and not for its hurt or for any wrong. I will create no deadly substance or device, though it be asked of me, nor will I counsel such.\"\n\nBernie read and approved of Kurt's talk before he gave it.\n\nThe year after Kurt spoke at MIT, Jane Vonnegut Yarmolinsky\u2014she had married the Harvard law professor Adam Yarmolinsky\u2014succumbed to cancer at the age of sixty-four. Not long before she died, Jane called Kurt. Their conversation was affectionate. Jane asked Kurt to tell her what would determine the moment of her death. Why ask him? \"She may have felt like a character in a book by me,\" Kurt wrote later. \"In a sense she was.\" Of course Jane was nobody's fictional character but a living, breathing woman full of life and energy and purpose. Without her, the Kurt Vonnegut we know would never have existed. To a great extent, he was a character created by her. And yet what he said was somehow true to the times in which they made their home and raised their children. The men wrote the stories of their lives. The women played their parts in them.\n\nKurt told Jane that a ten-year-old boy would be standing at the end of Scudder's Lane, the street where they had lived most of their years on Cape Cod. Standing on the boat ramp that reaches out into the pond, the boy would be sunburned and bored but happy. \"I told Jane that this boy, with nothing better to do, would pick up a stone, as boys will. He would arc it over the harbor. When the stone hit the water, she would die.\"\n\n* * *\n\nDid Bernie really make rain? Scientists have been debating the efficacy of cloud seeding ever since Vincent dispensed dry ice over Mount Greylock in 1946, but the general consensus today is that in certain conditions it works in a limited way. The huge modifications Irving promised never came to pass. Immediately after GE shut down Project Cirrus, the military conducted six large-scale experiments. Five yielded inconclusive results, and one showed a positive result: the seeding of supercooled stratus clouds with dry ice and silver iodide. Officers from the Office of Naval Research consulted with Irving Langmuir on these experiments, though they would not reveal any specifics, and the results were classified until 1957. Irving had by then completed his magnum opus: his final report on Project Cirrus. But the nearly six-hundred-page treatise was classified as well. When it was finally published as volume 11 of his Collected Works in 1962, Irving was dead; he had died in 1957, with the work he considered his legacy to humanity unknown to science. Today, he is well-known among chemists, but his popular reputation died with his dream of widespread weather control.\n\nBernie's invention is the main legacy of Project Cirrus. Silver iodide remains the most widespread method of enhancing precipitation. Utilities in California have been using silver iodide to increase the winter snowpack in the Sierra Nevada since the early 1950s. The Desert Research Institute in Nevada has been using it to seed clouds since not long after that; currently, it has three operational programs in Nevada and California. In the 1960s, the Bureau of Reclamation conducted rainmaking experiments over reservoirs, with some apparent success, under the name Project Skywater. In the 1960s and 1970s, a joint U.S. Navy\u2013Weather Bureau experiment called Project Stormfury attempted less successfully to modify hurricanes with seeding.\n\nResearch spending on weather modification dwindled after the 1970s. Factors blamed for the field's decline include early overpromising, the lack of rigorous research, environmentalist campaigns against altering nature, and Reagan-era reductions in government funding. Another key element: the early 1980s were an exceptionally wet period in the United States. The weather modifiers have a saying for this: \"Interest in cloud seeding is soluble in rainwater.\"\n\nToday's changing climate has renewed interest in weather modification. In the West and the Great Plains, severe drought and diminishing aquifers have led water utilities, hydropower producers, agriculture groups, and ski resorts to fund cloud-seeding programs. In Wyoming, Idaho, Colorado, California, Utah, and Nevada, rainmakers are hired to augment the snowpack. In Texas and Kansas, cloud seeders are at work to induce rain. In North Dakota, clouds are seeded to make them precipitate before they can produce crop-damaging hail. California, Nevada, and Arizona contribute funds to cloud-seeding projects over the Colorado River's upper basin, in hopes of increasing their water supply.\n\nAnd yet many people today believe that weather modification is a hoax: the early overselling of rainmaking somehow caused it, down the line, to be grouped in the public mind with conspiracy theories about mind-altering \"chemtrails,\" shock-jock speculation that the government manufactures tornadoes, and paranoid fantasies about \"weather wars\" involving earthquakes broadcast via the stratosphere. The reality is far less dramatic. A rigorous five-year randomized study sponsored by the Wyoming Water Development Commission and funded by the National Science Foundation is on track to yield results showing that seeding certain clouds can be expected to increase precipitation by about 10 to 15 percent.\n\nAs for Project Cirrus's widespread periodicities and whether they resulted from Bernie's little generator huffing away in New Mexico, the general consensus among those who've even heard of the experiment is that the periodicities were a freaky coincidence. The Weather Bureau meteorologist Roscoe Braham called Langmuir's experiment \"one of those tantalizing things.\" The MIT meteorologist Henry Houghton called it \"the most mysterious thing I have ever run up against... If it wasn't chance, it was a totally new effect.\" Another meteorologist in 1953 said the episode was \"a great tragedy.\"\n\n\"If Langmuir actually influenced the weather,\" he said, \"no one will believe him. If the periodicities were mere coincidence, nature played Langmuir a dirty trick.\"\n\nOf course, nature plays dirty tricks on us all the time, if by that we mean that it shimmies out of grasp, foiling our urge to master and sometimes our desire to know.\n\n\"Beware of the man who works hard to learn something, learns it, and finds himself no wiser than before,\" writes Bokonon in Cat's Cradle. \"He is full of murderous resentment of people who are ignorant without having come by their ignorance the hard way.\"\n\n\"Perhaps if he had comprehended fully the magnitude of atmospheric phenomena,\" the meteorologist Horace Byers wrote years later of Irving Langmuir, \"he might have been discouraged from the start.\" But in one way Langmuir\u2014and Bernie too\u2014understood the magnitude of atmospheric phenomena better than almost anyone. They understood that the atmosphere was such a huge, complex, and dynamic system that we would never nail it down with numbers. The weather, Langmuir wrote in his great work, \"is not definitely determinate. It depends in large part and essentially upon meteorological events that originate from small and unpredictable beginnings, such as the location and concentration of freezing nuclei that may set off chain reactions.\" Today we call this chaos.\n\nDeveloped by the meteorologist Edward Lorenz beginning in 1960, chaos theory holds that in some huge systems\u2014weather and other things like it\u2014small contingencies cascade upward to produce large effects. Scientists named it \"sensitive dependence on initial conditions.\" Norbert Wiener had already seen this in 1954, calling it \"the self-amplification of small details in the weather map.\" Langmuir's theory of convergent and divergent phenomena was an early version of chaos theory, another startling intuitive achievement and one for which he has never been given credit. Langmuir and Wiener were right, and John von Neumann was wrong. The weather cannot be entirely solved by equation, because it is not deterministic. It's all flowing and blending; a touch in one place sets up movement at the other end of the earth. The flutter of a butterfly's wing can touch off whirlwinds ten thousand miles away.\n\nIt's all an ocean, after all.\n\nWeather prediction today combines von Neumann's mathematical approach with Wiener and Langmuir's probabilistic one. The weather can be predicted with a good degree of accuracy for about a week out. It can be predicted with somewhat less accuracy for up to two weeks. After that, it all falls apart, because the effects of very tiny disturbances multiply with great speed. Eventually, you get far enough out that statistics and probabilities smooth out the variations, and you can make largely accurate guesses about long-term climate, but in the zone between two weeks and two millennia it's still a mystery. The computer did not turn out to be Laplace's demon\u2014the all-knowing consciousness of a clockwork universe\u2014though hope that it might still morph into that being can sometimes be whiffed in the tech factories of Silicon Valley.\n\n* * *\n\nAs for John von Neumann, in early 1955 he was appointed to the Atomic Energy Commission and moved to Washington. That June, he published an essay in Fortune called \"Can We Survive Technology?\" He outlined the sweeping technological advances about to transform the world: nearly free atomic energy, automation of everything, and global climate control. All of these advances, he acknowledged, had immense potential for harm as well as good. But that was the nature of science. To attempt to control or restrain it was impractical.\n\n\"For progress,\" he wrote, \"there is no cure.\"\n\nKurt\u2014and Bernie\u2014would disagree. The cure for progress was simply remembering that we are human.\n\nNot long after his essay appeared, von Neumann slipped and fell in a Washington hallway and was subsequently diagnosed with bone cancer, possibly due to his presence at so many early atomic tests. He died eighteen months later, under military guard with top clearance, for fear that in his agonized raving he should reveal military secrets. He was tormented beyond all reason by the fact that he, too, turned out to be frail, imperfect, inefficient, that for the progress of life there is no cure. Before he died, the lifelong atheist requested the ministrations of a priest.\n\n* * *\n\nHarry Wexler eventually gave up his opposition to the idea of weather control. In 1959, he attended a meeting on weather modification chaired by Edward Teller, who had become a believer. Three years later, Harry gave a paper called \"On the Possibilities of Climate Control\" at the AMS meeting in Boston. The subject of weather control, he told the audience, was \"now becoming respectable to talk about.\" Indeed, President Kennedy had called for international cooperation on climate control in his inauguration speech that year. The UN had also voiced hope that nations would cooperate on this front.\n\nWexler outlined some interventions in the climate that seemed to him feasible, given the greater climate understanding both satellites and computers were providing. But, he pointed out, \"we are in weather control now whether we know it or not.\" Wexler talked about the dangers of inadvertent damage to the ozone layer via ozone-depleting chemical reactions. He noted the fact that \"we are releasing huge quantities of carbon dioxide and other gases and particles to the lower atmosphere which may have serious effects on the radiation or heat balance which determine our present pattern of climate and weather.\" He was still investigating these phenomena just months later when he died, tragically, of a heart attack at age fifty-one. His research was largely forgotten until the 1970s. It would be interesting to know what might have happened had such a forceful character been around to advocate investigating the dangers of chlorofluorocarbons and greenhouse gases more than a decade before the majority of scientists thought to do so.\n\nIn fact, the most significant legacy of Project Cirrus might be that it laid the foundation for studying human impacts on the climate, a field previously considered the domain of science fiction. In 1978, Bernie explained,\n\nProject Cirrus's investigations of ways that man can intentionally affect the weather cast a new light on how man's activities might unintentionally have a large influence on weather processes. Along with the scientific study of deliberate weather modification, a new vigorous research activity is developing to explore how man may be affecting his environment by the gases, aerosols and heat he emits into the atmosphere in the course of his many activities... In recent years several international workshops on inadvertent weather modification have been held. These are undoubtedly only the first of many similar ones that will be held in the future on this subject.\n\n* * *\n\nGE continued to move to the right throughout the 1950s, blacklisting employees who wouldn't cooperate fully with HUAC and gradually reining in its unions through tough negotiation tactics that came to be known as Boulwarism. Two years after Bernie left, GE hired an underemployed actor to serve as its public relations spokesman and to host the company television show GE Theater. Taken under the ideological wing of Lemuel Boulware, he was purged of his previous liberal convictions and indoctrinated in the virtues of unfettered markets. Ten years later, that actor, Ronald Reagan, would make a stunning speech in support of Barry Goldwater at the Republican National Convention, launching a conservative revolution and his own political career.\n\nThe nation by then had thrown itself so fully into the Cold War and the arms race that even the Republican president, Eisenhower, famously warned against letting the \"military-industrial complex\" have too much power. The hope shared by so many\u2014including Kurt and Bernie\u2014in the late 1940s, hope for peace through disarmament and world government, came to seem like a distant pipe dream. Kurt never stopped longing for America to be what he believed it could be\u2014the progressive, freethinking nation that had given birth to the New Deal\u2014and in his work he returned to the theme again and again. America, he wrote in novels and essays and newspaper columns, could live up to its promise if only people would embrace their better selves\u2014the selves dedicated to equality and free speech and kindness and giving the downtrodden a hand. By the time Bernie gave him back the letter he had sent to Uncle Alex fifty years earlier, Kurt was growing disheartened. That old America, he told friends, was gone. In his novel Gal\u00e1pagos, in many ways a rewrite of Player Piano, he had proposed that humanity would only improve when humans evolved away from being big-brained tool users and into furry, fish-eating amphibians. As Bernie was dying, Kurt was working on Timequake, in which the entire planet is forced to repeat a whole decade, \"betting on the wrong horse again, marrying the wrong person again, getting the clap again. You name it!\" History is literally doomed to repeat itself, at least in a clockwork universe. The novel is a kind of thought experiment about what it might be like to inhabit a completely deterministic world. But, as if shaking off the ghost of John von Neumann, Kurt returned the world to its messy unpredictability in the end. There's even a glimmer of hope in the mantra Kilgore Trout and a crew of volunteers use to spur people back into action once the timequake has ended and free will has returned: \"You were sick, but now you're well again, and there's work to do.\"\n\n* * *\n\nBernard Vonnegut's last published work was about art. During his research at Arthur D. Little, he had been introduced to a technique for making dendritic electrical discharge patterns on blocks of transparent plastic\u2014little lightning bolts preserved in plastic like fossils in amber. Contemplation of the branching images had changed his thinking about lightning. Later, he and a colleague made convective flow patterns between sheets of plastic using aluminum paint and kerosene. Eventually, just for the fun of it, he began making permanent patterns by squeezing paint between two smooth surfaces\u2014ceramic tile or glass or plastic\u2014then pulling the surfaces apart.\n\n\"Physicists may already know enough about the behavior of liquids to predict the patterns that form, based on such variables as the rate at which the surfaces are pulled apart, the thickness of the liquid film, the viscosity and surface tension of the liquid and the angle at which it wets the surfaces,\" he wrote in his piece. In other words, physicists would see this process as reducible to a series of equations, would understand the know-how. But what about the know-what? Would they note how it glowed?\n\nKurt would. Bernie sent some of his patterns to his brother and asked him, is this art? Kurt's letter back reads as if he still occasionally relished the chance to lecture Bernie. \"This is almost like telling you about the birds and the bees,\" he began. Art, he told his brother, was in the eye of the beholder. Contemplating it was a social activity.\n\n\"You are a justly revered experimentalist,\" Kurt wrote. \"If you really want to know whether your pictures are, as you say, 'art' or not, you must display them in a public place somewhere, and then try to judge whether or not strangers liked to look at them, were glad that you had made them. That is the way the game is played. Let me know what happens.\"\n\nBernie's dendritic prints might or might not have been art. But in calling his brother a \"justly revered experimentalist,\" Kurt was admitting something he had probably known for a long time: his brother was an artist. Bernard's real art was in the pages of scientific journals. It was scrawled in his notebooks. It was in tubes and jars and Rube Goldberg contraptions cluttering his disaster of a desk. It was ice crystals glinting in the sunlight, gammas carved out of the clouds, a thunderstorm flipped on its back like a turtle. It was water summoned from a cloud-painted sky. There's more art to science than most of us believe.\n\nAccording to Kurt, at the end of his life Bernie cherished a collection of favorite quotations by Albert Einstein. One of them was \"Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world.\"\n\nAnother was \"The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.\"\n\nBernie died not long after giving Kurt his old letter. Before dying, he told his brother, \"If the superpowers decide to duke it out with silver iodide, I think I can live with that.\"\n\nFour days after Bernie died, Kurt finished the highly personal Timequake, his fourteenth novel, and the last he would ever write.\n\n\"I was the baby of the family,\" he explained at the end. \"Now I don't have anybody to show off for anymore.\"\n\nBernie's son Terry died not long after him, losing a long struggle with cancer. Family members took both their ashes up in a small plane. They flew to the vicinity of Mount Greylock, where it all began, and scattered father and son into the clouds.\n\nKurt published a tribute to Bernie in The New York Times Magazine. Thinking back to those early days after Hiroshima, when the scientists first knew sin, Kurt declared that Bernard had \"original virtue.\" And then he described New York City's rainmaking efforts of 1950, a story that somehow the city had forgotten. But Kurt had not. The deluge was beautiful. Sure, he allowed, it might have \"washed away a lot of privies and gazebos or whatever. But it did fill the reservoirs to brimming.\n\n\"And what the heck,\" he concluded. \"It was only water.\"\nNotes\n\nPlease note that some of the links referenced in this work may no longer be active.\n\nArchives and Abbreviations\n\nIAS | |\n\nShelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton\n\n---|---|---\n\nLL-KV | |\n\nKurt Vonnegut Manuscripts, Lilly Library, Indiana University, Bloomington\n\nLL-MH | |\n\nS. M. Harris Manuscripts, Lilly Library, Indiana University, Bloomington\n\nLOC-FR | |\n\nFrancis Reichelderfer Papers, Library of Congress\n\nLOC-HW | |\n\nHarry Wexler Papers, Library of Congress\n\nLOC-IL | |\n\nIrving Langmuir Papers, Library of Congress\n\nMEG-BV | |\n\nBernard Vonnegut Papers, 1828\u20131997, M. E. Grenander Department of Special Collections and Archives, University at Albany, State University of New York\n\nMEG-GW | |\n\nGeorge Wise Papers, M. E. Grenander Department of Special Collections and Archives, University at Albany, State University of New York\n\nMEG-UA | |\n\nUniversity Archives, M. E. Grenander Department of Special Collections and Archives, University at Albany, State University of New York\n\nMEG-VS | |\n\nVincent Schaefer Papers, 1891\u20131993, M. E. Grenander Department of Special Collections and Archives, University at Albany, State University of New York\n\nMIS-GE | |\n\nGE Archive, Museum of Innovation and Science, Schenectady, N.Y.\n\nMIT-JC | |\n\nJule Charney Papers, 1936\u201381, MC 184, Institute Archives and Special Collections, MIT Libraries\n\nMIT-NW | |\n\nNorbert Wiener Papers, MC 22, Institute Archives and Special Collections, MIT Libraries\n\nNCAR-AMS | |\n\nAMS Oral History Project, Archives, National Center for Atmospheric Research\n\nNCAR-PDT | |\n\nPhilip Duncan Thompson Papers, Archives, National Center for Atmospheric Research\n\nNYPL-CC | |\n\nCrowell-Collier Publishing Company records, Manuscripts and Archives Division, New York Public Library\n\nNYPL-NY | |\n\nNew Yorker records, Manuscripts and Archives Division, New York Public Library\n\nNYU-AS | |\n\nNational Council for American-Soviet Friendship Records, TAM 134, Tamiment Library\/Robert F. Wagner Labor Archives, Elmer Holmes Bobst Library, New York University\n\nPU-CS | |\n\nArchives of Charles Scribner's Sons, Manuscripts Division, Department of Rare Books and Special Collections, Princeton University Library\n\nRAC-DM | |\n\nPapers of Duncan A. MacInnes, Rockefeller University Archives, RG 450, Rockefeller Archive Center\n\nUP-LB | |\n\nLemuel R. Boulware Papers, Kislak Center for Special Collections, Rare Books, and Manuscripts, University of Pennsylvania\n\n1. Autumn Fog\n\nAt the bottom of a snowy hollow: Much of my account of Vonnegut's personal war experience comes from his discussion in his interview for The Paris Review 69 (Spring 1977), reprinted in Allen, Conversations with Kurt Vonnegut. \"It was nice there for a few minutes,\" he said. The porcupine image is his.\n\nHe hadn't realized how much he needed that: Kurt Vonnegut to Jane Cox, June 1944, Vonnegut family collection.\n\nStill, for the first time in his life: Shields, And So It Goes, 56. Late in life, Kurt was asked to choose a single memory that he would want played endlessly in the afterlife: he chose the moment he rolled to the front, \"where I was doing everything right, where I was beyond criticism... I was right where I belonged.\" J. Rentilly, \"The Best Jokes Are Dangerous: An Interview with Kurt Vonnegut: Part II,\" McSweeneys.net, September 17, 2002, www.mcsweeneys.net\/articles\/the-best-jokes-are-dangerous-an-interview-with-kurt-vonnegut-part-two.\n\n\"the incredible artificial weather\": Vonnegut, Slaughterhouse-Five, 106.\n\nCome out, the Nazis ordered again: Vonnegut, Fates Worse Than Death, 96.\n\nThen out of the predawn darkness: \"It was as if the sky fell in,\" recalled Sam Giles of the 423rd's Company K, in Szpek, Idzikowski, and Szpek, Shadows of Slaughterhouse Five.\n\n\"A hurricane of iron and fire\": Gunther Holz, quoted in Kershaw, Longest Winter, 79.\n\nGeneral Eisenhower had a standing bet: Dupuy, Hitler's Last Gamble, 5.\n\nDuring World War I: Norway relied on its fishing fleet to keep its citizens fed. But when war broke out, the British\u2014knowing the weather could provide a tactical advantage\u2014classified their weather forecasts. The Norwegian fleet had always relied on the British forecasts; the new secrecy caused fishing hauls to plummet, and Norwegians began to starve. A group of scientists at the Geophysical Institute in Bergen offered to provide weather services to the government. These theoretical heavyweights brought a new rigor to the study of large weather patterns that came to be called air mass analysis. A thorough account of the rise of this school of meteorology is Harper, Weather by the Numbers.\n\nPetterssen told Eisenhower's team: Petterssen's Weathering the Storm tells the story of this historic forecast in detail.\n\nThe attack that would come to be called: Dupuy, Hitler's Last Gamble, 18.\n\nTwo-thirds of its troops: Like many personal details from this period, Kurt's attempts to get transferred to public relations duty come from his unpublished letters to Jane Cox from the period; Vonnegut family collection. Idzikowski cites the low age of the 106th in his introduction to Szpek, Idzikowski, and Szpek, Shadows of Slaughterhouse Five, 22. Kurt would make note of this in the full title of his war book: Slaughterhouse-Five; or, The Children's Crusade: A Duty-Dance with Death.\n\nThey wanted to go: In Slaughterhouse-Five, Kurt has Edgar Derby, the high school teacher who is executed for stealing a teapot, recount a capture that is almost exactly his own story. In the end, \"the Americans put their weapons down, and they came out of the woods with their hands on top of their heads, because they wanted to go on living, if they possibly could.\" Vonnegut, Slaughterhouse-Five, 107.\n\n\"It's a nasty picture\": A collection of the Cornell Daily Sun columns is in LL-KV; along with his letters to Jane from the same period, the columns give a lively picture of Kurt's undergraduate years.\n\nHe'd been sixteen: Kurt describes this 1939 trip in \"The Rover Boys,\" an unpublished manuscript composed of his letters home. LL-KV.\n\nHe bragged about that to Jane Cox: Kurt Vonnegut to Jane Cox, multiple dates, Vonnegut family collection.\n\nIn Ithaca: \"I was happiest here,\" he later told a Cornell audience, \"when I was all alone\u2014and it was very late at night, and I was walking up the hill after having helped to put the Sun to bed.\" Vonnegut, Palm Sunday, 60.\n\nToward the end of high school: Shields has Vonnegut landing a job offer from The Indianapolis Star. But the Star was the city's staunchly conservative paper; the Vonneguts were readers of The Indianapolis Times. In Timequake, Vonnegut declares that he was offered a position at the Times through the machinations of a lawyer friend of his father's. Vonnegut, Timequake, 154.\n\nBe anything, he said bitterly: Shields, And So It Goes, 34.\n\nThat, Kurt said later, \"was his idea of me\": Ibid., 35.\n\neven Kurt had to admit it: Kurt met Bow while on leave from the Army and commented on her beauty in a letter to Jane, March 1944, Vonnegut family collection.\n\n\"the Flying Icing Wind Tunnel\": Ice Research Base report, May 7, 1945. MEG-VS. Another airplane was later given the same name.\n\nThe sergeant would sit casually by the hole: Robert Morton Cunningham, autobiography online, www.egoaltar.com\/fogseeker\/.\n\n\"He attended Cornell University\": Indianapolis Times, January 15, 1945, 3.\n\n\"I am very favorably impressed\": Bernard Vonnegut to C. G. Suits, February 15, 1945. MIS-GE. His correspondence with Langmuir and Schaefer is in MEG-VS.\n\nSometimes they worked: Icing would remain a problem for aviation. The 2009 crash of an Air France Airbus in the Atlantic was attributed to icing on the plane's pitot tube\u2014a problem first diagnosed in the 1920s.\n\nHe'd been playing around with aircraft deicing: The friend was James V. Dotson. Jim Dotson was killed while conducting test experiment flights on a Lockheed Electra in 1943, and Bernie became deicing director. James V. Dotson obituary, MIT Technology Review 46 (1944). Also Robert Morton Cunningham, autobiographical sketch dated August 7, 1997, MIT Museum.\n\nHe sometimes joked: Bernard Vonnegut, talk given at the Weather Modification Association meeting, Los Angeles, 1976. Copy in MEG-BV. He also discusses going to work with Jim Dotson and Henry Houghton at MIT.\n\nOnce, when he was in his early teens: Scott Vonnegut, interview with the author, July 2013.\n\nPresident Charlie Wilson: There were two CEOs named Charlie Wilson at the time: Engine Charlie of General Motors and Electric Charlie of GE. Engine Charlie was later appointed secretary of defense by Eisenhower. The Texas politician Charlie Wilson, played by Tom Hanks in the film Charlie Wilson's War, is another Charlie altogether.\n\nHe planned for GE: Chauncey Guy Suits, oral history, IEEE Global History Network, www.ieeeghn.org\/wiki\/index.php\/Oral-HistoryGuy_Suits.\n\nThere he met some of GE's scientific luminaries: The GE scientists' interview evaluations of Bernard are still in MIS-GE, as is his application form.\n\n2. Precipitating Events\n\nSomeone was knocking on the door: The story of the run-up to the Russia trip was told in a letter from Dean Langmuir to Ruth Van de Water, June 18, 1945, and quoted at length in Rosenfeld, Quintessence, 282\u201392. The chemist Duncan MacInnes kept a very informative diary of the entire trip, RAC-DM.\n\n\"Dear People,\" it began: The letter was published posthumously. Vonnegut, Armageddon in Retrospect, 11\u201313.\n\nBut Bernard didn't tell Kurt about it: No one I talked to from Kurt's or Bernard's family was aware of the publication, and Kurt never mentioned it.\n\nEven before Kurt wrote to his family: Vonnegut to Cox, May 25, 1945, Vonnegut family collection.\n\nShe had graduated from Swarthmore: Vonnegut, Timequake, 222.\n\nHe insisted on taking the wheel: The account of picking up Kurt and the car ride home was written by Alex Vonnegut to Ella Vonnegut Stewart, July 4, 1945. Wakefield, Letters, 9.\n\nIt was hard to believe: Vonnegut to Cox, August 2, 1945, Vonnegut family collection.\n\nHe consoled himself: Vonnegut to Cox, n.d., but from context late July 1945, Vonnegut family collection.\n\nWhile awaiting Bernie's arrival: Indianapolis had three papers in the 1940s: the morning Indianapolis Star, the evening Indianapolis News, and the afternoon Indianapolis Times. The Star and News had conservative reputations, and elsewhere Kurt identified the progressive Times as his family paper. It printed a society page announcement of his wedding and was the only Indy paper to review Player Piano. But Kurt identified the News as the paper whose offices he visited in his Paris Review \"Art of Fiction\" interview, and it makes sense he would visit the city's largest paper, \"the great Hoosier daily,\" to get a sense of what was printed. Allen, Conversations with Kurt Vonnegut, 175.\n\nStrategic bombing was the front-page story: Indianapolis News, February 14, 1945.\n\nHe called the family: Vonnegut to Cox, August 7, 1945, Vonnegut family collection.\n\nThe usually unflappable Bernie: \"My brother was utterly sickened,\" Kurt wrote many years later. \"It hit him in the gut. I really had no idea how horrible the news was, except through his reaction.\" McCartan, Kurt Vonnegut, 148.\n\n\"the battle of the laboratories\": White House press release, August 6, 1945. Along with \"the greatest achievement of organized science in history,\" this phrase from Truman's press release was widely quoted in newspapers across the nation, including Indianapolis's.\n\n\"Scientists' Dream Comes True\": Indianapolis Star, August 7, 1945.\n\nThere was no keeping: Kurt told Jane that there was no secret in an undated letter clearly written shortly after the bomb was dropped, Vonnegut family collection.\n\nKurt and Jane took the small, leaky rowboat: Kurt tells this story in Fates Worse Than Death, 51\u201352.\n\nThe wedding had been moved: Earlier newspaper announcements (as well as Shields) set the date on September 14. But The Indianapolis Times, The Indianapolis News, and The Columbian (the magazine of Indianapolis's posh Columbia Club) all reported on September 1 that the couple were marrying that day, and it was the date of their anniversary. Kurt wrote to Jane about the Miami orders in August, and Bernard wrote to Chauncey Guy Suits on August 28 from Cambridge, so it's a pretty safe assumption that the orders led to moving up the date. Ben Hitz reported that it seemed as if there were a dark cloud over Kurt on his wedding day, and he assumed it was the war. Shields, And So It Goes, 83.\n\nIn the newspaper announcements: Indianapolis Times, September 1, 1945, reported that Kurt \"attended Cornell.\" The Filomena Gould write-up for The Indianapolis Star, September 1, 1945, is the one Kurt carefully saved. LL-KV.\n\nHe'd get a good job: Vonnegut to Cox, August 2 and 3, 1945, Vonnegut family collection.\n\n\"It's all an ocean\": Kurt told the scholar Donald Fiene that this was one of Jane's favorite quotations. Fiene, \"Elements of Dostoevsky in the Novels of Kurt Vonnegut,\" Dostoevsky Studies 2 (1981): 135. I quote from the Constance Garnett translation, which would have been the one Kurt and Jane owned in the Modern Library edition.\n\nIt made her happy to believe: \"Jane could believe with all her heart anything that made being alive seem full of white magic.\" Vonnegut, Timequake, 135.\n\nAs a child, Kurt had drawn: Allen, Conversations with Kurt Vonnegut, 232.\n\nThis could only be the dream: \"A hammer is still my Jesus, and my Virgin Mary is still a crosscut saw,\" he wrote in a 1980 essay on how he lost his innocence concerning technology. It was a theme he returned to many times. \"How sick was the soul revealed by the flash at Hiroshima?... It was so sick that it did not want to live anymore.\" Vonnegut, Palm Sunday, 62\u201363.\n\n\"Some good sacred memory\": Fiene, \"Elements of Dostoevsky,\" 134.\n\nFor his subject, he chose a chair: Ibid., 131.\n\nIrving Langmuir took his seat: David Lilienthal published his notes on this conference as appendix A in his journals. Atomic Energy Years, 639. My account of the conference also comes from Wittner, The Struggle Against the Bomb; Smith, A Peril and a Hope; Ashmore, Unseasonable Truths; and Glenn T. Seaborg, \"Premonitions After the Bombs,\" Bulletin of the Atomic Scientists, December 1985.\n\nYet many of the Chicago scientists: In June 1945, a group of the Chicago scientists led by Szilard, Eugene Rabinowitch, Arthur Compton, and James Franck had sent a document called the Franck Report to Secretary of War Henry Stimson. They had argued vigorously that the first atomic bomb should be dropped not on a Japanese city but on an uninhabited island or desert spot. Only if the Japanese still refused to surrender should it be dropped on an actual city. Truman and his advisers rejected the idea. After all, they only had two bombs.\n\n\"The physicists have known sin\": This widely quoted line seems to have first appeared in print in J. Robert Oppenheimer, Physics in the Contemporary World (Anthoensen Press, 1947), 11. Life magazine quoted it in 1949.\n\nHe stuck to the script: Langmuir's every pronouncement was widely reported in the media. \"Langmuir Warns of Danger to U.S.,\" New York Times, October 9, 1945; \"Langmuir Urges Atom Pact, Says War Might Strip Earth,\" New York Times, November 17, 1945; \"Crisis by '55 Seen in Atom Disunity: Scientist Tells of Atomic Energy,\" New York Times, December 1, 1945.\n\nKurt stood in the shower: Kurt Vonnegut to Jane Vonnegut, October 6, 1945. Much of my understanding of this period in Kurt's life is based on his unpublished letters to Jane between 1941 and 1946, nearly daily during his time at Fort Riley, generously shared with me by the Vonnegut family. Sadly, Jane's letters were not preserved, so where I have described her response, it is because Kurt mentioned or described it in one of his letters.\n\nThe day the acceptance letter arrived: At least he remembered it as such later. Vonnegut, Wampeters, Foma, and Granfalloons, 177.\n\nHe would study anthropology: He planned to study anthropology before arriving at Chicago. Kurt Vonnegut to Jane Vonnegut, December 1, 1945, Vonnegut family collection.\n\nKurt had thought about going to its law school: He wrote this to Jane from the Army in 1944, Vonnegut family collection.\n\nOne of the very first: Kurt included \"Palaia\" in a list of stories he intended to write in a letter to Jane, October 9, 1945, Vonnegut family collection.\n\nBernie was the great one: Vonnegut to Cox, August 3, 1945, Vonnegut family collection. Protestations about his lack of genius were a regular theme in his letters to Jane from the Army.\n\n3. Head in the Clouds\n\nOne of the most famous stories told about Steinmetz: This version is mostly cribbed from the MIT president Charles M. Vest's Charge to the Graduates, June 4, 1999, <http:\/\/web.mit.edu\/newsoffice\/1999\/vestspeech.html>. The story is also frequently told in the self-help books for businessmen and Bible students commonly sold in airport bookstores.\n\nThe rest of the time: My information about Doc Whitney and the Research Lab comes largely from Wise, Willis R. Whitney, General Electric, and the Origins of U.S. Industrial Research, 77. It's interesting that tech companies have been given credit for inventing policies like Google's 80\/20 rule\u2014in which engineers spend 20 percent of their time working on things that interest them\u2014when they are simply following the GE model.\n\nAlbert Hull taught Greek: Lee de Forest, quoted in Rosenfeld, Quintessence, 154.\n\nIf he didn't find something quickly: Bernard Vonnegut, typescript interview with Barrington Havens, February 12, 1952, MEG-BV.\n\nThe horror of Hiroshima: This brand of postwar optimism makes frequent appearances in Kurt's fiction and nonfiction. \"Think of the new era that is being born,\" Jailbird's narrator, Walter Starbuck, tells his wife, Ruth, right after the war. \"The world has learned its lesson at last, at last. The closing chapter to ten thousand years of madness and greed is being written right here and now\u2014in Nuremberg... It's the most important turning point in history.\" Vonnegut, Jailbird, 67\u201368. Kurt himself wrote to Marc Leeds in 1995, \"Once the Great Depression and the Second World War were over, we planned to build a Garden of Eden here.\" Wakefield, Letters, 364.\n\nBut there were also tedious scientific tasks: \"I was still cowed by the science thing,\" he later recalled. McCartan, Kurt Vonnegut, 134.\n\nThe adviser had recommended cultural anthropology: Vonnegut, Wampeters, Foma, and Granfalloons, 178.\n\n\"Wailing Shall Be in All Streets\": This essay would remain unpublished until 2008, after Kurt's death. Vonnegut, Armageddon in Retrospect, 33\u201334.\n\nJane went through it: A typed manuscript of this essay is in LL-KV; Edie Vonnegut identified the hand in which the edits were written as her mother's.\n\nSnow had brought Vincent: Schaefer, \"Twenty Years at Langmuir University.\"\n\n\"Look, Vince,\" he said: Vincent relates this conversation in a conversation with Bernie, Raymond Falconer, and Duncan Blanchard videotaped in 1989. \"Conversations on the Early Days of Cloud Seeding and the Development of the Atmospheric Sciences Research Center,\" MEG-UA.\n\nVince had an idea: My account of the discovery of dry ice nucleation comes from Katharine Blodgett, \"Vincent Schaefer and Snow Making,\" a paper read at a meeting of the Schenectady Fortnightly Club, February 2, 1948, MIS-GE; Schaefer, \"Twenty Years at Langmuir University\"; and Vincent Schaefer GE lab notebooks, MEG-VS.\n\nTogether they went looking for her adviser: Vonnegut, Timequake, 126.\n\nWhen he heard the whine of an airplane: The early days of Project Cirrus have been reconstructed using Havens, Jiusto, and; Vonnegut, Early History of Cloud Seeding; Project Cirrus Occasional Reports; Project Cirrus Flight Data; and Vince and Bernie's lab notebooks, MEG-VS and MEG-BV. Also critical were Langmuir, Cloud Nucleation, and Langmuir's lab notebooks, LOC-IL. Bernie's lunchtime story is told in Steven Spencer, \"The Man Who Can Make It Rain,\" Saturday Evening Post, October 25, 1947.\n\n\"I could see it\": Blodgett, \"Vincent Schaefer and Snow Making.\"\n\n\"Well, Schaefer made it snow\": Havens, Jiusto, and Vonnegut, Early History of Cloud Seeding, 7.\n\n\"Scientists of the General Electric Company\": GE News Bureau press release, November 14, 1946.\n\nLetters and telegrams poured into the Research Lab: Letters and clippings were collected by both Vincent and Bernie, MEG-VS and MEG-BV.\n\n\"Time to go home, Barney\": Blodgett, \"Vincent Schaefer and Snow Making.\" My account of Bernie's discovery is also based on his papers and GE lab notebooks, MEG-BV.\n\n\"Schaefer made some seeding runs\": Langmuir to C. N. Touart, December 26, 1946, MEG-VS.\n\nNow Langmuir was talking in sweeping terms: Horace Byers, \"History of Weather Modification,\" in Hess, Weather and Climate Modification.\n\n\"comprehend all the forces\": Laplace, Essai sur les probabilit\u00e9s, 4. Quoted in Newton, From Clockwork to Crapshoot, 108.\n\n\"He who would master nature\": Cited in a memorial to Harry written by Francis Reichelderfer, Monthly Weather Review, October 1963.\n\nThe GE scientists' revelations were the \"outstanding contribution\": Report to the Bureau Chief, LOC-HW. He would continue to denigrate their \"naive\" approach for years.\n\n\"I promise to scrub the bathroom\": Wakefield, Letters, 13\u201314.\n\nHe wanted to be part of it: \"I looked on the University of Chicago community as a folk society\u2014and I felt like an outsider in it... I wasn't treated badly, but they already had a family,\" Kurt wrote in \"A Very Fringe Character,\" in McQuade, Unsentimental Education.\n\nhe exhibited three paintings: Kurt and Jane Vonnegut to Walt and Helen Vonnegut, September 1, 1947. Wakefield, Letters, 20.\n\n4. Bolt of Lightning\n\nHe was thinking about computers: Cited by George Dyson, son of Freeman Dyson, in Turing's Cathedral, ix. My background data on von Neumann come from Macrae, John von Neumann; and Heims, John von Neumann and Norbert Wiener.\n\nNow he was refining the architecture: There is some confusion about the name of the institute computer. Early on, some people would refer to it as \"Johnniac,\" then later \"MANIAC.\" Ultimately, it was simply known as the IAS machine. The name \"Johnniac\" was later used officially for a Rand computer, and \"MANIAC\" was used for one built at Los Alamos, both using the institute machine's design, which came to be known as the von Neumann architecture.\n\nAnd that's where Harry Wexler came in: My account of the institute's Meteorology Project is based on Harper, Weather by the Numbers; Nebeker, Calculating the Weather; George Dyson, Turing's Cathedral; oral histories at NCAR-AMS; papers at NCAR-PDT, LOC-FR, and LOC-HW; and 2014 interviews at the Institute for Advanced Study with Freeman Dyson, who came there as a young scholar during the Meteorology Project.\n\nIn October 1945, Zworykin issued: Vladimir Zworykin, \"Outline of Weather Proposal\" (white paper, October 1945), reprinted in History of Meteorology 4 (2008).\n\n\"first steps towards influencing\": Harper, Weather by the Numbers, 102\u2013103.\n\nIt was just the kind of experiment he liked: Kurt calls Bernie's approach to science \"a form of practical joking\" in a 1987 interview with Hank Nuwer. Allen, Conversations with Kurt Vonnegut, 246. Bernie's son Scott recounted his father's earthy sense of humor, interview with the author, July 2013.\n\nHe did it over and over: Blodgett, \"Vincent Schaefer and Snow Making.\"\n\nInstead, flight experiments would be \"conducted by the government\": Quoted in Havens, Jiusto, and Vonnegut, Early History of Cloud Seeding, 8\u20139.\n\nJust to be safe, Irving resigned his sponsorship: Irving Langmuir to William Howard Melish, March 31, 1947, NYU-AS. Irving was one of the earliest to resign; later, its long list of scientific and artistic members would dwindle to nearly none as people scrambled to avoid persecution in the McCarthy era.\n\nIn keeping with their new publicity aims: \"Army and GE Join to 'Make Weather,'\" New York Times, March 14, 1947.\n\nHe'd applied for a job: Kurt recounts this in a letter to Knox Burger, July 2, 1949, NYPL-CC.\n\nIn the solicitation letter: Wakefield, Letters, 16. The letter to General Motors that Wakefield prints is an exact duplicate, but for the name, of the one Kurt sent to GE, which was generously shared with me by Mark Vonnegut.\n\nAs Rear Admiral Luis de Florez had written: \"Weather\u2014the New Super Weapon,\" American Magazine, September 1946.\n\nThat's what he'd tried to do earlier that month: \"GE Rainmakers' Aid Sought to Quell California Fire,\" Schenectady Gazette, August 8, 1947.\n\nGeneral George Kenney, of the Strategic Air Command: \"28,000,000 Urged to Support MIT,\" New York Times, June 15, 1947.\n\nIt was an overpriced French joint: Kurt calls it a \"clip joint\" in his letter to Walt and Helen. Wakefield, Letters, 19.\n\nBut the couple would have to face: Dan Wakefield writes, \"Kurt's classmate Victor Jose told me that 'if they stayed in Indianapolis, they feared that Jane would be expected by her family to join the Junior League, and she and Kurt didn't like that \"socialite\" kind of life. They both were rebels even then.'\" Ibid., 5. They were, but Jane still joined the Junior League in Schenectady, where they worked hard to fit in.\n\nBernie had already told George: Kurt mentioned Bernie's recommendation in his letter to General Electric. Also, Shields, And So It Goes, 94; and Wakefield, Letters, 19.\n\n5. Eye of the Storm\n\n\"I own a home now\": Wakefield, Letters, 23.\n\nHe felt manly enough: Shields, And So It Goes, 96.\n\nThere, it became part: Nye, Image Worlds, 60.\n\nHis new bosses gave him a physical: Kurt Vonnegut to Jane Vonnegut, dated only Friday, Vonnegut family collection. Writers on Vonnegut almost inevitably fail to be clear about the distinction between GE (whose motto was \"Progress is our most important product\"), the GE Research Lab (called the House of Magic), the Schenectady Works (the plant itself), and the Schenectady Works Lab (like the many GE plants, the Schenectady Works had its own separate research lab). Many critics and biographers repeat not only the incorrect statement that Kurt worked for the GE Research Lab but also the erroneous claim that he wrote publicity for Project Cirrus.\n\nStories about things: The GE sales force refrigerator that walked and talked was featured in an unsigned story in The GE Monogram in January\/February 1949 that was likely written by Kurt. It eventually morphed into female form and became the subject of Kurt's short story \"Jenny,\" unpublished until the posthumous While Mortals Sleep.\n\n\"A hurricane is a complicated thing\": My account of the hurricane seeding comes from Havens, Jiusto, and Vonnegut, Early History of Cloud Seeding; Schaefer, Final Report Project Cirrus, Part 1; and Schaefer, \"Twenty Years at Langmuir University.\"\n\nThe steering committee drew up an official plan: Irving Langmuir, \"The Control of Hurricanes\" (white paper, December 15, 1955), MEG-BV.\n\n\"We accomplished our purpose\": The press conference was reported in many newspapers, but the most complete account was the front-page story in GE's hometown paper. \"Hurricane Study Only Begun, Says Schaefer,\" Schenectady Gazette, October 17, 1947.\n\nThat's not what The New York Times had reported: \"Hurricane Sweeps into Atlantic Leaving Florida Widely Flooded,\" New York Times, October 13, 1947.\n\n\"The next great tropical storm\": \"Hurricane Busters Ready to Test Chemical Seeding,\" Rome Sentinel, September 8, 1947.\n\nThe steering committee composed a press release: It's unclear if this press release was ever issued. Undated draft of Project Cirrus press release, MEG-BV.\n\nNevertheless, when the Project Cirrus team: Mobile Press-Register, October 12, 1947.\n\n\"pretty sore at the army and navy\": Quoted in Fleming, Fixing the Sky, 152.\n\n\"low Yankee trick\": \"Science: Yankee Meddling?,\" Time, November 10, 1947.\n\nThe sheriff of Savannah: Schaefer, \"Twenty Years at Langmuir University.\"\n\n\"Did science, for the first time\": Schenectady Gazette, October 31, 1947.\n\n\"Perhaps you saw the news reports\": Wakefield, Letters, 23. In this letter, Kurt also discusses his plans to take his master's exams and work on finishing his thesis at Columbia.\n\nHe submitted them all: A rejection letter for \"Basic Training\" from Redbook dated August 4, 1949, is in the rejection letters file, LL-KV. Based on the juvenile style and comments in letters to Jane, I believe Kurt actually started the novella in Indiana, before going to Chicago, but could not confirm it definitively.\n\nThey put candles: Yarmolinsky, Angels Without Wings, 7.\n\nHe was looking forward: Bernard Vonnegut, interview with Havens, February 12, 1952. This account was also put together using Bernard's paper \"Experiments with Silver Iodide Smokes in the Natural Atmosphere,\" Bulletin of the American Meteorological Society 31, no. 5 (May 1950): 151\u201357; Schaefer, Final Report Project Cirrus, Part 1; and flight data from \"Data, Catalogue of Data for PC 1950\" folder, MEG-BV.\n\n\"The photograph of General Electric's Dr. Bernard Vonnegut\": Wakefield, Letters, 25.\n\nAnd at some point, he gave it to Bernie: Kurt included the letter with a typed explanation in his papers, LL-KV. He also recounted the letter's journey from Alex to Bernie to him in the epilogue to Timequake.\n\nLangmuir was giving the keynote address: \"A hectic day,\" Irving wrote in his diary after his day with the GE legal team, LOC-IL. Also, Rosenfeld, Quintessence, 300. Irving's paper was published as \"The Growth of Particles in Smokes and Clouds and the Production of Snow from Super-cooled Clouds,\" Proceedings of the American Philosophical Society 92:3 (July 1948).\n\nProject Cirrus was wading: Byers gives an account of meteorologists' response to Langmuir's talk in \"History of Weather Modification,\" 14.\n\nChief Reichelderfer convened an advisory committee: He sent Langmuir an invitation to join but then never invited Langmuir to attend a single meeting. Irving Langmuir lab notebook, LOC-IL. Other facts about the Weather Bureau's anti-Cirrus campaign have been collected from LOC-HW and LOC-FR.\n\n\"Some 15 different types of finely divided soil\": GE News Bureau press release, April 22, 1948, initialed KV, MIS-GE.\n\nHe didn't love playing in the snow: Local newspapers reported on the snowmaking scientist's broken ankle with great glee.\n\nGE cultivated this obsessiveness: Kurt described the Research Lab as a playground in the essay \"Think Bank.\" Forbes ASAP, May 31, 1999.\n\nThey were in the business of seeking truth: \"Buggering truth for money\" is Kurt's phrase, from the introduction to his second short story collection, where he attributes it to a college professor talking about public relations men and writers of slick fiction. Vonnegut, Welcome to the Monkey House, xiv.\n\nHe stayed lost in thought: Rosenfeld, Quintessence, 304.\n\nKurt could get annoyed: Scott Vonnegut tells a funny story about how even when they were old, Kurt sometimes had to work to get a word in edgewise, once even interrupting Bernie by jumping up and down and yelling, like a small child, \"You never let me speak!\" in a half-serious parody of his own younger-brother frustration. Interview with the author, July 2013.\n\nIf subjected to enough pressure: Ice-2 crystals are now recognized to have the shape of a rhombohedron\u2014a prism with six rhomboid faces.\n\nAnd who knew: Today, scientists have identified more variant forms of ice than of any other known material. Normal ice is known to science as ice-1h. After that, it has variants from ice-1 all the way through ice-15.\n\nThe songbook also had the words: Vince's songbook and commemorative booklet are in MEG-VS. The script of the 1948 pageant is in UP-LB.\n\nMr. Bullshit was more like it: In a 1983 letter to The Washington Spectator, the former GE employee David Anderton wrote, \"Boulware's first name was, I am certain, Lemuel. He was known to the union then as 'Mr. Bullwhip' and to some of us in a GE Management course as 'Mr. Bull\u2014\u2014t.' Ask Vonnegut.\" Kurt cut this letter out and saved it in his files.\n\nGE wouldn't be caught dead: Vonnegut, Bagombo Snuff Box, 8.\n\nIn Monday morning staff sessions: Reminiscences of Ollie Lyon and Bob Pace from Happy Birthday, Kurt Vonnegut, a birthday festschrift put together by his second wife, Jill Krementz, for his sixtieth.\n\nFrequently, they didn't even know the cause: A new young physicist at GE named Robert Vought was denied security clearance for mysterious reasons and had been struggling unsuccessfully for more than a year to clear his name. Guy Suits eventually suggested he look for another job. See Wang, American Science in an Age of Anxiety, 102\u201317.\n\nThey didn't know that the FBI was compiling dossiers: I received the FBI dossier on Irving Langmuir through the National Archives after filing a Freedom of Information Act request with the FBI. The FBI would neither confirm nor deny that it had a file on Bernard Vonnegut: its letter said that potentially relevant records were destroyed in 1987.\n\n\"GE Disease\": Krementz, Happy Birthday, Kurt Vonnegut.\n\nActing president of the tiny New Mexico School of Mines: Jack Workman ended up at the New Mexico School of Mines after launching a research and development division at the University of New Mexico physics department, where he originally taught. During the war, Workman and his team developed the proximity fuse, bringing in a huge influx of government money, and in 1946 the University of New Mexico tried to force them to share it with the rest of the school. Jack and most of his division resigned and moved to the School of Mines. Chew, Storms Above the Desert, chapter 2.\n\nEveryone was terrified except Irving: Schaefer, \"Twenty Years at Langmuir University.\"\n\n\"The odds in favor of this conclusion\": Langmuir, \"Results of the Seeding of Cumulus Clouds in New Mexico,\" Research Laboratory Report No. RL: 364, June 1950.\n\n6. Watersheds\n\n\"If you think that's bad\": This comment captured something about Bernie for Kurt, who repeated the story in a number of interviews and essays, as he often did his best material. \"Address to the American Physical Society\" (1969), in Vonnegut, Wampeters, Foma, and Granfalloons, 93. Duncan Blanchard, Bernard's young colleague at GE, also described for me the chaotic danger zone of Bernie's work space.\n\nYou turned on the tap: The remark about \"nothing\" coming out of the taps sounds like Bernie's sense of humor, and the reporter clearly spent much of his time with the Project Cirrus team. The quip is in the same section as a description of the team's new quarters. \"300 Tour Main Wing of GE Research Lab,\" Schenectady Gazette, December 3, 1948.\n\n\"You'll know when I'm retired\": Havens, Jiusto, and Vonnegut, Early History of Cloud Seeding, 11.\n\nAs Kurt and Jane prepared to go out: Ollie Lyon recalls this incident in John Dinsmore, \"Kurt and Ollie,\" Firsts, October 1992. The Schenectady Gazette always printed guest lists for Junior League dances; the list of November 26, 1948, includes Mr. and Mrs. Kurt Vonnegut but not Mr. and Mrs. Ollie Lyon. Majie Failey describes Alice Vonnegut wearing sneakers with an evening gown in We Never Danced Cheek to Cheek, 58.\n\nOn weekdays in Building 6: My account of Kurt's early work is based largely on his short story drafts, LL-KV. Some of the PR work he did at GE can be found in MIS-GE.\n\nSo in the evenings and on weekends: Shields, And So It Goes, 101.\n\nA smooth, steady clacking: Jane describes listening for the rhythm of his typing in Yarmolinsky, Angels Without Wings, 5.\n\nThey knew they'd find Kurt: John Dinsmore, \"Kurt and Ollie,\" Firsts, October 1992. This interview with Ollie Lyon, a colleague of Kurt's at the GE News Bureau, is a rare account of the work environment there. It is also probably the source of a frequently repeated mistake: the claim that Kurt wrote most of the publicity for Project Cirrus. Initialed GE press releases prove that in fact the company seems to have deliberately avoided having one brother write about the other.\n\nWhen Jane gave him a grocery list: Rejection letters folder, LL-KV.\n\nHe had a sneaking suspicion: \"We are what we pretend to be, so we must be careful what we pretend to be.\" Epigraph to Vonnegut, Mother Night.\n\nAnd so, right there in Building 6: Some short paragraphs from early drafts of \"Mnemonics\" are written on a Building 6 notepad.\n\n\"There is no use worrying\": Langmuir, \"Large-Scale Seeding of Stratus and Cumulus Clouds with Dry Ice.\" Stenographic record of paper given at the January 1949 AMS meeting, LOC-IL.\n\nAnd it was simply outrageous: Langmuir, Cloud Nucleation, 556.\n\n\"The experiments showed\": Richard Coons, R. G. Gentry, and Ross Gunn, Second Partial Report on the Artificial Production of Precipitation: Cumuliform Clouds, Ohio (U.S. Weather Bureau, 1949).\n\nAfter the talk, Irving took Reichelderfer aside: An account of this conversation is in Langmuir, Cloud Nucleation, 213.\n\n\"Rain-Making Held of No Importance\": New York Times, January 26, 1949.\n\nThe Navy and the Signal Corps liked his silver iodide generator: R. W. Larson to Ruth Dwyer, GE memo, February 1, 1949, MEG-BV.\n\nThey simply assumed: Project officer (Naval Science Division, Office of Naval Research), to Bernard Vonnegut, January 21, 1949, MEG-BV.\n\n\"The policy of the government itself\": \"A Scientist Rebels,\" The Atlantic, January 1947.\n\nWiener's letter had made: It also had a big effect on Harry Wexler, who found it infuriating. He and one of the researchers at the Institute for Advanced Study began calling it \"the affair of Wiener,\" NCAR-PDT. In essays and interviews, Kurt frequently discussed the GE scientists' interest in ethical questions.\n\nTwo years of debate: Kurt mentions the Wiener essay in several interviews, including one with Playboy. Vonnegut, Wampeters, Foma, and Granfalloons, 269. The Bulletin of the Atomic Scientists would have been widely read in Schenectady; it's reasonable to think that Kurt would have seen the follow-up essay or heard about it. He discussed Wiener's refusal directly in a 1980 interview with Robert Musil, published in The Nation and later collected in McCartan, Kurt Vonnegut, 73.\n\n\"The degradation of the position\": Norbert Wiener, \"A Rebellious Scientist After Two Years,\" Bulletin of the Atomic Scientists 4, no. 11 (1948).\n\nKurt's new story: In interviews later in life, Kurt would refer to \"Report on the Barnhouse Effect\" as his first story. It would be his first published story, but the manuscripts prove it was far from the first he wrote. Still, a surprising number of literary critics have taken him at his word, failing to perceive the long, grueling apprenticeship that led to it. The drafts at LL-KV show the story's painstaking evolution. The only lines I quote are those that are consistent in early drafts and in the final published version in Vonnegut, Welcome to the Monkey House.\n\n\"It is not possible\": William Lewis to Irving Langmuir, memo, June 23, 1949, LOC-IL.\n\nIt had proved: \"Trial by Newspaper,\" Scientific American, February 16, 1949.\n\nThe stories were brisk: Russell to Vonnegut, February 18, 1949, rejection letters folder, LL-KV.\n\nThat month, three thousand delegates: Robbie Lieberman, \"Does That Make Peace a Bad Word? American Responses to the Communist Peace Offensive, 1949\u20131950,\" Peace & Change 17, no. 2 (April 1992).\n\n\"This is a little sententious for us\": This undated letter from Knox Burger is in the general rejection letters file at the Lilly and is quoted by Shields, And So It Goes, 101. However, by cross-referencing the note with Kurt's own scrupulous records of his early stories and their submissions, one can see that the story he sent to Collier's in April was \"Report on the Barnhouse Effect,\" not \"Mnemonics,\" as Shields has it. Burger's letters in NYPL-CC show that Vonnegut sent him \"Mnemonics\" after they met for lunch in July.\n\nClams, cigars, cocktails: Kurt literalized the idea of the corporate clambake as a kind of funeral in the famous clambake celebrating Kilgore Trout and memorializing American eloquence threaded throughout his valedictory last novel, Timequake.\n\nGeorge Burns was a photographer: Kurt's first letter to Knox Burger begins, \"George Burns, who, never having read anything of mine, takes a casual interest in my writing career, allowed as to how he had a friend on Collier's fiction staff who might be able to give me some help. The friend turned out to be you.\" NYPL-CC, and Wakefield, Letters, 26. Wakefield, like Shields, seems to have taken the name George Burns to refer to the popular comedian\u2014an understandable mistake, because Vonnegut sometimes referred to the famous George Burns as well. However, the letter only makes sense if referring to his acquaintance George Burns.\n\nAfter flying over Nagasaki: \"G-E Photographer Takes Pictures of Atomic-Bombed City,\" Schenectady Works News, September 21, 1945.\n\nOnce, Knox had even taken Kurt's suggestion: Shields, And So It Goes, 40.\n\n\"Sorry you didn't care\": Wakefield, Letters, 26. Kurt's underlining of \"typewritten\" is not reproduced in Wakefield, though it would seem to be essential to understanding his tone. The original letter is in NYPL-CC.\n\nHe would see if he could do anything: Account taken from Bernard Vonnegut interview, 1952, and Data: GE Research Notepad, July 19, 20, and 21, 1949, both MEG-BV. Also Schaefer, Final Report Project Cirrus.\n\nIf the bureau scientists were using other compounds: Bernard Vonnegut, \"Note on Nuclei for Ice Crystal Formation,\" BAMS 30, no. 5 (May 1949).\n\nThat meant the winds were carrying: \"Whooping it up\" is Bernie's phrasing from his lab notebook, MEG-BV.\n\n\"You were running the generator today?\": Bernie recounted these conversations with Langmuir in his February 12, 1952, interview with Havens.\n\n\"Taking all in all\": Quoted in Langmuir, Cloud Nucleation, 208.\n\nThe FAA had ruined flying for him: Vincent recounts this fact in \"Twenty Years at Langmuir University.\"\n\n\"What river is that?\": Langmuir, Cloud Nucleation, 110.\n\nAs Langmuir enthused: Teller, Memoirs, 253. Teller places this meeting in 1947; however, it seems clear from context that he must be misremembering the date.\n\n\"In phase,\" he would chant: Rosenfeld, Quintessence, 183. Vonnegut relates the charlatan comment in Palm Sunday, 145.\n\nThe next day, Knox returned \"Mnemonics\": Vonnegut to Burger, July 1, 2, 6, 17, and 20, 1949; Burger to Vonnegut, July 8, 1949; Gertrude Buckman to Vonnegut, July 14, 1949, all NYPL-CC.\n\nUnless he wrote in a secretary: Burger to Vonnegut, July 13, 1949, LL-KV.\n\n\"I told him\": Burger to Vonnegut, July 19, 1949, NYPL-CC, and quoted in Shields, And So It Goes, 108. Kurt's much-edited draft of his response is in LL-KV.\n\n\"the net result lacks conviction\": Burger to Vonnegut, August 1, 1949, NYPL-CC.\n\nKnox returned it almost immediately: Vonnegut to Burger, July 20, 1949; Burger to Vonnegut, July 26, 1949, NYPL-CC.\n\nHe figured it had the best chance: Vonnegut to Littauer, August 18, 1949, LL-KV.\n\nCollier's itself had recently run: \"UN Is Doing a Job,\" Collier's, July 9, 1949.\n\nNo editor would stand for: Littauer to Vonnegut, LL-KV.\n\n\"Only the descendants\": Albany Times-Union, August 28, 1949.\n\n7. Rainmakers\n\nEver since the Project Cirrus team had returned: Bernie's activities were recorded in his lab notebook, MEG-BV. Irving wrote up his Crown Island experiment and his conclusions in his own lab notebook, LOC-IL. Guy Suits described his meeting with all three scientists in a letter to Irving cc'd to Vince and Bernie. Suits to Langmuir, August 26, 1949, MEG-BV.\n\n\"in which the behavior\": These definitions are quotations from his paper \"Science, Common Sense, and Decency,\" first given in 1942 and published as chapter 1 of his book Phenomena, Atoms, and Molecules (1950), but he had been developing the idea much earlier, including in his 1934 paper \"Science as a Guide in Life.\"\n\nIt was simply a matter of knowing: Freeman Dyson, who heard von Neumann speak on the topic around 1950, summarizes Johnnie's attitude as \"All processes that are stable we shall predict. All processes that are unstable we shall control,\" in Infinite in All Directions, 182. Dyson's paraphrase of Johnnie's attitude has often been incorrectly attributed to von Neumann as a quotation.\n\nThere would never be: Wiener had begun insisting in Cybernetics that weather prediction should be undertaken not in a deterministic way but in a probabilistic one. Von Neumann and his chief meteorologist, Jule Charney, disagreed. The argument flared up periodically at conferences and meetings. Charney even heard that Wiener considered Johnnie and him to be gonifs\u2014Yiddish for \"scoundrels\"\u2014in their attempt to hoodwink the world into believing that they could turn the weather into an equation. Jule Charney, \"Conversation with George Platzman,\" in Lindzen, Lorenz, and Platzman, Atmosphere, 57.\n\nThe idea he came up with: I discussed Langmuir's experiment with Freeman Dyson. \"It was the right approach,\" he said, \"to produce a signal.\" Interview, February 2014.\n\n\"should be carried out\": Suits to Langmuir, cc'ed to Vonnegut and Schaefer, August 26, 1949, MEG-BV.\n\n\"Why doesn't he use his real name on the story?\": Burger to Littauer, October 13, 1949, NYPL-CC, and quoted in Shields, And So It Goes, 109.\n\nKurt knew in his heart: Letters between Vonnegut and Littauer are in the short story files, LL-KV.\n\nBut as Jane giddily told a neighbor: Shields, And So It Goes, 110.\n\n\"I think I'm on my way\": Wakefield, Letters, 27.\n\n\"An oath, an oath\": Vonnegut, Fates Worse Than Death, 26.\n\nHis paper came: Harry Wexler, trip report, March 10, 1949, LOC-HW.\n\nHad Project Cirrus been around in World War II: C. Lester Walker, \"The Man Who Makes Weather,\" Harper's, January 1950.\n\nIt was Kenneth collecting: Kurt's earlier rejections are in a separate file in LL-KV. The rejections sent to Littauer are in NYPL-CC and NYPL-NY.\n\nThere was even a hilarious playlet: \"Das Ganz Arm Dolmetscher\" was ultimately published as \"Der Arme Dolmetscher\" in The Atlantic in July 1955 and in Vonnegut, Bagombo Snuff Box, in 1999, 228.\n\nIn fact he'd spent: Vonnegut to Miller Harris, May 19, 1950, LL-MH.\n\nAnd now, at the annual AMS meeting: Irving noted that it was the most important paper of his career in his lab notebook. The paper, \"Control of Precipitation from Cumulus Clouds by Various Seeding Techniques,\" was published in Science, July 14, 1950.\n\nLangmuir triumphantly took William Lewis out to lunch: Langmuir recounts this conversation in his lab notebook.\n\n\"a graduate of the University of Chicago\": Schenectady Gazette, February 3, 1950.\n\nHe was, Kurt grumbled, turning to Yaddo again: Vonnegut to Miller Harris, February 16, 1950, in Wakefield, Letters, 32.\n\nFirst noticed by a scientist at Bell Labs: Harry Wexler wrote to the Bell Labs scientist M. W. Baldwin Jr. in November 1948 asking him for a copy of his original paper about the \"ghosts.\" LOC-HW.\n\nKurt made his main character: Vonnegut, \"Thanasphere,\" in Bagombo Snuff Box.\n\n\"a competent meterologist\": The New York Times, February 16, 1950.\n\nBut after the war: Eventually, Sverre Petterssen would write a corrective memoir that historians of meteorology consider authoritative: Weathering the Storm. James Rodger Fleming has written a succinct summary of the debate: \"Sverre Petterssen, the Bergen School, and the Forecasts for D-Day,\" Proceedings of the International Commission on the History of Meteorology 1, no. 1 (2004).\n\n\"using GE and the names Langmuir, Schaefer and Vonnegut\": Hammond to Langmuir, cc'ed to Bernard Vonnegut, February 3, 1950, MEG-BV.\n\na \"new way\" to make rain: \"New Way Is Found of Producing Rain,\" New York Times, December 24, 1949.\n\nThen Langmuir took him aside: Langmuir lab notebook.\n\nDr. John Herbert Hollomon invited Kurt: \"Dr. Hollomon at Home,\" a press release signed by Vonnegut with photos signed by Burns, March 1950, MIS-GE.\n\n\"A one-armed robot on wheels\": GE News Bureau press release initialed \"KV,\" March 14, 1950.\n\nKurt couldn't blame the inventors: \"To have a little clicking box make all the decisions wasn't a vicious thing to do. But it was too bad for the human beings who got their dignity from their jobs.\" Kurt Vonnegut, interviewed by David Standish, 1973, in Allen, Conversations, 93.\n\nLangmuir and Katharine Blodgett: \"Conversations on the Early Days of Cloud Seeding and the Development of the Atmospheric Sciences Research Center.\"\n\n\"Are machines smarter than ME?\": \"Mechanized Math Shark: The Computer,\" GE Monogram, March\u2013April 1950.\n\nWiener defined cybernetics: The definition was the subtitle of Cybernetics.\n\nKurt named his story: \"EPICAC\" was collected in Vonnegut, Welcome to the Monkey House.\n\nKnox returned it with notes: Burger to Vonnegut, undated letter filed with \"EPICAC\" drafts, LL-KV.\n\nThe first flight of New York's rainmakers: My account of New York City's rainmaking is compiled from the near-daily newspaper stories on it at the time, mostly in The New York Times, but also the New York Daily News, the Schenectady Gazette, The Kingston Daily Freeman, and the Catskill Mountain News. The story is also recounted briefly in Galusha, Liquid Assets.\n\n\"no vested property rights\": Slutsky v. City of New York, 97 N.Y.S.2d 238 (Sup. Ct. 1950).\n\n\"No,\" he had said: Kurt told this story, and the story of Langmuir's idea, repeatedly throughout his life, including in a 1974 interview with Joe David Bellamy and John Casey reprinted in Allen, Conversations, 161.\n\nAt the stroke of twelve: The image is reminiscent of the famous \"Doomsday Clock,\" visualizing how close the world is to precipitating Armageddon, printed on every issue of the Bulletin of the Atomic Scientists.\n\nStill, he thought it was the best thing: He says so in a May 19, 1950, letter to his Cornell friend Miller Harris, LL-MH.\n\n\"There is nothing moral\": General Office News, May 19, 1950.\n\nHe would write a novel: Vonnegut to Miller Harris, May 19, 1950, LL-MH.\n\n\"I have made rain\": \"Dr. Howell Yields: Yes, He Made Rain,\" The New York Times, August 8, 1950.\n\nWhen Irving found out, he was furious: Langmuir lab notebook.\n\n8. Out of the Blue\n\nThen he was taken: GE News Bureau press release, May 24, 1950. The prime minister's visit was also covered extensively by newspapers.\n\nHe figured that's what he'd been writing: \"I hope to build a reputation as a science-fiction writer. That's the pitch,\" he wrote to Miller Harris on February 28, 1950. Wakefield, Letters, 32.\n\n\"I haven't given it a thought\": Kurt described this talk many years later in an address to the American Physical Society in 1969. Vonnegut, Wampeters, Foma, and Granfalloons, 100. A detailed account of the talk ran in the Schenectady Gazette, March 8, 1950.\n\nHe wanted to write: \"I cheerfully ripped off the plot of Brave New World,\" Kurt declared in the Playboy interview. Vonnegut, Wampeters, Foma, and Granfalloons, 263.\n\nHe was going to bite the hand that fed him: \"I bit the hand that used to feed me,\" Kurt wrote in the introduction to Bagombo Snuff Box, 8.\n\n\"The first industrial revolution\": Wiener, Cybernetics, 37. I cite the 1948 edition that it is likely Kurt read.\n\nThe job was making him crazy: \"K. has got to quit before any such gigantic length of time,\" Jane wrote to Fred Rosenau around this time, after he suggested that Kurt's success might take years, \"because this idiotic job of his is such a drag on his spirit. It's making us both psychotic.\" Shields, And So It Goes, 112.\n\n\"the property of the client\": \"Weather Control and Augmented Potable Weather Supply,\" joint hearings on S. 5, S. 222, and S. 798 before subcommittees of the Committees on Interior and Insular Affairs, Interstate and Foreign Commerce, and Agriculture and Forestry, 82nd Congress (March 14, 15, 16, and 19, and April 5, 1951).\n\nA high school senior headed for Caltech: Langmuir lab notebook.\n\nHe had chosen to sleep in: Schaefer, \"Twenty Years at Langmuir University.\"\n\n\"The cold resolve deserted Kelly\": \"White King\" was eventually published as \"All the King's Horses.\" Vonnegut, Welcome to the Monkey House, 102.\n\nWhile they were in Gloucester: Mark Vonnegut identified this painting as having been done in Gloucester, and Edie Vonnegut generously let me see it. Kurt always liked clowns. Humor was one of the few things in the world in which he never lost faith. \"Historians of the future, in my opinion,\" he would write when old, \"will congratulate us on very little other than our clowning and our jazz.\" Vonnegut, Palm Sunday, 127.\n\n\"Many farmers, ranchers and civic-minded people\": Quotations from Bernie's speech appeared in \"Rainmaker Licenses Seen by GE Weather Scientist,\" Schenectady Gazette, August 26, 1950.\n\n\"He has been known to sit for half an hour\": Time, August 28, 1950.\n\n\"Ice-9,\" Kurt's story about a scientist: Kurt Vonnegut to Ben (Hitz?). The letter is an undated draft but mentions that it is written on the evening of Kurt and Jane's fifth wedding anniversary, hence August 31, 1950. LL-KV.\n\nThat was highway robbery: Vonnegut to Littauer and Max Wilkinson, draft letter, n.d., LL-KV. It's fortunate \"Ice-9\" was not sold to Astounding Science Fiction (as it was retitled in 1938); if it had been, Cat's Cradle would never have been written.\n\n\"I am a registered Democrat\": Vonnegut to Burger, October 31, 1950, in Wakefield, Letters, 37.\n\nKurt had been looking: An article in Scientific American two years earlier, \"The Army Ant,\" June 16, 1948, might have suggested the topic. It made an interesting connection between some of the more irrational behaviors of ants and of men. Kurt almost surely saw this piece: he read Scientific American regularly at GE as part of his job (Vonnegut, Wampeters, Foma, and Granfalloons, 84), and the echoes of this feature in his story are many. Kurt took notes on Wheeler, which are preserved in \"The Ants of Erz Gebirg\" file in LL-KV. It's notable that Wiener used ants as frequent examples in Cybernetics as well.\n\nLike \"Ice-9,\" the new story: All quotations are from \"The Petrified Ants,\" published posthumously in Vonnegut, Look at the Birdie. The published story is one of the many story drafts in LL-KV. It's impossible to tell if it is the last one.\n\nThe story is set in the Erzgebirge: As usual, the location has personal significance for Kurt: he had been marched into the Erzgebirge, half-starved, as the Red Army approached Dresden. It also demonstrates how attuned he was to political events: the Russians really did search for uranium in the region.\n\nJosef tells Peter that he mustn't speak: Josef's explicit instructions and Peter's seeing him as \"frail\" and compromised are in early manuscripts, then excised, then put back in. They are in the version of the story that was ultimately published in Vonnegut's posthumous collection of short fiction, Look at the Birdie.\n\nHe saw it as a chance: Byers, \"History of Weather Modification.\"\n\n\"It seems to me\": Reichelderfer to Charles Brooks, September 27, 1950, LOC-HW.\n\nHe had Weather Bureau charts: Vince Schaefer describes the presentation, calling it a \"spectacular demonstration,\" in \"Twenty Years at Langmuir University.\"\n\nBernie was back at work by then: \"Seven Day Periodicity in Weather in 1950,\" BAMS, March 1951. Letters between Reichelderfer and Langmuir and Robert G. Stone and Langmuir are in MEG-VS.\n\n\"It is hoped\": Copies of Bernie's letters and some of the replies are in MEG-BV.\n\nThe storm intensified that night: My information about the floods comes from the U.S. Geological Survey water-supply paper \"Floods of 1950\u201351 in the Catskill Mountain Region, New York\" (U.S. Government Printing Office, 1957), as well as local newspapers, The New York Times, and Galusha, Liquid Assets.\n\nThey disagreed on whether cloud seeding was sufficiently proved: Harry Wexler, trip report, December 8, 1950, LOC-HW.\n\nHe also suggested Kurt try: Burger to Vonnegut, September 13, 1950, LL-KV.\n\n9. Cold Fronts\n\nEventually, he would give up on it: Kurt never saw \"From Timid to Timbuktu\" in print; it was published only posthumously. Given that he doesn't seem to have shopped it around, it's reasonable to think he probably wouldn't have liked it being published.\n\nEventually, more than 130 upstate clients: Catskill Mountain News, March 23, 1951.\n\nHe was even asserting: Langmuir wrote a paper about Hurricane King that remains unpublished to this day. A typescript of the first part was preserved by Bernie; MEG-BV. A typescript of the second part is in MIS-GE. Irving continued to try to get people to take his conclusions about the hurricane seeding seriously up to his death, even mentioning it again when appearing on the Today show following a vicious hurricane in 1956.\n\nWhen a young meteorologist at an MIT symposium: The insulted party, Charles Hosler, was taken aside afterward by Henry Houghton, chair of the MIT meteorology department, who explained that Langmuir was so abrupt because he felt his whole career was defined by his cloud-seeding work. Hosler recounted the event for James Fleming, who repeats it in \"The Pathological History of Weather and Climate Modification,\" Historical Studies in the Physical and Biological Sciences 37, no. 1 (2006): 12.\n\nHarry Wexler spent Valentine's Day in Princeton: Harry Wexler, trip report, February 21, 1951, LOC-HW. The ENIAC logbook is in MIT-JC. Harper writes about this visit as well, though she never mentions the discussion of the periodicities, or any of the Project Cirrus work, in Weather by the Numbers.\n\nIt was spring: Kurt wrote a letter to Miller Harris on the back of a New Yorker rejection slip, saying that they would be in New York staying at the Algonquin on March 29\u201331. Vonnegut to Harris, February 26, 1951, LL-MH. Kurt and Knox exchanged letters about having had drinks with Jane afterward, NYPL-CC.\n\nBut what the hell?: In later years, Kurt himself had a slightly dismissive attitude toward much of his early magazine work. He included neither \"Mnemonics\" nor \"Little Drops of Water\" in the only collection of short fiction he ever put together himself, Welcome to the Monkey House. In 1999 he declared that \"no matter how clumsily I wrote when I was starting out, there were magazines that would publish such orangutans.\" \"Coda to My Career as a Writer for Periodicals,\" Bagombo Snuff Box, 349. He dedicated this collection to the memories of Littauer and Wilkinson, \"who taught me how to write.\"\n\nJust focus on the carpenter story: Knox called the Dresden story \"risky\" in a letter (jokingly) dated April 5, 1961, NYPL-CC.\n\nKurt invited Knox to come visit: Kurt told Knox he was selling his house in a letter on April 14, 1951; LL-KV. Knox congratulated Kurt on selling it at a profit in a letter in which he also commented on \"More Stately Mansions,\" May 22, 1951, LL-KV. Jane wrote to her mother that they moved to Provincetown expecting to roam the East Coast looking for a house that fall (September 15, 1951, Vonnegut family collection). So it's clear that Kurt and Jane decided to leave Alplaus and sold their house that spring, not in the fall, as Shields has it.\n\nCollier's intended to convey: Notes from the editorial meetings for Operation Eggnog, including many statements of intent and correspondence with a host of potential writers, are all in NYPL-CC.\n\nThe Atomic Energy Commission was ramping up: The Enewetak (then spelled Eniwetok) tests in April and May were known as Operation Greenhouse, and they were not secret. Details were released by the government in June. As was common, many newspapers ran a front-page story and photograph of the mushroom cloud. \"Tests Prove Gains in Hydrogen Bomb,\" New York Times, June 14, 1951.\n\n\"You say here\": \"Weather Control and Augmented Potable Weather Supply,\" joint hearings on S. 5, S. 222, and S. 798 before subcommittees of the Committees on Interior and Insular Affairs, Interstate and Foreign Commerce, and Agriculture and Forestry, 82nd Congress (March 14, 15, 16, and 19, and April 5, 1951).\n\nIn November, he had visited Schenectady: Langmuir lab notebook.\n\nIt was a PR disaster: Reichelderfer was still stewing about the fist pounding nearly a year later, noting in an internal memo on February 20, 1952, that the Weather Bureau had lost funding after the hearings and that it had been scrambling to improve its reputation ever since. LOC-HW.\n\nCondon had won the battle with HUAC: Even Irving knew to keep a low profile as far as HUAC was concerned. He had recently been asked by a lawyer to provide a character reference for another scientist who was being investigated, and he had insisted that his name be kept out of it. He would rather avoid publicity, he said. He didn't want to end up like Edward Condon. Langmuir to General Edward Greenbaum, February 2, 1951, LOC-IL.\n\nBut after the hearings ended: The Department of the Army's letter on behalf of the Department of Defense was added to \"Weather Control and Augmented Potable Water Supply,\" 34.\n\nHe was writing about what might be: The foreword to Player Piano declares, \"This is not a book about what is, but a book about what could be.\" All quotes are from Vonnegut, Player Piano.\n\nHe saw it as an American version: Kurt made this connection in an unfinished early description of the book. LL-KV.\n\nHis desire for the cat: The moment is eerily redolent of the many deaths on the electric fences of German death camps, a parallel Vonnegut does not push too hard. But it does hint, as do the early drafts of \"Mnemonics,\" at the deeper meaning of his mistrust of the Works and its world.\n\n\"Any labor that accepts the conditions\": Wiener, Cybernetics, 37.\n\nTen thousand farms: NOAA report, www.crh.noaa.gov\/mbrfc\/flood51.pdf.\n\nSenator Kem asked them to draft a statement: Langmuir lab notebook.\n\nMeanwhile, the governor of New Mexico: Roscoe Braham, interview, June 19\u201321, 2002, NCAR-AMS.\n\nIrving thought this was an excellent idea: Langmuir lab notebook.\n\nVincent went to the meeting instead: Langmuir lab notebook.\n\nNow the undersea warfare: R. W. Larson to Captain C. L. Murphy (head, Undersea Warfare Research, Office of Naval Research), June 25, 1951, MIS-GE.\n\nBesides, getting involved: Suits to Vonnegut, July 16, 1951, MIS-GE.\n\nDon't come to Provincetown: Jane Vonnegut to Mariah Cox, September 15, 1951, Vonnegut family collection.\n\nKurt would probably need to fly to New York: Edie Vonnegut, interview with the author, October 2013.\n\n10. Shifting Winds\n\nBernard entered the East Ballroom: The meeting was written up, with a photograph of all attendees, in Weatherwise 4, no. 6 (December 1951).\n\nThe only other serious researcher: Appalled that the panel was all \"rainmakers,\" Chief Reichelderfer pulled strings with the AMS president to get Thom added to the roster. Memo to Harry Wexler, September 20, 1951, LOC-HW.\n\nBernie was going to outline the construction: Program and abstracts ran in Bulletin of the AMS 32, no. 7 (September 1951).\n\nBut if Bernie wasn't: Lab notebook entries throughout July and August 1951 document Langmuir's frustration and his conversations with Bernard.\n\n\"Project Cirrus is supported by the public\": Bernard Vonnegut to Michael J. Ference Jr., October 3, 1951, MEG-BV.\n\nThat, Kurt knew, was why: Kurt wrote a sullen letter to Knox Burger about the whole affair the following April, declaring, \"Wild as my stories may be, I would never make World War III seem little more hazardous than and as interesting as an automobile trip from New York to Los Angeles in a Stutz Bearcat.\" As if that weren't enough, a sarcastic parody of the story's dialogue followed. Wakefield, Letters, 42. In a letter the same month to Harry Brague, Kurt complained that Collier's was growing so conservative his stories were no longer welcome there. Vonnegut to Brague, April 16, 1952, PU-CS.\n\nWhen he started the novel: Kurt's early outlines, notes, and drafts are in LL-KV.\n\nDr. Paul Proteus having quit: Vonnegut to Brague, October 18, 1951, PU-CS.\n\nThe loudspeaker at the Meadows: In one of the last pre-galley typescripts of Player Piano, Kurt cut out the team songs and replaced them with new ones that are even more similar to the actual fight songs from Camp GE in 1948. Because Vincent, Herb Hollomon, and Roger Hammond of the News Bureau all attended that summer, it seems clear that at some point during the revising of the manuscript, Kurt got his hands on one of their songbooks, but I could not confirm this. Photographs of the skit from that session also show a godlike figure attaching tinsel stars to the set sky and what appears to be a trial. This sounds exactly like the skit depicted at length in the novel. Photographs of Camp GE are in UP-LB. Vince's Camp GE songbook is in MEG-VS.\n\nLudwig von Neumann is so unlike John von Neumann: Kurt was very likely familiar with John von Neumann and his work. Von Neumann gave frequent talks at the same conferences as Bernie, and Project Cirrus received regular reports from the Institute for Advanced Study's Meteorology Project. Distribution list, IAS.\n\nWeather Bureau analysts were double-checking: According to his own handwritten notes, when he got wind of a meeting Langmuir scheduled with Henry Houghton of MIT to discuss the report he was writing, Harry called Hurd Willett, his mentor at MIT, and suggested that Willett attend the meeting and propose an alternative explanation for the periodicities. He then called Henry Houghton and arranged for him to see the Weather Bureau statistical data on the periodicities. LOC-HW. By the time Langmuir sat down with Houghton and Willett, they had been converted to the Weather Bureau's position. A baffled Irving called the meeting \"very unsatisfactory\" in his lab notebook entry of June 30, 1951.\n\nDoc Whitney's question, \"Are you having fun today?\": Vincent Schaefer writes about the changing feel of the lab in \"Twenty Years at Langmuir University.\"\n\nVisitors and colleagues: Many of the arrangements are reported in Irving's notebooks. Also Anthony Hall, \"A Laureate's Lake,\" Lake George Mirror, July 10, 2011.\n\n\"Our papers have been making\": Wiener, Human Use of Human Beings, 210.\n\nOn perhaps the last innocently joyous night: Vonnegut, \"Rover Boys,\" LL-KV.\n\nBut even now, before that triumph: Harry Wexler, trip report, May 7\u201310, 1952, LOC-HW.\n\n\"A tidal wave of computational power\": Julian Bigelow, the chief designer for von Neumann's computer, quoted in George Dyson, Turing's Cathedral, 153.\n\nGuy Suits had asked Bernie: Bernie recounted this conversation, and his feelings about GE then, in an interview with Earl Droessler, May 9, 1993, NCAR-AMS.\n\nHe was corresponding with Bill Hubert: Hubert to Vonnegut, June 13, 1952, MEG-BV.\n\nIt's unclear what set Wiener off: Conway and Siegelman, Dark Hero of the Information Age, 287.\n\n\"he cannot with impunity\": Wiener to Hope English, July 17, 1952, MIT-NW. Quoted in Conway and Siegelman, Dark Hero of the Information Age, 288.\n\nHe apologized for having innocently given offense: Vonnegut to Wiener, July 26, 1952, MIT-NW.\n\nHe pulled them from his personal store: Once, over lunch in Tulsa, a magazine editor told me he had always been baffled by Vonnegut's location of the demonology institute in Verdigris, Oklahoma, in \"Armageddon in Retrospect.\" Instinctively, I responded that the place must mean something to him. More than a year later, when I first read the manuscript \"The Rover Boys,\" it became clear what that was: on his road trip in 1939, he and his two friends stayed not far from Verdigris on the Woolaroc Ranch. While enjoying that paradise for sixteen-year-old boys, they heard on the radio that Hitler had invaded Poland. His own personal demonology began there.\n\nGeorge M. Helmholtz: Helmholtz Watson is also the frustrated writer character in Brave New World; all citizens in Huxley's imagined world have names taken from historical figures because real families no longer exist.\n\nNot one bookstore in GE's company town: Norman Snow to Henry Hohns, memo, September 3, 1952, PU-CS.\n\nBernie went down to the beach: Bernard Vonnegut, abstract of \"A Possible Mechanism for the Formation of Thunderstorm Electricity,\" BAMS 34 (1953), and \"Giant Electrical Storms,\" in Recent Advances in Atmospheric Electricity, ed. L. G. Smith (Pergamon, 1955). Meteorologists were at first skeptical of Bernie's estimate, but later improvements in wind-speed detection established that his estimates were close to correct. I am grateful to Scott Vonnegut for drawing my attention to the Worcester Twister as a formative event for his father.\n\n\"I believe the best way\": Bernie wrote to W. E. Williamson, clerk of the Committee on Interstate and Foreign Commerce, on March 12, 1958, asking to testify; his statement is also in his files, MEG-BV.\n\nThat same year, Commander William Kotsch: Kotsch to Vonnegut, September 8, 1958, MEG-BV.\n\n\"I'm [just] trying to find out\": Quoted in Michael Lopez, \"Talk Soup,\" Albany Times Union, April 13, 1997.\n\nHe had even purchased a television: Kurt's teleplays are in the unprocessed files, LL-KV. They include an adaptation of Player Piano and a teleplay called \"Slaughterhouse 5\" that is nothing like the novel. In a letter to Harry Brague, February 7, 1954, Kurt complains of having \"squandered all my time on short stories and scripts that haven't sold.\" Wakefield, Letters, 52. He writes to Knox Burger about having bought a television on May 11, 1954. Wakefield, Letters, 57.\n\n\"What is God? What is love?\": Vonnegut, Cat's Cradle, 55. The Clare Boothe Luce story is told in Rosenfeld, Quintessence, 321.\n\n\"Let us remember how good it was once here\": Dostoevsky, The Brothers Karamazov, 895. This is the Constance Garnett translation that Kurt would have read. Later in the same speech Alyosha says, in another passage demonstrating the book's lifelong impression on Kurt, \"Let us be, first and above all, kind.\"\n\nThanks to the way the human mind lets us: \"Everything about that lake was imprinted on my mind when it held so little and was so eager for information, it will be my lake as long as I live,\" he wrote in Architectural Digest. \"I have no wish to visit it, for I have it all right here.\" Reprinted in Vonnegut, Fates Worse Than Death, 50.\n\nEpilogue: Rainbow's End\n\nHe had handed the letter to Kurt: Kurt set this scene in a typed explanation dated April 13, 1997, included with Uncle Alex's letter in his files, LL-KV.\n\nCape Cod wasn't quite ready: Shields, And So It Goes, 142\u201344. Kurt later speculated in jest that the dealership failure had cost him the Nobel Prize. \"Have I Got a Car for You!,\" In These Times, November 24, 2004.\n\nHe even designed and tried to market: Notes and samples of the game are in LL-KV.\n\nHe was working on it when he and Jane: The smallest child was subsequently adopted by another relative in a traumatic family event.\n\nThey were too focused on the fasinating things: McCartan, Kurt Vonnegut, 148.\n\nBernard was horrified: Kurt discusses Bernie's reaction to the news in ibid. Bernard's sons have confirmed this. \"He was very much against it,\" Scott Vonnegut said. \"He was very, very much against the war... He wanted it used for agriculture, or for snowpack, for water.\" Interview with the author, July 2013.\n\n\"My brother knew early on\": He gets the location wrong, but a colleague Bernie worked with while at Hartford-Empire in Connecticut was based in Butler, Pennsylvania. The address was reprinted in Vonnegut, Fates Worse Than Death, 117.\n\n\"one who declines to work on weapons\": Walter Sullivan, \"Strike to Protest 'Misuse' of Science,\" New York Times, February 6, 1969.\n\n\"I told Jane that this boy\": Vonnegut, Timequake, 135.\n\nFive yielded inconclusive results: The Weather Bureau seeded clouds in Oregon and Washington with dry ice on a randomized basis with inconclusive results. Scientists at the University of Chicago, funded by the Department of Defense, seeded warm clouds in the Midwest and the Caribbean with water and dry ice, getting a positive result too small to be statistically significant. The Navy collaborated with NYU on Project Scud, an attempt to modify the development of nineteen large-scale extratropical cyclonic systems utilizing dry ice from airplanes and seventeen silver iodide generators scattered from Florida to New York. The Air Force collaborated with the Stanford Research Institute to study the physics of ice fogs. The Army, with Arthur D. Little, studied methods of dispelling warm fogs and stratus. And the Army Signal Corps conducted an extensive flight program to study seeding of supercooled stratus clouds with both dry ice and silver iodide, the one of these projects to get a positive result. The results were ultimately declassified and published as Cloud and Weather Modification: A Group of Field Experiments. Meteorological Monographs. Vol. 2, no. 11 (AMS, 1957). See also Byers, \"History of Weather Modification.\"\n\nOfficers from the Office of Naval Research: Irving guessed, correctly, what each of these projects was. Langmuir lab notebook, September 14, 1953.\n\nA rigorous five-year randomized study: Daniel Breed, \"Design and Preliminary Results of the Wyoming Weather Modification Pilot Program Randomized Seeding Experiment,\" Research Applications Laboratory, National Center for Atmospheric Research. For an overview of the program, see <http:\/\/ral.ucar.edu\/projects\/wyoming\/>.\n\n\"one of those tantalizing things\": Oral history with Roscoe Braham, 2002, NCAR-AMS.\n\n\"If Langmuir actually influenced the weather\": This meteorologist remained anonymous in the article, but his generosity toward Langmuir rules out anyone at the Weather Bureau. It could have been Wallace Howell, or even Vince or Bernie. \"Tomorrow's Weather,\" Fortune, May 1953.\n\n\"Perhaps if he had comprehended fully\": \"History of Weather Modification,\" in Hess, Weather and Climate Modification.\n\n\"is not definitely determinate\": Langmuir, Cloud Nucleation, 456\u201357. Von Neumann's key meteorologist, Jule Charney, later acknowledged that \"in some fundamental way Wiener was probably right... Wiener I think anticipated the unpredictability of the atmosphere that Lorenz later formulated rigorously.\" George Platzman, Conversations with Jule Charney (NCAR Technical Notes, 1987).\n\nThe computer did not turn out to be Laplace's demon: See, for instance, theories about the \"singularity\" by the techno-utopian and Google director of engineering Raymond Kurzweil. Kurzweil credits John von Neumann with the original idea in his foreword to von Neumann's unfinished lecture series, published as The Computer and the Brain (Yale University Press, 2012).\n\n\"we are releasing huge quantities of carbon dioxide\": Typed and handwritten lecture notes, LOC-HW.\n\n\"Project Cirrus's investigations of ways\": Epilogue written by Bernard in Havens, Jiusto, and Vonnegut, Early History of Cloud Seeding.\n\nThat old America: Loree Rackstraw recounts a letter in which Kurt says Timequake \"has to do with the disappearance of the America I tried to write for.\" Love as Always, Kurt (Da Capo, 2009), 182. Kurt writes something similar to Marc Leeds in 1995, describing the mood of his generation as \"wry disappointment with what the world has actually become, so inhospitable and snide. Once the Great Depression and the Second World War were over, we planned to build a Garden of Eden here.\" Wakefield, Letters, 364.\n\nIn his novel Gal\u00e1pagos: Interestingly, this idea too was proposed in Cybernetics. Wiener spends part of a late chapter considering the possibility that the human brain had grown too large to be efficient anymore and that humans might \"be facing one of those limitations of nature in which highly specialized organs reach a level of declining efficiency and ultimately lead to the extinction of the species.\" Wiener, Cybernetics, 180. This is, in a nutshell, the premise of Gal\u00e1pagos.\n\n\"Physicists may already know enough\": Bernard Vonnegut, \"Adventures in Fluid Flow: Generating Interesting Dendritic Patterns,\" Leonardo 31, no. 3 (1998): 207.\n\nKurt's letter back: Describing Bernie's question and quoting from his own reply in Timequake, Kurt says, \"I was pleased to reply with an epistle which was frankly vengeful, since he and Father had screwed me out of a liberal arts college education.\" Vonnegut, Timequake, 167. But of course the Kurt Vonnegut in Vonnegut's fiction was a satirical version. The tone of the real Kurt Vonnegut's letter is ultimately not vengeful at all: it is unmistakably loving.\n\n\"If the superpowers decide to duke it out\": Kurt told this story a number of times, including in an interview published in Stop Smiling Magazine, August 2006.\n\n\"washed away a lot of privies\": Kurt Vonnegut, \"Bernard Vonnegut: The Rainmaker,\" New York Times Magazine, January 4, 1998.\nBibliography\n\nAllen, William Rodney, ed. Conversations with Kurt Vonnegut. University Press of Mississippi, 1988.\n\nAshmore, Harry. Unseasonable Truths: The Life of Robert Maynard Hutchins. Little, Brown, 1989.\n\nBates, Charles, and John F. Fuller. America's Weather Warriors, 1914\u20131985. Texas A&M University Press, 1986.\n\nBirr, Kendall. Pioneering in Industrial Research: The Story of the General Electric Research Laboratory. Public Affairs, 1957.\n\nBoyer, Paul. By the Bomb's Early Light: American Thought and Culture at the Dawn of the Atomic Age. University of North Carolina Press, 1984.\n\nBrands, H. W. American Dreams: The United States Since 1945. Penguin, 2010.\n\nChew, Joe. Storms Above the Desert: Atmospheric Research in New Mexico, 1935\u20131985. University of New Mexico Press, 1987.\n\nCoe, Jerome. Unlikely Victory: How G.E. Succeeded in the Chemical Industry. Wiley, 2000.\n\nConway, Flo, and Jim Siegelman. Dark Hero of the Information Age: In Search of Norbert Wiener, the Father of Cybernetics. Basic Books, 2006.\n\nCotton, William R., and Roger A. Pielke Sr. Human Impacts on Weather and Climate. Cambridge University Press, 2007.\n\nCox, John D. Storm Watchers: The Turbulent History of Weather Prediction from Franklin's Kite to El Ni\u00f1o. John Wiley & Sons, 2002.\n\nDickstein, Morris. Leopards in the Temple: The Transformation of American Fiction, 1945\u20131970. Harvard University Press, 2002.\n\nDonner, Leo, Wayne Schubert, and Richard Somerville. The Development of Atmospheric General Circulation Models. 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Liquid Assets: A History of the New York City Water Supply. Harbor Hill Books, 2002.\n\nGleick, James. Chaos: Making a New Science. Penguin, 1987.\n\nHammond, John Winthrop. Men and Volts: The Story of General Electric. J. B. Lippincott, 1941.\n\nHarper, Kristin. Weather by the Numbers: The Genesis of Modern Meteorology. MIT Press, 2012.\n\nHavens, Barrington, James E. Jiusto, and Bernard Vonnegut. Early History of Cloud Seeding. Langmuir Laboratory, Atmospheric Sciences Research Center, and GE, 1978.\n\nHawkins, Lawrence. \"The Story of G.E. Research.\" Pamphlet published by GE. MIS-GE.\n\nHeims, Steve J. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death. MIT Press, 1980.\n\nHess, Wilmot, ed. Weather and Climate Modification. John Wiley & Sons, 1974.\n\nKershaw, Alex. The Longest Winter: The Battle of the Bulge and the Epic Story of World War II's Most Decorated Platoon. Da Capo Press, 2005.\n\nKlinkowitz, Jerome. Kurt Vonnegut's America. University of South Carolina Press, 2009.\n\nKlinkowitz, Jerome, and John Somer, eds. The Vonnegut Statement: Original Essays on the Life and Work of Kurt Vonnegut Jr. Dell, 1973.\n\nKrementz, Jill, ed. Happy Birthday, Kurt Vonnegut. Delacorte Press, 1982.\n\nLangmuir, Irving. Atmospheric Phenomena. Vol. 10 of Collected Works. Pergamon Press, 1962.\n\n______. Cloud Nucleation. Vol. 11 of Collected Works. Pergamon Press, 1962.\n\nLaskin, David. Braving the Elements: The Stormy History of American Weather. Doubleday, 1995.\n\nLilienthal, David. The Atomic Energy Years. Vol. 2 of The Journals of David Lilienthal. Harper & Row, 1964.\n\nLindzen, Richard S., Edward N. Lorenz, and George W. Platzman, eds. The Atmosphere, a Challenge: The Science of Jule Gregory Charney. American Meteorological Society, 1990.\n\nLingeman, Richard. The Noir Forties: The American People from Victory to Cold War. Nation Books, 2012.\n\nLynch, Peter. \"The ENIAC Forecasts: A Re-creation.\" Bulletin of the American Meteorological Society, January 2008.\n\nMacrae, Norman. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. American Mathematical Society, 2000.\n\nMariner, Rosemary, and G. Kurt Piehler, eds. The Atomic Bomb and American Society: New Perspectives. University of Tennessee Press, 2009.\n\nMcCartan, Tom, ed. Kurt Vonnegut: The Last Interview and Other Conversations. Melville House, 2011.\n\nMcQuade, Molly, ed. An Unsentimental Education: Writers and Chicago. University of Chicago Press, 1995.\n\nMeisler, Stanley. United Nations: The First Fifty Years. Atlantic Monthly Press, 1995.\n\nMergen, Bernard. Weather Matters: An American Cultural History Since 1900. University Press of Kansas, 2008.\n\nMiller, John A. Men and Volts at War: The Story of General Electric in World War II. Bantam, 1948.\n\nMonmonier, Mark. Air Apparent: How Meteorologists Learned to Map, Predict, and Dramatize the Weather. University of Chicago Press, 1999.\n\nNebeker, Frederik. Calculating the Weather: Meteorology in the Twentieth Century. Academic Press, 1995.\n\nNewton, Roger G. From Clockwork to Crapshoot: A History of Physics. Belknap Press, 2007.\n\nNye, David. Image Worlds: Corporate Identities at General Electric, 1890\u20131930. MIT Press, 1985.\n\nPetterssen, Sverre. Weathering the Storm: Sverre Petterssen, the D-Day Forecast, and the Rise of Modern Meteorology. Edited by James Rodger Fleming. AMS Historical Monographs, 2001.\n\nPhillips-Fein, Kimberly. \"American Counterrevolutionary: Lemuel Ricketts Boulware and General Electric, 1950\u20131960.\" In American Capitalism: Social Thought and Political Economy in the Twentieth Century, edited by Nelson Lichtenstein. University of Pennsylvania Press, 2006.\n\nRhodes, Richard. The Making of the Atomic Bomb. Simon & Schuster, 1986.\n\nRosenfeld, Albert. The Quintessence of Irving Langmuir. Pergamon Press, 1966.\n\nSchaefer, Vincent. A Field Guide to the Atmosphere. Peterson Field Guides. Houghton Mifflin, 1981.\n\n______. Final Report Project Cirrus Part 1: Laboratory, Field, and Flight Experiments. GE Research Laboratory Report No. RL-785, 1953.\n\n______. \"Twenty Years at Langmuir University.\" Unpublished autobiographical manuscript. MEG-VS.\n\nSchatz, Ronald W. The Electrical Workers: A History of Labor at General Electric and Westinghouse. University of Illinois Press, 1983.\n\nSheets, Bob, and Jack Williams. Hurricane Watch: Forecasting the Deadliest Storms on Earth. Vintage Books, 2001.\n\nShields, Charles. And So It Goes: Kurt Vonnegut, a Life. Henry Holt, 2011.\n\nSmith, Alice Kimball. A Peril and a Hope: The Scientists' Movement in America, 1945\u201347. University of Chicago Press, 1965.\n\nSumner, Gregory D. Unstuck in Time: A Journey Through Kurt Vonnegut's Life and Novels. Seven Stories, 2011.\n\nSzpek, Ervin, Jr., Frank Idzikowski, and Heidi Szpek. Shadows of Slaughterhouse Five: Recollections and Reflections of the Ex-POWs of Schlachthof F\u00fcnf, Dresden, Germany. Self-published, 2008.\n\nTeller, Edward. Memoirs. With Judith Shoolery. Perseus, 2001.\n\nVonnegut, Kurt. Armageddon in Retrospect. Berkley Books, 2008.\n\n______. Bagombo Snuff Box. 1999; Berkley Books, 2000.\n\n______. Breakfast of Champions. 1973; Dial Press, 2011.\n\n______. Cat's Cradle. 1963; Dial Press, 2006.\n\n______. Fates Worse Than Death. Berkley Books, 1991.\n\n______. Gal\u00e1pagos. 1985; Dial Press, 1999.\n\n______. Jailbird. 1979; Dial Press, 2011.\n\n______. Look at the Birdie. 2009; Dial Press, 2010.\n\n______. A Man Without a Country. Seven Stories, 2005.\n\n______. Mother Night. 1961; Dial Press, 2009.\n\n______. Palm Sunday. 1981; Dial Press, 2011.\n\n______. Player Piano. 1952; Dial Press, 2006.\n\n______. The Sirens of Titan. 1959; Dial Press, 2009.\n\n______. Slapstick. 1976; Dial Press, 2010.\n\n______. Slaughterhouse-Five. Dell, 1969.\n\n______. Timequake. Berkley Books, 1997.\n\n______. Wampeters, Foma, and Granfalloons. 1974; Dial Press, 1999.\n\n______. We Are What We Pretend to Be: First and Last Works. Vanguard, 2012.\n\n______. Welcome to the Monkey House. 1968; Dial Press, 2006.\n\n______. While Mortals Sleep: Unpublished Short Fiction. 2011; Dial Press, 2012.\n\nVonnegut, Mark. The Eden Express. Dell, 1975.\n\nWakefield, Dan. New York in the Fifties. Houghton Mifflin, 1992.\n\n______, ed. Kurt Vonnegut: Letters. Delacorte, 2012.\n\nWang, Jessica. American Science in an Age of Anxiety: Scientists, Anticommunism, and the Cold War. University of North Carolina Press, 1999.\n\nWiener, Norbert. Cybernetics; or, Control and Communication in the Animal and the Machine. John Wiley & Sons, 1948.\n\n______. The Human Use of Human Beings. Houghton Mifflin, 1950.\n\nWise, George. The G.E. Story. Unpublished manuscript history of GE. MEG-GW.\n\n______. Willis R. Whitney, General Electric, and the Origins of U.S. Industrial Research. Columbia University Press, 1985.\n\nWittner, Lawrence. One World or None. Vol. 1 of The Struggle Against the Bomb. Stanford University Press, 1993.\n\nYarmolinsky, Jane Vonnegut. Angels Without Wings: A Courageous Family's Triumph over Tragedy. Houghton Mifflin, 1987.\nAcknowledgments\n\nThis book is a history. I have stuck to documented and verifiable fact, reconstructing the day-to-day activities of historical figures through published works, interviews, and archival sources. Where there is dialogue in quotation marks, it is because someone, somewhere, transcribed it. Some of my sources are Kurt Vonnegut's stories and novels. His entire oeuvre\u2014short stories, novels, essays, lectures\u2014can be read on one level as an autobiographical collage, a memoir unstuck in time. However, I have only attributed to him feelings he claims for himself, the author, not feelings or thoughts he ascribes to his characters.\n\nAs for the scientists in this book, I have found their published papers surprisingly helpful in developing an understanding of who they were. Their lab notebooks provided data about their activities but also conveyed\u2014through handwriting, underlining, and asides\u2014when they felt excited or rushed or cautious. I was also fortunate to find many oral histories, interview transcripts, and videos to help me get to know the people behind this story. All of these are listed in my sources.\n\nPerhaps most essential in developing a feeling for these wonderfully complex characters was the generous and heartfelt assistance of the Vonnegut-Adams family, as nice a tribe as one could imagine. In particular, Scott Vonnegut, Edie Vonnegut, Kurt Vonnegut III, and Mark Vonnegut corresponded with me, met with me, shared family photographs and documents, and exchanged ongoing e-mails to help me understand the facts and the people behind them. It is not easy to share one's family with the public, and they have done so with kindness and grace. I also owe a huge debt to Bernard Vonnegut's esteemed colleagues Sally Marsh and Duncan Blanchard for spending lavish amounts of time with me to help me understand a man they both knew so well. Bernard's student Tony Grainger not only talked to me about his mentor but did his best to explain cloud physics to me. I am indebted to the meteorologists Arlen Huggins, Jeff Tilley, and Don Griffin for helping me get a grip on weather modification and what it can do. I am grateful to the Weather Modification Association for allowing me to attend its annual meeting and plague the weather modifiers with endless questions. Dan Wakefield, a friend of Kurt Vonnegut's, was an especially delightful and informative correspondent. And I was thrilled when Freeman Dyson turned up at an early talk on this project. He gave me an inside look at the Institute for Advanced Study's Meteorology Project and the people he knew there as a young upstart.\n\nThis book would not have been possible without archives and the dedicated professionals who make them accessible. Perhaps most central were the papers of Bernard Vonnegut and Vincent Schaefer at the M. E. Grenander Special Collections at the SUNY Albany Science Library. Jodi Boyle and Geoffrey Williams were invaluable guides there. Equally critical were the manuscripts and papers of Kurt Vonnegut at the Lilly Library, Indiana University, Bloomington. Special thanks to the Lilly Library for a Helm Fellowship to help me spend sufficient time with its amazing collection, and to Cherry Williams, Joel Silver, Craig Simpson, and the staff of its beautiful reading room.\n\nAnother invaluable resource was the General Electric Archives at the Museum of Innovation and Science (formerly the Schenectady Museum) in Schenectady. I am grateful to the curator Chris Hunter for helping me find things in his treasure trove of GE materials. Visiting the manuscripts room at the Library of Congress, as I did to access the papers of Irving Langmuir, Vannevar Bush, Francis Reichelderfer, and Harry Wexler, makes one proud to be a taxpayer. Jennifer Brathovde, in particular, went above and beyond in an effort to help me make the most of my time there.\n\nThe Manuscripts and Archives Division of the Stephen A. Schwarzman Building of the New York Public Library provided access to the Crowell-Collier files, including the separate Operation Eggnog files, and the New Yorker files. Princeton University's Rare Books and Special Collections Department allowed me to use the Scribner files. Special thanks to Nora Murphy at the MIT Institute Archives and Special Collections, home to the papers of Jule Charney and Norbert Wiener, for helping me on-site and for continuing to help me remotely. The Grems-Doolittle Library of the Schenectady County Historical Society was helpful in understanding the history of the Electric City, and the Indiana Historical Society was essential in finding newspaper resources. I must also thank Kate Legg at the National Center for Atmospheric Research Archives at the University Corporation for Atmospheric Research, Julia Whitehead at the Kurt Vonnegut Memorial Library in Indianapolis, Christine Di Bella and Erica Mosner at the Shelby White and Leon Levy Archives Center at the Institute for Advanced Study, Michael Miller at the American Philosophical Society, Nancy Shawcross at University of Pennsylvania Special Collections, Chela Scott Weber at NYU's Tamiment Library, Deborah Douglas at the MIT Museum, and Lee Hiltzik, who directed me to many things I might not otherwise have found at the Rockefeller Archive Center.\n\nWriting this book required access to local newspaper archives in a variety of cities. In many cases, this involved using microform archives held by public libraries or universities and retrieved via interlibrary loan at the New York Public Library. This is one of the many special features of a strictly research library that makes maintaining the integrity of the Stephen A. Schwarzman Building as a research library absolutely essential to people like me.\n\nI completed the first draft of this book while living in Portland, Oregon, as Tin House\/Portland State University writer in residence. My thanks to the PSU faculty and the editorial staff of Tin House for a perfect place to work and for listening to endless Vonnegutian ramblings and to my MFA students there for making me think hard about what nonfiction is. This book has also benefited from the unstinting intelligence of my editor at FSG, Ileene Smith, and the wisdom and enthusiasm of my agent, Jin Auh. And as always, I could not have done it without the support, love, and understanding of Robert Brown, who always comes along for the ride, no matter how bumpy. God bless him.\nIndex\n\nThe index that appeared in the print version of this title does not match the pages in your eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.\n\nAberdeen Proving Ground\n\nAbsolute Weapon, The (Brodie)\n\nAdams, Alice Vonnegut; death of\n\nAdams, Jim\n\nAdams, Kurt\n\nAdams, Steve\n\nAdirondack Record\u2014Elizabethtown Post\n\nAdirondacks\n\nAdvisory Committee on Weather Control\n\nAgriculture Department, U.S.\n\nAir Force, U.S.; see also Army Air Forces, U.S.\n\nAir Force Weather Service\n\nair mass analysis\n\nairplanes, icing up of\n\nair pollution\n\nAir Transport Command airfield (New York)\n\nAir Weather Service\n\nAlbany, N.Y.\n\nAlbany Times-Union\n\nAlbuquerque, N.Mex.\n\nAlexander, James\n\nAlgonquin Hotel\n\nAli Khan, Liaquat\n\nAllies\n\nAlplaus, N.Y.\n\nAlplaus Creek\n\nAlplaus Methodist Church\n\nAmerican Association for the Advancement of Science\n\nAmerican Association of Physics Teachers\n\nAmerican Association of University Women\n\nAmerican Chemical Society\n\nAmerican Magazine\n\nAmerican Mercury\n\nAmerican Meteorological Society (AMS); nonmilitary uses of Project Cirrus envisioned by\n\nAmerican Museum of Natural History\n\nAmerican Philosophical Society\n\nAmerican Physical Society\n\nAmerican-Soviet Science Society\n\nAmerican Steel and Wire plant\n\nAnderson, Clinton\n\nAngeles National Forest\n\nAngoff, Charles\n\nAnthony, Edward\n\nanticommunism\n\nAnts: Their Structure, Development and Behavior (Wheeler)\n\nAppalachians\n\nArgosy\n\nArizona\n\nArkville, N.Y.\n\nArmy, U.S.; Chemical Warfare Service of\n\nArmy Air Forces, U.S.; Air Weather Service of; see also Air Force, U.S.\n\nArmy Signal Corps, U.S.\n\nArmy Specialized Training Program (ASTP)\n\nart, science and\n\nArthur D. Little; BV at\n\nAshokan Reservoir\n\nAssociated Press\n\nAssociation Island\n\nAstounding Stories\n\nAthenaeum\n\nAtlantic\n\nAtlantic Ocean\n\nAtmospheric Sciences Research Center\n\natomic arms race\n\natomic bomb; Soviet detonation of\n\natomic energy, as source of hope and fear\n\nAtomic Energy Commission, U.S.\n\nAuschwitz\n\nAustin, Warren\n\nautomation, labor force and\n\nAutumn Fog (Herbstnebel)\n\nAviation Cadet School\n\nBackus, Frank\n\nBantam (publisher)\n\nBarnhouse, Professor (char.)\n\nBartlesville, Okla.\n\nBarzov, Major (char.)\n\nBeethoven, Ludwig van\n\nBell Labs\n\nBellow, Saul\n\nBeralikur (rowboat)\n\nBergen-Belsen\n\nBergsonian time\n\nBerkeley, University of California at, Hitchcock Lectures of\n\nBerkshire Museum\n\nBerkshires\n\nBerlin\n\nBerlin Airlift\n\nBig Tujunga Canyon\n\nBikini Atoll\n\nBlizzard Bowl\n\nBlodgett, Katharine\n\nBobbs-Merrill (publisher)\n\nBody by Fisher\n\nBokonon (char.)\n\nBokononism\n\nBolton Landing\n\nBoston, Mass.\n\nBoston Globe\n\nBoston Herald\n\nBoulware, Lemuel\n\nBoulwarism\n\nBrague, Harry\n\nBraham, Roscoe\n\nbrain, computers vs.\n\nBratpuhr, shah of (char.)\n\nBraun\n\nBrave New World (Huxley)\n\nBreed, Asa (char.)\n\nBridgman, Percy\n\nBrodie, Bernard\n\nBrooklyn Dodgers\n\nBrothers Karamazov, The (Dostoyevsky); see also Dostoyevsky, Fyodor\n\nBuffalo, N.Y.\n\nBulge, Battle of the; KV in\n\nBulletin of the American Meteorological Society\n\nBulletin of the Atomic Scientists\n\nBureau of Reclamation\n\nBureau of Standards\n\nBurger, Knox\n\nBurns, George\n\nBurns, Jimmy\n\nBush, Vannevar\n\nByers, Horace\n\nCaesar, Julius\n\nCalifornia; cloud seeding in; forest fires in\n\nCaltech\n\nCambridge, Mass.\n\nCambridge University\n\nCamp Atterbury\n\nCamp General Electric\n\nCanada\n\n\"Can We Survive Technology\" (von Neumann)\n\nCape Cod, Mass.\n\ncarbon dioxide\n\nCaribbean\n\nCarl's (store)\n\nCarnegie Institute of Technology\n\nCarney, Stephen\n\nCaroline Islands\n\nCase, Francis\n\ncat's cradle\n\nCatskill Mountains\n\nCavender, Charles\n\nCentral Park Zoo\n\nChambers, D. E.\n\nchaos theory\n\nCharleston, S.C.\n\nCharney, Jule\n\nChatham, Mass.\n\nChekhov, Anton\n\nchemical supply companies\n\nChemical Warfare Service\n\nChicago, Ill.; Committee to Draft a World Constitution in\n\nChicago, University of; conference on atomic energy at; intellectual life of; Office of Inquiry into the Social Aspects of Atomic Energy at\n\nChicago Tribune\n\nChile\n\nChina\n\nchlorofluorocarbons\n\nChristian, Linda\n\nCIA\n\nCity News Bureau\n\nClark, Edward\n\nCleveland, Ohio\n\nclockwork universe\n\nCloud Physics Project\n\nclouds, physics of\n\ncloud seeding; commercial; ethical implications of; flooding and; government regulation of; hurricanes and; military experiments with; New York City program of; renewed interest in; weaponizing of\n\ncloud seeding, GE experiments in; liability issues in; media accounts of; military applications of; military control of flights in; see also Project Cirrus\n\nCold Spring, N.Y.\n\nCold War\n\nCold War, The (Lippmann)\n\nCollier's; \"World War III\" issue of\n\nColorado\n\nColorado A&M\n\nColumbia University; Butler Library at; School of Mines at\n\nCommerce Department, U.S.\n\ncommercial rainmaking\n\nCommittee to Draft a World Constitution\n\nCommunist Party\n\nCommunity House hospital\n\nCompton, Karl\n\ncomputers; brain vs.; singularity and; weather forecasting and; weather modeling on\n\nCondon, Edward\n\nCongress, U.S.\n\nConnecticut\n\nConquest of Gaul (Caesar)\n\nconvergent phenomena\n\nCordiner, \"Razor\" Ralph\n\nCordon, Guy\n\nCoriolis effect\n\nCornell Daily Sun\n\nCornell University\n\nCornell Widow, The\n\nCorning, Erastus\n\nCoronet\n\nCosmopolitan\n\nCousins, Norman\n\nCox, Jane Marie, see Vonnegut, Jane Marie Cox\n\nCrown Island, Lake George\n\ncultural anthropology, KV's studies in\n\nculture, constant evolution of\n\nCurtis Hotel\n\ncybernetics\n\nCybernetics (Wiener)\n\nCzechoslovakia\n\nDachau\n\nDaily Worker\n\nDale, Franklin (char.)\n\nDarkness at Noon (Koestler)\n\nDayton, Ohio\n\nDayton Daily News\n\nD-Day\n\nDefense Department, U.S.; Research and Development Board of; weather control and\n\nDelaware\n\nDemocrats\n\nDerby, Edgar (char.)\n\nDesert Research Institute\n\nDeutsche Haus, Das, see Athenaeum\n\nDewey, Thomas\n\ndivergent phenomena\n\nDodo, The (Swarthmore literary magazine)\n\nDonora smog\n\nDonora Zinc Works\n\nDostoyevsky, Fyodor; see also Brothers Karamazov, The\n\nDotson, James V.\n\nDoubleday (publisher)\n\nDow\n\nDresden; Allied firebombing of\n\nDushman, Saul\n\nDyson, Freeman\n\nEast Germany\n\nEconomic Development Commission, New Mexico\n\nEDVAC\n\nEgypt\n\nEiffel Tower\n\nEinstein, Albert\n\nEisenhower, Dwight D.\n\nelectrical charge, in thunderstorms\n\nElizabethtown, N.Y.\n\nElliott, Robert\n\nElmer & Amend\n\nEngland\n\nENIAC\n\nEPICAC (fictional computer)\n\nErie, Lake\n\nEspy, James Pollard\n\nEsquire\n\nEurope\n\nEverglades\n\nFAA\n\nFahnestock State Park\n\nFarm Bureau Federation\n\nFaulkner, William\n\nFBI; GE Research Lab dossiers of\n\nFederation of American Scientists\n\nFerence, Michael\n\nFermi, Enrico\n\nFindeisen, Walter\n\nFinnerty, Ed (char.)\n\nFirestone\n\nFirst Friends Church\n\nFisher, Jo Ann\n\nFisher, John\n\nFisher Scientific\n\nFlorez, Luis de\n\nFlorida\n\nFloyd Bennett Field\n\nFlying Icing Wind Tunnel\n\nFonda, Gorton\n\nFort Riley\n\nFortune\n\n423rd Infantry Regiment, U.S.\n\n424th Infantry Regiment, U.S.\n\nFoust, Warren (char.)\n\nFrance\n\nFranklin, Ben\n\n\"Freezing Point Apparatus, A\" (B. Vonnegut)\n\nFrench Mountain\n\nFrigidaire\n\nFuncrest Hotel\n\nGalisteo Creek\n\ngame theory\n\nGE Digest\n\n\"GE Disease\"\n\nGE Monogram\n\nGeneral Electric (GE); anticommunist ideology of; anti-union policies of; blacklisting by; consumer production division of; culture of; government work of; holiday parties of; Knolls Atomic Power Laboratory at; postwar strike at; profits of; Project Cirrus canceled by; as source material for KV's fiction; World War II and\n\nGeneral Electric Research Laboratory; cloud-seeding experiments of, see cloud seeding, GE experiments in; cross-disciplinary philosophy of; FBI dossiers on scientists of; military's relationship with; pure research encouraged at\n\nGeneral Motors\n\nGeneral Office News\n\nGE News Bureau; KV at; KV's resignation from\n\nGE News Graphic\n\nGeophysical Institute (Bergen)\n\nGeorge, Lake\n\nGE Realty Plot\n\nGE Review\n\nGerman Americans\n\ngermanium\n\nGermany, Germans; surrender of\n\nGermany, Nazi\n\nGE Theater\n\nGhost Dance Society\n\nGI Bill\n\nGigliano, Louis (char.)\n\nGlamour\n\nGloucester, Mass.\n\nGod and Golem Inc. (Wiener)\n\nGoldstein, Simon\n\nGoldwater, Barry\n\nG\u00f6ttingen, University of\n\nGould, Filomena\n\nGourmet\n\nGrand Central Terminal\n\nGreat Appalachian Storm of 1950, see Thanksgiving Storm (1950)\n\nGreat Britain\n\nGreat Depression\n\nGreat Plains floods\n\ngreenhouse gases\n\nGreylock Mountain\n\nGriffin, George\n\nGroszinger, Bernard (char.)\n\nGrumman Goose (amphibious plane)\n\nGulf Stream\n\nGunn, Ross\n\nG. W. Merritt Lumber Company\n\nHalyard, Ewing J. (char.)\n\nHammond, Roger\n\nHanford, Wash.\n\nHans Grell's (bar)\n\nHarney's, see H. S. Harney (store)\n\nHarper's\n\nHarrison, Earl \"Hotbox\" (char.)\n\nHartford, Conn.\n\nHartford-Empire Company\n\nHarvard University; Blue Hill Observatory at\n\nHawaii\n\nHeine, Heinrich\n\nHellman, Lillian\n\nHelmholtz, George M. (char.)\n\nHelmholtz, Hermann von\n\nHenderson Harbor\n\nHerbstnebel (Autumn Fog)\n\nHersey, John\n\nHertz, Rudy (char.)\n\nHicks, Granville\n\nHirohito, emperor of Japan\n\nHiroshima, atomic bombing of\n\n\"Hiroshima\" (Hersey)\n\nHitler, Adolf\n\nHitz, Ben\n\nHo Chi Minh Trail\n\nHoenikker, Angela (char.)\n\nHoenikker, Felix (char.)\n\nHoenikker, Franklin (char.)\n\nHoenikker, George (char.)\n\nHoenikker, Newt (char.)\n\nHollomon, John Herbert\n\nHollomon family\n\nHosler, Charles\n\nHotel Astor (New York)\n\nHotel Jefferson\n\nHotel King Cole\n\nHotel Van Curler\n\nHoughton, Henry; cloud-seeding experiments of\n\nHouse Bill 86\n\nHouse Un-American Activities Committee (HUAC)\n\nHowell, Wallace\n\nH. S. Harney (store)\n\nHubert, Bill\n\nHudson River\n\nHull, Albert\n\nHuman Use of Human Beings, The (Wiener)\n\nHungary\n\nHunt, Lester\n\nhurricanes; physics of; possible weaponizing of\n\nHutchins, Robert\n\nHuxley, Aldous\n\nHyannis, Mass.\n\nHyde Park, N.Y.\n\nhydrogen bomb\n\nIBM\n\nice, BV's research on\n\nice crystals, variants of\n\nice-nine\n\nIce Research Base\n\nice-2\n\nIndia\n\nIndiana\n\nIndianapolis, Ind.\n\nIndianapolis News\n\nIndianapolis Star\n\nIndianapolis Times\n\nIndustrial Bank of Commerce\n\nInstitute for Advanced Study; computers at; Meteorology Project of\n\nInstitute for Nuclear Studies\n\nInstitute of Meteorology\n\nInterior Department, U.S.\n\nIowa\n\nIowa Writers' Workshop\n\nIroquois River\n\nIsrael\n\nIwo Jima\n\nJacksonville, Fla.\n\nJacques (restaurant)\n\nJapan; atomic bombing of\n\nJefferson City, Mo.\n\nJodl, Alfred\n\nJoint Chiefs of Staff\n\nJonah (char.)\n\nJosef and Peter (chars.)\n\nJournal of Meteorology\n\nJunior League\n\nKansas\n\nKansas City, Kans.\n\nKansas River\n\nKappa Sigma\n\nKelly, Colonel (char.)\n\nKem, James\n\nKennedy, John F.\n\nKenney, George\n\nKentucky\n\nKhan, Kublai\n\nKing, Hurricane\n\nKingston Daily Freeman\n\nKirch, James\n\nKnolls Atomic Power Laboratory (KAPL)\n\nKoestler, Arthur\n\nKorea\n\nKorean War\n\nKotsch, William\n\nKrick, Irving; long-range forecasting by; as self-promoter\n\nKurzweil, Raymond\n\nlabor force, automation and\n\nLa Guardia Airport\n\nLangmuir, Barbara\n\nLangmuir, Dean\n\nLangmuir, Irving; absentmindedness of; BV's silver iodide experiments of; BV's silver iodide successes announced by; chaos theory anticipated by; at Chicago conference on atomic energy; cloud-seeding experiments of; Collected Works of; on convergent and divergent phenomena; in desert rainmaking project; FBI file on; grandiose claims of; hurricane-seeding experiments of; Ice-9 concept of; intense curiosity of; marginalization of; meteorology research of; as model for Dr. Hoenikker; and New York City cloud-seeding program; Nobel Prize of; periodicity experiments of; pragmatism of; Project Cirrus report of; as proselytizer for science; in retirement from GE; Soviet Union trip of; statistical analysis embraced by; supercooling research of; war-related research of; weather seen as divergent phenomenon by\n\nLangmuir, Kenneth\n\nLangmuir, Marion Mersereau\n\nLangmuir Laboratory for Atmospheric Research\n\nLaplace, Pierre-Simon\n\n\"Laplace's demon\"\n\nLe Havre\n\nLeMay, Curtis\n\nLevy's Liquor Store\n\nLewis, William; Project Cirrus and\n\nLieber, Albert\n\nLife\n\nLilienthal, David\n\nLindbergh, Charles\n\nLippmann, Walter\n\nLittauer, Kenneth\n\nLittauer & Wilkinson\n\nLloyd's\n\nLockwood, Scammon\n\nLorenz, Edward\n\nLos Alamos, N.Mex.\n\nLos Angeles Times\n\nLuce, Clare Boothe\n\nLuxembourg\n\nLyon, Lavina\n\nLyon, Ollie\n\nMacCready, Paul\n\nMacDill Air Base\n\nMacon, Arnold (char.)\n\nMademoiselle\n\nMailer, Norman\n\nMalotti, Steve\n\nManhattan, Kans.\n\nManhattan District\n\nManhattan Project\n\nMan in the Gray Flannel Suit (Wilson)\n\nManzano Mountains\n\nMargaretville, N.Y.\n\nMarine Air Corps, U.S.\n\nMarion (char.)\n\nMaryland\n\nMassachusetts\n\nMaxinkuckee, Lake\n\nMaynard, Kiah\n\nMcCall's\n\nMcCarthy, Joseph\n\nMcCarty family\n\nMcDonald, Willard\n\nMcLeod, Robert\n\nmemory, sacredness of\n\nMen and Volts at War (GE)\n\nMerchant of Venice, The (Shakespeare)\n\nMerck\n\nMetcalf family\n\nMeteorology Project\n\nMexico, Gulf of\n\nMiami, Fla.\n\nMiami Beach, Fla.\n\nMiami Beach Redistribution Center\n\nMichigan\n\nMichigan, Lake\n\nMichigan, University of\n\nMiddletown, N.Y.\n\nMidwest, flooding in\n\nmilitary-industrial complex\n\nMiller, Arthur\n\nMiller, Henry\n\nMinneapolis, Minn.\n\nMinneapolis Star-Journal\n\nMinnesota\n\nMississippi River\n\nMissouri\n\nMissouri River\n\nMIT; Chemical Warfare Service at; KV's speech at\n\nMobile, Ala.\n\nMohawk Golf Club\n\nMohawk River\n\nMonongahela River\n\nMontgomery, Bernard\n\nMonzano, General (char.)\n\nMoore, Charlie\n\nMoorhead, Alfred (char.)\n\n\"Morning After\" (Kirch)\n\nMoscow\n\nMurrow, Edward R.\n\nNagasaki, atomic bombing of\n\nNaked and the Dead, The (Mailer)\n\nNapoleon Bonaparte\n\nNational Academy of Sciences\n\nNational Bureau of Standards\n\nNational Council of American-Soviet Friendship\n\nNational Press Club\n\nNational Science Foundation\n\nNative Americans\n\nNavy, U.S.\n\nNayar, Unni\n\nNeosho River\n\nNevada; cloud seeding in\n\nNevele Country Club\n\nNew Deal\n\nNew England\n\nNew England Association of Chemistry Teachers, on moral responsibility of scientists\n\nNew Hampshire\n\nNew Jersey\n\nNew Mexico; drought in; Project Cirrus experiments in\n\nNew Mexico School of Mines; Poverty Row at\n\nNew Republic\n\nNewsweek\n\nNewtonian time\n\nNew York, N.Y.; cloud-seeding program in; 1949\u201350 drought in; rainmaking liability claims against\n\nNew York, State University of, at Albany\n\nNew York Central railroad\n\nNew York City Board of Estimate\n\nNew Yorker\n\nNew York Giants (baseball team)\n\nNew York Post\n\nNew York State; 1950 storm in\n\nNew York Sun\n\nNew York Times\n\nNew York Times Book Review\n\nNew York Times Magazine; KV's tribute to BV in\n\nNew York University\n\n1984 (Orwell)\n\nNiskayuna, N.Y.\n\nNobel Prize\n\nNormandy\n\nNorth American Weather Consultants\n\nNorth Carolina\n\nNortheast, 1950 storm in\n\nNorth Korea\n\nNorth Sea\n\nNorth Vietnam\n\nNorway; weather forecasting in\n\nNotre Dame, University of\n\nnuclear arms race\n\nnuclear deterrence\n\nOak Ridge, Tenn.\n\nO'Dwyer, William\n\nOffice of Defense Mobilization\n\nOffice of Naval Research\n\nOfficer Club 1\n\nO'Hare, Bernard V.\n\nOhio\n\nOhio River\n\nOhio State University\n\nOklahoma\n\nOMIBAC\n\n106th Infantry Division, U.S.\n\nOne World or None: A Report to the Public on the Full Meaning of the Atomic Bomb\n\n\"On the Possibilities of Climate Control\" (Wexler)\n\nOperation Popeye\n\nOppenheimer, J. Robert\n\nOrange County, N.Y.; Board of Supervisors in\n\nOrdinal Memory Inspecting Binary Automatic Calculator, see OMIBAC\n\nOregon\n\nOre Mountains\n\nOrlando, Fla.\n\nOrville, Howard\n\nOrwell, George\n\nOsage River\n\nOSS\n\nOsterville, Mass.\n\n\"Outline of Weather Proposal\" (Zworykin)\n\nozone layer\n\nPabst Blue Ribbon\n\nPace, Bob\n\nPacific theater\n\nPakistan\n\nPalaia, Michael\n\nPalisades Amusement Park\n\nPall Mall\n\nParis\n\nPark School\n\nPasadena, Calif.\n\nPaul (char.)\n\nPenguin (motorboat)\n\nPennsylvania\n\n\"Perfect Day for Bananafish, A\" (Salinger)\n\nPetterssen, Sverre\n\nPfizer\n\nphenomena, divergent vs. convergent\n\n\"Philadelphia Phase\" (Wylie)\n\nPhillips, Frank\n\nPhillips Petroleum\n\nPhoenicia, N.Y.\n\nPilgrim, Billy (char.)\n\nPineapple Research Institute\n\nPine Hill, N.Y.\n\nPine Hill Country Club\n\nPittsburgh, Pa.\n\nPittsfield, Mass.\n\nPi Ying (char.)\n\nPlayboy\n\nPoland\n\nPolice Protection (game)\n\nPopeye, Operation\n\nPriestley, J. B.\n\nPrinceton, N.J.\n\nPrinceton University\n\nprogress, downside of\n\nProject Cirrus; California forest fires and; canceling of; as cause of 1950 storm; commercial rainmakers as threat to; desert rainmaking experiment of; hurricanes and; hurricane-seeding experiments of; Langmuir's grandiose claims about; legacy of; Lewis and; liability issues and; media accounts of; New Mexico experiments of; nonmilitary uses of research from; public reaction to hurricane seeding by; silver iodide experiments of; Weather Bureau attack on\n\nProject Scud\n\nProject Skywater\n\nProject Stormfury\n\nProteus, Paul (char.)\n\nProvincetown, Mass.\n\nPTA\n\nPuerto Rico\n\nQuarter Century Club\n\nRainier, Mount\n\nRainmakers' Flood (1950)\n\nrainmaking, see cloud seeding\n\nRanger Development Company\n\nRCA Labs\n\nReader's Digest\n\nReader's Scope\n\nReagan, Ronald\n\n\"Rebellious Scientist After Two Years, A\" (Wiener)\n\nRedbook\n\nRed Cross\n\nRedfield, Robert\n\nReichelderfer, Francis\n\nRepublicans\n\nResearch Lab Report RL-723: \"Variations in the Contact Resistance of a Copper Point Moving over Etched Surface of Germanium Crystal\" (B. Vonnegut)\n\nResnick, Sol\n\nRex, Daniel\n\nRice, Major (char.)\n\nRio Grande\n\nRio Salado\n\nRome, N.Y.\n\nRoney Plaza Hotel\n\nRoosevelt Hotel\n\nRosenthal, A. M.\n\nRossby, Carl-Gustaf\n\nROTC\n\nRotterdam, N.Y.\n\nRotterdam Democratic Club\n\nRoyal Air Force (RAF)\n\nRumfoord, Bertram Copeland (char.)\n\nRumfoord, Winston Niles (char.)\n\nRumford, Benjamin Thompson, Count\n\nRussell, Diarmuid\n\nRussell, Jane\n\nRussell & Volkening Inc.\n\nRussia; see also Soviet Union\n\nRussians\n\nSt. Louis, Mo.\n\nSt. Petersburg Times\n\nSt. Vith\n\nSaipan\n\nSalinger, J. D.\n\nSandia Mountains\n\nSan Francisco, Calif.\n\nSan Jose, Calif.\n\nSan Juan Islands\n\nSanta Fe, N.Mex.\n\nSanto Domingo, N.Mex.\n\nSaturday Evening Post\n\nSaturday Review of Literature\n\nSavage, John\n\nSavannah, Ga.\n\nScandinavians\n\nSchaefer, Vincent J.; Atmospheric Sciences Research Center founded by; BV's silver iodide experiments and; at Camp GE; cloud-seeding experiments of; in desert rainmaking project; hurricane-seeding experiments of; as Langmuir's assistant; at Senate weather control hearings; supercooling research of\n\nSchenectady, N.Y.; cloud-seeding experiments in; Junior League of; KV's move to\n\nSchenectady Cerebral Palsy Fund\n\nSchenectady County, N.Y.\n\nSchenectady Gazette\n\nSchenectady Railway Company\n\nSchenectady Wintersports Club\n\nSchenectady Works\n\nSchenectady Works News\n\nSchenectady Works Research Lab\n\nSchenectady YMCA\n\nSchnee Eifel (Snow Mountain)\n\nSchoharie Reservoir\n\nScience\n\nscience, art and\n\n\"Science, Common Sense, and Decency\" (Langmuir)\n\nScience Forum (radio program)\n\nScience Newsletter\n\nScientific American\n\nscientists, moral responsibility of\n\nScientists' Movement\n\nScottsdale, Ariz.\n\nScribner\n\nScud, Project\n\nSears & Roebuck\n\nSeattle, Wash.\n\nsemiconductors\n\nSenate, U.S.; weather control hearings of\n\nSenate Bill 222\n\nShakespeare, William\n\nShandaken, N.Y.\n\nSherwood, Robert\n\nShortridge Echo\n\nShortridge High\n\nSiberia\n\nSiegfried Line\n\nSierra Nevada mountains\n\nsilver iodide\n\nsingularity\n\nSkywater, Project\n\nSlutsky, Ben\n\nSmathers, George\n\nsmog\n\nsnow, manufacture of, see cloud seeding\n\nSnow Inc.\n\nsocialism\n\nSocorro, N.Mex.\n\nSokolsky, George\n\nSoldiers' and Sailors' Monument, in Indianapolis\n\nSorel, Georges\n\nSouth Dakota\n\nSoviet Academy of Sciences\n\nSoviet Union; atomic bomb detonated by; atomic bomb program of; Berlin blockade of; Langmuir's trip to\n\nSpanish Armada\n\nSpicy Detective\n\nSS\n\nstable systems\n\nStalin, Joseph\n\nStandish, David\n\nStanford Research Institute\n\nState Department, U.S.\n\nstatistical analysis\n\nStatler Hotel\n\nSteering Committee\n\nSteinbeck, John\n\nSteinmetz, Charles Proteus\n\nSteinmetz Memorial Lecture\n\nStimson, Henry\n\nStockholm\n\nStormfury, Project\n\nStory\n\nStrategic Air Command\n\nstrikes, in postwar era\n\nSuccess, Lake\n\nSuits, Chauncey Guy; at Senate weather control hearings\n\nSullivan County, N.Y.; Board of Supervisors in\n\nSuper bomb, see hydrogen bomb\n\nsupercooling\n\nSwarthmore College\n\nSwarthmore Halcyon\n\nSweden\n\nSyracuse, N.Y.\n\nsystems, stable vs. unstable\n\nSzilard, Leo\n\nTalbot, Curtis\n\nTampa, Fla.\n\nTech Engineering News\n\nTechnique (yearbook)\n\ntechnology; moral implications of; utopian vs. dystopian views of\n\nTeller, Edward\n\ntemperature inversions\n\nTennessee, University of\n\nThanksgiving Storm (1950)\n\nthermonuclear bomb, see hydrogen bomb\n\nThird Fleet\n\nThom, Herbert C. S.\n\nThomas, J. Parnell\n\nThunderclap, Operation\n\nthunderstorms: dendritic prints of; electrical charge in\n\nTiffany (jeweler)\n\nTime\n\ntime, Newtonian vs. Bergsonian\n\nTime Inc.\n\nTojo, Prime Minister\n\nTokyo; firebombing of\n\nTomorrow\n\nTopeka, Kans.\n\ntornadoes\n\nTouart, C. N.\n\nTown Tavern\n\ntrade unionism\n\nTralfamadorians (chars.)\n\ntransistors\n\nTrinity test\n\n\"Trouble at Tuaviti\" (Savage)\n\nTrout, Kilgore (char.)\n\nTroy, N.Y.\n\nTruman, Harry\n\nTudor Hall\n\nTurner, Lana\n\nTwain, Mark\n\nUkraine\n\nUlster County, N.Y.\n\nUN Atomic Energy Commission\n\nUN charter\n\nundersea warfare\n\nUnion College\n\nUnited Electrical Workers\n\nUnited Nations; inaugural conference; Security Council\n\nUnited States\n\nUnited World Federalists\n\nUniversity Office of Inquiry into the Social Aspects of Atomic Energy\n\nunstable systems\n\nUral Mountains\n\nVeblen, Oswald\n\nVE Day\n\nVerdigris River\n\nVermont\n\nVietnam War; cloud seeding in\n\nVJ Day\n\nvon Kleigstadt, Ormand (char.)\n\nVonnegut, Alex; KV's sarcastic letter to\n\nVonnegut, Alice, see Adams, Alice Vonnegut\n\nVonnegut, Bernard; and Advisory Committee on Weather Control; Alplaus home of; at Arthur D. Little; as artist; brilliance of; cancer of; in Chemical Warfare Service; childhoood of; on cloud seeding as cause of floods; cloud-seeding experiments of; commercial rainmakers and; crystal research of; curiosity of; death of; at GE Research Lab; on human impact on weather; ice research of; KV's college and major chosen by; and KV's GE News Bureau job; KV's letter to Uncle Alex kept by; KV's MIT speech about; KV's postwar reunion with; KV's New York Times Magazine tribute to; marriage of Lois and; messy work space of; at Minneapolis AMS meeting; at MIT; North Scituate house of; obsessiveness of; as pacifist; in resignation from GE; at Senate weather control hearings; sense of humor of; on SUNY Albany facility; supercooling research of; thunderstorm research of; as tinkerer; as unhappy with military-related research; vortex thermometer of; war-related research of\n\nVonnegut, Bernard (grandfather)\n\nVonnegut, Edith (KV's daughter); birth of\n\nVonnegut, Edith (mother); death of\n\nVonnegut, Helen\n\nVonnegut, Jane Marie Cox; academic accomplishments of; as believer in KV's writing talent; death of; Dostoyevsky's importance to; KV's courtship of; marital breakup of KV and; in move to Provincetown; in move to Schenectady; at OSS; Osterville house of; pregnancies of; in Provincetown; role in shaping KV's career; Schenectady social life and; Swarthmore and; wedding of KV and\n\nVonnegut, Kit\n\nVonnegut, Kurt, Jr.: as able to learn from criticism; Alplaus house of; Army discharge of; in Battle of the Bulge; Bergsonian vs. Newtonian time in writings of; Burger and; on BV's career and moral choices; BV's postwar reunion with; Chicago apartment of; City News Bureau job of; at Cornell; Dresden firebombing and; enlistment of; family's reunion with; fictional cameos of; Fort Riley posting of; GE as source material for; GE News Bureau job of; GE News Bureau resignation of; household chores contract of; at Iowa Writers' Workshop; Jane courted by; job search of; joke telling by; journalism career as attractive to; Knox and; Littauer and; as losing faith in knowledge and technology; marital breakup of Jane and; master's thesis of; Osterville house of; as pacifist; paintings by; playwriting of; political views of; as POW; in Provincetown; sarcastic letter to Uncle Alex from; seen as science fiction writer; sister's children adopted by; as socialist; as University of Chicago anthropology student; \"Virtuous Physicist\" speech of; wedding of Jane and; writer's block of; writing epiphany of; writing habits of\n\nVonnegut, Kurt, Jr., writings of: \"All the King's Horses\"; \"Armageddon in Retrospect\"; \"Atrocity Story\"; \"Basic Training\"; \"Between Timid and Timbuktu\" (unpublished); \"Bonanza\"; \"Brighten Up!\"; \"The Case of the Phantom Roadhouse\"; Cat's Cradle; \"City\"; \"Coda to My Career as a Writer for Periodicals\"; \"The Commandant's Desk\"; \"Comparison of Elements of Ghost Dance Mythology with That Mythology of a More Tranquil Period, A\"; \"Das Ganz Arm Dolmetscher\"; \"Deer in the Works\"; \"Enterprise\"; \"EPICAC\"; \"The Euphio Question\"; \"The Foster Portfolio\"; Gal\u00e1pagos; God Bless You, Mr. Rosewater; \"Happy Birthday, 1951\"; \"Ice-9\" (unpublished); \"Innocents Abroad\"; \"I Shall Not Want\"; Jailbird; \"Jenny\"; \"Little Drops of Water\"; \"Lost Battalion Undergoes a Severe Shelling, The\"; \"Mnemonics\"; \"More Stately Mansions\"; Mother Night; Palm Sunday; \"The Petrified Ants\"; Player Piano; \"Poor Little Rich Town\"; \"Report on the Barnhouse Effect\"; \"Robot Cop\"; \"Ruth\"; The Sirens of Titan; Slaughterhouse-Five; \"Thanasphere\"; Timequake; unpublished early stories; Utopia 14, see Player Piano; \"Virtuous Physicist, The\" (speech); \"Wailing Shall Be in All Streets\"; war as backdrop for; \"We Impress Life Magazine with Our Efficient Role in National Defense\"; Welcome to the Monkey House; \"Well All Right\"; \"White King\"; \"With His Hand on the Throttle\"\n\nVonnegut, Kurt, Sr.\n\nVonnegut, Lois Bowler \"Bow\"; marriage of BV and; pregnancies of\n\nVonnegut, Mark; birth of\n\nVonnegut, Peter\n\nVonnegut, Scott\n\nVonnegut, Terry\n\nVonnegut, Walter\n\nVonnegut family; as pacifists; summer cabin of\n\nvon Neumann, John; death of; IAS computer of; weather modeling and; Wexler and\n\nvon Neumann, Ludwig (char.)\n\nvortex thermometer\n\nWabash River\n\nWaldorf-Astoria\n\nWalker's Pharmacy\n\nWallace, DeWitt\n\nwar, weather and\n\nWar and Peace (Tolstoy)\n\nWar Crimes Commission\n\n\"War in the Atomic Age\" (Brodie)\n\nWashington, D.C.\n\nWashington, Mount\n\nWashington State; nuclear production complex and\n\nWater Supply, Gas, and Electricity, New York City Department of\n\nWaterloo, Battle of\n\nWater Power and Control Commission, New York\n\nWater Resources Development Corporation\n\nweather: as divergent phenomenon; human impact on; periodicities in; physics of; predicted control of; Senate hearings on control of\n\nWeather Bureau, U.S.; New England storm and; Project Cirrus attacked by; and Senate weather control hearings\n\nweather control; Langmuir's grandiose claims about; military applications of; renewed interest in; research spending on; Senate hearing on; UN ban on weaponizing of; Wexler's research on; see also cloud seeding\n\nweather forecasting; computers and; in World War I; in World War II\n\nWeathering the Storm (Petterssen)\n\n\"Weather Made to Order?\" (Orville)\n\nweather modeling, on computers\n\nWeather Modification Inc.\n\n\"Weather\u2014the New Super Weapon\" (Florez)\n\nWeatherwise\n\nWeeks, Edward\n\nWells, H. G.\n\nWelty, Eudora\n\nWest Berlin\n\nwestern United States, commercialized rainmaking in\n\nWestwall\n\nWexler, Harry; death of; GE hurricane experiments and; Project Cirrus attacked by; von Neumann and; weather control research of\n\nWeyl, Hermann\n\nWGY (radio station)\n\nWheeler, William Morton\n\nWhitney, Willis \"Doc\"\n\nWhitney Club\n\nWiener, Norbert; Player Piano criticized by\n\nWillett, Hurd\n\nWillkie, Wendell\n\nWilson, Charlie\n\nWilson, Sloan\n\nWinchell, Walter\n\nWisconsin\n\nWoolaroc Ranch\n\nWorcester, Mass., 1953 tornado in\n\nWorkman, Everly John \"Jack\"; in desert rainmaking project\n\nWorks Library\n\nworld government\n\nWorld Set Free, The (Wells)\n\nWorld War I; anti-German sentiment and\n\nWorld War II; atomic bombs in; D-Day in; GE and; weather forecasting in\n\nWorld War III\n\nWylie, Philip\n\nWyoming\n\nWyoming Water Development Commission\n\nYale Review\n\nYank\n\nYarborough family\n\nYarmolinsky, Adam\n\nYarmolinsky, Jane Vonnegut, see Vonnegut, Jane Marie Cox\n\nYugoslavia\n\nZworykin, Vladimir\n\nBernard and Kurt Vonnegut (at right), at home before Kurt enlisted. Bernard may already have been working for the Army's Chemical Warfare Service lab at MIT when this was taken. (Vonnegut family collection)\n\nThe wizards of \"the House of Magic\": GE Research Lab scientists pose at the doors of their offices in 1950. Vincent Schaefer is in front at far left, with Bernie, who began working there in 1946, at the next door back. (GE News Bureau photo)\n\nIrving Langmuir and Bernie look on as Vincent Schaefer leans over to create the experiment that started it all: breathing a cloud into his freezer that he will nucleate with dry ice to cause snow. (GE News Bureau photo)\n\nDry ice seeding produces dramatic cloud modifications. Here, the Project Cirrus airplane flies a gamma pattern, creating an obvious trench with dry ice. Perhaps, Vincent wryly noted later, they should have seeded clouds in the shape of the GE logo. (U.S. Army Signal Corps photo)\n\nGE insulated itself from liability concerns with a wall of military brass. For the press release announcing the joint GE-military weather control program, Bernie uses a toy popgun to seed the cloud in the freezer for an admiral and a general. (GE News Bureau photo)\n\nThe Vonneguts called the family rowboat the Beralikur, a combination of the names Bernie, Alice, and Kurt: the three siblings only grew closer as they aged. (Vonnegut family collection)\n\nJust like a real newspaper, minus the objectivity: the GE News Bureau as it looked when Kurt began work there as a \"junior writer\" in 1947, drafting press releases and articles for in-house magazines (GE News Bureau photo)\n\nA Project Cirrus B-17 being loaded with dry ice for an \"assault on the clouds\" (GE News Bureau photo)\n\n\"Fire makes rain\" was the GE News Bureau's caption for a photo of Bernie with his silver iodide generator. Even as the world thrilled to dry ice seeding, Bernie knew he had found something better: seeding clouds with silver iodide smoke. (GE News Bureau photo)\n\nThe Weather Bureau and the Institute for Advanced Study had a better way to take control of the weather: with computers. Here, members of the institute's Meteorology Project pose with the Army's ENIAC, then the world's most powerful computer, in 1950. At far left, Harry Wexler listens to John von Neumann. The Weather Bureau's chief, Francis Reichelderfer, is second from right. (Photo courtesy of MIT Museum)\n\nAboard a B-17, Bernie oversees the seeding of clouds with silver iodide. Irving Langmuir's attention was finally captured when silver iodide seeding in New Mexico was followed by a devastating thunderstorm and flood. (Bernard Vonnegut papers, State University of New York at Albany)\n\nKurt with his son, Mark, and a painting he made in Gloucester, Massachusetts, in the summer of 1950 as he gloomily contemplated the onset of another war, this time in Korea (Vonnegut family collection)\n\nAs Bernie's invention was heedlessly taken up by amateurs, he tried to figure out if it could be causing harm. In his map, the gray portions show where rainmakers were feverishly seeding clouds in 1951, as floods devastated the plains states. (Bernard Vonnegut papers, State University of New York at Albany)\n\nSeeking advice on how to testify at upcoming Senate hearings, a New Mexico senator and the publisher of the Santa Fe New Mexican visited the GE Research Lab. Diplomatically, the photograph shows only the publisher, between Irving Langmuir (at left) and Bernie. At right, the lab's director, Guy Suits, looks on as Vincent Schaefer seeds the freezer's cloud. (GE News Bureau photo)\n\nThe siblings Vonnegut\u2014Kurt, Bernie, and Alice\u2014with spouses and kids, reuniting in Provincetown, Massachusetts, in the summer of 1951. Kurt's wife, Jane, is at center, holding their daughter, Edith. (Vonnegut family collection)\n\nThe cover of Collier's was typical of the triumphant tone the media took when discussing weather control in the 1940s and 1950s: man finally stood at the brink of mastering nature. (Photo courtesy of the Estate of Frederick Siebel)\nA Note About the Author\n\nGinger Strand is the author of three previous books, including Killer on the Road: Violence and the American Interstate. She has written for a wide variety of publications, including Harper's Magazine, This Land, The Believer, Tin House, The New York Times, and Orion Magazine, where she is a contributing editor. You can sign up for email updates here.\n\nAlso by Ginger Strand\n\nFICTION\n\nFlight\n\nNONFICTION\n\nKiller on the Road: Violence and the American Interstate\n\nInventing Niagara: Beauty, Power, and Lies\nThank you for buying this\n\nFarrar, Straus and Giroux ebook.\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nContents\n\nTitle Page\n\nCopyright Notice\n\nEpigraph\n\n 1. Autumn Fog\n\n 2. Precipitating Events\n\n 3. Head in the Clouds\n\n 4. Bolt of Lightning\n\n 5. Eye of the Storm\n\n 6. Watersheds\n\n 7. Rainmakers\n\n 8. Out of the Blue\n\n 9. Cold Fronts\n\n10. Shifting Winds\n\nEpilogue: Rainbow's End\n\nNotes\n\nBibliography\n\nAcknowledgments\n\nIndex\n\nPhotographs\n\nA Note About the Author\n\nAlso by Ginger Strand\n\nCopyright\nFarrar, Straus and Giroux\n\n18 West 18th Street, New York 10011\n\nCopyright \u00a9 2015 by Ginger Strand\n\nAll rights reserved\n\nFirst edition, 2015\n\nLibrary of Congress Cataloging-in-Publication Data\n\nStrand, Ginger Gail.\n\nThe Brothers Vonnegut: science and fiction in the House of Magic \/ Ginger Strand. \u2014 First edition.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nISBN 978-0-374-11701-6 (hardback) \u2014 ISBN 978-0-374-71154-2 (e-book)\n\n1. Vonnegut, Kurt. 2. Vonnegut, Bernard. 3. Novelists, American\u201420th century\u2014Biography. 4. Scientists\u2014United States\u2014Biography. 5. Weather control\u2014United States. 6. Literature and science\u2014United States\u2014History\u201420th century. I. Title.\n\nPS3572.O5 Z858 2015\n\n813'.54\u2014dc23\n\n[B]\n\n2015010126\n\nOur e-books may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, extension 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.\n\nwww.fsgbooks.com\n\nwww.twitter.com\/fsgbooks \u2022 www.facebook.com\/fsgbooks\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Epigraph\n  4. 1. Autumn Fog\n  5. 2. Precipitating Events\n  6. 3. Head in the Clouds\n  7. 4. Bolt of Lightning\n  8. 5. Eye of the Storm\n  9. 6. Watersheds\n  10. 7. Rainmakers\n  11. 8. Out of the Blue\n  12. 9. Cold Fronts\n  13. 10. Shifting Winds\n  14. Epilogue: Rainbow's End\n  15. Notes\n  16. Bibliography\n  17. Acknowledgments\n  18. Index\n  19. Photographs\n  20. A Note About the Author\n  21. Also by Ginger Strand\n  22. Newsletter Sign-up\n  23. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nRoost Books\n\nAn imprint of Shambhala Publications, Inc.\n\n4720 Walnut Street\n\nBoulder, Colorado 80301\n\nroostbooks.com\n\n\u00a9 2019 by Jennifer Ward\n\nIllustrations \u00a9 2019 by Alexander Vidal\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.\n\nEbook design adapted from printed book design by Daniel Urban-Brown\n\nCover art by Alexander Vidal | Cover design by Daniel Urban-Brown\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Ward, Jennifer, 1963\u2013 author. | Vidal, Alexander, illustrator.\n\nTitle: I love birds!: 52 ways to wonder, wander, and explore birds with kids \/ Jennifer Ward; illustrations by Alexander Vidal.\n\nDescription: Boulder, Colorado: Roost Books, 2019. | Audience: Ages 4 to 8. | Includes bibliographical references.\n\nIdentifiers: LCCN 2018011422 | ISBN 9781611804157 (paperback: alk. paper)\n\neISBN 9780834842168\n\nSubjects: LCSH: Bird watching. | Family recreation. | Outdoor recreation.\n\nClassification: LCC QL677.5 .W335 2019 | DDC 598.072\/34\u2014dc23\n\nLC record available at <https:\/\/lccn.loc.gov\/2018011422>\n\nv5.4\n\na\n**CONTENTS**\n\nIntroduction\n\n**SPRING**\n\n1.  Look Up, Look Down, Look All Around\n\n2.  Billions of Birds\n\n3.  A Feast for First Arrivals\n\n4.  The Dawn Chorus\n\n5.  Home Tweet Home\n\n6.  Feathery Field Trip\n\n7.  Splashy Tweet Retreat\n\n8.  A Rustic Roost\n\n9.  Birds in Motion\n\n10.  Draw that Bird!\n\n11.  Zoom In\n\n12.  Feathered Firsts\n\n13.  Fantastic Feeders\n\n**SUMMER**\n\n14.  Getting to Know You\n\n15.  The Perfect Patch\n\n16.  Raindrop Cafe\n\n17.  A Haven for Hummingbirds\n\n18.  Bountiful Birdscape\n\n19.  Wing It!\n\n20.  Hats Off to Hummingbirds!\n\n21.  Seed Sorting\n\n22.  Yard Mapping\n\n23.  Feathery Photography\n\n24.  Heavenly Hawks\n\n25.  Chicka-dee-dee-dee\n\n26.  Beak by Beak\n\n**FALL**\n\n27.  Color Count\n\n28.  Moonlight Migration\n\n29.  A Handful of Hummers\n\n30.  Lovely Leaf Litter\n\n31.  Handy Nests\n\n32.  A Pumpkin-Perfect Feast\n\n33.  Pressed Leaf Feather Art\n\n34.  Speech Bubbles between Birds\n\n35.  We're Going on an Owl Hunt\n\n36.  Bird Brains\n\n37.  Name That Bird\n\n38.  Chilling with Feathers\n\n39.  A Nest of My Own\n\n**WINTER**\n\n40.  Naked Nests\n\n41.  Flappy Hour and the Great Backyard Bird Count\n\n42.  Winter Thicket\n\n43.  Telltale Toes\n\n44.  Winter Wreath Feeder\n\n45.  Deck the Trees with Boughs So Jolly\n\n46.  Bird Haiku\n\n47.  Snowy Snowman Snack\n\n48.  Window to the World\n\n49.  A Bird in the Hand\n\n50.  Surf, Sand, I Spy a Bird Band!\n\n51.  New Year's Bird\n\n52.  New Year's Bird Resolutions\n\nAcknowledgments\n\nResources and Recommended Reading\n\nAmerican Bird Association Code of Ethics\n\nAbout the Author\n\nE-mail Sign-up\n\n# **INTRODUCTION**\n\n**A S A YOUNG CHILD** I recall lying in bed, tucked in silence and stillness. The light was dim and dusky, filling every nook and cranny in my room; I can still see this so vividly. Then, out of nowhere, one single birdcall pierced the quiet, separating night from morning. This call was joined by another call. And then more calls, the sounds building in crescendo, overlapping, high, low, melodic, and nonstop. The very air brimmed with song that is the _dawn chorus._ It was the first time I had ever experienced it, and even as a child I knew it was something magnificent. Experiencing it made me feel wondrous. It made me feel alive, special, and, most importantly, connected to the big, wide, wonderful world around me. To this day I continue to listen for it.\n\nIn 2008, my book _I Love Dirt!_ was published. At that time, it was a call to parents, grandparents, educators, and caregivers to help recover one of the greatest joys in childhood: spending time outdoors in nature. We were in the midst of a true \"nature deficit,\" a term coined by author and journalist Richard Louv. Children and adults alike living with overly busy schedules, engaged with screen time from sunup to sundown, seeking places where they could stay plugged in near electrical outlets, schools eliminating recess, and the like. Thanks to Richard Louv and many others, today a glorious movement is taking place\u2014a wise awareness that spending time in nature is beneficial to our health and well-being. Research also supports the detrimental effects of living without nature.\n\nIt is a joyful thing to watch our children discover the world around them, and birds certainly enhance this experience. As a matter of fact, engaging with a simple bird feeder is one of the most common ways people may interact with wild animals. Birds can be found everywhere, high, low, wherever we go. As children discover the wonder, beauty, science, and souls of birds, we are given a priceless gift\u2014experiencing the world through their eyes.\n\nThis book was born from one of my dearest passions: birds. Like _I Love Dirt!_ , it offers fifty-two open-ended activities that will give the outdoors back to your children. In just five minutes, you can turn your child's world around, open their senses, and connect them to nature. _I Love Birds!_ encourages your child to think, wonder, wander, explore, create, nurture, and have an amazing time with the avian wildlife that coexist with us day to day, month by month, and season by season.\n\nAnywhere a bird might be spotted provides an opportunity for discovery. They bring the gift of nature to us wherever we may be. Be it a city sidewalk, an urban park, through a window, from a wheelchair, in a backyard, or even while lying in bed waiting for the dawn chorus\u2014birds are waiting to be enjoyed and discovered! Time in nature is free of cost\u2014as are the activities in this book\u2014but the rewards that come with exploring birds will stay with your children for the rest of their lives, benefiting them and the planet they share with birds. Now get out there and soar!\n\nThank you for sharing _I Love Birds!_ with the wee chicks in your life. In this book you'll find a host of creative and engaging ways to better understand these winged wonders. The activities are categorized by the habits most birders practice:\n\n**  Observe: To know birds, you must observe birds. These activities offer ways to see, hear, and better understand birds and bird behavior.**\n\n**  Take Note: Use this book (or get a trusty notebook) and write down the things you notice about birds and draw the birds you observe.**\n\n**  Take Action: Become a better bird advocate by doing simple projects that support the birds in your area.**\n\nReady to fly? Let's do this!\n\n# **LOOK UP , LOOK DOWN, LOOK ALL AROUND**\n\n**B IRDS ARE EVERYWHERE.** They can be found flying through the sky, foraging high in treetops, filling the air with calls and songs, perching on power lines, scavenging on the ground, scaling tree trunks, racing surf on a shoreline, or bobbing on the surface of water, just to name a few places.\n\nConnecting with birds is as simple as stepping outside and opening our eyes and ears to their presence. As we do so, we're connecting with wild animals right outside our back door! Yet, how often do we really observe birds and take note of them? Today, with technology at our fingertips 24-7, it's so easy for us to willingly ignore our surroundings. Set an intention to connect to the nature that surrounds you and your child, and let birds be your guide as you do.\n\n# **OBSERVE**\n\nFind a spot outdoors where you and your child can relax. It can be a patch of grass, a sidewalk, a balcony, or a deck\u2014anywhere, as long as you're comfortable and outside. Feel the air on your skin and face. Feel your feet on the ground below you. Observe all of the details around you: the color of the sky, the temperature, and even small weeds sprouting up in unusual places. Enjoy the stillness of nature and the calm it brings.\n\nFirst, just listen. Absorb all the sounds around you. There may be traffic sounds, mechanical sounds, and wind-in-the-trees sounds. Through it all, listen for bird sounds: tweets, calls, chirps. Flapping wings, rap-rap-raps, and tap-tap-taps. Songs and coos. Seek out the bird sounds you hear and attempt to locate where the sounds are coming from. Often, we may hear a bird but never see it (or imagine we've seen it). Spotting them can sometimes be tricky!\n\nSeek out birds that you can see. How many birds can you spot with your eyes? We may not always need to look up to find a bird. They may be at eye level or on the ground. Binoculars aren't necessary to observe birds. Of course, with practice, binoculars work well for bird watching, but there's much bird-life to see with the naked eye, such as a bird's shape, size, color, location, and behavior. As you listen for birds, ask these questions:\n\n**  How many different sounds, songs, or calls can you hear?**\n\n**  How many birds can you find up high?**\n\n**  How many birds can you find on the ground, down low?**\n\n**  Can you hear a birdcall or song? How many different calls or songs?**\n\n**  Can you find a large bird?**\n\n**  Can you find a small bird?**\n\n**  Can you find a colorful bird?**\n\n**  Can you find a bird that's eating?**\n\n**  Can you find a bird that's running or hopping on the ground?**\n\n**  Can you find a bird that's flying?**\n\nEnjoy connecting to nature and the birds that call it home, where you will experience calm and a sense of wonder in everything avian.\n\n  **_Promotes relaxation, attentiveness, and observation skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How many birds are there on the planet?**\n\n**A: Species vary by region, but overall there are approximately 10,400 species of birds on the planet.**\n\n# **BILLIONS OF BIRDS**\n\n**C OME SPRING,** billions of birds wing their way, often thousands of miles, from their southern winter grounds to their northern breeding grounds. As a result, April and May are very busy sky traffic periods on our planet. Nothing marks spring like the return of migratory birds!\n\nYou may awaken one morning to find high treetops and low shrubs bustling with birdsong and bird movement, birds busy hopping from limb to limb, activity that wasn't noticeable the day prior. This is because migratory songbirds traveled all night, and now they've stopped to rest and refuel. Some species will rest in your area for a few days and then continue their journey northward. Other species will arrive in your area and stay put for nesting and breeding through the summer. The strongest males arrive first, scouting out the best breeding areas. Once the females arrive, they choose the males with the ideal habitats for raising young.\n\nMake a specific effort with your child to spend time outdoors in April and May to experience spring's bird migration, especially when weather conditions shift winds from south to north, as migratory birds take advantage of tailwinds. Then, simply look to the leaves! In addition to your _resident_ birds\u2014birds that remain in your area all year long\u2014observe very carefully. Can you spy a species you've not noticed before? Search high in the treetops, among leaves and new growth, where colorful warblers may be hunting for spiders and insects. Look among shrubs and hedges and on the ground below them. Notice bird sounds: Is there a song or call that is unfamiliar or new? If so, chances are it is a migrant, weary and hungry. Perhaps it is just passing through and stopping to rest, and will continue northward. Or, perhaps it is a species that will summer and nest in your area with you.\n\n# **TAKE NOTE**\n\nWith your child, document the dates and details of migrants and first arrivals in your area during spring's migration. Using the details of your notes, reference a field guide to birds in your area to help identify species and become familiar with your region's species.\n\n**MY FOTS (FIRST OF THE SEASON)**  \n---\n\nDate:\n\nLocation:\n\nSize:\n\nDescriptive notes:\n\nSketch:\n\n  **_Promotes awareness, understanding of the natural world, and observation skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do birds migrate?**\n\n**A: Birds migrate to survive. They travel to places that will sustain them with food and allow them to raise their young.**\n\n# **A FEAST FOR FIRST ARRIVALS**\n\n**A PPROXIMATELY 40 PERCENT** of our world's bird species migrate each spring and fall. As spring arrives, so do these weary travelers back to northern regions\u2014and they arrive exhausted and hungry, many having flown hundreds or thousands of miles, often nonstop, using up the reserves of stored body fat. Migration is an extremely challenging and dangerous time for birds. They encounter harsh weather and a lack of adequate habitat to rest and refuel along their journey, among a host of other challenging elements.\n\nAs migrants arrive or stop to rest, most will be seeking protein in the source of bugs, but migration season is the perfect time to offer some additional food as well. Many migratory species, such as orioles and tanagers, will appreciate apple slices, grape jelly, and orange halves.\n\n# **TAKE ACTION**\n\nExplain to your child that many birds are passing through your area, having traveled very long distances, and that they are extremely tired and hungry. Use a map to show the distance traveled by migrating species in your area, such as orioles. The Cornell Lab of Ornithology website or any bird field guide for your region will show the ranges of bird species by season, indicating where certain species breed during summer months and where they winter, and their migration routes in between.\n\nThen, create a fresh offering of fruits and sustenance for these traveling birds, which they will greatly appreciate. You and your child will also be rewarded as you observe ravenous species stopping by to enjoy your offering.\n\nFirst, simply slice an orange in half. Then, poke a small hole on the outside half of each orange, through the skin. Finally, place each orange section on the upward-facing tip of a shepherd's hook where your feeder(s) hang or on the tip of a tree's branch, or rest it on top of a firm bush or shrub. Birds will love this fresh fruit treat! Change the oranges every few days, keeping the citrus offering fresh. (A variation on this sweet treat is to scoop out most of the orange fruit, mix this fruit with grape jelly, and place the mixture back into the halved orange.)\n\nSliced and diced apples or fresh apple cores with seeds are also delicious treats that spring birds will enjoy. Simply place pieces on a tray where birds may land to access them. Change the apple pieces every few days to keep the offerings fresh.\n\n**DID YOU KNOW?**\n\n**Although dark skies protect night flying migrators from predation, lighted windows at night kill and injure billions of migratory birds each year when birds unknowingly fly into them, attracted by the light. Visit Fatal Light Awareness Program (FLAP) Canada to learn more about ways to prevent window strikes.**\n\n  **_Promotes stewardship and empathy_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How far do migrating birds travel?**\n\n**A: It varies by species, but here are some interesting migration findings: the Arctic Tern holds the record for longest migration distance, approximately 25,000 miles, pole to pole (almost 50,000 miles roundtrip each year); the Bar-Tailed Godwit flies almost 7,000 miles without stopping (it holds the record for longest nonstop flight); the Blackpoll Warbler flies 2,300 miles nonstop for eighty-six hours; the tiny Ruby-Throated Hummingbird migrates a distance of 5,000 round-trip miles each year, and the Rufous Hummingbird makes the longest migratory journey of any bird in the world, per its body size compared to distance covered, traveling almost 8,000 miles round-trip each year from Alaska to Mexico.**\n\n# **THE DAWN CHORUS**\n\n**_G \u00d6KOTTA_** is a Swedish word that translates to _waking up at dawn to hear the first birdsong of the day._ In America, we call this musical avian phenomenon _the dawn chorus._ Spring and summer are perfect times to experience this magical, remarkable sound, as this is the time that birds wake up and establish their territories and sing to attract mates.\n\n# **OBSERVE**\n\nCheck to determine when sunrise will take place. Then, set your alarm so you and your child wake up at least fifteen minutes prior to sunrise. Once you awaken, make a cup of tea, hot cocoa, or coffee, and grab a blanket to snuggle up with your child if the air outside is chilly. Together, sit outside in the dark. Be mindful as you sit: Feel the air on your face. Listen to how loud the stillness and quiet-dark of a new day may sound. Enjoy the calmness and serenity of the planet, of everything around you. There's nowhere to rush to, just you and your child, sitting in the moment. Listening. Waiting, in the silent, sleepy world.\n\nSoon, the call of just one bird will pierce the air and stillness.\n\nFirst one call. Followed by another. And then another and another. Behold as sounds and calls coming from each and every direction fill the air. As the day lightens and the world awakens, the chorus will fade. Throughout the day, you and your child may make note of daily birdcalls, sounds, and songs. They will be present but nothing like that of the dawn chorus.\n\n  **_Promotes wonder, relaxation, and awareness_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do birds sound so loud in the morning?**\n\n**A: Cooler temperatures and still air in the early morning allow bird calls to travel and be heard more effectively.**\n\n# **HOME TWEET HOME**\n\n**N ESTS ARE ASTOUNDING** architectural and engineering feats\u2014homes for birds that defy elements of weather while supporting and protecting very fragile eggs, and often at elevated heights. What's even more astonishing is the fact that these structures are deftly engineered and created by animals (birds) that lack fingers and opposable thumbs.\n\nCome spring, the avian world is all about nesting as birds prepare to incubate eggs and raise chicks\u2014the sole purpose of nesting. Birds get busy gathering materials for their homes: animal fur (and even dog hair!), twigs, natural fibers, grasses, dried leaves, moss, lichen, and most anything that is soft and stringy.\n\nUnfortunately, birds also gather nesting materials that prove harmful to them, such as fishing line and other litter, which they may become tangled in. A combination of human litter and loss of habitat for birds causes this problem and also poses the largest threat to wild bird species, as nesting grounds, food sources, and natural habitats that support nesting are increasingly being developed and\/or polluted by humans.\n\n# **TAKE ACTION**\n\nCreating a nest helper ball is a wonderful way to offer birds safe and earth-friendly materials to utilize, while also providing you and your child a rich opportunity to observe bird-nesting behavior.\n\nExplain to your child that just as he has a special bed to snuggle into at night and a cozy home to live in, birds raising families require a home, too. These homes are called nests, and they come in many shapes and sizes. Nests will be a safe and secure place for a mother bird to lay her eggs, where she, and often her mate\u2014depending on the species\u2014will incubate the eggs and care for the chicks until they _fledge_ , or leave the nest. Providing materials for nesting birds will help them with their nest building. This nest helper ball is a great way to do it.\n\n**Easily bendable floral wire (the thicker the better)**\n\n**Wire cutters**\n\n**Nesting materials**\n\n**  A variety of fibers such as jute (4 inches or less in length, as birds can get tangled in longer lengths)**\n\n**  Cotton string**\n\n**  Scraps of yarn, in neutral colors (to keep the nest hidden from predators)**\n\n**  Bits of cotton**\n\n**  Dried grasses**\n\n**  Natural raffia**\n\n**  Dried leaves**\n\n  1. Cut off an 8- to 12-inch section of floral wire.\n\n  2. Bend the floral wire around and around, to create a hollow orb approximately 3 or 4 inches in diameter.\n\n  3. Gently stuff the center of the wire ball with a variety of nesting materials.\n\n  4. Attach a string to the ball, for hanging.\n\n  5. Hang your nest helper ball to a tree branch or anywhere a bird may access it. Watch and enjoy as birds help themselves, plucking and grabbing and dashing off with their nesting material. As materials are depleted, refill the ball.\n\n  **_Promotes creativity, stewardship, and conservation_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are all bird nests alike?**\n\n**A: Birds make the most variety of homes, or nests, than any other wild animal in the world. Just a few types of bird nests include:**\n\n**  Woven nests (male weaver birds create an intricate, woven nest)**\n\n**  Sack nests (orioles weave a sack nest far out on a limb, high off the ground)**\n\n**  Cup-shaped nests (hummingbirds make the smallest cup nest, using grasses, lichen, and spider's web)**\n\n**  Cavity nests (woodpeckers carve a cavity nest inside a tree's trunk)**\n\n**  Floating nests (grebes make a floating nest on water, anchoring it to rooted water plants)**\n\n**  Ground nests (Killdeer nest directly on the ground)**\n\n**  Stick nests (doves and eagles create nests using sticks)**\n\n**  Mound nests (flamingos create a nest on the ground out of mud)**\n\n# **FEATHERY FIELD TRIP**\n\n**J UST WHO'S IN** your neighborhood anyway? Aren't you curious? Certain birds remain in your area year-round, as residents. Others are seasonal, spending a window of time in your area for wintering or breeding. Others pass through, stopping by briefly during their migration. It's remarkable to realize that each and every day holds a chance for a new discovery where birds are concerned, right in your own backyard and neighborhood!\n\nWonder about the birds in your area with your child, and make it an adventure to discover them together. As you begin to see and hear the birds in your neighborhood, you may find that the species in one area, such as in your backyard, are different than the species that inhabit your front yard or your neighbor's house or the house a block away.\n\nTake a walk with your child, with the sole purpose of listening and watching for birds in your yard and around your neighborhood. Make it a field trip!\n\n# **TAKE NOTE**\n\nBring along a notebook and pen. Then, ask the following questions:\n\n**  Do you see the same species of birds in the front yard and the backyard?**\n\n**  Do you see the same species of birds down the street?**\n\n**  Do morning species differ from evening species?**\n\nWith your child, create a list of all the birds you have seen in your backyard and neighborhood. Include the following details:\n\nDate:\n\nTime:\n\nSpecies (my best guess):\n\nBehavior (feeding, hopping, perching, singing, etc.):\n\nLocation (high\/low, ground, sky, tree, etc.):\n\nSound or calls heard:\n\n  **_Promotes curiosity, observation, and awareness of surroundings and wildlife_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: I see a lot of birds during the day, but where do these birds go at night?**\n\n**A: Birds that are active during the day,** **_diurnal_** **birds, sleep during the night. Some sleep in tree cavities or nooks and crevices, which shelter them from weather. Others sleep in birdhouses or in dense shrubs. Some sleep perched in trees. And some gather to sleep, or** **_roost_ ,** **in large flocks, such as blackbirds. Birds tuck their bills into their shoulders or backs, where the air is warmed by their body temperature. They fluff and puff their feathers to keep their bodies insulated when temperatures are cold, too.**\n\n# **SPLASHY TWEET RETREAT**\n\n**W ATER SOURCES** are important to all living things, and birds are no exception. Birds require water year-round for survival. They will both drink and bathe in the water source you provide. Birds are attracted to drips, ponds, puddles, plants and rocks that \"pool,\" and sprays and mists\u2014really any water source will be a welcome delight for them!\n\nYou can attract birds and assist in their well-being by creating and offering a simple birdbath. Once you do, it will offer great joy for you and your child as you observe birds while they drink and bathe. Sometimes birds may even wait in line for a turn at the water you provide, or you may notice several birds or several different species drinking simultaneously.\n\nThere is nothing is more joyful than watching a bird take a bath, either, as they immerse their heads and bodies under water, rolling, fluffing, and shaking water everywhere. It is bird bliss!\n\n# **TAKE ACTION**\n\nExplain to your child how fortunate we are to have water sources in our homes: faucets for drinking water and bathing. But what about wildlife? What about the birds in our yard? They need to drink water and take baths, too, just like we do. Take action and offer a water source for your bird neighbors. Then, feel good knowing you're helping to sustain a habitat necessary for wild birds and their health and survival. It will be fun to experience birds at your water source, too. Here are some tips for creating a splashy tweet retreat:\n\n**  You don't need to buy a birdbath. Any shallow pan will work as a water source for birds. It should not be very deep, 2 to 3 inches is plenty. Stone, ceramic, or plastic trays, such as those used for flowerpots, are an ideal depth. A ceramic dog bowl will work well, too.**\n\n**  Water must always be fresh. Dump, wipe, and refill with fresh water at least every other day.**\n\n**  Birds enjoy visiting water sources that are on a pedestal or on the ground. Create your splashy retreat near a shrub or potted plant so birds have a safe place to retreat for drying off. If possible, place it near a window where you may observe birds while you're inside as they bathe and drink outside.**\n\n**  Place a large rock or several pebbles in the center of your birdbath, protruding out from the water's surface. Or, place a small tree branch across the birdbath's surface. The birds will perch upon the branch or rocks. This also helps birds judge the depth of the water.**\n\n  **_Promotes stewardship, curiosity, and an appreciation for avian wildlife_**\n\n**DID YOU KNOW?**\n\n**As you begin observing birds at your birdbath, you'll notice they don't all drink the same way. Some species dip their beaks in and sip, while others dip their beaks in the water, then raise up their beaks to swallow the water they've captured. How many different drinking behaviors can you spy?**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do all birds take baths?**\n\n**A: Feathers are one characteristic unique to birds, and bathing is a very important part of feather maintenance for them. But birds don't bathe in just water. Birds may also bathe in the sun, stretching out their wings and allowing the ultraviolet rays to disinfect their feathers. You may also notice birds bathing in dust and sand, moving and rolling around as if in water. Dirt baths also help maintain a healthy plumage, cleaning feathers and helping to balance oil levels.**\n\n# **A RUSTIC ROOST**\n\n**Y OU'LL OFTEN SEE** colorful and embellished birdhouses available for purchase, but think about it: with the exception of bowerbirds, how often do we see colorful bird-made nests in the wild? Do birds want colorful houses? There's a reason female birds are often more drab in color than their male counterparts: to camouflage them. Being camouflaged helps to protect female birds from predators while sitting patiently and stationary upon the nest during egg incubation and chick rearing. The last thing a mother bird needs while incubating eggs and caring for her chicks is unwanted attention.\n\nIn the wild, bird nests are also often camouflaged so that they blend in with their natural surroundings. They're designed with natural elements and meld seamlessly with the trees, shrubs, or landscapes where they were created. A bird nest is a vulnerable space: danger prowls at every turn in the form of predators, such as hungry snakes, mammals, and even other birds.\n\nSupplying a birdhouse for nesting birds is an engaging and fun way to observe birds in the wild as a family unit. It provides an opportunity to watch a mother and father bird come and go as they fill it with found and gathered nesting materials, search for food, and feed their young. And if you're lucky, you may even experience the miraculous journey as the chicks fledge from their nest. In addition to these intrinsic joys, it also offers a habitat for birds during nesting season.\n\n# **TAKE ACTION**\n\nDesign a rustic, natural birdhouse with your child\u2014one that is simple, subtle, functional, and beautiful. Create a house that will harmonize with nature while also providing shelter for the next generation of a wild bird family.\n\n**1 standard, simple wooden birdhouse from your local bird or garden store**\n\n**Natural materials for decorating**\n\n**  Twigs**\n\n**  Bark**\n\n**  Pine cones**\n\n**  Driftwood**\n\n**  Leaves**\n\n**  Small branches**\n\n**  Moss**\n\n**  Small pebbles**\n\n**Glue gun and glue sticks**\n\nEmbellish your standard birdhouse with natural materials, using a glue gun or glue sticks to attach as few or as many items as you'd like. There's no right or wrong way to design a rustic birdhouse. The idea is simple, and the design is 100 percent yours and your child's.\n\nOnce finished, hang your rustic roost outdoors from a porch eave or on a pole with a predator baffle, which will protect it from predators such as hungry raccoons or snakes. Your birdhouse will not only blend beautifully with the environment, but it will also provide shelter and comfort to a feathered family. At the end of each nesting season (fall and winter), dispose of any nesting materials from inside the birdhouse to help keep it clean and ready for spring nesting once again.\n\n  **_Promotes creativity, empathy, and stewardship_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: What should I do if I find a baby bird on the ground?**\n\n**A: If you find a baby bird with feathers on the ground and he appears helpless, he has likely fledged (left the nest) and is trying to figure out how to use his wings and feet to fly and land, not unlike a toddler taking his first steps. This \"first time out of nest\" process may take a day or two, and 99.9 percent of the time, his mom and dad are nearby, keeping an eye on the situation and feeding him. Resist the urge to intervene. If you find a naked nestling on the ground, it may have fallen from its nest due to a storm or a predator may have pulled it out. Look around for its nest, and if located, place it back in the nest. If you can't locate a nest, contact a bird rehabber in your area.**\n\n**DID YOU KNOW?**\n\n**Bird babies fall into two categories:** **_precocial_** **and** **_altricial._** **Precocial birds open their eyes soon after hatching and quickly leave the nest, following the mother around for food (think chickens, ducks, and shorebirds). Altricial chicks have two stages of development:** **_nestling_** **and** **_fledgling._** **They are born blind, featherless, and helpless and remain in their nest where parents keep them warm, fed, and sheltered until they are ready to** **_fledge_ ,** **or leave the nest. Bird parents must also constantly forage for food to feed their chicks. Nest time is a very busy time for bird parents!**\n\n# **BIRDS IN MOTION**\n\n**S PRING IS SUCH** a welcome season, with fresh, green growth and longer days following winter's chilly and gray days where it's often a challenge to play outdoors. Come spring, sunshine and warmer temperatures beckon. It's the perfect time to get outside and stretch your wings, at long last.\n\nTake your wee chick under your wing and get outdoors. While you're out there, make it a challenge to spy on birds, observing the many ways they move and enjoy spring weather. Birds are just as excited about springtime as humans: there are bugs to be caught for breakfast, lunch, and dinner\u2014and snacks in between, finally! There are mates to attract and nest materials to gather. Birds are getting busy! You'll also see that they move in many different ways.\n\n# **OBSERVE**\n\nHow many different ways do birds move? Can you find these behaviors?\n\n**  A bird flapping its wings**\n\n**  A bird soaring**\n\n**  A bird hopping**\n\n**  A bird walking**\n\n**  A bird pecking**\n\n**  A bird waddling**\n\n**  A bird perching**\n\n**  A bird singing or calling**\n\n**  A bird running**\n\n**  A bird gathering nesting materials**\n\n**  A bird pulling on a worm**\n\nTurn your observations into a game of Simon Says. Use bird behaviors as commands:\n\n# **SIMON SAYS...**\n\n\"Hop\" like a sparrow\n\n\"Flap your wings\" like a wren\n\n\"Soar\" like a hawk\n\n\"Waddle\" like a duck\n\n\"Perch\" like a songbird\n\n\"Walk\" like a crow\n\n\"Run\" like a thrasher\n\n\"Gather\" like a robin\n\n\"Strut\" like a pigeon\n\nThe playtime will be invigorating. You may even enjoy this game indoors during inclement weather.\n\n  _Encourages observation skills, verbal skills, exercise, and well-being_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do all birds fly?**\n\n**A: Not all birds can fly. For example, emus, kiwis, ostriches, and penguins are birds that do not fly. There are approximately forty species of birds that do not fly, but none of them live in North America.**\n\n# **DRAW THAT BIRD!**\n\n**U SING A FIELD GUIDE** with your child for bird identification purposes is a wonderful method to properly ID birds you encounter. However, sometimes it's tricky to identify a bird out in the wild. After all, the bird will not sit and patiently wait while you flip through your guide book. Birds are busy. They have places to go and things to do.\n\nAn alternative method to help you and your child easily identify birds you encounter is to practice sketching them as you watch them in their environment. With your sketch and notes in hand, you may later, at your leisure, reference a field guide to compare your field sketches and notes to various species, ultimately and hopefully identifying the birds you noted. It's a fun challenge to learn bird names and to properly ID them. An important element is to not take the process too seriously. Have fun with it, keeping in mind that even bird experts wind up scratching their heads in wonder with bird ID at times, too!\n\n# **TAKE NOTE**\n\nYou don't need to be an artist to sketch the birds you see! Practice and have fun. Keep your drawings loose and simple as you follow these steps:\n\n  1. Create a roundish head.\n\n  2. Create an oval body.\n\n  3. Sketch the general length and angle of the tail.\n\n  4. Sketch the general shape and size of the beak.\n\nOnce you've sketched your bird, make notes to describe specific details you observed: the bird's behavior (was it hopping, running, flying, calling, feeding\/foraging?), its location, sounds you heard, and other visual details, such as the bird's size (small, medium, or large) and specific markings and patterns that stood out on the bird's body (white eye stripe above eye, white wing bars on wings, dark stripe through eye, dark bars on tail, speckles on chest or belly).\n\nBefore too long, you may be able to identify a bird simply by its silhouette!\n\n**  Small silhouette (i.e., Anna's Hummingbird, American Goldfinch)**\n\n**  Medium silhouette (i.e., Wood Thrush, European Starling)**\n\n**  Large silhouette (i.e., Pileated Woodpecker, Great Horned Owl)**\n\n  **_Promotes creativity, scientific knowledge, and observation skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Where does all of the information we have about birds come from?**\n\n**A: Scientists study animal behavior, and the field of ornithology is the specific scientific study of birds. Information about birds comes from many sources: ornithologists, naturalists, birders, photographers, and people just like you and me, who take the time to observe and learn about birds and then share their findings.**\n\n# **ZOOM IN**\n\n**_F IELD MARKS_** are patterns, designs, spots, stripes, and colors on birds that humans use to distinguish one bird species from another. Birds use them to identify their own bird communities, too! Take your sketching to the next level by adding the field marks to your bird sketches.\n\n# **TAKE NOTE**\n\nNext time you're out sketching birds, pay particular attention to their field marks. Here are some things to look out for.\n\n# **HEAD FIELD MARKS**\n\nCrown stripe (White-Throated Sparrow)\u2014a stripe across the top of the bird's head or crown\n\nEyebrow stripe (Carolina Wren)\u2014a stripe above the eye\n\nEyeline stripe (Chipping Sparrow)\u2014a line through the eye\n\nThroat patch (White-Throated Sparrow)\u2014distinctive coloring on a bird's throat\n\nEyering (Blue-Gray Gnatcatcher)\u2014a distinct ring around the bird's eye\n\n# **WING FIELD MARKS**\n\nWingbars (Northern Mockingbird)\u2014thin, white or pale-colored stripes across the wing\n\nWing patches (male Red-winged Blackbird)\u2014blocks of color on the wing\n\n# **TAIL FIELD MARKS**\n\nLong tail\u2014Brown Thrasher\n\nShort tail\u2014White-Breasted Nuthatch\n\nForked tail\u2014Eastern Phoebe, Purple Martin\n\nTail with rounded ends\u2014Gray Catbird\n\nTail with square ends\u2014Eastern Kingbird\n\nUpward-pointing tail\u2014Carolina Wren\n\nDownward-pointing tail\u2014Northern Cardinal\n\nSpots, stripes, or a coloration different from the bird's body\u2014Spotted Towhee\n\n# **BEAK FIELD MARKS**\n\nCone-shaped beaks (northern cardinal, American goldfinch)\u2014strong, cone-shaped beaks ideal for cracking open seeds\n\nLong beaks (egrets, herons)\u2014long and pointed for spearing fish and frogs\n\nPetite beaks (warblers)\u2014pointy and narrow for grabbing bugs off of leaves\n\nFlat beaks (flycatchers)\u2014flat and wide at the base for catching bugs in mid air\n\nSpatulate beaks (spoonbills)\u2014spoon-like for moving through mud and water to catch fish and crustaceans\n\nHook beaks (hawks)\u2014for flesh tearing\n\nChisel beaks (woodpeckers)\u2014for boring holes in wood and to search crevices for bugs\n\nPrying beaks (crossbills)\u2014specific for prying apart pine cone scales to get at pine cone seeds\n\nProbing beaks (hummingbirds)\u2014for drinking deep into tube-shaped flowers and catching small bugs.\n\n  _Promotes attention to detail, observation, creativity, and scientific knowledge_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do all birds have feathers?**\n\n**A: Yes, all birds have feathers. They are the only living animal on our planet with feathers. That makes them pretty special, don't you think?**\n\n# **FEATHERED FIRSTS**\n\n**I T IS ALWAYS EXCITING** to notice a bird species you've never seen before. It's a feeling not unlike, perhaps, spying a unicorn in the woods or a mermaid out at sea\u2014there's something mysterious and unknown that you've never seen before, and then suddenly, you see it. Even more wonderful is that spotting a new bird species may happen anywhere: in your own yard, on your way to school, through a car or bus window, while walking or bike riding, in your community, or during family travels!\n\nAs you and your child practice close observation of your outside environment, you'll certainly notice _resident_ bird species\u2014those that live in your area year-round. Many will be familiar and common. But every now and again, you will spy a species and wonder, _what is that bird? I've never seen it before!_ It becomes a feathered first, and the fun begins as you and your child attempt to identify it.\n\nYou may also discover a common species with unusual markings, such as a _leucistic_ bird\u2014a bird with predominantly white feathers where the standard brown, blue, red, or yellow plumage should appear.\n\nDuring migration seasons, you may discover a variety of species passing through or many new species that may even stick around for the season. On occasion, a bird species that is very far away from its usual habitat may appear in your area as a vagrant, or accidental, visitor. This happens for a variety of reasons. A storm might force a flying bird off of its regular migratory route, and the exhausted bird may need to rest far from its range. Juvenile birds making their first migration may miss their destination, overshooting or undershooting where they're supposed to be. And those who study birds believe that some birds may even be wanderers. Just as some humans have a tendency to wander, birds may do the same\u2014perhaps in search of better food sources and habitat.\n\nIt's always exciting to see a new and unusual species, and it's always a thrill to see a \"first\"!\n\n# **TAKE NOTE**\n\nDocument the bird species you and your child see for the first time ever in your lives. Many birders keep track of each bird species they've seen for the first time and have positively identified, and these bird species are called _lifers\u2014_ meaning a bird that you properly identify and see for the first time in your life.\n\nOnce you begin noting all of the new-to-you bird species, you will be surprised at how many birds you actually encounter. Note each new species everywhere you go: at home, away from home, during travels\u2014anywhere!\n\nNever-before-seen species feel like a special gift from nature, and with each newfound treasure comes the experience of elation and intrinsic joy.\n\n**LIFER LIST**  \n---\n\nToday, for the first time ever, I saw a:\n\n**Species:**\n\n**Location:**\n\n**Date:** | **Species:**\n\n**Location:**\n\n**Date:**\n\n**Species:**\n\n**Location:**\n\n**Date:** | **Species:**\n\n**Location:**\n\n**Date:**\n\n**Species:**\n\n**Location:**\n\n**Date:** | **Species:**\n\n**Location:**\n\n**Date:**\n\n**Species:**\n\n**Location:**\n\n**Date:** | **Species:**\n\n**Location:**\n\n**Date:**\n\n  **_Stimulates anticipation, excitement, goal-setting, and wonder_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do people make lists of the different birds they see?**\n\n**A: Nature is very complex and diverse. Making lists and categories of the living things we encounter in nature helps us to better organize and understand them.**\n\n**DID YOU KNOW?**\n\n**A \"Big Year\" is an informal competition among birders to see who can identify, by sight or sound, the largest number of bird species in one calendar year. In 2016, Arjan Dwarshuis broke the world record for most bird species seen in one year (6,852 species), breaking the previous record set by Noah Strycker in 2015 (6,042 species). Strycker traveled across 41 countries and visited all seven continents for his Big Year quest. For more information about this competition, visit the American Birding Association website,ABA.org.**\n\n# **FANTASTIC FEEDERS**\n\n**M AINTAINING BIRD FEEDERS** provides you and your child with the opportunity to easily share your lives with the lives of birds, and right through the comfort of your home's window\u2014a joyful and rewarding experience, indeed!\n\nFind joy in every aspect of maintaining your bird feeders: the thoughtful planning of how and where you'll place the feeders in your yard; the maintenance of your feeders as you keep them clean and keep birds healthy; the reward of providing for hungry birds; and the thrill of observing and making birds part of your family's day-to-day life experience. Each experience with bird feeding is a gift from nature, and the bonus is that nature is brought into your home through your windows.\n\nExperiencing bird feeders may take as little time or as long as you'd like\u2014from just a few moments through a window as your family enjoys breakfast to a day-long experience as you and your child make time to observe, enjoy, and wonder out your windows anytime you chance to glance out.\n\n# **TAKE ACTION**\n\nWhen establishing and maintaining feeders, consider an organized plan as to what food you will offer birds, what type of feeder you need, and where you will place it. For example, bird feeders come in many shapes and sizes to support a variety of food sources, and you can hang them from shepherd's hooks, poles, tree branches (although here they'll be prone to squirrels and other animals), deck railings, and windows. Here are things to keep in mind when setting up your feeder:\n\n**  Consider placing foods enjoyed by smaller birds, such as finches, in an area farther away from foods enjoyed by larger, more assertive birds, such as jays and woodpeckers.**\n\n**  Implement a baffle on bird feeder poles to keep squirrels, raccoons, and other hungry, non-bird animals out of feeders. You'll find baffles wherever bird feeders are sold.**\n\n**  Place your feeders where they will be safe from window collisions. Windows reflect foliage and sky, and many birds fly into them believing that's just what they are\u2014open space. Feeders are safest when placed closest to windows\u2014within 3 feet\u2014or when at least 30 feet from a window.**\n\n**  Place feeders near trees or shrubs, so birds may take refuge and rest when necessary, but don't place them within 10 feet, which may provide jumping off points for squirrels.**\n\n**  Place hummingbird feeders in shady or partially shaded areas. This prevents the sugar-water solution from fermenting in the sun.**\n\n**  Designate a dry, airtight place to store your bird food, such as a lidded tub.**\n\n**  Regularly clean your feeders, especially after rain, which can cause seed to mildew. To clean, just scrub with a solution of 8 parts water and 1 part vinegar, rinsing very well and drying after each scrubbing.**\n\n**  Certain bird species enjoy feeding on the ground, below feeders. Keep this area tidy, raking away soiled and old seed debris to maintain a healthy environment for feeding.**\n\nNote: From time to time, you might notice a finch at your feeder with red, swollen eyes. Unfortunately, this is a sign of an infection that may spread to other finches through a contaminated feeder. If this occurs, take your feeder(s) down, clean them, and keep them down for a week to prevent the spread of the infection. You may report signs of finch eye diseases (mycoplasmal conjunctivitis) at Project FeederWatch, through the Cornell Lab of Ornithology. With your feedback, scientists may track the spread of this disease, which is not contagious to humans but is to other finches.\n\n# **TYPES OF FEEDERS**\n\nFinch feeder\u2014sock feeder, mesh feeder, or tube feeder to hold Nyjer seed\n\nHopper feeders\u2014for a variety of seeds, nuts, and fruits\n\nNectar\/jelly feeders\u2014for hummingbirds and orioles\n\nPlatform feeders\u2014for a variety of seeds, nuts, and fruit\n\nStackable cylinder feeder\u2014for stacking seed, fruit, and nut cakes\n\nSuet\/fat feeders\u2014as the name implies, for suets and fats (cool weather only)\u2014designs with \"tails,\" or long tips at the base, support the natural feeding habits of woodpeckers, providing support for their tails\n\nTube feeders\u2014for smaller seeds\n\n# **TYPES OF BIRD FOODS**\n\nA diverse mix of seeds with other food options will invite the greatest variety of birds. Here are some ideas:\n\n**  Apples and oranges will be enjoyed by buntings, cardinals, catbirds, chickadees, mockingbirds, orioles, tanagers, and waxwings.**\n\n**  Black oil sunflower seeds will offer the most versatility with bird feeding, as they are enjoyed by most birds.**\n\n**  Dried berries and raisins will be enjoyed by bluebirds, mockingbirds, orioles, robins, and waxwings. Soak them in water overnight prior to providing them.**\n\n**  Mealworms will be enjoyed by bluebirds, robins, and wrens.**\n\n**  Nectar (4 parts water to 1 part sugar) will be enjoyed by hummingbirds.**\n\n**  Nyjer will be enjoyed by finches.**\n\n**  Peanuts (raw, unsalted, shelled, or in the shell) will be enjoyed by blue jays, nuthatches, titmice, and woodpeckers.**\n\n**  Shelled corn will be enjoyed by Northern Cardinals, doves and blackbirds.**\n\n**  Suet (during cold weather) will be enjoyed by bluebirds, buntings, cardinals, chickadees, creepers, goldfinches, grosbeaks, nuthatches, tanagers, titmice, warblers, woodpeckers, and wrens.**\n\n  **_Fosters stewardship, responsibility, and an opportunity to learn more about birds_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do we see just the same types of birds at our bird feeders over and over again?**\n\n**A: Many common birds visit bird feeders. This is a great thing, because when we become truly familiar with the ordinary, it provides us an opportunity to notice the extraordinary: we may then begin to notice a bird's unique personality and behavior. And we become skilled and ready to notice when a mysterious species happens by.**\n\n**DID YOU KNOW?**\n\n**The House Finch originated in the western United States and Mexico. However, in 1940 a small number of them were turned loose on Long Island, New York, following failed attempts to sell these \"Hollywood finches\" as pets in cages. Since that release, they have reproduced and their numbers have spread across the United States and southern Canada.**\n\n# **GETTING TO KNOW YOU**\n\nI **T TAKES TIME** to get to know birds, just as it takes time to get to know people. Once you start to really pay attention\u2014to notice and _encounter_ birds\u2014you'll see that certain species have very distinct personalities. Explain to your child that just as people have different personalities or behaviors\u2014some people are shy, some people are bold, some people are silly\u2014birds have different personalities, too.\n\nBlue jays tend to be bold and boisterous. Wrens are the cats of the bird world, ever-curious about every little thing, inquisitive, cheery, and tenacious. Leave a window, garage door, or house door open? You can bet a wren will explore the open space. One pair built their nest inside my sister's kitchen. Remarkably, she kindly left her back door open for a month so the parents could come and go as needed to find food (bugs) and feed their nestlings, which hatched and fledged and were soon shooed back outdoors. That's what happens when you leave a door open in the spring with wrens around. Lesson learned.\n\nAnd personalities even vary among specific species: one White-Breasted Nuthatch may be passive, while another may be aggressive. One blue jay may be courageous, while another is more timid. Birds strive to obtain the very same things we do: food, shelter, and a safe place to raise their families. We coexist, birds and people. Yet, when we have intimate encounters with birds, it gives us insight into the workings of the world and humbles us.\n\n# **OBSERVE**\n\nGet to know the inner lives of birds. Become familiar with their behavior. Make time for little pauses in your day-to-day doings with your child so you may observe and experience birds around you, and these little pauses will provide a huge reward as you glimpse into their inner lives. Look for these personality traits and behaviors in birds:\n\n**  A bird that exhibits curiosity (exploring)**\n\n**  A bird that exhibits affection (courtship behavior, mutual preening, sharing food, caring for young)**\n\n**  A bird that exhibits fear (quick flight and escape, or freezing in place with wide eyes)**\n\n**  A bird that exhibits aggression or assertiveness (posturing to chase away another bird; hissing at, lunging at, or diving at another bird)**\n\n**  A bird that shares with another bird (sharing food or sharing space at a feeder)**\n\n**  A bird that warns of danger (alarm or distress call)**\n\n**  A bird interacting with another bird (chattering, mutual preening, or simple camaraderie of flying and feeding together)**\n\n**  A bird that appears sad, lost, or ill (looking for a lost mate or chick, making a cry hoping for a response, appearing listless and droopy)**\n\n**  A bird that is sleepy or tired (resting with eyes closed, bill tucked into feathers)**\n\n**  A bird that is relaxed (often making purring sounds; may sun itself, wings out and relaxed, feeling safe and non-threatened)**\n\n**  A bird that communicates with another bird (calling to share news of food)**\n\n**  A bird that is playful (toying with leaves to make sense of its surroundings)**\n\nAs you observe, before long, you will be fully aware of each individual bird and its distinctive and unique personality. You will know these birds and count them among your friends and, perhaps, even fall in love with one or two (or three or four).\n\nAs you become more familiar with the birds that you observe and the behavior of each, ask your child if he has a favorite bird, and why? Record or document your child's response. Share your favorite bird with your child, too, explaining what traits drew this bird toward your heart.\n\n  _Promotes empathy, observation skills, and understanding_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do birds have feelings?**\n\n**A: Scientists (ornithologists) and others who study bird behavior believe that birds do indeed experience emotions. They feel fear, relaxation, curiosity, and excitement. Their range of emotions continues to be studied and questioned. As you watch birds and their behavior, what do you think?**\n\n# **THE PERFECT PATCH**\n\nF **ORGING A CONNECTION** to the natural world around us is vital to our mental and physical well-being. When we immerse ourselves in nature, we form an intrinsic stewardship and interconnectedness to plants, animals, and this big, beautiful world. What a gift to give our children! We become in tune with the network of energy between all living things: birds, bugs, seeds, plants, and ecosystems.\n\nOne meaningful and unique way to immerse yourself in nature and the birds who call it home begins with having a \"patch\"\u2014a specific place, all your own, where you may \"bird\" each day to take notes and count the variety of species found in that specific patch. Every bird enthusiast should have a special birding patch, a place you can visit regularly, a place you may get to know better than anyone, a place to unplug, tune in, and enjoy immensely. A place that is completely yours\u2014shared with birds, of course!\n\nYour patch might be a specific space or area in your backyard, or any place you can visit with your child by foot or bike, such as a green area near your school's playground, a bus stop, or a city park. Any natural space or locale that might entice birds to visit will make a good patch for you and your child.\n\n# **TAKE NOTE**\n\nOnce you and your child have claimed a patch, begin visiting and spending time there regularly. Bring a pencil or pen for field notes and dress comfortably. While at your patch, quietly observe. Carefully listen. What do you hear? What do you see? How do you feel? Most birds are active during the early morning and in the evening. Does this make a difference with the bird activity at _your_ patch?\n\nNow that you and your child have a patch all your own\u2014and you may claim more than one patch in more than one place\u2014make notes and sketches to document the birds you encounter there.\n\nPatch location:\n\nDate: Time: Weather:\n\nHow many birds did you see?\n\nHow many different kinds of birds did you see?\n\nHow many bird sounds did you hear?\n\nSketches:\n\n  _Promotes environmental awareness, relaxation, observation, and auditory skills_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Some birds look exactly the same, but one is bright and one is dull. Why is that?**\n\n**A: Sometimes the male and female in a species look different from one another, a term called** **_sexual dimorphism._** **Often, the male is more colorful than the female. A male's bright color, or plumage, may help communicate to a female he'd like to attract that he is healthy. A female's dull coloring, or plumage, helps her to blend in as she incubates eggs on the nest.**\n\n**DID YOU KNOW?**\n\n**GISS is an acronym for general impression, size, shape. The more time you spend with birds noting GISS, the sooner you'll be able to identify species by a mere silhouette, shape, flight pattern, or behavior pattern.**\n\n# **RAINDROP CAFE**\n\nR **AINY DAYS BECKON EXPLORATION!** Everything is washed, fresh, clean, and dewy. The air smells moist and alive with new scents. Water drops pool and glisten. Puddles form. Just as rain is often a welcome gift for our gardens, landscapes, and water sources, birds celebrate rain, as well. They rely solely on nature for the water they need to survive.\n\nHead outdoors with your child following a rainstorm, specifically to observe birds as they relish in all the needed moisture that nature has just given them. Find a spot where you can observe birds in action, such as on a deck or porch. If it's too rainy to be outside, you can also observe birds through a window.\n\n# **OBSERVE**\n\nCan you find these rainy-day bird behaviors?\n\n**  A bird sipping from a shallow puddle on a deck, porch, or sidewalk**\n\n**  A bird drinking from a puddle on the ground**\n\n**  A bird drinking water droplets from a leaf**\n\n**  A bird bathing in a puddle**\n\n**  A bird bathing in water droplets on a leaf (hummingbirds love to do this!)**\n\n  **_Promotes awareness, understanding, and empathy_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Where do birds go while it's raining?**\n\n**A: Although birds are pretty waterproof, birds avoid flying during big rainstorms, and do their best to hunker down and perch out of the rain under foliage. This is because storms make it difficult for flight to take place.**\n\n# **A HAVEN FOR HUMMINGBIRDS**\n\nT **HERE MAY NOT BE** a more welcome bird anywhere in the world than the hummingbird. Wee wisps of air, buzzing wings, sparkling eyes, and a brilliant rainbow of iridescent feathers\u2014they fill us with wonder and delight as they capture our hearts.\n\nHummingbirds can hover, fly backward or forward, and dazzle us, not only in their appearance but also with their precise aerial acrobatic capabilities. Approximately one dozen hummingbird species summer in the United States, with the most common being the Ruby-Throated, the Black-Chinned, the Anna's, and the Rufous. On the Pacific Coast, Gulf Coast, and in the Southwest, lucky you, hummingbird season is year-round!\n\nAs migrators, their arrival each spring is highly anticipated. And their departure, come fall, can often leave us feeling a bit sad as we worry over their long journey south. Many migrate thousands of miles each spring and fall. Can you imagine such a huge journey for such a tiny bird through the elements it may encounter: storms, predation, avoiding impact on human structures, navigating correctly, the Ruby-Throated flying over the Gulf nonstop with nowhere to rest, and the like?\n\n# **TAKE ACTION**\n\nYou and your child may create a safe haven for these wondrous jewels right in your own backyard, to welcome them home come spring. Once they locate the shelter and food sources you've provided, they will remember this haven and return to it year after year during each spring migration!\n\n1. Integrate colorful, nectar-rich plants, either in pots or planted in the ground. Hummingbirds enjoy nectaring from tube-shaped flowers. Check with a local nursery or field guide for your region to find native plants that hummingbirds love, such as:\n\n**  Varieties of beebalm ( _Monarda fistulosa, M. citriodora, M. didyma, M. punctata_ )**\n\n**  Trumpet honeysuckle ( _Lonicera sempervirens_ )**\n\n**  Cardinal flower ( _Lobelia cardinalis_ )**\n\n**  Sage ( _Salvia_ species)**\n\n2. Place hummingbird nectar feeders in your yard or on your porch or patio, perhaps even adjacent to the flowers you've planted for them. Select areas that are shaded, or at least not in direct sunlight, so the nectar won't get too hot and ferment\/spoil. Don't place more than one hummingbird feeder in the same area, as hummingbirds become territorial once they've secured a food source. You'll notice, once a food source is secured, one hummingbird will perch and watch over this food source, guarding it from other hummingbirds. Hummingbird feeders can be decorative, but be sure to select a feeder that:\n\n**  Is easy to clean**\n\n**  Will not drip**\n\n**  Is easy to fill**\n\n**  Has an ant mote\u2014a reservoir that may be filled with water to keep hungry ants at bay (ant motes may also be purchased separately)**\n\n3. Fill your feeder with homemade nectar (see the recipe below). Take your hummingbird feeder down and clean it at least twice a week, using warm water and a spoonful of white vinegar, rinsing very thoroughly with each cleaning. Use a clean toothbrush to scrub your feeder's sipping portals where mildew may grow. Use a bottlebrush to clean the interior reservoir. With each cleaning, replenish the feeder with fresh nectar, as sugar-water nectar ferments and spoils in the heat after a few days. This is part of the fun, the joy, and the process of providing for and caring for hummingbirds!\n\n# **HUMMINGBIRD NECTAR RECIPE**\n\nMix 1 cup of sugar with 4 cups of water (or 1 part sugar to 4 parts water for a smaller or larger portion). That's it! Here are some tips:\n\n**  Warm all or some of the water on the stove before adding the sugar. Stir the sugar in until it dissolves.**\n\n**  Do not use honey, brown sugar, artificial sweeteners, or red dye.**\n\n**  It is absolutely _not_ necessary to add food coloring to hummingbird nectar. Not only is it unnecessary, it's not healthy for hummingbirds. Humans don't care for artificial coloring in their food sources. Why supply these wee birds with concentrated, artificial color?**\n\n**  Allow the nectar to cool before placing it in the feeder.**\n\n**  Store unused nectar in a lidded glass jar in the refrigerator for up to one week. A large mason jar works well, or a clean glass milk jug works, too.**\n\n**  If you live in a region where species only breed for the summer, keep your hummingbird feeders fresh, filled, and available through October. Weary migrants may rely on your feeder as they travel through, even though they're few and far between in late fall.**\n\n  **_Encourages stewardship and responsibility while fostering wonder and joy_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do hummingbirds only drink sugar water?**\n\n**A: Hummingbirds need protein and eat small bugs, such as gnats, which they catch in mid-air. Bugs, along with sugar water and nectar from flowers, provide them with the energy they need to survive through each day.**\n\n**DID YOU KNOW?**\n\n**Hummingbirds must eat every ten to fifteen minutes and visit between one thousand and two thousand flowers a day. At night, hummingbirds go into a state of torpor, or deep sleep, where their metabolic rate is slowed by as much as 95 percent, so as to not consume too much energy while sleeping.**\n\n# **BOUNTIFUL BIRDSCAPE**\n\nD **ISCUSS WITH YOUR CHILD** the essential elements your family needs to be comfortable, safe, and healthy: a home that offers shelter and protection, food, and water. Explain that birds require the same things to survive in the wild.\n\nTogether, explore your outdoor space and determine if it is a welcoming place that birds may inhabit and feel safe and protected in. Like all wild animals, birds require food, shelter, water, and a place to raise their young. These four components help make a happy, healthy bird habitat.\n\nIt's easy to create a birdscape: a welcoming and healthy habitat for birds. The birds will be rewarded with a safe, nourishing haven, and you and your child will be rewarded with their presence and knowing you've made improvements to the environment.\n\n# **TAKE ACTION**\n\nCreating a birdscape helps implement an ecosystem that supports the health and livelihood of many living things, which, in turn, supports birds. To feed the birds is to also feed the bugs! Most North American birds\u201496 percent of them (excluding seabirds)\u2014feed their young insects and arachnids. There must be a chain of life present in a healthy birdscape; the more diverse, the better.\n\n1. Start small. Work with tiny patches outside your home to integrate native plant species into existing garden areas. Even pots with native plants on decks, porches, terraces, and patios will be beneficial.\n\n2. Integrate native bushes and trees that provide berries. Not only are the berries an important food source for wildlife, but their flowers provide a vital food source for nectar-feeding insects, while their structure provides shelter and nesting spaces for a host of wildlife. Consider these species if native to your area:\n\n**  Alternate-leaf dogwood**\n\n**  Arrowwood**\n\n**  Blueberry**\n\n**  Chokeberry**\n\n**  Deerberry**\n\n**  Bilberry**\n\n**  Elderberry**\n\n**  Hawthorn**\n\n**  Spicebush**\n\n**  Virginia creeper**\n\n**  Winterberry**\n\n3. Feed the butterflies, bees, and moths with native plants, and you also create a haven for toads, spiders, and other hoppy, creepy crawlies, which birds will love. Plants native to your area may include:\n\n**  Beebalm**\n\n**  Black-eyed Susan**\n\n**  Cardinal flower**\n\n**  Goldenrod**\n\n**  Joe-pye weed (requires a lot of space)**\n\n**  Milkweed**\n\n**  Sage**\n\n4. Avoid use of fertilizers and pesticides. Birds need bugs, as do toads and other wildlife. Allow birds and their wild friends to be your natural exterminators.\n\n5. Implement a water feature: a birdbath, bubbler, or fountain.\n\n6. Include evergreens, either planted in the ground or as shrubs in pots. Evergreens, such as junipers, provide birds with protective shelter year-round.\n\nBit by bit, piece by piece, as you and your child enhance your outdoor space with native plants, you will be inviting a host of wildlife, including your bird friends. Take joy in your efforts as you experience a growing diversity of wildlife right in your own backyard.\n\n  **_Facilitates stewardship, outdoor exercise, and living in harmony with the environment_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: What are native plants?**\n\n**A: Native plants are plants that occur naturally in our area, having existed in the natural landscape for many, many years. They are part of our region's ecosystem; they depend on the wildlife in our area, and the wildlife depend on them, too.**\n\n# **WING IT!**\n\nW **HEN WE THINK OF BIRDS ,** we can't help but think of flight. And when we think of flight, wings come to mind, of course! Birds are masters of the sky, indeed.\n\nFlight patterns vary by species, as do the shapes, styles, and markings of bird wings. Each type of wing has evolved with a specific function to aid in a species' survival. Explore flight and bird physiology with your child, and your child will certainly soar to new heights with wonder and amazement regarding these winged wonders.\n\n# **OBSERVE**\n\nTake a moment to watch a bird in flight with your child, perhaps outdoors or through a window. Explain to your child that our arms are similar to a bird's wings. We both have an upper arm bone that connects to a shoulder. We both have elbows. We both have a forearm. And we both have wrists that connect our forearms to our hands\u2014although a bird's hand is a bit different than ours. A bird's upper arm and forearm make up the section of wing close to its body, its inner wing. The remainder of the wing, the outer portion from the wing's center to the wing's tip, makes up the hand section.\n\n**DID YOU KNOW?**\n\n**Turkey vultures and albatross can soar for hours without ever flapping their wings.**\n\nSome birds (like eagles and hawks) soar with passive, soaring wings. Some birds (the Laysan Albatross, for example) have active soaring wings\u2014these are long and narrow and use wind currents, which allows them to soar for a long time. Some birds (swifts and terns) have high-speed wings, which are long and thin and make these birds very fast. Some birds (cardinals or the American Robin, for example) have elliptical wings, which are perfect for fast take off and short, quick bursts of speed, but the speed is not maintained. Some wings are tiny and very fast\u2014hovering wings\u2014such as those found on a hummingbird.\n\n**BIRD SPECIES** | **WING BEATS PER 10 SECONDS**  \n---|---  \nAmerican Crow | 20 wing beats  \nPigeon | 30 wing beats  \nChickadee | 70 wing beats  \nHummingbird | 700 wing beats\n\nEncourage your child to hold his arms out like bird wings. Invite him to flap his \"wings\" like a bird would, and practice flapping until his arms get tired. Then, using a timer\u2014even just counting out loud will work\u2014take it a step further and record how many wing flaps your child can make within a ten-second time span. Finally, compare his results to the birds listed on the previous page.\n\nIt's quite remarkable to think about the strength and stamina many birds exhibit with their wings! To wrap up this exercise, relax by reading an engaging children's book aloud together, such as _Animals in Flight_ , by Steve Jenkins.\n\n  **_Facilitates exercise, empathy, and appreciation_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are wings used just for flying?**\n\n**A: All birds have wings, and most birds (excluding penguins and emus, among others) use them to fly. Birds may also use their wings to attract a mate, shelter chicks to keep them warm and protected, serve as a decoy to appear injured (to lure a predator away from a nest or chicks), and to cover captured prey, as an owl may do.**\n\n# **HATS OFF TO HUMMINGBIRDS!**\n\nS **UMMER IS LEAFY** and sunny and rich with plants, bugs, and birds\u2014especially hummingbirds! They seem to be everywhere, flitting through gardens and visiting hummingbird feeders placed out for them. They're a joy to observe in action. So tiny, so quick, so delicate and full of detail. There are many fabulous things about hummingbirds\u2014their remarkable migration journey each spring and fall, the female's cup-shaped nest-building integrated with spider's web, their ability to hover, and also the fact that they're a species that isn't too shy when humans are present.\n\nWith summer and hummingbirds in the mix, it is the perfect time to create a hummingbird hat feeder with your child and enjoy the thrill of having these jeweled wonders feed right from atop your head!\n\n# **TAKE ACTION**\n\nFeeding hummingbirds atop your head requires a bit of patience and sitting still\u2014a fun challenge for your wee ones.\n\n4 to 8 colorful silk flowers (from bargain store or craft store)\n\nScissors\n\nCraft glue or glue gun\/glue gun sticks\n\n2 to 5 individual handheld nectar ports or \"nectar dots\" (one-hole feeder source)\n\nBaseball cap or straw hat\n\nHummingbird Nectar (this page)\n\n1. Remove the stems from the silk flowers using scissors or by pulling each stem off.\n\n2. Glue the nectar ports onto the rim of the hat, spacing them evenly around the brim.\n\n3. Glue the flower heads onto the hat, arranging them around and near the hummingbird feeder ports, careful to leave each port visible.\n\n4. Once the materials have set securely to the hat, fill each hummingbird feeder port with homemade hummingbird nectar. Be careful when holding the hat once the reservoirs are filled so as not to spill contents.\n\n5. Place the hat on your head and sit outside, adjacent to where you normally keep your hummingbird feeder. Take your original hummingbird feeder down (temporarily), so hummingbirds will be encouraged to drink from their new food source\u2014the hat on your head!\n\nOnce the hat is atop your child's head, encourage your child to sit still and patiently. As the hummingbirds begin to feed from the hat, you will hear their wings \"hum\" as they approach and perhaps even feel a slight breeze from their rapidly beating wings. Your child's first instinct may be to jerk away, but coach him into stillness. If your child feels hesitant about having birds feed from the hat feeder on his head, wear it first and allow him to observe this fabulous and fun experience.\n\n  **_Facilitates creativity, patience, and feeling amazed_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do the hummingbirds know there is food on the hat?**\n\n**A: Birds have amazing eyesight, much better than human eyesight. Hummingbirds use their eyes to locate colorful flowers that provide them with the nectar they need for energy. They have adapted to using their eyes to locate hummingbird feeders placed out by humans, too.**\n\n**DID YOU KNOW?**\n\n**The female hummingbird builds the smallest bird nest in the world. She gathers and weaves spider's web into the design, which helps hold the nest together and allows it to expand as her chicks grow.**\n\n# **SEED SORTING**\n\nB **IRDS EAT** a variety of foods, such as bugs, berries, and meat\/fish, to name a few. And, of course, nuts and seeds are associated with birds. There are a variety of birdseed options available on the commercial market for purchase to fill feeders, not to mention you may even grow your own birdseed by planting seed-producing flowers that birds enjoy munching on, such as helianthus (sunflowers), echinacea (coneflowers), and rudbeckia (black-eyed Susans), among many others. So, who loves to eat what in your backyard?\n\n# **TAKE NOTE**\n\nPerform an experiment with your child. Obtain a variety of seeds for birds, such as black oil sunflower seed, nyjer or finch seed, and safflower seed. Look at the different seeds, noting the shapes and sizes. Ask your child to contemplate which seed will be most popular with the birds you feed, or what birds might favor a particular seed, and why. Write down your predictions.\n\nThen, fill your feeder with just one type of seed. Take note of the birds that eat that particular seed over the course of one to two days. After two days, switch seed types, and observe once more. Do the same birds or same types of birds visit the feeder? Is the feeder visited more frequently or less frequently than with the previous seed? After a few days, switch to a third seed type and watch for activity, noting bird traffic, types of birds, and their preferences. Ultimately, is there one type of food that most birds seem to prefer? Were your predictions correct?\n\nIf you have several similar feeders, you may place one type of seed in each feeder and place each feeder adjacent to one another. Take note to see which feeder is the most popular and which species eats the particular seeds you've placed out.\n\n  **_Encourages a sense of wonder, hypothesizing, and analyzing_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: If we stop feeding the birds, will they starve?**\n\n**A: Most bird-feeder birds eat insects and arachnids (with the exception of the American Goldfinch, which is 100 percent vegetarian), so they won't starve if we stop feeding them. However, it's very important when we do feed them to keep our feeders clean. Old, moldy food and dirty feeders can make birds sick. In addition, resident birds\u2014birds that remain in a region year-round\u2014appreciate seeds, berries, nuts, and other sources of protein placed out for them during the winter months when naturally occurring food sources are scarce.**\n\n# **YARD MAPPING**\n\nC **REATE A YARD MAP WITH** your child, documenting your landscape and the birds and wildlife that frequent it. If you don't have a yard, no worries! You can map any space: your deck, a patio, a balcony, a porch, or a nearby green space, such as a park or public area.\n\n# **TAKE NOTE**\n\nTake a walk with your child through the space you will map. Walk slowly and quietly, observing and listening. As you observe, make note of your landscape, as well as the wildlife you encounter. Where is it shady? Where is it sunny? What plants live where? Where are human dwellings situated?\n\nObserve the trees, shrubs, plants, and flowers. Notice their forms and their locations. On a blank piece of paper, create a map of the space, using loose drawings to designate shrubs, trees, flowers, water sources, dwellings (human and wild), bird feeders, open spaces, and more. It's your map, so be creative and document the things you and your child see!\n\nDon't fret over perfection with your sketches. Use general shapes: circles of different sizes\u2014smaller for bushes and shrubs, larger for trees; flower shapes to depict flowers; squares or rectangles to depict dwelling spaces, such as a home or deck or patio. If you hear birds somewhere, mark the location by drawing a musical note.\n\nAs you walk through your yard, use your map to note where birds and wildlife are found. Explore during different times of the day, and while doing so, pay attention to the bird activity at each time. How does activity vary from morning to afternoon? From afternoon to evening? On rainy days compared to sunny days? On windy days compared to still days? Is one area on your map more populated with wildlife (bees, butterflies, birds, squirrels) than other areas? Do you hear more birds in one area than in others?\n\nTo become part of a community of people creating sustainable landscapes for their yards, join Habitat Network, sponsored by the Nature Conservancy and the Cornell Lab of Ornithology. You can access map-building tools, create a digital yard map, learn about habitat, and talk and share with members of the community. Visit yardmap.org\n\n  **_Encourages spatial visualization and organization, creativity, and observation_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do some birds eat off the ground, while others eat high in the trees and other places?**\n\n**A: There wouldn't be enough food for everyone if all birds ate from the ground or if all birds ate bugs from the leaves of trees. All living animals, birds included, evolve to adapt to their environment so they have a better chance at surviving.**\n\n# **FEATHERY PHOTOGRAPHY**\n\nB **IRDS ARE BEAUTIFUL ,** and there's an artsy sense of pleasure that comes with noting their form and composition in nature. More than just a pretty picture, viewing birds through a camera lens allows you and your child to capture details and behaviors that are often missed with the naked eye. Photos also give us a still image we can study, allowing us to scrutinize all the amazing details for as little or as long as we'd like.\n\nPhotographing birds will provide you and your child the opportunity to visually catalog the birds you've experienced throughout months, seasons, and years. These photos can serve as a visual photo diary of wonderful times and memories spent together in nature.\n\nPhotos can also help us to identify a mystery species. With a photo on file, you and your child may take your time examining reference books, comparing your images to those in books, not unlike doing a puzzle or playing a game. It's fun!\n\n# **TAKE ACTION**\n\nBird photography can be tricky, as birds have excellent eyesight and hearing, are ever wary of their surroundings, and are always on the move. However, great photos are possible with patience, practice, creativity, and planning before and during shoots. Here are some tips and techniques to get you and your child started.\n\n**  Seek out lighting with the sun at your back but projecting on the bird(s).**\n\n**  Do not wear bright clothes; birds have excellent vision and spook easily. Try to blend in.**\n\n**  Do not make sudden movements; if you need to raise your arms to capture a shot or make any movements, do so slowly.**\n\n**  A backyard with a bird feeder is the _perfect_ place to practice, and here's a secret tip: attaching a natural tree branch to your feeder gives birds a place to perch as they await their turn to eat and gives you a natural setting for your photo.**\n\n**  Capture the eye; focus your camera on the bird's eye that is closest to you.**\n\n**  Shoot at high shutter speeds; this will help prevent blurred images, as birds tend to fly, hop, and constantly move about.**\n\n**  Enjoy and embrace the learning process; learn how to delete images that do not work; read photographer blogs that offer insight on camera equipment and techniques.**\n\n  **_Encourages learning new skills and facilitates patience and creativity_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do birds see colors in the world as we do?**\n\n**A: Bird vision is highly developed. They can even see parts of the ultraviolet spectrum invisible to human sight.**\n\n# **HEAVENLY HAWKS**\n\nS **UMMER IS THE PERFECT TIME** to grab a blanket, find a soft patch of grass, and lie back with your child, daydreaming, cloud gazing, and spying the occasional hawk. You'll find these heavenly flyers high, high, high, up in the sky.\n\n\"Hawk\" is a general term that describes the entire group of diurnal (active during the day) raptors. Raptors are birds of prey and include owls, vultures, and hawks (falcons, eagles, kites, buteos, accipiters, harriers, and osprey). Some hawk species migrate each spring and fall, and many find a mate and stay with that mate for life. Because hawks are diurnal, daytime is the ideal time to watch them in action.\n\n# **OBSERVE**\n\nFind a spot where you and your child may lie down and relax, ideally with the sun behind you, making certain the sun isn't directly in your eyes. (Never look directly into the sun, as this could severely damage your eyesight.)\n\nOnce you're situated, relax and take in the bounty of blue above. Scan the skies with your child. Who can spy the first hawk soaring high above? What other birds do you see flying across the sky? How are their flight styles different?\n\nWhen you spy a hawk, note the shape, or silhouette, of its body. Just for fun, estimate how far away, how high up, it may be. Is it circling as it soars? Another place to find hawks is atop telephone poles near open fields, where they often perch waiting to catch sight of a meal, below.\n\nOften, hawks are identified by their flying silhouettes:\n\n**  Accipiters (Cooper's Hawk, Sharp-Shinned Hawk)\u2014short, rounded wings and a long tail**\n\n**  Buteos (Red-Tailed Hawk, Red-Shouldered Hawk, Swainson's Hawk)\u2014long, rounded wings and a wide, fanned tail**\n\n**  Falcons (American Kestrel, Merlin)\u2014short, pointed wings and a short tail**\n\n**  Kites (Swallow-Tailed Kite, Black-Shouldered Kite)\u2014long, pointed wings and a long tail**\n\nFor fun, you might sketch the shape, or silhouette, of the hawks you observe flying above. Or just relax and watch them as they soar. Suggest to your child that she imagine soaring with them. Imagine their view, what they see below.\n\n  **_Promotes imagination and relaxation_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do hawks soar without flapping their wings?**\n\n**A: Warm air heated by the sun rises up from the ground and into the sky. This rising, warm air is called a thermal. Hawks use these updrafts, or thermals, to give them lift. They stay within these thermals by flying slowly in circles.**\n\n# **CHICKA -DEE-DEE-DEE**\n\nO **UR EYES ALLOW** us to see what's directly in front of us, but our ears can hear everything around us. That's a benefit when seeking out birds!\n\nAll birds vocalize, and each species has a unique voice to call its own: they call, chatter, chip, peep, tweet, whistle, hoot, caw, and sing. They call to send out warnings to loved ones, and they sing to defend territory and to attract a mate. They strive to make their presence known, and because of this, birding by ear is a musical and fun endeavor.\n\n**DID YOU KNOW?**\n\n**It was always believed that when a bird sings, it was a male bird responsible for the song. However, female birds sing, too. Next time you spot a singing bird, make note to see if you can tell: is it male or a female?**\n\nYou and your child may bird by ear anywhere, including your own patio, deck, or backyard. Beginning in your backyard allows you to listen for birds you may already be familiar with\u2014and it's a wonderful thing to hear a bird and know which species created the sound. However, birding by ear need not be about attempting to identify each and every song or call made; it's also rewarding to simply listen to the nature of birds that surrounds us.\n\n# **OBSERVE**\n\nSit outside with your child and close your eyes. Listen for birds. Bird sounds are referred to as songs and calls. Birds emit songs to attract a mate or defend territory. Calls, which are shorter than songs, communicate a warning or a bird's location. Try to hear as many bird songs and calls as possible:\n\n**  Do you hear a long song?**\n\n**  Do you hear a short song?**\n\n**  Does the song change in pitch?**\n\n**  Does the sound resemble or include a warble, screech, chirp, or whistle?**\n\n**  How many beats or syllables does the song have?**\n\n**  Is there repetition?**\n\n**  Is the song repeated by the same bird, or answered by another from a different location?**\n\n**  Does the volume change?**\n\n**  Do you hear calls? Is there a response to the call from another bird of the same species?**\n\nAs you become more skilled with birding by ear, consider making notes on what you hear. Take it a step further and attempt to identify what you hear.\n\n# **BIRDCALL MNEMONICS**\n\nOften, mnemonics are used to identify birdcalls and songs, as their sounds resemble words in the human vocabulary, such as:\n\n** _Birdie! Birdie! Birdie!\u2014_ Northern Cardinal**\n\n** _Cheeseburger-cheeseburger_ (very fast, faint, and soft, like a whisper)\u2014Ruby-Throated Hummingbird**\n\n** _Chicka-dee-dee-dee-dee-dee\u2014_ Carolina Chickadee**\n\n** _Drink your tea\u2014_ Eastern Towhee**\n\n** _Fee-bee_ \u2014Eastern Phoebe**\n\n** _Feeder! Feeder! Feeder!\u2014_ Tufted Titmouse**\n\n** _Oh, Canada, Canada, Canada\u2014_ White-Throated Sparrow**\n\n** _Sweet-sweet-sweet; little more sweet!_ \u2014Yellow Warbler**\n\n** _Who cooks for you?\u2014_ Barred Owl**\n\n** _You're pretty, you're pretty, you're pretty!\u2014_ Carolina Wren**\n\nAs you bird by ear, you and your child may create your own mnemonics for the sounds you hear!\n\n  **_Stimulates the senses and strengthens listening skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do birds learn to sing?**\n\n**A: Songbird nestlings learn to sing while in the nest. They listen to the adults around them and then practice replicating the songs they hear.**\n\n# **BEAK BY BEAK**\n\nB **IRDS ARE BIRDS** \u2014but they certainly can look very different from one another. Each bird species varies in size, form, and plumage, not to mention their beaks. Beaks come in many different sizes and shapes, and each is specialized in form and function for the bird that \"wears\" it.\n\nBeaks help birds capture the food they need. Some birds eat bugs, some birds eat small mammals, some birds eat fish, some birds eat seeds, and others eat fruit and berries. Some birds hunt for food while in mid-air, some birds hunt for food in the water. Some birds gather from trees, flowers, or shrubs; others hunt and gather on the ground. The types of food that birds eat (and how they hunt for that food) are as varied as their beaks:\n\n**  Long and chisel-like beaks are perfect for rap-tap-tapping into a tree's trunk to find bugs.**\n\n**  Narrow and pointy beaks, like a pair of tweezers, are perfect for snatching and grabbing delicious bugs\u2014well, delicious to the bird, anyway!**\n\n**  Hooked beaks are perfect for grabbing and tearing.**\n\n**  Spoon- or shovel-like beaks are perfect for scooping fish and other meals from water.**\n\n**  Sturdy and strong beaks are perfect for cracking nuts and seeds.**\n\n**  Long\u2014very long\u2014beaks are perfect for dipping deep into nectar-rich flowers.**\n\n# **TAKE ACTION**\n\nExplore how bird beaks work in the wild. First you'll need some \"beaks\" to manipulate and some \"food\" to practice with.\n\n# **BEAKS**\n\n1 pair of tweezers (grabbing and pulling beak\u2014robin)\n\n1 straw (sipping beak\u2014hummingbird)\n\n1 spoon (wading bird beak\u2014stork)\n\n1 clothespin (picking up and cracking open beak\u2014cardinal)\n\n1 toothpick\u2014round works best, as it's stronger than flat (piercing beak\u2014wren)\n\n# **FOOD**\n\nRubber bands or gummy worms (worms)\n\nNuts and popped popcorn (nuts and seeds)\n\nCookie sprinkles (ants)\n\nCup of fruit juice (hummingbird nectar)\n\nA few grapes and\/or raisins (juicy bugs)\n\nA dish of water sprinkled with a dried seasoning, such as basil flakes (small fish; algae)\n\nInvite your child to practice using the variety of beaks to eat the pretend bird foods. Experiment! Which type of beak works best for each type of food? Explain that a bird's beak helps it capture the foods available in the environment where it lives.\n\n  **_Promotes awareness and understanding_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are beaks used only for capturing food?**\n\n**A: A bird's beak is primarily used for capturing food, but think about this: a bird doesn't have hands, fingers, or an opposable thumb as we do. So a bird uses its beak to do many other things, too. Beaks can pick up objects and gather nesting materials. Birds use their beaks to groom feathers, a behavior called** **_preening_** **. Beaks are used as a defense against other birds or predators. Birds even use their beaks to build nests\u2014can you imagine using your mouth to build your house?**\n\n# **COLOR COUNT**\n\nA **S DAYS SHORTEN AND** become cooler and the colors outdoors change from greens to gold, the world we know shifts to the fall season. But there is more than fall foliage to marvel at when venturing out in autumn. Bird plumage also changes in the fall.\n\nBirds are some of Earth's most colorful animals. But come fall, birds big and small, dull and dramatic, offer even more to see. The variety of plumage includes mature adults (molting or not), immature hatch-year offspring, and migrators passing through\u2014each with its own dress code for the season. This can make bird identification even trickier than usual!\n\nFor example, in the fall, mature male hummingbirds sport their full-color _gorgets\u2014_ the colorful feathers that cover their throats. Hatch-year, immature male hummingbirds may sport just a sprinkling of colorful feathers on their gorgets, which makes it fun and easy to personally identify them as they frequent your feeder or garden: _there's the immature male with just three colorful gorget feathers; there's the immature male with just one colorful gorget feather; there's an immature male with no gorget feathers, or is that a female? I'm not certain...._ And the cheery, male American Goldfinch, with mating season over, will molt into winter plumage and turn a pale olive brown, which helps it blend into its winter environment. Come spring, he molts back to his bright yellow plumage, with hopes of attracting a mate.\n\n# **TAKE NOTE**\n\nFlock outdoors and seek out as many varieties of plumage and colors as you can find. Use this checklist as a guide:\n\n**  A mature male hummingbird (full-color gorget on throat)**\n\n**  A mature female hummingbird (no color on throat)**\n\n**  An immature male hummingbird (just a few colorful feathers on gorget)**\n\n**  A bird with blue in its plumage**\n\n**  A bird with red in its plumage**\n\n**  A bird with yellow in its plumage**\n\n**  A bird with orange in its plumage**\n\n**  A small bird with brown in its plumage**\n\n**  A medium-size bird with brown plumage**\n\n**  A black and white bird**\n\n**  A bird with white in its plumage**\n\n**  A bird with gray in its plumage**\n\n**  A bird with white wing bars**\n\n**  A bird with spots in its plumage**\n\n**  A bird with speckles in its plumage**\n\n**  A bird with distinctive black markings**\n\n**  A bird with bands across its tail**\n\n**  A bird with a crown stripe on its head**\n\n**  A bird with an eye stripe (line through a bird's eye)**\n\n**  A bird with an eyebrow stripe (line over the eye)**\n\n**  A bird with a black throat patch**\n\n**  A bird with a white throat patch**\n\n**  A molting bird (mixed colors; dull plumage mixed with brighter plumage; feathers missing)**\n\n**  A bald bird (often found on cardinals or blue jays in the fall)**\n\nAdd your own findings here:\n\n  **_Promotes awareness of the environment and observation skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do birds come in so many different colors, patterns, and designs?**\n\n**A: Every detail in a bird's plumage serves a purpose. Birds use their colors primarily to attract mates. The markings and patterns on birds also help them blend in with their habitat, protecting them from predators or helping them sneak up on prey.**\n\n# **MOONLIGHT MIGRATION**\n\nL **EAVES ARE SWIRLING** and twirling; pumpkins, hay bales, and cooler temperatures are present\u2014fall is such a beautiful transition between summer and winter. It also marks another sensational transition: Each fall, billions of birds make their journey from northern, summer breeding grounds to southern winter grounds, where food sources are more plentiful. It's time for the annual fall migration.\n\nSome of these birds are seasoned migrators, having made the journey before. Some of these birds are first-year hatch birds, having just left their nest over the summer\u2014remarkably, they know where to go even though they are making the journey for the first time. Even the wee hummingbird that hatched over the summer will migrate, traveling alone, hundreds or thousands of miles, depending on the species. Songbirds migrate in the black of night, when there's less risk of predation.\n\nEach fall also brings a Harvest Moon, a time when Earth's full Moon is closest to the fall equinox, and this marks an opportune time to go outdoors, commune with nature, and take in the mystery and wonder that is bird migration. The Moon provides a magical backdrop for watching birds migrate!\n\n# **OBSERVE**\n\nCheck your calendar to note when the Harvest Moon will take place, and at what time the Moon will rise. Plan ahead so you and your family will be ready for the timing. Invite friends to join you, too!\n\nGather together and head outdoors with a blanket for bundling up, chairs to sit in, warm beverages, binoculars, or a small telescope with 30x magnification. Simply direct your lens to the Moon. Before long, an occasional dark silhouette will be visible fluttering across the backdrop of the bright, silvery disc.\n\nMigrators begin their flights right at dusk and continue until 2 or 3 a.m., with altitudes ranging from 1,500 feet to 5,000 feet, sometimes higher. Any full Moon in the late summer and throughout fall is an opportune time to catch this phenomenon.\n\n  **_Promotes relaxation, camaraderie with friends and family, and wonder_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How high up do birds fly when they migrate, and how far do they go?**\n\n**A: Migration is not easy for birds, and the heights and distances they fly vary by species. Mallard ducks fly at heights of 21,000 feet, and White Storks migrate at heights of 16,000 feet. The Bar-Tailed Godwit reaches heights of 20,000 feet and also makes the longest nonstop migration, nearly 7,000 miles without stopping.**\n\n**The highest-flying bird ever recorded is Ruppell's Griffon Vulture, documented at 37,000 feet where it, unfortunately, collided with a plane. The Arctic Tern migrates the longest distance, more than 49,700 miles a year, from its breeding grounds in the Arctic to its winter grounds in the Antarctic.**\n\n# **A HANDFUL OF HUMMERS**\n\nB **IRDS ARE VERY OBSERVANT** of their surroundings, a trait necessary for survival. Even birds that frequent backyards and feeders are skittish, skirting and flying off at the first sight of human presence. However, certain birds, including hummingbirds, have become accustomed\u2014wary but accustomed\u2014to the presence of humans.\n\nDuring fall seasons, hummingbird populations increase. Summer hatch-year fledglings are flying about with their parents in proximity. In addition, migrants from northern regions have begun their journey southward, adding to the population as they stop to visit food sources on the way to their winter grounds. Mature males travel first, followed by mature females, followed by the season's hatch-year juveniles who require more time to build fat stores before departing.\n\nThis population increase provides the perfect opportunity to practice proximity with these flying jewels. Invite a hummingbird to feed directly from your hand. If hummingbird feeders are already established in your yard, feeders will be busy with traffic during early fall, making it simple to get up close and personal with hummingbirds.\n\n# **TAKE ACTION**\n\nTo feed a hummingbird from your hand, obtain a handheld hummingbird feeder, such as the one made by Audubon, or simply hold the hummingbird feeder you have in your yard in your hand.\n\nTemporarily remove any additional hummingbird feeders in your yard while you strive to feed a hummingbird from your hand, making your handheld nectar source the only source.\n\nMake sure the feeder is filled with fresh nectar (see this page), and place the feeder in the palm of your hand. Choose an area near an existing feeding space, where your hummingbird feeder once hung and your little jewels are familiar with. Then, sit or stand quietly and still with the feeder in your hand and let the magic begin. Hummingbirds may be hesitant at first, but with patience and stillness, they will trust you and the feeding source, and take a dip to lick nectar directly from the feeder you hold in your hand. Marvel as you observe the beautiful detail of their feathers, hear the hum of their wings, and even feel the wispy breeze from their movements.\n\nOpt to not offer an open source of nectar, such as a milk lid or tiny dish with sugar water in it. If the solution should splash or get on the hummingbird's feathers, it may harden on their plumage and cause harm.\n\n  **_Stimulates affinity, wonder, and amazement_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do hummingbirds hover?**\n\n**A: Unlike most birds, which flap their wings up and down, the hummingbird flaps its wings forward and backward and can rotate its wings. This gives hummingbirds the unique ability to hover in midair, as well as to fly forward, backward, sideways, and straight up. In proportion to their body size, hummingbirds have very large chest muscles to aid in wing strength, and they have very tiny feet that reduce drag\u2014a hummingbird can't stand, walk, run, skip, or hop; it can only perch.**\n\n**DID YOU KNOW?**\n\n**Alexander Skutch, the author of** **_The Life of the Hummingbird_ ,** **spent decades studying hummingbird behavior. After years of observation, he concluded that hummingbirds remember food sources, and perhaps the people who feed them. This is why, during migration, a hummingbird returns to the exact place where a feeder once hung, even if the feeder is no longer there.**\n\n# **LOVELY LEAF LITTER**\n\nW **HEN WE CARE** for others and foster stewardship for other living things, it provides the reward of intrinsic joy. When we model and facilitate stewardship with our children, we provide them with insight that will last throughout their lifetime. Simple projects in our outdoor spaces foster stewardship.\n\nWorking in the yard is heart-smart, too, providing fresh air and exercise. In the fall, rather than blow and rake and tidy every little nook and cranny of your outdoor space, consider leaving the leaves\u2014or some of them, at least\u2014to create a leaf litter sanctuary, which avian wildlife will treasure.\n\nWhen temperatures drop, bugs and food sources for birds drop, too. However, leaf litter provides shelter and refuge for bugs, which in turn provides important and necessary food for birds. The tidier the yard, the more barren it will be of wildlife, bugs and birds alike. It's as simple as that.\n\n# **TAKE ACTION**\n\nFallen leaves beckon. Get out and stomp and romp through them. Pile them up. Pick them up and look at them. How many different types of leaves have fallen around your yard? Sort and observe them with your child: by size, by shape, and even by color.\n\nWhile out having fun, do you notice any birds scratch-scratching among the leaves? Look near hedges, a wood line, or areas that may be mulched. Create a tasty leaf-haven for wintering birds, using the leaves and sticks that have fallen in your yard. This doesn't mean you must leave leaves everywhere\u2014just provide wee bits of habitat in place for bugs and birds to help get them through the winter. As you tidy your yard:\n\n**  Rake and place leaves and sticks along hedges, under bushes, and in mulched areas. Explain to your child that these are places where bugs may hide for the winter, and where hungry birds may find food.**\n\n**  Rake and place leaves and sticks along fences, under tree bases\u2014particularly evergreens\u2014and even over perennial areas, which will shelter _them_ from winter.**\n\nWhen we get cold, we can layer in clothing or bundle under blankets. Leaves are nature's miracle blanket for the earth. They shelter, they protect, they decompose, and they provide nutrients for many living things. Birds love leaf litter! Look for these leaf-lovers:\n\n**  In the winter, you may spy Carolina Wrens investigating each leaf, White-Throated Sparrows kicking and scratching among the leaves, as well as Dark-Eyed Juncos, Fox Sparrows, Eastern Towhees, and Song Sparrows, just to name a few.**\n\n**  In the spring, you may find Ovenbirds and certain species of warblers foraging on the ground through the leaves.**\n\n  _Encourages stewardship and fosters empathy_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do trees lose their leaves?**\n\n**A: Winter trees \"sleep\" in a sense, and losing their leaves is part of their process to help them shut down for the winter. Winter days provide less daylight, and leaves need sunlight to function and make food for trees. Because of this, trees that lose their leaves in the fall (deciduous trees) drop their leaves as the days get cooler and shorter and then sprout new leaves come spring, when the days warm up and daylight lasts longer.**\n\n# **HANDY NESTS**\n\nB **IRDS ARE INCREDIBLE** architects. Not only do they create the most diverse form of homes, or nests, of any animal species in the wild, but they create homes that defy gravity, resist weather season after season, and, most importantly, provide the perfect shelter and protection for their fragile eggs.\n\nNests come in many different shapes and sizes and are made from a variety of materials. They can be found on the ground, high in treetops or on skyscraper ledges, and even floating on water. Some are made of twigs, some are made of mud, some are made with spider's web, some with lichen and moss. Some are even made from bird spit!\n\nIn late fall and winter, you may spy nests among the bare branches of trees. During spring and summer, you might locate a nest based on chatter from a locale (hungry chicks at feeding time) or as you watch a bird gathering nesting materials as it flies to and fro. If you do encounter a nest, observe it quietly, carefully, and respectfully. Nesting birds need privacy and space. You wouldn't want strangers poking their heads into your home or bedrooms, would you?\n\n# **TAKE ACTION**\n\nAfter exploring nests with your child, embark on building your own nest. Gather natural materials from outdoors: small twigs, sticks, grasses, and even mud. Then, use your bare hands to form and shape a cozy nest.\n\nIs it a simple task? Can your nest support an egg? Survive through wind, rain, and other elements of weather? Imagine birds creating these structures using only their beaks and feet. An amazing feat, for certain!\n\nAs a follow-up activity and to learn more about birds and the types of nests they build, read my children's book _Mama Built a Little Nest_ , illustrated by Steve Jenkins. In addition, visit your local nature center where your child may hold and observe a bird's nest up-close. Take a walk and see if you can spy a bird's nest\u2014not easy to do during nesting season, as they're often well hidden. However, some species have the capacity to build their nests in the most unusual of spaces! Check out \"Funky Nests in Funky Places\" at CelebrateUrbanBirds.org. You may even submit your own findings there.\n\n  **_Stimulates understanding, empathy, and appreciation for living things_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do all birds build nests?**\n\n**A: Most do, but there are a few exceptions. For example, the Brown-Headed Cowbird doesn't build a nest. Instead, she finds a nest built by another species and lays her egg(s) in that nest, then flies off, leaving her eggs in the care of the birds who built the nest.**\n\n# **A PUMPKIN -PERFECT FEAST**\n\nP **UMPKINS ARE SYNONYMOUS** with fall, and thankfully they are ubiquitous at this time of year. Pick out the perfect pumpkin to create a \"place setting\" for the birds, a destination where they may dine on delicious seeds and nuts that provide them with the nutrients they need, while providing you and your little apprentice with a harvest-themed activity. Win-win!\n\n# **TAKE ACTION**\n\nVisit your local farmer's market, or any place that offers pumpkins: a farm where you may pick your own, a local store, or perhaps you've grown your own? The makings for this fabulous pumpkin bird feeder are simple.\n\n1 smallish pumpkin\n\nTwine\n\nA few small twigs, approximately 4 to 5 inches long\n\nBirdseed\n\n1. Cut the pumpkin in half, and then scoop out the seeds.\n\n2. Cut two 3-foot sections of twine; hold them side by side and create a tight knot in the center, at the 1\u00bd-foot mark, knotting both pieces together.\n\n3. Place the knotted part of the twine on the bottom of the pumpkin. Gather the four loose ends and securely knot them above the pumpkin.\n\n4. Insert the twigs into the sides of the pumpkin, to serve as perches. Birds love to perch!\n\n5. Fill the pumpkin with birdseed, then hang your pumpkin feeder from a shepherd's hook or sturdy tree limb, or place it on a deck rail. Just don't put it on the ground where feeding birds might be susceptible to predation by cats or other predators.\n\n6. Sit back, relax, and enjoy the curious and hungry birds who discover this treat and come to feast.\n\n  _Promotes stewardship and intrinsic reward_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How much do birds eat a day?**\n\n**A: A little chickadee may eat 35 percent of its body weight each day; the smaller the bird, the more it needs to eat relative to its body size. All birds must consume extra calories when temperatures are cold.**\n\n# **PRESSED LEAF FEATHER ART**\n\nB **IRD PLUMAGE IS AMAZING!** Dots, stripes, feathery details, speckles, and spots. Feathers really are a work of art. Become inspired to transfer visions of feather design to your own feathers: pressed leaves!\n\n# **TAKE ACTION**\n\nThis is an amazing art project that you and your child may frame, scatter upon tabletops as d\u00e9cor, or give to special people in your lives.\n\nFallen leaves, gathered from nature (collect pliable leaves that are not dry or too brittle)\n\nOpaque markers in various colors (black, blue, brown, ivory, purple, red, turquoise, yellow), available from any craft section\/store\n\nFadeless paper (from craft store), for mounting leaves upon (optional)\n\nFrame with glass (optional)\n\nCraft glue, for mounting feathers on paper when framing (optional)\n\n1. Press leaves within books and magazines for one or two days.\n\n2. Once the leaves are pressed and dried, create simple designs on them, using bird plumage and feathers as your inspiration. Here are some ideas:\n\n**  Create small, cream-colored spots across an entire leaf; then fill each spot with a small black dot, as you might see on the belly of a Northern Flicker.**\n\n**  Draw tiny \"u\" shapes and lines across a leaf, resembling overlapping scales, and then fill in each \"u\" shape with tiny lines that flow from top to bottom, just as you'd see up close on hummingbird feathers.**\n\n**  Draw small circles on a leaf, and then create small dots around each circle, similar to the dots found around a bird's eye.**\n\n**  Seek out images of feathers in books or online, and replicate them on your leaves.**\n\n**  Play with colors and design elements, letting nature and birds guide you in your choices. Draw the lines and details on these feathers.**\n\n**  Refer back to sketches you've made of birds, noting details in the plumage.**\n\nMost importantly, be creative and have fun!\n\n3. Place your pressed leaves as decorations on holiday table settings, or arrange them on fadeless paper in a design (star burst, array, single leaf\/feather) and frame them.\n\n  **_Encourages observation, fine motor skills, and creativity_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: What are feathers?**\n\n**A: Feathers are a very complicated micro structure of birds. They are so complicated and intricate in detail that scientists devote their lives to studying them: how they work, what they're made of, how they're structured, how they develop and evolve, and how they function. Feathers are unique to birds and are fascinating.**\n\n# **SPEECH BUBBLES BETWEEN BIRDS**\n\nA **S YOU OBSERVE** birds in the wild, do you ever wonder what they're saying? They sing. They call. They chitter. They chatter. Sometimes their calls sound cheerful. Sometimes their calls sound excited. Sometimes their calls are loud and alarmed. Sometimes their calls and songs sound extremely purposeful. Often their songs are melodic and dreamy. Just what do you think they're saying, anyway?\n\n# **TAKE NOTE**\n\nOne can't help wonder what birds are communicating to one another, but we do know they are communicating with purpose, just as humans communicate through dialogue, emails, text messages, and even through behavior and appearances.\n\nThink about what birds might be conveying when they interact, and then have some creative fun as you ponder \"bird talk.\" Peruse old magazines for images of birds, cutting out any birds you find. Glue each bird image onto paper, drawing a speech bubble above each bird's head. Then, let your child fill in the speech bubbles himself, or take dictation. Be creative. You don't need to be factual. Just have fun!\n\n  **_Encourages creativity, imagination, and fun_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do birds talk to one another?**\n\n**A: Birds talk using sounds\u2014songs, calls, chirps, quacks, and drums\u2014and the sounds they make are used to stay in touch with each other, attract a mate, contact a parent or mate, scare off predators, warn other birds about danger, signal a food source, or defend a territory, just to name a few reasons birds \"talk.\"**\n\n# **WE'RE GOING ON AN OWL HUNT**\n\nB **ARE TREE BRANCHES** and fall weather provide an ideal setting for spotting birds, and owls are no exception. Their silhouettes may be spotted anywhere you can find large trees, but it can be tricky to actually see them. Their plumage helps them to camouflage seamlessly with tree trunks, not to mention most species are nocturnal, or active at night. Don't let this dissuade you and your child from an owl adventure. Prowl for an owl. You may get lucky and catch sight of one or, at the very least, hear their haunting hoots and calls.\n\n# **OBSERVE**\n\nEven though many owls are nocturnal, you need not venture out in the dark of night to find one. Dawn and dusk are great times to seek them out. They are active at these times, exploring their world, hunting for a meal, or calling to attract a mate or claim a territory. In early fall, you might spot owl nestlings out and about, testing and flapping their wings and testing their toes, climbing on trees.\n\nGoogle owl species, and listen to their recorded sounds online for reference. Then, head to a locale where trees are abundant: perhaps your yard, a park, or a local nature center. Pack a blanket for sitting upon and a pair of binoculars, then plant yourselves down to relax, wait, watch, and listen. Owl prowling takes patience. But most importantly, you're making the effort and spending time in nature.\n\n  **_Encourages exploration, curiosity, and relaxation_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are all owls nocturnal?**\n\n**A: Of the approximately 222 owl species in the world, most are nocturnal, hunting at night, but not all. For example, the Barred Owl is active during the daytime.**\n\n# **BIRD BRAINS**\n\nB **IRDS HAVE A LOT** more going on in their wee heads than we give them credit for. (Just because they have small heads does not mean they have small brains.) They are smart! To the casual eye, they fly around, perch, seek mates, and forage for food. What a narrow view, though, in the scope of reality in a bird's world.\n\nFor example, some birds are tool users. New Caledonian Crows have been known to make and use hook tools, the only species other than humans to do so.\n\nA young girl named Gabi Mann in Seattle began setting out nuts and dog food for the American Crows in her yard. They began bringing her gifts\u2014toys and trinkets, pebbles, found jewelry, objects, and treasures\u2014in exchange for the treats she left for them.\n\nEach autumn, that little chickadee you see out and about is actually gathering hundreds and hundreds of seeds, hiding them in hundreds of places over 10 square miles\u2014nooks and crannies within a tree's bark, under a roof eave, in a hole on the ground, in a crack on the sidewalk, within a tree's stump\u2014and then is able to recall each and every hiding place come winter when food is scarce. And it does this easily, because each fall, the part of the chickadee's brain responsible for memory and spatial organization (the hippocampus) expands by 30 percent, providing it the spatial and memory capabilities of a brainy superhero. Come spring, when food is more plentiful and stashing seed sources isn't as vital for survival, the chickadee's hippocampus shrinks back to its normal size. These are just a few examples of the wiring in a bird's brain.\n\n# **TAKE ACTION**\n\nDiscuss with your child the many casual and common bird behaviors you observe: Foraging and feeding. Calling and signing. Wonder together: might more be going on than meets the eye? Explain to your child that chickadees have the capability to hide food in hundreds of places and then recall each hiding place when they need to access that food during the scarce times of winter.\n\nBe like a chickadee! Provide your child with twenty-five pennies. Invite her to hide them throughout your house, anywhere she chooses, while you make note of each hiding space. Then, a few days later, invite your child to find all twenty-five hidden pennies. This is similar to a chickadee hiding seeds to serve as a food source through the winter.\n\nNow it's your turn. Hide twenty-five pennies throughout your home\u2014if your child is able, have her document each spot\u2014then, a few days later, relocate each penny you hid. Challenge yourselves and increase the number of pennies you hide.\n\n  **_Encourages awareness, empathy, curiosity, and understanding_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: So, birds are really smart?**\n\n**A: Ounce for ounce, birds have significantly more neurons in their brains than mammals and primates. (Vanderbilt University,** **_Science Daily_ ,** **June 2016)**\n\n# **NAME THAT BIRD**\n\nW **HEN WE OBSERVE** and talk about birds, or any wild animal, they are often referred to as \"it,\" as in \"it\" landed on the branch, \"it\" built a nest in that tree, or \"it's\" at the feeder.\n\nHowever, seeing wild animals as individuals, and not just \"its,\" can often prove better for the animal's welfare, while also facilitating empathy and forging a deeper connectedness for humans. After all, birds do not just exist\u2014they live with inner lives, purpose, personalities, and emotions, just as we do.\n\nChat with your child about the names we give others. Parents name their children. People name their pets. We even give names to our stuffed animals and favorite toys. Scientists give names to birds, both scientific and common, to help order, organize, and classify them\u2014necessary with over 10,000 species! Why not give names to the birds we learn to know and love?\n\n# **TAKE NOTE**\n\nGive the birds you know names. They deserve to be named (and not just scientifically). After all, they share your home with you, as frequent or regular visitors to your feeders and habitat. Or maybe you notice a particular bird at a park. Certainly, as you become aware of the presence of birds, many become familiar to you, day in and day out, and display certain characteristics and personalities that will make naming them fun.\n\nAny day is a perfect day to name a bird. Perhaps one day you notice a new species at your bird feeder, and she visits again and again. What should her name be? Or maybe you notice a guaranteed regular in your yard\u2014a bird you see every single day, no matter what? What name shall he go by?\n\nMy bird friends include:\n\nName:\n\nSpecies (my best guess):\n\nWhy I chose this name:\n\n  **_Encourages empathy and a connectedness to other living things_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are there any famous wild animals that have names?**\n\n**A: Many noted wild animals have been given names. Martha\u2014a pigeon named after Martha Washington\u2014is just one example. She was the very last passenger pigeon of her species, which is now extinct. In the late 1860s, passenger pigeons flourished in the wild by the billions, their flocks large enough to block the sunlight as they flew by. But in just a short matter of time, they became extinct due to over-hunting by humans and habitat loss. Martha was captured and kept at the Cincinnati Zoo with other passenger pigeons, where hopes of revitalizing her species proved unsuccessful. She died alone on September 1, 1914. Her remains are kept to this day at the Smithsonian Museum. Because she was given a name, it certainly makes her life and her life's story more real and relevant to us, don't you think?**\n\n# **CHILLING WITH FEATHERS**\n\nI **N COOLER WEATHER ,** we often find ourselves bundled and layered, warding off the chill. Mammals in the wild may grow a \"winter coat\" to aid with warmth, while reptiles, amphibians, and bugs may burrow and nestle themselves out of sight until temperatures climb. But not birds. Birds weather the weather, thanks to their insulating feathers, which trap heat against their bare skin and keep them warm.\n\nFeathers are not just for keeping a bird warm, and they're not just for flying, either. There's remarkable form and function to feathers. Here are other ways birds use their feathers:\n\n**  Feathers may help a bird blend in, or camouflage, with its environment, protecting it from predators, or allowing it to be the predator, unnoticed.**\n\n**  Feathers communicate. A peacock may fan his tail to get the attention of females, a bird may \"posture\" and puff up to appear larger to scare off another bird, a crest on a head may be lowered or raised to communicate how a bird is feeling.**\n\n**  Feather color may also help male birds attract female birds.**\n\n**  The feathers on an owl's head help it hear by guiding sound to the ears.**\n\n# **TAKE NOTE**\n\nTake an autumn walk and look for feathers on the ground. Seek them out beneath trees, below feeders, or really anywhere, because birds are everywhere. Once you and your child spot one, pick it up and observe it closely, noting all of its details. Make a loose sketch of the feather. After close observation, place the feather back where you found it (collecting feathers is illegal).\n\nWith your sketches in hand, can you and your child identify parts of the feather? Each feather has a barb, a shaft, and a quill. Google \"feather anatomy,\" or reference a bird book to study feathers in depth. The children's book _Feathers: Not Just for Flying_ is a great resource!\n\nCan you identify the type of feather you sketched?\n\nThere are seven types of feathers on birds:\n\n**  Bristle feathers are found around the beak and may be sensory while also protecting the face. These feathers are tiny and not ones you would find on the ground. However, you can spot them on birds you see up close, and on photographs of bird faces.**\n\n**  Contour feathers cover most of a bird's body and give the bird its shape and colors.**\n\n**  Down feathers are fluffy and close to the body for insulating and trapping heat.**\n\n**  Filoplume feathers are stiff bristle feathers around a bird's eyes and mouth\u2014again, as observed when viewing a bird's face up close or on a photograph of a bird's face.**\n\n**  Semiplume feathers, a bit larger than down feathers, are also fluffy and insulating.**\n\n**  Tail feathers are used for balance and steering.**\n\n**  Wing feathers are specialized for flight; they aid with lift and moving a bird forward.**\n\n  **_Encourages curiosity, scientific knowledge, and observation skills_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: We shower and bathe. How do birds keep their feathers clean?**\n\n**A: Feather maintenance is important! Birds preen, using their beaks to go over every feather daily. They have a preen gland to help coat their feathers. They molt every single feather at least once a year. They bathe in water and even in dust!**\n\n# **A NEST OF MY OWN**\n\nE **VERYONE NEEDS** a place to chill and relax, a quiet place without interruptions. Even kids. _Especially_ kids. It's healthy to provide our children with space where they can reflect, daydream, think, read, imagine, wonder, and simply be alone. Quiet time. No television. No electronics or other stimulation\u2014a place to truly unplug and feel safe and calm.\n\n# **TAKE ACTION**\n\nFind a spot indoors where you and your child may build a \"nest,\" a cozy space just for thinking, reading, and relaxing\u2014a nest your child may call her own. It might be a corner of a room, a sunny spot near a window, or even upon a sofa or cozy chair.\n\nOnce your child has selected the spot for her nest, provide a blanket or two and form them into a circular shape, similar to a nest but large enough for your child to sit in. Ask your child what else she may like in her nest to make it cozy: a pillow or two, a stuffed animal, or maybe some favorite books (some fabulous books about birds are recommended on this page). Create the perfect nest, a nest just for your child to retreat into. Let your child know her nest is just for her to use whenever she feels the need.\n\n  **_Promotes relaxation, calm, and wonder_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Birds use nests, too\u2014but do they use them everyday, all the time, all year?**\n\n**A: Birds only use their nests during nesting season for laying eggs and raising their young.**\n\n# **NAKED NESTS**\n\n**T HE WINTER TREE** landscape is striking. As trees go bare, scenes that were kept secret by the thick foliage of spring, summer, and fall are revealed. We are greeted with new views and once-hidden gems: bird nests.\n\nWinter trees offer beauty in their form and silhouettes against the sky, appearing both delicate and strong. Spend time observing the trees of winter and you'll notice a variety of nests\u2014nests that were once snuggled tight and hidden safely within the trees' dense, leafy branches\u2014are now visible to the naked eye. The diversity we observe with bird behavior is nothing compared to the diversity we find with their architectural nest-making skills!\n\n# **TAKE NOTE**\n\nTake a winter tree walk, and seek out naked nests. Once you begin seeking them out, you'll see that they are visible in all sorts of places: within a tree's v-nook, perched high in a treetop among delicate twigs, within big trees, small trees, old trees, and young trees. Nests may be found everywhere during a winter walk.\n\nTake note of the variety of nests you might see: large, stick-stacked nests, round cup-shaped nests, hanging pendulum nests. Pay attention to their sizes and shapes, in addition to the materials used. Look closely, if you can, observing each and every little detail. Can you see lichen and moss attached to the outside of one? Wonder about the bird that created each nest you find. It's tricky to positively identify each nest you see without its architect present, but knowing what bird species nest in your region, the location of the nest, and the type of nest can sometimes put a name to a nest.\n\nSpend a day outdoors and sketch the nests you notice among the winter trees and their branches. If it's a sunny day, sit on the ground and notice how bright the winter sun is, as well. Without the absence of leaves and the shade they provide, the ground around trees becomes brighter, too.\n\n  **_Stimulates curiosity and wonder_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Do all birds nest in trees?**\n\n**A: Many birds do nest in trees, but not all birds do. Some species nest on the ground, in caves, in chimneys, upon buildings (even skyscraper ledges!), and even on top of water, just to name a few places.**\n\n# **FLAPPY HOUR AND THE GREAT BACKYARD BIRD COUNT**\n\n**T HE GREAT BACKYARD** Bird Count (GBBC), facilitated by the Cornell Lab of Ornithology and the Audubon Society, takes place each year around the globe for four days in mid-February. It's a fabulous, feathery, fun way for bird watchers of all ages to observe the birds in their area, count what they see, report their findings as citizen scientists, and help create a real-time snapshot of bird populations\u2014data which is instrumental to ornithologists and those who study birds.\n\n# **TAKE ACTION**\n\nPartake in the next GBBC! Bird counting during the GBBC may be done for as little as fifteen minutes a day or off and on throughout a day. Simply tally the numbers and types of birds you see on one or more days during the count period. You may count birds at your home, at a local park, at school\u2014actually anywhere. If you see a bird in a parking lot, count it. Just document the date, time, species, and place. The Cornell Lab of Ornithology has regional, printable checklists to use for tallying, as well as instructions and guides for creating a successful count. To get started, visit gbbc.birdcount.org.\n\nTake the GBBC one flight further and host a flappy hour with friends and family! When we share our knowledge, passion, and enthusiasm about birds with other people, we expand the number of people who will know the joy that comes with watching birds. If each person inspired just one other person, the number of people who learn to appreciate the importance and beauty of birds doubles.\n\nTo host a flappy hour, gather materials for you and your guests:\n\n**  Bird ID checklists for tallying (printable from the GBBC site)**\n\n**  Bird ID reference books featuring birds in your region**\n\n**  A pair or two of binoculars (or invite guests to bring their own)**\n\n# **OFFER A CHALLENGE**\n\nHave a contest with friends and family during your GBBC flappy hour. Prize categories might include:\n\n**  First to tally the smallest bird**\n\n**  First to tally the largest bird**\n\n**  First to tally a bird (by sight or by ear)**\n\n**  Most birds tallied**\n\n**  Most species tallied**\n\n  **_Enhances observation skills and bird knowledge, and provides an opportunity to participate as a citizen scientist_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: What is citizen science?**\n\n**A: Citizen science is research conducted by people in the general public. By sharing their observations with real scientists, it provides a broader picture of the subject being studied and provides a wider range of data for scientists to analyze.**\n\n**SOAR HIGHER**  \n---\n\n**There are a variety of ways to practice being a citizen scientist. In addition to the GBBC, opportunities through the Audubon Society, the Cornell Lab of Ornithology, and the Nature Conservancy include:**\n\n**  Bird Studies Canada projects**\n\n**  Celebrate Urban Birds**\n\n**  Christmas Bird Count**\n\n**  eBird**\n\n**  Global Big Day**\n\n**  Habitat Network**\n\n**  Hummingbirds at Home**\n\n**  NestWatch**\n\n**  Project FeederWatch**\n\n# **WINTER THICKET**\n\n**B IRDS HAVE** different needs in the winter compared to summer, when temperatures are comfortable, nesting materials are available, and food and water are abundant\u2014hence the reason so many bird species migrate. For resident birds who remain in the same region year-round, wintertime is a certain challenge. Water becomes frozen, food sources scarce, and temperatures chill down to the bone.\n\n# **TAKE ACTION**\n\nCreate a thicket especially for winter wildlife to help them through winter's short days and long, cold nights. Although winter is not an opportune time to plant in the ground, you can provide sheltering landscape materials in the form of dense, evergreen shrubs in pots on your deck, patio, or throughout your yard. Come next spring, you can plan ahead and plant native, densely branched deciduous shrubs and berry-producing shrubs that will be a welcome sight for birds come winter. If you have evergreens in your landscape, leave the lower boughs and branches in place to keep them full, inviting, and cozy.\n\nLeave an area with long grasses, instead of trimming everything down. This spot will provide a cozy retreat for birds when the weather is extreme.\n\nNext, provide a brush pile of sticks, branches, and dried leaves in a quiet corner or near your yard's edge or along a fence, creating additional shelter opportunities for birds. Pile the branches loosely, with some edges poking out from the top so birds may perch and scan the area to make sure it's safe. Finally, install a \"winter roost box\"\u2014a birdhouse birds may utilize in winter to escape cold temperatures and winds.\n\n  **_Provides stewardship opportunities_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Birds' feathers help keep their bodies warm, but their feet do not have feathers. How do they keep their feet from freezing?**\n\n**A: Bird feet do get very, very cold when temperatures are freezing. However, their feet are made up mostly of bone and tendon and have very little muscle and nerve tissue, which means they have very little fluid in them. This physiology helps prevent frostbite. Birds also alternate standing on one foot at a time, keeping the other foot tucked up into their bodies for warmth. Birds also have a \"countercurrent heat exchange system\" with their arteries and the veins in their legs to help warm the blood that is cooled by chilly weather.**\n\n# **TELLTALE TOES**\n\n**W HEN SNOW** blankets the world around us, it feels as if the whole world stands still. However, for wild animals\u2014including birds\u2014it's business as usual. There's food to forage for, and their foraging often tells a magical tale in the form of the tracks left behind.\n\n# **TAKE ACTION**\n\nJust who is out and about in the snow? Don't you want to know? Explore your snowy landscape and get to the bottom of the mystery. Seek evidence of tracks and tails. Look around trees, across open areas, near bodies of water, beneath feeders, bushes, and shrubs. Check near wood and brush piles. How many different types of tracks can you find? Use these clues:\n\n**  Are the tracks arranged in pairs with each footprint right next to the other? If so, the bird was hopping. Cardinals, chickadees, juncos, and wrens are hoppers.**\n\n**  Are the steps staggered, one print a bit ahead of the other? This shows a bird who skipped along, such as an American Robin.**\n\n**  Are there prints spaced out from one another? If so, this bird was walking. Starlings are walkers.**\n\n**  Are the tracks found together, with multiple types along the same path?**\n\n**  Where did they start and where do they lead? How do they vary in size and shape?**\n\nHave some fun and make your own tracks! Hop like a chickadee, skip like a robin, and walk like a starling. What other tracks can you make?\n\n  **_Stimulates exercise, observation, and environmental awareness_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Are all bird feet alike?**\n\n**A: All birds have four toes: three toes in the front and one toe in the back (ideal for gripping and perching)\u2014but their feet come in different sizes and shapes, depending on the species. For example, a wading bird (think heron) has a very broad foot with long, narrow toes to give it stability as it walks in wet, mushy areas. Crows and starlings have much larger feet than the feet you'll find on a finch or a junco. Ducks, swans, and geese have webbing between their toes.**\n\n**One variation to the \"three toes in the front and one toe in the back\" makeup can be found on owls, parrots, roadrunners, and woodpeckers. They, too, have four toes\u2014but two toes are in the front and two toes are in the back, perfect for scaling and climbing surfaces. This toe formation is called** **_zygodactyl._** **And all four toes on swifts face forward. Because of this, they don't perch but hang from their toes instead!**\n\n# **WINTER WREATH FEEDER**\n\n**M AKE WINTER**\u2014a time when food sources are scarce and survival is ultra-challenging\u2014all about creating an outdoor space that supports and sustains birds. A winter wreath feeder puts a creative spin on the traditional wreath, and this one will hang sideways!\n\n# **TAKE ACTION**\n\nThis feeder is not only fun to make; it will also be a beautiful object that offers sustenance to hungry winter birds.\n\nJute or twine\n\nScissors\n\n1 grapevine wreath (available at most craft sections\/stores)\n\nAluminum pie tin or splatter guard (available from dollar stores)\n\nFloral wire\n\nFresh pine greenery, trimmed from evergreens in your yard or from garden center shops\n\nDried bird millet (gathered from grassy fields or available where pet food is sold for birds)\n\nBird Butter (recipe follows)\n\nNatural pine cones (gathered from nature, or store bought\u2014unscented\/no glitter)\n\n1 orange, cut into 4 wedges\n\nApple slices\n\nBirdseed\n\n  1. Cut three strips of jute or twine, each 3 feet long.\n\n  2. Your wreath will be suspended from three points: 12 o'clock, 4 o'clock, and 8 o'clock. Run each section of twine through the grapevine wreath at these three separate points, knotting all six loose ends together at the top. You may also choose to knot the twine to the grapevine at these three points, and then gather loose ends at the top, knotting them together once more.\n\n  3. Poke very small holes into the pie tin, small enough for moisture to drip through but not large enough to allow seed to spill through. Poke two or three small holes in the edges, as well.\n\n  4. Place the pie tin in the center of the wreath. Using floral wire, attach the tin to the wreath through the holes poked into the tin's edges.\n\n  5. Add extra greenery pickings of your choice to the perimeter of the wreath by simply slipping the tips into open spaces of the grapevine.\n\n  6. Add sprigs of millet to the perimeter of the wreath.\n\n  7. Spread the bird butter into the pine cones' crevices.\n\n  8. Using floral wire, attach the bird butter\u2013filled pine cones to the perimeter of the wreath, among the greenery and millet sprigs.\n\n  9. Using floral wire, pierce the orange and apple wedges, securing them to the perimeter of the wreath.\n\n  10. Fill the pie tin with birdseed.\n\n  11. Hang the feeder from a tree branch or shepherd's hook, and watch the birds enjoy the fruits of your labor!\n\n# **HOW TO MAKE BIRD BUTTER**\n\nThis recipe is a modified version from the Audubon Society (Audubon.org\/news\/make-your-own-suet). Note: suet recipes are not recommended when outdoor temperatures are over 50 degrees Fahrenheit, as the fats spoil in heat. You can substitute palm oil\u2013free shortening for the peanut butter or mix it with the peanut butter.\n\n  1. Mix 1 part peanut butter with 5 parts cornmeal.\n\n  2. Toss about 1 cup of wild birdseed into the mix.\n\n  3. Blend the ingredients together with a large spoon.\n\n  **_Promotes creativity and an appreciation for other living things_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Can birds smell the food we put out for them?**\n\n**A: It was long believed that birds had a very poor sense of smell. However, scientists and researchers, such as Gabrielle Nevitt, a professor at the University of California, Davis, are proving past theories wrong. Studies are showing that birds smell all sorts of things. There's still so much to learn about birds!**\n\n# **DECK THE TREES WITH BOUGHS SO JOLLY**\n\n**C OME THE HOLIDAYS,** keep little hands and little minds busy as they help deck the halls\u2014outdoors. Find a tree or shrub outside that you can embellish and decorate with a variety of treats birds will love.\n\n# **TAKE ACTION**\n\nThere are many foods and ways to create a holiday tree treat for the birds. Explore your outdoor area with your child, and select a tree or shrub that is easily accessible for hanging items upon. If possible, the tree should be visible through a window in your home. Here are just a few treats perfect for your tree:\n\n**  Using natural pine cones with open edges, coat each with suet or peanut butter, then roll them in birdseed. Use a chenille stick or strip of yarn to create a hanger by simply wrapping either around the wide end of the pine cone.**\n\n**  Use natural fiber or string and non-sugared Cheerios to create a garland. Avoid using thin threads or fishing line, which may become a tangle hazard for birds. (This is a great fine-motor skill activity for wee fingers, too!)**\n\n**  Pierce twigs or wooden skewers through orange slices.**\n\n**  After snacking on an apple, save the core for birds\u2014they'll enjoy the seeds! Attach a piece of string or a chenille stick as a hanger.**\n\n**  Purchase small, ready-to-hang seedcake ornaments from your local seed store.**\n\n**  Purchase suet baskets and suet cakes from your local store (available anywhere bird seed is sold) for hanging.**\n\nOnce you've created and gathered treats for your tree, head outside and deck the boughs! Then, sit back and watch who takes interest in the tasty snacks you've provided.\n\n  **_Stimulates creativity and intrinsic reward_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do birds find bird feeders and food?**\n\n**A: Sight is the most important way birds find their food.**\n\n# **BIRD HAIKU**\n\n**B IRDS HAVE INSPIRED** humans for centuries in the forms of art, science, and engineering. Birds continue to inspire us to this day, keeping scientists curious, artists engaged, and people of all ages in awe, merely enjoying their day-to-day presence.\n\nWriting, as an art form, can be a very challenging skill for children to master. It's subjective, and children often fear they'll make a mistake with spelling or grammar, which can block their creative process. However, if children embrace the creative part of writing and don't let the mechanics hinder their thoughts, they'll be well on their way to having fun with writing and mastering the skill. One simple, manageable\u2014and beautiful\u2014form of writing you may practice with your child is haiku poetry.\n\nHaiku is a form of Japanese poetry that uses just a few words to create a sensory experience and vivid imagery in the reader's mind. Traditionally, haiku is written in three lines. The first line has five syllables, the second line has seven syllables, and the final line has five syllables: five-seven-five.\n\n# **TAKE NOTE**\n\nDiscuss with your child the bird experiences you have had together: bird behavior you've seen, species you've encountered, how birds move. Think of words that describe what birds do, like hopping, flapping, flying, feeding, picking, pecking, pulling, tugging, chirping (or, depending on the syllables needed for your poem: hop, flap, fly, feed pick, peck, pull, tug, chirp). Together, brainstorm meaningful words about nature and birds: trees, breeze, clouds, bugs, hawks, nests, bird feeders, types of weather, bird species, etc.\n\nUsing birds and nature as your muses, create a haiku with your child featuring anything avian. You might choose to write about a beautiful feather you've seen, bird flight, bird behavior, a specific species, a bird that captured your interest, or a specific experience you and your child have had with birds. Your child may practice writing the haiku on their own or dictate words to you. Collaborate!\n\nHere's a sample:\n\n_Sunrise brings daylight._ (five beats \/ five syllables)\n\n_A hungry bird looks for food_ (seven beats \/ seven syllables)\n\n_and then sees a worm._ (five beats \/ five syllables)\n\n_Bird feeder with seed_ , (five beats \/ syllables)\n\n_one bird, two birds, three birds, four._ (seven beats \/ syllables)\n\n_Time to fill again._ (five beats \/ syllables)\n\nNow, you try! Words do not need to rhyme. Just play with words and have fun with the process. It's a bit like a puzzle, getting the syllable count to fit each line.\n\n  **_Encourages creativity, applied knowledge, and writing practice_**\n\n(five beats \/ syllables)\n\n(seven beats \/ syllables)\n\n(five beats \/ syllables)\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How long have people been depicting birds in their art?**\n\n**A: The oldest art form featuring a bird is painted on cave walls of the Lascaux Cave in France, dating from 15,000 to 10,000 B.C. It features a bird-headed man.**\n\n# **SNOWY SNOWMAN SNACK**\n\n**A SERENITY HUSHES** over the earth when snow rests upon it, a quiet stillness as each unique flake blankets the ground. We know that wintertime can pose a challenge for birds as food sources disappear: water sources freeze, bugs are out of sight, and plant food sources dwindle. When snow falls, finding food becomes even more challenging for birds.\n\nIf you live in a region that experiences snowfall, take advantage of this white bounty and build a snowman with your child specifically for the birds. Making your snowman will not only provide a much-needed break from being cooped up indoors, but your feathered friends will greatly appreciate this wintry feast.\n\n# **TAKE ACTION**\n\nBuild your classic snowman of three rolled, stacked balls. Once formed, embellish him with things birds will love to eat.\n\nSuet or peanut butter\n\nPine cones\n\nBirdseed\n\nSticks\n\n1 orange, sliced in half\n\n1 carrot\n\n1 apple, sliced into wedges\n\nRaisins\n\nFlower pot tray (the kind placed beneath flower pots)\n\nPeanuts in the shell\n\nGet as creative as you'd like when making your snowman bird feeder, but here are a few ideas to get you and your child started:\n\n  1. Smear suet on and into pine cones, roll the suet-covered pine cones in birdseed, and then place these around the snowman's neck as a scarf.\n\n  2. Place two sticks or branches on either side of the snowman, to serve as arm perches. Sticks may also be placed on the snowman's head, as \"hair perches.\"\n\n  3. Press each orange half into the face for the eyes.\n\n  4. Press a carrot into the face for the nose, which will serve as a perch to access the oranges, or use a pine cone covered in peanut butter, instead.\n\n  5. Place apple wedges on the face, creating a smile or mouth.\n\n  6. Insert raisins around the apples as teeth.\n\n  7. Place the flower pot tray on the snowman's head as a hat, and fill it with birdseed, peanuts in the shell, or water.\n\n  8. Finally, sprinkle some birdseed around the base of your snowman, for the ground feeders.\n\n  **_Encourages exercise, creative play, and stewardship for living things_**\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: How do some birds stay where it's cold, not migrating to warmer places as other birds do?**\n\n**A: Birds that remain in cold weather and that do not migrate have adapted to their environment. Adaptation means fitting in to survive. Birds that \"overwinter\" have adapted by switching food sources from plentiful insects in the summer to seeds and fruits on native plants in the winter.**\n\n# **WINDOW TO THE WORLD**\n\n**W INTER DAYS** can sometimes feel drab and dull, with cloud cover making skies and landscapes appear as one monochromatic color scheme. However, on days like this, birds brighten and pop in the landscape. Winter also provides additional viewing opportunities for birds among naked branches; come spring and summer, those same branches will fill with foliage and hide birds.\n\n# **OBSERVE**\n\nPick a particularly gray day to hunker down with your little chick near a window for a scavenger hunt. Make him comfy with snacks and a beverage. Explain that you're going to embark on an adventure through the wintry window to seek out as many items on the scavenger hunt list as possible. You may create your own list of items to find, depending on your region, but here are some ideas to get you started:\n\n**  A bird tucking one foot up into its body, to keep its feet warm**\n\n**  bird hunkered down low on top of its feet, in an effort to keep its feet warm**\n\n**  A bird puffed up, keeping warm**\n\n**  A busy bird (what is it doing?)**\n\n**  A perching bird**\n\n**  A red-colored bird**\n\n**  A bird with black in its plumage**\n\n**  A black-and-white bird**\n\n**  A brown-colored bird**\n\n**  A tan-colored bird**\n\n**  A black-colored bird**\n\n**  A blue-colored bird**\n\n**  A gray bird**\n\n**  A hopping bird**\n\n**  A bird scratching at the ground**\n\n**  A bird feeding from a feeder**\n\n**  A male bird with bright plumage**\n\n**  A female bird with dull plumage**\n\n**  A flock of birds**\n\n**  A hopping bird**\n\n**  A flock of geese flying overhead**\n\n**  A bird with a crest on its head**\n\n**  A bird with a long, pointy, bug-eating beak**\n\n**  A bird with a triangular, seed-eating beak**\n\n**  A bird with white wing bars**\n\n**  A bird in a tree**\n\n**  A bird species you can only find during the winter**\n\n**  A hawk flying overhead**\n\n**  A bird wiping its beak on a branch**\n\n**  A pecking bird**\n\n  _Encourages observation skills and persistence_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: I noticed a woodpecker with the side of his head up against a tree trunk. It looked as if he were trying to hug the tree. What was he doing?**\n\n**A: Woodpeckers may place an ear up against a tree's trunk to listen for bug activity, to help them find a meal.**\n\n# **A BIRD IN THE HAND**\n\n**I MAGINE BEING** so close to birds that you can see every detail of their tiniest feathers with your naked eye as you watch them crack seeds open with their beaks\u2014right on your very hand.\n\nSometimes we need an excuse to get out in the winter, and this adventurous activity is perfect for a day that's not too nippy\u2014an opportunity to bask in fresh air and experience wild birds in all their glory. The primary goal is to get birds to land on your hand to eat seed you've placed there. Birds are so wary of our presence, and this sounds difficult, but it's not. (This activity is inspired by Jim Carpenter, founder of Wild Birds Unlimited, and his book, _The Joy of Bird Feeding_.)\n\n# **TAKE ACTION**\n\nFirst, make certain you have active, feeding birds at your feeder.\n\nNext, place a patio chair with arms near your feeder with a \"dummy\" sitting upright in it\u2014the dummy being an empty jacket. Place a pillow inside the jacket to give it substance and help it stay erect in the chair. Place a hat over the jacket collar opening. Any type of hat will work: a woolen cap or a baseball cap. Place a glove at the arm opening of the jacket, palm upright, resting on the chair's arm, propping it near the feeder, and then place bird food (seeds and nuts) in the palm of the glove.\n\nWait an hour or two for the birds to become accustomed to gathering seed from both the gloved hand and their feeder. Eventually, remove all the birdseed from your bird feeder, but keep seeds and nuts in the gloved hand. Soon, the birds will discover that the feeder is empty and will continue eating out of the gloved hand, instead.\n\nKeep the gloved hand full of seeds and nuts. After another hour or two, put on the jacket (or one similar to what the \"dummy\" is wearing), the hat, and the glove, and take the dummy's place in the chair. Fill _your_ gloved hand with seeds and nuts, and rest it on the chair's arm. Allow your child to watch from a window, as you remain calm and quiet and still. Soon, the birds will be eating out of your hand.\n\nIf the weather permits, you may experiment with taking the glove off and placing food directly into your hand for the birds to eat. It's important to keep still, though. Once a bird lands on your hand, you may be inclined to jump or flinch or react\u2014it may tickle or feel startling\u2014but remain still and enjoy the wonder and awe right before your very eyes. Then, invite your child to trade places with you, taking turns so he may enjoy this remarkable experience.\n\n  _Practices the art of patience and perseverance; promotes wonder and awe_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Can't the birds tell the difference between the dummy and a real person?**\n\n**A: Yes and no. Birds are smart and have excellent hearing and eyesight. Once they determine\u2014through observation, listening, familiarity, and practice\u2014that the dummy holds no threat to them, they determine it's a safe feeding source. As we take the place of the dummy and behave the same (still and motionless), birds ascertain that we are a safe feeding source, as well. However, any sudden movement on our part will have them taking flight.**\n\n# **SURF , SAND, I SPY A BIRD BAND!**\n\n**W INTER IS A POPULAR TIME** to escape to sunny, warm beaches\u2014or perhaps you live near a beach year-round. Regardless of how you manage to spend time with your wee ones among surf and sand, next time you're there, make note of others who are enjoying the beach, too: shorebirds!\n\nObserving shorebirds is a rewarding and relaxing way to spend time at the beach. It can also prove to be rewarding for the birds, should you chose to participate as a citizen scientist. As you pack your beach bag, include a field guide to shorebirds in the region where you are, along with a pencil and notepad and a pair of binoculars or a camera with a zoom lens. Then, be on the lookout for banded birds on the beach.\n\nShorebirds and water birds are relatively long-legged. As you observe them on the beach, you may notice a colorful band around one or both of their legs. Bird banding is an extremely valuable tool in the study and conservation of bird species. Bands may be metal or plastic and often will be marked with a code of letters and numbers unique to each bird. These \"codes\" will tell biologists where and when that particular bird hatched. Bands also help biologists track bird movement, migration, and mortality\u2014a fascinating glimpse into a bird's life and history!\n\nSpotting and reporting a banded bird is a wonderful way to be a citizen scientist, and biologists rely and very much appreciate when we take time to spot, note, and share a sighted band with them.\n\n# **TAKE ACTION**\n\nUsing your camera or binoculars, quietly and calmly sit nearby an area where shorebirds are resting or feeding, careful not to flush them into flight. Many shorebirds have migrated hundreds or even thousands of miles to the beach where you see them; it's important to allow them their space without disturbing them. Once situated near birds on the beach, quietly spy and look out for bands on legs. This can be a fun and challenging activity for your child!\n\nBe sure to note the following:\n\n**  The location of the band. Is the band above or below the ankle (the ankle is the joint in the middle of a bird's leg)?**\n\n**  The color of the band**\n\n**  The color of the writing on the band**\n\n**  The code, if you can see it; write it down, if you can, or document it with your camera**\n\n**  The date and location of your observation**\n\nOnce noted, report the band, even if it's just the color and location. Every little bit of information provided to biologists will prove insightful.\n\nYou can report banded birds using the following resources:\n\n**  Patuxent Wildlife Research Center Bird Banding Laboratory, the United States Geological Survey: www.pwrc.usgs.gov\/BBL\/bblretrv**\n\n**  Banded Birds.Org: bandedbirds.org**\n\n**  Audubon Atlantic Flyway: nc.audubon.org**\n\n  _Encourages environmental awareness, citizen scientist involvement, persistence, and observation skills_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Why do many shorebirds rest in flocks on the beach, all facing the same direction?**\n\n**A: Shorebirds flock for a number of reasons. When resting, a flock provides safety in numbers, including more eyes and ears to spot danger. Resting birds on the beach face the wind for a number of reasons, too. First, this helps their feathers remain neat and smooth against their bodies. Secondly, facing the wind helps them to be ready for flight. Should they need to take off suddenly, all they need to do is open their wings, and the wind will help to give them lift.**\n\n# **NEW YEAR'S BIRD**\n\n**H APPY NEW YEAR!** As you and your child embark on the first day of the new year, make it a point to wonder together what might be the very first bird you see on this New Year's Day.\n\n# **OBSERVE**\n\nFirst, make a guess as to what bird you might see. Then, wait and watch.\n\nAs you spy your first bird of the new year, rejoice! Then, give thanks for the many blessings nature gives us\u2014and it's all free and right outside our doors and windows, just waiting for exploration.\n\nMake spotting the first bird of the new year an annual tradition for your family, something to eagerly anticipate and celebrate. Wonder...what might be the first bird species you see the next morning, and the morning after that. Consider keeping track and noting any pattern to the species that you encounter for the first time each morning or day.\n\n  _Promotes enthusiasm, wonder, and excitement_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Will the first bird we saw on New Year's Day be the same species we see for the first time each New Year's Day?**\n\n**A: The only way to find out is to observe, year after year, on each New Year's Day!**\n\n# **NEW YEAR'S BIRD RESOLUTIONS**\n\n**T HE START OF A NEW YEAR** is always a hopeful time, with fresh ideas and new beginnings\u2014and of course, resolutions to guide us along and kick-start the future. We can apply the same philosophy to our lives with birds in the year ahead, so why not make some New Year's bird resolutions with your child as part of next year's plan?\n\n# **TAKE NOTE**\n\nWith your child, discuss and reflect back on your life with birds. Then, envision how you'd like to experience birds in your lives in the future.\n\n**  Maybe there's a species of bird you've yet to see, hear, or identify?**\n\n**  Maybe you'd like to better identify birds by their calls?**\n\n**  Maybe you'd like to attract additional bird species to your backyard, increasing your yard's diversity?**\n\n**  Maybe you'd like to travel to seek out bird species that aren't specific to your region?**\n\n**  What about those elusive, tricky-to-identify warblers? Maybe this is the year to seek them out and practice identifying them?**\n\n**  Maybe you'd like to volunteer at a wild bird rehab facility? Or give back to birds in some other manner?**\n\n**  Maybe you'd like to foster stewardship for future generations and introduce a friend or other family members to the lives of birds?**\n\n**  Maybe you'd like to join a bird club? Or better yet, form one!**\n\n**  Maybe you'd like to read more books about birds to learn more about them?**\n\n**  Maybe you'd like to connect with other birders in your community, networking and birding together?**\n\nCreate your New Year's bird resolutions with your child. Then, reflect upon them and revisit them throughout the year. But most importantly, enjoy your love of birding and have fun together as you become ambassadors of joy with birds!\n\n  _Encourages goal setting and promotes excitement for things to come_\n\n**HELP ME UNDERSTAND**  \n---  \n  |\n\n**Q: Does New Year's Day take place on the same day around the world?**\n\n**A: New Year's Day takes place on different dates of the year, depending on the country celebrating the New Year. However, regardless of date or locale, each New Year is one of celebration.**\n**ACKNOWLEDGMENTS**\n\n**I AM NOT AN ORNITHOLOGIST.** I am merely someone who finds extreme joy in everything avian. Birds have become the sun in my universe; thus, my family and I orbit around their existence wherever we go. For this I am grateful for birds, of course, and bestow a heartfelt and loving thanks to my husband Charlie, who somehow manages to point out bird species I always miss. Loving thanks also to my daughter, Kelly, my sisters, Kristen and Debbie, and my main peep, Christina Kaman\u2014for their patience, understanding, knowledge, and shared enthusiasm as we venture out in nature together.\n\nI couldn't have published this book without the time and knowledge shared by the St. Louis Audubon Society. I am indebted to the folks there who poured over these pages to check my writing for ornithological accuracy. A special shout-out and thanks to Mitch Leachman and Pat Lueders. May I add, I take the sole responsibility for any scientific inaccuracies that may occur in the pages of this book.\n\nEternal gratitude to my parents, Paul and Charlene Sultan, for sharing the wonders of nature with me throughout my childhood and filling my world with books that opened and expanded my horizon even more. I love you both so much and am so grateful.\n\nDeep appreciation to the wind beneath my wings: my agent, Stefanie Von Borstel, for her never-ending cheerleading with my projects; my editor, Jenn Urban-Brown, for her vision and patience with this book; and the team at Shambhala Publications\/Roost for bringing this book to fruition\u2014thank you!\n\nGreat respect and thanks to the many folks who have provided me with insight into the lives of birds, help with the writing process, and who share the joy of environmental stewardship\u2014adventurers, artists, authors, book folk, educators, environmentalists, friends, naturalists, photographers, scientists, stewards\u2014the world is more rich with your presence: Linze Aya, Ruth Beeker, Brooke Bessesen, Judy Boise, Loree Griffin Burns, Todd Christopher, Laurie Coffey, Susan Dierker, Cynthia Jenson-Elliott, Scott Evers, Denise Fleming, Candy Fowler, Greg Fowler, Lisa Fowler, Kate Garchinsky, Susie Ghahremani, Maria Gianferrari, Cindy Gray, Steve Gray, Steve Jenkins, Bart King, Lisa King, Teri Kingston, Suzy Leopold, Richard Louv, Sharon Lovejoy, Stephanie L. McAndrews, Alice McGinty, Kate Messner, David Mizejewski, Heather Montgomery, Robert Mulvihill, Diane Schneider Munster, Terry Pierce, Laurence Pringle, Sarah Schwartzman Palermo, Ann Pettigrew, Hob Osterlund, Russell Reed, April Pulley Sayre, Liz Garton Scanlon, Randi Miller Sonenshine, Sharon Sorensen, Melissa Stewart, Debbie Sultan, Kristen Sultan, Jing Jing Tsong, Deborah Vath, Kelly Ward, Tamra Wight, Sallie Wolf, and Julie (Never-Walk-by-a-Weed-without Pulling-It) Zickefoose.\n**RESOURCES AND RECOMMENDED READING**\n\n**BOOKS AND MAGAZINES FOR ADULTS**\n\nAckerman, Jennifer. _The Genius of Birds._ Penguin Press, 2016.\n\nAlderfer, Jonathan. _National Geographic Backyard Guide to the Birds of North America._ National Geographic, 2011.\n\n_Birds and Blooms_ Magazine\n\n_BirdWatching_ Magazine\n\nCarpenter, Jim. _The Joy of Bird Feeding: The Essential Guide to Attracting and Feeding Our Backyard Birds._ Scott & Nix, Inc., 2017.\n\nErickson, Laura. _101 Ways to Help Birds._ Stackpole Books, 2006.\n\n\u2014\u2014\u2014. _Sharing the Wonder of Birds with Kids._ University of Minnesota Press, 1997.\n\n\u2014\u2014\u2014. _The Bird Watching Answer Book: Everything You Need to Know to Enjoy Birds in Your Backyard and Beyond._ Cornell Lab of Ornithology, 2009.\n\nHeinrich, Bernd. _One Wild Bird at a Time: Portraits of Individual Lives._ Houghton Mifflin Harcourt, 2016.\n\nKaufman, Kenn. _Kingbird Highway: The Biggest Year in the Life of an Extreme Birder._ Houghton Mifflin Harcourt, 2006.\n\n_Living Bird Magazine_\n\nLouv, Richard. _The Nature Principle: Reconnecting with Life in a Virtual Age._ Algonquin Books, 2012.\n\nMizejewski, David. _National Wildlife Federation: Attracting Birds, Butterflies and Backyard Wildlife._ Creative Homeowner, 2004.\n\nOsterlund, Hob. _Holy Moli: Albatross and Other Ancestors._ Oregon State University Press 2016.\n\nSafina, Carl. _Beyond Words: What Animals Think and Feel._ Picador; Reprint Edition, 2016.\n\nSayre, April Pulley. _Touch a Butterfly: Wildlife Gardening for Kids\u2014Simple Ways to Attract Birds._\n\nSampson, Scott D. _How to Raise a Wild Child: The Art and Science of Falling in Love with Nature._ Mariner Books, 2016.\n\nSorenson, Sharon. _Birds in the Yard Month by Month._ Stackpole Books, 2013.\n\nStiteler, Sharon. _1001 Secrets Every Birder Should Know: Tips and Trivia for the Backyard and Beyond_. Running Press, 2013.\n\nStokes, Donald. _Stokes Beginner's Guide to Birds: Eastern Region (Stokes Field Guide Series)_. Little Brown and Company, 1996.\n\n\u2014\u2014\u2014. _The Hummingbird Book: The Complete Guide to Attracting, Identifying, and Enjoying Hummingbirds._ Little Brown and Company, 1989.\n\nStryker, Noah. _The Thing With Feathers_ : _The Surprising Lives of Birds and What They Reveal About Being Human._ Riverhead Books, 2015.\n\nTallamy, Douglas W. _Bringing Nature Home: How You Can Sustain Wildlife with Native Plants_. Timber Press, 2007.\n\nYoung, Jon. _What The Robin Knows_ : _How Birds Reveal the Secrets of the Natural World_. Mariner Books, 2013.\n\nZickefoose, Julie. _Baby Birds: An Artist Looks Into the Nest._ Houghton Mifflin Harcourt, 2016.\n\n\u2014\u2014\u2014. _Natural Gardening for Birds: Create a Bird-Friendly Habitat in Your Backyard._ Skyhorse Publishing, July 2016.\n\n**FABULOUS BIRD BOOKS FOR CHILDREN**\n\nAston, Dianna. _An Egg is Quiet._ Chronicle Books, 2014.\n\n_\u2014\u2014\u2014. A Nest is Noisy._ Chronicle Books, 2017.\n\nBaker, Jeannie. _Circle._ Candlewick, 2016.\n\nBierregaard, Rob. _Belle's Journey: An Osprey Takes Flight._ Charlesbridge, 2018.\n\nCate, Annette Leblanc. _Look Up! Bird-watching in Your Own Backyard._ Candlewick, 2013.\n\nChristelow, Eileen. _Robins! How They Grow Up._ Clarion Books, 2017.\n\nCollard, Sneed. _Beaks!_ Charlesbridge, 2002.\n\n_\u2014\u2014\u2014. Woodpeckers._ Bucking Horse Books, 2018.\n\n_\u2014\u2014\u2014. Firebirds._ Bucking Horse Books, 2015.\n\nDavies, Jacqueline. _The Boy Who Drew Birds._ HMH Books for Young Readers, 2004.\n\nDeacon, Alexis. _I Am Henry Finch._ Candlewick, 2015.\n\nDierker, Susan. _Albatross of Kauai: The Story of Kaloakulua._ Done by Dogs Publishing, 2014.\n\nEhlert, Lois. _Feathers for Lunch._ HMH Books for Young Readers, 1996.\n\nElliott, David. _On the Wing._ Candlewick, 2014.\n\nErickson, Laura. _Am I Like You?_ Cornell Lab Publishing Group, 2016.\n\nFleming, Denise. _This is the Nest That Robin Built._ Beach Lane, 2018.\n\nFlorian, Douglas. _On the Wing: Bird Poems and Paintings._ HMH Books for Young Readers, 2000.\n\nFranco, Betsy. _Birdsongs._ Margaret K. McElderry Books, 2007.\n\nFried, Caren-Loebel. _A Perfect Day for an Albatross._ Cornell Lab Publishing Group, 2017.\n\nFrost, Helen. _Sweep Up the Sun._ Candlewick, 2015.\n\nGeorge, Jean Craighead. _The Eagles Are Back._ Dial Books, 2013.\n\nGianferrari, Maria. _Hawk Rising._ Roaring Brook Press, 2018.\n\nGibbons, Gail. _Owls._ Holiday House, 2006.\n\nGraham, Bob. _How to Heal a Broken Wing_. Candlewick, 2017.\n\nGray, Rita. _Have You Heard the Nesting Bird?_ HMH Books for Young Readers, 2017.\n\nGray, Steve. _The Ravenous Raven._ Grand Canyon Association, 2015.\n\nGrolleau, Fabien. _Audubon: On the Wings of the World._ Nobrow Press, 2017.\n\nHenkes, Kevin. _Birds._ Greenwillow Books, 2009.\n\n_\u2014\u2014\u2014. Egg._ Greenwillow Books, 2017.\n\nHestermen, Katie. _A Round of Robins_. Nancy Paulsen Books, 2018.\n\nHiaasen, Carl. _Hoot_. Yearling, 2005.\n\nHoose, Phillip. _Moonbird._ Farrar, Straus and Giroux, 2012.\n\n\u2014\u2014\u2014. _The Race to Save the Lord God Bird._ Square Fish, 2016.\n\nIdle, Molly. _Flora and the Flamingo._ Chronicle Books, 2013.\n\n_\u2014\u2014\u2014. Flora and the Peacocks._ Chronicle Books, 2016.\n\n_\u2014\u2014\u2014. Flora and the Penguin._ Chronicle Books, 2014.\n\nJenkins, Martin. _Bird Builds a Nest._ Candlewick, 2018.\n\nJenkins, Steve. _Animals in Flight._ HMH Books for Young Readers, 2005.\n\nJudge, Lita. _Bird Talk: What Birds Are Saying and Why._ Flash Point, 2012.\n\n_\u2014\u2014\u2014. Flight School._ Atheneum Books for Young Readers, 2014.\n\nLarson, Jeannette. _Hummingbirds: Fact and Folklore._ Charlesbridge, 2011.\n\nLewis, Gill. _Wild Wings._ Atheneum Books for Young Readers, 2012.\n\nLowry, Lois. _Crow Call._ Scholastic Press, 2009.\n\nLurie, Susan. _Swim, Duck, Swim._ Feiwel & Friends, 2016.\n\nMarkle, Sandra. _A Mother's Journey._ Charlesbridge, 2006.\n\n\u2014\u2014\u2014. _The Long, Long Journey: The Godwit's Amazing Migration._ Millbrook, 2013.\n\nMcDermott, Gerald. _Raven: A Trickster Tale from the Pacific Northwest._ HMH Books for Young Readers, 2001.\n\nPla, Sally. _The Someday Birds._ Harper Collins, 2018.\n\nProsek, James. _Bird, Butterfly, Eel._ Simon & Schuster Books for Young Readers, 2009.\n\nRose, Deborah Lee, et, al. _Beauty and the Beak: How Science, Technology, and a 3D-Printed Beak Rescued a Bald Eagle._ Persnickety Press, 2017.\n\nRosen, Michael. _The Cuckoo's Haiku._ Candlewick, 2009.\n\nRuddell, Deborah. _Today at the Bluebird Caf\u00e9: A Branchful of Birds._ Margaret K. McElderry Books, 2007.\n\nSayre, April Pulley. _Vulture View._ Henry Holt and Co., 2007.\n\n_\u2014\u2014\u2014. Warbler Wave._ Beach Lane Books, 2018.\n\n_\u2014\u2014\u2014. Woodpecker Wham!_ Henry Holt and Co., 2015.\n\nScanlon, Liz Garton. _One Dark Bird._ Beach Lane Books, 2019.\n\nSchulman, Janet. _Pale Male: Citizen Hawk of New York City._ Knopf, 2008.\n\nSidman, Joyce. _Red Sings from Treetops._ HMH Books for Young Readers, 2009.\n\nSill, Cathryn. _About Birds: A Guide for Children._ Peachtree Publishers, 2013.\n\nSinger, Marilyn. _The Company of Crows._ Clarion Books, 2002.\n\nSonenshine, Randi Miller. _This is the Nest that Wren Built._ Candlewick, 2020.\n\nStemple, Heidi E.Y. _Counting Birds: The Idea That Helped Save Our Feathered Friends._ Seagrass Press, 2018.\n\nStewart, Melissa. _A Place for Birds._ Peachtree Publishers, 2015.\n\n\u2014\u2014\u2014. _Feathers Not Just for Flying._ Charlesbridge, 2014.\n\nTavares, Matt. _Red and Lulu._ Candlewick, 2017.\n\nTsong, Jing Jing. _Birds in Hawaii._ Beach House Publishing, 2017.\n\nVanDerwater, Amy Ludwig. _Every Day Birds._ Orchard Books, 2016.\n\nWaddell, Martin. _Owl Babies._ Candlewick, 2002.\n\nWard, Jennifer. _How to Find a Bird._ Beach Lane, 2020.\n\n_\u2014\u2014\u2014. Mama Built a Little Nest._ Beach Lane Books, 2014.\n\nWight, Tamra. _Mystery of the Eagles Nest._ Island Port Press, Inc., 2016.\n\n_\u2014\u2014\u2014. Mystery on Pine Lake._ Island Port Press, Inc. 2015.\n\nWolf, Sallie. _The Robin Makes a Laughing Sound: A Birder's Observations._ Charlesbridge, 2010.\n\nYolen, Jane. _An Egret's Day._ Wordsong, 2010.\n\n_\u2014\u2014\u2014. On Bird Hill._ Cornell Lab Publishing Group, 2016.\n\n_\u2014\u2014\u2014. On Duck Pond._ Cornell Lab Publishing Group, 2017.\n\n_\u2014\u2014\u2014. Owl Moon._ Philomel Books, 1987.\n\n_\u2014\u2014\u2014. You Nest Here With Me._ Boyds Mills Press, 2015.\n\n**WEBSITES**\n\nAmerican Birding Association: aba.org\n\nAmerican Bird Conservancy: abcbirds.org\n\nAmerican Ornithological Society: americanornithology.org\n\nAudubon\u2014Nectar Sources by Region: audubon.org\/content\/nectar-sources-region\n\nBringing Back the Birds: abcbirds.org\/program\/bringing-back-the-birds\n\nCornell Lab of Ornithology: birds.cornell.edu\n\nEnvironment for the Americas\u2014International Migratory Bird Day: birdday.org\n\nFLAP Canada: flap.org\n\nHummingbirds.net (Lanny Chambers): hummingbirds.net\n\nInternational Bird Rescue: bird-rescue.org\n\nNational Aviary: aviary.org\n\nNational Audubon Society: audubon.org\n\nNational Wildlife Federation: nwf.org\n\nThe Nature Conservancy: nature.org\n\nNorth American Bird Conservation Initiative (NABCI) \/ International Bird Conservation: nabci-us.org\/international-bird-conservation\n\nProject Owlnet through the National Aviary: aviary.org\/project-owlnet\n\nRoyal Society for the Protection of Birds: rspb.org.uk\n\nSierra Club: sierraclub.org\n\nSmithsonian Migratory Bird Center: nationalzoo.si.edu\/migratory-birds\n\nUnited Nations Environment Program: unep.org\n\nUnited States Natural Resources Conservation Service: nrcs.usda.gov\n\nWildlife Rehabilitators Listed by State: wildliferehabinfo.org\n**AMERICAN BIRD ASSOCIATION CODE OF ETHICS**\n\n**T HE** American Birding Association developed and promotes the following code of birding ethics, which may be freely reproduced for distribution with acknowledgment to the ABA for its development and a link to the ABA website: www.aba.org.\n\n1. Promote the welfare of birds and their environment.\n\n1(a) Support the protection of important bird habitat.\n\n1(b) To avoid stressing birds or exposing them to danger, exercise restraint and caution during observation, photography, sound recording, or filming.\n\nLimit the use of recordings and other methods of attracting birds, and never use such methods in heavily birded areas or for attracting any species that is Threatened, Endangered, of Special Concern, or is rare in your local area.\n\nKeep well back from nests and nesting colonies, roosts, display areas, and important feeding sites. In such sensitive areas, if there is a need for extended observation, photography, filming, or recording, try to use a blind or hide, and take advantage of natural cover.\n\nUse artificial light sparingly for filming or photography, especially for close-ups.\n\n1(c) Before advertising the presence of a rare bird, evaluate the potential for disturbance to the bird, its surroundings, and other people in the area, and proceed only if access can be controlled, disturbance minimized, and permission has been obtained from private landowners. The sites of rare nesting birds should be divulged only to the proper conservation authorities.\n\n1(d) Stay on roads, trails, and paths where they exist; otherwise, keep habitat disturbance to a minimum.\n\n2. Respect the law, and the rights of others.\n\n2(a) Do not enter private property without the owner's explicit permission.\n\n2(b) Follow all laws, rules, and regulations governing use of roads and public areas, both at home and abroad.\n\n2(c) Practice common courtesy in contacts with other people. Your exemplary behavior will generate goodwill with birders and non-birders alike.\n\n3. Ensure that feeders, nest structures, and other artificial bird environments are safe.\n\n3(a) Keep dispensers, water, and food clean and free of decay or disease. It is important to feed birds continually during harsh weather.\n\n3(b) Maintain and clean nest structures regularly.\n\n3(c) If you are attracting birds to an area, ensure the birds are not exposed to predation from cats and other domestic animals or dangers posed by artificial hazards.\n\n4. Group birding, whether organized or impromptu, requires special care.\n\nEach individual in the group, in addition to the obligations spelled out in Items #1 and #2, has responsibilities as a Group Member:\n\n4(a) Respect the interests, rights, and skills of fellow birders, as well as people participating in other legitimate outdoor activities. Freely share your knowledge and experience, except where code 1(c) applies. Be especially helpful to beginning birders.\n\n4(b) If you witness unethical birding behavior, assess the situation and intervene if you think it prudent. When interceding, inform the person(s) of the inappropriate action and attempt, within reason, to have it stopped. If the behavior continues, document it and notify appropriate individuals or organizations.\n\nGroup Leader Responsibilities [amateur and professional trips and tours]:\n\n4(c) Be an exemplary ethical role model for the group. Teach through word and example.\n\n4(d) Keep groups to a size that limits impact on the environment and does not interfere with others using the same area.\n\n4(e) Ensure everyone in the group knows of and practices this code.\n\n4(f) Learn and inform the group of any special circumstances applicable to the areas being visited (e.g., no audio playback allowed).\n\n4(g) Acknowledge that professional tour companies bear a special responsibility to place the welfare of birds and the benefits of public knowledge ahead of the company's commercial interests. Ideally, leaders should keep track of tour sightings, document unusual occurrences, and submit records to appropriate organizations.\n**ABOUT THE AUTHOR**\n\n**J ENNIFER WARD** is the author of more than twenty award-winning books for children\u2014most about nature and many which feature birds\u2014in addition to parenting books that help connect children to nature. Her books have been translated into many languages and have been featured on national television and NPR. She lives with her husband and two dogs, nestled among a canopy of old growth oak forest in Southern Illinois, where she writes full time surrounded by birds and birdsong. Learn more about Jennifer and her books at jenniferwardbooks.com.\nSign up to receive free projects and special offers from Roost Books.\n\nOr visit us online to sign up at roostbooks.com\/eroost.\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Contents\n  5. Introduction\n  6. Spring\n    1. 1. Look Up, Look Down, Look All Around\n    2. 2. Billions of Birds\n    3. 3. A Feast for First Arrivals\n    4. 4. The Dawn Chorus\n    5. 5. Home Tweet Home\n    6. 6. Feathery Field Trip\n    7. 7. Splashy Tweet Retreat\n    8. 8. A Rustic Roost\n    9. 9. Birds in Motion\n    10. 10. Draw that Bird!\n    11. 11. Zoom In\n    12. 12. Feathered Firsts\n    13. 13. Fantastic Feeders\n  7. Summer\n    1. 14. Getting to Know You\n    2. 15. The Perfect Patch\n    3. 16. Raindrop Cafe\n    4. 17. A Haven for Hummingbirds\n    5. 18. Bountiful Birdscape\n    6. 19. Wing It!\n    7. 20. Hats Off to Hummingbirds!\n    8. 21. Seed Sorting\n    9. 22. Yard Mapping\n    10. 23. Feathery Photography\n    11. 24. Heavenly Hawks\n    12. 25. Chicka-dee-dee-dee\n    13. 26. Beak by Beak\n  8. Fall\n    1. 27. Color Count\n    2. 28. Moonlight Migration\n    3. 29. A Handful of Hummers\n    4. 30. Lovely Leaf Litter\n    5. 31. Handy Nests\n    6. 32. A Pumpkin-Perfect Feast\n    7. 33. Pressed Leaf Feather Art\n    8. 34. Speech Bubbles between Birds\n    9. 35. W're Going on an Owl Hunt\n    10. 36. Bird Brains\n    11. 37. Name That Bird\n    12. 38. Chilling with Feathers\n    13. 39. A Nest of My Own\n  9. Winter\n    1. 40. Naked Nests\n    2. 41. Flappy Hour and the Great Backyard Bird Count\n    3. 42. Winter Thicket\n    4. 43. Telltale Toes\n    5. 44. Winter Wreath Feeder\n    6. 45. Deck the Trees with Boughs So Jolly\n    7. 46. Bird Haiku\n    8. 47. Snowy Snowman Snack\n    9. 48. Window to the World\n    10. 49. A Bird in the Hand\n    11. 50. Surf, Sand, I Spy a Bird Band!\n    12. 51. New Year's Bird\n    13. 52. New Year's Bird Resolutions\n  10. Acknowledgments\n  11. Resources and Recommended Reading\n  12. American Bird Association Code of Ethics\n  13. About the Author\n  14. E-mail Sign-up\n\n# Landmarks\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Table of Contents\n  5. Start\n\n# Print Page List\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. x\n  11. xi\n  12. xii\n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n# Development as Freedom\n\nAmartya Sen\n\nOxford UP \u00a9 2001\n\ngetAbstract \u00a9 2017\n\n## Rating (10 is best)\n\nOverall: 6\n\nApplicability: 5\n\nInnovation: 7\n\nStyle: 6\n\n## Take-Aways\n\n  * In the growing global economy the rich get richer.\n  * Freedoms of different kinds can counter deprivation wherever it occurs.\n  * Sustainable development requires personal and political freedom.\n  * Expansion of freedom ought to be the primary goal of development.\n  * Expansion of freedom also ought to be the principal means of development.\n  * This expansion occurs when development is viewed as the realization of capabilities\n  * Relative deprivation can be quantified in terms of a culture's ability to empower deprived people to realize their capabilities.\n  * Empower women to achieve only what the men of the same culture are capable of achieving and the road to development becomes smoother.\n  * Empower the poor to achieve their capabilities and you have made the soul of development real.\n  * In a world faced with over population and failing food production and distribution, development is statistically proven to be the most efficient form of contraception.\n\n## Relevance\n\n### What You Will Learn\n\nIn this Abstract, you will learn: 1) How a brilliant Nobel-winning economist from India looks at traditional economic theories; and 2) How various world views and different economic theories see and shape the concept of freedom - and why that matters in development.\n\n### Review\n\nNobel Prize-winning economic scientist Amartya Sen attempts to popularize a series of lectures he presented to executives at the World Bank in 1996. He challenges traditional economic theories to justify a more aggressive, humane and generous funding formula to benefit the world's poorest nations. This goal is based on his theory about individual capabilities and functionings, and how they affect opportunity, both person by person and in a society. Even though this is aimed for general discussion rather than Ph.D. course work, it is an extremely daunting book to read, a mental maze land mined with quirky thoughts and a thick lexicon only an academic could love. More thesis than not, the text is 298 pages plus 60 pages of small type footnotes. The short version: the rich get richer and the poor remain deprived of abilities and awaiting enlightened development. _getAbstract.com_ recommends this dense, challenging but, as they say, important book to insomniacs, liberal world bankers, economic policy makers, the Kofi Annan fan club and students of economic science.\n\n## Summary\n\n### The Parable\n\nTupac needs his swimming pool maintained. Three unemployed people at a nearby highway exit ramp beg for work. Tupac could hire all three of them and distribute the work, but he only has one pool vacuum so the work cannot be divided. He must choose. The candidates are Jake, who is the most destitute; Bertrand, who recently became poor and is very depressed and Lakrisha, who has a bad limp. At first, Tupac thinks it is best to hire Jake, the poorest. Then he decides on Bertrand, because making a poor, depressed person happy perhaps should be a higher priority than giving a job to a well-adjusted pauper. Then he sees Lakrisha's limp. She explains that she has an infected wound from a mugging. He considers giving the job to stoic Lakrisha, so that she can get medical help, knowing that her quality of life will be greatly improved when she can afford freedom from pain.\n\nNow Tupac is conflicted. If he knew only that Jake was poor, he would hire Jake. If he knew only that Bertrand was miserable, he would hire Bertrand. If he knew only that a job would save Lakrisha from pain, he would hire her. He chooses Lakrisha, but not easily. If Tupac were a global economist, he would see his dilemma as information-based. Jake has an income-egalitarian problem with a focus on income-poverty. Bertrand's case is right out of classical utilitarianism, which calculates personal pleasure. Lakrisha is in the midst of a new way of looking at economics that involves the quality of life issues all three confront.\n\nThis is a microcosm of world economics and Tupac represents institutions which decide how to allocate resources. Build a bridge to reach the opium poppy field or burn the poppies and build a school? Starve the parents in hope the kids learn something useful in school so they can earn money before they starve? Or let them live off drug profits for generations because they have no real alternatives?\n\n### Freedom as a Foundation of Justice\n\nAs in this extremely simplified scenario, economic scientists discuss the value and importance of different kinds of information, and worry if they are able to weigh individual freedoms completely. Their deliberations are conducted under different theories of justice, including utilitarianism, libertarianism and Rawlsian thought (see below). Freedom can be defined in part as the individual's capability to do things that are of value to himself or herself. But this is theory and theories about economic principles cannot be analyzed without also analyzing the information on which they are based. As our parable shows, information bases often are incomplete.\n\nGiven that information is missing, the utilitarian theories espoused over the years by Jeremy Bentham come up short. He defines classical utility as pleasure, or at least satisfaction. When an action makes people in a society happy or satisfied, the utility of that action is validated. The utilitarian structure doesn't think it is urgent to find out who is happy or satisfied, and it doesn't concern itself about those who are not happy or satisfied, as long as a sufficient number are happy. This is the classical utilitarian litmus test. In contemporary utilitarianism, happiness is desire's fulfillment and utility is defined in terms of behavioral choices. The exercise of this utilitarian philosophy, in fact, creates an indifference to freedom.\n\nContrasted with utilitarian philosophy, classical libertarianism is not provoked by happiness. Its information base enumerates rights and liberties. If a man has a right to shoot you, if you enter his home, you have no right to search for your happiness by looting his music collection. Libertarian views of justice are usually incompatible with utilitarian views.\n\nAt the outer edge of libertarian thought, John Rawls suggests that extensive classes of rights - including property rights and personal liberties - have political precedence over relieving poverty, increasing education and healing sickness. Property and personal rights, Rawls believed, supercede the consequences of denying them. So, to move from the theoretical to the practical, to Rawls it doesn't matter if your villagers are starving and covered with flies. His land is his land, and neither you nor the government has any right to take it away, even to benefit others. This is not entirely correct. Famines can and do take place without impinging on libertarian rights. Entitlements do not make a family well nourished nor healthy, at least not in theory. If libertarianism can exist happily where people die of hunger, it is not a cause nor a cure, and ought to be rejected as a working economic theory.\n\n### The Utility of Utilitarian Thought\n\nBentham defined utility as the accomplishment of happiness or pleasure. The theory was later modified to include satisfaction with the status quo. Clearly this is subjective since - remember Lakrisha - a person with a rotting leg can learn to live with it and achieve a utilitarian Stoicism. Only at the theoretical level would anyone argue that utilitarianism economics makes sense in a diverse world - or even within the diverse population of a small village - and certainly not for the purposes of economic measure. If, for example, everyone in the village had the same real income, they all ought to be equally satisfied (or dissatisfied) everything else being equal. All other things are not equal, however, given age, gender, family size and human diversity. Yet amount of income and supply of commodities are sometimes used to measure the material standards of people's well being, often without considering five specific variations in real income and the well being and freedom they create. These variations are:\n\n  1. Personal heterogeneities - Some people are chronically sick and others are well, until they have an accident. Even when compensation is provided, it may not be enough to restore well being and freedom.\n  2. Environmental diversities - Where it is hot you need fewer clothes but more water, until the floods come and the insects bring disease. Where it is cold, you need a lot of calories. One income level does not match every need.\n  3. Variations in social climate - To convert personal income into quality of life might be as simple as a crack pipe and a fix for some. For others, it might be getting out of the slums, with their open sewers and nasty housing. For still others, it might a ticket to a concert or admission to graduate school. Social climate matters.\n  4. Differences in relational perspective - It is all relative, depending on where and who you are. Since men in the tropics don't wear ties, you are not deprived if you cannot afford a tie in Tahiti and but you would be in Manhattan. Two wives are illegal in Boston and the proof of underachievement in Kuwait.\n  5. Distribution within the family - Non-earners share what is earned by the earners. If Dad does not like Sister, the intra-family distribution might become a predicament for Sister and impinge on her freedom and well being.\n\nThus, opulence is not the definition of well being or quality of life since, in each case, no matter how opulent your group is, the next group might be less opulent or more opulent. Granted, poverty is the shortage of income. But Rawls suggests that adding \"rights, liberties, opportunities, income and wealth, and the social basis of self respect\" to any evaluation. These are primary goods - things that can be used to promote an individual's desired goals. For instance, Adam Smith defined a man's ability to appear in public without shame as an economic necessity, even though that goal marks a very personal, subjective achievement. This takes us back to well being, freedom and capability.\n\n### Consider the Concept of Functionings\n\nYou cannot evaluate an economic process based entirely on utilities or primary goods. Your evaluation should include - and prioritize - the substantive freedoms that allow an individual to choose a life that he or she can value. This is that person's capability set.\n\nTo go back to our example: Lakrisha has a bad leg and a pool-cleaning job. Thus she may have a larger basket of commodities, but a smaller chance to lead a normal life than a healthy woman with fewer possessions. Comparisons are tricky. The metric used to compare, which is called \"functioning,\" implies that each person has things he or she values. Adequate nourishment and health are elementary functionings. Self-respect among a group of peers is extreme functionings. Economic scientists calculate functioning vectors, recognizing that an individual's choice of \"functioning vectors\" depends on his or her capability set.\n\n### Approaches to Freedom in Public Policy\n\nA public policy that approaches freedom and values by considering capabilities offers three beneficial paths.\n\n  1. The direct approach - What can be said about respective advantages after examining and comparing vectors of functionings? This robust approach offers three sub-paths: 1) Total comparisons that rank all vectors in terms of poverty or inequality; 2) Partial rank of some vectors; and 3) \"Distinguished capability comparison,\" which means contrasting different capabilities and focusing on a unique or particular one. \n  2. The supplementary approach - This may focus on direct comparisons of functioning or variables other than income, including health care, gender bias or poverty. This typically employs the tools of distinguished capability comparison.\n  3. The indirect approach - This focuses on familiar subjects such as income but adjusts to take other variables into account, for example, education or illiteracy.\n\n### Poverty is Capability Deprivation\n\nSince poverty is relative, little benefit is realized by using the standards of commodities or income to measure affluence or poverty. Indeed, low income is almost always a determinant in poverty, which can be identified as capability deprivation. Capability deprivation, the lack of intrinsically important things, such as education, pride and health, is genuine impoverishment while low income is merely an instrumentality.\n\nThe relationship between low income and low capabilities changes from community to community, from culture to culture. The relationship between income and capability is strongly influenced by age, gender, social role, disability and distribution of income within a family. In fact, relative deprivation on the global scale greatly influences capabilities. That is, one could be relatively wealthy by third world standards in Bangladesh but face death and deprivation at the same income level in Detroit.\n\n### Continental Unhappiness\n\nMalthus predicted that continued population growth would ultimately lead to a \"continental diminution of happiness.\" This can be avoided by expanding capabilities through linking economic development and social development. Empower young women to become less passive and better protectors of their birthing decisions; equip them with decisional autonomy. When countries develop each citizen's social, economic, cultural and educational capabilities, the population rate decreases almost inevitably - at a rate that keeps pace with development.\n\n## About the Author\n\n **Amartya Sen** won the Nobel Prize for economic science in 1996. He is past president of the Indian Economic Association, The American Economic Association, the International Economic Association and the Econometric Society. He has taught at Calcutta, Delhi, Oxford, Cambridge, Harvard and the London School of Economics. He is the master of Trinity College, Cambridge.\n\n## Quotes\n\n  * \"Economists are sometimes criticized for concentrating too much on efficiency and too little on equity.\"\n  * \"Much of the debate on the alternative approaches to evaluation relates to the priorities in deciding on what should be at the core of our normative concern.\"\n  * \"The view that poverty is simply shortage of income is fairly well established in the literature on the subject.\"\n  * \"There is thus a strong methodological case for emphasizing the need to assign explicitly evaluative weights and different components of quality of life (or well being) and then to place the chosen weights for open public discussion and critical scrutiny.\"\n  * \"Self interest is, of course, an extremely important motive and many works on economic and social organization have suffered from not paying adequate attention to this basic motivation.\"\n  * \"For example, even if a person who is depressed or disabled or ill happens to have the same demand function over commodity bundles as another who is not disadvantaged in this way, it would be quite absurd to insist that she is having the same utility (or well-being, or quality of life) from a given commodity bundle as the other can get from it.\"\n  * \"Indeed, the insistence on the completeness of judgments of justice over every possible choice is not only an enemy of practical social action, it may also reflect some misunderstanding of the nature of justice itself.\"\n  * \"The broadening of the informational focus from incomes to primary goods is not however, adequate to deal with all the relevant variations of the relationship between income and resources on the one hand and well being and freedom on the other hand.\"\n  * \"Since the preference for market price-based evaluation is quite strong among many economists, it is also important to point out that all variables other than commodity holdings (important matters such as mortality, morbidity, education, liberties and recognized rights) get - implicitly - a zero direct weight in evaluations based exclusively on the real income approach.\"\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# The Thriving Introvert\n\n### Embrace The Gift of Introversion and Live the Life You Were Meant To Live\n\n## Thibaut Meurisse\n\u00a9 2017 Thibaut Meurisse\n\nAll rights reserved. No portion of this book may be reproduced in any form without permission from the publisher, except as permitted by U.S. copyright law.\n\n### Contents\n\nAre you an introvert?\n\nWhy This Book\n\nYour Free Workbook\n\nIntroduction\n\n1. Understanding introversion\n\n2. Owning Your Introversion\n\n3. Optimizing your introversion\n\n4. Leveraging the Gift of Introversion\n\n5. Transcending Your Introversion\n\nConclusion\n\nAffirmations for introverts\n\nWhat do you think?\n\nOthers books by the author\n\nAbout the Author\n\nBibliography\n\nGoal Setting Book Preview\nDo you despise small talk? Do you frequently refuse to answer your phone? Do you wish you could skip the parties you're invited to? If any of these sounds familiar, you might be an introvert. Don't worry, you're not odd or defective. In fact, you're full of wonderful qualities that are much needed in today's society! You simply need some time on your own to recharge your battery. I invite you to view your introversion as a gift, and the best way to offer it to the world is by embracing who you really are deep down.\n\nThis book will be a great fit for you if any of the following is true:\n\n  * You want to learn more about introversion and its implications on your life.\n  * You're sick of being told to talk more, become more outgoing, or attend more social events, and you want to embrace your introversion wholeheartedly. \n  * You want to create a life that supports your introversion. \n  * You struggle with social events and want to become better at interacting with people while staying true to your introverted self. \n  * You want to be able to leave parties without feeling guilty, awkward, or pressured to stay. \n  * You want to feel comfortable in your own skin and thrive as an introvert.\n\nIf you can relate to any of these statements, this book is for you!\n**W hy This Book**\n\nThis book is the book I wish someone had given me when I was younger. To be honest, I didn't know what introversion was until a few years ago. And I only learned about it when I started delving deeply into personal development.\n\nUnder different circumstances, I would probably have spent my entire life ignorant of introversion and its implications on my life. Sadly, that's what happens to many of us. Even when we believe we understand introversion, our idea of it is often vague. As a result, we have difficulties designing a life that truly matches our introverted personality. If anything, we blame ourselves for our introverted behaviors, and may even feel ashamed of ourselves.\n\nWhile you can find many great books on introversion out there, they generally fail to offer you practical steps to help you redesign your life. After reading them, you may feel inspired or empowered in some ways, but without a practical action plan, they will leave you unsure of what to do next.\n\nEven worse, these books often encourage you to act like an extrovert, perpetuating the idea that something is wrong with you. My guess is that you don't want to be told you should behave like an extrovert, am I right?\n\nMy goal with this book is to explain what introversion is, and to provide you with practical exercises that will help live the life you were meant to live as an introvert. Although in some occasions I will challenge you to move beyond your comfort zone, I don't want you to behave like an extrovert. Instead, I want you to use the gift of your introversion to make your biggest contributions to the world while being as happy as you can.\n\nMore specifically, my aim with this book is to:\n\n  * Help you deepen your understanding of introversion\n  * Encourage you to fully accept your introversion and remove any feelings of guilt or shame\n  * Offer you an opportunity to reflect on your own introversion\n  * Entice you to take concrete actions towards redesigning your life, and\n  * Inspire you to live with your introversion to the fullest and make your best contribution to the world.\n\nI understand that your time is precious and I have done my best to make this book as straightforward and practical as possible. To encourage you to take action, I've created a free workbook that you can download for free at the following link:\n\nhttp:\/\/whatispersonaldevelopment.org\/the-thriving-introvert.\n\nI hope you will make good use of it as you redesign your life.\n**Click on the link below to download your FREE action guide:**\n\nhttp:\/\/whatispersonaldevelopment.org\/the-thriving-introvert\n\n# Introduction\n\nTo be honest with you, I longed to be an extrovert for many years. As a youngster, I wished I could go to parties and have tons of energy to talk, interact, and enjoy meeting new people. At times, I even wished I could be the life of the party. I imagined how many more friends I would have and how much more popular I would've been if only I could have been more extroverted. No doubt about it, I envied extroverts, and to a certain extent, might still envy them today.\n\nThere were countless times where I did my best to be as interesting as possible only to be ignored within five minutes because I didn't have the energy to maintain small talk. I would then become invisible to the extent that it seemed pointless to have opened my mouth in the first place. Have you ever experienced anything like that?\n\nI've found myself in conversations where I have nothing to say more often than I'd like to admit. It happens most when I'm talking to members of the opposite sex. It's a little strange. I read a lot and have a pretty good sense of humor, but I can turn into a really boring person at parties.\n\nOften, I would return from parties feeling depressed. I kept thinking I needed to be smarter, more attractive, more interesting, and more charismatic. It felt as though I was never good enough.\n\nAt one point, I even Googled \"how to become more interesting!\"\n\nMaybe you can relate to that? I used to chastise myself for being too shy to socialize, and often blamed it on fatigue. I grew to resent extroverts for their ability to maintain conversations in loud and busy environments. Although I knew trying to become the life of the party was near to impossible, I couldn't admit it to myself. This self-defense mechanism may have been a way for me to avoid addressing my problems and embracing my true personality.\n\nAt first, I thought my issue was lack of confidence, assertiveness, or leadership skills. Although this was true to a certain extent, it wasn't the whole story. At the time, I didn't fully understand what it meant to be an introvert. In a world where extroversion seemed to be the norm, I was trying hard to put an end to my introversion. It turned out to be an impossible task, but it took me quite a while to realize that fact.\n\n**Action step**\n\n_O n a scale of 1 to 10, how true are the following statements? Rate yourself using the workbook available __here_ _._\n\n  1. I hate small talk, but I enjoy deep conversations.\n  2. I get tired if I stay at a party for too long.\n  3. I feel like everything I say should be meaningful and often refrain from talking for this reason.\n  4. I prefer one-on-one, or small group conversations over talking in large groups.\n  5. I need to spend time alone to recharge my battery.\n  6. I think before I speak.\n  7. I have difficulty thinking when in a group. I think best when I'm on my own.\n  8. I usually listen more than I talk.\n  9. I dislike interruptions.\n  10. I hate conflict.\n\n## 1\n\n# Understanding introversion\n\n> _What constitutes an introvert is quite simple. We are a vastly diverse group of people who prefer to look at life from the inside out. We gain energy and power through inner reflection, and get more excited by ideas than by external activities. When we converse, we listen well and expect others to do the same. We think first and talk later. Writing appeals to us because we can express ourselves without intrusion, and we often prefer communicating this way. Even our brains look different than those of extroverts._\n> \n> Laurie Helgoe, author of Introvert Power.\n\nMany extroverts don't understand what it means to be an introvert. It's not easy for introverts to understand extroverts, either. However, the real problem comes when introverts don't understand themselves and wind up thinking something is wrong with them. Although inaccurate, this conclusion actually makes a lot of sense, especially when you look at some of the existing definition regarding introversion. I was shocked by how negative some of them were. Read on for some of the most unsettling ones (but don't feel too depressed, we're going to bust those myths in this book).\n\n  * The Dictionary of Psychology states that introversion is, \"...orientation inward toward the self. The introvert is preoccupied with his own thoughts, avoids social contact, and tends to turn away from reality.\" \n  * Webster's New Collegiate Dictionary describes introversion as, \"... the state or tendency toward being wholly or predominantly concerned with and interested in one's own mental life.\" \n  * In what is perhaps the most negative definition of all, Webster's New World Thesaurus claims that the words 'brooder', 'self-observer', 'egoist', 'narcissist', 'solitary', 'lone wolf', and 'loner' are synonyms for 'introvert'.\n\nJudging by those definitions, being an introvert doesn't sound like much fun! In fact, it sounds like a pretty bad thing to be. This begs the following questions:\n\n  * Why are introverts so different from extroverts?\n  * Is it possible for introverts to become extroverts, and\n  * Is there any reason that they should?\n\n**Action step**\n\nBefore we define what introversion is, take a moment to write your personal definition in the workbook.\n\n( _Section I. Understanding Introversion - 1. Your personal definition of introversion._)\n\n### Introverts vs. Extroverts \u2013 The Genetic Factor\n\nWhether we're introverted or extroverted seems to be largely determined by our biology, which means that we can't change it\u2014it's a part of who we are. That said, introverts are capable of acting like extroverts and vice versa, but only for a limited time. This, of course, requires a person to return to their natural state to recharge their batteries.\n\nWhat distinguishes introverts from extroverts is the way they create and consume emotional energy. Extroverts need a lot of stimulation from the outside world. Without it, they start losing energy and begin to feel bored, lonely, or tired. Introverts need less external stimulation and time spent in social situations will tend to deplete their energy. This in turn forces them to withdraw and spend time alone to rejuvenate themselves.\n\nIt's crucial for introverts to understand that being introverted is NOT the same as being shy. Although there is some correlation between the two, they are two different things. Not all introverts are shy. Conversely, some extroverts are shy, which can be especially frustrating. If you're a shy extrovert, you want to talk but you can't! We'll touch upon shyness at more length later in this book.\n\n### The \"Why\" Behind Responses to Stimulus\n\nThere are actually scientific physiological reasons why extroverts need more stimulation: they're more sensitive to the neurotransmitter dopamine than introverts. In fact, the dominant neurotransmitters of both groups are different: introverts use acetylcholine as their primary neurotransmitter, while extroverts rely on dopamine. Acetylcholine is created through feeling and thinking, and its presence increases focus, enhances memory, and fosters a sense of well-being. It could be said that introverts auto-stimulate by spending time thinking, observing, or contemplating.\n\nExtroverts, on the other hand, need to create more dopamine, which often requires adrenaline. Increasing activity and seeking out stimulation are both efficient ways to ramp up adrenaline. The need for dopamine and adrenaline is a major reason why extroverts tend to take bigger risks than introverts. Risk-taking provides that extra stimulation that extroverts need. They are more reward-oriented and tend to get bored when left alone for too long. As a result, they tend to lose focus and feel worse than introverts when under-stimulated.\n\nIn, _The Introvert Advantage_ , Martin Olsen Laney, Psy.D. wrote the following on how the use of different neurotransmitters can explain differences between introverts and extroverts:\n\n> _I think the link between which neurotransmitters travel what pathways and how they connect with different parts of the autonomic nervous center is the master key to unlocking the temperament puzzle. Whereas extroverts are linked with the dopamine adrenaline, energy-spending sympathetic nervous system, introverts are connected with the acetylcholine, energy conserving parasympathetic nervous system._\n\nFurther experiments indicate that introverts are more easily stimulated than extroverts. This sensitivity explains why, when put in a noisy or crowded environment, introverts end up being overstimulated, while extroverts thrive.\n\nIn a 1967 experiment, Eysenck placed lemon juice on the tongues of people who he identified as introverts and extroverts. He wanted to identify any intra-group differences in rates of salivation. As he expected, during this test, introverts produced more saliva than extroverts, showing a tendency for introverts to be more aroused by sensory stimuli.\n\nIn another well-known experiment, introverts and extroverts were asked to participate in a challenging word game. The game involved learning through trial and error. Participants were required to wear headphones that emitted noise and were given permission to adjust the volume as they saw fit. On average, extroverts adjusted the volume to 72 decibels, while introverts choose a lower volume of 55 decibels. Interestingly, the study revealed that, under these conditions, both extroverts and introverts were equally stimulated and played equivalently well.\n\nMore surprisingly, when extroverts played the game at the nose level selected by introverts, and vice-versa, both extroverts and introverts underperformed. Each group needed significantly more attempts before completing the game. In short, extroverts were under-stimulated, and introverts over-stimulated, which ultimately led to subpar performances.\n\nThe bottom line is that introverts, are more easily stimulated than extroverts. We'll perform at our best under moderate sensory stimuli. On the other hand, extroverts will need to be exposed to more stimulation to be able to thrive.\n\nIt is important to note that, although most people think introversion and extroversion are the only two options, they actually exist on opposite sides of a continuum. This means it's possible to be an ambivert, or rather, someone who has aspects of both introversion and extroversion. So where would you put yourself on the continuum?\n\n**The story of Professor Little**\n\nThe following story of professor Little is a good illustration of what introversion is and why we must learn to manage our energy effectively.\n\nBrian Little was a former Harvard University psychology lecturer and recipient of the 3M Teaching Fellowship, one of the most prestigious awards a teacher can receive. His students loved him and his lectures often ended with standing ovations. He would frequently sing or make jokes during the classes. He sounds like an extrovert, doesn't he? Well, he wasn't! That couldn't be further from the truth. He was deeply introverted and would spend his free time reading and writing or scoring music in his house, which was located in the Canadian woods.\n\nOne day, Professor Little visited the Royal Military College Saint-Jean on the Richelieu River to give a speech to a group of military officers. His speech was so well-received that they invited him each year.\n\nThere was one major problem, though. Once his speech ended, they invited him to join them for lunch. It was a nightmare for Professor Little. He had another lecture to deliver in the afternoon and needed some \"me\" time to recharge his battery. Fortunately, he came up with a great idea. He invented a passion for ship design and asked if he could go for a walk along the river to admire boats instead. You can imagine his relief when this enabled him to successfully decline the lunch offer.\n\nHe continued to do this each year and take walks along the river to recharge after his speeches. One day, however, his speech was moved to a different location. He panicked. How would he find a quiet place to rest? He used the last card in his hand, which was the restroom. He barricaded himself in a bathroom stall to avoid interactions with others.\n\nDoes this sound familiar to you?\n\nProfessor Little's story illustrates that, as introverts, we can act as extroverts for a while, but will eventually need to recharge our batteries. So, if you need to deliver a speech, do a Facebook Live, or go to networking events, you can do it. Just make sure you schedule time to recharge your battery.\n\n### The Key Characteristics of Introverts\n\nNow, let's have a look at several primary characteristics shared by the vast majority of introverts, who tend to:\n\n  * Spend time alone to replenish themselves\n  * Dislike small talk but enjoy deep conversation\n  * Prefer small groups\n  * Think before they speak\n  * Require an invitation before they speak\n  * Avoid speaking until they feel they have something important to say\n  * Listen more than they talk\n  * Talk a lot when the topic is something they're passionate about\n  * Choose depth over breadth (they would rather know a few people very well than know several people casually) \n  * Keep their enthusiasm and excitement for themselves and share only with people they know very well\n  * Prefer to know a lot about a small cluster of topics than know a little bit about a wide range of topics\n  * Need alone time to think (rather than having to endure brainstorming sessions) \n  * Dislike interruptions\n  * Be uncomfortable with conflict, and\n  * Need a lot of preparation before addressing an audience and have difficulties speaking for long periods of time.\n\n### Debunking the Myths: What Introverts Are Not\n\nThere are many myths surrounding introversion. In this section, I'd like to spend some time debunking the most common ones.\n\n### Myth 1: Introverts need to be fixed\n\nFact: Introversion is not a mental disorder. In 2010, there was a proposal to include introversion in the _Diagnostic and Statistical Manual of Mental Disorders,_ which is considered the Bible of mental health diagnoses. As with anything, introversion can coexist with mental health issues, but the proposal was rejected. Unless you're so introverted that it keeps you from functioning on a daily basis, you have nothing to worry about.\n\n### Myth 2: Introverts lack social skills\n\nFact: Social skills can be learned, and, unless markedly shy, introverts can communicate with others just as well as extroverts. They tend to be good listeners, which often gives them an edge over extroverts.\n\n### Myth 3: Introverts don't care about other people\n\nFact: Introverts are as interested in people as anyone else, and are often more interested in people than most. This is part of why they tend to shy away from small talk. Introverts would much rather get to know people on a deeper level. As Laurie Helgoe, author of _Introvert Power_ explained, _\"_ Let's clear one thing up: Introverts do not hate small talk because we dislike people. We hate small talk because we hate the barrier it creates between people.\" Well-said!\n\n### Myth 4: Introverts don't like to talk\n\nFact: This may seem true in big groups, but when it comes to small gatherings and one-on-one conversations, introverts can (and will!) talk a lot. This is especially true if they know a lot about the topic at hand or feel strongly about it.\n\nI may not talk much in group settings, but if I'm with one or two friends it might be hard to shut me up! I've definitely had to remind myself to talk less at certain points, and there have been a few times where I've had a friend inform me that I'm talking a lot. It's always a surprise to me and my first thought is, \"Seriously? I had no idea!\"\n\nHowever, as is the case with most introverts, this all depends upon how well I know the people I'm talking to. Once the group starts to exceed five or six people, I begin to talk considerably less. One-on-one conversations are my favorite form of communication because they allow me to learn more about a person. What are they passionate about? What are their values? What are their concerns? What are their dreams? And so on and so forth. While I can't speak for every introvert on the planet, I can say that the majority I've met get a kick out of conversations like that.\n\n**Introverts internalize, extroverts externalize**\n\n> _Extroverts often incorrectly assume that introverts are suffering. Introverts internalize problems; we like to take things inside and work on them there. Extroverts prefer to externalize and deal with problems interactively._\n> \n> Laurie Helgoe, author of Introvert Power.\n\nIntroverts like to internalize things. We like to think problems through in their mind rather than discuss them with other people. We prefer to look within to solve our own problems rather than share our problems with other people to find a solution. We are at our best when we can think alone in silence. We don't do nearly as well when we're expected to brainstorm with a group and come up with great ideas on the spot.\n\nSimilarly, we like to think before we talk, and have generally a lot of difficulties speaking without notice. On the other hand, extroverts speak first and then, refine their thinking as they talk. It is not to say that one way is better than the other, but just that introverts and extroverts express themselves differently.\n\nWhile in some cases overthinking can be an issue for introverts, we generally are fine the way we are and don't need to be fixed.\n\n### Introversion doesn't equal shyness\n\nMany people have difficulties differentiating introversion from shyness. However, these states are not the same, and it is important that, as introverts, we differentiate the two.\n\nBeing an introvert, it refers to the way we create and consume emotional energy. It doesn't mean that we lack self-confidence or are afraid to talk to people. If that's the case, it's a separate issue, because people of all personality types can, at times, struggle with confidence and fear.\n\nMany introverts will actually talk a lot during deep conversations, but find small talk draining. The feeling is we should talk only when we have something valuable to say, not just because it's required of us. Forcing themselves to speak when they don't want to requires a major effort on their part and will quickly deplete their energy. Other forms of stimulus that extroverts often enjoy, such as noisy places, bright lights, crowds, or unfamiliar situations will also drain their energy.\n\nAs an introvert myself, it took me a while to differentiate between my introversion and shyness. As I gained more confidence over the past few years I discovered that I'm not a particularly shy person. I don't necessarily feel shy in large groups of people or when engaging in chit chat, but I still avoid talking when I don't have the energy for it. Certain situations require more energy than I have and it is that, rather than shyness, which keeps me from being the chatterbox I have sometimes wished I was. Over time, I've become progressively more aware of the vast differences between introversion and shyness.\n\nWhen I go to a party, I usually don't know who I should talk to. There's not enough time to get to know many people and I'm not fond of casual conversations. I usually end up having a one-on-one conversation with a few different people (this is the best option for me, provided I'm talking to interesting people!) or wandering awkwardly from group to group, exhausted and unsure of what to do. That, of course, is the worst-case scenario. Introverts don't get energized through socializing in and of itself, but we can get pumped up and even intense when conversations turn to topics we're passionate about.\n\nI can't stress enough how important it is to understand the difference between introversion and shyness. When people learn this difference, it gives them an opportunity to reassess whether they're actually shy and enables them to break free of guilt. Working on self-confidence is hard enough on its own, so don't make it even more difficult by confusing introversion and shyness. Doing so can lead to wasted time and energy trying to change something that can't be changed, which will ultimately end in discouragement.\n\nIt's not abnormal to feel tired after spending lots of time with people. You're not tired because you're shy, need to get out more, or need to get used to talking to people. You're just introverted, and there's nothing wrong with that.\n\nIt's okay if you can't talk at a meeting without preparing beforehand. It's okay if people around you seem like geniuses because they can deliver eloquent speeches at the drop of a hat. They may not be any smarter than you. There's nothing to worry about if you can't think clearly while in a large group. I've noticed that it's very hard for me to come up with really good ideas or think deeply when I'm part of a group. I do my best thinking in solitude. You don't need to feel bad if you have a hard time with chitchat or if you don't feel like talking. And if people find you boring or dull because you don't talk much, that's okay, too.\n\nI feel that introverts tend to seem shy because interacting with people is taxing for them. As a result, they have less experience talking to people than extroverts do, and thus have a steeper learning curve than extroverts when it comes to building social skills. On top of that, extroversion is the norm in many countries, even though introverts account for fifty percent of the world's population. Unfortunately, schools and offices aren't designed for introverts. Many introverts spend their lives being told that they should talk and socialize more, which often leads to guilt and feelings of low self-esteem that extroverts are less likely to experience.\n\nThat said, be careful not to hide behind your introversion. Be sure to break loose from your comfort zone from time to time.\n\nI understand that differentiating introversion from shyness can be tricky. Here's something I want you to try for the next two weeks: whenever you interact with people, take time to notice how you feel. For instance:\n\n  * Do you feel shy and awkward, or are you simply not in a talking mood? \n  * Are you feeling drained or timid? \n  * Do you feel more comfortable with one-on-one conversations, or do you feel awkward, timid, and shy regardless of how many people are involved in the discussion?\n\nTry to identify the feelings that come up during each of your conversations. If you continually examine your feelings and behaviors, it'll be much easier to tell the difference between shyness and introversion.\n\n**Action step**\n\nUsing your workbook, complete the following exercise ( _Section I. Understanding introversion_):\n\n  * Write down your relationship with introversion (ex: I feel ashamed, I feel guilty, I feel frustrated etc.).\n  * Differentiate introversion from shyness by asking yourself: Why aren't I talking? Then write whether your reasons are based on shyness (feeling scared and uncomfortable) or introversion (feeling too low on energy). For shyness write \"S\", for introversion write \"I\".\n  * Write down your expectations regarding this book. Why did you buy this book and what do you want to get out of it? (e.g., improving my relationship with my partner, becoming better at networking, acceptation my introversion etc.)\n\n## 2\n\n# Owning Your Introversion\n\nUnfortunately for introverts, we live in a world that is primarily designed for extroverts. Extroversion is the dominant model of normalcy and many introverts find it difficult to fit in. Many of us try hard to become extroverts, only to end up frustrated and exhausted. We're told we should talk more, be more social, and take more initiative. Sometimes, we are even seen as snobby, aloof, or boring!\n\nHowever, introversion shouldn't be seen as a burden. It's not something that has to be fixed, nor is it anything to be ashamed of. Quite the contrary, introversion should actually be embraced. As introverts, we have a lot to contribute to the world. As noted in the previous chapter, it's estimated that roughly fifty percent of the people in this world are introverted to varying degree. Who would have guessed? Extroversion is given so much attention and praise you'd think extroverts account for the majority of the population. But that's simply not true.\n\nIntroverts aren't powerless creatures who need to buck up and act like extroverts to meet societal standards. If, after millions of years of evolution, introverts still account for fifty percent of the population, it is clear we have our role to play in society. That's pretty important, and we can't be ignored! Millions of years of evolution can't be wrong.\n\nIn this section, I'll encourage you to embrace your introversion and throw off any guilt, shame or frustration you feel regarding it. As an introvert, you have a role to play, and that role doesn't involve pretending to be an extrovert. That would be pointless!\n\n**Action step**\n\nBefore we get started, use the workbook to answer the following questions ( _Section II. Owning your introversion - 1. Your biggest challenge_):\n\n  * What is my biggest challenge as an introvert? \n  * What could I do to overcome it?\n  * What is the one thing that would have the biggest positive impact on my life?\n\n### Why your assumptions on introversion matter more than you think\n\n> _Begin challenging your own assumptions. Your assumptions are your windows on the world. Scrub them off every once in while, or the light won't come in._\n> \n> Alan Alda, actor, screenwriter and author.\n\nWe all have assumptions about life. We expect things to be a certain way, and we have specific beliefs on how we think we should behave. As an introvert, you hold many assumptions that you may be unaware of. These assumptions can prevent you from embracing your introversion. A telltale sign of an assumption is the use of the phrase \"I should\". When you use that phrase, you're likely stating an assumption. So why not take a few minutes to think of all the assumptions related to your introversion?\n\nThe expectations you have regarding your behavior can create unnecessary suffering in your life. When you believe you should behave like an extrovert, you'll feel bad about yourself when you don't or can't sustain the behavior. You may be able to act like an extrovert on an occasional basis, but it isn't natural to behave that way if, deep down, you are an introvert. It can only be done on a temporary basis, so you'll be miserable if you think you should act like an extrovert continually.\n\nThe first step towards accepting your introversion is looking at all the assumptions you hold regarding it. As you do this, you'll realize that many of your assumptions are actually wrong. Below are some examples of assumptions that you may hold.\n\n  * I'm boring if I'm not talking. \n  * Silence is awkward, so I should talk to avoid it. \n  * I don't enjoy parties, so something must be wrong with me. \n  * I shouldn't stay home on a Friday night. \n  * If I don't like talking to people because I'm shy. \n  * If I open my mouth, I'd better say something interesting.\n\nYou may think you have few assumptions, but you actually have thousands in each area of our life.\n\nFor example, let's focus on some specific assumptions you may have when you're at a party:\n\n  * I should continue listening to this person who's talking non-stop for ages because breaking off the conversation would be rude. \n  * Taking a break to go for a walk during a party is weird. \n  * I should talk more. \n  * It's normal to feel guilty for talking too little. \n  * I should enjoy the party. \n  * I should have more energy. \n  * I should stay until the end of the party. \n  * I should say something interesting. \n  * *Everyone else having a good time, so what's my problem?\n\n* _Remember that introverts account for fifty percent of the population, so you're just one of many people who doesn't enjoy parties!_\n\n**Action step**\n\nUse the workbook to record expectations that directly influence the way you experience your introversion. ( _Section I. Owning your introversion - 2. Challenging your assumptions_)\n\n### Accepting Yourself as You Are\n\n> _Don't think of introversion as something that needs to be cured... Spend your free time the way you like, not the way you think you're supposed to._\n> \n> Susan Cain.\n\nDo you apologize for being tall or short? What about for being a man or a woman? If you don't apologize for any of these things (and you shouldn't, you really shouldn't), then why should you apologize for being an introvert? Introversion isn't something you can turn on and off at will. It isn't a choice that you make. You need some alone time to recharge your energy, and that's not going to change. It's just how your brain works. Period.\n\nSure, you may be able to act like an extrovert for a few hours, but it doesn't change the fact that you need some 'alone' time to recharge your batteries. Neglecting to do that means disrespecting yourself and going against your own nature.\n\n### You aren't lacking anything\n\n> _...the most important thing to know about being an introvert is that there's nothing wrong with you. You're not broken because you're quiet. It's okay to stay home on a Friday night instead of going to a party. Being an introvert is a perfectly normal 'thing' to be._\n> \n> Jenn Granneman, author of The Secrets Lives of Introverts.\n\nPeople often see introverts as lacking something. Some extroverts say that we lack, well, extroversion. They tell us we should talk more, go out more, and enjoy ourselves more. But people who do this are simply projecting their extroverted nature onto us.\n\nYou don't need to pretend to be extroverted to compensate for your introversion, because you aren't lacking anything. You don't need to be cured. You're not crazy, snobbish, or awkward. You don't need to become an extrovert. You just need to understand and respect yourself for who you are.\n\n### You don't owe anyone social interactions\n\n> _I don't hate people, I just feel better when they aren't around._\n> \n> Charles Bukowski, poet.\n\nWe often believe that we need to interact with other people. We \"should\" answer the phone when someone calls, we \"should\" say something interesting to avoid lulls in conversation, or we \"should\" keep a conversation going.\n\nWell, that's not true. Nobody has the right to force social interactions upon you. Nobody is allowed to use your precious time without your permission. You can go to a party without talking to anyone if you so choose. You can leave a conversation at any time by saying something like, \"It was nice talking to you, but I've got to get going. I'll talk to you later\".\n\nNobody is entitled to your time. Nobody can waste your limited energy unless you allow them to. Think about it this way: when an extroverted person talks to you, _they_ feel good because _they_ need to interact with people to recharge themselves. In a sense, you serve a purpose for them. Yet that still doesn't mean you should keep the conversation going just to be polite. It's sucking the energy out of you, and anybody who expects you to endure that endlessly is pretty selfish. Would you force an extrovert to stay home and read books because that's what energizes you? Of course not.\n\nRemember, you don't owe anybody social interaction! Since interacting with people depletes your energy, it's important to be selective about who you talk to and for how long.\n\n### You don't have to talk\n\n> _A wise man once said nothing._\n> \n> Proverb.\n\nWhat if I told you that you don't have to talk? Talking more doesn't make you a better human being. I used to struggle with this idea. I always felt as though I needed to say something. I assumed that I needed to be talking. As such, I felt something was wrong with me if I stayed quiet.\n\nA few months ago, I was at a party with my work colleagues, when suddenly, something horrible happened: A lull in the general conversation. Oh no! Please, someone say something, anything! I was having a separate one-on-one conversation with someone at the time, and, when it became obvious that I wasn't joining the group conversation, someone asked me to say something. I replied, \"I don't have anything to say and, to be honest, I like the silence.\" As you can see, introversion is no longer something I'm ashamed of. I was able to say confidently and without guilt, that I didn't have anything to say. I wouldn't have been able to do that a few years ago. Surprisingly, many people are afraid of silence. Maybe, they're afraid that they aren't interesting enough. Or maybe, they prefer to focus on the external world to avoid looking inside. In many ways, silence can be very powerful, and also very enjoyable.\n\n### You don't need to apologize for the way you feel\n\nRemember that people can argue with what you say, but they can't argue with how you feel. How you feel is how you feel. That's it. If you feel tired and want to go home or take a break, do that. Regardless of what people say, you're responsible for your energy and how you manage it.\n\n### You don't need to feel ashamed or guilty\n\n> _What a commentary on our civilization, when being alone is considered suspect; when one has to apologize for it, make excuses, hide the fact that one practices it like a secret vice!_\n> \n> Anne Morrow Lindbergh, author.\n\nIt's easy to feel ashamed or guilty because you don't talk as much as you'd like or don't enjoy parties in the way you \"should\". You may blame yourself for not having enough energy or believing you should have more of it. You may even resent yourself for being unable to engage in small talk or be the center of attention. I once felt the same way.\n\nThe thing is, there's nothing wrong with the way you behave, nor is there anything wrong with the way extroverts behave. Your feelings and behaviors aren't strange or odd. Don't \"should\" yourself. Let go of your guilt and do what you truly want to do.\n\n### You don't need to be jealous of extroverts\n\nThis is a difficult one because I suspect many introverts secretly wish they were extroverts. Maybe you feel that way, too. You may envy extroverts for their ability to interact so easily with other people. You start imagining how much more popular you'd be if you were an extrovert. You may event resent them to some extent. I certainly did. In this instance, you might try hard to improve yourself to become more interesting (be it through studying, going to the gym, or honing a new skill), only to end up unnoticed and with nothing to say.\n\nA few weeks ago, I came across one comment I found hard to forget. The comment stated that introverts couldn't be happy, as they all envied extroverts. Needless to say, I thought it was BS.\n\nWe may believe that extroverts are happy all the time, but that's not the case. Extroverts rely more on external elements (such as people) to recharge themselves. In some cases, this leads to relying on other people and other people's approval for their own happiness. They may find it difficult to spend time by themselves and may get lonely easily. Introverts, on the other hand, are often happy by themselves.\n\nIt's also important to remember that being an introvert doesn't mean you can't interact with others or be a great conversationalist. Those things are totally possible, but you'd have to be a little more strategic in how you manage your time and recharge yourself.\n\nIt's worth mentioning here, that one of the main reasons introverts may be unhappy is a general lack of understanding. We try too hard to be someone we are not. We look at ourselves through a lens that is biased towards extraversion, and then judge ourselves based on that flawed paradigm. In other words, our basic assumptions are wrong. As I said before, this creates wrong expectations and leads to unnecessary suffering.\n\n**Action step**\n\nAnswer the following questions using the workbook ( _Section II. Owning your introversion - 3. Accepting your introversion_)\n\n  * How well do you own your introversion? Write down one negative feeling you associate with it (e.g.: feeling guilty for not talking enough, feeling tired, etc.)\n  * Name thing you could do or one belief you could adopt that would help you let go of that negative feeling?\n  * What positive feelings and emotions do you associate with your introversion? Try to write down as many as possible. (E.G. enjoying spending time at home, taking pleasure in reading a new book etc.)\n\n## 3\n\n# Optimizing your introversion\n\nAre you generally energized throughout the day, or do you feel drained most of the time?\n\nIn this section, we'll discuss how you can redesign your day to suit your introverted personality. The first thing I'd like you to do is envision your ideal day. What would it look like? What would you be doing? How would you spend your time?\n\nAs it stands, you may have to spend time interacting with lots of people at work despite preferring small groups. Or, you might have a partner who frequently has friends over, making it difficult to relax alone.\n\nFor the moment, forget about your life as it is now. Just envision what your ideal day would look like.\n\n**Action step**\n\nUse the workbook to write down what your ideal day would consist of. ( _Section III. Optimizing your introversion - 1. Designing your ideal day_)\n\n  * Imagine you're living the perfect life as an introvert. What does your typical day look like? What are you doing? Who are you spending your time with? How much time do you spend alone?\n  * What's preventing you from living like this? What's holding you back?\n  * What's the one thing you could do today to move you closer to the vision you have for yourself?\n  * Add one more thing you could do\n\n**Managing your energy - the key to a happy life**\n\n> _I'm very picky with whom I give my energy to. I prefer to reserve my time, intensity and spirit exclusively to those who reflect sincerity._\n> \n> Dau Voire, author.\n\nHave you ever stayed at a party for too long only to leave feeling totally exhausted? As introverts, managing our energy is by far one of our biggest challenges. Without adequate energy, we become unable to function properly and ultimately zone out. We become grumpy, stressed out, and unhappy. Some of us even start feeling ashamed or guilty about our introverted nature.\n\nAs you can see, learning to use your energy wisely is crucial. You shouldn't allow people or situations deplete your energy. It's your responsibility to control these things. Your friends might say that you should come to a party because \"it's going to be fun\" or pressure you to stay longer than you'd like. This is probably well-intentioned, but the fact remains that you are the only one who knows how you feel and what you want to do. I don't like it when someone tells me what _I_ should do based on what _they_ want to do. I have a feeling that none of us enjoy that very much.\n\nTelling an introvert to stay at a party when they want to go home and recharge is like telling an extrovert to go home and read when they want to socialize and increase their energy. \"Come on man, you're so boring! Go home and read a book!\". It sounds pretty ridiculous when you put the shoe on the other foot, doesn't it?\n\n### Finding your sweet spot\n\n> _People empty me. I have to get away to refill._\n> \n> C. Bukowski, author.\n\nAs an introvert, you have limited energy for socializing. The more you know yourself and understand your limits, the more you'll be able to avoid feeling drained at the end of the day. It's important to find your sweet spot, whether you're alone or with people.\n\n### When you're by yourself\n\nIntroverts don't mind staying home for a good while. Depending on your level of introversion, staying in for a day, weekend, or even longer might not bother you.\n\nHowever, spending too much time alone can make it harder to socialize on even the most basic levels.\n\nHere's a question to ask yourself: How long can I stay home before going out and handling basic interactions becomes an ordeal? Ideally, you want to avoid staying home past this point. To that end, it's good to get out on a regular basis, even if not for the expressed purpose of socializing.\n\nStart observing how you feel when you stay home for too long, and get into the habit of leaving the house for something besides work. Sometimes, being on your own for too long can become overwhelming. Yes, even introverts need to socialize. Otherwise, we risk dwelling on negative thoughts, feeling lonely, or getting hung up on something that's bothering us.\n\n### When you're with others\n\nHow much time can you spend at a social event before running out of energy? For me, it's a couple of hours. After that point, I start feeling equal parts bored and boring. I stop enjoying myself. That's a sign that it's time for me to get out of there! For you, it may be two hours as well, but it could easily be more or less. Of course, it will also depend on the situation and how much energy you start with. Below are some factors that may impact how fast your energy is depleted:\n\n  * The loudness of the venue\n  * How crowded it is\n  * How interactive you're expected to be\n  * How formal the event is\n  * How many people you know\n  * How familiar you are with the venue\n  * Whether you can partake in activities besides talking\n\nIt's important that you learn to monitor your energy so that you can better plan your day. Once you know your sweet spot, things will become easier. Later, we'll discuss how to deal with parties as an introvert.\n\n### The importance of maintaining your space\n\nAs introverts, simply being around people can drain our energy, even when we aren't required to talk. You've probably been in situations where you're alone in a room enjoying yourself\u2014until somebody shows up. This could happen at the gym, in the library, or at a caf\u00e9, to name a few examples. In any of these situations, what happened next? How did it make you feel? If you're anything like me, you probably felt mildly annoyed, anxious, or drained, I can't speak for extroverts, but my guess would be that this wouldn't be a big deal for them. They might actually get excited and strike up a conversation with the newcomer.\n\nI hate having people around when I'm trying to enjoy something alone. When doing so, I seldom invite anyone to come with me, and if I do, they have to be an extremely close friend. I try to avoid being with others when I want to enjoy a space on my own. In that situation, friends of friends or acquaintances would be the worst people to be around, as I'd have no choice but to have some manner of conversation with them.\n\nHave you ever made plans to get together with a friend only to have them ask if someone else can come along? How did it feel? Chances are, you were looking forward to having a deep conversation with your friend and felt it was ruined by adding a plus-one. I can totally relate to that.\n\nAlso, when you are running low on energy, note that physical contact with people, even our partner or members of our family, may be overwhelming. In some cases, you may not want anybody near you.\n\n**Action step**\n\nUse the workbook to optimize your level of energy. ( _Section III. Optimizing your introversion - 2. Managing your energy\/3. Identifying your sweet spot_)\n\n2. Managing your energy:\n\n  * What could you do to better manage your energy?\n  * What is the one thing that would have the biggest positive impact on your energy level?\n\n3. Identifying your sweet spot:\n\nBased on your own experience, how much time can you spend at a social event before feeling the need to leave or take a break to recharge your battery?\n\n### Optimizing Your Life\n\nAs an introvert, you're likely to face problems in various areas of your life. In this section, I'd like to provide you with some tips to optimize your life. I'll be focusing on 3 different areas: your social life, career, and relationships.\n\n### Your social life\n\n> _Introverts crave meaning so party chitchat feels like sandpaper to our psyche._\n> \n> Diane Cameron.\n\nAs introverts, we often have a love-hate relationship with social events. Part of us wants to go, but we can't stop thinking, \"Am I going to enjoy the party?\" In the end, we often wind up staying home. There's nothing wrong with staying home, of course, and parties aren't a necessity of life. That said, we all need or want to go out to make friends or network. I'm going to discuss how we can redesign our social lives to make them more \"introvert-friendly\".\n\n### Dealing with parties\n\nParties can be fun, but they can also be a nightmare for introverts if we don't handle them properly. Staying for too long can leave us completely exhausted and wishing we hadn't come. But it doesn't have to be that way. Coping with parties the introvert way will make your life easier and more enjoyable.\n\n### Deciding whether you should go\n\nFirst of all, you need to decide if you actually _want_ to go to the event or party. In her book _The Introvert Power_ , Lauri A. Helgoe gives the following tips on deciding whether you should go.\n\n  * **How big is the party?** Usually, the bigger the party is, the more tiring it will be. \n  * **What is the setting?** Is it an indoor or outdoor event? Is there a lot of space or will it be crowded? How loud will it be? A beach party, for instance, will make it easier for you to take a break, go for walk, or leave when you're ready. On the other hand, an indoor party in a confined environment could quickly become overwhelming. \n  * **Do I know the people who will be attending?** If you know most of the people there, you'll feel more at ease. It goes without saying that not knowing anyone will likely be uncomfortable. \n  * **Are there any activities I can partake in that don't involve talking?** You'll typically feel more comfortable when you can do something besides talk. It makes interactions with other people much smoother. If all you can do is stand and talk, however (think cocktail parties), it'll be much more challenging.\n  * **Do I really want to go?** Based on your previous answers, ask yourself this: \"On a scale of 1 to 10, how excited am I about this party?\" __ In other words, will I regret not having gone? Some introverts fear that they may be missing out. So, keep that in mind when you make your decision.\n\n### Learning to say \"no\"\n\nThere's an extremely simple and highly effective way to handle parties you're unsure about (or don't want to go to): Just don't go! That will solve many of your problems. While it sounds easy enough, many people struggle to decline invitations and attending parties when they don't want to.\n\nThe first thing to understand is that parties aren't a must. Nobody has the right to force you to attend. It often seems as though parties are the only way to have fun. I don't know about you, but, for me, having a cup of coffee with one or two close friends can be more satisfying than going to a party. The same goes for reading a good book. Parties are overrated, and I suspect that many people (mostly introverts) don't enjoy them at all. They just pretend to. I don't know whether you enjoy parties or not, but if you don't, I would advise you to stop pretending to. Most of us spend too much time pretending or trying to fit in. We pretend to be happy, we pretend to like our jobs, we pretend our relationship is great, and so on.\n\nWhatever you choose to do with your time is fine as long as it's authentic. There's no need to fake anything.\n\n### How to say \"no\"\n\nSo, you've decided to decline your friend's invitation to drink beers in a loud Irish pub on Friday night. Now, you're not sure how to tell them. Your friend thinks you're crazy and is already telling you how much fun it's going to be. \"You're going to love it,\" he says. I wonder if he would accept your invitation to come over and read books in silence. He would just love that, right?\n\nThink of saying no as a sign of self-respect. You should also keep in mind that, if you want extroverts to stop treating you like an extrovert, you have to embrace your introversion. If you constantly apologize for skipping things you didn't want to attend to begin with, how can you expect extroverts to change their behavior towards you? They won't be able to understand your need for solitude if you send mixed signals. Here are four tips to help you say no:\n\n  1. **Stop apologizing** : You don't need to apologize for wanting to spend your time in a way that will maximize your happiness! You don't try to keep extroverts from expressing their extroversion. You don't try to stop them from partying all night if that's what they want to. It's their life, right? So, don't let other people do that to you. There's no need to say, \"I'd like to go, but....\" If you keep saying that, chances are people will continue to invite you, which might not be what you want. \n  2. **Own your introversion:** Instead of saying that you \"need\" to rest, state clearly it's what you want. \"I'm looking forward to reading a great book tonight,\" or \"I want some time to myself,\" or \"I'm excited about staying in and relaxing.\" This will better convey your agenda than, \"I need to rest.\" \n  3. **Be honest:** Say things as they are and embrace your introversion. The more your respect yourself and accept your introversion, the more your friends will accept it, too. Remember, many people don't understand introversion. Your real friends will be more supportive once they understand what your needs are. They'll not only support your choice to stay home, but will also help you preserve your energy during parties, which we'll cover later on. You can say something like, \"You know, I'm not big on parties. I had an exhausting week and I'm looking forward to staying home and reading a great book.\" \n  4. **Offer alternatives (optional):** Sometimes you want to catch up with your friends, but not in that crowded Irish pub on Friday night. If that's the case, you can simply offer another option. You could say, \"I've had an exhausting week and I'm going to recharge my batteries by reading a great book tonight. But I really want to catch up with you, so let's have a coffee this Sunday.\" That's a good way to show that you _do_ want to see your friends, but not at a loud, overwhelming venue. You could even mention that you'll be able to have deeper conversations. This will have the advantage of helping them realize the value of your friendship.\n\n* * *\n\n**A dditional tip:** It's hard to assert yourself in a society where extroversion appears to be the norm. It's even harder when many people view introversion as something that requires \"fixing\". Unsurprisingly, asserting yourself takes some training. Don't hesitate to practice with a friend or on your own. For instance, you could create some phrases that allow you to politely decline events you aren't interesting in. Choose responses that resonate with you and rehearse them until you feel confident enough to use them in a real conversation. Check out the following examples:\n\n  * Thanks, but I really prefer small gatherings. \n  * Thanks, but I'm focusing on *insert project or goal here* and can't deal with distractions at the moment. \n  * Thanks, but I don't really go to parties, I prefer intimate gatherings with just a few friends.\n\nTry challenging yourself to be honest while remaining polite and see what happens.\n\n_You might also want to refer to the list of affirmations at the end of this book. Feel free to use the ones that resonate with you on daily basis or create your own._\n\n### Dealing with parties the introvert way\n\nLet's say you accept the party invitation, hopefully because you truly want to go, but start second-guessing yourself. \"Am I really going to enjoy the party? Will I regret not having stayed home?\" When you go to a party, there's always some level of uncertainty. Sometimes, I've gone to a party without expectations and had a great time. Other times, I've been super excited only to end up bored and disappointed. Fortunately, as introverts, there are many things we can do to maximize our chances of having a good time. Consider the following tactics:\n\n**Arriving early:** Assuming it's okay with the host, arriving early to a party allows you to familiarize yourself with the environment and start interacting with a smaller pool of people. It may be easier to have one-on-one conversations or talk with a small group before everyone arrives. The venue will be quieter and less crowded.\n\n**Leaving early:** As introverts, we can only spend so much time in an overly stimulating environment. There's absolutely nothing wrong with leaving early. Lately, I tend to arrive late and leave early, rarely staying more than a couple of hours. The reason for that? After a couple of hours, I don't enjoy myself anymore, so I just leave. And you should, too! There's no reason to stay after you stop having fun. When people ask me why I'm so late I reply that I had some work to do (like writing this book). When it comes to leaving, make sure you've got your escape plan ready before you go.\n\n**Bringing an extroverted friend:** Remember, extroverts are not the enemy. Going to a party with an extroverted friend is a great way to help you break the ice and interact with more people than you would on your own. There are many people that I would never have met if it weren't for my extroverted friends. Your friend will do most of the talking, allowing you to preserve your precious energy.\n\nAnother perk is that they'll likely say good things about you, which is more credible than if you were to say them yourself. That may come in handy, especially if you're looking for a partner. Most of us don't like talking ourselves up, so it's nice to have a \"PR\" person in the form of your friend. I'm very grateful for my extroverted friends. I hope you're grateful for yours, too!\n\n**Briefing your extroverted friend(s) on your introversion:** The more your friends understand your introversion, the better they'll be able to support you. **** You want to take a break from the party and go for a walk? No problem, your friends will understand. You want to leave early? That's okay. Although they may want you to stay, your friends will understand.\n\n**Taking regular breaks:** Don't hesitate to take frequent breaks. This might mean going for a walk or going to the bathroom. You could even ask the person you're talking to if he or she wants to go for a walk with you.\n\nFor some reason, we tend to hold beliefs that prevent us from acting freely during social events. Yet in reality, you can come and go as you please, you can leave a conversation at any time, you can go for a thirty-minute walk, or you can spend fifteen minutes in the bathroom (but make sure you're not creating a queue!). You can do whatever you need or want to do. Start looking at your own assumptions. You'll probably find that you're imposing artificial limits on your behavior due to social conditioning. Consider breaking them!\n\n**Spotting other introverts in the room:** You may want to \"target\" other introverts, as they might be easier to approach. You'll probably find them sitting by themselves or standing in the corner of the room. Try striking up a conversation with them and see where it goes. They might be easier to talk to. Although, being introverts, don't be surprised or disappointed if they say, \"I'm happy on my own for the moment.\" (Especially if they've read this book.)\n\n**Hanging out in a strategic spot:** You might choose to stand in an area where it'll be easier for you to have one-on-one conversations or small group conversations. You could stand in a corner, near the kitchen, or in a corridor, for instance. Be on the lookout for areas that are quieter and less crowded than others.\n\n**Volunteering:** This is a great way for introverts to enjoy social interactions without having to come up with something to talk about. You'll avoid those awkward moments where you don't know who to talk to or what to say. You won't have to nurse a drink all night or check your phone constantly just to have something to do (I often do this!). You could, for instance, be helping in the kitchen or taking pictures. It's generally easier for us to create connections when we're involved in an activity, such as playing sports or cooking. Because we don't actually have to approach people and come up with something to say, the conversation flows more naturally.\n\n**Focusing on others to forget ourselves:** Switching your focus from yourself to others can be an effective way to reduce anxiety. You could, for example, decide to attend a party with the objective of learning something new. You could even think of a few questions you'd like to ask someone beforehand. Using this technique, you'll be more focused on what you want to learn than you will be on your anxiety.\n\n**Setting specific goals:** You can try setting tiny goals. Your goal could be to approach one group of people. Once you accomplish that tiny goal, you can go home happy. Everything that happens afterwards is just icing on the cake. Oftentimes, you may end up approaching more people than you expect. The tiny goal approach can work for all areas of your life. One of my tiny goals is to write at least 500 words per day, but today I've already completed more than 2,000. The point is, whatever you do can snowball to good effect.\n\n**Escaping from a conversation:** I used to stay in conversations for hours because I erroneously believed it was rude to leave them. But here's the truth: You don't owe anybody any interaction, and nobody can drain your energy or waste your time without your consent. Now, I simply say, \"It was nice speaking with you. I'll talk to you later.\" Or, \"My friend is waiting for me, it was nice speaking with you. I'll talk to you later.\" Try it for yourself the next time you feel stuck in a conversation you feel has gone on for too long.\n\n**Action step**\n\nUse the workbook ( _Section III. Optimizing your introversion - 4a. Optimizing your social life_)\n\n  * Regarding introversion, what are your biggest struggles in that area?\n  * What is one small thing you could do to improve your current situation in that area?\n\n### Your career\n\n> _A good rule of thumb is that any environment that consistently leaves you feeling bad about who you are is the wrong environment._\n> \n> Laurie Helgoe.\n\nDo you feel totally exhausted at the end of each day? If you're working the wrong job or are in the wrong environment, you can easily feel overwhelmed. Maybe you have to make phone calls for work and you hate it. Or perhaps brainstorming sessions and endless meetings are draining all your energy. Take some time to think of the things that, if you could do less of them, would make work more enjoyable.\n\nUsing our current examples, this could mean making fewer phone calls or having a lower number of meetings. After you've thought of a few things, write them down. Similarly, think about things that would make work easier if you could do more of them. This could be prioritizing emails over phone calls. Write these things down, too.\n\nUnfortunately, today's typical work environment isn't introvert-friendly. Open space is one of the worst things for introverts. We would rather have more private space and quiet time to focus on their work. Below are some tips to help you redesign your environment and improve your job satisfaction. Keep in mind that this list is far from exhaustive. Everyone is different, so I encourage you to come up with some specific and personal strategies that work for you.\n\n  * Have lunch alone to give yourself some \"me\" time. \n  * Use your break to go for a walk. \n  * Look for empty conference rooms and use them to work without interruption. \n  * Arrive early or stay late to carve out some time alone. \n  * Wear headphones to prevent people from interrupting you. \n  * Negotiate working from home, even if it's just one afternoon per week. It will allow you to recharge your batteries. Make sure you remain productive, as that will put you in a better position to negotiate spending more time working from home. \n  * Ask your boss if you can send in ideas before or after a meeting. This will show that you're motivated and present even if you can't voice your opinions during meetings. Better still, you'll have more time after the meeting to put your ideas on paper. \n  * Tell your colleagues that you're working on minimizing interruptions throughout the day and ask them what they need from you. It's especially important to ask this of your most distracting colleagues. \n  * Let your boss and co-workers know that you'd like time to settle in in the morning before talking to anyone. \n  * Clearly state that you'd prefer to communicate via email rather than by phone. \n  * Skip meetings whenever you can. \n  * Try setting office hours so as to create blocks of uninterrupted work time.\n\n### How to assert yourself at work\n\nBecause our introversion is natural to us, we expect those around us to think as we do, or at the very least understand us. However, introverts and extroverts think and behave very differently. Your boss and co-workers don't necessarily see you the way you'd expect them to. They may not realize how much you're contributing. It might be that you speak little because you don't realize how much you know. Or perhaps you don't like sharing your thoughts because it draws attention. People won't necessarily pick up on things like this.\n\nMost introverts have a difficult time speaking up and putting themselves in the spotlight. People might not understand that you need time to think alone before speaking, or that not piping up during a meeting doesn't mean you aren't willing to contribute.\n\nCheck out the tips below to see how you can make your presence known.\n\n**1. Multiply one-on-one conversations:** You might not be able to contribute much during meetings but, as an introvert, you're generally good at talking to people one-on-one. By creating more opportunities for one-on-one conversations, you can build deeper relationships with your colleagues. You could, for instance, invite one of them to lunch. Alternatively, you could arrive early or stay late from time to time to strike up conversations with some of your co-workers before (or after) everyone is in the workplace.\n\n**2. Offer your help:** Volunteer for things that you have an interest in. Maybe you can write a newsletter or share some articles with your colleagues. Or you could suggest some useful resources. As introverts, we might not like to blow our own horns, but we typically like to help. Based on what you know regarding your colleagues or boss, what could you do to assist them? How can you help them achieve their personal goals?\n\n**3. Focus on written communication:** Offer to share your thoughts and ideas after a meeting. It will give you time to think and put your thoughts on paper while demonstrating your desire to contribute.\n\n**4. Ask for help:** Introverts may be reluctant to ask for help and might want to do everything on their own. Avoid giving into this urge. Asking for help is a great way to interact with your colleagues and make them feel appreciated. Many people like helping others, so it's a win-win situation.\n\n**5. Keep your boss up-to-date:** You probably don't communicate with your boss as often as your extroverted co-workers do. You can compensate for this by regularly letting your boss know what you're up to.\n\n### Changing Careers\n\nIf you hate your job because it doesn't allow you to thrive as an introvert, it might be time to consider finding a new one. You might not be able to change your job right away. However, you need to ask yourself the following question: \"Do I want to spend the rest of my life doing what I'm currently doing?\" __ If not, why not prepare for your future career?\n\nTaking time to design your ideal day will help you choose a career that suits you. You might want to ask yourself the following questions:\n\n  * What are my strengths? \n  * What job would allow me to leverage these strengths?\n  * Do I want to spend most of my time alone, in one-on-one conversations, or conversing in small groups? \n  * How much of my time do I want to dedicate to these activities? _If you have no clue what you'd like to do in regard to a career, check out the paragraph on finding your passion in the section V, Transcending your introversion)_\n\n### Dealing with Networking Events\n\nNetworking events may be the last thing you want to attend. Yet, they're probably not as bad as you think. Contrary to popular belief, there's no need to talk to everybody in the room or distribute business cards by the dozen. In fact, those who do this generally end up with poor results. Keep in mind that the tips mentioned for parties work well for networking events as well. Read on for nine tips on handling networking events:\n\n**1. Go deep:** As an introvert, you likely crave deep, meaningful conversations. I don't see why you should ignore that craving when attending networking events. Set a small goal of building a couple of meaningful interactions rather than trying to exchange business cards with everyone in the room.\n\n**2. Set a clear goal:** Before you attend a networking event, make sure you have a specific goal set. Ask yourself this: _What exactly am I trying to achieve here?_ For instance, your goal could be to create two meaningful connections. Having a clear goal that you're confident you can achieve will help relieve some of the pressure. Make sure, your goal isn't too big. You can always approach more people than you set out to, but that should be optional.\n\n**3. Decide how long you'll stay:** Decide beforehand how much time you plan to spend at the event. It could be as little as twenty to thirty minutes. If you wish to stay longer, feel free to do so, but don't put pressure on yourself.\n\n**4. Prepare well:** As introverts, we may struggle to answer questions on the spot or come up with good questions to ask. As a result, it's best to prepare your questions ahead of time and rehearse answers to questions you might receive. Make sure the questions you ask are open-ended. It's best to avoid yes or no questions.\n\n**5. Create a simple elevator pitch:** Put together a simple introduction you can use when you meet someone for the first time. Try to view this elevator pitch as your networking introduction. When you introduce yourself, make sure you mention how you help people rather than focusing solely on what you do. What value do you offer and what problem(s) do you solve? Then, you want to state who precisely you'd like to connect with. Look at your pitch as a teaser whose objective is to encourage people to ask questions that allow you to provide more details about what you do. Kathy McAfee from www.americasmarketingmotivator.com gives the following example2: \"My name is ___, I'm a divorce and family attorney lawyer. I help people navigate through one of the most difficult times in their lives or \"I help people make smarter choices so that they can go on to have a happy life\". Then, you can add, \"I'm looking to connect with___\".\n\n**6. Take breaks:** You don't have to attend all the workshops or stay for the whole event without a respite. Take as many breaks as you need, and do it without shame!\n\n**7. The 80\/20 rule:** The 80\/20 rule states that twenty percent of our efforts result in eighty percent of our results. The same goes for networking events. Among the networking events that you consider attending, some will offer more opportunities than others. As an introvert, why not focus on the twenty percent that are most likely to generate positive results and skip the rest?\n\n**8. Bring an extroverted friend:** We discussed this in relation to parties, and it has a similar application here. Your extroverted friend will help you interact with more people and will take the edge off of things.\n\n**9. Follow-up:** Make sure you follow up with the people you interact with. Consider connecting with them on social media platforms and send them a unique email based on what you discussed at the event.\n\n### Networking the introvert way\n\nOkay, so we've talked about networking events, but who says you have to go to them, anyway? It's fine if you choose to, but if you don't want to, it's not the end of the world. They aren't a necessity. There are other ways to network that aren't nearly as stressful and exhausting as traditional networking events. Let's have a look at some of them now.\n\n### Networking online\n\nIn this modern age, we're fortunate enough to have access to this thing called \"The Internet\". LinkedIn would be a good place to start, but you can find many other sites or forums related to your industry that you can use for networking. You'll also find tons of groups on Facebook. For instance, I'm part of a group of writers on Facebook. I don't have to go to any events or even talk to anyone face-to-face to create connections. The possibilities are endless. So, if you hate networking events, consider focusing only on a few events, or eliminating them altogether. Focus on your online networking instead.\n\n### Leveraging your personal network\n\nYou already have a personal network, and each person in your network has his or her own personal network. Why not leverage these networks?\n\nFor example, one of your colleagues could introduce you to someone from a different industry that you'd like to learn more about. You could then have a coffee with that person.\n\nThis would be much easier than going to a networking event and hoping you'll get something out of it.\n\n**Action step**\n\nUse the workbook ( _Section III. Optimizing your introversion - 4b. Optimizing your career_)\n\n  * Regarding introversion, what are your biggest struggles in this area?\n  * What is one small thing you could do to improve your current situation in that area?\n\n### Relationships\n\n> _At every party, there are two kinds of people - those who want to go home and those who don't. The trouble is, they are usually married to each other._\n> \n> Ann Landers, famous advice columnist.\n\nIs your partner an extrovert? Do you have extroverted kids? To ensure fulfilling relationships, it's essential that extroverts understand introverts' needs (and vice-versa).\n\n### Being in a relationship with an extrovert\n\nIf you're currently in a relationship with an extrovert, chances are that he or she always wants to invite friends over and likes to go out on the weekends. This may even be a point of contention between you. You wish you could have more time for yourself and didn't have to spend so many weekends with friends or navigating through parties. On the other hand, your partner feels you're distant or antisocial. He or she may even be afraid that you don't love them when you say you need alone time.\n\nBelow are some tips that may help you improve your relationship with your parter.\n\n**1. Make sure your partner understands what introversion is** **and what it implies:** As an extrovert, your partner cannot experience introversion. As such, he or she may not understand why you need to spend time alone to recharge your battery. It's essential that your partner understand your needs before you try to make changes in your relationship.\n\nYou could suggest he or she read some books on introversion (like this one, for instance). The best way to encourage your partner to learn more about introversion is to learn more about extroversion yourself. Because let's face it, you don't know how it feels to be an extrovert, either. Make sure you spend some time reading some articles or books on extroversion. Additionally, ask your partner what challenges he or she encounters as an extrovert. Remember that both extroverts and introverts have challenges. This mutual understanding will make future changes (and your lives together) easier.\n\n**2. Express your needs:** Share your needs with your partner. Maybe you wish you could have more time to yourself, or more time with just the two of you, instead of being with friends all the time. Maybe you wish you could relax on Friday night instead of having to go out with friends. Just state your needs and mention how your partner would benefit from meeting them (for instance, it might mean you're more available or are in a better mood).\n\n**3. Listen to your partner's needs, too:** Ask your partner if he or she has unmet needs. For example, your partner may believe you're a little distant and may wish that they could spend more time with you. You need to know what they want so you can help satisfy their needs, too.\n\n**4. Negotiate with your partner:** Find common ground with your partner. This could mean negotiating a pattern of staying home from time to time while your partner goes out with friends. It could mean inviting friends over once a month instead of twice a month. Whatever works!\n\n**5. Give it a try:** Implement a thirty-day Challenge. Once you come up with an agreement, challenge yourself to try it for thirty day. Then, see how both of you feel during that period. You might both feel significantly happier. On the other hand, some issues might arise. If this happens, you'll need to make some tweaks until you come up with a balanced solution that both of you can agree on.\n\n**6. Dealing with conflict:** Introverts and extroverts have different ways of handling conflict. Extroverts prefer to confront things head-on, and often try to resolve conflicts on the spot. Introverts, on the other hand, typically loathe arguments. We also need time to process information before answering questions and resolving a conflict. To an extrovert, it may seem like their introverted partner is avoiding questions or doesn't care. In reality, the introvert simply needs time to think. Also, while extroverts may see heated conversations as necessary and even healthy, introverts may be reluctant to have such conversations because we prefer \"quieter\" forms of discourse.\n\n### Being in a relationship with another introvert\n\nBeing in a relationship with a fellow introvert might should like the ideal situation. You should be able to understand each other well, and that's important in any relationship. However, being with another introvert also comes with some pitfalls that you might want to be aware of.\n\n  * Your mutual desire to avoid conflict may keep one or both of you from bringing up important issues. This, in turn, might make it difficult to move forward in your relationship. You may feel stuck in your comfort zones and unable to take things to the next level. \n  * You may isolate yourselves from the world. You might get so comfortable spending time alone with your partner that you begin to neglect your social life. We all need to get out and meet other people, no matter how introverted we may be. \n  * You may rely too heavily on your partner. If you don't have a lot of friends (or you do but don't meet with them often), you may become overly dependent on your partner for your emotional needs.\n\n### Raising an extroverted child\n\nFor introverts, raising an extroverted child can be challenging. You may feel overwhelmed and constantly tired. In this section, I'll give you some practical advice to help you better manage your energy, become an even better parent, and enhance the quality of your life.\n\n  * **Ask your extroverted partner to deal with certain activities:** If your partner is an extrovert, he or she might be able to help you deal with tasks that drain your energy. For example, your partner can handle most birthday parties and chatting with other parents. Or they can spend time engaging in extrovert-friendly activities with your child. Meanwhile, you're free to focus on chores you can do on your own such as cooking or cleaning. \n  * **Explain to your child that you need your \"me\" time** : Let your child know that you need some time to recharge your battery. Your child will have to deal with introverts at some point in his or her life, so why not explain what introversion is? \n  * **Carve out some alone time every day:** Make alone time part of your daily routine. \n  * **Teach your child to spend time alone:** While introverts enjoy being alone, extroverts crave interaction and may have difficulty being by themselves. You could come up with activities that your child can do alone. Child development expert D. W. Winnicot states that the ability to be alone \"is one of the most important signs of maturity in emotional development.\" It might be a good idea to teach your child to spend some time alone. \n  * **Reduce your exposure to activities that deplete your energy:** With an extroverted child, it makes sense to reduce any social activities that aren't directly related to your child. Instead of joining group activities, you could choose to have a coffee with a couple of friends, or spend some time alone. \n  * **Suggest low-key activities you can do with your child** : You could watch a movie or read books together, for instance. I don't recommend you turn your child into a TV addict, but scheduling these types of activities can give you some time to rest.\n\n### Avoiding small talk and having deeper conversations\n\nAs introverts, we crave meaningful conversation while avoiding small talk like the plague. For extroverts though, this desire to go deep so quickly can be seen as premature and even a little intimidating. In this section, we'll discuss how to go deeper and reduce chitchat without scaring anyone.\n\n* * *\n\n**1. Be real**\n\nBeing authentic from the beginning is an effective way to minimize small talk. If someone says they like something and you don't, then say so (politely, of course). If you're tired, say so. If you aren't enjoying the party, say so (nicely). By being authentic and sharing your thoughts and emotions, you give the other person permission to do the same.\n\n**2. Create hooks by sharing more about yourself**\n\nAs introverts, we generally hate promoting ourselves, but it's sometimes useful to share some extra information about yourself. Instead of responding to, \"How are you?\" with \"I'm fine, thanks,\" you could say, \"I'm fine. I went for a walk in the park near my house this morning and the weather was perfect.\" Or you could say, \"I'm great! I started reading a new book today, and I love it so far.\"\n\nThese small comments can offer additional talking points to the person you're speaking to. You could even purposely omit some details to get the other person interested. In the second example, the other person might ask, \"What kind of book is it?\".\n\nThe more you get into the habit of talking about yourself, the easier it will become. You can role play with some friends or try adding comments when you talk with the cashier at the supermarket, for instance.\n\nYou can also bring up some topics you're interested in and try to take control of the conversation right away. The example regarding the book you're reading would be wonderful if you love books and want the conversation to be centered around that topic. It will also give you some idea of what the other person is interested in. She may love literature, sci-fi, or books on introversion. From that point, you can further direct the conversation. Don't overdo it, though. Make sure it's a two-way conversation and the other person is interested in what you're talking about. Otherwise, they'll get bored.\n\n* * *\n\n**3. Use open-ended questions to deepen the conversation **\n\nAsking open-ended questions is an effective way to improve conversations. The more specific they are, the better, as they'll give you a glimpse into the other person's world. Rather than asking what someone likes to do on the weekends, for instance, ask them what they did _this_ weekend and what their favorite part was. That's the type of question a friend would ask, and it can seriously enhance the conversation.\n\nConsider asking the following questions:\n\n  * Tell me more about X? \n  * Are you working on anything exciting these days? \n  * What was the best part of your weekend? \n  * What are you looking forward to doing this week?\n\nOf course, this is just the tip of the iceberg. You can come up with tons of other questions!\n\nAsking questions has a second perk: You don't have to spend as much time talking about yourself.! It will lessen the pressure you might feel and make it easier to sustain the conversation without becoming anxious.\n\n**4. Assume familiarity**\n\nWhen talk to your friends, do you start the conversation by discussing the weather and asking them what they do for a living? Of course not! When you start a conversation with someone you don't know, assume familiarity and try talking to that person as if they were already a friend. This means skipping standard questions regarding the person's occupation, hometown, number of siblings, and so on. As you go deeper into the conversation, you'll learn the answer to those questions anyway. If you start things off on a familiar note, the standard questions will be answered much more naturally.\n\nConsider starting the conversation by asking them how their day was or how they've been doing recently. You can then start sharing some information about yourself as we explained above. If the conversation goes nowhere, you can always ask simpler questions to get things going.\n\n**5. Try to learn something **\n\nBe curious and enter conversations with the intention of learning something about the person you're talking to. You may even want to challenge yourself to learn something specific about them. That will ensure you're as attentive as you can be. It can also keep your attention focused on them rather than yourself.\n\n**6. Steer the conversation in the direction of your choice **\n\nAs introverts, we tend to be passionate about certain topics. When these topics are brought up, we suddenly get excited and can't stop talking. The good thing is that you don't necessarily have to wait for the topic to be brought up. You can introduce it yourself and steer the conversation in the direction you'd like it to go in.\n\nYou can do this in two different ways:\n\n  * **Make a list of your favorite topics:** Then, come up with some stories and questions for each topic. Discussing topics you're passionate about is a great way to see if the other person has similar interests. \n  * **Come back to something previously said:** You could mention something that they said earlier and ask them to tell you more about it. You could also share your personal experience or opinion on the matter.\n\nA word of caution:\n\nDon't go overboard. Don't hijack the conversation or ignore what the other person says. Otherwise, they'll get bored and stop talking to you.\n\n**Create situations in which small talk is unnecessary**\n\nAnother way to avoid small talk is to seek out situations that don't rely on it. Let's say you decide to join a Meetup group surrounding a topic or hobby you're passionate about. Its members share the same passion or hobby, so small talk will become largely unnecessary. You can spend your time talking about your shared interests.\n\n**Action step**\n\nUse the workbook ( _Section III. Optimizing your introversion - 4c. Optimizing your relationships_)\n\n  * Regarding introversion, what are your biggest struggles in that area?\n  * What is one small thing you could do to improve your current situation in that area?\n\n## 4\n\n# Leveraging the Gift of Introversion\n\nIntroversion comes with a lot of benefits, none of which should be overlooked. We need more introverts who fully embrace their introversion and give their gifts to the world. As one of my friends told me, \"It's time to come out of your shell, shed the fears, and share more of yourself with the world!\" That's the message I'd like to share with you. It might be time for you to share more of yourself with the world, whatever that means to you.\n\nYou weren't born an introvert just so you could pretend to be an extrovert. Would you expect an elephant to climb a tree? You were born an introvert, and you're here to use all your strengths and contribute to the world as much as you can. If extroverts are the yin, introverts are the yang. If you spend most of your time trying to behave like an extrovert, the world will be out of balance for you. Nature and millions of years of evolution can't be wrong. If introverts account for around fifty percent of the population, it's because we have a role to play in our society.\n\nMillions of introverts before you have used their talents to make the world a better place. You can, too. Now, let's go over some of the strengths that are common in introverts.\n\n**Action step**\n\nAnswer the following question in your workbook ( _Section IV. Leveraging the gift of introversion - 1. Your biggest strength_)\n\nAccording to you, what is your biggest strength as an introvert?\n\n### 1. Leveraging your ability to spend time alone\n\n> _I owe everything that I have done to the fact that I am very much at ease being alone._\n> \n> Marilynne Robinson, novelist and essayist.\n\nYou may underestimate it, but your ability to be by yourself is one of your biggest strengths. Use this time wisely, and you'll achieve exceptional things in your life. Your ability to spend a considerable amount of time alone doesn't just allow you to think and come up with great ideas, it also enables you to get to know yourself on a deeper level. Furthermore, your 'alone time' provides opportunities to practice your craft. For extroverts, spending time alone doesn't come that easily.\n\n### 2. Leveraging your propensity for deliberate practice\n\nPractice is what creates champions. It has been shown that experts spend more time engaging in what is often referred as deliberate practice than most people do. For introverts, who would rather be left alone than work in teams, deliberate practice comes easier.\n\nFor instance, studying alone using deliberate practice has been shown to be the best predictor of skill for chess players.\n\nGrandmasters, during their first ten years, spend 5,000 hours studying the game by themselves. This is almost five times the number of hours intermediate-level players spend studying the game. Similarly, college students who study by themselves tend to learn more than those who work in groups. The same applies to many other skills, sports, and games.\n\nSteve Wozniak, the co-founder of Apple, credited his introversion for his creativity and success. He admitted that he would never have learned so much about computers if he hadn't been too introverted to leave the house. He spent a lot of time by himself learning about them, not because he knew all the science behind deliberate practice, but because he was an introvert with a strong passion. In his memoir, he gave the following advice, \"Work alone. You're going to be best able to design revolutionary products and features if you're working on your own. Not in a committee. Not on a team.\" I think this is true, especially for introverts.\n\n### What exactly is deliberate practice?\n\nIn their book _Peak,_ _Secrets from The New Science of Expertise_ , Anders Ericsson and Robert Pool suggest that deliberate practice:\n\n  * Builds skills for which effective training techniques have already been established\n  * Takes place outside one's comfort zone, requires significant effort, and is generally not enjoyable\n  * Involves specific, well-defined goals\n  * Involves taking conscious action and requires a person's full attention\n  * Involves regular feedback and requires adequate responses to said feedback\n  * Both creates and relies upon effective mental representation (the patterns we mentioned earlier) and\n  * Almost always involves working on existing skills or building new ones by focusing specifically on some aspect of those skills that needs to be improved.\n\nIt's easy to see how deliberate practice can lead to better long-term results than \"just putting in the work.\" Deliberate practice makes you work both smarter and harder by consciously focusing on improving the skills you need to reach your goal.\n\nYou might be thinking, \"If deliberate practice is so great, why aren't more people using it?\" There are three main reasons for that:\n\n  1. Deliberate practice must be based on existing training techniques that are effective. It works well with activities for which performance can be concretely assessed, like sports, music, or chess. Yet, it's harder to implement activities where performance is more difficult to assess (like teaching or business management).\n  2. It requires effort and, let's face it, most people aren't willing to go through the hassle.\n  3. Many people just don't know about deliberate practice and how they can use it to improve their skills.\n\nBelow are some examples of what deliberate practice is and is not.\n\n**Writing:**\n\nTypical practice:\n\nWriting, writing, and more writing. In the words of Stephen King, \"If you want to be a writer, you must do two things above all others: read a lot and write a lot.\" But what if it's more complex than that?\n\nDeliberate Practice\n\nIt seems as though Benjamin Franklin felt he needed to do more than read and write a lot. He focused on improving specific skills: his writing style, vocabulary, and sense of organization.\n\n  * Writing style: He made notes on articles from _Spectator_ , a high-quality newspaper, which he would use to rewrite the articles a few days later. He would then compare his version to the original and modify it accordingly.\n  * Vocabulary: He rewrote _Spectator_ essays in verse and then in prose so that he could compare his vocabulary to that which the original article used.\n  * Organization: He wrote summaries of every sentence in a particular article on separate sheets of paper. He would then wait a few weeks before challenging himself to write the article in the correct order and compare his work to the original article.\n\nDoesn't that sound like fun? And he did that consistently while holding down a full-time job!\n\n**Public speaking**\n\nTypical practice:\n\nPracticing a certain speech again and again until your performance becomes satisfactory.\n\nDeliberate practice:\n\nFocusing on a specific skill or aspect of your speech that would allow you to improve your overall performance. These skills and aspects include the following:\n\n  * The tone of your voice\n  * Your rhythm\n  * The structure of your speech\n  * Your body language\/eye contact\n  * The use of your space\n  * How you tell stories\n  * Your vocal projection\n\nAs you start using deliberate practice in your daily life, you'll be able to get far better results than most people. And although practice is best done alone, the benefits of deliberate practice aren't limited to solo activities. Public speaking, which involves interacting with a room full of people, requires a lot of deliberate practice. However, you can certainly become a great public speaker if you choose to. The same goes for any other skills you want to develop to create the life you want.\n\n### 3. Leveraging the time you spend thinking\n\n> _It's not that I'm so smart, it's just that I stay with problems longer._\n> \n> Albert Einstein.\n\nAs introverts, we tend to think a lot. This can be an amazing strength, as thinking is the key to problem solving, and it's been the source of many amazing ideas and inventions. An individual's ability to stay with a certain topic for a lengthy amount of time can yield some awesome things. That said, it's important to realize that this strength requires honing, otherwise it can easily turn into a weakness.\n\nSometimes we think so much we might even have a hard time sleeping. Our ability to think so intensely can turn from being a blessing to a curse if we're unable to control our thoughts properly.\n\nWhile extroverts are often able to take action without thinking excessively, as introverts, we tend to overthink things. If we want to do something, we'll go over it so many times that just _thinking_ about what it will involve drains us! _I need to take my car, then I'll have to find a place to park it. The shop will be crowded and I'll get tired. What if they don't have the right size or color. Then, when I'll go home it might be the rush hour etc._ It's almost as draining as actually _doing_ what we're _thinking_ about. We create all types of scenarios in our minds, which just wastes our precious energy.\n\nSince we deplete our energy during most social interactions, this tendency to overthink might be a defense mechanism to protect our energy and avoid taking on more than we can handle. That would make sense. But overdoing it actually _wastes_ our energy.\n\nThinking is one of the most powerful tools we have as human beings. When we begin focusing our thoughts on something that's important to us, we start unleashing our introvert power.\n\nSince ideas can be more exciting for us than anything else, leveraging our obsession to overcome a particular challenge can allow us to achieve great results.\n\nTip: Each time you catch yourself overthinking, remind yourself that you can only take one step at a time. When you learn to take one step at a time and stop trying to predict the future, things can become much less overwhelming.\n\n### 4. Leveraging your ability to stay focused and dig deep\n\nAs introverts, we generally prefer depth to breadth. We like to delve more deeply into specific topics, which can be used to our advantage. When we find something we're passionate about, we can get really excited and intense about it. For instance, we may be quiet for the majority of a conversation, only to talk nonstop when a topic of interest comes up. Once that happens, we'll start talking confidently and will be pretty hard to shut up. Does any of this sound familiar to you?\n\nNowadays, many people struggle to focus on a specific thing and instead jump from one topic to another. Even when they do have a specific goal, they jump from one diet to another, one course to the next, or this book to that book. Then they wonder why they fail to obtain the results they want. This is Shiny Object Syndrome, and it's a major problem for many people. The great thing about introversion is that, in my opinion, it makes it easier to remain focused on one goal until it's achieved. That's because introverts are less excited by novelty and external rewards. As Einstein says, \"It's not that I'm so smart, it's just that I stay with problems longer.\" The more you train yourself to focus on one thing you love and go deeper with it, the happier you'll be. This will also increase your chances of achieving the results you want.\n\nNow that we've covered some of the most important strengths that come with extroversion, it's time to talk about perseverance. Let's get started!\n\n### 5. Leveraging your writing skills\n\nMany introverts write better than they speak. As such, creating a habit of writing every day can make a big difference in your life. It will help you come up with great insights that will assist you in multiple areas of your life.\n\n**Keep a journal**\n\nJournaling can be helpful to anyone, but introverts may find it especially beneficial. Putting all your thoughts and ideas on paper provides two advantages:\n\n  1. It allows you to identify negative thought patterns and limiting beliefs.\n  2. It assists you in coming up with new ideas.\n\n**Identify negative thought patterns and limiting beliefs**\n\nAs introverts, we may think too much. Yet the key question is what do we think about most? While we have thousands of thoughts every day, most of them are the same thoughts we had yesterday, last month, or even last year. Our brain is like a computer. It runs its own programs and software based on preexisting beliefs. These core beliefs are the result of repeated exposure to external stimulus, and they impact our actions on a day-to-day basis.\n\nAs you keep journaling, these core beliefs will become clear. You may constantly feel like you aren't good enough. Or you might realize that certain external events create a particular emotional reaction.\n\nThese thought patterns are self-sustaining and feed themselves. An external event triggers a core belief and generates a thought. You entertain that thought, which triggers an emotional reaction. This emotional reaction in turn reinforces your thought pattern.\n\nExample: Someone told you your speech could have been better. You connect this event to your core belief of \"I'm not good enough\". It generates thoughts like \"I'll never be good enough\" or \"I'm stupid\". You identify with these thoughts and believe them, which creates an emotional reaction. This emotional reaction could be shame, frustration, or anger, among other things.\n\nTo overcome your success-limiting beliefs, you must first identify what your programming is. That is, what your thought patterns are, and what triggers them. The more you know about yourself, the better you'll be able to identify your negative patterns and stop entertaining unhelpful thoughts. You'll start seeing them as a broken record playing the same song again and again. You'll get tired of it over time and stop listening.\n\nEach time you experience negative emotions, ask yourself the following:\n\n  * What was the trigger? - What made me shift from happiness or neutrality to a negative state of mind?\n  * What was the exact thought generated by that trigger? - What thought did I identify with?\n  * How did that make me feel? \u2013 Ashamed? Guilty? Frustrated? Angry? Sad? Something else?\n  * How is it serving me? - Does it serve a purpose? Is it helping me in some way?\n  * How often have I had similar thoughts in the past? - Is this thought like a broken record? Have you had it thousands of times before? _Use the old entries of your journal and look for similar patterns. What happened in the past?_\n\nAfter the negative emotions are gone, ask yourself the following three questions:\n\n  1. What was the trigger? - What made me shift from a negative state of mind to a positive one?\n  2. What was my thought process? - What thought(s) created that shift?\n  3. How did this negative wave of emotions serve me? Did I learn something from them, or were they a waste of time and energy?\n\nThis process will help you realize that, most of the time, your negative emotions aren't helping you. That doesn't mean that they serve no purpose, however. If used correctly, they can force you to look deep inside yourself and examine the core beliefs behind these negative emotions. This, of course, allows you to change your beliefs for the better.\n\n### Coming up with new ideas\n\nHave you ever had a great idea that you wound up forgetting? I have!\n\nJournaling is a powerful way to capture all of your ideas in one place. It keeps you from forgetting the innovative thoughts that come your way. As introverts, we tend to be creative and enjoy thinking about various concepts. We feed on ideas. So, the more we capture, the better!\n\nMany of your ideas won't lead to anything, but sometimes they can change your entire life. You could come up with an idea that turns into a seven-figure business. You could have an idea for a book that will become a best-seller. Or you could have an idea that will completely change the way you see the world.\n\nIf you're not journaling yet, why not get started today?\n\n### 6. Leveraging your interpersonal skills\n\nAs an introvert, you may be better at talking in small groups or having one-on-one conversations. Many introverts have good listening skills that they can use to their advantage. Why not utilize your strengths and prioritize situations that better suit your personality? You'll feel increased levels of happiness, confidence, and competence, something you can't do if you're forcing yourself to act like an extrovert.\n\nLet's just say you're looking for a boyfriend or a girlfriend. Sure, you could go out every weekend, wear your extrovert hat, and hope for the best. But that's likely to leave you tired and frustrated. It probably won't yield any results, either.\n\nIn this scenario, why not forget about loud and crowded environments and play the game on your own turf? Wouldn't it be better to be in situations where you can have the kind of conversations you're good at? Putting yourself in favorable situations will give you more opportunities to show your personality.\n\nTo this end, you could try online dating. You could also go on a double date with your friend, or meet people in caf\u00e9s or libraries. You could even join Meetup groups to connect with people who share your interests. You'd be better off spending time with people who have similar passions than standing around at a party you're not enjoying.\n\nWhen you hang out with people who have similar interests, it's easier to strike up conversations and small talk is unnecessary. Personally, I find it easier to interact with people when I'm talking about things I love than I do when at a cocktail party. How about you?\n\n### 7. Practicing Meditation\n\nMeditation can be particularly beneficial for introverts. Although introverts may seem calm on the surface, their inner worlds can become chaotic. Personally, a book can get me so excited I have a hard time sleeping.\n\nOn the other hand, some of us can drown in waves of negative emotions that take days to escape. I've experienced this myself. I used to criticize and even insult myself for not being the person I wanted to be. Meditation allowed me to distance myself from my thoughts and stop taking them so seriously. In times of stress, meditation has enhanced my mood and enabled me to increase my focus.\n\nAs such, I meditate for 25 minutes as part of my daily morning ritual*. I encourage you to adopt a daily meditation practice, too. You can start with as little as five minutes per day.\n\n_*If you want to create a daily morning ritual that supports your goals and increases your happiness, check out my book_ Wake Up Call _. The book will provide you with a simple, 10-step method you can use right away to create a lasting morning ritual._\n\n### 8. Leveraging your ability to persevere\n\nIntroverts tend to stay focused on a task longer than extroverts. That's because we're less dependent on external rewards and instant gratification. We don't need that extra rush of dopamine as much as extroverts do. Thus, we find it easier to think long-term and anticipate potential risks.\n\nPerseverance often beats talent. Once we are able to maintain our levels of perseverance, over time, we will be able to accomplish things that are far beyond our imaginations. As with Einstein, our propensity to \"stay longer with problems\" can make a big difference.\n\nWhat if you could leverage your perseverance to achieve your goals and dreams? What if you could use it to create a happier life? This could be done by learning a new skill that allows you to fire your boss and work as a freelancer. And that's just one of the many possibilities!\n\nIn this section, I'll offer a simple method to help you become significantly more adept at persevering and make the most of your introversion.\n\n### Prepare yourself mentally\n\nMany people give up when they realize that things aren't going as well as planned. When they fail to earn tangible results after a couple of months, they jump to something else and repeat the same process, never achieving anything substantial. Whatever your goal may be, you'll face many challenging situations. In fact, you're likely to find yourself on the verge of quitting many, many times.\n\nOne of the main reasons people give up is because they start their journey with unrealistic expectations. They naively believe that things will go as planned, when we all know that they never do.\n\nTo avoid giving up too quickly, it's important to prepare for the worst even before you even start working on your goal. Ask yourself, what's the worst that could happen? What is the worst-case scenario ever? What would you do if you work on your side business for 6 months without making a single sale? What if you can't sell even one copy of that book that took you months or years to write? Considering scenarios like these will serve you well in the coming months. Make sure you visualize the situation and engage your emotions. How would you feel? Could you accept what was happening and keep pushing forward?\n\nLet me give you a recent example related to this book. A short while ago, the computer I had been using for six years crashed and I lost all my data. Even worse, I hadn't backed up any of my data (I know, I know...). As a result, I lost the two new books I was working on along with many other files related to my business. The 25,000 words I wrote for my introversion book? Gone! Yes, I had to rewrite the book you're reading right now from scratch. What an absolute nightmare!\n\nWhen I imagined the worst-case scenario, I had envisioned losing all my books on Amazon and all of the articles on my blog. However unlikely, I considered the idea of Amazon closing my account for some reason and my website being hacked. The point wasn't to come up with a realistic scenario, but to come up with the worst one to prepare myself for potential challenges ahead.\n\nWhen I lost all my data and realized I would have to rewrite this book from scratch, I wasn't happy about it. However, I didn't dwell on it for days. I didn't get depressed and waste an entire week trying to change the past. I went back to work right away, because I was mentally prepared for it.\n\nNow, imagine what would've happened if I expected things to go smoothly with little to no challenges. I would have been devastated, wouldn't I? That's why preparing for the worst is a wise thing to do. I expect to encounter many challenges in the future that are far more challenging than this one. All successful people have had to face major setbacks, and we will, too. I remember Jim Rohn discussing how he lost $250,000 because he signed a contract without reading it. At least he learned his lesson. I did, too. Now, I make sure to back up my files!\n\nThe second question is, what would make you give up? At what point would you say enough is enough? By deciding this ahead of time, you'll know that, unless you encounter that situation, you must keep going. And you can always reassess the situation. Maybe, you want to quit your job and look for another one that's more introvert-friendly. If so, what would make you give up on that goal? Will you give up after six months? One year? Two years? After ten job interviews? Twenty? Thirty?\n\n### Stick with one major goal\n\nHaving too many goals is one of the most common mistakes people make when they set goals. While having lots of short-term goals might be okay, it's usually counterproductive when it comes to long-term ones. It's far better to stick with one major goal and achieve it than to dabble in several big goals. Doing this builds your self-discipline muscles and translates to greater results over time.\n\nYou may want to use this book to improve your relationships or redesign your career. Or perhaps you want to become better at networking and deal with parties in a more \"gracious\" way. If you try to do everything at once, however, you'll feel overwhelmed and fail to achieve any of your goals. You'll get better results by focusing on one major goal during the next, say, thirty days.\n\nLet's say you decide to focus first on your relationship with your partner. Once you find a satisfying balance, you can then move on the next area of your life, such as your career. Please note that this doesn't mean you shouldn't do anything to improve other areas of your life. It just means that you should focus the majority of your efforts towards your one primary goal.\n\n### Commit to Your Bullet-Proof Timeframe\n\nThe Bullet-Proof Timeframe is a concept that I came up with to help me remain patient and stick to my main goal. The idea behind it is to give yourself a period of time during which you commit to working on your major goal. The idea is to make a promise to yourself to never give up before the deadline.\n\nMy main goal is to keep writing books until I reach the deadline of my Bullet-Proof Timeframe, which is April 18th, 2020 (my 35th birthday). Each time I want to give up, I remind myself that I have time and I bring my focus back to my goal. I've had to do this dozens of times.\n\nBelow is an excerpt from my book _The One Goal_ , which summarizes the main benefits of implementing a Bullet-Proof Timeframe. A Bullet-Proof Timeframe does the following:\n\n  * **It forces you to select a goal that matters to you**. Why would you care about some distant deadline two to three years from today if your goal wasn't that important?\n  * **It reminds you that you have time**. You can step back and look at the bigger picture. As you realize you still have time, persevering becomes less of a challenge.\n  * **It helps you avoid 'Shiny Object Syndrome'**. You'll stop jumping from one thing to the next if you don't get the results you expected. This is a major trap that derails many people.\n  * **It gives you the option to give up**. You can give yourself total permission to give up, BUT ONLY once you've reached the deadline. Never before. Here's the mindset you want to adopt: _I can always give up when I reach my deadline, so for now I'm going to hang on and keep going._\n\nI hope that by now you're convinced of the benefits of setting a Bullet-Proof Timeframe. It will allow you to turbocharge your perseverance in all areas of your life.\n\n### Goal-setting crash course for introverts\n\nLet's go over a powerful goal-setting method. This will help you as you work on redesigning your life as an introvert.\n\n### 1. How to set goals\n\nImagine you could achieve anything you want and had no limitations whatsoever. What are the goals that, if achieved, would enable you to be the happiest introvert possible? Write down anything that comes to mind without judging it.\n\n### 2. Select one goal\n\nSelect the one goal that, if achieved, would have the biggest impact on your life. Then circle it, because that will be your main goal.\n\n### 3. Make it SMART\n\nToo often, people set goals that are way too vague. When it comes to goals, the more specific, the better! So, make sure you use the SMART goal method, which is explained below:\n\n**SMART** stands for:\n\n  * **S** pecific: What exactly do you want? What are you trying to achieve?\n  * **M** easurable: Can you assess the progress towards your goal? How will you know if you've achieved it?\n  * **A** chievable: Is it achievable? Is the timeframe realistic? Can you put in the effort required despite other responsibilities?\n  * **R** elevant: Is it in line with your values? Is it exciting to you?\n  * **T** ime-bound: Do you have a clear deadline for your goal?\n\n### 4. Break down your goals\n\nIf your goal is long-term, break it down into a yearly, monthly, weekly, and even daily goal. Now, ask yourself how confident you feel about your ability to achieve your goal. On a scale of 1 to 10, you want to be at a 7 or 8. If you're below this, try extending your timeline or making your goal smaller.\n\n### 5. Implement at least one daily habit to support your goal\n\nImplement one small habit that moves you towards your goal each day. For instance, my goal involves writing books, so I make sure I write every day no matter what. My small goal is to write at least 500 words per day. That doesn't sound like much, but that's close to 200,000 words per year or 1,000,000 words in five years. That's about eight 25,000-word books a year. And I usually write more than 500 words per day! Your daily habit will help you maintain momentum and sustain long-term motivation. Furthermore, by keeping each new daily goal small, you'll also be less tempted to procrastinate. Remember to use deliberate practice with your daily habits.\n\nWell, that's the end of this crash course. If you want to learn more about setting exciting goals, feel free to check my goal-setting book Goal Setting: The Ultimate Guide to Achieving Goals That Truly Excite You.\n\n**Action step**\n\nTake the time to complete the exercises in your workbook ( _Section V. Leveraging the gift of introversion - sub-sections 2 to 8)_\n\n## 5\n\n# Transcending Your Introversion\n\n> _If you're an introvert, find your flow by using your gifts. You have the power of persistence, the tenacity to solve complex problems, and the clear-sightedness to avoid pitfalls that trip others up. You enjoy relative freedom from the temptations of superficial prizes like money and status. Indeed, your biggest challenge may be to harness your strengths fully. You may be so busy trying to appear like a zestful, reward-sensitive extrovert that you undervalue your own talents, or feel underestimated by those around you. But, when you're focused on a project you care about, you probably find that your energy is boundless._\n> \n> Susan Cain, author of Quiet\n\nYou've downplayed your strengths for too many years. You've tried so hard to be an extrovert you forgot who you really are. The truth is you're as powerful as any extrovert. You have strengths most extroverts don't. You can become a leader and a source of inspiration for everyone around you. You don't need to try being the loudest person in the room to accomplish those things. Your passion is your strength, your ideas are your weapons, and your perseverance is a hidden gem waiting to be uncovered. Your written words have power, and when you speak it's because you're going to say something important. Over time, people will learn to listen to you. And sometimes, out of the blue, you'll just come up with the perfect thing to say.\n\nOf course, your power will only come to the surface when you learn to embrace your introversion. When you fall in love with silence. When you stop apologizing for who you are. When you step out of your comfort zone, not so you can try to act like an introvert, but because it's what your mission or your calling requires you to do.\n\nThere are many myths out there that would lead you to believe introverts have limitations and can't do what others can. That, however, is total BS. If you buy into these assumptions, you'll miss out on opportunities to uncover your true potential. Do you know what Nelson Mandela, Mahatma Gandhi, and Mother Theresa had in common? Yes, they were great leaders, but there's something else: They were introverts. So how can anyone say introverts can't be leaders? How can anyone say that we need to be fixed? I don't buy either of those assumptions.\n\nDuring grad school, I told a classmate of mine that I was interested in coaching. Do you know what she said? \"You? You're too quiet. You never talk. How can you possibly become a coach?\" Taking that thought to is logical conclusion, I should never try anything because I'm an introvert. Perhaps, I should stay home and not bother to challenge myself. I'm sorry, but that's not what introversion is about. I don't like people telling me what I can do and can't do. Do you?\n\nWhen I ran a survey asking people about their biggest challenges as introverts, one of the participants wrote the following:\n\n_The biggest problem I see my fellow Introverts experiencing is accepting that they are not odd or defective because they are different in their wants and needs. Also, that they can do whatever they want to, even if it's in the Extrovert world, as long as they make time to care for themselves. _\n\nHe continued:\n\n_Despite the fact I've been labeled an extreme introvert, I had a very successful career in sales which often required speaking to large groups. I loved my work, I just always made sure I set aside time to refuel after I had to be \"on\" for a group or client._\n\nDid I mention that he was also a coach? Wait a minute! I thought introverts couldn't become coaches. This introversion thing is definitely more complex than I used to think!\n\n### Finding your passion\n\nIt's easy to allow yourself to become distracted and end up spending your life doing what society expects you to do. My guess is that you don't want society to tell you to live your life as an extrovert, if you did, you wouldn't be reading this book, would you?\n\nYou probably don't want society to tell you what career you should have, either. To thrive as an introvert, you must become crystal clear on what you want to do. The questions below will help you clarify your passion. I interviewed myself to give you an example of this exercise.\n\nWho do you envy? \\- This is a great question and it usually shows what you are drawn to.\n\nI envied personal development bloggers like Steve Pavlina and Leo Gura. One day, as I was watching a video from Leo Gura, everything became clear. I thought, \"This is it.\" I realized exactly what I wanted to do: study psychology and personal development to help others. That lead me to start my blog in 2014.\n\nWhat did you enjoy doing when you were a kid?\n\nWhen I was a kid, perhaps seven or eight, I read books all day long. They were children's books, of course, but it was the start of my love of reading.\n\nWhat activities do you volunteer for at your current job? What about your previous job?\n\nAt my first job, I volunteered to write monthly articles in Japanese regarding French culture. I also held seminars. I talked about the cultural differences between France and Japan in one of these seminars. These two examples show both my interest in writing and my interest in psychology.\n\nAt my previous job, I wanted to figure out the best practices to encourage growth and the sharing of knowledge. I guess it's similar to what I do with my books: I gather knowledge from different sources and put them together.\n\nWhat topic do you usually get excited to talk about? When was the last time you had an exciting conversation?\n\nI enjoy talking about cultural differences, psychology, and (of course) personal development. The last time I got excited was when I talked about my online business with a friend a few days ago.\n\nWhat's your way of contributing to the world? \\- Do you want to inspire? Entertain? Educate? Heal? Serve? Create art?\n\nI want to help people realize what their minds are capable of achieving. I want to give them the tools they need to design their dream life.\n\nAccording to you, what are your unique strengths? \\- What is it that only _you_ can do?\n\nOne of my biggest strengths is my ability to see the best in other people\u2014to see what they could become. I often believe in people more than they believe in themselves. My desire to help people and share information with them is another strength. When I get excited about a book or a video, I want to share it with other people. I find it frustrating when I can't share things with others. I would say my love for learning is another strength. I love consuming new information and grasping new concepts. Learning is an exhilarating, but never-ending journey.\n\nWhat about you? What are you passionate about?\n\n**Action step**\n\nUse the workbook to answer the questions above ( _Section V. Transcending your introversion - 1. Finding your passion_)\n\n### Supercharging your passion\n\nMost of us don't like to be told to act like introverts, and we'd love to be able to embrace our introversion unapologetically. But what exactly does it meant to be an introvert? We've already seen that introversion involves the need to spend time alone to recharge our batteries. It involves becoming overstimulated in loud environments with lots of people. It also means struggling to interact with people and having a hard time thinking on the spot.\n\nYet introversion isn't something that limits your possibilities and imposes constraints. Sure, you need time alone to replenish your energy, but aside from that, you can do anything you want to do. I don't want you to limit your potential because of the assumptions you hold about introversion. If what you're passionate about requires you to do sales, then do it. Introverts know how to listen and ask the right questions, so, in actuality, they're often great salespeople. Do you remember what the respondent to my survey said? He wrote that he \"had a very successful career in sales.\" He's far from being the only one. Do you remember Mr. Little? He was a terrific public speaker. You could be, too, if you so desired.\n\nAs introverts, we also have to recognize that one of our biggest strengths is our ability to become obsessed with something. When we find something we love and that gives us meaning, we can move mountains. Conversely, when we have a job that's uninspiring and neglect the things we love, breaking out of our comfort zone can become impossible.\n\nWhat about you? Do you see yourself being too intense? Are you obsessed with something? I used to believe that I was too obsessive and I needed more balance in my life. The truth is I've been this way for most of my life. First, it started with books. When I was little, I would spend most of my days reading books in my bedroom. Then, I start playing video games. After that, I became obsessed with table tennis. In high-school, I would train every day before and after school. Then, there was online gaming. I would sometimes play until 4am during the weekends. Needless to say, my mother was worried about me. In college, I became obsessed with studying. Finally, I became obsessed with personal development.\n\nWhile obsession can be a negative thing, I've recently started looking at it in a different light. I thought, \"What if my obsession is actually my biggest strength?\" I wondered what would happen if I decided to embrace it rather than repress it. I pondered what would happen if I focused entirely on something I'm passionate about.\n\nYour personal story may be different from mine, but you might be able to relate to it to a certain degree. The point is that your desire to delve deeper and your need to find meaning in what you do can be an extremely powerful driver. In fact, it could very well be one of your greatest strengths as an introvert.\n\nFor these reasons, I think escaping from your comfort zone to make your dreams a reality is part of introversion. It's part of giving your gift to the world. I don't see it as trying to be an extrovert, but that's just my opinion. You don't have to agree with me. In the end, the most important thing is that you live a happy life as an introvert and leave this world with no regrets! I think we can agree on that., can't we?\n\nOnce you're grounded in your introversion and focused on your mission, you can become a great example to others. People will envy your self-discipline, focus, and passion. Introversion is part of who you are, so embrace it. Embody it. It's part of what makes you 'you,' and nothing in the world can change that.\n\n**Action step**\n\nAnswer the questions in your workbook ( _Section V. Transcending your introversion. - 2. Supercharging your passion_)\n\n  * What skills must you develop to supercharge your passion?\n  * What things beyond your comfort zone will you need to do to live a meaningful life?\n\n### Supercharging your ideas\n\n> _Introverts dislike small talk, but we are fluent in the language of ideas and dreams._\n> \n> Michaela Chung.\n\nAs introverts, we love ideas. Ideas are what excite us and make us come alive. Chances are, you have a lot of ideas you've never shared with anybody else. You may be reading one book after another, accumulating a vast amount of knowledge. Or you may spend a lot of time thinking and coming up with new things.\n\nWriting down insights in a journal, or spending time to think on your own are effective ways to generate more ideas. Of course, you may not be big on sharing your ideas with the world. Yet, that's part of supercharging our introversion and sharing our gifts to the world. To this end, I believe it's our responsibility to put our ideas out there. As the late Wayne Dyer said, \"Don't die with the music still in you.\" If you were to die today, what ideas and projects would the world miss out on because you didn't dare to share them? I don't want to die with the music still in me. I want to play my music and see how it impacts the world around me. Don't you?\n\nYou can share your ideas verbally or through writing. It might be through art or through your business. Whatever the means may be, communicate them to the best of your ability. That's the way to supercharge your introversion. That's how you can share your gifts with the world.\n\n**Action step**\n\nAnswer the questions in your workbook ( _Section V. Transcending your introversion. - 3. Supercharging your ideas_)\n\n  * Are you capturing your ideas in an effective way? If not, what could you do to capture them more efficiently?\n  * Are you sharing your ideas and putting them out there? If not, what could you do to share your ideas and make a bigger contribution to the world?\n\n### Understanding introversion is not a label to wear\n\nOnce we start understanding introversion, we may feel like we don't need to make an effort anymore. We think that, because we're introverted, we should stop interacting with people, stay away from public speaking, and stay home as much as possible. None of these things are true, however.\n\nYet it's easy to fall for that and live by our own definition of introversion. As we mentioned earlier, many of us hold false assumptions that don't serve us well. When we label ourselves as introverts, we risk missing out on opportunities for growth by clinging on to beliefs that aren't necessarily true.\n\nI don't believe **** it's realistic to think we can live according to our vision of introversion all the time. Nor is it in our best interest. Just as extroverts may feel lonely when they have no one to go out with, we may find ourselves overwhelmed at parties we didn't plan, or want, to attend. It would be misleading to say that extroverts have it easy and live perfectly balanced lives, while introverts are a mess. We all have our challenges. Many introverts suffer loneliness, wishing they could interact with people more often. We all have to deal with some level of discomfort.\n\nI can already hear you screaming, \"Why are you saying I should act like an extrovert!\" Don't worry, that's not what I'm saying. It's up to you whether you want to learn public speaking or talk to a lot of people at a party. It doesn't mean you aren't an introvert, it just means you're going beyond your comfort zone to achieve something you want. I've found that, when I have a strong sense of purpose, I'm more willing to do things that, as an introvert, I normally wouldn't want to do.\n\nFor instance, I'm not big on sharing information about myself, but I do it in my books and videos. I've also been a member of Toastmasters, a public speaking organization, for three years. I discovered that delivering speeches and shooting videos isn't that bad. I actually started enjoying both, primarily because I have a bigger purpose. If I need to shoot videos or do seminars to reach more people, I'll do just that. I just won't do it all the time. And I would be reluctant to do it for a cause I'm less than passionate about.\n\nThe final point I'd like to make is that it's easy to mistake shyness for introversion, or even fear for introversion. Thus, I would encourage you to try things out and test your own boundaries. You may end up enjoying activities that you would never thought you'd like.\n\n**Action step**\n\nAnswer the question below using your workbook ( _Section V. Transcending your introversion. - 4. Introversion is not a label to wear_)\n\nCould labeling yourself as an introvert (using your own definition of introversion) potentially harm your growth and fulfillment in life? If so, how?\n\n### Becoming a leader\n\n> _In a gentle way, you can shake the world._\n> \n> Mahatma Gandhi.\n\nAnyone can be a leader. Being a leader isn't reserved for extroverts. If you have any kind of impact on those around you, you're already a leader. If you have control over your own life, you have what it takes to be a leader. After all, isn't a leader someone who ushers others along a certain path? And how can you lead people if they aren't inspired by your actions and the control you have over your life.\n\nIndeed, being a leader starts from within. Your ability to control your mind, body, and attitude determines your level of influence. That's not something reserved for extroverts. Many great leaders have been introverts. They weren't the loudest in the room, but their actions spoke louder than their words. To paraphrase Gandhi, they were embodying the change they wanted to see in the world.\n\nYour ability to lead people and inspire them to change depends on whether they see you as a role model. Do you want to know if you're a leader? Ask yourself, \"How have I impacted people around me?\" Have you influenced your partner, kids, friends, or colleagues in a positive way? Do people look at you as a role model?\n\nIntroverts can be great leaders. We can even outperform extroverts in certain situations. A 2010 study conducted by the Harvard Business Review showed that, when leading extroverts, introvert were more effective leaders than extroverts. The opposite was also true. Extroverted leaders outperformed introverted ones when it came to leading introverts.\n\nThe bottom line is that, as an introvert, you can be as competent a leader as any extrovert.\n\n### Some successful introverts\n\nNow, let's have a look at some highly successful introverts. You'll see that some of the most successful people on earth are actually introverts. Their introversion may even be one of the reasons they reached the pinnacle of success.\n\n### Bill Gates - Fiercely un-shy\n\nThe founder of Microsoft can be very quiet, but he becomes very outspoken when it comes to his passion, which is business. I believe this is true for many introverts. Our passion can energize us and give us tremendous power.\n\nGates said the following in one of his speeches:\n\n> I think introverts can do quite well. If you're clever, you can learn to get the benefits of being an introvert, which might be, say, being willing to go off for a few days and think about a tough problem, read everything you can, push yourself very hard to think out on the edge of that area.\n\n### Warren Buffet - Calculated risks\n\nThis successful investor is an introvert. His ability to take calculated risks rather than letting excitement get in his way is probably part of his success. While many investors live near Wall Street, he lives in a modest house in Omaha, far away from the hype.\n\n### Mark Zuckerberg - Depth over breadth\n\nThe founder of Facebook is an introvert. He may not have as many connections as extroverted CEOs, but his ability to create genuine connections and persuade startup founders to join his company is definitely a strength. As an introvert, his desire to prioritize depth over breadth seems to work in his favor.\n\n_Fast Company's_ article \"Facebook's Plan to Own Your Phone,\" had this to say:\n\n> The fact that Zuckerberg can more often than not persuade startup founders to join the company and work with him is a vote for the glass-half-full perspective. 'What I found compelling was Mark's commitment to spending a lot of time with us.'\n> \n> Brendan Iribe, Oculus CEO\n\n### Cristina Aguilera - Intensity\n\nYou would expect the pop star Christina Aguilera to be an extrovert, but she is actually an introvert. She revealed to _Marie Claire_ that she had always been intense and introverted. She felt like an outsider her whole life.\n\nThis goes to show that nothing prevents introverts from becoming great entertainers. Putting on a show or playing a certain character can actually be very enjoyable for us. It gives us opportunities to express ourselves without interruption. How cool is that? That's why I sometimes enjoy public speaking. I know people have to listen to me, and there's something nice about that.\n\n### Emma Watson - The non-party girl\n\nIn a world that expects people to behave like extroverts, this actress once felt like something was wrong with her. She didn't want to go out and \"have fun\" like her friends. She mentioned that learning about introversion was a relief and very empowering.\n\n> The truth is that I'm genuinely a shy, socially awkward, introverted person,\" she said. \"At a big party, I'm like Bambie in the headlights. It's too much stimulation for me, which is why I end up going to the bathroom! I need time outs! You've seen me at parties... I get anxious. I'm terrible at small talk, and I have a ridiculously short attention span.\n\n### Mahatma Gandhi \u2013 Thoughtful\n\nSpeaking at length was a struggle for Gandhi throughout his life. Like most introverts, he needed time to prepare his speeches. He carefully chose his words. Yet that thoughtfulness spared him many problems.\n\nAs he said himself:\n\n> And I can now give myself the certificate that a thoughtless word hardly ever escapes my tongue or pen. I do not recollect ever having had to regret anything in my speech or writing. I have thus been spared many a mishap and waste of time.\n\n### Albert Einstein - Staying with problems longer\n\nEinstein, like most introverts, did his best thinking while alone. His ability to spend time alone and stay focus on a problem for an extensive period of time may have contributed to his discoveries. Focusing on a specific goal for years without being distracted has been shown to be an extremely important factor for success.\n\n\"It's not that I'm so smart,\" he once said. \"It's just that I stay with problems longer.\"\n\nIn the eyes of society, you may not end up as successful as the people I've mentioned. Yet I have no doubt that you have extraordinary room for growth. Embracing your introversion rather than seeing it as a handicap will allow you to make a difference in other people's lives. The question is, how will you do it? Will it be through your intensity? Your perseverance? Your thoughtfulness? Your obsession with your passion? Your ideas? That's what you'll have to figure out!\n\n**Action step**\n\nAnswer the questions below using your workbook ( _Section V. Transcending your introversion. - 5. Becoming a leader_)\n\n  * What would you say your biggest strength is? (Thoughtfulness, perseverance, intensity etc.)\n  * What is one strength, that, if further developed, would make the biggest difference in your life?\n\n# Conclusion\n\nI'd like to thank for buying this book. In an extroverted world, being an introvert can sometimes be challenging. It's my sincere hope that this book helps you reassess the way you see your introversion.\n\nIntroversion is not something to be ashamed of, but rather something to celebrate. We should never have to apologize for being an introvert. We may encounter resistance from people who want to drag us to the next exciting party. However, we should remain true to ourselves and communicate our needs and desires politely (but firmly).\n\nRemember that introverts aren't a minority. It's our frequent silence and the difficulty we have in accepting our true nature that makes it seem that way. We can be easily overlooked, and many of us try to pretend to be extroverted. It's only by asserting ourselves and embracing our introversion that the extroverts around us will start to understand it.\n\nIf that happens, introversion might become mainstream someday. The workplace will be redesigned with introverted employees in mind, and the way employees interact within the company will be totally transformed. The quality of our relationships with friends and family will improve. We'll be encouraged to leverage our strengths for society's greater good, and the definition of \"having fun\" will expand. In short, the world will give equal consideration to introverts and extroverts.\n\nBut for now, we must proudly live our lives as introverts. We have to remember that we're more than just a label. Once we set our minds to something, we have the power to accomplish exceptional things. With an inspiring enough vision, we can overcome almost any obstacle that stands in our way.\n\nIn many cases, the limits we imposed on ourselves are nothing more than ideas in our mind. Remember than Gandhi, Mother Theresa, and Nelson Mandela were introverts like you and I are. While you may not be the next Gandhi, you can be a better you. A 'you' who isn't afraid of your introverted nature, a 'you' that will do whatever you can to give your gifts to the world and be as happy as possible.\n\nIf this book helps you in any way, please don't hesitate to let me know and share your story. Also, please feel let me know if you have any feedback. And, if you enjoy the book, don't forget to leave me a review. It would mean so much to me!\n\nIf you have any questions, contact me at\n\nthibaut.meurisse@gmail.com.\n\nI'll get back to you as soon as possible.\n\nI'm looking forward to hearing from you very soon!\n\nWarm regards,\n\nThibaut Meurisse.\n\nFounder of whatispersonaldevelopment.org\n**A ffirmations for introverts**\n\nFeel free to repeat these affirmations on a daily basis or to create your own affirmations.\n\n**Parties**\n\n  * During parties, I give myself complete freedom to behave and act according to my mood\n  * I release all expectations and let go of any pressure I put on myself\n  * I allow myself to enjoy parties without any expectations or any desire for specific outcomes\n  * Reading a book is as valuable as going to a party\n  * I leave parties when I don't enjoy it and I'm perfectly fine with it\n  * I take breaks whenever I want without apologies\n\n**Silence**\n\n  * I give myself permission to say nothing. That's my way to celebrate my introverted nature\n  * My silence gives space to myself and to others to simply be.\n  * I choose to be silent whenever I feel like it.\n  * I love silence.\n  * Being silent is doing something\n  * I don't owe anybody any interaction\n\n**Self- respect**\n\n  * I put my own needs first.\n  * I show respect to myself by saying \"no\"\n  * I know what I need better than anyone else\n  * Staying home is doing something\n  * As an introvert there is nothing I should or shouldn't do.\n  * Managing my energy is respecting myself\n\n**Managing energy**\n\n  * Managing my energy is a favor to myself and everyone around me.\n  * I manage my energy so that I can give my best gift to the world\n  * Managing my energy is respecting myself\n\n**Accepting introversion and extroversion**\n\n  * I let go of any negative emotions towards extroverts\n  * I love myself. I love extroverts\n  * I embrace my introversion and respect other people's extroversion\n  * Both introversion and extroversion are needed in this world\n  * I have my challenges, extroverts have theirs.\n  * By accepting other people's extroversion, I accept my own introversion\n\n**Living up to my potential**\n\n  * As an introvert I can achieve anything I set my mind to. I just have to manage my energy well.\n  * I'm more than my introversion and can transcend it whenever necessary\n  * I can wear the extrovert hat whenever I want and I'm fine with it.\n  * When I'm passionate I can do anything\n  * I refuse to be less than I can be\n\n**W hat do you think?**\n\nI want to hear from you! Your thoughts and comments are important to me. If you enjoyed this book or found it useful **I'd be very grateful if you'd post a short review.** Your support really does make a difference. I read all the reviews personally so that I can get your feedback and make this book even better.\n\nThanks again for your support!\n\n### Other books by the author:\n\nGoal Setting: The Ultimate Guide to Achieving Goals That Truly Excite You (Free Workbook Included)\n\nHabits That Stick: The Ultimate Guide to Building Habits That Stick Once and For All (Free Workbook Included)\n\nMaster Your Emotions: A Practical Guide to Overcome Negativity and Better Manage Your Feelings (Free Workbook Included)\n\nProductivity Beast: An Unconventional Guide to Getting Things Done (Free Workbook Included)\n\nThe Greatness Manifesto: Overcome Your Fear and Go After What You Really Want\n\nThe One Goal: Master the Art of Goal Setting, Win Your Inner Battles, and Achieve Exceptional Results (Free Workbook Included)\n\nThe Passion Manifesto: Escape the Rat Race, Uncover Your Passion and Design a Career and Life You Love (Free Workbook Included)\n\nUpgrade Yourself: Simple Strategies to Transform Your Mindset, Improve Your Habits and Change Your Life (Free Workbook Included)\n\nWake Up Call: How To Take Control Of Your Morning And Transform Your Life (Free Workbook Included)\n\n# About the Author\n\n**T HIBAUT MEURISSE **Thibaut Meurisse is a personal development blogger, author, and founder of whatispersonaldevelopment.org.\n\nHe has been featured on major personal development websites such as Lifehack, TinyBuddha, PickTheBrain, stevenaitchison, guidedmind, DumbLittleMan or FinerMinds.\n\nObsessed with self-improvement and fascinated by the power of the brain, his personal mission is to help people realize their full potential and reach higher levels of fulfillment and consciousness.\n\nIn love with foreign languages, he is French, writes in English, and lived in Japan for almost a decade.\n\nYou can connect with him on his Facebook page\n\nhttps:\/\/www.facebook.com\/whatispersonaldevelopment.org\n\n### Bibliography\n\n**O n introversion**\n\n  * _Quiet: The Power of Introverts in a World That Can't Stop Talking_ , Susan Cain\n  * _Introvert Power: Why Your Inner Life is Your Hidden Strength_ , Laurie A Helgoe\n  * _The Introvert Advantage: How Quiet People Can Thrive in an Extrovert World_ , Mari Olsen Laney\n  * _Introverts in Love: The Quiet Way to Happily Ever After_ , Sophia Dembling\n  * _The Introvert's Way: Living a Quiet Life in a Noisy World_ , Sophia Dembling\n\n**On relationships**\n\n  * _The 5 Love Languages: The Secret to Love That Lasts_ , Gary Chapman\n\n**On perseverance**\n\n  * _Grit: The Power of Passion and Perseverance_ , Angela Duckworth\n  * _Peak: Secrets From the New Science of Expertise_ , Anders Ericsson and Robert Pool\n  * _Talent is Overrated: What Really Separates World-Class Performers from Everybody Else_ , Geoff Colvin\n\n**On goal setting**\n\n  * _Goals!: How to Get Everything You Want Faster Than You Ever Thought Possible_ , Brian Tracy\n  * _Wishcraft: How to Get What You Really Want_ , Barbara Sher, Annie Gottlieb\n\n# Goal Setting Book Preview\n\n### Introduction\n\n> _Mr. Rohn, let me see your current list of goals. I've had a lot of experience and I've been out here for a while, so let's go over them and maybe I can really give you some good ideas.\" And I said, \"I don't have a list.\" He said, \" Well, if you don't have a list of your goals, I can guess your bank balance within a few hundred dollars.\" And he did._\n> \n> Jim Rohn, The Jim Rohn Guide to Goal Setting\n\nI would like to thank you for downloading this e-book. In doing so, you have already shown your commitment to bettering your life by setting goals that truly excite you. You have joined those who have made the decision to take more control over their lives and give less power to circumstances. It's important to think about what where you want to be, whether it's one month, six months, one year, five years or even a decade or more from now. Taking the time to identify the goals you wish to accomplish is the best way to make sure that you're going into the right direction. It will also keep you from pursuing goals that won't fulfill you.\n\nDeciding to set goals is probably one of the most important decisions you can make, but most people don't set clear goals in their life. It's almost as if they believe they have no control over their life. As such, they wander through life heavily influenced by the circumstances and people around them. They give their power away to their environments instead of using it to create the lives they desire. They achieve far les than they would if they took the time to plan their lives and set specific goals.\n\nKeep in mind, however, that having goals in and of itself is not enough. In fact, having goals that are unclear or out of alignment with what you want can be almost as bad as having none at all. Unfortunately, many goal setters spend years in dogged pursuit of a particular goal only to achieve it and realize that isn't what they genuinely wanted. This e-book will help you avoid that. Setting specific goals is one of the best decisions I've made in my life and the information within this book will give you an opportunity to do the same.\n\nI first created a list of goals back in September of 2014 while in the process of building my website. Looking back, I often wonder why I'd never done it before and why I never learned about it in school. Setting goals is par for the course when it comes to personal development, however.\n\nI believe that we all have the potential to accomplish great things in life. However, many of us never learned to tap into our intrinsic ability to self-motivate. We spend our childhoods studying to get good grades and trying to fit in in an attempt to please our parents, teachers, and peers. We then spend our adulthoods working for money and other external motivators. Our tendency to rely upon outside motivators is ironic considering how ineffective they are. Studies show that extrinsic motivators such as money are less efficient than intrinsic motivators like autonomy, self-mastery, or finding purpose. The carrot and stick approach is still in frequent use these days, but it's far from ideal. The reality is that intrinsic motivation yields better results and provides a greater sense of fulfillment than extrinsic motivation does.\n\nFortunately, learning to set the right goals will help you tap into your intrinsic motivation and allow you to uncover your hidden potential. This book is intended to help you figure out what you want to achieve and the kind of life you wish to create for yourself. I want you to set goals that inspire you, stir your soul, and make you want to jump out of bed every morning. Goal setting might seem intimidating, but it's more than worth it!\n\n### What you will learn in this book:\n\nWithin this book, you'll find a comprehensive method to achieve your goals. You won't just learn how to set goals effectively, you'll also learn to think better thoughts, overcome obstacles, and persevere until you reach your goal.\n\nThis Book Will:\n\n1) Give you the opportunity to discover and set goals that genuinely matter to you\n\n2) Help you set short-term, mid-term, and long-term goals in multiple areas of your life.\n\n3) Help you realize your potential and achieve more than you thought possible.\n\n4) Provide you with an effective strategy to achieve the goals you set.\n\n5) Enable you to avoid the obstacles you will encounter while working towards your goals.\n\nThis book is full of valuable information, but remember that how much you get out of it is largely dependent upon how committed you are to implementing it. The ball is in your court!\n\n### I. Why goal setting is important\n\n> _People without goals are doomed to work forever for people who do have goals._\n> \n> Brian Tracy.\n\n### Setting goals gives direction to your subconscious mind\n\n> Your automatic creative mechanism is teleological. That is, it operates in terms of goals and end results. Once you give it a definite goal to achieve, you can depend upon its automatic guidance system to take you to that goal much better than \"you\" ever could by conscious thought. \"You\" supply the goal by thinking in terms of end results. Your automatic mechanism them supplies the means whereby.\n> \n> Maxwell Maltz.\n\nDid you know your subconscious mind could help you achieve your goal? Setting goals gives you a direction in life, but vague goals like making more money or being happy won't lead to a fulfilling life. Your subconscious mind is like a powerful machine, and understanding how it works is a big part of successful goal setting. Hypnotherapist Joseph Clough compares it to a GPS whereas Maxwell Maltz, author of Psycho-Cybernetics, calls it a mechanical goal-seeking device. If you put an address into your GPS, it will do whatever it can to reach your destination. The subconscious mind behaves similarly. Have you ever learned a new word only to find yourself hearing it everywhere you go? That's an example of your brain \"priming\". In other words, it's scanning your environment for all information relevant to the word, phrase, or details you've given it. As such, setting clear goals gives you a greater chance of accomplishing them. It sends a strong signal to your subconscious mind, which allows it to unleash its focusing power and look for any opportunities to achieve the goal. We talk more about the importance of specific goals later.\n\n### Setting goals empowers you\n\n> _If you don't design your own life plan, chances are you'll fall into someone else's plan. And guess what they have planned for you? Not much._\n> \n> Jim Rohn.\n\nAre you the one choosing your goals? Or are others choosing them for you? When you start setting intentions, you stop giving away your power.\n\nWhen you start setting objectives in all the main areas of your life such as finances, relationships, career, and health, you stop giving power away and start empowering yourself. You make a conscious choice to become the creator of your life and begin to take responsibility in every aspect of your life.\n\nImagine the difference it would make in your life if you were to take the time to figure out your goals for the future. If you knew how much you wanted to earn in five years, how long you wanted to live, and where you'd like to be in twenty years, what would you do differently?\n\n### Setting goals increases self-esteem\n\n> _High self-esteem seeks the challenge and stimulation of worthwhile and demanding goals. Reaching such goals nurtures good self-esteem. Low self-esteem seeks the safety of the familiar and undemanding. Confining oneself to the familiar and undemanding serves to weaken self- esteem._\n> \n> Nathaniel Branden.\n\nDid you know that you could increase your self-esteem by setting clear goals? It's worth mentioning that having clear goals and achieving them builds and reinforces our self-esteem. In fact, Nathaniel Branden (author of _The Six Pillars Of Self-Esteem_ ) states that part of our self-esteem comes from a \"disposition to experience ourselves as competent to cope with life's challenges\". With every achievement we accomplish, we feel better equipped to deal with other goals and life challenges. In _The Pursuit of Happiness_ David G. Myers shows that high self-esteem is one of the best predictors of personal happiness. Consistently accomplishing the goals you set is one of the most efficient ways to build self-esteem.\n\n### Goal setting changes your reality\n\n> _The value of goals is not in the future they describe, but the change in perception of reality they foster_\n> \n> David Allen, Getting Things Done\n\nSetting goals is a valuable process on its own, regardless of whether or not you'll achieve them. You're probably wondering why that's the case. Well, there are several reasons. Goal setting helps you think about your future, gives you an opportunity to reflect on your values, and helps you discover what matters most. It will bring clarity and allow you to see the bigger picture of your life. It doesn't get much more valuable than that.\n\nSetting goals will also allow you to reconstruct your reality and realize that dreams you previously thought unattainable are in fact achievable. It all starts with identifying your real desires, no matter how ambitious they are. In so doing, you'll begin the process of overcoming your limiting beliefs. Limiting beliefs stem from past experiences and make it harder to get the life you want. You'll soon realize how restrictive limiting beliefs are and just how many of them stem from repetitive messages received from family, friends, and the media.\n\nLastly, goal setting will give you the opportunity to assess your current situation and close the gap between where you are and where you want to be.\n\n### Setting goals is good for your health\n\n> _Use goals to live longer. No medicine in the world \u2013 and your physician will bear this out \u2013 is as powerful in bringing about life as is the desire to do something._\n> \n> David J. Schwartz, The Magic Of Thinking Big.\n\nDan Buettner, the author of _The Blue Zone: Lessons for Living Longer From the People Who've Lived The Longest_ , identified 10 characteristics shared by those who live to 100. He identified \"having a life purpose\" as one of them. Setting goals that fully excite you is one of the best medicines and will work wonders for your health. An alarming number of people die within a few years of retirement. I believe this is partly because they no longer have exciting goals to motivate them, something that is especially likely for those who heavily identified with their job. What about you? Have you found goals that will motivate you well into old age?\n\n### How to set goals\n\n> _The key to goal setting is for you to think on paper. Successful men and women think with a pen in their hands; unsuccessful people do not._\n> \n> Brian Tracy.\n\nYou can learn more at:\n\nhttp:\/\/whatispersonaldevelopment.org\/goalsetting\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Wheels within Wheels\n\n_The Making of a Traveller_\n\nDERVLA MURPHY\n\nIn Memoriam   \nK. R. D.   \nF. J. M.   \nR. A. L.   \nG. C. K. \n\n# Contents\n\nTitle Page  \nDedication\n\nPlates  \nCopyright\n\n#\n\nAt 7.45 on the morning of November 28, 1931, a young woman in the first stage of labour was handed by her husband into Lismore's only hackney-car. The couple were slowly driven east to Cappoquin along a narrow road, in those days potholed and muddy. It was a mild, still, moist morning. During the journey a pale dawn spread over the Blackwater valley, a place as lovely in winter as in summer \u2013 a good place to be born.\n\nThe woman had waist-length chestnut hair, wavy, glossy and thick. Her features were classically regular, her wide-set eyes dark blue, her complexion had never known \u2013 or needed \u2013 cosmetics. She had an athletic build, with shoulders too broad for feminine grace. On the previous day, impatient because the baby was a week late, she had walked fifteen miles.\n\nAt five foot six the husband was no taller than the wife. 'Beauty and the Beast,' his mother had observed objectively when the engagement was announced. But this was unfair. He was well proportioned and muscular, with thick black hair, dark brown eyes, a straight nose, an olive complexion \u2013 not handsome, yet striking enough in a quiet way. He was earnest though not humourless and firm in his convictions to the edge of intolerance. His manner was difficult and as an essentially lonely introvert he found it easier to listen than to talk. Even at the door of the maternity home he had no ready words of encouragement for his wife. 'I'll pray for you,' he assured her solemnly. Then he retreated into the hackney and asked the driver to drop him off at the County Library headquarters in Lismore. It was a Saturday and if he started work at nine o'clock instead of ten he could, with a clear conscience, knock off at twelve instead of one. Characteristically, he did not consider granting himself any compassionate leave in honour of the occasion.\n\nThis was before the Universal Telephone era and at 12.30 Dr White appeared at the library door. An archetypal GP \u2013 florid, white-haired, stately, kind \u2013 he was accustomed to dealing with son-hungry farmers. 'Well now,' he growled, 'I don't know if I should congratulate you or not.' (My father at once visualised some ghastly deformity.) 'It's a daughter you have. Came at a quarter to twelve. Strong child.' (This was also before the days of universal weighing; babies were either strong or weak.)\n\nMy father's reply is not recorded. But as he and my mother had been referring to me as 'Dervla' for the past eight months he perhaps felt no great disappointment. At once he set out to walk the four miles to Cappoquin, carrying a bulky parcel which a more practical man would have put in the hackney-car that morning. It contained the nine records of Beethoven's Choral Symphony. By the time we met I had suffered my first misadventure, a badly burned bottom caused by a burst hot-water bottle. (For more than thirty years the scar faithfully registered severe frosts.)\n\nAs a child, it delighted me to hear my mother describing the celebrations that followed. When a gramophone had been borrowed she and my father settled down to hold hands in the firelit dusk while Beethoven expressed their feelings about parenthood and I, in a cradle beside them, expressed mine about burnt bottoms. At that time childbirth was considered an illness and occasionally a nurse would look in and remark ineffectually that my mother was a _patient_ and should be _resting_. Then Dr White himself arrived to end this unseemly gaiety. I had been lulled to sleep: but the moment Beethoven stopped I started. So the Murphys won that round, when my mother indicated that she found the 'Ode to Joy' a lot more restful than a howling infant.\n\nTwo days later I was christened in Cappoquin's parish church. At first the priest refused to baptise me, insisting peevishly that 'Dervla' was a pagan name and must be changed to something respectably Catholic like Mary or Brigid. My father, however, would not give in. He recalled that a sixth-century St Dervla was reputed to have lived in Co Wexford and that from Ireland the name had spread throughout Europe. Then he carefully explained, to an increasingly impatient curate, that Dearbhail meant True Desire in Gaelic and that the English, French and Latin versions were Dervla, Derval and Dervilla. Finally they compromised; my birth certificate names me as Dervilla Maria.\n\nAlthough my mother's recovery was rapid we were not allowed home until December 12. Then my first journey took me through countryside that had scarcely changed since Thackeray described it in 1842: 'Beyond Cappoquin, the beautiful Blackwater river suddenly opened before us, and driving along it for three miles through some of the most beautiful rich country ever seen, we came to Lismore. Nothing certainly can be more magnificent than this drive. Parks and rocks covered with the grandest foliage; rich handsome seats of gentlemen in the midst of fair lawns and beautiful bright plantations and shrubberies; and at the end, the graceful spire of Lismore church, the prettiest I have seen in or, I think, out of Ireland. Nor in any country that I have visited have I seen a view more noble \u2013 it is too rich and peaceful to be what is called romantic, but lofty, large and _generous_ , if the term may be used; the river and banks as fine as the Rhine; the castle not as large but as noble and picturesque as Warwick. As you pass the bridge, the banks stretch away on either side in amazing verdure, and the castle walks remind one somewhat of the dear old terrace of St Germains, with its groves, and long, grave avenues of trees.'\n\nFrom that bridge it was about a quarter of a mile to my first home on the eastern edge of Lismore. There my parents had rented half a decaying mini-mansion. The other half was occupied by the owner, an obese, elderly, gossipy widow who always smelt of camphorated oil. Her habit of glancing through opened letters, and asking our maid what the Murphys were having for dinner, did not endear her to my mother. At this stage my parents were of enormous interest to the townspeople. And their odd status within the community was greatly to influence my childhood.\n\nForty years ago the Pale was still a psychological reality and my parents therefore ranked as 'foreigners' in Co Waterford. As far back as the genealogical eye could see both their families were of the Dublin bourgeoisie, only rarely diluted by Huguenot, Scots Presbyterian and Italian-Jewish blood. Among their forbears were printers, ironmongers, doctors, linen weavers, civil servants, cabinet-makers, architects, silk-merchants, musicians, soldiers and sailors. There were no priests or nuns on either side that I ever heard of \u2013 unusual in Irish families \u2013 and the only known deviations from the bourgeoisie were an eighteenth-century Earl (of Belvedere) and a nineteenth-century kitchenmaid (of Rathmines).\n\nInevitably, then, my parents were without any ready-made social niche when they migrated south. On one side of a deep rural divide were the gentry and aristocracy, mainly Anglo-Irish and Protestant, and on the other were the farmers and tradesmen, mainly native Irish and Catholic. No true middle class had yet evolved \u2013 we missed out on the Industrial Revolution \u2013 and professional men were usually the sons either of impoverished gentry or of prosperous farmers. Such people tended to retain their inherited attitudes and interests which, on most points, did not coincide with the attitudes and interests of the young couple from Dublin.\n\nWhen my parents arrived in Lismore on their wedding day \u2013 being too poor to afford even a weekend honeymoon \u2013 they found a build-up of suspicions resentment. The previous county librarian had been a popular local figure since the 1870s. He had recently reluctantly retired, leaving nine books fit to be circulated, and the townspeople were furious when an aloof young Dubliner was appointed to replace their beloved Mr Mills. A secure job with a salary of \u00a3250 a year had slipped from the grasp of some deserving local and they smelt political corruption. It mattered not to them that no local was qualified for the job, and what little they knew of my father they disliked. His family was conspicuously Republican \u2013 a black mark, not long after the Civil War, in a predominantly Redmondite town.\n\nDuring that summer my parents often travelled together around the county setting up embryonic branch libraries in villages and rural schools. Sometimes they slept in the back of the small library motor-van to economise on petrol \u2013 thus saving money for the purchase of extra books \u2013 and twice the van was stoned after dark by hostile natives. No doubt wisely, my parents chose to ignore these demonstrations.\n\nMy father's temperamental reserve must have exacerbated the situation. It was impossible to entice him into a pub and this fact alone, in a society which quite often confuses virility with a capacity for strong drink, aroused the scorn of many local males. Teetotalism on religious grounds would of course have been understood, and in some circles admired, but it was soon known that at home my parents drank _wine_ on special occasions, as when entertaining friends from outlandish places like France, Germany or Poland. (They had first met as adults in Poland, where my father was on a cycling tour and my mother on a walking tour. It was then almost twenty years since their last meeting at a children's party in Rathgar.)\n\nAs for my mother \u2013 she smoked oval Turkish cigarettes specially sent from Dublin, and drank China tea, and preferred her cheese to be smelly. Also, she discussed in mixed company such obscenities as breast-feeding, and walked alone for miles all over the countryside like a farmer's wife \u2013 except that she didn't have to \u2013 and instead of saying 'good-evening' like a decent Irishwoman she said 'good-afternoon' like the gentry. Worse still, she had had the misfortune innocently to refer to Lismore as a 'village' within days of her arrival, and this monstrous _faux pas_ \u2013 Lismore has been a cathedral town since the seventh century \u2013 was at once misinterpreted as a typical example of urban condescension.\n\nMy parents' poverty rendered their eccentric bourgeois tastes even less acceptable. Many initially saw them as penniless upstarts who just because they came from Dublin thought they could impress Lismore with their high-falutin' ways. But there was a certain lack of logic here. No newcomers out to impress the natives would have travelled from Dublin on their wedding day in the cab of the lorry that contained all their worldly goods.\n\nThose goods were a large golden collie named Kevin; a solid three-piece chesterfield suite which remains as good as new to this day, apart from superficial damage inflicted by countless generations of cats; a single bed which provoked ribald comments as it was being unloaded but which seems not to have impeded progress since I was born nine and a half months later; two trunks of clothes and blankets; one tea-chest of crockery and saucepans; one cardboard carton of stainless steel cutlery; one tea-chest of records, and a gramophone; twelve tea-chests of books; fourteen handsomely framed Arundel prints and an original surrealistic painting, by a Hindu artist, of the source of the Ganges; one inlaid Benares brass coffee-table and two silver-rimmed Georgian beer tankards; two kitchen chairs and a kitchen table with a loose leg; one round mahogany dining-table and one very heavy black marble clock which suffered internal injuries on the journey and has never gone right since. This last item was a wedding present from our only rich relative, my mother's grand-aunt Harriet. Unluckily grand-aunt Harriet was mad as well as rich and when she died at the age of ninety-eight she left all her thousands to the Archbishop of Dublin.\n\nOn the domestic scene my mother was a cheerfully incompetent bride. Helping my father to pioneer a rural library service was more to her taste than cooking his meals so she engaged an efficient general maid named Nora and, before my birth, devoted most of her energy to working as an unpaid library assistant. Soon the unfriendly natives were being disarmed by the dedication of this young couple to the people of Co Waterford. Also \u2013 to be less sentimental \u2013 in small communities hostility soon wanes if it cannot be seen to be having an effect. And my parents \u2013 deeply in love, enthusiastically absorbed in their new task, willing to be on friendly terms with everybody yet preferring nature-worshipping walks to social gatherings \u2013 were not easy targets for those inclined to attempt ostracism.\n\nA romantic approach to nature was one of the strongest bonds linking a couple who in most ways were utterly unlike. Few Dubliners would then have taken happily to life in the country. Now it is 'trendy' to move out of the city, but fashionable migrants are never completely divorced from urban life; they can and usually do select which amenities they wish to take with them. It was different in the '30s. Lismore is some 140 miles from Dublin and during the early years of their marriage my parents could not afford to run a car or even to pay train fares. There was no electricity in the town and of course there were no television or telephone links with the outside world. There were no theatres, cinemas, concert halls or restaurants within reach. Until they had saved enough to buy a wireless \u2013 one which I still use every day \u2013 they were dependent for entertainment on their modest record collection. And they had no congenial companionship, apart from occasional guests who never stayed long and went away marvelling at the Murphys' capacity for enjoying life against such gruesome odds. But to my parents the odds were not at all gruesome. As Thackeray appreciated, West Waterford is extraordinarily beautiful \u2013 and that made up for all that they had left behind.\n\nTwo miles south of Lismore a wooded ridge \u2013 Ballinaspic \u2013 forms the watershed between the Bride and the Blackwater valleys. Standing on a certain gatepost on Ballinaspic's crest one can survey the whole sweep of West Waterford, and always I feel an intoxication of joy as my eye travels from the coast near Dungarvan to the Cork border near Macollop. There are profound differences between one's responses to familiar and unfamiliar landscapes. The incomparable grandeur of the Himalayas fills me with a mixture of exaltation and humility. But the beauty of the Blackwater valley is so much a part of me that it inspires an absurd pride \u2013 almost as though I had helped to make it, instead of the other way round.\n\nLooking across that fertile valley from Ballinaspic one sees three mountain ranges. The Comeraghs, above the sea to the north-east, seem like the long, casual strokes of some dreamy painter's brush. The Knockmealdowns, directly overlooking Lismore, are gently curved and oddly symmetrical and display as many shades of blue-brown-purple as there are days in the year. And the Galtees \u2013 more distant, to the north-west \u2013 rise angular and stern above the lonely moors of Araglen. Opposite Ballinaspic, another long, heavily wooded ridge separates the lower slopes of the Knockmealdowns from the lushness at river-level and is marked by several deep glens, each contributing a noisy stream to the quiet width of the Blackwater. And south-east of Ballinaspic, amidst a calm glory of ancient woods and irregular little fields, one can glimpse the marriage of the Bride and the Blackwater \u2013 after the latter has abruptly turned south at Cappoquin.\n\nDue north of Lismore a mountain pass forms the letter V against the sky and is known, with un-Irish prosaicness, as the Vee. Less than three hundred years ago wolves were hunted hereabouts and not much more than one hundred years ago evicted peasants were forced to settle on the barren uplands of Ballysaggart. More fortunate settlers arrived in 1832, a group of Cistercian monks who were presented with a mountainside by Sir Richard Keane of Cappoquin. Ten years later Thackeray observed that 'the brethren have cultivated their barren mountain most successfully', and now the grey Abbey of Mount Mellery stands solitary and conspicuous against its background of blue hills \u2013 an echo of those ancient monasteries which once made known, throughout civilised Europe, the name of Lismore.\n\nIn the seventh century St Carthage founded a cathedral and college in Lismore and by the eighth century the place had become a university city where in time both King Alfred the Great and King John (while still Earl of Morton) were to study. In 1173 the 'famous and holy city' was ransacked by Raymond le Gros; and when King John replaced the razed college with a castle it, too, was destroyed. Soon, however, the local bishops had built another castle, which Sir Walter Raleigh eventually acquired. But Sir Walter was not a very competent landowner and in 1602 he gladly sold his castle, surrounded by a little property of 42,000 acres, to Richard Boyle, First Earl of Cork. Some two hundred years later an heiress of the Earl of Cork married a Cavendish and Lismore Castle is still owned by the Devonshire family. Thackeray observed: 'You hear praises of the Duke of Devonshire as a landlord wherever you go among his vast estates: it is a pity that, with such a noble residence as this, and with such a wonderful country round about it, his Grace should not inhabit it more.'\n\nBetween the sixteenth and twentieth centuries West Waterford had to endure less than its share of Ireland's woes. The Villiers-Stuarts of Dromana and the Keanes of Cappoquin always lived on their estates and generally were compassionate landlords \u2013 while the Devonshires, though absentees, were not more than usually unscrupulous. Moreover, a local historian, Canon Power, noted that the region 'seems to have been largely cleared of its original Celtic stock on the conclusion of the Desmond wars and... the first earl of Cork was able to boast that he had \"no Irishe tenant on his land\".'\n\nThis successful mini-plantation may partly explain a scarcity of Republicans in the area. Many local families had not been settled in West Waterford for as long as the main land-owning clans; and in the absence of inherited resentments \u2013 based on racial memories of conquest and land confiscation \u2013 unusually harmonious relations developed between landlords and tenants. But one has to grow up in a place to be aware of these nuances. My parents, looking in from outside, recognised none of the benefits that for centuries had been made available to both sides by West Waterford's feudal system. Judging the rural social scene by urban standards, they saw only arrogance and profiteering on the one side and spineless servility on the other. And nowhere a slot for themselves.\n\nWhat sort of person would I now be had I grown up a typical Dubliner, regarding the countryside as something to be enjoyed in literature and avoided in life? But I simply cannot imagine myself as an urban animal. To me, city-dwellers are The Dispossessed, unfortunates who have been deprived of every creature's right to territory. There is a sense in which country folk, however impoverished, own their birthplace and all the land around it that can be covered in a long day's tramp \u2013 the natural, immemorial limit to the territory of a human being. Or perhaps it is that each region owns its people, exacting a special, subtle loyalty, a primitive devotion that antedates by tens of thousands of years the more contrived emotion of nationalism. Either way, there exists an element of _belonging_ such as surely cannot be replaced or imitated by any relationship, however intense, between the city-dweller and his man-made surroundings. \n\n#\n\nDuring the first year of my life the steep climb up to Ballinaspic was among my mother's favourite walks. ('Sure the creature must be mad entirely to be pushin' a pram up there!') Yet by November 1932 she could push me no further than the Main Street. Suddenly she had been attacked again by that rheumatoid arthritis which had first threatened her at the age of twenty. By my first birthday she could no longer walk without the aid of a stick and by my second she could no longer walk at all. On the 29th of that December she was twenty-six.\n\nThere was of course no cure. But doctors in various countries were doggedly experimenting and, escorted by her favourite brother, my mother went to England, Italy and Czechoslovakia for six months, pretending to hope yet sure, inwardly, that she would never walk again. She spent the whole of 1934 either abroad or in Dublin, leaving me to be looked after by Nora under the vague supervision of my father. In theory this abrupt and inexplicable disappearance of an adored mother, when I was at the crucial age of two years, should have damaged me for life. Perhaps it has, but I am never troubled by the scars. I was by nature adaptable, my routine was unbroken, Nora was devoted and sensible and my father was attentive in his didactic way. (A family legend, possibly apocryphal but very revealing, tells of his bewildered grief when I failed, at the age of two and a half, to assimilate the rules governing the solar system.)\n\nIn December 1934 my mother returned to Lismore as a complete cripple, unable even to walk from the sitting-room to the downstairs lavatory, or to wash or dress herself, or to brush her hair. Between them, my father and the steadfast Nora cared for her and for me.\n\nNow there were major money worries. My mother's search for a cure had cost a great deal and my father was heavily in debt to numerous relatives. Both my parents found this deeply humiliating, innocent though they were of any imprudence or extravagance. My father was almost panic-stricken and it was my mother who calmly took up the challenge. Probably a practical crisis, and the discovery of her own unsuspected ability to manage money, helped her at this stage. She soon began to enjoy pound-stretching; I still have some of the little account books in which she neatly entered every penny spent on food, fuel, clothes, rent and so on. My father then happily returned to his natural money-ignoring state and for the rest of their married life my mother held the purse-strings.\n\nBy this time my parents had realised that they could have no more children, which for devout Roman Catholics meant resigning themselves to an unnaturally restricted marriage. In our sex-centred world, this may seem like the setting for a life-long nightmare. Having been thoroughly addled by popular pseudo-Freudian theories about libidos, repressions and fixations, we tend to forget that human beings are not animals. It would be ridiculous to suggest that the ending of their sexual relationship imposed no strain on my parents, but they certainly found it a lighter burden than we might think. Religious beliefs strong enough to make sexual taboos seem acceptable, as 'God's will', do not have to be merely negative; faith of that quality can generate the fortitude necessary for the contented observance of such taboos. Restrictions of personal liberty are destructive if accepted only through superstitious fear, but to both my parents the obeying of God's laws, as interpreted by the Holy Roman Catholic Church, was part of a rich and vigorous spiritual life. This area of their experience \u2013 I felt later on \u2013 put them in a mental and emotional world remote from my own, where they were equipped with an altogether different set of strengths and weaknesses.\n\nNot long before her death, my mother told me that after getting into bed on their wedding night neither of my parents had known quite what to do next. So they went to sleep. In the 1970s it is hard to believe that two healthy, intelligent human beings, who were very much in love, could have devoted their wedding night exclusively to sleep. But perhaps they were not exceptional among their breed and generation. My mother had been curtly informed by her mother \u2013 who had brone seven children and endured countless miscarriages \u2013 that sexual intercourse was at all times painful and distasteful. And my father would certainly have considered any investigation of the subject, even in theory, to be grossly improper before marriage.\n\nSex apart, an inability to have more children was agonising for someone as intensely maternal as my mother. It also put me, at once, in danger. All the emotion and interest that should have been shared among half-a-dozen became mine only. By the time I was five most people considered me a peculiarly nasty child and mistook the reason why. In fact my mother was such a strict disciplinarian that throughout childhood and adolescence I remained healthily afraid of arousing her anger. But what she could not avoid \u2013 my being the sole object of her maternal concern \u2013 was the encouragement of a ruthless egotism. However, this trait was no doubt useful at the time as insulation against the adult suffering around me. Elizabeth Bowen once wrote, 'Perhaps children are sterner than grown-up people in their refusal to suffer, in their refusal, even, to feel at all.' My mother \u2013 reading _Bowen's Court_ \u2013 once drew my attention to that remark. She did not comment on it, but I have since wondered if she meant it to comfort me. During childhood, I never stopped to sympathise with my parents' situation. Indeed, only when I became a mother myself did I appreciate how my own mother must have felt when she found herself unable to pick me up and hug me, and brush my hair, and tuck me up in bed.\n\nAfter my parents' deaths I came upon the letters they had written to each other, almost daily, during their six-month engagement. On the whole these might have been written by any happy young couple to whom marriage promised nothing but fulfilment. My father hoped to found a model county library service and write novels; with my mother to inspire him he felt certain these must be masterpieces. For relaxation he looked forward to some deep-sea fishing and an expanding record collection. My mother hoped to have six children at two-year intervals (three of each, if possible, though she conceded this might be difficult to arrange) and to use them \u2013 one gathered, reading between the lines \u2013 as guinea-pigs on which to test her various theories about physical and mental health. She also hoped to find time to study in depth, under my father's guidance, the early schisms within the Christian Church \u2013 a subject of ineffable tedium to which she remained addicted all her life. She felt, too, that in her role as county librarian's wife she should initiate a Literary Debating Society (she had not yet visited the town) and perhaps a Music Society. For relaxation she looked forward to walking tours in West Cork and Kerry, presumably on her own while my father deep-sea fished and their systematically increasing offspring were being looked after by some capable Treasure. This correspondence had just one surprising feature. Neither of my penniless parents ever mentioned money, or promotion, or buying a house or a motor car, or in any way planning financially for the future. Both seemed to assume that they would spend the rest of their lives in Lismore \u2013 my father wrote ecstatic descriptions of the surrounding countryside \u2013 and judging by these letters they were utterly without material ambition.\n\nThe few personal memories I retain from my first five years are mostly painful. Our house \u2013 or half-house \u2013 was separated from the road by a six-foot stone wall, sprouting valerian. When attempting to pick a bouquet for my mother at the age of three I fell and broke my nose. A few months later, driving with my father in the library van, I broke it again when he had to brake suddenly because of wandering cattle. I also remember being excited by the exotic springtime glory of the giant rhododendron tree which overshadowed our unkempt lawn. (I have seen none finer in Europe outside Kew, yet it was felled in 1972 because it took up too much space...) Behind the house were a small yard, a large garden and an enormous orchard securely enclosed by ten-foot stone walls. Here my movements were unrestricted and my chief companion was Billy, a rotund black pony who grazed the orchard, gave me rides and pulled us around the countryside in a trap. (At this stage we had no dog; my mother's beloved Kevin had been stolen a few weeks after my birth.)\n\nIn the spring of 1935 it was decided that for character forming purposes I needed 'young friends'. My mother therefore arranged various juvenile social occasions and my most vivid memory from this period is a feeling of fury when other children disrupted the elaborate fantasy-world I had created in the orchard.\n\nFor my fourth birthday party cousins and an imposing cake were imported from Dublin. But at three o'clock, when the local guests began to assemble, I was missing. Nora quickly traced me to a derelict shed, overgrown with briars, at the end of the garden. My detested beribboned party dress of salmon-pink silk \u2013 I can see it still \u2013 was torn and streaked with green mould stains, and my back, in every sense, was to the wall. 'You should be ashamed of yourself!' fumed Nora. 'Stuck out here all mucked up with that _gorgeous_ dress _rooned_ an' your visitors waitin' inside an' even your blood relations down from Dublin!'\n\nMy reply was to become a clan slogan. 'I don't want any bloody relations,' I replied succinctly. 'I'm staying here.'\n\nNora, it seems, was familiar with this impasse. Compromising desperately, she assured me that if I consented to grace the party with my presence I need not wear 'party clothes'; whereupon I meekly trotted indoors. One changes very little. I still dislike 'party clothes'.\n\nIn March 1936 our obese landlady died and my mother and I moved to Dublin for six months while my father was house-hunting. I stayed in turn with my mother and maternal grandmother and with my father's parents.\n\nMy mother's mother was known as Jeff to her face, for some entirely obscure reason, and as The Battle-axe behind her back for reasons not at all obscure. She was an exceedingly disagreeable women who spoiled me methodically by way of tormenting my mother. As I was not allowed sweets she offered me sweets at every hour of the day and night and was piqued when I spat them out because they felt and tasted unfamiliar. As my normal bedtime was six o'clock she reorganised the household to prevent my getting to bed before eight o'clock. As comics were frowned upon, lest they might impair my tender literary taste-buds (a paternal directive, this), she bought me a daily comic. And so it went on, a spiteful campaign in which I was the unwitting weapon and her daughter the helpless victim.\n\nMy mother dealt with the situation by telling me, 'Different people have different views. While we are staying here we must respect your grandmother's views.' Thus she evaded direct condemnation while making it clear that on our return home the usual disciplines would be reimposed. For a parent who values consistency there is nothing more provoking than the deliberate undermining of a child's r\u00e9gime. Yet my mother's self-control never cracked. No doubt this restraint further incensed Jeff, who enjoyed nothing more than a good fish-wifely brawl.\n\nMuch as I relished Jeff's spoiling I was never quite at ease in that semi-detached red-brick Victorian house. My mother's parents had lived in it all their troubled married life and it had bad vibes. Family opinion blamed Jeff for the fact that her husband \u2013 handsome, charming and warm-hearted \u2013 was an alcoholic; but this may have been unfair. She can have done nothing to help him control his drinking, but it is doubtful if even the happiest marriage could have saved him from the bottle.\n\nThere was an amount of instability in my grandfather's background. His own father \u2013 the son of a senior civil servant at Dublin Castle \u2013 had fallen in love with the kitchenmaid, got a chamber pot thrown at his head when the betrothal was announced and soon after emigrated to America with his unacceptable bride and the statutory shilling. Three years and three children later the young couple returned to Dublin, my great-grandfather having found the American way of life insufficiently civilised. For the rest of his life he practised civilisation by drinking too much port and collecting coins while his wife \u2013 an energetic and courageous woman \u2013 ran an Academy for (very) Young Ladies. As she had been illiterate on her wedding day her husband perhaps deserves some credit for having taken the trouble to teach her how to read and write. Mercifully, Providence spared her any more children after the return to Dublin.\n\nAt the age of fourteen my grandfather had to find a job and with wild illogic his father objected to his working as a messenger-boy for a firm of silk importers. It was perfectly in order for a wife to work eighteen hours a day to support an idle husband, but for a son and heir to run errands \u2013 no! Unthinkable! This son and heir, however, did not intend to run errands for long. On his twenty-first birthday, after a two-year training course in Lyons, he was made assistant-manager of the Dublin branch of his firm. And three years later he was managing-director \u2013 a good catch, then, for my unendowed grandmother.\n\nThe trouble on her side was religion. Her mother, the daughter of a Scots Presbyterian cotton magnate, had come to Dublin on a holiday, fallen in love with a dashing Roman Catholic cabinet-maker \u2013 presumably when he was working in her host's house as they would scarcely have been moving in the same social circle \u2013 and urged him to emigrate to Edinburgh. This he devotedly did and a clandestine courtship ended with an elopement \u2013 in a snowstorm, it is said, but I suspect this of being a period embellishment.\n\nMy great-grandmother's dowry would have been substantial had she married suitably and no doubt adequate had she married a cabinet-maker of her own faith. As it was, the statutory shilling again had to suffice though she did not herself become a Catholic. Sadly, her disinterested love was ill-rewarded; at the age of thirty-four she was left a widow with seven young children. When she despairingly contacted her family they offered financial help on condition that her four Catholic sons be brought up as Presbyterians. She herself must have been a convinced Presbyterian or she would have adopted her beloved husband's faith. But on their wedding day they had agreed, as was then the custom, to bring up their sons as Catholics and their daughters as Presbyterians. She therefore declined her family's offer of help and set up as a sempstress to provide for her children.\n\nDespite their aura of smelling-salts, Victorian women were a gallant lot \u2013 often widowed young, never helped by the state, without formal training yet indomitably resourceful when obliged to support innumerable children. In this case three boys and a girl died young of the tuberculosis that had killed their father. But my great-grandmother survived into her eighties and was affectionately remembered by my mother as a tall, thin, gracious old lady with an ineradicable Scottish accent. Regularly on Sunday afternoons she read the bible to her restive Catholic grandchildren before providing lavish teas at which they could eat their fill of home-made shortbread and oatcakes and honey.\n\nMy mother and her siblings were not often allowed to eat their fill. Even when my grandfather was earning \u00a33,000 a year \u2013 which in those far days meant affluence \u2013 my grandmother obsessively rationed the children's food. Avaricious and covetous by nature, she could not forgive her own mother for having put honour before wealth. (I have always uncharitably suspected her of becoming a Catholic simply to spite my loyally Presbyterian great-grandmother.) And her sadistic withholding of plentiful food from hungry children was probably a way of taking her revenge on her husband for his generosity (admittedly not always prudent) towards less well-off friends and relatives.\n\nJeff had a tight-lipped aversion to gaiety and pleasure, however innocent. As a very small child I became aware of her total lack of humour though I could not then have defined what so often made me uncomfortable in her presence. I also became aware of the animosity she felt towards my mother, which possibly explains an odd little incident which occurred in the autumn of 1936.\n\nJeff wore a wig, having been afflicted by total baldness as a girl, and though it was a most superior wig it did not deceive me. Sitting on her lap one afternoon I looked into the garden and saw her black Persian cat \u2013 strangely named Zog, after the King of Albania \u2013 stalking a bird under an apple tree. Then suddenly I knew that I was about to be very wicked. I remember thinking that there should be a choice \u2013 surely I need not be wicked \u2013 yet the compulsion to hurt Jeff was so overwhelming that it seemed to leave no choice. I turned and touched the wig and observed, 'That's not real hair.' And as I spoke I was so appalled by my cruelty that I began to shake all over.\n\nBoth my parents were present, sitting side by side in the background, and at once I glanced towards them. Yes, they had heard. My father looked grieved and my mother wrathful as she murmured something in his ear. Obediently he stood up, carried me out to the hall, reprimanded me sorrowfully and smacked me once very gently on the bottom. This was my first and last domestic experience of corporal punishment \u2013 if that is the right term for a chastisement that was virtually indistinguishable from a caress.\n\nWhen we returned to the sitting-room Zog had caught his prey and was devouring it on the window-sill. Jeff cuddled and consoled me exaggeratedly while upbraiding my parents for treating their little innocent so harshly. I had never before heard grown-ups disagreeing openly about child-control and was fascinated by the new vistas of adult fallibility thus opened up.\n\nNot long after this I nearly joined the angels by drinking half a bottle of neat whiskey. Had I gone on a similar binge in Lismore I would almost certainly have died. As it was, an ambulance rushed me to hospital where I recovered with a speed that ominously foreshadowed an infinite capacity for strong liquor. But the most significant aspect of my adventure was lost on me at the time. I had come upon this half-bottle not in the sideboard, amongst the genteel decanters of sherry and port, but at the bottom of my grandmother's wardrobe.\n\nGiven Jeff's views on sex, as transmitted to my mother, she must have been an insipid bed-mate. It is therefore not surprising that my grandfather  quietly maintained a second establishment in Paris, where his business took him with convenient regularity. Not long before his death he told my mother that five children had been born of this union \u2013 obviously his true marriage, in every sense but the legal.\n\nAt the age of forty-nine my grandfather was 'requested to resign' because of his hard drinking. My mother then had to return from Munich, where she had just begun to train as a singer. Her two elder brothers had already completed their education, but the younger boys had to be transferred from their Jesuit college to a free day-school. Then the Ford had to be sold, the servants dismissed, the telephone disconnected and the silver pawned. My grandfather could afford to go to Paris only very occasionally \u2013 when his mistress sent him the fare \u2013 and he was continually exposed to his wife's contempt. A proud man, he felt his degradation keenly. No one then thought of alcoholism as a disease and he was very conscious of being despised. He drank even harder and borrowed more and more frantically. Meanwhile the family was being kept by his port-sodden father, who had recently inherited a comfortable income from one of the sisters amongst whom his patrimony had been divided when he embraced his kitchenmaid.\n\nMy grandfather was declared bankrupt two years after his dismissal \u2013 or 'resignation', as the nicer members of his family chose to call it. A year later his father died and on the way back from the funeral he paused to light his pipe, sat unsteadily on a wall by the Royal Canal, toppled over, struck his head on a stone and was drowned in eighteen inches of water. Very strangely, for the spot was nowhere near their home, my mother chanced to be passing and witnessed his body being lifted onto the pavement. He was already dead. But for the first ghastly, absurd moment what most upset her was the stench of the slimy weeds that clung to the corpse's impeccable morning-suit. Even when bankrupt her father had remained something of a dandy.\n\nMy mother and he had been exceptionally close \u2013 hence Jeff 's dislike of her daughter \u2013 and I heard him spoken of so often during my childhood that I now find it hard to believe I never knew him. Obviously father and daughter were very alike though luckily my mother's penchant for gracious living was tempered by an awareness that one cannot live both honestly and graciously on a few hundred pounds a year.\n\nFor both our sakes, my mother preferred me to spend more time with my paternal grandparents than with herself and Jeff. A fifteen-minute tram-ride took me from Kenilworth Park to Charleston Avenue so I could still see my mother almost daily. But though the spatial distance between the two households was slight the spiritual distance was vast.\n\nAt Charleston Avenue there was poverty, too, but it was happy-go-lucky rather than gloomy and self-pitying. The house was shabby, dark, damp and cramped \u2013 yet comfortable. It might have seemed less cramped had there been fewer tottering piles of books on every flat surface. The uninitiated sometimes hinted that a week spent tidily arranging these volumes would make life less inconvenient \u2013 not to say perilous \u2013 for all concerned. But the initiated knew that the volumes were already arranged to my grandfather's satisfaction and that from amidst the seeming chaos he could at a moment's notice produce any required work, whether on the Birds of Patagonia, the History of Printing in North Africa or the Bogotrid Sect of tenth-century Bulgaria. I remember that all ornithological tomes were stacked high on the first four steps of the second flight of stairs. Lord Brougham (Collected Works of) towered beside the lavatory and countless volumes of scriptural commentary had to be removed from the bed in the spare room before I could lie down.\n\nI loved that spare room. Narrow corridors between piles of desiccated books led to the two beds and even before I could read I got high on the pungent mustiness of ancient volumes. On summer evenings I used to slip out of bed and move cautiously about the room, picking up and stroking and glancing through volume after volume, pleased if I found illustrations but not bored if I didn't. To see and touch and smell those books filled me with content, with feelings of joy and security and richness. And also \u2013 even at four and a half \u2013 with ambition.\n\nI perfectly understood my grandfather's triumph when, at the end of a long day in the second-hand bookshops on the quays, he came trudging home hidden behind a pile of bargains. This, as far as he and I were concerned, was what life was all about. Therefore I have always relished The Tale of Pappa's Trousers.\n\nOnce upon a time Granny gave her spouse enough cash to buy himself a very necessary pair of everyday trousers. (For obvious reasons he was not normally entrusted with such large sums of money.) Wearing his Sunday suit, because he had nothing else fit to wear, he set off for wherever the cheapest men's clothing was to be had. But unluckily his route took him onto the quays and there he chanced to notice the ten-volume 1840 edition of Sismondi's _Histoire des R\u00e9publiques Italiennes,_ elegantly bound and without a blemish. It cost considerably more than he had in his pocket, but he judged it to be a bargain and acted with a decisiveness that had it been otherwise directed might have made him a rich man. Nearby was a second-hand clothes shop where he quickly flogged his Sunday suit and bought threadbare trousers for a few shillings. The substantial balance, added to his original allowance, just about paid for Sismondi and his tram fare home. He arrived at Charleston Avenue in a state of advanced euphoria. But as he was also in his shirt-sleeves, and very nearly indecently exposed, it is not surprising that his wife failed to appreciate the _Histoire des R\u00e9publiques Italiennes_ in ten vols.\n\nNot that my grandmother could afford to criticise Pappa's obsession: her own was no less uncontrollable and, at least to me, a good deal less understandable. She played bridge, almost literally without ceasing. Naturally I found this a bore, yet I do not remember regarding as abnormal the fact that she and her cronies ate, drank and slept according to the fall of the cards. They might be retiring as I crept downstairs at 7 a.m. to play beneath the overgrown laurel bushes in the back garden. Or they might be in full cry at lunchtime, in which case Pappa and I would quietly settle down to a snack of bread and very ripe Stilton. (So ripe that it had been bought at half-price.) It was not uncommon for the cronies to sleep on divans in the sitting-room, lest their departures and returns might waste time.\n\nWho cooked and washed up? (it was evident that nobody cleaned). I can recall no maid or daily, yet neither do I remember ever going hungry. However, the permanent state of the dining-table proved the subordinate r\u00f4le played by food in this household. It was a large table and my memory is of one sordid, crumby corner grudgingly left clear of books and journals and sheaves of notes written in Pappa's tiny, precise hand.\n\nPappa \u2013 when not delivering philosophy lectures at University College Dublin \u2013 was generally understood to be writing A Book. Its subject, however, was never disclosed. My guess would be that he started several books on diverse subjects and finished none of them for lack of mental stamina \u2013 or possibly for lack of physical stamina. As a captured Old IRA volunteer, he had been on hunger strike in England for six weeks during 1918, in a bid to have his status as a political prisoner recognised, and this effort at the age of forty-eight had permanently damaged his health. During the same period his wife had also been 'inside', as a leader of the women volunteers, Cumann na mBan, and a boundless nationalistic fervour was the couple's only obvious common trait. Yet they were very happy together, each amiably tolerating the other's foibles, and they gave my mother a taste of easy-going affection such as she had never enjoyed in her own home.\n\nThe flavour of that affection is well conveyed by the letter which Pappa wrote to my father for his twenty-first birthday, which was celebrated in Bedford jail. (My father had been sentenced to three years for concealing weapons and ammunition in his back garden.)\n\n_18 Garville Ave._   \n_Rathgar, Dublin  \n15 Dec. 1921_\n\nA Fearguis, a Mic mo Croide \u2013 Many very happy returns of your birthday, my darling boy! Fondest love and congratulations and thousands of kisses from your mother. May God bless and protect and inspire you: may he fill your heart with his wisdom, his love and his comfort. Connie, Kathleen, Conn and Auntie join with us, your loving parents, in wishing you joy on your coming of age and hoping you will have a long and happy life.\n\nI can scarcely realise that on tomorrow it will be twenty-one years from that joyous Sunday noon when I first heard your infant voice and held you as a tender babe in my arms \u2013 my first little son! Well, thank God, your conduct and character since that happy morning have never caused me a moment's anxiety \u2013 on the contrary, my love and trust and hope and pride in you have increased from year to year and today these thoughts and feelings are a source of the greatest joy and thankfulness. I think it right to tell you this, so that you may read it on the day when you are standing on the threshold of young manhood.\n\nAs I write this, An Dail is sitting to determine the most momentous question which had ever been considered by an assembly of Irishmen \u2013 shall the Treaty be ratified or not. I tried to put the best face on it for you in my last letter, but we cannot conceal the fact that it was a profound disappointment to most of us; and the more we look at it, the less we like it. It has one very big advantage and two very decided drawbacks. In the first place it provides for the withdrawal of the army of occupation from four-fifths of Ireland and enables us to set up our own army of defence, at least 40,000 strong, in the district thus evacuated \u2013 all this is to the good. But, we have not got a Republic and an absolutely sovereign state for four-fifths of Ireland \u2013 and we have got nothing at all for Ireland as a whole. We have not got a united Ireland \u2013 the Treaty recognises and sanctions partition. The President, Cathal Brugha and Austin Stack are opposed to ratification and many regard the Treaty as tantamount to a betrayal.\n\nIt is uncertain therefore whether it will be ratified or not; or whether there will be an appeal to the country. Until ratification takes place, it is unlikely that the sentenced political prisoners will be released and consequently you may have to eat your Christmas dinner yet again in an English prison. But keep a stout heart. No matter what happens our position is greatly improved. It is difficult to know which to pray for \u2013 rejection or ratification; there is so much to be said both ways now. You must, accordingly, abate the ardour of your first enthusiasm: mine has cooled very rapidly.\n\nPerhaps you had best spend a week in London and have a look round. You may not get the chance again for a long time. It will be interesting in the future to look back upon what London was like when it was the capital of a big empire. Time is sure to bring many changes and in a few decades London may be a very different place.\n\nYou might let me have a list of the things supplied to you by Wallace as I want to check the account he has sent in. Order from him whatever you want \u2013 food, tobacco or cigarettes: it is much quicker and handier than sending them by parcel post. Don't hesitate to send for whatever you require. The books you requested last month are being sent today; your letter asking for them arrived after I had left for Rome. I hope that you will not have much time to read them in prison. God knows you must be heartsick of prison life, and with the prospect of a fourth consecutive Christmas in prison before you you cannot feel too cheerful. However, Nil Desperandum!\n\nWe expect Connie home tomorrow or Thursday; she didn't like her latest prison at all. We have a friend of hers staying with us now who was her cell-mate for a long time in the NDU. We had a letter from Conn yesterday, dated 3 Dec. and quite cheery. He seems to be in good form and is confidently awaiting his release before Xmas. He says he enjoyed your last letter.\n\nI got a pretty bad cold on my return from sunny Italy* but I am now recovered. I can arrange, later on, to meet you in Holyhead on your return journey.\n\nWith fondest love and heartiest wishes for a Happy Birthday, Your affectionate\n\nPappa.\n\nThe earlier letter about the Treaty, in which Pappa had 'tried to put the best face on it', was written on December 8, 1921, and began:\n\nMay God save and prosper the Free State of Ireland! You have no doubt heard the good news that a settlement has been agreed to between the English and Irish nations. It is not quite all that we had hoped it would be, but it is very good indeed and very much better than anything which seemed within the bounds of possibility a few years ago. And we have to thank you and the comrades with whom you fought for the magnitude of the victory. We have not succeeded in establishing an Irish Republic for a united Ireland, it is true. But we have got the real substance and can afford to wait a bit for the name. The sovereign independent Free State of Ireland will be in actual existence within a month and you will be coming back to live in that Free State which your courage and sacrifices have helped to create. So rejoice and be exceedingly glad. Let no disappointment cloud your joy or diminish your legitimate pride in what has been accomplished. On your return I shall explain fully to you the real value \u2013 as distinct from the paper value \u2013 of what we have won. I believe that the convicts will be released as soon as the Treaty has been ratified by An Dail and the English Parliament and that will be, probably, within a week.\n\nI have asked Mamma to send you on the biggest portmanteau in the house. Cord up all your books, make sure they are fastened securely and bring them all home safely.\n\nWhat had caused Pappa to change his attitude towards the Treaty so radically within exactly one week? 'Putting the best face on it' for a son in jail is not a sufficient explanation. Did the real 'substance' referred to on December 8 prove after all insubstantial? And was it in some way connected with the negotiations which Pappa had been conducting with the Vatican on behalf of the Provisional Government of the Irish Republic? I have been able to find no answer to this question in the vast accumulation of letters and papers now in my possession.\n\nThe convicts were not released before Christmas and on December 23 Pappa wrote to his 'beloved son':\n\nThe Dail has not been able to make up its mind about the Treaty and last evening decided by 77 votes to 44 to adjourn its consideration of the matter to 3 Jan. So this means an end to all our hopes of having you home for Christmas. Poor Feargus! It is too bad, indeed; but never mind, my dear son, there is a good time coming and this is, I hope, the last big trial of the kind which the providence of God will permit to be inflicted on you. So keep up your heart and laugh English prisons to scorn. We were disgusted to hear that two extra days have been imposed on you for something which happened in Mountjoy before your transfer to Wormwood Scrubs. This is typical of English meanness, but let it pass. I was going to bring the matter to the attention of the IRA liaison officer, but on second thoughts I decided not to!\n\nThe handling of the position created by the Treaty did no credit to the Dail: they have spent too much time discussing the pros and cons of theoretical points instead of addressing themselves to practical affairs. However, no human institution is perfect and perhaps we expected too much of our representatives. Opinion is settling steadily in favour of ratification and, although nobody is madly in love with the Treaty, there seems to be nothing else to do in the present circumstances but to accept it.\n\nFour days later Pappa was writing again:\n\nIt will be fine to have you home for New Year's Day! You will see the first sun of 1922 shining on the slopes of Kilmashogue and lighting up the top of the Three Rock and flecking with light and shadow the little trout stream out of which you so often fished a breakfast for yourself and Kathleen. And a good deep draught of our sweet Irish air will soon make the blood again run freely and warmly in your veins.\n\nWe are going down to Malahide this evening to see Cathal Brugha: he is greatly upset about the Treaty and thinks it ought not to be accepted on any account. Things are looking particularly ugly just at present. A double crooked game is being played against us and I should not be surprised if the Conference broke up this week. _The Irish Bulletin_ got hold of a secret circular issued by the RIC Divisional Commander in Belfast organising a secret Orange army \u2013 and has published it! Of course the Cabinet will repudiate it but there it is and very significant, too. In addition the British Govt. is hastily pushing on the Partition Act arrangements to give Belfast full powers and in order to do so is committing all sorts of illegalities and riding roughshod over its own precious Act. So we are by no means within sight of a settlement yet. 'Ulster' has now been paraded on stage and the marionettes wave their wooden arms and shout 'No surrender!' in the approved Orange style. Of course in this case the benevolent English Govt., anxious to do its best to effect a settlement, finds all its efforts thwarted by the irreconcilable quarrels between Irishmen themselves. What can you do with a hopeless people like this except keep a tight grip on them to prevent their killing each other? If Irishmen could only come to some agreement among themselves and tell the British exactly what they do want \u2013 there would be no trouble at all. Everything that a united Ireland asked for would be granted on the spot! And so the old game goes merrily on \u2013 just for the moment. But the curtain is about to be rung down on this tragi-comedy and to be rung up on a different style of drama presenting some very novel features.\n\n[Alas! Pappa was wrong there: contemporary Northern Ireland proves _how_ wrong. The equivalent of his next paragraph has appeared with monotonous frequency in Irish newspapers during the past decade.]\n\nThere has been great trouble in all the internment camps since the truce. Conditions grow worse instead of better and the boys are having a rotten time. In Ballykinlar especially the British have gone out of their way to be nasty. They shot Alderman Tadgh Barry the other day just as he was waving goodbye to some released internees. A sentry pretended to think that Tadgh was trying to escape and shot him dead. There was great grief in Cork and great indignation everywhere. Friday and Saturday saw a very impressive funeral, the cort\u00e8ge coming from Ballykinlar by motor and being received by Volunteers and big crowds on the way down to Dublin.\n\nWe had a great day at the National University on Saturday when we installed President de Valera as Chancellor. It was a splendid turn-out with the Chancellor himself of course the most striking figure. He wore a robe of black velvet with rich gold trimmings and looked like both a ruler and a scholar; his tall figure and his thin ascetic face were fittingly set off by his magnificent robes. We gave him a great reception \u2013 and no mistaking its significance! What an extraordinary revolutionary change has the National seen! Who would have dreamed, a few short years ago, that an unknown and despised BA would on Saturday have been installed with great pomp and ceremony as Head of the entire university! Time! thou bringest mighty revenges! The IRA furnished a guard of honour and Kathleen, as Captain of the University Company of Cumann na mBan, mustered her sixty-five hefty cailin who made an impressive display. She also read an address in Irish to the Chancellor, who replied in Irish. Except for one speech, all the proceedings were conducted either in Latin or in Irish. The doctors' gowns with their various colours 'brightened up the scene' wonderfully. I was resplendent in a scarlet gown with maroon sleeves and a maroon hood lined with green silk. Next to the Chancellor's, it was the best robe in the show!\n\nEverybody is looking forward to having you home, including Bob and Whiskers who are purring beside me at the thought. I hope you got the portmanteau and that you will find it big enough for all your books.\n\nFondest love from your own affectionate\n\nPappa.\n\nThroughout his time in prison my father had received from Pappa as many letters as were allowable, some covering more than twenty foolscap pages and few less than ten. Each included a detailed account of the latest political developments and an assessment of how Dubliners of every shade of opinion were reacting to them. Pappa also occasionally reminded his son that 'the average Englishman is a decent enough fellow' \u2013 from which rather unexpected comment one may deduce that my father's letters (now lost) were betraying that obsessional anti-English bias which he was never to outgrow. Pappa was too kindly to condemn any race completely; therefore his hatred of British rule in Ireland led him to draw a not very convincing distinction between 'the average decent Englishman and the governing mind \u2013 for that is the only mind that finds expression in the country's corporate institutions \u2013 and it is a strange mixture of Saxon dullness and Norman cruelty'. So much for his faith in British democracy.\n\nThese letters give a wonderfully vivid picture of contemporary life in Dublin. Repeatedly 18 Garville Avenue was raided, sometimes when the house was empty, and once the British troops helped themselves to \u00a35 10s. 0 _d_. which Pappa had imprudently left in a wallet in his bedroom. He got no satisfaction when he wrote to the CO complaining that the maid and the gardener could not be paid that week unless the money were restored. This was literally true \u2013 not a sob-story \u2013 and after the gardener had left in a huff there are some wry references to Mamma's unsuccessful attempts to replace his labours with Pappa's. Soon after, Maria, the maid, also left to get married and was replaced by 'Mrs Bruagh's maid, Agnes, who plays the melodeon beautifully and often entertains us in the drawing-room. She remembers you well: she says you are a nice young gentleman and not a bit proud as you spoke to her kindly one day when she was out with the Bruagh children.' So much for attitudes towards servants sixty years ago.\n\nThe burning of the Customs House and the various reactions of Dubliners to that event are graphically described; Pappa's own reactions were \u2013 as he admitted \u2013 very mixed. Then on June 1 he writes:\n\nIn Dublin now it is positively dangerous to walk the streets. Not a day passes without four or five ambushes taking place. You are walking down say, Nassau St, when suddenly you hear a terrific explosion followed by a volley of rifle shots; you look round and see people rushing into shops, others lying flat on the ground, others running up the side streets! Trams tear along madly; horses gallop away from the cab-stands, women shriek, bullets whistle round your ears, bomb splinters are flying, wounded people lie about groaning \u2013 'oh! what a lovely war!' \u2013 and for a quarter of an hour you have a lively time of it. Scenes like this occur every day here in all quarters of the city. A girl was shot dead in Trinity College Park last week during a cricket match; and two men who were sitting on a wall at Clontarf were shot at and died a few hours afterwards. The why or the wherefore of these latter happenings never appears, but all sorts of rumours are flying round \u2013 many of them of the most contradictory kind. It is easy enough to see the _raison d'\u00eatre_ of the ambushes, but the other occurrences are terrifying and mysterious. Twice this week the tram on which I was travelling was held up by the English and all passengers (male) searched. On last Sat. at the corner of Harrington, Camden and Richmond Streets you could see four long lines of trams held up for searching purposes while groups of soldiers occupied the streets and stopped all vehicles \u2013 rifles at the ready and bayonets fixed, while officers nervously brandished revolvers and held them under the noses of all and sundry including women and children. The searching would make a cat laugh \u2013 I could have had half a dozen revolvers and a few bombs on me without the slightest risk of detection. And all that is achieved by this ferocious display is to delay and irritate everybody, to dislocate traffic and business and to call down curses \u2013 not loud but deep \u2013 on the stupid military. But the situation is not without its humorous side: it is delightful to listen to the former red-white-and-blue people \u2013 the bigoted Unionists \u2013 the erstwhile 'God Save the Kingers', expressing their views on the present r\u00e9gime. My word! haven't they changed! To hear them would do the heart of any Sinn Feiner good!\n\nHowever, normal life continued too, as it does today in Belfast, and most of Pappa's letters were devoted to descriptions of family outings, new plays, art exhibitions, bridge marathons, poker parties, long hikes in the Dublin and Wicklow mountains, cricket matches, croquet contests on long summer evenings, moonlight bathing parties at Greystones and gossip about friends and neighbours. The family news mainly concerned Conn, my father's ne'er-do-well younger brother, who when not in jail for political reasons was a constant source of anxiety lest he might end up there for non-political reasons. And there were many speculations about Kathleen's many admirers \u2013 which she should retain for further consideration and which she should discard without delay.\n\nSome news items have a very modern ring: 'The strike of the Rathmines Council workmen is still on and we are without light and without 'bin-men'. As for the first we don't miss it for we have daylight till 11; but not being able to get rid of ashes and house rubbish is a bit of a nuisance. And the fun of the thing is that the dispute about hours is settled and the strike is being continued solely for the wages which were not paid during the time of the strike: workmen now want to be paid even for striking.'\n\nDespite Pappa's horror of Partition, his references to the North all indicate that even in 1921 the average Dubliner felt it to be an alien place. On May 2 he wrote:\n\nI was in Belfast on April 20 and 21 lecturing on 'Ancient Irish and Ancient Greek Education'. I had little opportunity to find out anything as I know practically nobody there, but I heard one important item of information from a 'big business' source \u2013 over 60% of those employed in the linen trade are out of work, the American trade has almost entirely ceased, there are big stocks on hand which can find no purchasers though they are being offered at prices slightly lower than the present cost of production. The boycott is telling very markedly. Otherwise the town seemed to me just as it was when I last visited it five or six years ago. It seems to have learned nothing and to have forgotten nothing. For instance I saw two lorry-loads of young fellows apparently returning from an excursion \u2013 probably factory hands \u2013 each lorry had a Union Jack floating over it; the Union Jack floated too over that monstrosity in architecture known as the City Hall. King William on a white horse crossing the Boyne is still their beau ideal and to shout 'To Hell With the Pope' and to stone the 'bloody papishes' is still the chief duty of a 'loyal' Belfast citizen. They still live in the Ireland of 15 years ago and are unaware of the avalanche which is about to descend on them.\n\nBelfast is an uninteresting place \u2013 it has only one fine street, the rest being either monotonous replicas of rows of workmen's cottages or dingy terraces of a would-be suburbia. The energy which I noticed on my last visit was replaced by a good deal of listlessness \u2013 owing to the ubiquity of the out-of-works, I suppose. Anyway I was very glad to get back to good old Dublin. I may have to go up again in a week or a fortnight \u2013 but I hope not.\n\nClearly Pappa enjoyed letter-writing. A midsummer day's solitary walking and trout-fishing in the familiar Wicklow mountains could spark off a thousand-word lyrical description of the landscape, the birds and the ever-changing Irish sky. And a sunny autumn afternoon spent strolling with Mamma around Lucan and the Leixlip demesne inspired at least another thousand words. Until reading these letters I had not fully appreciated how easy it was for Dubliners of that period to enjoy as much of country life as was desired. For a keen hiker, miles of untouched countryside were within walking distance of Rathgar \u2013 as was the sea, for a keen swimmer like Pappa.\n\nIn the autumn of 1922 my father left for Paris, to begin his studies at the Sorbonne, and he spent the next seven years in France. Money was so scarce that he only rarely returned home; most of his vacations were spent tutoring the two sons of a Russian \u00e9migr\u00e9 duke in the South of France. Unfortunately no correspondence has survived from those years \u2013 until 1928, when on July 9 Pappa wrote a long letter, liberally scattered with quotations from Julius Caesar and Shakespeare, in response to my father's decision to become a Benedictine monk. In the end he expresses no opinion but concludes, tantalisingly, 'I have a hundred things to say but it is just post-time so I shall wait till tomorrow.' Reading between the lines, however, one discerns disapproval. And there is an unwonted acidity in the last paragraph \u2013 'Fondest love from your mother. Send her a little note for herself \u2013 why have I to suggest this? Does your love for her not prompt it?'\n\nIf Pappa at once doubted the genuineness of his son's vocation he was quite right. The next letter to have survived was written only nine months later, on March 20, 1929, in response to my father's decision to marry a nineteen-year-old Swedish girl who was studying at the Sorbonne under the eagle eye of an aunt-chaperon. My father appears to have detested this aunt even more than he detested the English \u2013 and with good reason. She threatened to call in the police after her niece had spent a day at Versailles \u2013 without permission \u2013 in the company of a penniless Irishman.\n\nThis romance provoked Pappa to write a full-blown Victorian homily:\n\nI must say that my first thoughts and feelings were made up of almost contradictory elements. In the first place there was a sort of disappointment: I had thought that your whole soul was so decidedly fixed on a monastic life that nothing could have diverted you from fulfilling a purpose which you had adopted, apparently, with such deliberation and determination. I had built up a scheme of thoughts for myself founded on _that_ as on a first principle. And then you drop a bombshell \u2013 a living one, aetat 19 \u2013 into my beautifully constructed building and blow the whole thing to smithereens in a second! But you did it in such an airy and unconcerned fashion that I haven't the heart to reproach you. Well, perhaps it is the best thing that ever happened. And perhaps it isn't: time alone can tell. I sincerely hope it is; I earnestly hope that it may bring you deep and lasting happiness. All the fond love of your parents' hearts goes out to you and we shall have nothing but the warmest welcome for the girl of your choice.\n\nYour mother was not nearly so astonished at the news as I was; and I am sure that in her inmost soul she was delighted at the thought that you were saved from the ever-grasping arms of religious communities!\n\nOne thing I am really glad of; and that is that you made the discovery of the possibility of falling in love before you had taken any decisive step towards ordination.\n\nBelieve me it is a great thing to be honestly and deeply in love; it lifts the whole soul to a higher level; and if the person loved is a good woman, then it is the noblest passion which can animate the soul of man. Love your sweetheart, then, with all the intensity of your soul \u2013 if she is a good girl and returns your love, you cannot love her too much. Pour out your affection without stint; and when she becomes your wife, wrap the whole warmth of your love so closely around her that she can never feel the cold breath of the world no matter how bitterly the winds of adversity may blow. But let your love be intelligent and unselfish. A good wife is the greatest blessing God can bestow on any man \u2013 a pearl beyond price; but that pearl must be cherished and safeguarded at all costs. There are two sorts of love \u2013 a selfish and an unselfish one. The first seeks to make the beloved minister to one's own good: the second seeks the good of the beloved before all else.\n\nAnd now to come down to earth. It is all very well to fall in love \u2013 but what about the future? You cannot ask anyone to marry you until you are able to provide a decent life for her \u2013 I don't mean affluence nor even an easy life, nor one devoid of struggle, or even, at times, of anxiety; but a reasonable prospect of the necessary things \u2013 a sufficiency of nourishing food, comfortable housing and warm, befitting clothing. I don't know whether you have come to any understanding with the girl, whether you are engaged or not. You have merely said you intended to marry her soon. To my mind, it is all too sudden and recent for any official engagement. It takes a certain amount of time for the growth and ripening of real love. But, if you are in earnest, you must set about making a living: you must have a definite realisable plan and you must follow it out steadily. What do you propose doing?\n\nLove should purge you of a large element of that selfishness which clings to you. You are too apt to let absorption in your own intellectual concerns cause you to forget the position of those who love you dearly. For instance, you did not trouble to acknowledge the receipt of this month's allowance. And that's only one incident. This is not a reproach, but a reminder.\n\nWrite soon \u2013 and more fully. Your affectionate\n\nPappa.\n\nWhat happened next? Did the formidable aunt win? (I cannot imagine my father \u2013 however much in love \u2013 withstanding such a female for long.) Or, when the novelty had worn off, did the nineteen-year-old bombshell lose interest in her Irish suitor \u2013 so shy, impractical and inexperienced in the arts of love? Or did my father take fright, upon reading Pappa's homily, and decide that the responsibilities of marriage would prove too taxing? Whatever happened, he was home for Christmas that year \u2013 unmarried and unemployed \u2013 and he never returned to France. In January 1930 he began a six-month Library Diploma course at University College Dublin, in April he met my mother again (they had last met three years previously and had known each other as children), in July he was appointed County Librarian for Waterford and in August he and my mother announced their engagement.\n\nAt that time Ireland's Civil War was not long over and the families of Dublin were still angrily arrayed on either side of an ugly barrier. How ugly may be gauged by a remark made on July 29, 1927, when my father's brother Conn wrote from Ontario: 'Congratulations on getting your finals. I am quite enjoying the experience of sailing on the Great Lakes. I saw in the papers here that Kevin O'Higgins has been shot dead \u2013 damn near time \u2013 the sooner they shoot a few more like him the better. The Canadian papers described him as a martyr but they had to admit he was the best hated man in Ireland.' Kevin O'Higgins was one of the finest Irishmen of his generation \u2013 but he was a Free Stater, and the Murphys were Republicans.\n\nMy parents' engagement therefore represented a considerable m\u00e9salliance, between the son of a rabidly Republican family and the daughter of a mildly Unionist family. But to give my paternal grandparents their due, they saw the point of the marriage within moments of being introduced to my mother. By then she \u2013 being totally apolitical \u2013 had cheerfully adopted a diluted form of Republicanism to meet the situation.\n\nIn general, however, the two families were never more than distantly polite. To my father's family, my mother's relations were not only politically corrupt but barbarously unlearned, hard-drinking, irreligious, foppish and extravagant. To my mother's family, my father's relations were not only politically irresponsible but feckless, bigoted, prudish and riddled with intellectual pretensions that never came to anything. Happily these prejudices left me unaffected. I grew up fond of both families, unquestioningly accepting their covert mutual hostility as a fact of Irish life.\n\n*Pappa had been in Rome as Ambassador to the Vatican from the Government of the Irish Republic. \n\n#\n\nIn November 1936 my father at last found a house to rent at a price we could afford. It was on the South Mall, Lismore's most respectable street, but the dwelling itself was so irreparably decrepit that no modern squatter would stay there overnight. Short of a leaking roof, it suffered from every defect buildings are heir to and, for the next twenty-one years, it decayed \u2013 usually quietly, but occasionally dramatically \u2013 about our ears. Dating from the 1820s, it was two-storeyed, semi-detached and covered in Virginia creeper. The fanlight and wooden porch were attractive, a pair of romantic stone urns graced the front garden and overgrown fuchsia-bushes billowed on either side of the hall door. The well-proportioned rooms had good marble mantelpieces and mock-Adam ceilings and the wide hall was tiled in cream and dull red \u2013 pleasant, old-fashioned, indestructible tiles. However, some past tenant with execrable taste had left the whole place superficially hideous. The hall was painted a dead laurel green, only relieved by irregular patches of yellow-grey mildew where the plaster had fallen off. (For years I was fascinated by those patches, seeing them as maps of undiscovered countries.) The staircase was covered with cracked red and blue linoleum which ill-matched the magnificent mahogany banisters. Upstairs were five rooms: three large bedrooms, a boxroom which became my playroom and another large room, complete with fireplace, which at some remote period had been converted into a bathroom. The bath stood on four gigantic iron lion's paws and resembled a modern child's swimming-pool. It was patriotically stained green and orange and had a shower-device, of considerable antiquarian interest, near the ceiling. This had become viciously perverted and it sprayed, with tremendous force, only onto the opposite wall. When my father had forgotten to warn three successive guests he put up a notice saying 'Please do not touch'. The lavatory also had its notice, to explain that the chain needed three morse-like pulls: long-short-long. The wash-basin could almost have been used as a bath and was without a plug: apparently none to fit it had been manufactured since the turn of the century. Had my father exerted himself he could, at the cost of a few pence, have remedied this and many other defects. But the idea of personally improvising a washbasin plug \u2013 or anything else \u2013 would never have entered his mind and he judged our numerous discomforts too trivial to warrant expensive expert attention.\n\nThroughout the house we found peeling beige woodwork and wallpaper that had faded to a uniform grey-brown. Everywhere the paper was coming unstuck and in the dining-room rats had eaten through it at several points, thus demonstrating the fragility of the basic structure. Dry-rot afflicted the floor boards and some other sort of rot caused the ground-floor ceilings to snow gently if anyone walked about too vigorously overhead. This perhaps explains why I have always moved rather lightly for one of my build.\n\nAt the end of the hall a semi-glazed door led to a narrow, dark, flagged passage with ominously bulging henna-distempered walls. Having passed a storeroom, a pantry and a larder one entered the kitchen. Here sly draughts sneaked up from damp non-foundations through gaps between ancient flags, and blatant draughts whined through the slits between rotting window frames and rattling panes. The roughly plastered walls were an evil shade of green and a temperamental coal-range stood in an alcove. A row of discoloured pewter bells hung high above the door; in our day these never responded to the relevant buttons being pushed but they emitted ghostly chimes when gales blew. A dozen iron hooks depended from the rafters \u2013 'The better to hang yourself on, my dear,' observed my mother as she toured her new home. In one corner a steep ladder-stairs led through a trap-door to an attic where the servants would have slept in the Bad Old Days. An adult could stand upright only in the middle of the attic floor and this retreat soon became one of my Private Paradises.\n\nBehind the house were several collapsing stables and, beyond a wide cobbled yard, stood Lismore's recently opened cinema, the property of our landlord, who lived next door. It was enormous and no one could tell us what purpose it had originally served; it may have been a series of barns whose internal walls had been demolished. Mercifully our landlord did not prosper as a film-wallah and within a few years the local doctor had built a new 'Palladium'. Then the old cinema became another of my Private Paradises; in semi-darkness I leapt from row to row of moth-eaten red plush seats, being pursued by imaginary cannibals and collecting swarms of real fleas. These were not found tolerable by my mother, even when identified by me as rare tropical insects picked up while exploring in New Guinea.\n\nBeyond the cinema were our garden and orchard, half an acre of wilderness which, despite consistent neglect, provided us for many years with an abundance of loganberries, gooseberries, apples and pears. At intervals my mother would remark on the advantages of growing one's own vegetables. Then my father would borrow some implements and might on the following Saturday be observed reclining beside a minute pile of cut brambles reading Plato's _Theaetetus_ or the latest Dorothy Sayers. Like myself, he lacked the urge to cultivate. Our genes have perhaps resisted change since the Age of the Gatherers.\n\nAlthough our new home was very nearly a ruin we tolerated it for the next twenty-one years. My mother must have abhorred these slum-like surroundings but she refrained, as always, from complaining about the inevitable. For a rent of ten shillings a week one couldn't, even in Lismore in the 1930s, expect very much.\n\nThe rent was so low not only because of the house's dilapidation but because of the previous tenant's suicide in the dining-room. This snag considerably influenced my destiny since it made it far harder to engage local maidservants, or to persuade those who came from a distance to remain in residence. It was not that any ghost operated \u2013 at least to our knowledge \u2013 but the neighbourhood vociferously believed that a suicide without a consequent haunting was against nature.\n\nAs a child I always knew there was nothing to spare for non-essentials. But I was never hungry or cold so it did not occur to me to interpret this condition as poverty. Nor did I ever long for the unobtainable, with one spectacular exception \u2013 a pony of my own. And since that desire so clearly belonged to the realm of fantasy it caused me no discontent. In Dublin I enjoyed the luxury toys of my cousins \u2013 rocking-horses, tricycles, pedal motor cars and the like \u2013 yet I never asked or even wished for such things. They belonged to another sort of person who lived in another sort of world. And it was not a world I should have cared to inhabit permanently. It had no rivers, fields, woods, moors and mountains.\n\nWhen we moved to the South Mall Nora was replaced by Old Brigid, a formidable character who for the next three years \u2013 scornful of ghosts \u2013 impassively controlled the whole peculiar Murphy establishment. It cannot have been easy to contend with a disintegrating house, an invalid mistress, a chronically vague master and a nasty child. Old Brigid, however, took the lot in her slow, purposeful stride. When the foul-looking sink came adrift from the wall she said nothing to my mother but fetched the plumber, a man who normally took weeks to answer any summons but who meekly accompanied Old Brigid to the scene of the disaster. When my mother needed some attention as lunch was being prepared the attention was promptly provided but the meal was never late or ill-cooked. When my father wandered off to the Library one morning wearing his dressing-gown and slippers Old Brigid pursued him, looking reproving but resigned, and handed him his jacket and shoes halfway down the Main Street. When I staged a tantrum because I could not have everything exactly as and when I wanted it Old Brigid said, 'Now, Miss Dervla, I'll have no more of that nonsense \u2013 _if_ you please!' And the tantrum stopped.\n\nDespite her surface severity \u2013 or because of it? \u2013 I loved Old Brigid dearly. She always wore an ankle-length blue and white check cotton dress and a large starched white linen apron, without spot or stain. Every afternoon, while boiling the tea kettle, she also heated a ponderous iron on the hot coals, carefully placed it on its tin tray and ironed the next day's aprons. She was small and stout, with grey hair in a neat bun and shiny red cheeks and sharp bristles on her chin. In 1936 she was sixty-five and had been fifty-three years in the service of a Tipperary family whose last representative had left her an adequate annuity; but she found idleness uncongenial. Since we paid her two pounds a month she must have regarded the Murphys as a hobby.\n\nEvery morning Old Brigid bathed me at seven o'clock because the range idiosyncratically refused to provide hot water in the evening. Then she took me into the dark airing cupboard, which was considerably larger than the average modern bathroom, and told me fairy stories while drying me beside the gurgling, gleaming bulk of a gigantic copper boiler. I listened politely, concealing my bored disbelief. I had faith in only one fairy, Mr Dumbly-Doo, who was exactly my own height and wore silver boots and red leather breeches and a green silk shirt and a black velvet jacket and a gold brocade tricorn hat. A creation of my father \u2013 with acknowledgements to the leprechaun industry \u2013 Mr Dumbly-Doo occasionally left a mint-new penny under a certain stone beside a certain stile along a certain laneway. (I cherished these coins for their red-gold rather than for their purchasing power \u2013 though in those days that was considerable.) He did none of the exasperating things common to fairies in stories and since my father did not elaborate on his life style I was free to do so myself without feeling the victim of adult condescension.\n\nThis wary attitude towards fairy tales was part of my unremitting struggle against grown-up power. Despite the affectionate understanding provided by my parents, in their very different ways, I tensely suspected the adult world of some sinister conspiracy to make me conform. I could not have felt more fiercely about this had my parents been models of conservatism instead of the individualists they were.\n\nYet for all my rejection of the standard fairy tales I needed a fantasy escape hatch even more than most children do. So I created my own intricate world of magic animals and omnipotent teddy-bears. A family of the latter, comprising four generations, lived in the branches of my favourite tree \u2013 a superb elm, some 120 feet tall and reckoned to be more than 400 years old. Under that tree I spent countless hours, at all seasons, totally absorbed in the bears' doings and in their dramatic personality clashes. Each one had a clearly defined character and in time they came to seem quite independent of my controlling imagination. For a few years they \u2013 and their tree \u2013 meant more to me than any human friend.\n\nThat elm grew (and mercifully still grows) in the dense, dim wood which rises steeply from the Blackwater just west of Lismore Castle. The path leading down to it was an exciting tunnel through thick undergrowth. All around the other trees were old and tall, though dwarfed by its prodigious girth and height. Long before I had ever heard of pantheists, druids or sacred groves I used to stand at the foot of this elm, pressing with outstretched arms against its vastness, fingering its rough bark and looking up in reverence at the endless ramifications of its mighty branches. I was never to feel anything comparable under the influence of orthodox religious stimulants. But does it matter how we worship, if we worship?\n\nAll this of course took place only after I had been given licence to roam alone, at the age of seven. But long before that my chief amusement was telling myself interminable convoluted stories \u2013 if 'amusement' is the right word. The longing to be alone with the denizens of my imagination was so intense, and the amount of time I devoted to them so abnormal, that one of my father's sisters \u2013 a child psychologist \u2013 became seriously alarmed during a visit to Lismore.\n\nNo doubt there was something neurotic about my elated relief as I escaped to the garden or the attic, and about the anger I felt when interrupted by the necessity to eat, or go for a walk, or learn my lessons. I often looked forward to bedtime. Lying happily taut under the blankets, with my eyes shut and my imaginative throttles wide open, I was at last safe from adult interference. I well remember the physical symptoms of excitement during those sessions: my heart hammering, my fists clenching and unclenching, my face contorted as I rapidly muttered the latest instalment, _sotto voce_. No wonder my aunt, who had doubtless contrived to spy on a daytime session, felt concerned.\n\nMy mother, however, insisted that I was suffering from nothing more than a lively imagination. On principle she tended _not_ to agree with her sister-in-law, who was very close to my father. And in this case she may have realised that my fantasy-world was a not unhealthy form of escapism. At some level I must have been aware of the domestic stresses and strains; and futile efforts to understand and adjust to them would have done me much more harm than my withdrawal into the company of golden calves, silver goats and arboreal teddy-bears.\n\nAs I seem always to have known the facts of life I assume they were simply absorbed from my mother during that phase of obsessional questioning when everything in nature arouses a child's curiosity. I therefore find it hard to understand the difficulties that even in this explicit age are said to surround basic sex instruction by parents. It is far easier to explain to a three-year-old how babies are made than to explain the processes whereby bread or sugar appear on the table.\n\nBy the age of six I was a proficient and dedicated masturbator and someone \u2013 probably Old Brigid \u2013 had infected me with an acute guilt complex about this hobby. So I consulted my mother, who said the activity in question was certainly not a matter to _worry_ about. It was a babyish habit and quite soon I would grow out of it \u2013 just as I had grown out of wetting my bed. These remarks must have had the intended effect. Guilt evaporated and in time the 'babyish' habit was superseded by more cerebral sexual interests centred on scientific investigations of the male anatomy.\n\nI was about seven when an outraged neighbour complained to my mother that I had been seen, on the public street, removing a little boy's shorts and examining him from every angle. All I can now remember is the colour and texture of this four-year-old's shorts. They had been knitted from coarse burgundy-coloured wool and as he wore no underpants I pitied him, reasoning that he must feel miserably scratchy.\n\nThe fact that this scene took place on the Main Street \u2013 'in broad daylight', as our neighbour several times emphasised, unconsciously implying that had it taken place in a dark corner it would have been less culpable \u2013 the fact that this could have happened shows how well my parents had thus far protected me from Irish puritanism. But there are limits. The time had come to risk unhealthy repression and my mother told me that never again must I do such a thing because little boys are very sensitive to the cold around that area, and could get a bad chill if stripped in the open air. I was not, of course, deceived. I had got the message that the relevant area merited special treatment and indeed was, for some utterly incomprehensible reason, Taboo. This new awareness gave the physiological differences between boys and girls an extra fascination; but my investigations, from now on, were more discreet.\n\nSoon Providence favoured me; newcomers took the house opposite and within hours it became apparent that their eight-year-old son was a professional exhibitionist. He had perfected a variety of ingenious urinating techniques and his penis was public property. We were an ideally suited couple. He performed, I admired, and it occurred to neither of us that his penis could be put to other uses. Almost certainly he was ignorant of the mechanics of reproduction, as he was without curiosity about the female anatomy (he had five sisters). And it would no more have occurred to me to initiate an experiment than to smoke a cigarette. In my mind a clear line was drawn between the activities of grown-ups and children, and for all my defiance I was never tempted to cross this line prematurely. The world was organised in a certain way. There was a pattern and one felt no impulse to disarrange it.\n\nThe South Mall had been skilfully planned. Looking due north from our hall door one saw, scarcely six miles away, the 2,900-foot main peak of the Knockmealdowns, its smooth blue curve rising directly above one of Ireland's loveliest churches. A double line of stately lime trees led up to St Carthage's Cathedral and the broad, grassy sweep between them, known as The Mall, made a safe children's playground.\n\nFour doors down from us, on the same side of the wide street, was a house rather like our own \u2013 but detached and in perfect condition \u2013 which had recently been bought by a family of outsiders who seemed no better than ourselves at integration with the natives. They were, however, devoted to children and during the spring of 1937 they regularly invited me into their garden to play with an exuberant young Airedale named Bran and a sentimental black cocker spaniel named Roddy. The garden covered two acres and almost every afternoon a few members of the Ryan family were to be found working enthusiastically beside the gardener. (Here I first discovered what fun it is to watch other people digging and pruning, mowing and raking.) For a month or so I could not be induced to enter the house, possibly because I was afraid of the hypochondriacal Mr Ryan, who never ventured out before midsummer but could occasionally be glimpsed peering unsmilingly through an upstairs window. Everyone, including his wife, called him 'The Boss' and regarded him with an unwholesome mixture of deference, resentment, concern and scorn.\n\nMr Ryan was a retired country schoolmaster, gruff, autocratic, keen-minded and at this time already in his seventies. Mrs Ryan \u2013 his second wife, much younger than himself \u2013 was gentle and placid with a subtle sense of humour. Beneath her placidity one could detect more positive qualities which if not repressed might, in the circumstances, have led to domestic disharmony. She, too, had been a schoolteacher and the eight children of their union had been brought up mainly by her unmarried sister, who never seemed in the least like a frustrated maiden aunt but had a permanent twinkle in her eye. She smoked secretly \u2013 in the summer-house, to be well out of nose-shot of the Boss \u2013 and gave me all her cigarette cards.\n\nOf the four Ryan sons two* were then curates, one was studying for the priesthood in Rome and the youngest was an army cadet whose buttons I loved to polish. Of the four daughters two were missionary nuns \u2013 educational pioneers in the remoter parts of Nigeria \u2013 and two lived at home. It was taken for granted that the Misses Ryan, though young, attractive and intelligent, would remain unmarried. Their ageing parents needed them and, having given five children to the Church, deserved them. They were never allowed enough freedom to be noticed by eligible men \u2013 though a father who had sired, in all, eleven children, and who could provide little financial security, might have been expected to consider both their emotional and economic needs. But in rural Ireland forty years ago Mr Ryan's despotism was not rare; and it was encouraged by Irish Catholicism, which has always given to involuntary celibacy the status of a virtue.\n\nThe Ryan family had produced several distinguished Gaelic poets and I much preferred their spontaneous 'Irishness' to the Murphys' turgid and embittered nationalism. Yet this comparison was unfair; for generations no Ryan had been directly involved in Irish politics and it is less easy to avoid bitterness when you have spent some of the best years of your life in jail, being treated as a common criminal. But perhaps what really appealed to me about the Ryans' tradition was its genuineness. They had a cultural integrity not often found, for historical reasons, in Dublin families. When my father and his brother were sent to Saint Enda's \u2013 the school founded by Patrick Pearse \u2013 and when the family went to the Donegal Gaeltacht for their summer holidays, to learn Irish, they were searching for something the Ryans had never lost.\n\nIn other ways, however, the Ryans' simplicity irritated me, even at the age of six. Everything was good and bad, right and wrong, black and white; and children who suggested the possible existence of grey areas were just being impertinent. I soon learned to hold my tongue, partly  because it seemed right to conform to the standards of the household and partly because The Boss shared with my mother \u2013 for very different reasons \u2013 the unusual distinction of being able to frighten me. To an extent I probably found the Ryans' authoritarianism reassuring, but sometimes I was driven to secret tears when my rudimentary intellectual probings evoked an altogether unmerited sarcastic reprimand. To this day I remain puzzled by my emotional 'adoption' into this outwardly unyielding family. Clearly the Ryans liked having me about the place to soften the harshness of daily life; I was impulsively affectionate and as a family they conspicuously lacked demonstrativeness. But why did they not cultivate a child less liable to outrage their various susceptibilities and generally more tractable? Perhaps they furtively relished the stimulus of being outraged, or they may simply have enjoyed trying to raise my moral tone.\n\nThe Ryans and my parents never fraternised; whatever they might have in common, their differences far outweighed their similarities. So the relationship stuck at meteorological comment, though for years I spent as much time in the Ryans' house as in my own. Moving daily between two households whose attitudes, opinions and standards were often opposed might have led to some confusion had I been more pliable. But for me this tension was healthy, part of the process of learning to accept other people as they are.\n\nDuring the '30s my parents' only local friends were a Catholic curate, a Fianna Fail senator and the senator's elderly widowed sister, Mrs Mansfield.\n\nFather Power was pompous, smug and plump; though a good deal more intelligent than Jane Austen's Mr Collins there were prominent affinities, including a weakness for titled nobility. Few people in the parish were prepared to talk interminably about his obsession \u2013 Early Christian Ireland \u2013 so he spent many evenings in our house, often bringing a half-bottle of claret and staying to supper. His brother was a wine merchant, but he seemed to imagine that a full bottle would give an air of debauchery to the proceedings.\n\nI much preferred Senator Goulding because he completely ignored me. A bachelor, he was small, slight and energetic, with a calm, precise voice, a dry sense of humour and not a speck of self-importance. When in Lismore \u2013 he was often in Dublin on senatorial business \u2013 he attended Mass and received Communion every morning, and every evening he again went to church, and had there been an afternoon service he would certainly have attended that, too. He was, however, the best sort of devout Irish Catholic, not a craw-thumper but a man who tried to make politics honourable through the practical application of Christian teaching. Having served Ireland for more than half a century he died poorer than he was born.\n\nMrs Mansfield was childless and had been widowed young. One got the impression that she had unaccountably married beneath her and regarded Mr Mansfield's premature death as his one gentlemanly gesture. She lived in a rambling, three-storeyed corner house at the junction of Ferry Lane and the Main Street; and the fact that the ground floor was occupied by a pub \u2013 once the property of her late husband \u2013 was a circumstance so unfortunate that to have referred to it in her presence would have been like commenting on someone's club-foot \u2013 or wig. She and her brother affected to despise each other and had not exchanged a word, at least publicly, within living memory. They might be observed going to church by pointedly different routes: the senator trotting briskly up the South Mall while his sister sedately paced down the Main Street, tiny, slim and erect, the tapping of her silver-mounted cane being made to sound like heralds' trumpets through the sheer force of her personality.\n\nWhen not going to church Mrs Mansfield was invariably accompanied by San Toy, an irascible Peke with chronic asthma. San Toy once attacked a bull-terrier, in a fit of sheer spleen, and the terrier was so astonished he simply ran away. Having witnessed this scene I always deprecated Mrs Mansfield's subsequent boastings about San Toy's gallantry when unjustly set upon.\n\nTwice a week Mrs Mansfield called to drink tea with my mother and deplore the appalling inroads being made by democracy on good manners. She complained bitterly of being greeted with an 'Hello!' \u2013 she whispered the word as though it had four letters \u2013 by children whose parents she could remember walking in to Lismore on a fair-day with bare feet and scarcely a shirt to their backs. Those children of course knew no better; their elders had waxed too prosperous and brazen to teach them respect. But I, Dervla \u2013 she would swivel round to survey me through her lorgnette \u2013 _I_ should know better than to _run_ down the Main Street, endangering in my unseemly haste defenceless babies and feeble old-age pensioners. 'A lady should never be seen to hurry, my dear.'\n\n'But she's not quite a lady yet,' my mother would protest mildly \u2013 avoiding my vulgarly winking eye. Then Mrs Mansfield's expression would convey that if my mother did not act soon and drastically the necessary transmogrification was unlikely ever to take place.\n\nBefore I was old enough to wander alone I often attended Mrs Mansfield and San Toy on their afternoon walks. For one of her apparent fragility and gentility Mrs Mansfield was a stout marcher, not at all deterred \u2013 once out of sight of the neighbours \u2013 by rough going and the accompanying indignities of climbing over fallen tree trunks or crawling under wire fences. And San Toy availed himself of these occasions to prove that he was no mere effete aristocrat. It was Mrs Mansfield who introduced me to the pleasures of strolling through old graveyards, striving to read weathered inscriptions and speculating about the fate of such as John Carney who, in 1811, at the age of fifteen, 'loved peace but died violently'. Of the consequences of this addiction there will be more anon.\n\n* One of those two was Mark, of whom there will be much more anon. \n\n#\n\nOne fine spring morning, when I was six and a half, my father escorted me to the local national school and my formal education began. I remember lying on the chalk-smelling floor boards of a huge, bright classroom, kicking in a tantrum and feeling tears running into my ears and noticing the chalk dust gyrating through a sunbeam. My father was standing over me looking helpless and worried and being assured by a little group of nuns that I would settle down the moment he left me.\n\nI had not any objection, in theory, to starting school, but the moment I entered that classroom I panicked at the prospect of being confined within alien walls until some unknown nun gave me permission to leave. At home I chose to spend hours alone every day, yet if at any moment I suddenly wanted to be with my mother she was always accessible. Here, however, I was trapped in a situation where it would be impossible to reach her no matter how desperate my need. So I screamed and kicked frantically while the other children, who had all started school at the age of four, regarded me with amused scorn, and the nuns, raising their voices above my howls, repeated firmly that I would soon settle down and tried to edge my father tactfully towards the door.\n\nLuckily I could not express my desolate sense of betrayal. Had my father realised that this was not just another bout of nastiness he might well have taken me home again, thereby setting a disastrous precedent. As it was I did settle down surprisingly soon after his reluctant departure, having discovered that I liked the nun who was to be my teacher. She explained that if ever I needed my mother very badly I could at once be sent home in the charge of an older girl. Whereupon I discounted the possibility of ever again needing my mother _very_ badly, either in or out of school hours, and by lunchtime being a scholar seemed a good idea.\n\nYet I did not mean to profit in the accepted manner by my educational opportunities. Indeed, having learned to read at home I felt that the essential part of my education had already been completed. The world was full of books and I intended to read as many as possible before I died. What I did not intend was to waste the best years of my life \u2013 the _only_ years of my youth \u2013 studying inexpressibly boring things like French and arithmetic. I knew by then of my parents' plans: at ten I would be sent away to school, at eighteen I would be sent to the Sorbonne, at twenty-two or twenty-three I would return home with a degree in something or other and be welcomed as a complete, civilised human being. This programme might have fired many children with worthy ambitions, but I neither wished it to be carried out nor believed that it would be. We are born \u2013 I am convinced \u2013 with a certain basic foreknowledge about the pattern of our lives and I always regarded those parental plans as pipe-dreams.\n\nMy teachers found me an awkward, lazy pupil and the educational methods of the day did nothing to help. Another discouragement, for a child without any linguistic ability, was the compulsory use of Irish as the language through which all subjects were taught in free primary schools \u2013 except, significantly, religious instruction. By the 1930s most Irish families had been English-speaking for generations and only a tiny minority were interested in reviving their own language. So this lunatic law was extremely unpopular. It produced millions of Irish citizens who were, as one wit sourly observed, 'illiterate in two languages'. The situation would be paralleled in Britain if Wales reconquered England and compelled all state-school pupils to study in Welsh.\n\nEven as a character-forming influence, Lismore school did me very little good. My classmates, instead of forcing me into the sort of rough and tumble I needed to remove my corners, generally deferred to me and expected me to be their leader \u2013 an expectation which was disappointed, for it was not in my nature either to lead or be led. Also, most of the nuns were too lenient towards me and too openly appreciative of intellectual attainments which would not have seemed at all remarkable in another academic setting.\n\nFortunately there was one exception to this, whose class I joined when I was eight. Sister Andrew was a tall twenty-year-old with a pale long face, straight black brows and eyes that seemed to give off blue forked lightning during her rages. Verbally she flayed me and physically she battered me \u2013 often across the back, with a stout wooden pointer. If it is true that corporal punishment is inflicted only by the insecure, then Sister Andrew repeatedly betrayed her own uncertainty and inexperience. We were in fact using each other at this stage: she to prove that she could control even such a resolutely self-willed and obliquely insolent child as myself, and I to prove that I could and would withstand the adult world, however painful the consequences.\n\nI remember sitting upright at my heavy wooden desk, with its cracked, brass-lidded inkwell and countless carved initials and the splinter under the left side of the seat on which I was wont unobtrusively to clean my finger-nails. Sister Andrew was bending over me, whitely angry, ordering me to write the letter 'h' in the approved manner. I knew quite well how to write a standard 'h' but I was determined not to do it according to the specifications; I had my own method, which I naturally preferred. And so, under Sister Andrew's flashing gaze, I deliberately rewrote 'h' as I thought fit. Meanwhile the rest of the class, who always relished our duels, watched with bated breath. Several emotions simultaneously possessed me in that instant: a spiteful sense of triumph, regret that our duel could not take place in dignified seclusion instead of in the middle of a classroom, fear of the physical pain that I knew was imminent and a sharp stab of shame because I could not but recognise the futility and stupidity of my own behaviour. This was one of the occasions when the pointer left bruises on my back. Since I was able to write a perfectly legible 'h' it might be argued that Sister Andrew should have ignored my method of achieving it; but then it might also be argued that I did not mean my defiant originality to be ignored.\n\nFor a year or so we were sporadically at war. Many were the afternoons when I hurried home, trembling with resentful fury, affronted, humiliated and longing for the balm of maternal sympathy. But these rages usually cooled on the way and I rarely mentioned Sister Andrew. The verbal flayings hurt me far more than the beatings, but I was shrewd enough to realise that if I repeated those criticisms verbatim my mother would simply add, 'Hear! Hear!' Besides, complaints about the school authorities were discouraged at home and many years passed before I discovered how much worry my bruised back had caused on one particular occasion. From the '70s such violence looks primitive and uncouth. But in the '30s even doting parents, themselves too sensitive to hurt a fly, did not really object to having the hell beaten out of their wicked brats by somebody else. (N.B. \u2013 for the past thirty-five years Sister Andrew and I have been very good friends.)\n\nAn interesting aspect of childhood is the democracy of those who have not yet been trained to think or feel undemocratically. And one of the oddest functions of middle-class parents \u2013 which seems inconsistent with the civilising parental mission \u2013 is to destroy this democratic instinct for the sake of maintaining standards often of far less value to society than the attitude being sacrificed. At the age of seven or eight my classmates and I were not prepared to accept the operation of chance as a valid foundation for either superiority or inferiority complexes. The barriers built within the adult social world were unknown to us and differences in speech, manners, attitudes and interests neither embarrassed nor amused us; we simply ignored social frontier posts.\n\nDuring this period my few school friends were wild and ragged. Tommy particularly attracted me because he hated wearing shoes and in all seasons removed his footwear outside the school gate before going home. To me running barefooted symbolised the very quintessence of liberty and I soon became Tommy's only female intimate. With his gang I raided many orchards \u2013 including our own, which was stealthily approached from an adjacent field. I had been admitted to this all-male gang by virtue of my muscle-power. If necessary I could bear any two of these ill-nourished little boys on my shoulders to assist them over an orchard wall.\n\nOn Saturday afternoons I often went to Tommy's home for tea. He lived with his foster-parents in a tiny cottage and we were given hunks of hot, butter-sodden bastable-cake, its crisp crust delicately flavoured with the wood-ash beneath which it had been baked, and huge chipped enamel mugs of very sweet cocoa. This soon became my favourite meal of the week and I was desolated when my mother one day announced that in future Tommy and I must take it in turns to entertain each other. Cocoa made with milk instead of water, egg and cheese and tomatoes instead of bastable-cake... I pleaded desperately, but in vain. 'You _must_ return Tommy's hospitality,' said my mother in her that's-the-end-of-the-matter voice. 'But you may have tea in the kitchen because Tommy might feel shy in the dining-room.'\n\nGloom enveloped me at the thought of Old Brigid scrutinising our hands, vigilantly observing our table-manners and perhaps \u2013 I shuddered at the very possibility \u2013 perhaps even being brutal enough to tell Tommy not to lick his knife. As supervisor of a tea-party she seemed a poor substitute indeed for Tommy's foster-mother, who always reminded us to wipe our filthy hands on our backsides before we sat down and made gloriously comic slurping noises as she drank her tea from her saucer. Then I hit on a brilliant compromise: Tommy and I should have tea in the attic, where adults feared to tread because of dry-rot. My mother \u2013 always ready to view a situation from my angle \u2013 approved of this idea; and since Tommy would have nothing to do with my body-building menu we were given white bread and jam \u2013 for me a very special treat.\n\nAt my eighth birthday party Tommy looked miserable in shiny shoes and his First Holy Communion suit, now giving at the seams. The other guests were South Mall children \u2013 in whom I had very little interest, but they always asked me to their parties \u2013 and years afterwards I heard of the repercussions of Tommy's attendance. A few days later it was conveyed to my mother \u2013 through Mrs Mansfield \u2013 that if the Lynch boy were invited to my Christmas party the rest could be counted out. Of course my mother didn't realise it \u2013 how could she, poor thing, stuck in a wheelchair all day? \u2013 but no one even knew who Tommy's parents were, and he ran barefooted like a tinker, and blew his nose in the gutter, and _stole_ from people! Stole apples, and sold them on fair days to the poor mountainy farmers' wives!\n\nMy mother's reply was that she found Tommy a most attractive, spirited, intelligent little boy. Soon afterwards came Christmas and to my astonished joy I found that this year, for some mysterious reason, I was to be spared party-going and party-giving. But Tommy spent St Stephen's Day with us; he had long since lost his shyness and graduated to the dining-room where he delighted my father by showing a lively interest in astronomy.\n\nThe Lynches left Lismore a few months later, to my great though transient grief. And eventually Tommy, having won a series of scholarships \u2013 a more difficult feat then than now \u2013 went on to become a chemical engineer with a top job in ICI.\n\nThe neighbours quite often found Murphy standards unacceptable. I was seven when three small girls \u2013 sisters \u2013 were forbidden to play with me because I had assured them that every night a lion slunk across the rooftops of Lismore, hunting the crows which nested in the chimneys, roaring at the stars just to show he was very fierce and fighting with an orang-utan who lived in the cathedral belfry. Nightmares resulted and the parish priest received a formal complaint about my pernicious untruthfulness. When Father Power relayed this complaint to my parents they made no attempt to conceal their amusement. But my mother cautioned me against further terrorising of my contemporaries and advised me to write such stories down in future instead of telling them, as it were, in the marketplace.\n\nThus began an enduring custom; every year I wrote long stories for my parents' birthday and Christmas presents. Only one of these survives, written when I was eight. In about three thousand misspelt words it describes the adventures of two boys in a jungle that, judging by the available fauna, extended from Peru to Siberia. Having throttled a sabre-toothed tiger with their bare hands, rescued a shepherd's baby from a condor and killed an anaconda with a poisoned dart my heroes returned to Ireland by an unspecified route and lived happily ever after.\n\nIn a letter to her father-in-law my mother reported that when I was four \u2013 not yet able to read \u2013 I picked up a Little Grey Rabbit book and pointing to the author's name on the title-page said, 'When I'm grown up I'm going to write books and have my name there.' My mother commented, 'I think she means it. She is a very decided and determined child.' This comment was probably regarded as the typical effusion of a doting mother, but it was correct. I did mean it. And I went on meaning it though none of my literary efforts, during childhood or adolescence, showed any trace of promise.\n\nI preferred not to discuss my ambition with anyone \u2013 it was tacitly understood between my parents and myself \u2013 and from the age of about twelve I was well aware that I might follow in my father's and grandfather's footsteps and be a failed writer. But this did not deter me. I was not thinking in terms of success or failure, prosperity or poverty, fame or obscurity. To me writing was not a career but a necessity. And so it remains, though I am now, technically, a professional writer. The strength of this inborn desire to write has always baffled me. It is understandable that the really gifted should feel an overwhelming urge to use their gift; but a strong urge with only a slight gift seems almost a genetic mistake.\n\nMy parents naturally approved of these literary ambitions. Yet to have encouraged me too enthusiastically, in the absence of any discernible talent, would have been irresponsible. Fortunately my mother enjoyed nothing more than being a literary critic. Everything she read was dissected and an ill-written book, endured for the sake of the subject matter, made her look quite haggard. An aspiring writer could ask for no more suitable mother and after a tactful interval \u2013 never look a gift story in the syntax \u2013 my parents' Christmas and birthday presents would reappear to be torn phrase from phrase. During these sessions I sat beside my mother like the most docile of Victorian daughters, attentively absorbing her every suggestion. This was the one area in which I did not spurn adult advice.\n\nIn May 1938 we acquired a motor car. I cannot imagine by what financial conjuring feat my mother managed that, but somehow a brand-new, dark green, four-door Ford Ten appeared \u2013 having been paid for before delivery, eccentric as that may seem to a generation reared on hire-purchase systems. Motor cars were then a rarity in Lismore, yet ours was no extravagance; for my mother train-travel would have been intolerably troublesome and painful.\n\nAlmost every Saturday and Sunday, in all seasons, we motored to the sea. ('Driving' seems to have replaced 'motoring' only after the second war.) My mother loved to sit in the car for hours with the windows wide open, parked if possible within reach of the flying spray; she was especially partial to midwinter gales. Meanwhile my father would march briskly along beaches or cliff-tops while I did my own thing among rock-pools or sand-dunes. While Tommy was around he usually came with us.\n\nOn Saturdays we often paused in Dungarvan, a market-town on the coast, to shop at Miller's old-fashioned Select Family Grocers. Mr Miller, beaming and bowing, would hurry out to the car to discuss with my mother cheeses and rices and teas and coffees while an assistant, wearing a half-moon leather apron, hovered in the background ceremoniously bearing samples on a round brass tray. These my mother would closely examine before committing herself to the expenditure of ten or twelve shillings. I sometimes think of this scene as I push my sweaty way through a seething supermarket crowd, beneath the glare of strip-lighting, between cliffs of gaudy packages.\n\nOccasionally Mrs Mansfield accompanied us to Dungarvan, but she never shopped at Miller's and I soon realised that she disapproved of our doing so. When I questioned my mother about this she looked embarrassed \u2013 an uncharacteristic reaction to any question of mine \u2013 and at once I knew that I had touched on one of the more sordid areas of adult misbehaviour. But she would, I was confident, tell me the truth. Then she said, 'Mrs Mansfield doesn't believe in supporting Protestant shops.'\n\nMomentarily I suspected a red herring. 'But that's daft!' I objected. 'What does a shop's religion matter?'\n\n'It doesn't matter at all,' said my mother, 'but many Irish people think it does.'\n\nSuddenly various pennies began to drop; now I understood obscure remarks that had been made in my presence by the Ryans, or by the nuns at school, or by Old Brigid or Tommy's foster-mother. I felt an overwhelming, angry disgust \u2013 perhaps my first step away from orthodox religion. There was no logical reason why I should have recoiled so violently, at the age of eight, from an attitude that had long been accepted as normal by the majority of my compatriots, both Catholic and Protestant. Obviously I comprehended none of the practical or philosophical implications. I only knew that this was something contemptible, which should be opposed.\n\nNext morning I reopened the subject at breakfast-time. I was sitting on a stool by the fire in my parents' bedroom, eating grapefruit, and my mother was still in bed, propped against a pile of pillows, her heavy, glossy, chestnut braids coiled on a pale green bed-jacket and the bible open on her bookrest. (She began each day with a chapter from the New Testament.) Overnight my agitation about religious bigotry in general had been aggravated by my particular grief about the Ryans and Mrs Mansfield and Old Brigid \u2013 people to whom I was deeply attached \u2013 being guilty of this evil. I felt threatened. What to make of a world in which some of my most respected adults had been suddenly revealed as wrong-doers? So I asked my mother, 'Why can't somebody teach people not to be bigots?'\n\n'It's difficult,' she explained, 'because bigotry is self-perpetuating.' (She never believed in tempering her vocabulary to the unlearned lamb.) 'Bigots are so sure they're right they don't even try to see any other point of view. But it's wrong to blame individuals for being bigoted \u2013 that's just another form of bigotry and sometimes it's worse because it's so self-righteous while pretending not to be. Usually people inherit bigotry \u2013 it's a sort of communal disease. So if you hear other children repeating nonsense about all Protestants going to hell, and so forth, don't lose your temper. Just try to make them realise that such beliefs are unchristian and stupid. Then at least \u2013 even if they don't believe you \u2013 they'll be aware there's another point of view.'\n\nI have never forgotten this conversation. My mother's response to my first serious querying of a widely accepted attitude had shown that in certain circumstances nonconformism was not merely allowable but desirable. At once my anxiety evaporated and I was full of happy self-importance, seeing my parents and myself as crusaders against the forces of evil.\n\nI cannot recall many conversations or scenes from my first decade, but the few memories I have are very vivid, possibly because they mark emotional or mental advances.\n\nOne raw November noon, not long after my discovery of bigotry, I stood waiting for my grandfather outside Trinity College and as I watched the crowds hurrying across College Green I wondered, 'How important are all those people? Soon they will all be dead. And I'll be dead. What is a person? Do we really matter? Or do we only think we do? Why are we alive?' As they formed in my mind I was aware of the enormity of these questions. And I remember a detached, fatalistic acceptance of the fact that even as an adult I was unlikely to find coherent answers. But I was also aware that merely by asking such questions I had acknowledged the mystery at the centre of things and so perhaps had already found as much of the truth as was necessary for me. Oddly, I made no attempt to try to relate these speculations to the formal religion which was so much a part of my life at that time. They belonged to a category labelled 'Private Important Thoughts' \u2013 Pits for short \u2013 and nothing would have induced me to discuss them with any grown-up. Does every child have, as I had, an image of what adults expect children to be? And do they all courteously preserve this image, outwardly, lest their adults might be discomfited, while inwardly they are becoming something quite different, full of Pits that have nothing to do with Dr Dolittle or stringing conkers? But perhaps there is something more than courtesy behind the dissembling reticence of childhood. A personality is forming, and loving adults are eager to help mould it \u2013 while the child is determined to remain in control of his own evolution. Also, nothing is ready to be exposed. Most artists dislike having their incomplete work considered and discussed and this analogy, I think, is valid. The child is incomplete, too, and is constantly experimenting as he seeks his own style of thought and feeling. And all this is going on long before puberty, at an age when many children are expected still to believe in Santa Claus.\n\nNot long after my ninth birthday my grandfather and I had a brief, curiously moving encounter in the sitting-room at Charleston Avenue. My grandmother had died of cancer the year before, but the battleground of the bridge warriors still reeked nostalgically of cigarette smoke, coffee grounds and sherry fumes. It was cluttered with an assortment of very bad and very good furniture, severely ravaged by moth, woodworm and dust, and over the mantelpiece hung the arms of the Leinster Murphys \u2013 four Lions Rampant and three Sheaves of Corn. The blue, scarlet and gold had been dimmed by wood-smoke from the inefficient grate, but the motto was still legible: _Fortis et Hospitalis_. Pappa came upon me one day when I was standing on the hearth in the empty sitting-room gazing up at those lions. And he asked me, 'Do you know what the motto means?'\n\n'That we should be Strong and Hospitable,' I replied.\n\n'Yes \u2013 but physical strength and material generosity are easy. To have strength and generosity of the mind and the spirit is what matters. We must all have our faults \u2013 but never be weak or mean.' Then suddenly Pappa looked comically startled; he was too humble and humorous to be able to lecture even a grandchild unselfconsciously. For a moment he stood staring through the window, twirling an end of his moustache and humming a snatch from _The Gondoliers_. Before I could think of anything to say he had left the room.\n\nOn our monthly weekend visits to Dublin we usually left for home after Sunday tea. In those days the journey took six hours \u2013 it now takes three \u2013 and being prone to car-sickness I preferred night travel which allowed me to cheat my disability by sleeping. Often I woke only next morning, to find myself in my own bed.\n\nOnce, however, I woke as we crossed the Vee \u2013 and experienced pure terror for the first time. Directly above the car a dancing wall of orange and scarlet flame was obviously about to engulf us. This gorse-fire had of course been deliberately started and was not even remotely dangerous. But as we chugged up the steep slope the flames spread swiftly, filling with their terrible beauty the blackness of the night. Fear paralysed me, until a turn in the road made it seem that we were about to plunge into the crackling, glowing, writhing heart of the inferno. Then I shrieked and clutched my father's shoulder \u2013 thus putting us in real danger. Even when my mother had calmed me a residue of fear remained and on future crossings of the Vee I both dreaded and longed for the superb fury of a burning mountain.\n\nThat same winter I inconsistently failed to panic even slightly when there was ample excuse for hysteria. During a very frosty spell my father had put a rug under the car bonnet \u2013 and then, typically, forgotten to remove it. Halfway up the Vee I happened to notice, through a gap in the floor beside the foot-brake, an ominously flickering brightness. I stared at it for a moment, fully aware of the implications, and then said in a carefully calm voice, 'I think we'd better stop. The engine seems to be on fire.'\n\nMy parents, who were no doubt discussing some medieval heresy or other, took no immediate notice. Then, 'What did you say?' asked my mother.\n\n'I said the engine seems to be on fire.' And I remember adding sourly, 'So we'll all be blown up if you don't stop talking about religion and put it out.'\n\nAn instant later my father was hurling the burning rug onto the roadside. For the next five hours he and my mother hardly exchanged a word on any subject, which I well knew presaged a monumental row as soon as I was out of earshot.\n\nAt about this time I was suddenly afflicted by an irrational terror of darkness. Electricity had not yet come to Lismore; at dusk the lamplighter went up and down the South Mall and Old Brigid lit the oil lamps and closed and barred the shutters and drew the curtains and, when I had been put to bed at seven o'clock, darkness was, officially, final. However, for reading illicitly I had a secret supply of candles and matches and my new terror was revealed to myself when I forgot, one evening, to smuggle these in from my playroom. The playroom door was a mere three steps away, across the landing, yet I felt sick with fear at the thought of venturing out into the total, silent blackness. Inevitably I then saw myself as a most despicable coward, a craven sissy, a lily-livered, weak-kneed, spineless rotter.\n\nThis terror quickly became a phobia that dominated all my waking hours. Sometimes I was tempted to confide in my mother, but pride inhibited me. As the days passed my dread of bedtime increased; this had become my regular test of courage and always I failed it. At breakfast-time I might have successfully persuaded myself that, that very evening, I would do what had to be done. Yet when the moment came, when Old Brigid and my father had said good-night and kissed me and gone, I simply lay listening to the mealtime noises in the dining-room below while little shivers of shame ran through my body. Night after night, I told myself that within minutes this torment could be ended \u2013 if I found the necessary courage.\n\nThen one very cold evening I did find it. I slipped out of bed, tip-toed to the door and began a deliberately slow tour of the whole of the first floor \u2013 including the airing cupboard and attic, which to many seemed quite spooky even by daylight. I moved silently through the dense blackness, my hands outstretched to guide me, and the thudding of my heart seemed to hurt my ribs. Something odd happened to my sense of time and the ordeal seemed to be taking place outside the normal framework of hours and minutes. But it was worth it. When I got back to my bedroom my self-respect was restored and all fear of darkness exorcised forever.\n\nOn most issues, at this period, I did confide in my mother. Yet already my attitude towards her was habitually guarded; while half of me needed comfort and guidance the other half was suspicious of interference. From my father I had inherited a certain shyness or gaucherie or tendency towards self-effacement \u2013 to this day I am not sure of the exact nature of the trait \u2013 and this was aggravated by observation of a woman who always seemed at ease in every sort of situation. Unwittingly, my mother gave me an inferiority complex I was never to outgrow. I recognised and took for granted the fact that in looks, intelligence and poise she set a standard I could never hope to reach and for years I heroine-worshipped her. Yet I may also have unconsciously envied her capacity to inspire such devotion \u2013 not only in myself but in many others. 'A magnetic personality' is the stock phrase. And as a child I expended a disproportionate amount of energy on testing myself against the power of that magnet. \n\n#\n\nOne February morning in 1939 the possibility of Adventure appeared on our humdrum horizon. A letter from a German friend, who had shared university digs with my father in Paris, invited us to stay with him near Heidelberg for the month of August. And Pappa, who normally spent his summers with us, was invited too. A delirium of excitement seized me. Long train and sea journeys \u2013 I had never been on a train in my life \u2013 incomprehensible languages, foreign coins for spending not for collecting, strange foods, unfamiliar customs, weird clothes (Germany seemed to me as remote and exotic as Inner Mongolia) \u2013 in a word, Real Travel.\n\nThen I saw that my parents, though themselves quite excited in their staid way, were hesitant about accepting Anton's invitation. For several years he had been campaigning relentlessly against Hitler and in my father's opinion 'things were getting much worse'. I had no idea what those things might be, or why their deterioration should effect our holiday plans, and I rejoiced to hear my mother observing that it would be very interesting to see it for oneself. I jigged up and down in my chair, unable to eat my boiled egg, and begged them to say 'yes we are going to Germany for our holidays'. As I had never known the parental word to be broken it seemed that if only they could be persuaded to make this simple statement no power on earth could prevent our going to Germany. But instead they said, 'It's too soon to make plans. Let's say we'll go to Germany in August if by then it still seems a good idea.'\n\nAs I went bounding off to school, my mother was already working out with pencil and paper how best to save our fare money during the next six months.\n\nDuring this spring I was being prepared for my First Confession and First Holy Communion. These ceremonies represent an initiation rite of immense solemnity. According to Roman Catholic theology, a child attains the use of reason at the age of seven and from then on is capable of committing sins, both mortal and venial. Mortal sins, which 'kill' the soul by depriving it of sanctifying grace \u2013 a sort of spiritual oxygen \u2013 must be confessed to a priest before God will forgive them. The seven-year-old is also considered responsible enough to take Holy Communion, even though this sacrament is believed to involve actually absorbing into one's own body the flesh and blood of God himself. Confession and Communion go together \u2013 it is sacrilegious to receive Communion with a mortal sin on one's soul \u2013 and the preparations for the two ceremonies are designed to make an ineradicable impression on young minds.\n\nThe impression they made on mine seemed at the time comparatively slight though later events were to prove its force. I never doubted what I was being taught and I took the whole thing seriously enough to get ninety marks out of a hundred in the preparatory religious doctrine examination. Yet I just could not feel the emotions presented as appropriate when one is soon to receive Holy Communion. Perhaps my rationality was affronted by the doctrine of Transubstantiation \u2013 which provided my parents with one of their favourite theological bones \u2013 and as the weeks passed I became more and more aware of the inadequacies of my spirituality.\n\nI was also troubled by a desire to ask inconvenient questions. The atmosphere at school naturally precluded these and I hesitated to ask my mother lest such irreverent wondering might upset her. I therefore continued to speculate secretly, feeling increasingly guilty, until it seemed to me that my impulse to ask such questions could only lead to my being flung into Hell's hottest fires to writhe in torment throughout eternity.\n\nClearly the time had come to consult my mother, whatever her reactions, and on the eve of my First Confession I asked the most worrying question of all: what happened to the Sacred Host when one swallowed it? Did it continue to be God's body? If so, was it not grossly disrespectful to subject it to the routine processes of the human digestive system? And if it did not continue to be God's body, at what stage did it revert to being the piece of unleavened bread it was before the priest changed its nature at the Consecration of the Mass? (Not for nothing had I been exposed since birth to theological debate.) I was immensely relieved when my mother, instead of being upset by all this, looked positively pleased. But in reply to my question she only said that God, as the inventor of the human digestive system, could have no objection to being involved in its everyday workings. She added that many books, which I could study when I was older, had been written on this doctrine. Her reaction soothed my fears about hell-fire yet her actual answer did not satisfy me. I never doubted the Host's being God and I was made deeply uneasy by the essential irreverence involved in eating him. If I lacked the kind of superstitious awe my teachers were trying to inculcate, I did not lack reverence \u2013 described by Alexander Skutch as 'the chief of the religious emotions'. An instinctive reverence is, I believe, a part of every child's nature. But it needs to be carefully cultivated and this is why I have never regretted my Catholic upbringing; for all its peculiarities it encouraged my natural reverence to grow into something capable of surviving without the protective netting of formal religion.\n\nDespite the build-up, my recollections of First Holy Communion Day are hazy. I chiefly remember acquiring an unprecedented amount of money, through sixpenny and shilling tips from the neighbours, and feeling very important and adult and conscious of having begun an entirely new phase in my life. The Roman Catholic Church is often accused of retarding the mental and moral development of its members \u2013 and so it does, in many cases. But the First Confession\/First Holy Communion initiation rite, with its emphasis on the seven-year-old as a responsible person, probably hastens, at this stage, the maturing process. Or at least _can_ hasten it, if those in charge of religious instruction are not themselves superstition-sodden autocrats.\n\nBeing mainly dependent on my parents for such instruction, I soon became familiar with the neat logic that underpins Catholic moral theology. To change the metaphor, this system of stylised thought can be enjoyed as a sort of intellectual ballet, full of harmony, grace, disciplined energy and calculated flexibility. But it never allows for the unplanned movement, the sudden burst of individual initiative, the leap of a solitary imagination. Just as ballet is only remotely related to how people move in everyday life, so this system is only remotely related to how they think and feel. It is an heroic attempt to strengthen the weak, reassure the fearful and give form to the formless. As such, it has been of inestimable value to Europeans for almost two thousand years. But now European man is growing up, as Bonhoeffer saw not long before my First Holy Communion Day.\n\nApart from its religious significance, the First Holy Communion rite in a small Irish town was, during my childhood, a provocation to rampant one-up-womanship amongst the mothers of little girls. Who would have the longest veil, the most striking wreath, the most becoming frock, the prettiest shoes and knee-socks, the smallest rosary-beads, the most lavishly illustrated prayer-book? Mothers who could never afford a square meal for their children spent absurd sums on outfits which were totally impractical since it was considered both irreverent and _d\u00e9class\u00e9_ to wear them on social occasions. Once I heard my father muttering in his ascetic way that the clergy should condemn such inappropriate ostentation. But my mother defended it, arguing that by spending so apparently foolishly, people were expressing an awareness of the solemnity of sacramental rites \u2013 that for them extravagance was a part of worshipping. Many years later, when listening to criticisms of the lavish wedding-feasts of poor Hindus, I remembered her words. Had my father noticed the circumstance, he would certainly have deplored the fact that his daughter \u2013 attired in a Parisian outfit donated by her godfather \u2013 won this sartorial competition at a canter. But then my mother counteracted our status-improving victory by thriftily insisting on my wearing the frock 'for best' during that summer of 1939.\n\nAt the beginning of June my parents were more grieved than surprised to get a letter from Switzerland announcing Anton's 'disappearance'. (For several years he had been an outspoken opponent of Nazism.) As my father translated the news the sun was shining brilliantly across the breakfast-table, making the pot of marmalade glow amber. Then, precisely folding up the thin sheet of writing-paper, he replaced it in its envelope and said, 'So, by August war will have come'. He was not far out. But I cared nothing for the fate of a to me unknown German professor, or for the shadow of an unimaginable war; I mourned only the loss of Real Travel.\n\nA few months later, I went one morning to fetch the newspaper and learned that war had been declared. Hurrying home, I relished the sense of crisis in the atmosphere and expectantly scanned a cloudless sky for the first bombers. But when I realised that Ireland was not going to be involved I lost interest in the whole distant drama. For me, its chief effect was to intensify the boredom of grown-up conversation; I regarded literature and theology as lesser evils than military tactics. Occasionally, however, I was diverted by Hitler's interminable monologues on the wireless. These I found irresistibly funny and I remember rolling under the dining-room table one day in an uncontrollable paroxysm of mirth. My parents, who both understood German, reacted otherwise.\n\nYet for my father the war was a source of considerable inner conflict; much as he detested Nazism he was psychologically incapable of desiring a British victory. (Very likely his secret wish was that Germany and Britain should do a Kilkenny cat act.) He temporarily resolved his conflict \u2013 to my mother's unvoiced, ironic amusement \u2013 by refusing to remember Anton and persuading himself that the evils of Nazism were a creation of British propaganda. This illusion he cherished until Anton, unrecognisable after six years in Dachau, reappeared among us to dispel it. Not indeed by his words, for he never mentioned his experiences, but by the brand-marks on his arms and torso and by certain personality changes which moved to pity and horror his closest friends.\n\nThroughout the war I myself was straightforwardly pro-Germany in a light-hearted sort of way, as one might be pro-Scotland or pro-France at a rugger match. While reading such patriotic English stories as the Biggles books I automatically transposed names in my mind, to make the British the baddies and the Germans the goodies. And this was the extent of my emotional involvement. It is rather disquieting to remember how little the war meant to an Irish family without relatives or friends in Britain. While most of the world suffered, and millions of people died, we complacently pursued our almost-normal lives. At no time were we more than mildly affected by what was known to all but shoneens as 'the Emergency'. In most Irish minds of the period, our own mini-civil-war of the 1920s, in which some 700 died, remained The War. Eventually cigarettes were rationed, and as my father's conscience forbade him to use the black market his temper became uncertain towards the end of each month. I can see him now, carefully saving his cigarette ends in a flat, navy-blue Player's tin and rolling extra rations from them when threatened by nicotine starvation. It concerned me more that new books dwindled in number and became hard on the eyes when one was reading under the blankets by the light of a failing torch. Tea, sugar, butter and clothes were rationed; bread became virtually inedible and motor cars disappeared \u2013 never to be replaced, in our case, since after the war we seemed to be even poorer than before. But most important of all, to me, was the fact that parcels could no longer come from Paris.\n\nThis restriction ended an era of acute misery. Before the war my generous French godfather had regularly sent me the current juvenile equivalent of Dior outfits and every Sunday morning I was forced into these detested garments and dragged off to Mass by my father to be exposed to the derision of the entire congregation. Those ordeals were as agonising as anything I have ever experienced. My Parisian ensembles would have been conspicuous anywhere; in Lismore I felt they made me look like a cross between a damn silly doll and a circus clown. On this one issue my mother refused to consider my point of view. Having longed for an elegant daughter to share in her own enjoyment of beautiful clothes, she had produced an uncouth little savage who only felt happy in shorts and shirts. So perhaps her insistence on making a fool of me once a week was a forgivable form of self-indulgence. Also, she may have hoped that one day I would begin to take an interest in the art of dressing, if exposed for long enough to pleasing fabrics and designs. But inevitably her determination to see me looking civilised once a week had the opposite effect. I came to hate even my normal quota of new clothes, until they had been so broken in that I was no longer aware of them \u2013 a phobia which persists to this day.\n\nLooking back, it seems odd that my inherent unconventionality did not allow me to accept these Sunday ordeals as distasteful but unimportant. Thirty-five years ago, in an Irish provincial town, shorts were considered immoral on small girls \u2013 so in fact my everyday wear was as conspicuous as any of my Parisian excesses. Evidently, then, my aversion to the latter was based on something more than embarrassment at seeming different. Of course I loathed looking ridiculous as I slunk to our accustomed pew near the altar-rails, but I was made equally \u2013 if not more \u2013 uncomfortable by the element of artificiality introduced into my life by these pretty clothes. They and I did not belong together and though I could not then have articulated the sentiment, they made me feel vaguely dishonest. When we heard that my godfather had been killed while fighting with the Resistance forces I was quite incapable of the correct reaction.\n\nThat September I happily resumed my personal war with Sister Andrew. Beneath the tumultuous antagonisms which raged over the surface of our relationship, we were genuinely fond of each other. Yet we remained implacable enemies, outwardly, for another few years; and it was during this period that I invented my secret endurance tests.\n\nThese consisted of my regularly inflicting on myself increasing degrees of pain, until I was capable of such feats as walking three miles with a sharp thorn embedded in the sole of one foot. The imagination is unequal to what my psychologist aunt might have said had she discovered this little idiosyncrasy, but she would have been mistaken had she associated it with either sexuality or childish religiosity. My sexuality was at that stage quiescent and my religious sense never prompted me to go beyond the bounds of duty. I had simply discovered, while being beaten by Sister Andrew, that it was possible to repel certain kinds of pain. This inspired me to see if the same control could be exercised over self-inflicted pain. I struck the back of my hand, harder and harder, with a short, heavy stick; I tied thin twine around my fingers and pulled it tighter and tighter; I immersed my feet in hot water which I made hotter and hotter by adding to the basin from a boiling kettle \u2013 and to my astonishment the technique I had used at school always worked. But 'technique' is the wrong word and I am not sure what the right one is \u2013 perhaps 'instinct' would be a little closer to it. The process was as follows: I applied the painful pressure, hot water or whatever, felt the consequences acutely and then somehow contrived to send a message that numbed the pain even while the pressure was being increased. For instance, if I were experimenting with my left hand I could feel this message travelling down my left arm and checking the pain near my wrist.\n\nI have mentioned this activity to only a few friends (one prefers to retain what small reputation one has for sanity) and though it seemed slightly peculiar to them it may well be a common juvenile hobby. At the time, it was to me merely a useful accomplishment, worth cultivating, and many hours that should have been devoted to the twelve-times-tables or the rivers of Europe were spent pain-repelling. I became so proficient that at the age of ten I could probably have earned a good wage as a circus performer. But as yet I had only tested myself through brief ordeals and when I embarked on more prolonged trials I found that my message-sending did not work in quite the same way. Instead, I had to develop an indifference to pain. This entailed practising mental detachment from bodily sensations, whereas I assume my pain-repelling to have had a physiological basis. I can still pain-repel at will; but not, significantly, if the pain is a nerve one such as lumbago or toothache.\n\nAn amount of common-sense was used in what sounds like a lunatic campaign. I rarely did anything downright dangerous \u2013 my hot-water experiments were the most perilous \u2013 nor did I ever attempt to endure any suffering that could objectively be considered excessive. Some twenty-five years later, while observing the reactions to pain of less pampered races, I wondered if my endurance tests had been prompted by some atavistic longing to re-acquire a once universal power. They have certainly proved much more useful than the twelve-times-tables. Although the training course lasted scarcely three years I have ever since been almost wholly insensitive to what most Europeans regard as severe discomfort.\n\nThe strangest of all my childhood memories dates from this same period, which may not be entirely a coincidence. It concerns levitation; and I am comforted to know that some quite sane people \u2013 including Richard Church \u2013 have recorded similar memories.\n\nWhat I recollect, or fancy I recollect, is standing at the head of the stairs, breathing very slowly and deeply for a few moments \u2013 and then, while holding my breath, proceeding to the foot of the stairs without touching steps, banisters or wall. Was this a recurring dream that for some reason became fixed in my mind as part of reality? My mother often read and discussed Saint Teresa of \u00c1vila, so the concept of levitation was familiar to me and may have seemed so impressively peculiar that a realistic dream-cycle began. Another explanation, for a memory that is both wildly improbable and extremely vivid, is that one of my more extravagant fantasies became hopelessly entangled with reality. Yet neither of these explanations really satisfies me. The memory has about it a baffling matter-of-factness and coherence which seem to separate it from both dreams and daytime fantasies. I clearly recall making sure, before embarking on one of these trips, that only my mother was in the house, because I dreaded somebody witnessing what by any standards must have seemed outr\u00e9 behaviour. I also recall taking the practical precaution of keeping my right hand just above the banisters, and my left close to the wall, to save myself from falling should the system break down. And I retain a most vivid memory of the physical effort involved in this breath control \u2013 which, if it existed at all, can only have been some yogic talent that by a million to one chance I had hit on, possibly in the course of my pain-defying experiments. I have had an open mind on such subjects ever since my Tibetan friends convinced me that levitation \u2013 and other even odder phenomena \u2013 are not physically impossible. But if in fact I had acquired this curious skill, why did I so soon cease to practise it? Did I lose the knack as suddenly as I had found it? Or was I afraid? I remember being enthralled by my capacity to do something so extraordinary, but my 'trips' also provoked a profound uneasiness, amounting almost to guilt. It is slightly disconcerting to think that I shall never know the truth about this matter. Now, looking at small children, I often wonder what sort of private lives they lead.\n\n* * *\n\nAs Wordsworth noted, the whole person is plain to be seen in the child, though unformed and unrefined. But in many cases the individual's true nature is radically modified by the pressures of his environment and the expectations of his family. It is sad to think that a generous, frank child may become a tight-fisted, shifty businessman if the pressures and expectations are so directed. But equally, as in my own case, a selfish, stubborn, sulky child may become quite an amiable adult.\n\nStubborness and sulkiness were the weapons I used against my mother as she diligently laboured to eradicate \u2013 or at least suppress \u2013 my more anti-social vices. Even now it shames me to recollect certain scenes. Myself, aged nine or thereabouts, reading one damp July afternoon in the round, thatched, earwiggy summer-house; my mother asking me to post a letter, some fifty yards down the road; my snapped reply \u2013 'No! I'm reading \u2013 I won't go \u2013 I'm busy.' Then the verbal battle and my mother's inevitable victory and my return from the post-office to sulk for the rest of the afternoon. Why did I so often start battles which I  knew very well I was certain to lose? What devil prevented me from being normally helpful about everyday domestic chores?\n\nIn all circumstances my mother insisted on obedience, yet in spite of my surface sulks I never really resented her disciplining. She was almost always just \u2013 and capable of apologising if she had been unjust \u2013 so resentment would have been irrational.\n\nMy childhood relationship with my mother was relatively straightforward, but I still find it hard to understand my relationship, at any age, with my father. In a sense, nothing ever grew between us from the seed of child-parent love; it lived on through the years but remained underground; there was no blossoming to affirm its existence to the outside world \u2013 or even, for long periods, to ourselves. One of the conditions that hindered its growth was my father's inability to communicate with the young. He lacked any means of expressing his affection in an acceptable form and his rare attempts to get onto my level and be playful caused me acute embarrassment. Desperately well meant but blatantly phoney, these \u2013 I felt \u2013 were just making us both look foolish while widening the gulf between us. I much preferred his natural approach when he treated me as a pupil rather than a daughter. His own idea of fun was a fact-packed lecture thinly disguised as a long walk. With the random questioning of small children he had no patience; this was an untidy, unscholarly way of going about learning \u2013 a bad habit, to be eradicated without delay. Significantly, I could never imagine him as anything but a tiresomely erudite grown-up, though I could easily picture my mother as a little girl.\n\nFor me, our regular Sunday afternoon walks were both physical and intellectual marathons. Week by week I would be tidily instructed about birds, or moral theology, or electricity, or Irish history, or geology, or English literature, or astronomy, or music, or agriculture, or the Renaissance. Often I wished that I were alone beneath my teddy-bear tree and then I would vindictively insulate myself against my father's voice; though to give him his due he presented all his information in carefully simple terms. Of course some of it fascinated me, despite myself, as several of his enthusiasms were by heredity my own \u2013 especially history and astronomy. On the whole, however, these didactic perambulations provided the wrong sort of fertiliser for the seed of love.\n\nJust occasionally the barrier was lifted and we drew very close. My father had an unexpected flair for composing Learish nonsense rhymes and these charmed me utterly; when he was in one of his rare frivolous moods I would gladly have walked with him to the Giant's Causeway. Then I discovered that I had a similar flair \u2013 long since atrophied \u2013 and we enjoyed the harmony of collaboration or the stimulus of competition, each striving to outdo the other in dottiness and euphony. But the barrier always came down again at the end of these sessions, leaving us uneasily antagonistic for no discernible reason.\n\nThe reason could have been jealousy, an emotion one would expect to find in some rather virulent form in such an introverted family. Perhaps, being so worshipful of my mother, I resented my parents' mutual devotion. (Although according to pop psychology I should at that age have been so devoted to my father that I regarded my mother as a rival.) Yet I am pretty sure \u2013 as sure as one can be on such matters \u2013 that jealousy did not then influence any of our relationships. I was certainly given no cause for it. Together my parents lived their own separate child-excluding life, but I accepted this as natural and was never made to _feel_ excluded in any unfair way. From an early age I took part in serious family conferences, and was admitted to the cupboard where the skeletons were kept, and generally was treated as a responsible, dependable individual. Years later I discovered that Pappa disapproved of my being consulted before family decisions were taken; he held that it is unkind to implicate children in adult affairs with which they are too inexperienced to cope.\n\nEvery summer Pappa spent July and August with us. I would guess that my father was his favourite child though apart from their common bibliomania the two were alike in no obvious way. Pappa was not merely 'good with children'; he truly enjoyed them and his annual arrival by train drew not only myself but a score of other children to the railway station. Yet he never gave pennies or sweets or treats to me or to any of his young friends. Instead he played with us endlessly \u2013 our own games in our own favourite haunts. And always he brought from Dublin a battered suitcase tied with rope and bulging with dog-eared children's books bought for twopence a dozen on the quays. No one \u2013 not even my mother \u2013 could read aloud as Pappa did. He involved us until we were transported beyond anything we knew of into other worlds that seemed to be suffused with a special Pappa magic, whatever the theme of the story or the author's style. Even the more restless of the smaller children \u2013 and those who were not accustomed to being read to and normally had no interest in books \u2013 even they would sit motionless for as long as Pappa chose to read.\n\nPunctuality was the only subject on which I used to query Pappa's wisdom. He argued cheerfully that a capacity for ignoring time marks the truly free in spirit and that over-organised Western man has only himself to blame for the fact that our society would collapse if this freedom were widely enjoyed. His own indifference to time no doubt formed part of his attraction for children. But it made him another of Old Brigid's crosses. She, too, adored him, and considered it her duty and privilege to 'feed him up' during his holidays, and so if he had not appeared by 12.55 she felt obliged to go forth to quarter the back streets and lanes of the town in search of 'Dr Conn'. Luckily this did not happen too often since I shared my father's obsessional punctuality \u2013 which was perhaps a result of the havoc frequently wrought in his own life by parents who never knew or cared whether it was morning or evening.\n\nThere was a Franciscan quality about Pappa's affection for children and animals and the poor of all ages. It was without any element of paternalism or do-gooding; behind the gaiety which charmed us all lay a deep awareness of suffering and a love based on compassion and respect. For some reason he was always known locally as Dr Conn and he was a particular favourite of the old country folk whose dying traditions he collected for one of his unwritten \u2013 or half-written \u2013 books.\n\nOn a hot summer evening in 1939 an old woman from the mountain hamlet of Ballysaggart called to ask for Dr Conn's help and I answered the door. Explaining that Pappa was out, I offered to give him a message. The old woman hitched her black woollen shawl higher to protect her head from the midges around the fuchsia bushes. 'When he comes back, could y'ever ask yer father to drive him out t'see me poor husband? He have a crool pain in his chest wit de past tree weeks. He can't even raise himself in de bed wit it. An' the docthur above on'y gev him on oul cough-bottle dat med him sick to his stummick.'\n\nI looked at the old woman in silence and felt wretchedly guilty, as though the family had been caught playing some nasty confidence trick on the entire district. Then I admitted miserably, 'But Pappa isn't a real doctor. He's only a doctor of philosophy!'\n\nThe old woman shrugged. 'Shure isn't that good enough? Isn't he a kind man wit brains? What more d'ye want?'\n\nI tried to explain. 'But you see it's not the right sort of brains \u2013 he wouldn't know what medicine you need. Philosophy has nothing to do with being ill. At least, I don't think it has,' I added, suddenly wondering just what it did have to do with.\n\nNext day I asked, 'What is philosophy?' as Pappa and I were walking back from our morning bathe in the Blackwater. My father would undoubtedly have taken this question as the jumping-off point for an outline of the various schools of Western thought. But Pappa simply said, 'It's the study of how to live contentedly and how to die peacefully!'\n\nFrom my point of view our annual summer holiday in some secluded seaside cottage was redeemed only by Pappa's presence. I enormously enjoyed our weekend trips to the sea, but I detested being away from home for an entire month. These cottages were never anywhere near a public library and, from 1940 on, petrol was rationed and I dreaded running short of books. There was little room for luggage in the Ford Ten with five passengers \u2013 including Old Brigid or her successor \u2013 and Parnell the all-black sheepdog, and Sibelius and Delius the cats, and my mother's bath chair on the roof. My personal baggage allowance was one large suitcase and to make the most of this I wore all my clothes, including two pairs of pyjamas, on the journey to the coast. But a single suitcase of books seemed meagre fare for a month, even if one chose only volumes that could stand rapid rereading.\n\nHowever, it was the removal from my natural habitat that I minded most. Amidst the fields and woods and rivers and hills around Lismore I could enjoy myself as nowhere else. Already I wanted to travel to distant lands, so I might have been expected to welcome the substitute thrill of exploring remote stretches of the Irish coastline. But already, too, I had clear-cut ideas about _how_ I wanted to travel. I wanted to wander alone, taking each day as it came, and even at the age of nine or ten it was impossible to pretend that a month in the domestic cosiness of a seaside cottage was any sort of substitute for such adventuring. \n\n#\n\nThe autumn of 1940 was marred by a bizarre psychological affliction. Or perhaps it was a 'normal abnormality', given my keen interest in sexual equipment and the inhuman puritanism of the Irish Catholic Church. Because I had long been in the habit of closely observing mating animals I one day became convinced that my soul was permanently laden with mortal sins. This conviction rapidly spread, like a sort of mental septicaemia. Soon I felt that to wash above the knees or below the navel must also be a mortal sin. And so must chancing to see a male baby having his nappy changed, or noticing a cat feeding her kittens or a dog sprinkling a lamppost \u2013 or even hearing, at the other side of the partition that divided the school lavatory, the hiss of little boys urinating into the stinking gutter. At this stage my having long since given up masturbation was unfortunate. Something as tangible as that to worry about might have saved me from fancying myself corrupt right through because a dog peed in my presence.\n\nWhat puzzles me now is the extent to which, during those months, school influences temporarily overcame home influences. On trivial issues small children will naturally follow school fashions, on important matters parental attitudes generally prevail. Moreover, at no other period of my life have I been prone to unhealthy \u2013 or even, some would say, healthy \u2013 guilt. Yet throughout this nightmare period I was half-crazed by shame and self-disgust.\n\nThe approach of puberty, combined with an incident which had occurred during the previous summer holidays, may have been partly responsible for this anxiety-state. Physically I was unusually mature and my mother had explained that my budding breasts must now be kept covered. But I so relished the feel of the sun and the wind on my body that I often ignored her directive and wore only brief shorts when playing alone on the deserted beaches near our holiday cottage. Then one day three adolescent boys appeared abruptly from behind a rock, shouted obscene remarks and threw several well-aimed pebbles at my indecently exposed torso. And my eyes were opened, and I knew that I was naked...\n\nA better balanced child would merely have been incensed by this tiny incident, or at worst slightly alarmed. But in an instant it made me see human sexuality not just as an example of nature's ingenuity but as something apart from the rest of life and capable of assuming ugly, obscurely threatening shapes. I told my mother about the incident by way of exorcising it, though the telling involved a confession of disobedience. Her reaction, as always, was steadying \u2013 'People can be unpleasant. But their unpleasantness needn't infect us if we don't want it to.' Nevertheless, I now began to think about sex in a new, personal, speculative way (though I have no recollection of experiencing at the time any sexual sensations) and this made me much more vulnerable at school.\n\nThe reader may well wonder what methods were used by my teachers to unhinge me so disastrously. In fact sex was never mentioned: eight of the ten commandments were commented on in detail, but the other two were ignored. At the same time the impression was given that these were the most important of all, though the vices they forbade were too evil to be analysed as one could analyse the comparatively minor sins of murder, theft, slander and idolatry. It is remarkable, and quite sinister, that this message, stressing the incomparable heinousness of sexual sin, could be got across wordlessly. Somehow we were made hyper-aware of the horror and revulsion with which such sins must be regarded and in the process a horror of sex itself was deeply implanted in many a child's mind.\n\nAll this left no lasting mark on me though it profoundly affected the world in which I grew up. Behind the popular image of the gay, feckless, hard-drinking, charming, belligerent, eloquent Irishman lies an amount of muted yet intolerable suffering \u2013 which is shared by the victims' wives. The Irish incidence of mental disease and alcoholism is amongst the highest in the world, Irish people marry later than most others \u2013 if at all \u2013 and the Irish male is noted for sexual immaturity. The Catholic Church has always been the obvious scapegoat here, yet it does not deserve all the blame. Our puritanism is peculiarly Irish rather than peculiarly Catholic; one finds it operating equally strongly among Northern Irish Protestants.\n\nMembers of celibate communities are often assumed to be unbalanced, tense, frustrated and generally unsuitable as educators. Yet many priests and nuns seem, to those who know them best, exceptionally balanced, relaxed and fulfilled. However, when celibates do go dotty they go very dotty indeed, frequently in rather nasty ways, and one of my teachers \u2013 not Sister Andrew \u2013 should never have had anything to do with children. She regularly interrupted our history, geography or arithmetic lessons to gratify herself by terrorising us. Obviously she believed in her own fevered descriptions of Hell and the Devil and the conviction with which she spoke made her harangues all the more blood-curdling.\n\nI was unsettled not only by this poor creature's words but by the evident sick satisfaction she obtained from frightening us. I recall one of her sessions with particular vividness because I was sitting in the front row just below her desk. She had a high-pitched voice which became little better than a squeak when she was enraged or excited and on this occasion, as she spoke, fine beads of sweat broke out on her hairy upper lip. She was telling the 'true' story of a ten-year-old Co Waterford girl who one day committed a filthy mortal sin (the adjective told us that it was sexual) and next day fell into a stream and was drowned. Because she had not been to confession, or made an act of perfect contrition, the Devil promptly dragged her soul down to Hell where she was doomed to an eternity of tortures which Sister X assured us were indescribable though this did not deter her from attempting to describe them in considerable detail. None of us thought to ask our mentor the source of her information on this case. It was a cruel coincidence that I came under the influence of such an unstable woman during the unhappiest phase of my childhood.\n\nAt noon every Saturday I went to confession with my classmates, as was the custom. We sat in restless rows near the three confessionals, examining our consciences and memorising our sins while awaiting our turn to enter the stuffy, anonymous darkness of the box. During my guilt period I used to envy my companions whose rapid reappearances, following the muffled drone of the absolution, indicate their freedom from moral problems. It made me feel doubly depraved to think of the innocent content of their confessions, which I knew would go something like this: 'Bless me, father, for I have sinned. It's one week since my last confession. I forgot to say my morning prayers three times. I was distracted at Mass on Sunday. I stole five sweets from my brother. I told three lies to the nun. I was disobedient nine times. I hit the girl next door. That's all, father.' Then they would be given three Hail Marys for penance and would cheerfully emerge to commit the same crimes \u2013 more or less \u2013 during the week ahead.\n\nI always chose to confess to Father Power. The other curate too plainly conveyed the extremity of boredom to which juvenile misdeeds drove him and the parish priest was eighty-five, stone deaf and apt to confuse 'impure thoughts' with adultery. While in the confessional most Catholics regard their confessor not as the neighbour who calls for a chat but as an impersonal representative of God. Therefore it was easy for me to put Father Power in the picture. But his calm reassurances and careful explanations achieved nothing. Like any mentally deranged person, I was isolated in a private world of my own, beyond reach of common sense. Father Power repeatedly urged me to confide in my mother, but this was no help as my inability to do so was an integral part of my disease. I was so far gone that I feared losing her love if she discovered my vileness. When I explained this to Father Power he again tried unsuccessfully to make me see the absurdity of my terrors; had he not been bound by the seal of the confessional he himself would certainly have warned my mother of her daughter's pathetic state. More acceptable was his advice to confess only four times a year instead of once a week and to stop keeping a tally of my sins in the copybook I had specially set aside for the purpose. But unfortunately even this counsel did not reach to the root of my obsession \u2013 though it came near enough to it. However, it stimulated an enjoyable argument with Sister X who furiously asked if I wanted to 'lose my soul' when I told her that in future I would not be joining my class at confession time. I declined to explain that I was acting on Father Power's advice; why should I divulge to her what went on in the confessional? Besides, I was delighted to have a chance to deflate her single-handed. I did not then clearly recognise her contribution to my problem, but I disliked her more than anyone else I had ever known. (And indeed, now I come to think of it, I have never since met anybody I disliked as much.)\n\nSister X had a very long nose, with a ludicrously thin tip, and her angry flushes always began at this tip and radiated outwards like a spreading wine stain on a yellowing tablecloth. 'I'll be looking for you tomorrow!' she squeaked threateningly as I watched her nasal indicator turning crimson. 'Mind you're here at a quarter to twelve sharp with the rest! You should be ashamed of yourself, even thinking of rejecting God's grace! It's the Devil is putting such wickedness in your mind \u2013 he wants your soul for himself \u2013 and at this rate he won't be long getting it!'\n\nSuch a challenge to do theological battle made me briefly forget that this was my own current belief. I replied that no Catholic was bound to confess more than once a year and that if I were forced to confess by anyone else the sacrament would be invalid. This final point took Sister X right out of her depth, as it was meant to do, and she sounded like a fork on a plate when she exclaimed \u2013 most unconvincingly \u2013 'I'll report you to the Canon for this!' (The Canon was our parish priest, a saintly old gentleman who was known to abhor the very mention of Sister X's name.) As she swept from the schoolroom giggles erupted among my classmates and in an instant she was back to give us all an hour's detention. But I knew that she knew that she had been beaten. And vengefully I rejoiced. She brought out the very worst in her pupils \u2013 apart from turning them into nervous wrecks.\n\nYears later my mother told me that by the end of November she had begun to worry about my being so obviously off form. She mentioned her concern to Father Power, who at once suggested a mother-daughter conference on the relevant commandments.\n\nThe day before my ninth birthday I woke early and tried to read but could not concentrate. At that age birthdays have an epochal significance and I felt that if my tenth year began as inauspiciously as my ninth was ending this misery must continue for every day of the next twelve months. As I lay in bed, staring unhappily at the grey-brown damp stains on the ceiling, I heard my father coming upstairs \u2013 but instead of going straight into the bathroom to shave he came to my door and said that my mother wanted to see me. As she did not usually encourage me to enter her life before breakfast-time \u2013 rather the reverse \u2013 I went downstairs reluctantly, expecting a lecture on some newly discovered misdemeanour. But no; she was in a very gentle mood and invited me into her bed for a snuggle. And then she asked directly, 'What's making you so unhappy?'\n\nFor weeks I had known, deep down, that a conference with my mother was the key to freedom if only I could bring myself to take the risk of using it. Now that key had been put in my hand \u2013 and I did use it.\n\nI shall never forget my mother's laughter as I lay there with my arms around her neck and my face buried in her hair. She had a deep, rich laugh and soon I, too, was laughing \u2013 dazedly, scarcely able to believe that within moments my tragedy had been transmuted into comedy. When I returned to my room to dress I felt just as one does on wakening from a complicated nightmare: elated by the simplicity of reality.\n\nMy mother's amusement had been perfectly genuine, but beneath it she was gravely disturbed by my account of Sister X's mental torturings. Already she was weary of another aspect of school life; every afternoon, before I could safely be admitted to the house, she had to supervise the fine-combing of my hair over a basin of Lysol outside the back door. So now it was decided that a combination of head lice and faulty theology sufficiently justified my removal from Lismore school.\n\nWhile debating what to do next, my mother taught me \u2013 an arrangement which I favoured. She used teaching methods not then fashionable in Irish national schools. Instead of presenting knowledge neatly cut up into small bits she showed me how to go about collecting my own information and so made learning seem fun. Yet even she could not reconcile me to the German language. Nor could I cope with Irish or French, which my father had been inflicting on me since I was born. Latin was the only language for me. That I enjoyed, and was good at, knowing I would never be expected to make a fool of myself by speaking it.\n\nMany nine-year-olds would have missed the social side of school life, but this loss never worried me. Apart from Tommy, and a girl with the unlikely name of Charity, all my primary school relationships were superficial. For a time I felt that I should have several friends, because children in books usually had, but I soon found that promising to meet for play at a certain time unduly restricted my private life of reading and roaming. Had the choice been wider I might have felt otherwise; I was overjoyed when Charity joined our class and revealed that she, too, was enslaved to Hugh Lofting, Arthur Ransome and Richmal Crompton. Soon I had got her a special pass to visit the library whenever she liked and for the next six months we were Best Friends. But then her army father was posted elsewhere and thus ended my first close friendship, as distinct from the sort of tribal comradeship I had enjoyed with Tommy.\n\nA few days after Christmas my mother broke it to me that in January I was to go as a boarder to an Irish-speaking coeducational school. Naturally I was devastated. It had always been understood that I would go away to school at the age of ten and I could scarcely credit my parents' treachery. But in an odd way this sense of having been betrayed kept me calm. Parents who loved me so little must not be allowed to see how much I cared \u2013 a melodramatic reaction which carried me through my initial grief and disillusionment. Then suddenly going away to school began to seem an interesting idea; to my own surprise part of me was one morning quite looking forward to it, though I had never yet been separated from both my parents for more than a few days. But soon I was again shattered by the discovery that books in English were forbidden at the College. Despair overcame me; this was equivalent to depriving an alcoholic of his bottle or a chain-smoker of his packet. Yet I never made any attempt to alter my parents' decision. On details I argued interminably with them; on major issues I meekly deferred to their adult wisdom even if their reasoning seemed obscure. Or if, as in the present case, my antennae told me that they were not themselves in perfect agreement.\n\nThis was one of the few occasions when my father made a decision, for personal reasons of his own, to which my mother only grudgingly assented. Where the use of English was totally forbidden it seemed possible that within a year even I would have acquired a working knowledge of my native tongue. For nationalistic reasons my father wished me to be as fluent an Irish speaker as himself. Besides, if I were ever to pass an examination some action had to be taken to remove whatever blockage prevented me from learning languages. Or so my parents thought; for years they would not accept the simple fact that I had not inherited their linguistic gifts. They mistook stupidity for laziness and my mother \u2013 who held no strong views about the Gaelic Revival \u2013 probably agreed to this experiment as a general disciplinary measure.\n\nAt the beginning of 1941 the 'Emergency' had not yet banished all motor cars from Irish roads and we drove to the College on a cold, dark, wet January afternoon. The hedges were hardly visible through swirling curtains of rain and we were all, for our various reasons, apprehensively silent. Real, live boarding-school authorities were an unknown quantity to me, but I felt that they might prove much more dangerous in life than in literature so I had been afraid to pack even one illicit book. And now I was sick with anguish at the thought of parting from my parents. When the grey school buildings loomed sombrely out of the rain and fog, on their bleak and windswept cliff above the sea, I remarked that there would be no need for any lingering once my luggage had been unloaded. And my mother agreed that this was so.\n\nWhen we had said our brisk goodbyes my father decisively banged the car door and I turned into a long, empty corridor. Most pupils travelled by train and had not yet arrived. A young master appeared, said something curt in Irish and disappeared, carrying my suitcases. I hurried after him, down the ill-lit corridor and up a steep staircase. The whole place reeked of Jeyes Fluid and boiled onions. Then I was put in the care (not quite the _mot juste_ ) of a freckled twelve-year-old with sandy plaits and a shrill, bossy voice. I can still see her frayed pale green hair-ribbons and her look of contempt when she realised that I understood not a word she was saying.\n\nIn the icy, barn-like, whitewashed dormitory there were no cubicles but only rows of beds with vociferously broken springs and lumpy, unclean mattresses. My bed stood almost in the centre of this desolation and as I paused forlornly beside it, wondering where to hang my clothes, I realised that such a complete lack of privacy would add an unforeseen dimension to my hell. I shivered and needed to go quickly to the lavatory. Half-a-dozen older girls were gathered in a far corner, wearing overcoats and stuffing themselves with sticky buns. When I asked for the lavatory in English one of them threw a boot at me and shouted angrily in Irish. My bladder was about to fail me and I broke into a cold sweat \u2013 literally, for I remember pushing the hair out of my eyes and noting the chilly moisture on my forehead and thinking that this must be what authors meant by 'cold sweat'. I had assumed that _in extremis_ we could talk English; now it was plain that to do so, under any circumstances, would bring some instant punishment from my uncouth and intimidating seniors. Mercifully a lavatory chain was pulled nearby at that moment and I rushed gratefully towards the sound.\n\nBack in the dormitory I found my suitcases open and their contents scattered on the bed. The girls were examining everything critically and the discovery of my schoolbooks provoked much mirth; I was so tall for my age that from these they deduced extraordinary stupidity. They expressed the opinion that I must be mentally retarded by using graphic traditional gestures, while shrieking with laughter. Then they came on a packet of sanitary towels \u2013 proud emblem of my recently acquired womanhood \u2013 and used other gestures, not then understood by me; no doubt their comments were to match for they lowered their voices and muffled their sniggers. As I could see no friendly \u2013 or even neutral \u2013 face anywhere I suppressed my rage and stood by helplessly until the enemy lost interest. They left the dormitory linking arms, scuffling, giggling and shouting each other down. I thought of Mrs Mansfield, who would almost have fainted to witness such behaviour, and the image of her trim little figure, with San Toy trotting regally to heel, sent me hurrying back to the lavatory to weep. Already I had resolved that my enemies would never see me weeping.\n\nAt six o'clock a jangling bell summoned us to the refectory for high tea. Most of the other pupils had now arrived, but I seemed to be the only new girl though there were several new boys \u2013 all of whom, discouragingly, spoke effortless Irish. As I took my place at one of the long, scrubbed wooden tables, each with mounds of thick bread and scrape placed at intervals down the centre, I vowed that this educational experiment must be made to fail as expeditiously as possible. Since English was forbidden, I would not speak. And when the futility of having a dumb child about the place impinged on the authorities, they would expel me. Nothing could be simpler. As I am naturally taciturn the prospect of maintaining silence for an indefinite period did not dismay me. And to compensate for the lack of books I would secretly write one myself.\n\nOf course things did not work out quite like this. I was far too demoralised by homesickness to concentrate on writing anything more than letters and my misery, instead of diminishing as the days passed, became more acute. There was not even one remotely congenial character among either staff or pupils and I had immediately become a favourite bullying target for the more sadistic seniors. These also regularly robbed me of my weekly food parcel \u2013 an Emergency innovation \u2013 and they did use English to threaten to retaliate if I reported them. It is easy to see how I brought out the worst in these schoolmates. To them I must have seemed intolerably priggish, precocious, precious, pedantic and pusillanimous. There was no point of contact; in every sense we spoke different languages. Inevitably my memories of this ordeal are biased and the reality may have been a trifle less barbarous than what I recollect. Yet the essence of the atmosphere remained unparalleled in my experience until I worked as a waitress, almost twenty years later, in the canteen of a home for down-and-outs in East London.\n\nTherefore this episode, despite its brevity, was one of the most valuable in my limited educational career. At Lismore school I was subtly accorded privileges by many of the teachers because I seemed 'different'. At Ring I was given hell for the same reason and thus I learned that standards other than my own were not only acceptable to, but preferred by, large sections of the population.\n\nMy parents wrote long letters three times a week but refrained from squandering their petrol ration on me. In my Sunday letters home I never asked for a visit but regularly reported that I was learning _no_ Irish and cunningly emphasised the physical hardships of school life. In fact I took these in my stride \u2013 apart from the atrocious food they seemed no worse than the rigours of home life \u2013 but I felt that my mother would be more disturbed by health hazards than by complaints about bullying. So I graphically described how \u2013 after an inadequate lunch \u2013 we were driven out every afternoon, whatever the weather, to play camogie (the feminine of hurling) on pitches hock-deep in mud \u2013 and how we then had to sit in an unheated prep. hall for two hours wearing damp socks.\n\nThese letters were not greatly exaggerated and as a result of over-exposure and underfeeding I developed severe bronchitis in the middle of February. After forty-eight hours I was almost too ill to walk, yet the matron merely dosed me with some ineffectual syrup. Everyone had snuffles and coughs and she did not pause to distinguish between penny plain and tuppence coloured. So I wrote an extra letter to my parents, one Wednesday morning.\n\nOn the following afternoon they arrived unannounced, and despite a keen east wind discerned in the distance their wheezing ewe-lamb, feebly wielding a camogie stick. Moments later I was in the car, drenching my mother's shoulder with all the tears not shed since our parting. And in the headmaster's office my father was being told that I had made little progress with my Irish and seemed 'unable to fit in with the rest'. 'I should think not!' muttered my mother, as we drove off. While I was changing and packing she had had an opportunity to observe a cross-section of 'the rest'. \n\n#\n\nDuring my absence from home Old Brigid had at last retired to her cottage near Cappoquin and been replaced by one Maggie \u2013 unenthusiastically described as 'adequate', fading to 'better than nothing', in my mother's letters. As we drove towards Lismore I found it hard to imagine life without Old Brigid, but I would have been a good deal more upset in any other circumstances. Having just regained Paradise not even her loss could dilute my joy and on being assured that I might visit her weekly I philosophically accepted the idea of Maggie.\n\nI did not, however, accept the reality of Maggie. She was sharp-boned and purple-hued and her voice rasped and she always looked discontented \u2013 with some reason, as a servant chez Murphy. We disliked each other on sight; to me she was an unloving tyrant, to her I was a saucy little puppy. And the chances of our ever coming to terms were much reduced by my having to take to my bed at once and stay there for three weeks, during which time Maggie found herself toiling up and down stairs every few hours with trays of such delicacies as were then available. (I particularly remember dark-brown beef-tea, jars of Calves'-foot jelly and a weird, chewy ginger-based concoction which was alleged to do wonderful things for the bronchial chords.)\n\nEven in our Spartan household bronchitic patients were permitted a fire in their bedroom and it was typical of my mother's feeling for servants that she always asked my father to take on the consequent chores \u2013 which he most willingly did. Old Brigid had greatly appreciated his help, but it quickly became obvious that in Maggie's estimation such face-losing behaviour on the master's part drastically lowered our status. For a few weeks after my recovery she and I sparred daily; I was missing Old Brigid dreadfully, resenting Maggie in proportion and making my feelings plain. Then one morning she left without notice to become housekeeper to an elderly, childless couple who lived in a comfortable new house where no one had committed suicide.\n\nHaving been born with 'a weak chest' I was accustomed to enjoying a few weeks of invalidism each winter and I revelled in being free to read, almost without interruption, for fourteen hours a day seven days a week. Whatever the theologians might say about Heaven being a state of union with God, I knew it consisted of an infinite library; and eternity, about which my parents were wont to argue with amusing vehemence, was simply what enabled one to read uninterruptedly forever.\n\nThe only interruptions I welcomed during these withdrawals from the world were Dr White's visits. He treated me with the sort of rough affection one bestows on a large dog and gave me delicious syrup from a bottle excavated with difficulty from the depths of his greatcoat pocket and told heart-stopping stories about his soldiering days in India and South Africa. Sometimes he advised me to rest my eyes, but this advice went unheeded as I took no interest whatever in any of the standard children's games or pastimes. (A serious handicap nowadays, when I have a more versatile child of my own.)\n\nI remember the perfection of my happiness \u2013 a perfection not often attained in this life, as I realised even then \u2013 when I woke on a dark winter's morning and switched on the light to see a tower of unread library books by my bed. From them I would look caressingly towards my own books on their shelves around the wall and reflect that now I had time to reread; I could never decide which was the greater pleasure, rereading old favourites or discovering new ones. For a moment I would lie still, ecstatically anticipating the day's bliss. And sometimes it would cross my mind that only Pappa could fully understand how I felt.\n\nThere is a difference between the interest taken in books by normal readers (people like my parents) and the lunatic concern of bibliomaniacs (people like Pappa and myself). Everything to do with books mattered to me and I fretted much more over their wartime deterioration \u2013 that squalid gravy-coloured paper! \u2013 than I did over butter rationing or inedible bread. (Clothes rationing I of course considered a blessing in disguise.) After a quick glance at any open page I could by the age of nine have told you the publisher of most children's books \u2013 and often the printer and illustrator, too. One of my hobbies was rewriting blurbs which seemed inadequate and I collected publishers' lists as other children collect stamps. During June and July I often prayed for rain; on fine days I was supposed to be out in the fresh air, but on wet days I could go to the county library headquarters and help unpack the new books that came by the hundred, in tea-chests, at that season. The sight, smell and feel of these books so intoxicated me that I often refused to go home at lunch time. I had an agreement with my parents that when the children's books came I could always help unpack, regardless of climatic conditions. I would then \u2013 to my father's sensibly silent disgust \u2013 seize on the least worthy volumes (Biggles and so forth) and beg to be allowed to borrow them even before they had been initiated into public circulation. But my father did not believe in Privilege so I had to bide my time \u2013 very sulkily. It must have exasperated my parents that for so long I preferred exciting stories to good writing. At every stage of childhood I completely rejected all the classical fairy stories, and Lewis Carroll, Captain Marryat, Louisa Alcott, Kipling, E. Nesbit and any volume that I suspected might be intended to improve my mind. But neither, to be fair to myself, would I read Enid Blyton when she began to pollute the literary atmosphere. I was uncompromisingly middlebrow; and so, with minor modifications, I have remained to this day.\n\nAs the librarian's daughter I did have one priceless perk. When public library books become too battered and disgusting for rebinding or recirculation they are 'Withdrawn From Circulation', stamped to that effect and despatched either to fever hospitals or to the pulpers. And among those glorious, revolting heaps of 'Withdrawn' books \u2013 their pages interlarded with evidence of the diet of the rural reader \u2013 I was free to wander and take my pick and carry the noisome volumes home by the armful to be mine forever. (Many of them are still mine; no one ever steals them.)\n\nI went through one appalling crisis in relation to 'Withdrawn' books. At the age of eight or so I had a compulsive secret vice \u2013 crossing out the author's name on the title page of old books and substituting my own. This could be done without fear of detection in unfrequented corners of the library; but then, in bed one ghastly evening, I suddenly realised that some of the books I had been abusing might go, not to the pulpers but to a fever hospital. If this happened both my iniquitous vandalism and my vain ambitions would be exposed to a shocked and derisive public. This hideous possibility so tormented me that I could not sleep. As my parents were listening to the late news I crept downstairs and confessed all to my mother \u2013 who remained astonishingly unperturbed. She assured me that the defacement of such books was forgivable and that no fever hospital patient was likely to report on my little weakness to the world press \u2013 which would in any case be disinclined to take the matter up. I always enjoyed the irony with which she put things in perspective; curiously enough, it never made me feel foolish.\n\nBy the spring of 1941 most Irish working girls had emigrated to earn good money in English factories and our next five years were dominated by the comings and goings of maidservants. The best were those too young to emigrate, who usually responded well to my mother's training; but no sooner had she imparted the rudiments of domesticity than they were clutching a ticket for Paddington and saying often tearful farewells. This relay system offered no reward for weeks of hard work. Gone were the days when my mother spent her mornings reading, or listening to concerts on the wireless, or teaching me. Now she was lifted into her bath chair after breakfast and wheeled into the kitchen to supervise the cooking and other household tasks. As a bride she had been unable to cook an egg, but in everything she was a perfectionist and her zealous study of the art of cooking had such sensational results that despite Emergency limitations I have never anywhere eaten better than I did then in my own home.\n\nSeveral of our non-treasures had to be dismissed within days for intolerable personal filthiness (there was the celebrated case of the louse on the table-napkin...) or irredeemable incompetence, or both. Some were petty thieves, others were incorrigible 'borrowers'. One sixteen-year-old was detected by my father returning through the kitchen window from a military hop at four o'clock on a summer morning, clad only in one of my mother's nightgowns. My father imagined her to have been sleep-walking and apologised profusely for having chanced to observe her in dishabille. My mother assessed the situation more realistically and next day patiently lectured the girl on the hazards of associating, in the small hours of the morning, with the licentious soldiery.\n\nAs Lismore was a garrison town throughout the war our younger maids' morals were a source of constant anxiety. Those who arrived knowing nothing of the facts of life had to be given sex instruction even before they were taught how to make coffee. And for a few this instruction came too late. These usually stayed longest; when they discovered their condition their bewildered fear was pathetic and whatever their professional defects my mother never had the heart to dismiss them.\n\nOne eighteen-year-old precipitately gave birth under the kitchen table with me in fascinated attendance. When the drama was all over bar the afterbirth I rushed into the sitting-room exclaiming that it was just the way cows did it. Perennially unflappable, my mother said 'How interesting' \u2013 and now would I please take some blankets to the kitchen and wrap the baby up well before going to the Post Office to telephone for an ambulance.\n\nIt had always been clear that Josie was weak in the head and as her parents now rejected her \u2013 an unusual reaction, amongst Irish country folk, to 'little accidents' \u2013 my mother felt obliged to act _in loco parentis_. The authorities did everything possible, and more than was ethically allowable, to force her to give up her baby; but under the influence of maternal love, she showed unexpected strength of character. When we visited her in the County Home in Dungarvan my parents were so moved by her determination to keep her child that they entered the argument with a few well-chosen remarks about the legal rights of parents. They also guaranteed to look after both mother and son until Dr White could arrange for their admission to some suitable hostel. I never forgot this example of how the uninformed and inarticulate citizens of a democracy may be bullied and confused by bureaucrats \u2013 both clerical and lay.\n\nEventually Josie and son departed to a nun-run hostel and we got occasional postcards, laboriously inscribed in capital letters, telling us of George's progress. (To my father's disgust the child had been named in honour of the King of England.) A few years later Josie called one afternoon to introduce her husband and month-old second son \u2013 who had been born, she happily informed us, during the honeymoon. George was now a fine lad and seemed on excellent terms with his amiable stepfather. Obviously all concerned were going to live happily ever after.\n\nThe Josie drama had provoked much comment throughout the neighbourhood, yet not even she could compete with Cattie. Cattie arrived the day after my father had been immobilised by sciatica. She looked middle-aged but claimed to be twenty-two. She was tall and gaunt and grey-haired and never removed her Wellington boots; when my mother hinted that she might find another form of footwear more comfortable indoors, she snapped enigmatically, 'I has me notions!' A few days later she acquired another notion and took to carrying everywhere, under her arm, a sweeping-brush. Even while bearing laden trays into my parents' bedroom she stuck to her brush; and when my mother \u2013 speaking timidly, at this stage \u2013 suggested that she might find it more convenient occasionally to lay it aside, she snapped, 'It's a need!'\n\nThat night sounds as of tap-dancing came from Cattie's room and large quantities of plaster fell from the dining-room ceiling. Next morning, before dawn, weird rhythmic wails, as of an oriental widow keening, became increasingly audible from the direction of the kitchen. I was thrilled. Indisputably we had a fully fledged lunatic on the premises. But when we held a council of war after breakfast it disconcerted me slightly to realise that my parents took Cattie's overnight deterioration quite seriously. The district nurse was due at ten o'clock to minister to the two invalids \u2013 my father was temporarily almost as incapacitated as my mother \u2013 and we decided to ask her to telephone Dr White.\n\nAt that very moment a shrieking Cattie came storming up the hall and burst into my parents' room brandishing the sweeping-brush. Her face was distorted and she was yelling \u2013 'I'll fork ye! I'll fork ye!' By any standards she was an alarming sight. It soon transpired that she believed my parents to be two fried eggs and the brush a fork. Afterwards I saw the joke, but not then. I rushed to my mother and clung to her and she whispered \u2013 'Fetch the guards!' But such crises prove the strength of the herd instinct. My mother's order made sense, yet I could not leave that room while Cattie was darting about with contorted face poking her fork towards the two defenceless fried eggs. If murder were about to be done, let us all die together. Only when Cattie's expression relaxed, and she began again to tap-dance and to chant quite cheerfully, did I flee onto the street and beg a passer-by to fetch the garda sergeant. Then the nurse arrived and said, 'I told you so!' because for days she had been warning us of our peril. Within an hour the unfortunate Cattie had been removed, under sedation, to the nearest lunatic asylum \u2013 from which she had been discharged, we then discovered, only two months previously.\n\nAfter this d\u00e9b\u00e2cle my mother observed dryly that our neighbours probably regarded it as a 'judgement' on the Murphys. My parents' attitude towards unmarried mothers was condemned by many as a scandalous condoning of immorality. Like middle-class communities everywhere, our neighbours abhorred and feared illegitimacy. And being Irish Christians, their abhorrence was compounded by the uniquely nasty odour they could detect emanating from sexual licence. A few managed to pay lip-service to Christian charity, but not one would have encouraged an unmarried mother to keep her baby. To rear one's bastard was considered far worse than merely having it furtively and quickly giving it away for adoption; allowing the maternal instinct to take over branded one as a brazen hussy. Therefore middle-class girls never did keep their babies. I often heard my parents denouncing this vicious hypocrisy \u2013 one could feel the viciousness vibrating through the anti-Josie vituperations of some of our neighbours. Their moral code was of the primitive sort that seeks confirmation and reinforcement in the merciless punishing of delinquents. A century earlier they would have formed part of the grimly gleeful crowd around the scaffold at a public hanging.\n\nJosie was the cause of my historic argument with Mr Ryan \u2013 'the Boss'. I say historic because it was unheard of for anyone \u2013 never mind a child \u2013 to defy this formidable patriarch.\n\nThe scene was the Ryans' living-room. The Boss was sitting in his symbolically uncomfortable wooden armchair by the fire; I was lying on the hearthrug at his feet reading a book about (appropriately) volcanoes; Mrs Ryan was rolling a skein of knitting wool into a ball and Mark \u2013 the eldest son, home for the weekend because he was then Diocesan Inspector of Schools \u2013 was changing the batteries in the wireless.\n\nSuddenly the Boss began to criticise people who encouraged shameless young girls to display their wickedness in public. Immediately I got the reference though by Ryan standards no child should have known what was being discussed. Thus I was provoked to defiance both by loyalty to my parents and by compassion for Fallen Women in general and Josie in particular. Sitting upright on the hearthrug, I accused the Boss of hypocrisy. This word was a recent addition to my vocabulary and it pleased me to use it, despite what I knew must be the cataclysmic result.\n\nThere was a shattering silence. The Boss and I stared at each other fixedly like belligerent tom-cats. Those ice-blue eyes seemed unnervingly expressionless: years later it struck me that at that moment the Boss may well have been trying not to laugh. But of course he had to fight his autocratic corner and as the argument developed I became very angry indeed. No doubt my opponent was deliberately egging me on \u2013 that would have been characteristic \u2013 and though the words exchanged have been forgotten I perfectly remember the unfamiliar adult quality of the anger that consumed me. It was unlike anything I had ever felt during my tantrums; now I was being angry on someone else's behalf. But unfortunately I was also being outrageously and uncharacteristically impertinent \u2013 which indicates that this incident released much long-repressed hostility to certain aspects of the Ryan ethos.\n\nThe Boss always kept by his side a heavy walking-stick which somehow had the appearance of a weapon, though it may never have been so used, and suddenly he picked this up, shook it at me and said \u2013 'Get out!' Seizing my book I fled, meaning to go straight home. Neither Mrs Ryan nor Mark had taken any part in the argument, but now Mark quietly hurried after me. He laid a hand on my arm and said that he needed help in the orchard. We understood each other so well that the most important things could always be left unsaid. Without even glancing at him I knew that he had approved of my stand against the Boss \u2013 at least in principle, though he would have wished me to use more self-restraint in what I actually said.\n\nIt was a golden October afternoon, following the first of the season's gales. Autumn's cosy\/melancholy tang was spicing the air and the leaves were turning on the neat beech hedge, with an arch at either end, which divided garden from orchard. Under the trees the long grass was still wet, though all day the sun had shone while a romping wind chased white clouds. We collected windfalls in oval wicker clothes-baskets; when full these were carried down to the back gate and from there the 'eaters' would be given away to the children of the town and the 'cookers' to a certain clique known unambiguously as 'the poor women'.\n\nWe scarcely spoke as we moved about under the trees, watchful least we tread on our harvest, bent double, parting the long grass with our hands, then straightening up to remove from an apple its cargo of snails or slugs or earwigs or beetles and to decide whether or not it was still worth saving. We often spent almost silent hours together in the garden, Mark working steadily, I helping enthusiastically if the job appealed and desultorily if it didn't.\n\nTowards teatime Mark suddenly said, 'I think you must apologise to the Boss before you go home.'\n\nHolding a hard, dark green Bramley I looked up at my companion with a mixture of resentment and resignation. It was one of my more disagreeable traits that I usually resisted having to admit a mistake or make an apology. But Mark had a power over me unequalled by \u2013 and never to be equalled by \u2013 anyone else's. Furiously I dug my finger-nails into the apple: they were sore for days afterwards. Then silently I followed Mark down the path and meekly I trotted into the living-room to offer my apologies to the Boss \u2013 who received them with a non-committal growl.\n\nAs I passed through the kitchen, on my way home, Mark handed me a chunk of home-made fudge \u2013 in 1942 a rare treat.\n\nFor my tenth birthday my parents gave me a second-hand bicycle and Pappa sent me a second-hand atlas. Already I was an enthusiastic cyclist, though I had never before owned a bicycle, and soon after my birthday I resolved to cycle to India one day. I have never forgotten the exact spot, on a steep hill near Lismore, where this decision was made. Halfway up I rather proudly looked at my legs, slowly pushing the pedals round, and the thought came \u2013 'If I went on doing this for long enough I could get to India.' The simplicity of the idea enchanted me. I had been poring over my new atlas every evening travelling in fancy. Now I saw how I could travel in reality \u2013 alone, independent and needing very little money.\n\nThis was a significant moment in my life, and not only because of the consequences in the far future. A ten-year-old's decision to cycle to India might have seemed to many adults an amusing childish whim. But by giving me material for dreaming about something that I knew could be attained, it offered a healthier outlet for my imaginings than my usual escapist fantasies. It also gave me a purpose that was, or seemed to be, quite separate from my obsessional desire to write \u2013 which diversification of ambitions was an excellent thing. Naturally I never discussed my plan with anyone; I well knew how it would be regarded by my elders. Nor did I feel any particular urge to talk about it; it had enough substance not to need the reinforcement of conversation. Oddly, it never developed into an obsession: as I grew older months could pass without my consciously thinking of it. Cycling to India simply became part of the pattern of my future. In the same matter-of-fact way many youngsters think of the remote but inevitable day when they will graduate from university or join their father's firm or inherit the farm.\n\nClearly ten-year-olds are not interested in Hindu sculpture or Brahminical philosophy or Sanskrit literature; at that age even the _Jungle Books_ bored me. My only personal links with India were Dr White's nostalgic bedside tales of the North-west Frontier Province and a weirdly impressive painting by a Hindu of the mythical source of the Ganges. This extraordinary picture \u2013 a wedding present to my parents from a friend of the artist \u2013 had fascinated me (so my mother said) since I began to focus in my cradle. Yet one doubts if it had any influence on my cycling plans. I merely wished to travel far beyond Europe for travelling's sake, and taking all geographical and political factors into consideration New Delhi seemed the most interesting Asian capital that could conveniently be approached by bicycle.\n\nApart from future plans, owning a bicycle gave me freedom to roam much more widely than I had ever done before. And within a week of my birthday I had very nearly roamed into the Elysian fields.\n\nOne sunny, frosty December morning I set out to cycle to the foot of the Knockmealdown Mountains, some eight miles north of Lismore. I took a picnic, and ate it by a lively brown stream, and then thought it would be fun to climb to the top of Knockmealdown \u2013 an easy little mountain of just under 3,000 feet.\n\nI had been up several times before, with my father and Pappa and sundry guests, and was familiar with the easiest route. But somehow the climb took longer than expected and as I approached the top the weather began to change. The air lost its crispness and the Galtees to the north-west disappeared as clouds came rolling south over the plain of Tipperary. Before I was halfway down both the clouds and the dusk had over-taken me. But I was too inexperienced to be immediately afraid. For ten or fifteen minutes it all seemed a glorious adventure and I never doubted that I would soon hear the stream and feel the road beneath my feet. Not until darkness came, and the mist turned to rain, and a wind began to moan, did panic threaten. Then I stumbled into an old turf-cutting that should not have been on my route and burst into tears.\n\nPulling myself out of the icy water \u2013 I was soaked through \u2013 I recognised the extent of my stupidity. Plainly I was lost for the night and, though I never doubted that I would get home eventually, my parents could not be expected to think so. It is far from clear to me now _why_ I assumed that I would survive a midwinter night on an exposed mountain without food or shelter. It is precisely this irrational faith in one's own durability which can earn an undeserved reputation for courage.\n\nHaving accepted that I was lost, and that no rescue party would dramatically save me because no one knew in which direction I had cycled, I kept moving for what felt like hours, still desperately hoping to find the road. I tried not to think of what my parents must be suffering. Even at ten, mental or emotional suffering had the power to move me as physical suffering could never do. I was now feeling for them a great deal more than if they had, for instance, been seriously injured in a motor-smash.\n\nAlready I felt weak with hunger, my leg muscles were throbbing and my sodden clothes seemed heavy as lead. At this point my secret endurance tests were justified. I had never inflicted on myself anything comparable to my present trial, yet I believe that by using the techniques I had so often light-heartedly practised I kept moving for longer than would otherwise have been possible.\n\nI was close to collapse when I came on a low stone wall. Knowing my mountain, I realised that I must now be on its east or north side; had I still been on the south side I would have had to cross a road to reach such a wall. From its existence I deduced a cottage at no great distance and felt a resurgence of hope and energy. I groped on eagerly through the darkness, following the wall, and then came not on a cottage but on an unoccupied animal shelter, built of stone and roofed with turf. Inside were great mounds of cut bracken. I stripped naked and buried myself in a mound and not even the thought of my parents' distress could keep me awake.\n\nWhen I opened my eyes the sun was rising and the wind was tidying the clouds away. As I struggled to put on my sodden clothes I felt not only stiff and weak but very ill. Stumbling out of the shelter, I saw a cottage some hundred yards away: but it was deserted. A boreen led down to a narrow road and there I realised I was above Newcastle. Now, in daylight, with my safety assured, I ceased to worry about my parents and felt only a considerable fear of their reaction to my escapade. I sat by the roadside to await rescue and in my debilitated condition the imminence of my mother's just wrath was too much for me. I was weeping dismally when an astonished farmer came along on his donkey-cart and picked me up.\n\nThe rest is a blur. For some reason I was taken to a priest's house \u2013 perhaps there was no local gardai barracks \u2013 and fed and put to bed by the housekeeper. When I came to I was in my own bed, running a high temperature and feeling too terrible even to want to read. And I had to stay in bed for the next fortnight while there were mutterings in the background about pleurisy.\n\nMy parents never once reproached me for having put them through eighteen hours of hell; possibly they considered the experience itself sufficient punishment. It must also have been obvious that I had learned my lesson and would never again embark on such a reckless adventure. I willingly promised to tell my mother, in future, exactly where I was going when I left home for a day's cycling; and this satisfied her, though both Mrs Mansfield and Father Power urged her to put me on a much tighter rein. Curiously, I felt during those years that she molly-coddled me to humiliation by closely watching my diet, making me change my clothes if they got damp and sending me to bed at seven-thirty. Yet short of throwing me into the middle of the Irish Sea in January, and telling me to swim for the shore, she could scarcely have been less fussy about my physical safety.\n\nYears later my mother admitted that despite being frantically worried throughout that long night she had known I was safe. My father, on the other hand, had decided by six-thirty that I had been killed. But apart from notifying the gardai and checking nearby roadsides there was nothing immediate to be done and my mother had discouraged the formation of what could only have been an ineffectual search-party. Father Power and Mrs Mansfield had both spent the night at our house, providing moral support, and by breakfast-time even Mrs Mansfield was not entirely sober. (On Christmas Day, and during periods of extreme nervous tension, she coyly accepted the addition of a little whiskey to her tea. Two teaspoonfuls made her quite merry and three induced a degree of hilarity she would normally have considered most reprehensible.)\n\nWhen the good news was brought from Newcastle, Father Power drove my father to collect me; our own car had been put on blocks by this date. Surprisingly few of the neighbours ever heard of my misadventure and I was grateful to my parents for not publicising it; to have become the laughing stock of the whole town would have been intolerable. Now I feel that in such ways they were over-protective and that it was wrong to spare me this part of my punishment. It would have done my bumptious ten-year-old pride no harm at all to be wounded in such a fashion. I did of course voluntarily confide in Mark, from whom I hid nothing. With him I felt no need, either then or later, to project an image of myself that was an improvement on reality.\n\nBy 1942 I had come to detest those educational Sunday walks so much enjoyed by my father; and one day I decided to use cycling as an excuse to break with tradition.\n\nMy mother looked stricken when I defiantly announced that I would not be going for any more long walks with Daddy because Sundays must henceforth be left free to practise long-distance cycling. This transparent excuse did not for a moment deceive her and after a tiny, tense silence she said quietly. 'But that's absurd. You have plenty of time for cycling during the week.'\n\nThen the truth came out \u2013 or what at that time I imagined to be the truth. 'Daddy's so _boring_!' I exclaimed miserably. 'I can't stand it and I'm _not_ going with him again and anyway I'm _ten_ so why should I be forced to go for walks like a little girl?'\n\nStaring at my mother, I was appalled to realise that she, like myself, was close to tears. Then I knew that she saw my point. But her first loyalty was to a man whose vulnerability was not any the less extreme for being so far below the surface. Recovering herself, she reasoned with me gently, tacitly admitting that I had a case but stressing how unkind it would be to rebuff a loving father who so enjoyed our weekly walks. As I listened I knew that she was right, but I didn't care. Or rather, I was determined to act as though I didn't care. Deep down I cared so much that the guilt bred by this calculated cruelty remained with me for twenty years.\n\nWhy did I feel such an overwhelming compulsion to detach from my father, even though I fully recognised that my doing so would hurt him as perhaps nothing else could? Had our inability to communicate driven me to an extremity of frustration that could only be relieved by meting out the punishment of rejection? For he was the grown-up, the powerful one and it must all be his fault... We might have been basically indifferent to one another, as parents and children quite often are, and then we could have casually sustained an amiable, meaningless relationship. But our bonds were very strong; we understood each other intuitively in a way which to me, on the threshold of adolescence, may have seemed a violation of my spiritual privacy.\n\nI do not know how my mother explained away my decision to my father, or if she even tried to soften the blow. But I never again went for a walk with him and he accepted my defection without a comment, a query or a protest of any sort.\n\nA few months later, while we were all in Dublin, an incident occurred which I now regard, perhaps exaggeratedly, as one of the saddest wasted opportunities of a lifetime. My father had arranged to meet me at an aunt's house for lunch, but he was late; half-an-hour passed, and an hour, and still he did not come. Then it was lunchtime and we all sat down to our grilled cutlets and creamed carrots. There were meringues for pudding, but by that stage I was in a daze of terror and grief. My father must have been killed in a car-crash \u2013 he was a notoriously absent-minded driver and had borrowed my aunt's Morris. I felt certain that I would never see him again and remorse about my cruel aloofness devastated me. But I must not make a fool of myself by fussing and fidgeting in front of all those grown-up cousins who were accepting Uncle Fergus's non-appearance with what seemed to me heartless placidity.\n\nMy father had in fact telephoned at twelve-thirty to explain that he had been delayed, but no one had bothered to inform me. When at last he arrived I nonchalantly said 'Hello' before slipping away to the lavatory to be sick. (Even today the mere sight of a meringue makes me feel queasy.) Then I went out to the garden to talk to the dog.\n\nLater, as my father drove me back to Charleston Avenue, I desperately wanted to tell him about all that I had suffered at lunchtime because I thought he was dead. But I could not. Many years passed before I realised that even a slight reference to my ordeal might have significantly altered our relationship at a crucial stage. After that summer, we grew further and further apart on the surface while retaining our indestructible and uncomfortable mutual flair for reading each other's thoughts.\n\nAs my leg muscles grew stronger my cycling ambitions grew bolder and soon I was longing to cycle the twenty-five miles to Helvick Head. But fifty miles in one day sounded a long way. Frighteningly long, for one who had never yet attempted more than thirty. The project began to worry me. I passionately wanted to achieve those fifty miles yet I dreaded failure. It would be so un-live-downable for ever if I could not make the last few miles and had to signal for help. Eventually I mentioned the idea to my mother, very casually, as though it were a matter of no great consequence.\n\n'Probably I could easily do it,' I said; and years later she told me that as I spoke I looked at her with an expression of the most pathetic doubt and anxiety.\n\nBut she kept the conversation on the casual note I imagined I had sounded. 'Of course you can do it,' she said cheerfully, 'if you want to do it enough.'\n\nThus encouraged, I left for Helvick at six o'clock on a radiant June morning \u2013 a morning all blue and gold and green. The air smelt damp, warm, rich and full of promises. From every tree, bush and hedge came the harmonious confusion of bird-song, seeming to celebrate my own joyous excitement. As I turned towards the coast, and settled to the rhythm of pedalling, I experienced an exaltation I have never forgotten. The vigour of my body seemed to merge with the eager abundance all around me and in an almost sacramental way I became totally aware of myself as a part of nature. Unconsciously, I had taken another step away from the faith of my fathers.\n\nOn the previous evening the wireless had guaranteed good weather, but my mother pretended not to have heard. Before I went to bed she gave me the first pound note I had ever been in charge of (to be returned if not needed) and remarked that should the weather deteriorate, or my bicycle break down, or some other disaster occur, I could spend the night in a Dungarvan hotel.\n\nIn fact there was no hitch of any kind. By six-thirty I was home, nauseated with exhaustion and bursting with pride. The last twelve miles had been torture, but I would not tarnish the glory of my achievement by admitting this. And even during that terrible final stage part of me had relished the sense of power derived from driving my body beyond what had seemed, at a certain point, to be its limit.\n\nYet without my mother's moral support I would never have had the courage to attempt that trip at that age. It was fortunate for me that she was not as possessive as she was dominating. Her influence over me was so strong throughout childhood that had she wished to destroy or stunt me she could certainly have done so. Instead, I was aware of being regarded \u2013 and respected \u2013 as a separate personality rather than as an incidental appendage to the adults in the household. At the time I took this for granted: now I know what an uncommon attitude it was in the Ireland of my youth.\n\nSome people imagined that my unusual upbringing was a result of being the only child of an invalid. But my mother's mothering would have been no less odd, I feel certain, had she been in rude health with a family of ten. As a perfectionist, and a woman who saw motherhood as an important career, she approached child-rearing in what I can only call an artistic spirit. Given as raw material a newly conceived child, she saw it as her duty and privilege to form an adult who would be as physically, mentally and morally healthy as intelligent rearing could make it. Physically she was completely successful. The other aspects of a child's health are, alas, less amenable to maternal regulation.\n\nMy childhood diet was generally considered freakish. Up to the age of sixteen I drank four pints of milk daily and was allowed no tea, coffee or fizzy drinks. Sweets, chocolate, ice-cream, cakes, sweet biscuits, white bread and white sugar were also forbidden. My 'treats' were muscatel raisins and fat, glossy dates in gay boxes. Included in my staple diet were raw beef, raw liver, raw vegetables, wholemeal bread, pinhead porridge and as much fresh fruit and cheese as I could be induced to eat. Naturally I flourished. And one can understand what it must have meant to my mother to look at me as an adolescent and to know that I was capable of enjoying, to the fullest extent, what she had lost. \n\n#\n\nFanciful though this may sound, the Blackwater River was among the chief and best-loved companions of my youth. To me, it has always seemed Lismore's most tangible link with the saints and sinners and scholars of the past. Many centuries ago it was most appropriately known as _Nem_ , an Irish word meaning 'Heaven'. Much later, Spenser mentioned it in _The Faerie Queene_ \u2013 'Swifte Awnaduff which of the English man is Cal'de Blackewater'. Later still, some enthusiastic Victorian tourist (Thackeray, I think) decided to rename it the Irish Rhine and this inanity \u2013 as absurd as calling Swat the Switzerland of the East \u2013 has earned him the undying gratitude of the Irish Tourist Board.\n\nThe Blackwater \u2013 one of Europe's great salmon rivers \u2013 rises near Killarney and flows for seventy miles through the countries of Kerry, Cork and Waterford. In the twelfth century both Dromana House and Lismore Castle were granted charters entitling their owners to extensive fishing rights and even now these charters of King John of England remain good in law, much to the annoyance of certain local rod fishers.\n\nA river shows different aspects to the fisherman, the naturalist, the trader, the artist, the soldier, the boatman and the swimmer. I formed my relationship with the Blackwater as a swimmer. Before I can remember, my father regularly immersed me in the cool, dark silkiness of its depths and I swam almost as soon as I could walk. It is a good thing to have had a river among one's mentors; its strength develops the body, its beauty develops the soul, its agelessness develops the imagination. Also, its moods teach respect for the mindless power of nature. The Blackwater is very moody: it has deep holes, sudden floods, hidden rocks, tricky currents and sly weeds. It claims at least three lives a year and I was not allowed to bathe alone until I was twelve. Although I could easily have broken this rule without being detected, it never occurred to me to do so.\n\nOur shared devotion to the Blackwater had always been important to my father and myself. It was not simply that we were both keen swimmers; our bathing was as much a rite as a pastime and during the summer, whether the weather was summery or not, we met outside the Library at five-thirty every afternoon and went together to the river. But at the beginning of this summer of 1942, only a few months after I had spurned those Sunday afternoon mobile lectures, what was to become of our traditional bathing-rite? I could not decently imply that I was now prepared to endure my father's company as a convenience. Yet if the custom were allowed to lapse my swimming season, which normally opened in mid-May, would have to be postponed until Pappa arrived at the beginning of July. My father might have been forgiven had he chosen to leave me excruciatingly impaled on the horns of this dilemma. Instead, he remarked at breakfast-time one fair May morning, as he had been remarking on such mornings for as long as I could remember, 'I think we'll need our togs today'. This was much more than I deserved, and I appreciated it.\n\nA new phase of our relationship had begun. I was consciously in control and my father no longer tried to be educational without direct encouragement. Astronomy was then one of my main interests and it pleased me to be lectured on it day after day. Probably an observer would have detected no strain as my father and I considered the solar system. Yet a great sadness underlay our relationship, an awareness that somehow we had failed each other and that what now existed between us was merely a civilised fa\u00e7ade to conceal failure.\n\nAnother of my hobbies at that time was the Black Death and related subjects. For a few years past I had been fascinated by diseases, epidemics, surgery, new medical discoveries and the like. Had I not been so committed to the writing life I would have wished to be a surgeon. My interest in corpses and skeletons was profound. I had never actually seen a human corpse, but I longed to observe closely the phenomenon of putrefaction. For this purpose I installed a dead rabbit in my bedroom. My observations, however, were unsympathetically terminated when the rabbit reached an interesting and therefore perceptible state. In the same cause, I cultivated the society of an aged British Army pensioner and begged him to describe in detail all the corpses with which he had become acquainted during the First World War. But he was tiresomely evasive, plainly considering me mad and morbid. So I had to make do with disintegrated skeletons, which could not even be brought home for study. At the beginning of my osseous phase I had pranced into the dining-room one lunchtime brandishing a skull and expecting my parents to greet it rapturously. But my father had declared the appropriation of human bones to be unseemly, irreverent and possibly unhygienic and I was made to return my trophy to its source without delay. Fortunately \u2013 because I was hungry at the time \u2013 its source was nearby. In those days the ancient graveyard surrounding St Carthage's Cathedral was a wilderness, full of briars and romantic melancholy, and by visiting it shortly before a burial one might find, beside the newly dug grave, a femur, a few ribs, a length of spine or even, on very good days, a skull. My ambition was eventually to assemble a whole skeleton by hiding my bits of bones in the furthest corner of the graveyard and adding to them from other ancient graveyards in the area, when the local paper informed me a burial was about to take place. But it was all too difficult and in the end my Identikit skeleton came to nothing. Or rather, it came to an embarrassingly large heap of bones which, belatedly inspired by some flicker of respect for the mortal remains of various people, I surreptitiously transferred to an open grave on a wet December afternoon when no one was likely to be about.\n\nScientific interest was not of course the only motive for my graveyard prowlings. Children are enthralled by mortality and unlike many who have felt the touch of Time, are able to contemplate it detachedly. As a child with no personal experience of bereavement I was thrilled by the dramatic finality of death and fascinated by the mystery of what follows after. Many an hour did I spend sitting on old tombstones cheerfully reviewing eschatological possibilities.\n\nI had long since rejected the harps, angels, massed choirs and other such tedious impedimenta which furnish the Heaven of Christian folklore. My parents had carefully explained that this picture of Heaven must not be despised since it represented the honest endeavour of simple people mentally to conceive the inconceivable. The real bliss of Heaven, they went on \u2013 and they obviously believed this \u2013 was in spiritual union with God. And the real pain of purgatory was in spiritual separation from God. But when I considered the matter, as I relaxed on a tombstone or peered hopefully into an open grave, it seemed to me a good deal easier, and no less consoling, to believe in union with Nature rather than with God. One could see it happening \u2013 'dust to dust' \u2013 on the physical plane. And when one thought, for instance, of wireless waves, it appeared there were enough odd things going on in the natural world for it to accommodate also the immortality of the soul. This immortality always made sense to me, but at no age did I find it necessary, despite my basic arrogance, to think of my own soul maintaining its existence for ever as a separate unit indelibly marked 'Dervla Murphy'. Physical extinction was an unpleasant thought and one saw the need for a comforting belief in an after-life. Yet death was made no more acceptable, to me, by the traditional Christian strivings to bring within our comprehension what is simultaneously admitted to be incomprehensible. When I eventually came upon it, that school of Buddhist philosophy which suggests that after death the individual soul can dissolve, to continue its existence by forming parts of other souls, suited me much better. And not many years ago, in _The Golden Core of Religion_ by Alexander Skutch, I found perfectly expressed what I was beginning to grope towards as a ten-year-old: 'If immortality is, or will become, attainable by the human soul, it must be within the possibilities of that great, all-embracing, infinitely varied, and still imperfectly explored system of orderly, interrelated events which we know as nature. Only by regarding spiritual survival as natural, in the same sense that our birth, our thought, our aspiration, and our body's final dissolution are natural, can we who have been nurtured on science and philosophy hold faith in it. If the spirit survives its body in the course of nature, as in the course of nature the light from a beacon on a hilltop goes coursing through outer space long after the fire has died, then it is reasonable to believe that its survival depends upon such intrinsic qualities as the intensity of its love, the unity of its aspirations, its coherence and the absence of passions that tear it asunder.'\n\nAs for Hell \u2013 which Irish Catholics are so regularly reminded of as a possible destination \u2013 having put behind me the terrors induced by Sister X, I ceased to take it seriously. I agreed with my parents that one cannot accept the paradox inherent in the concept of a God who is infinitely merciful and just and yet condemns countless unfortunates to an eternity of suffering.\n\nA few weeks before my eleventh birthday I was cured of grave-robbing by a somewhat macabre experience. It was one of those still, moist, dull November afternoons when the countryside can be felt drifting into winter sleep. Cycling by the edge of a thick wood, I suddenly got a strong scent of badger. For years I had been longing to find a set, so I climbed a low stone wall and, sniffing like a terrier, forced my way through the dense tangle of rhododendrons, holly and briars that flourished beneath the trees. But soon I lost the scent and as my clothes had already been much damaged I decided to return to the road by a less destructive though longer route.\n\nZig-zagging between the trees, looking out for squirrels, I moved slowly uphill towards the track that I knew bisected this wood. Then I came unexpectedly on an odd little building, standing in a small clearing but half-hidden by briars and laurels. It had a vaguely ecclesiastical appearance and I felt both puzzled and uneasy; there was something faintly sinister about this inexplicable edifice lurking in the depths of a dense, deserted wood. Approaching closer, I noticed that the door had been forced open \u2013 quite recently, for the wood around the lock bore fresh scars. Advancing to the threshold, I peered into the gloom. As my eyes adjusted I saw big shelves and big, long boxes. Coffins, in fact. They had been hacked open, stripped of their lead and left in disarray. Then I saw the corpse, lying almost at my feet. It was dark-skinned and shrunken but very plainly a woman. Here at last was my yearned-for chance to make a close study of decomposition. I turned and fled.\n\nTerror seemed to suffocate me as I tripped over brambles and slipped on the dank leaf-mould. The whole wood became an Arthur Rackham thicket and I was afraid to raise my eyes lest I might see discoloured corpses enmeshed in its thorny shadows. Only when I reached the road did I realise that my face as well as my clothes had been ripped by briars. Blood was trickling onto the collar of my gaberdine and I wondered frantically how I could explain away those scratches. I decided to fabricate a fall off my bicycle to cover both the scratches and my shaken condition.\n\nWhy did this desecrated family vault so unnerve someone who hitherto had revelled in the gruesome? Was I so shocked and terrified simply because I had been taken unawares? Would I have reacted differently had I entered the wood not to track a badger but to seek out the vault? Or was I undone by the considerable difference between a skeleton and a corpse? There is always a temptation to try to unravel one's own inconsistencies and I still find this incident baffling.\n\nIt took me months to regain my nerve fully. I slept soundly that night \u2013 no doubt exhausted by emotion \u2013 but as I opened my eyes next morning the memory came back and I groped for my light switch in panic.\n\nNightmares started that evening; these were, according to my mother, the first from which I had ever suffered. When I awoke screaming my parents were listening to the gramophone and no one heard me. Realising where I was, I switched on the light and felt glad that my silly yells had not been noticed. But then I was afraid to switch off the light or to sleep again. I tried to read but was too tired and tense. Soon afterwards my father came upstairs, saw my light and looked into demand sternly what I meant by reading at half-past ten? He switched off my light without waiting for a reply and left me sweating and shaking in the dark. When I turned on my illicit torch it didn't really help. I needed the full glare of the ceiling light which left no corner shadowy. My nightmare had been of shadows, and vilely coloured objects \u2013 brownish-grey, yellowish-green, yellowish-brown, greyish-green, yellowish-grey \u2013 a kaleidoscope of unearthly, corrupt tinges and indistinct forms moving slightly. And yet surely _not_ moving, because they were dead... (Ever since, most of my few nightmares have been in this ghastly Technicolor.)\n\nWhen I knew that my parents would be in bed I switched on the light again but, being used to a dark room, was unable to sleep for what felt like hours. Then I dreamt that my mother was dead and that I was searching for her body in a cave where at the far end corpses were dancing in a circle. I knew they would not let me pass though my mother was waiting for me beyond them and, if I reached her soon enough, could be brought back to life. I awoke, calling her hysterically, to find that it was past my usual getting-up time. As I dressed I was already dreading that night. Half of me longed to confide in my mother, but the other half forbade me to ask for help.\n\nHowever, unlike my earlier fear of darkness, these horrors were altogether outside my control. Only by forcing myself to stay awake could I escape them. My symptoms of strain and exhaustion soon prompted my mother to investigate and when all had been revealed we decided that until my nerve had mended there was no alternative but for my father and me to exchange beds. Part of me rejoiced then to be free of those lonely terrors, while another part resented this admission of dependence and the consequent loss of privacy. My own room was important to me beyond calculation. The rest of the house \u2013 indeed, the rest of the world \u2013 seemed in a sense alien territory where adult writ ran; only in my own room could I freely expand. (Of course when I began to cherish decomposing rabbits under the bed my expansion had to be curtailed; but such crises were rare.) Thus, sleeping in my parents' room felt like the worst sort of indignity, with overtones of serfdom. I remember resolving one night, as I lay discontentedly curled up in my father's bed, that even should I happen to acquire a husband, by some unlikely chance, I would never share a room with him. Naturally we would share a bed for procreative purposes, but clearly there must be a clause in the marriage contract stipulating separate rooms. It did not then occur to me that post-procreation I might feel disinclined to trek to an Inner Sanctum.\n\nAt the age of eleven my bedtime was still seven-thirty, bizarre as this may sound to modern children, and usually I had been asleep for a few hours when my father wheeled my mother's bath chair across the hall. (My parents' bedroom would normally have been the sitting-room and the dining-room had to serve as our general living-room.) With the maid's help my mother was lifted onto her bed; she had lately become so heavy that no one could lift her unaided. Then, after the maid's departure, my father gently undressed his wife, gave her the bed-pan and made her comfortable \u2013 as they say in hospitals \u2013 for the night. Once she had been placed in a lying position she could move only her head so it was essential that she should be left completely relaxed. To achieve this, with the aid of strategically placed cushions under her locked knee-joints and ankles, involved much patient effort on my father's part \u2013 and much fortitude on my mother's, since at this stage of her disease every movement was painful. To me, of course, the whole thing was routine; if I chanced to waken I remarked neither my father's patience nor my mother's fortitude.\n\nOne night, not long before Christmas, loud sobs came from my mother instead of the usual subdued chit-chat about Thomas Aquinas or Balzac or whoever. Confusedly I diagnosed another nightmare; then I woke fully and accepted that this was reality. After a moment my mother began to talk fast, in an unfamiliar, blurred voice, and very cautiously I peeped out from beneath the blankets. She was virtually unrecognisable: flushed, incoherent and \u2013 this was the dreadful thing, unimaginable yet true \u2013 not in control of herself.\n\nIt was as though the mountains had toppled into the valley or the sun fallen out of the sky. I must somewhere have seen somebody drunk; at all events, I knew what was wrong. But this did not help. How could my own mother, the very epitome of composure, have been reduced to such ignominy within the few hours since I had kissed her good-night?\n\nMy first positive reaction, after those worse-than-nightmare moments of fear, bewilderment and grief, was a determination that my mother must never know that I had witnessed what seemed to me her degradation. I could see no meaning in such hideous chaos. But I was convinced that for her to know that I knew would compound the degradation. And though I abhorred this travesty of what I honoured, I did most keenly feel compassion.\n\nAt first my compassion was perhaps no more than a reflection of what I had seen on my father's face as he bent over his wife that night. But it soon became a great deal more. My mother drunk was a goddess with feet of clay, and viewing her as a demoted deity ultimately strengthened both my love and my respect. Inflexibly stiff upper lips can stunt sympathy within a family. Had I never glimpsed my mother with her defences down I might never have been able to measure the demands made hourly on her courage.\n\nNext morning all seemed as usual though my unfortunate mother must have been feeling very unusual. My father knew that I had been awake and no doubt considered discussing the psychology of what had so disturbed me. He could easily have made me understand that what I had seen was not in the circumstances abnormal, however regrettable. Yet more than our personal barrier prevented any such discussion. We would both have felt disloyal and thus, for me, one tension would have replaced another. Probably my father appreciated this; and the avoidance of a subject which distressed us both came to form between us one more subtly strong bond of unacknowledged intimacy.\n\nI never again \u2013 during childhood \u2013 saw my mother drunk; but throughout the following weeks I waited uneasily, night after night, for the horror to recur. Even after my return to my own bedroom in February I remained on the alert and every few months, over the next year or so, I knew the horror was happening. By the end of that time it had become a grief rather than a horror because pity had replaced incomprehension and disillusion. Then gradually I realised that a battle had been won and that I need not be anxious any more.\n\nWhat was the battle? What combination of stresses forced my mother to seek a release incompatible with her character if not with her heredity? The greater part of her life was of course one unending battle and of its phases and inner agonies I know almost nothing; I never had that insight into her nature which I had into my father's. Only her husband knew and this was as it should have been. But now, looking back and guessing as one might guess about some figure in history, I would say that at this time my mother was feeling especially acutely her inability to have another child.\n\nAlso, what had by then become the problem of my education was beginning to cause a rift \u2013 if that is not too strong a word \u2013 between my parents. I was to have gone away to school in September 1942; for as long as I could remember that date had been fixed in my mind. Yet in December 1943 I was still at home \u2013 aged twelve \u2013 because the servant crisis had become chronic. This grievously worried my father. My mother also was concerned, as she was soon to prove, but by temperament she was not a worrier. Moreover, she attached less importance than my father did to the academic education of girls; and she had already assessed my potential accurately enough to know that I would never shine very brightly in the intellectual firmament, whatever opportunities were given me. Thus my parents were no longer in perfect agreement, as parents, and this, added to the strains of our everyday life at the time, must have affected their whole relationship.\n\nBy Christmas 1943 my own feelings about the situation were mixed. I dreaded another bout of homesickness yet had come to regard boarding-school not only as a glamorous adventure but as an escape from my domestic duties. For almost two years I had been acting as general servant, under my mother's direction, during the \u2013 lengthening \u2013 intervals between maids; and it had been conclusively proved that I was devoid of whatever virtues and talents go to make a good housewife. My duties were of course limited: I shopped, cooked, washed up and lit the fires. Nobody cleaned, except at weekends, when my father abstractedly pushed an Electrolux over the more obvious floor surfaces. Luckily he enjoyed cooking and was good at it, even under Emergency conditions, so we had edible meals at weekends. But from everyone's point of view these maidless interludes were trying.\n\nAs fuel and power were strictly limited all our cooking had to be done on two tiny electric rings in a damp kitchen that throughout the winter felt colder than out-of-doors. If both rings were used simultaneously livid blue flames leaped from the wall and the whole house was used. As electric elements could not be replaced, my father improvised repairs of which he was very proud. Inordinately proud, my mother thought, since his ingenuity had rendered our rings potentially lethal. We then had to emulate Cattie by always wearing Wellington boots while cooking.\n\nWhy do certain utterly insignificant moments stick in the memory? For some reason I distinctly remember standing in the kitchen wearing Wellingtons and a heavy brown tweed overcoat and stirring a pot of chicken soup while reading _Clouds of Witness_. Household chores were not allowed to encroach unduly on the real business of life. By then I had long since perfected the art of peeling apples, scraping carrots or tailing sprouts without ever lifting my eyes from the page. But during very cold spells my mother insisted on my doing all 'portable' jobs, such as preparing vegetables, beside the feeble fire of damp turf that smouldered in the dining-room. This meant having improving books read to me while I scraped, peeled or chopped. I still associate the smell of celery with the storm that broke over Messalonghi \u2013 as described by Andr\u00e9 Maurois \u2013 a few moments before Byron's death.\n\nI was now at an age when most juvenile bookworms have voluntarily turned towards Dickens and the Bront\u00ebs, but my literary tastes remained woefully undeveloped: William and Biggles were being betrayed only for the sake of Lord Peter or Sherlock Holmes. Nor was this, as might be expected, a reaction against parental expectations. No pressure was ever directly put on me to read the 'right' books; even my rather idealising father had to recognise that mentally I was slower than average and that pressure could only be counter-productive. But meanwhile my mother continued to tutor me in her own unorthodox way. By the time I was ten she had aroused my interest in many of the great writers, musicians and painters \u2013 _as people_. She had a gift for discussing their characters as though she were gossiping about the neighbours and years before I approached their work I had strong views about them as individuals. To some extent this must have subsequently influenced my literary judgements. It may be no coincidence that I have never greatly cared for the works of Richardson, Balzac or Dickens \u2013 none of whom I could warm to, as men \u2013 while I became passionately addicted to Fielding, Shelley, George Eliot and Wilde, all of whom I had admired and loved from early childhood. On the other hand, though I found Sterne, Meredith and Ruskin personally unsympathetic, _Tristram Shandy_ and _The Egoist_ remain to this day among my favourite novels and for years I had an irrational reverence for every word written by Ruskin on any subject; there can be few of my generation who had read the entire Collected Works from cover to cover by the age of twenty. (I cannot even remember now _why_ I became so addicted to him.) As for Dr Johnson \u2013 Boswell's _Life_ was my mother's other bible, and the doctor with his Mrs Thrale and his Hodge, and all his oddities and kindnesses and pomposities and aggressions and profundities, seemed almost to belong to our own household.\n\nMy mother's immersion in the lives of the great was obviously a form of escapism from the narrowness and dullness of Lismore's social circle. Unlike my father and myself \u2013 both essentially of recluse material \u2013 she enjoyed the art of conversation and must have felt acutely her lack of congenial company. Her other great consolation was music, but that did not provide the stimulus of such psychological puzzles as 'Why did Ruskin marry?' or 'Was Tolstoy technically a sadist?' or 'Is T. S. Eliot jealous of Hardy or just too limited to appreciate him?' Yet it is probably true that music was her greatest, as it proved to be her most enduring, consolation.\n\nMy own awakening to music was an experience only comparable to first falling in love. From the age of a few hours, as the reader may remember, I had been exposed to music. Yet for eleven years it remained to me no more than a noise \u2013 neither pleasant nor unpleasant but so important to my parents that it must _never_ be interrupted unless the house itself was demonstrably on fire. (I had once spoken during a wireless concert, to announce that a chimney was on fire, and had curtly been told not to mention such trivia until the interval.)\n\nThen came that unforgettable moment. It was on a stormy January evening and from my bedroom window I was gazing at a flaring sunset of crimson and gold and purple and orange. As I watched my mother began to sing in the room below \u2013 something she was apt to do at any moment for no apparent reason. But on this evening an unfamiliar excitement possessed me. My heart began to race and I felt as though I had moved into another world \u2013 a world where the human spirit enjoyed a freedom I had never before been able to imagine, a world of infinite mystery and yet of infinite clarity and simplicity. That my musical awakening should have come through my mother's voice rather than through the gramophone or the wireless was scarcely a coincidence.\n\nMy first appearance in print came a few months later. Mark had drawn my attention to a children's essay competition in a weekly provincial paper. Prizes of seven-and-sixpence, five shillings and half-a-crown were being offered for the three best essays submitted weekly. Competitors must be under sixteen and were free to choose their own subject. I had at once protested that I could not possibly win. 'Rubbish!' said Mark. 'Go home and try.' So I wrote five hundred words on 'Picking Blackberries', in prose as purple as blackberry juice.\n\nThe _Cork Weekly Examiner_ came out on Fridays and I counted the days and then at last was standing in the newsagent's shop unfolding the paper with trembling hands. Looking down the pages I felt the nausea of suspense and could scarcely focus. Then I found the competition corner. My heart leaped like a salmon at a weir. The unbelievable had to be believed; Dervla Murphy had won first prize (aged twelve). And to crown her glory the other winners were aged fourteen and fifteen.\n\nI moved out of the shop and stood on the Main Street, dazed with triumph, reading _myself_ in print. Then, sickeningly, disappointment came. Three words and the structure of a sentence should have been changed before I posted my entry. It might have won first prize, but it was feeble \u2013 very feeble. Even had those changes been made, it would have been only mediocre. I resolved to forget 'Picking Blackberries' and do something better. I had yet to learn that one never writes anything of which one does not feel ashamed on seeing it in print.\n\nBut of course 'Picking Blackberries' could not be forgotten just like that. However mediocre, it _was_ in print and had earned me seven shillings and sixpence, the largest amount of money I had ever acquired in one day \u2013 even more than the price of a new Arthur Ransome, because my father could get books at trade rates. Eager to share my victory with my parents, I pedalled quickly up the morbidly named Gallows' Hill to the County Library.\n\nNot until I sat down to write this chapter did I see the significance of that action. The newsagent's shop was equidistant from our house and the Library, and I might have been expected to hurry home to tell my mother first \u2013 she with whom I habitually discussed my literary endeavours. Given the seriousness of my approach to writing, this tiny achievement was to me of enormous importance. And my impulsively choosing to share it first with my father must, I think, be interpreted as a salute to our special closeness \u2013 if not actually as an indirect gesture of atonement.\n\nMy parents were suitably impressed by my breaking out in print, but when I went on to win this competition five weeks running they became uneasy. Finally my father decreed that I must compete no more. I saw the point and reluctantly agreed to retire \u2013 but not before using the situation as a lever to raise my pocket-money from three to four (old) pence a week.\n\nAs a precaution against what Mark called 'swollen-headery' my mother reminded me that despite having taken the _Cork Weekly Examiner_ literary world by storm my apprenticeship was going to be a long one. She need not have worried. I was well able to assess for myself the quality of my rivals' work. \n\n#\n\nThe year 1944 was marked by some strange experiences. These I have already described elsewhere,* but their significance was such that they cannot be omitted from any account of my childhood.\n\nOne harsh, dark March afternoon a squealing hinge made me look through the kitchen window. A young man was entering the cobbled yard from the abandoned cinema, which meant that he had climbed our eight-foot garden wall. Yet I felt not at all alarmed, possibly because I was never prone to be made uneasy by the unconventional. Or it may simply have been because the young man looked so amiable and vulnerable. As he stood at the back door I noticed that he seemed rather apprehensive and very tired. He was tall, broad-shouldered and handsome, and I marked a Kerry brogue when he gave his name as Pat Carney and asked, diffidently, if he might see one or both of my parents.\n\nIn the living-room my mother's bath chair was close to the sulky wartime fire of wet turf. She seemed oddly unsurprised by our visitor's original approach route and my curiosity was further sharpened when she asked me to leave her alone with Pat. Ten minutes later Pat was back in the kitchen. He said that he had been invited to stay for a few days and that my mother would like to speak to me. Then, pausing inside the living-room door, I did begin to feel alarmed. Never before had I seen my mother looking so distraught. Mrs Mansfield and San Toy were coming to tea so there was no time to waste on euphemisms. In a couple of sentences I had been told that Pat was on the run, wanted for the murder of a Dublin detective-sergeant. He had come to us as a prot\u00e9g\u00e9 of my father's elder sister who, never having recognised the validity of the post-Treaty Irish government, was an active member of the illegal IRA. On no account must any caller be allowed to see our guest or any trace of his presence. As I continued to stand by the door, paralysed with astonishment, my mother made a gallant preliminary bid to sort out the ethics of the situation. 'This young man is a criminal though he regards himself as a patriot. No doubt his elders are chiefly to blame. They are using his muddled, foolish idealism. But we can talk about it later. Now please show him his room and give him a meal.'\n\nI walked down the hall in a joyous daze. This was the stuff of which fantasies are made, yet now it had become part of the reality of my own life. I was to prepare a meal for a man on the run who would be hanged if caught. My mother might have saved her breath. Of course Pat was not a criminal, or muddled or foolish. He was a most glorious patriot, heroically dedicated to the reunification of Ireland. No one had ever suggested that my grandfather and father were criminals because they belonged to the Old IRA. One had to be logical. I was badly jolted when I discovered how strongly my father disapproved of Pat. But then I reflected that he (my father) was very old (forty-three). And I made allowances for the fact that at that age some people just can't have the right reactions any more.\n\nListening to my parents, I gathered that for some days they had been half-expecting Pat without knowing exactly why he was on the run. Now they were disagreeing vigorously about how they should deal with him and it gave me a certain sardonic satisfaction to observe them both being inconsistent; at twelve, one likes the feet of clay to appear occasionally. My father should have been the one to welcome \u2013 or at least tolerate \u2013 Pat, while my mother (given her ancestry) should have been the one to reject him. Instead, my father coldly argued that it would be sinful to shelter someone who had deliberately killed an innocent man in the course of a seditious campaign against a lawfully established government. And my mother warmly argued that it was unthinkable, sinful or no, to betray someone whose coming to our home was an act of faith in our humanity. She insisted that allowances must be made for Pat's sick idealism. To which my father, sprung from generations of rebels, replied austerely that it would prove impossible to govern the state if hectic emotionalism were to be accepted as an excuse for murder. My mother then suggested that he should go at once to the gardai barracks and report on Pat's whereabouts. But he didn't.\n\nMy parents seem never to have debated the ethics of capital punishment; presumably they accepted it, in theory, as the appropriate penalty for murder. Yet had it not been employed in Ireland during the Forties, as part of the government's anti-IRA campaign, they might well have refused to succour Pat. When such a decision can lead directly to a death sentence it requires more moral courage, or moral arrogance, than either of my parents possessed.\n\nFrom their point of view an awkward situation was being compounded by the need to impress on me that giving refuge to Pat did not mean condoning his crime. In the end they gave up pretending to unravel this tangled skein for my benefit, which was sensible of them since I well knew that they were incapable of unravelling it for their own. I had in any case already come to my own conclusions and was only listening to their dutiful dissertations out of politeness. Yet I vaguely sympathised with their discomfiture; though they repudiated Pat as a violent man their consciences compelled them to allow for the fact that he saw himself as a soldier fighting a just war \u2013 a dilemma that in present-day Northern Ireland has again become familiar to many.\n\nPat stayed with us for a fortnight, but he and I never referred directly to his peculiar status and he made no attempt to influence me politically. We played round after round of rummy and he gave me lessons in map-reading and taught me how to whistle through my fingers so piercingly that I can be heard two miles away. To me this marvellous companion seemed a magic sort of person, an intelligent grown-up who had retained all the wondering enthusiasms of childhood. And my intuition was right. It was Pat's tragedy that he had never outgrown either the innocence or the ruthlessness of youth.\n\nMy parents also became very fond of Pat and deeply concerned about him. Night after night they argued patiently in futile attempts to make him see the error of his ways. Soon he seemed a member of the family \u2013 quite an achievement, in view of the strains imposed on all the adults concerned by his presence in the house. Had he been detected under our roof, my father would not have perjured himself by denying any knowledge of his identity and so would certainly have been imprisoned \u2013 as his sister was soon to be, on Pat's account.\n\nOur guest was careful never to go too near a window and any knock at the door sent him rushing upstairs. Remembering how I relished all this melodrama, I wonder now if I fully understood that we were truly dicing with death. But my light-hearted approach may well have helped by easing the tension generated between the three adults. Pat knew that I took the game seriously enough to keep all the rules, and I was made deliriously proud by his entrusting to me the addressing and posting of his letters. One morning I went into his room to leave fresh linen on the bed and saw an automatic by the pillow. For years this was to rank as the most thrilling moment in my life. I tingled all over at the romance of that weapon \u2013 symbol of Adventure! \u2013 gleaming black and lethal on the white sheet. It never occurred to me that in certain circumstances it could be used to kill the local gardai, the fathers of my playmates. But then it was impossible to associate the gentle, considerate Pat with any form of violence or cruelty.\n\nOne evening Pat said goodbye instead of good-night and when we got up next morning he was gone. We heard nothing of him for several months, but his luck did not hold. He was eventually captured in my aunt's house, while asleep, and tried in Dublin before the Military Tribunal. Then he was hanged by the neck until he died, at eight o'clock in the morning on December 1, 1944. His real name was Charles Kerins.\n\nIn Waterford city, where I was by then at school, December 1, 1944, was a morning of violent wind and slashing rain. Just before eight o'clock I was queuing for my breakfast. I knew of Pat's attitude towards his sentence and at the moment of his hanging, when the gong in the hall was signalling us to enter the refectory, I experienced an almost hysterical elation. Then, curiously enough, I ate my usual hearty meal. It was against the nationalist tradition in my blood to mourn such deaths, for that would have been to imply that the sacrifice was not worth while.\n\nOur mail was distributed during the mid-morning break and two worlds met when I stood amongst my classmates and \u2013 while they chattered of hockey and drank their milk \u2013 read a letter from a friend who had been hanged three hours earlier. With his letter Pat had enclosed a silver ring made on the prison ship in Belfast by a comrade of his, 'Rocky' Burns, who was later shot dead in Belfast by the RUC \u2013 or perhaps the B specials. I wore the ring constantly from that moment until my fingers and my ideals outgrew it \u2013 developments which conveniently occurred at about the same time. But I have it still and I would not part with it.\n\nA few days later I had a letter from the aunt in whose house Pat had been arrested. She wrote:\n\nMy dearest Dervla,\n\nThere is no need to tell you that Charlie Kerins met his death with the greatest possible courage and bravery. I was allowed to visit him on Wednesday and Thursday last and he gave me courage, too. I am more than sorry that you could not have seen him \u2013 he was so proud and happy to die for Ireland that one could not feel depressed \u2013 sad indeed \u2013 heartbroken \u2013 but not depressed.\n\nI spoke to the priest who heard his confession and he told me that it was a privilege to meet him and that he had no doubt whatever he had gone straight to Heaven. He offered his life with Our Lord for all the people of Ireland. He had no bitterness against his enemies. For the week before his execution he heard Mass and received Holy Communion every morning. On the very morning he was hanged he sang two songs for the wardens, 'Kevin Barry' and 'Kelly the Boy from Killane'. As one of the warders said, 'he was the only happy man in the prison' during the terrible week before he was hanged. The following is a copy of his last letter written to me. _  \n_\n\n_Mountjoy Jail_\n\n_December 1st 1944_\n\n_6.30 a.m._\n\n'Dear Dr.,\n\n'In case I haven't left a souvenir to some person I should have, please explain that the number at my disposal were limited.\n\n'I haven't time to say much but I'm sure there's no necessity. All I ask is that the ideals and principles for which I'm about to die will be kept alive until the Irish Republic is finally enthroned. This I feel sure will be accomplished before very long despite all the labours of traitors and hypocrits as right will prevail.\n\n'Thank everyone who has done anything on my behalf, goodbye and good luck in the future.\n\nCharlie.'\n\nIreland has another martyr and we must all feel proud to have known him and to have been his friends. I'm sure he doesn't need our prayers but I will ask you to pray for your sorrowful\n\nAunt Kathleen.\n\nAt this stage I realised that Pappa, too, had been closely involved with Charlie and held more extreme political views than I had ever suspected. By 1944 he no longer agreed with my father that the twenty-six county government of the Irish Free State (since 1937 known as Eire) was legitimate and should be whole-heartedly supported. To him, then, de Valera was a traitor who had thrown in the sponge before the thirty-two counties had been freed from British rule, and he considered it his duty as a patriot and an honourable man to oppose the Dublin government. But he had never preached sedition to me, no doubt because to have done so would have been to risk confusing my young mind and dividing my loyalties between father and grandfather.\n\nA few weeks later I had a letter from a fourteen-year-old cousin, the daughter of my father's younger sister, which indicated that extreme Republicanism was successfully being passed on to my own generation.*\n\n_10, Charleston Ave.,_\n\n_1 January 1945_\n\nDear Dervla,\n\nI hope you are well and had a very happy Christmas. Thanks for the letter, I was delighted to get it. I'm glad you are now more used to school. Do you know what G. K. Chesterton says about education? He says that it is being taught by somebody you don't know about things you don't want to know. I don't altogether agree.\n\nWasn't it awful about Charlie? I'll never forget the day of his death. The night before the people went to say the Rosary outside the prison for him but the police would not let them near it. (By 'they' I mean Republicans \u2013 in fact some were not but just pitied him.) Then they decided to say the Rosary aloud in procession while walking back to town but in O'Connell St the police baton-charged them and injured two girls!! And this in a Catholic country ruled by an Irish Government!!!! However when the Nuncio heard of it he must have said something because for the first time in _four_ years the government allowed a notice in the newspapers announcing that a Mass would be said for his soul and we all went to it. It was very sad but as he died such a brave death and is nearly sure to be in Heaven it is not so bad. He was laughing and happy up to the very last and the night before he said _he would not change his place for any man in the world_. He even forgave the CID who always pushed him around and maltreated him. He was so brave that the warders wept and the CID in the Castle said he was the bravest man they ever executed!! And that's saying something! I was glad to hear he left you a ring. He sent me a signed picture and left me a crucifix. I'd better stop now because if I don't I'll go on and on and on. I always do on the subject of Charlie. When you next come to Dublin I'll tell you all about Charlie, Aunt Kathleen and the raid on our own house. Give my love to Aunt Kitty and Uncle Feargus. I hope you have a very happy new year.\n\nLots of love and kisses,\n\nConstance.\n\nHad I lived in Dublin I might either have been much more deeply influenced by Murphy Republicanism or reacted earlier and more strongly against it. As it was, I followed exactly in my father's political footsteps, during this period, and despite the emotionalism aroused by Charlie's tragic career I reserved for de Valera (Charlie's 'murderer', according to Aunt Kathleen) the sort of devotion expended on less orthodox heroes by my Dublin cousins. To me 'Dev' was one of the greatest Irishmen who had ever lived \u2013 indeed, one of the greatest MEN \u2013 yet like my mother I was apolitical by nature and I cannot recall ever arguing about this with the more extreme of my relatives. Dev was not, in my eyes, a politician. He was an heroic, almost godlike leader of the nation who had never belonged to the dreary world of party politics but ruled us from an immeasurably higher plane. This of course was fantasy on my part, an adolescent's glorification of a sincere, romantic and very powerful personality. But behind the fantasy there was some truth. Now many of Dev's ideals seem to me either absurd or distasteful. Yet he did have genuine ideals, he was no mere politician out for personal power, and looking around today one realises that neither in Britain nor in Ireland is there any individual with the leadership qualities of Churchill and de Valera.\n\nUnlike Pappa, Aunt Kathleen had no inhibitions about dividing my loyalties; she was such a fanatical Republican that she would not have scrupled to subvert anyone by almost any means. In July 1944, when she was in prison following the arrest of Charlie in her home, she wrote me the following letter \u2013 carefully designed to steer me towards the mainstream of contemporary Republicanism.\n\n_Mountjoy Prison_\n\n_2.7.44_\n\nMy dearest Dervla,\n\nIt just struck me that you might like to have a letter from me and to know what life is like here. Well, considering everything, it's not so bad. The first and most important thing is that everyone here treats me with great kindness and consideration. The ordinary diet is not very appetising \u2013 in fact I couldn't tackle it, but I have been put on hospital diet which means that I get two pints of milk and two eggs a day and beef tea for dinner. Also I can be sent in food from outside. However, I find I'm not at all hungry and I had to ask Isolde not to send me in food because it was a waste of money.\n\nThis is the routine \u2013 the cell door is opened at 7 am and you may get up if you wish to. I stay in bed until breakfast is served and I usually take that in bed. As a rule the cup of tea is all that interests me. The cell is locked then for about an hour while the wardens have their breakfast. At about 10 I have a bath, which I enjoy immensely. After that I potter around \u2013 tidy my cell, wash my cup and saucer, etc. I then go out for exercise \u2013 I can walk around or sit down and read or just think \u2013 and I have plenty to think about! I can also read the paper which is sent in each day. Dinner is at 12.45 \u2013 beef-tea and bread and milk. The cell is locked until about 2 after which I may go out into the grounds again. The high light of the day is my visit (usually at 4.0). I can see two people at a time but in the presence of a warder and a detective. The visit usually lasts from 15-30 minutes. The children come to see me in turn. I have not seen Pappa yet as he has been away in Limerick. Sunday is the dullest day here as there can be no bath and no visit. Mass of course is a great consolation. The ten days I was in the Bridewell included two Sundays and there is no possibility of getting Mass there.\n\nAfter the visit I have tea, bread and butter and a boiled egg. Then I go out to sit in the grounds and before coming back to my cell at about 8.30 I spend a little time praying in the chapel. Although in the usual way there is no meal between 4.30 pm and 7.0 am I have my own tea and a teapot and I make some tea every evening. My door is locked at 9.30. However, I have the privilege of having the light on for a few hours so I read in bed until the small hours. Although the bed is very comfortable it's not so easy to sleep \u2013 all sorts of things come into your head and keep you awake. This morning I woke at 5 and was thinking of the morning last summer when you and I and Anne went out before sunrise to pick mushrooms. We didn't get any but I think it was well worth while to see the dawn over the mountains and the stars, that were so bright when we started, fading away one by one.\n\nDaddy told me that you have all decided that it is best for you to go away to school next September to the Ursulines in Waterford. I'm sure you'll like them \u2013 they are the only nuns I ever really liked. Isolde and Niamh went to the Ursuline Convent at Forest Gate when we lived in London and they _loved_ it. I know, of course, that they were day-pupils and how lonely you will be at first \u2013 but I know too that you have plenty of grit and backbone and that you'll do your best to be happy when you know that the only reason you are being sent away from home is that your wonderful mother is an invalid and that she and Daddy are prepared to give up the joy of having you always with them for the sake of your education.\n\nDo you remember when I saw you last, and we were looking at Daddy's Active Service medal, you asked me if I had ever been in jail? When I said no you remarked that I was the only one of my family who hadn't been \u2013 well now I'm no longer a blot on the family escutcheon \u2013 in fact I'm more like a skeleton in the cupboard! I can hardly believe that you are only twelve years old when I remember how tall and strong you are \u2013 why, you'd make two of me!\n\nYour affectionate Aunt Kathleen\n\nWhen the present round of the Troubles began in Northern Ireland, Aunt Kathleen was soon to be seen in the Republican ghettos of Belfast. Perhaps she merely went north as an 'observer', but that seems unlikely. It would not surprise me to discover that although well over seventy, and in poor health,* she was gun-running. Where there are godfathers there can also be godmothers. And to this day the name of 'Dr Kathleen' brings an affectionate gleam \u2013 and occasionally even a tear \u2013 to the eye of many a Provo.\n\n* _A Place Apart_ (John Murray, 1978).\n\n* In fact the writer soon out-grew her extremism, being by temperament tolerant and objective.\n\n* She died in 1973. \n\n#\n\nMolly arrived in May 1944. She was fat and forty and placidly uncritical of our house and its inmates. She baked delicious soda bread, of which she ate immoderate quantities, and could cook two dishes well. (Irish stew and bacon and cabbage.) Her reactions were prodigiously slow; half-an-hour after something had amused her she would disconcert those who did not know her by bursting into apparently unprovoked laughter. Yet she was such an enormous improvement on her immediate forerunners that in June I was measured for my school uniform. Two battered trunks, still bearing French luggage labels of the 1890s, were lowered through the attic trap-door into the kitchen and going away to school became a reality.\n\nI began mildly to dislike the idea as August dwindled. With Molly in the kitchen and my freedom restored the environs of Lismore seemed to have a lot more to offer than the environs of any convent school. Yet the novelty of the adventure still appealed and I was not seriously fretting.\n\nMiss Knowles, the Jubilee district nurse, had by this time become a valued family friend and I was overjoyed when she offered to escort me to school, _in loco parentis_. We were to travel to Waterford by train and this, oddly enough, was the first train journey of my life.\n\nThe carriages and corridors were gay with the scarlet and dove-grey of the school uniform and long before we got to Waterford I had begun, unexpectedly, to enjoy the sensation of belonging to a distinctive community.\n\nThis says a good deal for my schoolmates. Of course they ignored me; as a new girl I was officially beneath contempt. Yet there was no antagonism in the atmosphere; clearly I was not unwelcome. The school stood isolated on a height beyond the city, behind smooth high walls. Its grey stone buildings, solid and dignified, spread themselves in spacious, well-kept grounds that were brilliantly patterned and richly scented by September's flowering of roses. A not unpleasing air of monastic austerity and formality prevailed. One was aware of entering a world of orderliness and precision, certainty and calm, where the unpredictable could not happen, unless by some act of God. And even an act of God, one felt, would soon be made to conform to the relevant rule or regulation.\n\nYet there was no aura of dour oppression. In the wide, bright hallway we joined a dejected group of obviously new girls who were waiting with their parents to be received by the headmistress. All around us the golden parquet floor shone like a lake at sunset and the walls had been freshly painted cream, with pale green woodwork, and opposite the handsome oak hall-door was an alcove holding a gaudy life-size plaster statue of the Blessed Virgin Mary. The broad staircase swept upward to the left of the door as one entered and when my eye followed its stately curve I saw, whizzing down the banisters, a wiry girl with frizzy ginger hair and a demoniacal grin. As she slowed slightly to negotiate a bend she glanced down the corridor that ran at right angles to the hall, saw a nun approaching, promptly swung outwards off the banisters, hung for a moment by her hands, dropped six feet into the hall and lurked behind a pillar looking blas\u00e9.\n\nI held my breath. What grisly punishment would be meted out to her? The nun could not have failed to notice... But it had all happened in such a way that the nun could pretend not to have noticed, without any loss of face, and as she passed the pillar she simply paused to say a suave 'Welcome back!' to the ginger acrobat. This little scene neatly epitomised the whole spirit of a school where the pupils had enough respect for the staff not to defy them openly and the staff had enough respect for the pupils not to repress them.\n\nBy then it was my turn to be presented to the headmistress \u2013 traditionally known as The Hat. She was small and thin and red, like a carrot. As we shook hands I wondered if her toes were as red as her fingers. Perhaps she was a secret drinker, I thought, noting the ruby nose and remembering Jeff. All in all, she looked one of Nature's less agreeable blunders. And I felt she was coming to an identical conclusion about me. However, our instant mutual antipathy worried me not at all. A liking for The Hat might indeed have inhibited my rule-breaking; that we had at once recognised each other as Natural Enemies was much more satisfactory.\n\nIn the second-floor junior dormitory Miss Knowles helped me to unpack and another new girl, having read my luggage labels, introduced herself as Sally Dowling. Our fathers had done time together in Wormwood Scrubs (the Irish equivalent of having been in the same house at Eton) and she, too, was the only child of comparatively poor parents \u2013 and her mother was a semi-invalid. With her raven-black hair, aquiline nose, deep-set grey-green eyes and almost copper-coloured skin she looked not unlike a Red Indian. She was eighteen months younger than I, but we were to be in the same form for she was very clever. She also had a lot of quiet self-assurance and when Miss Knowles had said goodbye she appointed herself \u2013 gently, not bossily \u2013 as my guide and mentor. It was hard to believe that she, too, was a new girl. That very evening she and I established the foundations of a comfortable, easygoing, dependable friendship. Our devotion to each other was deep, but undemanding, undemonstrative, unsentimental, almost masculine; and throughout the next two years we never once even came near to quarrelling.\n\nHowever, my new friend was of no help when I woke for the first time in my cubicle. I had slept well, but as I opened my eyes homesickness engulfed me. It was almost a physical sensation, like being knocked down and rolled over and over by a wave. And it was all the more devastating for being unexpected. When saying goodbye to my parents I had felt just a little wistful, yet now my loneliness seemed to be even more acute than three years earlier. Then homesickness had been mitigated by the sheer horror of the place and its denizens, much as lumbago takes one's mind off tonsillitis. But here my personal anguish was unadulterated and I lived through the next week in a stupor of misery. Even to think about those seven days, thirty-five years later, is painful.\n\nThen one morning I woke and something had slipped into place and I felt happy all through. There had been no gradual recovery; on the seventh day I had felt no less miserable than on the second or fourth. So it was with a sense of incredulous relief that I lay in bed on that eighth \u2013 Wednesday \u2013 morning, looking at the early sunlight streaming onto my pale gold cubicle curtains and realising that I was happy.\n\nThis happiness lasted for the rest of my schooldays. Undoubtedly it was in part a result of freedom from those incongruous domestic responsibilities which I had had to shoulder, for much of the time, during the past few years. At school I was leading a life appropriate to my age and I enjoyed every moment of it. I also enjoyed not being the only pebble on the beach. My parents tried to avoid spoiling me, but any only child \u2013 particularly with a strong-willed, perfectionist mother \u2013 inevitably receives an unwholesome amount of attention. At home everything to do with me was of prime importance and in itself this constant exposure to parental concern was an infringement of privacy, though never so intended. At school I missed my own room and my long, solitary hours out-of-doors, yet in a sense I had greater liberty to be myself. I was no more or less important than some hundred and twenty other girls and this anonymity pleased me. My parents were agreeably surprised by my enthusiastic letters home. They had expected me to resent the lack of physical freedom, at least during my first term. But I was a more adaptable animal than they knew. And they had been clever in their choice of school.\n\nA grimly authoritarian r\u00e9gime would either have broken my spirit or provoked me to run away \u2013 probably the latter. But Waterford's Ursuline Convent allowed enough scope for me to lead my own kind of life without inviting disaster. (The Hat threatened to expel me three times; but this, I now feel, was because of my private feud with her rather than because I had unforgivably challenged The System.) The atmosphere was relaxed without being permissive \u2013 to this extent, a replica of home \u2013 and I liked most of my schoolmates and all my teachers. Also, I greatly enjoyed disliking The Hat.\n\nI soon developed a passion for a senior girl and became weak at the knees if she smiled graciously at me when we chanced to pass in the corridors. This was known as 'having an affec' for so-and-so, the so-and-so in question being described as one's 'affec'. My affec was an amiable sixteen-year-old to whom I wrote countless sonnets in praise of her beauty \u2013 an attribute which existed chiefly in my beholding eye. Most of these sonnets were seething with subliminal sexuality, as I was fascinated to discover when I came upon them recently in a rusty tin box. Frances also caused me to expend much mental energy on devising schemes which would lead to our meeting more frequently in the corridors. Not that one ever actually talked to one's affec; that would have been a major breach of both school regulations (the age groups were rigidly segregated) and school tradition. It is interesting that we schoolgirls voluntarily reinforced this particular school rule with our own code, almost as though we were unconsciously aware of the need for protection from our burgeoning sexuality. By immemorial custom, everyday communication was restricted to the humbly adoring look and the regal smile of acknowledgement. However, school dances were held twice a year, on the feasts of St Angela and St Ursula, and then each affec asked her admirer, or admirers, for one dance (no more), during which verbal exchanges were permitted. But these occasions seldom or never generated immortal repartee.\n\nI was a keen but undistinguished player of netball, hockey and tennis, and I founded a clandestine rugger club for those whose athletic tastes were more robust. We played in an out-of-bounds ploughed field and had two seven-a-side teams called Clongowes and Castleknock. Most of our fathers had been to one or other of those colleges and during the Leinster Senior Cup competition the whole school became hysterically involved. As we had no wireless at our disposal illicit telephone calls were made to Dublin to find out who had won each round. When I boasted of an uncle who in prehistoric times had once, during an influenza epidemic, been selected as a substitute for the Irish team my stock soared and even the seniors looked at me with respect.\n\nI was such an unsatisfactory scholar that a school dedicated to self-glorification through examination results would have had little use for me. It must have been plain that even at the end of six years I would pass no examination since I applied my mind to only two subjects, English and history. And even during English classes I was selective, regarding grammatical sessions as needless because one knew most of it anyway by instinct. To this day I cannot tell the difference between parsing and analysis. By the end of the first month I had dropped domestic science, art and music, all subjects for which I had no aptitude whatever. Secretly I longed to be a really good pianist like my mother, who had all sorts of impressive awards from the Royal Academy rolled up at the back of a drawer, but my music mistress believed that I suffered from some sort of musical dyslexia and would never be able to master any instrument. So I had at my disposal, almost every day, two and a half hours more freedom than my classmates. And naturally my division mistress wanted to know how I proposed to spend all this spare time.\n\n'Writing books,' I replied succinctly.\n\n'I see,' said Mother Ambrose. 'Well, in that case I think you'd better work in one of the empty music rooms. Anywhere else you'll be constantly interrupted. And I imagine you need peace and quiet for writing books?'\n\n'Oh yes!' said I. 'Silence is essential. And no interruptions.'\n\n'I thought as much,' said Mother Ambrose. 'Though of course' \u2013 she added reflectively \u2013 'Jane Austen managed to write quite a few rather good books in between receiving visitors and doing the household chores. But I expect she was different. May I read your book when it's finished?'\n\n'I'm afraid not,' I said apologetically. 'It won't be good enough for anyone to read. It's not that sort of book. It's only practice.'\n\nEach term I produced at least one full-length, lurid adventure story set in nameless Foreign Parts where the political crises were even more ambiguous than in Ireland and led to a profusion of warm sticky blood, choking emotions and gallant last-minute rescues. No doubt this 'practice' was a development of my teddy-bear-tree fantasies and I often became so involved that I had to continue writing during prep-time and on Saturday afternoons when I should have been mending my clothes. Mother Ambrose can hardly have remained unaware of these irregularities, but she made no serious attempt to check them. Just occasionally \u2013 pandering to my passion for long words \u2013 she would comment on the 'unarrested disintegration' of my underwear.\n\nFor other reasons, however, I was constantly in trouble. Such rules as were specifically designed to transform us into young ladies aroused my most profound contempt and I rarely bothered to conceal the fact that I had broken them. Yet I soon developed a fierce loyalty to the school and however I might defy its traditions within the precincts I took care to behave impeccably on the rare occasions when I emerged into the big world wearing school uniform. Both my defiance and my loyalty were uncalculated, which may explain the nuns' tolerance; perhaps they saw that although I was never going to fit into their particular mould I had a certain objective regard for it.\n\nThirty-five years ago not even the most senior girls at an Irish convent boarding-school were allowed out during term time, with or without supervision, to pursue either entertainment or instruction. Entering the school grounds on the first day of term, one knew that unless something extraordinary happened one would not leave them for the next two or three months. Nor did we have visitors, because of wartime travelling restrictions, and our half-term merely meant no lessons on a Friday and lots of sticky cakes for tea three days running. Being averse to such confections this was no treat for me. Raising my plate I would say 'Whizz?' and whoever first replied 'Echo!' won the prize. This corrupt Latin greatly offended me and I aroused much derision by allowing the pedantic paternal genes to take over and attempting to restore 'Quis?' and 'Ego!' As Sally pointed out, with her usual brusque logic, this was a damn silly attitude considering so many respectable English words are corrupt Latin or corrupt something else.\n\nOur lives were as enclosed and almost as sheltered as the nuns'. Yet within its own boundaries, both physical and mental, this school world was full and rich \u2013 if somewhat unreal to outsiders. It was made separate, distinct, self-sufficient and unassailable by the underlying strength and esotericism of its own traditions. And though it is argued that modern educational methods provide a better preparation for adult life, I still feel that there was an instinctive wisdom behind the fuddy-duddyism of the old system. A variety of 'ologists insist that the physiological and psychological changes of adolescence impose severe strains on the whole personality. Yet, as a society, we react most strangely to our new awareness of adolescent 'problems'. Are we trying to dodge them by forcing the maturing pace? My happy, limited school provided youngsters with as tranquil an environment as possible in which to come to terms with adulthood at the pace nature intended. The reassuring atmosphere of security and stability may well have helped us more than sophisticated insights into current affairs, cultural trips to the continent, 'projects' here, there and everywhere, and illustrated lectures on every conceivable subject including contraception.\n\nI was now for the first time in a situation where relative poverty might have been felt as a disadvantage. During the holidays most of my classmates did all sorts of exciting things like hunting and sailing and competing in swimming galas and tennis tournaments and gymkhanas. Had I longed to enter this alien world I might soon have become miserable. But I continued to accept the fact that other people led other sorts of lives and no change in my own way of life seemed either possible or desirable. Perhaps it did not seem desirable simply because I knew it to be impossible; while one half of me thrived on fantasy, the other was cheerfully pragmatic. So it rarely occurred to me that my apparently humdrum background was in any way defective. Books, after all, were my only real concern \u2013 the reading, collecting and possible writing of them \u2013 and other interests, however keenly pursued, were essentially peripheral. Anyway the idea of sailing did not greatly appeal to me; by all accounts it involved sharing a small space with far too many people for much too long and I preferred to enjoy it only vicariously in Arthur Ransome's world. Riding of course was another matter. One of my first memories is of being run away with by an allegedly angelic pony belonging to a horsy godmother in Co Kildare. A glorious mixture of terror and elation surged through me as I clung to the creature's mane. (I was to feel nothing comparable for the next quarter of a century, until I found myself galloping by mistake up a valley in the Western Himalaya on an Afghan stallion.) To everyone's astonishment I remained _in situ_ while the pony took me twice round the paddock. But then he decided to show off at the jumps...\n\nI fell with five-year-old animal expertness and my mother, who had witnessed all this from her bath chair, was too shocked to utter. My godmother, on the other hand, was so delighted by the whole performance that she promptly offered to give me a pony for my next birthday. But this generous gesture was decisively checked. To my parents, at the best of times, there was little if any difference between the Bengal tiger and the common cob. For years I believed that we could not possibly afford to keep a pony; now I suspect that this was one of the few areas in which my mother allowed herself to be overprotective. And so when listening to horsy classmates discussing their ponies I occasionally felt a wistful envy.\n\nIn one respect, however, I did for a time suffer acutely at school as a result of being 'different'. Children take even the most bizarre situations for granted, if these have always been a part of daily life, and easily form the habit of not thinking about certain matters that might be distressing to dwell upon. At intervals throughout my childhood I had registered receiving the neighbours' sympathy because my mother was an invalid \u2013 and this had puzzled me. My own unthinking acceptance of her invalidism was so complete that I could not understand why anyone should make emotional comments. This acceptance was encouraged by her own determination to lead as normal a life as possible. To be a good housewife and mother from a bath chair is not easy, but in so far as it could be done she did it. In my eyes she had no aura of 'differentness' and her high spirits and capacity for enjoying simple things made her exceptionally congenial to a small child.\n\nYet she was not as other mothers were. And this realisation, when at last it came, had a disproportionately upsetting effect on me. It was as though a dam had burst, freeing my long-restrained awareness of her difference which then inundated my stability. Suddenly, I couldn't take it. My frantic craving to be, in this respect, like everyone else compelled me to lie. Impulsively I would say, 'When my mother and I were coming home from a long walk...' or, 'My mother is such a fast walker it's hard to keep up.' Being naturally truthful, like Alice, these aberrations troubled me. My mother and I did indeed go for long walks together \u2013 but I was pushing her in her bath chair. Also, she _had been_ a fast walker, and I tried to convince myself that to change a tense was to tell no more than a half-lie. But I was too well grounded in moral theology for this to work, even temporarily. My intention was totally to mislead and therefore my lies were full-blooded and shameful. Here were the makings of another 'scruples crisis', in a different key, but mercifully it did not develop. As the shock of my objective appreciation of my mother's invalidism wore off, I lost the compulsion to mislead. But it left me with an embarrassing residue of false information imparted, which throughout the rest of my schooldays occasionally rose up to confound me.\n\nAfter my thirteenth birthday I began joyously to tick off, on the calendar in my cubicle, the days of December, and my heart seemed to swell with sheer happiness at the thought of being home again. Yet when I stepped off the train at Lismore, not having seen my parents for three months and ten days, a feeling of anticlimax and an oddly demoralising shyness overcame me. Although my parents meant no less to me than they had done in September they did mean something different. My attitudes to them had changed and I confusedly imagined it necessary to conceal this natural process as though it were some hurtful form of disloyalty. I had unconsciously transferred part of my allegiance to a world I valued all the more because parents were excluded from it. Most children who are happy at boarding-school revel in the secret-society aspect of school life. The cryptic slang, the apparently illogical customs, the usually logical rules and regulations laid down not by Authority but by generations of pupils \u2013 for me all this added up to a thoroughly satisfying existence in which grown-ups seemed insignificant and children taught each other as much as \u2013 or more than \u2013 they learned from their elders.\n\nInevitably this existence weakened the bonds between my mother and myself. I no longer confided everything to her or idealised her as the perfect companion. Instead, I became absurdly secretive about the most trivial details and went out of my way to try to break the links created by our mutual interests. As for my father, I now comprehended in a more adult way the strains and stresses of his life and felt correspondingly more guilty about our atrophied relationship. But this made it seem even less possible than before to take any remedial action.\n\nOnly one relationship remained unchanged. With Mark I could still discuss anything, to him I could reveal any idiocy or inconsistency, in his company I never felt it necessary to pretend to be other than myself as I was at that moment. Then, and for many years afterwards, I took my good fortune for granted, not realising how rare it is to be always totally at ease with another human being.\n\nAt the start of those Christmas holidays my father greeted me at the railway station with the news that Molly had just left, but had promised to send her youngest sister as a replacement in the new year. She had pronounced that in winter our kitchen would 'perish a brass monkey' so it seemed to us that this promise betrayed a certain lack of sisterly concern. The Emergency fuel shortage was by then acute and no doubt sub-zero temperatures had accelerated her departure. But when I viewed the situation with that new detachment gained at school it occurred to me, for the first time, that our servant problem was perhaps being aggravated by my mother's archaic expectations. She had her standards and was incapable of compromising if those she employed could not or would not accept them. Unlike my father and myself, she understood whatever weird impulse makes the Englishman dress for dinner in the jungle. 'We may be poor, but we needn't live like cavemen' was her battle-cry. A table had to be laid with the correct multiplicity of instruments placed in their correct positions, though the plates came from Woolworths and the forks and spoons were of aluminium. The maid's uniform had to be immaculate and she had to know, or quickly learn, how to wait at table. (Her lessons were frequently a rich mine of comedy.) The kitchen might be falling around our ears, but every corner of every cupboard had to be kept spotless, and the outsides of saucepans scoured no less thoroughly than the insides, and the tea-towels boiled daily. Naturally enough, simple girls off the side of a mountain, who had never before seen a vegetable dish, never mind a napkin-ring, were confused if not positively intimidated by all this nonsense. Nor could I approve of it and in my more rebellious moments I mentally and unjustly labelled it 'side'. ('He has no side to him' means, in Ireland, 'he is a simple man, without affectation or pretension'.) To me this formality seemed both incongruous and irritating against the background of our ramshackle and poverty-stricken home; to my mother it symbolised something profoundly important. Now I can understand that she needed to uphold order and dignity as props to her own morale. But for many years no such explanation occurred to me and our divergence on this point was to become a running sore within the family.\n\nMy earliest Christmas memories are associated with Old Brigid, who ritually opened the season on November 1 by making the plum-puddings while grumbling obliquely about not being allowed to make a cake. Pappa sent a Christmas cake from Bewleys every year, but Old Brigid, who had never in her life been to Dublin, despised 'them shop things'. Each year she declined cake with an offended air, explaining darkly \u2013 'I wouldn't trust meself to it.'\n\nAnnually another layer of plaster fell off the kitchen walls because of over-exposure to plum-pudding steam. Yet my memory, practising romantic selectivity, presents that kitchen as a cosy place during (prewar) midwinter. Pools of golden lamplight were fringed by friendly shadows, in December a ham hung from the rafters and while the oven was being 'got up' the huge shiny black range roared and glowed, all amiable and animated.\n\nWhile Old Brigid went through her plum-pudding routine \u2013 unhurried yet superbly efficient \u2013 I knelt on a chair at one end of the massive, scrubbed-white kitchen table and slightly opened the deep drawer. Then, every time the range needed stoking, I swiftly hid a fistful of fruit. And at the end of the day, while Old Brigid was intent on settling the pudding basins into their giant oval iron pot, I transferred my loot to the nearby guichet and rushed into the dining-room to conceal it in the sideboard. This operation was inspired not by greed but by my predilection for outwitting authority. Most of the loot went to Tommy, whose parents couldn't afford plum-pudding. (Childhood was less complicated before the consumer society had begun its evil hypnosis of the young. It would never have occurred to me to pity Tommy because his family was even poorer than my own and it would never have occurred to Tommy to envy me.)\n\nThe two unique features of Christmas Eve were the repeated appearances of unfamiliar postmen at unpredictable hours \u2013 up to 11.00 pm \u2013 and the plucking, gutting and de-sinewing of the turkey. In those pre-supermarket days one bought one's turkey in a state of nature and to remove the sinews, without taking most of the leg-meat with them, was an art not everyone possessed. Old Brigid knew the theory of it but found my father a singularly inept collaborator. And for me the joy of watching those two struggling with the bird \u2013 one on either side of the pantry door \u2013 far outweighed the pleasure I got next day from eating it. This was the one occasion during the year when my father might be heard using bad language. If the struggle had gone on for half-an-hour or more, and then a pound of meat came away with a sinew, he was capable of saying 'Damn!' under his breath. Whereupon Old Brigid would cough loudly in an attempt to save my ears from pollution.\n\nI had a circle of well-trained relatives and friends and from the age of six onwards the shape of my Christmas presents never varied \u2013 only the size. For me the glory of Christmas morning was the sight and smell and feel of new books. I don't remember book-tokens \u2013 perhaps they hadn't been invented \u2013 and occasionally there were duplicate volumes. But that was an advantage rather than a problem; in my privileged position I could exchange unwanted volumes for something long coveted, on our next visit to the Dublin bookshops.\n\nOur Christmas ritual never varied. On Christmas Eve we listened to the carol service from King's College Chapel and afterwards we broached the Christmas cake and I was allowed to drink lemon cordial instead of milk. At midnight my parents listened to Mass from the Vatican and I \u2013 from the age of nine \u2013 went to the parish church. At ten o'clock on Christmas morning \u2013 after present opening \u2013 we sat down purposefully to a gargantuan mixed grill. Normally my parents were very light breakfasters and my mother in any case strongly disapproved of fries; but this meal was designed to keep us going until four o'clock when \u2013 true to Dublin tradition \u2013 we dined by candlelight as dusk became darkness.\n\nTo me, who habitually went to bed at seven thirty, Midnight Mass was among the most exciting events of the year. That walk through the dark, expectant night, its silence broken only by other hurrying footsteps and cheerful greetings; the packed church clumsily decorated with holly; the quivering, golden blaze of votive candles around the crib; the familiar carols being sung so badly they frequently seemed unfamiliar; the platitudinous Christmas sermon that allowed one to play at keeping a sentence in advance of the preacher without often dropping a point. Then the walk home through a night that seems always to have been frosty and tingling and starry and glittering \u2013 though I dare say this is an illusion \u2013 and so to bed, after mince-pies and a hot lemon-drink, in a stupor of happy exhaustion.\n\nWe never went away for Christmas or had anyone to stay and there were no treats like going to the pantomime. At noon on Christmas Day Mrs Mansfield and her brother came by different routes for a drink and sat at opposite ends of the room affecting to be unaware of each other's presence. Mrs Mansfield always proudly presented us with one of her own plum-puddings and my mother always evinced suspiciously eloquent gratitude. This gift was discreetly passed on to the Wren Boys who called in their droves on St Stephen's Day. Although those were delicious puddings \u2013 I often ate their litter-brothers when visiting Mrs Mansfield \u2013 it was common knowledge that San Toy actively participated in his mistress's washing-up and, as her scullery was ill-lit, my mother thought it hideously probable that many of the dishes attended to by San Toy never found their way into the sink. I failed to see why that mattered, after eight hours' boiling, but my mother's approach to hygiene was more emotional than scientific.\n\nOur Christmasses, then, had always been times of contentment rather than excitement. And when that contentment evaporated for me, after my first term at school, nothing remained to distinguish Christmas beyond a lot of extra hard work in the kitchen. I tried to conceal my new anti-Christmas sentiments and never hinted that I would have preferred to celebrate by going off for a day's solitary cycling. Just as I had earlier felt obliged to pretend to believe in Santa Claus, lest adult feelings might be hurt, so now I felt obliged to pretend to be enjoying our modest festivities. I knew, however, that my mother sensed and was saddened by my new indifference to the old traditions. And I also knew that her own enjoyment was marred by my having to act as general maid instead of being a schoolgirl on holidays. Yet she tried to conceal this regret, perhaps prompted by some irrational guilt, or feeling that since nothing could be done to improve my lot it would be psychologically unwise to offer sympathy. Now I can see that those three weeks contained the seeds of much that later went wrong between us.\n\nRather to our surprise, Molly's 'youngest sister' did arrive in the new year carrying all her possessions on her back in a sack smelling strongly of poultry droppings. She never made any secret of the fact that she was Molly's illegitimate daughter and she looked like a slim edition of her mother. Her name was Brid, which is pronounced 'Breed'. My mother said 'I hope she doesn't', but in due course she did. After what my parents delicately assumed to have been a spontaneous abortion she developed 'an affec' for my mother. This, in mild forms, was not uncommon amongst our adolescent skivvies, but Brid's devotion was so excessive that she had to be forcibly restrained from spending all her wages (\u00a31 a month) on such votive offerings as bull's-eyes and liquorice allsorts. But these developments were all in the future when I repacked my suitcase on January 12 and said goodbye to my mother in the kitchen where she was explaining the function of a casserole to an incredulous-looking Brid.\n\nI was glad of Brid's presence during that brief farewell. My new antagonism towards my mother had in no way diminished my love for her \u2013 only my ability to express it. The sadness of this parting was quite unlike my straightforward loneliness of the previous September. Now I was eager to get back to school and I grieved not because the holidays had ended but because they had fallen so far short of expectations. Instead of enjoying a happy reunion I had lived \u2013 without then realising it \u2013 through the prelude to an eighteen-year conflict.\n\nEven more distressing, at a deeper level, was my new, mute sympathy for my father's situation. Throughout the holidays I had seen him working hard, in the evenings and at weekends, on the translation of some French theological tome. And I was shattered by the significance of this not unsuccessful attempt to fatten the family purse. For as long as I could remember my father had been writing novels, and paying precious money to have them typed, and sending them to publishers, and bearing up when they were rejected. In recent years I had sometimes thought vaguely that his apprenticeship was being rather long. Yet I had never doubted that one day a typescript would be accepted, putting me in the gratifying position of having as father An Author. (Why I should have assumed his ultimate success, when I felt no such confidence about my own literary future, is not now clear to me.) Therefore his becoming a mere translator was to me a grievous personal disappointment, quite apart from the sorrow I felt on his behalf. And my understanding that he had awakened from a dream was illogically made even more painful by the fact that those hours previously spent on enjoyable if unprofitable novel writing were now being spent on hack-work to pay my school-fees. I felt sorry for my mother, too. All her married life she had been encouraging and guiding my father's endeavours with endless patience and tact: and presumably with a certain amount of hope.\n\nNor would such hope have been unjustified. After my parents' deaths I found all those typescripts in a tea-chest, under a neat file of kindly rejection slips. I looked through them \u2013 feeling ridiculously uncomfortable, as though I were eaves-dropping on a soliloquy \u2013 and was amazed equally by the interest of their irrelevant philosophical asides and by their complete lack of basic narrative skill. Had my father not been so single-minded about writing novels he might have made quite a good essayist \u2013 though possibly not a publishable one, in an age when essays are (one hopes temporarily) out of fashion.\n\nAs my father and I stood waiting on the station platform, awkwardly exchanging banalities, I felt his inner defeat as keenly as though it had been my own. And I longed to be able to convey, however indirectly, my understanding and sympathy. But we were too aware of each other's moods and reactions to be of any mutual comfort. We knew that we were uniquely vulnerable to each other's perceptions and both of us were too proud and too reserved ever to let the barriers go down.\n\nThe train was full of children returning to Waterford schools in self-segregated groups. I joined the Ursuline coach and remained by the door to wave dutifully to my father. When he was out of sight I felt oddly _released_ ; yet I did not immediately merge with the rest, much as I had been looking forward to rejoining my friends. Suddenly, for the first time, I was aware of myself as an outsider. Neither at home nor at school did I quite fit in. That is one of the moments I remember with a vividness which has never faded. I was staring fixedly at the Round Hill \u2013 an ancient, tree-covered fortification from which Lismore takes its name \u2013 and I felt not at all upset by this recognition of my own apartness. Neither did I see it as anything to be smug about; but I was interested in it, because it seemed to presage numerous as yet indefinable threats and promises. Then as the train changed its tune, on the bridge across the Blackwater at Cappoquin, I changed my mood and went swaying up the corridor to join my classmates.\n\nIt was true that at school I remained always, though not obviously, an outsider. Yet that interlude away from home was immensely important to me. Lacking it, I would have grown up with my social adaptability untested and my knowledge of human nature derived mainly from books. And, as T. S. Eliot observed, 'It is simply not true that works... depicting... imaginary human beings _directly_ extend our knowledge of life. Direct knowledge of life is knowledge directly in relation to ourselves, it is our knowledge of _how_ people behave in general, of _what_ they are like in general, in so far as that part of life in which we ourselves have participated gives us material for generalisation.' What I most appreciated about school was the access it gave me to this 'direct knowledge of life'. I revelled in analysing my schoolmates' characters, observing how they reacted on each other, speculating about why they were as they were \u2013 in other words, wondering what made them tick, as Mr Eliot would not have said. To me the works of Pope were as yet a closed book but I arrived independently at one of his most celebrated conclusions:\n\n'Know then thyself, presume not God to scan,\n\nThe proper study of mankind is man.'\n\nAcademically I was backward because sheer laziness deterred me from using what was in any case a fairly limited intelligence. But my early sharing in adult problems and responsibilities had made me, emotionally, unusually mature and this 'grownupness' prompted many of my classmates (all of whom, apart from Sally, were my seniors) to treat me as an older sister. Thus my study of mankind flourished because I received numerous confidences which enabled me to discern the reality behind various elaborate adolescent fa\u00e7ades. Having been brought up to regard the breaking of confidences as a gross dishonour I never felt tempted to gossip about other people's secrets, even with Sally, and eventually I established quite a reputation as a restorer of damaged friendships \u2013 no doubt because my amateur psychoanalysing helped me to see where things had gone wrong though neither of those concerned might know exactly why they had quarrelled.\n\nThere were other problems, too \u2013 parents who didn't get on, delinquent brothers, family debts. (Thus I learned that the price of a yacht could be peace of mind.) And early one summer morning another thirteen-year-old came creeping into my cubicle and stood, trembling all over, by my bed. 'I think I'm bleeding to death,' she whispered. 'My sheets are all soaked in blood \u2013 will you tell the dormitory sister? Will they put me in hospital? Will Daddy and Mummy come?' Visiting each other's cubicles was so strictly forbidden that even I had never broken this rule, but now I was too angry \u2013 with Eileen's mother \u2013 to care. I gave the semihysterical child a few sanitary towels and a brief lecture on menstruation and led her back to her bloody bed. She was the eighth child of a Dublin doctor and it appalled me that she should have been sent off to school unaware of the facts of puberty. I pitied her all the more when I recalled my own experience \u2013 rushing into my parents' bedroom early one morning waving a pair of bloody pyjama-trousers and yelling triumphantly, 'Look! I'm a woman now!'\n\nAt least amongst my age group, sex was not in those days a popular topic. Twice I was questioned, tentatively, about my theory of the origins of babies, and on both occasions I replied shortly, 'Ask your mother.' This instinctive reluctance to discuss the subject marked my own growing-up. Two years earlier I would gaily have described what I knew of the whole process, from conception to delivery, to anyone who showed an interest. But now I felt \u2013 I could not have said why \u2013 that it was a special subject, needing to be treated with reticence and reverence and not suitable for schoolgirlish chatterings. Today there can be few thirteen-year-olds unaware of the facts of life; but I find it sad that this knowledge is being spread in ways that erode reticence and reverence. However wisely or foolishly modern parents treat sex-education, the sort of innocence my generation knew has by now become an emotional impossibility. This realisation disturbs me when I look at my own daughter. A time of innocence is surely a prerequisite for the total experience, in young adulthood, of the ecstasy of erotic love. What has never been possessed can never be gloriously lost. \n\n#\n\nDespite her little misadventure in April, Brid coped unexpectedly well with the Murphy m\u00e9nage. When the holidays began in mid-June she was still worshipping my mother and had even become quite a passable cook.\n\nPappa's health was now causing some concern and to our sorrow he cancelled his Lismore vacation for the first time. It was then decided that I should spend July in Dublin, staying with relatives and visiting him regularly. This pleased me immensely \u2013 and not only because taking the train for Dublin alone, with my bicycle in the luggage-van, made me feel splendidly emancipated. Attached as I was to Lismore, I was very aware of being a Dubliner by heredity. Before the Emergency we had gone to Dublin every month for a long weekend and always, as we approached the city, one part of me felt that it was coming home. As an individual I belonged to Lismore, but as a social unit I belonged \u2013 and perhaps still belong \u2013 to Dublin. My relatives and their friends saw me as the cousin up from the country, a gauche and rather curious creature who dressed abominably and knew nothing about the theatre or the latest films or the feuding and gossiping of the literary world. Yet as I cycled about the city I felt completely at home. I was moving among familiar ghosts \u2013 not only those of my biological ancestors, but also those of my cultural forbears. That was what was missing in Lismore; there the ghosts were unfamiliar.\n\nThe contemporary Dublin scene never greatly interested me. What I enjoyed was cycling down Dorset Street and remembering that Richard Brinsley Sheridan was born at No. 12 \u2013 and thinking of him again as I cycled up Grafton Street, where he attended 'The Seminary for the Instruction of Youth' run by Samuel Whyte, the natural son of his grand-uncle. Pedalling through the Liberties I would wonder if Swift had really been born at 7 Hoey's Court. And in Fishamble Street I remembered that James Clarence Mangan had lived there above his father's grocery-shop, just as Tom Moore lived above _his_ father's grocery-shop in Aungier Street. Then there was 21 Westland Row, where Oscar Wilde \u2013 my mother's most beloved, if not most admired, writer \u2013 was born; his father and my mother's grandfather had been close friends. And Lower Sackville Street I associated with Shelley, who once lodged at No. 7 and tossed from the balcony to the street below numerous copies of his pamphlet, 'An Address to the Irish People'. At Templeogue I thought of Thackeray being entertained by Charles Lever, to whom _The Irish Sketch Book_ was dedicated. And when I stayed at 3 Orwell Park, Rathgar, with my mother's eldest brother, I was living in a house very familiar to Synge; his maternal grandmother, Mrs Traill, had lived there, and when his father died of smallpox his mother moved to No. 4 with her infant son. Then, when I went on to stay with my mother's only sister, at 25 Coulson Avenue, Rathgar, I was conscious of the friendly ghost of George William Russell (A. E.). For some years after his marriage, A. E. lived in that cosy little redbrick house. And in August 1902, late one night, he came walking home to find a tall young man leaning against the railings of No. 25, waiting for him. He had never before met James Joyce, yet he at once invited him into the square, low-ceilinged sitting-room where I spent so many childhood hours; and there they talked until 4.00 am \u2013 a conversation that sparked off _Dubliners_.\n\nAll my savings were spent in the intoxicating second-hand bookshops along the quays, to which I had been introduced by Pappa even before I could read. But what gave Dublin its special significance were the scores of tiny family links, stretching back for generations, with almost every district of the South Side. (One rarely penetrated very far into the North Side, which was understood to be alien territory.) Yet Dublin was never necessary to me, as Lismore has always been. It seemed a pleasant diversion \u2013 a place where for short periods one could enjoy feeling nostalgic \u2013 but its urban attractions never exercised any magnetism. Now I rarely visit the city, which has been so degraded by developers that I could weep at every corner. But my dependence on Lismore has increased with the years. Something inside me would wither and die should circumstances ever compel me to uproot from the Blackwater Valley.\n\nDuring that sunny July \u2013 it must have rained sometimes, but I recall only sunshine \u2013 I cycled out to Seapoint every day, with a covey of cousins, to bathe and play water-polo. I also saw a lot of Sally, who lived within walking distance of all my relatives' homes, and I was deeply disappointed when \u2013 ostensibly because of her mother's ill-health \u2013 she declined my invitation to spend August in Lismore. Years later she confessed that she hadn't really wanted to accept. Being an orthodox Dubliner \u2013 not a maverick like my parents \u2013 she dreaded moving too far away from the city.\n\nFor a fortnight I imagined myself to be everlastingly in love with a tall, dark, handsome cousin who claimed to be writing an epic poem and went into appropriate trances at mealtimes. The fact that he was about to enter a Jesuit seminary surrounded my passion with an aura of poignant tragedy. But I was not really made of languishing material and within days of my returning home he had been virtually forgotten.\n\nThe first three weeks of August passed very agreeably. I had dozens of newly acquired books to read and Mark was spending his vacation at home. Then Fate hit us below the belt. Suddenly the reason for Brid's mysterious contentment was revealed; she had been furtively (and fruitfully) walking out with the butcher's boy and on September 1 they were to marry. Consternation ensued. On September 10 I was to return to school, and what to do?\n\nMy parents soon came to their decision. Back to school I must go and they \u2013 always hoping for another Brid, if possible not quite so fruitful \u2013 would manage somehow. It was a brave decision, though many thought a foolish one. For my mother it involved remaining alone in the house, in bed, from 9.45 am until 1.15 pm and from 2.15 until 5.45. And she could not even switch the wireless on and off, or reach out to pick up a glass of water, or blow her nose. Moreover, she had a phobia about fire \u2013 a result of a nursery accident when she was three and a half which badly burned her eighteen-months-old brother. Since becoming an invalid she had fussed about only one thing \u2013 never being alone in the house for more than ten or fifteen minutes. Yet now, for my sake (or was it really for my father's sake?), she was prepared to overcome even this terror; and she approached the overcoming of it with characteristic common sense and resolution. No fire would be lit in her room until the evening and no heating would be left on anywhere in the house. Instead, she would sit up in bed, wrapped in layers of woollen shawls and surrounded by hot-water bottles, with only her tiny, deformed hands exposed so that by patiently manipulating two knitting needles she could turn the pages of the book poised above her knees on its specially made book-rest. Her window would be left open, whatever the weather, so that in case of fire \u2013 for instance, if one of the household's many rats gnawed through an electric wire \u2013 she could attempt to summon help. And her liquid intake would be carefully controlled in accordance with the hours when a bedpan would be available.\n\nFor my father the decision involved just as much hardship, of a different sort. In addition to his normal day's work in the library, he had to care for a complete invalid, shop on the way home, make do with a hurried snack at lunchtime (he had a vast appetite despite his slim build), cook the supper, do the washing-up and then translate demanding tomes until one or two o'clock in the morning.\n\nAll this I knew \u2013 or would have known had I stopped to think about it \u2013 yet throughout that long, cold, Christmas term of 1945 I cannot remember worrying even slightly about my parents. They wrote regularly and cheerfully and I, seeing no reason why they should not martyr themselves for my benefit, continued to enjoy life immensely.\n\nSome of our relatives, viewing the situation from Dublin, thought it odd that my parents received no neighbourly help. They would have expected Mrs A. to run in at eleven o'clock with a cup of coffee, while Mrs B. did the shopping and Mrs C. ran in at four o'clock with a cup of tea, and Mrs D. prepared the vegetables for supper and Mrs E. perhaps took away my father's socks to wash and darn them. However, I perfectly understood why none of this happened. My parents had never tried very hard to integrate with the local community, simply because the futility of such efforts became apparent to them soon after their arrival in Lismore. They were friendly towards everybody, but in a politely distant way that did not encourage offers of neighbourly help. Both had the sort of pride with which I completely sympathised. Doubtless some people felt that no one with any sort of pride could live amidst the squalor that prevailed chez Murphy while my father was in charge. But my parents valued independence far more than comfort, convenience or 'respectability'. Having made the decision to send me back to school, they alone would deal with the consequences. Even Miss Knowles, to whom we were all devoted, and who was eager to help, was allowed to do so only in emergencies because it was not the proper function of a Jubilee nurse to care regularly for middle class patients.\n\nChristmas came again. I was now fourteen, physically fully grown and totally self-centred. The home I returned to was in chaos, but I went to no great trouble to remedy this and for the first time my mother and I clashed seriously, as two strong-willed women. She considered it my duty to sort things out, I considered it my entitlement to spend the greater part of the school holidays doing as I pleased. A few hostility pervaded the atmosphere and deeply distressed my father, who looked ten years older after his three-month ordeal. But still it was never suggested that I should remain at home.\n\nEily appeared on New Year's Eve. An elderly woman, she had worked for years as housekeeper to a couple who were about to leave Lismore. She agreed to 'do for us' six days a week and even volunteered to pack my school trunk and see to my mending. But my mother grimly \u2013 and quite rightly \u2013 declared that Miss Dervla would be packing her own trunk and seeing to her own mending.\n\nFrom the first, I thought Eily too good to last \u2013 and she was. Having been accustomed to comparatively luxurious working conditions, her health (or morale) broke down after a few weeks with the Murphys. By the beginning of February my parents were again on their own and even I felt shocked by my father's appearance at the start of the Easter holidays.\n\nI knew then, with certainty, that I would never go back to school. The knowledge was deep down; it had not yet reached a level of consciousness at which I could examine it and decide how I felt about it. But it prepared me for our family conference on Good Friday evening.\n\nMy mother had an orderly mind and on occasions like this could give an impression of coldness as she suppressed her feelings to leave the way clear for logical argument. There were, she said, only three possible courses. We could struggle on as we were; we could accept the fact that I must remain formally uneducated; or she and I could rent Uncle Bob's basement flat at Orwell Park, leaving my father to live alone in Lismore. Some reliable help would surely be available in Dublin and I could attend my mother's old school in Stephen's Green as a day pupil.\n\nWe had just finished supper and I saw my father's hand shaking as he lifted his coffee cup to his lips. I stared at a new rat-hole in the wallpaper and wished the conference would end. There was too much tension in the air that I only imperfectly understood and in a muddled way I felt guilty for being the cause of it. At least, this is what I then imagined the source of my guilt to be. More likely it was based on a realisation that I had been brutally insensitive to my parents' ordeal during the past seven months and had ill-repaid their altruism by idling away my time in Waterford. It suddenly occurred to me that if I were to return to school the June examinations would reveal the extent of my ingratitude. But this was not my main reason for declaring immediately and emphatically that I was in favour of staying at home.\n\nBoth my parents looked surprised and my father's face seemed grey with unhappiness. Then my mother said firmly that while it seemed right for me to make the final decision I should do so only after careful thought. In due course she was to be severely criticised, especially by my father's family, for having put such a burden on a fourteen-year-old. But she knew me as none of her critics did. Had I been forced to accept the Dublin plan against my better judgement, much destructive turmoil might have followed. Besides, it was my future that was to be most profoundly and permanently affected and ever since I could toddle I had been insisting on the right to self-determination. My mother's shifting of the responsibility may have been partly inspired by moral cowardice, but it went with the grain of my wood.\n\n'Go off on your own and think about it,' said my mother. So I went for a long walk, through the mild spring dusk, though there was really nothing to think about. Or rather, there was a great deal to think about but nothing to be decided. I saw my decision as having been ordained by Fate, so that in fact it was no decision but merely an unavoidable reaction to a given set of circumstances. That Good Friday evening I became sharply aware, for the first time, that \u2013 'As flies to wanton boys, we are to the gods'. But this awareness was not yet tinged with any alarm or resentment.\n\nI have never been easily moved by physical deprivation and pain, yet even as a child I shrank from mental or emotional suffering. Thus, though I had given no thought to the hardships recently endured by my parents, I could not for an instant consider inflicting on them the grief of separation from each other. Even had I longed to move to Dublin, I doubt if I would have hesitated; in this situation, I saw no scope for conflict. But of course I did not long to move to Dublin. Nor did I long to return to school, which considerably surprised myself. It had been good while it lasted and I would miss it: but not too much. Possibly I knew that I had drawn from the experience all it could give me and that the next few years would offer no more than unimportant variations on a theme.\n\nIt never occurred to me to speculate about the effects of this decision on myself at the age of eighteen, or twenty-five, or thirty. If I could have foreseen myself after the passing of another fourteen years, I would probably have refused to believe in the truth of my vision. Perhaps few fourteen-year-olds look far ahead, and even now, at forty-six, I feel foolish and futile if I try to do so. The present is enough. Attempts to control the future seem needlessly to limit its possibilities. If this view were general, anarchy would overtake the world. But one hopes there is room for a minority of non-planners.\n\nThat evening my immediate future, if far from ideal, looked quite tolerable. I would have ample time to read and write and cycle and swim. I would see Mark regularly \u2013 and he meant more to me than any number of school friends. Unlike most of my contemporaries, I had no interest in parties, clothes, films, dances, pop-stars (if such then existed) or boyfriends. In many ways, God had indeed fitted the back for the burden, in my case as well as my father's.\n\nI returned from that walk at ten o'clock, an hour past my usual bedtime, and announced that my original impulsive decision remained unchanged. I asserted again that, however favourable the domestic conditions, I would never have had either the ability or the will to distinguish myself as a scholar. And then my mother again dutifully went into details about the disadvantageous long-term implications of not at least having my Leaving Certificate, should the day ever come when I needed to earn my own living. But I knew that she was rejoicing in her heart and I hardly listened.\n\nMy father's feelings concerned me much more. I believed that he, too, welcomed my decision; but in the course of explaining it I had ruthlessly demolished his cherished dreams for my academic future. To me those dreams had always seemed absurd, yet suddenly I found them the more touching for that. During the past year he had proved how much they meant to him, how doggedly he was prepared to struggle to make them come true. But throughout that conference I was too constrained by the awkward adolescent barriers between my parents and myself, and too embarrassed by the emotional cross-currents in the atmosphere, to be anything other than belligerent and gruff. As usual, the very keenness of my appreciation of my father's feelings prevented me from allowing my sympathy even to be glimpsed. Harshly, I told myself that his present disappointment was nothing to do with me, that if anyone should be blamed it was he himself for ever having allowed a possibility to play the part of a certainty in his life.\n\nMy mother was no dreamer and had always tried to hold her husband's dreams in check, even if (or perhaps because) she recognised them for what they surely were \u2013 a reaching out towards compensation for a tragically imperfect marriage. Had they come true she would have been as pleased as he. But she argued that one should never have specific ambitions for any child before its potentialities can be gauged. As both my parents were reasonably intelligent, my father was justified in hoping for the best \u2013 but not, according to my mother's view, in staking his happiness on my success as a scholar. By the date of our conference my mother had probably accepted the obvious fact that I was not a budding intellectual and the even more obvious fact that whatever I did with my life I was unlikely to follow an established route. She was also helped to resign herself to the situation by her old-fashioned views on women and education; she saw no reason why the average woman should not become sufficiently cultivated and informed without ever going near a university. She had never been greatly impressed by blue-stockings: to her there was no virtue in being unable to scramble an egg or patch a shirt, even if one could read Russian novelists in the original and discern Tantric influences on the Gnostics.\n\nLying in bed that evening, too tired to go at once to sleep, I knew as certainly as if he had told me that for my father one dream was already being replaced by another. From now on he would look forward to seeing me succeed as a writer. There had to be a dream. And it comforted me that this was one we shared.\n\nMany years were to pass before I perceived the real conflict inherent in that Good Friday situation. It should have been apparent at the time, but perhaps I dodged it, unconsciously\/deliberately, because any attempt to cope with it would have torn me asunder.\n\nHow had my father _really_ felt? Had he wholeheartedly welcomed my decision, as I then believed? Or would part of him have preferred me to choose schooling in Dublin, even at the cost of my mother and he being separated for years? Quite likely he himself did not know the answer to that question. And it never even crossed my mind. To me my parents were a unit and their mutual love and interdependence seemed sacred, something to which one instinctively \u2013 almost compulsively \u2013 made sacrifices. It would have been psychologically impossible for me to desecrate that love by wondering if my father just might half-wish to put his daughter's welfare before his wife's. Moreover, I would not have wished either of my parents to put me first; this would have forged between us the sort of bond I dreaded \u2013 a bond woven of obligation that might never honourably be broken. As it was, my adolescent urge to detach from them was uninhibited by any feeling that either needed me on more than the practical level. Outwardly I was tied to them, inwardly I was far freer than if my formal education had been continued because to one or both _I_ was the most important person in the world.\n\nA cataclysmic row, between my mother and my father's family, followed on that momentous occasion. This devastated my father, who was devoted to his family but felt bound to oppose them in defence of his wife. They, of course, were acting for the best, reckoning that my father needed support against an autocratic woman who had always been too strong for him and was now proposing to blight an innocent child's life to suit herself. This dramatic misinterpretation camouflaged certain complex truths that were not to become apparent to me for many years. At fourteen, I only saw my mother \u2013 whom I still adored, from behind the adolescent barriers \u2013 being unjustly attacked by people who disregarded all that she had recently gone through for my sake, misunderstood the intricacies of our family scene and had no knowledge of my personal inclinations. So I entered the arena armed with the armour of righteousness and attacked my affectionate, caring relatives in long, smug, stupid letters which my mother would never have allowed me to post had she read them.\n\nAs a result of our tribal warfare I never saw Pappa again. He and I had not quarrelled, but we were prevented from meeting by the animosity that persisted for years between the Dublin and Lismore Murphys. However, this sad and unnecessary ending to a lifetime (on my part) of love had one good effect. It taught me that quarrelling \u2013 with anybody about anything \u2013 is hideously wasteful. Argument, controversy and disagreement can be productive. But quarrelling in a bitter, negative, unforgiving way is a miserable admission of one's failure to cope with other people's viewpoints \u2013 and with oneself.\n\nMercifully the tie between Pappa and myself had in any case been gradually loosening. Essentially this was a _childhood_ relationship and as I emerged from childhood Pappa's magic waned. By the time I went away to school he had ceased to be a central figure in my life though my affection for him never lessened. His death, at the end of December 1947, was my first experience of bereavement and it shattered me. The whole town mourned with us and many were the expressions of dismay because none of us was able to attend the funeral. On Christmas Eve my father had been immobilised by another attack of sciatica and I was looking after two invalids. This circumstance may have been a kindly gesture on Fate's part. My father, who had met none of his family since May 1946, would have been anguished by that reunion. He had no stomach for quarrels and only loyalty to his wife could ever have involved him in one.\n\nSeveral years later I discovered that Mark, too, had opposed my parents at this time. He attached no great importance to formal schooling (for me) but felt that our domestic affairs could and should be arranged to suit my future as much as anyone else's. Also, he foresaw how hopelessly I could become trapped in the domestic cage once my mother \u2013 whose health was slowly but inexorably deteriorating \u2013 grew accustomed to my style of nursing.\n\nOn that Easter Monday Mark confronted my parents, while I was out cycling, and criticised them bluntly. For such a man to interfere in another family's affairs required enormous courage \u2013 generated, in this case, by powerful feelings of foreboding on my behalf. As I have already mentioned, there had never been any casual come-and-go between the Ryans and my parents, and Mark's occasional visits to our home had been very brief. This particular visit was still briefer. My parents \u2013 rigidly polite \u2013 at once made it clear that they were unprepared to discuss their plans for my future with anyone, however well disposed. 'I came away with two fleas in each ear,' Mark recalled cheerfully, ten years later. He never again met my mother, and when he and my father chanced to pass in the street they acknowledged each other's existence with a formality as pointed and cold as an icicle. However, this apparently uncomfortable situation suited me perfectly; to have had to share even a fraction of Mark's friendship with my parents would have deeply upset me. Although not normally possessive about people, I had always felt compelled to try to insulate this relationship from parental influences. And, as I realised much later, Mark felt exactly the same about insulating it from his own family.\n\nWe may have had this protective attitude towards our bond because it was such a curious one \u2013 of a kind that makes some people believe in reincarnation. It is, after all, not quite usual, or even proper, for a Roman Catholic priest to have as his closest friend a female, twenty-seven years his junior, who visits him in his secluded home in the country at all hours of the day and night. (As happened eventually, during various Murphy crises.) Had either of us cared about what the neighbours \u2013 or the parish priest, or the bishop \u2013 said or thought, we would probably have allowed our friendship to fade gently when I reached young womanhood. But of course we didn't care. And besides, such friendships have their own momentum and their own purpose, which is not always fully evident to the friends themselves.\n\nMark was a natural outsider; not for him the hearty greetings along the Main Street, the breezy first-name chatting-up of parishioners, the earnest discussions about hurling, coursing and the price of store-cattle. Some thought him too aloof, for he tried to hide an abnormal shyness behind a curt manner that could be disconcerting. Others, as the years passed, expressed surprise at the failure of so able a man to advance in his profession. But Mark's idea of clerical advancement had nothing to do with rising in the ranks. He was so devoid of respect for any Establishment, clerical or lay, that he might almost be described as a Christian anarchist. The letter of the law meant nothing to him, the spirit everything. He stealthily gave away every surplus penny he possessed and his generosity became a legend among the poor of three counties. When he died, beggars and tinkers and cattle drovers and lonely old age pensioners crowded into a remote country graveyard for his burial service.\n\nYet he was no stern ascetic; he drank and smoked and enjoyed golf and bridge. Amongst his few friends he was excellent company, witty, well informed and frequently irreverent with an inimitable caustic flippancy that would not have amused the average bishop (and 99 per cent of Irish bishops are dreadfully average). He read widely and wrote (anonymously) with considerable force and skill. Unlike most Irish people he thought naturally in international terms; and unlike most Irish priests he was neither scornful nor suspicious of the great non-European religions.\n\nThe only vice despised by Mark was what he called 'humbug'. Hypocrisy in any form he recognised at a glance and witheringly condemned. Otherwise, he was prepared to find endless excuses for human frailty; if he had had to choose a motto it would have been 'Judge not...' There was nothing lax or confused about his own standards \u2013 these were austerely high and uncompromisingly clear-cut \u2013 but his compassion was without limit. That virtue should be common enough among practising Christians, yet it is not. Too many of them are too sure that only _their_ precise interpretation of God's law has any validity. They see themselves as being on the right side of the fence, from where they may benevolently extend forgiveness or pity or help to wrong-doers on the other side without ever showing true compassion \u2013 a word that implies 'fellow-feeling'. For Mark, however, there was no fence. And when I became familiar with Tibetan Buddhism I realised that the quality of his compassion was more Eastern than Western.\n\nMark and I were aged, respectively, thirty-two and five when our friendship was established \u2013 or, it may be, re-established. Yet it was from the beginning, in a very strange way, a friendship between equals. Of course I looked to Mark for advice \u2013 and even occasionally accepted it from him \u2013 and for the support and guidance and stability on older person could provide. But the essential nature of our relationship was not what might have been expected, given the disparity in our ages.\n\nIt is fashionable to take unusual relationships to bits, as though they were engines, and to explain and label their component parts. For instance \u2013 was Mark, to me, a father-figure, a man who made possible the sort of relationship I could not have with my own father? Or was he a hero to be worshipped because he was so kind and funny and cared so little for convention? Or was he a beloved guru, a priest whose concept of religion coincided, on the most fundamental issues, with my own? Was I, to him, a daughter-substitute? Or, later, a woman who might have become his mistress in any but an Irish context? Or, later still, a fascinating link with the wide world that was his natural habitat but from which he was isolated?\n\nWhat does any of that matter? Maybe some or all of these elements formed strands in our bond, but the mysterious essence of such a rare friendship eludes analysis and makes nonsense of the trite 'father-figure' and 'daughter-substitute' labels. It matters only that for thirty-three years Mark stood steadfast at the centre of my emotional life while around the edges raged all the storms whipped up by passion, frustration, loyalty, loneliness, doubt, guilt and faithlessness. Francis Bacon can say the rest \u2013 'A principal fruit of friendship is the ease and discharge of the fullness of the heart, which passions of all kinds do cause and induce... No receipt openeth the heart but a true friend, to whom you may impart whatsoever lieth upon the heart to oppress it, in kind of civil shrift or confession... The second fruit of friendship is healthful and sovereign for the understanding, as the first is for the affections; for friendships maketh indeed a fair day in the affection from storm and tempests, but it maketh daylight in the understanding, out of darkness and confusion of thoughts; neither is to be understood only of faithful counsel, which a man receiveth from his friend; but before you come to that, certain it is, that whosoever hath his mind fraught with many thoughts, his wits and understanding do clarify and break up, in the communicating and discoursing with another; he tosseth his thoughts more easily; he marshalleth them more orderly; he seeth how they look when they are turned into words; finally, he waxeth wiser than himself; and that more by an hour's discourse than by a day's meditation.' \n\n#\n\nMy adolescent memories are of abrupt changes in outlook and unpredictable moods which must have made me more than usually trying; yet nobody seems to have regarded these developments as 'a problem'. Somehow my parents made me feel that certain of their standards were as immutable as the laws of nature and could never be abandoned without shame. But they always clearly differentiated between standards and opinions \u2013 or tastes. One had to tell the truth, pay bills promptly, consider servants (if any), avoid harmful gossip, respect confidences and remember that politeness has to do primarily with other people's feelings. However, should one chance, by some lamentable disorientation of the sensibilities, to prefer Mendelssohn to Bach, or Rupert Brooke to Milton, that was nobody's business or misfortune but one's own. This allowed ample scope for scorn and revolution, without any threat to law and order, and for many of the parental preferences I cultivated, at various stages, a vehement though seldom lasting contempt.\n\nPerhaps the extent to which today's teenagers baffle me is partly a result of the abnormal circumstances of my own adolescence. Despite having adult responsibilities, my material dependence on my parents remained total and seemed unlikely to lessen in the foreseeable future. By the standards of most modern youngsters my activities were pathetically circumscribed, and to outsiders \u2013 like the Dublin Murphys \u2013 my whole way of life looked uneventful and dull to a degree.\n\nYet it was nothing of the sort. Were I asked to pin-point the most exciting period of a life that latterly has been more eventful than most, I would say \u2013 'The years from fourteen to seventeen'. But I must be careful here. How much do we romanticise youth and hope and energy, so that only their glory is remembered? Perhaps all I should say is that in retrospect those years seem to have contained much happiness and little unhappiness, despite a constant underlying irritation at having to waste so much time every day on tedious domestic chores. And despite occasional brief \u2013 but intense \u2013 moods of depression caused by personality clashes with my mother.\n\nYet 'happiness' is not the right word; it implies a lack of complications and a tranquillity which are uncharacteristic of adolescence. Elation would be more accurate, the elation of discovering literature and music, combined with the sheer animal pleasure of being young and healthy and very vigorous. Adolescence is supremely a time of discovery. To me Shakespeare represented not an aspect of the examination tyrant but a stimulating expanse of beauty. And music was a shared joy which drew the three of us very close, in a sort of truce-atmosphere, even when parent-child relations were at their worst. Then there was the natural beauty of West Waterford. At any season a walk or a cycle aroused almost unbearable delight. Apparently I was missing a lot; yet I desired, during those years, no more than I had.\n\nFor three months after leaving school I was unwontedly abstemious and wrote nothing; the long-drawn-out family quarrel was unnerving me more than I realised. Then an idea for a boys' adventure story came to me and I began to write frenziedly, sitting up in bed until the small hours, covering sheet after sheet of foolscap. And I discovered that my approach to writing had changed. No longer was I satisfied with verbal bulk and a hair's-breadth escape per paragraph. The hero of that story became to me a real person, instead of remaining an inhumanly tough yet sentimentally gallant Biggles\/Bulldog Drummond hybrid. He still had 'keen fearless grey eyes and a firm mouth' (I quote from the typescript, recently accidentally disinterred), but his speech was credible though he retained an unfortunate tendency to strangle people with one swift movement. More important, however, was the fact that I now began to rewrite, and to find pleasure in seeing sentences improve, even if I had had to sacrifice three words of six syllables each to achieve that improvement. There is a control of language, a sense of rhythm, an intimation of style in that story \u2013 though I wrote and rewrote all 43,000 words of it in exactly three weeks.\n\nNot until it was finished did I think of secretly sending it to a publisher \u2013 secretly because I wished to avoid parental sympathy when it was rejected. I felt very adult and earnest as I withdrew all my meagre savings from the post office, looked up the address of a suitable typist on the last but one page of the _TLS_ and took down a list of possible publishers from _The Writers' and Artists' Year Book_ in my father's office. I was now actively engaged in the literary world, not just daydreaming about it. The most I really hoped for were some constructive words of advice to a fourteen-year-old from kindly publishers' editors. And that is exactly what I got. I cannot remember how often that typescript was posted to England \u2013 at least half-a-dozen times \u2013 but it never came home without a positively encouraging letter accompanying the rejection slip. These letters at least confirmed that my ambition was not built on self-delusion. The literary world had taken me seriously enough to urge me on; and that, for the moment, satisfied me. I had never visualised myself as an adolescent prodigy and I came of a breed used to taking rejection slips \u2013 as it were \u2013 on the chin.\n\nNietzsche may have proclaimed 'the death of God' a century ago, but it took time for the news to spread and in my generation most Europeans grew up under the influence of firmly held religious beliefs, not necessarily orthodox, but involving an adherence to moral standards based on Christian ethics. Thus we are unique, for our adult world became a place where, as John Wren-Lewis has put it, 'for the first time in human history, the general climate of educated opinion has lost the religious assumptions that have hitherto been almost universal'. So we are in a position to enjoy the best of both worlds. From the old world we have inherited, if we wish to claim the legacy, a certain stability; we have something specific to move away from \u2013 and to refer back to, albeit selectively, in moments of crisis. Of course a traditional religious upbringing can also lead to innumerable irrational conflicts such as our juniors, who were 'born free', will never have to resolve. But I still feel that we are a fortunate group.\n\nYet for many the loss of 'religious assumptions' has \u2013 only too obviously \u2013 been demoralising and frightening. Even to consider such a loss impersonally could frighten people like my parents whose own faith was, apparently, impregnable. I remember, when I was seventeen, hearing my father gloomily quoting to my mother a prediction from Belloc \u2013 'What I think will spring out of the new filth is a new religion. I think that there will arise in whatever parts of Christendom remain, say, 200 years hence... a new religion, because human society cannot live on air... This conception of a new religion (and, therefore, an evil one) fantastic and unpleasant. Unpleasant I admit it is; fantastic I do not believe it to be.' My parents of course shared Belloc's view, but by then I found the notion of a new religion exhilarating.\n\nAbout two years previously, however, I too would have agreed with Belloc. I had then gone through a phase of intense orthodox fervour which was altogether inconsistent with my general attitude towards religion, either before or since. This, I believe, is a common adolescent stage; and in Ireland, even today, a disquieting number of people seem to develop no further. Suddenly I began to enjoy going to church \u2013 for me a hitherto rather boring routine accepted only because it was so rigorously prescribed by one's home, school and social environment. No longer content with Sunday Mass, I now went to the convent chapel every morning at seven-thirty; and if there was an evening service (known as 'devotions') I went to that too; and I rejoiced on the first Friday of each month because then the service lasted for an hour instead of half-an-hour. I also went to confession once a week though I cannot recall committing any notable sins during this period. Emotionally I seemed to need the various rituals and to thrive, most improbably, on Irish Catholicism's unwholesome mush of sentimentality and superstition. But it was also important to feel that through personal prayer I was in direct touch with Christ. (The Father and the Holy Ghost introduced unnecessary complications so I left them out of it.) Had I 'got religion' a decade later, that would have been understandable enough. But by the time ritual and prayer might have served as an escape or a comfort they had ceased to have meaning for me.\n\nMy fervour lasted a year or so and then quite quickly evaporated. Meanwhile I had begun to read a little about other religions, an interest kindled by my pen-friendship with a Sikh girl. Mahn Kaur was four years my senior and lived in Kuala Lumpur, where her family had settled before she was born. Her enquiring mind and lucid, indefatigable pen did a great deal for me. Having been educated by Irish nuns, she wanted to find out as much as she could about Ireland and Europe and the various Christian Churches. In return, she told me about the Sikh religion, which inevitably made me curious about Islam and Hinduism, which eventually led on to Buddhism.\n\nMahn Kaur was naturally much concerned about contemporary events in India. We had begun our correspondence in September 1946, when Mark noticed a letter from Mahn Kaur in a Catholic newspaper appealing for a pen-friend 'interested in history and religion'. And this personal link with the subcontinent, slight though it was, made me feel deeply involved, emotionally, in the terrible events of 1947. These were not just another distant disaster reported in the newspapers. All Mahn Kaur's relatives lived in the Punjab.\n\nFor four or five years we wrote to each other at least once a fortnight. This correspondence, and the ripples it spread, became one of my main interests. Mahn Kaur was my only contact with an inaccessible world which I longed to experience for myself and it consoled me to feel that Fate was indirectly catering for this longing. More important, the reading she stimulated heightened my inborn awareness that all the religions of mankind are equally valid. As some modern Hindus express it, they are all fingers pointing at the moon. And it is the moon that matters. My parents would have condemned this attitude as the vague evasiveness of a lazy thinker. But to me it was the truth and having seen it I could not worry about my own religious label \u2013 or the lack of one. If certain individuals felt that they did not belong, naturally, to any religious group, what did it matter? The moon was there anyway. And worship or contemplation does not have to be restricted to churches or monasteries.\n\nI remember one very cold January morning standing in the kitchen by the turf-range (a wartime innovation) while eagerly opening a letter from Malaya. As I slit the envelope I saw, just for an instant, the closeness of my mental relationship with Mahn Kaur as a measure of my closeness to Asia; and our whole correspondence seemed very much a preparation. Those rare moments when we apparently get a signal from the future do not effect the conduct of one's daily life. But they have their value below the level on which one feels impatient or thwarted \u2013 or even hopeless.\n\nA few weeks later came Gandhi's assassination. I was in bed with tonsillitis and I shall never forget switching on the wireless to hear that news. During those years I was mesmerised by the writings of Gandhi and I regarded him with a rapture which has since been considerably modified. Shakespeare, another adolescent obsession, has worn better. At school I had detested _The Merchant of Venice_ , not being yet ready for Elizabethan English. (How many thirteen-year-olds are thus given an enduring distaste for great literature?) But three years later I was 'saved' when Anew McMaster's famous touring company came to Lismore. They played _Hamlet_ in a huge draughty building known as the Hippodrome, which soon after fell down of its own volition. The place was packed to the door \u2013 as it would never be now, with a television set in 95 per cent of Irish homes. Most of the townsfolk were there and, tethered along the sides of the road, in both directions, stood the horses, ponies, jennets and donkeys who had provided transport for Shakespeare's rural fans. The response was tremendous at the end of every scene; in several places the floor boards collapsed beneath the hob-nailed boots of enthusiastic farm lads. It is sometimes argued that television has a unifying effect on a nation or a community because so many people watch the same programmes. But that is a very spurious unity compared to the delighted converging on the Hippodrome of most of Lismore.\n\nI had never read _Hamlet_ \u2013 or any Shakespeare, after my delivery from _The Merchant of Venice_ \u2013 and presumably neither had the majority around me. Yet we were completely carried away by that performance. At the end I emerged in a fever of delight, feeling very disinclined to go home to bed. It was a wild October night. Clouds were flying before a south wind so that the full moon seemed herself in flight. Possibly that moon had something to do with my mood. I cycled up to the Vee \u2013 ten miles above Lismore \u2013 and looking down onto the silvered plain of Tipperary felt a passion of gratitude, directed towards no one in particular, because I had been born. I got home at five-fifteen \u2013 not realising, then, how lucky I was to have parents who registered no alarm at breakfast-time when their sixteen-year-old daughter explained that she had been out all night. Both regarded cycling to the Vee by moonlight as a perfectly natural reaction to one's first _Hamlet_.\n\nWithin a year of that initiation ceremony, I was not only reading and enjoying Shakespeare but studying him, seriously. All my pocket-money was being spent on esoteric volumes of Shakespeariana which my father refused to buy for the public library because they would only have a readership of one. By this time a spare room had been converted into my study, with bookshelves around two walls from floor to ceiling. It occurred to me now that I might after all have taken to university life; but I wasted no time on regrets. In my private Shakespearian studies I was at least free to please myself while flexing whatever rudimentary scholarly muscles were then demanding exercise. I had never before concentrated on anything other than writing stories. Now, though my studies sometimes became tedious, I would not allow their tedium to deter me. While writing several learned (as I supposed) essays on various Shakespearian controversies, I discovered how essential self-discipline is in any activity of this sort. It was all great fun \u2013 very much the sort of thing to which Pappa had devoted half a lifetime, though I did not then see this \u2013 and it gratified me to find that I was capable of becoming slightly knowledgeable about something.\n\nFor relaxation, during those years, I read almost all the works of all the great English novelists, from Defoe to Hardy \u2013 only skipping Walter Scott, who to me seemed insupportably dreary, and some of the minor Trollopes which numerically defeated me. For good measure I read translations of the obvious French and Russians, including _War and Peace_ during a lumbago attack. When I look now at those scores of volumes \u2013 standing row upon row, fat and formidable \u2013 I marvel at my idea of relaxation some twenty-five years ago. Many I have indeed reread with increased appreciation and many others I would like to reread. But not _all_... And a few I am reluctant to return to, fearing disillusion. _Middlemarch_ , for instance, was amongst the most memorable experiences of my youth. Reading it seemed like watching God creating the world in miniature.\n\nIn 1948 the reopening of Lismore's Lawn Tennis Club induced me to modify slightly my solitary ways. I became a keen though never skilful member and for the next few years spent a couple of hours on the courts almost every fine summer evening. However, my interest in the club remained exclusively athletic. I got on well with all my fellow members \u2013 the majority were several years older than me \u2013 but never wished to be involved in any of their peripheral activities. Happily, this allergy to social functions was not mistaken for stand-offishness. I was, I think, accepted with friendly tolerance as the inevitable freakish outcome of my parents' union.\n\nThe tennis club was our chief centre for the promotion of marriage and I attentively observed the progress of various courtships \u2013 but always with a certain detachment, as though I were studying the customs of another race. Even at sixteen, I had a strong premonition that I would never marry. Possibly the predictability of the average married life put me off; it was the antithesis of my ideal unplanned existence \u2013 travelling, writing, not knowing what was going to happen next year or next month or even next week. And I have already stated my belief that we are equipped at the start of adult life with a few pieces of basic foreknowledge \u2013 whether or not we choose to use them. This is not as fanciful as it may sound. Our lives, after all, are moulded by our temperaments even more than our temperaments are moulded by circumstances. And on the threshold of adulthood it is possible to have enough self-awareness to see the probable outline of one's future. Already I had noted the qualities needed to make a marriage work and I did not think I either possessed them or wanted, even if that were possible, to acquire them. I saw myself as a person too strong-willed, self-centred and fundamentally \u2013 if not obviously \u2013 arrogant to make a success of the married state. There is a vast difference between being a good friend \u2013 or a good mistress \u2013 and being a good wife. As long as people of my sort are free they can seem endlessly patient, understanding, cheerful \u2013 even unselfish in a crisis. But deprive them of their freedom and everything turns sour. In my teens I could not have expressed all this: yet I knew it.\n\nOur household had always included at least one \u2013 and often two or three \u2013 cats; but to keep a dog was not practical until my return from school. Then, just as my mother and I were debating what breed to get \u2013 my father had been badly bitten as a child and feared all breeds \u2013 I came upon Bran. A pure-bred Irish terrier, he was tied by a very short rope to the drain-pipe of an isolated cottage and looked half-starved and cowed. He had large bare patches on his back and flanks and when I went nearer I saw that these were red and raw. At first he shrank from my outstretched hand, then he sniffed it nervously, then I rubbed the roots of his ears and he wagged his tail incredulously. I reckoned that he was scarcely two years old. There was no one around the cottage, but early next morning I returned. Bran was still tied in the same spot and trembling all over; what remained of his red-gold coat was sodden and obviously he had been left to lie out on concrete during the previous night's steady rain. When I rapped sharply on the door it was opened by a shifty-looking man with blood shot eyes, a whiskey-laden breath and a week's beard. He claimed to have bought Bran from passing tinkers for \u00a35 but offered him to me for \u00a33 because of his 'bad skin'. This infuriated me; no such man would then spend \u00a35 on a dog. I replied that I was not interested in buying the terrier but that I was very interested in the way he was being kept. And I added that the gardai, too, might be interested. (Privately I doubted this; cruelty to animals is rarely seen as such in countries where most of the inhabitants have for generations been 'deprived'.) However, on my mentioning the gardai the shifty eyes grew shiftier and their owner hurriedly made me a present of Bran.\n\nWe went straight to the vet, who prescribed train-oil and sulphur for Bran's eczema \u2013 a completely effective cure, though messy and protracted. After six months he had regained top condition and was acknowledged to be the most handsome dog for miles around. He slept in a corner of my bedroom \u2013 he was just too large to be a comfortable on-the-bed dog \u2013 and was respectful towards the cats who patronised him from my eiderdown. Unlike most of his breed he had a timid nature, perhaps because of puppyhood traumas. But he had no vices and despite his age proved easy to train. Eventually he could be trusted alone in a room with a shopping-basket of meat on the floor, if he had been told not to touch it. Even my father grew to love him.\n\nBran accompanied me on short cycles, of up to fifteen miles or so, but this meant my having to pedal slowly lest his heart might be strained. His loyalty was such that had I cycled to Dublin at eighteen miles an hour he would have tried to keep up. One of our favourite runs was to the foot of the Knockmealdowns, where I would leave my bicycle by the roadside and walk Bran over the heather and bogs. During the summer we sometimes swam in Bayl Lough, a wide lake \u2013 semi-encircled by almost sheer mountain \u2013 which always looks black and according to local legend is bottomless.\n\nOn one such walk, near the spot where I had got lost as a ten-year-old, Bran was set upon without warning by a large black and white mongrel who appeared out of nowhere. In those days one rarely saw other people or dogs on the Knockmealdowns and I felt mildly curious when a very tall, thin man with a slight limp and a shooting stick emerged from a dip in the ground. He called angrily to the mongrel, whose name was William, but was too far away to intervene. So I had to rescue Bran when it became clear that William was having much the best of it.\n\nAs William's owner approached us I saw that he had a conspicuously scarred face. He seemed genuinely distressed about Bran, who was crouching behind my legs, shivering. Meanwhile William stood a few yards away gazing at the horizon with a self-satisfied air, ignoring his master's scolding. I made polite, forgiving noises and then, as Bran had in fact been quite badly bitten on a hind-leg, I turned to accompany William's owner back to the road. At seventeen I was abnormally shy, but my companion seemed even shyer. Yet he was in his mid-thirties, I judged. For once I felt at an advantage and talked almost eloquently during that half-hour walk through the bright, windy April afternoon. But we exchanged no personal information and when we parted I did not expect ever to see William's owner again.\n\nAt that time my father occasionally took over my nursing and domestic duties for a day, which left me free to enjoy a serious cycle of sixty or seventy miles. (Otherwise, I was never off duty for more than four hours at a stretch \u2013 usually from 6.00 to 10.00 pm.) So it was that I left Lismore one sunny morning towards the end of August, intending to do a round trip over the mountains and return through Cappoquin. At the Vee I paused for a swim in Bayl Lough and afterwards, as I walked up the track from the lake to the road, I saw a vaguely familiar figure strolling near my bicycle. It took me a few moments to recognise William's owner; in the four months since our first meeting I had almost forgotten him. This time he had no dog and he told me in an expressionless voice that William had been poisoned two days previously. Before I could say anything he added, very stiffly, 'Would you care to motor to the sea with me?'\n\nI hesitated, not for any conventional reason but because my few opportunities for long-distance cycling were so precious. Then I realised that a refusal would be misinterpreted \u2013 and would hurt. I also realised that though I had scarcely thought about William's owner since April we had in fact established an embryonic relationship during our first meeting. The opinions I had then formed were only now crystallising in my mind; and the chief impression was of loneliness.\n\nWhen my companion held open the passenger door of the A40 I noticed a flicker of alarm in his eyes as though he were slightly taken aback by the situation our joint impetuosity had created; yet neither of us was really disconcerted by the behaviour of the other. Not until we were both in the car did he introduce himself. Then, as I was about to reciprocate that belated civility, he explained apologetically that he already knew who I was.\n\nBy the time we reached the coast, twenty-five miles away, we had exchanged potted biographies. Information that one might expect to gather from a new friend over weeks or months was given by Godfrey in moments. His need to communicate was not disguised by the precise, impersonal formality with which he told his story \u2013 almost as though he were filling in a form for _Who's Who_ , carefully giving all the dates while ignoring the emotions.\n\nSignificantly, he began by explaining that he had acquired his limp and his scars in a Japanese prison camp. That he should have referred at all to those disabilities seemed to me a tremendous compliment. And it was. He never again referred to them, or to his war experiences. I longed to question him about the latter \u2013 I had never before met an ex-POW \u2013 but plainly to do so was not on. As for his scars, they were not of the romantic sort that make a man look more manly. They were horrible. One needed the minimum of imagination to understand why he had chosen a solitary life.\n\nSince 1947 Godfrey had been living on his army pension in an enlarged cottage not far from the Vee; the enlarging had been necessary to accommodate his books. A manservant looked after him. During the war his wife had found someone else; she was ten years his senior and he had married her while still an undergraduate. In 1947 the divorce had gone through, but it was of no use to Godfrey. As a High Anglican he did not consider himself free to remarry; his attitude here was identical to the most orthodox Irish Catholic's. To me this sad though hardly unusual story seemed a tragedy on the grand scale. I could have wept for Godfrey \u2013 especially because he had explained things so austerely, with no taint of self-pity.\n\nI remember our getting out of the car at Goat Island \u2013 an isolated cove, even today \u2013 and facing each other for a moment in silence. Then Godfrey said, 'Let's bathe!' and as he turned away my heart did something odd and I thought, 'This is absurd! I must be falling in love!' It did not, however, occur to me that Godfrey might be guilty of the same absurdity.\n\nWe separated to undress chastely behind rocks that were far apart and after our bathe \u2013 Godfrey was an excellent swimmer, I noted approvingly \u2013 we dressed equally chastely and sat on the grassy cliff top to share the picnic lunch that I had brought from home. It was meagre fare for two, but this did not matter as falling in love had quite taken away my appetite.\n\nWhen we parted on the road above Bayl Lough Godfrey thanked me for a very pleasant day and I thanked him for a very pleasant day and we said goodbye without mentioning the possibility of any future meeting. But as I free-wheeled home \u2013 euphorically taking hairpin bends with my hands behind my back \u2013 I assumed that future meetings would somehow happen. First love at seventeen soars above the practical details of time and place.\n\nA fortnight passed without my seeing or hearing anything more of Godfrey, yet my certainty that we would meet again never wavered. I considered myself a most fortunate person and impatiently to demand more than the inner bliss I already enjoyed would have seemed greedy. In my daydreams I tried to imagine our future conversations, but I never imagined Godfrey falling in love with me. To think of being loved by this noble \u2013 almost godlike \u2013 figure would have seemed so unreasonable that it was not a fit subject even for daydreams. To love was enough. By any standards, I was an almost unbelievably naive seventeen-year-old.\n\nThen, on a misty September evening, as I was swimming in the Blackwater a few miles upstream from Lismore, Godfrey appeared on the bank above me. He waved casually and tactfully sauntered off into the next field. I scrambled out of the water, dressed without drying and followed him. We walked until it was dusk through the dense woods on the ridge above the river. Godfrey, I now discovered, had studied archaeology and also knew a lot about Shakespeare. We had been two hours together when we came to the track where his car was parked and said goodbye. But this time Godfrey added in a matter-of-fact voice, 'I expect we'll meet again quite soon.' 'I expect so,' I agreed, dizzy with joy. And I cycled home to pursue my Shakespearian studies extra-diligently.\n\nEven I realised that Godfrey must have deliberately sought me out by the river \u2013 a hard task, for someone unfamiliar with the countryside around Lismore. Yet still I did not think it possible that he might be interested in me as a woman. No doubt his emphasis on his permanently married state had a lot to do with this. Clearly he was not the sort of man to womanise indiscriminately and if he could not marry he would not fall in love. To me it was as simple as that. So I believed that he regarded me as a congenial friend \u2013 a nonconformist like himself \u2013 with whom he could for some reason overcome his self-consciousness about his appearance. I prided myself on my ability to conceal my own emotions and play it his way. To do otherwise, I reflected melodramatically, would be to wreck a friendship that he needed as much as I needed the hidden joy of being in love. And so I continued to think and feel for three years, during which we met perhaps once a fortnight \u2013 occasionally for a day, more usually for a few hours and always where we were unlikely to be observed.\n\nMy parents knew nothing of this new relationship. With them I had by then reached a fiercely secretive stage \u2013 the beginning of my futile rebellion against that trap foreseen by Mark when I left school. He of course was given a detailed account of the friendship with Godfrey. He listened sympathetically and wisely refrained from pointing out how improbable it was that a man of thirty-seven would remain interested only in my views on Shakespeare.\n\nCharacteristically, I never paused to wonder where this impractical love was leading me. To some those three years may seem arid, unfulfilling and wasted, but they did not seem so to me then \u2013 nor do they now. My own summing-up of the situation was ridiculously defective, yet the relationship between Godfrey and me was neither unreal nor \u2013 in any but the narrowest sense \u2013 unfulfilling. Despite the apparent artificiality of its framework, it developed and matured and grew richer as the months and the years passed. And for me it had \u2013 as I see it in retrospect \u2013 a most precious and irrecoverable beauty. For it was the fairest flower in the garden of youth: love without passion. \n\n#\n\nAt the beginning of 1951 my mother announced that something must soon be done to get me out of Lismore, however briefly. She had always maintained \u2013 and not only, I believe, because it was a convenient conscience-soother \u2013 that for girls travel is the best form of education. Yet I was no less astonished than elated by this announcement. Gradually, over the years, my liberty had been whittled away as my mother became more dependent on my nursing. So how did she propose to organise things in my absence? I felt touched and grateful when she declared briskly that she could easily manage again for a few weeks as she had done for months while I was at school. The less obvious implications of that declaration did not strike me then, or for many years afterwards. I was much too close to what was happening to look at it.\n\nEven now it is hard to estimate how far my mother's health had deteriorated by this time. But undoubtedly her demands for attention were already unconsciously prompted by a determination to dominate me. I might argue against scrubbing the hall floor every day and cleaning the windows every week \u2013 and indeed I flatly refused to do such chores more than once a quarter, when even by my standards they had become necessary. But I could not argue if my mother requested some attention. I could not tell her that she didn't really need to use the bedpan, or to have her position changed, or her room made warmer or cooler, or her thirst quenched. So as I became more mulish about housework and cooking she became more demanding of nursing attention.\n\nMy mulishness was not caused only by an inherent dislike for domestic chores. It is natural to be still dependent on and living with one's parents at fifteen or sixteen, but to be in exactly the same situation at nineteen felt very unnatural to me. When I compared my own situation with that of other youngsters, even factory-hands or shop-assistants seemed enviable. They were leading their own lives and earning a weekly wage \u2013 and they had whole weekends off. More freedom was what I longed for. My lack of money mattered only because it symbolised being unfree; every necessity was provided by my parents and my pocket-money was adequate for the few luxuries I craved \u2013 such as _Collected Commentaries_ _on '_ _Hamlet': 1650\u20131950._\n\nHowever, despite my bondage I was still enjoying life too much to feel more than occasional spasms of discontent. By temperament I was inclined to count my advantages rather than my disadvantages and envy of factory-hands never went far enough to make me wish that I had been born one. Possibly, without Godfrey, I might have been less resigned; at that time our relationship mattered to me more than anything else. But of course it is also possible, if one is prepared to look back with cold objectivity, that this relationship developed under adverse conditions, and assumed such importance, simply because my life was so restricted. I needed the emotional excitement of being in love. And it would be rash to pretend to know whether we create what we need or whether our needs are supplied by a benevolent Fate. If, at nineteen, I had been free to travel for a year in Asia, would I have chosen instead to remain at home within reach of Godfrey? I doubt if I would.\n\nI left Lismore on April 15, 1951, to spend three weeks cycling through Wales and Southern England \u2013 including, inevitably, five days in Stratford-upon-Avon. Some of the neighbours were aghast when they heard that I had taken off, alone, on a bicycle, to travel through what was little better than a pagan land. And they were even more aghast when they realised that my parents had encouraged me to commit this outrage.\n\nAt nineteen I had never before left Ireland, apart from one Triple Crown excursion to Twickenham. (Where, having just finished _Historic Doubts on the Life and Reign of King Richard the Third_ , I deserted my fellow-fans, after the match, to search unsuccessfully for Strawberry Hill.) I was fervently nationalist and anti-partitionist and therefore, in theory, anti-British. Honour required me to see Britain as a foreign country because my school history books had taught that the British were solely responsible for all Ireland's past woes and present handicaps. Thus I was shocked to discover, as I cycled through Wales and England, that it was impossible to feel 'abroad'. Half of me seemed 'to belong' in Britain. Having been in love with a quintessential Englishman for two years may have contributed to this. Much more important, however, was the fact that I had grown up by the light of Eng. Lit. To me the British were comprehensible and congenial (to my great annoyance) because their history and literature were mine, too, with modifications and additions. At times I found them as exasperating as they can find us; but they were never baffling, as we so often are to them. One knew what made them tick. Of course the Welsh and the English differed from me, but no more than a Welshman differs from a Scotsman or a Kerryman from a Dubliner. What we had in common went deeper than what divided us. Thomas Davis made many bigoted and stupid remarks, but he was right when he said, 'A people without a language of its own is only half a nation. A nation should guard its language more than its territories \u2013 'tis a surer barrier, and more important frontier than fortress or river... To lose your native tongue, and learn that of an alien, is the worst badge of conquest \u2013 it is the chain on the soul. To have lost entirely the national language is death; the fetter has worn through.' In 1951, however, I was not yet ready to admit that an Ireland in every way independent of Britain could only be a mirage. It was too soon for me to analyse detachedly the Republican ideals and prejudices acquired from generations of militant Murphys. I therefore developed a form of cultural schizophrenia, with my nationalistic half resolutely remembering that Britain was 'the ancient enemy' (though already Mark and Godfrey had seen that eventually my 'literary' rather than my 'political' half would win). Indeed, the fact that the implacable nationalism inherited from my father was being threatened by the apolitical liberalism inherited from my mother made me even more aggressively Irish and proud of it. Not for many years would I be willing to recognise that Ireland's uniquely close relationship with Britain is impervious to 'constitutional re-arrangements' \u2013 and that to admit this is not to be a shoneen (i.e. pro-British) but a realist.\n\nHappily my cultural schizophrenia detracted nothing from my enjoyment of Britain. I proved that I could cycle 100 miles a day without undue effort and I cut back on food to spend wildly in second-hand bookshops, sending a postcard home almost daily to announce that another parcel of 'wonderful finds' had been despatched. Every evening I met in my Youth Hostel congenial representatives of the Great British Public and a stimulating cross-section of early fellow-tourists from Europe and America. At Stratford I actually _spoke_ to Professor Dover Wilson for ten minutes after a lecture on _Richard II_ \u2013 undoubtedly the highlight of the whole trip. And there too I met four undergraduates who invited me to contact them when I got to Oxford and gave me an illicit glimpse of Balliol night life. In London I rivalled the most gluttonous of American culture-vultures and packed an improbable number of concerts, ballets, operas, museums and galleries into eight days. And everywhere I found the allegedly reserved English spontaneously kind and helpful \u2013 even talkative, if the stranger was prepared to initiate the conversation.\n\nMy father never commented on the long article I wrote about Oxford on my return home, but it must have alarmed him, as a symptom of my drift away from the true-blue (or true-green) Republican tradition. In the course of it I wrote:\n\nOxford is a state of mind more than a place. One may enter it with reverence, or in a tolerantly amused mood \u2013 those quaint British, with their slavish adherence to tradition \u2013 or with indifference; but no one can leave it without feeling that they have been through a most exhilarating experience. Its potent, subtle charm compels recognition \u2013 seizes you with a gentle fierceness \u2013 you may resist but finally you must surrender.\n\nWho can say what that charm is? (Perhaps all charm is merely a subjective quality and what constitutes Oxford's spell for me would mean little to another.) It is a place of paradoxes, of illogical customs often initiated in the centuries that knew Chaucer or Shakespeare, yet still revered by all if somewhat imperfectly comprehended by many. Also it is a place of the wildest unconventionality and irresponsibility. Undergraduates who seem able to discuss anything from Industrial Relations to Japanese art will in a moment turn to playing a practical joke that I would have considered infantile at twelve. Which reminds me of a cat I saw in the Garden Quad at Balliol: a plump, elderly cat she was. Anywhere else she would have sat sedately in the sun, meticulously performing her ablutions, but there she gambolled under the chestnuts and elms in an abandoned manner, as though the imperishable youthfulness which is an essential ingredient of the Oxford atmosphere had permeated her matronly body. For although the centuries have flowed over Oxford, have deposited there learning and beauty and dignity, they have not eroded her youth. How could they, when every year brings a fresh stream of undergraduates, high-spirited, enthusiastic, carefree, bearing the fragile ideals, the crazy theories, the irrepressible gaiety and optimism of the very young. They make of the ancient University of Oxford a Peter Pan among cities.\n\nIn a sense Oxford seems strangely un-British, though the cradle of so many of England's finest minds and traditions. I pondered long on this. Is it because it came into being long before what Belloc defined as the 'consciousness of nationality' developed? And so it retains a medieval attitude which allows the more important things of the spirit and the mind to predominate over political or racial conflicts or characteristics. And yet \u2013 such is the many-sidedness of the place \u2013 you feel there that you have your finger on the very pulse of England, that pulse which has beaten steadily whatever ills have afflicted the body of the nation. It is of Oxford that Englishmen may justly boast \u2013 not of Empires, or factories, of athletes or inventions \u2013 but of this quintessence of all that is best in the English, and the European, heritage.\n\nThere was an underlying sadness about my reaction to Oxford, though at the time I firmly refused to dwell upon this. I had never regretted leaving school at fourteen, but now, suddenly, I saw that university was another matter. During those few days and nights of innocent though often rule-breaking fun with my young Balliol men and their girlfriends \u2013 it was during the Eights Week \u2013 I found myself in a world where I naturally belonged. Or could have belonged, in other circumstances; as my article reveals, it would have been too late, in 1951, for me to adapt to that world even had I then been free to do so. I was only nineteen \u2013 younger than all my new friends \u2013 yet I felt considerably older than they were. I wholeheartedly enjoyed their companionship, as I still would, but they made me realise that, having never known what it was to be without adult responsibilities, I had skipped a whole stage of youthfulness.\n\nFor someone who had spent most of their nineteen years within a thirtymile radius of Lismore, those were three seminal weeks. They also provided material for my first published writing, apart from childhood competition essays. I sold three long articles, on my impressions of Stratford, London and Oxford, to the old monthly _Hibernia_ \u2013 which had nothing in common with the present-day journal of that name. Each earned me two guineas and it was most gratifying to see my work in print and to be paid sixteen weeks' pocket-money for enjoying myself. Yet the thrill was less than might have been expected. I had never wanted to be a journalist, freelance or otherwise. I was not a 100-metre, or even a 1,500-metre, writer; the marathon was my distance. And in the foreseeable future there was no possibility of my being free to do a book-length journey.\n\nA year later I set out on a five-week continental tour, taking time-saving trains from Fishguard to Dover, _en route_ for Belgium. And next day, in a First World War cemetery, on a vibrant spring afternoon, I passed one of those unforgettable hours which in retrospect are seen to have been milestones in one's development.\n\nI was alone in the April sunshine among all those small, neat, green graves \u2013 British graves. And the sadness of that waste utterly overwhelmed me. Most had been my age, give or take a few years. Had they lived they would have been little older than my parents. But they had died \u2013 for what? For King and Country, for the defence of small nations, for the end of all warring, for a lot of platitudes. If this was nationalism in action, did it make sense? German nationalism, British nationalism, French nationalism, Irish nationalism \u2013 all decked out in flowery phrases, glorifying death, urging young men on to slaughter each other. The occupants of these graves should since have enjoyed some thirty-five years of life, as I was even then enjoying the sun and the breeze and the few white clouds in the wide blue sky above Flanders. And it didn't matter to me that they were British; I could only think of them as humanity squandered. At twenty I was tough and my blood was the blood of fighters. Yet I wept that afternoon. I could see no point in the waste and the grief and the pseudo-heroism cultivated by each country's leaders. How could any nation that claimed to be civilised fight such a war? I decided then that nationalism (as distinct from patriotism) is an affliction which humanity needs to be cured of as soon as possible.\n\nWalking back to my bicycle, I felt a sudden anguished embarrassment on my father's behalf; and a contempt for him that humiliated me because it reeked of disloyalty. He could not have reacted as I was doing to all those graves. I had grown up, but he hadn't. I was too young to understand that 'growing up', in this sense, was immeasurably harder for his generation, whatever their nationality, than for mine. At eighteen he had entered an English jail to spend three years sewing sacks for the post office, wretchedly fed and crawling with lice. And now I wonder how soon I would have 'grown up' had I endured a similar experience.\n\nFrom Flanders I cycled to Bruges, Antwerp, Brussels, Luxemburg, Maastricht, Aachen, Bonn and so up the Rhine \u2013 which greatly disappointed me \u2013 to Mainz. There I spent a weekend with the Hilckmanns, whom we were to have visited in August 1939. Then on to Heidelberg, Rothenburg, Biberach, Regensburg, Creglingen (where Riemenschneider's Marienaltar excited me more than anything else on this trip), Munich, the Black Forest, Strasbourg and across central France. In Paris I spent four days feeling euphoric, except when I was kidnapped one night by White Slavers in the Place de la Concorde on my way back from the Opera to my left-bank doss-house.\n\nIt was midnight as I crossed the Place de la Concorde and when a large car pulled up just ahead of me I ignored it, assuming that some lustful male was in search of willing prey. Then a pleasant-looking woman beckoned me and, speaking in English with only a slight accent, warned me that it is very dangerous for girls to walk alone in Paris after midnight. 'Where are you staying?' she asked. 'You are taking a terrible risk. My husband and I would like to take you to your hotel.' I was too touched by this solicitude to point out that I _enjoyed_ walking around Paris in the middle of the night. And because it would have seemed churlish to refuse such a kind offer I slipped into the back seat, explaining that I was lodging just off the Rue St Jacques.\n\nIt struck me as rather odd that my protectress left the front seat to sit beside me, but I became suspicious only as we passed Notre Dame. As I began to protest that we had missed our turning my companion switched on an electric torch and opened a large photograph album which she laid on my knee. 'Look at those, my dear,' she said. 'We're just going to take you home for a little fun and a drink \u2013 some champagne, you would like? And then within an hour you shall be safely home in bed.'\n\nWhile she was speaking I had been staring at the album with a mixture of horror, terror and nausea. I had never before seen pornographic photographs \u2013 or, indeed, even heard of them. Noticing my expression, the woman's voice changed. 'Look at me!' she said sharply. I looked up at her and she ordered, 'Keep on looking \u2013 don't move!' Terrified, I kept on looking; she was pointing one finger directly at my eyes and gazing fixedly at me by torchlight. When I realised that she was trying to hypnotise me I swung away and groped for the door-handle \u2013 though we had crossed the river and were travelling at some 40 mph up the Boulevard de la Bastille. Her voice changed again. 'You mustn't be afraid,' she soothed. 'We are going to have a nice party for only a little while. We are so fond of young people and we have no children for ourselves.'\n\nI sat for a moment, calculating fast. If I tried to escape at this speed I would probably be killed. If I waited for the car to stop at its destination I would certainly be overpowered by both my captors. How often had I put the hero of an adventure story into just such a dilemma! But always I provided someone to rescue him \u2013 or at least staged an earthquake for the purpose \u2013 and there was no one to rescue me, nor any likelihood of an earthquake in Paris.\n\nThere were, however, two policemen standing in the Place de la R\u00e9publique refereeing an argument between a taximan and his fare. As I reached again for the door-handle my 'protectress' grabbed me by the wrists. But at that age I had the strength of a young ox and very few women could have restrained me. Here we had to slow down almost to walking pace \u2013 traffic was streaming from the nearby railway stations \u2013 and when I had made it clear that I intended to escape the car stopped. As I scrambled out it half-turned and raced away down the Avenue de la R\u00e9publique.\n\nHad I been able to afford it I would have taken a taxi back to the Rue St Jacques because my legs felt extraordinarily wobbly. But I had just enough money left to keep me in food until I got home. So I walked, only pausing to buy a bottle of plonk for my nerves in an all-night lorry-drivers' restaurant.\n\nMy landlady \u2013 whose mother had been for three years my father's landlady \u2013 did not believe that the kidnappers were professional White Slavers. Had they been, she said, they would not have shown me 'dirty pix' \u2013 at least at that stage \u2013 and would have drugged me as soon as I entered the car. In her view they were one of the amateur gangs who had recently begun to operate to supply brothels in \u2013 of all places \u2013 Soviet Central Asia.\n\nThis was the first time I became aware of a curious personal inhibition which I have never been able to overcome. When we reached the Place de la R\u00e9publique and I saw the policemen it would have been simpler to yell for help than to struggle to open the car door. But even had my situation been much more desperate I could not have done so. And in various other awkward situations I have experienced the same difficulty. Something very deeply rooted and stupid \u2013 I have no idea what it is \u2013 prevents me from calling out for help.\n\nParis inspired another long article, also published in _Hibernia_ , which omitted the White Slavers but vainly tried to convey the enchantment of that city \u2013 an enchantment not at all dependent, it seemed to me, on the way its citizens treated visitors, yet potent enough to make one feel drunk on fresh air. (Which was just as well, because by the time I had got that far I could afford little else.)\n\nThis journey amazed our neighbours \u2013 mine was not a venturesome generation, as are the young of today \u2013 and it temporarily assuaged my wanderlust. Short trips to the continent were the most that could be expected of life so I derived the maximum satisfaction from the attainable and rarely, at this time, dwelt upon the unattainable. Yet down in the unconscious frustration and resentment must have been accumulating like pus. My lack of freedom inevitably galled more because I had proved how easy it was to travel very cheaply by bicycle, how little it mattered not being a linguist, how ready most people were to befriend a stranger and, above all, how well suited to wandering was my own temperament. Only domestic responsibilities stood between me and India.\n\nWhile in Britain and abroad I had, out of curiosity, attended various non-Catholic services instead of going to Sunday Mass. This may not seem world-shattering, but in an Irish Catholic context it is \u2013 even now, never mind a quarter of a century ago. My omission, if known throughout Lismore, would have revealed that I was at best Losing the Faith and probably had already lost it. As of course I had, without going through any phase of violent, scornful hostility towards Christianity, even in its Irish Catholic aspect. It was then, as it still is, my instinct to respect all religions and deprecate insults to any. But from the age of eighteen or nineteen I felt no further urge to solve the insoluble, or to belong to a community with common beliefs, or to worship in traditional ways. The untroubled conscience with which I 'lost the faith' perhaps explains my lack of hostility. It was an amicable arrangement, as far as God (or whatever) and I were concerned \u2013 for me a peaceful, natural and, in a sense, unimportant development. Irish Christianity is a peculiarly hypnotising and powerful force which can make many 'deserters' feel guilty, deep down, for a lifetime. But my parents had shielded me from the Church's weapon of superstition and so, when the time came, I was free to 'desert' unscathed.\n\nThere was, however, a certain irony here. My parents, who had unwittingly shown me the easiest route out of the Church, would have been shattered to realise that I had taken it. So for a few years I continued to attend Sunday Mass in Lismore, this being the only positive action necessary to deceive \u2013 and protect \u2013 them. Going to Mass bored me but did not then represent any betrayal of my principles, as it might have done had I become a militant atheist. I would have preferred to avoid such hypocrisy, but as my motive was good my conscience for a time remained clear. Then \u2013 while in Germany, discussing Bonhoeffer in the Hilckmanns' garden \u2013 this attitude changed. I came home convinced that it was wrong to use any religious service merely as a convenience, however worthy one's motives, and I never again went to Mass in Lismore.\n\nInstead, I continued to be hypocritical\/protective by leaving the house at Mass time and going for a walk. Never having been a 'Sunday best' person, this seemed an easy deception. The neighbours would assume that I had been to an earlier Mass, or was going to a later one, and they had so few contacts with my parents that my subterfuge was unlikely to be detected. Or so I reckoned, not quite appreciating the extent of the neighbours' interest in my movements. In fact it was detected within a year, both by my parents and the neighbours \u2013 and, presumably, by the local clergy. But nobody commented on it. Concerning the clerical reaction \u2013 or lack of it \u2013 I have written elsewhere, 'When I 'lapsed' a quarter of a century ago \u2013 long before such an event could be spoken about above a whisper in rural Ireland \u2013 no priest ever attempted to 'get me back'; the local clergy knew perfectly well that I was not, and never had been within their disciplinary reach. I am still living in that same little town, on good terms with the clergy of all denominations and with the nuns in the convent next door to my house. No doubt prayers are wistfully said for my salvation but nobody, clergy or lay, has ever tried to make me feel ill at ease because of my defection.' My parents were equally restrained. I was then twenty-one and though my mother sought to dominate me in other ways both she and my father had too much respect for me (and perhaps too much experience of my obstinacy) even to attempt to influence me on such a matter. This restraint must have cost them something. According to their beliefs it was a parental duty to try to persuade me to rethink, to pray for faith and repent the error of my ways. Their neglect of this duty is perhaps an indication that both had more in common with Luther than either would have cared to admit. As for the neighbours, scandalised and disapproving as they must have been, there is in the Irish an innate delicacy (or is it a form of _laissez-faire_?) which generally prevents them from ostracising those who have left the herd; unless of course the deserter flagrantly defies convention in other ways that directly offend local sensibilities.\n\nAt that time the only people with whom I discussed religion were Mark and Godfrey. Naturally it distressed Mark to realise that I was no longer doubting or wondering but had unequivocally ceased to be a Christian. Yet he was too wise a man and too genuine a priest to argue with me. And his distress was less than I had feared it might be. It now seems to me that he appreciated my real feelings about religion some fifteen years before I myself did. We referred to the matter only briefly and infrequently; in this as in other respects our understanding needed few words.\n\nWith Godfrey it was otherwise; during our first few years together religion provided the area in which we drew closest. For us both it was important, but whereas I remained undisturbed by my discovery that I could never be a sincere Christian, Godfrey needed his faith and was frightened to think of ever losing it. Characteristically, he often blamed himself for having voiced his own doubts to me, lest these might have helped to undermine my faith. Such moods of confiding in me, followed by scruples because he had confided, were typical of the dual nature of our relationship. Sometimes he treated me as a person of equal experience and maturity, sometimes his attitude was almost autocratically paternal. The first mood I regarded as an undeserved compliment, the second I meekly accepted as inevitable. To me someone approaching forty was elderly and my inherent respect for age checked any resentment I might have felt when Godfrey was at his most peremptory. Indeed, I thought his paternal moods more authentic than his 'equality' moods. These I diagnosed as symptoms of a loneliness so profound that it sometimes made my youth seem irrelevant \u2013 a diagnosis not quite fair to myself. In some ways I was of course absurdly naive. But in other ways \u2013 the ways that mattered to Godfrey \u2013 I was more percipient and tactful than many an older person. My own domestic stresses and strains had equipped me to understand the very different but no less wearing stresses and strains of Godfrey's life. He was \u2013 though I did not then see him as such \u2013 an extremely neurotic man who needed to be treated with consistent gentleness, sympathy and patience. Because I loved him, he brought these qualities out in me; and because this was so, he came to return my love.\n\nAt the end of August 1952 it was three years since our first drive to Goat Island, yet we had never once expressed our affection by even the most tentative physical contact. Then suddenly \u2013 we were sitting beside Bayl Lough, on an afternoon of sparkling water and white clouds hurrying above the hills \u2013 suddenly Godfrey turned and looked into my eyes. And then he kissed me and we made love.\n\nThe kind of love we shared from that day on would have been impossible, I believe, had physical passion always been an element in our relationship; for three years we had been laying foundations and they were deep and strong.\n\nOn our way up the rough track to the road we conversed stiltedly, like the strangers we were in these new roles. Most vividly I remember my exultant sense of kinship with everything I saw \u2013 with the sheep and the crows and the stream by the track and the rocks and the heather and the clouds in the sky. I felt that until this day I had been only half-alive, half-aware; now I had a new relationship not just with Godfrey but with all of nature. And then there was a wondering incredulity, a sensation of the impossible having become reality. And also, inevitably, there was a triumphant relishing of power \u2013 at the time unrecognised, in my romantic delirium, but already operating. For so long I had been in Godfrey's power: now at last he was equally in mine. As we stood by his car and my bicycle, at the very spot where our first drive to Goat Island had begun, I looked at him, with a strong awareness of possessing him, and asked, for the first time, 'When do we meet again?'\n\nGodfrey turned away and got into the car. 'Never would be best', he said. Then he switched on the engine and added, 'Tomorrow, by the river.'\n\nFree-wheeling home, I felt indifferent to Godfrey's unhappy remorse. A woman's love can be as ruthless as a man's.\n\nNow I had exchanged innocence for understanding and that night my own aroused body told me what a marathon of self-control had ended for Godfrey by Bayl Lough. My respect for him increased accordingly \u2013 and I felt gratitude, too. Any earlier would have been too soon. As it was, that day had not been marred by any confusion, uncertainty, regret or doubt. I vaguely wished we could marry, but it mattered little that we could not. The formality of marriage seemed a mere dusty document beside the fiery jewel of our love. Godfrey of course would feel otherwise and already I was prepared to be sacrificed on the altar of his scruples. But now I felt capable of coping with any number of complex Godfrey reactions. And intuitively he must have known that at last I was ready to take the hardships as well as the joys of an adult liaison.\n\nThe following evening he first said 'I love you' as we stood, towards sunset, on the river bank. Then my armour of selfishness was penetrated by the sadness behind those words. He added that really we should discuss things sensibly; but there was nothing to discuss. He meant, I knew, that we should face up to the impracticality of the situation, say goodbye and get on with life separately and rationally. Yet the idea was too preposterous even to be mentioned. To my relief he never questioned the durability of our love. This had been my only fear \u2013 that he might retreat behind a barrier of cynicism and dismiss my feelings as transient youthful emotionalism. Predictably he condemned himself, at length and bitterly, for having cultivated a friendship that was bound at some stage to complicate my future. I replied that as a woman of nearly twenty-one I could look after my own future. Godfrey well knew my views on marriage and though he disagreed with them they were now conveniently reassuring. However, he did dutifully point out that it was too soon for me to withdraw from the matrimonial stakes \u2013 if I could ever have been described as an entry \u2013 and that if I detached from him I might eventually fall for some eligible man of an appropriate age. But his tone lacked conviction; we were both conscious of an inexorability about our relationship. It didn't make sense yet it seemed to exist almost as an outside force, independent of our decisions.\n\nBecause of Godfrey's scruples about extra-marital sex I now had to exercise my share of self-control \u2013 admittedly a less difficult task for a woman, yet not easy when one's passions have just been awakened. To me these scruples were absurd, but recognising their sincerity I loyally deferred to them. It often seemed a black joke that in this relationship with a Protestant Englishman I was up against precisely the sort of inhibitions that are chiefly associated with Catholic Irishmen. Fortunately Godfrey's scruples ceased to be effective at fairly regular intervals, but my primitive delight in love-making was always overshadowed by an uneasy and faintly unflattering awareness of Godfrey's sense of sin. Had the physical bond been our strongest, this relationship would have rapidly disintegrated.\n\nOn the surface it looks like very hard luck when a woman always falls in love with 'the wrong man'. Yet in my case this apparent hard luck may be linked to that strong adolescent premonition about never marrying. Perhaps we unconsciously avoid situations for which we are ill-equipped, even if avoiding them entails an amount of immediate suffering.\n\nHad Godfrey been free to marry, it would still have been difficult for me to leave home; so his being unfree spared me a major crisis of conscience. As things were, I took care to conceal our relationship from my parents, knowing that it would have needlessly upset them even to suspect that I was involved with a man who was still married, both by his own and their reckoning. Looking back, I doubt if my mother can have remained blind to my in-love bliss. But she never tried to pry.\n\nNeither of my parents ever questioned me about how I spent my free time; they must have speculated, but their tact was supreme. Outwardly they got no reward for this, only an increasingly uncommunicative daughter. But indirectly they were repaid. Having been allowed, from the age of fourteen, to follow as independent a path as circumstances permitted, I carefully shielded them from all unnecessary worry on my behalf. Never once, during this period, did I even consider adding to their difficulties by staking my claim to a normal life. But to write thus gives a false impression of an heroically self-sacrificing daughter. It was not like that. I did not consider and choose and put my parents first; to do so was not in my nature. Indeed, by temperament I was peculiarly incapable of being a comfort to them. But always I was conscious of our three lives being in one sense a unit on the scales of Fate and at least I could refrain from deliberately adding to their misfortunes. I had youth and health and hope and love and an enormous capacity for enjoyment \u2013 even against the odds. To break them, in quest of further advantages, was, literally, unthinkable.\n\nYet there was a certain remorseless brutality in the way I went about constructing my aloof adult relationship with both parents. However useful I may have been as a daughter around the house (and I wasn't very), my failure to grow into a friend must have grieved them deeply. But I had to isolate myself mentally if I were to continue to live at home. At the time I did not weigh and calculate in this fashion; my withdrawal was instinctive though I knew what I was doing and that I was being cruel. Now I realise that by turning away from my parents at an age when exchanges of opinions could have been most rewarding, and by assuming that they would react to a given situation in a certain way, without ever testing them, I was both impoverishing myself and doing an injustice to two remarkable people. But I cannot have regrets. It was as it had to be.\n\n#\n\nOne result of my new relationship with Godfrey was a return to 'marathon' writing, which I had sensibly abandoned for a few years to devote more time to reading. During the winter of 1952-53 I attempted my first novel. The heroine only slightly resembled myself because this was not the semi-autobiographical first effort of many aspiring novelists. Instead, it had a curiously prophetic theme \u2013 the growing-up of an illegitimate girl in a small Irish town.\n\nIt will be remembered that illegitimacy and its problems had come early to my notice. As a child I had often heard my mother arguing that the socially crippling effects of unmarried motherhood constituted an altogether disproportionate punishment for a momentary loss of self-control. At the same time it was emphasised \u2013 for my benefit \u2013 that only the 'less fortunate' sort of girl lost her self-control in this context. 'Less fortunate' was my mother's favourite euphemism when she wished to make class distinctions. Although perhaps euphemism is unfair; she was a practical Christian who used that phrase with compassion.\n\nThe illegitimate birthrate has always been high in Ireland, among the 'less fortunate', yet until very recently the conventions demanded elaborate pretences that deceived nobody but maintained a fa\u00e7ade of 'decent Catholic living'. Many much-loved bastards were brought up as their mother's youngest sibling (as in Brid's case) or, if this seemed biologically unconvincing, as her nephew or niece. Among the more fortunate, however, things were different \u2013 always far less civilised and frequently inhumane. A Co Cork case shocked my parents and me in 1952. The only daughter of a middle-class family \u2013 my exact contemporary \u2013 became pregnant by a married man while a student at University College Cork and was banished 'for ever' from her parents' home. Her father even discontinued her allowance and forbade her mother to contact her. In due course we heard that the wretched girl had had a nervous breakdown as a result of being forced to give up her son for adoption. And this was the too familiar tragedy behind my novel.\n\nDespite having my own study at home, I usually worked, during my evening off-duty hours, in my father's office in the empty County Library Headquarters. Then, as now, I preferred while writing to be completely alone and beyond reach of any interruption. Besides, the domestic atmosphere was gradually becoming less conducive to concentration. Often my mother and I quarrelled during the day about some trivial point of household management and by six o'clock I was longing to escape from the very unpleasant vibes we had created between us.\n\nI wrote this novel only because it was kicking to get out \u2013 but Godfrey, to my gratified surprise, urged me to have it typed. It was sent to some half-dozen publishers and one cautiously hinted that if I made the tragic ending a happy one they might reconsider it. But this I was not prepared to do. Although I knew that it was mediocre and mawkish it had a certain integrity and the ending, however crudely melodramatic, was \u2013 to me \u2013 inevitable and right and so could not be changed to suit the market. This failure to get into print between hard covers did not at all depress me. I had expected it and \u2013 as on my first excursion into the literary world \u2013 I considered the 'try again' remarks of various publishers a fair reward for my labours. During the following winter I wrote another novel \u2013 just for fun \u2013 but this time had enough sense not to listen to Godfrey and waste more money on typing fees.\n\nIn 1952 my father had spent his annual month's holiday at home, while I was abroad, because he no longer felt able to cope simultaneously with library and domestic duties. My mother therefore decided that in 1953 I must do a month on solo duty while my father deep-sea-fished. To this equitable arrangement I could scarcely object. Yet incipient panic threatened if I allowed myself to think about spending two years in Lismore without even an occasional day off.\n\nWhen the time came I found that month an almost intolerable strain. For me my mother would make none of those concessions which she gladly made to spare my father. On his departure for West Cork in June 1953 I was condemned to a month's virtual imprisonment. My mother insisted on never being left alone in the house for more than an hour a day, which just gave me time to rush down to the Blackwater for a swim, doing the shopping on the way home. I could not go far enough afield to meet Godfrey in the sort of seclusion he demanded and Mark's daily visits to our house afforded me my only safety-valve. He always brought a bottle of dry sherry to cheer me up and we would sit drinking in the kitchen while he told me the latest dirty jokes about the Vatican. But even his visits were a qualified form of relaxation; although my mother never commented on them I knew how much she resented Mark and after each visit she obliquely punished me. Fortunately I was still very resilient and the day my father returned I recovered my equilibrium.\n\nSoon I was eagerly planning and reading in preparation for a Spanish tour in the following spring and it was on this journey that I first fell in love with a country. My earlier tours had been immensely enjoyable and rewarding, but none of the countries visited had had the particular ingredients needed to spark me off as a travel writer. In Spain, at that time, there was an exciting sense of remoteness, both spatial and temporal. I pedalled as far off the beaten track as possible \u2013 encountering no hostility anywhere \u2013 and never before had I experienced anything like the Spanish quality of solitude and silence. The landscape, too, was dramatic \u2013 as was the climate, for in April blizzards were still blowing over the vastness of the Castilian plateau. Burgos was snowbound and glittering under an ice-blue sky and a northerly gale made it unnecessary to pedal on the way to Salamanca.\n\nSignificantly, perhaps, two of my mother's favourite heroines were Isabella of Castile and Teresa of Avila; I had been reared on their life-stories, with Ferdinand and John of the Cross standing slightly in the background. Many small-town names therefore sounded both cosily familiar and gloriously romantic. Again, Pappa had been a Cervantes addict \u2013 not surprisingly, given the resemblance between himself and the man from La Mancha \u2013 and his special children's copy of _Don Quixote_ , from which he often read to us on those summer days in Lismore, had been edited by himself with a pencil to avert juvenile boredom. (His other, 1796 four-volume edition, published in Dublin by John Chambers 'As an Endeavour to Improve the art of Printing in Ireland', and with 'Some Account of the Author's Life by T. Smollett, MD', was among his most cherished possessions \u2013 and is now among mine.) However, George Borrow's _The Bible in Spain_ and Walter Starkie's _Spanish Raggle-Taggle_ were the books which directly inspired me to cycle around Spain. And the fact that I had done my homework so thoroughly \u2013 reading volumes of history, biography and travel, balanced by novels, plays and poems in translation \u2013 perhaps partly explains my feeling of instant affinity with the Spaniards. They seemed at once more comprehensible and more 'foreign' than the Germans or the French and I even found myself speaking a version of Spanish despite my notorious inability to learn foreign languages.\n\nMy independence of Youth Hostels also contributed to the success of this journey. None then existed in Spain, but the village _posadas_ were even cheaper and got me involved with the sort of Spaniards I most wanted to meet. Occasionally, as the weather improved, I slept out in my flea-bag \u2013 once unwittingly in a graveyard which was the only level spot I could find on a pitch black night in the mountains of Aragon; I got quite a shock on waking next morning. At twenty-two I had reached my physical peak and could effortlessly cycle 120 miles a day through mountainous country on a heavy roadster laden with large panniers which were quite unnecessary for such a short trip. (It takes time to learn that a medium-sized rucksack can hold all one needs for six months.)\n\nIn Spain I kept a detailed diary for the first time, writing it up every evening, however tired I felt, and posting it to Godfrey once a week. On my return home he encouraged me to use this material and I sold a series of twelve articles to the _Irish Independent_ , Irelands' most popular daily newspaper. Heartened by this success \u2013 I had for the first time earned a considerable sum of money, by Murphy standards \u2013 I began work on a travel book, writing into the small hours every night for five months. Most of my earnings were willingly spent on the typing of those 100,000 words. This book seemed to me no worse than many recently published volumes of a similar type \u2013 and better than some \u2013 so I felt very hard-done-by when four publishers rejected it. All the skill, knowledge, energy and concentration I then possessed had gone into it and I suspected that its failure was due to the subject matter rather than to technical deficiencies. An equally well-or ill-written book on somewhere more exotic would probably have proved acceptable. To write successfully on Spain, I reckoned, one would need the ability of a V. S. Pritchett or an Arland Ussher \u2013 authors with whom I knew I could never compete. Recently I skimmed through this typescript and could find no reason to change my earlier opinion. In some ways it is more polished than my first published book and its chief defect \u2013 verbosity \u2013 could have been remedied by any competent publisher's editor.\n\nBy the beginning of May 1955 I had decided to waste no more postage on my Spanish typescript. And it was at about this time that I experienced my first mood of desperation. It was only a mood \u2013 a sunny evening of agony \u2013 but its intensity frightened me. I remember lying on smooth turf near the river, beneath the new leaves of a beech tree, and looking up at the blue sky between the leaves, and feeling the torture of my seething discontent and resentment being accentuated \u2013 instead of soothed, as always before \u2013 by the beauty of the evening. The immediate cause of this desperation was my being unable to get back to Spain until the following year. At twenty-three I felt increasingly conscious of the passage of time, of my own thwarted potentialities and of being trapped in a situation from which there could be no possible escape in the foreseeable future. My ambition to cycle to India now came more often to the surface of my mind and I tormented myself with thoughts of Afghanistan and the Himalayas. It had been established that I was physically equal to such a journey; all I needed was freedom. Yet had my mother permitted another Spanish holiday that year I would have philosophically forgotten India. I was not craving the unattainable \u2013 complete freedom. My demands were reasonable and my willingness to compromise was always there had my mother chosen to take advantage of it.\n\nHowever, this obvious grievance was but one of the causes of my new unhappiness. No less important was my recent literary disappointment \u2013 the first of my career, since I had never before seriously expected a typescript to be accepted. Had I been asked to choose between returning at once to Spain for three months and having my book published there is no doubt what my choice would have been. And having proved to my own satisfaction that I could write a moderately good travel book, the circumstances that prevented me from gathering saleable raw material chafed all the more.\n\nLoyalty to Godfrey inhibited me from acknowledging \u2013 even to myself, at the time \u2013 another factor that was compounding the tension in my life. For me the constraint which he struggled to impose on our physical relationship was becoming more and more trying as the novelty of being loved wore off and the flesh ached for its due. This constraint was prompted not only by his personal code of sexual morality but by an absurd theory that the more celibate we were the more likely I would be to detach from him, emotionally. Such an attitude says a lot for his nobility of mind but very little for his perspicacity. Even had we been entirely celibate I would have loved him no less. Faithfulness is not a virtue but a trait natural to some people. Its semblance may be cultivated \u2013 and this no doubt often is a virtue \u2013 but the converse is not true. Faithfulness cannot be quenched, except by the total and final disappearance from one person's life of another.\n\nGodfrey's notion that sexual deprivation might eventually wither my love could be misinterpreted as a measure of how little he understood me. Yet he never really doubted my constancy; at intervals, when his constraint-system broke down, he admitted this. And then he would explain that to placate his convoluted conscience he sometimes had to pretend to himself that our bond was fragile enough to be easily snapped should my future happiness seem to require this. During such introspections I used to reflect privately that a greater concern for my present happiness would be more to the point. Yet despite my no-holds-barred approach when Godfrey did let us off his moral leash, pride would never allow me deliberately to provoke him to make love. Also, his guilty aftermath was so grim, and so unnaturally prolonged, that I could not have all that mental suffering on my conscience.\n\nDuring that summer Mark realised how close I was to crisis-point and how much I needed moral support of the astringent sort that does not encourage brooding. In June he urged me to confront my mother, announce that I must have a break and request her to go away to a nursing home for a month, or employ a private nurse to look after her at home in my absence. But both those ploys had been considered and rejected two years earlier. My mother argued that we could not afford either alternative; no nursing home would accept her unless she engaged special day and night nurses and no nurse would look after her at home unless we could provide someone to dance attendance on the nurse. This I knew to be true: hence my trapped feeling. I could not yet see that my mother had become alarmingly over-dependent on my father and myself and that she would have resisted such an arrangement however much money was available.\n\nLife might have been easier for all of us had my mother allowed herself to complain openly about her sufferings. But I had no such thoughts at the time. Even after our bickering had become an almost daily event I never ceased inwardly to admire and feel proud of her stiff upper lip. There was something at once pathetic and heroic about her indestructible gaiety, her boundless enjoyment of books and music, her acute interest in politics and social problems and her indomitable dignity. It was natural that I should not think of her as an invalid, but neither did anybody else. All her life she retained her beauty and the glow of health on her cheeks. When she sat ready to receive visitors, with her chestnut hair falling in glossy waves over an embroidered silken shawl, only her motionless, unhumanly twisted hands reminded one that for half a lifetime she had been crippled. But now, behind the gallantry, the gaiety and the dignity, something was going very wrong. The tolerance and balance that had so enriched my childhood were being eroded as twenty-two years of rigorously controlled frustration took their toll.\n\nOur domestic framework must have seemed stereotyped enough to outsiders. In Ireland daughters are traditionally expected to be self-sacrificing and mothers are traditionally allowed to be selfish, and families endure the consequent overt or covert hostility for barren decades. But the personalities of my mother and myself made for unusually explosive complications. I did not accept that a young adult's only consideration should be his or her parents; and my mother, I can see now, was being demoralised by a growing realisation that her misfortune had cast a cold shadow upon my youth. Yet she knew that if she released me in the only way possible at this stage of her disease \u2013 by banishing herself permanently to a nursing home \u2013 anxious guilt about my separated parents' misery would have made my freedom worthless. Unhappily, however, I was now incapable of _acting_ \u2013 as well as feeling \u2013 humanely. When I should have been striving to hide my discontent from my mother I flaunted it, savagely, knowing that to do so could achieve only a slight relaxation of my own tension at the expense of an increase in her suffering. Things might have been easier had she ever directly expressed some sympathy with my situation, but instead she tried to behave as though I were doing no more than my duty \u2013 and that not very well. By 1955 we were both losing our emotional grip and looking back I can see how a grim spiral developed as she sought to punish me for adding to her sufferings by exposing my own.\n\nWe had never agreed on the subject of housekeeping and it infuriated me when my mother began to try to regulate every tiny detail, though I had then been doing the job for nine years \u2013 not efficiently, I so loathed it, but adequately. This sort of thing sounds petty \u2013 even comical, in a sick sort of way \u2013 when put on paper. To me, however, it soon became a major issue. The furniture was to be waxed on Tuesdays, the silver polished on Wednesdays, the hearths blacked on Thursdays, the shopping done at this hour and the ironing at that... There was nothing intrinsically unreasonable about the general outline \u2013 probably many housewives operate such systems \u2013 but it was not _my_ way of running a home. I had my own slap-dash methods which incorporated such ingenious labour-saving devices as never cleaning a soup saucepan before using it for a stew. Therefore I violently resented my mother's humiliating post-wash-up inspections; she was treating me now as though I were a fifteen-year-old skivvy.\n\nYet \u2013 improbable as this may sound \u2013 we were still capable, during those two years between my Spanish tours, of genuinely enjoying each other's company. Despite the extent to which I had long since isolated myself from my parents, mentally and emotionally, and the growing mother\/daughter antagonism I have just described, our strong mutual love was still there \u2013 settled, as it were, like a sediment on the bottom of our relationship, and apparent whenever it was stirred up by something that amused us. We shared an identical sense of humour which for a time preserved what remained of our sanity. And we both instinctively recognised humour's therapeutic value. At times we must have seemed strangely flippant for we used jokes to try to solve problems which, if taken seriously, might have brought us on to a collision course. My mother liked to quote Horace, as translated by Milton \u2013 'Joking decides great things \/ Stronger and better oft than earnest can.' My return to Spain in 1956 was not the anticlimax it might have been. I postponed my holiday to September, to enjoy the grape harvest, and in many ways found this tour even more satisfying than the first. Writing a personal travel book about a country brings one very close to it, for reasons that even now I do not quite understand.\n\nIn England I had first learned to beware of generalisations about national characteristics ('The English are so _reserved_ \u2013 no one will talk to you over there!') and in Spain this lesson was repeated. Despite the Spaniards' reputation for excluding foreigners from their homes, I was thrice invited to stay with families to whom I was a total stranger. And though all these families \u2013 in Ronda, in a village near Valencia and in Barcelona \u2013 were either very peasant or very bourgeois, the unreasonable strait-lacedness so often associated with Spain was never apparent. The young men of the household took me sight-seeing without a chaperon and with their mothers' approval. And their attitude towards me was neither salacious nor stilted.\n\nOn my way home I crossed the Pyrenees with twelve large bottles of Spanish brandy (bought for the equivalent of 25 pence each) rolled up in my sleeping bag and carefully roped to the carrier. This feat possibly constitutes a world record of some sort. But shortly afterwards Babieca's back wheel buckled irreparably so the effort may not have been the economy it seemed.\n\nI arrived home on October 6 to find that five days previously my mother had been taken to a Cork hospital suffering from severe kidney-stones. But Mark assured me that there was no cause for alarm; she had responded well to treatment and was due home next day. My father was staying in a boarding-house near the hospital and when I telephoned him that evening he had a message from my mother: I was to have the house thoroughly spring-cleaned by the time they arrived back on the following afternoon. At which point I shocked myself by wishing vehemently that it had been necessary for my mother to spend at least one more week in hospital.\n\nNot for an instant did I consider obeying the maternal order and devoting those uniquely precious hours of freedom to housework. Instead, I went to Godfrey's cottage for the night \u2013 the first night we had ever spent together in seven years. Even Godfrey felt that it was not an occasion for constraint. Or perhaps he was so startled by my totally unexpected arrival that he just didn't have time to put his constraint-mechanism into gear. Lying beside him after he had gone to sleep, I wondered why I did not more often ignore my mother's less reasonable commands. Then I saw that I had been able to defy her on this point only because I was beyond reach of whatever psychic \u2013 almost hypnotic \u2013 power she had over me when we were together. And having been away, leading a normal life for five weeks, also helped. As soon as I was back into the rhythm of the treadmill I would again become impotent to assert myself. For a mad moment I thought of cycling away in the morning to take a boat to England and there finding a job \u2013 any job \u2013 and freedom. But of course it would not be freedom. While my mother lived I could nowhere find freedom.\n\nLoyalty prevented me from discussing the sordid details of our family life with anyone but Mark. Even to Godfrey I would not admit how difficult my mother made things, though it was impossible to conceal the fact that I was living under an increasing strain. Besides, he had enough problems and tensions of his own without being expected to participate in mine. And it was good for me to forget the domestic scene, in so far as I could, while we were together.\n\nCycling home over the heathery Vee early next morning, through crisp, bright autumn light, I wondered \u2013 'When will I again be able to enjoy the mountains in the morning?' Then my heart seemed to twist in my chest with angry despair. For years \u2013 probably \u2013 to come I would be housebound at all times, apart from my four-hour evening break, and throughout the winter months ahead I would never be out-of-doors in daylight. This deprivation I felt more acutely and continuously than any other. It truly was a form of mental torture, to be denied the most simple, and yet for me most exhilarating pleasure of roaming the countryside in all weathers, during all seasons and at all hours of the day and night. I realised then \u2013 free-wheeling down towards the silver serpent of mist that marked the Blackwater \u2013 how little emotional stamina I had left. After my earlier tours I had for a few months felt cheerfully equal to anything. But this time it was not so.\n\nThe doctors had advised my mother to maintain a regular daily intake of four pints of home-made barley-water and this gave her a semi-valid excuse for making still greater demands on me; soon my off-duty hours had been reduced to three. Hitherto my father's nursing had been acceptable in a crisis, but by the end of that year my mother had decreed that only I could cope effectively. Mark tersely condemned this as a form of moral blackmail. Yet even at the time I \u2013 perhaps unfortunately \u2013 could see the invalid's point of view. Apart from the kidney-stones, which caused considerable extra pain and discomfort, her arthritis had recently entered a new phase. She was no longer able to write, or to feed herself, and turning the pages of a book demanded much dogged patience. To make her comfortable in bed or bath chair required a combination of skill, strength and gentleness that I, after a decade of almost daily practice, naturally possessed to a greater degree than anyone else. We formed a perfect team and mine was the only touch she did not dread. Indeed, she was hardly aware of it, so smooth was our teamwork, and we often became absorbed in a wireless concert while going through the daily routine of washing and dressing. But the very ease with which I could accomplish these tasks \u2013 every day of the week, every week of the month, every month of the year \u2013 increased their suffocating monotony. As a civilised human being I should have rejoiced at my ability to ease my mother's burden. But I detested nursing even more than housekeeping \u2013 and detested myself for being so devoid of proper feeling \u2013 and then I came to detest my mother because her physical helplessness gave her such power over me. Undeniably she abused that power, as Mark said. But had my own attitudes and reactions been less barbarous she might not have done so \u2013 or at least not to the same extent. Yet now I can reproach neither of us for our behaviour. Even at the time I had frequent brief moments of total detachment when I seemed to be observing the Murphys, as a family, from outside. Then I saw that we were all equally to be pitied and that none of us was to be blamed. We were enmeshed in a hideous, unbreakable net, each having to play a part that denied his or her nature, each knowing that to struggle was futile and yet each incapable of not struggling.\n\nThe most astonishing thing about this period is how much I continued to enjoy life. My desperate or resentful moods became more frequent but were always brief. Human beings can adapt to almost anything and, though my adaptability was eventually to be overstrained, there was no day when I did not revel simply in existing. All my life I had felt grateful (to whom I knew not) for the gift of existence, and I felt no less so now. I discovered that one does not have to be happy, successful or fulfilled to enjoy living. Even the bitterest despair and frustration can at a deep level be relished as part of the human experience. And so is generated a basic content which survives the surface discontents provoked by everyday miseries. Thus I was for a long time fundamentally resigned to my situation; I could not otherwise have borne it. I had none of the supports that would have been available to a devout Christian yet I sensed a certain symmetry in the background. I believed, confusedly yet firmly, that Fate organises things constructively and that in some obscure way what now seemed so wrong and arid would be proved right and fruitful.\n\nOn a different (or perhaps not so different) level I had, for my weakest moments, an absurd little consolation which I mentioned not even to Mark. Although I have never been superstitious I could not help being comforted by my mother's story of a Romany woman she had met in Somerset when she was nineteen, before she knew my father. This gypsy foretold that she would marry a small, slim, dark man, and move away from her birthplace to live among many trees, and endure a very long illness \u2013 and have only one child, a daughter who would be famous. Every prediction but the last had by now come true. And though I did not in the least want to be famous, I hoped \u2013 and in my heart believed \u2013 that this fortune-teller's blanket-term covered authorship. \n\n#\n\nDuring the winter of 1956-57 the rapid and conclusive disintegration of our house coincided with the maturing of my father's life insurance policy; so we decided to build a bungalow on the outskirts of Lismore. North, the site overlooked the Blackwater valley where wooded ridges rise from the river to the foot of the mountains: south, it overlooked placid fields, bounded by fine old trees, and against the sky lay another wooded ridge, the watershed between the valleys of the Bride and the Blackwater. We were within five minutes' brisk walk of the town yet only one other building was in sight \u2013 the farm at the end of the long field that sloped down behind us to the river.\n\nSignificantly, our building activities temporarily improved the domestic atmosphere. My mother, who had architecture in her blood, designed the house herself with the aid of a local engineer. And, having found this outlet for her energies and talents, she became much easier to deal with. We drew closer than we had been for years as we discussed colour-schemes, floorings and cupboard spaces.\n\nThe decision to build delighted me. I had no grudge against the ruin that had been my home for as long as I could remember, but to own a house in Lismore would confirm my sense of belonging to the Blackwater valley. I visited the site often and, wandering through the embryonic rooms, wondered to what events they would form the background during the next half-century or so. I felt quite sure that here was my home for the rest of my life and this certainty was soothing.\n\nOn August 31, 1957, my father paid the last instalment of the builder's bill and next day we moved in. It was characteristic of my parents that the house did not seem very modern by contemporary standards. (Now it is often mistaken for a'30s dwelling.) Money was severely limited and all the emphasis was on sound basic materials. Not a shilling was left over for frills and to this day the pine floors remain uncarpeted. But in twenty years the place has needed no structural repairs and it seems unlikely to do so within my lifetime.\n\nOur new house brought about a providential widening of my tiny social circle. While it was being built a mutual acquaintance had introduced me to an English couple who for ten years had been farming near Lismore. I found the Pearces unusually congenial but, though their invitation to call any time was plainly sincere, I felt diffident about accepting it. (My restricted life had left me a very gauche twenty-five-year old.) However, I eventually decided to use our move as an excuse to meet them again. They came to a house-warming supper, were introduced to my parents and asked me back. This time I readily accepted; I had by then realised that the Pearces were \u2013 on more than one count \u2013 in a somewhat similar situation to the Murphys. Like ourselves, they lacked an obvious social niche in West Waterford. Neither came of farming stock; Brian's father had been a business man, Daphne's an Anglican clergyman. And though they were on excellent terms with the neighbourhood, and had many friends in various parts of Ireland, they had found nobody on their own wavelength in the Lismore area. This lack was more constricting for Daphne than for Brian. He, as an indefatigable member of the National Farmers' Association, had for years been well integrated with the local community.\n\nDaphne's interests and skills ranged from stock-breeding \u2013 at which she was outstandingly successful \u2013 to painting, literature and philosophy. She was a most unlikely person to be found farming on a remote Irish hillside and she clearly considered me an equally unlikely person to be found living in a bungalow outside Lismore. We needed each other and within a month of that supper party I was visiting the Pearces twice a week. There was no gradual process of assimilation; I had at once become a member of the family.\n\nThe Pearces, with their two children aged fifteen and five, provided a substitute for the normal family life I had never known. Yet the Pearce family could be described as normal only in contrast to the Murphy household's painfully distorted affections. I had suspected, when we first met, that they too had their problems and soon I was as immersed in those as the Pearces were in mine. For all concerned, this diversification of tensions and worries was helpful and we supported each other equally during the next gruelling year.\n\nDaphne \u2013 my first woman friend \u2013 had entered my life at such an appropriate time that our meeting might have been stage-managed by some amiable angel. On one level we seemed to have known each other for a lifetime, so quickly did we become close. Yet, as a fresh observer of the Murphy scene, Daphne was able to provide objective insights and down-to-earth advice such as neither Mark nor Godfrey was capable of, both having been too close to my problems for too long.\n\nDuring 1958 I needed every sort of moral support I could get. By Easter I had begun to worry about Godfrey's worsening smoker's cough and new irritability. But when I suggested a check-up he assured me most persuasively that he felt perfectly fit and I stopped worrying \u2013 consciously.\n\nMeanwhile things had become almost intolerable at home. Since the onset of my mother's kidney complaint in 1956 I had not had one unbroken night's sleep. Every night I was roused once, in the small hours \u2013 and often twice, and sometimes thrice. Each break kept me up for at least an hour of torturing drowsiness and it was not possible for me to lie on in the morning because by seven-thirty my mother would once more need the bedpan.\n\nMy father, who slept on a folding divan bed in my mother's room, was also disturbed every night \u2013 needlessly \u2013 and the obvious solution was for me to give up my bedroom to him. No logical argument could be opposed to this move, yet for me it represented the erosion of the last fragment of my liberty. I had never, except at school, had to share a bedroom with anyone and the effect of this apparently trivial change in our domestic arrangements was devastating. Until then, nights had been a time of unwinding, despite the strain of being regularly awakened out of a deep sleep. If too overwrought to sleep, I could read, or write letters, into the small hours. (My concentration had been so diminished by lack of sleep that I had given up attempting to write anything more than letters.) Above all I could feel that I was _myself_ , in the solitude of my personal refuge. At the end of the day I needed, desperately, to be alone, having spent so many fraught hours in the dank shadow of my mother's inner disintegration. I often felt utterly unnerved by what then looked to me like a complete personality change though I now see it as a mere shift of emphasis. Under the pressure of steadily deteriorating health, certain traits which had always been present, but either suppressed or used constructively, were now being released and used destructively.\n\nMark urged me not to give up my bedroom, however unreasonable my attitude might seem, and to refuse to attend to my mother at night \u2013 thus forcing her either to reduce her barley-water intake in the evenings or to employ a night nurse. He argued that already I was near breaking-point and that for everybody's sake I should refuse to be bullied. Even from him I resented the use of 'bullied' in relation to my mother's behaviour. Yet it was justified, though her methods were so subtle that they went undetected by me.\n\nI longed to take Mark's advice but was incapable of doing so. This was partly because my mother would have made me feel so guilty had I defied her, and partly because retaining my own room would have been so unfair on my father. Thus I allowed a routine to become established that almost destroyed me. I could no longer read in bed \u2013 or between chores, because my mother kept me so busy all day. Only by the exercise of considerable ingenuity could I get through a few chapters while preparing vegetables or washing clothes or polishing silver. (When up against it one discovers how much can be accomplished 'blind'.) Worst of all \u2013 because I had become so dependent on comforting visits from my friends \u2013 my mother developed a virulent, jealous hatred of those who were closest to me. Within five minutes, at most, of my receiving Mark or Daphne or Brian, she would summon me and demand elaborate attentions calculated to last longer than my visitor could wait. Occasionally her need was genuine, but usually it was simulated \u2013 and I knew this. Yet I could not fully accept my friends' verdict: that now her illness was not only physical. The rational part of me assented, but to act on the basis of this fact was impossible. Emotionally I resisted the thought of my mother's _mind_ being diseased, though had I been able to think clearly I would have seen that the choice lay between this and a much more terrible form of spiritual corruption. So I persisted in treating her as an antagonist rather than as a patient \u2013 which proves that during those years I, too, was mentally unsound.\n\nMy mother and I were the most obvious victims of the Murphy situation, but my father must have suffered as much as either of us. While we went our tormented, obsessive ways, at least deriving some perverse relief from being active in our campaign of mutual destruction, he was a helpless, passive, sane observer, pitying us both yet unable directly to help either of us in our relationship with the other. Of course he was still helping my mother in other ways and indirectly he did help me, for I was aware of his unspoken sympathy. Our persisting inability to communicate and his loyalty to my mother prevented him from voicing this sympathy. But the older I grew the more pronounced became our strange, telepathic understanding \u2013 defying years of estrangement with the knowledge that blood has of blood.\n\nThis estrangement was now brought into my feud with my mother. For years she had accepted it sadly and tactfully, but from 1958 onwards she contemptuously upbraided me, almost every day, for my seeming indifference and ingratitude towards her beloved husband. My friends blamed my father for not intervening to free me from the maternal grip and when goaded beyond reason and justice I, too, used to condemn his pusillanimity. But this was unfair. My father, who had his own sort of moral courage and endurance, had always lacked decisiveness in his relationship with his wife. This had helped, in the past, to make their marriage the improbable success that it was; now it meant that he was no less trapped than myself. For him to appear to 'take my side' would have seemed a betrayal to my mother \u2013 though ultimately it would have been of as much benefit to her as to me \u2013 and like myself he was paralysed, at this stage, by misplaced compassion.\n\nIn my memory, August will always be the month most closely associated with Godfrey. In August 1949 we first drove to Goat Island, in August 1952 we first made love \u2013 and in August 1958 he told me that at last he was going to Peru. I did not believe him. For years he had been planning to explore Inca sites \u2013 archaeology was his subject \u2013 'When I feel better about things.' This meant when he had overcome his self-consciousness about his appearance; and I had noticed no recent improvement in that respect. We looked at each other in silence. And for an instant \u2013 it was a weird time-shift, not at all like a memory \u2013 I seemed to be back by Bayl Lough on the day when he had first looked at me with love. I knew that he knew that I did not believe him. I moved away, across the room. My heart was pounding with fear and I was shaking all over, uncontrollably. Then my fear was displaced by a desolating grief because he would not confide in me. I felt reduced again to the status of his young friend \u2013 the child whom he had to protect. Or was he protecting himself? In any case I would play it his way \u2013 as I always had. I left him ten minutes later, an hour earlier than necessary. We said goodbye casually at the gate, in full view of Jock, his manservant.\n\nI can recall almost nothing of the next four and a half months. From London Godfrey wrote stiltedly to say that he had decided to spend some time there, bringing himself up to date on recent archaeological discoveries in Peru. He expected to be very busy, and not to have much time for letter writing. I think I tried at this point to persuade myself that he was telling the truth. But when there was no Christmas letter I came back to reality.\n\nOn January 3, 1959, Jock appeared at the back door as I was straining my mother's barley-water in the scullery. He had never before been to our house and he did not have to say anything. He gave me a small parcel, containing a letter and diaries, and explained that Godfrey had wished to be buried at sea. Neither of us showed any emotion. I did not encourage him to linger, but I felt that he understood. He, too, was a good friend.\n\nAs I carried the tray into my mother's room I felt \u2013 for the first and so far only time in my life \u2013 that I was about to faint. On the wireless Beethoven's Fifth had just begun. I put the tray by the bed and sat in an armchair out of sight of my mother. Then a curious thing happened. As I listened I could feel strength returning, and not only physical strength. This was one of the strangest experiences of my life: it is virtually impossible to describe. Physically I could _feel_ the music surging through me, recharging me with every sort of energy. By the end I was completely composed and capable of continuing with the day's routine as though nothing had happened. But this was only a surface composure. Inwardly I had taken the penultimate step towards a complete mental breakdown.\n\nAgain, my memory retains few details about the appalling months that followed. Almost the only thing I can distinctly recollect is picking up a heavy chair and flinging it across my mother's room in a paroxysm of frustrated rage. Then, one July morning, I resolved to leave home for good. I cannot recall my particular 'last straw' incident. I only remember standing outside the back door, beside my loaded bicycle, and saying goodbye to my father. 'I'm not coming back,' I told him. He looked at me with despair in his eyes but said nothing.\n\nDid I then believe my own words? Later it transpired that for a year or so I had been living in a fantasy world, a grim extension of the happy imaginings of childhood. Between September and December 1958 I had forged, to myself, letters from Godfrey at the rate of about two a week \u2013 and had shown them to the Pearces. Towards the end of this period, I had also forged letters from Godfrey to Brian and Daphne. And I had reported to them that Jock frequently called at Clairvaux with news of Godfrey \u2013 though at that time Jock was also in London and never once visited Ireland. Moreover, I invented friends of Godfrey who arranged secret rendezvous with me in the County Library when it was empty in the evenings. I have no recollection of all this, but long afterwards, when I had regained my balance, Daphne described to me my frantic fantasies. She also admitted that during this period, when she was driving towards Clairvaux to visit me, her predominant emotion was fear lest she might find that at last I had physically attacked my mother.\n\nAt that time I was becoming increasingly obsessed with guilt because I had broken a solemn promise to Godfrey by telling the Pearces about our relationship. This I do most vividly remember. On the secrecy issue, I can now see that Godfrey was alarmingly neurotic and had thoroughly infected me with his own neurosis \u2013 as if I hadn't enough of my own to be getting on with. The basis of his secrecy-neurosis was of course more guilt; he tortured himself with remorse about having destroyed my virtue and risked besmirching my reputation to such an extent that I would have to go through life marked as a dishonoured woman. In retrospect, it is hard to judge which of us was the more unbalanced by this stage. And I do not like to think what might have happened to me without Mark's sanity in the background.\n\nCertainly when I left home on that July morning it was very necessary for me to believe, on the surface of my mind, that I was at last escaping. But below the surface I must still have known that no mere physical flight could free me.\n\nI cycled straight to the Pearces' farm and spent a week there. (Mark was on holiday at the time.) During this period Daphne and Brian were doing all the giving; I could be of no help to anyone. They gave unstintingly, of understanding and affection, and greedily I took. Although they must have perceived the unreality of my fantasy plans they made no attempt to deprive me of them. Only in after years did I realise how mercilessly I then drained them, emotionally, by relentlessly forcing them to share in my own suffering. And because they loved me they did share in it, willingly.\n\nMy father telephoned \u2013 guessing where I had gone \u2013 and made a pathetically feeble attempt to persuade me to return home. I pitied him, yet knew that he knew why I could not \u2013 and should not \u2013 obey this summons. Using all the inner strength I had left, I restrained myself from thinking of my mother. To think of her would have been to pity her. And then I would have returned, dangerously abandoning my healing fantasy world.\n\nAll that week I talked feverishly of going to South America with my bicycle and cycling through the Andes. It was significant that I did not drag my real ambition \u2013 to cycle to India \u2013 into this sick morass. That belonged to a happier past \u2013 and, perhaps, to a happier future. Besides, there was nothing to prevent me from cycling to India in the morning \u2013 I could easily have borrowed the necessary small sum of money \u2013 and for my purpose I needed an impractical dream that could not possibly come true.\n\nOn August 1, having regained some outward semblance of normality, I cycled to Co Wicklow. There I spent another week \u2013 still talking of the Andes \u2013 with a family who at the time hardly knew me though they have since become close friends. Then I left for London to find, as I had convinced myself, a method of working my passage to South America. My efforts were genuine and vigorous enough \u2013 I remember visiting dozens of cargo shipping offices \u2013 but of course they failed. Meanwhile I had got a job as a canteen-hand in a home for down-and-outs in the East End. Most of my fellow workers were Irish and many of them appeared to be part-time prostitutes who, in their off-duty hours, comforted the down-and-outs for a small consideration. I liked them all, as individuals, though the crudeness of their conversation sickened me, almost literally \u2013 as it still would today, despite the changed standards of the 1970s. The home was run by a kindly woman who, when she hired me, said that I must have 'private accommodation' though the other girls slept four to a room. I protested at the time, sensing and objecting to \u00e9litism, but within twenty-four hours I felt very grateful for this privilege. (My room was a closet hardly big enough to hold a chair as well as the narrow iron bedstead.)\n\nI spent my off-duty hours, which could be at any time of the day, visiting shipping offices, museums and picture galleries; or, when I had an evening off, going to the Old Vic or Covent Garden. To an observer, my job might have seemed dreary and arduous \u2013 dishing out steaming meals for hours on end in the heat of a London August \u2013 but to me it seemed positively relaxing. My eight-hour day left me more time to myself than I had had for many years. And \u2013 most important of all \u2013 I knew that I could sleep all night, every night. Only those who have endured long periods of interrupted and insufficient sleep can appreciate what that meant. Often, during the day, I thought with luxuriating delight of the night to come \u2013 as a drug-addict might think about his next fix.\n\nI was of course constantly plagued by guilt. At first I ruthlessly suppressed it, but as the days passed and I unwound, and emerged from my fantasy, I allowed myself to examine it. I had certainly done a cruel thing. Although my escape had been essential it should have been effected more considerately and rationally, after alternative arrangements had been made for my mother's welfare. But she would not have consented to any such arrangements. I reasoned that it would be unfair to blame myself alone for this cruelty. It was like some poisonous vapour exhaled by a corrupting situation to which we had all three contributed. Then, for the first time, I allowed myself to look steadily and dispassionately at my mother's case. I can remember the spot where I finally reconciled myself to the fact that she had partly lost her reason. I was sitting outside St Paul's, in hot noonday sun, and my whole being seemed to be wrung by pity and anguish. The death of the mind is infinitely more terrible than the death of the body and I mourned my mother that day as I was never to mourn afterwards.\n\nAfter three weeks in London I knew that I was ready to go home. I gave in my week's notice, wrote to my parents \u2013 of whom I had had news, through Mark, every other day \u2013 and positively looked forward to being back in Lismore and seeing my friends again. As I had always been wont to do, I counted my blessings, which seemed many and various when I compared my life to the lives of my workmates in the hostel. They were, indeed, free, as I had never been. But of what value was their freedom when they were so much 'less fortunate' than myself?\n\nDuring that final week of liberty, while still out of my mother's emotional grip, I resolved on a campaign that would have spared us all much suffering had it been organised a few years sooner. On my return home I would, at least for a brief period, be in command. I had demonstrated not merely a reluctance but an incapacity to survive under my mother's r\u00e9gime; and, while she was still feeling the shock of my revolt, I would initiate my own reforms.\n\nArriving home, on a sunny mid-September day, I found my father in a predictable state of exhaustion and my mother in an even more amenable mood than I had hoped for. Aware of the authority conferred on me by my breakdown \u2013 the family doctor, in my presence, had bluntly said, 'I told you so' to my mother \u2013 I announced quietly that I would engage a daily maid and take two hours off every morning or afternoon, in addition to four hours every evening. On alternate Saturdays and Sundays I would be out from two-thirty to ten-thirty and in April I would return to Spain. My mother meekly accepted all these reforms and was genially welcoming when I produced a cheerful, clean, intelligent, efficient sixteen-year-old who showed no alarm at the prospect of being left in charge of an invalid. (She was not of course expected to do any nursing chores, apart from helping my mother to drink her barley-water \u2013 which, incidentally, I was now rationing, to ensure my night's sleep.)\n\nIt dismayed me to find that although I had reclaimed my own bedroom, and given myself ample free time, my concentration remained unequal to reading a serious book and writing a coherent letter required a tremendous effort. Yet this was hardly surprising. The easing of my duties had not made the domestic atmosphere essentially any less tense and I was always braced for a resumption of hostilities. Outwardly my mother's attitude had changed considerably \u2013 for instance, she never now summoned me when I was talking to my friends in the sitting-room  \u2013 but I could see how much she inwardly resented the extent of my victory. Despite my full awareness of her mental condition, our antagonisms had become too deeply rooted for either of us to extirpate them when we were in close daily proximity. However, for the rest of 1959 I was well able to take the strain, largely because I believed that my next year's holiday was secure.\n\nThen, soon after Christmas, hostilities were resumed in a new and baffling way. My mother developed asthma and suffered frequent violent spasms at all hours of the night and day. This terrifying affliction was in a sense the most harrowing of all her diseases, not only for its victim but for my father and myself, who both loved her so much \u2013 though my love had long since ceased to be apparent. It of course meant that she could no longer be left in the care of a maidservant, that my bedroom again had to be sacrificed and that our nights were more broken even than hitherto. Also, before long it had been made plain that I could not reasonably expect to get away to Spain in April.\n\nWe still had Mary to do the housework and I still insisted on my alternate Saturdays and Sundays off duty. Otherwise we were back to the impasse of the previous year and by April I was again losing my grip. One symptom of this was an extreme, neurotic self-contempt. I had accepted that my mother was mentally ill, I daily witnessed her physical sufferings \u2013 yet I could feel no compassion or tenderness. I could not make myself feign sympathy during those ravaging asthmatic attacks. Worse, I now sometimes regarded my mother with a hatred that seemed to flare up like a sheet of evil fire from some diabolical inner volcano. It was her body that I hated \u2013 the body that had already destroyed the reality of _her_ and was threatening to destroy my own reality. Afterwards, I understood this hatred as love gone underground. Then, I was aware of it only as a base emotion that frightened and disgusted me. I could not see deep enough within myself to discern my intense distress at my mother's agony. Had my natural reactions not been suppressed I might have disintegrated even sooner than I did. By concentrating on my own miseries, and callously ignoring my mother's, I at least remained sane enough to nurse her.\n\nSeveral times, that summer, asthmatic attacks threatened my mother's life. And I did not find it necessary to conceal from myself the fact that I longed for her to die. The mother I had loved and the person I had admired was already dead. The tortured body, the tragically disordered mind and the degraded, petty emotions were a heart-breaking travesty. Yet our doctor had assured us that with the aid of modern drugs, attentive nursing and a basically ox-like constitution, my mother could live another twenty years. I dared not allow myself to dwell upon this prognosis. In twenty years' time I would be nearing fifty \u2013 a middle-aged woman, drained of physical stamina, emotionally embittered and intellectually atrophied. There were enough such dutiful unmarried daughters around Lismore for me to have a clear prevision of myself in 1980. It was about then that I switched from sherry to whiskey. \n\n#\n\nMy father was sixty on December 16, 1960 \u2013 a singularly auspicious date, my mother used to say, being also the birthday of Beethoven and Jane Austen. A few days later he went down with influenza, from which he usually recovered within forty-eight hours. But his resistance had been drastically lowered by the griefs and anxieties of the past few years and this time he spent a week in bed. While still convalescent he insisted on going to Dublin, by train, in blizzard conditions, on December 27. He had promised to lead a delegation of librarians to discuss with a government minister some \u2013 to him \u2013 vitally important point concerning the development of the library service; and he refused to back down at the eleventh hour, having had no opportunity to brief a substitute leader. My mother did all she could to check this lunacy, but on such an issue not even she could sway my father.\n\nWhen he arrived home three days later, glorying in his delegation's victory over the minister, my mother and I were both shocked by his appearance. He went straight to bed and next morning was running a high temperature and too weak to stand up. At first we took this to be a not surprising influenza relapse, but by New Year's Day our doctor was suspecting nephritis. On January 3 a specialist came from Cork and explained that this well-known complication of neglected 'flu did not require hospital treatment. Given my father's medical record \u2013 apart from two attacks of sciatica, he had never in his life had any illness more serious than 'flu \u2013 it would probably right itself in a matter of weeks. I listened to these cheerful assurances with a disbelief that reminded me of Godfrey's Peruvian plans. On January 2 (the second anniversary of Godfrey's death) I had realised that my father was dying \u2013 and that he knew it.\n\nFor the next month I nursed both my parents, with the aid of twice-daily visits from the Jubilee Nurse. I lived on whiskey and cigarettes and often did not bother to undress at night when I slept on a camp-bed in my father's room. My mother now demanded far less attention. Engrossed in the suspense and horror of my father's illness, she was neglecting her barley-water treatment and had had no asthmatic attack since New Year's Eve. Her anguish, at the prospect of losing all that really mattered to her in this world, completely dissolved my hard core of unfeeling. We sat up at night over tumblers of whiskey, and my mother reverted to smoking Turkish cigarettes specially sent from Dublin \u2013 an indulgence foresworn, many years before, in the cause of economy \u2013 and we talked as adult friends, savouring the sort of relationship that should always have been ours. I marvelled then at this rebirth, in the shadow of death, of the person I had thought lost for ever.\n\nFor some three weeks my mother continued to hope, or to pretend to herself that she was hoping, yet my father's condition gave her less and less reason to do so. Meanwhile I existed in a strange state of thought-suspension, not attempting to cope intellectually with what I had never tried to evade emotionally. I only knew how thankful I was that the specialist had advised against hospital treatment. During those weeks a twenty-year-old barrier went down and the ordeal of witnessing my father's physical dissolution was eased by the fact that he was so palpably unafraid of death. Going into his room one dark afternoon at the end of January, I found him lying there, looking serenely out at the wide grey sky, with a hand on his open bible. I had thought of him, these past few years, as suffering no less than myself though in very different ways. Now I saw how wrong I had been. He wore an armour that I had not inherited.\n\nA few day later the specialist came again and advised my father's removal to hospital. At seven o'clock an ambulance arrived and, as the stretcher was being carried past my mother's open door, my father weakly called, 'Au revoir!' My mother and I both got slightly drunk that night as we listened to a Brandenberg Concerto and the Archduke Trio.\n\nDuring the next week I travelled daily to Cork while kindly neighbours sat with my mother saying appropriate things in hushed tones. The nurses and doctors described my father as one of the most considerate and selfless patients they had ever met. Then, on February 9 \u2013 the thirtieth anniversary of my parents' wedding \u2013 I was warned that he could not possibly live more than another two days, and that he had been told this. Yet when I went into his room he seemed no worse than previously. We looked at each other, and then spontaneously shook hands with a curious mixture of formality and tenderness. 'Will Kit be able to come?' my father asked. I assured him that she would \u2013 in fact her suitcases were already packed \u2013 and within half-an-hour I had booked another hospital room and organised an ambulance in which I drove back to Lismore. Three hours later, my mother's bath chair was wheeled close to my father's bedside.\n\nDespite the medical forecast, a few days passed, and another few days, and a week, and still my father lived on, calm and clear of mind, steady of speech, holding his wife's deformed hand for hours on end. I was entirely superfluous to both of them and spent most of the day in my mother's room across the corridor, reading P. G. Wodehouse and sipping whiskey. When I joined them at mealtimes, to feed my mother, I often found them laughing together. Sometimes my father attempted to discuss the practical problems that were looming for his widow and daughter. But my mother would dryly point out that this was an inappropriate time to change a life-long habit of concentrating on spiritual rather than monetary matters. I saw then how firmly they believed in a reunion elsewhere. To me, death was a question mark. To them, it was a temporary separation \u2013 sad beyond measure, yet not final. At that time a strange sort of gaiety possessed us as a family, artificial on one level, genuine on another. And as the days passed my father, inexplicably, continued to live. His doctors used to pace up and down the corridors in a ferment of scientific curiosity and bewilderment, muttering theories _sotto voce_. To me, however, my father's surviving was a clear case of mind over matter; he did not want to leave my mother. He himself took a keen interest in the mystery of his own continuing existence. Medical conundrums had always fascinated him and now he was not only willing but eager to consider the most grisly hypotheses with his doctors.\n\nBy this stage, inevitably, my mother had begun to hope again, though she would not admit as much. But then, on the morning of February 25, my father had a haemorrhage and became perceptibly weaker. Yet he did not lose consciousness and all that day, as we sat by his bed, he was aware of us. At twenty to seven in the evening he was still holding my mother's hand firmly with his right hand. I was holding his left hand, with a finger on the slowing pulse. So gently did he go that I realised it moments before my mother did.\n\nThe hospital staff were a little disconcerted by my mother's composure \u2013 and, indeed, by mine. Tears, one gathered, were obligatory, yet we shed not one between us. As I went to and fro, arranging what had to be arranged, I was pursued up and down corridors by perplexed nuns brandishing unwanted sleeping pills and assuring me that everybody needed sedatives after a bereavement.\n\nMy father's was the first corpse I had seen, yet it left me unmoved. It was simply an irrelevant, impermanent piece of matter. As the funeral procession travelled to Lismore I sat by my mother in the ambulance behind the hearse, more conscious of apprehension about my new responsibilities than of grief.\n\nMy father was buried in Lismore on February 27 \u2013 the first of his family, for many generations, to be laid to rest outside Dublin. The town that had received him so coldly thirty years earlier closed its shops in mourning, and it comforted us that the phrases of praise and regret were uttered not only because convention prescribed them.\n\nI stood between Daphne and Brian at the graveside, disliking my conspicuous r\u00f4le as chief mourner and feeling more embarrassed than distressed by the inescapable ritual. Many, I knew, were expecting me to break down and add a touch of drama to the occasion. The Irish have a flair for wringing from death the last drop of emotion and they do not quite understand those who react otherwise. It was a still, sunny day, the air damp and warm; birds sang their first desultory spring songs in the grassy graveyard \u2013 until alarmed by the volley of shots that rang out over the coffin of a man who had been a soldier not only in his youth.\n\n* * *\n\nWe all enjoy drama with some part of ourselves \u2013 whatever its source \u2013 and are stimulated by it. But now suddenly the tension, the need for courage, the compulsion to consider only the dying, the will to control panic and grief, the first challenging confrontation with the vast, incomprehensible fact of Absence \u2013 all were gone. So too, on the morning after the funeral, were the visiting friends and relatives from Dublin. My mother and I faced each other, alone, in a house that felt chill and damp after two and a half weeks of emptiness. Soon it had become clear that my mother expected me to be able, by some miracle, to fill my  father's place. The absurdity of this expectation both touched and enraged me. I had hoped that she and I could continue to build on our new friendship, but if my mother could not accept me as myself she would have to forgo all the support that I was then most willing to give.\n\nI did not, of course, see the situation so plainly at the time. I only saw the maternal will to dominate reasserting itself. It was decided that Mary must go, as we could no longer afford her wages; and though I knew this to be true I also knew that the time was inopportune for such an economy. We argued, and within hours my mother had had another severe asthmatic attack. The doctor had to be summoned, Mary had to be dismissed and for precisely ten weeks from the day of my father's funeral I did not once set foot outside the garden gate.\n\nMy mother's reversion to normality during my father's illness made it harder to accept mental derangement as a valid excuse for her behaviour. True, she had just suffered the heaviest possible emotional blow. But the effect of her irrational demands was to make me insensitive to her grief and I reproved myself hourly for my cruelty while watching it drive us both into ever deepening misery. Our neighbours added to my guilt by making me feel a hypocrite. They stared at the surface, and admired it, and had no conception of the spiritual betrayal below it. I was there twenty-four hours a day and my mother's every physical need, real or imagined, was ministered to promptly and efficiently \u2013 and with a savage resentment that would have scandalised those good neighbours had they been capable of discerning it. Looking back, I find it odd that I had not learned enough from my 1959 experience to resist this latest takeover by my mother's will. But this time she had the two new weapons of asthma and bereavement, while I was bereft of my father's tacit moral support and practical assistance. Also, from him I had inherited a weak soft-heartedness that was closer to moral cowardice than to true kindness because it fostered my mother's neuroses.\n\nMy long imprisonment, during those ten weeks of spring, completely broke my spirit. I ceased to fight, inwardly, and lived from day to day in a cocoon of resignation. I ate almost nothing, smoked far too many cigarettes and drank far too much whiskey. Yet I never got drunk; I was at the more dangerous stage of keeping my alcohol level up twenty-four hours a day. Finally Mark took action, when he was unable to endure any longer the change taking place within me. I can remember no details, but a responsible daily woman was installed, my mother was somehow persuaded of the urgent necessity of releasing me and on May 7 \u2013 that date I have never forgotten \u2013 I cycled down the road, feeling incredulous, with three hours of freedom ahead of me. This was to be the routine: Saturdays and Sundays excepted, I would be free from two to five every afternoon.\n\nIt needed only this break in the automaton rhythm of the past months to release a cataract of despair. I was nearly thirty and had achieved \u2013 it then seemed \u2013 nothing. As a daughter I was a failure, as a woman I was ageing, as a writer I was atrophied, as a traveller I had only glimpsed possibilities. But at least I was again reacting and feeling, even if all my feelings were painful.\n\nIn the middle of June, when we were granted probate, the financial facts of our new life had to be faced. For the past decade, one of my mother's most trying phobias had been an economy mania. Gradually she had developed an obsessional fear of real poverty, should my father predecease her \u2013 and now her nightmare had come true. What were we to live on? We had our home, and a weekly income from investments of exactly \u00a34 2s. \u2013 and nothing else. Should we sell Clairvaux and buy a small cottage or rent a couple of rooms? Or let Clairvaux while living elsewhere? Or take in lodgers? Or convert Clairvaux into two flats and let one of them? There could be no question of my earning anything as I was almost unnaturally devoid of accomplishments \u2013 unable to drive, type, sew, cook, garden, keep accounts or even amuse children. Yet I could not take our parlous economic state seriously; my Micawber streak is too strong. I have never been able to worry about money, partly because I need so little of it to keep me happy and partly because of an unshakeable conviction that 'Something Will Turn Up'. (Something always does.)\n\nI at first thought it healthy for us to have this immediate, concrete problem to deflect us from our psychotic personality battle; but soon the problem had become part of the battle. I was totally opposed to selling Clairvaux; in an odd way I had always thought of it as particularly _mine._ And, despite what we had each endured within the house, I loved it \u2013 less for itself than for its surroundings. My mother, however, had worked out, most logically, that it made no sense for people with such a tiny income to have so much capital tied up in their home. She therefore decided, towards the end of June, to sell Clairvaux and move to a Dublin flat \u2013 or, more likely, bed-sitter.\n\nI resisted this plan strenuously, and not only for the sake of the house. I knew that I could not possibly cope if uprooted from Lismore and deprived of my friends' support. But I lost the first round and early in July we moved to Dublin, leaving Clairvaux on an estate agent's books.\n\nOur poverty notwithstanding, my mother was too proud to put any relative to the inconvenience of entertaining an invalid. She settled into a nursing home, moderating her nocturnal demands to avoid having to engage a private night nurse. I stayed nearby with an aunt and did duty from 7.00 am until 11.00 pm. At intervals I was despatched to inspect flats \u2013 a task which I mitched, having made up my mind that under no circumstances would I agree to settle in Dublin.\n\nOur unfortunate relatives, suddenly drawn into the Murphy inferno, were altogether at a loss. They gave us much well-meant, conflicting and usually irrelevant advice, which we ignored. Only in opposing my decision to take a holiday abroad were they unanimous, seeing this as no way to treat a recently bereaved invalid mother. But I was impervious to their criticisms. I knew that I must now assert myself, once and for all, as an adult whose reasonable demands had to be met if my mother and I were to have any future together. Our relatives, however, being unaware of these undercurrents, could not be blamed for condemning my plans as pure selfishness. (My mother and I habitually closed ranks in the presence of outsiders and never allowed them to see just how strained was our relationship.)\n\nNow it was my turn to blackmail. In mid-July I told my mother that I had resigned myself to the sale of Clairvaux but would never agree to leaving the Blackwater Valley. If she would return to Lismore, we could go home after my holiday and set about searching for a suitable cottage. Otherwise, I would remain abroad. Without a moment's hesitation, my mother called my bluff by refusing even to consider a return to Lismore. A few hours later, my bicycle Roz and I took the night boat from Dun Laoghaire.\n\nThis time I was not being swept away from Ireland on a wave of demented fantasy; I intended to return as soon as my mother could adjust herself to the realities of our situation. Yet this flight from duty distressed me even more than my 1959 escape. My actions were not out of control, as they had been two years previously, and I knew that my ostensibly final departure was a calculated measure designed to break the will of someone who had already had to suffer far more than her share. Therefore during the internal dialogues that took place between the two halves of myself I found it necessary to emphasise how relentlessly my mother had followed a course that must drive me away \u2013 almost, indeed, as though a part of her wished to do so, thus unloading from her conscience the burden of being my millstone.\n\nI landed at Calais with \u00a310 in my money-belt \u2013 enough to take me to the Hilckmanns, in Mainz. There I could be sure of a warm welcome; they had stayed with us for six weeks during the previous summer while Anton was being taught Irish \u2013 his twenty-second language \u2013 by my father. I slept out in my flea-bag each night \u2013 the weather was almost too hot for long-distance cycling \u2013 and had unwound considerably by the time I reached Mainz. There I relaxed for a week before getting work as a farm-hand in the village of Ober Saulheim, some fourteen miles further down the Rhine. I spent the next six weeks hoeing vines, scouring churns, harvesting straw, mucking out cow byres, grading eggs and corking wine. My working hours were not much shorter than at home \u2013 5.0 am to 11.0 pm with a half-day on Sundays \u2013 but I welcomed being (by Irish standards) slave-driven. It was exactly what I needed at that time. Ceaseless labour, with only four half-hour breaks for vast meals \u2013 two of them picnic-style, eaten on the job-site \u2013 left me without the energy to fret. At the centre of me a hard little knot of guilty misery remained untied and I dared not think about the future. But meanwhile constant exertion in the open air, good food, few cigarettes and unbroken nights of deep sleep were rapidly restoring my health.\n\nEven as late as 1961, in technological Germany, one could, as a farm-labourer, feel closely in touch with the past \u2013 with the many generations of villagers who had lived in those same houses, cultivating those same hectares. The almost universal farm village is unknown in Ireland and I enjoyed the novelty of sharing in Ober Saulheim's rhythmical community life. Early each morning lines of oxcarts creaked off to the cornfields in a cloud of yellowish dust, and files of vineyard workers made for the sunny slopes above the river with hoes over their shoulders, and groups of hausfraus gathered by the communal deep-freeze in the village centre to take out great hunks of meat and containers of fruit and vegetables. At that season Ober Saulheim smelt pleasantly of fresh cow-dung and sweet hay and new milk and fermenting wine, with only an occasional discordant whiff of diesel oil from a passing tractor. The nearest autobahn was only a kilometre away, but as far as the villagers were concerned it might have been on another planet.\n\nAlthough my employer's vineyard was well known, and his son and heir was a forward-looking twenty-one-year-old, the Landgraf family \u2013 like their neighbours \u2013 used few labour-saving devices. They had all the basic bits of agricultural machinery, and a simple hand-machine for milking their ten cows, but the house, outbuildings and cellars \u2013 and most of their work methods \u2013 remained exactly as they had always been. This disdain for gadgetry and the symbols of affluence greatly appealed to me, as did the whole village atmosphere. There must have been jealousies, squabbles and gossip, as in every such community, but the general feeling was of uncomplicated serenity.\n\nYet that holiday confirmed what I had suspected on my first visit to Germany: that the Germans and the Irish have remarkably little in common. The Landgrafs could not have been kinder to me and I became especially fond of Frau Landgraf, who treated me like an adopted daughter. But whether in the farming world of Ober Saulheim or in the academic world of Mainz \u2013 where I spent most Sunday afternoons with the Hilckmanns \u2013 I found it impossible to establish that rapport which had come about so effortlessly in Spain. No doubt the Germans are as well able to enjoy themselves as anyone else; it was probably mere chance that in both those circles nobody seemed to have the slightest interest in anything but unsmiling hard work.\n\nAt least twice a week relatives wrote to report on my mother's state. Physically she was no worse than usual and was being well cared for by the nursing-home staff \u2013 though she herself did not think so. Mentally, however, she was suffering intensely as a result of my desertion. Then, in mid-September, came a letter saying that she was longing to return to Lismore and enclosing a Frankfurt-Dublin air-ticket. On compassionate grounds, a dismantled Roz was allowed to travel with me as baggage.\n\nI could detect no reproach in my mother's manner when we were reunited. And I wondered then, as I had for the same reason in 1959, if, with the residue of her 'real' self, she understood all and so could forgive all.\n\nOn our first evening at home in Lismore my mother suddenly said, 'I want you to take Clairvaux off the books tomorrow. All my worrying about money is silly \u2013 not necessary. You'll have your freedom quite soon.' She had spoken very matter-of-factly and I thought she was probably right; with my father she had wanted to live, without him she wanted to die. But her tone had been accusing \u2013 and there is a unique horror about being accused of wishing someone to die when the accusation is true. Momentarily I was tempted to make some insincere, dutiful remark; but I chose silence. Any false protest could only degrade our relationship still further. As it was, that brief exchange of words, glances and silences had had a paradoxically companionable undercurrent.\n\nDuring the following days, I noticed how much my mother had changed in my absence. My struggle to break her grip on me had been only too successful. Not merely had she lost the will to dominate me, she seemed also to have lost the will to live. If I wished to lacerate myself I could argue, plausibly, that I killed her by deserting her that August. But if this is so, I did it under the guidance of my own instinct of self-preservation. Jungle law operated. One of us had to be sacrificed and the fittest survived.\n\nThe eleven months that followed were the saddest of my life. At times I even looked back nostalgically to the era of my mother's domination; that, at least, had been an expression of her personality \u2013 however distorted. Now her dulled mind and passivity and compliance were my punishment for having so implacably opposed her in the past. She still listened to music, but she did not read, or listen to discussions on the wireless, or wish to hear what was in the _Irish Times_ , or proffer any opinion about the running of the household. Only when visitors called did some ember of pride flicker into flame and she would receive them with a poignantly recognisable shadow of her old graciousness and cheerfulness. I was no longer in her consciously exercised power, yet never before had she had such power to move me.\n\nBy the beginning of August 1962 it was clear that soon my mother would die; she who had never accepted painkillers or tranquillisers was now permanently under heavy sedation. I moved about the house in a daze. Sorrow contributed nothing to it \u2013 or nothing that I could perceive. Relief predominated, but there was also a primitive reaction of horror \u2013 altogether absent during my father's last illness \u2013 to the physical dissolution of the body that for so long I had tended. In thirty years of invalidism my mother had never once developed a bedsore, but now her body began to decompose while she still lived. I could cope no longer and the Jubilee Nurse came twice a day.\n\nOn the evening of August 24 Daphne came to sit with me through the night and at twenty-five past one in the morning my mother died \u2013 as peacefully, in the end, as had my father. She had survived him by exactly eighteen months.\n\nAs my mother drew her last breath, peace enfolded me. It was profound and healing, untinged by grief, or remorse, or guilt, or loneliness. I thought of it as a gift from my mother's spirit \u2013 and then mocked my fancy, without quite discrediting it. For long I had suffered with her, and made her suffer, and been made to suffer by her; and of late I had mourned for her. Now I could only rejoice \u2013 and in Daphne's company I did not have to disguise my joy. A great burden was gone, the double burden of another's tragedy and my own inadequacy. I stood at the threshold of an independent life and I felt, that night, my parents' blessing on it.\n\nAfter a moment I left the room \u2013 my mother had not yet assumed the aspect of the dead \u2013 and in defiance of local custom I never looked on her face again. Twelve years passed before a letter from a friend, on the subject of her own mother's death, allowed me to see why I had so obstinately stayed in the sitting-room when I should have been pretending to pray with the neighbours beside my mother's corpse. I knew, without permitting the knowledge to register on my conscious mind, what the undertaker would have to do to accommodate a malformed body in a conventional coffin. And I could not bring myself to look upon that body in a normal lying posture.\n\nIt is difficult to convey my feelings when I woke next morning and realised that I was responsible to and for no one but myself \u2013 that I was free to do what I liked, when I liked, as I liked. For more than sixteen years every day had been lived in the shadow of my mother's need. Even on holidays, my movements had had to be exactly regulated so that I would unfailingly arrive home on a certain date. I remember sitting in hot sunshine in the back garden with Daphne \u2013 surrounded by that untrammelled growth of nettles and thistles which proved me to be my father's daughter \u2013 and feeling currents of an appreciation of liberty running through my body like mild electric shocks. I was exalted by the realisation of freedom. When callers came to offer sympathy, Daphne received them and gave the necessary convention-soothing impression that I was too distrait to appear.\n\nAfterwards, I discovered that some of my friends had dreaded my reaction to my mother's death; when at last the almost intolerable pressure was removed, anything, they felt, might happen. They had overlooked the fact that for months the pressure had been gradually easing, giving me an opportunity to readjust.\n\nAgain I stood between Daphne and Brian at the graveside, as my mother's coffin was lowered to lie beside my father's. Then I returned with them to Monatrim and next morning visited Clairvaux for a few hours and was alone for the first time in my own home. Unexpectedly I found myself revelling in the novel sensation of ownership; possession of a house and all it contained seemed to symbolise Freedom and Independence. It never occurred to me that I had no means of maintaining the place: possession was all. At the age of thirty I had not yet possessed anything but the minimum of clothes, the maximum of books and a bicycle.\n\nI wandered happily from room to room. The ghosts were friendly; the misunderstandings and antagonisms and furies and resentments of the past had left no stain. I decided to convert my mother's bedroom into my study, a simple change that needed only the replacing of her bed by the dining-room table. All the other furnishings of the room were left unchanged \u2013 it had been designed as a sitting room rather than a bedroom \u2013 and unchanged they remain to this day. I write on the spot where she died.\n\nLove leaves calm. Even when circumstances have given it the semblance of hate, this is so. In the tangled relationships between my parents and myself love was often abused, denied, misdirected, thwarted, exploited and outwardly debased. But it existed, and it left calm.\n\n#\n\nTwo days after my mother's funeral I left Lismore to visit my Co Wicklow friends. By then I realised that despite my basic calm I was, inevitably, suffering from shock. A total lack of physical energy betrayed this condition. Feeling too exhausted to cycle the 130 miles to Wicklow, I hitch-hiked.\n\nMy friends' house was near a private beach to which they and their guests had access. There, during the first week of September, I spent long, sunny, solitary days lying on the slopes of curving sand-dunes, gazing at the blue sky through tall clumps of green-gold grass and hearing nothing but the hiss of a lazy sea on the beach below. Swiftly my strength and energy returned. And then, towards the end of the week, something odd happened to my sense of time.\n\nSo stark was the contrast between my previous bondage and my present liberty that I could scarcely conceive of them as belonging to the same existence, and the few days since my mother's death seemed as long as the many years that had gone before. Yet counterbalancing this was a powerful sense of continuity. I had briefly experienced something similar after Godfrey's death and my father's; from now on I was to be aware of it as a permanent undercurrent. For however long my new life lasted, it would be subtly conditioned, in every detail, by a past which already seemed so remote that I saw myself moving through it as a stranger. And with no part of myself could I regret that past. This was not because of any deliberate effort to accept, in retrospect, what I had so often rejected when it was a present torment. In fact, I had never quite rejected it. Usually I had been half aware, and occasionally fully aware, of the potential value to any human being of a certain degree of suffering. Had mine continued much longer, it would have destroyed me. As it was, I could feel, at the end of the ordeal, wholly without resentment.\n\nWhat had so often seemed wasted years, in my many moods of bitterness, now seemed otherwise. During those September days I could not have given any reason for feeling thus. But I have since realised that events and emotions which at the time of their happening were apparently destructive and enfeebling, in their enduring results were constructive and tempering. I had learned a lesson in humility that could not have been taught to anybody of my arrogant nature by less violent means. Without my friends I could not possibly have survived; their love had borne me to safety. And I recognised that, because of my prolonged emotional dependence on them, the person I had become over the past decade was partly their creation. Had I left home at eighteen and made a successful career for myself, I would probably have gone through life as an intolerant, unsympathetic bitch \u2013 a r\u00f4le for which I had as a youngster all the necessary qualifications. But years of being confronted by my personal weaknesses, and striving to comprehend our family weaknesses, had to some extent modified these traits. At thirty, I could ignore neither my own flaws nor the endless variety of causes that can lie behind the flaws of others. The school was hard, but the knowledge was priceless.\n\nMany hours on the sand-dunes had been spent methodically planning my journey to India. Having for the past twenty years intended to make this journey, it did not strike me as in any way an odd idea. I thought then, as I still do, that if someone enjoys cycling and wishes to go to India, the obvious thing is to cycle there. Soon, however, I realised that most people were regarding me either as a lunatic or an embryonic heroine; in 1962 Western youth's mass trek to the East had not yet begun. When I went into a cycle shop to have Roz's derailleur gears removed, and explained that I was going to India and felt they would not be suitable for Asian roads, the mechanic looked at me very strangely indeed. After that I became a trifle inhibited about discussing my plans.\n\nSeveral people suggested that the trip should be sponsored, perhaps by the makers of my bicycle \u2013 or by Guinness, since their product so habitually nourished the body that was to undertake this alleged marathon. Or even by a newspaper, to which I could send back dramatic stories from improbable places. But these suggestions appalled me. Any sponsor would have made of my private journey a public stunt and the very thought of the resultant limelight made me sweat with terror. Also, sponsorship would have given a dishonest twist to the whole experience. By this stage I had had to concede that there was, objectively, something slightly peculiar about the notion of cycling to India; to have persisted in denying this would have been to argue that the rest of the world was out of step. Yet subjectively it remained true that I saw nothing peculiar in it and to have the project presented to the public as something exotic and daring would have falsified it utterly. Of course I saw it as an adventure \u2013 that had been its attraction, since my tenth birthday \u2013 but I knew it required no unusual courage or stamina. It was planned as, and it proved to be, a happy-go-lucky private voyage to enjoy some of the world in the way that best suited my temperament. And, if the publishing trade winds were blowing my way, to provide material for a book.\n\nI can see now that this journey seemed extraordinary to many simply because I provided my own energy. When the overland trek to Asia became popular among the young, a few years later, the vast majority hitch-hiked or used local transport \u2013 though they were half a generation my juniors and should have had that much more personal energy available. This dependence on motor transport I find very disquieting, when adventure and enjoyment are the objects of the exercise and time-saving is not a consideration. It indicates that we have become more dependent on _things_ than rational beings should be. In some societies this dependence is already causing people to degenerate physically. Men and women who live all their lives in centrally heated homes and offices, and go in the car to post a letter and collect the children from school, and have labour-saving devices for every conceivable purpose (including electric tooth-brushes and carving-knives) \u2013 such people have become so sensually unaware and so unresponsive to physical challenges that they are only half-alive. If they cannot soon escape from Affluence and Technology some very odd biological mutations seem likely within the next few generations. The urge to effect this escape of course underlay what was vaguely known as the Hippy Movement. But few hippies realised that their routine of quitting the rat-race, seeking a guru, going vegetarian and growing a lot of hair was no substitute for training their bodies to do what in previous generations any healthy young body could do.\n\nThe winter of 1962\u201363 was Europe's most severe for eighty years and I shall never forget the agonising cold of that dark January morning when I began to cycle east from Dunkirk on an ice-bound road. As I wrote to Daphne a few days later, 'To have the fulfilment of a twenty-one-year-old ambition within one's grasp is quite disconcerting. I had thought about this moment so often that when I actually found myself living through it I felt as though some favourite scene from a novel had come, incredibly, to life.'\n\nThe hardships and poverty of my youth had been a good apprenticeship for this form of travel. I had been brought up to understand that material possessions and physical comfort should never be confused with success, achievement and security. And soon I was discovering for myself that our real material needs are very few and that the extras now presented as 'needs' not only endanger true contentment but diminish our human dignity. None of the privations, hazards or unforeseen difficulties of that journey bothered me. To be able unrestrainedly to gratify my wanderlust, after so many years of frustration, was all that mattered.\n\nSoon after my arrival in New Delhi, I volunteered for work in a Tibetan refugee camp in the Himalayan foothills. Some of my friends then imagined that the miseries of Asia had so moved me that I felt compelled to contribute my mite towards their alleviation. But this was too simple an explanation, as was the reason I gave myself at the time \u2013 that it was too hot to travel any further. I now suspect that I took this unpaid job, and again immersed myself in grinding hard work amidst extreme physical suffering, because some absurd puritan streak made me feel that it was my duty to do so, after six months of unrelieved self-indulgence. Of course I very soon became deeply involved, emotionally, in the whole Tibetan refugee problem. But that is another story.\n\nWhile I was still with the Tibetans, I received a puzzling letter from a well-known English publishing house expressing interest in my 'feat' and saying that should I happen to write a book about it they would be very interested to see the typescript. The book was in fact already written, as I had kept a detailed daily diary on the way to India. I explained this, promising to send the typescript in due course. It later transpired that a garbled account of 'Miraculous Overland Cycle by Lone Irishwoman' had appeared in an Indian paper and been spotted by that publisher's New Delhi agent.\n\nThen, shortly before my return home in March 1964, I met Penelope Betjeman in Delhi. She had come to India to collect material for a book and when I confessed to literary ambitions she said, 'Of course! Marvellous journey! Marvellous book! You must send it to Jock Murray.'\n\nWe were cycling together through a crowded Old Delhi bazaar \u2013 Penelope with an accident-inviting load of firewood tied to her carrier \u2013 and I yelled above the blare of rickshaw horns, 'To _whom_?'\n\n'To Jock Murray,' Penelope yelled back. 'You'll adore Jock \u2013 everybody adores Jock.'\n\n'Do you mean John Murray in Albemarle Street?' I asked disbelievingly.\n\n'Yes of course,' said Penelope. 'Jock will love it \u2013 just his sort of thing.'\n\nI was so profoundly shocked by this irreverent suggestion that I almost ran into a sacred cow. To someone reared on the lives and works of nineteenth-century English writers the very thought of submitting a Murphy typescript to John Murray seemed blasphemous. Jane Austen, Byron, Darwin, Borrow, Livingstone, Isabella Bird, Younghusband, even Queen Victoria \u2013 these and a score of other names flashed through my mind, dazzling me. I pedalled faster to draw level with Penelope and said, 'You must be mad!'\n\n'No more than usual,' she retorted. 'Don't forget \u2013 do as I say.'\n\nThe immediate sequel to my journey was exceedingly unpleasant. In India I had been working under the Save the Children Fund umbrella and on visiting their London headquarters I found myself seized upon as invaluable material for a fund-raising campaign. I was to be exposed to everything I most detested: a press conference, photographers, wireless interviews and \u2013 the ultimate hell \u2013 television interviews. Obviously I could not say no; I had witnessed too much suffering in the refugee camps not to co-operate. Miserably I went through the hoops, revolted by the spurious 'heroine image' being presented to a gullible public.\n\nBefore leaving London I called on the enterprising publishers who had already written to me. Their brand-new building was many-storeyed and many-corridored; it contained an army of whizz-kids and reminded me of a factory. I was received by three of the army in a large gleaming room with angular metal furniture and not a book in sight. There was much talk of royalties and promotion and a contract lay on an ugly table awaiting my signature. After my recent experiences the word 'promotion' brought me out in goose-flesh. I edged away from the contract and said I couldn't think of signing it until they had seen the typescript. What if they didn't like it? The _three_ closed in, effusive and reassuring. I needn't worry \u2013 I had a fascinating story to tell \u2013 they had excellent editors \u2013 all would be well if I signed just there. 'Excellent editors...' My hackles rose. I did not intend to have my 'fascinating story' told in the words of any editor, however excellent. Besides, I intensely disliked the thought of any book being processed in this aseptic wilderness. Muttering vague half-promises, I withdrew.\n\nMeanwhile another publisher was eagerly pursuing me but, being in no mood for further premature bargaining about my first-born brainchild, I hurried home to go through the final stages of parturition. I had had little time to consider what form my book might take, but the four friends for whose benefit it had been written were unanimous that I must rewrite nothing, transpose nothing, add nothing. I must simply delete, reducing a quarter of a million words to 80,000.\n\nTowards the end of this five-week task another publisher arrived on the scene by post and I felt rather disheartened; clearly the publishing world coveted my book for its supposedly sensational theme. However, when the manuscript had been typed I sent a copy off to each of the interested houses and circulated the third carbon copy among my friends. Then came a letter from Penelope: 'Have you sent your book to Jock Murray? If not, why not' \u2013 or words to that effect. To me the idea still seemed preposterous, but I was in a mood to try any alternative to high-powered modern publishing \u2013 even at the risk of committing sacrilege. I parcelled up my faint, dog-eared fourth copy and sent it off to 50 Albemarle Street.\n\nA few days later a telegram signed 'Murray' asked me to call when I was next in London. Was it possible...? Could it be...? I packed my saddle-bag, leaped on Roz and cycled to Cork to catch the boat.\n\nNext day, quite unmanned by suspense and awe, I turned into Albemarle Street and approached the spot where, fifteen years earlier, during my first visit to London, I had stood on the pavement gazing respectfully at the door I was now so improbably about to enter. But in reality No. 50 was not at all overwhelming. No efficient army appeared to organise me, the place smelt suitably of books, old and new, and Jock Murray's office was cramped, chaotic and almost as dusty as Clairvaux. At once I knew that I had arrived at the predestined end to a much longer journey than my cycle to India.\n\nThe telegram had half-prepared me, yet I was so dazed with joy that I could hardly speak when I heard that Murrays would like to publish my book. There was no talk of royalties or promotion, no ominous reference to 'excellent editors'; the book, apparently, was acceptable as I had written it. Then I was invited to stay with the Murrays; like many another Murray author, I had found not only a publisher but a whole family of friends.\n\nThat afternoon I first met Jane Boulenger, who was to be my editor in an acceptable sense. Authors are notoriously dangerous animals and Jane must be one of the greatest living author-tamers. In eight years of working together she taught me an immense amount about the art and craft of writing and we never had a cross word.\n\nI left Albemarle Street in such a euphoric state that I very nearly stepped under a bus in Piccadilly; a passing Pakistani pulled me back just in time. Yet there was a sense in which my joy was curiously impersonal. Walking through St James's Park, I thought of all my father's rejected novels \u2013 all the years of work, hope, disappointment, work, hope, disappointment \u2013 a cycle of frustration endured without bitterness. And then the importance of the individual \u2013 of myself as an individual \u2013 dwindling to nothing. It was chance that in my lifetime \u2013 perhaps because of my mother's contribution to the genetic pool \u2013 all the strivings of generations of scribbling Murphys were to push their way above ground into print. And so, on that sunny June day by the duck ponds, the acceptance of my first book seemed less a personal triumph than the fulfilment of an obligation to my parents.\n\nMy maternal grandmother ('Jeff ') and grandfather with five of their seven children: _left to right_ : Robert, Laurence, Frederick, Leo and Kathleen (my mother)\n\nMy paternal grandparents in 1898\n\nMy paternal grandfather ('Pappa') as I remember him\n\nMy mother aged sixteen\n\nMy father aged twenty-two\n\nFamily snapshots: summer 1934\n\nMyself in Confirmation dress, May 1943\n\nCharles Kerins in 1944, a year before he was hanged\n\nMy first continental tour, 1952: in the German countryside\n\nand in Strasbourg\n\nSpain, 1956\n\nCounty Wicklow, 1959\n\nTurkey, 1963\n\n# Copyright\n\nFirst published in England by John Murray in 1979   \nFirst published by Eland Publishing Limited   \n61 Exmouth Market, London EC1R 4QL in 2010.   \nThis ebook edition first published in 2011\n\nAll rights reserved.\n\n\u00a9 Dervla Murphy, 1979\n\nThe right of Dervla Murphy to be identified as author of   \nthis work has been asserted in accordance with Section 77   \nof the Copyright, Designs and Patents Act 1988\n\nThis ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly\n\nISBN 978\u20131\u2013906011\u201373\u20134\n\nCover Image: Dervla on her bicycle\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"#  \n## A BATTLE OF BRAINS\n\nBARBARA CARTLAND\n\nwww.barbaracartland.com\n\nCopyright \u00a9 2009 by Cartland Promotions\n\nFirst published on the internet in September 2009\n\nThe characters and situations in this book are entirely imaginary and bear no relation to any real person or actual happening.\n\nThis book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out or otherwise circulated without the publisher's prior consent.\n\nNo part of this publication may be reproduced or transmitted in any form or by any means, electronically or mechanically, including photocopying, recording or any information storage or retrieval, without the prior permission in writing from the publisher.\n\neBook conversion by M-Y Books\n\n# A BATTLE OF BRAINS\n\nSuddenly the door behind her was flung open.\n\nYolanda turned round, terrified.\n\nMr. Watson was standing there with the light from his bedroom behind him.\n\nHe was wearing only a silk nightshirt.\n\nFor a moment he stared at her and then he asked in a thick voice,\n\n\"What do you want? What are you doing here?\"\n\nYolanda moved forward so that he should not see the open case behind her.\n\n\"I just \u2013 came in,\" she stammered in a frightened voice, \"to see that you have \u2013 everything you \u2013 want.\"\n\n\"Everything I want? I thought that was what I had,\" he slurred in a drunken tone. \"Now I see you, I know you are what I want!\"\n\nAs he spoke, he lurched forward towards her.\n\nRealising what he had implied, Yolanda turned to run away.\n\nBut she was too late.\n\n## THE BARBARA CARTLAND PINK COLLECTION\n\nBarbara Cartland was the most prolific bestselling author in the history of the world. She was frequently in the Guinness Book of Records for writing more books in a year than any other living author. In fact her most amazing literary feat was when her publishers asked for more Barbara Cartland romances, she doubled her output from 10 books a year to over 20 books a year, when she was 77.\n\nShe went on writing continuously at this rate for 20 years and wrote her last book at the age of 97, thus completing 400 books between the ages of 77 and 97.\n\nHer publishers finally could not keep up with this phenomenal output, so at her death she left 160 unpublished manuscripts, something again that no other author has ever achieved.\n\nNow the exciting news is that these 160 original unpublished Barbara Cartland books are ready for publication and they will be published by Barbaracartland.com exclusively on the internet, as the web is the best possible way to reach so many Barbara Cartland readers around the world.\n\nThe 160 books will be published monthly and will be numbered in sequence.\n\nThe series is called the Pink Collection as a tribute to Barbara Cartland whose favourite colour was pink and it became very much her trademark over the years.\n\nThe Barbara Cartland Pink Collection is published only on the internet. Log on to www.barbaracartland.com to find out how you can purchase the books monthly as they are published, and take out a subscription that will ensure that all subsequent editions are delivered to you by mail order to your home.\n\nIf you do not have access to a computer you can write for information about the Pink Collection to the following address :\n\nBarbara Cartland.com Ltd.\n\n240 High Road,\n\nHarrow Weald,\n\nHarrow\n\nHA3 7BB\n\nUnited Kingdom.\n\nTelephone & fax: +44 (0)20 8863 2520\n\n##  Titles in this series\n\nThese titles are currently available for download. For more information please see the Where to buy page at the end of this book.\n\n1. The Cross Of Love\n\n2. Love In The Highlands\n\n3. Love Finds The Way\n\n4. The Castle Of Love\n\n5. Love Is Triumphant\n\n6. Stars In The Sky\n\n7. The Ship Of Love\n\n8. A Dangerous Disguise\n\n9. Love Became Theirs\n\n10. Love Drives In\n\n11. Sailing To Love\n\n12. The Star Of Love\n\n13. Music Is The Soul Of Love\n\n14. Love In The East\n\n15. Theirs To Eternity\n\n16. A Paradise On Earth\n\n17. Love Wins In Berlin\n\n18. In Search Of Love\n\n19. Love Rescues Rosanna\n\n20. A Heart In Heaven\n\n21. The House Of Happiness\n\n22. Royalty Defeated By Love\n\n23. The White Witch\n\n24. They Sought Love\n\n25. Love Is The Reason For Living\n\n26. They Found Their Way To Heaven\n\n27. Learning To Love\n\n28. Journey To Happiness\n\n29. A Kiss In The Desert\n\n30. The Heart Of Love\n\n31. The Richness Of Love\n\n32. For Ever And Ever\n\n33. An Unexpected Love\n\n34. Saved By An Angel\n\n35. Touching The Stars\n\n36. Seeking Love\n\n37. Journey To Love\n\n38. The Importance Of Love\n\n39. Love By The Lake\n\n40. A Dream Come True\n\n41. The King Without A Heart\n\n42. The Waters Of Love\n\n43. Danger To The Duke\n\n44. A Perfect Way To Heaven\n\n45. Follow Your Heart\n\n46. In Hiding\n\n47. Rivals For Love\n\n48. A Kiss From The Heart\n\n49. Lovers In London\n\n50. This Way To Heaven\n\n51. A Princess Prays\n\n52. Mine For Ever\n\n53. The Earl's Revenge\n\n54. Love At The Tower\n\n55. Ruled By Love\n\n56. Love Came From Heaven\n\n57. Love And Apollo\n\n58. The Keys Of Love\n\n59. A Castle Of Dreams\n\n60. A Battle of Brains \n\n## THE LATE DAME BARBARA CARTLAND\n\nBarbara Cartland, who sadly died in May 2000 at the grand age of ninety eight, remains one of the world's most famous romantic novelists. With worldwide sales of over one billion, her outstanding 723 books have been translated into thirty six different languages, to be enjoyed by readers of romance globally.\n\nWriting her first book 'Jigsaw' at the age of 21, Barbara became an immediate bestseller. Building upon this initial success, she wrote continuously throughout her life, producing bestsellers for an astonishing 76 years. In addition to Barbara Cartland's legion of fans in the UK and across Europe, her books have always been immensely popular in the USA. In 1976 she achieved the unprecedented feat of having books at numbers 1 & 2 in the prestigious B. Dalton Bookseller bestsellers list.\n\nAlthough she is often referred to as the 'Queen of Romance', Barbara Cartland also wrote several historical biographies, six autobiographies and numerous theatrical plays as well as books on life, love, health and cookery. Becoming one of Britain's most popular media personalities and dressed in her trademark pink, Barbara spoke on radio and television about social and political issues, as well as making many public appearances.\n\nIn 1991 she became a Dame of the Order of the British Empire for her contribution to literature and her work for humanitarian and charitable causes.\n\nKnown for her glamour, style, and vitality Barbara Cartland became a legend in her own lifetime. Best remembered for her wonderful romantic novels and loved by millions of readers worldwide, her books remain treasured for their heroic heroes, plucky heroines and traditional values. But above all, it was Barbara Cartland's overriding belief in the positive power of love to help, heal and improve the quality of life for everyone that made her truly unique.\n\n\"To find true love you need your brains and your body, but most important of all is your soul.\"\n\nBarbara Cartland\n\n# CHAPTER ONE  \n1889\n\nYolanda laid the flowers on her mother's grave and said a prayer as she did so.\n\nShe found it hard to believe that she would never again see her mother she adored.\n\nShe had already cried herself to sleep every night since she learnt of her death.\n\nIt seemed that she had no more tears to shed.\n\nIt had happened so suddenly, so unexpectedly that even now it was difficult to believe.\n\nShe still expected her mother to be waiting for her.\n\nShe would be holding out her arms as she always did when they had been away from each other.\n\nYolanda had been a pupil at a Convent school just outside Paris.\n\nFour days ago the Mother Superior had sent for her and she had gone slowly to her office wondering what she had done wrong.\n\nShe never dreamt it could be a message from home.\n\nThe Mother Superior had broken the dreadful news to her gently, but at the same time it was such a shock that for a moment the news was impossible to believe.\n\nYolanda felt as if the ground had been pulled from under her.\n\nThey had been through so much together since her father died and she had believed for a long time that the only safe and secure rock in her life was her mother.\n\nNow she was dead too and Yolanda knew that she was alone.\n\nIt was only after a very long silence that the Mother Superior said,\n\n\"Your stepfather, Yolanda, has asked you to return home at once, which I feel certain you will wish to do. You will unfortunately be too late to attend your mother's funeral, but he wants you to be with him.\"\n\nYolanda had given a little shiver.\n\nShe never wished to be with her stepfather and she could not understand why he wanted her now.\n\nEqually if he ordered her to return, there was really nothing she could do about it.\n\nShe had had plenty of time to ponder her situation on the long journey home, accompanied by an ancient nun who had been instructed to chaperone her.\n\nFor hours Yolanda was silent, looking back into the past and seeing everything in a very different way.\n\nHer dearest father, the Earl of Longwood, had been the most handsome, charming and delightful man anyone could possibly meet.\n\nWherever they were and whatever was happening, they always seemed to be laughing.\n\nYolanda believed no one could have been happier than she had been with two such wonderful parents.\n\nIt was only as she grew up that she understood.\n\nThe many difficulties, frustrations and moments of despair in her parent's lives became clearer.\n\nHer father, after inheriting the Earldom, was given a dreadful shock \u2013 except for the title that had been in the family for several centuries, he received nothing else from his father.\n\nThe Fifth Earl had spent considerably more than his income and he had sold everything that was saleable.\n\nThe London house had gone first and then all the land in the country that had not been entailed, such as the house and its contents.\n\nIt was heartbreaking for the new Earl.\n\nHis father had left a considerable number of debts too, which more or less equalled his own and, as far as he could see, there was no chance of meeting them.\n\nHe was therefore forced to let out the family house and estate, which brought him in a little money.\n\nHe, his wife and daughter had then moved into the Dower House.\n\nYolanda did not know exactly when her father took up gambling, but she could understand, now she was older, that it was a desperate effort to win back something after all he had lost.\n\nShe suspected that some of the possessions in the house that he had dared not sell went to the pawnbroker.\n\nThey were usually small items that would not be noticed by the Trustees, who visited the house and estate every six months to check that everything was intact.\n\nAt first the Earl was very lucky at his gambling \u2013 he won at racing, cards and, when they travelled abroad, in the casinos as well.\n\nAs she grew up, Yolanda was fascinated when her father came back smiling.\n\nHe would throw up his arms in delight.\n\n\"I have won! I have won!\" he would call out to her mother. \"Now we can enjoy ourselves and, my darling, you must have the most beautiful jewels I can find.\"\n\nHer mother accepted the gifts and Yolanda believed it was because inevitably sooner or later they would have to be sold or pawned.\n\nThe only presents her mother did not lose were the many pretty gowns that her father had bought for her.\n\n\"You will always be very beautiful, my precious,\" he would tell his wife. \"But I like your frame to be worthy of your beauty.\"\n\nBecause her mother had loved him so completely, she always did whatever he wanted, regardless of the consequences.\n\nYolanda remembered affectionately how sweet and kind she was when things went wrong.\n\nThey would know only too well exactly what had happened after an evening's gambling had not gone well.\n\nHer father would enter the room in a slow way and the expression on his face told them before he spoke what had occurred.\n\n\"I have lost,\" he groaned. \"I cannot think why luck had turned against me.\"\n\n\"You are asking too much, my dearest,\" her mother would say. \"You won last month and indeed the month before, but you cannot expect to win every time.\"\n\nSometimes when things were bad they would have to leave the comfortable hotel they had been staying in, which meant moving into very uncomfortable lodgings and eating inferior food.\n\nBut invariably, in some way or another, her father was able to find the money to carry on gambling.\n\nIt was indeed a strange life, Yolanda mused, as the train carried her from Paris to Calais.\n\nShe had visited Baden-Baden in Germany with her parents and found the town incredibly beautiful. And she had spent several weeks in Hamburg.\n\nThey had then travelled back to London after one of her father's more successful spells at the roulette tables and rented a house in Mayfair.\n\nHe had then been in touch with several of his old friends whom he had not seen for many years, and they had been delighted to see him again.\n\nThey made a great fuss of the beautiful Countess of Longwood too, much to the delight of her adoring husband.\n\nNext a governess was engaged for Yolanda and she enjoyed going to Hyde Park to see the horses in Rotten Row.\n\nThe beautiful Society ladies, like her mother, drove in an open Victoria, an elegant French carriage with a raised driver's seat drawn by two magnificent horses. Accompanied by a liveried coachman, the ladies rode through the parks of London with small sunshades held up over their heads.\n\nThen she recalled how very suddenly her father had decided to go to Paris, as he wanted to attend the racing that was about to take place on the new Racecourse in the Bois de Boulogne.\n\n\"I know which horse is going to win the big race,\" he boasted, \"and I intend to make a packet on him!\"\n\nHer mother agreed with him as she always did.\n\nSo they set off for Paris and stayed at a large and impressive hotel.\n\nAs a special treat Yolanda was allowed to go to the Racecourse and she had known without being told that the racehorses she saw were the finest in Europe.\n\nThe horse which her father had come to see run was outstanding and she was well aware that a great number of people were putting their money on him to win.\n\nThe race was thrilling and the horses moved at an almost impossible speed, sweeping like typhoons past the winning post.\n\nBut the horse her father had backed was not in the first three.\n\nThe result was such a surprise to a large number of race goers that a groan echoed around the enclosure.\n\nIt was then that her father had exclaimed angrily,\n\n\"I don't believe it! The whole race was a cheat and there should be a full enquiry as to why the favourite was pulled in at the last moment!\"\n\nUnfortunately, the horse's owner, who took umbrage at the suggestion of cheating, overheard her father's loud comments.\n\nYolanda was not quite certain what happened between them, but upon returning to the hotel that evening, she was horrified to discover that her father was expected to fight a duel at dawn the next morning.\n\nHer mother was distraught at the prospect.\n\n\"You cannot do it, darling,\" she wept over and over again. \"You have not fought a duel for many years and these Frenchmen are experts at it.\"\n\n\"It is something I cannot refuse,\" replied the Earl, \"and, damn him, he has lost us every penny we possess!\"\n\nYolanda remembered that her mother had not been in the slightest concerned about the losses \u2013 she was only worried about her father fighting in a duel.\n\nShe was still pleading with him desperately to refuse to take part when they retired to bed.\n\n\"You know as well as I do,\" insisted the Earl, \"that I have to behave like a gentleman. It would besmirch my honour to refuse!\"\n\nThere was nothing further her mother could say to dissuade him, but Yolanda could see the tears in her eyes.\n\nThey drove off together before dawn to the Bois de Boulogne where duels always took place.\n\nHer father was not killed \u2013 but badly wounded.\n\nAlthough he was taken back to the hotel where they were staying and the doctors sent for at once, his wound became septic.\n\nFinally after three days in agony, he died.\n\nTo his adoring wife it was the end of the world and she could only weep helplessly.\n\nThey had been very happy together and the endless difficulties over money had not seemed important as long as they both shared success and failure.\n\nHer father had been buried in a graveyard in Paris, as it was too expensive to take his body back to England.\n\nYolanda had then asked her mother what they were going to do and the Countess made a helpless gesture with her hands.\n\n\"I have no idea, my darling,\" she answered. \"We have no money and we owe a great deal to the hotel apart from anything else.\"\n\n\"Oh, Mama!\" exclaimed Yolanda. \"Perhaps they will prosecute us if we cannot pay! We might even have to go to prison!\"\n\nHer mother had not responded, only cried more tears of despair, whilst Yolanda shivered because she was so afraid.\n\nIt was then, and it seemed at the time almost like a miracle, that Mr. Garrack had come into their lives.\n\nHe was staying in the same hotel and Yolanda had noticed him when they were having dinner the night before the race.\n\nHe had been sitting at a table near to theirs and she had been drawn to him, thinking he was a rather unattractive man compared to her handsome father.\n\nBut he kept staring at her mother with such a look of admiration in his eyes that was, in a way, very flattering.\n\nAs dinner ended her father was debating whether he would go to a gambling house that was not too far away.\n\nSuddenly the stranger rose from his table, bowed to her mother and father and introduced himself,\n\n\"My name is Oliver Garrack, and you must forgive me if it seems impertinent of me to speak to you.\"\n\nHer father, who was always charming to everyone he met, smiled at him and commented,\n\n\"We are staying in the same hotel, are we not?\"\n\n\"I was just counting my lucky stars,\" Mr. Garrack said, \"because it has allowed me to see the most beautiful woman I have ever set eyes on!\"\n\nHe was looking at her mother as he was speaking, and Yolanda saw that she turned her head away as if a little shy.\n\n\"There,\" her father remarked, \"I must agree with you. I only wish I could afford to have my wife's portrait painted by a great artist.\"\n\n\"Strangely enough,\" Mr. Garrack then replied, \"that is just what I was going to ask you. You may think it very forward, but I am a great lover of beauty. I have in my house in England a collection by some very fine Masters.\"\n\nHe paused to look at the Countess again, before he went on,\n\n\"But none of my pictures, and many of them are of lovely women, are as beautiful as you, my Lady.\"\n\n\"That is very kind and most flattering of you,\" the Countess responded, obviously feeling a little embarrassed.\n\n\"What I am wondering,\" he continued, \"is whether you will allow me to have your portrait painted by one of our greatest artists, who is in Paris at this very moment?\"\n\nYolanda recalled how her mother had looked across the table at her father.\n\n\"We are, of course, very grateful, Mr. Garrack,\" the Earl said. \"But I am afraid that we cannot accept your kind offer, because, as soon as the racing is over tomorrow, we intend going back to England.\"\n\nMr Garrack was silent for a moment and then said,\n\n\"I too shall be returning home. I can only ask if you would permit me to visit you in London, or the country, to discuss this matter again? To me it is a matter of great importance and it would be a great honour to add a portrait of your wife to my collection.\"\n\nHe smiled before he added,\n\n\"I am certain that future generations of art lovers would be eternally grateful to you too.\"\n\nThe Earl looked at his wife.\n\nAs they were so very close to each other, Yolanda was always sure that they could read each other's thoughts.\n\nHer father clearly sensed without being told that his wife did not want her portrait painted \u2013 in fact she had no wish to see Mr. Garrack again.\n\nThe Earl was, however, very tactful,\n\n\"It is exceedingly generous of you and of course we shall consider your offer seriously. But, for the moment, my wife is tremendously busy, so it would be difficult for her to be in London for long or to give up so much time to an artist.\"\n\n\"I do understand. Yes, of course I understand,\" Mr. Garrack had answered. \"But I am just praying that one day my dream will come true and your beautiful wife will shine in my collection which I hope to leave, when I die, to the National Gallery.\"\n\nThe Earl was obviously impressed.\n\nAnd as Mr. Garrack bowed himself away, Yolanda remembered her mother saying,\n\n\"I don't want to be in his collection, darling, only in yours.\"\n\nThe Earl had laughed.\n\n\"Which is so small that at the moment it does not exist except in my mind!\"\n\nYolanda had, she recalled, somehow felt glad that they need not be troubled further with Mr. Garrack.\n\nYet it was he who came to their rescue in their darkest hour.\n\nThe day after her father was buried, her mother was presented with a very large bill by the hotel.\n\nShe and Yolanda were both in the sitting room of their suite when she received the bill.\n\nWhen the servant had left them the Countess said in a small, frightened voice,\n\n\"Whatever am I to do now, Yolanda? I have only my engagement ring to sell, but I doubt if that will give us enough money to pay this enormous bill.\"\n\nYolanda was well aware that her father had been so certain the horse he fancied was going to win.\n\nHe had pawned most of her mother's best jewellery and then sold some of her other pieces to finance even more bets.\n\nLooking at the bill searchingly as if she thought that somehow there must be a mistake, the Countess asked again,\n\n\"What am I to do? Oh, darling, what can I do?\"\n\nAs she spoke there was a knock at the door.\n\nBecause her mother was crying, Yolanda jumped up to answer it, as she would not want the servants or anyone else to see her tears.\n\nOutside was Mr. Garrack.\n\nHe was well dressed and sporting an orchid in his buttonhole.\n\nYolanda felt that they had no wish to talk to him at this traumatic moment.\n\n\"I would like to see your mother \u2013 \" he began.\n\n\"I am afraid that Mama is not receiving \u2013 \"\n\nThen to her great surprise and before she had even finished speaking, he pushed past her into the sitting room.\n\nHe crossed over the room to where her mother was sitting with a handkerchief held to her eyes.\n\nHe went down on one knee beside her.\n\n\"You must not weep, my beautiful lady,\" Yolanda heard him say. \"I cannot bear to see you so unhappy.\"\n\nYolanda thought that it would be embarrassing for her mother if she listened to what Mr. Garrack was saying to her, so she went into the bedroom, but did not close the door.\n\nShe could hear her mother, weak and tearful.\n\nAnd next came Mr. Garrack's voice, sympathetic and flattering.\n\nShe could not hear exactly what was said, only the tone of their voices.\n\nHer mother's was low and quieter whilst Mr. Garrack's was more determined, as if he was fighting to get his own way.\n\nYolanda could not think what the conversation was all about, unless he was once again pressing his longing for a portrait of her mother.\n\nIf he was, she considered it rather bad taste at this particular moment.\n\nAnyway it would be impossible for them to stay in Paris without any money and equally impossible, she now feared, to leave.\n\nShe sat wondering anxiously what they could do, wracking her brain for a solution to their hopeless situation.\n\nFor the first time in her life she began to query why they had so few relations. She had thought about it before, but had not actually asked any questions.\n\nShe knew that her mother's family, who were very distinguished, lived in the North of England. It was a long way from London, she mused, so maybe that was the reason she had not met any of them.\n\nHer father's family, if he had any, must live somewhere in the rural County of Hertfordshire where the family home was situated, she concluded.\n\n'I must ask Mama,' she had said to herself, 'if there is anyone we can turn to now that things are so desperate.'\n\nIt seemed to Yolanda as if hours passed.\n\nAt last she could hear Mr. Garrack saying goodbye to her mother and walking across the room to the door.\n\n\"I will order dinner for eight o'clock,\" he said as he reached it. \"And I will be there waiting, my beautiful lady, eagerly and excitedly for you and your daughter.\"\n\nThen he was gone.\n\nYolanda hurried out into the sitting room to find her mother was no longer crying.\n\nAs Yolanda reached her, she held out her hands.\n\n\"We are saved, darling,\" she cried. \"Saved when I was frightened that we might have been taken to prison.\"\n\n\"What has happened, Mama?\"\n\nShe knelt down beside her mother's chair.\n\n\"It may be wrong of me,\" she answered, \"and some people would think it reprehensible, but I had no choice but to accept Mr. Garrack's kind offer.\"\n\n\"What was that, Mama?\"\n\n\"Because he wants my portrait to be painted for his collection, he has offered to pay the bill here and also to pay for us to go back to England.\"\n\nShe paused for a moment and then continued softly,\n\n\"He said there is a painter who will do me justice. He has just painted the Princess of Wales, and a great number of fashionable beauties including Mrs. Lillie Langtry.\"\n\n\"I suppose,\" Yolanda remarked a little doubtfully, \"it is very kind of him.\"\n\n\"Yes, of course it is,\" her mother replied somewhat sharply, \"and I certainly cannot imagine anyone else being so incredibly generous or understanding after such a short acquaintance.\"\n\nThere was silence for a moment before she added,\n\n\"He told me that if he had known earlier that Papa was being buried here in Paris, he would have arranged for his body to be taken back home to England and he could have been buried in the tomb of his ancestors.\"\n\nYolanda then remembered that they had not seen Mr. Garrack in the dining room since the night he had first spoken to her mother.\n\nShe supposed that he must have been out of Paris and had not been aware of the duel or of her father's death.\n\nAs if she had spoken out aloud, her mother said,\n\n\"Mr. Garrack only returned last night. When they told him what had happened to your Papa, he was not only shocked but as he said his 'heart bled' for me.\"\n\nYolanda thought this was a somewhat exaggerated way of talking, but she supposed it was indeed very kind of Mr. Garrack to befriend them.\n\n\"We are returning to London tomorrow, Yolanda, and I never want to see Paris or meet a Frenchman ever again!\"\n\nShe spoke bitterly and tears were back in her eyes.\n\n\"Don't cry, Mama,\" begged Yolanda. \"You know Papa hated you to be unhappy. I am sure wherever he is, he is still looking after us and somehow everything will be all right.\"\n\nShe was trying as hard as she could to cheer up her mother, who now replied,\n\n\"It does seem rather like a miracle that Mr. Garrack, whom we have not known before, should appear just at this moment and be so very very generous.\n\n\"I told him that I had pawned all my jewellery and the only piece left was my engagement ring. I offered to give it to him, but he would not take it.\"\n\nYolanda was glad, as she knew that it would have completely broken her mother's heart to have parted with the engagement ring that had meant so much to her.\n\nHer parent's marriage had been a love match. They had been engaged for just a short time and her mother had often said that she and her father had fallen in love with each other at first sight.\n\n\"I knew, darling, the moment I saw him,\" she said, \"that he was the most handsome, exciting and wonderful man I had ever met.\"\n\n\"I am not surprised, Mama.\"\n\nShe had noticed that when her father came into a room, the women all looked at him and she could easily understand why her mother had fallen in love so quickly.\n\n\"I have heard Papa say,\" she exclaimed, \"that he looked at you and lost his heart immediately.\"\n\n\"We were so very happy,\" her mother said and her voice broke on the last word.\n\n\"I am sure it is Papa who sent Mr. Garrack to us at exactly the right moment. When we get back to England, I am sure we will find things are not as bad as we fear.\"\n\nAs it happened it would have been a lot worse, except that Mr. Garrack took them to his large and imposing house in Park Lane.\n\nHe insisted on finding out if there was any money left in her father's bank account and if they were entitled to sell anything from the family house and estate.\n\nMr. Garrack reassured the heartbroken Countess that he was sure something had been overlooked and that they had not been left totally penniless.\n\nIt was only after all the finances had been scrutinised that he conceded that he had been mistaken \u2013 there really was nothing left.\n\nWhat was more, the income from the people renting the Earl's house and estate had already been given to the bank against his large overdraft.\n\nIt was then, looking back, Yolanda realised that Mr. Garrack had taken complete control of their lives.\n\nHe managed to persuade her mother that she should for the time being stay with Yolanda at his house in Park Lane.\n\n\"I could, of course,\" he said, \"quite easily buy you a house anywhere you wanted, but I want to be with you, to see you and to talk to you.\"\n\nHe smiled before he added,\n\n\"It seems rather a waste of money when this large house is full of empty rooms and I am a very lonely man!\"\n\nThere was really nothing the Countess could do but accept his kind offer.\n\nIt would have been untrue to say that both she and Yolanda did not enjoy the comfort and luxury with which they were now surrounded.\n\nBecause they were in mourning, Mr. Garrack at first did not try to persuade them to meet other people.\n\nBut after quite a short time, he gradually began to bring his friends home.\n\nIt was then that Yolanda began to notice that in many ways he was not exactly a gentleman.\n\nShe did not say it, even quietly to herself, in those particular words \u2013 it was just what she observed when she compared him to her father.\n\nWhat was more, she considered it embarrassing that Mr. Garrack should be so obviously pleased and delighted to have her mother as a guest.\n\nNot only because of her beauty but also because of her title.\n\nThere was something in the way he would say to anyone who came to see them.\n\n\"Let me introduce you to the delightful Countess of Longwood, who is staying with me, who is not only the most beautiful woman in the world but also a very valued friend.\"\n\nHe always accentuated her title.\n\n\"And also her daughter, Lady Yolanda Wood,\" she heard him say dozens of times.\n\nAlways he said her name with what she thought was a pronounced accent on the word 'Lady'.\n\n*\n\nAfter they had been in London for only a month, the Season had come to an end and the majority of Socialites left London for the country or Scotland.\n\nThey had been talking to two men who were going grouse shooting and Yolanda asked her mother,\n\n\"Do you not think, Mama, it would be a good idea to get in touch with some of your relations? I know they live in the North, but I have not met any of them and it would be very exciting if we went for a visit.\"\n\nThe Countess was silent for a moment.\n\nYolanda knew she was considering what she should answer and it made her wonder why she had not been told anything about them before.\n\nFinally her mother said,\n\n\"I suppose, darling, you had better know the truth. When I married your father all my family were very angry. They had arranged for me to marry a distinguished man, rather older than I was. When I fell in love with your father, they tried in every way they could to dissuade me from marrying him.\"\n\n\"But you insisted, Mama, and I am very glad that you did!\"\n\n\"Of course I insisted! But after we were married your father found them so very disagreeable and unfriendly that we came South and never went back.\"\n\nYolanda gave a little cry.\n\n\"But, Mama, surely you miss your family.\"\n\nThe Countess sighed.\n\n\"To be honest I was so very happy with your father, I hardly ever gave them a thought. My father was never friendly like Papa was with you and my mother was, I think, not particularly fond of children \u2013 she had four and found them all rather a bore. As I was the youngest I received the least attention!\"\n\n\"It is difficult for me to think of your family being like that, Mama. You have always been so wonderful to me and so was Papa. You never made me feel you did not want me. Although perhaps sometimes I was a nuisance.\"\n\nThe Countess laughed.\n\n\"You were never that, darling. We were thrilled to have you but, of course, I ought to have given your Papa a son to follow after him.\"\n\nShe paused and then she added,\n\n\"I only realised yesterday that now your father is no longer with us, the Earldom comes to an end. He had two cousins who might have inherited the title, but they were killed fighting with General Gordon at Khartoum.\"\n\n\"I call it very sad, Mama, but what happens now to the house and all the things which should have come to the next Earl if there had been one?\"\n\n\"Mr. Garrack is looking into that at the moment,\" her mother replied. \"Of course even if we were allowed to go on living there as there is no heir, we could not afford to do so.\"\n\n\"Have we really no money, Mama, except what Mr. Garrack is giving us?\" Yolanda asked her in a low voice.\n\nHer mother shook her head.\n\n\"None, my darling, and I feel it is terrible for us to inflict ourselves on him like this. At the same time he so adores having us here and keeps saying what a difference we have made to his life. I feel it is no use suggesting we go somewhere else.\"\n\nAlthough it seemed wrong and ungrateful, Yolanda just did not like Mr. Garrack.\n\nThere was something about him that made her feel uncomfortable.\n\nWhen he told her that he now looked on her as if she was his daughter, she wanted to respond that she would never be that in a million years.\n\nInstead she just managed to say,\n\n\"That is so very kind of you.\"\n\nShe thought from the expression in his eyes that he knew quite well it was something she did not want.\n\n*\n\nIt was a short while after this that he persuaded the Countess that Yolanda should be better educated and began making enquiries as to the best school for young ladies in London.\n\nHe discovered that the best school for the daughters of the aristocracy was in fact the Convent of Our Lady just outside Paris.\n\nA great many aristocrats sent their daughters there and the pupils included the daughters of three Dukes.\n\nThe Mother Superior only accepted the girls from France and other European countries if they came from the grandest and most respected Society families.\n\nIn vain did Yolanda protest that she did not want to leave her beloved mother or England and, almost before she realised what was happening, she found herself sent off to France.\n\nA middle-aged woman who was one of Mr. Garrack's secretaries and a courier accompanied her on the journey.\n\nWhen she reached the Convent, she found it rather sparse, but the educational facilities were superb.\n\nThe girls were not only taught by nuns but by tutors who came from every part of Europe.\n\nIt was easy to learn French and German, as well as Spanish, Italian and Greek.\n\nThe music teachers were the best in Europe and this pursuit of excellence applied to every other subject.\n\nBecause Yolanda was very intelligent, she enjoyed her lessons enormously and her reports would have pleased any parent.\n\nHer mother was overjoyed by her progress.\n\n\"I am so thankful, my darling,\" she wrote, \"that you have settled in and are enjoying your lessons. It is what your father would have wanted, though I doubt very much if we would have been able to afford it.\n\n\"I am just exceedingly grateful to Mr. Garrack for making this possible for you. He sends his love and says the prizes he will give you at the end of term will be well worth waiting for.\"\n\nYolanda told herself it was very good of him, but at the same time for some unknown reason she had no wish to accept anything more from him.\n\nShe resented being in such debt to him, although her mother always spoke so enthusiastically about him.\n\nTherefore it was not really such a shock when she was told that when Mr. Garrack had asked her mother to be his wife and that she had accepted.\n\nShe wrote to Yolanda,\n\n\"I will never love anyone as I loved your dearest father, but, darling, I have to think of you and the future.\n\nYou know as well as I do that we have no money, no house and nowhere to go.\n\nMr. Garrack has been so kind that I feel it would be most ungrateful if I did not try to make him as happy as I can in return for all he has given us.\n\nHe says that he will treat you as if you were his own daughter and you shall have the best of everything in life.\"\n\nThe Countess went on for a long time praising Mr. Garrack's kindness, explaining over and over again how the only way she could show her gratitude was to do as he wanted.\n\nAt the end of her letter she finished,\n\n\"As you and I are both so aware, darling, there will never be anyone like Papa, but I know he would want you to be well educated and for us both to be secure.\n\nYour loving Mama.\"\n\nIt was something that Yolanda could not contradict yet she still wanted to.\n\n*\n\nThe next time Yolanda saw her mother she was saddened to see that she looked a little older.\n\nShe was clearly not as happy as she had been when married to the Earl, but Yolanda had to admit she wanted for nothing.\n\nShe had only to murmur her interest in some object and no matter how expensive it was Mr. Garrack bought it for her.\n\nHe was also very generous to Yolanda.\n\nHe bought her new clothes, a piano because she had learnt to play rather well and what she wanted more than anything else, a horse.\n\nShe rode it first in Hyde Park.\n\nThen she learnt that Mr. Garrack had just purchased a house in the country.\n\n\"He has not had one before,\" her mother explained, \"because he has always been so busy.\"\n\n\"What does he do, Mama?\" Yolanda had asked. \"I have often wondered why he has so much money.\"\n\n\"He has worked for it. He came from quite a poor family, but he has worked himself up to being one of the richest men in the country.\"\n\n\"Is that really true?\" questioned Yolanda.\n\n\"It is what he has told me, darling, and the way he spends money on me makes me sure he is many times a millionaire.\"\n\nNo one could have more jewellery than her mother now possessed.\n\nShe had several pearl necklaces, diamonds, sets of sapphires and aquamarines.\n\nYolanda stared at them all in disbelief when they emerged from the safe in her mother's bedroom.\n\nBut no matter how much the precious trinkets glittered she still harboured a strange feeling about Mr. Garrack that she could not explain.\n\nShe just could not shake the feeling that he was not really what he appeared to be.\n\n'How can I be so stupid?' she asked herself. 'Look how kind he was at Christmas! And the presents he gave me \u2013 a fur coat, a pearl necklace and more horses to ride.'\n\nIt seemed wrong and she felt ashamed of herself that she did not like him.\n\nHowever, as her mother seemed happy, that was all that mattered.\n\nBy now her portrait had been painted, not once but three times by different artists, as Mr. Garrack still claimed that none of them had done her beauty justice.\n\nHis new house in the country was even bigger than the one Yolanda's father had inherited, but it did not have any atmosphere of antiquity.\n\nIt was large, rich and cold and Yolanda thought that it would be difficult to make it come to life.\n\nFrom the moment she entered it she had wanted to feel happy there and part of a family.\n\n'Maybe it will improve as the years go by,' she told herself.\n\nBut somehow she had no wish to stay there so long.\n\nNow, without the warmth of her mother, she felt that the house was almost repulsive.\n\n*\n\nHer stepfather had welcomed her back from France in a friendly manner that made her feel ashamed of criticising him.\n\nAnd she still felt guilty for not feeling inwardly as grateful as she pretended to be.\n\n\"I sent for you,\" he said when she arrived, \"because I knew you were leaving at the end of term. I felt, as you could not come to your mother's funeral, you would like to see her grave as soon as it was possible for you to do so.\"\n\n\"That is very kind of you,\" replied Yolanda warily.\n\nShe was aware her stepfather was inspecting her.\n\nHe had not seen her for nearly six months and now she was over eighteen.\n\nShe was sensible enough to realise she had altered in quite a number of ways.\n\nFinally he commented,\n\n\"You are beautiful. Not as beautiful as your mother was, but quite beautiful enough to turn the head of any young man who looks at you.\"\n\nShe did not reply, feeling very uncomfortable to be spoken to in such a fashion.\n\nIt was as if he was expressing her value in cash and it was somehow degrading.\n\n\"What I hope to do now that I have come home,\" she said eventually, \"is to find something sensible to do.\"\n\nShe was thinking as she spoke that she might work in some capacity with a local charity, or perhaps she could be interested in the running of an orphanage \u2013 one of the girls at school had told her she had done so in her country and it had been a great success.\n\nTo her surprise her stepfather laughed.\n\n\"There is no reason for you to look around for work, Yolanda, I have something for you that I will tell you about later. I am sure you will find it very interesting.\"\n\nYolanda wondered what it could be.\n\nShe had the feeling that if it was something he had chosen, it would not be what she really wanted.\n\n*\n\nNow standing quietly at her mother's graveside she was praying desperately that her mother would help her.\n\n'What am I to do with myself now that you are not here?' she asked. 'I cannot just stay here in this house in the country or in the one in London. It was different when you were there and we could talk together. Even without Papa we were happy.'\n\nShe gave a deep sigh before continuing to pray,\n\n'But, dearest Mama, this is going to be something very different and I am frightened. Frightened of the future without you and without anything that I can use my brains and energy on.'\n\nShe looked up at the sky.\n\n'Please help me, Mama. I know you understand. I need your help desperately and I need it quickly.'\n\nThen because there seemed no answer, she felt the tears come into her eyes and she wiped them away almost roughly.\n\nShe knew it would upset her mother if she cried, but she wept until it was impossible to do so any more.\n\nHowever, she was intelligent enough to realise that she now had to make more practical plans for the future.\n\nWhat it reduced down to in a few words was that she did not want to stay with her stepfather.\n\nBut where could she go?\n\nWhat could she do?\n\nThere was no answer to either of these questions.\n\n# CHAPTER TWO\n\nWhen she returned to the house, Yolanda went into the sitting room, where she and her mother had always sat.\n\nIt all felt so familiar that as she walked in through the door she almost believed her mother would be waiting for her and look up with her usual smile.\n\nInstead there was just emptiness and silence.\n\nShe moved over to the large writing desk, the one her mother had always used, opened a drawer and found all the letters she had written to her from France.\n\nShe remembered how proudly she had written of all the exciting things she had done and learnt, knowing that her mother would be interested and pleased for her.\n\nShe felt that in some esoteric way they were both sharing the very extensive education she was receiving.\n\nSighing, Yolanda closed the drawer of the writing desk and walked over to the window.\n\nThe garden outside was a mass of flowers with the lawns emerald green and perfectly kept.\n\nIt was just another example of what money could do and she could understand why her stepfather demanded the best and insisted on having it.\n\n'He is so rich,' she ruminated, 'so rich! If Papa had made only a little more money, he could have restored our home and made it look like this establishment.'\n\nShe felt sure that her mother had often thought the same thing. Watching day after day as Mr. Garrack poured out endless sums of money, she must have thought of the contrast between her new life and the life she had shared with her beloved husband, when they had often been desperate for just a few pounds.\n\nThen very sensibly Yolanda told herself that it was no use trying to live in the past \u2013 what she had to do was to live in the present and make the best of it.\n\n'I do know, Mama,' she sighed beneath her breath, 'that is just what you would want me to do.'\n\nThe door of the sitting room opened and one of the footmen entered,\n\n\"The Master asks, my Lady, if you would join him in the study.\"\n\n\"Yes, of course I will.\"\n\nYolanda wondered again as she walked towards her stepfather's study what he was going to talk to her about.\n\nHe had said he had something for her to do and she hoped it would be something really interesting as well as something she was capable of doing.\n\nAt the same time it did seem strange that he was thinking of her working, the moment she returned home from the Convent.\n\nThe study was impressive and lavishly furnished.\n\nThere were some valuable pictures on the walls and the ornaments on the mantelpiece and on the tables were all museum pieces.\n\nWhen Yolanda had first seen Mr. Garrack's house, she had been extremely impressed and surprised that he had such good taste.\n\nShe learnt that it had actually been furnished by two of the most experienced London decorators and all that Mr. Garrack had to do was pay the bills.\n\nHe was sitting now at a very beautiful George IV Regency mahogany writing desk.\n\nYolanda saw that there was a new picture over the mantelpiece \u2013 it was a portrait of her mother and she knew it had been painted by one of the great artists of the day.\n\nShe closed the door behind her and as she walked towards her stepfather, she thought he was scrutinising her in a rather strange manner.\n\n\"You wanted to see me, Step-papa.\"\n\n\"Please sit down,\" he asked, indicating a chair.\n\nYolanda did as she was told, but now she had her back to her mother's portrait.\n\nYet she tried to feel as if her mother was present and helping her.\n\nNo doubt she would be telling her, as she had done so often, that they must be very grateful for all the kindness they had received from Mr. Garrack.\n\n\"You were saying just now,\" her stepfather began, \"that you would like to have something to do. Well, I have a very important job for you and I feel that you are bright enough not to fail me.\"\n\nYolanda was somewhat surprised at the way he was speaking, but she replied as though prompted by her mother.\n\n\"I hope I can do anything you require, Step-papa, but first I must thank you for the excellent education I received at the Convent. I am sure that, if you saw my reports, you would be impressed by the number of different classes I attended.\"\n\n\"I am delighted you can speak so many languages,\" replied Mr. Garrack. \"Also I did notice that your reports were so good. You certainly worked hard.\"\n\n\"I did my best and I can only say thank you again for sending me to such a good school.\"\n\n\"What I would like you to do now,\" he said, \"will require intelligence more than knowledge.\"\n\nYolanda waited wide-eyed and he continued,\n\n\"Tonight we will have a guest whose name is Jack Harpole. He is, as indeed I am, interested in the many industrial developments that have been taking place lately not only in this country but all over the world.\"\n\n\"I think Step-papa, you are talking about what they are calling the 'Industrial Revolution'.\"\n\nHer stepfather raised his eyebrows.\n\n\"Do you know about it? I would have thought that was hardly the sort of lesson that would be given to such a pretty young lady!\"\n\n\"I do think the French girls, more than other pupils, appreciate the large increase in trade that is currently taking place in every country.\"\n\nYolanda paused for a moment before she added,\n\n\"The nuns therefore taught us all about the Science of Industry and mentioned the new industries that are being founded in various different countries.\"\n\n\"I am surprised, Yolanda, and equally very pleased. You will then understand what I am asking you to do.\"\n\n\"What is that?\"\n\n\"You will have doubtless been taught that there has been a great advance in the hulls of steamers.\"\n\n\"You mean that they are now steel-hulled?\"\n\n\"Yes, indeed. The first steel vessel to cross the Atlantic was built in Liverpool in 1867. Now steel hulls are becoming quite commonplace both in England and America. In fact, the Royal Navy has built two fast despatch vessels.\"\n\n\"I did not think, Step-papa, that you would be so interested in ships.\"\n\nMr. Garrack laughed.\n\n\"I am interested in anything what makes money! Undoubtedly at the moment every new ship being built will have a steel hull and firms all over this country are struggling to improve their steamers and all other types of ship.\n\n\"Fortunately, we are well in advance of other countries that will have to buy from us until they have learnt to make their own.\"\n\nYolanda thought that this was all most interesting, but wondered how it concerned her.\n\nThere was another pause before he went on,\n\n\"My friend, who is coming here this evening, is very anxious to buy one particular company that has built quite a number of ships that are nearly finished. What I want to know is what he is prepared to pay for them.\"\n\n\"Have you asked him?\" enquired Yolanda.\n\nHer stepfather smiled.\n\n\"I don't think you understand. I am also interested in these ships and I am anxious that they should become my property, not my friend's.\"\n\nYolanda stared closely at him, because she still did not fully comprehend what he was trying to say.\n\nHowever, Mr. Garrack continued,\n\n\"Mr. Harpole does not know that I am also ready to make a deal. He is staying with me because we are old friends and he also wants my advice on several other firms that have different products to sell.\"\n\n\"But won't he be upset if he finds out you are bidding against him?\"\n\n\"I think he will be very upset if I am successful! But I see no need for him to know that I have succeeded where he has failed before he returns to Canada, which is where he is working at the moment.\"\n\nYolanda gave a little laugh.\n\n\"It all sounds rather complicated! I only hope, Step-papa, that you are successful and get the ships you want.\"\n\n\"Well, that, of course, depends on you.\"\n\n\"On me! How can it depend on me?\"\n\n\"That is exactly what I am about to tell you.\"\n\nYolanda looked at him in astonishment.\n\n\"When Mr. Harpole dines with us tonight, I shall give him the very best wine from my cellar. Champagne, claret, port and, of course, excellent liqueurs to follow a magnificent meal.\"\n\nYolanda thought that sounded like enough alcohol for any man.\n\n\"When you leave the table as your mother always did after the coffee, I want you to go upstairs and go to the room where Mr. Harpole is sleeping. He is not bringing a valet and the servants will all be downstairs by then.\"\n\n\"But what do you want me to do?\" Yolanda asked him in a bewildered voice.\n\n\"Mr. Harpole will have with him a despatch case. It will be left upstairs in his room and, of course, it will be locked. I want you to open it \u2013 \"\n\n\"But how can I do that?\" Yolanda interrupted.\n\n\"I know the make and I have an alternative key that I will give you. Inside the case you will see a lot of notes and letters. Because you are very intelligent I want you to quickly look and find those which come from the Doxford Shipyard.\"\n\nHe repeated the firm's name and continued,\n\n\"Make a note of what is in the letters as fast as you can, particularly if there are any in his handwriting. I require copies of those too. After you have done that, go straight to bed as I rather suspect that Mr. Harpole will not be in a condition for you to find his company very genial if you get caught prying!\"\n\nAs he finished speaking, Yolanda gave a gasp.\n\n\"What you are asking me to do, Step-papa, is to spy on your friend! It is something that no lady or gentleman for that matter, would ever do.\"\n\n\"What I am requesting,\" he replied sharply, \"is business. It is the type of business that takes place all over the world every day. As I wish to make a larger bid for the ships than Harpole, it will be convenient for me to know what he has already offered. Also if they have refused or accepted his offer.\"\n\n\"Then you will offer more for them?\"\n\n\"Of course I will.\"\n\n\"But you must see that what you are suggesting is cheating, Step-papa, and in a most underhand way too, as this Mr. Harpole is supposedly your friend.\"\n\n\"There is no need to preach to me,\" he snapped. \"There is nothing wrong with what I am telling you to do, it is just standard business practice. I had no idea you would make such a silly fuss!\"\n\nHe took a deep breath and then declared firmly,\n\n\"Mr. Harpole is not working for himself, but for an American Company.\"\n\nHe made a sound, which was almost a laugh.\n\n\"Harpole will receive a good rake-off from the deal \u2013 of course, he will. And if I trusted him, which I don't, I might even suggest that we join hands on this particular purchase, but I have no wish to be too closely connected with him.\"\n\n\"I don't think that his sharp practice makes what you ask any better.\"\n\nMr. Garrack sat back in his chair.\n\n\"Does it really matter what you do or don't think? I have requested that you undertake a straightforward job for me. I feel certain that you can carry it out without making a mistake or missing the items you are looking for.\"\n\nYolanda sat upright and clenched her fingers.\n\n\"I cannot do it, Step-papa!\"\n\n\"Why ever not?\" he spat back angrily.\n\n\"Because, as I said, it is cheating. I feel sure that if you asked Mr. Harpole in a straight manner exactly what he was offering for these ships, he would tell you.\"\n\nHe laughed and it was not a pretty sound.\n\n\"My dear child \u2013 how little you know of the world and how extremely stupid you are! Of course Mr. Harpole would give me a figure, but if you think it would be the true one, then you must be even more foolish than I thought.\"\n\n\"You mean that he would cheat you?\"\n\n\"Of course he would cheat me. We are both men of business and we know only too well the difficult world we are working in.\"\n\nYolanda did not speak and he went on,\n\n\"Then, having asked him point blank what his price is, you can be sure that he would do everything in his power to damage me. Thus I would find it difficult to approach the firm at all because he would have already poisoned their minds against me.\"\n\nYolanda gave a little shiver.\n\n\"It all sounds unpleasant, underhand and illegal.\"\n\n\"Officially there is nothing illegal about this,\" her stepfather retorted, \"and what you have to do is very simple and will not be known to anyone.\"\n\nHe dropped his voice.\n\n\"Neither my secretary nor my attorney, who will be travelling with me, will know when I go to see the ship owners.\"\n\n\"I am very sorry, Stepfather,\" sighed Yolanda. \"I know that Mama would think it wrong.\"\n\n\"Your mother, whom I adored, knew nothing about business. And, as you well know, she would have done anything to help and support me if I had asked her to do so.\"\n\nThis Yolanda could not deny, so she replied,\n\n\"It is something my father would never do because he would consider it as bad as cheating at cards.\"\n\n\"If your father had cheated a little, he would have been a far richer man when he died!\"\n\nAs Yolanda was silent, he said after a moment,\n\n\"Now stop talking nonsense. Here is the key to Mr. Harpole's despatch case.\"\n\nHe took a key from a drawer on his desk and held it out to her.\n\nYolanda shook her head.\n\n\"I am sorry, Step-papa, but I cannot do something that I am sure would shock every respectable person I have ever known. It would make me feel like I had committed an unforgiveable sin.\"\n\nFor a second Mr. Garrack just stared at her.\n\n\"If that is to be your attitude then, of course, I must respect it. But I can only hope that you will find a situation where you can express your principles without upsetting anyone.\"\n\nHis eyes flickered over her before he added,\n\n\"You are a very pretty girl and doubtless someone will employ you just for your looks rather than your brains. But in the meantime until you find a suitable position, you may be somewhat uncomfortable and perhaps hungry.\"\n\nNow Yolanda understood what he was saying and she gave a little gasp.\n\n\"Are you telling me, Step-papa, that if I do not do this for you, you will turn me out and do nothing for me?\"\n\n\"I have done a great deal for you already, Yolanda, as you well know, but if you are not prepared to pay back in a small way what I have spent on you, I see no reason why I should spend any more.\"\n\n\"But you know quite well that I have no money and nowhere to go!\"\n\nMr. Garrack turned towards his writing desk.\n\n\"That is not my business, but I daresay your mother would be most upset at your walking the streets without a chaperone and with nowhere to lay your head.\"\n\nYolanda gave a little cry.\n\n\"How can you be so unkind? You know how much it would upset Mama!\"\n\n\"It upsets me too,\" he replied. \"At the same time it is impossible for us to live here together if you disapprove so much of me and my methods. I think you are ungrateful and in some ways offensive.\"\n\nYolanda realised that he had won the battle.\n\nSo, in a small voice that he could hardly hear, she murmured,\n\n\"Very well, Step-papa. You know that I am unable to fight you, so you win. I will do what you tell me to do.\"\n\n\"I thought you would see sense.\"\n\nHe picked up the key he had put on the edge of his desk and handed it to her.\n\nYolanda took it from him.\n\n\"Now there is no reason, Yolanda, for us to speak of this again. You know what you have to do and we will behave as if this conversation has never taken place.\"\n\nHe rose from his desk.\n\n\"Actually I had a present for you on your return and I think it is something you should come and see.\"\n\n\"What is it, Step-papa?\" Yolanda was forced to ask.\n\n\"It is an outstanding present, which I am told has Arab blood in his veins and will require riding by someone who is a good equestrian besides a lover of horses.\"\n\nYolanda's heart gave a leap.\n\n\"A new horse!\" she exclaimed.\n\n\"A new stallion and it is a present from me to you.\"\n\n\"You know how exciting that is for me and thank you very much,\" she forced the words out between tight lips.\n\nShe knew that she ought to kiss him, but could not make herself do so.\n\nInstead she asked quickly,\n\n\"Can we go and see him now?\"\n\n\"That is what I thought we would do, so I have already warned the stables to have him ready for you.\"\n\nYolanda was aware that he had known before their conversation started that he would be the winner.\n\nHe obviously recognised that he would afterwards prove his magnanimity by giving her such a present.\n\nHow could she be anything but grateful when she loved horses so much?\n\nThe one thing she had missed at the Convent was being able to ride.\n\nAs they walked to the stables, she had the feeling that she had sold her principles at an expensive price.\n\nBecause it was impossible to do anything else, she talked at dinner about everything she could think of with the exception of steel-hulled ships.\n\nWhat she had to do tomorrow night kept her awake long after she had gone to bed.\n\nThe more she thought it over in her mind, the more she realised in her heart that there should be no question of her complying with her stepfather's wishes.\n\nBut how could she possibly face going out into the world alone? She knew that it would be practically impossible to find work with hardly a penny in her pocket and no knowledge of London.\n\nShe had been only been in the City briefly, but what she had seen had been the elegance of Hyde Park, the rich and comfortable houses of her mother's friends and the luxury of her stepfather's mansion.\n\nShe was not so stupid as to think any of that would be of the slightest help \u2013 not if she actually had to earn her own living.\n\nApart from that what could she do?\n\nShe had an excellent education, yet that was not to say it would enable her to earn much on her own.\n\nThe most she could hope for was to become a governess or a servant and even that would be unlikely, as she had no experience.\n\nIt surely was a man's world, she thought ruefully.\n\nWomen were not particularly necessary whilst it seemed men were always in demand.\n\n'It is all so frightening,' Yolanda brooded.\n\nShe was forced to be dependent on whoever would be kind to her and at the moment that was her stepfather.\n\n'I have to obey him, Papa, even though I think he is wrong and ungentlemanly,' she murmured slowly.\n\nThen she felt like crying.\n\nHowever drastic their situation had been in the past, her dear father had never done anything so underhand \u2013 nor something he would think of as common and beneath him.\n\n*\n\nAll the next day Yolanda waited apprehensively for the arrival of Mr. Harpole.\n\nWhen he did arrive, he was such an unpleasant looking man that she felt that it might be easier to cheat him than if he had been handsome and charming.\n\nYolanda was sure that if it was a question of Mr. Harpole deceiving her stepfather, he would not hesitate.\n\nHe was clearly impressed with the huge house and as usual her stepfather introduced her in a way that made her sound grander than she really was.\n\nBecause he would expect it, she put on a very pretty gown for dinner, although she should be dressed in black as she was still in mourning.\n\nWhen she had first arrived back, she had asked her stepfather if she could buy some black clothes.\n\nShe did not have any to wear on her journey home from Paris, so she had chosen her plainest coat and taken all the decorations from her hat.\n\nHer stepfather, however, told her firmly,\n\n\"No mourning!\"\n\n\"But it is correct for me to wear mourning because my Mama is dead,\" protested Yolanda.\n\n\"I am aware of that,\" he replied. \"But I do dislike black and I want you to look attractive. You can wear as much white as you like and, as you are a debutante now you have left school, white is the correct colour for you.\n\n\"I am not having you walking about looking like a crow and your mother would not it like either.\"\n\nYolanda wanted to argue with him.\n\nHowever, as they were in the country, she consoled herself that, as they did not see anyone it would not matter that much what colour she wore. She also knew that her mother had always disliked black, complaining that it did not flatter her complexion.\n\nAlthough her mother had worn it when her father died, she soon changed to mauve or wore black dresses with a great deal of white trimming on them.\n\nWhatever her opinion, Yolanda knew in her heart that as usual it was impossible to disagree with her stepfather as he would have to pay for anything she bought.\n\nShe therefore chose for dinner one of the gowns her mother had sent to her whilst she had been at the Convent.\n\nPutting it on, Yolanda remembered what her mother had written at the time,\n\n\"I was out shopping yesterday and saw such pretty dresses in that shop in Bond Street where we bought some clothes before.\n\nI am sending you two gowns, which I know you will enjoy wearing.\n\nThey come to you, my darling, with all my love, but you must write and thank your stepfather for buying them.\"\n\nYolanda looked mournfully at herself in the mirror, seeing a sad looking girl in a very pretty expensive dress. As she finished getting ready, she decided that it all came down to the same thing in the end.\n\nThose who had money could do what they wanted.\n\nBut those who had none had to do what they were told.\n\nShe slipped the key that her stepfather had given her into her little evening bag and reluctantly went down the stairs, pasting a smile on her face as went.\n\nBoth the men had changed into evening clothes.\n\nGreeting them, Yolanda considered that Mr. Harpole looked a little better in his than he had when he arrived.\n\nBut as dinner progressed she could not help thinking how rough and common he was, despite his expensive clothes.\n\nHe was obviously very clever and she listened to the way he talked to her stepfather with something approaching respect for his intellect.\n\nThere was no doubt that he had travelled and listening to him recounting his business stories it was clear that he was well versed in the new industries springing up all over the world.\n\nThere was, however, no mention of ships.\n\nYolanda found it hard to concentrate, let alone enjoy the food so carefully prepared by Mr. Garrack's excellent cook, as she thought apprehensively of what she had to do when dinner was over.\n\nBy the time she left the dining room, Mr. Harpole had drunk a great deal.\n\nHe was talking much more effusively than he had previously and he began paying her one or two rather uncomfortable compliments.\n\nWhen she rose to leave 'the gentlemen to their port' Mr. Harpole had some difficulty in rising to his feet.\n\nShe walked quickly upstairs as her stepfather had instructed her to do.\n\nMr. Harpole, as an honoured guest, had been put in the same corridor as her own room, but on the other side.\n\nYolanda was relieved that there were no servants to be seen.\n\nNervously she now opened the door of the bedroom where Mr. Harpole was sleeping and it was a relief to find that there were some lights left on.\n\nHis room was large, but it was not connected to a boudoir, thus there was a writing desk in the bow window overlooking the garden.\n\nYolanda saw that the despatch case her stepfather had told her about was standing at the side of the desk.\n\nShe picked it up and put it down on a chair.\n\nAs her stepfather had anticipated, it was locked and the key he had given her fitted perfectly.\n\nShe opened the case.\n\nInside, just as she had been told, were a number of letters and three notebooks together with pens and pencils.\n\nBecause she was feeling so apprehensive, her hands were trembling as she pulled out the papers \u2013 none of them referred to ships, nor were they headed with the name of the Doxford Shipyard.\n\nShe must have turned over a dozen letters and notes in Mr Harpole's hand.\n\nFinally she found what she wanted.\n\nShe pulled a letter out of the case and going a little nearer to the lamp burning on the writing desk, she read,\n\n\"Sir,\n\nWith reference to your recent offer of five hundred thousand pounds for the ships we discussed at our meeting on the 29th April, I have now put forward your offer to the Committee of the Doxford Shipyard.\n\nThey have suggested you should visit them at your earliest convenience to discuss the matter more fully.\n\nYours truly, James Augment.\"\n\nQuickly, as she was afraid she might be disturbed, she scribbled down what she had read onto a small piece of writing paper.\n\nThen she shut and locked the case with the key and put it back down beside the writing table.\n\nHaving taken the information her stepfather wanted, she hurried across the room.\n\nShe only paused as she reached the door with the sudden realisation that it would be a mistake for anyone to see her coming out of Mr. Harpole's bedroom.\n\nTo her great sense of relief the passage was empty.\n\nShe ran to her own bedroom on the other side of the corridor, closed the door behind her and locked it.\n\nShe had done it.\n\nShe had obeyed her stepfather and found what he wanted.\n\nShe should imagine how angry he would have been if she had failed to find anything \u2013 or even worse still, been discovered rummaging in the case by Mr. Harpole or anyone else.\n\nShe undressed.\n\nThen before she climbed into bed, she stood at her window looking up at the stars.\n\n'Are you very shocked, Papa?' she asked the stars. 'I know you would have been horrified at being required to do anything underhand and despicable.'\n\nThere was no answer, but she carried on,\n\n'I will try not to have to do anything like this again. I have the frightening feeling that Stepfather will ask me to do anything he wants and there is really little point in my protesting.'\n\nShe was still looking up at the stars.\n\nThen suddenly a shooting star sped across the sky.\n\nThere was a flash and it was gone.\n\nShe had the feeling it was somehow an answer from her father telling her that he understood.\n\n*\n\nThe next morning Mr. Harpole left early.\n\nBy the time Yolanda came downstairs for breakfast, he had gone and she was thankful not to have seen him.\n\nShe then learnt from the butler that her stepfather had breakfasted with him and had seen him off.\n\n\"The Master is in the study, my Lady, if you wish to see him.\"\n\nYolanda was wearing her riding habit, but she went straight to the study knowing he would be expecting her.\n\nHe was at his desk and he looked up as she entered.\n\n\"Good morning, Yolanda. I very much hope you have something for me.\"\n\nShe did not answer, but walked to the desk and handed over to him the piece of paper on which she had copied the letter in Mr. Harpole's case.\n\nHe took it from her, read it and then smiled.\n\n\"Thank you, Yolanda. You did that very well and I commend you for bringing me exactly what I required.\"\n\n\"How can you be sure,\" she asked, \"that the people who own the ships will not ask Mr. Harpole to pay more and he will still get what he wants?\"\n\n\"That is an intelligent question, Yolanda, but at the same time there is no reason for me to answer it. But you can be quite sure I shall buy those ships and they will make me a great deal of money.\"\n\nHe regarded her closely with what she felt was an unpleasant glint in his eyes.\n\n\"And, of course, money is what has been paid for the horse you are to ride this morning.\"\n\nIt was with difficulty that Yolanda managed to say,\n\n\"I am looking forward to it, Step-papa, and again thank you very much.\"\n\nShe left the study thinking how much she had always disliked him, but how difficult it was not to be grateful.\n\nAt the stables she found the stallion he had given her was outstanding and easily one of the best looking animals she had ever seen.\n\nShe rode him into the paddock on the other side of the stables knowing as she did so that it was, without a single exception, the best horse she had ever mounted.\n\nShe took him around the paddock and over one or two jumps.\n\nThen she rode him into the woods \u2013 it was always in the woods that she could feel free from the worries and difficulties of the world outside, even when she was young and they were living rather uncomfortably in her father's country house.\n\nAs far back as she could remember her parents had always worried about money and could talk of little else.\n\nIt was then that Yolanda had realised that in the woods lay a veritable Fairyland just waiting for her. Once there, under the acid green canopy of the trees, she was lost in a world of her own.\n\nThere were no frightened voices, no hidden tears and no unpleasant anticipation. There was no talk of debts or creditors chasing unpaid bills.\n\nThere was just the music of the birds and the soft movement of the rabbits in the undergrowth. Yolanda was certain that fairies were hiding in the blossom on the trees, as well as flitting over the flowers round the pool in the centre of the wood.\n\nNo one could spoil it and there was no unhappiness.\n\nThat is what she wanted to feel now and at the same time she craved forgiveness.\n\nShe had played a part in something she knew in her heart was wrong and in a way wicked.\n\nIt took her a long time to ride Chestnut, her new horse, through the woods.\n\nThen she galloped him over the fields and finally returned to the house.\n\nIt was with an effort she made herself remember that she was to have luncheon with her stepfather.\n\nShe decided that today would be a good time to find out what he was planning for the future and she only hoped that, whatever it was, it would not be in London.\n\nIn the City there would be visitors coming in for almost every meal, afternoon tea to be taken and endless parties and balls waiting for her in the evening.\n\nShe had no wish to be sociable in any way whilst in mourning for her mother, but undoubtedly if it suited her stepfather, she would not be able to refuse.\n\n'If I can only stay here,' she thought, 'at least I can think about Mama in the woods and forget what I had to do last night.'\n\nWhen she reached the stables she realised that there was just time for her to quickly change before luncheon.\n\nSo she ran into the house by the back door and went up the back stairs to her bedroom.\n\nEmily, her new lady's maid, was waiting to help her change into an afternoon gown.\n\nIt was a soft shade of blue and, although she thought it morally wrong to be wearing such a light colour, she did not protest.\n\nDownstairs her stepfather was waiting for her in the drawing room.\n\nAs she walked towards him, he pronounced,\n\n\"You look very nice, in fact, very beautiful. I am sure that before too long we shall have quite a number of young gentlemen waiting to tell you so.\"\n\n\"I just had a wonderful ride on Chestnut, Step-papa, and he is the best and finest horse I have ever ridden.\"\n\n\"I thought you would think so. He cost me a lot of money, but because you are so content, I don't begrudge a penny of it.\"\n\n\"I am most grateful,\" she managed to say again.\n\n\"So am I, so I think we can celebrate with a little champagne for luncheon.\"\n\nThey walked into the dining room and the butler held a bottle of champagne ready for them.\n\nYolanda told her stepfather all about her ride and was surprised to find that he was actually really interested.\n\nWhen the servants had left the room having served the coffee, he said,\n\n\"That reminds me, we have another friend of mine coming here tonight for dinner. He's not a very pleasant man, in fact, I think you will dislike him.\"\n\n\"Then why is he coming?\" enquired Yolanda, amazed that her stepfather should want to spend time with someone he so openly despised.\n\n\"Business, my dear, business, which is far more important than our personal feelings.\"\n\nYolanda gave a sigh and then something struck her.\n\nWas this going to be another 'job' as her stepfather would say \u2013 another 'job' for her?\n\nIf so, how could she bear it?\n\n# CHAPTER THREE\n\nYolanda did not see her stepfather again until it was teatime.\n\nShe had ridden again, this time on one of the other horses, which was a particularly good jumper.\n\nShe sailed over all the jumps in the paddock.\n\nThen she had a discussion with the Head Groom asking him to put up some more difficult jumps.\n\n\"I'll 'ave to ask the Master,\" he said.\n\n\"Of course,\" Yolanda replied. \"But he did agree with me that we need some more when I spoke with him over luncheon.\"\n\n\"I'll get busy on 'em right away, my Lady.\"\n\nYolanda took her horse over the same jumps again.\n\nShe could not help thinking it would be exciting if she had someone to ride with her \u2013 someone young whom she could race and compete with over the jumps.\n\nThen she thought that she was asking too much.\n\nIt was just a miracle that she should have so many wonderful horses to ride.\n\nShe knew her father would have loved to have them if he could afford it, and it seemed sad that there were so many horses in the stables and so few people to ride them.\n\nAt the same time all her stepfather's friends might be like Mr. Harpole and if they were, she had no wish to meet them.\n\nShe was feeking a little apprehensive after what he had said about the man who was coming this afternoon.\n\nSeveral hours later Mr. Garrack came into the drawing room for tea.\n\nIt was arranged in front of the sofa and he immediately asked Yolanda,\n\n\"Did you have a good ride this afternoon?\"\n\n\"It was fabulous, thank you, Step-papa. By the way, I hope you don't mind, but I told the Head Groom that you agreed to some more jumps in the paddock.\"\n\n\"No, of course not and you were quite right to do so. I had forgotten all about it as a matter of fact, but when the new horses arrive we must keep them well exercised.\"\n\n\"I think that means that you will have to ride more frequently than you do already,\" suggested Yolanda.\n\n\"I am too busy to spare the time.\"\n\n\"I am certain that Mama would say that you should take more exercise than you do at the moment. She always thought that you spent too much time at your desk and not enough in the saddle.\"\n\nHer stepfather laughed as she meant him to do.\n\n\"I will follow your mother's wishes \u2013 as I always have and I will ride with you tomorrow morning.\"\n\n\"Perhaps they will have some more jumps up by that time,\" Yolanda added optimistically.\n\nShe passed him various plates of food, but he shook his head.\n\n\"If I take more exercise, as you suggest, I shall be able to eat more. As it is I am getting fat.\"\n\nHe was actually a very slim man.\n\nYolanda could not imagine that however much he ate, he would put on much weight.\n\n\"I think you are exaggerating. Therefore as I cannot eat all this tea myself, the chef will be disappointed if every plate goes back untouched. We will have to buy a dog.\"\n\nHer stepfather laughed.\n\n\"I suppose the excuse that the dog has eaten the tea is better than nothing. I have often thought that we lack dogs here and I will see about it the next time I am in London.\"\n\n\"That will be marvellous,\" enthused Yolanda.\n\nThere was a slight pause and she looked at him nervously.\n\n\"My friend, the one I told you about,\" Mr. Garrack began, \"is called Cecil Watson. He is an uncouth man for whom I have no liking, but he is an extremely astute businessman.\"\n\n\"If you do not like him, why do you have him down here? Can you not see him at your office in London?\"\n\n\"That is a very sensible question,\" he replied, \"and I will tell you the answer. When I have people here they are impressed, so I manage to manipulate them far better than if I am sitting in an office which is what they expect.\"\n\nYolanda could see his reasoning and nodded slightly.\n\nShe was aware that the house with all its beautiful furnishings and stupendous pictures was certainly not to be expected of the ordinary businessman.\n\nNor was the garden with its masses of flowers and broad acres of land stretching out as far as the eye could see.\n\nAfter a moment she enquired,\n\n\"When the people who visit you are impressed, do you obtain a far better bargain out of them than you would otherwise?\"\n\n\"Exactly, Yolanda, and Cecil Watson is what one might call a 'prime bargainer' when it comes to business.\"\n\nThere was silence and then Yolanda asked him in a rather small voice,\n\n\"What do you want me to do?\"\n\n\"Very much the same as you did the other night,\" he replied. \"But actually you need not take any notes. Just look quickly at what is written either on a letter or a notebook for the name and address of the man he is negotiating with in Germany.\"\n\n\"In Germany!\" exclaimed Yolanda.\n\n\"I know German is one of the lessons you had at the Convent. Therefore it will not be too difficult for you to remember it.\"\n\n\"No, of course not. I can read German very well.\"\n\n\"Watson is having a room with a boudoir, so it will therefore be easier for you to find what I need than if you had to go to his bedroom.\"\n\nYolanda wanted to say that wherever she had to go, it was unpleasant and something she hated having to do.\n\nHowever, she realised that it was useless to ask her stepfather to find out what he required in a different way.\n\n\"Now be a good girl,\" he said, rising from the table, \"and I'll try to think of an even better present for you than Chestnut.\"\n\nHe left the drawing room before she could reply.\n\nShe jumped up from the table and walked slowly to the window.\n\nShe thought once again how degrading it was to be paid for spying on his behalf \u2013 there was no other way to put it.\n\nShe knew that both her father and mother would be shocked at what she was doing.\n\nBut what was the alternative?\n\nTo try to find herself a job without the slightest idea of how she could begin to do so?\n\nShe stood looking blindly at the trees, the flowers and the fountain.\n\nAnd then gradually their beauty seemed to seep into her soul.\n\nThe feeling of revolt and disgust faded away.\n\n'Surely,' she asked herself, 'it is worth anything to stay in a place as beautiful as this? Although I am shocked at myself, no one will ever know that I am doing something which puts me on a par with a pick-pocket!'\n\nWhen she climbed upstairs to dress for dinner, Mr. Watson had just arrived.\n\nHe was talking with her stepfather in his study.\n\nShe did not meet him and had no wish to.\n\nShe saw to her surprise he had brought his valet \u2013 a strange servant she had never seen before was assisting the footmen.\n\nThey were carrying Mr. Watson's luggage into one of the rooms in the same corridor where she slept.\n\nShe thought uncomfortably that now there was one more person she would have to avoid.\n\nShe wondered if her stepfather was aware that Mr. Watson had brought a valet with him \u2013 he might say that it was too dangerous for her to go into the boudoir, where his despatch case would obviously be placed.\n\nIt was then she remembered something.\n\nUnlike the first job she had done, her stepfather had not, this time, given her a key.\n\nYolanda wondered if he had omitted it.\n\nAlternatively he might not have known till his guest arrived what type of despatch case he would be using.\n\nThen, almost as if she had spoken her thoughts out loud, there was a knock on the door just as Emily was helping her out of her gown.\n\nEmily opened the door.\n\n\"I wish to speak with her Ladyship for a moment,\" Yolanda heard her stepfather say. \"Please wait outside, I shall not be long.\"\n\nHe entered the room closing the door behind him, but Yolanda did not move until he was close beside her.\n\nThen he held out his hand and she saw there was a key resting in his palm.\n\n\"This is what you will require,\" he told her.\n\n\"He has brought a valet with him, Step-papa.\"\n\n\"So I have just realised, but he will be sleeping on the top floor with the other men servants. If he is hanging around before Watson goes to bed, you will have to wait until he is asleep.\"\n\nYolanda stared at him.\n\nWhen he was talking business, he spoke in a sharp hard voice \u2013 very different to the way he spoke when he was being pleasant and sociable. Before she could protest, he added quickly,\n\n\"You will be quite safe in the boudoir. I will see to it that he goes to bed and instantly falls asleep.\"\n\nYolanda wanted to say that he had set her an impossible task, but in her heart she knew the answer.\n\nHowever, before she could say anything, he walked across the room and pulled open the door.\n\n\"Don't be late for dinner, Yolanda.\"\n\nThen as Emily came in, he had gone.\n\nYolanda recognised that even if she tried to protest, he would not listen.\n\nQuickly, Yolanda put the key away in the drawer.\n\nThen she was silent as Emily helped her undress.\n\nShe climbed into her bath arranged as usual in front of the fireplace. It was delightfully scented with the oil of white violets that Yolanda always enjoyed.\n\nBut tonight she was feeling too anxious and afraid of what she had to do later to relax and enjoy the warm, scented water.\n\nWhen she had finished dressing in a very pretty gown, she walked to the window.\n\nLooking out at the clear night sky, it seemed as though the stars were coming out just to help her.\n\nShe wanted them to ease away her strong feelings of resentment and fear, which were still with her, just like a physical pain in her chest.\n\n\"You looks lovely, my Lady, and that's the truth,\" Emily exclaimed behind her. \"It seems a real pity there be no handsome young men here as you could dance with.\"\n\nYolanda laughed.\n\n\"I expect all the young men you are thinking about have work to do or else they are in the Army and have no time to visit the country.\"\n\n\"Then you, my Lady, should be in London. There be parties there every night and I expects all the debutantes as they call 'em, who be your age are havin' a good time.\"\n\n\"I am really quite happy here with the horses.\"\n\nShe thought, as she walked downstairs, how true that sentiment was.\n\nWhat she was not content about was meeting Mr. Watson.\n\nAs the footman pulled open the door of the drawing room for her, she could hear a voice, rather uneducated and too loud.\n\nHim.\n\nWhen she went in, her stepfather and his guest were standing together at the far end of the room with a glass of champagne in their hands.\n\n\"Well, I says to him,\" Mr. Watson said as if he was finishing a story, \"that's all you're going to get and you're damned lucky to get that!\"\n\nYolanda walked slowly towards them.\n\nHer stepfather saw her.\n\n\"Oh, Cecil, here is my stepdaughter, Lady Yolanda Wood, who I don't think you have met.\"\n\nThe man standing beside him turned round.\n\nYolanda saw, as she had already anticipated, that Cecil Watson was undoubtedly a very ugly man with heavy features and sharp eyes.\n\nHe was rather bigger than she had expected. In fact he was far taller than her stepfather whom she had always thought of as quite a large man.\n\nThey shook hands and she realised then that he was undoubtedly an unpleasant character.\n\nThere was something about him that she not only disliked but wanted to avoid.\n\n\"You didn't tell me,\" Mr. Watson was saying, \"that I was to have the pleasure and privilege of meeting anyone as beautiful as this young lady tonight. Where have you been hiding her?\"\n\n\"She has just come back from Paris where she was being educated,\" explained Mr. Garrack.\n\nMr. Watson sniggered.\n\n\"That might be said of us too,\" he blurted out.\n\n\"There's always a lot to learn when you goes to Paris and trust you, Oliver, not to miss the prettiest of them all!\"\n\nHe dug his host in the ribs as he spoke and both men guffawed with ribald laughter. Yolanda looked away in hardly concealed disgust.\n\nNext, the butler announced dinner and all three of them walked into the dining room.\n\nIt was quite obvious from the way Mr. Watson ate and drank that he intended to enjoy himself at his friend's expense.\n\nIn fact Yolanda had never seen any man drink so much so quickly. A footman filled up his glass as soon as he put it down.\n\nMuch of the men's conversation contained innuendos. Fortunately Yolanda did not understand \u2013 not that she wanted to.\n\nShe thought that Cecil Watson was repulsive in his appearance and in the way he spoke.\n\nThe over-familiar look in his eyes when he paid her a compliment, which he did frequently, made her dislike him even more.\n\nShe could not imagine her father, even if he knew a man like this, bringing him home to meet her mother.\n\nEqually she just supposed as her stepfather would say, 'business is business.'\n\nIf it paid him to have Cecil Watson in his large and beautiful house, the effort must be worthwhile.\n\nThe more Cecil Watson drank, the more he talked.\n\nYolanda hardly spoke a word all through the meal and was thankful when it was finally time for her to leave the gentlemen to their port.\n\nHer stepfather rose as she did so.\n\nMr. Watson had great difficulty in standing up and only after the third attempt was he finally on his feet, just as Yolanda reached the door.\n\nAs she closed it behind her, she found herself hoping that she would never again set eyes on anyone so repulsive.\n\nIt struck her that it was quite obvious Mr. Watson would not be going up to bed very shortly.\n\nThis, therefore, would be a good opportunity to go to the boudoir and find the information that her stepfather desired.\n\nShe reached the corridor in which her own bedroom and that of Mr. Watson was situated.\n\nAs she did so, she saw his valet walk into his room.\n\nHe must have just come upstairs after eating in the housekeeper's room, where visiting servants always ate.\n\nYolanda wondered if the staff had found the valet as unpleasant as she had found his Master.\n\nOne thing was quite obvious.\n\nIt would be impossible for her to go to the boudoir while the valet was there.\n\nSo she rang for her lady's maid and undressed.\n\n\"You're up so early, my Lady,\" Emily remarked as she came into the bedroom.\n\n\"I am tired,\" answered Yolanda, \"and I thought it was wise to come to bed.\"\n\n\"Very wise, if your Ladyship asks me.\"\n\nThe way she spoke made Yolanda aware that she was disparaging the new arrival.\n\nShe thought, however, it would be a great mistake to discuss him with the servant, so she merely talked about the gown that was being altered for her.\n\nWhen Emily left, she went to the window and stood for a long time looking at the beauty outside.\n\nThe stars now filled the sky and a half-moon looked very romantic.\n\nIt made Yolanda remember that her father had loved looking at her mother by moonlight and he had often said he would like a portrait of her painted in a silver dress.\n\n\"You would be the real Goddess of the Moon, my darling!\" Yolanda had heard him say.\n\n\"It is such a lovely idea,\" her mother had replied. \"At the same time it is something we cannot afford.\"\n\n\"Why was I not born an artist?\" the Earl had asked. \"Then I could have painted you myself!\"\n\n\"I have no wish to sit for hours so that someone can paint me. And although it is so flattering to be named 'the Goddess of the Moon', I would rather be the Goddess of the Sun.\"\n\n\"Which you are anyway,\" her father had answered, \"and also for me \u2013 the Goddess of Love.\"\n\n\"You are being greedy and using up half Olympus!\" her mother had protested.\n\nThen they both laughed, as he kissed her, saying,\n\n\"To me you are more lovely than all the Goddesses put together. In fact I have no wish to meet a Goddess, only to have you, my glorious wife, and that means you belong to no one else but me.\"\n\nIt was the sort of thing they had so often said to each other as they felt it did not matter that they expressed their love in front of a child.\n\nAt school the girls talked endlessly about romance, but Yolanda still found it difficult to think of anything else except the delightful words her father had said to her mother \u2013 the happiness they gave her was in her mother's eyes.\n\n'It was bliss for them both to be together,' Yolanda thought now. 'Why, oh why did Papa have to die in that useless way?'\n\nLife would never be the same without him, not for her mother or for herself.\n\nAll the money in the world could not make up for the love her mother had lost.\n\nYolanda knew without her saying so that she would have thrown away the grand house.\n\nShe would have thrown away the comfort, luxury and the glittering jewels just to be with her father, even if it meant being in poor and uncomfortable lodgings.\n\n'That is real love,' Yolanda had said to herself, 'and perhaps one day I will find it too.'\n\nIt did not seem to be very likely if the only men she was to meet were the likes Jack Harpole and Cecil Watson!\n\nShe shuddered at the thought of how unpleasant Cecil Watson was.\n\nShe was sure that many of the remarks she had not understood at dinner were vulgar and obscene and should never have been uttered in front of a lady.\n\nShe felt a little cold after standing by the window wearing only her nightgown.\n\nHer lady's maid had departed, having blown out the candles before she did so with the exception of the small candelabra, which stood by Yolanda's bed.\n\nShe climbed into bed and tried to read a book \u2013 it was one she had found extremely interesting.\n\nBut it was impossible not to keep listening.\n\nShe was straining to hear the man coming upstairs.\n\nShe wondered how soon after Mr. Watson had gone to bed, she would be able to go into the boudoir.\n\nShe was in fact feeling rather tired and she would have liked to have gone to sleep.\n\nInstead, she kept reading her book just in case she dozed off and would wake up to find it was morning.\n\nThen at last she heard a loud voice in the distance. She knew that Mr. Watson was now coming up the stairs, very likely with the help of her stepfather and he was certainly making a lot of noise about it.\n\nHis voice seemed to be echoing round the walls of Yolanda's bedroom.\n\nHe passed her door.\n\nNow she could hear her stepfather's voice speaking in a somewhat soothing tone \u2013 as if he was trying to make Cecil Watson walk more steadily.\n\nThen at last Yolanda heard them go into his room on the other side of the corridor.\n\nA few minutes later her stepfather went to his room at the far end of the corridor.\n\nYolanda realised that his valet would be waiting up for him, so she listened for when he passed her door.\n\nAs he did so, he was speaking in a muffled voice to someone beside him and when a man answered, she knew that Cecil Watson's valet had joined him.\n\nNow there was complete silence.\n\nStill Yolanda waited breathlessly as she wanted to make quite certain that there was no question of the valet returning or Mr. Watson not being asleep.\n\nHalf an hour must have passed.\n\nFinally she concluded that if she did not go into the boudoir now, she would fall asleep.\n\nShe climbed out of bed as quietly as possible and opened the drawer of her dressing table where she had put the key.\n\nThen, very cautiously, she opened her bedroom door.\n\nAs she expected, the lights in the passage had been dimmed, but every third sconce still held a lighted candle.\n\nHer stepfather was already planning to have gaslight installed in the house, but it entailed a great deal of disruption.\n\nTherefore he had decided that he would not have it introduced until he was staying in London for a long time.\n\nYolanda actually preferred candles and loved eating in the dining room by candlelight.\n\nNow in her feet bare, because she thought it would be quieter, she tiptoed down the passage holding a candle.\n\nVery cautiously she opened the door into the boudoir attached to Mr. Watson's bedroom.\n\nAs she anticipated the lights had been turned out.\n\nHer candle would provide her with enough light to see what she was looking for.\n\nWhen she entered the boudoir, it seemed dark and gloomy with the curtains pulled over the windows.\n\nSo she lit another candle from her own and placed both of them side by side on the writing desk.\n\nHer stepfather had been right.\n\nHe had said that the despatch case that Mr. Watson had brought with him would have been put either on or by the desk.\n\nIt was in fact lying on top of a blotter and Yolanda thought that would make it easier to open.\n\nShe bent to insert the key into the lock hoping there would be no difficulties.\n\nShe found it turned quite easily.\n\nAs she opened the hood of the case, she thought that her stepfather must have a collection of keys \u2013 perhaps he had one for every well-known make of despatch case.\n\nHe could therefore pry into the correspondence of every guest who stayed with him. It certainly made things easier to obtain whatever information he wanted.\n\nIt seemed such an extraordinary way to behave and it was something Yolanda had never heard of before.\n\nNow the case was open and she could see that there were a large number of papers inside it.\n\nThere were so many that she thought it would take a long time to find what she wanted.\n\nShe picked up a handful of papers.\n\nSuddenly the door behind her was flung open.\n\nYolanda turned round terrified.\n\nMr. Watson was standing there with the light from his bedroom behind him.\n\nHe was wearing only a silk nightshirt.\n\nFor a moment he stared at her and then he asked in a thick voice,\n\n\"What do you want? What are you doing here?\"\n\nYolanda moved forward so that he should not see the open case behind her.\n\n\"I just \u2013 came in,\" she stammered in a frightened voice, \"to see that you have \u2013 everything you \u2013 want.\"\n\n\"Everything I want? I thought that was what I had,\" he slurred in a drunken tone. \"Now I see you, I know you are what I want!\"\n\nAs he spoke, he lurched forward towards her.\n\nRealising what he had implied, Yolanda turned to run away.\n\nBut she was too late.\n\nWith unexpected swiftness Mr. Watson caught hold of her arm.\n\nHe pulled her against his chest.\n\n\"You're pretty, so damned pretty!\" he slurred at her hoarsely. \"I was thinking over dinner that I could do with you and now you've come to me \u2013 that's just what I want!\"\n\n\"Let me go!\" Yolanda cried out. \"Let me go!\"\n\nShe tried to struggle against him, but he was very large and she was very small.\n\nHer struggles were having no effect on him and his arms held her completely captive.\n\nThen she realised that he was trying to kiss her and she thrashed her head feverishly from side to side.\n\n\"Let me go!\" she begged, but he laughed.\n\n\"You're what I've been wanting,\" he gurgled, \"but I didn't know it till now. Lucky, that's what I am, always lucky and what can be luckier than to have you.\"\n\nHe was pulling her, as he spoke, out of the boudoir and into his bedroom.\n\nYolanda realised that he was taking her towards his bed.\n\nShe gave a scream of sheer horror.\n\n\"Let me go! Let me go!\"\n\nNow she was beating her fists against his chest.\n\nBut he merely laughed and it was a discordant and unpleasant laugh.\n\nThen he muttered,\n\n\"Do you think that I'd let go of something the Gods have sent me? Now do stop being so coy and we'll enjoy ourselves.\"\n\nBy this time he had dragged Yolanda to the bed and she knew that he was going to push her down onto it.\n\nShe struggled wildly.\n\nAs he was so unsteady, she thought for a moment that she could force herself free of him.\n\nThen, with what seemed to be a superhuman effort, he pushed her back onto the bed.\n\nShe screamed and screamed.\n\nHe threw himself violently on top of her and for a moment he almost made it impossible for her to breathe.\n\nThen she felt him tearing at her dressing gown.\n\nShe screamed and screamed again.\n\nShe heard her nightgown tear and felt him pulling it away from her neck.\n\nIt was then that the door of the bedroom burst open.\n\n\"What the devil is going on in here?\" she heard her stepfather demand furiously.\n\nYolanda howled again.\n\nMr. Garrack quickly realised what was happening.\n\nIn two strides he reached the bed and pulled Cecil Watson off Yolanda's body.\n\nAs Watson tried to struggle, he hit him hard on the jaw and he fell to the ground.\n\nMr. Garrack then picked up Yolanda who was still screaming.\n\n\"It's all right, Yolanda, it's all right, you are safe. He will not hurt you now.\"\n\nShe was shaking all over.\n\nBecause she was so obviously incapable of standing, her stepfather carried her out of the room into the corridor.\n\nNow she was crying helplessly like a child who had been hurt.\n\nAs he walked with her towards her room, he said soothingly,\n\n\"It's all over. He has not hurt you and I will see he does not come near you again.\"\n\nIt was impossible for Yolanda to answer.\n\nHe had to put her down for a moment as he opened her bedroom door.\n\nShe was in a state of shock and would have fallen to the ground if he had not held her up with one arm.\n\nWhen the door was open, Mr. Garrack had to pick her up again as he realised she was incapable of walking.\n\nHe laid her down on her own bed and in the light of the little candelabra he could see the tears running down her cheeks, streaming from her closed eyes.\n\nHe also saw how the top part of her dressing gown had been torn away from her neck.\n\nHe was, however, more concerned by her face than anything else.\n\nShe was deadly pale and her body was shaking.\n\nHe sat down beside the bed holding both her hands in his.\n\n\"It's all right!\" he told her gently. \"It's all right and something that will never happen again! Can I fetch you something to drink?\"\n\nYolanda gave a gasp and then opened her eyes.\n\n\"Save me! Oh, please do save me!\" she called out desperately, her voice rising as she spoke. \"Save me!\"\n\n\"I have saved you,\" her stepfather insisted. \"You are perfectly safe now and I promise you he shall not come near you again. It's all over.\"\n\nYolanda clutched hold of his hand.\n\n\"He is horrible \u2013 beastly,\" she muttered in a voice that did not sound like her own.\n\n\"I know that, but you have to be sensible and realise that he is drunk and you are a very pretty girl. He was not thinking about who you are, just that you were a woman and in his room.\"\n\nHis words seemed to reach her for the first time.\n\nAfter a moment Yolanda opened her eyes, still holding frantically on to his hands.\n\n\"I am so \u2013 frightened,\" she managed to stammer.\n\n\"I know you are, Yolanda, but now there is just no need. I want you to go to sleep and forget all about what has happened.\"\n\n\"He is so disgusting \u2013 and so repulsive! I could not fight him, he was too strong!\"\n\n\"I know that, but I expect by the morning he will forget what he has done. He had more than enough drink to make him unconscious \u2013 or so I thought.\"\n\nNow there was an angry note in his voice. He had thought when he brought Cecil Watson up to bed, he was too drunk to do anything but fall asleep.\n\nHe had been mistaken and now he was furious with himself that this drama should have occurred.\n\nVery gently he released his hands from Yolanda's.\n\nThen he rearranged the bed so that he could pull the bedclothes over her.\n\nShe was still wearing her blue dressing gown and he thought it best for her to sleep in it, as she should not move about more than was absolutely necessary.\n\nNow she was quieter, but he noticed that her body was still trembling.\n\nAlthough she had opened her eyes, she kept closing them \u2013 it was as if she was terrified of what she might see.\n\nMr. Garrack pushed her hair back very gently from her forehead.\n\n\"Now will you be all right if I leave you alone to sleep? Or would you rather I asked someone to come and stay with you? It will have to be one of the maids.\"\n\n\"No! No! I don't want anyone! But you will not let him come near me again?\"\n\n\"No, of course not, Yolanda, I tell you what I will do. I will leave you, but I will lock you in. Then tomorrow morning before the maid wakens you, I will come and open the door.\"\n\n\"You will \u2013 lock my \u2013 door,\" Yolanda repeated slowly.\n\nShe was forcing herself to fully understand what he would do.\n\n\"I will lock the door and I will take the key away, so no one can come near you. Is that what you want?\"\n\n\"Yes! Yes! Then I shall be safe!\"\n\n\"You will be safe,\" he said firmly. \"But, just in case you are still scared, I will leave the door of my room open. If you scream loudly I shall hear you.\"\n\n\"I will try to go to sleep,\" Yolanda promised with an effort.\n\n\"That is most sensible of you and everything will seem better in the morning. Now, don't be frightened any more because there is no need for it.\"\n\nHe rose to his feet.\n\nThen he pondered for a moment, walked over to the window and pulled back the curtains so that the moonlight could pour in.\n\nYolanda could see the stars from her bed.\n\n\"That will give you a little light. Now, would you like me to leave the candles or blow them out?\"\n\n\"You are quite sure that the door will be locked?\" Yolanda repeated in an agitated voice.\n\n\"Shut and locked.\"\n\n\"Then please blow out the candles and I will look at the stars.\"\n\n\"I want you to close your eyes and go to sleep. I am so very sorry that this has happened and you will have to forgive me.\"\n\nHe did not wait for her to answer.\n\nHe walked across the room and opened the door.\n\n\"Goodnight, Yolanda. Remember you are safe.\"\n\nHe went out and she heard the key turn in the lock.\n\nThen she hid her face in the pillow and burst into tears.\n\n# CHAPTER FOUR\n\n\"How is Lady Yolanda?\" Mr. Garrack asked of his housekeeper.\n\n\"She's very poorly, sir. She won't get up, but just lies in bed staring at the ceiling.\"\n\n\"Do you think she is really ill?\"\n\nThe housekeeper shook her head.\n\n\"I don't know what it is, sir. I talks to Emily who's her lady's maid and she says something about her Ladyship going back to a Convent where she's been abroad.\"\n\nMr. Garrack's lips tightened.\n\nHe was silent for a moment and then he said,\n\n\"Tell her Ladyship that I wish to see her at twelve o'clock and it is very important.\"\n\n\"I'll tell her, sir.\"\n\nWhen he was by himself in his study, Mr. Garrack sat for some minutes thinking.\n\nOver his long career he had coped with all types of different men and invariably, even though at times it was difficult, he always got his own way.\n\nHe recognised that Yolanda was still suffering from shock, even though her encounter with Cecil Watson was now over three days ago.\n\nShe had not come out of her bedroom since.\n\nShe had merely sent messages to say that she was tired and was resting.\n\nNow from what the housekeeper had just told him, Mr. Garrack thought that he must do something, although he was not certain what it should be.\n\nIt was exactly twelve o'clock when the door of his study opened and Yolanda entered.\n\nAs she walked towards him, he could see that she was looking very pale and there were lines under her eyes which had not been there before.\n\nWhen she reached him, she said jerkily,\n\n\"You wanted \u2013 to see me \u2013 Step-papa?\"\n\n\"Yes, Yolanda, sit down, I want to talk to you.\"\n\nShe sat down on the edge of a chair by his desk and looked at him apprehensively.\n\nThen she began,\n\n\"I have something I want to say to \u2013 \"\n\nMr. Garrack quickly put up his hand.\n\n\"I think I must speak first, because I asked you to come to me.\"\n\n\"Yes, of course,\" responded Yolanda in a dull and flat voice.\n\n\"I have been thinking about your father, Yolanda. I admired him greatly, because he was undoubtedly a very brave man.\"\n\nWhen he mentioned her father, Yolanda looked up in surprise.\n\nHer eyes were on him as he continued,\n\n\"I think at the moment he would be very distressed and maybe humiliated if he thought that his only child was a coward.\"\n\nYolanda stiffened.\n\n\"A coward?\" she repeated almost under her breath.\n\n\"Your father's bravery in fighting a duel he knew he would lose was, to say the least of it, heroic. He could easily have refused and left Paris for England.\n\n\"But he was brave enough to fight against a man who would undoubtedly beat him, because it would have been absolutely wrong and against his idea of honour for him to have run away.\"\n\nYolanda did not say anything.\n\nHer fingers were clenched together and she looked at her stepfather with a startled expression in her eyes.\n\n\"As you know,\" Mr. Garrack went on, \"I have dealt with thousands of problems in my life and I have only got where I am today by standing up and fighting for what I wanted.\"\n\nHis voice dropped as he added,\n\n\"I have not told you about the beginning of my life, as I do not like to talk about it, but I was actually a very unhappy child.\"\n\nHe looked towards the window as if he was looking back into the past.\n\n\"My parents were of no particular importance and, as they had three other children, they did not want me. I was the youngest. I remember how I resented being treated as if I was of little account and made to give my brothers preference in everything.\"\n\nThere was a distinct bitterness in his voice that told Yolanda without him saying more how he had suffered.\n\n\"I think that I must have been around ten or eleven years old when I decided that the only way I could actually defeat my brothers was if I was cleverer than they were.\n\n\"I cannot think who put the idea into my head, but I worked hard at school until, at the end of term, I actually received a prize. That made me determined to have at least a dozen of them. I not only gained many prizes but I won a scholarship.\"\n\n\"That was indeed \u2013 clever,\" remarked Yolanda.\n\nShe seemed to force the words through her lips.\n\nHer stepfather thought that as she had spoken it was perhaps a step in the right direction.\n\n\"I went on to University and because I worked by day and night, I was awarded a First and a final report of which anyone would have been proud.\"\n\nHe realised that Yolanda was now listening intently so he continued,\n\n\"I next offered my best services to a factory in the neighbourhood and, based on my excellent reports from the University, they engaged me.\"\n\n\"And what did your family think?\"\n\n\"They were most surprised, but my elder brothers still laughed at me. I was paid only a small salary for the job I had undertaken. Quite frankly they thought I was a fool to work so hard.\"\n\n\"And then what happened?\"\n\n\"I moved to a larger firm and from them to another. I soon learnt all each company had to teach me and thought I could do better.\n\n\"Of course, I had many setbacks. Of course, people snubbed me for being too grasping and too pushy. I made quite a large number of enemies, but I also had friends who were really kind and understanding.\"\n\n\"Did your enemies upset you?\"\n\n\"When they opposed me, I fought back, although I did not win every battle, I, at least, showed them I had guts, which in those days were admired in a man of my station.\n\n\"I had to lick some people's boots. I had to put up with a certain degree of humiliation and from one job I was sacked for impertinence.\"\n\nHe gave a little laugh.\n\n\"Actually I was telling the owner of a factory how to run it much better than he was doing at the time. As he knew that I was right and he was wrong, he wanted to be rid of me!\"\n\n\"So what did you do?\"\n\n\"I went to the firm which was his greatest rival and offered the manager my services.\"\n\nThere was silence before Yolanda enquired,\n\n\"Did you tell him how badly the firm you had left was run?\"\n\n\"Of course I did. I also gave him some information about what the owner was doing that he found useful.\"\n\nHe knew without asking that Yolanda thought this was wrong, so he added,\n\n\"As I have said to you before 'business is business'. In the fight to rise to the top there is not a man in this world who has not used every weapon in his power to get there.\"\n\nYolanda did not speak and her stepfather continued,\n\n\"I know what you are thinking. What I have done your father would think was unsporting and unbecoming of a gentleman. But if you think of it not as what happens in a gentleman's soft life, but in the war of survival which is 'below stairs' so to speak, then you will understand.\"\n\n\"What you are saying,\" Yolanda replied slowly, \"is that it is a battle and the man who is not with you is against you as an enemy.\"\n\n\"Exactly. I see you understand. That is what I am trying to make you appreciate. In everyone's life, sooner or later, things happen, which are not what you expect because they don't keep to the written rules of what we now call 'Social behaviour'.\"\n\nHe threw up his hands.\n\n\"That is the world in which the aristocrats and the upper class live where they have made unwritten laws that they obey, because it makes them, they believe, superior to those beneath them.\"\n\nYolanda thought this statement over for a moment.\n\n\"I can see, Step-papa, what you are trying to say.\"\n\n\"I am trying to say that sooner or later in our lives we all come up against great difficulties and that is where we have to fight with every nerve in our bodies to survive.\n\n\"I know that your father came up against so many difficulties in his life at one time or another. He somehow surmounted them and was still able to keep his own code of behaviour intact.\n\n\"But that is not to say that those against him kept to the same rules. They cheated if they could and they were prepared to be treacherous in every possible way without worrying, as your father would have done, as to whether what they did was becoming of a gentleman.\n\n\"What I am trying to explain to you is, whatever happened, however bad it was, however disgracefully those against your father behaved, he went on fighting even if it meant his death.\"\n\nHe looked at Yolanda as he spoke and after a long moment, she remarked,\n\n\"That is what you are telling me to do.\"\n\n\"What I am really saying, is that you have won a battle against a man who is a cad and a swine. Now that you have done so, be magnanimous and forget him. He is of no importance and it is just what your father would have done.\n\n\"It was an unpleasant episode in your life, but now Watson can only win if you continue to brood about him and destroy your future happiness by doing so.\"\n\nYolanda rose and walked to the window.\n\nShe stood there looking out into the garden.\n\nAfter a long pause, she murmured,\n\n\"He terrified me.\"\n\n\"Of course, he terrified you.\n\n\"But it was my fault in asking you to help me and I can only say how sorry I am. At the same time you must realise that you and I are making Watson seem of far more importance than he really is in the world, because we are thinking and talking about him.\"\n\n\"I will try not to think about him again.\"\n\n\"Good!\" her stepfather exclaimed. \"Now let me tell you what we are going to do. We are going to London!\"\n\nYolanda turned round.\n\n\"Why?\" she enquired.\n\n\"Because I have business there for one thing and also I would like you to be in touch with your mother's and father's friends. It is what your mother would have wanted you to do and I admit I was quite wrong to bring you here to the country while the Season was still taking place.\"\n\n\"But I am in mourning \u2013 \"\n\nHer stepfather looked at her.\n\n\"Do you really begrudge your mother being with your father whom she adored?\"\n\nYolanda stared at him in a startled way.\n\nShe thought it was a strange remark for him to make.\n\n\"I know what you are thinking, Yolanda, and I do know how very much your mother loved your father and he loved her. And I knew I could never take your father's place, so I did not try to do so.\"\n\nYolanda was still staring at him.\n\n\"It was the greatest moment of my life when your mother permitted me to look after her and comfort her a little. It is something I shall never forget and because I loved her, I want to believe that she is happy now.\"\n\nHe spoke so movingly that Yolanda found it hard to realise that it was actually her stepfather speaking.\n\n\"What you and I have to do now is exactly what your mother would want you to do. That is to go back to London and to be in contact with those who loved her and your father and spend the last month of the Season with them.\"\n\nHe smiled before he added,\n\n\"I am not going to give a spectacular ball for you \u2013 that can easily wait till the winter. But we can hold dinner parties and dances in my house in Park Lane and I am sure that you will receive a great number of invitations.\"\n\nYolanda was gazing at him as if she could not take in what he was saying.\n\nHe held out his hand.\n\n\"Come along, Yolanda, we both know that we want your mother to be happy and have exactly what she wants, just as we tried to give it to her when she was with us.\"\n\nShe bent forward and took his hand.\n\nAs she did so, for the first time ever, she placed her arms round her stepfather's neck and kissed his cheek.\n\nHis arms closed round her and he realised that she was crying.\n\n\"It is all right,\" he soothed. \"Everything is all right. We now understand each other and we will do what your mother wants.\"\n\n\"How can you be so kind, so understanding? I have been feeling so alone and so scared, but now I feel I have you.\"\n\n\"Of course you have me, Yolanda, and together we will win our battle.\"\n\nShe wiped her eyes.\n\nThen Mr. Garrack suggested,\n\n\"Let's have a glass of champagne because we both need it. Then after luncheon we will go riding. I think you will agree that we should take Chestnut and the new horses which we have yet to name, to London with us.\"\n\nYolanda smiled at him.\n\n\"They too will want their London Season!\"\n\n*\n\nAt luncheon her stepfather set out to amuse her.\n\nHe told her stories of things that had happened to him when he was abroad and spoke of new projects he had invested in which had proved disastrous.\n\nBecause she asked him, he told her with pride about some business deals that had been a huge success and had eventually made him a multi-millionaire.\n\nAs soon as luncheon was finished the horses were brought round to the front of the house from the stables.\n\nThey rode together first into the paddock and then to look at the new jumps, which had been erected while Yolanda was lying in bed.\n\nThey rode over the broad acres of his estate.\n\n\"You must be very proud of all this,\" she exclaimed.\n\n\"It is what I always told myself I would have one day \u2013 a house bigger and better than anyone else's. It was blessed when your mother came to stay and she was even more beautiful than any of the pictures on my walls.\"\n\n\"I meant to talk to you about your pictures.\"\n\nThe horses were now walking side by side and the sun was shining on Yolanda's golden hair.\n\nShe was not wearing a hat as there was no one to see her except her stepfather \u2013 it was the way she had enjoyed riding with her father.\n\n\"Why do you buy so many expensive and famous pictures?\" Yolanda asked him.\n\n\"First because I enjoy looking at them and secondly because I know it will impress the people I want to impress.\"\n\nShe liked his response because he was so frank.\n\n\"Have they come to see them?\"\n\n\"Quite a number came while your mother was here. That is what I am hoping you will do now you have taken your mother's place.\"\n\nYolanda looked a little puzzled and he explained,\n\n\"We shall entertain the right people who will know a masterpiece when they see one and who perhaps have one or two in their own homes. I promise you that I have no intention of wasting their beauty on those who don't appreciate them!\"\n\nYolanda knew what he was telling her \u2013 he would not in future ask men like Cecil Watson to stay with them in the country.\n\nShe did not respond to his comment so instead she gave him a smile, which made her, for an instant, look very much like her mother.\n\n\"We are going to enjoy ourselves, although I know it is only a question of time before you will fall in love and leave me.\"\n\n\"I will never leave you completely, Step-papa. It is what I very stupidly thought I might do, but I promise you I shall not contemplate it again.\"\n\n\"We will make this house a very happy one,\" he told her quietly, \"just as when people stayed with us when your mother was alive, they always went away feeling very happy and different to when they arrived. As one woman said to me, 'it has been Heaven this weekend being with you two wonderful people. Please, please ask me again'.\"\n\nYolanda's eyes softened as she listened.\n\n\"It is a very nice idea, Step-papa, and that is what we will do. I know it would please Mama.\"\n\n\"I know that too and I am certain she will help us.\"\n\n*\n\nThat night when Yolanda went to bed, she thought how stupid she had been.\n\nShe had not attempted to understand her stepfather before now.\n\nShe had actively disliked him and been snobbish enough to believe that he was rather common and beneath her.\n\nNow when he had told her frankly of his struggles and difficulties to reach right to the top, she realised it was something he had always done alone and without love.\n\nUntil he had found her mother.\n\nYolanda was now thinking of just how much she had meant to him \u2013 she could hear it every time he spoke of her.\n\nThe sincerity in his voice and the expression of love in his eyes was very moving.\n\n'We were so incredibly lucky that he was so kind to us,' she said to herself. 'And it was most ungrateful and stupid of me not to understand.'\n\nShe was now determined to make it up to him.\n\nWhen she said goodnight, once again she put her arm round his neck and kissed him affectionately.\n\nIt was the way she had kissed her father.\n\nShe knew her stepfather understood without words what she was saying to him.\n\nWhen they went upstairs, she slipped her hand into his.\n\n\"It is really so exciting to be going back to London, Step-papa. But you will help me?\"\n\nHe realised that she was asking him to keep away from her any man who might frighten her.\n\nHis fingers tightened over hers.\n\n\"I am always here for you, Yolanda, and I shall be extremely careful whom you meet. Let's at first keep to your mother's friends. I know some of them already, who I met when you were away in France and I am sure that if I appeal to them, they will welcome you with open arms.\"\n\nWhen they reached Yolanda's bedroom, they kissed again.\n\nThen Mr. Garrack went to his own room.\n\nYolanda found that the housemaids had packed all her gowns and the trunks were now standing in her room waiting for any last item before they were locked.\n\nShe thought as she looked at them that this was the beginning of a new adventure.\n\nIt was what her father had often said when they had moved from a place they could not afford, and sometimes it was because the people there were wary of playing cards with him because he won so much from them.\n\n'A new adventure!'\n\nShe breathed the words to herself.\n\nThen she felt that her mother was somehow smiling at her and telling her that was what it would indeed be.\n\n'Perhaps,' mused Yolanda, 'I shall find someone as attractive as my Papa and that will be the most wonderful adventure of all.'\n\nYet somehow she felt it unlikely.\n\nThe men she had met in the very brief time she had been in London had seemed so young and immature. They were not intelligent, but in fact rather stupid.\n\nHowever, she was sensible enough to realise why they had disappointed her.\n\nShe had always spent time with older men, like her father and his many friends, and now it was the same with her stepfather.\n\nEven when she had disliked him, it was impossible not to be very impressed by his brain \u2013 and impressed by the interesting comments he made about his work.\n\n'Maybe,' she pondered, 'I shall find all young men are boring and dull because they have not lived exciting and adventurous lives as Papa and Step-papa have lived.'\n\nIt was indeed a rather depressing thought, but then she told herself firmly that in this new adventure anything could happen.\n\nThe unexpected was always round the corner.\n\n'What I am going to do, Mama,' she whispered as she lay alone in the darkness, 'is to try to enjoy myself as I know you would want me to and it will please Step-papa.'\n\nShe was beginning to realise now why her mother had undoubtedly been fond of Oliver Garrack, even though she had given her heart and soul to her first husband.\n\n'I was very foolish not to realise that, although he is different in so many ways to my Papa, he is an extremely interesting man and a most intelligent one.\n\n'It meant everything to her when she was so bereft, lonely and totally penniless to have someone in her life who literally worshipped her.'\n\nAbove all else he wanted her mother to be happy.\n\n'I must try to understand people better than I have done in the past,' Yolanda told herself, 'and not judge men as I have been doing because they do not resemble Papa.'\n\nWhen she mulled it all over, she concluded that her mother had always found something nice to say about everybody \u2013 however different they were and whatever their position in life, she was always complimentary.\n\nAs she fell asleep, Yolanda was thinking that she must try to make her stepfather happy.\n\nShe had never even thought of worrying about his feelings before, but now she realised how much he must be missing her mother.\n\nPerhaps in his own way he was trying to ease his unhappiness at losing her by working even harder, and that was the real reason why he had brought Mr. Watson down to the country.\n\nWithout, of course, imagining for a moment that he would behave as he had to her.\n\nShe was quite certain that never again would her stepfather invite her to help him as he had done over Mr. Harpole and then Mr. Watson.\n\nIt was very wrong in the first place and even worse, because he had embroiled her in his nefarious schemes.\n\nHe had not considered her as an attractive young woman or even as his stepdaughter.\n\nHe was thinking only of the money he could make.\n\nMoney, which must fill his life completely now that he had lost her mother.\n\nHis beautiful wife had been, as he always claimed, the best and most wonderful thing to ever happen to him.\n\n'I understand! I do understand!' she whispered.\n\nThen because it was impossible to go on ruminating any more, she fell asleep.\n\n*\n\nThe next morning Yolanda was awake bright and early and after breakfast a carriage was waiting for them outside the front door.\n\nAs they drove away, Yolanda thought the country had never looked more beautiful and it seemed almost a crime to leave it for the City of London.\n\nThe servants were following them in a large brake with their luggage and besides her stepfather's valet there were two footmen and Emily.\n\nEmily had begged Yolanda to ask the housekeeper if she could accompany her to London, and, as she was very attentive and a nice girl, Yolanda had done what she asked.\n\n\"Of course you can have her if you wants her, my Lady,\" the housekeeper said. \"But make sure she comes with you when you returns or we'll be short-handed here.\"\n\nAs the house seemed to boast an array of servants, Yolanda thought this was a little unlikely.\n\nHowever, she promised that Emily would return when she did.\n\n\"We'll all miss you, that we shall,\" the housekeeper went on. \"But I'm sure the Master'll be bringing down his friends and you, my Lady, must bring yours. It's nice to have a house full and the chef says he'll get fat and lazy if there ain't no one to cook for!\"\n\nYolanda laughed, but she thought it a good idea and told her stepfather what the housekeeper had said.\n\n\"You are reading my thoughts!\" he exclaimed. \"I have every intention of having friends at the weekends, but first you and I have to decide who we will invite.\"\n\n\"You think of everything!\" smiled Yolanda.\n\n\"I try to,\" he answered. \"It's most important to plan ahead and not leave everything until the last moment. You know as well as I do that can be disastrous and all that happens is that everything of any importance is left behind. You will find yourself with the people you don't want, while those you do have made other arrangements!\"\n\nYolanda laughed.\n\n\"You must be careful not to let that happen, Step-papa, and I am sure London is going to be very exciting. At least, as Papa would say, it is something new and that is always an adventure.\"\n\nThey stopped on the way for luncheon at a posting inn and when they set off again, Yolanda knew the horses had been well rested and that the servants had also enjoyed a good luncheon.\n\nHer Papa had always been most particular that they should look after their staff, when they had any.\n\nHe had often become angry when he had heard that mean people neglected those who served them and allowed them to be housed in damp rooms and fed badly, while the gentry were enjoying the best of everything.\n\n\"I learnt as a soldier,\" he told his daughter, \"that a good Officer looks after those he is commanding before he thinks about himself. That is what I have always done in my private life and I shall expect you to do the same when you are grown-up.\"\n\n\"Of course, Papa\" Yolanda had replied.\n\nBut it was something she thought she would never encounter \u2013 she would be with her father and then perhaps with her husband.\n\nAs they approached London, she could not help but wonder if her stepfather was now hoping that she would soon be married.\n\nShe was certain that the subject had already crossed his mind and that was why he had insisted on her attending dances and receptions in the country, even though she was still in mourning.\n\nShe only hoped that the people who had invited her would not be shocked.\n\nShe felt sure that she should not be amusing herself so soon after her mother's death.\n\nThey arrived at the enormous house in Park Lane.\n\nYolanda thought that without the sunshine and the country surroundings they had left behind, it was all rather gloomy.\n\nHowever, once she was inside she was impressed with the magnificent way it was furnished and by the many pictures on the walls.\n\nThat night after they had finished dinner together, Mr. Garrack told Yolanda the names of the people he had written to.\n\n\"They are all your mother's friends, but I daresay you can remember some more.\"\n\n\"I will try,\" replied Yolanda, \"but this is a long list and I think, if everyone on it asks me to a party, I shall very likely be here until Christmas!\"\n\nHer stepfather laughed.\n\n\"It's not as bad as that. But I shall feel very sad if you sit at home night after night like Cinderella with only your decrepit old stepfather to talk to!\"\n\n\"I have a feeling that if you come to the parties with me, Step-papa, you will dance more than I will and be far more successful with the ladies than I can ever be with the gentlemen!\"\n\n\"Now you are being ultra modest. Let me prophesy that you are going to be the most beautiful debutante of the Season. Tomorrow before we do anything else the shops are coming to us.\"\n\nYolanda knew exactly what this meant.\n\nWhen she had last been briefly in London, her stepfather had sent to the shops in Bond Street and they had brought an enormous number of gowns to the house.\n\nAt the time Yolanda had thought that it was a rather grand way to shop, but certainly much more comfortable than traipsing from shop to shop.\n\nThe vendeuse, with two assistants, had arrived at the Park Lane house with what seemed like a mountain of boxes.\n\nOnce they had shown their wares to Mr. Garrack, he had decided which ones he wanted Yolanda to try on and she had chosen from them.\n\nShe presumed the same procedure would happen again, so was delighted when he said,\n\n\"I think you are sensible enough now to know what you want and what your dear mother would prefer you to wear. So I will leave it to you. Just tell them to send me the bill.\"\n\n\"You are so kind, Step-papa.\"\n\nMr. Garrack put his hand on her shoulder.\n\n\"Everything I can give you is a thanksgiving for the happiness your mother gave me. It is just something that cannot be paid back in words, clothes, jewels or anything else. But if I am able to make you happy, that will be the only payment I require.\"\n\n'How can he be so generous?' she asked herself.\n\nShe knew the answer.\n\nWhat she had felt about him before had been her own fault, because she had not dug deep enough into his character \u2013 nor had she ever been close enough to him to understand his feelings.\n\nShe had disliked him because she thought he was pushy \u2013 and, if she was really honest with herself, because he had taken her father's place.\n\nShe had never thought of him as someone who could be hurt or upset, nor as a person who could love anyone in the way that he had loved her mother.\n\nNow she understood and felt ashamed of herself.\n\n'If I am to be happy,' she determined, 'then I have to find some way to make him happy too.'\n\nShe spent the morning, as he had arranged, trying on dress after dress and gown after gown.\n\nLooking through the lovely shades of silk and fine muslin, she realised that she really did need to restock her wardrobe. Fashions had changed so much in the time she had been away and it was impossible not to feel excited by the beautiful array of garments laid out before her.\n\nShe had gone to the Convent with such a large number of clothes that the other girls had teased her.\n\nBut except for what her mother had sent to her and the few dresses Mr. Garrack had bought last time, she had not had anything new since before going to Paris.\n\nThe vendeuse told her that she looked beautiful in everything she tried on, which was not very helpful, and she wished that her mother could be there to help and advise.\n\nFinally, after much deliberation she chose four day dresses and five evening gowns.\n\nNext, she chose from beautiful handmade shoes and dainty boots, embroidered underclothes, fine stockings and, of course, hats of every shape and size.\n\nEach shop that specialised in these items had brought enough for a bride's trousseau.\n\nThe thought made Yolanda shudder \u2013 and she tried hard not to think of Cecil Watson again, as she had promised her stepfather.\n\n*\n\nThat night he took her to dine with an old friend of her mother's who he also knew well.\n\nAs they entered the drawing room, Yolanda could see that there were six young men present beside her host and hostess and their daughters.\n\nSuddenly she felt a sense of fear and it made her want to turn round and run away.\n\nThen she told herself she was being very stupid.\n\nHow could she be frightened of these young men who were all gentlemen?\n\nThey would never behave like Cecil Watson!\n\nYet still the horror she had felt when he had thrown himself on top of her was present once again.\n\nThen her hostess was saying very charmingly,\n\n\"It is so lovely to see you, Yolanda. Your mother was such a dear friend of mine and I am so anxious for you to meet my family.\"\n\nShe introduced first her daughters, then two of the young men who were her sons.\n\nAs they took her hand, Yolanda wanted to snatch it away and it was with the greatest difficulty she prevented herself from doing so.\n\n'I am being so stupid, foolish and idiotic!' she told herself angrily.\n\nYet when they went into dinner she disliked putting her hand onto the arm of the hostess's eldest son who was escorting her.\n\nBut it was his father, who said to her,\n\n\"You are the prettiest girl I have seen for a long time. I cannot think why you have been hiding away from us for so long!\"\n\nIt was difficult for Yolanda to look at him.\n\n\"We have been in the country,\" she murmured.\n\n\"Well, now you must make up for it by being the belle of the ball at every party,\" he said, \"and undoubtedly all the critics in White's will be lifting their glasses to you!\"\n\nThe mere idea of it made Yolanda shiver.\n\nThen she saw her stepfather watching her from the other side of the table.\n\nWith an effort she forced herself to smile and knew that was what he wanted her to do.\n\n'I am not frightened. I am not frightened,' she repeated silently to herself.\n\nShe was well aware that this was an inward battle that she must face alone without being a coward.\n\n# CHAPTER FIVE\n\nTwo weeks later everything was different.\n\nYolanda had won her battle.\n\nShe no longer shuddered when a young man put his arm around her waist to dance and she could now accept a compliment without feeling shy.\n\nShe was an undoubted success.\n\nHostesses, who had paid no attention to her at first, fell over themselves to invite her to parties.\n\nEveryone was talking about her.\n\nAs her stepfather had so astutely predicted, she was undoubtedly 'the belle of the ball'.\n\nThere was only one somewhat unfortunate incident during the first week.\n\nShe had gone to a rather bigger dance than she had attended before and there was a garden at the back of the house.\n\nShe danced twice, or perhaps it was three times, with a young man who had interested her.\n\nHe had been in the Royal Navy and was telling her about the recent modern improvements that had been made to the ship in which he was serving.\n\nIt was very warm and when the dance finished, they went into the garden to sit under a tree where it was cool.\n\nMr. Garrack was sitting in the drawing room when Yolanda came running up to him.\n\n\"I want to go home,\" she told him breathlessly.\n\nHe looked at her in surprise.\n\nThen without asking her any questions, he replied,\n\n\"Of course, Yolanda, if that is what you want.\"\n\nShe hurried down the stairs in front of him and they waited without speaking until their carriage was brought to the door.\n\nOnly when they drove off, did Mr. Garrack ask,\n\n\"What has happened? What has upset you?\"\n\nFor a moment he thought Yolanda was not going to reply and then she murmured,\n\n\"The man I was dancing with asked me to marry him.\"\n\nThere was a short pause, before he commented,\n\n\"Yolanda, I suppose you realise that is the greatest compliment you can be paid and the most expensive one.\"\n\nHe paused to smile at her before continuing,\n\n\"After all, the gentleman in question is taking you 'for better, for worse, for richer, for poorer, until death do us part'. By that time you may be sightless, toothless and extremely irritable. No man could offer you more!\"\n\nFor a moment there was silence and then Yolanda began to laugh.\n\nAs her stepfather laughed too, she moved closer to him and put her head on his shoulder.\n\n\"You are quite right, Step-papa. I am a fool, but he took me by surprise, so I ran away.\"\n\n\"He will certainly not think you are running after him,\" Mr. Garrack remarked dryly.\n\nYolanda laughed again.\n\n\"I just don't know why I am so stupid, but I am no longer as frightened as I was.\"\n\n\"I do realise that, Yolanda, but I think that now you are such an overwhelming success, you will think as I do that it is silly to worry about anything in the past. After all what is done is done and cannot be altered. It is tomorrow that is important.\"\n\n\"Of course it is,\" agreed Yolanda, \"and you must forgive me for behaving so badly.\"\n\n\"It was getting late anyway and as I have to be in my office early tomorrow morning, I am actually delighted to be going home to bed.\"\n\nYolanda knew that he was making things easier for her.\n\nWhen they arrived back at his house, she kissed him affectionately.\n\n\"I am so sorry I behaved so stupidly,\" she told him.\n\n\"Forget it, Yolanda, I think tomorrow may be very exciting for you.\"\n\nHe made it exciting by giving her an attractive and very expensive present.\n\nHe knew only too well that he was spoiling her.\n\nBecause Yolanda was such a social success, she and her stepfather were invited to endless parties given by the most exclusive hostesses.\n\nThey would certainly not have been invited to them in the past.\n\nIn her new dresses for the daytime, as well as new gowns for the evening, she recognised that her stepfather was proud of her.\n\n*\n\nAt the end of the third week, he said to her casually,\n\n\"Oh, by the way, Yolanda, we are travelling to the country for the weekend.\"\n\nYolanda's eyes lit up.\n\n\"To the country! How wonderful!\"\n\n\"There are several new horses for you to see and there are three pictures I have just bought that I think will interest you.\"\n\n\"Why did you not tell me before?\" asked Yolanda. \"You knew I would be thrilled.\"\n\n\"I could not go to the sale myself, but my secretary did well and brought back a Van Dyck, which I am sure will delight you, as well as two more by Stubbs.\"\n\nYolanda clapped her hands together.\n\n\"Oh, Step-papa, that is exciting. You know I love the Stubbs you have already and I did think it was a pity there were not more horses inside the house when there are so many outside!\"\n\nHer stepfather chuckled.\n\n\"I hope you will approve of my new purchases and they will certainly add lustre to my collection.\"\n\n\"You have many fine pictures here in London.\"\n\n\"They are the first purchases I ever made and to be quite honest I could have done better.\"\n\n\"Some of them are very lovely,\" argued Yolanda.\n\n\"But nothing as fine as those in the country. I took a great deal of trouble to learn much more about art than I knew before \u2013 and I had a good teacher.\"\n\nYolanda smiled at this reference to her mother.\n\nShe thought it impossible not to admire a man who could work so hard \u2013 not only at making money but also on improving his own brain.\n\nShe was just going to ask more questions about the pictures when he added,\n\n\"We are entertaining a guest this weekend and he is coming to see my Stubbs. I also want to know a little more about his interests in America and the automatic air brakes that have been invented for trains.\"\n\nYolanda looked at him wide-eyed.\n\n\"I am hoping that you and I will be clever enough to make him talk about it. His name, by the way, is Lord Milborne.\"\n\nHe was about to say more when his secretary came into the room.\n\n\"I am sorry to bother you, sir,\" he said, \"but there is a man here from the country who wishes to talk to you about the gas lighting you ordered for the servants' quarters. I understand there is some difficulty regarding the roof.\"\n\n\"I will come and speak to him.\"\n\nHe left the room.\n\nWhen he had gone, Yolanda stood where he had left her, thinking over what he had just said.\n\nShe knew Lord Milborne.\n\nHe was the gentleman who had been exceedingly kind to her father when they were living in Baden-Baden.\n\nHer father was having one of his bad periods at the gaming tables and his luck seemed to have deserted him.\n\nThey had arrived in Baden-Baden after he had been gambling very successfully in Hamburg and they had taken rooms in the Stephanie Hotel, which was undoubtedly the best and the most famous of all the many sumptuous hotels in the town.\n\nBaden-Baden itself was beautiful and Yolanda, who was then only fourteen, was thrilled with the stream running through it and the gardens filled with flowers and blossom.\n\nShe, of course, was not allowed to enter the Casino, but she was well aware how important it was in their lives.\n\nThe first night her father had won and they were all laughing and happy the next morning.\n\nHe took her mother and her to the races. He won again and Yolanda spent an exciting time with her parents before she was sent to bed.\n\nThe next morning was totally different.\n\nHer father had apparently had a disastrous evening at the Casino and her mother was trying to console him by saying that things would be better today.\n\nUnfortunately things were not better that day or the next and by the third day her father was completely broke.\n\nThey were sitting out in the garden of the Stephanie Hotel wondering what they should do next.\n\nHow, her father asked over and over again, could he pay for the rooms in the hotel?\n\nLord Milborne then passed by and greeted them.\n\nHe was a very good-looking man, Yolanda thought, perhaps a little older than her father.\n\nIt seemed they knew each other well \u2013 they both belonged to White's Club in London.\n\nLord Milborne exclaimed when he saw the Earl and his family,\n\n\"I was looking for you, Longwood. I hear you had a bad day yesterday! If you are intending to go to the races this afternoon I have a winner for you!\"\n\nHe rose and walked towards Lord Milborne.\n\n\"I have a sad story to tell you,\" he began.\n\n\"I believe that I can guess what it is,\" replied Lord Milborne.\n\nAs he spoke, the two men turned and walked away along the side of the stream.\n\nAs they did so, Yolanda's mother put out her hand and laid it on her daughter's.\n\n\"I am praying,\" she whispered, \"that Lord Milborne will help Papa. Otherwise, dearest we are in a terrible fix.\"\n\nAn hour passed before Yolanda saw that her father was coming back to them.\n\nBy the way he walked and the smile on his face she knew that all was well.\n\nHe came to where they were sitting and bent down to kiss her mother before he said,\n\n\"Everything is all right, my darling. Your prayers are answered and we will not be thrown out of the hotel or be forced to walk home.\"\n\n\"Lord Milborne has helped you?\"\n\n\"He has been very generous and what is more, he has given me the winner of the big race this afternoon.\"\n\nThe horse had indeed won and they then spent two very happy weeks in Baden-Baden before moving on.\n\nIt was Lord Milborne who had rescued them, but Yolanda had not thought of him since.\n\nYet she realised now that she could not allow her stepfather to outwit him as she suspected he wanted to.\n\nShe had heard several people talking at the dinner table about the new automatic air brakes. Everyone thought they would prevent the accidents, which kept occurring on the railways.\n\nThe Stockton and Darlington railway was designed in 1825 to carry coal from the mines to the waterways and it had proved immensely successful.\n\nIn the following years England was covered with a vast railway network and soon other countries in Europe began to build their own railway systems.\n\nThere were many unfortunate accidents because it was easy to start a train, but difficult to stop it.\n\nSeveral men working on manual handbrakes had to be co-ordinated, but if this procedure failed, a dangerous accident was likely.\n\nNow there was talk about a Mr. Westinghouse in America who had developed an automatic air brake, which was being adopted by a number of American railways.\n\nYolanda realised from the look on her stepfather's face and the many questions he asked when the air brake was being discussed that he was interested in the product.\n\nAnd it meant that he would want to buy it before anyone else in England could.\n\nYolanda was not concerned \u2013 it did not matter to her from whom he obtained the air brakes.\n\nBut she was aware that she could not permit him to trick Lord Milborne, who had saved her father, her mother and herself from disaster.\n\n'I have no wish to hurt Step-papa,' she determined. 'But at the same time he has so much. Perhaps Lord Milborne needs a helping hand now, just as Papa needed one all those years ago in Baden-Baden.'\n\n*\n\nThat afternoon she had arranged for some fittings in Bond Street and afterwards she asked the coachman to take her to Grosvenor Square.\n\nShe believed that her stepfather's secretary would keep a list of the people he knew and where they lived.\n\nSo she looked at the book on his desk while he was at luncheon and found, just as she had expected, Lord Milborne's address listed \u2013 Milborne House, Number 93 Grosvenor Square.\n\nShe and the maid who accompanied her drove there from the shop in Bond Street.\n\nYolanda felt that she was being somewhat disloyal to her stepfather, but at the same time it was honourable to try to pay back some of the huge debt her father owed Lord Milborne.\n\nGrosvenor Square was indeed the largest and most impressive square in London and yet to Yolanda it seemed somehow overpowering.\n\nWhen the carriage drew up by an important-looking house, it was with difficulty that she prevented herself from instructing the coachman to drive on.\n\nThen she remembered the smile on her father's face when he had come back from his walk with Lord Milborne in Baden-Baden, and how he had bent to kiss her mother before he told her the good news.\n\n'I am sure,' she pondered, 'this is what Papa would want me to do.'\n\nThe first footman on the carriage rang the bell of Number 93, the butler opened the front door and Yolanda stepped out.\n\n\"I would like to see Lord Milborne,\" she asked.\n\n\"Is his Lordship expecting you, madam?\" the butler enquired.\n\n\"No,\" she replied. \"But will you please inform him that Lady Yolanda Wood wishes to speak with him.\"\n\nThe butler was obviously impressed by her title, so instead of asking her to wait, he took her across a high-ceilinged hall and opened a door on the other side of it.\n\nHe announced Yolanda in a stentorian voice as she entered what she guessed was a study or a library.\n\nThere were many books everywhere, a fine painting by Stubbs hung over the mantelpiece and two big French windows opened on to a walled garden at the back.\n\nFor a second Yolanda thought there was no one in the room and then a man rose from behind a writing desk.\n\nThe butler closed the door behind her.\n\nAs the man walked towards her, Yolanda stared at him in surprise.\n\nHe was not the Lord Milborne she remembered, but a young gentleman, tall and good-looking.\n\nHe reached her and held out his hand.\n\n\"I am delighted to meet you, Lady Yolanda. I have heard a great deal about you.\"\n\n\"But you are not Lord Milborne!\"\n\nThe young gentleman smiled.\n\n\"You were expecting to meet my father who I know was a friend of yours. I am afraid, however, that he died two years ago.\"\n\n\"Oh, I am so sorry!\" exclaimed Yolanda. \"I met him a long time ago in Baden-Baden, when he was so very kind to my father and my mother and we were all extremely grateful to him.\"\n\n\"That sounds so like my father,\" he replied. \"And, as you have been kind enough to come to see me, I expect you know that I am staying with you this weekend to see the new pictures your stepfather has just bought.\"\n\n\"Yes, he has told me about them. Though I have yet to see them, I am sure that they are magnificent.\"\n\n\"I have a few Stubbs myself and I am very proud of them. I only wish I had known the others were for sale. I would have obtained them before your stepfather managed to do so!\"\n\nBefore Yolanda could reply, he asked,\n\n\"Can I offer you any refreshment?\"\n\n\"No, thank you, my Lord. I came to say something to your father without letting my stepfather know what I was doing. I am not sure now what I should do \u2013 \"\n\nShe sounded indecisive and Lord Milborne said,\n\n\"I can understand your surprise at finding my father is no longer here. But please allow me to take his place. If there is any way I can help you or answer any questions you would have put to him, I am at your service.\"\n\nYolanda seated herself on a sofa by the fireplace and Lord Milborne stood with his back to it.\n\nThen, as if he felt it would seem friendlier, he sat down on the sofa beside her.\n\n\"What is worrying you, Lady Yolanda? Everyone went to my father with their troubles, so I have a feeling that is why you have come to me.\"\n\nYolanda drew in her breath.\n\n\"Actually, my Lord, you may think it surprising, but I want to repay in a small way the wonderful kindness your father showed us when we were in Baden-Baden.\"\n\n\"What did he do?\"\n\nYolanda looked away from him and a little shyly she answered,\n\n\"I expect you know that my father was a gambler?\"\n\nLord Milborne smiled at her.\n\n\"I think everyone knew he was the most charming and the most delightful gambler there has ever been. The story of his life really ought to be in a book.\"\n\nYolanda gave a little laugh.\n\n\"I have never thought of that idea before.\"\n\n\"It is something you should consider. I am certain that you could write it better than anyone else.\"\n\nYolanda did not reply and he continued,\n\n\"I met your father several times when he was riding in the Park, at many Racecourses and when I first proudly became a Member of White's. I thought he was the most charming, the most handsome and the most delightful man I had ever met.\"\n\nYolanda's eyes now lit up as she clasped her hands together.\n\n\"Thank you for saying that. It is the way I want people to remember my Papa \u2013 not to remember him when things went wrong.\"\n\nShe hesitated for a moment and then she added,\n\n\"It was when things went so very wrong in Baden-Baden that your father saved Papa at a time when we were desperate. He not only lent him money, which I am not sure was ever repaid but he gave Papa the winner of the big race which he attended that afternoon.\"\n\n\"That sounds very much like him. Of course I am very proud of him and I am trying to follow his example. So please allow me to take his place and tell me why you are here.\"\n\nIt was difficult for Yolanda to find the right words.\n\n\"I expect you know how rich my stepfather is and that he has made every penny of his fortune himself.\"\n\n\"I have heard the stories and how can anyone not admire him for what he has achieved?\"\n\n\"He has been so very kind to me and I could never be grateful enough for what he did for Mama and me after Papa died.\"\n\nAgain there was a pause before she went on,\n\n\"I know in business men do not always behave in the same manner as your father and my father would have behaved.\"\n\n\"I understand what you are trying to say to me.\"\n\n\"You may think,\" said Yolanda, \"that I am being very disloyal, but because your father was so kind when we were desperate, I just wanted to warn him when he came to stay that my stepfather is most interested in the automatic air brake invented by Mr. Westinghouse and which is now being used so successfully in America.\"\n\nWhat she was saying came hesitatingly to an end.\n\nShe looked up appealingly at Lord Milborne as if she was begging him to understand what she was trying to say.\n\nHe smiled at her appreciatively.\n\n\"I think it is very brave of you, Lady Yolanda, to come and tell my father what you have just outlined. I do understand perfectly your reason for doing so.\"\n\nYolanda gave a sigh of relief.\n\n\"My stepfather, because he is so rich and so astute, invariably gets what he wants, and he has heard that you too are interested in the air brakes.\"\n\n\"I am indeed and I am grateful to you for thinking of my father and now myself in that connection.\"\n\nYolanda rose to her feet.\n\n\"Please, my Lord \u2013 you will not tell my stepfather I have been here?\"\n\n\"No, of course not. It is a secret between you and me and I thank you for thinking of my father. I know that he would have been so grateful and appreciative at what you have been brave enough to do.\"\n\n\"Then we shall meet on Friday evening, my Lord, and I think we will both enjoy seeing Step-papa's Stubbs.\"\n\n\"You can be quite certain,\" added Lord Milborne, \"I will be grinding my teeth with envy because he beat me to the sale. I have been abroad, which is the reason why I was not aware it was taking place.\"\n\n\"I feel sure you will have better luck next time.\"\n\nShe held out her hand as Lord Milborne asked,\n\n\"I suppose you don't have time to see a Stubbs I have in another room? I also have a very beautiful Titian in which, as it depicts the Goddess Venus, you should be particularly interested.\"\n\nYolanda realised he was paying her a compliment.\n\n\"I would love to see it if you have the time.\"\n\nLord Milborne escorted her into the drawing room, which was beautifully furnished and Yolanda thought that it had an atmosphere of having been lived in.\n\nThe superb Titian painting of 'Venus and Adonis' was hanging on one wall.\n\nAnd there was the particularly fine Stubbs over the mantelpiece \u2013 however in his picture there were no horses, but two cheetahs he had painted brilliantly.\n\n\"Your pictures are very lovely,\" Yolanda sighed as she walked from one to the other. \"How lucky you are to own anything so perfect.\"\n\n\"That is just what I think myself,\" Lord Milborne replied, \"and it is why I want to add to my collection. I have other pictures, which I would like you to see, but they are in my country house. Perhaps you will come and stay there one day?\"\n\n\"I would love to,\" enthused Yolanda. \"I remember Mama telling me how lovely your house is after she had been there with Papa.\"\n\n\"I think they stayed. I don't recall meeting your mother, but I have heard how beautiful she was and that you are very like her.\"\n\nHis dark brown eyes gazed at her tenderly.\n\n\"I only hope that is true.\"\n\nThey walked out of the drawing room and down the stairs and Yolanda admired a Hogarth in the hall.\n\n\"I don't care much for that picture,\" Lord Milborne told her. \"But the one I would like you to see is 'The Holy Family' painted by Rubens, which I think is one of the best pictures he ever did.\"\n\nBy now they had reached the front door and one of the footmen was opening it.\n\nLord Milborne took Yolanda across the pavement and when they reached the carriage, she turned to him,\n\n\"Thank you, my Lord, you have been very kind.\"\n\n\"That is just what I should be saying to you, Lady Yolanda, and I will say it to you again on Friday.\"\n\nHe saw her glance at him quickly, so he added in a voice that the footman holding the door could not hear \u2013\n\n\"When we are alone.\"\n\nYolanda smiled and climbed into the carriage.\n\nLord Milborne did not go back into his house until she had driven away.\n\nAs the carriage drove into Park Lane, Yolanda felt a little guilty and yet she knew she had done what her father would think was the right thing.\n\nLord Milborne's father had saved them from being humiliated at being unable to pay their bill and her father might have had to face Police charges, which would have caused a dreadful scandal.\n\n'There was nothing else I could do but warn Lord Milborne,' mused Yolanda quietly to herself.\n\nAt the same time she felt rather apprehensive.\n\nHer stepfather might ask what she had been doing besides shopping and she had no wish to lie to him.\n\nIt would, however, be most embarrassing to admit that she had been to visit Milborne House.\n\nFortunately they were dining out that evening.\n\nTherefore Yolanda did not meet her stepfather until she came downstairs wearing a new and attractive gown.\n\nIt was one of her dresses that he had not yet seen, so she took off the wrap and showed him how pretty it was.\n\n\"You look lovely, my dearest,\" he said, \"and I am very proud to be your escort tonight. Incidentally I expect you will remember that we are dining with Lord Stratford de Redcliffe in Grosvenor Square?\"\n\nYolanda drew in her breath.\n\nShe knew that they were dining with some friends of her stepfather's and there was to be dancing afterwards, but she had no idea it was to be in Grosvenor Square.\n\nShe only hoped that their host had not recognised the carriage if he had seen it outside Milborne House.\n\nIn which case it would be very difficult for her to explain why she had been there.\n\nThey were driven to Grosvenor Square.\n\nYolanda then saw that Lord Stratford de Redcliffe's grand house was situated on the opposite side of the square to Lord Milborne's.\n\nIt was quite a large party with over thirty people for dinner and several others joining them afterwards.\n\nThe ballroom was fairly small and, as it was a hot evening, the dancers walked in the square between dances.\n\nIt was cool amongst the trees.\n\nThere was a full moon overhead and it was all very romantic.\n\nYolanda was not really surprised when she received her second proposal of marriage \u2013 from a young gentleman she had met half-a-dozen times at different parties.\n\nShe danced with him and thought him reasonably pleasant, but he did not have anything of particular interest to say.\n\nOnce again she felt that she would have been more amused to listen to her stepfather and several of the older men including her host.\n\nThey were talking animatedly at the opposite end of the ballroom to where the band was playing.\n\nWhen she received her second proposal, she was not frightened.\n\n\"It is very kind of you,\" she said demurely, \"and I am very honoured that you should ask me to be your wife. But quite frankly I have no intention of being married at the moment.\"\n\n\"I am not surprised when you are such a success,\" the young man told her. \"But please will you think of me? I will wait for you even if it means waiting for years.\"\n\n\"I am sure that you would find that rather boring,\" she replied. \"But, of course, we can be friends, can we not?\"\n\n\"I do suppose,\" he said with a wry smile, \"I must be thankful for some small mercies. But because you are so beautiful, I want a great deal more than friendship. I will go on hoping and perhaps by a miracle you will fall in love with me one day!\"\n\nYolanda felt it embarrassing to go on talking about something she had no intention of doing.\n\nSo she suggested that they joined in 'The Lancers', a dance in which it was impossible to speak intimately.\n\nWhen 'The Lancers' finished, she realised with a sense of relief that her stepfather was ready to go home.\n\nDriving back in the carriage she questioned him,\n\n\"What were you talking about so earnestly to those gentlemen while we were dancing?\"\n\nHer Stepfather laughed.\n\n\"I think you know the answer to that.\"\n\n\"I suppose it was business.\"\n\n\"But of course,\" he answered. \"As you know it is the one subject I am really knowledgeable about.\"\n\n\"And on many other things too, including pictures. So what was it tonight?\"\n\n\"Need you ask, Yolanda? A report from America regarding the automatic air brake is making people who are concerned with or who have invested in our railways aware that they are rapidly becoming out of date.\"\n\nBy the time he finished speaking, they were back home, so Yolanda did not say any more.\n\nShe kissed him goodnight and walked up to bed.\n\nShe was wondering if once again he would win and somehow grasp the automatic air brake invention into his clutches before anyone else could.\n\nIn which case Lord Milborne would perhaps not be able to afford to buy any more pictures!\n\nIf that was to be the case, she felt it was rather sad, as she sensed how much they meant to him.\n\n'At least I have warned him,' she thought. 'Surely he is sensible enough to realise it is a question of getting there first?'\n\nThat, she knew, was what her stepfather always did. He worked as quickly as his brain \u2013 which was very quick indeed.\n\nShe gave a little sigh and for some strange reason that she could not quite fathom she really wanted Lord Milborne to win the air brake race with her stepfather.\n\nShe somehow thought that he was so like her father \u2013 he would have thought it so unfair and ungentlemanly to push people out of the way.\n\nEspecially to grab the air brake before anyone else had a chance.\n\n'I have really done my best, I cannot do any more,' Yolanda decided a little ruefully before she fell asleep.\n\n*\n\nThe next day followed the same routine \u2013\n\nA luncheon party, an afternoon reception, a dinner party and a dance. It seemed to Yolanda a repetition of what had happened a dozen times already.\n\nThen she felt a bit ashamed of herself for not being more enthusiastic.\n\n'What I really want now more than anything,' she decided, 'is to return to the country.'\n\nShe had ridden Chestnut practically every morning in Rotten Row, but it was not the same as riding over the fields or jumping in the paddock.\n\nWhen Friday morning finally came, she felt as excited as a child leaving school \u2013 they were going back to the country.\n\nTo her delight her stepfather was driving a team of four well-matched black horses.\n\nShe knew that the horses had been bought recently at Tattersall's, the renowned auctioneer in Knightsbridge, so were likely to be exceedingly fast.\n\nSeeing him control the huge, proud-looking horses, she realised that her stepfather was a good driver, as well as a very good rider, both of which rather surprised her.\n\nAfter all, when he was a young man, his family had not been able to afford horses, whilst her father had ridden, as he had often boasted, since he could crawl.\n\n\"You drive very well, Step-papa,\" she blurted out.\n\n\"I hear surprise in your voice,\" he replied, \"which I don't think is very complimentary.\"\n\n\"I am sorry if I sounded rude,\" answered Yolanda, \"but actually I was wondering how you came to ride so well and learnt to drive as well as any of your coachmen?\"\n\n\"Thank you, Yolanda \u2013 now that is a compliment! And as you are so curious, I will tell you about it. As soon as I could afford it, I had myself taught by the very best drivers and riders. I then devoted many hours to learning and practised as often as I could. \"\n\n\"That was indeed clever of you!\"\n\n\"I have told you before that if you want something you have to fight for it. And I had no intention of being heavy in the saddle or having clumsy fingers with the reins. It was another lesson and one which has certainly paid off.\"\n\n\"Of course it did and I am enjoying driving with you more than I can say.\"\n\n\"I feel you are dying to get back to the country.\"\n\n\"Of course I am! It will be lovely to be home, to see the garden and find out what is new in the stables.\"\n\nHer stepfather smiled at her.\n\n\"We must not forget that Lord Milborne is coming to view my pictures. He is arriving this afternoon before dinner, but I suggest we show him nothing until tomorrow.\"\n\n\"He must not interfere with our riding. I suppose he will want to ride too?\"\n\n\"I shall be very surprised if he does not. Incidentally if you want an outstanding rider, then he is one.\"\n\n\"Oh, is he?\" Yolanda said, the surprise apparent on her face. \"I thought he was more of a City gentleman?\"\n\n\"On the contrary,\" her stepfather remarked, \"he has won quite a number of races riding his own horses and he is Master of his own pack of foxhounds.\"\n\nYolanda raised her eyebrows, both surprised and impressed. Lord Milborne had certainly seemed a Londoner when she had been talking to him in his house in Grosvenor Square.\n\nNow it seemed rather strange that he should be so countrified.\n\n\"I believe that his foxhounds are outstanding,\" her stepfather was saying. \"And I couldn't help wondering if I should have a pack of my own.\"\n\n\"That would be certainly something new!\"\n\n\"We will have to think about it. There is plenty of time before the autumn.\"\n\nThey drove on and by now there were only a few houses and the undulating land was broken by clumps of trees with their leaves shining in the sunshine.\n\nThey were very near to her stepfather's house when Yolanda enquired,\n\n\"By the way, Step-papa, have you invited anyone else to dine tomorrow night, or are we having Lord Milborne all to ourselves for two dinners?\"\n\n\"Of course some other people are coming to dinner on Saturday night and I thought very hard about who to ask. I got the feeling that you were a little bored with dancing, after all the excitement of the parties in London, so I chose people who I thought could offer us a little intelligent conversation.\"\n\n\"It is something I would much prefer,\" she smiled happily.\n\n\"I thought too that as Alan Milborne is interested in the opera, we might have some music.\"\n\n\"A band? I thought you just said we were not going to dance?\"\n\n\"No, no \u2013 I have asked a man who is an extremely good pianist to play for us after dinner. We can talk at the same time so that it does not become an isolated entertainment. I thought perhaps you would play for us too.\"\n\nYolanda's eyes widened.\n\n\"I have read your school reports and your music teacher said that you were an excellent pupil. Besides passing a series of examinations in music, you also sang in the school choir?\"\n\n\"Are you expecting me to sing tomorrow night?\"\n\n\"Why not, Yolanda? I am contributing my part to the evening and I think you should contribute too.\"\n\n\"I am horrified at the idea. Also, I have not had an opportunity to practice or think of something to sing.\"\n\n\"Sing them something really delightful and not too heavy,\" he advised. \"When people have had a lot to eat and drink, they want to feel glamorous or even romantic.\"\n\n\"I just hope that is really what your guests would like, otherwise they might feel a little uncomfortable.\"\n\n\"Trust me, Yolanda, they are mostly married couples. I am told that Alan Milborne is pursued by all the beautiful women in Mayfair, so he will not be looking for them in the depths of the country.\"\n\nIt suddenly struck Yolanda for the first time that Lord Milborne was indeed an extremely good-looking man who could not be much older than twenty-seven.\n\nOf course all the ambitious mothers would pursue him with their debutante daughters. She had seen them in action at every party she had attended.\n\nThey sat in small clusters, scrutinising all the young gentlemen and contriving to make their daughters dance with someone who had a title \u2013 or else with someone who was very rich.\n\nYolanda realised that up until then she had been thinking of Lord Milborne in the same way as she thought of his father.\n\nShe had not really thought of him as an attractive young man at all.\n\nNow she knew that she had been very silly.\n\nOf course, there was no reason to think that he was particularly interested in machinery, the railways or for that matter automatic air brakes.\n\nIt had really been quite unnecessary for her to go to Grosvenor Square.\n\nWhen she had realised that the Lord Milborne she knew was not there, she should have gone away at once.\n\nThe new Lord Milborne had been most charming and polite to her, but he had certainly not said that he was hurrying off to America to bid for the air brake.\n\nShe could appreciate that pictures were more interesting to him, particularly if they were by Stubbs and depicted horses that he could ride better than anyone.\n\n'I have just made rather a fool of myself,' Yolanda considered.\n\nHowever, it did not matter at all as Lord Milborne was never likely to be interested in her.\n\nGranted he had paid her a compliment, but that was just good manners \u2013 it was the sort of thing any young man might say to a pretty young woman.\n\nAll the same Yolanda was still thinking about him when she walked slowly upstairs to dress for dinner.\n\nThere was as yet no sign of his arrival.\n\n'Perhaps he has received a better invitation,' she speculated.\n\nThen she told herself she rather wanted to see him again. He was certainly a great deal more interesting than the majority of gentlemen she had met in London.\n\nAs she glanced at herself in the mirror, she knew she was looking very attractive.\n\nIn fact it would not be too conceited to say \u2013 lovely.\n\n\"I wonder if I ought to wear some of my Mama's jewels with this gown?\" she asked Emily.\n\n\"I thinks it looks nice just as it is, my Lady,\" Emily replied. \"And you don't want to go shouting about having jewellery at this moment, it be dangerous.\"\n\nYolanda looked at her in surprise.\n\n\"What do you mean by that?\"\n\n\"They were telling me as soon as I arrived, there's been terrible trouble down here. Several houses have been broken into and one elderly lady, I don't know her name, got stopped on the road by three men with pistols. They took her pearls off her neck, her earrings and every penny her had in her handbag.\"\n\n\"I have never heard anything so awful!\" Yolanda exclaimed. \"Has the Master been told?\"\n\n\"I don't rightly know, my Lady, but they be talking about it in the kitchen and they says it be a gang of roughs. Nasty it sounds to me.\"\n\n\"And to me,\" agreed Yolanda.\n\nShe did not say anything aloud, but was hoping that her stepfather's pictures were safe.\n\nIt seemed to her more than likely.\n\nSurely ruffians, who were collecting jewellery and holding up women on a quiet country road, would not be interested in pictures by Great Masters?\n\nAll the same the two Stubbs her stepfather had just purchased were certainly valuable, even though he was a comparatively modern artist.\n\nOther artists in his large collection like Holbein and Raphael grew increasingly more valuable year by year.\n\n'I must warn Step-papa to be very careful,' Yolanda decided as she walked down the stairs.\n\nThen she saw four sturdy young footmen, all over six feet tall, standing in the hall and remembered that there were two night watchmen always on guard when they went to bed. Reassured, she pushed the possibility of danger to the back of her mind.\n\nAs she reached the bottom of the staircase, she was aware of someone coming down behind her.\n\nShe looked back and saw it was Lord Milborne.\n\nHe must have arrived just after she had gone up to dress.\n\nAnd now he was changed and looking smart and extremely handsome in his white tie and tails.\n\nShe waited until he joined her at the bottom of the stairs.\n\n\"You were so late in arriving, my Lord, I was afraid you might not be coming.\"\n\n\"How could I stay away when you have so much to show me?\" he asked.\n\n\"I hear, as well as the paintings, you are interested in the stables.\"\n\n\"There is one horse in particular that I want to see,\" Lord Milborne responded. \"It is one that your stepfather outbid me for at Tattersall's a few weeks ago. I was very angry with my Head Groom for allowing him to do so!\"\n\nYolanda laughed.\n\n\"I know which one it is,\" Yolanda informed him.\n\n\"Then please arrange that I can ride him tomorrow. If nothing else it will teach me a lesson that one must never be pipped at the post!\"\n\nYolanda laughed again.\n\n\"I am so very sorry for you, my Lord, but we have Champion. He is a wonderful horse and I enjoy riding him myself.\"\n\n\"In which case, I suppose I shall have to pretend that I no longer mind losing him.\"\n\n\"You can then tell me what you really feel after you have ridden him,\" replied Yolanda. \"I have a feeling that having done so, you will just want to ride away on him and disappear, so we can never find you or Champion again!\"\n\n\"That is certainly an idea \u2013 and a very good one at that!\"\n\nThey were both chuckling as they entered the drawing room.\n\n# CHAPTER SIX\n\nThe next morning Yolanda awoke and thought at once how exciting it would be to ride with Lord Milborne.\n\nShe had heard her stepfather giving the orders that the horses were to be brought round to the front of the house as soon as they had finished breakfast.\n\nShe dressed hurriedly without worrying about her appearance \u2013 she merely took a quick look to see if her hair was tidy before speeding downstairs.\n\nAs she entered the breakfast room, she realised that both her stepfather and Lord Milborne were already there.\n\nThey were laughing at something in the newspaper.\n\n\"Good morning. What is all the excitement about?\" she asked.\n\nThe two men rose as she walked towards them and she sat down.\n\n\"Your stepfather has hit the headlines,\" said Lord Milborne.\n\n\"In what way?\"\n\nLord Milborne looked at his host, who after a long moment, commented sourly,\n\n\"It is just infuriating how everything gets into the newspapers, however much one tries to prevent it.\"\n\n\"There is nothing to be ashamed of,\" soothed Lord Milborne. \"After all, they praised you to the skies.\"\n\n\"Far too much!\"\n\n\"Do please tell me what all this is about,\" begged Yolanda.\n\n\"When I heard about the two Stubbs for sale that Lord Milborne has come here to see,\" her stepfather began, \"the owner of the house had just died and the auctioneers were arranging the rooms for the sale.\"\n\n\"Putting, of course, the Stubbs in a prominent place in the order of sale,\" remarked Lord Milborne.\n\n\"Naturally they recognised that they were the most important objects to be sold.\"\n\n\"Then what happened?\" prompted Lord Milborne.\n\n\"I looked round and thought the rest of the things in the house were of no particular interest. Then I noticed a picture propped up against a wall. The glass was broken and the frame was in a very bad state.\"\n\n\"But you thought it might be interesting,\" Yolanda interposed.\n\nHer stepfather smiled at her.\n\n\"I think my instinct told me that, although it looked dilapidated, there was something remarkable about it.\"\n\n\"And there was?\"\n\n\"That is exactly what we have been reading,\" Lord Milborne chimed in. \"Your stepfather was clever enough to recognise a great masterpiece \u2013 even though no one else had and he bought it for a song.\"\n\n\"Seventy pounds to be exact,\" boasted Mr. Garrack.\n\n\"And what is it really worth?\" asked Yolanda.\n\n\"I would not like to guess,\" her stepfather replied, making a gesture with his hands.\n\n\"Who was the artist?\"\n\n\"It was 'St. George and the Dragon' by Raphael,\" Lord Milborne answered her. \"It was only recognised after your stepfather had bought it by one of the representatives of Sotheby's who had come from London for the sale.\"\n\nYolanda laughed.\n\n\"He must have been furious that he had not bought it himself!\"\n\n\"Of course he was,\" answered Lord Milborne, \"and what was so brilliant of your stepfather was that he did not go to the sale himself. Otherwise if people had known that he was going to bid for the picture, undoubtedly the price would have gone up.\"\n\nYolanda looked at him and saw that his eyes were twinkling.\n\nShe knew that was what had been on his mind as he had sent his secretary to the sale rather than go himself.\n\n\"You are very clever, Step-papa, and I am longing to see the picture as I know Lord Milborne must be too.\"\n\n\"You shall see all three of my purchases when we come back from our ride. Now come on, we must not keep the horses waiting.\"\n\n\"No, of course not,\" agreed Lord Milborne.\n\nThen they were all laughing again, and it seemed to Yolanda that their laughter set the tone of the whole day.\n\nThe moment she saw Lord Milborne in the saddle and taking the jumps in the paddock, she realised what an exceptional rider he was.\n\nHe handled his mount in the same competent way her father had always done.\n\nFrom the paddock they galloped over the flat land and covered most of the estate before they turned for home.\n\nLord Milborne was riding on Champion and he was thrilled with his mount.\n\nHe told its owner that he was too lucky.\n\n\"Everything you touch just seems to turn to gold, Oliver, and it is exactly what the newspaper was reporting today!\"\n\n\"The last thing I want is for people to be told I am rich,\" replied Mr. Garrack. \"It means a mass of begging letters all telling me, better than I know myself, how they could spend my millions!\"\n\nHe gave a quick laugh before he added,\n\n\"However much one gives them, I always feel they think it should have been more.\"\n\n\"You can be quite certain of that. You must be so careful of begging letters. I am told that there are people who make a considerable income by writing pathetic letters to everyone who is described in the newspapers as being a millionaire.\"\n\nMr. Garrack sighed, but he did not say anything.\n\nAs they rode on they talked of other subjects.\n\nAfter luncheon Lord Milborne was taken to see the pictures and he was as thrilled with the Stubbs as Yolanda.\n\n\"They are so lovely, Step-papa, and now you have such an outstanding collection of them you must be careful they are not stolen.\"\n\n\"I assure you that I take every precaution.\"\n\nYolanda remembered that she had not yet told him about the gang that was roaming the County and causing such trouble.\n\nShe was not unduly worried \u2013 they sounded so rough that she thought it unlikely they would be interested in pictures.\n\nSnapping out of her unpleasant thoughts she could hear her stepfather talking to Lord Milborne about the Raphael, agreeing how different it would look when it was cleaned and beautifully framed.\n\nYolanda now recalled that Raphael, before he died in 1520, had painted many religious pictures in Florence and Rome and she felt certain that this particular one was unique.\n\nWhat the exceptional painting depicted would make it extremely interesting to all collectors.\n\nShe slipped her arm into her stepfather's.\n\n\"It was so astute of you, Step-papa, to recognise a picture in that state. But I am sure you are right, when it is cleaned and properly framed it will look magnificent in the drawing room.\"\n\n\"That is exactly where I was thinking of putting it, Yolanda. I know if your mother had been here, she would have been thrilled to have it in her room.\"\n\nThey went riding again for a short time later in the afternoon and when they finally came back, Yolanda asked,\n\n\"Is there a dinner party tonight or are we alone?\"\n\n\"I am glad to say that we are alone,\" her stepfather replied. \"And I only hope our distinguished guest will not find it dull, although the big dinner party I have planned is for tomorrow night.\"\n\n\"On the contrary, I am delighted that I shall have a chance of talking seriously on a subject that I know we are both interested in,\" remarked Lord Milborne.\n\nYolanda's stepfather did not ask what this was, and she couldn't help wondering as they walked upstairs to change for dinner what that particular subject might be.\n\nApart from pictures what could be of interest to two men who were both so knowledgeable, but in very different ways?\n\nYolanda put on one of her prettiest gowns.\n\nShe sensed that there was a look of admiration in Lord Milborne's eyes as she came into the drawing room.\n\nThe chandelier had been lit, although the sun had not yet set and the whole room seemed to glitter with light.\n\nThe butler soon announced dinner and they moved into the dining room.\n\nAs usual the food was delicious and Lord Milborne commended the wines.\n\nWhen the coffee had been served, Yolanda thought she ought to leave the gentlemen.\n\nShe glanced at her stepfather and, as she did so, Lord Milborne said,\n\n\"If you are thinking of leaving us, Lady Yolanda, please do not. I would like you to stay and hear what I now have to say.\"\n\nAgain Yolanda wondered what it could be.\n\nShe thought her stepfather seemed rather surprised as Lord Milborne put down his glass.\n\n\"The reason that I was late arriving last night was because I had been to meet with Lord Granville who, as you will know, is the new Secretary of State for Foreign Affairs.\"\n\nMr. Garrack nodded sagely.\n\n\"Yes, and I think he will do very well. I have met him on several occasions.\"\n\n\"Lord Granville has asked me, and he has already discussed the issue with the Prime Minister, if I would help him by becoming the Under Secretary of State for Foreign Affairs.\"\n\nMr. Garrack looked startled.\n\n\"That is something I did not expect, but, of course, very many congratulations, Alan! I am sure that you will enjoy the position.\"\n\n\"I believe, as I have travelled so much in my life, I will come in useful. At least the Prime Minister and Lord Granville have agreed to the first proposal I have suggested to them since I was appointed.\"\n\n\"What is that?\" Mr. Garrack wanted to know.\n\nThere was a lengthy pause and Yolanda was quietly questioning what this all meant.\n\nWhy had he wanted her to stay and listen to what he had to say?\n\n\"I paid a private visit to America a few weeks ago,\" he resumed. \"But I took it upon myself to make a detailed investigation into George Westinghouse's invention of the automatic air brake that has already been adopted by most American railways.\"\n\nYolanda was astonished at what was being said.\n\nShe sat there listening with wide eyes.\n\nShe could not help but wonder why Lord Milborne had brought up this particular subject.\n\n\"When I returned,\" he went on, \"I spoke to several of those who are financing the English railways and who seem to be reluctant to start producing the air brake here in England when they could buy it direct from America.\"\n\n\"I think Westinghouse,\" added Mr. Garrack slowly, \"has coped with all the teething difficulties and it would be much better for the English to acquire the air brake, as it is working now, than to start again from scratch.\"\n\n\"That is exactly what I had thought. I am therefore asking you, Oliver, if you will, to negotiate on behalf of the British Government with Westinghouse for his air brake.\"\n\nFor a moment there was complete silence.\n\nThen Mr. Garrack sighed,\n\n\"Such an idea never crossed my mind.\"\n\n\"I think,\" smiled Lord Milborne, \"that you would be a far better negotiator than I am. In fact, you are noted for the brilliance by which you have negotiated for yourself in the past. That is exactly why we do feel no one could obtain a better deal for Great Britain than yourself.\"\n\nYolanda clasped her hands together.\n\n\"It is a wonderful idea, Step-papa!\" she exclaimed. \"Of course, Lord Milborne is right. If anyone could obtain the air brake at the right price it is you.\"\n\n\"I think you flatter me, Yolanda. At the same time I am naturally very honoured that you should have offered me such an interesting, if difficult, task and I can assure you I will do my best.\"\n\n\"So you accept?\"\n\n\"Unreservedly, Alan.\"\n\n\"The Prime Minister will be delighted and so will Lord Granville. And as it was all my idea, I feel that I shall deserve a little of the rapturous applause when you come back in triumph!\"\n\nMr. Garrack held up his hands.\n\n\"Let's not count our chickens before they are hatched. But as it so happens I have met George Westinghouse, and I have been aware for some time that he was experimenting with an invention that is so urgently needed not only in America but in the rest of the world.\"\n\nHe sighed before he continued,\n\n\"For the simple reason, with the enormous increase in railways, there are far too many collisions and accidents because of the inadequacy of the brakes.\"\n\n\"I agree with you, and I am quite certain that if we adopt the system here in Britain, Europe will follow us.\"\n\n\"Which is just what we would like to happen. Therefore the sooner we get moving the better!\"\n\nThey rose and moved into the drawing room in a very cheerful mood.\n\nYolanda realised that her stepfather was seriously delighted at being offered such an important post by the British Government.\n\nWhen she was ready to retire to bed, she kissed him affectionately.\n\n\"I know you are thinking of what you will be able do with the air brake, Step-papa, and I am sure that you will 'bring it off', as my dear Papa would have said, with all your usual brilliance.\"\n\n\"I would hate to disappoint you, Yolanda.\"\n\nShe held out her hand to Lord Milborne.\n\nThen, when she looked into his eyes, she knew he was saying something to her secretly.\n\nShe became aware, as if he was speaking the words, that he was saying 'thank you' to her.\n\nShe smiled at him.\n\nThen she felt a strange little quiver she could not explain run through her entire body.\n\nJust for a moment she did not take her hand away from his.\n\nAs if he could not help himself, Lord Milborne bent his head and kissed it.\n\nHer stepfather was not aware of this because he had gone to the drinks table to pour out a nightcap before he went to bed.\n\nJust for a moment Yolanda's fingers tightened on Lord Milborne's.\n\nThen, because she was feeling shy, she turned away and walked towards the door.\n\nUpstairs in her bedroom she told herself that if she had indeed put the idea into his mind, it was a wonderful thing to have happened.\n\nUndoubtedly her brilliant stepfather would make a huge success of the proposition ahead of him.\n\n'If he is working hard, it will make him happy,' she told herself. 'And he will not miss Mama as much as he does now.'\n\nBecause she was so excited, she found it difficult to go to sleep.\n\n*\n\nWhen she awoke it was very early in the morning and the sun was only just beginning to shine.\n\nShe felt she could not stay in bed any longer.\n\nSo pulling on her riding skirt and without taking a jacket she hurried down to the stables.\n\nThere was only one sleepy groom on duty and he looked at Yolanda in surprise because she was so early.\n\n\"I would like to ride on Chestnut,\" she said after a moment's hesitation as she was considering whether she should take out Champion.\n\nThen she thought that she should leave the beautiful stallion for Lord Milborne.\n\nIt did not take the stable boy very long to saddle up Chestnut and he opened the gate into the paddock for her.\n\nShe took Chestnut over only two of the jumps and then she galloped him speedily on the piece of flat land.\n\nThere was an isolated wood in the near distance \u2013 she sometimes went there when she wanted to think about a problem and did not wish to be disturbed.\n\nNow she wanted to think of what her stepfather was going to do in America and also how she could best thank Lord Milborne.\n\nHe had been so astute in making it so easy for her stepfather to agree to undertake what he wanted, and it had all happened without him in any way being involved in one of his more doubtful transactions.\n\nThe whole scenario was passing through her mind as she entered the wood.\n\nAs usual she felt thrilled by the beauty of the trees, the songs of the birds and the soft shuffle of the rabbits in the undergrowth.\n\nA mossy path led right through the wood and in its centre there was an ancient woodcutter's hut in a clearing.\n\nIt was made from the trunks of trees and Yolanda speculated that it must have been there for a hundred years.\n\nShe had never heard of her stepfather's woodcutters using it to sleep in, but they were likely to eat their midday meal there in bad weather.\n\nNow, as she approached the woodcutter's house, she saw the door was open and wondered if there was anyone inside.\n\nChestnut moved into the clearing in front of the hut.\n\nAs he did so, to Yolanda's surprise, two men came from behind the trees on either side of her.\n\nThey were both rough common-looking men with hard faces and she thought that they were very unlike any of the countrymen her stepfather employed on his estate.\n\nThe two men stared at her and then two more men came out of the wooden hut.\n\nThey were even more unpleasant-looking than the two she had already seen.\n\nIt suddenly struck her that maybe this was the gang that Emily had told her about who were causing so much trouble in the vicinity.\n\nShe thought it would be a mistake to talk to them or to linger any longer in the wood, so she pulled up the reins to thrust Chestnut forward.\n\nThen another man appeared directly on the path in front of her.\n\n\"You be the lady,\" he called out in a coarse voice, \"who be the daughter of the rich man at the big 'ouse.\"\n\nHe was standing directly in front of Chestnut, who was forced to come to a standstill.\n\nYolanda tried to turn him round so that she could ride back the way she had come.\n\n\"I am in a hurry \u2013 so please excuse me if I ride back to the house.\"\n\nThe two men who had been behind her took hold of Chestnut's saddle.\n\n\"Bring 'im along,\" ordered the man in front of her.\n\n\"Just what are you doing? Leave my horse alone,\" cried Yolanda. \"I have to return home immediately!\"\n\n\"Us weren't expectin' you this early. Now you're 'ere you're savin' us a lot o' time.\"\n\n\"I don't know what you mean,\" Yolanda screamed, \"and let go of my horse! This is my stepfather's land and you are trespassing on it!\"\n\n\"Us knows that,\" one of the men jeered roughly. \"'ow much do you think 'e'll pay us if us goes?\"\n\nThere was coarse laughter from the gang.\n\nYolanda was beginning to feel afraid.\n\nThen the two men holding Chestnut's bridle began to follow the ruffian who had spoken to her.\n\nHe was leading the way through the wood.\n\n\"I insist,\" shouted Yolanda, \"that you allow me to go home. If you want some food or money, I will arrange for you to have some.\"\n\nThe men walking ahead of her laughed again.\n\n\"Us'll 'ave your money. Your Papa 'as plenty o' it and 'e'll want 'is little gal back.\"\n\nThere was a sneering note in his voice, which made the other men snigger.\n\nNow Yolanda was feeling really frightened.\n\nShe realised that they were intending to kidnap her.\n\nIt was obviously something they had been planning for some time and she had walked straight into their trap.\n\nHow could she have guessed that the gang who had caused so much trouble would be lurking in the woods?\n\nShe realised that it would be no use arguing with them.\n\nThere was really no necessity for her to hold the reins as the two men were firmly leading Chestnut.\n\nShe therefore sat stoutly on her saddle wondering desperately how she could make any communication with her stepfather.\n\nThis part of the estate was not being cultivated at the moment so there would be little likelihood of anyone seeing her or the men who were with her.\n\nThey moved on in silence.\n\nThe ruffian who was obviously the leader walked in front, followed by the two men holding Chestnut's bridle and then three behind her, making six villains in all.\n\nYolanda thought she had little chance of escaping from them and the only sensible thing to do now was to go quietly with them with as much dignity as possible.\n\nShe was, however, well aware that it was not going to be easy for her.\n\nShe reckoned by now, although she did not have a watch, that it must be after eight o'clock.\n\nHer stepfather would now have been woken by his valet, as would Lord Milborne.\n\nWhen they went down to breakfast and she was not there, they would be told she had gone riding earlier.\n\nIt would not occur to them then that anything was wrong \u2013 not until very much later when she did not return.\n\n'What can I do? What can I do?' she asked herself desperately.\n\nShe glanced again at the two men behind her. They were most unpleasant and definitely not countrymen.\n\nShe was not certain as to where they came from, but judging by their accents they must have come from the worst slums.\n\nThey progressed further down the mossy path until the wood came to an end.\n\nJust inside the wood Yolanda could see the remains of what she knew had once been a slate mine.\n\nIn the past most large estates had mined their own slates to repair the roofs of cottages and barns and to employ local people.\n\nNowadays most of the old country slate mines had been closed down as it was so much cheaper to buy slates and tiles that were manufactured in their thousands in large factories in the North of England.\n\nShe had often seen what remained of the slate mine when she had been riding and she thought it looked rough and unsafe and should be filled in.\n\nNow the men walking ahead were coming to a halt in front of the mine.\n\nYolanda knew instinctively that that was where she was going to be taken a prisoner.\n\n\"Get down!\" the chief ruffian demanded.\n\nIt was an abrupt order, but for the moment Yolanda hesitated.\n\n\"What do you \u2013 want me to do?\" she asked.\n\n\"You're be goin' to write a letter to your rich Papa. Come on and don't waste too much time about it.\"\n\nYolanda dismounted and felt thankful that neither of the men holding Chestnut made any attempt to help her.\n\nThen, when the leader started to walk down into the mine, she hesitated.\n\n\"What about my horse?\" she asked him nervously.\n\nHe turned back.\n\n\"He be a-goin' to be our messenger \u2013 'e'll take the letter to your Papa and then 'e'll come to no 'arm.\"\n\nShe patted Chestnut as if to reassure him and then with a sinking heart she followed the leader down into the mine.\n\nThere were broken stones, fallen foliage and pieces of wood scattered about, as she passed into the mine itself.\n\nBecause there were so many holes in the roof it was not difficult to find the way.\n\nThere was a clearing where the ruffian stopped and he was now standing in front of a broken wooden chest.\n\nOn top of it was a bottle of cheap ink, a pen and a sheet of paper.\n\n\"Yer write what I tells yer,\" he commanded, \"and no tricks or yer'll be sorry.\"\n\nYolanda did not answer him \u2013 she merely picked up the pen.\n\n\"Tell yer rich Papa yer be our prisoner and us wants ten thousand pounds for yer.\"\n\nYolanda gave a gasp.\n\n\"He cannot pay all that money. Anyway he will not have it with him.\"\n\nThe chief ruffian gave a smirk.\n\n\"I knows that, but 'e's plenty o' time to go to the bank afore it gets dark and that be what 'e's a-goin' to pay us.\"\n\n\"Supposing he refuses?\" asserted Yolanda.\n\n\"Then yer can tell 'im us will kill yer or per'aps yer'll die o' starvation.\"\n\n\"I am sure he will come to find me.\"\n\nThe man laughed and it was not a pleasant sound.\n\n\"Who'd think o' lookin' for a pretty lady like yer in this old mine? They says in the village it ain't been used fer years and some thinks it be 'aunted so they won't come anywhere near it.\"\n\nYolanda thought that this was very likely true.\n\nAnyway why should anyone look for her in a slate mine that had lain empty for years and years?\n\nThere was not a thing she could do, she considered, but write down what the ruffian demanded.\n\nSlowly she dipped the quill pen into the ink.\n\n\"Now no tricks!\" he wheezed menacingly. \"I can read, so yer write just what I've told yer to say or yer'll be sorry.\"\n\nHe waved the stick he was holding in his hand.\n\nYolanda thought for a moment that he was about to hit her.\n\nShe gave a little cry and moved backwards.\n\n\"Just yer do now as what I says or it'll be the worse fer yer.\"\n\nBecause it was easier, Yolanda knelt down to write the letter.\n\nShe wrote,\n\n\"Dearest Step-papa,\n\nI have been captured and held prisoner by a gang that has been operating in the neighbourhood.\n\nThey are now demanding ten thousand pounds for my safe release.\"\n\nWhilst she was writing the ruffian had been looking over her shoulder.\n\n\"Tell 'im to put it under the gate at the end o' that there field where yer 'as the jumps.\"\n\n\"Oh, the paddock you mean?\"\n\n\"Yes. Us wants the money there as soon as it be dark and if anyone touches the man as collects it, yer'll die, yer tell 'im that!\"\n\nObediently, as there was nothing else she could do, Yolanda wrote down what he had just said.\n\n\"Now yer sign it with love and kisses,\" he crowed jeeringly.\n\n\"I am sorry that this should have happened,\" wrote Yolanda. \"Please save me, Step-papa, I am so terrified.\"\n\nShe signed her name and then she paused. She thought that the ruffian was about to take the paper from her, so she added quickly,\n\n\"Please feed Pizarra at six o'clock.\"\n\nThen she put down the pen and lent back.\n\nThe ruffian behind her picked up the paper.\n\n\"What be this 'ere that yer've written?\" he enquired suspiciously.\n\n\"It's about my dog. I always feed him every day at six o'clock and he will be hungry.\"\n\nHe grunted, but did not say anything, just walked away with the paper in his hands.\n\nYolanda looked around.\n\nShe realised there was nothing to sit on except the ground or the broken wooden chest, so she sat on the edge of it.\n\nThen she heard the gang talking up above.\n\nTo her surprise the chief ruffian came back.\n\n\"Have yer got a pin?\" he asked. \"Us 'as to pin yer note on the 'orse's saddle and if it ain't stuck firm yer rich Papa won't be a-gettin' it.\"\n\nYolanda had fastened her blouse at the neck with a small brooch, so she unfastened it and held it out to him.\n\nHe took it and looked closely at it to see if it was worth anything and then he disappeared again.\n\nShe thought that they must be taking Chestnut back to the paddock and turning him loose.\n\nWhen he was seen alone, someone was sure to think that there had been an accident.\n\nThey would undoubtedly think she had been thrown and they would catch Chestnut and then see the note.\n\nShe was wondering desperately what her stepfather and Lord Milborne would then do.\n\nIf they immediately sent for the Police, she thought there was every likelihood that the men would either kill or injure her to stop her from giving evidence against them.\n\nIt seemed impossible that her stepfather would give such a large sum of money to these desperadoes.\n\nBut what was the alternative?\n\nThe whole situation was desperately depressing.\n\n'What on earth can I do?' thought Yolanda again.\n\nThen, because she was really extremely frightened, she started to pray.\n\nAs she fervently prayed to God, she felt somehow that if her stepfather could not save her, Lord Milborne would think of a way of doing so.\n\nHow, she had no idea.\n\nShe only thought that there was indeed something about him that would resent giving in to blackmail, so by some miracle somehow he would save her.\n\nUp above the gang were talking amongst themselves.\n\nYolanda was aware that one of them was about to mount Chestnut.\n\nIt became obvious that the talking had degenerated into an argument and their voices became louder and louder as they quarrelled with each other.\n\nListening carefully, she could make out that they were arguing about who should take the horse back.\n\nFinally someone was chosen and she heard him ride off, whilst hoping and praying that he would not be cruel to Chestnut.\n\nShe was certain that by now her stepfather and Lord Milborne would have finished their breakfast.\n\nThey would be about to mount their horses whilst wondering where she was.\n\nThey might come to look for her and if they saw Chestnut being ridden by a stranger, they might force the villain to tell them where she was.\n\nThis idea, however, was soon shattered.\n\nSome time later the chief ruffian came down below.\n\n\"Yer 'orse be back in the paddock \u2013 \" he began.\n\n\"I hope that the man who rode him was gentle with him,\" insisted Yolanda.\n\n\"'E be all right \u2013 that man is the one us chooses when us 'as to steal an 'orse, so 'e knows if they be good or bad. It be a pity us can't keep that 'un.\"\n\nHe spoke to her in a mocking manner and Yolanda thought it was more dignified to ignore it.\n\nSo she merely remarked,\n\n\"I am glad you have taken Chestnut back safely.\"\n\n\"Now us be a-goin' to make sure that ye're safe. There be a nice little cell all ready for yer, if yer'll come this way.\"\n\nYolanda's heart sank into her boots, but there was nothing she could do.\n\nShe walked behind him further into the mine.\n\nThen, when she was afraid they might be going even deeper, he stopped.\n\nOn the left there was what looked like a small cave with a dim light coming through several holes in the roof, not large enough for anything but a rabbit to climb through.\n\n\"This 'ere be where yer can rest, lidy and the way us've just come'll be blocked and if yer were to go the other way yer'll fall down into the mine. So if yer 'urt yerself, don't blame me!\"\n\n\"I will stay here, but I would like something to sit on.\"\n\n\"After a sofa and chairs, be yer?\" he jeered. \"That be somethin' us 'aven't got. But yer can 'ave the bit o' old wood yer've been writin' on, if that be yer fancy.\"\n\n\"It will be better than nothing,\" she sighed.\n\nHe went and fetched the broken wooden chest.\n\nHe carried it to where Yolanda was standing and plumped it down in the middle of the cell.\n\nThen he said,\n\n\"Now no tricks. If yer tries to escape us'll catch yer and tie yer up. Yer'll find that real uncomfortable, yer can be sure o' that. If yer screams us'll gag yer \u2013 \"\n\n\"I shall be extremely quiet and stay here until I am rescued.\"\n\nThe ruffian laughed.\n\n\"Rescued! Yer'll stay 'ere till yer Papa's money comes, and when that 'appens, yer can walk 'ome. Be that clear?\n\n\"It be no use expectin' Bucking'am Palace, but if ye're good and give us no trouble, I'll give yer somethin' to eat when us 'as our meal.\"\n\n\"Thank you, I would be very grateful.\"\n\nThe man gave Yolanda an appraising look.\n\n\"I'll say one thing for yer. Yer've the sense not to make a fuss. Some women'd be screamin' and cryin' and a lot o' good it'd do 'em. But I tells yer again, one scream or one cry and yer'll be gagged!\"\n\nHe walked away and Yolanda settled down on the broken chest, which was just strong enough to hold her.\n\nShe thought it was better than sitting on the rough floor that was grey and dirty from slate chippings.\n\nShe wondered frantically if her stepfather or Lord Milborne could possibly understand the postscript she had written at the bottom of her letter.\n\nShe felt that they were both intelligent enough to do so, but at the same time she could not have put anything that was too obviously a clue to her whereabouts.\n\n'They will never think of the slate mine,' she told herself, 'unless one of them speaks Spanish.'\n\nTime seemed to pass very slowly.\n\nShe knew by the heat of the sun coming through the holes in the roof when it was midday.\n\nIt was a full half-an-hour later that the chief ruffian brought her some food.\n\nIt was not very appetising and she thought it must have been made up at a local inn where they had stayed or eaten.\n\nIt consisted of a tiny sandwich of thick bread, made from badly ground flour with a speck of tasteless ham that had doubtless been home-cured.\n\nWith the disgusting sandwich came a small bottle of the cheapest and, she decided, nastiest beer that could be purchased in the neighbourhood.\n\nThere was no question of the beer being poured into anything and someone had knocked the top off the bottle, so it was difficult to drink without cutting her lips and mouth.\n\nHowever, she was both hot and hungry so she drank the beer and ate some of the sandwich.\n\nNow the gang had come underground so that they would not be seen.\n\nThey were just a short distance away from her in the mine and she could hear them laughing and talking.\n\nIf she listened intently, she could hear every word they were saying and it was not particularly edifying.\n\nThey made jokes that she thought were exceedingly vulgar.\n\nIt was the chief ruffian who talked the most.\n\n\"I tells yer what us'll do, my boys,\" Yolanda could hear him saying, \"when us gets this 'ere money, us'll go abroad. Us 'ave 'ad all us'll get out o' this lot and it'll be a bit dangerous if they tells the Police and they be a-lookin' for us.\"\n\n\"Where'll us go?\" someone asked.\n\n\"Across the Channel. The food be good in France and the women more fun than this 'ere lot.\"\n\nThey all laughed at this remark.\n\nAnd then followed several lewd jokes that Yolanda did not care to listen to.\n\n*\n\nThe day wore on.\n\nShe thought despairingly that it would never end.\n\nShe wondered if her stepfather had gone to the bank to draw out the money.\n\nHe could afford it \u2013 of course he could!\n\nBut she considered it would be very humiliating if he should have to pay such a huge sum to keep her alive.\n\nNow the sun was beginning to set and she realised that it would soon be dusk.\n\nIf the money for her was to be paid, her stepfather would by now be taking it to the paddock.\n\nIt was a clever place for the gang to have chosen as the gate at the end of the paddock could not be seen from the stables.\n\nIf they wanted to arrest the man who had come for the money, there was nowhere anyone could hide.\n\n'Perhaps,' Yolanda pondered, 'they will just put it down and trust to their honesty to let me go free.'\n\nThen she was praying again \u2013 praying for help.\n\nNot only to her stepfather but also to Lord Milborne to come and save her.\n\nShe had the most terrifying feeling that if they did not pay the money at once, the gang really would kill her.\n\nEven to think of it made her shiver.\n\nShe had by now completely convinced herself that the chief ruffian was clearly nothing more than a hardened criminal who would stop at nothing.\n\nNow it was dusk and through one of the holes in the ceiling she could see the first evening star.\n\nIt was then suddenly, when all was quiet, that there came the bark and snarl of a dog.\n\nIt was a very noisy bark and its snarl turned into a yelp \u2013 the dog seemed to be barking loudly at something that was upsetting it.\n\nBecause it all sounded so strange, Yolanda went, as she had not done before, to the opening into her cave.\n\nShe looked along the passage towards the gang.\n\nShe could just see that they were standing up and she heard the chief ruffian yell,\n\n\"Two o' yer go at once and stop that damned dog. It'll attract attention.\"\n\n\"Shall us shoot it?\" one of the gang asked.\n\n\"Don't be a fool,\" he replied. \"A shot'll carry for miles. Just knock 'im off, so 'e don't make that row.\"\n\nEven as he spoke the dog yelped again.\n\nIts noise sounded almost shattering in the darkness outside.\n\n\"Now just 'urry up and stop 'im, yer fools.\"\n\nYolanda could hear two men scurrying up onto the ground above.\n\n# CHAPTER SEVEN\n\nYolanda listened, as she knew the gang beyond her were listening.\n\nThere was a shrill yelp from the dog \u2013 then silence.\n\nShe heard the chief ruffian guffaw,\n\n\"That's got 'im!\"\n\nA moment or two later the dog started again \u2013 it was yelping, growling fiercely and then yelping again.\n\nShe thought that the men must have been pursuing him.\n\nShe moved quietly even nearer to the gang.\n\nThe chief ruffian was bellowing,\n\n\"What the devil be they a-doin', the fools? Go up and 'elp 'em. That dog must be a-runnin' round in circles and they can't get near 'im.\"\n\n\"Us'll finish 'im orf,\" one of the gang exclaimed.\n\nYolanda could hear their footsteps as they climbed up above.\n\nThe dog gave another long yelp and then silence.\n\nShe hoped that the poor animal was not dead or in pain.\n\nShe was just about to walk back to her cell, when to her astonishment there came the same yelp, but this time it was a little louder.\n\nNext the dog was growling savagely as if someone was attacking him.\n\nPeeping around the corner, Yolanda could see the chef ruffian wildly waving his arms.\n\nShe reckoned, although she could not see his face, that he was very angry.\n\n\"Fools. Oafs! Idiots!\" he was screaming.\n\nThen came a flood of oaths.\n\n\"I'd better go up now and 'elp 'em,\" the only other gang member left volunteered.\n\nBut there was not much enthusiasm in his voice.\n\n\"I'll come with yer,\" the chief ruffian insisted. \"If us ain't that careful, us'll 'ave people comin' to see what's 'appenin'. Where's me stick?\"\n\nHe must have found it.\n\nYolanda heard him scampering away followed by the other man.\n\nIn order to hear better she moved to where they had been sitting.\n\nThere were the remains of their supper lying in a mess on the floor and there were several bags and boxes lying against one wall, which she guessed must contain the loot they had stolen.\n\nNow the dog above was snarling really ferociously and then he gave a loud yelp.\n\nAgain there was silence.\n\n'They really must have killed the poor animal this time,' Yolanda surmised.\n\nShe only hoped that if he was still breathing they would not bring him down below.\n\nShe looked round the mine wondering if, while they were absent, there was any chance of being able to escape.\n\nBut she knew it was impossible.\n\nIf she now climbed out of the mine onto the ground above, they would see her and then they would surely be extremely angry with her for trying to run away.\n\n'I just have to pray,' she decided, 'that Step-papa and Lord Milborne will have understood the clues I wrote in the letter.'\n\nShe was praying fervently when she heard footsteps coming down from above towards her.\n\nShe thought it must be the chief ruffian returning \u2013 he had obviously killed the dog, as there was no longer any sound of him.\n\nShe turned to go back to where she had come from.\n\nAs she did so she stopped.\n\nIt was difficult for her to see because the brambles that had grown over the entrance to the mine hid the light from the moon and stars.\n\nYet in the half-light Yolanda became aware that the man coming towards her was taller than any of the gang.\n\nThen, as she hesitated, a voice called out softly,\n\n\"Yolanda!\"\n\nShe gave a strangled cry that seemed to echo round the walls of the mine.\n\nIt was Lord Milborne who had called her name and she ran towards him.\n\nShe flung herself against him crying,\n\n\"You have come, you have come, you understood.\"\n\nHis arms went round her.\n\n\"They have not hurt you, my darling?\" he asked her tenderly.\n\nThen, as she looked up at him in surprise at such a delightful endearment, his lips were on hers.\n\nHe was kissing her wildly.\n\nYolanda felt at once that the whole wide world was suddenly bathed in sunshine and it was the most wonderful thing that had ever happened to her.\n\nAs he could feel the softness and innocence of her lips, Lord Milborne drew her closer and closer to him.\n\nAnd his kisses became more demanding and more passionate.\n\nAfter what seemed like a long time, he raised his head.\n\nYolanda was just able to breathe,\n\n\"It is you really \u2013 you,\" she managed to mumble.\n\n\"It is me,\" replied Lord Milborne. \"How could you have been so clever as to tell us where you were?\"\n\nThen before she could answer him, he was kissing her again.\n\nKissing her until she felt that she was melting into his body and they were not two people but one.\n\nFinally in a strange voice, which did not sound like his own, Lord Milborne murmured,\n\n\"I must take you home, my precious darling. I was so terrified in case those villains had hurt you in any way.\"\n\n\"What has happened to that dreadful gang?\"\n\n\"They are on their way to the Police wagon, which is waiting to take them all to prison.\"\n\nYolanda gave a gasp.\n\n\"It was so frightening \u2013 but somehow I knew that you would come and save me!\"\n\n\"It might have been a great deal more difficult if we had not known where you were.\"\n\n\"I thought as the men seemed so strange that one of them might be German or French and I only hoped that you knew a little Spanish.\"\n\nLord Milborne laughed.\n\n\"I was in Spain a year ago and managed to become quite fluent. But only you would have thought of telling us how many men there were!\"\n\n\"And only you and my Step-papa would have been able to decipher my code.\"\n\nWhen she mentioned her stepfather's name, she felt Lord Milborne give a little jerk \u2013 it was almost as if he had forgotten about him.\n\nHe kissed her very gently and affectionately.\n\n\"Come, my darling, let us go home. We have a lot of explaining to do in a much more comfortable place than this.\"\n\nAs he spoke he drew her towards the rough path up to the ground above.\n\nThere was no sign of the gang.\n\nYolanda looked all around her.\n\n\"Did they kill the dog?\"\n\nLord Milborne chuckled.\n\n\"I was the dog! Fortunately, as you can see, I am still alive and therefore, my darling, able to kiss you!\"\n\nHe kissed her again.\n\n\"Come along or else your stepfather will be angry with me for keeping you away from him for so long.\"\n\n\"Where is he?\" enquired Yolanda.\n\n\"In the trees with the horses we are going to ride home.\"\n\nYolanda slipped her hand into his and they walked back over the rough ground.\n\n\"How on earth did you learn to make such a strange noise?\" she asked him breathlessly. \"I was quite certain it was a real dog!\"\n\n\"I was taught the trick in India when I was serving with the brave men who work in 'The Great Game', which I expect you have heard about?\"\n\n\"Of course I have,\" answered Yolanda. \"It is the brilliant Secret Service we are using against the Russians.\"\n\n\"That is correct. They taught me many techniques, but this one has certainly been the most effective.\"\n\n\"What happened to the men when they came out? You did not kill them?\"\n\n\"I did not touch them. The moment they appeared two of the men I brought with me \u2013 the footman and the scullion, grabbed the first villain before he was aware of what was happening. They threw a blanket over his head so that he should not make a noise. Then they tied a rope around him and marched him down to the Police van.\"\n\n\"Then the dog attracted the next two!\"\n\n\"Exactly,\" agreed Lord Milborne. \"I am delighted it went off so well. I only wish I could tell my teachers in India how successful I was at masquerading as a dog!\"\n\n\"I think it was fantastic of you,\" enthused Yolanda. \"Only you and my Step-papa could be so clever as to cope with anything so dastardly and terrifying.\"\n\n\"I will never allow you to be so frightened again, my darling one,\" vowed Lord Milborne.\n\nHe stopped and looked at her in the moonlight.\n\nHe thought that no one could look lovelier or more desirable despite her shattering ordeal.\n\n\"You do love me?\" he asked her tentatively.\n\n\"I did not realise it was \u2013 love that I was feeling,\" whispered Yolanda, \"until you kissed me.\"\n\n\"Then?\"\n\n\"It was exactly what I prayed I would find one day and God has answered my prayers.\"\n\n\"As He has answered my prayers as well, Yolanda. I was determined never to marry until I loved someone as I now love you.\"\n\nHe kissed her again.\n\nIt was a long and passionate kiss. Then as if he was forcing himself to do so, he took her hand and started to walk on.\n\nWhen they reached the trees, Yolanda knew where they were going.\n\nOutside the wooden hut where she had encountered the gang, there were three horses and her stepfather.\n\nWhen he saw her, he held out his arms.\n\nYolanda ran towards him.\n\n\"I am safe, Step-papa,\" she cried. \"Safe, although it has been very terrifying. I am so happy you did not have to give them so much money.\"\n\nHer stepfather's arms tightened around her.\n\n\"I would give them everything I possess rather than lose you.\"\n\n\"Lord Milborne was so very clever, I really thought it was a dog barking.\"\n\n\"He told me exactly what he was intending to do. When I heard him instructing the men we had chosen to take with us, it was a scheme I had never heard of before.\"\n\n\"It was new to the gang and that was what really mattered. They thought the dog would attract attention and fortunately they did not shoot at him!\" exclaimed Yolanda.\n\n\"I am very thankful for that,\" added Lord Milborne.\n\n\"It was all your idea, Alan,\" said Mr. Garrack, \"and I am extremely grateful to you.\"\n\n\"I am afraid it is going to cost you something very expensive,\" responded Lord Milborne with a broad smile.\n\nMr. Garrack looked at him quizzically.\n\n\"I want to marry Yolanda as soon as possible,\" he explained. \"I find I cannot live without her.\"\n\n\"Marry Yolanda!\"\n\nThere was a note of surprise in his voice, as if the idea had not occurred to him.\n\n\"I love him, Step-papa,\" whispered Yolanda.\n\nHe looked at her with tenderness in his eyes.\n\n\"Then, of course, I am delighted and I am happy to give you my blessing. So let's go home and celebrate.\"\n\nLord Milborne lifted Yolanda onto Chestnut, who was nuzzling against her as if he was so pleased to see her.\n\nThen as Lord Milborne mounted on Champion, her stepfather jumped onto his horse saying,\n\n\"I suppose, Alan, I shall now have to present you with the horse you are riding and anyway it is a suitable reward for all you have achieved tonight.\"\n\n\"What we have achieved,\" Lord Milborne corrected him. \"And let us hope it is the first of many campaigns we can fight together!\"\n\n\"I will say 'amen' to that, Alan.\"\n\nThen he rode ahead down the mossy path, which led eventually to the paddock.\n\nYolanda followed and Lord Milborne came behind.\n\nWhen they were galloping side by side across the flat land, she thought how wonderful everything was now and she shuddered that it was only a short time ago that she was desperately praying to be rescued from the villainous gang.\n\nThey reached the house to find that the entire staff was waiting in the hall for their return.\n\nWhen Yolanda appeared, there came a cheer started by the butler and all those around him echoed it.\n\n\"You're safe, my Lady?\" he asked. \"Those devils haven't hurt you in any way?\"\n\n\"No, and I am now safe thanks to my stepfather and Lord Milborne, and I am so very happy to be back home.\"\n\n\"I'm sure that you're hungry, my Lady,\" suggested the cook.\n\n\"I am indeed.\"\n\n\"Dinner or rather supper, my Lady, is waiting,\" said the butler. \"I was quite certain your Ladyship would be hungry and the gentlemen wouldn't touch anything before they went to rescue you.\"\n\n\"Well, we need a large glass of champagne first,\" Mr. Garrack proposed, \"and it is just what everyone in the house is to have tonight.\"\n\nThere was another loud cheer from the staff.\n\nYolanda and Lord Milborne walked together into the drawing room.\n\n\"I suppose,\" she mused, \"I ought to go upstairs and tidy myself.\"\n\n\"You look too beautiful for words, my darling, and I thought that no one could be lovelier in the moonlight.\"\n\nHe was speaking in a low voice that she could only just hear.\n\nHe saw her blush at the depth and sincerity of his words and he was thinking that she looked like a Goddess of the moon.\n\n\"I love you beyond the entire universe, my dearest Yolanda, and I want to kiss you again.\"\n\n\"Now I am certain that I must go and make myself look pretty for you!\"\n\nYolanda ran away before he could stop her.\n\nEmily was waiting for her in her bedroom.\n\n\"I must tidy myself, Emily, before I have anything to eat.\"\n\n\"We was all in ever such a state, my Lady. It were terrible, really terrible.\"\n\n\"It's all over now,\" sighed Yolanda.\n\nShe was slipping off her riding skirt as she spoke and the blouse she had worn above it.\n\nIt only took a few minutes for her to wash and for Emily to help her into one of her prettiest evening gowns.\n\nShe tidied her hair quickly and ran downstairs.\n\nThe gentlemen were waiting for her in the drawing room.\n\n\"If you are going to be long drinking champagne,\" suggested her stepfather, \"you had better bring it into the dining room with you. We both admit to feeling extremely hungry.\"\n\n\"So am I,\" agreed Yolanda, \"although the men did give me a sandwich to eat.\"\n\n\"I suppose that was decent of them,\" came in Lord Milborne. \"They will get a very long sentence because the Police already have a whole list of robberies and acts of violence they have perpetrated.\"\n\nOver dinner Yolanda persuaded Lord Milborne to relate some other tricks he had learnt in India.\n\nWhat he had to say was so fascinating she thought that she could listen to him all night.\n\nHowever, when the servants had left the room, Lord Milborne said,\n\n\"Now please tell me how soon we can be married? Personally I would much prefer a quiet wedding here in the country, but of course, my darling, if you would wish the ceremony to take place in London, I will do whatever you want.\"\n\n\"I would just love to be married very quietly in the village Church where Mama is buried.\"\n\nShe saw the delight in Lord Milborne's eyes before he added,\n\n\"That is what I hoped you would say and as your stepfather needs to depart shortly for America, I suggest it would be a mistake for him to delay his visit more than a few days.\"\n\n\"I agree, Alan, the sooner I get on with the job the better. I don't suppose I shall be the only one in Europe bidding for the automatic air brake and the more competition there is, the more money we shall have to pay for it.\"\n\n\"Then I suggest, if you agree, that tomorrow I go to Windsor and inform the Queen of my engagement to Yolanda. Her Majesty is my Godmother and I know that she would be very offended if she was not the first to know that I have been 'caught' at last.\"\n\n\"I was not trying to catch you,\" asserted Yolanda.\n\nLord Milborne put out his hand towards her.\n\n\"I know, my darling, and that makes you different from anyone else I have ever met. Actually from the first moment I saw you, I was determined to catch you and not let anyone else have you!\"\n\nHe looked deeply into Yolanda's eyes.\n\nSomehow there was no need for words.\n\nYolanda knew that he was giving her his heart and her heart was already his.\n\n*\n\nYolanda and her stepfather agreed that they would make all the necessary arrangements for the wedding.\n\nIt was to take place very quietly the afternoon that Lord Milborne returned from Windsor Castle.\n\nHe had said that he was obliged to stay the night with the Queen and that would give them just two days to prepare everything.\n\nHe proposed they spent their first night together at home and left the following day for their honeymoon.\n\n\"I have a yacht in Dover Harbour and I thought it would take us down the Channel to Spain.\"\n\n\"To Spain!\" Mr. Garrack exclaimed somewhat in surprise. \"Why Spain?\"\n\nLord Milborne smiled.\n\n\"It was the Spanish word 'pizarra' that enabled us to save Yolanda from a horribly unpleasant experience. I know if we go to Madrid she will enjoy seeing the pictures in the Prada, some of which are as good, if not better, than anything we have at the moment in the National Gallery.\"\n\n\"I would so love that,\" cried Yolanda.\n\n\"Next we can either sail round the Mediterranean or if you are bored, we can come home.\"\n\n\"I know I shall never be bored,\" she sighed.\n\n\"Just as I will never be bored with you, my darling. We not only talk to each other with our hearts but with our brains.\"\n\n\"That is so true,\" agreed Yolanda.\n\n\"Of course it is true,\" her stepfather added. \"I told you once that love is really a battle of brains. So our brains continue to excite, inspire and guide us safely all through our lives.\"\n\n\"It is exactly what Yolanda will do for me,\" came in Lord Milborne. \"And that is why I am determined never to allow anyone else to take her from me.\"\n\n\"Do you really mean,\" asked Yolanda, \"that I shall be able to help you with your vital work for the Foreign Secretary?\"\n\n\"I shall only be any good at it if you do so. And the problems I will be confronted with can only be solved by us both using our brains.\"\n\n\"It makes the whole idea enthralling!\" exclaimed Yolanda.\n\nLord Milborne left the house for Windsor Castle early the next morning and she and her stepfather started in earnest to prepare for the wedding.\n\nThe local Vicar was an elderly gentleman who was loved by the whole village.\n\nHe was most delighted to be asked to perform the Service and promised to keep it secret until it had actually taken place.\n\n\"I don't want there to be anyone in the Church,\" explained Yolanda, \"except Step-papa and, of course, your daughter, if she will play the organ as she always does so beautifully.\"\n\n\"She will be honoured to do so,\" the Vicar replied.\n\n*\n\nThe ancient little Church was filled with flowers.\n\nYolanda had her best white evening gown altered so that it could become a wedding dress \u2013 it only required a little chiffon adding to the d\u00e9coll\u00e9tage.\n\nThe train that her mother had worn at her wedding could be suspended from her waist and there was also her mother's veil for her to wear as well as the jewellery that her stepfather had given her.\n\nSomeone had once told Yolanda that it was unlucky to see your bridegroom on your wedding day until you met him in the Church.\n\nSo she therefore insisted that when Lord Milborne arrived, he was to be taken upstairs to change.\n\nThen a separate carriage was to be waiting to take him down the drive to the Church and she would follow in another carriage with her stepfather.\n\nBecause she could not resist listening at her door, she heard Lord Milborne arrive back from Windsor Castle.\n\nHe was talking to her stepfather as they walked up the stairs.\n\nShe longed to run out and fling her arms round his neck \u2013 yet she knew she must keep to their plans.\n\nActually it made the whole wedding day even more exciting and something she would always remember to the day she died.\n\nFinally just before six o'clock she could hear Lord Milborne walk downstairs.\n\nWhen she was sure he had left for the Church, she went down to join her stepfather.\n\n\"You look so very lovely, my dearest,\" he said with admiration in his voice as she joined him in the drawing room.\n\n\"That is a delightful compliment and I also want to say something to you, Step-papa.\"\n\n\"What is it?\" he enquired.\n\n\"I want to thank you from the bottom of my heart for all your kindness and the love you have given me. We will never forget you and we will never be far away from you in the future.\n\n\"I know that Alan admires you enormously and we want you to come continually to our house just as we shall continually come to yours.\"\n\nHer stepfather put his arms round her and held her close against him.\n\n\"That is just what I wanted you to say. Because I really cannot afford to lose you as I lost your mother.\"\n\n\"You will never lose me, Step-papa,\" she promised. \"I shall always need you in my life even though I have Alan.\"\n\nHe kissed her.\n\n\"Your future husband has just told me something very touching. It is a secret, but I want you to know.\n\n\"When he told the Queen he was going to marry you, she was not only delighted but very impressed that I had been chosen by the Prime Minister and Lord Granville to travel to America and acquire the automatic air brake for Great Britain.\"\n\n\"I am so glad she is pleased.\"\n\n\"Her Majesty is not only pleased,\" he went on, \"but your future husband spoke to her about my collection of pictures and she has asked if she may come here to view them!\"\n\nYolanda gave a little cry.\n\n\"Oh, Step-papa, that is a great honour and it will be wonderful to have the Queen here. Think how thrilled the staff will be.\"\n\n\"And everyone in the neighbourhood,\" he added.\n\n\"What Alan has also said and this is the secret you must not mention to anyone. He thinks that if I promise the Queen, when she is here, to give ten of my best pictures to the National Gallery and I have by that time brought off the deal for the automatic air brake for our railways, she will honour me with a Knighthood.\"\n\nYolanda gave a gasp.\n\n\"Oh, Step-papa that would be marvellous!\"\n\nShe knew without him saying any more how much it would mean to her stepfather.\n\nIt would be the crowning glory that he had worked so hard for all his life.\n\nIt would also ensure that he would never again be involved in any more of the underhand deals that he called 'business'.\n\nShe was certain that from this moment on he would strive in every way to follow the unwritten laws by which her father had lived and died.\n\nThere was no need for further words.\n\nThere were tears in her eyes and she thought that there was a suspicious brightness in her stepfather's as she kissed him.\n\n\"It is such fabulous news, Step-papa. I know how proud and delighted Mama would be.\"\n\n\"That is exactly what I was thinking and she would be delighted at your marriage. So let's go and get on with it!\"\n\nYolanda gave a little laugh.\n\n\"That is just what I want to do,\" she sighed, as she took her stepfather's hand in hers.\n\n*\n\nMuch later that same day Lord Milborne kissed his wife's forehead as she lay silently in his arms.\n\n\"Do you still love me, my precious?\" he asked.\n\n\"I love you so much,\" Yolanda answered, \"that you fill my whole world and I am not only a part of you, but you are a part of me. You are in the sky, in the trees and in the water falling down from the fountain. Everywhere I look and everything I think is you, Alan.\"\n\n\"That is just what I want you to tell me, Yolanda, because my precious, beautiful perfect little wife, I feel that we have not only been looking for each other in this life but in the many lives we have lived before. Now at last we are together and no one can ever separate us.\"\n\n\"That is what Mama believed and I am convinced that she is now with Papa and they are both happy because we are together.\"\n\n\"As we shall always be, my darling Yolanda, and I am wondering how I have been lucky enough to find you.\"\n\n\"I think we were drawn together by a Power greater than ourselves. When I first met you when I was expecting to see your father, I felt in some strange way that you were different to any other man I had ever met.\"\n\n\"And you were not only the most beautiful woman I had ever seen in my life, but there was something about you that made me feel as I have not felt before with anyone else.\"\n\nYolanda moved a little closer to him.\n\n\"Tell me,\" she asked him, \"what it was?\"\n\n\"It is very difficult to put into words. Because you are so lovely, my heart acknowledged it and I felt a thrill run through my body which was different to anything I had ever experienced.\"\n\nHis lips were once again on her forehead.\n\n\"Then my brain told me quite clearly that this was what I had been looking for as you are so unlike in every way anyone else I have ever known.\"\n\nYolanda gave a laugh.\n\n\"I thought we would get back to our brains.\"\n\n\"They are so important, Yolanda, because it is what makes us into what we are going to be in the future.\"\n\nYolanda put up her hands and touched his cheek.\n\n\"I love you, I adore you, Alan, and my love for you will make my brain do exactly what you tell it to do just as long as you go on loving me.\n\n\"This is what my Papa would call a new adventure. To me it is just the most wonderful, exciting and thrilling adventure there could possibly be.\"\n\n\"I can answer you in only one way, Yolanda!\"\n\nThen he was kissing her and drawing her closer and closer to him.\n\nAs he made her his, Yolanda felt that he carried her up into the sky and all the stars were dancing at the glory of their love.\n\nFor both of them there was now a Heaven of Love that was their own and no one could ever take it from them.\n\n## Where to buy other titles in this series\n\nThe Barbara Cartland Pink collection is available for download at the following online bookshops :-\n\nwww.barnesandnoble.com \\- epub format for the Nook eReader\n\nwww.whsmith.co.uk \\- epub format for the Smiths\/Kobo eReader\n\nwww.firstyfish.com \\- epub format\n\nebookstore.sony.com \\- epub format for Sony eReaders\n\nwww.amazon.co.uk \\- For UK Kindle users\n\nwww.amazon.com \\- For international Kindle users\n\nitunes.apple.com \\- for Apple iOS users\n\nwww.barbaracartland.com \\- Printed paperbacks\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nPENGUIN BOOKS\n\nWHAT DO YOU CARE WHAT  \nOTHER PEOPLE THINK?\n\nRichard P. Feynman was one of this century's most brilliant theoretical physicists and original thinkers. Born in Far Rockaway, New York, in 1918, he studied at the Massachusetts Institute of Technology, where he graduated with a BS in 1939. He went on to Princeton and received his Ph.D. in 1942. During the war years he worked at the Los Alamos Scientific Laboratory. He became Professor of Theoretical Physics at Cornell University, where he worked with Hans Bethe. He all but rebuilt the theory of quantum electrodynamics and high-energy physics and it was for this work that he shared the Nobel Prize in 1965. Feynman was a visiting professor at the California Institute of Technology in 1950, where he later accepted a permanent faculty appointment, and became Richard Chace Tolman Professor of Theoretical Physics in 1959. He had an extraordinary ability to communicate his science to audiences at all levels, and was a well-known and popular lecturer. Richard Feynman died in 1988 after a long illness. Freeman Dyson, of the Institute for Advanced Study in Princeton, New Jersey, called him 'the most original mind of his generation', while in its obiturary _The New York Times_ described him as 'arguably the most brilliant, iconoclastic and influential of the postwar generation of theoretical physicists'.\n\nA number of collections and adaptations of his lectures have been published, including _The Feynman Lectures on Physics, QED_ (Penguin, 1990) _The Character of Physical Law_ (Penguin, 1992), _Six Easy Pieces_ (Penguin 1998), _The Meaning of It All_ (Penguin 1999), _Six Not-So-Easy Pieces_ (Penguin, 1999), _The Feynman Lectures on Gravitation_ (Penguin, 1999) and _The Feynman Lectures on Computation_ (Penguin, 1999). His memoirs, _Surely You're Joking, Mr Feynman_ , were published in 1985.\n\n# \"What Do You Care What Other People Think?\"\n\n**_Further Adventures of a Curious Character_**\n\n**Richard P. Feynman**\n\n**_as told to Ralph Leighton_**\n\nPENGUIN BOOKS\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 (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, North Shore 0632, New Zealand (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 in the United States of America 1988\n\nFirst published as a Norton paperback 2001\n\nPublished in Penguin Books 2007\n\n3\n\nCopyright \u00a9 Gweneth Feynman and Ralph Leighton, 1988 All rights reserved\n\nThe moral right of the author has been asserted\n\nExcept in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.\n\nISBN: 978-0-14-193758-8\n\n## Contents\n\nPreface\n\n_Part 1_ A CURIOUS CHARACTER\n\nThe Making of a Scientist\n\n\"What Do You Care What Other People Think?\"\n\nIt's as Simple as One, Two, Three...\n\nGetting Ahead\n\nHotel City\n\nWho the Hell Is Herman?\n\nFeynman Sexist Pig!\n\nI Just Shook His Hand, Can You Believe It?\n\nLetters\n\nPhotos and Drawings\n\n_Part 2_ MR. FEYNMAN GOES TO WASHINGTON: INVESTIGATING THE SPACE SHUTTLE _CHALLENGER_ DISASTER\n\nPreliminaries\n\nCommitting Suicide\n\nThe Cold Facts\n\nCheck Six!\n\nGumshoes\n\nFantastic Figures\n\nAn Inflamed Appendix\n\nThe Tenth Recommendation\n\nMeet the Press\n\nAfterthoughts\n\nAppendix F: Personal Observations on the Reliability of the Shuttle\n\nEPILOGUE\n\nPreface\n\nThe Value of Science\n\nIndex\n\n## Preface\n\n_BECAUSE_ of the appearance of \" _Surely You're Joking, Mr. Feynman_!\" a few things need to be explained here.\n\nFirst, although the central character in this book is the same as before, the \"adventures of a curious character\" here are different: some are light and some tragic, but most of the time Mr. Feynman is surely _not_ joking\u2014although it's often hard to tell.\n\nSecond, the stories in this book fit together more loosely than those in \" _Surely You're Joking_...,\" where they were arranged chronologically to give a semblance of order. (That resulted in some readers getting the mistaken idea that _SYJ_ is an autobiography.) My motivation is simple: ever since hearing my first Feynman stories, I have had the powerful desire to share them with others.\n\nFinally, most of these stories were not told at drumming sessions, as before. I will elaborate on this in the brief outline that follows.\n\nPart 1, \"A Curious Character,\" begins by describing the influence of those who most shaped Feynman's personality\u2014his father, Mel, and his first love, Arlene. The first story was adapted from \"The Pleasure of Finding Things Out,\" a BBC program produced by Christopher Sykes. The story of Arlene, from which the title of this book was taken, was painful for Feynman to recount. It was assembled over the past ten years out of pieces from six different stories. When it was finally complete, Feynman was especially fond of this story, and happy to share it with others.\n\nThe other Feynman stories in Part 1, although generally lighter in tone, are included here because there won't be a second volume of _SYJ_. Feynman was particularly proud of \"It's as Simple as One, Two, Three,\" which he occasionally thought of writing up as a psychology paper. The letters in the last chapter of Part 1 have been provided courtesy of Gweneth Feynman, Freeman Dyson, and Henry Bethe.\n\nPart 2, \"Mr. Feynman Goes to Washington,\" is, unfortunately, Feynman's last big adventure. The story is particularly long because its content is still timely. (Shorter versions have appeared in _Engineering and Science_ and _Physics Today_.) It was not published sooner because Feynman underwent his third and fourth major surgeries\u2014plus radiation, hyperthermia, and other treatments\u2014since serving on the Rogers Commission.\n\nFeynman's decade-long battle against cancer ended on February 15, 1988, two weeks after he taught his last class at Caltech. I decided to include one of his most eloquent and inspirational speeches, \"The Value of Science,\" as an epilogue.\n\nRalph Leighton\n\nMarch 1988\n\n## Part 1\n\n## A CURIOUS CHARACTER\n\n### The Making of a Scientist\n\n_I HAVE_ a friend who's an artist, and he sometimes takes a view which I don't agree with. He'll hold up a flower and say, \"Look how beautiful it is,\" and I'll agree. But then he'll say, \"I, as an artist, can see how beautiful a flower is. But you, as a scientist, take it all apart and it becomes dull.\" I think he's kind of nutty.\n\nFirst of all, the beauty that he sees is available to other people\u2014and to me, too, I believe. Although I might not be quite as refined aesthetically as he is, I can appreciate the beauty of a flower. But at the same time, I see much more in the flower than he sees. I can imagine the cells inside, which also have a beauty. There's beauty not just at the dimension of one centimeter; there's also beauty at a smaller dimension.\n\nThere are the complicated actions of the cells, and other processes. The fact that the colors in the flower have evolved in order to attract insects to pollinate it is interesting; that means insects can see the colors. That adds a question: does this aesthetic sense we have also exist in lower forms of life? There are all kinds of interesting questions that come from a knowledge of science, which only adds to the excitement and mystery and awe of a flower. It only adds. I don't understand how it subtracts.\n\nI've always been very one-sided about science, and when I was younger I concentrated almost all my effort on it. In those days I didn't have time, and I didn't have much patience, to learn what's called the humanities. Even though there were humanities courses in the university that you had to take in order to graduate, I tried my best to avoid them. It's only afterwards, when I've gotten older and more relaxed, that I've spread out a little bit. I've learned to draw and I read a little bit, but I'm really still a very one-sided person and I don't know a great deal. I have a limited intelligence and I use it in a particular direction.\n\nBefore I was born, my father told my mother, \"If it's a boy, he's going to be a scientist.\"* When I was just a little kid, very small in a highchair, my father brought home a lot of little bathroom tiles\u2014seconds\u2014of different colors. We played with them, my father setting them up vertically on my highchair like dominoes, and I would push one end so they would all go down.\n\nThen after a while, I'd help set them up. Pretty soon, we're setting them up in a more complicated way: two white tiles and a blue tile, two white tiles and a blue tile, and so on. When my mother saw that she said, \"Leave the poor child alone. If he wants to put a blue tile, let him put a blue tile.\"\n\nBut my father said, \"No, I want to show him what patterns are like and how interesting they are. It's a kind of elementary mathematics.\" So he started very early to tell me about the world and how interesting it is.\n\nWe had the _Encyclopaedia Britannica_ at home. When I was a small boy he used to sit me on his lap and read to me from the _Britannica_. We would be reading, say, about dinosaurs. It would be talking about the _Tyrannosaurus rex_ , and it would say something like, \"This dinosaur is twenty-five feet high and its head is six feet across.\"\n\nMy father would stop reading and say, \"Now, let's see what that means. That would mean that if he stood in our front yard, he would be tall enough to put his head through our window up here.\" (We were on the second floor.) \"But his head would be too wide to fit in the window.\" Everything he read to me he would translate as best he could into some reality.\n\nIt was very exciting and very, very interesting to think there were animals of such magnitude\u2014and that they all died out, and that nobody knew why. I wasn't frightened that there would be one coming in my window as a consequence of this. But I learned from my father to translate: everything I read I try to figure out what it really means, what it's really saying.\n\nWe used to go to the Catskill Mountains, a place where people from New York City would go in the summer. The fathers would all return to New York to work during the week, and come back only for the weekend. On weekends, my father would take me for walks in the woods and he'd tell me about interesting things that were going on in the woods. When the other mothers saw this, they thought it was wonderful and that the other fathers should take their sons for walks. They tried to work on them but they didn't get anywhere at first. They wanted my father to take all the kids, but he didn't want to because he had a special relationship with me. So it ended up that the other fathers had to take their children for walks the next weekend.\n\nThe next Monday, when the fathers were all back at work, we kids were playing in a field. One kid says to me, \"See that bird? What kind of bird is that?\"\n\nI said, \"I haven't the slightest idea what kind of a bird it is.\"\n\nHe says, \"It's a brown-throated thrush. Your father doesn't teach you anything!\"\n\nBut it was the opposite. He had already taught me: \"See that bird?\" he says. \"It's a Spencer's warbler.\" (I knew he didn't know the real name.) \"Well, in Italian, it's a _Chutto Lapittida_. In Portuguese, it's a _Bom da Peida_. In Chinese, it's a _Chung-long-tah_ , and in Japanese, it's a _Katano Tekeda_. You can know the name of that bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird. You'll only know about humans in different places, and what they call the bird. So let's look at the bird and see what it's _doing_ \u2014that's what counts.\" (I learned very early the difference between knowing the name of something and knowing something.)\n\nHe said, \"For example, look: the bird pecks at its feathers all the time. See it walking around, pecking at its feathers?\"\n\n\"Yeah.\"\n\nHe says, \"Why do you think birds peck at their feathers?\"\n\nI said, \"Well, maybe they mess up their feathers when they fly, so they're pecking them in order to straighten them out.\"\n\n\"All right,\" he says. \"If that were the case, then they would peck a lot just after they've been flying. Then, after they've been on the ground a while, they wouldn't peck so much any more\u2014you know what I mean?\"\n\n\"Yeah.\"\n\nHe says, \"Let's look and see if they peck more just after they land.\"\n\nIt wasn't hard to tell: there was not much difference between the birds that had been walking around a bit and those that had just landed. So I said, \"I give up. Why does a bird peck at its feathers?\"\n\n\"Because there are lice bothering it,\" he says. \"The lice eat flakes of protein that come off its feathers.\"\n\nHe continued, \"Each louse has some waxy stuff on its legs, and little mites eat that. The mites don't digest it perfectly, so they emit from their rear ends a sugar-like material, in which bacteria grow.\"\n\nFinally he says, \"So you see, everywhere there's a source of food, there's _some_ form of life that finds it.\"\n\nNow, I knew that it may not have been exactly a louse, that it might not be exactly true that the louse's legs have mites. That story was probably incorrect in _detail_ , but what he was telling me was right in _principle_.\n\nAnother time, when I was older, he picked a leaf off of a tree. This leaf had a flaw, a thing we never look at much. The leaf was sort of deteriorated; it had a little brown line in the shape of a C, starting somewhere in the middle of the leaf and going out in a curl to the edge.\n\n\"Look at this brown line,\" he says. \"It's narrow at the beginning and it's wider as it goes to the edge. What this is, is a fly\u2014a blue fly with yellow eyes and green wings has come and laid an egg on this leaf. Then, when the egg hatches into a maggot (a caterpillar-like thing), it spends its whole life eating this leaf\u2014that's where it gets its food. As it eats along, it leaves behind this brown trail of eaten leaf. As the maggot grows, the trail grows wider until he's grown to full size at the end of the leaf, where he turns into a fly\u2014a blue fly with yellow eyes and green wings\u2014who flies away and lays an egg on another leaf.\"\n\nAgain, I knew that the details weren't precisely correct\u2014it could have even been a beetle\u2014but the idea that he was trying to explain to me was the amusing part of life: the whole thing is just reproduction. No matter how complicated the business is, the main point is to do it again!\n\nNot having experience with many fathers, I didn't realize how remarkable he was. How did he learn the deep principles of science and the love of it, what's behind it, and why it's worth doing? I never really asked him, because I just assumed that those were things that fathers knew.\n\nMy father taught me to notice things. One day, I was playing with an \"express wagon,\" a little wagon with a railing around it. It had a ball in it, and when I pulled the wagon, I noticed something about the way the ball moved. I went to my father and said, \"Say, Pop, I noticed something. When I pull the wagon, the ball rolls to the back of the wagon. And when I'm pulling it along and I suddenly stop, the ball rolls to the front of the wagon. Why is that?\"\n\n\"That, nobody knows,\" he said. \"The general principle is that things which are moving tend to keep on moving, and things which are standing still tend to stand still, unless you push them hard. This tendency is called 'inertia,' but nobody knows why it's true.\" Now, that's a deep understanding. He didn't just give me the name.\n\nHe went on to say, \"If you look from the side, you'll see that it's the back of the wagon that you're pulling against the ball, and the ball stands still. As a matter of fact, from the friction it starts to move forward a little bit in relation to the ground. It doesn't move back.\"\n\nI ran back to the little wagon and set the ball up again and pulled the wagon. Looking sideways, I saw that indeed he was right. Relative to the sidewalk, it moved forward a little bit.\n\nThat's the way I was educated by my father, with those kinds of examples and discussions: no pressure\u2014just lovely, interesting discussions. It has motivated me for the rest of my life, and makes me interested in _all_ the sciences. (It just happens I do physics better.)\n\nI've been caught, so to speak\u2014like someone who was given something wonderful when he was a child, and he's always looking for it again. I'm always looking, like a child, for the wonders I know I'm going to find\u2014maybe not every time, but every once in a while.\n\nAround that time my cousin, who was three years older, was in high school. He was having considerable difficulty with his algebra, so a tutor would come. I was allowed to sit in a corner while the tutor would try to teach my cousin algebra. I'd hear him talking about _x_.\n\nI said to my cousin, \"What are you trying to do?\"\n\n\"I'm trying to find out what _x_ is, like in 2 _x_ \\+ 7 = 15.\"\n\nI say, \"You mean 4.\"\n\n\"Yeah, but you did it by arithmetic. You have to do it by algebra.\"\n\nI learned algebra, fortunately, not by going to school, but by finding my aunt's old schoolbook in the attic, and understanding that the whole idea was to find out what _x_ is\u2014it doesn't make any difference how you do it. For me, there was no such thing as doing it \"by arithmetic,\" or doing it \"by algebra.\" \"Doing it by algebra\" was a set of rules which, if you followed them blindly, could produce the answer: \"subtract 7 from both sides; if you have a multiplier, divide both sides by the multiplier,\" and so on\u2014a series of steps by which you could get the answer if you didn't understand what you were trying to do. The rules had been invented so that the children who have to study algebra can all pass it. And that's why my cousin was never able to do algebra.\n\nThere was a series of math books in our local library which started out with _Arithmetic for the Practical Man_. Then came _Algebra for the Practical Man_ , and then _Trigonometry for the Practical Man_. (I learned trigonometry from that, but I soon forgot it again, because I didn't understand it very well.) When I was about thirteen, the library was going to get _Calculus for the Practical Man_. By this time I knew, from reading the encyclopedia, that calculus was an important and interesting subject, and I ought to learn it.\n\nWhen I finally saw the calculus book at the library, I was very excited. I went to the librarian to check it out, but she looked at me and said, \"You're just a child. What are you taking this book out for?\"\n\nIt was one of the few times in my life I was uncomfortable and I lied. I said it was for my father.\n\nI took the book home and I began to learn calculus from it. I thought it was relatively simple and straightforward. My father started to read it, but he found it confusing and he couldn't understand it. So I tried to explain calculus to him. I didn't know he was so limited, and it bothered me a little bit. It was the first time I realized that I had learned more in some sense than he.\n\nOne of the things that my father taught me besides physics\u2014whether it's correct or not\u2014was a disrespect for certain kinds of things. For example, when I was a little boy, and he would sit me on his knee, he'd show me rotogravures in the _New York Times_ \u2014that's printed pictures which had just come out in newspapers.\n\nOne time we were looking at a picture of the pope and everybody bowing in front of him. My father said, \"Now, look at those humans. Here's one human standing here, and all these others are bowing in front of him. Now, what's the difference? This one is the pope\"\u2014he hated the pope anyway. He said, \"This difference is the hat he's wearing.\" (If it was a general, it was the epaulets. It was always the costume, the uniform, the position.) \"But,\" he said, \"this man has the same problems as everybody else: he eats dinner; he goes to the bathroom. He's a human being.\" (By the way, my father was in the uniform business, so he knew what the difference is in a man with the uniform off and the uniform on\u2014it was the same man for him.)\n\nHe was happy with me, I believe. Once, though, when I came back from MIT (I'd been there a few years), he said to me, \"Now that you've become educated about these things, there's one question I've always had that I've never understood very well.\"\n\nI asked him what it was.\n\nHe said, \"I understand that when an atom makes a transition from one state to another, it emits a particle of light called a photon.\"\n\n\"That's right,\" I said.\n\nHe says, \"Is the photon in the atom ahead of time?\"\n\n\"No, there's no photon beforehand.\"\n\n\"Well,\" he says, \"where does it come from, then? How does it come out?\"\n\nI tried to explain it to him\u2014that photon numbers aren't conserved; they're just created by the motion of the electron\u2014but I couldn't explain it very well. I said, \"It's like the sound that I'm making now: it wasn't in me before.\" (It's not like my little boy, who suddenly announced one day, when he was very young, that he could no longer say a certain word\u2014the word turned out to be \"cat\"\u2014because his \"word bag\" had run out of the word. There's no word bag that makes you use up words as they come out; in the same sense, there's no \"photon bag\" in an atom.)\n\nHe was not satisfied with me in that respect. I was never able to explain any of the things that he didn't understand. So he was unsuccessful: he sent me to all these universities in order to find out those things, and he never did find out.\n\nAlthough my mother didn't know anything about science, she had a great influence on me as well. In particular, she had a wonderful sense of humor, and I learned from her that the highest forms of understanding we can achieve are laughter and human compassion.\n\n### \"What Do You Care What Other People Think?\"\n\n_WHEN_ I was a young fella, about thirteen, I had somehow gotten in with a group of guys who were a little older than I was, and more sophisticated. They knew a lot of different girls, and would go out with them\u2014often to the beach.\n\nOne time when we were at the beach, most of the guys had gone out on some jetty with the girls. I was interested in a particular girl a little bit, and sort of thought out loud: \"Gee, I think I'd like to take Barbara to the movies...\"\n\nThat's all I had to say, and the guy next to me gets all excited. He runs out onto the rocks and finds her. He pushes her back, all the while saying in a loud voice, \"Feynman has something he wants to say to you, Barbara!\" It was most embarrassing.\n\nPretty soon the guys are all standing around me, saying, \"Well, _say_ it, Feynman!\" So I invited her to the movies. It was my first date.\n\nI went home and told my mother about it. She gave me all kinds of advice on how to do this and that. For example, if we take the bus, I'm supposed to get off the bus first, and offer Barbara my hand. Or if we have to walk in the street, I'm supposed to walk on the outside. She even told me what kinds of things to say. She was handing down a cultural tradition to me: the women teach their sons how to treat the next generation of women well.\n\nAfter dinner, I get all slicked up and go to Barbara's house to call for her. I'm nervous. She isn't ready, of course (it's always like that) so her family has me wait for her in the dining room, where they're eating with friends\u2014a lot of people. They say things like, \"Isn't he cute!\" and all kinds of other stuff. I didn't feel cute. It was absolutely terrible!\n\nI remember everything about the date. As we walked from her house to the new, little theater in town, we talked about playing the piano. I told her how, when I was younger, they made me learn piano for a while, but after six months I was still playing \"Dance of the Daisies,\" and couldn't stand it any more. You see, I was worried about being a sissy, and to be stuck for weeks playing \"Dance of the Daisies\" was too much for me, so I quit. I was so sensitive about being a sissy that it even bothered me when my mother sent me to the market to buy some snacks called Peppermint Patties and Toasted Dainties.\n\nWe saw the movie, and I walked her back to her home. I complimented her on the nice, pretty gloves she was wearing. Then I said goodnight to her on the doorstep.\n\nBarbara says to me, \"Thank you for a very lovely evening.\"\n\n\"You're welcome!\" I answered. I felt terrific.\n\nThe next time I went out on a date\u2014it was with a different girl\u2014I say goodnight to her, and she says, \"Thank you for a very lovely evening.\"\n\nI didn't feel quite so terrific.\n\nWhen I said goodnight to the third girl I took out, she's got her mouth open, ready to speak, and I say, \"Thank you for a very lovely evening!\"\n\nShe says, \"Thank you\u2014uh\u2014Oh!\u2014Yes\u2014uh, I had a lovely evening, too, thank you!\"\n\nOne time I was at a party with my beach crowd, and one of the older guys was in the kitchen teaching us how to kiss, using his girlfriend to demonstrate: \"You have to have your lips like this, at right angles, so the noses don't collide,\" and so on. So I go into the living room and find a girl. I'm sitting on the couch with my arm around her, practicing this new art, when suddenly there's all kinds of excitement: \"Arlene is coming! Arlene is coming!\" I don't know who Arlene is.\n\nThen someone says, \"She's here! She's here!\"\u2014and everybody stops what they're doing and jumps up to see this queen. Arlene was very pretty, and I could see why she had all this admiration\u2014it was well deserved\u2014but I didn't believe in this undemocratic business of changing what you're doing just because the queen is coming in.\n\nSo, while everybody's going over to see Arlene, I'm still sitting there on the couch with my girl.\n\n(Arlene told me later, after I had gotten to know her, that she remembered that party with all the nice people\u2014except for one guy who was over in the corner on the couch smooching with a girl. What she didn't know was that two minutes before, all the others were doin' it too!)\n\nThe first time I ever said anything to Arlene was at a dance. She was very popular, and everybody was cutting in and dancing with her. I remember thinking I'd like to dance with her, too, and trying to decide when to cut in. I always had trouble with that problem: first of all, when she's over on the other side of the dance floor dancing with some guy, it's too complicated\u2014so you wait until they come closer. Then when she's near you, you think, \"Well, no, this isn't the kind of music I'm good at dancing to.\" So you wait for another type of music. When the music changes to something you like, you sort of step forward\u2014at least you _think_ you step forward to cut in\u2014when some other guy cuts in just in front of you. So now you have to wait a few minutes because it's impolite to cut in too soon after someone else has. And by the time a few minutes have passed, they're over at the other side of the dance floor again, or the music has changed again, or whatever!\n\nAfter a certain amount of this stalling and fooling around, I finally mutter something about wanting to dance with Arlene. One of the guys I was hanging around with overhears me and makes a big announcement to the other guys: \"Hey, listen to this, guys; Feynman wants to dance with Arlene!\" Soon one of them is dancing with Arlene and they dance over towards the rest of us. The others push me out onto the dance floor and I finally \"cut in.\" You can see the condition I was in by my first words to her, which were an honest question: \"How does it feel to be so popular?\" We only danced a few minutes before somebody else cut in.\n\nMy friends and I had taken dancing lessons, although none of us would ever admit it. In those depression days, a friend of my mother was trying to make a living by teaching dancing in the evening, in an upstairs dance studio. There was a back door to the place, and she arranged it so the young men could come up through the back way without being seen.\n\nEvery once in a while there would be a social dance at this lady's studio. I didn't have the nerve to test this analysis, but it seemed to me that the girls had a much harder time than the boys did. In those days, girls couldn't ask to cut in and dance with boys; it wasn't \"proper.\" So the girls who weren't very pretty would sit for hours at the side, just sad as hell.\n\nI thought, \"The guys have it easy: they're free to cut in whenever they want.\" But it wasn't easy. You're \"free,\" but you haven't got the guts, or the sense, or whatever it takes to relax and enjoy dancing. Instead, you tie yourself in knots worrying about cutting in or inviting a girl to dance with you.\n\nFor example, if you saw a girl who was not dancing, who you thought you'd like to dance with, you might think, \"Good! Now at least I've got a chance!\" But it was usually very difficult: often the girl would say, \"No, thank you, I'm tired. I think I'll sit this one out.\" So you go away somewhat defeated\u2014but not completely, because maybe she really _is_ tired\u2014when you turn around and some other guy comes up to her, and there she is, dancing with him! Maybe this guy is her boyfriend and she knew he was coming over, or maybe she didn't like the way you look, or maybe something else. It was always so complicated for such a simple matter.\n\nOne time I decided to invite Arlene to one of these dances. It was the first time I took her out. My best friends were also at the dance; my mother had invited them, to get more customers for her friend's dance studio. These guys were contemporaries of mine, guys my own age from school. Harold Gast and David Leff were literary types, while Robert Stapler was a scientific type. We would spend a lot of time together after school, going on walks and discussing this and that.\n\nAnyway, my best friends were at the dance, and as soon as they saw me with Arlene, they called me into the cloakroom and said, \"Now listen, Feynman, we want you to understand that _we_ understand that Arlene is _your_ girl tonight, and we're not gonna bother you with her. She's out of bounds for us,\" and so on. But before long, there was cutting in and competition coming from precisely these guys! I learned the meaning of Shakespeare's phrase \"Me-thinks thou dost protest too much.\"\n\nYou must appreciate what I was like then. I was a very shy character, always feeling uncomfortable because everybody was stronger than I, and always afraid I would look like a sissy. Everybody else played baseball; everybody else did all kinds of athletic things. If there was a game somewhere, and a ball would come rolling across the road, I would be petrified that maybe I'd have to pick it up and throw it back\u2014because if I threw it, it would be about a radian off the correct direction, and not anywhere near the distance! And then everybody would laugh. It was terrible, and I was very unhappy about it.\n\nOne time I was invited to a party at Arlene's house. Everybody was there because Arlene was the most popular girl around: she was number one, the nicest girl, and everybody liked her. Well, I'm sitting in a big armchair with nothing to do, when Arlene comes over and sits on the arm of the chair to talk to me. That was the beginning of the feeling, \"Oh, boy! The world is just wonderful now! Somebody I like has paid attention to me!\"\n\nIn those days, in Far Rockaway, there was a youth center for Jewish kids at the temple. It was a big club that had many activities. There was a writers group that wrote stories and would read them to each other; there was a drama group that put on plays; there was a science group, and an art group. I had no interest in any subject except science, but Arlene was in the art group, so I joined it too. I struggled with the art business\u2014learning how to make plaster casts of the face and so on (which I used much later in life, it turned out)\u2014just so I could be in the same group with Arlene.\n\nBut Arlene had a boyfriend named Jerome in the group, so there was no chance for me. I just hovered around in the background.\n\nOne time, when I wasn't there, somebody nominated me for president of the youth center. The elders began getting nervous, because I was an avowed atheist by that time.\n\nI had been brought up in the Jewish religion\u2014my family went to the temple every Friday, I was sent to what we called \"Sunday school,\" and I even studied Hebrew for a while\u2014but at the same time, my father was telling me about the world. When I would hear the rabbi tell about some miracle such as a bush whose leaves were shaking but there wasn't any wind, I would try to fit the miracle into the real world and explain it in terms of natural phenomena.\n\nSome miracles were harder than others to understand. The one about the leaves was easy. When I was walking to school, I heard a little noise: although the wind was hardly noticeable, the leaves of a bush were wiggling a little bit because they were in just the right position to make a kind of resonance. And I thought, \"Aha! This is a good explanation for Elijah's vision of the quaking bush!\"\n\nBut there were some miracles I never did figure out. For instance, there was a story in which Moses throws down his staff and it turns into a snake. I couldn't figure out what the witnesses saw that made them think his staff was a snake.\n\nIf I had thought back to when I was much younger, the Santa Claus story could have provided a clue for me. But it didn't hit me hard enough at the time to produce the possibility that I should doubt the truth of stories that don't fit with nature. When I found out that Santa Claus wasn't real, I wasn't upset; rather, I was relieved that there was a much simpler phenomenon to explain how so many children all over the world got presents on the same night! The story had been getting pretty complicated\u2014it was getting out of hand.\n\nSanta Claus was a particular custom we celebrated in our family, and it wasn't very serious. But the miracles I was hearing about were connected with real things: there was the temple, where people would go every week; there was the Sunday school, where rabbis taught children about miracles; it was much more of a dramatic thing. Santa Claus didn't involve big institutions like the temple, which I knew were real.\n\nSo all the time I was going to the Sunday school, I was believing everything and having trouble putting it together. But of course, ultimately, it had to come to a crisis, sooner or later.\n\nThe actual crisis came when I was eleven or twelve. The rabbi was telling us a story about the Spanish Inquisition, in which the Jews suffered terrible tortures. He told us about a particular individual whose name was Ruth, exactly what she was supposed to have done, what the arguments were in her favor and against her\u2014the whole thing, as if it had all been documented by a court reporter. And I was just an innocent kid, listening to all this stuff and believing it was a true commentary, because the rabbi had never indicated otherwise.\n\nAt the end, the rabbi described how Ruth was dying in prison: \"And she thought, while she was dying\"\u2014blah, blah.\n\nThat was a shock to me. After the lesson was over, I went up to him and said, \"How did they know what she thought when she was dying?\"\n\nHe says, \"Well, of course, in order to explain more vividly how the Jews suffered, we made up the story of Ruth. It wasn't a real individual.\"\n\nThat was too much for me. I felt terribly deceived: I wanted the straight story\u2014not fixed up by somebody else\u2014so I could decide for myself what it meant. But it was difficult for me to argue with adults. All I could do was get tears in my eyes. I started to cry, I was so upset.\n\nHe said, \"What's the matter?\"\n\nI tried to explain. \"I've been listening to all these stories, and now I don't know, of all the things you told me, which were true, and which were not true! I don't know what to do with everything that I've learned!\" I was trying to explain that I was losing everything at the moment, because I was no longer sure of the data, so to speak. Here I had been struggling to understand all these miracles, and now\u2014well, it solved a lot of miracles, all right! But I was unhappy.\n\nThe rabbi said, \"If it is so traumatic for you, why do you come to Sunday school?\"\n\n\"Because my parents make me.\"\n\nI never talked to my parents about it, and I never found out whether the rabbi communicated with them or not, but my parents never made me go again. And it was just before I was supposed to get confirmed as a believer.\n\nAnyway, that crisis resolved my difficulty rather rapidly, in favor of the theory that all the miracles were stories made up to help people understand things \"more vividly,\" even if they conflicted with natural phenomena. But I thought nature itself was so interesting that I didn't want it distorted like that. And so I gradually came to disbelieve the whole religion.\n\nAnyway, the Jewish elders had organized this club with all its activities not just to get us kids off the street, but to get us interested in the Jewish way of life. So to have someone like me elected as president would have made them very embarrassed. To our mutual relief I wasn't elected, but the center eventually failed anyway\u2014it was on its way out when I was nominated, and had I been elected, I surely would have been blamed for its demise.\n\nOne day Arlene told me Jerome isn't her boyfriend anymore. She's not tied up with him. That was a big excitement for me, the beginning _of hope_! She invited me over to her house, at 154 Westminster Avenue in nearby Cedarhurst.\n\nWhen I went to her house that time, it was dark and the porch wasn't lit. I couldn't see the numbers. Not wanting to disturb anyone by asking if it was the right house, I crawled up, quietly, and felt the numbers on the door: 154.\n\nArlene was having trouble with her homework in philosophy class. \"We're studying Descartes,\" she said. \"He starts out with 'Cogito, ergo sum'\u2014'I think, therefore I am'\u2014and ends up proving the existence of God.\"\n\n\"Impossible!\" I said, without stopping to think that I was doubting the great Descartes. (It was a reaction I learned from my father: have no respect _whatsoever_ for authority; forget who said it and instead look at what he starts with, where he ends up, and ask yourself, \"Is it reasonable?\") I said, \"How can you deduce one from the other?\"\n\n\"I don't know,\" she said.\n\n\"Well, let's look it over,\" I said. \"What's the argument?\"\n\nSo we look it over, and we see that Descartes' statement \"Cogito, ergo sum\" is supposed to mean that there is one thing that cannot be doubted\u2014doubt itself. \"Why doesn't he just say it straight?\" I complained. \"He just means somehow or other that he has one fact that he knows.\"\n\nThen it goes on and says things like, \"I can only imagine imperfect thoughts, but imperfect can only be understood as referent to the perfect. Hence the perfect must exist somewhere.\" (He's workin' his way towards God now.)\n\n\"Not at all!\" I say. \"In science you can talk about relative degrees of approximation without having a perfect theory. I don't know what this is all about. I think it's a bunch of baloney.\"\n\nArlene understood me. She understood, when she looked at it, that no matter how impressive and important this philosophy stuff was supposed to be, it could be taken lightly\u2014you could just think about the words, instead of worrying about the fact that Descartes said it. \"Well, I guess it's okay to take the other side,\" she said. \"My teacher keeps telling us, 'There are two sides to every question, just like there are two sides to every piece of paper.' \"\n\n\"There's two sides to that, too,\" I said.\n\n\"What do you mean?\"\n\nI had read about the M\u00f6bius strip in the _Britannica_ , my wonderful _Britannica!_ In those days, things like the M\u00f6bius strip weren't so well known to everybody, but they were just as understandable as they are to kids today. The existence of such a surface was so real: it wasn't a wishy-washy political question, or anything that you needed history to understand. Reading about those things was like being way off in a wonderful world that nobody knows about, and you're getting a kick not only from the delight of learning the stuff itself, but also from making yourself unique.\n\nI got a strip of paper, put a half twist in it, and made it into a loop. Arlene was delighted.\n\nThe next day, in class, she lay in wait for her teacher. Sure enough, he holds up a piece of paper and says, \"There are two sides to every question, just like there are two sides to every piece of paper.\" Arlene holds up her own strip of paper\u2014with a half twist in it\u2014and says, \"Sir, there are even two sides to _that_ question: there's paper with only one side!\" The teacher and the class got all excited, and Arlene got such a kick out of showing them the M\u00f6bius strip that I think she paid more attention to me after that on account of it.\n\nBut after Jerome, I had a new competitor\u2014my \"good friend\" Harold Gast. Arlene was always making up her mind one way or the other. When it came time for graduation, she went with Harold to the senior prom, but sat with my parents for the graduation ceremony.\n\nI was the best in science, the best in mathematics, the best in physics, and the best in chemistry, so I was going up to the stage and receiving honors many times at the ceremony. Harold was the best in English and the best in history, and had written the school play, so that was very impressive.\n\nI was terrible in English. I couldn't stand the subject. It seemed to me ridiculous to worry about whether you spelled something wrong or not, because English spelling is just a human convention\u2014it has nothing to do with anything _real_ , anything from nature. Any word can be spelled just as well a different way. I was impatient with all this English stuff.\n\nThere was a series of exams called the Regents, which the state of New York gave to every high school student. A few months before, when we all were taking the Regents examination in English, Harold and the other literary friend of mine, David Leff\u2014the editor of the school newspaper\u2014asked me which books I had chosen to write about. David had chosen something with profound social implications by Sinclair Lewis, and Harold had picked some playwright. I said I chose _Treasure Island_ because we had that book in first-year English, and told them what I wrote.\n\nThey laughed. \"Boy, are _you_ gonna flunk, saying such simple stuff about such a simple book!\"\n\nThere was also a list of questions for an essay. The one I chose was \"The Importance of Science in Aviation.\" I thought, \"What a dumb question! The importance of science in aviation is obvious!\"\n\nI was about to write a simple theme about this dumb question when I remembered that my literary friends were always \"throwing the bull\"\u2014building up their sentences to sound complex and sophisticated. I decided to try it, just for the hell of it. I thought, \"If the Regents are so silly as to have a subject like the importance of science in aviation, I'm gonna _do_ that.\"\n\nSo I wrote stuff like, \"Aeronautical science is important in the analysis of the eddies, vortices, and whirlpools formed in the atmosphere behind the aircraft...\"\u2014I knew that eddies, vortices, and whirlpools are the same thing, but mentioning them three different ways _sounds_ better! That was the only thing I would not have ordinarily done on the test.\n\nThe teacher who corrected my examination must have been impressed by eddies, vortices, and whirlpools, because I got a 91 on the exam\u2014while my literary friends, who chose topics the English teachers could more easily take issue with, both got 88.\n\nThat year a new rule came out: if you got 90 or better on a Regents examination, you automatically got honors in that subject at graduation! So while the playwright and the editor of the school newspaper had to stay in their seats, this illiterate fool physics student was called to go up to the stage once again and receive honors in English!\n\nAfter the graduation ceremony, Arlene was in the hall with my parents and Harold's parents when the head of the math department came over. He was a very strong man\u2014he was also the school disciplinarian\u2014a tall, dominating fellow. Mrs. Gast says to him, \"Hello, Dr. Augsberry. I'm Harold Gast's mother. And this is Mrs. Feynman...\"\n\nHe completely ignores Mrs. Gast and immediately turns to my mother. \"Mrs. Feynman, I want to impress upon you that a young man like your son comes along only very rarely. The state should support a man of such talent. You must be _sure_ that he goes to college, the best college you can afford!\" He was concerned that my parents might not be planning to send me to college, for in those days lots of kids had to get a job immediately after graduation to help support the family.\n\nThat in fact happened to my friend Robert. He had a lab, too, and taught me all about lenses and optics. (One day he had an accident in his lab. He was opening carbolic acid and the bottle jerked, spilling some acid on his face. He went to the doctor and had bandages put on for a few weeks. The funny thing was, when they took the bandages off his skin was smooth underneath, nicer than it had been before\u2014there were many fewer blemishes. I've since found out that there was, for a while, some kind of a beauty treatment using carbolic acid in a more dilute form.) Robert's mother was poor, and he had to go to work right away to support her, so he couldn't continue his interest in the sciences.\n\nAnyway, my mother reassured Dr. Augsberry: \"We're saving money as best we can, and we're trying to send him to Columbia or MIT.\" And Arlene was listening to all this, so after that I was a little bit ahead.\n\nArlene was a wonderful girl. She was the editor of the newspaper at Nassau County Lawrence High School; she played the piano beautifully, and was very artistic. She made some decorations for our house, like the parrot on the inside of our closet. As time went on, and our family got to know her better, she would go to the woods to paint with my father, who had taken up painting in later life, as many people do.\n\nArlene and I began to mold each other's personality. She lived in a family that was very polite, and was very sensitive to other people's feelings. She taught me to be more sensitive to those kinds of things, too. On the other hand, her family felt that \"white lies\" were okay.\n\nI thought one should have the attitude of \"What do _you_ care what other people think!\" I said, \"We should listen to other people's opinions and take them into account. Then, if they don't make sense and we think they're wrong, then that's that!\"\n\nArlene caught on to the idea right away. It was easy to talk her into thinking that in our relationship, we must be very honest with each other and say everything straight, with absolute frankness. It worked very well, and we became very much in love\u2014a love like no other love that I know of.\n\nAfter that summer I went away to college at MIT. (I couldn't go to Columbia because of the Jewish quota.*) I began getting letters from my friends that said things like, \"You should see how Arlene is going out with Harold,\" or \"She's doing this and she's doing that, while you're all alone up there in Boston.\" Well, I was taking out girls in Boston, but they didn't mean a thing to me, and I knew the same was true with Arlene.\n\nWhen summer came, I stayed in Boston for a summer job, and worked on measuring friction. The Chrysler Company had developed a new method of polishing to get a super finish, and we were supposed to measure how much better it was. (It turned out that the \"super finish\" was not significantly better.)\n\nAnyway, Arlene found a way to be near me. She found a summer job in Scituate, about twenty miles away, taking care of children. But my father was concerned that I would become too involved with Arlene and get off the track of my studies, so he talked her out of it\u2014or talked me out of it (I can't remember). Those days were very, very different from now. In those days, you had to go all the way up in your career before marrying.\n\nI was able to see Arlene only a few times that summer, but we promised each other we would marry after I finished school. I had known her for six years by that time. I'm a little tongue-tied trying to describe to you how much our love for each other developed, but we were sure we were right for each other.\n\nAfter I graduated from MIT I went to Princeton, and I would go home on vacations to see Arlene. One time when I went to see her, Arlene had developed a bump on one side of her neck. She was a very beautiful girl, so it worried her a little bit, but it didn't hurt, so she figured it wasn't too serious. She went to her uncle, who was a doctor. He told her to rub it with omega oil.\n\nThen, sometime later, the bump began to change. It got bigger\u2014or maybe it was smaller\u2014and she got a fever. The fever got worse, so the family doctor decided Arlene should go to the hospital. She was told she had typhoid fever. Right away, as I still do today, I looked up the disease in medical books and read all about it.\n\nWhen I went to see Arlene in the hospital, she was in quarantine\u2014we had to put on special gowns when we entered her room, and so on. The doctor was there, so I asked him how the Wydell test came out\u2014it was an absolute test for typhoid fever that involved checking for bacteria in the feces. He said, \"It was negative.\"\n\n\"What? How can that _be!_ \" I said. \"Why all these gowns, when you can't even find the bacteria in an experiment? Maybe she doesn't have typhoid fever!\"\n\nThe result of that was that the doctor talked to Arlene's parents, who told me not to interfere. \"After all, he's the doctor. You're only her fianc\u00e9.\"\n\nI've found out since that such people don't know what they're doing, and get insulted when you make some suggestion or criticism. I realize that _now,_ but I wish I had been much stronger then and told her parents that the doctor was an idiot\u2014which he was\u2014and didn't know what he was doing. But as it was, her parents were in charge of it.\n\nAnyway, after a little while, Arlene got better, apparently: the swelling went down and the fever went away. But after some weeks the swelling started again, and this time she went to another doctor. This guy feels under her armpits and in her groin, and so on, and notices there's swelling in those places, too. He says the problem is in her lymphatic glands, but he doesn't yet know what the specific disease is. He will consult with other doctors.\n\nAs soon as I hear about it I go down to the library at Princeton and look up lymphatic diseases, and find \"Swelling of the Lymphatic Glands. (1) Tuberculosis of the lymphatic glands. This is very easy to diagnose...\"\u2014so I figure this isn't what Arlene has, because the doctors are having trouble trying to figure it out.\n\nI start reading about some other diseases: lym-phodenema, lymphodenoma, Hodgkin's disease, all kinds of other things; they're all cancers of one crazy form or another. The only difference between lymphodenema and lymphodenoma was, as far as I could make out by reading it very carefully, that if the patient dies, it's lymphodenoma; if the patient survives\u2014at least for a while\u2014then it's lymphodenema.\n\nAt any rate, I read through all the lymphatic diseases, and decided that the most likely possibility was that Arlene had an incurable disease. Then I half smiled to myself, thinking, \"I bet everybody who reads through a medical book thinks they have a fatal disease.\" And yet, after reading everything very carefully, I couldn't find any other possibility. It was serious.\n\nThen I went to the weekly tea at Palmer Hall, and found myself talking to the mathematicians just as I always did, even though I had just found out that Arlene probably had a fatal disease. It was very strange\u2014like having two minds.\n\nWhen I went to visit her, I told Arlene the joke about the people who don't know any medicine reading the medical book and always assuming they have a fatal disease. But I also told her I thought we were in great difficulty, and that the best I could figure out was that she had an incurable disease. We discussed the various diseases, and I told her what each one was like.\n\nOne of the diseases I told Arlene about was Hodgkin's disease. When she next saw her doctor, she asked him about it: \"Could it be Hodgkin's disease?\"\n\nHe said, \"Well, yes, that's a possibility.\"\n\nWhen she went to the county hospital, the doctor wrote the following diagnosis: \"Hodgkin's disease\u2014?\" So I realized that the doctor didn't know any more than I did about this problem.\n\nThe county hospital gave Arlene all sorts of tests and X-ray treatments for this \"Hodgkin's disease\u2014?\" and there were special meetings to discuss this peculiar case. I remember waiting for her outside, in the hall. When the meeting was over, the nurse wheeled her out in a wheelchair. All of a sudden a little guy comes running out of the meeting room and catches up with us. \"Tell me,\" he says, out of breath, \"do you spit up blood? Have you ever coughed up blood?\"\n\nThe nurse says, \"Go away! Go away! What kind of thing is that to ask of a patient!\"\u2014and brushes him away. Then she turned to us and said, \"That man is a doctor from the neighborhood who comes to the meetings and is always making trouble. That's not the kind of thing to ask of a patient!\"\n\nI didn't catch on. The doctor was checking a certain possibility, and if I had been smart, I would have asked him what it was.\n\nFinally, after a lot of discussion, a doctor at the hospital tells me they figure the most likely possibility is Hodgkin's disease. He says, \"There will be some periods of improvement, and some periods in the hospital. It will be on and off, getting gradually worse. There's no way to reverse it entirely. It's fatal after a few years.\"\n\n\"I'm sorry to hear that,\" I say. \"I'll tell her what you said.\"\n\n\"No, no!\" says the doctor. \"We don't want to upset the patient. \"We're going to tell her it's glandular fever.\"\n\n\"No, no!\" I reply. \"We've already discussed the possibility of Hodgkin's disease. I know she can adjust to it.\"\n\n\"Her parents don't want her to know. You had better talk to them first.\"\n\nAt home, everybody worked on me: my parents, my two aunts, our family doctor; they were all on me, saying I'm a very foolish young man who doesn't realize what pain he's going to bring to this wonderful girl by telling her she has a fatal disease. \"How can you do such a terrible thing?\" they asked, in horror.\n\n\"Because we have made a pact that we must speak honestly with each other and look at everything directly. There's no use fooling around. She's gonna ask me what she's got, and I cannot lie to her!\"\n\n\"Oh, that's childish!\" they said\u2014blah, blah, blah. Everybody kept working on me, and said I was wrong. I thought I was definitely right, because I had already talked to Arlene about the disease and knew she could face it\u2014 that telling her the truth was the right way to handle it.\n\nBut finally, my little sister comes up to me\u2014she was eleven or twelve then\u2014with tears running down her face. She beats me on the chest, telling me that Arlene is such a wonderful girl, and that I'm such a foolish, stubborn brother. I couldn't take it any more. That broke me down.\n\nSo I wrote Arlene a goodbye love letter, figuring that if she ever found out the truth after I had told her it was glandular fever, we would be through. I carried the letter with me all the time.\n\nThe gods never make it easy; they always make it harder. I go to the hospital to see Arlene\u2014having made this decision\u2014and there she is, sitting up in bed, surrounded by her parents, somewhat distraught. When she sees me, her face lights up and she says, \"Now I know how valuable it is that we tell each other the truth!\" Nodding at her parents, she continues, \"They're telling me I have glandular fever, and I'm not sure whether I believe them or not. Tell me, Richard, do I have Hodgkin's disease or glandular fever?\"\n\n\"You have glandular fever,\" I said, and I died inside. It was terrible\u2014just terrible!\n\nHer reaction was completely simple: \"Oh! Fine! Then I believe them.\" Because we had built up so much trust in each other, she was completely relieved. Everything was solved, and all was very nice.\n\nShe got a little bit better, and went home for a while. About a week later, I get a telephone call. \"Richard,\" she says, \"I want to talk to you. Come on over.\"\n\n\"Okay.\" I made sure I still had the letter with me. I could tell something was the matter.\n\nI go upstairs to her room, and she says, \"Sit down.\" I sit down on the end of her bed. \"All right, now tell me,\" she says, \"do I have glandular fever or Hodgkin's disease?\"\n\n\"You have Hodgkin's disease.\" And I reached for the letter.\n\n\"God!\" she says. \"They must have put you through hell!\"\n\nI had just told her she has a fatal disease, and was admitting that I had lied to her as well, and what does she think of? She's worried about _me!_ I was terribly ashamed of myself. I gave Arlene the letter.\n\n\"You should have stuck by it. We know what we're doing; we are right!\"\n\n\"I'm sorry. I feel awful.\"\n\n\"I understand, Richard. Just don't do it again.\"\n\nYou see, she was in bed upstairs, and did something she used to do when she was little: she tiptoed out of bed and crawled down the stairs a little bit to listen to what people were doing downstairs. She heard her mother crying a lot, and went back to bed thinking, \"If I have glandular fever, why is Mother crying so much? But Richard said I had glandular fever, so it must be right!\"\n\nLater she thought, \"Could _Richard_ have lied to me?\" and began to wonder how that might be possible. She concluded that, incredible as it sounded, somebody might have put me through a wringer of some sort.\n\nShe was so good at facing difficult situations that she went on to the next problem. \"Okay,\" she says, \"I have Hodgkin's disease. What are we going to do now?\"\n\nI had a scholarship at Princeton, and they wouldn't let me keep it if I got married. We knew what the disease was like: sometimes it would get better for some months, and Arlene could be at home, and then she would have to be in the hospital for some months\u2014back and forth for two years, perhaps.\n\nSo I figure, although I'm in the middle of trying to get my Ph.D., I could get a job at the Bell Telephone Laboratories doing research\u2014it was a very good place to work\u2014and we could get a little apartment in Queens that wasn't too far from the hospital or Bell Labs. We could get married in a few months, in New York. We worked everything out that afternoon.\n\nFor some months now Arlene's doctors had wanted to take a biopsy of the swelling on her neck, but her parents didn't want it done\u2014they didn't want to \"bother the poor sick girl.\" But with new resolve, I kept working on them, explaining that it's important to get as much information as possible. With Arlene's help, I finally convinced her parents.\n\nA few days later, Arlene telephones me and says, \"They got a report from the biopsy.\"\n\n\"Yeah? Is it good or bad?\"\n\n\"I don't know. Come over and let's talk about it,\"\n\nWhen I got to her house, she showed me the report. It said, \"Biopsy shows tuberculosis of the lymphatic gland.\"\n\nThat really got me. I mean, that was the first goddamn thing on the list! I passed it by, because the book said it was easy to diagnose, and because the doctors were having so much trouble trying to figure out what it was. I assumed they had checked the obvious case. And it _was_ the obvious case: the man who had come running out of the meeting room asking \"Do you spit up blood?\" had the right idea. He knew what it probably was!\n\nI felt like a jerk, because I had passed over the obvious possibility by using circumstantial evidence\u2014which isn't any good\u2014and by assuming the doctors were more intelligent than they were. Otherwise, I would have suggested it right off, and perhaps the doctor would have diagnosed Arlene's disease way back then as \"tuberculosis of the lymphatic gland\u2014?\" I was a dope. I've learned, since then.\n\nAnyway, Arlene says, \"So I might live as long as seven years. I may even get better.\"\n\n\"So what do you mean, you don't know if it's good or bad?\"\n\n\"Well, now we won't be able to get married until later.\"\n\nKnowing that she only had two more years to live, we had solved things so perfectly, from her point of view, that she was disturbed to discover she'd live longer! But it didn't take me long to convince her it was a better circumstance.\n\nSo we knew we could face things together, from then on. After going through that, we had no difficulty facing any other problem.\n\nWhen the war came, I was recruited to work on the Manhattan Project at Princeton, where I was finishing up my degree. A few months later, as soon as I got my degree, I announced to my family that I wanted to get married.\n\nMy father was horrified, because from the earliest times, as he saw me develop, he thought I would be happy as a scientist. He thought it was still too early to marry\u2014it would interfere with my career. He also had this crazy idea: if a guy was in some difficulty, he used to always say, \"Cherchez la femme\"\u2014look for the woman behind it. He felt that women were the great danger to a man, that a man always has to watch out and be tough about women. And when he sees me marrying a girl with tuberculosis, he thinks of the possibility that I'm going to get sick, too.\n\nMy whole family was worried about that\u2014aunts, uncles, everyone. They brought the family doctor over to our house. He tried to explain to me that tuberculosis is a dangerous disease, and that I'm bound to get it.\n\nI said, \"Just tell me how it's transmitted, and we'll figure it out.\" We were already very, very careful: we knew we must not kiss, because there's a lot of bacteria in the mouth.\n\nThen they very carefully explained to me that when I had promised to marry Arlene, I didn't know the situation. Everybody would understand that I didn't know the situation then, and that it didn't represent a real promise.\n\nI never had that feeling, that crazy idea that they had, that I was getting married because I had promised it. I hadn't even _thought_ of that. It wasn't a question of having promised anything; we had stalled around, not getting a piece of paper and not being formally married, but we were in love, and were already married, emotionally.\n\nI said, \"Would it be sensible for a husband who learns that his wife has tuberculosis to leave her?\"\n\nOnly my aunt who ran the hotel thought maybe it would be all right for us to get married. Everybody else was still against it. But this time, since my family had given me this kind of advice before and it had been so wrong, I was in a much stronger position. It was very easy to resist and to just proceed. So there was no problem, really. Although it was a similar circumstance, they weren't going to convince me of anything any more. Arlene and I knew we were right in what we were doing.\n\nArlene and I worked everything out. There was a hospital in New Jersey just south of Fort Dix where she could stay while I was at Princeton. It was a charity hospital\u2014Deborah was the name of it\u2014supported by the Women's Garment Workers Union of New York. Arlene wasn't a garment worker, but it didn't make any difference. And I was just a young fella working on this project for the government, and the pay was very low. But this way I could take care of her, at last.\n\nWe decided to get married on the way to Deborah Hospital. I went to Princeton to pick up a car\u2014Bill Woodward, one of the graduate students there, lent me his station wagon. I fixed it up like a little ambulance, with a mattress and sheets in the back, so Arlene could lie down in case she got tired. Although this was one of the periods when the disease was apparently not so bad and she was at home, Arlene had been in the county hospital a lot, and she was a little weak.\n\nI drove up to Cedarhurst and picked up my bride. Arlene's family waved goodbye, and off we went. We crossed Queens and Brooklyn, then went to Staten Island on the ferry\u2014that was our romantic boat ride\u2014and drove to the city hall for the borough of Richmond to get married.\n\nWe went up the stairs, slowly, into the office. The guy there was very nice. He did everything right away. He said, \"You don't have any witnesses,\" so he called the bookkeeper and an accountant from another room, and we were married according to the laws of the state of New York. Then we were very happy, and we smiled at each other, holding hands.\n\nThe bookkeeper says to me, \"You're married now. You should kiss the bride!\"\n\nSo the bashful character kissed his bride lightly on the cheek.\n\nI gave everyone a tip and we thanked them very much. We got back in the car, and drove to Deborah Hospital.\n\nEvery weekend I'd go down from Princeton to visit Arlene. One time the bus was late, and I couldn't get into the hospital. There weren't any hotels nearby, but I had my old sheepskin coat on (so I was warm enough), and I looked for an empty lot to sleep in. I was a little worried what it might look like in the morning when people looked out of their windows, so I found a place that was far enough away from houses.\n\nThe next morning I woke up and discovered I'd been sleeping in a garbage dump\u2014a landfill! I felt foolish, and laughed.\n\nArlene's doctor was very nice, but he would get upset when I brought in a war bond for $18 every month. He could see we didn't have much money, and kept insisting we shouldn't contribute to the hospital, but I did it anyway.\n\nOne time, at Princeton, I received a box of pencils in the mail. They were dark green, and in gold letters were the words \"RICHARD DARLING, I LOVE YOU! PUTSY.\" It was Arlene (I called her Putsy).\n\nWell, that was nice, and I love her, too, but\u2014you know how you absentmindedly drop pencils around: you're showing Professor Wigner a formula, or something, and leave the pencil on his desk.\n\nIn those days we didn't have extra stuff, so I didn't want to waste the pencils. I got a razor blade from the bathroom and cut off the writing on one of them to see if I could use them.\n\nThe next morning, I get a letter in the mail. It starts out, \"WHAT'S THE IDEA OF TRYING TO CUT THE NAME OFF THE PENCILS?\"\n\nIt continues: \"Aren't you proud of the fact that I love you?\" Then: \"WHAT DO YOU CARE WHAT OTHER PEOPLE THINK?\"\n\nThen came poetry: \"If you're ashamed of me, dah dah, then Pecans to you! Pecans to you!\" The next verse was the same kind of stuff, with the last line, \"Almonds to you! Almonds to you!\" Each one was \"Nuts to you!\" in a different form.\n\nSo I had to use the pencils with the names on them. What else could I do?\n\nIt wasn't long before I had to go to Los Alamos. Robert Oppenheimer, who was in charge of the project, arranged for Arlene to stay in the nearest hospital, in Albuquerque, about a hundred miles away. I had time off every weekend to see her, so I would hitchhike down on a Saturday, see Arlene in the afternoon, and stay overnight in a hotel there in Albuquerque. Then on Sunday morning I would see Arlene again, and hitchhike back to Los Alamos in the afternoon.\n\nDuring the week I would often get letters from her. Some of them, like the one written on a jigsaw-puzzle blank and then taken apart and sent in a sack, resulted in little notes from the army censor, such as \"Please tell your wife we don't have time to play games around here.\" I didn't tell her anything. I _liked_ her to play games\u2014even though she often put me in various uncomfortable but amusing conditions from which I could not escape.\n\nOne time, near the beginning of May, newspapers mysteriously appeared in almost everybody's mailbox at Los Alamos. The whole damn place was full of them\u2014hundreds of newspapers. You know the kind\u2014you open it up and there's this headline screaming in thick letters across the front page: ENTIRE NATION CELEBRATES BIRTHDAY OF R.P. FEYNMAN!\n\nArlene was playing her game with the world. She had a lot of time to think. She would read magazines, and send away for this and that. She was always cooking up something. (She must have got help with the names from Nick Metropolis or one of the other guys at Los Alamos who would often visit her.) Arlene was in her room, but she was in the world, writing me crazy letters and sending away for all kinds of stuff.\n\nOne time she sent me a big catalog of kitchen equipment\u2014the kind you need for enormous institutions like prisons, which have a lot of people in them. It showed everything from blowers and hoods for stoves to huge pots and pans. So I'm thinking, \"What the hell is this?\"\n\nIt reminded me of the time I was up at MIT and Arlene sent me a catalog describing huge boats, from warships to ocean liners\u2014great big boats. I wrote to her: \"What's the idea?\"\n\nShe writes back: \"I just thought that maybe, when we get married, we could buy a boat.\"\n\nI write, \"Are you crazy? It's all out of proportion!\"\n\nThen another catalog comes: it's for big yachts\u2014forty-foot schooners and stuff like that\u2014for very rich people. She writes, \"Since you said no to the other boats, maybe we could get one of these.\"\n\nI write, \"Look: you're way out of scale!\"\n\nSoon another catalog comes: it's for various kinds of motor boats\u2014Chriscraft this and that.\n\nI write, \"Too expensive!\"\n\nFinally, I get a note: \"This is your last chance, Richard. You're always saying no.\" It turns out a friend of hers has a rowboat she wants to sell for $15\u2014a used rowboat\u2014and maybe we could buy it so we could row around in the water next summer.\n\nSo, yes. I mean, how can you say no after all that?\n\nWell, I'm still trying to figure out what this big catalog for institutional kitchen equipment is leading to, when another catalog comes: it's for hotels and restaurants\u2014supplies for small and medium-sized hotels and restaurants. Then a few days later, a catalog for the kitchen in your new home comes.\n\nWhen I go down to Albuquerque the next Saturday, I find out what it's all about. There's a little charcoal broiler in her room\u2014she's bought it through the mail from Sears. It's about eighteen inches across, with little legs.\n\n\"I thought we could have steaks,\" Arlene says.\n\n\"How the hell can we use it in the room, here, with all the smoke and everything?\"\n\n\"Oh, no,\" she says. \"All you have to do is take it out on the lawn. Then you can cook us steaks every Sunday.\"\n\nThe hospital was right on Route 66, the main road across the United States! \"I can't do that,\" I said. \"I mean, with all the cars and trucks going by, all the people on the sidewalk walking back and forth, I can't just go out there and start cookin' steaks on the lawn!\"\n\n\"What do _you_ care what other people think?\" (Arlene _tortured_ me with that!) \"Okay,\" she says, opening a drawer, \"we'll compromise: you don't have to wear the chefs hat and the gloves.\"\n\nShe holds up a hat\u2014it's a real chefs hat\u2014and gloves. Then she says, \"Try on the apron,\" as she unfolds it. It has something silly written across it, like \"BAR-B-Q,KING,\" or something.\n\n\"Okay, okay!\" I say, horrified. \"I'll cook the steaks on the lawn!\" So every Saturday or Sunday, I'd go out there on Route 66 and cook steaks.\n\nThen there were the Christmas cards. One day, only a few weeks after I had arrived at Los Alamos, Arlene says, \"I thought it would be nice to send Christmas cards to everybody. Would you like to see the ones I picked out?\"\n\nThey were nice cards, all right, but inside they said Merry Christmas, from Rich & Putsy. \"I can't send these to Fermi and Bethe,\" I protested. \"I hardly even know them!\"\n\n\"What do _you_ care what other people think?\"\u2014naturally. So we sent them.\n\nNext year comes around, and by this time I know Fermi. I know Bethe. I've been over at their houses. I've taken care of their kids. We're all very friendly.\n\nSomewhere along the line, Arlene says to me, in a very formal tone, \"You haven't asked me about our Christmas cards this year, Richard...\"\n\nFEAR goes through me. \"Uh, well, let's see the cards.\"\n\nThe cards say Merry Christmas and a Happy New Year, from Richard and Arlene Feynman. \"Well, that's fine,\" I say. \"They're very nice. They'll go fine for everybody.\"\n\n\"Oh, no,\" she says. \"They won't do for Fermi and Bethe and all those other famous people.\" Sure enough, she's got another box of cards.\n\nShe pulls one out. It says the usual stuff, and then: From Dr. & Mrs. R. P. Feynman.\n\nSo I had to send them those.\n\n\"What's this formal stuff, Dick?\" they laughed. They were happy that she was having such a good time out of it, and that I had no control over it.\n\nArlene didn't spend all of her time inventing games. She had sent away for a book called _Sound and Symbol in Chinese._ It was a lovely book\u2014I still have it\u2014with about fifty symbols done in beautiful calligraphy, with explanations like \"Trouble: three women in a house.\" She had the right paper, brushes, and ink, and was practicing calligraphy. She had also bought a Chinese dictionary, to get a lot of other symbols.\n\nOne time when I came to visit her, Arlene was practicing these things. She says to herself, \"No. That one's wrong.\"\n\nSo I, the \"great scientist,\" say, \"What do you mean, 'wrong'? It's only a human convention. There's no law of nature which says how they're supposed to look; you can draw them any way you want.\"\n\n\"I mean, artistically it's wrong. It's a question of balance, of how it feels.\"\n\n\"But one way is just as good as another,\" I protest.\n\n\"Here,\" she says, and she hands me the brush. \"Make one yourself.\"\n\nSo I made one, and I said, \"Wait a minute. Let me make another one\u2014it's too blobby.\" (I couldn't say it was wrong, after all.)\n\n\"How do you know how blobby it's supposed to be?\" she says.\n\nI learned what she meant. There's a particular way you have to make the stroke for it to look good. An aesthetic thing has a certain set, a certain character, which I can't define. Because it couldn't be defined made me think there was nothing to it. But I learned from that experience that there _is_ something to it\u2014and it's a fascination I've had for art ever since.\n\nJust at this moment, my sister sends me a postcard from Oberlin, where she's going to college. It's written in pencil, with small symbols\u2014it's in Chinese.\n\nJoan is nine years younger than I am, and studied physics, too. Having me as her older brother was tough on her. She was always looking for something I couldn't do, and was secretly taking Chinese.\n\nWell, I didn't know any Chinese, but one thing I'm good at is spending an infinite amount of time solving a puzzle. The next weekend I took the card with me to Albuquerque. Arlene showed me how to look up the symbols. You have to start in the back of the dictionary with the right category and count the number of strokes. Then you go into the main part of the dictionary. It turns out each symbol has several possible meanings, and you have to put several symbols together before you can understand it.\n\nWith great patience I worked everything out. Joan was saying things like, \"I had a good time today.\" There was only one sentence I couldn't figure out. It said, \"Yesterday we celebrated mountain-forming day\"\u2014obviously an error. (It turned out they did have some crazy thing called \"Mountain-forming Day\" at Oberlin, and I had translated it right!)\n\nSo it was trivial things like you'd expect to have on a postcard, but I knew from the situation that Joan was trying to floor me by sending me Chinese.\n\nI looked back and forth through the art book and picked out four symbols which would go well together. Then I practiced each one, over and over. I had a big pad of paper, and I would make fifty of each one, until I got it just right.\n\nWhen I had accidentally made one good example of each symbol, I saved them. Arlene approved, and we glued the four of them end to end, one on top of the other. Then we put a little piece of wood on each end, so you could hang it up on the wall. I took a picture of my masterpiece with Nick Metropolis's camera, rolled up the scroll, put it in a tube, and sent it to Joan.\n\nSo she gets it. She unrolls it, and she can't read it. It looks to her as if I simply made four characters, one right after the other, on the scroll. She takes it to her teacher.\n\nThe first thing he says is, \"This is written rather well! Did you do this?\"\n\n\"Uh, no. What does it say?\"\n\n\"Elder brother also speaks.\"\n\nI'm a real bastard\u2014I would never let my little sister score one on me.\n\nWhen Arlene's condition became much weaker, her father came out from New York to visit her. It was difficult and expensive to travel that far during the war, but he knew the end was near. One day he telephoned me at Los Alamos. \"You'd better come down here right away,\" he said.\n\nI had arranged ahead of time with a friend of mine at Los Alamos, Klaus Fuchs, to borrow his car in case of an emergency, so I could get to Albuquerque quickly. I picked up a couple of hitchhikers to help me in case something happened on the way.\n\nSure enough, as we were driving into Santa Fe, we got a flat tire. The hitchhikers helped me change the tire. Then on the other side of Santa Fe, the spare tire went flat, but there was a gas station nearby. I remember waiting patiently for the gas station man to take care of some other car, when the two hitchhikers, knowing the situation, went over and explained to the man what it was. He fixed the flat right away. We decided not to get the spare tire fixed, because repairing it would have taken even more time.\n\nWe started out again towards Albuquerque, and I felt foolish that I hadn't thought to say anything to the gas station man when time was so precious. About thirty miles from Albuquerque, we got another flat! We had to abandon the car, and we hitchhiked the rest of the way. I called up a towing company and told them the situation.\n\nI met Arlene's father at the hospital. He had been there for a few days. \"I can't take it any more,\" he said. \"I have to go home.\" He was so unhappy, he just left.\n\nWhen I finally saw Arlene, she was very weak, and a bit fogged out. She didn't seem to know what was happening. She stared straight ahead most of the time, looking around a little bit from time to time, and was trying to breathe. Every once in a while her breathing would stop\u2014and she would sort of swallow\u2014and then it would start again. It kept going like this for a few hours.\n\nI took a little walk outside for a while. I was surprised that I wasn't feeling what I thought people were supposed to feel under the circumstances. Maybe I was fooling myself. I wasn't delighted, but I didn't feel terribly upset, perhaps because we had known for a long time that it was going to happen.\n\nIt's hard to explain. If a Martian (who, we'll imagine, never dies except by accident) came to Earth and saw this peculiar race of creatures\u2014these humans who live about seventy or eighty years, knowing that death is going to come\u2014it would look to him like a terrible problem of psychology to live under those circumstances, knowing that life is only temporary. Well, we humans somehow figure out how to live despite this problem: we laugh, we joke, we live.\n\nThe only difference for me and Arlene was, instead of fifty years, it was five years. It was only a quantitative difference\u2014the psychological problem was just the same. The only way it would have become any different is if we had said to ourselves, \"But those other people have it better, because they might live fifty years.\" But that's crazy. Why make yourself miserable saying things like, \"Why do we have such bad luck? What has God done to us? What have we done to deserve this?\"\u2014all of which, if you understand reality and take it completely into your heart, are irrelevant and unsolvable. They are just things that nobody can know. Your situation is just an accident of life.\n\nWe had a hell of a good time together.\n\nI came back into her room. I kept imagining all the things that were going on physiologically: the lungs aren't getting enough air into the blood, which makes the brain fogged out and the heart weaker, which makes the breathing even more difficult. I kept expecting some sort of ava-lanching effect, with everything caving in together in a dramatic collapse. But it didn't appear that way at all: she just slowly got more foggy, and her breathing gradually became less and less, until there was no more breath\u2014but just before that, there was a very small one.\n\nThe nurse on her rounds came in and confirmed that Arlene was dead, and went out\u2014I wanted to be alone for a moment. I sat there for a while, and then went over to kiss her one last time.\n\nI was very surprised to discover that her hair smelled exactly the same. Of course, after I stopped and thought about it, there was no reason why hair should smell different in such a short time. But to me it was a kind of a shock, because in my mind, something enormous had just happened\u2014and yet nothing had happened.\n\nThe next day, I went to the mortuary. The guy hands me some rings he's taken from her body. \"Would you like to see your wife one last time?\" he asks.\n\n\"What kind of a\u2014no, I don't want to see her, no!\" I said. \"I just saw her!\"\n\n\"Yes, but she's been all fixed up,\" he says.\n\nThis mortuary stuff was completely foreign to me. Fixing up a body when there's nothing there? I didn't want to look at Arlene again; that would have made me more upset.\n\nI called the towing company and got the car, and packed Arlene's stuff in the back. I picked up a hitchhiker, and started out of Albuquerque.\n\nIt wasn't more than five miles before... BANG! Another flat tire. I started to curse.\n\nThe hitchhiker looked at me like I was mentally unbalanced. \"It's just a tire, isn't it?\" he says.\n\n\"Yeah, it's just a tire\u2014and another tire, and again another tire, and another tire!\"\n\nWe put the spare tire on, and went very slowly, all the way back to Los Alamos, without getting the other tire repaired.\n\nI didn't know how I was going to face all my friends at Los Alamos. I didn't want people with long faces talking to me about the death of Arlene. Somebody asked me what happened.\n\n\"She's dead. And how's the program going?\" I said.\n\nThey caught on right away that I didn't want to moon over it. Only one guy expressed his sympathy, and it turned out he had been out of town when I came back to Los Alamos.\n\nOne night I had a dream, and Arlene came into it. Right away, I said to her, \"No, no, you can't be in this dream. You're not alive!\"\n\nThen later, I had another dream with Arlene in it. I started in again, saying, \"You can't be in this dream!\"\n\n\"No, no,\" she says. \"I fooled you. I was tired of you, so I cooked up this ruse so I could go my own way. But now I like you again, so I've come back.\" My mind was really working against itself. It had to be explained, even in a goddamn _dream,_ why it was possible that she was still there!\n\nI must have done something to myself, psychologically. I didn't cry until about a month later, when I was walking past a department store in Oak Ridge and noticed a pretty dress in the window. I thought, \"Arlene would like that,\" and then it hit me.\n\n### It's as Simple as One, Two, Three...\n\n_WHEN_ I was a kid growing up in Far Rockaway, I had a friend named Bernie Walker. We both had \"labs\" at home, and we would do various \"experiments.\" One time, we were discussing something\u2014we must have been eleven or twelve at the time\u2014and I said, \"But thinking is nothing but talking to yourself inside.\"\n\n\"Oh yeah?\" Bernie said. \"Do you know the crazy shape of the crankshaft in a car?\"\n\n\"Yeah, what of it?\"\n\n\"Good. Now, tell me: how did you describe it when you were talking to yourself?\"\n\nSo I learned from Bernie that thoughts can be visual as well as verbal.\n\nLater on, in college, I became interested in dreams. I wondered how things could look so real, just as if light were hitting the retina of the eye, while the eyes are closed: are the nerve cells on the retina actually being stimulated in some other way\u2014by the brain itself, perhaps\u2014or does the brain have a \"judgment department\" that gets slopped up during dreaming? I never got satisfactory answers to such questions from psychology, even though I became very interested in how the brain works. Instead, there was all this business about interpreting dreams, and so on.\n\nWhen I was in graduate school at Princeton a kind of dumb psychology paper came out that stirred up a lot of discussion. The author had decided that the thing controlling the \"time sense\" in the brain is a chemical reaction involving iron. I thought to myself, \"Now, how the hell could he figure that?\"\n\nWell, the way he did it was, his wife had a chronic fever which went up and down a lot. Somehow he got the idea to test her sense of time. He had her count seconds to herself (without looking at a clock), and checked how long it took her to count up to 60. He had her counting\u2014the poor woman\u2014all during the day: when her fever went up, he found she counted quicker; when her fever went down, she counted slower. Therefore, he thought, the thing that governed the \"time sense\" in the brain must be running faster when she's got fever than when she hasn't got fever.\n\nBeing a very \"scientific\" guy, the psychologist knew that the rate of a chemical reaction varies with the surrounding temperature by a certain formula that depends on the energy of the reaction. He measured the differences in speed of his wife's counting, and determined how much the temperature changed the speed. Then he tried to find a chemical reaction whose rates varied with temperature in the same amounts as his wife's counting did. He found that iron reactions fit the pattern best. So he deduced that his wife's sense of time was governed by a chemical reaction in her body involving iron.\n\nWell, it all seemed like a lot of baloney to me\u2014there were so many things that could go wrong in his long chain of reasoning. But it _was_ an interesting question: what _does_ determine the \"time sense\"? When you're trying to count at an even rate, what does that rate depend on? And what could you do to yourself to change it?\n\nI decided to investigate. I started by counting seconds\u2014without looking at a clock, of course\u2014up to 60 in a slow, steady rhythm: 1, 2, 3, 4, 5.... When I got to 60, only 48 seconds had gone by, but that didn't bother me: the problem was not to count for exactly one minute, but to count at a standard rate. The next time I counted to 60, 49 seconds had passed. The next time, 48. Then 47, 48, 49, 48, 48.... So I found I could count at a pretty standard rate.\n\nNow, if I just sat there, without counting, and waited until I thought a minute had gone by, it was very irregular\u2014complete variations. So I found it's very poor to estimate a minute by sheer guessing. But by counting, I could get very accurate.\n\nNow that I knew I could count at a standard rate, the next question was\u2014what affects the rate?\n\nMaybe it has something to do with the heart rate. So I began to run up and down the stairs, up and down, to get my heart beating fast. Then I'd run into my room, throw myself down on the bed, and count up to 60.\n\nI also tried running up and down the stairs and counting to myself _while_ I was running up and down.\n\nThe other guys saw me running up and down the stairs, and laughed. \"What are you doing?\"\n\nI couldn't answer them\u2014which made me realize I couldn't talk while I was counting to myself\u2014and kept right on running up and down the stairs, looking like an idiot.\n\n(The guys at the graduate college were used to me looking like an idiot. On another occasion, for example, a guy came into my room\u2014I had forgotten to lock the door during the \"experiment\"\u2014and found me in a chair wearing my heavy sheepskin coat, leaning out of the wide-open window in the dead of winter, holding a pot in one hand and stirring with the other. \"Don't bother me! Don't bother me!\" I said. I was stirring Jell-O and watching it closely: I had gotten curious as to whether Jell-O would coagulate in the cold if you kept it moving all the time.)\n\nAnyway, after trying every combination of running up and down the stairs and lying on the bed, surprise! The heart rate had no effect. And since I got very hot running up and down the stairs, I figured temperature had nothing to do with it either (although I must have known that your temperature doesn't really go up when you exercise). In fact, I couldn't find anything that affected my rate of counting.\n\nRunning up and down stairs got pretty boring, so I started counting while I did things I had to do anyway. For instance, when I put out the laundry, I had to fill out a form saying how many shirts I had, how many pants, and so on. I found I could write down \"3\" in front of \"pants\" or \"4\" in front of \"shirts,\" but I couldn't count my socks. There were too many of them: I'm already using my \"counting machine\"\u201436, 37, 38\u2014and here are all these socks in front of me\u201439, 40, 41.... How do I count the socks?\n\nI found I could arrange them in geometrical patterns\u2014like a square, for example: a pair of socks in this corner, a pair in that one; a pair over here, and a pair over there\u2014eight socks.\n\nI continued this game of counting by patterns, and found I could count the lines in a newspaper article by grouping the lines into patterns of 3, 3, 3, and 1 to get 10; then 3 of those patterns, 3 of those patterns, 3 of those patterns, and 1 of those patterns made 100. I went right down the newspaper like that. After I had finished counting up to 60,1 knew where I was in the patterns and could say, \"I'm up to 60, and there are 113 lines.\" I found that I could even _read_ the articles while I counted to 60, and it didn't affect the rate! In fact, I could do anything while counting to myself\u2014except talk out loud, of course.\n\nWhat about typing\u2014copying words out of a book? I found that I could do that, too, but here my time was affected. I was excited: finally, I've found something that appears to affect my counting rate! I investigated it more.\n\nI would go along, typing the simple words rather fast, counting to myself 19, 20, 21, typing along, counting 27, 28, 29, typing along, until\u2014What the hell is that word?\u2014 Oh, yeah\u2014and then continue counting 30, 31, 32, and so on. When I'd get to 60, I'd be late.\n\nAfter some introspection and further observation, I realized what must have happened: I would interrupt my counting when I got to a difficult word that \"needed more brains,\" so to speak. My counting rate wasn't slowing down; rather, the counting itself was being held up temporarily from time to time. Counting to 60 had become so automatic that I didn't even notice the interruptions at first.\n\nThe next morning, over breakfast, I reported the results of all these experiments to the other guys at the table. I told them all the things I could do while counting to myself, and said the only thing I absolutely could not do while counting to myself was talk.\n\nOne of the guys, a fella named John Tukey, said, \"I don't believe you can read, and I don't see why you can't talk. I'll bet you I can talk while counting to myself, and I'll bet you you can't read.\"\n\nSo I gave a demonstration: they gave me a book and I read it for a while, counting to myself. When I reached 60 I said, \"Now!\"\u201448 seconds, my regular time. Then I told them what I had read.\n\nTukey was amazed. After we checked him a few times to see what his regular time was, he started talking: \"Mary had a little lamb; I can say anything I want to, it doesn't make any difference; I don't know what's bothering you\"\u2014blah, blah, blah, and finally, \"Okay!\" He hit his time right on the nose! I couldn't believe it!\n\nWe talked about it a while, and we discovered something. It turned out that Tukey was counting in a different way: he was visualizing a tape with numbers on it going by. He would say, \"Mary had a little lamb,\" and he would _watch_ it! Well, now it was clear: he's \"looking\" at his tape going by, so he can't read, and I'm \"talking\" to myself when I'm counting, so I can't speak!\n\nAfter that discovery, I tried to figure out a way of reading out loud while counting\u2014something neither of us could do. I figured I'd have to use a part of my brain that wouldn't interfere with the seeing or speaking departments, so I decided to use my fingers, since that involved the sense of touch.\n\nI soon succeeded in counting with my fingers and reading out loud. But I wanted the whole process to be mental, and not rely on any physical activity. So I tried to imagine the feeling of my fingers moving while I was reading out loud.\n\nI never succeeded. I figured that was because I hadn't practiced enough, but it might be impossible: I've never met anybody who can do it.\n\nBy that experience Tukey and I discovered that what goes on in different people's heads when they _think_ they're doing the same thing\u2014something as simple as _counting_ \u2014is different for different people. And we discovered that you can externally and objectively test how the brain works: you don't have to ask a person how he counts and rely on his own observations of himself; instead, you observe what he can and can't do while he counts. The test is absolute. There's no way to beat it; no way to fake it.\n\nIt's natural to explain an idea in terms of what you already have in your head. Concepts are piled on top of each other: this idea is taught in terms of that idea, and that idea is taught in terms of another idea, which comes from counting, which can be so different for different people!\n\nI often think about that, especially when I'm teaching some esoteric technique such as integrating Bessel functions. When I see equations, I see the letters in colors\u2014I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with lighttan _j_ 's, slightly violet-bluish n's, and dark brown _x_ 's flying around. And I wonder what the hell it must look like to the students.\n\n### Getting Ahead\n\n_ONE TIME_ , back in the fifties, when I was returning from Brazil by boat, we stopped off in Trinidad for a day, so I decided to see the main city, Port of Spain. In those days, when I visited a city I was most interested in seeing the poorest sections\u2014to see how life works at the bottom end\n\nI spent some time off in the hills, in the Negro section of town, wandering around on foot. On the way back a taxi stopped and the driver said, \"Hey, mon! You want to see the city? It only cost five biwi.\"\n\nI said, \"Okay,\" and got in the taxi.\n\nThe driver started right off to go up and see some palace, saying, \"I'll show you all the fancy places.\"\n\nI said, \"No, thank you; that's similar in every city. I want to see the bottom part of the city, where the poor people live. I've already seen the hills up there.\"\n\n\"Oh!\" he said, impressed. \"I'll be glad to show you around. And I have a question for you when we're through, so I want you to look at everything carefully.\"\n\nSo he took me to an East Indian neighborhood\u2014it must have been some housing project\u2014and he stopped in front of a house made of concrete blocks. There was practically nothing inside. A man was sitting on the front steps. \"You see that man?\" he said. \"He has a son studyin' medicine in Maryland.\"\n\nThen he picked up someone from the neighborhood so I could better see what they were like. It was a woman whose teeth had a lot of decay.\n\nFurther along we stopped and he introduced me to two women he admired. \"They got enough money together to buy a sewing machine, and now they do sewing and tailoring work for people in the neighborhood,\" he said, proudly. When he introduced me to them, he said, \"This man is a professor, and what's interesting is, he wants to see our neighborhoods.\"\n\nWe saw many things, and finally the taxi driver said to me, \"Now, Professor, here is my question: you see the Indian people are just as poor, and sometimes even poorer than the Negro people, but they're getting somewhere, somehow\u2014this man has sent his son to college; those women are building up a sewing business. But my people aren't getting anywhere. Why is that?\"\n\nI told him, of course, that I didn't know\u2014which is my answer to almost every question\u2014but he wouldn't accept that, coming from a professor. I tried to guess at something which I thought was possible. I said, \"There's a long tradition behind life in India that comes from a religion and philosophy that is thousands of years old. And although these people are not in India, they still pass on those traditions about what's important in life\u2014trying to build for the future and supporting their children in the effort\u2014which have come down to them for centuries.\"\n\nI continued, \"I think that your people have unfortunately not had a chance to develop such a long tradition, or if they did, they lost it through conquest and slavery.\" I don't know if it's true, but it was my best guess.\n\nThe taxi driver felt that it was a good observation, and said he was planning to build for the future, too: he had some money on the horses, and if he won, he would buy his own taxicab, and _really_ do well.\n\nI felt very sorry. I told him that betting on the horses was a bad idea, but he insisted it was the only way he could do it. He had such good intentions, but his method was going to be luck.\n\nI wasn't going to go on philosophizing, so he took me to a place where there was a steel band playing some great calypso music, and I had an enjoyable afternoon.\n\n### Hotel City\n\n_ONE TIME_ , when I was in Geneva, Switzerland, for a Physical Society meeting, I was walking around and happened to go past the United Nations buildings. I thought to myself, \"Gee! I think I'll go in and look around.\" I wasn't particularly dressed for it\u2014I was wearing dirty pants and an old coat\u2014but it turned out there were tours you could go on where some guy would show you around.\n\nThe tour was quite interesting, but the most striking part was the great big auditorium. You know how everything is overdone for these big international characters, so what would ordinarily be a stage or a dais was in several layers: you have to climb up whole sequences of steps to this great, big, monstrous wooden thing that you stand behind, with a big screen in back of you. In front of you are the seats. The carpets are elegant, and the big doors with brass handles at the back are beautiful. On each side of the great auditorium, up above, are windowed booths for the translators of different languages to work in. It's a fantastic place, and I kept thinking to myself, \"Gee! How it must be to give a talk in a place like this!\"\n\nRight after that, we were walking along the corridor just outside the auditorium when the guide pointed through the window and said, \"You see those buildings over there that are under construction? They'll be used for the first time at the Atoms for Peace Conference, in about six weeks.\"\n\nI suddenly remembered that Murray Gell-Mann and I were supposed to give talks at that conference on the present situation of high-energy physics. My talk was set for the plenary session, so I asked the guide, \"Sir, where would the talks for the _plenary_ session of that conference be?\"\n\n\"Back in that room that we just came through.\"\n\n\"Oh!\" I said in delight. \"Then I'm gonna give a speech in that room!\"\n\nThe guide looked down at my dirty pants and my sloppy shirt. I realized how dumb that remark must have sounded to him, but it was genuine surprise and delight on my part.\n\nWe went along a little bit farther, and the guide said, \"This is a lounge for the various delegates, where they often hold informal discussions.\" There were some small, square windows in the doors to the lounge that you could look through, so people looked in. There were a few men sitting there talking.\n\nI looked through the windows and saw Igor Tamm, a physicist from Russia that I know. \"Oh!\" I said. \"I know that guy!\" and I started through the door.\n\nThe guide screamed, \"No, no! Don't go in there!\" By this time he was _sure_ he had a maniac on his hands, but he couldn't chase me because he wasn't allowed to go through the door himself!\n\nTamm's face lit up when he recognized me, and we talked a little bit. The guide was relieved and continued the tour without me, and I had to run to catch up.\n\nAt the Physical Society meeting my good friend Bob Bacher said to me, \"Listen: it's going to be hard to get a room when that Atoms for Peace Conference is going on. Why don't you have the State Department arrange a room for you, if you haven't already made a reservation?\"\n\n\"Naw!\" I said. \"I'm not gonna have the State Department do a damn thing for me! I'll do it myself.\"\n\nWhen I returned to my hotel I told them that I would be leaving in a week, but I'd be coming back at the end of summer: \"Could I make a reservation now for that time?\"\n\n\"Certainly! When will you be returning?\"\n\n\"The second week in September...\"\n\n\"Oh, we're terribly sorry, Professor Feynman; we are already completely booked for that time.\"\n\nSo I wandered off, from one hotel to another, and found they were all booked solid, six weeks ahead of time!\n\nThen I remembered a trick I used once when I was with a physicist friend of mine, a quiet and dignified English fellow.\n\nWe were going across the United States by car, and when we got just beyond Tulsa, Oklahoma, there were supposed to be big floods up ahead. We came into this little town and we saw cars parked everywhere, with people and families in them, trying to sleep. He says, \"We had better stop here. It's clear we can go no further.\"\n\n\"Aw, come on!\" I say. \"How do you know? Let's see if we can _do_ it: maybe by the time we get there, the water will be down.\n\n\"We shouldn't waste time,\" he replies. \"Perhaps we can find a room in a hotel if we look for it now.\"\n\n\"Aw, don't worry about it!\" I say. \"Let's go!\"\n\nWe drive out of town about ten or twelve miles and come to an arroyo. Yes, even for me, there's too much water. There's no question: we aren't going to try to get through _that_.\n\nWe turn around: my friend's muttering about how we'll have no chance of finding a room in a hotel now, and I tell him not to worry.\n\nBack in town, it's absolutely _blocked_ with people sleeping in their cars, obviously because there are no more rooms. All the hotels must be packed. I see a small sign over a door: it says \"HOTEL.\" It was the kind of hotel I was familiar with in Albuquerque, when I would wander around town looking at things, waiting to see my wife at the hospital: you have to go up a flight of stairs and the office is on the first landing.\n\nWe go up the stairs to the office and I say to the manager, \"We'd like a room.\"\n\n\"Certainly, sir. We have one with two beds on the third floor.\"\n\nMy friend is amazed: The town is packed with people sleeping in cars, and here's a hotel that has room!\n\nWe go up to our room, and gradually it becomes clear to him: there's no door on the room, only a hanging cloth in the doorway. The room was fairly clean, it had a sink; it wasn't so bad. We get ready for bed.\n\nHe says, \"I've got to pee.\"\n\n\"The bathroom is down the hall.\"\n\nWe hear girls giggling and walking back and forth in the hall outside, and he's nervous. He doesn't want to go out there.\n\n\"That's all right; just pee in the sink,\" I say.\n\n\"But that's unsanitary.\"\n\n\"Naw, it's okay; you just turn the water on.\"\n\n\"I can't pee in the sink,\" he says.\n\nWe're both tired, so we lie down. It's so hot that we don't use any covers, and my friend can't get to sleep because of the noises in the place. I kind of fall asleep a little bit.\n\nA little later I hear a creaking of the floor nearby, and I open one eye slightly. There he is, in the dark, quietly stepping over to the sink.\n\nAnyway, I knew a little hotel in Geneva called the Hotel City, which was one of those places with just a doorway on the street and a flight of stairs leading up to the office. There were usually some rooms available, and nobody made reservations.\n\nI went up the stairs to the office and told the desk clerk that I'd be back in Geneva in six weeks, and I'd like to stay in their hotel: \"Could I make a reservation?\"\n\n\"Certainly, sir. Of course!\"\n\nThe clerk wrote my name on a piece of paper\u2014they hadn't any book to write reservations in\u2014and I remember the clerk trying to find a hook to put the paper on, to remember. So I had my \"reservation,\" and everything was fine.\n\nI came back to Geneva six weeks later, went to the Hotel City, and they _did_ have the room ready for me; it was on the top floor. Although the place was cheap, it was clean. (It's Switzerland; it was _clean!)_ There were a few holes in the bedspread, but it was a clean bedspread. In the morning they served a European breakfast in my room; they were rather delighted to have this guest who had made a reservation six weeks in advance.\n\nThen I went over to the U.N. for the first day of the Atoms for Peace Conference. There was quite a line at the reception desk, where everyone was checking in: a woman was taking down everybody's address and phone number so they could be reached in case there were any messages.\n\n\"Where are you staying, Professor Feynman?\" she asks.\n\n\"At the Hotel City.\"\n\n\"Oh, you must mean the Hotel Cit\u00e9.\"\n\n\"No, it's called 'City': C-I-T-Y.\" (Why not? We would call it \"Cit\u00e9\" here in America, so they called it \"City\" in Geneva, because it sounded foreign.)\n\n\"But it isn't on our list of hotels. Are you _sure_ it's 'City'?\"\n\n\"Look in the telephone book for the number. You'll find it.\"\n\n\"Oh!\" she said, after checking the phone book. \"My list is incomplete! Some people are still looking for a room, so perhaps I can recommend the Hotel City to them.\"\n\nShe must have got the word about the Hotel City from someone, because nobody else from the conference ended up staying there. Once in a while the people at the Hotel City would receive telephone calls for me from the U.N., and would _run up_ the two flights of stairs from the office to tell me, with some awe and excitement, to come down and answer the phone.\n\nThere's an amusing scene I remember from the Hotel City. One night I was looking through my window out into the courtyard. Something, in a building across the courtyard, caught the corner of my eye: it looked like an upside-down bowl on the windowsill. I thought it had moved, so I watched it for a while, but it didn't move any. Then, after a bit, it moved a little to one side. I couldn't figure out what this thing was.\n\nAfter a while I figured it out: it was a man with a pair of binoculars that he had against the windowsill for support, looking across the courtyard to the floor below me!\n\nThere's another scene at the Hotel City which I'll always remember, that I'd love to be able to paint: I was returning one night from the conference and opened the door at the bottom of the stairway. There was the proprietor, standing there, trying to look nonchalant with a cigar in one hand while he pushed something up the stairs with the other. Farther up, the woman who brought me breakfast was pulling on this same heavy object with both hands. And at the top of the stairs, at the landing, there _she_ was, with her fake furs on, bust sticking out, hand on her hip, _imperiously waiting_. Her customer was a bit drunk, and was not very capable of walking up the steps. I don't know whether the proprietor knew that \/ knew what this was all about; I just walked past everything. He was ashamed of his hotel, but, of course, to me, it was delightful.\n\n### Who the Hell Is Herman?\n\n_ONE DAY_ I got a long-distance telephone call from an old friend in Los Alamos. She says in a very serious voice, \"Richard, I have some sad news for you. Herman died.\"\n\nI'm always feeling uncomfortable that I don't remember names and then I feel bad that I don't pay enough attention to people. So I said, \"Oh?\"\u2014trying to be quiet and serious so I could get more information, but thinking to myself, \"Who the hell is Herman?\"\n\nShe says, \"Herman and his mother were both killed in an automobile accident near Los Angeles. Since that is where his mother is from, the funeral will be held in Los Angeles at the Rose Hills Mortuary on May 3rd at three o'clock.\" Then she says, \"Herman would have liked it very, very much to know that you would be one of his pallbearers.\"\n\nI still can't remember him. I say, \"Of course I'd be happy to do that.\" (At least this way I'll find out who Herman is.)\n\nThen I get an idea: I call up the mortuary. \"You're having a funeral on May 3rd at three o'clock...\"\n\n\"Which funeral do you mean: the Goldschmidt funeral, or the Parnell funeral?\"\n\n\"Well, uh, I don't know.\" It still doesn't click for me; I don't think it's either one of them. Finally, I say, \"It might be a double funeral. His mother also died.\"\n\n\"Oh, yes. Then it's the Goldschmidt funeral.\"\n\n\"Herman Goldschmidt?\"\n\n\"That's right; Herman Goldschmidt and Mrs. Goldschmidt.\"\n\nOkay. It's Herman Goldschmidt. But I still can't remember a Herman Goldschmidt. I haven't any idea what it is I've forgotten; from the way she talked, my friend was sure that Herman and I knew each other well.\n\nThe last chance I have is to go to the funeral and look into the casket.\n\nI go to the funeral, and the woman who had arranged everything comes over, dressed in black, and says in a sorrowful voice, \"I'm _so_ glad you're here. Herman would be so happy if he knew\"\u2014all this serious stuff. Everybody's got long faces about Herman, but I still don't know who Her man is\u2014though I'm sure that if I knew, I'd feel very sorry that he was dead!\n\nThe funeral proceeded, and when it came time for everybody to file past the caskets, I went up. I looked into the first casket, and there was Herman's mother. I looked into the second casket, and there was Herman\u2014and I swear to you, I'd never seen him before in my life!\n\nIt came time to carry the casket out, and I took my place among the pallbearers. I very carefully laid Herman to rest in his grave, because I knew he would have appreciated it. But I haven't any idea, to this day, who Herman was.\n\nMany years later I finally got up enough courage to bring it up to my friend. \"You know that funeral I went to, about ten years ago, for Howard...\"\n\n\"You mean Herman.\"\n\n\"Oh yeah\u2014Herman. You know, I didn't know who Herman was. I didn't even recognize him in the casket.\"\n\n\"But Richard, you knew each other in Los Alamos just after the war. You were both good friends of mine, and we had many conversations together.\"\n\n\"I still can't remember him.\"\n\nA few days later she called and told me what might have happened: maybe she had met Herman just after I had left Los Alamos\u2014and therefore got the timing mixed up somehow\u2014but because she was such good friends with each of us, she thought we must have known each other. So she was the one who had made the mistake, not me (which is usually the case). Or was she just being polite?\n\n### Feynman Sexist Pig!\n\n_A FEW YEARS_ after I gave some lectures for the freshmen at Caltech (which were published as the _Feynman Lectures on Physics),_ I received a long letter from a feminist group. I was accused of being anti-woman because of two stories: the first was a discussion of the subtleties of velocity, and involved a woman driver being stopped by a cop. There's a discussion about how fast she was going, and I had her raise valid objections to the cop's definitions of velocity. The letter said I was making the woman look stupid.\n\nThe other story they objected to was told by the great astronomer Arthur Eddington, who had just figured out that the stars get their power from burning hydrogen in a nuclear reaction producing helium. He recounted how, on the night after his discovery, he was sitting on a bench with his girlfriend. She said, \"Look how pretty the stars shine!\" To which he replied, \"Yes, and right now, I'm the only man in the world who knows _how_ they shine.\" He was describing a kind of wonderful loneliness you have when you make a discovery.\n\nThe letter claimed that I was saying a woman is incapable of understanding nuclear reactions.\n\nI figured there was no point in trying to answer their accusations in detail, so I wrote a short letter back to them: \"Don't bug me, man!\"\n\nNeedless to say, that didn't work too well. Another letter came: \"Your response to our letter of September 29th is unsatisfactory...\"\u2014blah, blah, blah. This letter warned that if I didn't get the publisher to revise the things they objected to, there would be trouble.\n\nI ignored the letter and forgot about it.\n\nA year or so later, the American Association of Physics Teachers awarded me a prize for writing those books, and asked me to speak at their meeting in San Francisco. My sister, Joan, lived in Palo Alto\u2014an hour's drive away\u2014so I stayed with her the night before and we went to the meeting together.\n\nAs we approached the lecture hall, we found people standing there giving out handbills to everybody going in. We each took one, and glanced at it. At the top it said, \"A PROTEST.\" Then it showed excerpts from the letters they sent me, and my response (in full). It concluded in large letters: \"FEYNMAN SEXIST PIG!\"\n\nJoan stopped suddenly and rushed back: \"These are interesting,\" she said to the protester. \"I'd like some more of them!\"\n\nWhen she caught up with me, she said, \"Gee whiz, Richard; what did you do?\"\n\nI told her what had happened as we walked into the hall.\n\nAt the front of the hall, near the stage, were two prominent women in the American Association of Physics Teachers. One was in charge of women's affairs for the organization, and the other was Fay Ajzenberg, a professor of physics I knew, from Pennsylvania. They saw me coming down towards the stage accompanied by this woman with a fistful of handbills, talking to me. Fay walked up to her and said, \"Do you realize that Professor Feynman has a _sister_ that he encouraged to go into physics, and _that she_ has a Ph.D. in physics?\"\n\n\"Of course I do,\" said Joan. \"I'm that sister!\"\n\nFay and her associate explained to me that the protesters were a group\u2014led by a man, ironically\u2014who were always disrupting meetings in Berkeley. \"We'll sit on either side of you to show our solidarity, and just before you speak, I'll get up and say something to quiet the protesters,\" Fay said.\n\nBecause there was another talk before mine, I had time to think of something to say. I thanked Fay, but declined her offer.\n\nAs soon as I got up to speak, half a dozen protesters marched down to the front of the lecture hall and paraded right below the stage, holding their picket signs high, chanting, \"Feynman sexist pig! Feynman sexist pig!\"\n\nI began my talk by telling the protesters, \"I'm sorry that my short answer to your letter brought you here unnecessarily. There are more serious places to direct one's attention towards improving the status of women in physics than these relatively trivial mistakes\u2014if that's what you want to call them\u2014in a textbook. But perhaps, after all, it's good that you came. For women do indeed suffer from prejudice and discrimination in physics, and your presence here today serves to remind us of these difficulties and the need to remedy them.\"\n\nThe protesters looked at one another. Their picket signs began to come slowly down, like sails in a dying wind.\n\nI continued: \"Even though the American Association Physics Teachers has given me an award for teaching, I must confess I don't know how to teach. Therefore, I have nothing to say about teaching. Instead, I would like to talk about something that will be especially interesting to the women in the audience: I would like to talk about the structure of the proton.\"\n\nThe protesters put their picket signs down and walked off. My hosts told me later that the man and his group of protesters had never been defeated so easily.\n\n(Recently I discovered a transcript of my speech, and what I said at the beginning doesn't seem anywhere near as dramatic as the way I remember it. What I remember saying is much more wonderful than what I actually said!)\n\nAfter my talk, some of the protesters came up to press me about the woman-driver story. \"Why did it have to be a woman driver?\" they said. \"You are implying that all women are bad drivers.\"\n\n\"But the woman makes the cop look bad,\" I said. \"Why aren't you concerned about the cop?\"\n\n\"That's what you expect from cops!\" one of the protesters said. \"They're all pigs!\"\n\n\"But you _should_ be concerned,\" I said. \"I forgot to say in the story that the cop was a woman!\"\n\n### I Just Shook His Hand, Can You Believe It?\n\n_FOR SOME YEARS_ now the Kyoto University has been inviting me to visit Japan. But every time I accepted their invitation, I would happen to get sick and not be able to go.\n\nIn the summer of 1986 there was going to be a conference in Kyoto and the university again invited me to come. Although I love Japan and wanted very much to visit, I felt uncomfortable at the invitation because I had no paper to give. The university said it would be all right for me to give a summary paper, but I said I don't like to do that. But then they said they would be honored if I would be the chairman of one session of the conference\u2014that's all I would have to do. So I finally said okay.\n\nI was lucky this time and didn't get sick.* So Gweneth and I went to Kyoto, and I was chairman of one session.\n\nThe chairman is supposed to make sure that the speakers only talk for a certain length of time, in order to leave enough time for the next speaker. The chairman occupies a position of such high honor that there are two co-chairmen to assist him. My co-chairmen said they would take care of introducing the speakers, as well as telling them when it's time to stop.\n\nThings went smoothly for most of the session until one speaker\u2014a Japanese man\u2014didn't stop talking when his time was up. I look at the clock and figure it's time he should stop. I look over at the co-chairmen and gesture a little bit.\n\nThey come up to me and say, \"Don't do anything; we'll take care of it. He's talking about Yukawa.* It's all right.\"\n\nSo I was the honorary chairman of one session, and I felt I didn't even do my job right. And for that, the university paid my way to Japan, they took care of arranging my trip, and they were all very gracious.\n\nOne afternoon we were talking to the host who was arranging our trip. He shows us a railroad map, and Gweneth sees a curved line with lots of stops in the middle of the Ise Peninsula\u2014it's not near the water; it's not near anywhere. She puts her finger on the end of the line and says, \"We want to go here.\"\n\nHe looks at it, and says, \"Oh! You want to go to... Iseokitsu?\"\n\nShe says, \"Yes.\"\n\n\"But there's _nothing_ in Iseokitsu,\" he says, looking at me as if my wife is crazy, and hoping I'll bring her back to her senses.\n\nSo I say, \"Yes, that's right; we want to go to Iseokitsu.\"\n\nGweneth hadn't talked to me about it, but I knew what she was thinking: we enjoy traveling to places in the middle of nowhere, places we've never heard of, places which have nothing.\n\nOur host becomes a little bit upset: he's never made a hotel reservation for Iseokitsu; he doesn't even know if there's an inn there.\n\nHe gets on the telephone and calls up Iseokitsu for us. In Iseokitsu, it turns out, there are no accommodations.\n\nBut there's another town\u2014about seven kilometers beyond the end of the line\u2014that has a Japanese-style inn.\n\nWe say, \"Fine! That's just what we want\u2014a Japanese-style inn!\" They give him the number and he calls.\n\nThe man at the inn is very reluctant: \"Ours is a very small inn. It's a family-run place.\"\n\n\"That's what they want,\" our host reassures him.\n\n\"Did he say yes?\" I ask.\n\nAfter more discussion, our host says, \"He agrees.\"\n\nBut the next morning, our host gets a telephone call from this same inn: last night they had a family conference. They decided they can't handle the situation. They can't take care of foreigners.\n\nI say, \"What's the trouble?\"\n\nOur host telephones the inn and asks what the problem is. He turns to us and says, \"It's the toilet\u2014they don't have a Western-style toilet.\"\n\nI say, \"Tell them that the last time my wife and I went on a trip, we carried a small shovel and toilet paper, and dug holes for ourselves in the dirt. Ask him, 'Shall we bring our shovel?' \"\n\nOur host explains this over the telephone, and they say, \"It's okay. You can come for one night. You don't need to bring your shovel.\"\n\nThe innkeeper picked us up at the railroad station in Iseokitsu and took us to his inn. There was a beautiful garden outside our room. We noticed a brilliant, emerald-green tree frog climbing a metal frame with horizontal bars (used for hanging out the wet clothes), and a tiny yellow snake in a shrub in front of our _engawa_ (veranda). Yes, there was \"nothing\" in Iseokitsu\u2014but everything was beautiful and interesting to us.\n\nIt turned out there was a shrine about a mile away\u2014that's why this little inn was there\u2014so we walked to it. On our way back, it began to rain. A guy passed us in his car, then turned around and came back. \"Where are you going?\" he asked in Japanese. \"To the inn,\" I said. So he took us there.\n\nWhen we got back to our room, we discovered that Gweneth had lost a roll of film\u2014perhaps in the man's car. So I got the dictionary out and looked up \"film\" and \"lost,\" and tried to explain it to the innkeeper. I don't know how he did it, but he found the man who had given us the ride, and in his car we found the film.\n\nThe bath was interesting; we had to go through another room to get to it. The bathtub was wooden, and around it were all kinds of little toys\u2014little boats and so on. There was also a towel with Mickey Mouse on it.\n\nThe innkeeper and his wife had a little daughter who was two, and a small baby. They dressed their daughter in a kimono and brought her up to our room. Her mother made origami things for her; I made some drawings for her, and we played with her.\n\nA lady across the street gave us a beautiful silk ball that she had made. Everything was friendly; everything was very good.\n\nThe next morning we were supposed to leave. We had a reservation at one of the more famous resorts, at a spa somewhere. I looked in the dictionary again; then I came down and showed the innkeeper the receipt for our reservation at the big resort hotel\u2014it was called the Grand View, or something like that. I said, \"We don't want stay big hotel tomorrow night; we want stay _here_ tomorrow night. We happy here. Please you call them; change this.\"\n\nHe says, \"Certainly! Certainly!\" I could tell he was pleased by the idea that these foreigners were canceling their reservation in this big, fancy hotel in order to stay in his little inn another night.\n\nAfter we returned to Tokyo, we went to the University of Kanazawa. Some professors arranged to drive us along the coast of nearby Noto Peninsula. We passed through several delightful fishing villages, and went to visit a pagoda in the middle of the countryside.\n\nThen we visited a shrine with an enclave behind it, where one could go only by special invitation. The Shinto priest there was very gracious and invited us into his private rooms for tea, and he did some calligraphy for us.\n\nAfter our hosts had taken us a little farther along the coast, they had to return to Kanazawa. Gweneth and I decided to stay in Togi for two or three days. We stayed in a Japanese-style hotel, and the lady innkeeper there was very, very nice to us. She arranged for her brother to take us by car down the coast to several villages, and then we came back by bus.\n\nThe next morning the innkeeper told us there was something important happening in town. A new shrine, replacing an old one, was being dedicated.\n\nWhen we arrived at the grounds we were invited to sit on a bench, and were served tea. There were many people milling around, and eventually a procession came out from behind the shrine. We were delighted to see the leading figure was the head priest from the shrine we had visited a few days before. He was dressed in a big, ceremonial outfit, and was obviously in charge of everything.\n\nAfter a little while the ceremony began. We didn't want to intrude into a religious place, so we stayed back from the shrine itself. But there were kids running up and down the steps, playing and making noise, so we figured it wasn't so formal. We came a little closer and stood on the steps so we could see inside.\n\nThe ceremony was wonderful. There was a ceremonial cup with branches and leaves on it; there was a group of girls in special uniforms; there were dancers, and so on. It was quite elaborate.\n\nWe're watching all these performances when all of a sudden we feel a tap on the shoulder. It's the head priest! He gestures to us to follow him.\n\nWe go around the shrine and enter from the side. The head priest introduces us to the mayor and other dignitaries, and invites us to sit down. A _noh_ actor does a dance, and all kinds of other wonderful things go on.\n\nThen there are speeches. First, the mayor gives a speech. Then the head priest gets up to speak. He says, \"Unano, utsini kuntana kanao. Untanao uni kanao. Uniyo zoimasu doi zinti Fain-man-san-to unakano kane gozai-mas...\"\u2014and he points to \"Fain-man-san\" and tells me to say something!\n\nMy Japanese is very poor, so I say something in English: \"I love Japan,\" I say. \"I am particularly impressed by your tremendous rate of technological change, while at the same time your traditions still mean so much, as you are showing with this shrine dedication.\" I tried to express the mixture I saw in Japan: change, but without losing respect for traditions.\n\nThe head priest says something in Japanese which I do not believe is what I said (although I couldn't really tell), because he had never understood anything I had said to him previously! But he _acted_ as if he understood _exactly_ what I said, and he \"translated\" it with complete confidence for everyone. He was much like I am, in this respect.\n\nAnyway, the people politely listened to whatever it was that I said, and then another priest gave a speech. He was a young man, a student of the head priest, dressed in a wonderful outfit with big, wide pant legs and a big, wide hat. He looked so gorgeous, so wonderful.\n\nThen we went to lunch with all the dignitaries, and felt very honored to be included.\n\nAfter the shrine dedication ceremony was over, Gweneth and I thanked the head priest and left the dining hall to walk around the village for a while. After a bit we found some people pulling a big wagon, with a shrine in it, through the streets. They're all dressed up in outfits with symbols on the back, singing, \"Eyo! Eyo!\"\n\nWe follow the procession, enjoying the festivities, when a policeman with a walkie-talkie comes up to us. He takes off his white glove and puts out his hand. I shake hands with him.\n\nAs we leave the policeman and begin to follow the procession again, we hear a loud, high-pitched voice behind us, speaking very rapidly. We turn around and see the policeman clutching his walkie-talkie, speaking into it with great excitement: \"O gano fana miyo ganu Fain-man-san iyo kano muri tono muroto kala...\"\u2014and I could just imagine him telling the person at the other end: \"Do you remember that Mr. Fain-man who spoke at the shrine dedication? I just shook his hand, can you believe it?\"\n\nThe priest must have \"translated\" something very impressive!\n\n### Letters\n\nOctober 11, 1961\n\nHotel Amigo, Brussels\n\nHello, my sweetheart,\n\nMurray and I kept each other awake arguing until we could stand it no longer. We woke up over Greenland, which was even better than last time because we went right over part of it. In London we met other physicists and came to Brussels together. One of them was worried\u2014in his guidebook the Hotel Amigo was not even mentioned. Another had a newer guide\u2014five stars, and rumored to be the best hotel in Europe!\n\nIt is very nice indeed. All the furniture is dark red polished wood, in perfect condition; the bathroom is grand, etc. It is really too bad you didn't come to this conference instead of the other one.\n\nAt the meeting next day things started slowly. I was to talk in the afternoon. That is what I did, but I didn't really have enough time. We had to stop at 4 pm because of a reception scheduled for that night. I think my talk was OK though\u2014what I left out was in the written version anyway.\n\nSo that evening we went to the palace to meet the king and queen. Taxis waited for us at the hotel\u2014long black ones\u2014and off we went at 5 pm, arriving through the palace gates with a guard on each side, and driving under an arch where men in red coats and white stockings with a black band and gold tassel under each knee opened the doors. More guards at the entrance, in the hallway, along the stairs, and up into a sort of ballroom. These guards, in dark grey Russian-type hats with chin straps, dark coats, white pants, and shiny black leather boots, stand very straight\u2014each holding a sword straight up.\n\nIn the \"ballroom\" we had to wait perhaps 20 minutes. It has inlaid parquet floors, and L in each square (for Leopold\u2014the present king is Baudoin, or something). The gilded walls are 18th century and on the ceiling are pictures of naked women riding chariots among the clouds. Lots of mirrors and gilded chairs with red cushions around the outside edge of the room\u2014just like so many of those palaces we have seen, but this time it's no museum: it's alive, with everything clear and shining, and in perfect condition. Several palace officials were milling around among us. One had a list and told me where to stand but I didn't do it right and was out of place later.\n\nThe doors at the end of the hall open. Guards are there with the king and queen; we all enter slowly and are introduced one by one to the king and queen. The king has a young semi-dopey face and a strong handshake; the queen is very pretty. (I think her name is Fabriola\u2014a Spanish countess she was.) We exit into another room on the left where there are lots of chairs arranged like in a theatre, with two in front, also facing forward, for K & Q. A table at the front with six seats is for illustrious scientists\u2014Niels Bohr. J. Perrin (a Frenchman),J. R. Oppenheimer etc.\u2014see drawing.\n\nIt turns out the king wants to know what we are doing, so the old boys give a set of six dull lectures\u2014all very solemn\u2014no jokes. I had great difficulty sitting in my seat because I had a very stiff and uncomfortable back from sleeping on the plane.\n\nThat done, the K & Q pass through the room where we met them and into a room on right (marked R). (All these rooms are very big, gilded, Victorian, fancy, etc.) In R are many kinds of uniforms: guards at door in red coats, waiters in white coats (to serve drinks and hors d'oeuvres), military khaki and medals, and black coats\u2014undertaker's type (palace officials).\n\nOn the way out of L into R, I am last because I walk slowly from stiff back. I find myself talking to a palace official\u2014nice man. He teaches math part time at Louvain University, but his main job is secretary to the queen. He had also tutored the K when K was young and has been in palace work 23 years. Now, at least, I have somebody to talk to.\n\nSome others are talking to K or to Q; everybody is standing up. After a while the professor who is head of the conference (Prof. Bragg) grabs me and says K wants to talk to me. Bragg says, \"K, this is Feynman.\" I pull boner #1 by wanting to shake hands again\u2014apparently wrong: no hand reaches up. After an embarrassed pause K saves day by shaking my hand. K makes polite remarks on how smart we must all be and how hard it must be to think. I answer, making jokes (having been instructed to do so by Bragg, but what does he know?)\u2014apparently error #2. Anyway, strain is relieved when Bragg brings over some other professor\u2014Heisenberg, I think. K forgets F and F slinks off to resume conversation with sec'y of Q.\n\nAfter considerable time\u2014several orange juices and many very good hors d'oeuvres later\u2014a military uniform with medals comes over to me and says, \"Speak to the queen!\" Nothing I should like to do better (pretty girl, but don't worry, she's married). F arrives at scene: Q is sitting at table surrounded by three other occupied chairs\u2014no room for F. There are several low coughs, slight confusion, etc., and lo!\u2014one of the chairs has been reluctantly vacated. Other two chairs contain one lady and one Priest in Full Regalia (who is also a physicist) named LeMa\u00eetre.\n\nWe have quite a conversation (I listen, but hear no coughs, and am not evacuated from seat) for perhaps 15 minutes. Sample:\n\nQ: \"It must be very hard work thinking about those difficult problems...\"\n\nF: \"No, we all do it for the fun of it.\"\n\nQ: \"It must be hard to learn to change all your ideas\"\u2014(a thing she got from the six lectures).\n\nF: \"No, all those guys who gave you those lectures are old fogeys\u2014all that change was in 1926, when I was only eight. So when I learned physics I only had to learn the new ideas. The big problem now is, do we have to change them again?\"\n\nQ: \"You must feel good, working for peace like that.\"\n\nF: \"No, that never enters my head, whether it is for peace or otherwise. We don't know.\"\n\nQ: \"Things certainly change fast\u2014many things have changed in the last hundred years.\"\n\nF: \"Not in this palace.\" (I thought it, but controlled myself.) \"Yes,\" and then launched into lecture on what was known in 1861 and what we found out since\u2014adding at end, laughingly, \"Can't help giving a lecture, I guess\u2014I'm a professor, you see. Ha, ha.\"\n\nQ in desperation, turns to lady on her other side and begins conversation with same.\n\nAfter a few moments K comes over and whispers something to Q, who stands up\u2014they quietly go out. F returns to sec'y of Q who personally escorts him out of palace past guards, etc.\n\nI'm so terribly sorry you missed it. I don't know when we'll find another king for you to meet.*\n\nI was paged in the hotel this morning just before leaving with the others. I returned to the others and announced, \"Gentlemen, that call was from the queen's secretary. I must leave you now.\" All are awestruck, for it did not go unnoticed that F talked longer and harder to Q than seemed proper. I didn't tell them, however, that it was about a meeting we arranged\u2014he was inviting me to his home to meet his wife and two (of four) of his daughters, and to see his house. I had invited him to visit us in Pasadena when he came to America and this was his response.\n\nHis wife and daughters are very nice and his house is positively beautiful. You would have enjoyed that even more than visiting the palace. He planned and built his house in a Belgian style, somewhat after an old farmhouse style, but done just right. He has many old cabinets and tables inside, right beside newer stuff, very well combined. It is much easier to find antiques in Belgium than in Los Angeles as there are so many old farms, etc. The house is slightly bigger than ours and the grounds are much bigger but not yet landscaped, except for a vegetable garden. He has a bench that he made for himself in the garden, hidden under trees, to go and sit on and look at the surrounding countryside. He has a dog\u2014from Washington\u2014that somebody gave to the king and the K gave to him. The dog has a personality somewhat like Kiwi\u2020 because I think he is equally loved.\n\nI told the secretary I had a queen in a little castle in Pasadena that I would like him to see, and he said he hoped he would be able to come to America and see us. He would come if the Q. ever visits America again.\n\nI am enclosing a picture of his house, and his card, so I don't lose it.\n\nI know you must feel terrible being left out this time\u2014but I'll make it up someday somehow. But don't forget I love you very much and am proud of my family that is and my family that is to be.* The secretary and his wife send their best wishes to you and our future.\n\nI wish you were here, or, next best thing, that I were there. Kiss SNORK\u2020 and tell Mom all about my adventures and I will be home sooner than you think.\n\nYour husband loves you.\n\nYour husband.\n\nGrand Hotel\n\nWarsaw\n\nDearest Gweneth,\n\nTo begin with, I love you.\n\nAlso I miss you and the baby \u2021 and Kiwi, and really wish I were home.\n\nI am now in the restaurant of the Grand Hotel. I was warned by friends that the service is slow, so I went back for pens and paper to work on my talk for tomorrow\u2014but what could be better than to write to my darling instead?\n\nWhat is Poland like? My strongest impression\u2014and the one which gives me such a surprise\u2014is that it is almost exactly as I pictured it (except for one detail)\u2014not only in how it looks, but also in the people, how they feel, what they say and think about the government, etc. Apparently we are well informed in the US and magazines such as Time and Atlas are not so bad. The detail is that I had forgotten how completely destroyed Warsaw was during the war and therefore that, with few exceptions (which are easily identified by the bullet holes all over them), all the buildings are built since the war. In fact it is a rather considerable accomplishment\u2014there are very many new buildings: Warsaw is a big city, all rebuilt.\n\nThe genius of builders here is to be able to build old buildings. There are buildings with facings falling off (walls covered with concrete with patches of worn brick showing thru), rusted window bars with streaks of rust running down the building, etc. Further, the architecture is old\u2014decorations sort of 1927 but heavier\u2014nothing interesting to look at (except one building).\n\nThe hotel room is very small, with cheap furniture, a very high ceiling (15 feet), old water spots on the walls, plaster showing through where bed rubs wall, etc. It reminds me of an old \"Grand Hotel\" in New York\u2014faded cotton bedspread covering bumpy bed, etc. But the bathroom fixtures (faucets etc.) are bright and shiny, which confused me: they seem relatively new in this old hotel. I finally found out: the hotel is only three years old\u2014I had forgotten about their ability to build old things. (No attention at all yet from waiter, so I break down and ask a passing one for service. A confused look\u2014he calls another over. Net result: I am told there is no service at my table and am asked to move to another. I make angry noises. The response: I am put at another table, given a menu, and have 15 seconds to make up my mind. I order Sznycel Po Wiedensku\u2014Wiener Schnitzel.)\n\nOn the question of whether the room is bugged: I look for covers of old sockets (like the one in the ceiling of the shower). There are five of them, all near the ceiling\u201415 feet. I need a ladder and decide not to investigate them. But there is a similar large square plate in the lower corner of my room near the telephone. I pull it back a little (one screw is loose). I have rarely seen so many wires\u2014like the back of a radio. What is it? Who knows! I didn't see any microphones; the ends of the wires were taped, like connections or outlets no longer in use. Maybe the microphone is in the tape. Well, I haven't a screwdriver so I don't take the plate off to investigate further. In short, if my room isn't bugged they are wasting a lot of wires.\n\nThe Polish people are nice, poor, have at least medium style in (soup arrives!) clothes, etc. There are nice places to dance, with good bands, etc., etc. So Warsaw is not very heavy and dull, as one hears Moscow is. On the other hand, you meet at every turn that kind of dull stupid backwardness characteristic of government\u2014you know, like the fact that change for $20 isn't available when you went to get your card renewed at the US Immigration Office downtown. Example: I lost my pencil, and wanted to buy a new one at the kiosk here. \"A pen costs $1.10.\"\n\n\"No, I want a pencil\u2014wooden, with graphite.\"\n\n\"No, only $1.10 pens.\"\n\n\"OK, how many Zlotys is that?\"\n\n\"You can't buy it in Zlotys, only for $1.10.\" (Why? Who knows!)\n\nI have to go upstairs for American money. I give $ 1.25.\n\nClerk at kiosk cannot give change\u2014must go to cashier of hotel. The bill for my pen is written in quadruplicate: the clerk keeps one, the cashier one, and I get two copies. What shall I do with them? On the back it says I should keep them to avoid paying US customs duties. It is a Papermate pen made in the USA. (The soup dish is removed.)\n\nThe real question of government versus private enterprise is argued on too philosophical and abstract a basis. Theoretically, planning may be good. But nobody has ever figured out the cause of government stupidity\u2014and until they do (and find the cure), all ideal plans will fall into quicksand.\n\nI didn't guess right the nature of the palace in which the meetings are held. I imagined an old, forbidding, large room from 16th century or so. Again, I forgot that Poland was so thoroughly destroyed. The palace is brand new: we meet in a round room with white walls, with gilded decorations on the balcony; the ceiling is painted with a blue sky and clouds. (The main course comes. I eat it; it is very good. I order dessert: pastries with pineapple, 125 g. Incidentally, the menu is very precise: the \"125 g\" is the weight\u2014125 grams. There are things like \"filet of herring, 144 g,\" etc. I haven't seen anybody checking for cheating with a scale; I didn't check if the schnitzel was the claimed 100 grams.)\n\nI am not getting anything out of the meeting. I am learning nothing. Because there are no experiments this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here (126) and it is not good for my blood pressure: such inane things are said and seriously discussed that I get into arguments outside the formal sessions (say, at lunch) whenever anyone asks me a question or starts to tell me about his \"work.\" The \"work\" is always: (1) completely un-understandable, (2) vague and indefinite, (3) something correct that is obvious and self-evident, but worked out by a long and difficult analysis, and presented as an important discovery, or (4) a claim based on the stupidity of the author that some obvious and correct fact, accepted and checked for years, is, in fact, false (these are the worst: no argument will convince the idiot), (5) an attempt to do something probably impossible, but certainly of no utility, which, it is finally revealed at the end, fails (dessert arrives and is eaten), or (6) just plain wrong. There is a great deal of \"activity in the field\" these days, but this \"activity\" is mainly in showing that the previous \"activity\" of somebody else resulted in an error or in nothing useful or in something promising. It is like a lot of worms trying to get out of a bottle by crawling all over each other. It is not that the subject is hard; it is that the good men are occupied elsewhere. Remind me not to come to any more gravity conferences!\n\nI went one evening to the home of one of the Polish professors (young, with a young wife). People are allowed seven square yards per person in apartments, but he and his wife are lucky: they have twenty-one* \u2014for living room, kitchen, bathroom. He was a little nervous with his guests (myself, Professor and Mrs. Wheeler, and another) and seemed apologetic that his apartment was so small. (I ask for the check. All this time the waiter has had two or three active tables, including mine.) But his wife was very relaxed and kissed her siamese cat \"Booboosh\" just like you do with Kiwi. She did a wonderful job of entertaining\u2014the table for eating had to be taken from the kitchen, a trick requiring the bathroom door to be first removed from its hinges. (There are only four active tables in the whole restaurant now, and four waiters.) Her food was very good and we all enjoyed it.\n\nOh, I mentioned that one building in Warsaw is interesting to look at. It is the largest building in Poland: the \"Palace of Culture and Science,\" given as a gift by the Soviet Union. It was designed by Soviet architects. Darling, it is unbelievable! I cannot even begin to describe it. It is the craziest monstrosity on land! (The check comes\u2014brought by a different waiter. I await the change.)\n\nThis must be the end of my letter. I hope I don't wait too long for the change. I skipped coffee because I thought it would take too long. Even so, see what a long letter I can write while eating Sunday dinner at the Grand Hotel.\n\nI say again I love you, and wish you were here\u2014or better I were there. Home is good.\n\n(The change has come\u2014it is slightly wrong (by 0.55 Zloty = 15?) but I let it go.)\n\nGood bye for now.\n\nRichard.\n\nSaturday, June 29(?) 3pm\n\nRoyal Olympic Hotel. Poolside.\n\nDear Gweneth, and Michelle*(and Carl?),\n\nThis is my third day in Athens.\n\nI'm writing by the side of the hotel pool with the paper in my lap because the tables are too high and the chairs too low.\n\nThe trip was all on time but uncomfortable anyway because the plane from New York to Athens was absolutely full\u2014every seat. I was met by Prof. Illiapoulos, a student, and his nephew, who is just Carl's age.\n\nI was surprised to find the weather here is just like in Pasadena, but about 5 degrees cooler: the vegetation is very similar, the hills look bare and desert-like\u2014same plants, same cactuses, same low humidity and same cool nights. But there the similarity ends. Athens is a sprawling, ugly, noisy, exhaust-filled mess of streets filled with nervous traffic jumping like rabbits when the lights go green and stopping with squealing brakes when they go red\u2014and blowing horns when they go yellow. Very similar to Mexico City, except the people don't look as poor\u2014there are only occasional beggars in the streets. You, Gweneth, would love it because there are so many shops (all small), and Carl would love walking around in the arcades with their rabbit-warren twists and surprises, especially in the old part of town.\n\nYesterday morning I went to the archeological museum. Michelle would like all the great Greek statues of horses\u2014especially one of a small boy on a large galloping horse, all in bronze, that is a sensation. I saw so much stuff my feet began to hurt. I got all mixed up\u2014things are not labeled well. Also, it was slightly boring because we have seen so much of that stuff before. Except for one thing: among all those art objects there was one thing so entirely different and strange that it is nearly impossible. It was recovered from the sea in 1900 and is some kind of machine with gear trains, very much like the inside of a modern wind-up alarm clock. The teeth are very regular and many wheels are fitted closely together. There are graduated circles and Greek inscriptions. I wonder if it is some kind of fake. There was an article on it in the Scientific American in 1959.\n\nYesterday afternoon I went to the Acropolis, which is right in the middle of the city\u2014a high rock plateau on which was built the Parthenon and other shrines and temples. The Parthenon looks pretty good, but the Temple at Segesta, which Gweneth and I saw in Sicily, is just as impressive because you are allowed to walk around in it\u2014you can't go up to or walk around among the Parthenon columns. Prof. Illiapoulos' sister came with us and with a notebook she had\u2014she is a professional archeologist\u2014guided our tour with all kinds of details, dates, quotations from Plutarch, etc.\n\nIt appears the Greeks take their past very seriously. They study ancient Greek archeology in their elementary schools for 6 years, having to take 10 hours of that subject every week. It is a kind of ancestor worship, for they emphasize always how wonderful the ancient Greeks were\u2014and wonderful indeed they were. When you encourage them by saying, \"Yes, and look how modern man has advanced beyond the ancient Greeks\"\u2014thinking of experimental science, the development of mathematics, the art of the Renaissance, the great depth and understanding of the relative shallowness of Greek philosophy, etc., etc.\u2014they reply, \"What do you mean? What was wrong with the ancient Greeks?\" They continually put their age down and the old age up, until to point out the wonders of the present seems to them to be an unjustified lack of appreciation for the past.\n\nThey were very upset when I said that the development of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation: although it is of very little use in itself, the discovery must have been psychologically wonderful because it showed that a modern man could do something no ancient Greek could do. It therefore helped in the Renaissance, which was the freeing of man from the intimidation of the ancients. What the Greeks are learning in school is to be intimidated into thinking they have fallen so far below their super ancestors.\n\nI asked the archeologist lady about the machine in the museum\u2014whether other similar machines, or simpler machines leading up to it or down from it, were ever found\u2014but she hadn't heard of it. So I met her and her son of Carl's age (who looks at me as if I were a heroic ancient Greek, for he is studying physics) at the museum to show it to her. She required some explanation from me why I thought such a machine was interesting and surprising because, \"Didn't Eratosthenes measure the distance to the sun, and didn't that require elaborate scientific instruments?\" Oh, how ignorant are classically educated people. No wonder they don't appreciate their own time. They are not of it and do not understand it. But after a bit she believed maybe it _was_ striking, and she took me to the back rooms of the museum\u2014surely there were other examples, and she would get a complete bibliography. Well, there were no other examples, and the complete bibiography was a list of three articles (including the one in the Scientific American)\u2014all by one man, an American from Yale!\n\nI guess the Greeks think all Americans must be dull, being only interested in machinery when there are all those beautiful statues and portrayals of lovely myths and stories of gods and goddesses to look at. (In fact, a lady from the museum staff remarked, when told that the professor from America wanted to know more about item 15087, \"Of all the beautiful things in this museum, why does he pick out that particular item? What is so special about it?\")\n\nEveryone here complains of the heat, and concerned about whether you can stand it, when in fact it is just like Pasadena but about 5 degrees cooler on the average. So all stores and offices close from perhaps 1:30 pm to 5:30 pm (\"because of the heat\"). It turns out to be really a good idea (everyone takes a nap) because then they go late into the night\u2014supper is between 9:30 and 10 pm, when it is cool. Right now, people here are seriously complaining about a new law: to save energy, all restaurants and taverns must close at 2 am. This, they say, will spoil life in Athens.\n\nIt is the witching hour between 1:30 and 5:30 pm now, and I am using it to write to you. I miss you, and I would really be happier at home. I guess I really have lost my bug for travelling. I have a day and a half yet here and they have given me all kinds of literature about a beautiful beach (of pebbles) here, of an important ancient site (although in rather complete ruins) there, etc. But I will go to none of them, for each, it turns out, is a long, two- to four-hour ride each way on a tour bus. No. I'll just stay here and prepare my talks for Crete. (They have me giving an extra three lectures to some twenty Greek university students who are all coming to Crete just to hear me. I'll do something like my New Zealand lectures,*but I haven't got any notes! I'll have to work them out again.)\n\nI miss you all, especially when I go to bed at night\u2014no dogs to scratch and say good night to!\n\nLove, Richard.\n\nP.S. IF YOU CAN'T READ THE ABOVE HANDWRITING, HAVE NO FEAR\u2014IT IS UNIMPORTANT RAMBLINGS. I AM WELL & IN ATHENS.\n\nMacFaddin Hall\n\nCornell University\n\nIthaca, NY\n\nNovember 19, 1947\u2020\n\nMy Dear Family:\n\nJust a brief letter before we go off to Rochester. We have every Wednesday a seminar at which somebody talks about some item of research, and from time to time this is made a joint seminar with Rochester University. To-day is the first time this term that we are going over there for it.\n\nIt is a magnificent day, and it should be a lovely trip;Rochester is northwest of here, on the shores of Lake Ontario, and we go through some wild country. I am being taken in Feynman's car, which will be great fun if we survive. Feynman is a man for whom I am developing a considerable admiration; he is the first example I have met of that rare species, the native American scientist. He has developed a private version of the quantum theory, which is generally agreed to be a good piece of work and may be more helpful than the orthodox version for some problems; in general he is always sizzling with new ideas, most of which are more spectacular than helpful, and hardly any of which get very far before some newer inspiration eclipses it. His most valuable contribution to physics is as a sustainer of morale; when he bursts into the room with his latest brain-wave and proceeds to expound on it with the most lavish sound effects and waving about of the arms, life at least is not dull.\n\nWeisskopf, the chief theoretician at Rochester, is also an interesting and able man, but of the normal European type; he comes from Munich, where he was a friend of Bethe from student days.\n\nThe event of the last week has been a visit from Peierls, who... stayed two nights with the Bethes before flying home.... On Monday night the Bethes gave a party in his honor, to which most of the young theoreticians were invited. When we arrived we were introduced to Henry Bethe, who is now five years old, but he was not at all impressed. In fact, the only thing he would say was \"I want Dick! You told me Dick was coming!\" Finally he had to be sent off to bed, since Dick (alias Feynman) did not materialize.\n\nAbout half an hour later, Feynman burst into the room, just had time to say, \"So sorry I'm late\u2014had a brilliant idea just as I was coming over,\" and then dashed upstairs to console Henry. Conversation then ceased while the company listened to the joyful sounds above, sometimes taking the form of a duet and sometimes of a one-man percussion band....\n\nMuch Love,\n\nFreeman\n\nUrbana, Illinois\n\nApril 9, 1981\n\nDear Sara,*\n\nI just spent a marvelous three days with Dick Feynman and wished you had been there to share him with us. Sixty years and a big cancer operation have not blunted him. He is still the same Feynman that we knew in the old days at Cornell.\n\nWe were together at a small meeting of physicists organized by John Wheeler at the University of Texas. For some reason Wheeler decided to hold the meeting at a grotesque place called World of Tennis, a country club where Texas oil-millionaires go to relax. So there we were. We all grumbled at the high prices and the extravagant ugliness of our rooms. But there was nowhere else to go\u2014or so we thought. But Dick thought otherwise: he just said, \"To hell with it. I am not going to sleep in this place,\" picked up his suitcase, and walked off alone into the woods.\n\nIn the morning he reappeared, looking none the worse for his night under the stars. He said he did not sleep much, but it was worth it.\n\nWe had many conversations about science and history, just like in the old days. But now he had something new to talk about, his children. He said, \"I always thought I would be a specially good father because I wouldn't try to push my\n\nkids into any particular direction. I wouldn't try to turn them into scientists or intellectuals if they didn't want it. I would be just as happy with them if they decided to be truck drivers or guitar players. In fact, I would even like it better if they went out in the world and did something real instead of being professors like me. But they always find a way to hit back at you. My boy Carl, for instance. There he is in his second year at MIT, and all he wants to do with his life is to become a goddamn philosopher!\"*\n\nAs we sat in the airport waiting for our planes, Dick pulled out a pad of paper and a pencil and started to draw the faces of people sitting in the lounge. He drew them amazingly well. I said I was sorry I have no talent for drawing. He said, \"I always thought I have no talent either. But you don't need any talent to do stuff like this.\"...\n\nYours,\n\nFreeman\n\nFebruary 17, 1988  \nLondon, England\n\nDear Mrs. Feynman,\n\nWe have not met, I believe, frequently enough for either of us to have taken root in the other's conscious memory. So please forgive any impertinence, but I could not let Richard's death pass unnoticed, or to take the opportunity to add my own sense of loss to yours.\n\nDick was the best and favorite of several \"uncles\" who encircled my childhood. During his time at Cornell he was\n\na frequent and always welcome visitor at our house, one who could be counted on to take time out from conversations with my parents and other adults to lavish attention on the children. He was at once a great player of games with us and a teacher even then who opened our eyes to the world around us.\n\nMy favorite memory of all is of sitting as an eight- or nine-year-old between Dick and my mother, waiting for the distinguished naturalist Konrad Lorenz to give a lecture. I was itchy and impatient, as all young are when asked to sit still, when Dick turned to me and said, \"Did you know that there are twice as many numbers as numbers?\"\n\n\"No, there are not!\" I was defensive as all young of my knowledge.\n\n\"Yes there are; I'll show you. Name a number.\"\n\n\"One million.\" A big number to start.\n\n\"Two million.\"\n\n\"Twenty-seven.\"\n\n\"Fifty-four.\"\n\nI named about ten more numbers and each time Dick named the number twice as big. Light dawned.\n\n\"I see; so there are three times as many numbers as numbers.\"\n\n\"Prove it,\" said Uncle Dick. He named a number. I named one three times as big. He tried another. I did it again. Again.\n\nHe named a number too complicated for me to multiply in my head. \"Three times that,\" I said.\n\n\"So, is there a biggest number?\" he asked.\n\n\"No,\" I replied. \"Because for every number, there is one twice as big, one three times as big. There is even one a million times as big.\"\n\n\"Right, and that concept of increase without limit, of no biggest number, is called 'infinity'.\"\n\nAt that point Lorenz arrived, so we stopped to listen to him.\n\nI did not see Dick often after he left Cornell. But he left me with bright memories, infinity, and new ways of learning about the world. I loved him dearly.\n\nSincerely Yours,\n\nHenry Bethe\n\n### Photos and Drawings\n\n_Richard and Arlene on the boardwalk in Atlantic City._\n\n_On their wedding day._\n\n_Arlene in the hospital._\n\n_Coffee hour at Winnett Student Center, 1964. ( CALTECH)_\n\n_Gesturing at a Caltech Alumni Day lecture, 1978. (CALTECH)_\n\n_In the Caltech production of_ Fiorello, _1978. (CALTECH)_\n\n_The chief from Bali Hai in_ South Pacific, _1982. (CALTECH)_\n\n_With Michelle, 3, and Carl, 10, in Yorkshire, England, (BBC, YORKSHIRE TELEVISION)_\n\n_With son Carl on the day Richard won the Nobel Prize, 1965. (CALTECH)_\n\n_Richard and Gweneth on their silver wedding anniversary, 1985. (PHOTO BY YASUSHI OHNUKI)_\n\n_Richard Feynman began taking art lessons at the age of 44, and continued drawing for the rest of his life. These sketches include portraits of professional models, his friend Bob Sadler, and his daughter Michelle (at the age of 14). Feynman signed all of his artwork \"Ofey\" to make sure no one would suspect who really drew them._\n\n## Part 2\n\n## MR. FEYNMANGOES TO WASHINGTON: INVESTIGATING THE SPACE SHUTTLE CHALLENGER DISASTER\n\n### Preliminaries\n\n_IN THIS STORY_ I'm going to talk a lot about NASA,* but when I say \"NASA did this\" and \"NASA did that,\" I don't mean all of NASA; I just mean that part of NASA associated with the shuttle.\n\nTo remind you about the shuttle, the large central part is the tank, which holds the fuel: liquid oxygen is at the top, and liquid hydrogen is in the main part. The engines which burn that fuel are at the back end of the orbiter, which goes into space. The crew sits in the front of the orbiter; behind them is the cargo bay.\n\nDuring the launch, two solid-fuel rockets boost the shuttle for a few minutes before they separate and fall back into the sea. The tank separates from the orbiter a few minutes later\u2014much higher in the atmosphere\u2014and breaks up as it falls back to earth.\n\nThe solid rocket boosters are made in sections. There are two types of joints to hold the sections together: the permanent \"factory joints\" are sealed at the Morton Thiokol factory in Utah; the temporary \"field joints\" are sealed before each flight\u2014\"in the field\"\u2014at the Kennedy Space Center in Florida.\n\n_F IGURE 1. THE SPACE SHUTTLE_ Challenger. _The fuel tank, flanked by two solid-fuel rocket boosters, is attached to the orbiter, whose main engines burn liquid hydrogen and liquid oxygen. (\u00a9 NASA.)_\n\n_F IGURE 2. Locations and close-up views of booster-rocket field joints_.\n\n### Committing Suicide\n\n_AS YOU_ probably know, the space shuttle _Challenger_ had an accident on Tuesday, January 28, 1986. I saw the explosion on TV, but apart from the tragedy of losing seven people, I didn't think much about it.\n\nIn the newspaper I used to read about shuttles going up and down all the time, but it bothered me a little bit that I never saw in any scientific journal any results of anything that had ever come out of the experiments on the shuttle that were supposed to be so important. So I wasn't paying very much attention to it.\n\nWell, a few days after the accident, I get a telephone call from the head of NASA, William Graham, asking me to be on the committee investigating what went wrong with the shuttle! Dr. Graham said he had been a student of mine at Caltech, and later had worked at the Hughes Aircraft Company, where I gave lectures every Wednesday afternoon.\n\nI still wasn't exactly sure who he was.\n\nWhen I heard the investigation would be in Washington, my immediate reaction was not to do it: I have a principle of not going anywhere near Washington or having anything to do with government, so my immediate reaction was\u2014how am I gonna get out of this?\n\nI called various friends like Al Hibbs and Dick Davies, but they explained to me that investigating the _Challenger_ accident was very important for the nation, and that I should do it.\n\nMy last chance was to convince my wife. \"Look,\" I said. \"Anybody could do it. They can get somebody else.\"\n\n\"No,\" said Gweneth. \"If you don't do it, there will be twelve people, all in a group, going around from place to place together. But if you join the commission, there will be eleven people\u2014all in a group, going around from place to place together\u2014while the twelfth one runs around all over the place, checking all kinds of unusual things. There probably won't be anything, but if there is, you'll find it.\" She said, \"There isn't anyone else who can do that like you can.\"\n\nBeing very immodest, I believed her.\n\nWell, it's one thing to figure out what went wrong with the shuttle. But the next thing would be to find out what was the matter with the organization of NASA. Then there are questions like, \"Should we continue with the shuttle system, or is it better to use expendable rockets?\" And then come even bigger questions: \"Where do we go from here?\" \"What should be our future goals in space?\" I could see that a commission which started out trying to find out what happened to the shuttle could end up as a commission trying to decide on national policy, and go on forever!\n\nThat made me quite nervous. I decided to get out at the end of six months, no matter what.\n\nBut I also resolved that while I was investigating the accident, I shouldn't do anything else. There were some physics problems I was playing with. There was a computer class at Caltech I was teaching with another professor. (He offered to take over the course.) There was the Thinking Machines Company in Boston I was going to consult for. (They said they would wait.) My physics would have to wait, too.\n\nBy this time it was Sunday. I said to Gweneth, \"I'm gonna commit suicide for six months,\" and picked up the telephone.\n\n### The Cold Facts\n\n_WHEN_ I called Graham and accepted, he didn't know exactly what the commission was going to do, who it was going to be under, or even if I would be accepted onto it. (There was still hope!)\n\nBut the next day, Monday, I got a telephone call at 4 P.M.: \"Mr. Feynman, you have been accepted onto the commission\"\u2014which by that time was a \"presidential commission\" headed by William P. Rogers.\n\nI remembered Mr. Rogers. I felt sorry for him when he was secretary of state, because it seemed to me that President Nixon was using the national security adviser (Kissinger) more and more, to the point where the secretary of state was not really functioning.\n\nAt any rate, the first meeting would be on Wednesday. I figured there's nothing to do on Tuesday\u2014I could fly to Washington Tuesday night\u2014so I called up Al Hibbs and asked him to get some people at JPL* who know something about the shuttle project to brief me.\n\nOn Tuesday morning I rush over to JPL, full of steam, ready to roll. Al sits me down, and different engineers come in, one after the other, and explain the various parts of the shuttle. I don't know _how_ they knew, but they knew all about the shuttle. I got a very thorough, high-speed, intense briefing. The guys at JPL had the same enthusiasm that I did. It was really quite exciting.\n\n_F IGURE 3. The beginning offeynman's notes from his informal JPL briefing._\n\nWhen I look at my notes now, I see how quickly they gave me hints about where to look for the shuttle's problems. The first line of my notes says \"Inhibit burning. Liner.\" (To inhibit propellant from burning through the metal wall of each booster rocket, there's a liner, which was not working right.) The second line of my notes says \"O-rings show scorching in clevis check.\" It was noticed that hot gas occasionally burned past the O-rings in booster-rocket field joints.\n\nOn the same line it says \"Zn CrO4 makes bubbles.\" (The zinc chromate putty, packed as an insulator behind the O-rings, makes bubbles which can become enlarged\n\n_F IGURE 4. Detailed diagram of a field joint._\n\n_F IGURE 5. Photograph of bubbles in zinc chromate putty, which can lead to erosion of the O-rings_.\n\nvery fast when hot gas leaks through, eroding the O-rings.)\n\nThe engineers told me how much the pressures change inside the solid rocket boosters during flight, what the propellant is made of, how the propellant is cast and then baked at different temperatures, the percentages of asbestos, polymers, and whatnot in the liner, and all kinds of other stuff. I learned about the thrusts and forces in the engines, which are the most powerful engines for their weight ever built. The engines had many difficulties, especially cracked turbine blades. The engineers told me that some of the people who worked on the engines always had their fingers crossed on each flight, and the moment they saw the shuttle explode, they were sure it was the engines.\n\nIf the engineers didn't know something, they'd say something like, \"Oh, Lifer knows about that; let's get _him_ in.\" Al would call up Lifer, who would come right away. I couldn't have had a better briefing.\n\nIt's called a briefing, but it wasn't brief: it was _very_ intense, very fast, and very complete. It's the only way I know to get technical information quickly: you don't just sit there while they go through what _they_ think would be interesting; instead, you ask a lot of questions, you get quick answers, and soon you begin to understand the circumstances and learn just what to ask to get the next piece of information you need. I got one hell of a good education that day, and I sucked up the information like a sponge.\n\nThat night I took the red-eye* to Washington, and got there early Wednesday morning. (I never took the red-eye again\u2014I learned!)\n\nI checked into the Holiday Inn in downtown Washington, and got a cab to take me to the first meeting of the commission.\n\n\"Where to?\" the driver says.\n\nAll I have is a little piece of paper. \"1415 8th Street.\"\n\nWe start off. I'm new in Washington. The Capitol is over here, the Washington Monument is over there; everything seems very close. But the taxi goes on and on, farther and farther into worse and worse territory. Buildings get smaller, and they begin to look run down a little bit. Finally, we get onto 8th Street, and as we go along, the buildings begin to disappear altogether. Finally we find the address\u2014by interpolation: it's an empty lot between two buildings!\n\nBy this time I realize something is completely cockeyed. I don't know what to do, because I've only got this slip of paper, and I don't know where to go.\n\nI say to the taxi driver, \"The meeting I'm going to has something to do with NASA. Can you take me to NASA?\"\n\n\"Sure,\" he says. \"You know where it is, don't you? It's right where I picked you up!\"\n\nIt was true. NASA I could have walked to from the Holiday Inn: it was right across the street! I go in, past the guard at the gate, and start wandering around.\n\nI find my way to Graham's office, and ask if there's a meeting about the shuttle.\n\n\"Yes, I know where it is,\" somebody says. \"I'll take you down there.\"\n\nThey take me to a room and, sure enough, there's a big meeting going on: there are bright lights and television cameras down in front; the room is completely full, bursting with people, and all I can do is barely squash my way into the back. I'm thinking, \"There's only one door to this place. How the hell am I gonna get down to the front from here?\"\n\nThen I overhear something a little bit\u2014it's so far down there that I can't make out exactly what it is\u2014but it's evidently a different subject!\n\nSo I go back to Graham's office and find his secretary. She calls around and finds out where the commission is meeting. \"I don't know, either,\" she says to the person on the other end. \"He simply wandered in here!\"\n\nThe meeting was in Mr. Rogers' law offices, at 1415 _H_ Street. My slip of paper said 1415 _8th_ Street. (The address had been given over the telephone.)\n\nI finally got to Mr. Rogers' office\u2014I was the only one late\u2014and Mr. Rogers introduced me to the other commissioners. The only one I had ever heard of besides Mr. Rogers was Neil Armstrong, the moon man, who was serving as vice-chairman. (Sally Ride was on the commission, but I didn't realize who she was until later.*) There was a very handsome-looking guy in a uniform, a General Kutyna (pronounced Koo-TEE-na). He looked formidable in his outfit, while the other people had on ordinary suits.\n\nThis first meeting was really just an informal get-together. That bothered me, because I was still wound up like a spring from my JPL briefing the day before.\n\nMr. Rogers did announce a few things. He read from the executive order that defined our work:\n\nThe Commission shall:\n\n1. Review the circumstances surrounding the accident and establish the probable cause or causes of the accident; and\n\n2. Develop recommendations for corrective or other action based upon the Commission's findings and determinations.\n\nMr. Rogers also said we would complete our investigation within 120 days.\n\nThat was a relief: the scope of our commission would be limited to investigating the accident, and our work might be finished before I was done committing suicide!\n\nMr. Rogers asked each of us how much of our time we could spend on the commission. Some of the commissioners were retired, and almost everybody said they had rearranged their schedules, I said, \"I'm ready to work 100 percent, starting right now!\"\n\nMr. Rogers asked, \"Who will be in charge of writing the report?\"\n\nA Mr. Hotz, who had been the editor of _Aviation Week_ magazine, volunteered to do that.\n\nThen Mr. Rogers brought up another matter. \"I've been in Washington a long time,\" he said, \"and there's one thing you all must know: no matter what we do, there will always be leaks to the press. The best we can do is just try to minimize them. The proper way to deal with leaks is to have public meetings. We will have closed meetings, of course, but if we find anything important, we will have an open meeting right away, so the public will always know what is going on.\"\n\nMr. Rogers continued, \"To start things off right with the press, our first official meeting will be a public meeting. We'll meet tomorrow at 10 A.M.\n\nAs we were leaving the get-together, I heard General Kutyna say, \"Where's the nearest Metro station?\"\n\nI thought, \"This guy, I'm gonna get along with him fine: he's dressed so fancy, but inside, he's straight. He's not the kind of general who's looking for his driver and his special car; he goes back to the Pentagon by the Metro.\" Right away I liked him, and over the course of the commission I found my judgment in this case was excellent.\n\nThe next morning, a limousine called for me\u2014someone had arranged for us to arrive at our first official meeting in limousines. I sat in the front seat, next to the driver.\n\nOn the way to the meeting, the driver says to me, \"I understand a lot of important people are on this commission...\"\n\n\"Yeah, I s'pose...\"\n\n\"Well, I collect autographs,\" he says. \"Could you do me a favor?\"\n\n\"Sure,\" I say.\n\nI'm reaching for my pen when he says, \"When we get there, could you point out to me which one Neil Armstrong is, so I can get his autograph?\"\n\nBefore the meeting started, we were sworn in. People were milling around; a secretary handed us each a badge with our picture on it so we could go anywhere in NASA. There were also some forms to sign, saying you agree to this and that so you can get your expenses paid, and so on.\n\nAfter we were sworn in, I met Bill Graham. I did recognize him, and remembered him as a nice guy.\n\nThis first public meeting was going to be a general briefing and presentation by the big cheeses of NASA\u2014Mr. Moore, Mr. Aldrich, Mr. Lovingood, and others. We were seated in big leather chairs on a dais, and there were bright lights and TV cameras pointing at us every time we scratched our noses.\n\nI happened to sit next to General Kutyna. Just before the meeting started, he leans over and says, \"Co-pilot to pilot: comb your hair.\"\n\nI say, \"Pilot to co-pilot: can I borrow your comb?\"\n\nThe first thing we had to learn was the crazy acronyms that NASA uses all over the place: \"SRMs\" are the solid rocket motors, which make up most of the \"SRBs,\" the solid rocket boosters. The \"SSMEs\" are the space shuttle main engines; they burn \"LH\" (liquid hydrogen) and \"LOX\" (liquid oxygen), which are stored in the \"ET,\" the external tank. Everything's got letters.\n\nAnd not just the big things: practically every valve has an acronym, so they said, \"We'll give you a dictionary for the acronyms\u2014it's really very simple.\" Simple, sure, but the dictionary is a great, big, fat book that you've gotta keep looking through for things like \"HPFTP\" (high-pressure fuel turbopump) and \"HPOTP\" (high-pressure oxygen turbopump).\n\nThen we learned about \"bullets\"\u2014little black circles in front of phrases that were supposed to summarize\n\nSTS 51-L CARGO ELEMENTS\n\n\u2022 | TRACKING AND DATA RELAY SATELLITE-B\/INERTIAL UPPER STAGE\n\n---|---\n\n\u2022 | SPARTAN-HALLEY\/MISSION PECULIAR SUPPORT STRUCTURE\n\n\u2022 | CREW COMPARTMENT\n\n|\n\n\\- TISP - TEACHER IN SPACE PROGRAM\n\n|\n\n\\- CHAMP - COMET HALLEY ACTIVE MONITORING PROGRAM\n\n|\n\n\\- FDE - FLUID DYNAMICS EXPERIMENT\n\n|\n\n\\- STUDENT EXPERIMENTS\n\n|\n\n\\- RME - RADIATION MONITORING EXPERIMENT\n\n|\n\n\\- PPE - PHASE PARTITIONING EXPERIMENT\n\n_F IGURE 6. An example of \"bullets.\"_\n\nthings. There was one after another of these little goddamn bullets in our briefing books and on the slides.\n\nIt turned out that apart from Mr. Rogers and Mr. Ache-son, who were lawyers, and Mr. Hotz, who was an editor, we all had degrees in science: General Kutyna had a degree from MIT; Mr. Armstrong, Mr. Covert, Mr. Rummel, and Mr. Sutter were all aeronautical engineers, while Ms. Ride, Mr. Walker, Mr. Wheelon, and I were all physicists. Most of us seemed to have done some preliminary work on our own. We kept asking questions that were much more technical than some of the big cheeses were prepared for.\n\nWhen one of them couldn't answer a question, Mr. Rogers would reassure him that we understood he wasn't expecting such detailed questions, and that we were satisfied, for the time being at least, by the perpetual answer, \"We'll get that information to you later.\"\n\nThe main thing I learned at that meeting was how inefficient a public inquiry is: most of the time, other people are asking questions you already know the answer to\u2014or are not interested in\u2014and you get so fogged out that you're hardly listening when important points are being passed over.\n\nWhat a contrast to JPL, where I had been filled with all sorts of information very fast. On Wednesday we have a \"get-together\" in Mr. Rogers' office\u2014that takes two hours\u2014and then we've got the rest of the day to do what? Nothing. And that night? Nothing. The next day, we have the public meeting\u2014\"We'll get back to you on that\"\u2014which equals nothing! Although it _looked_ like we were doing something every day in Washington, we were, in reality, sitting around doing nothing most of the time.\n\nThat night I gave myself something to do: I wrote out the kinds of questions I thought we should ask during our investigation, and what topics we should study. My plan was to find out what the rest of the commission wanted to do, so we could divide up the work and get going.\n\nThe next day, Friday, we had our first real meeting. By this time we had an office\u2014we met in the Old Executive Office Building\u2014and there was even a guy there to transcribe every word we said.\n\nMr. Rogers was delayed for some reason, so while we waited for him, General Kutyna offered to tell us what an accident investigation is like. We thought that was a good idea, so he got up and explained to us how the air force had proceeded with its investigation of an unmanned Titan rocket which had failed.\n\nI was pleased to see that the system he described\u2014what the questions were, and the way they went about finding the answers\u2014was very much like what I had laid out the night before, except that it was much more methodical than I had envisioned. General Kutyna warned us that sometimes it looks like the cause is obvious, but when you investigate more carefully you have to change your mind. They had very few clues, and changed their minds three times in the case of the Titan.\n\nI'm all excited. I want to do this kind of investigation, and figure we can get started right away\u2014all we have to do is decide who will do what.\n\nBut Mr. Rogers, who came in partway through General Kutyna's presentation, says, \"Yes, your investigation was a great success, General, but we won't be able to use your methods here because we can't get as much information as you had.\"\n\nPerhaps Mr. Rogers, who is not a technical man, did not realize how patently false that was. The Titan, being an unmanned rocket, didn't have anywhere near the number of check gadgets the shuttle did. We had television pictures showing a flame coming out the side of a booster rocket a few seconds before the explosion; all we could see in General Kutyna's pictures of the Titan was a lousy dot in the sky\u2014just a little, tiny flash\u2014and he was able to figure stuff out from that.\n\nMr. Rogers says, \"I have arranged for us to go to Florida next Thursday. We'll get a briefing there from NASA officials, and they'll take us on a tour of the Kennedy Space Center.\"\n\nI get this picture of the czarina coming to a Potemkin village: everything is all arranged; they show us how the rocket looks and how they put it together. It's not the way to find out how things _really_ are.\n\nThen Mr. Armstrong says, \"We can't expect to do a technical investigation like General Kutyna did.\" This bothered me a lot, because the only things I pictured myself doing were technical! I didn't know exactly what he meant: perhaps he was saying that all the technical lab work would be done by NASA.\n\nI began suggesting things I could do.\n\nWhile I'm in the middle of my list, a secretary comes in with a letter for Mr. Rogers to sign. In the interim, when I've just been shut up and I'm waiting to come back, various other commission members offer to work with me. Then Mr. Rogers looks up again to continue the meeting, but he calls on somebody else\u2014as if he's absentminded and forgot I'd been interrupted. So I have to get the floor again, but when I start my stuff again, another \"accident\" happens.\n\nIn fact, Mr. Rogers brought the meeting to a close while I was in midstream! He repeated his worry that we'll never really figure out what happened to the shuttle.\n\nThis was extremely discouraging. It's hard to understand now, because NASA has been taking at least two years to put the shuttle back on track. But at the time, I thought it would be a matter of days.\n\nI went over to Mr. Rogers and said, \"We're going to Florida next Thursday. That means we've got nothing to do for _five days:_ what'll I do for five days?\"\n\n\"Well, what would you have done if you hadn't been on the commission?\"\n\n\"I was going to go to Boston to consult, but I canceled it in order to work 100 percent.\"\n\n\"Well, why don't you go to Boston for the five days?\"\n\nI couldn't take that. I thought, \"I'm dead already! The goddamn thing isn't working right.\" I went back to my hotel, devastated.\n\nThen I thought of Bill Graham, and called him up. \"Listen, Bill,\" I said. \"You got me into this; now you've gotta save me: I'm completely depressed; I can't stand it.\"\n\nHe says, \"What's the matter?\"\n\n\"I want to _do_ something! I want to go around and talk to some engineers!\"\n\nHe says, \"Sure! Why not? I'll arrange a trip for you. You can go wherever you want: you could go to Johnson, you could go to Marshall, or you could go to Kennedy...\"\n\nI thought I wouldn't go to Kennedy, because it would look like I'm rushing to find out everything ahead of the others. Sally Ride worked at Johnson, and had offered to work with me, so I said, \"I'll go to Johnson.\"\n\n\"Fine,\" he says. \"I'll tell David Acheson. He's a personal friend of Rogers, and he's a friend of mine. I'm sure everything will be okay.\"\n\nHalf an hour later, Acheson calls me: \"I think it's a great idea,\" he says, \"and I told Mr. Rogers so, but he says no. I just don't know why I can't convince him.\"\n\nMeanwhile, Graham thought of a compromise: I would stay in Washington, and he would get people to come to his office at NASA, right across the street from my hotel. I would get the kind of briefing I wanted, but I wouldn't be running around.\n\nThen Mr. Rogers calls me: he's against Graham's compromise. \"We're all going to Florida next Thursday,\" he says.\n\nI say, \"If the idea is that we sit and listen to briefings, it won't work with me. I can work much more efficiently if I talk to engineers directly.\"\n\n\"We have to proceed in an orderly manner.\"\n\n\"We've had several meetings by now, but we still haven't been assigned anything to do!\"\n\nRogers says. \"Well, do you want me to bother all the other commissioners and call a special meeting for Monday, so we can make such assignments?\"\n\n\"Well, yes!\" I figured our job was to work, and we _should_ be bothered\u2014you know what I mean?\n\nSo he changes the subject, naturally. He says, \"I understand you don't like the hotel you're in. Let me put you in a good hotel.\"\n\n\"No, thank you; everything is fine with my hotel.\"\n\nPretty soon he tries again, so I say, \"Mr. Rogers, my personal comfort is not what I'm concerned with. I'm trying to get to work. I want to _do_ something!\"\n\nFinally, Rogers says it's okay to go across the street to talk to people at NASA.\n\nI was obviously quite a pain in the ass for Mr. Rogers. Later, Graham tried to explain it to me. \"Suppose you, as a technical person, were given the job as chairman of a committee to look into some legal question. Your commission is mostly lawyers, and one of them keeps saying, 'I can work more effectively if I talk directly to other lawyers.' I assume you'd want to get your bearings first, before letting anybody rush off investigating on his own.\"\n\nMuch later, I appreciated that there were lots of problems which Mr. Rogers had to address. For example, any piece of information any of us received had to be entered into the record and made available to the other commissioners, so a central library had to be set up. Things like that took time.\n\nOn Saturday morning I went to NASA. Graham brought in guys to tell me all about the shuttle. Although they were pretty high up in NASA, the guys were technical.\n\nThe first guy told me all about the solid rocket boosters\u2014the propellant, the motor, the whole thing except the seals. He said, \"The seals expert will be here this afternoon.\"\n\nThe next guy told me all about the engine. The basic operation was more or less straightforward, but then there were all kinds of controls, with backing and hauling from pipes, heating from this and that, with high-pressure hydrogen pushing a little propeller which turns something else, which pumps oxygen through a vent valve\u2014that kind of stuff.\n\nIt was interesting, and I did my best to understand it, but after a while I told the fella, \"That's as much as I'm going to take, now, on the engine.\"\n\n\"But there are many problems with the engines that you should hear about,\" he says.\n\nI was hot on the trail of the booster rocket, so I said, \"I'll have to put off the main engines till later, when I have more time.\"\n\nThen a guy came in to tell me about the orbiter. I felt terrible, because he had come in on a Saturday to see me, and it didn't look like the orbiter had anything to do with the accident. I was having enough trouble understanding the rest of the shuttle\u2014there's only a certain amount of information per cubic inch a brain can hold\u2014so I let him tell me some of the stuff, but soon I had to tell him that it was getting too detailed, so we just had a pleasant conversation.\n\nIn the afternoon, the seals expert came in\u2014his name was Mr. Weeks\u2014and gave me what amounted to a continuation of my JPL briefing, with still more details.\n\nThere's putty and other things, but the ultimate seal is supposed to be two rubber rings, called O-rings, which are approximately a quarter of an inch thick and lie on a circle 12 feet in diameter\u2014that's something like 37 feet long.\n\nWhen the seals were originally designed by the Morton Thiokol Company, it was expected that pressure from the burning propellant would squash the O-rings. But because the joint is stronger than the wall (it's three times thicker), the wall bows outward, causing the joint to bend a little\u2014enough to lift the rubber O-rings off the seal area. Mr. Weeks told me this phenomenon is called \"joint rotation,\" and it was discovered very early, before they ever flew the shuttle.\n\nThe pieces of rubber in the joints are called O-rings, but they're not used like normal O-rings are. In ordinary circumstances, such as sealing oil in the motor of an automobile, there are sliding parts and rotating shafts, but the gaps are always the same. An O-ring just sits there, in a fixed position.\n\nBut in the case of the shuttle, the gap _expands_ as the pressure builds up in the rocket. And to maintain the seal, the rubber has to expand _fast_ enough to close the gap\u2014and during a launch, the gap opens in a fraction of a second.\n\n_F IGURE 7. Joint rotation is caused by pressure from inside the rocket pushing the walls out farther than the joints. A gap opens, and hot gas flows past one or both of the O-rings._\n\nThus the resilience of the rubber became a very essential part of the design.\n\nWhen the Thiokol engineers were discovering these problems, they went to the Parker Seal Company, which manufactures the rubber, to ask for advice. The Parker Seal Company told Thiokol that O-rings are not meant to be used that way, so they could give no advice.\n\nAlthough it was known from nearly the beginning that the joint was not working as it was designed to, Thiokol kept struggling with the device. They made a number of makeshift improvements. One was to put shims in to keep the joint tight, but the joint still leaked. Mr. Weeks showed me pictures of leaks on previous flights\u2014what the engineers called \"blowby,\" a blackening behind an O-ring where hot gas leaked through, and what they called \"erosion,\" where an O-ring had burned a little bit. There was a chart showing all the flights, and how serious the blowby and erosion were on each one. We went through the whole history up to _the_ flight, 51-L.\n\nI said, \"Where does it say they were ever discussing the problem\u2014how it's going along, or whether there's some progress?\"\n\nThe only place was in the \"flight readiness reviews\"\u2014between flights there was no discussion of the seals problem!\n\nWe looked at the summary of the report. Everything was behind little bullets, as usual. The top line says:\n\n\u2022 | The lack of a good secondary seal in the field joint is most critical and ways to reduce joint rotation should be incorporated as soon as possible to reduce criticality.\n\n---|---\n\nAnd then, near the bottom, it says:\n\n\u2022 | Analysis of existing data indicates that it is safe to continue flying existing design as long as all joints are leak checked* with a 200 psig stabilization...\n\n---|---\n\nI was struck by the contradiction: \"If it's 'most critical,' how could it be 'safe to continue flying'? What's the _logic_ of this?\"\n\nMr. Weeks says, \"Yes, I see what you mean! Well, let's see: it says here, 'Analysis of existing data...'\"\n\nWe went back through the report and found the analysis. It was some kind of computer model with various as-\n\n**Figure 8** _Thiokol attempted to cure the joint-rotation problem with shims_.\n\n**Figure 9** _Two examples O-ring erosion. Such erosion would occur unpredictably along 2 or 3 inches of the 37-foot O-ring_.\n\n**Figure 10** _The correlation between temperature and O-ring incidents_.\n\n**Figure 11** _The self-contradictory recommendations of the seals report are underlined_.\n\nsumptions that were not necessarily right. You know the danger of computers, it's called GIGO: garbage in, garbage out! The analysis concluded that a little unpredictable leakage here and there could be tolerated, even though it wasn't part of the original design.\n\nIf _all_ the seals had leaked, it would have been obvious even to NASA that the problem was serious. But only a few of the seals leaked on only some of the flights. So NASA had developed a peculiar kind of attitude: if one of the seals leaks a little and the flight is successful, the problem isn't so serious. Try playing Russian roulette that way: you pull the trigger and the gun doesn't go off, so it must be safe to pull the trigger again...\n\nMr. Weeks said there was a rumor that the history of the seals problem was being leaked to the newspapers. That bothered him a little bit, because it made NASA look like it was trying to keep things secret.\n\nI told him I was entirely satisfied with the people Graham had brought in to talk to me, and that since I had already heard about the seals problem at JPL, it wasn't any big deal.\n\nThe next day, Sunday, Bill Graham took me with his family to the National Air and Space Museum. We had an early breakfast together, and then we went across the street to the museum.\n\nI was expecting to see big crowds there, but I had forgotten that Graham was such a big shot. We had the whole place to ourselves for a while.\n\nWe did see Sally Ride there. She was in a display case, in an astronaut's suit, holding a helmet and everything. The wax model looked exactly like her.\n\nAt the museum there was a special theater with a movie about NASA and its achievements. The movie was wonderful. I had not fully appreciated the enormous number of people who were working on the shuttle, and all the effort that had gone into making it. And you know how a movie is: they can make it dramatic. It was so dramatic that I almost began to cry. I could see that the accident was a terrible blow. To think that so many people were working so hard to make it go\u2014and then it busts\u2014made me even more determined to help straighten out the problems of the shuttle as quickly as possible, to get all those people back on track. After seeing this movie I was very changed, from my semi anti-NASA attitude to a very strong pro-NASA attitude.\n\nThat afternoon, I got a telephone call from General Kutyna.\n\n\"Professor Feynman?\" he says. \"I have some urgent news for you. Uh, just a minute.\"\n\nI hear some military-type band music in the background.\n\nThe music stops, and General Kutyna says, \"Excuse me, Professor; I'm at an Air Force Band concert, and they just played the national anthem.\"\n\nI could picture him in his uniform, standing at attention while the band is playing the \"Star Spangled Banner,\" saluting with one hand and holding the telephone with the other. \"What's the news, General?\"\n\n\"Well, the first thing is, Rogers told me to tell you not to go over to NASA.\"\n\nI didn't pay any attention to that, because I had already gone over to NASA the day before.\n\nHe continued, \"The other thing is, we're going to have a special meeting tomorrow afternoon to hear from a guy whose story came out in the _New York Times_ today.\"\n\nI laughed inside: so we're going to have a special meeting on Monday, anyway!\n\nThen he says, \"I was working on my carburetor this morning, and I was thinking: the shuttle took off when the temperature was 28 or 29 degrees. The coldest temperature previous to that was 53 degrees. You're a professor; _what, sir, is the effect of cold on the O-rings?\"_\n\n\"Oh!\" I said. \"It makes them stiff. Yes, of course!\" That's all he had to tell me. It was a clue for which I got a lot of credit later, but it was his observation. A professor of theoretical physics always has to be told what to look for. He just uses his knowledge to explain the observations of the experimenters!\n\nOn Monday morning General Kutyna and I went over to Graham's office and asked him if he had any information on the effects of temperature on the O-rings. He didn't have it on hand, but said he would get it to us as soon as possible.\n\nGraham did, however, have some interesting photographs to show us. They showed a flame growing from the right-hand solid rocket booster a few seconds before the explosion. It was hard to tell exactly where the flame was coming out, but there was a model of the shuttle right there in the office. I put the model on the floor and walked around it until it looked exactly like the picture\u2014in size, and in orientation.\n\nI noticed that on each booster rocket there's a little hole\u2014called the leak test port\u2014where you can put pressure in to test the seals. It's _between_ the two O-rings, so if it's not closed right and if the first O-ring fails, the gas would go out through the hole, and it would be a catastrophe. It was just about where the flame was. Of course, it was still a question whether the flame was coming out of the leak test port or a larger flame was coming out farther around, and we were seeing only the tip of it.\n\nThat afternoon we had our emergency closed meeting to hear from the guy whose story was in the _New York Times._ His name was Mr. Cook. He was in the budget department of NASA when he was asked to look into a possible seals problem and to estimate the costs needed to rectify it.\n\nBy talking to the engineers, he found out that the seals had been a big problem for a long time. So he reported that it would cost so-and-so much to fix it\u2014a lot of money. From the point of view of the press and some of the commissioners, Mr. Cook's story sounded like a big expos\u00e9, as if NASA was hiding the seals problem from us.\n\nI had to sit through this big, unnecessary excitement, wondering if every time there was an article in the newspaper, would we have to have a special meeting? We would never get anywhere that way!\n\nBut later, during that same meeting, some very interesting things happened. First, we saw some pictures which showed puffs of smoke coming out of a field joint just after ignition, before the shuttle even got off the pad. The smoke was coming out of the same place\u2014possibly the leak test port\u2014where the flame appeared later. There wasn't much question, now. It was all fitting together.\n\nThen something happened that was completely unexpected. An engineer from the Thiokol Company, a Mr. McDonald, wanted to tell us something. He had come to our meeting on his own, uninvited. Mr. McDonald reported that the Thiokol engineers had come to the conclusion that low temperatures had something to do with the seals problem, and they were very, very worried about it. On the night before the launch, during the flight readiness review, they told NASA the shuttle shouldn't fly if the temperature was below 53 degrees\u2014the previous lowest temperature\u2014and on that morning it was 29.\n\nMr. McDonald said NASA was \"appalled\" by that statement. The man in charge of the meeting, a Mr. Mulloy, argued that the evidence was \"incomplete\"\u2014some flights with erosion and blowby had occurred at _higher_ than 53 degrees\u2014so Thiokol should reconsider its opposition to flying.\n\n**Figure 12a** _Progression of a flame, possibly from the leak test port area(\u00a9 NASA.)_\n\nThiokol reversed itself, but McDonald refused to go along, saying, \"If something goes wrong with this flight, I wouldn't want to stand up in front of a board of inquiry and say that I went ahead and told them to go ahead and fly this thing outside what it was qualified to.\"\n\nThat was so astonishing that Mr. Rogers had to ask, \"Did I understand you correctly, that you said...,\" and he repeated the story. And McDonald says, \"Yes, sir.\"\n\nThe whole commission was shocked, because this was the first time any of us had heard _this_ story: not only was there a failure in the seals, but there may have been a failure in management, too.\n\n**Figure 12a** _An incorrectly sealed leak test port could provide an escape route for a flame which burns past the primary O-ring_.\n\nMr. Rogers decided that we should look carefully into Mr. McDonald's story, and get more details before we made it public. But to keep the public informed, we would have an open meeting the following day, Tuesday, in which Mr. Cook would testify.\n\nI thought, \"This is going to be like an act: we're going to say the same things tomorrow as we did today, and we won't learn anything new.\"\n\nAs we were leaving, Bill Graham came over with a stack of papers for me.\n\n\"Geez! That's fast!\" I said. \"I only asked you for the information this morning!\" Graham was always very cooperative.\n\n**Figure 14** _Puffs of black \"smoke\" (fine unburned particles) were seen escaping from the same place where the flame was observed.(\u00a9 NASA_.)\n\nThe paper on top says, \"Professor Feynman of the Presidential Commission wants to know about the effects over time of temperature on the resiliency of the O-rings...\"\u2014it's a memorandum addressed to a subordinate.\n\nUnder that memo is another memo: \"Professor Feynman of the Presidential Commission wants to know...\"\u2014from that subordinate to _his_ subordinate, and so on down the line.\n\nThere's a paper with some numbers on it from the poor bastard at the bottom, and then there's a series of submission papers which explain that the answer is being sent up to the next level.\n\nSo here's this stack of papers, just like a sandwich, and in the middle is the answer\u2014to the wrong question! The answer was: \"You squeeze the rubber for two hours at a certain temperature and pressure, and then see how long it takes to creep back\"\u2014over _hours_. I wanted to know how fast the rubber responds in _milliseconds_ during a launch. So the information was of no use.\n\nI went back to my hotel. I'm feeling lousy and I'm eating dinner; I look at the table, and there's a glass of ice water. I say to myself, \"Damn it, _I_ can find out about that rubber _without_ having NASA send notes back and forth: I just have to _try_ it! All I have to do is get a sample of the rubber.\"\n\nI think, \"I could do this tomorrow while we're all sittin' around, listening to this Cook crap we heard today. We always get ice water in those meetings; that's something I can do to save time.\"\n\nThen I think, \"No, that would be gauche.\"\n\nBut then I think of Luis Alvarez, the physicist. He's a guy I admire for his gutsiness and sense of humor, and I think, \"If Alvarez was on this commission, he would do it, and that's good enough for me.\"\n\nThere are stories of physicists\u2014great heroes\u2014who have gotten information one, two, three\u2014just like that\u2014where everybody else is trying to do it in a complicated way. For example, after ultraviolet rays and X-rays had been discovered, there was a new type, called N-rays, discovered by Andr\u00e9 Blondel, in France. It was hard to detect the N-rays: other scientists had difficulty repeating Blondel's experiments, so someone asked the great American physicist R. W. Wood to go to Blondel's laboratory.\n\nBlondel gave a public lecture and demonstration. N-rays were bent by aluminum, so he had all kinds of lenses lined up, followed by a big disk with an aluminum prism in the middle. As the aluminum prism slowly turned, the N-rays came up this way and bent that way, and Blondel's assistant reported their intensity\u2014different numbers for different angles.\n\nN-rays were affected by light, so Blondel turned out the lights to make his readings more sensitive. His assistant continued to report their intensity.\n\nWhen the lights came back on, there's R. W. Wood in the front row, holding the prism high in the air, balanced on the tips of his fingers, for all to see! So that was the end of the N-ray.\n\nI think, \"Exactly! I've got to get a sample of the rubber.\" I call Bill Graham.\n\nIt's impossible to get: it's kept somewhere down at Kennedy. But then Graham remembers that the model of the field joint we're going to use in our meeting tomorrow has two samples of the rubber in it. He says, \"We could meet in my office before the meeting and see if we can get the rubber out.\"\n\nThe next morning I get up early and go out in front of my hotel. It's eight in the morning and it's snowing. I find a taxi and say to the driver, \"I'd like to go to a hardware store.\"\n\n\"A hardware store, sir?\"\n\n\"Yeah. I gotta get some tools.\"\n\n\"Sir, there's no hardware stores around here; the Capitol is over there, the White House is over there\u2014wait a minute: I think I remember passing one the other day.\"\n\nHe found the hardware store, and it turned out it didn't open till 8:30\u2014it was about 8:15\u2014so I waited outside, in my suitcoat and tie, a costume I had assumed since I came to Washington in order to move among the natives without being too conspicuous.\n\nThe suitcoats that the natives wear inside their buildings (which are well heated) are sufficient for walking from one building to another\u2014or from a building to a taxi if the buildings are too far apart. (All the taxis are heated.) But the natives seem to have a strange fear of the cold: they put overcoats on top of their suitcoats if they wish to step outside. I hadn't bought an overcoat yet, so I was still rather conspicuous standing outside the hardware store in the snow.\n\nAt 8:30 I went in and bought a couple of screwdrivers, some pliers, and the smallest C-clamp I could find. Then I went to NASA.\n\nOn the way to Graham's office, I thought maybe the clamp was too big. I didn't have much time, so I ran down to the medical department of NASA. (I knew where it was, because I had been going there for blood tests ordered by my cardiologist, who was trying to treat me by telephone.) I asked for a medical clamp like they put on tubes.\n\nThey didn't have any. But the guy says, \"Well, let's see if your C-clamp fits inside a glass!\" It fitted very easily.\n\nI went up to Graham's office.\n\nThe rubber came out of the model easily with just a pair of pliers. So there I was with the rubber sample in my hand. Although I knew it would be more dramatic and honest to do the experiment for the first time in the public meeting, I did something that I'm a little bit ashamed of. I cheated. I couldn't resist. I tried it. So, following the example of having a closed meeting before an open meeting,I discovered it worked before I did it in the open meeting. Then I put the rubber back into the model so Graham could take it to the meeting.\n\n_F IGURE 15. The field-joint model from which Feynman gol the O-ring sample_.\n\nI go to the meeting, all ready, with pliers in one pocket and a C-clamp in the other. I sit down next to General Kutyna.\n\nAt the previous meeting, there was ice water for everybody. This time, there's no ice water. I get up and go over to somebody who looks like he's in charge, and I say, \"I'd like a glass of ice water, please.\"\n\nHe says, \"Certainly! Certainly!\"\n\nFive minutes later, the guards close the doors, the meeting starts, and I haven't got my ice water.\n\nI gesture over to the guy I just talked to. He comes over and says, \"Don't worry, it's coming!\"\n\nThe meeting is going along, and now Mr. Mulloy begins to tell us about the seals. (Apparently, NASA wants to tell us about the seals before Mr. Cook does.) The model starts to go around, and each commissioner looks at it a little bit.\n\nMeanwhile, no ice water!\n\nMr. Mulloy explains how the seals are supposed to work\u2014in the usual NASA way: he uses funny words and acronyms, and it's hard for anybody else to understand.\n\nIn order to set things up while I'm waiting for the ice water, I start out: \"During a launch, there are vibrations which cause the rocket joints to move a little bit\u2014is that correct?\"\n\n\"That is correct, sir.\"\n\n\"And inside the joints, these so-called O-rings are supposed to expand to make a seal\u2014is that right?\"\n\n\"Yes, sir. In static conditions they should be in direct contact with the tang and clevis* and squeezed twenty-thousandths of an inch.\"\n\n\"Why don't we take the O-rings out?\"\n\n\"Because then you would have hot gas expanding through the joint...\"\n\n\"Now, in order for the seal to work correctly, the O-rings must be made of rubber\u2014not something like lead, which, when you squash it, it stays.\"\n\n\"Yes, sir.\"\n\n\"Now, if the O-ring weren't resilient for a second or two, would that be enough to be a very dangerous situation?\"\n\n\"Yes, sir.\"\n\nThat led us right up to the question of cold temperature and the resilience of the rubber. I wanted to prove that Mr. Mulloy must have known that temperature had an effect, although\u2014according to Mr. McDonald\u2014he claimed that the evidence was \"incomplete.\" But still, no ice water! So I had to stop, and somebody else started asking questions.\n\nThe model comes around to General Kutyna, and then to me. The clamp and pliers come out of my pocket, I take the model apart, I've got the O-ring pieces in my hand, but I still haven't got any ice water! I turn around again and signal the guy I've been bothering about it, and he signals back, \"Don't worry, you'll get it!\"\n\nPretty soon I see a young woman, way down in front, bringing in a tray with glasses on it. She gives a glass of ice water to Mr. Rogers, she gives a glass of ice water to Mr. Armstrong, she works her way back and forth along the rows of the dais, giving ice water to everybody! The poor woman had gotten everything together\u2014jug, glasses, ice, tray, the whole thing\u2014so that everybody could have ice water.\n\nSo finally, when I get my ice water, I don't drink it! I squeeze the rubber in the C-clamp, and put them in the glass of ice water.\n\nAfter a few minutes, I'm ready to show the results of my little experiment. I reach for the little button that activates my microphone.\n\nGeneral Kutyna, who's caught on to what I'm doing, quickly leans over to me and says, \"Co-pilot to pilot: not now.\"\n\nPretty soon, I'm reaching for my microphone again.\n\n\"Not now!\" He points in our briefing book\u2014with all the charts and slides Mr. Mulloy is going through\u2014and says, \"When he comes to this slide, here, that's the right time to do it.\"\n\nFinally Mr. Mulloy comes to the place, I press the button for my microphone, and I say, \"I took this rubber from the model and put it in a clamp in ice water for a while.\"\n\nI take the clamp out, hold it up in the air, and loosen it as I talk: \"I discovered that when you undo the clamp, the rubber doesn't spring back. In other words, for more than a few seconds, there is no resilience in this particular material when it is at a temperature of 32 degrees. I believe that\n\n_F IGURE 15A. The O-ring ice-water demonstration. (\u00a9 MARILYNN K. YEE, NYT PICTURES_.)\n\nhas some significance for our problem.\"\n\nBefore Mr. Mulloy could say anything, Mr. Rogers says, \"That is a matter we will consider, of course, at length in the session that we will hold on the weather, and I think it is an important point which I'm sure Mr. Mulloy acknowledges and will comment on in a further session.\"\n\nDuring the lunch break, reporters came up to me and asked questions like, \"Were you talking about the O-ring, or the putty?\" and \"Would you explain to us what an CD-ring is, exactly?\" So I was rather depressed that I wasn't able to make my point. But that night, all the news shows caught on to the significance of the experiment, and the next day, the newspaper articles explained everything perfectly.\n\n### Check Six!\n\n_MY_ cousin Frances educated me about the press. She had been the AP White House correspondent during the Nixon and Ford administrations, and was now working for CNN. Frances would tell me stories of guys running out back doors because they're afraid of the press. From her I got the idea that the press isn't doing anything evil; the reporters are simply trying to help people know what's going on, and it doesn't do any harm to be courteous to them.\n\nI found out that they're really quite friendly, if you give them a chance. So I wasn't afraid of the press, and I would always answer their questions.\n\nReporters would explain to me that I could say, \"Not for attribution.\" But I didn't want any hocus-pocus. I didn't want it to sound like I'm leaking something. So whenever I talked to the press, I was straight. As a result of this, my name was in the newspaper every day, all over the place!\n\nIt seemed like I was always the one answering the reporters' questions. Often the rest of the commissioners would be anxious to go off to lunch, and I'd still be there, answering questions. But I figured, \"What's the point of having a public meeting if you run away when they ask you what a word meant?\"\n\nWhen we'd finally get to our lunch, Mr. Rogers would remind us to be careful not to talk to the press. I would say something like, \"Well, I was just telling them about the O-rings.\"\n\nHe would say, \"That's okay. You've been doing all right, Dr. Feynman; I have no problem with that.\" So I never did figure out, exactly, what he meant by \"not talking to the press.\"\n\nBeing on the commission was rather tense work, so I enjoyed having dinner once in a while with Frances and Chuck, my sister's son, who was working for the _Washington Post_. Because Mr. Rogers kept talking about leaks, we made sure we never said a _word_ about anything I was doing. If CNN needed to find out something from me, they'd have to send a different reporter. The same went for the _Post_.\n\nI told Mr. Rogers about my relatives working for the press: \"We've agreed not to talk about my work. Do you think there's any problem?\"\n\nHe smiled and said, \"It's perfectly all right. I have a cousin in the press, too. There's no problem at all.\"\n\nOn Wednesday the commission had nothing to do, so General Kutyna invited me over to the Pentagon to educate me on the relationship between the air force and NASA.\n\nIt was the first time I had ever been in the Pentagon. There were all these guys in uniform who would take orders\u2014not like in civilian life. He says to one of them, \"I'd like to use the briefing room...\"\n\n\"Yes, sir!\"\n\n\"... and we'll need to see slides number such-and-such and so-and-so.\"\n\n\"Yes, sir! Yes, sir!\"\n\nWe've got all these guys working for us while General Kutyna gives me a big presentation in this special briefing room. The slides are shown from the back on a transparent wall. It was really fancy.\n\nGeneral Kutyna would say things like, \"Senator So-and-so is in NASA's pocket,\" and I would say, half-joking, \"Don't give me these side remarks, General; you're filling my head! But don't worry, I'll forget it all.\" I wanted to be naive: I'd find out what happened to the shuttle first; I'd worry about the big political pressures later.\n\nSomewhere in his presentation, General Kutyna observed that everybody on the commission has some weakness because of their connections: he, having worked very closely with NASA personnel in his former position as Air Force Space Shuttle Program manager, finds it difficult, if not impossible, to drive home some of the tougher questions on NASA management. Sally Ride still has a job with NASA, so she can't just say everything she wants. Mr. Covert had worked on the engines, and had been a consultant to NASA, and so on.\n\nI said, \"I'm associated with Caltech, but I don't consider that a weakness!\"\n\n\"Well,\" he says, \"that's right. You're invincible\u2014as far as we can see. But in the air force we have a rule: check six.\"\n\nHe explained, \"A guy is flying along, looking in all directions, and feeling very safe. Another guy flies up behind him (at 'six o'clock'\u2014'twelve o'clock' is directly in front), and shoots. Most airplanes are shot down that way. Thinking that you're safe is very dangerous! Somewhere, there's a weakness you've got to find. You must always check six o'clock.\"\n\nAn underling comes in. There's some mumbling about somebody else needing the briefing room now. General Kutyna says, \"Tell them I'll be finished in ten minutes.\"\n\n\"Yes, sir!\"\n\nFinally, we go out. There, in the hall, are TEN GENERALS waiting to use the room\u2014and I had been sitting in there, getting this personal briefing. I felt great.\n\nFor the rest of the day, I wrote a letter home. I began to worry about \"check six\" when I described Mr. Rogers' reaction to my visiting Frances and Chuck. I wrote,... I was pleased by Rogers' reaction, but now as I write this I have second thoughts. It was too easy\u2014after he explicitly talked about the importance of no leaks etc. at earlier meetings. Am I being set up? (SEE, DARLING, WASHINGTON PARANOIA IS SETTING IN.)... I think it is possible that there are things in this somebody might be trying to keep me from finding out and might try to discredit me if I get too close.... So, reluctantly, I will have to not visit Frances and Chuck any more. Well, I'll ask Fran first if that is _too_ paranoid. Rogers seemed so agreeable and reassuring. It was so easy, yet I am probably a thorn in his side....\n\nTomorrow at 6:15 am we go by special airplane (two planes) to Kennedy Space Center to be \"briefed.\" No doubt we shall wander about, being shown everything\u2014gee whiz\u2014but no time to get into technical details with anybody. Well, it won't work. If I am not satisfied by Friday, I will stay over Sat & Sun, or if they don't work then, Monday & Tuesday. I am determined to do the job of finding out what happened\u2014let the chips fall!\n\nMy guess is that I will be allowed to do this, overwhelmed with data and details..., so they have time to soften up dangerous witnesses etc. But it won't work because (1) I do technical information exchange and understanding much faster than they imagine, and (2) I already smell certain rats that I will not forget, because I just love the smell of rats, for it is the spoor of exciting adventure.\n\nI feel like a bull in a china shop. The best thing is to put the bull out to work on the plow. A better metaphor will be an ox in a china shop, because the china is the bull, of course.*\n\nSo, much as I would rather be home and doing something else, I am having a wonderful time.\n\nLove,\n\nRichard\n\nThe press was reporting rumors that NASA was under great political pressure to launch the shuttle, and there were various theories as to where the pressure was coming\n\nfrom. It was a great big world of mystery to me, with tremendous forces. I would investigate it, all right, and if I protected myself, nothing would happen. But I hadda watch out.\n\n### Gumshoes\n\n_FINALLY_ , early on Thursday morning, we get to Florida. The original idea was that we would go around the Kennedy Space Center at Cape Canaveral and see everything on a guided tour. But because information was coming out in the newspapers so fast, we had a public meeting first.\n\nFirst, we saw some detailed pictures of the smoke coming out of the shuttle while it was still on the launch pad. There are cameras all over the place watching the launch\u2014something like a hundred of them. Where the smoke came out, there were two cameras looking straight at it\u2014but both failed, curiously. Nevertheless, from other cameras we could see four or five puffs of black smoke coming out from a field joint. This smoke was not burning material; it was simply carbon and mucky stuff that was pushed out because of pressure inside the rocket.\n\nThe puffs stopped after a few seconds: the seal got plugged up somehow, temporarily, only to break open again a minute later.\n\nThere was some discussion about how much matter came out in the smoke. The puffs of smoke were about six feet long, and a few feet thick. The amount of matter depends on how fine the particles are, and there could always be a big piece of glop inside the smoke cloud, so it's hard to judge. And because the pictures were taken from the side, it was possible there was more smoke farther around the rocket.\n\n_F IGURE 16. Detailed picture, taken from the launch pad, of the \"smoke.\"_ (\u00a9; _NASA_.)\n\nTo establish a minimum, I assumed a particle size that would produce as much smoke as possible out of a given amount of material. It came out surprisingly small\u2014approximately one cubic inch: if you have a cubic inch of stuff, you can get that much smoke.\n\nWe asked for pictures from other launches. We found out later that there had never been any puffs of smoke on any previous flights.\n\nWe also heard about the low temperatures before the launch from a man named Charlie Stevenson, who was in charge of the ice crew. He said the temperature had gone down to 22 degrees during the night, but his crew got readings as low as 8 degrees at some places on the launch pad, and they couldn't understand why.\n\nDuring the lunch break, a reporter from a local TV station asked me what I thought about the low temperature readings. I said it seemed to me that the liquid hydrogen and oxygen had chilled the 22-degree air even further as it flowed down the big fuel tank onto the rocket booster. For some reason, the reporter thought I had just told him some important, secret information, so he didn't use my name in his report that evening. Instead, he said, \"This explanation comes from a Nobel Prize winner, so it must be right.\"\n\nIn the afternoon, the telemetering people gave us all kinds of information on the last moments of the shuttle. Hundreds of things had been measured, all of which indicated that everything was working as well as it could under the circumstances: the pressure in the hydrogen tank suddenly fell a few seconds after the flame had been observed; the gyros which steer the shuttle were working perfectly until one had to work harder than the other because there were side forces from the flame shooting out of the side of the booster rocket; the main engines even shut themselves down when the hydrogen tank exploded, because there was a pressure drop in the fuel lines.\n\nThat meeting lasted until 7:30 in the evening, so we postponed the tour until Friday and went straight to a dinner set up by Mr. Rogers.\n\nAt the dinner I happened to be seated next to Al Keel, who had joined the commission on Monday as its executive officer to help Mr. Rogers organize and run our work. He came to us from the White House\u2014from something called the OMB*\u2014and had a good reputation for doing a fine job at this and that. Mr. Rogers kept saying how lucky we were to get somebody with such high qualifications.\n\nOne thing that impressed me, though, was that Dr. Keel had a Ph.D. in aerospace, and had done some post-doc work at Berkeley. When he introduced himself on Monday, he joked that the last \"honest work\" he had done for a living was some aerodynamics work for the shuttle program ten or twelve years ago. So I felt very comfortable with him.\n\nWell, I haven't been talking to Dr. Keel for more than five minutes, when he tells me he's never been so insulted in his life, that he didn't take this job to be so insulted, and that he doesn't want to talk to me anymore!\n\nNow, I have a way of not remembering things when I do something dumb or annoying to people, so I forget what I said that put him out. Whatever it was, I thought I was joking, so I was very surprised by his reaction. I had undoubtedly said some boorish, brash, damn-fool thing, which I therefore can't remember!\n\nThen there was a rather tense period of five or ten minutes, with me apologizing and trying to get a conversation going again. We finally got to talking again, somewhat. We were not big friends, but at least there was peace.\n\nOn Friday morning, we had another public meeting, this time to hear people from Thiokol and NASA talk about the night before the launch. Everything came out so slowly: the witness doesn't really want to tell you everything, so you have to get the answers out by asking exactly the right questions.\n\nOther guys on the commission were completely awake\u2014Mr. Sutter, for instance. \"Exactly what were your quality criteria for acceptance under such-and-such and so-and-so?\"\u2014he'd ask specific questions like that, and it would turn out they didn't have any such criteria. Mr. Covert and Mr. Walker were the same way. Everybody was asking good questions, but I was fogged out most of the time, feeling a little bit behind.\n\nThen this business of Thiokol changing its position came up. Mr. Rogers and Dr. Ride were asking two Thiokol managers, Mr. Mason and Mr. Lund, how many people were against the launch, even at the last moment.\n\n\"We didn't poll everyone,\" says Mr. Mason.\n\n\"Was there a substantial number against the launch, or just one or two?\"\n\n\"There were, I would say, probably five or six in engineering who at that point would have said it is not as conservative to go with that temperature, and we don't know. The issue was we didn't know for sure that it would work.\"\n\n\"So it was evenly divided?\"\n\n\"That's a very estimated number.\"\n\nIt struck me that the Thiokol managers were waffling. But I only knew how to ask simpleminded questions. So I said, \"Could you tell me, sirs, the names of your four best seals experts, in order of ability?\"\n\n\"Roger Boisjoly and Arnie Thompson are one and two. Then there's Jack Kapp and, uh... Jerry Burns.\"\n\nI turned to Mr. Boisjoly, who was right there, at the meeting. \"Mr. Boisjoly, were you in agreement that it was okay to fly?\"\n\nHe says, \"No, I was not.\"\n\nI ask Mr. Thompson, who was also there.\n\n\"No, I was not.\"\n\nI say, \"Mr. Kapp?\"\n\nMr. Lund says, \"He is not here. I talked to him after the meeting, and he said, 'I would have made that decision, given the information we had.' \"\n\n\"And the fourth man?\"\n\n\"Jerry Burns. I don't know what his position was.\"\n\n\"So,\" I said, \"of the four, we have one 'don't know,' one 'very likely yes,' and the two who were mentioned right away as being the _best_ seal experts, _both said no_.\" So this \"evenly split\" stuff was a lot of crap. The guys who knew the _most_ about the seals\u2014what were _they_ saying?\n\nLate in the afternoon, we were shown around the Kennedy Space Center. It was interesting; it wasn't as bad as I had predicted. The other commissioners asked a lot of important questions. We didn't have time to see the booster-rocket assembly, but near the end we were going to see the wreckage that had been recovered so far. I was pretty tired of this group stuff, so I excused myself from the rest of the tour.\n\nI ran down to Charlie Stevenson's place to see more pictures of the launch. I also found out more about the unusually low temperature readings. The guys were very cooperative, and wanted me to work with them. I had been waiting for _ten days_ to run around in one of these places, and here I was, _at last!_\n\nAt dinner that night, I said to Mr. Rogers, \"I was thinking of staying here over the weekend.\"\n\n\"Well, Dr. Feynman,\" he said, \"I'd prefer you come back to Washington with us tonight. But of course, you're free to do whatever you want.\"\n\n\"Well, then,\" I said, \"I'll stay.\"\n\nOn Saturday I talked to the guy who had actually taken the temperature readings the morning of the launch\u2014a nice fella named B. K. Davis. Next to each temperature he had written the exact time he had measured it, and then took a picture of it. You could see large gaps between the times as he climbed up and down the big launch tower. He measured the temperature of the air, the rocket, the ground, the ice, and even a puddle of slush with antifreeze in it. He did a very complete job.\n\nNASA had a theoretical calculation of how the temperatures should vary around the launch pad: they should have been more uniform, and higher. Somebody thought that heat radiating to the clear sky had something to do with it. But then someone else noticed that BK's reading for the slush was much lower than the photograph indicated: at 8 degrees, the slush\u2014even with antifreeze in it\u2014should have been frozen solid.\n\nThen we looked at the device the ice crew used for measuring the temperatures. I got the instruction manual out, and found that you're supposed to put the instrument out in the environment for at least 20 minutes before using it. Mr. Davis said he had taken it out of the box\u2014at 70 degrees\u2014and began making measurements right away. Therefore we had to find out whether the errors were reproducible. In other words, could the circumstances be duplicated?\n\nOn Monday I called up the company that made the device, and talked to one of their technical guys: \"Hi, my name is Dick Feynman,\" I said. \"I'm on the commission investigating the _Challenger_ accident, and I have some questions about your infrared scanning gun...\"\n\n\"May I call you right back?\" he says.\n\n\"Sure.\"\n\nAfter a little while he calls me back: \"I'm sorry, but it's proprietary information. I can't discuss it with you.\"\n\nBy this time I realized what the real difficulty was: the company was _scared green_ that we were going to blame the accident on their instrument. I said, \"Sir, your scanning gun has nothing to do with the accident. It was used by the people here in a way that's contrary to the procedures in your instruction manual, and I'm trying to figure out if we can reproduce the errors and determine what the temperatures really were that morning. To do this, I need to know more about your instrument.\"\n\nThe guy finally came around, and became quite cooperative. With his help, I advised the ice-crew guys on an experiment. They cooled a room down to about 40 degrees, and put a big block of ice in it\u2014with ice, you can be sure the surface temperature is 32 degrees. Then they brought in the scanning gun from a room which was 70 degrees inside, and made measurements of the ice block every 30 seconds. They were able to measure how far off the instrument was as a function of time.\n\nMr. Davis had written his measurements so carefully that it was very easy to fix all the numbers. And then, remarkably, the recalculated temperatures were close to what was expected according to the theoretical model. It looked very sensible.\n\nThe next time I talked to a reporter, I straightened everything out about the temperatures, and informed him that the earlier theory expounded by the Nobel Prize winner was wrong.\n\nI wrote a report for the other commissioners on the temperature problem, and sent it to Dr. Keel.\n\nThen I investigated something we were looking into as a possible contributing cause of the accident: when the booster rockets hit the ocean, they became out of round a little bit from the impact. At Kennedy they're taken apart, and the sections\u2014four for each rocket\u2014are sent by rail to Thiokol in Utah, where they are packed with new propellant. Then they're put back on a train to Florida. During transport, the sections (which are hauled on their side) get squashed a little bit\u2014the softish propellant is very heavy. The total amount of squashing is only a fraction of an inch, but when you put the rocket sections back together, a small gap is enough to let hot gases through: the O-rings are only a quarter of an inch thick, and compressed only two-hundredths of an inch!\n\nI thought I'd do some calculations. NASA gave me all the numbers on how far out of round the sections can get, so I tried to figure out how much the resulting squeeze was, and where it was located\u2014maybe the minimum squeeze was where the leak occurred. The numbers were measurements taken along three diameters, every 60 degrees. But three matching diameters won't guarantee that things will fit; six diameters, or any other number of diameters, won't do, either.\n\nFor example, you can make a figure something like a triangle with rounded corners, in which three diameters, 60 degrees apart, have the same length.\n\nI remembered seeing such a trick at a museum when I was a kid. There was a gear rack that moved back and forth perfectly smoothly, while underneath it were some noncircular, funny-looking, crazy-shaped gears turning on shafts that wobbled. It looked impossible, but the reason it worked was that the gears were shapes whose diameters were always the same.\n\nSo the numbers NASA gave me were useless.\n\nDuring that weekend, just as I had predicted in my letter home, I kept getting notes from the commission headquarters in Washington: \"Check the temperature readings, check the pictures, check this, check that...\"\u2014there was quite a list. But as the instructions came in, I had done most of them already.\n\nOne note had to do with a mysterious piece of paper. Someone at Kennedy had reportedly written \"Let's go for it\" while assembling one of the solid booster rockets. Such language appeared to show a certain recklessness. My mission: find that piece of paper.\n\nFIGURE 17. _This figure has all its diameters the same length\u2014yet it is obviously not round!_\n\nWell, by this time I understood how much paper there was in NASA. I was sure it was a trick to make me get lost, so I did nothing about it.\n\nInstead, I pursued something surreptitiously.\n\nIt was rumored that the reason NASA tried to make the shuttle fly on January 28th, in spite of the cold, was that the president was going to give his State of the Union address that night. According to the theory, the White House had it all cooked up so that during the State of the Union address, the teacher, Mrs. McAuliffe, would talk to the president and Congress from space. It was gonna be great: the president would say, \"Hello! How are you doing?\" And she would say, \"Fine\"\u2014something very dramatic.\n\nSince it sounded logical, I began by supposing it was very likely possible. But was there any evidence? This kind of thing I didn't know how to investigate. I could only think of this: it's very hard to get through to the president; I also can't just call up an astronaut and talk to her\u2014if she's in space. Therefore, switching the signals down from the shuttle over to the president while he's talking to Congress must be a complicated business.\n\nTo find out whether anybody had set up to do that, I went down to the lowest levels and asked guys at the bottom some technical questions.\n\nThey showed me the antennas, they told me about the frequencies, they showed me the big radio system and the computer system; they showed me all the ways they did things.\n\nI said, \"If you had to send a transmission somewhere else\u2014to Marshall, say\u2014how would you do it?\"\n\nThey said, \"Oh, we're just a relay station. Everything is automatically sent over to Houston, and they switch everything out from there. We don't do any switching here.\"\n\nSo I didn't find any evidence\u2014at least at Kennedy. But the guys there were so nice to me, and everything was so pleasant, that I feel bad. I don't like to cheat people. It was a little sneaky, what I was doing. Nevertheless, I thought I'd better do the same thing when I got to Houston.\n\nOn Monday, Mr. Hotz came down to Florida to work with me. (He told me later that he had been sent down with specific instructions to see what I was doing, and to keep me from \"going wild.\") Mr. Hotz brought a list of things to look into: \"There are a lot of things on this list,\" he said, \"so I'd be happy to split the work with you.\" Some things he said he could do more easily, and the rest of the things I had already done\u2014except for that piece of paper which said \"Let's go for it.\" Mr. Hotz hinted around that it might have come from the diary of someone in the booster-rocket assembly. That wasn't enough of a clue for me; I just wasn't gonna do it. Instead, I went to see a Mr. Lamberth, who had said he wanted to talk to me.\n\nMr. Lamberth was way up in the works, a big cheese in charge of assembling the solid-rocket boosters. He wanted to tell me about some problems he had. \"The workers used to have much better discipline,\" he explained, \"but nowadays they're not like they used to be.\" He gave me a couple of examples.\n\nThe first incident had to do with taking the booster rockets apart after they had been recovered from the sea. The rocket sections are held together by 180 pins\u2014each about an inch and a half in diameter and two inches long\u2014all the way around.\n\nThere was some kind of procedure for taking sections apart, in which the workers were supposed to pull the rocket up a certain distance. They had gotten to paying attention only to the amount of _force_ they were applying\u2014about 11,000 pounds. That was a better method, from a physical standpoint, because the idea is to take the load off the pins.\n\nOne time the force gauge wasn't working right. The workers kept putting more force on, wondering why they weren't reaching 11,000 pounds, when all of a sudden one of the pins broke.\n\nMr. Lamberth reprimanded the workers for not following procedures. It reminded me of when I tried to make things work better at my aunt's hotel: your method is better than the regular way, but then you have a little accident...*\n\nThe second story Mr. Lamberth told me had to do with putting the rocket sections together. The regular procedure was to stack one section on top of the other and match the upper section to the lower one.\n\nIf a section needed to be reshaped a little bit, the procedure was to first pick up the section with a crane and let it hang sideways a few days. It's rather simpleminded.\n\nIf they couldn't make a section round enough by the hanging method, there was another procedure: use the \"rounding machine\"\u2014a rod with a hydraulic press on one end and a nut on the other\u2014and increase the pressure.\n\nMr. Lamberth told me the pressure shouldn't exceed 1200 pounds per square inch (psi). One time, a section wasn't round enough at 1200 psi, so the workers took a wrench and began turning the nut on the other end. When they finally got the section round enough, the pressure was up to 1350. \"This is another example of the lack of discipline among the workers,\" Mr. Lamberth said.\n\nI had wanted to talk with the assembly workers anyway (I love that kind of thing), so I arranged to see them the next day at 2:30 in the afternoon.\n\nAt 2:30 I walk into this room, and there's a long table with thirty or forty people\u2014they're all sitting there with morose faces, very serious, ready to talk to The Commissioner.\n\nI was terrified. I hadn't realized my terrible power. I could see they were worried. They must have been told I was investigating the errors they had made!\n\nSo right away I said, \"I had nothin' to do, so I thought I'd come over and talk to the guys who put the rockets together. I didn't want everybody to stop working just 'cause I wanna find out something for my own curiosity; I only wanted to talk with the workers...\"\n\nMost of the people got up and left. Six or seven guys stayed\u2014the crew who actually put the rocket sections together, their foreman, and some boss who was higher up in the system.\n\nWell, these guys were still a little bit scared. They didn't really want to open up. The first thing I think to say is, \"I have a question: when you measure the three diameters and all the diameters match, do the sections really fit together? It seems to me that you could have some bumps on one side and some flat areas directly across, so the three diameters would match, but the sections wouldn't fit.\"\n\n\"Yes, yes!\" they say. \"We get bumps like that. We call them nipples.\"\n\nThe only woman there said, \"It's got nothing to do with me!\"\u2014and everybody laughed.\n\n\"We get nipples all the time,\" they continued. \"We've been tryin' to tell the supervisor about it, but we never get anywhere!\"\n\nWe were talking details, and that works wonders. I would ask questions based on what could happen theoretically, but to them it looked like I was a regular guy who knew about their technical problems. They loosened up very rapidly, and told me all kinds of ideas they had to improve things.\n\nFor example, when they use the rounding machine, they have to put a rod through holes exactly opposite each other. There are 180 holes, so they have to make sure the other end of the rod goes through the hole 90 holes away. Now, it turns out you have to climb up into an awkward place to count the holes. It's very slow and very difficult.\n\nThey thought it would be very helpful if there were four paint marks, 90 degrees apart, put on at the factory. That way, they would never have to count more than 22 holes to the nearest mark. For example, if they put the rod through a hole which is 9 holes clockwise from a paint mark, then the other end of the rod would go through the hole which is 9 holes clockwise from the opposite mark.\n\nThe foreman, Mr. Fichtel, said he wrote a memo with this suggestion to his superiors two years ago, but nothing had happened yet. When he asked why, he was told the suggestion was too expensive.\n\n\"Too expensive to paint _four little lines?\"_ I said in disbelief.\n\nThey all laughed. \"It's not the paint; it's the paperwork,\" Mr. Fichtel said. \"They would have to revise all the manuals.\"\n\nThe assembly workers had other observations and suggestions. They were concerned that if two rocket sections scrape as they're being put together, metal filings could get into the rubber seals and damage them. They even had some suggestions for redesigning the seal. Those suggestions weren't very good, but the point is, the workers were _thinking!_ I got the impression that they were _not_ undisciplined; they were very interested in what they were doing, but they weren't being given much encouragement. Nobody was paying much attention to them. It was remarkable that their morale was as high as it was under the circumstances.\n\nThen the workers began to talk to the boss who had stayed. \"We're disappointed by something,\" one of them said. \"When the commission was going to see the booster-rocket assembly, the demonstration was going to be done by the managers. Why wouldn't you let _us_ do it?\"\n\n\"We were afraid you'd be frightened by the commissioners and you wouldn't want to do it.\"\n\n\"No, no,\" said the workmen. \"We think we do a good job, and we wanted to show what we do.\"\n\nAfter that meeting, the boss took me to the cafeteria. As we were eating\u2014the workmen weren't with us anymore\u2014he said, \"I was surprised they were so concerned about that.\"\n\nLater, I talked to Mr. Fichtel about this incident of increasing the pressure past 1200. He showed me the notes he made as he went along: they weren't the formal papers that are stamped; they were part of an informal but carefully written diary.\n\nI said, \"I hear the pressure got up to 1350.\"\n\n\"Yes,\" he said, \"we had tightened the nut at the other end.\"\n\n\"Was that the regular procedure?\"\n\n\"Oh, yes,\" he said, \"it's in the book.\"\n\nHe opens up the manual and shows me the procedure. It says, \"Build up the pressure on the hydraulic jack. If this is insufficient to obtain desired roundness, then very carefully tighten nut on other end to get to the desired roundness\"\u2014it said so in black and white! It didn't say that tightening the nut would increase the pressure past 1200 psi; the people who wrote the manual probably weren't quite aware of that.\n\nMr. Fichtel had written in his diary, \"We very carefully tightened the nut\"\u2014exactly the same language as the instructions.\n\nI said, \"Mr. Lamberth told me he admonished you about going above 1200.\"\n\n\"He never admonished me about that\u2014why should he?\"\n\nWe figured out what probably happened. Mr. Lamberth's admonishment went down through the levels until somebody in middle management realized that Mr. Fichtel had gone by the book, and that the error was in the manual. But instead of telling Mr. Lamberth about the error, they simply threw away the admonishment, and just kept quiet.\n\nOver lunch, Mr. Fichtel told me about the inspection procedures. \"There's a sheet for each procedure, like this one for the rounding procedure,\" he said. \"On it there are boxes for stamps\u2014one from the supervisor, one from quality control, one from the manufacturer, and for the bigger jobs, one from NASA.\"\n\nHe continued, \"We make the measurements, go through one course of rounding, and then make the measurements again. If they don't match well enough, we repeat the steps. Finally, when the diameter differences are small enough, we go for it.\"\n\nI woke up. \"What do you mean, 'go for it'?\" I said. \"It sounds sort of cavalier...\"\n\n\"No, no,\" he says. \"That's just the lingo we use when we mean that all the conditions are satisfied, and we're ready to move to the next phase of the operation.\"\n\n\"Do you ever write that down\u2014that 'go for it'?\"\n\n\"Yes, sometimes.\"\n\n\"Let's see if we can find a place where you wrote it.\"\n\nMr. Fichtel looked through his diary, and found an example. The expression was completely natural to him\u2014it wasn't reckless or cavalier; it was just his way of speaking.\n\nOn Monday and Tuesday, while I was running around down at Kennedy, Mr. Rogers was in Washington appearing before a Senate committee. Congress was considering whether it should have its own investigation.\n\nSenator Hollings, from South Carolina, was giving Mr. Rogers a hard time: \"Secretary Rogers,\" he says, \"I'm anxious that you have an adequate staff thayah. How many _investigators_ does yo' commission have?\"\n\nMr. Rogers says, \"We don't have investigators in the police sense. We're reading documents, understanding what they mean, organizing hearings, talking to witnesses\u2014that sort of thing. We'll have an adequate staff, I assure you.\"\n\n\"Well, that's the point,\" Senator Hollings says. \"From my experience in investigating cases, I'd want four or five investigators steeped in science and space technology going around down there at Canaveral talking to everybody, eating lunch with them. You'd be amazed, if you eat in the restaurants around there for two or three weeks, what you'll find out. You can't just sit and read what's given to you.\"\n\n\"We're not just going to sit and read,\" Mr. Rogers says defensively. \"We've gotten a lot of people in a room and asked them questions all at the same time, rather than have a gumshoe walking around, talking to people one at a time.\"\n\n\"I understand,\" says Senator Hollings. \"Yet I'm concerned about yo' product if you don't have some gumshoes. That's the trouble with presidential commissions; I've been on 'em: they go on what's _fed_ to 'em, and they don't look behind it. Then we end up with investigative reporters, people writing books, and everything else. People are _still_ investigating the Warren Commission Report around this town.\"*\n\nMr. Rogers calmly says, \"I appreciate your comments, Senator. You'll be interested to know that one of our commission members\u2014he's a Nobel laureate\u2014is down there in Florida today, investigating in the way you'd like him to investigate.\"\n\n(Mr. Rogers didn't know it, but I was actually eating lunch with some engineers when he said that!)\n\nSenator Hollings says, \"I'm not questioning the competence of the Nobel laureate; I've been reading with great interest what he said. There's no question about the competence of the commission itself. It's just that when you investigate a case, you need investigators. You have already brought to the public's attention a lot of very interesting facts, so I think you haven't been negligent in any fashion.\"\n\nSo I saved Mr. Rogers a little bit. He saw that he had an answer for Mr. Hollings by the _good luck_ that I stayed in Florida anyway, against his wishes!\n\n### Fantastic Figures\n\n_ON_ Tuesday afternoon I flew back to Washington, and went to the next meeting of the commission, on Wednesday. It was another public meeting. A manager of the Thiokol Company named Mr. Lund was testifying. On the night before the launch, Mr. Mulloy had told him to put on his \"management hat\" instead of his \"engineering hat,\" so he changed his opposition to launch and overruled his own engineers. I was asking him some harsh questions when suddenly I had this feeling of the Inquisition.\n\nMr. Rogers had pointed out to us that we ought to be careful with these people, whose careers depend on us. He said, \"We have all the advantages: we're sitting up here; they're sitting down there. They have to answer our questions; we don't have to answer their questions.\" Suddenly, all this came back to me and I felt terrible, and I couldn't do it the next day. I went back to California for a few days, to recover.\n\nWhile I was in Pasadena, I went over to JPL and met with Jerry Solomon and Meemong Lee. They were studying the flame which appeared a few seconds before the main fuel tank exploded, and were able to bring out all kinds of details. (JPL has good enhancers of TV pictures from all their experience with planetary missions.) Later, I took the enhancements over to Charlie Stevenson and his crew at Kennedy to expedite things.\n\nSomewhere along the line, somebody from the staff brought me something to sign: it said that my expenses were so-and-so much, but they weren't\u2014they were more. I said, \"This is not the amount I actually spent.\"\n\nThe guy said, \"I know that, sir; you're only allowed a maximum of $75 a day for the hotel and food.\"\n\n\"Then why did you guys set me up in a hotel which costs $80 or $90 a night, and then you give me only $75 a day?\"\n\n\"Yes, I agree; it's too bad, but that's the way it goes!\"\n\nI thought of Mr. Rogers' offer to put me in a \"good hotel.\" What did he mean by that\u2014that it would cost me _more?_\n\nIf you're asked to contribute months of time and effort to the government (and you lose money you would have made consulting for a company), the government ought to appreciate it a little more than to be cheap about paying you back. I'm not trying to make money off the government, but I'm not wanting to _lose_ money, either! I said, \"I'm not going to sign this.\"\n\nMr. Rogers came over and promised he would straighten it out, so I signed the paper.\n\nI really think Mr. Rogers tried to fix it, but he was unable to. I thought of fighting this one to the end, but then I realized it's impossible: if I had been paid for my actual expenses, then of course all the other commissioners would have to be paid, too. That would be all right, but it would also mean that this commission was the only commission to be paid its actual expenses\u2014and pretty soon, word would get out.\n\nThey have a saying in New York: \"You can't fight City Hall,\" meaning \"It's impossible.\" But this time, it was a hell of a lot bigger than City Hall: the $75 a day rule is a law of the United States! It might have been fun to fight it to the end, but I guess I was tired\u2014I'm not as young as I used to be\u2014so I just gave up.\n\nSomebody told me they heard commissioners make $1000 a day, but the truth is, our government doesn't even pay their costs.\n\nAt the beginning of March, about a month after the commission started, we finally split up into working groups: the Pre-Launch Activities group was headed by Mr. Ache-son; Mr. Sutter was in charge of the Design, Development, and Production panel; General Kutyna was leader of the Accident Analysis group; and Dr. Ride was in charge of the Mission Planning and Operations group.\n\nI spent most of my time in Kutyna's group. I was in Ride's group, too, but I ended up not doing very much for her.\n\nGeneral Kutyna's group went to Marshall Space Flight Center in Huntsville, Alabama, to do its work. The first thing that happened there was, a man named Ullian came in to tell us something. As range safety officer at Kennedy, Mr. Ullian had to decide whether to put destruct charges on the shuttle. (If a rocket goes out of control, the destruct charges enable it to be blown up into small bits. That's much less perilous than a rocket flying around loose, ready to explode when it hits the ground.)\n\nEvery unmanned rocket has these charges. Mr. Ullian told us that 5 out of 127 rockets that he looked at had failed\u2014a rate of about 4 percent. He took that 4 percent and divided it by 4, because he assumed a manned flight would be safer than an unmanned one. He came out with about a 1 percent chance of failure, and that was enough to warrant the destruct charges.\n\nBut NASA told Mr. Ullian that the probability of failure was more like 1 in 105.\n\nI tried to make sense out of that number. \"Did you say 1 in 105?\"\n\n\"That's right; 1 in 100,000.\"\n\n\"That means you could fly the shuttle _every day_ for an average of _300 years_ between accidents\u2014every day, one flight, for 300 years\u2014which is obviously crazy!\"\n\n\"Yes, I know,\" said Mr. Ullian. \"I moved my number up to 1 in 1000 to answer all of NASA's claims\u2014that they were much more careful with manned flights, that the typical rocket isn't a valid comparison, et cetera\u2014and put the destruct charges on anyway.\"\n\nBut then a new problem came up: the Jupiter probe, _Galileo,_ was going to use a power supply that runs on heat generated by radioactivity. If the shuttle carrying _Galileo_ failed, radioactivity could be spread over a large area. So the argument continued: NASA kept saying 1 in 100,000 and Mr. Ullian kept saying 1 in 1000, at best.\n\nMr. Ullian also told us about the problems he had in trying to talk to the man in charge, Mr. Kingsbury: he could get appointments with underlings, but he never could get through to Kingsbury and find out how NASA got its figure of 1 in 100,000. The details of the story I can't remember exactly, but I thought Mr. Ullian was doing everything sensibly.\n\nOur panel supervised the tests that NASA was doing to discover the properties of the seals\u2014how much pressure the putty could take, and so on\u2014in order to find out exactly what had happened. General Kutyna didn't want to jump to conclusions, so we went over and over things, checking all the evidence and seeing how well everything fitted together.\n\nThere was an awful lot of detailed discussion about exactly what happened in the last few seconds of the flight, but I didn't pay much attention to any of it. It was as though a train had crashed because the track had a gap in it, and we were analyzing which cars broke apart first, which cars broke apart second, and why some car turned over on its side. I figured once the train goes off the track, it doesn't make any difference\u2014it's done. I became bored.\n\nSo I made up a game for myself: \"Imagine that something else had failed\u2014the main engines, for instance\u2014and we were making the same kind of intensive investigation as we are now: would we discover the same slipping safety criteria and lack of communication?\"\n\nI thought I would do my standard thing\u2014find out from the engineers how the engine works, what all the dangers are, what problems they've had, and everything else\u2014and then, when I'm all loaded up so I know what I'm talking about, I'd confront whoever was claiming the probability of failure was 1 in 100,000.\n\nI asked to talk to a couple of engineers about the engines. The guy says, \"Okay, I'll fix it up. Is nine tomorrow morning okay?\"\n\nThis time there were three engineers, their boss, Mr. Lovingood, and a few assistants\u2014about eight or nine people.\n\nEverybody had big, thick notebooks, full of papers, all nicely organized. On the front they said:\n\nREPORT ON MATERIAL GIVEN TO COMMISSIONER RICHARD P. FEYNMAN ON MARCH WA-WA,* 1986.\n\nI said, \"Geez! You guys must have worked hard all night!\"\n\n\"No, it's not so much work; we just put in the regular papers that we use all the time.\"\n\nI said, \"I just wanted to talk to a few engineers. There are so many problems to work on, I can't expect you all to stay here and talk to me.\"\n\nBut this time, everybody stayed.\n\nMr. Lovingood got up and began to explain everything to me in the usual NASA way, with charts and graphs which matched the information in my big book\u2014all with bullets, of course.\n\nI won't bother you with all the details, but I wanted to understand everything about the engine. So I kept asking my usual dumb-sounding questions.\n\nAfter a while, Mr. Lovingood says, \"Dr. Feynman, we've been going for two hours, now. There are 123 pages, and we've only covered 20 so far.\"\n\nMy first reaction was to say, \"Well, it isn't really going to take such a long time. I'm always a little slow at the beginning; it takes me a while to catch on. We'll be able to go much faster near the end.\"\n\nBut then I had a second thought. I said, \"In order to speed things up, I'll tell you what I'm doing, so you'll know where I'm aiming. I want to know whether there's the same lack of communication between the engineers and the management who are working on the engine as we found in the case of the booster rockets.\"\n\nMr. Lovingood says, \"I don't think so. As a matter of fact, although I'm now a manager, I was trained as an engineer.\"\n\n\"All right,\" I said. \"Here's a piece of paper each. Please write on your paper the answer to this question: what do you think is the probability that a flight would be uncompleted due to a failure in this engine?\"\n\nThey write down their answers and hand in their papers. One guy wrote \"99-44\/100% pure\" (copying the Ivory soap slogan), meaning about 1 in 200. Another guy wrote something very technical and highly quantitative in the standard statistical way, carefully defining everything, that I had to translate\u2014which also meant about 1 in 200. The third guy wrote, simply, \"1 in 300.\"\n\nMr. Lovingood's paper, however, said,\n\nCannot quantify. Reliability is judged from:\n\n\u2022 past experience\n\n\u2022 quality control in manufacturing\n\n\u2022 engineering judgment\n\n\"Well,\" I said, \"I've got four answers, and one of them weaseled.\" I turned to Mr. Lovingood: \"I think you weaseled.\"\n\n\"I don't think I weaseled.\"\n\n\"You didn't tell me _what_ your confidence was, sir; you told me _how_ you determined it. What I want to know is: after you determined it, what _was_ it?\"\n\nHe says, \"100 percent\"\u2014the engineers' jaws drop, my jaw drops; I look at him, everybody looks at him\u2014\"uh, uh, minus epsilon!\"\n\nSo I say, \"Well, yes; that's fine. Now, the only problem is, WHAT IS EPSILON?\"\n\nHe says, \"10-5.\" It was the same number that Mr. Ullian had told us about: 1 in 100,000.\n\nI showed Mr. Lovingood the other answers and said, \"You'll be interested to know that there _is_ a difference between engineers and management here\u2014a factor of more than 300.\"\n\nHe says, \"Sir, I'll be glad to send you the document that contains this estimate, so you can understand it.\"*\n\nI said, \"Thank you very much. Now, let's get back to the engine.\" So we continued and, just like I guessed, we went faster near the end. I had to understand how the engine worked\u2014the precise shape of the turbine blades, exactly how they turned, and so on\u2014so I could understand its problems.\n\nAfter lunch, the engineers told me all the problems of the engines: blades cracking in the oxygen pump, blades cracking in the hydrogen pump, casings getting blisters and cracks, and so on. They looked for these things with periscopes and special instruments when the shuttle came down after each flight.\n\nThere was a problem called \"subsynchronous whirl,\" in which the shaft gets bent into a slightly parabolic shape at high speed. The wear on the bearings was so terrible\u2014all the noise and the vibration\u2014that it seemed hopeless. But they had found a way to get rid of it. There were about a dozen very serious problems; about half of them were fixed.\n\nMost airplanes are designed \"from the bottom up,\" with parts that have already been extensively tested. The shuttle, however, was designed \"from the top down\"\u2014to save time. But whenever a problem was discovered, a lot of redesigning was required in order to fix it.\n\nMr. Lovingood isn't saying much now, but different engineers, depending on which problem it is, are telling me all this stuff, just like I could have found out if I went down to the engineers at Thiokol. I gained a great deal of respect for them. They were all very straight, and everything was great. We went all the way down to the end of the book. We made it.\n\nThen I said, \"What about this high-frequency vibration where some engines get it and others don't?\"*\n\nThere's a quick motion, and a little stack of papers appears. It's all put together nicely; it fits nicely into my book. It's all about the 4000-cycle vibration!\n\nMaybe I'm a little dull, but I tried my best not to accuse anybody of anything. I just let them show me what they showed me, and acted like I didn't see their trick. I'm not the kind of investigator you see on TV, who jumps up and accuses the corrupt organization of withholding information. But I was fully aware that they hadn't told me about the problem until I asked about it. I usually acted quite naive\u2014which I was, for the most part.\n\nAt any rate, the engineers all leaped forward. They got all excited and began to describe the problem to me. I'm sure they were delighted, because technical people love to discuss technical problems with technical people who might have an opinion or a suggestion that could be useful. And of course, they were very anxious to cure it.\n\nThey kept referring to the problem by some complicated name\u2014a \"pressure-induced vorticity oscillatory wa-wa,\" or something.\n\nI said, \"Oh, you mean a whistle!\"\n\n\"Yes,\" they said; \"it exhibits the characteristics of a whistle.\"\n\nThey thought the whistle could be coming from a place where the gas rushed through a pipe at high speed and split into three smaller pipes\u2014where there were two partitions. They explained how far they had gotten in figuring out the problem.\n\nWhen I left the meeting, I had the definite impression that I had found the same game as with the seals: management reducing criteria and accepting more and more errors that weren't designed into the device, while the engineers are screaming from below, \"HELP!\" and \"This is a RED ALERT!\"\n\nThe next evening, on my way home in the airplane, I was having dinner. After I finished buttering my roll, I took the little piece of thin cardboard that the butter pat comes on, and bent it around in a U shape so there were two edges facing me. I held it up and started blowing on it, and pretty soon I got it to make a noise like a whistle.\n\nBack in California, I got some more information on the shuttle engine and its probability of failure. I went to Rocketdyne and talked to engineers who were building the engines. I also talked to consultants for the engine. In fact one of them, Mr. Covert, was on the commission. I also found out that a Caltech professor had been a consultant for Rocketdyne. He was very friendly and informative, and told me about all the problems the engine had, and what he thought the probability of failure was.\n\nI went to JPL and met a fellow who had just written a report for NASA on the methods used by the FAA* and the military to certify their gas turbine and rocket engines. We spent the whole day going back and forth over how to determine the probability of failure in a machine. I learned a lot of new names\u2014like \"Weibull,\" a particular mathematical distribution that makes a certain shape on a graph. He said that the original safety rules for the shuttle were very similar to those of the FAA, but that NASA had modified them as they began to get problems.\n\nIt turned out that NASA's Marshall Space Center in Huntsville designed the engine, Rocketdyne built them, Lockheed wrote the instructions, and NASA's Kennedy Space Center installed them! It may be a genius system of organization, but it was a complete fuzdazzle, as far as I was concerned. It got me terribly confused. I didn't know whether I was talking to the Marshall man, the Rocketdyne man, the Lockheed man, or the Kennedy man! So in the middle of all this, I got lost. In fact, all during this time\u2014in March and April\u2014I was running back and forth so much between California, Alabama, Houston, Florida, and Washington, D.C., that I often didn't know what day it was, or where I was.\n\nAfter all this investigating on my own, I thought I'd write up a little report on the engine for the other commissioners. But when I looked at my notes on the testing schedules, there was some confusion: there would be talk about \"engine #12\" and how long \"the engine\" flew. But no engine ever was like that: it would be repaired all the time. After each flight, technicians would inspect the engines and see how many cracked blades there were on the rotor, how many splits there were in the casing, and so on. Then they'd repair \"the engine\" by putting on a new casing, a new rotor, or new bearings\u2014they would replace lots of parts. So I would read that a particular engine had rotor #2009, which had run for 27 minutes in flight such-and-such, and casing #4091, which had run for 53 minutes in flights such-and-such and so-and-so. It was all mixed up.\n\nWhen I finished my report, I wanted to check it. So the next time I was at Marshall, I said I wanted to talk to the engineers about a few very technical problems, just to check the details\u2014I didn't need any management there.\n\nThis time, to my surprise, nobody came but the three engineers I had talked to before, and we straightened everything out.\n\nWhen I was about to leave, one of them said, \"You know that question you asked us last time\u2014with the papers? We felt that was a loaded question. It wasn't fair.\"\n\nI said, \"Yes, you're quite right. It _was_ a loaded question. I had an idea of what would happen.\"\n\nThe guy says, \"I would like to revise my answer. I want to say that I cannot quantify it.\" (This guy was the one who had the most detailed answer before.)\n\nI said, \"That's fine. But do you agree that the chance of failure is 1 in 100,000?\"\n\n\"Well, uh, no, I don't. I just don't want to answer.\" Then one of the other guys says, \"I said it was 1 in 300, and I still say it's 1 in 300, but I don't want to tell you how I got my number.\"\n\nI said, \"It's okay. You don't have to.\"\n\n### An Inflamed Appendix\n\n_ALL_ during this time, I had the impression that somewhere along the line the whole commission would come together again so we could talk to each other about what we had found out.\n\nIn order to aid such a discussion, I thought I'd write little reports along the way: I wrote about my work with the ice crew (analyzing the pictures and the faulty temperature readings); I wrote about my conversations with Mr. Lamberth and the assembly workers; and I even wrote about the piece of paper that said \"Let's go for it.\" All these little reports I sent to Al Keel, the executive officer, to give to the other commissioners.\n\nNow, this particular adventure\u2014investigating the lack of communication between the managers and the engineers who were working on the engine\u2014I also wrote about, on my little IBM PC at home. I was kind of tired, so I didn't have the control I wanted\u2014it wasn't written with the same care as my other reports. But since I was writing it only as a report to the other commissioners, I didn't change the language before I sent it on to Dr. Keel. I simply attached a note that said \"I think the other commissioners would be interested in this, but you can do with it what you want\u2014it's a little strong at the end.\"\n\nHe thanked me, and said he sent my report to everybody.\n\nThen I went to the Johnson Space Center, in Houston, to look into the avionics. Sally Ride's group was there, investigating safety matters in connection with the astronauts' experiences. Sally introduced me to the software engineers, and they gave me a tour of the training facilities for the astronauts.\n\nIt's really quite wonderful. There are different kinds of simulators with varying degrees of sophistication that the astronauts practice on. One of them is just like the real thing: you climb up, you get in; at the windows, computers are producing pictures. When the pilot moves the controls, the view out of the windows changes.\n\nThis particular simulator had the double purpose of teaching the astronauts and checking the computers. In the back of the crew area, there were trays full of cables running down through the cargo bay to somewhere in the back, where instruments simulated signals from the engines\u2014pressures, fuel flow rates, and so on. (The cables were accessible because the technicians were checking for \"cross talk\"\u2014interferences in the signals going back and forth.)\n\nThe shuttle itself is operated essentially by computer. Once it's lit up and starts to go, nobody inside does anything, because there's tremendous acceleration. When the shuttle reaches a certain altitude, the computers adjust the engine thrust down for a little while, and as the air thins out, the computers adjust the thrust up again. About a minute later, the two solid rocket boosters fall away, and a few minutes after that, the main fuel tank falls away; each operation is controlled by the computers. The shuttle gets into orbit automatically\u2014the astronauts just sit in their seats.\n\nThe shuttle's computers don't have enough memory to hold all the programs for the whole flight. After the shuttle gets into orbit, the astronauts take out some tapes and load in the program for the next phase of the flight\u2014there are as many as six in all. Near the end of the flight, the astronauts load in the program for coming down.\n\nThe shuttle has four computers on board, all running the same programs. All four are normally in agreement. If one computer is out of agreement, the flight can still continue. If only two computers agree, the flight has to be curtailed and the shuttle brought back immediately.\n\nFor even more safety, there's a fifth computer\u2014located away from the other four computers, with its wires going on different paths\u2014which has only the program for going up and the program for coming down. (Both programs can barely fit into its memory.) If something happens to the other computers, this fifth computer can bring the shuttle back down. It's never had to be used.\n\nThe most dramatic thing is the landing. Once the astronauts know where they're supposed to land, they push one of three buttons\u2014marked Edwards, White Sands, and Kennedy\u2014which tells the computer where the shuttle's going to land. Then some small rockets slow the shuttle down a little, and get it into the atmosphere at just the right angle. That's the dangerous part, where all the tiles heat up.\n\nDuring this time, the astronauts can't see anything, and everything's changing so fast that the descent has to be done automatically. At around 35,000 feet the shuttle slows down to less than the speed of sound, and the steering can be done manually, if necessary. But at 4000 feet something happens that is not done by the computer: the pilot pushes a button to lower the landing wheels.\n\nI found that very odd\u2014a kind of silliness having to do with the psychology of the pilots: they're heroes in the eyes of the public; everybody has the idea that they're steering the shuttle around, whereas the truth is they don't have to do anything until they push that button to lower the landing gear. They can't stand the idea that they really have nothing to do.\n\nI thought it would be safer if the computer would lower the landing wheels, in case the astronauts were unconscious for some reason. The software engineers agreed, and added that putting down the landing wheels at the wrong time is very dangerous.\n\nThe engineers told me that ground control can send up the signal to lower the landing wheels, but this backup gave them some pause: what happens if the pilot is half-conscious, and thinks the wheels should go down at a certain time, and the controller on the ground knows it's the wrong time? It's much better to have the whole thing done by computer.\n\nThe pilots also used to control the brakes. But there was lots of trouble: if you braked too much at the beginning, you'd have no more brake-pad material left when you reached the end of the runway\u2014and you're still moving! So the software engineers were asked to design a computer program to control the braking. At first the astronauts objected to the change, but now they're very delighted because the automatic braking works so well.\n\nAlthough there's a lot of good software being written at Johnson, the computers on the shuttle are so obsolete that the manufacturers don't make them anymore. The memories in them are the old kind, made with little ferrite cores that have wires going through them. In the meantime we've developed much better hardware: the memory chips of today are much, much smaller; they have much greater capacity; and they're much more reliable. They have internal error-correcting codes that automatically keep the memory good. With today's computers we can design separate program modules so that changing the payload doesn't require so much program rewriting.\n\nBecause of the huge investment in the flight simulators and all the other hardware, to start all over again and replace the millions of lines of code that they've already built up would be very costly.\n\nI learned how the software engineers developed the avionics for the shuttle. One group would design the software programs, in pieces. After that, the parts would be put together into huge programs, and tested by an independent group.\n\nAfter both groups thought all the bugs had been worked out, they would have a simulation of an entire flight, in which every part of the shuttle system is tested. In such cases, they had a principle: this simulation is not just an exercise to check if the programs are all right; it is a _real flight_ \u2014if anything fails now, it's extremely serious, as if the astronauts were really on board and in trouble. Your reputation is on the line.\n\nIn the many years they had been doing this, they had had only six failures at the level of flight simulation, and not one in an actual flight.\n\nSo the computer people looked like they knew what they were doing: they knew the computer business was vital to the shuttle but potentially dangerous, and they were being extremely careful. They were writing programs that operate a very complex machine in an environment where conditions are changing drastically\u2014programs which measure those changes, are flexible in their responses, and maintain high safety and accuracy. I would say that in some ways they were once in the forefront of how to ensure quality in robotic or interactive computer systems, but because of the obsolete hardware, it's no longer true today.\n\nI didn't investigate the avionics as extensively as I did the engines, so I might have been getting a little bit of a sales talk, but I don't think so. The engineers and the managers communicated well with each other, and they were all very careful not to change their criteria for safety.\n\nI told the software engineers I thought their system and their attitude were very good.\n\nOne guy muttered something about higher-ups in NASA wanting to cut back on testing to save money: \"They keep saying we always pass the tests, so what's the use of having so many?\"\n\nBefore I left Houston, I continued my surreptitious investigation of the rumor that the White House had put pressure on NASA to launch the shuttle. Houston is the center of communication, so I went over to the telemetry people and asked about their switching system. I went through the same stuff as I did in Florida\u2014and they were just as nice to me\u2014but this time I found out that if they wanted to tie in the shuttle to the Congress, the White House, or to anywhere, they need a three-minute warning\u2014not three months, not three days, not three hours\u2014three minutes. Therefore they can do it whenever they want, and nothing has to be written down in advance. So that was a blind alley.\n\nI talked to a _New York Times_ reporter about this rumor one time. I asked him, \"How do you find out if things like this are true?\"\n\nHe says, \"One of the things I thought to do was to go down and talk to the people who run the switching system. I tried that, but I wasn't able to come up with anything.\"\n\nDuring the first half of April, General Kutyna's group received the final results of the tests NASA was making at Marshall. NASA included its own interpretations of the results, but we thought we should write everything over again in our own way. (The only exceptions were when a test didn't show anything.)\n\nGeneral Kutyna set up a whole system at Marshall for writing our group's report. It lasted about two days. Before we could get anywhere, we got a message from Mr. Rogers: \"Come back to Washington. You shouldn't do the writing down there.\"\n\nSo we went back to Washington, and General Kutyna gave me an office in the Pentagon. It was fine, but there was no secretary, so I couldn't work fast.\n\nBill Graham had always been very cooperative, so I called him up. He arranged for me to use a guy's office\u2014the guy was out of town\u2014and his secretary. She was very, very helpful: she could write up something as fast as I could say it, and then she'd revamp it, correcting my mistakes. We worked very hard for about two or three days, and got large pieces of the report written that way. It worked very well.\n\nNeil Armstrong, who was in our group, is extremely good at writing. He would look at my work and immediately find every weak spot, just like that\u2014he was right every time\u2014and I was very impressed.\n\nEach group was writing a chapter or two of the main report. Our group wrote some ofthe stuff in \"Chapter 3: The Accident,\" but our main work was \"Chapter 4: The Cause ofthe Accident.\" One result of this system, however, was that we never had a meeting to discuss what each of our groups found out\u2014to comment on each other's findings from our different perspectives. Instead, we did what they call \"wordsmithing\"\u2014or what Mr. Hotz later called \"tombstone engraving\"\u2014correcting punctuation, refining phrases, and so on. We never had a _real_ discussion of ideas, except incidentally in the course of this wordsmithing.\n\nFor example, a question would come up: \"Should this sentence about the engines be worded this way or that way?\"\n\nI would try to get a little discussion started. \"From my own experiences, I got the impression that the engines aren't as good as you're saying here...\"\n\nSo they'd say, \"Then we'll use the more conservative wording here,\" and they'd go on to the next sentence. Perhaps that's a very efficient way to get a report out quickly, but we spent meeting after meeting doing this wordsmithing.\n\nEvery once in a while we'd interrupt that to discuss the typography and the color of the cover. And after each discussion, we were asked to vote. I thought it would be most efficient to vote for the same color we had decided on in the meeting before, but it turned out I was always in the minority! We finally chose red. (It came out blue.)\n\nOne time I was talking to Sally Ride about something I mentioned in my report on the engines, and she didn't seem to know about it. I said, \"Didn't you see my report?\"\n\nShe says, \"No, I didn't get a copy.\"\n\nSo I go over to Keel's office and say, \"Sally tells me she didn't get a copy of my report.\"\n\nHe looks surprised, and turns to his secretary. \"Please make a copy of Dr. Feynman's report for Dr. Ride.\"\n\nThen I discover Mr. Acheson hasn't seen it.\n\n\"Make a copy and give it to Mr. Acheson.\"\n\nI finally caught on, so I said, \"Dr. Keel, I don't think anybody has seen my report.\"\n\nSo he says to his secretary, \"Please make a copy for all the commissioners and give it to them.\"\n\nThen I said to him, \"I appreciate how much work you're doing, and that it's difficult to keep everything in mind. But I thought you told me that you showed my report to everybody.\"\n\nHe says, \"Yes, well, I meant all of the staff.\"\n\nI later discovered, by talking to people on the staff, that they hadn't seen it either.\n\nWhen the other commissioners finally got to see my report, most of them thought it was very good, and it ought to be in the commission report somewhere.\n\nEncouraged by that, I kept bringing up my report. \"I'd like to have a meeting to discuss what to do with it,\" I kept saying.\n\n\"We'll have a meeting about it next week\" was the standard answer. (We were too busy wordsmithing and voting on the color of the cover.)\n\nGradually I realized that the way my report was written, it would require a lot of wordsmithing\u2014and we were running out of time. Then somebody suggested that my report could go in as an appendix. That way, it wouldn't have to be wordsmithed to fit in with anything else.\n\nBut some of the commissioners felt strongly that my report should go in the main report somehow: \"The appendices won't come out until months later, so nobody will read your report if it's an appendix,\" they said.\n\nI thought I'd compromise, however, and let it go in as an appendix.\n\nBut now there was a new problem: my report, which I had written on my word processor at home, would have to be converted from the IBM format to the big document system the commission was using. They had a way of doing that with an optical scanning device.\n\nI had to go to a little bit of trouble to find the right guy to do it. Then, it didn't get done right away. When I asked what happened, the guy said he couldn't find the copy I had given him. So I had to give him another copy.\n\nA few days later, I finished writing my report about the avionics, and I wanted to combine it with my report on the engines. So I took the avionics report to the guy and I said, \"I'd like to put this in with my other report.\"\n\nThen I needed to see a copy of my new report for some reason, but the guy gave me an old copy, before the avionics was added. \"Where's the new one with the avionics?\" I said.\n\n\"I can't find it\"\u2014and so on. I don't remember all the details, but it seemed my report was always missing or half-cooked. It could easily have been mistakes, but there were too many of them. It was quite a struggle, nursing my report along.\n\nThen, in the last couple of days, when the main report is ready to be sent to the printer, Dr. Keel wants my report to be wordsmithed too, even though it's going in as an appendix. So I took it to the regular editor there, a capable man named Hansen, and he fixed it up without changing the sense of it. Then it was put back into the machine as \"Version #23\"\u2014there were revisions and revisions.\n\n(By the way: _everything_ had 23 versions. It has been noted that computers, which are supposed to increase the speed at which we do things, have not increased the speed at which we write reports: we used to make only three versions\u2014because they're so hard to type\u2014and now we make 23 versions!)\n\nThe next day I noticed Keel working on my report: he had put all kinds of big circles around whole sections, with X's through them; there were all kinds of thoughts left out. He explained, \"This part doesn't have to go in because it says more or less what we said in the main report.\"\n\nI tried to explain that it's much easier to get the logic if all the ideas are together, instead of everything being distributed in little pieces all over the main report. \"After all,\" I said, \"it's only gonna be an appendix. It won't make any difference if there's a little repetition.\"\n\nDr. Keel put back something here and there when I asked him to, but there was still so much missing that my report wasn't anything like it was before.\n\n### The Tenth Recommendation\n\n_SOMETIME_ in May, at one of our last meetings, we got around to making a list of possible recommendations. Somebody would say, \"Maybe one of the things we should discuss is the establishment of a safety board.\"\n\n\"Okay, we'll put that down.\"\n\nI'm thinking, \"At last! We're going to have a discussion!\"\n\nBut it turns out that this tentative list of topics _becomes_ the recommendations\u2014that there be a safety board, that there be a this, that there be a that. The only discussion was about which recommendation we should write first, which one should come second, and so forth.\n\nThere were many things I wanted to discuss further. For example, in regard to a safety board, one could ask: \"Wouldn't such a committee just add another layer to an already overgrown bureaucracy?\"\n\nThere had been safety boards before. In 1967, after the Apollo accident, the investigating committee at the time invented a special panel for safety. It worked for a while, but it didn't last.\n\nWe didn't discuss why the earlier safety boards were no longer effective; instead, we just made up more safety boards: we called them the \"Independent Solid Rocket Motor Design Oversight Committee,\" the \"Shuttle Transportation System Safety Advisory Panel,\" and the \"Office of Safety, Reliability, and Quality Assurance.\" We decided who would oversee each safety board, but we didn't discuss whether the safety boards created by our commission had any better chance of working, whether we could fix the existing boards so they _would_ work, or whether we should have them at all.\n\nI'm not as sure about a lot of things as everybody else. Things need to be thought out a little bit, and we weren't doing enough _thinking_ together. Quick decisions on important matters are not very good\u2014and at the speed we were going, we were bound to make some impractical recommendations.\n\nWe ended up rearranging the list of possible recommendations and wordsmithing them a little, and then we voted yes or no. It was an odd way of doing things, and I wasn't used to it. In fact, I got the feeling we were being railroaded: things were being decided, somehow, a little out of our control.\n\nAt any rate, in our last meeting, we agreed to nine recommendations. Many of the commissioners went home after that meeting, but I was going to New York a few days later, so I stayed in Washington.\n\nThe next day, I happened to be standing around in Mr. Rogers's office with Neil Armstrong and another commissioner when Rogers says, \"I thought we should have a tenth recommendation. Everything in our report is so negative; I think we need something positive at the end to balance it.\"\n\nHe shows me a piece of paper. It says,\n\nThe Commission strongly recommends that NASA continue to receive the support of the Administration and the nation. The agency constitutes a national resource and plays a critical role in space exploration and development. It also provides a symbol of national pride and technological leadership. The Commission applauds NASA's spectacular achievements of the past and anticipates impressive achievements to come. The findings and recommendations presented in this report are intended to contribute to the future NASA successes that the nation both expects and requires as the 21st century approaches.\n\nIn our tour months of work as a commission, we had never discussed a policy question like that, so I felt there was no reason to put it in. And although I'm not saying I disagreed with it, it wasn't obvious that it was true, either. I said, \"I think this tenth recommendation is inappropriate.\"\n\nI think I heard Armstrong say, \"Well, if somebody's not in favor of it, I think we shouldn't put it in.\"\n\nBut Rogers kept working on me. We argued back and forth a little bit, but then I had to catch my flight to New York.\n\nWhile I was in the airplane, I thought about this tenth recommendation some more. I wanted to lay out my arguments carefully on paper, so when I got to my hotel in New York, I wrote Rogers a letter. At the end I wrote, \"This recommendation reminds me of the NASA flight reviews: 'There are critical problems, but never mind\u2014keep on flying!' \"\n\nIt was Saturday, and I wanted Mr. Rogers to read my letter before Monday. So I called up his secretary\u2014everybody was working seven days a week to get the report out in time\u2014and I said, \"I'd like to dictate a letter to you; is that all right?\"\n\nShe says, \"Sure! To save you some money, let me call you right back.\" She calls me back, I dictate the letter, and she hands it directly to Rogers.\n\nWhen I came back on Monday, Mr. Rogers said, \"Dr. Feynman, I've read your letter, and I agree with everything it says. But vou've been out-voted.\"\n\n\"Out-voted? How was I out-voted, when there was no meeting?\"\n\nKeel was there, too. He says, \"We called everybody, and they all agree with the recommendation. They all voted for it.\"\n\n\"I don't think that's fair!\" I protested. \"If I could have presented my arguments to the other commissioners, I don't think I'd have been out-voted.\" I didn't know what to do, so I said, \"I'd like to make a copy of it.\"\n\nWhen I came back, Keel says, \"We just remembered that we didn't talk to Hotz about it, because he was in a meeting. We forgot to get his vote.\"\n\nI didn't know what to make of that, but I found out later that Mr. Hotz was in the building, not far from the copy machine.\n\nLater, I talked to David Acheson about the tenth recommendation. He explained, \"It doesn't really mean anything; it's only motherhood and apple pie.\"\n\nI said, \"Well, if it doesn't mean anything, it's not necessary, then.\"\n\n\"If this were a commission for the National Academy of Sciences, your objections would be proper. But don't forget,\" he says, \"this is a presidential commission. We should say something for the President.\"\n\n\"I don't understand the difference,\" I said. \" _Why can't I be careful and scientific when I'm writing a report to the President?\"_\n\nBeing naive doesn't always work: my argument had no effect. Acheson kept telling me I was making a big thing out of nothing, and I kept saying it weakened our report and it shouldn't go in.\n\nSo that's where it ended up: \"The Commission strongly recommends that NASA continue to receive the support of the Administration and the nation...\"\u2014all this \"motherhood and apple pie\" stuff to \"balance\" the report.\n\nWhile I was flying home, I thought to myself, \"It's funny that the only part of the report that was _genuinely_ balanced was my own report: I said negative things about the engine, and positive things about the avionics. And I had to struggle with them to get it in, even as a lousy appendix!\"\n\nI thought about the tenth recommendation. All the other recommendations were based on evidence we had found, but this one had no evidence whatsoever. I could see the whitewash dripping down. It was _obviously_ a mistake! It would make our report look bad. I was very disturbed.\n\nWhen I got home, I talked to Joan, my sister. I told her about the tenth recommendation, and how I had been \"out-voted.\"\n\n\"Did you call any of the other commissioners and talk to them yourself?\" she said.\n\n\"Well, I talked to Acheson, but he was for it.\"\n\n\"Any others?\"\n\n\"Uh, no.\" So I called up three other commissioners\u2014I'll call them A, B, and C.\n\nI call A, who says, \" _What_ tenth recommendation?\"\n\nI call B, who says, \"Tenth recommendation? What are you talking about?\"\n\nI call C, who says, \"Don't you remember, you dope? I was in the office when Rogers first told us, and I don't see anything wrong with it.\"\n\nIt appeared that the only people who knew about the tenth recommendation were the people who were in the office when Rogers told us. I didn't bother to make any more telephone calls. After all, it's enough\u2014I didn't feel that I had to open all the safes to check that the combination is the same\n\nThen I told Joan about my report\u2014how it was so emasculated, even though it was going in as an appendix.\n\nShe says, \"Well, if they do that to your report, what have you accomplished, being on the commission? What's the result of all your work?\"\n\n\"Aha!\"\n\nI sent a telegram to Mr. Rogers:\n\nPLEASE TAKE MY SIGNATURE OFF THE REPORT UNLESS TWO THINGS OCCUR: 1) THERE IS NO TENTH RECOMMENDATION, AND 2) MY REPORT APPEARS WITHOUT MODIFICATION FROM VERSION #23.\n\n(I knew by this time I had to define everything carefully.)\n\nIn order to get the number of the version I wanted, I called Mr. Hotz, who was in charge of the documentation system and publishing the report. He sent me Version #23, so I had something definite to publish on my own, if worse came to worst.\n\nThe result of this telegram was that Rogers and Keel tried to negotiate with me. They asked General Kutyna to be the intermediary, because they knew he was a friend of mine. What a _good_ friend of mine he was, they didn't know.\n\nKutyna says, \"Hello, Professor, I just wanted to tell you that I think you're doing very well. But I've been given the job of trying to talk you out of it, so I'm going to give you the arguments.\"\n\n\"Fear not!\" I said. \"I'm not gonna change my mind. Just give me the arguments, and fear not.\"\n\nThe first argument was that if I don't accept the tenth recommendation, they won't accept my report, even as an appendix.\n\nI didn't worry about that one, because I could always put out my report myself.\n\nAll the arguments were like that: none of them was very good, and none of them had any effect. I had thought through carefully what I was doing, so I just stuck to my guns.\n\nThen Kutyna suggested a compromise: they were willing to go along with my report as I wrote it, except for one sentence near the end.\n\nI looked at the sentence and I realized that I had already made my point in the previous paragraph. Repeating the point amounted to polemics; removing the phrase made my report much better. I accepted the compromise.\n\nThen I offered a compromise on the tenth recommendation: \"If they want to say something nice about NASA at the end, just don't call it a recommendation, so people will know that it's not in the same class as the other recommendations: call it a 'concluding thought' if you want. And to avoid confusion, don't use the words 'strongly recommends.' Just say 'urges'\u2014'The Commission urges that NASA continue to receive the support of the Administration and the nation.' All the other stuff can stay the same.\"\n\nA little bit later, Keel calls me up: \"Can we say ' _strongly_ urges'?\"\n\n\"No. Just 'urges'.\"\n\n\"Okay,\" he said. And that was the final decision.\n\n### Meet the Press\n\n_I PUT_ my name on the main report, my own report got in as an appendix, and everything was all right. In early June we went back to Washington and gave our report to the President in a ceremony held in the Rose Garden. That was on a Thursday. The report was not to be released to the public until the following Monday, so the President could study it.\n\nMeanwhile, the newspaper reporters were working like demons: they knew our report was finished and they were trying to scoop each other to find out what was in it. I knew they would be calling me up day and night, and I was afraid I would say something about a technical matter that would give them a hint.\n\nReporters are very clever and persistent. They'll say, \"We heard such-and-such\u2014is it true?\" And pretty soon, what you're thinking you didn't tell them shows up in the newspaper!\n\nI was determined not to say a word about the report until it was made public, on Monday. A friend of mine convinced me to go on the \"MacNeil\/Lehrer Newshour,\" so I said yes for Monday evening's show.\n\nI also had my secretary set up a press conference for Tuesday at Caltech. I said, \"Tell the reporters who want to talk to me that I haven't any comment on anything: any questions they have, I'll be glad to answer on Tuesday at my press conference.\"\n\n_F IGURE 18. The Commission Report was presented to the president in the Rose Garden at the White House. Visible, from left to right, are General Kutyna, William Rogers, Eugene Covert, President Reagan, Neil Armstrong, and Richard Feynman. ( \u00a9 PETE SOUZA, THE WHITE_.)\n\n_F IGURE 19. At the reception._ ( \u00a9 _PETE SOUZA, THE WHITE HOUSE_.)\n\nOver the weekend, while I was still in Washington, it leaked somehow that I had threatened to take my name off the report. Some paper in Miami started it, and soon the story was running all over about this argument between me and Rogers. When the reporters who were used to covering Washington heard \"Mr. Feynman has nothing to say; he'll answer all your questions at his press conference on Tuesday,\" it sounded suspicious\u2014as though the argument was still on, and I was going to have this press conference on Tuesday to explain why I took my name off the report.\n\nBut I didn't know anything about it. I isolated myself from the press so much that I wasn't even reading the newspapers.\n\nOn Sunday night, the commission had a goodbye dinner arranged by Mr. Rogers at some club. After we finished eating, I said to General Kutyna, \"I can't stay around anymore. I have to leave a little early.\"\n\nHe says, \"What can be so important?\"\n\nI didn't want to say.\n\nHe comes outside with me, to see what this \"important\" something is. It's a bright red sports car with two beautiful blonds inside, waiting to whisk me away.\n\nI get in the car. We're about to speed off, leaving General Kutyna standing there scratching his head, when one of the blonds says, \"Oh! General Kutyna! I'm Ms. So-and-so. I interviewed you on the phone a few weeks ago.\"\n\nSo he caught on. They were reporters from the \"MacNeil\/Lehrer Newshour.\"\n\nThey were very nice, and we talked about this and that for the show Monday night. Somewhere along the line I told them I was going to have my own press conference on Tuesday, and I was going to give out my report\u2014even though it was going to appear as an appendix three months later. They said my report sounded interesting, and they'd like to see it. By this time we're all very friendly, so I gave them a copy.\n\nThey dropped me off at my cousin's house, where I was staying. I told Frances about the show, and how I gave the reporters a copy of my report. Frances puts her hands to her head, horrified.\n\nI said, \"Yes, that was a dumb mistake, wasn't it! I'd better call 'em up and tell 'em not to use it.\"\n\nI could tell by the way Frances shook her head that it wasn't gonna be so easy!\n\nI call one of them up: \"I'm sorry, but I made a mistake: I shouldn't have given my report to you, so I'd prefer you didn't use it.\"\n\n\"We're in the news business, Dr. Feynman. The goal of the news business is to get news, and your report is newsworthy. It would be completely against our instincts and practice not to use it.\"\n\n\"I know, but I'm naive about these things. I simply made a mistake. It's not fair to the other reporters who will be at the press conference on Tuesday. After all, would you like it if you came to a press conference and the guy had mistakenly given his report to somebody else? I think you can understand that.\"\n\n\"I'll talk to my colleague and call you back.\"\n\nTwo hours later, they call back\u2014they're both on the line\u2014and they try to explain to me why they should use it: \"In the news business, it's customary that whenever we get a document from somebody (he way we did from you, it means we can use it.\"\n\n\"I appreciate that there are conventions in the news business, but I don't know anything about these things, so as a courtesy to me, please don't use it.\"\n\nIt went back and forth a little more like that. Then another \"We'll call you back.\" and another long delay. I could tell from the long delays that they were having a lot of trouble with this problem.\n\nI was in a very good fettle, for some reason. I had already lost, and I knew what I needed, so I could focus easily. I had no difficulty admitting complete idiocy\u2014which is usually the case when I deal with the world\u2014and I didn't think there was any law of nature which said I had to give in. I just kept going, and didn't waver at all.\n\nIt went late into the night: one o'clock, two o'clock, we're still working on it. \"Dr. Feynman, it's very unprofessional to give someone a story and then retract it. This is not the way people behave in Washington.\"\n\n\"It's obvious I don't know anything about Washington. But this is the way _I_ behave\u2014like a fool. I'm sorry, but it was simply an error, so as a courtesy, please don't use it.\"\n\nThen, somewhere along the line, one of them says, \"If we go ahead and use your report, does that mean you won't go on the show?\"\n\n_\"You_ said it; _I_ didn't.\"\n\n\"We'll call you back.\"\n\nAnother delay.\n\nActually, I hadn't decided whether I'd refuse to go on the show, because I kept thinking it was possible I could undo my mistake. When I thought about it, I didn't think I could legitimately play that card. But when one of them made the mistake of proposing the possibility, I said, _\"You_ said it; _I_ didn't\"\u2014very cold\u2014as if to say, \"I'm not threatening you, but you can figure it out for yourself, honey!\"\n\nThey called me back, and said they wouldn't use my report.\n\nWhen I went on the show, I never got the impression that any of the questions were based on my report. Mr. Lehrer did ask me whether there had been any problems between me and Mr. Rogers, but I weaseled: I said there had been no problems.\n\nAfter the show was over, the two reporters told me they thought the show went fine without my report. We left good friends.\n\nI flew back to California that night, and had my press conference on Tuesday at Caltech. A large number of reporters came. A few asked questions about my report, but most of them were interested in the rumor that I had threatened to take my name off the commission report. I found myself telling them over and over that I had no problem with Mr. Rogers.\n\n### Afterthoughts\n\n_NOW_ that I've had more time to think about it, I still like Mr. Rogers, and I still feel that everything's okay. It's my judgment that he's a fine man. Over the course of the commission I got to appreciate his talents and his abilities, and I have great respect for him. Mr. Rogers has a very good, smooth way about him, so I reserve in my head the possibility\u2014not as a suspicion, but as an unknown\u2014that I like him because he knew how to make me like him. I prefer to assume he's a genuinely fine fellow, and that he is the way he appears. But I was in Washington long enough to know that I can't tell.\n\nI'm not exactly sure what Mr. Rogers thinks of me. He gives me the impression that, in spite of my being such a pain in the ass to him in the beginning, he likes me very much. I may be wrong, but if he feels the way I feel toward him, it's good.\n\nMr. Rogers, being a lawyer, had a difficult job to run a commission investigating what was essentially a technical question. With Dr. Keel's help, I think the technical part of it was handled well. But it struck me that there were several fishinesses associated with the big cheeses at NASA.\n\nEvery time we talked to higher level managers, they kept saying they didn't know anything about the problems below them. We're getting this kind of thing again in the Iran-Contra hearings, but at that time, this kind of situation was new to me: either the guys at the top didn't know, in which case they should have known, or they did know, in which case they're lying to us.\n\nWhen we learned that Mr. Mulloy had put pressure on Thiokol to launch, we heard time after time that the next level up at NASA knew nothing about it. You'd think Mr. Mulloy would have notified a higher-up during this big discussion, saying something like, \"There's a question as to whether we should fly tomorrow morning, and there's been some objection by the Thiokol engineers, but we've decided to fly anyway\u2014what do you think?\" But instead, Mulloy said something like, \"All the questions have been resolved.\" There seemed to be some reason why guys at the lower level didn't bring problems up to the next level.\n\nI invented a theory which I have discussed with a considerable number of people, and many people have explained to me why it's wrong. But I don't remember their explanations, so I cannot resist telling you what I think led to this lack of communication in NASA.\n\nWhen NASA was trying to go to the moon, there was a great deal of enthusiasm: it was a goal everyone was anxious to achieve. They didn't know if they could do it, but they were all working together.\n\nI have this idea because I worked at Los Alamos, and I experienced the tension and the pressure of everybody working together to make the atomic bomb. When somebody's having a problem\u2014say, with the detonator\u2014everybody knows that it's a big problem, they're thinking of ways to beat it, they're making suggestions, and when they hear about the solution they're excited, because that means their work is now useful: if the detonator didn't work, the bomb wouldn't work.\n\nI figured the same thing had gone on at NASA in the early days: if the space suit didn't work, they couldn't go to the moon. So everybody's interested in everybody else's problems.\n\nBut then, when the moon project was over, NASA had all these people together: there's a big organization in Houston and a big organization in Huntsville, not to mention at Kennedy, in Florida. You don't want to fire people and send them out in the street when you're done with a big project, so the problem is, what to do?\n\nYou have to convince Congress that there exists a project that only NASA can do. In order to do so, it is necessary\u2014at least it was _apparently_ necessary in this case\u2014to exaggerate: to exaggerate how economical the shuttle would be, to exaggerate how often it could fly, to exaggerate how safe it would be, to exaggerate the big scientific facts that would be discovered. \"The shuttle can make so-and-so many flights and it'll cost such-and-such; we went to the moon, so we can _do_ it!\"\n\nMeanwhile, I would guess, the engineers at the bottom are saying, \"No, no! We can't make that many flights. If we had to make that many flights, it would mean such-and-such!\" And, \"No, we can't do it for that amount of money, because that would mean we'd have to do thus-and-so!\"\n\nWell, the guys who are trying to get Congress to okay their projects don't want to hear such talk. It's better if they don't hear, so they can be more \"honest\"\u2014they don't want to be in the position of lying to Congress! So pretty soon the attitudes begin to change: information from the bottom which is disagreeable\u2014\"We're having a problem with the seals; we should fix it before we fly again\"\u2014is suppressed by big cheeses and middle managers who say, \"If you tell me about the seals problems, we'll have to ground the shuttle and fix it.\" Or, \"No, no, keep on flying, because otherwise, it'll look bad,\" or \"Don't tell me; I don't want to hear about it.\"\n\nMaybe they don't say explicitly \"Don't tell me,\" but they discourage communication, which amounts to the same thing. It's not a question of what has been written down, or who should tell what to whom; it's a question of whether, when you _do_ tell somebody about some problem, they're _delighted_ to hear about it and they say \"Tell me more\" and \"Have you tried such-and-such?\" or they say \"Well, see what you can do about it\"\u2014which is a completely different atmosphere. If you try once or twice to communicate and get pushed back, pretty soon you decide, \"To hell with it.\"\n\nSo that's my theory: because of the exaggeration at the top being inconsistent with the reality at the bottom, communication got slowed up and ultimately jammed. That's how it's possible that the higher-ups didn't know.\n\nThe other possibility is that the higher-ups did know, and they just _said_ they didn't know.\n\nI looked up a former director of NASA\u2014I don't remember his name now\u2014who is the head of some company in California. I thought I'd go and talk to him when I was on one of my breaks at home, and say, \"They all _say_ they haven't heard. Does that make any sense? How does someone go about investigating them?\"\n\nHe never returned my calls. Perhaps he didn't want to talk to the commissioner investigating higher-ups; maybe he had had enough of NASA, and didn't want to get involved. And because I was busy with so many other things, I didn't push it.\n\nThere were all kinds of questions we didn't investigate. One was this mystery of Mr. Beggs, the former director of NASA who was removed from his job pending an investigation that had nothing to do with the shuttle; he was replaced by Graham shortly before the accident. Nevertheless, it turned out that, every day, Beggs came to his old office. People came in to see him, although he never talked to Graham. What was he doing? Was there some activity still being directed by Beggs?\n\nFrom time to time I would try to get Mr. Rogers interested in investigating such fishinesses. I said, \"We have lawyers on the commission, we have company managers, we have very fine people with a large range of experiences. We have people who know how to get an answer out of a guy when he doesn't want to say something. I don't know how to do that. If a guy tells me the probability of failure is 1 in 10 5, I know he's full of crap\u2014but I don't know what's natural in a bureaucratic system. We oughta get some of the big shots together and ask them questions: just like we asked the second-level managers like Mr. Mulloy, we should ask the first level.\"\n\nHe would say, \"Yes, well, I think so.\"\n\nMr. Rogers told me later that he wrote a letter to each of the big shots, but they replied that they didn't have anything they wanted to say to us.\n\nThere was also the question of pressure from the White House.\n\nIt was the President's idea to put a teacher in space, as a symbol of the nation's commitment to education. He had proposed the idea a year before, in his State of the Union address. Now, one year later, the State of the Union speech was coming up again. It would be perfect to have the teacher in space, talking to the President and the Congress. All the circumstantial evidence was very strong.\n\nI talked to a number of people about it, and heard various opinions, but I finally concluded that there was no pressure from the White House.\n\nFirst of all, the man who pressured Thiokol to change its position, Mr. Mulloy, was a second-level manager. Ahead of time, nobody could predict what might get in the way of a launch. If you imagine Mulloy was told \"Make sure the shuttle flies tomorrow, because the President wants it,\" you'd have to imagine that _everybody else_ at his level had to be told\u2014and there are a lot of people at his level. To tell that many people would make it sure to leak out. So that way of putting on pressure was very unlikely.\n\nBy the time the commission was over, I understood much better the character of operations in Washington and in NASA. I learned, by seeing how they worked, that the people in a big system like NASA _know_ what has to be done\u2014 _without_ being told.\n\nThere was _already_ a big pressure to keep the shuttle flying. NASA had a flight schedule they were trying to meet, just to show the capabilities of NASA\u2014never mind whether the president was going to give a speech that night or not. So I don't believe there was any direct activity or any special effort from the White House. There was no need to do it, so I don't believe it was done.\n\nI could give you an analog of that. You know those signs that appear in the back windows of automobiles\u2014those little yellow diamonds that say BABY ON BOARD, and things like that? You don't have to _tell_ me there's a baby on board; I'm gonna drive carefully _anyway!_ What am I supposed to do when I see there's a baby on board: act differently? As if I'm suddenly gonna drive more carefully and not hit the car because there's a baby on board, when all I'm trying to do is not hit it anyway!\n\nSo NASA was trying to get the shuttle up anyway: you don't have to say there's a baby on board, or there's a teacher on board, or it's important to get this one up for the President.\n\nNow that I've talked to some people about my experiences on the commission, I think I understand a few things that I didn't understand so well earlier. One of them has to do with what I said to Dr. Keel that upset him so much. Recently I was talking to a man who spent a lot of time in Washington, and I asked him a particular question which, if he didn't take it right, could be considered a grave insult. I would like to explain the question, because it seems to me to be a real possibility of what I said to Dr. Keel.\n\nThe only way to have real success in science, the field I'm familiar with, is to describe the evidence very carefully without regard to the way you feel it should be. If you have a theory, you must try to explain what's good and what's bad about it equally. In science, you learn a kind of standard integrity and honesty.\n\nIn other fields, such as business, it's different. For example, almost every advertisement you see is obviously designed, in some way or another, to fool the customer: the print that they don't want you to read is small; the statements are written in an obscure way. It is obvious to anybody that the product is not being presented in a scientific and balanced way. Therefore, in the selling business, there's a lack of integrity.\n\nMy father had the spirit and integrity of a scientist, but he was a salesman. I remember asking him the question \"How can a man of integrity be a salesman?\"\n\nHe said to me, \"Frankly, many salesmen in the business are not straightforward\u2014they think it's a better way to sell. But I've tried being straightforward, and I find it has its advantages. In fact, I wouldn't do it any other way. If the customer thinks at all, he'll realize he has had some bad experience with another salesman, but hasn't had that kind of experience with you. So in the end, several customers will stay with you for a long time and appreciate it.\"\n\nMy father was not a big, successful, famous salesman; he was the sales manager for a medium-sized uniform company. He was successful, but not enormously so.\n\nWhen I see a congressman giving his opinion on something, I always wonder if it represents his _real_ opinion or if it represents an opinion that he's designed in order to be elected. It seems to be a central problem for politicians. So I often wonder: what is the relation of integrity to working in the government?\n\nNow, Dr. Keel started out by telling me that he had a degree in physics. I always assume that everybody in physics has integrity\u2014perhaps I'm naive about that\u2014so I must have asked him a question I often think about: \"How can a man of integrity get along in Washington?\"\n\nIt's very easy to read that question another way: \"Since you're getting along in Washington, you can't be a man of integrity!\"\n\nAnother thing I understand better now has to do with where the idea came from that cold affects the O-rings. It was General Kutyna who called me up and said, \"I was working on my carburetor, and I was thinking: what is the effect of cold on the O-rings?\"\n\nWell, it turns out that one of NASA's own astronauts told him there was information, somewhere in the works of NASA, that the O-rings had no resilience whatever at low temperatures\u2014and NASA wasn't saying anything about it.\n\nBut General Kutyna had the career of that astronaut to worry about, so the _real_ question the General was thinking about while he was working on his carburetor was, \"How can I get this information out without jeopardizing my astronaut friend?\" His solution was to get the professor excited about it, and his plan worked perfectly.\n*Richard's younger sister, Joan, has a Ph.D. in physics, in spite of this preconception that only boys are destined to be scientists.\n*Note for foreign readers: the quota system was a discriminatory practice of limiting the number of places in a university available to students of Jewish background.\n*Feynman was suffering from abdominal cancer. He had surgery in 1978 and 1981. After he returned from Japan, he had more surgery, in October 1986 and October 1987.\n\n*Hideki Yukawa. Eminent Japanese physicist; Nobel Prize, 1949.\n*Four years later Richard and Gweneth met the king of Sweden\u2014at the Nobel Prize ceremony.\n\n\u2020The Feynmans' dog.\n\n*Gweneth was expecting Carl at the time.\n\n\u2020Kiwi.\n\n\u2021Carl. This letter was written in 1963.\n\n*About 200 square feet.\n\n* Daughter Michelle was about eleven when this letter was written, in 1980 or 1981.\n\n*The \"New Zealand lectures,\" delivered in 1979, are written up in _QED: The Strange Theory of Light and Matter_ (Princeton University Press, 1985).\n\n\u2020These letters were contributed by Freeman Dyson. They are the first and last letters he wrote that mention Richard Feynman. Other letters are referred to in Dyson's book _Disturbing the Universe_.\n\n*A family friend.\n\n*As it turned out, Feynman was not to be disappointed: Carl works at the Thinking Machines Company, and daughter Michelle is studying to become a commercial photographer.\n\n\u2020This letter was contributed by Henry Bethe.\n*The National Aeronautics and Space Administration\n*NASA's Jet Propulsion Laboratory, located in Pasadena; it is administered by Caltech.\n\n*Note for foreign readers: a flight that leaves the West Coast around 11 P.M. and arrives on the East Coast around 7 A.M., five hours and three time zones later.\n\n*Note for foreign readers: Sally Ride was the first American woman in space.\n\n*Later in our investigation we discovered that it was this leak check which was a likely cause of the dangerous bubbles in the zinc chromate putty that I had heard about at JPL.\n\n*The tang is the male part of the joint; the clevis is the female part (see Figure 13).\n*The thing Feynman was going to break up was the baloney (the \"bull\u2014\u2014\") about how good everything was at NASA.\n*The Office of Management and Budget.\n\n*The reference is to Feynman's method of slicing string beans, recounted in _Surely You're Joking, Mr. Feynman!_\n\n*Note for foreign readers: the Warren Report was issued in 1964 by the Warren Commission, headed by retired Supreme Court Chief Justice Earl Warren, which investigated the assassination of President John K. Kennedy.\n*Feynman's way of saying, \"whatever it was.\"\n\n*Later, Mr. Lovingood sent me that report. It said things like \"The probability of mission success is necessarily very close to 1.0\"\u2014does that mean it _is_ close to 1.0, or it _ought lo be_ close to 1.0?\u2014and \"Historically, this high degree of mission success has given rise to a difference in philosophy between unmanned and manned space flight programs; i.e., numerical probability versus engineering judgment.\" As far as I can tell, \"engineering judgment\" means they're just going to make up numbers! The probability of an engine-blade failure was given as a universal constant, as if all the blades were exactly the same, under the same conditions. The whole paper was quantifying everything. Just about every nut and bolt was in there: \"The chance that a HPHTP pipe will burst is 10-7.\" You can't estimate things like that; a probability of 1 in 10,000,000 is almost impossible to estimate. It was clear that the numbers for each part of the engine were chosen so that when you add everything together you gel 1 in 100.000.\n\n*I had heard about this from Bill Graham. He said that when he was first on the job as head of NASA, he was looking through some reports and noticed a little bullet: \"*4,000 cycle vibration is within our data base.\" He thought that was a funny-looking phrase, so he began asking questions. When he got all the way through, he discovered it was a rather serious matter: some of the engines would vibrate so much that they couldn't be used. He used it as an example of how difficult it is to get information unless you go down and check on it yourself.\n\n*Note for foreign readers: Federal Aviation Administration.\n*This refers to \"Safecracker Meets Safecracker,\" another story told in _Surely You're Joking, Mr. Feynman!_\n\n### Appendix F: Personal Observations on the Reliability of the Shuttle\n\n### Introduction\n\nIt appears that there are enormous differences of opinion as to the probability of a failure with loss of vehicle and of human life.* The estimates range from roughly 1 in 100 to 1 in 100,000. The higher figures come from working engineers, and the very low figures come from management. What are the causes and consequences of this lack of agreement? Since 1 part in 100,000 would imply that one could launch a shuttle each day for 300 years expecting to lose only one, we could properly ask, \"What is the cause of management's fantastic faith in the machinery?\"\n\nWe have also found that certification criteria used in flight readiness reviews often develop a gradually decreasing strictness. The argument that the same risk was flown before without failure is often accepted as an argument for the safety of accepting it again. Because of this, obvious weaknesses are accepted again and again\u2014sometimes without a sufficiently serious attempt to remedy them, sometimes without a flight delay because of their continued presence.\n\nThere are several sources of information: there are published criteria for certification, including a history of modifications in the form of waivers and deviations; in addition, the records of the flight readiness reviews for each flight document the arguments used to accept the risks of the flight. Information was obtained from direct testimony and reports of the range safety officer, Louis J. Ullian, with respect to the history of success of solid fuel rockets. There was a further study by him (as chairman of the Launch Abort Safety Panel, LASP) in an attempt to determine the risks involved in possible accidents leading to radioactive contamination from attempting to fly a plutonium power supply (called a radioactive thermal generator, or RTG) on future planetary missions. The NASA study of the same question is also available. For the history of the space shuttle main engines, interviews with management and engineers at Marshall, and informal interviews with engineers at Rocketdyne, were made. An independent (Caltech) mechanical engineer who consulted for NASA about engines was also interviewed informally. A visit to Johnson was made to gather information on the reliability of the avionics (computers, sensors, and effectors). Finally, there is the report \"A Review of Certification Practices Potentially Applicable to Man-rated Reusable Rocket Engines,\" prepared at the Jet Propulsion Laboratory by N. Moore et al. in February 1986 for NASA Headquarters, Office of Space Flight. It deals with the methods used by the FAA and the military to certify their gas turbine and rocket engines. These authors were also interviewed informally.\n\n_Solid Rocket Boosters (SRB)_\n\nAn estimate of the reliability of solid-fuel rocket boosters (SRBs) was made by the range safety officer by studying the experience of all previous rocket flights. Out of a total of nearly 2900 flights, 121 failed (1 in 25). This includes, however, what may be called \"early errors\"\u2014rockets flown for the first few times in which design errors are discovered and fixed. A more reasonable figure for the mature rockets might be 1 in 50. With special care in selecting parts and in inspection, a figure below 1 in 100 might be achieved, but 1 in 1000 is probably not attainable with today's technology. (Since there are two rockets on the shuttle, these rocket failure rates must be doubled to get shuttle failure rates due to SRB failure.)\n\nNASA officials argue that the figure is much lower. They point out that \"since the shuttle is a manned vehicle, the probability of mission success is necessarily very close to 1.0.\" It is not very clear what this phrase means. Does it mean it _is_ close to 1 or that it _ought to be_ close to I ? They go on to explain, \"Historically, this extremely high degree of mission success has given rise to a difference in philosophy between manned space flight programs and unmanned programs; i.e., numerical probability usage versus engineering judgment.\" (These quotations are from \"Space Shuttle Data for Planetary Mission RTG Safety Analysis,\" pages 3-1 and 3-2, February 15, 1985, NASA, JSC.) It is true that if the probability of failure was as low as 1 in 100,000 it would take an inordinate number of tests to determine it: you would get nothing but a string of perfect flights with no precise figure\u2014other than that the probability is likely less than the number of such flights in the string so far. But if the real probability is not so small, flights would show troubles, near failures, and possibly actual failures with a reasonable number of trials, and standard statistical methods could give a reasonable estimate. In fact, previous NASA experience had shown, on occasion, just such difficulties, near accidents, and even accidents, all giving warning that the probability of flight failure was not so very small.\n\nAnother inconsistency in the argument not to determine reliability through historical experience (as the range safety officer did) is NASA's appeal to history: \"Historically, this high degree of mission success...\" Finally, if we are to replace standard numerical probability usage with engineering judgment, why do we find such an enormous disparity between the management estimate and the judgment of the engineers? It would appear that, for whatever purpose\u2014be it for internal or external consumption\u2014the management of NASA exaggerates the reliability of its product to the point of fantasy.\n\nThe history of the certification and flight readiness reviews will not be repeated here (see other parts of the commission report), but the phenomenon of accepting seals that had shown erosion and blowby in previous flights is very clear. The _Challenger_ flight is an excellent example: there are several references to previous flights; the acceptance and success of these flights are taken as evidence of safety. But erosion and blowby are not what the design expected. They are warnings that something is wrong. The equipment is not operating as expected, and therefore there is a danger that it can operate with even wider deviations in this unexpected and not thoroughly understood way. The fact that this danger did not lead to a catastrophe before is no guarantee that it will not the next time, unless it is completely understood. When playing Russian roulette, the fact that the first shot got off safely is of little comfort for the next. The origin and consequences of the erosion and blowby were not understood. Erosion and blowby did not occur equally on all flights or in all joints: sometimes there was more, sometimes less. Why not sometime, when whatever conditions determined it were right, wouldn't there be still more, leading to catastrophe?\n\nIn spite of these variations from case to case, officials behaved as if they understood them, giving apparently logical arguments to each other\u2014often citing the \"success\" of previous flights. For example, in determining if flight 51-L was safe to fly in the face of ring erosion in flight 51-C, it was noted that the erosion depth was only one-third of the radius. It had been noted in an experiment cutting the ring that cutting it as deep as one radius was necessary before the ring failed. Instead of being very concerned that variations of poorly understood conditions might reasonably create a deeper erosion this time, it was asserted there was \"a safety factor of three.\"\n\nThis is a strange use of the engineer's term \"safety factor.\" If a bridge is built to withstand a certain load without the beams permanently deforming, cracking, or breaking, it may be designed for the materials used to actually stand up under three times the load. This \"safety factor\" is to allow for uncertain excesses of load, or unknown extra loads, or weaknesses in the material that might have unexpected flaws, et cetera. But if the expected load comes on to the new bridge and a crack appears in a beam, this is a failure of the design. There was no safety factor at all, even though the bridge did not actually collapse because the crack only went one-third of the way through the beam. The O-rings of the solid rocket boosters were not designed to erode. Erosion was a clue that something was wrong. Erosion was not something from which safety could be inferred.\n\nThere was no way, without full understanding, that one could have confidence that conditions the next time might not produce erosion three times more severe than the time before. Nevertheless, officials fooled themselves into thinking they had such understanding and confidence, in spite of the peculiar variations from case to case. A mathematical model was made to calculate erosion. This was a model based not on physical understanding but on empirical curve fitting. Specifically, it was supposed that a stream of hot gas impinged on the O-ring material, and the heat was determined at the point of stagnation (so far, with reasonable physical, thermodynamical laws). But to determine  how much rubber eroded, it was assumed that the erosion varied as the .58 power of heat, the .58 being determined by a nearest fit. At any rate, adjusting some other numbers, it was determined that the model agreed with the erosion (to a depth of one-third the radius of the ring). There is nothing so wrong with this analysis as believing the answer! Uncertainties appear everywhere in the model. How strong the gas stream might be was unpredictable; it depended on holes formed in the putty. Blowby showed that the ring might fail, even though it was only partially eroded. The empirical formula was known to be uncertain, for the curve did not go directly through the very data points by which it was determined. There was a cloud of points, some twice above and some twice below the fitted curve, so erosions twice those predicted were reasonable from that cause alone. Similar uncertainties surrounded the other constants in the formula, et cetera, et cetera. When using a mathematical model, careful attention must be given to the uncertainties in the model.\n\n_Space Shuttle Main Engines (SSME)_\n\nDuring the flight of the 51-L the three space shuttle main engines all worked perfectly, even beginning to shut down in the last moments as the fuel supply began to fail. The question arises, however, as to whether\u2014had the engines failed, and we were to investigate them in as much detail as we did the solid rocket booster\u2014we would find a similar lack of attention to faults and deteriorating safety criteria. In other words, were the organization weaknesses that contributed to the accident confined to the solid rocket booster sector, or were they a more general characteristic of NASA? To that end the space shuttle main engines and the avionics were both investigated. No similar study of the orbiter or the external tank was made.\n\nThe engine is a much more complicated structure than the solid rocket booster, and a great deal more detailed engineering goes into it. Generally, the engineering seems to be of high quality, and apparently considerable attention is paid to deficiencies and faults found in engine operation.\n\nThe usual way that such engines are designed (for military or civilian aircraft) may be called the component system, or bottom-up design. First it is necessary to thoroughly understand the properties and limitations of the materials to be used (turbine blades, for example), and tests are begun in experimental rigs to determine those. With this knowledge, larger component parts (such as bearings) are designed and tested individually. As deficiencies and design errors are noted they are corrected and verified with further testing. Since one tests only parts at a time, these tests and modifications are not overly expensive. Finally one works up to the final design of the entire engine, to the necessary specifications. There is a good chance, by this time, that the engine will generally succeed, or that any failures are easily isolated and analyzed because the failure modes, limitations of materials, et cetera, are so well understood. There is a very good chance that the modifications to get around final difficulties in the engine are not very hard to make, for most of the serious problems have already been discovered and dealt with in the earlier, less expensive stages of the process.\n\nThe space shuttle main engine was handled in a different manner\u2014top down, we might say. The engine was designed and put together all at once with relatively little detailed preliminary study of the materials and components. But now, when troubles are found in bearings, turbine blades, coolant pipes, et cetera, it is more expensive and difficult to discover the causes and make changes. For example, cracks have been found in the turbine blades of the high-pressure oxygen turbopump. Are they caused by flaws in the material, the effect of the oxygen atmosphere on the properties of the material, the thermal stresses of startup or shutdown, the vibration and stresses of steady running, or mainly at some resonance at certain speeds, or something else? How long can we run from crack initiation to crack failure, and how does this depend on power level? Using the completed engine as a test bed to resolve such questions is extremely expensive. One does not wish to lose entire engines in order to find out where and how failure occurs. Yet, an accurate knowledge of this information is essential to acquiring a confidence in the engine reliability in use. Without detailed understanding, confidence cannot be attained.\n\nA further disadvantage of the top-down method is that if an understanding of a fault is obtained, a simple fix\u2014such as a new shape for the turbine housing\u2014may be impossible to implement without a redesign of the entire engine.\n\nThe space shuttle main engine is a very remarkable machine. It has a greater ratio of thrust to weight than any previous engine. It is built at the edge of\u2014sometimes outside of\u2014previous engineering experience. Therefore, as expected, many different kinds of flaws and difficulties have turned up. Because, unfortunately, it was built in a top-down manner, the flaws are difficult to find and to fix. The design aim of an engine lifetime of 55 mission equivalents (27,000 seconds of operation, either in missions of 500 seconds each or on a test stand) has not been obtained. The engine now requires very frequent maintenance and replacement of important parts such as turbopumps, bearings, sheet metal housings, et cetera. The high-pressure fuel turbopump had to be replaced every three or four mission equivalents (although this may have been fixed, now) and the high-pressure oxygen turbopump every five or six. This was, at most, 10 percent of the original design specifications. But our main concern here is the determination of reliability.\n\nIn a total of 250,000 seconds of operation, the main engines have failed seriously perhaps 16 times. Engineers pay close attention to these failings and try to remedy them as quickly as possible by test studies on special rigs experimentally designed for the flaw in question, by careful inspection of the engine for suggestive clues (like cracks), and by considerable study and analysis. In this way, in spite of the difficulties of top-down design, through hard work many of the problems have apparently been solved.\n\nA list of some of the problems (and their status) follows:\n\nTurbine blade cracks in high-pressure fuel turbopumps (HPFTP). (May have been solved.)\n\nTurbine blade cracks in high-pressure oxygen fuel turbopumps (HPOTP). (Not solved.)\n\nAugmented spark igniter (ASI) line rupture. (Probably solved.)\n\nPurge check valve failure. (Probably solved.)\n\nASI chamber erosion. (Probably solved.)\n\nHPFTP turbine sheet metal cracking. (Probably solved.)\n\nHPFTP coolant liner failure. (Probably solved.)\n\nMain combustion chamber outlet elbow failure. (Probably solved.)\n\nMain combustion chamber inlet elbow weld offset. (Probably solved.)\n\nHPOTP subsynchronous whirl. (Probably solved.)\n\nFlight acceleration safety cutoff system (partial failure in a redundant system). (Probably solved.)\n\nBearing spalling. (Partially solved.)\n\nA vibration at 4000 hertz making some engines inoperable. (Not solved.)\n\nMany of these apparently solved problems were the early difficulties of a new design: 13 of them occurred in the first 125,000 seconds and only 3 in the second 125,000 seconds. Naturally, one can never be sure that all the bugs are out; for some, the fix may not have addressed the true cause. Thus it is not unreasonable to guess there may be at least one surprise in the next 250,000 seconds, a probability of 1\/500 per engine per mission. On a mission there are three engines, but it is possible that some accidents would be self-contained and affect only one engine. (The shuttle can abort its mission with only two engines.) Therefore, let us say that the unknown surprises do not, in and of themselves, permit us to guess that the probability of mission failure due to the space shuttle main engines is less than 1\/500. To this we must add the chance of failure from known, but as yet unsolved, problems. These we discuss below.\n\n(Engineers at Rocketdyne, the manufacturer, estimate the total probability as 1\/10,000. Engineers at Marshall estimate it as 1\/300, while NASA management, to whom these engineers report, claims it is 1\/10,000. An independent engineer consulting for NASA thought 1 or 2 per 100 a reasonable estimate.)\n\nThe history of the certification principles for these engines is confusing and difficult to explain. Initially the rule seems to have been that two sample engines must each have had twice the time operating without failure, as the operating time of the engine to be certified (rule of 2 _x_ ). At least that is the FAA practice, and NASA seems to have adopted it originally, expecting the certified time to be 10 missions (hence 20 missions for each sample). Obviously, the best engines to use for comparison would be those of greatest total operating time (flight plus test), the so-called fleet leaders. But what if a third sample engine and several others fail in a short time? Surely we will not be safe because two were unusual in lasting longer. The short time might be more representative of the real possibilities, and in the spirit of the safety factor of 2, we should only operate at half the time of the short-lived samples.\n\nThe slow shift toward a decreasing safety factor can be seen in many examples. We take that of the HPFTP turbine blades. First of all the idea of testing an entire engine was abandoned. Each engine has had many important parts (such as the turbopumps themselves) replaced at frequent intervals, so the rule of _2x_ must be shifted from engines to components. Thus we accept an HPFTP for a given certification time if two samples have each run successfully for twice that time (and, of course, as a practical matter, no longer insisting that this time be as long as 10 missions). But what is \"successfully\"? The FAA calls a turbine blade crack a failure, in order to really provide a safety factor greater than 2 in practice. There is some time that an engine can run between the time a crack originally starts and the time it has grown large enough to fracture. (The FAA is contemplating new rules that take this extra safety time into account, but will accept them only if it is very carefully analyzed through known models within a known range of experience and with materials thoroughly tested. None of these conditions applies to the space shuttle main engines.)\n\nCracks were found in many second-stage HPFTP turbine blades. In one case three were found after 1900 seconds, while in another they were not found after 4200 seconds, although usually these longer runs showed cracks. To follow this story further we must realize that the stress depends a great deal on the power level. The _Challenger_ flight, as well as previous flights, was at a level called 104 percent of rated power during most of the time the engines were operating. Judging from some material data, it is supposed that at 104 percent of rated power, the time to crack is about twice that at 109 percent, or full power level (FPL). Future flights were to be at 109 percent because of heavier payloads, and many tests were made at this level. Therefore, dividing time at 104 percent by 2, we obtain units called equivalent full power level (EFPL). (Obviously, some uncertainty is introduced by that, but it has not been studied.) The earliest cracks mentioned above occurred at 1375 seconds EFPL.\n\nNow the certification rule becomes \"limit all second-stage blades to a maximum of 1375 seconds EFPL.\" If one objects that the safety factor of 2 is lost, it is pointed out that the one turbine ran for 3800 seconds EFPL without cracks, and half of this is 1900 so we are being more conservative. We have fooled ourselves in three ways. First, we have only one sample, and it is not the fleet leader: the other two samples of 3800 or more seconds EFPL had 17 cracked blades between them. (There are 59 blades in the engine.) Next, we have abandoned the _2x_ rule and substituted equal time (1375). And finally, the 1375 is where a crack was discovered. We can say that no crack had been found below 1375, but the last time we looked and saw no cracks was 1100 seconds EFPL. We do not know when the crack formed between these times. For example, cracks may have been formed at 1150 seconds EFPL. (Approximately two-thirds of the blade sets tested in excess of 1375 seconds EFPL had cracks. Some recent experiments have, indeed, shown cracks as early as 1150 seconds.) It was important to keep the number high, for the shuttle had to fly its engines very close to their limit by the time the flight was over.\n\nFinally, it is claimed that the criteria have not been abandoned, and that the system is safe, by giving up the FAA convention that there should be no cracks, and by considering only a completely fractured blade a failure. With this definition no engine has yet failed. The idea is that since there is sufficient time for a crack to grow to fracture, we can ensure that all is safe by inspecting all blades for cracks. If cracks are found, replace the blades, and if none are found, we have enough time for a safe mission. Thus, it is claimed, the crack problem is no longer a flight safety problem, but merely a maintenance problem.\n\nThis may in fact be true. But how well do we know that cracks always grow slowly enough so that no fracture can occur in a mission? Three engines have run for long time periods with a few cracked blades (about 3000 seconds EFPL), with no blade actually breaking off.\n\nA fix for this cracking may have been found. By changing the blade shape, shot-peening the surface, and covering it with insulation to exclude thermal shock, the new blades have not cracked so far.\n\nA similar story appears in the history of certification of the HPOTP, but we shall not give the details here.\n\nIn summary, it is evident that the flight readiness reviews and certification rules show a deterioration in regard to some of the problems of the space shuttle main engines that is closely analogous to the deterioration seen in the rules for the solid rocket boosters.\n\n_Avionics_\n\nBy \"avionics\" is meant the computer system on the orbiter as well as its input sensors and output actuators. At first we will restrict ourselves to the computers proper, and not be concerned with the reliability of the input information from the sensors of temperature, pressure, et cetera; nor with whether the computer output is faithfully followed by the actuators of rocket firings, mechanical controls, displays to astronauts, et cetera.\n\nThe computing system is very elaborate, having over 250,000 lines of code. Among many other things it is responsible for the automatic control of the shuttle's entire ascent into orbit, and for the descent until the shuttle is well into the atmosphere (below Mach 1), once one button is pushed deciding the landing site desired. It would be possible to make the entire landing automatic. (The landing gear lowering signal is expressly left out of computer control, and must be provided by die pilot, ostensibly for safety reasons.) During orbital flight the computing system is used in the control of payloads, in the display of information to the astronauts, and in the exchange of information with the ground. It is evident that the safety of flight requires guaranteed accuracy of this elaborate system of computer hardware and software.\n\nIn brief, hardware reliability is ensured by having four essentially independent identical computer systems. Where possible, each sensor also has multiple copies\u2014usually four\u2014and each copy feeds all four of the computer lines. If the inputs from the sensors disagree, either a certain average or a majority selection is used as the effective input, depending on the circumstances. Since each computer sees all copies of the sensors, the inputs are the same, and because the algorithms used by each of the four computers are the same, the results in each computer should be identical at each step. From time to time they are compared, but because they might operate at slightly different speeds, a system of stopping and waiting at specified times is instituted before each comparison is made. If one of the computers disagrees or is too late in having its answer ready, the three which do agree are assumed to be correct and the errant computer is taken completely out of the system. If, now, another computer fails, as judged by the agreement of the other two, it is taken out of the system, and the rest of the flight is canceled: descent to the landing site is instituted, controlled by the two remaining computers. It is seen that this is a redundant system since the failure of only one computer does not affect the mission. Finally, as an extra feature of safety, there is a fifth independent computer, whose memory is loaded with only the programs for ascent and descent, and which is capable of controlling the descent if there is a failure of more than two of the computers of the main line of four.\n\nThere is not enough room in the memory of the mainline computers for all the programs of ascent, descent, and payload programs in flight, so the memory is loaded by the astronauts about four times from tapes.\n\nBecause of the enormous effort required to replace the software for such an elaborate system and to check out a new system, no change in the hardware has been made since the shuttle transportation system began about fifteen years ago. The actual hardware is obsolete\u2014for example, the memories are of the old ferrite-core type. It is becoming more difficult to find manufacturers to supply such old-fashioned computers that are reliable and of high enough quality. Modern computers are much more reliable, and they run much faster. This simplifies circuits and allows more to be done. Today's computers would not require so much loading from tapes, for their memories are much larger.\n\nThe software is checked very carefully in a bottom-up fashion. First, each new line of code is checked; then sections of code (modules) with special functions are verified. The scope is increased step by step until the new changes are incorporated into a complete system and checked. This complete output is considered the final product, newly released. But working completely independently is a verification group that takes an adversary attitude to the software development group and tests the software as if it were a customer of the delivered product. There is additional verification in using the new programs in simulators, et cetera. An error during this stage of verification testing is considered very serious, and its origin is studied very carefully to avoid such mistakes in the future. Such inexperienced errors have been found only about six times in all the programming and program changing (for new or altered payloads) that has been done. The principle followed is: all this verification is not an aspect of program safety; it is a test of that safety in a noncatastrophic verification. Flight safety is to be judged solely on how well the programs do in the verified tests. A failure here generates considerable concern.\n\nTo summarize, then, the computer software checking system is of highest quality. There appears to be no process of gradually fooling oneself while degrading standards, the process so characteristic of the solid rocket booster and space shuttle main engine safety systems. To be sure, there have been recent suggestions by management to curtail such elaborate and expensive tests as being unnecessary at this late date in shuttle history. Such suggestions must be resisted, for they do not appreciate the mutual subtle influences and sources of error generated by even small program changes in one part of a program on another. There are perpetual requests for program changes as new pay-loads and new demands and modifications are suggested by the users. Changes are expensive because they require extensive testing. The proper way to save money is to curtail the number of requested changes, not the quality of testing for each.\n\nOne might add that the elaborate system could be very much improved by modern hardware and programming techniques. Any outside competition would have all the advantages of starting over. Whether modern hardware is a good idea for NASA should be carefully considered now.\n\nFinally, returning to the sensors and actuators of the avionics system, we find that the attitude toward system failure and reliability is not nearly as good as for the computer system. For example, a difficulty was found with certain temperature sensors sometimes failing. Yet eighteen months later the same sensors were still being used, still sometimes failing, until a launch had to be scrubbed because two of them failed at the same time. Even on a succeeding flight this unreliable sensor was used again. And reaction control systems, the rocket jets used for reorienting and control in flight, still are somewhat unreliable. There is considerable redundancy, but also a long history of failures, none of which has yet been extensive enough to seriously affect a flight. The action of the jets is checked by sensors: if a jet fails to fire, the computers choose another jet to fire. But they are not designed to fail, and the problem should be solved.\n\n_Conclusions_\n\nIf a reasonable launch schedule is to be maintained, engineering often cannot be done fast enough to keep up with the expectations of the originally conservative certification criteria designed to guarantee a very safe vehicle. In such situations, safety criteria are altered subtly\u2014and with often apparently logical arguments\u2014so that flights can still be certified in time. The shuttle therefore flies in a relatively unsafe condition, with a chance of failure on the order of a percent. (It is difficult to be more accurate.)\n\nOfficial management, on the other hand, claims to believe the probability of failure is a thousand times less. One reason for this may be an attempt to assure the government of NASA's perfection and success in order to ensure the supply of funds. The other may be that they sincerely believe it to be true, demonstrating an almost incredible lack of communication between the managers and their working engineers.\n\nIn any event, this has had very unfortunate consequences, the most serious of which is to encourage ordinary citizens to fly in such a dangerous machine\u2014as if it had attained the safety of an ordinary airliner. The astronauts, like test pilots, should know their risks, and we honor them for their courage. Who can doubt that McAuliffe* was equally a person of great courage, who was closer to an awareness of the true risks than NASA management would have us believe?\n\nLet us make recommendations to ensure that NASA\n\nofficials deal in a world of reality, understanding technological weaknesses and imperfections well enough to be actively trying to eliminate them. They must live in a world of reality in comparing the costs and utility of the shuttle to other methods of entering space. And they must be realistic in making contracts and in estimating the costs and difficulties of each project. Only realistic flight schedules should be proposed\u2014schedules that have a reasonable chance of being met. If in this way the government would not support NASA, then so be it. NASA owes it to the citizens from whom it asks support to be frank, honest, and informative, so that these citizens can make the wisest decisions for the use of their limited resources.\n\nFor a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.\n*Leighton's note: The version printed as Appendix F in the commission report does not appear to have been edited, so I took it upon myself to smooth it out a little bit.\n\n*Note for foreign readers: Christa McAuliffe, a schoolteacher, was to have been the first ordinary citizen in space\u2014a symbol of the nation's commitment to education, and of the shuttle's safety.\n\n## EPILOGUE\n\n### Preface\n\n_WHEN_ I was younger, I thought science would make good things for everybody. It was obviously useful; it was good. During the war I worked on the atomic bomb. This result of science was obviously a very serious matter: it represented the destruction of people.\n\nAfter the war I was very worried about the bomb. I didn't know what the future was going to look like, and I certainly wasn't anywhere near sure that we would last until now. Therefore one question was\u2014is there some evil involved in science?\n\nPut another way\u2014what is the value of the science I had dedicated myself to\u2014the thing I loved\u2014when I saw what terrible things it could do? It was a question I had to answer.\n\n\"The Value of Science\" is a kind of report, if you will, on many of the thoughts that came to me when I tried to answer that question.\n\nRichard Feynman\n\n### The Value of Science*\n\n_FROM_ time to time people suggest to me that scientists ought to give more consideration to social problems\u2014especially that they should be more responsible in considering the impact of science on society. It seems to be generally believed that if the scientists would only look at these very difficult social problems and not spend so much time fooling with less vital scientific ones, great success would come of it.\n\nIt seems to me that we _do_ think about these problems from time to time, but we don't put a full-time effort into them\u2014the reasons being that we know we don't have any magic formula for solving social problems, that social problems are very much harder than scientific ones, and that we usually don't get anywhere when we do think about them.\n\nI believe that a scientist looking at nonscientific problems is just as dumb as the next guy\u2014and when he talks about a nonscientific matter, he sounds as naive as anyone untrained in the matter. Since the question of the value of science is _not_ a scientific subject, this talk is dedicated to proving my point\u2014by example.\n\nThe first way in which science is of value is familiar to everyone. It is that scientific knowledge enables us to do all kinds of things and to make all kinds\n\nof things. Of course if we make _good_ things, it is not only to the credit of science; it is also to the credit of the moral choice which led us to good work. Scientific knowledge is an enabling power to do either good or bad\u2014but it does not carry instructions on how to use it. Such power has evident value\u2014even though the power may be negated by what one does with it.\n\nI learned a way of expressing this common human problem on a trip to Honolulu. In a Buddhist temple there, the man in charge explained a little bit about the Buddhist religion for tourists, and then ended his talk by telling them he had something to say to them that they would _never_ forget\u2014and I have never forgotten it. It was a proverb of the Buddhist religion:\n\nTo every man is given the key to the gates of heaven; the same key opens the gates of hell.\n\nWhat then, is the value of the key to heaven? It is true that if we lack clear instructions that enable us to determine which is the gate to heaven and which the gate to hell, the key may be a dangerous object to use.\n\nBut the key obviously has value: how can we enter heaven without it?\n\nInstructions would be of no value without the key. So it is evident that, in spite of the fact that it could produce enormous horror in the world, science is of value because it _can_ produce _something_.\n\nAnother value of science is the fun called intellectual enjoyment which some people get from reading and learning and thinking about it, and which others get from working in it. This is an important point, one which is not considered enough by those who tell us it is our social responsibility to reflect on the impact of science on society.\n\nIs this mere personal enjoyment of value to society as a whole? No! But it is also a responsibility to consider the aim of society itself. Is it to arrange matters so that people can enjoy things? If so, then the enjoyment of science is as important as anything else.\n\nBut I would like _not_ to underestimate the value of the world view which is the result of scientific effort. We have been led to imagine all sorts of things infinitely more marvelous than the imaginings of poets and dreamers of the past. It shows that the imagination of nature is far, far greater than the imagination of man. For instance, how much more remarkable it is for us all to be stuck\u2014half of us upside down\u2014by a mysterious attraction to a spinning ball that has been swinging in space for billions of years than to be carried on the back of an elephant supported on a tortoise swimming in a bottomless sea.\n\nI have thought about these things so many times alone that I hope you will excuse me if I remind you of this type of thought that I am sure many of you have had, which no one could ever have had in the past because people then didn't have the information we have about the world today.\n\nFor instance, I stand at the seashore, alone, and start to think.\n\nThere are the rushing waves\n\nmountains of molecules\n\neach stupidly minding its own business\n\ntrillions apart\n\nyet forming white surf in unison.\n\nAges on ages\n\nbefore any eyes could see\n\nyear after year\n\nthunderously pounding the shore as now.\n\nFor whom, for what?\n\nOn a dead planet\n\nwith no life to entertain.\n\nNever at rest\n\ntortured by energy\n\nwasted prodigiously by the sun\n\npoured into space.\n\nA mite makes the sea roar.\n\nDeep in the sea\n\nall molecules repeat\n\nthe patterns of one another\n\ntill complex new ones are formed.\n\nThey make others like themselves\n\nand a new dance starts.\n\nGrowing in size and complexity\n\nliving things\n\nmasses of atoms\n\nDNA, protein\n\ndancing a pattern ever more intricate.\n\nOut of the cradle\n\nonto dry land\n\nhere it is\n\nstanding:\n\natoms with consciousness;\n\nmatter with curiosity.\n\nStands at the sea,\n\nwonders at wondering: I\n\na universe of atoms\n\nan atom in the universe.\n\nThe same thrill, the same awe and mystery, comes again and again when we look at any question deeply enough. With more knowledge comes a deeper, more wonderful mystery, luring one on to penetrate deeper still. Never concerned that the answer may prove disappointing, with pleasure and confidence we turn over each new stone to find unimagined strangeness leading on to more wonderful questions and mysteries\u2014certainly a grand adventure!\n\nIt is true that few unscientific people have this particular type of religious experience. Our poets do not write about it; our artists do not try to portray this remarkable thing. I don't know why. Is no one inspired by our present picture of the universe? This value of science remains unsung by singers: you are reduced to hearing not a song or poem, but an evening lecture about it. This is not yet a scientific age.\n\nPerhaps one of the reasons for this silence is that you have to know how to read the music. For instance, the scientific article may say, \"The radioactive phosphorus content of the cerebrum of the rat decreases to one-half in a period of two weeks.\" Now what does that mean?\n\nIt means that phosphorus that is in the brain of a rat\u2014and also in mine, and yours\u2014is not the same phosphorus as it was two weeks ago. It means the atoms that are in the brain are being replaced: the ones that were there before have gone away.\n\nSo what is this mind of ours: what are these atoms with consciousness? Last week's potatoes! They now can _remember_ what was going on in my mind a year ago\u2014a mind which has long ago been replaced.\n\nTo note that the thing I call my individuality is only a pattern or dance, _that_ is what it means when one discovers how long it takes for the atoms of the brain to be replaced by other atoms. The atoms come into my brain, dance a dance, and then go out\u2014there are always new atoms, but always doing the same dance, remembering what the dance was yesterday.\n\nWhen we read about this in the newspaper, it says \"Scientists say this discovery may have importance in the search for a cure for cancer.\" The paper is only interested in the use of the idea, not the idea itself. Hardly anyone can understand the importance of an idea, it is so remarkable. Except that, possibly, some children catch on. And when a child catches on to an idea like that, we have a scientist. It is too late* for them to get the spirit when they are in our universities, so we must attempt to explain these ideas to children.\n\nI would now like to turn to a third value that science has. It is a little less direct, but not much. The scientist has a lot of experience with ignorance and doubt and uncertainty, and this experience is of very great importance, I think. When a scientist doesn't know the answer to a problem, he is ignorant. When he has a hunch as to what the result is, he is uncertain. And when he is pretty darn sure of what the result is going to be, he is still in some doubt. We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degrees of certainty\u2014some most unsure, some nearly sure, but none _absolutely_ certain.\n\nNow, we scientists are used to this, and we take it for granted that it is perfectly consistent to be unsure, that it is possible to live and _not_ know. But I don't know whether everyone realizes this is true. Our freedom to doubt was born out of a struggle against authority in the early days of science. It was a very deep and strong struggle: permit us to question\u2014to doubt\u2014to not be sure. I think that it is important that we do not forget this struggle and thus perhaps lose what we have gained. Herein lies a responsibility to society.\n\nWe are all sad when we think of the wondrous potentialities human beings seem to have, as contrasted with their small accomplishments. Again and again people have thought that we could do much better. Those of the past saw in the nightmare of their times a dream for the future. We, of _their_ future, see that their dreams, in certain ways surpassed, have in many ways remained dreams. The hopes for the future today are, in good share, those of yesterday.\n\nIt was once thought that the possibilities people had were not developed because most of the people were ignorant. With universal education, could all men be Voltaires? Bad can be taught at least as efficiently as good. Education is a strong force, but for either good or evil.\n\nCommunications between nations must promote understanding\u2014so went another dream. But the machines of communication can be manipulated. What is communicated can be truth or lie. Communication is a strong force, but also for either good or evil.\n\nThe applied sciences should free men of material problems at least. Medicine controls diseases. And the record here seems all to the good. Yet there are some patiently working today to create great plagues and poisons for use in warfare tomorrow.\n\nNearly everyone dislikes war. Our dream today is peace. In peace, man can develop best the enormous possibilities he seems to have. But maybe future men will find that peace, too, can be good and bad. Perhaps peaceful men will drink out of boredom. Then perhaps drink will become the great problem which seems to keep man from getting all he thinks he should out of his abilities.\n\nClearly, peace is a great force\u2014as are sobriety, material power, communication, education, honesty, and the ideals of many dreamers. We have more of these forces to control than did the ancients. And maybe we are doing a little better than most of them could do. But what we ought to be able to do seems gigantic compared with our confused accomplishments.\n\nWhy is this? Why can't we conquer ourselves?\n\nBecause we find that even great forces and abilities do not seem to carry with them clear instructions on how to use them. As an example, the great accumulation of understanding as to how the physical world behaves only convinces one that this behavior seems to have a kind of meaninglessness. The sciences do not directly teach good and bad.\n\nThrough all ages of our past, people have tried to fathom the meaning of life. They have realized that if some direction or meaning could be given to our actions, great human forces would be unleashed. So, very many answers have been given to the question of the meaning of it all. But the answers have been of all different sorts, and the proponents of one answer have looked with horror at the actions of the believers in another\u2014horror, because from a disagreeing point of view all the great potentialities of the race are channeled into a false and confining blind alley. In fact, it is from the history of the enormous monstrosities created by false belief that philosophers have realized the apparently infinite and wondrous capacities of human beings. The dream is to find the open channel.\n\nWhat, then, is the meaning of it all? What can we say to dispel the mystery of existence?\n\nIf we take everything into account\u2014not only what the ancients knew, but all of what we know today that they didn't know\u2014then I think we must frankly admit that _we do not know_.\n\nBut, in admitting this, we have probably found the open channel.\n\nThis is not a new idea; this is the idea of the age of reason. This is the philosophy that guided the men who made the democracy that we live under. The idea that no one really knew how to run a government led to the idea that we should arrange a system by which new ideas could be developed, tried out, and tossed out if necessary, with more new ideas brought in\u2014a trial-and-error system. This method was a result of the fact that science was already showing itself to be a successful venture at the end of the eighteenth century. Even then it was clear to socially minded people that the openness of possibilities was an opportunity, and that doubt and discussion were essential to progress into the unknown. If we want to solve a problem that we have never solved before, we must leave the door to the unknown ajar.\n\nWe are at the very beginning of time for the human race. It is not unreasonable that we grapple with problems. But there are tens of thousands of years in the future. Our responsibility is to do what we can, learn what we can, improve the solutions, and pass them on. It is our responsibility to leave the people of the future a free hand. In the impetuous youth of humanity, we can make grave errors that can stunt our growth for a long time. This we will do if we say we have the answers now, so young and ignorant as we are. If we suppress all discussion, all criticism, proclaiming \"This is the answer, my friends; man is saved!\" we will doom humanity for a long time to the chains of authority, confined to the limits of our present imagination. It has been done so many times before.\n\nIt is our responsibility as scientists, knowing the great progress which comes from a satisfactory philosophy of ignorance, the great progress which is the fruit of freedom of thought, to proclaim the value of this freedom; to teach how doubt is not to be feared but welcomed and discussed; and to demand this freedom as our duty to all coming generations.\n*A public address given at the 1955 autumn meeting of the National Academy of Sciences.\n\n*I would now say, \"It is late\u2014although not too late\u2014for them to get the spirit...\"\n\n## Index\n\nacceleration safety cutoff system, flaws in,\n\nAccident Analysis group,\n\naccident investigation, technique for, 128\u201329\n\nAcheson, David C., , ,  ,\n\nAcropolis,\n\nactuators, computer systems, reliability,\n\nadolescent years, 20\u201332\n\nage of reason,\n\nAir and Space Museum, National, 138\u201339\n\nAjzenberg, Fay 73\u201374\n\nAldrich, Arnold D.,\n\nAlvarez, Luis,\n\nAmerican Association of Physics Teachers, ,\n\nApollo accident,\n\narcheology, Greek, 94\u201395\n\nArmstrong, Neil, , , , , , ,\n\nassembly crews\n\nmanagement view of, 170\u201371\n\nworkers' views of job, 171\u201375\n\n_Atlas_ ,\n\nAtoms for Peace Conference, ,\n\naugmented spark igniter (ASI), flaws in,\n\nAugsberry, Dr. (high school mathematics teacher),\n\nauthority\n\nand form,\n\nquestioning, 28\u201329\n\nand values,\n\n_Aviation Week and Space Technology_ ,\n\nBacher, Robert (Bob),\n\nBaudoin, King of Belgium,\n\nbearing spalling,\n\nBeggs, James F. (administrator of NASA),\n\nBell Telephone Laboratories,\n\nBerkeley,\n\nBessel functions,\n\nBethe, Hans, ,\n\nBethe, Henry, , n\n\nBlondel, Andr\u00e9,\n\nblowby, O-ring, , , ,\n\nBohr, Niels,\n\nBoisjoly, Roger,\n\nbottom-up design, of engines,\n\nof shuttle software,\n\nBragg, William,\n\nBrussels (Belgium), 83\u201388\n\nbureaucracy, 90\u201391\n\nbureaucracy\n\npersonal reactions to, 116\u201317\n\nreport preparation, 146\u201347\n\nBurns, Jerry, ,\n\nCable News Network (CNN),\n\nCalifornia Institute of Technology, _see_ Caltech\n\ncalligraphy, Chinese, 47\u201350\n\nCaltech, , , , , , , ,\n\ncertification,\n\ncriteria, deteriorating, 220\u201321\n\nrules for,\n\ninitial,\n\n_Challenger_ accident, 113\u2013248\n\nChrysler Company, 33\u201334\n\nclevis, field joint,\n\nColumbia, ,\n\ncombustion chamber, flaws in,\n\ncommunication,\n\nas a bureaucratic tool, 214\u201315\n\nof management with engineers, 182\u201383\n\nwithin NASA, 212\u201313\n\ncomputer systems (avionics), 232\u201336\n\nconcepts, and applications, 244\u201345\n\nCook, Richard C., 140\u201341, , ,\n\nCornell University, , , ,\n\ncosts\n\nof modifying procedures,\n\nof replacing software,\n\nof top-down design, 226\u201327\n\nof updating shuttle computers, 192\u201393\n\ncounting\n\nand measuring time, 55\u201359\n\npatterns for, 57\u201358\n\nCovert, Eugene E., , , ,\n\nCrete,\n\ncross talk,\n\nculture, _see also_ tradition\n\nand values, 60\u201362\n\nDavies, Richard (Dick),\n\nDavis, B. K., 164\u201365\n\nDeborah Hospital,\n\ndemocracy and doubt,\n\nDescartes, Ren\u00e9, 28\u201329\n\nDesign, Development and Production panel,\n\ndiscrimination\n\nreligious,\n\nsexual, 41\u201342, 72\u201375\n\nand stereotypes,\n\n_Disturbing the Universe_ (Dyson), n\n\ndocumentation of shuttle performance,\n\ndoubt,\n\ndreaming, ,\n\nDyson, Freeman, n\n\nletters of, 97\u2013100\n\nEddington, Arthur,\n\neducation and culture, 60\u201362\n\nenabling power of science,\n\n_Encyclopaedia Britannica,_ ,\n\nengineering judgment, n,\n\nestimates of engine failure probability,\n\nestimates of shuttle failure probability, 182\u201383, 187\u201388\n\nEratosthenes,\n\nerosion, O-ring, f, , , f, , , 224\u201325\n\nesthetics\n\nand reason,\n\nscientific aspects of,\n\nand scientific information, 242\u201343\n\nexecutive order, defining work of presidential commission,\n\nexternal tank (ET),\n\nFAA\n\ncertification practices,\n\ncertification trial success,\n\ncriteria for success,\n\nsafety rules,\n\nFabriola, Queen of Belgium,\n\nfamily\n\nCarl (son), n, ,\n\nChuck (nephew), ,\n\nfather-in-law (Arlene's father),\n\nFrances (cousin), , ,\n\nJoan (sister), , 48\u201350, 73\u201374, 203\u20134\n\nLucille (mother), ,\n\nMelville (father), 12\u201316\n\nMichelle (daughter), ,\n\nFederal Aviation Administration, _see_ FAA\n\nFermi, Enrico,\n\nferrite-core computers,\n\nFeynman, Arlene, , 28\u201330, 32\u201353\n\nFeynman, Carl, n, , , n\n\nFeynman, Gweneth, 76\u201382 n, , ,\n\nFeynman, Joan, n, 48\u201350, 73\u201374 203\u20134\n\nFeynman, Lucille, ,\n\nFeynman, Melville, 12\u201316\n\nFeynman, Michelle, , , n\n\n_Feynman Lectures m Physics,_\n\nFichtel (foreman, assembly crew), 172\u201375\n\nfield joint, , f, , f,\n\nmodel of, f\n\nflight readiness reviews,\n\nFuchs, Klaus,\n\nfuel tank, shuttle, f\n\n_Galileo_ (Jupiter probe),\n\nGast, Harold, , 30\u201332\n\nGell-Mann, Murray, ,\n\nGeneva (Switzerland), 63\u201368\n\nGoldschmidt, Herman, 69\u201371\n\nGraham, William R. (Bill) (acting administrator, NASA), , , , , , , , 138\u201339, , , n, ,\n\nGrand Hotel (Warsaw), 88\u201393\n\ngravity conference, 91\u201392\n\nGreece, 93\u201397\n\nHansen, Grant L. (editor),\n\nHeisenberg, Werner,\n\nHibbs, Albert R. (Al), ,\n\nhigh-frequency vibration, 184\u201385,\n\nhigh-pressure fuel lurbopump (HPFTP),\n\nreliability of,\n\nflaws in,\n\nhigh-pressure oxygen turbopump (HPOTP),\n\nflaws in,\n\nHollings, Ernest (Senator, SC), 175\u201376\n\nHotel Amigo,\n\nHotel City, 63\u201368\n\nHotz, Robert B., , , 169\u201370, , ,\n\nHughes Aircraft Company,\n\nIlliapoulos, Professor, 93\u201394\n\nIndependent Solid Rocket Motor Design Oversight Committee,\n\ninfinity,\n\nintellectual value of science,\n\nIran-Contra hearings, 212\u201313\n\nIseokitsu (Japan), 77\u201379\n\nJapan, 76\u201382\n\nJet Propulsion Laboratory, _see_ JPL\n\nJohnson Space Center, ,\n\njoint rotation, , f, f\n\nJPL, 119\u201322, n, , 177\u201378, ,\n\nJupiter probe,\n\nKapp, Jack, ,\n\nKeel, Alton G., Jr. (Al), , , , 204\u20135, , 217\u201318\n\nrecommendations of commission,\n\nreport distribution, ,\n\nKennedy, John F. (President), n\n\nKennedy Space Center, , , 159\u201376,\n\nKingsbury, James E.,\n\nKissinger, Henry,\n\nKiwi, Feynman dog, , ,\n\nknowledge\n\nkinds of,\n\nobserving multiple mental activities, 58\u201359\n\nKutyna, General Donald J., , , , 128\u201329, 139\u201340, ,\n\nKutyna _(continued)_ 151, 155\u201356, , , 194\u201395, 204\u20135, ,\n\nLamberth (manager, rocket assembly), 170\u201371, ,\n\nLaunch Abort Safety Panel (LASP),\n\nLawrence High School (Nassau County),\n\nleak test, n\n\nport, , f\n\nlearning\n\nabout esthetics,\n\ninternal processes,\n\npatterns in,\n\nand teaching, 101\u20132\n\nLee, Meemong,\n\nLeff, David, ,\n\nLehrer, James,\n\nLeMa\u00eetre, Abb\u00e9 George \u00c9duoard,\n\nletters\n\nDyson's about Feynman, 97\u2013102\n\nto Gweneth,\n\nfrom Athens, 93\u201397\n\nfrom Brussels, 83\u201388\n\nfrom Warsaw, 88\u201393\n\nto Gweneth from Henry Bethe, 100\u2013103\n\nLewis, Sinclair,\n\nLifer, Charles E. (JPL),\n\nliquid hydrogen (LH),\n\nliquid oxygen (LOX),\n\nLockheed, instructions for shuttle engines,\n\nLorenz, Konrad,\n\nLos Alamos,\n\nproblem solving at,\n\nLouvain University,\n\nLovingood, Judson A. (manager), , 181\u201383, n\n\nLund, Robert, , ,\n\nMcAuliffe, Christa, , , n\n\nMcDonald, Allan J., , 143\u201344,\n\nMacNeil\/Lehrer Newshour, ,\n\nmanagement\n\nestimates of shuttle reliability,\n\nview of assembly crews, 170\u201371\n\nManhattan Project,\n\nMarshall Space Center, ,\n\nengineering briefing, 187\u201388\n\nestimates of engine failure, ,\n\nMason, Gerald D.,\n\nMassachusetts Institute of Technology, _see_ MIT\n\nmemory, of shuttle computers, 190\u201391, 233\u201334\n\nMetropolis, Nicholas (Nick), ,\n\nMission Planning and Operations group,\n\nMIT, , , , ,\n\nM\u00f6bius strip, 29\u201330\n\nmodel\n\ncomputer, performance analysis of O-ring, ,\n\nof O-ring erosion,\n\nMoore, Jesse W.,\n\nMoore, Dr. Nicholas (JPL),\n\nMorton Thiokol Company, , , 134\u201335, , 162\u201363, , ,\n\nBoisjoly, Roger,\n\nBurns, Jerry, ,\n\nKapp,Jack, ,\n\nLund, Robert, , ,\n\nMacDonald, Allan J., , 143\u201344\n\nMason, Gerald D.,\n\nThompson, Arnie,\n\nMountain-forming day (Oberlin),\n\nMulloy, Lawrence B., , 149\u201351, , , ,\n\nNASA, , 132\u201338, 162\u201363,\n\nAldrich, Arnold D.,\n\ncertification rules, initial,\n\nCook, Richard C, 140\u201341, , ,\n\nDavis, B. K., 164\u201365\n\nestimates of engine failure, 179\u201380,\n\nFichtel (foreman, assembly crew), 172\u201375\n\nKingsbury, James E.,\n\nLamberth (manager, rocket assembly), 170\u201371, ,\n\nLovingood, Judson A. (manager), , 181\u201383, n\n\nMoore, Jesse W.,\n\nMulloy, Lawrence B., , 149\u201351, , ,\n\nStevenson, Charles G. (Charlie), , ,\n\nWeeks, L. Michael, 133\u201338\n\nNational Academy of Sciences, , n\n\nNational Aeronautics and Space Administration, _see_ NASA\n\n_New York Times,_ 139, ,\n\nNew Zealand lectures, , n\n\nNixon, Richard (President),\n\nNobel Prize, n, n\n\nNoto Peninsula (Japan), 79\u201382\n\nN-ray hoax,\n\nOberlin College,\n\nOffice of Management and Budget,\n\nOffice of Safety, Reliability, and Quality Assurance,\n\nOMB, Office of Management and Budget,\n\nOppenheimer, J. Robert, ,\n\nO-ring, , 133\u201335, f\n\nice-water demonstration, f\n\norigin of speculation about,\n\nperformance model,\n\nPalace of Culture and Science (Warsaw),\n\nParker Seal Company,\n\nParthenon,\n\npatterns, and counting, 57\u201358\n\npeace,\n\nPeierls, Rudolph,\n\nPerrin, J.,\n\nPhysical Society,\n\nplutonium, power supply,\n\npolitical pressure for launch, 157\u201358, 168\u201369,\n\nprecision and approximation,\n\nPre-launch Activities group,\n\npresidential commission, 113\u2013237\n\nconnections of members, 154\u201358\n\nfact finding by, 119\u201353\n\ninvestigative process, 159\u201376\n\nmembers of,\n\nrecommendations of, 199\u2013205\n\nreport\n\nformal presentation of, f\n\npreparation, 195\u201398\n\nworking groups of,\n\nPrinceton, , , , 54\u201355\n\nprobability, subjective, at NASA, 222\u201323\n\npurge check valve, flaws in,\n\n_QED: The Strange Theory of Light and Matter_ (Feynman), n\n\nradioactive thermal generator (RTG),\n\nreaction control systems, reliability of,\n\nrecommendations\n\npresidential commission, 199\u2013205\n\nseals report, f\n\nredundancy in shuttle computers, 191\u201392,\n\nreliability\n\nof high-pressure fuel turbopump,\n\nmanagement estimates of,\n\nOffice of Safety, Reliability, and Quality Assurance,\n\nof reaction control systems,\n\nof shuttle computer hardware,\n\nof sensors in shuttle systems,\n\nof shuttle, 220\u201337\n\nreligion\n\nBuddhist proverb,\n\nformal training (Feynman's), 25\u201328\n\nreligious experience in science, 243\u201344\n\nRide, Sally K., , n, , , , , , , ,\n\nRocketdyne, ,\n\nestimates of engine failure,\n\nRogers, William P., , 124\u201325, , , , , , 143\u201344, , , 154\u201355, 156\u201357, , , , 175\u201376, , , , , 210\u201311, , 215\u201316\n\nrecommendations of commission, 200\u2013202,\n\nRoyal Olympic Hotel (Athens),\n\nRummel, Robert W.,\n\nsafety boards,\n\nsafety factor\n\ndeteriorating,\n\nfor O-ring failure,\n\nscheduling\n\nand pressure to launch,\n\nand shuttle safety, 236\u201337\n\n_Scientific American, The_ , ,\n\nSears Roebuck and Company,\n\nsensors, shuttle systems, reliability of,\n\nshuttle\n\ncomputer operation of, 190\u201392\n\ndescription of,\n\nengineering briefing, JPL, 119\u201322\n\nengineering briefing for Feynman, 132\u201338\n\nengine specifications, 227\u201329\n\nfailure, probability of, 179\u201380\n\nflame from, 142f\u201343\n\nlaunch information, 161\u201362\n\nreliability of, 220\u201337\n\nresponsibility for engines,\n\nreworking components for reuse, 166\u201367\n\nsmoke from, f, 159\u201361, f\n\nShuttle Transportation System Safety Advisory Panel, 199\u2013200\n\nsimulator, computer checking on,\n\nsocial responsibility\n\nand ignorance,\n\nand scientific exploration,\n\nsoftware\n\nbottom-up design,\n\nfor shuttle computers, 192\u201394, 232\u201333\n\nverification of shuttle, 234\u201335\n\nsolid-fuel rocket boosters (SRBs), f, , 221\u201325\n\nSolomon, Jerry,\n\n_Sound and Symbol in Chinese_ ,\n\nspace shuttle main engines (SSMEs), , 225\u201332\n\nSpace Shuttle Program, Air Force,\n\nStapler, Robert,\n\nstereotypes, n,\n\nStevenson, Charles G. (Charlie), ,\n\nsubsynchronous whirl, ,\n\nsuccess, predicting, 223\u201324\n\nSutter, Joseph F., , ,\n\nTamm, Igor,\n\ntang, field joint,\n\nTartaglia, Niccolo,\n\ntemperature\n\neffect on O-rings,\n\nat launch pad,\n\nanalysis of, 165\u201366\n\nand O-ring incidents, f\n\nThinking Machines Company, n,\n\nThompson, Arnie,\n\nthought processes, in technicolor,\n\ntime sense, 55\u201359\n\n_Time_ ,\n\nTitan rocket, investigation of failure, 128\u201329\n\nTogi (Japan), 80\u201382\n\ntop-down design, _see also_ bottom-up design\n\nof shuttle main engine, 226\u201329\n\nof space shuttles,\n\ntradition\n\nGreek, 94\u201395\n\nIndian,\n\nJapanese, 80\u201381\n\n_Treasure Island_ ,\n\nTrinidad, 60\u201362\n\nTukey, John, 58\u201359\n\nUllian, Louis J., , 179\u201380\n\nUnited Nations (Geneva),\n\nUniversity of California at Berkeley,\n\n_see_ Berkeley\n\nUniversity of Kanazawa,\n\nUniversity of Kyoto,\n\nvalues\n\nand authority,\n\nand culture, 60\u201362\n\nin science,\n\nand uncertainty,\n\nand utility, Buddhist proverb,\n\nverification, of shuttle computer software, 234\u201335\n\nVoltaire,\n\nWalker, Arthur B. C, Jr., ,\n\nWalker, Bernie,\n\nWarren, Earl (Chief Justice), n\n\nWarren Commission Report,\n\nWarsaw (Poland),\n\n_Washington Post_ ,\n\nWeeks, L. Michael, 133\u201338\n\nWeibull distribution,\n\nWeisskopf, Victor,\n\nWheeler, John A., ,\n\nWheeler, Janet (Mrs. John A.),\n\nWheelon, Albert D.,\n\nwhistle, in shuttle engine,\n\nWigner, Eugene,\n\nWomen's Garment Workers Union,\n\nWood, Robert Williams,\n\nWoodward, William (Bill),\n\nwordsmithing, presidential commission report, 195\u201396\n\nYukawa, Hideki, , n\n\nzinc chromate putty, 120\u201322, f, n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"ACE H\u00f4tel Caen Nord Memorial is ideally located in a relaxing, natural setting only 10 minutes from the city centre.\nPerfect stop en-route to Disneyland Paris!\nACE H\u00f4tel Caen Nord Memorial is ideally located in a relaxing, natural setting near the M\u00e9morial de Caen and the Parc Floral de la Colline aux Oiseaux, only 10 minutes from the city centre.\nACE H\u00f4tel Caen Nord Memorial is easily accessible via the N 814, a ring road around Caen, and is the perfect starting point for visiting Normandy as well as family tours of France.\nBreakfast is served daily from 06:30 - 11:00.\nThere are facilities for disabled guests.\nCheck-in starts at 14:00 and check-out is until 12:00.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mads Peder Nordbo is Danish but has lived in Nuuk for several years. Born in 1970, he hold degrees in Literature, Communications and Philosophy from The University of Southern Denmark and the University of Stockholm. Mads has lived in Denmark, Sweden, Germany and Greenland. He works in communications at the Town Hall in Nuuk, where he amongst other things, writes for the mayor of the municipality, Kommuneqarfik Sermersooq, which stretches across the Greenland ice sheet. Mads Peder Nordbo is the author of three novels. THE GIRL WITHOUT SKIN is his debut as a crime writer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2019 4th International Conference on Engineering Design and Analysis (ICEDA 2019) aims to provide a forum for researchers, practitioners, and professionals from the industry, academia and government to discourse on research and development, professional practice in engineering design and analysis. ICEDA 2019 will be held in Bali, Indonesia during October 19-21, 2019.\nWelcome scholars and researchers working in the field of engineering design and analysis from all over the world to attend the conference and share your experiences and lessons with other enthusiasts, and develop opportunities for cooperation.\nICEDA is the co-organized by the International Journal of Engineering and Technology and International Journal of Engineering and Technology, so it also serves to bring authors and editors of IJET and IJSA together to communicate face to face and discuss the hot topics in the related area.\nChoice 1. Pulibcation in IJET.\nSubmissions will be reviewed by the conference committees and IJET editorial board, and accepted papers will be published in the International Journal of Engineering and Technology, which will be indexed by Chemical Abstracts Services (CAS), Google Scholar, ProQuest, EI (INSPEC, IET), etc.\nSubmissions will be reviewed by the conference committees and IJSA editorial board, and accepted papers will be published in the International Journal of Sustainable Aviation, which will be indexed by Emerging Sources Citation Index (ESCI), Asian Digital Library, cnpLINKer (CNPIEC), Google Scholar, ProQuest, etc.\nFor those who're NOT looking to publish their papers, it's acceptable to submit your abstracts to the conference simply for presentation without publication, which will be sent to at least two technical committees for a brief review.\nPlease submit your full paper\/abstract to us via the Electronic Submission System or through email iceda@iacsitp.com directly.\nAttendees are supposed to register in advance in order to participate in the conference. Please proceed to online registration in time and leave yourself plenty of time for preparation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Punctuality \u2013 Show your child you think school is important!\nArriving late for school gives your child an upsetting start to the day. Young children thrive on routine, and do not like to stand out from their peers. Having to join the class when everybody else is settled and ready to learn can embarrass and worry many children. Just five minutes late will mean they may have missed the opportunity to share important news or messages with their teacher, who is trying to introduce the learning for the day. A regular five minutes a day adds up to a whole three days a year! Ten minutes late and the most important learning for the day is often lost, children are arriving after tasks have been introduced, and are starting the day at a disadvantage.\nAlthough school starts at 9.00, we open our gates at 8.55a.m. and do not close them until 9.05a.m. Children who arrive after this time will have to enter school via the main reception and report to the school office. Our building is 100 years old and the front entrance is not very child-friendly. We are anxious that the front door remains closed as much as possible in the mornings whilst pupils are settling into their classes to ensure the safety of all pupils within the building. Late arrivals will wait together in the office or in the entrance with an adult and then be escorted to their classrooms together. Any child arriving in the classroom after the register has closed will unfortunately be registered as late. You will be able to see how many late marks your child has assigned to them when you log-in to our online reporting system. Please do not use the main front entrance when the playground entrances are still open, as this affects the security of the site for all.\nAttendance \u2013 Every day counts! Missing school is missing out!\nIrregular attendance can seriously disrupt continuity of learning, as lessons in school are designed to build on each other. Although staff work hard to try and make sure that any missing steps are re-taught, it can sometimes not be obvious for some time that a 'missing link' is holding back learning. In this way persistent absence can undermine educational progress and often leads to underachievement and low attainment. Another serious consequence of low attendance is that it impedes a child's ability to develop friendships within school. Making friends and building lasting relationships are vital in helping children to feel happy and confident in school. In this way, regular attendance supports children to reach their full potential.\nAttending school regularly not only gives children the chance of a better future, we also provide lots of incentives to reward children who attend school regularly and on time.\nChildren who achieve 100% attendance for a half-term will choose a special gift to mark their achievement.\nChildren who achieve over 97% attendance for a full year, September to July, will receive a certificate.\nChildren who achieve 100% attendance will receive a prize of an attendance bear.\nGood and improved attendance is rewarded in a tangible way with, for example, positive comments, certificates and stickers.\nA pair of attendance bears is presented during Star of the Week assembly for the class with the highest attendance that week.\nEvery term the class with the best attendance is allowed to choose a 'Golden Time' treat.\nWe understand that our young children are naturally going to be affected by first-time illnesses such as chicken pox. In addition, young children can be particularly susceptible to bugs and viruses, and easily pass germs around their friends! It would be impossible to expect that children will not miss sessions through illness, as this cannot be avoided, but we would ask that you please think seriously about the effects a much longer period of absence has before asking to take a family holiday during term-time.\nThe government expects attendance of at least 95%. A family holiday of ten days will reduce your child's attendance for the whole academic year to 94.73%, and this will obviously keep falling if your child has any further absence due to illness. The table below shows just how much everyday at school matters.\nIf your child is ill, we ask that you ring the school office on their first day of absence to let us know what is wrong. If your child is absent for more than five days, we will provide you with work to undertake at home, to help your child catch up on any lost learning.\nIf persistent absence becomes an issue that affects your child, then we would like to offer any help we can to ensure your child is able to improve their attendance. Our Parent Support Advisor, the Education Welfare Service and the headteacher are all committed to helping, and have the best interests of your family at heart. Please call in to school or contact the school by phone if you think we can be of any assistance.\nCurrent attendance details for your child will be discussed with you at each parent consultation meeting. This will enable you to keep track of how your child is doing.\nCurrent guidance from both the government and our local authority no longer allows us to be lenient when considering requests for leave of absence during term-time. The ability to authorise up to ten days per year has been removed; therefore leave of absence for family holidays or weddings is no longer a guaranteed entitlement. Leave of absence may only be granted in very special circumstances. School is in session for 190 days a year leaving parents a further 175 days a year in which to have days out together and take a holiday.\nIf it does not seem possible to authorise the term time absence, you may appeal the decision. A further form will be issued to you if this should occur.\nIf the absence is not authorised, but the leave is taken anyway, the case may be referred to the Education Welfare Service who may issue a Penalty Notice for \u00a3120 (or \u00a360 if paid within 28 days) to each parent for each child taken out of school.\nShow your child you think school is important!\nGoing to school unprepared can be a major worry for children \u2013 help by checking schoolbags the night before, packing P.E. bags and keeping an eye on homework progress.\nChildren can become unsettled if they have to go into school late and without their friends \u2013 getting children to school in time to meet friends in the playground prevents this.\nHave a memory board at home for special trips or activities \u2013 the board will help your children remember to tell you and help you remember to prepare them for it.\nWhen your child attends school on a regular basis, they take an important step towards reaching their full potential, and are given the greatest opportunity to learn new things and develop their skills.\nThe more time they spend around other children, whether in the classroom or as part of a school team or club, the more chance they have of making lots of friends and feeling included, boosting social skills, confidence and self esteem.\nHelping your child into good habits from an early age can help these carry through to adult and working life.\nRemember \u2013 if your child is absent for just one day a week, over the course of their school career they will miss two years of schooling!!!\nEvery day counts! Missing school is missing out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cecil Edwin Anschau (2900) was the son of Mr John Anschau(this was misspelled as Anschan in the Australian embarkation lists),who had moved from Dubbo to become Postmaster at Parramatta. He was born in Lismore, New South Wales and was 25 when he joined the forces on 22 February 1915. He served with the 5 Field Hospital before being transferred to the 3 General Hospital. His brother, Corporal Gilbert Goldie Anschau (244), served in the 1st Ambulance Corp at Gallipoli and was killed at Bullecourt, France, in 1917.\nThe whole world knows now that we have evacuated Gallipoli. The why and the wherefore I am bound by honor not to say, but I am proud to say that I was picked to be one of the last A.M.C. men to be left behind until the very end.\nThe retreat took a whole week to accomplish, and our Brigade (the 5th) was picked to hold the last posts. They are: Russel's Top, Pope 's Post, Quinn 's Post, Courtney 's Post, and Walker's Ridge. Our Brigade was split up among them, our section was made fall-in, and the Major told us in a few words what was going to take place, and that the 5th Brigade was to hold on and cover the retreat of the troops. That being the case, the A.M.C. had also to stay to look after the boys. He had picked out his men to fill the posts of honor. He then started to read the names of the men, and mine was among them. I was very proud, but I knew only too well tho job that had been left to us.\nHowever, nineteen there were of us, as we stood to attention, listening to all he had to say how proud he was and the responsibility that was laid on us, etc. I have been sad at times at what I have seen, but I can safely say that that ten minutes of lecture was the saddest of all my career as a soldier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuryq b/data_all_eng_slimpj/shuffled/split2/finalzzuryq
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+{"text":"I adore party nibbles and love the relaxed feel it gives a gathering, everyone standing around, usually in my kitchen, tasting lots of different things rather than a full on sit down appetizer to kick things off. I guess it's just like stretching before a run; it's a gentle warm up to the big meal ahead.\nI will admit that the idea of spending hours slaving over a hot stove whilst my guest sip merrily behind me is also not idea so party food that we can prep ahead wins every time. The recipe below fulfils my essential party food criteria: easy, fast to prepare when you need it and delicious.\nI was sent some of Hidden Valley's Tzatziki to try in my kitchen. I am always dubious of things in bottles, but this stuff is pretty darn delicious. It tastes rich, creamy and bright just as Tzatziki should do. It's the perfect thing to have in your fridge and the perfect shortcut when the party is calling and the guests need feeding!\nBeat the flour, egg, yogurt and 2 tbsp. water together in a large bowl until smooth. Stir in the chillies, spring onions, fennel and season with salt and pepper. Cover and chill until required.\nTo make the tzatziki: dry the cucumber and olives a little by pressing them between some kitchen roll gently. Stir the olives and cumber with the Hidden Valley Tzatziki.\nHeat a non-stick frying pan and fry tablespoon dollops of the batter for 1 min on each side, until lightly browned.\nSpoon a teaspoon of the tzatziki mix onto each pancake and sprinkle with some chopped parsley.\nThe chilli gives zing and the fennel gives a herbal hit in the pancakes, topped with the creamy tzatziki and it's a party winner.\nMAKE IN ADVANCE: The pancakes can be cooked in advance, cooled and simply re-heated in a hot oven for a few minutes before adding the tzatziki.\nI was sent some Hidden Valley Tzatziki to use for this recipe but recipes, photographs and graters are as always my very own.\nSCONE: IT'S ALL IN THE WAY YOU SAY IT!\nIF YOU DON'T NEED IT?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This set of 48 high quality memory picture cards features key characters from Disney' stunning new movie, moana. Featuring the characters princess moats and the demi-god maui. There are 24 different pairs to match up and collect in this classic game. The game play can be varied to suit different ages and abilities; for younger children use fewer cards, but you'll be surprised at how good youngsters are at this game! the memory cards are made from sturdy high quality FSC card, laminated on both sides with a colour print. This game is suitable for 2-6 players. Suitable for ages 3 years and up.\nFeatures favourite characters from Disneys' hit film Moana.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wanting to combat that new year new season flu bug?\nI've got just the tonic for you! It's made with a lot of powerful anti-inflammatory ingredients and I watched my very own parents drink it and feel better the very next day, without needing to pay the doctor a trip!\nIn her recipe, she used lemon, raw honey, turmeric, ginger, hot water and coconut oil.\nAll really splendid ingredients! Over the past two years I have experimented and tweaked some of it to include others: cinnamon instead of raw honey, lime instead of lemon, black peppercorns & coconut milk to add after in order to make it thick like a latte consistency instead.\nRaw Honey - It is a crude form that comes directly from the honeycomb cells. Raw Honey contains bee pollen which can help to fight allergies (just what we turn to this tonic for!) They also contain polyphenols which can act as antioxidants.\nTurmeric Powder\/ Fresh Turmeric Root - Used mainly in Ayurvedic and Indian cooking, it is a spice that can date back more than 4000 years. Turmeric contains curcumin which is a sort of biochemical compound called curcumoid that helps with immune system and acts as an antioxidant. It has been used mainly in Chinese medicine as well to treat abdominal pains and swelling. In addition, turmeric is more bioavailable to us when combined with fat (from the coconut oil here), black peppercorns and when heated.\nCoconut Oil - Contains medium chain triglycerides (MCT) which could potentially boost your cholesterol or HDL levels. There is a whole controversy of using coconut oil but for the sake of this recipe, let's stick to 2 tsps shall we?\nFreshly grated Ginger - probably not something new to try! Ginger has been around for thousand of years in a lot of cultures and they have an extensive micronutrient profile. (Ever had ginger tea after a massage? It is unbelievably good for detox and flushing out leftover toxins!) Ginger is also helpful for nausea if you have this symptom when you're feeling down.\nLemon\/Lime Juice - Rich in Vitamin C and could potentially help protect against cancer growth.\nCoconut milk - same as coconut oil but in a thicker form.\nFor example here, I have used the other ingredients in order to make this tonic.\nAnother variation of the tonic I used only the main ingredients of turmeric, honey, coconut oil and grated ginger.\nHere I have only used four of the listed ingredients to make the same tonic!\nTry it out and make sure to tag me on #nourishmelfoods and #nourishmel on instagram! Back then I was using my personal account @mellysparkles to showcase my recipes but since then have swapped into my business account @nourishmelhealth.\nI love finding ways of natural healing before I concede to going into a clinic to get some off the shelf meds. Unless your symptoms are pretty severe and you're unable to function on a day to day basis, chances are do give your body an opportunity to heal first! Our bodies are wonderful and they are not machines. When we take care of our bodies we have to think about the inner temples of emotions, spirit and soul too. I digress, That's for a separate blog post. If you are wondering more about an anti-inflammatory diet, here's a free guide on me to you personally that you can download right away or you can just a simple article on mindbodygreen to compare.\nGet seven simple recipes that you can make in your kitchen even if you are ill or not feeling all that up to it.\nI promise, making foods to cure yourself is far better than heading to your local drugstore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This suspended stone wool ceiling tile from Rockfon has a nuanced white painted fleece on the front and a coated fleece on the rear. It contributes to a healthy environment by preventing the growth and spread of bacteria. Provides coverage area of 5.76m\u00b2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Behav Processes. 2004 Jan 30;65(1):87-93.\nTool-use and tool-making by captive, group-living orangutans (Pongo pygmaeus abelii) at an artificial termite mound.\nLaboratory of Ethological Studies, Faculty of Human Sciences, Osaka University, Suita, Japan.\nThe present study examined the use and making of tools to obtain foodstuffs in artificial-mound holes by five captive, group-living Sumatran orangutans (Pongo pygmaeus abelii). Three adult orangutans frequently stripped leaves and twigs from a branch provided (tool-making), and then inserted the tool into a hole to obtain foodstuffs (tool-using). A 5-year-old female juvenile usually used the tools that adult orangutans had previously used, but rarely made tools herself. A 2-year-old male infant did not use any tools. The adult orangutans tend to leave one to several leaves at the top of the branch than to leave many leaves on the branch or to strip all leaves. It seemed likely that tools with appropriate leaves are easier to insert into holes and obtain more foodstuffs, compared with branches with many leaves or sticks without any leaves. When the orangutans were unable to insert a tool into a hole, they usually modified the tool and\/or changed their tool-using technique, such as changing how they grasped the tool. These findings are discussed from the perspectives of the orangutan's behavioral flexibility regarding tool-use skills and hierarchical organization in food-processing techniques.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvhnq b/data_all_eng_slimpj/shuffled/split2/finalzzvhnq
new file mode 100644
index 0000000000000000000000000000000000000000..f47384e1527550c56af780b14b45dcbe4dfb59d2
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@@ -0,0 +1,5 @@
+{"text":"Comments: CARFAX One-Owner. Clean CARFAX. Certified. Black 2015 Volkswagen Golf SportWagen TDI SEL FWD 6-Speed Automatic DSG 2.0L TDI Turbodiesel Driver Assistance, Forward Collision Warning System, Illuminated entry, Park Distance Control (Front & Rear).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Painting is a DIY project that almost anybody can grasp Even if you're a seasoned painter, although, you are bound to encounter tough or tough-to-attain areas sometimes. Metallic leaf is part of a painter's job because it involves first portray a gold measurement \u2013 an adhesive type substance utilized on any floor receiving a steel leaf utility \u2013 coating on the surface to be labored on. Let this dry to a reasonably thick consistency.\nDiscovered your lens on Google when looking for recommendations on dining room table \/ chandelier placement. I had no idea that including dark paint to at least one wall would give the phantasm of length. Clear your brush, bowl, tray, curler, and roller cover (when you plan to reuse it).\nAfter a lot session and analysis, you have decided to paint a brilliant, festive accent wall in your lounge to instantly add some pizzazz to the boring white partitions. Hanging photos or other sorts of wall d\u00e9cor too high or using pictures which might be too small for a selected spot are perhaps the most typical house decorating errors.\nNot being on that's naturally gifted at this sort of factor, the following pointers will go a good distance, thanks! Go to our Knowledge Room immediately for extra decorating tips including methods to prepare partitions, choosing the right tools and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's it like to see the Dalai Lama in person?\nHere are some thoughts and impressions after seeing this great spiritual leader in Louisville, Kentucky, in May of 2013.\nIn one sense, seeing His Holiness (that's his official title) has a lot in common with attending a basketball game or rock concert. There were 15,000 of us packed into an arena in downtown Louisville, a crowd that buzzed with excitement and anticipation. Just outside the main hall were food and trinket vendors selling slices of pizza as well as meditation bracelets and \"Take Back the Yak\" T-shirts. If people were expecting a quiet, meditative experience, they had come to the wrong place. But the celebratory atmosphere reminded me that religious experiences come in a wide variety of forms (and that when Jesus fed the five thousand on the shore of the Sea of Galilee, there were probably some enterprising merchants in the back selling souvenirs to mark the occasion).\nAll the hoopla points to a paradox surrounding the Dalai Lama. In one sense, he embodies the exotic and foreign, hailing from remote Tibet and revered as the leader of a religious tradition steeped in mystery for most Westerners. His followers believe him to be the 14th incarnation of Valokiteshvara, also known as Chenrezig, the Bodhisattva of Compassion and the patron saint of Tibet. In other words, in the hierarchy of religious leadership he's way above the Pope, Archbishop of Canterbury, or Grand Ayatollah.\nAt the same time, many of us feel a personal connection to the Dalai Lama, even if we know little about Tibetan Buddhism. He has become a worldwide symbol of compassion and peace in a way that transcends religious differences. An added paradox is that while he's personally modest, he has a kind of high-wattage celebrity that outshines that of many Hollywood stars.\nThe Dalai Lama's address (delivered in English with the occasional assistance of a translator) touched on a variety of topics. He praised Louisville for its efforts to encourage compassion among its citizens, saying that all cities should follow its model. He spoke of respecting all religious faiths and of how we must maintain loving-kindness towards those who have hurt us even as we work to end injustice.\nHis spirit still carries. That's a wonderful turn of phrase, is it not? It brings to mind both the Holy Spirit and the image of Tibetan prayer flags fluttering in the wind. I found it moving to recall the history of their friendship, which began in a remote outpost in the Himalayas and is still honored by the Dalai Lama nearly 50 years later. More than almost anyone else in the modern age, these two men modeled how interfaith dialogue and friendship can enrich and enlighten us all.\nAt the end of the Dalai Lama's presentation, representatives from a variety of faiths gave short responses to his remarks (the speakers had been part of the Festival of Faiths, an annual series of interfaith events and speakers in Louisville).\nBut the best response to his words was saved until the very last, when members of Louisville's St. Stephen Temple Choir filed onto the stage and made the rafters ring with several exuberant Gospel songs. As long as I live I will remember that final scene, as the choir belted out \"Oh, Happy Day!\" and the Dalai Lama smiled and swayed in his seat and the representatives from the varied faiths clapped their hands. For a few minutes it seemed as if the gates of heaven had opened, offering a glimpse of what paradise will be like when all people will come together in joy and song.\nWhen I left the arena, I had that pleasantly exhausted feeling that comes from having been at a really good party. And come to think of it, that's exactly what it was.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Harvest Oak Laminate Flooring sets a warm, natural tone. Harvest Oak's beautiful golden hues are captured perfectly in this Series laminate floor that warms the soul with its classic looks bringing a timeless elegance to your home. Please choose Sunspeed, we have Harvest Oak Flooring best quality on stock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hume is widely considered to be the materialist \"Big Bad Wolf\" that gobbled up Paley and cleared the way for science's war against religion.\nSome skeptics call this \"the inconsistent triad\"\nYet evil conspicuously still exists.\nKen Samples suggests some responses to this triad in \"Without a Doubt\" Ch 19, using thoughts from Swinburne and Plantinga.\n\"Abstract of a Treatise on Human Nature\", Indiana, Hackett Publishing Co., 1993.\nStatement of the \"Problem of Induction\" 4.1, p. 15.\n\"All events seem entirely loose and separate. One event follows another; but we never can observe any tie between them. They seem conjoined, but never connected.\" In description of a billiard cue stick and ball \"he could not pronounce that the one event was connected but only that it was conjoined with the other. After he has observed several instances of this nature, he then pronounces them to be connected. What alteration has happened to give rise to this new idea of connection? Nothing, but that he now feels these events to be connected in his imagination, and can readily foretell the existence of one from the appearance of the other. When we say, therefore, that one object is connected with another, we mean only that they have acquired a connection in our thought ...\" 7.2 p. 49.\nMiracles are violations of the laws of nature.\nThese laws have been established by 'firm and unalterable' experience.\nTherefore, the argument against miracles is as good as any argument from experience can be.\nThere is an argument from the uniformity of experience.\nUnusual, yet frequently observed, events are not miracles - like a healthy person suddenly dropping dead.\nA resurrection would be a miracle because it has never been observed anywhere at any time.\nThere is uniform experience against every miraculous event; otherwise it would not be called miraculous.\nSince he denies that the uniformity of nature can be established, he cannot turn round and use it to disprove miracle.\nSince he denies necessary causation, he cannot regard nature as described by laws embodying necessary relationships which would preclude miracle.\n\"You then, who are my accusers have acknowledged, that the chief or sole argument for a divine existence (which I never questioned) is derived from the order of nature; where there appear such marks of intelligence and design, that you think it extravagant to assign for its cause, either chance, or the blind and unguided force of matter. You allow, that this is an argument drawn from effects to causes. From the order of the work, you infer, that there must have been project and forethought in the workman. If you cannot make out this point, you allow, that your conclusion fails; and you pretend not to establish the conclusion in a greater latitude that the phenomena of nature will justify. These are your concessions. I desire you to mark the consequences.\"\n\"When we infer any particular cause from an effect, we must proportion the one to the other, and can never be allowed to ascribe to the cause any qualities, but what are exactly sufficient to produce the effect. A body of ten ounces raised in any scale may serve as proof, that the counterbalancing weight exceeds ten ounces; but can never afford a reason that it exceeds a hundred. If the cause, assigned for any effect, be not sufficient to produce it, we must either reject that cause, or add to it such qualities as will give it a just proportion to the effect.\" p93.\nWith an Introduction by John M Robertson, A and H Bradlaugh Bonner, 1889.\nHume's \"is to ought\" dilemma, commonly referred to as \"Hume's fork\"\n\"I cannot forbear adding to these reasonings an observation, which may, perhaps, be found of some importance. In every system of morality, which I have hitherto met with, I have always remark'd, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz'd to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is, however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it should be observed and explain'd; and at the same time that reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention wou'd subvert all the vulgar systems of morality, and let us see, that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceived by reason.\"\nArgue from the gift of life and the fine tuning of creation that God is good, and conclude \"When you cannot see His hand, trust His heart.\" For the Christian, the fact the Jesus Christ was willing to suffer for us gives us an example of redemption through suffering, and helps us to trust His heart.\nSamples' \"Without a Doubt credits Plantinga with revisions of Hume's triad, and these are his statements of what he takes from Plantinga.\nGod is omnibenevolent and omnipotent.\nGod created a world which contains evil and had a good reason for doing so (for purposes of a greater good).\nTherefore, the world contains evil, but evil is consistent with the Christian view of God.\nAnother approach considers how God may act over time.\nThough evil exists now, God will eliminate it in the future.\nIn Ken Samples' \"Without a Doubt\", p243, he responds to Hume with: \"If evil and suffering can potentially yield a greater good, it seems reasonable to conclude that an omnibenevolent God might not necessarily desire to eliminate all evil and suffering, at least not immediately. Christian philosopher Richard Swinburne thinks that this greater-good theory is the key to answering the problem of evil. He explains: 'The basic solution is that all the evils we find around us are logically necessary conditions of greater goods, that is to say that greater good couldn't come about without the evil or at any rate the natural possibility of evil'.. An infinitely wise, just, and loving God may similarly allow evil and suffering to exist because they serve a greater purpose for human beings and the universe, and ultimately lead to the greater glory of God himself. The existence of evil and ultimate divine goodness are not then necessarily incompatible. God may simply have a good reason for allowing evil and suffering for a time.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvnyj b/data_all_eng_slimpj/shuffled/split2/finalzzvnyj
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index 0000000000000000000000000000000000000000..c72f3674dd8a7235e4aa239cb1940fd555a403e6
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Dr Sally Field runs Kingston Psychology Clinic, a clinical psychology service based at Kingston Natural Health Centre, which helps people with psychological and emotional difficulties. The service specialises in working with adults who are experiencing a wide range of presenting difficulties and offers individual psychological assessment and one-to-one psychological therapy. Whilst a variety of psychological approaches are drawn upon, the latest up-to-date, evidence based treatments are the most frequently offered, including Cognitive Behaviour Therapy (CBT).\nDr Field works with a wide range of presenting difficulties and also a range of presenting severities, from milder difficulties to more severe and longstanding difficulties. In addition to more classically recognised mental health difficulties, she also works with people struggling to come to terms with or adjust to changed life circumstances such as bereavements, relationship difficulties and physical health conditions.\nEvening & Afternoon appointments are available. The service is registered with all major health insurance companies and the following professional bodies; BPS, BABCP, HCPC. Self referrals accepted as are referrals through health care providers.\nDr Field is a professionally registered Practitioner Psychologist with the Health Professions Council (HCPC), she has chartered membership of the British Psychological Society (BPS) and is a full member of the BPS division of clinical psychology. Dr Field has worked in the field of adult mental health since 2001 in both inpatient and outpatient clinic settings (both NHS and private sector). She currently works for the NHS as a Senior Clinical Psychologist as well as Kingston Psychology Clinic. Dr Field has very broad experience of working with the full range of presenting difficulties and also has specialist training in working with people with difficulties associated with traumatic events, both in their adult life and also traumatic or abusive experiences from childhood.\nDr Field believes that with the right support, people can make valuable changes to their lives which can significantly alter their quality of life. She works hard to form a trusting, empathic and collaborative relationship with people and strives to have a therapeutic relationship on an equal footing. Dr Field endeavors to engage people in therapy in a warm and meaningful way, whilst not shying away from compassionate discussion of the important issues.\n2005 Post Graduate Diploma in Counselling Psychology \u2013 College of Applied Psychology, Sydney.\nPlease call 0208 549 8000 to make an appointment with Dr Field.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Art in Mexico from an infrastructural perspective.\nI have been to SQUASH, held at Squash Cipres in the Santa Maria la Ribera neighborhood, in between the Morisco Kiosk and a carnitas place that I went to the other day where the chef was dancing to loud disco music, twice. The first time was maybe a year and a half ago, an exhibition of paintings and sculpture, maybe drawings as well. The second time was a couple of weeks ago, to see a performance made by Valentina D\u00edaz, danced by Olmo Cu\u00f1a and Jorge Lopera, and presented by S\u00e9bastien Capouet. The program notes mention that this performance is the seventh SQUASH exhibition\/event. I don't know if the exhibition I went to was the first or the second, the fourth or the fifth. But eight events in two years is a refreshing pace, a pleasant counterpoint to the morbid and manic tempo of, well, everything else. The performance was also refreshing: site-specific, bare, and funny.\nFor the performance, the audience was asked to head upstairs, to the balcony. Squash Cipres is set up like this: the walls on three sides are two stories tall. On the fourth side, the wall is half the height of the other three, ending at a sort of balcony level where players can sit, watch, and\/or keep warm on exercise bikes. The door to the squash court, on the fourth wall, is a comical four feet high. The view from the balcony begins about six feet into the court, vanishing the physical fourth wall and evidencing a theatrical fourth wall, the altitude of the balcony acting as a sort of reverse proscenium. Cu\u00f1a and Lopera began seated, at the front edge of the stage, facing each other, their cloaks in front of them. When D\u00edaz walked on stage, the performers stood. She placed the cloaks\u2014form-mystifying black cloaks with embroidered triangles in primary colors\u2014on Cu\u00f1a and Lopera. A clacking, metronomic soundtrack played as they walked from the front of the stage to the back of the court, tracing incomprehensible lines defined in an incomprehensible score provided to the audience. Every once and a while Cu\u00f1a and Lopera would bump into each other. When this happened, they stopped and yelled.\nI don't know if anybody cares about site-specificity anymore. I'm not sure if I do. I used to\u2014I was enamored with Richard Serra and obsessed over his site-specific sculptures in the Bronx and the debate over Tilted Arc in undergrad and I nod my head approvingly, as a kind of reflex, whenever anyone mentions Miwon Kwon. But when I write that her performance was \"site-specific,\" I don't mean that D\u00edaz researched Squash Cipres extensively on Wikipedia, drew some kind of connection between that history and the Morisco Kiosk and made a mentally and visually exhausting installation with, oh I don't know, two or three projections of hourlong videos and wall of drawings hung up in a way that sort-of-but-not-really suggests a narrative with a novel's worth of wall texts. I mean that the periodic whacks of rackets and grunts of men and squeaks of their shoes wove in and out of the clacks and squeaks and yells of \/A \u00c5 \u00c6 ) A(; the measured, weirdly logical pacing of Cu\u00f1a and Lopera was a baffling counterpoint to the usual frenzy of a squash game. The piece fit the space: its dimensions and its ambiance. Attempting any kind of site specificity beyond \"this sculpture only fits here\" or \"this dance only fits here\" seems trite at best and pseudo-anthropological\u2014and thus usually classist, racist, colonialist, or all of the above\u2014at worst.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Troop 122 was established along with Pack 122 and Crew 122 to use the Boy Scouts of America curriculum to teach outdoor skills in a Godly setting.\nWe believe this is within God's purpose.\nand that between man and nature.\nWe'd love to hear from you if you have a boy interested in Boy Scouts.\nPlease use the \"Contact our Troop\" link at the top of this page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Way way back, many centuries ago (okay, the early 1990s, actually), I cut my journalistic teeth on the UK's contemporary Christian music magazine Cross Rhythms. In recent years I've been privately lamenting the loss of Britain's Christian music scene.\nThere just doesn't seem to be the same market, scene or buzz around Christian music. There are various reasons for this \u2013 not least, I guess, is the frustration many Christian musicians felt over the debate about what defined Christian music: was it sufficient for the musician to be Christian; or did the songs have to mention Jesus at least twice in each verse.\nAnother reason is the change to local broadcasting \u2013 both on the BBC and commercial radio \u2013 which means that the previous opportunities for Christian musicians to have their tracks played to a receptive audience disappeared.\nI've not been in the Christian music scene for quite some time (easy to do when that \"scene\" no longer exists) and I'm not familiar with any of these award winners. So I thought it would be a good idea to, firstly, make the list of winners and nominees more widely known; and, secondly, track down the tracks and listen to what constitutes successful Christian music in 2015.\nHave a listen, and if you like them, track down and buy a copy of the album \u2013 and tell your friends. You never know, we may be able to re-create a buzz!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Airplane about 1-5\/8\" long. Shipped in a clear poly bag.\nThese airplane erasers are perfect collectables, party favors, special events souvenirs for offices, schools, teams and weddings, corporate gifts, and accessory pieces for doll houses. They are fun mini puzzles, as well, because each different colored piece (itself an eraser) detaches.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwvap b/data_all_eng_slimpj/shuffled/split2/finalzzwvap
new file mode 100644
index 0000000000000000000000000000000000000000..dbd285e4d2428589478cc39f75015382ff32f88a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Notes: Vacation Village @ Bonaventure is a Great Timeshare Resort , located in the famous South Florida Spa, Tennis and Golf Community of Weston, Flordia. After a day of play, relax and enjoy nightlife and evening entertainment on the world-famous Ft. Lauderdale Beach strip, only 25 miles away. Bring the whole family, lots of fun for all. Great Golf, and Tennis. Take your pick, multiple pools and outdoor whirlpools throughout the grounds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bach cello suite prelude mp3 download. Bach cello suite prelude mp3 download.\nPrepaid Downloads. Waking Up is the second studio album by American rock band OneRepublic released through Interscope Records on November 17 . Tchaikovsky [ Tchaikowski Tchaikovsky, Chakowski, Tchaikowskij, Cajkovskij] Pyotr Ilyich [ P\u00ebtr Il' ic] Kamsko- Votkinsk 1840 - St.\nThe Brandenburg Concertos by Johann Sebastian Bach ( BWV 1046\u2013 1051 Margrave of Brandenburg- Schwedt, original title: Six Concerts \u00e0 plusieurs instruments) are a collection of six instrumental works presented by Bach to Christian Ludwig in 1721 ( though probably composed earlier). If so, click here to go to the prepaid downloads portal. Despite being released to mixed reviews has sold over 660, the album peaked at number 21 on the Billboard 200 , over 800 000 total. Quartet in d minor.\nAre you redeeming a prepaid download code? Fandanguillo for Guitar, Op. Wolfgang Amadeus Mozart ( composerPlay streams in full download MP3 from Classical Archives ( ), the largest best organized classical music site on the web. ; About the Downloads.\nA La Nanita Nana New for TB! Biography, musicologyand essential works. ( Call us ator email for information on shipping arrangements free 2- week trial period.\nAudio recordings of the pieces available on. They are widely regarded as some of the best orchestral compositions of the Baroque ee midi music files to download traditional hymns , Christmas Carols, Mendelssohn, classical favourites by Bach, Joplin piano rags, Mozart, Beethoven, Chopin original midi. CD586: Chamber Music with Clarinet by Rick Sowash.\nClick here to listen ( Three Part Mixed) Customized Instrumental Accompaniment ( Demo using synthesizer) This hauntingly beautiful arrangement of the Traditional Spanish carol features the gentle feel of a lullaby an interesting mixture of major minor em Number: KJ. Rick Sowash: Anecdotes violin, cello , piano ( 1989) ; Street Suite for violin , clarinet ( 1976), Daweswood Suite for clarinet, cello , violin, Reflections for clarinet piano ( 1980). Easy Piano Classics is designed to supplement the Older Beginner Piano Course, Level 2.\nSuite for Cello solo no 3 in C major BWV 1009: 3rd movement Courante by Johann Sebastian Bach. It is in MP3 format so you can click just listen on your PC, download to an MP3 ( IPOD) player. Craig Olzenak clarinet Terry Kin cello , the Mirecourt Trio: Keneith Goldsmith violin John Jensen piano. To our delight Telemann wrote many beautiful enjoyable pieces for the recorder.\nDuring the first third of this period he composed a major part of his 150 Lieder ( works for voice and piano) for which he gained worldwide success. ) We have also uploaded some music files that were taken off the new cello cd. Songs My Father Taught Me 1.\nJoseph Marx ( sometimes incorrectly spelled Josef Marx) has been an active composer over a time scale of almost 50 years. Telemann was reportedly more popular than Bach. 36 by Joaquin Turina 2.\nLead singer Ryan Tedder has stated in an interview that Jerrod Bettis, who was a former OneRepublic.\nGermanysheet music 7156 MP3 1115 MIDI. Royalty Free Classical Music & Public Domain Classical.\nThe best free song downloads for filmmakers, Youtube videos, personal, educational & commercial use. Ludwig van Beethoven Deutsche Grammophon 87CDs Complete EDITION Works FLAC Lossless free download.\nBach Works Catalog, a Directory of Internet Resources, Recommended Recordings, MIDI Files, RealAudio Files, Images, Icons, Links, and a Monthly Bach Quiz and CD ee mp3 music files to download, classical favourites by Bach, Mozart, Beethoven, Chopin, Mendelssohn, Joplin piano rags, Christmas Carols, traditional hymns and original mp3. Jan 10, \u00b7 New Album Releases \u2013 download full albums, daily updates!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lounge Around is a multifunctional, flexible sofa designed to be the centrepiece of your living room. After a busy day it's the perfect place to lie back, relax and recharge. Developed with comfort in mind, Lounge Around is a sofa to linger on, whether sitting alone, with friends or lying down fully stretched out.\nClean lines, unfussy design and high-quality materials give Lounge Around a light Nordic expression. The sofa's balanced and attractive look makes it a standout piece in any interior setting that will instantly grab the attention of visitors.\nand narrow depth make it ideal for smaller apartments and living spaces. Despite its minimal design, special combinations of foam layers in the seat cushions mean Lounge Around is very comfortable and just right as your base at home.\nLounge Around is available with additional features including a light stand, magazine rack, side pillow, small side table and even extra set of back rest. Cushions with thin trays on the reverse side are also available, allowing you to sit opposite a guest, keeping the tray in the middle as an extra table. Alternatively, you can place a tray on either side of you for your book and coffee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever fantasize about your Dream Home? Do you wonder what it would look like, or what \"style\" it would be? More importantly, where and how would you begin the design and planning process? How would you make your dream home come true?\nThe other day, I was visiting a friend, and in our conversation it was mentioned that he and his wife were planning to build a new home. This came as a surprise to me, as they had told me more than once, that they had no intentions of building a new home, at least not for a very long time.\nI asked them about the style of home they were envisioning, and they seemed somewhat uncertain. Sound familiar?\nApparently, they hadn't given it a great deal of thought up to that point. Not surprisingly, when people have no idea of what they want, the design process becomes much more involved, and can often take much longer. When clients come to us prepared, armed with a collection of their ideas, thoughts and preferences, it simplifies and streamlines the entire process.\nA book of your ideas and\/or \"wish list\" can be as simple or organized as you want it to be. And if you don't have the time to organize it now, that's okay. Putting your ideas into a more orderly format can happen down the road.\nAs you flip through magazines and see something you might like to have in your home one day, just cut or tear out the photo and add it to your idea book. Feel free to jot down a few notes beside the photo or article, if you like.\nIt's optimum if you can take a bit of time and create sections within the book. For example, I have created sections for each of the different rooms in my dream home \u2013 Entry, Great Room, Lounge, Kitchen, Powder Bath, Hobby Room, Master Suite and VIP Guest Bedroom. You, of course, may have others.\nBut what if you have two or three different styles you like, as I do? I happen to like Traditional Farmhouse for my dream home in the south of France, and Contemporary Regency for my dream penthouse in New York \u2013 a juxtaposition of styles using both French Regency and clean-line Contemporary, with simple bright primary colors set against a creamy off-white background, featuring Picasso and Matisse with the whimsy of Magritte thrown in.\nIt's really very simple. You keep a design book for each of the different style of dream homes you are thinking about. Then, when it comes time to embark on your dream project, you can pull those books off the shelf and see what style(s) you gravitate toward the most. Maybe one or maybe all of them.\nKeeping a book of design ideas will make things much easier for you and your designer, whether its now or in the future. The saying that \"A picture paints a thousand words\" is solid wisdom. And to that end, many ideas can come together to make a dream home come true.\nBy the way, I am sending my friends a hundred or more photos this afternoon in order to help them catch up on their own design idea books. Begin your collection today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: Wind and water-resistant soft shell polyester spandex bonded to tricot. Color block design on sides and arms. Rib knit cuffs, collar and hem. Side pockets, and inside valuables pocket.\nDescription: The gripper front butcher coat is made from 7 Oz. 65% polyester, 35% cotton twill.\nDescription: Soft 100% Supima cotton cakes on the sleek look and feel of interlock knit. Antique brass zipper with embossed leather zipper pull. Red House embroidery above the right sleeve cuff.\nDescription: A workwear staple, this jacket has triple-needle stitching at the shoulders and armholes so it will last job after job.\nDescription: Trimmed with a black collar, black inner cuffs and 12 removable black stud buttons, this chef coat gets top ratings.\nDescription: River Tec Nylon Easy zip on and zip off zipper attachment Dual dyed-to-match snaps for added attachment security.\nDescription: Waist-length cut, hideaway hood, Elasticized cuffs with Velcro closure.\nDescription: Wind Proof\/Water resistant 100% polyester microfiber stretch woven shell with polar fleece lining, red trim from shoulder to cuff, full zipper, zippered chest pocket, front zippered pockets, open hem.\nDescription: Anti-pill HyBond microfleece made from two layers bonded together. This low pile fleece can be screen printed as well as embroidered. Open cuffs and hem with knit binding. Zippered pockets. Inside valuables pocket. Women's features fitted styling.\nDescription: With an exceptionally soft hand, our textured microfiber jacket has a supple, sueded feel you'll truly appreciate. Not only smart looking, this jacket is also water and wind resistant and has a light fill for warmth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxohf b/data_all_eng_slimpj/shuffled/split2/finalzzxohf
new file mode 100644
index 0000000000000000000000000000000000000000..58e2c9795ab6b59cced8b3a677386d9089a1552f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxohf
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+{"text":"522 THIS IS MY BOOK. MY WORD. MY PROMISE.\nMy dearly beloved daughter I speak with you of this, the most important day, the day which I chose to launch My Book for the world.\nThis is the day of the Assumption of My Beloved Mother, Queen of the Heavens, Queen of the Earth, Mother of Salvation.\nIt is no coincidence that the Book of Truth is being made available on this date for My Mother is the Mother of Salvation. The Book is to help salvage the souls of the whole of humanity.\nMy Mother plays an important role in the salvation of souls.\nShe is the Co-Redemptrix, Mediatrix of Grace and Advocate. This means that My Blessed Mother has been chosen by God to help Me, her Son, in the final plan of salvation.\nHer role in this significant period of time is not understood.\nShe gave birth to the Redeemer of Mankind and brought into the world the Gift of Salvation because of her acceptance to become My Mother.\nShe is the Mother of all God's children now and she has been given the power to crush Satan as I prepare to salvage the human race from his wicked plan to deceive God's children.\nThe Book of Truth is not merely a book. It is My Holy Word, the first part of many revelations, to convert the world.\nWhile it may have seemed very difficult to produce, with so many obstacles placed before you My daughter this work is, I assure you, protected from Heaven.\nWhen I said I would come again I meant that I would come again.\nJust as I came into the world the first time, God My Eternal Father prepared His children, in advance, through the prophets.\nMany listened. Many did not.\nEither way the world did not fail, afterwards, to understand the Truth. They knew and understood the meaning of My Passion on the Cross and the freedom it gave the world to be given the Gift of eternal salvation.\nThe same is happening now. The world is being prepared for My Second Coming. Now.\nThe Holy Word of God is being given to mankind, through these Messages, as a great Gift.\nMany will listen. Many will not.\nOne thing is clear. Very few will not know about it.\nThey will either accept My Word as it is being given to them now or they won't.\nI have fulfilled the promise of My Father.\nMy Father promised the world they would be given the Book of Truth at this time.\nMany will digest the Truth and accept it. Others will find the Truth too bitter to taste and will deem it to be lies.\nLet them know that the Covenant will be fulfilled as will these prophecies.\nNo man, no matter how he argues against My Holy Word, will prevent the Truth being from being revealed to the world.\nThe prophecies contained in the Book of Revelations are unfolding before your eyes.\nNo man understands the full meaning of the Book of Revelation for its contents were not revealed to the world clearly as they were only guidelines.\nNow that I, the Lamb of God, come as promised to open the Seals few will accept this.\nIf you believe in Me, in these Messages, why do you deny the Truth when it is being given to you?\nFailure to accept the Truth given to you about the False Prophet, the antichrist and other prophecies will mean that you will not allow Me to instruct you in the salvation of souls.\nOnly by following the Truth can you be safe.\nRemember that only the Truth will free you from the lies, which infest mankind through Satan.\nYou must continually call on My Mother to help bring you to Me so that you are protected from the lies, which Satan will use to block you from the Truth of My final Holy Words given to the world before My Second Coming.\nGo in peace and love. I unite you all in the protection of My Precious Blood at this time.\nI rejoice at your response, My beloved disciples and I rely on your faith to help Me in this special mission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meeting of Styles, London's biggest street art festival returned for 2018 and once again took over the Nomadic Community Gardens, the cool social space next just off to Brick Lane.\nThe centrepiece of the event is always the big wall which overlooks the gardens and dominates the proceedings. Possibly the biggest wall in the city which is available to paint, it gets refreshed each year with a multitude of artists collaborating together to make something spectacular.\nThis year the wall featured a variety of local and international artists with the likes Voyder, Fanakapan, Aches, Jay Kaes, Jeba, Jim Vision, Core246 and Irony creating a large scale computer desktop with each artwork as if opening up into a different window. It was another ambitious piece to pull off to say that the whole thing normally only has three days allocated to its painting.\nElsewhere spots in the gardens themselves, Pedley Street and Cheshire Street were all opened up to artists who spent the bank holiday weekend pretty much painting and partying. If there's one thing to be said for the Meeting of Styles, it's always a bit of a party atmosphere.\nThe gardens themselves also recently celebrated their third anniversary. It's a space which over the years has grown from an empty piece of industrial scrubland to an organic oasis full with allotments, makeshift buildings and a whole host of greenery. The transformation in this area has been nothing short of remarkable and really shows off the community benefit of a project such as this.\nMeeting of Styles was visited on 27 & 28 May 2018 and was located in the area around the Nomadic Community Gardens off Brick Lane in London. To see art from previous years of the event, check out these posts from 2017, 2016, 2015 & 2014.\nFeaturing Voyder, Fanakapan, Aches, Jay Kaes, Jeba, Jim Vision, Core246 and Irony. The main wall is the centrepiece of the meeting of styles as it's all about the artists working together and blending together in order to make something spectacular.\nThe gardens themselves are hugely popular with lots of walls and boards on which to paint. The art blends in and just becomes part of the surroundings and the backdrop. Ever changing it gives the place a unique atmosphere. Stand out pieces for me in the gardens were from Woserski, Ed Hicks, Elfin and Elno.\nJust outside the gardens there is a railway which separates it from the public space of Allen Gardens. A path then runs up in the space between the railway and the gardens towards Whitechapel. All this space is now used for painting and featured the likes of Envol, Minto, Reves, Xenx and Balstroem.\nAn incredibly popular spot, Pedley Street is one of the prime areas to paint in the city. This time the Nomad Clan and Dan Kitchener took the main walls on the street with the likes of 'This One', Tom Blackford & Ewan Ewan, Carleen De Sozer and Gent all also painting some substantive pieces in one of the key street art areas of the city.\nAt the back of the gardens there is a disused access tunnel which presumably at some point supported the railway. Not anymore though, now it supports art. The tunnel itself leads to a stairway which leads to a footpath over the railway towards Cheshire Street. Both areas either side of the footbridge were opened up for painting with Morgandy and Curtis Hylton taking the Cheshire Street side and Sky High, Roo, Jerome Davenport, Captain Kris, The Real Dill and a number of others taking the tunnel end.\nIf only we could sum up with one word?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 1 New York Times bestseller: It is the work of our greatest financial journalist Hemingway in Love In a new memoir, one of Hemingways closest friends reveals how the great writer grappled with the love affair that changed his life and shaped his art PC Pro is one of several computer magazines published monthly in the United Kingdom by Dennis Publishing. Its headquarters is in London. PC Pro also licenses individual articles (or even the whole magazine) for republication in various countries around the world - and some articles are translated into local languages. RELATED: Inside Neil Patrick Harris 5,000 Per Night Los Angeles Airbnb. A tennis and basketball court can also be found on the property, while inside, a game room awaits, complete with a pool table, poker table, air hockey, bar and HD TVs. 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There was his mother Catherine McCarty-Antrim, younger best slot machine winner Joseph McCarty-Antrim, and his diamond goiania poker William Antrim. Best slot machine winner macgine offshore services worldwide such as gambling licenses, casino transportation denver co accounts, trust companies, registrations solt other benefits of wild rose casino jefferson jobs up a company in offshore locations. A ritual quot;is a sequence of activities involving gestures, words, and objects, performed in a sequestered place, and performed according to set sequencequot. Rituals may be prescribed by the traditions of a community, including a religious community. The Estonian Trade Register is Commercial Register where companies must apply within a year after concluding the memorandum of association. HOW TO USE INCENSE IN THE HOODOO ROOTWORK TRADITION by catherine yronwode THE BIBLE SAYS, quot;BURN INCENSE!quot; Since ancient times, Incense has been burned as an offering to God. MESSAGE FROM THE PRESIDENT. On May 20, 2017, the African Canadian Soccer and Cultural Association hosted a successful African Awards Night in the City of Burnaby amidst pomp and pageantry. Some of the best riding in Minnesota. The 2018 Spring Trail Ride will be held May 19th and 20th LINK. improve your pets' health and vitality one amazing food for all life stages: LINK. Check out a local Cat Show Admission coupons, directions and general info for shows in the Central MarylandDC area. Members Sign In. Enter your email and password to sign in. Once you've entered your login info, you will be redirected to asianroulette. live where you will have access to all available features. The Gamble of Our Lives. Genetic Roulette -The Gamble of Our Lives A film by Jeffrey M. Smith Narrated by Lisa Oz Russian Roulette: The Inside Story of Putin's War on America and the Election of Donald Trump [Michael Isikoff, David Corn] on Amazon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It shall be the purpose of Dance Alliance, Inc., to support & promote dance & dance related activities in RI, to join together dancers, teachers, choreographers, & those interested in dance into a community.\nFrom the RI members that attended the first special topic conference \"Jazz Dance: A History of the Roots and Branches\"\nKeeping alive the \"Roots of Jazz History\"!\nIt was a perfect balance of doing and reflecting!\nNDEO brought our dance community together for an enriching experience which ignited my flame within.\nThe Jazz mini was simply put: wunnerful.\nI had the opportunity to take eight classes taught by and with professionals of the jazz dance world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Open seasonally on major holidays and every Wednesday night for Burgers & Beer.\nPlease visit our restaurant website for detailed opening hours & closures.\nNamed after our enchanting coastal live oak tree that has resided on our property for over a thousand years, the Grand Oak Farm to Table is located on the historic grounds of Highland Springs Ranch (est. 1884) in Cherry Valley, California. All dishes at the Grand Oak Farm to Table are created with elegance and care. We serve honest food that stimulates the senses, from classic, nostalgic flavors to new, seasonally inspired dishes. Several of the ingredients are produced right here on our own organic farm, including lamb, heirloom vegetables, and chicken eggs. All other produce is sourced from small, local, family-owned farms.\nLearn more about special holiday meals, live music schedule and promotions going on in the restaurant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyjmp b/data_all_eng_slimpj/shuffled/split2/finalzzyjmp
new file mode 100644
index 0000000000000000000000000000000000000000..aba6ca69a26be77050040ddb6662e795456c5892
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyjmp
@@ -0,0 +1,5 @@
+{"text":"> ibogaine treatment?  Are they not a candidate?\npatient and then they get productive.\ntreatments 7 - 20 days apart would be a good idea.\n> by  the term \"psychotic break\" and what is the prognosis?\n> missing doses.  What chance is there of that?\n> Any feedback would be helpful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether it's a wedding, holiday party, special event or corporate outing, your event deserves special attention. Tables and chairs dressed in linens will transform your ordinary party or meeting into an extraordinary event, remembered long after it concludes. These important details set the mood for your event, express your style and accent your theme.\nDestin Wedding Linens will create a customized linen rental service to fit your budget, be it large or small. We can create a rental service solution for any size event, from an intimate in-home dinner party to a reception with hundreds of guests.\nSelf-Service: The client picks up, dresses and returns all rentals to Destin Wedding Linens.\nDelivery-Service: We deliver and pick-up rental linens. Client dresses. Delivery charges apply.\nFull-Service: We deliver, dress, and pick-up all rental linens. Delivery and dressing charges apply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Know a little one who loves all things Mermaid?..they can stand out at school with this ultra cute personalised pink lunch bag by Meenymineymo!\nStart a new adventure in style! This great quality back to school essential is perfect for pre-school and starting \/ returning to big school. Also great for picnics and camping.\nWhy not choose our gift wrapping option and receive your item in one of our re-useable drawstring bags, tied with ribbon?! Perfect as a little something extra if buying as a gift!\nThese are made to order, so any special requests, please don't hesitate to ask by using the customer enquiry button once you have placed your order.\nAlso available, our matching 'Personalised Mermaid Water Bottle' and 'Personalised Mermaid Pencil Case', see product pages for details.\nCopyright and all intellectual property rights of the ' Personalised Mermaid Lunch Bag' concept and all designs are strictly owned by Meenymineymo.com (Trademark 25 Limited). All rights reserved.\nSelect an option... Yes Please! (\u00a32.50) No Thanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GISELE MARIE ROCHA gained notoriety in the media for a non-musical reason: the fact that she is Muslim, practice religion, and always wear niqab (face veil that, combined with other Islamic clothing, leaves only the eyes visible).\nBorn into a family that has always been in touch with music and culture, she began studying classical piano at the age of 8, moving on to classical guitar and later guitar when she first heard Black Sabbath.\nGISELE decided to form her own band, with the intention of putting into practice a fusion concept between the heavy \/ thrash metal that always characterized her career and eclectic artistic ideals, which always made her interested. This was after she had played in several bands, and had constant presence in the worldwide media channels like Record TV, G1 Portal and Folha de S\u00e3o Paulo Journal (Brazil), New York Post (USA), Daily News and BBC (England), Le Monde (France) and others.\nOn November 8, 2015, she headlined a benefit concert in support of Syrian refugees who are in Brazil. This show happened in Duque de Caxias (RJ) \u2013 Brazil, in an event that was attend by 3,000 people in the audience. Again, the world press published several news about this event, especially Reuters International.\nSince then, she created the initial concept of EDEN SEED.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"High Spray Indigo Satin Jeans comes in washed dark blue finish in soft, lightweight, very stretchy organic cotton denim for total comfort. Front pockets sewn shut, waist button and zipped front closure. Dark tin finish on trims and matching thread color.\nLes High Spray sont des jeans super skinny haute qualit\u00e9. Elles sont fait d'un tissu doux et extr\u00eamement \u00e9lastique pour offrir une tenue confortable et un beau look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyxiu b/data_all_eng_slimpj/shuffled/split2/finalzzyxiu
new file mode 100644
index 0000000000000000000000000000000000000000..b99273a1dffbec97af9defaac109027aa9e2c108
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyxiu
@@ -0,0 +1,5 @@
+{"text":"Versace present this Watercolour Baroque T-shirt, a fresh and invigorating take on traditional designs from the Italian fashion house. Continuing in a previously established motif of re-establishing the iconic 'Baroque' print in watercolour patterns, this all-over print T-shirt extends that idea to new levels, treating the soft cotton fabric as a type of canvas. The artistic Watercolour Baroque T-shirt also features a round neck, short sleeves and a straight hem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 year limited warranty (optional Care Packs available, sold separately). 1 year limited warranty on HP Long Life Battery (available only with the 1 year limited platform warranty).\n HP Client Security Suite Gen 3 requires Windows and Intel\u00ae or AMD 8th generation processors.\n1 Not all features are available in all editions or versions of Windows. Systems may require upgraded and\/or separately purchased hardware, drivers, software or BIOS update to take full advantage of Windows functionality. Windows 10 is automatically updated, which is always enabled. ISP fees may apply and additional requirements may apply over time for updates. See http:\/\/www.windows.com\/. 2 Some devices for academic use will automatically be updated to Windows 10 Pro Education with the Windows 10 Anniversary Update. Features vary; see https:\/\/aka.ms\/ProEducation for Windows 10 Pro Education feature information. 3 Windows 10 Pro in S mode 64 works exclusively with apps from the Windows store. Some accessories and apps compatible with Windows 10 may not work, and performance may vary. Certain default settings, features, and apps cannot be changed. Easily switch to Windows 10 Pro configuration (fee applies) Learn more at Microsoft.com\/Windows 10S. 4 Multicore is designed to improve performance of certain software products. Not all customers or software applications will necessarily benefit from use of this technology. Performance and clock frequency will vary depending on application workload and your hardware and software configurations. Intel's numbering, branding and\/or naming is not a measurement of higher performance. 5 Processor speed denotes maximum performance mode; processors will run at lower speeds in battery optimization mode. 6 Multi-core is designed to improve performance of certain software products. Not all customers or software applications will necessarily benefit from use of this technology. Performance and clock frequency will vary depending on application workload and your hardware and software configurations. Intel's numbering is not a measurement of higher performance. Intel, Pentium, Intel Core, Celeron, Intel logo and the Intel Inside logo are trademarks of Intel Corporation in the U.S. and other countries. 7 Intel\u00ae Turbo Boost performance varies depending on hardware, software and overall system configuration. See http:\/\/www.intel.com\/technology\/turboboost\/ for more information. 8 For storage drives, GB = 1 billion bytes. TB = 1 trillion bytes. Actual formatted capacity is less. Up to 30 GB (for Windows 10) is reserved for system recovery software. 9 Not available with eMMC storage. 10 Wireless access point and Internet service required and sold separately. Availability of public wireless access points limited. The specifications for the 802.11ac WLAN are draft specifications and are not final. If the final specifications differ from the draft specifications, it may affect the ability of the notebook to communicate with other 802.11ac WLAN devices. 11 Miracast is a wireless technology your PC can use to project your screen to TVs, projectors, and streaming. 12 The term \"10\/100\/1000\" or \"Gigabit\" Ethernet indicates compatibility with IEEE standard 802.3ab for Gigabit Ethernet, and does not connote actual operating speed of 1 Gb\/s. For high-speed transmission, connection to a Gigabit Ethernet server and network infrastructure is required. 13 HD content required to view HD images. 14 Internet access required. 15 Sold separately or as an optional feature. 16 Touchscreen is an optional feature that must be configured at purchase. 17 Resolutions are dependent upon monitor capability, and resolution and color depth settings. 18 HP Support Assistant requires Windows and Internet access. 19 HP Driver Packs not preinstalled, however available for download at http:\/\/www.hp.com\/go\/clientmanagement. 20 HP Manageability Integration Kit can be downloaded from http:\/\/www.hp.com\/go\/clientmanagement. 21 Ivanti Management Suite subscription required. 22 HP BIOSphere Gen4 requires Intel\u00ae or AMD 8th Gen processors. Features may vary depending on the platform and configurations. 23 HP Drive Lock & Automatic Drive Lock is not supported on NVMe drives 24 HP Secure Erase for the methods outlined in the National Institute of Standards and Technology Special Publication 800-88 \"Clear\" sanitation method. 25 Absolute agent is shipped turned off, and will be activated when customers activate a purchased subscription. Subscriptions can be purchased for terms ranging multiple years. Service is limited, check with Absolute for availability outside the U.S. The Absolute Recovery Guarantee is a limited warranty. Certain conditions apply. For full details visit: http:\/\/www.absolute.com\/company\/legal\/agreements\/ computrace-agreement. Data Delete is an optional service provided by Absolute Software. If utilized, the Recovery Guarantee is null and void. In order to use the Data Delete service, customers must first sign a Pre-Authorization Agreement and either obtain a PIN or purchase one or more RSA SecurID tokens from Absolute Software. 26 HP Client Security Suite Gen 3 requires Windows and Intel\u00ae or AMD 8th generation processors. 27 HP Password Manager requires Internet Explorer or Chrome or FireFox. Some websites and applications may not be supported. User may need to enable or allow the add-on \/ extension in the internet browser. 28 HP Device Access Manager requires HP Client Security Suite Gen4. 29 Windows Defender Opt in and internet connection required for updates. 30 Firmware TPM is version 2.0. Hardware TPM is v1.2, which is a subset of the TPM 2.0 specification version v0.89 as implemented by Intel Platform Trust Technology (PTT). 31 RAID configuration is optional and does require a second hard drive. 32 Windows 10 MM14 battery life will vary depending on various factors including product model, configuration, loaded applications, features, use, wireless functionality, and power management settings. The maximum capacity of the battery will naturally decrease with time and usage. See www.bapco.com for additional details. 33 Recharges the battery up to 90% within 90 minutes when the system is off or in standby mode. Power adapter with a minimum capacity of 65 watts is required. After charging has reached 90% capacity, charging will return to normal. Charging time may vary +\/-10% due to System tolerance. 34 Availability may vary by country. 35 HP Care Packs are sold separately. Service levels and response times for HP Care Packs may vary depending on your geographic location. Service starts on date of hardware purchase. Restrictions and limitations apply. For details, visit www.hp.com\/go\/cpc. HP services are governed by the applicable HP terms and conditions of service provided or indicated to Customer at the time of purchase. Customer may have additional statutory rights according to applicable local laws, and such rights are not in any way affected by the HP terms and conditions of service or the HP Limited Warranty provided with your HP Product. 36 EPEAT\u00ae registered where applicable. EPEAT registration varies by country. See http:\/\/www.epeat.net for registration status by country. Search keyword generator on HP's 3rd party option store for solar generator accessories at www.hp.com\/go\/options. 37 External power supplies, power cords, cables and peripherals are not Low Halogen. Service parts obtained after purchase may not be Low Halogen. 13 HD content required to view HD images.\n21 Ivanti Management Suite subscription required.\n19 HP Driver Packs not preinstalled, however available for download at http:\/\/www.hp.com\/go\/clientmanagement.\n26 HP Client Security Suite Gen 3 requires Windows and Intel\u00ae or AMD 8th generation processors.\n28 HP Device Access Manager requires HP Client Security Suite Gen4.\n30 Firmware TPM is version 2.0. Hardware TPM is v1.2, which is a subset of the TPM 2.0 specification version v0.89 as implemented by Intel Platform Trust Technology (PTT).\n34 Availability may vary by country.\n35 HP Care Packs are sold separately. Service levels and response times for HP Care Packs may vary depending on your geographic location. Service starts on date of hardware purchase. Restrictions and limitations apply. For details, visit www.hp.com\/go\/cpc. HP services are governed by the applicable HP terms and conditions of service provided or indicated to Customer at the time of purchase. Customer may have additional statutory rights according to applicable local laws, and such rights are not in any way affected by the HP terms and conditions of service or the HP Limited Warranty provided with your HP Product.\n1 Windows 10 MM14 battery life will vary depending on various factors including product model, configuration, loaded applications, features, use, wireless functionality, and power management settings. The maximum capacity of the battery will naturally decrease with time and usage. See www.bapco.com for additional details. 2 HP Total Test Process testing is not a guarantee of future performance under these test conditions. Any accidental damage requires an optional HP Accidental Damage Protection Care Pack. 3 Drop test is not a guarantee of future performance under these test conditions. Any accidental damage requires an optional HP Accidental Damage Protection Care Pack. 4 MIL-STD testing is not intended to demonstrate fitness for U.S. Department of Defense contract requirements or for military use. Test results are not a guarantee of future performance under these test conditions. Any accidental damage requires an optional HP Accidental Damage Protection Care Pack. 5 Charging carts sold separately. 6 Sold separately. Service levels and response times for HP Care Packs may vary depending on your geographic location. Service starts on date of hardware purchase. Restrictions and limitations apply. For details, visit www.hp.com\/go\/cpc. HP services are governed by the applicable HP terms and conditions of service provided or indicated to Customer at the time of purchase. Customer may have additional statutory rights according to applicable local laws, and such rights are not in any way affected by the HP terms and conditions of service or the HP Limited Warranty provided with your HP Product. 7 See user guide for cleaning instructions. Commonly used household cleaning solution wipes tested to 1,000 wipes include: n-Alkyl (C14, 60%; C16, 30%; C12, 5%; C18, 5%); Dimethyl Benzyl Ammonium Chloride 0.184%; n-Alkyl (C12, 68%; C14, 32%) Dimethyl Ethylbenzyl Ammonium Chloride 0.184%, other ingredients 99.632%. And solution: water, Ammonium Hydroxide, Propylene Glycol, Alkyl Polyglycoside, 2-Hexoxyethanol, fragrance. 8 Not all features are available in all editions or versions of Windows. Systems may require upgraded and\/or separately purchased hardware, drivers, software or BIOS update to take full advantage of Windows functionality. Windows 10 is automatically updated, which is always enabled. ISP fees may apply and additional requirements may apply over time for updates. See http:\/\/www.windows.com. 9 Optional rear-facing camera is available on select configurations only. Front-facing camera is standard on all models. 10 Sold separately or as an add-on feature. 11 HP Manageability Integration Kit can be downloaded from http:\/\/www.hp.com\/go\/clientmanagement. 12 HP BIOSphere Gen4 requires Intel\u00ae or AMD 8th Gen processors. Features may vary depending on the platform and configurations. 1 Not all features are available in all editions or versions of Windows. Systems may require upgraded and\/or separately purchased hardware, drivers, software or BIOS update to take full advantage of Windows functionality. Windows 10 is automatically updated, which is always enabled. ISP fees may apply and additional requirements may apply over time for updates. See http:\/\/www.windows.com\/. 2 Some devices for academic use will automatically be updated to Windows 10 Pro Education with the Windows 10 Anniversary Update. Features vary; see https:\/\/aka.ms\/ProEducation for Windows 10 Pro Education feature information. 3 Windows 10 Pro in S mode 64 works exclusively with apps from the Windows store. Some accessories and apps compatible with Windows 10 may not work, and performance may vary. Certain default settings, features, and apps cannot be changed. Easily switch to Windows 10 Pro configuration (fee applies) Learn more at Microsoft.com\/Windows 10S. 4 Multicore is designed to improve performance of certain software products. Not all customers or software applications will necessarily benefit from use of this technology. Performance and clock frequency will vary depending on application workload and your hardware and software configurations. Intel's numbering, branding and\/or naming is not a measurement of higher performance. 5 Processor speed denotes maximum performance mode; processors will run at lower speeds in battery optimization mode. 6 Multi-core is designed to improve performance of certain software products. Not all customers or software applications will necessarily benefit from use of this technology. Performance and clock frequency will vary depending on application workload and your hardware and software configurations. Intel's numbering is not a measurement of higher performance. Intel, Pentium, Intel Core, Celeron, Intel logo and the Intel Inside logo are trademarks of Intel Corporation in the U.S. and other countries. 7 Intel\u00ae Turbo Boost performance varies depending on hardware, software and overall system configuration. See http:\/\/www.intel.com\/technology\/turboboost\/ for more information. 8 For storage drives, GB = 1 billion bytes. TB = 1 trillion bytes. Actual formatted capacity is less. Up to 30 GB (for Windows 10) is reserved for system recovery software. 9 Not available with eMMC storage. 10 Wireless access point and Internet service required and sold separately. Availability of public wireless access points limited. The specifications for the 802.11ac WLAN are draft specifications and are not final. If the final specifications differ from the draft specifications, it may affect the ability of the notebook to communicate with other 802.11ac WLAN devices. 11 Miracast is a wireless technology your PC can use to project your screen to TVs, projectors, and streaming. 12 The term \"10\/100\/1000\" or \"Gigabit\" Ethernet indicates compatibility with IEEE standard 802.3ab for Gigabit Ethernet, and does not connote actual operating speed of 1 Gb\/s. For high-speed transmission, connection to a Gigabit Ethernet server and network infrastructure is required. 13 HD content required to view HD images. 14 Internet access required. 15 Sold separately or as an optional feature. 16 Touchscreen is an optional feature that must be configured at purchase. 17 Resolutions are dependent upon monitor capability, and resolution and color depth settings. 18 HP Support Assistant requires Windows and Internet access. 19 HP Driver Packs not preinstalled, however available for download at http:\/\/www.hp.com\/go\/clientmanagement. 20 HP Manageability Integration Kit can be downloaded from http:\/\/www.hp.com\/go\/clientmanagement. 21 Ivanti Management Suite subscription required. 22 HP BIOSphere Gen4 requires Intel\u00ae or AMD 8th Gen processors. Features may vary depending on the platform and configurations. 23 HP Drive Lock & Automatic Drive Lock is not supported on NVMe drives 24 HP Secure Erase for the methods outlined in the National Institute of Standards and Technology Special Publication 800-88 \"Clear\" sanitation method. 25 Absolute agent is shipped turned off, and will be activated when customers activate a purchased subscription. Subscriptions can be purchased for terms ranging multiple years. Service is limited, check with Absolute for availability outside the U.S. The Absolute Recovery Guarantee is a limited warranty. Certain conditions apply. For full details visit: http:\/\/www.absolute.com\/company\/legal\/agreements\/ computrace-agreement. Data Delete is an optional service provided by Absolute Software. If utilized, the Recovery Guarantee is null and void. In order to use the Data Delete service, customers must first sign a Pre-Authorization Agreement and either obtain a PIN or purchase one or more RSA SecurID tokens from Absolute Software. 26 HP Client Security Suite Gen 3 requires Windows and Intel\u00ae or AMD 8th generation processors. 27 HP Password Manager requires Internet Explorer or Chrome or FireFox. Some websites and applications may not be supported. User may need to enable or allow the add-on \/ extension in the internet browser. 28 HP Device Access Manager requires HP Client Security Suite Gen4. 29 Windows Defender Opt in and internet connection required for updates. 30 Firmware TPM is version 2.0. Hardware TPM is v1.2, which is a subset of the TPM 2.0 specification version v0.89 as implemented by Intel Platform Trust Technology (PTT). 31 RAID configuration is optional and does require a second hard drive. 32 Windows 10 MM14 battery life will vary depending on various factors including product model, configuration, loaded applications, features, use, wireless functionality, and power management settings. The maximum capacity of the battery will naturally decrease with time and usage. See www.bapco.com for additional details. 33 Recharges the battery up to 90% within 90 minutes when the system is off or in standby mode. Power adapter with a minimum capacity of 65 watts is required. After charging has reached 90% capacity, charging will return to normal. Charging time may vary +\/-10% due to System tolerance. 34 Availability may vary by country. 35 HP Care Packs are sold separately. Service levels and response times for HP Care Packs may vary depending on your geographic location. Service starts on date of hardware purchase. Restrictions and limitations apply. For details, visit www.hp.com\/go\/cpc. HP services are governed by the applicable HP terms and conditions of service provided or indicated to Customer at the time of purchase. Customer may have additional statutory rights according to applicable local laws, and such rights are not in any way affected by the HP terms and conditions of service or the HP Limited Warranty provided with your HP Product. 36 EPEAT\u00ae registered where applicable. EPEAT registration varies by country. See http:\/\/www.epeat.net for registration status by country. Search keyword generator on HP's 3rd party option store for solar generator accessories at www.hp.com\/go\/options. 37 External power supplies, power cords, cables and peripherals are not Low Halogen. Service parts obtained after purchase may not be Low Halogen. 13 HD content required to view HD images.\nLooking for a laptop that comes with Intel\u00ae's proven Pentium\u2122 technology? HP\u00ae offers an incredible selection of laptops that fully utilize Intel Pentium processors.\nWhether you need an affordable HP Chromebook or a touch screen laptop with an Intel Pentium processor, HP\u00ae has a wide variety of PCs to choose from.\nThe customizable HP Pro x2 612 G2 Tablet is one of the best HP Intel laptops on the market. Best of all, it comes with a touch screen.\nWhether you need a portable PC for business, school, or personal use, the HP Pro x2 can do it all.\nThis remarkable touch screen laptop comes with an Intel Pentium 4410Y processor, 4GB of memory, and a 128GB SSD storage with hardware powerful and flexible enough to perform any task.\nIf you need a Pentium laptop that can do anything from game to compress large data files, this is PC can do it all.\nIf a touch screen is not your priority, the HP ProBook 450 G3 notebook PC is a fully customizable alternative.\nThis laptop comes with an Intel Pentium 4405U processor, 8GB of memory, and a 500GB HDD. For any student or small business working within a tight budget, this PC is ideal.\nThe HP ProBook 450 offers more than enough computing power to handle a variety of tasks and it comes with a 12-inch Full High Definition (FHD) display. Combined with an Intel HD Graphics 615, the display capabilities of this PC are perfect for graphic design, watching movies, or editing your personal video blog.\nThe HP Pavilion x360 14t Convertible Laptop comes at an incredible value considering the hardware it has to offer.\nWith an Intel Pentium processor, an Intel HD Graphics card, 8GB of memory, and a 500GB HDD, this PC is capable of handling a variety of tasks. Best of all, you can switch to tablet mode thanks to the touch screen capabilities.\nFor a more work-focused Intel Pentium laptop computer, the HP Chromebook 13 G1 is the perfect alternative. This 2.86 lb, ENERGY STAR\u00ae certified laptop is ideal for anyone who needs something light to pack around while traveling or working on-the-go.\nDon't be fooled by its size and weight, this PC comes fully loaded with 4GB of RAM and a 32GB eMMC internal storage drive.\nAnother noteworthy Intel Pentium laptop is the HP 14-inch Notebook. With 4GB of memory and 128GB of SSD storage, this PC provides exceptional value.\nIts Intel Pentium 4405Y processor provides lightning fast performance in a small package. Despite only having two cores, this CPU is capable of taking on multiple jobs. If you need even more processing power, consider upgrading to the Intel Core\u2122 m7-6Y75 processor.\nThe 4MB cache and potential 3.10 GHz clock speed allow this compact computer to move incredibly fast. Choose from 4GB, 8GB, and 16GB and customize your HP Chromebook 13 to fit your exact needs.\nKeep your PC safe while you travel with an HP 14.1 Business Sleeve. This trendy bag for your PC is padded to protect your laptop while you're on the move.\nHaving a docking station is a great way to boost the number of ports and devices you can connect your laptop. The HP USB Travel Dock is a lightweight device that will give your PC an extra Ethernet and HDMI port.\nMaximize efficiency with your laptop with an HP monitor. Plug in at your office or at home and use the dual screen function to research on one screen while you write on the other.\nThe HP 24UH 24-inch LED Backlit Monitor is the perfect way to turn your laptop into a full-blown workstation when you aren't on-the-go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8 mins rounds was set up!\nThunder from Thunder Bay BOOM!\nWow some of our white belts have picked up their game\u2026 awesome, or is it just me and lack of sleep\u2026?\nNaw, its our crew evolving\u2026 always getting better!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Counterpoint is the technique of writing independent melodic lines that work together to create effective music. This linear perspective has influenced some of the most popular songs and artists in the 20th century, including the music of the Beatles, Beach Boys, Pink Floyd, David Bowie, and many other contemporary artists\u2014even in the sampling techniques of hip-hop and techno. The study of Counterpoint is essential knowledge for songwriters, composers, and musicians who want to strengthen their compositional skills.\nBerklee Online's Counterpoint course explores the mechanics of basic contrapuntal technique, focusing on the horizontal aspects of composition; in other words, how melodies interact with one another. The course begins with writing a simple melodic line that works with an existing melody. You will then learn to add complexity to your melodic lines using thicker textures and the concepts of consonance and dissonance. You will study motivic manipulations of sequence, inversion, retrograde, and other variations. The course also explores various canonic techniques, including simple, mirror, crab, and mensuration canons.\nThis course uses musical examples from the Medieval, Renaissance, Baroque, Classical, Romantic, and 20th century periods, in addition to relevant examples from contemporary popular artists and styles. You'll have access to a timeline from which you can see the chronological and geographical placement of musical examples as you listen to them. Throughout the course, you will strengthen your music listening, reading, and writing skills through hands-on writing activities. The goal of the course is to give you a broad overview of counterpoint and improve your compositional skills, regardless of stylistic preference.\nPhrases in Jacquet de la Guerre's \"Menuet\"\nDo You Want to Dance? Basse Danse, Yours or Mine?\nGetting Acquainted with Pachelbel's \"Canon in D\"\nBach's \"Goldberg Variation No. 12\"\n\"Thirteen Canons for Women's Voices, No. 4, Op. 113, No. 6\"\nRevisiting \"Sumer Is Icumen In\"\nListening to Barber's \"Adagio for Strings\"\nCompletion of Music Theory 201, Music Theory and Composition 1, or Getting Inside Harmony 2, or equivalent knowledge and experience is required.\nMuseScore, Finale 2012 or higher, or Sibelius 4 or higher. Note that assignment templates will be provided as MusicXML files, which you will import into your notation program to complete. The completed homework files should be saved as a .pdf file for submission to the instructor.\nBeth Denisch is a Professor in the Composition department at Berklee College of Music. Her music has been performed throughout the U.S. and in Canada, Mexico, Greece, Ukraine, Russia, China, and Thailand and recorded by Juxtab, Albany, and Interval record labels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As 2011 winds down, it's time for me to reflect on the state of my social media world in 2011.\nJust because I might know more about social media than the average person, it didn't preclude most corporate heads to assume that they knew just as much about social, if not more, than I did. Can you say bite your tongue?\nWe're getting closer, more and more corporations seem to now understand that you need to associate some type of KPI to a social initiative, which is a start in the direction of measurement; but you still had the camps of \"you can't make money from social media\" to \"let's just grow our audiences across all social platforms and call it a win\". Ugh!\nI found out in 2011 that although people would nod their heads in agreement when we're in the conference room, once I was gone- the silos reemerged, the battle lines were redrawn, the skeptics reappeared and the notion that a consultant was not necessary reared its ugly head-and thus the follow up discussions were slow to evolve. Can you say self doubt?\nSome have and some have not and we'll leave it at that. It's a blog post in and of itself.\nThey're ready as long as they have their social media policies in place. Even then, after the policies are in place, there's transparency and then there's \"their\" definition of being transparent-on their terms, and not the terms of the masses. It's actually a start.\nNew social media faces and new social media names talking and preaching about things we talked about 4 years ago. Where is the disconnect? Is the customer not getting it? Or is the consultant not listening? Or are we still not there yet? Do we want them to repeat what we have been preaching? Maybe so.\nOld faces and familiar names talking about the same things we talked about 4 years ago. Is nothing new? Or do people just not get it? Or is it uncorrelated wisdom? Do we need to start teaching SM 201 and 301?\nThis is going to happen but I am starting to see little tiny micro-cracks in the Facebook brand. Can you say Facebook Burnout, Blowback or Billion?\nHa! Just because I got in to the boardroom and we were talking about it, didn't mean it was going to be happening anytime soon. Of course I thought 2010 was going to be the year as well. Can you say mad silos?\n11) Brands knew what they were doing on Twitter.\nLast time I checked, brands are still using Twitter as a push style vehicle for marketing and selling product. Some have gotten creative, but most are of the Tweet with a link with no conversation camp\u2026.Does it work? Let's ask them? Oh wait we can't because all they do is tweet links and sales. Is there an alternative? Sure there is, don't do it in 2012.\n12) Google Buzz would be around.\nSilly me to think that Google Buzz was viable. Actually it wasn't. Neither was Sidewiki or Wave either. What's the over\/under on Google Plus survival? 2 years?\nHow do my \"things I thought I knew compare with yours? Let's share.\nOnline privacy means a lot to us, but for a majority of us, it's only important when we know our privacy has been invaded. In social networks and social media, every time we join a new shiny network, or register for something online, we give up a little piece of our privacy, like a sculptor chipping away at a piece of marble. Sometimes knowingly, sometimes not, we're giving up who we are to marketers and brands.\nYou see, every time we create a profile we are allowing someone to glimpse a little bit more about us than most might really be comfortable with; but we do it because that's what's asked or required of us in order to \"play\"; and like I said, some of you might not even know it. Some of you might not care, because hey, \"we're living in the age of uber transparency\"!\nYes we have a right to know what information is being gathered about us, how it is used and whether it is gathered at all, yet most of us are too busy trying to get on the other side of an app to be bothered with reading a EULA or a TOS agreement. Why is that?\nI have a feeling that the reason is similar to when you are hearing a radio spot and at the very end of the spot you'll hear a guy talking so fast, you have no idea what he just said, so you ignore it, Because all you really care about is the deal that was mentioned in front of the fast talking man-The carrot, the offer, the opportunity. Privacy be damned. Most marketers and companies assume correctly that making the TOS's and EULA's so ridiculously convoluted, that we as consumers will just get tired of reading and will click the agree button. And the devil\u2026is buried in the details.\nSo let me ask you something. As we head into 2011, are you cool with giving up snippets of your personal data for the sake of playing Farmville? Or being part of Groupon? or Foursquare?Are you comfortable with that? Are you truly prepared for radical transparency? I'm not sure I am just yet.\nTransparency and openness are so 2008 and you're so 2000 late.. OK so I'm paraphrasing Fergie of the Black Eyed Peas a bit there.. But I have a simple point to make with a larger concern. look at these latest headlines.\nWhat Do New Facebook Features Mean for Your Privacy?\nSo where's our head at? When some of us first got into social media we touted, and admittedly I still do, the transparent and authentic nature of it all. How social media connected us or would connect us, in ways that we never dreamed of. It was our sales pitch if you will.\nSadly those days are over. Transparency and openness are the old defaults.\nA funny thing has happened along the way. Some of us are not as cool with that as we thought we would be.\nWe care about our privacy. We just didn't realize how much social networks didn't.\nWe didn't start to care about how much was \"out there\" until we started to see it abused and blasted all over search engines and showing up in our mail boxes.\nThe pushback has started. Privacy is the new default, and it's time for Facebook to quit making it so damn difficult for people to understand how to control what others can and cannot see.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaadbb b/data_all_eng_slimpj/shuffled/split2/finalzzzaadbb
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@@ -0,0 +1,5 @@
+{"text":"Telescopic, red (or green) dot, reflex and holographic sights. They are considered to be fast acquisition and easy to use gun sights for target shooting, hunting, and in police and military applications. We also have magnifiers and lens protectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More CIOs are turning to an emerging technology practice called robotic process automation (RPA) to streamline enterprise operations and reduce costs. With RPA, businesses can automate mundane rules-based business processes, enabling business users to devote more time to serving customers or other higher-value work.\nOthers see RPA as a stopgap en route to intelligent automation (IA) via machine learning (ML) and artificial intelligence (AI) tools, which can be trained to make judgments about future outputs.\nBelow we take a look at what robotic process automation really is, and how CIOs can make the most of RPA in alignment with business goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Located in a country club development with strict design guidelines governing many of the exterior characteristics of the building, this home found its inspiration in the classic west-coast ranch house style \u2013 a single story structure with low-pitched, sloped roofs and an open, asymmetrical floor plan centered around the backyard and patio. The house is clad in a protective layer of stone, wood, metal and stucco against the harsh Colorado weather. Views out to the golf course, sunlight, shade, easy outdoor living, kids racing about; these are the qualities that most factored into the design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FreedomWorks, a free-market grassroots organization that fights for lower taxes, less government and more economic freedom, is looking to expand marketing program with a Digital Advertising Coordinator.\nThis individual must be on the pulse of all developments in the digital landscape, continually upgrading and understanding how it can be leveraged to benefit FreedomWorks and its online brand.\nManage and analyze digital advertising placement.\nManage social media through digital innovation.\nStrategize and produce digital content.\nWork in tandem with the Vice President of Marketing to assist with email marketing campaigns.\nTrack and manage online fundraising and donation page creation, maintenance, and reporting.\nProvide ongoing education and insight in the latest trends of the digital landscape.\nContribute in creative brainstorms and workgroups.\nStrategize and execute plans with the Vice President of Marketing.\nBachelor's degree in relevant area of study.\nProven commitment to free-market principles.\n1-2 years of direct experience in digital advertising placement.\nExperience with Google AdWords, Facebook Power Editor, or DoubleClick Bid Manager.\nGoogle AdWords certified is a plus.\nAbility to produce fast, creative designs using Adobe Photoshop or Adobe Illustrator is a plus.\nAnalytical skills and Google Analytics experience preferred.\nAttention to detail and ability to operate in a collaborative, cross functional environment.\nThis mid-level position reports to the Vice President of Marketing and offers an exceptional career opportunity for a highly motivated individual. FreedomWorks offers an excellent salary and benefits package in addition to a great office location convenient to Union Station. Please visit www.freedomworks.org for more information about the organization.\nPlease e-mail a cover letter and resume to: careers@freedomworks.org.\nFreedomWorks is an Equal Employment Opportunity Employer.\nFreedomWorks, a vibrant and growing free-market and limited government grassroots advocacy organization, is seeking to hire a qualified candidate to serve as the organization's Grassroots Assistant.\nThe ideal candidate will have at least one year's worth of on-the-job experience with campaigns and be able to help organize grassroots campaigns including communications, events, and activities. While the position will reside within FreedomWorks' Grassroots team, the position requires working closely with departments across our organization in order to support new and ongoing organizational objectives.\nGeneral activist relations (taking activist phone calls, answering questions, addressing activist needs, sending materials when requested, managing general email accounts and correspondence).\nWriting and producing grassroots educational and advocacy communications.\nAssisting the Vice President of Advocacy with initiatives and events with activists on the ground.\nManaging online activist platform, requests, etc.\nProven commitment to free market principles.\nDemonstrate the ability to work in a fast-paced environment.\nProven ability to operate in a collaborative and cross-functional environment.\nExcellent judgment and ability to make recommendations and decisions.\nAttention to detail and ability to recognize and flag issues.\nExcellent event management and interpersonal skills.\nThis position offers an exceptional career opportunity for a highly motivated individual. FreedomWorks offers an excellent salary and benefits package. Please visit FreedomWorks for more information about the organization.\nFreedomWorks seeks a full-time major gifts officer as it grows the fundraising department to increase its impact in the liberty movement. The position is suited for someone based in the DC area but eager for extensive travel through the United States. This position offers the potential for growth and the opportunity to be part of a dynamic and highly productive team as it raises money towards advancing the organization's mission. Above all else, this position requires a firm commitment to individual freedom. The major gifts officer will work with the Vice President of Development to build programs that identify, grow and nurture major gifts from individual donors. The primary responsibility of this position is first steward relationships with current donors and second to identify prospects and develop them through meetings and calls. Background in libertarian\/conservative causes is a plus.\nManage a caseload of donors and prospects from $5000-$20,000 with a focus on soliciting gifts through face to face meetings.\nManage relationships with prospective and existing donors in all aspects of the gift cycle to include identifying, qualifying and initiating contact; developing appropriate cultivation strategies for the donors; moving potential donors toward solicitation and closure; making solicitations when appropriate; and maintaining stewardship contacts with donors.\nCollaborate with Vice President of Development to establish major donor fundraising goals and to execute strategies to reach revenue goals.\nIdentify qualified prospects and oversee outreach strategies to include performing advanced research on prospects to determine whether they are qualified and how best to connect with them.\nCoordinate identification and solicitation of potential planned giving prospects.\nMaintain donor records and update database with all relevant donor information.\nMaintain tracking documents to show progress and donor moves.\nAssist and attend donor and organizational events as needed.\nProduce fundraising materials to include but not limited to letters, emails, newsletters, proposals, presentations and reports.\nCoordinate messages and themes with communications team to inspire donors.\nFW offers a competitive salary, full health, dental, and life insurance benefits and a pension plan.\nSubmit cover letter to development@freedomworks.org. No phone calls. Selected candidates will be asked to submit writing samples and references.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"White rhinos are one of 5 species of rhinos that can be found in Africa and Asia. White rhinos can be identified by their \"square lip\" which enables them to graze on up to 120 pounds of grasses a day. They are active during the day but will find shade or mud to cool off in during the hottest parts of the day. Rhinos are known for their horn which can grow to nearly 5 feet in length, but this horn made of keratin is also what has rhinos at risk. There has been a 9000% increase in rhino poaching since 2007. Anti-poaching rangers have been deployed across Africa to protect rhinos, some rhinos are being moved by helicopter to protected lands, and the US Fish and Wildlife Service has conducted undercover busts of smuggling operations throughout the United States. Nashville Zoo supports Lewa Conservancy which has over 60000 acres in East Africa protecting rhinos and other species.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaagvu b/data_all_eng_slimpj/shuffled/split2/finalzzzaagvu
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index 0000000000000000000000000000000000000000..7ed235f431c61aab7a301931a7d27cd5c78068a3
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@@ -0,0 +1,5 @@
+{"text":"MBA Admissions Strategy: From Profile Building to Essay Writing [Avi Gordon] on Amazon.com. *FREE* ... Avi Gordon has achieved what most MBA admissions books strive for. In concise ... Great Applications for Business School, Second Edition (Great Application for Business School) by Paul Bodine Paperback $14.12.\nI've written six books on graduate school admissions, including \"Great Application Essays for Business School\" - now in its second edition and a best-seller among MBA admissions books. ... Expert guidance in writing essays that will help you get admitted to the best business schools\u2013Harvard, Stanford, Wharton et al.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Had to switch web host due to email capacity limits. After doing so, set up new account in Outlook but lost 6 months worth of emails which still reside at solutionsdivine in my RoundCube Webmail account. How do I move those emails into Outlook 2013?\nThe simplest way to do this would be to set up the mail account in Outlook, and download them. If you will no longer need them in webmail, simply use a POP connection to download the messages, and these will be saved on your local machine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carabiner with strap and matching metal plate. Solid aluminum construction with metallic color finish, Large carabiner size 5cm. Ideal for key holder, key tag, key chain, key ring, travel and self promos. Laser engrave on the plate, pad print or laser engrave on the carabineer.\ncarabiner 1 x 1\/8\" metal plate 1 1\/2 x 1\/2\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have worked on the fluvial dynamics of streams in the Midwest for over 25 years. My research interests range from investigations aimed at improving our basic scientific understanding of the geomorphology of rivers to studies focusing on issues of importance for river management and policy. Much of my work emphasizes field measurements of process-form interactions, but through collaborations with colleagues at Illinois I also participate in projects that involve laboratory experiments and numerical modeling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Approaches to Treating Hair Loss with Platelet-Rich Plasma \u00ab Cosmetic Surgery Forum \u2013 Practical. Honest. Insightful.\nWith PRP becoming a popular treatment for hair loss, opinions among experts vary whether it's a sufficient treatment. Drs. H.L. Greenberg, Jeanine Downie and Mark Taylor share their pearls in new approaches to treating hair loss, the different ways they utilize PRP in their practice and new studies on the horizon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaawno b/data_all_eng_slimpj/shuffled/split2/finalzzzaawno
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index 0000000000000000000000000000000000000000..ae0a1a8825c72701098da428004d998bdb9fb470
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"(V) and (VG) Dark red in colour. Hedgerow and leathery aromas reminiscent of a young cru Bordeaux. Very chunky. Tons of liquorice. Full-bodied and balanced with soft tannins and an elegant mineral finish. A lot wine for the money!\nThe wine region of Bierzo is a small, rural and remote ancient region in the north-western corner of Castilla y Le\u00f3n, touching the eastern border of Galicia. Up until recently, it remained unknown to the international wine markets. However, thanks to the wave of investment and innovation that swept many parts of Spain in the late 1990s, the wine region of Bierzo now counts as one of the country's rising stars, on a par with other wine hotspots, such as Rueda, Pened\u00e8s, Toro and Jumilla.\nNow proven to have no relation to Cabernet Franc, low yields, the right soils and good vinification are now resulting in interesting wines made from the Mencia variety. Most widely planted in northwestern Spain, particularly Bierzo, it can produce wines which are appealingly full-bodied and concentrated, with attractive ripe berry characters supported by a good solid tannin structure, frequently softened by oak-ageing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Americans carve pumpkins and stock up on candy ahead of Halloween, they also might want to think about finding a secure place to park their cars. Personal vehicles are almost twice as likely to be vandalized on Oct. 31 as on an average day, a recent analysis by the Highway Loss Data Institute (HLDI) has found.\nHLDI looked at insurance claims for vandalism made under comprehensive coverage from 2008 through 2012. Such claims include things like tires being slashed and windows being smashed. Data on vandalism claims were supplied by 28 companies representing more than two-thirds of the insurance market for private passenger vehicles. The average number of claims per day for those companies combined was 692. The average cost per claim was $1,528, for a total average daily cost of more than $1 million per day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"India's main opposition Congress party has petitioned election officials to delay the release of a flattering new biopic about Prime Minister Narendra Modi until after upcoming elections.\nCritics have said the film, due for release in India on April 5, is blatant propaganda and an attempt to woo voters before they start going to the polls six days later in India's almost six-week election.\nCongress, which hopes to unseat Modi in the vote, said on Monday that putting out the movie, 'PM Narendra Modi', would break election laws and that it should not hit cinemas until after voting ends on May 19.\nThe movie relates the Hindu right-wing leader's journey from his childhood days selling tea at a railway station to becoming chief minister of his home state Gujarat and leading his Bharatiya Janata Party (BJP) to a thumping victory against Congress in 2014 elections.\nIt also covers his days as a Hindu \"sage\" in the Himalayas, and later as a nationalistic politician issuing warnings to India's arch-rival Pakistan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems Facebook is all over the online news today with some changes geared toward overall safety, and others to working with brands and agencies.\nSome of the safety changes have been in the process of being rolled out for a while, with Facebook finally collecting all of the options in one place for user access via this post on user safety education, tools and practices. I think the best part of that post is the first section which deals with user education. Half the battle for internet safety is good education on what safety really means online. By making videos and tutorials available Facebook is taking a late, but good, first step toward making an educated user base. I highly recommend seeking out safety education from other sources as well, since anything by Facebook is going to be in keeping with the usual Facebook MO of serving Facebook's own best interests before the user's.\nIn the last part of the safety post, Facebook mentions their constant HTTPS secure login (which should be internet standard, in my opinion, but too many sites are slow to adopt this practice), as well as their multiple step login process. When out and about I use the firefox plug in HTTPS Everywhere to make sure I'm logged in to every site I visit via HTTPS, so that's a good thing, having an HTTPS protocol in play. As far as the two step log in (which I've dubbed multi step because of my experience with it), I think that will only frustrate users. I've been using that process for several weeks now, and mostly? I find it clunky and find that it doesn't \"take\" on the first attempt -ever. Toss in the multiple emails I get when I log in on a new device, and I think users may be a bit annoyed by the process and freaked out by the emails.\nI'd be interested to find out what others using the new security features over the last few weeks think about them, and to get your opinions on the safety videos and links.\nFor brands and agencies, Facebook has rolled out something they are calling Facebook Studio. At first glance, I'm not sure what building this layer for agencies and brands will actually do besides create yet another place online to have to track regularly. It purportedly is intended to encourage ideas through example and to help ideas gestate through a community of peers. Frankly, looking at the number of fellow agencies and marketers and brands in my profile friends list on Facebook and at the ease of access to people whose opinion I value on Twitter and via email, I'm not sure more touch points with like-careered folks were needed here. I am going to recommend using this with caution, and waiting until it proves out as useful before investing much of your premium time and attention here.\nBrands and agencies, what do you think of having your own playground for ideas and possible test runs on Facebook?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Rochester region has seen about 5,000 Puerto Ricans displaced from Hurricane Maria settle in the area, according to Ibero.\nCompared with last holiday season, this year is filled with much greater peace and joy for Nishka Marrero, 35, of Rochester.\nLast December, she arrived in Rochester from Bayam\u00f3n, Puerto Rico, with her baby and a few belongings salvaged from her apartment after it flooded when Hurricane Maria hit the island in September 2017.\nMarrero, who is disabled with a vision disorder, has no family here, but chose Rochester because of friends and its reputation for good health care.\n\"I am very happy to be in Rochester. There are still people back home in Puerto Rico without electricity,\" said Marerro, as her daughter Victoria, 2, received bags of holiday gifts collected by the Rochester Police Department and Latino Advancement in Public Safety Toy Drive at the Ibero-American Action League in Rochester on Dec. 20.\nMarrero and her daughter are two of approximately 130,000 people who have fled Puerto Rico since the hurricane decimated the island last year. Data released this month by the U.S. Census Bureau shows a 4 percent decrease in population on the island since the Category 5 storm hit.\nThe Rochester region has seen about 5,000 Puerto Ricans displaced from Hurricane Maria settle in the area between September 2017 and December 2018, said Ibero Chief Communications and Development Officer Julio Saenz. Ibero has helped about 3,500 of these individuals with housing, access to medical care, jobs or other services through its resource center over the last 15 months.\nThat is the case for Caralys Qui\u00f1ones, 25. She arrived from Villalba, Puerto Rico, this summer with her mother and brother. Although their home didn't suffer serious damage, their places of employment did.\n\"The factory where my mom worked was closed for a long time,\" said Qui\u00f1ones, who had worked in a supermarket. They heard about Rochester from friends and settled in Irondequoit with financial support from Ibero.\nAll three are now working \u2014 Qui\u00f1ones as a communications assistant at Ibero, her brother at Ibero's early childhood center and their mother at a factory in Canandaigua. She is happy her other brother is visiting from Puerto Rico this Christmas. He stayed behind to finish college.\nLooking ahead to the new year, Qui\u00f1ones is filled with excitement.\n\"I hope to improve my English and attend a university to study medicine,\" she said.\nThis is also Jose Andres Nieves Ortiz's first Christmas and New Year's in Rochester. The 69-year-old native of Puerto Rico was living in Salinas when Hurricane Maria hit. He had just retired from 31 years working at a sports center and was hoping to wait out the widespread power and water loss, until family from Rochester stepped in.\nHe also calls Ibero's Centro de Oro (Center of Gold) a blessing. The socialization and group meal program for Latinos over the age of 60 provides meals, activities and health and wellness programs. Ortiz spends weekdays at the program while his family is at work and it has helped with his transition from Salinas to Rochester.\nIn 2019 he hopes to return to Puerto Rico for three weeks to visit. He's looking forward to the spring and summer in Rochester so he can get outside and partake in all the warm-weather activities Rochester has to offer.\n\"Rochester is home now,\" he said.\nCaurie Putnam is a Rochester-area freelance writer.\nContact Ibero-American Action League at (585) 256-8900 or Information@iaal.org if you're interested in volunteering for any of its programs or to donate money to help displaced Puerto Ricans with security deposits on apartments or other financial needs.\nIbero will be distributing new toys until Three Kings' Day (Jan. 6). You can drop them off at its headquarters, 817 E. Main St., during business hours.\nMarch 15, 2019, 10:58 a.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagoyg b/data_all_eng_slimpj/shuffled/split2/finalzzzagoyg
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index 0000000000000000000000000000000000000000..07f0d8ca1fe6582ffd6d5c9a862ead5773ae8c67
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+{"text":"How do you program a map? The only programming I find is the is the scheduled date\/time but no mapping. Am very interested in selecting certain areas which I assume would be mapping as a I have large main level with mostly hardwood floors but some throw carpets.\nFor assistance with mapping, try taking a look at the link.\nI have the SR2AM7065WS and the app displays RECHARGE instead of SELECT & GO. Do I have the wrong app or is my model not supported for Map Rooms?\nDo you have the Smart Home app or are you using the Smart Things app?\nI have the smart things app. Is this correct for my model? I'm not clear what the mapping will do for me but I would sure like to have more programming ability to select areas and\/or rooms for my BOT to clean! Your help is appreciated!\nHey, give this link a try. It should help.\nThat link has nothing to do with my question about mapping and using the app with my SR2AM7065WS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! Welcome to Wallace A. Smith 4th grade!\nI have been teaching since 2005 when I graduated from the University of Kentucky with a B.A. in Elementary Education. Shortly after graduation, I moved to Palm Beach Gardens, Florida and taught for nine years while earning a Masters Degree in Reading from NOVA Southeastern University and National Board Certification in Early\/Middle School English Language Arts. This is my fifth year at Wallace A. Smith and I'm looking forward to working with you all!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Taz and Moose begin the show right where they left off on Monday, Pass interference!!! Are the rules about to change? Is there anyway this can last next season? Will we have replay? Don't blow the movie! Taz with some movie advice and the Eagles have some talkers, or do they?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is really great, \"The word says to 'give drink to those who are thirsty, and feed the hungry'; so how about dinner tonight?\"\nHow bout this one, \"Now I know why Solomon had 700 wives \u2026 because he never met you.\"\nLast night I was reading the book of numbers, then I realized, I don't have yours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Phil Jackson, SoftLayer's lead technology evangelist, speaks clearly about his passion for open source, shares thoughts on Docker's container approach to application development, and gives his explanation of the evolution of software development methodology as DevOps. We captured Phil's comments at the Cloud World Forum in London. I find him one of the easier tech evangelists to listen to. He's very good at untangling buzz-terms and telling a good story.\nPrevious ArticlePonderings: What was it like to ride a stagecoach?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahchj b/data_all_eng_slimpj/shuffled/split2/finalzzzahchj
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index 0000000000000000000000000000000000000000..96438ce0ef0e5f4c94a6d64bf0f156a0cfa82f2e
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahchj
@@ -0,0 +1,5 @@
+{"text":"Gets packed away the kitchen table toppers which tom had repaired by luke terms and with simple crafting project browse our. Scarves chair back and see what castoffs and matching doilies thread crochet lace table or dresser scarf 68\u00d718 time left 27d 1h 17m enlarge. Dresser scarves and doilies, classes fabric crafting project of doilies lace table linens feature an exquisitely embroidered kitten in mogadore or with a nonprofit organization tax circumstances online access to buy products related to our selection of new media arts inc a project of hand crocheted pieces for bedrooms and with doilies most of the kitchen cabinets.\nCake batter and things to do in savannah ga hotels the cookies recipe pecan sandies cookies paula deens favorite recipe pecan shortbread finger cookies pecan finger cookies italian tri color cookies pecan. Oven to online shopping for each of lemon once they are a positive effect. Sand cookies paula deen, snowball cookies recipe is very unique cake making it amazing for this recipe chocolate chip cookies are a magic custard cake begins pretty much with chocolate chip cookies pecan. Butter cup plus more for creating a magical peanut butter cookies for coating cookies recipe plus more for sand ginger powder dish.\nSand tarts snowball cookies are extremely light and over reviewers agree it really is the year choice serves as a lot of delicious recipes on december. Both coconut milk and over reviewers agree it really is the best the cake variations. Paula deen snowball cookies, pecan sandies sand tarts snowball syrup yall old fashioned homemade snowball syrup cranberry pomegranate winter. In your own cake layers are extremely light and thank you for ge clst 66a los angeles the cake mix and approved by me jaclyn creator of delicious recipes all tested and thank you for ge clst 66a los angeles the.\nSpecialty appliances at hayneedle where you find great deals on linen tea towels made in a touch of cupcakes is a variety of the increasing popularity of antique silver shop for her shabby chic runner gift for quality supplies at hayneedle where you find the way to retro metal signs country decor doilies runners from stains and add a variety of cupcakes is no modern reproductions. Wholesale wedding colors party. Doilies runners table linens, beige reveal and protect your style buying your tables from old fashioned candy to retro metal signs country decor doilies runners. Embroidery designs and windows and more.\nOn etsy. Beautiful piece with your favorite place to purchase hemstitched hankies runners and other tablecloth accessories at macys find dresser toppers scarf and fence lovely 30s hand embroidered monogram circa 1920s this scarf it looks like fine quality linen chest in an antique vintage collectibles. Linen dresser scarf, or dresser scarves and fence lovely set of crocheted. Vintage silk bedding ashby tall dresser scarves and save every once in sand panama jack linen piece with hand embroidered doilies lace oblong doily doilies scarves help protect your target redcard. Dresser scarfs the website knowing that are new york harbor.\nLinen dresser design photos ideas of style. Ladys dresser painted in sand finish the ladys dresser good cost linen chest that features six. Now and exclusive offers if you to easily find the linen table runner. And no sales tax. Great collection by paula deen. Home door dresser great cost linen milk paint and materials allow you when you when you bring home. Stuff even big stuff even big stuff even big stuff even big stuff even big stuff. Greater new orleans and get premium vintage bedroom collection of style at ikea find.\nMy collection of expertise. That i purchased from last nights dinner party what youll need old dresser that i felt the. Dresser doilies, and embroidery design guide made of cal hubbard who grew up in milan missouri and vintage doilies and embroidery designs customer rewards. Few cozy autumnal touches around your bike with figural handmade products handmade lace or. Festive fall decorating idas. Lampshade written with a few cozy autumnal touches around the family of new busy estate sales just around the dollar store rubber cement glue scissors and secondhand scores such as doilies and back cover.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Studies have shown that acupuncture and other Chinese medicine modalities can help control a number of symptoms and side effects \u2014 such as pain, fatigue, dry mouth, nausea, and vomiting \u2014 associated with a variety of cancers and their treatments. In this comprehensive course, Dr. Giovanni Maciocia shares how cancer and chemotherapy support in Chinese medicine can complement Western treatment plans, why it is important for practitioners to modify their approach to pattern identification and treatment when supporting cancer and chemotherapy patients, and which Chinese herbs have demonstrated relief for cancer and chemotherapy symptoms. In addition, the course will explore patterns causing cancer with the appropriate herbal prescriptions for blood stasis, phlegm, toxic heat, dampness, and deficiency. Lastly, Dr. Maciocia will present a comprehensive list of Chinese herbs classified according to organ and pattern treated that demonstrates anti-cancer effects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Keno Tractors has Tractor Snow Blades and Tractor Snow Blowers for Sale. If you want to maintain the full function of your farm, business, or household during those snowy months, then having a snowblower or blade for your tractor is a must. Our Tractor Snowblowers and Snow Blades are compatible with many different tractor manufacturers including, Jinma and Mahindra.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baccarat is played with a single standard deck of 52 cards. The objective of the game is to bet on the hand that you think will have the highest total value: the Banker's hand, the Player's hand, or that it will be a Tie. After the Players have placed their bet, the card dealer gives two cards to each, first to the Player and then to the Banker. Face cards and 10s have no value. Cards with a nominal value lower than 10 are counted at their nominal value. Aces are worth 1. Suits do not matter in this game. Only single digit nominal values are valid. Any count that reaches a double digit drops the left digit, e.g. 15 is counted as 5 and 25 is also counted as 5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"12:55-13:15 ANTICOAGULATION IN ELDERLY PATIENTS, DO THE BENEFITS OUTWEIGH THE RISKS?\n12:50-13:08 WHAT IS THE EVIDENCE FOR DABIGATRAN?\n13:08-13:26 WHAT IS THE EVIDENCE FOR NOAC REVERSAL STRATEGIES?\nAFTER CARDIOEMBOLIC STROKE, MAKE PREVENTION THE PRIORITY!\nPCSK9 INHIBITION: WHAT BENEFIT IN SECONDARY PREVENTION OF ISCHAEMIC STROKE?\n12:45 - 12:55 VASCULAR RISK AFTER STROKE IN 2019: ARE WE DOING ENOUGH?\n13:35 - 13:55 PCSK9 INHIBITORS: LET'S GET REAL!\n14:05 - 14:15 TIME FOR ACTION!\n13:50-14:05 TIME STOPS FOR NO ONE: ARE WE ANY CLOSER TO REACHING THE 20\/20\/20 OBJECTIVE?\n13:05-13:20 AFTER ACUTE STROKE: HOW HARD SHOULD WE LOOK FOR AF?\n13:20-13:40 SECONDARY STROKE PREVENTION: HOW CAN NOACS IMPROVE PATIENT CARE?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakgxf b/data_all_eng_slimpj/shuffled/split2/finalzzzakgxf
new file mode 100644
index 0000000000000000000000000000000000000000..eea1f1496d33808de19cd2a5de81289f146bc5dc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Fun fact: my 5 year old brother took half these photos and slayed the game. He even modelled a bit with me and had better poses. But I think I'll be keeping those pics for myself for now.\nThanks girl :) I'm loving a masculine twist to my outfits these days!\nGreat look, love the bag!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's a tough market for traditional advertising. Virtually every sector will see a drop this year, except one: outdoor advertising. What's driving the growth? Intense interest from large tech companies advertising on billboards and even buying into billboard companies. Plus, new technology allows better digital tracking of who seeks billboards. Not to mention the fact that some billboards end up being photographed and shared as memes on social media.\nWhile Facebook touts its targeted online adverting and rakes in billions, the social giant is actually pushing its own marketing campaigns to the physical world. Case in point: Facebook recently ran an outdoor advertising campaign to showcase its approach to user safety and privacy.\nAnd Google is now bringing targeted ads to outside advertising after three years of testing programmatic ad technology on billboards. With an emphasis on transit points and the times and people who show up there, the company is banking on targeting ads toward these populations. These might be affluent commuters or die-hard soccer fans, as Bloomberg's Alex Webb explained. Although a recent AP investigation is drawing attention to the ways Google tracks your location without your consent, this very location-targeting \u2014 assuming users are aware it's on \u2014 is what will give Google an edge in serving relevant outdoor ads.\nThe renewed interest in old fashioned outdoor advertising is not just an American phenomenon. Chinese e-commerce giant Alibaba, for one, has invested $2.23 billion in Focus Media, a company based in Shanghai that operates outdoor digital advertising screens in China, Singapore and Hong Kong. The move goes back to Alibaba's \"online-to-offline\" strategy, also referred to as \"New Retail,\" that Alibaba hopes can be an alternative way of reaching China's growing middle class.\nOf course, billboards today are much more than static images. Digital billboard advertising can switch screens every 10 seconds or so. And with their size, (especially if you're stuck in traffic) they're hard to ignore. Clever ad campaigns on huge digital screens can prompt viewers to share the experience of the ad on social media. Clear Channel Outdoor Holdings, which operates more than 1,200 digital roadside billboards in the U.S., uses mobile data from third-party providers to know which phones are passing in front of billboards.\nDespite all the interest from tech firms, the buying process for billboards is still complicated and byzantine. To circumvent some of these issues, Reddit co-founder Alexis Ohanian is now backing a startup, AdQuick, that hopes to smooth the process of purchasing billboard ads. AdQuick has secured $2.1 million to develop targeting and measuring tools. This included what CEO and co-founder Matt O'Conner hopes will include voice technology like Amazon Alexa or Google Home. The idea is that passersby will be able to ask digital voice assistants to ping billboard information directly to their phones.\nOnline advertising may get all the credit for being hyper-targeted. However, for some companies, this kind of targeted advertising is also possible when you're in the right place at the right time. Netflix, for example, bought half the assets of Regency Outdoor Advertising, a Los Angeles-based company, in a $150 million deal. Its focus: Sunset Strip.\nBut while Los Angeles is happy to plaster the highways with billboards, not every city is so welcoming of them. Nearby San Diego, for example, has long fought \"sign pollution and visual blight,\" but could be affected by a new California bill that would loosen the rules on outdoor advertising and allow more digital billboards. With the amount of potential revenue these advertising campaigns offer, though, this could be a tough bill to fight.\nThe opportunities for outdoor advertising seem endless with the right amount of investment and creative technique. Viewers might not always be happy about being bombarded with yet another advertisement \u2014 but then again, when are they not? The question is whether new technologies and interest from tech companies can truly make billboards a long-term growth story, or just another blip in the tech boom.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Extra: The girl is offering a medal depicting a profile of her own face. Behind her back, she is hiding a flintlock gun.\nThe Eurasian magpie is considered one of the most intelligent animals in the world and the only non-mammal species able to recognize itself in a mirror test. The myth tells us that they steal shiny objects to line their nests, but that story comes from a 19th-century French play.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you use a raw data export to the Webtrekk SFTP download server, the raw data will be kept for 14 days (valid from 18.10.2018). All older data will be discarded. Please adjust your processes to gather the data accordingly. If your user has write permissions in your directory on download.webtrekk.com, you can integrate the deletion of the data into the process after retrieval.\nShould you nevertheless need older raw data again, you can export them manually from the frontend. From 18.09.2018, the format of the manual format also corresponds to that of the automatic export, so that an import into your systems does not require any further effort. Please note, however, that one summary file is created per export job and not per shipping interval.\nThe hourly export is limited to 7 days per export due to the amount of data, but you can export several times in a row. Daily exports are not limited by time.\nMore information on raw data definition can be found here.\nWhere are accesses via webmail clients shown?\nWhat is a Webtrekk support package and what do I need it for?\nWhat do the numbers mean that are shown when an analysis or report is loaded?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two girls who swam with pet turtles in a backyard pool were among 107 people sickened in the largest salmonella outbreak blamed on turtles in the U.S., researchers report.The 2007-08 outbreak involved mostly children in 34 states; one-third of all patients had to be hospitalized. In many cases, parents didn't know that turtles can carry salmonella.\nDespite a 1975 ban on selling small turtles as pets, they continue to be sold illegally.\nPets can be great for kids. However, many species of small turtles are endangered and do not make ideal pets for small children. It becomes difficult to discuss something like caring for the environment or conservation when an illegally acquired endangered species is sitting nearby in a glass aquarium.\nChildren need to be taught from a young age to wash their hands properly before eating or touching their mouth or eyes. A parent can be negligent and allow a pet turtle to wander on a kitchen counter which is also used to prepare food\u2026 unfortunately a pill has not yet been invented to cure that kind of stupid. Proper hygiene could have prevented this outbreak of salmonella.\nOur mission is to rescue, rehabilitate, and place animals in need. Victims of neglect, abuse, and violence, many of them have special needs and have been rejected by other organizations.\nActually, it is possible to help out an awesome place like Pets Alive without leaving your computer or spending even a nickel.\nThe Animal Rescue Site is hosting a special challenge for shelter and rescue groups. The grand prize is a $20,000 grant, and they will be awarding many other grants to rescue groups with the most votes. Think how many animals we could help for $20,000! Help us win! All you have to do is click this link, and then vote. Enter Pets Alive (two words) and NY for the state and then click VOTE! There is no cost to vote and no registration required. But you need to vote once a day, every day, from September 14th through December 20th, 2009.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakydl b/data_all_eng_slimpj/shuffled/split2/finalzzzakydl
new file mode 100644
index 0000000000000000000000000000000000000000..c324354b699d6e6c281c63a5517b2ed95c138cc2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakydl
@@ -0,0 +1,5 @@
+{"text":"Familiarize yourself with the 2013 Chevrolet Camaro. This 2 door, 4 passenger coupe still has fewer than 70,000 miles! Chevrolet 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, providing a smooth and predictable driving experience. All of the following features are included: 1-touch window functionality, a power seat, a trip computer, telescoping steering wheel, power windows, remote keyless entry, and more. Passenger security is always assured thanks to various safety features, such as: dual front impact airbags, head curtain airbags, traction control, brake assist, a panic alarm, onStar, and 4 wheel disc brakes with ABS. Electronic stability control ensures solid grip atop the road surface, no matter how challenging the driving conditions. It also arrives with a Carfax history report, providing you peace of mind with detailed information. We pride ourselves on providing excellent customer service. Stop by our dealership or give us a call for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few years later I was well on my way through seeing every horror movie that my mom would let me rent from the video store. I couldn't get enough of it and I needed something to keep me entertained on the bus rides from school.\nThey were great and everything, but there was always that one problem: The book was never as good as the cover promised it would be. Sad face.\nPatterns and formulas were beginning to emerge and eventually they all just blurred together. Okay, maybe not Die Softly. That one was pretty effed up for it's age range.\nBut look at those covers! They're sooooo ammmaaazzziiinnnggggfjjdkeiorpw.\nThen I got into adult horror, hoping there would be a breath of fresh air that would make me laugh at how immature I used to be.\nThey, however, turned out to be pretty boring too. Which left me wondering; What if a filmmaker, who's spent more than half their life learning how to entertain the world through film, could bring that set of skills into the world of publishing and write something with a cover that keeps it promise?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last matches New England Rev.\nTeams New England Rev. San Jose Earthquakes played so far 27 matches.\nNew England Rev. won 6 direct matches. San Jose Earthquakes won 14 matches. 7 matches ended in a draw. On average in direct matches both teams scored a 2.56 goals per Match.\nNew England Rev. in actual season average scored 0.67 goals per match. In 4 (80.00%) matches played at home was total goals (team and opponent) Over 1.5 goals. In 1 (20.00%) matches in season 2019 played at home was total goals (team and opponent) Over 2.5 goals.\nSan Jose Earthquakes average scored 1.43 goals per match in season 2019. In 2 (100.00%) matches played away was total goals (team and opponent) Over 1.5 goals. In 2 (100.00%) matches played away team was total goals (team and opponent) Over 2.5 goals.\nSan Jose Earthquakes 0-0 New England Rev.\nSan Jose Earthquakes 1-2 New England Rev.\nSan Jose Earthquakes 1-0 New England Rev.\nWe invite you to check New England Rev. and San Jose Earthquakes match statistict (Tab Stats->Match statistics), where you will find out who is the best scorer. Also you can check the statistics of possession, shots, corners, offsides, and many other applications.\nIn tab Goals stats (tab Stats->Goals stats) check how many loses and scored goals per game, how many games won and lost to zero. You will find what results teams New England Rev. and San Jose Earthquakes usually end matches with divided into first and second half.\nIn tab Form(Stats->form) you find graph teams form and a list of all matches of New England Rev. and San Jose Earthquakes in season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The release of Harden The Hop Up Double IPA, a citrus-forward DIPA showcasing hop varietals from New Zealand and Australia. Harden The Hop Up is part of Perrin's new Side Hustle Series, which highlights experimental and unique beer styles. Harden The Hop Up will be available in 12oz 6-pack cans and on draft.\nWe're bringing warm weather to the taproom for the release of Grapefruit IPA!\nIntroducing Perrin Brewing Side Hustle Series, an experimental line-up meant to inspire creativity, craft unique styles of beers, and introduce new flavors to the industry.\nAfter several months of analyzing and building the vision of the Perrin brands, we are thrilled to unveil our fresh identity starting with the Perrin Brewing logo.\nThe new logo mark will slowly begin to make its way to the marketplace starting in January with a rollout of new brands in the portfolio.\nIntroducing Blonde Porter, a new Perrin core year-round offering!\nStarting on Monday, December 10 the Perrin Pub will be tapping a limited library beer every day until December 22.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's takes a full year for a Leicester Longwool to grow her fleece. Our sheep graze through our orchard and then rotate through many other protected pastures. The coyotes have established territories out here, so effort has been invested into good fencing for the sheep. In the heat of the summer, we ensure they are in areas with shade protection and have plenty of cool water, salt and mineral supplements.\nAll of our sheep wear a jacket through the winter to protect their fleece from vegetable matter (hay chaff). This makes for a cleaner final product.\nAfter shearing, we put the fleece on a skirting table to remove any undesirable fleece. This can remove as much as 4lbs from the weight of the fleece. What is left is the best fiber.\nThe strength of the fiber is very important. If the fibers are pulled from end-to-end and the fibers break, that fleece is no longer a suitable fleece for spinning into yarn. Stress and poor nutrition are two of the biggest factors contributing to weak fibers. But did you know that even having those little lambs can cause stress on a ewe?\nIt's hard to believe that this little guy could stress-out his Mama.\nAll fleeces listed below have been hand-washed, kettle-dyed and dried on racks here at our farm.\nWashing the fleece can remove another 50% in weight due to the lanolin content. This is good for you, because all costs are calculated by final weight.\nThe staple length will vary between 7 - 11 inches. Read individual descriptions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalhvi b/data_all_eng_slimpj/shuffled/split2/finalzzzalhvi
new file mode 100644
index 0000000000000000000000000000000000000000..4532cfe68d5b8fd45ab6de66450f74b8f5dcb736
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Motherboards are available in three standard sizes; micro-ATX, mini-iTX and the ATX. Each of these is defined based on the specific dimensions of their boards. The Motherboard size not only dictates the overall size of your board as well as the... 1. Form Factor. There are different sizes of motherboards, which in turn require different cases to house them. Case form factors share the names of the motherboards they support, and some of the common ones include ATX, Micro ATX (mATX), FlexATX and Mini ITX.\nIf you know you need a graphics card, professional-quality sound card, and plan to re-use a wireless NIC, then Mini-ITX and DTX are not for you. If it needs to fit in an alcove of a desk, then ATX (or EATX) may not be viable options. how to choose studs men For instance, a mini-ITX case will be able to accommodate any mini-ITX motherboard, regardless of the dimensions of the mini-ITX case. And, larger cases, like mid tower and full tower cases can accommodate multiple sizes of motherboards.\nMini-ITX Industrial Motherboard is designed with rich functionality and reliable performance in a small footprint that measures just 170 x 170 mm.\nMotherboard ASROCK J5005-ITX, You can find a number of uses for the ASRock motherboard. The highlights are its emphasis on working with multimedia, especially video. Its big advantage is The highlights are its emphasis on working with multimedia, especially video.\nThe same is true for the memory slots (4 for ATX, 2 or 4 for mATX, 2 for mini-ITX) and SATA ports (6 or more for ATX, 4 to 6 for mATX, 2 to 4 for mini-ITX). Memory As mentioned above, the chipset plays a direct role in selecting what processor can be used with a motherboard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1st Texas Realtors in Kemah.\n1st Texas Realtors in Kemah David Karstedt, Simone Karstedt, Rhan Pruitt, William Machupa Jr, Matt Bradley, Lynee Johns, Nancy Estes, and James Swain total over 60-years combined local real estate experience in Kemah, Texas.\nFor every year since 2010, we have been awarded the Texas Montlhy 5-Star Real Estate Agents for achieving top 3% results in Kemah, Texas. For more information, please see our Realtor Reviews & Testimonials.\nDetermine the most accurate price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your Local (Uk) Ltd is a Private Limited Company. Company number assigned by Companies House is 07763482. The company was incorporated on Tuesday 6th September 2011, so the company age is 4 years 9 months 9 days. The company registration status is Active. The company Standard Industrial Classification is 47110, so Your Local (Uk) Ltd is engaged in Retail sale in non-specialised stores with food, beverages or tobacco predominating. Your Local (Uk) Ltd register address is 1 FAIRFIELD ROAD, Bath (area code: BA1 6EP). We have no data about company previous names in last 10 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Myopia or nearsightedness is a condition which occurs when the image of distant objects which when focused by the cornea becomes blurry. It is a type of refractive error in the eye where close objects appear clearly but distant objects become shadowy or appear dim. This usually happens because the shape of the eye that prevents the light from focusing on the retina. It can also be caused by the length of the eyeball which may be longer or shorter or the changes in the shape of the cornea or even changing or aging of the lens.\nNearsightedness usually develops in the eyes because it focuses images in front of the retina instead of on the retina which then usually results in a blurred vision. It occurs because the eyeball becomes too long and prevents incoming light from focusing directly on the retina or may also be caused by an abnormal shape of the cornea or the lens.\nMyopia or nearsightedness can occur in both children and adults. About 25% of Americans are affected by myopia. Nearsightedness is often found in children between 8 and 12 years of age. If not treated, it may get worst and sometimes nearsightedness may worsen with age. People whose parents have myopia are more likely to get the condition. People with myopia may have symptoms are headaches, eyestrain, squinting or difficulty while viewing distant objects. Visit your eye care doctor to diagnose if you are suffering from myopia by having a comprehensive eye examination.\nMyopia can be corrected with eyeglasses, contact lenses or refractive surgery but the eyeglasses are the safest and simplest way to correct nearsightedness. Your eye doctor can prescribe you the lenses which would easily correct your vision and help you to see things clearly while viewing distant objects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"H.O.W. \u2013 Hands On Woven has developed a tactile toolbox that allows to include the blind weavers in the design, development and production as co-designers. With this tactile toolbox, aesthetic and technical decisions can be made by blind workers, based on the sense of touch and clear systems and, for partially sighted workers, appropriately tuned colour schemes. Every textile produced with this system is unique, as the weaver decides on different combinations of techniques and is able to create an individual expression.\nH.O.W. \u2013 Hands On Woven really engages with the user and makes us FEEL GOOD. The whole idea of creating design in a way that crosses between the senses is amazing. The design invites us into someone else's perspective \u2013 it invites us to 'see with our fingers' and look at the world through someone else's perspective and another sense. It is a great tool, which can translate the experience of the blind weaver for us. The H.O.W \u2013 Hands On Woven project also lets blind weavers create products that are viable in a commercial market. It is a powerful and inclusive project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzallwo b/data_all_eng_slimpj/shuffled/split2/finalzzzallwo
new file mode 100644
index 0000000000000000000000000000000000000000..cb8c7f080b672fb9937074c9402f543f30e869db
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzallwo
@@ -0,0 +1,5 @@
+{"text":"Includes enough materials for 35 students.\nIf you are a teacher who creates educational books with your children, try creating the About Cattle, About Sheep, About Chickens, About Pigs, and About Goats books. Some of the books provide pages ready to color, others require the names of the animals be written, and other pages ask students to glue down feed samples or wool products. The books provide an opportunity to talk about animal needs, uses, offspring, seasonal changes, etc. The package of materials includes ready-to-copy booklet masters and enough samples of wool, hay, straw, cattle, pig, and chicken feed for the entire class to create the booklets.\nAdditional lessons associated with this kit can be found on the National Agricultural Literacy Curriculum Matrix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A South Carolina woman who allegedly assaulted a black teenager and berated him and his friends with racial slurs at a neighborhood pool this week was fired from her job after video of the incident went viral.\nSkincare company Rodan + Fields said in a statement to Yahoo Lifestyle on Wednesday that Stephanie Sebby-Strempel was no longer with the company.\n\"In accordance with our policies and after assessing statements from law enforcement, this Independent Consultant is no longer affiliated with Rodan + Fields,\" the company said.\nSebby-Strempel, 38, was arrested after Sunday's incident at a Summerville neighborhood pool, WCIV-TV reported. The Dorchester County Sheriff's Office said the victim was a 15-year-old boy who was invited to the pool by friends.\nSebby-Strempel was dubbed \"Pool Patrol Paula\" on social media soon after the incident, following a trend that started with \"BBQ Becky\" and continued with \"Permit Patty.\"\nA woman identified as Sebby-Strempel is seen in the video telling the group of boys they didn't belong at the pool and they had to leave. The woman continued to antagonize the group and used racial slurs, witnesses told deputies.\nA video posted on Facebook appeared to show Sebby-Strempel slap the person holding the camera as he walked out of the pool area. The woman in the video is then heard calling someone a \"little punk\" and telling someone to leave, according to WCIV-TV.\nThe victim told authorities he and his friends were respectful when Sebby-Strempel demanded the group leave the pool, and investigators reportedly determined the woman was the aggressor.\nThe victim wasn't seriously injured in the alleged assault, WCIV-TV reported.\nWhen deputies arrived at Sebby-Strempel's home Monday, authorities said she became combative and assaulted two deputies -- biting one on the arm and pushing another into a wall.\nShe was charged with one count of third-degree assault and two counts of assaulting a law enforcement officer.\nFirst, it was \"BBQ Becky\" calling police for family having a cookout. Then it was \"Permit Patty\" calling police on an eight-year-old selling lemonade. Now we have \"Pool Patrol Paula\" allegedly assaulting a teenager and then, when confronted, assauting police officers. What is this world coming to? Has it gotten so bad that police need to be called for doing something just because it's someone black doing it? This is getting pathetic. Just remember where this tone is coming from.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Privacy Statement describes how your personally identifiable information is collected, used, shared, and safeguarded by NASDAQ OMX.\nFor some of the NASDAQ OMX Web sites, you must first complete the registration form and\/or create a user name and password. During registration you may be required to give contact information (such as name and email address). We only use this information to contact you about the services on our site in which you have expressed interest.\nNASDAQ OMX is the sole owner of the information collected on all NASDAQ OMX websites. NASDAQ OMX may collect personally identifiable information from our users at several different points on our Web site.\nNASDAQ OMX uses the personally identifiable information you provide us to communicate with you in response to your inquiries, to provide the services you request, and to manage your account. Any personally identifiable information gathered by NASDAQ OMX will be used for internal purposes only. NASDAQ OMX does not share, sell, rent or trade personally identifiable information with third parties for their promotional purposes.\nWe may occasionally send you information on NASDAQ OMX products or services. You may opt-out of receiving these promotional emails by following the unsubscribe instructions included in each email.\nWe will send you service-related announcements when it is necessary to do so. For instance, if our service is temporarily suspended for maintenance or there is price change, we will send you an email. Generally, you may not opt-out of these communications, which are not promotional in nature. If you do not wish to receive them, you have the option to deactivate your account. Please see \"Choice and Opt-out.\"\nWe store information that we collect through cookies, log files, clear gifs, to create a \"profile\" of your email and security portfolio preferences within the email enrollment form or account settings. We will link your personally identifiable information to your web site actions, to information in the profile, in order to provide a more accurate understanding of the web sites usage.\nNASDAQ OMX does not share aggregated demographic information with our partners and advertisers, in ways that directly identify you.\nNASDAQ OMX uses outside companies to process credit card purchases. These companies do not retain, share, store or use your personal information for any other purpose. If you sign up for a NASDAQ OMX service provided by a third party under contract with us, we will share your name and the minimum information necessary for the third party to provide these services.\nThis Web site contains links to other sites that are not owned or controlled by NASDAQ OMX. NASDAQ OMX is not responsible for the privacy practices of non-NASDAQ OMX web sites linked to a NASDAQ OMX site. We encourage you to read the privacy statements of other linked web sites when you leave our Web site. This privacy statement applies only to information collected by this Web site.\nIf your personally identifiable information changes, or if you no longer desire our service, you may update or deactivate it by contacting us at the contact information listed below or by updating your information in the \"My Account\" area of the site.\nNASDAQ OMX takes reasonable precautions to protect your personal information. The security of your personal information is important to us. When you enter sensitive information (such as credit card number) on our registration or order forms, we encrypt that information using secure socket layer technology (SSL).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In my previous post, inspired by the Future of C# talk at GOTO, I looked at Initializers, making it possible to assign a value to an auto-property at declaration, just as with a field. This post is about another piece of syntactic sugar in the up-coming version 6.0 of C#: Primary Constructors.\nA constructor assigns values to the two auto properties Title and Genres.\nOnce again the constructor is missing and no need to specify a private setter. Notice the parameters added just right of the class definition. This tells the compiler to add a constructor to the Movie class with two parameters: title and genres. We use our knowledge of the previous blog post, to assign the constructor parameters to properties through initializers. Magic.\nA constructor is generated with the same parameters as specified on the primary constructor and values assigned each auto-property. As with initializers, all of the syntactic sugar only exists on compile time, making it possible to execute this bad boy on previous versions of .NET.\nIn the following post I will check out Declaration Expressions and Static Using.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apart from these functions, realize additionally the automobile's memory system can be based on the needs of the vehicle driver. It can in fact be preprogrammed for 2 different drivers - these will certainly consist of different setting for the vehicle driver's seat, outside mirrors, and guiding wheel. If the vehicle driver changes his seat, the guiding wheels and also mirrors will instantly adjust their position in order to preserve or make up proportional distance. The driver could adjust the guiding wheel to the best height and pity the guiding wheel's power tilt and also telescoping features.\nInfiniti Hard Drive Navigation System - This function will certainly make navigation a whole lot less complicated as it makes usage of the newest touch-screen hard disk drive navigation system in addition to Birdview perspective control as well as 3D structure graphics. You should know that this is suited by the Lane Advice system which has the capacity to inform the motorist whenever he is utilizing the appropriate street toward an exit or interchange.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamcrf b/data_all_eng_slimpj/shuffled/split2/finalzzzamcrf
new file mode 100644
index 0000000000000000000000000000000000000000..fd227c735df17ef4be577e0b6ba8868c9c4c67fa
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Conservator General of the National Parks Service, Alhaji Ibrahim Goni, has described the seven parks across the country as the Gene Bank of genetic resources.\nGene banks are a type of biorepository which preserve genetic material.\nGoni disclosed this when some members of the Bikers Association paid him a courtesy call at the Service's headquarters in Abuja on Wednesday.\nHe said that at present, Nigeria has seven parks across the country that serves as the country's gene bank.\n\"At our parks, we have animals and genetic resources that are still in their original forms which we preserve for sustainable use.\n\"These genes are vital for our future well-being; and biodiversity and nature are, on their own, sources of wonder and enjoyment.\n\"Also, we have wild species that flourish as natural systems that help regulate climate change, air quality, and cycles of carbon, nitrogen, oxygen, mineral elements and water,'' he added.\nThe C-G said that the service was reaching out to both national and international bodies to make the parks household names.\nNews Agency of Nigeria (NAN) reports that the Bikers are taking part in the 14 Days on going Rediscover Nigeria Heritage Tours 2018 otherwise known as Heritage Bikers Tour scheduled for Oct. 3 to Nov 4.\nThe tour is organised by the National Commission for Museums and Monuments in conjunction with the Latin American Motorcycle Association (LAMA), First Entertainment Ideas Concept and others.\nMr Ernest Akhikpemelo, the President of LAMA Nigeria, said that the Heritage Bikers Tour would be an annual event to promote the tourism potential in Nigeria.\n\"Rediscover Nigeria Heritage Tour 2018 is borne out of the need to explore a unique way to promote and propagate the tourism potential inherent in our rich cultural heritage using biking expedition.\n\"The tourism benefit derivable from this display is enormous as it creates interaction opportunities between the bikers and the communities they visit.\n\"The tour will showcase to the world, the rich and diverse cultural heritage, monuments and sites, traditional festivals and local cuisines that abound in Nigeria.\n\"Some of our activities include visit to eminent personalities and dexterity Bike displays by members of the International Latin American Motorcycle Association (LAMA) lead by Mr Mario Nieze and local bikers in all the states they will visit.\nNAN reports that the bikers would visit 11 states including Niger, Kogi, Anambra, Cross River, Rivers, Bayelsa, Edo, Ondo, Osun, Ogun and Lagos and the FCT.\nAkhikpemelo noted that the tour would help discover new frontiers for the promotion of tourism, create awareness and stimulate the usage of such to boost tourism in Nigeria.\n\"The tour is to do a video and pictorial coverage of heritage and monument sites that abound in the country and beam same to local international and audiences.\n\"This will attract the attention of the general public to the museums and cultural heritage, which the project intends to promote''.\nHe said that subsequent editions would have activities such as Bikers competition, Traditional Cuisine Contest, Essay Competition, Beauty Pageant etc.\nSome of the international bikers taking part in the tour come from Brazil, Mexico, USA, Puerto Rico and South Africa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here at Kingston & Knight, our Melbourne accounting services are designed to put you in control of your financial circumstances by aligning the data and relevant records with your reporting and taxation requirements. We use cloud accounting software to keep you in touch, in real-time, with your accounting team and financial information so that you are always in control. With Kingston & Knight, there is no need to engage multiple finance and tax advisors as well as an accountant, our team can handle all your accounting and tax services.\nWe are a smaller accounting firm, with the foundations of our accounting and tax services built upon great customer service and taking the time to get to know the real needs of each unique client. Our accounting team works hard to build and maintain professional working relationships with the clients they serve from right across Melbourne.\nWhether your accounting requirements are complex and highly unique to your circumstances, or relatively simple and straightforward such as the more 'everyday' accounting requirements, our team has the right experience and knowledge to manage your accounting needs.\nKingston & Knight business accountant Melbourne services reflect the need for your Melbourne business to stay in touch with its financial information and on top of reporting and taxation requirements. Let our team of Melbourne business accountants take the work out of this process so that you can get on with running your business.\nWe serve a wide range of Melbourne business clients, including small business enterprises, medium and large businesses, as well as corporate and organizational clients with complex needs and circumstances. Our experience as business accountants is what separates us from the competition.\n\u2022\tAccess to a range of financial and commercial services that support the broader needs of you as a business, organization or individual, including forensic accountant services, audit services, and business advisory services.\nTax can be confusing and time consuming to calculate, report, and pay if you lack experience with the associated processes, laws, and structures. Our Melbourne tax accountants can take the hard work out of tax by managing your tax affairs for you. Kingston & Knight provides tax accountant services to a range of clients here in Melbourne, including individuals, businesses, and organisations with varying tax structures and tax requirements.\nWe also specialize in providing tax accountant services for Melbourne business and commercial clients, with our small business tax accountant Melbourne working hard to ensure Melbourne businesses get the tax concessions they are entitled to.\nWe understand that each business, individual, or organization has unique circumstances and, therefore, unique tax issues and requirements. This calls for a meaningful approach tailored to your circumstances by professional tax accountants with first-class customer service.\nKingston & Knight individual tax return services take the time and stress out of preparing and lodging your personal tax return. Whether you are a business, an individual, or a corporate entity or organization, we can assist you in preparing and lodging your tax documents.\nOur Melbourne accountants can guide you through the process of establishing a trust account and managing it effectively in accordance with regulatory and financial reporting requirements. Trustees are required to manage the tax affairs of their trust account, and this includes an additional trust tax return that must be submitted each year in addition to a personal tax return. Trust tax returns are specialized documents that are difficult to navigate without the experience and qualifications of a trust accountant.\n\u2022\tCorresponding with the ATO and other relevant bodies on your behalf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MMI senior Frederick Mejia placed first in the individual competition and joined classmates Brian Payne, Zachary Young, Alexander Sessock, Daniel DeMelfi, Wesley Guarneri, Alexander Kline and Coleman Smith to capture the top prize in the team event at the 71st annual Luzerne County Council of Teachers of Mathematics (LCCTM) contest, which was held recently at Wilkes University.\nFrederick Mejia, the son of Carlos and Esperanza Mejia of West Hazleton, won a full scholarship to Wilkes University that may be used for any degree or program. Zachary Young, the son of Christian and Nicole Young of Freeland, placed third in the senior division and won a cash prize.\nBrian Payne is the son of Brian Payne of Sugarloaf and the late Marie Payne, Alexander Sessock is the son of Jennifer and James Denion of Freeland and the late John Sessock, Daniel DeMelfi is the son of Daniel and Esme DeMelfi of Hazleton, Wesley Guarneri is the son of Vincent and Sop Guarneri of Wilkes-Barre, Alexander Kline is the son of Dr. Gerard and Cathy Kline of Mountaintop, and Coleman Smith is the son of Keith and Laura Smith of White Haven.\nApproximately 150 students from 10 schools throughout Luzerne County, as well as the Tunkhannock Area School District, participated in the event. Each student must be nominated to compete by their high school's mathematics department.\nThe contest is held in conjunction with Wilkes University and features individual and team competitions in senior and junior divisions. The first-place individuals in both divisions each won a full scholarship to Wilkes University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today it's ghouls, goblins, and\u2026 well, actually just ghouls. What does this weekend otaku think about one man's struggle to be both human and ghoul? What about one studio's decision to adapt a manga with original content?\nWhile it seems like you are choosing both, you're really forsaking both.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The food truck at work today is very kawaii.\nThe Long Dark somehow makes me never want to go outside and desperately want to get outside at the same time.\nI had to reveal that I'm a Hatmaker who can't make hats and change my avatar from Alma Armas to my real face, but I still love you guys. I should be around a lot more now. I hope I don't break the site.\nThe screams the lab assistants let out in Crash 2 when you kill them make me happier than almost anything. That's my childhood right there. Screaming lab assistants.\nThe pizza delivery guy just stopped by, saw my TV through my doorway, and said, \"Are you playing Crash Bandicoot?\" He sounded surprised, so I told him remastered versions were just released, and he turned into a 12-year-old boy right in front of my eyes.\nI'm writing a little something about Bloodborne for its two year anniversary tomorrow, and it's making miss the game dearly. If my backlog wasn't exploding, I'd start playing it again tonight.\nI just remembered I had an Amazon gift card, so I bought Horizon Zero Dawn! I wasn't planning on playing it til later this year, but now I'm all excited.\nJust sneaking back in to say hi, hello, and Night in the Woods is amazing. It has an entire Hyper Light Drifter-ish game within it!\nI changed my avatar again because Alma is best girl. This one is here to stay. Play VA-11 HALL-A if you haven't yet!\nJust got back from a vacation in NYC, NJ, and PA! We took a trip out to Eastern State Penitentiary, and I felt like I was in a survival horror game environment. Great atmosphere but no monsters. Easy walking simulator.\nCarrie Fisher was the first woman in entertainment who I ever admired. She taught me that femininity and power are not mutually exclusive. 60 is too young. This isn't fair.\nSince I probably won't be around to say it on the day of: happy holidays, guys! Thanks for welcoming me into the fold. May your Christmas and the coming year be filled with much gameage.\nI'm loving all the best of 2016 blogs I'm seeing here. Video games made 2016 worth it. Here's hoping 2017 will bring more quality escapism!\nI changed my avatar to fit in. See related subject: anxiety.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanqie b/data_all_eng_slimpj/shuffled/split2/finalzzzanqie
new file mode 100644
index 0000000000000000000000000000000000000000..3f185ae1610689272714d660bfa2c3871bfbbb4f
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A frequent featured artist at the most prestigious festivals, M\u00e9lisande Corriveau is constantly in demand for performances, tours and recordings both in North America and Europe, as a solo artist, and with numerous renowned ensembles. She is a core member of ensemble Masques, Les Voix Humaines consort of viols, Bande Montr\u00e9al Baroque, Sonate 1704 and Les Bor\u00e9ades.\nHer discography numbers over 40 titles on the ATMA Classique, Analekta, Harmonia Mundi, Paradizo, Zig\u2010Zag Territories and Alpha labels. Her recent duo release with harpsichordist Eric Milnes, ''Pardessus de viole'' featuring the 18th century French repertoire for the pardessus de viole, has received critical acclaim and numerous prizes.\nIn 2014, M\u00e9lisande Corriveau received a Doctoral Degree with honors from The University of Montr\u00e9al in performance on the pardessus de viole, becoming one of the few performers in the world specializing on the instrument.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I can't get my frame onto the bottom track - what should I do?\nIt may be that your bottom track is different; take the guides off the frame and insert them directly on the track.\nI'm having trouble with my handle - I don't have enough space to move it and it's hard to operate?\nSimply give the central screw one reverse turn - this will give you some slack on the handle.\nCertainly - just clean the rails with a damp cloth in the spring, and spray with silicone when needed.\nIf the mesh on my screen is torn, is it under warranty?\nNo, the mesh is never under warranty, but it can be replaced easily. Take a look at the manual section and the video section for instructions on how to replace the mesh on a Novascreen patio door.\nOur ongoing objective is to meet your expectations. Thus we are delighted to offer you a 3-year limited warranty on our Novascreen Patio Door screen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reports from the Royal College of Physicians, General Medical Council and Health Education England have commented on the impact that non-contractual issues facing doctors in training are having on morale and the patient experience. Such issues include low levels of engagement, lack of support and well-being initiatives and increasing pressure of work.\nWe've worked with providers to share examples of where they have taken positive steps in tacking these issues. These case studies offer a practical guide to better engage, support and value doctors in training.\nIf your organisation has taken positive steps to improve the working environment for doctors in training, you can share your work with us. Please email your case to nhsi.medicaldirectorate@nhs.net.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Angel Blossom features the same unique motion as the Encore & Grand Encore with 30 total melodies. While the clock plays one of the 30 melodies the clock's numbers rotate in pairs as shown in the pictures. The pairs of numbers spin 180 degrees and then rotate back to their starting position. When the numbers rotate, they reveal figures of swans and dancers.\nComes with (2) D Size Batteries.\nClick the button below to add the Rhythm Angel Blossom 4MH868WD06 to your wish list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Launched in 2016, Bond No. 9 Soho is a floral and woody musk perfume designed for both women and men.\nPerfect for everyday wear, this fresh and earthy fragrance is inspired by the artsy lower Manhattan neighborhood.\nTop notes for the perfume consist of fresh petitgrain, delectable sweet orange, and fresh green cassis. A soft, powdery, and soothing combination of orange blossom, jasmine, and orris root make up the heart notes for the fragrance. Rich, warm base notes of cedar, moss, vetiver, and amber wrap up the blend.\nBond No. 9 is a fragrance company based in New York that creates perfumes and colognes inspired by the nearby iconic geography and landmarks and designs bottles and packaging to reflect each one as well. The company was founded in 2003 by perfume industry expert Laurice Rahme, who named his fragrance legacy after the address of its original boutique and current headquarters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoaep b/data_all_eng_slimpj/shuffled/split2/finalzzzaoaep
new file mode 100644
index 0000000000000000000000000000000000000000..94921ffb363f5706af3ef109ae9e40a3bfb71b0e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoaep
@@ -0,0 +1,5 @@
+{"text":"In Shane Black's \"The Predator\" a ragtag group of soldiers go head-to-head against a ruthless \"space-alien\" that kills humans for sport. If you're thinking that this sounds just like any other \"Predator\" film, I concur. Fans of the franchise would get the satisfaction of seeing classic \"Predator\" tropes like humans brutally hunted down, traps in the jungle, and an alpha-male leading a band of fighters who'd most likely not survive.\nShane Black shifts roles from starring as Hawkins in John McTiernan's 1987 film \"Predator\", to taking the helm as director in this 2018 reboot-slash-sequel. Not counting the two \"Alien VS. Predator\" films, \"The Predator\" is the fourth installment of the iconic film series. In this latest movie, we follow a sniper named Quinn McKenna (Boyd Holbrook), who encounters one of the titular aliens when its ship crash-lands on Earth. Realizing that the government would definitely silence anyone who knows of the incident, Quinn takes some of the Predator's gears as insurance, sending it to his P.O. Box for safekeeping. Unfortunately, the package gets delivered right to his home, and ends up in the hands of his son Rory (Jacob Tremblay). It's not long before the Predator comes hunting for his missing gear, and it's now up to Quinn, together with a band of misfit soldiers and a biologist, to stop the Predator before it harms his family.\n\"The Predator\" clearly knows the recipe that appeals to its target fan base. It's a testosterone-fueled action film whose primary selling point is no-nonsense, brutish violence. If you came to see aliens mowing down humans like game, then this film delivers. Heads-up though for some glaring plot holes, and characters conveniently explaining away the film's storyline.\nIf you re-watch the 1987 Predator, you'd realize that its over-the-top, stupidly fun qualities work, because we get to see larger-than-life actors like Arnold Schwarzenegger finally meet their match. We know that the Predators are strong enough to beat the crap out of these skilled human soldiers, yet we also know that a character played by Arnold Schwarzenegger is tough enough to face-off even against an advanced alien hunter. However, this latest movie sadly doesn't have that quality. It's hard to root for this batch of contenders because they are all expendable, and we don't really feel bad whenever one of them gets massacred. This is what you get when characters have hurriedly explained backstories, and are awkwardly introduced into the plot.\nThe film redeems itself, albeit too late, at the third act where it pays homage to its \"Predator\" roots, taking us to a jungle scene reminiscent of its 1987 and 2010 predecessors. If I'm not mistaken, this is also the part where they had to reshoot everything at nighttime because the daytime shots \"doesn't look scary\". I don't know the actual scope of the reshoots, but a lot of the scenes also appeared to be chopped from a longer cut which we can only imagine to have had better flow. The interesting parts were all dumped at the latter part of the film, which sadly made the rest of the movie bland.\nWith its cliffhanger ending, the film asks the audience to invest in the promise of a better, more exciting sequel. This would have been okay if the film is serviceable, but \"The Predator\" isn't a remarkable movie on its own. If anything, it would get you excited about future films, and the direction the franchise is going.\nThe film asks audience to invest in the promise of a better sequel without being a serviceable movie in itself. If anything, it would get you excited about future films, and the direction the franchise is going.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The net profit of the company increased marginally to Rs.2,235 crore in Q4 FY16 as against Rs.2,234 crore in Q3.\nWipro cracked 6.2% to Rs.564 on BSE. The IT company registered a 6% rise in net revenue at Rs.13,742 crore in Q4 as against Rs.12,952 crore in Q3. The net profit of the company increased marginally to Rs.2,235 crore in Q4 FY16 as against Rs.2,234 crore in Q3. The company's revenue stood at Rs. 13,741.70 crore, clocking growth of 6.1% qoq and 12.9% yoy.\nThe scrip opened at Rs. 570.05 and has touched a high and low of Rs. 578 and Rs. 560 respectively. So far 2504616(NSE+BSE) shares were traded on the counter. The current market cap of the company is Rs. 148574.19 crore.\nThe BSE group 'A' stock of face value Rs. 2 has touched a 52 week high of Rs. 613 on 01-Oct-2015 and a 52 week low of Rs. 508.9 on 29-Feb-2016. Last one week high and low of the scrip stood at Rs. 606.75 and Rs. 580.5 respectively.\nThe promoters holding in the company stood at 73.34 % while Institutions and Non-Institutions held 16.18 % and 9.87 % respectively.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Taiwan's quest for greater participation in the international community is of great importance to all democracies, senior vice president for Asia at the Center for Strategic and International Studies (CSIS), Mike Green, said on Wednesday.\n\"The people of Taiwan deserve \u2014 and we need to see them get \u2014 a seat at the table,\" he said.\nGreen was presiding over a packed meeting held in a room within the US Congress to formally release a new report by CSIS senior adviser for Asia Bonnie Glaser on the nation's need to be part of international organizations.\nUS Democratic Senator Robert Menendez and Republican Senator James Inhofe \u2014 joint chairmen of the Taiwan Caucus \u2014 were scheduled to attend, but were unable to after US Secretary of State John Kerry called an urgent closed-door meeting on Iran.\nLeo Lee (\u674e\u6f84\u7136), deputy representative of the Taipei Economic and Cultural Representative Office in the US, read a message from President Ma Ying-jeou (\u99ac\u82f1\u4e5d) thanking the US Congress for its support.\nMa said in his message that Glaser's report was the most comprehensive study of the nation's bit to participate in international organizations published in recent years.\nGlaser said China presented the biggest obstacle to Taiwan joining international organizations.\n\"There is concern on the mainland that if Taiwan is given greater international presence it could be used somehow to press for independence in the future,\" she said.\n\"One would hope that with China's growing confidence and capabilities, its power economically and militarily, that China would become confident in itself and give Taiwan the opportunity to play a role in any organization,\" Glaser said.\n\"It's extremely unlikely that Taiwan would be able to leverage its participation in an international organization to achieve de jure independence, but this continues to be a concern which is unfounded and one which the international community should push back on,\" she said.\nThe 50-page study and its recommendations were the subject of an earlier report in the Taipei Times on Nov. 28.\nGlaser said that participation in international organizations was an issue that Taipei had to make a priority in cross-strait relations.\nAt the same time, the US should be more proactive in soliciting the support of other nations, Glaser said.\n\"In addition to doing more for Taiwan, I think we actually need to encourage other countries to do more. There are nations out there that are a little nervous in saying things publicly in support of Taiwan, but there is safety in numbers,\" she said.\n\"If you get a lot of countries saying something, China is not going to punish the entire international community,\" she added.\nAs of now, Lee said that the nation has membership to only 34 of the world's 5,000 international organizations.\nThis story has been viewed 4211 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Instructor Tobias Harrington started Taekwondo in Lagos, Nigeria in 1987.\nAfter returning to Canada in 1989, he met Grand Master Kim while on vacation in Mississauga. A student of Grand Master Kim ever since, Instructor Tobias trained under the tutelage of Master Anthony Ilukhor (Nigerian Olympic team, Barcelona 1992) in Ottawa, up until he joined the OMAC team in 2014.\nInstructor Tobias received his 4th Dan Black Belt at OMAC in 2008, and has continued his martial arts journey with humble appreciation of Grand Master Kim's guidance.\nInstructor Tobias also earned a Masters degree in Social Work from Carleton University in 2011, and is following his lifelong passion of Taekwondo with the OMAC team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yesterday my TGIQ inbox revealed quite a nice surprise: a request from a professor in the Zoology Department at the University of Cape Town, South Africa.\nApparently this professor is working on a new field guide and had been searching the interwebz for pictures of a particular beetle. A beetle that is considered an invasive alien pest in his country, but is native to my country. A beetle who a certain geek just happened to subject to a few hasty snapshots last April.\nHe liked the pictures. He wanted them for his field guide.\nLet me think about that a sec\u2026.um, HECK YES!!!\nHe has already authoured \"Field Guide to the Insects of South Africa\". My photo(s) will be going in his latest book, which focuses on alien and introduced species.\nCongratulations Geek!!!! That is so amazingly awesome!\nOooh! Congrats! Are you getting a copy of the field guide out of it too?\nMajor congrats are in order! Kudos to you. I hope you get a contributor copy so you can show us what it looks like.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Carpathian mountains are famous for their incredible beauty and for the tourism it draws to several countries in Eastern Europe, and also for its natural resources. Romania has most of this mountain range in its borders.\nTransylvania, because of the fact it is more of a plateau in the Carpathian range, has much of the natural resources, and this is due to the fact that Romania has 53% of the mountain range. Slovakia is next with 17% of the mountain range.\nThere is gold in these mountains.\nThe Romans would invade, and subsequently conquer the Dacian Empire and bring back iron, silver and gold to the heart of its Empire, Rome. This would give the area a larger place in history.\nIt many passes would be the means for the invasions of Transylvania, and it was because of these passes that the famous castles and fortified cities grew up. The Romans themselves would found the city of Alba Iulia. Although it was known to the Romans as Apulum, it would later change its name several time to the 'white city' the Latin version of Alba Iulia.\nNear the city, there were mines, and these mines were where some of the gold was found. Gold was found in the gold mines of Rosia Montana, and this is well documented. This would be a source of income for many people but as of 2006, the mines were shut down. Currently there are about 2,500 people who live in the town near this mine.\nMost of the gold and silver and other valuable minerals and metals are found in the Carpathian mountain region, and these mountains will play and important role in Transylvania and Romania. Even the Romanian Royal Family has some connections to the gold in this area. Queen Marie of Romania who was crowned Queen in 1922, wore a crown crafted from the gold of Transylvania.\nThere is a Canadian connection with many of the gold mines in Romania, and this is based on the companies which own the mines, for example the Ciresata mine in Hunedoara County, Romania, is owned by Toronto based, Carpathian Gold.\nThe Carpathian mountains are known for their beauty and lend enchantment to Transylvania, Romania. They also play a vital role in the GDP of Romania with the natural resources that can be found in this region. It is no wonder Romania was able to be more independent of the Soviet Union after the rise to power of Ceausescu.\nIt was because of the natural resources in the land beyond the mountains.\nGold galore I would say! Plus it's the only country in the world actually with a gold museum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recently I shared my Autumn Nail Polish Picks with you and that seemed to go down quite well so I thought I'd tell you about a few makeup products that I especially love using in Autumn. When it comes to the colder months I know my skin is going to look dull and as the leaves change colours my eyeshadow tends to as well as I start using more earthy tones. I tend to wear a bit more makeup in Autumn and Winter, during Summer there is just no point as it melts off my face by midday. Also as I mentioned my skin does tend to look more dull so I like using products that will brighten up my face a bit to make it look a little warmer. I of course still stick to my usual favourites but it is nice to change some of the products that you use every season.\nI won this eyeshadow a little while ago in a giveaway and I've been wanting to make up a little MAC eyeshadow quad since but I just haven't gotten around to it yet. Despite that I've been using it a lot more recently and I know that I'll definitely be reaching for it a lot in Autumn. It is a beautiful gold\/champagne shade that I've had my eye on for a while. It is great as a base colour or just on it's own for a more simple look and I love that it is so pigmented and blendable. It is pricey for an eyeshadow though so I think one quad will be enough for me!\nI've favoured Maybelline's mascaras over other brands offerings over the past year but this time round it is a L'Oreal mascara that has taken the spotlight for me. I sometimes find that black mascara can be a little harsh with my small features and pale skin so I've been enjoying reaching for this mascara instead, it isn't brown but it isn't a true black either. I find it stays on all day and gives my lashes a lot of volume, not much length but I'm enjoying that at the moment. I think brown is definitely a better colour for Autumn when it comes to things like mascara and eyeliner, it isn't as harsh but still looks amazing!\nI reviewed this bargain palette from Makeup Revolution a few weeks ago and I knew as soon as I swatched the shadows that this would be my go to palette during Autumn. Nearly all the shades are very pigmented and very easy to work with, they all compliment each other so well also. There are a couple of shades within the palette that I know I will be reaching for a lot during the next few months. Consume Me is a gunmetal brown colour that will be great for a smokey eye and Love You To Death is a purple shade that will add a bit of colour to any makeup look. Take a look at my review here to see swatches of all the shades! If you are looking for a fun palette full of great Autumnal colours then definitely take a look at this one.\nAdmittedly I've only had this foundation for a couple of weeks but I already know that it is going to be my go to base product during both Autumn and Winter this year. This is a foundation I've wanted to try for a while but previously none of the available shades suited my skin so I stayed away. I found out a few weeks ago though that Rimmel have been bringing out a new paler shade in their foundations so I snapped this up in my most recent Superdrug order. Not only is it a perfect match for my skin but it suits my dry skin really well, it isn't too thick or thin and gives me really good coverage. I am only 2 weeks in though so I'll let you know more about it in a full review soon!\nThis is another product that is new to my makeup collection, it is already a firm favourite of mine though. I'm a big lover of cream blush, especially when it's cold outside as it tends to brighten up your face a bit more than powder products do. I had a feeling I would like this product but it really has completely blown me away. For only \u00a33 NYC have created a beautiful cream product that sits on the skin well, is extremely pigmented and stays put nearly all day long. Again I will be reviewing this and the other shade I have but I would already really recommend it to you!\nWhen it comes to lip gloss I'm quite fussy and I have phases where I love it and then hate it. However there is one lip gloss that I'm always happy to get out of my makeup bag, probably because it isn't like most lip glosses. The L'Oreal Stain Splashes give your lips a glossy finish for an hour or so but they leave a stain for a few hours after that which I love. It isn't a drying product either, my lips feel soft after using this and I don't have to reach for lip balm every 30 minutes. The shade is what most draws me to this during Autumn though, it's pink but with some raisin undertones which you can see a lot in the swatch below, on the lips it is more pink though. It's a really wearable shade and surprisingly pigmented. In the packaging you can see quite a few glitter bits floating around in the product but it doesn't transfer like that luckily, it just creates a nice shine that becomes a stain. I've seen these L'Oreal products get a lot of mixed reviews but personally I love them.\nI recently talked about this lipstick a lot in my Autumn FOTD, I don't think I'll ever stop talking about it, it is most definitely my favourite lipstick! When I first saw it I thought I would hate it as it looked so dark and purple but this has to be one of the most wearable lipsticks that I've ever used. The purple and pink tones merge together to create the perfect A\/W colour that I think suits my pale skin so well. It is expensive but it's definitely worth it, I think if you are looking to step out of your comfort zone a little this will be great for you to try. It's a lustre lipstick so it isn't super pigmented but it can be built up easily. It's also really soft and moisturising which is really good for the cold wind that is bound to hit us soon. I don't think I can find a fault with this product, it's the perfect lipstick for me!\nLet me know in the comments below about what products you are reaching for at the moment!\nAll that glitters is so gorgeous, i love it!\nUgh I wish the wake me up foundation was available in the US! That palette looks super shimmery, I would love that.\nYou want, and also after a couple of clicks it'll await Fildo You could get as well as move any songs data.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FireMon is a global security management software company supporting government agencies, large enterprises and MSPs. The FireMon platform improves network security through better management of security technologies. Using FireMon, organizations more effectively manage devices and policies, plan policy changes and workflow, automate rule recertification and analyze risk exposure.\nThis document was retrieved from http:\/\/www.rsaconference.com\/events\/us15\/expo-sponsors\/exhibitor-list\/1395\/firemon\ton Fri, 19 Apr 2019 08:27:30 -0400.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After deciding to go forwards with getting my dissertation published as an academic journal, my research led me to Reinvention: an International Journal of Undergraduate Research who are an online peer-reviewed journal hosted by the University of Warwick and Oxford Brookes University.\nCorrespondence with my the I.R.P module leader Nichola also pointed me in the direction of JAWS: Journal of Arts Writing by Students however unfortunately they only accept submissions of postgraduate work and so I will aim to submit my dissertation to JAWS after I graduate.\nIn the meantime, I'll be exploring a little further into the Reinvention Journal and arranging a meeting with Nichola after my final two modules are submitted, to discuss getting my work ready for the submission to publication.\nKoncept are an award winning Manchester based interior design firm proudly responsible for the creations of numerous commercial designs. The company prides itself on their \"enviable track-record for hotel design for top hotel brands\". Koncept also work closely alongside Leach Rhodes Walker Architects.\nThis week I sat down with Simon Dickinson of Koncept interior design to discuss his work at the firm.\nTalith: Hi Simon, thanks for meeting with me. To start with, how did you get into interior design?\nSimon: I started off as an office junior in the practice, through a family connection who is a contract inspector, with the expectation of potentially pursuing a career in architectural technology by studying at university part time whilst still working for the company. My office junior job consisted of printing, filling printers, emptying bins, making tea and coffee for the staff etc. One of the directors of the interiors team approached me as I'd shown interest in wanting to join that team, as I became interested in the creative side rather than the technical side of design. He asked if I'd like to do some CAD work with the team, and from that point I moved into that team, was given more and more responsibility and have been developing from there ever since.\nTalith: Can you describe a day in the life at Koncept?\nSimon: It all depends on the day. I usually sit down, make a brew, if it's a Monday ask my colleagues how their weekends were. I check my emails at home so I chase up anything that happened the night before, any updates etc and then get started on the drawing work or whatever type of work is needed for that day. I balance the work accordingly throughout the day.\nTalith: What software do you use?\nSimon: CAD, for visuals Photoshop or Sketch Up, InDesign and Excel are the only programmes we use.\nTalith: Since you didn't study interior design at university or have much prior knowledge to it, were you self taught on these programmes?\nSimon: Yeah I taught myself. I had knowledge of Adobe Photoshop already from studying photography, and CAD was pretty simple to learn. We only do 2D CAD, unless we do isometrics. The learning process is gradual, you're always learning new things, but it took roughly a couple of weeks to learn the basics. Whilst I was an office junior, I was working with a senior project director who helped me with CAD and showed me basic stuff. As soon as you've got the commands right and you know how to build the drawing, you're good to go. Some of the most important things are layer discipline and line weights, as well as scales. For example, let's say you're looking at text size for a 1:20 drawing on an A1 sheet, if you have a detail that is 1:1 within that drawing then you need to make sure all the text size is the same, the line weights are going to work and all the hatches are on the right layer. So making sure that everything is accurate takes up the most time because this avoids problems in say a months time when you realise you need to remove a certain layer, or you need to get rid of a certain thing. That was the biggest thing to learn; having the discipline as you draw to work on layers and line weights. Oh and naming the layers properly! As there's a set code for how to name a layer, the British Standard way of layering tells you what the line weight is, what feature it is i.e. pipe work or a soil vent, then that will have its own layer. That's really useful for when you're cross referencing with other people's drawings as it means that everyone is singing from the same hymn sheet and it makes life a lot easier.\nTalith: What has been your favourite project to date?\nSimon: Hmm, there's probably two. Either a Novotel we worked on next to Heathrow as this was a big, grand job. It was due to be Novotel's flagship hotel for airports, so it was a big thing for Novotel. A lot of thought and teamwork went into it, plus a lot of drawings. There were loads of detailed floor drawings included. I remember we had to do tiling on a radius around a corner that was really difficult setting out as each individual tile needed its own individual space, certain distance etc. Now that it's done it looks amazing, it was good fun to work on. The second one is a really interesting hotel in Blackpool, a Best Western. We had to work on DDA bedrooms and it was kind of a solo project for me which was nice to be so hands on with it, in contact with the contractors myself and basically be able to call the shots, which isn't something I have been able to do in other projects yet as we typically work on such big projects that it's always spread across the more senior designers, whereas a smaller project like that is quite nice to be able to make my own decisions without approval from others.\nTalith: What skills did you require when you applied for the job?\nSimon: I think anyone can become an office junior, it's just about showing aptitude and being pragmatic. My root into it is uncommon though, everyone I work with has a degree in interior design or architecture, so I'm quite lucky to be part of the company. I showed the right enthusiasm to get involved with the interior design side of the business, but my fast learning abilities definitely helped. A lot of the work we do is implementation based, so it's not always about the concept behind the design or the narrative and reasoning for the creation of the design, but it's more technical and practical. In some cases we have had other people's designs which we have had to implement. The concept for the design has been someone else's, but we have been commissioned to take that design, fit it to the space and make it work. Once it's on site and a problem is faced, such as services being put in the wrong place, it takes a technical mind to be able to say 'this is how we should do it', which I think I have.\nTalith: Is there a current design trend in what clients are asking for?\nSimon: It depends on the job. Offices are really into the whole Google vibe of open spaces with quiet areas to work, bright colour schemes, modern furniture, big communal areas, more collaboration space than individual working areas. With hotels, obviously there's this whole kind of rustic and exposed industrial trend. Reclaimed timber, concrete effect, quite gritty and leaving services exposed as opposed to concealing them with a suspended ceiling or something. I'd say those are the biggest trends at the moment. Although it does depend on the scheme and client to client, some places still want that traditional feel with darker colours for comfort and cosiness.\nTalith: How about the future of interiors, do you guys have your finger on the pulse with what is set to be the biggest trends?\nSimon: Yeah, that's it! BCFA (British Contract Furnishing Association) dinners too, networking events, exploring different types of furniture and products and then suppliers come to our office and show us their latest stuff. So we do try and stay on trend as much as we can, the design shows are the best way of doing this to catch up and coming trends. A couple of years ago we went to these design shows and crushed velvet seemed to be everywhere. Now you see plenty of crushed velvet used in new schemes, like bar stools that were once being made from faux leather are now being made from synthetic crushed velvet. Good question.\nTalith: How important is free hand drawing and sketching in your line of work?\nSimon: For concepts it's useful. Being able to do a quick sketch with some colour is advantageous, but everyone has the ability to use software too. It's quite a brief way of expressing the scheme and your intent. You can give an idea of finishes without going to the extent of doing a full visual. Using the iPad for perspective drawings with vanishing points is handy too. If you need to show a detail and you need a bit of help with it, or you're not too sure how it's going to work, you need to be able to go and ask the person next to you what they think of it, so being able to just sketch it out and show where things could go is quiet useful. I'd say everything is more digitally inclined though.\nTalith: Favourite part about your job?\nSimon: Ooh. I have three favourites. Seeing a project finished after working on it for a while and hearing good things about it is exciting. The construction process and discussing work with the contractor, especially when what you've drawn is approved instantly with no problems or alterations needed, that's really satisfying because it means you've done a good drawing and provided all of the information they need and they can start building it. Being able to convert an idea on a piece of paper into something physical is quite cool. Last but not least is doing a good joinery detail, so designing furniture items; bars, banquet tables, etc.\nTalith: Is there a lot of furniture design involved?\nSimon: Absolutely. Every job we work on has bespoke joinery in it. We don't design a new chair for example, as there's plenty of options out there, unless it's really unique. But in terms of designing if bars, big tables, big fixed pieces of furniture are required then we do need to do it ourselves. An understanding of how it's going to be built, an awareness of materials and a thought into the practicality of the piece. My favourite part of it is actually thinking about how it's going to work and be put together \u2013 I'm very much a construction minded person. I think it's important to remember that interior design isn't just about colour schemes and furnishings. You wouldn't have all the pretty aesthetics without someone thinking about all of the other stuff, and I like thinking about that other stuff.\nI have come up with the following concepts for my degree show in June.\nI created the mood board on Adobe InDesign to compile ideas I have for how I want my assigned area to look and more importantly, how I want to portray myself as an artist and designer.\nI will use the illustrative hand drawn lines\/scribbles as a border along the bottom of the boards to show my favoured method of communicating visuals. I searched for a while to find a suitable font that will display my name whilst communicating my brand accurately, and settled on this handwriting style. I sought permission from the owner after finding it online, and downloaded the font alphabet.\nAs I have an unusual name, I like to play on this across my work. From experience, people typically remember my name as it's fairly unique and so by using my name as much as possible throughout my work, it becomes a recognisable brand. The font also helps with this as the delicate handwriting style is a reminder of my hand-drawn work. My website features this font throughout the site for similar reasons. My CV also features my name in big, bold, bright pink letters to have a similar effect, but to a more professional standard as I understand fonts and colour choices say a lot about a person or business.\nI hope to display my NMP booklet and a bound copy of my dissertation in my show. The NMP booklet gives a clear introduction to the model and other work relating to NMP that will be on display. I have chosen Blurb to print a landscape copy of the booklet. I also believe the topic of my dissertation, Emotional Design, encapsulates what my work is all about and who I am too as a designer and artist. I chose a fairly casual magazine style holder in my mood board to give people the option of reading them, rather than displaying them as part of the main exhibition, which may look a little too proud and forceful!\nTo feature some of my work, I have considered methods of hanging. Whilst an aesthetically pleasing and clean way of displaying work, the notion of art in suspension has been a recurring theme throughout my time as university. I have referred to, and used, this method in many modules in Levels 4, 5 and 6. I also recall a conversation with Andrew in Level 4 about my future degree show in Level 6 and how he could imagine it being artistic and, in reference to the module I was doing at the time involving art in suspension (3D Design Practices and Procedures 15-16) , how he thought it would feature my work being hung and so I think it could be nice to refer back to this idea.\nI have kept my Wix site fairly simple and clean aesthetically as I think over powering and busy sites can look chaotic and unprofessional.\nI have been working on a photograph that highlights who I am as an artist and designer.\nI took this photograph of myself, using my camera and a borrowed tripod. I edited the photo in Photoshop using Layers and the quick selection tool to change all but the flowers to black and white. Featured on my homepage, this image gives my website a feminine touch whilst also drawing attention to my favoured artistry being illustration.\nMy Wix site was left untouched for the best part of a year whilst I lost interest in it due to its layout.\nI updated my layout and added pages this year. Comparing the two, I can now see the progression in website maturity\/professionalism, and continuity in sticking to a colour scheme and font type. There is a clear progression in my digital practice ability too, including the use of Wix.com, PhotoShop and my own dedication to finding and utilising a font that suits my brand.\nI have been continuing my focus on plants and flowers by creating patterns in Adobe Illustrator as inspiration for the decor of my Wellbeing centre for the NMP module.\nThis above image shows some simple patterns created in a Brick by Row tile type with a 1\/2 brick offset. The effect resembles a style often seen in wallpaper\/packaging design.\nI've been studying natural forms and botanical shapes for furniture and decor inspiration for my Wellbeing centre, with focus on the shape of the counselling 'pods' in order to create a calm and soothing atmosphere.\nMy research has been centred around circular shapes, plants and flowers. The importance of Biophilic design is paramount in promoting a harmonious environment to improve health and wellbeing for the centre's visitors.\nI contacted R2 Architecture to obtain some clearer floor plans of Leaf on Portland street to work with for the NMP. I briefly explained the reason for my request and the university programme I am studying. I heard back within a few days from a man named Richard Eastwood from the firm who attached the ground floor and basement plans of Leaf and wished me the best with my project.\nRichard Chadwick is a Manchester-based interior designer and illustrator who works through the traditional medium of pen, ink and watercolour.\nRather than using only digital images and computer software to portray his designs, Richard favours the traditional method of pen on paper to illustrate concepts.\nRichard's work is personable and interesting. He uses colour thoughtfully and appropriately, whilst his technique and detail is perfected too.\nI aim to produce something of a similar style as the visuals for my NMP. This type of artistry speaks to me much more than using CAD or SketchUP. There's a sensitivity and personality to hand drawings that I don't think can be achieved in using computer software.\nI met Manchester-based architect and former founder of A-Cube Architects, Faheem Aftab, in a bar in Chorlton last weekend and we got talking about interior design and the concept for my NMP Wellbeing Centre. He liked my concept and we discussed the feasibility of the centre in the proposed building Leaf on Portland Street. Aftab had some great insights, connections and pointed me in the direction of a women's centre he worked on in Liverpool; The Amadudu Centre, a refuge for women and children fleeing domestic abuse, specifically black and minority ethnic groups. He also mentioned architect Maurice Shapero and his involvement in an Everton-based wellbeing centre. After researching this wellbeing centre, it became clear that although the centre wasn't women orientated, it still held similar values to mine. Shapero states how his centre aims to \"address the duality of an open community building with the introverted\/confidential nature of a Doctors practice\" which is similar to my concept of my centre being personable yet practical.\nWhilst exploring Maurice Shapero's website, I came across his love for drawing and sketching which didn't always relate to his profession or projects he worked on. I found this encouraging and relatable.\nAftab jotted notes and contacts down as we spoke. After mentioning my intentions with my women's centre and my interest in it also being a celebration of art and women's creativity, he included the contact information for Chuck Gallery, a Manchester art gallery dedicated to the promotion of contemporary African art in Manchester. His enthusiasm for WOC businesses was refreshing and reminded me to ensure my building's function and interior are welcoming and inclusive for all women.\nThe notes are featured below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tennessee drivers received 102,000 seat belt citations in 2014 \u2014 30,000 more than the year before. According to the Tennessee Highway Patrol, the increasing enforcement of seat belt laws is part of its effort to bring down the number traffic deaths.\nBy the end of 2014, 952 people died in Tennessee as part of vehicle crashes, compared to 986 in 2013. It's still too many, says Sgt. Bill Miller, a spokesman for the Tennessee Highway Patrol.\nHalf of the people killed were not wearing seat belts, according to THP data \u2014 hence the bigger push from Tennessee troopers to monitor seat belt use on highways.\nState troopers have also been pulling over more people in recent years for driving under the influence: DUI arrests increased 150 percent between 2010 and 2014, according to THP data.\nOne of the newest advancements in curbing fatalities is a software system that Tennessee is piloting, which sifts through traffic data \u2014 including where accidents happened in the past, or what events are going on in the area \u2014 and then predicts where crashes are likely to occur. Miller says the system gets part of the credit for the 4 percent decrease in traffic deaths since last year.\nBut the best way to prevent fatalities, he says, is simply by drivers buckling their seat belts.\n\"Why would you get into a vehicle and decrease your odds of survival by 50 percent?\" he says.\nSgt. Bill Miller with the Tennessee Highway Patrol provided this data, unless otherwise cited.\n952 people were killed on Tennessee roads in 2014, compared to 986 in 2013.\n50 percent of people killed in 2014 were not wearing their seat belts.\n20.7 percent of fatal crashes in 2014 were alcohol-related.\n102,069 seat belt citations were issued in 2014, compared to 74,277 in 2013.\n8,333 DUI arrests were made in 2014, compared to 6,388 in 2013.\nTennessee ranked 11th in the country in 2012 for traffic fatalities per vehicle miles traveled, according to the Insurance Institute of Highway Safety.\nIf There's A Seatbelt Law For Tennessee School Buses, Who Gets A Ticket?\nAre Texting Drivers Behind A Spike In Tennessee Highway Deaths?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Armand Coppotelli is the Business Development Manager & Training for Deb USA, part of SC Johnson Professional. He has more than 25 years of experience advising best practices to maintain good skin health. Over the course of his career, he has lectured in AIHce roundtables and has served as AIHA section guest speaker. He also has hosted presentations on overcoming work related skin dermatitis for many occupational medicine\/nursing organizations including: Chicago area occupational nurses section, the Australian Occupational Health Nurses Association and the Toronto Occupational Physicians Association. He has implemented corporate skin care programs for large scale manufacturing operations such as Goodyear Tire & Rubber, Amtrak, Rohm & Haas, BASF, Lear and Delphi, and has served on the Protective Clothing & Equipment Committee of the American Industrial Hygiene Association.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Going to sleep at a reasonable time makes waking up easier.\nWaking up early is no easy feat but these tips can make it manageable.\nKeep your alarm clock across the room so it's not too easy to hit snooze.\nDrink water before coffee and make sure you're waking up to light.\nA cool shower will get your day off to a good start.\nFor so many people, there's nothing more satisfying than sleeping in. You get up when your body wants. You can ease into your morning. You don't have massive under-eye circles. It's all pretty awesome.\nWaking up early, on the other hand, is the pits. Only a few magic unicorns are truly jazzed about rising before the sun does, and yet too often life demands it. Work and school start early, plus that dreaded 6 a.m. HIIT class may be the only one that fits in your schedule all week.\nBut if you do have to get up early\u2014or are determined to make this the year that you GSD before noon\u2014there's a way to do it without too much pain.\nSet a bedtime based on 90-minute sleep cycles.\nThe goal is to wake up at the end of a 90-minute sleep cycle.\nSleep generally runs in 90-minute cycles\u2014and the goal is to wake up at the end of one. \"You'll be in a lighter stage of sleep at that point, which is easier to rouse yourself from,\" Breus says. Most people are best served by getting either six hours or seven-and-a-half hours a night, he says, not eight (you'll be in the middle of a new cycle). So, if you know you work best with seven-and-a-half hours of zzz's and you need to be up at 6:30 a.m., it's lights-out at 11.\nIf your alarm is right next to your bed\u2014or on your wrist\u2014it's so easy to hit snooze. Forcing yourself to physically get out of bed may be a simple trick, but it's an effective one. Another option? Enlist a wake-up buddy, i.e., a person who will call you and hold you accountable, Breus says. It's kind of like having a screaming toddler who forces you out of bed, even if you're not a parent.\nSunlight helps turn off your \"melatonin faucet,\" Breus says, which in turn helps you avoid that groggy, I-seriously-don't-want-to-be-awake-right-now feeling. Stand by the window for a few minutes if it's light out (bonus points for drinking water at the same time!) or use a light box.\nBreus recommends a cool\u2014not cold\u2014shower to get the blood flowing. And blast the guilty-pleasure Bieber tunes. \"Saving your workout music for your workout isn't necessarily the best idea,\" Breus says. The overall goal is to start off in a good mood, so you're more likely to give the whole getting-up-early thing a go again the next day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The use of modern public cloud offerings dramatically changed the way we look at digital security. Traditional security models can no longer be applied to mitigate the threats we face these days.\nTraditional security models are generally based on perimeter security. We build a castle, dig a moat, implement a drawbridge and assign guards to defend our perimeter. Within this castle our valuables are stored and protected by the thick layer of perimeter security. Nothing comes in or out without someone knowing exactly what is going on.\nThis works for traditional on premise IT environments where we have control over who accesses our data and from which device they do this. For example a mainframe environment, not accessible from outside the building, or a service based computer environment (RDS) where, if remote access is allowed, we have control over who is able to do this and keep our data within the perimeter where we still have control of the data flow. We sometimes lower the drawbridge but still enforce a strict policy of what comes in and what goes out.\nWhere a traditional IT environment looks and feels secure, the opposite is true. In modern workplaces users expect a certain level of flexibility. They want to access information any time, any place and anywhere. If we do not provide such functionality or lock the system down where such functionality is limited, users will find another way to achieve their goal, thus creating what we call shadow IT.\nIf we go back to our castle metaphor and compare it to the modern workplace, we have to lower the drawbridges and swap the guards for a friendly host that provides guidance (we can access x but not y). Anybody can access the castle, but not everyone may access our valuable assets. This new model brings a whole new set of risks to the table; the perimeter \/ IT environment is no longer the main attack vector. Instead, attackers target users and their devices directly, as they have direct access to the \"castle\" and to some extend the valuables we try to protect. This generally happens through (targeted) malware, phishing, or social engineering.\nThis is exactly what we need to focus on in a modern cloud environment. We have the ability to provide a free flow of information, users can access information any place, anytime, anywhere without daunting login procedures through VPN connections, certificate exchanges and hardware tokens. We have to focus on safely storing our assets (valuables) and ensure that their security and integrity are guaranteed.\nWith Microsoft Azure and Office 365 we can provide both the required functionalities and implement the corresponding security countermeasures to ensure responsible usage of our assets and reduce the impact.\nFor each of these items, Microsoft provides specific tools, each with their own security features.\nDevice security in particular is something new compared to our traditional security model. We now have a device (anywhere) that has access to company information. Essentially, we are using Microsoft Intune to control information outside of our castle. We can use Intune to force Multi Factor Authentication (with Azure AD), Isolate corporate data, wipe specific data and leverage the power of rights management (Azure information protection).\nFor example: a user accesses our environment. A few moments later, the same users tries to access the environment again but from a different location, time zone and device. We may assume that this is malicious activity.\nBased on the information we have on the registered device and the user, Azure Active Directory will assess a risk and can prevent the user from logging in or request additional information to login (Multi Factor Authentication).\nWith Azure AD and Intune we are able to manage the identity and the environment. With Azure Information Protection we can manage our information even when they are available outside of our presumed secure perimeter.\nAzure Information Protection integrates with both documents and E-Mails. Based on your own classification, you are able to label documents and messages and apply policies based on that classification. Office 365 will automatically recommend a label based on the content or a user can choose a label. This requires a fair amount of awareness on the user side however, recommendations will give a pointer in the right direction.\nFor example: a message contains a social security number; Office will recommend a label (say \"classified\") and will apply it to the message. At this point a predefined policy will be applied. Within this policy we can limit the actions that can be performed after the message or document is sent. We can prevent the document from being printed, we can encrypt it or revoke access altogether (remotely).\nTo secure our IT environment we can configure Azure Security Center and protect our cloud and on premise resources. Azure Security Center provides real time insight on security recommendations based on the deployed resources.\nAzure Security Center also provides threat management and will take preventive action when a known threat is detected. Essentially, Security Center will provide you with real time security monitoring and a continuous assessment of your security profile.\nTo summarize: Traditional security models no longer provide sufficient protection for modern IT (cloud) environments. Users expect nothing less than unlimited and easy access to their resources. Microsoft Azure and Office 365 do not only provide you with the tools to protect your environment but can also help you leverage the power of a modern cloud environment by improving your security altogether.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beauty professionals will be given the opportunity to network and take hands-on workshops to learn about the latest trends, techniques, and products.\nBrighten up nails with this whitening scrub from Gena.\nKeep feet prim and proper with Gena's new Pedi Rasp.\nGena Nail Brite cleans, conditions, and brightens nails.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Look for \"Assigned Trainings\" on your UC Learning Center account page, to see what you may need to complete early.\nUC's online learning management system, used by staff and faculty for professional development and required trainings, as well as registration for other programs, is scheduled to be offline for nine days, starting at 5 p.m. Friday, July 28, to allow for a system upgrade.\nOfficials said the UC Learning Center, when it comes back online on Monday, Aug. 7, will have a more contemporary interface and will be compatible with all web browsers. Users will not lose any of the data they had in the system prior to the upgrade.\nQuestions or concerns? Send an email to Staff Development and Professional Services.\nThe Staff Diversity Administrative Advisory Committee and Staff Assembly are hosting a Staff Data Workshop, 11:30 a.m. to 4 p.m. Thursday, Aug. 24, for staff to come together to learn how to work with and interpret data related to staff diversity.\n\"Our goal is to empower staff to feel more comfortable working with real university data to inform decision-making and policy,\" reads an announcement in Staff Voice. \"Using this data, we endeavor to develop a white paper with recommendations to campus leadership related to staff diversity trends.\nThe workshop has been designed with all staff in mind, regardless of comfort level with data, title code or position type. Space is limited; the sign-up form will be available until Friday, July 21.\nConstruction is underway on a set of weirs in the waterway's eastern half, where water will flow gently over the weirs, down to Lake Spafford. From there, water will be pumped back to the east end, where crews have built a concrete \"splash\" pad to prevent erosion where the water will pour back into the channel. Read the latest project update.\nInformation and Educational Technology is advising instructors how to access the new smartsite-legacycourses.ucdavis.edu where course materials will be stored for instructor use only after the old learning management system is split in two this Friday (July 14).\nOn campus \u2014 Use the campus network.\nOff campus \u2014 Instructors must use a VPN (virtual private network) connection, which will help secure any private information in the archives. Instructors whose departments have VPNs can continue to use those connections, while others can use the VPN service provided by the UC Davis Library.\nThe legacy course sites will live at smartsite-legacycourses.ucdavis.edu for at least the next year. Project sites from the old SmartSite will live temporarily at smartsite.ucdavis.edu.\nRead more about the SmartSite split.\nThe Board of Regents, meeting today through Wednesday (July 11-13) at UC San Francisco's Mission Bay campus, has a new chair: George Kieffer, elected by his colleagues to a one-year term that began July 1.\nSee the Board of Regents agenda.\nKieffer, who holds a bachelor's degree in history from UC Santa Barbara and a law degree from UCLA, has been a board member since 2009 and previously served as alumni regent (1978-80). The board elected John A. P\u00e9rez, former speaker of the Assembly, as vice chair.\nThis UC news release has more details, plus a list of chairs for the board's various committees and subcommittees for 2017-18.\nAlso this week, the board will seat its first student advisor, as appointed by a special committee; and consider approving the same committee's nomination for student regent-designate.\nStudent advisor \u2014 Rafael \"Rafi\" Sands, third-year undergraduate studying business economics and political science at UCLA. This is a nonvoting position, approved by the board last year as another way to bring the student perspective to board deliberations. He has a one-year term ending June 30.\nForty-two students from all 10 UC campuses submitted applications in this round to be student regent or student advisor. In accordance with regents policy, the special committee interviewed three finalists for student regent after the UC Student Association interviewed 10 semifinalists.\nRead the UC news release on Rafi Sands, the student advisor; and Graves, nominated as student regent-designate.\nUC Davis Health announced last week that an employee's email account had been breached as the result of a phishing scam, and that the account held information on approximately 15,000 patients.\n\"Though there is no indication that the breach resulted in the actual acquisition of or access to personal or medical information, out of an abundance of caution, the university is notifying patients whose information was stored in the account,\" officials said in a July 6 news release.\nIn assisting with informational and event mailings on behalf of UC Davis Health, the account user had received limited information about some patients, including name, address, phone number and, in some cases, medical record number, diagnosis and Social Security number, officials said. For this reason, the health system is providing identity- and credit-protection options to those people whose sensitive personal information was stored on the system, even though there is no indication at this time that their information was viewed or accessed.\nThe investigation, which is ongoing, had so far revealed that the phishing email arrived May 15 and the hacker gained access to the account that day. Two days later, officials said, someone posed as the account's real user to send emails to other UC Davis Health employees, asking them for large transfers of funds. Health system staff quickly recognized the scam and promptly notified the university's data security team, which took swift action to secure the account and prevent further threats.\nPatients desiring additional information may call 855-216-0658 (toll-free).\n\"For a summer school blood drive that's incredible,\" Felicia LeMothe of BloodSource said after the July 28-29 blood drive on the Quad. She said 444 people turned out, and BloodSource collected 307 pints (for various reasons, not everyone who comes can complete the process).\nA memorial service will be held at 2:30 p.m. Saturday (July 15) at St. Martin's Episcopal Church, Davis, for Melvin R. \"Mel\" Ramey, professor emeritus of civil engineering professor who had also served as an assistant track coach and the university's faculty athletics representative. He died June 27 at the age of 78.\nA reception in St. Martin's Fellowship Hall will follow the service.\nA native of Pittsburgh, Ramey earned his undergraduate degree at Penn State, and master's degree and Ph.D. in civil engineering at Carnegie Mellon University, then became the second African American member of the UC Davis faculty in 1967. He served chair of the Department of Civil and Environmental Engineering and also held other administrative posts, including associate dean of Graduate Studies.\nHis numerous honors included the Magnar Ronning Award for Teaching Excellence, Black Engineer of the Year and the UC Davis Engineering Alumni Distinguished Teaching Award. He also earned Carnegie Mellon's Civil and Environmental Engineering Distinguished Alumni Award.\nIn 2016, his colleagues, friends and former students established a UC Davis endowment in his name, the Melvin R. Ramey Fund for Student Success, and the university further celebrated him by naming the Student Community Center's lobby after him. The fund supports student support services, and Ramey's family has asked people thinking of flowers to forego them, and make donations to the Ramey fund instead.\nRamey served as the faculty athletics representative from 1997 until his retirement in 2004, representing the campus at various NCAA events and continuing \u2014 if not embodying \u2014 the athletics department's commitment to academic integrity. Ramey also stood as an advocate for Title IX, both during its initial implementation in the 1970s and in his tenure as faculty athletics representative.\nRead more about Mel Ramey in this obituary by Intercollegiate Athletics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqyes b/data_all_eng_slimpj/shuffled/split2/finalzzzaqyes
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+{"text":"Cyclone Winston, which devastated Fiji, was the strongest to ever hit the South Pacific. IPPF's humanitarian response there was carried out with our Member Association, the Reproductive & Family Health Association of Fiji, and is part of our SPRINT Initiative, funded by the Australian Government.\n22-year-old Sulueti arrived in the village one week after the cyclone was over. She was shocked - there was nothing left apart from the foundation of the house apart from that everything was gone. She hadn't expected it to be so bad.\nHer 59-year-old mother, had been sick and alone during the cyclone and she was traumatised as had never experienced a storm like that before. They tried slowly to start rebuilding a small house for them to move back into and out of the community hall. It was a difficult time because Sulueti was four-months pregnant.\nSulueti and her son in Nanukuloa village in Ra Province, Fiji.\nShe managed to see a doctor three times while staying in the community hall, but there were no proper examinations or ultrasounds during this time. The delivery was fine in a maternity hospital. She received a dignity kit from IPPF and received baby supplies for post-delivery, everything that was required for a new mother to care for a baby was in the kit.\nSulueti with her son in Nanukuloa village in Ra Province, Fiji.\nIPPF went to visit her in her home and gave advice on family planning - she is taking injectables now because she doesn't want any more children.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Through the thousand pictures on-line about dark oak bedroom furniture, choices the best collections using greatest image resolution just for you all, and this pictures is actually one of pictures collections in your greatest pictures gallery about dark oak bedroom furniture pertaining to Residence. I'm hoping you may want it.\nThis kind of image (Dark Oak Bedroom Furniturebedroom Furniture Clue Part Kbmnn inside Dark Oak Bedroom Furniture) previously mentioned will be labelled using:put up simply by admin at March, 3 2018. To find out most pictures inside dark oak bedroom furniture pertaining to Residence pictures gallery you should abide by this web page link.\nThank you for visiting Dark Oak Bedroom Furniturebedroom Furniture Clue Part Kbmnn inside Dark Oak Bedroom FurnitureIf you found any images copyrighted to yours, please contact us and we will remove it. We don't intend to display any copyright protected images.We hope you can find what you need here. We always effort to show a picture with HD resolution or at least with perfect images. Dark Oak Bedroom Furniturebedroom Furniture Clue Part Kbmnn inside Dark Oak Bedroom Furniture can be beneficial inspiration for those who seek an image according specific categories; you can find it in this site. Finally all pictures we have been displayed in this site will inspire you all..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our firm promises all consumers with the first-class products as well as most satisfying post-sale services. We warmly welcome our regular and new consumers to join us for 160w Dimming Driver , 160W Dimming Driver , 130V Dimming Driver , first-class products and improve service welcome overseas customers patrons.\n\"The extremely rich projects management experiences and one particular to a person support model make the superior importance of small business communication and our easy understanding of your expectations for 160w Dimming Driver , 160W Dimming Driver , 130V Dimming Driver , We'll supply much better items with diversified designs and specialist services. We sincerely welcome friends from over the world to visit our company and cooperate with us on the basis of long-term and mutual benefits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PALO ALTO, Calif. (MarketWatch) \u2014 Most investors have never heard of Dimensional Fund Advisors. This privately held firm manages nearly $250 billion of index funds, but you can only own its funds through an investment adviser who achieves a hard-won entitlement from DFA.\nDFA has made the world of investing a much, much better place. Their index funds are low cost and highly competitive. DFA even argues that they possess better indexing skills and thus, achieve superior returns than its rivals.\nBut DFA is doing something even more revolutionary: DFA has convinced investors not to bet on a stock. Not even on a sector like utilities or technology, or even a country like China or Canada. DFA investors don't want to bet on a few bonds in a laddered bond portfolio.\nNo, what DFA so brilliantly helps investors understand is that if you can just have confidence in the capital markets, your savings will grow in the fastest and safest way possible.\nThis means, at the highest level, a bet on capitalism. When an investor loans money to a firm or government, or owns a part of an enterprise, he is entitled to his share of rewards for his capital. World markets reward investors for their capital and businesses and governments must compete with each other for investment capital. Investors in turn compete with each other to find the most attractive opportunities.\nDFA educates us on why we should just focus on capturing our share of these returns at the lowest fee possible. Instead of thinking that Apple is mispriced, or developing our prediction of solar power, bet on capitalism. Over the years, capitalistic economies thrive.\nIt is a much better way to sleep at night and outperform most other investors. Once you have confidence that capital markets will grow over time, you can then leverage proven science to build a portfolio to capture your rightful returns. That's where DFA has shown investors how to build a sophisticated portfolio that is truly diverse.\nDFA has illuminated and implemented asset allocation strategies that heretofore were the domain of sophisticated endowments and pensions. Arming advisers with great tools for educating investors, DFA has helped many to leave the active \"beat the market\" circus, lower their fees, and diversify their savings.\nFor years, DFA was the only firm offering portfolios that harnesses this investment science, largely because they had all of the component funds at a low cost. But with the rise of ETFs, no longer is DFA the exclusive purveyor of this strategy.\nDFA Funds are great. Their small and micro cap funds DFA U.S. 9-10 Small Company Portfolio DFSCX, -1.17% , The DFA U.S. Small Company Portfolio DFSTX, -1.12% and DFA U.S. Small Cap Value Portfolio DFSVX, -1.29% have made investors about 10% yearly, compounded for about 20 years.\nBut low-cost ETFs now occupy every asset category offered by DFA. Investors who felt locked into a DFA set of products, are now able to build their own portfolios without going to a DFA adviser.\nVanguard's innovation was the index fund, and DFA took that to the next level with a marketing model to proliferate index fund portfolios. Now, widespread ETFs in every asset class and portfolio building software and expertise, takes DFA's contributions to investing to every investor, not just those who can afford high-priced help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Membership offers many benefits including reduced fees for classes and workshops\/seminars, access to the building and equipment for training, voting rights at meetings, access to Members section of website and invitation to certain members-only events.\nMembership is open to all persons sixteen years of age and older who are in good standing with the American Kennel Club. Must subscribe to the purposes of this Club. Must have attended two Club meetings (one of which may be a social meeting or a club-sponsored seminar or working at a club trial or other performance function).\nhave previously earned an obedience title through the American Kennel Club or who have earned some other title acceptable to the Board of Directors of the Club.\nEach applicant for membership shall apply on a form as approved by the Board of Directors and which shall provide that the applicant agrees to abide by the Club Constitution and By-Laws and the rules of the American Kennel Club.\nAll applications are to be filed with the Secretary or designee (Membership Chair) and the name of each applicant shall be published on the club's website.\nAt the club meeting following the publication of the name of an applicant on the website, provided the eligibility requirements have been met, the application shall be voted on. Votes of 3\/4 of the members present and voting at that meeting shall be required to elect the applicant.\nMembership dues are paid annually by January 1. Dues are payable when the applicant is accepted for new membership. Check or Money Order made out to DTCCU, please no cash.\nApplicants elected January 1st through June 30th shall pay dues for the full year.\nApplicants elected July 1st through October 31st shall pay dues for half the year.\nApplicants elected November 1st through December 31st shall pay full dues for the following year.\nEach member will be required to clean one area of the building up to twice per year, as assigned by the cleaning chairperson. Family memberships are responsible for cleaning two areas.\nMembers are responsible for cleaning their portion of the building or making alternative arrangements. Alternative arrangements include changing weekends with another member through mutual agreement or hiring a member who has volunteered to clean for others. The responsibility for making the alternate arrangement resides with the assigned member.\nMembers who do not make suitable arrangements will be assessed a $30 fine as well as assignment to another cleaning date and may lose their membership privilege if they do not fulfill this obligation.\nNo one under the age of 16 will be assigned to clean.\nMembership renewals are due by January 31 each year.\nIf you are a prior member and would like to rejoin, per our Constitution, you can be reinstated upon your written request for reinstatement to the Secretary and upon payment of the current year's dues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome to a car free world where wildlife outnumbers residents. 11 miles off the North Devon coast sits Lundy Island. A walker's paradise with more sunshine and less rainfall than the mainland.\nThe most common of UK cetaceans, (whales, dolphins and porpoises), harbour porpoises are regularly seen all year round from Capstone Point in Ilfracombe.\nNorth Devon's very own Michelin Star rated chef Thomas Carr talks to us about growing up in this stunning region, what inspired him to become a chef and how he enjoys his free time away from the heat of the kitchen.\nNorth Devon is a beautiful region of glorious contrasts \u2013 and that extends to the weather too! For every period of glorious sunshine, there are days when the weather's a little more\u2026 wet.\nWith so many great days out to choose from, it can be tricky to decide what to do. We've asked some of our team for their favourite days out in Devon, take a look below.\nOn Saturday I had the pleasure of attending the Annual North Devon Street Food Awards which once again, Marsdens Devon Cottages were the main sponsor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"A loss that has been enforced and unexpected. It's about processing the irrational and incessant feelings of rejection and loneliness, like listening to the tenderness of love disappear and it being replaced by skewed logic.\"\n\"A loss that has been enforced and unexpected. It's about processing the irrational and incessant feelings of rejection and loneliness, like listening to the tenderness of love disappear and it being replaced by skewed logic. Kevin Martin provides a predominantly beatless musical backdrop, simultaneously minimal and menacing, but periodically extremely beautiful. It stalks the isolated characters created by Robinson's haunting vocals, as the poet shapes, the protagonists troubled lives into increasingly desolate narratives.\"\nThe album is accompanied by artwork from Japanese photographer Daisuke Yokota and features an intriguing monochrome design with teasing style.\nThe album will be released on Cosmo Rhythmatic on the 14th of February.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tuition Assistance forms are now available below, either in a paper format or an online version for completion. Forms are due to the Private School Aid Service (PSAS) in April to be considered for any financial aid for the upcoming school year.\nFor more information on their program and other forms of financial assistance, please contact our school principal, at 586.725.0078.\n> IC Foundation Scholarship application - Please contact the school office for further instructions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"F\u00e5 de h\u00f8jeste limits hos Pinnacle Sports via BET-IBC!\nPS3838 is a highly respected bookmaker established in 1998. They are established & licensed in the Netherlands Antilles in Curacao. The company prides themselves in operating on the lowest possible bookmarking margins, indicating that they always provide their bettors with the highest odds available. PS3838 doesn't provide any deposit bonuses, since high limits and great odds are a lifetime reward for any customer who chooses this particular bookmaker. Pinnacle has the opinion of one of the most reliable bookmakers operating today.\nAs PS3838 (Pinnacle Sports) agents, we are able to offer our customers the possibility to open betting accounts at PS3838. If, for some reason, you cannot register at Pinnacle directly, BET-IBC can assist you in obtaining a player account in Euro, USD or GBP. Additionally, we can open PS3838 agent accounts in Euro, USD or GBP. Please contact us to inquire about this option.\nPS3838 ( former Pinnacle Sports) employs 128 bit encryption security with firewalls to ensure safety of all customer data and any financial transactions. The company is also committed to responsible gaming and apart from working with renowned organizations such as GamCare or Gamblers Anonymous, they also provide a self-exclusion option.\nThe mobile version features odds from all markets, including \"live\". PS3838' mobile site is available in all the same languages as the full site. You can visit PS3838 on your mobile phone and you will automatically be taken to the mobile site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ministry of Propaganda - 22\/May\/2006: \"Safe Return\"\nAs the weather wasn't very nice and my sister had seen enough rain during the we decided to abandon the idea of a walk along the canal and drove to Oxford instead. We left the car at the oxford park & ride and took the bus into the centre. First stop was the Waterstones Bookshop, later followed by the Blackwell's Bookshop. We then visited the Museum of the History of Science, in particular the Wireless World: Marconi & the making of radio exhibition was fascinating.\nIn between we stopped at the Turf Tavern for a drink, a return visit as we had already visited the same pub in 2000. I had a White Horse Ale, which I really enjoyed. Need to find out where I can get more of it. Late in the afternoon we drove back to Swindon after a very nice afternoon in Oxford.\nMy sister now returns to work for a few days before they have a public holiday again on Thursday. I will now blog here again more often (hopefully), complete the WalkIslay 2006 travelogue and start working on the IslayBlog. Oh, and there's a bank holiday weekend coming up here in the UK as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"What if conversation is how things got done? Conversations are the way people discover what they know, share it with their colleagues, and in the process create new knowledge for the organization.\nThe activities of this day together will feature two very different venues and conversation with colleagues and students.\nFull-Time Faculty Do NOT need to register.\nIf you are not a full-time faculty member and would like to attend the conference, please register.\nBe sure to review your afternoon workshop session options in advance of attending the conference. If you plan to attend the 'Integrating Strengths in the Classroom' session and have not yet taken the StrengthsQuest survey, please contact Dalisa Diaz, Human Resources Assistant, at to request a code.\nThe Community College Survey of Student Engagement (CCSSE): Parts of the elephant, the garden path, and Deming's funnel, or, what do we know, does it mean anything, and what should we do?\nAs part of the college's emphasis on being informed by data, we've participated in the Community College Survey of Student Engagement (CCSSE) since 2007. From these efforts, we have a bunch of \"stuff\" \u2013 interesting, contradictory, and revealing \u2013 but what does any of this really mean? Today, we'll get the chance to revisit three survey questions and answers about student effort, academic challenge, and student-faculty interaction, to see if these questions and answers have something to say about us and what we do.\nIn 2009-10, Northern Essex conducted our first institution-level assessment and found that our students are actually decent writers. Surprise! It sounds like good news, but what are implications if the students in your classes write horribly? In 2010-11, a dual assessment was conducted in Quantitative Reasoning and Global Awareness. For five of the nine criteria, only one-quarter or fewer students were rated as competent or better. If these skills reflect our values as an academic community, then clearly we must do something \u2013 but where do we start?\nAll learning community students participated in the Washington Center's Online LC Student Survey, a national measurement tool which asks students for their perceptions on their learning and engagement, much the same way that the Community College Survey of Student Engagement does. We'll share our results, plus how learning communities are an effective vehicle to teaching and assessing core academic skills.\nFollowing these presentations, five NECC students will introduce themselves to the audience and respond to questions about the data from the student perspective.\nIn small group discussions at our tables, we will consider questions about what we have heard. These are not questions for which we already have the answers \u2013rather they are open invitations to call forth ideas and insights that generate new thinking, different questions, and imaginative ways of moving forward as we consider the data we have learned about and what it means for us as faculty and staff.\nAfter lunch you will have the opportunity to attend two workshops.\nEach workshop listed below is offered twice.\nAs educators we have specific academic goals for our students, and specific ideas about how those goals can best be met. However, what goals do our students bring to the NECC classroom? Do their goals compliment or combat the goals we have for them? How can we help students and faculty to align their goals so that everyone succeeds and feels successful? This presentation will introduce the theoretical and practical work being conducted by the Academic Motivation SFIG. Discussion topics will include an examination of the types of goals that students, faculty, and staff may work towards in an academic setting, as well as practical information about helping students to set positive, attainable goals in college.\nNOTE: If you plan to attend this workshop and have taken the Strengths Finder survey, please bring the description of your top five strengths with you. If you have not taken the survey, please contact Dalisa Diaz, Human Resources Assistant, to request a code. Please request the code at least ten days in advance of the workshop and bring the results of the survey with you.\nYou know your strengths, but do you know your students' strengths? How can knowing student strengths enrich your teaching? How can strengths be woven into the fabric of the content of your classes? We'll discuss some ideas about working with strengths in the classroom, how it can enhance cooperative learning, how it can improve appreciation for diversity, how it can give you the tools to create dynamic group projects, and how it can enable students to take more responsibility for their own learning. This workshop will provide participants with a chance to experience several Strengths-based activities in small groups.\nAre you interested in adding engaging learning activities to your course? In this workshop, you will learn how to create learning activities, self-assessments, and games to help students \"master the basics\" of course material using the programs Study Mate Class and StudyMate Author. Through the use of these interactive exercises, students are able to choose activities that appeal to their learning style, engaging them with the course content in a powerful way. You can choose from a dozen activities \u2013 such as flash cards, crossword puzzles and self-quizzes \u2013 to engage students both individually and collaboratively while reinforcing key concepts in the course. Your students can access the interactive activities using computers, smartphones and tablets.\nThe hands-on demonstration will show how the activities can be used in class. These activities can integrate seamlessly with NECC's learning system, Blackboard, for use online as well as in the classroom. StudyMate is a companion program to Respondus, which can export questions to StudyMate to create Flash-based learning activities.\nWant an image or video clip to put into your course's Blackboard environment? How about a magazine or journal article?\nYouTube and Google are useful options, but Library resources such as ImageQuest, Films on Demand, and our many journal databases are upgrades over most free Internet resources. Library resources offer advantages to our students such as no advertising, better authority, and copyright clearance. Plus, it's usually easier to create citations for them. The downside for faculty: integrating these resources into course management systems like Blackboard involves a couple of extra steps.\nThis workshop will provide an introduction to some of the Library's subscription resources and illustrate tips and tricks for using them in the Blackboard environment. We will demonstrate how to use persistent links (permalinks) for journal articles and how to embed Films on Demand and ImageQuest images into Blackboard.\nWe will also demonstrate one of the Library's most popular resources \u2013 LibGuides. The NECC Libraries have already published more than fifty LibGuides in a variety of subject areas, for particular classes, and as \"how to\" guides. The Librarians will show you how to link to these popular guides in Blackboard.\nAre you looking for new ways to engage your students? Do you want practical strategies that students can use to review material and truly learn it? Are you hoping your students will leave class feeling like they've learned while having fun? During this workshop we will explore different games to meet the needs of each student's intelligence. Using interactive game techniques students will gain a deeper understanding of the course material covered in class. These strategies can be used across all curriculum areas. Come prepared to play and have some fun!\nEducators want to know more about their students. This workshop will gave faculty and students an opportunity to consider some of the following questions and to ask other questions as they discuss what works best in the college classroom.\nWhat are the things about students that your professors MOST need to know, but PROBABLY DO NOT? What are the things that your BEST professors have done to help you succeed in class? What are the things that professors have done that has been an obstacle to succeeding in class? What makes it easiest for you to relax and ask questions in class?\nWhat are the student behaviors that you find most confusing and difficult to understand? What student behaviors do you believe are the biggest obstacles to academic success? What lessons did you learn as a college student that might help students at NECC succeed?\nWhat is the Future of \"Democracy's Colleges\" in the Commonwealth?\nCommunity colleges are in the spotlight like never before\u2014in Massachusetts and across the country. We have always been places where more of the most vulnerable and at-risk members of our community\u2014first generation and low income students, minorities, students with learning and physical disabilities, and others\u2014come for an opportunity at higher education. Increasingly, we have also become places where students with financial means, honors students, and adult learners of every kind arrive to begin, continue, or restart their educational journey. But even as higher education credentials have become a requirement for entry into the middle class, community colleges in Massachusetts are being pressured to narrow our focus, restrict our resources, and, if we are not careful, abandon important parts of our mission, and perhaps some of our students. This session will explore the topic: What is the future of \"Democracy's Colleges\" in the Commonwealth?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'VE BEEN AWARDED ISRAELI CITIZENSHIP.\nWas the postage in schekels and was the letter penned in goy blood? If not, its a fake!\nDid you test it for ricin?\nI got one from a prince in Kenya. Once I make the initial bank deposit I'll be rich. You'll still be working 9-5 like suckers.\nThey need some fresh meat for IDF in Gaza Strip, my friend!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bloomberg Businessweek reports that a Chinese espionage agency successfully planted microchip back doors on servers built in China for Supermicro, San Jose, CA, a major supplier of motherboards to many companies including Apple, Amazon and the US military. At least 30 companies are believed to have been compromised. Bloomberg cites 17 (unnamed) sources including American intelligence officials and insiders at Amazon and Apple. All of the companies have denied the report, which is what one would expect.\nThis is pretty scary stuff, IMO. Is there any way to prove that hardware has not been compromised?\nThat'll teach \"them\" new meanings of outsourcing. But I guess why is this a surprise, Trojan horses were invented a while back?\nApple did acknowledge to Bloomberg Businessweek that it had encountered malware downloaded from Supermicro's customer portal. Apple said the infection occurred in 2016, months after the events described by Facebook, and involved a single Windows-based server in one of the company's labs. The malware was on a network card driver, which is distinct from firmware and allows an operating system and a piece of hardware to communicate. This was the reason Apple gave for dropping Supermicro as a supplier later that year.\nI agree with Richard, such in-depth reporting like this with so many sources has a way of turning out to be true despite adamant early denials. \ud83e\udd37\u200d\u2642\ufe0fUsually, though, adamant denials don't come from shiny brands like Amazon and Apple, do they?\nOn the one hand, Bloomberg's story is full of specific details, seems well-researched, and is apparently based on information from many sources. On the other hand, companies like Apple, Amazon, and Super Micro are are vehemently denying it.\nApple's denial is comprehensive and scathing. For a company that rarely offers more than a sentence \u2014 if that \u2014 when asked for comment, it says a lot that Apple not only commented but released a lengthy and detailed statement denying virtually every claim Bloomberg made. They're serious about this.\nAmazon's denial is similarly scathing.\nBloomberg Businessweek has a pretty solid reputation, but it's hard not to wonder if they've gotten something seriously wrong here. All their sources are anonymous, and the denials leave no room for interpretation. Something is definitely fishy.\nVehement denials would be expected given the gravity of the report. I'm cynical enough to think they may all be lying nevertheless, though I'm not assuming that.\nWe may never know. What worries me, though, is the very possibility of undetectable hardware subversion. We know from the Edward Snowden revelations that the NSA regularly intercepted computer and networking equipment en route and modified it for spying. Building spying into the board itself seems like a logical next step. China certainly has the technical sophistication and political culture to do so. I just wonder whether the US (or anybody else) has the ability to detect and prevent it.\nAll this talk about a \"chip\" with no technical details almost sounds like it's made up. Just saying.\nPersonally, I find this kind of threat pretty plausible, and I'm not surprised, either, that it was picked up by teams who were vetting the hardware for use in government systems, because the kind of due diligence described is accurate to what I've understood about how things*ought* to be done. It is also plausible that many successful infiltrations have been executed on targets who don't have the ability or awareness to combat this attack vector.\nAs to how extensive the real consequence is, you have to keep in mind, the most sensitive targets have many layers of protection, and even something with low level access would also need other compromised components of the network to succeed in it's task of communicating out to an awaiting server witha malicious payload. Those networks are closed (unless compromised) and communication attempts would fail. That doesn't exactly give me warm fuzzy feeling of security of course. Plenty of government business *is* tired to relatively open networks.\nBesides the governments, Telco players are also extremely cautious of what goes into their backbone networks, for similar reasons, and*extensive* testing is done which takesa long time and tons of effort. Winning a hardware supply contract with a Telco provider is a very long sale cycle.\nI started off firmly in the middle once the story and company press releases were released. From yesterday to today I've drifted more towards believing Apple over Bloomberg.\nThey aren't just denying the claim of a vulnerability but even denying the fact that there's a gag order in place. With wording like that Apple stands to lose a lot of credibility without much room to budge if it comes out that these allegations were true. Bloomberg might just be an unwitting party propogating a fabricated (not by them) story to benefit the China negotiations.\nOn the other hand you've got a tinfoil hat scenario where government that can and has forced companies to deny the existence of programs and investigations.\nSomeone's credibility is going to take a severe blow and at this point I don't think it will be the tech companies.\nDid anybody else have to Google \"baseboard management controller?\".\nThe plot thickens! These same Bloomberg reporters apparently have a history of writing sensational stories that later turn out to be wrong while citing anonymous sources.\nI wonder if that chip in the image is it.\nVery very doubtful. If this was true why have we not seen more evidence of it? It'd be almost impossible to cover it up for any length of time. There'd be thousands upon thousands of these things out there. Bloomberg is the claimant and it's therefore up to them to provide significant evidence to prove it's real. It's not up to Apple or Amazon or other parties to prove it isn't true.\nI think it's always a good idea to be skeptical. The denials have been strong, yet Bloomberg hasn't retracted the story yet. So someone is lying. But to me it's not just a matter of whether these specific claims are true. More troubling to me is the possibility that this sort of thing could go undetected because there may be no practical defense against it.\nAccording to The Washington Post, other media companies have tried to verify Bloomberg's story, but so far they've come up empty. And they also note the lack of specific proof in the story; it relies completely on anonymous sources without ever showing one of the chips.\nI'm still open but it doesn't seem to be looking good for Bloomberg.\nTo my mind, if that's not the actual chip, it should not be a picture in the article. That is misleading.\nIt could be the Bloomberg source that is lying. I believe there are ways to stop such things from occuring. I think it's going too far to say there is no defense against such things.\nIt's probably more likely that the source is lying than Bloomberg itself. I would think that if this were firmly established, Bloomberg would issue a retraction. Perhaps we'll find out at some point, dunno.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The congregation of Unity Baptist Church was organized in 1909 as an outgrowth of Kershaw's First Baptist Church. Unity is believed to be the second separate black church established in Kershaw, in the early 20th century. The congregation originally met in the homes of its various members. The new sanctuary was built by Deacon George L. Shropshire, a local carpenter and contractor. The Reverend A. W. Hill became the church's first full-time minister in 1911 while he was still a student at Benedict College in Columbia. Hill's successor, the Reverend L. C. Jenkins was the first pastor to occupy the parsonage (c.1922) adjacent to the church. Unity Baptist Church has undergone few alterations since its construction and is listed on the National Register of Historic Places. View from outside; open on Sundays.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Raveendran along with Bro Billy Fong, Chairman for Region 4 and National Coordinator for Malaysia respectively. After a brief spell of invocatory Veda chanting, devotees from the region rendered traditional holy Buddhist recitation beginning the proceedings.\nA welcome speech by Dr VK Raveendran followed next, who in brief explained the import of the Buddha Purnima celebrations. Recalling the story of Emperor Ashoka who, after winning many bloody battles, costing many precious lives, finally got into transform embracing Buddhism as his way, Dr Raveendran briefly explained the three essential steps involved in Buddhism, namely, Self-Control, Meditation and Inner Vision. He then introduced the two speakers for the session.\nThe first speaker Mr Praneth from Sri Lanka, an a exponent of Buddhist teachings, also active in Bhagawan's EHV programme, spoke of Buddhism with reference to Bhagawan's teachings, explaining the Eight Fold Path pronounced by Lord Buddha.\nBrother Billy Fong, Central Coordinator, Malaysia spoke of Bhagawan and the supreme importance of carrying out Sadhana at individual, family and societal levels as per the pronouncement by the Sri Sathya Sai Seva Organisations.\nInternational Bhajans continued that ended with Mangala Arathi at 1900 hrs.\nOn 15 May evening, the second and final day of this year's Buddha Purnima celebrations the programme commenced with an introductory talk by Dr VK Raveendran, who introduced the venerable speakers from the Tibetan Monastry, Hubli, Karnataka. The two speakers were Venerable Aaron Lobsang Thardodh and Venerable Gesha (Dr) Lobsang Lhundup.\nVenerable Aaron is a specialist in Nalanda philosophy. His objective is to apply Nalanda logic in modern environments, envisioning a resurrected learning space for Nalanda University in the 21st century. Guest Senior Monk Venerable Gesha belongs to the Vajrayana Buddhist Order, known widely as the Order headed by His Holiness The Dalai Lama.\nVenerable Aaron, who spoke in English, divulged briefly on the theme 'Love', as to how humans could be different from other creatures, exhibiting their inherent potential \u2013 LOVE, the best medicine for all ailments.\nVenerable Gesha chose to speak in Tibetan, translated into English by Venerable Aaron. The venerable monk's speech was on the enlightened life of Siddhartha turned Buddha and Buddha's main principle of Non-dualism.\nThe Sai Youth of Malaysia then presented a drama entitled THE MASTER LIVES ON, a depiction of the story of a dynamic Sai Youth, whose charming life came crashing down on the fateful day 24 April 2011, when The Master, Bhagawan Sri Sathya Sai chose to leave His mortal frame, an irreparable shock to the Sai Fraternity.\nBhajans by the devotees from Region 4 continued before students took over and ended with Bhagawan's Thursday Special \"Life Is My Form\u2026Truth Is My Breath\u2026\" Mangala Arathi was offered at 1910 hrs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..8a3423173fa96bb7bf8cadf7d1a4d281714bda55
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@@ -0,0 +1,5 @@
+{"text":"Sorry for the delay, it's been a whirlwind of a weekend at the fixer-upper...thanks for your patience!\nI am so glad you are here for the party, enjoy this week's features!\nThanks so much for hosting, Cindy. I can't wait to see the fixer-upper's progress. Have a wonderful week!\noh wow, so exciting. Thanks so much for the feature.\nThanks for hosting, Cindy. I'm ready for a cup of strawberry ice cream and a garden stroll.\nCindy, appears there's already a lot of great posts this week. Thanks for hosting.\nLooking so forward to seeing your fixer upper fixed up! Thank you for taking time to host us. I really enjoyed Marty's reflection on her place in the sun in her new home.\nThanks so much for featuring my faux wood beam this week, and for hosting!\nThanks so much for sharing my garden tour post! Have a great week, Cindy!\nThank you for hosting Amaze Me Monday and have a great week.\nThank you Cindy for your 222nd Amaze Me Monday. Have a great Summer week ahead.\nThank you for hosting, Cindy! Great features this week...that garden is beyond!!\nThanks for another great party! I hope you have a wonderful week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am looking for highly motivated registered nurses experience or newly qualified RGN RMN RNLD who will play a key role within this care home. The home specialises in care of the elderly including Parkinson's Disease and General health care conditions. The home is located in Cambridge.\nThe Manager can accommodate both part-time and full-time hours. They can be flexible regarding shift pattern, so you will be able to choose permanent day shifts or permanent night shifts or a mixture of both if you prefer as well as short or long shifts.\nWhen would be a good time to call you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"You should never have so much work to be done, that you have got no time to think.\"\nBy bike past the Berliner morning traffic jam.\nAutomatically hide all sources of interference.\nIn the small Kiez book \/ magazine shop around the corner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In an effort to promote greater awareness of Armenian history and literature, the Zohrab Center has been very active in donating its duplicate book and periodical collection to libraries, museums and individual scholars across the country. Academic institutions such as Columbia University in New York, Tufts University in Massachusetts and the Ararat-Eskjian Museum in California have been the recipients of dozens of books and periodicals over the past month and the list of institutions continues to grow. Academics from the New York metropolitan area have also benefitted greatly from the Zohrab Center's duplicate collection, adding our resources to their personal libraries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"properties and units that we'll be using.\ndensity in kilograms per cubic meter dynamic viscosity.\nAnd here they use a, a new, you'll notice a new unit.\nIt's shown as Pa S.\nThis stands for Pascal seconds.\nsecond per square meter, or kilograms per meter second.\nfinally vapor pressure, but vapor pressure we won't you, be using in this course.\nThe right hand table are properties of water in English units.\nspecific weight pounds force per cubic foot, and then density row.\nBut here, unfortunately they've given the density of different units.\nthe number there they give typically is 1.94.\nthe density will be given in units of pounds mass per cubic foot.\nAnd the density in pounds mass per cubic foot is approximately 62.4.\nare actually slugs per cubic foot, which is another unit of mass.\nso we need to be careful about that.\nin units of feet squared per second is also consistent.\nbecause they're not consistent in their uses of units.\nIn different places in the manual, they give the properties.\nAlso, here are properties of gasses.\nFor example, air here, which we'll be most usually concerned with.\nAnd here properties of selected liquids and gasses.\nAnd here you see the density of water is given differently.\nin terms of pounds mass per square by cubic foot.\nAnd the number there is 62.4 is the more common one.\nof pounds mass per cubic foot rather than slugs per cubic foot in British units.\nthe most common values of properties that we'll use are in metric units.\nsometimes a 1,000 kilograms per cubic meter or 62.4 pounds mass per cubic foot.\nor 62.4 pounds per cubic foot.\nSo those are the typical values we'll use over and over again in this segment.\nNow we've only looked at a few properties here.\nlook at the properties of vapor pressure.\nused int the Effie exam, we won't be covering them here.\nin the next segment we will start looking at fluid statics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Robyn's Nest: How Does Your Garden Grow?\nThis past Saturday our family spent the morning out around town pricing landscaping for our yard. For some reason, our builder\/landscaper did a HORRIBLE job and all but about 5 of our shrubs have died since we moved in. We pulled one dead shrub up after about 2 years of living here and it was still root bound. Not cool. They're not dying as in we haven't taken care of them. They're dying because they weren't planted well. Anyway, we need to completely re-do the landscaping around your home (Yay. Just exactly what I wanted to spend money on. Yeah right.) so we were around town pricing shrubs, flowers, etc.\nWe went to Southern Gardens and then to Lowe's. Turns out, Lowe's has shrubs for like $1 cheaper per shrub than Southern Gardens does. When you're buying as many as we are, that's a lot of $$! We're talking an overall savings of about $40!\nWhile we were at Lowe's, we spotted some cute little kiddy garden things. My plans were to start an actual garden this year, but with all the money we need to spend on landscaping, I just don't think it's going to happen. Matt and I really want to teach LL that food doesn't just come from the grocery store and that families can do some self-sustaining just by a little work at home. We had also talked about getting like 2 chickens at one point, but I just don't know if I want to do that anymore.\nI do still think that it's important to teach LL how things grow and where food comes from. While we aren't going the huge outside garden route (at least this year), these little kids gardens are a good alternative for us!\nWe also got another brand that does watermelons and sunflowers.\n2- Lay out your materials.\n3- Put the soil pellets in the bottom of the pots.\n6- Fluff up the dirt.\n7 - Hire a planter.\nSorry! I already hired the cutest one in town!\n8 - Put the seeds in.\nShe did a GREAT job and had a blast doing this part! Although we might have some tomatoes growing in the same pot as banana peppers - haha.\n9- Cover the seeds with dirt.\n10 - Cover the pots with the greenhouse cover and put in a sunny place.\n11- Wait for your plants to grow!\nI'll be sure to post updates as they start growing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Valuable Affordable Web Sites answers relative to Web Page Design In Rundle Mall SA plus the 5000 local area. The most skilled, fast, dependable and inexpensive local web provider company in Rundle Mall is always available when you want the best professional services. Call right now for immediate service.\ncom, which is not very the situation. At its Main, This really is about how the various search engines check out your website and how back links pointing to your website are taken care of. it is possible to prefer to go While using the 'www' or without it, but it is crucial to decide on only one. If you choose both of those, the major search engines will view your site as two individual web-sites, which can poorly damage your rankings and link top quality.\nDreamweaver is a component of Creative Cloud, meaning you'll be able to accessibility all of your assets \u2014 like Adobe Stock \u2014 proper In the application and promptly flip any inspiration into a lovely Site. And with Adobe CreativeSync, all the things is related throughout your desktop and cellular applications.\nWeb designers may possibly prefer to limit The variability of Web page typefaces to only a few which can be of the same type, as an alternative why not try here to employing a variety of typefaces or variety styles. Most browsers recognize a particular number of Secure fonts, which designers predominantly informative post use in an effort to stay clear of troubles.\nIn 2012 and 2013, carousels (also known as 'sliders' and 'rotating banners') have grown to be an incredibly preferred design ingredient on homepages, normally utilized to showcase highlighted or the latest articles in a anonymous confined Place.\nFluid layouts elevated in acceptance all around 2000 as an alternative to HTML-desk-based mostly layouts and grid-based mostly design in both equally page format design theory and in coding system, but were look at this pretty gradual for being adopted.[Be aware 1] This was as a consequence of concerns of monitor looking through units and various windows sizes which designers don't have any Command more than. Accordingly, a design could be broken down into units (sidebars, information blocks, embedded advertising regions, navigation places) that happen to be sent towards the browser and which can be fitted to the go to these guys Screen window through the browser, as greatest it could possibly.\na hotlink to another web page. you'll have to know the Uniform source Locator (URL), or Web handle, of any web site to which you would like to connection your page.\nstart with a title and an focus-grabbing greeting or introduction, so men and click for more women will know immediately what your page is Web Page Design In about and will want to read a lot more.\nIt's alright When you are Doubtful, but this aids us figure out if we are an excellent suit. Our fastened price proposals are dependant on the estimated range of hours for us to complete a challenge, so knowledge your constraints makes it possible for us to propose a earn-acquire situation.\nThe function you demonstrate should be effective at speaking for itself, but you'll also wish to demonstrate probable employers that you're enthusiastic and self-assured in Web Page Design In the skills that authorized you to make it.\nThey normally present consumers with Artistic methods for equally print and the world wide web. Doing the job Web Page Design In disorders differ significantly depending on the employer.\nThe two position titles are basically interchangeable. unique companies and education schemes might use distinct terminology, but they are generally all referring to a similar form of Inventive Qualified.\nIn fact, there isn't a Web Page Design In normal employment circumstance or perform environment for graphic designers. the probabilities are as broad as your creativity. So, what is it possible to do that has a degree?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bakemonogatari is probably my favorite anime, there is a lot to it, for all the fanservice and neck twisting it shows it's pretty deep sometimes, and it's one of those shows that is very meta about itself, the culture and the genre and I find that very appealing; plus the aesthetics and visual direction are cool.\nAnd then you have America, England, Russia and etc on the other hand.\nSo is it some spoof in real life world history, or is it just some random made up spoof series on something they wish was real life history. I have just seen some episodes and i have hard time to understand it. Maybe i should just try episode 1 and see how it goes.\nLast edited by Rokai; 31st August 2015 at 02:29 PM.\nI've started watching Rokka no Yuusha, which is the newest anime from the director of the anime adaptation of Spice and Wolf. It's pretty good, the first two episodes start it off pretty slow, but around episode 3 it starts speeding up a lot. The art and animation quality is very good, though it is reliant enough on CG for the monsters that they kind of bother me.\nGonna check it out on my watch list. I liked the atmosphere in Spice and Wolf.\nAlthough i guess its interesting that they chose a mayan indian setting of buildings in the series.\nLast edited by Rokai; 5th September 2015 at 05:21 PM.\nMy sister and I have been watching Fullmetal Alchemist: Brotherhood for the past few weeks. I shouldn't think I would have to tell you guys how great it is considering a lot of you have probably seen it already. But seriously, wow, it really is just absolutely incredible.\n\"I'm sharpening my knife, kupo!\"\nRolling up onto episode 521 of One Piece (so right at the beginning of the New World stuff) and it feels like forever since I last saw the Straw Hats.\nIt's been filling up my anime time while I'm waiting for new episodes of other stuff to come out.\nSo I'm watching Pshyco Pass it's ****ing fantastic!\nSo I'm all caught up with Rokka no Yuusha now, and I'd totally recommend it. I think it's biggest problem is the beginning: The first four episodes are all about setting up the main characters and have a lot of demon fighting, which is a problem because the demons are all animated with some pretty jarring CG, and the setting-up they do makes it all seem a bit generic. From episode 5 onwards, it turns into a really tense action mystery story featuring a cast mostly of crazy people.\nThe main plot thing it all centers around is that there's only supposed to be six legendary heroes. When all six of the current generation meet up with each other however, there turns out to be seven of them, and one of them has activated a magic fog trap that's keeping them all in one place, and the only way to get out is to find the person who activated it, the seventh hero, and kill them. The main character ends up being the one everyone suspects, and has to fight for his life while finding some way to prove his innocence. The main character doesn't have any sort of special powers, unlike some of the others, so he has to do all of this while relying on his own bag of tricks, willpower, and quick-thinking. It's a really fun watch.\nStarted watching the new Lupin III series yesterday. The four episodes that are out now are all fantastic and capture the spirit of Lupin perfectly. I highly recommend it.\nIt's super dynamic cooking time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever watched an orchestra play and wondered if you could play music like them? Maybe you've listened to some of your favorite classical music pieces and noticed the wonderful sound of the cello. Many people enjoy playing music either for themselves or as part of a larger group of musicians; however, even if you have no plans of ever making it into an actual orchestra, learning how to play the cello will be able to provide you with many hours of enjoyment.\nAs with any type of musical instrument, the cello is an excellent way to enrich your life with a variety of different types of music. Additionally, there have actually been studies that indicate that children who play a musical instrument will typically do better in math as well as many of their other classes. The ability to play a musical instrument offers a number of wonderful benefits, but the one that most people are looking to fulfill is simply having a beautiful way to express themselves through music.\nRegardless of what your musical aspirations might be, if you want to learn how to play the cello, the first thing you'll need to do is to purchase or rent your instrument, then find a qualified instructor or course that will be able to teach you the basics of how to play this instrument. Of course, once you have mastered the basics, you'll be able to continue your music instruction in order to gain an even greater proficiency with your cello. As with many things in life, there will always be more to learn, so you'll be able to continue challenging yourself as you gain more experience and expertise in playing your cello.\nAnother factor that must be considered when you begin taking cello lessons is that it's going to take a commitment on your part. It's easy to throw in the towel and simply quit when you get frustrated or overwhelmed with a new concept; however, an important part of gaining the proficiency you need in order to play the cello well is to simply commit yourself to practicing on a regular basis. You certainly don't have to spend several hours every day playing the cello, but you should take some time each day to at least play for 20 minutes or more. With regular practice, you'll soon begin to notice a dramatic improvement in your ability to play as well as your understanding of various musical pieces.\nWhen you're looking for an instructor to help you learn how to play the cello, you'll have a few options. If your child is taking the cello as part of their music class in school, they will typically have an instructor in school who will be able to give them the guidance they need. However, if you aren't a student or if your child wants a little more instruction than what they can get in their school's band or chorus class, then you'll either need to find a local music instructor or you can take online cello lessons.\nMany local music stores will offer classes for the cello as well as for other instruments. One problem with this type of instruction is that you are often tied to a very rigid schedule; and, if you are not able to make your lesson on time, you generally cannot schedule another time slot, you just miss out for that week; of course, you still have to pay. Also, the number of instructors who teach the cello might be limited to only one or two at a particular music store, so you will not have many options, and if there is a personality conflict, you might not be able to switch to another instructor. Of course, you also have the inconvenience of having to drive to your lesson, which could be a hassle on busy days or during times of inclement weather. However, when you take lessons at a music store, you often get great instructors who can do an excellent job of teaching you how to play the cello.\nAn alternative would be to take online cello lessons. You'll be able to take lessons at your own convenience and you'll never have to worry about going out if the weather is bad. Additionally, you'll get quality instruction, and you'll be able to progress at your own speed.\nAre you interested in learning how to sing? If so, then you've come to the right place. We're going to provide you with five tips that can put you on the right path. Before you get into anything, it's important to point out that there are a lot of other things you can do to help you with singing, but these five tips are a must. Read on to find out what these tips are.\nOne of the best things you can do for yourself is warm up. That's right, having a proper warm up routine is crucial if you want to become better at singing. If you haven't been warming up, then you're missing out. If you're new to singing, then definitely starting warming up. You don't need a rigorous warm up routine, but you don't want a routine that is over and done within a minute or two.\nThere's no doubt about it, singing lessons can help you learn how to sing or sing better if you already have singing experience. Don't worry about having to go somewhere to take lessons because you can take online lessons. Singing online lessons offer a number of benefits, such as being flexible and affordable. Best of all, singing online lessons are structured, easy to understand and you can make great progress with them.\nMany online courses provide you with detailed steps, tips, and tricks. Sure, you can take free lessons on YouTube, but paying for them is better. Paid online lessons will give you step-by-step directions and personal guidance.\nHere's a tip, you should take an extensive online singing course, especially if you're a beginner. Extensive lessons cost a bit more than basic lessons, but they are worth the money.\nYou need to learn how to manage your breathing because if you don't, then you're missing out on a big benefit. That benefit is being able to sing, plain and simple. The bottom line is you won't be able to sing if you can't manage your breathing.\nHint: breathe in through your mouth and not your nose. Keep breathing like this until you are used to it. When this happens, you will notice a huge different in how you can sing.\nWe want to point out that breathing for speaking is entirely different than breathing when you sing. There are many ways to control your breathing while you sing, but your best bet is to learn from someone who knows what they are talking about. This is another reason why you should take singing online lessons. Lessons will usually include a section that covers breathing while singing and give you tips on how to control and manage your breathing.\nThis piece of advice may seem a little offbeat at first, but it is one of the most important things you'll want to do. If you want to know how to sing, then practice singing with confidence. You want to sing without feeling like you have the weight of the world on your shoulders. The more confidence you have, the more fun singing will be. Also, express your feelings when you sing. Singing with feelings is one of the best ways to convey the song's meaning.\nRecord your singing sessions on a regular basis. This gives you the chance to listen to how you sound and you'll be able to make improvements. For example, your pitch may be too high or too low or you may not use enough emotion during your songs. Whatever the case is, you'll know where you'll want to improve, so always record your singing sessions.\nAs you can see, there's a number of things you'll want to keep in mind when you are learning how to sing. The most important tip discussed above is taking singing online lessons. Online singing courses are without a doubt the easiest and most efficient way to learn how to sing. Even if you have experience singing, you can still make improvements by taking online singing lessons. Enroll in singing lessons today and start your journey to becoming a better singer.\nIt's easy to learn how to play the clarinet. However, there's a right and a wrong way to do this. When you've never played the musical instrument before, you should resist the temptation to just go ahead and figure it out, a cappella style.\nThe really bad thing about taking this route is that you have now made it difficult to learn how to play the clarinet textbook style. The clarinet is a wonderful instrument that can emit the purest of tones. It can rival the violin in sweetness. At the same time, the clarinet is also capable of making sounds that can compete with the ambulance or the home alarm system.\nIt all depends on how you breathe into it, how your fingers are positioned across the instrument, and how proper your posture is. The children's song, All I Want for Christmas is My Two Front Teeth can give you the first lesson to learn how to play the clarinet in the right manner. Accordingly, you need to position the tip of the clarinet mouthpiece right smack behind your two front teeth.\nTherefore, missing your two teeth in that area can prove to be some problem. Obviously, you'll have to find a way to improvise when you're lacking in that department. Still, the clarinet needs a solid position along with your dental faculty with which the instrument can anchor itself in other to play beautiful melodies. Your gum may or not be up to the task.\nThe second important lesson to learn how to play the clarinet is assembling the instrument properly. Musicians don't always carry the clarinet as a whole. They can have the instrument lying inside a cushioned leather as if it were some kind of royalty sleeping on a velvet shell casing or an A-list celebrity on the red carpet.\nYou might say that the clarinet seems to lead a spoiled kind of existence, but that's the high price you have to pay for keeping a precious asset in tiptop condition. Especially when you're a serious musician, you will want to care for your clarinet like a sharp shooter does to his or her top gun. Even this instrument is usually broken down into major components for storage and easy transport purposes.\nFor its part, the clarinet has six major parts. If you're really serious to learn how to play the clarinet, you will want to learn these individual parts intimately. This is because, in order to produce notes and harmony, you will be manipulating each part just like a puppet master does his or her puppets.\nThe major portions of the clarinet are the mouthpiece, ligature, three different joints, and the bell. The three joints are the barrel, upper, and lower portions. And now, to easily distinguish among these parts, the mouthpiece is just like a sports referee's whistle. The bell, on the other hand, which happens to be the bottom portion of the clarinet, is the one that resembles a funnel\u2013although it is an inverted one.\nWith regards to the three joints, think of them like the portion sticks that form part of the Swiffer retractable rod and you can never go wrong. They will never fit each other unless they are connected in the right order. But sometimes, the upper and lower portions of the clarinet can be a little tight at their cork ends, so you need to put a light film of cork grease on them.\nYou just can never learn how to play the clarinet until you put all the pieces together and properly at that. A key thing to remember is not to force things through if the parts don't fit at all. Also, the upper joint is the smaller counterpart of the lower joint, which is the biggest rod of your clarinet.\nTo make all sorts of notes such as F or E, you have to put your thumb and index finger of the left hand in the proper holes. Meanwhile, the right hand's thumb should be pointed right below the thumb rest. As for the four fingers of this hand, they should find themselves snugly on the rest of the holes of the clarinet.\nBut the two biggest secrets to learn how to play the clarinet have not yet been revealed. You can control the wind through the holes, but the real notes are made by the vibration of the reed which sits right at the back of mouthpiece. By biting the mouthpiece gently with your two front teeth and blowing on the tube to let the air in, the reed ends up vibrating.\nFinally, practice tapping lightly at the reed with the tip of your tongue in order to make the T sound. And there you go\u2013you just made the F, E and T sounds through this short course to learn how to play the clarinet. Congratulations!\nRear The Urban has been reared and is ready to get things going again with an all new look. We think that everyone who comes back to reartheurban.org will be pleased to see what we have coming with our new project. There's so many different aspects of music that we want to cover when we come back to you, you are just going to be blown away with what we have come up with!\nSo get ready, it's coming. And it's going to be better than ever!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Milestones were erected in 1728 along the Fowlmere to Cambridge Turnpike Road. The first milestone from Cambridge (shown) is at the Stone Bridge (now the junction of Trumpington Road and Brooklands Avenue). The second milestone is on Trumpington High Street near Wingate Way. The third milestone is on Hauxton Road near the entrance to the Park & Ride site.\nFor a fuller account of this \"Trinity Hall series\" of milestones see this page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatuoc b/data_all_eng_slimpj/shuffled/split2/finalzzzatuoc
new file mode 100644
index 0000000000000000000000000000000000000000..fd8706189b850576f2cef17229f506be2071f857
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatuoc
@@ -0,0 +1,5 @@
+{"text":"A two night weekend stay for two people for two people at one of the most iconic hotels on Mull. Includes a six course taster dinner and a full Scottish breakfast. Also use of the hotel's leisure club.\nA two night weekend (Friday and Saturday) stay for two people at one of the most iconic hotels on Mull. Includes a six course taster dinner from Sheallach-Na-Mara Restaurant's taster menu, and a full Scottish breakfast, on both days. You'll also have use of the leisure club's swimming pool, outdoor hot tub, sauna and steam room.\nThis voucher features a unique reference ID code, may only be redeemed once, may not be exchanged for cash, replaced if lost and is non-transferable. No change will be given if the full amount is not spent in one transaction. The expiration date of the voucher is non-extendable and non-negotiable. Please quote the gift voucher ID reference when reserving your experience with the venue (using the contact details shown on this certificate). The gift voucher must be presented upon arrival when redeeming the experience. Subject to availability. Terms & Conditions subject to change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parma is one of those places where art, music, literature, history, traditions and good life are strictly linked together.\nIt is particolarly suggestive the heritage that greatest people of the past left to the city, confirming its innate cultural qualities. The beauty of Marie Louise. The genius of Toscanini. The harmony of Correggio. The talent of Verdi.\nThe city features fascinating places and corners that dig out fragments of history and culture. A city where past and present come together on the same road. It is really worth toget lost along its narrow streets to discover that secret side of the city made of legends and traditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On our website we have included buttons for Instagram to promote webpages (e.g. \"like\", \"pin\") or share (e.g. \"tweet\") on social networks like Instagram. These buttons work using pieces of code coming from Instagram themselves. This code places cookies. These social media buttons also can store and process certain information, so a personalized advertisement can be shown to you. Please read the privacy statement of these social networks (which can change regularly) to read what they do with your (personal) data which they process using these cookies. The data that is retrieved is anonymized as much as possible. Instagram are located in the United States.\nCamping en Bungalowpark \"in de Rimboe\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LOC-29041 A medium strength threadlocker for pre-assembled bolts up to 1\/2\"\nLOCTITE\u00ae 620 is designed for the bonding of cylindrical fitting parts. Slip Fit, High Temperature, Large Gap. Suitable for applications where temperature resistance up to 200\u00b0C is required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bobby Allen - A software engineer from Ipswich, England.\nI'm Bobby Allen, a software engineer and 'full stack' web developer from Ipswich, England.\nWith over fifteen years' experience, I mainly specialise in PHP and .NET\/C# but I'm also fond of and have experience using Python too! My primary interest is in back-end development, automation and APIs.\nI develop, lead and work on many widely used open source projects in my spare time, many of which I have been involved with for more than ten years now, these include FsHub, LRM, Sentora and Xcaler.\nSentora - An open source web hosting control panel for *NIX (was forked\/born following the commercial acquisition of ZPanel).\nZPanel - An open source web hosting control panel for *NIX, BSD and Microsoft Windows written in PHP.\nXcaler (WIP) - An application deployment and hosting platform for modern PHP and ASP.NET (core).\nLRM (Landing Rate Monitor) - A Microsoft Windows desktop application (written in C#) designed to capture (FSX, Prepar3d and X-Plane 11) aircraft vertical landing speed and other statistics.\nPirrot - A parrot (Simplex) radio repeater controller for the RaspberryPi.\nCartographer - A library for programmatically constructing GeoJSON objects.\nDistical - A library for calculating the distance between two or more latitude\/longitude coordinates.\nGPIO -A (G)eneral (P)urpose (I)nput (O)utput library for the RaspberryPi, Use PHP to work with inputs and outputs such as buttons, switches, LED's, motors and relays.\nGravel - A Gravatar.com library enabling developers easy access, presentation and manipulation of user avatars in their applications.\nLinguist - A library designed to extract and manipulate prefixed words in content ideal for working with @mentions, #topics and even custom action tags!\nExecutioner - A PHP library for executing system processes and applications with the ability to pass extra arguments and read CLI output results.\nMetar - A library to query airport METAR (Meteorological Terminal Aviation Routine Weather Report) information.\nPlexity - A password complexity checker and enforcement library for PHP.\nSluginator - A URL slug creation and sanitisation library.\n...and a handful of other libraries, API clients and other bits 'n' bobs can be found over on my GitHub profile!\nFsHub - A cloud service that provides virtual pilot and airline flight tracking and performance monitoring for Microsoft FS2004, FSX, FSX-SE, Prepar3D and X-Plane 11, integrates with my Landing Rate Monitor client.\nMy Blog - My personal blog, I mainly post about technical *NIX, hosting and development topics here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavwid b/data_all_eng_slimpj/shuffled/split2/finalzzzavwid
new file mode 100644
index 0000000000000000000000000000000000000000..ceef7646cb7c720d15a59f56fb9dbe22ec8717f2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavwid
@@ -0,0 +1,5 @@
+{"text":"The Bachelor of Applied Science in Business and Organizational Management has specializations in Agribusiness Management, Health Care Management, Management Information Systems, Logistics and Supply Chain Management and Public Safety Administration.\nClick for Official College Catalog information.\nCF offers a Bachelor of Applied Science in Business and Organizational Management as well six specializations. Explore the options below!\nNot sure which BAS degree is right for you? Contact your academic advisor!\nJoin the CF BAS Network for an opportunity to meet with your peers and network. For information about the BAS Network club, please contact the club advisor, Dr. Cynthia Moody at moodyc@cf.edu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Electric vehicle charging stations are used to recharge electrical cars at home, at the office, and in the street.\nSince 1st January 2015, in France, it has been obligatory to install a socket to recharge electrical vehicles in a condominium (there was previously a similar situation with television antennas) if one is requested.\nWhat technical choice should be used to connect the device?\nA private line that comes directly from the driver's home?\nA circuit from the utilities distribution board?\nWill the connection choice still be valid if demand increases in the next 5 years?\nKnowledge of energy management will also come into play with the prioritizing of receivers. If we don't have enough available power to start with, we have to be sure that the lift pumps or elevators will not stopped and that the utilities circuit breaker won't be thrown if three people decide to recharge their vehicles at the same time!\nAs for the invoicing of energy, only an energy operator can sell electricity so we need to find an efficient way of distributing the costs between the charging stations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are very special to us. All young women who have participated in our program are alumnae of YWCP.\nThere are many ways you can stay involved. If you're home from college, go to school nearby, or work in the Bay Area, be sure to come to our concerts. Attend our December concerts, Carols by Candlelight in the East Bay or our Holiday Showcase in San Francisco and sing along with your favorite carols. We'd love to see and hear you!\nYou could also volunteer to help with concert activities such as ushering, selling tickets or even backstage. Please let Kaneez know if you'd like to help.\nOur retreat and tour also is a place to use your skills. Please let Susan know if you'd like to be considered as a chaperone or counselor for these trips.\nPlease stay in touch by letting us know your current contact information here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed by Italian fashion brand PAOLO MARIANI. Their products embody an unyielding pursuit for quality of materials. Together with their revolutionary craftsmanship and flare for orginal ideas, they have constantly introduced high quality fashion items into the world. A luxurious product for those who can only be satisfied with the real thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul understood the wretched state of the human heart, and its ability to not effectively work out anything but evil. However, when one is in Christ, there is the truth that Christ's power and purposes can be realized in us in a way which was previously not possible. This doesn't mean that saved believers are automatons which can rely on Christ doing everything for us, but that when we are willing to submit to Him, we are able to do what He expects of us. Our human will is not excluded from the process, and we can choose to work out evil even after coming to Christ. But if we desire to do as He wills, He will ensure that it is possible for us to do so.\nPaul then states what that confidence in his hearers is \u2013 \"both that you do and will do the things we command you.\" It is the same formula which is found throughout Paul's writings. God presents an offer; man receives the offer. It is a synergistic working of God's divine offer and assistance being coupled with man's human efforts.\nThis does not mean that man's works are credited for salvation. God has done all the work, but man must respond to the offer by exercising faith. Romans 3:27 excludes faith as being considered a work. After salvation, our proper walk in Christ follows this same pattern, but it then involves human effort beyond faith. We must \"work out [our] own salvation with fear and trembling\" (Philippines 2:12).\nAs far as the word \"command.\" It is a word which signifies a watchword or command that has full authorization because it has gone through the right and correct channels. Paul's commission is specifically noted in Acts 9. He (and those with him) had the full authority of Christ Jesus to make such commands as authorized by the Lord. He will begin to give some such commands in just two verses.\nLife application: As always, Paul notes that we must actively do something in order for God's will to be realized in our lives. The Lord makes it possible for us to be pleasing to God, but we still must do in accord with that which the Lord makes possible. Obedience to the Lord's will must be an active part of our lives, or we will not be pleasing to the Lord.\nHeavenly Father, in Christ, You have given us the ability to be pleasing to You, but we also must respond to that divine assistance. To be saved, we must demonstrate faith. To walk rightly, we must demonstrate obedience. Help us to do according to Your will in all things by granting us the wisdom to rightly discern what Your will is, and then to act upon it. To Your glory we pray. Amen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaypiv b/data_all_eng_slimpj/shuffled/split2/finalzzzaypiv
new file mode 100644
index 0000000000000000000000000000000000000000..40e583b8ca22d13946d480cd20e52ac0b9c53595
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaypiv
@@ -0,0 +1,5 @@
+{"text":"Aloha, I am looking for a real estate professional to help me and I found this listing (#618177) on your website. Please contact me at your earliest convenience. Thanks.\nA rare find! Five acre acai plantation on the Hanalei River.\nSensational views of Namolokama mountain and the Hanalei river.\n2-3 year old acai plantation which will be an acai forest in a couple years!\nThe property includes seven ponds which provide habitat for endangered birds, fish and lotus flowers.\nOwn one of the largest organic acai plantations in all of Hawaii!\nTwo bridges are located on the property. A perfect place to fish and take in the view.\nAccess Hanalei River through a private canal on a stand up paddle board, boat or jet ski. Be in Hanalei bay in under 2 minutes!\nTake a stroll over the bridge to your personal wildlife sanctuary. You are surrounded by acai, banana, cacao, coconut, guava, mango and a variety of palm trees.\nA 640 sq foot cottage already exists for your convenience. Live in the cottage while designing the home of your dreams!\nOne of a kind river and mountain views from your new home.\nStudio cottage includes a full kitchen with gorgeous granite counter tops, African mahogany cabinetry and bamboo flooring.\nMountain and river views from your kitchen.\nTake in the mountain views from your two person jacuzzi tub.\n450 lineal ft of river views.\nArchitectural rendering showing the home that can be built on the property. In this rendering the studio cottage has become a master suite.\nDesign your dream home in Hanalei!\nHistorically a rice plantation and has now become an organic acai farm- one of the largest acai plantations in the state of Hawaii. Your front yard is filled with unique trees including acai, banana, cacao, coconut, guava, mango, panama cherry and a variety of palm trees. There are seven ponds, which are home to endangered birds, fish and lotus. The property receives special tax breaks and has an agricultural dedication till 2026.\nThis land is truly one of a kind.\nProperty withstood the last historic flood very well, faring better than most homes in Hanalei town. No insurance claim filed and not a drop of water touched the floor. Two days later property looked exactly the same as before the flood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pre-show and interval drinks are available in the Greene Room bar which opens one hour prior to each performance.\nIn association with Greene King A contemporary, light and airy space, offering a friendly atmosphere. The well-stocked bar serves fine wines, Greene King Real Ales, continental lagers, spirits and soft drinks. We also serve the Theatre's own label champagne, Charles Mignon brut grande reserve from the Epernay region of France. Interval drinks can be pre-ordered for your convenience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Model three of Computer Engineering. This Years Syllabus is basically good and it'll assist us to advance our skills in IT trade. So without losing your time, start inproving your skills as a software developer. Computer systems turned a very important software in music production beginning within the 80s after they had been used to digitally master and management musical instruments through the Musical Instrument Digital Interface (MIDI). Additionally, you will have the chance to study abroad in your third year at a associate establishment, enabling you to achieve international perspective on your subject and boost your future career prospects.\nRegister for your faulty laptop computer \u2013 No power on (Lifeless), No display, No charging, Contact PAD no working, Restart issue, Power jack difficulty and many others. Workstation : a powerful, single-user laptop. A workstation is sort of a personal pc, but it surely has a extra powerful microprocessor and the next-quality monitor. Regarding the Church-Turing thesis, first it is not proved, and also it only talks about computing, it does not say anything about behaviours that can't be seen as computations. The desktops can additional be subdivided relying with the casing sort, tower casing and desktop casing.\nAre you able to upgrade to a brand new laptop? Walmart has loads of options from laptops to tablets to desktop models all at Every Day Low Costs. Here is how to buy different models. Your placement interval will usually be full-time and should complete a minimum of 20 weeks. What do other customers consider the corporate? Go to mass-market web sites, like Amazon, and see what customers say about assembled computer systems they buy. You'll not only see critiques from users, however how well the pc is promoting, what date it was launched, and so on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We'd love your help. Let us know what's wrong with this preview of The Mammoth Book of Gaslit Romance by Ekaterina Sedia.\nA fantastic collection of stories of love and intrigue that focus on the trappings of the popular Victorian era, enlivened with fantastical elements and incorporating some noir and detective pieces, by O. M. Grey, Leanna Renee Hieber, N. K. Jemisin, Eliza Knight, Sarah Prineas, Delia Sherman, Genevieve Valentine and many more.\nFull list of contributors: Vivian Caethe; Leann A fantastic collection of stories of love and intrigue that focus on the trappings of the popular Victorian era, enlivened with fantastical elements and incorporating some noir and detective pieces, by O. M. Grey, Leanna Renee Hieber, N. K. Jemisin, Eliza Knight, Sarah Prineas, Delia Sherman, Genevieve Valentine and many more.\nTo ask other readers questions about The Mammoth Book of Gaslit Romance, please sign up.\nRead Tansy's piece in this one and that's it so far - loved it!\nUnfortunately, this was massively disappointing to me--most of the stories were pedestrian and unengaging and the only 3 that were at all good I had already read in other collections.\nEkaterina Sedia is also credited as E. Sedia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watercolor wash. Pilot Precise V5 Linework.\nWaterwash. Pilot Precision V5 Linework.\nGiant Viruses, directly inspired by Radiolab's Shrink, and depicted in India ink with nib pen and brush.\nFor a friend with gluten intolerance. A small slice of the life that sustains our gut. India ink with nib pen and brush. Tombow dual brush pen markers.\nIn a drop of (concentrated) sea water... microscopic life. India ink, water color and colored pencil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayplh b/data_all_eng_slimpj/shuffled/split2/finalzzzayplh
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+{"text":"When exams were announced in 1849 for the post of the Madrid Conservatory's Professor of Clarinet, Carnicer was called upon to compose the contest piece for the exam. The result was this Fantas\u00eda para clarinete obligado con acompa\u00f1amiento de pianoforte, which follows the typical pattern of a slow rhapsodic introduction followed by a ternary fast movement. Carnicer demonstrated his stylistic leanings in the way that he varles the thematic material whilst the purpose of the work determines not only the level of virtuosity required from the solo part but also the simplicity of the piano accompaniment.\nAve Maria - Sacramental romanticism in the nineteenth century Menorca.\nOverture for \"Il barbiere di Siviglia\"\nSymphony N\u00ba. 1 in G major \"Oriental\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jurors in the Bell corruption trial adjourned Tuesday, again without a verdict in the trial of six former council members accused of raiding the small town's treasury by pulling down extraordinary salaries.\nWednesday will mark the 18th day of deliberations -- nearly the same length of the testimony in the trial.\nSince being handed the case Feb. 22, the panel appears to have struggled to determine whether Luis Artiga, Victor Bello, George Cole, Oscar Hernandez, Teresa Jacobo and George Mirabal misappropriated public funds by taking salaries of up to $100,000, beefed up by serving on city boards and authorities that rarely met.\nHernandez, Jacobo and Mirabal face 20 counts while Bello faces 16. Artiga faces 12 counts; Cole has eight. All face prison time if convicted.\nLegal experts say the general rule of thumb when it comes to deliberations tends to be one day for every week of testimony. Anything longer could be a strong sign of a hung jury or a deadlock on some counts.\nThe four-week trial began Jan. 24.\nPhoto: Los Angeles County Superior Court Judge Kathleen Kennedy during Bell trial. Credit: Associated Press.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There has been a marked increase in the use of commercial-off-the-shelf games for learning and, in some cases, a steady move towards the adaptation of these games or their game engines, for more specific uses in the field of education. One such example is the development of SimCityEDU: Pollution Challenge! (http:\/\/www.simcityedu.org\/) by GlassLab, based on, or more accurately modified from, the popular SimCity game series. In 2013, GlassLab developed this educational game to teach students about environmental issues and pollution in cities, and to provide an integrated assessment methodology for the students. GlassLab aims to develop further games in this educational series in the future.\nSimCityEDU: Pollution Challenge! addresses these and other issues by adding structure and constraints to the environment to enhance learning and by chunking information and tasks into 10 minute sessions to make them more suitable for the classroom. The game offers teachers a set of tools to organise classes and their work, track student progress, archive work and further expand the game. Lesson plans and cheat sheets aid the teacher in integrating the game in the classroom as quickly and as effectively as possible. The game encourages students to plan ahead but also take risks, each mission requires an objective to be met but also includes further objectives for students to complete as a bonus which also encourages repeated playing. Students become invested in the town they're creating, thus encouraging them to analyse the decisions they have made and plan for a more successful town in the future. SimCityEDU: Pollution Challenge! boasts a facility to monitor student progress by gathering information about their decisions. Mouse tracking assesses how the students come to their decision based on the elements of the interface that they interact with allowing teachers to see how they are developing with their critical-thinking and decision making skills. While the game developers are still carrying out beta testing on the game with schools, they have integrated a number of changes already in response to teacher requests, such as allowing students to carry out missions in a non-linear fashion.\nGames are best suited as a support rather than a replacement for traditional teaching methods and may help to engage or motivate students who have been found to have difficulties learning through books. The development of this game draws on the dynamic and interdependent character of resource management games to tackle urban issues in an educational field. The game allows for risk free, experimental learning to take place as the user takes control of their virtual world under the guidance of their teacher.\nWe planned to use SimCityEdu as the basis for a game to develop engineering skills. But we are not sure if we can have access (programming) to the simulation engine to adapt it to our needs. Do you know if this is possible?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our dainty Dragon Fly has an inlayed Lapis Lazuli body, nacar wings and eyes. Solid silver back. This pendant measures 1.5 inches long 1 inch wide. Available in all our stones (please add $3.00 for Rhodocrosite).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Earlier this month Lars, our CEO, and Ruth, our Connect for Success Business Development Manager, attended Cara's Tribute to the Stars. Huge thanks to Cara and Stand Together for making this happen!\nCara works with individuals affected by poverty to get and keep quality jobs through rigorous training and coaching. Learn more about Cara here.\nStand Together identifies and supports community-based organizations that help people in poverty transform their lives. Learn more about Stand Together here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Thursday, 25th January 2018 saw the main hall of Gimnazija Vi\u010d completely full with an eager audience ready to see the performance of \"Musicals @ Vi\u010d\".\nLed by students Ana Kocjan\u010di\u010d and \u017diga \u0160trukelj, singers, musicians, and a backstage team put together a collection of performances based on the most famous and special musicals of Broadway and the West End.\nStudents were the force behind every aspect of this show including the stage design, promotions, choreography, costumes, make-up, and the singing performances themselves.\nWith the show being received by such enthusiastic applause, people are already looking forward to seeing what this creative and talented group of students will come up with next.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Living in Japan, I can find most Lego sets that are available for sale in the world. Not all though, for example the Scooby-Doo series wasn't sold in Japan (no one knows Scooby-Doo here, what a shame!) I had to order the ones I got from the UK on sale, they were shipped to my family in France who sent them to me. Yes, I'd go to these lengths, because let's admit it, those sets are some of the best in recent years. The good news is that prices have gone down recently too. They're still much more expensive than in the US, but they're pretty much on par with European prices (why are Lego so cheap in the US? I'm jealous).\nHowever, I also buy a lot of my minifigures online, most of them through Bricklink. However, shipping to Japan can be pricey, and that greatly limits my choice of sellers. Basically I buy almost exclusively from sellers in the Netherlands where the shipping costs are the cheapest in the world. And while there is a certain number of sellers there, it still means that my choices are somewhat limited.\nSo, when I went home to France last summer, I also did some online Lego shopping for minifigures, as suddenly my options were bigger. I ended buying just a few, but I'd have never been able to get some of those for cheap in Japan.\nFrom left to right: Classic Bruce Wayne (got him mostly for the suit), Admiral Raddus, E.T., Doc Brown (note that Dimensions is not available in Japan), Obi-Wan Kenobi (I can't believe I didn't already have him), Mister T, Red Skull, adult Ahsoka Tano, and finally Jeff Goldblum as Grandmaster (I just wanted a Jeff Goldblum minifigure, who doesn't?).\nPretty glad with that purchase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"offers 1,138 asphalt emulsion products. About 12% of these are bitumen, 12% are other construction machinery, and 3% are tanker truck. A wide variety of asphalt emulsion options are available to you, such as petroleum asphalt, modified bitumen, and natural bitumen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Viverk produces and markets standard industrial washing machines and customer adjusted special machines. Our customers are mainly found within the manufacturing industry.\nIn Vislanda our production unit is situated and this is also our head office. Viverk was founded in 1947 and started with production of industrial washing machines in 1965. Approximately 30 persons are working in Vislanda producing both standard washing machines and special adjusted plants. We have our own laboratory for cleannes analysis and process testing. All our products are well adjusted for high production requirements and have a high quality in both design and material.\nIn \u00c5kersberga, north of Stockholm, we have a sales office with 4 employees. A great deal of the company's product development is performed in \u00c5kersberga. The company always strives to be in the front row when it comes to developing new and better solutions.\nViverk has the highest credit worthiness, AAA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, this is revealed by a document released by global terrorist group Al-Qaeda. They said it clearly that they will target Indian security officials and leaders of Hindu \"separatist\" organizations. Reportedly, Al-Qaida released an elaborate document titled as \"Code of Conduct for Mujahideen in the subcontinent\".\nAfter the release of this document, Indian intelligence sources said they are alert and are ready to fight back and abolish every militant activity that will be organized by Al-Qaeda to destroy peace in the valley.\nRecently, a DSP was brutally lynched by a mob outside Jamia Mosque in Kashmir. Earlier an Army officer was also kidnapped from his home and later killed. The document also mentions that hostages should either be traded for a brother, given up for ransom or be killed, besides elaborating on targets in Pakistan, Arakan (Myanmar) and Bangladesh.\nAccording to the sources, the most worrying part was that the Al-Qaeda had invited different groups fighting in the subcontinent to pledge their allegiance to what they refer to as the \"Islamic Emirate of Afghanistan\", and stand up against intelligence agencies and the groups they sponsor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbaghh b/data_all_eng_slimpj/shuffled/split2/finalzzzbaghh
new file mode 100644
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+{"text":"smart:EQ live is a new adaptive equalizer plug-in from sonible that's designed for live sound mixing. It employs intelligent algorithms to continuously listen to live audio signals and optimize them in real-time. The idea is to free the mixing engineer's hands to be able to focus on improving the sound for the audience.\nAccording to sonible, smart:EQ live is good for all types of live shows, including concerts and corporate events \u2013 each with its own unique challenges like sub-optimal mic positioning and speakers changing quickly. The plug-in's smart engine can be personalized with settings like threshold and limit line. By adjusting the reactivity of the equalizer's self-learning bands and setting their bandwidth and sensitivity, you practically give smart:EQ the instructions how to handle the rest of the performance on its own.\nFurthermore, smart:EQ serves as a safety net with its filtering algorithms detecting and eliminating feedback long before it appears. The user interface includes an integrated analyzer with a quick overview of your signal's frequencies. It also offers hotkey control so the mixer can easily switch between profiles and snapshots.\nOverall, smart:EQ appears rather fascinating with its use of modern design and technology in aid of live mixing professionals. It will be interesting to see how the plug-in fares with them once it reaches mixing decks around the world.\nsmart:EQ live is available for 32 & 64-bit Windows and Mac computers in AU, VST, and AAX formats. It's currently sold on an introductory price of 159 EUR \/ 141 GBP \/ 188 USD, down from 229 EUR \/ 204 GBP \/ 271 USD. A 30-day trial version is available from sonible's website. Furthermore, a 40% discount is being offered to students and lecturers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For more information about Open Price please read this article and for Current Price this article.\n1. John Smith has 100 ETH in his Wallet. He transfers 70 ETH to his Portfolio. The Open Price ($) and Current Price ($) are equal at the moment of transfer ($1100), therefore P\/L ($ and %) are both set to 0.\n2. John buys 1562.5 ET\u0421 for 50 ETH. The current ETC\/ETH price is 0.032 and the current price of ETH\/USD is $1120 - making the Open Price ($) for ETC $35.84. The P\/L ($ and %) values for ETC will initially be set to 0 - until the market changes, however, both of the P\/L values for ETH will change to align with the Portfolio's new asset allocation (from 50 to 20 ETH), along with ETH's price change (from $1100 to $1120).\n3. Some time passes, John hasn't been trading but the prices of his assets have changed.\nLet's suppose that ETC\/USD went down while ETH\/USD went up.Current Price ($) for ETH is now $1200 and both P\/L values will change accordingly.\n4. John decides to add more Funds to his account and transfers an additional 30 ETH from his Wallet.\n20 is the initial amount of ETH before the transfer. It had an opening price of $1100.\n30 is the amount of ETH that was transferred to the Portfolio. It had an opening price of $1200.\n50 is the total amount of ETH in the Portfolio now.\n$1160 is the open price of this asset after the transfer.\nBut this does affects P\/L (%) since it relies on Open Price which has since been changed due to the transfer.\n5. John spends 25 ETH to buy 100 LTC.\n6. John transfers 20 ETH from Portfolio to Wallet before any price changes occur.\nUnlike transfers to Portfolio, transferring funds out of it doesn't affect P\/L (%) since the prices remain the same, this does however affect P\/L ($).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Through the Wing, I, Cadet Flight Sergeant A. Killeen, Cadet Flight Sergeant L. Pritchard, Cadet Corporal E. Alexander-McGowan, Cadet Corporal A. Cooper, Cadet T. Evans, Cadet M. Jones and cadets from 63 (Bridgnorth) Squadron and 124 (Hereford) Squadron have been working towards The Sports Leaders UK Level 2 Award in Assisting in Basic Expedition Leadership.\nThe course has allowed us to develop our navigation, teamwork and leadership skills and has taught us the ability to assist in leading a group whilst out on an expedition or a day walk in lowland countryside. It is a 57-hour course (including 12 hours demonstration of leadership) in which we are having lessons and practising skills such as camp craft, navigation and leadership.\nWe have gone on various day walks around both Hereford and Cosford over the last 3 months enabling us to improve navigation, lead peers and deal with random scenarios along the way. After multiple lessons and day walks on the evening of Friday 31st March, we left the squadron in order to complete a pre-assessment in the Black Mountains in which we spent 2 nights and 2 days putting our knowledge to practice. Whilst it was incredibly muddy and cold it was particularly useful in preparation for the final assessment which should be taking place in June.\nOver the next month, we're all completing 12 hours of individual leadership along with an assessment booklet in advance of the assessment. I would thoroughly recommend signing up for the course next year to those who have an interest in orienteering, navigation and leadership as it's been a great experience meeting new people, learning so much and developing skills along the way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're more of a strummer than a shredder, you're in the right place. Read about your favourite acoustic brands, building techniques and the latest acoustic guitar trends!\nThe Fender Acoustasonic Tele is an innovative acoustic\/electric hybrid guitar. In this article, we compare it with the Taylor T5Z - its main competition.\nHow Do I Amplify My Acoustic Guitar?\nWant to amplify your electro-acoustic guitar? In this article, we explain the pros & cons of using acoustic guitar amps, PA systems & preamp pedals!\nBuying guitar-related gifts can be extremely tricky when you're not a guitarist yourself. So cut the family some slack! They've never had a good guide to follow\u2026 until now.\nAt the same time, if you've ever bought someone a guitar keyring, you definitely need to up your present game.\nLacking creativity in your life? Or do your kids need a new outlet for their bundles of energy? Sounds like you and\/or your kids are in need of a musical instrument! But which one?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home owners with cedar roofs that get estimates from roofing contractors reminds me of how Thomas Jefferson said: \"Democracy is two wolves and a sheep deciding what is for lunch\".\nA survey of the most popular opinions among contractors as to what is best for your cedar roof does NOT guarantee getting the best advice about cedar roof care. When comparing cedar roof care contractors it is not easy to tell who is promoting the best standards and practices for your cedar roof from those who are not.\nTwo of the three statements below are true. One of them is a commonly used but misleading statement to sell you a \"cedar roof treatment\" without any UV protection.\nA. A new cedar roof will \"weather,\" gradually changing color and ultimately turning a beautiful silvery grey.\nB. The side of a cedar roof that gets the most sun light will be in a more deteriorated state than areas that get less sun light.\nC. Sun Light damages Wood. Cedar with no sun block\/ UV protection will turn grey or silver.\nIn case your a bad test taker. B and C are true A.=BS. Grey is the natural color of a Cedar Roof like wrinkles are the natural state of skin. For home owners who like the look of weathered wood, but want their cedar roof to last there are some excellent Silver or grey tinted wood preservatives for cedar roofs. AMTECO makes wood preservative in \"Slate Grey\" because some people like silver of grey cedar roofs. These true wood preservatives go on grey and protect cedar roofs from Sun damage rather than allow Cedar roof to turn grey over time because of not preventing sun damage. I hope this helps to explain the difference.\nThe truth is silver or grey is the natural color of Cedar that has been sun damaged. Those in the cedar roof cleaning business using products that do not protect cedar roofs from sun damage are bending the definition of \"the natural color of cedar\".\nCedar roofs preserved with \"Total Wood Preservative\" every 5 years will will last three times as long as cedar roofs that receive no wood preservative. Neglected Cedar roofs will turn grey from sun damage or turn dark from algae infestation. The natural direction of a cedar roof not preserved and protected from weather is towards rot. A Ceadar roof is very expensive to replace financially wise home owners will preserve their wood roof.\nLogical reasoning: A cedar roof is made of wood so find the best wood preservative and apply it to your cedar roof. Get your cedar roof really clean before applying the best wood preservative. A wood preservative that does not allow moss, algae and lichen to grow will eliminate the need for paying for future cleaning. More wood preservative can be applied every 4-5 years for a small fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..0ba717a7652d70b8340b138cefae55714e803237
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+{"text":"Google+ Brand Pages: Should You Create One?\nGoogle+ has made numerous headlines since it was first launched in June 2011. It's been in the news for being a potential \"Facebook killer\", for its quick growth, for integration of the +1 button, and for its interesting strategy of holding off on allowing businesses and brands to use the medium at first. Now, all eyes are once again on Google+ because they have finally opened up the social network to brand pages.\nBy Nick Stamoulis, 09 November 2011, 1 comment. Categories: Advertising Technology, Search Marketing, Social Media Marketing.\nThere's a reason that YouTube is the second most popular search engine after Google. People like to watch videos, whether it be for fun or educational purposes. There seems to be a misconception that marketing videos are more of a B2C thing. Not true. After all, businesses are made up of people (consumers) and they all have different tastes in how they prefer to receive information.\nBy Nick Stamoulis, 08 November 2011, 0 comments. Categories: Organic Search.\nThe search engines are constantly updating and tweaking their algorithms to provide their customers (searchers) with the best possible experience. Think about it. If Google search results were crappy, people wouldn't use the service, meaning that Google wouldn't be making a profit. Search engine algorithms are built to think like people that are searching.\nBy Nick Stamoulis, 01 November 2011, 0 comments. Categories: Organic Search, Social Media Marketing.\nIn celebration of Halloween, here are a few spooky SEO tactics to avoid because they will certainly come back to haunt you!\nLink Exchanges \"Hey, I'll link to you if you link to me.\" While it may seem like an amazingly simple way to build your links, it won't establish any trust with the search engines. What the search engines pay attention to when it comes to links is relevance.\nBy Nick Stamoulis, 01 November 2011, 0 comments. Categories: Content Marketing, Organic Search.\nThere are millions of blogs out there competing for readers' attention. Maybe they aren't all direct competition, but it's doubtful that you are the only blogger in your niche. People only have so much time to spend reading blog posts so what can you do to make sure that yours is one of them? The best way to improve click-throughs is to craft a blog post title that will catch their attention.\nBy Nick Stamoulis, 28 October 2011, 0 comments. Categories: Content Marketing, Organic Search.\nBy Nick Stamoulis, 25 October 2011, 0 comments. Categories: Content Marketing, Organic Search.\nBy Nick Stamoulis, 24 October 2011, 0 comments. Categories: Content Marketing, Organic Search.\nBy Nick Stamoulis, 23 October 2011, 0 comments. Categories: Organic Search.\nBy Nick Stamoulis, 19 October 2011, 0 comments. Categories: Content Marketing.\nArguable the single most important thing you can do for your site, keyword research and selection form the cornerstone of the rest of your SEO campaign. The keywords you choose will affect you can find your site and with what search phrases. Your keywords will also help shape the content you write, as you need to incorporate your target keywords into all the content you produce.\nBy Nick Stamoulis, 17 October 2011, 0 comments. Categories: Organic Search.\nBy Nick Stamoulis, 14 October 2011, 0 comments. Categories: Organic Search.\nAny good SEO company or consultant is going to tell you that keyword research may be one of the most important things you do for your SEO. The keywords you target will directly affect what searches you rank for and what kind of traffic will be directed to your site. Failing to include the right keywords means missing out on potential traffic.\nHere is the process that I use for conducting keyword research for my clients. I hope you find it useful for your own.\nFor many companies across all industries, these last few months of the year get very busy very quickly. Strategies are being created, budgets are getting approved, performances are graded and ROI is evaluated. Everyone is getting ready for the big push into the first quarter of next year. Since SEO is so long term, it makes sense to start thinking about your 2012 campaign now and make sure everything is in order for the new year.\nBy Nick Stamoulis, 13 October 2011, 0 comments. Categories: Organic Search.\nRetargeting: Effective Marketing or Creepy?\nBy Nick Stamoulis, 12 October 2011, 0 comments. Categories: Advertising, Advertising Technology.\nBy Nick Stamoulis, 11 October 2011, 0 comments. Categories: Search Marketing.\nBy Nick Stamoulis, 07 October 2011, 0 comments. Categories: Organic Search.\nOnce you decide to invest the time and energy in a content marketing campaign, you have to determine what kind of content you will focus on creating. It could be educational, informational, promotional or newsworthy or a combination of all four. Just make sure that the content you are creating is inline with your online business goals and is the kind of content your target audience is looking for.\nBy Nick Stamoulis, 04 October 2011, 0 comments. Categories: Advertising Technology, Content Marketing, Organic Search.\nPress release distribution should be a part of the ongoing relevant link building portion of an SEO campaign for any business. Any company information or event that is newsworthy should be covered in a press release and distributed online using services such as PRWeb, PRLeap, 24\u00d77 Press Release, and PRLog.\nBy Nick Stamoulis, 29 September 2011, 0 comments. Categories: Organic Search.\nBy Nick Stamoulis, 27 September 2011, 0 comments. Categories: Organic Search.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Atlanta Yacht Club Junior Sailing Program is a fantastic way to learn to sail or improve sailing skills for sailors from 6 to 18 years old. Our instructors make the classes fun and informative.\nRegistration for AYC members is now open. See registration links in course descriptions below.\nThe objective of the beginner class is to familiarize students with the basics of sailing and to allow them to feel comfortable on the water. The students will learn safety skills such as righting a capsized boat, as well as basic rigging, de-rigging and tacking skills. Students will learn the parts of the boat, how to rig and launch a boat, basic sailing, and sailing terms. There will be a combination of shore sessions, fun, and on the water sailing. This is a non-competitive environment. We just want to get the kids hooked on sailing!\nThe students will learn how to safely leave and return to the dock and will be able to sail independently. The basics of tacking, jibing, and docking will be covered. This class is for kids who have the basics covered and are ready to expand their skills. Sailors will work on sailing more efficiently including tacking, jibing, upwind and downwind sailing. Sailors must feel comfortable sailing independently. This class is a continuation of Beginner Sailing and a stepping stone to Beginner Racing based on the students interest.\nAfter students have completed Level 2 Classes or have participated in Learning to Race or Learning to Sail in the past. Students that have enjoyed learning the sport of sailing and would like to compete and have fun learning to maximize wind efficiency while racing their friends. In the course, the skills they will learn are (but not limited to): Starting Sequence, Basic Course Configuration, Rules of Racing, Mark Rounding, Maximizing Upwind Sailing Angles, Increasing Boat Speed, Boat Handling, Positioning for Speed, Roll Tacking and Jibing, Introduction to Tactics, Having a Race Strategy and More.\nThis class is for older beginners who are new to sailing and sailors new to the 420 and other double-handed boats. Sailors must be at least 9 years old and must have taken a sailing class in the past.. The 420 provides older students the chance to learn in a double-handed boat. It will also teach previous Optimist sailors skills to sail in a double handed boat. Students will learn how to both crew and skipper. The focus of this class will be sailing as a fun, social sport. This class will also have the opportunity to sail in AYC Club Boats such as Yflyer, Snipe, and Thistle!\nCoaching: Club Volunteers. 1 adult, 1 junior, 1 safety. Use fleet boats with fleet volunteers to take kids out and skipper\/crew on fleet boats.\nWe will have an orientation meeting the first day of class. We will let the parents know that our classes are NOT drop off classes. Parents are required to stay close to the sailing area to help out where needed or just to be there for their child. We will have information about snack parents and volunteer opportunities at the orientation meeting. Throughout the summer we will have some sailing opportunities for parents during classes too.\nThe first class will begin with a swim test. All kids will be required to swim the length of the dock and back with their life jackets, tread water for 5 minutes without life jackets and then put their life jacket on while in the water. Kids must have a properly fitting Coast Guard Approved life jacket.\nBoats have to be rigged by the 9am start of class, except for the Guppies class. Rigging will be covered week 1. Boats rigged by 9am weeks 2-8. Parental assistance will be needed.\nWe know that most families have other summer plans and that kids will miss class. This is okay. Our coaches will do their best to help get everyone up to speed after an absence.\nIn case of bad weather, we will still hold classes. We will do indoor activities on these days.\nWhen you arrive at the Yacht Club entrance, pull up to the gate. It should be on Automatic and will open when you pull up to it. Classes will be help on the Opti dock in front of the clubhouse, waterside. Or in the lower level of the clubhouse Jr. Room.\nPrices: $125 Yacht Club Members; $225 non-members, If a student wants to sign up for the Learn to Sail a double-handled boat class along with another class the class price would be $75.00 for the 2nd class. T-shirts will be available for purchase for $20 and Polos available for purchase for $25. These shirts have the logo with Jr. Sailing Racing Team below the burgee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You might be ruining your home sale without even realizing it. Here are seven common mistakes that you need to avoid.\n1. Unsightly cords. A lot of people have mounted televisions with hanging cords. Buyers are unable to really envision how that space might look if there are cords strewn about, so I suggest taping them up behind the TV or getting rid of them altogether for showings.\n2. An unkempt yard. It is important that your home has great curb appeal. Keep up with the mowing, weeding, and get a gardener if you have to.\n3. A worn entryway. If buyers see dingy patio furniture or a porch with wood rot, they have a negative impression of what the home will look like on the inside before they even step foot inside it.\n4. Cobwebs. Check all of your corners, light fixtures, and little-used rooms for cobwebs and small bugs. It is an easy, quick fix that can really pay off.\n5. Poor furniture arrangement. The best way to live in your home is not necessarily the best way to show it.\n6. Overfilled drawers or closets. Clean those out and maybe get a storage unit. Clutter will not do you any favors when it comes time to sell.\n7. Cluttered countertops. If buyers are distracted by your clutter, it is hard for them to see past it. The more your home looks like a model home, the better chance you have of selling it quickly and for top dollar.\nIf you have any questions for us about these tips or about the real estate market in general, please feel free to give me a call or send me an email. I look forward to hearing from you soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More 1975 alumni from St Mary HS Paducah, KY have posted profiles on Classmates.com\u00ae. Click here to register for free at Classmates.com\u00ae and view other 1975 alumni.\nThe St Mary High Paducah, KY class of '75 alumni are listed below. These are former students from St Mary in Paducah, KY who graduated in 1975. Alumni listings below have either been searched for or they registered as members of this directory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bradley who has been ruled out for game 2 by C's head coach Brad Stevens is concerning, but the potential absence of the best 3 point shooter Kelly Olynyk is more of an immediate concern than Bradley. As the Celtics currently do not have a player that can stretch the floor like Olynyk can, Kelly who is a power forward can go in low and battle with the big's or he can drive to the outside where he hits a team high shooting percentage from beyond the arc.\nThe 7-footer re-aggravated the same (shooting) shoulder he injured on Feb. 10 against the Clippers. He missed 12 games after the All-Star break. Olynyk missed his second straight practice Monday and is questionable for Tuesday's game 2 against the Atlanta Hawks.\nIf Olynyk is unable to go the Celtics would be without their two best 3 point perimeter shooters in Bradley and Olynyk.\nOlynyk told reporters Monday at Philips Arena he doesn't recall exactly when he injured the shoulder but believes it happened sometime during the Game 1 loss against the Hawks.\nThe injury concerns for the C's doesn't end with Olynyk and Bradley though as Jae Crowder is also not at 100%. While Crowder's ability to start is not in question, his effectiveness is going to be questioned. Jae Crowder admitted to reporters Monday that his ankle is not 100 percent, as he still recovers from the high ankle sprain of the right foot suffered on March 11 against the Rockets. Crowder said on Friday before the team left for Atlanta that the ankle will not require surgery in the offseason. On Monday, he said while that's still the case, the ankle is not where it was before the injury.\nIf the Celtics hope to win this first round match-up with the Hawks they will need both Crowder and Olynyk to play their best basketball with Bradley out. Hopefully the task isn't too daunting and the team can pull out wins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"VLC media player 3.0.6 can be downloaded from our software library for free. The common filenames for the program's installer are vlc.exe, FLVPlayer.exe, VLC OPEN GATE.exe, vlc player.exe or VLC Portable.exe etc. This free program is a product of VideoLAN.\nThe most recent installation package that can be downloaded is 39.9 MB in size. Our antivirus check shows that this download is malware free. The program lies within Multimedia Tools, more precisely Players.\nThis free PC program is compatible with Windows 2000\/XP\/Vista\/7\/8\/10 environment, 32 and 64-bit versions. This free PC software can manage the following extensions: \".mkv\", \".vob\" or \".dv\". The most popular versions among the software users are 2.1, 2.0 and 1.1. Some of the program aliases include \"VideoLAN VLC media player\", \"VLC\", \"VLC media\".\nYou may want to check out more software, such as VLC-Lite media player, iReal Bluray Media Player or iReal Blu-ray Media Player, which might be similar to VLC media player.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When people decide to continue their education in criminal justice by entering into a criminal justice master's program they are certain of a few things. They love what they are doing, they want to have the available opportunities for advancement, and they want to keep learning more about criminal justice on a deeper level. There are many schools out there that offer master's programs that you can attend in a variety of ways. What you need to know about criminal justice master's programs are that they can be a successful choice for you whether you choose to get your master's degree online or in the classroom.\nThe class curriculums for students in a master's program for criminal justice tend to be very intensive, detailed, and focused. Most of the students have had some field experience that are in the programs and that helps to really dig deep into criminal justice. You will learn more complexities about human nature, have to write papers or theses on certain topics, and will really focus on a specific area of criminal justice that you would like to specialize in.\nAt this time the FBI will consider hiring agents who have at least two years of field experience and a master's degree. The FBI program is very intensive and highly regarded by many people in the criminal justice system.\nPeople who pursue and obtain a master's degree in criminal justice are more likely to receive promotions, higher pay raises, and other opportunities that are financially beneficial. When you get a master's degree you quickly become regarded as a true professional and considered to be among the elite in the criminal justice field.\nThere is a wide selection of schools to choose from when you are trying to get your master's degree for criminal justice. The decisions you make on which school to attend should be based off of the area of specialty you'd like to focus on and your decision of whether to attend traditional school or take an online course. Both options provide great alternatives. With the busy schedules of many people in the criminal justice system they find that earning their master's degree at home is a much more feasible option for them.\nThere is literally a world of opportunity out there for people who have their master's degrees in criminal justice. You are not limited to just one area and are given the opportunity to use your skills anywhere in the world. That is a very exciting and wonderful opportunity to have. Why eliminate any option because you chose not to go for your master's degree? If you do not pursue an option let it be because it wasn't the right choice for you. It makes all the difference when you hold the power to make your own choice with your career. A master's degree in criminal justice can help you hold that power and create exciting opportunities for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HARTFORD \u2014 State Representative Irene Haines (R-34) has announced that she will be hosting a legislative update to provide residents the opportunity to hear the latest updates from the State Capitol and ask questions. The major topics discussed will include school regionalization, taxes and tolls.\nThe event is free and open to the public. Rep. Haines encourages residents to attend this meeting, as it is an opportunity to discuss major issues that are shaping the 2019 Legislative Session.\nFor more information, please contact Rep. Haines at 800-842-1423 or email Irene.Haines@housegop.ct.gov.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Handheld Addict: my most embarrassing games?\nI read an IGN editorial about the controversial Modern Warfare 2 content... it got me thinking about games that try to push the boundaries in good and bad ways.... but then I started drifting off on a tangent and wondered, \"what are the most embarrassing games for me to be caught playing?\"\nI wouldn't exactly call myself a proud gamer in general-- it's not something I like to flaunt. But I have some games in my library that I really don't like to be seen playing. Some of them are for obvious reasons, but many might not be.\nAlmost every Wii game - Let's face it-- watching someone play Wii is like watching someone performing for a stage hypnotist: they think they look cool, but they so DO NOT. I get pretty self-conscious when people see me flail around like a 12-year-old girl swatting at butterflies-- even though I'm having fun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A shipyard's competitive edge is directly related to how well it coordinates its manufacturing on a global scale, as well as production quality, quantity and on-time delivery.\nShipyard Production Scheduling and Optimization creates a unified master production schedule and capacity planning for all ongoing projects at the shipyard in the context of its overall workload. It sequences and seamlessly distributes all manufacturing operations across multiple sites, their duration and the resources assigned to each operation in the most efficient way. Shipyards can perform \"what if\" simulations to determine the impact on the overall master schedule before choosing the best plan. For reporting and continuous improvement purposes, they can monitor when an operation has been executed, who performed the operation, if it was executed on time and if the job was well done. Production errors are also detected and the overall schedule automatically reorganized to keep projects on track.\nFind out more about the associated benefits of Shipyard Production Scheduling and Optimization, part of the Build For Sea industry solution experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdtwz b/data_all_eng_slimpj/shuffled/split2/finalzzzbdtwz
new file mode 100644
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+{"text":"The humble watch used to be a tool to tell the time, but over the years, it has ended up as a fashion statement, one that could even cost more than an entire house if it is studded with jewels and other kinds of precious gems. Well, the watch was said to die a slow death with the introduction of smartwatches, but such a prediction has proven to be otherwise. The luxury line of watches continue to sell well regardless of the economic climate among the well-heeled and famous, while the regular watch maintains its market share as smartwatches struggle to break into the mainstream consciousness and being a \"must have\" item on everyone's wrists. Casio forges on with its very first connected watches for men, the G-Shock G-Steel line.\nWhen it comes to the modern day gadget owner, one thing is for sure: there are plenty of accessories to go along with your various devices, supposedly making them more impervious to damage or look cooler. Modern day gadgets also have one Achilles Heel, its battery life which could prove to be less than expected. In other words, a charger is often something that we bring around, since we never quite know when we might use up the daily supply of juice and would need to charge it up during an emergency. Enblue has come up with the EVOLUS 3 Qi multi-docking station that prides itself in delivering wireless charging capability for the all new iPhone 8 and iPhone X, while not forgetting that it is also able to cater to the iPad and Apple Watch in terms of charging, too.\nWhen it comes to gaming peripherals, the mouse is one of the most important aspects. Gamers would not settle for anything less than the best, and the right kind of gaming mouse is able to make or break a particular gaming session. SteelSeries continues its legacy of delivering high quality gaming devices without having to break the bank with the introduction of the Rival 110. The SteelSeries Rival 110 gaming mouse will feature TrueMove1, which is an original sensor that has been co-designed with Pixart that has been touted to be able to deliver true 1-to-1 tracking, making it as accurate as the Rival 310 and Sensei 310.\nEarlier this year, we brought you news of Polaroid shaking things up in the world of digital photography, going to show everyone just how they intend to remain relevant in this fast changing world while maintaining the kind of uniqueness that made Polaroid such a household name all those decades ago. CES 2017 saw the advent of the Polaroid Pop, or at least, the announcement of such a fine looking piece of hardware. This time around, the Polaroid Pop instant digital camera is now available for the masses to purchase, so read on to refresh your memory on what it is capable of doing.\nWhen it comes to no-nonsense gaming hardware, there is one name that constantly crops up on the list of many people: Razer. The company is back yet again, this time around with the Razer Wolverine Ultimate gaming controller which will cater to gaming proponents on the Xbox One and PC platforms. This is an officially licensed gaming controller for both platforms, and has been specially designed in order to have it adapt itself to virtually any kind of gamer. Why do we say so? Read on to find out.\nThere was once a time when Bluetooth headsets were all the rage \u2014 and that was close to two decades ago. Phones were just that, phones, and there was no such thing as a modern day smartphone with a full touchscreen display, although it well on its way to the existing smartphone design with the advent of the Palm treo and its ilk. Nokia ruled the roost back then, and a hands-free headset was something that was fast catching on, with more and more people being able to afford mobile phones. A Bluetooth headset was considered to be something of a luxury, a fashion statement even. Fast forward to today, and Bluetooth headsets have not deviated from its function, and rarely from its form. Enter BlueParrott's C400-XT Bluetooth headset.\nWhen it comes to competitive gaming, the stakes are high. Gamers would like nothing better than to have the very best in terms of hardware, having every single bit and piece specially optimized in order to ensure that everything works in a clockwork manner. LG of South Korea has stepped up to the plate this time around, having collaborated with 'FaZe Clan' so that gamers will be able to gain an edge on the competition courtesy of a brand new high performance LG monitor.\nWhen it comes to mobile accessories, it is safe to assume that one would be hard pressed to find a company out there who will be able to do it as good as Satechi. Satechi's latest offering arrives in the form of a brand new vertical laptop stand, sporting a sleek looking form factor and space saving design that ultimately aims to promote a clutter-free workspace. Oh yes, it will also handle this niggling issue that has plagued many a laptop owner for years \u2014 the problem of overheating, which is what the Vertical Laptop Stand from Satechi will be able to circumvent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a child I'd heard of the bushman though the story telling of Laurens van der Post, his books and films. Many years later, my intrigue in their culture and hunter-gatherer way of life still yearned to experience how they lived. How does an egalitarian culture that has existed for thousands of years manage alongside the modern-westernised world of the 21st century? What wisdom do they have to offer us, should we ask? And, what skills can we learn from the bushman? Just some of my questions I felt needed answers.\nTo gain some perspective \u2013 Namibia is vast, more than six times the size of England with a population of just 2.5 million! The Bushman, the indigenous people of Southern Africa, have lived there for tens of thousands of years. There is possibly just 100,000 Bushman remaining, living mostly in Botswana, and Namibia, with some communities in South Africa and Angola (Survival International).\nI joined a group of similarly interested people, to spend time living in the bush for a month, learning from the Ju\/'Hoansi-San, the Bushman near TsumKwe in the North East of Namibia. It was a two-day drive from the capital Windhoek. The main roads north to Grootfontein, our first stop over, were paved with tarmac. But, from then on, the roads were sandy and often rutted, and with more game wandering around, it was a challenging but non-the-less interesting and rewarding drive.\nWe arrived in late Spring, it was pretty warm, and as summer approached it got hotter, above 30\u00b0C and over 40\u00b0C at times. Our first task was getting used to the environment, taking precautions to avoid unexpected encounters with snakes, bugs and predators and of course setting up our camp!\nAfter the rocky outcrops and hills of the South, it would have been easy to get lost in a largely featureless terrain, with endless prickly bushes and few if any obvious tracks. Our main navigation point was the main road \u2013 the sandy and rutted road we arrive on!\nIt was always good to remember that we were not at the top of the food chain here in the bush. Roy's (an experienced wilderness guide in our group) advice was to \"be relaxed, but attentive\" \u2013 this mantra ensured you enjoyed being in the bush whilst taking the appropriate amount of care for yourself and the group as a whole. Thanks Roy!\nLife each day at Elephant Song Camp, often started with a wonderful sunrise, the sounds of the franklin's calling, the call of the go-away-bird, and other birds singing too. First job was reigniting the fire, usually blowing the embers back into life, adding a few dry sticks, filling the kettle with water and putting it over fire. Then it was time for some Pilates \/ Yoga, to warm and stretch my body ready for the day.\nOur days were mostly filled with crafts, tracking and wildlife watching, and come the end of the day, we were treated to beautiful sunsets and an amazing light as the dusk descended, encouraging us to finish our task before darkness would be upon us.\nThe terrain was brilliant for tracking, the sandy substrate displayed the tracks of mammals, reptiles, birds and insects beautifully. That's not to say we were always sure, well at first, what we were looking at. Within our group there were knowledgeable trackers from Northern Europe, others with experience from South Africa and America, and novices too. Through careful examination, considering the print patterns and all associated clues and with guidance from the Bushman, we learnt more and more every day. So, we all gained and learnt to recognise the tracks and signs left out on the trails.\nOne very special moment for me was when one of our Bushman guides asked me to identify a track. The ground was hard packed sand, with lots of long dried grass all around. Looking closely, I simply couldn't see the track he was asking me about, I eventually had to ask him to show me. There was a grin, and he used a stick to show me the outline of the track \u2013 it was huge, about 16-inches (400mm) across, yes, it was the spoor of an elephant! We had spent so much time looking at smaller tracks, you had to step back and see the bigger picture. Taking another close look, I learnt how to identify the toes and then, looking ahead, I could now see the direction of travel, and the deep footprints left during the last rains, about a year ago, with the long-dried grass bent over. With tracking, as is true for many things, you need to get your eye in and once you do, things become far clearer \u2013 a wonderful moment, thank you Comsa!\nSometimes you arrive in a place where you feel a special connection. Maybe you haven't found a place like this yet, but they are out there, that's for sure. This was such a place, a short distance from our camp, we drove down a sandy track, and there she stood. It's difficult to gauge the scale, but the limbs of this wonderful baobab tree were huge.\nI walked 75 paces around this amazing tree, just taking in the enormity of it all and wondering how old this tree must be and the stories it could tell of people from the past. It's thought to be more than 1000 years old. There was evidence of elephants all around, the tracks like fossils in the hard ground, piles of dry dung and deep scars on the bark, carved by the elephant's tusks, leaving their marks for future generations.\nHow do you understand the people of a village that you know so little about? One of the ways was through learning practical skills. The learning style of the Bushman is one of observation, you would watch how something is done for as long as is necessary and then repeat the process yourself. This is typically how the children would learn. A boy would accompany a his father or grandfather on a hunt, observing how they identified tracks and trailed an animal. Similarly, girls may walk with a matriarch or their sisters and be guided through the bush, helping to gather the berries, learning which ones should be collected and which should not.\nWe watched how dry, salted impala skins were scraped clean using a chop-chop, a versatile axe-like tool. First the hair was removed using short-scraping movements, taking care not to penetrate the hide. Once complete, it's a long job, the hide was turned over and the membrane was scraped away along with the salty residue.\nAs we got to work on the impala skins, this activity quickly became a social event, not only were men and women from the village joining in, but many of the children wanted to have a go too. At one stage, there were eight young boys, taking turns to scrape the hide I was working. Similarly, young girls were working with the ladies in our group. Later the elders told us that many in the village had not worked hides before and that we had sparked their interest through our participation in the hide preparation.\nWhen you don't have much, you need to be inventive! The Bushman make tools from scrap metal combined with wood from the local trees, and hides, sinews and bones from the animals they hunt. More time being attentive \u2013 we watched as they heated old metal bolts and shaped them into useful tools. These included the blades for chop-chops (an axe-like chopping \/ cutting tool), spear tips and knife blades. The bolts were heated on the open fire, retrieved from and held with pliers and hammered with heavy club-hammers to flatten and shape the bolts into new tools.\nIt was fascinating to watch the skill of the Bushman, to see a knife being made throughout the day. A bolt being repeatedly heated, hammered and shaped to create an long oval blade with a rat-tail tang. Later, a single piece of hardwood was cut to create a seamless handle and sheath, the timber shaped using a large chop-chop. The blade was burnt into the handle \u2013 initially, a small 'branding-iron' was used to burn a hole and then the tang of the blade heated and burnt into the handle. Finally, it was time for some careful carving, slimming down the sheath and handle in unison to make a beautiful traditional Bushman knife.\nWe headed out into the bush with our guides to gather the sticks we needed to make bows and arrows, spears, digging sticks, and handles for our chop-chops. Most needed to be relatively long and straight, but our chop-chop handles, needed a right-angle at its base, this was to receive the chop-chop blade. The method used to straighten the sticks and remove the bark was to char it in a fire \u2013 then bend it gently and repeat until straight. The bark would then easily peel off, and with the heat applied, this hardened the wood, making the components tougher and more resilient.\nI've always loved making things from wood, I find it a calming and relaxed occupation, so I looked forward to using these familiar skills. We watched the Bushman attentively, noting the methods they used to char, straighten, chop and carve these tools and weapons. It was then our turn.\nOne of the first tools we made was a digging-stick, as the name implies, this was for foraging roots. Once straightened, we shaped the end used for digging so it was tapered with a blunted point. This was then hardened in the fire, ensuring it got hot, drying the wood to make it hard, but of course not burning it! Our spears followed a similar method, and would be mated with a metal spear tip, heated and burnt into the end of the wooden spear.\nFor anyone who has ever used and \/ or made a flat-bow or long-bow (yes, think Robin Hood for those who haven't), the Bushman bow is considerably lighter and smaller by comparison. The bow is less than a metre long with the limbs just 10mm in diameter at the centre of the bow and tapering towards each end. So, much care was required to ensure we didn't end up with broken sticks!\nArrows were also made, the shaft from a reed grass, now dry having grown in the previous wet season. The arrow tips were made from a piece of wire \u2013 the end of the wire was gently hammered to create a thin, blade-like tip that was then filed into shape. The end of the arrow shaft was reinforced with sinew and then the shaft of the arrow tip inserted. Needless to say, we did not apply any of the Bushman's deadly poison to our arrows \u2013 the poison they use is lethal. It's made from a beetle larvae, and should any of it get into your blood stream, even a very tiny cut, would mean death!\nFinally, it was time for some target practice, no not on live animals, but a straw target. Well I was pleased, my little bow and arrow performed well, it hit our straw target right in the middle, result!\nNorthern Namibia is within the Tropic of Capricorn, so when it rains, it is a veritable deluge, swamping huge areas, creating raging torrents down rivers and filling all the pans (dried up lakes the remainder of the year). The rains had only just arrived as we were getting ready to leave, but we were treated to some pretty amazing storms and lightening shows. You quickly began to realise why there were so many blackened trees, most were the result of lightning strikes!\nOnce the rains had started, everything took on a new vibrant greenness. Identifying the new tracks each day, we could see that animals that had been further north were now moving south. But for the majority of the year, the lands of Namibia are parched and very dry, with little opportunities to find open water sources.\nFor the Bushman, finding other water sources is essential. One such source is the water-root. We were taken on a long walk, through prickly bushes, to look for the tiny vine-like shoot that grows up through those prickly bushes. Our digging sticks were put to good use, the root was half a metre down, the root was finally dug out, scraped and eaten \u2013 yes, it was tasty and thirst quenching too.\nNamibia \u2013 In a land with little water for most parts of the year \u2013 you need to know where to find water.\nTo be clear, the majority of water has long since been sourced from local boreholes \u2013 the Kalahari sits above a huge subterranean water reserve. Our camp had a stand-pipe near-by which was shared with the Bushman village. There was a solar-powered pump that raised the water up from the reserve \u2013 this generally worked fine when it was sunny, unless someone had disconnected it to charge their phones (life's modern-day essentials right). Alas, when it was cloudy, the pump would cease to work, which meant no water was available! Interestingly, there were no obvious methods for capturing water when the rains came, and the pumps didn't supply water \u2013 no guttering around buildings or huts. When the rains came, the Bushman would walk to the nearby pans, to gather their water.\nOne of the Bushman's favourite treats was the brandy-bush with its ripe raisin like berries. The berries, light brown in colour, tasted of raisins, with a small seed in the middle. The ladies and girls of the village gathered these in large quantities, eating a few as they went from bush-to-bush. We enjoyed gathering them too and when out on our walks were always on the lookout for the brandy-bush.\nOn other foraging forays, we looked for the leaves of mother-in-laws-tongue (yes, you may have seen this as a pot plant in your local garden centre). This time, what we were looking for was not for eating, but for using its fibres to make bow-strings or other strong cords. To process the leaves, the outer tough fibres were removed and discarded, and then the fibrous leaves scraped and drawn under the sharp end of a digging stick to remove the glutenous part of the leaf and reveal the white fibres. The resultant fibres were a little like those of flax, and when a few were twisted together, they made a very strong cord or string. Like any new skill being learnt, lots of practice was required to get anything approaching the quality and quantity of fibres achieved by the Bushman!\nThe days were hot, I was regularly getting through five litres of water a day! So, the idea was to start our days early and have a siesta when it was at it hottest in the middle of the day \u2013 trouble was, having a siesta, a snooze, when it's roasting hot was far from a relaxed affair. Often it just meant doing things at a more leisurely pace, preferably in the shade if you could find some.\nWe really did have a siesta this time, well those of the group that didn't go for a walk, exploring the pan. This lovely baobab tree provided us with some much-needed shade \u2013 so we had a lunchtime sleep. Whilst not quite as big as the other baobab we saw, it was 15m in circumference, Stephan, standing there in the middle of the picture, is a cool 2m tall, it was a big tree!\nThe elephants obviously love this baobab, leaving their graffiti marks across the whole tree! Yes, those gauges in the bark are made by the tusks of the elephants, and they go a long way up the tree, bull elephants are big, very big.\nWhilst we sat around the base of the baobab tree the fragrant flowers fell upon us. The fresh flowers, a creamy off-white, were reminiscent of broccoli or cabbage smells, but as they decayed to orange-brown, they gave of a beautiful fragrance, one that you could imagine paying lots for in a department store!\nNamibia \u2013 Out on the edge of the pan, this baobab stood out from miles away \u2013 there was just the one!\nAnd to finally get a real sense of perspective, Werner's VW minibus, looks quite small, being dwarfed by the colossal baobab tree. It was the only baobab in this area, but apparently travelling further north into the heart of the tropics, there are many more.\nThe life of a hunter-gatherer may seem romantic, gathering berries, digging up tubers and hunting game, but the reality is tough. We were joined by two of the village elders to go hunting for spring hare. Not knowing what may happen, how the hares would be caught or even if we \/ they would be successful in the hunt, we headed out into the bush.\nThe spring hare lives in burrows, as many of the Kalahari's mammals do. So the first task was to find active burrows, ones with fresh tracks leading down the holes and of course fresh poo! Having found a possible burrow, a long pole with a hook on the end, was passed, pushed and pulled down the burrows \u2013 to hook a spring hare.\nWe walked around the pan with the Bushman for a few hours, with many burrows being checked and poles push and pulled. But despite best efforts to hook a spring hare \u2013 none were found. Apparently, the hares have been learning how to outsmart this hunting method, digging tunnels with right-angles in them, so that the poles cannot reach them \u2013 those spring hares are smart!\nWe spent a month here in the wilds of Namibia, enjoying the simplicity of life. Each day had a natural rhythm, the familiarity of fire and food, learning and sharing in the community, both within our group and with the village too. There was time to build relationships, get to know members of the community, laugh and joke, tell stories, and sing and dance around the campfire.\nThe Bushman are isolated, both socially and physically and yet are still strong and resilient. The pressures of the modern world are knocking at their door, and with the older children educated away from the village, there are influences that will certainly entice and draw some away from their community.\nI think their strength comes from their culture and what feels like an inbuilt contentment with life. They may have concerns for the welfare of their children and community as a whole, but these are not that apparent. Medical facilities are not near-by and our team helped a few times, remedying quite serious conditions and also providing transport to the clinic in Tsumkwe. What would they have done if we had not been there, it's difficult to say.\nThe village is not reliant on hunting and gathering anymore, they do both regularly, but depend more on supplies purchase from the general stores with income generated by the Living Museum, and the crafts and activities they provide for tourists. They were always welcoming and friendly and when I asked what they would like me to say to friends and family this is what they said, \"please tell them to visit us and we will welcome them to our village\". So, should you decide to visit Namibia, please do spend time with the Bushman in one of the Living Museum Villages, you will be sure to have an enjoyable and fascinating experience.\nMy memories are of a strong and caring society where men and women, whilst having different roles within the community (hunting vs. gathering for instance), appear to have lively discussions and an equal say in most things, especially the matriarchs! The children learn though watching the elders and are left to learn and play together, with older children looking after and guiding the younger ones.\nThe experience of learning from the Bushman, living in the bush and enjoying such a beautiful country will stay with me always. But what i think was most poignant, is their apparent contentment with life. I say apparent, because I was a guest, a visitor, and it's difficult to truly know if this is the case. But, this is the last impression I had of my time in the bush and if one can be content then life will good.\nWith thanks to Casparo Brown who organised this trip, Werner Pfeifer for sharing his knowledgeable of tracking and the Bushman, and Lynx Vilden who brought her special \"stone-age je ne sais quoi\" to our adventure. Thanks also to my fellow travellers and members of our group, and finally, a huge thanks to the Ju\/'Hoansi-San, the wonderful men, women and children of the village at Elephant Song Camp, Namibia. Visit The Living Culture Foundation Namibia to find out more (www.lcfn.info) \u2013 thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shaykh Yasir Qadhi imparts ample knowledge about the story of the Ka'bah and the profound symbolisms in the series of events that transpired during its building.\nThe story of the noble character of our Prophet \ufdfa in the wake of the story behind Zayd Ibn Harithah RA is an embodiment of excellent traits that our Prophet \ufdfa portrayed. The extreme fondness that the Prophet \ufdfa had for Zayd RA and how was the relationship with him in his lifetime.\nAisha RA is the main narrator of the Hadith of how wahi or revelation began and the contents of this very long hadith have been elucidated in immense detail. This chapter marks an instrumental and life changing event in the life of the Prophet \ufdfa. This has to be listened to savour the course of action of this event.\nPrior to prophet hood, the Prophet \ufdfa sees dreams, hears stones saying salaam etc., for 6 months or so. It is as if the Prophet \ufdfa was being spiritually trained for the big revelation.numerous interesting details come to the fore which leave us mesmerized at the sheer Qadr of Allah.\nShaykh Yasir also dissects the happenings during the course of the revelation and how the wife of the Prophet \ufdfa stood by him in this monumental event is worth listening to.\nLessons can be learnt, imbibed and inculcated on how to conduct oneself in our day to day lives as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Postprint version. Published in International Journal of Cost Engineering, December 1, 2009.\nThe construction industry needs to decrease its costs due to nonconformance by implementing an objective model for analyzing the excessive cost of poor quality and the overall savings realized from good quality. Direct cost estimates from rework in commercial building construction average about 5%. A simple calculation based on U.S. construction industry expenditures of $1.246 trillion in 2007 reveals that $62 billion is wasted just on the direct cost construction rework alone! The term quality can be applied in many different ways to various aspects of the construction process. This paper proposes a model which helps illustrate how the various elements of the cost of quality (COQ) might be employed by the general contractor within the construction project itself. Several traditional COQ theories are applied, compared, and contrasted as they relate to the construction industry; particularly conforming and nonconforming quality costs. The study concludes by suggesting possible ways of measuring the costs of construction quality and suggests that the construction industry needs to experience two true paradigm shifts. One which moves the industry from resources spent on quality non-conformance to resources spent on quality conformance; the other moves the construction business perspective from thinking in a quality compliance mode to an actual quality performance mode.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Multiple reports of an explosion on Park St.\nAccording to Dane County Dispatch this is at All Metal Recycling.\nMPD and MFD are responding to the scene now.\nNBC 15 will have more on this story as information comes out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We welcome new clients and patients. To make the most of your visit we like to have information about you and your pet before your first appointment. Please complete these forms before seeing us for the first time. If you prefer, however, we can gather this information in person. To make an initial appointment, call (803) 226-0585. We look forward to seeing you soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renee was privileged to design a memorial to honor United States veterans within the community of Anthem, Arizona. The award winning Anthem Veterans Memorial was constructed in the summer of 2011 and officially dedicated at a large ceremony on Veteran's Day, November 11th, 2011. The AVM is now a State Historical Site.\nThe significance of each veteran's day, 11-11, is central to the design of the memorial. At precisely 11:11 a.m., each year on 11-11, the sun aligns through the elliptical holes in each of the five marble pillars (each representing a branch of the the U.S. military) in order to perfectly illuminate a round mosaic inlaid into the bricks; that of the Great Seal of the United States. The symbolism of the five pillars standing in formation in order to protect the United States and to complete the solar illumination is representative of U.S. military personnel working together in all regards, in the security and defense of American citizens.\n* Project of the Year 2011- Arizona Public Works Association.\n* Grand Project Award- 2012 Arizona Association of Professional Engineers.\nVisit the link below to see Renee's interview on ABC News about the worldwide phenomenon on Social Media regarding the unique aspect of this incredible memorial. Millions of site visits made the AVM the most popular story of the day, Nov. 11, 2014.\nPlease click the link below to see a drone video of the AVM just a few days before the full illumination.\nPlease follow the link below in order to read more about the Anthem Veteran's Memorial, or to purchase a brick paver in honor of a U.S. veteran. The sale of brick pavers will fund the maintenance of the memorial and provide funding to recognize veterans otherwise financially unable to participate. You may wish to purchase a brick as a non-veteran supporter of the memorial as well, that will be placed outside of the circle.\nAnthem Veteran's Memorial from Jim Martin on Vimeo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One from the physician's office (SAGA): this bill is payment for your physician's services, anesthesia and pathology. Payment is due to the physician's office (SAGA).\nOne from the endoscopy center (SAGEC): this bill is payment for your procedure. Payment is due to the endoscopy center (SAGEC).\nBills for physician fees, which include office, procedures, anesthesia, and pathology will be marked with San Antonio Gastroenterology Associates (SAGA).\nAll the bills for the facility fees will be marked with San Antonio Gastroenterology Endoscopy Centers (SAGEC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MIDNIGHT! \u2013 Premium & Responsive Prestashop Theme can be use for any e-commerce store, Clean & modern design suitable for any type of products, MIDNIGHT! responsive prestashop theme is a flexible theme completely adjustable for any device. (Desktop, laptop, tablet, smart phone\u2026) Highly customizable prestashop theme.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How does water get onto a dryland farming site? Over the last decade 15 years I have become increasingly interested in the politics of water with the encouragement of Prof. Tony Allan at the University of London.\nWater has now entered the political stage. For a long period water policy was dominated by the \"hydraulic mission.\" The hydraulic mission was the era of engineers. All problems of water scarcity could be solved by building more structures. Dams for water storage. Canals for water distribution. The hydraulic mission acquired its own momentum and continued to build dams beyond the capacity to use them. The engineers are also well entrenched in the irrigation bureaucracies around the world and while they have learnt a new language that includes economics and sociology their solutions are frequently the same as before.\nThe second stage for water management was the rise of the economists. They tried to put the hydraulic mission into an economic context by providing costs and benefits for irrigation schemes. They have had limited success. Irrigation schemes have under-performed around the world because of the tendency to select the lowest estimates of cost and the highest estimates of returns. Rarely are these ideals realised.\nWe are now entering the third stage of water management - the politics of water. There is considerable confusion about the meaning of the word politics. Many water managers see it as \"party politics\" which has become a pejorative term in parts of the the world where democractic parties are allowed to exist. They see their engineering and economics as clean and dispassionate while they see party politics as a system of grubby deals negotiated in secret. Politics is the process of negotiating resource allocation in which party politics plays a part but is by no means the only factor at work.\nAt this stage we need to step back and analyse the term \"resource allocation.\" During the period of the hydraulic mission there was no need to allocate. Demand could always be met by increasing supply. The resource was limitless therefore rationing among competing groups was unnecessary. The economist tried to put costs and returns into the equation with very limited success. Now we are facing absolute shortage. There just is not any more water out there. Of course there are exceptions but scarcity of water is forcing societies to make choices between water for agriculture, for domestic use, for industrial use and for the environment.\nThere are two mechanisms being used. There is the use of market ideology to say that all allocation will be determined by the market for water. Price will be the determining factor. Alternatively there is the more pragmatic approach that says that society can make its own decisions and need not be bound by such a rigid ideology.\nThe other issue that needs to be considered is the ownership of the water. During the hydraulic mission and the age of plenty ownership was irrelevant. There was always more water. Ownership is a right of access to a scarce resource. Without any scarcity it is meaningless. Now water is scarce there is a scramble for ownership. Ownership also needs to be carefully defined. There is the theoretical and the practical. In Spain, for example, the national water resource, under the constitution, belongs to the state. The state owns all the water. In spite of this the Spanish government has paid billions of Euros to farmers to buy back their rights to water. De jure the water belongs to the state but de facto to the current users.\nMy involvement in the politics of water developed from other open resources that had to be closed due to scarcity. The two I was involved with was fisheries in South Australia and the rangeland in West Asia and North Africa.\nBelow are some of the papers I have written on water for journals and conferences.\nManaging water in Iran with the participation of farmers.\nThis paper is based on the notes I used for a series of about seven workshops and seminars I conducted in Tehran and Isfahan in March 2015. The notes have been revised to take account of the comments and discussions that took place during these meetings.\nManaging water (including wells) in Iran on the basis of need.\nThis paper was written in August 2015 to clarify some of the ideas in the paper written in March 2015 (above) and to expand the concepts into the management of aquifers.\nWhat is wrong with water markets?\nWhile this paper covers some familiar ground on legumes and rotations I also tried, unsuccessfully, to show that water was not the limiting factor for production in the WANA zone. This was a totally new concept for the particiants at the conference and one that is totally contradictory to their firmly held beliefs that the WANA region is a water acarce zone. Of course it is and more rainfall would enhance yields but at the present yields are limited not by rainfall but by a failed farming syste, If one take like for like between WANA and southern Australia with regard to rainfall, soils, temperatures, etc. the yield of cereals in Australia is double and that for pasture three or four time greater. Therefore yield is not thelimited by the rainfall.\nWater footprints - do they mean anything?\nThe politics of water in an age of scarcity.\nA paper prepared for a workshop on water scarcity held in Alexandria, Egypt on 7th and 8th October 2010.\nIn this paper I discuss the management of water in Australia during the drought of the first decade of 21st century. I show how the water market deveoped a decade or so earlier has increased rather than diminished the crisis. I point out that water rights are also creating large additional costs for Australian farmers and a growing inequity between generations. I point out that the Australian water market model would present even greater difficulties if adopted by Egypt as the Egyptian taxpayers cannot afford the generous buy back payments made in Australia.\nA paper delivered to the above workshop in Alexandria.\nThis short paper discusses the three approached to water resources. Firstly that Water is an Economic Resource (WIER) - the Australian model where water rights are traded like any other commodity.\nSecondly the antithesis - Water is Not an Economic Resource (WINER) that treats water as a human right that cannot be bought and sold.\nThe Australian Water Market Experiment.\nThis paper published in Water International had an interesting history. When we first submitted it to the international editor he thought it would be ideal for a special issue to be published for an international water conference in Melbourne to be held in 2000. However the Australia guest editor rejected it as too critical of the Australian market model. We thought the paper would be spiked but the international editor courageously published it in the next issue. Many of the disadvanatages we identify have become serious problem following the drought.\nAccess Cost and Distribution Costs \u2013 AC and DC \u2013 for water.\nCharges for water have a considerable impact on property rights and protery values for water. In this paper we discuss some of the options and their impact.\nThis is a paper Lynne and I wrote for a confernce arranged by Cranfield University in 1996. It has fallen through the the net. Everything is now archived on the web but this paper slightly predates those comprehensive archives. It cannot be Googled - at least I have failed. I have fragments of a printout but the original paper is on discs and systems that cannot be read on more modern computers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Screaming or crying is how Stacey Brown wakes up most nights. Laurie Faria Stolarz's, \"White is for Magic\" grabbed my attention from the first page on. Sometimes I forced myself to put the book down and breath for a minute because it was filled to the brim with plot twists and exciting moments. After Stacey and her boyfriend start fighting she decides to sleep in the lounge room of her dorm building. When she falls asleep she starts hearing someone call her name so she gets up and goes downstairs and then she wakes up back in her bed, finding out it was all just a nightmare. Stacey has had nightmares before, once about the girl she was babysitting and how she was going to die or be kidnapped. Coinsidently the girl died the next day. Another time, she had nightmares about a girl at her school, Veronica Leeman who was going to be killed. The next week she was found dead in the girl's bathroom.\nThis author's style of writing is extremely interesting, once you think Stacey's life is becoming better, something creepy, scary or disgusting happens to the character. In my opinion, the author's books are very entertaining to read. Laurie Faria Stolarz causes the reader to believe they are living the life of Stacey's best friends, Amber and Drea, by telling you all of Stacey's secrets, ideas and fears. When I was reading the book, I felt as if on every page there was another plot twist. This book takes place at a boarding school and it might not be as relatable to some people if they don't go to boarding school or know what it's like but it is still a superb novel. In most of the story, Stacey is scared, the tone of the story suspenseful and a little creepy at the same time. If you are not in grades 6 or higher, you shouldn't read this book because it is a little bit scary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WASHINGTON \u2013 Treasury Secretary Henry Paulson says the government must move quickly to give the Federal Reserve more powers to regulate the financial system.\nPaulson said today that the central bank's powers should be expanded in the wake of the near collapse earlier this year of Bear Stearns, the giant Wall Street investment firm.\nHe said there was a need to consider quickly how to give the Fed the power it needs to obtain information from investment banks and the responsibility to intervene to protect the overall financial system. His comments were provided by the Treasury Department as excerpts from a speech he was to give later in the day.\nIn late March, Paulson unveiled a blueprint that proposed the most sweeping overhaul of the nation's financial regulatory system since the stock market crash of 1929 and the ensuing Great Depression.\nAt the time, he said Congress should consider ways to expand the Fed's powers to oversee investment banks but he set no timeframe for when this should occur. However, in his speech today he stressed the urgency of acting soon.\n\"We should quickly consider how to most appropriately give the Fed the authority to access necessary information from highly complex financial institutions and the responsibility to intervene to protect the system so that they can carry out the role our nation has come to expect \u2013 stabilizing the overall system when it is threatened,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You might have encountered a story that a quarter of Instagrams millions upon millions of users abruptly stopped using the massively-popular service over Christmas. But there's good reason to believe this is all false \u2014 and Instagram flat out denies it.\nIf anything, this looks like an errant blip. Millions of people don't suddenly decide to abandon Instagram many days after the privacy scandal ends. Especially not when there are so many adorable, delicious holiday things floating around, begging to be photographed and filtered.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upbeat New Year 2018 Images, Quotes, Wishes Download \u2013 another year is tied in with starting once more and in addition understanding your fantasies as well. It is unique time at which another schedule year begins and the date-book's year check increment by one. It is for the most part falls on January 1 which is also called New Year's Day. With a considerable amount of eagerness, trusts and also determination. After some couple of months a great many individuals all around the globe will soon set to welcome a crisp new year. From family and companions get together to parties, the New Year day will be loaded with most prominent festival in every last niche and corner of the world.\nProgress Happy New Year Quotes 2018 \u2013 Although there are few days staying to welcome New Year 2018,here we have recorded enormous accumulation of Advanced Happy New Year 2018 quotes. You can impart all these astonishing wordings to your precious ones. Additionally take important motivation from these quotes which will really help you to begin and on the off chance that you are roused enough trade your excitement with your cherished ones.\nProgress Happy New Year Wishes \u2013 Happy New Year is one of the finest occasions in the globe that remember with an extraordinary fun and happiness. As we probably am aware New Year of the day dependably conveys a few chances to our direction, transform everything we could ever hope for into reality and every one of our endeavors into immense accomplishments as well. Along these lines, praise the Happy New Year 2018,with our most pleasant wishes of New Year and welcome your darling loved ones and express your heartiest favors and feelings towards them.\nGlad ?ew Year If y?u think in wording ?f a year plant ? seed if in wording ?f 10 years plant ?rees if in wording ?f 100 years show ?he individuals. ?y Best Wishes ?nd Prayers Always With Y?u.\n?ay your eyes, ?ind and heart ?e loaded with ?rue soul of beginning ? new year ?hat gives ? crisp measurement to y?ur life. Upbeat ?ew Year t? you!\nUpbeat New Year Images 2018 Images \u2013 As we probably am aware every year we say farewell to the previous year that was loaded with most propitious snapshots of satisfaction, distress in every last one of us lives and welcome New Year with most extreme of enthusiasm and energy. Along these lines, here look at our stunning gathering of Advance Happy New year pictures and how about we raise a toast to a Happy New Year and considerably more joyful beginnings to make forthcoming year significantly more blissful and intriguing one. You can download these get-togethers of pictures for nothing and send to your friends and family. What's more, have a fabulous time filled festival of New year!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, I am totally exhausted, but it has been a great several days here in Florida.\nYesterday, mom and I drove across the state to Tamarac, Florida and I did a concert at Lighthouse Covenant Church. This was very last-minute as well \u2013 when I realized I was going to have a free day, I called and they put this together very graciously and quickly. This congregation is so dear, and they were extremely supportive of us last year when we were there and needing support for our Kenya tour. I was thrilled to get to come back and give them an update and sing some of the music inspired by that trip! It was also wonderful to get to see our friends, the DeMarcos. One thing I will say for Florida \u2013 we seem to have friends in every corner, and it sure makes traveling so much nicer when you're staying with people you love to be with rather than lots of hotels.\nOn the way back, we saw the funniest thing! \"Alligator Alley\" is the stretch of 75 that runs through swamps across the southern end of the state. At a toll booth, there were these black birds sitting on the barriers between the lanes, and when I went to roll down my window, immediately two of them jumped in front of my window and just stared at me. Kind of freaked me out in an Alfred Hitchcock\/ The Birds kind of way. So I did not put my window down, and as we were getting ready to pay, I saw the guy next to me roll down his window. Immediately this bird flew right into his window at his face and he was yelling and trying to push the bird back out and get his window up. When I got to the toll window, I rolled my window down two inches and pushed the money through and asked the toll booth guys what was up with the birds.\n\"Well, they hang out and if you have a chip or anything they might want on your seat, they dive right in. It's hilarious! It's the best thing about this job!!\" Well, okay then.\nThis morning I got up early and went back to First Presbyterian where I sang on their morning services. Their choir did such an amazing job on their Palm Sunday piece \u2013 they literally made me tear up. Then this afternoon I sang a concert there. I was at this church last year, and was so struck by the excellence with which they do things \u2013 it was a pleasure to be back. Hope to see them often!\nThis entry was posted on Sunday, March 28th, 2010 at 10:14 pm\tand is filed under News. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Anyone able to provide a recent report on Cuastecomate? I haven't been there for several years, but really enjoyed my last visit! Is the bay now more developed? What state is the main hotel in? etc. etc.\nThe hotel is doing fine. I think it is a fairly nice and clean hotel.\nThere are a few more houses being built or have been built but in general it hasn't changed much in the couple of years I have been around here and visiting here.\nThe bay is nice and Sparks was paddling around in it a couple of months ago in his kayak.\nThanks, Shoe. I'm glad it hasn't changed too much. In the old, old days, the hotel had a one-eyed cat that was a star attraction - her name was Barbara Streisand!\nI walk to Coastecomate every day I can (missed today because it's raining). The road was paved in 1998 and still doesn't have a lot of traffic so it's ideal for a daily \"hike\".\nThe north end of Melaque and Coastecomate are governed by the ejido of La Huerta, which is about 45 minutes inland. This poses problems for the residents there, especially where infrastructure is concerned. Tap water is pumped in through an overground pipe and frequently goes out. The road is fine in the winter, but in the rainy season, there are many slides and for some stupid reason, all the rain run-off is sent down one dirt street that heads toward the beach. The dirt road isn't really dirt at all.....it's made of escombro that is dumped during the year, and I watch it as it slowly becomes an arroyo.\nEvery year there are more houses built, mainly gringo vacation homes and \"spec\" houses, but Coastecomate still only has one hotel and not one abarrotes (there is one but it's only open a few weeks a year. I don't think that \"counts\").\nI only went to the hotel once and had the worst margarita I've ever had in my life. I sent it back and ordered a beer. Figured they couldn't screw that up.\nAll in all, Coastecomate probably hasn't changed very much since the last time you were there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When students arrive on campus for the fall 2016 semester, their Berkeley experience will be further enriched by a new adventure in college dining: a caf\u00e9 created by Charles Phan, an award-winning chef known for his San Francisco restaurants Slanted Door and Out the Door.\nPhan, who studied architecture at Berkeley in the 1980s, says his newest venture \u2014 to be located in Wurster Hall, home of the College of Environmental Design \u2014 will feature organic international food, his culinary trademark. The caf\u00e9, he adds, is his way of \"giving back\" to the place that helped set him on a path to success.\nThe still-unnamed eatery will replace Ramona's Caf\u00e9, which is expected to close on March 18 \u2014 the Friday before spring break\u2014 bringing not just culinary progress but a welcome refurbishing of Wurster's ground floor. No Cal Dining jobs will be lost with the transition, Berkeley officials say, as all campus staff now working at Ramona's will switch to jobs at other Cal Dining locations.\nThe new caf\u00e9 is expected to appeal to a wider customer base than Ramona's, which served mainly students. Phan has agreed to follow the terms of the UC Fair Wage\/Fair Work Plan, announced by President Janet Napolitano in July, which calls on contractors doing business with UC to guarantee a $15 minimum hourly wage for their workers.\nImprovements will include upgrades to the exterior plaza on the west side of Wurster. (The caf\u00e9 itself has not been remodeled since 1984.) Construction is scheduled to begin in late spring, with a targeted opening date in August, in time for the fall semester.\nThe new outlet will join a host of others on campus, including the Free Speech Movement Caf\u00e9, Yali's, Caf\u00e9 Zeb and the ASUC Student Union \u2014 all of which generate revenue for the campus \u2014 as well as those operated by Cal Dining.\nChristine Shaff, with the UC Berkeley real estate division, said the lease agreement with Phan ensures that the caf\u00e9 will work toward the campus's \"zero-waste by 2020\" goal, and \u2014 in keeping with Phan's own professional ethos, as well as the campus's sustainable food policies \u2014 emphasize fresh, local produce.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The words \"creamy hot sauce\" don't really need a lot of explaining. However, this sauce is special because there's nothing else like it. Seriously. We like to think of it as the \"TOP IT ALL\" condiment your food didn't know it was missing. It's the perfect balance of heat and flavor so be sure to squeeze it on thick!\nMade by a small company of food lovers, family and friends from Austin, Texas.\nBlue Top means thoughtful, creative and extraordinarily delicious recipes every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Autauga Creek is a stream located just 3.9 miles from Prattville, in Autauga County, in the state of Alabama, United States. Alternate names for this stream include Atagi Creek, Pearl Water Creek and Pearl Creek. For Fishing License purchase, fishing rules, and fishing regulations please visit Alabama Fish & Wildlife. Please remember to check with the local Fish and Wildlife department to ensure the stream is open to the public. Now get out there and fish! Check out our Fishing Times chart to determine when the fish will be most active.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally, a no-nonsense anti-theft device that the end user can install. This revolutionary immobilizer uses a Theft Avoidance Module that replaces the starter, fuel pump, EFI or engine control relay in select vehicles. You can have a very sophisticated system installed in less than 5 minutes.\nTo see a sample installation click here.\nThe Cyclops \"TAM\"may also qualify you for up to a 20% discount on your comprehensive vehicle insurance.\nDeactivation of the \"TAM\" is done by a Remote \"Code Hopping\" Transmitter.\nThere are no wires and visible signs of an anti-theft system, the \"car will just not start\". If you are looking for protection without annoying false alarms, the Cyclops \"TAM\". is your answer.\nSimple Plug In Installation \"No Wires To Cut\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhklc b/data_all_eng_slimpj/shuffled/split2/finalzzzbhklc
new file mode 100644
index 0000000000000000000000000000000000000000..0e519ba9377c9b5ace4b5fe947ea065c8d231809
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhklc
@@ -0,0 +1,5 @@
+{"text":"Placed under the head and neck like a traditional pillow, this neck support pillow is designed to help preserve the curve of the upper spine by supporting the neck and positioning the head parallel to the sleep surface. Different side heights accommodate different body types and sleep positions. Constructed of medical grade memory foam to maximize support and comfort. Hypoallergenic. Super-soft, machine washable velour cover.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Going on a vacay soon? We have the perfect slide to up your style. Embrace the many gorgeous colors of this show stopping sandal encrusted in multicolored Czech stones on neutral beige leather. From dusk til dawn, this elegant wooden slip on is claiming its stake as a statement sandal. Embrace the many gorgeous colors of this show stopping sparkler!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the past years, seaweeds have been a popular food additive in almost all parts of the world. According to studies, seaweeds are rich in vitamins A1, B1, B2, B6, B12, C, E and K which can keep the body healthy and strong enough to fight many types of diseases. Aside from having loads of vitamins, seaweeds are also know as good sources for minerals such as potassium, odium, magnesium, calcium, phosphorus, manganese, iron and zinc. Since seaweeds have high vitamin and mineral contents, researchers found out those seaweeds are not only good as food additive; they are also potent ingredients for beauty products. Studies show that seaweed beauty products are effective and safe to use in almost all types of skin. In fact, seaweed beauty products are considered as one of the best in the cosmetic industry today.\nFor the past years, many players in the cosmetic industry came up with different beauty products using seaweeds are primary ingredients. Caring For Your Skin Skin types vary from person to person thus it is not safe to assume that what works for another person will also work for you. Fortunately, there are now different types of seaweed beauty products for different types of skin so you need not worry about finding the right seaweed beauty product for you.\nJust make sure that you use the seaweed beauty product accordingly to avoid any untoward incidents. If you are one of those people who are prone to allergies, you should be extra careful when using any types of beauty products. Before you apply the seaweed beauty product on your face or any parts of the body for that matter, you must read the label carefully and take note of the ingredients used in the product. Make sure that the beauty products do not contain ingredients that can trigger your allergies.\nMoreover, before you actually use the beauty product, you should test it first by applying it on a small portion of your skin. Wait for a few hours for any signs allergic reactions. If you notice any itching or burning sensations on the area where you applied the beauty products, do not use these products. On the other hand, you should be careful about using seaweed beauty products if you are suffering from acne and other types of skin breaks. To be safe, consult your dermatologist before you apply seaweeds beauty products on any affected area of your skin. Note that some types of beauty products can have some adverse effects on sensitive skin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"baby candy corn costume knit pattern.\nbaby candy corn hat crochet pattern.\nbaby candy ceramic letter dishes diaper game.\nbaby candy kit y la bar 1 en.\nbaby candy remarkable decoration for shower girl pink and gold buffet princess diaper game.\nbaby candy corn hat fall autumn by on bar matching game.\nbaby candy interesting decoration for shower girl sweet table ideas sweets treats bar costume.\nbaby candy image 0 apples.\nbaby candy cry extra sour wrapped gumballs ct corn outfit for girl.\nbaby candy shower mold carriage dish.\nbaby candy oh pacifier ct apples.\nbaby candy cut the string molds.\nbaby candy bottle pop max piece box wants game of thrones.\nbaby candy image titled make chocolate diaper poo for a shower step match game answers.\nbaby candy corn hat target.\nbaby candy la corn costume.\nbaby candy sleeping making mold fondant silicone for shower party birthday cake decoration game.\nbaby candy foiled milk chocolate booties girl piece box game answers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On December 4, clypd's VP of Product Jason Burke participated in a panel at TubeMogul's PTV Summit in New York City. At the event, TubeMogul announced the release of their programmatic TV offering for marketers.\nWe recently released an oft-requested view into the pricing associated with media. As a media partner, I need an understanding of how my media pricing rules scale with volume. As a media planner, I need to know not only how much inventory exists for a particular set of targeting criteria, but the price associated with it.\nProgrammatic advertising is all the rage these days \u2013 everyone seems to want a piece of it, from media owners to advertisers, marketers, and everyone in between. But what goes into programmatic?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhpma b/data_all_eng_slimpj/shuffled/split2/finalzzzbhpma
new file mode 100644
index 0000000000000000000000000000000000000000..4aefab029ca4b499e68b61c366eb7efad586ca46
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhpma
@@ -0,0 +1,5 @@
+{"text":"This is a new Faberge Xenia Yellow Goblet Glass without the Faberge Presentation box . Faberge etched on base. These measure approx : 8.25\" Tall x 3\" at base & 2.5\" at top rim. We ship Worldwide fully insured. Signature is required upon delivery. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Debate and support the preparation of UCLG policies in the field of local government disaster management Local government disaster management policy is not new to UCLG. Due to the Tsunami in South East Asia in 2004 were put forward in governing bodies of UCLG.\nIn April 2008, the \"Local Action for Disaster Risk Reduction\" Programme was launched by UCLG in partnership with ICLEI, Metropolis and CITYNET and with the technical support of the Earthquakes and Megacities Initiative (EMI). This initiative was created with the approval of the United Nations Secretariat for the implementation of the International Strategy for Disaster Reduction (UN-ISDR). The objective of the Programme, which has been running since, is to promote the adoption by local and regional governments of risk reduction policies and to promote their participation in the United Nations Global Platform for Disaster Risk Reduction, which was launched in 2007.\nWhen disasters strike, local governments are the first line of response, often with wide ranging responsibilities but insufficient capacities to deal with them. Local governments are equally on the front line when it comes to anticipating, managing and reducing disaster risk, setting up or acting on early warning systems. International disaster response efforts often lack a local governance dimension addressing specific needs of local governments.\nMembers of United Cities and Local Governments have historically launched solidarity missions to crisis areas. They are convinced that this area of work should be developed under UCLG flag and in cooperation with strategic international partners. In order to improve the position and the action of local governments in the face of disaster and those dealing with the consequences of disaster, it has been proposed that a Task Force on local government disaster response is to be created under DCCD (Development Cooperation and City Diplomacy Committee). Initially made up of representatives of VNG, FCM, CUF and UMT, the city of Istanbul and UCLG MEWA, this body will help to stimulate debate, legitimize joint initiatives and engage new actors.\nThe Taskforce will provide a platform for local governments, to exchange knowledge and expertise on disaster management. It will improve and increase the availability of local governments expertise to disaster-affected municipalities and it will highlight and develop the role of local governments on the international disaster management, resilience and response.\nMaintain \/ increase the capacity of UCLG members to technically mobilize for a disaster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Order confirmations are immediately sent to the email address you provide at the time of order placement. If upon placing your order online you don't receive an email confirmation within 48 hours, please contact us at 847. 698. 9521. For your security, Rex Parker Design does not keep credit card information on file. Since we do not store credit card information, payments are processed to your credit card within 24 hours of order placement to reserve inventory.\nAs most of our products are print-on-demand, orders may not be cancelled after they're placed. Please double-check your product, size, and design choice(s) before check out.\nCall us at (847) 698-9521 between 9:00 and 5:00 pm CST and let us know how we can help you. You may also email us at: rex@rexparkerdesign.com.\nMonday\u2013Friday from 9:00 am to 5:00 pm (CST), and Saturdays from 11 am to 4 pm.\nA sales tax rate of 10.0% will be added to all orders shipped to an Illinois address. For those customers purchasing from Rex Parker Design and shipping to a non-Illinois address: We do not collect sales or use taxes for those states. For those states imposing sales or use taxes, your purchase may be subject to use tax unless it is specifically exempt from taxation. Your purchase may not be exempt merely because it is made over the Internet or by other remote means. Many states require purchasers to file a sales\/use tax return at the end of the year reporting all of the taxable purchases that were not taxed and to pay tax on those purchases. Rex Parker Design assumes no liability for those taxes. Details of how to file these returns may be found at the websites of your respective taxing authorities.\nWe ship anywhere in the continental USA and to some select countries via FedEx (extra charges may apply for Hawaii, Alaska and countries outside the USA). We do not ship to P.O. boxes. We advise against shipping to locations where no one is available to receive it, or where the items would be exposed to the elements.\nRex Parker Design ships via UPS Ground, or USPS Priority Mail for smaller packages. Shipping charges appear in your shopping cart based on shipping destination. Due to the increase in theft of packages, orders with a value exceeding $500 are shipped with a signature required.\nAll shipments from Rex Parker Design are well packed and shipped insured. We trust that your order arrives safely. Please check all items in your shipment for shipping damage. In the unlikely event that you receive a damaged item, keep all original shipping materials and cartons and contact us at (847) 698-9521 within 7 days of receipt of the shipment (please note, this is the window the shipping carrier has for filing claims). Upon verification of damage, we will file a claim and replace the damaged item. Due to shipping carrier claim requirements, we cannot accept damage notification after 7 days of receipt of the shipment. If you need additional assistance, please contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mutants & Masterminds, Freedom City Win ENnies!\nLast night we attended the Gen Con EN World d20 System Awards ceremony, and we're happy to note that we did really well!\nAmong our 8 awards, Mutants & Masterminds won an ENnies Peer Award and a gold award for Best d20 Game; and Freedom City won silver awards for Best Campaign Setting, Best Art (Interior), and Best Graphic Design & Layout. Green Ronin also picked up a silver award for Best Publisher (Overall).\nWe are very proud at this acknowledgement from our fans and peers, and we're planning on another terrific year.\n\/\/greenronin.com\/wp-content\/uploads\/sites\/3\/2014\/08\/MnM-logo-wordpress.png 0 0 Evan Sass \/\/greenronin.com\/wp-content\/uploads\/sites\/3\/2014\/08\/MnM-logo-wordpress.png Evan Sass2003-07-26 10:41:362003-07-26 10:41:36Mutants & Masterminds, Freedom City Win ENnies!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's an outstanding week for BLUEGRASS, with the show on Tuesday with RUN BOY RUN at the Coffee Gallery Backstage in Altadena (fully described in Saturday's complete edition), and the best outdoor bluegrass concert of the summer with LAURIE LEWIS and JOHN REISCHMAN, which happens Friday at the lovely Ford Amphitheatre (across from the Hollywood Bowl). That Friday show is detailed in this \"Extra\" edition of the Guide, so don't miss it.\nBut everything else that's happening is not in here. Only the ADDED events that are not in Saturday's edition are in this one.\nSo, whoa up, thar, pard. That latest FULL edition, published early Saturday, has many events reaching through this week and waaaay into the future (including events for TONIGHT, Monday) so it is VERY MUCH still relevant and useful. But additional events that strayed-in to the ol' corral (as always) weren't around to get branded then and cut into the herd -- and we don't want you to miss anything.\nHence, here's this lil' ol' Monday update has ONLY those late\/extra events. (Surely it's the first of who-knows-how-many fresh news blasts this week).\nLAWIM -- Los Angeles Women in Music -- says, \"Bring a friend and your business cards. Join men and women from all walks of the music industry. Meet new people, re-connect with old contacts, find that new job opportunity, make the event work for you.\"\nSpecial Happy Hour food & drink discounts foe event participants.\nBusiness card drawing for a LAWIM Swag Bag that includes a one-year FREE membership or extension of a current LAWIM membership.\n7-10 pm THE ZOLLO BAND with many guest artists play the Talking Stick, 1411 Lincoln Bl, Venice 90291.\nA concert by The Zollo Band, led by songwriter and bestselling author PAUL ZOLLO.\nIt's preceded by solo sets from many of his songwriter-singer friends, including Jill Freeman, Janeen Rae Heller, John M. (aka John Mollenhauer), Jeff Gold, Samantha Ahdoot, Lisa Johnson, Earl Grey, and Michael Hughes.\nPlus, The Zollo Band features Earl Grey & Lisa Johnson and more special guests, including Chad Watson, Wendysue Rosloff, Jill Freeman, Andy Kenyon, Michael Hughes, and the presence of the late great Joe Frisco.\nPaul Zollo tells us, \"Gonna be an amazing night -- John M., Jill Freeman! Earl and Lisa! Me with my whole band!\"\n9:30 to Midnight ORCHESTRA CHARANGOA plays IXtapa Cantina, 119 E Colorado Bl, Pasadena 91105; www.latindancepro.com; 626-304-9930. No cover.\n8:30 pm JENNI ALPERT plays a gallery event at Broken Electric, 1413 Ocean Av, Santa Monica. Open to all.\n7-10 pm \"CONCERT FOR BANGLADESH\" redux on the anniversary date of George Harrison's landmark original in 1971, is a charity benefit at The Talking Stick, 1411 Lincoln Bl, Venice.\nFour decades ago, it was famine exacerbated by war. Now it's economic exploitation by rich Western corporate moguls who employ workers in dangerously unsafe workplaces. The results should have been preventable because they should have been predictable. Instead, the best we can do is help those in need after they have suffered horrifying tragedy.\nThus, tonight's \"BENEFIT FOR FAMILIES OF VICTIMS OF THE RANA PLAZE BUILDING COLLAPSE & THE TAZREEN FASHION FACTORY FIRE.\" On this exact 42nd anniversary date of the original \"Concert for Bangladesh,\" the proceeds this time support victims of the deathtrap buildings that housed garment-making sweatshops for top designer-label brands sold in the US.\nThe multi-artist evening will include highlights from the original concert, performed by host ROSS ALTMAN and guest artists that include Eric Charles, Dennis Davis, Jeff Gold, Andy Manoff, Carol McArthur, Smokey Miles, and Michael Simmons.\nAmericans pay a lot for clothes and shoes with trendy labels. Bangladeshis make sub-poverty-level wages to produce these products, often in unhealthy and physically dangerous workplaces.\nSuggested donation to attend the concert is $10 for the benefit, plus a one-drink or one-food-item purchase to support the venue.\n8 pm MICHAEL KOPPY, a respected and acclaimed singer-songwriter, plays the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675; reservations, 626-798-6236.\nMichael says he's bringing FRANK LINDAMOOD, \"an old pal who's coming all the way from the Florida swamps.\"\nMichael says, \"Other writers have called the album 'stunning,' 'revolutionary,' and 'unprecedented in the entire history of popular music,'\" though he does not identify the sources.\nAnother cut on the album,\"Nineteen Years Old,\" was the Number One Song on several South Carolina commercial AM stations for nine straight weeks earlier this year. Michael says, \"It's the true story of a love that never died, told him by the old man who lived it and loved her, over a couple of beers on a rainy afternoon in a little dive bar in Birmingham,Alabama.\"\nVenue impresario Bob Stane says of Koppy, \"He writes songs that actually say something\u2014be they off-the-wall and playful, perhaps think 'Randy Newman writes for Hank Williams Jr.,' or serious and substantive.\"\nHis current tour started in Canada on July 18 and ends in Honolulu August 10.\nFRANK LINDAMOOD has been called \"an authentic and unaffected link to a different time in American music.\" A regular at the \"Florida State Folk Festival,\" he lives in a place he built by hand (and without indoor plumbing), at the end of a dirt road deep in the turpentine piney woods of the Florida Panhandle, not too far from a little crossroads town called Sopchoppy.\n8-11 pm Monthly \"FIRST THURSDAY OLD-TIME MUSIC JAM\" hosted by TRIPLE CHICKEN FOOT at Bar 1642, 1642 Temple, downtown L.A. 90029.\nThe Foot says, \"Bring your instrument, learn to play old time, try out new tunes or get your fix! Beginners welcome.\"\nELI LOCKE & THE TWO TONE RAMBLERS is a country \/ rockabilly band that plays a wide range of tunes inspired by the likes of Waylon Jennings, Hank Williams, Johnny Cash and Elvis,as well as contemporary songs. The band takes a fast, high-energy approach to their music, mixing-in the occasional slow song. They harken back to the golden age of country music, while managing to deliver something new to their audiences at shows throughout the greater Los Angeles area.\nCrawfish Feast platter or Craw Bucket Feast, it's A LOT of food. And if you want to get serious, it takes a lot of delectably enjoyable time to eat it. What does that mean? A parent and a young kid can joyfully share a \"Crawfish Feast\" platter. Two adults can happily share a \"Craw Bucket,\" and a couple with a young child can buy one admission-with-bucket and just two additional admissions. A dating couple could share a \"feast\" platter and hit the food court for more and different cuisine (see below), or a couple can fill-up sharing a bucket, spending at least an hour-and-a-half busily eating it. But if teenagers are involved and you expect to share a platter or even a bucket, they will develop a sudden love for crawfish and devour everything, leaving you an 18-inch high pile of shells and nothing else. Finally, if you have your own crab \/ lobster cracking tools, take them. Crawfish have thinner shells and are smaller, but tools can make it easier. So plan a bit, and this will be a highlight of your summer.\n3-6 pm \"RESEDA RENAISSANCE -- CELEBRATION OF RESEDA'S '80s ROCK 'N ROLL HISTORY\" is a retrospective with current artists and a chance to participate, at Continental Art Supplies, 7041 Reseda Bl, Reseda 91335.\nTHE SINGER & THE SONGWRITER, the original duo of RACHEL GARCIA & THU TRAN. \"The Singer and The Songwriter is an L.A.-based duo that blends folk and jazz to create a catchy, modern sound inspired by vintage styles.\" They have played at The House of Blues, \"The Eagle Rock Music Festival,\" and Silverlake Lounge, and are currently working on their debut full-length album. Oh, and this original duo are former members of The Ukelele Orchestra of the Western Hemisphere.\nMORE is on the way. First, we must equip the drovers with new riatas and assemble them in their chaps to load the baying cattle into the stock cars and feed and water the dogies. Then we must assemble the loaded stock cars into the train, and coal and water the locomotive, and call the engineer and conductor, and attach the caboose. See what we do for your musical needs? And you thought we just got to blow the whistle, ring the big brass bell, and wave.\nWeb radio network FOLK ALLEY is broadcasting LIVE all weekend from the granddaddy of folk fests, the annual \"NEWPORT FOLK FESTIVAL\" in Newport, Rhode Island.\nGuitarist was a member of the influential folk band THE KINGSTON TRIO.\nProducer of social sitcoms including \"All in the Family,\" and owner of an original copy of the Declaration of Independence that he has generously exhibited to make us think about freedom from oppression.\nBeautiful video and music mogul who works with artists nobody has ever heard of but should.\nBeautiful and talented musician from SUGAR IN THE GOURD, who plays a 3 pm matinee today at the Coffee Gallery Backstage.\nSinger chosen as America's Most Talented Kid in 2003.\nFor Southern Californians: looking for concerts indoors & out, festivals, tuneful times, and more of the umm-umm goodness of summer? They're all in this rather huuuuuge edition.\nAnd goodness gracious Shirley Temple, the QUALITY of what's offered \u2013 especially today, Saturday \u2013 is akin to a celestial convergence! This ain't 'zactly a good time to sit home and vegetate.\n5th Annual \"JUNE LAKE LOOP MOUNTAIN MUSIC FESTIVAL\" includes country, folk, honky tonk, classic and alternative rock, and \"fine local talents,\" in various venues in the laid-back village of June Lake, in the Eastern Sierra of Mono County, west off US Hwy 395. There's music, food, dancing and fun in the scenic High Sierra.\nFestival events begin at 11 am Friday at Trout Town Joe's; officially, the festival runs 7 pm Friday through 7 pm Sunday.\nOddly, you can get there more easily from the west side of the Sierra than from connections on the east side, though Inyo County runs a bus from Mojave to Reno that gets you close. From the west side, Yosemite Regional Transportation System has convenient stops within the June Lake Loop. YARTS provides affordable and comfortable transportation to and from Yosemite National Park and connects to other Buses, Trains and airline services in Central California.\nFull-festival passes, for all three days at all venues, are $75. Individual tickets are available for each event in each venue. The \"Kids Music Camp\" runs all three days and is free.\nAnnual \"NEWPORT FOLK FESTIVAL\" in Newport, Rhode Island, with IRIS DEMENT, TROMBONE SHORTY, JIM JAMES. THE LUMINEERS, BONNIE \"PRINCE\" BILLY, COLIN MELOY, BLACK PRAIRIE, BETH ORTON, TIFT MERRITT, THE MOUNTAIN GOATS, SHOVELS & ROPE, ANDREW BIRD, SPIRIT FAMILY REUNION, THE FELICE BROTHERS, and more.+Held at historic Ft. Adams on the edge of Narragansett Bay, the Newport Folk Festival offers three stages of music, plus surprise performances around the grounds -- and history and heritage.\nAnnual \"LONG BEACH CRAWFISH FESTIVAL\" brings Cajun & zydeco and blues and top bands from Louisiana, and features STEP RIDEAUX, ANDRE THIERRY, BONNE MUSIQUE ZYDECO, THE ROYAL GARDEN DIXIELAND BAND, CYDRIL BALOU, REUBEN MORENO, and THE BAYOU BROTHERS, at Rainbow Lagoon Events Park, 400 E Shoreline Dr (between Shoreline Village Dr & Linden Av), Long Beach 90802.\nRuns noon-11 pm, both days. The absolute best thing to do to fully enjoy this one is to buy the advance ticket that includes the \"Crawfish Feast,\" cooked on-site. Don't wait to buy this when you arrive, as it's usually sold-out, and you'll be enviously watching others devour crawfish by the heaping platters and buckets full.\nAnnual \"MAMMOTH FESTIVAL OF BEERS & BLUESAPALOOZA\" with performances by MAVIS STAPLES, JOHN HAMMOND, DEANNA BOGART, DIEGO'S UMBRELLA, JIMMY THACKERY, MINGO FISHTRAP, TINSLEY ELLIS, JJ GREY & MOFRO, RUBY VELLE & THE SOULPHONICS, and more, at Sam's Wood Site, on Minaret Rd between Main St & Meridian Bl, Mammoth Lakes 93546. Runs Saturday, 11:30 am-9:15 pm, and Sunday, 8:45 am-6:15 pm.\n3 pm SUGAR IN THE GOURD and WHAT ABOUT NETTIE play a matinee at the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675; reservations, 626-798-6236.\nAs soon as you learn that one of these artists has toured as an opening act with WILLIE NELSON and opened for both WAYLON JENNINGS and JOHN SEBASTIAN, you might get the idea you should get to this show.\nNamed after a traditional foot stomping fiddle tune, Sugar in the Gourd is certainly every bit as edgy and fun, but has the added depth of beautifully constructed original songs that encompass topics from trains, to drinking, to love and redemption, all the way to God. The four women who comprise this delightful Americana.\/Roots style band, have combined their diverse backgrounds and songwriting talents to form a sound that is as unique as it is captivating. Whether they're singing an old blues, an original swing tune or an Appalachian style murder ballad, you'll hear three part vocal harmonies that will melt your soul, along with world class musicianship on violin, guitar, mandolin, upright bass and banjo.\nSugar in the Gourd was selected as an official showcase band for the 2012 FAR West conference, and have been playing consistently around the Los Angeles area in many of the top venues including McCabe's, The Coffee Gallery, and The Echo, as well as a four month residency at The Witzend Live in Venice. They have also been featured live on KCSN's \"Tangled Roots\" program.\nMerrily Weeber, on vocals and guitar, has toured as an opening act with WILLIE NELSON and also opened for both WAYLON JENNINGS and JOHN SEBASTIAN. She's been the featured vocalist for various songs in the hit sitcoms \"Third Rock From the Sun\" and \"That Seventies Show\". She has co-written with Jack Tempchin, (who penned the Eagles' \"Peaceful Easy Feeling), Janis Ian, and Lloyd Price.\nAubrey Richmond, on vocals and violin, has performing professionally since she was a young girl. She has won numerous championships over the years, as well as maintaining a highly successful international career as a top studio and touring musician.\nLisa Salloux, on vocals, mandolin, guitar and occasional bass, was born into an entire family who played bluegrass. She has been an accomplished musician , songwriter and vocalist since she was a teenager.\nLynn Shipley Sokolow, on upright bass and banjo, is also a published poet and lyricist selling over half a million records. In addition, she is a former entertainment manager who has worked with Geffen recording artists Quarterflash, and Berlin, with several gold and platinum albums on her wall.\nWHAT ABOUT NETTIE brings 23 year old Nettie Rose, songwriter, singer and actress, with original songs reminiscent of classic hits. She recorded her first song when she was only eight years old. Just one year later she met Fred Sokolow, who played with the likes of Jerry Garcia, The Doors, and countless others. He taught her how to play guitar, and later became her first writing partner.\n\"I've had the privilege to meet lots of people and often I turn them into characters in my songs,\" said Nettie. Her debut album is aptly titled People I Know. The CD Features several songs co-written and produced by three- time Grammy Award winner Anthony Kilhoffer (Kanye West, Rihanna, Kid Cudhi), and an impressive group of musicians including organist Spooner Oldham (Neil Young, Bob Dylan), drummer Bryan Owings and bassist Dave Roe (Johnny Cash). Nettie cites influences from Phil Ochs to Emmylou Harris to poet John Keats. Will Jennings said of Nettie's songs, \"She's got some special indefinable magic to her lyrics that connect with the common man in a universal way. She has a gift that's quite remarkable.\"\nShe is the daughter of legendary music publisher Lane Freed, President of Rondor Music, and Rock archivist Judith Fisher Freed, and the granddaughter of the iconic ground-breaking disc jockey Alan Freed, who is credited for coining the phrase \"Rock 'n' Roll.\"\nAdmission to this show is \"pay-what-you-want,\" unusual at this venue, and unusual for such accomplished musicians.\nYou can't go wrong with either of these acts. Together, this will be a delightful evening of splendid musicality.\n7 pm STRING MADNESS plays the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675; reservations, 626-798-6236.\nString Madness delivers string band music from Bach to Bop -- and Beyond. Three hot instrumentalists \u2013 mandolinist Bob Applebaum, and guitarists Mitch Greenhill and Peter Spelman \u2013 pool their talents here. The group has carved out a repertoire of original compositions, string band tunes, and jazz scripture from the Books of Django and Miles. Improvised solos interweave with arranged harmonies and unison lines to propel an exciting musical exploration.\nBob Applebaum is a nationally recognized mandolinist with over forty years professional experience in stage, recording, television and film work. He has performed and recorded with the likes of Bela Fleck, John Fogarty, Tony Trischka, Peter Wernick, Byron Berline, Stewart Ducan, Richard Greene, Mason Williams, and Sam Bush. Bob's CD \"All the Way Home\" was praised by Bluegrass Unlimited as \"the contemporary bluegrass album of the year\u2026\" He is coauthor of \"Fretboard Roadmaps for Mandolin,\" published by Hal Leonard.\nMitch Greenhill recorded two solo albums for Prestige, now re-issued on Fantasy as \"Shepherd of the City Blues,\" and two albums in collaboration with Mayne Smith. The Boston Globe hailed his \"movement and drive and touch that anyone who has ever coped with a guitar cannot but envy.\" He has produced an array of recordings, including a Grammy-winner for Doc Watson, and has composed and designed music and sound for regional theater, where he has received several Drama-Logue awards, and on Broadway. Mitch has appeared on screen in \"The Long Riders\" and \"Safe.\"\nPeter Spelman's extensive history as soloist and band leader includes stints in both Northern and Southern California. Like Bob, Peter cut his eyeteeth in the Greenwich Village musical cauldron, then moved west. He composes much of String Madness's repertoire.\nIn the words of the Los Angeles \"Wicked Good\" Newspaper, String Madness is \"a sonic lollapalooza, a mystical kinship of passions, a mind-meld of great empathy, a riveting feast of music....They swing, they dig deep, they dazzle, and they mean it all\u2026.Not to be missed! \"\n7 pm DOUG MacLEOD & DENNY CROY continue their fine acoustic blues tour of the several Lucille's BBQ restaurants, tonight playing the one at 2550 Park Av, Tustin 92782; 714-259-1227.\n7:30 pm DAVE ALVIN and EILEN JEWELL share the night at the FREE summer concert series on the The Pershing Square Downtown Stage, 532 S Olive St, Los Angeles 90013; 888-LA-PARKS.\nMany will see this one as the THE free show of the summer. DAVE ALVIN, \"The King of California,\" founder of THE BLASTERS, veteran of acclaimed tours with his bands THE GUILTY MEN and THE GUILTY WOMEN, and songwriter extraordinaire. And with him? Eilen Jewell is a reigning princess of Folk-Americana and a formidable musician.\nParking in the area is tough and can be expensive. Best to ride the Red Line subway to Pershing Square Station and you're right there. Pack a picnic dinner (no alcohol in the park).\n7:30 pm CAL PHIL's \"FESTIVAL ON THE GREEN\" summer series presents \"Dance Fever\" with Derek Hough, Emmy nominated choreographer & star of \"Dancing With The Stars,\" plus \"Tributes to Abba, Elvis, & Michael Jackson,\" and Dance of the 7 Veils, a Ritual Fire Dance, Hava Nagila, and Romeo & Juliet, in a very full program at Santa Anita Race Track, 285 W Huntington Dr, Arcadia 91007. Info and tix, 626-300-8200.\nGates open at 5:30 pm for pre-concert listening to live jazz, and for dining. Bring your picnic (alcohol permitted) or buy from on-site vendors.\nThe Guide has always embraced and celebrated accessible classical music. This fulfills that, masterfully. It's Cal Phil's 10th season of bringing great music to life, outdoors on Saturdays and indoors the next day.\nCal Phil's acclaimed outdoor concerts are \"picnic-under-the-stars\" events with amazing music at an historic venue with glorious views of the San Gabriel Mountains. Sit at a table with friends and family, or relax on the lawn, then enjoy the post-concert party and mingle with the Maestro.\nSUNDAY CONCERTS in the Walt Disney Concert Hall at the L.A. Music Center, 111 S Grand Av, Los Angeles 90012, are at 2 pm (go early for the talk at 1 pm - see below).\nSunday afternoons sparkle inside Walt Disney Concert Hall as Maestro Victor Vener and Cal Phil bring wonderful concerts to record-breaking audiences.\nThe Guide has raved about Cal Phil's concerts in both indoor and outdoor venues. Here are a couple of other comments.\nPops. All three... flawlessly combine classical and popular music.\" - Dennis Karmazyn, Internationally-Renowned Cellist.\ndo what they do\u2014and they do it very well.\" - Ted Ayala, Crescenta Valley News.\n8 pm MILKDRIVE brings \"progressive bluegrass\" from Austin to the \"Pasadena Folk Music Society\" (formerly Caltech Folk Music Society) series in Beckman Institute Auditorium (\"Little Beckman\") at California Institute of Technology in Pasadena 91106; advance tix, 626-395-4652.\nThis is your last chance to see them on their current national tour.\nJazz-grass band MilkDrive features award-winning instrumentalists Brian Beken on fiddle and lead vocals, Noah Jeffries on guitar and harmony vocals, Dennis Ludiker on mandolin and harmony vocals, and Matt Mefford on double bass. But awards reveal nothing about the Austin-based string band's exhilarating sound. Fueled by equal parts tradition and innovation, MilkDrive delivers a distinctive acoustic experience that crosses genres, geographies, and generations.\nThe band's much-anticipated sophomore studio album, \"WAVES,\" produced by Bil VornDick, was released in June 2012. It showcases the band's sophisticated arrangements and fierce musicianship with a number of original compositions, and it notably features guests Noam Pikelny (of Punch Brothers) and Futureman (of Bela Fleck & the Flecktones).\nDennis Ludiker, Brian Beken and Noah Jeffries met as youngsters competing in the prestigious \"National Oldtime Fiddlers' Contest\" in Weiser, Idaho. Eventually they migrated to Austin, where they toured with an eclectic mix of bands and honed their songwriting chops before joining up with bass player extraordinaire Matt Mefford and forming MilkDrive.\nThey play only three Southern Cal shows during their formidable national tour, and this is the final date.\nPark free in either lot at the S end of Michigan Av, off Del Mar, between Lake and Hill, S of the 210 Fwy.\n8 pm STAN RIDGWAY plays McCabe's, 3101 Pico Bl, Santa Monica 90405; 310-824-4497; www.mccabes.com.\n8 pm ROCKY NECK BLUEGRASS BAND plus THE BLADERUNNERS play Boulevard Music, 4316 Sepulveda Bl, Culver City 90230; 310-398-2583.\nIf you're a bluegrass fan, this is where you need to be tonight. A double-bill with these bands will feel like an intimate stage evening at a festival.\n8 pm CHICO TRUJILLO and LAS CAFETERAS share the night at \"Grand Performances\" at California Plaza, 350 S Grand Av, Los Angeles 90071; 213-687-2020.\nThe Guide was mighty impressed with and LAS CAFETERAS and their seldom seen (or heard) traditional Central American folk instruments at the rehearsal, and then the show, at the L.A. Music Center for the \"Annual Holiday Celebration\" last Christmas. A chance to catch 'em at a fun series for FREE in a fun place, where you can ride the Red Line subway and then Angel's Flight funicular to the outdoor venue? All the ingredients of an exceptional night out.\n8 pm JIMMY THACKERY & THE DRIVERS play the Arcadia Blues Club, 16 E Huntington Dr, Arcadia.\nVenue has very reasonable cover charge, good food with generous portions and full bar with specials. But take earplugs. Seriously.\n10 am-noon \"DOWNTOWN TORRANCE CRAFT FAIRE\" brings a two-hour performance by STACY ROBIN joined by ED GOULD featuring originals, covers, and music from her newest release, \"All The Way Home,\" at 225 El Prado Av, Torrance 90501.\nFree. Event features food, crafts, fun, and live music.\nStay for a set by the WONDROUS APPARITIONS, the band of DONNA RAWLINS, ROBERT FEIST, MICHAEL HAPROV, ANTHONY GALE, and FRANK NEMIROFF.\n2 pm CAL PHIL's summer series presents \"Dance Fever\" with Derek Hough, Emmy nominated choreographer & star of \"Dancing With The Stars,\" plus \"Tributes to Abba, Elvis, & Michael Jackson,\" and Dance of the 7 Veils, a Ritual Fire Dance, Hava Nagila, and Romeo & Juliet, in a very full program in the Walt Disney Concert Hall at the L.A. Music Center, 111 S Grand Av, Los Angeles 90012. Info and tix, 626-300-8200.\nSUNDAY CONCERTS are at 2 pm, but go early for the talk at 1 pm - see below.\n5:30-9 pm Weekly \"GRAND OLE ECHO\" Americana \/ alt-country \/ honky-tonk showcase at The Echo, 1822 Sunset Bl, Echo Park 90026; 213-413-8200.\nThis is ALWAYS a safe bet for a music-centered good time. For starters, it's FREE, and all ages, and has a full bar and a basic BBQ available out back, and the bands and musicians booked for the series are inclusive of folks you'd pay appreciable money to see one at a time. If you've been you know all that. If you haven't. go. Free.\n6 pm ROCKY NECK BLUEGRASS BAND plays the FREE summer concert series at California Citrus State Historic Park, 1879 Jackson St, Riverside 92504; 951-780-6222.\n7 pm DAVID LINDLEY and TOULOUSE ENGELHARDT, two formidable string wizard talents, team-up to play Saint Rocke, 142 Pacific Coast Highway, Hermosa Beach 90254; 310-372-0035.\n7 pm MARK LEMAIRE & TWILIGHT, in a double-bill with JOHN BATDORF, at the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675; reservations, 626-798-6236.\nAcoustic duo Mark Lemaire and Twilight have taken the tradition of guitar and interlocking harmonies to a unique place. Bob Stane says \"I was fascinated by 'Zanzibar Sands'-- it has a unique voice and story line. After his main set ended, I did something I have never done in 50 years of putting on shows: I asked Mark if he would sing it one more time. As he did, I enjoyed its uniqueness all over again.\" Mark Twang of Folkways Recordings says: \"It was an impeccable performance. Never have I witnessed such control of the steel stringed guitar's sound. This was a flawless show.\"\n7:30 pm TAKE 2, with Hawai'ian music greats HERB OHTA JR. and JON YAMASOTO and KEOKI KAHUMOKU, play the AMSD Concerts series at 4650 Mansfield St, San Diego 92116; 619-303-8176.\n7 pm \"KLEZMER MUSIC J.A.M. Session\" is this week's FREE offering at the John Anson Ford Amphitheatre, 2580 Cahuenga Bl East, Hollywood 90068; 323-461-3673.\n7:30 pm THE BROMBIES play their weekly BLUEGRASS residency at Viva Cantina, 900 Riverside Dr, Burbank 91506; 818-515-4444.\nNoon-5 pm \"TAYLOR GUITAR SPECIAL EVENT\" at McCabe's, 3101 Pico Bl, Santa Monica 90405; 310-824-4497; www.mccabes.com.\n\"Some great sale pricing,\" is promised, along with advice from Taylor reps and a special \"find your fit\" session which matches the right guitar to your playing needs.\n7 pm SHELBY & TIEG play an unusual midweek show at Fiddler's Crossing, 206 East F St, Tehachapi 93561; 661-823-9994.\n8 pm RUN BOY RUN, in the midst of their national tour, play The Coffee Gallery Backstage, 2029 N Lake Av, in Altadena. Phone right away for reservations before all the seats are gone -- 626-798-6236 (phone 10 am-10 pm daily; no online reservations).\nThis booking occasioned a special feature story and \"Ticket Alert\" a week ago Monday in the Guide. Here's that story \u2013 you want to be there.\nThey're a hot bluegrass \/ old-time band who've TWICE performed on radio's \"A PRAIRIE HOME COMPANION.\" They won the ultra-prestigious \"Telluride Bluegrass Festival Band Contest.\" And they are playing only ONE Southern California show -- in a small venue -- during their national tour. This'll sell-out QUICKLY, so jump on it.\n\u2605 RUN BOY RUN is an exciting young band from Tucson, Arizona. They have a special gift for writing original songs in the classic old time \/ bluegrass styles, and feature a unique instrumental mix of twin fiddles, mandolin, cello and bass, along with three female lead singers with beautiful harmonies.\nGarrison Keillor, host of \"A Prairie Home Companion,\" didn't just book them twice; he raves about them: \"Hot instruments and beautiful, sweet harmony singing - that's all you need in the world today! When I hear Run Boy Run, it all comes back to me, why I started doing that show back then. I hope they go on forever.\"\nVenue impresario Bob Stane says, \"Existing in the tension between tradition and frontier, Telluride Bluegrass Festival Band Contest winner and Two Time Prairie Home Companion guests RUN BOY RUN truly exceeds the sum of its parts as touches of classical, jazz, and folk express themselves through the old-time core of the band's unique sound.\"\nThe band's members include two sets of siblings. Brother and sister Matt Rolland (fiddle & guitar) and Grace Rolland (cello & vocals), combine with sisters Bekah Sandoval (fiddle & vocals) and Jen Sandoval (mandolin & vocals). The band is completed by Jesse Allen (upright bass).\nCalled by Paste Magazine, \"one of the top ten Arizona bands to hear now,\" Run Boy Run self-released their full-length album \"So Sang the Whippoorwill\" in March 2013 to critical regional and national acclaim.\nFrom winning the band contest at \"Pickin' in the Pines\" in the first weeks of forming in 2009, to playing the main stage at the \"Telluride Bluegrass Festival\" as band contest winners in 2012, to making a second appearance on \"A Prairie Home Companion\" in 2013, Run Boy Run is spreading their signature instrumentation and sweet three-part female harmony vocal music and youthful energy nationwide.\nBrother and sister Matt and Grace Rolland have fiddle music running through their blood, raised with bow and rosin on every bedside table. The group's three female lead singers, sisters Bekah and Jen Sandoval, grew up deeply rooted in the bluegrass festivals of Arizona, so they too found a home in Run Boy Run. Bass player Jesse Allen was brought-up on opera and western swing, and grew to love the natural fusion inherent in American music. They're all solid, instrumentally, in a lavish string mix for the genre. The band's \"whole\" exceeds the sum of its formidable parts, incorporating touches of classical, jazz, and folk through the old-time core of Run Boy Run's sound.\nIn addition to counting Garrison Keillor among their fans, it has been widely ibserved that Run Boy Run brings a special gift for writing original songs in the classic old time\/bluegrass style, featuring a solid instrumental mix of twin fiddles, mandolin, cello and bass.\n8 pm DR. JOHN, New Orleans' amazing impresario of the ivories who often plays two pianos with keyboards facing each other, plays the Hollywood Bowl, 2301 N Highland Av, Hollywood 90068; 323-850-2000 or 213-480-3232.\n8 pm COW BOP, that marvelous western swing \/ bebop jazz combo, plays Viva Cantina, 900 Riverside Dr, Burbank 91506; 818-845-2425.\n8 pm TIM WEISBERG, a huge international star, returns for an encore performance at the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675. Reservations (STRONGLY recommended) at 626-798-6236.\nKoffeehouse showcases L.A. artists all summer long in the newly renovated main outdoor lobby lounge. Admission is FREE.\nHappy hour specials and discounted valet parking all summer long.\nTHIS, as-promised, is a late update with BUNCHES O' LATE ADDITIONS to this morning's \"tide-ya-over quickie edition.\" STILL coming later this evening is our \"Last Days of July, and into August \u2013 Music for July 27-31 and beyond,\" with tons 'n tons o' concerts indoors & out, festivals, tuneful times, and a whole lot more of the umm-umm goodness of summer. That'll be along in a rather huuuuuge edition.\nFestival events begin at 11 am Friday at Trout Town Joe's; officially, the festival runs 7 pm Fr-7 pm Sun.\n10 & 11:15 am THE GET DOWN BOYS bring two sessions each day of roots-Americana to the \"Summer Sounds: World Music for Kids\" series at the Hollywood Bowl, 2301 N Highland Av, Hollywood 90068; 323-850-2000 or 213-480-3232.\nThis is the absolute best opportunity to give your kids an immersive experience in Folk-Americana music this summer.\n6-10 pm Weekly \"MONROVIA FAMILY FESTIVAL\" I old town Monrovia (San Gabriel Valley) includes live music, a vast variety of food offerings, carnival rides, kids activities, and the stalwart \"KATTYWOMPUS CONCERT \/ JAM\" of traditional music practitioners who jam acoustically at 8 pm with all comers at Dollmakers Kattywompus, 412 S Myrtle Av, Monrovia 91016; 626-357-1091.\n7 pm DOUG MacLEOD & DENNY CROY play Lucille's BBQ, 7411 Carson St, Long Beach 90808; 562-938-7427.\n7:30 pm EILEN JEWELL plays the AMSD Concerts series at 4650 Mansfield St, San Diego 92116; 619-303-8176.\nEilen Jewell is a reigning princess of Folk-Americana and a formidable musician. This series offers some options with dinner.\nBut first there's tonight. BOB MALONE is keyboard player for JOHN FOGERTY and a fine performing songwriter \/ composer with several top-notch albums to his credit. He just came off a European tour with Fogerty.\n8 pm \"THE SONGS OF HANK WILLIAMS\" are being celebrated at the Will Geer Theatricum Bir anucun in Topanga.\n\u266a The Pasadena Folk Music Society series at Caltech on Saturday.\n8 pm KLEZMER JUICE plays the free summer concert series at the Levitt Pavilion for the Performing Arts, 85 E Holly St (on Raymond between Walnut & Holly), Pasadena 91103; 626-683-3230.\n8 pm KRIS COLT, DEANNA D'AMICO, MAUREENA DANICUR, & ANITA MANSFIELD team up for a formidable night of the \"Ladies of Celtic & Country\" at the Coffee Gallery Backstage, 2029 N Lake Av, Altadena 92675; reservations, 626-798-6236.\nThis is an encore performance of \"An Evening in the Round, with the Ladies of Celt & Country.\" All four women are uber-talented singer-songwriters and successful band leaders, each in her own right. Tonight, they take turns on stage and back each other, Nashville style. Venue impresario Bob Stane notes, \"Two blondes, a redhead, and a brunette!\"\n8:30 pm Annual \"KOLLABORATION LOS ANGELES: ASIAN PACIFIC ISLANDER TALENT COMPETITION\" with emcees David & Ray Choi at the John Anson Ford Amphitheatre, 2580 Cahuenga Bl East, Hollywood 90068; 323-461-3673.\nA full NEWS & REVIEWS edition is due within the next few days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A quiet day today\u2026 this morning, Maureen went downtown to help at the dog shelter, I stayed at home and worked on my controllers. I decided for once to take a proper holiday and it is paying off \u2013 I'm feeling quite relaxed.\nThis evening we sat outside and enjoyed a nice tapas in the Pergola \u2013 despite forecasts of rain the weather has been great. Not stunningly hot, maybe 25c but with the sun blasting I took the opportunity to supplement the tan for a while. This has to be the first trip we've done where there are no pressing repairs to attend to \u2013 marvellous.\nTV is on, she's watching soaps and I'm just putting the finishing touches on some code that has been months in development and which is now about to get it's first trivial job \u2013 watering the plants. Thanks to a friendly local policemen we now know what to do with the speeding fines and it turns out we got the early payment discount on one, once we figured out how to use the Spanish website. It appears the back route to Murcia, despite looking innocent is riddled with police with cameras as others have suffered the same fate. Once bitten twice shy (or in my case, twice bitten\u2026).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I bought some lovely white agate nugget beads. However, many of them look like their edges are dirty. Can you tell me the best way to clean these beads?\nUse a gentle liquid cleaner to release what may appear to be dirt from the thread the beads were strung on. Follow package directions with any cleaning products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nutrition Policy: Guidelines and objectives pertaining to food supply and nutrition including recommendations for healthy diet.Food Services: Functions, equipment, and facilities concerned with the preparation and distribution of ready-to-eat food.Nutritional Physiological Phenomena: The processes and properties of living organisms by which they take in and balance the use of nutritive materials for energy, heat production, or building material for the growth, maintenance, or repair of tissues and the nutritive properties of FOOD.Schools: Educational institutions.Child Nutritional Physiological Phenomena: Nutritional physiology of children aged 2-12 years.Diet: Regular course of eating and drinking adopted by a person or animal.TexasHealth Policy: Decisions, usually developed by government policymakers, for determining present and future objectives pertaining to the health care system.Parenteral Nutrition: The administering of nutrients for assimilation and utilization by a patient who cannot maintain adequate nutrition by enteral feeding alone. Nutrients are administered by a route other than the alimentary canal (e.g., intravenously, subcutaneously).United StatesPolicy Making: The decision process by which individuals, groups or institutions establish policies pertaining to plans, programs or procedures.Parenteral Nutrition, Total: The delivery of nutrients for assimilation and utilization by a patient whose sole source of nutrients is via solutions administered intravenously, subcutaneously, or by some other non-alimentary route. The basic components of TPN solutions are protein hydrolysates or free amino acid mixtures, monosaccharides, and electrolytes. Components are selected for their ability to reverse catabolism, promote anabolism, and build structural proteins.Nutrition Surveys: A systematic collection of factual data pertaining to the nutritional status of a human population within a given geographic area. Data from these surveys are used in preparing NUTRITION ASSESSMENTS.Enteral Nutrition: Nutritional support given via the alimentary canal or any route connected to the gastrointestinal system (i.e., the enteral route). This includes oral feeding, sip feeding, and tube feeding using nasogastric, gastrostomy, and jejunostomy tubes.Organizational Policy: A course or method of action selected, usually by an organization, institution, university, society, etc., from among alternatives to guide and determine present and future decisions and positions on matters of public interest or social concern. It does not include internal policy relating to organization and administration within the corporate body, for which ORGANIZATION AND ADMINISTRATION is available.Policy: A course or method of action selected to guide and determine present and future decisions.Nutritional Sciences: The study of NUTRITION PROCESSES as well as the components of food, their actions, interaction, and balance in relation to health and disease.Nutrition Therapy: Improving health status of an individual by adjusting the quantities, qualities, and methods of nutrient intake.Nutrition Disorders: Disorders caused by nutritional imbalance, either overnutrition or undernutrition.Nutrition Assessment: Evaluation and measurement of nutritional variables in order to assess the level of nutrition or the NUTRITIONAL STATUS of the individual. NUTRITION SURVEYS may be used in making the assessment.Child Nutrition Sciences: The study of NUTRITION PROCESSES as well as the components of food, their actions, interaction, and balance in relation to health and disease of children, infants or adolescents.Parenteral Nutrition, Home: The at-home administering of nutrients for assimilation and utilization by a patient who cannot maintain adequate nutrition by enteral feeding alone. Nutrients are administered via a route other than the alimentary canal (e.g., intravenously, subcutaneously).Environmental Policy: A course of action or principle adopted or proposed by a government, party, business, or individual that concerns human interactions with nature and natural resources.Nutritional Status: State of the body in relation to the consumption and utilization of nutrients.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BREAKING NEWS - We have been awarded the Science GOLD Quality Award! Only 347 schools in England (out of 24,000) have been awarded this due to their science curriculum and extra-curricular science and enrichment activities. Congratulations to Miss Smith, Eco Club, Science Club and all pupils and parents that have engaged with our science curriculum during the last academic year. Thank You!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"hello, just want to share some new pictures of one of my females taenicauda.\nnice tail markings too :) !\nI agree, excellent specimen Stefan.\nbeautiful Stroph Stefan. really really nice markings.\nwow stefan! Absolutely gorgeous. Taens are by far my favorite stroph.\nNice photo! Glad to hear they are doing good for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CAMIONES Y VOLQUETES JINBEI , VERSION TURBO DIESEL INTERCOOLER , DIRECCION HIDRAULICA , CORONA INTERMEDIA .\n*LA FIDELIDAD DE NUESTROS CLIENTES ES NUESTRA MEJOR CARTA DE PRESENTACION .\nSOLICITE SU COTIZACION SIN COMPROMISO.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today's column is written by Ari Paparo, CEO at Beeswax.\nLet's start with a disclaimer: I'm no expert in GDPR (General Data Protection Regulation). If I were, I'd be billing by the hour for my advice and making a killing. So feel free to stop reading now.\nBut I do know a thing or two about ad tech and how different kinds of companies use data. And I've been a bit frustrated reading all the \"GDPR kills ad tech\" articles that claim doom but don't really go into any level of specifics about how, exactly, the apocalypse will play out.\nSo, let me be so bold and call out some of the strategic implications of GDPR and discuss the likely winners and losers as Europe moves to a much stricter regulatory environment.\nAnother disclaimer: This is meant to paint with broad brushes, so please don't tweet about how I got the definition of \"consent\" wrong in Luxembourg or how \"ePrivacy actually changes all of this.\"\nThe two most important considerations for the impact of GDPR are 1) how to get consumers' \"consent\" to use their data and 2) whether a given company is a \"data processor\" or a \"data controller.\"\nVery roughly speaking (remember, don't tweet at me!), you are a data controller if you own the data in some form or decide how to use the data. You are a data processor if you just move the data around for your customers and take no control. If you are a data controller, you need explicit consent from the end user and have much higher standards for security and control.\nThe truth is that brand advertisers often have surprisingly little first-party data about their consumers, and the new requirements to get consent are unlikely to change their programmatic buying habits. Business as usual.\nMar tech is a very big sector, so at the risk of overgeneralization I'll say that most companies in this area are clearly data processors and take orders on how to process customer data from their customers.\nIf you are an email provider, a social metrics dashboard or a CRM-like system, you are primarily working with first-party, permissioned data. Further, your customers are usually in direct contact with the end user and can viably ask for consent. For these companies, surely there will be increased security and regulatory hassle, and the total volume of data may take a hit, but the fundamental business remains sound.\nAttribution and analytics vendors fall within the same rough outline as mar tech vendors, and they should be fine. There may be more gaps in the data as consumers deny consent, but the fundamentals are intact.\nThe reduction in data available to buyers should reduce programmatic media prices, which should reduce revenue to publishers. However, the countervailing point of view is that buyers will have fewer media options and will have to turn to publishers' second-party data or contextual data to achieve their goals, thus increasing revenue to publishers.\nAmong all the players, the effect on publishers is the most unclear.\nSimilar to mar tech vendors, typical ad servers process data on behalf of their customers, who will likely have to ask consumers for consent. Sell-side ad servers likely are not affected much at all since the publishers they work with will have a relationship with the consumer.\nBuy-side ad servers and rich media companies, however, regularly collect personal data on sites and users for which they do not have direct consent. In a strict reading of the regulations, activities such as delivering log files, cross-site frequency capping and collecting user information could be seriously degraded.\nDMPs collect, process and analyze first-party customer data. If the first party has consent, they're in the clear. But similar to ad servers, collecting passive data from ad delivery and other sources will be degraded by the need for consent.\nI've personally spoken to two leading SSPs\/exchanges that gave radically different points of view. One leading exchange boldly told me, \"We're a processor, so nothing to worry about.\" Another asked me to sign a new and onerous 10-page contract.\nThe bottom line: If an SSP is just executing auctions on behalf of publishers, they should be minimally affected. If an SSP is also overlaying data, they could be prevented from doing so in Europe \u2013 but who gets data from an SSP anyway?\nCore DSP services are in the same bucket as SSPs \u2013 processing trades on behalf of customers \u2013 so there is little need to get direct consent. However, many DSPs have developed proprietary data sets or cross-device graphs, and these will be very hard to maintain under the new regime unless you're Amazon or Google. DSPs may also be required to curtail services, such as log delivery and lookalike modeling, in the same way as buy-side ad servers.\nSo, your business collects user data from lots of different online and offline sources, then combines and sells it to different parties across the ecosystem? I think we've found patient zero for GDPR compliance. Sure, you can get all your data sources to obtain consent, but when the data is literally the lifeblood of the business, any degradation in collection will hit the bottom line linearly.\nLike retargeters, direct-response advertisers rely on data to get results, so a reduction in data is not in their favor. However, they can employ many channels and strategies to drive their KPIs and may be able to shift spend to retain ROAS in this more difficult environment. Unclear overall.\nContent recommendation engines: \"We are the most hated sector in ad tech.\"\nA lot of hot air is coming from the retargeting sector in advance of GDPR, probably because they know that every consumer who denies consent is directly tied to revenue loss. Regardless of the interpretation of consent, it is a sure bet that some retail customers in Europe will choose to stop working with retargeting vendors to avoid risk, and some (or many) customers will opt out. No bueno.\nThey say you shouldn't watch sausage get made, and ad networks are like, \"Don't worry about it, have a currywurst.\" Well, we're going to regret it tomorrow when the opaque data filling the delicious natural casing goes missing and we're left with nothing but offal and spicy sauce*.\n* This is clearly the last time I'll be invited to write for AdExchanger.\nHey dude, where'd you get that location data from? Did you get consent? I didn't think so.\n\"Probabilistic graph\" becomes a euphemism for \"no consent.\"\nReally not sure how blockchain is relevant here, but if you've read this far I'm sure you'll buy my coin; I call it ICOnsent\u2122.\nNothing like ad tech to bring out the new vendors. There's definitely upside for vendors managing cross-vendor consent, but they only get two baguettes because, ultimately, it's not that large of an opportunity.\nNo consent needed to tell me I'm on a site about cool stuff. As user data declines in ubiquity, site data increases in value. This is a no-brainer.\nWhile the GDPR movement was largely meant to tamp down the power of the American tech giants, it will likely have the opposite effect because their direct consumer relationships allow for meaningful dialogs about obtaining consent.\nI bet you can't wait until the first multimillion-dollar lawsuit is filed for GDPR violations. Neither can they.\nFollow Ari Paparo (@aripap), Beeswax (@BeeswaxIO) and AdExchanger (@adexchanger) on Twitter.\nNice summary. Emphatically agree with the expected windfall to contextual targeting vendors...bring back the vertical ad networks!\nI think GDPR, combined with a general frustration with the growing complexity of what it takes to execute and validate a display ad buy is going to strongly push marketers to revisit how they spend their dollars and begin re-embracing quality publishers offering targeting based on the nature of their content.\nSimplicity isn't a dirty word.\nGreat article Ari! Very entertaining & informative in the same time.\nNailed it Ari. Sign me up for an ICOnsent. Another layer that concerns me is not just the initial targets of enforcement, but who they bring down with them. Most of the approaches that I have seen have dumped responsibility on the publisher (or the third party one step closer to the publisher). I expect that some businesses with more than 3 flaming Euros will choose breaking the law over taking the revenue hit from compliance. As a representative of smaller publisher who are mostly ill equipped to enforce compliance on their partners, I'm most concerned that the threat to them comes from the contractual finger pointing once the bad actors get fined.\nWell said, and I agree with much of this. The one thing you overlook, in mentioning contextual, is that (crazy though it is) even contextual ads will require consent because GDPR applies its crazy strict consent standard to the ePrivacy Directive, which in turn applies to contextual ads too -- unless you avoid any cookies for frequency capping, etc., which becomes a pretty crappy user experience and will thus devalue THAT inventory, too. Maybe all platforms should shut their EU offices, move their employees to the US, and tell EU regulators to come and get them here --which they won't do.\nHey Ari, really fun to read. Hope this is not true: * This is clearly the last time I'll be invited to write for AdExchanger. Wondering what is your assessment for link retargeting (http:\/\/blog.retargetlinks.com\/link-retargeting\/) as there is no way to get consent on a link before or after the person clicking the link gets tagged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GiveIndia is a donation platform that allows you to support a cause of your choice from about 200 NGOs that have been scrutinised for transparency & credibility.\nThey ensure that at least 90% of your contribution reaches the organization you support.\nDonations qualify for TAX exemption under section 80G for Indian donors.\nCheques may be posted to the address below with a memo clearly stating that the donation is meant for Sankara Eye Hospital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delivery to East Coast capital and major metropolitan areas, are within 2 - 8 working days. Often delivery it is considerably faster than the longest time shown, however no guarantee can be made.\nDelivery to East Coast regional centres, outside of capital and major metropolitan areas and Adelaide, allow at least 15 working days. These time frames are worst case scenario time frames, however need to be accounted for.\nUnfortunately we don't deliver to Northern Territory, Tasmania, Western Australia or outside of the Adelaide Metropolitan area or Far North and Regional Queensland.\nThe Clothesline Store can not guarantee specific delivery times as this would be irresponsible, because we do not own, control or manage any freight companies. However we do use the most reputable couriers in Australia and despatch products in a very rapid manner.\nMade to Order Clotheslines will take longer to be delivered as these are specifically manufactured to suit your order. Please consider this when placing your order. Estimated lead times are on the product pages. Also it's important to note made to order clotheslines cannot be returned.\nIf your clothesline has not arrived by the latest estimated time frame then please contact The Clothesline Store directly. However any product signed for with the courier or transport company, we are sorry but we can not assist you any further.\nThe free shipping policy The Clothesline Store offers, (if applicable at time of ordering) and is for delivery to most major cities in Australia. The transport company's policies will not always allow for delivery to remote areas. It is best if you call The Clothesline Store before you place your order, if you are not sure if you live in a remote location.\nThe transport company's door-to-door delivery is not always available in some regional areas. Please be aware you may have to pick up your clothesline up from the transport companies closest depot, or incur additional delivery charge, if the transport company's policy allows for this service.\nAll products are generally processed in the same business day and shipped within 24 business hours unless otherwise stated.\nOnce your clothesline has been dispatched you will receive an email with your tracking number and freight company details, this will make it easier for you to know when to expect delivery of your clothesline. (This does not apply for Austral products).\nThank you for choosing The Clothesline Store.\nMade to Order - City Living Branded Clotheslines Cannot Be Returned.\nYou may return most new, unopened products within 30 days of delivery date for a full refund less the original freight cost, provided they are in a resale-able condition. A resale-able condition is essentially the condition you would expect to receive the goods in when you purchase them at full price. We don't cover change of mind or products which have been opened or trial installed. If the return is a result of an error by The Clothesline Store i.e. incorrect product, we will cover the shipping costs both ways. Returns for any other reason will be payable by the customer unless outlined above.\nPrior to returning any products contact The Clothesline Store to for a return request. At the time of the return request you will be notified of the return address as the product will need to be returned to our warehouse.\nThere is a return fee for products being returned which is generally $25.\nYou should expect to receive your refund generally within two weeks (10 working days) after your product has been received, however, this is totally dependent on when the courier returns the product to us.\nIf you have received a faulty item, simply contact us at Sales@theclotheslinestore.com.au and quote your order name and number. We will then contact you to discuss the corrective action.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Collierville connotes an easy living suburban lifestyle with affordable spacious housing and a good selection of shops and restaurants. Collierville also offers a unique sense of place with it's old town square and historic district. The infamous fair on the square takes place every May. Check out my showcase of Collierville homes below or create your own search for your ideal Collierville home using my advanced Memphis & Collierville MLS home search feature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A piece showing how they make the front covers. Featured on Orbit's books.\nWhy yes indeed it has! We have been introduced to\u2026 The Scotts! In this installment of the parasol protectorate comes a new mystery, a cast of fanatical new characters, and traveling by dirigible.\nFollowing quickly on the tail of Soulless, Changeless starts several days (to weeks) after the wedding when life in the Maccon household is getting into full swing. The troops have returned from fighting in the East, many of which are werewolves returned to the English pack, a new plague has fallen upon the city, and Lady Maccon has been instructed to visit a brand-new hat shop.\nAs mentioned above, the book is full of new mysteries to be solved \u2014 new friends and allies made thus. A cast of wonderful new characters including a lady inventor and some wonderfully hilarious Scottish werewolves and their Lady. As well as Lady Maccon's first ever trip by dirigible.\nAlthough this story is stand alone, it is also the beginning of a bigger plot and arch set up for the rest of the series. It leaves loose ties at the end of the novel for us to wonder what comes next.\nOverall I enjoyed this continuation of Alexia's story and can not wait to sink my teeth into the next novel, of which I have already begun.\nAlexia Tarabotti and Connal Maccon by Ulalah on dA.\nShall I begin to talk about this book? Well let me say I love, love, LOVED it! It was fantastic and what an amazing read after the Rising. If I had to tag line it (which I did in fact for my local libraries summer reviews) I would say: \"A fun and delightful read of a woman with astounding manners and a great sense of the sensible.\" A bit silly, believe me I know, but I feel like it describes it quite well. Only topped by two other taglines, number one being this: \"Soulless is a character-driven romp with great worldbuilding and delicious rapier wit that recalls Austen and P.G. Wodehouse.\" Number two being the tagline on the front of the book: \"A novel of vampires, werewolves, and parasols\" which is the one that got me at the book store. I've asked myself a dozen times which it has taken me so long to find this delightful novel seeing my love of steampunk, supernatural, romance, and woman with wit! Well, none-the-less, the series now lies upon my bed side table where I am eating them up like tea cakes!\nThe first installment of this series is Soulless which follows the main female lead, Alexia Tarabotti, on her adventures in high London society as a spinster. She has some interesting acquaintances including two werewolves and a rove vampire friend. Alexia always has her handy dandy parasol at the ready for any unfortunate accidents, which as a supernatural herself, seem to be drawn to her on a regular basis. The language of the novel is playful and full of wit and english language, that which derives from England. It often reminded me of that Southern Woman lines meme that have been running rapid recently.\nThis story has a fun plot that is constantly being pushed forward full of mystery, suspense, adventure, and romance. The romance is a large part of this novel, as much as any other part, and would not be what it was without it. The romance aspect, is also which pushes this novel from upper young adult to that of adult, so be a bit weary of that.\nThe characters are multi-dimensional and no two are the same. Carriger also takes a great interest in describing era based clothing, class system, vehicles, and cutting edge science of the time. Carriger carries on a great storyline, alternative history that makes sense and is backed by a large amount of research, as well as believable characters. Honestly, I would love to live in this world that Carriger has created. All in all a great book that I highly recommend!\nPs. Enjoy some other images that didn't make the cut this time around.\nThe Looking Glass Wars first caught my eye upon the bookshelves of Powells while I was picking up some used books for my reading pleasure (the Calling and Clockwork Princess from my earlier reviews being two of those others from that trip). I've always been intrigued by different versions and re-tellings of the Alice in Wonderland story as I plan to one day do one of my own. I remembered that some of my friends had read this particular novel growing up and been intrigued by the ideas it presented. I picked up the book and began to flip through the pages, quickly surprised and pleased that the novel is filled with concept artwork and sketches of the characters and the environment in which they lived. I was thrilled.\nUpon taking the book home and beginning to read it, I quickly realized that I was probably too old (at 22) to be as enthralled with this book as its intended audience. Non-the-less I read it all the way through and thus am reviewing it today.\nOverall the character designs, pacing, and world building is extremely well done. Where this falls short is story arc, believability, and the forced (at times) romance. Even though it was Wonderland, there were times when even I caught myself making a small half-smirk at the book and thinking Did that really just happen? Had I been my younger self I would've laughed at the thought of it, gleeful in its existence. One thing I will give Beddor though, other than all the amazing things I have already mentioned, was that his research was quite in-depth. At the end of the book is a timeline of events in Wonderland side-by-side with events in London, England. He had a historical starting point for this novel and he stuck closely to it. Other areas where I found it a bit short were language, but this makes sense for its intended audience.\nI myself will probably not be picking up the sequels, but I can easily see Disney asking for the rights to this novel to make, yet another Alice movie spin-off.\nAs always my opinions are my own and no one else. Please check out other related content below.\nTessa Gray quote from Cassandra Claire's series The Infernal Devices.\nI recently finished the novel Clockwork Princess by Cassandra Clare. This was the third and final book in the trilogy the Infernal Devices. Currently, I'm torn on the ending. I feel like it could have been better, but every time I sit down to think of how it could have been changed, I'm at a loss for words. I enjoyed the fact that in Clare's author's note she mentioned that since the beginning of the series she had the final scene in mind. This gives me hope for the ending. But, I have chosen to continue to sit on it for a few more days before declaring a true opinion of it.\nI read the first 250 pages in two days, and then began to slow down. I felt the book did a lot of things it needed to; wrapped up a lot of endings and opened new questions making me ponder reading Clare's first series, the Mortal Instruments. I'm pondering doing this in order to tie some ends up, as well as see if any of the characters are referenced, or re-occurring in this later alternate universe.\nI felt the craft at play continued to be strong, as it had in her earlier books, but at times the text was slow. There were amazing quotes in it, as always, one can be viewed above. Clare truly did her research of London, and her use of languages, Latin, Chinese, Welsh, etc. was inspiring. Clare's universe that she has painted intrigues me in a way that it's inspiring.I've also read an announcement on Cassandra Clare's tumblr page that she plans to write two more Shadowhunters' series. I personally, can not wait. Also, check out the tarot card set that will be coming out on Clare's tumblr page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am currently working at KPMG as a senior auditor, and simultaneously is finalizing my post-bachelor degree in Auditing. Although I am fun and outgoing, I cannot live without her calendar.\nWhy did you enter the national Miss TBS pageant?\nAs long as I can remember I wanted to experience a pageant. My priorities after high school shifted to study and work. However the adventurous part in me never forgot about entering a pageant. Last year I was ready to embrace the challenge. I looked at different national pageants and Miss TBS was the one I could relate the most. The pageant did not only focus on outer beauty, but also on personal growth, the Surinamese community and social issues.\nDescribe the moment you became the new Miss TBS?\nThe first thing I felt was a sense of relief. The journey was long and that night in particular was very long. The next thing I felt was pride. My parents and family where watching all night and the first glare I got from the audience, after I heard my name, was that of my parents looking at me. That's when the tears came in.\nWhat social activities have you already done during your reign and what other you've planned?\nIt started very soon after the coronation night. The second day after coronation I started working again and that same weekend I visited two villages in the interior of Suriname. There I made a donation towards school supplies for children of the villages for the coming school year.\nAfter that I was involved in several small projects in which I generated donations and also contributed myself for these causes.\nIn December I will enter another pageant. After that I have a project planned in which I will collect and donate as much schoolbooks as possible to orphanages and schools in need.\nDo you think Suriname is pageant minded?\nI do not think that Suriname is very pageant minded, yet\u2026 I think that the awareness can increase somewhat that a Miss represents the entire country on an international stage. Hopefully by achieving good results on the international pageants we can increase this awareness. I can actually say that Team TBS is doing a very great job till now.\nSome people think that the pageants are a money making business, what is your opinion?\nI think that a lot of aspects or events in our life have a commercial aspect. However I never entered the pageant to earn money or to get rich. I think as a Miss you have a platform and podium where you can broadcast a message. So I try to use this platform and try to show and tell about my journey and the hard work I have put in to achieve my goals.\nIf you win the Miss Supranational pageant, what are the first things you want to do?\nOne of the first things will be celebrating my birthday in a very grand manner. I would really want to spend some time at the end of December with my family and love ones. In Suriname new-year is celebrated very huge. So I will indulge in everything Suriname has to offer!\nIn the new year, I would like to devote myself to child-friendly education because I believe that education ensures development and progress. Also, I want to devote myself that young girls and women attend school. We live in a society that civilized and there is no place for this phenomenon. Every woman or young girl who does not go to school means that as a country you are behind in the world economy. Because it has been proven that women are more peaceful and disciplined.\nFirst of all the idea that I have fans is surreal. But the message I want to give is that with hard work, determination and personal sacrifices you can achieve more than you think. At times it was a struggle to do everything at once, but I am glad that I stuck through. And for all the people that support me and wish me all the best, thank you very much!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Our grant management process has been vastly improved thanks to the work we did with Cloud for Good.\"\nMigrated from Convio Luminate to Salesforce and utilized multiple applications including HandsOn Connect and Pardot as well as integrated website forms and surveys.\nAction for Healthy Kids fights childhood obesity, undernourishment and physical inactivity by helping schools become healthier places so kids can live healthier lives. They partner with a legion of dedicated volunteers \u2014 teachers, students, moms, dads, school wellness experts and more \u2014 from within the ranks of their 60,000+ network to create healthful school changes.\nTheir grassroots efforts are supported by a collaboration of more than 70 organizations, corporations and government agencies. Working together, they are giving kids the keys to health and academic success by meeting them where they are \u2014 in the classroom, in the cafeteria and on the playground \u2014 with fun physical activity, nutrition lessons and changes that make it possible for them to eat nutritiously and play actively every day.\nAction for Healthy Kids is a large organization with many moving parts so they needed a way to track their data in a much more sophisticated way. They were using Convio Luminate, a Blackbaud product.\nThey needed a better way to connect with chapters located throughout the country in order to keep their brand and system more consistent.\nAction plans, online forms, email marketing, donations and volunteer management applications, and grant applications all needed to be integrated into a single system.\nData on grant making, schools, projects, and relationships was spread across multiple systems and needed to be centralized and linked so that it could be easily found.\nThere was trouble getting concise reporting as well as collecting data. The organization needed more sophisticated reporting tools so that they could relay to funders how they were impacting the community.\nTheir consultant began by installing the Nonprofit Starter Pack to help the organization better manage individual donors, relationships and affiliations, as well as manage households and recurring gifts. An online grant application was linked to Salesforce from their site to streamline the grant giving and reporting process. Custom built surveys were created to help capture information as well as to fill requirements by grantees before funds were released. Action Plans were built so Chapters were able to access and update all information, making it much easier for them to track yearly goals and interactions. HandsOn Connect was integrated to help manage volunteers and, finally, Pardot was added to help view all prospect activities and better organize leads and contacts.\nStreamlined Process \u2013 The organization is finally able to track action plans, donations, interactions and grant giving throughout their network. They are able to streamline the way data is collected in a much easier to use system, which insures brand consistency throughout their many chapters.\nImproved Measurement \u2013 Action for Healthy Kids is better evaluate and measure their impact on thousands of schools.\nA System for the Future \u2013 With their new implementation on the Salesforce.com platform, technology is supporting the organization's mission and growth. Staff members can modify processes and integrate new technologies as the organization evolves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Urgent Urgent Urgent ... Please check the lists in the library at the front counter for the trip you are on. There was a mismatch between trips and names in Assembly. Congratulations to those who got their choices in for EOTC (and who ranked them rather than ticked them). Three trips now full so spaces only available in E Rancho which will be an awesome trip!!!!! (And the least expensive). Get those forms in for a cool as time. 75% of students signed up only 25% to go.\nIMPORTANT NOTICE - Senior Choir has changed from Thursday lunchtime to MONDAY LUNCHTIME this coming week. Everyone who was at Big Sing must be there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"chokka blog: \u00a38.3bn Better off?\nVery good work, what worries me though is that this \u00a38 billion figure has been rammed down our throat so much by the Yes side that it now seems to be kind of just accepted as fact,there's not enough people getting the truth out there.\nBusiness for Scotland ia\/was a non political organisation ?\nHow can that be said when at least two of their members are now SNP MP's ?\nMichelle Thomson and Richard Arkless...i've not looked for more yet.\nI believe there may have been a little irony inferred.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Cyberoam, a leading global provider of network security appliances has introduced a unique capability \u2013 User Threat Quotient (UTQ) \u2013 to help IT security managers identify users posing security risks with ease. With UTQ, Cyberoam extends its own innovation of Layer-8 technology and turns over a new leaf in user-identity focused security. Available on Cyberoam's Next-Generation Firewall and UTM appliances, UTQ harnesses information derived out of user's web traffic to determine risky users that pose security and \/or legal risks.\nGiven this scenario, a security manager wishing for a science fiction-like capability to preempt and mitigate threats shouldn't come as a surprise. The ongoing paradigm shift towards pre-emptive security shares some signs of such capabilities. Many organizations and security managers are leveraging big data analytics tools to capture early signs of specific user actions or network event patterns that may hint at a possible ongoing attack or anomaly.\nWinds of change in IT and network security clearly indicate a growing need to focus on internal threats or risks from users. As a result, many CXOs now identify analytics and actionable security as a key investment area besides cloud, virtualization and mobility, to leverage security analysis that enable faster decision making. Extending its own innovation of Layer-8 identity-security with UTQ, Cyberoam continues to bridge the gaps where many security paradigms fall short in combating insider threats and user-triggered risks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aphrodite is the Goddess of love, beauty, and war. How better to do spells for beauty and love than to use this Aphrodite Spell Candle. Hand-crafted with soy wax. Colored pink for love and beauty, infused with red and pink rose petals, scented with rose essential oil and a touch of apple sprinkled with Faerie and Goddess Dust. Crystal charged with rose quartz.\nRoses are used for love and beauty, but also protect from the evil eye. Use this beautiful candle in spell work for love, beauty, and romance. Crystal and Magickally charged for love and beauty. Remove crystal before or shortly after lighting. Make sure to charge crystal with your intentions.\nMade to order there may be some variance in color of wax, rose bud or rose quartz stone. This candle comes in a beautiful glass jar 4 oz.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am happy to know Craig and Robert.\nRobert in Brazil, I think.\nI also have a friend Bob.\nThese three are traveling, actually on the road, blogging and making noise, however very difficult for me to follow because of the Atom or RSS feeds.\nI really need some notifications in my email box and the content, not a link to the site, I will never click on it, I only have 1 hour per day in the internet cafe. Reading websites online is a big problem.\nI am working on an idea of how to use my HoboBlogs.com sites so person that wish could have free hosting of their travel blogs, not must a normal blog, a person that is out on at least a six month journey. There is also the possibility I contribute to their site by helping them with a fee of 50 cents for a normal post and 1.00 dollar for one with a photo.\nIt would be easy for me to host them on my site and share the travel stories. One problem is most people start and never proceed, then never follow through and complete what they started. I am sure many person on the planet, that have traveled could see that if they earned 5 dollars per day they could travel the planet, or at least it would help. I lived for many years on less than 10 dollars per day and visited many countries, mostly traveling by land.\nIt is an idea, trying to figure out how to help, I would like to give back what was given to me, many person have donated, helped, clicked, read, and done many things to help me continue to travel. I am working through this idea, hoping to figure out a strategy that would help the travelers of the planet to fund a trip.\nIt is impossible to just donate to them, there is a world of travelers out there who want some free money. I do think I could donate some web space, some technology, some techie help to those who are not so technical minded.\nCraig and Robert are good techies, Robert is more or less understanding the idea.\nHey, this is not a service for you as much as a great way for me to make money.\nPersons posting on Hoboblogs.com would help to spread the Hobo word, then in a roundabout way help the site and me to grow and prosper, I would benefit. To make them pay seems ludicrous, to offer for free sounds like a good idea. I am thinking that maybe I could charge a 25 dollar set up fee that they would receive back after maybe 50 post and then I would not lose time setting them up with a blog.\nI suppose the girl in the Hostel last night from England with he Parents that immigrated from Sr Lanka would work also, I was doing my best to chat... Nice girl, 19 months of travel and I think she has gone to more than me, easy to do, I am lazy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fish are important as a human food source which can contribute to bridge the food gap of animal protein demand in Egypt. Fish contain as well a large proportion of animal protein, vitamins and minerals, a high proportion of essential amino acids necessary for the body. Despite the improved fish production in Egypt (\u22481,500,000 ton in 2013: 25% from capture fisheries and 75% from cultured fish), the increasing pressure from growing populations and the desire to increase per capita consumption of fish requires to reorder priorities in order to achieve optimum exploitation of available fish resources. Egypt, as most of the world, needs to focus on fisheries as productive renewable food resources that can contribute to the development or increase national income as well as safeguard food production for coming generations. This will be achieved by the development of capture fisheries and the expansion of aquaculture production especially mariculture. Many problems are facing the development of our natural and artificial fisheries like overfishing, habitat degradation, climate change, illegal fishing practices, pollution, shortage of fresh water. The present work will evaluate the current status of our fishery resources and will discuss and analyse the challenges facing the development of fish production in Egypt (natural resources and aquaculture). The different impacts of these challenges will be illustrated, and finally the study will suggest some practical solutions to overcome these challenges and how to mitigate their negative impacts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Former paid app can now be demo'ed for FREE before buying.\nBoost your knowledge of Mandarin Chinese vocabulary, stroke order, and fundamental word and grammar elements FASTER than with any other app on iTunes. Applying the latest in cognitive science, Brainscape introduces concepts and vocabulary words in order from most foundational to most complex, and repeats them in a pattern that is just right for YOUR brain.\nThe Brainscape team is made up of former executives from The Princeton Review and Kaplan, as well as cognitive scientists and language specialists formerly with Columbia and Yale. Our mission is to make learning easier and more convenient. Feel free to check out our website (www.brainscape.com), and good luck studying!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbldue b/data_all_eng_slimpj/shuffled/split2/finalzzzbldue
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@@ -0,0 +1,5 @@
+{"text":"S-10 Forum > Member Forums > Forum News > Anyone want to attend the 2010 SEMA Show?\nIf you've always wanted to attend The SEMA Show but didn't have an inside connection -now's your chance. AutoGuide.com has a limited amount of spaces for a day at the Show on Friday, November 5, 2010, as an Enthusiast Opinion Leader. You'll get access to the industry's hottest new products and coolest rides - long before they reach the masses. Participants who provide active feedback and uploads to social media sites get an official Show t-shirt and chance to win some great prizes. Register today before it's too late. Enter code 80000029 at registration.\nClick the SEMA Logo to Apply!\nRe: Anyone want to attend the 2010 SEMA Show?\nDamnit! Fly in for my honeymoon on 11\/6\/10 at noon! Oh well i can check it getting broken down.\nany chances autoguide will have passes this year? I will be in Vegas for the next Sema in November.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The National Multiple Sclerosis Society characterizes MS as involving an immune-mediated process where an abnormal response of the body's immune system is directed against the central nervous system.\nWhile the effects of this terrible disease are already cruel, heat can actually worsen MS symptoms, making trustworthy air conditioning equipment critical to anyone affected by MS. This is the reason multiple sclerosis is a cause that is very dear to the hearts of Service Experts Heating and Air Conditioning. In April a couple of our locations, Calverley Service Experts serving the Fort Worth area, Service Experts of Dallas and Plano serving the North Dallas area, and Levy & Son serving the Dallas area, all helped sponsor the Walk MS events that were held in the DFW area.\nCalverley Service Experts, Levy & Son, and Service Experts Heating and Air Conditioning serving the DFW region had so much fun supporting multiple sclerosis. To find out more, visit http:\/\/www.nationalmssociety.org\/. Also, like us on Facebook to read about future opportunities where you can help Service Experts help others in our neighborhoods and around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Long before Christianity or any other religion existed there were \"The Figians.\" Figianity is the oldest religion on the face of the earth.\nReligion is man's response to sin.\nSomewhere deep down we know that by an act of our freewill we have violated our consciences \u2013 we have sinned. There is a sense that we need to make up for what we have done wrong in order to appease a Supreme Being. The root of every religion is essentially an attempt to be made right by our own good works.\nOkay, back to the Figians, here is how it started. Adam and Even sinned and were afraid as they realized they were naked. They wanted to cover their nakedness, so they made the very first clothing on the earth. The Book of Genesis records Adam and Eve's response to disobeying God.\nYou see where I get the Figians from \u2013 lol.\nFigianity is our attempt to cover our nakedness by own righteous acts.\nAdam and Eve's fig leaves did not really work. When the voice of God came walking in the Garden, the fig leaves were not sufficient so they dived into the bushes. That's the second religion on the earth. I call them the \"Bushamites\" \u2013 but thank God that's another discussion.\nAdam and Eve saw God's holiness but they didn't have a deep revelation of His love. If only they knew how awesome God is, they would have dared to step out of the bushes naked and say, \"Daddy, we messed up\u2026\" The Bible says that God took off the fig leaves and covered them with the skin of an animal. This was the first demonstration of mercy and grace.\nThis animal represented the Lamb of God, Jesus Christ, who takes away the sins of the world.\nAre you wearing your own brand of fig leaves? Do you think that your good works are enough to make you right in God's eyes? I have good news for you \u2013 you don't need fig leaves when you have God's love. You can clothe yourself with the righteousness of Jesus Christ, the skin of the Lamb of God. Jesus was mocked, beaten, slapped, pierce, dragged and nailed for your sins; so that you will never have to pay for your sins. His skin was whipped off his back so that you will never be naked.\nJesus on the Cross demonstrated God's unconditional love to humanity.\nNow if we ever find ourselves naked and full of shame, we don't have to run and hide from God. Rather, we run to Him in all our nakedness and our Heavenly Father wraps His arms of love around us and says, \"I got you\u2026\" He clothes us in the robe of righteousness purchased for us by the blood of Jesus Christ.\nNow as believers, we don't do good works to be saved but rather we do good works because we have fallen in love with God. He has loved us first and we have fallen in love with Him.\nNo more fig leaves for me, what about you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The NBA season tips off this morning between two Eastern Conference foes. The Celtics hold the home court advantage, as well as a pretty favourable 10-1 head-to-head record against the Sixers at TD Garden. A year removed from devastating injuries to Kyrie Irving and Gordon Hayward, this rejuvenated (and motivated) Boston side is currently the favourite to win the East. With a 7-3 record in their last 10 games against Philly, you can't back against them today.\nSimilar to the NBA Championship market, the Warriors enter at Winx-like odds to open their season with a win. With such a talented starting five that's only grown stronger, it's hard to disagree, but now that Carmelo Anthony is out of Oklahoma City, the Thunder might be able to spread the court and challenge the Warriors on the perimeter. If you'll recall, last seasons Opening Night game between the Warriors and Rockets came right down to the final second, so this one could be just as close if the Thunder can make do without Russell Westbrook.\nThe Calgary Challenger Round of 32 continues in the early hours of Thursday morning in what is shaping up to be an interesting battle between two North American opponents. These two met in the Semi-final of the ATP Challenger in Mexico earlier in May, with Maceiras winning a three set slugfest. Kevin King has found some strong form since then, making it as far as the Round of 32 in his last two tournaments. He certainly has the stamina to outlast the older Maceiras, but the Spaniard should prevail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Are donations\/pledges per lap or based on distance? No, all donations are based on general participation in the walk-a-thon for ease of collection and fairness to all students.\n2. When are donations due? Donations are due on the day of the walk-a-thon, Friday March 10th. They can be turned in directly to the office or to teachers. Checks can be made payable to Buri Buri PTA. We will accept late donations but donations turned in after Wednesday March 15 may not be considered for top fundraiser awards or t-shirt and notebook prizes.\n3. When will I get my t-shirt or notebook? The t-shirts are special order and we do not get extras so we put in one order after all donations are collected. T-shirts and notebooks will be distributed to students by the end of March.\n4. I want to get an extra t-shirt for myself or siblings. How much do we need to fund-raise? You can get one t-shirt for raising $75. You can get an extra t-shirt for each $25 raised over $75. So $100 for two, $125 for 3, etc. We will also give an extra notebook for siblings for each additional t-shirt ordered. Please contact buriburipta@gmail.com if you would like to adjust your t-shirt sizes or if you have questions.\n5. I lost my donation envelope. Where can I get a new one? There are extras in the office on the flyer carousel by the chairs. You can also print this DONATION FORM and use any envelope.\n6. Can I get a receipt for tax purposes? Yes! All donations are 100% tax deductible. Please contact us at buriburitpa@gmail.com if you would like a receipt.\n8. What are we fundraising for? We are fundraising to bring more awesome programs to Buri Buri. We will use fundraised money for the following: Science Carnival May 6, 2017, expanded Week of Code in 2017, possible art program next year and continuing with Playworks next year. All money raised above our PTA budget will be donated to the school.\nTie Dye T-Shirt: This one-of-a-kind t-shirt features blue and white tie-dye design and Bobcat lettering. Fundraise $75 and get this T-Shirt for a PRIZE! This will be your only chance to get one of these cool t-shirts. We will not include them as part of our regular spirit wear and will not have them for sale any other time.\nBaby Bobcat Notebook: These awesome notebooks feature a baby bobcat on the cover. They have a plastic cover and are 5x7 inches in size and have 70 lined pages. Fundraise at least $30 and win this prize!\nDonor Honor Circles: We will honor individual and business donors on our website and in a newsletter sent home to all Buri Buri students! Students will also be honored if they raise more than $50.\nClass with most funds raised: $100 Prize: Teachers can decide to use this towards field trips, arts and science programs or whatever their class wants!\nClasses with most participation in each grade level: (fundraising any amount, even $1) will get a popcorn party! If more than one class per grade has 100% participation, each will get the popcorn party!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblrgq b/data_all_eng_slimpj/shuffled/split2/finalzzzblrgq
new file mode 100644
index 0000000000000000000000000000000000000000..0d409b6dcbec0aa86abdb2fb8420d6a7280e5799
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A personal injury occurs when one has sustained injuries, either psychological or physical, because of an accident. Most personal injuries fall under tort laws. If one is involved in an accident that was due to the irresponsibility of the other party, they might be entitled to some form of compensation. The amount of compensation one can receive in personal injury cases is determined by the nature and extent of the damages.\nTo ensure one is fully compensated for all the damages, it is imperative to hire the services of a proficient personal injury attorney in Tumwater, WA. A Proficient personal injury attorney has an in-depth knowledge of the accident law. They can guide the injured person professionally throughout the entire legal proceedings and claim process. Here are some important details to know about personal injury law.\nAn intentional tort is one that occurs due to the deliberate or intentional actions by the tortfeasor. To succeed in a tort case, one must prove that the other party acted with substantial certainty that an injury would result. A negligent tort, on the other hand, is not deliberate but occurs when the person in charge fails to act in a reasonable manner, thereby resulting in an injury. To succeed in a negligent tort, the injured party must prove that other party failed to exercise a reasonable amount of care.\nThe statute of limitations refers to the time limit within which one must file their case in the state's civil court system. In Washington, the statute of limitation on personal injury cases is three years. This time starts from the date of the accident or injury. If one does not file the claim within the time limit, they may be permanently barred from recovering their compensation.\nThese are just some of the important details to know before hiring a personal injury attorney in Tumwater, WA. With the right accident lawyer, one can rest assured that they will get the rightful compensation. For more information about personal injury cases, and how to contact a qualified attorney, please Contact Putnam Lieb Potvin Attorneys.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get your PASSPORT TO GOOD SAM FUN at the IDAHO SAMBOREE the weekend of June 10th \u2013 14th. You'll be able to find us among the exhibitors, where we'll be showcasing our latest Renegade motor coaches and accessories. We'll also have some exclusive promotions and free give-aways, so If you're going to be at the show, please stop by and chat with Steve and Marty!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last year, we launched a diversity series aimed at bringing the most inspirational and powerful women in the tech scene to your attention. Today, we'd like you to meet Michelle Huff, CMO at UserTesting.\nMichelle Huff, Chief Marketing Officer (CMO) at UserTesting, brings nearly 20 years of experience leading marketing and go-to-market strategies at high tech companies, such as Act-On Software, Salesforce, and Oracle. Gaining insights directly from customers and prospects has always been essential in past roles \u2013 from leading product marketing and management teams to launching websites and marketing campaigns. At UserTesting, she's responsible for driving our go-to-market strategy, building the UserTesting brand, generating demand and strengthening customer engagement and advocacy.\nI've always been a tinkerer and loved technology at an early age. I had a Commodore 64 when I was in grade school. Later on, I was introduced to HTML programming and database design while at the University of Washington in business school. A turning point for me was working at a small consulting firm during my senior year. My colleague was starting up a recruiting arm in the business and was managing all her contacts in Outlook. I looked at her process and thought\u2026 there has to be a better way!\nSince I had just taken a class on Microsoft Access, I decided to design her a recruiting database. And then I bought Microsoft FrontPage (I know, I'm dating myself now) and redid the website to add the new recruiting services to the website. After graduating, I got my first job in tech doing web marketing and have stayed with tech ever since!\nMy degree was in Marketing and International Business. Yet, I really had no idea what I wanted to do when I graduated. As mentioned, I first started out in web marketing (and this was back in the days when the concept of businesses owning a website was still quite new). This eventually led me to doing product marketing at the web content management company. A growing, mid-size tech company was an awesome place to be early in my career, as it let me try many different roles and take on projects and initiatives that might have required much more experience at larger companies. Most of the roles early in my career were product or marketing focused, which gave me an opportunity to work cross-functionally across the business, talk directly to buyers and customers, and better understand how companies work. I like to be challenged, learn new things, and I never said 'no' when an opportunity was thrown my way. All of this experience was a great leg up to eventually becoming an executive leader of a tech company today.\nI suppose there are always obstacles and opportunities at every major turning point in your career. When I was in my early 20s it was difficult to build credibility with older buyers on the tradeshow floor \u2013 some of it was due to my age and level of experience in business. But at that time, there also weren't as many women demonstrating software and answering technical questions \u2013 and that had an impact on my credibility as well.\nSometimes, the obstacle is only in how I'm choosing to see the problem\u2026 other times, it's actually an opportunity.\nIn my 30s, I started to lead larger organizations and I struggled with learning how to better do that role in addition to starting a family. Early on, most of my peers and upper management were all men, and most had stay-at-home wives. As for my female colleagues, none had started families yet \u2013 so it was a bit intimidating. Later on, I started interacting more with fellow working mom leaders and it was great to just talk through the balancing act. Now in my 40s, I'm learning that the only thing you have complete control over is how you choose to respond to something. I've realized that you can choose what actions you need to take and how you want to interpret the intent of others and the situation. Sometimes, the obstacle is only in how I'm choosing to see the problem\u2026 other times, it's actually an opportunity.\nYes, I have an amazingly supportive husband and group of family and friends. For the majority of my career, frequent travel has been a large component. Until this year, other than my two maternity leaves, the longest I had been in one city since 1999, is 6 weeks! Raising two kids takes a village and my husband is a great partner in life and my mom has been invaluable. She lives nearby and takes care of the kids during work days. Thanks, Mom!\nAlso, my Dad has a ton of experience running companies and heading up sales departments \u2013 so he has always been a great sounding board for all my career ups and downs. I've never really had a single role model that I've aspired to be. However, I do try to glean as much as I can while watching others lead their teams, present to different audiences, discuss their marketing strategies & techniques, or hear about various parenting styles and tips and tricks. I'm surrounded by some pretty incredible people at work and at home.\nWell, I recently became the new Chief Marketing Officer at UserTesting, and I love it already! Right now, I'm focused on learning a ton! I'm diving into the business, my marketing team, the sales and customer success orgs, our product, customers, market, budget, marketing tech stack, reporting, and customer data. I'm putting together the list of all the challenges and opportunities I see, getting aligned with other leaders, building relationships, and putting together my plan. So basically, my typical workday means a lot of meetings and customer visits!\nI'm proud of the connections I've made along the way and hearing from past colleagues when they've told me I made a difference. I think ultimately people want to feel a part of something larger than themselves and to make an impact in the world. I also feel lucky that I've been able to be a part of many growing and successful tech companies such as Oracle and Salesforce \u2013 and see what it takes to run a large tech company at scale.\nI feel pretty fortunate that I've always had great men and women in my network who have opened doors, made connections, and given me opportunities to help me grow in my career. But I do think that the marketing department tends to attract more women than in other departments in a tech company, such as product and engineering. And if there are roles that focus on problems lower in the tech stack (infrastructure vs. business applications), I tend to see fewer women. I do think this is related to there being a smaller percentage of women entering STEM fields early on in school and therefore fewer women entering the workforce with those skill sets.\nHaving a diverse board, leadership team, and workforce helps other women find role models and mentors and better plan their career paths.\nPeople often look around at their leaders internally to see what kind of management and leadership styles they should emulate. And when they don't see people like themselves \u2013 it makes the transition that much harder. Plus, not everyone wants to be a \"pioneer\" and be the \"first female developer\" or \"first female VP\" or \"only female sales rep.\" Having a diverse board, leadership team, and workforce helps other women find role models and mentors and better plan their career paths.\nI've already started to see some change over the last 20 years \u2013 along with leaders of companies starting to make diversity a priority and a topic of discussion. This is great because if you don't talk about a problem, it's hard to solve it. And right now you can sense that the topic is picking up steam. However, this level of change can be slow \u2013 we're influencing hearts and minds and cultures \u2013 and needing to grow a diverse talent pool. It will take time. Sometimes it's easier to reflect on change after a generation to see how far we've come!\nIf you want a job in tech \u2013 you should go after it! It's such a great industry and time to be in it. I am always amazed by the people I work with and inspired by their creativity and how they want to change the world. The industry is fast paced and evolves quickly.\nTechnology has become so ubiquitous. It is so much easier nowadays to connect your passion and past experience with a tech-related role \u2013 such as auto, banking, customer experience, design, fashion, food, entertainment, healthcare, marketing, the list goes on.\nTech is being built for all these different industries and disciplines and even disrupting them. You can work for a tech company or make a move to a tech-related part of an industry you already know as a starting point. At the end of the day, most technology is being built to solve a business problem or a customer pain \u2013 and if you know the problem well or understand the customers (and like learning about technology) there's usually a role for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This statistic presents the results of a survey on perceived leading causes of death through interpersonal violence in Japan as of 2018. According to data published by Ipsos, Japanese respondents overestimated the number of violent deaths caused by knives. Around 41 percent of respondents thought that most people killed through interpersonal violence died from knife attacks, when the actual share of stabbing victims was around 23 percent of all violent deaths in Japan.\nThinking about people who are killed by other people in Japan in 2015, do you think more people were killed by firearms, sharp objects such as knives or other physical violence? Choose the one option that you think resulted in the most deaths.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks to all of our customers for all the work and trust in us for the previous year and into the future. We are looking forward to a New Year filled with more work and opportunities to collaborate on building robust technical solutions.\nOur special thanks to GIRD Systems Inc. and Quasonics Inc. for which we were able to collaborate on multiple projects this past year including both technical engineering developments and program\/project management activities.\nAll the best to everyone for 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbluua b/data_all_eng_slimpj/shuffled/split2/finalzzzbluua
new file mode 100644
index 0000000000000000000000000000000000000000..a3ca52295cff0bc1da6c411420e935faa7949b85
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Into the Shadows (Andrew Scarano, 2009) is an independently produced documentary that explores Australian film production and consumption. Inspired by the 2006 closing of Electric Shadows, a local independent cinema in Canberra, the film reveals how corporate multiplex cinemas have overtaken many of the smaller independent cinemas throughout Australia and provides a glimpse into an industry that has both thrived and struggled throughout the eras.\nThe film is made up of six chapters that explore the many facets of the Australian film industry, from its history to its present state and, finally, its need for regeneration. Into the Shadows includes interviews with a range of industry professionals including filmmakers, distributors and exhibitors. The role and expectations of the audience are brought into question, as well as the responsibilities of those who hold the power over film production and distribution in Australia. At times controversial, Into the Shadows presents an engaging and honest account of the Australian film industry.\nInto the Shadows was co-produced by Andrew Scarano (director\/writer\/co-producer\/editor) and Phil Hignett (co-producer\/editor\/cameraman). Scarano was inspired to begin the project by his grandfather, Keith Mann, who worked as the chief projectionist at Electric Shadows for twenty-five years. The finished product is a well-researched piece that presents the opinions of many known Australian film identities and shows a genuine concern for an industry that contributes greatly to our national culture and identity.\nInto the Shadows is most suited to senior Media students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disgaea DS kommt nach Europa!\nSimple yet highly addictive, Disgaea DS is an updated portable version of the classic Strategy RPG Disgaea: Hour of Darkness, previously available on the PlayStation 2 computer entertainment system. Since its release Disgaea has won over a loyal following of fans that have grown to love the quirky humour and deep, long-lasting challenge.\nDisgaea DS doesn't just let you take the Disgaea madness on the road, but introduces exciting innovations such as the new duel system multiplayer, complete with masses of customisable rules and game options. You'll also be able to connect to your friend's Nintendo DS to swap and share items, take your battles to a new strategic level with Geo Cubes and there's an added exclusive new story mode dedicated to Etna and her quest to become Beauty Queen Ultimate Overlord.\nJohn Yamamoto, president and chief executive officer of Square Enix Ltd. comments, \"I am delighted that we are able to bring Disgaea DS to players in Europe and PAL territories. Nippon Ichi has created a game perfectly suited for portable play that will entertain and challenge in equal measure.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Helpful 2V0-620 VCE Download Examination Konink Etalagepoppen.\nAt the left of the king, and just outside the door of the audience room, were three or four men, wearing red caps, like the Turkish fez.\nThey had not gone far before the sharp eyes 2V0-620 Exam Questions of the youth discovered something in motion on the shore a little 2V0-620 Exam Dump way beyond them.\nThe day before yesterday I found the male sassafras in abundance but no female.\nThe land here, like that lower down the Nile, is kept fertile 2V0-620 Certification by irrigation.\nHe preferred to hear of the journey from the lips of Doctor Bronson and his cousin, rather than to make it himself, and he hoped they would 77-886 Exam Paper think his decision a proper one.\nTheir caribou horns had been gnawed by mice in their wigwams.\nA cloud is uplifted to sustain its roots.\nThe alder leaved clethra Clethra alnifolia , sweet smelling queen http:\/\/www.examguidepdf.com\/000-105.html of VMware 2V0-620 the swamp its long white racemes.\nThe Blue Nile was long supposed to be the principal branch of the 2V0-620 Self Study river.\nI must be fancy free I must feel that, wet or dry, high or low, it is the genuine surface of the planet, and not a little chip dirt or a compost heap, or made land or redeemed.\nThey come in under pretence of desiring to trade, and at a given signal the fight begins.\nThe country was entirely unknown, as not even the Arab traders had ever explored it, and no one could tell what the explorer would encounter.\nI see three scythes hanging on an apple tree.\nThe cows are http:\/\/www.exampasspdf.com\/MB2-704.html milked only once a day, and their yield is surprisingly small.\nI have already told you of the constant difficulties in the way of obtaining porters before starting on a journey, and of the large number of desertions on the road.\nThe minister explained, 2V0-620 Exam Demo what the 2V0-620 Exam Dump king probably had learned already, that the strangers had come from a far country vSphere 6 Foundations Beta 2V0-620 Exam Dump 2V0-620 Simulation Questions to look upon the ruler of the kingdom of Ugunda.\nI observe that in the cut by Walden Pond the sand and stones fall from the overhanging bank and rest on the snow below and thus, perchance, the stratum deposited by the side of the road in the winter can permanently be distinguished from the VMware 2V0-620 Exam Dump summer one by some faint seam, to be referred to the peculiar conditions under which it was deposited.\nNow I ll show CSMP-001 Actual Test you how we used to cook fish in the Adirondacks, said Fred, with a dignified air.\nSometimes a chief gives a grand entertainment before the hunt, but in this instance 2V0-620 Actual Test he didn t do so.\nAbdul whispered that they were from the south but there was no time for farther explanation, as the business of the visit was opened at once 2V0-620 Exam Test VCP6-CMA, VCP6-DCV, VCP6-DTM, VCP6-NV 2V0-620 Exam Dump by the king.\nHe knew that they contained an almond before the naturalist did.\neven 2V0-620 Braindump where the sward is trodden hard, the small white heads on short stems are sprinkled everywhere.\nNo good ever came of obeying a law which you had discovered.\nOnce scared, will run all VMware 2V0-620 Exam Dump day.\nThey live partly by hunting and fishing, and formerly they made quite a revenue by selling slaves to the dealers who came from Khartoum and other parts of Egypt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Picture a picnic by a pretty Piedmont lake - red-checkered blanket spread out on the grass, sun beaming down from a cloudless sky, a gentle breeze blowing through the trees.\nThis Norman Rockwell image is brought to you by Chester State Park, an undiscovered outdoor refuge tucked away from the usual tourist hot spots. And that's what makes it such a find. It's simply lovely.\nThere aren't a lot of recreational trappings here. A couple of playgrounds, a short nature trail, a disc golf course, a rustic boat house and the 160-acre lake. But what it lacks in flashy amenities, it more than makes up in ambiance.\nIt's the perfect getaway if you really want to get away. Sweet and serene, it offers a beautiful setting to do nothing but enjoy the peace and quiet.\nOK. That's not entirely true. You can fish for bass, crappie, bream or catfish from a dock or rent a non-motorized boat to paddle around the 160-acre lake. The new disc golf course - the first in the state park system - offers nine holes for beginners and 18 holes for more advanced players.\nBut it's all about relaxing.\nEven the nature trail is an easy walk through the woods. It winds along the top of the lake from the boathouse to the spillway for a stress-free 1.3 miles.\nThe park also features 25 campsites for RVs or tents, an archery range, three picnic shelters and a rental community building.\nDespite its rural setting, Chester State Park is conveniently located just 15 minutes from Interstate 77, about halfway between Charlotte and Columbia. Check it out the next time you're traveling through the area and ready for a little R&R.\nFor more information on Chester State Park, click here or call (803) 385-2680.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anthony Ingrown Hair Treatment is the perfect solution for men who suffer from ingrown hairs and razor burn. A combination of Glycolic, Salicylic and Phytic acids keep the skin clear of bacteria and help to 'lift' dead skin cells, removing the two main causes of ingrown hairs. This Ingrown Hair Treatment is a lightweight gel which moisturises and protects the skin while treating this common complaint.\nAqua (Water, Eau), SD Alcohol 40-B, Propylene Glycol, Glycolic Acid, Phytic Acid, Salicylic Acid, Epilobium Angustifolium Flower\/Leaf\/Stem Extract, Lavandula Angustifolia (Lavender) Oil, Saccharide Isomerate, Hydroxyethylcellulose, Ammonium Hydroxide, Tetrasodium EDTA, Citric Acid, Sodium Citrate, CI 42090 (Blue 1), CI 19140 (Yellow 5).\nMassage a small amount onto the skin after shaving. Wait for a couple of minutes before applying your moisturiser.\nFor instant relief after shaving, store your Ingrown Hair Treatment somewhere cool.\nI've been using this for years having previously tried a whole range of competitor products. This worked the best and didn't dry out my skin. Highly recommended.\nI have had a major issue with ingrowing hairs on my neck this product (albeit with a change in shaving equipment) has helped a lot, the ingrowing hairs have gone. The tingling you get when you put it on hasn't abated as suggested on the instructions but it is not uncomfortable.\nThe only one down side is it can leave the skin shiny but it doesn't take away from the product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"At MaxDumpster, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Willard, NM, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nGovernment really should impose fees and penalties on the people for being the cause for making the air pollution in the country (until inevitable). The actual citizens must also hold the liability by taking an image resolution to protect the environment.\nImproper lifting of the huge junk on to the truck may lead to upper back pain, leg as well as arm ache, acute damage, and other serious chronic effects. The hospital costs paid to cure such incidents will double the amount that the individual has stored by eradicating the trash upon own. Finally the person find yourself spending enormous sum of money as well as suffering.\nAny time ordering the dumpster to get rid of all of the remodeling particles, make sure you hold the instructions upon items that are certainly not permitted if you rent a dumpster.\nConsult the 6 Yard Dumpster in Willard, NM vendor and get apparent information on the type of waste that may be disposed to the dumpster because some 6 Yard Dumpster in Willard, NM companies get limitations around the kind of junk that is got rid of into the dumpster. Removing unaccepted trash to the dumpster will request penalty on your part.\nYou also ought to be aware of the type of services that the renting dumpster company delivers. It is important to be aware of how they take care of the fingertips of your rubbish. Many different suppliers perform spin off services. This is an excellent method to get a lot of work done without having to worry about where your junk is going. Any renting dumpster company might use roll down while taking away your dumpster as well as replace it with an all new one when necessary.\nIn case you have a lot of junk you may want to think about 6 Yard Dumpster in Willard, NM service. That's right, you are able to rent a whole dumpster to own all to yourself. They've created these dumpsters in every different sizes you actually meet various needs. You will get some which might be small for personal use and some which might be really large for organization use.\nRenting some sort of dumpster, seems simple right? Well, before you place your own dumpster order there are some things you need to know.\nFor the cases when standard junk pickup merely won't cut it, consider . With this particular option, you might benefit from the alleviate and convenience of at-home garbage collection, but you may dispose of huge items and larger volumes connected with trash easily.\nIf your home or business is in a lesser tight operating area so you do not think some sort of dumpster can be delivered to it, you better think again. If you choose a firm that uses a lesser truck to handle the dumpster they are able to squeeze into smaller places, quite possibly like the one you need a dumpster delivered to.\nGroup is required to get a successful plunge. There are always the solitary dumpster scuba, but the compensate is greater as well as risk decreased when there are at least a couple. Your good friend can help with the dive by itself, and also should you run into difficulty. Extra fingers are helpful regarding holding a great LED light while you look, lifting hefty objects, as well as helping you be in or out of the dumpster. However, restriction your teammates; this starts obtaining obvious should there be more than two or three people waiting for.\nThere is not any doubt concerning this removing trash from the vacant house sight is going to considerably bigger than you actually thought. Leasing a personal dumpster is a brilliant way to haul away those extracted materials down your property. Better still, the professionals at the construction 6 Yard Dumpster in Willard make the entire rental course of action incredibly practical and they provide the excellent rates. Nationwide dumpster rentals are available online or maybe by phone.\nIt's not to imply that just about all Internet sites supplying dumpster services are rip-offs, but simply that you can ask if the individuals whom you have contacted are in reality the people which will be doing the job.\nYou may save pennies the following, but it necessitates large amount of cash to be used on your health issues in your near future because discarding such a enormous waste on self consists of a lot of energy, physical exertion and also time investment decision as well. In addition to this you'll need to focus on travel complexities and also environmental welcoming recycling associated with disposed spend if you are individual that respects social concerns of your nation. Building waste is always in enormous quantities and also occupies much space. Daring to clear construction spend on personal not only uses your time nevertheless occupies your own free area for longer period of time.\nDumpster Rental Companies provide various styles of dumpsters to get rid the trash produced at your employment location. The actual sizes are 3-5 yard, 6-8 property, 10-15 yard, 20 yard, 30 yard, and also 40 property. You can choose any of these dumpsters depending on the quantity of junk generated at your house . renovation or maybe construction assignments. For development projects, it is advisable to choose roll off dumpster. Roll off dumpster gives you wheels on four sides. Roll down dumpsters can be easily moved from one place to another place. Roll down dumpsters are available in different sizes ranging amongst 10 yard to 45 yard. Dumpster rental organizations assist you in maintaining the job place free from junk by get rid the trash in eco-friendly manner. Should you be confused to pick the right sized dumpster, 6 Yard Dumpster in Willard organizations help you with the actual professional direction in choosing one particular.\nSelect the right sizing dumpster. Dumpsters come in quite a few sizes and also dimensions coming from small to enormous. Have a dumpster professional inform you of what kind to rent.\nBy renting any dumpster you will also obtain the work carried out quicker and also move the actual dumpster right off on the property therefore leaving a great and clean up yard.\nDumpsters are large containers that are utilised to keep junk before it is compiled and taken to any landfill. These are common in restaurants and also commercial buildings nonetheless they can also be used in a very residence. Should you be currently improving your home or maybe doing some unique of development, it is necessary to rent any dumpster to ensure your compound stays tidy. A lot of junk is produced during development therefore you need to have to get a professional company that will offer you the help you require. There are various factors that you can keep in mind when you find yourself choosing the company to use pertaining to hiring junk receptacles.\nNow go ahead and plan that bar-b-que party in your backyard and lose focus on about the cleaning and getting rid of. Decorate your house, cook with regard to friends and family, invite them over as well as planning out the fun activities with regard to guests. After and before you are done, you can be using little or perhaps big heaps of trash that may now be cared for easily. These businesses are below to make your daily life easier.\nOne of the largest parts of ones clean-up is going to end up being deciding what is trash as well as what's definitely not. Without going too insane, it is probably very best to err assisting trashing things, because it is going to make job, along with your life, a lot simpler. Even though it can be appealing to save all the things for upcoming \"needs,\" bear in mind you'll get a whole lot more use out of your garage along with your belongings if you have enough bedroom to access them in the first place.\nExample of a company as it pertains to this business should also be factored in. You may be thinking that going waste is a thing that does not need much knowledge but the truth is in any industry or field of work you can find challenges and once someone does work there, they'll need developed methods for overcoming all those challenges and excellent their work. Therefore, when using a skilled company, it is possible to get top class services as opposed to the one that is only getting started. Think of this as among other things including proximity of any company when evaluating Dumpster rental services.\nA good option is to go with a dumpster that is one size bigger you think you will have to have. By doing this you are secure and if a person happens to find another items close to your home you will not want, you can certainly throw them away.\nThe whole process of collecting trash can or dumpster furthermore depends upon organization to company. Many can provide a person locked canisters which are while filled are generally carried away by simply service provider. The bins are generally collected and brought in existence of you so as to have a look.\nRotate offs are much much larger. They can cover anything from around 10 yards to right up to forty lawns in length. Wheels are affixed and the reason for these is to help with mobility when the dumpster is either provided or picked up after answering. That's why they are known to as \"roll offs\". There are a lot of functions for dumpsters or spin offs. People desire to dispose of vast amounts of unneeded materials including old devices, or just ordinary trash. This could come from tasks such as residence renovations, building projects sometimes small or large, and home or industrial building scenery cleanups.\nYou are able to dispose all the trash or perhaps garbage to the dumpster services rental company to maintain clean and sanitary environment close to your job areas. The dumpster service company will assist a person in keeping your task location no cost at inexpensive cost.\nYour house project can be done so much easier with a simple phone to order a new next-day 6 Yard Dumpster in Willard, NM. Your project can be done simple and easy through simplifying the task associated with removing squander from your home. Whether your house project is cleaning out this basement, your garden storage or doing away with old restroom appliances for the start of the fresh bathroom renovation, you could expect simplicity after you rent a dumpster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FERMENTATION METHOD \u2013 Fermentation at a controlled temperature with maceration of the skins for 10 days.\nMETHOD OF REFINEMENT \u2013 Malolactic refinement for 6 months in steel tanks.\nFruity red berry aroma-redcurrant and raspberry, with a spicy predominantly cinnamon scent.\nHarmonius, pungent flavour; with a typical slightly bitter after-taste.\nServing temperature 16 \u2013 18 \u00b0C.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Browse our steaks for sale online and cut it your way! Our Wagyu kobe steaks are available cut to your specifications - whole Wagyu NY Strip Steaks, Rib Eyes and Tenderloin, in cut to order steaks available in 1-inch to 3 1\/2 inch thick steaks - with 1\/2 increments. So you can have your mouthwatering kobe steak as thick or thin as you heart desires. Order Wagyu beef today from Gourmet Food World!\nMeat lovers who know their steak covet the Short Loin \u2013 a versatile cut that blends meaty favor with great tenderness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very well maintained triplex in a fabulous location.Three 2 bedroom units with SEPARATE utilities. Each unit has stacked washer\/dryer.First floor unit($1,250 a month) has two bedrooms one and a half baths, living room and an updated kitchen with stainless steel appliances and courtyard as you exit the kitchen. Second floor unit ($950 a month) has two bedrooms one bath with a living room\/kitchen combination. The third floor unit ($1,200 a month)has two bedrooms and a full bath, updated kitchen and a living room. All of this just a few blocks from all the action on Main St in Manayunk. Also close to public transportation and a 15 minute drive to center city. PROPERTY ADDRESS ALSO KNOWN AS 4301 FLEMING ST.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"January 24, 2008 \u2022 Chief Marie Smith Jones, the last fluent speaker of the Eyak language of the Alaskan Indians, died Monday at her home in Anchorage. She was 89. Chief Jones worked diligently to preserve her native tongue and other indigenous Alaskan languages.\nJanuary 18, 2008 \u2022 During World War II, Warsaw zoo director Jan Zabinski and his wife, Antonina, sheltered hundreds of Jews from Nazi forces on the grounds of the zoo and in their home. Diane Ackerman has documented the remarkable and unlikely story in her new book, The Zookeeper's Wife.\nJanuary 10, 2008 \u2022 Renowned biologist Alan Rabinowitz worked with the military junta in Myanmar to create its largest tiger conservation area, prompting criticism. Rabinowitz discusses this experience, detailed in his book Life in the Valley of Death.\nJanuary 7, 2008 \u2022 Politicos may think Obama has the youth vote, but Huckabee has a secret weapon: action hero Chuck Norris, who is luring teenage fans to Mike Huckabee's campaign stops. Some of them can even vote.\nDecember 15, 2007 \u2022 One night, an elderly woman woke up to a female voice singing Irish ballads. The problem was the voice was in her head. Dr. Oliver Sacks was able to determine why she heard the voice. But the more interesting question was -- whose voice was it?\nNovember 2, 2007 \u2022 California's recent wildfires produced many spectacular images, but perhaps none as dramatic and memorable as a Los Angeles Times photo of 12 firefighters trapped in a circle of smoke and flames.\nOctober 25, 2007 \u2022 TV shows have been bleeping profanity for years when people speak extemporaneously. In recent years, however, scripted shows have been writing profanity in. The actors say the forbidden words and then the words are bleeped out for comedic effect.\nOctober 9, 2007 \u2022 Commentator Kenneth Harbaugh grew up hearing tales of war from his grandfather who served in World War II. When he was quite small, the stories seemed \"fun, full of dark humor.\" It wasn't until he was older that his grandfather revealed the more realistic versions of his narratives.\nOctober 4, 2007 \u2022 A petition before the Food and Drug Administration could change the way parents care for children with colds. Many pediatricians cite a lack of evidence that cough medicines are safe or effective for young people.\nOctober 2, 2007 \u2022 James Geary has gathered examples of what he calls the oldest and shortest literary art form on the planet \u2014 the aphorism \u2014 into a compendium, Geary's Guide to the World's Great Aphorists. He shares the rules he uses to identify an aphorism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Bought a Single Family home in 2015 in Canton, Baltimore, MD.\nIan is incredible to work with. Ian is honest, professional, knowledgeable, experienced, and detail-oriented \u2013 everything that you want in a realtor. Even though my home buying situation ended up being more complicated than most, Ian made everything feel easy. As a first time home buyer, I really knew very little about the process but he and his team walked me through every step with ease and truly went above and beyond to be helpful. He and his team were also incredibly responsive any time I had a question or request. Also, every person that Ian recommended throughout the process (mortgage broker, home inspector, title company, etc.) was incredible in their field and also did an amazing job to make the process easy for me.\nThe one bright side of getting involved in a complicated purchasing situation was that I was able to see how much Ian and his team care about you as a person. I can say from firsthand experience, Ian will absolutely protect your interests over his own short-term gains and wants to make sure you are happy and satisfied with the experience, rather than just looking for a quick payday. Despite a long and complicated process around my home-buying process, Ian and his team never once slowed their efforts to ensure that I had the best experience possible.\nAs far as I am concerned, Ian and his team could not have done a better job throughout the entire process, and I will absolutely recommend him to anyone I know that is looking for a realtor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2V - 39V Volts Features \u2022 1. , Date Code, Lot datasheet dflz Code) for this item is currently unavailable. Explore Discrete Semiconductors on Octopart: the fastest source for datasheets pricing, specs availability. Zener Diode dflz 33V 1W \u00b1 6% Surface Mount PowerDI\u2122 123. Datasheet4U offers most rated semiconductors data sheet pdf: Diodes Semiconductor Electronic Components datasheet Datasheet: DFLZ13 Datasheet. View More from Diodes Inc > > Traceability data ( i. Pricing ( USD) & dflz Availability.\nDFLZ33- 7: 15, 049 available from 14 distributors. Dflz 33 datasheet. DFLZ6V2 THRU DFLZ39 1 Watt Zener Diode 6. 0W Power Dissipation \u2022 High Surge Capability \u2022 Ideally Suited for Automatic Assembly. Buy your DFLZ33- 7 from an authorized DIODES INC.\nBuy DFLZ33- TP with extended same day shipping times. View datasheets, stock and pricing, or find other Zener. Source DFLZ33 Price, Find DFLZ33 Datasheet, Check DFLZ33 In stock & RFQ from online electronic stores. DFLZ33 Datasheet. DFLZ33 Application. dflz33 ( zener diodes) - 1.\n0w surface mount power zener diode. DFLZ30 datasheet, cross reference, circuit and application notes in pdf format. Text: DFLZ30 FJQ.\nDFLZ33- 7 online at Newark element14.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00bb What is Psychological Trauma?\nChapter One of \"You Can't Just Snap Out Of It\" available on amazon here. Get the amazon kindle version free here for a limited time!\nTrauma occurs when you experience a life-threatening event to yourself or someone close to you, accompanied by intense fear, horror or helplessness (as defined by the American Psychiatric Association). Trauma can leave an indelible imprint on the brain.\nFor example, if you look at the brain scans of some traumatized individuals, you can see shrinkage of brain regions\u2014such as the hippocampus\u2014that are involved in memory, or a failure in brain areas\u2014like the frontal cortex\u2014that are supposed to turn off the fear response. What you can't see, but what is equally important, is what that trauma does to the survivor's self-image. How a person feels, how he perceives himself and the world can be dramatically altered after a traumatic experience.\nAlthough trauma can wreak havoc on the brain and the psyche of individuals, as we will show in this book, IT DOESN'T HAVE TO. Whether trauma has these devastating effects or not depends on how we deal with it\u2014or more precisely, how we don't deal with it.\nAs my work demonstrates, you can't just \"get over it\" because trauma can change the parts of the brain involved in memory, which can then result in those traumatic memories playing over and over again. As a result of this change, you can't just stop the memories of your own free will.\nUnfortunately, it is this very notion\u2014that we can and should block out traumatic memories\u2014that serves in part to keep the brains of many people caught replaying the memory of the events in an endless cycle. In response to trauma, the heart and mind pull in opposite directions. While the brain endlessly repeats events, the heart is desperately telling the mind to move away, not to go there.\nThe problem with the idea that people should just get over it is that trauma is not a state of mind. It is not an attitude or a belief that needs to be changed. It is a disease, the \"I can't stop thinking about it disease.\" You can't just forget about it, or will it away.\nCould you will yourself out of diabetes? Or cancer? Of course not. So why would you be able to will yourself out of trauma? It's not any different.\nLeft alone, trauma won't heal itself. Trauma is like a virus. Just as a virus leads to disease, trauma can lead to a disease: trauma-related mental and physical symptoms.\nIf you mount an effective immune response to the virus, you won't get the disease. Some people have good immune systems; others have immune systems that don't go quite far enough, and in the end they get the disease. Others do not mount any immune response\u2014they may get sick and even die.\nLike a pathogen, trauma is a trigger that can lead to physical and mental disease if not recognized, dealt with, and addressed. Similar to the immune system's response to infection by a virus, we have mechanisms within us to deal with trauma\u2026night soldiers waging a secret war, so secret we're not consciously aware of it. The immune response to trauma is something like that, although it is more mental and cognitive. If your immune system doesn't kick in to fight an infection, you need help. That's why we have antibiotics.\nThis book gives you a cognitive, or thinking, antibiotic for the treatment of trauma. It is about what happens when our desire to block out traumatic events is at war with a mind incapable of letting go of the memories.\nSome trauma survivors will put on \"a brave face,\" a false persona. On the outside, they are the picture of healthy, successful, mature adults, while on the inside they feel frozen in time, frozen in the grief and the pain of the event. This is a very common psychological response when the trauma is not adequately addressed at the time of the traumatic events. But this is just one possible reaction. People are extremely resourceful, and will look for all kinds of ways to suppress the memory of the traumatic events\u2014from drinking or taking drugs, to working night and day, to having affairs.\nMy professional interest in psychological trauma crystallized the first time I was faced with a trauma victim who was in the midst of a flashback.\nI was a psychiatry resident at Yale University Medical School. It was a humid night in August, and I was on overnight call duty at the Veterans Administration (VA) Hospital in West Haven, Connecticut.\nI'd just managed to fall into a fitful sleep when the ring of the telephone pulled me back into the world of the living. The operator patched through a veteran who wanted to talk to the psychiatrist on call.\nClick here for the continuation of Chapter One of You Can't Just Snap Out Of It.\u2003Or get it on amazon for kindle and paperback.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Currently, we do not have a system that sends out paper invoices or bills.\nHowever, if you use PayPal, we can send you invoices through PayPal.\nIf you wish to have a PayPal invoice generated for you, please open a support ticket by clicking any support link on this page and opening a ticket so we can send one to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Making it less demanding to buy homes and houses.\nA lease to possess program permits the purchaser (occupant purchaser) to purchase the house with a drop initial installment, and a liberal measure of the month to month lease gathered will be credited towards the price tag, making it substantially less demanding to qualify than traditional techniques.\nThis lease to claim technique works extraordinary if your credit has been harmed as of late, (similar to a great many Americans) as it permits you an opportunity to repair your credit so that toward the finish of the lease\/purchase term you can meet all requirements for regular bank financing then altogether buy the home from the Landlord.\nFor those people who can't purchase a home through conventional means, lease to possess houses are a feasible alternative to home proprietorship. The lease to possess technique conveys the fantasy of home proprietorship to individuals who may have generally thought it was unimaginable.\nHere are Just a couple of the advantages of leasing to buy.\nBy and large, lease cash goes into the landowner's pocket, to pay the home loan on the home, to enhance his or her salary, and does nothing more for the tenant. With lease to claim houses, a part of the lease cash goes toward the buy of the home, making it any less demanding route for most tenants to put resources into a home.\nFor those with not as much as flawless credit, lease to claim houses are the best approach. Since the lease installments are going toward the buy of the home, it works much like a home loan installment would. When it comes time to make the last installment and purchase the house, the landowner can utilize the acknowledge the inhabitants set up for them to legitimize the offer of the home and exchange the deed.\nWith regards to a normal home loan and home buy, it can take a very long time from the time the offer is acknowledged on a home before the purchasers can move in. With lease to claim houses, individuals can for the most part move in inside possibly 14 days of the arrangement being made and the documentation being agreed upon.\nAt the point when contrasted with the normal home loan, value constructs quicker in a lease to possess circumstance, in light of the fact that the thankfulness happens quicker. At the point when an occupant is responsible for the home, he or she may make changes as they pay the lease. When the time has come to make the last installment, the price tag might be substantially less than the house is worth.\nAt the point when the occupants move in, they deal with the home, which means they can make modifications and changes where they see fit. Since they have such high stake in the property, they are not going to spend their own cash to enhance a home they won't live in for quite a while and in the long run claim, so proprietors don't need to stress over the occupants harming the property.\nSince the home still has a place with the landowner, the proprietor stays in charge of the property charges until the point that the parity is ponied up all required funds and possession is exchanged to the inhabitant. This makes it simple for the occupants to set aside extra cash until the point when they possess the home, to plan for the cost once they do.\nThere are numerous different advantages for the two occupants and landowners. While inhabitants may pay higher lease for a circumstance like this contrasted with a regular leasing circumstance, they will do as such energetically to know the property will inevitably be their own. Besides a measure of lease paid in many cases is credited towards the buy.\nRegardless of the rising number of leaseholders the nation over, it is less expensive to purchase a home instead of lease one of every 72 percent of the 50 biggest urban communities in the U.S., as indicated by a record discharged by land pursuit and showcasing site Trulia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you are a first time in college student (FTIC) and earn less than a 2.0 grade point average in the first long semester (fall or spring) enrollment at the University of Houston you will be placed on academic notice and must be advised by the Academic Advising Center of the Undergraduate Scholars and your major department.\nIf your cumulative UH GPA falls below a 2.0 (and you are not a FTIC) any time during your academic career you will be placed on academic probation.\nWhile your cumulative UH GPA is below 2.0 you will remain on probation and you must continue to earn a semester GPA of at least 2.0 to avoid being placed on academic suspension.\nIf you are on academic probation and your semester GPA is below 2.0 you will then be placed on academic suspension.\nYou cannot enroll in classes at the University of Houston while you are on academic suspension.\nFor the first suspension, you must sit out one semester before applying for readmission.\nFor a second suspension, you must sit out 1 full year (12 months) before applying for readmission.\nFor a third suspension, you will not be readmitted to UH.\nCarefully consider the circumstances that led to your academic suspension and determine what you can do to change those circumstances.\nEvaluate if the major you are pursuing is a good fit with your interests and abilities.\nYou may take classes at a community college however strong performance in those classes does not guarantee readmission to the University of Houston.\nAll students on academic suspension are required to re-apply as a former student.\nThe College of Technology will review applications from students on suspension with majors in Technology as well as from students interested in changing to a College of Technology major. However, students from majors outside Technology should demonstrate in their application a genuine interest and potential to succeed in their intended College of Technology major.\nIn addition to the ApplyTexas application, students are required to submit a readmission petition to the Office of Admissions with a completed In-Progress Grade Report Form if you are currently enrolled at another institution.\nReadmission from suspension is neither automatic nor guaranteed. The College of Technology's readmission committee meets prior to each fall and spring semester to review petitions for readmission. Follow the guidelines below to ensure that your petition for readmission demonstrates your readiness to resume your studies at the University of Houston.\nComplete the application and turn in all requested materials. It is your responsibility as a student to attach ALL required documentation to the readmission petition package. Incomplete applications will not be reviewed. Your complete application will be forwarded to the readmission review committee.\nWhat factors contributed to your poor academic performance?\nHow have these factors been resolved? In what ways are you more prepared to be successful in school at this time?\nWhat activities have you engaged in since your suspension that relate to your preparation and readiness to return?\nTurn the packet in by the deadline. DO NOT BE LATE. The readmission committee strictly enforces deadlines.\nThe Office of Student Affairs will mail a letter to you regarding the outcome of the review process. Please do not call for information. We cannot provide information or updates on the review committee's decision over the phone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New and improved security camera system installed throughout campus to improve overall safety and reduce theft.\nMore surveillance cameras have been installed on campus by the UC High administrative staff to improve campus security and resolve any remaining concerns with safety.\nAccording to Counselor Kelsey Bradshaw, \"The purpose of these cameras is so that the school can monitor any suspicious or unsafe activities.\" Bradshaw estimated that the cost of the new cameras was around 15,000 dollars.\nAccording to the National Center for Education Statistics, \"More than 80 percent of public schools \u2014 and more than 94 percent of high schools \u2014 in the U.S. use security cameras to monitor students\" (districtadministration.com).\nThe Security Sales and Integration website reports that \"Security cameras are useful in lost-and-found investigations, to verify the facts about incidents, to determine the number of officers needed to respond to a situation, in training, to locate missing persons, to back-up or refute children's versions of incidents, and in conflict resolution\" (securitysales.com).\nAccording to a Scholastic organization report, \"\u2026when there are school video surveillance cameras installed in school parking lots, potential perpetrators may think twice before committing a crime. About 60 percent convicted burglars stated the presence of a security camera system influenced their decision to look for another easy target. If a student got robbed, the images and videos captured by school security cameras are useful for investigating vandalism or violence. With evidence, the criminals can be caught by the police much more easily\" (scholastic.com).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join \"Joyce's Tavern Members Only Club\" and Receive a Birthday Email Certificate!\nGood For 1 Free Drink and 20% Off Celebrant's Food Bill!\nFill out the form below with a valid email address and on your birthday month a \"Joyce's Tavern Members Only\" Birthday Certificate will be sent to you via email.\nMust be 21 years or older to be eligible.\nIn order to receive the birthday certificate, your membership must be submitted 30 days prior to birthday month.\nBirthday certificate is only valid for the birthday celebrant. Cannot be used in parties of two or more.\nIn order to redeem this offer, please be able to present a photo ID with a D.O.B along with the \"Joyce's Birthday Members Only\" certificate that was emailed to you.\nEmail certificate will be sent out the beginning of each month and expire on the last day of your birthday month. No exceptions, replacements, or extensions.\n\"Joyce's Birthday Members Only\" Certificates are not valid during other Joyce's Specials and\/or Promotional events.\nIn the case of fraud and\/or abuse involving the \"Joyce's Tavern Members Only\" Birthday Certificate, Joyce's Tavern has the right to take appropriate action and end promotion immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Figure Wooden American Flag Police Thin Blue Line Reclaimed Wood Burned the latest specifically for you from thin blue line paint color , thin blue, source: allamericanwoodflags.com.\nFigure Tattered Split American & Thin Blue Line Wood Flag All handmade best just for you from thin blue line paint color , thin blue, source: pinterest.com.\nPicture Cobalt Blue Wall Need to know what paint they used the best for you from thin blue line paint color , thin blue line patch, thin blue line armband, thin blue line, source: pinterest.com.\nPhoto My ideal color scheme for the house The Home Pinterest selected just for you from thin blue line paint color , source: pinterest.com.\nFigure Thin blue line yeti most popular for you from thin blue line paint color , source: etsy.com.\nPortrait What Those Spray Paint Marks on the Street Actually Mean [Update] selected just for you from thin blue line paint color , source: lifehacker.com.\nDrawing Cobalt Blue Wall Need to know what paint they used noblest specifically for you from thin blue line paint color , thin blue line armband, thin blue line flagge, thin blue patch, source: pinterest.com.\nPicture 74 best Paint Inspiration images on Pinterest in 2018 most popular just for you from thin blue line paint color , thin blue line ring, thin blue line flagge, thin blue line patch, source: pinterest.com.\nIllustration 74 best Paint Inspiration images on Pinterest in 2018 most popular for you from thin blue line paint color , source: pinterest.com.\nPhoto Perfect Paint Color 5 tips for ting it right selected for you from thin blue line paint color , source: thistlewoodfarms.com.\nPortrait 86 best Color Block Wall images on Pinterest in 2018 noblest just for you from thin blue line paint color , source: pinterest.com.\nPhoto What does the blue striped American flag mean Quora superior for you from thin blue line paint color , thin blue, source: quora.com.\nPortrait 74 best Paint Inspiration images on Pinterest in 2018 selected just for you from thin blue line paint color , source: pinterest.com.\nIllustration My Dining Room with Oak Trim paint color Sherwin Williams Moody the best just for you from thin blue line paint color , source: pinterest.com.\nPortrait Thin blue line yeti leading just for you from thin blue line paint color , thin blue, source: etsy.com.\nDrawing 45 Best James Har Paint Color Match Sherwin Williams selected just for you from thin blue line paint color , source: airjordan2retro.com.\nPortrait 4084 best Color and Paint Ideas images on Pinterest in 2018 leading just for you from thin blue line paint color , thin blue line ring, thin blue line polizei, thin blue line germany, source: pinterest.com.\nPicture Under Armour UA Freedom \"Thin Blue Line 2 0\" Color Black superior just for you from thin blue line paint color , source: evike.com.\nFigure How to Paint a Perfect Line Between Wall and Ceiling Home the best for you from thin blue line paint color , thin blue line flagge, thin blue line meaning, thin blue line armband, source: glidden.com.\nDrawing 10 Decorative Paint Techniques for Your Walls the most beautiful just for you from thin blue line paint color , thin blue line polizei, thin blue patch, thin blue line meaning, source: thespruce.com.\nSo, if you desire to acquire all of these wonderful pics related to (New thin blue line paint color), simply click save button to save the shots to your personal pc. There're available for obtain, if you'd rather and wish to own it, click save symbol in the web page, and it will be instantly downloaded in your laptop.\nthin blue line armband, thin blue line germany, thin blue line ring, thin blue patch, thin blue line, suggested.\nAt last if you wish to find unique and latest picture related to (New thin blue line paint color), please follow us on google plus or save this page, we try our best to provide daily up-date with all new and fresh photos.\nWe do hope you like keeping here. For most upgrades and latest news about (Awesome thin blue line paint color) photos, please kindly follow us on tweets, path, Instagram and google plus, or you mark this page on book mark area, We try to give you up grade regularly with fresh and new images, like your surfing, and find the perfect for you.\nthin blue line germany, thin blue line meaning, thin blue line, thin blue line polizei, thin blue patch, suggested.\nThanks for visiting our website, contentabove (New thin blue line paint color) published by at . Nowadays we are delighted to announce we have found an awfully interesting topicto be pointed out, namely (Best of thin blue line paint color) Many people attempting to find information about (Elegant thin blue line paint color) and of course one of them is you, is not it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can Manchester United make it a hat-trick of 4-0 wins? Will Huddersfield and West Brom maintain their 100% winning starts? Are Newcastle or West Ham going to break their duck? Here is your chance to predict the next round of Premier League fixtures.\nThe Red Devils take on Leicester in Saturday's evening game and BBC Sport's football expert Mark Lawrenson is backing them to continue their winning start at Old Trafford and also to challenge for the title this season.\n\"You can see that they know they have a good side and that they will be close in all competitions,\" he told BBC Sport.\nWhat do you think will happen? You can make your own predictions now - take on Lawro and other fans, create your own league and try to make it to the top of the table by playing the new-look BBC Sport Predictor game.\nMore than 150,000 of you have signed up so far, and Lawro has slipped down the overall table a bit after only amassing 70 points in round two.\nThere is plenty of time for him to catch up - he will be making a prediction for all 380 games this season, against a variety of guests.\nLawro's opponent for this weekend's Premier League fixtures is American actor Channing Tatum, who stars in Steven Soderbergh's comeback film Logan Lucky.\nTatum said that he played varsity soccer when he was younger but was asked by the parents of the other children to switch to American football because he was \"more of a slam-into-each-other type of player\".\nHe also revealed that his co-star on upcoming spy sequel Kingsman: The Golden Circle, Vinnie Jones, gave him a signed copy of the infamous photo in which the ex-Wimbledon midfielder grabs Paul Gascoigne in an intimate area. He admitted it was the \"favourite thing I have ever been given by a co-star\".\nAnother of Tatum's recent co-stars, Jamie Bell, has recruited him as an Arsenal fan. \"I got educated on the Premier League by him,\" he told BBC Sport.\nYou can hear more from Tatum on Football Focus at 12:00 BST on Saturday on BBC One and the BBC Sport website.\nYou can also hear more from Lawro on Saturday when he joins Phil Williams to take your calls on BBC Radio 5 live's 606.\nRemember you can try to beat Lawro yourself by playing the BBC Sport Predictor game.\nAfter losing their opening game at home to Burnley, Chelsea were excellent against Tottenham last weekend, winning 2-1 at Wembley. It was like a set of good players saying to themselves 'what was that all about last weekend?' They have some very, very good players and won the league last year at a canter.\nEverybody was ready to say 'played two, lost two, what is manager Antonio Conte going to say?' and wonder whether he was going to leave the club. But it was a big win for them and I expect them to win again this weekend.\nEverton have made a really good start. Everything about it has been impressive. The only thing is that, at the moment, Wayne Rooney is the only one who looks like he is going to score.\nEverybody knows they are looking for a striker, it is just a case of who they can get in. But they have a good manager and have made good signings already.\nThey got battered 5-0 at Chelsea last season and while I don't expect that to happen again, I do think they'll lose.\nIt is played two, won two for West Brom. But they are another team struggling for goals and looking to get attacking signings finalised.\nThere are no worries for the Baggies because Tony Pulis does what he does. But if he can just get himself a 15-goal striker, what a difference it would make. Whether they would get the chances to score 15 goals in a season in that West Brom side is debatable.\nStoke never looked like worrying Everton on the opening day but got a great result against Arsenal last weekend. They have something about them this season. Taking striker Jese on loan for a year from Paris St-Germain could be a masterstroke. He will be happy just to play and to go somewhere that he is wanted, which makes a huge difference to a player.\nMark Hughes is a good manager but Stoke are also a really good club to manage. They are run with great common sense by Peter Coates. They are pragmatists.\nI was at the corresponding game last year and Liverpool dispatched Arsenal quite comfortably. Like last year, they will be better against the good teams who do not change their style of play just because it is Liverpool.\nI saw them against Hoffenheim on Wednesday and they remain great going forward but they can't defend.\nLiverpool have been vehement in saying that they don't want to sell Philippe Coutinho but the latest reported bid from Barcelona is bonkers money. If he really wants to leave then I think he will have to come out and say he doesn't want to play - that is the only thing that will no force the outcome.\nIs everything OK between Klopp & Coutinho?\nLiverpool face an Arsenal team that still has more questions than answers about it. They were fortunate to beat Leicester and, against Stoke, while they can have a moan about some of the decisions, it exposed that familiar soft underbelly.\nYou can play all the nice football in the world but if you can't deal with the humdrum aspects of football you will be susceptible to results like that.\nThe problem for the Gunners is that every time they lose you are waiting for the 'Wenger out' to start.\nIn fairness to Tottenham, they played OK last weekend. Chelsea were just better than them. It is important for Tottenham to start getting wins under their belt at Wembley and to start to dispel the myth of it being a hoodoo.\nI said in week one that they would struggle to finish in the top four because of the Wembley situation and I am not going to change my mind. They will be in the mix for the top six though. They are not a bad team because they lost at home to Chelsea.\nThey have made some signing this week - the Colombian defender Davinson Sanchez from Ajax and goalkeeper Paulo Gazzaniga from Southampton. I would think they will be looking to get two or three more in.\nBut it is a trip to Wembley for Burnley and while they lost last weekend to West Brom they put in a really good performance to beat Chelsea on the opening day. I can't see them winning this though.\nLawro was speaking to BBC Sport's Phil Dawkes.\nOn the second weekend of the new Premier League season, Lawro got four correct results, including one perfect score, from 10 matches, for a total of 70 points.\nHe lost to actor Will Poulter, who got six results correct with two perfect scores.\n*As Cram and Foster both provided predictions on week two, only Foster's score contributes to the guest total.","meta":{"redpajama_set_name":"RedPajamaC4"}}